diff --git "a/Saturation Magnetization/2.json" "b/Saturation Magnetization/2.json" new file mode 100644--- /dev/null +++ "b/Saturation Magnetization/2.json" @@ -0,0 +1 @@ +[ { "title": "2401.14170v1.No_go_guide_for_the_Hubble_tension__late_time_or_local_scale_new_physics.pdf", "content": "No-go guide for the Hubble tension: late-time or local-scale new physics\nLu Huang,1,∗Shao-Jiang Wang,1, 2,†and Wang-Wei Yu3, 4,‡\n1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,\nChinese Academy of Sciences (CAS), Beijing 100190, China\n2Asia Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea\n3Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, D-30167 Hannover, Germany\n4Leibniz Universit¨ at Hannover, 30167 Hannover, Germany\nThe standard model of modern cosmology might be cracked by the recent persistent hot debate\non the Hubble-constant ( H0) tension, which manifests itself as the sound-horizon ( rs) tension or\nabsolute-magnitude ( MB) tension if deeming the origin of the Hubble tension from modifying the\nearly or late Universe, respectively. In this Letter, we achieve a fully model-independent constraint\n(fitting a model-independent global parameterization to a model-independent inverse distant lad-\nder with a model-independent high-redshift calibration) on late-time models with strong evidence\nagainst homogeneous new physics over the Λ-cold-dark (ΛCDM) model. Further using this model-\nindependent constraint to calibrate sufficiently local supernovae with corresponding late-time models\nextrapolated below the homogeneity scale, we find surprisingly that, although both H0tension and\nMBtension are absent in our local Universe, a combination of H0andMBas the intercept aBof\nthe magnitude-redshift relation exhibits 3 ∼7σtension even for the ΛCDM model. This aBtension\nseems to call for local-scale inhomogeneous new physics disguised as local observational systematics.\nIntroduction.— The Λ-cold-dark-matter (ΛCDM)\nmodel as the standard model of modern cosmology has\nbeen well established as a phenomenologically concor-\ndant model [1], fitting simultaneously into the current\nprecision cosmology [2] including cosmic microwave back-\nground (CMB), baryon acoustic oscillation (BAO), and\ntype Ia supernovae (SNe Ia). This concordant model\nmight be cracked by an ever-enlarging tension [3–7] on\nthe Hubble constant H0between the global fitting of\nPlanck 2018 observation with ΛCDM extrapolation [8]\nand the local quasi-direct measurements from Pantheon+\ncompilation [9] with SH0ES (Supernovae and H0for the\nEquation of State of dark energy) calibration [10]. If not\ncaused by any of the early-time or late-time systemat-\nics, the above Hubble-constant tension [11–15] could be\na promising clue to the new physics. However, despite nu-\nmerous attempts claiming to relieve or even resolve this\nH0tension [16–20], various no-go arguments have been\nsuggested against new physics from either the early Uni-\nverse [21–25] or the late Universe [26–30]. On the other\nhand, if the ΛCDM model should indeed be modified, the\nHubble tension cannot be the only tension that manifests\nthe breakdown of the current standard model of cosmol-\nogy. Below we only mention the most relevant ones to\ntheH0tension but refer most of the other tensions to the\nmost recent update [31].\nrstension.— One such tension comes from tracing the\norigin of the Hubble tension back to the early Universe\nbut assuming the ΛCDM model at the late time. After\nextrapolated into higher redshifts with the aid of BAO\ndata [32–34], the SNe Ia sample calibrated by SH0ES\n∗huanglu@itp.ac.cn\n†schwang@itp.ac.cn (corresponding author)\n‡wangwei.yu@aei.mpg.demeasurements usually infers a smaller sound horizon rs\nthan that constrained by the CMB data alone [35–38].\nAs BAO data apparently maintains a nearly constant\nproduct rsH0, this rstension [11–14] also serves as an\nearly-Universe reflection of the H0tension, whose res-\nolutions naively require a reduced sound horizon from\nmodifying either early expansion or recombination his-\ntories [21]. However, such early resolutions necessarily\neither suppress or shorten the growth rate or growing\ntime for matter perturbations, which can only be com-\npensated with a larger matter fraction to match the late-\ntime galaxy clustering or lensing data [21, 22] but yet\ncommonly worsen the dubbed S8tension [20, 31, 39],\ncalling for additional decaying matter mechanisms to res-\ncue at the late time [40–42]. Besides, these early resolu-\ntions also intimately tend to prefer an extremely scale-\ninvariant scalar spectrum index ns≈1 [43–50], calling\nfor even more exotic inflationary model buildings [51–\n56]. Therefore, if the Hubble tension indeed originates\nfrom the early Universe, then it is never enough just to\nalter the early Universe alone [57], but to specifically fine-\ntune both the late and primordial Universe at the same\ntime, which is way too much price to pay to rebuild the\nconcordant cosmology.\nMBtension.— The other tension comes from locat-\ning the cause of the Hubble tension in the late Uni-\nverse but assuming the ΛCDM model at the early time.\nAfter calibrated by the sound horizon from CMB con-\nstraints, the inverse distance ladder (IDL) [33, 37, 58–\n64] consisting of SNe Ia and BAO data usually leads\nto a fainter absolute magnitude MBthan SH0ES mea-\nsurements for the SNe Ia. As MBandH0jointly form\nthe intercept aBof the magnitude-redshift relation con-\nstrained by SNe Ia data, this MBtension also serves as\na late-Universe reflection of the Hubble tension, whosearXiv:2401.14170v1 [astro-ph.CO] 25 Jan 20242\nresolutions seem to require either a fainter absolute mag-\nnitude from local-Universe systematics or a faster expan-\nsion rate of late/local Universe, for example, a phantom-\nlike dark energy (PDE) transition [65]. However, such a\nlate-Universe PDE transition is strongly constrained by\nthe current IDL data calibrated by the CMB prior on the\nsound horizon [26–28]. What’s more, other late-Universe\ndeviations from the ΛCDM model are also strongly dis-\nfavored by an improved IDL method [29, 30], where the\nmodel-independent fitting model adopts a global, eco-\nnomic, and precise Parameterization based on the cosmic\nAge (PAge) [66–74] while high-redshift calibrators em-\nploy cosmologically model-independent Hubble parame-\nters from cosmic chronometer (CC) [75]. In this Letter ,\nwe will first further strengthen this late-time “no-go the-\norem” [26–30] by using not only the model-independent\nPAge model and model-independent CC calibration but\nalso a fully model-independent IDL dataset consisting of\nHubble-flow SNe Ia and two-dimensional BAO data.\naBtension.— The above-mentioned rstension and the\nassociated S8tension as well as the MBtension have nar-\nrowed down a satisfactory resolution to the local Universe\nfrom either local systematic errors or local new physics.\nThe local systematics have been thoroughly investigated\nby the SH0ES team [10], finding no significant contribu-\ntion to the current MBtension. Therefore, it seems to\nfurther pin down the local-Universe resolutions to local\nnew physics, which will be the main target of investiga-\ntions in the remaining part of this Letter . Without the\nuse of the SH0ES measurement on MBthat brings in the\nHubble tension in the first place, we will use the above-\nmentioned fully model-independent IDL constraints to\ncalibrate the sufficiently local SNe Ia, leading to a new\ntension in the intercept aBof magnitude-redshift relation\neven without appearances of both H0andMBtensions.\nFully model-independent IDL method.— A local\ndistance ladder is used for calibrating a distant sample\nof distance indicators (e.g. SNe Ia), while the “inverse”\ndistance ladder (IDL) refers to a reverse process by cali-\nbrating a combined distant sample of distance indicators\n(i.e. SNe+BAO) with more distant objects, for example,\nthe sound horizon rsfrom CMB [35–38] or time-delay dis-\ntance D∆t[76] from strong lensing time delay [77], which\nstill separately depends on early or late/local Universe\nvia integrating over the expansion history in rsorD∆t.\nWe propose here a fully model-independent IDL method\nfrom the following fully model-independent datasets:\nHF SNe Ia data.— The Hubble-flow (HF) SNe Ia are\nusually selected in the redshift range 0 .0233 < z < 0.15,\nwhere the lower bound zmin= 0.0233 corresponds to the\nhomogeneity scale Rhomo≈70 Mpc /h[78], and it is re-\nquired to suppress the cosmic variance [79–82] from our\nlocal matter density contrast [83] within Rhomo [84–87],\nwhile the upper bound zmax= 0.15 is required to de-\ntach any cosmological dependence on the late-Universe\nevolution. Specifically, we select 490 HF SNe Ia outof a state-of-art sample from the Pantheon+ compi-\nlation [9, 10, 88–92], which mainly adds low-redshift\n(z < 0.1) SNe from extra low- zsurveys (see, e.g. [9])\ncompared to the original Pantheon sample [93]. The full\ndata releases of Pantheon+ sample are publicly available\nat https://pantheonplussh0es.github.io/. The likelihood\nlnLHFSN =−1\n2∆T\nHFSN C−1\nstat.+syst .∆HFSN is constructed\nfrom the magnitude-redshift residual vector ∆ HFSN ,i=\nmB(zi)−MB−µmodel (zi), where the apparent B-band\npeak magnitude mB(zi) of i-th SN at redshift zihas\nbeen corrected for the stretch, color, simulation bias, and\nmass-step effects, and the absolute B-band peak mag-\nnitude MBwill be left free or calibrated by either the\nSH0ES measurement or the IDL constraint later, while\nthe distance modulus µmodel (zi) at corresponding red-\nshift is modeled as\nµmodel (zi) = 5 lg dL(zi) + 5 lgc\nH0·Mpc+ 25, (1)\nwhere the dimensionless luminosity distance dL(z)≡\nDL(z)/(c/H 0) = (1 + zhel)RzHD\n0dz′/E(z′) is computed\ngiven a cosmological model for the dimensionless Hub-\nble parameter E(z)≡H(z)/H0. Here zhelis the he-\nliocentric redshift and zHDis the Hubble-diagram red-\nshift with CMB and peculiar velocity corrections in Pan-\ntheon+ samples. The covariance matrix Cstat.+syst .in-\ncludes all statistical and systematic uncertainties as well\nas the expected light-curve correlations in the sample.\n2D BAO dataset.— The BAO from the early Universe\nleaves permanent imprints on the galaxy clustering in the\nlate Universe, extremalizing locally the two-point corre-\nlation function (2PCF) of galaxy pairs with a comoving\nseparation of sound-horizon size rdat drag epoch. Due to\nthe redshift-space distortion [94] and Alcock–Paczynski\neffect [95] for a mismatched fiducial cosmology, the usual\nanisotropic fitting to the measured BAO signals with\ntemplates [96] for the monopole and quadrupole moments\nof 2PCF can provide independent measurements [97–100]\nfor the Hubble parameter H(z) and angular diameter\ndistance DA(z)≡DL(z)/(1 + z)2with respect to the\nfiducial cosmology. Although this kind of relative mea-\nsurements can largely eliminate the explicit dependence\non the fiducial cosmology, the model-dependent bias can\nstill sneak into analysis in following three aspects: First,\ndifferences in the distance–redshift relation used for cal-\nculating the galactic separation between the true and\nfiducial cosmologies [101] can scale non-linearly in some\ninhomogeneous models or with backreaction; Second, dif-\nferences in modeling the comoving clustering between the\ntrue and fiducial cosmologies [102] might not be fully ab-\nsorbed into the same set of nuisance parameters that are\nused to marginalize over any non-BAO signal [103] in the\nfitting template; Third, differences in estimating the co-\nvariance matrix from mock random samples [103] can be\nquite different from numerical simulations of large scale\nstructures beyond the standard cosmological model [104].3\nHowever, unlike the conventional BAO that explicitly\ndepends on the fiducial cosmology, the two-dimensional\n(2D) BAO [105] from measuring the two-point angular\ncorrelation function relies only on the angular separa-\ntion between pairs of galaxies without assuming a fidu-\ncial cosmology, rendering the most model-independent\nconstraint on angular BAO scales θBAO(zi) modeled as\nθmodel (zi) =rd\n(1 +zi)DA(zi), (2)\nwhich are totally uncorrelated between any two differ-\nent redshifts zisince they are obtained from totally dis-\njoint spherical shells with the redshift thickness ∆ zjust\nenough to extract significant transversal BAO signals,\nand hence the covariance matrix includes only diagonal\nerrors σBAO(zi). Therefore, the likelihood is estimated\nas lnL2DBAO =−1\n2(θBAO(zi)−θmodel (zi))2/σBAO(zi)2\nfrom 15 measurements of 2D BAO data collected from\nRefs. [105–109]. See, for example, Refs. [110, 111] for\nrecent studies using 2D BAO data.\nCC calibration.— The CC data as high-redshift cali-\nbrations to inversely calibrate the late-Universe distance\nladder HFSN+2DBAO is completely independent of both\nearly and late/local Universe as the Hubble parameter\nH(z) is directly measured by [75]\nH(z) =−1\n1 +zdz\ndt=−1\n1 +zeff∆z\n∆t(3)\nfor passively-evolving early-time massive galaxies that\nformed around the same time with much shorter age\ndifferences ∆ tthan their evolving time scales, and are\nseparated by a small redshift interval ∆ zaround zeff.\nThe likelihood ln LCC=−1\n2∆T\nCCC−1\nstat.+syst .∆CCis con-\nstructed from the residual vector ∆ CC,i=Hobs(zi)−\nHmodel (zi) and a completed covariance matrix [2, 112]\nincluding both statistical and systematic uncertainties,\nwhere the observed Hubble parameters Hobs(zi) are the\nsame as those in Table I of Ref. [30] from Refs. [113–121],\nwhile the full covariance matrix Cstat.+syst .can be found\nat https://gitlab.com/mmoresco/CCcovariance.\nModel-independent parameterization.— Another\nindispensable pillar of the fully model-independent IDL\nmethod is to fit with a model-independent parameteriza-\ntion. As shown in Refs. [29, 30], the traditional Taylor\nexpansion in redshift zory-redshift y≡1−a=z/(1+z)\neven up to the fifth orders cannot faithfully approximate\nthe model it claims to parameterize at redshift z≳1 even\nfor the model as simple as the ΛCDM model. Hence, we\nadopt a better parameterization dubbed PAge model [66]\nas a collection of relatively gentle modifications to the\nΛCDM model, while using the PDE transition model as\na typical example of more violent saltation in the late\nand local Universe. We sketch these two models below:\nPAge model.— The spirit of PAge model is to param-\neterize the accumulated quantity (like the cosmic age)instead of the instant quantity (like the expansion rate)\nas the latter one can change more dramatically and hence\nmore difficult to parameterize than the former one. For a\npure matter Universe, it holds Ht−2/3 = 0 = H0t0−2/3\nfor the cosmic age tand its current value t0. Since the\nearly radiation duration is negligible and the matter du-\nration contributes most of the cosmic age, it is natural to\nparameterize various late-time homogeneous dark energy\nmodels as deviations from the pure matter Universe by\nTaylor expansions in the cosmic age as [66, 67]\nHt−2\n3=\u0012\nH0t0−2\n3(1 +η)\u0013\u0012t\nt0\u0013\n+2\n3η\u0012t\nt0\u00132\n,(4)\nwhere a free parameter ηis introduced to characterize\nsuch a deviation from a pure matter Universe with η= 0,\nwhile H0t0≡pageis another free parameter in the PAge\nmodel together with ηto solve for E(z)≡H(z)/H0with\nthe help of the definition H(z)≡ −dz/dt/(1+z). Higher-\norder expansions like the cubic term in cosmic age [69]\nare not necessary as shown in Ref. [30] as the quadratic\norder is precise enough to parameterize the usual late-\ntime homogeneous models [66, 67] even up to a higher\nredshift z≳1 [29, 30] where some of our CC calibrators\nlive. For a given model to be parameterized by the PAge\nmodel, one is free to choose a moment when these two\nmodels coincide, which is usually convenient to be fixed\nat our present day when the two PAge parameters can be\nrelated by η= 1−3\n2p2\nage(1 +q0) via the current value q0\nof the deceleration parameter q(t)≡ −¨aa/˙a2. Therefore,\nthe PAge model serves as a global, economic, and precise\nparameterization in the late/local Universe for any homo-\ngeneous modification gently beyond the ΛCDM model.\nPDE model.— For those more abrupt changes in the\nexpansion rate that cannot be well captured by the PAge\nmodel, we can use the PDE model following Ref. [28] as a\nrepresentative illustration. The Hubble parameter reads\nH(z) =Hf\n0r\nΩm(1 +z)3+ (1−Ωm)\u0010\n1 + ∆ e−(z\nzc)β\u0011\n,\n(5)\nwhere the phantom-like transition occurs at a transition\nreshift zcwith a strength ∆ and an index β, leading to\na Hubble constant H0≡H(z= 0) generally larger than\nthe fiducial one Hf\n0while still keeping its high-redshift\nbehavior at z≫zcas if it goes along with the ΛCDM\nmodel with a naive extrapolation at H(z= 0) = Hf\n0.\nLate-time or local-scale new physics?.— Using\nthe fully model-independent IDL method to constrain the\nlate or local Universe can be divided into two steps, be-\ntween which the (in)consistency reveals whether or not\nthere is late or local new physics beyond the ΛCDM\nmodel with H(z)2/H2\n0= Ω m(1 + z)3+ 1−Ωm. We\nfind that the late Universe still prefers the ΛCDM model,\nwhich, however, admits a new tension in local Universe.4\nTABLE I. Parameters and priors\nModel parameters Parameter priors\nΛCDM PAge PDE Local Universe Late Universe\nMB (-21, -18)\nrd (100, 200)\nH0 Hf\n0 (40, 90)\nΩm — Ωm (0, 1)\n— page — (0.7, 1.2)\n— η — (-1, 1)\n— — ∆ (-1, 1)\n— — zc (0, 0.0233) (0.0233, 1)\n— — β (-4, 4)\n60657075\nH0140150160170180rd19.6\n19.4\n19.2\nMB\n19.6\n19.4\n19.2\nMB140150160170\nrd60657075H0LCDM PAge PDE\nFIG. 1. The marginalized 1 σand 2 σconstraints with the 1D\nposteriors on H0,MB, and rdfrom fitting ΛCDM, PAge and\nPDE models with the fully model-independent IDL dataset.\nLate Universe.— Fitting the fully model-independent\nIDL dataset from HF SN+2D BAO+CC to the ΛCDM,\nPAge, and PDE models in the late Universe ( z >0.0233)\nwith the Markov Chain Monte Carlo code EMCEE [122],\nwe can constrain their model parameters with flat priors\npresented in Table. I from the joint likelihood ln Ltot=\nlnLHFSN + lnL2DBAO + lnLCC. In particular, the\nmarginalized 1 σand 2 σconstraints with 1D posteriors on\nparameters H01,MB, and rdare presented in Fig. 1 with\nalmost indistinguishable differences for both PAge and\nPDE models with respect to the ΛCDM model. This can\nbe drawn more quantitatively from the Bayesian infor-\n1For PDE model the true Hubble constant H0≡H(z= 0) is\nobtained from all five model parameters Hf\n0, Ωm, ∆,zc, and β.\n586266707478\nH019.6\n19.4\n19.2\n19.0\nMBCDM\nIDL=HFSN+2DBAO+CC\nMB(IDL)+LocalSN\n(MB, m)(IDL)+LocalSN\nMB(SH0ES)+HFSN\nMB(SH0ES)+LocalSNFIG. 2. Marginalized 1 σand 2 σcontours of H0andMBfrom\nIDL-calibrated local SNe (yellow and blue) compared to the\npure IDL constraint (red). The SH0ES-calibrated local SNe\nand HF SNe are shown in purple and green, respectively.\nmation criterion (BIC) [123] BIC = −2 lnLmax+klnN\ndefined from a Gaussian approximation to the Bayesian\nevidence in the limit of large sample size, where the\nnumber of free parameters takes k= 4,5,7 for ΛCDM,\nPAge, and PDE models, respectively, and the total num-\nber of data points N= 537 consists of 490 HF SNe\nIa, 15 2D BAO and 32 CC data points. We therefore\nfind a strong evidence (∆BIC = 6 .27) against the PAge\nmodel and a decisive evidence (∆BIC = 18 .42) against\nthe PDE model over the ΛCDM model, further strength-\nening the previous no-go arguments [29, 30] but with a\nmuch more model-independent manner against any ho-\nmogeneous new physics in the late Universe.\nLocal Universe.— Having deduced the late Universe\ncannot deviate even mildly away from the ΛCDM model,\nit is then tempting to further pin down the possible new\nphysics within the local Universe ( z < 0.0233) by fit-\nting to a local SN sample from 336 SNe Ia below the\nhomogeneity scale in the Pantheon+ sample with well\nseparated peculiar-velocity corrections [90, 91]. As it is\nthe SH0ES calibration on MBthat raises the issue of\nthe Hubble tension in the first place, we will instead cal-\nibrate these local SNe Ia from the previously obtained\nfully model-independent IDL constraint on MB. Taking\nthe ΛCDM model as a typical example, we can infer the\nposterior of H0from fitting this IDL-calibrated local SN\nsample with and without the IDL posterior on Ω mas\nshown in blue and yellow in Fig. 2, respectively, where\nthe pure IDL constraint is shown in red for comparison.\nIt is then intriguing to observe that, although both MB\nandH0are consistent between IDL constraints and IDL-\ncalibrated local constraints, they are actually separated\nsharply along the diagonal direction. This diagonal sep-\naration seems to be independent of calibrations as it also\noccurs even between the SH0ES-calibrated local SN and\nHF SN samples shown in purple and green, respectively.5\n4.79\n 4.78\n 4.77\n 4.76\n 4.75\naB4.2\n7.3\n3.6\nCDM\nIDL=HFSN+2DBAO+CC\nMB(IDL)+LocalSN\nm(IDL)+LocalSN\nMB(SH0ES)+HFSN\nMB(SH0ES)+LocalSN\n4.79\n 4.78\n 4.77\n 4.76\n 4.75\naB5\n4\n3.9\nPAge\nIDL=HFSN+2DBAO+CC\nMB(IDL)+LocalSN\n(page,)(IDL)+LocalSN\nMB(SH0ES)+HFSN\nMB(SH0ES)+LocalSN\n5.00\n 4.95\n 4.90\n 4.85\n 4.80\n 4.75\n 4.70\naB3.4\n3.4\nPDE\nIDL=HFSN+2DBAO+CC\nMB(IDL)+LocalSN\n(m,,zc,)(IDL)+LocalSN\nMB(SH0ES)+HFSN\nMB(SH0ES)+LocalSN\nFIG. 3. 1D posteriors of aBfor ΛCDM, PAge, and PDE models from the pure IDL constraint (red), IDL-calibrated local direct\n(blue) and indirect (yellow) constraints, and SH0ES-calibrated constraints from HF SN (green) and local SN (purple).\nThis diagonal direction is in fact the intercept aBin the\nmagnitude-redshift relation mB(zi) = 5 lg dL(zi)−5aB\nthat is degenerated in both H0andMBby\naB≡ −1\n5\u0012\nMB+ 5 lgc\nH0·Mpc+ 25\u0013\n, (6)\nwhich can also be inferred from a SN sample mB(zi) with\na covariance matrix Cijfor a given model dL(z) by [28]\naB=\nX\nijC−1\nij\u0012\nlgdL(zi)−1\n5mB(zi)\u0013\n\u001eX\nijC−1\nij.\n(7)\nThen, this diagonal separation leads surprisingly to an\naBtension as shown explicitly in Fig. 3 for the ΛCDM\n(left), PAge (medium), and PDE (right) models but with\nthe vertical axis not to scale for clarity. The pure IDL\nconstraints on aBare all shown in red via (6) from IDL\nposteriors of MBandH0, which are in 3 ∼7σtensions\nwith the IDL-calibrated local constraints on aBfor both\nΛCDM and PAge models. Here, the IDL-calibrated local\nconstraint on aBcan be implemented either directly by\nthe IDL posterior(s) on the model parameter(s) via (7) as\nshown in blue or indirectly by the IDL posterior on MB\nto first constrain H0and then aBvia (6) as shown in\nyellow. This aBtension occurs for both IDL-calibrated\nlocal direct and indirect constraints with respect to the\npure IDL constraint except for the PDE case with more\nmodel parameters to enlarge the uncertainty in the IDL-\ncalibrated local direct constraint. Intriguingly, this aB\ntension seems to be independent of calibrations as it also\noccurs even between the SH0ES-calibrated local SN and\nHF SN samples shown in purple and green, respectively.\nAlthough the PAge model is comparable to the ΛCDM\nmodel in fitting the local Universe, we find decisive ev-\nidence ∆BIC = −36.9 and Bayesian evidence [124, 125]\n∆ lnZ= 16.4 with DYNESTY [126, 127] supporting a PDE\ntransition at zc= 0.00218+0.00063\n−0.00073 over the ΛCDM model.Conclusions and discussions.— In this Letter , we\nhave first strengthened the so-called [57] late-time “no-\ngo theorem” [26–30] with a fully model-independent IDL\nconstraint against any homogeneous new physics beyond\nthe ΛCDM model in the late Universe. We then use\nthis fully model-independent IDL constraint to calibrate\nthe local SN sample, which is in direct tension with the\nIDL constraint on the intercept aBof the magnitude-\nredshift relation. Future independent investigations with\nother distance calibrators and distance indicators would\nbe more appealing to confirmatively claim the aBtension\nas a reflection of the H0tension in the local Universe.\nAlthough an ultra-low-redshift PDE transition seems\nto be favored over the ΛCDM model, it does not solve\ntheaBtension nor explain its origin. Besides, the Hub-\nble diagram found in the Carnegie-Chicago Hubble Pro-\ngram VIII [128] admits very little scatter in their veloc-\nity flow corrected distances of the SN host galaxies with\nredshift z≲0.007, therefore, there seems no room for\nan abrupt change to the equation of state at very low\nredshifts [28]. Therefore, this preference for a local PDE\ntransition might as well be an illusion of some unknown\nlocal systematics. And because our aBtension is iden-\ntified independent of the use of SH0ES calibrations, this\naBtension could be originated from some local-scale new\nphysics disguised as our local systematics.\nLocal systematics do not need to come from our own\nlocal Universe but can also refer to a distant local Uni-\nverse that impacts our own local measurements. Such\nan impact should be necessarily small, but the observed\ncorrelation between the amount of such an impact and\nthe property of that distant local Universe has recently\nbeen found to be in direct tension with the perturbative\nprediction from the ΛCDM model. This newly discov-\nered δH0tension [129] therefore signifies a cosmological\nbreakdown of the ΛCDM model at perturbative level in\nthe local-scale Universe, hinting for some local-scale new\nphysics disguised as local systematics that would dim dis-\ntance indicators and calibrators in denser regions. Yet its\nrelation to the aBtension merits further future studies.6\nWe thank Sunny Vagnozzi for an early correspondence\non the model dependency of 3D BAO analysis. This work\nis supported by the National Key Research and Develop-\nment Program of China Grants No. 2021YFA0718304,\nNo. 2021YFC2203004, and No. 2020YFC2201501, the\nNational Natural Science Foundation of China Grants\nNo. 12105344, No. 12235019, and No. 12047503, the\nScience Research Grants from the China Manned Space\nProject with No. CMS-CSST-2021-B01, and the Post-\ndoctoral Fellowship Program of CPSF. We also acknowl-\nedge the use of the HPC Cluster of ITP-CAS.\n[1] P. A. R. Ade etal. (Planck), “Planck 2013 results.\nXVI. Cosmological parameters,” Astron. Astrophys.\n571, A16 (2014), arXiv:1303.5076 [astro-ph.CO].\n[2] Michele Moresco etal., “Unveiling the Universe with\nemerging cosmological probes,” Living Rev. Rel. 25, 6\n(2022), arXiv:2201.07241 [astro-ph.CO].\n[3] Adam G. Riess etal., “A 2.4% Determination of the Lo-\ncal Value of the Hubble Constant,” Astrophys. J. 826,\n56 (2016), arXiv:1604.01424 [astro-ph.CO].\n[4] Adam G. Riess etal., “Milky Way Cepheid Standards\nfor Measuring Cosmic Distances and Application to\nGaia DR2: Implications for the Hubble Constant,” As-\ntrophys. J. 861, 126 (2018), arXiv:1804.10655 [astro-\nph.CO].\n[5] Adam G. Riess etal., “New Parallaxes of Galactic\nCepheids from Spatially Scanning the Hubble Space\nTelescope: Implications for the Hubble Constant,” As-\ntrophys. J. 855, 136 (2018), arXiv:1801.01120 [astro-\nph.SR].\n[6] Adam G. Riess, Stefano Casertano, Wenlong Yuan,\nLucas M. Macri, and Dan Scolnic, “Large Magel-\nlanic Cloud Cepheid Standards Provide a 1% Founda-\ntion for the Determination of the Hubble Constant and\nStronger Evidence for Physics Beyond LambdaCDM,”\nAstrophys. J. 876, 85 (2019), arXiv:1903.07603 [astro-\nph.CO].\n[7] Adam G. Riess, Stefano Casertano, Wenlong Yuan,\nJ. Bradley Bowers, Lucas Macri, Joel C. Zinn, and\nDan Scolnic, “Cosmic Distances Calibrated to 1% Pre-\ncision with Gaia EDR3 Parallaxes and Hubble Space\nTelescope Photometry of 75 Milky Way Cepheids Con-\nfirm Tension with ΛCDM,” Astrophys. J. Lett. 908, L6\n(2021), arXiv:2012.08534 [astro-ph.CO].\n[8] N. Aghanim etal. (Planck), “Planck 2018 results. VI.\nCosmological parameters,” Astron. Astrophys. 641, A6\n(2020), [Erratum: Astron.Astrophys. 652, C4 (2021)],\narXiv:1807.06209 [astro-ph.CO].\n[9] Dan Scolnic etal., “The Pantheon+ Analysis: The Full\nData Set and Light-curve Release,” Astrophys. J. 938,\n113 (2022), arXiv:2112.03863 [astro-ph.CO].\n[10] Adam G. Riess etal., “A Comprehensive Measure-\nment of the Local Value of the Hubble Constant with\n1 km/s/Mpc Uncertainty from the Hubble Space Tele-\nscope and the SH0ES Team,” Astrophys. J. Lett. 934,\nL7 (2022), arXiv:2112.04510 [astro-ph.CO].\n[11] Jose Luis Bernal, Licia Verde, and Adam G. Riess,\n“The trouble with H0,” JCAP 1610 , 019 (2016),arXiv:1607.05617 [astro-ph.CO].\n[12] L. Verde, T. Treu, and A. G. Riess, “Tensions between\nthe Early and the Late Universe,” Nature Astron. 3,\n891 (2019), arXiv:1907.10625 [astro-ph.CO].\n[13] Lloyd Knox and Marius Millea, “Hubble constant\nhunter’s guide,” Phys. Rev. D101 , 043533 (2020),\narXiv:1908.03663 [astro-ph.CO].\n[14] Adam G. Riess, “The Expansion of the Universe is\nFaster than Expected,” Nature Rev. Phys. 2, 10–12\n(2019), arXiv:2001.03624 [astro-ph.CO].\n[15] Wendy L. Freedman, “Measurements of the Hubble\nConstant: Tensions in Perspective,” Astrophys. J. 919,\n16 (2021), arXiv:2106.15656 [astro-ph.CO].\n[16] Eleonora Di Valentino etal., “Snowmass2021 - Letter\nof interest cosmology intertwined II: The hubble con-\nstant tension,” Astropart. Phys. 131, 102605 (2021),\narXiv:2008.11284 [astro-ph.CO].\n[17] Eleonora Di Valentino, Olga Mena, Supriya Pan,\nLuca Visinelli, Weiqiang Yang, Alessandro Melchiorri,\nDavid F. Mota, Adam G. Riess, and Joseph Silk,\n“In the realm of the Hubble tension—a review of\nsolutions,” Class. Quant. Grav. 38, 153001 (2021),\narXiv:2103.01183 [astro-ph.CO].\n[18] Nils Sch¨ oneberg, Guillermo Franco Abell´ an, Andrea\nP´ erez S´ anchez, Samuel J. Witte, Vivian Poulin, and\nJulien Lesgourgues, “The H0 Olympics: A fair rank-\ning of proposed models,” Phys. Rept. 984, 1–55 (2022),\narXiv:2107.10291 [astro-ph.CO].\n[19] Paul Shah, Pablo Lemos, and Ofer Lahav, “A buyer’s\nguide to the Hubble constant,” Astron. Astrophys. Rev.\n29, 9 (2021), arXiv:2109.01161 [astro-ph.CO].\n[20] Elcio Abdalla etal., “Cosmology intertwined: A review\nof the particle physics, astrophysics, and cosmology as-\nsociated with the cosmological tensions and anomalies,”\nJHEAp 34, 49–211 (2022), arXiv:2203.06142 [astro-\nph.CO].\n[21] Karsten Jedamzik, Levon Pogosian, and Gong-Bo\nZhao, “Why reducing the cosmic sound horizon alone\ncan not fully resolve the Hubble tension,” Commun. in\nPhys. 4, 123 (2021), arXiv:2010.04158 [astro-ph.CO].\n[22] J. Colin Hill, Evan McDonough, Michael W. Toomey,\nand Stephon Alexander, “Early dark energy does not\nrestore cosmological concordance,” Phys. Rev. D102 ,\n043507 (2020), arXiv:2003.07355 [astro-ph.CO].\n[23] Weikang Lin, Xingang Chen, and Katherine J. Mack,\n“Early Universe Physics Insensitive and Uncalibrated\nCosmic Standards: Constraints on Ωm and Implications\nfor the Hubble Tension,” Astrophys. J. 920, 159 (2021),\narXiv:2102.05701 [astro-ph.CO].\n[24] Sunny Vagnozzi, “Consistency tests of ΛCDM from the\nearly integrated Sachs-Wolfe effect: Implications for\nearly-time new physics and the Hubble tension,” Phys.\nRev. D 104, 063524 (2021), arXiv:2105.10425 [astro-\nph.CO].\n[25] Oliver H. E. Philcox, Gerrit S. Farren, Blake D. Sher-\nwin, Eric J. Baxter, and Dillon J. Brout, “Determin-\ning the Hubble constant without the sound horizon: A\n3.6% constraint on H0 from galaxy surveys, CMB lens-\ning, and supernovae,” Phys. Rev. D 106, 063530 (2022),\narXiv:2204.02984 [astro-ph.CO].\n[26] Giampaolo Benevento, Wayne Hu, and Marco Raveri,\n“Can Late Dark Energy Transitions Raise the Hub-\nble constant?” Phys. Rev. D101 , 103517 (2020),\narXiv:2002.11707 [astro-ph.CO].7\n[27] David Camarena and Valerio Marra, “On the use of\nthe local prior on the absolute magnitude of Type Ia\nsupernovae in cosmological inference,” Mon. Not. Roy.\nAstron. Soc. 504, 5164–5171 (2021), arXiv:2101.08641\n[astro-ph.CO].\n[28] George Efstathiou, “To H0 or not to H0?” Mon.\nNot. Roy. Astron. Soc. 505, 3866–3872 (2021),\narXiv:2103.08723 [astro-ph.CO].\n[29] Rong-Gen Cai, Zong-Kuan Guo, Shao-Jiang Wang,\nWang-Wei Yu, and Yong Zhou, “No-go guide for the\nHubble tension: Late-time solutions,” Phys. Rev. D\n105, L021301 (2022), arXiv:2107.13286 [astro-ph.CO].\n[30] Rong-Gen Cai, Zong-Kuan Guo, Shao-Jiang Wang,\nWang-Wei Yu, and Yong Zhou, “No-go guide for\nlate-time solutions to the Hubble tension: Matter\nperturbations,” Phys. Rev. D 106, 063519 (2022),\narXiv:2202.12214 [astro-ph.CO].\n[31] Leandros Perivolaropoulos and Foteini Skara, “Chal-\nlenges for ΛCDM: An update,” New Astron. Rev. 95,\n101659 (2022), arXiv:2105.05208 [astro-ph.CO].\n[32] Shadab Alam etal. (BOSS), “The clustering of galaxies\nin the completed SDSS-III Baryon Oscillation Spectro-\nscopic Survey: cosmological analysis of the DR12 galaxy\nsample,” Mon. Not. Roy. Astron. Soc. 470, 2617–2652\n(2017), arXiv:1607.03155 [astro-ph.CO].\n[33] Shadab Alam etal. (eBOSS), “Completed SDSS-IV ex-\ntended Baryon Oscillation Spectroscopic Survey: Cos-\nmological implications from two decades of spectro-\nscopic surveys at the Apache Point Observatory,” Phys.\nRev. D 103, 083533 (2021), arXiv:2007.08991 [astro-\nph.CO].\n[34] T. M. C. Abbott etal. (DES), “Dark Energy Survey\nYear 3 Results: A 2.7% measurement of Baryon Acous-\ntic Oscillation distance scale at redshift 0.835,” (2021),\narXiv:2107.04646 [astro-ph.CO].\n[35] Marc Vonlanthen, Syksy R¨ as¨ anen, and Ruth Durrer,\n“Model-independent cosmological constraints from the\nCMB,” JCAP 1008 , 023 (2010), arXiv:1003.0810 [astro-\nph.CO].\n[36] Benjamin Audren, Julien Lesgourgues, Karim Benabed,\nand Simon Prunet, “Conservative Constraints on Early\nCosmology: an illustration of the Monte Python cos-\nmological parameter inference code,” JCAP 1302 , 001\n(2013), arXiv:1210.7183 [astro-ph.CO].\n[37] Licia Verde, Emilio Bellini, Cassio Pigozzo, Alan F.\nHeavens, and Raul Jimenez, “Early Cosmology Con-\nstrained,” JCAP 1704 , 023 (2017), arXiv:1611.00376\n[astro-ph.CO].\n[38] Kevin Aylor, MacKenzie Joy, Lloyd Knox, Marius\nMillea, Srinivasan Raghunathan, and W. L. Kimmy\nWu, “Sounds Discordant: Classical Distance Ladder\n\\& ΛCDM -based Determinations of the Cosmolog-\nical Sound Horizon,” Astrophys. J. 874, 4 (2019),\narXiv:1811.00537 [astro-ph.CO].\n[39] Eleonora Di Valentino etal., “Cosmology intertwined\nIII:fσ8andS8,” Astropart. Phys. 131, 102604 (2021),\narXiv:2008.11285 [astro-ph.CO].\n[40] Kyriakos Vattis, Savvas M. Koushiappas, and Abra-\nham Loeb, “Late universe decaying dark matter can re-\nlieve the H0tension,” Phys. Rev. D99, 121302 (2019),\narXiv:1903.06220 [astro-ph.CO].\n[41] Kanhaiya L. Pandey, Tanvi Karwal, and Subinoy\nDas, “Alleviating the H0and σ8anomalies with a\ndecaying dark matter model,” JCAP 07, 026 (2020),arXiv:1902.10636 [astro-ph.CO].\n[42] Steven J. Clark, Kyriakos Vattis, and Sav-\nvas M. Koushiappas, “Cosmological constraints on\nlate-universe decaying dark matter as a solution to\ntheH0tension,” Phys. Rev. D 103, 043014 (2021),\narXiv:2006.03678 [astro-ph.CO].\n[43] Vivian Poulin, Tristan L. Smith, Tanvi Karwal, and\nMarc Kamionkowski, “Early Dark Energy Can Resolve\nThe Hubble Tension,” Phys. Rev. Lett. 122, 221301\n(2019), arXiv:1811.04083 [astro-ph.CO].\n[44] Florian Niedermann and Martin S. Sloth, “New early\ndark energy,” Phys. Rev. D 103, L041303 (2021),\narXiv:1910.10739 [astro-ph.CO].\n[45] Florian Niedermann and Martin S. Sloth, “Resolving\nthe Hubble tension with new early dark energy,” Phys.\nRev. D102 , 063527 (2020), arXiv:2006.06686 [astro-\nph.CO].\n[46] Gen Ye and Yun-Song Piao, “Is the Hubble tension a\nhint of AdS phase around recombination?” Phys. Rev.\nD101, 083507 (2020), arXiv:2001.02451 [astro-ph.CO].\n[47] Gen Ye, Bin Hu, and Yun-Song Piao, “Implication of\nthe Hubble tension for the primordial Universe in light\nof recent cosmological data,” Phys. Rev. D 104, 063510\n(2021), arXiv:2103.09729 [astro-ph.CO].\n[48] Jun-Qian Jiang and Yun-Song Piao, “Toward early\ndark energy and ns=1 with Planck, ACT, and SPT\nobservations,” Phys. Rev. D 105, 103514 (2022),\narXiv:2202.13379 [astro-ph.CO].\n[49] Jun-Qian Jiang, Gen Ye, and Yun-Song Piao, “Return\nof Harrison-Zeldovich spectrum in light of recent cos-\nmological tensions,” (2022), arXiv:2210.06125 [astro-\nph.CO].\n[50] Ze-Yu Peng and Yun-Song Piao, “Testing the ns−H0\nscaling relation with Planck-independent CMB data,”\n(2023), arXiv:2308.01012 [astro-ph.CO].\n[51] Matteo Braglia, William T. Emond, Fabio Finelli,\nA. Emir Gumrukcuoglu, and Kazuya Koyama, “Uni-\nfied framework for early dark energy from α-attractors,”\nPhys. Rev. D 102, 083513 (2020), arXiv:2005.14053\n[astro-ph.CO].\n[52] Fuminobu Takahashi and Wen Yin, “Cosmological im-\nplications of ns=1 in light of the Hubble tension,” Phys.\nLett. B 830, 137143 (2022), arXiv:2112.06710 [astro-\nph.CO].\n[53] Gen Ye, Jun-Qian Jiang, and Yun-Song Piao, “Toward\ninflation with ns=1 in light of the Hubble tension and\nimplications for primordial gravitational waves,” Phys.\nRev. D 106, 103528 (2022), arXiv:2205.02478 [astro-\nph.CO].\n[54] Matteo Braglia, Andrei Linde, Renata Kallosh, and\nFabio Finelli, “Hybrid α-attractors, primordial black\nholes and gravitational wave backgrounds,” JCAP 04,\n033 (2023), arXiv:2211.14262 [astro-ph.CO].\n[55] Chia-Min Lin, “D-term inflation in braneworld models:\nConsistency with cosmic-string bounds and early-time\nHubble tension resolving models,” Phys. Rev. D 106,\n103511 (2022), arXiv:2204.10475 [hep-th].\n[56] Chengjie Fu and Shao-Jiang Wang, “Landing early\ndark energy on Harrison-Zeldovich Universe,” (2023),\narXiv:2310.12932 [astro-ph.CO].\n[57] Sunny Vagnozzi, “Seven Hints That Early-Time New\nPhysics Alone Is Not Sufficient to Solve the Hubble Ten-\nsion,” Universe 9, 393 (2023), arXiv:2308.16628 [astro-\nph.CO].8\n[58] Antonio J. Cuesta, Licia Verde, Adam Riess, and Raul\nJimenez, “Calibrating the cosmic distance scale lad-\nder: the role of the sound horizon scale and the local\nexpansion rate as distance anchors,” Mon. Not. Roy.\nAstron. Soc. 448, 3463–3471 (2015), arXiv:1411.1094\n[astro-ph.CO].\n[59] Alan Heavens, Raul Jimenez, and Licia Verde,\n“Standard rulers, candles, and clocks from the low-\nredshift Universe,” Phys. Rev. Lett. 113, 241302 (2014),\narXiv:1409.6217 [astro-ph.CO].\n[60] Eric Aubourg etal., “Cosmological implications of\nbaryon acoustic oscillation measurements,” Phys. Rev.\nD92, 123516 (2015), arXiv:1411.1074 [astro-ph.CO].\n[61] Licia Verde, Jose Luis Bernal, Alan F. Heavens, and\nRaul Jimenez, “The length of the low-redshift stan-\ndard ruler,” Mon. Not. Roy. Astron. Soc. 467, 731–736\n(2017), arXiv:1607.05297 [astro-ph.CO].\n[62] Shadab Alam etal. (BOSS), “The clustering of galaxies\nin the completed SDSS-III Baryon Oscillation Spectro-\nscopic Survey: cosmological analysis of the DR12 galaxy\nsample,” Mon. Not. Roy. Astron. Soc. 470, 2617–2652\n(2017), arXiv:1607.03155 [astro-ph.CO].\n[63] E. Macaulay etal. (DES), “First Cosmological Results\nusing Type Ia Supernovae from the Dark Energy Survey:\nMeasurement of the Hubble Constant,” Mon. Not. Roy.\nAstron. Soc. 486, 2184–2196 (2019), arXiv:1811.02376\n[astro-ph.CO].\n[64] Stephen M. Feeney, Hiranya V. Peiris, Andrew R.\nWilliamson, Samaya M. Nissanke, Daniel J. Mortlock,\nJustin Alsing, and Dan Scolnic, “Prospects for resolv-\ning the Hubble constant tension with standard sirens,”\nPhys. Rev. Lett. 122, 061105 (2019), arXiv:1802.03404\n[astro-ph.CO].\n[65] Michael J. Mortonson, Wayne Hu, and Dragan Huterer,\n“Hiding dark energy transitions at low redshift,” Phys.\nRev. D 80, 067301 (2009), arXiv:0908.1408 [astro-\nph.CO].\n[66] Zhiqi Huang, “Supernova Magnitude Evolution and\nPAge Approximation,” Astrophys. J. Lett. 892, L28\n(2020), arXiv:2001.06926 [astro-ph.CO].\n[67] Xiaolin Luo, Zhiqi Huang, Qiyue Qian, and Lu Huang,\n“Reaffirming the Cosmic Acceleration without Super-\nnovae and the Cosmic Microwave Background,” As-\ntrophys. J. 905, 53 (2020), arXiv:2008.00487 [astro-\nph.CO].\n[68] Lu Huang, Zhiqi Huang, Xiaolin Luo, Xinbo He, and\nYuhong Fang, “Reconciling low and high redshift GRB\nluminosity correlations,” Phys. Rev. D 103, 123521\n(2021), arXiv:2012.02474 [astro-ph.CO].\n[69] Lu Huang, Zhiqi Huang, Zhuoyang Li, and Huan\nZhou, “A More Accurate Parameterization based\non cosmic Age (MAPAge),” (2021), 10.1088/1674-\n4527/21/11/277, arXiv:2108.03959 [astro-ph.CO].\n[70] Lu Huang, Zhiqi Huang, Huan Zhou, and Zhuoyang Li,\n“The S 8tension in light of updated redshift-space dis-\ntortion data and PAge approximation,” Sci. China Phys.\nMech. Astron. 65, 239512 (2022), arXiv:2110.08498\n[astro-ph.CO].\n[71] Zhiqi Huang, “Thawing k-essence dark energy in the\nPAge space,” Commun. Theor. Phys. 74, 095404 (2022),\narXiv:2204.09713 [astro-ph.CO].\n[72] Zhuoyang Li, Lu Huang, and Junchao Wang, “Redshift\nevolution and non-universal dispersion of quasar lumi-\nnosity correlation,” Mon. Not. Roy. Astron. Soc. 517,1901–1906 (2022), arXiv:2210.02816 [astro-ph.CO].\n[73] Junchao Wang, Zhiqi Huang, and Lu Huang, “Revis-\niting progenitor-age dependence of type Ia supernova\nluminosity standardization process,” Sci. China Phys.\nMech. Astron. 66, 129511 (2023), arXiv:2303.15267\n[astro-ph.CO].\n[74] Bao Wang, Yang Liu, Hongwei Yu, and Puxun\nWu, “Observations favor the redshift-evolutionary LX-\nLUV relation of quasars from copula,” (2024),\narXiv:2401.01540 [astro-ph.CO].\n[75] Raul Jimenez and Abraham Loeb, “Constraining\ncosmological parameters based on relative galaxy\nages,” Astrophys. J. 573, 37–42 (2002), arXiv:astro-\nph/0106145.\n[76] Nikki Arendse etal., “Cosmic dissonance: are new\nphysics or systematics behind a short sound horizon?”\nAstron. Astrophys. 639, A57 (2020), arXiv:1909.07986\n[astro-ph.CO].\n[77] Kenneth C. Wong etal., “H0LiCOW - XIII. A 2.4\nper cent measurement of H0 from lensed quasars:\n5.3σtension between early- and late-Universe probes,”\nMon. Not. Roy. Astron. Soc. 498, 1420–1439 (2020),\narXiv:1907.04869 [astro-ph.CO].\n[78] Morag Scrimgeour etal., “The WiggleZ Dark Energy\nSurvey: the transition to large-scale cosmic homogene-\nity,” Mon. Not. Roy. Astron. Soc. 425, 116–134 (2012),\narXiv:1205.6812 [astro-ph.CO].\n[79] Edwin L. Turner, Renyue Cen, and Jeremiah P. Os-\ntriker, “The Relationship of Local measures of Hubble’s\nConstant to its Global Value,” Astronomical Journal\n103, 1427 (1992).\n[80] Xiangdong Shi, Lawrence M. Widrow, and L. Jonathan\nDursi, “Measuring hubble’s constant in our inhomoge-\nneous universe,” Mon. Not. Roy. Astron. Soc. 281, 565\n(1996), arXiv:astro-ph/9506120.\n[81] Xiang-Dong Shi and Michael S. Turner, “Expectations\nfor the difference between local and global measure-\nments of the Hubble constant,” Astrophys. J. 493, 519\n(1998), arXiv:astro-ph/9707101.\n[82] Yun Wang, David N. Spergel, and Edwin L.\nTurner, “Implications of cosmic microwave background\nanisotropies for large scale variations in Hubble’s\nconstant,” Astrophys. J. 498, 1 (1998), arXiv:astro-\nph/9708014.\n[83] W. D’Arcy Kenworthy, Dan Scolnic, and Adam Riess,\n“The Local Perspective on the Hubble Tension: Lo-\ncal Structure Does Not Impact Measurement of the\nHubble Constant,” Astrophys. J. 875, 145 (2019),\narXiv:1901.08681 [astro-ph.CO].\n[84] Benjamin Sinclair, Tamara M. Davis, and Troels\nHaugbolle, “Residual Hubble-bubble effects on super-\nnova cosmology,” Astrophys. J. 718, 1445–1455 (2010),\narXiv:1006.0911 [astro-ph.CO].\n[85] Valerio Marra, Luca Amendola, Ignacy Sawicki, and\nWessel Valkenburg, “Cosmic variance and the measure-\nment of the local Hubble parameter,” Phys. Rev. Lett.\n110, 241305 (2013), arXiv:1303.3121 [astro-ph.CO].\n[86] Ido Ben-Dayan, Ruth Durrer, Giovanni Marozzi, and\nDominik J. Schwarz, “The value of H0in the inhomoge-\nneous Universe,” Phys. Rev. Lett. 112, 221301 (2014),\narXiv:1401.7973 [astro-ph.CO].\n[87] David Camarena and Valerio Marra, “Impact of the cos-\nmic variance on H0on cosmological analyses,” Phys.\nRev. D 98, 023537 (2018), arXiv:1805.09900 [astro-9\nph.CO].\n[88] Dillon Brout etal., “The Pantheon+ Analysis: Cos-\nmological Constraints,” Astrophys. J. 938, 110 (2022),\narXiv:2202.04077 [astro-ph.CO].\n[89] Dillon Brout etal., “The Pantheon+ Analysis:\nSuperCal-fragilistic Cross Calibration, Retrained\nSALT2 Light-curve Model, and Calibration System-\natic Uncertainty,” Astrophys. J. 938, 111 (2022),\narXiv:2112.03864 [astro-ph.CO].\n[90] Erik R. Peterson etal., “The Pantheon+ Analysis:\nEvaluating Peculiar Velocity Corrections in Cosmologi-\ncal Analyses with Nearby Type Ia Supernovae,” (2021),\narXiv:2110.03487 [astro-ph.CO].\n[91] Anthony Carr, Tamara M. Davis, Dan Scolnic, Daniel\nScolnic, Khaled Said, Dillon Brout, Erik R. Peterson,\nand Richard Kessler, “The Pantheon+ analysis: Im-\nproving the redshifts and peculiar velocities of Type Ia\nsupernovae used in cosmological analyses,” Publ. As-\ntron. Soc. Austral. 39, e046 (2022), arXiv:2112.01471\n[astro-ph.CO].\n[92] Brodie Popovic, Dillon Brout, Richard Kessler, and\nDaniel Scolnic, “The Pantheon+ Analysis: Forward\nModeling the Dust and Intrinsic Color Distributions of\nType Ia Supernovae, and Quantifying Their Impact on\nCosmological Inferences,” Astrophys. J. 945, 84 (2023),\narXiv:2112.04456 [astro-ph.CO].\n[93] D. M. Scolnic etal., “The Complete Light-curve Sam-\nple of Spectroscopically Confirmed SNe Ia from Pan-\nSTARRS1 and Cosmological Constraints from the Com-\nbined Pantheon Sample,” Astrophys. J. 859, 101 (2018),\narXiv:1710.00845 [astro-ph.CO].\n[94] N. Kaiser, “Clustering in real space and in redshift\nspace,” Mon. Not. Roy. Astron. Soc. 227, 1–27 (1987).\n[95] C. Alcock and B. Paczynski, “An evolution free test for\nnon-zero cosmological constant,” Nature 281, 358–359\n(1979).\n[96] Xiaoying Xu, Antonio J. Cuesta, Nikhil Padmanabhan,\nDaniel J. Eisenstein, and Cameron K. McBride, “Mea-\nsuring D A and H at z=0.35 from the SDSS DR7 LRGs\nusing baryon acoustic oscillations,” Mon. Not. Roy. As-\ntron. Soc. 431, 2834 (2013), arXiv:1206.6732 [astro-\nph.CO].\n[97] Teppei Okumura, Takahiko Matsubara, Daniel J. Eisen-\nstein, Issha Kayo, Chiaki Hikage, Alexander S. Szalay,\nand Donald P. Schneider, “Large-Scale Anisotropic Cor-\nrelation Function of SDSS Luminous Red Galaxies,” As-\ntrophys. J. 676, 889–898 (2008), arXiv:0711.3640 [astro-\nph].\n[98] Enrique Gaztanaga, Anna Cabre, and Lam Hui, “Clus-\ntering of Luminous Red Galaxies IV: Baryon Acoustic\nPeak in the Line-of-Sight Direction and a Direct Mea-\nsurement of H(z),” Mon. Not. Roy. Astron. Soc. 399,\n1663–1680 (2009), arXiv:0807.3551 [astro-ph].\n[99] Chris Blake etal., “The WiggleZ Dark Energy Sur-\nvey: measuring the cosmic expansion history using\nthe Alcock-Paczynski test and distant supernovae,”\nMon. Not. Roy. Astron. Soc. 418, 1725–1735 (2011),\narXiv:1108.2637 [astro-ph.CO].\n[100] Chia-Hsun Chuang and Yun Wang, “Measurements of\nH(z) and D A(z) from the Two-Dimensional Two-Point\nCorrelation Function of Sloan Digital Sky Survey Lumi-\nnous Red Galaxies,” Mon. Not. Roy. Astron. Soc. 426,\n226 (2012), arXiv:1102.2251 [astro-ph.CO].\n[101] Asta Heinesen, Chris Blake, and David L. Wilt-shire, “Quantifying the accuracy of the Alcock-\nPaczyński scaling of baryon acoustic os-\ncillation measurements,” JCAP 01, 038 (2020),\narXiv:1908.11508 [astro-ph.CO].\n[102] Paul Carter, Florian Beutler, Will J. Percival, Joseph\nDeRose, Risa H. Wechsler, and Cheng Zhao, “The\nimpact of the fiducial cosmology assumption on BAO\ndistance scale measurements,” Mon. Not. Roy. Astron.\nSoc.494, 2076–2089 (2020), arXiv:1906.03035 [astro-\nph.CO].\n[103] Shao-Jiang Wang, Qi Guo, and Rong-Gen Cai,\n“Smoothing the redshift distributions of random sam-\nples for the baryon acoustic oscillations : applications\nto the SDSS-III BOSS DR12 and QPM mock samples,”\nMon. Not. Roy. Astron. Soc. 472, 2869–2876 (2017),\narXiv:1701.02427 [astro-ph.CO].\n[104] Krzysztof Bolejko, “Emerging spatial curvature can\nresolve the tension between high-redshift CMB and\nlow-redshift distance ladder measurements of the Hub-\nble constant,” Phys. Rev. D 97, 103529 (2018),\narXiv:1712.02967 [astro-ph.CO].\n[105] G. C. Carvalho, A. Bernui, M. Benetti, J. C. Carvalho,\nand J. S. Alcaniz, “Baryon Acoustic Oscillations from\nthe SDSS DR10 galaxies angular correlation function,”\nPhys. Rev. D 93, 023530 (2016), arXiv:1507.08972\n[astro-ph.CO].\n[106] E. de Carvalho, A. Bernui, F. Avila, C. P. Novaes, and\nJ. P. Nogueira-Cavalcante, “BAO angular scale at zeff =\n0.11 with the SDSS blue galaxies,” Astron. Astrophys.\n649, A20 (2021), arXiv:2103.14121 [astro-ph.CO].\n[107] Jailson S. Alcaniz, Gabriela C. Carvalho, Armando\nBernui, Joel C. Carvalho, and Micol Benetti, “Measur-\ning baryon acoustic oscillations with angular two-point\ncorrelation function,” Fundam. Theor. Phys. 187, 11–19\n(2017), arXiv:1611.08458 [astro-ph.CO].\n[108] G. C. Carvalho, A. Bernui, M. Benetti, J. C. Carvalho,\nE. de Carvalho, and J. S. Alcaniz, “The transverse\nbaryonic acoustic scale from the SDSS DR11 galaxies,”\nAstropart. Phys. 119, 102432 (2020), arXiv:1709.00271\n[astro-ph.CO].\n[109] E. de Carvalho, A. Bernui, H. S. Xavier, and C. P.\nNovaes, “Baryon acoustic oscillations signature in the\nthree-point angular correlation function from the SDSS-\nDR12 quasar survey,” Mon. Not. Roy. Astron. Soc. 492,\n4469–4476 (2020), arXiv:2002.01109 [astro-ph.CO].\n[110] Armando Bernui, Eleonora Di Valentino, William Giar` e,\nSuresh Kumar, and Rafael C. Nunes, “Exploring the\nH0 tension and the evidence for dark sector interac-\ntions from 2D BAO measurements,” Phys. Rev. D 107,\n103531 (2023), arXiv:2301.06097 [astro-ph.CO].\n[111] Thais Lemos, Ruchika, Joel C. Carvalho, and Jailson\nAlcaniz, “Low-redshift estimates of the absolute scale of\nbaryon acoustic oscillations,” Eur. Phys. J. C 83, 495\n(2023), arXiv:2303.15066 [astro-ph.CO].\n[112] Michele Moresco, Raul Jimenez, Licia Verde, Andrea\nCimatti, and Lucia Pozzetti, “Setting the Stage for\nCosmic Chronometers. II. Impact of Stellar Population\nSynthesis Models Systematics and Full Covariance Ma-\ntrix,” Astrophys. J. 898, 82 (2020), arXiv:2003.07362\n[astro-ph.GA].\n[113] Raul Jimenez, Licia Verde, Tommaso Treu, and Daniel\nStern, “Constraints on the equation of state of dark en-\nergy and the Hubble constant from stellar ages and the\nCMB,” Astrophys. J. 593, 622–629 (2003), arXiv:astro-10\nph/0302560.\n[114] Joan Simon, Licia Verde, and Raul Jimenez, “Con-\nstraints on the redshift dependence of the dark energy\npotential,” Phys. Rev. D 71, 123001 (2005), arXiv:astro-\nph/0412269.\n[115] Daniel Stern, Raul Jimenez, Licia Verde, Marc\nKamionkowski, and S. Adam Stanford, “Cosmic\nChronometers: Constraining the Equation of State of\nDark Energy. I: H(z) Measurements,” JCAP 02, 008\n(2010), arXiv:0907.3149 [astro-ph.CO].\n[116] M. Moresco etal., “Improved constraints on the expan-\nsion rate of the Universe up to z˜1.1 from the spec-\ntroscopic evolution of cosmic chronometers,” JCAP 08,\n006 (2012), arXiv:1201.3609 [astro-ph.CO].\n[117] Cong Zhang, Han Zhang, Shuo Yuan, Tong-Jie Zhang,\nand Yan-Chun Sun, “Four new observational H(z) data\nfrom luminous red galaxies in the Sloan Digital Sky Sur-\nvey data release seven,” Res. Astron. Astrophys. 14,\n1221–1233 (2014), arXiv:1207.4541 [astro-ph.CO].\n[118] Michele Moresco, “Raising the bar: new constraints on\nthe Hubble parameter with cosmic chronometers at z ∼\n2,” Mon. Not. Roy. Astron. Soc. 450, L16–L20 (2015),\narXiv:1503.01116 [astro-ph.CO].\n[119] Michele Moresco, Lucia Pozzetti, Andrea Cimatti,\nRaul Jimenez, Claudia Maraston, Licia Verde, Daniel\nThomas, Annalisa Citro, Rita Tojeiro, and David\nWilkinson, “A 6% measurement of the Hubble pa-\nrameter at z∼0.45: direct evidence of the epoch\nof cosmic re-acceleration,” JCAP 05, 014 (2016),\narXiv:1601.01701 [astro-ph.CO].\n[120] A. L. Ratsimbazafy, S. I. Loubser, S. M. Crawford,\nC. M. Cress, B. A. Bassett, R. C. Nichol, andP. V¨ ais¨ anen, “Age-dating Luminous Red Galaxies ob-\nserved with the Southern African Large Telescope,”\nMon. Not. Roy. Astron. Soc. 467, 3239–3254 (2017),\narXiv:1702.00418 [astro-ph.CO].\n[121] Nicola Borghi, Michele Moresco, and Andrea Cimatti,\n“Towards a Better Understanding of Cosmic Chronome-\nters: A new measurement of H(z) atz∼0.7,” (2021),\narXiv:2110.04304 [astro-ph.CO].\n[122] Daniel Foreman-Mackey, David W. Hogg, Dustin Lang,\nand Jonathan Goodman, “emcee: The MCMC Ham-\nmer,” Publ. Astron. Soc. Pac. 125, 306–312 (2013),\narXiv:1202.3665 [astro-ph.IM].\n[123] Gideon Schwarz, “Estimating the Dimension of a\nModel,” Annals Statist. 6, 461–464 (1978).\n[124] Roberto Trotta, “Bayesian Methods in Cosmology,”\n(2017) arXiv:1701.01467 [astro-ph.CO].\n[125] Harold Jeffreys, The Theory of Probability, Oxford\nClassic Texts in the Physical Sciences (1939).\n[126] Joshua S. Speagle, “dynesty: a dynamic nested sam-\npling package for estimating Bayesian posteriors and ev-\nidences,” Mon. Not. Roy. Astron. Soc. 493, 3132–3158\n(2020), arXiv:1904.02180 [astro-ph.IM].\n[127] John Skilling, “Nested Sampling,” AIP Conf. Proc. 735,\n395 (2004).\n[128] Wendy L. Freedman etal., “The Carnegie-Chicago\nHubble Program. VIII. An Independent Determina-\ntion of the Hubble Constant Based on the Tip of\nthe Red Giant Branch,” Astrophys. J. 882, 34 (2019),\narXiv:1907.05922 [astro-ph.CO].\n[129] Wang-Wei Yu, Li Li, and Shao-Jiang Wang, “First de-\ntection of the Hubble variation correlation and its scale\ndependence,” (2022), arXiv:2209.14732 [astro-ph.CO]." }, { "title": "2401.14173v2.Multicasting_Optical_Reconfigurable_Switch.pdf", "content": "Multicast ing Optical Reconfigurable Switch \n \nNiyazi Ulas Dinc1, †, Mustafa Yildirim1, †, Ilker Oguz1, Christophe Moser1, and Demetri Psaltis2 \n1 Laboratory of Applied Photonics Devices, Ecole Polytechnique Fédérale de Lausanne (EPFL), \nSwitzerland \n2 Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland \n† Equally contributing authors . \n \nAbstract \nArtificial Intelligence (AI ) demand s large data flows within datacenters, heavily relying \non multicasting data transfer s. As AI models scale, the requi rement for high -bandwidth \nand low -latency networking compounds . The common use of electrical packet \nswitching faces limitations due to optical -electrical -optical conversion bottleneck s. \nOptical switches, while bandwidth -agnostic and low -latency, suffer from having only \nunicast or non -scalable multicasting capability . This paper introduces an optical \nswitching technique addressing th is challenge . Our approach enables arbitrarily \nprogrammable simultaneous unicast and multicast connectivity, eliminating the need \nfor optical splitters that hinder scalability due to optical power loss. We use phase \nmodulation in multiple layers , tailored to implement any multicast connectivit y map . \nPhase modulation also enables wavelength selectivity on top of spatial selectivity, \nresulting in an optical switch that implements space -wavelength routing . We \nconducted simulations and experiments to validate our approach . Our r esults affirm \nthe concept's feasibility , effectiveness , and scalability , as a multicasting switch by \nexperimentally demonstrating 16 spatial ports using 2 wavelength channels . \nNumerically , 64 spatial ports with 4 wavelength channels each were simulated , with \napproximately constant efficien cy (< 3 dB) as ports and wavelength channels scale . \n Datacenters manage and process large volumes of data , comprising thousands of \ncomputing nodes alongside networking and storage systems . The n etworking fabric \nenables interconnection across many devices to execut e workloads efficiently . Data \nrouting within a datacenter relies primari ly on electrical packet switching through fiber \noptic cables , where an electronic switch receives a data packet , processes, and \nregenerate s it for the target port . In traditional network architectures, network devices \nthemselves make decisions about where to forward data, based on pre -defined \nprotocols . Meanwhi le, network management has evolved, and software -defined \nnetwork ing (SDN) has been established to manage the datacenter network more \nefficiently , optimizing global traffic across the entire data center [1]. SDN separates the \ncontrol plane (deciding where traffic should be sent) from the data plane (the actual \nforwarding of the traffic). Therefore , it is not necessary anymore to read packets , \ninstead, the centralized controller can communicate the connectivity map to the switch . \nThis evolution enable d the adoption of unicast (one-to-one) optical circuit switch es \n(OCS) in datacenters [2-4]. \nLarge AI models can occupy thousands of compute r nodes , with model parameters \nand dataset are multicast among nodes iteratively during training [5, 6]. As the volume \nof multicast communication patterns increases along with the scale of operations, \nnetwork congestion become s unavoidable . Current optical switches cater solely to \nunicast, managing multicast via message replication with multiple electronic switch es \nor passive power splitters coupled with programmable unicasting OCSs [4, 6-9]. \nHowever, the latter's power loss (due to m ultiple fixed divisions ) curbs scalability, \nleaving a need for scalable and reconfigurable OCSs equipped with both unicast and \nmulticast functionalities . \n \nOCS based on microelectromechanical systems ( MEMS ) mirrors [ 10] is a deployed \noptical interconnect technology . While MEMS switches can provide energy -efficient \noptical routing, they cannot multicast a single input to many output channels. This \narises from the ray -optic characteristics when using millimeter -sized, two -axis \ncontinuously rotating mirrors. Moreover, the se systems are not able to switch among \nwavelength channels. [3, 4]. Another solution, Wavelength -selective switches (WSSs) \nare devices capable of redirecting optical signals to different output ports based on \ntheir wavelengths. Such switches typically consist of a one -dimensional (1D) diffraction grating and a beam -steering mechanism. The diffraction grating disperses \nthe optical signals into their constituent wavelengths, while the beam -steering \nmechanism directs them to specific output ports based on their wavelengths. In many \nWSSs, L iquid -Crystal -on-Silicon (LCoS) spatial light modulators are utilized as beam -\nsteering devices [11, 12]. However, the limitation of WSSs is that they can only redirect \none input signal to one output port at a time. Consequently, WSSs are also unsuitable \nfor multicasting. Furthermore , the 1D input/output configuration also limits the \nscalability of WSSs unless it is combined with MEMS -based re -routing at the cost of \nincreased complexity [ 12]. Fixed interconnect methodologies, such as waveguides \nand volume hologr ams [14-18] are studied to address some of the mentioned \nchallenges. However, they are non-programmable, restricting their applicability in \ndynamic environments like datacenters . \n \nIn this work, we introduce a multicasting optical reconfigurable switch (MORS ) capable \nof multicast and wavelength selectivity. Our method leverages spatial light modulation \nto enable programming of optical paths consisting of both unicast and multicast \nconnections (Fig. 1a) . The current approach in industry for multicast employs fixed \npower splitter s combined with optical unicast switches (Fig. 1b ). An optical unicast \nswitch, like MEMS -based mirror arrays, acts as a shutter to control light passage. In \nthis approach, a switch having Ni = No = N input/output ports splits each input light into \nN paths, regardless of the demanded multicast port count (N m). Consequently, when \nthe number of demanded multicast receivers is less than the output port count (Nm400 GHz at \n850 nm central wavelength. For the experiments, we show the wavelength selectivity \nfor two wavelengths at 835 nm and 865 nm and provide the experimentally obtained \nefficiency -crosstalk plot in Fig. 3 f. In these experiments, w e obtained -2.10 dB insertion \nloss and -19.12 dB crosstalk. \n \nFigure 3: Efficiency and crosstalk results of MORS for broadband and wavelength -selective \noperation . a) Schematic representation of MORS showcasing wavelength selective unicast and \nmulticast connections . b) The efficiency -crosstalk plots (b, d, f) show the correctly routed powers on the \ndiagona l from the nth spatial input port 𝜎𝑖𝑛 to the corresponding nth spatial output port 𝜎𝑜𝑛 for each \nwavelength . Off -diagonal elements are crosstalk between the target and other ports. Simulated \nefficiency -crosstalk plot for a single wavelength with four spatial ports is depicted, providing insights \ninto the connectivity map. c) Average optical power transmission for broadband application (covering \nthe same connectivity over a 100nm range) is illustrated, we present power transmissions for both single \nand multiple wavelength training cases . d) The simulated efficiency -crosstalk plot s for four different \nwavelengths are presented, show ing distinct connectivity maps for each wavelength channel, all \nsharing spatial ports. e) Power transmission of four different wavelengths is shown while each \nwavelength has a different connectivity map. f) Experimentally measured efficiency -crosstalk plots for \ntwo wavelength channels (835nm, 865nm ), demonstrating the practical application of MORS . \n \n3. Scaling study \nFor the implementation of large networks, the scalability of MORS is of paramount \nimportance. Therefore, we conducted a scaling study based on simulations by \ndevising a switch t hat has 64 spatial input ports and 64 spatial output ports operating \nwith four different wavelength channels . This result s in 256 spatiotemporal input and \n256 spatiotemporal output ports. We calculate the parameter count of MORS by \nmultiplying the layer number by the number of pixels used in one layer , which results \nin the total number of pixels (see Methods). In Fig. 4a, we plot the mean efficiency \nresults for different c onfigurations . We clearly see that the increased degrees of \nfreedom improve the overall performance. We observe that while the 2 -layer \nconfiguration performs poorly, 6 -layer and 8 -layer configurations settle in a \ncompara ble scaling trend, showing a direct link between employed parameters and \nachievable efficiency once the system has sufficient depth to use diffraction to \nseparate and route different wavelengths to their corresponding spatial output ports. \nIn Fig. 4b, we present signal to noise ratio (SNR) which is coupled optical power over \ncrosstalk in dB scale. We observe that although the coupling efficiency level s off \naround three million parameters/pixels , SNR continues to improve with the additional \nparameters. \n \nNext, we investigate how parameters /pixels are shared among the different \nconnections, where a connection is a link from a specific input to one or more desired \noutput ports. We define allocated pixels by counting how many pixels are used when \ninput light is routed to its target port(s). In Fig. 4c, we illustrate an example of allocated \npixels corresponding to the multicasting configuration up to 64 (the last row of Fig. 2d ). \nThe pixel values of Fig. 4c indicate how many connections go through each pixel. In \nFig. 4 d, we show how well each connection works based on how many pixels it uses \nfor each output port . For example, a connection multicasting to 64 ports uses around \n400,000 pixels, which results in roughly 6 ,000 pixels per output port as shown in Fig. \n4d. The data reveals an inverse relationship between the connection efficiency and its \nmulticast port count. While connections with higher multicasting use more pixels \noverall , the number of allocated pixels per output port is lower. \n \nFurthermore, significant variations arise between different random mappings for large \nmulticast counts (i.e. 32 or 64) indicated by the same color -code in Fig. 4 d. This \nsuggests that efficiency is impacted not only by the number of allocated pixels but also \nby their overlap. We define overlap as the degree to which parameters /pixels allocated \nby a specific connection are shared with other connections (see Methods). In Fig. 4 e, \nwe plot efficiency against the overlap calculated f rom the data points of Fig. 4 d. The results demonstrate that higher overlap (more shared pixels) leads to a lower \nconnection efficiency. This trend becomes increasingly pronounced for connections \nwith larger multicast port counts. For example, the yellow data point with the efficiency \nof >70% shown in Fig. 4 d is located on the left -most side of Fig. 4c, indicating a small \namount of overlap for this particular connection. \n \n \n \n \nFigure 4: Efficiency trend of MORS with respect to the number of pixels (parameter count) . a) \nAverage coupling efficiencies for various number of layers are plotted while parameter count is varied \nby changing layer width. b) Scaling of SNR for the same configuration as in a. c) An example of allocated \npixels over six layers. The color code measures how many connections pass through each pixel. d) The \nefficiency of individual connections for a single wavelength switch having 128 ports . The number of \nmulticasting outputs is up to 64. The data for ten different (randomly picked) connectivit y are present. \nX-axis denotes the number of pixels that are actively used by these connections . c) The efficiency \nversus overlap (shared pixels) plot for the same device as in d. Efficiency follows a decreasing trend \nwith higher overlap. In d and e, color code represents multicast port count. \n \n \nDiscussion \nMORS is different from conventional methods by introduc ing a novel optical path \nreconfiguration employing spatial light modulation across multiple planes . This \napproach inherently incorpora tes multicasting capabilities with wavelength selectivity . \nThe p romising scaling trends we obtained from numerical studies were substantiated \nby experimental validation. It is noted that free -space techniques inherently lead to \nefficient throughput , a key advantage over integrated solutions where factors like \nwaveguide loss and coupling losses can impact scaling . Moreover, the ability to \noperate on the wavefront facilitates optimization with respect to the étendue of the \noutput fibers . For simplicity , this aspect is ignored in the drawings show n in Fig. 1c and \n1d. MEMS -based systems require two separate mirror arrays, the first one to define \nthe output position and the second one to correct the beam angle to couple light . As \nthe beam size gets larger along with the device scale , respecting the étendue \nbecomes challenging without precise angle control of the mirrors [4]. This is one of the \nreason s limiting the scalability of MEMS switches [19]. MORS on the other hand has \nmore degrees of freedom to shape the wavefront , optimizing étendue of the beam for \ncoupling . \n \nOur scaling analysis demonstrated a consistent improvement in the SNR with an \nincreasing number of pixels. Notably, at around a million pixels, deeper architectures \nlead to enhanced performance . However, our findings also revealed a performance \nbottleneck stemming from shared pixel usage among connections, highlighting the \nlimitations of solely increasing parameter count for scaling. \n \nWith MORS , the efficiency of multicasting remains consistent, offering a potential \nresolution to longstanding challenges for upscaling multicasting optical switching . Our \napproach also has the capability to manage both space and wavelength . To conclude, \nMORS offers a new perspective in network design, potentially bridging the efficiency \ngap in multicasting present in today's optical interconnect technologies with additional \nflexibility by possessing space -wavelength granularity. \n \n \nMethods \n \nExperimental setup: For our experiments, a continuous wave Solstis M2 laser was \nemployed. The selected mirror, measuring 11.6 mm in width and 17 .1 mm away from \nthe SLM display , accommodates the four reflections. To channel the input beam to the \nSLM, 4F imaging was applied, relaying the beam that was reflected off a digital \nmicromirror device (DMD). This DMD acts to simulate the input grid by activating \nspecific sub -regions. For the modulation layers, we designated patches of 280 by 280 \npixels on the SLM. The SLM used in our arrangement has a pixel pitch of Λ=9.2 μ m. Following the fourth reflection, the beam is imaged, with the resultant output intensity \ncaptured by a CMOS camera. For the wavelength -selective experiments, the tunable \nlaser is utilized to test the results at two wavelengths that are 30 nm apart (835 nm \nand 865 nm). \n \nMulticast and space -wavelength granularity : In numerical studies showing \nmulticast in the first part of the Results section , we employed six diffractive layer s. The \nlayer width is set to 640 pixels for the multicast case. To achieve wavelength selectivity, \nwe execute the forward model for each desired central wavelength, optimizing the \npixel phase based on the aggregated backpropagated error signals. We refer to these \nwavelengths as training wavelengths. For wavelength -selective numerical \nexperiments, we also used six diffractive layers having a layer width of 360 pixels . \nLayer -to-layer distance is set to four millimeters for both cases. The d etails about the \ntraining method can be found in [27]. \n \nFor the scaling study, we used the layer width values 307, 538, 768, and 1036. The \nnumber of layers is swept from 2 to 8. We activated 20 input ports for each wavelength \nchannel randomly . By exceeding 64 available spatial ports, we ensured that some of \nthe ports transmit multiple wavelengths. For the corresponding outputs, we populated \nall the possibilities, me aning that there are 256 active outputs distributed among 64 \nspatial ports and 4 wavelength channels. Each input is mapped randomly to the output \nports including unicast and multicast connections, where the number of multicasts for \neach input is also arranged randomly. In other words, a single input is mapped to \noutput ports where the output port number is in the range of 1 to 44. These selections \nare restricted to the available number of output ports and channels. \n \nOverlap (shared pixels) : We calculate the overlap amount to indicate shared pixels \nby tracing each input over the modulation planes and label pixels where the light \nintensity is higher than a certain threshold as a “allocated pixel” for that input. We \niterate this procedure for al l the inputs and obtain allocated pixels for every connection. \nNext, we assign a degree to pixels according to the number of connections that it \ncontributes to. For a pixel that only contributes to a single connection, this degree is \nzero. The overlap is defined as sum of the assigned degrees of allocated pixels, \ndivided by thousand. \nReferences \n \n[1] McKeown, N., Anderson, T., Balakrishnan, H., Parulkar, G., Peterson, L., Rexford, J., Shenker, S., & \nTurner, J. (2008). OpenFlow: enabling innovation in campus networks. ACM SIGCOMM computer \ncommunication review , 38(2), 69 -74. \n[2] Farrington, N., Porter, G., Radhakrishnan, S., Bazzaz, H. H., Subramanya, V., Fainman, Y., Papen, \nG., & Vahdat, A. (2010, August). Helios: a hybrid electrical/optical switch architecture for modular data \ncenters. In Proceedings of the ACM SIGCOMM 2010 Conference (pp. 339 -350). \n[3] Poutievski, L., Mashayekhi, O., Ong, J., Singh, A., Tariq, M., Wang, R., Zhang, J., Beauregard, V., \nConner, P., Gribble, S., Kapoor, R., Kratzer, S., Li, N., Liu, H., Nagaraj, K., Ornstein, J., Sawhney, S., \nUrata, R., Vicisano, L., Yasamura, K., Zhang, S., Zhou, J., & Vahdat, A. (2022, August). Jupiter evolving: \ntransforming google's datacenter network via optical circuit switches and software -defined networking. \nIn Proceedings of the ACM SIGCOMM 2022 Conference (pp. 66 -85). \n[4] Urata, R., Liu, H., Yasumura, K., Mao, E., Berger, J., Zhou, X., Lam, C., Bannon, R., Hutchinson, D., \nNelson, D., Poutievski, L., Singh, A.,Ong, J.,& Vahdat, A. (2022). Mission Apollo: Landing optical circuit \nswitching at datacenter scale. arXiv preprint arXiv:2208.10041. \n[5] Shahbaz, M., Suresh, L., Rexford, J., Feamster, N., Rottenstreich, O., & Hira, M. (2019). Elmo: \nSource routed multicast for public clouds. In Proceedings of the ACM Special Interest Group on Data \nCommunication (pp. 458 -471). \n[6] Xia, Y., Ng, T. E., & Sun, X. S. (2015, April). Blast: Accelerating high -performance data analytics \napplications by optical multicast. In 2015 IEEE Conference on Computer Communications (INFOCOM) \n(pp. 1930 -1938). IEEE. \n[7] Das, S., Rahbar, A., Wu, X. C., Wang, Z., Wang, W., Chen, A., & Ng, T. E. (2022). Shufflecast: An \noptical, data -rate agnostic and low -power multicast architecture for next -generation compute clusters. \nIEEE/ACM Transactions on Networking. \n[8] Samadi, P., Gupta, V., Xu, J., Wang, H., Zussman, G., & Bergman, K. (2015). Optical multicast \nsystem for data center networks. Optics express, 23(17), 22162 -22180. \n[9] Barry, L. P., Wang, J., McArdle, C., & Kilper, D. (2018). Optical switching in datacenters: architectures \nbased on optical circuit switching. Optical switching in next generation data centers , 23-44. \n[10] Stepanovsky, M. (2019). A comparative review of MEMS -based optical cross -connects for all -\noptical networks from the past to the present day. IEEE Communications Surveys & Tutorials, 21(3), \n2928 -2946. [11] D. M. Marom, Y. Miyamoto, D. T. Neilson and I. Tomkos, \"Optical Switching in Future Fiber -Optic \nNetworks Utilizing Spectral and Spatial Degrees of Freedom,\" in Proceedings of the IEEE, vol. 110, no. \n11, pp. 1835 -1852, Nov. 2022, doi: 10.1109/JPROC.2022.3 207576. \n[12] Marom, D. M., Colbourne, P. D., D’errico, A., Fontaine, N. K., Ikuma, Y., Proietti, R., ... & Tomkos, \nI. (2017). Survey of photonic switching architectures and technologies in support of spatially and \nspectrally flexible optical networking. Journal of Optical Communications and Networking, 9(1), 1 -26. \n[13] Li, J., Gan, T., Bai, B., Luo, Y., Jarrahi, M., & Ozcan, A. (2023). Massively parallel universal linear \ntransformations using a wavelength -multiplexed diffractive optical network. Advanced Photonics, 5(1), \n016003. \n[14] N. U. Dinc, J. Lim, E. Kakkava, C. Moser, and D. Psaltis, “Computer generated optical volume \nelements by ad -ditive manufacturing,” Nanophotonics , vol. 9, no. 13, pp. 4173 –4181, Jun. 2020, doi: \n10.1515/nanoph -2020 -0196. \n[15] Goodman, J. W., Leonberger, F. J., Kung, S. Y., & Athale, R. A. (1984). Optical interconnections for \nVLSI systems. Proceedings of the IEEE, 72(7), 850 -866. \n[16] Xu, Y., Huang, J., Yang, L. et al. Inverse -designed ultra -compact high efficiency and low crosstalk \noptical interconnect based on waveguide crossing and wavelength demultiplexer. Sci Rep 11, 12842 \n(2021). https://doi.org/10.1038/s41598 -021-92038 -w \n[17] Lee, H., Gu, X. G., & Psaltis, D. (1989). Volume holographic interconnections with maximal capacity \nand minimal cross talk. Journal of applied physics, 65(6), 2191 -2194. \n[18] Dinc, N. U., Moser, C., & Psaltis, D. (2024). Volume holograms with linear diffraction efficiency \nrelation by (3+ 1) D printing. Optics Letters, 49(2), 322 -325. \n[19] Sato, Ken -ichi. \"Optical switching will innovate intra data center networks [Invited Tutorial].\" Journal \nof Optical Communications and Networking 16, no. 1 (2023): A1 -A23. \n[20] Morizur, J. F., Nicholls, L., Jian, P., Armstrong, S., Treps, N., Hage, B., ... & Bachor, H. A. (2010). \nProgrammable unitary spatial mode manipulation. JOSA A , 27(11), 2524 -2531. \n[21] Labroille, G., Denolle, B., Jian, P., Genevaux, P., Treps , N., & Morizur, J. F. (2014). Efficient and \nmode selective spatial mode multiplexer based on multi -plane light conversion. Optics express, 22(13), \n15599 -15607 \n[22] Wang, H., & Piestun, R. (2018). Dynamic 2D implementation of 3D diffractive optics. Optica , 5(10), \n1220 -1228. \n[23] N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre –Gaussian mode \nsorter,” Nature Commun., vol. 10, no. 1, pp. 1 –7, Dec. 2019. [24] Ramachandran , S., Gregg, P., Kristensen, P., & Golowich, S. E. (2015). On the scalability of ring \nfiber designs for OAM multiplexing. Optics express, 23(3), 3721 -3730. \n[25] Wang, H., Ai, J., Ma, Z., Ramachandran, S., & Wang, J. (2023). Finding the superior mode basis \nfor mode -division multiplexing: a comparison of spatial modes in air -core fiber. Advanced Photonics, \n5(5), 056003 -056003. \n[26] Kupianskyi, H., Horsley, S. A., & Phillips, D. B. (2024). All -optically untangling light propagation \nthrough multimode fibers. Optica , 11(1), 101 -112. \n[27] Yildirim, M., Dinc, N. U., Oguz, I., Psaltis, D., & Moser, C. (2023). Nonlinear Processing with Linear \nOptics. arXiv preprint arXiv:2307.08533. \n[28] U. S. Kamilov et al., “Learning approach to optical tomography,” Optica, vol. 2, no. 6, p. 517, Jun. \n2015, doi: 10.1364/OPTICA.2.000517. \n[29] L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED \narray micro -scope,” Optica, vol. 2, no. 2, p. 104, Feb. 2015, doi: 10.1364/OPTICA.2.000104. \n[30] X. Lin et al., “All -optical machine learning using diffractive deep neural networks,” Science, vol. 361, \nno. 6406, pp. 1004 –1008, Sep. 2018, doi: 10.1126/science.aat8084. \n[31] J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High -fidelity optical diffraction tomography of \nmultiple scattering samples,” Light Sci Appl, vol. 8, no. 1, p. 82, Dec. 2019, doi: 10.1038/s41377 -019-\n0195 -1. \n \n " }, { "title": "2401.14174v1.The_Boundaries_of_Tractability_in_Hierarchical_Task_Network_Planning.pdf", "content": "The Boundaries of Tractability in Hierarchical Task Network Planning\nCornelius Brand1,Robert Ganian2,Fionn Mc Inerney2, and Simon Wietheger2\n1Algorithms & Complexity Theory Group, Regensburg University, Germany\n2Algorithms and Complexity Group, TU Wien, Austria\ncornelius.brand@ur.de, {rganian, fmcinern }@gmail.com, swietheger@ac.tuwien.ac.at\nAbstract\nWe study the complexity-theoretic boundaries of\ntractability for three classical problems in the con-\ntext of Hierarchical Task Network Planning: the\nvalidation of a provided plan, whether an exe-\ncutable plan exists, and whether a given state can be\nreached by some plan. We show that all three prob-\nlems can be solved in polynomial time on primitive\ntask networks of constant partial order width (and\na generalization thereof), whereas for the latter two\nproblems this holds only under a provably neces-\nsary restriction to the state space. Next, we ob-\ntain an algorithmic meta-theorem along with cor-\nresponding lower bounds to identify tight condi-\ntions under which general polynomial-time solv-\nability results can be lifted from primitive to gen-\neral task networks. Finally, we enrich our inves-\ntigation by analyzing the parameterized complex-\nity of the three considered problems, and show that\n(1) fixed-parameter tractability for all three prob-\nlems can be achieved by replacing the partial or-\nder width with the vertex cover number of the\nnetwork as the parameter, and (2) other classical\ngraph-theoretic parameters of the network (includ-\ning treewidth, treedepth, and the aforementioned\npartial order width) do not yield fixed-parameter\ntractability for any of the three problems.\n1 Introduction\nAutomated Planning is a core research topic whose goal is\nto devise computational methods capable of finding solutions\nto prominent planning tasks. In this work, we consider auto-\nmated planning in the context of Hierarchical Task Networks\n(HTNs) , which have received significant attention in the ar-\ntificial intelligence research community. HTNs are capable\nof incorporating the compound, hierarchical structure of real-\nworld tasks, as opposed to the fine-grained view of classical\nplanning that operates directly on the state space of the sys-\ntem under consideration, and are one of the best-established\nplanning formalisms in the literature [Tsuneto et al. , 1996;\nKuter et al. , 2005; Li et al. , 2009; Hogg et al. , 2009; Parunak\net al. , 2009; Sohrabi et al. , 2009; Geier and Bercher, 2011;Alford et al. , 2015; Xiao et al. , 2017; H ¨oller et al. , 2019;\nBehnke et al. , 2019; Olz et al. , 2021; Lin and Bercher, 2023 ].\nOn a high level,1HTNs can be informally described and\nillustrated as follows: at the top level of the task hierarchy\nis a sequence of tasks that is to be executed in a (partially\nspecified) order. For instance, the task of going to work may\nbe decomposed into three subtasks: leaving the house, going\nthe actual distance, and getting set up at the office. Now, each\nof these tasks can in turn be decomposed into a set of smaller\ntasks, like climbing the stairs or taking a bus, which again can\nbe decomposed, and so forth. This continues until we reach\nan elementary level of tasks that are no longer decomposable,\nwhich correspond to single executable actions .\nThe crucial feature of the model is that it separates the user-\nfacing description of a task (what we want to do) from its in-\nternal implementation (how to do it). For instance, leaving\nthe house always needs to be done before going to work, re-\ngardless of how it is done. However, in an HTN, we may have\nseveral options of how to decompose this task: the agent can,\ne.g., either climb the stairs or take the elevator. Similarly, the\nagent might opt for walking to work instead of going by bus.\nIn the literature, HTN planning is usually considered\nthrough the lens of heuristic approaches, which are evalu-\nated against real-world problems. On the other hand, much\nless is known about the complexity-theoretic foundations\nof planning problems associated with HTNs. This lack of\nunderstanding contrasts with the major advances that have\nbeen made towards obtaining a thorough comprehension of\nthe complexity of other fundamental problems arising in\nthe context of artificial intelligence, such as Bayesian Net-\nwork Learning [Ordyniak and Szeider, 2013; Gr ¨uttemeier et\nal., 2021a; Gr ¨uttemeier et al. , 2021b; Gr ¨uttemeier and Ko-\nmusiewicz, 2022 ], Clustering [Ganian et al. , 2020b; Ganian\net al. , 2022; Kellerhals et al. , 2023; Bandyapadhyay et al. ,\n2023 ], Resource Allocation [Bliem et al. , 2016; Deligkas et\nal., 2022; Eiben et al. , 2023 ], and Scheduling [Ganian et al. ,\n2020a; Bentert et al. , 2023; Heeger et al. , 2023 ]. The aim of\nthis article is to fill this gap by identifying the precise bound-\naries of tractability for the central computational tasks that\narise in the context of HTN planning.\nConsidered Problems. Perhaps the most natural algorith-\nmic question associated with HTNs is the following: given\n1Formal definitions are provided in the Preliminaries.arXiv:2401.14174v1 [cs.CC] 25 Jan 2024an HTN, an initial state, and a plan, is the plan executable\non the HTN? This question has been extensively investigated\nin the literature, where authors typically consider a plan to\nbe formalized as either a sequence of tasks [Behnke et al. ,\n2015; Bart ´aket al. , 2021 ], or actions [H¨oller et al. , 2022;\nLin and Bercher, 2023 ]. Here, we adopt the latter ap-\nproach and note that the complexity of this problem—denoted\nPLAN VALIDATION —has been studied, e.g., in the context\nof post-optimization and repairing of solutions to HTN prob-\nlems [Lin and Bercher, 2023 ].\nThe second problem we consider is P LAN EXISTENCE :\ngiven an HTN and some initial state, is there a way to ex-\necute all the tasks in the network? The complexity of this\nfundamental problem has been studied in several variants of\nHTN planning problems [Erol et al. , 1996; Geier and Bercher,\n2011; Alford et al. , 2015; Chen and Bercher, 2021 ].\nThird, we consider the question of whether a specified\ntarget state can be reached from an initial state of the sys-\ntem, formalized as the S TATE REACHABILITY problem. This\nis the typical goal in classical planning [Fikes and Nilsson,\n1971; Chapman, 1987; B ¨ackstr ¨om and Nebel, 1995 ], so it\nshould come as no surprise that HTN planning problems with\nspecified target states have attracted interest as well [H¨oller\nand Bercher, 2021; Olz et al. , 2021 ].\nMotivated by the above considerations of states and tasks,\nwe also define the corresponding problem of constructing a\nplan executing (at least) a specified set of actions. However,\nthis A CTION EXECUTABILITY problem is in some sense a\ngeneralization of P LAN EXISTENCE , and, in fact, all results\nthat we prove for one of the problems also hold for the other.\nThus, we omit A CTION EXECUTABILITY from the discus-\nsion of our results below.\nOur Contribution. The lack of results on the computa-\ntional complexity of P LAN VALIDATION , PLAN EXISTENCE ,\nand S TATE REACHABILITY may at least partially stem from\nthe fact that they remain computationally intractable (in par-\nticular, NP-hard) even in severely restricted settings. While\nthe emerging intractability might seem difficult to navigate,\nwe identify interesting and natural special cases where pock-\nets of polynomial-time solvability show themselves. More-\nover, we prove that these pockets give rise to a surprisingly\nrich complexity-theoretic landscape.\nTo be more precise, we embark by considering primitive\ntask networks, i.e., HTNs in which there are no compound\ntasks. While this may seem restrictive, it is a natural place\nto start from a complexity-theoretic perspective: all three of\nthe considered problems remain NP-hard on primitive task\nnetworks. In fact, as our first results, we establish the NP-\nhardness of P LAN VALIDATION , PLAN EXISTENCE , and\nSTATE REACHABILITY even on networks which consist of\ncollections of stars (Theorems 2 and 5). Moreover, unlike\nPLAN VALIDATION , we show that the latter two problems\nremain NP-hard even on trivial (in particular, edgeless) net-\nworks when the state space is not of fixed size (Theorem 1);\nthis means that in order to avoid immediate intractability, we\nhenceforth consider P LAN EXISTENCE and S TATE REACH -\nABILITY exclusively in the setting of constant-sized state\nspaces. This requirement matches analogous restrictions thatwere used in other planning contexts [Kronegger et al. , 2013;\nB¨ackstr ¨omet al. , 2015 ].\nWhile these initial results may seem discouraging, we\nprove that all three problems are polynomial-time solvable\non networks which are either chains or antichains (Observa-\ntions 6 and 7, Theorems 8 and 10). These tractable cases hint\nat cracks in the wall of computational intractability. We pry\nthese cracks open by showing that these two restrictions actu-\nally represent special cases in a spectrum of tractable special\ncases: P LAN VALIDATION , PLAN EXISTENCE , and S TATE\nREACHABILITY are polynomial-time solvable on networks\nof bounded “generalized partial order width”, that is, the par-\ntial order width of the network when ignoring isolated ele-\nments (Theorems 11, 13 and 14). This non-trivial result, es-\ntablished via a combination of branching techniques and deep\ninsights into the behavior of optimal plans in the state space,\nunifies and generalizes the aforementioned tractability results\non chains and antichains.\nBuilding on the above results for the primitive case, in the\nsecond part of the paper, we establish an algorithmic meta-\ntheorem that allows us to precisely describe the circumstances\nunder which general tractability results for primitive task net-\nworks can be lifted to the compound case. Essentially, if\n1. the total number of compound tasks in the network,\n2. the maximum size of a network that a task can be de-\ncomposed into, and\n3. the maximum depth of a decomposition\nare all bounded by some constant, then anypolynomial-time\nsolvability result for any“reasonable” HTN planning prob-\nlem on primitive task networks can be lifted to compound\nnetworks (Theorem 16). Crucially, we prove that this result\nis tight in the sense that tractability cannot be guaranteed if\nany of the three restrictions are dropped (Theorems 17 to 20).\nIn the third and final part of the paper, we push be-\nyond classical complexity and analyze P LAN VALIDATION ,\nPLAN EXISTENCE , and S TATE REACHABILITY from the\nperspective of the more fine-grained parameterized com-\nplexity paradigm [Downey and Fellows, 2013; Cygan et\nal., 2015 ]. In parameterized complexity, one considers the\ntractability of problems not only w.r.t. the input size n, but\nalso w.r.t. a specified numerical parameter k, where the aim\nis to obtain algorithms which run in time at most f(k)·nO(1)\n(so-called fixed-parameter algorithms ); problems admitting\nsuch algorithms are called fixed-parameter tractable .\nThe natural question from the parameterized perspective\nis whether the aforementioned tractability results for HTNs\nwith constant-sized structural measures (Theorems 11, 13\nand 14) can be lifted to fixed-parameter tractability when pa-\nrameterized by the same measures. While analogous ques-\ntions have been thoroughly explored—and often answered in\nthe affirmative—in many other areas of artificial intelligence\nresearch [Ordyniak and Szeider, 2013; Eiben et al. , 2021;\nGanian and Korchemna, 2021; Froese et al. , 2022; Heeger\net al. , 2023 ], here we prove that the generalized partial or-\nder width does not yield fixed-parameter tractability for any\nof our problems on primitive task networks (Corollaries 21\nand 22), at least under well-established complexity assump-\ntions. This means that different structural parameterizationsare required to obtain fixed-parameter tractability.\nFurther, Theorems 2 and 5 can already be seen to rule\nout fixed-parameter tractability for the considered problems\non primitive task networks w.r.t. classical graph-theoretic\nmeasures such as treewidth [Robertson and Seymour, 1986 ],\ntreedepth [Nesetril and de Mendez, 2012 ], and even the feed-\nback edge number [Uhlmann and Weller, 2013; Ganian and\nKorchemna, 2021 ]. However, fixed-parameter tractability can\nstill be achieved under the right parameterization: in Theo-\nrems 23 and 24, we obtain highly non-trivial fixed-parameter\nalgorithms for P LAN VALIDATION , PLAN EXISTENCE , and\nSTATE REACHABILITY when parameterized by the vertex\ncover number of the input primitive task network. Finally,\nfor compound HTNs, we lift our general algorithmic meta-\ntheorem for polynomial-time solvability (Theorem 16) to the\nsetting of fixed-parameter tractability when additionally pa-\nrameterizing by the “breadth” of an HTN, i.e., the maximum\nnumber of pairwise non-isomorphic networks that a task can\nbe decomposed into (Theorem 25).\nRelated Work. HTN planning has been extensively stud-\nied in the artificial intelligence research community and\nhas found applications in areas such as healthcare [Garc ´ıa-\nRemesal, 2008; Gonz ´alez-Ferrer et al. , 2011; Gonz ´alez-\nFerrer et al. , 2013b; Bihlmaier et al. , 2014 ], business pro-\ncesses [Gonz ´alez-Ferrer et al. , 2013a; Ko et al. , 2009; Ko et\nal., 2012 ], human-robot interaction [Hayes and Scassellati,\n2016; Tewari and Persiani, 2021 ], real-time games [Onta ˜n´on\nand Buro, 2015; Namiki et al. , 2021; Soemers and Winands,\n2016 ], emergency response [Liuet al. , 2016; Zhao et al. ,\n2017 ], and visualization [Padia et al. , 2019 ]. For an overview\nof the current state of hierarchical planning at large, we\nrefer the reader to the survey of Bercher, Alford, and\nHoeller (2019) and the overview article of Georgievski and\nAiello (2015).\nThe works most closely related to ours are those of Lin\nand Bercher (2023) and Olz, Biundo, and Bercher (2021),\nwhich also established some of the formalisms that we build\non. In a similar vein, questions on the complexity and\neven computability of many tasks associated with HTNs\nhave been treated in the literature [Geier and Bercher, 2011;\nAlford et al. , 2015 ], where precursors of the abovementioned\nformal models were defined. Beyond these recent works,\nthere are also classical investigations of computational ques-\ntions related to HTN planning in slightly different settings\nthan the ones we look at here, such as the early influential\nworks by Erol, Hendler, and Nau (1994; 1996).\nAlbeit carried out in a different context, the work of Kro-\nnegger, Pfandler, and Pichler (2013) on propositional strips\nwith negation (PSN) studies a problem ( k-PSN) which is re-\nlated to S TATE REACHABILITY . The main difference is that\nwhile PSN includes a broader definition of actions, it does\nnot consider any primitive or hierarchical network structure\nwhatsoever. Finally, we note that the computational complex-\nity of planning has also been studied in a variety of other set-\ntings beyond HTNs [Bylander, 1994; B ¨ackstr ¨omet al. , 2012;\nB¨ackstr ¨omet al. , 2013; B ¨ackstr ¨omet al. , 2015 ].2 Preliminaries\nFor a positive integer n, let[n] ={1, . . . , n }. We employ\nstandard graph-theoretic terminology [Diestel, 2012 ]. The\nunderlying undirected graph of a digraph Dis the graph ob-\ntained by replacing each arc in Dwith an edge (discarding\nloops and multiple edges).\nFor some V′⊆V, the subgraph induced by V′isG[V′] =\n(V′, E∩(V′×V′)). Awalk inGis a sequence of vertices\nv1, . . . , v ℓsuch that (vi, vi+1)∈Efor all i∈[ℓ−1]. As\nthe walk contains ℓ−1edges, we say that it has length ℓ−1.\nA walk is a cycle ifv1=vℓ. A graph is acyclic if it does\nnot contain any cycles. A (simple) path is a walk such that\nvi̸=vjfor all i̸=j, i, j∈[ℓ]. A simple cycle is a walk\nwhere this restrictions holds except for v1andvℓ.\nAmultigraph is a graph that allows for multiple edges be-\ntween a pair of vertices. Formally, we define a multigraph\nbyG= (V, E, β :E→(V×V)). Avertex walk inGis a\nsequence of vertices v1, . . . , v ℓsuch that there exists ei∈E\nwithβ(ei) = ( vi, vi+1)for all i∈[ℓ−1]. An edge walk\ninGis a sequence of edges e1, . . . , e ℓsuch that ei∈Efor\nalli∈[ℓ]and if β(ei) = (v, v′), then β(ei+1) = (v′, v′′)for\nsome v′′∈V. Note that an edge walk corresponds to a vertex\nwalk by considering the vertices it passes through. We trans-\nfer the notion of (simple) vertex paths and cycles and (simple)\nedge paths and cycles from non-multigraphs, where an edge\nwalk is simple if and only if its corresponding vertex walk is\nsimple.\nThe partial order width wof a partially ordered set ≲is\nthe maximum number of pairwise-disjoint maximal chains\ncontained in ≲or, equivalently, the size of its largest an-\ntichain [Dilworth, 1950 ]. An element in ≲isisolated if it\nis incomparable to every other element in the set. We de-\nfine the generalized partial order width of a partially ordered\nset≲as the maximum number of pairwise-disjoint maximal\nchains contained in ≲which are not isolated elements (or,\nequivalently, the size of the largest antichain not containing\nany isolated element).\nHierarchical Task Networks. Our formalization of HTNs\nfollows the terminology employed in recent works [Olzet al. ,\n2021; Lin and Bercher, 2023 ]. The domain of a planning\nproblem is a tuple (F, A,C, δ, M ), where Fis a finite set of\npropositions, Ais a set of actions (orprimitive task names ),\nandCis a set of compound task names such that A∩ C=∅.\nThe function δ:A→2F×2F×2Fmaps each action to\na set prec( a)of preconditions, a set del(a)of propositions\nto be removed, and a set add(a)of propositions to be added\nto the current state. Lastly, Mrepresents the decomposition\nmethods and is a mapping from each c∈ Cto a set Zof task\nnetworks (defined below) which ccan be decomposed into.\nAtask network in the domain (F, A,C, δ, M )is defined as\na tuple (T,≺+, α)that consists of a set Tof task identifiers\nwith a partial order ≺+⊆:T×Tand a function α:T→\nA∪C. The usual representation considered for a task network\nis its cover graph D≺= (T,≺), where ≺is the cover of ≺+\n(in particular, (a, b)∈≺if and only if a≺+band there is no\nelement csuch that a≺+c≺+b). When analyzing the struc-\nture of directed graphs, it is often useful to also consider their\nunderlying undirected graph [Ordyniak and Szeider, 2013;t2:a2\nt1:a1\nt3:a1\nt4:a3Action prec del add\na1 ∅ ∅{1}\na2 ∅ ∅{2}\na3 {2}{1}∅\nFigure 1: The digraph D≺for a primitive HTN with tasks T=\n{t1, t2, t3, t4}, actions A={a1, a2, a3}, propositions F={1,2},\nand initial state s0=∅. Each task’s action is given in its node. A\ndirected edge from a task tto a task t′corresponds to (t, t′)∈≺+.\nt2\nt1\nt3;3\nt4\nt3;2\nt3;1\ntn0\nt3\nt2\nt1\ntnm\nFigure 2: Example for the decomposition of a compound task net-\nwork. Let tnbe the compound task network that is depicted in\nFig. 1, where t3is a compound task instead of a primitive task with\naction a1. If(α(t3), tnm)∈M, then tndecomposes to tn′.\nGanian et al. , 2021 ]; we hence use G≺to denote the under-\nlying undirected graph of D≺. An example of a simple task\nnetwork is provided in Fig. 1.\nA set of tasks T′⊆Tis a chain if≺+defines a total\norder on T′, and an antichain if all tasks in T′are pairwise\nincomparable w.r.t. ≺+. Two task networks tn= (T,≺+, α)\nandtn′= (T′,≺+′, α′)are isomorphic if there is a bi-\njection ϕ:T→T′such that (t, t′)∈≺+if and only if\n(ϕ(t), ϕ(t′))∈ ≺+′andα(t) =α′(ϕ(t))for all t, t′∈T.\nIntuitively, two task networks are isomorphic if they describe\nthe same structure, but using different task names.\nA task tisprimitive ifα(t)∈Aandcompound otherwise.\nA task network is primitive if it contains only primitive tasks.\nNon-primitive task networks can be decomposed into primi-\ntive ones by applying the decomposition methods in M. More\nprecisely, let there be a task network tn= (T,≺+, α),t∈T,\nandm= (α(t), tnm)∈M. Let tn′\nm= (T′\nm,≺+\nm′, α′\nm)\nwithT′\nm∩T=∅being isomorphic to tnm. Then, tintn\ncan be decomposed by mto obtain the new task network\ntn′= (T′,≺+′, α′)with\nT′=T\\ {t} ∪T′\nm,\n≺+′=≺+∩(T′×T′)∪ ≺+\nm′∪ {(t1, t2)|(t1, t)∈≺+,\nt2∈T′\nm} ∪ {(t2, t1)|(t, t1)∈≺+, t2∈T′\nm},\nα′=α\\ {(t, α(t))} ∪α′\nm.\nIntuitively, decomposing a task tto a subnetwork tnmmeans\nreplacing tbytnmsuch that each task in tnmis connected\nto tasks outside tnmin the same way twas before. The extra\nstep using the isomorphic tn′\nmis required to ensure unique\ntask identifiers after the decomposition. An example is pro-\nvided in Fig. 2.\nAnHTN planning problem is a problem whose input in-\ncludes a domain (F, A,C, δ, M ), an initial task network tn\nfrom that domain, and an initial state s0⊆F. An action a\nisexecutable in a state s⊆Fifprec( a)⊆s. If executed, it\nchanges the state stos′= (s\\del(a))∪add(a). A plan is asequence of actions. A plan is executable from an initial state\ns0if all actions in the plan can be executed in sequence (one\nafter the other) starting from s0. A sequence t1, t2, . . . , t ℓ\nof tasks is executable in a task network tniftncan be de-\ncomposed into a primitive task network tn′such that the plan\nα(t1), . . . , α (tℓ)is executable, ti̸=tj, andti̸≺+tjifi > j .\nFor a sequence pof tasks or actions, we refer to its ithelement\nbyp[i]and to its length by |p|.\nExample 1. Consider the primitive HTN depicted in\nFig. 1. The task sequence t1, t2, t4is not executable since\n(t3, t4)∈≺+means that t3must be executed before t4. The\ntask sequence t1, t3,t4is not executable either, as after ex-\necuting t1andt3, the state is {1}, so the precondition of a3\nis not fulfilled when executing t4. On the other hand, the\ntask sequence t1, t2, t3, t4is executable. It conforms with\nthe partial order, and for the corresponding action sequence\na1, a2, a1, a3, after executing any prefix of the sequence, the\nstate is a superset of the precondition of the next action.\nWe are now ready to define our problems of interest.\nPLAN VALIDATION\nInput: A task network tn= (T,≺+, α)over some\ndomain (F, A,C, δ, M ), an initial state s0⊆\nF, and a plan poverA.\nQuestion: Is there a task sequence that corresponds to\nplanp, is executable in tnfrom s0, and ex-\necutes all the tasks in tn?\nAs mentioned in the introduction, authors have also some-\ntimes considered a version of P LAN VALIDATION where we\nare provided a sequence of tasks on the input as opposed to a\nplan. We remark that this would give rise to a problem which\nis trivially linear-time solvable on allprimitive task networks,\nand is also fully captured by our algorithmic meta-theorems\n(Theorems 16 and 25) on compound networks. In view of\nthis, we exclude this problem from our further considerations.\nPLAN EXISTENCE\nInput: A task network tnover some domain\n(F, A,C, δ, M )and an initial state s0⊆F.\nQuestion: Is there an executable plan starting in s0that\nexecutes all the tasks in tn?\nACTION EXECUTABILITY\nInput: A task network tnover some domain\n(F, A,C, δ, M ), an initial state s0⊆F, and\na multiset Sof actions in A.\nQuestion: Is there a plan (not necessarily using all\navailable tasks) that is executable in tnfrom\ns0and that covers S?\nACTION EXECUTABILITY is a generalization of P LAN\nEXISTENCE in the sense that, on primitive task networks,\nPLAN EXISTENCE simply equals A CTION EXECUTABILITY\nifSis the multiset of actions obtained by mapping each task\nbyα. On general networks, any P LAN EXISTENCE instance\ncan be reduced to A CTION EXECUTABILITY by adding a newtasktwith new action a, letting S={a}, and adding (t′, t)\nto≺for each sink t′in the task network. As this may change\nsome structural parameters associated to the task network, al-\nternatively one can create a new task tt′with action afor each\nsinkt′, only add the edges (t′, tt′)to≺, and let Scontain a\nas many times as there are sinks. Therefore, algorithmic and\nhardness results transfer in the respective directions; in our\nsetting, we establish all of our hardness results for the “sim-\npler” P LAN EXISTENCE , while all of our algorithms can deal\nwith the “harder” A CTION EXECUTABILITY .\nSTATE REACHABILITY\nInput: A task network tnover some domain\n(F, A,C, δ, M ), an initial state s0⊆F, and\na target state sg⊆F.\nQuestion: Is there a plan (not necessarily using all\navailable tasks) that is executable in tnfrom\ns0and that results in a state s⊇sg?\nWe remark that, while P LAN VALIDATION , PLAN EX-\nISTENCE , ACTION EXECUTABILITY , and S TATE REACH -\nABILITY are stated as decision problems for complexity-\ntheoretic purposes, all of our obtained algorithms are con-\nstructive and can output a task sequence or plan as a witness.\nFor each of these problems, we define the input size to simply\nbe the sum of the sizes of elements occurring on the input; for\nexample, for an instance Iof P LAN VALIDATION this would\nbe|I|=|T|+| ≺+|+|α|+|F|+|A|+|C|+|δ|+|M|.\nFor some of our algorithms, we employ a state transition\ngraph . For a set of actions Aand an initial state s0, we define\nthe state transition graph to be a multigraph G= (S,T, β)\nthat describes the set Sof states reachable from s0and\nhow actions transform one state into another. We construct\nGinductively as follows. First, initialize S={s0}and\nT=∅. Then, update from s0. To update from a state\ns, iterate through all actions a∈A. Ifprec( a)⊆s, let\ns′= (s\\del(a))∪add(a). Ifs′/∈ S , add s′toSand\nupdate from s′. In any case, add e= (a, s, s′)toTand\nletβ(e) = ( s, s′). The computation is completed once no\nmore updates occur. An example is provided in Fig. 3. This\nconstruction takes at most O\u0000\n|I|2\u0001\ntime. Observe that Gcon-\ntains exactly all states that s0can be transformed into by ap-\nplying sequences of actions in A. Further, a pair of states\nis connected by directed edges representing all actions that\ntransform one of the states into the other. It is known that\nthe number of states |S|can be upper-bounded by |A|!·2|A|\n[Kronegger et al. , 2013 ].\nParameterized Complexity. Parameterized complex-\nity[Downey and Fellows, 2013; Cygan et al. , 2015 ]analyzes\nthe running time of algorithms with respect to a parameter\nk∈Nand input size n. The high-level idea is to find a pa-\nrameter that describes the structure of the instance such that\nthe combinatorial explosion can be confined to this param-\neter. In this respect, the most favorable complexity class is\nFPT (fixed-parameter tractable ) which contains all problems\nthat can be decided by an algorithm running in f(k)·nO(1)\ntime, where fis a computable function. Algorithms with this\nrunning-time are called fixed-parameter algorithms .\n;\nf1g\nf1;2;3g\nf1;2g\na1\na1\na1; a2\na4; a5\na4\na1; a2; a4; a5\na2\na3\na3ActionProp. setprec del add\na1 ∅ ∅ {1}\na2 {1} ∅ {2}\na3 {2}{2,3} ∅\na4 {1} ∅ {2,3}\na5 {2} ∅ {3}\nFigure 3: Example of a state transition graph with initial state s0=∅\nand actions A={a1, a2, a3, a4, a5}. An edge labeled with aithat\nconnects some state sto some state s′refers to the edge (ai, s, s′)\nin the state transition graph. An edge labeled with multiple actions\nrefers to multiple edges, one for each action, in the multigraph.\nShowing that a problem is hard for the complexity classes\nW[1]orW[2]rules out the existence of a fixed-parameter al-\ngorithm under the well-established assumption that W[1]̸=\nFPT . This is done via a parameterized reduction [Downey\nand Fellows, 2013; Cygan et al. , 2015 ]from some known\nW[1]-hard ( W[2]-hard, respectively) problem. A parameter-\nized reduction from a parameterized problem Pto a parame-\nterized problem Qis a function:\n• which maps yes-instances to yes-instances and no-\ninstances to no-instances,\n• which can be computed in f(k)·nO(1)time, where fis\na computable function, and\n• where the parameter of the output instance can be upper-\nbounded by some function of the parameter of the input\ninstance.\n3 Setting the Stage: The State Space\nAs a basic precondition for solving our problems on more ad-\nvanced task networks, we need to ensure tractability at least\non “trivial” primitive task networks where ≺+=∅. Unfor-\ntunately, in their most general form, all but the first of our\nproblems of interest turn out to be intractable even on these\ndegenerate networks via a simple reduction from 3-SAT.\nTheorem 1. STATE REACHABILITY andPLAN EXISTENCE\nareNP-hard on primitive task networks with ≺+=∅.\nProof. We give a reduction from 3-SAT to S TATE REACHA -\nBILITY . Given an instance ϕof 3-SAT, we create an instance\n(tn, s 0, sg)of S TATE REACHABILITY as follows. For each\nclause Cjinϕ,j∈[m], we create a proposition CjinF. For\neach variable xiinϕ,i∈[n], we create a proposition xiin\nF, as well as two actions axi,axiand two tasks txi,txithat\nare mapped to axiandaxi, respectively. For all i∈[n], letCxi(Cxi, respectively) be the set of all propositions of the\nform Cjthat correspond to clauses in ϕthat are satisfied by\nsetting the variable xito TRUE (FALSE, respectively). For\nalli∈[n], we set prec( axi) = prec( axi) = del( axi) =\ndel(axi) ={xi},add(axi) =Cxi, and add(axi) =Cxi.\nFinally, we set s0={x1, . . . , x n}andsg={C1, . . . , C m}.\nNote that tnis primitive with ≺+=∅.\nThe equivalence between the two instances is easy to\nsee. Indeed, since s0={x1, . . . , x n}andprec( axi) =\nprec( axi) = del( axi) = del( axi) ={xi}, only one of the\ntwo tasks txi,tximay be executed for each i∈[n]. Further,\nthe execution of txi(txi, respectively) causes the execution of\naxi(axi, respectively), which adds, to the current state, all the\npropositions corresponding to clauses in ϕthat are satisfied\nby setting the variable xito TRUE (FALSE, respectively). Fi-\nnally, sgcontains all the clause propositions. Hence, ϕis sat-\nisfiable if and only if (tn, s 0, sg)is a yes-instance of S TATE\nREACHABILITY .\nWe can also show that the same holds for A CTION EX-\nECUTABILITY , even if |S|= 1; indeed, it suffices to use\nthe same construction with the addition of a new task ta∗\nthat is mapped to a new action a∗with S={a∗}and\nprec( a∗) ={C1, . . . , C m}.\nFinally, for P LAN EXISTENCE , we begin by recalling that\nthe problem is NP-hard on primitive task networks [Erol et\nal., 1996, Theorem 8 ]. We can reduce from P LAN EXIS-\nTENCE on primitive task networks to P LAN EXISTENCE on\nprimitive task networks with ≺+=∅by encoding the par-\ntial order into the preconditions. Let there be a task network\ntn= (T,≺+, α)with action set Aand set of propositions\nF. Then, construct a new task network tn′= (T,∅, α′)\nwith action set A′and proposition set F′as follows. Let\nF′=F∪{ft|t∈T}. For each t∈T, create an action at∈\nA′such that prec( at) = prec( α(t))∪ {ft′|(t′, t)∈≺+},\ndel(at) = del( α(t)), and add(at) = add( α(t))∪ {ft}. Let\nα′(t) =atfor all t∈T. Observe that this way, a task in\ntn′can only be executed if the respective preconditions as\ndefined for the original instance are fulfilled and all its pre-\ndecessors in ≺+are executed. Thus, any task sequence is\nexecutable in tn′from some initial state s0if and only if it is\nexecutable in tnfrom s0.\nA crucial requirement for the reductions underlying Theo-\nrem 1 is that the state space (i.e., the number of states) can\nbe large, and in particular not bounded by any constant. As\nwe will see in the next sections, tractability can be achieved\nif the state space is assumed to be fixed (that is, bounded by\na constant that is independent of the input). Hence, we will\nhereinafter analyze the affected problems (S TATE REACHA -\nBILITY , PLAN EXISTENCE , and its generalization A CTION\nEXECUTABILITY ) in the bounded state-space setting.\n4 Primitive Task Networks\nOur first set of results focuses on the classical complexity of\nthe targeted problems on primitive task networks. We begin\nwith a series of lower bounds which rule out polynomial-time\nalgorithms even for some very simple “tree-like” networks.The Intractability of Tree-like HTNs. First, note that the\nintractability of P LAN VALIDATION on tree-like networks\nwas already established by previous work of Lin and Bercher:\nTheorem 2 ([Lin and Bercher, 2023 ], Theorem 3) .PLAN\nVALIDATION on primitive task networks is NP-hard even if\nG≺is a forest of stars and prec( a) = del( a) = add( a) =∅\nfor all actions a∈A.\nFurther, P LAN VALIDATION remains intractable even if\n≺+describes a set of total orders that are independent from\none another (i.e., if D≺is a disjoint union of paths). We\nestablish this via a reduction from the S HUFFLE PRODUCT\nproblem:\nSHUFFLE PRODUCT\nInput: Words u, c1, . . . , c wover an alphabet Σ.\nQuestion: Can ube generated by interlacing the\nwords in c1, . . . , c w, i.e., can the letters of\nc1, . . . , c wbe rearranged to form uwhile\npreserving the order the letters appear in for\neach individual word ci,i∈[w]?\nSHUFFLE PRODUCT is known to be NP-hard even when\nrestricted to the case where |Σ|= 2[Warmuth and Haussler,\n1984 ].\nTheorem 3. PLAN VALIDATION isNP-hard, even when re-\nstricted to primitive task networks such that ≺+is a collec-\ntion of chains, |A|= 2, and prec( a) = del( a) = add( a) =∅\nfor all actions a∈A.\nProof. We reduce from S HUFFLE PRODUCT with the alpha-\nbetΣ ={a, b}. Let A= Σ. Construct a task for each letter\nin the words c1, . . . , c w. In particular, let the task ti,jrefer\nto the jthletter of the word ci,i∈[w]. Let≺+be such that\nthe tasks for each word form a chain in the respective order,\ni.e., let ≺consist of the edges (ti,j, ti,j+1), for all i∈[w]\nandj∈[|ci| −1]. Then, ≺+partitions Tintowchains (see\nFig. 4). Now, set α(ti,j)to the action representing the jth\nletter of ci. Let the input plan pbe such that p=u. The con-\nstructed instance consists of wchains and the construction\ntakes polynomial time.\nThe correctness of the reduction follows from the fact that\nverifying whether the tasks in this P LAN VALIDATION in-\nstance can be arranged to match the input plan corresponds to\ninterlacing the words to form u.\nA different reduction from S HUFFLE PRODUCT yields sim-\nilar hardness results for the other problems; it will be useful\nto recall that hardness for P LAN EXISTENCE directly carries\nover to hardness for A CTION EXECUTABILITY .\nTheorem 4. STATE REACHABILITY andPLAN EXISTENCE\nareNP-hard, even when restricted to primitive task networks\nsuch that ≺+is a collection of chains, |A| ≤5, and the state\ntransition graph has at most 4states.\nProof. We reduce from S HUFFLE PRODUCT with the al-\nphabet Σ = {a, b}. Construct a task for each letter in\nthe words u, c1, . . . , c w. In particular, for each word x∈\n{u, c1, c2, . . . , c w}, let the task tx,jrefer to its jthletter. LetActionProp. setprec del add\naL\na {LEFT}{LEFT} {a}\naL\nb {LEFT}{LEFT} {b}\naR\na {a} {a} {LEFT}\naR\nb {b} {b} {LEFT}\nag ∅ ∅ {GOAL }\nTable 1: Preconditions and effects of actions in the proof of Thm. 4.\nTcbe the set of tasks created for letters in c1, . . . , c w, and Tu\nthe set of tasks tu,j,j∈ |u|. Let≺+be such that the tasks for\neach word form a chain in the respective order, i.e., let ≺con-\nsist of the edges (tx,j, tx,j+1)for each x∈ {u, c1, . . . , c w}\nandj∈[|x|−1]. Further, add a task tgand let (tu,|u|, tg)∈≺.\nThen,≺+partitions Tintow+ 1chains (see Fig. 4).\nFor the sake of visualization and the corresponding naming\nof actions and propositional states, we consider the chains for\nc1, . . . , c wto be on some leftside and the chain for uto be on\nsome right side. Let A={aL\na, aL\nb, aR\na, aR\nb, ag}with the pre-\nconditions and effects of these actions described in Table 1. If\nthejthletter of uis ana, then we set α(tu,j) =aR\na, and oth-\nerwise, α(tu,j) =aR\nb. Similarly, for each i∈[w], if the jth\nletter in ciis ana, then we set α(tci,j) =aL\na, and otherwise,\nα(tu,j) =aL\nb. Further, let α(tg) =agands0={LEFT}.\nFor S TATE REACHABILITY , we let sg={GOAL }.\nWe prove that these instances are yes-instances if and only\nif the given S HUFFLE PRODUCT instance is a yes-instance.\nSuppose uis a shuffle product of c1, . . . , c w. Then, each\nletter in the words c1, . . . , c wis assigned a unique posi-\ntion with a matching letter in usuch that the assigned po-\nsitions of the letters in each word ci,i∈[w], are in or-\nder. Let tℓrefer to the task that corresponds to the letter\nthat is assigned to position ℓ. Consider the task sequence\nt1, tu,1, t2, tu,2, . . . , t |u|, tu,|u|, tg. This sequence does not\nconflict with ≺+. Further, each task is executable in its re-\nspective state since the state alternates between {LEFT}and\neither{a}or{b}, depending on which letter was seen last.\nAs the letters in c1, . . . , c ware assigned to positions with a\nmatching letter in u, the task tu,jis always executable after\nthe execution of tℓ, for all ℓ∈[|u|], and in turn enables the\nexecution of tℓ+1. After the execution of tg, the target state\nsgis reached and all the tasks are executed, and thus, the re-\nduction instance is a yes-instance.\nFor the other direction, suppose there is a task sequence\nsolving the S TATE REACHABILITY or P LAN EXISTENCE in-\nstance. Consider the first part of the solution sequence up to\nthe task tg. All the tasks in Tuare executed before tg, and\n≺+ensures that they are executed in the order of their corre-\nsponding letters in u. Note that, by the effects and precondi-\ntions of the actions, the sequence starts with a task in Tcand\nalternates between tasks in TcandTu. Further, a task in Tcis\nalways followed by a task in Tuthat corresponds to the same\nletter. Thus, listing all the letters in the words c1, . . . , c win\nthe order their corresponding tasks are executed in the task\nsequence, gives u. As≺+ensures that the letters from each\nword ci,i∈[w], are listed in the order they appear in ci, it is\nRIGHT\nLEFT\nt1;1\nt1;2\nt1;jc1j\n...\ntw;1\ntw;2\ntw;jcwj\n...\n: : :\ntu;1\ntu;2\ntu;juj\n...\ntgFigure 4: Constructed task network for reductions from S HUFFLE\nPRODUCT with words c1, . . . , c wandu. Each word is represented\nby a chain of tasks where each task represents a letter in that word.\nThe colors orange and blue represent the two different letters in the\nalphabet. For the reduction to P LAN VALIDATION (Theorem 3), the\ntask network only consists of vertices on the LEFT side. For the\nreductions to S TATE REACHABILITY and P LAN EXISTENCE (The-\norem 4), a task’s action is defined by its color and whether it is on\nthe LEFT or RIGHT side.\na yes-instance of S HUFFLE PRODUCT .\nNote that the only states in the state transition graph are\n{LEFT},{a},{b}, and{LEFT ,GOAL }. Also, the instance\nis constructed in polynomial time and has w+ 1chains.\nHaving established the intractability of all three problems\non collections of paths, it is natural to wonder whether S TATE\nREACHABILITY and P LAN EXISTENCE can at least be solved\non collections of stars. Here, too, we provide a negative an-\nswer via a reduction from the strongly NP-hard 3-P ARTITION\nproblem [Garey and Johnson, 1979 ]:\n3-PARTITION\nInput: m, B ∈Nand3mpositive integers\nn1, . . . , n 3msuch thatP3m\ni=1ni=mB.\nQuestion: Can the integers nibe partitioned into m\ntriples such that each triple sums up to B?\nTheorem 5. STATE REACHABILITY andPLAN EXISTENCE\nareNP-hard on primitive task networks that are forests of\nstars, even if |A|= 8 and the state transition graph has at\nmost9states.\nProof. We reduce from 3-P ARTITION . We employ the set of\nactions A={a∅, aβ, aˆβ, aγ, aˆγ, aδ, aˆδ, ag}. The precondi-\ntions and effects of these actions are described in Figure 5,\nwhich also illustrates the construction of the following task\nnetwork. For each i∈[3m], create a star representing the in-\nteger nias follows. Create a task tiand two task sets Ti, T′\ni,\neach of cardinality ni. Let α(ti) =a∅. For all t∈Ti, let\nα(t) =aˆβand(t, ti)∈≺. For all t∈T′\ni, letα(t) =aγand\n(ti, t)∈≺. Next, create mmore gadgets representing the m\ntriples to be built, where each triple is represented by a star\nas follows. For all i∈[m−1], the ithtriple has a center\nvertex tδ,i, a set TˆγofBtasks, and a set TβofBtasks. Let\nα(tδ,i) =aδ. For all t∈Tˆγ, letα(t) =aˆγand(t, tδ,i)∈≺.\nFor all t∈Tβ, letα(t) =aβand(tδ,i, t)∈≺. The star repre-\nsenting the last triple differs from the above construction. The\nlast triple has a task tδ,mand a set TˆγofBtasks as describedtg:ag\na^\u000e\na^\u000e\nt\u000e;i:a\u000e\na^\r\na^\r\n:::\na\f\na\f\n:::\na\f\na\f\ni2[m\u00001]\ni2[3m]\nBtasks\nBtasks\nnitasks\n:::\nBtasks\nti:a;\na^\f\na^\f\n:::\na\r\na\r\nnitasks\n:::\nt\u000e;m:a\u000e\na^\r\na^\r\n:::\nBtasks\n:::\nmtasksActionProp. setprec del add\na∅ ∅ ∅ ∅\naβ {ˆβ}{ˆβ} {β}\naˆβ{β}{β} {ˆβ}\naγ {ˆγ}{ˆγ} {γ}\naˆγ {γ}{γ} {ˆγ}\naδ ∅ ∅ {δ}\naˆδ{δ}{δ} ∅\nag ∅{ˆβ,ˆγ}{GOAL }\nFigure 5: Actions and constructed task network for the reduction in\nthe proof of Theorem 5. The label at each task teither refers to both\nits name and action ( t:α(t)) or, for legibility, just its action.\nabove, but no set Tβ. To initiate things, there is also an in-\ndependent set of Btasks all with action aβ. Lastly, create\na star to count the completed triples. This star consists of a\ncenter vertex tgand a set Tˆδofmleaf-tasks. Let α(tg) =ag\nand, for each t∈Tˆδ, letα(t) = aˆδand(t, tg)∈≺. Let\nthe initial state be s0={ˆβ,ˆγ}. For S TATE REACHABIL -\nITY, letsg={GOAL }. We prove that these instances are\nyes-instances if and only if the 3-P ARTITION instance is a\nyes-instance.\nSuppose there is a 3-P ARTITION of the integers ni. We\nbuild a task sequence that enumerates these triples. Let\n(nx, ny, nz)be the first triple in the solution. Initially, there\nareBtasks with action aβthat are executable. Alternate ex-\necuting these tasks and leaf-tasks with action aˆβof the stars\nwith centers tx,ty, andtz. This is possible as nx+ny+nz=\nBand the preconditions and effects of aβandaˆβenforce this\nalternating behavior. Now, execute tx,ty, and tz. Then, al-\nternate between executing one of the Btasks with action aγ\nin the stars of tx,ty, andtz, and the Btasks with action aˆγin\nthe star of tδ,1. Then, execute tδ,1, followed by one of the not\nyet executed tasks of action tˆδ. After executing this sequence,\nwe are in a similar state as in the beginning. Once more, there\nareBtasks with action aβexecutable. However, one of the\ntasks with action aˆδand all of the tasks in the star with center\ntδ,1, as well as the three stars corresponding to nx,ny, andnz\nare executed. After repeating the above procedure for all m\ntriples, the only task remaining is tg(since the last star rep-\nresenting a triple does not have a set Tβ) and it is executable,\nwhich results in a yes-instance for S TATE REACHABILITYand P LAN EXISTENCE .\nFor the other direction, suppose we have a yes-instance\nof S TATE REACHABILITY or P LAN EXISTENCE . This im-\nplies that there is a sequence executing tg. This task is\nonly executable after the execution of all mtasks with\naction aˆδ, which in turn requires the execution of all m\ntasks tδ,ifori∈[m]. Without loss of generality, let\nthese tasks be executed in the order tδ,1, . . . , t δ,m. We de-\nfine multisets Sifori∈[m]as follows. Let S1=\n{nj|tjwas executed before tδ,1}and, for i∈[m−1], let\nSi+1={nj|tjwas executed after tδ,iand before tδ,i+1}.\nObserve that P={Si|i∈[m]}partitions the integers\nn1, . . . , n 3m. In order to execute a task tjforj∈[3m],\nfirst its njin-neighbors and, by their preconditions, njtasks\nof action aβhave to be executed. The execution of the task\ntjthen enables the execution of njtasks of action aγ. Note\nthat, for the execution of tδ,1, exactly Btasks with action aβ\nare available, and hence,P\nn∈S1≤B, and the execution of\nBtasks with action aγis required, and hence,P\nn∈S1≥B.\nThus,P\nn∈S1=Band, after the execution of tδ,1, there are\nonce more exactly Btasks of action aβavailable and each\nof the not yet executed tasks tjstill has njnot yet executed\nin-neighbors. Hence, we can repeatedly apply our argument\nand obtainP\nn∈Si=Bfor all i∈[m]. Thereby, Pis a 3-\nPARTITION . Lastly, observe that, for each reachable state s\nfor which GOAL /∈s, either β∈sorˆβ∈s, either γ∈sor\nˆγ∈s, and either δ∈sorδ /∈s. Since s={GOAL }when\nGOAL ∈s, this gives k≤23+ 1 = 9 reachable states.\nChains and Antichains. Given the seemingly discourag-\ning lower bounds obtained so far in this section, one might\nrightfully wonder whether there are reasonable structural re-\nstrictions on the network which can be exploited to obtain\npolynomial-time algorithms. In this subsection, we chart\nthe route towards identifying these by first establishing the\ntractability of all considered problems (under the state-space\nrestrictions justified in Section 3) in the extremes of ≺+,\nthat is, where Tis a chain (i.e., total order) or an antichain\n(≺+=∅).\nWe begin by showing that the problems are easily solved on\nchains. Recall from our discussion in Section 2 that on prim-\nitive task networks, polynomial-time solvability results for\nACTION EXECUTABILITY carry over to P LAN EXISTENCE .\nObservation 6. PLAN VALIDATION , STATE REACHABIL -\nITY, and A CTION EXECUTABILITY are linear-time solvable\non primitive task networks such that ≺+defines a total order.\nProof. Note that ≺+defines the exact order in which the\ntasks in Tmust be executed and that in order to execute a\ncertain task, all tasks that precede it in the order must be ex-\necuted first. It thus suffices to iteratively execute the tasks in\nthe order given by ≺+and at each step check whether the re-\nspective action given by αcan be executed in the current state\nand otherwise abort. For S TATE REACHABILITY and A C-\nTION EXECUTABILITY , we have a yes-instance if after some\nstep the target state is reached or the action set is covered.\nFor P LAN VALIDATION with plan p, we have a yes-instance\nif, for the ithtaskti, we have α(ti) =p[i]for all i∈[|T|].It is also trivial to solve P LAN VALIDATION on antichains.\nObservation 7. PLAN VALIDATION is linear-time solvable\non primitive task networks such that ≺+=∅.\nProof. Since≺+=≺=∅, there are no precedence constraints\non the order the tasks may be executed with respect to task\nnames, and thus, we can simply map the tasks to their cor-\nresponding multiset of actions. Deciding whether a plan is\nexecutable in the task network is achieved by first checking\nwhether the number of action occurrences in the sequence\nmatches the multiset of actions and then simulating the se-\nquence to check whether the preconditions of each action are\nfulfilled.\nHandling antichains for the remaining problems already re-\nquires more effort. For S TATE REACHABILITY , we can ob-\ntain an algorithm via an analysis of how a hypothetical solu-\ntion traverses the state graph.\nTheorem 8. STATE REACHABILITY is polynomial-time\nsolvable on primitive antichain task networks with constantly\nmany states.\nProof. Letkbe the number of states, s0be the initial state and\nsgthe target state. We begin by computing the state transition\ngraphGofIin at most quadratic time. In G, if there are more\nthankedges between any pair of states, arbitrarily select k\nof them and discard the rest. Then, consider all simple edge\npaths from s0tosg. Note that there are O\u0000\n2k·kk−1\u0001\nsimple\nedge paths, as there are O\u0000\n2k\u0001\nsimple vertex paths, each of\nlength at most k−1, and each step might choose between\nup to kdifferent edges. For each simple edge path, check\nwhether there is any action that is used more times in the path\nthan there are tasks of that action in T. If there exists a sim-\nple edge path for which the number of required tasks does\nnot exceed the number of available ones, then give a positive\nanswer, and otherwise return that it is a no-instance.\nIf such a path is found, the instance is indeed a yes-instance\nas there is a task sequence that results in a path to sginG. For\nthe other direction, let there be a task sequence that leads to\nthe state sg. Consider the subsequence obtained by exhaus-\ntively removing cycles in Gfrom that sequence. The resulting\ntask sequence describes a simple edge path from s0tosg. It\nmight employ edges that were discarded from Gby the algo-\nrithm. However, in this case there are kother edges connect-\ning the respective states in G. Thus, one can choose to re-\nplace deleted edges by existing ones in such a way that each\nreplacement action occurs at most once in the path, which\nensures that there are enough tasks to actually walk this path.\nHence, there is a path from s0tosgthat only uses edges that\nare present in Gafter the deletion and for which the number of\nrequired tasks does not exceed the number of available ones.\nAs the algorithm will find this path, it will correctly give a\npositive output.\nReducing the number of edges in GtakesO(|A| ·k)time,\nwhile the number of tasks for each action can be counted\ninO(|T|)time. For each path of length at most k, it\ntakesO(k)time to check whether there are enough tasks\nfor each action in the path. This gives a total runtime of\nO\u0000\n|I|2+|A|k+|T|+ 2k·kk\u0001\n, where the middle two termsare always dominated by either the first or the last term.\nHence, any instance Iof S TATE REACHABILITY in which\nthe task network is primitive and ≺+=∅can be solved in\nO\u0000\n|I|2+ 2k·kk\u0001\ntime, where kis the number of states.\nObserve that when trying to adapt the above approach for\nPLAN EXISTENCE or, more generally, for A CTION EXE-\nCUTABILITY , we might now have to visit a state multiple\ntimes as multiple tasks or actions might be executable in that\nstate alone. The number of times a state is visited is bounded\nby the size of the action set S, however this insight only\nyields a polynomial-time algorithm if k+|S|is fixed. In\nparticular, we note that for the reduction from P LAN EXIS-\nTENCE to A CTION EXECUTABILITY , it may possibly happen\nthat|S|= Θ(|T|). Nevertheless, we obtain a polynomial-\ntime algorithm for both problems on instances with fixed k\nby a more complex approach which exploits the fact that suf-\nficiently long walks in the state transition graph will revisit\nthe same simple cycle multiple times.\nBefore we present the algorithms, we will need to intro-\nduce two reduction rules and the notion of equivalent actions.\nDefinition 4.1. For an HTN planning instance with state tran-\nsition graph G= (S,T, β), let a1, a2∈Aand, for all\ns, s′∈ S, let(a1, s, s′)∈ T if and only if (a2, s, s′)∈ T .\nThen, a1anda2areequivalent actions.\nWe note that there are at most (k+ 1)kequivalence classes\nbecause each class is identified by a vector Dof length k=\n|S|overS ∪ {∅} . Thereby, D[i] =sis to be interpreted\nas the equivalence class transforms the ithstate into state s,\nandD[i] =∅implies the class does not transform the ith\nstate into anything as its preconditions are not fulfilled. This\nallows us to define a reduction rule reducing the number of\ndifferent actions. A reduction rule is safeif the rule preserves\n(or correctly identifies) yes- and no-instances.\nFor any primitive task network tn= (T,≺+, α)and action\na∈A, letTa={t∈T|α(t) =a}andmax( a) =|Ta|. For\nACTION EXECUTABILITY instances with action multiset S,\nletmin(a)denote the number of occurrences of ainS. Note\nthat a primitive A CTION EXECUTABILITY instance has a so-\nlution if and only if there is a task sequence that is executable\nintnands0in which each action aoccurs at least min(a)and\nat most max( a)times. We now provide the reduction rules\nwe will use in our algorithm for A CTION EXECUTABILITY .\nReduction Rule R0: For a primitive A CTION EXE-\nCUTABILITY instance, if there is an action a∈Asuch that\nmin(a)>max( a), then output “no”.\nReduction Rule R1: Consider a primitive A CTION EXE-\nCUTABILITY instance such that Reduction Rule R0 cannot be\napplied. Let a1, a2∈Awitha1̸=a2be equivalent actions.\nIf, for all t1∈Ta1, t2∈Ta2, t∈T, the following holds:\n•(t, t1)∈≺+if and only if (t, t2)∈≺+and\n•(t1, t)∈≺+if and only if (t2, t)∈≺+,\nthen change αsuch that, for all t2∈Ta2, now α(t2) =a1,\nand replace all occurrences of a2inSbya1.\nLemma 9. Reduction Rule R1 is safe.Proof. Letmax r(a)andminr(a)be defined as above, but\nfor the reduced instance. Hence, max r(a1) = max( a1) +\nmax( a2)andminr(a1) = min( a1) + min( a2). Thus, any\ntask sequence that solves the original instance also solves\nthe reduced instance. Now assume there is a task sequence\nthat solves the reduced instance. If the same task sequence\nsolves the original instance, we are done. Otherwise, let x1\nandx2be the number of occurrences of a1- and a2-tasks,\nrespectively, in the sequence resolving the reduced instance\naccording to the original mapping α. Note that max r(a1) =\nmax( a1) + max( a2)≥x1+x2≥min(a1) + min( a2).\nDue to the equivalence in their actions and neighborhoods\nin≺+,a1- and a2-tasks can be used interchangeably in the\nsequence. Thus, any a1-task can be replaced by an a2-\ntask, and vice versa, which, due to the above inequalities,\nmakes it possible to find a valid solution for the original in-\nstance, i.e., one in which max( a1)≥x1≥min(a1)and\nmax( a2)≥x2≥min(a2).\nNote that an instance of A CTION EXECUTABILITY with\n|S|>|T|is a trivial no-instance, so we assume |S| ≤ |T|. By\ncombining the above reduction rules with layered branching\ntechniques and integer programming, we prove:\nTheorem 10. ACTION EXECUTABILITY is polynomial-time\nsolvable on primitive antichain task networks with constantly\nmany states.\nProof. We begin by exhaustively applying Reduction Rules\nR0 and R1, which takes at most O(|I|3)time. Afterwards,\nwe have either correctly solved the instance, or—since the\nnetwork is an antichain—obtained a new equivalent instance\nwhere the number of actions is bounded by a function of the\nnumber kof states, and in particular |A| ≤(k+ 1)k.\nThe algorithm now branches over the at most 2k· |A|k\nsimple edge paths in Gthat start at s0. Let Cbe the set of\ndistinct simple edge cycles in G, where two cycles are con-\nsidered distinct if and only if they do not contain the exact\nsame vertices, i.e., cycles obtained from other cycles by in-\nverting their directions are not counted as different cycles.\nThen,|C| ≤2k· |A|k. The algorithm branches on all possi-\nble subsets C′⊆Cto decide which of these cycles are used\nat least once by a solution sequence. To ensure that there is a\nwalk, set all cycles in C′as unmarked and exhaustively mark\ncycles that either share a vertex with the chosen path or an\nalready marked cycle. Proceed with C′only if all cycles are\nmarked by this procedure.\nWe write an Integer Linear Program (ILP) as follows. Let\nthe ILP have a variable xcfor every c∈C′. For each action a\nandc∈C′, letc(a)denote the number of occurrences of ain\nc. Further, let p(a)denote the number of occurrences of ain\nthe chosen path. For each a∈A, require the two constraints\np(a) +c1(a)xc1+c2(a)xc2+. . . c|C′|(a)xc|C′|≤max( a)\np(a) +c1(a)xc1+c2(a)xc2+. . . c|C′|(a)xc|C′|≥min(a),\nwhere c1, . . . , c |C′|are the elements of C′. Further, for all\nc∈C′, add a constraint xc≥1to ensure that each of the\nselected cycles is walked at least once. The algorithm decides\nthat the given instance is a yes-instance if and only if there is\na solution to the ILP.If the ILP has a solution, construct a plan pas follows. Start\nwith the chosen path. For each cycle c∈C′, add xcrepeti-\ntions of that cycle at the first time a vertex of cappears in the\npath. As the cycles end in the same state they started in, the\nresulting action sequence is executable. Possibly, some cy-\ncles cannot be added directly, but only after other cycles have\nbeen added. Nevertheless, as all cycles in C′were marked\nby the algorithm, and the ILP ensures that each cycle in C′\nis executed at least once, all cycles can be added in this way.\nFurther, the ILP ensures that each action aappears at least\nmin(a)and at most max( a)times in the resulting executable\naction sequence. Thus, the given instance is a yes-instance.\nFor the other direction, suppose there is an executable plan\npsuch that each action aoccurs at least min(a)and at most\nmax( a)times in p. Consider an initially empty set of cycles\nC∗. Then, find a substring of pthat describes a simple edge\ncycle in G. Add this cycle to C∗while disregarding its start-\ning vertex and its direction, remove the respective substring\nfrom p, and repeat until no more such cycles exist. Observe\nthatpthen describes a simple path in G, and C∗exclusively\ncontains simple edge cycles in G. There is a branch of the\nalgorithm that chooses this simple path and has C′=C∗.\nFurther, as the cycles were iteratively removed from a walk\nin the state transition graph, the algorithm will properly mark\nall cycles in C′. Also, the ILP has a valid solution by, for each\ncycle c∈C′, setting xcto the number of times this cycle was\nremoved from p. Thus, in this case the algorithm will decide\nthat the given instance is a yes-instance.\nComputing min(a)andmax( a)for all a∈Atakes\nO(|T|+|S|)time. Combining the choices for the path and\nC′gives at most 2k· |A|k·22k·|A|kbranches. Testing a set\nC′by the marking procedure takes O\u0000\n|C|2k\u0001\ntime, but this\ntime is dominated by the time to solve the ILP. The ILP has at\nmost|C| ≤2k·|A|kvariables and 2|A|+|C|=O\u0000\n2k· |A|k\u0001\nconstraints, in which the absolute values of all coefficients are\nbounded by |T|.\nBy the result of [Lenstra, 1983 ]and its subsequent\nimprovements [Kannan, 1987; Frank and Tardos, 1987 ]\nan ILP instance IILPwith nvariables can be solved in\nO\u0000\nnO(n)· |I ILP|\u0001\ntime. We note that, for the given ILP,\n|IILP|=O\u0000\n2k· |A|k·log(|T|)\u0001\n. Thus, solving the ILP\ntakesO\u0010\n(2k· |A|k)O(2k·|A|k)·22k· |A|2k·log(|T|)\u0011\ntime.\nHence, the total runtime can be upper-bounded by\nO\u0010\n2k· |A|k·22k·|A|k·(2k· |A|k)O(2k·|A|k)·22k· |A|2k·\nlog(|T|) +|T|+|S|\u0011\n=O\u0010\n22k·|A|k·(2|A|)O(k·2k·|A|k)·log(|T|) +|T|\u0011\n.\nRecalling the bound on |A|argued in the first paragraph of the\nproof, we obtain that any instance of A CTION EXECUTABIL -\nITYin which the task network is primitive and ≺+=∅can be\nsolved in time O(f(k)·log(|I|)+|I|3), for some computable\nfunction fof the number kof states.\nUsing Generalized Partial Order Width. At this point,\nwe have seen that all of the considered problems on HTNsbecome polynomial-time solvable on chains and antichains.\nIn this subsection, we unify and extend these positive results\nby establishing tractability for all HTNs of constant general-\nized partial order width (whereas chains and antichains have\na generalized partial order width of 1and0, respectively). In-\ntuitively speaking, the following three theorems are obtained\nby “supercharging” the proof techniques used for chains and\nantichains via a careful dynamic programming routine.\nTheorem 11. PLAN VALIDATION is polynomial-time solv-\nable on primitive task networks of constant generalized par-\ntial order width.\nProof. Letwbe the generalized partial order width of the\ninput network, Uthe set of isolated elements in T, and re-\ncall that the set Vof all remaining elements in Tinduces a\npartial order of width at most w. First, check whether pis\nan executable sequence with respect to the preconditions of\neach action. If not, then return that it is a no-instance. Other-\nwise, decompose D≺[V]intowchains c1, . . . , c w. This can\nbe done in O\u0000\n|V|2.5\u0001\ntime [Fellows and McCartin, 2003 ].\nFor0≤i≤ |T|, letRi[h1, . . . , h w]be a boolean variable\nthat is TRUE if and only if the sequence of the first iactions\ninpis executable in tnsuch that exactly the first hjtasks\nare used in each chain cj, j∈[w]. We compute this variable\nfor all possible assignments in a dynamic programming man-\nner. Initialize all values to FALSE except for R0[0, . . . , 0],\nwhich is initialized to TRUE. This completes the calculations\nfori= 0. Assume that the calculations for some arbitrary,\nbut fixed iare complete.\nThen, update the values for i+ 1as follows. Iterate over\nall assignments Ri[h1, . . . , h w]that are TRUE. Let T′⊆U∪\nVbe such that t∈T′if and only if α(t) =p[i+ 1] and,\nfor each chain cj, all the predecessors of tin≺+incjare\namong the first hjtasks in cj. Intuitively, T′contains the\ntasks that are executable as the (i+ 1)thtask in the plan given\nthat the first hjtasks are used in each chain cj. Further, let\nV′⊆Vbe such that t∈V′if and only if α(t) =p[i+ 1]\nandtis among the first hjtasks in its chain cj, that is, V′\ncontains the already executed tasks of the current action from\nthe chains. Let odenote the number of occurrences of the\naction p[i+1]in the first ielements of the plan p. If|U∩T′|>\no− |V′|, then set Ri+1[h1, . . . , h w]to be TRUE. Intuitively,\nthis means that the next task can be taken from Usince there\nis an available task in Uthat is not already required earlier\nin the plan. Also, the (i+ 1)thtask may come from a chain.\nThus, for each chain cj, j∈[w],such that the (hj+ 1)th\ntask is in T′, setRi+1[h1, . . . , h j−1, hj+ 1, hj+1, . . . , h w]\nto TRUE. Continue until the computations for i=|T|are\ncompleted. Then, the given instance is a yes-instance if and\nonly if there is R|T|[h1, . . . , h w]that is TRUE.\nWe show by induction over ithat the algorithm sets all\nvariables correctly. For i= 0 , the algorithm sets all\nR0[h1, . . . , h w]correctly as the only way to build an empty\nplan is to take no tasks from any chain. Now, assume that\nthe algorithm correctly assigned all variables up to some ar-\nbitrary, but fixed i.\nIf the algorithm sets some value Ri+1[h1, . . . , h w]to\nTRUE, then we distinguish between two cases. First, assume\nit does so starting from Ri[h1, . . . , h w]being TRUE. Due tothe induction hypothesis, there is an executable task sequence\nthat uses the first hjtasks of each chain cj,j∈[w], and\nwhose actions correspond to the first iactions of p. Action\np[i+1] occurs otimes among these. As |V′|contains all used\ntasks with this action from the chains, exactly o− |V′|exe-\ncuted tasks of this action are in U. As T′∩Ucontains all\nexecutable tasks in U, if|T′∩U|> o− |V|, then there is a\ntaskt∈Uthat is executable for action p[i+ 1] and has not\nalready been used. Thus, it is correct to set Ri+1[h1, . . . , h w]\nto TRUE. In the other case, there is j∈[w]such that\nRi[h1, . . . , h j−1, hj−1, hj+1, . . . , h w]is TRUE and the hth\nj\naction of the chain cjis executable. Thus, in this case it is\nalso correct to set Ri+1[h1, . . . , h w]to TRUE.\nFor the other direction, suppose there is an executable se-\nquence rofidifferent tasks in Tof the required actions that\nuses the first hjtasks of each chain cj, that is, Ri[h1, . . . , h w]\nshould be TRUE. Let the last task in this sequence be tand\nletr′denote the sequence rwithout t. Ift∈U, then the\nexistence of r′implies that Ri−1[h1, . . . , h w]is TRUE, and\nin the computations starting from there we have |T′∩U|>\no− |V|. Ift∈V, then tis the hth\njtask of a chain cj,\nRi−1[h1, . . . , h j−1, hj−1, hj+1, . . . , h w]is TRUE, and in\nthe computations starting from there we have t∈T′. In both\ncases, the algorithm correctly sets Ri[h1, . . . , h w]to TRUE.\nThis procedure sets O(|T| · |T|w)values to TRUE. For\neach of these, computing T′, V′,andotakesO(|T|+| ≺ |)\ntime and it can update at most w+1variables. Hence, setting\nall variables takes O\u0000\n|T|w+1(|T|+| ≺ |+w+ 1)\u0001\n=\nO\u0000\n|T|w+1(|T|+| ≺ |)\u0001\ntime, which dominates the\nO\u0000\n|V|2.5\u0001\ntime it takes to decompose D≺[V]into w\nchains. In particular, the running time can be upper-bounded\nby|I|O(w).\nWe note that the above algorithm is able to also solve the\nproblem on more general networks, specifically even if the\nisolated vertices in Uare allowed to have in-edges from V.\nNext, we set our sights on S TATE REACHABILITY . While\nthis problem could be solved via a stand-alone argument on\nantichains in Theorem 8, in the more general setting consid-\nered here, we will first need to apply a reduction rule that can\nbe seen as a simpler version of Reduction Rule 1 used for\nACTION EXECUTABILITY .\nReduction Rule R2: Assume we are given a primitive\nSTATE REACHABILITY instance. Let a1anda2be two equiv-\nalent actions. For all t∈Tsuch that α(t) = a2, set\nα(t) =a1.\nObservation 12. Reduction Rule R2 is safe.\nProof. Assume there is a task sequence t1, . . . , t ℓsolving the\noriginal instance. Then, the same sequence solves the reduced\ninstance as it gives the same path in the state transition graph\nin either instance. The same holds for the other direction.\nWith this, we are now ready to also establish tractability\nfor S TATE REACHABILITY .\nTheorem 13. STATE REACHABILITY is polynomial-time\nsolvable on primitive task networks with constantly many\nstates and constant generalized partial order width.Proof. Letkbe the number of states and wthe generalized\npartial order width of the input instance. Set Uto be the set of\nisolated elements in T, and recall that the set Vof all remain-\ning elements in Tinduces a partial order of width at most\nw. We begin by exhaustively applying Reduction Rule R2,\nwhich can be done in time at most O\u0000\n|I|3k2\u0001\n. Afterwards,\nwe note that there is at most one action for each of the at most\n(k+ 1)kequivalence classes.\nThe algorithm now decomposes D≺[V]intowchains\nc1, . . . , c w. This can be done in O\u0000\n|V|2.5\u0001\ntime [Fellows and\nMcCartin, 2003 ]. Let A={a1, . . . , a |A|}. For each state s\nin the state transition graph, let Rs[h1, . . . , h w][u1, . . . , u |A|]\nbe a boolean variable that is TRUE if and only if there is an\nexecutable task sequence in tnstarting from s0and ending in\nstatesthat employs exactly the first hjtasks of each chain cj,\nj∈[w], and contains exactly uitasks with action ai, for all\ni∈[|A|].\nWe compute this variable for all possible assignments in\na dynamic programming manner. Initialize all values to\nFALSE except for Rs0[0, . . . , 0][0, . . . , 0], which is initialized\nto TRUE. Whenever a variable Rs[h1, . . . , h w][u1, . . . , u |A|]\nthat was FALSE before is set to TRUE, also perform the\nfollowing update step. Let T′⊆Tbe such that t∈T′\nif, for each chain cj, all the predecessors of tin≺+in\ncjare among the first hjtasks in cj. Consider all the\nstates s′that are adjacent to sinG. If there is j∈[w]\nsuch that the (hj+ 1)thtask of the chain cjis in T′,\nhas action ai, and aiconnects sands′inG, then set\nRs′[h1, . . . , h j−1, hj+ 1, hj+1, . . . , h w][u1, . . . , u i��1, ui+\n1, ui+1, . . . , u |A|]to TRUE. This corresponds to taking the\nnext task from a chain and adding it to a given sequence. An-\nother update rule corresponds to adding a task from Uin-\nstead. If there is an action aithat connects sands′inG,\nthen define the following. Let Ti={t∈T|α(t) =ai}and\nletvidenote the number of tasks in V∩Tithat are among\nthe first hjtasks of a chain cj. Ifui−vi<|Ti∩T′∩U|,\nthen set Rs′[h1, . . . , h w][u1, . . . , u i−1, ui+1, ui+1, . . . , u |A|]\nto TRUE. Once no more variables are updated and set to\nTRUE, the computation is complete. Then, a given S TATE\nREACHABILITY instance is a yes-instance if and only if there\nisRsg[h1, . . . , h w][u1, . . . , u w]that is TRUE. A given A C-\nTION EXECUTABILITY instance is a yes-instance if and only\nif there is Rs[h1, . . . , h w][u1, . . . , u w]that is TRUE such that,\nfor each action ai∈S, we have min(a)≥ui.\nWe show that the algorithm sets all variables correctly by\nan inductive argument overP|A|\ni=1ui. Note that when one\nTRUE variable sets another variable to TRUE, this increases\nthis sum by exactly 1. ForP|A|\ni=1ui= 0, the algorithm sets\nall variables correctly as the only way to build an empty plan\nis to take no tasks from any chain and remain in the ini-\ntial state, and thus, the only TRUE variable in this case is\nRs0[0, . . . , 0][0, . . . , 0]. Now, assume that the algorithm cor-\nrectly assigned all variables up to some arbitrary, but fixedP|A|\ni=1ui.\nIf some value Rs[h1, . . . , h w][u1, . . . , u w]is set to\nTRUE by the algorithm, then we distinguish between\ntwo cases. First, assume it does so starting from some\nRs′[h1, . . . , h w][u1, . . . , u i−1, ui−1, ui+1, . . . , u |A|]beingTRUE with i∈[|A|]. Then, aiis executable in state s′and\ntransforms it into s. Further, due to the induction hypothe-\nsis, there is an executable task sequence that uses the first hj\ntasks of each chain cj,j∈[w], and consists of uℓtasks of\naction aℓ, for each ℓ∈[|A|]\\ {i}, and ui−1tasks of ac-\ntionai. Thus, it contains ui−vitasks in Uwith action ai.\nThere are more than ui−viexecutable tasks of action aiin\nU, and thus, one of the remaining tasks of action aican be ap-\npended to the sequence. The resulting sequence is executable\nintnstarting from s0, uses the same number of tasks of each\nchain and action, except one more task with action ai. Thus,\nit is correct to set Rs′[h1, . . . , h w][u1, . . . , u w]to TRUE. In\nthe other case, it sets the variable to TRUE starting from\nsome Rs′[h1, . . . , h j−1, hj−1, hj+1, hw][u1, . . . , u i−1, ui−\n1, ui+1, . . . , u |A|]being TRUE with j∈[w],i∈[|A|]. Then,\ndue to the induction hypothesis, there is an executable task\nsequence that uses the first hℓ′tasks of each chain cℓ′, for\neachℓ′∈[w]\\ {j}, but only uses the first hj−1tasks of\nthe chain cj, and it consists of uℓtasks of action aℓ, for each\nℓ∈[|A|]\\ {i}, and ui−1tasks of action ai. Further, the hth\nj\ntask of chain hjhas action ai, is executable at the end of this\nsequence, and transforms state s′intos. Thus, in this case it\nis also correct to set the variable to TRUE.\nFor the other direction, suppose there is an executable se-\nquence rofP|A|\ni=1uidifferent tasks in Tsuch that, for each\nℓ∈[|A|], there are uℓtasks with action aℓthat use the first\nhjtasks of each chain cj. Suppose rends in state s, that is,\nRs[h1, . . . , h w][u1, . . . , u |A|]should be TRUE. Let the last\ntask in this sequence be twithα(t) =aiand let r′denote the\nsequence rwithout t. Letr′end in state s′. Ift∈U, then the\nexistence of r′implies Rs′[h1, . . . , h w][u1, . . . , u i−1, ui−\n1, ui+1, . . . , u |A|]is TRUE. In the update step of this vari-\nable,tis executable, and thus, ui−vi<|Ti∩T′∩U|. Then,\nthe algorithm correctly sets Rs[h1, . . . , h w][u1, . . . , u |A|]to\nTRUE. If t∈V, then tis the hth\njtask of a chain cj,\nRs′[h1, . . . , h j−1, hj−1, hj+1, . . . , h w][u1, . . . , u i−1, ui−\n1, ui+1, . . . , u |A|]is TRUE, and in the computations starting\nfrom there t∈T′. In both cases, the algorithm correctly sets\nthe variable to TRUE.\nFor each variable that is TRUE, we have that sis\namong the kstates in G,0≤hj≤ |T|forj∈[w], and\n0≤P|A|\ni=1ui≤ |T|. Thus, at most O\u0000\n|T|w·k· |T||A|\u0001\nvariables are considered. For each of these, computing\nT′, the ui, and vitakesO(|T|+| ≺ |)time, it can update\nat most (w+ 1)· |A|variables, and it can be checked in\nO(|A|)time if it yields a yes-instance. Hence, the algorithm\ntakes O\u0000\n|T|w+|A|·k·(|T|+| ≺ |+ (w+ 1)· |A|)\u0001\n=\nO\u0000\n|T|w+|A|·k·(|T|+| ≺ |+w|A|)\u0001\ntime. Once again,\nthe theorem now follows by the bound on |A|established in\nthe first paragraph of the proof.\nWe observe that, once more, the above algorithm is actu-\nally able to solve the more general problem where the isolated\nvertices in Uare allowed to have in-edges from V. Finally,\nwe note that Reduction Rule R1 by itself does not suffice to\ntransfer this result to A CTION EXECUTABILITY as the num-\nber of neighborhoods of tasks is not bounded (as it was forTheorem 10). Nevertheless, the desired result can be obtained\nby carefully adapting the dynamic program from Theorem 13\nand incorporating the idea of the reduction rule on the (iso-\nlated) tasks in U.\nTheorem 14. ACTION EXECUTABILITY is polynomial-time\nsolvable on primitive task networks with constantly many\nstates and constant generalized partial order width.\nProof. Letkbe the number of states and wthe generalized\npartial order width of the input instance. Set Uto be the set of\nisolated elements in T, and recall that the set Vof all remain-\ning elements in Tinduces a partial order of width at most w.\nThe algorithm is similar to the one presented for Theorem 13.\nFirst, test whether Reduction Rule R0 can be applied, and if\nyes we use it to terminate. Otherwise, decompose D≺[V]\nintowchains c1, . . . , c w. This can be done in O\u0000\n|V|2.5\u0001\ntime [Fellows and McCartin, 2003 ]. LetA={a1, . . . , a |A|}.\nLetE={e1, . . . , e |E|}denote the set of equivalence classes\namong all actions. For each state sin the state transition\ngraph, let Rs[h1, . . . , h w][r1, . . . , r |E|]be a boolean variable\nthat is TRUE if and only if there is an executable task se-\nquence in tnstarting from s0and ending in state sthat em-\nploys exactly the first hjtasks of each chain cj,j∈[w], and\ncontains exactly ritasks from Uwith action equivalence class\nei, for all i∈[|E|].\nWe compute this variable for all possible assignments in\na dynamic programming manner. Initialize all values to\nFALSE except for Rs0[0, . . . , 0][0, . . . , 0], which is initialized\nto TRUE. Whenever a variable Rs[h1, . . . , h w][r1, . . . , r |E|]\nthat was FALSE before is set to TRUE, also perform the fol-\nlowing update step. Let T′⊆Tbe such that t∈T′if,\nfor each chain cj, all the predecessors of tin≺+incjare\namong the first hjtasks in cj. Consider all the states s′that\nare adjacent to sinG. If there is j∈[w]such that the\n(hj+ 1)thtask of the chain cjis inT′, has action ai, and\naiconnects sands′inG, then set Rs′[h1, . . . , h j−1, hj+\n1, hj+1, . . . , h w][r1, . . . , r |E|]to TRUE. This corresponds to\ntaking the next task from a chain and adding it to a given se-\nquence. Another update rule corresponds to adding a task\nfrom Uinstead. If there is an action equivalence class ei\nthat connects sands′inG, then define the following. Let\nTi={t∈T|α(t)∈ei}and let videnote the number of\ntasks in V∩Tithat are among the first hjtasks of a chain cj. If\nri−vi<|Ti∩U|, then set Rs′[h1, . . . , h w][r1, . . . , r i−1, ri+\n1, ri+1, . . . , r |E|]to TRUE. Once no more variables are up-\ndated and set to TRUE, the computation is complete.\nThe algorithm sets all variables correctly by the same\nreasoning as in the proof of Theorem 13. For any\nRs[h1, . . . , h w][r1, . . . , r |E|]and action a∈A, let c(a)\ndenote the number of tasks in Vwith action athat are\namong the first hjtasks of a chain cj. Then, the al-\ngorithm returns that it is a yes-instance if and only if\nthere is Rs[h1, . . . , h w][r1, . . . , r |E|]that is TRUE such\nthat, for each action equivalence class ei∈E, we haveP\na∈eimax{0,min(a)−c(a)} ≤ ri, and, for each action\na∈A, there are at least min(a)−c(a)tasks of action ainU.\nSuppose there is such a TRUE variable. Then, there is an\nexecutable task sequence that uses c(a)tasks in Vwith actionafor each a∈A, and uses ritasks from Uwith action in\nequivalence class eifor all i∈[|E|]. As all tasks in Uare\nisolated, two tasks from Ucan be used interchangeably in a\ntask sequence if they are from the same action equivalence\nclass. Tasks in the sequence can be replaced by tasks from\nUof the same equivalence class such that there are at least\nmin(a)−c(a)tasks with action afrom Uin the sequence\nfor all a∈A. Then, the sequence is a solution to A CTION\nEXECUTABILITY .\nFor the other direction, suppose there is a task sequence\nthat solves the A CTION EXECUTABILITY instance. Let\nit result in some state s, let it use the first hjtasks\nof each chain cj, and let it employ ritasks of equiv-\nalence class eifor all i∈[|E|]. Then, the variable\nRs[h1, . . . , h w][r1, . . . , r |E|]is TRUE. Further, as the se-\nquence contains at least min(a)tasks of each action a∈A,\nand only c(a)of these are tasks in V, it contains at least\nmin(a)−c(a)tasks of action afrom U. In particular, this\nalso givesP\na∈ei(max{0,min(a)−c(a)})≤ri, so the al-\ngorithm correctly returns that it is a yes-instance.\nComputing the equivalence classes takes O\u0000\n|I|3\u0001\ntime.\nAs there are |E| ≤(k+ 1)kequivalence classes, at most\nO\u0010\n|T|w·k· |T|(k+1)k\u0011\nvariables are considered. For each\nof these, computing all vitakesO(|T|)time by iterating\nthrough all chains time and it can update at most w+|E|\nvariables. Further, it takes O(|T|)time to check whether a\nvariable yields a yes-instance. Hence, the algorithm takes\nO\u0010\n|T|w+(k+1)k·k·(|T||+w+|E|) +|V|2.5+| ≺ |\u0011\n=\nO\u0010\n|T|w+(k+1)k+1·k\u0011\ntime.\nWe recall that Theorem 14 also immediately implies the\nanalogous result for P LAN EXISTENCE .\n5 Hierarchical Task Networks\nConsidering HTNs with compound tasks adds another, seem-\ningly opaque layer of difficulty to the considered planning\nproblems. In this section, we show that—at least in terms of\nclassical complexity—we can cleanly characterize the jump\nfrom primitive to compound through three measures on com-\npound task networks: we obtain an algorithmic meta-theorem\nthat allows to lift polynomial-time solvability from primitive\nto compound task networks, if all three of these measures are\nbounded. Moreover, this result is tight in the sense that none\nof the three measures can be dropped from the meta-theorem.\nThe three measures we need are C#,Cs, and Cd:\nDefinition 5.1. For an HTN planning problem with initial\ntask network tn= (T,≺+, α), we refer to the number\nof compound tasks in tnbyC#=|{t∈T|α(t)∈ C}| .\nThe maximum size of a network that a task can be decom-\nposed into is Cs= max {|Tm| |(c,(Tm,≺+\nm, αm))∈M}.\nThe maximum depth Cdof a network is defined recursively:\nCd= 0 for primitive task networks, and a compound task\nnetwork has Cd=iifiis the smallest integer such that every\ndecomposition of all compound tasks in the initial network T\nresults in a new network where Cd≤i−1.For our later considerations, it will be useful to addition-\nally define the maximum number of pairwise non-isomorphic\nnetworks that a task can be decomposed into; we denote that\nbyCc= max c∈C|{tnm|(c, tn m)∈M}|. The next tech-\nnical lemma shows how these measures help us bound the\n“decomposition complexity” of compound HTNs.\nLemma 15. A task network tn= (T,≺+, α)can be decom-\nposed into at most CcPCd−1\ni=0C#·Csipairwise non-isomorphic\nprimitive task networks tn′= (T′,≺+′, α′). For each of\nthese,|T′| ≤ |T|+C#·(CsCd−1).\nProof. We first show the second part of the statement. The\n|T|−C#primitive tasks in tnare still present in any network\ntn′thattncan be decomposed into. A compound task in tn\ncan be decomposed into at most Cscompound tasks in one\ndecomposition step and there are at most Cddecomposition\nsteps. Thus, each compound task in tncan create at most\nCsCdtasks in tn′. Hence, |T′| ≤ |T| −C#+C#·CsCd.\nFor the first statement, note that decomposing each of the\nC#compound tasks once in a task network of depth Cdcan\nresult in at most CcC#pairwise non-isomorphic task net-\nworks of depth at most Cd−1. Further, each of these re-\nsulting networks has at most C#·Cscompound tasks. Thus,\ntncan be decomposed into at most\nCcC#·CcC#·Cs·. . .·CcC#·CsCd−1=CcPCd−1\ni=0C#·Csi\npairwise non-isomorphic task networks.\nHaving defined our measures of interest, we proceed to for-\nmalize the set of problems our meta-theorem will capture. In-\ntuitively, the aim here is to formalize the set of all problems\nwhich deal with compound HTNs by decomposing them.\nDefinition 5.2. Let PR be an arbitrary HTN planning prob-\nlem. Then, PR is decomposable if and only if, for all non-\nprimitive task networks tn, PR on tnis a yes-instance if and\nonly if there is a primitive task network tn′such that tncan\nbe decomposed into tn′andtn′is a yes-instance of PR.\nNote that all problems discussed in this work are decom-\nposable. Next, in order to capture the general notions of\n“tractability” for various HTN problems on networks where\ncertain measures are bounded, we need to slightly restrict the\nmeasures in question, to avoid entirely degenerate measures.\nDefinition 5.3. Letκbe any numerical measure of HTNs,\ni.e., a mapping that assigns each HTN a non-negative number.\nWe say that κisstable if there exists a computable function\nfwith the following property: for each primitive HTN tn\nand each primitive HTN tn′obtained from tnby adding or\nremoving a single task, κ(tn′)≤f(κ(tn)).\nIn particular, all measures considered here are stable. The\nnumber of states in the state transition graph only depends\non the initial state s0and the set of actions A, so it does not\nchange when applying decomposition methods. Moreover,\nadding an element into the network can only increase the gen-\neralized partial order width (as well as all the other structural\nmeasures mentioned in the manuscript) by at most 1.Theorem 16. LetPRbe a decomposable HTN planning\nproblem and κa stable measure. Assume PRis polynomial-\ntime solvable when restricted to primitive networks of con-\nstant κ. Then, PRis polynomial-time solvable when re-\nstricted to networks of constant κ,Cd, C#, Cs.\nProof. LetIhave task network tn= (T,≺+, α). If there are\nno compound tasks in T, i.e., Cd= 0, the statement immedi-\nately holds by applying an algorithm for primitive networks.\nThus, assume Cd≥1. Then, PR in tncan be solved by\nlisting all the primitive instances tncan be decomposed into\nand solving PR in each of these. By Lemma 15, there are at\nmost Ccg(Cd,C#,Cs)≤ |I|g(Cd,C#,Cs)such instances Ipfor\ng(Cd, C#, Cs) =PCd−1\ni=0C#·Csiand, for each of these,\n|Ip| ≤ |I| +f′′(Cd, C#, Cs)for some computable function\nf′′. Further, for each task network tnpof an instance Ip,\nwe have κ(tnp)≤f′′′(κ, Cd, C#, Cs), for some computable\nfunction f′′′asκis stable and tnpis obtained from tnby\nremoving C#compound tasks and adding f′′(Cd, C#, Cs)\nprimitive tasks. Thus, if primitive instances Ipcan be solved\nin time at most |Ip|f(κ(tnp)), then the total runtime is\nO\u0010\n(|I|+f′′(Cd, C#, Cs))f(f′′′(κ,Cd,C#,Cs))\n·Ccg(Cd,C#,Cs)\u0011\n=O\u0010\n|I|f′(κ,Cd,C#,Cs)\u0011\n,\nwhere f′(κ, Cd, C#, Cs) =g(Cd, C#, Cs) +\nf(f′′′(κ, Cd, C#, Cs)) +f′′(Cd, C#, Cs)f(f′′′(κ,Cd,C#,Cs)).\nWe complement Theorem 16 by providing lower bounds\nwhich show that bounding Cd,C#, andCsis necessary in or-\nder to obtain the result. In particular, we show that if any one\nof these measures is not bounded by a constant, then some of\nthe problems considered in this article become NP-hard on\ncompound networks, even for settings that were easily solv-\nable on primitive networks. We note that there are several\nother existing reductions showing the hardness that derives\nfrom decomposing non-primitive networks, see, e.g., [Erol et\nal., 1996; Behnke et al. , 2015 ]. However, our reductions are\nstronger in the sense that they bound all but one of the param-\netersCd,C#,Cs.\nRecall that, by Theorem 11, P LAN VALIDATION is solv-\nable in polynomial time on primitive networks of constant\ngeneralized partial order width. Moreover, as per Theo-\nrem 10, the same also holds for A CTION EXECUTABILITY\nwhen additionally bounding the number of states. In con-\ntrast, in Theorems 17 to 19, we prove P LAN VALIDATION\n(and also A CTION EXECUTABILITY ) to be NP-hard on gen-\neral networks even if:\n• any two of Cd,C#,Cs, are constant,\n• all possible decompositions of the network have a gen-\neralized partial order width of 0,\n• there is only a single state, and\n•Ccis also constant.Theorem 17. PLAN VALIDATION and ACTION EXE-\nCUTABILITY areNP-hard even if Cc= 7,Cd= 1,Cs= 2,\n≺+=∅, and prec( a)=del(a)=add(a)=∅for every a∈A.\nProof. We prove hardness by reducing from 3-SAT-(2,2),\nanNP-hard 3-SAT variant, where among all clauses each\nvariable appears exactly 4 times: 2 times as a positive literal\nand 2 times as a negated literal [Darmann and D ¨ocker, 2021 ].\nWe construct an HTN planning instance where the set of\nactions A={x1,x1, . . . , x n,xn}contains an action for each\nliteral. The initial task network contains a compound task tx\nfor each variable xand a compound task tcfor each clause.\nA variable compound task txcan be decomposed into either\na task network containing two tasks with action xor a task\nnetwork containing two tasks with action x. Decomposing tx\ninto the first (second, respectively) one of these is interpreted\nas setting xto TRUE (FALSE, respectively). A clause com-\npound task tcwhere the clause ccontains literals ℓ1, ℓ2, ℓ3can\nbe decomposed into one of 7 subnetworks, one with no tasks\nat all, one with a single task with action ℓ1, ℓ2,orℓ3, or one\nwith two tasks with actions ℓ1andℓ2, actions ℓ1andℓ3, or ac-\ntions ℓ2andℓ3. Decomposing a clause task is interpreted as\nacknowledging that exactly the described literals are FALSE.\nThis gives an instance with ≺+=∅. For P LAN VALIDATION ,\nlet the input plan pbe any sequence that contains each literal\nexactly twice. For A CTION EXECUTABILITY , letScontain\neach literal exactly twice.\nIf we are given a yes-instance of 3-SAT-(2,2), con-\nsider a satisfying variable assignment and decompose the\ncompound tasks accordingly using the interpretations given\nabove. Then, in the resulting primitive task network, each lit-\neral appears exactly twice as follows. If a variable xis set\nto TRUE, then txcreates exactly 2 tasks with action xand\nthere is no clause task creating a task with action x. At the\nsame time, the literal xappears in exactly 2 clauses and, as\nit is FALSE, each of these clause tasks creates a task with\naction x. The case in which the variable xis set to FALSE\nis analogous. As the tasks are executable in any order, this\nyields a yes-instance for P LAN VALIDATION and A CTION\nEXECUTABILITY .\nFor the other direction, suppose we have a yes-instance for\nPLAN VALIDATION or A CTION EXECUTABILITY . Then, the\nconstructed HTN can be decomposed into a primitive task\nnetwork with an executable plan such that each literal appears\nat least twice. Consider the variable assignment given by in-\nterpreting the decomposition of the variable tasks. Towards\na contradiction, assume that there is a clause cwith literals\nℓ1, ℓ2, ℓ3that is not satisfied by the assignment. Then, none\nof these literals is represented by a task decomposed from a\nvariable task. Further, the clause task ccan only be decom-\nposed into at most 2 tasks with different actions from ℓ1,ℓ2,\nandℓ3. Without loss of generality, let the decomposition of\nthis clause task not produce a task with action ���1. This lit-\neral occurs in exactly one other clause and the correspond-\ning clause task can produce at most one task with action ℓ1.\nThis contradicts having 2 occurrences of action ℓ1in the plan.\nThus, all the clauses are satisfied.\nObserve that the created task network has depth Cd= 1\nand each compound task can be decomposed into one of atmost Cc= 7 different task networks which each contain at\nmost Cs= 2tasks.\nTheorem 18. PLAN VALIDATION and ACTION EXE-\nCUTABILITY areNP-hard even if Cc= 7,Cd= 2,C#= 1,\n≺+=∅, and prec( a)=del(a)=add(a)=∅for every a∈A.\nProof. Consider the task network tnconstructed in the proof\nof Theorem 17. Build a new initial task network tn′= (T=\n{t},≺+=∅, α), and let (α(t), tn)∈M. Then, C#= 1,\nCd= 2,Cc= 7, and after decomposing tintotn, the reduc-\ntion works exactly as in the proof of Theorem 17.\nTheorem 19. PLAN VALIDATION and ACTION EXE-\nCUTABILITY areNP-hard even if Cc= 7,C#= 1,Cs= 3,\n≺+=∅, and prec( a)=del(a)=add(a)=∅for every a∈A.\nProof. We adapt our construction used in the proof of The-\norem 17. Let the 3-SAT-(2,2) instance have variables\nx1, . . . , x nand clauses C1, . . . , C m. Once more, let the set\nof actions A={x1,x1, . . . , x n,xn}contain an action for\neach literal, and let there be a compound task for each vari-\nable and clause, i.e., C={X1, . . . , X n, C1, . . . , C m}. For\neachi∈[n−1], the variable task Xican be decomposed\ninto one of two subnetworks. Both these subnetworks con-\ntain the compound task Xi+1, and one of them contains two\ntasks with action xi, while the other contains two tasks with\naction xi. The decomposition for Xnis similar, but instead\nof a compound task Xn+1, it adds the compound task C1in\nboth subnetworks. Just like before, the compound tasks for\nclauses can be decomposed into one of the 7 subnetworks\nrepresenting proper subsets of the literals in the clause. For\ni∈[m−1], they additionally contain the compound task\nCi+1. The initial network now contains only a single com-\npound task X1, giving C#= 1. As before, each compound\ntask can be decomposed into one of at most Cc= 7 dif-\nferent task networks which, in this case, each contain at most\nCs= 3tasks. However, the decomposition depth is no longer\nbounded by a constant. For P LAN VALIDATION , let the in-\nput plan pbe any sequence that contains each literal exactly\ntwice. For A CTION EXECUTABILITY , letScontain each lit-\neral exactly twice. The correctness of the reduction follows\nfrom the same arguments that were used in the proof of The-\norem 17.\nWe conclude this section with an analogous intractability\nresult complementing Theorem 16, even if we consider the\n“classical” partial order width as our stable measure, as op-\nposed to the generalized variant considered here:\nTheorem 20. PLAN EXISTENCE ,ACTION EXECUTABIL -\nITY, and STATE REACHABILITY areNP-hard in task net-\nworks where ≺+describes a total order on Tand all subnet-\nworks in Mare total orders even if Cc= 2,Cd= 1, and\nCs= 1. The same holds if Cc= 2,Cd= 2, and C#= 1, or\nCc= 2,C#= 1, and Cs= 2.\nProof. We reduce from k-MULTICOLORED -CLIQUE on a\ngraph G= (V, E), which is NP-hard for unbounded k[Fel-\nlows et al. , 2009 ]. Let tn(p, d, a )withp, d, a ⊆Frefer\nto a task network with a single task twith action α(t)such\nthatprec( α(t)) = p,del(α(t)) = d, and add(α(t)) = a.For each vertex vi,i∈ |V|, create a compound task Vi,\nand, for each edge ei,i∈ |E|, create a compound task\nEi. Create a primitive task tgwith action agsuch that\nprec( ag) ={EDGE j,j′|j, j′∈[k], j < j′},del(ag) =∅,\nandadd(ag) ={GOAL }. The initial network tncontains\nexactly these tasks in a total order where (Vi, Vi+1)∈≺\nfor all i∈[|V| −1],(V|V|, E1)∈≺,(Ei, Ei+1)∈≺for\nalli∈[|E| −1], and (E|E|, tg)∈≺. Each vertex and\nedge compound task can be decomposed into a primitive\ntask with no effect, that is, (Vi, tn(∅,∅,∅))∈Mfor all\ni∈[|V|], and (Ei, tn(∅,∅,∅))∈Mfor all i∈[|E|].\nThis corresponds to not selecting respective vertices and\nedges as part of the clique. Further, a vertex can be cho-\nsen for color jif no other vertex has been chosen for this\ncolor (represented by having COLOR jin the state). Thus,\n(Vi, tn({COLOR j},{COLOR j},{vi}))∈Mfori∈[|V|]\nand the color jof the vertex vi. An edge can be chosen\nas the jthedge in the clique if both its vertices are chosen.\nThus, for all i∈[|E|], ifei= (v, v′),vhas color j, and\nv′has color j′, then (Ei, tn({v, v′},∅,{EDGE j,j′}))∈M.\nThe construction is concluded by setting the initial state s0=\n{COLOR i|i∈[k]}, setting S={ag}for A CTION EXE-\nCUTABILITY , and setting sg={GOAL }for S TATE REACH -\nABILITY . The construction clearly takes polynomial time.\nSuppose there is a multicolored clique of size k. Decom-\npose according to the above interpretation. We obtain a prim-\nitive network with a total order on the tasks. The corre-\nsponding task sequence is executable in tnfrom s0, and as\nit executes all tasks including tg, it yields a yes-instance for\nPLAN EXISTENCE , ACTION EXECUTABILITY , and S TATE\nREACHABILITY .\nFor the other direction, suppose any one of the three HTN\ninstances has a solution. This implies that the task tgcan be\nexecuted. The precondition of tgensures that an edge is se-\nlected for each pair of colors. The preconditions on the edge\ntasks ensure that all the vertices incident to the selected edges\nhave been selected. The preconditions on the vertex tasks en-\nsure that at most 1 vertex of each color is selected. Thus, we\nhave a yes-instance of k-MULTICOLORED -CLIQUE .\nFor the constructed instances, Cd=Cs= 1 andCc= 2.\nFor the two other combinations of parameters for which the\nstatement holds, we employ the same approaches we used to\nadapt Theorem 17. In particular, to prove the statement for\nCc= 2,Cd= 2, andC#= 1, we let the initial network con-\ntain a single compound task that is decomposed into a net-\nwork with the above structure. To prove the statement for\nCc= 2,C#= 1, and Cs= 2, we let the initial network only\ncontain the compound task V1. Then, we adapt all the decom-\nposition methods. For each method (c, tn m)∈M, the net-\nwork tnmconsisted of a single primitive task tm. Let t′de-\nnote the task identifier that followed cin the total order in the\ninitial network. Instead of decomposing into a single primi-\ntive task tm, let the method now decompose into a network\nwith tasks tmandt′with(tm, t′)∈≺+. Note that, in this\nconstruction, exactly the same set of primitive networks can\nbe created as in the original construction, but now C#= 1\nandCdis not bounded.6 A Parameterized Analysis of HTNs\nIn this final section, we ask whether—and under which\nparameterizations—the fundamental problems on HTNs ad-\nmit fixed-parameter algorithms.\nWe begin by noting that while there is by now a very broad\nrange of graph-theoretic parameters which can typically be\nused to achieve fixed-parameter tractability, none of the\n“usual suspects” in that regard will help when dealing with\nany of the four problems of interest here. In particular, Theo-\nrems 2 and 5 immediately exclude the algorithmic application\nof not only treewidth [Robertson and Seymour, 1986 ]—the\nby far most commonly used structural graph parameter—but\nalso treedepth [Nesetril and de Mendez, 2012 ], the feedback\nedge number [Ganian and Korchemna, 2021 ]and even the\nbroad family of analogous measures defined specifically on\ndirected graphs [Ganian et al. , 2016 ].\nMoreover, while the positive results obtained in Section 4\nestablish that our problems of interest are polynomial-time\nsolvable when the generalized partial order width is fixed,\nnone of the problems is fixed-parameter tractable when pa-\nrameterized by the generalized partial order width (or even\nby the “usual” partial order width). Indeed, since the S HUF-\nFLEPRODUCT problem used in the reductions underlying the\nproofs of Theorems 3 and 4 is also known to be W[2]-hard\nwhen parameterized by the width w[van Bevern et al. , 2016,\nTheorem 2.4 ], it turns out that the proofs of these two theo-\nrems also immediately establish the following:\nCorollary 21. PLAN VALIDATION on primitive networks\nwhere ≺+partitions Tintowchains such that there are no\ndependencies between the chains is W[2]-hard parameterized\nbyw, even if |A|= 2 andprec( a) = del( a) = add( a) =∅\nfor all actions a∈A.\nCorollary 22. STATE REACHABILITY and PLAN EXIS-\nTENCE on primitive networks where ≺+partitions Tintow\nchains are W[2]-hard parameterized by w, even if |A|= 5\nand the state transition graph has k= 4states.\nFixed-Parameter Tractability via Vertex Cover. In this\nsubsection, we contrast the above lower bounds with a fixed-\nparameter algorithm that exploits a different parameterization\nof the network: we show that P LAN VALIDATION is fixed-\nparameter tractable when parameterized by the vertex cover\nnumber of G≺. Recall that a vertex cover of G≺is a subset\nof tasks V⊆Tsuch that, for all t≺t′,t∈Vand/or t′∈V;\nthe vertex cover number is the minimum size of a vertex\ncover of G≺. We remark that, while the vertex cover number\nhas also been used to obtain fixed-parameter algorithms for\nmany other problems [Bhore et al. , 2020; Balko et al. , 2022;\nBlazej et al. , 2023 ], the techniques from previous works do\nnot easily translate to HTN planning problems. Instead, our\nalgorithms make use of a delicate branching routine whose\ncorrectness requires a careful analysis of the state space.\nTheorem 23. PLAN VALIDATION is fixed-parameter\ntractable parameterized by the vertex cover number of G≺.\nProof. By definition, pis executable if and only if the ac-\ntions in the plan have their preconditions fulfilled and each\nt∈Tcan be assigned a unique index in the sequence such\nthat the assignment according to αmatches the plan and theimposed order does not conflict with ≺+. Observe that sim-\nulating the action sequence suffices to check whether all pre-\nconditions are met at the respective step, so it suffices to han-\ndle the assignment of tasks to indices. We branch over all\npossible orderings of the vertices in V. Orderings that do\nnot respect an edge in ≺+∩(V×V)are discarded immedi-\nately. For i∈[|V|], letvidenote the ithtask in the ordering.\nThen, for all t∈T\\V, let the priority of tbe defined as\nψ(t) = min (t,vi)∈≺iandψ(t) =∞if there is no such edge.\nFor all vi∈V, letψ(vi) =i+ 0.5. We now iterate through\npand assign tasks to each index in a greedy manner. Assume\nwe have already assigned tasks to the first i−1actions for\nsome i∈[|T|]. Then, we say that t∈Tis executable as the\nithaction if α(t) =p[i], it is not yet assigned an index, and\nall its predecessors in ≺+are assigned to indices less than i.\nAmong all executable tasks at the index i, we choose the one\nwith the lowest priority according to ψ. If there is no exe-\ncutable task for some index, then we return that this branch\ndoes not yield a yes-instance. The algorithm returns that it is\na yes-instance if any branch manages to assign all the tasks\ntop, and otherwise, returns that it is a no-instance.\nIf a branch assigns each task to an index in the plan, it is in-\ndeed a yes-instance as the algorithm only assigns executable\ntasks at each step. For the other direction, assume there is a\nbijective assignment a∗:T→[|T|]such that, for all t∈T,\nwe have α(t) =p[a∗(t)]and the corresponding ordering of\nthe tasks according to a∗does not conflict with ≺+. Con-\nsider the branch in which the tasks in Vare ordered according\ntoa∗. Asa∗is a valid assignment, this branch is not imme-\ndiately discarded. Further, this branch successfully assigns a\ntask to each index by the following inductive argument. We\nclaim that, for each 0≤i≤ |T|, there is a valid assignment\nthat assigns the indices 1 to iexactly like the algorithm does.\nAsa∗is a valid assignment, this holds for i= 0. Suppose the\nclaim holds for an arbitrary but fixed iand the algorithm is\nabout to assign index i+ 1. Then, by the induction hypothe-\nsis, there is a valid assignment a∗\nithat assigns all tasks up to\nithe same as the algorithm has. Let tandt∗be the tasks that\nare assigned to i+ 1by the algorithm and a∗\ni, respectively. If\nt=t∗, then the induction claim trivially holds for i+1. Thus,\nassume that t̸=t∗. Asa∗\ni(t∗) =i+ 1,t∗is executable when\nthe algorithm assigns ttoi+ 1. Since the algorithm chooses\ntinstead, we have ψ(t)≤ψ(t∗). We show that a∗\ni+1, the\nassignment obtained from a∗\niby switching only tandt∗, is a\nvalid assignment. Note that the algorithm chooses tat index\ni+ 1, and so, tis executable at this point. It remains to show\nthat there is no task t′such that a∗\ni(t∗)< a∗\ni(t′)< a∗\ni(t)\nand(t∗, t′)∈≺+. Observe that when a task from Vis exe-\ncutable, it is the only executable task with the lowest priority,\nand thus, since ψ(t)≤ψ(t∗), we get that t∗/∈V. Hence, as\nVis a vertex cover, we only have to consider the case where\nt′∈V. This implies that if ψ(t∗) =∞, then no conflicting\nedge (t∗, t′)exists, and so, we only have to consider the case\nwhere ψ(t∗)̸=∞, and thus, ψ(t)̸=∞. Since a∗\niis a valid\nassignment, either tis placed before vψ(t)ort=vψ(t)−0.5.\nNow, ψ(t)≤ψ(t∗)immediately yields that, by swapping t\nandt∗,t∗is still placed before vψ(t∗), and hence, no conflict-\ning edge (t∗, t′)exists. Thus, a∗\ni+1is a valid assignment andthe induction claim holds for i+ 1. By the inductive argu-\nment, the algorithm finds a valid assignment if there is one,\nand hence, correctly solves the decision problem.\nThe algorithm branches over the |V|!possible orderings of\ntasks in the vertex cover. For each ordering, the priorities can\nbe computed in O(| ≺ |+|T|)time. By employing a Hollow\nHeap [Hansen et al. , 2017 ]to implement a priority queue,\niterating through the sequence, extracting the minimum pri-\nority executable task, and inserting all then executable tasks\nto the queue takes O(| ≺ |+|T|log|T|)total time, yielding\na runtime of O(|V|!(| ≺ |+|T|log|T|))for the entire algo-\nrithm.\nBy Theorem 1, tackling the remaining problems requires\nadditional restrictions on the state space. We show that here,\ntoo, we can achieve tractability, albeit the arguments are even\nmore involved than in the previous case.\nTheorem 24. PLAN VALIDATION and ACTION EXE-\nCUTABILITY are fixed-parameter tractable when parameter-\nized by the vertex cover number of G≺plus the number of\nstates in the state transition graph.\nProof. Let the input instance consist of the primitive network\ntn= (T,≺+, α), an initial state s0, and either a target state\nsg(for S TATE REACHABILITY ) or an action set S(for A C-\nTION EXECUTABILITY ). For A CTION EXECUTABILITY , as-\nsume that Reduction Rule R0 cannot be applied (as otherwise\nwe can solve the instance directly). Begin by computing a\nminimum-size vertex cover VofG≺by the classical fixed-\nparameter algorithm [Downey and Fellows, 2013 ], and let K\ndenote the number of states in the state transition graph G.\nThe algorithm proceeds analogously for both problems, so\nwe provide only a single description while making explicit\nthe few areas where the techniques diverge. The algorithm\nbranches over all possibilities for the set of employed vertex\ncover tasks V′⊆V, orders v1, . . . , v |V′|of the tasks in V′,\nand states s1, . . . , s |V′|from which the tasks are executed,\nrespectively. This gives O\u0000\n2|V|· |V|!·k|V|\u0001\nbranches.\nFor each branch, we define a strong equivalence relation\non the tasks. For all t∈T\\V, the admissible interval is\ndefined as [max{i|vi≺t},min{i|t≺vi} −1].2Then,\nt1, t2∈Tare strongly equivalent if and only if α(t1)and\nα(t2)are equivalent and have the same admissible interval.\nThere are at most E= (k+1)k·(|V′|2+1) strong equivalence\nclasses. Define the augmented state transition graph G′over\nthe same set of vertices as the regular state transition graph,\nbut let it have no edges for actions. Instead, let there be an\nedge (e, s, s′)for all strong equivalence classes eand states\ns, s′such that etransforms sintos′. For each such edge, let\nβ((e, s, s′)) = ( s, s′).\nThen, split the branch for each of the possible |V′|+1sim-\nple edge paths in the augmented state transition graph G′that\nstart and end in the following vertices. A path W0from s0to\ns1, a path W1from s1\\del(v1)∪add(v1)tos2, and so on.\nFor A CTION EXECUTABILITY , also branch on all possibili-\nties for a path W|V′|starting at s|V′|\\del(v|V′|)∪add(v|V′|)\n2If the first set is empty, then the interval starts at 0. If the second\nset is empty, then it ends at |V′|.and ending at any state. For S TATE REACHABILITY , only\nconsider paths W|V′|that end in sg. If one of these paths can-\nnot be constructed as there is no directed path between the\nrespective vertices, then discard the branch. This splits each\nbranch into O\u0010\n(2E)k(|V′|+1)\u0011\nnew branches.\nFurther, let Cbe the set of distinct simple edge cycles in G,\nwhere two cycles are considered distinct if and only if they do\nnot contain the exact same vertices, i.e., cycles obtained from\nother cycles by inverting their directions are not counted as\ndifferent cycles. Then, |C| ≤(2E)k. For each 0≤i≤ |V′|,\nwe branch over all possible sets Ci⊆C, the cycles that\nare walked as a detour from path Wi. This splits each prior\nbranch into O\u0010\n(2|C|)|V′|+1\u0011\n≤ O\u0010\n2(2E)k·(|V′|+1)\u0011\nnew\nbranches. Thus, the total number of branches is\nO\u0010\n2|V|· |V|!·k|V|·(2E)k(|V′|+1)·2(2E)k·(|V′|+1)\u0011\n.\nBranches where there is a task visuch that prec( vi)⊈si\nare discarded. Further, a branch is discarded if, for some 0≤\ni≤ |V′|, there is an equivalence class in Wior in a cycle c∈\nCisuch that iis outside the admissible interval of that class.\nBranches where there is 0≤i≤ |V′|such that Wiand its\nassociated cycles do not describe a connected component are\ndiscarded as well (as can be tested by the marking procedure\ndescribed in the proof of Theorem 10).\nWe build a solution plan that walks each of the paths Wi,\n0≤i≤ |V′|, and additionally walks through the associ-\nated cycles. The exact number of times a cycle is walked is\ndetermined by an ILP. This ILP has a variable xcfor every\nc∈C. For each strong equivalence class e, letmax′(e)de-\nnote the number of tasks in T\\Vin equivalence class e. For\nallc∈C, letc(e)denote the number of occurrences of ein\nc. Further, let wi(e)denote the number of occurrences of ein\nWifor0≤i≤ |V′|. Note that, in any plan that is a solution\nto the problem instance, two strongly equivalent tasks can be\nswapped and the plan will still be valid. Thus, to ensure no\ntask in T\\Vis executed more than once, for each equivalence\nclass ewe have a constraint\nmax′(e)≥w0(e) +···+w|V′|(e)\n+c1(e)xc1+···+c|C|(e)xc|C|,\nwhere c1, . . . , c |C|are the elements of C. Further, to ensure\nthat each selected cycle is executed at least once for each path\nit is associated with, for all cycles c∈C, add a constraint\nxc≥ |{0≤i≤ |V′| |c∈Ci}|.\nOnly for A CTION EXECUTABILITY instances, for each ac-\ntion equivalence class ea, letmin′(ea)be the number of oc-\ncurrences of eainS, and let w′\ni(ea)andc′\ni(ea)be the number\noccurrences of eain path Wiand cycle ci, respectively. Then,\nadd a constraint\nmin′(ea)≥w′\n0(ea) +···+w′\n|V′|(ea)\n+c′\n1(ea)xc1+···+c′\n|C|(ea)xc|C|.\nFor all instances, the algorithm decides that the given instance\nis a yes-instance if and only if there is a solution to the ILP\nfor any non-discarded branch.If the ILP has a solution, then construct a sequence of\nequivalence classes as follows. For each 0≤i≤ |V′|, take\nany cycle c∈Cithat shares a vertex swith the current path\nWi. Delete cfromCiand augment Wiby inserting cycle cat\nthat position once. If there is no more 0≤j≤ |V|′,j̸=i,\nsuch that c∈Cj, instead add as many repetitions of ctoWi\nas the number of times that cycle cwas added to the path ex-\nactlyxctimes in total. As the cycles end in the same state they\nstarted in, the resulting action sequence is executable. Repeat\nuntilCiis empty. As CiandWiform a connected compo-\nnent, all cycles in Cican be added this way. Construct a\nsolution task sequence by concatenating the augmented paths\nW0, . . . , W |V′|and, for a step by a certain equivalence class,\nemploy any task of that class. For A CTION EXECUTABIL -\nITY, choose tasks with a not yet covered action in Swhenever\npossible. Right before each path Wi,i∈[|V′|], execute task\nvi. Observe that this way the preconditions of each task are\nfulfilled when it is about to be executed. Further, branches\nin which there are tasks that are executed before their prede-\ncessors or after their successors in ≺+are already discarded.\nThe ILP ensures that each equivalence class eappears at most\nmax′(e)times. As there are max′(e)tasks of that class and\nthey can be used interchangeably, the sequence is a valid con-\nstruction. For S TATE REACHABILITY , only paths that end in\nsgare considered. For A CTION EXECUTABILITY , the addi-\ntional set of constraints ensures that Sis covered as there are\nenough tasks of each action and tasks from the same strong\nequivalence class can be used interchangeably. Thus, the\ngiven instance is a yes-instance.\nFor the other direction, suppose there is a task sequence\nsolving the instance. There is a branch of the algorithm that\nconsiders exactly the set V′of vertex cover tasks used in the\nsequence in the order v1, . . . , v |V|and executes each task vi\nfrom state si. Split the sequence at the vertex cover tasks into\nsubwalks W∗\n0, . . . , W∗\n|V|. Then, for all 0≤i≤ |V′|, con-\nsider an initially empty set of cycles C∗\ni. Find any substring\nW∗\nithat describes a simple edge cycle in G′. Add this cycle\ntoC∗\ni(disregarding direction and starting vertex) and remove\nthe respective substring from W∗\ni. Repeat until no more such\ncycles exist. Observe that each W∗\nithen describes a simple\nedge path in G′, and all C∗\niexclusively contain simple edge\ncycles in G′. We write W′\nifor these reduced W∗\ni. There is\na branch of the algorithm such that Wi=W′\niandCi=C∗\ni\nfor all 0≤i≤ |V′|. This branch is not discarded because\nthe task sequence is valid, so prec( vi)⊆sifor all i∈[|V′|],\nand the paths Wiand cycles in Ciare such that they do not\ndepend on any vertex cover task after vi, and no vertex cover\ntask up to videpends on them. Further, as the cycles in C∗\ni\nwere iteratively removed from the path W∗\ni, the paths and cy-\ncles in WiandCiform a connected component in G′. For\neach cycle c∈C, letx∗\ncbe the total number of times chas\nbeen removed from paths W∗\ni. Then, the ILP has a solution\nifxc=x∗\ncfor all c∈Cas follows. As the x∗\ncare com-\nputed from a valid sequence, summing over all occurrences\nof an equivalence class ein paths Wiand cycles in Cidoes\nnot exceed max′(e). Further, by construction, x∗\ncis at least\nthe number of sets Ciwithc∈Ci. In the case of a valid A C-\nTION EXECUTABILITY solution, the number of occurrences\nof each action equivalence class eais at least min′(ea), andthus, the additional constraints are fulfilled. Hence, the ILP\nhas a solution and the algorithm correctly decides for a yes-\ninstance.\nEmploying reduction rule R0 and building the aug-\nmented state transition graph takes O(|A|k+|T|)\ntime. To decide whether a branch is discarded\ntakes O\u0000\n| ≺ |+|T|+|V||C|2k\u0001\ntime. Com-\nputing c(e),wi(e),max′(e),c′(ea),w′\ni(ea),\nmin′(ea), and |{0≤i≤ |V′| |c∈Ci}| takes\nO(|T|+ (|C|+|V|+ 1)k+|C| ·(|V|+ 1)) time. The\nILP has |C|variables and at most E+|C|+ (k+ 1)k=\nO(E+|C|)constraints, in which the absolute values of\nall coefficients are bounded by max{k,|T|}. As noted\nbefore, an ILP instance IILPwithnvariables can be solved\ninO\u0000\nnO(n)· |I ILP|\u0001\ntime. Thus, the ILP is solved in\nO\u0010\n|C|O(|C|)· |C| ·(E+|C|)·log(max {(k,|T|)})\u0011\ntime. As |C| ≤(2E)k, the total runtime is at most f(k,|V|)·\n|T|for some computable function f. Multiplying the number\nof branches with the per-branch runtime yields a quantity that\ncan be upper bounded by 2kO(k2)|V|O(k)·(|T|2).\nA Meta-Theorem for Fixed-Parameter Tractability. As\nour final contribution, we show that the algorithmic meta-\ntheorem for polynomial-time solvability can be lifted to the\nsetting of fixed-parameter tractability if we additionally re-\nstrict the maximum “breadth” of a compound task, i.e., Cc.\nTheorem 25. LetPRbe a decomposable HTN planning\nproblem and κa stable measure. Assume PRis fixed-\nparameter tractable on primitive task networks when param-\neterized by κ. Then, PRis fixed-parameter tractable when\nparameterized by κ+Cd+C#+Cs+Cc.\nProof. LetIhave task network tn= (T,≺+, α). If there\nare no compound tasks in T, i.e., Cd= 0, both statements\nimmediately hold by applying an algorithm for primitive net-\nworks. Thus, assume Cd≥1. Then, PR in tnis solved\nby listing all primitive instances tncan be decomposed into\nand solving PR in each of these. By Lemma 15, there are\nat most CcPCd−1\ni=0C#·Csisuch instances Ipand, for each of\nthese, |Ip| ≤ |I| +f′′(Cd, C#, Cs)for some computable\nfunction f′′. Further, for κ′, the parameter κonIp, we have\nκ′≤f′′′(κ, Cd, C#, Cs), for some computable function f′′′\nasκis stable. Thus, if primitive instances Ipcan be solved in\ntimef(κ)· |Ip|O(1), then the total runtime is\nO\u0010\nCcg(Cd,C#,Cs)·f(f′′′(κ, Cd, C#, Cs))·\n(|I|+f′′(Cd, C#, Cs))O(1)\u0011\n=O\u0010\nCcg(Cd,C#,Cs)·f′(κ, Cd, C#, Cs)· |I|O(1)\u0011\n,\nwhere g(Cd, C#, Cs) =PCd−1\ni=0C#·Csiand\nf′(κ, Cd, C#, Cs) = f(f′′′(κ, Cd, C#, Cs))·\nf′′(Cd, C#, Cs)O(1).7 Concluding Remarks\nThis article provides a comprehensive understanding of\nthe complexity-theoretic boundaries for several fundamental\nproblems on hierarchical task networks. Our results include\nstronger algorithmic lower bounds as well as complementary\npositive results—not only for specific problems of interest,\nbut also in the form of meta-theorems that can be used for\nany other hierarchical task network planning problem.\nIn terms of open questions, we note that while we have pro-\nvided lower bounds justifying all the restrictions applied in\nthe algorithmic meta-theorem for polynomial-time solvability\n(Theorem 16), in the more refined fixed-parameter tractability\nsetting, we leave it open whether the “hierarchical breadth”\nCcneeds to be part of the parameterization in order to estab-\nlish Theorem 25. While we suspect this to be necessary for at\nleast some of the considered problems, we conclude by show-\ning that it may be omitted in the case of P LAN VALIDATION .\nTheorem 26. Letκbe a stable measure of HTNs such that\nPLAN VALIDATION is fixed-parameter tractable on primi-\ntive networks when parameterized by κ. Then PRis fixed-\nparameter tractable parameterized by κ+Cd+C#+Cs.\nProof. The algorithm employs the same approach as the one\nused for Theorem 16, but first employs a reduction rule that\nbounds the number of choices of compound tasks by a func-\ntion of Cd, C#, Cs.\nThe input plan pgives exactly the multiset of actions that\nare executed by any solution task sequence. Out of this mul-\ntiset, exactly |T| −C#actions are covered by primitive tasks\nin the initial network. Let Sdenote the multiset of actions\nthat are not covered by primitive tasks. By Lemma 15, any\nprimitive network into which the initial network can be de-\ncomposed has at most C#·CsCdadditional tasks. Thus, if\n|S|=|p| −(|T| −C#)> C #·CsCd, the instance is a no-\ninstance. Otherwise, the C#compound tasks in the initial\nnetwork have to be decomposed to exactly cover S. Thus,\nwe can disregard all methods (c, tn m)∈Mwhere the prim-\nitive tasks in tnmare not a subset of S. Out of the remain-\ning methods (c, tn m)∈Mwith non-primitive task network\ntnm, we only require those for which tnmcan be decom-\nposed into one of the remaining primitive networks, each of\nwhich has depth at most Cd−1. There are at most 2|S|pos-\nsibilities for which actions the primitive networks capture, so\nby enumerating all possible partial orders, there are less than\n2|S|·2|S|2= 2|S|2+|S|pairwise non-isomorphic primitive net-\nworks left. As we assumed that |S| ≤C#·CsCd, this upper\nbounds the number of pairwise non-isomorphic primitive net-\nworks we have to consider by a function of Cd, C#, Cs. Fur-\nther, the number of methods with pairwise non-isomorphic\ntask networks of depth at most Cd−1whose primitive tasks\nare subsets of Sand that can be decomposed into one of these\nat most 2|S|2+|S|primitive networks is upper bounded by a\nfunction of Cd, C#, Csas well. This bounds the number of\nrelevant subnetworks in M. As there are at most C#·CsCd\ncompound tasks that are either in the initial network or any\nrelevant subnetwork, the number of relevant compound task\nnames is bounded by a function of Cd, C#, Csas well. Bydiscarding all methods (c, tn m)for which not both candtnm\nare relevant, we have Cc≤ |M| ≤f(Cd, C#, Cs)for some\ncomputable function f. Then, employing Theorem 25 gives\nthe desired result.\nReferences\n[Alford et al. , 2015 ]Ron Alford, Pascal Bercher, and\nDavid W. Aha. Tight bounds for HTN planning with task\ninsertion. In Proc. of the 24th International Joint Confer-\nence on Artificial Intelligence , pages 1502–1508, 2015.\n[B¨ackstr ¨omet al. , 2012 ]Christer B ¨ackstr ¨om, Yue Chen, Pe-\nter Jonsson, Sebastian Ordyniak, and Stefan Szeider. The\ncomplexity of planning revisited - A parameterized anal-\nysis. In Proc. of the 26th AAAI Conference on Artificial\nIntelligence , pages 1735–1741, 2012.\n[B¨ackstr ¨omet al. , 2013 ]Christer B ¨ackstr ¨om, Peter Jonsson,\nSebastian Ordyniak, and Stefan Szeider. Parameterized\ncomplexity and kernel bounds for hard planning prob-\nlems. In Proc. of the 8th International Conference on Al-\ngorithms and Complexity , volume 7878 of LNCS , pages\n13–24. Springer, 2013.\n[Balko et al. , 2022 ]Martin Balko, Steven Chaplick, Robert\nGanian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr,\nand Alexander Wolff. Bounding and computing obstacle\nnumbers of graphs. In Proc. of the 30th Annual European\nSymposium on Algorithms , volume 244 of LIPIcs , pages\n11:1–11:13. Schloss Dagstuhl - Leibniz-Zentrum f ¨ur In-\nformatik, 2022.\n[Bandyapadhyay et al. , 2023 ]Sayan Bandyapadhyay, Fe-\ndor V . Fomin, Petr A. Golovach, William Lochet, Nidhi\nPurohit, and Kirill Simonov. How to find a good explana-\ntion for clustering? Artif. Intel. , 322:103948, 2023.\n[Bart´aket al. , 2021 ]Roman Bart ´ak, Simona Ondr ˇckov ´a,\nGregor Behnke, and Pascal Bercher. On the verification\nof totally-ordered HTN plans. In Proc. of the 33rd IEEE\nInternational Conference on Tools with Artificial Intelli-\ngence , pages 263–267, 2021.\n[Behnke et al. , 2015 ]Gregor Behnke, Daniel H ¨oller, and Su-\nsanne Biundo. On the complexity of HTN plan verifica-\ntion and its implications for plan recognition. In Proc. of\nthe 25th International Conference on Automated Planning\nand Scheduling , volume 25, pages 25–33, 2015.\n[Behnke et al. , 2019 ]Gregor Behnke, Daniel H ¨oller, and Su-\nsanne Biundo. Finding optimal solutions in HTN planning\n- A SAT-based approach. In Proc. of the 28th International\nJoint Conference on Artificial Intelligence , pages 5500–\n5508, 2019.\n[Bentert et al. , 2023 ]Matthias Bentert, Robert Bredereck,\nP´eter Gy ¨orgyi, Andrzej Kaczmarczyk, and Rolf Nieder-\nmeier. A multivariate complexity analysis of the material\nconsumption scheduling problem. J. Sched. , 26:369–382,\n2023.\n[Bercher et al. , 2019 ]Pascal Bercher, Ron Alford, and\nDaniel H ¨oller. A survey on hierarchical planning – one\nabstract idea, many concrete realizations. In Proc. of the28th International Joint Conference on Artificial Intelli-\ngence , pages 6267–6275, 2019.\n[Bhore et al. , 2020 ]Sujoy Bhore, Robert Ganian, Fabrizio\nMontecchiani, and Martin N ¨ollenburg. Parameterized al-\ngorithms for book embedding problems. J. Graph Algo-\nrithms Appl. , 24(4):603–620, 2020.\n[Bihlmaier et al. , 2014 ]Andreas Bihlmaier, Luzie Schreiter,\nJ¨org Raczkowsky, and Heinz W ¨orn. Hierarchical task net-\nworks as domain-specific language for planning surgical\ninterventions. In Proc. of the 13th International Con-\nference on Intelligent Autonomous Systems , volume 302\nofAdvances in Intelligent Systems and Computing , pages\n1095–1105. Springer, 2014.\n[Blazej et al. , 2023 ]V´aclav Blazej, Robert Ganian, Dusan\nKnop, Jan Pokorn ´y, Simon Schierreich, and Kirill Si-\nmonov. The parameterized complexity of network mi-\ncroaggregation. In Proc. of the 37th AAAI Conference on\nArtificial Intelligence , pages 6262–6270, 2023.\n[Bliem et al. , 2016 ]Bernhard Bliem, Robert Bredereck, and\nRolf Niedermeier. Complexity of efficient and envy-free\nresource allocation: Few agents, resources, or utility lev-\nels. In Proc. of the 25th International Joint Conference on\nArtificial Intelligence , pages 102–108, 2016.\n[Bylander, 1994 ]Tom Bylander. The computational com-\nplexity of propositional STRIPS planning. Artif. Intel. ,\n69(1-2):165–204, 1994.\n[B¨ackstr ¨om and Nebel, 1995 ]Christer B ¨ackstr ¨om and Bern-\nhard Nebel. Complexity results for SAS+planning. Com-\nputational Intelligence , 11(4):625–655, 1995.\n[B¨ackstr ¨omet al. , 2015 ]Christer B ¨ackstr ¨om, Peter Jonsson,\nSebastian Ordyniak, and Stefan Szeider. A complete pa-\nrameterized complexity analysis of bounded planning. J.\nComput. Syst. Sci. , 81(7):1311–1332, 2015.\n[Chapman, 1987 ]David Chapman. Planning for conjunctive\ngoals. Artif. Intel. , 32(3):333–377, 1987.\n[Chen and Bercher, 2021 ]Dillon Chen and Pascal Bercher.\nFully observable nondeterministic HTN planning – For-\nmalisation and complexity results. In Proc. of the 31st\nInternational Conference on Automated Planning and\nScheduling , volume 31, pages 74–84, 2021.\n[Cygan et al. , 2015 ]Marek Cygan, Fedor V . Fomin, Łukasz\nKowalik, Daniel Lokshtanov, D ´aniel Marx, Marcin\nPilipczuk, Michał Pilipczuk, and Saket Saurabh. Parame-\nterized Algorithms . Springer, Cham, 2015.\n[Darmann and D ¨ocker, 2021 ]Andreas Darmann and\nJanosch D ¨ocker. On simplified NP-complete variants of\nmonotone 3-SAT. Discr. Appl. Math. , 292:45–58, 2021.\n[Deligkas et al. , 2022 ]Argyrios Deligkas, Eduard Eiben,\nRobert Ganian, Thekla Hamm, and Sebastian Ordyniak.\nThe complexity of envy-free graph cutting. In Proc. of\nthe 31st International Joint Conference on Artificial Intel-\nligence , pages 237–243, 2022.\n[Diestel, 2012 ]Reinhard Diestel. Graph Theory, 4th Edition ,\nvolume 173 of Graduate texts in math. Springer, 2012.[Dilworth, 1950 ]Robert P. Dilworth. A decomposition theo-\nrem for partially ordered sets. Annals of Math. , 51(1):161–\n166, 1950.\n[Downey and Fellows, 2013 ]Rodney G. Downey and\nMichael R. Fellows. Fundamentals of Parameterized\nComplexity . Texts in Computer Science. Springer, 2013.\n[Eiben et al. , 2021 ]Eduard Eiben, Robert Ganian, Iyad\nKanj, Sebastian Ordyniak, and Stefan Szeider. The pa-\nrameterized complexity of clustering incomplete data. In\nProc. of the 35th AAAI Conference on Artificial Intelli-\ngence , pages 7296–7304, 2021.\n[Eiben et al. , 2023 ]Eduard Eiben, Robert Ganian, Thekla\nHamm, and Sebastian Ordyniak. Parameterized complex-\nity of envy-free resource allocation in social networks. Ar-\ntif. Intel. , 315:103826, 2023.\n[Erol et al. , 1994 ]Kutluhan Erol, James A. Hendler, and\nDana S. Nau. HTN planning: Complexity and expressiv-\nity. In Proc. of the 12th National Conference on Artificial\nIntelligence , volume 2, pages 1123–1128, 1994.\n[Erol et al. , 1996 ]Kutluhan Erol, James Hendler, and Dana\nNau. Complexity results for HTN planning. Annals of\nMathematics and Artificial Intelligence , 18:69–93, 1996.\n[Fellows and McCartin, 2003 ]Michael R. Fellows and\nCatherine McCartin. On the parametric complexity of\nschedules to minimize tardy tasks. Theor. Comput. Sci. ,\n298(2):317–324, 2003.\n[Fellows et al. , 2009 ]Michael R. Fellows, Danny Hermelin,\nFrances Rosamond, and St ´ephane Vialette. On the param-\neterized complexity of multiple-interval graph problems.\nTheor. Comput. Sci. , 410(1):53–61, 2009.\n[Fikes and Nilsson, 1971 ]Richard E. Fikes and Nils J. Nils-\nson. Strips: A new approach to the application of theorem\nproving to problem solving. Artif. Intel. , 2(3):189–208,\n1971.\n[Frank and Tardos, 1987 ]Andr ´as Frank and ´Eva Tardos. An\napplication of simultaneous diophantine approximation in\ncombinatorial optimization. Combinatorica , 7(1):49–65,\n1987.\n[Froese et al. , 2022 ]Vincent Froese, Leon Kellerhals, and\nRolf Niedermeier. Modification-fair cluster editing. In\nProc. of the 36th AAAI Conference on Artificial Intelli-\ngence , pages 6631–6638, 2022.\n[Ganian and Korchemna, 2021 ]Robert Ganian and Viktoriia\nKorchemna. The complexity of Bayesian network learn-\ning: Revisiting the superstructure. In Advances in Neural\nInformation Processing Systems 34 (NeurIPS 2021) , pages\n430–442, 2021.\n[Ganian et al. , 2016 ]Robert Ganian, Petr Hlinen ´y, Joachim\nKneis, Daniel Meister, Jan Obdrz ´alek, Peter Rossmanith,\nand Somnath Sikdar. Are there any good digraph width\nmeasures? J. Comb. Theory, Ser. B , 116:250–286, 2016.\n[Ganian et al. , 2020a ]Robert Ganian, Thekla Hamm, and\nGuillaume Mescoff. The complexity landscape of\nresource-constrained scheduling. In Proc. of the 29thInternational Joint Conference on Artificial Intelligence ,\npages 1741–1747, 2020.\n[Ganian et al. , 2020b ]Robert Ganian, Iyad Kanj, Sebastian\nOrdyniak, and Stefan Szeider. On the parameterized com-\nplexity of clustering incomplete data into subspaces of\nsmall rank. In Proc. of the 34th AAAI Conference on Arti-\nficial Intelligence , pages 3906–3913, 2020.\n[Ganian et al. , 2021 ]Robert Ganian, Thekla Hamm, and Se-\nbastian Ordyniak. The complexity of object association in\nmultiple object tracking. In Proc. of the 35th AAAI Con-\nference on Artificial Intelligence , pages 1388–1396, 2021.\n[Ganian et al. , 2022 ]Robert Ganian, Thekla Hamm, Vik-\ntoriia Korchemna, Karolina Okrasa, and Kirill Simonov.\nThe complexity of k-means clustering when little is\nknown. In Proc. of the 39th International Conference on\nMachine Learning , volume 162 of PMLR , pages 6960–\n6987, 2022.\n[Garc ´ıa-Remesal, 2008 ]Miguel Garc ´ıa-Remesal. Using hi-\nerarchical task network planning techniques to create\ncustom web search services over multiple biomedical\ndatabases. In Proc. of the 12th International Confer-\nence on Knowledge-Based Intelligent Information and En-\ngineering Systems , volume 5178 of LNCS , pages 42–49.\nSpringer, 2008.\n[Garey and Johnson, 1979 ]Michael R. Garey and David S.\nJohnson. Computers and intractability: A guide to the\ntheory of NP-completeness . W. H. Freeman, N.Y ., 1979.\n[Geier and Bercher, 2011 ]Thomas Geier and Pascal\nBercher. On the decidability of HTN planning with\ntask insertion. In Proc. of the 22nd International Joint\nConference on Artificial Intelligence , pages 1955–1961,\n2011.\n[Georgievski and Aiello, 2015 ]Ilche Georgievski and\nMarco Aiello. HTN planning: Overview, comparison, and\nbeyond. Artif. Intel. , 222:124–156, 2015.\n[Gonz ´alez-Ferrer et al. , 2011 ]Arturo Gonz ´alez-Ferrer, An-\nnette ten Teije, Juan Fern ´andez-Olivares, and Krystyna\nMilian. Careflow planning: From time-annotated clini-\ncal guidelines to temporal hierarchical task networks. In\nProc. of the 13th Conference on Artificial Intelligence in\nMedicine in Europe , volume 6747 of LNCS , pages 265–\n275. Springer, 2011.\n[Gonz ´alez-Ferrer et al. , 2013a ]Arturo Gonz ´alez-Ferrer,\nJuan Fern ´andez-Olivares, and Luis A. Castillo. From\nbusiness process models to hierarchical task network\nplanning domains. Knowl. Eng. Rev. , 28(2):175–193,\n2013.\n[Gonz ´alez-Ferrer et al. , 2013b ]Arturo Gonz ´alez-Ferrer, An-\nnette ten Teije, Juan Fern ´andez-Olivares, and Krystyna\nMilian. Automated generation of patient-tailored elec-\ntronic care pathways by translating computer-interpretable\nguidelines into hierarchical task networks. Artif. Intel.\nMedicine , 57(2):91���109, 2013.\n[Gr¨uttemeier and Komusiewicz, 2022 ]Niels Gr ¨uttemeier\nand Christian Komusiewicz. Learning Bayesian networksunder sparsity constraints: A parameterized complexity\nanalysis. J. Artif. Intel. Res. , 74:1225–1267, 2022.\n[Gr¨uttemeier et al. , 2021a ]Niels Gr ¨uttemeier, Christian Ko-\nmusiewicz, and Nils Morawietz. Efficient Bayesian net-\nwork structure learning via parameterized local search on\ntopological orderings. In Proc. of the 35th AAAI Confer-\nence on Artificial Intelligence , pages 12328–12335, 2021.\n[Gr¨uttemeier et al. , 2021b ]Niels Gr ¨uttemeier, Christian Ko-\nmusiewicz, and Nils Morawietz. On the parameterized\ncomplexity of polytree learning. In Proc. of the 30th Inter-\nnational Joint Conference on Artificial Intelligence , pages\n4221–4227, 2021.\n[Hansen et al. , 2017 ]Thomas Dueholm Hansen, Haim Ka-\nplan, Robert E. Tarjan, and Uri Zwick. Hollow\nheaps. ACM Transactions on Algorithms , 13(3):42:1–\n42:27, 2017.\n[Hayes and Scassellati, 2016 ]Bradley Hayes and Brian\nScassellati. Autonomously constructing hierarchical task\nnetworks for planning and human-robot collaboration. In\nProc. of the IEEE International Conference on Robotics\nand Automation , pages 5469–5476, 2016.\n[Heeger et al. , 2023 ]Klaus Heeger, Danny Hermelin,\nGeorge B. Mertzios, Hendrik Molter, Rolf Niedermeier,\nand Dvir Shabtay. Equitable scheduling on a single\nmachine. J. Sched. , 26:209–225, 2023.\n[Hogg et al. , 2009 ]Chad Hogg, Ugur Kuter, and H ´ector\nMu˜noz-Avila. Learning hierarchical task networks for\nnondeterministic planning domains. In Proc. of the 21st\nInternational Joint Conference on Artificial Intelligence ,\npages 1708–1714, 2009.\n[H¨oller et al. , 2019 ]Daniel H ¨oller, Pascal Bercher, Gregor\nBehnke, and Susanne Biundo. On guiding search in HTN\nplanning with classical planning heuristics. In Proc. of the\n28th International Joint Conference on Artificial Intelli-\ngence , pages 6171–6175, 2019.\n[H¨oller and Bercher, 2021 ]Daniel H ¨oller and Pascal\nBercher. Landmark generation in HTN planning. In Proc.\nof the 35th AAAI Conference on Artificial Intelligence ,\nvolume 35, pages 11826–11834, 2021.\n[H¨oller et al. , 2022 ]Daniel H ¨oller, Julia Wichlacz, Pascal\nBercher, and Gregor Behnke. Compiling HTN plan ver-\nification problems into HTN planning problems. In Proc.\nof the 32nd International Conference on Automated Plan-\nning and Scheduling , pages 145–150, 2022.\n[Kannan, 1987 ]Ravi Kannan. Minkowski’s convex body\ntheorem and integer programming. Math. of Operations\nResearch , 12(3):415––440, 1987.\n[Kellerhals et al. , 2023 ]Leon Kellerhals, Tomohiro Koana,\nPascal Kunz, and Rolf Niedermeier. Parameterized algo-\nrithms for colored clustering. In Proc. of the 37th AAAI\nConference on Artificial Intelligence , pages 4400–4408,\n2023.\n[Koet al. , 2009 ]Ryan K. L. Ko, Stephen Siang Guan Lee,\nEng Wah Lee, and Andre Jusuf. Dynamic collaborative\nbusiness process formulation via ontologised hierarchicaltask network (HTN) planning. In Proc. of the IEEE In-\nternational Conference on Web Services , pages 959–967,\n2009.\n[Koet al. , 2012 ]Ryan K. L. Ko, Eng Wah Lee, and Stephen\nSiang Guan Lee. Business-owl (BOWL) - A hierarchical\ntask network ontology for dynamic business process de-\ncomposition and formulation. IEEE Trans. Serv. Comput. ,\n5(2):246–259, 2012.\n[Kronegger et al. , 2013 ]Martin Kronegger, Andreas Pfan-\ndler, and Reinhard Pichler. Parameterized complexity of\noptimal planning: A detailed map. In Proc. of the 23rd\nInternational Joint Conference on Artificial Intelligence ,\npages 954–961, 2013.\n[Kuter et al. , 2005 ]Ugur Kuter, Dana S. Nau, Marco Pis-\ntore, and Paolo Traverso. A hierarchical task-network\nplanner based on symbolic model checking. In Proc. of\nthe 15th International Conference on Automated Planning\nand Scheduling , pages 300–309, 2005.\n[Lenstra, 1983 ]H. W. Lenstra. Integer programming with a\nfixed number of variables. Math. of Operations Research ,\n8(4):538—-548, 1983.\n[Liet al. , 2009 ]Nan Li, Subbarao Kambhampati, and\nSung Wook Yoon. Learning probabilistic hierarchical task\nnetworks to capture user preferences. In Proc. of the 21st\nInternational Joint Conference on Artificial Intelligence ,\npages 1754–1759, 2009.\n[Lin and Bercher, 2023 ]Songtuan Lin and Pascal Bercher.\nWas fixing this really that hard? On the complexity of cor-\nrecting HTN domains. In Proc. of the 37th AAAI Confer-\nence on Artificial Intelligence , volume 37, pages 12032–\n12040, 2023.\n[Liuet al. , 2016 ]Dian Liu, Hongwei Wang, Chao Qi, Peng\nZhao, and Jian Wang. Hierarchical task network-based\nemergency task planning with incomplete information,\nconcurrency and uncertain duration. Knowl. Based Syst. ,\n112:67–79, 2016.\n[Namiki et al. , 2021 ]Kousuke Namiki, Tomohiro Mori,\nYouichiro Miyake, Shinpei Sakata, and Gustavo Martins.\nAdvanced real-time hierarchical task network: Long-term\nbehavior in real-time games. In Proc. of the 17th AAAI\nConference on Artificial Intelligence and Interactive Digi-\ntal Entertainment , volume 17, pages 208–212, 2021.\n[Nesetril and de Mendez, 2012 ]Jaroslav Nesetril and\nPatrice Ossona de Mendez. Sparsity: Graphs, Struc-\ntures, and Algorithms , volume 28 of Algorithms and\ncombinatorics . Springer, 2012.\n[Olzet al. , 2021 ]Conny Olz, Susanne Biundo, and Pascal\nBercher. Revealing hidden preconditions and effects of\ncompound HTN planning tasks – A complexity analysis.\nInProc. of the 35th AAAI Conference on Artificial Intelli-\ngence , volume 35, pages 11903–11912, 2021.\n[Onta ˜n´on and Buro, 2015 ]Santiago Onta ˜n´on and Michael\nBuro. Adversarial hierarchical-task network planning for\ncomplex real-time games. In Proc. of the 24th Interna-\ntional Joint Conference on Artificial Intelligence , pages\n1652–1658, 2015.[Ordyniak and Szeider, 2013 ]Sebastian Ordyniak and Ste-\nfan Szeider. Parameterized complexity results for exact\nBayesian network structure learning. J. Artif. Intel. Res. ,\n46:263–302, 2013.\n[Padia et al. , 2019 ]Kalpesh Padia, Kaveen Herath Bandara,\nand Christopher G. Healey. A system for generating sto-\nryline visualizations using hierarchical task network plan-\nning. Comput. Graph. , 78:64–75, 2019.\n[Parunak et al. , 2009 ]H. Van Dyke Parunak, Theodore C.\nBelding, Robert Bisson, Sven A. Brueckner, Elizabeth\nDowns, and Rainer Hilscher. Stigmergic reasoning over\nhierarchical task networks. In Proc. of the 8th Interna-\ntional Joint Conference on Autonomous Agents and Multi-\nagent Systems , volume 2, pages 1195–1196, 2009.\n[Robertson and Seymour, 1986 ]Neil Robertson and Paul D.\nSeymour. Graph minors. II. Algorithmic aspects of tree-\nwidth. J. Algorithms , 7(3):309–322, 1986.\n[Soemers and Winands, 2016 ]Dennis J. N. J. Soemers and\nMark H. M. Winands. Hierarchical task network plan\nreuse for video games. In Proc. of the IEEE Conference on\nComputational Intelligence and Games , pages 1–8, 2016.\n[Sohrabi et al. , 2009 ]Shirin Sohrabi, Jorge A. Baier, and\nSheila A. McIlraith. HTN planning with preferences. In\nProc. of the 21st International Joint Conference on Artifi-\ncial Intelligence , pages 1790–1797, 2009.\n[Tewari and Persiani, 2021 ]Maitreyee Tewari and Michele\nPersiani. Towards we-intentional human-robot interac-\ntion using theory of mind and hierarchical task net-\nwork. In Proc. of the 5th International Conference on\nComputer-Human Interaction Research and Applications ,\npages 291–299. SCITEPRESS, 2021.\n[Tsuneto et al. , 1996 ]Reiko Tsuneto, Kutluhan Erol,\nJames A. Hendler, and Dana S. Nau. Commitment\nstrategies in hierarchical task network planning. In Proc.\nof the 13th National Conference on Artificial Intelligence ,\nvolume 1, pages 536–542, 1996.\n[Uhlmann and Weller, 2013 ]Johannes Uhlmann and Math-\nias Weller. Two-layer planarization parameterized by feed-\nback edge set. Theor. Comput. Sci. , 494:99–111, 2013.\n[van Bevern et al. , 2016 ]Ren´e van Bevern, Robert Bred-\nereck, Laurent Bulteau, Christian Komusiewicz, Nim-\nrod Talmon, and Gerhard J. Woeginger. Precedence-\nconstrained scheduling problems parameterized by partial\norder width. In Discrete Optimization and Operations Re-\nsearch , volume 9869 of LNCS , pages 105–120, 2016.\n[Warmuth and Haussler, 1984 ]Manfred K. Warmuth and\nDavid Haussler. On the complexity of iterated shuffle. J.\nComput. Syst. Sci. , 28(3):345–358, 1984.\n[Xiao et al. , 2017 ]Zhanhao Xiao, Andreas Herzig, Laurent\nPerrussel, Hai Wan, and Xiaoheng Su. Hierarchical task\nnetwork planning with task insertion and state constraints.\nInProc. of the 26th International Joint Conference on Ar-\ntificial Intelligence , pages 4463–4469, 2017.\n[Zhao et al. , 2017 ]Peng Zhao, Chao Qi, and Dian Liu.\nResource-constrained hierarchical task network planningunder uncontrollable durations for emergency decision-\nmaking. J. Intel. Fuzzy Syst. , 33(6):3819–3834, 2017." }, { "title": "2401.14253v3.Vapor_compression_and_energy_dissipation_in_a_collapsing_laser_induced_bubble.pdf", "content": "Vapor compression and energy dissipation in a collapsing laser-induced bubble AIP/123-QED\nVapor compression and energy dissipation in a collapsing laser-induced bubble\nD. B. Preso,1D. Fuster,2A. B. Sieber,1D. Obreschkow,3and M. Farhat1\n1)Institute of Mechanical Engineering, École Polytechnique Fédérale de Lausanne,\nAvenue de Cour 33 Bis, 1007 Lausanne, Switzerland\n2)Sorbonne Université, Centre National de la Recherche Scientifique,\nUMR 7190, Institut Jean Le Rond ∂’Alembert, F-75005 Paris,\nFrance\n3)International Centre for Radio Astronomy Research (ICRAR),\nUniversity of Western Australia, Crawley, WA 6009, Australia\n(*Electronic mail: davide.preso@epfl.ch)\n(Dated: 25 March 2024)\nThe composition of the gaseous phase of cavitation bubbles and its role on the collapse re-\nmains to date poorly understood. In this work, experiments of single cavitation bubbles in\naqueous ammonia serve as a novel approach to investigate the effect of the vapor contained\nin a bubble on its collapse. We find that the higher vapor pressure of more concentrated\naqueous ammonia acts as a resistance to the collapse, reducing the total energy dissipation.\nIn line with visual observation, acoustic measurements, and luminescence recordings, it is\nalso observed that higher vapor pressures contribute to a more spherical collapse, likely\nhindering the growth of interface instabilities by decreasing the collapse velocities and ac-\ncelerations. Remarkably, we evidence a strong difference between the effective damping\nand the energy of the shock emission, suggesting that the latter is not the dominant dis-\nsipation mechanism at collapse as predicted from classical correction models accounting\nfor slightly compressible liquids. Furthermore, our results suggest that the vapor inside\ncollapsing bubbles gets compressed, consistently with previous studies performed in the\ncontext of single bubble sonoluminescence, addressing the question about the ability of\nvapors to readily condense during a bubble collapse in similar regimes. These findings\nprovide insights into the identification of the influence of the bubble content and the en-\nergy exchanges of the bubble with its surrounding media, eventually paving the way to a\nmore efficient use of cavitation in engineering and biomedical applications.\n1arXiv:2401.14253v3 [physics.flu-dyn] 22 Mar 2024Vapor compression and energy dissipation in a collapsing laser-induced bubble\nI. INTRODUCTION\nThe collapse of cavitation bubbles often leads to shock waves emission, light radiation, and\nrebound bubbles. The occurrence of these phenomena suggests the presence of a gaseous phase\nwithin the bubble, which is highly compressed during the collapse. However, its nature and influ-\nence on the bubble dynamics is to date still a subject of debate. Furthermore, although it is widely\naccepted that the non-condensable gas within the bubble undergo adiabatic compression, the role\nof condensable vapors remains vague as equilibrium conditions are at best only satisfied at the\nbubble interface but not in the bubble interior.1–3This problem was already introduced in the last\ncentury by Plesset4, who speculated about the inability of vapor to change phase at the same rate\nof the bubble shrinkage. Successively, several sophisticated numerical models have been proposed\nto capture the influence of phase change on the bubble dynamics. Fujikawa and Akamatsu5, and\nlater Akhatov et al.6, developed numerical models which include liquid compressibility, and heat\nand mass transfer. They investigated the incidence of non-equilibrium processes at the bubble wall\ndue to thermal inertia of condensing vapor, concluding that this could lead to the occurrence of\nsupercritical conditions at the final stage of the collapse. They therefore highlighted the possibility\nof the vapor to behave as a non-condensable gas in the case where the volume reduction rate of\nthe bubble was much higher than the condensation rate. In addition, Akhatov et al.6introduced a\nsticking coefficient of water vapor, which played as a tuning parameter to fit experimental data and\npredicted the condensed vapor at the bubble-liquid interface. Their work was further endorsed by\nSzeri et al.7, who studied the heat and mass transfer during cavitation bubbles collapse, concluding\nthat the latter occurs so fast that thermal diffusion and phase change effects are nearly obviated,\nas the vapor condensation rate is much slower than the bubble volume reduction rate. More re-\ncently, Magaletti, Marino, and Casciola8conducted a similar numerical investigation considering\nphase change, occurrence of supercritical conditions, thermal conduction, and liquid compress-\nibility effects, reporting the disappearance and reappearance of liquid-vapor interface during the\nfinal stage of the collapse because of a transition to super-critical conditions of the vapor. They\nalso concluded that, in agreement with Fujikawa and Akamatsu5, purely vapor bubble may be\nable to emit shock waves at collapse. Lately, Liang et al.9studied the transition from nonlinear to\nlinear oscillations of collapsing cavitation bubbles. They developed a novel approach based on the\nGilmore10model, with which they could fit the progressive condensation of water vapor during\nnonlinear bubble oscillations from experimental data by means of a tuning parameter, obtaining\n2Vapor compression and energy dissipation in a collapsing laser-induced bubble\nin turn the partial pressure of condensable vapor and non-condensable gas within the bubble. The\nresults were in good agreement with the ones of Akhatov et al.6. Following this approach, Wen\net al.11were able to track the bubble dynamics of millimeter-sized spherical cavitation bubbles\nup to the fourth oscillation. The relevance of phase change on the bubble motion has been also\ndiscussed theoretically. Already for linear oscillations it is possible to distinguish regimes where\nthe vapor is trapped inside the bubble while keeping equilibrium conditions at the interface12,13.\nFor strongly non-linear oscillations, Fuster, Hauke, and Dopazo3have shown that this asymptotic\nlimit is reached for large values of the accommodation coefficient, where the net flux across the\ninterface is eventually dominated by diffusion effects7. Thus, the relevance of non-equilibrium\nconditions at the interface and the consequences of it on the bubble motion remain an open prob-\nlem that needs of careful experimental investigations.\nIrrespective from the fact that equilibrium conditions are sustained at the interface or not, the\ninfluence of phase change on the dynamics of bubbles has been experimentally confirmed in sev-\neral works. In relation to single bubble sonoluminescence, Vazquez and Putterman14observed an\nincreased collapse cushioning and decreased light emission at increasing water temperature. The\nlatter observation was further confirmed by Toegel et al.15and Hopkins et al.16, who highlighted\nthe importance of the partial pressure of vapor trapped within the bubble during the collapse on the\nintensity of light emission. Later, Tinguely17and Phan et al.18investigated the dynamics of laser-\ninduced cavitation bubbles in water at different temperatures, showing that the higher the water\ntemperature, hence the vapor pressure, the larger the rebound bubble. To explain these effects,\nnumerical models based on the slightly compressible versions of the Rayleigh-Plesset equation\nproposed by Keller and Miksis19and Gilmore10, use the bubble internal pressure to fit the bubble\nradius evolution9,20,21. However, due to technological challenges involved in measuring the bubble\ncontents at the sub-millimeter and sub-millisecond scale within the bubbles, direct probing of the\ninner bubble pressure remains uncertain22and may hinder some limitations of the model when us-\ning experimental data to fit the evolution of the bubble radius. Among others, some limitations of\nthese models are that numerically describe the dynamics of perfectly-spherical cavitation bubbles\nneglecting effects such as chemical reactions, phase change or strongly non-linear effects related\nto liquid and gas compressibility. In addition to the aforementioned uncertainties regarding the\nmodelling of phase change processes, the influence of non-spherical deformations deserves par-\nticular attention as its effect has been indeed observed in various studies. Brennen23investigated\nthe bubble fission process due to bubble shape instabilities, concluding that the energy dissipated\n3Vapor compression and energy dissipation in a collapsing laser-induced bubble\nby the mixing and turbulence due to bubble fission may be preponderant compared to the conven-\ntional viscous and acoustic damping. Delale and Tunç24developed a numerical model also ac-\ncounting for deviations from sphericity, confirming the results of Brennen23. Moreover, Supponen\net al.25experimentally investigated deformed cavitation bubbles and reported that bubbles expe-\nrience weak jetting phenomena even with reduced anisotropy. Bubble shape perturbations have\nbeen also largely investigated in relation to single bubble sonoluminescence, where the develop-\nment of hydrodynamic instabilities at the bubble-liquid interface has been shown to determine the\nstability diagrams in which light emission is observed26.\nIn this work we investigate the influence of the vapor content during the collapse of laser-\ngenerated single cavitation bubbles in aqueous ammonia by systematically varying the ammonia\nmass fraction wNH3in solution. The latter two-component solutions have similar densities, but\ndifferent p∗\nvvalues, which we use to investigate the influence of the bubble internal composition.\nCompared to single bubble sonoluminescence experiments, the main difference is that it is pos-\nsible to investigate the influence of the vapor content in regimes in conditions where the bubble\noscillation is not stable gaining further insights about the role of phase change on extremely vio-\nlent transient collapses. In addition, because the system’s temperature is kept constant, we avoid\nsome problems related to thermal expansion-related misalignment of optical components, sensi-\ntivity variation of measuring instruments, and change of laser energy absorbed by the liquid at\nbubble generation. These experiments allow us to investigate the influence of vapor content on\nvarious variables including (i) the rebound size and collapse time, (ii) microscopic shape of re-\nbound bubble, (iii) luminescence, (iv) radiated shock at collapse, and (v) liquid pressure build up\nprior to final collapse. Our findings show that all five vary significantly with the NH 3concentra-\ntion in solution, evidencing the significant role of the latter on the bubble collapse and supporting\nthe notion of vapor compression. Furthermore, our measurements provide evidence of the effect\nof the bubble contents on the acoustic emission at bubble collapse.\nII. MATERIALS AND METHODS\nA. Laser-induced cavitation bubbles\nThe experimental apparatus relies on a laser-based technique for the generation of single cav-\nitation bubbles27–30schematically shown in Figure 1. A bubble arises from the plasma generated\n4Vapor compression and energy dissipation in a collapsing laser-induced bubble\nFIG. 1. Top-view schematic of the experimental apparatus.\nby a 9-ns Nd:YAG laser pulse (Quantel CFR 400, 532 nm) focused into a point at the centre of an\nextended volume of liquid (aqueous ammonia in this work). The laser beam, redirected with a set\nof high-intensity mirrors, is enlarged tenfold with a beam expander and focused with an off-axis\nparabolic mirror, immersed into the liquid in the test chamber. The anisotropy parameter ζfor the\ngenerated bubbles was kept below the topological limit between spherical and toroidal collapse\n(ζ<4×10−4), such that any re-entrant jet did not pierce the bubble at collapse (a detailed de-\nscription of ζis given by Obreschkow et al.31and Supponen et al.25). To this end, we generated\nbubbles with a maximum radius R0≈1.5 mm. This size was achieved by adjusting the laser beam\nenergy with a neutral-density filter. Aqueous ammonia was contained in a transparent gas-tight\nbox (cubic shape, 18 cm edge length), which prevented ammonia leaks and hence concentration\nvariations. A total of six ammonia mass fractions wNH3were exploited in the experiments, ranging\nfrom 0 (pure water) to 0.05. The pressure p∞and the temperature T∞of the liquid at rest were kept\n5Vapor compression and energy dissipation in a collapsing laser-induced bubble\nFIG. 2. (a) Total vapor pressure of aqueous ammonia solutions as a function of the mass fraction of ammonia\nwNH 3(and relative water mass fraction wH2O) at 21◦C from Green and Perry32. The plot also displays the\npartial vapor pressure of the solution’s components. (b) Phase diagram of pure water and pure ammonia\nfrom Green and Perry32. The solid lines show the liquid-vapor phase boundary for both substances. The\nblack circles indicate the triple point, whereas the black squares indicate the critical point. L, V , and G\nstands for liquid, vapor, and gas, respectively. The triangle indicates the experimental conditions.\nconstant and equal to the atmospheric pressure ( ≈97 kPa) and ambient temperature ( ≈20◦C),\nrespectively.\nA high-speed camera (Shimadzu HPV-X2) filming at up to 10 million frames per second, back-\nlighted by a collimated LED, recorded shadowgrams of the bubble. The camera is equipped with a\n105 mm objective (Nikon AF-S Micro 1:2.8 GED), and a 2x teleconverter (Nikon AF-S TC-20E).\nA safety filter (532 nm high-pass filter) is mounted in front of the objective to prevent reflected\nlaser beams to accidentally reach the camera sensor. The camera records 256 frames per film at a\nfixed resolution of 400x250 pixels at all frame rates below 10 million. At the latter, the resolution\nis halved. The results of bubble dynamics presented in this work were obtained from high-speed\nrecordings at 500’000 fps, whereas luminescence was recorded at 10 million fps. The instan-\ntaneous radius Rof the bubble was obtained with automated image processing from high-speed\nrecordings by retrieving the equivalent radius Reqof the bubble axial cross sectional area Aassum-\ning spherical symmetry: R=Req=p\nA/π. The bubble cross sectional area is represented by the\nblack area in the bubble shadowgram recorded by the camera.\n6Vapor compression and energy dissipation in a collapsing laser-induced bubble\nA needle-hydrophone placed perpendicularly to the bubble walls and 32.9 mm away from its\ncenter recorded the shock waves generated upon bubble generation and collapse. At the instant of\nbubble maximum expansion, the potential energy of the bubble Ep0can be written as\nEp0=V0(p∞−pv), (1)\nwhere V0is the volume of the bubble at maximum expansion, pvis the vapor partial pressure\nwithin the bubble, conventionally equal to p∗\nv(T∞), and p∞is the liquid pressure in the far field\nequal to the atmospheric pressure. Accordingly, the potential energy of the rebound bubble at the\ninstant of maximum expansion is Ep1=V1(p∞−pv), where V1is the maximum volume of the first\nrebound bubble. Potential energy loss\u0000\nEp11\n[see Fig. 1 (b)]. This e ffectively expands the original equilib-\nrium Mpemba regime. An example demonstrating the shifts\nof the boundary is shown in Fig. 1 (b). Notably, the bound-\nary shift diverges as the chemical potential bias ∆µis en-\nlarged beyond certain threshold. One can numerically deter-\nmine that the threshold value of the chemical potential bias\nis∆µ∗≈3.2. This suggests that when strong nonequilibrium\nconditions are introduced, i.e., ∆µ >∆µ∗, an arbitrary large\n˜µ4will induce the nonequilibrium QMPE. Consequently, the\nphase space of the Mpemba regime is massively extended. As\nshown in Fig. 1 (c), the equilibrium QMPE emerges within\nthe regime bounded by the blue and the black curves, and\nthe nonequilibrium condition of the two baths expands the\nMpemba regime into one bounded by the red and the black\ncurves. Interestingly, the red curve, which one of the bound-\naries of the Mpemba regime, ceases to extend beyond a fi-\nnite ˜µ2and leaves the entire regimes on top of the black curve\nsuccumbed to the QMPE. This opens up a much broader pa-\nrameter window for experimental investigations of the QMPE.\nThe evolution of the density matrix elements in the nonequi-\nlibruim QMPE regimes exemplified by the black dot in Fig. 1\n(c) is demonstrated in the Supplemental Material [22].\nEmergence of MPEs in quantum correlations. –Quantum\ncorrelations are often viewed as resources for faster quantum\nprocessing and play a central role in the quantum thermody-\nnamics and quantum computing [23]. The dynamics of quan-\ntum correlations such as the entanglement have been shown\nto demonstrate peculiar features such as anomalous symmetry\nrestorations and dynamic phase transitions [14, 24–27], and\nhave been one of the focal points of research.\nTo investigate the dynamics of quantum correlations, we\nconsider a simple bipartite model with two sites. An N-level\nquantum system truncated to two levels can be easily related\nto an anti-commuting fermionic system through the Jordan-\nWigner transformation [28]. We study the entanglement dy-\nnamics of the two-site fermionic system with each site cou-\npled to its own bath. Each site is either occupied by a fermion\nor vacant and the fermions can tunnel between the two sites.\nThe Hamiltonians of the system HSand of the reservoirs HR\nFIG. 2. (a) Evolution of entanglement between the two sites as a\nfunction of time. The red curve represents the evolution of the initial\nstate|ρI(0)⟩={0,0.2,0.7,0.1}and the orange curve represents the\ninitial state|ρII(0)⟩={0.1,0.7,0.1,0.1}. Here,µ1=µ2=3. (b) The\ntime of Mpemba crossing as a function of chemical potential bias ∆µ.\nHere,µ1=¯µ+ ∆µ, µ 2=¯µ−∆µ. For both, the parameters used are:\nT1=T2=1,Γ =0.05, ω 1=ω2=1,∆ =0.2.\nare given as follows:\nHS=2X\ni=1ωiη†\niηi+ ∆(η†\n1η2+η†\n2η1),\nHR=X\nk,pωk(a†\nkpakp)+X\nq,sωq(b†\nqsbqs),(6)\nwhere ∆is the hopping rate between the two sites, η†\ni(ηi) is the\ncreation (annihilation) operator on the i-th site which follows\nthe standard fermionic statistics. The interaction Hamiltonian\nbetween the system and the two reservoirs is given by\nHint=X\nk,pλk(η†\n1akp+η1a†\nkp)+X\nq,sλq(η†\n2bqs+η2b†\nqs),(7)\nwhereλis the interaction strength between the system and the\nreservoir and a†\nkp(b†\nkp) is the creation operator for a particle of\nmomentum k, polarization pfrom the reservoir.\nFor simplicity, we consider the symmetric case ω1=ω2=\n1. In this case, the eigenvalues of the transition matrix of its\nLindblad equation are independent of the tunneling rate and\nare given by\nλ1=−4Γ, λ 2=λ3=−2Γ, λ 4=0. (8)\nThe transition matrix Mis diagonalizable and the vectorized\ndensity operator can be expressed as\n⃗ρ(t)=X\nieλitαi(0)⃗vi, (9)\nwhere⃗viis the i-th eigenvector of the transition matrix and\nαi(0) is the coe fficient of the i-th eigenvector defined by\nαi(0)=⟨⃗δi,⃗ρ(0)⟩. The matrix of vectors {⃗δi}is the inverse\nof the matrix of eigenvectors {⃗vi}.\nUsually, to have the “strong Mpemba e ffect”, the coe fficient\nof the slowest decaying mode is required to vanish for the par-\nticularly chosen initial conditions. In this model, both α2(0)4\nFIG. 3. MPE and inverse MPE in the evolutions of the QMI between\nthe two sites. (a) The red curve represents the evolution of the initial\nstate|ρI(0)⟩={0.1,0.1,0.7,0.1}and the orange curve is for the initial\nstate|ρII(0)⟩={0.1,0.65,0.1,0.15}. Parameters used are: T1=T2=\n1, µ1=µ2=3. (b) The red curve represents the evolution of the\ninitial state|ρ(0)⟩={0.4,0.1,0.2,0.3}and the orange curve is for the\ninitial state|ρ(0)⟩={0.3,0.3,0.2,0.2}. Parameters used are: T1=\nT2=0.1, µ1=µ2=1.2. For both, Γ =0.05, ω 1=ω2=1,∆ =0.2.\nandα3(0) need to vanish due to the degeneracy of the eigen-\nvalue. This gives the condition ⟨⃗δ2,⃗ρ(0)⟩=⟨⃗δ3,⃗ρ(0)⟩=0\nwhere the vector ⃗δ2and⃗δ3are given in the Supplemental\nMaterial. Notably, the strong MPE refers the exponentially\nfaster equilibration towards the equilibrium state, but does not\nguarantee the emergence of the anomalous crossing. For the\nnonequilibrium systems with multiple baths, a similar “strong\nMpemba e ffect” can be defined for systems that show expo-\nnentially faster decay to the NESS with vanishing coe fficients\nfor the slowest decaying mode. This coe fficient now depends\non the parameters of both of the two baths. Generally, it is not\nnecessary for a system to satisfy the strong Mpemba condition\nto manifest crossings of interested physical quantities during\nthe relaxation.\nTo demonstrate the dynamical behaviors of quantum corre-\nlations in the system, we introduce the concurrence and the\nquantum mutual information (QMI) between the two subsys-\ntems. The concurrence is a entanglement monotone derived\nfrom the entanglement formation [29, 30]. For the system we\nconsider, the concurrence can be simplified to\nE(ρ)=2 max(0,|ρl\n22|−q\nρl\n11ρl\n44), (10)\nwhereρl\ni jare the entries of the density matrix in local ba-\nsis. Another widely-used measure of correlations is the QMI,\nwhich is a direct generalization of classical mutual informa-\ntion and quantifies the maximum amount of information that\ncan be securely transferred between two parties [31, 32]. For\na bipartite system ABwith the density operator ρABand the\nsubsystem A(B) with the reduced density operator ρA(B)=\nTrB(A)(ρAB), the QMI is defined as\nI(ρAB)=S(ρA)+S(ρB)−S(ρAB), (11)\nwhere S(ρ)=−tr (ρlog2ρ) is the von Neumann entropy.\nThe dynamics of the entanglement and the QMI for our sys-\ntem are shown in Fig. 2 and 3. In Fig. 2, we demonstrate the\nFIG. 4. Evolution of the density matrix element ρ33as a function\nof time with and without coherence terms. (a) The population dy-\nnamics without the coherence terms. (b) The population dynam-\nics with the coherence terms. The o ff-diagonal terms in the ini-\ntial condition are chosen at ρ23=ρ32=0.2 for the red curve and\nρ23=ρ32=−0.1 for the orange curve. For both (a) and (b), the pop-\nulation terms are identical. The red curve represents the evolution of\nthe initial state with the population terms |ρI(0)⟩={0.1,0.25,0.65,0}\nand the orange curve is for the initial state with the population terms\n|ρII(0)⟩={0.1,0.2,0.6,0.1}. (c) The shaded region in orange repre-\nsents the MPE regime of (b) with µ1,2=¯µ±∆µ. Parameters used are:\nT1=T2=1, µ1=0.1, µ2=3,Γ =0.05, ω 1=ω2=1,∆ =0.05.\nemergence of the MPE in the entanglement evolution from\ntwo di fferent initial states. The initially more entangled state,\nrepresented by the red curve in Fig. 2 (a), has a faster rate\nof disentanglement and disentangles earlier than the state that\nis initially less entangled. This represents possible tradeo ffs\nbetween the entangling time and the entangling strength for\nquantum states. In addition, the vanishing of entanglement\nwithin a finite time has been observed in various systems\nand is termed the “sudden death” in contrast to the smooth\nasymptotic decay observed in the dynamics of QMI [33]. As\nshown in Fig. 2 (b), the nonequilibrium condition can signif-\nicantly influence the crossing time of the entanglements and\nadvance it when the average chemical potential of the baths\nis large. This e ffect can be intuitively understood as follows.\nWhen the baths are set at large µ’s, the quantum states are ap-\nproximately fully occupied. Increasing the bias in this case\nmeans lowering the occupation of one of the bath while keep-\ning the occupation of the other bath roughly unchanged. In\nthis case, the state ρI(0) which has a higher excitation energy\nobtains a higher rate of decay, resulting in an earlier cross-\ning time. On the other hand, for very low chemical potentials\nthe fermion states are approximately vacant. Enlarging ∆µ5\neffectively raises the occupation number of one of the baths\nwhile leaves the other bath unchanged. This increase of the\nchemical potential causes a slower decay and a delayed cross-\ning time. The same argument explains why the crossing time\napproaches a constant for extremely large chemical potential\nbiases regardless of the average potentials. In Fig. 3, we show\nthe MPE and the inverse MPE in the evolution of the QMI.\nThese two figures jointly demonstrate the ubiquity of Mpemba\ncrossings in the evolution of quantum systems.\nInfluence from the quantum coherence. –One of the es-\nsential features of a quantum system is the existence of the o ff-\ndiagonal coherence terms, which di fferentiate its density ma-\ntrix from a classical one [34]. In the conventional Lindbladian\ntreatment, the coherence terms represented by the o ff-diagonal\nelements in the density matrix decouple with the population\nterms in the dynamics. Under the wide-band approximation\nfrequently used in quantum dot systems, the coherence terms\nare erased completely. For a quantum system out of equilib-\nrium and in contact with multiple baths, these ignored quan-\ntum coherence can play a significant role in the system dy-\nnamics, especially when the tunneling rate is comparable to\nthe decoherence rate [35–37]. We show that in weak tun-\nneling regimes, the quantum coherence can give rise to the\nemergence of QMPE while the population dynamics without\nit predicts the otherwise. This distinctive feature caused by\ncoherence can only take place when strong nonequilibrium\nconditions are introduced.\nIn Fig. 4 (a) and (b), we show explicitly that in the regime\nwhere no sign of the QMPE is witnessed when the coherence\nis ignored, the QMPE emerges when coherence is considered.\nFurthermore, the emergence of QMPE is only possible when\nan intrinsic nonequilibrium condition is introduced. As shown\nin Fig. 4 (c), for larger average potentials ¯ µ’s, larger ∆µ’s\nare required for the emergence the QMPE. This is due to the\nfact that larger biases are necessary to substantially alter theoccupation number of the baths at high chemical potentials.\nSmall biases fail to create enough di fferences from the equi-\nlibrium solutions, consequently, do not generate enough co-\nherence necessary to trigger the QMPE. This is a demonstra-\ntion that quantum coherence can have qualitative influence on\nthe dynamics of the system. Importantly, the magnitude of the\nquantum coherence is amplified in the weak tunneling regimes\nand its asymptotic value is approximately proportional to the\nbias between the baths occupation numbers [35]. Therefore,\nthis “coherence-induced MPE” is an intrinsic nonequilibrium\nphenomenon that only emerges when the multiple supporting\nbaths of subsystems are di fferent and it is most conspicuous\nwhen the tunneling rate of the fermions is not significantly\nlarger than the decay rate ( ∆≲2Γ).\nConclusion. –In this study, we investigated the quantum\nMpemba e ffect in the quantum dot and the two-site fermion\nsystems coupled with two di fferent baths. Firstly, we showed\nthat nonequilibrium conditions can dramatically expand the\nparameter space where the MPE emerges. This opens up a\nmuch wider window both conceptually and also practically\nfor experimental investigations. Secondly, we investigated the\ndynamics of quantum correlations in the two-fermionic sys-\ntem coupled with two di fferent baths and showed that anoma-\nlous decays of MPEs and inverse MPEs emerge in the evo-\nlution of the entanglement and the QMI. We demonstrated\nthat nonequilibrium conditions can significantly influence the\ntimes of the Mpemba crossings. Thirdly, we studied the pos-\nsible influence on the dynamics due to the nonequilibrium-\ninduced quantum coherence which is absent in the conven-\ntional Lindbladian dynamics. Our results show that in the\nweak tunneling regimes, the quantum coherence, which is\nsupported by the nonequilibrium conditions of the two baths,\nhas nontrivial influence the population dynamics and can in-\nduce the emergence of the QMPE which the population dy-\nnamics alone fail to predict.\nSUPPLEMENTAL MATERIAL\nIn the Supplemental Material, we provide certain calculation details of the quantum dot and the two fermion models in the\nmain text as well as extra figures on the MPEs in the quantum dot system.\nMPE in the quantum dot model\nThe Lindblad equation of the quantum dot model has been studied in many previous papers [12, 38, 39]. The results we\nused in this study are summarized in this section which can be found in Refs. [12, 13, 40]. The four spin configurations –\nthe doubly-occupied state, two singly-occupied states, and empty state – are labeled by α=1,2,3,4, respectively. Under the\nwide-band approximation, the density matrix ρreduces to a diagonal form with the four matrix elements whose vectorized form\nis represented by ρα. The quantum Master equation of the vector is given by\nd\ndtρi=X\njMi jρj, (12)6\nFIG. 5. Example of the evolution of the density matrix element ρ22at the (˜µ2,˜µ4)=(2,6) indicated by the black dot in Fig. 1 (c). Parameters\nused are T1=T2=˜T1=˜T2=˜T3=˜T4=1,¯µ=3,∆µ=4,˜µ1=2,˜µ3=1.\nwhere the transition matrix is given by [12]\nMi j=−2(2−f(1)) f(1)f(1)0\n(2−f(1))−(2−f(0))−f(1)0 f(0)\n(2−f(1)) 0 −(2−f(0))−f(1)f(0)\n0 (2−f(0)) (2−f(0))−2f(0). (13)\nHere, we have set the decay rate to Γ =1.f(j)with j=0,1 is defined as the sum of fermionic occupation numbers of the two\nbaths f(j)=1\n1+e(ϵ0+jU−µL)/T+1\n1+e(ϵ0+jU−µR)/T, whereµL(R)represents the chemical potential of the left (right) bath. The time\nevolution of the density matrix element ρα(t) is given by\nρα(t)=4X\nn=1eλntRαnan, (14)\nwhere an=P4\nm=1Lnmρm(0),λnrepresents the eigenvalue of the transition matrix M, and Li j(Ri j) represents the matrix of left\n(right) eigenvectors the transition matrix given as follows:\nR=f(0)f(1)\n4+2(f(0)−f(1))02f(0)f(1)\n−4+(f(0)−f(1))2f(0)f(1)\n4−2(f(0)−f(1))\nf(0)(2−f(1))\n4+2(f(0)−f(1))−1\n2−f(0)(2−f(0)−f(1))\n−4+(f(0)−f(1))2−f(0)f(1)\n4−2(f(0)−f(1))\nf(0)(2−f(1))\n4+2(f(0)−f(1))1\n2−f(0)(2−f(0)−f(1))\n−4+(f(0)−f(1))2−f(0)f(1)\n4−2(f(0)−f(1))\n(2−f(0))(2−f(1))\n4+2(f(0)−f(1))0−2f(0)(2−f(0))\n−4+(f(0)−f(1))2f(0)f(1)\n4−2(f(0)−f(1)), (15)\nand\nL=1 1 1 1\n0−1 1 0\n−2−f(1)\nf(0)−2−f(0)−f(1)\n2f(0)−2−f(0)−f(1)\n2f(0) 1\n(2−f(0))(2−f(1))\nf(0)f(1)−2−f(0)\nf(0)−2−f(0)\nf(0) 1. (16)\nIn Fig. 5, we show the example of the QMPE at (˜ µ2,˜µ4)=(2,6) indicated by the black dot in Fig. 1 (c) in the main body. In\nFig. 6, we show that the vast extension of parameter space of the Mpemba regime also exists for the other singly-occupied state\nρ3. For the doubly-occupied and vacant states, the nonequilibrium condition has no influence on the Mpemba regime.\nMPE in the two fermion model\nIn the energy eigenbasis, the master equation for the density matrix is\nd\ndtρi j=X\nklMkl\ni jρkl. (17)7\nFIG. 6. The boundaries of parameter regime that show QMPE for the third density matrix element ρ3(−10there is an O/parenleftbigg\nnO(dlog(d)/ǫ2)/parenrightbigg\n-time algorithm which, given a collection of vectors v1, . . . , vn∈\nRdand a matroidM= ([n],I)returns a set B∈I such that with probability at least 1−d−4\nλmin/parenleftBigg\n∑\ni∈Bviv⊤\ni/parenrightBigg\n≥(1−ǫ)·max\nB⋆∈Iλmin/parenleftBigg\n∑\ni∈B⋆viv⊤\ni/parenrightBigg\n.\nOur result generalizes to give a PTAS (for constant dimensio n) when the objective is a general matrix function\nsatisfying certain technical properties. In particular, t his implies that a similar result as in Theorem 1 is achievabl e\nwhen the objective is to maximize the determinant of ∑i∈Bviv⊤\nior to minimize any norm of the eigenvalues of\n(∑i∈Bviv⊤\ni)−1.\nTheorem 2 Suppose we have a collection of vectors V= (v1, . . . , vn)∈Rd, and a matroidM= ([n],I). Let f :S+\nd:→R\nbe a concave, monotone, and homogeneous function given with a value and first order oracle. For any ǫ>0, there is an\nO/parenleftbigg\nnO(dlog(d)/ǫ2)/parenrightbigg\n-time randomized algorithm, which takes (V,M,f)as input and returns a set B ∈I such that with\n1probability at least 1−d−4,\nf/parenleftBigg\n∑\ni∈Bviv⊤\ni/parenrightBigg\n≥(1−ǫ)·max\nB⋆∈If/parenleftBigg\n∑\ni∈B⋆viv⊤\ni/parenrightBigg\n.\nAlthough Theorem 2 is stated in terms of maximizing concave f unctions, our algorithm can also be applied to min-\nimize monotone and homogeneous convex functions (e.g., tra ce(∑i∈Bviv⊤\ni)−1) by considering the natural convex\nrelaxation of the function over the matroid base polytope an d using the same rounding strategy.\nTechnical Overview. The first natural direction is to construct a convex programm ing relaxation for the problem\nand aim to apply randomized rounding methods to it.\nmax λmin(X)\nX=n\n∑\ni=1xi·viv⊤\ni\nx∈P(M)\nx≥0(CP)\nHere,P(M)denotes the matroid base polytope of M. Unfortunately, this direct approach faces problems as thi s\nnatural relaxation has an unbounded integrality gap even in very special cases (see Appendix A.1). The main\nchallenge is the presence of long vectors that contribute significantly towards the optimum s olution. A natural\nway to formalize the contribution of a vector is to consider i tsleverage score . Indeed, if Tdenotes the optimum\nsolution and AT=∑i∈Tviv⊤\ni, let li=v⊤\niA−1\nTvibe the leverage score of viand let S={i∈T:li≥ǫ2\nlogd}be the\nset of vectors in the optimum solution with large leverage sc ores. The boundǫ2\nlogdis chosen to allow randomized\nrounding methods to work (see Lemma 2 for details). Using the simple fact that the sum of leverage scores of\nall vectors in the optimum solution is exactly d, it follows that|S|≤dlogd\nǫ2. Thus we could easily enumerate all\nsuch subsets Sin time nO(dlogd/ǫ2). For each such guess S, we include Sin our solution and solve the convex\nprogram. We then apply the randomized rounding method to the solution of the convex program. Unfortunately,\nthe challenge lies in ensuring that the convex program not on ly selects the vectors in S(this can be easily done by\nsetting their indicator variable to one) but also avoids sel ecting all vectors notinTthat have a large leverage score.\nThe latter is crucial for the randomized rounding approach t o work effectively. Unfortunately, since we did not\nguess T, we have no way to insist that we do not pick these vectors in th e convex program.\nTo address this problem, we present a new structural lemma th at enables us to compute the leverage score as\ngiven by matrix AS=∑i∈SvivT\ni. The lemma shows that there are few vectors with large levera ge scores, even\nwhen using ASinstead of AT. Observe that A−1\nS/{ollowsequalA−1\nTand therefore, the leverage scores with respect to ASare\nlarger. Nevertheless, we still show a similar bound in the fo llowing lemma.\nLemma 1 For any set T and a set of vectors {vi:i∈T}inRdsuch that ∑i∈Tviv⊤\niis invertible, there exists a subset S ⊆T\nsuch that|S|=O(dlog(d)/ǫ2), AS=∑i∈Sviv⊤\niis invertible, and for all i ∈T\\S,\nv⊤\niA−1\nSvi≤ǫ2\n10 log(d).\nWith the help of Lemma 1, we can now guess the set Sand insist that the convex program includes all these vector s\nin the chosen subset. More importantly, it allows us to insis t that all vectors vinot in Ssuch that v⊤\niA−1\nSvi>ǫ2\n10 log(d)\nnot be included in the chosen solution. The last step can be do ne since we have guessed the set S. This allows us\nto apply the randomized rounding approach to the convex prog ramming solution.\nThere are some points worth mentioning about the randomized rounding approach. When the constraint matroid\nis a partition matroid, randomized rounding is a natural cho ice: for each part, the convex programming solution\ncan be interpreted as a probability distribution over vecto rs in that part. Independently, for each part, pick one\nof the vectors with probability given by the convex programm ing solution. A simple application of the matrix\nChernoff bound and the fact that leverage scores are all smal l due to Lemma 1 gives us the desired result. Due\n2to the simplicity of the approach for partition matroids as w ell as the applicability of these constraints, we first\nprove the result for partition matroids in Section 2. We also show the application of our result to obtain an algo-\nrithmic version of the Kadison-Singer problem [MSS15] for c onstant dimension. We slightly improve the run time\ncompared to the recent work [JMS22].\nCorollary 1 Suppose we are given collection of vectors U= (u1, . . . , un)∈Rdwith/bardblui/bardbl2≤αfor any i∈[n]and\n∑n\ni=1uiu⊤\ni=Idand a constant c >0such that there exists a set T∗satisfying\n/parenleftbigg1\n2−c√\nα/parenrightbigg\nId/√recedesequal∑\ni∈T∗uiu⊤\ni/√recedesequal/parenleftbigg1\n2+c√\nα/parenrightbigg\nId.\nFor any ǫ>0, there exists a randomized algorithm such which given Uand c as input, returns a set T such that\n(1−ǫ)·/parenleftbigg1\n2−c√\nα/parenrightbigg\nId/√recedesequal∑\ni∈Tuiu⊤\ni/√recedesequal(1+ǫ)·/parenleftbigg1\n2+c√\nα/parenrightbigg\nId,\nwith probability at least 1−O(d−4). The run time of the algorithm is O (nO(dlogd/ǫ2)).\nFor general matroids, a straightforward application of ran domized rounding does not work since it will not ensure\nthat the chosen set is an independent set in the matroid. Inst ead, we use pipage rounding for general matroids, which\ninvolves randomly walking in the matroid polytope to return a vertex while ensuring that the output solution\nhas even better concentration than is given by independent r andomized rounding. To show these concentration\nresults, we build on the work of Harvey and Olver [HO14] and gi ve lower tail bounds on the distribution obtained\nvia pipage rounding in Lemma 4.\n1.2 Related Work\nThe minimum eigenvalue problem with partition constraints can be interpreted as a generalization of the max-min\nallocation problem. In the case of cardinality constraints , it can also model problems from experimental design\nand spectral sparsification. We give an overview of prior wor k for these special cases.\nMax-min allocation and Santa Claus: In the max-min allocation problem, we are given a set [d]of agents and a\nset[n]of items where agent j∈[d]has valuation hij≥0 for item i. The goal is to select an assignment σ:[n]→[d]\nwhich maximizes\nmin\nj∈[d]∑\ni:σ(i)=jhij.\nThis can be seen as a special case of the minimum eigenvalue pr oblem with partition constraints.\nBansal and Sviridenko [BS06] introduced the configuration L P as a relaxation for the max-min allocation problem\nbut showed that it has an integrality gap of Ω(√n)[BS06]. Asadpour and Saberi [AS07] gave a rounding scheme\nfor the same LP , which achieves an O(√nlog3n)-approximation. This was later improved by Chakrabarty et\nal. [CCK09] to an ˜O(nǫ)-approximation for any ǫ∈Ω(log log n/ log n)by iteratively constructing new instances\nwith smaller integrality gap.\nA further special case is the Santa Claus problem where each i tem ihas an intrinsic value Hi≥0 such that\nhij∈{0,Hi}for all players j∈[d]. Here, Bansal and Sviridenko [BS06] used the configuration L P to find an\nO(log log n/ log log log n)-approximation. Feige [Fei08] non-constructively showed a constant upper bound on\nthe integrality gap of the configuration LP for the Santa Clau s problem by iteratively applying the Lov´ asz Local\nLemma. The current best bound is due to Haxell and Szab´ o [HS2 3], who used new topological techniques to\nshow that the integrality gap is at most 3.534. Bounds on the i ntegrality gap do not immediately lead to efficient\napproximation algorithms, but Davies et al. [DRZ20] recent ly gave an algorithm for a more general setting that\ncan be used to achieve a (4+ǫ)-approximation for the Santa Claus problem.\nExperimental Design (E-optimal Design): Even with cardinality constraints (uniform matroid of rank k), the\nminimum eigenvalue problem is NP-hard [cMI09]. Allen-Zhu e t al. [AZLSW17] showed that it is possible to\n3deterministically find a (1−ǫ)-approximation so long as k≥Ω(d/ǫ2)by rounding the natural convex relaxation.\nThey also conjectured that this requirement was necessary. This conjecture was confirmed in [NST19], where\nthey showed an integrality gap instance for the convex relax ation. Recently Lau and Zhou [LZ21] have built\non the regret minimization framework from [AZLSW17] to show that a modified local search algorithm with a\n“smoothed” objective works as long as there is a near-optima l solution with a good condition number.\nSpectral Sparsification and Kadison-Singer. The problem of rounding the natural convex programming rela x-\nation for the minimum eigenvalue problem is closely related to spectral sparsification [BSS09] and the Kadison-\nSinger problem [MSS15]. In spectral sparsification [BSS09] , the goal is to pick a small subset of vectors S⊆[n]\nsuch that ∑i∈Swiviv⊤\nispectrally approximates ∑i∈[n]viv⊤\nifor some weights wi. In the cardinality constrained\nminimum eigenvalue problem, rounding the convex programmi ng solution involves finding a small set S, such\nthat ∑i∈Sviv⊤\nispectrally approximates ∑i∈[n]xiviv⊤\ni, where the weights xiform the solution to the convex relax-\nation. Indeed [AZLSW17] essentially build on this connecti on to obtain their results for the E-design problem\ndiscussed earlier. The Kadison-Singer problem [MSS15] is c losely related to the minimum eigenvalue problem\nunder a partition matroid constraint. We utilize this conne ction in Corollary 1 to give an algorithmic version of\nthe Kadison-Singer problem for constant dimensions. More g enerally, the Kadison-Singer problem can be refor-\nmulated as showing that the integrality gap of the natural re laxation of the minimum eigenvalue problem under\npartition matroid constraints is at most 1/ (1−ǫ)if the length of each vector is at most O(ǫ). We discuss this\nconnection in Section 4.\n2 The Algorithm for Partition Matroids\nTo highlight the main idea of our algorithm, we first prove The orem 1 for the special case of partition matroid. Let\nM= (E,I)be a partition matroid where E=P1∪···∪ Pkbe a disjoint union of parts with each part containing\nnelements, and we have a collection of vectors vijfori∈[k]and j∈Pi. The goal is to select an element σ(i)∈Pi\nfor each ito maximize λmin/parenleftBig\n∑k\ni=1viσ(i)v⊤\niσ(i)/parenrightBig\n.\nWe can construct the natural convex relaxation of this probl em as follows. For each i∈[k]and j∈Pi, we add a\ndecision variable xijwhich represents whether we select the vector vjfrom part Pi, i.e., if σ(i) =j. Then we get the\nconvex program\nmax λmin(X)\nX=k\n∑\ni=1∑\nj∈Pixij·vijv⊤\nij\n∑\nj∈Pixij=1,∀i∈[k]\nx≥0\nThe constraint ∑j∈Pixij=1 ensures that we have a probability distribution over the po ssible assignments within\neach part in the optimal solution.\nGiven an optimal solution x⋆with value OPT , a natural rounding strategy is to round independently with in each\npart. Following this rounding strategy, we get a rank 1 rando m matrix Mifor each part Piwith\nPr(Mi=vijv⊤\nij) =x⋆\nij,∀j∈Pi.\nThe following matrix concentration inequality bounds the p robability of failure of this rounding strategy.\nTheorem 3 [Tro15, Theorem 5.1.1] Consider independent random matric es M1, . . . , Mk∈S+\nd. Set\nµmin=λmin/parenleftBigg\nE/bracketleftBigg\nk\n∑\ni=1Mi/bracketrightBigg/parenrightBigg\n.\n4Ifλmax(Mi)≤R for all i∈[k]a.s. then\nPr/parenleftBigg\nλmin/parenleftBigg\nk\n∑\ni=1Mi/parenrightBigg\n<(1−ǫ)µmin/parenrightBigg\n≤d·exp/parenleftBigg\n−ǫ2µmin\n2R/parenrightBigg\n.\nIf we round according to the optimal solution x⋆then E/bracketleftBig\n∑k\ni=1Mi/bracketrightBig\n=∑k\ni=1∑j∈Pix⋆\nijvijv⊤\nij.\nSoµmin=OPT , and since for our particular case Miare rank 1, R=maxiλmax(Mi) =maxij/bardblvij/bardbl2. To bound the\nfailure probability, we want R≈ǫ2/ log(d), which in turn requires that maxij/bardblvij/bardbl2=O(ǫ2/ log(d)). This is a\nvery strong assumption on an instance.\nThe plan is to “guess” a suitable change of basis such that all the vectors in the support of our optimal solution\nhave a small norm. This will be useful because of the followin g standard, but slightly more flexible, version of the\npreceding matrix concentration inequality.\nCorollary 2 Consider independent random matrices M1, . . . , Mk∈S+\ndand let A be an arbitrary positive definite matrix.\nDefine µmin:=λmin/parenleftBig\nA−1/2E/bracketleftBig\n∑k\ni=1Mi/bracketrightBig\nA−1/2/parenrightBig\n.Ifλmax(A−1/2MiA−1/2)≤R for all i∈[k]a.s. then\nPr/parenleftBigg\nk\n∑\ni=1Mi/notfollowsoreql(1−ǫ)µmin·A/parenrightBigg\n≤d·exp/parenleftBigg\n−ǫ2µmin\n2R/parenrightBigg\n.\nAgain, since Miis rank 1 for our case, we have R=maxi∈[k]λmax(A−1/2MiA−1/2) =maxi,jv⊤\nijA−1vij. So, to use\nthis corollary, we first need to find a matrix Asuch that v⊤\nijA−1vij=O(ǫ2/ log(d))for all[i]∈[k],j∈Pi. We will\nonly need to consider matrices Aof a specific form that uses the input vectors.\nGiven a subset S⊆E, we define AS:=∑(i,j)∈Svijv⊤\nij, and consider the set of long vectors in the norm induced by\nAS:L(S):=/braceleftBig\n(i,j)∈E\\S:v⊤\nijA−1\nSvij>ǫ2\n10 log(d)/bracerightBig\n. For a fixed set S, the following convex program ensures that S\nis included in the solution and no “long” vectors from L(S)are included in the solution.\nmax λmin(X)\nX=k\n∑\ni=1∑\nj∈Pixij·vijv⊤\nij\n∑\nj∈Pixij=1,∀i∈[k]\nxij=0,∀(i,j)∈L(S)\nxij=1,∀(i,j)∈S\nx≥0(CP(S))\nBecause of the extra constraints excluding “long” vectors, we could now use the flexible matrix concentration\ninequalities to randomly round the optimal solution.\nBut, it is not clear that there is a good choice of Sfor which the convex program CP(S) is still a relaxation of th e\noriginal problem. Lemma 1, which we restate here for the read er’s convenience, shows that there exists a suitable\nsetSthat is not too large.\nLemma 1 For any set T and a set of vectors {vi:i∈T}inRdsuch that ∑i∈Tviv⊤\niis invertible, there exists a subset S ⊆T\nsuch that|S|=O(dlog(d)/ǫ2), AS=∑i∈Sviv⊤\niis invertible, and for all i ∈T\\S,\nv⊤\niA−1\nSvi≤ǫ2\n10 log(d).\nThe proof of this lemma is inspired by the local search algori thm of [MSTX19].\nAt first glance, it may not be apparent why a subset satisfying the conditions of Lemma 1 should exist. However,\n5in the proof, we show that any subset of Tthat is locally optimal with respect to a local search criter ia indeed\nsatisfies the guarantees of Lemma 1.\nProof (of Lemma 1) We consider the local search process of [MSTX19] . Starting with a set Sof sizeℓsuch that\nA=∑i∈Sviv⊤\niis invertible, we apply the following update rule. For any j∈T\\Sand i∈S, if det(A)<\ndet(A−viv⊤\ni+vjv⊤\nj), update S={S\\{i}}∪{ j}and iterate.\nLetS⊆Tbe a locally optimal (under single element swaps) solution f or this process (such an Scorresponds to\nthe locally optimal solution determinant maximization pro blem subject to the cardinality constraint |S|≤ℓ), and\nletA=∑i∈Sviv⊤\ni. More concretely, this means that for all i∈Sand j∈T\\S,\ndet(A)≥det(A−viv⊤\ni+vjv⊤\nj).\nWe calculate the determinant on the right-hand side using th e matrix determinant lemma,\ndet(A−viv⊤\ni+vjv⊤\nj) =det/parenleftBig\nA+/bracketleftbigvivj/bracketrightbig/bracketleftbig−vivj/bracketrightbig⊤/parenrightBig\n=det(A)·det/parenleftBig\nI2+/bracketleftbig−vivj/bracketrightbig⊤A−1/bracketleftbigvivj/bracketrightbig/parenrightBig\n=det(A)·/parenleftBig\n(1−v⊤\niA−1vi)(1+v⊤\njA−1vj)+( v⊤\niA−1vj)2/parenrightBig\n.\nSo local optimality implies that for every i∈Sand j/∈S,(1−v⊤\niA−1vi)(1+v⊤\njA−1vj) + ( v⊤\niA−1vj)2≤1.\nRearranging this inequality we get\nv⊤\njA−1vj−(v⊤\niA−1vi)·(v⊤\njA−1vj)+( v⊤\niA−1vj)2≤v⊤\niA−1vi. (1)\nNote that ∑i∈Sv⊤\niA−1vi=/a\\}bracketle{tA,A−1/a\\}bracketri}ht=dand ∑i∈S(v⊤\niA−1vj)2=v⊤\njA−1vj. So for a fixed j∈T\\S, summing\nequation (1) over all i∈Simpliesℓ·v⊤\njA−1vj−d·v⊤\njA−1vj+v⊤\njA−1vj≤d. Rearranging, we see that for any\nj∈T\\S,\nv⊤\njA−1vj≤d\nℓ−d+1=ǫ2\n10 log(d),\nwhere the last equality follows by choosing ℓ=10dlog(d)/ǫ2+d−1. /square\nWe will apply this lemma to the case when T={viσ⋆(i):i∈[k]}m where σ∗is the choice function that maximizes\nthe minimum eigenvalue, i.e., when Tcontains the vectors from an optimal integral assignment. I n particular, we\nget the following corollary.\nLemma 2 There is a subset S ⊆E such that|S|=O(dlog(d)/ǫ2), and the convex program CP(S) is a relaxation for the\nminimum eigenvalue problem.\nAsdis a constant, the size of the set Swe search for is also constant. Thus, there are at most O(nO(dlog(d)/ǫ2))\npossible choices for S. We will consider each choice in turn to guess the correct set . Note that trying every set of\nthe appropriate size will be the dominant factor in determin ing the algorithm’s runtime.\nThe following lemma proves that for any fixed subset S, rounding the optimal solution to CP(S) gives a good\napproximation to the optimal value of CP(S).\nLemma 3 Let S⊆E be an independent set, and let x⋆be the optimal solution to CP(S). Then rounding randomly in e ach\npart outputs an assignment σ:[k]→E with σ(i)∈Pisuch that\nPr/bracketleftBigg\nλmin/parenleftBigg\nk\n∑\ni=1viσ(i)v⊤\niσ(i)/parenrightBigg\n<(1−ǫ)·λmin/parenleftBigg\nk\n∑\ni=1∑\nj∈Pix⋆\nijvijv⊤\nij/parenrightBigg/bracketrightBigg\n0 and a set of vectors u1, . . . , um∈Rdsuch that/bardblui/bardbl2≤α,∑m\ni=1uiu⊤\ni=I\nwhere there exists a subset T⊆[m]satisfying\n/parenleftbigg1\n2−c√\nα/parenrightbigg\nI/√recedesequal∑\ni∈Tuiu⊤\ni/√recedesequal/parenleftbigg1\n2+c√\nα/parenrightbigg\n, (2)\nthe goal is actually to find a set T⊆[m]which satisfies the above condition. This problem is FNP-har d when\nc=1/(4√\n2)for general values of d[JMS22, Theorem 2].\nTheir main result [JMS22, Theorem 1] is an algorithm with run ning time\nO/parenleftbigg/parenleftbiggm\nk/parenrightbigg\n·poly(m,d)/parenrightbigg\nfork=O/parenleftbiggd\nǫ2log(d)log/parenleftbigg1\nc√α/parenrightbigg/parenrightbigg\n,\nwhich returns a set T′⊆[m]such that\n(1−ǫ)/parenleftbigg1\n2−c√\nα/parenrightbigg\nI/√recedesequal∑\ni∈T′uiu⊤\ni/√recedesequal(1+ǫ)/parenleftbigg1\n2+c√\nα/parenrightbigg\nI, (3)\nIn this section, we will show how to use the rounding techniqu e for partition matroids to give a simpler algorithm\nthat achieves the same guarantee with the same run time, exce pt we save the small dependence on log (1/c√α)in\nthe exponent.\nProof (of Corollary 1) Given vectors u1, . . . , um∈Rd, we construct an instance of the minimum eigenvalue with\npartition constraints as follows. Let E={1, 2}×[m], with mparts P1, . . . , Pmso that Pi={(i, 1),(i, 2)}fori∈[m].\nFor each i∈[m]define the vectors\nvi1=/bracketleftbiggui\n0/bracketrightbigg\n∈R2d, and vi2=/bracketleftbigg0\nui/bracketrightbigg\n∈R2d.\nTo see how vand uare related, note that for any δ∈[0, 1/2)there is a choice function σ:[m]→{1, 2}such that\n/parenleftbigg1\n2−δ/parenrightbigg\nI2d/√recedesequalm\n∑\ni=1viσ(i)v⊤\niσ(i) (4)\nif and only if there is a set T⊆[m]such that\n/parenleftbigg1\n2−δ/parenrightbigg\nI/√recedesequal∑\ni∈Tuiu⊤\ni/√recedesequal/parenleftbigg1\n2+δ/parenrightbigg\nI. (5)\nGiven σsatisfying (4), let X1:=∑i:σ(i)=1uiu⊤\niand X2:=∑i:σ(i)=2uiu⊤\ni. Then X1and X2are respectively the\nfirst and second diagonal d×dblock of ∑m\ni=1viσ(i)v⊤\niσ(i). Therefore/parenleftBig\n1\n2−δ/parenrightBig\nI2d/√recedesequal∑m\ni=1viσ(i)v⊤\niσ(i)if and only if\nX1/{ollowsequal/parenleftBig\n1\n2−δ/parenrightBig\nIdand X2/{ollowsequal/parenleftBig\n1\n2−δ/parenrightBig\nId. In addition, since X1+X2=Id, this is equivalent to\n/parenleftbigg1\n2−δ/parenrightbigg\nId/√recedesequal∑\ni:σ(i)=1uiui=Id−X2/√recedesequal/parenleftbigg1\n2+δ/parenrightbigg\nId.\n8We then use Algorithm 1 to find a (1−ǫ)approximate solution σ:[m]→{ 1, 2}to inputMand vectors vij. Since\nwe assume there is a set Tsatisfying (2), Theorem 1 implies that with probability at l east 1−O(d−4), Algorithm\n1 will return a choice function σ∗such that (1−ǫ)/parenleftBig\n1\n2−c√α/parenrightBig\nI2d/√recedesequal∑m\ni=1viσ∗(i)v⊤\niσ∗(i), and we will return the set\nT′={i∈[m]:σ∗(i) =1}.\nFrom the equivalence between (4) and (5), the set T′={i∈[m]:σ(i) =1}satisfies (3)\n(1−ǫ)/parenleftbigg1\n2−c√\nα/parenrightbigg\nId/√recedesequal∑\ni∈T′uiui/√recedesequal(1+ǫ)/parenleftbigg1\n2+c√\nα/parenrightbigg\nId.\n/square\n3 General Matroid Constraints\nIn the general form of the problem, we are given a collection o f vectors v1, . . . , vn∈Rdand a matroidM=\n([n],I), and the goal is to find a basis B∈I which maximizes λmin/parenleftBig\n∑i∈Bviv⊤\ni/parenrightBig\n. For background on matroids,\nsee Appendix B.1.\nFor a general matroid, the idea of finding a linear transforma tion under which all elements in the optimal solution\nhave a small norm generalizes easily. So we can use the same ap proach of first guessing a set S⊆Eon a reasonable\nsize and then solving the convex relaxation of the problem co nditioned on Sbeing included in the solution.\nGiven a subset S⊆[n], we can again set AS=∑i∈Sviv⊤\ni, and consider the set of long vectors:\nL(S) =/braceleftBigg\ni∈[n]\\S:v⊤\niA−1\nSvi>ǫ2\n10 log(d)/bracerightBigg\n.\nFor a matroidM, letP(M)⊆[0, 1]ndenote the matroid base polytope. Then the following is the n atural convex\nprogramming relaxation which excludes the “long” vectors.\nmax λmin(X)\nX=n\n∑\ni=1xi·vijv⊤\nij\nx∈P(M)\nxi=0,∀i∈L(S)\nxi=1,∀i∈S\nx≥0(CP(S))\nThis convex program can be solved in polynomial time (see App endix B.1). Just like in the partition case, Lemma 1\nguarantees that there is a set Sfor which CP(S) is a relaxation for the minimum eigenvalue pr oblem. As before,\nafter solving CP(S), we can guarantee that all the vectors in the fractional support of the optimal solution will have\na small norm with respect to AS.\nThe challenge in extending the earlier approach to general m atroid constraints comes from the rounding step. For\na partition matroid, we could simply round the fractional op timum of CP(S) independently in each part to obtain\na basis. However, for more general constraints, it is not so c lear how to round a fractional solution to a basis.\nInstead of rounding independently, we will use the techniqu e of pipage rounding to find a basis. The following\nlemma is the lower-tail version of the same concentration in equality proved in [HO14]. For completeness, we will\ninclude a proof of the version we need in Appendix B.\nLemma 4 LetP(M)be a matroid base polytope and x ∈P(M). Let M1, . . . , Mmbe self-adjoint matrices that satisfy\nλmax(Mi)≤R. Let µ=λmin/parenleftBig\n∑i∈[n]xiMi/parenrightBig\n. If randomized pipage rounding (Algorithm 3) starts at x and outputs the\n9extreme point ˆx=χ(B)ofP(M), then we have\nPr/bracketleftBigg\n∑\ni∈BMi≤(1−ǫ)·µ/bracketrightBigg\n≤d·exp/parenleftBigg\n−ǫ2µ\n2R/parenrightBigg\n.\nWe use this lemma to generalize our earlier approach to all ma troids.\nLemma 5 Let S⊆E be an independent set in Mand let x⋆be the optimal solution to CP(S). Then pipage rounding start ing\nat x⋆outputs a basis B such that\nPr\nλmin/parenleftBigg\n∑\ni∈Bviv⊤\ni/parenrightBigg\n<(1−ǫ)λmin\n∑\ni∈[n]x⋆\niviv⊤\ni\n\n0, define\npǫ(x):= Pr\nB∼D(x)\nλmin(∑\ni∈Bviv⊤\ni)≤(1−ǫ)·λmin\n∑\ni∈[n]xiviv⊤\ni\n\n.\nSopǫ(x)is the failure probability of getting a (1−ǫ)-approximation when rounding independently at point x.\n[HO14] showed that there exists a function gǫ(x)s.t.pǫ(x)≤gǫ(x)≤d·exp/parenleftbigg\n−ǫ2µmin\n2R/parenrightbigg\nand gǫis concave under\nswaps, i.e., for all a,b∈[n]and x∈P(M)the map z/ma√sto→gǫ(x+z(ea−eb))is concave.\nSo, if xis not an extreme point of P(M), then there exist a,b∈[n]and ǫ>0 such that x±ǫ(ea+eb)∈P(M).\nLetl=min{z:x+z(ea−eb)∈P(M)}and u=max{z:x+z(ea−eb)∈P(M).\nWith this, we can define xl=x+l(ea−eb)and xu=x+u(ea−eb). Since g(x+z(ea−eb))is concave as a function\nofz, we know that either g(xl)≤g(x)org(xu)≤g(x). Moreover, both xland xuare on a lower dimensional face\nthan the initial point x. Thus, for any initial point x0∈P(M), a total of miterations suffice to find an extreme\npoint with ˆxwith g(ˆx)≤g(x0).\nIn randomized pipage starting at x∈P(M), our next iterate x′of the rounding procedure will be xlwith prob-\n11abilityu\nu−land xuwith probability−l\nu−l. This ensures that E(x′) = x, and the concavity under swaps guarantees\nthat E[gǫ(x′)]≤gǫ(x)by Jensen’s in the variable z. If we start at a point x0∈P(M))and iterate this random\nprocedure mtimes, we get an extreme point ˆxwhich satisfies E[ˆx] =x0and E[g(ˆx)]≤g(x0).\nThis gives the intuition behind the proof of Lemma 6, and lead s to the following algorithm.\nAlgorithm 3 Randomized Pipage Rounding\n1:Input : Point x∈P(M), whereP(M)is a matroid base polytope\n2:while xis not integral do\n3: a,b←distinct elements of [n]s.t.∃ǫ>0 with x±ǫ(ea−eb)∈P(M)\n4:ℓ←min{y≥0 :x−y(ea−eb)∈P(M)}\n5: h←max{y≥0 :x+y(ea−eb)∈P(M)}\n6: x←/braceleftBigg\nx−ℓ(ea−eb)w.p.ℓ/(ℓ+h)\nx+h(ea−eb)w.p. h/(ℓ+h)\n7:end while\n8:Return basis B∈P(M)with indicator vector x\n4 Conclusion and Remarks\nThe resolution of the Kadison-Singer problem in [MSS15] usi ng the interlacing families of polynomials implies the\nfollowing existential result about maximizing the minimum eigenvalue under partition matroid constraints.\nTheorem 5 [MSS15, Theorem 1.4] For ǫ>0and vectors{vij}i∈[k],j∈[n]∈Rdwith/bardblvij/bardbl2≤ǫfor all i∈[k],j∈[n], if\nthere exist xij≥0such that\nk\n∑\ni=1n\n∑\nj=1xij·vijv⊤\nij=Idandn\n∑\nj=1xij=1for all i∈[k],\nthen there exists a choice function σ:[k]→[n]such that\n(1−√\nǫ)2·Id/√recedesequalk\n∑\ni=1viσ(i)v⊤\niσ(i)/√recedesequal(1+√\nǫ)2·Id.\nWe can state this result equivalently as an “existential” ro unding result. When /bardblvij/bardbl2≤ǫ, Theorem 5 implies\nthat the integrality gap of the natural convex relaxation (C P) for the minimum eigenvalue problem with partition\nconstraints is only 1/ (1−√ǫ)2. It is an open problem to efficiently round the solution to the convex relaxation\nwith comparable guarantees for any dimension d.\nMore generally, the problem of designing an approximation a lgorithm for the minimum eigenvalue problem under\npartition or matroid constraints in arbitrary dimensions r emains wide open. However, checking whether there is\na solution with a non-zero objective can be solved in polynom ial time solvable through matroid intersection.\nRecently, there has been significant progress in the case of m aximizing the determinant [Nik15, SEFM15, NS16,\nAGV18, SX18, MNST20, BLP+22], but it remains open whether those techniques can be util ized for the minimum\neigenvalue problem.\n12References\n[AB13] Haim Avron and Christos Boutsidis. Faster subset sel ection for matrices and applications. SIAM\nJournal on Matrix Analysis and Applications , 34(4):1464–1499, 2013.\n[AFS12] Arash Asadpour, Uriel Feige, and Amin Saberi. Santa claus meets hypergraph matchings. ACM\nTrans. Algorithms , 8(3), jul 2012.\n[AGV18] Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzan t. Log-concave polynomials, entropy, and\na deterministic approximation algorithm for counting base s of matroids. In 2018 IEEE 59th Annual\nSymposium on Foundations of Computer Science (FOCS) , pages 35–46. IEEE, 2018.\n[AS07] Arash Asadpour and Amin Saberi. An approximation alg orithm for max-min fair allocation of indi-\nvisible goods. In Proceedings of the Thirty-Ninth Annual ACM Symposium on The ory of Computing , STOC\n’07, page 114–121, New York, NY, USA, 2007. Association for C omputing Machinery.\n[AZLSW17] Zeyuan Allen-Zhu, Yuanzhi Li, Aarti Singh, and Yi ning Wang. Near-optimal discrete optimization\nfor experimental design: A regret minimization approach. Mathematical Programming , 186, 11 2017.\n[BLP+22] Adam Brown, Aditi Laddha, Madhusudhan Pittu, Mohit Sing h, and Prasad Tetali. Determinant max-\nimization via matroid intersection algorithms. In 2022 IEEE 63rd Annual Symposium on Foundations of\nComputer Science (FOCS) , pages 255–266, Los Alamitos, CA, USA, nov 2022. IEEE Comput er Society.\n[BS06] Nikhil Bansal and Maxim Sviridenko. The santa claus p roblem. In Proceedings of the Thirty-Eighth\nAnnual ACM Symposium on Theory of Computing , STOC ’06, page 31–40, New York, NY, USA, 2006.\nAssociation for Computing Machinery.\n[BSS09] Joshua D. Batson, Daniel A. Spielman, and Nikhil Sri vastava. Twice-ramanujan sparsifiers. In Pro-\nceedings of the Forty-First Annual ACM Symposium on Theory o f Computing , STOC ’09, page 255–262,\nNew York, NY, USA, 2009. Association for Computing Machiner y.\n[CCK09] Deeparnab Chakrabarty, Julia Chuzhoy, and Sanjeev Khanna. On allocating goods to maximize fair-\nness. In Proceedings of the 2009 50th Annual IEEE Symposium on Founda tions of Computer Science , FOCS\n’09, page 107–116, USA, 2009. IEEE Computer Society.\n[cMI09] Ali C ¸ ivril and Malik Magdon-Ismail. On selecting a maximum volume sub-matrix of a matrix and\nrelated problems. Theoretical Computer Science , 410:4801–4811, 2009.\n[Cun84] W. H. Cunningham. Testing membership in matroid pol yhedra. Journal of Combinatorial Theory, Series\nB, 36:161–188, 1984.\n[DRZ20] Sami Davies, Thomas Rothvoss, and Yihao Zhang. A tal e of santa claus, hypergraphs and matroids.\nInProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms , SODA ’20, page\n2748���2757, USA, 2020. Society for Industrial and Applied Ma thematics.\n[Fei08] Uriel Feige. On allocations that maximize fairness . InProceedings of the Nineteenth Annual ACM-SIAM\nSymposium on Discrete Algorithms , SODA ’08, page 287–293, USA, 2008. Society for Industrial a nd\nApplied Mathematics.\n[HO14] Nicholas JA Harvey and Neil Olver. Pipage rounding, p essimistic estimators and matrix concen-\ntration. In Proceedings of the twenty-fifth annual ACM-SIAM symposium o n Discrete algorithms , pages\n926–945. SIAM, 2014.\n[HS23] Penny Haxell and Tibor Szab´ o. Improved integrality gap in max-min allocation: or topology at the\nnorth pole. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discre te Algorithms (SODA) ,\npages 2875–2897, 2023.\n[JMS22] Ben Jourdan, Peter Macgregor, and He Sun. Is the algo rithmic kadison-singer problem hard?, 2022.\narxiv.org:2205.02161.\n[LZ21] Lap Chi Lau and Hong Zhou. A local search framework for experimental design. In Proceedings of the\nThirty-Second Annual ACM-SIAM Symposium on Discrete Algor ithms , SODA ’21, page 1039–1058, USA,\n2021. Society for Industrial and Applied Mathematics.\n[MNST20] Vivek Madan, Aleksandar Nikolov, Mohit Singh, and Uthaipon Tantipongpipat. Maximizing deter-\nminants under matroid constraints. In 2020 IEEE 61st Annual Symposium on Foundations of Computer\nScience (FOCS) , pages 565–576, 2020.\n13[MSS15] Adam W. Marcus, Daniel A. Spielman, and Nikhil Sriva stava. Interlacing families ii: Mixed charac-\nteristic polynomials and the kadison–singer problem. Annals of Mathematics , 182:327–350, 2015.\n[MSTX19] Vivek Madan, Mohit Singh, Uthaipon Tantipongpipa t, and Weijun Xie. Combinatorial algorithms for\noptimal design. In Proceedings of the Thirty-Second Conference on Learning Th eory, 06 2019.\n[Nik15] Aleksandar Nikolov. Randomized rounding for the la rgest simplex problem. In Proceedings of the\nForty-Seventh Annual ACM Symposium on Theory of Computing , STOC ’15, page 861–870, New York,\nNY, USA, 2015. Association for Computing Machinery.\n[NS16] Aleksandar Nikolov and Mohit Singh. Maximizing dete rminants under partition constraints. In Pro-\nceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing , STOC ’16, page 192–201,\nNew York, NY, USA, 2016. Association for Computing Machiner y.\n[NST19] Aleksandar Nikolov, Mohit Singh, and Uthaipon Tao T antipongpipat. Proportional volume sampling\nand approximation algorithms for a-optimal design. Proceedings of the Thirtieth Annual ACM-SIAM\nSymposium on Discrete Algorithms , pages 1369–1386, 2019.\n[Puk06] Friedrich Pukelsheim. Optimal design of experiments . SIAM, 2006.\n[SEFM15] Marco Di Summa, Friedrich Eisenbrand, Yuri Faenza , and Carsten Moldenhauer. On largest volume\nsimplices and sub-determinants. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on\nDiscrete Algorithms , SODA ’15, page 315–323, USA, 2015. Society for Industrial a nd Applied Mathe-\nmatics.\n[SX18] Mohit Singh and Weijun Xie. Approximate positive cor related distributions and approximation algo-\nrithms for D-optimal design. In Proceedings of SODA , 2018.\n[Tro15] Joel A. Tropp. An introduction to matrix concentrat ion inequalities. Foundations and Trends® in Ma-\nchine Learning , 8(1-2):1–230, 2015.\n[Wea04] Nik Weaver. The kadison–singer problem in discrepa ncy theory. Discrete mathematics , 278(1-3):227–\n239, 2004.\nA Omitted proofs\nProof (of Corollary 2) This is a simple calculation, using the fact the the semidefinite order is preserved under\nconjugation.\nPr/parenleftBigg\nk\n∑\ni=1Mi/notfollowsoreql(1−ǫ)µmin·A/parenrightBigg\n=Pr/parenleftBigg\nk\n∑\ni=1A−1/2MiA−1/2/notfollowsoreql(1−ǫ)µmin·I/parenrightBigg\n=Pr/parenleftBigg\nλmin/parenleftBigg\nk\n∑\ni=1A−1/2MiA−1/2/parenrightBigg\n<(1−ǫ)µmin/parenrightBigg\n≤d·exp/parenleftBigg\n−ǫ2µmin\n2R/parenrightBigg\n.\n/square\nA.1 Integrality Gap Example\nConsider the vectors v1=e1,v2=e1,v3=e2,v4=e3inR3and a partition matroid M= ([ 4],I)defined by the\nbases{1, 2, 3},{1, 2, 4}. The optimal value of maximizing the minimum eigenvalue for this instance is 0 as we are\nforced to pick v1and v2in any basis and they are linearly dependent.\n14The convex relaxation of maximizing the minimum eigenvalue for this instance is given by\nmax λmin(X)\nX=x1·v1v⊤\n1+x2·v2v⊤\n2+x3·v3v⊤\n3+x4·v4v⊤\n4\nx1=1,∀i∈[k]\nx2=1,∀i∈[k]\nx3+x4=1,∀i∈[k]\nx≥0(CP)\nThe optimum of (CP) is attained when x1=x2=1 and x3=x4which gives\nX=2e1e⊤\n1+1\n2e2e⊤\n2+1\n2e3e⊤\n3.\nSo the optimal value of (CP) is 1/2, whereas the true optimal i s 0.\nB Matroids and Pipage Rounding\nIn this section, we provide the necessary background on matr oids, as well as the lower tail versions of lemmas\nfrom [HO14], which let us prove Lemma 4.\nB.1 Matroids\nA pairM= (E,I)is a matroid if Eis a finite set andIis a collection of subsets of Esatisfying\n(1) If I∈I and J⊆Ithen J∈I, and\n(2) If I,J∈I and|I|<|J|then there is e∈J\\Isuch that I∪{e}∈I .\nThe sets inIare referred to as the independent sets of the matroidM. The maximal sets in Iare called bases , and\nit is a consequence of the matroid axioms that all bases have t he same cardinality. For a subset U⊆E, we denote\nmyr(U)the maximum size of an independent set in Uand call this the rank of U. In this notation, we can say that\nevery basis ofMhas cardinality exactly r(E). Given a matroid M, the matroid base polytope is the convex hull of\nindicator vectors of the bases of M, and is denotedP(M). The base polytope has the following linear description\nP(M) =conv{χ(B):Ba basis ofM}\n=/braceleftBigg\nx∈RE:∑\ne∈Exe=r(E),∑\ne∈Uxe≤r(U)∀U⊆E,x≥0/bracerightBigg\n.\nCunningham [Cun84] showed that given x∈RE\n+, it is possible to find a violated constraint for P(M)in strongly\npolynomial time using only an independence oracle for the ma troidM.\nB.2 Pipage Rounding\nThe following theorem follows from the discussion in Sectio n 3.2.\nTheorem 6 [HO14] There is a randomized polynomial time algorithm that , given x0∈P(M), outputs an extreme point ˆx\nofP(M)with E[ˆx] =x0and such that for any g concave under swaps E[g(ˆx)]≤g(x).\nWe will mainly make use of this theorem through the following claim. The conditions of the claim come from\npessimistic estimators, but not all of them are strictly nec essary. For a point x∈[0, 1]n, let D(x)represent the\ncorresponding product distribution over {0, 1}nwith marginals given by x.\n15Lemma 6 [HO14] LetE⊆{ 0, 1}nand g :P(M)→Rsatisfy\nPrx∼D(x)[x∈E]≤g(x),and\nmin{g(x−xiei),g(x+(1−xi)ei)}≤ g(x)\nfor all x∈[0, 1]n, and g be concave under swaps. If pipage rounding is started a t an initial point x0∈P and ˆx is the random\nextreme point, then Pr[ˆx∈E]≤g(x0).\nEssentially, this lemma says that if we have a pessimistic es timator which is concave under swaps, then pipage\nrounding has the same type of concentration behavior as inde pendent rounding, but will actually return a vertex\nof the matroid polytope.\nFor our particular application, we will be choosing the func tion gto be an estimator for matrix concentration due\nto Tropp [Tro15].\nTheorem 7 [Tro15, Theorem 5.1.1] Let M1, . . . , Mnbe self-adjoint matrices with λmax(Mi)≤R for all i∈[n]and let\nµmin=λmin(Ex∼D(x)[∑i∈[n]xiMi]. For t∈R, we have the bound\nPrx∼D(x)\nλmin\n∑\ni∈[n]xiMi\n≤t\n≤inf\nθ<0gt,θ(x)\nwhere gt,θ(x) =e−θt·tr exp/parenleftBig\n∑i∈[n]logEeθxi·Mi/parenrightBig\n. Furthermore, for t = (1−ǫ)µmin,\ngt,θ(x)≤d·/parenleftBigg\ne−ǫ\n(1−ǫ)1−ǫ/parenrightBiggµmin/R\n.\nThis is the lower-tail version of the same concentration ine quality which was used in [HO14]. In that paper, they\nprovide an upper-tail version of Lemma 4 using a new generali zation of Lieb’s concavity theorem, stated below.\nLemma 7 [HO14] Let L∈Sd, C1,C2∈S++\nd, and K1,K2∈S+\nd. Then the univariate function\nz→tr exp(L+log(C1+zK1)+log(C2−zK2))\nis concave in a neighborhood of 0.\nAs a consequence, we get the following lemma.\nLemma 8 Forθ<0, all x∈[0, 1]m, the function\ngt,θ(x) =e−θt·tr exp\n∑\ni∈[m]logEx∼D(x)eθxi·Mi\n\nis concave under swaps.\nProof (of Lemma 8) Let Ci:=Ex∼D(x)[eθxi·Mi] =xi·eθMi+(1−xi)·I≻0, and for any i∈[n]\nEx∼D(x+zei)[eθxi·Mi] = ( xi+z)eθMi+(1−xi−z)·I=Ci−z·(I−eθMi).\n16Then∀a,b∈[n],\ngt,θ(x+z(ea−eb))\n=e−θt·tr exp\n∑\ni∈[n]\\{a,b}logCi+log/parenleftBig\nCb+z·(I−eθMb)/parenrightBig\n+log/parenleftBig\nCa−z·(I−eθMa)/parenrightBig\n\n=e−θt·tr exp(L+log(Cb+z·Kb)+log(Ca−z·Ka)),\nwhere Ka= (I−eθMa),Kb= (I−eθMb), and L=∑i∈[n]\\{a,b}logCi∈Sd. Ifθ≤0, then Ka/{ollowsequal0 and Kb/{ollowsequal0. Using\nLemma 7, z→gt,θ(x+z(ea−eb))is concave in z, and the result follows. /square\nCombining Lemma 8 and Theorem 7 with Lemma 6, we obtain Lemma 4 .\n17" }, { "title": "2401.14363v1.An_analytic_version_of_stable_arithmetic_regularity.pdf", "content": "arXiv:2401.14363v1 [math.LO] 25 Jan 2024AN ANALYTIC VERSION OF STABLE ARITHMETIC\nREGULARITY\nGABRIEL CONANT AND ANAND PILLAY\nAbstract. We prove a structure theorem for stable functions on amenabl e\ngroups, which extends the arithmetic regularity lemma for s table subsets of\nfinite groups. Given a group G, a function f:G→[-1,1] is called stable if\nthe binary function f(x·y) is stable in the sense of continuous logic. Roughly\nspeaking, our main result says that if Gis amenable, then any stable function\nonGis almost constant on all translates of a unitary Bohr set in Gof bounded\ncomplexity. The proof uses ingredients from topological dy namics and contin-\nuous model theory. We also discuss some applications, inclu ding a short proof\nof the noncommutative analogue of Bogolyubov’s Lemma for am enable groups.\n1.Introduction\nIn [40], Terryand Wolfprovean arithmetic regularitylemma for “stab le”subsets\nof (Z/pZ)n. Their result compares to Green’s [20] arithmetic regularity lemma in\n(Z/2Z)nin direct analogy to how Malliaris and Shelah’s [29] regularity lemma\nfor stable graphs compares to Szemer´ edi’s [38] original result for all finite graphs.\nShortly after [40], a non-quantitative generalization to arbitrary fi nite groups was\nproved by the authors and Terry [15], followed by a quantitative ver sion for finite\nabelian groups by Terry and Wolf [41]. A quantitative generalization to all finite\ngroups was later obtained by the first author [12].\nThemaingoalofthispaperistoextendtheseresultstofunctions. Thiscontinues\na line started in [9], where the authors and Chavarria proved a regula rity lemma\nfor stable binary functions on sets in analogy to Malliaris and Shelah’s r esult for\ngraphs (but without explicit quantitative bounds). Roughly stated , the regularity\nlemma for a stable binary relation E⊆V×V(withVfinite) yields a partition\n{Vi}ofV, with bounded size, such that the relation induced by Eon each pair\nVi×Vjis almost complete or almost empty. For a stable function f:V×V→[0,1],\nthe result in [9] is a natural generalization, and says that fis almost constant on\nalmost all of each pair Vi×Vj.\nFor a finite group G, the results from [40, 41, 15, 12] can be interpreted as saying\nthat ifA⊆Gis stable then the relation on G×Gdefined by xy∈Aadmits a\npartitionasabove, but alsowith the piecesgivenbythe cosetsofan ormalsubgroup\nofG. Following the analogy, one then might guess that a stable function o nGis\nalmost constant on almost all of each coset of some subgroup. How ever, this turns\nouttobefalse(seeExample2.12)and, asisoftenthecaseinadditive combinatorics,\nwe will need to abandon the graph-theoretic focus on partitions an d involve Bohr\nneighborhoods. The model-theoretic explanation for this is that ta me arithmetic\nDate: January 25, 2024.\nPartially supported by NSF grants DMS-1855503, DMS-220478 7 (Conant) and DMS-1665035,\nDMS-1760212, DMS-2054271 (Pillay).\n12 G. CONANT AND A. PILLAY\nregularityresults correspondto “domination” by a certain compac t groupK(along\nthe lines of G/G00). Arithmetic regularity in terms of coset partitions then arises\nfrom situations where Kis profinite. For the case of stable sets, Kturns out to be\na closed subgroup of a topological semigroup on a type space, which is of course\nprofinite (in classical first-order logic). For stable functions, we w ill obtainKfrom\na type space in a similar way, but in the context of continuous logic whe re type\nspaces need not be profinite.\nWe now describe our main results, which require the following terminolo gy. Let\nGbe a group and fix a function f:G→R. GivenB⊆G, we sayfisǫ-constant\nonBif|f(x)−f(y)|<ǫfor allx,y∈B. IfGis finite, then we say fisζ-almost\nǫ-constant on Bif it isǫ-constant on a set B′⊆Bwith|B′|≥|B|−ζ|G|.1\nFinally, given a function k:R+→Z+, we say that fisk-stableif for allǫ >0,\nthere do not exist a1,...,a k(ǫ),b1,...,b k(ǫ)∈Gsuch that|f(aibj)−f(ajbi)|≥ǫ\nfor all 1≤i0,k:R+→Z+, andζ:R+×N2→R+. Suppose G\nis a finite group and f:G→[-1,1]isk-stable. Then there is a normal subgroup\nH≤G, with index m≤Ok,ζ,ǫ(1), and a(δ,T(n))-Bohr neighborhood BinH, with\nδ-1,n≤Ok,ζ,ǫ(1), such thatfisζ(δ,n,m)-almostǫ-constant on all translates of B.\nIn the previous statement, T( n) denotes the real n-dimensional torus, and a\n(δ,T(n))-Bohr neighborhood is a homomorphic preimage of the identity neig hbor-\nhood of radius δin T(n). See Definition 2.1 for details.\nRoughly speaking, Theorem 1.1 says that a stable function on a finite group\nbehaves similarly to a continuous function on a compact group. Since continuous\nfunctions are canonical examples of stable functions, one can thu s interpret The-\norem 1.1 as an “inverse theorem” in the sense of [39]. See Section 2.3 f or further\ndiscussion along these lines.\nThe main result of the paper will in fact be a more general version of T heorem\n1.1 suitable for any amenable (discrete) group. Recall that a group Gisamenable if\nthere is a left-invariant finitely additive probability measure on the Bo olean algebra\nofsubsets of G. Forbrevity, we will simply say left-invariant measure when working\nin the context of an amenable group. With such a measure on Gfixed, we have the\nobvious analogue of what it means for a real-valued function on Gto be “almost\nconstant” as defined for finite groups above (see also Remark 2.3) .\nTheorem 1.2 (main result) .Fixǫ >0,k:R+→Z+, andζ:R+×N→R+.\nSupposeGis an amenable group with left-invariant measure µ, andf:G→[-1,1]\nisk-stable. Then there is a (δ,U(n))-Bohr neighborhood BinG, withδ-1,n≤\nOk,ζ,ǫ(1), such that fisζ(δ,n)-almostǫ-constant on all translates of B.\nHere we have replaced the torus T( n) with the unitary group U( n). WhenG\nis finite, one can recover T( n) as in Theorem 1.1 (after passing to a subgroup H)\nthanks to Jordan’s Theorem. In fact, Theorem 1.1 holds for any am enable torsion\ngroup by Schur’s generalization of Jordan’s result (see Corollary 4.3 ).\nTo proveTheorem 1.2, we will first obtain a correspondingmodel the oretic result\nin continuouslogic, and then apply anultraproduct argument. The m odel-theoretic\ncompanionresult isgiveninTheorem3.9, and concernsametric struc tureMwith a\nsortGfor a group, along with a left-invariant stable formula ϕ(x,z) (withxof sort\n1Remark 2.3 explains why we have not chosen to write ζ|B|here instead.AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 3\nG). The key consequence of stability is weak almost periodicity of the t ype space\nSϕ(M) asaG(M)-flow. By workofEllis and Nerurkar[18] in topologicaldynamics,\nit follows that the Ellis semigroup of Sϕ(M) has a unique minimal subflow K,\nwhich is a compact Hausdorff group. We also give a precise description of the\nEllis semigroup of Sϕ(M) as a certain type space (see Lemma 3.1), which allows\nus to view Kas a definable compactification of G. This also opens the route\nto the main model theoretic result (Theorem 3.9), which says rough ly that any\nϕ-formulaθ(x) is almost constant on all translates of a preimage of an identity\nneighborhood in K. Finally, we use the Peter-Weyl Theorem to replace Kwith a\nunitary group (see Corollary 3.10). All of this sets the stage for pr oving Theorem\n1.2with ultraproducts, althoughsomenontrivialsteps arerequire dcomparedtothe\ncase of [15] for stable sets. For example, we will use recent work wit h Hrushovski\n[13] to facilitate the use of Bohr neighborhoods in ultraproducts. T his step of\nthe proof represents a key use of the amenability assumption thro ugh a result of\nKazhdan [26] on “approximate homomorphisms”. See the start of S ection 4.1 for\nfurther discussion.\nAfter the proof of Theorem 1.2, we will state some refinements in sp ecial cases\nsuch as groups of bounded exponent, abelian groups, and torsion groups (see Corol-\nlaries 4.1, 4.2, and 4.3). We will then turn to applications of Theorem 1.2 which\nexploit the fact that the continuous theory of Hilbert spaces is sta ble. For example,\nthis fact implies that any function on an amenable group obtained as c onvolution\nof two functions is k-stable for some absolute k(see Definition 2.10 and Corollary\n2.11). Combined with Theorem 1.2, we have the following conclusion.\nCorollary 1.3. Fixǫ>0andζ:R+×N→R+. SupposeGis an amenable group\nwith left-invariant measure µ, andf,g:G→[-1,1]are arbitrary functions. Then\nthere is a (δ,U(n))-Bohr neighborhood BinG, withδ-1,n≤Oζ,ǫ(1), such thatf∗g\nisζ(δ,n)-almostǫ-constant on all translates of B.\nIn Section 4.3, we will use this corollary to give a short proof of Bogoly ubov’s\nLemma for amenable groups.\nFinally, in Section 5, we will return to the general model-theoretic se tting of\nTheorem 3.9 (discussed above), and prove some further results a bout generic types,\nstabilizers, and connected components. For the most part, thes e will be natural\ngeneralizationsof known results in classical logic (e.g., from [15] and [1 1]), although\nwe will make note of some crucial differences.\nNotation. Givenr,s∈Randǫ>0, we write r≈ǫsto denote|r−s|≤ǫ.\n2.Preliminaries\n2.1.Bohr neighborhoods.\nDefinition 2.1. LetKbe a metric group. Fix a (discrete) group Gand a real\nnumberδ >0. Then a ( δ,K)-Bohr neighborhood in Gis a set of the form\nτ-1(U), whereτ:G→Kis a group homomorphism and Uis the open identity\nneighborhood in Kof radiusδ.\nWe will only need the above definition in the case that Kis either the unitary\ngroup U(n) of degree n, or the real n-dimensional torus T( n). We equip U( n) with\nthe metric induced by the operator norm on GL( n); and we equip T( n) with the\nmetric induced from U( n) when viewing T( n) as the maximal torus consisting of4 G. CONANT AND A. PILLAY\ndiagonal matrices. In particular, the metric on T( n) is the Cartesian product of\nthe complex distance metric on the unit circle.\nNote that the above metric on U( n) is bi-invariant. Consequently, if Bis a\n(δ,U(n))-Bohr neighborhood in a group G, thenB=B-1andgB=Bgfor any\ng∈G. The following additional facts are proved in [13, Section 5].\nProposition 2.2. LetBbe a(δ,U(n))-Bohr neighborhood in a group G.\n(a)Gcan be covered by (c/δ)n2translates of B, wherec>0is an absolute constant.\n(b)IfGhas exponent randδ≤Or(1), thenBis a normal subgroup of G.\n(c)IfGis abelian then Bis a(δ,T(n))-Bohr neighborhood.\n(d)IfGis a torsion group then there is a normal subgroup H≤Gof indexOn(1)\nsuch thatB∩His a(δ,T(n))-Bohr neighborhood in H.\nRemark 2.3. SupposeGis amenable with left-invariant measure µ. Then Propo-\nsition 2.2(a) implies that any ( δ,U(n))-Bohr neighborhood in Ghas measure at\nleast (δ/c)n2. Now recall that a function f:G→Risζ-almostǫ-constant on a set\nB⊆Gif it isǫ-constant on some B′⊆Bwithµ(B′)≥µ(B)−ζ. This statement\nloses potency if ζis large relative to µ(B). However our main result (Theorem 1.2)\nis formulated with ζafunction of the parameters δandnassociated to a unitary\nBohr set. So, for example, suppose f:G→Risρ(c/δ)n2-almostǫ-constant on a\n(δ,U(n))-Bohr neighborhood B⊆Gfor someρ>0. Thenfisǫ-constant on some\nB′⊆Bwithµ(B′)≥(1−ρ)µ(B). In other words, requiring µ(B′)≥(1−ζ)µ(B)\nin the definition of “ ζ-almost” would have no effect on the validity of our results.\nThat being said, we have chosen to not use this latter formulation in t he actual\ndefinition in orderto removeirrelevant(and distracting)steps in ce rtain arguments.\n2.2.Stable binary functions on sets.\nDefinition 2.4. Letf:V×W→Rbe a function, where VandWare arbitrary\nsets. Given an integer k≥1 and a real number ǫ>0, we say that fis (k,ǫ)-stable\nif there do not exist a1,...,ak∈Vandb1,...,bk∈Wsuch that\n|f(ai,bj)−f(aj,bi)|≥ǫfor all 1≤i0.\nFor a binary relation E⊆V×W, stability of the indicator function 1Erecovers\nthe standard definition of stability for E, up to a uniform change in the parameters.\nSee [9, Remark 1.2] for precise details.\nWe will now discuss two families of stable functions, which can both be v iewed\nas coming from stability of a certain continuous theory.\nExample 2.5. SupposeXandYare compact spaces and f:X×Y→Ris a\ncontinuous function. Then fisk-stable for some k:R+→Z+.\nWhenXandYare metrizable, this is the continuous logic analogueof the trivial\nfactthatanybinaryrelationonfinitesetsisstable. Thatbeingsaid, adirectproofis\nstill an informative exercise (which we leave to the reader). Moreov er, as discussed\nin the next subsection, this example will provide some useful intuition for our main\narithmetic regularity results.\nThe next example is much less trivial, but equally well known (and previo usly\ndiscussed in [9, Section 1]).AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 5\nFact 2.6. There is a function k:R+→Z+such that if Bis the unit ball of a\nHilbert space, then the inner product is k-stable as a function from B×Bto[-1,1].\nWhile this fact follows from stability of Hilbert spaces in continuous logic , which\nis proved in [5], it is also evident from much earlier literature.2In particular, a\nslightlyweakerversionofstability(known as“stabilityin amodel”) can be deduced\nfor the inner product from a result of Grothendieck [21]. Alternat ively, stability (in\na model) of the function /b⊣r⌈blx+y/b⊣r⌈blis proved by Krivine and Maurey [28], which then\npasses to the inner product using the parallelogram law. In either ca se, to obtain\na uniform stability function k, one can use the fact that Hilbert spaces are closed\nunder ultrapowers, which was first established in Krivine’s thesis.\nOne takeaway from Fact 2.6 is that stable functions arise quite natu rally across\nmathematics and perhaps are more common than stable graphs. In deed, this fact\nhas previously led to remarkable applications of stability in many areas . Further\ndiscussion can be found in [9, Section 1].\n2.3.Stable unary functions on groups. In a group G, stability (for relations)\ncan be viewed as a property of a (unary) set A⊆Gby considering the binary\nrelationx·y∈A. This is the context of the arithmetic regularity results in [40, 41,\n15, 12]. For functions on groups, we proceed analogously.\nDefinition 2.7. Fixk:R+→Z+. Given a group G, we say that a function\nf:G→Risk-stable iff(x·y):G×G→Risk-stable.\nAs previously observed, if Ais a stable subset of a group G, then1Ais a stable\nfunction on G. For a different kind of example, it follows from Example 2.5 that if\nGisa compactHausdorffgroup, then anycontinuousfunction f:G→Risk-stable\nfor some k. As alluded to in the introduction, this connects to our main results v ia\nthe following standard exercise in topological groups.\nProposition 2.8. LetGbe a compact Hausdorff group. Then a function f:G→R\nis continuous if and only if for every ǫ>0, there is an open identity neighborhood\nU⊆Ksuch thatfisǫ-constant on all translates of U.\nInlightofthepreviousfact, Theorem1.2saysthatifafunction fonanamenable\ngroup behaves like a continuous function on a compact group in the s ense of being\nstable, then fis structurally similar to such a function.\nRemark 2.9. Thepreviousdiscussioncan be made moreprecisethroughthe use o f\nthe Bohr topology on a group G(i.e., the possibly non-Hausdorff topology induced\nfrom the Bohr compactification of G). Givenf:G→R, if for allǫ >0 there are\nδ,nsuch thatfisǫ-constant on all translates of a ( δ,U(n))-Bohr neighborhood\ninG, thenfis uniformly continuous with respect to the Bohr topology on G. So\nTheorem 1.2 says that stable functions exhibit an approximate form of uniform\nBohr-continuity. Thus the role of uniformly Bohr-continuous func tions onGin\narithmetic regularity for stable functions is roughly analogous to th e role of unions\nof cosets of subgroups of Gin arithmetic regularity for stable sets.\nRecall that if Gis amenable, then any left-invariant measure µonGuniquely\ndetermines a positive linear functional/integraltext\nfdµon bounded real-valued functions f\nonG, which satisfies µ(A) =/integraltext\n1Adµfor allA⊆G.\n2Thanks to Ward Henson for communicating the following summa ry.6 G. CONANT AND A. PILLAY\nDefinition2.10. LetGbeanamenablegroupwithleft-invariantmeasure µ. Given\nbounded functions f,g:G→R, theconvolution of fandgis the function\nf∗g:G→Rsuch that for x∈G,\n(f∗g)(x) =/integraldisplay\nf(t)g(t-1x)dµ(t).\nThenextcorollaryisessentiallythesameas[9, Corollary1.4( ii)], butinaslightly\ndifferent setting.\nCorollary 2.11. Letk:R+→Z+be as in Fact 2.6. Suppose Gis an amenable\ngroup andf,g:G→[-1,1]are arbitrary functions. Then f∗gisk-stable.\nProof.Letµbe the implied measure on Gfrom the statement of the corollary,\nwhich we view as a regular Borel probability measure on βG. Consider the Hilbert\nspaceH=L2(βG,µ) with inner product /⊣\\}br⊣ck⌉tl⌉{tf,g/⊣\\}br⊣ck⌉tri}ht=/integraltext\nfgdµ, and letBdenote the\nunit ball. Note that any function f:G→[-1,1] extends uniquely to a continuous\nfunction on βGinB. Givenf,g:G→[-1,1], definef1,g2:G→Bsuch that\nf1(x) =f(xt) andg2(y) =g(t-1y). Then (f∗g)(xy) =/⊣\\}br⊣ck⌉tl⌉{tf1(x),g2(y)/⊣\\}br⊣ck⌉tri}ht. Sof∗gis\nk-stable (as a function on the group G) by Fact 2.6. /square\nUsing Corollary 2.11, we can give a counterexample to the naive stren gthening\nof Theorem 1.2 obtained by replacing Bohr neighborhoods with subgr oups.\nExample 2.12. LetG=Z/pZfor some prime p >2, and letA={0,1,...,p\n2}.\nDefinef:G→[0,1] such that f(x) =1\np|A∩(x+A)|. Thenf=1A∗1-A, and\nsofisk-stable where kis as in Fact 2.6. However, Ghas no proper nontrivial\nsubgroups and, as pincreases, the image of fcontains values arbitrarily close to 0\nand to1\n2. So for sufficiently small ǫandζ,fis notζ-almostǫ-constant on G(which\nis the only subgroup of index independent of p).\nOn the other hand, if one restricts to groups of a fixed finite expon ent, then such\na strengthening of Theorem 1.2 does follow; see Corollary 4.1.\n2.4.Topological dynamics. LetGbe a group. A G-flowis a compact Hausdorff\nspaceStogetherwithaleftactionof Gbyhomeomorphisms. For g∈G,let ˘gdenote\nthe corresponding homeomorphism of S(following the notation from Glasner’s\nsurvey [19]). As shown by Ellis [17], the closure of {˘g:g∈G}in the space of\nfunctions from StoSis a semigroup under composition, called the Ellis semigroup\nE(S) ofS.3Note thatE(S) is itself a G-flow under the natural action of G. One\ncan also check that for a fixed σ∈E(S), the right composition map τ/m⊣psto→τ◦σ\nis continuous; so E(S) is aright topological semigroup . The following fact is a\nstandard exercise.\nFact 2.13. LetSbeG-flow. Then for any σ∈E(S), the left composition map\nLσ:τ/m⊣psto→σ◦τis inE(E(S)). Moreover, σ/m⊣psto→Lσis an isomorphism (of G-flows\nand right topological semigroups) from E(S)toE(E(S)).\nThe next result is the key ingredient from topological dynamics need ed for our\nproof of Theorem 1.2.\n3We caution the reader that [17] works in the context of right a ctions (as does [18] cited below),\nwhereas we are following the model-theoretic tradition of l eft actions (as does [19]).AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 7\nTheorem 2.14 (Ellis & Nerurkar [18]) .LetSbe aG-flow with Ellis semigroup E,\nand assume that every function in Eis continuous. Then:\n(a)Ehas a unique minimal subflow K.\n(b)Kis the unique left ideal in (E,◦).\n(c) (K,◦)is a compact Hausdorff group.\n(d)The identity in Kcommutes with every element of E.\n(e)Eisuniquely ergodic , i.e., there is a unique G-invariant regular Borel prob-\nability measure on E.\nAG-flow satisfying the assumptions of the previous theorem is called weakly\nalmost periodic (although this is not the official definition; see [18] for further\ndetails). It is worth noting that the proof of Theorem 2.14 relies on t he same result\nof Grothendieck [21] mentioned above in the context of Fact 2.6. As shown by Ben\nYaacov [4], this result can also be used to give a quick proof of Theore m 2.18 below,\nwhich is the main model-theoretic ingredient in our proof of Theorem 1 .2.\n2.5.Continuous logic and stability. We will workin the standardframeworkof\ncontinuous logic as in the monograph [5]. The reader is also referred to [9, Section\n2] for further details of certain aspects not directly discussed in [5 ]. Throughout\nthis subsection,Ldenotes a first-order language in continuous logic.\nDefinition 2.15. LetMbe anL-structure and let Φ be a collection of partitioned\nL-formulas of the form ϕ(x,y) wherexandyare fixed tuples of variables. Then a\nΦ-formula over Mis a continuous function from SΦ(M) toR, whereSΦ(M) is\nthe compact Hausdorff space of complete Φ-types over M.\nSee[9, Section2]and[7,Section6]forfurtherdetailsinthecasewh enΦisasingle\nformulaϕ(x,y) (the situation for arbitrary Φ generalizes directly). Recall that in\ngeneral, a Φ-formula is built from formulas in Φ using an infinitary continuous\nconnective (see [7, Fact 6.1]). For this reason, what we call Φ-form ula would be\nreferred to as a “Φ-predicate” according to [7], where “formula” is reserved for the\ncase of finitary connectives. Note that this discrepancy does not arise in classical\nlogic, where a Φ-formula (as we have defined it) is always given by a finit e Boolean\ncombination of instances from Φ.\nWe will assume familiarity with Keisler measures in continuous logic, viewe d\neither as linear functionals on formulas or as regular Borel probabilit y measures\non type spaces. See [9, Section 2.3]. However, we will diverge from [9] in that\nwe will use the same notation for functionals and measures. To clarif y, letMbe\nanL-structure, and suppose µis a Keisler measure over Min variables x, i.e., a\nregular Borel probability measure on Sx(M). Given anL-formulaθ(x) overM, we\nletµ(θ(x)) denote/integraltext\nSx(M)θ(x)dµ. Given anL-formulaθ(x) and a Borel set B⊆R,\nwe writeµ(θ(x)∈B) forµ(θ-1(B)). We will also use this notation in the local\nsituation of KeislerΦ-measures over M(again, see [9, Section 2.3]).\nWe now recall the definition of stability for formulas in continuous logic .4\nDefinition 2.16. LetTbe a completeL-theory and fix an L-formulaϕ(x,y).\n(a) Givenk≥1 andǫ >0, we sayϕ(x,y) is (k,ǫ)-stable (in T)if for every\nM|=T, the function ϕ:Mx×My→Ris (k,ǫ)-stable.\n4This definition is formulated specifically for our purposes, but one can easily check it is\nequivalent to the standard definition from other sources (e. g., [7]).8 G. CONANT AND A. PILLAY\n(b) We say that ϕ(x,y) isstable (in T)there is some k:R+→Z+such that for\nallǫ >0,ϕ(x,y) is (k(ǫ),ǫ)-stable (in T). In this case, we also say ϕ(x,y) is\nk-stable (in T).\nProposition 2.17. Let(Mi)i∈Ibe a family ofL-structures and let Ube an ultra-\nfilter onI. Fix anL-formulaϕ(x,y)and a function k:R+→Z+, and assume that\nϕ(x,y)isk-stable in Th(Mi)for alli∈I. Then there is some k′:R+→Z+such\nthatϕ(x,y)isk′-stable in Th(/producttext\nUMi).\nProof.More precisely, it is not hard to check that for any ǫ >0 and anyǫ′< ǫ,\nϕ(x,y) is (k(ǫ′),ǫ)-stable in Th(/producttext\nUMi). See the proofof[9, Lemma5.5]for details.\nSo, e.g., one can take k′(ǫ) =k(ǫ/2). /square\nThe next theorem is an important result from the foundations of st ability theory\n(see [37, ChapterII] and [33, Chapter 1]). The generalizationto c ontinuoustheories\nwasdonebyBenYaacovandUsvyatsov(seePropositions7.6and7.1 6in[7]). Given\nanL-formulaϕ(x,y), we letϕ∗(y,x) denote the same formula, but with the roles\nof object and parameter variables exchanged.\nTheorem 2.18. LetTbe a completeL-theory and suppose ϕ(x,y)is anL-formula\nthat is stable in T.\n(a)For anyM|=T, ifp∈Sϕ(M)then the map b/m⊣psto→ϕ(p,b)fromMytoRis given\nby aϕ∗-formula over M, which we denote by dϕ\np(y).\n(b)For anyM|=T, ifp∈Sϕ(M)andq∈Sϕ∗(M), thendϕ\np(q) =dϕ∗\nq(p).\nWe will frequently use the notation dϕ\np(y) in situations where pis a type in a\nlarger fragment of formulas that includes all ϕ-formulas (over some fixed model).\nIn such cases, it is understood that dϕ\np(y) meansdϕ\np|ϕ(y).\n3.Main model theoretic result\nThroughout this section, we let Tbe a continuousL-theory with a sort Gex-\npanding a group. We also fix a model M|=Tand identify GwithG(M). Letxbe\nof sortG, and letϕ(x,z) be anL-formula that is left-invariant , i.e., for any b∈Mz\nandg∈Gthere isc∈Mzsuch thatϕ(gx,b) =ϕ(x,c). From this assumption it\nfollows that the type space Sϕ(M) is aG-flow under the natural action.\nFor the entirety of this section, we assume ϕ(x,z)is stable in T.Our\nmain goal is Theorem 3.9 below, which provides an arithmetic regularity statement\nforϕ-formulas in terms of a canonical definable compactification of G. The rough\nstrategy will follow the methods in [11] using topological dynamics. I n particular,\nwe will use Theorem 2.18 to give a concrete description of the Ellis semig roup of\nSϕ(M), and then apply the results of Ellis and Nerurkar in Theorem 2.14.\nGivenb∈Mz, theformula ϕ(y·x,b) isitselfstablebyleft-invarianceandstability\nofϕ(x,z). So by Theorem 2.18( a), for every b∈Mzandp∈Sϕ(M), we have a\ndefining formula db\np(y):=dϕ(y·x,b)\np(y).\nNow letϕ♯(x;y,z) denote the bi-invariant formula ϕ(x·y,z).We emphasize\nthat this formula need not be stable (see [14, Example 3.7] and [11, Example 5.11]).\nNevertheless, given b∈Mzandp∈Sϕ♯(M), we have a ϕ♯-formuladb\np(y) as above.\nNote that for any b∈Mzand anypinSϕ(M) orSϕ♯(M), the formula db\np(y) is a\nϕ♯-formula over Mby Theorem 2.18( a).\nFor the rest of this section, let S=Sϕ(M) andS♯=Sϕ♯(M).AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 9\nLemma 3.1. S♯is isomorphic as a G-flow to the Ellis semigroup of S. Moreover:\n(i)The induced semigroup operation ∗onS♯is given by ϕ((p∗q)g,b) =db\nqg(p)\nwherep,q∈S♯,g∈G, andb∈Mz.\n(ii)For anyp∈S♯, the mapq/m⊣psto→p∗qfromS♯toS♯is continuous.\nProof.LetXbe the space of real-valued functions on Mz. Note that Scan be\nidentified as a closed subset of X. SoSSis closed in XS, andE(S) is the closure\nof{˘g:g∈G}inXS(recall ˘gdenotes the g-action map on S). Define a function\nΣ:p/m⊣psto→σpfromS♯toXSso that, given q∈Sandb∈Mz,σp(q)[b] =db\nq(p).\nWe first show Σ is continuous. By definition of the topology on XS, it suffices\nto consider an open set W⊆XSof the form{σ∈XS:σ(q)[b]∈I}for some fixed\nq∈S,b∈Mz, and open interval I⊆R. Forp∈S♯, we haveσp∈Wif and only\nifdb\nq(p)∈I, and hence Σ-1(W) is the open set in S♯determined by db\nq(x)∈I.\nNow note that Σ(tpϕ♯(g/M)) = ˘gfor anyg∈G. Since the realized types\nare dense in S♯, and Σ is continuous from the compact space S♯to the Hausdorff\nspaceXS, it follows that Σ( S♯) is the closure of {˘g:g∈G}, i.e., Σ(S♯) =E(S).\nIn particular, if p∈S♯andq∈Sthenσp(q) is the unique type in Ssatisfying\nϕ(σp(q),b) =db\nq(p) for allb∈Mz.\nNext we show Σ is injective, and thus a homeomorphism from S♯toE(S) (since\nboth spaces are compact Hausdorff). Fix p∈S♯. Then for any g∈Gandb∈Mz,\nifq= tpϕ(g/M) thenϕ(pg,b) =db\nq(p) =ϕ(σp(q),b) (the first equality is clear when\npis a realized type in S♯, and thus holds for all pby continuity of db\nq(y)). Hencep\nis uniquely determined by σp.\nFinally, we showthat Σ preservesthe action of G. Fixg∈G,q∈S, andb∈Mz.\nWe need to show ϕ(σgp(q),b) =ϕ(gσp(q),b) for anyp∈S♯. Fixc∈Mzsuch that\nϕ(gx,b) =ϕ(x,c). Thendb\nq(gy) =dc\nq(y), and so for any p∈S♯, we have\nϕ(σgp(q),b) =db\nq(gp) =dc\nq(p) =ϕ(σp(q),c) =ϕ(gσp(q),b).\nWe have now shown that Σ is the desired G-flow isomorphism from S♯toE(S).\nIt remains to verify ( i) and (ii). Statement ( i) is a straightforward calculation. For\n(ii), fixp∈S♯and letLpdenoteq/m⊣psto→p∗q. Fixg∈G,b∈Mz, and an open\nintervalI⊆R, and consider the sub-basic open set W={q∈S♯:ϕ(qg,b)∈I}.\nLetψ(x,y) =ϕ(y·x,b), and recall that ψ(x,y) is stable. Applying Theorem 2.18( b)\nand (i), we have that for q∈S♯,\nLp(q)∈W⇔ϕ((p∗q)g,b)∈I⇔dψ\nqg(p)∈I⇔dψ∗\np(qg)∈I.\nSoL-1\np(W) is the open set in S♯determined by dψ∗\np(xg)∈I. /square\nCorollary 3.2.\n(a)S♯has a unique minimal subflow K.\n(b)Kis the unique left ideal in (S♯,∗)\n(c) (K,∗)is a compact Hausdorff group.\n(d)The identity in Kcommutes with every element of S♯.\n(e)There is a unique left-invariant ϕ♯-Keisler measure on G.\nProof.By Lemma 3.1 and Fact 2.13, ( S♯,∗) is isomorphic to ( E(S♯),◦) via the map\ntakingpinS♯toq/m⊣psto→p∗qinE(S♯). So every function in E(S♯) is continuous by\nLemma 3.1( ii). The claims now follow from Theorem 2.14. /square10 G. CONANT AND A. PILLAY\nRemark 3.3. AssuggestedbyCorollary3.2anditsproof, theresultsofthissect ion\nwill really only rely on the fact that S♯is isomorphic to its own Ellis semigroup.\nThus Lemma 3.1 could be replaced by this statement (whose proof wo uld be very\nsimilar). Onthe otherhand, the preciseidentificationof S♯asthe Ellissemigroupof\nSdemonstrates the importance of the formula ϕ♯(x;y,z) in the analysis of ϕ(x,z).\nMoreover, Lemma 3.1 will be the starting point for the more thoroug h treatment\nof local stability for ϕ(x,z) given in Section 5.\nRemark 3.4. The previous arguments establish weak almost periodicity of S♯via\nwork of Ellis and Nerurkar [18]. However, as previously mentioned, th e general\nconnection between stability and weak almost periodicity is well known in the\nmodel theoryliterature (e.g., [4, 6, 24]) and goes back to Grothen dieck [21]. Indeed,\none can directly verify the definition of weak almost periodicity for S♯using [21,\nTh´ eor` eme 6] and the small exercise that θ(x·y) is stable for any ϕ♯-formulaθ(x).\nBy going this route, one obtains a proof of part ( ii) of Lemma 3.1 using [18] and\n[21] instead of Harrington’s Lemma (Theorem 2.18( b)).\nNote that the same methods establish weak almost periodicity of S. Indeed, one\ncaneithercheckthis directly usingthe definition and Grothendieck, oruseTheorem\n2.18(b) in the samewaytoprovecontinuity ofthe maps σpfromthe proofofLemma\n3.1. More generally, if f:S1→S2is a surjective homomorphism of G-flows, and\nS1is weakly almost periodic, then so is S2. In our case, the restriction map from\nS♯toSis such a homomorphism. (In fact, any G-flow with a dense orbit admits a\nsurjective homomorphism from its Ellis semigroup.)\nFor the rest of this section, we let Kdenote the unique minimal subflow of S♯.\nDefinition 3.5. Letube the identity in K, and define π:G→Kbyπ(g) =gu.\nProposition 3.6. The mapπ:G→Kis aϕ♯-definable compactification of G.\nMoreover,p/m⊣psto→p∗uis the unique continuous extension of πtoS♯, and this map is\na semigroup homomorphism.\nProof.By Corollary 3.2( b),p/m⊣psto→p∗uis a well-defined map from S♯toK, which\nclearlyextends π. This mapisalsocontinuous(just because S♯isarighttopological\nsemigroup). So πisϕ♯-definable. Note that π(G) is the orbit of u, which is dense\ninKby Corollary3.2( a). So to finish the proof, we just need to show that p/m⊣psto→p∗u\nis a semigroup homomorphism. For this, fix p,q∈S♯. Then\np∗q∗u=p∗q∗u∗u=p∗u∗q∗u\nwhere the second equality uses Corollary 3.2( d). /square\nIn light of the previous corollary, we will use πto denote the map p/m⊣psto→p∗ufrom\nS♯toK. We also let µdenote the unique left-invariant Keisler ϕ♯-measure on G\n(which exists by Corollary 3.2( e)).\nProposition 3.7. Letηbe the Haar functional on K. Thenµ(θ(x)) =η(θ|K)for\nanyϕ♯-formulaθ(x). In particular, µ(K) = 1.\nProof.The second claim follows from the first by regularity of µand Urysohn’s\nLemma. For the first claim, it suffices by uniqueness of µto show that the map\nθ(x)/m⊣psto→η(θ|K) is a left-invariant Keisler ϕ♯-measure on G. It is clear that this map\nis a linear functional. Left-invariance follows from left-invariance of η, and since\n(gθ)|K=gu∗θ|Kfor anyϕ♯-formulaθ(x) and anyg∈G. /squareAN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 11\nThe next definition is a variation of “almost constant” (defined befo re Theorem\n1.1), which will be more manageable in later arguments with ultraprodu cts.\nDefinition 3.8. Letθ(x) be aϕ♯-formula. Fix a Borel set B⊆Sϕ♯(M) and some\nr∈R. Thenθ(x) isµ-nearlyǫ-close toronBifµ({p∈B:θ(p)/\\⌉}⊣tio\\sl⊣sh≈ǫr}) = 0.\nWe can now prove the main result of this section.\nTheorem 3.9. Letθ(x)be aϕ♯-formula and fix some ǫ>0. Then there is an open\nidentity neighborhood U⊆Ksuch that for any g∈G,θ(x)isµ-nearlyǫ-close to\nθ(gu)ongπ-1(U).\nProof.Recall that θ(x) is a continuous function on Sϕ♯(M) and hence restricts to\na continuous function the compact Hausdorff group K. So by Proposition 2.8 there\nis an open identity neighborhood U⊆Ksuch thatθ(x) isǫ-constant on all left\ntranslates of UinK. Now fixg∈G. We show that θ(x) isµ-nearlyǫ-close toθ(gu)\nongπ-1(U). By Proposition 3.7, it suffices to fix p∈gπ-1(U) withθ(p)/\\⌉}⊣tio\\sl⊣sh≈ǫθ(gu),\nand showp/\\⌉}⊣tio\\sl⊣sh∈K. So fix such p. Then, using Proposition 3.6, we have\np∗u=π(p) =π(g)∗π(g-1p)∈π(g)∗Uandgu=π(g)∗u∈π(g)∗U.\nThereforeθ(p∗u)≈ǫθ(gu) by choice of U, and soθ(p∗u)/\\⌉}⊣tio\\sl⊣sh=θ(p) by choice of p.\nIn particular, p∗u/\\⌉}⊣tio\\sl⊣sh=p, and hence p/\\⌉}⊣tio\\sl⊣sh∈Ksinceuis the identity in K. /square\nNext we use the Peter-Weyl Theorem to specialize the previous res ult to unitary\ngroups. For some ad hoc notation, let Bn\nδdenote the open identity neighborhood\nof radiusδin U(n).\nCorollary 3.10. Letθ(x)be aϕ♯-formula and fix some ǫ >0. Then there are\nn≥0,δ >0, and aϕ♯-definable homomorphism τ:G→U(n)such that for any\ng∈G,θ(x)isµ-nearlyǫ-close toθ(gu)ongτ-1(Bn\nδ).5\nProof.LetU⊆Kbe an open identity neighborhood in Kas in Theorem 3.9.\nRecall that Kis a compact Hausdorff group. By the Peter-Weyl Theorem, Kis an\ninverse limit lim←−IKiof compact Lie groups (see [22, Corollary 2.43]). For i∈I, let\nρi:K→Kibe the projection map. Then there is some i∈Iand an open identity\nneighborhood V⊆Kisuch thatρ-1\ni(V)⊆U(see [34, Exercise 1.1.15]). By Peter-\nWeyl again, we may fix some n≥0 and a topological embedding ι:Ki→U(n) of\nKias a closed subgroup of U( n) (see [27, Theorem 6.1.2]). Pick δ >0 such that\nι-1(Bn\nδ)⊆V. Thenτ:=ιρiπ:G→U(n) is aϕ♯-definable homomorphism and\nτ-1(Bn\nδ)⊆π-1(U). The conclusion now follows from the choice of U. /square\nRemark 3.11. In analogy to [11], all of the material in this section can be gen-\neralized to the following abstract setting. Let Gbe a group (without any explicit\nmodel-theoretic context). Suppose Fis a collection of real-valued bounded stable\nfunctions on G, which is closed under left-translation. Then we have the associate d\n“type space” S(F) (i.e., the space of functions p:F→Rthat are finitely approx-\nimated inG) and notion of an “ F-formula” (i.e., a continuous function on S(F)).\nMoreover,S(F) is aG-flow by left-invariance of F. The same arguments show that\nS(F) is weakly almost periodic with Ellis semigroup S(F♯), whereF♯is the set of\nfunctions of the form f(xg) forf(x)∈Fandg∈G. Moreover, Theorem 3.9 holds\nfor anyF♯-formula. In fact, since Theorem 2.18 holds under the weaker notio n of\n5As with the map πin Theorem 3.9, here we identify τwith its unique continuous extension\ntoSϕ♯(M) so that τ-1(Bn\nδ) make sense in the context of Definition 3.8.12 G. CONANT AND A. PILLAY\n“stability in a model” (see, e.g., [4] or [7, Appendix B]), it suffices to just assume\nthat eachf∈Fisstable inG, i.e., for all ǫ>0 there do not exist infinite sequences\n(ai)i<ωand (bi)i<ωinGsuch that|f(aibj)−f(ajbi)|≥ǫfor alli0, and aϕ♯-definable homomorphism τ:G→U(n)\nsuch that for any g∈G, there is some h∈Gsuch thatf(x) isµϕ-nearlyǫ\n4-close\ntof(h) ongτ-1(Bδ\nn).6\nProof.Corollary 3.10 yields this statement but with haϕ♯-type (namely gu). We\ncan fix this by applying Corollary 3.10 withǫ\n8, and then for each g∈Gfinding\nh∈Gwithf(h)≈ǫ/8f(gu). ⊣claim\nFollowing [13, Section 4.3], we view ( G,U(n),τ) as a structure in a two-sorted\nlanguageLwith the original L0-structure on G(i.e.,Lis the language denoted\nL0\nU(n),τin [13]). By [13, Corollary 4.12], for each s≥1, there is a homomorphism\nτs:Gs→U(n) such that theL-structure ( G,U(n),τ) is (canonically isomorphic\n6Recall that Bδ\nndenotes the open identity neighborhood of radius δin T(n).AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 13\nto)/producttext\nU(Gs,U(n),τs). Viewing each µsas a positive linear functional on Gs, define\nthe ultralimit functional µ= limUµs(see [9, Proposition 2.15]). Then µis a left-\ninvariant Keisler measure on the L-structure ( G,U(n),τ) in the sort for G.\nRecall the continuous connective r.−s= max{r−s,0}. Define theL-formula\nψ(x,y,z) = min{δ.−d(τ(x),τ(y)),|f(x)−f(z)|.−ǫ\n4}.\nClaim 2. For anyg∈Gthere is some h∈Gsuch thatµ(ψ(x,g,h)>0) = 0.\nProof.By Claim 1, it suffices to fix g,h∈Gand show\nµ(ψ(x,g,h)>0) =µϕ({p∈gτ-1(Bδ\nn) :|f(p)−f(h)|>ǫ\n4}).\nLet Φ consist of the L-formulasϕ♯(x;y,z) andχ(x;y) =d(τ(x),τ(y)). So any\ninstance of ψ(x;y,z) is a Φ-formula. Since τisϕ♯-definable, every type in Sϕ♯(G)\nhas a unique extension to a type in SΦ(G). So the restriction of µto Φ-formulas\ncan be identified with µϕ, which gives us the desired result. ⊣claim\nIn order to transfer Claim 2 through the ultraproduct, we will need to further\nexpand the language as in [9, Section 5]. Define L+to consist ofLtogether with,\nfor eachη>0, a new [0,1]-valued predicate symbol Qη(y,z) with a trivial modulus\nofuniform continuity. Expand( Gs,U(n),τs) to anL+-structureG+\nsbyinterpreting\nQηas the function ( g,h)/m⊣psto→µs(ψ(x,g,h)≥η). LetG+=/producttext\nUG+\ns, and note that\nthe reduct of G+toLis (G,U(n),τ). To ease notation, we use QηandQs\nηfor the\ninterpretations of QηinG+andG+\ns, respectively.\nClaim 3. For anyη>0 andg,h∈G,Qη(g,h)≤µ(ψ(x,g,h)≥η).\nProof.See [9, Lemma 5.4], which is written for more general families of formula s,\nbut in the contextof ultraproductsoffinite structureswith the n ormalizedcounting\nmeasure. The proof works verbatim in our setting, and involves exp anding by addi-\ntionalpredicate symbols(see the start of[9, Section 5.1]; in our se tting the family of\nL-formulasFcan be taken as the closure of {ψ(x,y,z)}under the connectives αD\nmentioned there). So here we are again using the fact that ultrapr oducts commute\nwith reducts (or one can just work in a larger language). ⊣claim\nNow letcbe the absolute constant from Proposition 2.2( a), and define\nη= min/braceleftbigδ\n2,ǫ\n4,ζ(δ\n2,n)/bracerightbig\n.\nCombining Claims 2 and 3, we have supxinfyQη(x,y) = 0. Let I⊆Z+be the\nset ofs≥1 such that supxinfyQs\nη(x,y)< η. ThenI∈Uand so, sinceUis\nnonprincipal, we may fix some s∈Isuch thatn,(δ/2)-1≤s. LetB=τ-1\ns(Bδ/2\nn),\nand note that Bis a (δ\n2,U(n))-Bohr neighborhood in Gs. To finish the proof, we\nwill show that fsisζ(δ\n2,n)-almostǫ-constant on all translates of B, contradicting\nthe initial choice of Gsandfs.\nFixg∈Gs. We need to show that fsisζ(δ\n2,n)-almostǫ-constant on gB. Since\nsupxinfyQs\nη(x,y)<η, there is some h∈Gssuch thatµs(ψ(x,g,h)≥η)<η. Let\nZdenote the set in Gsdefined byψ(x,g,h)≥η. Soµs(Z)<η≤ζ(δ\n2,n). Thus it\nsuffices to show that fsisǫ-constant on gB\\Z.\nRecall from the definition of ψ(x,y,z) that\nZ={x∈Gs:d(τs(x),τs(g))≤δ−ηand|fs(x)−fs(h)|≥ǫ\n4+η}.\nNote also that if x∈gBthend(τs(x),τs(g))<δ\n2≤δ−η, and hence if x∈gB\\Z\nthen we must have |fs(x)−fs(h)|<ǫ\n4+η≤ǫ\n2. Sofsisǫ-constant on gB\\Zby\nthe triangle inequality, as desired. /square14 G. CONANT AND A. PILLAY\n4.2.Corollaries. Inthis subsection, weshowthat whenrestrictingTheorem1.2to\ncertain subclasses of amenable groups, one can replace unitary Bo hr neighborhoods\nby “nicer” objects. These subclasses will correspond to parts ( b), (c), and (d) of\nProposition 2.2. We start with amenable groups of bounded exponen t.\nCorollary 4.1. Fixr≥1,ǫ >0, and functions k:R+→Z+andζ:N→R+.\nSupposeGis an amenable group of exponent r, with left-invariant measure µ, and\nf:G→[-1,1]isk-stable. Then there is a normal subgroup H≤Gof index\nm≤Ok,ζ,ǫ,r(1)such thatfisζ(m)-almostǫ-constant on all cosets of H.\nProof.Letcbe the absolute constant from Proposition 2.2( a), and letγr>0 be\ntheOr(1) parameter from Proposition 2.2( b). Defineζ∗:R+×N→R+so that\nζ∗(δ,n) = min{ζ(m) :m≤(c/min{δ,γr})n2}.\nNow fixG,µ, andfas in the statement of the corollary. Apply Theorem 1.2 with\nparameters k,ζ∗, andǫto obtain a ( δ,U(n))-Bohr neighborhood BinG, with\nδ-1,n≤Ok,ζ,ǫ,r(1), such that fisζ∗(δ,n)-almostǫ-constant on all translates of\nB. Letδ∗= min{δ,γr}and note that we still have δ-1\n∗≤Ok,ζ,ǫ,r(1). LetHbe\nthe (δ∗,U(n))-Bohr neighborhood in Gdefined using the same homomorphism to\nU(n) that yields B. ThenHis a normal subgroup of Gby Proposition 2.2( b), and\nH⊆B. Moreover, Hhas indexm≤(c/δ∗)n2inGby Proposition 2.2( a). So\nζ∗(δ,n)≤ζ(m).\nNow fixg∈G. Then there is a set Z⊆gB, withµ(Z)<ζ∗(δ,n) such that fis\nǫ-constant on gB\\Z. SetW=Z∩gH. ThengH\\W⊆gB\\Z, sofisǫ-constant on\ngH\\W. Moreover, µ(W)≤µ(Z)< ζ∗(δ,n)≤ζ(m). Therefore fisζ(m)-almost\nǫ-constant on all cosets of H. /square\nNext, recallthat any abelian groupis amenable (see, e.g., [32, Propo sition0.15]).\nSo Theorem 1.2 and Proposition 2.2( c) yield the following conclusion.\nCorollary 4.2. Fixǫ >0and functions k:R+→Z+andζ:R+×N→R+.\nSupposeGis an abelian group, with left-invariant measure µ, andf:G→[-1,1]\nisk-stable. Then there is a (δ,T(n))-Bohr neighborhood BinG, withδ-1,n≤\nOk,ζ,ǫ(1), such that fisζ(δ,n)-almostǫ-constant on all translates of B.\nFinally, we consider amenable torsion groups.\nCorollary 4.3. Fixǫ >0and functions k:R+→Z+andζ:R+×N2→R+.\nSupposeGis an amenable torsion group, with left-invariant measure µ, andf:G→\n[-1,1]isk-stable. Then there is a normal subgroup H≤G, with index m≤\nOk,ζ,ǫ(1), and a(δ,T(n))-Bohr neighborhood BinH, withδ-1,n≤Ok,ζ,ǫ(1), such\nthatfisζ(δ,n,m)-almostǫ-constant on all translates of B.\nProof.Letcbe the absolute constant from Proposition 2.2( a), and letm:N→N\nbe the function given by On(1) in Proposition 2.2( d). Defineζ∗:R+×N→R+so\nthatζ∗(δ,n) =ζ(δ,n,m(n)).\nNow fixG,µ, andfas in the statement of the corollary. Apply Theorem 1.2\nwith parameters k,ζ∗, andǫto obtaina ( δ,U(n))-Bohrneighborhood B0inG, with\nδ-1,n≤Ok,ζ,ǫ(1), such that fisζ∗(δ,n)-almostǫ-constant on all translates of B0.\nBy Proposition 2.2( d), there a normal subgroup H≤Gof indexm(n)≤Ok,ζ,ǫ(1)\nsuch thatB:=B0∩His a (δ,T(n))-Bohr neighborhood in H. Arguing as in the\nend of the proof of Corollary 4.1, it follows that fisζ(δ,n,m(n))-almostǫ-constant\non all translates of B. /squareAN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 15\nNote that Theorem 1.1 is a special case of Corollary 4.3.\n4.3.Bogolyubov’s Lemma. In this subsection we use Theorem 1.2 to deduce\nBogolyubov’s Lemma for amenable groups, which says the following.\nTheorem 4.4. LetGbe an amenable group with left-invariant measure µ, and fix\nα >0. Suppose A⊆Gis such that µ(A)≥α. Then there is a (δ,U(n))-Bohr\nneighborhood B⊆G, withδ-1,n≤Oα(1), such that B⊆(AA-1)2.\nThis result was first proved for finite abelian groups by Ruzsa [35] wit h explicit\nbounds. A qualitative analogue for arbitrary finite groups was given by the first\nauthor in [10]. This was further generalized to arbitrary amenable gr oups by the\nauthorsandHrushovskiin[13](wherewein factproveastronger result; seeRemark\n4.6). A non-uniform version of Theorem 4.4 for a single countable ame nable group\nfollows from results of Beiglb¨ ock, Bergelson, and Fish [2].\nThe proof in [10] for finite groups was based on work of Sanders [36 ], which was\nalso a key ingredient in results of Breuillard, Green, and Tao [8] on app roximate\ngroups. The proof in [13] for amenable groups used a variation of Hr ushovski’s\nstabilizer theorem [23] due to Montenegro, Onshuus, and Simon [30]. As is now\nwell understood, certain aspects of the stabilizer theorem can be accounted for by\nstability of Hilbert spaces in continuouslogic (Fact 2.6). This provides a connection\nto the proof here, which will use Fact 2.6 (via Corollary 2.11) in order t o leverage\nTheorem 1.2. The only other tools needed for the proof are Propos ition 2.2(a) and\na variation of Ruzsa’s Covering Lemma, which we now describe.\nFor the rest of this section, let Gbe an amenable group with left-invariant\nmeasureµ. Ruzsa’s Covering Lemma states that if A⊆Gandµ(A)>0, thenG\ncan be covered by at most µ(A)-1left translates of AA-1. The proof is elementary\nand relies on the observation that AA-1is precisely the set of x∈Gsuch that\nA∩xAis nonempty. By taking into account the measure of A∩xA, we obtain the\nfollowing “density version” of Ruzsa’s result.\nLemma 4.5. SupposeA⊆Gis such that µ(A)>0. Define\nX={x∈G:µ(A∩xA)>1\n2µ(A)2}.\nThenGcan be covered by at most 2µ(A)-1left translates of X.\nProof.Setα=µ(A). Call a set F⊆Gseparated ifµ(gA∩hA)≤1\n2α2for all\ndistinctg,h∈F. We first show that any separated set Fhas size at most 2 α-1.\nFor a contradiction, suppose F⊆Gis separated with |F|>2α-1. Then we may\npick pairwise distinct g1,...,gn∈F, withn=⌈2\nα⌉. So2\nα≤n<2+α\nα. Moreover,\n1≥µ/parenleftBiggn/uniondisplay\ni=1giA/parenrightBigg\n≥n/summationdisplay\ni=1µ(giA)−/summationdisplay\n1≤i1\n2α2, i.e.,\nµ(h-1gA∩A)>1\n2α2, i.e.,h-1g∈X, i.e.,g∈hX. /square16 G. CONANT AND A. PILLAY\nProof of Theorem 4.4 .Setǫ=1\n2α2. Letcbe the absolute constant from Propo-\nsition 2.2(a), and define ζ:R+×Z+→R+so that\nζ(δ,n) = min{α\n2(δ\nc)n2,1\n2(δ\n2c)n2}.\nNow define f:G→[0,1] such that f(x) =µ(A∩xA). Thenf=1A∗1A-1, so\nwe can apply Corollary 1.3 with parameters ǫandζ. This yields a ( δ,U(n))-Bohr\nneighborhood CinG, withδ-1,n≤Oα(1), such that fisζ(δ,n)-almostǫ-constant\non all translates of C.\nClaim.There is some g∈Gsuch thatµ(gC\\AA-1)<ζ(δ,n).\nProof.By Proposition 2.2( a), there is a set F⊆Gof size at most ( c/δ)n2such\nthatG=FC. For each g∈F, fixZg⊆gCsuch thatµ(Zg)< ζ(δ,n) andfis\nǫ-constant on gC\\Zg. LetZ=/uniontext\ng∈FZg. Thenµ(Z)<|F|ζ(δ,n)≤α\n2, henceG\ncannot be covered by at most 2 α-1left translates of Z. By Lemma 4.5, we may\nfix somex∈G\\Zwithf(x)> ǫ. Chooseg∈Fsuch thatx∈gC. Thenx/\\⌉}⊣tio\\sl⊣sh∈Zg\nsincex/\\⌉}⊣tio\\sl⊣sh∈Z. So given any y∈gC\\Zg, we havef(x)≈ǫf(y), and hence f(y)>0\nsincef(x)>ǫ. In particular, A∩yAis nonempty for all y∈gC\\Zg, which implies\ngC\\Zg⊆AA-1, i.e.,gC\\AA-1⊆Zg. This yields the claim by choice of Zg.⊣claim\nLetBbe the (δ\n2,U(n))-Bohr neighborhood in Gdefined from the same homo-\nmorphism to U( n) that yields C. ThenB⊆B2⊆C, andµ(gC\\AA-1)<1\n2µ(B)\nby the claim, Proposition 2.2( a), and the choice of ζ. We show B⊆(AA-1)2.\nTo ease notation, set U=g-1AA-1and letV=G\\(C\\U). ThenV∩C⊆Uand\nµ(V)>1−1\n2µ(B). Now fixx∈B. Then\nµ(V∩xV) = 2µ(V)−µ(V∪xV)≥2µ(V)−1>1−µ(B) = 1−µ(xB),\nwhich implies V∩xV∩xB/\\⌉}⊣tio\\sl⊣sh=∅. Now observe\nV∩xV∩xB= (V∩xB)∩x(V∩B)⊆(V∩C)∩x(V∩C)⊆U∩xU.\nThereforeU∩xU/\\⌉}⊣tio\\sl⊣sh=∅, i.e.,x∈UU-1. This establishes B⊆UU-1=g-1(AA-1)2g.\nSoB⊆(AA-1)2sinceB=gBg-1. /square\nRemark 4.6. The end of the previous proof is based on item (1) of [13, Remark\n5.11]. Onecanuseitem (2)ofthesameremarktofurther showthat AAA-1contains\na translate of B(after a suitable change to ζ(δ,n)). This recovers the strong form\nof Bogolyubov’s Lemma in [13, Theorem 5.9], except with AA-1almost containing\na translate of a Bohr neighborhood, rather than the Bohr neighbo rhood itself.\n5.Further model-theoretic results\nWe return to the model-theoretic setting of Section 3. Let Tbe a continuous\ntheory with a sort Gexpanding a group. Fix a left-invariant formula ϕ(x,z), with\nxof sortG. Assume that ϕ(x,z) is stable in T. Letϕ♯(x;y,z) =ϕ(x·y,z).\n5.1.Summary. We first recall what was established in Section 3.\nTheorem 5.1. LetMbe a model of T.\n(a)The Ellis semigroup of Sϕ(M)is(Sϕ♯(M),∗)(defined in Lemma 3.1).\n(b)Sϕ(M)andSϕ♯(M)are both weakly almost periodic, uniquely ergodic, and have\nunique minimal subflows, denoted MϕandMϕ♯, respectively.\n(c) (Mϕ��,∗)is a compact group.\n(d)Mϕis a compact homogeneous Mϕ♯-space.AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 17\nPart (d) wasn’t proved explicitly, but follows from parts ( a), (b), and (c) through\npurely topological means (see [11, Lemma 4.10]).\nFollowing standard terminology, we will refer to elements of Mϕ(resp.,Mϕ♯)\nasgenericϕ-types (resp., ϕ♯-types) over M. This follows the use of “generic” in\nthe general setting of topological dynamics, e.g., as in [31]. In part icular, ifSis\nanyG-flow, then a point p∈Sis generic if and only if S=GUfor any open\nneighborhood Uofp(and hence S=FUfor some finite F⊆Gby compactness).\nSee also [11, Proposition 2.7]. Consequently, our use of “generic” is a lso consistent\nwith the work of Ben Yaacov in [3, Section 6.1], which we will use below.\nThe main goal of this section is to establish results on the relationship between\ngeneric types, stabilizers, and (model theoretic) connected com ponents. In partic-\nular, we will prove the following theorem.\nTheorem 5.2. LetM|=Tbe sufficiently saturated, and set G=G(M). Letube\nthe identity inMϕ♯, and letuϕdenoteu|ϕ(which is inMϕ).\n(a)Gacts transitively on MϕandMϕ♯.\n(b)There is a smallest ϕ-type-definable subgroup of G, denotedG00\nϕ. Moreover,\nG00\nϕ= Stab(uϕ)andG/G00\nϕ∼=Mϕ(as topological homogeneous G-spaces) via\nthe mapaG00\nϕ/m⊣psto→auϕ.\n(c)There is a smallest ϕ♯-type-definable subgroup of G, denotedG00\nϕ♯. Moreover,\nG00\nϕ♯is normal,G00\nϕ♯= Stab(u), andG/G00\nϕ♯∼=Mϕ♯(as topological groups) via\nthe mapaG00\nϕ♯/m⊣psto→au.\n(d)Ifa∈Gthenauϕis the unique type in Mϕconcentrating on aG00\nϕ, andauis\nthe unique type in Mϕ♯concentrating on aG00\nϕ♯.\n(e)G00\nϕ♯is the intersection of all conjugates of G00\nϕ.\n(f)Ifp∈Mϕ♯thenStab(p) =G00\nϕ♯; and ifp∈MϕthenStab(p) =aG00\nϕa-1where\np|=aG00\nϕ. In particular, G00\nϕ♯=/intersectiontext\np∈MϕStab(p).\nWhenTis a classical first-order theory, the group G00\nϕisϕ-definable of finite\nindex (and typically denoted G0\nϕ), and the saturation assumption on Mis unnec-\nessary. In this case, the correspondence between generic ϕ-types and cosets of G0\nϕ\nwas established by the authors and Terry in [15] using earlier work o f Hrushovski\nand the second author [25]. The role of ϕ♯was made explicit by the authors in [14],\nand then a complete account of the above theorem (in the classical case) was done\nby the first author in [11].\nRecall from Remark 3.11 that Theorem 5.1 holds when ϕ(x,z) is only stable in\nM(using the same proofs). The same is true for Theorem 5.2 in the clas sical case\nsince this is the setting considered in [11]. However, if Tis continuous then the\nminimal subflow in Sϕ(M) need not be finite (or even profinite), and some amount\nof saturation is needed to make sense of the type-definable group s in Theorem\n5.2. Thus the the analogue of Theorem 5.2 when ϕ(x,z) is only stable in a single\n(non-saturated) model is not addressed here.\nFinally, we note that the local stability for groups in continuous logic w as ini-\ntiated by Ben Yaacov in [3], with a focus on the connection between fo rking and\ngenericity (following the classical case in [25]). This work will be used in o ur proof\nof Proposition 5.3. Further results on generic types and connecte d components in\nthe continuous setting are given in [3], but under a global assumptio n of stability.18 G. CONANT AND A. PILLAY\n5.2.Absoluteness of the generic types. As discussed above, when Tis a clas-\nsical first-order theory, the space Mϕin Theorem 5.2 is finite (see also Remark 5.12\nbelow) and hence the isomorphism type of Mϕdoes not depend on the choice of\nmodelM. However, this fact is not necessary for the proof of Theorem 5.2 in the\nclassical case (at least the one given in [11]). By contrast, to prove Theorem 5.2\nfor continuous T, we will firstestablish thatMϕis an invariant of the theory.\nThroughout this subsection, we will use the notation Mϕ(M) (resp.,Mϕ♯(M))\nfor the unique minimal flow in Sϕ(M) (resp.,Sϕ♯(M)), whereM|=T.\nProposition 5.3. FixM|=TandN/{ollows⌉qu⊣lM. Givenp∈Sϕ(M), letp∗∈Sϕ(N)be\nthe uniqueM-definable extension of p. ThenMϕ(N) ={p∗:p∈Mϕ(M)}.\nProof.First suppose ˆ p∈Mϕ(N). Since ˆpis generic, no condition in ˆ pforks over\n∅by [3, Lemma 6.6] (where “forking” is in the sense of [3]). It follows (us ing, e.g.,\n[3, Proposition 2.6]) that ˆ pdoes not fork over ∅, and thus ˆ pis definable over any\nsubmodel of N(e.g.,M). Therefore ˆ p=p∗wherep= ˆp|M∈Mϕ(M).\nThe converse direction is similar to [3, Proposition 6.8], which makes a glo bal\nassumption of stability. We sketch a proof in the local case. Fix p∈Mϕ(M).\nWithout loss of generality, assume Nis sufficiently saturated (relative to M). Fix\nsome element of Mϕ(N) which, by the above, we may write as q∗for someq∈\nMϕ(M). Sincepis in theG(M)-orbit closure of q, it follows by saturation that\nthereis some g∈G(N) suchthat ( gq∗)|M=p. Therefore gq∗=p∗by the argument\nabove (applied with ˆ p=gq∗). Sop∗∈Mϕ(N), as desired. /square\nThe desired absoluteness result for generic ϕ-types now follows.\nTheorem 5.4. IfM,N|=TthenMϕ(M)is homeomorphic to Mϕ(N). In par-\nticular, ifM/pr⌉c⌉⌈⌉s⌉qu⊣lNthen the restriction map is a homeomorphism from Mϕ(N)to\nMϕ(M)whose inverse is given by definitional extension.\nWe also have the following corollary of the proof of Proposition 5.3.\nCorollary 5.5. IfM|=Tisω-saturated, then G(M)acts transitively on Mϕ(M).\nProof.Fixp,q∈Mϕ(M). By definability of pandq, we have formulas ψ1(y,z) =\nϕ(y·p,z) andψ2(z) =ϕ(q,z) overM. Now letN≻Mbe sufficiently saturated\n(relative to M), and letp∗,q∗∈Sϕ(N) be theM-definable extensions of pandq,\nrespectively. As in the proof of Proposition 5.3, there is some g∈G(N) such that\ngp∗=q∗. SoN|= infysupz|ψ1(y,z)−ψ2(z)|= 0. SinceMisω-saturated, there is\ng∈G(M) such that M|= supz|ψ1(g,z)−ψ2(z)|= 0, i.e.,gp=q. /square\nNext we establish absoluteness of the compact group Mϕ♯(M).7Most of the\nlegwork will be handled by the following general topological remark.\nRemark 5.6. SupposeSis a weakly almost periodic G-flow with a dense G-orbit,\nsay given by x∈S. LetEbe the Ellis semigroup of S, and letKbe the unique\nminimal subflow of E. Then one can show that C:={σ(x) :σ∈K}is the unique\nminimal subflow of S, and the Ellis semigroup E(C) is a compact group isomorphic\ntoK. In particular, it is not hard to check that any function in KmapsCtoC(in\nfact, any function in Kmaps all of StoCby [18, Proposition II.5]). So we obtain\na well-defined map from KtoCCsendingσ∈Ktoσ↾C. This map is the desired\nisomorphism between KandE(C).\n7At this point it is worth reminding the reader that ϕ♯(x;y,z) need not be stable.AN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 19\nTheorem 5.7. IfM,N|=TthenMϕ♯(M)∼=Mϕ♯(N)(as topological groups). In\nparticular, if M/pr⌉c⌉⌈⌉s⌉qu⊣lNthen the restriction map is a topological group isomorphism\nfromMϕ♯(N)toMϕ♯(M).\nProof.It suffices to assume N/{ollows⌉qu⊣lM. To ease notation and line up with Remark 5.6,\nwe letK=Mϕ♯(M),K∗=Mϕ♯(N),C=Mϕ(M), andC∗=Mϕ(N). Applying\nRemark5.6, we havethat Cis aG(M)-flow with E(C)∼=K, andC∗is aG(N)-flow\nwithE(C∗)∼=K∗. But note that C∗is also aG(M)-flow, whose Ellis semigroup\nwe denote by EM(C∗). By Theorem 5.4, restriction from Sϕ(N) toSϕ(M) induces\na homeomorphism ρ:C∗→C. Also, ifg∈G(M) andp∈Sϕ(N) then (gp)|M=\ng(p|M). Therefore, ρis aG(M)-flow isomorphism, and thus canonically induces an\nisomorphism (of G(M)-flows and topological groups) from EM(C∗) toE(C). To\nsummarize, we have isomorphisms K∗∼=E(C∗) andEM(C∗)∼=E(C)∼=K. So to\nestablishK∼=K∗, it suffices to show E(C∗) =EM(C∗).\nForg∈G(N), let ˘g:C∗→C∗denote the action by g. We fixa∈G(N) and\nshow that ˘a∈EM(C∗) ={˘g:g∈G(M)}. In light of the homeomorphism ρ, it\nsuffices to fix b1,...,bn∈Mz,p1,...,pn∈C∗, andǫ>0, and find g∈G(M) such\nthat|ϕ(api,bi)−ϕ(gpi,bi)|<ǫfor all 1≤i≤n. Such agexists since M/pr⌉c⌉⌈⌉s⌉qu⊣lNand\neachpiis definable over M(by Proposition 5.3).\nWe havenowshownthat K∗andKareisomorphicastopologicalgroups. Byun-\nravelingall of the maps used above, one can check that the underly ing isomorphism\nis precisely the restriction map. /square\n5.3.Proof of Theorem 5.2. Throughout this section we let M|=Tbe a suffi-\nciently saturated model, and set G=G(M). We follow the notation laid out in\nSection 5.1 and in the statement of Theorem 5.2. We will frequently us e the fact\nthat the identity uinMϕ♯commutes with every element of Sϕ♯(M) (see Corollary\n3.2(d)). In particular, gu=ugfor allg∈G.\nRecall that for a point pin aG-flowS, the stabilizer Stab( p) ={g∈G:gp=p}\nis a subgroup of G. Note also that Stab( gp) =gStab(p)g-1.\nProposition 5.8.\n(a) Stab(u)is normal and ϕ♯-type-definable of bounded index, and G/Stab(u)∼=\nMϕ♯(as topological groups) via gStab(u)/m⊣psto→gu.\n(b) Stab(uϕ)isϕ-type-definable of bounded index, and G/Stab(uϕ)∼=Mϕ(as\ntopological homogeneous G-spaces) via gStab(uϕ)/m⊣psto→guϕ.\n(c)u|= Stab(u)anduϕ|= Stab(uϕ).\nProof.Part (a). Defineπ:G→Mϕ♯so thatπ(g) =gu. By Proposition 3.6, πis a\nϕ♯-definablehomomorphismwith dense image. Since Mϕ♯is bounded (by Theorem\n5.7) andMis sufficiently saturated, πinduces an isomorphismoftopologicalgroups\nfromG/ker(π) toMϕ♯. Finally, note that ker( π) = Stab(u).\nPart (b). SinceGacts transitively on Mϕ(by Corollary 5.5), it makes sense\nto viewMϕas a homogeneous G-space. To prove the result, it suffices to show\nthat the map g/m⊣psto→guϕisϕ-definable, since then we can argue as in part ( a) (using\nTheorem 5.4 in place of Theorem 5.7). Consider the map σu∈E(Sϕ(M)) from the\nproof of Lemma 3.1. Recall from Remark 3.4 that σuis continuous. One can also\ncheck thatσu(p|ϕ) = (p∗u)|ϕfor anyp∈Sϕ♯(M). HenceσumapsSϕ(M) toMϕ\n(recall Corollary 3.2( b)) and extends the map g/m⊣psto→guϕ.\nPart (c). By the proof of part ( a) (and Proposition 3.6), a type p∈Sϕ♯(M)\nconcentrates on Stab( u) if and only if p∗u=u. Sou|= Stab(u). Similarly, from20 G. CONANT AND A. PILLAY\nthe proof of part ( b), we see that a type p∈Sϕ(M) concentrates on Stab( uϕ) if and\nonly ifσu(p) =uϕ. Moreover, σu(uϕ) = (u∗u)|ϕ=uϕ. Souϕ|= Stab(uϕ)./square\nNote that the proof of Proposition 5.8( a) establishes surjectivity of the map\ng/m⊣psto→gufromGtoMϕ♯. So we also have the following conclusion.\nCorollary 5.9. Gacts transitively on Mϕ♯.\nTo finish the proof of Theorem 5.2, we will need the following basic fact , whose\nproof is left as an exercise (and is a routine adaptation of the classic al case).\nFact 5.10. Letψ(x,w)be a left-invariant formula (with xof sortG), and assume\nthatSψ(M)has a unique minimal subflow (i.e., there is a generic ψ-type overM).\nLetH≤Gbe aψ-type-definable bounded-index subgroup of G. Then there is some\ngenericp∈Sψ(M)concentrating on H.\nCorollary 5.11.\n(a) Stab(u)is the smallest ϕ♯-type-definable bounded-index subgroup of G.\n(b) Stab(uϕ)is the smallest ϕ-type-definable bounded-index subgroup of G.\nProof.Part (a). LetHbe aϕ♯-type-definable bounded-index subgroup of G. By\nCorollary 5.9 and Fact 5.10 there is some g∈Gsuch thatgu|=H. Now if\na∈Stab(u) thenagu=aug=ug=gu, soagu|=H. Sinceagu|=aH, we\nconcludeH=aH, i.e.,a∈H.\nPart (b). LetHbe aϕ-type-definable bounded-index subgroup of G. Then\nStab(u)⊆Hby part (a), sou|=Hby Proposition 5.8( c). SinceHisϕ-type-\ndefinable, we have uϕ|=H. So ifg∈Stab(uϕ), thenuϕalso concentrates on gH,\nhenceH=gH, i.e.,g∈H. /square\nThe previous corollary justifies writing G00\nϕ♯andG00\nϕfor Stab(u) and Stab( uϕ),\nrespectively. Combining the above results, we now have proved par ts (a), (b), and\n(c) of Theorem 5.2, which leaves parts ( d), (e), and (f).\nForpart(d), fixsomea∈G. Thenau|=aG00\nϕ♯byProposition5.8( c). Conversely,\nsupposep∈Mϕ♯concentrates on aG00\nϕ♯. Fixg∈Gsuch thatp=gu. Then\np|=gG00\nϕ♯, thusaG00\nϕ♯=gG00\nϕ♯, i.e.,au=gu=p. By a similar argument, auϕis the\nunique type inMϕconcentrating on aG00\nϕ.\nFor part (e), supposeg/\\⌉}⊣tio\\sl⊣sh∈G00\nϕ♯. Thengu/\\⌉}⊣tio\\sl⊣sh=u, so there is some instance ϕ♯(x;a,b)\nsuch thatϕ♯(u;a,b)/\\⌉}⊣tio\\sl⊣sh=ϕ♯(gu;a,b). Sinceua=au, it follows that ϕ(auϕ,b)/\\⌉}⊣tio\\sl⊣sh=\nϕ(gauϕ,b). Sog/\\⌉}⊣tio\\sl⊣sh∈Stab(auϕ) =aG00\nϕa-1.\nPart (f) now follows immediately from the previous statements, which finishe s\nthe proof of Theorem 5.2.\nRemark 5.12. For the sake of completeness, we sketch how to recover finitenes s of\nMϕwhenthetopologyon Sϕ(M)inducedfromthecanonicallocalmetricisdiscrete\n(see [7, Definition 6.1]). Let µbe the unique left-invariant Keisler ϕ-functional over\nM. Then, assuming discreteness of Sϕ(M), we can write µas a weighted sum of\nDirac measures, i.e., µ=/summationtext\ni∈Iαipiwhereαi>0 andpi∈Sϕ(M). This can be\nproved as in case of classical logic, but also follows immediately from [9 , Theorem\n3.12] which gives an analogous result for any local Keisler functional defined from\na continuous stable formula. Now, by Proposition 3.7 and the relation ship between\nthe Haar functionals on Mϕ♯and onMϕ(recall Theorem 5.1( d)), eachpiabove isAN ANALYTIC VERSION OF STABLE ARITHMETIC REGULARITY 21\ninMϕand is a point of positive Haar measure. But it is a standard exercise t hat\nany compact homogeneous space with such a point must be finite.\nReferences\n[1] M.A. Alekseev, L. Y. Glebski˘ ı, and E. I. Gordon, On approximations of groups, group actions\nand Hopf algebras , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov . (POMI) 256\n(1999), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metod y. 3, 224–262, 268.\n[2] M. Beiglb¨ ock, V. Bergelson, and A. Fish, Sumset phenomenon in countable amenable groups ,\nAdv. Math. 223(2010), no. 2, 416–432.\n[3] I. Ben Yaacov, Stability and stable groups in continuous logic , J. Symbolic Logic 75(2010),\nno. 3, 1111–1136.\n[4] ,Model theoretic stability and definability of types, after A . Grothendieck , Bull. Symb.\nLog.20(2014), no. 4, 491–496.\n[5] I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyats ov,Model theory for metric\nstructures , Model theory with applications to algebra and analysis. Vo l. 2, London Math.\nSoc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cam bridge, 2008, pp. 315–427.\n[6] I. Ben Yaacov and T. Tsankov, Weakly almost periodic functions, model-theoretic stabil ity,\nand minimality of topological groups , Trans.Amer.Math.Soc. 368(2016), no.11, 8267–8294.\n[7] I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability , Trans. Amer.\nMath. Soc. 362(2010), no. 10, 5213–5259.\n[8] E. Breuillard, B. Green, and T. Tao, The structure of approximate groups , Publ. Math. Inst.\nHautes´Etudes Sci. 116(2012), 115–221.\n[9] N. Chavarria, G. Conant, and A. Pillay, Continuous stable regularity , arXiv:2111.05435, to\nappear in J. London Math. Soc.\n[10] G. Conant, On finite sets of small tripling or small alternation in arbit rary groups , Combin.\nProbab. Comput. 29(2020), no. 6, 807–829.\n[11] ,Stability in a group , Groups Geom. Dyn. 15(2021), no. 4, 1297–1330.\n[12] ,Quantitative structure of stable sets in arbitrary finite gr oups, Proc. Amer. Math.\nSoc.149(2021), no. 9, 4015–4028.\n[13] G. Conant, E. Hrushovski, and A. Pillay, Compactifications of pseudofinite and pseudo-\namenable groups , arXiv:2308.08440, 2023.\n[14] G. Conant and A. Pillay, Pseudofinite groups and VC-dimension , J. Math. Log. 21(2021),\nno. 2, Paper No. 2150009, 23.\n[15] G. Conant, A. Pillay, and C. Terry, A group version of stable regularity , Math. Proc. Cam-\nbridge Philos. Soc. 168(2020), no. 2, 405–413.\n[16] ,Structure and regularity for subsets of groups with finite VC -dimension , J. Eur.\nMath. Soc. (JEMS) 24(2022), no. 2, 583–621.\n[17] R. Ellis, Lectures on topological dynamics , W. A. Benjamin, Inc., New York, 1969.\n[18] R. Ellis and M. Nerurkar, Weakly almost periodic flows , Trans. Amer. Math. Soc. 313(1989),\nno. 1, 103–119.\n[19] E. Glasner, Enveloping semigroups in topological dynamics , Topology Appl. 154(2007),\nno. 11, 2344–2363.\n[20] B. Green, A Szemer´ edi-type regularity lemma in abelian groups, with applications , Geom.\nFunct. Anal. 15(2005), no. 2, 340–376.\n[21] A. Grothendieck, Crit` eres de compacit´ e dans les espaces fonctionnels g´ en ´ eraux, Amer. J.\nMath.74(1952), 168–186.\n[22] K. H. Hofmann and S. A. Morris, The structure of compact groups , De Gruyter Studies in\nMathematics, vol. 25, De Gruyter, Berlin, 2013, A primer for the student—a handbook for\nthe expert, Third edition, revised and augmented.\n[23] E. Hrushovski, Stable group theory and approximate subgroups , J. Amer. Math. Soc. 25\n(2012), no. 1, 189–243.\n[24] E. Hrushovski, K. Krupi´ nski, and A. Pillay, Amenability, connected components, and defin-\nable actions , Selecta Math. (N.S.) 28(2022), no. 1, Paper No. 16, 56.\n[25] E. Hrushovski and A. Pillay, Groups definable in local fields and pseudo-finite fields , Israel\nJ. Math. 85(1994), no. 1-3, 203–262.\n[26] D. Kazhdan, Onε-representations , Israel J. Math. 43(1982), no. 4, 315–323.22 G. CONANT AND A. PILLAY\n[27] E. Kowalski, An introduction to the representation theory of groups , Graduate Studies in\nMathematics, vol. 155, American Mathematical Society, Pro vidence, RI, 2014.\n[28] J.-L. Krivine and B. Maurey, Espaces de Banach stables , Israel J. Math. 39(1981), no. 4,\n273–295.\n[29] M. Malliaris and S. Shelah, Regularity lemmas for stable graphs , Trans. Amer. Math. Soc.\n366(2014), no. 3, 1551–1585.\n[30] S. Montenegro, A. Onshuus, and P. Simon, Stabilizers, NTP2groups with f-generics, and\nPRC fields , J. Inst. Math. Jussieu 19(2020), no. 3, 821–853.\n[31] L. Newelski, Topological dynamics of definable group actions , J. Symbolic Logic 74(2009),\nno. 1, 50–72.\n[32] A. L. T. Paterson, Amenability , Mathematical Surveys and Monographs, vol. 29, American\nMathematical Society, Providence, RI, 1988.\n[33] A. Pillay, An introduction to stability theory , Oxford Logic Guides, vol. 8, The Clarendon\nPress Oxford University Press, New York, 1983.\n[34] L. Ribes and P. Zalesskii, Profinite groups , second ed., Ergebnisse der Mathematik und ihrer\nGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathem atics [Results in Mathematics\nand Related Areas. 3rd Series. A Series of Modern Surveys in M athematics], vol. 40, Springer-\nVerlag, Berlin, 2010.\n[35] I. Z. Ruzsa, Generalized arithmetical progressions and sumsets , Acta Math. Hungar. 65\n(1994), no. 4, 379–388.\n[36] T. Sanders, On a nonabelian Balog-Szemer´ edi-type lemma , J. Aust. Math. Soc. 89(2010),\nno. 1, 127–132.\n[37] S. Shelah, Classification theory and the number of nonisomorphic model s, second ed., Stud-\nies in Logic and the Foundations of Mathematics, vol. 92, Nor th-Holland Publishing Co.,\nAmsterdam, 1990.\n[38] E. Szemer´ edi, Regular partitions of graphs , Probl` emes combinatoires et th´ eorie des graphes\n(Colloq. Internat. CNRS,Univ. Orsay, Orsay, 1976), Colloq .Internat. CNRS, vol. 260, CNRS,\nParis, 1978, pp. 399–401.\n[39] T. Tao, The dichotomy between structure and randomness , presented at\nthe 2006 International Congress of Mathematicians, Madrid , available at:\nhttps://www.math.ucla.edu/ ~tao/preprints/Slides/icmslides2.pdf .\n[40] C. Terry and J. Wolf, Stable arithmetic regularity in the finite field model , Bull. Lond. Math.\nSoc.51(2019), no. 1, 70–88.\n[41] ,Quantitative structure of stable sets in finite abelian grou ps, Trans. Amer. Math.\nSoc.373(2020), no. 6, 3885–3903.\nDepartment of Mathematics, The Ohio State University, Colum bus, OH 43201, USA\nEmail address :conant.38@osu.edu\nDepartment Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA\nEmail address :apillay@nd.edu" }, { "title": "2401.14385v1.Entropic_Quantum_Central_Limit_Theorem_and_Quantum_Inverse_Sumset_Theorem.pdf", "content": "ENTROPIC QUANTUM CENTRAL LIMIT THEOREM\nAND QUANTUM INVERSE SUMSET THEOREM\nKAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nABSTRACT . We establish an entropic, quantum central limit theorem and quantum\ninverse sumset theorem in discrete-variable quantum systems describing qudits or\nqubits. Both results are enabled by using our recently-discovered quantum convolu-\ntion. We show that the exponential rate of convergence of the entropic central limit\ntheorem is bounded by the magic gap. We also establish an “quantum, entropic in-\nverse sumset theorem,” by introducing a quantum doubling constant. Furthermore,\nwe introduce a “quantum Ruzsa divergence”, and we pose a conjecture called “convo-\nlutional strong subaddivity,” which leads to the triangle inequality for the quantum\nRuzsa divergence. A byproduct of this work is a magic measure to quantify the non-\nstabilizer nature of a state, based on the quantum Ruzsa divergence.\nCONTENTS\n1. Introduction 1\n2. Background 3\n3. The entropic, quantum, central-limit theorem 6\n4. Entropic quantum inverse sumset theorem 10\n5. Quantum Ruzsa divergence 12\n6. Conclusion 20\n7. Acknowledgement 20\n8. Appendix 21\nReferences 21\n1. I NTRODUCTION\nIn this paper, we generalize some central results in modern, classical Fourier\nanalysis to a quantum framework. This new framework is based on a quantum\nconvolution that we introduced earlier in [1, 2]. In Theorems 12 and 13 we prove\nan entropic, quantum, central-limit theorem (entropic q-CLT) for discrete-variable\n(DV) quantum systems. We introduce a quantum doubling constant in Definition 20,\nwhich enables us to prove an entropic, quantum inverse-sumset in Theorem 22. We\nalso introduce a quantum version of the Ruzsa divergence in Definition 26. We prove\na number of its properties in Theorem 28 and provide a characterization of stabilizer\nstates via quantum Ruzsa divergence in Proposition 35.\n1.1. Background. The central limit theorem (CLT) is a fundamental result in prob-\nability theory. Given independent, identically distributed random variables Xiwith\nzero mean, and finite variance σ, the normalized sum ZN=PN\ni=1Xip\nNconverges to a\n1arXiv:2401.14385v1 [quant-ph] 25 Jan 20242 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nGaussian random variable Zwith the same mean value and variance. The study\nof entropic central limit theorem has a long history, back to the work of Linnik [3],\nwhere the entropy h(ZN)= −R\nfZNlogfZNfor the probability density function fZN.\nBarron later showed that h(ZN) converges to the h(Z) asN→ ∞ , where Zis the cor-\nrepsonding Gaussian random variable [4]. Furthermore, the rate of convergence in\nthe entropic central limit theorem has attracted much attention [5–7]. In the case of\ndiscrete random variables, the entropic CLT has also been considered by Gavalakis\nand Kontoyiannis [8], based on the normalized sum of nindependent and identically\ndistributed lattice random variables and an appropriately discretized Gaussian.\nIn the case of continuous variable (CV) quantum systems, various central limit\ntheorems also have an interesting history including Cushen and Hudson [9], and\nrelated work of Hepp and Lieb [10, 11]. Many other quantum or noncommutative\nversions of the central limit theorem appeared later, see [12–28]. For the entropic\nq-CLT in CV systems, the rate of convergence was given for an m-mode quantum\nstate under some technical assumption [26].\nHowever, little is known about the entropic q-CLT for qudits or qubits. This is\none of the foci of the current paper, which is based on the quantum convolution re-\ncently proposed by the authors [1, 2, 29]. This quantum convolutional framework\nprovides a new method to understand and study stabilizer states. Stabilizer states\nwere first introduced by Gottesman [30], and now have many applications includ-\ning quantum error correction codes [31, 32], and the classical simulation of quan-\ntum circuits, known as Gottesman-Knill theorem [33]. These applications indicate\nthat nonstabilizer states and circuits are necessary to achieve the quantum com-\nputational advantage. Later, the extension of the Gottesman-Knill theorem beyond\nstabilizer circuits was further studied [34–43]. The term “magic” was introduced by\nBravyi and Kitaev [44] to express the property that a state is not a stabilizer.\nOne consequence of our convolutional framework is a quantum central limit the-\norem for DV quantum systems. This means that repeated quantum convolution\nwith any given, mean-zero state converges to a stabilizer state. Therefore we iden-\ntify the set of stabilizer states as the set of “quantum-Gaussian” states. Some con-\nvolutional inequalities for the quantum entropies and other information measures\nhave also been proved, with stabilizer states being the only extremizers of these\ninequalities [1, 2, 45]. Furthermore, based on purity invariance of stabilizer states\nunder quantum convolution, a convolution-swap test has been proposed to determine\nwhether a state is a stabilizer [29]. This can be regarded as a quantum-state version\nof linearity testing.\nFinally we comment that (to our knowledge) the only earlier proposal for a good\nconvolution for qudit states was given by Audenaert, Datta, and Ozols [46] and stud-\nied by Carlen, Lieb, and Loss [47]. Their convolution is a modified convex combina-\ntion of the input states ρandσ, namely λρ+(1−λ)σ−ip\nλ(1−λ)[ρ,σ]. Unfortunately,\nthis convolution does not lead to an interesting central limit theorem.\nIn the setting of free probability theory, a free convolution was introduced by\nVoiculescu; this also leads to a free central limit theorem [48]. But this framework\nis different from what we consider here.\nA related problem for the classical CLT is the inverse sumset theory, which is an\nimportant topic in additive combinatorics. The inverse sumset theory, also known\nas Freiman-Ruzsa inverse sumset theory, explores properties of sets Asuch that theENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 3\nsize (or the Shannon entropy) of A+Ais close to that of A. In this theory, the dou-\nbling constant |A+A|/|A|and its entropic analog exp( H(X+X′)−H(X)) with i.i.d.\ncopies of random variables XandX′, have been used to investigate the properties of\nthe subset Aand the random variable X[23,49–53]. For example, by Tao’s work [50],\ngiven i.i.d. copies XandX′of a discrete random variables, if exp( H(X+X′)−H(X))\nis small, then the distribution of Xis close to the uniform distribution on a gener-\nalized arithmetic progression. Recently, there is a breakthrough in this theory by\nGowers, Green, Manners, and Tao, which proves the Marton’s conjecture (or polyno-\nmial Freiman–Ruzsa conjecture) [53]. These results involved an important measure,\ncalled Ruzsa distance [50], which was later generalized to Ruzsa divergence [54]. The\nproperties of Ruzsa distance depends on some inequalities about the sum of sets or\nrandom variables, known as sumset inequalities. They also provide some connection\nbetween the size (or the Shannon entropy) of the sumset A+B(or random variables\nXandY) and that of sets AandB(orXandY) [50,54–57].\nIn this paper, we generalize many of these ideas to a DV quantum setting, and\nfind that the quantum features will lead to some new phenomenon.\n1.2. Outline and summary of our main results.\n1. In §3, we establish an entropic q-CLT for DV quantum systems. We show that the\nquantum relative entropy between the n-th repetition of our quantum convolution\nand the mean state converges to zero at an exponential rate. This rate is bounded\nfrom below by the “magic gap” defined by the state.\n2. In §4, we establish an entropic, quantum inverse-sumset theorem. We study the\nquantum-doubling constant for a state ρ. This is the difference between the von\nNeumann entropy S(ρ⊠ρ) of the self-convolution of ρand the entropy S(ρ) of the\nstate itself. The quantum-doubling constant and the magic gap provide an upper\nbound on the quantum relative entropy between the given state and its mean\nstate.\n3. In §5, we generalize the classical Ruzsa divergence to a quantum divergence.\nWe use quantum subset inequalities to establish some properties of the quantum\nRuzsa divergence, which also provide a good characterization of stabilizer states.\nWhile sub-additivity holds for the classical convolution, i.e., H(X+Y)ÉH(X)+\nH(Y) for random variables XandY[49, 58], we prove that quantum features\nprevent sub-additivity from holding in general. We conjecture a convolutional\nstrong subaddivity of the quantum entropy, which is different from the strong\nsub-additivity proved by Lieb and Ruskai [59]. We prove this convolutional strong\nsubadditivity for two special cases. Moreover, in §5.2, we define a magic measure\nvia the quantum Ruzsa divergence to quantify the magic of states.\n2. B ACKGROUND\nFix natural numbers n(the number of qudits) and d(the degree of each qudit)\nand study the Hilbert space H⊗n, where H=Cd. Consider an orthonormal basis\n{|k〉:k∈Zd}for the Hilbert space H; here Zddenotes the cyclic group of order d.\nOne calls these vectors the computational basis . The Pauli operators Xand Zare\ndefined by\n(1) X:|k〉 7→ |k+1〉, Z:|k〉 7→ωk\nd|k〉,∀k∈Zd,\nwhere ωd=exp(2 πi/d) is the primitive d-th root of unity. We restrict dto be prime\nin order to define our quantum convolution.4 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nThe local Weyl operators (or generalized Pauli operators) are\nw(p,q)=ζ−pqZpXq,where ζ=(\nω(d+1)/2\nd,fordodd\neiπ/2, ford=2.\nFor the n-qudit system on H⊗n, the Weyl operators are defined as\nw(⃗p,⃗q)=w(p1,q1)⊗...⊗w(pn,qn)=w(−⃗p,−⃗q)†, (2)\nwith⃗p=(p1,p2,...,pn)∈Zn\nd,⃗q=(q1,...,qn)∈Zn\nd. Let us denote Pnas the group gen-\nerated by the Weyl operators and phase ζ. The Weyl operators are an orthonormal\nbasis for the space of linear operators on H⊗nwith respect to the inner product\n(3) 〈A,B〉 =1\ndnTrh\nA†Bi\n.\nDenote Vn:=Zn\nd×Zn\nd; this represents the phase space for n-qudit systems, as was\nstudied in [60]. Let D(H⊗n) denote the set of all quantum states on H⊗n, namely\npositive matrices with unit trace.\nDefinition 1 (Characteristic function). The characteristic function Ξ(⃗p,⃗q)of a\nstate ρ∈D(H⊗n)is the coefficient of ρin the Weyl basis,\nΞρ(⃗p,⃗q)=Tr£\nρw(−⃗p,−⃗q)¤\n,and ρ=1\ndnX\n(⃗p,⃗q)∈VnΞρ(⃗p,⃗q)w(⃗p,⃗q).\nThe process of taking characteristic functions is the quantum Fourier transform\nthat we consider. More details about the properties of the characteristic functions\ncan be found in [2, 60, 61]. In this work, we also use Ξρ(⃗x) with ⃗x=(⃗p,⃗q)∈Vnand\nthe expectation\nEki∈Zd(·) :=1\ndX\nki∈Zd(·).\nDefinition 2 (Stabilizer states [30, 62]). A pure stabilizer state |ψ〉〈ψ|for an n-\nqudit system is the projection onto a stabilizer vector |ψ〉, namely a common unit\neigenvector of an abelian (stabilizer) subgroup of the Weyl operators of size dn. If the\ngenerators of the stabilizer group are {g1,...,gn}i∈[n]with gi∈Pn, then\n|ψ〉〈ψ| =Πn\ni=1Eki∈Zdgki\ni.\nA general stabilizer state is a mixed state ρobtained as a convex linear combination\nof pure stabilizer states; we denote the set of stabilizer states by STAB.\nDefinition 3 (Minimal stabilizer-projection state). A quantum state ρis a mini-\nmal stabilizer-projection state (MSPS) associated with an abelian subgroup generated\nby{gi}i∈[r]with gi∈Pn, if it has the following form\nρ=1\ndn−rΠr\ni=1Eki∈Zdgki\ni.\nFor example, in an n-qudit system and with the abelian group with the generators\n{Z1,...,Zn−1}, the states {1\nd|⃗j〉〈⃗j|⊗I}⃗j∈Zn−1\ndare MSPS.\nDefinition 4 (Clifford unitary). Ann-qudit unitary Uis a Clifford unitary, if\nUw(⃗x)U†is also a Weyl operator up to a phase for any ⃗x∈Vn.\nIt is easy to see that Clifford unitaries map stabilizer states to stabilizer states.ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 5\nDefinition 5 (Mean state (MS) [1, 2]). Given an n-qudit state ρ, the mean state\nM(ρ)is the operator with the characteristic function:\nΞM(ρ)(⃗x) :=(Ξρ(⃗x),|Ξρ(⃗x)| =1,\n0, |Ξρ(⃗x)| <1.(4)\nThe mean state M(ρ)is a stabilizer state.\nIn addition, M(ρ) has a stabilizer group, i.e., the abelian group generated by the\nPauli operator w(⃗x) such that w(⃗x)M(ρ)w(⃗x)†=M(ρ). For simplicity, we denote it as\nGρ.\nDefinition 6 (Zero-mean state [1,2]). A given n-qudit state ρhas zero-mean, if the\ncharacteristic function of M(ρ)takes values in {0,1}.\nNote that, if ρis not a zero-mean state, there exists a Weyl operator w(⃗x) such\nthat w(⃗x)ρw(⃗x)†is a zero-mean state [2].\nDefinition 7 (Magic gap [1,2]). Given an n-qudit state ρ∈D(H⊗n)for any integer\ndÊ2, the magic gap of ρis\nMG(ρ)=1− max\n⃗x∈Supp (Ξρ):|Ξρ(⃗x)|̸=1|Ξρ(⃗x)|.\nIf{⃗x∈Supp (Ξρ) :|Ξρ(⃗x)| ̸=1}= ;, define MG(ρ)=0, i.e., there is no gap on the support\nof the characteristic function.\nDefinition 8 (Quantum convoltuion [1, 2]). Lets2+t2≡1 mod d, with s,t̸=0,\nand let Us,tbe the unitary\nUs,t=X\n⃗i,⃗j∈Zn\nd|s⃗i+t⃗j〉〈⃗i|⊗|− t⃗i+s⃗j〉〈⃗j|, (5)\nacting on the 2n-qudit sytems HA⊗HBwithHA=HB=H⊗n, and the vector |⃗i〉 =\n|i1〉⊗···⊗| in〉 ∈H⊗n. The convolution of two n-qudit states ρandσis\nρ⊠s,tσ=TrBh\nUs,t(ρ⊗σ)U†\ns,ti\n. (6)\n2.1. Basic properties of entropy. We review some basic properties of quantum\nentropy and relative entropy.\nDefinition 9 (Quantum entropy). Given a quantum state ρ, the von Neuman en-\ntropy is\nS(ρ) := −Tr£\nρlogρ¤\n. (7)\nGiven a parameter α∈[0,+∞], the quantum Rényi entropy is\nSα(ρ) :=1\n1−αlogTr£\nρα¤\n. (8)\nNote that lim α→1Sα(ρ)=S(ρ). Also S∞(ρ)=limα→∞Sα(ρ)= −logλmax, where\nλmaxis the largest eigenvalue of ρ.\nDefinition 10 (Quantum relative entropy). The relative entropy of ρwith respect\ntoσis\nD(ρ||σ) :=Tr£\nρ(logρ−logσ)¤\n. (9)\nGiven a parameter α∈[0,+∞], the quantum Rényi relative entropy Dαis\nDα(ρ||σ) :=1\nα−1logTrh³\nσ1−α\n2αρσ1−α\n2α´αi\n.6 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nNote that lim α→1Dα(ρ||σ)=D(ρ||σ), and\nD∞(ρ||σ)=lim\nα→+∞Dα(ρ||σ)=minlog {λ:ρÉλσ}.\nOne fundamental result is that the mean state M(ρ) is the closest MSPS to the given\nstate ρby using the quantum relative entropy as a distance measure.\nLemma 11 ( [1,2]) .Given an n-qudit state ρ, we have\nmin\nσ∈MSPSDα(ρ||σ)=Dα(ρ||M(ρ))=Sα(M(ρ))−Sα(ρ),\nfor any αÊ1.\n3. T HE ENTROPIC ,QUANTUM ,CENTRAL -LIMIT THEOREM\nHere we give our main results on the entropic q-CLT. In the earlier work we estab-\nlished and studied a quantum, central limit theorem [1,2,29], where the exponential\nrate of convergence in L2norm is bounded below by the magic gap. Here let us\nconsider the entropic q-CLT. There are two entropic measures in the study of the en-\ntropic q-CLT, one is the quantum relative entropy D(ρ||M(ρ)) , and the other is the\ndifference of quantum entropies S(M(ρ))��S(ρ). These two measures are shown to\nbe equivalent by Lemma 11. Hence, we will use these entropic measures to establish\nan entropic q-CLT.\nTheorem 12 (Entropic q-CLT). Given an n-qudit state ρ, the quantum entropy\nof the Nthrepetition of the quantum convolution S(⊠Nρ)increases monotonically to\nS(M(ρ)), namely\nS(⊠Nρ)↗S(M(ρ)),asN→ ∞ . (10)\nThis is equivalent to\nD(⊠Nρ||M(⊠Nρ))→0,asN→ ∞ . (11)\nFor a state ρwith zero-mean,\nD(⊠Nρ||M(ρ))→0,asN→ ∞ . (12)\nProof. The monotonicity of the quantum entropy S(⊠Nρ) with respect to Nin the q-\nCLT has been proved by the authors; see Theorem 57 in [1] and also [2,29], namely\nS(⊠Nρ)ÉS(⊠N+1ρ),∀NÊ1.\nMoreover, as ⊠Nρ→M(ρ) for N→ ∞ , by the continuity of quantum entropy, we\nhave\nS(⊠Nρ)→S(M(ρ)),asN→ ∞ .\nSinceM(⊠Nρ) is equivalent to M(ρ) up to a Clifford unitary, then S(M(⊠Nρ))=\nS(M(ρ)). Thus, by the equality D(ρ||M(ρ))=S(M(ρ))−S(ρ) in Lemma 11, we have\nD(⊠Nρ||M(⊠Nρ))=S(M(⊠Nρ))−S(⊠Nρ)\n=S(M(ρ))−S(⊠Nρ)\n→0,asN→ ∞ .\nFor a zero-mean state ρ,M(⊠Nρ)=M(ρ). Hence, we have\nD(⊠Nρ||M(ρ))=D(⊠Nρ||M(⊠Nρ))→0,asN→ ∞ .\n□ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 7\nWe can also provide an upper bound on the rate of convergence in the entropic\nq-CLT in terms of magic gap, where the rate of convergence is exponentially small\nwith respect to the number of convolutions.\nTheorem 13 (Rate of convergence). Given an n-qudit state ρ, the quantum rela-\ntive entropy of ⊠Nρwith respect to the mean state M(⊠Nρ)has the following bound,\nD(⊠Nρ||M(⊠Nρ))=S(M(ρ))−S(⊠Nρ)\nÉlogh\n1+(1−MG(ρ))2N−2¡\nTr£\nρ2¤\nR(ρ)−1¢i\n(13)\nÉ(1−MG(ρ))2N−2¡\nTr£\nρ2¤\nR(ρ)−1¢\n,\n→0,asN→ ∞ ,\nwhere R(ρ)is the rank of the state M(ρ).\nProof. Let us assume that Gρis the stabilizer group of M(ρ), then R(ρ)=dn\n|Gρ|, and\nthus S(M(ρ))=logR(ρ)=logdn\n|Gρ|. For simplicity, let us take λ=(1−MG(ρ))2=\nmax⃗x∈Supp(Ξρ):|Ξρ(⃗x)|̸=1|Ξρ(⃗x)|2. Hence\nD(⊠Nρ||M(⊠Nρ))\n=S(M(ρ))−S(⊠Nρ)\nÉS(M(ρ))−S2(⊠Nρ)\n=logdn\n|Gρ|+logÃ\n|Gρ|\ndn+1\ndnX\n⃗x∉G|Ξ⊠Nρ(⃗x)|2!\nÉlogdn\n|Gρ|+logÃ\n|Gρ|\ndn+λN−1\ndnX\n⃗x∉G|Ξρ(⃗x)|2!\n=logdn\n|Gρ|+log·|Gρ|\ndn+λN−1µ\nTr£\nρ2¤\n−|Gρ|\ndn¶¸\n=log·\n1+λN−1µ\nTr£\nρ2¤dn\n|Gρ|−1¶¸\n,\nwhere the second line comes from Lemma 11, the third line comes from the mono-\ntonicity of Rényi entropy S(ρ)ÊS2(ρ), the fifth line comes from the definition of λ,\nand the sixth line comes from the fact that\nTr£\nρ2¤\n=1\ndnX\n⃗x|Ξρ(⃗x)|2=|Gρ|\ndn+1\ndnX\n⃗x∉Gρ|Ξρ(⃗x)|2.\n□\nBy the entropic q-CLT, we have the following q-CLT based on trace distance as a\ncorollary.\nCorollary 14. Given an n-qudit state ρwith zero mean, we have\n°°°⊠Nρ−M(ρ)°°°\n1Ép\n2(1−MG(ρ))N−1q¡\nTr£\nρ2¤\nR(ρ)−1¢\n,asN→ ∞ . (14)\nProof. This is a direct corollary of the quantum Pinsker inequality1\n2°°ρ−σ°°2\n1É\nD(ρ||σ), and the Theorem 13. □8 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nFIGURE 1. A diagram of the repeated quantum convolution in the\nq-CLT.\nNow, let us generalize the entropic q-CLT from von Neumann entropy to Rényi\nentropy.\nTheorem 15 (Rényi entropic q-CLT via magic gap). Given an n-qudit state ρ\nand any parameter α∈[1,∞], the α-quantum Rényi relative entropy of ⊠Nρwith\nrespect to the mean state M(⊠Nρ)satisfies the bound,\nDα(⊠Nρ||M(⊠Nρ))=Sα(M(ρ))−Sα(⊠Nρ)\nÉlogÃ\n1+(1−MG(ρ))N−1R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ)!\n(15)\nÉ(1−MG(ρ))N−1R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ)\n→0,asN→ ∞ .\nProof. Based on the monotonicity of Rényi relative entropy DαÉD∞for any αÊ0,\nwe only need to prove the statement for the max-relative entropy D∞. For simplicity,\nlet us take λ=(1−MG(ρ))2=max⃗x∈Supp(Ξρ):|Ξρ(⃗x)|̸=1|Ξρ(⃗x)|2. First, we have\n|〈ψ|ρ|ψ〉|\nÉ|〈ψ|ρ−M(ρ)|ψ〉|+|〈 ψ|M(ρ)|ψ〉|\nÉq\nTr£\n(ρ−M(ρ))2¤\n+1\nR(ρ)\n=vuut1\ndnX\n⃗x∉Gρ|Ξρ(⃗x)|2+1\nR(ρ),\nwhere the second line comes from the triangle inequality, the third line comes from\nthe Cauchy-Schwarz inequality and the last line comes from the fact that ρ−M(ρ)=\n1\ndnP\n⃗x∉GρΞρ(⃗x)w(⃗x). Hence, we have\nλmax(⊠Nρ)\nÉvuut1\ndnX\n⃗x∉Gρ|Ξ⊠Nρ(⃗x)|2+1\nR(ρ)\nÉvuutλN−1\ndnX\n⃗x∉Gρ|Ξρ(⃗x)|2+1\nR(ρ)(16)\n=λ(N−1)/2s\nTr£\nρ2¤\n−1\nR(ρ)+1\nR(ρ),ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 9\nwhere the third line comes from the definition of λ, and the last line comes from the\nfact that\nTr£\nρ2¤\n=1\ndnX\n⃗x|Ξρ(⃗x)|2=|Gρ|\ndn+1\ndnX\n⃗x∉Gρ|Ξρ(⃗x)|2.\nHence,\nD∞(⊠Nρ||M(ρ))\n=S∞(M(ρ))−S∞(⊠Nρ)\n=logR(ρ)−log1\nλmax(⊠Nρ)\n=logh\nR(ρ)λmax(⊠Nρ)i\nÉlog\"\n1+λ(N−1)/2R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ)#\nÉλ(N−1)/2R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ),\n→0,asN→ ∞ .\nwhere the second line comes from the Lemma 11, and the fifth line comes from the\ninequality (16). □\n3.1. Entropic q-CLT for qubit systems. Now, let us consider the entropic q-CLT\nfor the qubit systems, where we need to change the definition of quantum convolution\nto the one used in [29].\nDefinition 16 (Key Unitary). The key unitary VforKquantum systems, with each\nsystem containing nqubits, is\nV:=U⊗n=U1,n+1,...,(K−1)n+1⊗U2,n+2,...,(K−1)n+2⊗...⊗Un,2n,...,Kn. (17)\nHere Uis aK-qubit unitary constructed using CNOT gates:\nU:=Ã\nKY\nj=2CNOT j→1!Ã\nKY\ni=2CNOT 1→i!\n, (18)\nandCNOT 2→1|x〉|y〉 = |x+y〉|y〉for any x,y∈Z2.\nDefinition 17 (Convolution of multiple states). Given Kstates ρ1,ρ2,...,ρK, each\nwith n-qubits, the multiple convolution ⊠Kofρ1,ρ2,...,ρKmaps to an n-qubit state:\n⊠K(ρ1,ρ2,...,ρK)=⊠K(⊗K\ni=1ρi)=Tr1ch\nV⊗K\ni=1ρiV†i\n. (19)\nHere Vis the key unitary in Definition 16, and Tr1c[·]denotes the partial trace taken\non the subsystem 2,3...,K, i.e., Tr1c[·]=Tr2,3,...,K[·].\nTheorem 18 (Entropic q-CLT for qubits). Given an n-qubit state ρ, and N=\n2K+1for any integer KÊ1,\nD(⊠Nρ||M(⊠Nρ))=S(M(ρ))−S(⊠Nρ)\nÉlogh\n1+(1−MG(ρ))2N−2¡\nTr£\nρ2¤\nR(ρ)−1¢i\n(20)\nÉ(1−MG(ρ))2N−2¡\nTr£\nρ2¤\nR(ρ)−1¢\n→0,asN→ ∞ .10 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nProof. The proof is the same as the qudit case. □\nTheorem 19 (Rényi entropic q-CLT via magic gap). Given an n-qubit state ρ,\nN=2K+1for any integer KÊ1, and any parameter α∈[1,∞], we have\nDα(⊠Nρ||M(⊠Nρ))=Sα(M(ρ))−Sα(⊠Nρ)\nÉlogÃ\n1+(1−MG(ρ))N−1R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ)!\n(21)\nÉ(1−MG(ρ))N−1R(ρ)s\nTr£\nρ2¤\n−1\nR(ρ),\n→0,asN→ ∞ .\nProof. The proof is the same as the qudit case. □\n4. E NTROPIC QUANTUM INVERSE SUMSET THEOREM\nIn this section, we study the quantum inverse sumset theory in DV quantum\nsystem. For which, we need to define a quantum doubling constant, and then use it\nto quantify the distance between the given state and the stabilizer states.\nDefinition 20 (Quantum-doubling constant). Given an n-qudit state ρ, the quantum-\ndoubling constant is\nδq[ρ]=exp(S(ρ⊠ρ)−S(ρ)). (22)\nIn general, the α-order quantum-doubling constant is\nδq,α[ρ]=exp(Sα(ρ⊠ρ)−Sα(ρ)). (23)\nBased on the definition, the quantum doubling constant is the entropy difference\nof the first step in the q-CLT. Moreover, for a pure state ψ, the quantum-doubling\nconstant δ[ψ] is equal to the magic entropy ME(ψ)=S(ψ⊠ψ) defined in [29] up to a\nlogarithm.\nRemark 21. Similar to the quantum-doubling constant, we can also define the quantum-\ndifference constant is\nδ−\nq[ρ]=exp(S(ρ⊟ρ)−S(ρ)), (24)\nwhere ρ⊟ρ=TrAh\nUs,t(ρ⊗ρ)U†\ns,ti\n, i.e., the complementary channel of the quantum\nchannel ⊠.\nIn this work, we consider the following problem: given a quantum state ρ, how\ncould the quantum-doubling constant δq[ρ] tell the structure of the state ρ, that is,\nhow close the state is to the set of MSPS. We call this the quantum inverse sumset\nproblem. Here, we focus on the pure state case, for which we have the following\nresult.\nTheorem 22 (Quantum inverse sumset theorem using magic gap). Given an\nn-qudit pure state ψ,\n(1)δq[ψ]Ê1, with equality iff ψ∈STAB.\n(2) If 1<δq[ψ]ÉC, then\nD(ψ||M(ψ))ÉlogR(ψ)\nlogR(ψ)−log[1+λ(R(ψ)−1)]logC. (25)ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 11\nProof. (1) It comes directly from the entropy inequality for quantum convolution\nin [1,2].\n(2) First, by the monotonicity of relative entropy under the quantum channel,\nthere exists some factor κψÉ1 such that\nD(ψ⊠ψ||M(ψ)⊠ψ)=D(ψ||M(ψ))κψ.\nBy Lemma 11 and the fact that S(M(ψ)⊠ψ)=S(M(ψ)), it can be rewritten as\nS(M(ψ))−S(ψ⊠ψ)=(S(M(ψ))−S(ψ))κψ.\nSince δq[ψ]>1,ψis not a stabilizer state, and thus κψ<1. Hence\nS(ψ⊠ψ)−S(ψ)=(1−κψ)(S(M(ψ))−S(ψ)),\nwhich implies that\nD(ψ||M(ψ))=1\n1−κψ£\nS(ψ⊠ψ)−S(ψ)¤\nÉ1\n1−κψlogC.\nNow, let us provide an upper bound on the factor κψusing magic gap.\nκψ=D(ψ⊠ψ||M(ψ))\nD(ψ||M(ψ))=S(M(ψ))−S(ψ⊠ψ)\nS(M(ψ))\nÉS(M(ψ))−S2(ψ⊠ψ)\nS(M(ψ)),\nwhere the inequality comes from the fact that Sαis nonincreasing with respect to α.\nLet us assume that Gψis the stabilizer group of ψ, then R(ψ)=dn\n|Gψ|, and thus\nS(M(ψ))=logR(ψ)=logdn\n|Gψ|. Hence\nS(M(ψ))−S2(ψ⊠ψ)\n=logdn\n|Gψ|+logÃ\n|Gψ|\ndn+1\ndnX\n⃗x∉G|Ξψ(⃗x)|4!\nÉlogdn\n|Gψ|+logÃ\n|Gψ|\ndn+λ\ndnX\n⃗x∉G|Ξψ(⃗x)|2!\n=logdn\n|Gψ|+log·|Gψ|\ndn+λµ\n1−|Gψ|\ndn¶¸\n=log·\n1+λµdn\n|Gψ|−1¶¸\n,\nwhere the third line comes the from definition of λ, and the fourth line comes from\nthe fact that\n1=Tr£\nψ2¤\n=1\ndnX\n⃗x|Ξψ(⃗x)|2=|Gψ|\ndn+1\ndnX\n⃗x∉Gψ|Ξψ(⃗x)|2(26)\n□\nHere we consider the inverse sumset theorem for the pure states. For mixed\nstates, the above method cannot provide a good estimate for the property (2), which\nmay require new techniques. We leave it for a future study. However, we can still\nprove the following result for the n-qudit mixed states.12 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nProposition 23. Given an n-qudit state ρ, the quantum-doubling constant satisfies\nthe following properties:\n(1)Positivity: δq[ρ]Ê1, with equality iff ρ∈MSPS.\n(2)Additivity under tensor product: δq[ρ1⊗ρ2]=δq[ρ1]δq[ρ2].\n(3)Invariance under Clifford unitary: δq£\nUρU†¤\n=δq[ρ]for any Clifford uni-\ntary U.\n(4)Monotonicity under partial trace: δq£\nTri£\nρ¤¤\nÉδq[ρ], where Tri[·]denotes\nthe partial trace on the i-th qudit for any i∈[n].\nProof. These results come directly from the properties of quantum Ruzsa divergence\nin Theorem 28. □\nFor the qubit case, we can generalize the quantum doubling constant to the quan-\ntum tripling constant as follows.\nDefinition 24 (Quantum tripling constant). Given an n-qubit state ρ, the quan-\ntum tripling constant is\n˜δq[ρ]=S(⊠3ρ)−S(ρ). (27)\nwhere the quantum convolution ⊠3is defined in (19).\nSimilar to the qudit case, we also have the following result for the n-qubit pure\nstate.\nProposition 25 (Quantum inverse sumset theorem for qubits). Given an n-\nqubit pure state ψ,\n(1)˜δq[ψ]Ê1, with equality iff ψ∈STAB.\n(2) If 1<˜δq[ψ]ÉC, then\nD(ψ||M(ψ))ÉlogR(ψ)\nlogR(ψ)−log[1+λ2(R(ψ)−1)]logC. (28)\nProof. The proof is similar to that of qudit case. □\nNote that, the properties of the quantum-doubling constant in qudits in Proposi-\ntion 23 also hold for the quantum-tripling constant in qubits by using the properties\nof⊠3in [29].\n5. Q UANTUM RUZSA DIVERGENCE\nIn this section, we introduce a quantum Ruzsa divergence. To study its properties,\nwe discuss the quantum sumset inequalities from the point of view of our quantum\nconvolutional framework.\n5.1. Definition and properties of the quantum Ruzsa divergence.\nDefinition 26 (Quantum Ruzsa Divergence). Given a quantum convolution ⊠,\nthe quantum Ruzsa divergence of a state ρwith respect to the state σis\nDRz(ρ||σ) :=S(ρ⊠σ)−S(ρ). (29)\nTheα-order quantum Ruzsa divergence of ρwith respect to σis\nDα,Rz(ρ||σ) :=Sα(ρ⊠σ)−Sα(ρ). (30)ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 13\nDefinition 27 (Symmertized Quantum Ruzsa Divergence). Given two quantum\nstates ρandσ, the symmertized quantum Ruzsa divergence between ρandσis\ndRz(ρ,σ) :=1\n2¡\nS(ρ⊠σ)+S(σ⊠ρ)−S(ρ)−S(σ)¢\n. (31)\nTheα-order quantum Ruzsa divergence between ρandσis\ndα,Rz(ρ,σ) :=1\n2¡\nSα(ρ⊠σ)+Sα(σ⊠ρ)−Sα(ρ)−Sα(σ)¢\n. (32)\nTheorem 28. The quantum Ruzsa divergence satisfies the following properties:\n(1)Positivity: DRz(ρ||σ)Ê0, . Also DRz(ρ||σ)=0iff the state ρis in the abelian\nC*-algebra generated by stabilizer group Sofσ, i.e., ρis a convex sum of MSPSs\nassociated with S. From this we infer that DRz(ρ||ρ)=0, iffρis an MSPS.\n(2)Additivity under tensor product: DRz(ρ1⊗ρ2||σ1⊗σ2)=DRz(ρ1||σ1)+\nDRz(ρ2||σ2).\n(3)Invariance under Clifford unitary: DRz(UρU†||UσU†)=DRz(ρ||σ)for any\nClifford unitary U.\n(4)Monotonicity under partial trace: DRz(Tri£\nρ¤\n||Tri[σ])ÉDRz(ρ||σ), where\nTri[·]denotes the partial trace on the i-th qudit for any i∈[n].\n(5)Convexity in the first term and concavity in the second: DRz(P\nipiρi||σ)ÉP\nipiDRz(ρi||σ), and DRz(ρ||P\niqiσi)ÊP\niqiDRz(ρ||σi), where {pi}iand {qi}iare\nclassical probability distributions.\nProof. (1)DRz(ρ||σ)Ê0 comes from the entropy inequality under convolution, i.e.,\nS(ρ⊠σ)Êmax {S(ρ),S(σ)}[1,2], and the condition for DRz(ρ||σ)=0 is the condition\nfor the equality S(ρ⊠σ)=S(ρ). (See Theorem 58 in [2].)\n(2) This follows from the fact that ( ρ1⊗ρ2)⊠(σ1⊗σ2)=(ρ1⊠σ1)⊗(ρ2⊠σ2).\n(3) This is the commutativity of Clifford unitaries and quantum convolution; see\nLemma 85 in [2]. In other words, there always exists some Clifford unitary U′such\nthat ( UρU†)⊠(UσU†)=U′(ρ⊠σ)U′†for Clifford unitary U.\n(4) First, we have\nS³\n(E⃗xw(⃗x)ρw(⃗x)†)⊠σ´\n−S³\nE⃗xw(⃗x)ρw(⃗x)†´\nÉE⃗xh\nS³\n(w(⃗x)ρw(⃗x)†)⊠σ´\n−S³\nw(⃗x)ρw(⃗x)†´i\n=E⃗xh\nS³\nw(s⃗x)(ρ⊠σ)w(s⃗x)†´\n−S³\nw(⃗x)ρw(⃗x)†´i\n=S(ρ⊠σ)−S(ρ).\nHere the second line comes from the joint convexity of the quantum relative entropy\nD(Us,tρ⊗σU†\ns,t||ρ⊠σ⊗I\ndn), i.e.,\nDµ\nUs,tE⃗xw(⃗x)ρw(⃗x)†⊗σU†\ns,t||E⃗xw(⃗x)ρw(⃗x)†⊠σ⊗I\ndn¶\nÉE⃗xDµ\nUs,tw(⃗x)ρw(⃗x)†⊗σU†\ns,t||w(⃗x)ρw(⃗x)†⊠σ⊗I\ndn¶\n,\nwhere D(Us,tρ⊗σU†\ns,t||ρ⊠σ⊗I\ndn)=S(ρ⊠σ)+nlogd−S(ρ)−S(σ). And the third line\nis a consequence of Proposition 41 in [2]. In fact,\nE³\nw(⃗x)⊗w(⃗y)ρABw(⃗x)†⊗w(⃗y)†´\n=w(s⃗x+t⃗y)ρABw(s⃗x+t⃗y)†,14 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nwhere E(ρAB)=TrBh\nUs,tρABU†\ns,ti\n. Consequently, we only need to prove that\nDRz¡\nTri£\nρ¤\n||Tri[σ]¢\n=S³\n(E⃗xw(⃗x)ρw(⃗x)†)⊠σ´\n−S³\nE⃗xw(⃗x)ρw(⃗x)†´\n. (33)\nTo prove this statement, note that\nµ\nTri£\nρ¤\n⊗Ii\nd¶\n⊠σ\n=³\nE⃗xi∈Vw(⃗xi)ρw(⃗xi)†´\n⊠σ\n=E⃗xi∈Vw(s⃗xi)(ρ⊠σ)w(s⃗xi)†\n=Tri£\nρ⊠σ¤\n⊗Ii\nd,\nwhere the first and last lines come from the fact that E⃗xi∈Vw(⃗xi)(·)w(⃗xi)†=Tri[·]⊗Ii\nd,\nand the second line comes from the following property (See Proposition 41 in [2])\nE(w(⃗x)⊗w(⃗y)ρABw(⃗x)†⊗w(⃗y)†)=w(s⃗x+t⃗y)ρABw(s⃗x+t⃗y)†,\nwhere E(ρAB)=TrBh\nUs,tρABU†\ns,ti\n. Repeating the above process for σ, we obtain the\nfollowing result\nµ\nTri£\nρ¤\n⊗Ii\nd¶\n⊠σ=Tri£\nρ⊠σ¤\n⊗Ii\nd=Tri£\nρ¤⊠Tri[σ]⊗Ii\nd. (34)\nHence, we have\nS³\n(E⃗xw(⃗x)ρw(⃗x)†)⊠σ´\n=Sµ\nTri£\nρ¤⊠Tri[σ]⊗Ii\nd¶\n=S¡\nTri£\nρ¤⊠Tri[σ]¢\n+logd,\nS³\nE⃗xw(⃗x)ρw(⃗x)†´\n=Sµ\nTri£\nρ¤\n⊗Ii\nd¶\n=S¡\nTri£\nρ¤¢\n+logd.\nHence, we have\nS³\n(E⃗xw(⃗x)ρw(⃗x)†)⊠σ´\n−S³\nE⃗xw(⃗x)ρw(⃗x)†´\n=DRz¡\nTri£\nρ¤\n||Tri[σ]¢\n.\nTherefore, we obtain the result.\n(5) The convexity of DRz(ρ||σ) with respect to the state ρcomes directly from the\njoint convexity of the quantum relative entropy D(Us,tρ⊗σU†\ns,t||ρ⊠σ⊗I\ndn). That is,\nD(X\nipiUs,tρi⊗σU†\ns,t||X\nipiρi⊠σ⊗I\ndn)ÉX\nipiD(Us,tρi⊗σU†\ns,t||ρi⊠σ⊗I\ndn).\nAnd, the concavity of DRz(ρ||σ) with respect to the state σcomes directly from the\nconcavity of the von Neuman entropy S(·). That is,\nS(X\niqiρ⊠σi)ÊX\niqiS(ρ⊠σi).\n□\nNote that the quantum Ruzsa divergence differs significantly from the quantum\nrelative entropy. For instance, the quantum relative entropy for identical states is\nalways zero, i.e., D(ρ||ρ)=0, which does not hold for quantum Ruzsa divergence.\nThe symmetrized quantum Ruzsa divergence also satisfies the properties of The-\norem 28; one can repeat the proof above. Moreover, the properties (1)-(3) also hold\nfor the α-order quantum Ruzsa divergence for 1 Éα< +∞ ; this is a consequence of\nthe property of quantum Rényi entropy and Lemma 57 in [2].ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 15\nConjecture 29 (Triangle inequality for the quantum Ruzsa divergence). Given\na balanced convolution ⊠s,t, i.e., s≡tmod d, the quantum Ruzsa divergence satisfies\nthe triangle inequality\nDRz(ρ||τ)ÉDRz(ρ||σ)+DRz(σ||τ), (35)\nwhich is equivalent to\nS(ρ⊠τ)+S(σ)ÉS(ρ⊠σ)+S(σ⊠τ). (36)\nNote that the inequality (36) is not balanced; the state σappears once on the\nleft-hand side but twice on the right-hand side. Hence, we proposed the following\nbalanced version, which we call \"convolutional strong sub-additivity\".\nConjecture 30 (Convolutional strong subadditivity). Given three n-qudit quan-\ntum states ρ,σ,τ, we have\nS(ρ⊠τ⊠σ)+S(σ)ÉS(ρ⊠σ)+S(σ⊠τ), (37)\nwhere ρ⊠τ⊠σ:=(ρ⊠s,tτ)⊠l,mσ,ρ⊠σ:=ρ⊠s,tσ,σ⊠τ:=σ⊠s,tτ,s2+t2≡1 mod s,\ns≡tmod d,m≡ls, and l2+m2≡1 mod d.\nThe reason that we choose the parameters in this way is to generalize the classical\nbalanced convolution on three random variablesX+Y+Zp\n3=q\n2\n3³X+Yp\n2´\n+1p\n3Zto the\nquantum case.\nLemma 31. If the convolutional strong subadditivity holds, then the triangle in-\nequality of quantum Ruzsa divergence also holds.\nProof. This comes from the entropy inequality for quantum convolution S(ρ⊠τ⊠σ)Ê\nmax {S(ρ⊠τ),S(σ)}[1,2]. □\nHere, let us show that the convolutional strong subadditivity holds with some\nadditional assumption on the states.\nProposition 32. The convolutional strong subadditivity holds for the following two\ncases,\n(1) the quantum states ρ,σ, and τare all stabilizer states;\n(2) the quantum states ρ,σ, and τare diagonal states in the computational basis.\nProof. (1) Let us assume that quantum states ρ,σ, and τare all stabilizer states\nwith stabilizer group Gρ,GσandGτ. then the absolute value of the characteristic\nΞρis either to 1 or 0, and Ξρ(⃗x) equals 1, iff the Weyl operator w(⃗x) belongs to the\nstabilizer group. Similar arguments also work for σandτ.\nSince\nΞρ⊠τ⊠σ(⃗x)=Ξρ(ls⃗x)Ξτ(lt⃗x)Ξσ(m⃗x),\nwith s2+t2≡1 mod d,l2+m2≡1 mod d, then |Ξρ⊠τ⊠σ(⃗x)|is equal to 1 or 0, and\n|Ξρ⊠τ⊠σ(⃗x)| =1,iffw(⃗x)∈Gρ∩Gσ∩Gσ.\nThat is, ρ⊠τ⊠σis a stabilizer state with Gρ∩Gσ∩Gτas the stabilizer group, and\nthe quantum entropy is\nS(ρ⊠τ⊠σ)=logdn\n|Gρ∩Gσ∩Gτ|.16 KAIFENG BU, WEICHEN GU, AND ARTHUR JAFFE\nUsing the same reasoning, we can also prove that ρ⊠σis a stabilizer state with\nGρ∩Gσas the stabilizer group, and the quantum entropy is\nS(ρ⊠σ)=logdn\n|Gρ∩Gσ|.\nσ⊠τis a stabilizer state with Gσ∩Gτas the stabilizer group, and the quantum\nentropy is\nS(σ⊠τ)=logdn\n|Gσ∩Gτ|.\nBesides, the quantum entropy of the stabilizer state σis\nS(σ)=logdn\n|Gσ|.\nHence, to prove the convolutional strong sub-additivity, it is equivalent to proving\nthe following statement,\n1\n|Gρ∩Gτ∩Gσ||Gσ|É1\n|Gρ∩Gσ||Gσ∩Gτ|.\nThat is\n|Gρ∩Gσ|\n|Gρ∩Gσ∩Gτ|É|Gσ|\n|Gσ∩Gτ|.\nSince Gρ,Gσ,Gτare all finte abelian group, then above statement is equivalent to\nthe\n|Gρ∩Gσ:Gρ∩Gσ∩Gτ| É |Gσ:Gσ∩Gτ|,\nwhere the [ G:H] is the index of a subgroup H in a group G. This comes from the\nfollowing property of the index of the group: if H,Kare subgroups of G, then\n|H:H∩K| É |G:K|.\n(2) Let us consider that 2 s2≡1 mod d,m≡ls, and l2+m2≡1 mod d. Since\nρ,σ,τare diagonal in the computational basis, they can be written as follows\nρ=X\n⃗x∈Zn\ndp(⃗x)|⃗x〉〈⃗x|,σ=X\n⃗x∈Zn\ndq(⃗x)|⃗x〉〈⃗x|,τ=X\n⃗x∈Zn\ndr(⃗x)|⃗x〉〈⃗x|, (38)\nwhere {p(⃗x)},{q(⃗x)},{r(⃗x)}are probability distributions on Zn\nd.\nLet us consider three random variables X,Y,Z, which takes values in Zn\ndas fol-\nlows\nPr[X=⃗x]=p(⃗x),Pr[Y=⃗x]=q(⃗x),Pr[Z=⃗x]=r(⃗x).\nThen, after some calculation, we find that\nS(ρ⊠τ⊠σ)=H(X+Y+Z),S(σ)=H(Y);\nS(ρ⊠σ)=H(X+Y),S(σ⊠τ)=H(Y+Z),\nwhere H(X) is the Shannon entropy of the discrete random variable X, andX+Yis\nthe sum of random variables mod d. Hence, the convolutional strong-subadditivity\nin this case is reduced to the classical case\nH(X+Y+Z)+H(Y)ÉH(X+Y)+H(Y+Z) (39)\nwhich comes directly from the data-processing inequality I(X:X+Y+Z)ÉI(X:\nX+Y).ENTROPIC QUANTUM CENTRAL LIMIT THEOREM AND QUANTUM INVERSE SUMSET THEOREM 17\n□\nNote that, in the convolutional strong subadditivity (37), we consider the quantum\nconvolution of three states as ( ρ⊠τ)⊠σ, where s2+t2≡1 mod s,s≡tmod d,m≡ls,\nand l2+m2≡1 mod d. We may also consider other possible quantum convolutions\non three input states, like the one in (19) defined on qubit-systems.\nMoreover, the usual strong subadditivity in quantum information theory is: given\na tripartite state ρABC, it holds that S(ρABC)+S(ρC)ÉS(ρAC)+S(ρBC), which was\nconjectured by Robinson, Ruelle [63] and Lanford, Robinson [64] and later proved\nby Lieb and Ruskai [59]. Our inequality (37) shares a similar form, so that we call\n(37) as \"Convolutional strong subadditivity\". Besides the strong subadditivty, there\nis a subadditivity for bipartite states, i.e., S(ρAB)ÉS(ρA)+S(ρB). It is natural to\nconsider the convolutional subaddivity or supaddivity. However, neither of them\nholds, as we can give some counterexamples in the following theorem.\nTheorem 33 (No subaddivity or supaddivity for the quantum convolution).\n(1) There exist quantum states ρandσsuch that\nS(ρ⊠σ)>S(ρ)+S(σ). (40)\n(2) There also exist quantum states ρ,σsuch that\nS(ρ⊠σ)S(ρ)+S(σ)=0.\n(2) Let us take both ρandσto be the maximally mixed state I/dn, then\nS(ρ)=S(σ)=nlogd.\nMoreover, ρ⊠σ=I/d, then\nS(ρ⊠σ)=nlogd0 initially. We plot\nin Fig. 1 CVs with donor masses and periods, estimated from\nsuperhump periods1assuming M1=0.75M⊙by Knigge (2006,\ntheir Table 1)2. There is quite a bit of scatter among these points\nand they do not seem to converge on a unique evolutionary track.\nThis illustrates that varying strengths of MB likely operate be-\nlow the period gap3. However, studying the significance of this\neffect is beyond the scope of this letter. The track with κ=15\nmatches with the systems with the biggest Porbfor a given M2\nin the catalog of Knigge (2006). So, hereinafter, we assume that\nthe tracks with κ=15 andκ=0 exhibit respectively the upper\nand lower limits of Porbfor a given M2.\nConsider the other set of tracks where κvaries with stellar\nparameters. This set illustrates the behavior of the system when\nthe strength of MB depends on stellar structure and changes as\nthe donor evolves. We use the result of the strong- and weak-field\ndynamo for fully convective low-mass stars proposed by Morin\net al. (2011) to model such a MB strength. They argue, based\non spectropolarimetric observations by Morin et al. (2010), that\ntwo di fferent magnetic field profiles exist in isolated fully con-\nvective stars with similar rotation rates and masses. The first is\na strong and steady axial dipole field and the second is a weak\nmultipolar non-axisymmetric field that is changing rapidly. Be-\n1We do not see any CVs in the catalog of Knigge (2006) following\nthe zero MB track in Fig. 1. This is because superhumps are a result\nof donor-induced eccentricity in their accretion disk and polars do not\nform an accretion disk.\n2These are pre-bounce CVs. The M2for post-bounce CVs is calcu-\nlated di fferently (Sect. 2.3).\n3We note that the scatter of the systems in the catalog of Knigge\n(2006) may be the result of other processes, such as irradiation of the\ndonor by the WD, which can alter the thermal timescale of the donor.\nArticle number, page 2 of 8Sarkar et al.: Weakening of magnetic braking explains the dearth of period bouncers\ncause donors in short-period CVs are fully convective, it is pos-\nsible that a strong-field dynamo also operates in such CV donors\nwhere it drives MB. So, we use the formula for the magnetic field\ngiven by Morin et al. (2011, their Eq. (2)) to define κ. For details\non how they derive their magnetic field expression, we urge the\nreader to refer to their Sect. 4.2. Other physical mechanisms that\nmay lead to a stellar-dependent κare discussed in Sect. 3.\nWe define a dimensionless quantity Bas a proxy for the mag-\nnetic field as4\nB=6 kG\n19.5 kG M2\nM⊙!1/2 R2\nR⊙!−1 L2\nL⊙!1/6\u0012Porb\nd\u0013−1/2\n, (1)\nwhere R2andL2are the radius and the luminosity of the donor.\nWe compute these using MESA. The last term in Eq. (2) of\nMorin et al. (2011) is ( Pspin/d)−1/2, where Pspinis the spin pe-\nriod of the M-dwarf. This becomes ( Porb/d)−1/2in our Eq. (1)\nbecause of tidal locking. The denominator 19.5 kG is the dipolar\nfield at the time of the commencement of Roche lobe overflow\n(RLOF). This ensures that B<1 throughout the evolution. We\nplot two tracks where κ=15B2andκ=15B4. The exponents\nare ad hoc but highlight the varying degrees of the dependence of\nMB strength on the magnetic field, and hence, the stellar struc-\nture. They also lead to the system attaining Pminat 86 and 80\nmin respectively (Fig. 1), which are the upper and lower limit of\nthe observed period minimum spike reported by Gänsicke et al.\n(2009). The behavior of Bcan be understood as follows. Be-\ncause of RLOF and the fact that the donors are close to ther-\nmal equilibrium, R2,L2andPorbare functions of M2, and so\nB≡B(M2).Porb∝R3/2\n2M−1/2\n2. For our donors L2∝Mβ\n2, where\n2≲β≲4. If we define R2∝Mα\n2, we get B∝M3/4+β/6−7α/4\n2.\nWe haveα > 0 pre-bounce and α≲0 post-bounce. Choosing\nβ=3,α=0.6 pre-bounce and α=0.3 post bounce (similar to\neq. (16) of Knigge et al. 2011), we get B∝M0.2\n2pre-bounce and\nB∝M0.725\n2post-bounce. So, post-bounce Bdecreases strongly\nbecause of a change in the M2−R2relation of the donor. The\nevolution of Bis shown in Fig. 1.\n2.3. Results\nWe follow the evolution of the models with κ=15B2and\nκ=15B4in the M2−Porbplane. At M2≈0.2M⊙, these systems\nare driven by strong MB so they follow the track with κ=15.\nHowever, Bstarts decreasing gradually at M2≈0.125M⊙and\nsubstantially when M2≲0.05M⊙. This leads to the weakening\nof the MB strength in these systems. We note importantly that for\nall our models, the absolute value of AML decreases as the CV\nevolves (see Appendix B). So, when we say ‘weakening’ of MB,\nwe mean the additional weakening of the MB strength caused\nbyB(Fig. B.1). The weakening of MB is such that the donor\nstar always adjusts to it on its thermal timescale. The extent of\nthe weakening depends on the power of B. Close to their respec-\ntivePmin, MB becomes negligible. This can be understood with\nEq. (1)–further evolution decreases M2and increases R2and, as\na consequence, Porb. These systems, now only driven by GWR,\nevolve further to join the κ=0 track. This causes their evolution\ntimescale to drastically increase around and beyond their Pmin.\nOwing to their long evolutionary timescales, these systems stall\n4There is an additional term ( η⊙/ηref)1/2in the expression of the mag-\nnetic field in Morin et al. (2011). Here ηref≡1011cm2s−1is the mag-\nnetic di ffusivity and η⊙is the reference magnetic di ffusivity. Studying\nhow this term varies for our CVs is beyond the scope of this work. So\nwe set (η⊙/ηref)=1.in the period minimum spike and spend a lot of time there com-\npared to systems evolved with a constant κ. This can be seen\nwith the pluses marked on each track in Fig. 1 that denote a time\ninterval of 100 Myr. Around their period minimum, the pluses\nare concentrated much more in the κ=15B4track than in the\nκ=4 track. We define period bouncer CVs as systems with a\nbrown dwarf donor ( M2≤0.07M⊙) with Porb≥80 min (Pala\net al. 2020) and calculate the amount of time each of our models\nspends in the period minimum spike as a period bouncer. Al-\nthough the tracks with κ=4 andκ=15B4have approximately\nthe same Pmin, the latter spends about 1.2 Gyr as a bouncer, while\nthe former spends only about 0.44 Gyr. Because the systems are\nclustered around the period minimum spike, it is very di fficult\nto distinguish between pre-bounce and post-bounce systems ob-\nservationally (Pala et al. 2018). We highlight that the weaken-\ning MB models also reproduce the period minimum reported by\nKnigge et al. (2011) but that the M2at which Pminis attained is\nmuch smaller than the 0.069 M⊙reported by Knigge et al. (2011).\nSo, if MB weakens in near- PminCVs, our models suggest that\nmost of the period bouncer candidates in Fig. 1 are pre-bounce\nCVs.\nWe also analyse our models in the ˙M2−Porbplane in Fig. 2.\nWe compute the accretion rate of a system ˙M2using the relation\nderived by Townsley & Bildsten (2004),\nLWD=10−3L⊙ ˙M2\n10−10M⊙yr−1! MWD\n0.9M⊙!0.4\n(2)\nwhich relates the WD mass, radius, and temperature to the ac-\ncretion rate (Table A.1). It is also possible to estimate accretion\nrates from the X-ray luminosity or disk luminosity. The former\nrequires a model of the X-ray emission mechanism, and the latter\na model of disk geometry. Both require an estimate of accretion\nefficiency, which is often parameterized as ηin the following:\nL=η\n2GM WD˙M2\nRWD(3)\nwhere RWDis the WD radius and Lis the observed accretion lu-\nminosity (either from the disk or from the boundary layer, in X-\nrays). However, the range of ηin CVs is a subject of current de-\nbate (see Sect. 6.1 of Mukai 2017 for a thorough explanation). As\nan example, one model of accretion is advective dominated ac-\ncretion flow (ADAF), which was first applied to explain the hard\nX-ray spectra of CVs in Narayan & Popham (1993). It was later\nextended in Narayan et al. (1996) to X-ray binaries observed in\na low accretion state. In this work, accretion e fficiencies were\nshown to be very low, with ηbetween 10−3to 10−4. From Eq. (3),\nit is clear how failing to incorporate low e fficiencies could lead\nto an underestimate of accretion rate, given an observed lumi-\nnosity. More recently, Liu et al. (2008) applied the ADAF model\nto X-ray spectra of CVs and found good agreement. Neverthe-\nless, Mukai (2017) warns that a complete analysis of accretion\nefficiency in CVs, which takes into account interactions between\ndisk annuli, is still needed. Later we demonstrate the sensitivity\nof our predictions to the method of obtaining ˙M2.\nAll our candidate bouncer CVs (Table. A.1) with mass trans-\nfer via RLOF5have ˙M2(estimated by Eq. (2)) about a few\ntimes 10−11M⊙yr−1(also see Pala et al. 2022, where all ˙M2>\n5Three magnetic CVs have been discovered with ˙M2≈10−14M⊙yr−1\n(Muñoz-Giraldo et al. 2023). These mass-transfer rates are two orders\nof magnitude smaller than that predicted by the model evolved with GR.\nThis may be because the mass-transfer rates are underestimated (see the\ndiscussion on Eq. 3).\nArticle number, page 3 of 8A&A proofs: manuscript no. aanda\n10−1210−1110−1010−9\n˙M2/M⊙yr−16080100120140160Porb/min\nEZ Lyn∗\nSRGeJ0411 + 6853∗\nEZ Lyn\nSDSSJ1035 + 0551\nSDSSJ1057 + 2759\nSDSSJ1433 + 1011\nSDSSJ1501 + 5501\nSRGeJ0411 + 6853\nEG Cnc\nGD 552\n1RXSJ105010−1404\nQZ Lib\nSDSSJ1435 + 2336\nBW Scl\nGW Lib\nWZ Sge\nFig. 2. The evolution of CVs below the period gap. Solid lines show evolution in the Porb−˙M2plane for the same tracks as in Fig. 1. Pluses on\neach track denote timesteps of 100 Myr. The dotted vertical line denotes ˙M2=10−11M⊙yr−1. Observed period bouncer candidates from Table A.1\nare also plotted. Eclipsing systems are plotted as circles while non-eclipsing systems are plotted as triangles. The systems labeled and marked with\nstars have their ˙M2derived from X-ray luminosity (see the discussion in Sect. 2.3). The horizontal shaded region is the observed period minimum\nspike (80 ≲Porb/min≲86) reported by Gänsicke et al. (2009).\n10−11M⊙yr−1). So we assume an optimistic detection threshold\nof˙M2=10−11M⊙yr−1such that any system below this limit is\nundetectable. This is likely to change with emerging data from\noptical and X-ray surveys, such as SDSS-V (Kollmeier et al.\n2017) and SRG /eROSITA (Predehl et al. 2021; Sunyaev et al.\n2021), respectively. The former has already led to the discovery\nof new period bouncer candidates, optically fainter than much\nof the population (Inight et al. 2023). The latter is 5 to 15 times\ndeeper than the last all-sky X-ray survey, potentially revealing\nsystems with lower accretion rates, for instance, the bouncer can-\ndidate reported by Galiullin et al. (2024). In addition, Rodriguez\n(2024) has presented a method to discover optically faint CVs\nsuch as period minimum and period bouncer systems using their\nX-ray /optical properties. This could reveal new systems in up-\ncoming SRG /eROSITA data.\nUnder the above assumption on detection, the system with\nκ=15B4never emerges from the period minimum spike as a\ndetectable period bouncer. This track explains observed candi-\ndates clustered at the lower end of the period minimum spike in\nFig. 2. The track with κ=15B2bounces at about 86 min but\nbecomes undetectable at about 90 min. This track explains ob-\nserved candidates clustered at the upper end of the period mini-\nmum spike. The system with κ=4 emerges from the period min-\nimum spike with ˙M2>10−11M⊙yr−1. So, if such a constant κis\nat play post bounce, there should be systems populating the re-\ngion with 86 ≲Porb/min≲105 and ˙M2≳10−11M⊙yr−1. These\nare not observed, indicating further that MB weakens post-\nperiod minimum. Once SRG /eROSITA unveils systems with\n˙M2≈10−12M⊙yr−1, theκ=15B4track indicates a population\nof systems upto Porb≈110 min and the κ=15B2track upto\nPorb≈115 min.It is important to note that our accretion rate estimates, based\non WD properties (Eq. (2)), place the accretion rates of systems\nsuch as EZ Lyn (Amantayeva et al. 2021) and SRGeJ0411 +6853\n(Galiullin et al. 2024) nearly an order of magnitude higher than\nthat reported by authors using X-ray or disk luminosities. Aman-\ntayeva et al. (2021) estimated the accretion rate based on the\noptical disk luminosity, and assumed η=1 in Eq. (3) to ob-\ntain ˙M2≈3×10−12M⊙yr−1(EZ Lyn∗in Fig. 2). Galiullin\net al. (2024) incorporated a bolometric correction to the X-ray\nluminosity, which assumed a thermal bremsstrahlung model for\nthe emission, to obtain ˙M2≈(1.7−7.8)×10−12M⊙yr−1\n(SRGeJ0411 +6853∗in Fig. 2). However, they did not explore\na range of radiative e fficiencies. In both cases, the accretion\nrates could have been underestimated. Another reason why\nthese may be underestimated is because their ˙M2(lower end of\nSRGeJ0411 +6853∗) are smaller than that by our κ=0 model.\nAssuming that the CV remains semidetached, the estimates by\ntheκ=0 model set the minimum accretion rate post-bounce.\nRegardless, the ˙M2of EZ Lyn∗is only a factor of 2 smaller than\nthat predicted by our κ=15B4model. It will agree with our\nmodel if we choose η=0.5 in Eq. (3) to calculate ˙M2. The ˙M2\nof SRGeJ0411 +6853∗is already in general agreement with both\ntheκ=15B2andκ=15B4models. Our model tracks agree\nwell with several systems in Fig. 2, but notably our κ=15B4\nmodel is in good agreement with all the estimates of SDSSJ1501\nand SDSSJ1035, namely Porb,M2and ˙M2, while our κ=15B2\nmodel is in agreement with the PorbandM2estimate of EZ Lyn\nand within a factor of 2 of its ˙M2estimate.\nArticle number, page 4 of 8Sarkar et al.: Weakening of magnetic braking explains the dearth of period bouncers\n3. Physical processes driving the weakening of\nmagnetic braking\nWe highlight a few physical processes that may cause the weak-\nening of magnetic braking in short-period CVs. We note that this\nlist is not exhaustive and there can be additional mechanisms\ndriving such a weakening.\n3.1. Dynamo action in cool stars\nIn Sect. 2.2 we showed that if the strong-field dynamo pro-\nposed by Morin et al. (2011) operates in short-period CV donors,\nEq. (1) causes Bto reduce significantly for M2≲0.07M⊙. There\nis observational evidence to suggest that stars with Teff≲2200 K\nsuch as L-dwarfs experience a significant decrease in their chro-\nmospheric activity despite being rapid rotators (Mohanty &\nBasri 2003). This means that the magnetic field strength drops\nfrom fully convective M-dwarfs to brown dwarfs. In Sect. 2.2\nwe show that this drop is due to the change in the mass-radius\nrelation of the star. The results of the α2dynamo model proposed\nby Chabrier & Küker (2006) also show, in accordance with that\nby Morin et al. (2011), that there is a transition in the magnetic\nfield structure from a steady, large-scale field in late M-dwarfs\nto a toroidal, oscillatory field in brown dwarfs. In addition, the\nconductivity of the atmosphere of cool objects such as brown\ndwarfs decreases greatly, thereby hampering the formation of a\nhot corona which drives stellar winds. The combined e ffect of\nweaker stellar winds and reduced magnetic field strength drives\na weaker MB in brown dwarfs (Mohanty & Basri 2003; Chabrier\n& Küker 2006). In other words, if such a dynamo operates in\nshort-period CV donors, MB reduces significantly as the donor\nenters the brown dwarf regime ( M2≲0.07M⊙).\n3.2. White dwarf magnetism\nIsern et al. (2017) suggested that cool WDs generate strong mag-\nnetic fields by a crystallization-driven dynamo. Schreiber et al.\n(2021) showed that magnetic CVs can be explained by the rapid\nrotation and crystallization of the WD accretors, which can gen-\nerate fields of several MG (Ginzburg et al. 2022). Schreiber et al.\n(2023) recently proposed that such fields are generated in the ac-\ncretor of short-period CVs post period minimum. This field con-\nnects with that of the donor star, resulting in the detachment of\nperiod bouncers for several Gyr. They argue that this can lead to\nabout a 60% reduction in the observed period-bouncer CVs.\nWe illustrate a variation in their analysis where the CV may\nremain semidetached. Schreiber et al. (2023) assume that the\ndiffusion timescale of the magnetic field to the WD surface is\n100 Myr (Fig. 3 of Ginzburg et al. 2022). However, recently\nBlatman & Ginzburg (2023) showed that the magnetic field on\nthe WD surface gradually emerges on a Gyr timescale (their\nbottom right subplot in Fig. 1). By consistently taking into ac-\ncount phase separation, they find that the magnetic di ffusion time\nis about a Gyr at the time of breakout and shortly afterward\n(this also depends on the WD mass).The donor has its thermal\ntimescale about a few Gyr depending on the mass-transfer rate.\nSince the thermal timescale of the donor is comparable to the\ndiffusion timescale of the WD magnetic field, there is a possi-\nbility that the donor adjusts to the reduction in MB because of\nmagnetic reconnection post-period minimum while continually\nfilling its Roche lobe. In such a case, the evolution will be simi-\nlar to that given in Sect. 2. However, such a weakening depends\non the properties of the WD accretor, such as its mass and tem-\nperature, but is independent of the donor star transitioning froman M-dwarf to a brown dwarf. So, such systems would not nec-\nessarily experience a MB weakening at M2≈0.07M⊙but when\nthe WD becomes magnetic (Schreiber et al. 2023).\n3.3. Dead zones\nThe dead zone is the region around a spinning magnetized star\nwhere the stellar wind is captured and forced to corotate along\nits magnetic field lines (Mestel & Spruit 1987). This leads to a\nreduction of wind mass loss and, as a consequence, the strength\nof MB. Dead zones were first studied by Mestel & Spruit (1987)\nwho gave a simple description for isolated solar-like stars with\ndifferent rotation rates. Subsequently, several groups have im-\nplemented the e ffects of dead zones in their calculations of MB\ntorque in stellar spin-down (Réville et al. 2015; Garra ffo et al.\n2015). Because dead zones arise through the interplay of grav-\nity, centrifugal force, and magnetism in the star, they should be\nat play in every system undergoing MB. This includes the donor\nstars in CVs. The only di fference here is that, owing to tidal lock-\ning,Porbgoverns the behavior of the dead zone. We calculate the\nevolution of dead zones using the simple treatment of Mestel\n& Spruit (1987, their Eqs (8) and (9)), adopting solar parame-\nters for the coronal temperature and mean molecular weight. The\nchoice of these parameters does not alter the qualitative behavior\nof our dead zone calculation.\nFor the expression of the ratio of the magnetic pressure and\nthe thermal pressure at the base of the dead zone ζd, we study\nthe behavior of two cases: ζd=60(Ω/Ω⊙) andζd=60(Ω/Ω⊙)2,\nin their Table 1. Here Ωis the orbital angular velocity of the CV .\nThe evolution of the dead zone of the donor star for the models\nwith constant κin Fig. 1 is shown in Fig. 3. Here fDZ=R2/RDZ\nis the fraction of field lines contributing to MB in the system,\nwhere RDZis the equatorial radius of the dead zone. With no\ndead zones fDZ=1. The value fDZ,idenotes the contribution\nof dead zones at the time of commencement of RLOF6. These\ntracks demonstrate how the dead zones would behave in a short-\nperiod CV . We see that when ζd∝Ω,fDZchanges very little\nthroughout the evolution. However, the dead zones grow ( fDZbe-\ncomes smaller) with decreasing M2whenζd∝Ω2, with the drop\nbecoming steep at M2≈0.05M⊙. A stronger dependence of ζd\nonΩyields a steeper drop in fDZ. One way in which MB a ffects\ndead zones is through the generated magnetic field in the donor\n(say, by a strong-field dynamo or an α2dynamo) which governs\nthe magnetic pressure outside the star (through ζd). Dead zones\nwork as an additional mechanism of MB alteration which is al-\nways at play regardless of the physical mechanism that drives\nMB. It can further weaken MB if d ln ζd/d lnΩ≳2 (Fig. 3).\n4. Conclusion\nIn this Letter, we have shown that the weakening of magnetic\nbraking in short-period CVs can explain the current dearth of\nobserved period bouncers. We find that the weakening of mag-\nnetic braking around the period minimum stalls the evolution\nof CVs around the observed period minimum spike between 80\nand 86 min. This makes them di fficult to distinguish from pre-\nbounce systems. When they evolve post-period minimum, their\nmass-transfer rate decreases below the current detectable thresh-\nold. We predict the system properties of fainter period bouncer\ncandidates that upcoming surveys such as SRG /eROSITA can\n6We note that this plot is made post-evolution so that dead zones do\nnot alter the MB strength of these models.\nArticle number, page 5 of 8A&A proofs: manuscript no. aanda\n0.05 0.10 0.15 0.20\nM2/M⊙0.700.750.800.850.900.951.00fDZ/fDZ,i\nζd= 60(Ω/Ω⊙)2\nζd= 60(Ω/Ω⊙)1\nFig. 3. The evolution of the dead zone relative to that at the beginning\nof RLOF fDZ/fDZ,iwith M2for the models with constant κ. The colors\ndenote the same models as in Figs 1 and 2, with κ=0 shown in blue,\nκ=4 in green and κ=15 in black. The line styles denote the choice\nofζd. The dead zone for each track is calculated post-evolution by the\nmethod of Mestel & Spruit (1987).\nlikely detect. We discuss how the weakening of magnetic brak-\ning can be caused by physical processes such as a change in the\ndynamo action in the donor, the emergence of magnetism in the\nwhite dwarf accretor and dead zones in the donor trapping stel-\nlar winds. An accurate estimate of the relative fraction of period\nbouncers with this formalism will be obtained with a population\nsynthesis study which we shall undertake in the future.\nAcknowledgements. AS thanks the Gates Cambridge Trust for his scholarship.\nAS also thanks Ken Shen and Elmé Breedt for discussions on the nature of short-\nperiod cataclysmic variables. ACR acknowledges support from an NSF Gradu-\nate Research Fellowship. AS and ACR are grateful to Franco Giovanelli and\nthe Golden Age of Cataclysmic Variables and Related Objects VI Workshop for\nfacilitating fruitful conversations. SG acknowledges support from the Israel Min-\nistry of Innovation, Science, and Technology (grant No. 1001572596), and from\nthe U.S. – Israel Binational Science Foundation (BSF; grant No. 2022175). AS\nand SG thank Daniel Blatman for the discussion on the emergence of magnetic\nfields in white dwarfs. CAT thanks Churchill College for his fellowship.\nReferences\nAmantayeva, A., Zharikov, S., Page, K. L., et al. 2021, ApJ, 918, 58\nBlatman, D. & Ginzburg, S. 2023, arXiv e-prints, arXiv:2311.09299\nChabrier, G. & Küker, M. 2006, A&A, 446, 1027\nGaliullin, I., Rodriguez, A. C., Kulkarni, S. R., et al. 2024, arXiv e-prints,\narXiv:2401.04178\nGänsicke, B. T., Dillon, M., Southworth, J., et al. 2009, MNRAS, 397, 2170\nGarra ffo, C., Drake, J. J., & Cohen, O. 2015, ApJ, 813, 40\nGinzburg, S., Fuller, J., Kawka, A., & Caiazzo, I. 2022, MNRAS, 514,\n4111–4119\nGoliasch, J. & Nelson, L. 2015, ApJ, 809, 80\nInight, K., Gänsicke, B. T., Schwope, A., et al. 2023, MNRAS, 525, 3597\nIsern, J., García-Berro, E., Külebi, B., & Lorén-Aguilar, P. 2017, ApJ, 836, L28\nJermyn, A. S., Bauer, E. B., Schwab, J., et al. 2023, ApJS, 265, 15\nKnigge, C. 2006, MNRAS, 373, 484\nKnigge, C., Bara ffe, I., & Patterson, J. 2011, ApJS, 194, 28\nKolb, U. 1993, A&A, 271, 149\nKollmeier, J. A., Zasowski, G., Rix, H.-W., et al. 2017, arXiv e-prints,\narXiv:1711.03234\nLi, J. K., Wu, K. W., & Wickramasinghe, D. T. 1994, MNRAS, 268, 61\nLittlefair, S. P., Dhillon, V . S., Marsh, T. R., et al. 2008, MNRAS, 388, 1582\nLiu, F. K., Meyer, F., Meyer-Hofmeister, E., & Burwitz, V . 2008, A&A, 483, 231\nMcAllister, M. J., Littlefair, S. P., Dhillon, V . S., et al. 2017, MNRAS, 467, 1024\nMestel, L. & Spruit, H. C. 1987, MNRAS, 226, 57\nMohanty, S. & Basri, G. 2003, ApJ, 583, 451\nMorin, J., Donati, J. F., Petit, P., et al. 2010, MNRAS, 407, 2269\nMorin, J., Dormy, E., Schrinner, M., & Donati, J. F. 2011, MNRAS, 418, L133Muñoz-Giraldo, D., Stelzer, B., de Martino, D., & Schwope, A. 2023, A&A,\n676, A7\nMukai, K. 2017, PASP, 129, 062001\nNarayan, R., McClintock, J. E., & Yi, I. 1996, ApJ, 457, 821\nNarayan, R. & Popham, R. 1993, Nature, 362, 820\nNeustroev, V . V . & Mäntynen, I. 2023, MNRAS, 523, 6114\nPaczynski, B. & Sienkiewicz, R. 1981, ApJ, 248, L27\nPala, A. F., Gänsicke, B. T., Belloni, D., et al. 2022, MNRAS, 510, 6110\nPala, A. F., Gänsicke, B. T., Breedt, E., et al. 2020, MNRAS, 494, 3799\nPala, A. F., Schmidtobreick, L., Tappert, C., Gänsicke, B. T., & Mehner, A. 2018,\nMNRAS, 481, 2523\nPaxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3\nPaxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4\nPaxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15\nPaxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34\nPaxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10\nPredehl, P., Andritschke, R., Arefiev, V ., et al. 2021, A&A, 647, A1\nRéville, V ., Brun, A. S., Matt, S. P., Strugarek, A., & Pinto, R. F. 2015, ApJ, 798,\n116\nRodriguez, A. C. 2024, arXiv e-prints, arXiv:2401.09537\nSarkar, A. & Tout, C. A. 2022, MNRAS, 513, 4169\nSarkar, A., Yungelson, L., & Tout, C. A. 2023, MNRAS, 526, 870\nSchreiber, M. R., Belloni, D., Gänsicke, B. T., Parsons, S. G., & Zorotovic, M.\n2021, Nature Astronomy, 5, 648\nSchreiber, M. R., Belloni, D., & van Roestel, J. 2023, A&A, 679, L8\nSunyaev, R., Arefiev, V ., Babyshkin, V ., et al. 2021, A&A, 656, A132\nTownsley, D. M. & Bildsten, L. 2004, ApJ, 600, 390\nWarner, B. 2003, Cataclysmic Variable Stars (Cambridge University Press)\nArticle number, page 6 of 8Sarkar et al.: Weakening of magnetic braking explains the dearth of period bouncers\n0.05 0.10 0.15 0.20\nM2/M⊙1032103310341035−˙J/g2cm2s−2κ= 0\nκ= 4\nκ= 15\nκ= 15B2\nκ= 15B4\nFig. B.1. The evolution of ˙Jwith M2for the models in Fig. 1.\nAppendix A: A catalog of period bouncers\nWe present a catalog of all known period bouncers in Table A.1.\nWhile there are more (some 25 to 30 in total) such candidate\nsystems in the literature, we require there to be precise estimates\nof 1) an orbital period, 2) donor mass, 3) WD mass (and ra-\ndius) and 4) WD temperature for us to include it in our sample.\nIf a donor mass is not available, we ensure that WD properties\nare well-measured and that there is spectroscopic evidence for a\nbrown dwarf donor. We also indicate the few systems known to\nbe eclipsing, because those have, on the whole, more precisely\nmeasured donor star parameters.\nAppendix B: Evolution of angular momentum loss\nIn Fig. B.1 we plot the evolution of the total angular momentum\nloss ˙J≡AML GWR+AML MBwith the donor mass. Note that\nalthough−˙Jdecreases throughout the evolution for each model,\natM2≈0.07M⊙it decreases more steeply for the models where\nκdepends on stellar parameters.\nArticle number, page 7 of 8A&A proofs: manuscript no. aanda\nName Porb/hr Ecl.? M2/M⊙ MWD/M⊙ RWD/0.01R⊙Teff,WD/K ˙M2/10−10M⊙yr−1Ref.\nEZ Lyn 1.430 Yes 0.042 ±0.014 0.85±0.01 0.94 11250 ±40 0.242+0.003\n−0.0031\nSDSSJ1035 +0551 1.370 Yes 0.052 ±0.002 0.94±0.01 0.87±0.01 10100±200 0.12+0.005\n−0.0052\nSDSSJ1057 +2759 1.510 Yes 0.0436 ±0.002 0.80±0.015 1.04±0.017 13300±1100 0.54+0.14\n−0.173\nSDSSJ1433 +1011 1.300 Yes 0.06 ±0.003 0.868±0.007 0.958±0.008 12800±200 0.38+0.02\n−0.022\nSDSS J1501 +5501 1.364 Yes 0.053 ±0.003 0.80±0.03 1.04±0.04 12500±200 0.426+0.009\n−0.012\nSRGeJ0411 +6853 1.625 Yes 0.84 ±0.07 1.0±0.09 13790 ±500 0.56+0.04\n−0.044\nEG Cnc 1.439 No 1.03 ±0.05 0.77+0.06\n−0.0512290±55 0.2+0.03\n−0.035\nGD 552 1.712 No 0.78 ±0.04 1.07+0.05\n−0.0410760±40 0.25+0.02\n−0.035\n1RXS J1050–1404 1.476 No 0.77 ±0.03 1.08+0.04\n−0.0311520±50 0.34+0.02\n−0.035\nQZ Lib 1.539 No 0.82 ±0.19 1.01+0.23\n−0.1811420±200 0.28+0.1\n−0.25\nSDSS J1435 +2336 1.300 No 0.84 ±0.07 1.0+0.08\n−0.0912000±160 0.32+0.05\n−0.055\nBW Scl 1.304 No 0.051 ±0.006 1.007±0.01 0.8+0.014\n−0.01115145±50 0.51+0.01\n−0.0085, 6\nGW Lib 1.279 No 0.83 ±0.12 1.03+0.15\n−0.1016166±350 1.09+0.26\n−0.345\nWZ Sge 1.360 No 0.80 ±0.02 1.05+0.03\n−0.0313190±115 0.53+0.01\n−0.015\nTable A.1. Known period bouncers with either a well-measured donor mass or accretion rate (derived from WD mass and temperature estimates,\nEq. 2). Ecl. stands for eclipsing. References: (1) Amantayeva et al. (2021); (2) Littlefair et al. (2008); (3) McAllister et al. (2017); (4) Galiullin\net al. (2024); (5) Pala et al. (2022); (6) Neustroev & Mäntynen (2023).\nArticle number, page 8 of 8" }, { "title": "2401.14394v2.O_1__Insertion_for_Random_Walk_d_ary_Cuckoo_Hashing_up_to_the_Load_Threshold.pdf", "content": "arXiv:2401.14394v2 [cs.DS] 8 Feb 2024O(1) Insertion for Random Walk d-ary Cuckoo Hashing\nup to the Load Threshold\nTolson Bell∗and Alan Frieze†\nDepartment of Mathematical Sciences\nCarnegie Mellon University\nPittsburgh, PA 15213\nU.S.A.\nFebruary 8, 2024\nAbstract\nThe random walk d-ary cuckoo hashing algorithm was defined by Fotakis, Pagh,\nSanders, and Spirakis to generalize and improve upon the sta ndard cuckoo hashing\nalgorithm of Pagh and Rodler. Random walk d-ary cuckoo hashing has low space\noverhead, guaranteed fast access, and fast in practice inse rtion time. In this paper,\nwe give a theoretical insertion time bound for this algorith m. More precisely, for every\nd≥3hashes,let c∗\ndbethesharpthresholdfortheloadfactoratwhichavalidass ignment\nofcmobjects to a hash table of size mlikely exists. We show that for any d≥4 hashes\nand load factor c < c∗\nd, the expectation of the random walk insertion time is O(1), that\nis, a constant depending only on dandcbut notm.\n1 Introduction\n1.1 Random Walk d-ary Cuckoo Hashing\nIn random walk d-ary cuckoo hashing, the goal is to store objects Xin a hash table Ygiven\ndhash functions f1,...,f d:X→Y. Following previous literature, we will take each hash\nfunction to be an independent uniformly random function from XtoY. When a new object\nxis inserted, a uniformly random i∈[d] is chosen, and xis placed into position hi(x). If\nhi(x) was already occupied, we remove its previous occupant, x2, and reinsert x2by the same\nalgorithm (choosing a new i2∈[d] and putting x2intohi2(x2)). This iterative algorithm\nterminates when we insert an object into an empty slot.\nAn object xis accessed by checking h1(x),...,h d(x), which takes constant time for con-\nstantd. If we want to remove x, we simply delete it from its slot in the hash table. Thus\naccess and deletion are both guaranteed to be fast.\nLetn=|X|andm=|Y|. We represent the hash functions as a bipartite graph with\nvertex set ( X,Y ), and for each x∈X, edges from xtoh1(x),...,h d(x). For a set W⊆X, we\n∗thbell@cmu.edu. Research supported in part by NSF Graduate Rese arch Fellowship grant DGE 2140739.\n†frieze@cmu.edu. Research supported in part by NSF grant DMS 195 2285\n1letN(W) denotes its set of neighbors in Y. An analogous definition is assumed for Z⊆Y.\nFinally, we replace N({u}) byN(u) for singleton sets.\nFor this insertion to terminate, it must be true that there is an assig nment of every object\nto a slot such that no slot has more than one object and every obje ctxis assigned to hi(x)\nfor some 1 ≤i≤d. This can be represented as a matching of size nin the bipartite graph.\nWe know by Hall’s Theorem that such a matching exists if and only if |N(W)| ≥ |W|for\neveryW⊆X.\nUnless explicitly noted otherwise, all asymptotics in this paper are wr itten for m,n→ ∞\nwithn=cmfor fixed d∈Nand fixed load factor c∈(0,1). There is a sharp threshold c∗\nd,\ncalled the load threshold , for a matching of size nto exist in the bipartite graph; that is,\nthere is a constant c∗\ndsuch that as n,m→ ∞ withn=cm, ifc < c∗\ndthen the probability of\nsuch a matching goes to 1, and if c > c∗\ndthen the probability of such a matching goes to 0.\nOur result is the following:\nTheorem 1.1. Assume that we have d≥4,c < c∗\nd, andn=cm. Then with high probability\nover the random hash functions, we have that the expected ins ertion time for the random walk\ninsertion process is O(1).\nAdditionally, under the same conditions, there is a constan tC= Θ(1)such that for suffi-\nciently high nand allℓ∈N, the probability of the random walk taking more than ℓsteps is\nat mostCe−ℓ.009.\nIn other words, our main result is that the expected insertion time is a constant depending\nonly ondandcbut notnorm. We did not try to optimize the constant. By insertion time,\nwe mean the number of rehashes, which is what is the barrier in pract ice and has been\npreviously analyzed theoretically; naturally, the hash function mus t output O(log(n)) bits to\nrepresent a position in Y. Our result is also true beginning with any arbitrary assignment of\nobjects to slots in the hash table.\nNote that we are required to take our statement to only hold with hig h probability over\nthe choices of hash functions, as there is a non-zero chance that the hash functions will not\nhave any valid assignment of objects to slots (will fail Hall’s condition) and thus will have\ninfinite insertion time.\nThe second part of Theorem 1.1 gives super-polynomial tail bounds on the insertion time.\nThe double exponent 0.009 can be made to tend towards 1 as d→ ∞ .\n1.2 Applications and Relation to Previous Literature\nStandard cuckoo hashing was invented by Pagh and Rodler in 2001 [P R01] and has been\nwidely used in both theory and practice. Their formulation, though o riginally phrased with\ntwo hash tables, is essentially equivalent to the case d= 2 of the algorithm described here.\nThey showed that for all c < c∗\n2= 0.5, one can get O(1) expected insertion time, an analysis\nthat was extended by Devroye and Morin [PR01, DM03].\nd-ary cuckoo hashing was invented by Fotakis, Pagh, Sanders, and Spirakis in 2003\n[FPSS03]. The main advantage of increasing dabove 2 is that the load threshold increases.\nEven going from d= 2 tod= 3, the threshold c∗\ndgoes from 0.5 to ≈0.918, that is, with\njust one more hash function, we can utilize 91% of the hash table inst ead of 49%. The cor-\nresponding tradeoff is that the access time increases linearly with d.d-ary cuckoo hashing,\n2also called generalized cuckoo hashing or improved cuckoo hashing, “ has been widely used\nin real-world applications” [SHF+17].\nThe exact value for c∗\ndfor alld≥3 was discovered via independent works by a number\nof authors [DGM+10, FP10, FM12]. This combinatorial problem of finding the matching\nthreshold in these random bipartite graphs (which can also be viewed as random d-uniform\nhypergraphs) is directly related to other problems like d-XORSAT [DGM+10] and load bal-\nancing [GW10, FKP11].\nThe primary insertion algorithm analyzed by Fotakis, Pagh, Sanders , and Spirakis was not\nrandom walk insertion, but rather was BFS insertion. In BFS insertio n, instead of selecting\na random i∈[d] and hashing xtohi(x), the algorithm finds the insertion path minimizing\nthe number of rehashes. In other words, i1,...,i ℓ∈[d] are chosen such that ℓis minimized,\nwherexis to be hashed to hi1(x), the removed object x2is to be hashed to hi2(x2), and so\non untilhiℓ(xℓ) is an empty slot. While BFS insertion requires more overhead to comp ute,\nit is easier to analyze theoretically than random walk insertion. Fotak is, Pagh, Sanders, and\nSpirakis proved that BFS insertion is O(1) for load factor cwhend≥5 + 3 log( c/(1−c))\n[FPSS03]. Though not explicitly stated, the results of Fountoulakis, Panagiotou, and Steger\nimply that this result extends to all d≥3 andc < c∗\nd[FPS13].\nBFS insertion for cuckoo hashing is the “average-case” or “rando m graph” version of\nthe “online bipartite matching with replacements” problem. Take any bipartite graph with\nV= (X,Y ) that contains a matching of size |X|. If elements of Xand their incident edges\narrive online, the amortized BFS insertion time is O(log2(n)) [BHR18]. The lower bound is\nΩ(log(n)) [GKKV95], which is matched if the vertex arrival order is randomiz ed [CDKL09].\nThe previous paragraph shows that if the graph itself is random rat her than worst-case, this\nΘ(log(n)) insertion time bound is with high probability reduced to Θ(1).\nFotakis, Pagh, Sanders, and Spirakis also introduced the insertion algorithm we study,\nrandom walk insertion, describing it as “a variant that looks promising in practice”, since\nthey did not theoretically bound its insertion time but saw from exper iments that its in-\nsertion time was fast [FPSS03]. Random walk insertion requires no ext ra space overhead or\nprecomputation: very important for the use case of d-ary cuckoo hashing, situations requiring\nhigh load factor. In a 2009 survey on cuckoo hashing, Mitzenmache r raised the importance of\nproving theoretical bounds for random walk insertion, calling rando m walk insertion “much\nmore amenable to practical implementation” and “usually much faste r” than BFS insertion\n[Mit09]. Other insertion algorithms than random walk or BFS have been proposed, which\nhave provable O(1) insertion [KA19] or more evenly distributed memory usage [EGMP 14]\nwhile having lower memory overhead than BFS insertion. However, ra ndom walk insertion “is\ncurrently the state-of-art method and so far considered to be t he fastest algorithm” [KA19].\nFor load factors somewhat below the load threshold and d≥8, the random walk inser-\ntion time was proven to be polylogarithmic by Frieze, Melsted and Mitze nmacher in 2009\n[FMM09]. Fountoulakis, Panagiotou, and Steger then were able to sh ow polylogarithmic\ninsertion time for all d≥3 andc < c∗\nd. The exponent of their logarithm was anything greater\nthan 1 + bd, wherebd=d+log(d−1)\n(d−1)log(d−1)[FPS13]. Our proof uses some techniques and lemmas\nof these two papers.\nThe first O(1) random walk insertion bound was proven by Frieze and Johansso n, who\nshowed that for any load factor c, there exists some dsuch that there is O(1) insertion time\nfordhashes at load factor c[FJ17]. However, their bounds only hold for large dand load\n3factors significantly less than the load threshold, specifically, c= 1−Od→∞(log(d)/d), while\nwe know that c∗\nd= 1−(1 +od→∞(1))(e−d).\nFor lower d, Walzer used entirely different techniques to prove O(1) random walk insertion\nup to the “peeling threshold”. The strongest result here is in the ca sed= 3, where Walzer\ngetsO(1) insertion up to load factor c=.818, compared to the optimal value c∗\n3=.918.\nWalzer pointed out that there was no d≥3 for which O(1) insertion was known up to the\nload threshold, saying, “Given the widespread use of cuckoo hashin g to implement compact\ndictionaries and Bloom filter alternatives, closing this gap is an importa nt open problem for\ntheoreticians” [Wal22].\nTheorem 1.1 is the first result to get O(1) insertion up to the load threshold for any d≥3,\nand works for all d≥4. The state of the art results are summarized in the tables below:\nd c∗\nd Maximal load factor Insertion time\nforO(1) insertion atc= (1−ǫ)c∗\nd\n210.510.51O(1)\n320.91830.8184O(log3.664(n))\n420.97730.7724O(log2.547(n))\n520.99230.7024O(log2.152(n))\n620.99730.6374O(log1.946(n))\n720.99930.5824O(log1.818(n))\nLarge21−(1 +od→∞(1))(e−d)51−Od→∞(logd\nd)4O(log1+(logd)−1+Od→∞(1/d)(n))\nPrior work:1[PR01, DM03]2[DGM+10, FP10, FM12]3[Wal22]4[FPS13]5[FJ17]\nd c∗\nd Maximal load factor Insertion time\nforO(1) insertion atc= (1−ǫ)c∗\nd\n210.510.51O(1)\n320.91830.8187O(log2.509(n))\n420.97760.9776O(1)\n520.99260.9926O(1)\n620.99760.9976O(1)\n720.99960.9996O(1)\nLarge21−(1 +od→∞(1))(e−d)61−(1 +od→∞(1))(e−d)6O(1)\nBounds after our work:6Theorem 1.1 and7Also given in our proof\n1.3 Future Work\nThe central open question is to remove the restriction d≥4 from Theorem 1.1, that is, to\ngetO(1) insertion up to the load threshold for d= 3. We are hopeful that the techniques in\nour paper can be extended to finish this final case.\nThe super-polynomial tail bounds on the insertion time in Theorem 1.1 can be made\nto tend towards being exponential tail bounds as d→ ∞ . It would be interesting to show\nexponential tail bounds, as well as O(1) insertion, for all d≥3.\nIt would also be interesting to give a stronger bound on the o(1) term in our “with high\nprobability” statements. A careful analysis of our and previous wo rks ([FP10, FPS13]) shows\n4that this probability could currently be taken to be O(n−β) for some small β= Θ(1). By\na union bound, the failure probability also implies that the O(1) expected insertion time is\nrobust to O(nβ) non-hash-dependent deletions and insertions, as long as the load factor stays\nbelowc.\nNow that we have an insertion time independent of n, another avenue for future study is\nto optimize the insertion time in terms of d,c, and absolute constants.\nIt has been shown under some previous models of cuckoo hashing th at the assumption of\nuniformly random hash functions can be relaxed to families of efficient ly computable hash\nfunctions while retaining the theoretical insertion time guarantees [CK09, ADW14]. As our\nproof relies on similar “expansion-like” properties of the bipartite gr aph to previous work,\nwe believe that Theorem 1.1 should still hold under practically computa ble hash families.\nA different model for generalizing cuckoo hashing, proposed in 2007 , gives a capacity\ngreater than one to each hash table slot (element of Y), instead of (or in addition to) ad-\nditional hash functions [DW07]. The load thresholds for this model ar e known for both\ntwo hashes [CSW07, FR07] and d≥3 hashes [FKP11]. As in our model, O(1) expected\ntime for random walk insertion has been shown for some values below t he load threshold\n[FP18, Wal22], but it remains open for any capacities greater than on e to prove O(1) inser-\ntion up to the load thresholds.\nIn general, it would be nice to extend our random walk insertion time gu arantees to other\nmodifications of cuckoo hashing, such as those that get good load f actors with somewhat\nfewer hashes [Yeo23] or those that deal with the situation where a valid matching fails to\nexist [KMW09, MP23].\n2 Determining the “Bad” Sets\nOur techniques to prove Theorem 1.1 build off the techniques of Foun toulakis, Panagiotou,\nand Steger [FPS13], who showed expansion-like properties of the bip artite hashing graph\nthat hold with high probability. The main new ingredient is the introduct ion of specifically\ndefined “bad” sets B1⊇B2⊇.... In this section, we will give the definition of these bad\nsets and explain the overall proof structure. The main idea is that a random walk avoiding\nBiwill probably finish in O(i) steps.\nWe defer our most technical section, Section 5, to the end of the p aper. Section 5 shows\nthat the size of the Bidecline exponentially in i. In Section 3, we will show that reaching\na small set, such as the Bior a short cycle, is unlikely. In Section 4, we finish the proof of\nTheorem 1.1, accounting for how the matching changes over the co urse of the random walk.\nWe show that the probability of walking O(i) steps declines rapidly with i, as both reaching\nBiin the first O(i) steps and walking for O(i) steps while outside of Biare unlikely.\n2.1 The Matching and BFS Distance\nWe will study the form of the random walk where at each object remo val, we choose a random\none of the ( d−1) other hashes for the object that was just kicked out (not ret urning it to\nthe spot it was just kicked out of). Proving the expected run time o f this isO(1) also proves\nthe same of the run time of choosing a random one of the dhashes each time (including the\n5one it was just kicked from), as this just adds a delay of twice a Geom ((d−1)/d) random\nvariable at each step in the previous random walk, multiplying the expe ctation by 2 /(d−1).\nWe will only consider the insertion of one element into the hash table. A s the only “with\nhigh probability” statements in our proof are about the structure of the bipartite graph, this\nimpliesO(n) time with high probability to build the hash table of nelements online. Let\nMbe the starting matching of size n−1 just before we insert the nth element. Let U⊆Y\nbe the set of open spots in the hash table, which stays the same at e ach time step while the\nalgorithm is running (as the algorithm terminates when it hits an open s lot).\nOur proof only relies on expansion-like properties of the bipartite gr aph on (X,Y ) that\nhold with high probability. In particular, given the random bipartite gr aph, our result holds\nfor any arbitrary starting matching Mof objects to slots.\nLetWi(x)⊆Xbe the set of all possible endpoints of a walk of length at most istarting\nfromxand the matching M(so|Wi(x)| ≤/summationtexti\nj=0(d−1)j≤(d−1)i+1, as we do not allow\nelements to be rehashed back into the position they were just evict ed from). Note that we\nare referring to a walk of length ithroughout to refer to ireassignments, but really this is a\nwalk of length 2 iin the bipartite graph ( X,Y ).\nThe BFS distance of an object wfrom an object xunderMis the minimal isuch that\nw∈Wi(x). We can define BFS distances involving sets in the natural way, by m inimizing\nover elements of those sets. We can similarly define the BFS distance of a slot yfrom an\nobjectxas 1 plus the BFS distance from xtoN(y).\nFor every u∈Uthat is at BFS distance jfromxfor some j≤i, countu1,...,u (d−1)i−j\nas distinct elements of Wi(x). LetUi(x) then be the multiset of the u∈U∩Wi(x) with\nthese multiplicities. We also have W0(x) ={x}andx∈Wi(x)∀i∈N. ForS⊆X, we can\nsimilarly define Wi(S) =∪x∈SWi(x) andUi(S) being the disjoint union ⊔x∈SUi(x).\nLemma 2.1 (Corollary 2.3 of [FPS13]) .Assume n=cmforc < c∗\nd. Then with high\nprobability, we have that for any matching Mand anyα= Θ(1)>0, there exists M= Θ(1)\nsuch that for the unoccupied vertices UofY, we have that at most αnof the vertices of X\nare at BFS distance > MfromU.\n(Lemma 2.1 had initially been proven by the inventors of d-ary cuckoo hashing under the\nweaker condition d≥5 + 3 log( c/(1−c)) forn=cm[FPSS03]. Note that all logarithms in\nour paper are natural.)\nLetα >0 be sufficiently small (but still Θ(1), to be set later) and take the co rresponding\nM= Θ(1) as in Lemma 2.1. For any M, letGbe all vertices of Xof BFS distance at most\nMfromU. When we start at a vertex g∈G, we have at least a ( d−1)−Mchance that our\nrandom walk will finish in at most Mmore steps. (That is, there is at least a ( d−1)−Mchance\nthat our random walk will be the BFS path.) Intuitively, this gives that expected length on\na random walk that stays inside Gat every time tis at most ( d−1)M+M= Θ(1) (though\nsome technicalities arise due to the changing matching as the walk pro gresses). This shows\nintuitively that it suffices to only focus on the “worst” αnvertices for some α= Θ(1)>0.\n2.2 Definition of Bi\nWe will split up the bad set X\\Ginto further worse and worse subsets defined based on G.\n6From any x∈X, there are ( d−1)iequally likely walks of length i∈N, given that we\nhaveUi(x) with proper multiplicities. Take C0= Θ(1) to be fixed later. For any i∈N, we\ndefine\nGi=/braceleftbigg\nx∈X:|Wi(x)∩(G∪Ui(x))| ≥(d−1)i\nC0i.99/bracerightbigg\n(We define G0=G.) The definition of Giis useful for the following reason: if we have\na random walk starting at some x∈Gi, we have at least a ( C0)−1i−.99chance that the\nrandom walk will be in Gafter some j≤isteps, as for each w∈Wi(x), there is at least a\n(d−1)−j≥(d−1)−ichance we are at wafterj≤isteps.\nTherefore, for a random walk starting at some x∈Gi, we have at least a ( d−1)−MC−1\n0i−.99\nchance that the random walk will finish in at most i+Msteps, by reaching Ginj≤isteps\nand then taking the BFS path from there. This intuitively shows that the expected length\nof a random walk that stays within ∪i\nj=0Gjat each time tis at most C0(d−1)Mi.99+i+d:\nat each step, we are in some Gj, and thus by the previous paragraph have at least a ( d−\n1)−M(C0)−1i−.99chance of finishing in at most j+M≤i+Mfurther steps. The reason this\nis not rigorous is that the matching of objects to slots changes as t he walk progresses, but\nwe will show in Section 4 that these changes do not significantly affect this expected time.\nNow, define our bad sets to be the complement of these,\nBi=X\\(∪i\nj=0Gj).\nSo then we have X=B−1⊇B0⊇B1⊇....\n3 Probability of Reaching a Small Set\n3.1 Neighbors of a Small Set\nTo show that reaching some bad set is unlikely, we want to upper boun d the probability of\nreaching some small set. To accomplish this, we need to bound the nu mber of neighbors that\na small set can have.\nLemma 3.1. With high probability, there is not a set Z⊆Ywith|Z| ≤n/12such that\n|N(Z)| ≥3dlog/parenleftBig\nn\n|Z|/parenrightBig\n|Z|.\nProof. First, imagine fixing Z⊆Y, then randomly choosing the edges of our graph. Let\ne(Z) be the number of edges incident to Z. Our bipartite graph has dnedges, and each\nhas an independent |Z|/m≤ |Z|/nchance of landing in |Z|. Thus, we can assume e(Z)∼\nBin(dn,|Z|/n) andE(e(Z)) =d|Z|. By standard Chernoff bounds,\nP/parenleftbigg\ne(Z)≥3dlog/parenleftbiggn\n|Z|/parenrightbigg\n|Z|/parenrightbigg\n≤/parenleftbigge\n3 log(n/|Z|)/parenrightbigg3d|Z|log(n/|Z|)\n≤e−3d|Z|log(n/|Z|)=/parenleftbigg|Z|\nn/parenrightbigg3d|Z|\nThen\nP/parenleftbigg\n∃Z⊆Ys.t.|N(Z)| ≥3dlog/parenleftbiggn\n|Z|/parenrightbigg\n|Z|/parenrightbigg\n7≤n/12/summationdisplay\ni=1/parenleftbiggm\ni/parenrightbigg/parenleftbiggi\nn/parenrightbigg3di\n≤n/12/summationdisplay\ni=1/parenleftbigg2en\ni/parenrightbiggi/parenleftbiggi\nn/parenrightbigg3di\n=n/12/summationdisplay\ni=1/parenleftBigg\n2e/parenleftbiggi\nn/parenrightbigg3d−1/parenrightBiggi\n≤log2(n)/summationdisplay\ni=12e/parenleftbigglog2(n)\nn/parenrightbigg2\n+n/12/summationdisplay\ni=log2(n)/parenleftBigg\n2e/parenleftbigg1\n12/parenrightbigg2/parenrightBigglog2(n)\n=o(1/n).\nNow, for x∈Xandj∈N, letW−j(x)={w∈X:x∈Wj(w)}.\nLemma 3.2. ForanyS⊆X,|S| ≤n/12, andt∈N, we have |W−j(S)| ≤/parenleftBig\n3dlog/parenleftBig\nn\n|S|/parenrightBig/parenrightBigj\n|S|.\nProof. We can prove this inductively as a corollary of the lemma above. We see that it is\ntrue forj= 0. Then note that W−j(S) =W−1(W−j+1(S)) =N(Z) whereZ⊆Yis the spots\noccupied by W−j+1(S), which thus has the same cardinality of W−j+1(S).\nSo using Lemma 3.1, we have\n|W−j(S)| ≤3dlog/parenleftbiggn\n|W−j+1(S)|/parenrightbigg\n|W−j+1(S)| ≤3dlog/parenleftbiggn\n|S|/parenrightbigg\n|W−j+1(S)|\n≤/parenleftbigg\n3dlog/parenleftbiggn\n|S|/parenrightbigg/parenrightbiggj\n|S|\nas desired.\n(Note that if we ever have |W−j+1(S)| ≥n/12 (so Lemma 3.1 can’t be applied), then we\nhave|W−j(S)| ≤3dlog/parenleftBig\nn\n|S|/parenrightBig\n|W−j+1(S)|anyway, as the right side of the equation is then\nmore than n.)\n3.2 Applying Lemma 3.2\nIn this subsection, we will see two applications of Lemma 3.2 that we will need to complete\nthe proof. The first has the “small set” being the set of short cyc les, while the second has\nthe “small set” being the Bi.\nLemma 3.3. Letz= (10 log( n))0.9999and letSCyc⊆Xbe the set of vertices who are\non a cycle of length zor less. With high probability over the choice of random hash es,\n|W−z(SCyc)|< n0.3.\nProof. Fixℓ∈Nand consider the cycles of length 2 ℓin the bipartite graph. Each has the\nform (x1,y1,x2,y2,...,x ℓ,yℓ) for some x1,...,x ℓ∈Xandy1,...,y ℓ∈Y, wherexihashes to\nbothyiandyi−1(withx1also hashing to yℓ). There are at most nℓmℓordered sets of vertices\n(x1,y1,x2,y2,...,x ℓ,yℓ). The probability that all required hashes will be chosen is at most/parenleftBig\nd(d−1)\nm2/parenrightBigℓ\n≤d2ℓm−2ℓ. Thus, the expected number of cycles of length ℓin the bipartite graph\nis at most nℓmℓd2ℓm−2ℓ< d2ℓ.\n8Then the number of cycles of length at most zis at most/summationtextz\nℓ=1d2ℓ≤d2z+1=o(dlog(n)/(100d)) =o(n0.1). Markov’s inequality gives that with high\nprobability there are less than n0.1cycles of length at most z.\nEach of these cycles has at most zvertices on it, so |SCyc|< n0.1z < n0.2for sufficiently\nlargen.\nThen we apply Lemma 3.2 to say that\n|W−z(SCyc)| ≤/parenleftbigg\n3dlog/parenleftbiggn\n|SCyc|/parenrightbigg/parenrightbiggj\n|SCyc|<(3dlog(n))zn0.2< n0.3.\nIn Section 5, we will prove that the Bihave exponentially decreasing sizes, proving the\nfollowing lemma:\nLemma 3.4. With high probability over the choice of d≥4hashes, there is a C= Θ(1)\nsuch that |Bi| ≤Cn2−ifor any matching Mand for all i∈N.\nBecause the proof of Lemma 3.4 is a bit more technical, we defer it to t he end of our\npaper. We now put Lemma 3.2 and Lemma 3.4 together:\nLemma 3.5. There exists C1= Θ(1)such that |W−2C0(d−1)Mi.999(Bi)| ≤C1(1.9−i)n.\nProof.\n|W−2C0(d−1)Mi.999(Bi)| ≤/parenleftbigg\n3dlog/parenleftbiggn\n|Bi|/parenrightbigg/parenrightbigg2C0(d−1)Mi.999\n|Bi| by Lemma 3.2\n≤C/parenleftBig\n3dlog/parenleftBign\nC2−in/parenrightBig/parenrightBig2C0(d−1)Mi.999\n2−in by Lemma 3.4\n≤C(3d(i−log(C)))2C0(d−1)Mi.9992−in\n≤C(2o(i))2−in≤C1(1.9)−in\nfor sufficiently large C1= Θ(1).\n4 Proving Theorem 1.1 (assuming Small Bi)\nTo complete the proof of Theorem 1.1, it is necessary to show that t he change in the matching\nover the course of the insertion process does not have too large a n effect on properties of the\nX\\Bi. We will now generalize the definitions in Section 2 to account for how t he random\nwalk has changed the matching.\nLetx0be the starting object that we are inserting and iteratively define xtto be the\nobjected evicted by the hash of xt−1. LetMtbe the matching of size n−1 that exists while\nxtis being rehashed (so M0=M). LetW(t)\ni(xt)⊆Xbe the set of all possible endpoints of\na walk of length at most istarting from xtunder the matching Mt(defining U(t)\nias expected,\nand noting W(0)\ni(x0) =Wi(x0)). The BFS(t)distance of an x∈XfromUis the minimal i\n9such that W(t)\ni(x)∩U/\\e}a⊔io\\slash=∅. Using the same value Mas in Lemma 2.1, let G(t)be the subset\nofXof the elements at BFS(t)distance ≤MfromU. Let\nG(t)\ni=/braceleftbigg\nx∈X:|W(t)\ni(x)∩(G(t)∪U(t)\ni(x))| ≥(d−1)i\n2C0i.99/bracerightbigg\n.\nNote that we have put an extra factor of 2 into the denominator, s oG(0)\ni⊇Gi. As expected,\nwe define B(t)\ni=X\\(∪i\nj=0G(t)\nj).\nRecall the notation of Lemma 3.3 that z= (10 log( n))0.9999andSCyc⊆Xis the set of\nvertices who are on a cycle of length zor less. In essence, the following lemma shows that\nwhen considering X\\Bi, we need not worry about how the matching has changed in the first\ntsteps of the random walk.\nLemma 4.1. Assume that x0/∈W−z(SCyc). Fix any/radicalbig\nlog(n)≤i≤2 logd−1(n)and any\n0≤t≤i0.999. We have that xt/∈Bi=⇒xt/∈B(t)\ni.\nProof. Originally, xthad at least(d−1)i\nC0i.99elements in |Wi(xt)∩(G∪ Ui(xt))|. Our goal is to\nshow that at least half of these same elements remain in |W(t)\ni(xt)∩(G(t)∪U(t)\ni(xt))|, or in\nother words, at most half have been removed by the changing matc hing. Note that xthas\nnot been rehashed before step t(ast≤i0.999≤zandx0/∈W−z(SCyc)), so the hash it is\nbeing evicted from is the same as its matching under M.\nHow could an element x′be in|Wi(xt)∩(G∪Ui(xt))|but not in |W(t)\ni(xt)∩(G(t)∪U(t)\ni(xt))|?\nThis could happen only if one of the following three conditions hold: (i) x′was rehashed and\nthus occupies a new position in Y(that is, equals some xkfork≤t); (ii) any element on\nthe BFS path between xtandx′was rehashed and x′is no longer in W(t)\ni(xt); and (iii) any\nelement on the BFS path between x′andUwas rehashed and x′is no longer in G(t).\nSo, how many elements could a single rehash of an object xkremove? If the BFS distance\nfromxttoxkwerejfor some j≤i, then in particular, there could be at most/summationtexti−j\nf=0(d−1)f≤\n(d−1)i−j+1objects removed by conditions (i) and (ii).\nNote that t < z/ 2. Therefore, x0/∈W−z(SCyc) means that neither xt, nor any element of\nWz/2(xt), is on a cycle of length at most z, soWz/2(xt) =W(t)\nz/2(xt). Therefore, all rehashed\nelements must be at distance at least z/2≥i.999fromxt. So any rehashed element xkcan\nonly take out at most ( d−1)i−i.999+1elements through conditions (i) or (ii).\nBy Lemma 3.2, |W−M(xk)| ≤(3dlog(n))M, and being in W−M(xk) is a necessary condi-\ntion for an element to be removed by condition (iii), so rehashing xkcan remove at most\n(3dlog(n))Melements via condition (iii).\nTherefore, each element on its own can remove at most ( d−1)i−i.999+1+(3dlog(n))Mfrom\n|Wi(xt)∩(G∪Ui(xt))|, so all the rehashed elements together can only remove at most\nt/parenleftBig\n(d−1)i−i.999+1+ (3dlog(n))M/parenrightBig\n≤(d−1)i.999 (d−1)i\n(d−1)(i.999)+ (3d)MlogM+2(n)≤(d−1)i\n2C0i.999\nelements (using in the last step that/radicalbig\nlog(n)≤i), completing the proof.\nTo complete the proof of Theorem 1.1, we assume Lemma 3.4 and the f ollowing lemma,\nboth of which will be proven in Section 5.\n10Lemma 4.2 (Weaker version of Lemma 5.3) .With high probability over the choice of d≥4\nhashes, under any matching Mt,B(t)\n2logd−1(n)=∅.\nWe now have all the ingredients needed to complete the proof of The orem 1.1:\nLemma 4.3. Assume that we have d≥4,c < c∗\nd, andn=cm. With high probability over\nthe choice of hash functions, there is a constant C= Θ(1)such that for sufficiently high n\nand allℓ∈N, the probability of the random walk taking more than ℓsteps is at most Ce−ℓ.009.\nThis implies Theorem 1.1, as E(|RW|) =/summationtext∞\nℓ=1ℓP(|RW|=ℓ)≤/summationtext∞\nℓ=1ℓP(|RW| ≥ℓ)≤/summationtext∞\nℓ=1ℓCe−ℓ.009−O(1).\nProof of Lemma 4.3. Takei∈N. In order for the random walk to take at least 2 C0(d−\n1)Mi.99+i+Msteps, either we reach Biin at most 2 C0(d−1)Mi.99steps, or we walk outside\nofBifor at least 2 C0(d−1)Mi.99steps without choosing to finish in the next i+Msteps.\nWe claim that the probability of the former is O(1.9−i) and the probability of the latter is\nat moste−i.009.\nP(reachBiin≤2C0(d−1)Mi.99steps)\n≤P(x0hashed to W−2C0(d−1)Mi.99(Bi)) =1\nn|W−2C0(d−1)Mi.99(Bi)|\n≤1\nn(C1(1.9)−in) by Lemma 3.5\nFor the later probability, we split into three cases of i. We can ignore all i <(2C0(d−1)M)1.1\nby increasing the C= Θ(1) in Lemma 4.3. For i≤2 logd−1(n) (cases 1 and 2), we can\nassume that we do not start in W−z(SCyc), as the probability of starting in W−z(SCyc) is\n1\nn|W−z(SCyc)| ≤n−0.7=o(e−i0.009).\nCase 1: (2 C0(d−1)M)1.1≤i≤/radicalbig\nlog(ℓ). The fact that we are not in W−z(SCyc) and\nz > 2imeans that we will not rehash the same element at any point in the firs t 2C0(d−\n1)Mi.999+i+M≤2isteps. Thus, we can treat the matching as unchanging, so on any\nstep of our random walk, the fact that we are not in Bimeans we have probability at least\n(C0)−1(d−1)−Mi−.99of finishing in ≤i+Mfurther steps.\nCase 2:/radicalbig\nlog(ℓ)≤i≤2 logd−1(n). In this case, for all 1 ≤t≤2C0(d−1)Mi.999, we have\nby Lemma 4.1 that xt/∈B(t)\ni, and thus we have probability at least (2 C0)−1(d−1)−Mi−.99of\nfinishing in ≤i+Mfurther steps.\nCase 3:i≥2 logd−1(n). In this case, for all 1 ≤t≤2C0(d−1)Mi.999, we have by Lemma\n4.2 that xt/∈B(t)\ni, and thus we have probability at least (2 C0)−1(d−1)−Mi−.99of finishing\nin≤i+Mfurther steps.\nSo in any of the three cases, at each of the first 2 C0(d−1)Mi.999steps in our walk, we\nhave probability at least (2 C0)−1(d−1)−Mi−.99of finishing in ≤i+Mfurther steps. Thus,\nthe probability that we walk for at least C0(d−1)Mi.99steps without choosing to finish in\n≤i+Mfurther steps is at most (1 −(2C0)−1(d−1)−Mi−.99)2C0(d−1)Mi.999≤e−i.009\nTakingℓ= 2C0(d−1)Mi.99+i+M, the above shows that the probability of the random\nwalk lasting at least ℓsteps isO(1.9−i) +e−i.009=O(e−i.009) =O(e−ℓ.009).\n11This completes the proof of Theorem 1.1, except that it still remains to prove Lemmas\n3.4 and 4.2.\nIn fact, tracing through our proof, we see that .009 could be any v alue less than 1 −bdfor\na valuebd≤(d−1)+log(d−1)\n(d−1)log(d−1), and we have bd→0 asd→ ∞ . So in other words, the tail bounds\nare super-polynomially decreasing and tend towards an exponentia l decrease as d→ ∞ .\n5 Bounding the sizes of Bi\nThe remaining task is to show that the sizes of the Bidecline like O(2−i). The results in this\nsection rely heavily on results of Fountoulakis, Panagiotou, and Ste ger [FPS13].\nRecall that for any matching MandS⊆X, we have W1(S) =∪x∈SW1(x) is the set of all\nw∈Xthat we could reach by zero or one cuckoo iteration starting somew here inS, which\nis allw∈Xoccupying a position in N(S). Then |W1(S)|=|N(S)| ≤d|S|. The following\nlemma shows that for small S,|W1(S)|is close to its upper bound.\nLemma 5.1 (Proposition 2.4 of [FPS13]) .For any 1≤s≤ |X|/d, define\nxs=/braceleftBigg\n0 if|S| ≤log log(n)\nlogd((d−1)ed)\nlog(|X|/|S|)−1iflog log(n)≤ |S| ≤ |X|/d\nWith high probability, we have that for all S⊆Xwith|S| ≤ |X|/dthat\n|N(S)| ≥(d−1−x|S|)|S|.\nLemma 5.1 is sufficient for our proof to go through for d≥6. We defer the d≤5 cases\nto the computation-heavy Subsection 5.2, where we will prove a for m of Lemma 5.1 with\nstronger parameters.\n5.1 Bounding |Bi|ford≥6\nLetad= (d−1)ed. Now, following [FPS13], let s0= 1 and inductively set si= (d−1−\nxsi−1)si−1. We cite another lemma from [FPS13]:\nLemma 5.2 (Claim 4.5 of [FPS13]) .For every d≥3andγ >0there exists ǫ0=ǫ0(γ,d) =\nΘ(1)such that for all 0< ǫ < ǫ 0andnsufficiently large the following is true. Set\nT= logd−1(n) +/parenleftbigglog(ad)\n(d−1) log(d−1 +γ)/parenrightbigg\nlogd−1(logd−1(n)).\nThensT> ǫn.\nTakeγ= Θ(1) sufficiently small such thatlog(ad)\n(d−1)log(d−1+γ)< .98 (noting thatlog(ad)\n(d−1)log(d−1)<\n.98 ford≥6) and take the corresponding ǫ0= Θ(1). In Lemma 2.1, take α≤ǫ0/(2(d−1))).\nNow, clearly there is some Rsuch that 2 αn≤sR≤ǫ0n, as we multiply siby at\nmostd−1 at each step. Fix some such Rand note logd−1(2αn)≤R≤logd−1(n) +/parenleftBig\nlog(ad)\n(d−1)log(d−1+γ)/parenrightBig\nlogd−1(logd−1(n)) by Lemma 5.2.\n12Lemma 5.3. Under any matching M, we have BR=∅.\nProof. Assume there were some x∈BR. Then we would inductively get that |Wi(x)| ≥si\n(remembering to count the elements of Uwith proper multiplicities in Ui(x)), so in particular\n|WR(x)| ≥2αn=⇒ |WR(x)∩(G∪Ui(x))| ≥αn=⇒\n(d−1)−R|WR(x)∩(G∪Ui(x))| ≥/parenleftbigg1\nnlog.98(n)/parenrightbigg\nαn >1\nC0R.99\nwhen assuming C0> α−1. This, however, contradicts that we need by definition that |WR(x)∩\n(G∪Ui(x))| ≤(d−1)R\nC0R.99for allx∈BR.\nSo we have successfully shown that for i≥R,Biis empty. To bound the sizes of lower\nBi, we need to look closer at the proof of Lemma 5.2 and use some additio nal lemmas of\n[FPS13].\nLemma 5.4 (Claim 4.4 of [FPS13]) .Lett≥logd−1(log(log(n))) + 1. For every ǫ > 0\nsufficiently small, if st≤ǫn, then for all 0≤i≤t−logd−1(log(log(n)))−1, we have\nxst−i≤logd(ad)\nilogd(d−1−γ) + logd(1/ǫ)−1\nwhereγ=logd(ad)\nlogd(1/ǫ)−1.\nRecall that we have set ǫsmall enough such thatlog(ad)\n(d−1)log(d−1+γ)< .98.\nLemma 5.5 (Proposition 4.1 of [FPS13]) .For any constants ζ,η > 0we have that whenever\nD=D(ζ,η)is sufficiently large then\ni/productdisplay\nk=1/parenleftbigg\n1−ζ\nkη+D/parenrightbigg\n≥i−ζ/η(ηD)−ζ/ηe−ζ2/(ηD)for alli≥2/η\nThe previous two lemmas combine to prove the following:\nLemma 5.6 ([FPS13]) .For all 1≤i≤.99 logd−1(n), we have sR−i≤C0αn\n2(d−1)ii.99.\nProof. This is proved following the first half of the proof of Claim 4.5 of [FPS13].\nBecause sR≤ǫ0n, we can use the definition si= (d−1−xsi−1)si−1and Lemma 5.4 to\nget\nsR−i≤ǫ0n/producttexti\nk=1(d−1−xsR−i)≤ǫ0n\n(d−1)i/producttexti\nk=1/parenleftBig\n1−logd(ad)/(d−1)\nklogd(d−1−γ)+logd(1/ǫ0)−1/parenrightBig\nthen using Lemma 5.5 with ζ=logd(ad)\nd−1andη= logd(d−1−γ) we get for all i≥4≥2/η\nthat\nsR−i≤ǫ0n\n(d−1)i/producttexti\nk=1/parenleftBig\n1−logd(ad)/(d−1)\nklogd(d−1−γ)+logd(1/ǫ0)−1/parenrightBig≤Cǫ0,dǫ0n\n(d−1)i/parenleftbigg\ni/parenleftBiglogd(ad)\n(d−1)logd(d−1−γ)/parenrightBig/parenrightbigg\n,\nwhich is less than the desired quantity as long as we take C0>4(d−1)Cǫ0,d(recalling\nα=ǫ0/(2(d−1)) and we set γsuch thatlog(ad)\n(d−1)log(d−1−γ)< .98). Assuming C0>4 then also\nworks for 1 ≤i≤3.\n13Lemma 5.7. |Bi| ≤C0αn\n(d−1)ii.99for any matching Mand for all 1≤i≤.9 logd−1(n).\nProof. Note that we have for every x∈Bithat|Wi(x)∩(G∪ Ui(x))| ≤(d−1)i\nC0i.99, so we also\nknow\n|Wi(S)∩(G∪Ui(S))| ≤ |S|(d−1)i\nC0i.99for anyS⊆Bi. (1)\nAssume for contradiction that we had |Bi|> C0αn\n(d−1)ii.99. Then|Bi|>2sR−iby Lemma\n5.6. Then in particular, we could find a S⊆BiwithsR−i≤ |S| ≤2sR−i. Then|Wi(S)| ≥\nsR≥2αn, so\n|Wi(S)∩(G∪Ui(S))| ≥αn=/parenleftbigg\nC0αn\n(d−1)ii.99/parenrightbigg(d−1)i\nC0i.99≥ |S|(d−1)i\nC0i.99\ncontradicting (1).\nSo we now know by Lemma 5.7 that |Bi|declines exponentially for 2 ≤i≤.9 logd−1(n),\nand we know by Lemma 5.3 that Bi= 0 fori≥logd−1(n)+/parenleftBig\nlog(ad)\n(d−1)log(d−1+γ)/parenrightBig\nlogd−1(logd−1(n)).\nThis, plus knowing that |Bi|is monotone decreasing in i, gives us the result we want:\nLemma 3.4. There is a C= Θ(1)such that |Bi| ≤Cn2−ifor any matching Mand for all\ni∈N.\nProof. First, we can take C1= Θ(1) large enough such that\n|Bi| ≤C0αn\n(d−1)ii.99≤C1n2−i\nfor all 0≤i≤.9 logd−1(n).\nThen, for sufficiently large n, we have 2R≤21.1logd−1(n)=n1.1logd−1(2)≤n0.7ford≥4, so\nn2−R≥n0.3. Additionally, we have\nαn\n(d−1).9logd−1(n)(logd−1(n)/2).99=O(n0.2),\nso we can take C2= Θ(1) to be large enough such that for all .9 logd−1(n)≤i≤R,\n|Bi| ≤ |B.9logd−1(n)| ≤αn\n(d−1).9logd−1(n)(.9 logd−1(n)).99≤C2n2−R≤C2n2−i.\nAnd asBi=∅for alli≥R,C= max(C1,C2) works for all i∈N.\nThis completes the proof of Theorem 1.1 for all d≥6.\n5.2 Improved Expansion Properties for Smaller d\nJust to get the d= 4 and d= 5 cases of Theorem 1.1 as well (and to improve the exponent\nof the logarithm for d= 3), we need a more careful analysis. In this section, we will prove\nthe following stronger version of Lemma 5.1:\n14Lemma 5.8. There is a τ= Θ(1)such that the following holds. Let a3= 8.1,a4= 15,\na5= 24, andad= (d−1)ed−1for alld≥6. For any 1≤s≤τn, define\nxs=/braceleftBigg\n0 if|S| ≤log(n)/(2d)\nlogd(ad)\nlogd(|X|/|S|)−1iflog(n)/(2d)≤ |S| ≤τn\nWith high probability, we have that for all S⊆Xwith|S| ≤τnthat\n|N(S)| ≥(d−1−x|S|)|S|.\nThe exact value of adis never used in the proof of Lemma 5.2 and Lemma 5.4 in [FPS13]\nand we can assume |S| ≤τnby Lemma 2.1. Therefore, the proof in [FPS13] goes through to\ngive insertion time O(log1+bd(n)) for all d≥3. Letbd=log(ad)\n(d−1)log(d−1). When we have bd< .98,\nour proof in Subsection 5.1 goes through to prove Lemma 3.4 and finis h Theorem 1.1. We\ngetbd< .98 ford≥4, while we only get b3≤1.509.\nTo prove Lemma 5.8, need a more accurate count on the number of w ays that |N(S)|\ncould take on a given value, and thus we use Stirling numbers of the se cond kind,/braceleftbiga\nb/bracerightbig\n,\nwhereb!/braceleftbiga\nb/bracerightbig\ncounts the number of labelled surjections from [ a] into [b]. We use the following\napproximation for Stirling numbers of the second kind due to Moser a nd Wyman:\nLemma 5.9 (Equation (5.1) of [MW58]) .Ifa=bgfor some constant g >1, we have that\nb!/braceleftbigga\nb/bracerightbigg\n=/parenleftbigg\n1±O/parenleftbigg1\na/parenrightbigg/parenrightbigga!(er−1)b\n2ra√\nhb\nwhereris the solution tor\n1−e−r=gandh=πrer(er−1−r)\n2(er−1)2.\nLetxsbe as in Lemma 5.8. For S⊆Xwith|S| ≤τn, we say that Sis a failing set if\n|N(S)|<(d−1−xs)s.\nLemma 5.10. Letv3= 7.266,v4= 14.986,v5= 25.5, andvd= (d−1)ed−1for alld≥6.\nThere exists some τ,ζ= Θ(1)such that for all S⊆Xwith log log(n)≤ |S| ≤τn, for\nsufficiently large n\nP(Sis a failing set )≤ζm−xss−ssxss+s(vd)s.\nProof. FixS⊆Xwith log log( n)≤ |S| ≤τn. Lets=|S|and letσ=⌊(d−1−xs)s⌋. We\nwill assume that d≤5, as for d≥6 this follows from the proof of Lemma 5.1 (Proposition\n2.4 of [FPS13]). Then\nP/parenleftbig\n|N(S)|<(d−1−x|S|)|S|/parenrightbig\n=σ/summationdisplay\ni=0P/parenleftbigg\n∃R∈/parenleftbiggY\ni/parenrightbigg\ns.t.N(S) =R/parenrightbigg\n=m−dsσ/summationdisplay\ni=0/parenleftbiggm\ni/parenrightbigg\ni!/braceleftbiggds\ni/bracerightbigg\nWe will now show that the sum above is dominated by the i=σterm. Let a,b∈N\nwitha≥b+ 1 and let Θ( a,b) be the set of partitions of [ a] intobunlabelled parts (we have\n|Θ(a,b)|=/braceleftbiga\nb/bracerightbig\n). We consider pairs ( θ1,θ2) where the θ1∈Θ(a,b),θ2∈Θ(a,b+ 1) and the\nsecond partition is a refinement of the first, that is, is obtained fro m the first by splitting a\nset. Now, for θ1∈Θ(a,b), letdL(θ1) denote the number of times θ1occurs first in such a\n15pair and, analogously for θ2∈Θ(a,b+ 1), let dR(θ2) denote the number of times θ2occurs\nsecond in such a pair. Then\ndL(θ1)≥min/braceleftBigg/summationdisplay\nj2xj−2 :x1+···xb=a/bracerightBigg\n≥b(2a/b−2).\ndR(θ2)≤/parenleftbiggb+ 1\n2/parenrightbigg\nBecause/summationtext\nθ1∈Θ(a,b)dL(θ1) =/summationtext\nθ2∈Θ(a,b+1)dR(θ2), we have\nb(2a/b−2)/braceleftbigga\nb/bracerightbigg\n≤/parenleftbiggb+ 1\n2/parenrightbigg/braceleftbigga\nb+ 1/bracerightbigg\n.\nLetui=/parenleftbigm\ni/parenrightbig\ni!/braceleftbigds\ni/bracerightbig\nfor some 0 ≤i≤σ. Then we have\nui+1\nui≥m−i\ni+ 1·(i+ 1)·i(2ds/i−2)/parenleftbigi+1\n2/parenrightbig =2(m−i)(2ds/i−2)\ni+ 1\n≥2(m−(d−1)τcm)(2d/(d−1)−2)\n(d−1)τcmasi≤(d−1)s≤(d−1)τcm\n≥4(1−(d−1)τc)(21/(d−1)−1)\n(d−1)τc>1 if τ <1/(8c) andd≤5.\nThus/summationtextσ\ni=0ui≤ζusfor some constant ζ >0. So,\nP/parenleftbig\n|N(S)|<(d−1−x|S|)|S|/parenrightbig\n≤ζm−ds/parenleftbiggm\nσ/parenrightbigg\nσ!/braceleftbiggds\nσ/bracerightbigg\nThend\nd−1−0.00001|N(S)|−(d−1−xs)≥ |N(S)|−(d−1).\n(2)\nIn particular, this means that for Sto possibly be a minimal failing set, we must have\n|N(S)| ≥ |N(S\\{x})| ≥(d−1−xs)(s−1). So after casting the dsballs into the mbins,\nand thus determining |N(S)|, we can assume that we have ( d−1−xs)(s−1)≤ |N(S)|<\n(d−1−xs)s, that is, it suffices to show that in this case, the probability of Sbeing a minimal\nfailing set is at most ( qd)s, as in other cases Sis not a minimal failing set.\nLetA⊆[ds] be the set of balls that ended up in a bin with another ball.\n|A| ≤2(ds−|N(S)|)≤2(ds−(d−1−xs)(s−1)) = 2(1 + xs)(s−1) + 2d≤2.001s.\nNow, we go about assigning Ato a random subset A′ofS×[d]. If there is some x∈S\nfor which |(x×[d])∩A′|<2, then\n|N(S\\{x})| ≤N(S)−d+|(x×[d])∩A′| ≤N(S)−d+ 1\nwhich is a contradiction to Equation (2), that is, S\\{x}becomes a failing set.\nTherefore, the probability that Sis a minimal failing set is at most the probability that\n|(x×[d])∩A′| ≥2 for every x∈S. Clearly, this is impossible (probability 0) if |A|<2|S|,\nso it suffices to show that for every 2 s≤ |A| ≤ 2.001s, the probability of A′satisfying\n|(x×[d])∩A′| ≥2 for every x∈S, conditioned on |A|, is at most ( qd)s.\n17Assume that we have thrown the balls and thus fixed A. The total number of equally\nlikely possibilities for A′is, ford≤4,\n/parenleftbiggds\n|A|/parenrightbigg\n≥/parenleftbiggds\n2.001s/parenrightbigg\n≥2dsH(2.001/d)(ds+ 1)−1\n(whereH(p) =−plog2(p)−(1−p) log2(1−p)) or, for d≥5,\n/parenleftbiggds\n|A|/parenrightbigg\n≥/parenleftbiggds\n2s/parenrightbigg\n≥2dsH(2/d)(ds+ 1)−1\nThe number of possibilities for A′that satisfy the condition |(x×[d])∩A′| ≥2 for every\nx∈Sis at most\n/parenleftbiggd\n2/parenrightbiggs/parenleftbiggds\n|A|−2s/parenrightbigg\n≤/parenleftbiggd(d−1)\n2/parenrightbiggs/parenleftbiggds\n.001s/parenrightbigg\n≤/bracketleftbiggd(d−1)(1000ed).001\n2/bracketrightbiggs\nThus,\nP(Smin. failing set)\nP(Sfailing set)≤/bracketleftbiggd(d−1)(1000ed).001(ds+ 1)1/s\n21+d(max(H(2.001/d),H(2/d)))/bracketrightbiggs\nThis expression is less than qdfor all 3 ≤d≤10 and sufficiently large n. If we ignore the\n(1000ed).001(ds+ 1)1/sin the expression (which can be removed in the limit by making τ\ndepend on d), the limit of this expression as d→ ∞ is/parenleftbig2\ne2/parenrightbigs���0.271s.\nNow, we have all the ingredients we need to prove our improved expa nsion lemma.\nProof of Lemma 5.8. For Lemma 5.8 to fail, there must be some S⊆Xwith|S| ≤τnsuch\nthatSis a minimal failing set. Then\nP(Lemma 5.8 fails) ≤τn/summationdisplay\ns=1P(∃S∈/parenleftbiggX\ns/parenrightbigg\ns.t.Sis a minimal failing set)\n≤log(n)/(2d)/summationdisplay\ns=1P(∃S∈/parenleftbiggX\ns/parenrightbigg\ns.t.Sis a failing set)\n+τn/summationdisplay\ns=log(n)/(2d)P(∃S∈/parenleftbiggX\ns/parenrightbigg\ns.t.Sis a minimal failing set)\n≤log(n)/(2d)/summationdisplay\ns=1ds\nn/parenleftbig\nc∗\nd(d−1)ed/parenrightbigs\n+τn/summationdisplay\ns=log(n)/(2d)P(∃S∈/parenleftbiggX\ns/parenrightbigg\ns.t.Sis a minimal failing set)\n(by the proof of Proposition 2.4, [FPS13])\n≤O(n−1/5) +τn/summationdisplay\ns=log(n)/(2d)/parenleftbiggn\ns/parenrightbigg\n(qd)sP(Sis a failing set)\n18(by Lemma 5.11)\n≤O(n−1/5) +τn/summationdisplay\ns=log(n)/(2d)/parenleftbiggcm\ns/parenrightbigg\n(qd)sζm−xss−sss+xss(vd)s\n(by Lemma 5.10)\n≤O(n−1/5) +ζτn/summationdisplay\ns=log(n)/(2d)(c∗\nde)smss−sm−xss−sss+xss(qdvd)s\n≤O(n−1/5) +ζτn/summationdisplay\ns=log(n)/(2d)[(s/m)xsc∗\ndqdvde]s\n≤O(n−1/5) +ζτn/summationdisplay\ns=log(n)/(2d)0.999s\n=o(n−η) for some small η= Θ(1)\ntakingxs= logm/s(ad) =logd(ad)\nlogd(m/s)≤logd(ad)\nlogd(|X|/|S|)−1, as we have set ad> c∗\ndqdvde/0.999.\n5.3 Acknowledgement\nWe thank Stefan Walzer for discovering an issue with a previous vers ion.\nReferences\n[ADW14] Martin Aum¨ uller, Martin Dietzfelbinger, and Philipp Woelfel. Ex plicit and\nefficient hash families suffice for cuckoo hashing with a stash. Algorithmica ,\n70:428–456, 2014.\n[BHR18] Aaron Bernstein, Jacob Holm, and Eva Rotenberg. Online bip artite matching\nwith amortized O(log2(n))replacements. Proceedings of the 2018 Annual ACM-\nSIAM Symposium on Discrete Algorithms (SODA) , pages 947–959, 2018.\n[CDKL09] Kamalika Chaudhuri, Constantinos Daskalakis, Robert Klein berg, and Henry Lin.\nOnline bipartite perfect matching with augmentations. Proceedings of the 28th\nIEEE Conference on Computer Communications (IEEE INFOCOM) , pages 1044–\n1052, 2009.\n[CK09] Jeffrey S. Cohen and Daniel M. Kane. Bounds on the independ ence required for\ncuckoo hashing. 2009.\n[CSW07] Julie Anne Cain, Peter Sanders, and Nick Wormald. The rando m graph threshold\nfor k-orientiability and a fast algorithm for optimal multiple-choice allo cation.\nProceedings of the Eighteenth Annual ACM-SIAM Symposium on Di screte Algorithms\n(SODA) , page 469–476, 2007.\n19[DGM+10] Martin Dietzfelbinger, Andreas Goerdt, Michael Mitzenmacher , Andrea Monta-\nnari, Rasmus Pagh, and Michael Rink. Tight thresholds for cuckoo h ashing via\nxorsat. Proceedings of the 37th International Colloquium Conference o n Automata,\nLanguages and Programming (ICALP) , pages 213–225, 2010.\n[DM03] Luc Devroye and Pat Morin. Cuckoo hashing: Further analys is.Information\nProcessing Letters , 86(4):215–219, 2003.\n[DW07] Martin Dietzfelbinger and Christoph Weidling. Balanced allocatio n and dic-\ntionaries with tightly packed constant size bins. Theoretical Computer Science ,\n380(1):47–68, 2007.\n[EGMP14] David Eppstein, Michael T. Goodrich, Michael Mitzenmache r, and Pawe/suppress l Pszona.\nWear minimization for cuckoo hashing: How not to throw a lot of eggs in to\none basket. Proceedings of the International Symposium on Experimental Algorithms\n(SEA) , pages 162–173, 2014.\n[FJ17] Alan Frieze and Tony Johansson. On the insertion time of rand om walk cuckoo\nhashing. Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algo-\nrithms (SODA) , pages 1497–1502, 2017.\n[FKP11] Nikolaos Fountoulakis, Megha Khosla, and Konstantinos Pan agiotou. The\nmultiple-orientability thresholds for random hypergraphs. Proceedings of the 2017\nAnnual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 1222–1236,\n2011.\n[FM12] Alan Frieze and P´ all Melsted. Maximum matchings in random bipa rtite graphs\nand the space utilization of cuckoo hash tables. Random Structures & Algorithms ,\n41(3):334–364, 2012.\n[FMM09] Alan Frieze, P´ all Melsted, and Michael Mitzenmacher. An an alysis of random-\nwalk cuckoo hashing. Proceedings of the 2009 International Conference on Random-\nization and Computation (RANDOM) , 2009.\n[FP10] Nikolaos Fountoulakis and Konstantinos Panagiotou. Orienta bility of random\nhypergraphs and the power of multiple choices. Proceedings of the 37th International\nColloquium Conference on Automata, Languages and Programmin g (ICALP) , pages\n348–359, 2010.\n[FP18] Alan Frieze and Samantha Petti. Balanced allocation through r andom walk.\nInformation Processing Letters , 131:39–43, 2018.\n[FPS13] Nikolaos Fountoulakis, Konstantinos Panagiotou, and Ange lika Steger. On the\ninsertion time of cuckoo hashing. SIAM Journal on Computing , 42(6):2156–2181,\n2013.\n[FPSS03] Dimitris Fotakis, Rasmus Pagh, Peter Sanders, and Paul G . Spirakis. Space\nefficient hash tables with worst case constant access time. Proceedings of the\n2020th Annual Symposium on Theoretical Aspects of Computer Scie nce (STACS) , page\n271–282, 2003.\n[FR07] Daniel Fernholz and Vijaya Ramachandran. The k-orientability thresholds for\nGn,p.Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Di screte Al-\ngorithms (SODA) , pages 459–468, 2007.\n[GKKV95] Edward Grove, Ming-Yang Kao, P. Krishnan, and Jeffrey S cott Vitter. Online\nperfect matching and mobile computing. Proceedings of the 4th International Work-\nshop on Algorithms and Data Structures (WADS) , 955:194–205, 1995.\n[GW10] Pu Gao and Nicholas C. Wormald. Load balancing and orientability thresholds\nfor random hypergraphs. Proceedings of the 42nd ACM Symposium on Theory of\nComputing (STOC) , pages 97–104, 2010.\n[KA19] Megha Khosla and Avishek Anand. A faster algorithm for cuck oo insertion and\nbipartite matching in large graphs. Algorithmica , 81(9):3707–3724, 2019.\n[KMW09] Adam Kirsch, Michael Mitzenmacher, and Udi Wieder. More r obust hashing:\nCuckoo hashing with a stash. SIAM Journal on Computing , 39(4):1543–1561, 2009.\n[Mit09] Michael Mitzenmacher. Some open questions related to cuck oo hashing. Pro-\nceedings of the 17th Annual European Symposium on Algorithm s (ESA) , pages 1–10,\n2009.\n[MP23] Brice Minaud and Charalampos Papamanthou. Generalized cuc koo hashing with\na stash, revisited. Information Processing Letters , 181(106356), 2023.\n[MW58] L. Moser and M. Wyman. Stirling numbers of the second kind. Duke Mathematical\nJournal , 25(1):29–43, 1958.\n[PR01] Rasmus Pagh and Flemming Friche Rodler. Cuckoo hashing. Proceedings of the\n9th Annual European Symposium on Algorithms (ESA) , pages 121–133, 2001.\n[SHF+17] Yuanyuan Sun, Yu Hua, Dan Feng, Ling Yang, Pengfei Zuo, Shu nde Cao, and\nYuncheng Guo. A collision-mitigation cuckoo hashing scheme for large -scale stor-\nage systems. IEEE Transactions on Parallel and Distributed Systems , 28(3):619–632,\n2017.\n[Wal22] Stefan Walzer. Insertion time of random walk cuckoo hashing below the peeling\nthreshold. Proceedings of the 30th Annual European Symposium on Algori thms (ESA) ,\n244(87):1–11, 2022.\n[Yeo23] Kevin Yeo. Cuckoo hashing in cryptography: Optimal param eters, robustness\nand applications. 43rd Annual International Cryptology Conference (CRYPTO) , page\n197–230, 2023.\n21" }, { "title": "2401.14419v1.Constraining_the_mass_of_fermionic_dark_matter_from_its_feeble_interaction_with_hadronic_matter_via_dark_mediators_in_neutron_stars.pdf", "content": "Constraining the mass of fermionic dark matter from its feeble interaction with\nhadronic matter via dark mediators in neutron stars\nAtanu Guha1and Debashree Sen2\n1Department of Physics, Chungnam National University,\n99, Daehak-ro, Yuseong-gu, Daejeon-34134, South Korea and\n2Center for Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea∗\n(Dated: January 29, 2024)\nConsidering ten well-known relativistic mean field models, we invoke feeble interaction between\nhadronic matter and fermionic dark matter (DM) χvia new physics scalar ( ϕ) and vector ( ξ) media-\ntors in neutron star core, thereby forming DM admixed neutron stars (DMANSs). The chosen masses\nof the DM fermion ( mχ) and the mediators ( mϕandmξ) are consistent with the self-interaction\nconstraint from Bullet cluster while their respective couplings ( yϕandyξ) are also constrained\nby the present day relic abundance. Assuming that both ϕandξcontribute equally to the relic\nabundance, we compute the equation of state of the DMANSs and consequently their structural\nproperties. We found that for a particular (constant) DM density, the presence of lighter DM re-\nsults in more massive DMANSs with larger radius. In the light of the various recent constraints like\nthose from the massive pulsar PSR J0740+6620, the gravitational wave (GW170817) data and the\nresults of NICER experiments for PSR J0030+0451 and PSR J0740+6620, we provide a bound on\nmχwithin the framework of the present work as mχ≈(0.1−30) GeV for a wide range of fixed DM\nFermi momenta kχ\nF=(0.01 −0.06) GeV. In the case of the hadronic models that yield larger radii\ncorresponding to the low mass neutron stars in the no-DM scenario, interaction with comparatively\nheavier DM fermion is necessary in order to ensure that the DMANSs obtained with such models\nsatisfy the radius constraints from both GW170817 and NICER data for PSR J0030+0451.\nPACS numbers:\nI. INTRODUCTION\nNeutron star cores are one of the most interesting, exotic and complex systems to study. At present we lack concrete\nexperimental data at high density relevant to neutron stars and hence the knowledge regarding the composition,\ninteraction and the equation of state of neutron star matter is based on theoretical modeling along with the related\nuncertainties. However, several recent astrophysical observations like those from the massive pulsar PSR J0740+6620\n[1–3], gravitational wave (GW170817) data [4] and the NICER data for PSR J0030+0451 [5, 6] put certain constraints\non the equation of state of neutron stars.\nThe extreme conditions of the neutron star environment is not only related to density but also gravity. The strong\ngravity of the neutron stars gives rise to the phenomenon of accretion and the neutron stars accrete matter from its\nsurroundings which may include dark matter (DM) and thus forming DM admixed NSs (DMANSs). DM may also\nbe produced in neutron star cores from neutron decay [7, 8]. Observational evidences like the rotation curves of the\ngalaxies, gravitational lensing, X-ray analysis of Bullet cluster [9, 10] support the existence of DM in the Universe.\nThe Cosmic Microwave Background anisotropy maps, obtained from the Wilkinson Microwave Anisotropy Probe\n(WMAP) data [11–13], provides the present day thermal relic abundances of DM as ∼Ωh2≈0.12 [14–18]. Thus the\npresence of DM in neutron star surroundings and eventual accretion onto neutron stars may be possible. Although the\nproperties and the interaction of DM candidates are inconclusive, but literature suggests that the Weakly Interacting\nMassive Particles (WIMPs) are the most suitable DM particle candidates. Several direct detection experiments like\nsuperCDMS [19], XENON100 [20], XENON1T [21], LUX [22], PANDAX-II [23], DARKSIDE-50 [24], SENSEI [25]\nand very recently the LUX-ZEPLIN (LZ) [26] etc. for the WIMPs search are being attempted worldwide. Moreover,\nDM may be self-interacting and in such cases the masses of the DM fermion and the DM mediators are constrained\nby the self-interaction constraints from bullet cluster [27–31]. Also, the self-interaction couplings are constrained to\nreproduce the observed non-baryonic relic density [32–34].\nIf we consider the process of DM accretion as a source of presence of DM inside neutron star cores, then the accreted\nDM particles undergo collisions with the hadronic matter in the neutron stars and hence lose kinetic energy. With\ndue course of time, the DM particles end up being gravitationally bound to the star. Eventually, the accretion stops\n∗Electronic address: atanu@cnu.ac.kr, debashreesen88@gmail.comarXiv:2401.14419v1 [astro-ph.HE] 23 Jan 20242\nand the DM particles attain thermal equilibrium among themselves due to the self interactions [35, 36]. Due to this\nfact, it is justified to consider the DM particle density ρχto be almost constant [37–39] in the case of DMANSs.\nApart from accretion, there are other mechanisms which can be responsible for the presence of DM in neutron stars.\nFor example, DM can be inherited by neutron stars from progenitors during supernovae explosions [40, 41] or or DM\nmay be produced inside neutron stars via neutron decay [7, 8, 42]. In the later mechanism the DM number density\nis reasonably high, ρχ= (0.01−0.1)ρ, where, ρis total baryon number density [8, 42]. In the present work we do\nnot actually focus on any particular mechanism as the possible source of the presence of DM in neutron stars. We\nintend to show how the presence of DM, through any of the possible mechanisms as discussed above, can affect the\nstructural properties of neutron stars. Also in this context we investigate the possible range of DM mass in order to\nsatisfy the observational constraints on neutron star properties. Co-existing along with the baryonic matter inside\nneutron star cores, DM may or may not interact with hadronic matter. In case they do not interact, the two types\nof matter coexist in the two fluid form [43–62] whereas the interaction between the DM and the baryonic matter\nis also suggested by [37, 40, 63–68], mostly via the Higgs boson as mediator. The interaction between DM and the\nhadronic matter of the star must be extremely weak [69] to prevent the collapse of the star into a black hole due to\nheavy accretion of DM. Therefore in [39] we invoked feeble interaction between hadronic and fermionic DM χvia a\nnew scalar mediator ϕand a dark vector mediator ξin [38] in order to explain the possible existence of DMANSs.\nϕandξinteract with the hadronic matter ψwith a very feeble coupling strength. The masses of DM fermion mχ\nand the mediators ( mϕandmξ) and the couplings ( yϕandyξ) are consistent with the self-interaction constraint from\nthe Bullet cluster and from the present day relic abundance, respectively. In both [38, 39] we considered only the\neffective chiral model as the hadronic model to study the effects of DM interaction on the properties of DMANSs. We\nconcluded that mass of DM ( mχ) plays a very important role in determining the structural properties of DMANSs.\nThe massive the DM, the less are the maximum mass, radius and tidal deformability of the DMANSs. In the present\nwork we aim to constrain the value of mχor rather we seek a possible range of mχfor which the DMANSs satisfy the\nconstraints on the structural properties of compact stars obtained from PSR J0740+6620, GW170817 and the NICER\ndata for PSR J0030+0451. For the purpose we consider ten well-known relativistic mean field (RMF) models viz.\nTM1, GM1, NL3, PK1, DD-MEX, DD2, TW99, DD-ME2, PK-DD and DD-LZ1. In order to obtain the range of mχ,\nwe consider one minimum and another maximum value of the DM Fermi momentum kχ\nFwhich gives the maximum\nand minimum values of the constant DM particle density ρχ.\nThis paper is organized as follows. In the next section II, we briefly address the framework of the ten RMF hadronic\nmodels. In the same section, we also discuss the mechanism of invoking the DM interaction with hadronic matter\nvia the dark mediators ϕandξand the the structural properties of the DMANSs. We then present our results and\ncorresponding discussions in section III. We summarize and conclude in the final section IV of the paper.\nII. FORMALISM\nFollowing our previous works [38, 39] we introduce feeble interaction of the dark fermion ( χ) with the hadronic\nmatter ( ψ=n, p) through the scalar ( ϕ) and vector ( ξ) new physics mediators in neutron star core. For the pure\nhadronic matter sector we consider ten well-known RMF models of two different classes - i) models with non-linear\nself couplings like TM1 [70], GM1 [71], NL3 [72], and PK1 [73] and ii) models with density-dependent couplings like\nDD-MEX [74], DD2 [75], DD-ME2 [76], PK-DD [73], DD-LZ1 [77], and TW99 [78]. For the dark sector, we consider\nthe phenomenological treatment to describe the self-interaction of non-relativistic DM by a Yukawa potential [29]\nV(r) =±αχ\nre−mϕr(1)\nwhere, αχ=y2\n4πis the dark fine structure constant. We consider that ϕandξhave their respective couplings as yϕ\nandyξwith χ\nLint=(\nyϕϕ¯χχ\nyξ¯χγµχξµ(2)\nThe complete Lagrangian is given as3\nL=¯ψ[γµ(i∂µ−gωωµ−gρ⃗ ρµ·⃗ τ−gξξµ)−(M+gσσ+gϕϕ)]ψ+1\n2∂µσ∂µ−1\n2m2\nσσ2−1\n3g2σ3−c\n4g3σ4\n−1\n4ωµνωµν+1\n2m2\nωωµωµ+1\n4c3(ωµωµ)2−1\n4⃗Rµν·⃗Rµν+1\n2m2\nρ⃗ ρµ·⃗ρµ\n+1\n2∂µϕ∂µϕ−1\n2m2\nϕϕ2−1\n4VµνVµν+1\n2m2\nξξµξµ+ ¯χ[(iγµ∂µ−yξγµξµ)−(mχ+yϕϕ)]χ (3)\nIn the pure hadronic sector the nucleons interact via the scalar σ, the vector ωand iso-vector ρmesons. The vacuum\nexpectation values (VEVs) of the meson fields ( σ0,ω0andρ03) in RMF approximation remain unaffected due to the\npresence of DM and the expressions can be found in [79]. The mesons in the hadronic sector have density independent\ncouplings gσ,gωandgρwith the nucleons for the models like TM1, GM1, NL3, and PK1. For such models g2and\ng3are the higher order scalar field coefficients while c3is the higher order vector field coefficient. These non-linear\nself couplings are effectively considered in order to account for the in-medium effects. In the following table we first\nshow the density independent couplings ( gσ,gω,gρ,g2,g3, and c3) and the mass of mesons ( mσ,mωandmρ) and\nneutron ( mn) and proton ( mp) adopted in the models like TM1 [70], GM1 [71], NL3 [72], and PK1 [73] according to\nthe respective references.\nTABLE I: The density independent meson-nucleon couplings and parameters adopted in the models TM1 [70], GM1\n[71], NL3 [72], and PK1 [73].\nModel mn mp mσ mω mρ gσ gω gρ g2 g3 c3\n(MeV) (MeV) (MeV) (MeV) (MeV) (fm−1)\nTM1 938 938 511.198 783 770 10.0289 12.6139 4.6322 -7.2325 0.6183 71.3075\nGM1 938 938 510 783 770 8.874 43 10.60957 4.09772 -9.7908 -6.63661 0\nNL3 939 939 508.1941 782.501 763 10.2169 12.8675 4.4744 -10.4307 -28.8851 0\nPK1 939.5731 938.2796 514.0891 784.254 763 10.3222 13.0131 4.5297 -8.1688 -9.9976 55.636\nIn models like DD-MEX [74], DD2 [75], DD-ME2 [76], PK-DD [73], DD-LZ1 [77], and TW99 [78] g2=g3=c3=0 and\nthe in-medium effects are treated with the density-dependent couplings following the Typel-Wolter ansatz [78] as\ngi(ρ) =giai1 +bi(x+di)2\n1 +ci(x+di)2(4)\nwhere i=σ, ωandx=ρ/ρ 0while\ngρ(ρ) =gρexp[a ρ(x−1)] (5)\nAll the relevant masses and the parameters are listed in Tables I and II. The saturation properties like the saturation\ndensity ρ0, symmetry energy J0, slope L0, nuclear incompressibility K0, skewness coefficient S0, and the curvature\nparameter Ksymof the nuclear symmetry energy as obtained for all the above ten models considered in this present\nwork for the specific parameters can be found in the respective references and also in [79].\nThe dark bosons ϕandξinteract with the hadronic matter ψwith a very feeble coupling strength gϕ=gξ∼10−4.\nThe VEVs of the DM mediator fields in RMF approximation are\nϕ0=m⋆\nχ−mχ\nyϕ(6)\nand\nξ0=gξρ+yξρχ\nm2\nξ(7)\nThe modified effective mass due to DM interaction is\nm⋆\nB=MB+gσσ+gϕϕ (8)4\nTABLE II: The density dependent meson-nucleon couplings and parameters adopted in the models DD-MEX [74],\nDD2 [75], DD-ME2 [76], PK-DD [73], DD-LZ1 [77], and TW99 [78].\nModel mn mp mσ mω mρ gσ gω gρ\n(MeV) (MeV) (MeV) (MeV) (MeV)\nDD-MEX 938.5 938.5 547.3327 783 763 10.706722 13.338846 3.619020\nDD2 939.56536 938.27203 546.212459 783 763 10.686681 13.342362 3.626940\nDD-ME2 938.5 938.5 550.1238 783 763 10.5396 13.0189 3.6836\nPK-DD 939.5731 938.2796 555.5112 783 763 10.7385 13.1476 4.2998\nDD-LZ1 938.9 938.9 538.619216 783 763 12.001429 14.292525 7.575467\nTW99 939 939 550 783 763 10.7285 13.2902 3.6610\nModel aσ bσ cσ dσ aω bω cω dω aρ\nDD-MEX 1.397043 1.334964 2.067122 0.401565 1.393601 1.019082 1.605966 0.455586 0.620220\nDD2 1.357630 0.634442 1.005358 0.575810 1.369718 0.496475 0.817753 0.638452 0.983955\nDD-ME2 1.3881 1.0943 1.7057 0.4421 1.3892 0.9240 1.4620 0.4775 0.5647\nPK-DD 1.327423 0.435126 0.691666 0.694210 1.342170 0.371167 0.611397 0.738376 0.183305\nDD-LZ1 1.062748 1.763627 2.308928 0.379957 1.059181 0.418273 0.538663 0.786649 0.776095\nTW99 1.365469 0.226061 0.409704 0.901995 1.402488 0.172577 0.344293 0.983955 0.515000\nwhile the modified chemical potential is\nµB=q\nk2\nB+m⋆\nB2+gωω0+gρI3Bρ03+ ΣR+gξξ0 (9)\nwhere, the rearrangement term ΣR=0 for the models with density independent couplings and for the models with\ndensity dependent couplings it is given by [80] as\nΣR=dgσ(ρ)\ndρσ0ρSB+dgω(ρ)\ndρω0ρ+dgρ(ρ)\ndρρ03I3BρB (10)\nHere, B=n, p and ρSis the scalar density. I3Bis the third component of isospin for the individual nucleons.\nThe complete expressions for the equation of state is also modified due to the presence of DM. The energy density\nεis given as\nε=1\n2m2\nσσ2\n0+1\n3g2σ3\n0+1\n4g3σ4\n0+1\n2m2\nωω2\n0+3\n4c3ω4\n0+1\n2m2\nρρ2\n03\n+γ\n2π2X\nB=n,pZkF\n0q\nk2\nB+m⋆\nB2k2\nBdk+γ\n2π2X\nl=e,µZkl\n0q\nk2\nl+m2\nlk2\nldkl\n+1\n2m2\nϕϕ2\n0+1\n2m2\nξξ2\n0+γχ\n2π2Zkχ\nF\n0q\nk2χ+m⋆χ2k2\nχdkχ (11)\nand the pressure is given as\nP=−1\n2m2\nσσ2\n0−1\n3g2σ3\n0−1\n4g3σ4\n0+1\n2m2\nωω2\n0+1\n4c3ω4\n0+1\n2m2\nρρ2\n03\n+γ\n6π2X\nB=n,pZkF\n0k4\nBdkq\nk2\nB+m⋆\nB2+γ\n6π2X\nl=e,µZkl\n0k4\nldklp\nk2\nl+m2\nl\n+1\n2m2\nϕϕ2\n0+1\n2m2\nξξ2\n0+γχ\n6π2Zkχ\nF\n0k4\nχdkχq\nk2χ+m⋆χ2(12)\nAs mentioned in the Introduction section I that following our previous works [38, 39, 81], in the present work the\nvalues of mχ,mϕandmξare considered consistent with the self-interaction constraints from bullet cluster [27–31]\nwhile the self-interaction couplings are also chosen by reproducing the observed non-baryonic relic density [32–34].5\nThe permitted values of mϕandmξcorresponding to the range of mχare already shown in our previous works\n[38, 39, 81].\nWith the obtained DMANS equation of state, we compute the structural properties like the gravitational mass\n(M) and the radius ( R) of the DMANSs in static conditions by integrating the Tolman-Oppenheimer-Volkoff (TOV)\nequations [82, 83]. The dimensionless tidal deformability (Λ) is obtained in terms of the mass, radius and the tidal\nlove number ( k2) following [84, 85].\nIn the present work we have considered the DM number density ρχto be constant via constant DM Fermi momentum\nthroughout the radial profile of the star following [37] and our previous works [38, 39]. This number density is quite\nhigh compared to the density considered in [86] where the authors have shown that using the quark-meson coupling\n(QMC) model and considering local DM mass density (=0.3 GeV/cc), the capture rate of accreted DM can be ∼(1033\n- 1043) GeV s−1for DM mass mχ∼1 GeV with different operators and for different neutron star mass. Further from\n[86] we find that the DM capture rate is roughly directly proportional to the DM number density ρχ. In the present\nwork we have the DM number density as 4.4 ×10−6fm−3(mass density 4.4 ×1033GeV/cc) and 9.5 ×10−4fm−3(mass\ndensity 9.5 ×1035GeV/cc) for Fermi momentum kχ\nFto be 0.01 GeV and 0.06 GeV, respectively. So for mχ∼1 GeV\nthe DM mass density is very high compared to the local DM density. Therefore for the case where the accretion is\nthe only mechanism for presence of DM in neutron stars, the DM capture rate has to be enhanced compared to [86]\nroughly by an factor of ∼1034forkχ\nF=0.01 GeV and ∼1037forkχ\nF=0.06 GeV, to explain such a high density of DM\ninside neutron star. So in the present work the maximum DM capture rate is ∼1077GeV s−1forkχ\nF=0.01 GeV and\n∼1080GeV s−1forkχ\nF=0.06 GeV. Considering our estimate of DM capture rate, we find that it is largely inconsistent\nwith the results of [86]. The main reason is that our consideration of ρχis quite high which leads to high values of\nthe DM capture rate. A possible solution to fix this problem may be to consider the local DM density. It can be\nexpected that consideration of the local DM density can match the order of DM capture rate as obtained by [86].\nHowever, as mentioned in the introduction, there maybe other possible sources for the presence of DM inside\nneutron stars. So even if the DM density in the vicinity of the neutron star is considered to be the local DM density,\nwhich can make the capture rate to be consistent with [86], the DM density inside the neutron star can be quite high\ndue to the other mechanisms involved, as seen from [8, 42]. This maybe another feasible explanation for the high DM\ndensity inside neutron star along with the DM capture rate being consistent with [86].\nSo irrespective of the mechanism of the presence of DM inside neutron star, we proceed to study the effects of DM\non the structural properties of neutron stars in the next section.\nIII. RESULTS\nA. Neutron stars without dark matter\nWe first show the results of the structural properties of neutron stars obtained with the ten chosen RMF models in\nthe absence of DM in Fig. 1. Fig. 1a shows the variation of mass with radius and Fig. 1b shows the relation of tidal\ndeformability with mass of neutron stars without DM. It can be seen from Fig. 1a that among the ten chosen RMF\nmodels, NL3 yields the most massive neutron star with maximum radius while with the TW99 model we obtain the\nleast massive neutron star configuration with minimum radius compared to that obtained with the other models. The\nneutron star configurations obtained with all the chosen models satisfy the constraint on the mass-radius relationship\nof the neutron stars obtained from the most massive pulsar PSR J0740+6620 [1–3] and also the NICER data for PSR\nJ0030+0451 [5, 6]. However, it is well known that the constraints from GW170817 [4] both on the M−RandM−Λ\nplanes are not or barely satisfied by the results with the NL3, TM1, and the PK1 models. Our results in Figs. 1a\nand 1b support the same. The result of the DD-MEX model satisfies the bound from GW170817 in the M−Rplane\nbut not in the M−Λ plane. This subsection serves as an overview of the present literature. We present the existing\nresults of the structural properties of neutron star without the presence of DM with different models particularly for\nthe purpose of comparison.\nB. Dark matter admixed neutron stars with maximum kχ\nF=0.06 GeV\nWe next present our results of the structural properties of the DMANSs obtained with the ten chosen hadronic\nmodels, first considering maximum value of kχ\nF=0.06 GeV. This maximum value of kχ\nFimplies the maximum DM\ndensity ρmax\nχ(= 9.5 ×10−4fm−3) i.e., when DM populates the neutron star the maximum. The dark matter accreted\nby neutron stars affects the equation of state and consequently the structural properties of the dark matter admixed\nneutron stars. From equations 11 and 12 (last terms) it can be seen that the equation of state i.e. the total\nenergy density and pressure of the dark matter admixed neutron star is not only affected by the dark matter Fermi6\n0.811.21.41.61.822.22.42.62.8\n9 10 11 12 13 14 15 16 17Mass (M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451no DM\nTM1\nGM1\nNL3\nPK1\nDD-MEX\nDD2\nTW99\nDD-ME2\nPK-DD\nDD-LZ1\n(a)\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8Λ\nM (M⊙)GW170817no DM\nTM1\nGM1\nNL3\nPK1\nDD-MEX\nDD2\nTW99\nDD-ME2\nPK-DD\nDD-LZ1 (b)\nFIG. 1: Variation of (a) mass with radius and (b) tidal deformability with mass of neutron stars without dark matter\nwith hadronic models. In figure (a) the observational limits imposed from the most massive pulsar PSR J0740+6620\n(M= 2.08±0.07M⊙) [1] and R= 13.7+2.6\n−1.5km (yellow shaded region [2]) or R= 12.39+1.30\n−0.98km (shaded region with\ngreen diagonal lines [3]) are also indicated. The constraints on M−Rplane prescribed from GW170817 (pink and\norange shaded regions [4]) and the NICER experiment for PSR J0030+0451 (shaded regions with brown [5] and cyan\n[6] diagonal lines) are also compared. In figure (b) the constraint on Λ1.4from GW170817 [4] is also shown.\nmomentum but also the mass of the dark matter. Since the structural properties of the star like the mass, radius and\ntidal deformability are directly dictated by the equation of state, the presence of dark matter and its Fermi momentum\nand the mass play important role in determining the structural properties of the star. In our previous works [38, 39]\nwe found that lighter fermionic DM results in more massive DMANSs with larger radius. Therefore, we check with\neach model the suitable mass range of fermionic DM in order to obtain reasonable DMANSs configurations in the\nlight of the different astrophysical constraints on the structural properties of compact stars.\nIn Figs. 2a, 2b, 2c, 2d, 4a, 4b, 4c, 4d, 6a and 6b we show the maximum and minimum values of mχfor which\nthe DMANS configurations with maximum kχ\nF(ρχ) can satisfy all the astrophysical constraints on the mass-radius\nvariation for hadronic models TM1, GM1, NL3, PK1, DD-MEX, DD2, TW99, DD-ME2, PK-DD and DD-LZ1,\nrespectively. For better understanding we also show the results for two more values of mχ- one bellow the minimum\nand one above the maximum limits for each model in order to obtain a moderately clear range of mχ. It is seen that\nfor a value of mχbelow the minimum limit, the result is inconsistent with the GW170817 data while the choice of\nmχabove the maximum, leads to the violation of the NICER data for PSR J0030+0451. The obtained allowed range\nofmχfor the DMANSs is then tested in the M−Λ plane with respect to the constraint on the tidal deformability of\n1.4M⊙neutron star (Λ 1.4) obtained from the GW170817 data in Figs. 3a, 3b, 3c, 3d, 5a, 5b, 5c, 5d, 7a and 7b for\nthe hadronic models TM1, GM1, NL3, PK1, DD-MEX, DD2, TW99, DD-ME2, PK-DD and DD-LZ1, respectively.\nExcept for NL3, the obtained allowed range of mχ, in terms of the different astrophysical constraints, is same in both\ntheM−RandM−Λ planes. For the NL3 model the lower limit of mχ=500 MeV satisfy all the astrophysical\nconstraints in the M−Rplain as seen from Fig. 2c but Fig. 3c shows that with this value of mχ=500 MeV the result\noffshoots the upper bound on Λ 1.4obtained from GW170817 data and combining the joint results of Figs. 2c and 3c,\nwe find that the minimum value of mχfor maximum kχ\nF=0.06 GeV is 1 GeV in order to satisfy all the astrophysical\nconstraints.\nFor further understanding of the allowed range of mχthat yields reasonable DMANSs that can successfully satisfy\nthe constrained properties like maximum mass Mmax,R1.4and Λ 1.4, we depict the individual variation of these\nquantities with mχin Figs. 8a, 8b and 8c, respectively. For convenience we also compare the respective constraints7\n0.811.21.41.61.822.22.4\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nTM1\nno-DM\nmχ=300 MeV\n500 MeV\n25 GeV\n30 GeV\n(a) TM1\n0.811.21.41.61.822.22.42.6\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nGM1\nno-DM\nmχ=250 MeV\n300 MeV\n20 GeV\n25 GeV (b) GM1\n11.522.5\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nNL3\nno-DM\nmχ=300 MeV\n500 MeV\n30 GeV\n50 GeV\n(c) NL3\n0.811.21.41.61.822.22.4\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nPK1\nno-DM\nmχ=300 MeV\n500 MeV\n25 GeV\n50 GeV (d) PK1\nFIG. 2: Variation of mass with radius of dark matter admixed neutron stars for different values of mχand\nmaximum kχ\nFwith hadronic models (a) TM1 (b) GM1 and (c) NL3 and (d) PK1.\nin the same figure. In Figs. 8a, 8b and 8c we show with each model, the results for the two extreme values of mχ\nthat signifies the allowed range of mχfor which the DMANS satisfy all the astrophysical constraints. Consistent with\nour previous works [38, 39] we find that for any model, lighter fermionic DM results in more massive DMANSs with\nlarger radius. In Table III we present the allowed range of mχthus obtained for the maximum DM fraction with the\nten chosen models.8\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2Λ\nM (M⊙)GW170817TM1\nno-DM\nmχ=500 MeV\n25 GeV\n(a) TM1\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817GM1\nno-DM\nmχ=300 MeV\n20 GeV (b) GM1\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8Λ\nM (M⊙)GW170817NL3\nno-DM\nmχ=500 MeV\n1 GeV\n30 GeV\n(c) NL3\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817PK1\nno-DM\nmχ=500 MeV\n25 GeV (d) PK1\nFIG. 3: Variation of tidal deformability with mass of dark matter admixed neutron stars for different values of mχ\nand maximum kχ\nFwith hadronic models (a) TM1 (b) GM1 and (c) NL3 and (d) PK1.\nC. Dark matter admixed neutron stars with minimum kχ\nF=0.01 GeV\nWe now proceed to obtain our results with the minimum value of kχ\nF=0.01 GeV which corresponds to the minimum\nDM density ρmin\nχ(= 4.4 ×10−6fm−3). In the same way as in case of the maximum kχ\nF, we try to obtain the allowed\nrange of mχrequired to obtain reasonable DMANSs configurations for the minimum value of kχ\nF.9\n0.811.21.41.61.822.22.42.6\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nDD-MEX\nno-DM\nmχ=500 MeV\n1 GeV\n25 GeV\n30 GeV\n(a) DD-MEX\n0.811.21.41.61.822.22.42.6\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nDD2\nno-DM\nmχ=300 MeV\n500 MeV\n15 GeV\n20 GeV (b) DD2\n0.811.21.41.61.822.22.4\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nTW99\nno-DM\nmχ=75 MeV\n100 MeV\n10 GeV\n15 GeV\n(c) TW99\n0.811.21.41.61.822.22.42.6\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nDD-ME2\nno-DM\nmχ=300 MeV\n500 MeV\n15 GeV\n20 GeV (d) DD-ME2\nFIG. 4: Variation of mass with radius of dark matter admixed neutron stars for different values of mχand\nmaximum kχ\nFwith hadronic models (a) DD-MEX (b) DD2 and (c) TW99 and (d) DD-ME2.\nSimilar to Figs. 8a, 8b and 8c obtained for maximum kχ\nF, we present for the minimum kχ\nF=0.01 GeV, in Figs. 9a, 9b\nand 9c the dependence of Mmax,R1.4and Λ 1.4onmχwith respect to the constraints on these quantities. Interestingly,\nin the case of very low DM population, we find that the values of Mmax,R1.4and Λ 1.4saturate at a maximum value\nofmχ=10 GeV i.e, above this value of mχthe structural properties of the DMANSs do not change for any of the\nhadronic models considered in the present work. Therefore in this case of minimum kχ\nFwe do not obtain any particular\nupper bound on mχbut a saturation value msat\nχ=10 GeV irrespective of the hadronic model considered to obtain the10\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Λ\nM (M⊙)GW170817DD-MEX\nno-DM\nmχ=1 GeV\n10 GeV\n25 GeV\n(a) DD-MEX\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817DD2\nno-DM\nmχ=500 MeV\n15 GeV (b) DD2\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817TW99\nno-DM\nmχ=100 MeV\n10 GeV\n(c) TW99\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817DD-ME2\nno-DM\nmχ=500 MeV\n15 GeV (d) DD-ME2\nFIG. 5: Variation of tidal deformability with mass of dark matter admixed neutron stars for different values of mχ\nand maximum kχ\nFwith hadronic models (a) DD-MEX (b) DD2 and (c) TW99 and (d) DD-ME2.\nDMANS configurations. This is because with the lower DM population, the scenario is close to the no-DM case and\nunder such circumstances the low DM content cannot bring any perceptible change to the structural properties of\nthe star. For example the maximum mass of both the DMANS for mχ=10 GeV and the neutron star in the no-DM\nscenario is 2.32 M⊙for the PK1 model while it is 2.42 for the DD2 model. For lower kχ\nF,mχsaturates at a lower\nvalue compared to that for a higher value of kχ\nF. So for kχ\nF=0.06 GeV, the value of msat\nχis quite higher and beyond\nthe maximum value of mχrequired to satisfy all the astrophysical constraints. Therefore in Table IV we tabulate the11\n0.811.21.41.61.822.22.42.6\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nPK-DD\nno-DM\nmχ=75 MeV\n100 MeV\n15 GeV\n20 GeV\n(a) PK-DD\n0.811.21.41.61.822.22.42.62.8\n9 10 11 12 13 14 15 16Mass ( M⊙)\nRadius (kms)PSR J0740+6620\nGW170817 M1\nGW170817 M2PSR J0030+0451\nDD-LZ1\nno-DM\nmχ=75 MeV\n100 MeV\n20 GeV\n25 GeV (b) DD-LZ1\nFIG. 6: Variation of mass with radius of dark matter admixed neutron stars for different values of mχand\nmaximum kχ\nFwith hadronic models (a) PK-DD and (b) DD-LZ1.\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4Λ\nM (M⊙)GW170817PK-DD\nno-DM\n100 MeV\n15 GeV\n(a) PK-DD\n0100200300400500600700\n1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Λ\nM (M⊙)GW170817DD-LZ1\nno-DM\nmχ=100 MeV\n20 GeV (b) DD-LZ1\nFIG. 7: Variation of tidal deformability with mass of dark matter admixed neutron stars for different values of mχ\nand maximum kχ\nFwith hadronic models (a) PK-DD and (b) DD-LZ1.\nminimum values of mχfor which the DMANS at minimum kχ\nFsatisfy all the astrophysical constraints.12\n22.12.22.32.42.52.62.72.8\n0.1 1 10Mmax(M⊙)\nmχ(GeV)kχ\nF=0.06 GeV\n(a)Mmax vsmχ\n910111213141516\n0.1 1 10R1.4(km)\nmχ(GeV)kχ\nF=0.06 GeV\nGW170817\nTM1\nGM1\nNL3\nPK1\nDD-MEX\nDD2\nTW99\nDD-ME2\nPK-DD\nDD-LZ1 (b)R1.4vsmχ\n100200300400500600\n0.1 1 10Λ1.4\nmχ(GeV)kχ\nF=0.06 GeV\nGW170817\n (c) Λ 1.4vsmχ\nFIG. 8: Variation of (a) Mmax, (b) R1.4and (c) Λ1.4with mχof dark matter admixed neutron stars at maximum\nkχ\nFfor different hadronic models within the range of fulfillment of all the astrophysical constraints.\nTABLE III: The range of mχfor which the dark matter admixed neutron stars at maximum kχ\nFsatisfy all the\nastrophysical constraints on the structural properties of compact stars.\nModel mχ(GeV)\nTM1 0.5 −25\nGM1 0.3 −20\nNL3 1.0 −30\nPK1 0.5 −25\nDD-MEX 1.0 −25\nDD2 0.1 −15\nTW99 0.1 −10\nDD-ME2 0.1 −15\nPK-DD 0.1 −15\nDD-LZ1 0.1 −20\n2.12.22.32.42.52.62.72.8\n0.1 1 10Mmax(M⊙)\nmχ(GeV)kχ\nF=0.01 GeV\n(a)Mmax vsmχ\n910111213141516\n0.1 1 10R1.4(km)\nmχ(GeV)kχ\nF=0.01 GeV\nGW170817\nTM1\nGM1\nNL3\nPK1\nDD-MEX\nDD2\nTW99\nDD-ME2\nPK-DD\nDD-LZ1 (b)R1.4vsmχ\n100200300400500600\n0.1 1 10Λ1.4\nmχ(GeV)kχ\nF=0.01 GeV\nGW170817\n (c) Λ 1.4vsmχ\nFIG. 9: Variation of (a) Mmax, (b) R1.4and (c) Λ1.4with mχof dark matter admixed neutron stars at minimum\nkχ\nFfor different hadronic models within the range of fulfillment of all the astrophysical constraints. The black dashed\nvertical line indicate saturation of the values of maximum mass, R1.4andΛ1.4at 10 GeV for all the hadronic models.13\nTABLE IV: The minimum value of mχfor which the dark matter admixed neutron stars at minimum kχ\nFsatisfy all\nthe astrophysical constraints on the structural properties of compact stars.\nModel mmin\nχ(GeV)\nTM1 0.5\nGM1 0.3\nNL3 1.0\nPK1 0.5\nDD-MEX 1.0\nDD2 0.1\nTW99 0.1\nDD-ME2 0.1\nPK-DD 0.1\nDD-LZ1 0.1\nThus combining the results of the Tables III and IV we obtain a range of mχfor which the DMANS satisfy all the\nastrophysical constraints within a wide range of kχ\nF=(0.01 −0.06) GeV or wide range of DM fraction in neutron stars.\nWe present this combined range of mχin Fig. 10. It can be seen from Fig. 10 that for the models like NL3, TM1,\nPK1, and DD-MEX that do not or barely satisfy the constraints on R1.4and Λ 1.4from GW170817 in the absence of\nDM (Fig. 1), comparatively massive DM is required to obtain reasonable (with respect to the various astrophysical\nconstraints) DMANSs configurations. We also find that considering all the ten RMF hadronic models chosen for the\npresent work, the combined range of mχ≈(0.1−30) GeV for a wide range of kχ\nF=(0.01 −0.06) GeV.\nIV. SUMMARY AND CONCLUSION\nIn the present work we aim to study the effects of feeble interaction between hadronic matter and fermionic DM\nvia new physics scalar and vector mediators on the structural properties of the DMANSs in the light of the different\nastrophysical constraints. For the purpose we consider ten well-known RMF models to describe the pure hadronic\nmatter. mχ,mϕandmξare consistent with the self-interaction constraint from Bullet cluster while yϕandyξ\nare constrained by the present day relic abundance. We assume that both ϕandξcontribute equally to the relic\nabundance and compute the equation of state and the structural properties of the DMANSs. In order to satisfy the\nvarious recent constraints like those from the massive pulsar PSR J0348+0432, the gravitational wave (GW170817)\ndata and the results of NICER experiments for PSR J0030+0451 and PSR J0740+6620, we find that within the\nframework of the present work, the DMANSs may contain fermionic DM of mass in the range of mχ≈(0.1−30)\nGeV corresponding to a wide range of fixed kχ\nF=(0.01 −0.06) GeV. For the above mentioned mass range of DM, the\nDMANSs well satisfy the astrophysical constraints on structural properties of the compact stars. This range of mχ\ncan be considered to be potentially favorable in order to explain the possible existence of the DMANSs.\nFor the hadronic models that yield larger radii corresponding to the low mass neutron stars in the no-DM scenario,\ninteraction with comparatively heavier DM fermion is necessary in order to ensure that the DMANSs obtained with\nsuch models satisfy the radius constraints from both GW170817 and NICER data for PSR J0030+0451.\nAcknowledgements\nWork of A.G. is supported by the National Research Foundation of Korea (NRF-2019R1C1C1005073). Work of DS\nis supported by the NRF research Grants (No. 2018R1A5A1025563).\n[1] E. Fonseca et al., Astrophys. J. Lett. 915, L12 (2021), 2104.00880.\n[2] M. C. Miller et al., Astrophys. J. Lett. 918, L28 (2021), 2105.06979.\n[3] T. E. Riley et al., Astrophys. J. Lett. 918, L27 (2021), 2105.06980.\n[4] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett. 121, 161101 (2018), 1805.11581.\n[5] T. E. Riley et al., Astrophys. J. Lett. 887, L21 (2019), 1912.05702.\n[6] M. C. Miller et al., Astrophys. J. Lett. 887, L24 (2019), 1912.05705.\n[7] W. Husain, T. F. Motta, and A. W. Thomas, JCAP 10, 028 (2022), 2203.02758.14\n0.1 1 10\nmχ(GeV)TM1GM1NL3PK1DD-MEXDD2TW99DD-ME2PK-DDDD-LZ1\nFIG. 10: The combined range of mχfor which the dark matter admixed neutron stars satisfy all the astrophysical\nconstraints on the structural properties of compact stars for all the hadronic models.\n[8] W. Husain and A. W. Thomas, J. Phys. G 50, 015202 (2023), 2206.11262.\n[9] G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405, 279 (2005), hep-ph/0404175.\n[10] N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), 1807.06209.\n[11] C. L. Bennett et al. (WMAP), Astrophys. J. Suppl. 208, 20 (2013), 1212.5225.\n[12] P. A. R. Ade et al. (Planck), Astron. Astrophys. 571, A16 (2014), 1303.5076.\n[13] C. L. Bennett et al. (WMAP), Astrophys. J. Suppl. 208, 20 (2013), 1212.5225.\n[14] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).\n[15] M. Bauer and T. Plehn, Yet Another Introduction to Dark Matter: The Particle Physics Approach, vol. 959 of\nLecture Notes in Physics (Springer, 2019), 1705.01987.\n[16] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).\n[17] M. Bauer and T. Plehn, Yet Another Introduction to Dark Matter: The Particle Physics Approach, vol. 959 of\nLecture Notes in Physics (Springer, 2019), 1705.01987.\n[18] M. Cannoni, Eur. Phys. J. C 76, 137 (2016), 1506.07475.\n[19] R. Agnese et al. (SuperCDMS), Phys. Rev. Lett. 121, 051301 (2018), [Erratum: Phys.Rev.Lett. 122, 069901 (2019)],\n1804.10697.\n[20] E. Aprile et al. (XENON100), Phys. Rev. Lett. 109, 181301 (2012), 1207.5988.\n[21] E. Aprile et al. (XENON), Phys. Rev. Lett. 121, 111302 (2018), 1805.12562.\n[22] D. S. Akerib et al. (LUX), Astropart. Phys. 45, 34 (2013), 1210.4569.\n[23] Q. Wang et al. (PandaX-II), Chin. Phys. C 44, 125001 (2020), 2007.15469.15\n[24] P. Agnes et al. (DarkSide), Phys. Rev. Lett. 121, 111303 (2018), 1802.06998.\n[25] M. Crisler, R. Essig, J. Estrada, G. Fernandez, J. Tiffenberg, M. Sofo haro, T. Volansky, and T.-T. Yu (SENSEI), Phys.\nRev. Lett. 121, 061803 (2018), 1804.00088.\n[26] J. Aalbers et al. (LZ) (2022), 2207.03764.\n[27] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradac, Astrophys. J. 679, 1173 (2008), 0704.0261.\n[28] M. Bradac, D. Clowe, A. H. Gonzalez, P. Marshall, W. Forman, C. Jones, M. Markevitch, S. Randall, T. Schrabback, and\nD. Zaritsky, Astrophys. J. 652, 937 (2006), astro-ph/0608408.\n[29] S. Tulin, H.-B. Yu, and K. M. Zurek, Phys. Rev. D 87, 115007 (2013), 1302.3898.\n[30] S. Tulin and H.-B. Yu, Phys. Rept. 730, 1 (2018), 1705.02358.\n[31] T. Hambye and L. Vanderheyden, JCAP 05, 001 (2020), 1912.11708.\n[32] G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 185, 960 (2014), 1305.0237.\n[33] P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991).\n[34] A. Guha, P. S. B. Dev, and P. K. Das, JCAP 02, 032 (2019), 1810.00399.\n[35] N. F. Bell, G. Busoni, and S. Robles, JCAP 06, 054 (2019), 1904.09803.\n[36] N. F. Bell, G. Busoni, S. Robles, and M. Virgato, JCAP 03, 086 (2021), 2010.13257.\n[37] G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 083004 (2017), 1709.06312.\n[38] A. Guha and D. Sen, JCAP 09, 027 (2021), 2106.10353.\n[39] D. Sen and A. Guha, Mon. Not. Roy. Astron. Soc. 504, 3354 (2021), 2104.06141.\n[40] A. Nelson, S. Reddy, and D. Zhou, JCAP 07, 012 (2019), 1803.03266.\n[41] P. Ciarcelluti and F. Sandin, Phys. Lett. B 695, 19 (2011), 1005.0857.\n[42] G. Baym, D. H. Beck, P. Geltenbort, and J. Shelton, Phys. Rev. Lett. 121, 061801 (2018), 1802.08282.\n[43] I. Lopes and G. Panotopoulos, Phys. Rev. D 97, 024030 (2018), 1801.05031.\n[44] J. Ellis, G. H¨ utsi, K. Kannike, L. Marzola, M. Raidal, and V. Vaskonen, Phys. Rev. D 97, 123007 (2018), 1804.01418.\n[45] A. Li, F. Huang, and R.-X. Xu, Astropart. Phys. 37, 70 (2012), 1208.3722.\n[46] L. Tolos and J. Schaffner-Bielich, Phys. Rev. D 92, 123002 (2015), 1507.08197.\n[47] M. Deliyergiyev, A. Del Popolo, L. Tolos, M. Le Delliou, X. Lee, and F. Burgio, Phys. Rev. D 99, 063015 (2019), 1903.01183.\n[48] Z. Rezaei, Astrophys. J. 835, 33 (2017), 1612.02804.\n[49] S. Mukhopadhyay, D. Atta, K. Imam, D. N. Basu, and C. Samanta, Eur. Phys. J. C 77, 440 (2017), [Erratum: Eur.Phys.J.C\n77, 553 (2017)], 1612.07093.\n[50] P. Mukhopadhyay and J. Schaffner-Bielich, Phys. Rev. D 93, 083009 (2016), 1511.00238.\n[51] G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 023002 (2017), 1706.07272.\n[52] J. C. Jim´ enez and E. S. Fraga, Universe 8, 34 (2022), 2111.00091.\n[53] G. Panotopoulos and I. Lopes, Phys. Rev. D 98, 083001 (2018).\n[54] K.-L. Leung, M.-c. Chu, and L.-M. Lin, Phys. Rev. D 105, 123010 (2022), 2207.02433.\n[55] D. R. Karkevandi, S. Shakeri, V. Sagun, and O. Ivanytskyi, Phys. Rev. D 105, 023001 (2022), 2109.03801.\n[56] O. Louren¸ co, T. Frederico, and M. Dutra, Phys. Rev. D 105, 023008 (2022), 2112.07716.\n[57] T. Gleason, B. Brown, and B. Kain, Phys. Rev. D 105, 023010 (2022), 2201.02274.\n[58] Y. Dengler, J. Schaffner-Bielich, and L. Tolos, Phys. Rev. D 105, 043013 (2022), 2111.06197.\n[59] G. Panotopoulos and I. Lopes, Int. J. Mod. Phys. D 27, 1850093 (2018), 1804.05023.\n[60] G. Panotopoulos and I. Lopes, Phys. Rev. D 96, 083013 (2017), 1709.06643.\n[61] Z. Miao, Y. Zhu, A. Li, and F. Huang, Astrophys. J. 936, 69 (2022), 2204.05560.\n[62] N. Rutherford, G. Raaijmakers, C. Prescod-Weinstein, and A. Watts, Phys. Rev. D 107, 103051 (2023), 2208.03282.\n[63] B. Bertoni, A. E. Nelson, and S. Reddy, Phys. Rev. D 88, 123505 (2013), 1309.1721.\n[64] S. A. Bhat and A. Paul, Eur. Phys. J. C 80, 544 (2020), 1905.12483.\n[65] O. Louren¸ co, C. H. Lenzi, T. Frederico, and M. Dutra, Phys. Rev. D 106, 043010 (2022), 2208.06067.\n[66] M. Dutra, C. H. Lenzi, and O. Louren¸ co, Mon. Not. Roy. Astron. Soc. 517, 4265 (2022), 2211.10263.\n[67] B. Hong, Z. Ren, C. Wu, and X. Mu, Class. Quant. Grav. 40, 125007 (2023), 2305.14101.\n[68] X. Mu, B. Hong, X. Zhou, and Z. Feng, Astrophysics and Space Science 368, 67 (2023).\n[69] H. Zheng and L.-W. Chen, Astrophys. J. 831, 127 (2016), 1603.07518.\n[70] Y. Sugahara and H. Toki, Nucl. Phys. A 579, 557 (1994).\n[71] N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67, 2414 (1991).\n[72] G. A. Lalazissis, J. Konig, and P. Ring, Phys. Rev. C 55, 540 (1997), nucl-th/9607039.\n[73] W.-h. Long, J. Meng, N. Van Giai, and S.-G. Zhou, Phys. Rev. C 69, 034319 (2004), nucl-th/0311031.\n[74] A. Taninah, S. E. Agbemava, A. V. Afanasjev, and P. Ring, Phys. Lett. B 800, 135065 (2020), 1910.13007.\n[75] S. Typel, G. Ropke, T. Klahn, D. Blaschke, and H. H. Wolter, Phys. Rev. C 81, 015803 (2010), 0908.2344.\n[76] G. A. Lalazissis, T. Niksic, D. Vretenar, and P. Ring, Phys. Rev. C 71, 024312 (2005).\n[77] B. Wei, Q. Zhao, Z.-H. Wang, J. Geng, B.-Y. Sun, Y.-F. Niu, and W.-H. Long, Chin. Phys. C 44, 074107 (2020).\n[78] B.-N. Lu, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. C 84, 014328 (2011), 1104.4638.\n[79] C.-J. Xia, T. Maruyama, A. Li, B. Y. Sun, W.-H. Long, and Y.-X. Zhang, Commun. Theor. Phys. 74, 095303 (2022),\n2208.12893.\n[80] H. Lenske and C. Fuchs, Phys. Lett. B 345, 355 (1995).\n[81] D. Sen and A. Guha, Mon. Not. Roy. Astron. Soc. 517, 518 (2022), 2209.09021.\n[82] R. C. Tolman, Phys. Rev. 55, 364 (1939).\n[83] J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374 (1939).16\n[84] T. Hinderer, Astrophys. J. 677, 1216 (2008), 0711.2420.\n[85] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, Phys. Rev. D 81, 123016 (2010), 0911.3535.\n[86] N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. Thomas, and M. Virgato, Phys. Rev. Lett. 127, 111803 (2021),\n2012.08918." }, { "title": "2401.14455v2.Spin_drag_mechanism_of_giant_thermal_magnetoresistance.pdf", "content": "arXiv:2401.14455v2 [cond-mat.mes-hall] 1 Apr 2024Spin drag mechanism of giant thermal magnetoresistance\nAlex Levchenko1and A. V. Andreev2\n1Department of Physics, University of Wisconsin-Madison, M adison, Wisconsin 53706, USA\n2Department of Physics, University of Washington, Seattle, Washington 98195, USA\n(Dated: March 22, 2024)\nWe study hydrodynamic thermal transport in high-mobility t wo-dimensional electron systems\nplaced in an in-plane magnetic field, and identify a new mecha nism of thermal magnetotransport.\nThis mechanism is caused by drag between the electron popula tions with opposite spin polarization,\nwhich arises in the presence of a hydrodynamic flow of heat. In high mobility systems, spin drag\nresults in strong thermal magnetoresistance, which become s of the order of 100 % at relatively\nsmall spin polarization of the electron liquid. We express t he thermal magnetoresistance in terms\nof intrinsic dissipative coefficients of electron fluid and sh ow that it is primarily determined by the\nspin diffusion constant.\nTheroleofmutual dragin transportphenomenaispiv-\notal to understanding kinetic properties of metals, semi-\nconductors, and insulators. Perhaps the most prominent\nexamples of these effects are phonon drag [1], Coulomb\ndrag in bilayers [2], and magnon drag in magnetic sys-\ntems [3].\nPhonon drag, also known as the Gurevich effect, leads\nto significant deviations in the thermopower of various\nmaterials from the predictions based solely on electronic\ntheory. These deviations are caused by the transfer of\nmomentum from the electrons to the phonons, resulting\nin a substantial heat flow carried by phonons. Likewise,\nthe momentum transfer induced by interlayer electron\ncollisions, mediated by Coulomb interactions, gives rise\nto drag resistance [4–7]. This occurs when one layer (e.g.\na quantum wire or a two-dimensional electron system) is\ndriven out of equilibrium by a current, inducing a nonlo-\ncal voltage in the adjacent layer.\nIn ferromagnetic metals, electron-magnon scattering\nproduces thermoelectric anomalies similar to the phonon\ndrag effect [8–11]. In magnetic insulators, one can realize\na nonlocal magnon drag induced by magnetic dipolar in-\nteractionsbetweenthelayers[12]. Incompleteanalogyto\nthe Coulombdrag, a magnoncurrentin one layerinduces\na chemical potential gradient and/or a temperature gra-\ndient in the other layer, which are characterized by the\nmagnon current transresistivity and the magnon ther-\nmal transresistivity. The effect of mutual drag between\nphonons and spin excitations has been also discussed in\nthe literature in the context of the thermal conductivity\nof a quantum spin system [13].\nIn this work, we introduce a different drag-induced\nthermal effect. We show that at charge neutrality, spin\npolarization of the electrons strongly affects the thermal\nconductivity. The salient feature of heat transport at\ncharge neutrality is that it can proceed via the hydrody-\nnamic flow of the neutral electron-hole plasma. We thus\nfocus our consideration on the hydrodynamic regime,\nwhich attracted significant attention in recent years, see\nreviews [14–16] and references therein. In a pristine sys-\ntem, the thermal conductivity becomes infinite, and thusvery sensitive to disorder and other perturbations. In re-\nalisticsystems, the thermalconductivity islimited by the\ndisorder-induced friction, which can be made sufficiently\nweakinhigh mobilitysystems. Significanthydrodynamic\nenhancement of thermal conductivity in graphene sys-\ntems has been reported in Ref. [17]. Continuing progress\nin nanofabrication enables fabrication of samples with an\neven greater mobility, making the thermal conductivity\nof the system extremely sensitive to other perturbations.\nHere, we show that thermal transport properties of high\nmobility systems become very sensitive to spin polariza-\ntionoftheliquid, leadingtostrongthermalmagnetoresis-\ntivity. The microscopic mechanism of this phenomenon\nin the hydrodynamic regime involves spin diffusion (or\nspin drag). The physical reason for this effect can be un-\nderstood by observing that since thermal conductivity is\nmeasured at zero spin-current, the convective part of the\nspin current must be compensated by spin-diffusion rela-\ntive to the liquid. This dramaticallyincreases dissipation\nat nonzero spin polarization, resulting in giant thermal\nmagnetoresistivity. Our predictions may enable probing\nspin drag in electron-hole plasma in graphene by thermal\nmeasurements [18].\nThe hydrodynamic description of electron transport\nin a solid applies provided the rate of momentum- and\nenergy-conservingelectron-electroncollisionsexceeds the\nmomentumandenergyrelaxationratesonimpuritiesand\nphonons [19, 20]. Therefore macroscopic hydrodynamic\nequationsexpressconservationofthenumberofparticles,\nenergy, and momentum of the electron liquid. In addi-\ntion, in multivalley conductors or in the spin-polarized\nsystems additional approximate conservation laws are\npossible for (pseudo-) spin degrees of freedom.\nIn what follows we consider a two-dimensional system\nin which the electron fluid is partially spin polarized by\nan in-plane external magnetic field. We assume that the\nspin-orbit interaction is absent, so that the spin com-\nponent along the magnetic field is conserved. Experi-\nments show that even at room temperature spin trans-\nport in single-, bi-, and trilayer graphene devices ex-\nhibitnanosecondspinlifetimeswithspindiffusionlengths2\nreaching 10 µm [21, 22]. These observations justify our\nassumptions.\nWhen the electron system is tuned to the charge neu-\ntrality point by an applied gate voltage the hydrody-\nnamic flow is decoupled from charge current. Thus, the\nhydrodynamicequationsinvolveonlythe entropycurrent\ndensityjsand spin current density jσ. In a steady state\nand in the linearregimethese twocurrentsareconserved,\nwhich is expressed by the continuity equation of the form\n∇·/vectorJ= 0,/vectorJ=/vector xu−ˆΞ/vectorX. (1)\nHere we used column vector notations\n/vectorJ=/parenleftbiggjs\njσ/parenrightbigg\n, /vector x=/parenleftbiggs\nς/parenrightbigg\n,/vectorX=/parenleftbigg∇T\n∇µσ/parenrightbigg\n,(2)\nwheresandςare, respectively, entropy and spin densi-\nties, while /vectorXis the vector of conjugated thermodynamic\nforces defined by gradients of temperature Tand spin\nchemical potential µσ. It should be noted that ςrefers\nto the projection on the axis of the external field, and\nthe other spin components do not appear in the hydro-\ndynamic description because they are not conserved due\nto spin precession. The first term in Eq. (1) corresponds\nto the convective part of the current. It is worthwhile\nto stress that in the collision-dominated regime hydrody-\nnamic velocity u(r) is the same for all the spin compo-\nnents. This is different from the regime of spin Coulomb\ndrag [23, 24], where the populations of spin-up and spin-\ndown electrons have different drift velocities.\nThenet currentsofentropyandspinconsistofthe con-\nvectivecurrentsproduced bythe thermally-drivenflowof\nthe partially spin-polarized electron liquid, and the dis-\nsipative currents relative to the liquid described by the\nsecond term in Eq. (1). The latter are characterized by\nthe matrix of intrinsic kinetic coefficients\nˆΞ =/parenleftbiggκ/T γ σ/T\nγσ/T D σ/parenrightbigg\n, (3)\nwhich satisfies the Onsager symmetry principle [25, 26].\nThe diagonalelements contributing to dissipation arethe\nintrinsic thermal conductivity κand the spin diffusion\nconstant Dσof the electron liquid. The off-diagonal el-\nements describe the so-called spin Seebeck effect, which\nhas been studied in much detail for ferromagnets in the\nfield of spin caloritronics [27]. Since the spin density is\nodd under time reversal symmetry and energy is not,\nthe Onsager symmetry requires the intrinsic spin ther-\nmoconductivity to be odd function of the magnetic field\nγσ(H) =−γσ(−H).\nIn the stationary regime the force balance condition\nfor an element of the fluid can be expressed in the form\n∇·Σ−ku=/vector xT/vectorX, (4)\nwhere the first term in the left hand side represents the\ndivergence of the viscous stress tensor [28]\nΣij=η(∂iuj+∂jui)+(ζ−η)δij∂kuk(5)withηandζbeing, respectively, shear (first) and bulk\n(second) viscosity of the electron liquid. The force den-\nsity in the right hand side of Eq. (4) comes from the\nlocal gradients of pressure in the fluid P. To express\nit in this form we used the thermodynamic relation\n∇P=s∇T+ς∇µσ=/vector xT/vectorXand the column vector no-\ntations of Eq. (2). The superscript Tdenotes transposi-\ntion. The remaining term in Eq. (4) describes the generic\ndisorder-induced friction characterizedby the friction co-\nefficient k. Working under the assumption of smooth\ndisorder, namely weak disorder potential with the long\ncorrelation radius ξ, the coefficient of friction kcan be\nexpressed in terms of the local density variations δn(r)\ninduced by disorder potential and the intrinsic conduc-\ntivityσas follows [29]\nk=e2\n2σ/an}bracketle{tδn2/an}bracketri}ht, (6)\nwhere/an}bracketle{t.../an}bracketri}htdenotes spatial averaging. We recall that the\nintrinsic conductivity does not vanish in generic electron\nliquids which do not possess Galilean invariance. The\nassumed model is motivated by the experimental obser-\nvations of the long-range disorder in graphene devices in\nthe form of charge puddles [30–32] (with the typical scale\nofξ∼100 nm). The local form of Eq. (4) is supported\nby the recent analysis presented in Refs. [29, 33, 34],\nwhere it was shown that for a weakly-disordered system\none can develop an effective renormalized hydrodynamic\ndescription on length scales exceeding ξ.\nFor a given geometry of the sample and appropriate\nboundary conditions Eqs. (1) and (4) uniquely deter-\nmine the flow profile. The precise form of macroscopic\ntransport coefficients follows from the expression for the\nentropy production rate [35]\nT˙S=/angbracketleftBig\nΣij∂jui+/vectorXTˆΞ/vectorX+ku2/angbracketrightBig\n(7)\nthat should be equated to the Joule heating power P=\n/vectorJTˆR/vectorJ. The matrix elements of ˆRdefine thermal and\nspin resistances. Alternatively, one can proceed via the\nlinearresponsetorelatecurrentstoappliedgradientsand\nthus infer the effective matrix of conductivities whose\ninverse is ˆR. Below we use the second route as it is more\nstraightforward for the problem at hand.\nThe macroscopic thermal conductivity κis defined as\nthe proportionality coefficient between the entropy cur-\nrent and the temperature gradient at vanishing spin cur-\nrent\nκ=−T(js/∇T)jσ=0. (8)\nIn the absence of spin polarization, the thermal conduc-\ntivity of large systems at charge neutrality is determined\nby the friction coefficient and is independent of the liq-\nuid viscosity [17, 29, 33]. We show below that for spin-\npolarized systems the thermal resistivity remains inde-3\npendent from viscosity and is controlled by the spin dif-\nfusioncoefficient. Tothisend, wenoticethatthecompar-\nison of the gradient terms in Eq. (4) describing viscous\nstresses, ∇·Σ = (η+ζ)∇2u, to the friction term, ku,\nintroduces a characteristic length scale in the problem,\nwhich is the Gurzhi length [19]\nlG=/radicalbigg\nη+ζ\nk. (9)\nTherefore, if the sample size Lis smaller than lGthe flow\nprofile is essentially inhomogeneous (Poiseuille-like) and\nthus viscous effects play an important role. In the oppo-\nsite case of wide devices, L≫lG, the flow is mostly uni-\nform except in the boundary layer of thickness ∼lGnear\nthe sample edges. Based on this reasoning we assume\nthe following hierarchy of length scales ξ≪lG≪L. In\nthis limit we may neglect the gradient terms in Eq. (4)\nin the bulk of the sample, which significantly simplifies\nthe consideration. Then trivially solving for uwe find\nu=−(/vector xT/vectorX)/k. At the same time, the required condi-\ntion on the vanishing spin current gives us from Eq. (1)\nthatu=Dσ\nς∇µσ+γσ\nςT∇T. These two equations fix u\nand give a local relationship between ∇µσand∇T\n∇µσ=−∇Ts+γσk\nςT\nς+Dσk\nς. (10)\nHaving determined both uand∇µσin terms of ∇T, we\ninsertbothexpressionsintothefirstrowofEq. (1),which\ngives us the entropy current in the presence of the ther-\nmal spin drag. After straightforward algebra we obtain\nthe following result for the effective thermal conductivity\nfrom Eq. (8)\nκ(H) =κ+T/parenleftbigsDσ\nς−γσ\nT/parenrightbig/parenleftBig\ns+γσk\nTς/parenrightBig\nς+Dσk\nς−sγσ\nς.(11)\nFor small spin polarizations it is safe to assume that s≫\nmax{ςγσ\nTDσ,kγσ\nςT}, so that only s2Dσ/ςshould be retained\nin the numerator of the second term of Eq. (11), and the\nlast term can be dropped as well. Indeed, for example,\nfor the graphene monolayer s∼(T/v)2, wherevis the\nband velocity of graphene. For long range disorder we\nhaveξ≫lT≡v/T. Therefore, the above conditions\nare satisfied in the hydrodynamic regime. Furthermore,\nsince for weak disorder κ≪Ts2/k, our main result can\nbe simplified to\nκ(H)≈Ts2Dσ\nkDσ+ς2. (12)\nNote that in the absence of spin diffusion the thermal\nconductivity in Eq. (12) vanishes. This occurs because a\nhydrodynamic flow of spin-polarized liquid at vanishing\nspin currentis impossible in the absence of spin diffusion.Thus, in the ideal fluid limit, where both intrinsic ther-\nmalconductivityandspindiffusioncoefficientvanish, the\nsystem becomes a thermal insulator. This correspondsto\nspin-induced stagnationofthe electron liquid, which may\nbe used to create spin-actuated thermal valves. A sim-\nilar stagnation effect arises in hydrodynamic transport\nof charge away from charge neutrality. In that case si-\nmultaneous conservation of currents of charge, entropy,\nand (for a partially spin-polarized liquid) spin precludes\npotential flow of an ideal liquid in a smooth external po-\ntential [20, 36], resulting in diverging resistivity of 1D\nsystems in the ideal fluid limit [37, 38].\nWe note that the reduction of the thermal magneto-\nconductivity by spin polarization reaches ∼100% when\nthe spin density ς(H) becomes of the order of root mean\nsquare of the charge density fluctuations induced by dis-\norder, namely when ς∼/radicalBig\nDσe2\nσ/radicalbig\n/an}bracketle{tδn2/an}bracketri}ht. At such weak\nfields magnetic field dependence ofthe spin diffusion con-\nstantDσ(H) and intrinsic conductivity σ(H) can be ne-\nglected. Furthermore, equation (12) remains valid even\nin the case when spin polarization arises due to sponta-\nneous symmetry breaking as long as the hydrodynamic\nlimit can still be justified.\nIn the case of field-induced spin polarization Eq. (12)\ncan be used to obtain thermal magnetoconductivity at\nlow magnetic fields. Indeed, we write the spin density in\nthe form ς=χH, whereχdenotesthe spin susceptibility.\nIn this case Eq. (12) yields a Lorentzian dependence of\nthermal conductivity on H,\nκ(H)≈Ts2\nk1\n1+(H/Hσ)2, Hσ=√kDσ\nχ.(13)\nThe corresponding thermal resistivity ̺th=κ−1is thus\npositive and quadratic. It is of interest to note that the\nrelative thermal magnetoresistance,\n∆̺th(H)≡��th(H)−̺th(0)\n̺th(0)=ς2\nkDσ,(14)\nprovidesawayto extractthe spindiffusion coefficient Dσ\nfrom thermal transport measurements, since the degree\nof spin polarization and the strength of disorder can be\ndetermined from the independent experimental probes.\nAt stronger field, when spin density saturates, the resis-\ntivity̺thalso saturates to a constant value. The effect is\nanomalously strong since ∆ ̺th∼1 already for H∼Hσ.\nIn closing we note that our consideration focused on\nthebulkcontributiontothermalspindragmagnetotrans-\nport where momentum relaxation is driven by the disor-\nder potential and the hydrodynamic flow velocity is uni-\nform. In devices whose dimensions are smaller or compa-\nrable to the Gurzhi length there will be additional contri-\nbution to the thermal resistance, which is determined by\nthe viscous flow near sample boundaries. An extension\nof the present theory to the devices with Hall bar and\nCorbino geometry will be presented elsewhere.4\nThis research project was financially supported by the\nNational Science Foundation Grant No. DMR-2203411\nand H. I. Romnes Faculty Fellowship provided by the\nUniversityofWisconsin-MadisonOfficeoftheViceChan-\ncellor for Research and Graduate Education with fund-\ning from the Wisconsin Alumni Research Foundation\n(A. L.). A. V. A. acknowledges support from the Na-\ntional Science Foundation through the MRSEC Grant\nNo. DMR-1719797, the Thouless Institute for Quantum\nMatter (TIQM), and the College of Arts and Sciences at\nthe University of Washington.\n[1] R. Fletcher, E. Zaremba, and U. Zeitler, “Phonon\ndrag thermopower of low-dimensional systems,” in\nElectron-Phonon Interactions in Low-Dimensional Structu res\n(Oxford University Press, 2003).\n[2] B. N. Narozhny and A. Levchenko, “Coulomb drag,”\nRev. Mod. Phys. 88, 025003 (2016).\n[3] Marius V.Costache, GermanBridoux, IngmarNeumann,\nand Sergio O. Valenzuela, “Magnon-drag thermopile,”\nNature Materials 11, 199–202 (2012).\n[4] Lian Zheng and A. H. MacDonald, “Coulomb drag be-\ntween disordered two-dimensional electron-gas layers,”\nPhys. Rev. B 48, 8203–8209 (1993).\n[5] Antti-Pekka Jauho and Henrik Smith, “Coulomb drag\nbetween parallel two-dimensional electron systems,”\nPhys. Rev. B 47, 4420–4428 (1993).\n[6] Karsten Flensberg, Ben Yu-Kuang Hu, Antti-Pekka\nJauho, and Jari M. Kinaret, “Linear-response the-\nory of Coulomb drag in coupled electron systems,”\nPhys. Rev. B 52, 14761–14774 (1995).\n[7] AlexKamenevandYuvalOreg,“Coulomb draginnormal\nmetals and superconductors: Diagrammatic approach,”\nPhys. Rev. B 52, 7516–7527 (1995).\n[8] M. Bailyn, “Maximum variational principle for conduc-\ntion problems in a magnetic field, and the theory of\nmagnon drag,” Phys. Rev. 126, 2040–2054 (1962).\n[9] F. J. Blatt, D. J. Flood, V. Rowe, P. A. Schroeder,\nand J. E. Cox, “Magnon-drag thermopower in iron,”\nPhys. Rev. Lett. 18, 395–396 (1967).\n[10] G. N. Grannemann and L. Berger, “Magnon-\ndrag Peltier effect in a Ni-Cu alloy,”\nPhys. Rev. B 13, 2072–2079 (1976).\n[11] Daisuke Miura and Akimasa Sakuma, “Mi-\ncroscopic theory of magnon-drag thermo-\nelectric transport in ferromagnetic metals,”\nJournal of the Physical Society of Japan 81, 113602 (2012).\n[12] Tianyu Liu, G. Vignale, and Michael E. Flatt´ e, “Non-\nlocal drag of magnons in a ferromagnetic bilayer,”\nPhys. Rev. Lett. 116, 237202 (2016).\n[13] Suhas Gangadharaiah, A. L. Chernyshev, and Wol-\nfram Brenig, “Thermal drag revisited: Boltzmann versus\nKubo,” Phys. Rev. B 82, 134421 (2010).\n[14] Andrew Lucas and Kin Chung Fong, “Hy-\ndrodynamics of electrons in graphene,”\nJournal of Physics: Condensed Matter 30, 053001 (2018).\n[15] Alex Levchenko and J¨ org Schmalian, “Transport prop-\nerties of strongly coupled electron–phonon liquids,”\nAnnals of Physics 419, 168218 (2020).[16] Boris N. Narozhny, “Hydrodynamic ap-\nproach to two-dimensional electron systems,”\nLa Rivista del Nuovo Cimento 45, 661–736 (2022).\n[17] Jesse Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu,\nAchim Harzheim, Andrew Lucas, Subir Sachdev, Philip\nKim, Takashi Taniguchi, Kenji Watanabe, Thomas A.\nOhki, and Kin Chung Fong, “Observation of the Dirac\nfluid and the breakdown of the Wiedemann-Franz law in\ngraphene,” Science 351, 1058–1061 (2016).\n[18] Jonah Waissman, Laurel E. Anderson, Artem V. Ta-\nlanov, Zhongying Yan, Young J. Shin, Danial H. Na-\njafabadi, Mehdi Rezaee, Xiaowen Feng, Daniel G. No-\ncera, Takashi Taniguchi, Kenji Watanabe, Brian Skinner,\nKonstantin A. Matveev, and Philip Kim, “Electronic\nthermal transport measurement in low-dimensional ma-\nterials with graphene non-local noise thermometry,”\nNature Nanotechnology 17, 166–173 (2022).\n[19] R.N.Gurzhi,“Hydrodynamiceffectsinsolidsatlowtem-\nperature,” Soviet Physics Uspekhi 11, 255 (1968).\n[20] A. V. Andreev, Steven A. Kivelson, and\nB. Spivak, “Hydrodynamic description of trans-\nport in strongly correlated electron systems,”\nPhys. Rev. Lett. 106, 256804 (2011).\n[21] Marc Dr¨ ogeler, Frank Volmer, Maik Wolter, Bernat\nTerr´ es, Kenji Watanabe, Takashi Taniguchi, Gernot\nG¨ untherodt, Christoph Stampfer, and Bernd Beschoten,\n“Nanosecond spin lifetimes in single- and few-layer\ngraphene–hbn heterostructures at room temperature,”\nNano Letters , Nano Letters 14, 6050–6055 (2014).\n[22] Marc Dr¨ ogeler, Christopher Franzen, Frank Volmer, To -\nbias Pohlmann, Luca Banszerus, Maik Wolter, Kenji\nWatanabe, Takashi Taniguchi, Christoph Stampfer, and\nBernd Beschoten, “Spin lifetimes exceeding 12 ns in\ngraphene nonlocal spin valve devices,” Nano Letters ,\nNano Letters 16, 3533–3539 (2016).\n[23] Irene D’Amico and Giovanni Vignale, “Theory of\nspin Coulomb drag in spin-polarized transport,”\nPhys. Rev. B 62, 4853–4857 (2000).\n[24] Irene D’Amico and Giovanni Vignale, “Spin Coulomb\ndrag in the two-dimensional electron liquid,”\nPhys. Rev. B 68, 045307 (2003).\n[25] Lars Onsager, “Reciprocal relations in irreversible p ro-\ncesses. I.” Phys. Rev. 37, 405–426 (1931).\n[26] Lars Onsager, “Reciprocal relations in irreversible p ro-\ncesses. II.” Phys. Rev. 38, 2265–2279 (1931).\n[27] Gerrit E. W. Bauer, Eiji Saitoh, and\nBart J. van Wees, “Spin caloritronics,”\nNature Materials 11, 391–399 (2012).\n[28] L. D. Landau and E. M. Lifshitz, Fluid Mechanics ,\n2nd ed., Course of Theoretical Physics Series, Vol. 6\n(Butterworth-Heinemann, Oxford, 1987).\n[29] Songci Li, Alex Levchenko, and A. V. Andreev, “Hy-\ndrodynamic electron transport near charge neutrality,”\nPhys. Rev. B 102, 075305 (2020).\n[30] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann,\nJ. H. Smet, K. von Klitzing, and A. Ya-\ncoby, “Observation of electron–hole puddles in\ngraphene using a scanning single-electron transistor,”\nNature Physics 4, 144–148 (2008).\n[31] Yuanbo Zhang, Victor W. Brar, Caglar Girit,\nAlex Zettl, and Michael F. Crommie, “Ori-\ngin of spatial charge inhomogeneity in graphene,”\nNature Physics 5, 722–726 (2009).\n[32] Jiamin Xue, Javier Sanchez-Yamagishi, Danny Bul-5\nmash, Philippe Jacquod, Aparna Deshpande, K. Watan-\nabe, T. Taniguchi, Pablo Jarillo-Herrero, and Brian J.\nLeRoy, “Scanning tunnelling microscopy and spec-\ntroscopy of ultra-flat graphene on hexagonal boron ni-\ntride,” Nature Materials 10, 282–285 (2011).\n[33] Andrew Lucas, Jesse Crossno, Kin Chung Fong, Philip\nKim, and Subir Sachdev, “Transport in inhomoge-\nneous quantum critical fluids and in the Dirac fluid in\ngraphene,” Phys. Rev. B 93, 075426 (2016).\n[34] Alex Levchenko, Songci Li, and A. V. Andreev, “Giant\nmagnetoresistance in weakly disordered non-Galilean in-\nvariant conductors,” Phys. Rev. B 109, 075401 (2024).[35] See file of supplemental materials for further details.\n[36] Aaron Hui, Calvin Pozderac, and Brian Skinner,“2Dhy-\ndrodynamic electron flow through periodic and random\npotentials,” (2023), arXiv:2309.15917 [cond-mat.str-el ].\n[37] Alex Levchenko, Tobias Micklitz, J´ erˆ ome Rech,\nand K. A. Matveev, “Transport in partially\nequilibrated inhomogeneous quantum wires,”\nPhys. Rev. B 82, 115413 (2010).\n[38] Wade DeGottardi and K. A. Matveev, “Electrical and\nthermal transport in inhomogeneous Luttinger liquids,”\nPhys. Rev. Lett. 114, 236405 (2015)." }, { "title": "2401.14465v1.Order_chaos_transition_in_correlation_diagrams_and_quantization_of_period_orbits.pdf", "content": "Order-chaos transition in correlation diagrams and quantization of period orbits\nF. J. Arranz,1,∗J. Montes,1, 2,†and F. Borondo2,‡\n1Grupo de Sistemas Complejos, Universidad Polit´ ecnica de Madrid, av. Puerta de Hierro 2–4, 28040 Madrid, Spain\n2Departamento de Qu´ ımica, Universidad Aut´ onoma de Madrid, Cantoblanco, 28049 Madrid, Spain\n(Dated: January 29, 2024)\nEigenlevel correlation diagrams has proven to be a very useful tool to understand eigenstate\ncharacteristics of classically chaotic systems. In particular, we showed in a previous publication\n[Phys. Rev. Lett. 80, 944 (1998)] how to unveil the scarring mechanism, a cornerstone in the theory\nof quantum chaos, using the Planck constant as the correlation parameter. By increasing Planck\nconstant, we induced a transition from order to chaos, in which scarred wavefunctions appeared as\nthe interaction of pairs of eigenstates in broad avoided crossings, forming a well defined frontier in the\ncorrelation diagram. In this paper, we demonstrate that this frontier can be obtained by means of the\nsemiclassical quantization of the involved scarring periodic orbits. Additionally, in order to calculate\nthe Maslov index of each scarring periodic orbit, which is necessary for the semiclassical quantization\nprocedure, we introduce a novel straightforward method based on Lagrangian descriptors. We\nillustrate the theory using the vibrational eigenstates of the LiCN molecular system.\nI. INTRODUCTION\nA cornerstones in the development of quantum chaos\nis level statistics [1], i.e., the statistics of the spacing be-\ntween adjacent energy levels in the spectrum of a Hamil-\ntonian operator, which is mathematically based on the\nrandom matrix theory [2]. In this framework, two ex-\ntreme cases can be considered: (i) integrable systems\nwhere, as was shown by Berry and Tabor [3], the statis-\ntics of the energy spacing follows a exponential distribu-\ntion derived from a Poisson distribution of the eigenener-\ngies, and (ii) classically fully chaotic systems (K-systems)\nwhere, as was conjectured by Bohigas, Giannoni, and\nSchmit [4] and proved in a semiclassical context by M¨ uller\net al. [5], the statistics of the energy spacing follows the\ndistribution of the Gaussian orthogonal ensemble (GOE)\nof the random matrix theory. In the first case, since the\ndistribution of levels corresponds to a Poisson process,\nthere are no interaction between energy levels, such that\nsmall spacing is highly likely. While, in the second case,\nthe GOE distribution implies certain level repulsion , such\nthat small spacing is highly unlikely.\nHowever, in most cases, physical systems are not fully\nchaotic (K-systems), but systems with mixed dynamics,\ni.e., systems where chaotic regions and islands of reg-\nularity coexist in classical phase space. This case is an\nintermediate one where it can typically be found an order\nregion, with a Poisson-like statistics, and a mixed chaos\nregion, with an in-between Poisson-GOE statistics that\ncan be described by means of a Weibull distribution, as\nwas proposed by Brody [6] (therefore known as the Brody\ndistribution in this context), which smoothly interpolates\nbetween both extreme cases. In the mixed chaos region,\nthere are states with very little or no interaction, which\n∗fj.arranz@upm.es (corresponding author)\n†jmontes.3@alumni.unav.es\n‡f.borondo@uam.esare related to classical islands of stability, and also states\nwith level repulsion, which are related to classical chaotic\nregions. Molecular systems, in particular, mostly belong\nto this intermediate case.\nMoreover, notice that by taking the Planck constant\nas a varying parameter, thus obtaining a correlation di-\nagram of eigenenergies versus Planck constant, it can be\nused as an ideal tool to implement a kind of microscope\nthat focuses with varying resolution on the classical regu-\nlar structures existing in the phase space of systems with\nmixed chaos. In this way, it has been shown in the lit-\nerature [7–10] for different molecular systems, including\nHCN, LiCN, KCN, and HO 2, the existence of a singu-\nlar series of broad avoided crossings (ACs) in the corre-\nlation diagram of eigenenergies versus Planck constant,\nwhich constitutes the frontier that separates the order\nand mixed chaos regions. Namely, below this series of\nACs (the order region), no level repulsion is found, while\nabove the series of ACs (the mixed chaos region), ex-\ntensive level repulsion is found, as well as some states\nwith very little interaction related to classical islands\nof stability. Then, based on the level statistics results,\nthis series of ACs can be considered the frontier between\norder and chaos. Interestingly, the eigenstates involved\nin this frontier are scarred states, i.e., states where the\nextremes (maxima and minima) of their wavefunctions\nare distributed along an isolated unstable periodic or-\nbit (PO), phenomenon first studied by Heller [11] in the\nBunimovich stadium billiard. Notice that, as energy in-\ncreases, the eigenstates of the frontier are the first scarred\nstates to appear, such that this case adds some insight\ninto the scar formation in molecular systems.\nIn a previous work [12], we studied the correspondence\nbetween classical and quantum resonances in the order\nregion of the correlation diagram of the LiCN molecular\nsystem, leading to a semiclassical theory for this corre-\nspondence. In the present work, we further study the\nfrontier of scars quantitatively in the correlation diagramarXiv:2401.14465v1 [nlin.CD] 25 Jan 20242\nof the LiCN molecular system, addressing the semiclas-\nsical quantization of the scarring POs involved in this\nfrontier of scars, which will lead to obtaining a semiclas-\nsical frontier between order and chaos.\nThe semiclassical quantization of POs was addressed\nby Gutzwiller [13] in the seminal work where his cele-\nbrated trace formula was first obtained. The quantiza-\ntion condition obtained for the classical action depends\non the number of conjugate points of the PO over one\nperiod, an integer value also known as Maslov index.\nTherefore, in order to achieve the semiclassical quanti-\nzation of a PO, it is necessary to compute its Maslov\nindex, but a rigorous calculation using any of the differ-\nent methods described in the literature is mathematical\ncomplicated. In the present work, as an additional re-\nsult, we will introduce a novel straightforward method\nfor the calculation of the Maslov index of a PO, based on\nLagrangian descriptors [14, 15]. Lagrangian descriptors\nhave been shown to be a fruitful tool to study the com-\nplex invariant structures existing in the phase space of\nnon-linear systems. In recent years, these mathematical\nobjects have been applied to a multitude of cases, in-\ncluding the LiCN molecular system [16, 17]. Now, we in-\ntroduce this novel application of Lagrangian descriptors,\nwhile a detailed research will be presented in a further\nwork.\nThe organization of the paper is as follows. Section II is\ndevoted to the description of the Hamiltonian model used\nto represent the LiCN molecular system (Sec. II A), as\nwell as to the description of the calculations to obtain the\nclassical POs and a suitable Poincar´ e surface of section\n(Sec. II B), the quantum eigenenergies and eigenstates\nof the corresponding Hamiltonian operator (Sec. II C),\nand also the Lagrangian descriptors used to obtain the\nMaslov index (Sec. II D). Section III is devoted to the\njoint presentation and discussion of the obtained results.\nIn Sec. III A the obtained values of the parameters de-\ntermining the frontier of scars are listed, where certain\nlinear correlation is found. Also, as a representative ex-\nample, one of the cases in the frontier of scars is illus-\ntrated and discussed through the depiction of the scarred\nwavefunctions and the three scarring POs involved. In\nSec. III B the semiclassical quantization of the three scar-\nring POs is carried out. First, the Maslov index of each\nPO must be obtained (Sec. III B 1), which is trivially ob-\ntained by counting the number of turning points in two\nof the three POs, while this non-rigorous method fails\nin the third PO. Hence, a novel straightforward method\nbased on Lagrangian descriptors is introduced and used\nfor the third PO. Then, the quantization is performed\n(Sec. III B 2), such that, taking advantage of the linear\ncorrelation found in Sec. III A, a continuous semiclassi-\ncal frontier between order and chaos is obtained. Last,\nthe paper is summarized and the conclusions reached are\npresented in Sec. IV.II. SYSTEM DESCRIPTION AND\nCALCULATIONS\nA. Hamiltonian model\nThe system studied in this work corresponds to the vi-\nbrational dynamics of the most abundant isotopic com-\nbination of the lithium isocyanide molecule7Li12C14N.\nRegarding the Hamiltonian model used, some remarks\nare in order. On the one hand, the rotational motion\nis not considered, i.e., the model will account for the\npurely vibrational motion of the molecule. On the other\nhand, since the C-N bond is much stronger than the in-\nteractions with the Li atom, an adiabatic decoupling of\nthe corresponding degree of freedom is feasible, such that\nthe C-N bond length can be fixed at its equilibrium value,\ni.e., the model will describe the relative motion of the Li\natom and the CN group. These simplifications lead to\na chemically realistic model of the LiCN molecule with\nonly two degrees of freedom, which is suitable for our\npurposes.\nConsidering the above simplifications, the Li-CN\nmolecular system will be modeled by means of the Hamil-\ntonian function\nH=P2\nR\n2µ1+P2\nθ\n2\u00121\nµ1R2+1\nµ2r2eq\u0013\n+V(R, θ),(1)\nwhere µ1=mLi(mC+mN)/(mLi+mC+mN) and µ2=\nmCmN/(mC+mN) are reduced masses ( mLi,mC, and\nmNbeing the corresponding atomic masses), req= 2.19\na.u. is the fixed N-C equilibrium length, Ris the length\nbetween the CN group center of mass and the Li atom,\nandθis the angle formed by the corresponding reqand\nRdirections (i.e., N →C andC\nN→Li, respectively). Thus,\ne.g., θ= 0 corresponds to the linear configuration Li-\nCN, and θ=πrad to the linear configuration CN-Li.\nLast, PRandPθare the conjugate momenta correspond-\ning to Randθcoordinates, respectively, and V(R, θ) is\nthe potential energy function describing the interatomic\ninteraction.\nThe potential energy function V(R, θ) is taken from\nthe literature [18]. It presents two minima: a rela-\ntive minimum at ( R, θ) = (4 .79,0) (a.u., πrad) with\nV= 2281 cm−1, corresponding to the Li-CN isomer,\nand an absolute minimum at ( R, θ) = (4 .35,1) (a.u., π\nrad) with V= 0, corresponding to the most stable CN-\nLi isomer. Both minima are separated by a saddle at\n(R, θ) = (4 .22,0.29) (a.u., πrad) with V= 3455 cm−1.\nThese three characteristic points can be connected by the\nminimum energy path (MEP), i.e., the path connecting\nall characteristic points along which the variation of en-\nergy is minimal. Notice that, accordingly to the physics\nof the Li-CN molecular system, the potential energy func-\ntionV(R, θ) is periodic in the angular coordinate θ, with\nperiod 2 πrad, and has a symmetry line at each value\nθ=kπrad ( k= 0,±1,±2, . . .). Finally, it is worth not-\ning that the well around the absolute minimum (CN-Li3\nisomer) is very anharmonic and, consequently, the tran-\nsition from regular classical motion to chaos in this sys-\ntem [19, 20] takes place for energies around 1700 cm−1,\nwell below the isomerization barrier energy of 3455 cm−1.\nB. Classical trajectories\nClassical trajectories will be calculated by numeri-\ncally integrating the canonical equations of motion corre-\nsponding to the Hamiltonian function in Eq. (1), where\nstandard numerical methods will be used for the integra-\ntion.\nThe scarring POs involved in the frontier of scars will\nbe obtained by means of the systematic method described\nin Ref. [20], which is based on the propagation of the\nsymmetry line at θ=πrad.\nMoreover, in order to get an graphical representation of\nthe trajectories in phase space, which shows the different\nregular and chaotic regions, a suitable Poincar´ e surface\nof section (PSS) will be defined. For this purpose, the\nfollowing canonical transformation will be applied\nρ=R−Req(θ), P ρ=PR,\nϑ=θ, P ϑ=Pθ+PRdReq(θ)\ndθ, (2)\nwhere Req(θ) is a series expansion in θcoordinate that\nfits the MEP. Thus, for a given energy E, the PSS along\nthe MEP will be defined in ( ϑ, Pϑ) coordinates by taking\nρ= 0 and choosing an arbitrary branch (the negative\none in our case) in the second degree equation for Pρ\nthat arises from the Hamiltonian conservation condition\nH(ρ, ϑ, P ρ, Pϑ) =E.\nC. Eigenenergies and eigenstates\nIn order to calculate the eigenenergies and eigen-\nstates of the Hamiltonian operator corresponding to the\nHamiltonian function in Eq. (1), the discrete variable\nrepresentation-distributed Gaussian basis (DVR-DGB)\nmethod proposed by Baˇ ci´ c and Light [21] will be used. As\nshown by these authors, the DVR-DGB method provides\ngood accuracy for highly excited vibrational states, per-\nforming very well for the Li-CN molecular system, which\nwas used as test system in their paper.\nIt is relevant to note here that, as a consequence of the\nseparation of variables procedure for obtaining the vibra-\ntional Hamiltonian operator from the total Hamiltonian\noperator (see Ref. [21] and references therein), the angu-\nlar coordinate is defined in the range θ∈[0, π] rad, within\nwhich consecutive quantum numbers n2= 0,1,2, . . .can\nbe assigned, where n2represents excitation in the θco-\nordinate. However, in order to implement a suitable cor-\nrespondence with classical mechanics, the range of the\nangular coordinate will be extended to θ∈[0,2π] rad by\napplying the symmetry line θ=πrad, such that onlyeven quantum numbers n2= 0,2,4, . . .can be assigned.\nDue to this approach, the symmetry line θ=πrad plays\na singular role in the procedure of semiclassical quantiza-\ntion, as discussed in Sec. III B 1. Notice that, however, no\nrestrictions apply to the quantum numbers correspond-\ning to the radial coordinate, since in this case consecutive\nquantum numbers n1= 0,1,2, . . .can be assigned, where\nn1represents excitation in the Rcoordinate.\nMoreover, it has been shown in the literature [7, 8]\nthat by expanding the range of ℏvalues in the calcula-\ntions, thus obtaining a correlation diagram of eigenener-\ngies versus Planck constant, a conspicuous series of quan-\ntum resonances formed by broad ACs is observed, which\nconstitutes the frontier that separates the regions of order\nand chaos in the Li-CN molecular system. In this paper\nwe will calculate the position od this quantum frontier,\nwhere scarring phenomena first appear, demonstrating\nthat it can be obtained by semiclassical quantization of\nthe corresponding scarring POs.\nIn this way, the DVR-DGB method will be used at val-\nuesℏ={0.01,0.02, . . . , 3.00}a.u., obtaining the 130 low\nlying eigenstates for each value of ℏwith eigenenergies\nconverged to within 1 cm−1. It is worth noting that, in\norder to maintain accuracy, the number of rays(the fixed\nvalues of θ-coordinate taken in DVR-DGB method) must\nbe increased as ℏdecreases. Thus, a final basis set of 414-\n418 ray eigenvectors lying in 45 rays will be used in the\nrangeℏ∈[1.01,3.00] a.u., a basis set of 820-841 ray eigen-\nvectors lying in 90 rays in the range ℏ∈[0.31,1.00] a.u.,\nand a basis set of 1480-1710 ray eigenvectors lying in 180\nrays in the range ℏ∈[0.01,0.30] a.u.\nD. Lagrangian descriptors\nIntroducing a novel method, Lagrangian descriptors\nwill be used to compute the POs Maslov index, which\nare necessary for their semiclassical quantization. La-\ngrangian descriptors have been shown to be a very pow-\nerful tool to unveil the intricate invariant structures of\nthe phase space of chaotic dynamical systems. Note that\ndifferent definitions for the Lagrangian descriptors can\nbe used, each of them leading to slightly different re-\nsults [14, 15]. In our case, we will use the definition that\nhas been shown to be suitable for the Li-CN molecular\nsystem, in particular in Refs [16, 17].\nFor a system with Ndimensions, the Lagrangian de-\nscriptors Mare defined as follows,\nM±(z0;α, τ) =±2NX\nk=1Z±τ\n0|˙zk(t)|αdt, (3)\nwhere z= (z1, . . . , z 2N) is the vector formed by the Npo-\nsition variables and their corresponding Nconjugate mo-\nmenta, such that, Lagrangian descriptors are a function\ndepending on the initial condition z0= (z10, . . . , z 2N0),\nat time t= 0, and two fixed parameters, the expo-\nnent α∈(0,1] and the integration time τ∈(0,+∞).4\nNote that, in the case of an unstable PO, backward M−\nand forward M+forms in Eq. (3) will permit to obtain\nthe unstable and stable invariant manifolds, respectively.\nThe overall Lagrangian descriptors M, as commonly used\nin the literature, are given by the sum of both forms,\nnamely, M=M−+M+.\nFor the Li-CN molecular system, we have N= 2 and\nz= (R, θ, P R, Pθ). Additionally, we will take the value\nα= 1 for the exponent, which corresponds to the integra-\ntion of the so-called taxicab norm [22] of the Hamiltonian\nflow ˙z(t) in Eq. (3), and the value τ= 486 fs for the in-\ntegration time, which is large enough compared with the\ninverse of the stability exponent of the PO under study\n(namely, |λ−1|= 87.50 fs) as prescribed in Ref. [17]. In\nany case, note that the choice of these values is heuristic,\nand then it is necessary to probe with different guesses\nuntil obtaining the clearest picture of the invariant man-\nifolds.\nIn order to calculate the Maslov index of a PO, differ-\nent initial conditions z0= (R0, θ0, PR0, Pθ0) will be taken\nalong the PO in configuration space, exploring the ener-\ngetically accessible momentum space at each position, as\ndescribed in the discussion of the results in Sec. III B 1.\nIII. RESULTS AND DISCUSSION\nA. Quantum results\nAlthough an extensive correlation diagram within the\nranges given in Sec. II C for ℏand the corresponding\neigenenergies has been calculated, we will mainly focus\non the region where the quantum transition from order to\nchaos occurs, i.e., the frontier of scars. The whole correla-\ntion diagram is not shown here, but it is reported in the\nprevious article [12]. Instead, a magnification centered\non the frontier of scars, with the semiclassical results su-\nperimposed, will be shown below in Fig. 7.\nThe values of the parameters determining the posi-\ntion of the series of ACs that constitutes the frontier of\nscars are listed in Table I. The center point of each AC,\ngiven by the corresponding value of the Planck constant\nℏn, is defined as the value of ℏat which the coupling\n⟨ψi|∂/∂ℏ|ψj⟩between the two eigenstates |ψi⟩and|ψj⟩\ninvolved in the AC reaches its maximum. It is worth\nnoting that the mixing between both states is completely\ndetermined by the coupling ⟨ψi|∂/∂ℏ|ψj⟩[7, 8]. Accord-\ningly, the lower E−\nnand upper E+\nneigenenergy values in\nTable I correspond to the energy of the two eigenstates\ninvolved in the AC at Planck constant ℏn. Also, the cor-\nresponding state numbers N−\nnandN+\nnare listed, where\nN= 1 stands for the ground state. Observe that, due\nto the existence of ACs where the interaction of the in-\nvolved states is ostensibly small, this giving rise to very\nsharp ACs non-observable by naked-eye inspection in the\neigenenergies correlation diagram, the state numbers N−\nn\nandN+\nnare not consecutive in all cases. Notice that the\nwell known non-crossing rule ensures that, in this system,all eigenstates undergo ACs.\nAll parameters in Table I are labeled by the quantum\nnumber nassociated with each AC, which is obtained\nfrom the nodal pattern of the corresponding scarred\nwavefunction. Namely, the quantum number nis cal-\nculated by counting the number of times that the graph\nof the scarring PO crosses a nodal line of the scarred\nwavefunction. Note that, due to the extended range of\nthe angular coordinate θ∈[0,2π] rad mentioned above in\nSec. II C, all values of nare even numbers. The quantum\nnumber nrepresents excitation neither in the Rcoordi-\nnate nor in the θcoordinate, but in the coordinate defined\nalong the corresponding scarring PO. As was shown in\nRef. [23], this quantum number of scarred states is re-\nlated to the different bands appearing in the appropriate\nlow-resolution spectrum, such that each band in the spec-\ntrum is associated with the corresponding scarred state\nand its quantum number. Moreover, the low-resolution\nspectrum of the Li-CN molecular system, related to the\nscarred states involved in the frontier of scars, was stud-\nied in Ref. [24].\nAs a representative example, the case of the AC cor-\nresponding to the quantum number n= 16 is shown\nin Fig. 1, where the scarred wavefunctions are depicted\nwith their scarring POs superimposed on them. Ob-\nserve that, characterizing the scarring phenomenon, the\nextremes (maxima and minima) of the wavefunctions\nare distributed along the corresponding POs, such that\ncounting the number of times that the graph of a PO\ncrosses a nodal line of the scarred wavefunction, the\nquantum number n= 16 is obtained. Strictly speaking,\nthe scarring POs must be isolated unstable POs, other-\nwise we would have a localization phenomenon rather\nthan a scarring phenomenon. In order to show the iso-\nlated and unstable character of the involved POs, we have\nTABLE I. Values of the parameters determining the series of\navoided crossings that constitutes the frontier of scars separat-\ning the regions of order and chaos in the correlation diagram\nof eigenenergies versus Planck constant. For each avoided\ncrossing, the quantum number n, the Planck constant value\nℏn, the lower E−\nnand upper E+\nneigenenergy values, and their\ncorresponding state numbers, N−\nnandN+\nn, are listed.\nnℏn(a.u.) E−\nn(cm−1)N−\nn E+\nn(cm−1)N+\nn\n12 2.430 3601 10 3694 11\n14 1.930 3205 11 3274 13\n16 1.600 2944 13 3000 14\n18 1.370 2766 16 2814 17\n20 1.200 2638 18 2680 19\n22 1.062 2526 21 2564 22\n24 0.955 2443 24 2479 25\n26 0.867 2374 27 2407 29\n28 0.794 2318 31 2348 32\n30 0.733 2272 35 2301 36\n32 0.684 2242 39 2270 405\nR (a.u.)(a)\n2.53.54.55.56.5\nθ (π rad)R (a.u.)(b)\n0 0.5 1 1.5 22.53.54.55.56.5\nFIG. 1. Scarred wavefunctions corresponding to the upper (a)\nand lower (b) states involved in the avoided crossing with\nquantum number n= 16. The scarring periodic orbits, de-\npicted in thick line, have been superimposed on the wavefunc-\ntions. The energy contour corresponding to each eigenenergy\nhas also been included, depicted in thin line.\ndepicted in Fig. 2 a composite PSS for the middle energy\nat the AC with quantum number n= 16, where the\nperiodic points corresponding to the involved POs have\nalso been superimposed as open circles. As can be ob-\nserved, in all cases each periodic point (at the center of\nits open circle mark) is immersed in the chaotic region,\nevidencing the isolated and unstable character of the cor-\nresponding PO. Additionally, the stability of the involved\nPOs throughout the series of ACs have been determined\nby the calculation of the trace of the monodromy matrix.\nFor the n= 16 case shown in Fig. 2, all of the three\nPOs are unstable and isolated, however, this is not the\ncase for all instances in the series of the ACs. For the\nlower state, only one PO is involved (hereafter referred\nto as PO-C), which is unstable and isolated throughout\nthe series. For the upper state, two POs are involved, one\nmore and one less extended in the θ-coordinate [approxi-\nmately, θ∈[0.4,1.6]πrad and θ∈[0.5,1.5]πrad, respec-\ntively, in the case shown in Fig. 1 (a)]. The less extended\none (hereafter referred to as PO-B) is unstable and iso-\nlated throughout the series, while the more extended one\n(hereafter referred to as PO-A) is unstable and isolated\nfor the ACs where n= 12,14,16,18,20, but it is stable\nfor the ACs where n= 22,24,26,28,30,32. These facts\nwill be taken into account again in Sec. III B 2, where\nthe semiclassical quantization is discussed. In any case,\nthroughout the series of ACs, there exists at least one\nϑ (π rad)Pϑ (a.u.)\n0 0.5 1 1.5 2−40−2002040FIG. 2. Composite Poincar´ e surface of section, defined along\nthe minimum energy path, for energy E= 2972 cm−1, which\ncorresponds to the middle energy at the avoided crossing with\nquantum number n= 16. The periodic orbits referred to in\nthe text as PO-A, PO-B, and PO-C are marked with red\n(dark), cyan (light), and blue (darkest) open circles ( ⃝), re-\nspectively. Gray region represents the energetically forbidden\nregion.\nisolated unstable PO scarring the corresponding eigen-\nstate. Additional details about the characteristics of the\nPOs in connection with the onset of chaos of the LiCN\nmolecular system can be obtained in Ref. [20].\nMoreover, it is interesting to note that the values of the\nPlanck constant ℏn, i.e., those where the ACs are cen-\ntered, have a high linear correlation with the quantized\nnℏn, as quantitatively indicated by the Pearson correla-\ntion coefficient r= 0.99986. This linear correlation is\nshown graphically in Fig. 3, where the least-squares fit-\nting of a straight line to the data points is also depicted.\nThe fitted straight line\nnℏn=a+bℏn (4)\nhas intercept a= 18 .92±0.03 a.u. and slope b=\n4.20±0.02, with a mean squared error of 0 .00166 (a.u.)2.\nThis result will be used below in Sec. III B 2 in order\nto define a continuous curve, derived from semiclassical\nquantization, determining the frontier between order and\nchaos.\nB. Semiclassical results\nThe semiclassical quantization of an unstable PO, as\nobtained by Gutzwiller in the derivation of his trace for-6\nh (a.u.)−nnh (a.u.)−n\n0 0.5 1 1.5 2 2.5 310152025303540\nFIG. 3. Linear correlation in the frontier of scars. Shown is\nthe quantization nℏnversusℏn,nandℏnbeing the quantum\nnumbers and the Planck constant values, respectively, given\nin Table I. The least-squares fitting of a straight line to the\ndata points is also depicted.\nmula [13], is given by\nS=ℏ\u0010\nn+µ\n4\u0011\n, (5)\nwhere Sis the classical action over one period of the or-\nbit,1ℏis the Planck constant, nis the quantum number,\nandµis the Maslov index of the PO. This index is an in-\nvariant of the PO, which counts the number of conjugate\npoints over one period of the orbit. Then, in order to\naccomplish the quantization of the POs involved in the\nfrontier of scars, it is previously required the calculation\nof their Maslov indices.\n1. Maslov index\nAs pointed out in Sec. II C, the original range of the\nangular coordinate in the Li-CN molecular system is\nθ∈[0, π] rad, such that, due to the symmetry of the sys-\ntem, the line θ=πrad behaves as a hard-wall potential,\ni.e., at θ=πrad the incident angle of a classical trajec-\ntory is equal to the reflected angle. The prescription of\nthe semiclassical quantization when there is a hard-wall\nis to add a value of 2 to the Maslov index, accounting\nfor the phase loss in the semiclassical propagation of the\nwave along the classical trajectory. When the range of\nthe angular coordinate is extended to θ∈[0,2π] rad by\napplying the symmetry line θ=πrad, the prescription of\nadding a value of 2 to the Maslov index remains. Observe\nthat this prescription is consistent with the fact that, in\nthe extended range θ∈[0,2π] rad, all eigenfunctions are\nsymmetric (with respect to the line θ=πrad), otherwise\n1Notice that the factor (1 /2π) have been included in the definition\nof the action, otherwise the original Planck constant hshould be\nused instead of the reduced Planck constant ℏ.antisymmetric eigenfunctions, with odd quantum num-\nbers, should also exist. Accordingly, a value of 2 must be\nadded to the Maslov indices of the POs calculated in the\nextended range θ∈[0,2π] rad.\nA rigorous calculation of the Maslov index can be im-\nplemented by means of different techniques, from the\nlong-established method of Eckhardt and Wintgen [25],\nbased on the winding number of the invariant manifolds\nof the PO, to the most recent method of Vergel et al. [26],\nbased on the number of zeros of the Jacobi field of the\ngeodesic line corrersponding to the PO in the geometro-\ndynamic approach. In all cases a rigorous calculation\nrequires a demanding mathematical work.\nMoreover, in some cases the Maslov index of a PO can\nbe obtained by means of an easy method, as is the count-\ning of the number of turning points in each degree of free-\ndom over one period of the orbit. Thus, for example, if\nwe take Ref. [26], where Maslov indices are calculated in\nthe framework of the geometrodynamic approach, and we\nfocus on the unstable POs of the two-dimensional system\nrepresented in Fig. 9 of this reference, the counting of the\nnumber of turning points in the four cases represented\nin panels (b)-(e), which correspond to POs with more\nand less complex graphs, gives the correct Maslov indices\n(listed in Table II of Ref. [26]). However, the counting of\nthe number of turning points in the two cases represented\nin panels (a) and (f), which correspond to POs with ex-\ntremely simple graphs (oblique and horizontal straight\nlines, respectively), gives wrong Maslov indices. The case\nin panel (a), the oblique straight line graph, is rather triv-\nial. The correct Maslov index is 2 and the counting of the\nnumber of turning points is 4 (2 in each degree of free-\ndom), but a coordinate rotation leading from oblique to\neither horizontal or vertical straight line yields a counting\nof 2 turning points. On the contrary, the case in panel\n(f), the horizontal straight line graph, is amazing. In-\ndeed, as in the previous case, we would expect a Maslov\nindex of 2, however the correct Maslov index is 16. This\ncase exemplify the complex behavior that the Maslov in-\ndex can sometimes exhibit, such that easy methods as\nthe counting of the number of turning points should only\nbe used when the obtained results can be tested.\nReturning to the Li-CN molecular system, the graph\nof the three scarring POs involved in the frontier of scars\nhave been depicted separately in Fig. 4, where their turn-\ning points in each degree of freedom are highlighted. Ob-\nserve that by following each path over one period, i.e.,\ngoing and coming back to the initial point, the number\nof turning points is the same in the three cases. The num-\nber of turning points in the radial coordinate R, i.e., the\npoints where its conjugate momentum takes the value\nPR= 0 changing the sign, is 16. Also, the number of\nturning points in the angular coordinate θ, i.e., the points\nwhere its conjugate momentum takes the value Pθ= 0\nchanging the sign, is 2. Then, accounting the value 2 due\nto the hard-wall line at θ=πrad, the Maslov index ob-\ntained by counting the number of turning points in each\ndegree of freedom is µ= 16+2+2 = 20. As we will see in7\nR (a.u.)(a)\n2.53.54.55.56.5R (a.u.)(b)\n2.53.54.55.56.5\nθ (π rad)R (a.u.)(c)\n0 0.5 1 1.5 22.53.54.55.56.5\nFIG. 4. Turning points of the scarring periodic orbits asso-\nciated with the scarred wavefunctions corresponding to the\nupper [(a) and (b)] and lower (c) states involved in the series\nof ACs that constitutes the frontier of scars. These periodic\norbits are referred to in the text as PO-A (a), PO-B (b), and\nPO-C (c). The turning points in the radial coordinate Rare\nmarked with dots ( •) while the turning points in the angular\ncoordinate θare marked with open circles ( ⃝). The graph of\neach periodic orbit, the minimum energy path, and the cor-\nresponding energy contour are represented by thick magenta,\nmedium blue, and thin black lines, respectively.\nSec. III B 2, the semiclassical energies obtained from the\nquantization with the Maslov index µ= 20 are consis-\ntent with the eigenenergies obtained from the quantum\ncalculations for both PO-A and PO-B, i.e., those associ-\nated with the scarred wavefunctions corresponding to the\nupper states in the series of ACs. However, the results\nare inconsistent for PO-C, i.e., that one associated with\nthe scarred wavefunctions of the lower states in the series\nof ACs, suggesting that the Maslov index µ= 20 is not\ncorrect in this case.\nIn order to obtain the Maslov index for PO-C, we will\nintroduce a novel method, based on the rigorous (and\nmathematically demanding) technique of Eckhardt andWintgen [25], but making it easy through the use of\nLagrangian descriptors. Eckhardt and Wintgen showed\nthat the Maslov index of an unstable PO is given by the\nnumber of half-turns around the PO of the associated\ninvariant manifolds over one period. When this calcula-\ntion is implemented in configuration space, rather than in\nphase space, the existence of simultaneous turning points\n(i.e., points of the trajectory where all momentum val-\nues vanish at the same time value) must be taken into\naccount. This is the case when the path of the PO in\nconfiguration space is self-retracing. Due to the singular-\nities that appear at the simultaneous turning points in\nconfiguration space, the calculation fails at these points.\nThe solution, however, is straightforward: The value 1\nmust be added to the number of half-turns for each si-\nmultaneous turning point. In any case, the calculation of\nthe number of half-turns is mathematically demanding.\nThe novelty in our method is to calculate the number of\nhalf-turns of the invariant manifolds by means of a suit-\nable graphical representation of the (easy to calculate)\nLagrangian descriptors along the PO.\nLagrangian descriptors have been shown to be a\nstraightforward tool to depict the invariant manifolds of\nisolated POs embedded in the chaotic region of nonlinear\nsystems, in particular also in the Li-CN molecular sys-\ntem [16, 17]. In these works, the Lagrangian descriptors\nwere calculated in a typical PSS representation. In our\ncase, and for the sake of computing Maslov indices from\nthem, a little different surface of section will be used.\nThus, for a given total energy, we will consider a surface\nof section along the PO in configuration space, parame-\nterizing the position coordinates ( R, θ) by means of the\nnormalized length of the path, Q, such that Q= 0 cor-\nresponds to the left simultaneous turning point, Q= 0.5\ncorresponds to the right simultaneous turning point, and\nQ= 1 again corresponds to the left simultaneous turn-\ning point. Moreover, at each position Qin configuration\nspace, all energetically accessible momentum values will\nbe explored, parameterizing the momentum coordinates\n(PR, Pθ) by means of the form\nP=\u0000\nϕ−ϕPO\u0001\n∥P∥ϕ, (6)\nwhere ∥P∥ϕandϕ∈\u0002\nϕPO−π, ϕPO+π\u0003\nrad are the\nmodulus and angle, respectively, of the vector P=\n(PR, Pθ) in momentum space, ϕPObeing the angle of\nthe momentum corresponding to the PO for each given\nposition Q. Observe that, as follows from the form of the\nHamiltonian function in Eq. (1), the curve of the energet-\nically accessible values in momentum space (for a given\nposition Q) is a ellipse rather than a circle. Consequently,\nthe modulus ∥P∥ϕdepends on the angle ϕ, hence the no-\ntation used. In this way, for a given PO, the initial con-\ndition z0= (R0, θ0, PR0, Pθ0) of the Lagrangian descrip-\ntorsM±(z0) will be given by the parameterized position\ncoordinates ( R0, θ0) = ( R, θ)Qand the parameterized\nmomentum coordinates ( PR0, Pθ0) = ( PR, Pθ)Q,P, such\nthat the initial condition z0, as well as the Lagrangian\ndescriptors M±(z0), will be a function of ( Q,P). Fi-8\n Q P (a.u.)\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−50−40−30−20−1001020304050\nmin max\nFIG. 5. Color scale representation of the forward form M+(Q,P) of the Lagrangian descriptors calculated along the unstable\nperiodic orbit in Fig. 4 (c), referred to in the text as PO-C, over one period. Colorless white area represents energetically\ninaccessible region. The crossings of the stable invariant manifold with the line P= 0 are marked with white open circles ( ⃝).\nnally, in order to avoid a double counting of the number\nof half-turns around the PO of the associated invariant\nmanifolds, only one of the two (either the stable or the\nunstable) invariant manifolds will be obtained by taken\neither forward M+(Q,P) or backward M−(Q,P) form.\nThe forward form M+(Q,P) calculated for PO-C is de-\npicted in Fig. 5. On the one hand, throughout the range\nofQwe can observe a horizontal line for P= 0, which\ncorresponds exactly to the PO path. In fact, this line\nis the locus were stable and unstable invariant surfaces\nintersect at the origin of the tangent space of the PO.\nConsequently, this horizontal line should evidently ap-\npear in the Lagrangian descriptors. On the other hand,\nwe can observe a series of apparently different lines cross-\ning the line P= 0, which are approximately straight in\nthe neighborhood of the crossings while they stretch and\ntwist as recede from these ones. These crossing lines cor-\nrespond to the two branches of the stable invariant man-\nifold, hence they are not different lines but a single line\nwhich is given by the intersection of the stable invariant\nmanifold and our surface of section defined above. The\naforementioned stretching and twisting, resulting from\nthe nonlinear character of the Li-CN molecular system,\nhinder the visualization of this geometric object as a sin-\ngle line. Note that, since the value P= 0 corresponds\nto the momentum of the PO, at each crossing point the\ncorresponding branch of the invariant manifold coincides\nwith the PO, i.e., it determines a turn of the branch\naround the PO. Therefore, by counting the successive\ncrossing points we are counting the alternating turns of\neach branch, namely, we are counting the number of half-\nturns of the invariant manifold around the PO.\nIn this way, the correct Maslov index for PO-C wouldbe obtained as follows. The number of half-turns of the\ninvariant manifold around the PO over one period, cal-\nculated by counting the number of crossings of the in-\nvariant manifold with the line P= 0 in Fig. 5, is 18.\nThe number of simultaneous turning points of the PO in\nthe configuration space, which are clearly represented in\nFig. 5 as the two singularities at Q= 0,1 and Q= 0.5,\nis 2. Last, the hard-wall line at θ=πrad adds the value\n2. Eventually, the Maslov index of the third PO will be\nµ= 18 + 2 + 2 = 22.\n2. Quantization\nIn order to quantize the three scarring POs involved in\nthe frontier of scars, the quantization condition in Eq. (5)\nhas been applied, taking in each case the corresponding\nMaslov index calculated above, and calculating the clas-\nsical action Sas follows,\nS=1\n2πI\nPOP·dQ\n=1\n2πZT\n0\u0010\nPR˙R+Pθ˙θ\u0011\ndt\n=1\n2π \n1\nµ1ZT\n0P2\nRdt+1\nµ1ZT\n0P2\nθ\nR2dt+1\nµ2r2eqZT\n0P2\nθdt!\n,\n(7)\nwhere the Hamiltonian function of the system [Eq. (1)]\nand the canonical equations of Hamilton have been used.\nNote that the integration is performed over the period\nTof the corresponding PO. Also note that the obtained9\naction depends on the energy of the PO, such that, by in-\ntegrating in Eq. (7) for different energy values, the graph\nof the function S(E) is obtained. This graph, in the di-\nmensionless form S(E)/ℏ(withℏ= 1 a.u.), is depicted\nin Fig. 6 for the three POs, namely PO-A, PO-B, and\nPO-C. By taking the values of the Planck constant ℏn\ncorresponding to each AC with quantum number n(val-\nues given in Table I), the dimensionless action S(E)/ℏn\nfor each AC, also depicted in Fig. 6, is obtained. Thus,\nthe quantization condition in Eq. (5) can be written in\nthe form\nS(En)\nℏn=\u0010\nn+µ\n4\u0011\n, (8)\nwhere Enis the quantized energy corresponding to the\nAC with quantum number n. In Fig. 6, the left hand side\nin Eq. (8) is represented by the blue (dark) lines, whilst\nthe right hand side corresponds to the horizontal gray\nlines, where the value µ= 20 has been taken in panels\n(a) and (b) [cases PO-A and PO-B, respectively], and the\nvalue µ= 22 in panel (c) [case PO-C].\nThe quantized energies obtained from Eq. (8) are su-\nperimposed on the correlation diagram of eigenenergies\nversus Planck constant in Fig. 7, where the series of\nbroad ACs that constitutes the frontier of scars has been\nmarked with open circles. Notice that, from right to\nleft in the figure, the quantum number nincreases from\nn= 12 to n= 32 across the series, as indicated in Table I\nwhere all parameters determining the series are listed.\nAs was mentioned in Sec. III A, the figure shows how the\nstate numbers N−\nnandN+\nnare not consecutive in cases\nn= 14 and n= 26, due to the existence of sharp ACs.\nNotice also that the frontier of scars actually separates\nthe region of order (below the frontier), characterized\nby sharp ACs, and the region of chaos (above the fron-\ntier), characterized by overlapping ACs that lead to level\nrepulsion property. However, since the Li-CN molecular\nsystem exhibits mixed chaos, such that it can be observed\nthe existence of stability islands embedded in the classi-\ncal chaotic sea, then it can also be observed the existence\nof sharp ACs embedded in the quantum level repulsion\nsea.\nMoreover, observe that the semiclassical results are\nin good agreement with quantum results, demonstrating\nthat the frontier of scars can be obtained by means of\nthe semiclassical quantization of the corresponding scar-\nring POs. More specifically, it can be observed a good\nagreement throughout the series of ACs for cases PO-\nA and PO-C, while a rough agreement is observed at\nthe beginning of the series ( n= 12), which progressively\nbecomes a good agreement as nincreases, for the case\nPO-B. It is interesting to note that, as was mentioned\nin Sec. III A, PO-A is unstable for quantum numbers\nn= 12,14,16,18,20, which are the quantum numbers for\nwhich the agreement for PO-B is not good enough, whilst\nit is stable for quantum numbers n= 22,24,26,28,30,32,\nwhich are the quantum numbers for which the agreement\nis good enough. In other words, it seems that, when\n(a)\n \n(b)\n \n(c)\nE (cm−1)S/h− n\n 0 10 20 30 40 50 60\n2000 2500 3000 3500 4000010203040506070FIG. 6. Quantization of the scarring periodic orbits asso-\nciated with the scarred wavefunctions corresponding to the\nupper [(a) and (b)] and lower (c) states involved in the se-\nries of avoided crossings that constitutes the frontier of scars.\nThese periodic orbits are referred to in the text as PO-A (a),\nPO-B (b), and PO-C (c). The classical action S(E) obtained\nfrom Eq. (7), in the dimensionless form S(E)/ℏ(withℏ= 1\na.u.), is depicted in cyan (light) line. The dimensionless classi-\ncal action for each avoided crossing S(E)/ℏn(withℏnvalues\ngiven in Table I), is depicted in blue (dark) line. Horizon-\ntal gray lines represent the quantization condition in Eq. (5)\nfor the indicated quantum number n. The points where the\nquantization condition is satisfied for each avoided crossing\nare marked with dots ( •). The open circle ( ⃝) marks the\nrepresentative case with quantum number n= 16 discussed\nin Sec. III A. The axis in little panels (a) and (b) are the same\nas in the big panel (c).\nquantum number increases from n= 20 to n= 22, PO-B\nreplaces PO-A in the role of isolated unstable PO required\nfor scarring phenomena.\nFinally, we will take advantage of the linear relation-\nship in Eq. (4) to obtain a semiclassical continuous ex-\npression for the frontier between order and chaos. Notice\nthat from Eq. (4) we can write\nℏn=S∞\n(n−n0), (9)10\nh (a.u.)−E/h (cm−1/a.u.)−\n0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.614001600180020002200240026002800300032003400\nFIG. 7. Magnification of the correlation diagram of eigenenergies versus Planck constant centered on the quantum frontier of\nscars [marked with red open circles ( ⃝)], that separates the regions of order and chaos. The circled circle symbol ( ⃝⃝) marks\nthe representative case with quantum number n= 16 discussed in Sec. III A. On grounds of graphical clarity, energy is divided\nby Planck constant. The energies obtained from the semiclassical quantization of the three periodic orbits in Fig. 4 (a) PO-A,\n(b) PO-B, and (c) PO-C are marked with red (dark), cyan (light), and blue (darkest) dots ( •), respectively. The semiclassical\nfrontier between order and chaos is depicted in thick gray line.\nwhere S∞= 18 .92±0.03 a.u. and n0= 4.20±0.02\nare the intercept aand slope b, respectively, in Eq. (4).\nThe physical meaning of parameter n0in Eq. (9) is clear:\nSince lim n→n0ℏn= +∞, it is the lower limit of the open\ninterval defining the domain, i.e., n∈(n0,+∞)|n∈2N.\nIn other words, the quantum number ncould take any\neven value strictly greater than n0. Moreover, the phys-\nical meaning of parameter S∞is obtained by inserting\nEq. (9) into the quantization condition in Eq. (8), namely\nSn=ℏn\u0010\nn+µ\n4\u0011\n=S∞(n+µ/4)\n(n−n0), (10)\nwhere it is fulfilled that lim n→∞Sn=S∞. Then, the\nparameter S∞is the asymptotic value of the action in\nthe semiclassical limit n→ ∞ (i.e.,ℏn→0). Notice that\nEq. (9) accurately retrieves the values of the Planck con-\nstantℏnfor the corresponding quantum number nat each\nAC listed in Table I, such that by applying the quantiza-\ntion condition, which is represented in Fig. 6, the quan-\ntized energies Endepicted in Fig. 7 are also accurately\nobtained. Therefore, if in this process we consider a con-\ntinuous rather than discrete domain for the “quantum”\nnumber n∈(n0,+∞) when applying the “quantization”\ncondition, then an also continuous rather than discrete\nset for the energies will be obtained. This continuous set\nof energies constitutes the semiclassical frontier between\norder and chaos, which is depicted superimposed on thecorrelation diagram in Fig. 7 for the case PO-C.\nIt is worth noting that the linear correlation shown in\nFig. 3 and the corresponding fitted straight line given in\nEq. (4) should be an approximation, i.e., the behavior of\nnℏnversusℏnis nearly but not strictly linear. In partic-\nular, as the position of the point with highest quantum\nnumber ( n= 32) in Fig. 3 seems to indicate, the series\ncould deviate from the linear behavior as nincreases (i.e.,\nℏndecreases). Consequently, the relationships in Eqs. (9)\nand (10), both derived from Eq. (4), will also be approx-\nimate. However, we think that their qualitative behav-\nior, namely, monotonically decreasing functions (consid-\nering a continuous domain) with vertical asymptote at\nn=n0and horizontal asymptote at ℏ= 0 or S=S∞\nin each case, is the correct one. The value n0≈4 ob-\ntained from the fitting implies that the minimum value\nfor the quantum number is n= 6, since it must be an\neven integer strictly greater than n0. However, the AC\nwith the lowest possible quantum number, since there\nare not lower states leading to an AC, corresponds to\nn= 8 (see the previous article [12]), hence the vertical\nasymptote could be at n= 6 (due to the deviation from\nthe linear behavior) rather than at n= 4. Moreover,\nthe value S∞≈19 a.u. obtained from the fitting cor-\nresponds to an energy around E≈1700 cm−1in the\nclassical action function S(E) for PO-C. Note that, as\nwas pointed out in Sec. II A, this energy value also cor-\nresponds to the classical transition from order to chaos\nin the Li-CN molecular system. However, if we assume11\nthe deviation from the linear behavior suggested by the\npoint corresponding to n= 32 in Fig. 3, then the hori-\nzontal asymptote should be at a value greater than the\nfitted parameter, such that there would be no direct re-\nlation to the threshold energy of transition to classical\nchaos. As a conjecture connected with the scarring phe-\nnomena, perhaps the horizontal asymptote could be at\nS≈22 a.u., which corresponds to the energy E= 1958\ncm−1at which PO-C bifurcates becoming an isolated un-\nstable PO. In any case, the question of the semiclassical\nlimitℏn→0 of the series of ACs that constitutes the\nfrontier of scars remains an open question.\nIV. SUMMARY AND CONCLUSIONS\nWe have studied the frontier of scars, previously estab-\nlished in the literature [7, 8], that separates the regions of\norder and chaos in the correlation diagram of eigenener-\ngies versus Planck constant of the Li-CN molecular sys-\ntem, with the purpose of demonstrating that it can be\nobtained through the semiclassical quantization of the\ninvolved scarring POs. It should be remarked that, as\nshown by previous work of our group, this method is like\na microscope in the phase space, where by changing the\nmagnification power by decreasing the value of ℏ, many\nrelevant features of the vibrational states of the system.\nThree scarring POs, referred to as PO-A, PO-B, and\nPO-C, are involved in the frontier of scars, which is con-\nstituted by a series of broad ACs. The first two (PO-\nA and PO-B) are associated with the upper eigenstates\nin the series of ACs, while the third (PO-C) is associ-\nated with the lower eigenstates. Moreover, within the\nwhole energy range, the last two (PO-C and PO-B)\nare isolated unstable POs, while the first one (PO-A)\nchanges from isolated unstable to stable PO as quan-\ntum number increases (i.e., energy and Planck constant\ndecrease) throughout the series. When these POs are\nquantized, yielding the corresponding semiclassical en-\nergies, the cases PO-A and PO-C give throughout the\nseries a good agreement with the energies of the upper\nand lower eigenstates, respectively. However, the case\nPO-B evolves, as quantum number increases in the se-\nries, from a energy value close to the energy of the lower\neigenstate, until a energy value close to the energy of the\nupper eigenstate. Indeed, the energies of both eigenstates\ncorresponding to each AC in the series can be obtained\nthrough the semiclassical quantization of an isolated un-stable PO, namely, case PO-C for the lower eigenstate\nand cases PO-B or PO-A (depending on the quantum\nnumber) for the upper eigenstate. And this is the main\nresult of our work.\nAdditionally, we have found an approximate linear cor-\nrelation in the frontier of scars that relates quantum num-\nber and Planck constant value at which each AC takes\nplace. Extending the discrete domain of the quantum\nnumber in this relationship to a continuous domain, and\napplying the “quantization” condition, we have obtained\nthe continuous semiclassical frontier between order and\nchaos, which matches the quantum frontier at (positive\neven) integer quantum numbers. Moreover, although the\nrelationship is approximate, we can assume that the qual-\nitative behavior of the quantized action derived from it\nis correct. Namely, as quantum number increases, the\nquantized action monotonically decreases from a vertical\nasymptote towards a horizontal asymptote. Assuming a\npositive deviation from the linear behavior as quantum\nnumber increases, we have conjectured values n= 6 and\nS≈22 a.u. for the vertical and horizontal asymtote,\nrespectively (rather than values n0= 4 and S∞≈19\na.u. obtained from the linear fitting), which are related\nto the first possible AC in the frontier of scars observed in\nthe correlation diagram and the bifurcation where PO-C\nbecomes an isolated unstable PO.\nOn the other hand, in order to calculate the non-\ntrivial Maslov index of case PO-C, which is necessary\nfor the semiclassical quantization, we have introduced a\nnovel straightforward method based on Lagrangian de-\nscriptors [14], this been the second relevant contribution\nof this paper. Notice that, in the cases PO-B and PO-\nA, the Maslov index was trivially obtained by counting\nthe number of turning points in each degree of freedom.\nEckhardt and Wintgen [25] proved that the Maslov in-\ndex of a PO can be obtained by calculating the winding\nnumber of the invariant manifolds over one period, albeit\nthe direct calculation of this parameter is mathematical\ndemanding. However, we have shown how this winding\nnumber can be obtained by means of the easily calcu-\nlation and depiction of the Lagrangian descriptors on a\nsuitable surface of section along the corresponding PO.\nACKNOWLEDGMENTS\nThis research was supported by the Ministry of Sci-\nence and Innovation-Spain under Grant No. PID2021-\n122711NB-C21 ( ChaSisCOMA project).\n[1] F. Haake, Quantum Signatures of Chaos (Springer-\nVerlag, Berlin Heidelberg, 2010).\n[2] G. Akemann, J. Baik, and P. Di Francesco (Eds.), The\nOxford Handbook of Random Matrix Theory (Oxford Uni-\nversity Press, Oxford, 2015).\n[3] M. V. Berry and M. Tabor, Level clustering in the regularspectrum, Proc. R. Soc. Lond. A 356, 375 (1977).\n[4] O. Bohigas, M. J. Giannoni, and C. Schmit, Character-\nization of chaotic quantum spectra and universality of\nlevel fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).\n[5] S. M¨ uller, S. Heusler, A. Altland, P. Braun, and\nF. Haake, Periodic-orbit theory of universal level correla-12\ntions in quantum chaos, New J. Phys. 11, 103025 (2009).\n[6] T. A. Brody, A statistical measure for the repulsion of\nenergy levels, Lett. Nuovo Cimento, 7, 482 (1973).\n[7] F. J. Arranz, F. Borondo, and R. M. Benito, Avoided\ncrossings, scars, and transition to chaos, J. Chem. Phys.\n107, 2395 (1997).\n[8] F. J. Arranz, F. Borondo, and R. M. Benito, Scar forma-\ntion at the edge of the chaotic region, Phys. Rev. Lett.\n80, 944 (1998).\n[9] F. J. Arranz, L. Seidel, C. G. Giralda, R. M. Benito, and\nF. Borondo, Scars at the edge of the transition from or-\nder to chaos in the isomerizing molecular systems LiNC-\nLiCN and HCN-HNC, and HO 2, Phys. Rev. E 82, 026201\n(2010).\n[10] H. P´ arraga, F. J. Arranz, R. M. Benito, and F. Borondo,\nUsing correlation diagrams to study the vibrational spec-\ntrum of highly nonlinear floppy molecules: The K-CN\ncase, Phys. Rev. E 101, 062215 (2020).\n[11] E. J. Heller, Bound-state eigenfunctions of classically\nchaotic Hamiltonian systems: Scars of periodic orbits,\nPhys. Rev. Lett. 53, 1515 (1984).\n[12] F. J. Arranz, R. M. Benito, and F. Borondo, Correspon-\ndence between classical and quantum resonances, Phys.\nRev. E 103, 062207 (2021).\n[13] M. C. Gutzwiller, Periodic orbits and classical quantiza-\ntion conditions, J. Math. Phys. 12, 343 (1971).\n[14] A. M. Mancho, S. Wiggins, J. Curbelo, and C. Mendoza,\nLagrangian descriptors: A method for revealing phase\nspace structures of general time dependent dynamical\nsystems, Commun. Nonlinear Sci. Numer. Simulat. 18,\n3530 (2013).\n[15] C. Lopesino, F. Balibrea-Iniesta, V. J. Garcı´ a-Garrido,\nS. Wiggins, and A. M. Mancho, A theoretical framework\nfor Lagrangian descriptors, Int. J. Bifurcation Chaos 27,1730001 (2017).\n[16] F. Revuelta, R. M. Benito, and F. Borondo, Unveiling\nthe chaotic structure in phase space of molecular systems\nusing Lagrangian descriptors, Phys. Rev. E 99, 032221\n(2019).\n[17] F. Revuelta, R. M. Benito, and F. Borondo, Identifica-\ntion of the invariant manifolds of the LiCN molecule us-\ning Lagrangian descriptors, Phys. Rev. E 104, 044210\n(2021).\n[18] R. Essers, J. Tennyson, and P. E. S. Wormer, An SCF\npotential energy surface for lithium cyanide, Chem. Phys.\nLett. 89, 223 (1982).\n[19] F. J. Arranz, F. Borondo, and R. M. Benito, Transition\nfrom order to chaos in a floppy molecule: LiNC/LiCN,\nChem. Phys. Lett. 317, 451 (2000).\n[20] F. J. Arranz, R. M. Benito, and F. Borondo, The on-\nset of chaos in the vibrational dynamics of LiNC/LiCN,\nJ. Chem. Phys. 123, 134305 (2005).\n[21] Z. Baˇ ci´ c and J. C. Light, Highly excited vibrational\nlevels of “floppy” triatomic molecules: A discrete vari-\nable representation-distributed Gaussian basis approach,\nJ. Chem. Phys. 85, 4594 (1986).\n[22] E. F. Krause, Taxicab Geometry (Dover, New York,\n1986).\n[23] G. G. de Polavieja, F. Borondo, and R. M. Benito, Scar\nin groups of eigenstates in a classically chaotic system,\nPhys. Rev. Lett. 73, 1613 (1994).\n[24] F. J. Arranz, F. Borondo, and R. M. Benito, Molecular\nspectra, quantum chaos, and scars, Eur. Phys. J. D 4,\n181 (1998).\n[25] B. Eckhardt and D. Wintgen, Indices in classical mechan-\nics, J. Phys. A 24, 4335 (1991).\n[26] A. Vergel, J. Montes, and F. Borondo, Geometrody-\nnamic approach to conjugate points and the Maslov in-\ndex, Phys. Rev. E 106, 064213 (2022)." }, { "title": "2401.14508v1.Energy_Conservative_Relaxation_Free_Runge_Kutta_Schemes.pdf", "content": "Energy Conservative Relaxation-Free Runge-Kutta Schemes\nMohammad R. Najafian and Brian C. Vermeire\nDepartment of Mechanical, Industrial, and Aerospace Engineering\nConcordia University\nMontreal, QC, Canada\nJanuary 29, 2024\nAbstract\nA wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation\nof entropy or various energies, which arise implicitly under exact solution of their governing PDEs.\nHowever, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce\nthese constraints, leading to a loss of accuracy and stability. Previously, the Incremental Directional\nTechnique RK (IDT-RK) and Relaxation Runge-Kutta (R-RK) approaches have been proposed to\naddress this. However, these lead to a loss of accuracy in the case of IDT-RK, or a loss of step\nsize control in the case of R-RK. In the current work we propose Relaxation-Free Runge-Kutta\n(RF-RK) schemes, which conserve energy, maintain order of accuracy, and maintain a constant step\nsize, alleviating many of the limitations of the aforementioned techniques. Importantly, they do so\nwith minimal additional computational cost compared to the base RK scheme. Numerical results\ndemonstrate that these properties are observed in practice for a range of applications. Therefore, the\nproposed RF-RK framework is a promising approach for energy conservative time integration of\nsystems of PDEs.\n1 Introduction\nMany physical systems yield measurable quantities that either remain constant or evolve monotonically\nover time. Examples of such behavior include explicitly conserved quantities, such as mass, momentum,\nand energy, or implicitly conserved quantities such as entropy in the compressible Euler equations with\nsmooth solutions. When solving these Partial Di fferential Equations (PDEs) approximately, an important\nindicator of the quality of the numerical solution is preserving physical progression of these parameters [ 1].\nOften, these constraints are enforced explicitly through chosen conservation laws, such as conservation of\nmass, momentum, and total energy with the Euler equations.However, in some cases, such as entropy,\nthey are enforced indirectly under exact evolution of conservation laws. Failure to maintain these at\napproximate discrete solutions can result in non-physical solutions [ 2], stability issues for long-time\nintegration [ 3], or increased numerical error [ 4,5,6] .Therefore, numerical schemes that can provide\nstructure-preserving solutions, maintaining these implicitly conserved quantities, are of great importance.\nSolving time-dependent PDEs commonly involves two main steps: first, the domain undergoes spatial\ndiscretization, transforming the system into semi-discrete time-dependent Ordinary Di fferential Equations\n(ODEs). Subsequently, these ODEs are solved using a temporal integration scheme to obtain a numerical\nsolution that is fully-discrete in space and time. Each of these steps can potentially contaminate conserved\nquantities. For spatial discretization, there is a rich literature on stability-preserving techniques concerning\nkinetic energy and entropy for the Euler and Navier-Stokes equations, see e.g. [ 7,8,9,10]. Nevertheless,\nthe resulting semi-discrete ODEs should be coupled with a time integration scheme to proceed in time,\nand the important question remains: if the conserved quantities are maintained after numerical integration.\nTo answer that, stand-alone integration schemes can be studied in terms of conserving nonlinear stability\nproperties of the ODE system. Here we consider the case of energy conservation.\n1arXiv:2401.14508v1 [math.NA] 25 Jan 2024To proceed, consider the following ODE system, which can be a semi-discrete representation of an\ninitial PDE problem discretized via a spatial discretization\nu′(t)=f(t,u(t)), (1.1a)\nu(t0)=u0, (1.1b)\nwhere u(t)is am×1real vector of a solution in Hilbert space. A common quantity of interest is the L2\ninner product norm of the solution vector, or herein called energy: ⟨u,u⟩=||u||2. For the ODE system 1.1,\nenergy is a continuous function of time: ||u(t)||2, while for its numerical solution, energy is only available\nat each discrete time step n:||u(t0+n∆t)||2=||un||2. We call the semi-discrete system 1.1 energy dissipative\n(or strongly stable) if it satisfies an energy decay relation [11, 12]\nd\ndt||u(t)||2=2⟨u,f(t,u)⟩≤0. (1.2)\nMaintaining this decay in energy by an integration scheme implies that the energy at each time step should\nnot be increasing\n||un+1||2≤||un||2. (1.3)\nTime integration schemes that guarantee non-increasing energy at each time step for dissipative problems\nare called monotonicity preserving (also known as strong stability preserving) integration methods.\nMoreover, system 1.1 will be called energy conservative if the time rate change of energy is exactly\nzero\nd\ndt||u(t)||2=2⟨u,f(t,u)⟩=0. (1.4)\nSuch integration schemes that, for conservative systems, maintain energy unchanged at each step up to\nmachine precision, are called energy conservative integration schemes [11], yielding\n||un+1||2=||un||2. (1.5)\nAs previously mentioned, conservation and monotonicity preservation play a pivotal role in securing\naccurate and physically meaningful solutions [ 13,14]. However, many widely used integration schemes\nfall short of guaranteeing these properties. For dissipative systems, it has been shown that many explicit\nschemes cannot preserve monotonicity, even for linear autonomous systems. For instance, Sun & Shu\n[14] demonstrated that the classical fourth order RK method may adversely result in an increase in energy\nat the first time step, no matter how small the step size is. Moreover, no classical explicit RK method\ncan preserve energy up to machine precision for conservative systems. They all lead to spurious energy\nchange above their truncation error in the general case [15].\nThere have been attempts to modify explicit RK schemes to make them energy conservative while\nmaintaining favorable accuracy and e fficiency properties. In this regard, Calvo and coauthors [ 16]\ndeveloped a directional projection technique, in which at the end of each time step, the solution is\nprojected along an oblique direction to the conservative manifold. This technique has been further\ndeveloped by Ketcheson [ 11] and referred to as the Relaxation Runge-Kutta technique (R-RK). With\nthis method, the resulting explicit scheme is energy conservative, and the order of accuracy is retained.\nHowever, it comes at the expense of step size relaxation at each step. If instead one desires to keep the\nstep size unchanged, it can be done with the so-called Incremental Directional Technique (IDT-RK), but\nthe order of accuracy of the scheme will be decreased by one [16].\nTo address the limitations of both R-RK and IDT-RK techniques, this paper introduces a novel\napproach making explicit RK methods energy conservative, while the step size remains constant and\n2the order of accuracy is retained. We refer to this method as the Relaxation-Free Runge-Kutta approach\n(RF-RK). This method’s additional computational cost at each step remains minimal, and is comparable\nto that of R-RK and IDT-RK methods.\nThe rest of this paper is organized as follows. In Section 2 a brief description of explicit Runge-\nKutta schemes is provided. It then reviews relaxation Runge-Kutta methods, and the relaxation-free\nmethod is introduced. In Section 3 it will be demonstrated that the accuracy of the original RK scheme\nis preserved with the proposed relaxation-free technique. Then, the linear stability regions of RF-RK\nschemes compared to their base RK method will be studied. Finally, numerical examples and comparisons\nbetween these energy conserving methods and unmodified RK schemes have been provided in Section 4.\nFinally, conclusions and recommendations for future work are presented in Section 5.\n2 Methodology\nThis section describes classical RK methods and the two energy conserving techniques: the previous R-RK\napproach, and the proposed RF-RK approach. Section 2.1 briefly describes the classical RK framework,\nalong with the energy change introduced at each step. Section 2.2 briefly reviews how the relaxation\ntechnique is defined to cancel out the spurious energy change. Then, in Section 2.3 , our alternative\ntechnique, called a relaxation-free method, is presented.\n2.1 Classical Runge-Kutta Schemes\nGiven a solution vector at the time step n, denoted as un=u(tn), the next-step solution using a classical s\nstage Runge-Kutta scheme is approximated as\nyi=un+ ∆tsX\nj=1ai jfj,i=1, ..., s, (2.1a)\nu(tn+ ∆t)≈un+1=un+ ∆tsX\nj=1bjfj. (2.1b)\nwhere fjis the jth stage derivative\nfj=f(tn+cj∆t,yj),\nand the coe fficients ai j,bj, and cj=P\niai jare elements of the scheme’s Butcher tableau A,b, and c,\nrespectively. An RK scheme is explicit when its matrix Ais lower triangular.\nFollowing Ketcheson [11], the change in energy after each time step will be\n||un+1||2−||un||2=\r\r\r\run+ ∆tsX\nj=1bjfj\r\r\r\r2\n−||un||2,\n=2∆tsX\nj=1bj⟨un,fj⟩+ ∆t2sX\ni,j=1bibj⟨fi,fj⟩,\n=2∆tsX\nj=1bj⟨yj,fj⟩+2∆tsX\nj=1bj⟨un−yj,fj⟩+ ∆t2sX\ni,j=1bibj⟨fi,fj⟩.\nBy using Eq. (2.1a), this can be written as\n||un+1||2−||un||2=2∆tsX\nj=1bj⟨yj,fj⟩−2∆t2sX\ni,j=1biai j⟨fi,fj⟩+ ∆t2sX\ni,j=1bibj⟨fi,fj⟩. (2.2)\n3The first term on the right-hand side of Eq. (2.2) depends on the spatial semi-discretization, and will be\nzero for conservative systems, or negative for dissipative ones. On the other hand, the summation of the\nremaining two terms, denoted by ∆En\nt\n∆En\nt=−2∆t2sX\ni,j=1biai j⟨fi,fj⟩+ ∆t2sX\ni,j=1bibj⟨fi,fj⟩, (2.3)\nis the spurious energy introduced by numerical integration. If this is zero, then energy conservation, or\nmonotonicity, of the semi-discrete system will be preserved. However, when ∆En\ntdeviates from zero,\nwhich is possible for all explicit RK schemes, energy conservation can be lost, and monotonicity may not\nbe preserved [ 11]. The objective is to enforce ∆En\nt=0at each step by modifying either the parameters of\nthe integration scheme or the time step size at each step in Eq. (2.1b).\n2.2 Relaxation Runge-Kutta Schemes\nThe relaxation family of Runge-Kutta schemes of Ketcheson [ 11] is created by replacing Eq. (2.1b) with\nthe following expression\nu(tn+γn∆t)≈un+1=un+ ∆tγnsX\nj=1bjf(tn+cj∆t,yj). (2.4)\nThis means that while intermediate solutions and their derivatives in Eq. (2.1a) are computed using\nthe original step size, ∆t, the next step solution in Eq. (2.4) in fact falls at tn+γn∆t. This inherent\ninconsistency in the time step size provides freedom to have control over energy change at each step by\nfine-tuning the so-called relaxation variable γn.\nWith this relaxation method, the energy change at each step becomes\n||un+1\nγ||2−||un||2=2γn∆tsX\nj=1bj⟨yj,fj⟩−2γn∆t2sX\ni,j=1biai j⟨fi,fj��\n+γ2\nn∆t2sX\ni,j=1bibj⟨fi,fj⟩.(2.5)\nThe last two terms, which comprise ∆En\nt, can be eliminated by solving for γnvia\nγn=1||Ps\nj=1bjfj||2=0\n2Ps\ni,j=1biai j⟨fi,fj⟩Ps\ni,j=1bibj⟨fi,fj⟩||Ps\nj=1bjfj||2,0(2.6)\nKetcheson [ 11] demonstrated that while being energy conservative, this method retains the order of\naccuracy of the original RK scheme, and the additional cost is limited to the inexpensive calculation of\ns+1inner products. However, the drawback of this method persists: by relaxing the time step size, the\nuser will not have direct control over the actual step sizes taken. Opting for a fixed step size, denoted as\nthe IDT-RK approach, means that the resulting integration scheme has parameters γnbjinstead of bj. This\ncreates a local inconsistency since in general,Ps\njγnbj,1, which leads to reduced accuracy. Therefore,\nthe user has to decide between relaxing time steps or losing an order of accuracy.\nThe following section introduces our proposed technique, called here a relaxation-free RK scheme (RF-\nRK), enabling explicit RK schemes to be energy conservative without step size relaxation or decreasing\nthe scheme’s order of accuracy.\n42.3 Relaxation-Free Runge-Kutta Schemes\nWe introduce a family of energy conservative RK schemes, called here relaxation-free RK methods, as\nthey do not impose relaxation on the time step size. With this method, the coe fficients bjof the original\nRK scheme in Eq. (2.1b) are substituted with modified values, ˆbj\nu(tn+ ∆t)≈un+1=un+ ∆tsX\nj=1ˆbjfj, (2.7)\nˆbj=bj+kjϵn, (2.8)\nwhereϵnis a real-valued parameter to be calculated at each step nto enforce elimination of spurious energy\nproduction or dissipation induced by numerical integration, and kjare constant multipliers that must\nadhere to certain conditions to follow. First, similar to all consistent RK schemes that satisfyP\njbj=1,\nthis requirement needs to be followed by the new ˆbjparameters:P\njˆbj=1. With Eq. (2.8), this can be\nguaranteed ifP\njkj=0. Moreover, it will be shown in Section 3.1 that to keep the order of accuracy\nunchanged, these kjparameters need to be selected such thatP\njkjcj,0. So, to design a RF-RK method\nfrom a base RK scheme, we need the following\nsX\ni=1ki=0, (2.9a)\nsX\ni=1kici,0. (2.9b)\nWith this modified scheme, energy variation at each step takes the form:\n||un+1||2−||un||2=2∆tsX\nj=1(bj+kjϵn)⟨yj,fj⟩−2∆t2sX\ni,j=1(bi+kiϵn)ai j⟨fi,fj⟩\n+ ∆t2sX\ni,j=1(bi+kiϵn)(bj+kjϵn)⟨fi,fj⟩.(2.10)\nAgain, the summation of the last two terms is the numerically-induced energy change. Aiming to make\nthis zero, a quadratic equation has to be solved to find the appropriate value for ϵn\nA∗\nnϵ2\nn+B∗\nnϵn+C∗\nn=0, (2.11)\nwhere A∗,B∗, and C∗are\nA∗\nn=sX\ni,j=1kikj⟨fi,fj⟩, (2.12a)\nB∗\nn=−2sX\ni,j=1kiai j⟨fi,fj⟩+2sX\ni,j=1kibj⟨fi,fj⟩, (2.12b)\nC∗\nn=−2sX\ni,j=1biai j⟨fi,fj⟩+sX\ni,j=1bibj⟨fi,fj⟩. (2.12c)\nFor a special case of constant stage derivatives, fi=f, which happens in steady state solutions, we would\nhave A∗\nn=(P\niki)2⟨f,f⟩=0, and C∗\nn=−2P\nibici⟨f,f⟩+(P\nibi)2⟨f,f⟩=0for all RK schemes of second\norder or higher. In this case, the solution for Eq. (2.11) becomes ϵn=0, which aligns intuitively with the\n5notion that in a steady state solution, there is no change in energy with the original RK scheme. However,\nwhen the condition fi=fdoes not hold, the solution depends on the sign of parameter ˆ∆n\nˆ∆n=(B∗\nn)2−4A∗\nnC∗\nn. (2.13)\nWhen ˆ∆ntakes a positive value, there are two real solutions for Eq. (2.11), while it will be shown that\nonly one of them preserves the order of accuracy of the scheme\nϵn=−B∗\nn+q\nˆ∆n\n2A∗nfor A∗\nn,0||ˆ∆n≥0. (2.14)\nOn the contrary, when ˆ∆n<0, there are no real-valued solution for Eq. (2.11). So, ϵnat each step will\ntake one of the following values\nϵn=0 for A∗\nn=0\n−B∗\nn+q\nˆ∆n\n2A∗nfor A∗\nn,0||ˆ∆n≥0\nNo real value for A∗\nn,0||ˆ∆n<0(2.15)\nNote that similar to standard RK schemes that conserve linear invariants of the ODE system [ 1],\nRF-RK schemes also hold this property automatically.\nIn the section 3.1, it will be proven that, for su fficiently small step sizes, ˆ∆n≥0, and a real-valued\nsolution for 2.11 exists. Moreover, we will show with ϵnset by Eq. (2.15), this new energy conservative\nscheme has at least the same order of accuracy as the original RK method.\n3 Accuracy and stability of RFRK\nThis section demonstrates accuracy preservation of RF-RK schemes in Section 3.1, and stability properties\nof RF-RK schemes compared to unmodified RK schemes in Section 3.2.\n3.1 Accuracy\nIn this section, we show that for su fficiently small step sizes, there exists a real-valued ϵnat each step as\na solution for Eq. (2). Then, it will be proven that starting with a RK scheme of order p, the order of\naccuracy of the RF-RK method will be equal to, or higher than, p.\nLemma 1. For small enough time steps, ˆ∆n≥0. So there would be at least one real-valued solution for\nEq. (2.11)\nProof of Lemma 1. As indicated in Section 2.3, when stage derivatives are equal to each other, we have\na real-valued solution: ϵn=0. However, when stage derivatives are not identical, the existence of\nreal-valued solutions for Eq. (2.11) depends upon the sign of ˆ∆n. Since ˆ∆nis a function of A∗\nn,B∗\nn, and C∗\nn\n(see Eq. (2.13)), we start with the magnitude of these parameters. According to Ketheson [ 11] (proof of\nLemma 4), we know that C∗is of order p−1\nC∗=O(∆tp−1). (3.1)\nRegarding B∗,we first write the Taylor series expansion for stage derivatives fjup to their linear term\n[17]\nfj=f(tn+cj∆t,yj)=f0+f(1)\nj∆t+O(∆t2), (3.2)\nwhere f0=f(tn,un), and f(1)\njrepresents the linear coe fficient within the Taylor series expansion for fj.\nSo, the Taylor series for the inner product of stage derivatives fiandfjbecomes\n6⟨fi,fj⟩=⟨f0,f0⟩+\u0010\n⟨f(1)\ni,f0⟩+⟨f0,f(1)\nj⟩\u0011\n∆t+O(∆t2). (3.3)\nPutting this expression in the definition of B∗, Eq. (2.12b), the Taylor series expansion for this parameter\ncan be written as\nB∗=\u0010\n−2sX\ni,j=1kiai j+2sX\ni,j=1kibj\u0011\n⟨f0,f0⟩+O(∆t). (3.4)\nThe constant portion of this series is composed of two terms. The first, −2P\ni,jkiai j=−2P\nikici, is\nnonzero by the definition of kiterms, Eq. (2.9), while the second is zero, again by the conditions set on ki,\n2P\ni jkibj=2(P\niki=0)(P\njbj=1)=0. So, we are left with the following Taylor expansion for B∗\nB∗=\u0010\n−2sX\ni,j=1kici\u0011\n⟨f0,f0⟩+O(∆t). (3.5)\nThis means that B∗has a non-zero constant term in its Taylor series\nB∗=O(1). (3.6)\nLastly, it can be shown that A∗is of order of a positive integer m\nA∗=O(∆tm). (3.7)\nTherefore, by using the order conditions for A∗,B∗, and C∗, we can write the following for ˆ∆n\nˆ∆n=(B∗\nn)2−4A∗\nnC∗\nn\n=(O(1))2−4(O(∆tm))(O(∆tp−1)).(3.8)\nThis shows that ˆ∆nhas a positive constant term in its Taylor series, and it will take a positive value if the\nstep size is small enough. Therefore, according to Eq. (2.15), there is at least one real-valued solution for\nEq. (2.11) when the step size is su fficiently small.\n□\nHaving a positive ˆ∆nfor small step sizes, we can provide a statement about the magnitude of ϵn\ndefined in Eq. (2.15).\nLemma 2. Having a parent RK method of order p, ϵnin Eq. (2.15) converges to zero with a rate of p−1\nprovided that time steps are su fficiently small\nϵn=O(∆tp−1).\nProof of Lemma 2. When we have A∗\nn=0,ϵnis identically zero. Moving forward with A∗\nn,0, we saw\nthat having small enough step sizes, we will have ˆ∆n≥0. Since B∗has a non-zero constant term in its\nTaylor series, for the square root of ˆ∆nwe can write\nq\nˆ∆n=q\n(B∗n)2−4A∗nC∗n\n=q\n(B∗n)2−O(∆tm+p−1)\n=r\u0010\nB∗n−O(∆tm+p−1)\u00112\n=B∗\nn−O(∆tm+p−1).(3.9)\nPutting this expression in Eq. (2.15), we obtain the magnitude of ϵnas a function of the time step size\n7ϵn=−B∗\nn+q\nˆ∆n\n2A∗n,\n=−B∗\nn+\u0010\nB∗\nn−O(∆tm+p−1)\u0011\nO(∆tm),\n=O(∆tp−1),(3.10)\nwhich shows ϵnconverges to zero with a rate of p−1.\n□\nNow that we have obtained the convergence rate of ϵn, we can show the order of accuracy of RF-RK\nschemes.\nTheorem 3. If the parent /original RK method is of order p, the RF-RK scheme is of order p or higher.\nProof of Theorem 3. Herein, ( n+1)th step solution out of the parent RK method is denoted as un+1, while\nthe corresponding solution with the RF-RK scheme is denoted un+1\nRF. Having a parent RK scheme of order\npmeans that the Taylor series for the exact solution u(tn+ ∆t)and for un+1coincide up to the term ∆tp\n[17]\n||u(tn+ ∆t)−un+1||≤K∆tp+1. (3.11)\nForu(tn+ ∆t), it is possible to write a Taylor series [17]\nu(tn+ ∆t)=u(tn)+ ∆t f(tn,un)+∆t2\n2\u0010\nft+fuf\u0011\n(tn,un)\n+∆t3\n6\u0010\nftt+2ftuf+fuuf f+fuft+fufuf\u0011\n(tn,un)+O(∆t4).(3.12)\nAlso, it is possible to write a Taylor series for un+1\nun+1=u(tn)+ ∆tsX\ni=1bif(tn,un)+∆t2\n2sX\ni=12bici\u0010\nft+fuf\u0011\n(tn,un)\n+∆t3\n6\u0010\n3sX\ni=1bic2\ni(ftt+2ftuf+fuuf f)+6sX\ni,j=1biai jcj(fuft+fufuf)\u0011\n(tn,un)+O(∆t4).\n(3.13)\nFrom this, the following Taylor series for un+1\nRFcan be obtained easily by replacing biwith ˆbi\nun+1\nRF=u(tn)+ ∆tsX\ni=1ˆbif(tn,un)+∆t2\n2sX\ni=12ˆbici\u0010\nft+fuf\u0011\n(tn,un)\n+∆t3\n6\u0010\n3sX\ni=1ˆbic2\ni(ftt+2ftuf+fuuf f)+6sX\ni,j=1ˆbiai jcj(fuft+fufuf)\u0011\n(tn,un)+O(∆t4).\n(3.14)\nSince we know from Eq. (2.8) that ˆbi−bi=kiϵn, the di fference between the two Taylor series for un+1\nandun+1\nRFbecomes\n8un+1\nRF−un+1=∆tϵnhsX\ni=1kii\nf(tn,un)+∆t2\n2ϵnh\n2sX\ni=1kicii\u0010\nft+fuf\u0011\n(tn,un)\n+∆t3\n6ϵn\u0010\n3sX\ni=1kic2\ni(ftt+2ftuf+fuuf f)+6sX\ni,j=1kiai jcj(fuft+fufuf)\u0011\n(tn,un)+ϵnO(∆t4).\n(3.15)\nFrom Eq. (2.9) we already know thatP\niki=0andP\nikici,0. So, on the right hand side ϵn=O(∆tp−1)is\nmultiplied by ∆tm,m≥2. Therefore, the di fference between the Taylor series for RF-RK and RK becomes\nun+1\nRF−un+1=O(∆tp+1), (3.16)\nand from Eq. (3.11 ) we conclude that the order of accuracy of the RF-RK scheme is at least p\nu(tn+ ∆t)−un+1\nRF=O(∆tp+1). (3.17)\n□\n3.2 Stability\nIn this part, the linear stability of RF-RK schemes with respect to the original RK methods will be\nexamined. Assume the problem of interest satisfies\nu′(t)=λu(t), (3.18)\nλbeing a complex number. After a step of length ∆t, the exact solution will be multiplied by ez,z=λ∆t.\nHowever, the approximate solution by the RK integration scheme will be multiplied by its so-called\nstability polynomial, R(z)[17]. This stability polynomial also defines the region of linear stability, which\nis the area where the magnitude of R(z)is less than or equal one. So, we can compare the stability regions\nfrom the original RK method and its RF-RK counterpart, by comparing their stability polynomials.\nFor an RK integration scheme, the stability polynomial can be obtained from [18]\nR(z)=1+zbT(I−zA)−1e, (3.19)\nwhere eis a vector of ones. In an expanded polynomial form, this will be\nR(z)=1+zsX\ni=1bi+z2sX\ni=1bici+z3sX\ni,j=1biai jcj+...+zssX\ni,j=1bigs,\nWhere gsis a function of entries Aandconly. By substituting biwith ˆbi=bi+kiϵnwe can create the\nstability polynomial for an RF-RK scheme at the step n,RRF,n(z), as a function of R(z)\nRRF,n(z)=R(z)+ϵn\u0010\nzsX\ni=1ki+z2sX\ni=1kici+z3sX\ni,j=1kiai jcj+...+zssX\ni=1kigs(A,c)\u0011\n.\nSince we have setPs\ni=1ki=0,ϵnwhich isO(hp−1)gets multiplied by (z=λ∆t)m,m≥2. So, we can say\nthe stability polynomials for a RK method and its energy conservative RF-RK scheme at each point zare\nclose to each other with a di fference of orderO(∆tp+1)\n|RRF,n(z)−R(z)|=O(∆tp+1).\nWhile for non-zero ∆tthe stability limit for the RF-RK method can be di fferent from the original RK\nmethod, the two stability limits tend to converge as the step size decreases. To have better insight, Figure\n91 shows the change in stability regions after applying the RF-RK technique. Note that these plots are\nobtained with ϵnranging from−0.05 to0.05, which are much larger than practical values for ϵnas will be\nshown later. Clearly, the stability limit along both axes can be changed by the RF-RK method, and this\nchange gets amplified by higher values for ϵn. So, it can be expected that by using smaller time steps, ϵn\nin turn will be smaller in magnitude, and the linear stability region will be modified less.\n(a) 2nd order RF-RK method\n (b) 3rd-order RF-RK method\n (c) 4th-order RF-RK method\nFigure 1. Change in linear stability region for integration schemes after application of RF-RK method\nforϵnbetween−0.05 to0.05, where the black lines belong to ϵn=0, and the darker blue lines belong to\npositive values for ϵn. Note that these values for ϵnare chosen large to exhibit its dependence clearly.\n4 Numerical examples\nThis section presents the usage of RF-RK schemes for validation test cases. To better compare the results\nof RF-RK with the already developed R-RK technique, the example cases of Ketcheson [ 11] are followed\nhere. This entails using several types of base RK schemes for di fferent linear and non-linear illustrative\nproblems.\nTypes of base RK schemes that are used here are standard RK schemes, SSP methods of [ 19] called\nSSPRK, and fifth order method of [ 20] called BSRK [ 11]. The number of stages and the order of each\nscheme are indicated as (s,p). For instance, SSPRK(10,4) is the 4th-order SSPRK method with 10 stages.\nUsing this notation, the base integration schemes used here based on Kecheson [11] are\n•SSPRK(2,2) [19],\n•SSPRK(3,3) [19],\n•RK(4,4) [14],\n•BSRK(8,5) [11].\nWe will use the term \"R\" or \"RF\" before the names of base methods to represent their relaxation or\nrelaxation-free energy conservative ones, respectively. For example, RF-RK(4,4) is the relaxation-free\nversion of RK(4,4).\n10In the following examples, the kmultipliers which are needed to employ RF-RK technique are chosen\nto be [1,−1]for RF-SSPRK(2,2), [2,−1,−1]for RF-SSPRK(3,3), [1, 2,−2,−1]for RF-RK(4,4), and\n[2,−1,−1, 0, 0, 0, 0, 0] for RF-BSRK(8,5), although other consistent values may also be used.\n4.1 A normal, linear, autonomous problem\nThe first example case from [ 11] concerns the application of the Fourier spectral collocation technique to\ndiscretize space in a 1D linear advection problem. The domain, which is periodic with a length of 2π, is\ndiscretized with m=128 points. The resulting semi-discrete ODE system becomes\nu′(t)=−Du(t), (4.1)\nwhere Dis the m×mskew-Hermitian Fourier spectral di fferentiation matrix. Since this matrix is normal,\nwe can obtain the biggest linearly stable step size using its eigenvalues, called λ. For this problem, all the\neigenvalues lay on the imaginary axes\nλ=±ik,k=m\n2−1,m\n2−2, .., 0.\nTo have linear stability, all of these eigenvalues should fall within the stability region of the employed\nintegration scheme. By using a base RK scheme with an imaginary axis stability limit of I(A,b), the\nbiggest stable step size, ∆tmax, becomes\n∆tmax=I(A,b)\nmax|λ|=I(A,b)\nm/2−1.\nNote that relative to reference [ 11], a minus one is added to the denominator as the modes are mirrored\nabout theλ=0 mode.\nFrom the section 3.2, we saw that by modifying the bparameters to ˆbthrough the RF-RK technique,\nthe stability limit would change, which in turn a ffects the biggest linearly stable step size. This change will\nalso happen for the R-RK method [ 11]. Nevertheless, in this exercise the step size limit for the base RK\nscheme will be used as a reference, and each step size is defined using a multiplier µ, where ∆t=µ∆tmax.\nTo study the energy change over time, one may decompose the solution vector into its spectral\ncomponents and see the change in amplitude of each mode over time. Spectral decomposition can be\nperformed by discrete Fourier transformation, which gives the following for the nth step solution\nun\nj=m−1X\nk=0ˆun\nkeikxj,\nwhere ˆun\nkis a complex multiplier whose magnitude is the the amplitude of mode kat the time step n. With\nthe exact solution, the amplitude of each mode remains constant over time, so the total energy remains\nunchanged. However, numerical integration may result in dissipation or amplification of each mode,\nwhich can lead to a change in total energy. The change in amplitude of each mode kafter ntime steps can\nbe represented by their relative amplification factor [11]\n|ˆun\nk|−|ˆu0\nk|\n|ˆu0\nk|.\nNote that since ˆun\nkandˆun\nm−kare complex conjugate and their magnitudes are equal, it is su fficient to study\nthe amplitude change for half of the modes.\nFor the first case, the problem is initialized with a white noise input ( ˆu0\nk=eiθk,θkbeing random) to\ndistribute energy equally across all wave numbers. Then, we record the amplitude change for each mode\nafter a final time of tf=1, for each integration scheme and time step size. The results are depicted in\nFigure 2, where the output for standard RK(4,4) is compared with its two energy conserving counterparts.\nWith the RK(4,4) scheme, high wave number modes exhibit significant damping when increasing the\n11step size. So, the total energy will decrease more by using larger step sizes. On the other hand, with both\nenergy conserving schemes, energy loss at high wave numbers is compensated for amplification of lower\nmodes, such that the total energy remains unchanged for each step size. Moreover, a slight di fference\nin stability limits for energy conserving schemes and the original method is visible in the behavior of\nhigh wave number modes with the time step ∆t=0.99∆tmax. This time step is slightly larger than the\nstability limit for R-RK method, which is why the highest modes started to amplify sharply, overpassing 0.\nHowever, this step size is smaller than the stability limit for RF-RK method, and there is no such sharp\namplification. Finally, we note the similarity of the R-RK and RF-RK results, while the RF-RK scheme\ndid not require relaxation of the step size.\n(a) RK(4,4) method\n (b) R-RK(4,4) method\n (c) RF-RK(4,4) method\nFigure 2. Relative amplification for each wavelength when using white noise initial data for the problem\n4.1. It is integrated up to tf=1, with the step sizes ∆t=µ∆tmax. While with the RK(4,4) the total energy\nhas decreased, the two energy conserving schemes amplifyed lower modes to cancel out the energy loss\nby high wavenumber modes.\nFor the second case in this example, we instead distribute the initial energy mainly among the low\nwavenumber modes by employing a smooth initial data\nU0=sech2(7.5( x+1)). (4.2)\nFigure 3 shows the amplitude change of each mode with this smooth initial condition. For RK(4,4),\nlow wavenumber modes, which contain most of the energy, see a negligible change in their amplitude.\nTherefore, the change in total energy is very small, and the conservative schemes need to impose little\nmodification to cancel out energy loss. This results in similar behavior for energy conserving and\nnon-conserving schemes.\nSuggested by [ 11], the solution for the smooth data after a final time of tf=400πwithµ=0.99\nis provided in Figure 4. As indicated, for smooth initial data the energy conservative schemes behave\nsimilar to the unmodified RK(4,4) scheme. Moreover, for this time step size, the solutions from R-RK(4,4)\nand RF-RK(4,4) are nearly indistinguishable. Note that ϵnfor RF-RK(4,4) in this case remains less than\n1.25×10−3. For a slightly larger step size when µ=1.0001 , Figure 5 compares the behavior of the two\nenergy conserving schemes. While R-RK(4,4) sees linear instability with high amplitude oscillations,\nRF-RK(4,4) still remains stable. Again for this step size, ϵnfor RF-RK(4,4) remains less than 1.25×10−3.\nWith larger step sizes, the relaxation-free scheme may not find a real solution for ϵn, and the relaxation\ntechnique encounters problems in completing the simulation because γntends to zero. However, this is\nnot unexpected, as both have exceeded their linear stability limits.\n4.2 A linear energy-decaying system\nSun & Shu [ 14] demonstrated that for a dissipative semidiscrete system, integrating with RK(4,4) may\nadversely lead to an increase in energy after the first integration step, no matter how small the step size is.\n12(a) RK(4,4) method\n (b) R-RK(4,4) method\n (c) RF-RK(4,4) method\nFigure 3. Relative amplification for each wavelength with smooth initial data for the problem 4.1.\nIntegration is performed up to tf=1, with the step sizes ∆t=µ∆tmax. Since most of the initial energy is\ndistributed among the low wavenumber modes, energy change with RK(4,4) is small, and the curves for\nRK(4,4) and the energy conserving schemes are similar.\n(a) RK(4,4) vs. R-RK(4,4)\n (b) RK(4,4) vs. RF-RK(4,4)\nFigure 4. Final solutions for problem 4.1 with the smooth initial data, integrated up to tf=400π, with\nµ=0.99. The two energy conserving schemes, R-RK(4,4) and RF-RK(4,4) behave similarly for this step\nsize.\n(a) R-RK(4,4)\n (b) RF-RK(4,4)\nFigure 5. Comparison of the solutions out of the two energy conserving schemes for the problem 4.1\nwhen it is integrated up to tf=400πwithµ=0.99, andµ=1.0001 . While ∆t=1.0001 ∆tmaxis slightly\nhigher than the stability limit for R-RK(4,4), it is still within the stability limit for RF-RK(4,4).\n13An indicative example is a linear dissipative system in the form of u′(t)=Lu(t)\nu1\nu2\nu3′\n=−1−2−2\n0−1−2\n0 0−1u1\nu2\nu3, (4.3)\nwith an initial condition equal to the first right singular vector of R(0.5L), with R(z)being the stability\npolynomial of RK(4,4). Ketcheson [ 11] showed that while RK(4,4) cannot preserve monotonicity for\nthis problem, application of the relaxation technique allows it to be monotonicity preserving. Figure 6\nshows energy change after one step through each scheme, while using two di fferent step sizes: ∆t=0.5\nand∆t=0.7. As stated, with the standard RK(4,4) energy increases for both step sizes. In contrast,\nboth R-RK(4,4) and RF-RK(4,4) preserve monotonicity and make the energy decrease for each step size.\nThis monotonicity-preservation however comes with the cost of step size modification for the R-RK\ntechnique. Table 1 demonstrates these step size modifications. With ∆t=0.5, the actual first step size for\nR-RK becomes γ1∆t≃0.44, and for the larger step ∆t=0.7it becomesγ1∆t≃0.42, even smaller than\nbefore. However, using the RF-RK technique we achieve monotonicity-preservation without any step size\nmodification.\nFigure 6. Change in energy after one time step for a dissipative problem integrated with RK(4,4) and the\ntwo monotonicity-preserving techniques, using step sizes of ∆t=0.5and∆t=0.7. With RK(4,4) energy\nincreased at the first step, but R-RK and RF-RK methods preserved the monotonicity of the problem.\nHowever, R-RK has modified the input step size, while for RF-RK it is constant.\nTable 1. Comparison of assigned step sizes and actual step sizes for each monotonicity-preserving scheme.\nWhile for R-RK there is a noticeable di fference between the input and actual step size, the RF-RK method\nkeeps the step size unchanged.\nIntegration scheme Assigned ∆t Actual ∆t Relative ∆tchange\nR-RK(4,4) 0.50 0.44 12%\nR-RK(4,4) 0.70 0.42 40%\nRF-RK(4,4) 0.50 0.50 0%\nRF-RK(4,4) 0.70 0.70 0%\n4.3 A nonlinear oscillator\nFollowing another example from the work of Ketcheson [ 11], we test the integration schemes on a\nconservative, nonlinear oscillator problem\n\"u1\nu2#′\n=1\n||u||2\"−u2\nu1#\n,\"u1(0)\nu2(0)#\n=\"1\n0#\n, (4.4)\nwhich has the following exact analytical solution\n\"u1(t)\nu2(t)#\n=\"cos(t)\nsin(t)#\n.\n14(a) RK\n (b) R-RK\n (c) RF-RK\nFigure 7. Evolution of energy for the nonlinear oscillator problem, integrated with di fferent integration\nschemes and a time step of ∆t=0.1. All the original RK methods increase energy up to their truncation\nerror, but the R-RK and RF-RK techniques conserve energy up to machine precision, and the RF-RK\nmethod does so with a constant step size.\nFigure 8. Convergence study for the nonlinear oscillator problem. For base RK methods convergence is\ndemonstrated by solid lines, while for their relaxation-free counterpart it is depicted by dashed lines. With\nRF-RK, the order of accuracy is equal to, or higher than, the corresponding base RK scheme.\nFigure 7a shows that employing each of the unmodified RK schemes with a time step size of ∆t=0.1\nleads to a monotonic increase in energy. On the other hand, presented in Figures 7b and 7c, both R-RK\nand RF-RK techniques made these RK schemes conserve energy up to machine precision. Note that\nwhile using the R-RK method the actual time step ( γn∆t) is not exactly 0.1, but it is in the range of\n0.0995≤γn∆t≤0.1. On the other hand, with the RF-RK method the step size maintains exactly 0.1,\nwhile the variable ϵnremains in the range of −0.0015≤ϵn≤0.\nConcerning their accuracy, Figure 8 shows solution convergence for unmodified schemes with solid\nlines and the corresponding RF-RK schemes in dashed lines. It confirms that with the RF-RK method, the\norder of accuracy is either the same, or higher, than the base RK method.\n4.4 Burgers’ equation\nThe last example case which has been used for R-RK methods in [ 11] is an inviscid Burger’s problem on\na periodic interval of −1≤x≤1\nUt+1\n2(U2)x=0, (4.5)\n15(a) RK\n (b) R-RK\n (c) RF-RK\nFigure 9. Evolution of energy for the Burgers’ equation with di fferent integration schemes with a step size\nof∆t=0.3∆xup to a final time of tf=2. With each of the unmodified RK schemes energy increased\nmonotonically, while energy conserving counterparts conserved energy up to machine precision\nU(x, 0)=e−30x2.\nThis problem can be transformed to an energy conservative ODE system by discretizing the domain with\n50 equally-spaced points and using the second-order accurate symmetric flux [21]\nu′\ni(t)=−1\n∆x(Fi+1/2−Fi−1/2), Fi+1/2=u2\ni+uiui+1+u2\ni+1\n6. (4.6)\nThis problem is integrated with a time step of ∆t=0.3∆xup to a final time of tf=2using di fferent\nbase RK schemes and their energy conservative counterparts. Figure 9 shows that with the original RK\nschemes, there is a noticeable change in the energy of the system, while with both R-RK and RF-RK\nschemes, energy is conserved up to machine precision. Then, to visualize the order of accuracy of the\nschemes, convergence analysis has been performed at tf=0.2with a fixed spatial discretization and\ndifferent input time steps: ∆t=CFL×∆x,CFL =0.3×0.50,1,..,6. The results in Figure 10 confirm that\nwith R-RK order of accuracy is retained, but with IDT-RK order of accuracy decreases by one. RF-RK\nhowever, benefits from a fixed step size, similar to IDT-RK, and preserved accuracy, similar to R-RK\nschemes.\n5 Conclusions\nWe have proposed a new family of energy conservative Relaxation-Free Runge-Kutta schemes. These\nschemes introduce a simple modification to the Butcher tableau coe fficients that allow conservation\nof energy, preservation of the order of accuracy of the base RK scheme, while maintaining a constant\ntime-step size. This is in contrast to classical RK schemes, which are not energy conservative, IDT-RK\nschemes, which do not maintain their base scheme’s order of accuracy, and R-RK schemes, which do\nnot maintain a constant step size. In this sense, the proposed RF-RK schemes are superior to these\nprevious schemes in several respects. Numerical results demonstrate that these aforementioned properties\nare observed in practice for a normal, linear, autonomous problem, a linear energy-decaying system, a\nnon-linear oscillator, and Burgers equation. The RF-RK schemes consistently conserved energy while\nmaintaining a constant step size and the order of accuracy of their base RK scheme.\nThe proposed RK-RK schemes present a promising framework for non-linear stability of time\ndependent systems of PDEs. Future work will focus on extension to other forms of entropy, and\napplication to systems with multiple such constraints.\n16(a) IDT-RK\n (b) R-RK\n (c) RF-RK\nFigure 10. Convergence analysis for Burger’s equation integrated up to a final time of tf=0.2with the\ntime steps: ∆t=CFL×∆x. It confirms that while IDT-RK decreases the order of accuracy by one, both\nR-RK and RF-RK methods retain the order of accuracy of original RK method.\nAcknowledgments\nThe authors acknowledge financial support from the Natural Sciences and Engineering Research Council\nof Canada (NSERC) and the Fonds de Recherche du Québec - Nature et Technologies (FRQNT) via the\nNOV A program. Additionally, the authors would like to thank Siva Nadarajah, Alexander Cicchino, and\nCarolyn Pethrick for their helpful discussions at the early stages of this work.\nReferences\n[1]E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration , volume 31 of Springer\nSeries in Computational Mathematics . Springer-Verlag, Berlin /Heidelberg, 2006.\n[2]C.W. Gear. Invariants and numerical methods for ODEs. Physica D: Nonlinear Phenomena ,\n60(1):303–310, 1992.\n[3]A. Arakawa. Computational Design for Long-Term Numerical Integration of the Equations of\nFluid Motion: Two-Dimensional Incompressible Flow. Part I. Journal of Computational Physics ,\n135(2):103, 1997.\n[4]J. de Frutos and J.M. Sanz-Serna. Accuracy and conservation properties in numerical integration:\nThe case of the Korteweg-de Vries equation. Numerische Mathematik , 75(4):421–445, February\n1997.\n[5]A. Durán and J.M. Sanz-Serna. The numerical integration of relative equilibrium solutions. The\nnonlinear Schrödinger equation. IMA Journal of Numerical Analysis , 20(2):235–261, April 2000.\n[6]H. Ranocha, M.Q. de Luna, and D.I. Ketcheson. On the Rate of Error Growth in Time for Numerical\nSolutions of Nonlinear Dispersive Wave Equations. Partial Di fferential Equations and Applications ,\n2(6):76, December 2021.\n[7]T. Chen and C.W. Shu. Review of Entropy Stable Discontinuous Galerkin Methods for Systems of\nConservation Laws on Unstructured Simplex Meshes. CSIAM Transactions on Applied Mathematics ,\n1(1):1–52, June 2020.\n17[8]J. Chan. On discretely entropy conservative and entropy stable discontinuous Galerkin methods.\nJournal of Computational Physics , 362:346–374, June 2018.\n[9]D. Del Rey Fernández, J. Hicken, and D. Zingg. Review of summation-by-parts operators with simul-\ntaneous approximation terms for the numerical solution of partial di fferential equations. Computers\n&Fluids , 95:171–196, May 2014.\n[10] Y . Kuya and S. Kawai. High-order accurate kinetic-energy and entropy preserving (KEEP) schemes\non curvilinear grids. Journal of Computational Physics , 442:110482, October 2021.\n[11] D.I. Ketcheson. Relaxation Runge–Kutta Methods: Conservation and Stability for Inner-Product\nNorms. Society for Industrial and Applied Mathematics , 57(6):2850–2870, 2019.\n[12] Z. Sun and C.W. Shu. Strong stability of explicit Runge-Kutta time discretizations. SIAM Journal\non Numerical Analysis , 57(3):1158–1182, 2019.\n[13] A. Biswas and D.I. Ketcheson. Multiple-Relaxation Runge Kutta Methods for Conservative Dynam-\nical Systems. Journal of Scientific Computing , 97(1):4, October 2023.\n[14] Z. Sun and C.W. Shu. Stability of the fourth order Runge–Kutta method for time-dependent partial\ndifferential equations. Annals of Mathematical Sciences and Applications , 2(2):255–284, 2017.\n[15] C. Lozano. Entropy Production by Explicit Runge–Kutta Schemes. Journal of Scientific Computing ,\n76(1):521–564, July 2018.\n[16] M. Calvo, D. Hernandez-Abreu, J.I. Montijano, and L. Randez. On the Preservation of Invariants by\nExplicit Runge–Kutta Methods. SIAM Journal on Scientific Computing , 28(3):18, 2006.\n[17] E. Hairer, G. Wanner, and S. P. Nørsett. Solving Ordinary Di fferential Equations I: Nonsti ffProblems .\nSpringer, second edition, 1996.\n[18] J.C. Butcher. Numerical Methods for Ordinary Di fferential Equations . John Wiley & Sons, Ltd, 2nd\nedition, 2008.\n[19] D.I. Ketcheson. Highly e fficient strong stability-preserving Runge Kutta methods with low-storage\nimplementations. SIAM Journal on Scientific Computing , 30(4):2113–36, 2008.\n[20] P. Bogacki and L.F. Shampine. An e fficient Runge-Kutta (4,5) pair. Computers &Mathematics with\nApplications , 32(6):15–28, September 1996.\n[21] E. Tadmor. Entropy stability theory for di fference approximations of nonlinear conservation laws\nand related time-dependent problems. Acta Numerica , 12:451–512, May 2003.\n18" }, { "title": "2401.14568v2.A_counterexample_regarding_a_two_phase_problem_for_harmonic_measure_in_VMO.pdf", "content": "A COUNTEREXAMPLE REGARDING A TWO-PHASE PROBLEM FOR\nHARMONIC MEASURE IN VMO\nXAVIER TOLSA\nAbstract. Let Ω+⊂Rn+1be a vanishing Reifenberg flat domain such that Ω+and Ω−=\nRn+1\\Ω+have joint big pieces of chord-arc subdomains and the outer unit normal to ∂Ω+belongs\nto VMO( ω+), where ω±is the harmonic measure in Ω±. Up to now it was an open question if\nthese conditions imply that logdω−\ndω+∈VMO( ω+). In this paper we answer this question in the\nnegative by constructing an appropriate counterexample in R2, with the additional property that\nthe outer unit normal to ∂Ω+is constant ω+-a.e. in ∂Ω+.\n1.Introduction\nThis paper deals with a two-phase problem for harmonic measure. In such problems one\nconsiders two disjoint domains Ω+,Ω−⊂Rn+1whose boundaries have non-empty intersection, and\nwhose respective harmonic measures ω+, ω−are usually mutually absolutely continuous in some\nsubset of ∂Ω+∩∂Ω−. Then one has to relate the analytic properties of the densitydω−\ndω+|∂Ω+∩∂Ω−\nto the geometric properties of ∂Ω+∩∂Ω−. For example, in [AMT1] and [AMTV] it has been\nproved that if ω+andω−are mutually absolutely continuous in a subset E⊂∂Ω, then ω±|Eis\nconcentrated in an n-rectifiable set. For a previous related result see [KPT], and for a more recent\nwork involving elliptic measure, see [AM]. For other works of more quantitative nature where one\nassumes Ω+, Ω−to be complementary NTA domains and either that ω−∈A∞(ω+) or stronger\nconditions, see [KT3], [En], [AMT2], [PT], and [TT], for instance. See also [BET1] and [BET2]\nfor other recent results dealing with the structure of the singular set of the boundary.\nIn connection with the precise question studied in this paper, by combining works of Prats,\nTolsa, and Toro, the following is known:\nTheorem A. LetΩ+⊂Rn+1be a bounded NTA domain and let Ω−=Rn+1\\Ω+be an NTA\ndomain as well. Denote by ω+andω−the respective harmonic measures with poles p+∈Ω+and\np−∈Ω−. Suppose that Ω+is aδ-Reifenberg flat domain, with δ >0small enough. Then the\nfollowing conditions are equivalent:\n(a)ω+andω−are mutually absolutely continuous and logdω−\ndω+∈VMO( ω+).\n(b)Ω+is vanishing Reifenberg flat, Ω+andΩ−have joint big pieces of chord-arc subdomains,\nand\n(1.1) lim\nρ→0sup\nx∈∂Ω+\n00small enough. Denote by ωthe harmonic measure in Ωwith pole p∈Ωand\nwrite σ=Hn|∂Ω. Then the following conditions are equivalent:\n(i)logdω\ndσ∈VMO( σ).\n(ii)Ωis vanishing Reifenberg flat and the outer unit normal Nto∂Ωbelongs to VMO( σ).\n(iii) The outer unit normal Nto∂Ωexists σ-a.e. and it belongs to VMO( σ).\nIn view of the similarities between the statements (b) in Theorem A and (ii) in Theorem B, a\nnatural question (which is left open in the works [PT] and [TT]) is if under the assumptions in\nTheorem A, the statement (b) is equivalent to the following:\n(b’) Ω+is vanishing Reifenberg flat, Ω+and Ω−have joint big pieces of chord-arc subdomains,\nand the outer unit normal Nto∂Ω+belongs to VMO( ω+).\nNotice that (b’) is the same as (b), with NB(x,r)in (1.1) replaced by −´\nB(x,r)N dω+. In this paper\nwe provide a negative answer by constructing a suitable counterexample in R2. The precise result\nis the following:\nTheorem 1.1. There exists a bounded vanishing Reifenberg flat domain Ω+⊂R2such that Ω+\nandΩ−:=R2\\Ω+have joint big pieces of chord-arc subdomains, the outer unit normal Nis\nconstant ω+-a.e. in ∂Ω+, and such that (1.1) does not hold, and so logdω−\ndω+̸∈VMO( ω+).\nObserve that in the theorem, although logdω−\ndω+̸∈VMO( ω+), we still have ω−∈A∞(ω+),\nbecause Ω+and Ω−:=R2\\Ω+have joint big pieces of chord-arc subdomains. Of course, the\ntheorem also implies that in the statement (c) in Theorem A one cannot replace the assumption\n(1.1) by the fact that N∈VMO( ω+).COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 3\nTo prove Theorem 1.1 we will construct a snowflake-type domain satisfying the required proper-\nties. Its construction is vaguely inspired by the example in [AMT2, Section 8]. We will show that\nin this domain harmonic measure is concentrated in a countable collection of vertical segments by\na probabilistic argument which uses the central limit theorem (see Lemma 3.1) and which has its\nown interest.\n2.Preliminaries\nWe denote by Corcsome constants that may depend on the dimension and perhaps other\nfixed parameters. Their values may change at different occurrences. On the contrary, constants\nwith subscripts, like C0, retain their values. For a, b≥0, we write a≲bif there is C > 0 such\nthat a≤Cb. We write a≈bto mean a≲b≲a. If we want to emphasize that the implicit\nconstant depends on some parameter η∈R, we write a≈ηb.\n2.1.Ahlfors regularity and uniform rectifiability. All measures in this paper are assumed\nto be Borel measures. A measure µinRdis called doubling if there is some constant C >0 such\nthat\nµ(B(x,2r))≤C µ(B(x, r)) for all x∈supp µandr >0.\nA measure µinRdis called Ahlfors regular (orn-Ahlfors regular) if\n(2.1) C−1rn≤µ(B(x, r))≤Crnfor all x∈supp µand 0 < r≤diam(supp µ).\nA set E⊂Rdis Ahlfors regular (or n-Ahlfors regular) if Hn|Eis Ahlfors regular.\nA measure µinRdis called uniformly n-rectifiable if it is n-Ahlfors regular and there exist\nconstants θ, M > 0 such that for all x∈supp µand all 0 < r≤diam(supp µ) there is a Lipschitz\nmapping gfrom the ball Bn(0, r) inRntoRdwith Lip( g)≤Msuch that\nµ(B(x, r)∩g(Bn(0, r)))≥θrn.\nA set E⊂Rdis called uniformly n-rectifiable if the measure Hn|Eis uniformly n-rectifiable.\nRecall that Eis called n-rectifiable if there are Lipschitz maps gi:Rn→Rdsuch that\nHn\u0010\nE\\S\nigi(Rn)\u0011\n= 0.\nIt is easy to check that if Eis uniformly n-rectifiable, then it is n-rectifiable.\n2.2.NTA and Reifenberg flat domains. Given Ω ⊂Rn+1andC≥2, we say that Ω satisfies\ntheC-Harnack chain condition if for every ρ >0,k≥1, and every pair of points x, y∈Ω with\ndist(x, ∂Ω),dist(y, ∂Ω)≥ρand|x−y|<2kρ, there is a chain of open balls B1, . . . , B m⊂Ω,\nm≤Ck, with x∈B1, y∈Bm, Bk∩Bk+1̸=∅andC−1diam( Bk)≤dist(Bk, ∂Ω)≤Cdiam( Bk).\nThe chain of balls is called a Harnack chain .\nForC≥2, Ω is a C-corkscrew set if for all ξ∈∂Ω and r∈(0, R) there is a ball of radius r/C\ncontained in B(ξ, r)∩Ω. Finally, we say that Ω is C-non-tangentially accessible (or C-NTA, or\njust NTA) if it satisfies the Harnack chain condition and both Ω and ( Ω)careC-corkscrew sets.\nAlso, Ω is two-sided C-NTA if both Ω and ( Ω)careC-NTA. A chord-arc domain is an NTA domain\nΩ⊂Rn+1whose boundary is n-Ahlfors regular. NTA domains were introduced by Jerison and\nKenig in [JK]. In this type of domains, the authors showed that harmonic measure is doubling\nand it satisfies other remarkable properties, such as the following change of pole formula.4 XAVIER TOLSA\nTheorem 2.1 ([JK, Lemma 4.11]) .Letn≥1,Ωbe aC-NTA open set in Rn+1and let Bbe a\nball centered in ∂Ω. Let p1, p2∈Ωbe such that dist(pi, B∩∂Ω)≥c−1\n0r(B)fori= 1,2. Then, for\nany Borel set E⊂B∩∂Ω,\n(2.2)ωp1(E)\nωp1(B)≈ωp2(E)\nωp2(B),\nwith the implicit constant depending only on n,c0andC.\nGiven a set E⊂Rn+1,x∈Rn+1,r >0, and a hyperplane P, we set\n(2.3) DE(x, r, P ) =r−1max(\nsup\ny∈E∩B(x,r)dist(y, P),sup\ny∈P∩B(x,r)dist(y, E))\n.\nWe also define\n(2.4) DE(x, r) = inf\nPDE(x, r, P )\nwhere the infimum is taken over all hyperplanes P. Given δ, R > 0, a set E⊂Rn+1is (δ, R)-\nReifenberg flat (or just δ-Reifenberg flat) if DE(x, r)< δfor all x∈Eand 0 < r≤R, and it is\nvanishing Reifenberg flat if\nlim\nr→0sup\nx∈EDE(x, r) = 0 .\nLet Ω ⊂Rn+1be an open set, and let 0 < δ < 1/2. We say that Ω is a ( δ, R)-Reifenberg flat\ndomain (or just δ-Reifenberg flat) if it satisfies the following conditions:\n(a)∂Ω is ( δ, R)-Reifenberg flat.\n(b) For every x∈∂Ω and 0 < r≤R, denote by P(x, r) ann-plane that minimizes DE(x, r).\nThen one of the connected components of\nB(x, r)∩\b\nx∈Rn+1: dist( x, P(x, r))≥2δ r\t\nis contained in Ω and the other is contained in Rn+1\\Ω.\nIf, additionally, ∂Ω is vanishing Reifenberg flat, then Ω is said to be vanishing Reifenberg flat,\ntoo. It is well known that if Ω is a δ-Reifenberg flat domain, with δsmall enough, then it is also\nan NTA domain (see [KT1]).\nGiven two NTA domains Ω+⊂Rn+1and Ω−=Rn+1\\Ω+, we say that Ω+and Ω−have joint big\npieces of chord-arc subdomains if for any ball Bcentered in ∂Ω+with radius at most diam( ∂Ω+)\nthere are two chord-arc domains Ωs\nB⊂Ωs, with s= +,−, such that Hn(∂Ω+\nB∩∂Ω−\nB∩B)≳r(B)n.\n2.3.The space VMO .Given a Radon measure µinRn+1,f∈L1\nloc(µ), and A⊂Rn+1, we write\nmµ,A(f) =−ˆ\nAf dµ =1\nµ(A)ˆ\nAf dµ.\nAssume µto be doubling. We say that f∈VMO( µ) if\n(2.5) lim\nr→0sup\nx∈supp µ−ˆ\nB(x,r)\f\ff−mµ,B(x,r)(f)\f\f2dµ= 0.\nIt is well known that the space VMO( µ) coincides with the closure of the set of bounded uniformly\ncontinuous functions on supp µin the BMO norm.COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 5\nS1S2 S3\nS4\nFigure 1. The segments S1,S2,S3,S4, with α=π/6.\n3.Construction of the domain\nWe will construct a domain Ω = Ω+⊂R2by a limiting procedure. First we let Ω 0be the\ninterior of a regular polygon with 100 sides, say, with side length equal to 1. Given Ω n, to\nconstruct Ω n+1, we assume that Ω nis a Jordan domain such that its boundary is a piecewise\nlinear curve made up of mnclosed segments Ln\n1, . . . , Ln\nmnwith equal length ℓn. We suppose that\nthey are ordered clockwise, so that there are points zn\n1, . . . , zn\nmn∈∂Ωnsuch that Ln\nj= [zn\nj, zn\nj+1]\nfor each j= 1, . . . , m n, with zn\nmn+1=zn\n1. We assume also that, if Bn\njis a closed ball centered in\nthe mid-point of Ln\njwith radius ℓn/2, then one component of Bn\nj\\Ln\njis contained in Ω nand the\nother in R2\\Ωn.\nThe first step to construct Ω n+1consists in replacing each segment Ln\njby a piecewise linear\nJordan arc Γn\njwith the same end-points as Ln\nj.\n3.1.Construction of Γn\nj.To shorten notation, denote L=Ln\nj, and let N(L) be the outer unit\nnormal of Ω natL. Denote by γLthe angle between N(L) and the horizontal vector (1 ,0). We\nadopt the convention that γL∈(−π, π]. We will define Γn\njby a snowflake type construction. First\nwe denote\nα=αL=|γL|\nMn,\nwhere Mnis a big integer (say Mn≥20) which will be chosen inductively later. For the moment,\nlet us say that {Mn}nwill be a monotone sequence of natural numbers tending to ∞. Now we\nstart the usual construction of an arc of the Koch snowflake with angle αinstead of π/3. To this\nend, we denote ℓ=H1(L) (so ℓ=ℓn) and we replace Lby four segments S1, . . . , S 4with equal\nlength\ns:=1\n2(1 + cos α)ℓ,\nas in Figure 1. That is, in the case when L= [(0 ,0),(ℓ,0)] is a horizontal segment and N(L) =\n(0,1), we consider the points (in complex notation)\ny0= 0, y 1=s, y 2=ℓ\n2+i ssinα, y 3=ℓ−s, y 4=ℓ,\nand we let Si= [yi−1, yi].\nFor an arbitrary segment L, we define S1, . . . , S 4so that after a suitable translation and rotation\nwe are in the preceding situation. We denote by F(L) the curve generated in this way. That is,\nF(L) =S1∪S2∪S3∪S4.\nWe also denote by N(Si) the unit normal to each vector Si, so that the angle between N(L) and\nN(Si) is at most α. In other words, N(Si) is the outer unit normal at Siof the domain enclosed\nby the curve obtained by replacing LbyF(L) in∂Ωn.6 XAVIER TOLSA\nFigure 2. The curve Γn\njgenerated by a horizontal segment Ln\njwith parameters\nN(Ln\nj) = (1 ,0),α=π/4, and Mn= 2.\nIn the first iteration, we let Γ 1(L) =F(L). To construct, Γ 2(L) we iterate the construction in\nthe usual way: we let\nΓ2(L) =F(S1)∪F(S2)∪F(S3)∪F(S4),\nwith F(Si) defined in the same way as F(L), with Lreplaced by Si. We denote the four segments\nwhich compose F(Si) bySi,1, . . . , S i,4.To construct Γ 3(L) we proceed similarly.\nHowever, we introduce a special rule in the iteration of the next curves Γ k+1(L): if one of the\nsegments Si1,...,ikof Γ k(L) is vertical and the associated outer normal N(Si1,...,ik) is the vector\n(1,0), then in the construction of Γ k+1(L) we keep Si1,...,ikunchanged. That is, for a segment\nSi1,...,ikwe let\neF(Si1,...,ik) =\u001a\nF(Si1,...,ik) if N(Si1,...,ik)̸= (1,0),\nSi1,...,ik ifN(Si1,...,ik) = (1 ,0).\nIn the latter case, we let Si1,...,ik,1,···, Si1,...,ik,4be the four closed segments obtained by splitting\nSi1,...,ikinto four segments with disjoint interiors and the same length, and we let N(Si1,...,ij,h) =\n(1,0) for h= 1,2,3,4. Then we let\nΓk+1(L) =[\n1≤i1,...,ik≤4eF(Si1,...,ik).\nNotice that, by the definition of α, the first appearance of a vertical segment Si1,...,iksuch that\nN(Si1,...,ik) = (1 ,0) when α̸= 0 occurs for k=Mn≥20, and so the last definition is coherent\nwith the construction described for the first curves Γ 1(L),Γ2(L),Γ3(L).\nWe iterate this construction M2\nn≡(Mn)2times, and we let\nΓn\nj= ΓM2n(L).\nObserve that Γn\njis made up of 4M2\nnsegments Si1,...,iM2n. See Figure 2 (where we took Mn= 2 for\nan easy viewing of the vertical segments with associated normal (1 ,0)).\n3.2.The chord-arc property of Γn\nj.Notice that for 1 ≤k≤M2\nn,\n(3.1)1\n4H1(Si1,...,ik−1)≤ H1(Si1,...,ik)≤1\n2(1 + cos α)H1(Si1,...,ik−1).\nThus,\n4−kℓ≤ H1(Si1,...,ik)≤\u00121\n2(1 + cos α)\u0013k\nℓfor 1≤k≤M2\nn.COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 7\nWe can also write the second inequality as follows:\nH1(Si1,...,ik)≤4−k\u00122\n1 + cos α\u0013k\nℓ= 4−k\u0012\n1 +1−cosα\n1 + cos α\u0013k\nℓ.\nNext we use the fact that, by the definition of α,\n1 +1−cosα\n1 + cos α≤1 + (1 −cosα)≤1 +Cα2≤1 +C\nM2n.\nThus, using also that k≤M2\nn,\nH1(Si1,...,ik)≤4−k\u0012\n1 +1−cosα\n1 + cos α\u0013M2\nn\nℓ≤4−k\u0012\n1 +C\nM2n\u0013M2\nn\nℓ≲4−kℓ.\nConsequently, since Γn\njconsists of 4M2\nnsegments of the form Si1,...,iM2n, we derive\n(3.2) H1(Γn\nj)≤4M2\nnsup\ni1,...,iM2nH1(Si1,...,iM2n)≲4M2\nn4−M2\nnℓ=ℓ.\nBy the same arguments, the sub-arc of Γn\njgenerated by the segment Si1,...,ik(i.e., with the same\nend-points as Si1,...,ik), which we denote by bSi1,...,ik, satisfies\n(3.3) H1(bSi1,...,ik)≤4M2\nn−ksup\ni1,...,iM2nH1(Si1,...,iM2n)≲4M2\nn−k4−M2\nnℓ= 4−kℓ≤ H1(Si1,...,ik),\nwhere we used the first estimate in (3.1) in the last inequality. This shows that Γn\njis an Ahlfors\nregular curve (with regularity constant bounded above by an absolute constant).\nIt is also immediate to check that, for some fixed c >0,\n(3.4) Γn\nj⊂ Ucαℓ(Ln\nj)⊂ Ucℓn/Mn(Ln\nj),\nwhere Ur(A) stands for the r-neighborhood of A. In fact, by the same arguments that apply to\nthe von Koch snowflake, it follows easily that Γn\njis a quasi-arc (i.e., an arc of a quasi-circle).\nTogether with the Ahlfors regularity of Γn\nj, this means that Γn\njis a chord-arc curve.\n3.3.The abundance of vertical segments in the curve Γn\nj.Denote by V(Γn\nj) the subfamily of\nthe segments Si1,...,iM2n, with 1 ≤i1, . . . , i M2n≤4, which are vertical and such that N(Si1,...,iM2n) =\n(1,0). Before going on with the construction of Ω n+1, we will prove the abundance of that type\nof segments. This will be the key property that we will use below to show that the outer unit\nnormal to Ω equals (1 ,0)ω+-a.e. The precise result we need is the following:\nLemma 3.1. There exists an absolute constant c1>0such that, for any choice of Mnlarge\nenough,\n(3.5)X\nS∈V(Γn\nj)H1(S)≥c1H1(Γn\nj).\nProof. We will use a probabilistic argument. We can assume that α̸= 0, because otherwise all\nthe segments Si1,...,iM2nare vertical and their associated unit normal is (1 ,0). Consider the set of\ncodings of the segments Si1,...,iM2n, that is, In:={1,2,3,4}M2\nn. Let µbe the uniform probability8 XAVIER TOLSA\nmeasure on {1,2,3,4}(so that µ({1}) =···=µ({4}) = 1 /4), and let µIn=µ× ··· × µbe the\nproduct measure ( M2\nntimes) of µonIn. Consider the function g:{1,2,3,4} →Rdefined by\ng(1) = g(4) = 0 , g(2) = α, g (3) =−α,\nand consider the random variables X1, . . . , X M2nonIndefined by\nXj\u0000\n(i1, . . . , i M2n)\u0001\n=g(ij).\nIt is immediate to check that the variables X1, . . . , X M2nare independent and identically dis-\ntributed, and they have zero mean and variance σ2=α2/2.\nNotice that, for i∈Inwith Si̸∈ V(Γn\nj), we have that Σ M2n(i) :=X1(i) +. . .+XM2n(i) is the\nangle that the segment Sihas rotated with respect the initial segment1Ln\nj. Suppose for simplicity\nthat the angle γLbetween N(L) =N(Ln\nj) and the vector (1 ,0) is positive. Then, if for some given\ni= (i1, . . . , i M2n)∈In, we have Σ M2n(i1, . . . , i M2n)≤ −Mnα, this implies that some segment Si1,...,ik\nhas rotated clockwise by an angle equal to MNα=γLwith respect to L, so that it is vertical and\nN(Si1,...,ik) = (1 ,0). By construction this also ensures that N(Si1,...,iM2n) = (1 ,0). Consequently,\n#V(Γn\nj)≥#\b\ni∈In: ΣM2n(i)≤ −Mnα\t\nand thus\n(3.6)#V(Γn\nj)\n#In≥µIn\u0000\n{i∈In: ΣM2n(i)≤ −Mnα}\u0001\nObserve that\nΣM2n\nMnσ=√\n2 ΣM2n\nMnα.\nBy the central limit theorem, the random variableΣM2n\nMnσconverges in law to the standard normal\ndistribution as Mn→ ∞ . Therefore, for some absolute constant η > 0 (determined by the\nstandard normal distribution),\nµIn\u0000\n{i∈In: ΣM2n(i)≤ −Mnα}\u0001\n=µIn\u0010n\ni∈In:ΣM2n(i)\nMnσ≤ −√\n2o\u0011\n→ηasMn→ ∞ .\nConsequently, by (3.6),\n#V(Γn\nj)\n#In≥η\n2,\nforMnbig enough.\nTo finish the proof of (3.5), recall that any segment Si, with i∈In, satisfies H1(Si)≥\n4−M2\nnH1(Ln\nj). Thus,\nX\nS∈V(Γn\nj)H1(S)≥#V(Γn\nj) 4−M2\nnH1(Ln\nj)≥η\n2#In4−M2\nnH1(Ln\nj) =η\n2H1(Ln\nj)≥c ηH1(Γn\nj),\nusing (3.2) for the last inequality. □\n1In fact, ΣM2n(i) coincides also with the angle that any segment from {Si}i∈Inwould have rotated if we had not\napplied the special rule about the vertical segments in the construction of Γn\nj.COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 9\neTieTi+1\nCi\nFigure 3. The segments eTi,eTi+1, and the arc Cithat joins them. The length of\neach dotted segment is an+1/100.\n3.4.The domain Ωn+1.Recall that ∂Ωn=Smn\nj=1Ln\njand that each segment Ln\njhas the same\nend-points as the chord-arc curve Γn\nj. We consider the curve\nGn+1=mn[\nj=1Γn\nj,\nand we let Vn+1be the domain enclosed by Gn+1. This domain should be considered as a first\napproximation of Ω n+1. In order to guaranty the Reifenberg flatness property of the final domain\nΩ, we have to modify Vn+1. Indeed, it is easy to check that the angle between any two neighboring\nsegments contained in the same piecewise curve Γn\njis at most αLn\nj≤C/M n, and we will choose Mn\nso that Mn→ ∞ . However, the angle between two consecutive segments of the curve Gn+1which\nare contained in different (but neighboring) curves Γn\nj, Γn\nj+1is the same as the angle between Ln\nj\nandLn\nj+1, which is not convenient, because we want this to decrease.\nTo define a suitable smoothened version of Gn+1we need some additional notation. Let\nT1, . . . , T tn+1the segments which compose the piecewise curve Gn+1. That is to say,\n\b\nTi\t\n1≤i≤tn+1=\b\nSi1,...,iM2n(Γn\nj) : 1≤j≤mn,1≤i1, . . . , i M2n≤4},\nwhere Si1,...,iM2n(Γn\nj) stands for the segment Si1,...,iM2nappearing in the construction of Γn\nj. Suppose\nthatT1, . . . , T tn+1are ordered clockwise in the curve Gn+1. Let\n(3.7) an+1= min\n1≤i≤tn+1H1(Ti),\nand for each ileteTibe a closed segment such that eTi⊂Tiwith the same mid-point as Tiand\nsuch that\nH1(eTi) =H1(Ti)−an+1\n200.\nLetCibe a closed circular arc that joints eTitoeTi+1, so that Ciis tangent both to TiandTi+1\nand the tangency points coincide with the two closest end-points of eTiandeTi+1, as in Figure 3.\nThen we let\neGn+1=[\n1≤i≤tn+1(eTi∪Ci).\nThat is, roughly speaking, eGn+1is the curve obtained by erasing Ti\\eTi, for 1 ≤i≤tn+1and\nreplacing the erased parts by circular arcs, so that the resulting curve is of type C1.\nThe next step is to consider a family of points zn+1\n1, . . . , zn+1\nmn+1fromeGn+1ordered clockwise,\nwith mn+1≥tn+1, so that\n|zn+1\ni−zn+1\ni+1|=|zn+1\nj−zn+1\nj+1|=:ℓn+1for 1≤i, j≤mn+1,10 XAVIER TOLSA\nunderstanding zn+1\nmn+1+1=zn+1\n1. We denote Ln+1\ni= [zn+1\ni, zn+1\ni+1]. Further, we take mn+1large\nenough so that ℓn+1≤an+1\n200and so that the angle between two neighboring segments Ln+1\ni,Ln+1\ni+1\nis at most 2−n−4π(here we mean the angle equal to the smallest one of the two supplementary\nangles that form the two lines supporting Ln+1\niandLn+1\ni+1). By a continuity argument and the C1\ncharacter of eGn+1it is easy to prove the existence of the family of points zn+1\n1, . . . , zn+1\nmn+1.\nFinally, we let Ω n+1be the domain enclosed by the curve formed by the union of the segments\nLn+1\n1, . . . , Ln+1\nmn+1, so that\n∂Ωn+1=[\n1≤i≤mn+1Ln+1\ni.\n3.5.The domain Ω.It is easy to check that the curves ∂Ωnare quasi-circles which converge in\nHausdorff distance to another quasi-circle Γ ∞asn→ ∞ . We let Ω be the domain bounded by\nthe quasi-circle Γ ∞, and we also set Ω+= Ω, Ω−=R2\\Ω+.\nRecall that, for any n≥1, the curve ∂Ωnis piecewise linear and that the angles between two\nneighboring segments Ln+1\ni,Ln+1\ni+1which compose ∂Ωnis at most 2−n−4π. From this fact, and the\nconstruction above, one can check that Ω is a vanishing Reifenberg-flat domain. Further, from\n(3.4) and the construction above, it follows easily that\n(3.8) ∂Ω⊂ Ucℓn/Mn(∂Ωn),\nfor some fixed c >0.\nRecall that T1, . . . , T tn+1are the segments which compose the piecewise curve Gn+1. Denote by\nVn+1the subfamily of segments T∈ {T1, . . . , T tn+1}such that N(T) = (1 ,0), and let\n(3.9) F=[\nn≥1[\nT∈Vn+11\n2T.\nLemma 3.2. The set Fis contained in ∂Ω.\nProof. LetT∈ Vn+1. Then T⊂Gn+1and99\n100T⊂eGn+1. Since we chose ℓn+1≤an+1\n200, it follows\neasily that9\n10T⊂∂Ωn+1, and moreover9\n10Tis covered by a subfamily of segments from Ln+1\ni\nsuch that N(Ln+1\ni) = (1 ,0). We denote this family by {Ln+1\ni}i∈In+1(T). By inspection of the\nabove construction, it can be seen that (assuming the sequence {mn}nto growth fast enough\nif necessary) at most two segments from the family {Ln+1\ni}i∈In+1(T)are not included in ∂Ωn+2.\nIterating, we see for each m > n , there is a segment Tmconcentric with Tcontained in ∂Ωmwith\nlength\nH1(Tm)≥9\n10H1(T)−2mX\nj=n+1ℓj≥1\n2H1(T).\nLetting m→ ∞ , this implies that1\n2T⊂∂Ω. □\nLemma 3.3. For any ball Bcentered in ∂Ωsuch that diam( B)≤diam( ∂Ω)there are two chord-\narc subdomains Ω±\nB⊂Ω±such that\n(3.10) H1(F∩∂Ω+\nB∩∂Ω−\nB)≥cdiam( B),\nwhere c >0is an absolute constant and the chord-arc character of Ω±\nBdoes not depend on B.COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 11\nTo prove this lemma we will use following result of Jonas Azzam [A, Theorem 6.4]:\nTheorem 3.4. LetΩ+⊂Rn+1be a two-sided NTA domain and let Ω−= (Ω)c. Let E⊂∂Ω∩Z,\nwhere Zis a uniformly n-rectifiable set. Then there are two chord-arc domain Ω±\nEsuch that\nΩ±\nE⊂Ω±, with diam( ∂Ω±\nE)≳diam( ∂Ω+), such that E⊂∂Ω+∩∂Ω+\nE∩∂Ω−\nE. The chord-arc\nparameters of Ω±\nEdepend only on the NTA parameters of Ω±and on the uniform rectifiability\nparameters of Z.\nProof of Lemma 3.3. Given a ball Bas in the lemma, let nbe such that 10 ℓn≥r(B)>10ℓn+1.\nAs shown above, each curve Γn\njisC-Ahlfors regular, for some fixed absolute constant C. Since,\nfornfixed, only a finite number of the curves Γn\njintersect 2 B, it follows that Gn∩2Bis also\nAhlfors regular (perhaps with a slightly worse constant C′). By construction, this implies that\nalsoeGn∩1.5Band∂Ωn+1∩Bare upper C′′-Ahlfors regular.\nLetHbe the arc of B∩∂Ωn+1that passes through the center of B. Then His aC′′-Ahlfors\nregular curve with diam( H)≈ H1(H)≥2r(B). Denote by {Ln+1\ni}i∈JHthe family of segments\nLn+1\ni, 1≤i≤mn+1, such that Ln+1\ni⊂H. By the choice of n, it follows easily that\n(3.11)X\ni∈JHH1(Ln+1\ni)≈r(B).\nNext, form the curve Γ Hby replacing each Ln+1\ni,i∈JH, in the arc Hby the corresponding curve\nΓn+1\ni. Since H1(Γn+1\ni)≈ H1(Ln+1\ni) and Γn+1\niis Ahlfors regular for each i, we infer that Γ His a\nC′′′-Ahlfors regular curve, where C′′′is another absolute constant.\nBy Lemmas 3.1 and (3.2), we deduce that\nH1(Γn+1\ni∩F∩∂Ω)≳H1(Ln+1\ni) for all i∈JH.\nSumming on i∈JHand using (3.11), we deduce that\nH1(ΓH∩F∩∂Ω)≳r(B).\nFinally we apply Theorem 3.4 with E=F∩BandZ= ΓHto deduce the existence of the required\nchord-arc domains Ω±\nB⊂Ω±satisfying (3.10). □\n4.The set Fhas full harmonic measure and (1.1) does not hold\nLemma 4.1. The set Fdefined in (3.9) has full harmonic measure in Ω+.\nNotice that the outer unit normal of Ω+at every ξ∈Fequals (1 ,0). From the fact that F\nhas full harmonic measure, it follows then that the outer unit normal of Ω+is constant ω+-a.e.\nin∂Ω+.\nProof of Lemma 4.1. By standard arguments, it suffices to prove that there is a constant c >0\nsuch that for any ball Bcentered in ∂Ω+such that diam( B)≤diam( ∂Ω+), it holds\nω+(F∩B)≥c ω+(B).\nHere we assume that the pole for harmonic measure is a point p∈Ω such that dist( p, ∂Ω)≳\ndiam(Ω+).\nLet Ω±\nBbe as in Lemma 3.3 and let pB∈B∩Ω+\nBbe such that\ndist(pB, ∂Ω+\nB)≳diam( B),12 XAVIER TOLSA\nso that pBis also a corkscrew point for Bwith respect to Ω+. By the change of pole formula in\nTheorem 2.1, it suffices to show that\nω+,pB(F∩B)≳1.\nBy (3.10), we know that\nH1(F∩∂Ω+\nB)≥cdiam( B)≈ H1(∂Ω+\nB∩B).\nSince the harmonic measure ωpB\nΩ+\nBis an A∞weight with respect to H1|∂Ω+\nB, we deduce that\nωpB\nΩ+\nB(F∩B)≳1.\nThen, by the maximum principle it holds\nω+,pB(F∩B)≥ωpB\nΩ+\nB(F∩B)≳1,\nas wished. □\nTo complete the proof of Theorem 1.1 it only remains to show that the condition (1.1) does\nnot hold. To this end, it suffices to check that there are arbitrarily small balls Bcentered in ∂Ω+\nsuch that\n(4.1) |NB−(1,0)| ≥c2,\nfor some fixed c2>0.\nFor simplicity, suppose that one of the initial segments L0\njfrom Ω 0is horizontal and satisfies\nN(L0\nj) = (0 ,1). By construction, the segment with coding S1,...,1(with M2\n01’s) is also horizontal\nand its associated outer normal is (0 ,1). So one of the segments Ti, 1≤i≤t1, from the curve\nG1is also horizontal and N(Ti) = (0 ,1). By the smallness of the parameter a1defined in (3.7)\nand the fact that ℓ1≤a1/100, it follows easily that there exists at least one segment L1\nkfrom\n∂Ω1contained in Ti, so that N(L1\nk) = (0 ,1). Iterating, we deduce that for any n≥1, there is a\nhorizontal segment Ln\nknfrom ∂Ωnsuch that N(Ln\nkn) = (0 ,1).\nLetBnbe a ball centered in the mid-point of Ln\nknwith radius rn:=H1(Ln\nkn)/4. From (3.8), we\ndeduce that ∂Ω∩1\n2Bn̸=∅and∂Ω∩Bn⊂ U rn/100(Ln\nkn) for nbig enough. Now let eBn⊂Bnbe\na ball centered in ∂Ω such that diam( eBn)≥9\n10diam( Bn). Then the vector NeBnis close to being\nvertical and thus |N(eBn)−(1,0)|≳1. Hence there are arbitrary small balls satisfying (4.1).\nReferences\n[A] J. Azzam. Sets of absolute continuity for harmonic measure in NTA domains . Potential Anal. 45 (2016),\nno. 3, 403–433. 11\n[AM] J. Azzam and M. Mourgoglou. Tangent measures of elliptic measure and applications. Anal. PDE 12\n(2019), no. 8, 1891–1941. 1\n[AMT1] J. Azzam, M. Mourgoglou and X. Tolsa. Mutual absolute continuity of interior and exterior harmonic\nmeasure implies rectifiability. Comm. Pure Appl. Math. 71 (2017), no. 11, 2121–2163. 1, 2\n[AMT2] J. Azzam, M. Mourgoglou and X. Tolsa. A two-phase free boundary problem for harmonic measure and\nuniform rectifiability. Trans. Amer. Math. Soc. 373 (2020), no. 6, 4359–4388. 1, 2, 3\n[AMTV] J. Azzam, M. Mourgoglou, X. Tolsa, and A. Volberg. On a two-phase problem for harmonic measure\nin general domains. Amer. J. Math. 141 (2019), no. 5, 1259–1279. 1\n[BET1] M. Badger, M. Engelstein, and T. Toro. Structure of sets which are well approximated by zero sets of\nharmonic polynomials. Anal. PDE 10 (2017), no. 6, 1455–1495. 1COUNTEREXAMPLE FOR TWO-PHASE PROBLEM FOR HARMONIC MEASURE 13\n[BET2] M. Badger, M. Engelstein, and T. Toro. Regularity of the singular set in a two-phase problem for\nharmonic measure with H¨ older data. Rev. Mat. Iberoam. 36 (2020), no. 5, 1375–1408. 1\n[En] M. Engelstein. A two-phase free boundary problem for harmonic measure. Ann. Sci. ´Ec. Norm. Sup´ er.\n(4) 49 (2016), no. 4, 859–905. 1\n[JK] D. S. Jerison and C. E. Kenig. Boundary behavior of harmonic functions in nontangentially accessible\ndomains , Adv. in Math. 46 (1982), no. 1, 80–147. 3, 4\n[KPT] C. Kenig, D. Preiss, and T. Toro. Boundary structure and size in terms of interior and exterior harmonic\nmeasures in higher dimensions. J. Amer. Math. Soc. 22 (2009), no. 3, 771–796. 1\n[KT1] C.E. Kenig and T. Toro. Harmonic measure on locally flat domains . Duke Math. J. 87 (1997), no. 3,\n509–551. 2, 4\n[KT2] C. Kenig and T. Toro. Free boundary regularity for harmonic measures and Poisson kernels, Ann. of\nMath. 150 (1999) 369–454. 2\n[KT3] C. Kenig and T. Toro. Free boundary regularity below the continuous threshold: 2 phase problems . J.\nReine Angew Math. 596 (2006), 1–44. 1\n[KT4] C. Kenig and T. Toro. Poisson kernel characterization of Reifenberg flat chord arc domains , Ann. Sci.\nEcole Norm. Sup. (4) 36 (2003), no.3, 323–401. 2\n[PT] M. Prats and X. Tolsa. The two-phase problem for harmonic measure in VMO. Calc. Var. Partial\nDifferential Equations 59 (2020), no. 3, Paper No. 102, 58 pp. 1, 2\n[TT] X. Tolsa and T. Toro. The two-phase problem for harmonic measure in VMO and the chord-arc condi-\ntion. Preprint arXiv:2209.14346v1 (2022). 1, 2\nICREA, Barcelona, Dept. de Matem `atiques, Universitat Aut `onoma de Barcelona, and Centre de\nRecerca Matem `atica, Barcelona, Catalonia.\nEmail address :xavier.tolsa@uab.cat" }, { "title": "2401.14623v1.Structure_in_Communication_Complexity_and_Constant_Cost_Complexity_Classes.pdf", "content": "SIGACT News Complexity Theory Column, March 2024\nStructure in Communication Complexity and Constant-Cost\nComplexity Classes\nHamed Hatami1Pooya Hatami2\nAbstract\nSeveral theorems and conjectures in communication complexity state or speculate that the\ncomplexity of a matrix in a given communication model is controlled by a related analytic\nor algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction\nis typically easy as the structural implications of small complexity often imply a bound on\nsome matrix parameter. The challenge lies in establishing the reverse direction, which requires\nunderstanding the structure of Boolean matrices for which a given matrix parameter is small or\nlarge. We will discuss several research directions that align with this overarching theme.\n1 Introduction\nIn 1979, Yao [Yao79] introduced an abstract model for analyzing communication. It quickly became\napparent that the applications of this elegant paradigm go far beyond the concept of communica-\ntion. Many results in communication complexity have equivalent formulations in other fields that\nare equally natural, and the techniques developed within this framework have proven to be powerful\ntools applicable across various domains. Today, communication complexity is a vibrant research\narea with many connections across theoretical computer science and mathematics: in learning the-\nory, circuit design, pseudorandomness, data streaming, data structures, computational complexity,\ncomputer networks, time-space trade-offs, discrepancy theory, and property testing.\nIn this article, we focus on the most standard framework where a communication problem\nis simply a Boolean matrix. Formally, there are two parties, often called Alice and Bob, and a\ncommunication problem is defined by a matrix F∈ {0,1}X×Y. Alice receives a row index x∈ X,\nand Bob receives a column index y∈ Y. Together, they should compute the entry F(x, y) by\nexchanging bits of information according to a previously agreed-on protocol tailored to F. There is\nno restriction on their computational power; the only measure we care to minimize is the number\nof exchanged bits.\nMany questions in communication complexity concern the basic structural properties of Boolean\nmatrices and are relevant to any field that requires an in-depth analysis of them. Mathematicians,\n1McGill University, Montreal, QC, Canada . hatami@cs.mcgill.ca . Supported by an NSERC grant.\n2The Ohio State University, Columbus, OH, USA. pooyahat@gmail.com . Supported by NSF grant CCF-1947546.\n1arXiv:2401.14623v1 [cs.CC] 26 Jan 2024of course, have studied matrices for centuries through the lenses of linear algebra, geometry, and\nanalysis. They have produced an extensive collection of tools and theories that apply to any field\ndealing with these mathematical objects. However, the assumption of Booleanity introduces a novel\nangle and unveils new problems and challenges.\nConsider the example of rank. Elementary linear algebra provides several satisfactory structural\ndescriptions of small-rank matrices. For example, one could construct a small-rank matrix by\nsumming a few rank-one matrices. In contrast, the structure of small-rank Boolean matrices is the\nsubject of the most well-known conjecture in communication complexity, the log-rank conjecture .\nThere are many theorems and conjectures in communication complexity that fall into a similar\nparadigm as the log-rank conjecture: They state or speculate that the complexity of a matrix in\na given communication model is essentially determined by a related analytic or algebraic matrix\nparameter, e.g., rank, sign-rank, discrepancy, trace norm, approximate trace norm, γ2-factorization\nnorm, approximate factorization norm. The forward direction is typically easy as the structural\nimplications of small complexity often imply a bound on some matrix parameter. The challenge\nlies in establishing the reverse direction, which requires understanding the structure of Boolean\nmatrices for which a given matrix parameter is small or large.\nIn this article, we discuss some new research directions and open problems, as well as some\nclassical ones, that align with this overarching theme.\nNotation: All logarithms are in base 2. Sometimes, we use a≲bto denote a=O(b). Let Im\ndenote the m×midentity matrix and JX×Y denote the X × Y all-1 matrix. For a positive integer\nk, we denote [ k] ={1, . . . , k }. We often identify a Boolean matrix FX×Y with the corresponding\nfunction F:X × Y → { 0,1}defined as F: (x, y)7→F(x, y). We define the complement of a\ncommunication problem F=FX×Y as¬F=JX×Y−F. For two X × Y matrices AandB, we\ndenote their entry-wise product (i.e. Schur product ) byA◦B.\nCommunication complexity classes: We measure the communication complexity of a matrix\nFX×Y in relation to the number of input bits n(F):=⌈log max( |X|,|Y|)⌉. We often consider\n2n×2nmatrices Fn:{0,1}n× {0,1}n→ {0,1}, where the inputs of Alice and Bob are n-bit\nstrings. Moreover, similar to computational complexity, the goal is to understand the asymptotic\ncomplexity, and therefore, a communication problem typically refers to an infinite family of Boolean\nmatrices rather than a single matrix.\nFor example, Equality is the family of matrices EQn:{0,1}n×{0,1}n→ {0,1}with EQ(x, y) =\n1 iffx=y. Equivalently, EQnis the 2n×2nidentity matrix, where the rows and columns are labelled\nwith n-bit strings.\nIn the theory of Turing machines, a polynomial complexity is considered efficient, but this is not\na suitable criterion for communication since even in the deterministic model, communication com-\nplexity is at most n+ 1. In an influential paper, Babai, Frankl, and Simon [BFS86] proposed poly-\nlogarithmic complexity as the criteria for efficiency and used it to define the communication classes\nPcc,NPcc,RPcc,BPPcc,PPcc,UPPcc, analogous to the classical computational complexity classes.\nThe definition of communication classes by [BFS86] provides a formal paradigm to compare the\npower of different communication models.\nSince in this article, we only study communication classes, to make the notation less cumber-\nsome, from this point on, we will drop the superscript ccand denote these communication classes\nsimply as P,NP,RP,BPP,PP,UPP.\n2Constant-cost communication classes: For many communication problems, it may not be\ncompletely natural to allow the communication complexity to depend on the matrix size. For\nexample, consider the geometric problem where Alice receives a point x∈R2, and Bob receives\na closed half-space H⊆R2, and they wish to know whether x∈H. Alternatively, consider the\nproblem where Alice and Bob receive points x, y∈R2, respectively, and they wish to know whether\nthe distance between these points is at most 1. In these examples, the number of possible inputs\nis infinite, and even if we artificially restrict the number of possible inputs to a finite set, the most\nnatural way to represent the inputs is by vectors in R2. Therefore, it is more natural to consider\ncommunication protocols that solve the problem (in a certain communication model or under some\npromised guarantees about the inputs) using a bounded number of communicated bits, independent\nof the number of possible inputs.\nFurthermore, many parameters studied in communication complexity are also relevant to other\nareas of theoretical computer science, where the Boolean matrices are either infinite or their size\nis of lesser significance than other features. For example, in learning theory, notions such as VC\ndimension, Littlestone dimension, margin complexity, and Threshold dimension of binary concept\nclasses are of primary interest rather than the class size. Viewing binary concept classes as Boolean\nmatrices, these parameters are directly related to notions such as stability, discrepancy, and sign-\nrank that play a central role in communication complexity.\nThe definition of communication classes by [BFS86], no doubt, is natural and has successfully\nled to many fruitful lines of research seeking to prove separations between different communication\nclasses. However, if one wishes to view communication complexity in these broader contexts, it be-\ncomes essential also to analyze communication problems that have uniformly bounded complexity.\nIndeed, there is already a rapidly growing body of work dedicated to studying constant-cost commu-\nnication [LMSS07, LS09a, BCK14, HHL20, Har20, HWZ22, HHP+22, EHK22, HHH23, CHHS23,\nACHS23, HZ24].\nThese considerations motivate the definition of the constant-cost analogues of the communica-\ntion classes of [BFS86]. Here, the criterion of effectiveness is a O(1) complexity, independent of\nthe input size n. We will denote these classes by adding a 0 subscript to the corresponding poly-\nlogarithmic class, where 0 refers to the power din the polylogarithmic complexity O(logd(n)). For\nexample, a family of Boolean matrices Fnis in P0if their deterministic communication complexity\nis uniformly bounded by a constant C=O(1).\nWe will formally define and discuss the classes P0,NP0,RP0,BPP 0,PP0,UPP 0later, but for now,\nlet us mention that, unlike in the poly-logarithmic communication classes where P⊊RP⊊NPand\nBPP⊊PP, here we have P0=NP0⊊RP0andBPP 0=PP0. We also believe that BPP 0̸⊆UPP 0,\nwhile Newman’s lemma [New91] implies that BPP⊆UPP.\nMonochromatic rectangles: Amonochromatic rectangle in a Boolean matrix FX×Y refers to\na submatrix FS×T, where all entries share the same value. Monochromatic rectangles are the\nbuilding blocks of every communication protocol, and many results and problems in communication\ncomplexity focus on describing the structure of Boolean matrices in terms of these rectangles or\nshowing the existence of large monochromatic rectangles in them.\nDefine the rectangle ratio of a Boolean matrix FX×Y as\nrect(F):= max\nR|R|\n|X × Y|,\nwhere the maximum is over monochromatic rectangles in F.\n3Concerning communication complexity, the parameter rect( ·) has a few drawbacks: While\nadding new rows and columns to a matrix can only increase the communication complexity, it\ncan deteriorate rect( F) by creating large monochromatic rectangles. Moreover, repeating a row or\ncolumn of Fdoes not change its communication complexity, but it can affect rect( F). Therefore,\nit is more natural to consider the following version of rect that considers weighted monochromatic\nrectangles, first studied by Impagliazzo and Williams [IW10].\nDefinition 1.1 (Weighted Rectangle Ratio) .For every Boolean matrix F, define\nwrect( F):= inf\nµmax\nRµ(R),\nwhere the infimum is over all product probability measures µ=µX×µYonX × Y , and the\nmaximum is taken over all monochromatic rectangles in F.\nBoth rect( ·) and wrect( ·) are natural parameters that measure the existence of structure in\nBoolean matrices. As we will discuss throughout the paper, they play a crucial role in several\napplications in communication complexity, and some basic and consequential questions regarding\nthese quantities remain open.\n2 Deterministic communication complexity and rank\nAdeterministic communication protocol is defined by a binary tree, where every internal node v\nspecifies which player speaks at that node and what bit they must send. For example, if an internal\nnode vis associated with Alice, it is labelled with a function av:X → { 0,1}, which prescribes the\nbit sent by Alice at this node. After this bit is sent, the players move to the corresponding child\nofv: they move to the left child if the bit is 0 and to the right child if the bit is 1. They continue\nthis process until they reach a leaf. Every leaf is labelled with 0 or 1, which corresponds to the\noutput of the protocol. The cost of the protocol is the height of the tree, which is equal to the\nmaximum number of bits exchanged on any input. The deterministic communication complexity\nofF, denoted by D(F), is the smallest cost of a protocol that computes Fcorrectly on all inputs.\nLetPandP0be, respectively, the class of problems with poly-logarithmic and O(1) deterministic\ncommunication complexity.\nThe fact that the bit communicated at every node depends only on one of the inputs implies\nthe most useful fact in communication complexity: for every node v, the set of all inputs that lead\nthe protocol to vis acombinatorial rectangle , which is a product set Rv=Sv×Tvwith Sv⊆ X\nandTv⊆ Y.\nFor every b∈ {0,1}, a combinatorial rectangle Ris called a b-monochromatic rectangle if\nF(x, y) =bfor every entry ( x, y)∈R. Note that for every b-labelled leaf ℓ, the set Rℓmust be a\nb-monochromatic rectangle. Therefore, the leaves of a deterministic protocol partition the matrix\ninto monochromatic rectangles.\nA Boolean matrix Fwith small D(F) is highly structured, as it consists of at most 2D(F)\nmonochromatic rectangles. This partition of Finto at most 2D(F)disjoint monochromatic rectangles\nimplies\nD(F)≥log(1/rect(F)). (1)\nMoreover, since the real rank of a monochromatic rectangle is at most 1, we have rk( F)≤2D(F),\nwhere rk( F) denotes the real rank of F. Combined with the easy fact D(F)≤rk(F) + 1, we have\nlog rk( F)≤D(F)≤rk(F) + 1. (2)\n4Therefore, P0coincides with the class of problems with bounded rank. However, the exponential\ngap between the lower bound and the upper bound is too wide to provide a characterization of P\nin terms of rank. The log-rank conjecture speculates that the lower bound is essentially sharp and\nD(F) and log rk( F) are polynomially equivalent.\nConjecture 1 (The log-rank conjecture [LS88]) .There exists a universal constant C >0such that\nfor every Boolean matrix F,\nD(F)≤C(log rk( F))C.\nThe log-rank conjecture, if true, provides a structural description for Boolean matrices of small\nrank. Namely, they must have the same tree-like partition into monochromatic rectangles as ma-\ntrices with small deterministic communication complexity. Nisan and Widgerson [NW95] showed\nthat to prove the log-rank conjecture, it suffices to show the existence of one large monochromatic\nrectangle.\nConjecture 2 (The log-rank conjecture - equivalent formulation) .There exists a universal constant\nC >0such that every Boolean matrix Fsatisfies\nrect−1(F)≤ClogCrk(F).\nTo date, the strongest known bound is still exponentially far from that in Conjecture 1. Con-\ncretely, Lovett [Lov16] showed that D(f)≤O(p\nrk(f) log rk( f)), which was recently improved by\nSudakov and Tomon [ST23] to O(p\nrk(f)). On the lower bound side, a construction of [GPW18a]\nshows that the constant Ccannot be strictly less than 2. We refer to surveys [Lov14, LS23] for a\ndetailed discussion of the log-rank conjecture.\n3 Equality oracles and γ2-Factorization norm\nThe results discussed in this section mostly stem from joint work by Lianna Hambardzumyan and\nthe authors [HHH23]. Regarding communication complexity, we will focus on the deterministic\nmodel with access to an Equality oracle. We will show that this model has compelling ties to\nharmonic analysis and operator theory. In particular, we will discuss a conjecture in communication\ncomplexity with profound links to the characterization of the idempotents of the algebra of Schur\nmultipliers and Cohen’s celebrated idempotent theorem, a well-known and notable theorem in\nharmonic analysis.\n3.1 An analytic version of the rank lower-bound\nThe rank lower bound on D(F) follows from the observation that every Boolean matrix Fcan be\nwritten as F=P2D(F)\ni=1Bi, where each Biis a rank-one Boolean matrix. If instead of counting the\nnumber of rank-one Boolean matrices in this sum, we focus on the sum of the coefficients, we can\nestablish an alternative lower bound on D(F)—a lower bound with an analytical flavour.\nDefine the µ-norm of a realmatrix Aas\n∥A∥µ= inf\nλi,Bi(X\ni|λi|:A=X\niλiBi)\n,\n5where each Biis a rank-one Boolean matrix and each λiis a real number. Note that ∥ · ∥µsatisfies\nall the axioms of a norm, and we have\nlog∥F∥µ≤D(F). (3)\nTheµ-norm is not well-known in functional analysis, and its definition is tailored to its purpose\nas a lower bound in communication complexity. Fortunately, the Grothendieck inequality shows\nthat the µ-norm is equivalent to the well-studied γ2-factorization norm.\nDefinition 3.1. Theγ2-norm of a real matrix AX×Y, denoted by ∥A∥γ2, is the inifimum of c\nsuch that there exists a positive integer dand vectors ux, vy∈Rdforx∈ X andy∈ Y with\n⟨ux, yv⟩=A(x, y)and∥ux∥2· ∥vy∥2≤cfor all x, y.\nA key property of the γ2-norm is that for every two X × Y real matrices A1andA2, we have\n∥A1◦A2∥γ2≤ ∥A1∥γ2∥A2∥γ2, (4)\nwhere we recall that A1◦A2is the entry-wise product of A1andA2.\nThe Grothendieck inequality implies that for every real matrix A, we have\n∥A∥γ2≤ ∥A∥µ≤4K∥A∥γ2,\nwhere K≤π\n2 ln(1+√\n2)≈1.7822 is the so-called Grothendieck constant for real numbers. In light of\nthis equivalence, we will rephrase the lower bound Equation (3) in terms of the γ2-norm:\nlog∥F∥γ2≤D(F). (5)\nIf we compare this lower bound to Equation (2), it is natural to wonder whether we can bound\nD(F) from above by a function of ∥F∥γ2. The answer is negative. The γ2-norm of every identity\nmatrix is 1, but identity matrices can be of arbitrarily large rank and, therefore, of arbitrarily large\ndeterministic communication complexity.\nProposition 3.2. For every m, the γ2-norm of the m×midentity matrix Imis1,\nProof. Note that since the standard basis e1, . . . , e m∈Rmsatisfies\n⟨ei, ej⟩=®\n1i=j\n0i̸=j,\nby Definition 3.1, Imsatisfies ∥Im∥γ2≤1. Moreover, it is clear from the definition of the γ2-norm\nthat for every matrix A, we have ∥A∥γ2≥ ∥A∥∞:= max x,y|A(x, y)|, and therefore, ∥Im∥γ2≥1.\nWhat are the Boolean matrices with small γ2-norm then? It follows from ∥A∥γ2≥ ∥A∥∞\nthat the γ2-norm of every non-zero Boolean matrix is at least 1. Let us first study the Boolean\nmatrices whose γ2-norm is exactly 1.\nProposition 3.2 shows that all identity matrices have γ2-norm 1. Note that the proof of Propo-\nsition 3.2 generalizes to a larger class of matrices, which we call blocky matrices .\nDefinition 3.3 (Blocky matrices) .A Boolean matrix FX×Y isblocky if there exist disjoint sets\nXi⊆ X and disjoint sets Yi⊆ Y such that the support of Fis exactlyS\niXi× Yi.\n6LetBlocky denote the set of all blocky matrices. It turns out that Blocky is precisely the set of\nBoolean matrices with γ2-norm 1.\nProposition 3.4 (Livshits [Liv95]) .A Boolean matrix Fsatisfies ∥F∥γ2= 1iffF∈Blocky .\nProof. It is observed in [Liv95] that\n\r\r\r\r\r\r\n1 1\n0 1\n\r\r\r\r\r\r\nγ2=2√\n3>1.\nSince ∥ · ∥ γ2norm is invariant under row and column permutations, a Boolean matrix Fwith\n∥F∥γ2= 1 cannot have any 2 ×2 submatrices with exactly 3 ones. It is straightforward to verify\nthat a Boolean matrix satisfying this property must be blocky.\nSince every Boolean matrix with γ2-norm 1 is blocky, it is natural to ask whether Boolean\nmatrices of bounded γ2-norm can be characterized through blocky matrices.\nWe first show that if Fis generated by entry-wise operations from a few blocky matrices, it\nmust have a small γ2-norm.\nProposition 3.5. Consider X × Y blocky matrices B1, . . . , B rand a combining function Γ :\n{0,1}r→ {0,1}. The Boolean matrix FX×Y defined as\nF(x, y):= Γ(B1(x, y), . . . , B r(x, y)) (6)\nsatisfies ∥F∥γ2≤3r.\nProof. We prove the statement by induction on r. The base case r= 0 is trivial. Next, consider\na Boolean matrix Fsatisfying Equation (6), and write F= (B1◦F1) + ( J−B1)◦F2, where the\nentries of F1andF2depend only on B2, . . . , B randJis the all-1 matrix. Note that Jis a blocky\nmatrix and satisfies ∥J∥γ2= 1. We have\n∥F∥γ2≤ ∥B1◦F1∥γ2+∥(J−B1)◦F2∥γ2\n≤ ∥F1∥γ2+ (∥J∥γ2+∥B1∥γ2)∥F2∥γ2≤3r−1+ 2·3r−1= 3r.\nIn [HHH23], we conjectured that Boolean matrices of small γ2-norm are precisely those of the\nform Equation (6).\nConjecture 3. Suppose that Fis a Boolean matrix with ∥F∥γ2≤c. Then we may write\nF=LX\ni=1±Bi, (7)\nwhere Biare blocky matrices and L≤ℓ(c)for some integer ℓ(c)depending only on c.\nConjecture 3 is inspired by Cohen’s idempotent theorem, and it is known to be true for a large\nclass of Boolean matrices, including the xor-lifts of Boolean functions.\nRecall that the xor-lift of a function f:Zn\n2→ {0,1}is the matrix f⊕:Zn\n2×Zn\n2→ {0,1}\ndefined as f⊕(x, y) =f(x+y).\n7The sum of the absolute values of the Fourier coefficients of a function f:Fn\n2→Ris called the\nFourier algebra norm orspectral norm offand is denoted by\n∥f∥A:=∥bf∥1=X\nχ∈“G|bf(χ)|.\nThe following identity relating the Fourier algebra norm of fto the γ2-norm of its xor lift is due\nto [LS09b, Lemma 36].\n∥f∥A=∥f⊕∥γ2. (8)\nTherefore, in the case of xor-lifts, the assumption ∥f⊕∥γ2≤cof Conjecture 3 is equivalent ∥f∥A≤\nc. The structure of Boolean functions f:Zn\n2→ {0,1}with∥f∥A≤cis characterized by the\nquantitative version of the so-called Cohen’s idempotent theorem for Zn\n2.\nTheorem 3.6 (Quantitative Cohen’s theorem for Zn\n2[Coh60, GS08]) .IfS⊆Zn\n2satisfies ∥1S∥A≤\nc, we may write\n1S=LX\ni=1±1Hi+ai, (9)\nwhere Hi+aiare cosets and L≤ℓ(c)for some integer ℓ(c)depending only on c.\nNote that the xor-lift of every 1Hi+aiis a blocky matrix, and therefore, Theorem 3.6 verifies\nConjecture 3 for xor-lifts. In fact, these results extend to every finite group. Given a finite group\nG, we can generalize the notion of xor-lifts to group -lifts, where we define F(x, y) =f(y−1x) for\nf:G→C. The notion of algebra norm also generalizes to other finite groups. We will not give\nthe original definition of the algebra norm, but for the purposes of this paper, it suffices to know\nthat similar to Equation (8), we have ∥f∥A=∥F∥γ2. This identity was observed in [HHH23] based\non a result of Davidson and Donsig [DD07].\nTheorem 3.7 (Quantitative Cohen’s theorem for finite groups [San11]) .There exists a function\nℓ:N→Nsuch that the following holds. For every finite group Gand every S⊆G, ifFG×G(x, y) =\n1S(y−1x)satisfies ∥F∥γ2≤c, we may write\n1S=LX\ni=1±1Hiai, (10)\nwhere Hiaiare cosets and L≤ℓ(c). In particular, F=PL\ni=1±Bi, where Biare blocky matrices.\nTheorem 3.7 verifies Conjecture 3 for the adjacency matrix of every Cayley (directed) graph.\nOn the other hand, for general Boolean matrices, it is not even known whether ∥F∥γ2≤cimplies\nrect(F)≥κ(c) for some κ(c)>0, which would be an easy consequence of Conjecture 3.\nRegarding the bound in Theorem 3.7, for general finite groups, one can take ℓ(c) =A(6, O(c)),\nwhere Ais the Ackerman function [San11]. For the case of finite Abelian groups, a better bound\nofℓ(c) = 2O(c4polylog( c))is due to [San20].\nIn the special case of Zn\n2which corresponds to the xor-lifts, the best bound [San19] that\nappears in the literature is ℓ(c)≤2O(c3polylog( c)). However, recently, Gowers, Green, Manners,\nand Tao [GGMT23] announced a proof for Morton’s conjecture (aka polynomial Freiman–Ruzsa\nconjecture). Substituting this result in Sanders proof for [San19, Proposition 2] shows that in the\ncase of Zn\n2, one may take ℓ(c) = 2O(cpolylog( c)).\n83.2 Equality Oracle Protocols\nEquality is the canonical problem with the strongest possible separation between deterministic\nand randomized communication complexities. We have D(EQn) =n+1, which is the largest possible\nvalue for any n-bit communication problem. On the other hand, as we will discuss in Section 5.1,\nthe randomized communication complexity of EQnis only O(1).\nWe know that the deterministic model cannot solve Equality efficiently. What if we augment\nthe model with an equality oracle? Does this result in a significantly stronger model? Can this\nmodel efficiently solve every problem with small randomized communication complexity?\nFormally, in the deterministic communication model with access to an Equality oracle, a\nprotocol for a Boolean matrix FX×Y corresponds to a binary tree. Each non-leaf node vin the tree\nis labelled with two functions av:X → { 0,1}mandbv:Y → { 0,1}mfor some m. On this node, the\nplayers map their inputs to strings av(x) and bv(y), respectively, and the oracle will broadcast the\nvalue of EQm(av(x), bv(y)) to both players. This will contribute only 1 to the cost of the protocol.\nNote that the oracle queries can simulate sending one-bit messages from each party to the other\none. For example, if it is Alice’s turn to send a bit a, the query EQ1(a,1) can transmit it to Bob.\nHence, in this model, we can assume that all the communication is through oracle queries.\nLetDEQ(F) denote the smallest cost of a deterministic protocol with equality oracle for the\nmatrix F, and define PEQandPEQ\n0to be, respectively, the class of problems with poly-logarithmic\nand constant communication costs in this model.\nIn the same way that combinatorial rectangles are the building blocks of deterministic com-\nmunication protocols, blocky matrices serve as the foundational components of equality oracle\nprotocols. Indeed, every node vof an equality oracle protocol for computing F(x, y) corresponds\ntoBv(x, y) =EQ(av(x), bv(y)) where Bvis a blocky matrix.\nEquation (2) characterizes P0as the set of problems with O(1) rank. Can we obtain a similar\ncharacterization for PEQ\n0via the blocky matrices? To this end, let us define a notion of rank based\non blocky matrices.\nDefinition 3.8 (Blocky Rank) .The blocky rank of a real matrix A, denoted rkBlocky(A), is the\nsmallest integer rsuch that Ais a real linear combination of rblocky matrices.\nBlocky rank has interesting connections to circuit and communication complexity theory [AY22,\nHHH23]. The following proposition shows analogous bounds to Equation (5) on DEQ(F), and implies\nthat a matrix family {Fn}is in PEQ\n0iff rk Blocky(Fn) =O(1).\nProposition 3.9 ([HHH23]) .For every Boolean matrix FX×Y, we have\n1\n2log rk Blocky(F)≤DEQ(F)≤rkBlocky(F)\nand\nlog∥F∥γ2≤2·DEQ(F). (11)\nProof. We first prove DEQ(F)≤rkBlocky(F). Let k= rk Blocky(F). We construct an EQ-oracle protocol\nforF. In advance, Alice and Bob agree on a decomposition F=Pk\ni=1λiBi, where Biis a blocky\nmatrix and λi∈Rfori∈[k]. Since each blocky matrix Bicorresponds to an EQquery, for an input\n(x, y), Alice and Bob make kqueries to the oracle to determine F(x, y).\nFor the lower bounds, let d=DEQ(M). Consider a leaf ℓin the EQ-oracle protocol tree computing\nFand let Pℓdenote the path of length kℓ≤dfrom the root to ℓ. Note that each non-leaf node\n9vin the tree corresponds to a query to the equality oracle, and each such query corresponds to a\nblocky matrix Bv. Define B1\nv=BvandB0\nv=¬Bv=JX×Y−Bv.\nSuppose Pℓ=v1, v2, . . . , v kℓ, ℓ, and consider the matrix\nFPℓ:=Bσv1v1◦Bσv2v2◦. . .◦Bσvkℓvkℓ,\nwhere σvi∈ {0,1}andσvi= 1 iff the edge ( vi−1, vi) is labeled by 1. Hence, after simplification,\nFPℓcan be written as a sum of at most 2dsummands with ±1 coefficients, where each summand\nis a Schur product of at most kℓblocky matrices. Observe that the Schur product of two blocky\nmatrices is a blocky matrix. Thus, FPℓis a sum of at most 2dblocky matrices with ±1 coefficients.\nSumming over all the leaves that are labelled by 1, we get F=P\nℓis a 1-leaf FPℓ. As the number\nof leaves is bounded by 2d, and each FPℓis a±1 linear combination of at most 2dblocky matrices,\nwe have rk Blocky(F)≤22dand∥F∥γ2≤22d.\n3.3 Analogue of the log-rank conjecture for blocky rank is false\nThe log-rank conjecture speculates that the deterministic communication complexity is polynomi-\nally equivalent to the logarithm of the rank. In light of Proposition 3.9 it is natural to ask a similar\nquestion for DEQand log rk Blocky. Arkadev Chattopadhyay3observed that the recent counter-example\nto the so-called log-approximate-rank conjecture by Chattopadhyay, Mande, and Sherif [CMS19]\nimplies that the answer is negative.\nRecall that a node in a directed graph is called a sink if all of its adjacent edges are incoming.\nDefine a function SINK m:{0,1}(m\n2)→ {0,1}where the input of length\u0000m\n2\u0001specifies the orientation\nof the edges of the complete graph on mvertices. The function outputs 1 if there is a vertex that\nis a sink in the given orientation of edges and 0 otherwise.\nFixm, and for i∈[m], define ψi:{0,1}(m\n2)→ {0,1}to be the indicator function of whether i\nis a sink in the orientation given by ψi. Note that\nψi(x) = 1⇔xj,i= 1∀j̸=i, (12)\nwhere xj,i= 1 indicates that the edge between iandjis oriented towards i. Since no orientation\nof the complete graph has more than one sink, we have\nSINK m(x) =mX\ni=1ψi(x).\nWe will consider the family of xor-lifts of sink functions. Recall that the xor-lift of a function\nf:{0,1}n→Risf⊕:{0,1}n× {0,1}n→Rwith f⊕(x, y):=f(x⊕y). We have\nSINK⊕\nm=mX\ni=1ψ⊕\ni.\nIt follows from Equation (12) that each ψ⊕\niis a blocky matrix, and therefore rk Blocky(SINK⊕\nm)≤m.\nOn the other hand, Chattopadhyay, Mande, and Sherif [CMS19] prove that the R(SINK⊕\nm) = Θ( m).\nSince Rprovides a lower bound on DEQ(see Equation (15)), we obtain the following theorem.\nTheorem 3.10 (Chattopadhyay, Mande, and Sherif [CMS19]) .For the family of Boolean matrices\nFm=SINK⊕\nm, we have DEQ(Fm) =eΩ(m)andrkBlocky(Fm)≤m.\n3private communication, no pun intended!\n103.4 Blocky matrices and Idempotents of Schur Multipliers\nLetXandYbe countable sets, and let B(Y,X) denote the space of bounded linear operators\nA:ℓ2(Y)→ℓ2(X) endowed with the operator norm:\n∥A∥= sup\nx∈ℓ2(Y):∥x∥2=1∥Ax∥2.\nA matrix MX×Y is called a Schur multiplier if, for every A∈B(Y,X), we have M◦A∈B(Y,X).\nIn other words, ∥M◦A∥<∞for every A=AX×Ywith∥A∥<∞. Note that Schur multipliers form\nan algebra with addition and Schur product: If M1andM2are Schur multipliers, then M1+M2\nandM1◦M2are both Schur multipliers.\nEvery Schur multiplier Mdefines a map B(Y,X)→B(Y,X) via A7→M◦A, which assigns an\noperator norm to it:\n∥M∥m:=∥M∥B(Y,X)→B(Y,X)= sup\nA∈B(Y,X)\n∥A∥=1∥M◦A∥.\nNote that ∥ · ∥mis an algebra norm as for every M1andM2, we have\n∥M1◦M2∥m≤ ∥M1∥m∥M2∥m.\nIn other words, the algebra of Schur multipliers endowed with the norm ∥·∥mis a Banach algebra.\nA classical result, due to Grothendieck, shows that the multiplier norm coincides with the γ2-norm.\nProposition 3.11 (See [Pis96, Theorem 5.1]) .For every matrix A, we have ∥A∥m=∥A∥γ2.\nAn element aof a Banach algebra is said to be an idempotent (aka projection ) ifa2=a. The\nfollowing question arises naturally.\nWhat are the idempotents of the algebra of Schur multipliers?\nEvery idempotent Fof this algebra must satisfy F=F◦Fand, therefore, is a Boolean matrix.\nHowever, not every (infinite) Boolean matrix is a bounded Schur multiplier, as it is possible to\nhave∥F∥m=∞for a Boolean matrix F. Proposition 3.4 shows that blocky matrices are precisely\nthe set of all contractive idempotents. In other words, an idempotent Schur multiplier satisfies\n∥F∥m≤1 iff it is a blocky matrix.\nQuestion 4. Are the idempotent Schur multipliers precisely those Boolean matrices that can be\nwritten as a ±1-linear combination of finitely many contractive idempotents (equivalently, blocky\nmatrices)?\nA simple compactness argument, as outlined in [HHH23], shows that this problem is equivalent\nto Conjecture 3. Therefore, a positive to Conjecture 3 would characterize idempotents of Schur mul-\ntipliers, analogous to Cohen’s [Coh60] characterization of the idempotents of the Fourier–Stieltjes\nalgebra.\n114 Nondeterministic Model and PNP\nIn a nondeterministic protocol πfor a problem FX×Y, the parties receive a shared advice string a\nand use it in a standard deterministic protocol πa. We say that a protocol computes Fif\nF(x, y) = 1⇔ ∃a, πa(x, y) = 1 .\nThe cost of the protocol is the bit-length of aplus the maximum cost of πa(x, y) over all choices of\na, x, y . The nondeterministic communication complexity of F, denoted by N(F), is the minimum\ncost of such a protocol for F. A matrix family is in the class NPif they have poly-logarithmic\nnondeterministic communication complexity.\nUnlike in the Turing-Machine complexity, in the communication framework, it is known that\nP=NP∩coNP , which follows from D(F) =O(N(F)·N(¬F)); see [KN97, Theorem 2.11]. How-\never, nondeterministic protocols are provably more powerful than deterministic ones, as can be\ndemonstrated by the important example of the set intersection problem.\nTheSet-Int problem, INTn, is defined by INTn(x, y) = 1 if there exists a coordinate isuch that\nxi=yi= 1. Since the players can use their nondeterminism to guess the intersecting coordinate\ni, we have N(INTn) =O(logn). However, it is easy to see that D(INTn) =n+ 1. In fact, [BFS86]\nalready in the 1980s proved that Set-Int does not belong to PEQ.\nThe structural properties of NP:The nondeterministic communication complexity of a prob-\nlem is fully captured by its monochromatic rectangle covering number. Let C1(F) denote the\nminimum number of 1-monochromatic rectangles required to cover the 1 entries of F. It is easy to\nsee [KN97] that\nN(F) = log\u0000C1(F)\u0001+O(1). (13)\nCombined with D(F) =O(C1(F)), we have\nD(F)≤O(2N(F)). (14)\nTherefore, P0=NP0=coNP 0. The following proposition shows that nondeterministic protocols,\nwhile more powerful than deterministic ones, satisfy the same quantitative bound on wrect( ·).\nProposition 4.1. For every Boolean matrix F,\nN(F)≳log(1/wrect( F)).\nProof. LetFX×Y be a Boolean matrix, and c= C1(F) =O(2N(F)). Let µX×µYbe a product\nprobability measure on X ×Y , and let S1×T1, . . . , S c×Tcbe a 1-monochromatic rectangle covering\nofF. The case c≤1 is trivial, so assume c >1.\nIf there exists iwith µX(Si)·µY(Ti)≥1/4c2, then we are done. So, assume otherwise that for\nevery i, we have µX(Si)·µY(Ti)<1/4c2. Let Ito be the set of indices isuch that µX(Si)<1/2c.\nNote that, if i /∈I, then µY(Ti)<1/2c. Now define, A=X \\∪ i∈ISiandB=Y \\∪ j̸∈ITi. It is easy\nto see that A×Bis a 0-monochromatic rectangle of FandµX(A)·µY(B)>1/4.\nImpagliazzo and Williams [IW10] extended the bound in Proposition 4.1 to the more powerful\nmodel of deterministic communication with access to NPoracles. Let us first define this model\nformally.\n12Anoracle communication protocol for a communication problem Fis a protocol where each\nnode vis either a regular communication node or it is labelled with a triple ( Pv, av, bv) where Pv\nis Boolean matrix, and Pv(av(x), bv(y)) is used to decide whether to travel to the left or the right\nchild of v.\nThe complexity class PNP.The DNPcost of an oracle communication protocol is the largest\ncost of a path from the root to a leaf, which is the sum of the communicated bits plus the sum of\nN(Pv) for every von the path.\nDefine DNP(F) to be the smallest DNPcost of an oracle communication protocol for F. The\ncomplexity class PNPis the class of problems {Fn}with DNP(Fn) = polylog n.\nAmong the extensive list of complexity classes detailed in G¨ o¨ os, Pitassi, and Watson’s ar-\nticle [GPW18b], titled “ the landscape of communication complexity classes ”,PNPis the largest\nnon-probabilistic class for which an explicit lower bound is known. For example, consider the inner\nproduct problem IPn:{0,1}n× {0,1}n→ {0,1}defined as IPn(x, y) =x1y1+···+xnynmod 2.\nA simple argument, based on dimension and orthogonality (see [RY20, Claim 1.17]), shows that\nevery monochromatic rectangle in IPnis of size at most 2n, and therefore, wrect( IPn)−1≥2Ω(n).\nThe following theorem of [IW10] shows that DNP(IPn) = Ω( n).\nTheorem 4.2 (Impagliazzo and Williams [IW10]) .For every Boolean matrix F, we have\nDNP(F)≳log\u0000wrect( F)−1\u0001.\nOne might ask whether log\u0000wrect( F)−1\u0001andDNP(F) are polynomially equivalent. The answer\nis negative as [GKPW17] constructs an explicit family of Boolean matrices exhibiting a large gap\nbetween the two quantities.\nTheorem 4.3 (G¨ o¨ os, Kamath, Pitassi, and Watson [GKPW17]) .There exists a sequence of 2n×2n\nBoolean matrices Fnsatisfying DNP(Fn)≥nΩ(1)andlog\u0000wrect( Fn)−1\u0001≤logO(1)(n).\n5 Probabilistic Communication Models\nNext, we discuss probabilistic communication protocols where the players can act in a randomized\nfashion. Randomness can be introduced in two different ways: private randomness and public\nrandomness.\nIn aprivate-coin randomized protocol , each player has access to their own independent random\nbits and can use them to decide which bit to send next. More precisely, Alice and Bob have access\nto random strings RAandRB, respectively. These two strings are chosen independently, each\naccording to some probability distribution described by the protocol. The bit sent by Alice at\na node vis determined by a function avof both xandRA. Similarly, the bits sent by Bob are\ndetermined by functions of yandRB.\nIn the public-coin model, the players have access to a shared source of randomness. In other\nwords, Alice and Bob both receive the same random string R. The public-coin model is stronger\nthan the private-coin model as the former can simulate the latter by setting R= (RA, RB).\nThe cost of a randomized protocol is the maximum number of communicated bits over all inputs\nand all choices of random strings. A probabilistic protocol is allowed to make errors. It is common\nto consider three types of errors:\n13•Two-sided error ( BPP):For every x, y, the probability that the protocol makes an error\non (x, y) is at most ϵfor some ϵ <1/2. When ϵis a fixed constant strictly less than 1 /2,\nthe protocol is called a bounded-error protocol. The particular choice of ϵis unimportant\nas a simple error reduction shows that it affects the complexity by only a constant factor.\nTherefore, as it is common, we will fix the error parameter to ϵ= 1/3.\n•One-sided error ( RP):In this setting, the protocol can only make an error if F(x, y) = 1.\nIn other words, for every x, ywith F(x, y) = 0, the protocol must always correctly output 0,\nbut for every x, ywith F(x, y) = 1, it might output a wrong answer with probability at most\nϵfor some fixed ϵ <1. We will fix the error parameter to ϵ= 1/3.\n•Zero-error ( ZPP):In this case, the output of a protocol is 0, 1, or ⊥, where ⊥indicates a\nfailure to compute F(x, y). The protocol must never output 0 or 1 erroneously; however, on\nevery input, it is allowed to output ⊥with probability at most1\n2.\nA classical result in communication complexity, called Newman’s lemma, states that in the two-\nsided error, one-sided error, and zero-error settings, the following is true. The difference between\npublic-coin and private-coin randomized communication complexities of any n-bit communication\nproblem is O(log(n)).\nNewman’s lemma shows that when defining the poly-logarithmic communication complexity\nclasses BPP,RP,ZPP, it is unimportant whether we use shared randomness or private randomness.\nHowever, to define constant-cost classes BPP 0,RP0,ZPP 0, we need to make a choice. It turns out\nthat in the setting of private-coin, all these classes collapse to P0. Therefore, we shall define these\nclasses in the public-coin model.\nRegarding the zero-error protocols, the following theorem shows that even in the public-coin\nmodel, the zero-error randomized communication complexity of a matrix Fis polynomially equiv-\nalent to D(F) and in particular ZPP 0=P0.\nTheorem 5.1 ([DHP+22, Theorem 2.1]) .The public-coin zero-error randomized communication\ncomplexity of every Boolean matrix Fis at least Ω(D(F)1/4).\nIt is an open problem whether this bound can be improved to Ω(p\nD(F)), which, if true,\nwould be sharp. We will not further discuss the zero-error model and refer the interested reader\nto [DHP+22] for further reading.\n5.1 The power of randomness: BPP\nTherandomized communication complexity of a Boolean matrix F, denoted by R(F), is the min-\nimum cost of a public-coin randomized protocol with two-sided error ≤1/3. Let BPP andBPP 0\nbe, respectively, the class of problems with poly-logarithmic and O(1) randomized communication\ncomplexities.\nAre probabilistic protocols more powerful than deterministic protocols? The example of Equal-\nityshows that randomness can provide a significant advantage. To test whether x̸=y, Alice and\nBob can use their shared randomness to jointly sample a random subset S⊆ {0,1}nat no cost and\nthen, by exchanging two bits of information, indicate to each other whether their inputs belong to\nS. If they see a disparity, they can conclude confidently that x̸=y. They can run this test twice,\nand if they do not detect x̸=y, they declare x=y. Note that the probability of error is ≤1/4.\n14Therefore, R(EQn) =O(1), and Equality ∈BPP 0. In particular, we have the relations P⊊BPP\nandBPP 0̸⊆P. Also note that R(EQn) =O(1) implies via standard error-reduction that\nR(F)≲DEQ(F) logDEQ(F), (15)\nwhich establishes PEQ⊆BPP andPEQ\n0⊆BPP 0.\nWhich problems have efficient randomized protocols? A substantial portion of the literature in\ncommunication complexity is dedicated to lower-bound techniques against randomized communi-\ncation complexity, and many celebrated results establish such lower bounds for important concrete\nproblems, such as Set Disjointness [Raz92], Gap Hamming Distance [CR12], and Halfspace [She08].\nWhile it is possible to write a voluminous book about the lower bounds against randomized com-\nmunication complexity, we know very little about what is inside BPP and BPP 0. In fact, until\nrecently, it was not known whether there is any problem in BPP that is not in PEQ. Let us list some\nclassical problems in BPP.\n•Greater-Than is the family of communication problems GTn: [2n]×[2n]→ {0,1}where\nGTn(x, y) = 1 iff x≤y. It is known [Nis93, Vio15, RS15] that R(GTn) = Θ(log( n)), and\ntherefore,\nGreater-Than ∈BPP\\BPP 0.\n•Hypercube is the family of communication problems Qn:{0,1}n× {0,1}n→ {0,1}where\nQn(x, y) = 1 iff xandydiffer in exactly one coordinate. Given x, y∈ {0,1}n, Alice and Bob\ncan pick a uniform partition of [ n] into 8 sets S1, . . . , S 8and accept if for exactly one i∈[8],\nit holds that ( ⊕j∈Sixj)⊕(⊕j∈Siyj) = 1. It is easy to see that the communication cost of this\nprotocol is constant and that the error probability is at most 1 /3. Therefore, R(Qn) =O(1).\nOn the other hand, [HHH23, Lemma 2.15] and Equation (8) shows that ∥Qn∥γ2≥Ω(√n) and\ntherefore, DEQ(Qn)≥Ω(log n). We have\nHypercube ∈BPP 0\\PEQ\n0;\nsee also [HWZ22] for a different proof of this fact.\n•More generally, let ℓ(n)< n/ 2 be an integer, and Sn⊆ {0, . . . , ℓ (n)} ∪ {n−ℓ(n), . . . , n }and\ndenote the hamming weight of an x∈ {0,1}nby|x|. Ifℓ(n) = polylog( n), then the family of\nthexor-lifts1⊕\nSn:{0,1}n× {0,1}n→ {0,1}defined as 1⊕\nSn(x, y) =1Sn(|x⊕y|) is in BPP.\nIfℓ(n) =O(1), then this family is in BPP 0[Yao03]. Note that Hypercube corresponds to\nSn={1}.\n•Integer Inner product : Given a fixed positive integer t, the communication problem IIPt\nis the family of functions IIPt,n: [−2n,2n]t×[−2n,2n]t→ {0,1}with IIPt,n([x1, . . . , x t],[y1, . . . , y t]) =\n1 iffx1y1+. . .+xtyt= 0. To check the validity of the equation, Alice and Bob can choose a\nrandom prime p≈log(n) and exchange ximod pandyimod pfori= 1, . . . , t . This leads\nto a randomized protocol for IIPtwith cost O(log(n)), which shows IIPt∈BPP. On the\nother hand, for any fixed t >2, it was shown in [CLV19] that DEQ(IIPt,n)≥Ω(n). In fact, as\n[CHHS23] shows, one even has ∥IIPt,n∥γ2≥2Ω(n). Hence, for t >2,\nIIPt∈BPP\\PEQ.\n15We are unaware of any examples in BPP that fundamentally differ from those listed above.\nIn fact, IIPt, which was introduced by Chattopadhyay, Lovett, and Vinyals [CLV19], is the only\nknown example of a communication problem in BPP that is not in PEQ(see also [PSS23, CHHS23]).\nLet us mention a conjecture about IIPtbefore proceeding further. We do not know how to prove\nanyω(1) lower bound for R(IIPt,n).\nConjecture 5 (See [CHHS23, Conjecture 6.4]) .Fort >2,IIPt̸∈BPP 0.\nNote that disproving Conjecture 5 would imply that BPP 0̸⊆PEQ.\nThe communication problems in PEQare highly structured as they are linear combinations of\na few blocky matrices. On the other hand, the only known example in BPP\\PEQis the integer\ninner product , which has a low-dimensional geometric representation (i.e. bounded sign-rank)\nand enjoys nice structural properties. All these known examples contain large monochromatic rect-\nangles. Does every Boolean matrix with an efficient randomized communication protocol contain a\nlarge monochromatic rectangle? More specifically, G¨ o¨ os, Kamath, Pitassi, and Watson [GKPW17]\nasked the following question.\nQuestion 6. Is it the case that for every family of n-bit communication problems FninBPP, there\nexists c >0such that rect(Fn)≥2−c(logn)c?\nBy Theorem 4.2, a negative answer to Question 6 would imply that BPP̸⊆PNP, a relation that\nremains unknown.\nConjecture 7. BPP̸⊆PNP.\nIndependently from [GKPW17], and also motivated by Conjecture 7, [CLV19] asked whether\nthere exists a c >0 such that every communication problem Fsatisfies rect( F)≥2−cR(F)c. In fact,\nwe do not know whether there is a uniform lower bound on rect( F) depending only on R(F).\nQuestion 8 ([CLV19, HHH23]) .Is there a function κ:N→(0,1)such that rect(F)≥κ(R(F))?\nAs we shall see in Theorem 5.6, a negative answer to Question 8 would imply that BPP 0̸⊆UPP 0,\nwhich is currently unknown.\n5.2 Is two-sided error necessary?\nDefine the one-sided randomized communication complexity R1(F) and its corresponding complex-\nity classes RPandRP0analogous to the two-sided error counterparts R(F),BPP, and BPP 0.\nAre two-sided error protocols genuinely more powerful than one-sided error protocols?\nOne could give an affirmative answer to this question by referring to Equality , which satisfies\nR(EQn) =O(1) while R1(EQn) = Ω( n). This, however, is not a fully satisfactory separation. Indeed,\nsince R1(¬EQn) =O(1), we can solve Equality with a single oracle query to the Nonequality\nproblem which belongs to RP0. In other words, Equality ∈coRP 0.\nIf we examine all the known examples in BPP, we realize that they all essentially boil down to\nsolving problems with one-sided error in the sense that they are either in RP∪coRP , or they are\ncomposed of a few components, each belonging to RP∪coRP . We find this surprising, as we are not\naware of any evident reasons as to why a two-sided error protocol might be simulated by a series\n16of steps that can be performed by efficient one-sided error protocols. We suspect this phenomenon\nto be due to our limited knowledge of examples in BPP.\nDefine the class PRPsimilarly to PNP, except that the protocol is now charged R1(Pv) for its\noracle queries Pvat a node v. The simple inclusions of RP⊆BPP,PRP⊆BPP andPRP\n0⊆BPP 0\nare immediate from the definitions.\nQuestion 9. Is it true that BPP =PRP?\nIt is known that nondeterministic protocols can simulate one-sided protocols with a logarithmic\nloss,\nN(F)≤R1(F) +O(logn). (16)\nThis shows that RP⊆NPandPRP⊆PNP. In particular, Conjecture 7 would imply a negative\nanswer to Question 9.\nIt is interesting to ask the above questions in the constant-cost setting.\nQuestion 10. Is it true that BPP 0=PRP\n0?\nTheorem 5.2 below implies that for every communication problem in PRP\n0, we have wrect( F) =\nΩ(1). In particular, a negative answer to Question 8 would imply BPP 0⊊PRP\n0.\nTheorem 5.2 ([HHH23, Theorem 3.8]) .For every communication problem F,\nwrect( F)≥2−O(R1(F)).\n5.3 Sign-rank and UPP\nTheunbounded-error communication complexity ofF, denoted by U(F), is the smallest communi-\ncation cost of a private-coin randomized protocol πthat satisfies\nPr[π(x, y)̸=f(x, y)]<1\n2∀x, y.\nIn other words, the protocol is only required to outperform a random guess. The complexity classes\ncorresponding to this measure are UPP andUPP 0.\nIt is crucial that in this communication model, the players have only access to private ran-\ndomness. Otherwise, given access to shared randomness, they could jointly sample a random\ninput ( x0, y0) at no cost and use two bits of communication to verify whether ( x, y) = (x0, y0). If\n(x, y) = ( x0, y0), then they know the output F(x, y), and if it is not, they can output a random\nbit. This protocol has an error probability strictly less than 1 /2.\nPaturi and Simon [PS86] proved that the unbounded-error communication complexity is pre-\ncisely determined by an elegant matrix parameter called sign-rank .\nTo discuss sign-rank, it is more convenient to switch from Boolean matrices to sign matrices ,\nwhich are matrices with ±1 entries. The sign-rank rk±(F) of a sign matrix FX×Y is the smallest\nrank of a real matrix AX×Y such that the entries of Aare nonzero and have the same signs as their\ncorresponding entries in F. Geometrically, sign-rank corresponds to the smallest dimension where\nwe can represent Fas points and homogeneous half-spaces.\nWe can reformulate the definition of sign-rank as follows.\nDefinition 5.3 (Sign-rank) .The sign-rank of a sign matrix FX×Y is the smallest dsuch that there\nexist vectors ux, vy∈RdwithF(x, y) = sgn( ⟨ux, vy⟩)for all (x, y)∈ X × Y .\n17Recall that the log-rank conjecture speculates that for deterministic protocols, the communica-\ntion complexity is polynomially related to the logarithm of the rank of the corresponding matrix.\nPaturi and Simon proved that a similar and tighter connection is true for unbounded-error proto-\ncols, except that rank is replaced by sign-rank.\nTheorem 5.4 (Paturi and Simon [PS86]) .For every sign-matrix F, we have\nU(F) = log rk ±(F)±O(1).\nIn light of Theorem 5.4, to study U(F), one can set aside the intricacies of communication and\nfocus on the geometric notion of sign-rank.\nNumber of matrices of small sign-rank: Shortly after the introduction of sign-rank in [PS86],\nAlon, Frankl, and R¨ odl [AFR85] used results of [Mil64, Tho65, War68] on the number of connected\ncomponents of real algebraic varieties and obtained a linear lower bound on the sign-rank of random\nsign matrices. This argument was later refined in [AMY16, Lemma 24] to the following bound on\nthe number of low sign-rank matrices.\nLemma 5.5 (See [AMY16, Lemma 24]) .Ford≤m\n2, the number of m×msign matrices of\nsign-rank at most ddoes not exceed (O(m/d))2dm≤2O(dmlog(m)).\nLemma 5.5 shows that there are very few matrices with small sign-rank and that a typical\nm×msign matrix has sign-rank Ω( m). This scarcity of small-sign-rank matrices suggests that\nthey might possess strong structural properties.\nLarge monochromatic rectangles: [APP+05] used the geometric properties of sign-rank to\nprove that every X ×Y sign matrix of sign-rank dcontains an|X|\n2d+1×|Y|\n2d+1monochromatic rectangle.\nTheir result uses a theorem of Yao and Yao [YY85], which is based on the Borsuk-Ulam theorem, a\nresult in topology. Slightly different bounds are also obtained in [FPS16] using the cutting lemma of\nChazelle [Cha93]. In our notation, we have the following relation between sign-rank and wrect( F).\nTheorem 5.6 (See [APP+05, Theorem 1.3]) .For every sign-matrix F, we have\nrk±(F)≳log\u0000wrect( F)−1\u0001. (17)\nOn the other hand, [HHP+22] used a counting argument to show that there are matrices with\nwrect( A)−1=O(1) and very large sign-rank.\nTheorem 5.7 ([HHP+22, Theorem 3.2]) .There exists m×msign matrices Asuch that\nwrect( A)−1≤215, while rk±(A) = ΩÇ\nm1/3\nlog(m)å\n.\nIt is known that PNP⊊UPP (see [GPW18b]). Therefore, Theorem 5.7 is stronger than the sep-\naration of Theorem 4.3, as it shows the existence of communication problems with wrect( F)−1=\nO(1) that are not in UPP. This answers an open problem by G¨ o¨ os, Kamath, Pitassi, and Wat-\nson [GKPW17].\nThe caveat of Theorem 5.7 is that its existential proof does not provide any explicit construc-\ntion. In fact, regarding explicit examples that separate sign-rank and wrect( ·)−1, our knowledge is\nembarrassingly limited. The following problem is open.\n18Problem 11. Construct an explicit sequence of matrices Fnsuch that wrect( Fn)−1=O(1)and\nlim\nn→∞rk±(Fn) =∞.\nBy Theorem 5.7, we know such matrices exist, and in fact, with very large sign-ranks. On the\nother hand, none of the known lower bound techniques can directly imply a solution to Problem 11.\nIndeed, in addition to the monochromatic rectangle lower bound, there are only two other known\nmethods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is at least the\nVC-dimension: rk ±(A)≥VC(A); (ii) Forster’s method, which states that sign-rank is at least the\ninverse of the largest possible average margin among the representations of the matrix by points and\nhalf-spaces: rk ±(F)≥mavg(F)−1. We refer the reader to [HHP+22] for the definition of average\nmargin and a thorough discussion of these facts.\nQualitatively, Equation (17) is the strongest known method for proving lower bounds on the\nsign-rank of an explicit matrix. If it fails to provide a super-constant lower bound for the sign-rank\nof a matrix, then the other two methods will also fail. More precisely, we have\n»\nVC(A)≤mavg(A)−1≤wrect−1(A). (18)\nIn this sense, Problem 11 captures the limitation of the currently known lower bound techniques\nfor sign-rank.\nSign-rank of hypercubes and BPP 0vsUPP 0:Linial, Mendelson, Schechtman, and Shraib-\nman [LMSS07] asked whether sign-rank can be bounded from above by a function of the so-called\nmargin complexity. The relation between margin complexity, discrepancy, and randomized com-\nmunication complexity, which were discovered later, allows us to rephrase their question as follows.\nQuestion 12. Is it true that BPP 0⊆UPP 0? Equivalently, is it possible to upper bound rk±(F)by\na function of R(F)?\nWe believe the answer to this question to be negative. Consider the sign version of the Hy-\npercube problem, that is, let Qnbe the sign matrix whose rows and columns are indexed with the\nelements of {0,1}n, and Qn(x, y) =−1 ifxandydiffer in exactly one coordinate. As we discussed\nearlier, R(Qn) =O(1). We conjecture that the sign-rank of Qntends to infinity as ngrows, which,\nif true, would imply BPP 0̸⊆UPP 0.\nConjecture 13 (Sign-rank of Hypercube [HHP+22]).We have\nlim\nn→∞rk±(Qn) =∞.\nIt is worth pointing out that proving Conjecture 13 in the positive would likely require some\nnew lower bound techniques, as wrect( Qn)−1=O(1), shown in [HHP+22]. Note that, a positive\nanswer to Conjecture 13 would also solve Problem 11.\nEquality oracles, PEQ\n0⊊UPP 0:It is easy to show that the EQn∈UPP 0, as its sign-rank is 3.\nThe following theorem shows that, in fact, PEQ\n0⊆UPP 0. The example of Greater-Than shows\nthat this inclusion is strict. It is easy to see that Greater-Than ∈UPP 0as its sign-rank is 2.\nThis fact combined with R(GTn) = Θ(log n) shows that\nGreater-Than ∈UPP 0\\BPPEQ\n0⊆UPP 0\\PEQ\n0.\n19Theorem 5.8 ([HHP+22]).For every sign matrix FX×Y, we have rk±(F)≤4DEQ(F). In particular,\nU(F)≤2DEQ(F) +O(1).\nProof. We proceed by induction on d:=DEQ(A). When d= 1, Fcorresponds to a blocky matrix,\nwhich in fact has rk ±(F)≤3. For larger d, consider a cost dprotocol for a sign matrix F\nand suppose the equality query at the root of the tree is EQ(a(x), b(y)), where here we assume\nwithout loss of generality a(x) and b(y) take integer values. Let SX×Y be the matrix with entries\nSxy=1a(x)=b(y). We branch according to the output of the first query either to the left or the right\nsubtree of the root, each corresponding to a protocol with cost at most d−1. Let the corresponding\nmatrices for these protocols be Π 1and Π 2, and note that\nF=S◦Π1+ (J−S)◦Π2,\nwhere J:=JX×Y is the all-ones matrix. By the induction hypothesis, Π 1and Π 2have sign-rank at\nmost≤4d−1. LeteΠ1andeΠ2be real matrices with rank at most 4d−1that satisfy sgn( eΠ1) = Π 1\nand sgn( eΠ2) = Π 2. Let EX×Y be the rank-3 matrix with entries Exy= (a(x)−b(y))2. Note that\nfor a sufficiently large k, we have\nA= sgn( eΠ1+kE◦eΠ2).\nFinally, we have\nrk(eΠ1+keΠ2◦E)≤rk(eΠ1) + rk( eΠ2)·rk(E)≤4d−1+ 3·4d−1= 4d.\nThe above theorem combined with Equation (15) shows that PEQ\n0⊆UPP 0∩BPP 0. Both the\ninclusions PEQ\n0⊊UPP 0andPEQ\n0⊊BPP 0are strict; the former follows from the example of Greater-\nThan and the latter holds for Hypercube [HHH23, HWZ22]. The question of whether these\nseparations can be obtained simultaneously was asked recently by [HZ24].\nQuestion 14 ([HZ24]) .Is it the case that PEQ\n0=UPP 0∩BPP 0?\nSince Hypercube ̸∈PEQ\n0andHypercube ∈BPP 0, a positive answer to Question 14 would\nimply Hypercube ̸∈UPP 0and solve Conjecture 13.\nIn the converse direction, Conjecture 13 would imply a positive answer to Question 14 for the\nspecial case of xor-lifts. Indeed, if Hypercube ̸∈UPP 0, then the result of [CHZZ22] would imply\nthat every family of xor-liftf⊕\nn∈UPP 0∩BPP 0must satisfy ∥fn∥A=O(1) and therefore by\nTheorem 3.7 and Proposition 3.9, we must have f⊕\nn∈PEQ\n0.\n5.4 Weakly unbounded-error complexity, PP\nTheweakly unbounded-error communication complexity of a problem FX×Y is defined as\nPP(F):= min\nϵ<1/2Rϵ(F) + logÇ\n1\n1\n2−ϵå\n,\nwhere Rϵ(F) is the minimum cost of a public-coin randomized protocol with two-sided error at\nmost ϵ.\n20Define PPandPP0to be the class of families of n-bit problems Fnwith PP(Fn) = polylog nand\nPP(Fn) =O(1), respectively. It is immediate from the definitions and Newman’s lemma [New91]\nthat\nBPP⊆PP⊆UPP, and PP0=BPP 0.\nAs discussed before, the Greater-Than problem separates UPP 0from BPP 0=PP0. Babai,\nFrankl, and Simon [BFS86] asked whether UPP =PP. Their question remained unanswered for\nover two decades, until [BVdW07, She08] independently showed that there are 2n×2nsign matrices\nFwith rk ±(F) =nbutPP(F) =nΩ(1). The separation was strengthened in subsequent works to\nrk±(F) =nandPP(F) = Ω( n) in [She21].\nThe example proposed in [BVdW07], in fact, belongs to PNP, and therefore also shows that\nPNP̸⊆PP. It turns out that the opposite direction of this inclusion is not true either, as follows\nfrom an argument involving bounds on the rectangle ratio.\nTheorem 5.9. There exists a family of n-bit communication problems Fnwith PP(Fn) =O(log(n))\nandrect(Fn)−1= 2Ω(n). In particular, DNP(Fn) = Ω( n)andPP̸⊆PNP.\nProof. LetFn(x, y) = 1 iff xandydiffer on at least n/2 bits. It is known through classical results\nfrom combinatorics [FF81] that rect( Fn)−1= 2Ω(n). Thus by Theorem 4.2, we get DNP(Fn) = Ω( n).\nTo see the inclusion in PP, note that the simple protocol where two parties pick a random index\niuniformly at random and output 1 iff xi̸=yi, has cost O(logn).\nFinally, recent works have shown that PPdoes not even contain UPP 0. Indeed, [HHL20,\nACHS23] gave simple constructions of n-bit communication problems Fwith rk ±(F) = 3 and\nPP(F) = Ω( n).\n6 Final remarks\nWe discussed several open problems that indicate significant gaps in our understanding of commu-\nnication complexity and capture the limitations of the currently available techniques. For example,\ndisproving BPP =PRP,BPP⊆PNP, or giving a negative answer to Question 6 or Question 8,\nrequires constructing a family of matrices in BPP that is fundamentally different from all the cur-\nrently known examples. Conversely, proving that any of these statements is true would be a major\nstride toward achieving a structural description of BPP. Similarly, Conjecture 13 and Problem 11\nrequire a new lower-bound technique for sign-rank that can reach beyond the log wrect( ·)−1bound\nof Theorem 5.6.\nWe hope that similar to the introduction of communication classes by [BFS86], the formal\nparadigm of constant-cost communication classes will catalyze future research—That the efforts\nto establish separations between these classes will lead to the discovery of new examples and\nlower-bound techniques and give us a deeper understanding of communication models and their\nconnections to other areas of theoretical computer science.\nDue to space limitations, we did not discuss quantum communication models, multi-party mod-\nels, search problems, and various related query models, most notably parity decision trees. The\nquestions that are being discussed in this article can be asked in a similar way for these models.\nWe conclude by presenting, in Figure 1 and Figure 2, the known relations and separations among\nvarious classes discussed in this article. These figures include a selected list of classes, excluding\neasier-to-handle classes such as P,RP,NP,P0=NP0. We define the classes RectandRect 0to consist\n21of matrix families with wrect−1(·) bounded from above by 2polylog nandO(1), respectively. The\nclass Rect appears in [GKPW17] with the different name of PM(for Product Method).\nPEQ\n0PRP\n0BPP 0UPP 0Rect 0PEQPRPBPP PP UPP PNPRect\nPEQ\n0 =⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ ⊆\nPRP\n0 ̸⊆ = ⊆ ? ⊆ ?⊆ ⊆ ⊆ ⊆ ⊆ ⊆\nBPP 0̸⊆ ? = ? ? ? ?⊆ ⊆ ⊆ ? ?\nUPP 0̸⊆ ̸⊆ ̸⊆ = ⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ⊆ ?⊆\nRect 0̸⊆ ̸⊆ ̸⊆ ̸⊆ = ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ⊆\nPEQ̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ =⊆ ⊆ ⊆ ⊆ ⊆ ⊆\nPRP̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ = ⊆ ⊆ ⊆ ⊆ ⊆\nBPP ̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ? =⊆ ⊆ ? ?\nPP ̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ =⊆ ̸⊆ ̸⊆\nUPP ̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ = ̸⊆ ̸⊆\nPNP̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ⊆ = ⊆\nRect ̸⊈⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ ̸⊆ =\nFigure 1: The entry at a row Aand a column Bindicates whether A⊆BorA̸⊆B. A question\nmark indicates that the relationship is unknown. The separations in grey entries follow trivially\nvia padding.\nBPP 0\nPRP\n0\nPEQ\n0UPP 0Rect 0\nPEQBPPPPUPP Rect\nPRPPNP\nFigure 2: A→Bindicates A⊆B.\n22Acknowledgements. We wish to thank Tsun-Ming Cheung, Mika G¨ o¨ os, Lianna Hambardzumyan,\nNathan Harms, Shachar Lovett, and Alexander Razborov for their helpful comments on a draft of\nthis article.\nReferences\n[ACHS23] Manasseh Ahmed, Tsun-Ming Cheung, Hamed Hatami, and Kusha Sareen. Com-\nmunication complexity of half-plane membership. In Electron. Colloquium Comput.\nComplex. , 2023.\n[AFR85] Noga Alon, Peter Frankl, and Vojtech R¨ odl. Geometrical realization of set systems and\nprobabilistic communication complexity. In 26th Annual Symposium on Foundations\nof Computer Science, FOCS 1985 , pages 277–280. IEEE Computer Society, 1985.\n[AMY16] Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank versus VC dimension. In\nProceedings of the 29th Conference on Learning Theory, COLT 2016 , volume 49, pages\n47–80, 2016.\n[APP+05] Noga Alon, J´ anos Pach, Rom Pinchasi, Radoˇ s Radoiˇ ci´ c, and Micha Sharir. Crossing\npatterns of semi-algebraic sets. Journal of Combinatorial Theory, Series A , 111(2):310–\n326, 2005.\n[AY22] Daniel Avraham and Amir Yehudayoff. On blocky ranks of matrices. Electron. Collo-\nquium Comput. Complex. , TR22-137, 2022.\n[BCK14] Abhishek Bhrushundi, Sourav Chakraborty, and Raghav Kulkarni. Property test-\ning bounds for linear and quadratic functions via parity decision trees. In Computer\nscience—theory and applications , volume 8476 of Lecture Notes in Comput. Sci. , pages\n97–110. Springer, Cham, 2014.\n[BFS86] L´ aszl´ o Babai, P´ eter Frankl, and Janos Simon. Complexity classes in communication\ncomplexity theory. In 27th Annual Symposium on Foundations of Computer Science\n(SFCS 1986) , pages 337–347. IEEE, 1986.\n[BVdW07] Harry Buhrman, Nikolay Vereshchagin, and Ronald de Wolf. On computation and\ncommunication with small bias. In Twenty-Second Annual IEEE Conference on Com-\nputational Complexity (CCC’07) , pages 24–32. IEEE, 2007.\n[Cha93] Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput.\nGeom. , 9(2):145–158, 1993.\n[CHHS23] Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley. Separation\nof the factorization norm and randomized communication complexity. In Proceedings,\nIEEE Conference on Computational Complexity (CCC) , 2023.\n[CHZZ22] TsunMing Cheung, Hamed Hatami, Rosie Zhao, and Itai Zilberstein. Boolean functions\nwith small approximate spectral norm. In Electronic Colloquium on Computational\nComplexity (ECCC), TR22-041 , 2022.\n23[CLV19] Arkadev Chattopadhyay, Shachar Lovett, and Marc Vinyals. Equality alone does not\nsimulate randomness. In 34th Computational Complexity Conference (CCC 2019) .\nSchloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019.\n[CMS19] Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The log-approximate-\nrank conjecture is false. In Proceedings of the 51st Annual ACM SIGACT Symposium\non Theory of Computing , STOC 2019, pages 42–53, New York, NY, USA, 2019. ACM.\n[Coh60] Paul J. Cohen. On a conjecture of Littlewood and idempotent measures. Amer. J.\nMath. , 82:191–212, 1960.\n[CR12] Amit Chakrabarti and Oded Regev. An optimal lower bound on the communication\ncomplexity of gap-Hamming-distance. SIAM J. Comput. , 41(5):1299–1317, 2012.\n[DD07] Kenneth R. Davidson and Allan P. Donsig. Norms of Schur multipliers. Illinois J.\nMath. , 51(3):743–766, 2007.\n[DHP+22] Ben Davis, Hamed Hatami, William Pires, Ran Tao, and Hamza Usmani. On public-\ncoin zero-error randomized communication complexity. Information Processing Letters ,\n178:106293, 2022.\n[EHK22] Louis Esperet, Nathaniel Harms, and Andrey Kupavskii. Sketching distances in mono-\ntone graph classes. Approximation, Randomization, and Combinatorial Optimization.\nAlgorithms and Techniques , page 1, 2022.\n[FF81] Peter Frankl and Zolt´ an F¨ uredi. A short proof for a theorem of harper about hamming-\nspheres. Discrete Mathematics , 34(3):311–313, 1981.\n[FPS16] Jacob Fox, J´ anos Pach, and Andrew Suk. A polynomial regularity lemma for semi-\nalgebraic hypergraphs and its applications in geometry and property testing. SIAM\nJournal on Computing , 45(6):2199–2223, 2016.\n[GGMT23] W. T. Gowers, Ben Green, Freddie Manners, and Terence Tao. On a conjecture of\nMarton, 2023.\n[GKPW17] Mika G¨ o¨ os, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-\ncommunication lifting for PNP.32nd Computational Complexity Conference , 79:Art.\nNo. 12, 16, 2017.\n[GPW18a] Mika G¨ o¨ os, Toniann Pitassi, and Thomas Watson. Deterministic communication vs.\npartition number. SIAM Journal on Computing , 47(6):2435–2450, 2018.\n[GPW18b] Mika G¨ o¨ os, Toniann Pitassi, and Thomas Watson. The landscape of communication\ncomplexity classes. computational complexity , 27(2):245–304, 2018.\n[GS08] Ben Green and Tom Sanders. Boolean functions with small spectral norm. Geometric\nand Functional Analysis , 18(1):144–162, 2008.\n[Har20] Nathaniel Harms. Universal communication, universal graphs, and graph labeling. In\n11th Innovations in Theoretical Computer Science Conference , volume 151 of LIPIcs.\nLeibniz Int. Proc. Inform. , pages Art. No. 33, 27. Schloss Dagstuhl. Leibniz-Zent. In-\nform., Wadern, 2020.\n24[HHH23] Lianna Hambardzumyan, Hamed Hatami, and Pooya Hatami. Dimension-free bounds\nand structural results in communication complexity. Israel Journal of Mathematics ,\n253(2):555–616, 2023.\n[HHL20] Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign rank vs discrepancy. In\n35th Computational Complexity Conference , volume 169 of LIPIcs. Leibniz Int. Proc.\nInform. , pages Art. No. 18, 14. 2020.\n[HHP+22] Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, and Rosie Zhao. Lower bound\nmethods for sign-rank and their limitations. In Approximation, Randomization, and\nCombinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) .\nSchloss Dagstuhl-Leibniz-Zentrum f¨ ur Informatik, 2022.\n[HWZ22] Nathaniel Harms, Sebastian Wild, and Viktor Zamaraev. Randomized communication\nand implicit graph representations. In Proceedings, ACM Symposium on Theory of\nComputing (STOC) , 2022.\n[HZ24] Nathaniel Harms and Viktor Zamaraev. Randomized communication and implicit\nrepresentations for matrices and graphs of small sign-rank. In Proceedings, ACM-SIAM\nSymposium on Discrete Algorithms (SODA) , 2024.\n[IW10] Russell Impagliazzo and Ryan Williams. Communication complexity with synchronized\nclocks. In 2010 IEEE 25th Annual Conference on Computational Complexity , pages\n259–269. IEEE, 2010.\n[KN97] Eyal Kushilevitz and Noam Nisan. Communication complexity . Cambridge University\nPress, Cambridge, 1997.\n[Liv95] Leo Livshits. A note on 0-1 Schur multipliers. Linear Algebra Appl. , 222:15–22, 1995.\n[LMSS07] Nati Linial, Shahar Mendelson, Gideon Schechtman, and Adi Shraibman. Complexity\nmeasures of sign matrices. Combinatorica , 27(4):439–463, 2007.\n[Lov14] Shachar Lovett. Recent advances on the log-rank conjecture in communication com-\nplexity. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS , 112:18–35, 2014.\n[Lov16] Shachar Lovett. Communication is bounded by root of rank. Journal of the ACM\n(JACM) , 63(1):1–9, 2016.\n[LS88] L´ aszl´ o Lov´ asz and Michael Saks. Lattices, mobius functions and communications com-\nplexity. In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer\nScience , pages 81–90. IEEE Computer Society, 1988.\n[LS09a] Nati Linial and Adi Shraibman. Learning complexity vs. communication complexity.\nCombin. Probab. Comput. , 18(1-2):227–245, 2009.\n[LS09b] Nati Linial and Adi Shraibman. Lower bounds in communication complexity based on\nfactorization norms. Random Structures & Algorithms , 34(3):368–394, 2009.\n[LS23] Troy Lee and Adi Shraibman. Around the log-rank conjecture. Israel Journal of\nMathematics , 256(2):441–477, 2023.\n25[Mil64] J. Milnor. On the Betti numbers of real varieties. Proc. Amer. Math. Soc. , 15:275–280,\n1964.\n[New91] Ilan Newman. Private vs. common random bits in communication complexity. Inf.\nProcess. Lett. , 39(2):67–71, July 1991.\n[Nis93] Noam Nisan. The communication complexity of threshold gates. Combinatorics, Paul\nErd˝ os is Eighty , 1:301–315, 1993.\n[NW95] Noam Nisan and Avi Wigderson. On rank vs. communication complexity. Combina-\ntorica , 15(4):557–565, 1995.\n[Pis96] Gilles Pisier. Similarity problems and completely bounded maps , volume 1618 of Lecture\nNotes in Mathematics . Springer-Verlag, Berlin, 1996.\n[PS86] Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. Jour-\nnal of Computer and System Sciences , 33(1):106–123, 1986.\n[PSS23] Toniann Pitassi, Morgan Shirley, and Adi Shraibman. The strength of equality ora-\ncles in communication. In Proceedings, Innovations in Theoretical Computer Science\n(ITCS) , 2023.\n[Raz92] A. A. Razborov. On the distributional complexity of disjointness. Theoret. Comput.\nSci., 106(2):385–390, 1992.\n[RS15] Sivaramakrishnan Natarajan Ramamoorthy and Makrand Sinha. On the communica-\ntion complexity of greater-than. In 2015 53rd Annual Allerton Conference on Com-\nmunication, Control, and Computing (Allerton) , pages 442–444. IEEE, 2015.\n[RY20] Anup Rao and Amir Yehudayoff. Communication complexity and applications . Cam-\nbridge University Press, Cambridge, 2020.\n[San11] Tom Sanders. A quantitative version of the non-abelian idempotent theorem. Geom.\nFunct. Anal. , 21(1):141–221, 2011.\n[San19] Tom Sanders. Boolean functions with small spectral norm, revisited. Mathematical\nProceedings of the Cambridge Philosophical Society , 167(2):335–344, 2019.\n[San20] Tom Sanders. Bounds in Cohen’s idempotent theorem. Journal of Fourier Analysis\nand Applications , 26(2):1–64, 2020.\n[She08] Alexander A Sherstov. Halfspace matrices. Computational Complexity , 17:149–178,\n2008.\n[She21] Alexander A. Sherstov. The hardest halfspace. Comput. Complexity , 30(2):Paper No.\n11, 85, 2021.\n[ST23] Benny Sudakov and Istv´ an Tomon. Matrix discrepancy and the log-rank conjecture.\narXiv preprint arXiv:2311.18524 , 2023.\n26[Tho65] Ren´ e Thom. Sur l’homologie des vari´ et´ es alg´ ebriques r´ eelles. In Differential and Combi-\nnatorial Topology (A Symposium in Honor of Marston Morse) , pages 255–265. Prince-\nton Univ. Press, Princeton, N.J., 1965.\n[Vio15] Emanuele Viola. The communication complexity of addition. Combinatorica ,\n35(6):703–747, 2015.\n[War68] Hugh E. Warren. Lower bounds for approximation by nonlinear manifolds. Trans.\nAmer. Math. Soc. , 133:167–178, 1968.\n[Yao79] Andrew Chi-Chih Yao. Some complexity questions related to distributive computing\n(preliminary report). In Proceedings of the eleventh annual ACM symposium on Theory\nof computing , pages 209–213, 1979.\n[Yao03] Andrew Chi-Chih Yao. On the power of quantum fingerprinting. In Proceedings of the\nthirty-fifth annual ACM symposium on Theory of computing , pages 77–81, 2003.\n[YY85] A C Yao and F F Yao. A general approach to d-dimensional geometric queries. In\nProceedings of the Seventeenth Annual ACM Symposium on Theory of Computing ,\nSTOC ’85, page 163–168. Association for Computing Machinery, 1985.\n27" }, { "title": "2401.14639v1.Constructing_many_twist_Möbius_bands_with_small_aspect_ratios.pdf", "content": "Constructing many-twist M¨ obius bands with\nsmall aspect ratios\nAidan Hennessey\nDecember 2023\nAbstract\nI present a construction of a folded paper ribbon knot that provide\na constant upper bound on the infimal aspect ratio for paper M¨ obius\nbands and annuli with arbitrarily many half-twists. In particular, the\nconstruction shows that paper M¨ obius bands and annuli with any number\nof half-twists can be embedded with aspect ratio less than 8.\n1 Introduction\nIn 1977, Halpern and Weaver conjectured that the infimal aspect ratio of an\nembedded paper M¨ obius band is√\n3 [3]. This conjecture was recently proven by\nSchwartz in [4]. Shortly after, Schwartz proved that the infimal aspect ratio for\nthe 2-half-twist annulus is 2 [5]. Noah Montgomery independently showed this\nresult using alternative methods (unpublished). Moreover, Brown and Schwartz\nconjecture the infimal aspect ratio for the 3-half-twist embedded paper M¨ obius\nband is 3 [6]. Beyond these published results, unpublished experiments by\nBrown indicate the infimal aspect ratios for embedded paper M¨ obius bands\nwith 5 and 7 half-twists are each at most 5.\nThe seeming pattern of more twists requiring a longer band raises the question:\nWhat is the asymptotic growth of the minimal aspect ratio λnas a function of\nthe number of twists n? Noah Montgomery found a construction with length\ncomplexity O(√n) (unpublished), but did not produce any lower bound. My\nconstruction puts the asymptotic question to rest by constructing an O(1) so-\nlution. I do not show that the construction’s constant bound is tight. I.e.,\ndetermining the value of lim sup {λn}is still an open problem. The construction\nis actually a folded ribbon (un)knot which can be arbitrarily well-approximated\nby smooth paper band embeddings. Its folded ribbon knot form negatively\nanswers conjecture 39 in [2].\n1arXiv:2401.14639v1 [math.GT] 26 Jan 20242 Background\nFormally, a paper M¨ obius band is a smooth locally isometric embedding of the\nM¨ obius strip\n([0,1]×[0, λ])/∼; ( t,0)∼(1−t, λ)\nintoR3. Similarly, a paper annulus is a smooth locally isometric embedding\nof the cylinder ([0 ,1]×[0, λ])/((t,0)∼(t, λ)) into R3. Refer to these maps\ncollectively as paper bands. λis called the aspect ratio of the band.\nThe center line of a band is the image of {0.5} ×[0, λ] under the embedding.\nDefine an nhalf-twist paper band to be\n•A paper M¨ obius band for which the boundary and center line have linking\nnumber ±n(fornodd).\n•A paper annulus for which one of the boundaries and the center line have\nlinking number ±n/2 (for neven).\nMany1paper bands can be gently pressed down to lie in the plane, at which\npoint the image is a union of rectangles, parallelograms, and trapezoids, joined\nto one another at creases in sequence. Such an object is known as a folded\nribbon knot. For a formal definition, see [2].\nTo a folded ribbon knot we can associate a pre-bend diagram, which is a rectan-\ngle with non-intersecting solid and dotted line segments drawn on it. Each line\nsegment represents a fold, and the texture of the segment dictates which way\nthe fold goes. One can imagine the rectangle as a strip of paper, with the side\nfacing the viewer colored red, and the other side colored blue. Then, a solid line\nindicates folding so that the red side is on the inside, and a dotted line indicates\na fold which has a blue inside.\nFigure 1: Three different ribbon knots. Their prebend diagrams all have line\nsegments in the same places, but they differ in which segments are solid or\ndashed. Reused with permission from [1].\n1It is likely not all paper bands have this property. A particular likely counterexample is\nthe cap, an efficient 3-twist band featured in [6]. This counterexample was pointed out to me\nby Richard Schwartz.\n2Figure 2: The prebend diagrams for the above three ribbon knots. The top\nprebend diagram corresponds to the left ribbon knot, the middle with middle,\nand bottom with right.\n3 Construction\n3.1 Folded Ribbon Knot\nMost constructions aimed at this problem are centered around the following\nconcept: Coil a belt, and then pull the ends apart without allowing them to\nrotate. The coils turn into twists. This is useful because it means one can\nconstruct a many-twist band by tightly coiling the band, yielding very many\ntwists while using a small length of band. The issue with this is that one end of\nthe band ends up confined in a very small space, which prevents reconnection\nof the two ends without using a very large amount of band to “escape.”\nHere’s the key idea for this paper: If we wrap very tightly at a large angle, then\nwe can escape using a constant length of band. Construct an “escape accordion”\nby folding along parallel lines, 45 degrees rotated from the sides of the band.\nFigure 3: The prebend diagram for the escape accordion\nColor the front side of the band red and the back side blue. Then, after folding\nthe accordion, the band looks like this:\nFigure 4: The escape accordion made from colored paper\n3The key insight about the accordion is that its construction uses a parallelogram\nwith base 2, regardless of the distance between folds. Thus, if we want to achieve\nnhalf-twists using ϵadditional length of band, we can let there be ⌈n/2ϵ⌉folds\nin the accordion. Adding in the pre-bends for the wrapping step yields a new\nprebend diagram:\nFigure 5: The prebend diagram for the accordion and the wrapping\nWe can now fold this up, retaining the same red-blue coloring used in figure 5.\nδ0\nFigure 6: The full construction, up to reattaching the ends. Notice that for any\nfixed number of twists, the distance labeled δ0can be made arbitrarily small\nwith an adequately skinny accordion.\nAll that’s left in the construction is to reattach the ends, which requires a length\nof band independent of the number of twists.\n3.2 Linking Number\nThe ribbon linking number of a folded ribbon knot is the linking number of\nthe boundary with the centerline (in the case of a M¨ obius band) or the linking\nnumber of the centerline with one of the boundaries (in the case of an annulus).\n[2] gives an easy method for counting the linking number. The method involves\ncounting the contribution of each fold and crossing and then adding those up.\nBy a crossing I mean a place where the centerline crosses itself. General overlap\nof the band with itself does not count as a crossing.\nThis paper’s construction does not have any crossings, so the only informa-\ntion relevant to calculating the linking number is the number of each type of\nfold. There are four types to consider, each with their own contribution and\ncorresponding realization in the prebend diagram:\n1. Right underfolds, +1 - downward sloping dashed lines in prebend diagrams\n42. Right overfolds, −1 - downward sloping solid lines\n3. Left underfolds, −1 - upward sloping dashed lines\n4. Left overfolds, +1 - upward sloping solid lines\nNote that the above contributions are only for mobius strips. In a M¨ obius strip,\neither side of the center line is part of the same single boundary. Compared\nto an annulus, then, each fold creates twice as many intersections, and thus\ncontributes twice as much to the linking number in the M¨ obius band case versus\nthe annulus case. The distinct cases in the definition of an nhalf-twist paper\nband exist to counterbalance this effect.\nExample 1. Using the described method, calculate the ribbon linking number\nof the ribbons corresponding to the prebend diagrams in figure 2. Confirm that\nyour answer matches what you would visually infer from figure 1.\nUsing this counting method, we can see in figure 5 that the folds of the ac-\ncordion cancel out, while the wrapping around it causes the linking number to\naccumulate. It takes nconsecutive solid lines in the prebend diagram to create\na band with nhalf-twists (assuming no contribution from the reconnection).\nThe shortest reconnection methods may contribute some additional linking or\nunlinking, but any reasonably simple method will only contribute a constant\namount, so this does not matter.\n3.3 Smooth Approximation\nThe folded ribbon knot above can be well-approximated by smooth embedded\npaper bands. In particular, there exists a family of such bands which converges\npointwise to the folded ribbon knot, and whose aspect ratios converge to that\nof the folded ribbon knot. The main idea is to model each fold with a very\ntight turnaround, or pseudofold. For other examples of a similar procedure, see\n[3, 5, 6].\nAs defined in [3], proof of lemma 9.1, a pseudofold is based on a plane curve\nγ(δ, t) (parameterized by arc length t) with curvature κ(t) satisfying:\n•κ(t) is smooth and has compact support (bump function)\n•κ(t)≥0\n•R\nκ(t) =π\nSo in essence, γ(δ, t) follows the x-axis for some time, turns around smoothly,\nand then follows the line y=δis the other direction. The length of the curved\npart is cδfor some constant cdepending on the particular bump function chosen.\nLet the curved part correspond to t∈[0, cδ]\nThis curve is used to construct the pseudofold as a surface by placing it in\ntheyz-plane, sweeping it through the x-axis to get a surface, and shearing as\nnecessary (amount depending on the pseudofold angle).\n5δ\nFigure 7: The top and side views of a pseudofold. The side view is simply the\ngraph of γ(δ, t) in the plane.\nSeparate each layer of the folded ribbon knot vertically by some small distance\nδand then connect the layers with pseudofolds. Let the sizeof a pseudofold\nbe the height disparity between the layers it connects. The pseudofolds of the\naccordion all have size δ. The corresponding to the wrapping have sizes ( m+1)δ,\n(m+ 2)δ, ..., ( m+n)δ, assuming there are maccordion folds and nwrapping\nfolds. We can represent this in a prebend diagram. In this diagram, let green\nparallelograms represent pseduofolds which replace solid lines, and let purple\nparallelograms represent pseudofolds which replace dashed lines.\nFigure 8: The prebend diagram for the smooth approximation. The base of\neach parallelogram is√\n2 times the size of the corresponding pseudofold.\nEvery error in the approximation is proportional to δ. Thus, as δgoes to 0, the\nadditional band length and the distance between any particular point on the\nribbon and its approximating band go to 0.\nFigure 9: A side view of the complicated part of a smooth approximation\n64 Parity\nThe construction so far applies to both M¨ obius bands and annuli. Which one\nis constructed comes down to how the ends of the band are connected to one\nanother. Letting one side be colored red and the other blue, a M¨ obius band is\nobtained from taping red to blue, while an annulus is obtained by taping red to\nred. Here are two reasonably efficient ways to reconnect the ends for each type\nof band.\nFigure 10: A fully constructed paper M¨ obius band. The orange line in the\nmiddle indicates where the strip of paper is taped/glued to itself. The fact that\nthere are opposite colors on either side of the line corresponds to the fact that\nthis is a M¨ obius band, not an annulus.\nFigure 11: A fully connected many-twist paper annulus. The orange gluing line\nhas the same color (red) on each side of it, indicating that the band has an even\nnumber of half-twists.\nThe above constructions give many-twist M¨ obius bands with aspect ratio 6.25\nand many-twist annuli with aspect ratio 7.45. The disparity comes from the\nfact that the reconnection in the annulus case is less efficient.2\n5 Acknowledgements\nI would like to thank Richard Schwartz and Luke Briody for many helpful\ndiscussions around this topic.\n2The need for two distinct reconnection methods, and the particular lengths for each type,\nwere made clear in a conversation with Luke Briody.\n7References\n[1] Elizabeth Denne, Mary Kamp, Rebecca Terry, and Xichen Zhu. Ribbon-\nlength of folded ribbon unknots in the plane. Knots, links, spatial graphs,\nand algebraic invariants , 2017.\n[2] Elizabeth Denne and Troy Larsen. Linking number and folded ribbon un-\nknots. The Journal of Knot Theory and its Ramifications , 2023.\n[3] Halpern and Weaver. Inverting a cylinder through isometric immersions and\nembeddings. Transactions of the American Mathematical Society , 1977.\n[4] Richard E. Schwartz. The optimal paper moebius band. 2023.\n[5] Richard E. Schwartz. The optimal twisted paper cylinder. 2023.\n[6] Richard E. Schwartz and Brienne E. Brown. The crisscross and the cup:\nTwo short 3-twist paper moebius bands. 2023.\n8" }, { "title": "2401.14643v2.Deficit_Angles_in_4D_Spinfoam_with_Cosmological_Constant___Anti__de_Sitter_ness_and_More.pdf", "content": "Deficit Angles in 4D Spinfoam with Cosmological Constant: (Anti) de Sitter-ness and\nMore\nMuxin Han1, 2,∗andQiaoyin Pan1,†\n1Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA\n2Institut f¨ ur Quantengravitation, Universit¨ at Erlangen-N¨ urnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany\n(Dated: February 15, 2024)\nThis paper investigates the critical behaviors of the 4-dimensional spinfoam model with cosmo-\nlogical constant for a general 4-dimensional simplicial complex as the discretization of spacetime.\nWe find that, at the semi-classical regime, the spinfoam amplitude is peaked at the real critical\npoints that correspond to zero deficit angles (modulo 4 πZ/γ) hinged by internal triangles of the\n4-complex. Since the 4-simplices from the model are of constant curvature, the discrete geometry\nwith zero deficit angle manifests a de Sitter (dS) spacetime or an anti de Sitter (AdS) spacetime\ndepending on the sign of the cosmological constant fixed by the boundary condition. The non-\n(A)dS spacetimes emerge from the complex critical points by an analytic continuation to complex\nconfigurations.\nContents\nI. Introduction 2\nII. Preliminary: 4D spinfoam with Λ̸= 0from boundary Chern-Simons theory 3\nA. Classical theory 3\nB. Quantum theory – the vertex amplitude 5\nIII. Gluing S3\\Γ5’s – the edge amplitude 8\nIV. The full finite amplitude 10\nV. Stationary analysis for vertex amplitude and curved Regge action 11\nVI. Stationary analysis for spins and the critical deficit angle 14\nVII. Away from the (A)dS-ness 16\nVIII. An example: spinfoam amplitude of the ∆34-complex 17\nIX. Conclusion and outlook 19\nAcknowledgments 20\nA. Fock-Goncharov coordinates and the Fenchel-Nielsen coordinates 20\nB. Geometrical interpretations of the Fenchel-Nielson coordinates 20\nC. Fock-Goncharov coordinates on S2andS′\n1in∆34-complex 26\nD. Proof of Theorem VII.1 27\nReferences 28\n∗Electronic address: hanm@fau.edu\n†Electronic address: qpan@fau.eduarXiv:2401.14643v2 [gr-qc] 14 Feb 20242\nI. INTRODUCTION\nIn the research area of Loop Quantum Gravity (LQG), the understanding of quantum gravity without a cosmological\nconstant is far deeper than that of quantum gravity with a non-vanishing cosmological constant Λ. It is because the\nintroduction of Λ ̸= 0 brings complexity to the quantum geometry construction and it is commonly believed (and\nexemplified in lower dimensional LQG) that new mathematical tools need to be applied. In the covariant, also called\nthe spinfoam, LQG approach, this is especially the case. In 3+1 dimensions (4D), the investigations of spinfoam\nmodels are mostly on those with Λ = 0, among which the most studied one is the EPRL model [1]. It is hoped that\nan adequate understanding of spinfoam model(s) with Λ = 0 can shed light on constructing a Λ ̸= 0 version. In this\npaper, inversely, we present a result from the spinfoam model with Λ ̸= 0 that helps resolve an ambiguity in the\nEPRL model.\nOur analysis is based on the spinfoam model introduced in [2]. It describes Lorentzian 4D quantum gravity with a\neither positive or negative cosmological constant, which is taken as a global coupling constant and whose sign depends\non the boundary geometrical condition. One of the advantages of this spinfoam model compared to other existing\nones with Λ ̸= 0 ( e.g.[3–8]) lies in that it not only illustrates discrete curved geometry in its semi-classical regime but\nalso manifests the expected finiteness in the amplitude. In this paper, we give a complete description of the amplitude\nin this spinfoam model for a general 4-complex and show that it retains the finiteness property. We then focus on\nthe semi-classical regime of the amplitude and analyze its asymptotic behaviour and geometrical interpretation. As\ndesired, the peak of the amplitude can be interpreted as 4-simplices glued together by identifying boundary geometries\nto form a 4-complex. This is consistent with the preliminary result in the original paper [2].\nAt the semi-classical regime of the amplitude, an action of configuration variables can be constructed. Of particular\ninterest in this paper are the equations of motion concerning the internal spins jf’s that dress the internal triangles\nof the 4-complex. In the EPRL model, the use of Poisson resummation for jf’s in the semi-classical approximation of\nthe amplitude gives infinite sums in the following form (see e.g. [9])\nZEPRL =X\n{jf}∈N/2Y\nfA(jf)Z\ndµ(X)eP\nfjfFf(X)=X\n{uf}∈ZZY\nfA(jf)d(2jf)Z\ndµ(X)eP\nfjf(Ff(X)+4πiuf),(1)\nwhere Xdescribes all spinfoam integration variables other than jf’s,µ(X) the collection of their measures and Ff(X)\na function on these variables. By the stationary phase analysis in the large- jregime, the real critical point1describes\nthe deficit angles εf’s hinged by the internal triangles [10]. However, in this model, it is not clear how many uf’s\ncontribute to the critical point, so in principle, one needs to perform the stationary phase analysis for every {uf},\nalthough there are numerical evidences that only one ufcontributes in simple models [9]. The deficit angles may\ntake values εEPRL\nf = 4πZ/γ[11, 12], where γis the Barbero-Immirzi parameter. The critical point with εEPRL\nf = 0\ncorresponds to the smooth flat geometry, since the 4-simplices are endowed with the flat geometry.\nWe observe that the spinfoam model with Λ ̸= 0 involves a similar Poisson resummation, but there is only one{uf}\ncorresponding to the dominant contribution in the semi-classical approximation. In contrast to (1), the semiclassical\namplitude receives the dominant contribution from only one term:\nZΛ≃Z\nY\nfA(jf)d(2jf)\nZ\ndµ(X)eS({jf},X)+P\nf4πiufjf, u f∈Zfixed∀f . (2)\nWith a certain choice of face amplitude and a chosen lift of some phase space coordinate to its logarithmic correspon-\ndence, the critical deficit angle can take the value εf= 4πZ/γ, similar to the EPRL model. In the special case that\nεf= 0 for all f, it describes a smooth dS or AdS spacetime because the 4-simplices are all constantly curved with\nconsistent Λ.\nThe solution εf= 0, describing an (A)dS spacetime, only appears as the real critical solution, when we consider the\naction as a function of real configuration variables. By analytic continuation, we also find a complex critical solution\nthat gives a non-trivial deficit angle hinged by each internal triangle. This means the spinfoam model does not suffer\nfrom an “(A)dSness problem” but allows freedom of intrinsic curvature at the semi-classical regime, just as how the\nflatness problem in the EPRL model is resolved [9].\nThis paper is organized as follows. In Section II, we give a concise review of the spinfoam model with Λ ̸= 0\nintroduced in [2], focusing on constructing the vertex amplitude of the spinfoam model. In Section III, we propose a\n1The real critical point is inside the integration domain understood as a real manifold. The complex critical point is in the complexified\nintegration domain and in general away from the real integration domain.3\nprecise form of the edge amplitude that describes the gluing of 4-simplices. We complete the construction in Section\nIV by further fixing the face amplitudes, which allows us to write the full amplitude for any given 4-complex. We\nthen perform the stationary analysis on the spinfoam amplitude to find the critical solutions. This is done in two\nparts. Firstly, we derive in Section V the real critical solution that describes curved 4-simplices and their gluing.\nWe then focus on the real critical solution to the deficit angle in Section VI. In Section VII, we discuss the complex\ncritical solution and find a non-trivial critical deficit angle. After these general analyses, we give a concrete example\nwith a so-called ∆ 34-complex and illustrate the critical behaviour of the corresponding spinfoam amplitude. We\nconclude and give outlooks in Section IX. Some details and existing results supporting the analysis are supplied in\nthe Appendices.\nII. PRELIMINARY: 4D SPINFOAM WITH Λ̸= 0FROM BOUNDARY CHERN-SIMONS THEORY\nIn this section, we give a concise review of the spinfoam model introduced in [2] which describes 4D quantum gravity\nwith a non-vanishing cosmological constant Λ in Lorentzian signature. For more details, we refer to the original paper\n[2] and a more recent one [13].\nA. Classical theory\nThe starting point is the Plebanski action [14] of the first-order 4D gravity with Λ on a 4-ball B4:\nSPlebanski [e,A] =−1\n2Z\nB4Tr\u0014\u0012\n⋆+1\nγ\u0013\n(e∧e)∧\u0012\nF(A) +Λ\n6(e∧e)\u0013\u0015\n, (3)\nwhere eis the cotetrad one-form valuing in sl(2,C),Ais an sl(2,C) connection with F(A) being its curvature two-\nform, ⋆is the Hodge star operation and γis the Barbero-Immirzi parameter which takes a real value. (3) can be\nformulated as a BF action\nSBF[B,A] =−1\n2Z\nB4Tr\u0014\u0012\n⋆+1\nγ\u0013\nB∧\u0012\nF(A) +|Λ|\n6B\u0013\u0015\n(4)\nfollowed by imposing the simplicity constraint B∼=νe∧ewhich encodes the sign ν:= sgn(Λ) of Λ. Consider a path\nintegral of the BF action (4). The (Gaussian) integration in the Bfield reduces the exponent of the integrand to a\nsecond Chern-term, which can be written into two CS actions on the boundary ∂B4≡S3ofB4. That is,\nZ\ndAdB ei\nℓ2pSBF[B,A]=Z\ndAexp\u00123i\n4ℓ2p|Λ|Z\nB4Tr\u0014\u0012\n⋆+1\nγ\u0013\nF(A)∧ F(A)\u0015\u0013\n=Z\ndAd¯Aexp\u0000\nSCS[A] +SCS[¯A]\u0001\n,\n(5)\nwhere ℓp=p\n8πGℏ/c3is the Planck length and the Chern-Simons (CS) action SCS[A] (resp. SCS[¯A]) is a function\nof the self-dual connection A(resp. the anti-self-dual connection ¯A) with a complex coupling t(resp. ¯t). The actions\ntake the form\nSCS[A] =t\n8πZ\nS3Tr\u0014\nA∧dA+3\n2A∧A∧A\u0015\n, S CS[¯A] =¯t\n8πZ\nS3Tr\u0014\n¯A∧d¯A+3\n2¯A∧¯A∧¯A\u0015\n, (6)\nwhere t=k+isand¯t=k−iswith k=12π\nℓ2pγ|Λ|∈Z+, s=γk∈R.\nPerforming the Gaussian integral in Bis equivalent to imposing the constraint F=|Λ|\n3B, so the simplicity constraint\nnow relates the curvature to the cotetrad:\nF=Λ\n3e∧e . (7)\nIn order to impose this simplicity constraint later at the quantum level, we introduce defects on a graph in S3, denoted\nby Γ 5(see the middle graph in blue of fig.1), which contains 5 nodes and 10 links and can be viewed as the dual\ngraph of the triangulation of S3– the boundary of a 4-simplex. The defects, carrying the information of the simplicity\nconstraints, generate boundary conditions of the CS theory on the graph-complement S3\\Γ5and will be quantized to\nboundary states in the quantum theory.\nA set of phase space coordinates on the boundary of S3\\Γ5can be constructed based on the ideal triangulation of\nS3\\Γ5, denoted as T(S3\\Γ5)), as shown in fig.1. It contains 5 ideal octahedra , which are octahedra with truncated4\n¯5\n¯2¯3¯4¯1S3\\Γ5oct(1)¯5\n¯2¯3¯4yxzwoct(2)¯5\n¯3¯4¯1yxzw\noct(3)¯5\n¯2¯4¯1yxzwoct(4)¯5\n¯2¯3¯1yxzw\noct(5)¯4\n¯2¯3¯1yxzw\nFIG. 1: The decomposition of the ideal triangulation T(S3\\Γ5)ofS3\\Γ5into 5 ideal octahedra ( in red ), each\nof which can be decomposed into 4 ideal tetrahedra. The cusp boundaries of the ideal octahedra are shrunk to\nvertices in this figure. Numbers ¯1,¯2,¯3,¯4,¯5with bars denote the 4-holed spheres on ∂(S3\\Γ3). In each ideal\noctahedron, x, y, z, w (labelled in red ) are chosen to form the equator of the octahedron. The same figure\nappears in [2, 15].\nvertices as shown in fig.2b. Every truncated vertex produces a boundary denoted as a cusp boundary . By adding\nan internal edge, an ideal octahedron can be decomposed into 4 ideal tetrahedra , denoted by △, as shown in fig.2a\n(see [2, 13] for more details on such triangulation and see e.g.[16–22] for ideal triangulation on manifolds with other\ntopologies).\nOn the boundary ∂△of each △, the holomorphic CS phase space P∂△, which is the moduli space of framed flat\nconnection2on∂△, is given by a triple of Fock-Goncharov (FG) coordinates ( z, z′, z′′)∈(C∗)3dressing the edges of\n△as in fig.2a subject to a constraint:\nP∂△={z, z′, z′′∈C∗|zz′z′′=−1} ∈(C∗)2. (8)\nThe anti-holomorphic phase space and the corresponding symplectic form are defined in the same way in terms of the\ncomplex conjugated FG coordinates ( z,z′,z′′). The constraint as shown in (8) eliminates z′andz′from the phase\nspace coordinate. The symplectic form of the boundary CS phase space takes the form\nωk,s=t\n4πΩ +¯t\n4πΩ,with Ω =dz′′\nz′′∧dz\nz,Ω =dz′′\nz′′∧dz\nz, (9)\nwhich motivates us to take the logarithm of the FG coordinates, each with a randomly chosen but fixed lift: Z:=\nlog(z), Z′:= log( z′), Z′′:= log( z′′) and similarly for the anti-holomorphic counterparts, so that the canonical pairs\n2A framed flat connection is a flat connection with a flat section s, called the framing flag , in an associated CP1bundle over every cusp\nboundary, satisfying (d −A)s= 0 [16, 19, 23].5\nz′zz′′z′′zz′\n(a)\nx′′y′w′′x′z′′w′y′′z′xyx′y′′ww′x′′zz′w′′y′z′′xyzw\n(b)\nFIG. 2: (a)An ideal tetrahedron whose edges are dressed with FGcoordinates z, z′orz′′. Each pair of opposite\nedges are dressed with the same coordinate. The cusp boundaries are shown in gray .(b)An ideal octahedron.\nChoose the equator to be edges dressed with x, y, z, w . Adding an internal edge ( in red ) orthogonal to the\nequator separates the ideal octahedron into four ideal tetrahedra, each of which is dressed with different copies\nof coordinates (x, x′, x′′),(y, y′, y′′),(z, z′, z′′),(w, w′, w′′). For edges shared by different ideal tetrahedra,\ncoordinates are multiplied together.\nhave the standard Poisson brackets: {Z, Z′′}Ω={Z,Z′′}Ω= 1. We reparametrize these logarithmic coordinates in\nterms of a pair of real variables ( µ, ν)∈R2and a pair of periodic discrete variables ( m, n)∈(Z/kZ)2by\nZ=2πi\nk(−ibµ−m), Z′′=2πi\nk(−ibν−n),Z=2πi\nk\u0000\n−ib−1µ+m\u0001\n,Z′′=2πi\nk\u0000\n−ib−1ν+n\u0001\n, (10)\nwhere bis a phase related to the Barbero-Immirzi parameter by b2=1−iγ\n1+iγwith positive real part Re( b)>0 and\nnon-zero imaginary part Im( b)̸= 0.\nThe moduli space of flat connection on △, defined by L△={(z, z′′)∈ P ∂△|z−1+z′′−1 = 0}is a Lagrangian\nsubmanifold of P∂△3. The algebraic curve equation z−1+z′′−1 = 0, therefore, restricts the connection on △to be\nflat. This will play a key role in the critical property of the spinfoam amplitude we analyze in Section V.\nB. Quantum theory – the vertex amplitude\nThe new variables µ, ν, m, n are quantized into operators and their Poisson brackets are at the same time quantized\ninto commutation relations as follows.\n{µ, ν}ω={n, m}ω=k\n2π,{µ, n}ω={ν, m}ω= 0 −→ [µ,ν] = [n,m] =k\n2πi,[µ,n] = [ν,m] = 0 .(11)\nThe spectra of µ,νare real while those of m,nare discrete and bounded to be Z/kZ. It is then natural to define the\nkinematical Hilbert space to be Hkin\nk,s=L2(R)⊗CCkwhereCkis ak-dimensional vector space.\nThe building block of the vertex amplitude is provided by the CS partition function on each △, which is the quantum\n3xE=z, z′, z′′can be defined in terms of framing flags parallel transported from the four cusp boundaries of △(see (B2)). One can\ndirectly check that, for the case of nilpotent monodromies, the following equations are indeed satisfied.\nzz′z′′=−1, z−1+z′′−1 = ( z′′)−1+z′−1 = ( z′)−1+z−1 = 0 .6\ndilogarithm function Ψ△(µ|m) of the “position variables” ( µ, m) of the phase space coordinates\nΨ△(µ|m) =\n\n∞Q\nj=01−qj+1z−1\n1−˜q−j˜z−1,if|q|>1\n∞Q\nj=01−˜qj+1˜z−1\n1−q−jz−1,if|q|<1. (12)\nHere µis analytically continued to be a complex variable and zis changed to ˜ zaccordingly (as it is no longer the\ncomplex conjugate of z).qand ˜qencode the CS couplings and play the role of quantum parameters:\nq= exp\u00124πi\nt\u0013\n= exp\u00142πi\nk(1 +b2)\u0015\n≡eℏ,˜q= exp\u00124πi\n¯t\u0013\n= exp\u00142πi\nk(1 +b−2)\u0015\n≡e˜ℏ. (13)\nThe classical limit is at ℏ,˜ℏ→0 or equivalently k→ ∞ . Importantly, Ψ △(µ|m) is holomorphic only in the upper\nhalf-plane Im( µ)>0 whereas has simple poles at the origin and in the lower half-plane Im( µ)≤0. More precisely,\nthe poles are located at\nµpole=ibu+ib−1vwith u, v∈Z−∪ {0}and u−v=−m+kZ. (14)\nIn order to obtain an absolutely convergent integrals on (Fourier transform of) Ψ △(µ|m), which are essential for a\nfinite result of the spinfoam amplitude as we will show later, the integration contour of µneeds to be shifted to\navoid the poles. This was the motivation to introduce imaginary parts α= Im( µ) and β= Im( ν) to the continuous\nparameters µandνrespectively in the quantum theory. ( α, β) are chosen within a region called the positive angle\nstructure [20] of Ψ △, denoted as P△. However, only the real parts Re( µ) and Re( ν) are quantized while α, βare kept\nclassical.\nGluing △’s reflects symplectic transformations on the phase space coordinates. In addition, a constraint on the\nFG coordinates is imposed on each internal edge created from gluing △’s. Given the (logarithmic) FG coordinates\n{Z△\nE,eZ△\nE}△∋Eon an internal edge Efrom different △’s, such a constraint and its quantization take the following\nform.\nCE=X\n△∋EZ△\nE= 2πi ,eCE=X\n△∋EeZ△\nE= 2πi−→ CE= 2πi+ℏ,eCE= 2πi+˜ℏ. (15)\nInT(S3\\Γ5), internal edges are those added in the ideal octahedra to separate each ideal octahedron into 4 △’s (see\nfig.2b). Consider 4 copies of (logarithmic) FG coordinates ( X, X′, X′′),(Y, Y′, Y′′),(Z, Z′, Z′′),(W, W′, W′′), each\nfor one △in an ideal octahedron. The constraints and their quantizations are\nµX+µY+µZ+µW= 0\nmX+mY+mZ+mW= 0−→µX+µY+µZ+µW=iQ\nmX+mY+mZ+mW= 0, Q =b+b−1, (16)\nwhere {µi, mi}i=X,Y,Z,W are the parameters of different FG coordinate copies defined in the same way as in (10)\nand{µi,mi}i=X,Y,Z,W are their quantization respectively. Such constraints allow us to eliminate one set of FG\ncoordinates, say ( W, W′, W′′), by symplectic quotient. As a result, the CS partition function on an ideal octahedron\nis\nZoct(x, y, z ; ˜x,˜y,˜z) =∞Y\ni,j,k,l =01−qi+1x−1\n1−˜q−i˜x−11−qj+1y−1\n1−˜q−j˜y−11−qk+1z−1\n1−˜q−k˜z−11−qlxyz\n1−˜q−l−1˜x˜y˜z, (17)\nwhere ( x, y, z ; ˜x,˜y,˜z)≡exp[( X, Y, Z ;eX,eY ,eZ)]. The positive angle structure PoctofZoctis different from P△but is\nproven in [2] to be a non-empty region.\nGluing 5 ideal octahedra to form T(S3\\Γ5) does not introduce more internal edges but the partition function on\nS3\\Γ5is subject to a series of symplectic transformations on the FG coordinates which can be summarized in a\nsymplectic matrix Mdefined as follows.\n\u0012⃗Q\n⃗P\u0013\n=M\u0012⃗Φ\n⃗Π\u0013\n+\u0012\niπ⃗t\n0\u0013\n,M=\u0012A B\n−(B⊤)−10\u0013\n, (18)\nwhere AandBare 15 ×15 matrices with integer entries and ⃗tis a length-15 vector with integer elements. ( ⃗Φ,⃗Π)⊤is a\nvector of coordinates in the CS phase space P∂(S3\\Γ5)≡N5\ni=1P∂(oct) iof 5 copies of octahedron boundaries with ⃗Φ =7\n(Xi, Yi, Zi)⊤\ni=1,···,5being the position variables and ⃗Π = ( PXi:=X′′\ni−W′′\ni, PYi:=Y′′\ni−W′′\ni, PZi:=Z′′\ni−W′′\ni)⊤\ni=1,···,5\nbeing the conjugate momenta before the symplectic transformations. ⃗Q,⃗Pare the position and momentum variables\nafter the transformations respectively.\nParametrizing these coordinates as ⃗Q=2πi\nk(−ib⃗ µ−⃗ m) and ⃗P=2πi\nk(−ib⃗ ν−⃗ n) with ⃗ µ, ⃗ ν∈C15, ⃗ m,⃗ n ∈(Z/kZ)15,\nthe resulting partition function is written as [13]\nZS3\\Γ5(⃗ µ|⃗ m) =4i\nk15X\n⃗ n∈(Z/kZ)15Z\nC×15d15⃗ ν(−1)⃗t·⃗ neiπ\nk(−⃗ ν·AB⊤·⃗ ν+⃗ n·AB⊤·⃗ n)e2πi\nk[−⃗ ν·(⃗ µ−iQ\n2⃗t)+⃗ n·⃗ m]Z×(−B⊤⃗ ν|−B⊤⃗ n),(19)\nwhere the integration contour C×15is along ⃗β:= Im( ⃗ ν) which satisfies the positive angle structure P(S3\\Γ5) of\nZS3\\Γ5. See [2] for more details on P(S3\\Γ5).\nDifferent from the fact that elements of ⃗Φ and ⃗Π are coordinates on edges of ideal octahedra, elements of ⃗Qand\n⃗Pare coordinates on annuli, which are the boundaries created by removing the edges of Γ 5from S3, and coordinates\non 4-holed spheres {Sa}a=1,···,5, which are the boundaries created by removing the 4-valent-nodes of Γ 5from S3. We\ndenote these coordinates as follows.\n⃗Q= ({2Lab}a b . There is a constraint Lba=−Labon each ( ab) due to the gluing of ideal tetrahedra\nto form ideal octahedra [2].\nSimplicity constraints can be separated into first-class type, which we impose strongly, and second-class type, which\nwe impose weakly, at the quantum level as how we treat them in the EPRL model [24]. The first-class simplicity\nconstraints correspond to flat connections on the annuli while the second-class ones correspond to those on the 4-\nholed spheres. Parametrize each FN length as 2 Lab:=2πi\nk(−ibµab−mab). The first-class constraints require that\n2Lab∈iRhence Re( µab) = 0. Imposing this quantumly means that we require the partition function, or the\nquantum state, ZS3\\Γ5(⃗ µ|⃗ m) to satisfy Re( µab)ZS3\\Γ5(⃗ µ|⃗ m) = 0. Such quantum states are those labeled by “spins”\njab:=mab/2∈ {0,1\n2,···,k−1\n2}dressing the annuli (since αab= Im( µab) is not quantized):\nZS3\\Γ5({iαab}a0) or\nan AdS spacetime (when Λ <0) in the semiclassical regime. We call this the “(A)dSness” property of this spinfoam\nmodel.\n11We also have the symmetry Tab=−Tbaas the FN lengths. When one expresses Tv\nfandeTv\nfin (62) explicitly as {Tv\nab,eTv\nab}vwith a < b ,\na minus sign appears when two annuli of opposite orientations are glued.16\nVII. AWAY FROM THE (A)DS-NESS\nIn Sections V and VI, we have only considered the real critical points of the spinfoam amplitude. We can, neverthe-\nless, consider the complex critical points by extending µ, ν, m, n in the parametrizations of the phase space coordinates\nto complex variables12. A complex critical point can be seen as the shift of a real critical point from the real axes to\nthe complex (hyper-)plane. This is given by Hormander’s theorem (Theorem 7.7.12 of [31], see also Theorem 2.3 of\n[32]), which we formulate into Theorem VII.1 below as a special case.\nWe first express the spinfoam amplitude Z⃗ˆρ∂(⃗ α|⃗jb) (38) for a 4-manifold M4in the large- kregime:\nZ⃗ˆρ∂(⃗ α|⃗jb)k→∞− − − − → N totX\n⃗ p∈Z15VX\n⃗ u∈Z5VX\n⃗ uf∈ZF\nEinY\ne=1Z\nM⃗jvadˆρv∈e\naZ\nM⃗jv′\nbdˆρv′∈e\nb\n\nZ\nC×Fin\nLfFin^\nf=1d (i2Lf) (i2Lf)p\nI,\nwithI=Z\nC×40V\nMvVY\nv=1dMvexp (kS), (71)\nwhere Ntot=\u0000k\n2π\u0001Fin\u0000k\n4π2\u00012EinNV, and the action Sis defined in (45). For the integral I, where the measure d Mv\nis defined in (41), all the coherent state labels {ˆρv\na}and the FN lengths {2Lf}on the internal spinfoam faces are\nregarded as boundary data, which we collect in a vector ⃗ r∈Rmof real variables with length m= 10 V+Fin.S\nis a function of n= 40Vreal variables ⃗ x={{νv\nI, nv\nI}15\nI=1,{µv\na, mv\na}5\na=1}V\nv=1∈Rn, which are the parametrization of\n{{Pv\nI,ePv\nI}15\nI=1,{Xv\na,eXv\na}5\na=1}V\nv=1. Due to the use of Poisson resummation, {nv\nI, mv\na}are all continuous variables with\nintegration range [ −δ, k−δ]. The integral Ican be approximated in terms of complex critical points as follows.\nTheorem VII.1. Let⃗ x0∈Rnbe a real critical point of the action S(⃗ x,⃗ r)defined in (45) with⃗ xand⃗ rdefined above\nwhere the Hessian is non-degenerate at the critical points, i.e. det\u0000\n∂2\n⃗ x⃗ xS\u0001\f\f\n⃗ x=⃗ x0̸= 0, then\nRe (S(⃗ x,⃗ r))≤0,Re (S(⃗ x0(⃗ r),⃗ r)) = 0 ,∂S(⃗ x,⃗ r)\n∂⃗ x\f\f\f\f\n⃗ x=⃗ x0(⃗ r)= 0. (72)\nAnalytic continue ⃗ xto⃗ z=⃗ x+i⃗ y∈Cnnear the critical point ⃗ x0with|⃗ y|small, and solve∂S(⃗ z,⃗ r)\n∂⃗ z= 0for a complex\ncritical point ⃗ z0(⃗ r). Then at the critical point ⃗ z0(⃗ r), there exists some 0< C < ∞such that\nRe (S(⃗ z0(⃗ r),⃗ r))≤ −C|Im (⃗ z0(⃗ r))|. (73)\nSuppose that S(⃗ x,⃗ r)has finitely many real critical points {⃗ x(α)\n0}α, and{⃗ z(β)\n0}βis a collection of the complex critical\npoints at their neighbourhood ( βis not necessarily equal to α). Then the integral Idefined in (71)can be approximated\nas\nI=\u00121\nk\u0013n\n2X\nβekS(⃗ z(β)\n0,⃗ r)\nr\ndet\u0010\n−H⃗ z(β)\n0/(2π)\u0011(1 +O(1/k)),with H⃗ z0=∂2S(⃗ z,⃗ r)\n∂⃗ z2\f\f\f\f\n⃗ z=⃗ z(β)\n0. (74)\nThe proof follows [31]. For self-consistency, we provide proof in Appendix D. As analyzed in the previous sec-\ntion, a real critical point corresponds to a zero deficit angle hinged by an internal triangle (when sgn( V4) = 1 for\nall 4-simplices). In contrast, a complex critical point gives a non-zero deficit angle. At the semi-classical regime,\ntherefore, this theorem states that a real critical point corresponds to an (A)dS geometry while a complex critical\npoint corresponds to a non-(A)dS geometry.\nThe complex critical point ⃗ z0(⃗ r) is an analytic function of the boundary parameter ⃗ rwith a real-vector value\n⃗ z0(⃗ r0) =⃗ x0at⃗ r=⃗ r0.⃗ z0(⃗ r) deviates from the real space RntoCnwhen ⃗ rdeviates from ⃗ r0with a finite distance, as\nillustrated in fig.4. On the other hand, when the critical point is real, the critical action contributes to an oscillatory\nphase. Eq.(73) means that the amplitude decays when the critical point is complex, and that the further the complex\ncritical point is away from the real space, the faster the amplitude decays. At large k, the contribution from complex\ncritical points is dominated by the one closest to the real space as others exponentially decay much faster.\n12For clarity, the imaginary parts Im( µ) =αand Im( ν) =βin previous sections are fixed and are there only for convergent contour\nintegrations so µ= Re( µ) and ν= Re( ν) are still considered real variables in the amplitude. Here, we extend these real parameters to\ncomplex variables.17\nRe(z)Im(z)\nx0(r)z0(r)•\nFIG. 4: A complex critical point ⃗ z0(⃗ r)in the neighbourhood of a real critical point ⃗ x0(⃗ r).\nVIII. AN EXAMPLE: SPINFOAM AMPLITUDE OF THE ∆34-COMPLEX\nThe simplest example where one can apply the formalism (71) and the above theorem is the ∆ 34-complex, where\nthere is only one internal triangle and it is shared by three 4-simplices, denoted as v, v′andv′′. We will denote\nelements on v′with primes and those on v′′with double primes accordingly in this section.\nThe diagram of the 3-manifold corresponding to ∆ 34-complex is illustrated in fig.5a, which has a similar pattern\nas the cable diagram of the ∆ 34-complex (see e.g.[9]). The patterns (the way the annuli connect different 4-holed\nspheres) in different S3\\Γ5’s are identical. Denote the gluing, or identifying, of 4-holed spheres by ∼, then the internal\ntriangle comes from\nv∋ S2∼ S′\n1∈v′, v′∋ S′\n2∼ S′′\n1∈v′′, v′′∋ S′′\n2∼ S1∈v . (75)\nFig.5b illustrates the gluing of S2andS′\n1as an example. It identifies the FN and FG coordinates from the two\nsimplices as follows (we take κab= 1 for a < b andκab=−1 otherwise in this section).\neL21=e−L′\n12, eL23=e−L′\n14, eL24=e−L′\n15, eL25=e−L′\n13, eX2=eY′\n1, eY2=eX′\n1. (76)\nSimilar relations are true for the other two gluings S′\n2∼ S′′\n1andS′′\n2∼ S1. These constraints result from identifying\nthe framing flags on the glued holes. For instance, as illustrated in fig.5b, the exponential FG coordinates e−Y2onS2\nandeX′\n1onS′\n1are defined in terms of the framing flags as (see Appendix B, especially (B2) and fig.6)\ne−Y2=⟨s4∧s1⟩⟨s3∧s2⟩\n⟨s4∧s3⟩⟨s1∧s2⟩, eX′\n1=⟨s′\n3∧s′\n1⟩⟨s′\n2∧s′\n4⟩\n⟨s′\n3∧s′\n2⟩⟨s′\n1∧s′\n4⟩, (77)\nwhere siis the framing flag on hole iofS2parallel transported to a common point in S2ands′\njis the framing flag on\nholejofS′\n1parallel transported to a common point in S′\n1. The identification of framing flags s1∼s′\n1, s2∼s′\n3, s3∼\ns′\n2, s4∼s′\n4leads to the constraint eY2−X′\n1= 1 hence Y2− X′\n1= 0 with a chosen lift. Other constraints in (76) can be\nobtained in the same manner. We collect them in Appendix C.\nThe edge amplitudes for such a gluing is\nAe(ˆρv\n2,ˆρv′\n1|j12, j23, j24, j25, j′\n12, j′\n13, j′\n14, j′\n15) =δj12,j′\n12δj23,j′\n14δj24,j′\n15δj25,j′\n13exp[Se(ˆρv\n2,ˆρv′\n1)], (78)\nwhere ˆ ρv\n2= (ˆz2,ˆx2,ˆy2) and ˆ ρv′\n1= (ˆz′\n1,ˆx′\n1,ˆy′\n1). The action Se(ˆρv\n2,ˆρv′\n1) reads\nSe(ˆρv\n2,ˆρv′\n1) =−1\n4π\u0010\n(Re(ˆz2) + Im(ˆ z′\n1))2+ (Re(ˆ z′\n1) + Im(ˆ z2))2+ (ˆx2+ ˆy′\n1)2+ (ˆy2+ ˆx′\n1)2\u0011\n+i\n4π(4Im(ˆ z′\n1)Im(ˆz2) + ˆx2ˆy2+ ˆx′\n1ˆy′\n1+ 2ˆy2ˆy′\n1). (79)\nThe other two edge amplitudes take the same form except for changing the same elements in vtov′(resp. the same\nelements in vtov′′) and those in v′tov′′(resp. those in v′tov). At the large- kapproximation of the full amplitude18\nS2=S′\n1\nS′\n2=S′′\n1 S′′\n2=S1S3\nS4\nS5S′\n5\nS′\n4\nS′\n3\nS′′\n5\nS′′\n4S′′\n3v v′\nv′′\n(a)\n−Y2\nX2X′\n1\n−Y′\n1L25=−L′\n13\nL23=−L′\n14L21=−L′\n12\nL24=−L′\n153\n14\n22\n14\n3\nS2 S′\n1\ns1 s2s3 s4\ne−Y2\ns′\n1 s′\n4s′\n2 s′\n3\neX′\n1(b)\nFIG. 5: (a)Diagram of the 3-manifold corresponding to the ∆34-complex. The ambient 3-manifold ( in black )\nhas one non-contractable cycle. The (non-intersecting) blue lines denote the annuli and the red loop denotes\nthe torus corresponding to the internal triangle shared by three 4-simplices. The 3-manifold on which the CS\namplitude is defined is the graph (composed of the blue lines and the red loop) complement of the ambient\n3-manifold. ( b) The upper panel illustrates the gluing of S2andS′\n1. Numbers 1,2,3,4 label the holes and the\ndotted lines denote the annuli dressed with FN coordinates that identify the holes pairwise. Edges in thick are\ndressed with the FG coordinates on the 4-holed spheres. The lower panel illustrates the quadrilateral to define\ne−Y2andeX′\n1through (77).\nfor the ∆ 34-complex, the following constraints on the FG coordinates are obtained by solving the equations of motion\nas in Section III.\nX2− Y′\n1=X′\n2− Y′′\n1=X′′\n2− Y1= 0 = Y2− X′\n1=Y′\n2− X′′\n1=Y′′\n2− X 1. (80)\nWe can embed the phase spaces for different S3\\Γ5’s into the full phase space for the 3-manifold after gluing. Under\nthe standard Poisson bracket, the constraints (80) can be checked to be all first-class.\nWe are in particular interested in the equations of motion from the variation of the internal FN length 2 L12. It\nreads\n∂S\n∂(2L12)=−i\n2π(b2+ 1)h\u0010\nT12−b2eT12\u0011\n+\u0010\nT′\n12−b2eT′\n12\u0011\n+\u0010\nT′\n12−b2eT′\n12\u0011i\n+1\n2π(2a12·2L12+iπb12)−u12= 0,(81)\nwhere u12∈Zcomes from the Poisson resummation of j12anda12, b12∈Rare the coefficients of the face amplitude\nFf(2L12) =a12(2L12)2+iπb12·2L12+c12. The FN twist T12and its tilde sector eT12along the B-cycle of the\ntorus corresponding to this internal triangle are the linear combinations of 2 L12and its conjugate momenta on three\ndifferent 4-simplices. Explicitly13,\nT12=T12+T′\n12+T′′\n12+r12·2L12+iπs12,eT12=eT12+eT′\n12+eT′′\n12−r12·2L12−iπs12, r 12, s12∈R. (82)\n13When gluing the annuli to form the internal torus corresponding to the FN length L12, the orientations of the annuli are congruent, as\ncan be seen in fig.5a, hence there is no sign difference for T12,T′\n12andT′′\n12in (82).19\nParametrize the FN twist as\nT12=2πi\nk(−ibν12−n12),eT12=2πi\nk\u0000\n−ib−2ν12+n12\u0001\n, ν 12∈R, n 12∈Z/kZ. (83)\nT12andeT12are related to the dressed deficit angle ε(s)\n12hinged by the internal triangle through\nT12=−1\n2νε(s)\n12+ 2πiN 12,eT12=−1\n2νε(s)\n12−2πiN 12, N 12∈Z. (84)\nDefine the face amplitude Ff(2L12) by fixing a12=−1\n2r12andb12=−s12, then (81) can be simplified to be\n−n12\nk−u12= 0, (85)\nwhose only solution, when n12is restricted to [0 , k), is\nn12= 0, u 12= 0. (86)\nThis is the real critical solution to the action of (the large- kapproximation of) the amplitude for the ∆ 34-complex.\nEquating (83) with (84), one gets (recalling that Q= (b+b−1) = 2Re( b) and Im( b) =−γRe(b) =−γQ\n2)\n2πb\nkν12≡πQ\nkν12−iπγQ\nkν12=−1\n2νε(s)\n12+ 2πiN 12=⇒ γε(s)\n12= 4πνN 12∈4πZ. (87)\nExtending the variables {{νI, nI, ν′\nI, n′\nI, ν′′\nI, n′′\nI}15\nI=1,{µa, ma, µ′\na, m′\na, µ′′\na, m′′\na}5\na=1} ∈R120(at large- kregime) to C120,\nthe critical solution becomes complex by the Hormander’s theorem VII.1. The critical solution renders n12̸= 0, leading\nto a non-vanishing deficit angle. Its contribution to the full amplitude is small compared to the real critical solution\n(86) by Theorem VII.1.\nIX. CONCLUSION AND OUTLOOK\nIn this paper, we have, in a systematical way, given the complete spinfoam amplitude, composed by vertex am-\nplitudes, edge amplitudes and face amplitudes, for a general 4-complex as the triangulation of a spacetime manifold\nwhen a non-vanishing cosmological constant is present. It is formulated as finite sums and convergence integrals on\nthe symplectic coordinates of moduli space of SL(2 ,C) flat connection on copies of S3\\Γ5’s and coherent state labels.\nWe have analyzed the critical solutions to the equations of motion at the large- kregime of the full amplitude. The\nreal critical solutions give SU(2) flat connection on the graph complement of the 3-manifold after gluing different\nS3\\Γ5’s through boundary 4-holed spheres. Each such flat connection determines the geometry of all the 4-simplices\nas the sub-cells of the 4-complex under study, hence determining the geometry of the full 4-complex. This means that,\nwhen the 4-volume of all 4-simplices are positive, the amplitude of this spinfoam model peaks at an (A)dS spacetime\ndepending on the sign of the cosmological constant.\nWe have particularly focused on the critical solutions from varying the internal spins, each corresponding to an\ninternal triangle shared by tetrahedra from different 4-simplices, and we observe a similar result as in the EPRL model\nas follows. With the specific definition of the face amplitude, which may vary for different spinfoam faces, and at a\nspecific lift of the phase space coordinate, the real critical point gives a vanishing deficit angle εf= 0 hinged by each\ninternal triangle, and different lifts relate to different deficit angles separated by 4 π/γ. This separation matches that\nof the EPRL model.\nWe have also observed a technical advantage of studying this spinfoam model compared to the EPRL model:\nthe semi-classical approximation formula of the amplitude is simpler in that the infinite many summations coming\nfrom each Poisson resummation of internal spin is reduced to a single sum at the large- kregime. Apart from that,\nanother advantage of this spinfoam model is the finiteness of amplitude for a general 4-complex, which means no\nfurther regularization is needed. With these distinctive features, this spinfoam model extends an invitation for deeper\nexploration and investigation. We list some of the possible directions to look into below.\n•In this work, the full amplitude is constructed by grouping vertex amplitudes, edge amplitudes and face ampli-\ntudes by the local amplitude ansatz. Another way to construct the full amplitude is to first write down the CS\npartition function for the final 3-manifold that corresponds to the 4-complex under study, then couple it with\ncoherent states on the boundary to impose the second-class simplicity constraints. The interpretation of flat\nconnection at the critical points of the action would be better explained if the latter approach is used. However,\nthe difficulty lies in that a symplectic transformation from the FG coordinates on ideal octahedra to suitable\ncoordinates on the final 3-manifold might not exist for a complex 3-manifold. If it exists, it remains the question\nof whether there is a systematic way to perform such a symplectic transformation for a general 3-manifold.20\n•The complex critical deficit angle is only argued to exist in this paper. Having the complete and concrete\nspinfoam model, it is interesting the study the complex critical points numerically as is done in the EPRL model\n[9, 33], and investigate how these complex critical points contribute to the final amplitude. We expect that\nthe finiteness of amplitude would bring benefit to the numerical study. Furthermore, when it involves solving\ncritical solutions to the action, the feature that only polynomial equations are involved (see discussion in [13]\nfor more details) could also boost the numerical operation compared to that of the EPRL model.\n•The form of the face amplitude (36) is based on the conjecture that the boundary Hilbert space is spanned by\nsome q-deformed spin network states with qa root-of-unity. To investigate if it is true, one needs to construct\nexplicitly the coherent intertwiners spanning such Hilbert space and clarify if there is a canonical bijection\nbetween the coherent intertwiners and the boundary data in the spinfoam model. A first step to construct the\ncoherent intertwiners on a homogeneously curved tetrahedron has been initiated in [34], and these coherent\nintertwiners span the intertwiner Hilbert space on a curved tetrahedron defined in [35].\n•An important question is how this spinfoam model relates to the Hamiltonian constraint in the canonical\napproach. It would be a difficult task for the general setting. To begin with, one can study the truncated\nmodel. As the dS spacetime is at the critical points of the spinfoam model, it is interesting to apply it to the\ncosmological setting by imposing (discretized version of) isotropic and homogeneous conditions. The numerical\nmethod could be also helpful for the analysis.\nAcknowledgments\nThis work receives support from the National Science Foundation through grants PHY-2207763, PHY-2110234, the\nBlaumann Foundation, the Jumpstart Postdoctoral Program at FAU, and the College of Science Research Fellowship\nat FAU.\nAppendix A: Fock-Goncharov coordinates and the Fenchel-Nielsen coordinates\nThe FG coordinates {χ(a)\nij}that dress the edges in the ideal triangulation of 4-holed spheres on S3\\Γ5are related\nto the coordinates {{Lab}a0\nfor either ν[25].\nξiis in fact the (normalized) framing flag siparallel transported to the base point pai.e.ξi=si\n||si||. Therefore,\n{ξi}i=1,···,4can be used to define the FG coordinates as in (B2).\nThe FN lengths admit the symmetry Lab=−Lba, which geometrically means that t(a)\niand the triangle t(b)\njonTb\ncorresponding to some H(b)\nj, such that hole iofSaand hole jofSbare connected by annulus ( ab), share the same\nareaaab. We can also diagonalize this H(b)\nj:\nH(b)\nj=M(ξ′\nj) diag( e2πijab\nk, e−2πijab\nk)M(ξ′\nj)−1, M (ξ′\nj) =\u0000\nξ′\nj, Jξ′\nj\u0001\n, (B8)\nwhere |ξ′\nj⟩ ≡ξ′\njand|ξ′\nj]≡Jξ′\njare defined in the same way as ξiandJξiin (B5). The normal of t(b)\nj, which is defined\nas ˆnba=⟨ξ′\nj|⃗ σ|ξ′\nj⟩=νˆnbaifjab∈[0, k/4) while ˆ nba=−⟨ξ′\nj|⃗ σ|ξ′\nj⟩=νˆnbaifjab∈[k/4, k/2), is in general different from\nˆnabsince they are calculated in different tetrahedron local frames. We can also drop the label for holes and denote\nHab≡H(a)\niandHba≡H(b)−1\nj , whose parametrizations (B5) and (B8) can be equivalently written as [7]\nHab=eΛ\n3aabˆnab·⃗ τ, H ba=e−Λ\n3aabˆnba·⃗ τ, (B9)\nwhere ⃗ τ=1\n2i⃗ σ. (Note that eLbais the eigenvalue of H(b)\njinstead of the eigenvalue of Hba.)HabandHbaare related\nthrough conjugation by an SL(2 ,C) element, denoted as Gab:\nHab=GabHbaG−1\nab. (B10)\nGabdescribes the parallel transport of the reference frame of TatoTb. There is no canonical choice for Gaband each\ndescribes the parallel transport along a path on the annulus ( ab) of the reference frame of TatoTb15.\nBy the factorizations (B5) and (B8) of HabandHbarespectively, (B10) can be reformulated as\n\u0012λab0\n0λ−1\nab\u0013\nM−1\naGabMb=M−1\naGabMb\u0012λab0\n0λ−1\nab\u0013\n, λ ab=e−i|Λ|\n6aab, (B11)\n15An apparent example is Gab=M′(ξi)M(Jξ′\nj)−1∈SU(2). However, complex rescalings of ξiandξ′\njs.t.|ξi⟩ →λ|ξi⟩,|ξi]→λ−1|ξi]\nand|ξj⟩ →λ′|ξj⟩,|ξ′\nj]→λ′−1|ξ′\nj] with any λ, λ′∈C\\{0}preserve the relation (B10).23\nwhere Ma=M(ξi) and Mb=M(ξ′\nj) . This means M−1\naGabMb∈U(1) and can be parametrized as\nM−1\naGabMb=\u0012γ′\nab 0\n0γ′−1\nab\u0013\n, γ′\nab=(\nγab, ifjab∈[0,k\n4)\nγ−1\nab, ifjab∈[k\n4,k\n2), γ ab=eψab+iθab, ψ ab∈R, θab∈[0,2π).\n(B12)\nIn the rest of the derivation, we will eliminate the labels of holes on TaandTb. When hole iofTais glued to hole j\nofTbthrough annulus ( ab), we denote\nξab:=(\nξi, ifjab∈[0,k\n4)\nJξi, ifjab∈[k\n4,k\n2), ξ ba:=(\nξ′\nj, ifjab∈[0,k\n4)\nJξ′\nj, ifjab∈[k\n4,k\n2). (B13)\nThen (B12) means that the spinors ξbaandξabare related by parallel transportation in the manifold of SL(2 ,C)\nfollowed by a rescaling. Explicitly,\n|ξab⟩=γ−1\nabGab|ξba⟩,|ξab] =γabGab|ξba]. (B14)\nOne of these formula gives the parallel transport from ξitoξ′\nj, which means ξican be used as the framing flag to\ndefine flat connection on the whole boundary ∂(S3\\Γ5).\nDenote by eHab≡eHba∈SL(2,C) for the holonomy along the meridian loop of the annulus ( ab) in the fundamental\ngroup π1(S3\\Γ5). The set {eHab}a0\n(a)\nxtTaTbNaNb\nΘ<0 (b)\nFIG. 7: Two spacelike tetrahedra TaandTbforming a wedge (2 spacial dimensions are reduced). NaandNb\nare the outward-pointing normal to TaandTbrespectively. (a)A thin wedge with dihedral angle Θab>0.(b)\nA thick wedge with dihedral angle Θab<0.\nIt remains to fix the sign of the correspondence, which is done by the following consideration. Let NaandNb\nbe the outward-pointing normals of TaandTbrespectively in a common frame, which could be future-pointing or\npast-pointing. Denote UaandUbto the the corresponding future-pointing normals. Then Ua=±NaandUb=±Nb.\nThe boost from UatoUbencodes the hyper-dihedral angle in the transformation matrix Lab∈SO(1 ,3)+such that\nLabUa=Ub. Explicitly18,\nLab=e|Θab|Ua∧Ub\n|Ua∧Ub|≡e|Θab|U[I\naUJ]\nbJIJ\n|Ua∧Ub|,withJ0i=Ki,Jij=ϵk\nijJk. (B20)\nLet us check that Labdefined as such does transport UatoUb. With no loss of generality, choose the coordinate system\nsuch that Ua= (1,0,0,0)⊤andNbis on the tx-plane. Then Ub= (cosh Θ ab,sinh|Θab|,0,0)⊤19. AsUa∧(Ub+cUa)≡\nUa∧Ub,∀c∈R, we can choose a vector U′\nbas the (normalized) linear combination of UaandUband is orthogonal\ntoUa. That is, let U′\nb= (0,1,0,0)⊤and hence |Θab|Ua∧U′\nb=|Θab|U0\naU′\nb1J01=|Θab|K1, leading to\nLab=\ncosh Θ absinh|Θab|0 0\nsinh|Θab|cosh Θ ab0 0\n0 0 1 0\n0 0 0 1\n=⇒LabUa=Ub. (B21)\nNote that |Θab|Ua∧Ub=−ΘabNa∧Nbsince when TaandTbform a thin wedge, Θ ab>0 and the time component\nof either NaorNbis negative, while both time components take the same sign when TaandTbform a thick wedge\nand Θ ab<0 (r.f.fig.7).\nTo proceed, we first show identityNa∧Nb\n|Na∧Nb|=νsgn(V4)ˆnab·⃗Kin the following. Consider a homogeneously curved\nspacetime ( M4, gµν). Let eµ\nI(b) be a generic orthonormal frame at a vertex bof the triangle fabshared by TaandTb,\nandϵαβµν be an arbitrary volume element on M4compatible with gµν≡ηµνeµ\nIeν\nI. Then sgn( V4) is defined by the\ncompatibility between ϵαβµν andeI\nα:\nϵ= sgn( V4)e0∧e1∧e2∧e3. (B22)\nThe volume element of fabis then defined by ϵαβ=ϵαβµνNµ\na(b)Nν\nb(b) with Nµ=NIeµ\nI. Then the following relation\nholds.\n⋆(Na(b)∧Nb(b))\n|⋆(Na(b)∧Nb(b))|= sgn( V4)\u0000\nϵαβeαeβ\u0001\nab(b). (B23)\nIn the local frame of Tawhose timelike normal is u= (1,0,0,0)⊤,ϵαβeαeβ=ˆnab·⃗J, which can be viewed as an\nsl(2,C) element. sl(2,C) can be viewed as a 6D algebra with real generators ⃗J=⃗ τand⃗K=−i⃗ τ. Then the duality\n18KiandKiare undistinguished in this paper. Same for JiandJi.\n19The minus sign comes from our convention for the dihedral angle of a thick or thin wedge as illustrated in fig.7.25\nmap ⋆acts as ⋆⃗J=−⃗Kand⋆⃗K=⃗J. Therefore, in the frame of Ta,\n⋆\u0000\nϵαβeαeβ\u0001\n=−ˆnab·⃗K=−νˆnab·⃗K. (B24)\nCombining (B23) and (B24), we conclude that\nNa∧Nb\n|Na∧Nb|=νsgn(V4)ˆnab·⃗K. (B25)\nWe then can re-express (B20) as\nLab(a)≡exp\u0010\n−νsgn(V4)Θabˆnab·⃗K\u0011\n. (B26)\nOn the other hand, Λab(B16) can also be rewritten as\nΛab=\u0010\nRae2ψabK3R−1\na\u0011\u0010\nRae−2θabJ3R−1\nb\u0011\n= exp\u0010\n2ψabˆnab·⃗K\u0011\nR′, (B27)\nwhere we have used the fact that Raˆz= ˆnaband that R′=Rae−2θabJ3R−1\nbis a pure rotation. Both ΛabandLab,\nnow written in the frame of Tbcan transform the normal NbtoNa, which means their boost parts must agree, i.e.\nexp\u0010\n−νsgn(V4)Θabˆnab·⃗K\u0011\n= exp\u0010\n2ψabˆnab·⃗K\u0011\n=⇒ Θab=−2νsgn(V4)ψab. (B28)\nLet us finally relate the hyper-dihedral angle to the FN twist. The definition of the SL(2 ,C) FN twist along an\nannulus ( ab) depends on the choice of another two auxiliary holes on Saand another two auxiliary holes on Sb, or\neffectively depends on the choice of a path on ( ab) connecting SaandSb. Let sabbe the framing flag on ( ab) and\nsac, sad(resp. sbe, sbf) be the framing flags on the other two holes of Sa(resp.Sb) which connect to ScandSd(resp.\nSeandSf) respectively. Then the (exponential) PSL(2 ,C) FN twist is defined as\nτ2\nab=−⟨sbe(pb)∧sbf(pb)⟩\n⟨sbe(p)∧sab(p)⟩⟨sbf(p)∧sab(p)⟩⟨sac(p)∧sab(p)⟩⟨sad(p)∧sab(p)⟩\n⟨sac(pa)∧sad(pa)⟩, (B29)\nwhere pa∈ Sa,pb∈ Sb, and pis a common point for evaluating sab∧s′,∀s′.τ2\nabis indeed invariant under the\nrescaling of framing flags. As we have observed, the role of framing flags can be played by the spinors when they are\ndefined on a common point on the 4-holed sphere. Let us choose p=pb. In order to evaluate the second ratio in\n(B29) at pa, we need to parallel transport the framing flags with Gab:s(pb) =G−1\nabs(pa). Then the second ratio in\n(B29) can be re-expressed as\n⟨G−1\nabsac(pa)∧sab(pb)⟩⟨G−1\nabsad(pa)∧sab(pb)⟩\n⟨sac(pa)∧sad(pa)⟩=⟨G−1\nabξac∧ξba⟩⟨G−1\nabξad∧ξba⟩\n[ξac|ξad⟩=γ2\nab[ξac|ξab⟩[ξad|ξab⟩\n[ξac|ξad⟩, (B30)\nwhere we have used the fact that the produce ⟨· ∧ ·⟩ is SL(2 ,C) invariant hence ⟨G−1\nabξ′∧ξba⟩=⟨ξ′∧Gabξba⟩=\nγab[ξ′|ξab⟩for any ξ′by (B14). We then lift τ2\nabto an SL(2 ,C) FN twist by taking its positive square root τab. We\ncan, therefore, express τabin terms of the spinors as\nτab=γabp\nχab(ξ)≡e−1\n2νsgn(V4)Θab+iθabp\nχab(ξ), χ ab(ξ) =−[ξbe|ξbf⟩\n[ξbe|ξba⟩[ξbf|ξba⟩[ξac|ξab⟩[ξad|ξab⟩\n[ξac|ξad⟩. (B31)\nLetTab= log τabwith a chosen branch/lift. As an FN twist, Tabis conjugate to 2 Labin the sense that {2Lab, Tab}= 1\nand Poisson commutes with {2Lcd}(cd)̸=(ab)and{Xa,Ya}(but not necessarily commutes with Tab). This can be\nchecked by using the framing flag definitions of FN length and FN twist. On the other hand, Tabcan be obtained\nfrom the octahedron FG coordinates by symplectic transformation, which means Tabis a linear combination of the\nFG coordinates ( ⃗Ψ,⃗Π) just as ( ⃗Q,⃗P). We, therefore, conclude that Tabcan be expressed in terms of the canonical\npair (2 Lab,Tab) by linear transformation Tab=r·2Lab+Tab+iπswith r, s∈R. Such a relation makes sense also\ngeometrically: any path on the annulus ( ab) can be approximated by a piecewise smooth path composed of meridian\npieces, contributing some portion of 2 Lab, and longitudinal pieces, contributing some portion of Tab. Therefore,\nTabcorresponding the such a path can be expressed as a linear combination of 2 LabandTab.scomes from affine\ntranslation which does not affect the Poisson structure.\nFor each given boundary condition of the 4-simplex, one can find two solutions AandeAto flat connections which\ncorrespond to opposite 4-volume of the 4-simplex, and they are related by parity transformation, analogous to the26\nsituation in the EPRL model [36]. That is, sgn( V4)|A=−sgn(V4)|eA[6, 9]. Since Tabhas dependence on sgn( V4),\nthese two flat connections in turn gives two solutions to Tab:\nTab|A=−1\n2νsgn(V4)Θab+iπNA\nab+ζab,Tab|eA=1\n2νsgn(V4)Θab+iπNeA\nab+ζab, ζ ab=iθab+1\n2logχab−r·2Lab+iπs ,\n(B32)\nwhere NA\nab, NeA\nab∈Zcorrespond to different lifts whose parities match as they correspond to the same eζab. It leads to\nthe difference of the two momenta\nTab|A− Tab|eA=−νsgn(V4)Θab+ 2πiN ab,with 2 Nab=NA\nab−NeA\nab∈2Z. (B33)\nIn summary, from the above derivation, we have seen that each FN length 2 Labencodes the area of the triangle\ndual to the holes linked by the annulus ( ab) and that its dual FN twist Tabencodes the hyper-dihedral angle hinged\nby this triangle. Such a geometrical interpretation is useful for interpreting the critical solution to the equations of\nmotion for the total amplitude w.r.t. the internal FN lengths in Section VI.\nIt remains to figure out the geometrical interpretation of θabdefined in (B12). Consider again the 4-vector rep-\nresentation ΛabofGaband its action on the triangle fabshared by TaandTb. the plane of fabis spanned by the\nbivector ⋆(Na∧Nb) where ⋆is the Hodge star operator. Λabchanges the frame from TbtoTa. Consider a 4-vector\nVthat represents an edge of fabshared by TaandTb.Vis indeed in the plane of ⋆(Na∧Nb).Λabacts on the Vas\n(we omit the signs νsgn(V4) in the following for conciseness)\nΛabV≡Rae−2θabJ3e−ΘabK3R−1\nbV=\u0010\nRae−2θabJ3R−1\nb\u0011\u0010\nRbe−ΘabK3R−1\nb\u0011\nV=\u0010\nRae−2θabJ3R−1\nb\u0011\ne−Θabˆnba·⃗KV .\n(B34)\nThe boost generated by e−Θabˆnba·⃗Kis along the normal Na∧Nbtofabhence it keeps the plane spanned by the\nbivector ⋆(Na∧Nb), hence Von the plane, invariant. Therefore, (B34) can be simplified to be\nΛabV=Rae−2θabJ3R−1\nbV . (B35)\nR−1\nbrotates the vector ˆ zto−ˆnbain the frame of Tb,e−2θabJ3generates a rotation around the z-axis, and Rarotates\nthe vector ˆ zto ˆnabin the frame of Ta. Therefore, in general, Vis rotated to a different vector by Λab.\nWe are interested in a special case when the parallel transport is along a series of connected tetrahedra within\nthe triangulation of a 4-manifold whose trajectory forms a (non-self-interacting) loop. That is, the initial and final\ntetrahedron in the transportation are the same: Ta=Tb, and we denote Gab=GfandΛab=Λf. In this case, θf\ncan be determined in the following way.\nFirstly, the rotation matrices Ra=Rb=Rin (B34) as ξab=ξba=ξ.Λfmust keep the edge Vinvariant since fab\nremains the same, hence Re−2θfJ3R−1V≡e−2θfˆn·⃗J!=V. where ˆ n= ˆnab= ˆnba. Since e−2θfˆn·⃗Jgenerates a rotation\naround the normal ˆ ntofabby an angle −2θf,Vis kept invariant only when (recall the range θf∈[0,2π))20\n2θf= 0 or 2 π⇐⇒ θf= 0 or π . (B36)\nReturning to the fundamental representation (B12), the choice θf=πchanges γf=γabtoγ−1\nfcompared to the\nchoice θf= 0. The two solutions to θfcan be understood as different lifts from SO(1 ,3)+to SL(2 ,C). In other words,\nif we interpret the lift θf= 0 as a time-oriented map SO(1 ,3)+→SO(1 ,3)+, then the lift θf=πcan be interpreted\nas a time-flipping map SO(1 ,3)+→SO(1 ,3)−. Such an interpretation makes sense because the time-like normal to\na tetrahedron on the boundary of a 4-simplex can be future-pointing or past-pointing. When the 4-manifold, hence\nits triangulation, is globally time-oriented, the unique solution θf= 0 for all f’s is picked. We see in Section VI that\nsuch a solution leads to the uniqueness of the solution to the deficit angle.\nAppendix C: Fock-Goncharov coordinates on S2andS′\n1in∆34-complex\nDenote the framing flag parallel transported from hole iofS2(resp.S′\n1) to a common point on S2(resp.S′\n1) assi\n(resp. s′\ni). Denote the edge of the ideal triangulation of S2(resp.S′\n1) connecting hole iand hole jaseij(resp. e′\nij).\n20This result was also derived in Appendix F of [15] in a slightly different manner.27\nThen the FG coordinates on the 6 edges are summarized as follows.\ne13:eX(2)\n14=e−Y2=⟨s4∧s1⟩⟨s3∧s2⟩\n⟨s4∧s3⟩⟨s1∧s2⟩e′\n12:eX′(1)\n25=eX′\n1=⟨s′\n3∧s′\n1⟩⟨s′\n2∧s′\n4⟩\n⟨s′\n3∧s′\n2⟩⟨s′\n1∧s′\n4⟩=⟨s2∧s1⟩⟨s3∧s4⟩\n⟨s2∧s3⟩⟨s1∧s4⟩≡eY2\ne12:eX(2)\n15=eX2=⟨s3∧s1⟩⟨s2∧s4⟩\n⟨s3∧s2⟩⟨s1∧s4⟩e′\n13:eX′(1)\n23=e−Y′\n1=⟨s′\n4∧s′\n1⟩⟨s′\n3∧s′\n2⟩\n⟨s′\n4∧s′\n3⟩⟨s′\n1∧s′\n2⟩=⟨s4∧s1⟩⟨s2∧s3⟩\n⟨s4∧s2⟩⟨s1∧s3⟩≡e−X2\ne14:eX(2)\n45=⟨s2∧s1⟩⟨s4∧s3⟩\n⟨s2∧s4⟩⟨s1∧s3⟩e′\n14:eX′(1)\n35=⟨s′\n2∧s′\n1⟩⟨s′\n4∧s′\n3⟩\n⟨s′\n2∧s′\n4⟩⟨s′\n1∧s′\n3⟩=⟨s3∧s1⟩⟨s4∧s2⟩\n⟨s3∧s4⟩⟨s1∧s2⟩≡e−χ(2)\n45\ne23:eX(2)\n13=⟨s1∧s2⟩⟨s3∧s4⟩\n⟨s1∧s3⟩⟨s2∧s4⟩e′\n23:eX′(1)\n24=⟨s′\n1∧s′\n2⟩⟨s′\n3∧s′\n4⟩\n⟨s′\n1∧s′\n3⟩⟨s′\n2∧s′\n4⟩=⟨s1∧s3⟩⟨s2∧s4⟩\n⟨s1∧s2⟩⟨s3∧s4⟩≡e−χ(2)\n13\ne24:eX(2)\n35=⟨s3∧s2⟩⟨s4∧s1⟩\n⟨s3∧s4⟩⟨s2∧s1⟩e′\n34:eX′(1)\n34=⟨s′\n1∧s′\n3⟩⟨s′\n4∧s′\n2⟩\n⟨s′\n1∧s′\n4⟩⟨s′\n3∧s′\n2⟩=⟨s1∧s2⟩⟨s4∧s3⟩\n⟨s1∧s4⟩⟨s2∧s3⟩≡e−χ(2)\n35\ne34:eX(2)\n34=⟨s1∧s3⟩⟨s4∧s2⟩\n⟨s1∧s4⟩⟨s3∧s2⟩e′\n24:eX′(1)\n45=⟨s′\n3∧s′\n2⟩⟨s′\n4∧s′\n1⟩\n⟨s′\n3∧s′\n4⟩⟨s′\n2∧s′\n1⟩=⟨s2∧s3⟩⟨s4∧s1⟩\n⟨s2∧s4⟩⟨s3∧s1⟩≡e−χ(2)\n34. (C1)\nOne finds that, from the calculation point of view, the gluing of 4-holed spheres follows the same way as gluing ideal\ntetrahedra to form an ideal octahedron (it r.f. Section II). This is because, although we need to flipped the orientation\nofS′\n1, we read the labels of holes on the quadrilateral (lower panel of fig.5b) from the “inside” of S′\n1. Then this is the\nsame as reading the labels from the “outside” before flipping the orientation of S′\n1.\nAppendix D: Proof of Theorem VII.1\nPF\nf=1\u0000i\n2πFf(2Lf)−2ufLf\u0001\nin (45) comes from the face amplitudes and it is only imaginary since F(2Lf) is a real\nfunction of 2 Lfupon the imposition of simplicity constraints. We are left to consider each Sv\n⃗ pv,⃗ uv,⃗ˆρv(42) obtained from\nthe large- kapproximation of partition function (19) and coherent states (22) for a spinfoam vertex. We first observe\nthat the positive angles that contribute to the imaginary parts of {µI, νI}are not seen at the large- kapproximation\nof the action. Then each tilted variable is merely the complex conjugate of its non-tilted counterpart. Additionally,\nb−1is the complex conjugate of bas it is a phase. Therefore, Sv\n1+eSv\n1is pure imaginary obviously seen from their\nexpressions (43b) – (43c). We next consider the rest of the first line of (42), which can be rewritten as\nSv\n0−2πi\nk⃗ pv·⃗ nv=πi\nk2\u0014\n−2\u0012\n⃗ µv−iQ\n2⃗t\u0013\n·⃗ νv+ 2⃗ mv·⃗ nv−⃗ νv·AB⊤·⃗ νv+⃗ nv·AB⊤·⃗ nv+k⃗ nv·(⃗t+ 2⃗ pv)\u0015\n.(D1)\n⃗ µv, ⃗ νvcan be viewed as real variables at large khence the above expression is also pure imaginary. The second line\nof (42) contains the logarithms of coherent states and a term2πi\nkP5\na=1uv\namv\nafrom the Poisson resummation for mv\na.\nThe latter is apparently imaginary. All the real parts of (45), therefore, come from the coherent states. Due to the\nnature of coherent states (and is clear from the definitions (22)), the norms are Gaussian and hence must contribute\na non-positive real part for the action with zero obtained at the critical point. This proves the first two equations of\n(72). The last equation is the definition of a critical point hence is trivially satisfied.\nThe first two equations of (72) also imply that the real parts of the eigenvalues of the Hessian, denoted as Re( H⃗ x),\nsatisfy Re( H⃗ x)≤0 at the neighbourhood of the real critical point ⃗ x0.\nAsS(⃗ x,⃗ r) is apparently analytic near ⃗ x0, its analytic continuation S(⃗ z,⃗ r) is also analytic near the complex critical\npoint ⃗ z0(⃗ r). Then S(⃗ z,⃗ r) possess a convergent Taylor series at ⃗ z0(⃗ r):\nS(⃗ z,⃗ r) =S(⃗ z0(⃗ r),⃗ r) +X\n|α|=21\nα!DαS(⃗ z,⃗ r)|⃗ z=⃗ z0(⃗ r)(⃗ z−⃗ z0(⃗ r))α+O(|⃗ z−⃗ z0(⃗ r)|3), (D2)\nwhere Dαstands for the derivative of order αacting on a function f(⃗ z) with ⃗ z∈Cnas\nDαf=∂|α|f\n∂zα1\n1···∂zαnn,|α|=α1+···+αn (D3)\nandα! :=α1!···αn!.DαS(⃗ z,⃗ r) with |α|= 2 is simply the Hessian H⃗ zof the action.\nAs assumed, the complex critical point ⃗ z0is in the neighbourhood Uof the real critical point ⃗ x0, as illustrated in\nfig.4. ⃗ z0(⃗ r) is an analytic function in ⃗ r. Let ⃗ x0(⃗ r) =⃗ z0(⃗ r0). Then ⃗ z0(⃗ r) can be viewed as a path in Cnstarting at ⃗ x0.\nWithin the neighbourhood U, Re(H⃗ x)≤0 implies Re( H⃗ z0)≤0 by analyticity, which in turn implies S(⃗ z0,⃗ r)≤0. By\n(D2), we have\nRe (S(⃗ z0,⃗ r)) +X\n|α|=21\nα!Re (H⃗ z0(⃗ z−⃗ z0)α) + Re\u0000\nO(|⃗ z−⃗ z0|3\u0001\n≤0. (D4)28\nConsider ⃗ z= Re( ⃗ z0) +|Im(⃗ z0)|⃗ swith some ⃗ s∈Rn,|⃗ s|<1. When Im( ⃗ z0) is small, ⃗ zparametrized in this way is\nwithin Uhence (D2) is valid. Define ⃗ η= Im( ⃗ z0)/|Im(⃗ z0)|. Then (D4) can be reformulated as\nRe(S(⃗ z0,⃗ r))≤ −|Im(⃗ z0)|2\nsup\n|⃗ s|<1X\n|α|=21\nα!Re (H⃗ z0(⃗ s−i⃗ η)α) +C′|Im(⃗ z0)|\n, (D5)\nwhere 0 < C′<∞is some real constant. We are left to prove that the expression in the bracket above is non-negative\n(as it is indeed bounded). Firstly, Re( H⃗ z0)≤0 as observed above. We expand the termP\n|α|=21\nα!Re (H⃗ z0(⃗ s+i⃗ η)α):\n1\n2nX\ni,j=1[Re(H⃗ z0)ij(sisj−ηiηj)−2Im(H⃗ z0)ijsiηj] =:1\n2⟨⃗ s,Re(H⃗ z0)⃗ s⟩ −1\n2⟨⃗ η,Re(H⃗ z0)⃗ η⟩ − ⟨⃗ s,Im(H⃗ z0)⃗ η⟩. (D6)\nTo proceed, we only need to find an admissible ⃗ s(|⃗ s|<1) such that (D6) is positive. To this end, if ⟨⃗ η,Re(H⃗ z0)⃗ η⟩ ̸= 0,\nwe let ⃗ s= 0. Then (D5) is proven as ⟨⃗ η,Re(H⃗ z0)⃗ η⟩<0 is guaranteed by Re( H⃗ z0)≤0. If⟨⃗ η,Re(H⃗ z0)⃗ η⟩= 0, then\nRe(H⃗ z0) =⃗0. The assumption det( H⃗ z0)̸= 0 then implies that Im( H⃗ z0)̸=⃗0. In this case, we take ⃗ s=−ϵIm(H⃗ z0)⃗ η\nwith ϵ >0 being small so that |⃗ s|<1 is not violated. Then ⟨⃗ s,Im(H⃗ z0)⃗ η⟩>0 hence (D6) is positive. Therefore, (73)\nis proved. (74) is the result from stationary analysis with distinct critical points added. We refer to Theorem 7.7.12\nin [31] for a detailed proof. The distinctness of critical points is implied by det( H⃗ z0)̸= 0 as, otherwise, continuous\ncritical points would lead to degenerate directions of the Hessian.\n[1] J. Engle, E. Livine, R. Pereira, and C. Rovelli, “LQG vertex with finite Immirzi parameter,” Nucl. Phys. B 799(2008)\n136–149, arXiv:0711.0146 .\n[2] M. Han, “Four-dimensional spinfoam quantum gravity with a cosmological constant: Finiteness and semiclassical limit,”\nPhys. Rev. D 104(2021), no. 10, 104035, arXiv:2109.00034 .\n[3] K. Noui and P. Roche, “Cosmological deformation of Lorentzian spin foam models,” Class. Quant. Grav. 20(2003)\n3175–3214, arXiv:gr-qc/0211109 .\n[4] M. Han, “4-dimensional Spin-foam Model with Quantum Lorentz Group,” J. Math. Phys. 52(2011) 072501,\narXiv:1012.4216 .\n[5] W. J. Fairbairn and C. Meusburger, “Quantum deformation of two four-dimensional spin foam models,” J. Math. Phys.\n53(2012) 022501, arXiv:1012.4784 .\n[6] H. M. Haggard, M. Han, W. Kami´ nski, and A. Riello, “SL(2,C) Chern–Simons theory, a non-planar graph operator, and\n4D quantum gravity with a cosmological constant: Semiclassical geometry,” Nucl. Phys. B 900(2015) 1–79,\narXiv:1412.7546 .\n[7] H. M. Haggard, M. Han, W. Kaminski, and A. Riello, “SL(2 ,C) Chern-Simons theory, flat connections, and\nfour-dimensional quantum geometry,” Adv. Theor. Math. Phys. 23(2019), no. 4, 1067–1158, arXiv:1512.07690 .\n[8] H. M. Haggard, M. Han, W. Kami´ nski, and A. Riello, “SL (2, C) Chern–Simons theory, a non-planar graph operator,\nand 4D quantum gravity with a cosmological constant: Semiclassical geometry,” Nuclear Physics B 900(2015) 1–79.\n[9] M. Han, Z. Huang, H. Liu, and D. Qu, “Complex critical points and curved geometries in four-dimensional Lorentzian\nspinfoam quantum gravity,” Phys. Rev. D 106(2022), no. 4, 044005, arXiv:2110.10670 .\n[10] J. W. Barrett, R. J. Dowdall, W. J. Fairbairn, H. Gomes, and F. Hellmann, “Asymptotic analysis of the EPRL\nfour-simplex amplitude,” J. Math. Phys. 50(2009) 112504, arXiv:0902.1170 .\n[11] V. Bonzom, “Spin foam models for quantum gravity from lattice path integrals,” Phys. Rev. D 80(2009) 064028,\narXiv:0905.1501 .\n[12] M. Han, “On Spinfoam Models in Large Spin Regime,” Class. Quant. Grav. 31(2014) 015004, arXiv:1304.5627 .\n[13] M. Han and Q. Pan, “Melonic Radiative Correction in Four-Dimensional Spinfoam Model with Cosmological Constant,”\narXiv:2310.04537 .\n[14] J. F. Plebanski, “On the separation of Einsteinian substructures,” J. Math. Phys. 18(1977) 2511–2520.\n[15] M. Han, “4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block,” JHEP 01(2016)\n065, arXiv:1509.00466 .\n[16] D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKB Approximation,”\narXiv:0907.3987 .\n[17] T. Dimofte, “Quantum Riemann Surfaces in Chern-Simons Theory,” Adv. Theor. Math. Phys. 17(2013), no. 3, 479–599,\narXiv:1102.4847 .\n[18] T. Dimofte, D. Gaiotto, and S. Gukov, “Gauge Theories Labelled by Three-Manifolds,” Commun. Math. Phys. 325\n(2014) 367–419, arXiv:1108.4389 .\n[19] T. Dimofte, D. Gaiotto, and R. van der Veen, “RG Domain Walls and Hybrid Triangulations,” Adv. Theor. Math. Phys.\n19(2015) 137–276, arXiv:1304.6721 .29\n[20] T. Dimofte, “Complex Chern–Simons Theory at Level k via the 3d–3d Correspondence,” Commun. Math. Phys. 339\n(2015), no. 2, 619–662, arXiv:1409.0857 .\n[21] J. E. Andersen and R. Kashaev, “Complex Quantum Chern-Simons,” arXiv preprint arXiv:1409.1208 (2014).\n[22] T. D. Dimofte, Refined BPS Invariants, Chern-Simons Theory, and the Quantum Dilogarithm . PhD thesis, Caltech, 2010.\n[23] V. V. Fock and A. B. Goncharov, “Moduli spaces of local systems and higher Teichmuller theory,” 2003.\n[24] J. Engle, R. Pereira, and C. Rovelli, “Flipped spinfoam vertex and loop gravity,” Nucl. Phys. B 798(2008) 251–290,\narXiv:0708.1236 .\n[25] H. M. Haggard, M. Han, and A. Riello, “Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from\nGroup-Valued Moment Maps,” Annales Henri Poincare 17(2016), no. 8, 2001–2048, arXiv:1506.03053 .\n[26] M. Han and H. Liu, “Analytic continuation of spinfoam models,” Phys. Rev. D 105(2022), no. 2, 024012,\narXiv:2104.06902 .\n[27] E. Bianchi, D. Regoli, and C. Rovelli, “Face amplitude of spinfoam quantum gravity,” Class. Quant. Grav. 27(2010)\n185009, arXiv:1005.0764 .\n[28] A. Y. Alekseev, H. Grosse, and V. Schomerus, “Combinatorial quantization of the Hamiltonian Chern-Simons theory. 2.,”\nCommun. Math. Phys. 174(1995) 561–604, arXiv:hep-th/9408097 .\n[29] A. Y. Alekseev, H. Grosse, and V. Schomerus, “Combinatorial quantization of the Hamiltonian Chern-Simons theory,”\nCommun. Math. Phys. 172(1995) 317–358, arXiv:hep-th/9403066 .\n[30] A. Y. Alekseev and V. Schomerus, “Representation theory of Chern-Simons observables,” arXiv:q-alg/9503016 .\n[31] L. H¨ ormander, The analysis of linear partial differential operators I: Distribution theory and Fourier analysis . Springer,\n2015.\n[32] A. Melin and J. Sj¨ ostrand, “Fourier integral operators with complex-valued phase functions,” in Fourier Integral\nOperators and Partial Differential Equations: Colloque International, Universit´ e de Nice, 1974 , pp. 120–223, Springer.\n2006.\n[33] M. Han, H. Liu, and D. Qu, “Complex critical points in Lorentzian spinfoam quantum gravity: Four-simplex amplitude\nand effective dynamics on a double-∆3 complex,” Phys. Rev. D 108(2023), no. 2, 026010, arXiv:2301.02930 .\n[34] M. Han, C.-H. Hsiao, and Q. Pan, “Quantum Curved Tetrahedron, Quantum Group Intertwiner Space, and Coherent\nStates.” to appear.\n[35] M. Han, C.-H. Hsiao, and Q. Pan, “Quantum Group Intertwiner Space From Quantum Curved Tetrahedron,”\narXiv:2311.08587 .\n[36] J. W. Barrett, R. J. Dowdall, W. J. Fairbairn, F. Hellmann, and R. Pereira, “Lorentzian spin foam amplitudes:\nGraphical calculus and asymptotics,” Class. Quant. Grav. 27(2010) 165009, arXiv:0907.2440 ." }, { "title": "2401.14677v1.Maximum_plasmon_thermal_conductivity_of_a_thin_metal_film.pdf", "content": "Maximum plasmon thermal conductivity of a thin metal film\nKuk Hyun Yun,1, 2,∗Dong-min Kim,1, 2,∗and Bong Jae Lee1, 2,†\n1Department of Mechanical Engineering,\nKorea Advanced Institute of Science and Technology, Daejeon 34141, South Korea\n2Center for Extreme Thermal Physics and Manufacturing,\nKorea Advanced Institute of Science and Technology, Daejeon 34141, South Korea\n(Dated: January 29, 2024)\n1arXiv:2401.14677v1 [physics.app-ph] 26 Jan 2024ABSTRACT\nDue to their extremely long propagation lengths compared to the wavelengths, surface\nplasmon polaritons (SPPs) have been considered as a key in enhancing thermal conductiv-\nity in thin metal films. This study explores the conditions at which the plasmon thermal\nconductivity is maximized, considering the thickness-dependent metal permittivity. We de-\nrived the analytical solutions for the plasmon thermal conductivity in both the thin-film\nand thick-film limits to analyze the effect of the permittivities of metals and substrates.\nFrom the analytical solutions of plasmon thermal conductivity, we deduced that the plas-\nmon thermal conductivity is proportional to the electron thermal conductivity based on the\nWiedemann-Franz law. Additionally, we analyzed the conditions where the enhancement ra-\ntio of the thermal conductivity via SPPs is maximized. Metals with high plasma frequency\nand low damping coefficient are desirable for achieving the maximum plasmon thermal con-\nductivity as well as the maximum enhancement ratio of thermal conductivity among metals.\nSignificantly, 10-cm-long and 14-nm-thick Al film demonstrates most superior in-plane heat\ntransfer via SPPs, showing a 53.5% enhancement in thermal conductivity compared to its\nelectron thermal counterpart on a lossless glass substrate.\nI. INTRODUCTION\nWhen the characteristic length of thin films or the diameter of nanowires becomes compa-\nrable to the mean free path of basic energy carriers (e.g., phonons and electrons), boundary\nscattering becomes increasingly significant [1, 2]. This effect causes a reduction in the ef-\nfective mean free path of energy carriers compared to their bulk counterparts, leading to\na decrease in thermal conductivity [3–5]. This classical size effect of nanostructures can\ncause significant issues, such as performance degradation and reliability reduction in mod-\nern semiconductor devices, intensifying the focus on thermal management challenges [6, 7].\nIn response to these challenges, most research has been directed toward replacing conven-\ntional materials in modern devices with those having higher thermal conductivities [8]. Re-\ncently, two-dimensional (2-D) materials with pronounced anisotropic thermal conductivity\nhave drawn attention as heat spreaders [9–11]. They possess very high in-plane thermal\n∗These two authors contributed equally\n†bongjae.lee@kaist.ac.kr\n2conductivities ranging from hundreds to thousands W/m ·K, while their cross-plane thermal\nconductivities remain significantly lower. With their tens of nanometer thickness, such 2-D\nmaterials enable concentrated in-plane heat conduction, efficiently diffusing heat away from\nthe hot spots. However, as of now, the commercialization of 2-D material heat spreader is\nchallenging due to the challenges in fabrication and assembly on the devices [12].\nRecently, it has been reported that surface waves, such as surface phonon polaritons\n(SPhPs) and surface plasmon polaritons (SPPs) in nanoscale thin films, can compensate\nfor the classical size effect via additional heat conduction channels [13–19]. In fact, surface\nwaves can propagate over distances longer than a centimeter, significantly enhancing ther-\nmal conductivity [20–22]. Therefore, nanoscale metal films can maintain the high thermal\nconductivity comparable to their bulk values due to additional in-plane heat transfer via\nSPPs, suggesting that they can serve as heat spreaders similar to 2-D materials. Further-\nmore, unlike 2-D materials, thin metal films can be easily deposited on devices through a\nmicroelectromechanical systems process. Recently, Ordonez-Miranda et al. [22] theoretically\ndemonstrated an increase of 25% in the thermal conductivity of a metal due to long-range\nSPPs for a 1-cm-long gold film deposited on a Si substrate. However, this increase rate is\noverestimated to some extent because the plasmon thermal conductivity does not take into\naccount the thickness-dependent metal permittivity. More recently, Kim et al. [23] com-\nprehensively accounted for the size effect on the permittivity of metals, calculating and\nexperimentally verifying the plasmon thermal conductivity in Au and Ag thin films. They\nshowed that for 5-cm-long Au and Ag thin films, the plasmon thermal conductivity reaches\nabout 20% of its electron thermal conductivity.\nHowever, it remains unclear yet what optical properties of metals determine the plasmon\nthermal conductivity and what the optimal conditions are for maximizing it. To analyze the\neffect of the permittivities of metal film and dielectrics (i.e., substrate or superstrate) on\nplasmon thermal conductivity, it is necessary to analytically express the plasmon thermal\nconductivity as a function of the permittivities. This is because the properties of SPPs,\nsuch as the in-plane wavevector and propagation length, are determined by the permittiv-\nities of media. In other words, the material combinations of the metal film and dielectrics\nare a primary factor in determining the magnitude of plasmon thermal conductivity. But,\nthe analytical solution of dispersion relation of SPPs exhibits intricate dependence with\nthe permittivity of media [22], making it difficult to analyze its effect on plasmon thermal\n3conductivity.\nIn this study, we present criteria that maximize the plasmon thermal conductivity of\na thin metal film. This is achieved by analytically deriving expressions for the plasmon\nthermal conductivity as a function of the film thickness and the permittivity of a metal\nfilm and a substrate, considering the thickness-dependent metal permittivity using a mod-\nified Drude model. These analytic expressions demonstrate a monotonic dependence of the\nplasma frequency and damping coefficient on the plasmon thermal conductivity, through\napproximation with respect to the film thickness. We also investigated the ratio of the plas-\nmon thermal conductivity to the electron thermal conductivity in a metal film. Finally, we\nproposed which of the existing metals are most desirable for the SPP-mediated in-plane heat\ntransfer.\nII. THEORETICAL MODEL\nConsider the in-plane heat transfer via SPPs along an infinitely long metal film sur-\nrounded by dielectrics (i.e., substrate and superstrate). The plasmon thermal conductivity\ncan be calculated from kinetic theory with Boltzmann transport equation (BTE) under\nrelaxation time approximation and diffusion approximation [13, 14]:\nkSPP=1\n4πdmZ∞\n0ℏωβRΛSPP∂f0\n∂Tdω, (1)\nwhere dmis the thickness of the metal film, ℏis the Planck constant divided by 2 π,ωis the\nangular frequency, βRis the real part of in-plane wavevector of SPP (i.e., β=βR+iβI),\nΛSPPis the propagation length of SPP, f0is the Bose-Einstein distribution function, and\nTis the temperature. For a metal film with finite-length of Lm, the effective propaga-\ntion length derived from the BTE can be used to consider the boundary scattering, i.e.,\nΛeff= [1−4ψ(0)/(πµ)]ΛSPP[24]. Here, µ=Lm/ΛSPPandψ(ξ) = E5(ξ)−E5(µ−ξ),\nwhere ξ=z/ΛSPPandEn(x) =Rπ/2\n0(cosθ)n−2exp(−x/cosθ)dθwith θbeing the polar an-\ngle between the SPP propagation direction and global heat transfer direction. To calculate\nthe plasmon thermal conductivity, the in-plane wavevector and propagation length of SPPs\nshould first be obtained by solving the dispersion relation. In this study, the three-layer con-\nfiguration consisting of a substrate, metal film, and superstrate is considered. For simplicity,\nthe superstrate is set to be air, and the corresponding three-layer dispersion relation for\n4SPPs is given by [25]\ntanh( pmdm) =−pmεm(psεa+paεs)\np2\nmεsεa+pspaε2\nm, (2)\nwhere the subscripts ‘ m’, ‘s’, and ‘ a’ represents metal, substrate, and air, respectively. Addi-\ntionally, pn=p\nβ2−εnk2\n0denotes the cross-plane wavevector of SPPs for medium n=m, s,\nora, where εnis the relative permittivity of the corresponding medium, k0=ω/c 0is the\nwavevector in a vacuum with c0being the speed of light in a vacuum. In this study, we\nconsider only the metal as a lossy material (i.e., εm=εR+iεI), while treating the substrate\nas a lossless dielectric for simplicity.\nIt is well known that SPPs at both interfaces of a metal film become decoupled (i.e.,\nSPPs at two interfaces behave independently of each other) as the film thickness increases\nto the optically thick limit (i.e., thick-film limit) [26]. In the thick-film limit, the plasmon\nthermal conductivity increases as the film thickness decreases, as seen in Eq. (1). The plas-\nmon thermal conductivity reaches its peak at the thickness where the decoupling of SPPs\nbegins (i.e., Re( pm)dm≈1) [23]. When the film thickness is sufficiently thin, the SPPs at\nboth interfaces start to couple, resulting in the thickness-dependent SPP disperion relation.\nSuch intensification of SPPs coupling lead to increased energy losses within the metal film,\ncausing the plasmon thermal conductivity to decrease accordingly [22]. Therefore, to analyze\nthe peak of plasmon thermal conductivity, understanding the behavior of SPPs just before\ndecoupling occurs is crucial.\nIn the thin-film limit when |pm|dm≪1, the left side of Eq. (2) can be approximated to\ntanh( pmdm)≈pmdm. Therefore, the dispersion relation becomes\ndm(p2\nmεsεa+pspaε2\nm) =−εm(psεa+paεs), (3)\nwhich is an implicit equation for β, making it difficult to obtain an analytic solution. If the\ncross-plane wavevector for air is expressed as pa=√α=p\nβ2−εak2\n0, then the cross-plane\nwavevector for the substrate can be expressed as ps=p\nα−(εs−εa)k2\n0. Additionally, if\nthe magnitude of permittivity of the metal is much greater than that of the substrate and\nair (i.e., |εm| ≫ | εs|,|εa|), the cross-plane wavevector for metal becomes pm≈k0√−εm\n[22]. Consequently, with those approximations in the thin-film limit, the implicit dispersion\nrelation in Eq. (3) can be written to the explicit form for αas\nε4\nmd4\nmα4−2(εs−εa)ε4\nmk2\n0d4\nmα3+(εs−εa)2ε4\nmk4\n0d4\nmα2−(εs−εa)2ε2\nmε2\nak4\n0d2\nmα+(εs−εa)2ε4\nak4\n0= 0.\n(4)\n5The in-plane wavevector of SPPs can be easily determined from the explicit form of a\nfourth-order polynomial of α. For long-range SPPs with propagation lengths exceeding\ncentimeters, the imaginary part of the in-plane wavevector should be considerably small\n(ΛSPP= 1/(2βI)), and the real part of the in-plane wavevector is nearly identical to the\nlight line (i.e., βR≈k0√εa). This implies that the real part of the in-plane wavevector is\nsignificantly larger than its imaginary part ( |βR| ≫ |βI|), allowing the magnitude of the in-\nplane wavevector to be assumed equal to its real part. With this assumption, the magnitude\nof√αbecomes very small compared to the light lines (i.e.,√α=p\nβ2−εak2\n0≪k0√εa).\nTherefore, α≪(εs−εa)k2\n0is satisfied except in cases when the substrate is identical to\nthe superstrate. As a result, Eq. (4) can be further approximated to a second-order polyno-\nmial of α. The solution of second-order polynomial, the cross-plane wavevector for air (i.e.,\nsuperstrate), becomes\n√α=pa=εa\nεmdm. (5)\nEquation (5) indicates that the penetration depth of the SPPs into the air (i.e., δa=\n1/(2Re( pa))) increases as the film thickness increases. The frequency-dependent permittivity\nof metal can be described using the Drude model\nεm(ω, d m) = 1−ω2\np\nω2+iΓ(dm)ω, (6)\nwhere ωpis the plasma frequency and Γ is the thickness-dependent damping coefficient, which\nsignifies the collision of electrons with phonon, grain boundaries, and film boundaries [2].\nIn other words, the damping coefficient, as the inverse of the free electron’s relaxation time,\nenables the consideration of boundary scattering. By applying Matthiessen’s rule to account\nfor boundary scattering, the damping coefficient can be related to its bulk value (Γ ∞), as\nΓ = Γ ∞+vf/dm, where vfis the Fermi velocity [27, 28]. As the film thickness decreases, the\ndamping coefficient increases, leading to more losses in the metal. If the angular frequency\nof SPPs is much lower than the plasma frequency and is much higher than the damping\ncoefficient (i.e., Γ ≪ω≪ωp), the real and imaginary parts of the permittivity can be\napproximated to εR=−ω2\np/ω2andεI=ω2\npΓ/ω3, respectively. By substituting the metal\npermittivity derived from Drude model into Eq. (5), the analytical solution of the dispersion\n6relation in the thin-film limit can be expressed by\nβR=k0√εa+ε3/2\na\n2k0d2\nm\u0012ω\nωp\u00134\n, (7a)\nβI=c0ε3/2\na\nd2\nmΓ\nω2\np\u0012ω\nωp\u00132\n. (7b)\nEquation (7a) indicates that in the thin-film limit, the SPP dispersion curve moves further\naway from the light line as the film becomes thinner. Note that if the angular frequency is\nmuch lower than the plasma frequency, the dispersion curve becomes simply the light line\n(i.e., βR≈k0√εa). Concurrently, the imaginary part of the in-plane wavevector increases\nas the film thickness decreases, which reduces the propagation length of the SPPs. In the\nthinner film, the SPPs at both interfaces become coupled, resulting in increased energy\ndissipation within the metal film. This implies that the energy losses from coupled SPPs\nlead to a reduction in their propagation lengths. Such a decrease in SPP propagation lengths\nis responsible for the pronounced reduction in the plasmon thermal conductivity.\nEquations (4), (5), and (7) represent the solutions about the dispersion relation of SPPs\npropagating along the air/metal interface. Due to the symmetry of dispersion relation for\nair and substrate (see Eq. (2)), the in-plane wavevector for the substrate can be obtained\nsimply by replacing the subscript ‘ a’ with ‘ s’. By substituting Eqs. (7a) and (7b) into Eq.\n(1), the plasmon thermal conductivity in the thin-film limit can be written as\nkSPP,thin =X\nn=s,aℏdm\n8πϵnc2\n0ω4\np\nΓZ∞\n0∂f0\n∂Tdω. (8)\nThe above equation suggests that in the thin-film limit, the plasmon thermal conductivity\nincreases as the film thickness increases (i.e., kSPP,thin ∝dm). Note also that the damping\ncoefficient decreases as the film thickness increases, leading to an increase in the plasmon\nthermal conductivity. Equation (8) also suggests that kSPP,thin increases for metals with a\nhigher plasma frequency and a lower damping coefficient, and it also increases with lower\npermittivity of the substrate. For a given substrate and superstrate, the gradient of plasmon\nthermal conductivity with respect to film thickness is solely determined by the permittivity of\nmetal. That is, metals with a high plasma frequency and a low damping coefficient represent\na steeper increase in the plasmon thermal conductivity as dmincreases. For instance, for Au\n(ωp= 64660 cm−1, Γ∞= 252 cm−1, and vf= 13.82×105m/s) and Ag ( ωp= 72071 cm−1,\nΓ∞= 145 cm−1, and vf= 14.48×105m/s) films, Ag has a steeper increase in the plasmon\nthermal conductivity in the thin-film limit than that of Au [23].\n7On the other hand, when a metal film thickness becomes optically thick, the SPPs at\nboth interfaces of the metal film become completely decoupled and the corresponding SPP\ndispersion curve is independent of film thickness. The plasmon thermal conductivity using\nthe analytical solution for the dispersion relation of SPPs in the thick-film limit is given by\n[22]\nkSPP,thick =X\nn=s,aℏ\n4πdmεnω2\np\nΓ∞Z∞\n0∂f0\n∂Tdω. (9)\nIn the thick-film limit, kSPP,thick is inversely proportional to the film thickness, which is the\nopposite of what is observed in the thin-film limit. However, similarly to the thin-film limit\nkSPP,thick increases also for metals with a higher plasma frequency and a lower damping\ncoefficient, and for substrates with lower permittivity. Note that in both limits, the plasmon\nthermal conductivity has a factor of ω2\np/(Γεn). Based on the Drude model and Wiedemann-\nFranz law, the electron thermal conductivity ( ke) can be expressed by ke=ε0LTω2\np/Γ,\nwhere ε0is the vacuum permittivity and Lis the Lorenz number for each metal [29, 30].\nThis clearly implies that the plasmon thermal conductivity is proportional to the electron\nthermal conductivity in both limits. Metals with a high plasma frequency have a larger\nnumber density of free electrons [26]. This larger number density of free electrons facilitates\ngreater energy transfer, resulting in enhanced electrical and thermal conductivities. Likewise,\nthis principle is also applicable to SPPs, which are energy carriers formed by the coupling\nof free electrons and photons. In other words, an increase in the number density of free\nelectrons leads to the enhancement of energy transfer via SPPs, thereby improving the\nplasmon thermal conductivity.\nIII. RESULTS AND DISCUSSION\nWe intentionally selected five metals whose phonon contribution to the total intrinsic\nthermal conductivity (i.e., thermal conductivity via electron and phonon) is less than 10%\nto further investigate the relationship between the plasmon thermal conductivity and the\nelectron thermal conductivity, as listed in Table I. For each metal, its Drude parameters\n(i.e., plasma frequency and damping coefficient) for low frequencies were obtained by the\nleast-square fitting of experimental data in Refs. [31, 32], and for high frequencies, from\ntabulated data [33]. In addition, their Fermi velocities were taken from values predicted by\n8TABLE I. This table includes Drude parameters, Fermi velocities, and thermal conductivity for\nvarious metals. kintrin denotes the total intrinsic thermal conductivity of metal, keis the electron\nthermal conductivity predicted by Wiedemann-Franz law, kphis the phonon thermal conductivity,\nandktotal=ke+kphis the total thermal conductivity. L0= 2.44×10−8W·Ω/K2is the Sommerfeld\nvalue of Lorenz number ( L).\nMetalωp(cm−1)\n[31–33]Γ∞(cm−1)\n[31–33]vf(105m/s)\n[34]L/L 0\n[37]ke(W/m ·K)kintrin (W/m ·K)\n[35]kph/ktotal(%)\n[37]\nAg 72071 145 14.48 0.98 428.6 429 1.25\nCu 59022 76.1 11.09 0.94 525.3 401 4.60\nAu 64660 252 13.82 1.03 208.6 317 1.01\nAl 96627 437 15.99 0.94 245.2 237 3.71\nPt 41775 614 5.2 [38] 1.0 34.70 71.6 6.74\nthe density functional theory [34]. The electron thermal conductivity of Ag and Al, predicted\nusing the fitted Drude parameters and the Wiedemann-Franz law, showed good agreement\nwith the literature values [35]. However, for the remaining metals (Cu, Au, and Pt), the\ncalculated values deviate from the measurements by about 30 to 50%. Such discrepancy\ncan be attributed to the limitations of the Drude model, which simplifies the behavior of\nelectrons in metals to that of free particles and fails to account for complex interactions\nsuch as electron-electron interactions [36]. Note that the main focus of this study lies on\nanalyzing the effect of Drude parameters on the plasmon thermal conductivity. Therefore,\nin this study, the thermal conductivity of metals is simply considered based on the electron\nthermal conductivity predicted from the Drude model and the Wiedemann-Franz law.\nInitially, to observe the effect of metal’s permittivity on the plasmon thermal conductivity,\nthe superstrate and substrate were fixed as air and glass (amorphous SiO 2neglecting losses),\nrespectively. The permittivity of lossless glass was set to be 3.6955 for below 200 Trad/s,\nwhich dominantly contributes to the plasmon thermal conductivity [17]. This value is the\naverage obtained from tabulated data for frequency below 200 Trad/s [33]. In the calculation,\nwe fixed the lateral size of the metal film as Lm= 10 cm. As previously explained in Eq.\n(1), the plasmon thermal conductivity is calculated using the effective propagation length in\nfinite-length metal films. Since the effective propagation length is proportional to the intrinsic\n9propagation length of SPPs, Eqs. (8) and (9) for infinite-length film can adequately explain\nthe plasmon thermal conductivity for finite-length films with respect to the permittivity of\nthe metal. We will address this matter later in the discussion.\nFigures 1a and 1b respectively show the calculated real part of the in-plane wavevector\nβRand the propagation length of SPP Λ SPP, considering the size effect of permittivity of\n10-nm-thick metal films on a glass substrate. For both interfaces of the metal film, βR\naligns linearly with the light line except Pt, showing photon-like characteristics. Due to Pt’s\nlower plasma frequency compared to other metals, it diverges from the light line in the\nhigh-frequency region, as explained by Eq. (7a). This less photon-like behavior of SPPs in\nFIG. 1. (a) In-plane wavevector and (b) propagation length of SPP along the interface of 10-nm-\nthick metal film supported by lossless glass substrate. The solid lines represent the metal/glass\ninterface, while the dashed lines indicate the air/metal interface.\n10the Pt film is due to the strong coupling between SPPs at both interfaces. Metals with a\nhigher plasma frequency have a thinner penetration depth into the metal, leading to the\ndecoupling of SPPs at thinner film thickness [22]. Therefore, for the same 10-nm thickness,\nSPPs propagating along the Pt films experience increased energy losses due to their stronger\ncoupled nature, resulting in the shortest propagation length among selected metals, as shown\nin Fig. 1b. In the thin-film limit, the propagation length increases for metals with a higher\nplasma frequency and a lower damping coefficient, i.e., Λ SPP∝ω4\np/Γ, as shown in Eq.\n(7b). Therefore, the sequence of metals by longer propagation length is as follows: Al, Ag,\nCu, Au, and Pt. Here, Cu and Au exhibit small difference in their propagation lengths.\nSpecifically, while the fourth power of the plasma frequency for Au is 1.44 times greater\nthan that for Cu (( ωp,Au/ωp,Cu)4= 1.44), its damping coefficient is proportionally 1.48 times\nlarger (Γ Au/ΓCu= 1.48). This ultimately results in the propagation length in Au being\napproximately 0.97 times shorter than that in Cu.\nThe dispersion relation for thin metal films with Lm= 10 cm deposited on a glass\nsubstrate, as described in Eq. (2), was numerically solved. Using Eq. (1), the plasmon thermal\nconductivity as a function of film thickness was also calculated. To circumvent the non-\nlocal effect observed in extremely thin metallic layers, we focused on thicknesses exceeding\n10 nm [23]. Figure 2 shows the calculated plasmon thermal conductivity for each metal\nas a function of their thickness. When dm<20 nm, the plasmon thermal conductivity\nincreases as the film thickness increases, and the sequence of metals with high plasmon\nthermal conductivity is Al, Ag, Cu, Au, and Pt. This ordering mirrors the sequence observed\nfor the longest propagation lengths shown in Fig. 1b. When the film thickness increases, the\nplasmon thermal conductivity first increases and reaches a peak, and then decreases. The\nsequence of metals with higher plasmon thermal conductivity changes if the film thickness\nincreases to the optically thick limit. In the thick-film limit, the sequence of metals with\nhigh plasmon thermal conductivity changes to Cu, Ag, Al, Au, and Pt. This sequence is the\nsame as the sequence of bulk metals with high electron thermal conductivity predicted from\nWiedemann-Franz law, as shown in Table I.\nIn both the thin-film and thick-film limits, the plasmon thermal conductivity shares com-\nmon variables: the Drude parameters and film thickness. In the thin-film limit, plasmon ther-\nmal conductivity is given by kSPP,thin ∝ω4\npdmΓ−1, while in the thick-film limit, kSPP,thick ∝\nω2\npd−1\nmΓ−1\n∞. In the intermediate regime (i.e., thin-film limit < d m0\n0,ifx≤0.(1)4\nX1\n...\nXLEncoder Decoder X1,X2,···,XαW1\nW2...\nWL\nEaves-\ndropperno information of Xα ...\n|A| ≤[α−s]+|U|=α\nFig. 1. The sliding secure SMDC model.\nFor1≤α≤L, no matter which specific subset of encoders are accessible, the decoder should be able to\nreconstruct the first αsources if |U|=αand the source Xαshould be kept perfectly secure from the eavesdropper\nif|A| ≤[α−s]+.\nLetKbe a secret key taking values in some finite key space K. Note that we don’t limit the size of Kin this paper.\nThe secret key is shared by all the encoders but not the decode r or the eavesdropper. An (n,M1,M2,···,ML)\ncode is defined by Lencoding functions\nfl:L/productdisplay\ni=1Xn\ni×K → { 1,2,···,Ml},forl∈ L (2)\nand decoding functions\ngU:/productdisplay\nl∈U{1,2,···,Ml} →α/productdisplay\ni=1Xn\ni,forU ⊆ L s.t.|U|=α, (3)\nfor1≤α≤L. Let\nWl/definesfl(X1,X2,···,XL,K) (4)\nbe the output of Encoder- l. For any B ⊆ L , letWB={Wl:l∈ B} . A nonnegative rate tuple (R1,R2,···,RL)\nisadmissible for the(L,s)sliding secure SMDC problem if for any ǫ >0, there exists, for sufficiently large n, an\n(n,M1,M2,···,ML)code such that\n1\nnlogMl≤Rl+ǫ,∀l∈ L, (5)\nand for all α= 1,2,···,L,\ngU(WU) =X1,X2,···,Xα,∀ U ⊆ L s.t.|U|=α, (6)\nH(Xα|WA) =H(Xα),∀ A ⊆ L s.t.|A| ≤[α−s]+. (7)\nThe admissible rate region RL,sis the collection of all admissible rate tuples.\nWhens=L, there are no security constraints, and thus the (L,s)sliding secure SMDC problem becomes the\nclassicalL-channel SMDC problem.5\nB. Multilevel Secret Sharing\nThe(L,s)sliding secure SMDC problem reduces to the (L,s)multilevel secret sharing problem if the first s−1\nsources are constants, i.e.\nH(X1) =H(X2) =···=H(Xs−1) = 0. (8)\nSpecifically, a collection of L−s+1independent discrete memoryless sources Xs,Xs+1,···,XLare encoded by\nLencoders indexed by L. A decoder and an eavesdropper have access to a subset of U ⊆ L andA ⊆ L of encoders,\nrespectively. For s≤α≤L, the decoder is required to reconstruct sources Xs,Xs+1,···,Xαif|U|=αand the\nsourceXαshould be kept perfectly secure from the eavesdropper if |A| ≤α−s. The(n,M1,M2,···,ML)code\nreduces to the encoding functions\nfl:L/productdisplay\ni=sXn\ni×K → { 1,2,···,Ml},for alll∈ L (9)\nand decoding functions\ngU:/productdisplay\nl∈U{1,2,···,Ml} →α/productdisplay\ni=sXn\ni,for allU ⊆ L s.t.|U|=α, (10)\nfors≤α≤L. The output of Encoder- lbecomes\nWl=fl(Xs,Xs+1,···,XL,K). (11)\nA nonnegative rate tuple (R1,R2,···,RL)isadmissible for the(L,s)multilevel secret sharing problem if for any\nǫ >0, there exists, for sufficiently large n, an(n,M1,M2,···,ML)code such that\n1\nnlogMl≤Rl+ǫ,∀l∈ L, (12)\nand for all α=s,s+1,···,L,\ngU(WU) =Xs,Xs+1,···,Xα,∀ U ⊆ L s.t.|U|=α, (13)\nH(Xα|WA) =H(Xα),∀ A ⊆ L s.t.|A| ≤α−s. (14)\nThe admissible rate region Rmss\nL,sis the collection of all admissible rate tuples. The relatio n between sliding secure\nSMDC and multilevel secret sharing problems implies that\nRmss\nL,s={(R1,R2,···,RL)∈ RL,s:H(X1) =H(X2) =···=H(Xs−1) = 0}. (15)\nDenote the rate region of the classical L-channel SMDC problem by RL\nSMDC . Let\nRL∗\nSMDC={(R1,R2,···,RL)∈ RL\nSMDC:H(X1) =H(X2) =···=H(Xs−1) = 0}. (16)\nBefore the main results, we state some useful existing resul ts about ramp secret sharing in the following subsection.6\nC. Ramp Secret Sharing\nLetXbe a discrete memoryless source sequence encoded by Lencoders. For any 0≤c < k≤L, the(c,k,L)\nramp secret sharing problem requires that any subset of no mo re thancencoders provide no information about the\nsource and any subset of kencoders can losslessly reconstruct the source. The proble m is also known as the secure\nsymmetric single-level diversity coding (S-SSDC) in [12]. The admissible rate region Rrssis fully characterized in\n[17] [12]. The result in summarized is the following lemma.\nLetR(L,k,H)be the collection of rate tuples (R1,R2,···,RL)satisfying\n/summationdisplay\ni∈BRi≥H (17)\nfor any subset B ⊆ L such that |B|=k.\nLemma 1. For the(c,k,L)ramp secret sharing problem, the admissible rate region is a s follows,\nRrss=R(L,k−c,H(X)). (18)\nWhenk=c+1, the(c,k,L)ramp secret sharing problem becomes the (k,L)threshold secret sharing problem\nand the rate region reduces to R(L,1,H(X)).\nIII.(L,1)SLIDING SECURE SMDC\nWhens= 1, the(L,1)sliding secure SMDC problem and the (L,1)multilevel secret sharing problem coincide.\nFor1≤α≤L, the source Xαcan be losslessly reconstructed if the decoder accesses a su bset of any αencoders\nand should be kept perfectly secure if the eavesdropper acce sses no more than α−1encoders. The reconstruction\nand security constraints in (6)(7) become\nH(Xα|WU) = 0,∀ U ⊆ L s.t.|U|=α, (19)\nH(Xα|WA) =H(Xα),∀ A ⊆ L s.t.|A| ≤α−1. (20)\nA simple coding scheme for the (L,1)sliding secure SMDC problem is separately encoding the Lindependent\nsources, which is referred to as superposition coding . For each α∈ L, we use the (α,L)threshold secret sharing\nscheme to encode the single source Xα, which requires a rate of at least H(Xα)at each encoder,i.e., the rate\nregion is simply R/parenleftbig\nL,1,H(Xα)/parenrightbig\n.\nThe superposition region R1\nsupinduced by separately encoding the Lsources is the collection of nonnegative rate\ntuples(R1,R2,···,RL)such that\nRi=L/summationdisplay\nα=1rα\ni,fori∈ L (21)\nwhererα\ni≥0,1≤α≤L, and\n/parenleftbig\nrα\n1,rα\n2,···,rα\nL/parenrightbig\n∈ R/parenleftbig\nL,1,H(Xα)/parenrightbig\n. (22)\nIt is easy to eliminate rα\ni(i,α∈ L)and obtain the following equivalent characterization of th e superposition region,\nR1\nsup={(R1,R2,···,RL) :Ri≥L/summationdisplay\nα=1H(Xα),for alli∈ L}. (23)7\nThe following theorem states that superposition coding is o ptimal for the (L,1)sliding secure SMDC problem and\nalso the(L,1)multilevel secret sharing problem.\nTheorem 1. RL,1=R1\nsup.\nIn order to prove Theorem 1, we only need to show the converse p art.\nProof. For1≤α≤L, letWα\n1= (W1,W2,···,Wα). From the conditions in (19)(20), we have\nH(Xα) =H(Xα|Wα\n2)\n=I(Xα;W1|Wα\n2)+H(Xα|W1,Wα\n2)\n=I(Xα;W1|Wα\n2)\n=H(W1|Wα\n2)−H(W1|Wα\n2,Xα). (24)\nThus,\nn·/parenleftBiggL/summationdisplay\nα=1H(Xα)/parenrightBigg\n=L/summationdisplay\nα=1H(Xα)\n=L/summationdisplay\nα=1/bracketleftbig\nH(W1|Wα\n2)−H(W1|Wα\n2,Xα)/bracketrightbig\n=H(W1)−L−1/summationdisplay\nα=1/bracketleftbig\nH(W1|Wα\n2,Xα)−H(W1|Wα+1\n2)/bracketrightbig\n−H(W1|WL\n2,XL)\n=H(W1)−L−1/summationdisplay\nα=1I(W1;Wα+1|Wα\n2,Xα)−H(W1|WL\n2,XL) (25)\n≤H(W1)\n≤n(R1+ǫ) (26)\nwhere (25) is obtained by\nH(W1|Wα\n2,Xα)−H(W1|Wα+1\n2)\n=H(W1|Wα\n2,Xα)−/bracketleftbig\nH(W1|Wα+1\n2,Xα)+I(W1;Xα|Wα+1\n2)/bracketrightbig\n=H(W1|Wα\n2,Xα)−H(W1|Wα+1\n2,Xα)\n=I(W1;Wα+1|Wα\n2,Xα) (27)\nfor any1≤α≤L−1. Divide both sides of (26) by nand letǫ→0, we have\nR1≥L/summationdisplay\nα=1H(Xα). (28)\nSimilarly, for any i∈ L, we can prove that\nRi≥L/summationdisplay\nα=1H(Xα). (29)\nTherefore, Theorem 1 is proved.8\nRemark 1.For1≤α1< α2≤L, sinceXα2is always assumed to be more secure than Xα1, it is impossible to use\nXα2as a secret key for Xα1. For1≤α≤L, if we can access any αencoders, the first αsourcesX1,X2,···,Xα\nare losslessly reconstructed and Xα+1should be kept perfectly secure, thus it is also impossible t o use the first\nαsources as secret keys for Xα+1and the successive sources after Xα+1. The fact that any source can not be\nused as secret keys for other sources provides an intuition o f why superposition is optimal for the (L,1)multilevel\nsecret sharing problem.\nRemark 2.Fors= 1, the security constraint (7) reduces to\nH(Xα|WA) =H(Xα),∀ A ⊆ L s.t.|A| ≤[α−1]+. (30)\nThis coincides with the sum-rate optimality condition (28) in [13], which implies that superposition coding cannot\nbe optimal for 2≤s≤L.\nRemark 3.We can generalize the optimality of superposition to the fol lowing setup. Consider the (L,s)sliding\nsecure SMDC problem where sdividesLand\nH(Xj) = 0,forj/ne}ationslash= 0( mods). (31)\nThat is we only have a set ofL\nssourcesXs,X2s,···,XL. The sources are encoded by a set of Lencoders. For\n1≤α≤L\ns, the source Xα·sis required to be losslessly reconstructed by the decoder if any subset of α·sencoders\nare accessible and should be kept perfectly secure from the e avesdropper if no more than α·(s−1)encoders are\naccessible. Superposition remains to be optimal for this se tup. The proof is very similar to that of Theorem 1, we\nomit the proof here. This setup reduces to (L,1)sliding secure SMDC when s= 1.\nIV. M ULTILEVEL SECRET SHARING\nA.(3,2)Multilevel Secret Sharing\nConsider the (3,2)multilevel secret sharing problem. Now we have two independ ent sources X2andX3. For\nα= 2,3, the source Xαcan be losslessly reconstructed by the decoder if a subset of αencoders are accessible and\nXαshould be kept perfectly secure if the eavesdropper can acce ss no more than α−2encoders. The reconstruction\nand security constraints in (13)(14) become\nH(X2|Wi,Wj) = 0,for all1≤i < j≤3 (32)\nH(X3|W1,W2,W3) = 0, (33)\nH(X3|Wi) =H(X3),for alli= 1,2,3. (34)\nForx,y∈ {1,2,3}, define the operation ⊙by\nx⊙y=\n\nx+y, ifx+y≤3\nx+y−3ifx+y >3.(35)9\nLetR∗\n1be the collection of nonnegative rate tuple (R1,R2,R3)such that\n2Ri+Rj≥H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (36)\nRi+Rj≥1\n2H(X2)+H(X3),for1≤i < j≤3 (37)\nRi+Rj≥H(X2),for1≤i < j≤3 (38)\nR1+R2+R3≥3\n2H(X2)+H(X3), (39)\n2Ri+Ri⊙1+Ri⊙2≥2H(X2)+H(X3),for1≤i≤3. (40)\nFor notational simplicity, we will use the following abbrev iation in the sequel:\nm= max/braceleftbigg\nH(X2),1\n2H(X2)+H(X3)/bracerightbigg\n. (41)\nThe following theorem fully characterizes the rate region Rmss\n3,2of the(3,2)multilevel secret sharing problem.\nTheorem 2. Rmss\n3,2=R∗\n1.\nProof. The achievability and converse proofs are in Appendices A an d B.\nTheorem 2 indicates that superposition is suboptimal for the(3,2)multilevel secret sharing problem. In order\nto achieve the optimality, we need to jointly encode the two s ources. The main idea is that we can use X2as the\nsecret key for X3to reduce coding rates.\nConsider the (3,2)classical SMDC problem with H(X1) = 0 . We can see that the constraints (38)-(40) exactly\ncharacterize the rate region of the classical SMDC problem. The rate region R3,2of the multilevel secret sharing\nproblem can be obtained from the classical SMDC problem by ad ding two more constraints (36) and (37). This\nis reasonable since the reconstruction requirements are th e same for the two problems. The security constraints of\nmultilevel secret sharing problem require the additional c onstraints (36) and (37).\nB. The General (L,s)Multilevel Secret Sharing\nThroughout this subsection, let the first s−1sourcesX1,X2,···,Xs−1in the classical SMDC problem be\nconstants. For any (L,s), since the reconstruction requirements of the multilevel s ecret sharing and classical SMDC\nproblems are the same, the rate constraints of classical SMD C must be also satisfied by the multilevel secret sharing\nproblem. Thus the rate region of classical SMDC provides an o uter bound for the multilevel secret sharing problem.\nWe state the result in the following lemma.\nLemma 2. For any(L,s),Rmss\nL,s⊆ RL∗\nSMDC .\nProof. The containment is trivial.\nSince there are additional security requirements for the mu ltilevel secret sharing problem, the outer bound is not\ntight in general. However, from Section IV-A, we can see that the minimum sum rates for the (3,2)multilevel\nsecret sharing problem and the classical SMDC problem are th e same. It is natural to ask that whether the minimum10\nsum rates for the general (L,s)multilevel secret sharing problem and the classical SMDC pr oblem are always the\nsame. The answer is given in following theorem.\nTheorem 3. For any2≤s≤L, if a rate tuple (R1,R2,···,RL)is admissible for the (L,s)multilevel secret\nsharing problem, then\nR1+R2+···+RL≥L/summationdisplay\nα=sL\nαH(Xα). (42)\nMoreover, this lower bound is tight.\nProof. The converse part follows from Lemma 2. The details for provi ng the converse of the sum rate bound can\nbe found in [1].\nNext, we propose a coding scheme that achieves the minimum su m rate in (42). Let qbe the smallest prime\nsuch that q≥L. Forα∈ L, without loss of generality, we assume that Xαis a memoryless uniformly distributed\nsequence of length lαoverFq. For1≤i≤lα, denote the i-th symbol of XαbyXα,i. For sufficiently large n, the\nlengthlαis arbitrary close tonH(Xα)\nlog2q. Equipartition the sequence Xαintoαmutually independent pieces, denoted\nbyX1\nα,X2\nα,···,Xα\nα. Without loss of generality, assume\nXi\nα=/parenleftBig\nXα,(i−1)lα\nα+1,Xα,(i−1)lα\nα+2,···,Xα,i·lα\nα/parenrightBig\n(43)\nfor1≤i≤α.\nChoose any Ldistinct nonzero elements from Fq, denoted by {b1,b2,···,bL}. Define a Vandermond matrix by\nVα×L=\n1 1 ···1\nb1b2···bL\nb2\n1b2\n2···b2\nL\n............\nbα−1\n1bα−1\n2···bα−1\nL\n. (44)\nThen any αcolumns of Vα×Lare linearly independent. Thus we can construct an (L,α)MDS code with the\ngenerator matrix Vα×L. Let\n/parenleftbig\nY1\nα,Y2\nα,···,YL\nα/parenrightbig\n=/parenleftbig\nX1\nα,X2\nα,···,Xα\nα/parenrightbig\n·Vα×L. (45)\nFor anys≤α≤Landl∈ L, the length of Yl\nαislα\nαand we have\nH(Yl\nα)≤H(Xl\nα) =1\nαH(Xα). (46)\nFor1≤i≤lα\nα, denote the i-th symbol of Yl\nαbyYl\nα,i. Since the operation on Xi\nα(1≤i≤α)is symbolwise, all\nthe symbols of Yl\nαare mutually independent and thus\nH(Yl\nα) =lα/α/summationdisplay\ni=1H(Yl\nα,i). (47)\nThe MDS property of (45) ensures that any αof/braceleftbig\nY1\nα,Y2\nα,···,YL\nα/bracerightbig\ncan losslessly recover the sequence X1\nα,X2\nα,···,Xα\nα\nand then can recover the source Xα. This implies that\n/summationdisplay\nl∈BH(Yl\nα)≥H(Xα),∀B ⊆ L,|B|=α. (48)11\nFrom (46) and (48), we have\nH(Yl\nα) =1\nαH(Xα). (49)\nThus for l∈ L and1≤i≤lα\nα\nH(Yl\nα,i) =1\nlαH(Xα) = log2q, (50)\nwhich implies that Yl\nα,iis uniformly distributed.\nEquipartition Yl\nαintoL(α−s+1) mutually independent pieces, denoted by Yl,1\nα,Yl,2\nα,···,Yl,L(α−s+1)\nα . For\n1≤i≤L(α−s+1) , let\nYl,i\nα=/parenleftbigg\nYl\nα,(i−1)·lα\nαL(α−s+1)+1,Yl\nα,(i−1)·lα\nαL(α−s+1)+2,···,Yl\nα,i·lα\nαL(α−s+1)/parenrightbigg\n(51)\nfor1≤i≤L(α−s+1) .\nNow we construct the code that achieves the minimum sum rate. Since there is no ambiguity, we simply use\n“+” to denote the addition of two sequences over Fqand⊕to denote the modulo Laddition. We first consider\na special case that, for all s+1≤α≤L,\n1\nα−1H(Xα−1) =L(α−s)\nαH(Xα), (52)\nwhich is equivalent to\nlα−1\nα−1=L(α−s)·lα\nα. (53)\nThe encoding and decoding procedures are as follows.\n•Encoding: For l∈ L, the output of Encoder- lis\nWl=/parenleftBigg\nYl\ns,Yl\ns+1+Yl⊕1,l\ns,Yl\ns+2+2/summationdisplay\ni=1Yl⊕i,2(l−1)+i\ns+1 ,···,\nYl\nα+α−s/summationdisplay\ni=1Yl⊕i,(l−1)(α−s)+i\nα−1 ,···,Yl\nL+L−s/summationdisplay\ni=1Yl⊕i,(l−1)(L−s)+i\nL−1/parenrightBigg\n. (54)\n•Decoding: When receiving a subset Uof codewords such that |U|=α, we do the following:\ni. Ifα=s, the decoding of Xαis the trivial MDS decoding.\nii. Ifα > s , we first recover losslessly the sequence Xs. Then initialize j=s.\niii. Calculate {Y1\nj,Y2\nj,···,YL\nj}using (45), then recover {Yl\nj+1:l∈ U} from{Yl\nj+1+/summationtextj−s+1\ni=1Yl⊕i,(l−1)(j−s+1)+i\nj :\nl∈ U} . Sinceα≥j+1, the decoder can recover losslessly the sequence Xj+1and then the source Xj+1.\niv. Ifα=j+1, we are done. If α > j+1, setj=j+1and then go back to step (iii).\nNote: Fors+1≤α≤L, decoding of the source Xαshould base on total information of all the previous\nsourcesXs,Xs+1,···,Xα−1.\nNext we check the security requirements in (14). For s+1≤α≤L, the eavesdropper can access a subset A\nof encoders such that |A|=α−s. Without loss of generality, assume A={1,2,···,α−s}. The eavesdropper12\ncan recover at most Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1of the(α−1)-th source. Any α−1ofY1\nα−1,Y2\nα−1,···,YL\nα−1are\nmutually independent. Then we have\nH/parenleftbigα−s/summationdisplay\ni=1Y1⊕i,i\nα−1,α−s/summationdisplay\ni=1Y2⊕i,(α−s)+i\nα−1 ,···,α−s/summationdisplay\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 |Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1/parenrightbig\n=α−s/summationdisplay\nl=1H/parenleftbigα−s/summationdisplay\ni=1Yl⊕i,(l−1)(α−s)+i\nα−1 |Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1,α−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,α−s/summationdisplay\ni=1Y(l−1)⊕i,(l−2)(α−s)+i\nα−1/parenrightbig\n=α−s/summationdisplay\nl=1H/parenleftbig\nYα−s+1,l(α−s−1)+1\nα−1 +Yα−s+2,l(α−s−1)+2\nα−1 +···+Ymin{α−s+l,L},min{l(α−s),(l−1)(α−s)+L−l}\nα−1 |\nY1\nα−1,Y2\nα−1,···,Yα−s\nα−1,α−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,α−s/summationdisplay\ni=1Y(l−1)⊕i,(l−2)(α−s)+i\nα−1/parenrightbig\n(55)\n=α−s/summationdisplay\nl=1H/parenleftbig\nYα−s+1,l(α−s−1)+1\nα−1 +Yα−s+2,l(α−s−1)+2\nα−1 +···+Ymin{α−s+l,L},min{l(α−s),(l−1)(α−s)+L−l}\nα−1 |\nY1,l(α−s−1)+1\nα−1 ,Y2,l(α−s−1)+1\nα−1 ,···,Yα−s,l(α−s−1)+1\nα−1 ,Y1,l(α−s−1)+2\nα−1 ,Y2,l(α−s−1)+2\nα−1 ,···,Yα−s,l(α−s−1)+2\nα−1\nY1,min{l(α−s),(l−1)(α−s)+L−l}\nα−1 ,Y2,min{l(α−s),(l−1)(α−s)+L−l}\nα−1 ,···,Yα−s,min{l(α−s),(l−1)(α−s)+L−l}\nα−1/parenrightbig\n(56)\n=α−s/summationdisplay\nl=1H/parenleftbig\nYα−s+1,l(α−s−1)+1\nα−1 +Yα−s+2,l(α−s−1)+2\nα−1 +···+Ymin{α−s+l,L},min{l(α−s),(l−1)(α−s)+L−l}\nα−1/parenrightbig\n(57)\n=α−s/summationdisplay\nl=1H/parenleftbigα−s/summationdisplay\ni=1Yl⊕i,(l−1)(α−s)+i\ns/parenrightbig\n(58)\nwhere (55) follows from H(A+B|A) =H(B|A), (56) follows from the mutual independence of the partition s of\nYi\nα−1for anyi∈ L, (57) follows from the mutual independence of any α−1ofY1\nα−1,Y2\nα−1,···,YL\nα−1and (58)\nfollows from the fact that all symbols of Yα−1and the summations of any j(j≤α−1)of them are uniformly\ndistributed. Thus, we have\nH/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα,α−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,α−s/summationdisplay\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 ,Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1/parenrightbig\n=α−s/summationdisplay\nl=1H(Yl\nα−1)+H/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα|Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1/parenrightbig\n+H/parenleftbigα−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,α−s/summationdisplay\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 |Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1,Y1\nα,Y2\nα,···,Yα−s\nα/parenrightbig\n=α−s/summationdisplay\nl=1H(Yl\nα−1)+H/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα/parenrightbig\n+H/parenleftbigα−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,α−s/summationdisplay\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 |Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1/parenrightbig\n(59)\n=α−s/summationdisplay\nl=1H(Yl\nα−1)+α−s/summationdisplay\nl=1H/parenleftbig\nYl\nα/parenrightbig\n+α−s/summationdisplay\nl=1H/parenleftbigα−s/summationdisplay\ni=1Yl⊕i,(l−1)(α−s)+i\nα−1/parenrightbig\n(60)\nwhere (59) follows from the mutual independence of YαandYα−1and (60) follows from (58) and the mutual inde-\npendence of Y1\nα,Y1\nα,···,Yα\nα. This implies that all of Y1\nα,Y2\nα,···,Yα−s\nα,/summationtextα−s\ni=1Y1⊕i,i\nα−1,/summationtextα−s\ni=1Y2⊕i,(α−s)+i\nα−1 ,···,13\n/summationtextα−s\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 ,Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1are mutually independent. From Appendix D, we have\nH/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα|Y1\nα+α−s/summationdisplay\ni=1Y1⊕i,i\nα−1,···,Yα−s\nα+α−s/summationdisplay\ni=1Y(α−s)⊕i,(α−s−1)(α−s)+i\nα−1 ,Y1\nα−1,Y2\nα−1,···,Yα−s\nα−1/parenrightbig\n=H/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα/parenrightbig\n, (61)\nwhich implies that\nH/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα|W1,···,Wα−s/parenrightbig\n=H/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα/parenrightbig\n, (62)\nand thus\nH/parenleftbig\nXα|WA/parenrightbig\n=H/parenleftbig\nXα/parenrightbig\n. (63)\nfors+1≤α≤L. Thus, all security requirements in (14) are satisfied.\nFrom (54), we obtain the coding rates\nRi=L/summationdisplay\nα=s1\nαH(Xα),fori∈ L, (64)\nwhich implies that\nL/summationdisplay\ni=1Ri=L/summationdisplay\nα=sL\nαH(Xα). (65)\nThus, the coding scheme achieves the sum rate bound in (42) if\n1\nα−1H(Xα−1) =L(α−s)\nαH(Xα), (66)\nfor alls+1≤α≤L. For the case that\n1\nα−1H(Xα−1)≥L(α−s)\nαH(Xα) (67)\nfor alls+1≤α≤L, we modify the coding scheme in (54) as follows. Denote the fir stlα\nαsymbols of Yi\nα−1(i∈ L)\nbyYi∗\nα−1. Equipartition Yi∗\nα−1intoL(α−s)pieces, denoted by Yi∗,1\nα,Yi∗,2\nα,···,Yi∗,L(α−s)\nα . Then we get the new\ncodewords by replacing Yl\nα+/summationtextα−s\ni=1Yl⊕i,(l−1)(α−s)+i\nα−1 in (54) with Yl\nα+/summationtextα−s\ni=1Y(l⊕i)∗,(l−1)(α−s)+i\nα−1 for alll∈ L.\nIt is easy to check that the decoding and security requiremen ts are fulfilled and the same minimum sum rate is\nachieved.\nThus, we have designed a coding scheme that achieves the mini mum sum rate in (42) when\n1\nα−1H(Xα−1)≥L(α−s)\nαH(Xα) (68)\nfor alls+1≤α≤L. Therefore, we finish proving Theorem 3.\nRemark 4.The minimum sum rate in (42) can be achieved only for s≥2. Fors= 1, superposition coding is\nproved to be optimal in Theorem 1 and thus no source can perfor m as secret keys for other sources.\nRemark 5.In the proof, we derive a sufficient condition in (68) that the minimum sum rate can be achieved.\nHowever, the condition is not necessary in general. In the co ding scheme, the sequence Yl\nαis partitioned into many\npieces to ensure perfect secrecy. But on the other hand, too m any partitions result in that the sufficient condition14\n(68) is too tough. The necessary and sufficient condition can be far from that. The following lemma provides a\nmuch better sufficient condition for L <2s.\nLemma 3. IfL <2s, the minimum sum rate in (42) can be achieved when\nH(Xs)\ns≥H(Xs+1)\ns+1≥ ··· ≥H(XL)\nL. (69)\nProof. We first consider a special case that\nH(Xs)\ns=H(Xs+1)\ns+1=···=H(XL)\nL. (70)\nwhich is equivalent to\nls\ns=ls+1\ns+1=···=lL\nL. (71)\nThe encoding and decoding procedures are as follows.\n•Encoding: For l∈ L, the output of Encoder- lis\nWl=/parenleftbig\nYl\ns,Yl\ns+1+Yl⊕1\ns,Yl\ns+2+Yl⊕1\ns+1,···,Yl\nL+Yl⊕1\nL−1/parenrightbig\n. (72)\n•Decoding: For s+1≤α≤L, decoding of the source Xαshould base on the previous source Xα−1. When\nreceiving a subset Uof codewords such that |U|=α, we have the following decoding procedure:\ni. Ifα=s, the decoding of Xαis the trivial MDS decoding.\nii. Ifα > s , we first recover losslessly the sequence Xs. Initialize j=s.\niii. Calculate {Yl⊕1\nj:l∈ U} using (45), then recover {Yl\nj+1:l∈ U} from{Yl\nj+1+Yl⊕1\nj:l∈ U} . Since\nα≥j+1, the decoder can recover losslessly the sequence Xj+1.\niv. Ifα=j+1, we are done. If α > j+1, setj=j+1and then go back to step (iii).\nNext we check the security requirements in (14). For s+ 1≤α≤L, the eavesdropper can access a subset\nAof encoders such that |A|=α−s. SinceL <2s, we have α−s < s . Without loss of generality, assume\nA={1,2,···,α−s}.\nThe eavesdropper can recover Y1\ns,Y2\ns,···,Yα−s\ns. Using (72), it can then recover Y1\ns+1,Y2\ns+1,···,Yα−s−1\ns+1 from\nY1\ns+1+Y2\ns,Y2\ns+1+Y3\ns,···,Yα−s−1\ns+1+Yα−s\ns. SinceXsandXs+1are mutually independent and α−s+1≤s,\nwe have that Yα−s\ns+1,Y1\ns,Y2\ns,···,Yα−s\ns,Yα−s+1\ns are mutually independent. Thus, from Appendix section D we\nhave\nH/parenleftbig\nYα−s\ns+1|Yα−s\ns+1+Yα−s+1\ns,Y1\ns,Y2\ns,···,Yα−s\ns/parenrightbig\n=H/parenleftbig\nYα−s\ns+1/parenrightbig\n. (73)\nThis implies that\nH/parenleftbig\nYα−s\ns+1|WA/parenrightbig\n=H/parenleftbig\nYα−s\ns+1/parenrightbig\n. (74)\nSimilarly, for s+1≤j≤α, the eavesdropper can recover Y1\nj,Y2\nj,···,Yα−j\nj using (72). Since α−s+1≤j−1,\nY1\nj−1,Y2\nj−1,···,Yα−s+1\nj−1 are mutually independent. From Appendix D we have\nH/parenleftbig\nYα−j+1\nj,Yα−j+2\nj,···,Yα−s\nj|Yα−j+1\nj+Yα−j+2\nj−1,Yα−j+2\nj+Yα−j+3\nj−1,···,Yα−s\nj+Yα−s+1\nj−1,Y1\nj−1,Y2\nj−1,···,Yα−j+1\nj−1/parenrightbig\n=H/parenleftbig\nYα−j+1\nj,Yα−j+2\nj,···,Yα−s\nj/parenrightbig\n, (75)15\nwhich implies that\nH/parenleftbig\nYα−j+1\nj,Yα−j+2\nj,···,Yα−s\nj|WA/parenrightbig\n=H/parenleftbig\nYα−j+1\nj,Yα−j+2\nj,···,Yα−s\nj/parenrightbig\n. (76)\nThen for j=α, the eavesdropper can recover nothing and we have\nH/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα|WA/parenrightbig\n=H/parenleftbig\nY1\nα,Y2\nα,···,Yα−s\nα/parenrightbig\n, (77)\nwhich implies\nH(Xα|WA) =H(Xα). (78)\nThus, all security requirements in (14) are satisfied.\nFrom (72), we obtain the coding rates\nRi=L/summationdisplay\nα=s1\nαH(Xα),fori∈ L, (79)\nwhich implies that\nL/summationdisplay\ni=1Ri=L/summationdisplay\nα=sL\nαH(Xα). (80)\nThus, the coding scheme achieves the sum rate bound in (42) fo r the case thatH(Xs)\ns=H(Xs+1)\ns+1=···=H(XL)\nL.\nFor the other cases, we modify the coding scheme in (54) as fol lows.\n1) IfH(Xα−1)\nα−1≥H(Xα)\nαfor alls+1≤α≤L, denote the firstlα\nαsymbols of Yi\nα−1(i∈ L)byYi∗\nα−1. Then\nwe get the new codewords by replacing Yl\nα+Yl⊕1\nα−1in (72) with Yl\nα+Y(l⊕1)∗\nα−1. It is easy to check that the\ndecoding and security requirements are fulfilled and the sam e minimum sum rate is achieved.\n2) IfH(Xα−1)\nα−1L\nαH(Xα). (83)\nThus, the sum rate exceeds the bound in (42).16\nWe have designed a coding scheme that achieves the minimum su m rate in (42) for the case that\nH(Xs)\ns≥H(Xs+1)\ns+1≥ ··· ≥H(XL)\nL. (84)\nTherefore, Lemma 3 is proved.\nForL <2s, Lemma 3 provides a sufficient condition that achieves the mi nimum sum rate bound. We conjecture\nthat the condition in (69) is also necessary.\nConjecture 1. ForL <2s, the sum rate bound in (42) is inactive if\nH(Xα−1)\nα−1L/summationdisplay\nα=sL\nαH(Xα), (91)17\nwe conclude that superposition coding for (L,s)multilevel secret sharing problem is suboptimal. Thus, Cor ollary 3.1\nis proved.\nRemark 6.Fors= 1, superposition is proved to be optimal in Section III. For s=L, the problem reduces to the\nclassical SMDC problem with H(X1) =···=H(Xα−1) = 0 and thus superposition is optimal.\nV. S LIDING SECURE SMDC\nA.(3,2)Sliding Secure SMDC\nConsider the (3,2)sliding secure SMDC problem. Now we have three independent s ourcesX1,X2, andX3. For\nα= 1,2,3, the source Xαcan be losslessly reconstructed by the decoder if a subset of αencoders are accessible\nandXαshould be kept perfectly secure if the eavesdropper can acce ss no more than [α−2]+encoders. The\nreconstruction and security constraints in (6)(7) become\nH(X1|Wi) = 0,for alli= 1,2,3 (92)\nH(X2|Wi,Wj) = 0,for all1≤i < j≤3 (93)\nH(X3|W1,W2,W3) = 0, (94)\nH(X3|Wi) =H(X3),for alli= 1,2,3. (95)\nLetR∗\n2be the collection of nonnegative rate triples (R1,R2,R3)such that for i= 1,2,3\nRi=r0\ni+r1\ni (96)\nwherer0\ni,r1\ni>0and\nr1\ni≥H(X1) (97)\n(r0\n1,r0\n2,r0\n3)∈ Rmss\n3,2. (98)\nWe can see that R∗\n2is the superposition region induced by separately encoding two sets of sources X1and(X2,X3)\nwith rates r1\niandr0\ni, respectively. It is easy to write the region equivalently b y eliminating rj\ni, which is the set of\nrate tuples (R1,R2,R3)such that\nRi≥H(X1),for1≤i≤3 (99)\n2Ri+Rj≥3H(X1)+H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (100)\nRi+Rj≥2H(X1)+1\n2H(X2)+H(X3),for1≤i < j≤3 (101)\nRi+Rj≥2H(X1)+H(X2),for1≤i < j≤3 (102)\nR1+R2+R3≥3H(X1)+3\n2H(X2)+H(X3), (103)\n2Ri+Ri⊙1+Ri⊙2≥4H(X1)+2H(X2)+H(X3),for1≤i≤3. (104)\nThe following theorem characterizes the rate region R3,2of the(3,2)sliding secure SMDC problem.\nTheorem 4. R3,2=R∗\n2.18\nProof. SinceR∗\n2is the rate region induced by superposition coding of X1and(X2,X3), all the rates (R1,R2,R3)∈\nR∗\n2are achievable. We only need to show the converse. The detail s are in Appendix C.\nTheorem 4 indicates that separately encoding two sets of sou rcesX1and(X2,X3)is optimal for the (3,2)\nsliding secure SMDC problem even though superposition of th ree individual sources is suboptimal. The result\ncoincides with the region of (3,2)classical SMDC by setting X3to be constants for both problems.\nWe can see that the constraints in (99) and (102)-(104) are th e same constraints that defines the classical SMDC\nregion. Thus, the rate region R3,2can be obtained from the classical SMDC region by adding two m ore constraints\n(100) and (101).\nB. The General (L,s)Sliding Secure SMDC\nFor any(L,s), since the reconstruction requirements of the sliding secu re SMDC and classical SMDC problems\nare the same, the rate constraints of classical SMDC must be a lso satisfied by the sliding secure SMDC problem.\nThus the rate region of classical SMDC provides an outer boun d for the sliding secure SMDC problem. We state\nthe result in the following lemma.\nLemma 4. For any(L,s),RL,s⊆ RL\nSMDC .\nProof. The containment is trivial.\nSince there are additional security requirements for the sl iding secure SMDC problem, the outer bound is not\ntight in general. However, from Section V-A, we can see that t he(3,2)sliding secure SMDC problem achieves the\nsame minimum sum rate as the classical SMDC problem. Moreove r, in Section IV-B, we show that the multilevel\nsecret sharing problem achieves the sum rate bound of R∗\nSMDC . In the following theorem, we generalize the sum\nrate property in Sections IV-B and V-A to the general (L,s)sliding secure SMDC problem.\nTheorem 5. For any2≤s≤L−1, if a rate tuple (R1,R2,···,RL)is admissible for the (L,s)sliding secure\nSMDC problem, then\nR1+R2+···+RL≥L/summationdisplay\nα=1L\nαH(Xα). (105)\nMoreover, the lower bound is tight and can be achieved by supe rposition coding of the sets of sources X1,X2,\n···,Xs−1, and(Xs,Xs+1,···,XL).\nProof. The converse follows from Lemma 4.\nNext, we design a coding scheme that achieves that minimum su m rate in (105). For i∈ L, and0≤α≤s−1,\nlet\nrα\ni=1\nαH(Xα),for1≤α≤s−1 (106)\nr0\ni=L/summationdisplay\nα=s1\nαH(Xα). (107)19\nThen separately encode the ssets of sources X1,X2,···,Xs−1, and(Xs,Xs+1,···,XL)with rates r1\ni,r2\ni,···,rs−1\ni,\nandr0\ni, respectively for i∈ L. For any 1≤α≤s−1andB ⊆ L such that |B|=α, since\n/summationdisplay\ni∈Brα\ni=/summationdisplay\ni∈B1\nαH(Xα) =H(Xα), (108)\nwe can losslessly reconstruct the sources X1,X2,···,Xs−1. Since the rates r0\ni(i∈ L)in (107) are the same as\nrates in (64), we can use the coding scheme in Section IV-B to e ncode the set of sources (Xs,Xs+1,···,XL)if\n1\nα−1H(Xα−1)≥L(α−s)\nαH(Xα) (109)\nfor alls+1≤α≤L. Thus, the rate tuple (R1,R2,···,RL), where\nRi=s−1/summationdisplay\nα=0rα\ni,fori∈ L (110)\nis admissible and superposition of the ssets of sources achieves the sum rate\nR1+R2+···+RL=L/summationdisplay\ni=1/parenleftBiggs−1/summationdisplay\nα=0rα\ni/parenrightBigg\n=L/summationdisplay\ni=1L/summationdisplay\nα=11\nαH(Xα) =L/summationdisplay\nα=1L\nαH(Xα). (111)\nTherefore, the optimality of such superposition coding in t erms of achieving the minimum sum rate in (105) is\nproved. This completes the proof of Theorem 5.\nLetR3\nsupbe the superposition region of the (L,s)sliding secure SMDC problem induced by separately encoding\ntheLindependent sources. Then R3\nsupis the set of rate tuples (R1,R2,···,RL)such that for i∈ L\nRi=L/summationdisplay\nα=1rα\ni (112)\nwhererα\ni≥0,1≤α≤Land\n/summationdisplay\ni∈Brα\ni≥H(Xα),for1≤α≤s−1andB ⊆ L s.t.|B|=α (113)\n/summationdisplay\ni∈Brα\ni≥H(Xα),fors≤α≤LandB ⊆ L s.t.|B|=s. (114)\nIt is easy to characterize the sum rate bound of R3\nsupby\nR1+R2+···+RL≥s−1/summationdisplay\nα=1L\nαH(Xα)+L/summationdisplay\nα=sL\nsH(Xα). (115)\nCorollary 5.1. R3\nsup/subsetnoteqlRL,sfor2≤s≤L−1.\nProof. The sum rate bound of R3\nsupis in (115). For 2≤s≤L−1, since\ns−1/summationdisplay\nα=1L\nαH(Xα)+L/summationdisplay\nα=sL\nsH(Xα)>L/summationdisplay\nα=1L\nαH(Xα), (116)\nwe conclude that superposition coding for (L,s)sliding secure SMDC problem is suboptimal. This proves Coro l-\nlary 5.1.\nRemark 7.Fors= 1, superposition is proved to be optimal in Section III. For s=L, the problem reduces to the\nclassical SMDC problem and thus superposition of the Lsources is optimal.20\nEven though Corollary 5.1 states that superposition of X1,X2,···,XLis suboptimal, Theorem 5 tells us\nthat superposition of X1,X2,···,Xs−1,(Xs,Xs+1,···,XL)achieves the minimum sum rate. Denote the\nsuperposition region induced by separately encoding the sset of sources by R4\nsup. ThenR4\nsupis the set of\nnonnegative rate tuples (R1,R2,···,RL)such that\nRi=s−1/summationdisplay\nα=0rα\ni,fori∈ L (117)\nwhererα\ni≥0,0≤α≤s−1and\n/summationdisplay\ni∈Brα\ni≥H(Xα),for1≤α≤s−1andB ⊆ L s.t.|B|=α (118)\n/parenleftbig\nr0\n1,r0\n2,···,r0\nL/parenrightbig\n∈ Rmss\nL,s. (119)\nWe conjecture that superposition of X1,X2,···,Xs−1,(Xs,Xs+1,···,XL)can achieve the entire admissible\nrate region of the sliding secure SMDC problem.\nConjecture 2. RL,s=R4\nsupfor all(L,s).\nA trivial case that the conjecture is true is the classical SM DC problem with H(Xs+1) =···=H(XL) = 0 . The\nsimplest nontrivial example is the (3,2)problem whose rate region is characterized in Theorem 4. The conjecture\nis also true for the special cases that s= 1 ands=L.\nHowever, since superposition is suboptimal for general (L,s)multilevel secret sharing problem (with 2≤s≤\nL−1), it is difficult to characterize the rate region Rmss\nL,s. Thus, it is challenging to prove the conjecture at even\nthe first step. Similar as the rate region of classical SMDC pr oblem in [2], we may try to find some implicit ways\nto characterize the rate region of sliding secure SMDC in the future work.\nVI. C ONCLUSION\nThis paper considered the problem of sliding secure SMDC, wh ich is a generalization of the classical SMDC\nproblem to the security settings. The (L,s)sliding secure SMDC problem is specialized to the (L,s)multilevel\nsecret sharing problem when the first s−1sources are constants. We have fully characterized the rate regions of the\nsliding secure SMDC and multilevel secret sharing problems fors= 1 and(L,s) = (3,2). Fors= 1, separately\nencoding independent sources (superposition coding) is op timal for both sliding secure SMDC and multilevel secret\nsharing problems.\nFors≥2, it was shown that superposition coding of the independent s ources is suboptimal for the multilevel\nsecret sharing problem. The main idea that joint encoding ca n reduce coding rates is that we can use the previous\nsourceXα−1as the secret keys for source Xα. Based on this idea, we designed a coding scheme that achieve s\nthe entire rate region of the (3,2)problem. For the general case, we proposed a coding scheme th at achieves the\nminimum sum rate. However, it is difficult to determine wheth er such coding schemes can achieve the entire rate\nregion since it is really challenging to characterize the re gion of the general multilevel secret sharing problem.\nFor the(3,2)sliding secure SMDC problem, superposition of two sets of so urcesX1and(X2,X3)was shown\nto be optimal. For the general problem that s≥2, we have shown that superposition of the ssets of sources X1,21\nX2,···,Xs−1and(Xs,Xs+1,···,XL)achieves the minimum sum rate. To show the optimality of achi eving\nthe entire rate region, we need more efforts in the future wor k.\nAPPENDIX A\nACHIEVABILITY PROOF OF THEOREM 2\nIn this section, we will show the achievability of R∗\n1by designing coding schemes for the following four case,\nrespectively.\ni.m=1\n2H(X2)+H(X3),3\n2m >3\n2H(X2)+H(X3), and2m >2H(X2)+H(X3),\nii.m=1\n2H(X2)+H(X3),3\n2m≤3\n2H(X2)+H(X3), and2m >2H(X2)+H(X3),\niii.m=1\n2H(X2)+H(X3),3\n2m≤3\n2H(X2)+H(X3), and2m≤2H(X2)+H(X3),\niv.m=H(X2).\nFor each case, the rate region R∗\n1is a convex polyhedron specified by several hyperplanes. Due to time-sharing\narguments, we only need to show the achievability for the cor ner points. For α∈ L, without loss of generality, we\nassume that Xαis a memoryless binary symmetric sequence of length lα. For sufficiently large n,lαis arbitrary\nclose tonH(Xα).\nA. Case i\nThe set of conditions m=1\n2H(X2) +H(X3),3\n2m≥3\n2H(X2) +H(X3), and2m≥2H(X2) +H(X3)are\nequivalent to the constraint\nH(X2)<2\n3H(X3). (120)\nWe can check that, under the condition in (120), the second co nstraint (37) of R∗\n1implies the last two (39)(40), thus\nthe last two constraints are inactive. The rate region R∗\n1becomes the set of nonnegative rate tuples (R1,R2,R3)\nsuch that\n2Ri+Rj≥H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (121)\nRi+Rj≥1\n2H(X2)+H(X3),for1≤i < j≤3. (122)\nThe rate region is drawn in Fig. 2. For abbreviation, we denot eH(X2)andH(X3)byH2andH3for labeling\nin the figures. For sufficiently large n, the lengths l2andl3are large enough so that we can alway asymptotically\npartition the new source sequences X2andX3properly.\nDue to time sharing arguments and symmetry, we only need to sh ow the achievability for the corner points\nQ1= (0,H2+H3,H2+H3),P1= (1\n2H2,H3,H3)andO= (1\n4H2+1\n2H3,1\n4H2+1\n2H3,1\n4H2+1\n2H3).\n1) corner point Q1: Let the secret key Z1be a random binary sequence with the same length as X3.we use the\nfollowing scheme which works for all H2andH3(in particular for H2<2\n3H3)\n\n\nW1=∅\nW2= (X2,Z1)\nW3= (X2,X3+Z1).(123)22\nR1R2R3\nO= (1\n4H2+1\n2H3,1\n4H2+1\n2H3,1\n4H2+1\n2H3)Q1= (0,H2+H3,H2+H3)\nQ2= (H2+H3,H2+H3,0)Q3P1= (1\n2H2,H3,H3)\nP2= (H3,H3,1\n2H2)P3R1≥0\nR3≥0R2≥02R1+R3≥H2+H3\nR1+ 2R3≥H2+H32R1+R2≥H2+H3R1+ 2R2≥H2+H3\n2R2+R3≥H2+H3\nR2+ 2R3≥12H2+H3R1+R3≥1\n2H2+H3R1+R2≥1\n2H2+H3\nR2+R3≥12H2+H3\nFig. 2. rate region R∗\n1: case i ( H2<2\n3H3)\nIt is easy to check that\n(R1,R2,R3) = (0,H2+H3,H2+H3) (124)\nand any two channels can losslessly reconstruct X2, all the three channels can reconstruct X3, and any one\nchannel know nothing (information theoretic) about X3. We will only describe the coding schemes and omit\nwriting out such checks in the sequel.\n2) corner point P1: equipartition X2intoA2andB2with the same length1\n2l2. Partition X3intoA3,B3, and\nC3with lengths1\n2l2,1\n2l2, andl3−l2. The length l3−l2is nonnegative for all H2andH3such that H2≤H3,\nin particular for H2<2\n3H3. LetZ2be a random binary sequence with the same length as C3. Then we use\nthe following scheme which works for all H2andH3such that H2≤H3(in particular for H2<2\n3H3):\n\n\nW1= (A2+B2)\nW2= (A2,A3+B2,Z2)\nW3= (B2,B3+A2,C3+Z2).(125)\nWe can check that\n(R1,R2,R3) = (1\n2H2,H3,H3). (126)\n3) corner point O: equipartition X2intoA2andB2with the same length1\n2l2. Partition X3intoA3,B3,C3,\nD3, andE3with lengths1\n2l2,1\n2l2,1\n2l2,1\n2l3−3\n4l2, and1\n2l3−3\n4l2. The length1\n2l3−3\n4l2is positive since23\nH2<2\n3H3. LetZ3be a random binary sequence with the same length as D3andE3. Then we use the\nfollowing scheme:\n\nW1= (A2+B2,A3+A2,Z3)\nW2= (A2,B3+B2,D3+Z3)\nW3= (B2,C3+A2,E3+Z3).(127)\nWe can check that\nR1=R2=R3=1\n2H2+1\n2H2+(1\n2H3−3\n4H2)\n=1\n4H2+1\n2H3. (128)\nTherefore, we finish the achievability proof for the case tha tH2<2\n3H3.\nB. Case ii\nThe set of conditions m=1\n2H(X2) +H(X3),3\n2m≤3\n2H(X2) +H(X3), and2m >2H(X2) +H(X3)are\nequivalent to the constraint\n2\n3H(X3)≤H(X2)< H(X3). (129)\nWe can check that, under the condition in (129), the second co nstraint (37) of R∗\n1implies the last one (40), thus\nthe last constraint is inactive. The rate region R∗\n1becomes the set of nonnegative rate tuples (R1,R2,R3)such that\n2Ri+Rj≥H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (130)\nRi+Rj≥1\n2H(X2)+H(X3),for1≤i < j≤3 (131)\nR1+R2+R3≥3\n2H(X2)+H(X3). (132)\nThe rate region is depicted in Fig. 3. Due to time-sharing arg uments and symmetry, we only need to show the\nachievability for the corner points Q1= (0,H2+H3,H2+H3),P1= (1\n2H2,H3,H3)andS1= (H3−1\n2H2,H2,H2).\n1) corner point Q1: the same as Q1in (123)\n2) corner point P1: the same as P1in (125)\n3) corner point S1: equipartition X2intoA2andB2with the same length1\n2l2. Partition X3intoA3,B3, and\nC3with lengths1\n2l2,1\n2l2, andl3−l2. The condition H2< H3implies that the length l3−l2is positive. Since\n2\n3H3≤H2impliesl3−l2≤1\n2l2, we denote the first l3−l2bits ofA2byA1\n2. Then we use the following\nscheme: \n\nW1= (A2+B2,C3+A1\n2)\nW2= (A2,A3+B2)\nW3= (B2,B3+A2).(133)\nIt is easy to check that\nR1=1\n2H2+(H3−H2) =H3−1\n2H2 (134)\nR2=R3=1\n2H2+1\n2H2=H2. (135)24\nR1R2R3\nQ1= (0,H2+H3,H2+H3)\nQ2= (H2+H3,H2+H3,0)Q3S1= (H3−1\n2H2,H2,H2)\nS2= (H2,H2,H3−1\n2H2)S3P1= (1\n2H2,H3,H3)\nP2= (H3,H3,1\n2H2)P3R1≥0\nR3≥0R2≥02R1+R3≥H2+H3\nR1+ 2R3≥H2+H32R1+R2≥H2+H3R1+ 2R2≥H2+H3\n2R2+R3≥H2+H3\nR2+ 2R3≥12H2+H3R1+R3≥1\n2H2+H3R1+R2≥1\n2H2+H3\nR2+R3≥12H2+H3\nR1+R2+R3≥3\n2H2+H3\nFig. 3. rate region R∗\n1: case ii (2\n3H3≤H2< H3)\nIt may not be so obvious to see that (A2+B2,C3+A1\n2)provide no information about C3. So we state the\nresult in the following lemma, which ensures that W1provide no information about X3. The assumption that\nX2andX3are memoryless symmetric sources indicates all the bits of X2andX3are linearly independent.\nSince the addition operations A2+B2,C3+A1\n2are bitwise, we only state the claim for single bits A,B,\nandCin the lemma.\nLemma 5. LetA,B, andCbe three independent random variables taking values in the s ame finite field F2.\nIfX=A+CandY=A+B, then\nH(C|X,Y) =H(C). (136)\nProof. We try to show that I(X,Y;C) = 0 . Since mutual information is nonnegative, we only need to pr ove\nthatI(X,Y;C)≤0. We have the following.\n−I(X,Y;C) =H(C,X,Y)−H(X,Y)−H(C)\n=/bracketleftbig\nI(A;Y|C,X)+H(A,C,X,Y )−H(A,C,X)+H(C,X)/bracketrightbig\n+/bracketleftbig\nI(X;Y)−H(X)−H(Y)/bracketrightbig\n−H(C)\n=I(A;Y|C,X)+H(A,C,Y)−H(A,C)+/bracketleftbig\nH(C)+H(X)/bracketrightbig\n+I(X;Y)−H(X)−H(Y)−H(C) (137)25\n=I(A;Y|C,X)+I(X;Y)+H(A,C,Y)−H(A,C)−H(Y)\n=I(A;Y|C,X)+I(X;Y)−H(A,C)−H(Y)\n+/bracketleftbig\nI(B;C|A,Y)+H(A,B,C,Y )−H(A,B,Y)+H(A,Y)/bracketrightbig\n=I(A;Y|C,X)+I(X;Y)−H(A,C)−H(Y)\n+I(B;C|A,Y)+H(A,B,C)−H(A,B)+/bracketleftbig\nH(A)+H(Y)/bracketrightbig\n(138)\n=I(A;Y|C,X)+I(X;Y)+I(B;C|A,Y)\n+H(A,B,C)+H(A)−H(A,B)−H(A,C)\n=I(A;Y|C,X)+I(X;Y)+I(B;C|A,Y) (139)\n≥0, (140)\nwhere (137) follows from\nH(X|A,C) = 0 (141)\nand\nH(C,X) =H(C)+H(X), (142)\n(138) follows from\nH(Y|A,B) = 0 (143)\nand\nH(A,Y) =H(A)+H(Y), (144)\nand (139) follows from the mutual independence of A,B, andC. Then we conclude that I(X,Y;C) = 0 and\nthusH(C|X,Y) =H(C).\nTherefore, the achievability for the case that2\n3H3≤H2< H3is proved.\nC. Case iii\nThe set of conditions m=1\n2H(X2) +H(X3),3\n2m≤3\n2H(X2) +H(X3), and2m≤2H(X2) +H(X3)are\nequivalent to the constraint\nH(X3)≤H(X2)<2H(X3). (145)\nWe can see that, under the condition in (145), all the constra ints in (36)-(40) are active. The rate region R∗\n1becomes\nthe set of nonnegative rate tuples (R1,R2,R3)such that\n2Ri+Rj≥H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (146)\nRi+Rj≥1\n2H(X2)+H(X3),for1≤i < j≤3 (147)\nR1+R2+R3≥3\n2H(X2)+H(X3) (148)\n2Ri+Ri⊙1+Ri⊙2≥2H(X2)+H(X3),for1≤i≤3. (149)26\nThe rate region is depicted in Fig. 4. Due to time-sharing arg uments and symmetry, we only need to show the\nR1R2R3\nQ1= (0,H2+H3,H2+H3)\nQ2= (H2+H3,H2+H3,0)Q3T1= (1\n2H3,H2,H2)\nT2= (H2,H2,1\n2H3)T3\nS4= (1\n2H2,H2,H3)\nS5= (H3,H2,1\n2H2)\nS6S7S8S9R1≥0\nR2≥0\nR3≥02R1+R3≥H2+H3\nR1+ 2R3≥H2+H32R1+R2≥H2+H3R1+ 2R2≥H2+H3\n2R2+R3≥H2+H3\nR2+ 2R3≥12H2+H3R1+R2≥1\n2H2+H3\nR1+R3≥1\n2H2+H3\nR2+R3≥12H2+H3\nR1+R2+R3≥3\n2H2+H32R1+R2+R3≥2H2+H3\nR1+R2+ 2R3≥2H2+H3R1+ 2R2+R3≥2H2+H3\nFig. 4. rate region R∗\n1: case iii ( H3≤H2<2H3)\nachievability for the corner points Q1= (0,H2+H3,H2+H3),T1= (1\n2H3,H2,H2)andS4= (1\n2H2,H2,H3).\n1) corner point Q1: the same as Q1in (123)\n2) corner point T1: partition X2intoA2,B2, andC2with lengths1\n2l3,1\n2l3, andl2−l3. The length l2−l3is\nnonnegative for all pairs (H2,H3)such that H3≤H2, in particular for H3≤H2<2H3. Equipartition X3\nintoA3andB3with the same length1\n2l3. Then we use the following scheme which works for all H2and\nH3such that H3≤H2(in particular for H3≤H2<2H3):\n\n\nW1= (A2+B2)\nW2= (A2,C2,A3+B2)\nW3= (B2,C2,B3+A2).(150)\nWe can check that\n(R1,R2,R3) = (1\n2H3,H2,H2). (151)\n3) corner point S4: equipartition X2intoA2andB2with the same length1\n2l2. Partition X3intoA3andB3\nwith lengths1\n2l2andl3−1\n2l2. SinceH3≤H2impliesl3−1\n2l2≤1\n2l2, we denote the first l3−1\n2l2bits ofA227\nbyA1\n2. Then we use the following scheme:\n\n\nW1= (A2+B2)\nW2= (A2,A3+B2)\nW3= (B2,B3+A1\n2).(152)\nIt’s easy to check that\n(R1,R2,R3) = (1\n2H2,H2,H3). (153)\nTherefore, we prove the achievability for the case that H(X3)≤H(X2)<2H(X3).\nD. Case iv\nThe condition m=H(X2)is equivalent to the constraint\n2H(X3)≤H(X2). (154)\nWe can see that, under the condition in (154), all the constra ints in (36)-(40) are active. The rate region R∗\n1becomes\nthe set of nonnegative rate tuples (R1,R2,R3)such that\n2Ri+Rj≥H(X2)+H(X3),for1≤i,j≤3andi/ne}ationslash=j (155)\nRi+Rj≥H(X2),for1≤i < j≤3 (156)\nR1+R2+R3≥3\n2H(X2)+H(X3) (157)\n2Ri+Ri⊙1+Ri⊙2≥2H(X2)+H(X3),for1≤i≤3. (158)\nThe rate region is depicted in Fig. 5. Due to time-sharing arg uments and symmetry, we only need to show the\nachievability for the corner points Q1= (0,H2+H3,H2+H3),T1= (1\n2H3,H2,H2),T4= (H3,H2,H2−H3)\nandS10= (1\n2H2,1\n2H2+H3,1\n2H2).\n1) corner point Q1: the same as Q1in (123)\n2) corner point T1: the same as T1in (150)\n3) corner point T4: partition X2intoA2,B2, andC2with lengths l3,l3, andl2−2l3. The length l2−2l3is\nnonnegative since 2H3≤H2. Then we use the following scheme:\n\n\nW1= (A2+B2)\nW2= (A2,C2,X3+B2)\nW3= (B2,C2).(159)\nIt’s easy to check that\n(R1,R2,R3) = (H3,H2,H2−H3). (160)28\nR1R2R3\nQ1= (0,H2+H3,H2+H3)\nQ2= (H2+H3,H2+H3,0)Q3T1= (1\n2H3,H2,H2)\nT2= (H2,H2,1\n2H3)T3T4= (H3,H2,H2−H3)\nT5= (H2−H3,H2,H3)\nT6T7T8T9\nS10= (1\n2H2,1\n2H2+H3,1\n2H2)\nS11S12R1≥0\nR2≥0\nR3≥02R1+R3≥H2+H3\nR1+ 2R3≥H2+H32R1+R2≥H2+H3R1+ 2R2≥H2+H3\n2R2+R3≥H2+H3\nR2+ 2R3≥12H2+H3R1+R2≥1\n2H2+H3\nR1+R3≥1\n2H2+H3\nR2+R3≥12H2+H3\nR1+R2+R3≥3\n2H2+H32R1+R2+R3≥2H2+H3\nR1+R2+ 2R3≥2H2+H3R1+ 2R2+R3≥2H2+H3\nFig. 5. rate region R∗\n1: case iv ( 2H3≤H2)\n4) corner point S10: equipartition X2intoA2andB2with the same length1\n2l2. Since2H3≤H2implies\nl3≤1\n2l2, we denote the first l3bits ofB2byB1\n2. Then we use the following scheme:\n\n\nW1= (A2+B2)\nW2= (A2,X3+B1\n2)\nW3= (B2).(161)\nWe can check that\n(R1,R2,R3) = (1\n2H2,1\n2H2+H3,1\n2H2). (162)\nTherefore, the achievability for the case that 2H(X3)≤H(X2)is proved.\nTo sum up the four cases, we finish the achievability proof of T heorem 2 for all H(X2)andH(X3).\nAPPENDIX B\nCONVERSE PROOF OF THEOREM 2\nFrom conditions in (32)-(34), we can write the entropies of X2andX3as follows.\nH(X2) =I(X2;W1W2)+H(X2|W1W2)\n=I(X2;W1W2)29\n=H(W1W2)−H(W1W2|X2) (163)\nand\nH(X3) =I(X3;W1W2|W3)+H(X3|W1W2W3)\n=I(X3;W1W2|W3)\n=H(W1W2|W3)−H(W1W2|W3X3). (164)\nNext, we use (163) and (164) to show the bounds in (36)-(40) on e by one.\nA. Proof of 2Ri+Rj≥H(X2)+H(X3)\nDue to symmetry, we only need to show this for i= 1 andj= 2 which is 2R1+R2≥H(X2)+H(X3).\nn·/bracketleftbig\nH(X2)+H(X3)/bracketrightbig\n=H(X2)+H(X3)\n=/bracketleftbigg\nH(W1W2)−H(W1W2|X2)/bracketrightbigg\n+/bracketleftbigg\nH(W1W2|W3)−H(W1W2|W3X3)/bracketrightbigg\n(165)\n=H(W1W2)−/bracketleftbigg\nH(W1W2|X2)−H(W1W2|W3)/bracketrightbigg\n−H(W1W2|W3X3) (166)\n=H(W1W2)+I(W1;W2)−/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2|X2)−I(W2;W3|W1)−I(W1W3;X2|W2)/bracketrightbigg\n−H(W1W2|W3X3) (167)\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2|X2)−I(W2;W3|W1)/bracketrightbigg\n+I(W1W3;X2|W2)\n−H(W1W2|W3X3)\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2|X2)−I(W2;W3|W1)/bracketrightbigg\n+/bracketleftbigg\nH(W1)−H(W1|W2X2)+I(W3;W2X2|W1)−I(W1W3;W2)/bracketrightbigg\n−H(W1W2|W3X3)\n=/bracketleftbigg\n2H(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2)−I(W1;X2|W2)−I(W3;W2X2|W1)/bracketrightbigg\n−/bracketleftbigg\nH(W1|W2)−I(W2;W3|W1)/bracketrightbigg\n−/bracketleftbigg\nI(W1W3;W2|X2)+H(W1W2|W3X3)/bracketrightbigg\n=/bracketleftbigg\n2H(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nI(W2;W3)+I(W1;W2)+I(W1;W3|W2X2)/bracketrightbigg\n−/bracketleftbigg\nH(W1|W2)−I(W2;W3|W1)/bracketrightbigg\n−/bracketleftbigg\nI(W1W3;W2|X2)+H(W1W2|W3X3)/bracketrightbigg\n(168)\n=/bracketleftbigg\n2H(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nI(W2;W3)+I(W1;W2)+H(W1|W2)−I(W2;W3|W1)/bracketrightbigg\n−/bracketleftbigg\nI(W1;W3|W2X2)+I(W1W3;W2|X2)+H(W1W2|W3X3)/bracketrightbigg\n=/bracketleftbigg\n2H(W1)+H(W2)/bracketrightbigg\n−/bracketleftbigg\nH(W1|W2W3)+I(W1;W2)+I(W1;W3)/bracketrightbigg30\n−/bracketleftbigg\nI(W1;W3|W2X2)+I(W1W3;W2|X2)+H(W1W2|W3X3)/bracketrightbigg\n(169)\n≤2H(W1)+H(W2) (170)\n≤n·(2R1+R2+3ǫ) (171)\nwhere (167) is obtained by\nH(W1W2|X2)−H(W1W2|W3)\n=H(W1W2|X2)−H(W1W2|W3X2)−I(W1W2;X2|W3)\n=I(W1W2;W3|X2)−I(W1W2;X2|W3)\n=I(W1W2;W3)−I(W1W2;X2)\n=I(W1W2;W3)−H(X2)\n=I(W1W2;W3)−I(W1W3;X2)\n=I(W1W2;W3)−/bracketleftbig\nI(W1W3;X2W2)−I(W1W3;W2|X2)/bracketrightbig\n=I(W1W2;W3)−/bracketleftbig\nI(W1W3;W2)+I(W1W3;X2W2)−I(W1W3;W2|X2)/bracketrightbig\n=I(W1W2;W3)−/bracketleftbig\nI(W1;W2)+I(W2;W3|W1)+I(W1W3;X2W2)−I(W1W3;W2|X2)/bracketrightbig\n(172)\nequality (168) is from\nI(W1W2;W3)+I(W1W3;W2)−I(W1;X2|W2)−I(W3;W2X2|W1)\n=I(W1W2;W3)+I(W1W3;W2)−I(W1;X2|W2)−/bracketleftbigg\nI(W3;W1W2X2)−I(W1;W3)/bracketrightbigg\n=I(W1W2;W3)+I(W1W3;W2)−I(W1;X2|W2)−/bracketleftbigg\nI(W3;W1W2)+I(W3;X2|W1W2)−I(W1;W3)/bracketrightbigg\n=I(W1;W3)+I(W1W3;W2)−I(W1;X2|W2)\n=I(W1;W3)+I(W2;W3)+I(W1;W2|W3)−I(W1;X2|W2)\n=I(W2;W3)+I(W1;W2W3)−I(W1;X2|W2)\n=I(W2;W3)+I(W1;W2W3X2)−I(W1;X2|W2)\n=I(W2;W3)+I(W1;W2)+I(W1;W3X2|W2)−I(W1;X2|W2)\n=I(W2;W3)+I(W1;W2)+I(W1;W3|W2X2), (173)\nand the inequality (170) follows from the nonnegativity of S hannon’s information measures. Divide both sides of\n(171) by nand letǫ→0, we can obtain the desired bound\n2R1+R2≥H(X2)+H(X3). (174)31\nB. Proof of Ri+Rj≥m\nDue to symmetry, we only need to show this for i= 1 andj= 2 which is R1+R2≥m. Recall that\nm= max/braceleftbig\nH(X2),1\n2H(X2)+H(X3)/bracerightbig\n. (175)\nFirstly, it is easy to show that\nn·(R1+R2+2ǫ)≥H(M1)+H(M2) (176)\n≥H(M1,M2) (177)\n≥H(X2) (178)\n=n·H(X2). (179)\nTo show the other bound, we consider the following.\nn·/bracketleftbig1\n2H(X2)+H(X3)/bracketrightbig\n=1\n2H(X2)+H(X3)\n=1\n2/bracketleftbigg\nH(W1W2)−H(W1W2|X2)/bracketrightbigg\n+/bracketleftbigg\nH(W1W2|W3)−H(W1W2|W3X3)/bracketrightbigg\n(180)\n=1\n2H(W1W2)−1\n2/bracketleftbigg\nH(W1W2|X2)−H(W1W2|W3)/bracketrightbigg\n+1\n2H(W1W2|W3)−H(W1W2|W3X3)\n=1\n2H(W1W2)−1\n2/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2|X2)−I(W1;W2)−I(W2;W3|W1)−I(W1W3;X2|W2)/bracketrightbigg\n+1\n2H(W1W2|W3)−H(W1W2|W3X3) (181)\n=1\n2/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n+1\n2/bracketleftbigg\nI(W2;W3|W1)+I(W1W3;X2|W2)+H(W1W2|W3)/bracketrightbigg\n−/bracketleftbigg1\n2I(W1W2;W3)+H(W1W2|W3X3)+1\n2I(W1W3;W2|X2)/bracketrightbigg\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n+1\n2/bracketleftbigg\nI(W1;X2|W2)+I(W3;W2X2|W1)−I(W1W3;W2)/bracketrightbigg\n−1\n2/bracketleftbigg\nI(W1W2;W3)+2H(W1W2|W3X3)+I(W1W3;W2|X2)+I(W1;W3)/bracketrightbigg\n(182)\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n−1\n2/bracketleftbigg\nI(W1W2;W3)+I(W1W3;W2)−I(W1;X2|W2)−I(W3;W2X2|W1)/bracketrightbigg\n−1\n2/bracketleftbigg\n2H(W1W2|W3X3)+I(W1W3;W2|X2)+I(W1;W3)/bracketrightbigg\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n−1\n2/bracketleftbigg\nI(W1;W3|W2X2)+I(W1;W2)+I(W2;W3)/bracketrightbigg\n−1\n2/bracketleftbigg\n2H(W1W2|W3X3)+I(W1W3;W2|X2)+I(W1;W3)/bracketrightbigg\n(183)\n≤H(W1)+H(W2) (184)\n≤n·(R1+R2+2ǫ), (185)32\nwhere (181) follows from (166)-(167), (182) is obtained by\nI(W2;W3|W1)+I(W1W3;X2|W2)+H(W1W2|W3)\n=I(W1W2;W3)−I(W1;W3)+I(W1W3;X2|W2)+H(W1W2)−I(W1W2;W3)\n=H(W1W2)−I(W1;W3)+I(W1W3;X2|W2)\n=H(W1W2)−I(W1;W3)+I(W1W3;W2X2)−I(W1W3;W2)\n=H(W1W2)−I(W1;W3)+I(W1;W2X2)+I(W3;W2X2|W1)−I(W1W3;W2)\n=H(W1W2)−I(W1;W3)+I(W1;W2)+I(W1;X2|W2)+I(W3;W2X2|W1)−I(W1W3;W2)\n=/bracketleftbigg\nH(W1)+H(W2)/bracketrightbigg\n−I(W1;W3)+I(W1;X2|W2)+I(W3;W2X2|W1)−I(W1W3;W2), (186)\n(183) follows from (173) and the inequality (184) follows fr om the nonnegativity of Shannon’s information measures.\nDivide both sides of the two inequalities (179) and (185) by nand letǫ→0, we can obtain the desired bound\nR1+R2≥max{H(X2),1\n2H(X2)+H(X3)}. (187)\nC. Proof of R1+R2+R3≥3\n2H(X2)+H(X3)\nThis sum-rate bound is the same as the sum-rate bound for the c lassical SMDC problem (with H(X1) = 0 ) in\n[1]. We briefly write out the proof in the following.\nn·(R1+R2+R3+3ǫ)≥H(W1)+H(W2)+H(W3) (188)\n=1\n2/bracketleftbig\nH(W1)+H(W2)/bracketrightbig\n+1\n2/bracketleftbig\nH(W1)+H(W3)/bracketrightbig\n+1\n2/bracketleftbig\nH(W2)+H(W3)/bracketrightbig\n≥1\n2/bracketleftbig\nH(W1,W2)+H(W1,W3)+H(W2,W3)/bracketrightbig\n=3\n2H(X2)+1\n2/bracketleftbig\nH(W1,W2|X2)+H(W1,W3|X2)+H(W2,W3|X2)/bracketrightbig\n≥3\n2H(X2)+H(W1,W2,W3|X2) (189)\n≥3\n2H(X2)+H(X3) (190)\n=n·/bracketleftbig3\n2H(X2)+H(X3)/bracketrightbig\n(191)\nwhere (189) follows from Han’s inequality and the inequalit y (190) follows from\nH(W1,W2,W3|X2) =H(W1,W2,W3,X3|X2) (192)\n=H(X3|X2)+H(W1,W2,W3|X2,X3)\n≥H(X3) (193)\nwhere (193) follows from the independence of the two sources and the nonnegativity of Shannon’s information\nmeasures. Divide both sides of (191) by nand letǫ→0, we obtain the desired bound\nR1+R2+R3≥3\n2H(X2)+H(X3). (194)33\nNote that the proof in (188)-(191) only use the reconstructi on constraints in (32),(33) and the security constraints\nin (34) are not used. Thus we obtain the same bound as the class ical SMDC.\nD. Proof of 2Ri+Ri⊙1+Ri⊙2≥2H(X2)+H(X3)\nThis bound is also the same as the corresponding bound for the classical SMDC problem (with H(X1) = 0 ) in\n[1]. We briefly write out the proof in the following. Due to sym metry, we only need to show this for i= 1, which\nis\n2R1+R2+R3≥2H(X2)+H(X3). (195)\nTo show the bound, consider\nn·/bracketleftbig\n2R1+R2+R3+4ǫ/bracketrightbig\n≥2H(W1)+H(W2)+H(W3)\n=/bracketleftbig\nH(W1)+H(W2)/bracketrightbig\n+/bracketleftbig\nH(W1)+H(W3)/bracketrightbig\n≥H(W1,W2)+H(W1,W3)\n= 2H(X2)+H(W1,W2|X2)+H(W1,W3|X2)\n≥2H(X2)+H(W1,W2,W3|X2)\n≥2H(X2)+H(X3) (196)\n=n·/bracketleftbig\n2H(X2)+H(X3)/bracketrightbig\n(197)\nwhere (196) follows from (192)-(193). Divide both sides of ( 197) bynand letǫ→0, we have\n2R1+R2+R3≥2H(X2)+H(X3). (198)\nAPPENDIX C\nPROOF OF THEOREM 4\nIt is easy to see that every rate triple in R∗\n2is achievable by superposition coding of X1and(X2,X3). Next, we\nshow the converse of Theorem 4. The constraints (99)(102)(1 03)(104) are the same as that of the classical SMDC\nproblem. The proofs can be found in [1]. Thus, we only need to p rove (100) and (101).\nFrom the conditions in (92)-(95), we have\nH(X2) =I(X2;W1W2)+H(X2|W1W2) =I(X2;W1W2)\n=H(W1W2)−H(W1W2|X2) (199)\n=/bracketleftbigg\nH(W1W2|X1)+I(W1W2;X1)/bracketrightbigg\n−/bracketleftbigg\nH(W1W2|X1X2)+I(W1W2;X1|X2)/bracketrightbigg\n=/bracketleftbigg\nH(W1W2|X1)−H(W1W2|X1X2)/bracketrightbigg\n+/bracketleftbigg\nI(W1W2X2;X1)−I(W1W2;X1|X2)/bracketrightbigg\n=/bracketleftbigg\nH(W1W2|X1)−H(W1W2|X1X2)/bracketrightbigg\n+I(X1;X2)\n=H(W1W2|X1)−H(W1W2|X1X2) (200)\n(201)34\nand\nH(X3) =I(X3;W1W2|W3)+H(X3|W1W2W3) =I(X3;W1W2|W3)\n=H(W1W2|W3)−H(W1W2|W3X3) (202)\n=/bracketleftbigg\nH(W1W2|W3X1)+I(W1W2;X1|W3)/bracketrightbigg\n−/bracketleftbigg\nH(W1W2|W3X1X3)+I(W1W2;X1|W3X3)/bracketrightbigg\n=/bracketleftbigg\nH(W1W2|W3X1)−H(W1W2|W3X1X3)/bracketrightbigg\n+/bracketleftbigg\nI(W1W2;X1|W3)−I(W1W2;X1|W3X3)/bracketrightbigg\n=/bracketleftbigg\nH(W1W2|W3X1)−H(W1W2|W3X1X3)/bracketrightbigg\n+/bracketleftbigg\nI(W1W2W3;X1)−I(W1W2W3;X1|X3)/bracketrightbigg\n−/bracketleftbigg\nI(W3;X1)−I(W3;X1|X3)/bracketrightbigg\n=/bracketleftbigg\nH(W1W2|W3X1)−H(W1W2|W3X1X3)/bracketrightbigg\n+I(X1;X3)−/bracketleftbigg\nH(X1)−H(X1|X3)/bracketrightbigg\n=H(W1W2|W3X1)−H(W1W2|W3X1X3) (203)\nComparing (200)(203) and (163)(164), we can see that (165) c an be replaced by a conditional version that each\nterm additionally conditions on X1. Then we can check that each step between (165) and (171) can b e replaced\nby a corresponding conditional version that conditions on X1. This finally yields\nH(X2)+H(X3)\n=/bracketleftbigg\nH(W1W2|X1)−H(W1W2|X1,X2)/bracketrightbigg\n+/bracketleftbigg\nH(W1W2|W3,X1)−H(W1W2|W3,X1,X3)/bracketrightbigg\n(204)\n≤2H(W1|X1)+H(W2|X1). (205)\nThen we have\nn·/bracketleftbig\n3H(X1)+H(X2)+H(X3)/bracketrightbig\n= 3H(X1)+H(X2)+H(X3)\n≤3H(X1)+2H(W1|X1)+H(W2|X1) (206)\n= 2H(W1)+H(W2) (207)\n≤n·(2R1+R2+3ǫ) (208)\nwhere (206) follows from (205). Divide both sides of (208) by nand letǫ→0, we have\n2R1+R2≥3H(X1)+H(X2)+H(X3). (209)\nDue to symmetry, we can prove (100). Similarly, if we replace (180)-(185) by their conditional versions that\ncondition on X1, we can prove (101). Therefore, Theorem 4 is proved.\nAPPENDIX D\nWe state the following lemma which will be used in the proof of Theorem 3.35\nLemma 6. Forn≥1, letA1,A2,···,An,B1,B2,···,Bn, andCbe uniform random variables taking values in\nsome finite field Fq. All the random variables are mutually independent. If Di=Ai+Bifori= 1,2,···,n, then\nH(A1,A2,···,An|C,D1,D2,···,Dn) =H(A1,A2,···,An). (210)\nProof. Consider the following.\nH(A1,A2,···,An|C,D1,D2,···,Dn)\n=H(A1,A2,···,An|C)−I(A1,A2,···,An;D1,D2,···,Dn|C)\n=H(A1,A2,···,An)−I(A1,A2,···,An;D1,D2,···,Dn)\n=H(A1,A2,···,An)\n−/bracketleftbig\nH(A1,A2,···,An)+H(D1,D2,···,Dn)−H(A1,···,An,B1,···,Bn)/bracketrightbig\n=H(A1,A2,···,An)\n−/bracketleftbig\nH(A1,A2,···,An)+H(D1,D2,···,Dn)−H(A1,···,An)−H(B1,···,Bn)/bracketrightbig\n=H(A1,A2,···,An)−n/summationdisplay\ni=1/bracketleftbig\nH(Di)−H(Bi)/bracketrightbig\n=H(A1,A2,···,An), (211)\nwhere the last step follows from the uniform distribution of AiandBifori= 1,2,···,n.\nREFERENCES\n[1] J. R. Roche, R. W. Yeung, and K. P. Hau, “Symmetrical multi level diversity coding,” IEEE Transactions on Information Theory , vol. 43,\npp. 1059–1064, May 1997.\n[2] R. W. Yeung and Z. Zhang, “On symmetrical multilevel dive rsity coding,” IEEE Transactions on Information Theory , vol. 45, pp. 609–621,\nMar. 1999.\n[3] T. Guo and R. W. Yeung, “The explicit coding rate region of symmetric multilevel diversity coding,” IEEE Transactions on Information\nTheory , vol. 66, no. 2, pp. 1053–1077, 2020.\n[4] A. Albanese, J. Blomer, J. Edmonds, M. Luby, and M. Sudan, “Priority encoding transmission,” IEEE Transactions on Information Theory ,\nvol. 42, no. 6, pp. 1737–1744, 1996.\n[5] Z. Xiao, J. Chen, Y . Li, and J. Wang, “Distributed multile vel diversity coding,” IEEE Transactions on Information Theory , vol. 61, no. 11,\npp. 6368–6384, 2015.\n[6] C. Tian and T. Liu, “Multilevel diversity coding with reg eneration,” IEEE Transactions on Information Theory , vol. 62, no. 9, pp. 4833–\n4847, 2016.\n[7] S. Shao, T. Liu, C. Tian, and C. Shen, “Multilevel diversi ty coding with secure regeneration: Separate coding achiev es the MBR point,”\nEntropy , vol. 20, no. 10, p. 751, 2018.\n[8] S. Mohajer, C. Tian, and S. N. Diggavi, “Asymmetric multi level diversity coding and asymmetric gaussian multiple de scriptions,” IEEE\nTransactions on Information Theory , vol. 56, pp. 4367–4387, Sept 2010.\n[9] C. Li, S. Weber, and J. M. Walsh, “Multilevel diversity co ding systems: Rate regions, codes, computation, & forbidde n minors,” IEEE\nTransactions on Information Theory , vol. 63, no. 1, pp. 230–251, 2017.\n[10] C. Li, X. Guang, C. Tan, and R. W. Yeung, “Fundamental lim its on a class of secure asymmetric multilevel diversity cod ing systems,”\nIEEE Journal on Selected Areas of Communications , vol. 36, no. 4, pp. 737–747, 2018.\n[11] C. Li and X. Guang, “Asymmetric multilevel diversity co ding systems with perfect secrecy,” IEEE Transactions on Vehicular Technology ,\nvol. 66, no. 9, pp. 8558–8562, 2017.36\n[12] A. Balasubramanian, H. D. Ly, S. Li, T. Liu, and S. L. Mill er, “Secure symmetrical multilevel diversity coding,” IEEE Transactions on\nInformation Theory , vol. 59, pp. 3572–3581, June 2013.\n[13] T. Guo, C. Tian, T. Liu, and R. W. Yeung, “Weakly secure sy mmetric multilevel diversity coding,” IEEE Transactions on Information\nTheory , vol. 66, no. 11, pp. 7033–7055, 2020.\n[14] A. Shamir, “How to share a secret,” Commun. ACM , vol. 22, pp. 612–613, Nov. 1979.\n[15] H. Yamamoto, “Secret sharing system using (k,L,n)threshold scheme,” IEICE Trans. Fund.(Jpn. Edition) , vol. J68-A, Sep. 1985. (English\nTranslation: Scripta Technica, Inc., Electronics and Comm un. in Japan, Part I, vol. 69, pp. 46–54, 1986).\n[16] G. R. Blakley and C.Meadows, “Security of ramp type,” Proc. Adv. Cryptology , vol. LNCS-196, pp. 242–269, 1985.\n[17] W.-A. Jackson and K. M. Martin, “A combinatorial interp retation of ramp schemes,” Australasian Journal of Combinatorics , vol. 14,\npp. 51–60, 1996." }, { "title": "2401.14813v1.Symbol_Specific_Sparsification_of_Interprocedural_Distributive_Environment_Problems.pdf", "content": "Symbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\nKadiray Karakaya\nHeinz Nixdorf Institute\nPaderborn University\nPaderborn, Germany\nkadiray.karakaya@upb.deEric Bodden\nHeinz Nixdorf Institute\nPaderborn University & Fraunhofer IEM\nPaderborn, Germany\neric.bodden@upb.de\nABSTRACT\nPrevious work has shown that one can often greatly speed up\nstatic analysis by computing data flows not for every edge in the\nprogram’s control-flow graph but instead only along definition-use\nchains. This yields a so-called sparse static analysis. Recent work\nonSparseDroid has shown that specifically taint analysis can be\n“sparsified” with extraordinary effectiveness because the taint state\nof one variable does not depend on those of others. This allows\none to soundly omit more flow-function computations than in the\ngeneral case.\nIn this work, we now assess whether this result carries over to the\nmore generic setting of so-called Interprocedural Distributive Envi-\nronment (IDE) problems. Opposed to taint analysis, IDE comprises\ndistributive problems with large or even infinitely broad domains,\nsuch as typestate analysis or linear constant propagation. Specif-\nically, this paper presents Sparse IDE, a framework that realizes\nsparsification for any static analysis that fits the IDE framework.\nWe implement Sparse IDE in SparseHeros , as an extension to\nthe popular Heros IDE solver, and evaluate its performance on\nreal-world Java libraries by comparing it to the baseline IDE al-\ngorithm. To this end, we design, implement and evaluate a linear\nconstant propagation analysis client on top of SparseHeros . Our\nexperiments show that, although IDE analyses can only be sparsi-\nfied with respect to symbols and not (numeric) values, Sparse IDE\ncan nonetheless yield significantly lower runtimes and often also\nmemory consumptions compared to the original IDE.\nKEYWORDS\nstatic analysis, sparse analysis, IFDS, IDE, constant propagation\n1 INTRODUCTION\nStatic program analysis has proven useful for diverse purposes in-\ncluding compiler optimization [ 18], program comprehension [ 9]\nand developer assistance [ 38]. It is now an essential part of soft-\nware engineering for assuring bug-free [ 4], secure [ 22] and quality\nsoftware[ 12]. The key strength of static program analysis is to\naccount for all possible executions of a target program. But this\nimposes two often competing challenges: precision and scalability.\nStatic analyses yield more precise results by tracking statement\nordering and by distinguishing different calling contexts.\nIDE (Interprocedural Distributive Environment) [ 30], with its\nextensions [ 2,24,33], is a state-of-the-art precise interprocedu-\nral static analysis framework. It covers a wide class of data-flow\nproblems ranging from variations of classical taint analysis [ 16]\nto typestate [ 11,20] and constant propagation [ 25] analyses. IDE\nrepresents data-flow analysis problems on an exploded supergraphand models data-flow facts as environments. Environments are\nmappings from symbols (often program variables) to domain val-\nues. The exploded supergraph is a data-flow graph induced by the\ninter-procedural control-flow graph (ICFG) for the whole program.\nIts nodes are pairs(𝑠,𝑑)of program statements and data-flow facts.\nA data-flow fact 𝑑holds at a statement 𝑠if in the exploded super-\ngraph the corresponding node (𝑠,𝑑)is reachable from the start\nnode. The edges of the exploded supergraph represent the effects\nof program statements on a data-flow fact. IDE computes over the\nexploded supergraph by tracking all data-flow facts densely across\nall program points. As previous work [ 1,15,19,41] has shown, this\napproach does not scale well for large-scale real-world programs. A\nkey observation is, however, that in practice many program state-\nments do not affect the analysis result. Such statements thus can\nbe safely ignored, e.g. by sparsifying the exploded supergraph.\nSparsification is a well-known technique for scaling data-flow\nanalyses [ 13,14,26,31,35,36] while still maintaining their preci-\nsion. Sparsification approaches create sparse versions of the original\nCFGs of a target program by removing statements that are irrelevant\nto the analysis and then computing over the sparse CFGs. Recent\non-demand approaches take sparsification further by utilizing the\ninformation available during the analysis. SparseBoomerang [17]\naccelerates demand-driven pointer analysis by computing over\nsparse CFGs specialized to the alias queries. SparseDroid [15]\naccelerates taint analysis by computing over sparse CFGs special-\nized to individual data-flow facts. Both approaches demonstrate\nsparsification on IFDS-based problems, that focus on mere symbol\nreachability, without considering value computation.\nThe IFDS (Interprocedural Finite Distributive Subset) [ 29] frame-\nwork is the “small brother” of IDE. It reduces the data-flow analysis\nproblems to a pure graph reachability problem. Yet, IFDS is limited\nto data-flow problems with finite domains: all IFDS problems can\nbe encoded as IDE problems, but only a subset of IDE problems\ncan be encoded as IFDS problems [ 30]. As an example, consider\nthe statement a = a + 1 . Here, using IFDS one can encode a\nsimple taint analysis inferring that ais tainted/reachable after the\nstatement if and only if it was previously tainted/reachable. Ef-\nficient computation of a’s numeric value, however, requires one\ntocompute values within the infinitely broad domain of integers,\ngoing beyond pure reachability. As we show, this has implications\nfor sparsification: while the statement a = a + 1 can be safely\nconsidered irrelevant w.r.t. a’s reachability, and will be disregarded\nin sparsification approaches for IFDS [ 15,17], itisa relevant state-\nment when constant propagation is considered: it changes a’s value.\nThis observation is not limited to constant propagation analysis,\nit applies to other data-flow analysis problems that require valuearXiv:2401.14813v1 [cs.SE] 26 Jan 2024Karakaya and Bodden\nmappings. For instance, a sparse typestate analysis must retain\nstatements that alter a symbol’s associated state value. Based on\nthis observation, we generalize the recent work on SparseDroid ,\ni.e., on sparse IFDS [ 15]: we propose Sparse IDE , a symbol-specific\nsparsification of the IDE framework, that enables efficient sparsifi-\ncation, even in the presence of arbitrarily large value domains. In\naddition, we also show the limits of sparsification in IDE: while one\ncan effectively sparsify with respect to symbols, such sparsification\ncannot be performed with respect to values.\nWe formalize Sparse IDE, and show how this formalization cov-\ners also IFDS data-flow analysis problems as a special case. We\nimplement Sparse IDE in a tool SparseHeros , extending the pop-\nularHeros IDE solver [ 5]. We compare both implementations in\nterms of performance, and show that sparsification maintains cor-\nrectness. To this end, we implement a linear constant propagation\nanalysis client that uses both implementations. To validate Sparse-\nHeros ’s correctness, we run both on ConstantBench , a novel\nmicrobenchmark suite for integer linear constant propagation anal-\nysis. To evaluate its performance impact, we run the analysis client\non real-world Java libraries using both Heros andSparseHeros .\nThe analysis client produces the same results in both cases while\nterminating significantly faster when using SparseHeros .\nTo summarize, this paper presents the following original contri-\nbutions, whose implementations are open-sourced1:\n•A formalization of Sparse IDE and its implementation in\nSparseHeros on top of Heros andSoot [37],\n•its correctness evaluation on the ConstantBench microbench-\nmark suite for linear constant propagation analysis, and\n•its performance evaluation on real-world Java libraries.\nThe remainder of the paper is organized as follows. In Section 2,\nwe present the background. In Section 3, we introduce Sparse IDE\nand in Section 4, we instantiate it on linear constant propagation\nanalysis. In Section 5, we present the evaluation results. In Section 6,\nwe discuss the limitations of our approach and threats to its validity.\nIn Section 7, we discuss the related work and we conclude with\nSection 8.\n2 BACKGROUND\nThis section briefly introduces the background that our work builds\non. We begin with the IFDS and IDE frameworks. Then we intro-\nduce sparse data-flow analysis and discuss why it is an effective\nalternative. Finally, we explain how the recent approaches sparsify\nfurther by utilizing the information available during the analysis\nruntime.\n𝑓𝑖𝑑:𝜆𝑆.𝑆𝑓𝑔𝑒𝑛:𝜆𝑆.(𝑆∪{𝑎})𝑓𝑎𝑠:𝜆𝑆.if𝑎∈𝑆:(𝑆∪{𝑏})else(𝑆\\{𝑏})\nΛ\nΛa\naΛ\nΛa\naΛ\nΛa\nab\nb\nFigure 1: Flow functions (reproduced from [29]).\n1https://github.com/secure-software-engineering/SparseIDE2.1 IFDS and IDE\nIFDS [ 29] and IDE [ 30] are two frameworks for interprocedural flow-\nand context-sensitive data-flow analysis. IFDS represents data-flow\nanalysis problems as graph reachability on an exploded supergraph,\nwhose nodes are pairs of program statements and data-flow facts.\nThe individual edges in the exploded supergraph constitute flow\nfunctions ; they show each statement’s effect on each data-flow fact’s\nreachability. A flow function determines whether a data-flow fact is\nbeing generated, propagates to the next statement, spawns another\nfact, or gets killed.\nFigure 1 shows how the flow functions are represented as edges\nin the exploded supergraph. The data-flow fact above the edge\nmeans that it holds before applying the function; the fact below\nmeans that it holds after. A special fact, Λholds always. Facts\nconnected to it are newly generated. The identity function, 𝑓𝑖𝑑,\nleaves data-flow facts unchanged. The function 𝑓𝑔𝑒𝑛shows the case\nwhere data-flow fact ais being generated. The function 𝑓𝑎𝑠shows\nhow the existing fact, acreates another fact, b, e.g. at an assignment,\nb = a .\nIDE generalizes the IFDS framework by computing domain val-\nues that symbols map to. It does so in two phases: first it deter-\nmines whether symbols are reachable, just like IFDS, and then\ncomputes their values. IDE achieves this by annotating the individ-\nual exploded supergraph edges with so-called edge functions , which\nconstitute environment transformers.\n𝑒𝑖𝑑:𝜆𝑒𝑛𝑣.𝑒𝑛𝑣𝑒𝑣𝑎𝑙:𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→3]𝑒𝑜𝑝:𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑏↦→2∗𝑒𝑛𝑣(𝑎)+1]\nΛ\nΛ𝜆𝑙.𝑙a\na𝜆𝑙.𝑙Λ\nΛ𝜆𝑙.𝑙a\na𝜆𝑙.3Λ\nΛ𝜆𝑙.𝑙a\na𝜆𝑙.𝑙b\nb𝜆𝑙.2∗𝑙+1\nFigure 2: Edge functions (reproduced from [30]).\nFigure 2 shows how the edge functions are represented. The\nenvironment transformer 𝑒𝑖𝑑keeps the values as they are. 𝑒𝑣𝑎𝑙\nshows the case where data-flow fact, ais mapped to a domain value,\ne.g. through a constant assignment, a = 3 .𝑒𝑜𝑝shows how the value\nofbis calculated depending on the value of a, e.g. through a linear\narithmetic operation, b = 2*a + 1 . IDE can only compute linear\nequations precisely.\nIFDS and IDE apply to a wide class of data-flow analysis problems.\nIFDS requires data-flow problems to be defined with flow functions\nthat are distributive over the merge operator. Many reachability\nproblems such as taint, reaching definitions, or live variables anal-\nysis fall into this category. IDE, on the other hand, also requires\ndata-flow problems to be expressed with distributive environment\ntransformers. IFDS suits better the problems with a binary value\ndomain, e.g. taint analysis where the domain simply consists of\ntwo values, tainted ornot tainted [3]. It has been applied to more\ncomplex domains, e.g. for typestate analysis where the domain\ncontains arbitrary object states [ 23]. The drawback of IFDS is that\nit represents data-flow facts as symbol-value pairs, which blows\nup the data-flow fact space with increasing size of the domain. Be-\ncause of this representation, IFDS’s runtime performance depends\non the value domain’s size. Further, it may not terminate whenSymbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\nthe value domain is infinitely broad, e.g., in constant propagation\nanalysis, where the domain contains all integers. IDE, on the other\nhand, restricts data-flow facts to static symbols and computes their\n(approximated) runtime values using the edge functions along the\npath where the symbols are reachable in the exploded supergraph.\nTherefore, IDE can terminate efficiently even with infinitely broad\nvalue domains—only the set of symbols must be finite.\n2.2 Sparse Data-flow Analysis\nData-flow analysis techniques aim to produce precise results while\nremaining scalable within a reasonable time budget. Techniques\nthat prioritize scalability often resort to sacrificing precision aspects:\nflow-insensitive analyses ignore control-flow ordering [ 40], field-\ninsensitive analyses approximate field accesses[ 8], and context-\ninsensitive analyses do not distinguish different calling contexts\n[21]. Sparse data-flow analyses, on the other hand, often improve a\ndense data-flow analysis’ scalability while maintaining its precision.\nThey sparsify a target program’s control-flow graph by removing\nprogram statements that provably do not affect the analysis result.\nSparsification often uses a cheaper pre-analysis stage to aid a more\nexpensive analysis [ 14,31,35]. Recent on-demand sparse data-flow\nanalyses sparsify further by exploiting the information that is only\navailable during analysis runtime [15, 17].\n2.3 Fact-Specific On-Demand Sparsification\nWhen IFDS and IDE compute a data-flow fact’s reachability, starting\nfrom the statement that generates the data-flow fact, they propagate\nit along all statements as long as it is not killed. At each statement,\nthey check whether the statement is relevant for all the data-flow\nfacts that have reached it. Figure 3 shows how the reachability is\ncomputed for an example constant-propagation analysis setting.\nThe fact-specific id edges andnon-id edges show the edges which\nIFDS and IDE create when propagating data-flow facts. The data-\nflow facts actually only need to be propagated to the required nodes .\nFor instance, data-flow fact aonly needs to propagate to the state-\nment b = a; all other statements are redundant for a. Similarly, b\nonly needs to propagate to the statement, c = b + 1 . Based on this\nobservation, He et al. [ 15] introduced the sparse IFDS algorithm in\ntheir implementation SparseDroid . Instead of propagating all the\ndata-flow facts to the next statement, it propagates them simply\nto the next statement that uses the facts. Sparse IFDS keeps all\nnon-id edges and replaces the fact-specific id edges with sparse id\nedges , effectively keeping all required nodes and skipping over all\nredundant nodes .\nFact-specific on-demand sparsification allows effective propaga-\ntion of the data-flow facts along the sparse CFGs specific to them,\nwhich is not limited to data-flow analysis. Recent work [ 17] has\napplied it to pointer analysis, where the variable in alias queries is\ntreated as the initial data-flow fact and propagated along its query-\nspecific sparse CFGs. So far, however, fact-specific on-demand spar-\nsification has only been applied to the analysis problems that deal\nwith fact reachability. In this work, we expand the scope of fact-\nspecific on-demand sparsification to include the data-flow analyses\nthat compute over an additional value domain, specifically IDE.\nint foo(){\n \n a = 1\n b = a\n x = new X()\n c = b + 1\n x.f = \"_\"\n d = c\n return d\n}Λ a b c d\nfact-specific id edgesnon-id edge\nsparse id edgesrequired node\nredundant nodeFigure 3: Original and sparse propagations after applying\nfact-specific on-demand sparsification.\n3 SYMBOL-SPECIFIC ON-DEMAND\nSPARSIFICIATION WITH SPARSE IDE\nIn this section, we first explain the original IDE algorithm [ 30] in\ndetail. We then introduce the Sparse IDE algorithm by highlighting\nthe modifications to the original IDE algorithm.\n3.1 The Original IDE Algorithm\nSagiv et al. [ 30] define an IDE problem instance formally as 𝐼𝑃=\n(𝐺∗,𝐷,𝐿,𝑀), where\n•𝐺∗is the program supergraph (ICFG), which consists of\ncontrol flow graphs (CFG), 𝐺𝑝of individual procedures,\n•𝐷is a finite set of program symbols,\n•𝐿is a finite-height lattice (which can be infinitely broad),\nand\n•𝑀:𝐸∗𝑑− →(𝐸𝑛𝑣(𝐷, 𝐿)→𝐸𝑛𝑣(𝐷, 𝐿))is an assignment of\ndistributive environment transformers to the edges of 𝐺∗.\nThe original IDE algorithm [ 30] solves such an IDE problem,\n𝐼𝑃, in two phases. In Phase I, it creates the jump functions that\nshow the reachability of each 𝑑∈𝐷, by assuming that their initial\nmappings to 𝐿are always𝜆𝑙.⊤. In Phase II, it computes each 𝑑’s\nactual value mapping to 𝐿by evaluating the edge functions defined\nin𝑀.\nAccording to Sagiv et al. [ 30], the total cost of the IDE algorithm\nis bounded by 𝑂(|𝐸||𝐷|3), which is the cost of Phase I. Since 𝐷is the\nset of symbols, it should not change if correctness is preserved. We,\ntherefore, apply our sparsification approach in Phase I, where the\njump functions are created by reducing 𝐸, the set of edges. Phase II\nis oblivious to how the jump functions are created—it automatically\nbenefits from the sparsification of Phase I.\nFigure 4 shows the algorithm for Phase I. Each procedure 𝑝’s CFG,\n𝐺𝑝consists of a start node𝑠𝑝, anexitnode𝑒𝑝, and normal (non-call)\nnodes𝑚or𝑛. Procedure calls are represented with two nodes: the\ncall-site node𝑐denotes the point right before the procedure call, and\nthereturn-site node𝑟denotes the point right after. Program symbols,\ne.g. variables, access paths, etc., are denoted with 𝑑′,𝑑∈𝐷∪{Λ}Karakaya and Bodden\n1Function ForwardComputeJumpFunctionsSLRPs() :\n2for⟨𝑠𝑝,𝑑′⟩,⟨𝑚,𝑑⟩s.t.𝑚occurs in proc. 𝑝and𝑑′,𝑑∈𝐷∪{Λ}do\n3𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑝,𝑑′⟩→⟨𝑚,𝑑⟩)=𝜆𝑙.⊤\n4for corresponding call-return pairs (𝑐,𝑟)and𝑑′,𝑑∈𝐷∪{Λ}do\n5𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑐,𝑑′⟩→⟨𝑟,𝑑⟩)=𝜆𝑙.⊤\n6 PathWorkList B{⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩ →⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩}\n7𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩ →⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩)B𝑖𝑑\n8while PathWorkList ≠∅do\n9 Select and remove an item ⟨𝑠𝑝,𝑑1⟩→⟨𝑛,𝑑2⟩from PathWorkList\n10 let𝑓=𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑝,𝑑1⟩→⟨𝑛,𝑑2⟩)\n11 switch(𝑛)do\n12 case𝑛is a call node in 𝑝, calling a procedure 𝑞do\n13 for𝑑3s.t.⟨𝑛,𝑑2⟩→⟨𝑠𝑞,𝑑3⟩∈𝐸#do\n14 Propagate(��𝑠𝑞,𝑑3⟩→⟨𝑠𝑞,𝑑3⟩,𝑖𝑑)\n15 let𝑟be the return-site node that corresponds to 𝑛\n16 for𝑑3s.t.𝑒=⟨𝑛,𝑑2⟩→⟨𝑟,𝑑3⟩∈𝐸#do\n17 Propagate(⟨𝑠𝑝,𝑑1⟩→⟨𝑟,𝑑3⟩,𝐸𝑑𝑔𝑒𝐹𝑛(𝑒)◦𝑓)\n18 for𝑑3s.t.𝑓3=𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑛,𝑑2⟩→⟨𝑟,𝑑3⟩)≠𝜆𝑙.⊤do\n19 Propagate(⟨𝑠𝑝,𝑑1⟩→⟨𝑟,𝑑3⟩,𝑓3◦𝑓)\n20 case𝑛is the exit node of 𝑝do\n21 for call node𝑐that calls𝑝with corresponding return-site node 𝑟do\n22 for𝑑4,𝑑5s.t.⟨𝑐,𝑑4⟩→⟨𝑠𝑝,𝑑1⟩∈𝐸#and⟨𝑒𝑝,𝑑2⟩→⟨𝑟,𝑑5⟩∈𝐸#do\n23 let𝑓4=𝐸𝑑𝑔𝑒𝐹𝑛(⟨𝑐,𝑑4⟩→⟨𝑠𝑝,𝑑1⟩)and\n24 𝑓5=𝐸𝑑𝑔𝑒𝐹𝑛(⟨𝑒𝑝,𝑑2⟩→⟨𝑟,𝑑5⟩)and\n25 𝑓′=(𝑓5◦𝑓◦𝑓4)⊓𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑐,𝑑4⟩→⟨𝑟,𝑑5⟩)\n26 if𝑓′≠𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑐,𝑑4⟩→⟨𝑟,𝑑5⟩)then\n27 𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑐,𝑑4⟩→⟨𝑟,𝑑5⟩)B𝑓′\n28 let𝑠𝑞be the start node of 𝑐’s procedure\n29 for𝑑3s.t.𝑓3=𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑞,𝑑3⟩→⟨𝑐,𝑑4⟩)≠𝜆𝑙.⊤do\n30 Propagate(⟨𝑠𝑞,𝑑3⟩→⟨𝑟,𝑑5⟩,𝑓′◦𝑓3)\n31 case𝑛is an intraprocedural node in 𝑝do\n32 for⟨𝑚,𝑑 3⟩𝑠.𝑡.⟨𝑛,𝑑2⟩→⟨𝑚,𝑑 3⟩∈𝐸#do\n33 Propagate(⟨𝑠𝑝,𝑑1⟩→⟨𝑚,𝑑 3⟩,\n34 𝐸𝑑𝑔𝑒𝐹𝑛(⟨𝑛,𝑑2⟩→⟨𝑚,𝑑 3⟩)◦𝑓)\n35\n36Function Propagate( e, f):\n37 let𝑓′=𝑓⊓𝐽𝑢𝑚𝑝𝐹𝑛(𝑒)\n38 if𝑓′≠𝐽𝑢𝑚𝑝𝐹𝑛(𝑒)then\n39𝐽𝑢𝑚𝑝𝐹𝑛(𝑒)B𝑓′\n40 Insert𝑒into PathWorkList\nFigure 4: The original IDE algorithm for Phase I (repro-\nduced from [30]).\nincluding the special symbol Λ.Λis required for generating new\nsymbols at arbitrary program points.\nInitialization. In lines 2–5, jump and summary functions are\ninitialized. Jump functions, denoted by 𝐽𝑢𝑚𝑝𝐹𝑛 , correspond to\nthesame-level realizable paths (SLRPs) from the start node 𝑠𝑝of\na procedure 𝑝to a node𝑚in𝑝. Summary functions, denoted by\n𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛 , summarize the effect of a procedure call through same-\nlevel realizable paths from the call-site 𝑐to return-site 𝑟. In line 3,\n𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑝,𝑑′⟩→⟨𝑚,𝑑⟩)=𝜆𝑙.⊤states that the jump function\nfrom the node⟨𝑠𝑝,𝑑′⟩to each⟨𝑚,𝑑⟩is initialized to 𝜆𝑙.⊤. In line\n5,𝑆𝑢𝑚𝑚𝑎𝑟𝑦𝐹𝑛(⟨𝑐,𝑑′⟩→⟨𝑟,𝑑⟩)=𝜆𝑙.⊤states that the summary\nfunction from each call-site node ⟨𝑐,𝑑′⟩to its corresponding return-\nsite⟨𝑟,𝑑⟩is initialized to 𝜆𝑙.⊤. Line 6 initializes the PathWorkList\nto{⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩ → ⟨𝑠𝑚𝑎𝑖𝑛,Λ⟩}representing a self-loop edge on\nthe start node of the main procedure whose jump function is the\nidentity function, id. The jump function from the start node 𝑠𝑝until\nthe current statement 𝑛is denoted with 𝑓.1Function ForwardComputeSparseJumpFunctionsSLRPs() :\n2 . . .\n8while PathWorkList ≠∅do\n9 Select and remove an item ⟨𝑠𝑝,𝑑1⟩→⟨𝑛,𝑑2⟩from PathWorkList\n10 let𝑓=𝐽𝑢𝑚𝑝𝐹𝑛(⟨𝑠𝑝,𝑑1⟩→⟨𝑛,𝑑2⟩)\n11 switch(𝑛)do\n12 case𝑛is a call node in 𝑝, calling a procedure 𝑞do\n13 . . .\n15 let𝑟be the return-site node that corresponds to 𝑛\n16 for𝑑3s.t.𝑒=⟨𝑛,𝑑2⟩→⟨𝑟,𝑑3⟩∈𝐸#do\n17 let𝑟′=NextUse(𝑝,𝑑3,𝑟)\n18 Propagate(⟨𝑠𝑝,𝑑1⟩→⟨𝑟′,𝑑3⟩,\n19 𝐸𝑑𝑔𝑒𝐹𝑛(⟨𝑛,𝑑2⟩→⟨𝑟,𝑑3⟩)◦𝑓)\n20 . . .\n31 case𝑛is an intraprocedural node in 𝑝do\n32 for⟨𝑚,𝑑 3⟩𝑠.𝑡.⟨𝑛,𝑑2⟩→⟨𝑚,𝑑 3⟩∈𝐸#do\n33 let𝑚′=NextUse(𝑝,𝑑3,𝑛)\n34 Propagate(⟨𝑠𝑝,𝑑1⟩→⟨𝑚′,𝑑3⟩,\n35 𝐸𝑑𝑔𝑒𝐹𝑛(⟨𝑛,𝑑2⟩→⟨𝑚,𝑑 3⟩)◦𝑓)\n36\n41Function NextUse( p, d, n ):\n42 let𝐺𝑝,𝑑be the sparse CFG of 𝑑in procedure𝑝\n43 let𝐶be the sparse CFG cache with (𝑝,𝑑)typed keys and 𝐺𝑝,𝑑as values\n44 if𝐺𝑝,𝑑∉𝐶then\n45 construct𝐺𝑝,𝑑and add to𝐶\n46 return the next statement after 𝑛from𝐺𝑝,𝑑\nFigure 5: Modifications for Sparse IDE algorithm for\nPhase I (mirrors the design from [15]).\nCall nodes. Lines 12-19 handle the case where 𝑛is a call-site\nnode in𝑝, calling a procedure 𝑞. In line 14, the self-loop edge on\nthe start node of the callee procedure 𝑞is initialized with id. In line\n17, the edge from 𝑠𝑝the corresponding return-site 𝑟is computed by\ncomposing the 𝑓, the jump function until 𝑛and the edge function\nfrom𝑛to𝑟. In line 19, the edge from 𝑠𝑝the corresponding return-site\n𝑟is computed by composing 𝑓and𝑓3, the corresponding summary\nfunction when it is not mapping to ⊤.\nExit nodes. Lines 20-30 handle the case where 𝑛is the exit node\nof𝑝. Edges from each call-site node 𝑐to the start node 𝑠𝑝(shown\nwith𝑓4) and from the exit node, 𝑒𝑝to each caller’s return-site 𝑟\n(shown with 𝑓5) must be computed. In line 25, a new summary\nfunction𝑓′is computed by composing 𝑓5,𝑓, and𝑓4and merging\nthe existing summary function for the same 𝑐and𝑟. When it is a\nnew summary, a new jump function is computed from the caller\nprocedure’s start node 𝑠𝑞to the node return-site node 𝑟by compos-\ning the𝑓′with the existing jump function 𝑓3from𝑠𝑞to call-site\nnode𝑐.\nNormal nodes. Lines 31-33 handle the case where 𝑛is a non-call\nor intraprocedural node. Edges from the start node 𝑠𝑝to each node\n𝑚, which is the statement that appears directly after 𝑛in procedure\n𝑝, are computed by composing the edges from 𝑠𝑝to𝑛(shown with\n𝑓) and the edges from 𝑛to𝑚.\n3.2 The Sparse IDE Algorithm\nIn the original IDE algorithm, each symbol 𝑑∈𝐷∪{Λ}at a state-\nment𝑛is propagated to its direct successor statement 𝑚. As also\npointed out in previous work [ 15], this behavior is desired when\n𝑛is a call and exit node. For these nodes, the reachability of each\n𝑑in different contexts is left to the data-flow function definition.Symbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\ncall-flow functions propagate each 𝑑into the context of the callee\nprocedure. return-flow functions propagate each 𝑑back to the con-\ntext of the caller procedure. call-to-return-flow functions propagate\neach𝑑from before a procedure is called to after the procedure is\ncalled. However, when 𝑛is a non-call node, each 𝑑can safely be\npropagated to 𝑑’s next use statement.\nFigure 5 shows the modifications for the Sparse IDE algorithm\nfor Phase I. We replace line 17 from the original IDE algorithm\nwith lines 17-19 in the Sparse IDE algorithm. Instead of propagat-\ning𝑑3to the direct return site node 𝑟, we obtain 𝑟′which is the\nnext use statement of 𝑑3in its symbol-specific sparse control flow\ngraph. Similarly, we replace line 33 with lines 33-35, to propagate\n𝑑3to its next use statement 𝑚′its sparse control flow graph. Our\nsparsification approach mirrors that of sparse IFDS algorithm [ 15],\nhowever, since we generalize it to IDE, we also account for edge\nfunction composition.\n3.3 Sparse IFDS Revisited\nAs shown in Figure 3, a statement can behave as identity function ,\nmeaning it does not affect any data-flow fact, 𝑑∈𝐷. However, as\nshown by He et al. [ 15], many statements only affect a few data-\nflow facts, often even just a single fact. Their flow functions can be\nconsidered fact-specific identity functions for the facts that they do\nnot affect. Sparse IFDS defines fact-specific identity functions as\nfollows [15]:\nGiven a symbol, 𝑑∈𝐷and a flow function, 𝑓∈2𝐷→2𝐷,𝑓is a\nd-specific identity function if the following conditions hold:\n∀𝑋∈2𝐷:𝑑∈𝑋⇒𝑑∈𝑓(𝑋) (1.1)\n∀𝑋∈2𝐷\\{𝑑}:𝑓(𝑋)\\{𝑑}=𝑓(𝑋∪{𝑑})\\{𝑑} (1.2)\nCondition 1.1 states that 𝑑is not affected by other facts when\napplying𝑓, and 1.2 states that 𝑑does not affect the other facts\nwhen applying 𝑓. However, these conditions only apply to symbols\nfrom𝐷and ignore mappings from 𝐷to the value domain 𝐿, and,\nif applied to IDE problems, one would wrongly treat such flow\nfunctions that are annotated with non-identity edge functions as\n𝑑-specific identity functions as well.\nFigure 6 shows two important cases where sparse IFDS would\nsparsify incorrectly. First, reassignments: a = 3 reassigns a, but\nsparse IFDS recognizes that 𝑎already exists (is “tainted”), and there-\nfore it treats this statement as 𝑎-specific identity. Second, value\nupdates: a = a + 1 updates𝑎’s value, but sparse IFDS has no\nnotion of values, therefore, from its perspective, this statement is\n“identity” as well. Sparse IDE, on the other hand, is aware of the\neffects on the value domain and retains both statements.\n3.4 Fact-Specific Identity Transformers\nTo generalize fact-specific sparsification to the IDE framework,\nwe define symbol-specific identity transformers that take into ac-\ncount the environments that map the symbols from domain 𝐷to\nthe values from domain 𝐿. Given a symbol 𝑑∈𝐷and a value\n𝑙∈𝐿,𝑒𝑛𝑣=[𝑑↦→𝑙]is an environment 𝑒𝑛𝑣mapping from 𝑑to𝑙,\ni.e.,𝑒𝑛𝑣(𝑑)=𝑙. Then𝑒𝑛𝑣is an element of the set of environments\n𝐸𝑛𝑣(𝐷,𝐿). An environment transformer, 𝑡∈𝐸𝑛𝑣(𝐷,𝐿)→𝐸𝑛𝑣(𝐷,𝐿)\nint bar(){\n \n a = 2\n a = 3\n a = a + 1\n b = a\n return b\n}Λ a b\nfact-specific id edgesnon-id edge\nsparse id edgesrequired node\nredundant nodeΛ a b\nλl.2\nλl.l+1 λl.3\nλl.l(a) Sparse IFDS (b) Sparse IDE\nFigure 6: Comparison of the Sparsification Approaches of\nSparse IFDS and Sparse IDE\nis a𝑑-specific identity transformer , denoted by 𝑡≡𝑡𝑑, if the follow-\ning holds:\nFirst, the transformer 𝑡keeps all𝑑-specific mappings intact:\ngiven𝑑∈𝐷:∀𝑒𝑛𝑣∈𝐸𝑛𝑣(𝐷,𝐿):\n𝑒𝑛𝑣(𝑑)=𝑡(𝑒𝑛𝑣(𝑑)) (2.1)\nSecond, for all other mappings, 𝑡produces identical results no\nmatter whether or not 𝑑-specific mappings are present:\ngiven𝑑∈𝐷:∀𝑒𝑛𝑣∈𝐸𝑛𝑣(𝐷,𝐿).∀𝑑′∈𝐷\\{𝑑}.∀𝑙∈𝐿:\n𝑡(𝑒𝑛𝑣(𝑑′))=𝑡(𝑒𝑛𝑣[𝑑↦→𝑙](𝑑′)) (2.2)\nWe test the edge functions from Figure 2 on these conditions.\n𝑒𝑖𝑑is an𝑎-specific identity transformer ( 𝑒𝑖𝑑≡𝑒𝑎\n𝑖𝑑), because ap-\nplying𝜆𝑒𝑛𝑣.𝑒𝑛𝑣 does not change a’s previous mapping. 𝑒𝑣𝑎𝑙is not\nan𝑎-specific identity transformer ( 𝑒𝑣𝑎𝑙.𝑒𝑎\n𝑣𝑎𝑙), because applying\n𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→3]changes a’s previous mapping. 𝑒𝑜𝑝is also not\nan𝑎-specific identity transformer ( 𝑒𝑜𝑝.𝑒𝑎𝑜𝑝) because applying\n𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑏↦→2∗𝑒𝑛𝑣(𝑎)+1]changes another value’s mapping\n(for𝑏) depending on what 𝑎maps to, and because it changes b’s\nvalue𝑒𝑜𝑝is not a𝑏-specific identity transformer either ( 𝑒𝑜𝑝.𝑒𝑏𝑜𝑝).\nNote that, importantly, a transformer can only be considered a\n𝑑-identity transformer if the above restrictions hold irrespective of\nany concrete 𝑙∈𝐿that might be associated with 𝑏: (2.2) quantifies\nover all𝑙∈𝐿. This is necessary because IDE produces procedure\nsummaries that must be sound with respect to all 𝑙, and thus their\ncreation must not be made dependent on 𝑙. In other words, IDE can\nsupport symbol-specific but not value-specific sparsification!\n3.5 Determining symbol-specific identity\nWhen propagating fact 𝑑, we consider only those statements as\nirrelevant statements for 𝑑that fulfil conditions (2.1) and (2.2). But\nsince these conditions are value-agnostic—they quantify over all 𝑙∈\n𝐿, this allows one to determine ahead of time the statements whose\nenvironment transformers adhere to both conditions, structurally.\nFirst, by Condition 2.1, a statement’s corresponding environmentKarakaya and Bodden\ntransformer 𝑡isnota𝑑-specific identify transformer if 𝑡affects𝑑’s\nvalue mapping in any way, i.e., 𝑡=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝒅↦→_]. Second, by\nCondition 2.2, 𝑡isnota𝑑-specific identity transformer either, if 𝑡\nuses𝑑’s value mapping 𝑒𝑛𝑣(𝑑)to compute another fact’s value, i.e.\n𝑡=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[_↦→...𝑒𝑛𝑣(𝒅)...].\nNaturally, sparsification effectiveness is closely tied to the analysis-\nspecific environment-transformer definitions. The environment\ntransformer for the statement a = a + 1 is𝑡≡𝑡𝑎for taint analysis,\nwhere𝑡=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣 . For constant propagation analysis, however,\n𝑡.𝑡𝑎, where𝑡=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑒𝑛𝑣(𝑎)+1].\nSparse IDE strictly generalizes Sparse IFDS as presented in Sparse-\nDroid. One can easily define sparse IFDS as an instantiation of\nsparse IDE by restricting the value domain 𝐿to{⊥,⊤}, where sym-\nbols that map to⊥are considered reachable. In this setting, our\ndefinitions (2.1) and (2.2) become equivalent to (1.1) and (1.2).\n4 APPLICATION TO LINEAR CONSTANT\nPROPAGATION\nAs Sagiv, Reps and Horwitz explain in their seminal work [ 30],\nconstant propagation analysis is the perfect problem setting where\nIDE outperforms IFDS [ 29]. This is not only because the problem’s\nlattice is larger than the binary domain, but also it is infinitely\nbroad where IFDS cannot terminate. We are, therefore, motivated\nto apply the Sparse IDE framework to linear constant propagation\nanalysis. Heros , and thus SparseHeros , are generic tools and they\nare independent of the target language and their intermediate rep-\nresentations (IRs). In this work, we use Soot [37] static program\nanalysis framework for Java and its intermediate representation\nJimple . Therefore, in the following, we explain our implementation\nbased on the Jimple IR.\n4.1 Analysis Definition\nLinear constant propagation analysis handles the linear expressions\nthat generate a new data-flow fact by using just a single other fact,\ne.g.a = b ora = 2*b + 1 . Full constant propagation analysis\ninvolves statements such as a = b + c . Such a statement’s flow\nfunction is not distributive; it cannot be precisely computed within\nthe IDE framework. Our linear constant propagation analysis im-\nplementation handles the assignment statements shown in Table 1.\nIR. The IR always ensures binary operation ( binop ) representa-\ntion by reducing more complex operations to binary operations.\nFor instance, a = 2*b + 1 would be reduced to s1 = 2 * b and\na = s1 + 1 . The IR also reduces longer access paths to multiple as-\nsignments with a single access path ( n=1). For instance, a statement\nsuch as a = b.f1.f2 would be reduced to s1 = b.f1 ,s2 = s1.f2 ,\nanda = s2 . The same reduction applies to procedure invocations\nas well.\nFlow functions. Wegenerate a symbol when it is assigned with\naconstant . As discussed, we handle the binary operations in the\nlinear form. We distinguish between the assignments that require\nalias handling and the ones that do not. The assignments such as\nlocal, field load, static field load, andarray load , overwrite the local\nvariable,𝑎, on their left-hand side and therefore do not need to know\n𝑎’s aliases. The assignments such as field store, static field store, and\narray store , on the other hand, require handling the aliases of the\nbase variables or the array references. To handle aliasing we usetheBoomerang [34] demand-driven pointer analysis framework.\nWhen necessary, we query the aliases of the base variables and add\nthem to the set of propagated symbols. Note that in Table 1, the\nalias sets contain the query variable as well. The IDE framework\nrequires three types of flow functions to model the effects of invoke\nstatements. The callflow function propagates the symbol for the\nactual parameter to the context of the callee procedure, by mapping\nit to the procedure’s corresponding formal parameter. The return\nflow function propagates the symbol for the returned variable to\nthe context of the caller procedure, by mapping it to the symbol\non the left-hand side of the invoke expression. The call-to-return\nflow function propagates the symbols that are not passed to the\ncontext of the callee procedure, to the next statement after the\ninvoke statement.\nEdge functions. For most statements, the edge functions map\nthe target symbol to the value of the source symbol, acting as\nidentity transformers . The constant andbinop statements are the\nonly exceptions. The constant statement maps the target symbol,\n𝑎to the given constant value, 𝐶𝑜𝑛𝑠𝑡 . The binop statement maps\nthe target symbol, 𝑎to a new value. The value is computed by\nsimulating the operation ⊙using the source symbol’s value, 𝑒𝑛𝑣(𝑏)\nand the constant operand, 𝐶𝑜𝑛𝑠𝑡 . Edge functions must be composed\nand reduced to a simple value mapping when computing the actual\nvalues. Given 𝑓1,𝑓2∈𝐸𝑛𝑣(𝐷, 𝐿)and𝑓1appears before 𝑓2as an\nedge in the exploded supergraph, we compose the edge functions\nas follows:\n𝑓2◦𝑓1:= \n𝑓2 if𝑓1=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣\n𝑓1 if𝑓2=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣\n𝑓2 if𝑓2=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝐶𝑜𝑛𝑠𝑡]\n𝑓2(𝑓1)if𝑓2=𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝑏)ˆ⊙𝐶𝑜𝑛𝑠𝑡]\nIf an edge function is the identity transformer, we always apply\nthe other function by the first two conditions. We always apply\nthe subsequent edge function if it is a constant assignment, by the\nthird condition. If the subsequent edge is a binop, we compute its\nvalue immediately in place by applying the preceding edge first, as\nsuggested in previous work [5].\nLattice. We perform the linear constant propagation on integers.\nTherefore the lattice is Z⊤\n⊥. Given𝑙1,𝑙2∈Z⊤\n⊥, we define the meet\noperator as follows:\n𝑙1⊓𝑙2= \n𝑙1if𝑙2=⊤\n𝑙2if𝑙1=⊤\n⊥if𝑙1=⊥∨𝑙2=⊥\n⊤if𝑙1=⊤∧𝑙2=⊤\nIf a value is⊤, the meet operator yields the other value by the\nfirst two conditions. If either of the values is ⊥, the meet yields⊥,\nand if both of the values are ⊤it yields⊤by the third and fourth\nconditions respectively.\n4.2 Sparsification for Constant Propagation\nOur sparsification approach has much in common with the one\nproposed by He et al. [ 15], though modifications were necessary.\nWe build the sparse control flow graphs (CFGs) by ignoring symbol-\nspecific identity functions. Given a procedure, 𝑝,𝐺𝑝is its original\ndense CFG. We build sparse CFGs specific to each symbol, 𝑑in𝑝,Symbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\nTable 1: Statements for Linear Constant Propagation Analysis with Corresponding IRs and Flow/Edge Functions.\nStatement IR Flow Function Edge Function\nconstant 𝑎←𝐶𝑜𝑛𝑠𝑡 𝜆𝑆.{𝑆∪{𝑎}} 𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝐶𝑜𝑛𝑠𝑡]\nbinop 𝑎←𝑏⊙𝐶𝑜𝑛𝑠𝑡𝜆𝑆.\u001a𝑆∪{𝑎}if𝑏∈𝑆\n𝑆\\{𝑎}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝑏)ˆ⊙𝐶𝑜𝑛𝑠𝑡]\nlocal 𝑎←𝑏𝜆𝑆.\u001a𝑆∪{𝑎}if𝑏∈𝑆\n𝑆\\{𝑎}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝑏)]\nfield load 𝑎←𝑏.𝑓𝜆𝑆.\u001a𝑆∪{𝑎}if𝑏.𝑓∈𝑆\n𝑆\\{𝑎}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝑏.𝑓)]\nfield store 𝑎.𝑓←𝑏𝜆𝑆.\u001a𝑆∪{𝑝.𝑓|𝑝∈𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝑎)} if𝑏∈𝑆\n𝑆\\{𝑝.𝑓|𝑝��𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝑎)}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑝.𝑓↦→𝑒𝑛𝑣(𝑏)]\nstatic field load 𝑎←𝑇.𝑓𝜆𝑆.\u001a𝑆∪{𝑎}if𝑇.𝑓∈𝑆\n𝑆\\{𝑎}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝑇.𝑓)]\nstatic field store 𝑇.𝑓←𝑏𝜆𝑆.\u001a𝑆∪{𝑝.𝑓|𝑝∈𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝑇)} if𝑏∈𝑆\n𝑆\\{𝑝.𝑓|𝑝∈𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝑇)}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑝.𝑓↦→𝑒𝑛𝑣(𝑏)]\narray load 𝑎←𝐴[𝑖]𝜆𝑆.\u001a𝑆∪{𝑎}if𝐴[𝑖]∈𝑆\n𝑆\\{𝑎}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑎↦→𝑒𝑛𝑣(𝐴[𝑖])]\narray store 𝐴[𝑖]←𝑏𝜆𝑆.\u001a𝑆∪{𝑝[𝑖]|𝑝∈𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝐴)} if𝑏∈𝑆\n𝑆\\{𝑝[𝑖]|𝑝∈𝑎𝑙𝑖𝑎𝑠𝑒𝑠(𝐴)}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑝[𝑖]↦→𝑒𝑛𝑣(𝑏)]\ncall 𝑟←𝑏.𝑚(𝑎𝑖)𝜆𝑆.\u001a𝑆∪{𝑝𝑖}if𝑎𝑖∈𝑆∧𝑎𝑖↦→𝑝𝑖in𝑚\n𝑆\\{𝑝𝑖}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑝𝑖↦→𝑒𝑛𝑣(𝑎𝑖)]\nreturn 𝑟←𝑏.𝑚(𝑎𝑖)𝜆𝑆.(\n𝑆∪{𝑟}if𝑟′∈𝑆∧𝑚returns𝑟′\n𝑆\\{𝑟}𝜆𝑒𝑛𝑣.𝑒𝑛𝑣[𝑟↦→𝑒𝑛𝑣(𝑟′)]\ncall-to-return 𝑟←𝑏.𝑚(𝑎𝑖)𝜆𝑆.\u001a𝑆\\{𝑎𝑖}if𝑎𝑖∈𝑆∧𝑎𝑖↦→𝑝𝑖in𝑚\n𝑆𝜆𝑒𝑛𝑣.𝑒𝑛𝑣\ndenoted as𝐺𝑝,𝑑and propagate 𝑑across its own sparse CFG. As\nshown with the IR in Table 1, 𝑑can be a local, an instance field or\nstatic field, or an array access. 𝐺𝑝,𝑑is constructed by determining\nwhether each statement’s corresponding flow function in 𝐺𝑝is a\n𝑑-specific identity function.\nAs a major modification, and most importantly, we account for\na statement’s effect on the value domain. In addition to determin-\ning whether each statement’s corresponding flow function is a\nd-specific identity function , we determine whether its edge func-\ntion is a d-specific identity transformer with the assumptions ex-\nplained in Section 3.3. Further, we propagate the tautological fact,Λ,\n(sparsely) to the statements that can generate new data-flow facts,\ne.g.𝑎←𝐶𝑜𝑛𝑠𝑡 . Otherwise, it is impossible to generate new facts at\narbitrary program points. Finally, we soundly retain all branching\nstatements to keep the original CFGs’ control flow as it is.\n5 EVALUATION\nWe next explain the research questions that guide our evaluation\nand its experimental setup, and then we discuss the evaluation\nresults. Sparse data-flow analyses promise extensive performance\nimprovements, while still maintaining the precision of their non-\nsparse counterparts. Therefore, first, we compare the sparse analysis\nresults against the non-sparse analysis results. Second, we measure\nwhether the sparse analysis produces the promised performance\nbenefits. Third, we investigate the factors contributing to the per-\nformance impact. Therefore, we focus on the following research\nquestions:•RQ1: Does Sparse IDE produce the same results as the orig-\ninal IDE?\n•RQ2: How does the sparsification impact the performance\nin terms of runtime and memory?\n•RQ3: To what extent does the number of propagations cor-\nrelate with the performance impact?\n5.1 Experimental Setup\nWe implement the proposed approach in SparseHeros , by extend-\ning the open source Heros IDE solver’s latest version, at the time\nof writing ( e7e4a85 ) [32]. Using SparseHeros and the Soot static\nanalysis framework [ 37], we implement a linear constant propaga-\ntion analysis. To handle aliasing, we integrate our client analysis\nwith the Boomerang [34] demand-driven pointer analysis, using\nits latest version ( 1179227 ) [7].Heros , and thus SparseHeros , sup-\nport multi-threading, yet, because Boomerang is single-threaded,\nour client analysis uses a single-thread. Therefore, our evaluation\nresults present single-thread performance.\nAs benchmark subjects we use:\n•ConstantBench : A benchmark suite for constant propa-\ngation analysis targeting Java, did not previously exist. We,\ntherefore, created ConstantBench as a micro-benchmark\nsuite for integer linear constant propagation analysis. We\nrun both Heros andSparseHeros on this benchmark suite\nand compare the analysis results that they produce.\n•Real-world Libraries : We include real-world Java libraries\nto investigate the performance of our approach under theKarakaya and Bodden\n#1 #2 #3 #4 #5 #6 #7 #8 #9#10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24 #25 #26 #27 #28 #29 #300204060801002 2 2 3 3 3 4 4 5 14 19 26 49 56 71 79 130 161 172 207 289 301 317 423 432 476 540 843 1158 3251 2 2 2 3 3\n3\n345\n214\n6\n7\n4111421\n61625 3527 27 3848\n237188\n37186Baseline\nRuntime\n(s)IDE Sparse IDE\nFigure 7: Relative runtime of Sparse IDE compared to the baseline original IDE in %, annotated with exact runtimes in seconds,\nsorted by original IDE’s runtime\n#1 #2 #3 #4 #5 #6 #7 #8 #9#10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 #21 #22 #23 #24 #25 #26 #27 #28 #29 #30020406080100120\n0.08 0.16 0.16 0.1 0.19 0.17 0.33 0.1 0.28 0.52 2.83 2.29 2.0 2.68 3.24 2.46 7.1 5.74 6.95 6.7 6.15 6.92 6.77 4.81 4.19 5.94 2.53 5.32 8.76 7.19 0.08 0.16 0.16\n0.09\n0.14\n0.09\n0.170.090.28\n0.151.39\n0.82\n0.6\n0.160.891.65\n1.79\n0.351.73\n0.672.38\n1.451.092.162.97\n1.973.02\n1.5\n0.931.17Baseline\nMemory\n(GB)IDE Sparse IDE\nFigure 8: Memory consumption of Sparse IDE compared to the baseline original IDE in %, annotated with exact memory\nconsumptions in GB, using the same sorting as Figure 7\nworkload of large-scale and complex programs. As opposed\nto applications, libraries do not have a specific entry method.\nWe follow the closed package assumption [27] for analyzing\nlibrary code, and treat public methods of the libraries as\nentry methods. We consider a method as an entry method if\nit adheres to the following entry method selection criteria:\n–c1:The method is a public instance method that is not\nabstract, native or a constructor,\n–c2:The method contains an integer assignment state-\nment.\nWe selected the most downloaded (>5000) Java libraries\nfrom the maven repository [ 28]. We discarded the libraries\nthat do not contain any entry methods according to the\nselection criteria, and the ones that caused an error in the\nunderlying static analysis tool, Soot [37]. In the end, we\nretained 30 libraries.\n•Replication Package : We set up a replication package,\navailable at https://zenodo.org/records/10461449\nWe have performed the evaluations on an Intel i7 Quad-Core at\n2,3 GHz with 32GB memory. We configured the JVM with 25GB\nmaximum heap size ( -Xmx25g ) and 1GB stack size ( -Xss1g ).5.2 RQ1: Does Sparse IDE produce the same\nresults as the original IDE?\nConstantBench consists of 40 target programs with various pro-\ngram properties and sensitivity-testing edge cases, as listed in Table\n2.Assignment cases test possible flow and edge functions, as well as\nflow sensitivity. Branching andLoops cases test the meet operation.\nField sensitivity cases test field sensitivity and aliasing scenarios.\nContext sensitivity cases test various calling contexts. Array cases\ntest array handling and NonLinear cases test analysis’ behavior\nunder unanticipated non-linear operations. The results validate the\ncorrectness of Sparse IDE by showing that SparseHeros produces\nthe same outputs as the non-sparse Heros .\n5.3 RQ2: How does the sparsification impact the\nperformance in terms of runtime and\nmemory?\nFigure 7 shows the relative analysis runtime spent by Sparse IDE\nin comparison to the runtime of the baseline original IDE algo-\nrithm. We sorted the results for each library by the time spent by\nthe original IDE algorithm. Note that we keep the same sorting\nfor the rest of the paper. This sorting highlights the fact that our\nSparse IDE approach pays off better for the cases where the origi-\nnal IDE’s runtime is relatively larger. Sparse IDE, compared to the\noriginal IDE algorithm, performs up to 30.7x faster. We measureSymbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\nTable 2: ConstantBench Test Cases\nAssignment Field Sensitivity\nConstant LoadConstant\nConstantBinop StoreConstant\nLocalBinop StoreViaAlias\nLocalMultipleBinop StoreBinop\nOverwrite FieldToField\nIncrement StoreBinopViaAlias\nOperators StoreLocalViaAlias\nAssignmentChain Context Sensitivity\nStatic Id\nBranching Increment\nSameValueMergedAndUsed Add\nSameValueMergedNotUsed Nested\nSameValueMergedAndUsedInBinop AssignFieldInCallee\nDiffValuesMergedAndUsed AssignStaticInCallee\nDiffValuesMergedNotUsed Array\nDiffValuesMergedAndUsedInBinop LoadConstant\nLoops StoreConstant\nForLoopFixedBound ArrayToArray\nForLoopUnkownBound AliasedArrays\nWhileTrue LargeIndex\nWhileUnknown Non-Linear\nNestedLoops Binop\nHashCode\nthe mean speedup as 7.9x, and the median speedup as 6.7x. The\nconcrete measurements are presented in Table 3. Results show that,\nin terms of runtime, Sparse IDE outperforms the original IDE in\neach run, except for the libraries #1-#3 (jcl-over-slf4j, slf4j-api, lom-\nbok), which have the shortest analysis time. In each run, Sparse\nCFG construction overhead is lower than 1% of the Sparse IDE total\nanalysis runtime, which is substantially smaller than the achieved\nspeedups.\nFigure 8 shows the relative memory consumption of Sparse IDE\nin comparison to the memory consumption of the original IDE\nalgorithm. We have measured up to 94% reduction in memory\nconsumption in the best case, and up to a 19% increase in the\nworst. The Sparse IDE algorithm, compared to the original IDE,\nassociates data-flow facts with fewer statements, therefore, we\nanticipated memory improvements. On the other hand, because we\ncache sparse CFGs ( 𝐺𝑑,𝑝) per each symbol and procedure pair ( 𝑑,𝑝),\nfor some input programs memory consumption increases. However,\nas shown in Figure 8, these cases are limited to a few outliers.\nMoreover, the mean and median impacts on memory consumption\nare 51% and 63% reduction, respectively.\nWe statistically assess the significance of the Sparse IDE algo-\nrithm’s impact on runtime and memory improvements. According\nto Wilcoxon signed-rank test [ 39] at 0.05 significance level, Sparse\nIDE significantly improves both the runtime ( 𝑝=6.1e−08) and\nmemory consumption ( 𝑝=5.7e−07) of the original IDE algorithm.\n2 4 6 8\nIDE Propagation Count / Sparse IDE Propagation Count012345IDE Runtime / \n Sparse IDE Runtime\nFigure 9: Ratio of data-flow fact propagations and correspond-\ning speedup ratios, in log scale\n2 4 6 8\nIDE Propagation Count / Sparse IDE Propagation Count34567IDE Memory / \n Sparse IDE Memory\nFigure 10: Ratio of data-flow fact propagations and corre-\nsponding memory consumption ratios, in log scale\n5.4 RQ3: To what extent does the number of\npropagations correlate with the\nperformance impact?\nThe essence of the Sparse IDE approach is that, compared to the\noriginal IDE algorithm, it propagates data-flow facts to fewer state-\nments. We investigate to what extent this contributes to improving\nthe scalability of the original IDE algorithm. Figure 9, shows how\nthe ratio of data-flow fact propagations in IDE and Sparse IDE cor-\nrelate with the ratio of runtime speedups. We observe that reducing\nthe number of propagations is an effective approach to improving\nIDE’s scalability in terms of runtime. Similarly, Figure 10 correlates\nthe same with the ratio of memory consumptions in IDE and Sparse\nIDE. We observe a comparable trend but not to the same degree.\nGiven these findings, in the future, one could investigate the poten-\ntial synergies between our approach and recent approaches that\nimprove the scalability, in particular, in terms of memory [1, 19].\n6 LIMITATIONS AND THREATS TO VALIDITY\nBy definition, Sparse IDE can solve the same data-flow problems\nas the original IDE framework [ 30]. It requires data-flow analysis\nproblems to be expressible as distributive environment problems.\nMany popular static analyses, such as taint analysis for vulnera-\nbility detection [ 3] or typestate analysis for API misuse detection\n[10], are expressible as distributive environment problems. Just like\nother fact-specific sparsification approaches [ 15,17], Sparse IDE\nalso exploits analysis domain knowledge. Domain-specific analysisKarakaya and Bodden\nTable 3: Performance of Sparse IDE compared to the baseline original IDE algorithm\n# Library Version#Entry Runtime (s) Memory (GB) #Propagations Sparse CFG\nMethods IDE SP IDE/SP IDE SP SP/IDE (%) IDE SPIDE/SP Count Const. (ms) %Runtime\n1 jcl-over-slf4j 2.0.7 1 2 2 1.00 0.08 0.08 100.78 48 34 1.41 2 0 0.01\n2 slf4j-api 2.0.7 7 2 2 0.99 0.16 0.16 100.62 104 94 1.11 13 0 0.00\n3 lombok 1.18.26 5 2 2 0.99 0.16 0.16 99.40 894 227 3.94 13 0 0.02\n4 commons-logging 1.2 14 3 3 1.00 0.10 0.09 93.87 1,509 917 1.65 41 0 0.00\n5 junit-jupiter-api 5.9.2 10 3 3 1.01 0.19 0.14 75.39 182 158 1.15 20 0 0.00\n6 jackson-annotations 2.14.2 79 3 3 1.14 0.17 0.09 55.10 13,115 6,511 2.01 190 0 0.00\n7 maven-plugin-api 3.9.1 13 4 3 1.20 0.33 0.17 49.61 17,353 4,780 3.63 294 4 0.14\n8 junit-jupiter-engine 5.9.2 23 4 4 1.02 0.10 0.09 86.81 3,204 1,181 2.71 105 0 0.02\n9 osgi.core 8.0.0 124 5 5 1.04 0.28 0.28 100.83 58,675 28,247 2.08 664 7 0.15\n10 jakarta.servlet-api 6.0.0 12 14 2 5.25 0.52 0.15 29.28 126,656 341 371.43 33 0 0.00\n11 commons-io 2.11.0 178 19 14 1.30 2.83 1.39 48.94 156,595 15,290 10.24 1,279 116 0.78\n12 commons-codec 1.15 77 26 6 4.25 2.29 0.82 35.90 652,560 100,866 6.47 532 13 0.21\n13 json 20230227 33 49 7 6.88 2.00 0.60 30.24 1,071,045 10,846 98.75 407 0 0.00\n14 logback-classic 1.4.7 93 56 4 11.28 2.68 0.16 5.92 1,286,543 8,027 160.28 372 12 0.24\n15 logback-core 1.4.7 218 71 11 6.44 3.24 0.89 27.55 1,739,303 14,767 117.78 925 0 0.00\n16 gson 2.10.1 147 79 14 5.45 2.46 1.65 66.93 2,009,909 29,391 68.39 1,586 54 0.37\n17 commons-lang3 3.12.0 318 130 21 6.14 7.10 1.79 25.22 3,418,491 31,856 107.31 1,144 0 0.00\n18 commons-beanutils 1.9.4 109 161 6 25.97 5.74 0.35 6.15 5,855,012 20,640 283.67 648 2 0.04\n19 mockito-core 5.3.1 235 172 16 10.20 6.95 1.73 24.85 5,025,407 51,374 97.82 1,663 119 0.71\n20 junit-jupiter-params 5.9.2 293 207 25 8.22 6.70 0.67 10.03 6,266,620 99,285 63.12 1,506 109 0.43\n21 assertj-core 3.24.2 334 289 35 8.22 6.15 2.38 38.71 10,033,236 45,563 220.21 2,418 37 0.11\n22 commons-collections4 4.4 620 301 27 10.90 6.92 1.45 20.91 9,140,963 42,741 213.87 1,796 1 0.01\n23 testng 7.7.1 246 317 27 11.68 6.77 1.09 16.08 9,329,214 116,084 80.37 2,910 15 0.06\n24 joda-time 2.12.5 375 423 38 11.11 4.81 2.16 44.93 15,151,487 137,705 110.03 3,227 69 0.18\n25 guice 5.1.0 336 432 48 8.95 4.19 2.97 70.80 15,141,525 390,634 38.76 3,918 58 0.12\n26 hamcrest-all 1.3 290 476 23 20.48 5.94 1.97 33.10 17,953,051 71,200 252.15 1,105 28 0.12\n27 log4j-core 2.20.0 512 540 71 7.60 2.53 3.02 119.69 18,746,154 1,218,580 15.38 4,666 64 0.09\n28 jackson-databind 2.14.2 844 843 88 9.57 5.32 1.50 28.20 35,842,682 166,906 214.75 7,884 5 0.01\n29 okhttp 4.10.0 717 1,158 37 30.69 8.76 0.93 10.58 37,431,312 581,852 64.33 5,928 69 0.18\n30 guava-31.1 jre 1,332 3,251 186 17.43 7.19 1.17 16.31 131,993,565 239,589 550.92 12,200 4 0.00\nsemantics must be correctly encoded with flow and edge function\ndefinitions within the IDE framework.\nSparse IDE should theoretically lead to a similar performance\nimpact on other data-flow analysis problems where IDE is appli-\ncable. For instance, when performing a typestate analysis, Sparse\nIDE would safely omit the statements that have no impact on the\ntracked state. However, due to space constraints, we were not able\nto empirically show whether our evaluation results carry over to\nother analysis problems.\nThe reported evaluation results might depend on the selected\nset of Java libraries, and entry-method selection criteria. Neverthe-\nless, for real-world library selection, we followed the systematic\nprocedure described in Section 5.1.\nTo account for variations in runtime and memory measurements,\nwe conducted three runs and presented the average across these\nruns.\nA direct comparison to SparseDroid [15] was not possible for\nmany reasons. It extends an existing taint analysis client ( FlowDroid\n[3]) that has a basic integrated alias analysis, whereas our analysis\nclient utilizes a sophisticated external demand-driven pointer anal-\nysis [ 34]. Moreover, SparseDroid ’s implementation is not publicly\navailable, and most importantly, IFDS may not terminate when the\nvalue domain is infinitely broad.\n7 RELATED WORK\nThe IFDS [ 29] and IDE [ 30] frameworks enabled precise interproce-\ndural data-flow analyses that are flow- and context-sensitive. Pre-\nvious works have extended these frameworks with diverse goals.\nNaeem et al. [ 24] proposed four extensions to the IFDS framework,\nto improve its scalability and precision under certain practical anal-\nysis conditions. Heros [5] introduced a Java-based generic IFDS\nand IDE solver. Reviser [2] proposed an algorithm to adapt IFDSand IDE to incremental program updates. CleanDroid [1] intro-\nduced a technique for reducing the memory footprint of IFDS-based\ndata-flow analyses. DiskDroid [19] applied a disk-assisted com-\nputing approach for improving the scalability of IFDS-based taint\nanalysis.\nSparsification has been applied to improve the scalability of\nstatic analyses. Choi et al. [ 6] introduced sparse data-flow eval-\nuation graphs based on SSA (static-single assignment). Oh et al.\n[26] presented an abstract interpretation-based framework for de-\nsigning generic sparse analyses, which guarantees to preserve the\nprecision of the non-sparse analysis through data dependencies.\nPinpoint [31],SVF[35] and SFS[14] utilize cheaper pre-analyses\nto sparsify pointer analyses. Recent on-demand sparsification ap-\nproaches exploit the data-flow facts that become available during\nthe analysis runtime for further sparsification. SparseBoomerang\n[17] exploits the variables in alias queries during demand-driven\npointer analysis, to create query-specific sparse CFGs. The sparse\nIFDS algorithm [ 15] exploits data-flow facts to create fact-specific\nsparse CFGs and propagate each fact on its own sparse CFG. In\nthis work, we present the more generic Sparse IDE algorithm that\nefficiently solves not just IFDS-based reachability problems, but\nalso IDE problems that require value computation.\n8 CONCLUSION AND FUTURE WORK\nIn this work, we presented the Sparse IDE framework as a scalable\nalternative to the original IDE framework. Sparse IDE is the first\nfact-specific sparsification approach that allows for computations\non infinitely broad domains. The essence of Sparse IDE is creating\nsymbol-specific sparse control flow graphs on-demand, and propa-\ngating data-flow facts sparsely through these graphs. Sparse IDE\nproduces equally precise results as the original IDE, while signifi-\ncantly improving its scalability. We also explicitly discuss the limitsSymbol-Specific Sparsification\nof Interprocedural Distributive Environment Problems\nof sparsification for IDE: while symbol-specific sparsification is pos-\nsible and useful, one cannot sparsify with respect to the (typically\nnumeric and infinite) value domain.\nIn the future, we plan to apply the Sparse IDE framework to\nother data-flow analysis problems and investigate problem-specific\nrequirements for building sparse CFGs. We also plan to combine\nSparse IDE with other scalability-improving techniques that are\northogonal to our sparsification approach.\nACKNOWLEDGMENTS\nWe gratefully acknowledge the support of Martin Mory and Marcus\nHüwe in this work. We thank Martin for the enlightening discus-\nsions and for the encouragement to conclude this work. We thank\nMarcus for sharing his expertise on the formal notation.\nREFERENCES\n[1]Steven Arzt. 2021. Sustainable Solving: Reducing The Memory Footprint of\nIFDS-Based Data Flow Analyses Using Intelligent Garbage Collection. In 2021\nIEEE/ACM 43rd International Conference on Software Engineering (ICSE) . IEEE,\n1098–1110.\n[2] Steven Arzt and Eric Bodden. 2014. Reviser: efficiently updating IDE-/IFDS-based\ndata-flow analyses in response to incremental program changes. In Proceedings\nof the 36th International Conference on Software Engineering . 288–298.\n[3] Steven Arzt, Siegfried Rasthofer, Christian Fritz, Eric Bodden, Alexandre Bartel,\nJacques Klein, Yves Le Traon, Damien Octeau, and Patrick McDaniel. 2014. Flow-\nDroid: Precise Context, Flow, Field, Object-Sensitive and Lifecycle-Aware Taint\nAnalysis for Android Apps. In Proceedings of the 35th ACM SIGPLAN Conference\non Programming Language Design and Implementation (Edinburgh, United King-\ndom) (PLDI ’14) . Association for Computing Machinery, New York, NY, USA,\n259–269. https://doi.org/10.1145/2594291.2594299\n[4] Nathaniel Ayewah, William Pugh, David Hovemeyer, J. David Morgenthaler, and\nJohn Penix. 2008. Using Static Analysis to Find Bugs. IEEE Software 25, 5 (2008),\n22–29. https://doi.org/10.1109/MS.2008.130\n[5] Eric Bodden. 2012. Inter-Procedural Data-Flow Analysis with IFDS/IDE and Soot.\nInProceedings of the ACM SIGPLAN International Workshop on State of the Art in\nJava Program Analysis (Beijing, China) (SOAP ’12) . Association for Computing\nMachinery, New York, NY, USA, 3–8. https://doi.org/10.1145/2259051.2259052\n[6] Jong-Deok Choi, Ron Cytron, and Jeanne Ferrante. 1991. Automatic construction\nof sparse data flow evaluation graphs. In Proceedings of the 18th ACM SIGPLAN-\nSIGACT symposium on Principles of programming languages . 55–66.\n[7]CodeShield. [n. d.]. CodeShield-Security/SPDS: Efficient and Precise Pointer-\nTracking Data-Flow Framework. https://github.com/CodeShield-Security/SPDS.\n(Accessed on 03/30/2023).\n[8]Alain Deutsch. 1994. Interprocedural May-Alias Analysis for Pointers: Be-\nyond k-Limiting. In Proceedings of the ACM SIGPLAN 1994 Conference on Pro-\ngramming Language Design and Implementation (Orlando, Florida, USA) (PLDI\n’94). Association for Computing Machinery, New York, NY, USA, 230–241.\nhttps://doi.org/10.1145/178243.178263\n[9] T. Eisenbarth, R. Koschke, and D. Simon. 2001. Aiding program comprehension by\nstatic and dynamic feature analysis. In Proceedings IEEE International Conference\non Software Maintenance. ICSM 2001 . 602–611. https://doi.org/10.1109/ICSM.200\n1.972777\n[10] Michael Emmi, Liana Hadarean, Ranjit Jhala, Lee Pike, Nicolás Rosner, Martin\nSchäf, Aritra Sengupta, and Willem Visser. 2021. RAPID: checking API usage for\nthe cloud in the cloud. In Proceedings of the 29th ACM Joint Meeting on European\nSoftware Engineering Conference and Symposium on the Foundations of Software\nEngineering . 1416–1426.\n[11] Stephen J. Fink, Eran Yahav, Nurit Dor, G. Ramalingam, and Emmanuel Geay.\n2008. Effective Typestate Verification in the Presence of Aliasing. ACM Trans.\nSoftw. Eng. Methodol. 17, 2, Article 9 (may 2008), 34 pages. https://doi.org/10.1\n145/1348250.1348255\n[12] Jeffrey S Foster, Michael W Hicks, and William Pugh. 2007. Improving software\nquality with static analysis. In Proceedings of the 7th ACM SIGPLAN-SIGSOFT\nworkshop on Program analysis for software tools and engineering . 83–84.\n[13] Ben Hardekopf and Calvin Lin. 2009. Semi-Sparse Flow-Sensitive Pointer Analy-\nsis.SIGPLAN Not. 44, 1 (jan 2009), 226–238. https://doi.org/10.1145/1594834.14\n80911\n[14] Ben Hardekopf and Calvin Lin. 2011. Flow-sensitive pointer analysis for millions\nof lines of code. In International Symposium on Code Generation and Optimization\n(CGO 2011) . IEEE, 289–298.[15] Dongjie He, Haofeng Li, Lei Wang, Haining Meng, Hengjie Zheng, Jie Liu,\nShuangwei Hu, Lian Li, and Jingling Xue. 2019. Performance-Boosting Sparsifi-\ncation of the IFDS Algorithm with Applications to Taint Analysis. In 2019 34th\nIEEE/ACM International Conference on Automated Software Engineering (ASE) .\n267–279. https://doi.org/10.1109/ASE.2019.00034\n[16] N. Jovanovic, C. Kruegel, and E. Kirda. 2006. Pixy: a static analysis tool for\ndetecting Web application vulnerabilities. In 2006 IEEE Symposium on Security\nand Privacy (S&P’06) . 6 pp.–263. https://doi.org/10.1109/SP.2006.29\n[17] Kadiray Karakaya and Eric Bodden. 2023. Two Sparsification Strategies for\nAccelerating Demand-Driven Pointer Analysis. In IEEE International Conference\non Software Testing, Verification and Validation (ICST) .\n[18] Gary A. Kildall. 1973. A Unified Approach to Global Program Optimization\n(POPL ’73) . Association for Computing Machinery, New York, NY, USA, 194–206.\nhttps://doi.org/10.1145/512927.512945\n[19] Haofeng Li, Haining Meng, Hengjie Zheng, Liqing Cao, Jie Lu, Lian Li, and Lin\nGao. 2021. Scaling up the IFDS algorithm with efficient disk-assisted computing.\nIn2021 IEEE/ACM International Symposium on Code Generation and Optimization\n(CGO) . IEEE, 236–247.\n[20] Tuo Li, Jia-Ju Bai, Yulei Sui, and Shi-Min Hu. 2022. Path-sensitive and alias-\naware typestate analysis for detecting OS bugs. In Proceedings of the 27th ACM\nInternational Conference on Architectural Support for Programming Languages\nand Operating Systems . 859–872.\n[21] Yue Li, Tian Tan, Anders Møller, and Yannis Smaragdakis. 2020. A Principled\nApproach to Selective Context Sensitivity for Pointer Analysis. ACM Trans.\nProgram. Lang. Syst. 42, 2, Article 10 (may 2020), 40 pages. https://doi.org/10.1\n145/3381915\n[22] V Benjamin Livshits and Monica S Lam. 2005. Finding Security Vulnerabilities in\nJava Applications with Static Analysis.. In USENIX security symposium , Vol. 14.\n18–18.\n[23] Nomair A. Naeem and Ondrej Lhotak. 2008. Typestate-like Analysis of Multiple\nInteracting Objects. In Proceedings of the 23rd ACM SIGPLAN Conference on\nObject-Oriented Programming Systems Languages and Applications (Nashville,\nTN, USA) (OOPSLA ’08) . Association for Computing Machinery, New York, NY,\nUSA, 347–366. https://doi.org/10.1145/1449764.1449792\n[24] Nomair A Naeem, Ondřej Lhoták, and Jonathan Rodriguez. 2010. Practical\nextensions to the IFDS algorithm. In Compiler Construction: 19th International\nConference, CC 2010, Held as Part of the Joint European Conferences on Theory and\nPractice of Software, ETAPS 2010, Paphos, Cyprus, March 20-28, 2010. Proceedings\n19. Springer, 124–144.\n[25] Damien Octeau, Daniel Luchaup, Matthew Dering, Somesh Jha, and Patrick\nMcDaniel. 2015. Composite constant propagation: Application to android inter-\ncomponent communication analysis. In 2015 IEEE/ACM 37th IEEE International\nConference on Software Engineering , Vol. 1. IEEE, 77–88.\n[26] Hakjoo Oh, Kihong Heo, Wonchan Lee, Woosuk Lee, and Kwangkeun Yi. 2012.\nDesign and Implementation of Sparse Global Analyses for C-like Languages. In\nProceedings of the 33rd ACM SIGPLAN Conference on Programming Language\nDesign and Implementation (Beijing, China) (PLDI ’12) . Association for Computing\nMachinery, New York, NY, USA, 229–238. https://doi.org/10.1145/2254064.2254\n092\n[27] Michael Reif, Michael Eichberg, Ben Hermann, Johannes Lerch, and Mira Mezini.\n2016. Call Graph Construction for Java Libraries. In Proceedings of the 2016 24th\nACM SIGSOFT International Symposium on Foundations of Software Engineering\n(Seattle, WA, USA) (FSE 2016) . Association for Computing Machinery, New York,\nNY, USA, 474–486. https://doi.org/10.1145/2950290.2950312\n[28] Maven Repository. [n. d.]. Maven Repository: Search/Browse/Explore. https:\n//mvnrepository.com/. (Accessed on 03/30/2023).\n[29] Thomas Reps, Susan Horwitz, and Mooly Sagiv. 1995. Precise interprocedural\ndataflow analysis via graph reachability. In Proceedings of the 22nd ACM SIGPLAN-\nSIGACT symposium on Principles of programming languages . 49–61.\n[30] Mooly Sagiv, Thomas Reps, and Susan Horwitz. 1996. Precise interprocedu-\nral dataflow analysis with applications to constant propagation. Theoretical\nComputer Science 167, 1-2 (1996), 131–170.\n[31] Qingkai Shi, Xiao Xiao, Rongxin Wu, Jinguo Zhou, Gang Fan, and Charles\nZhang. 2018. Pinpoint: Fast and Precise Sparse Value Flow Analysis for Million\nLines of Code. In Proceedings of the 39th ACM SIGPLAN Conference on Pro-\ngramming Language Design and Implementation (Philadelphia, PA, USA) (PLDI\n2018) . Association for Computing Machinery, New York, NY, USA, 693–706.\nhttps://doi.org/10.1145/3192366.3192418\n[32] soot oss. [n. d.]. soot-oss/heros: IFDS/IDE Solver for Soot and other frameworks.\nhttps://github.com/soot-oss/heros. (Accessed on 03/30/2023).\n[33] Johannes Späth, Karim Ali, and Eric Bodden. 2017. Ide al: Efficient and precise\nalias-aware dataflow analysis. Proceedings of the ACM on Programming Languages\n1, OOPSLA (2017), 1–27.\n[34] Johannes Späth, Lisa Nguyen Quang Do, Karim Ali, and Eric Bodden. 2016.\nBoomerang: Demand-driven flow-and context-sensitive pointer analysis for java.\nIn30th European Conference on Object-Oriented Programming (ECOOP 2016) .\nSchloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Karakaya and Bodden\n[35] Yulei Sui and Jingling Xue. 2016. SVF: Interprocedural Static Value-Flow Anal-\nysis in LLVM. In Proceedings of the 25th International Conference on Compiler\nConstruction (Barcelona, Spain) (CC 2016) . Association for Computing Machinery,\nNew York, NY, USA, 265–266. https://doi.org/10.1145/2892208.2892235\n[36] Yulei Sui, Sen Ye, Jingling Xue, and Pen-Chung Yew. 2011. SPAS: Scalable Path-\nSensitive Pointer Analysis on Full-Sparse SSA. In Programming Languages and\nSystems , Hongseok Yang (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg,\n155–171.\n[37] Raja Vallée-Rai, Phong Co, Etienne Gagnon, Laurie Hendren, Patrick Lam, and\nVijay Sundaresan. 2010. Soot: A Java bytecode optimization framework. In\nCASCON First Decade High Impact Papers . 214–224.[38] Carmine Vassallo, Sebastian Proksch, Timothy Zemp, and Harald C Gall. 2020.\nEvery build you break: developer-oriented assistance for build failure resolution.\nEmpirical Software Engineering 25 (2020), 2218–2257.\n[39] Frank Wilcoxon. 1992. Individual comparisons by ranking methods . Springer.\n[40] Sen Ye, Yulei Sui, and Jingling Xue. 2014. Region-Based Selective Flow-Sensitive\nPointer Analysis. In Static Analysis , Markus Müller-Olm and Helmut Seidl (Eds.).\nSpringer International Publishing, Cham, 319–336.\n[41] Xiaodong Yu, Fengguo Wei, Xinming Ou, Michela Becchi, Tekin Bicer, and\nDanfeng Yao. 2020. GPU-Based Static Data-Flow Analysis for Fast and Scalable\nAndroid App Vetting. In 2020 IEEE International Parallel and Distributed Processing\nSymposium (IPDPS) . 274–284. https://doi.org/10.1109/IPDPS47924.2020.00037" }, { "title": "2401.14839v2.Well_posedness_and_stability_of_the_Navier_Stokes_Maxwell_equations.pdf", "content": "arXiv:2401.14839v2 [math.AP] 23 Mar 2024Well-posedness and stability of the Navier-Stokes-Maxwel l equations\nKyungkeun Kang∗Jihoon Lee†Dinh Duong Nguyen‡\nMarch 26, 2024\nAbstract\nThe paper is devoted to studying the well-posedness and stab ility of the generalized Navier-Stokes-\nMaxwell (NSM) equations with the standard Ohm’s law in Rdford∈ {2,3}. More precisely, the global\nwell-posedness is established in case of fractional Laplac ian velocity ( −∆)αvwithα=d\n2for suitable data.\nIn addition, the local well-posedness in the inviscid case i s also provided for sufficient smooth data, which\nallows us to study the inviscid limit of associated positive viscosity solutions in the case α= 1, where\nan explicit bound on the difference is given. On the other hand , in the case α= 0 the stability near a\nmagnetohydrostatic equilibrium with a constant (or equiva lently bounded) magnetic field is also obtained in\nwhich nonhomogeneous Sobolev norms of the velocity and elec tric fields, and the L∞norm of the magnetic\nfield converge to zero as time goes to infinity with an implicit rate. In this velocity damping case, the\nsituation is different both in case of the two and a half, and th ree-dimensional magnetohydrodynamics\n(MHD) system, where an explicit rate of convergence in infini te time is computed for both the velocity and\nmagnetic fields in nonhomogeneous Sobolev norms. Therefore , it seems that there is a gap between NSM\nand MHD in terms of the norm convergence of the magnetic field a nd the rate of decaying in time, even the\nlatter equations can be proved as a limiting system of the for mer one in the sense of distributions as the\nspeed of light tends to infinity.\nKeywords: Navier-Stokes-Maxwell, Well-posedness and stability, Magnetohyd rodynamics.\nMathematics Subject Classification: 35Q35, 35Q60, 76D03, 76W05, 78A25.\nContents\n1 Introduction 2\n1.1 The systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.2 The state of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.3.1 Global well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.3.2 Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n1.3.3 Stability and large-time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2 Proof of Theorem 1.1 13\n3 Proof of Theorem 1.2 28\n4 Proof of Theorem 1.3 36\n5 Proof of Theorem 1.4 38\n6 Proof of Theorem 1.5 43\n7 Appendix 49\n7.1 Appendix A: Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n7.2 Appendix B: Homogeneous Sobolev inequalities and proof of ( 2.4) . . . . . . . . . . . . . . . . . 50\n7.3 Appendix C: A logarithmic Gronwall inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\n7.4 Appendix D: Fractional heat equation and proof of ( 3.1) . . . . . . . . . . . . . . . . . . . . . . . 51\n7.5 Appendix E: Remarks on the Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\n7.6 Appendix F: Proof of Proposition 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56\n∗Yonsei University. E-mail address: kkang@yonsei.ac.kr\n†Chung-Ang University. E-mail address: jhleepde@cau.ac.kr\n‡Yonsei University and Chung-Ang University. E-mail addres s:dinhduongnguyen.math@gmail.com\n11 Introduction\n1.1 The systems\nLet us consider the one fluid incompressible Navier-Stokes-Maxwell equations with the standard Ohm’s law,\nwhich are given in the following form1\n\n\n∂tv+v·∇v+∇π=−ν(−∆)αv+j×B,\n1\nc∂tE−∇×B=−j,\n1\nc∂tB+∇×E= 0,\nσ(cE+v×B) =j,\ndivv= divB= 0,(NSM)\nwhereα≥0, ford∈ {2,3}, (v,E,B,j ) :Rd×(0,∞)→R3andπ:Rd×(0,∞)→Rare the velocity, electric,\nmagnetic and electric current fields, and scalar pressure of the flu id, respectively. The positive constants ν,σ\nandcdenote in order the viscosity, electric conductivity and speed of ligh t. We will denote the initial data\nby (v,E,B)|t=0= (v0,E0,B0). We note that the case d= 2 is also known as the 21\n2-dimensional version. Let\nus also quickly recall the standard meaning of the above system. In (NSM), through the Lorentz force j×B\n(under quasi-neutrality assumptions) the fluid equations (the firs t ones) are coupled to the Maxwell system,\nwhich consists of the Ampere’s equations + Maxwell’s correction (the second equations) and the Faraday’s law\n(the third equations). In addition, the fourth equations are the u sual Ohm’s law and the last one stands for\nthe incompressiblity of the velocity and magnetic fields. It can be see n that if the term1\nc∂tEis neglected\nformally for either large cor time-independent E, then (NSM) reduces to the usual 21\n2-dimensional fractional\nmagnetohydrodynamics (MHD) equations, i.e., ( H-MHD) withκ= 0 (for more physical introduction to the\nmagnetohydrodynamics, see [ 9,28]). Therefore, ( NSM) withα= 1 is also known as the full MHD system.\nIn fact, by ignoring thermal effects, ( NSM) withα= 1 can be derived from kinetic equations (see [ 41]). By\nconsidering solenoidal Ohm’s law2instead, it also can be formally obtained as a limiting system of a two-flu id\nincompressible Navier–Stokes–Maxwell system by taking the momen tum transfer coefficient ǫ >0 tends to zero\n(see [3]). More precisely, if v+andv−denote the cations and anions velocities, respectively, with the sam e\nviscosity µ >0 and the corresponding thermal pressures π+andπ−, then the scaled two-fluid incompressible\nNavier–Stokes–Maxwell equations were proposed in [ 34] and will be written in the following form3(we use the\nsame notation for the electric and magnetic fields as previously)\n\n\n∂tv++v+·∇v++1\n2σǫ2(v+−v−) =µ∆v+−∇π++1\nǫ(cE+v+×B),\n∂tv−+v−·∇v−−1\n2σǫ2(v+−v−) =µ∆v−−∇π−−1\nǫ(cE+v−×B),\n1\nc∂tE−∇×B=−1\n2ǫ(v+−v−),\n1\nc∂tB+∇×E= 0,\ndivv+= divv−= divE= divB= 0,(2-NSM)\nwhich models the motion of a plasma of positively (cations) and negativ ely (anions) charged particles under\nthe assumption of equal masses. In the above system, the condit ion divE= 0, which is known as a degenerate\nGauss’s law (see [ 3]) and follows from the charge neutrality and the incompressibility of t he plasma (see [ 34]),\nand the third term on the right-hand side of the second equation pr esents the momentum transfer between the\ntwo fluids. The existence and uniqueness of global energy solutions to (2-NSM) (for more general coefficients)\nhave recently obtained in [ 33] in two dimensions. In the three-dimensional case, they also showe d the existence\nof global energy solutions and local well-posedness (LWP) for initial data (v±\n0,E0,B0)∈H1\n2×L2×L2, and\nthis local solution can be globally extended for small v±\n0in˙H1\n2norm. It can be seen that the energy equality\nto (2-NSM) formally reads4\n1\n2d\ndt/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggv±\n√\n2,E,B/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+µ/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n2∇v±/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+1\nσ/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\n2ǫ(v+−v−)/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2= 0,\n1Here, the usual fractional Laplacian operator is defined in t erms of Fourier transform, i.e., for α∈R\nF((−∆)α(f))(ξ) :=|ξ|2αF(f)(ξ) where F(f)(ξ) :=/integraldisplay\nRdexp{−iξ·x}f(x)dxforξ∈Rd.\nIn the case α= 0, we use the standard convention that ( −∆)0is the identity operator.\n2In this case, j=σ(−∇¯π+cE+v×B) with div j= divE= 0 and for some additional pressure ¯ π, see (NSM-SO ).\n3In fact, the authors in [ 34] suggested a more general model with different coefficients ap pearing in the equations.\n4The notation /ba∇⌈bl(f1,...,fn)/ba∇⌈blm\nX:=/summationtextn\ni=1/ba∇⌈blfi/ba∇⌈blm\nXwill be used throughout the paper for n,m∈Nand some functional space X.\n2which suggests us to consider a system, which is satisfied by the follo wing quantities\nk:=1\n2ǫ(v+−v−), u:=1\n2(v++v−), p:=1\n2(π++π−) and ¯ p:=ǫ\n2(π+−π−)\nand is given by rewriting ( 2-NSM) as follows\n\n\n∂tu+u·∇u+ǫ2k·∇k=µ∆u−∇p+k×B,\nǫ2(∂tk+u·∇k+k·∇u)+1\nσk=ǫ2µ∆k−∇¯p+cE+u×B,\n1\nc∂tE−∇×B=−k,\n1\nc∂tB+∇×E= 0,\ndivu= divk= divE= divB= 0.\nAsǫ→0, the above system formally converges to ( NSM) with solenoidal Ohm’s law5instead of the usual one\n(see [3]), i.e., the following system (for simplicity we will replace u,k,p,¯pandµbyv,j,π,¯πandν, respectively)\n\n\n∂tv+v·∇v+∇π=ν∆v+j×B,\n1\nc∂tE−∇×B=−j,\n1\nc∂tB+∇×E= 0,\nσ(−∇¯π+cE+v×B) =j,\ndivv= divj= divE= divB= 0,(NSM-SO)\nthat shares a similar structure and mathematical difficulties to thos e of (NSM) withα= 1. In fact, we will\nlist known results to ( NSM) and it is possible to obtain similar ones to ( NSM-SO ). The rigorous proof of the\nconvergence from ( 2-NSM) to (NSM-SO ) asǫ→0 does not seem to be known for L2initial data. In [ 3], the\nauthors established the limit as first c→ ∞and then ǫ→0, where ( 2-NSM) converges weakly to the standard\n21\n2-dimensional MHD system, i.e., ( H-MHD) withα= 1 and κ= 0. They also pointed out that the same result\nalso holds in the case c→ ∞andǫ→0 at the same time, but with additional conditions on the relation betw een\nǫandcin which ǫis considered as a function of csatisfying further assumptions. However, the other order of\ntaking limit has not been confirmed yet, i.e., the limit as ǫ→0 first and then c→ ∞, where the limiting system\nis the same as the previous case. In addition, it is also very intereste d and much more complicated to consider\n(NSM) with a generalized Ohm’s law, which can be derived from either the two -fluid Navier-Stokes-Maxwell\nequations or kinetic models with different masses (see [ 1,41,61]) forα= 1, in particular, the new system takes\na more general form as follows\n\n\n∂tv+v·∇v+∇π=−ν(−∆)αv+j×B,\n1\nc∂tE−∇×B=−j,\n1\nc∂tB+∇×E= 0,\nσ(−∇¯π+cE+v×B) =j+κj×B,\ndivv= divj= divE= divB= 0,(NSM-GO)\nwhich takes into account of the Hall effect for some nonnegative co nstantκ. This new constant is corresponding\nto the magnitude of the Hall effect compared to the typical fluid leng th scale. Furthermore, by taking the limit\nasc→ ∞formally or ignoring the term1\nc∂tE(for example, Eis time-independent), ( NSM-GO ) reduces to the\nstandard Hall magnetohydrodynamics equations, i.e.,\n\n\n∂tv+v·∇v+∇π=−ν(−∆)αv+(∇×B)×B,\n∂tB−∇×(v×B) =1\nσ∆B−κ\nσ∇×((∇×B)×B),\ndivv= divB= 0.(H-MHD)\nIndeed, the Hallterm in ( H-MHD) playsan importantrolein magnetic reconnection, which cannot be e xplained\nby using ( MHD), and is derived from either two-fluid models or kinetic equations in [ 1]. The systematical\n5It is not clear (at least) to us whether or not should we keep th e pressure term ¯ p:=ǫ\n2(π+−π−) in (NSM-SO ) after taking the\nlimit formally.\n3study of the above equations was initiated in [ 54] long time ago. However, even in the case that d= 2 and\nα= 1, the global regularity issue of ( H-MHD) has not been fully established for general initial data. In this\ncase, the existence of global energy solutions has been provided in [16] in both two and three dimensions,\nbut it is not the case for ( NSM) and (NSM-GO ) as mentioned before. In addition, by using the convex\nintegration framework, the author in [ 25] proved the nonuniqueness of weak solutions in the Leray-Hopf cla ss\nford= 3. In the case of without the resistivity, illposedness results arou nd shear-type flows are also obtained\nin [42]. Furthermore, global small initial data solutions in both cases d= 2 and d= 3 have been provided in\n[4,16,18,26,27,57,64,68,69,70,76]. In the stationary case, regularity of weak solutions can be found in [21]\nin the two-dimensional case. As mentioned previously, ( H-MHD) can be obtained formally from ( NSM-GO ).\nTherefore, it is reasonableto considerthe conditional global well-p osedness(GWP) for ( NSM-GO ), for instance,\nunder smallness assumptions of initial data.\nFinally, it is also convenient to write down the standard fractional MH D system as follows\n\n\n∂tv+v·∇v+∇π=−ν(−∆)αv+B·∇B,\n∂tB+v·∇B=−µ(−∆)βB+B·∇v,\ndivv= divB= 0,(MHD)\nwhere (v,B) :Rd×(0,∞)→Rdandπ:Rd×(0,∞)→Rford∈ {2,3},β≥0 and the magnetic resistivity\nconstant µ >0. In three dimensions, it is well-known that by using some vector iden tities, (H-MHD) withκ= 0\nand (MHD) withβ= 1,µ=1\nσare equivalent to each other up to a modified pressure.\n1.2 The state of the art\nA. The case d= 2.Let us give a quick review on the study of ( NSM) in two dimensions with α= 1. Formally,\nits energy balance is given by (the same for ( NSM-SO ) and (NSM-GO ) in both cases d= 2 and d= 3)\n1\n2d\ndt/ba∇dbl(v,E,B)/ba∇dbl2\nL2+ν/ba∇dbl∇v/ba∇dbl2\nL2+1\nσ/ba∇dblj/ba∇dbl2\nL2= 0.\nThus, similar to in the case of the usual Navier-Stokes equations, w e could expect the existence of global energy\nsolutions (see [ 52,53]). However, it seems that this energy equality is not enough to obta in the existence of\nL2weak solutions, which is different to that of ( 2-NSM) in the two and three-dimentional cases as mentioned\npreviously. The main difficulty is the lack of compactness, due to the h yperbolicity of the Maxwell equations,\nwhich is needed to pass to the limit as n→ ∞of the term jn×Bn, especially for the one En×Bn, where\nnis the regularization parameter of a standard approximate system to (NSM) (for example, see the proof of\nTheorem 1.1). Therefore, higher regular data should be considered on the GWP issue. The first GWP result\nto (NSM) was obtained in [ 59] in the case where\n(v0,E0,B0)∈L2×Hs×Hsfors∈(0,1).\nIn addition, higher regular estimates are also provided in [ 59] in the case where ( v0,E0,B0)∈Hδ×Hs×Hs\nforδ≥0,s≥1,s−2< δ≤s6, see also [ 45] for another proof7and [30] for the case of bounded domains. The\nGWP is also obtained in [ 38] for small initial data satisfying8\n(v0,E0,B0)∈˙B0\n2,1×L2\nlog×L2\nlog,\nwhere we have the following relations ˙B0\n2,1⊂L2and∪s>0Hs⊂L2\nlog⊂L2. However, the LWP has not been\ncontructed for the above arbitrary large initial data. After that , the authors in [ 32] have been considered mild\nsolutions to ( NSM) and they obtained the LWP for possibly large initial data and the GWP for small initial\ndata under the assumption\n(v0,E0,B0)∈L2×L2\nlog×L2\nlog.\nHere, in the two previous results, in order to estimate the term E×Bcoming from j×Bin some homogeneous\nBesov spaces, the authors used the paraproduct estimate ( 7.1), and it is critical in two dimentions, thus the\nextra logarithmic regularityof ( E0,B0) is needed. Recently, the authors in [ 2] revisited ( NSM) in the case where\n6In the statement, the author assumed that δ≥0 ands≥1. However, it seems to us that he used conditions δ >0 ands >1\nduring the proof.\n7By using the standard Brezis-Gallouet inequality, the auth ors in [45] considered the case where δ=s= 2 and all the third\ncomponents is assumed to be zero, i.e., v,E,B,j :R2×(0,∞)→R2. However, the pure 2D flow assumption can be removed and\nthe assumption on the initial data can be improved when we con sider (NSM), see Theorem 1.1below.\n8The space ˙B0\n2,1is the usual homogeneous Besov space (see Appendix A) and L2\nlogis the set of tempered distributions fsatisfying\n/ba∇⌈blf/ba∇⌈bl2\nL2\nlog:=/summationdisplay\nq∈Z,q≤0/ba∇⌈bl˙∆qf/ba∇⌈bl2\nL2+/summationdisplay\nq∈Z,q>0q/ba∇⌈bl˙∆qf/ba∇⌈bl2\nL2<∞,\nwhere for each q∈Z,˙∆qis the homogeneous dyadic block (see Appendix A).\n4(v0,E0,B0)∈L2×Hs×Hsfors∈(0,1), as considered previously in [ 59], with providing further improvements,\nwhich include some c-independent estimates of ( v,E,B). That allowed them to investigate the asymptotic be-\nhavior as c→ ∞by proving the convergence of solutions to ( NSM) to that of the standard 21\n2-dimensional\nMHD equations, i.e., ( H-MHD) withα= 1 and κ= 0, in the sense of distributions.\nB. The case d= 3.As mentioned previously, the existence of energy solutions is unkno wn so far. We will\nshortly recall some results to ( NSM) in the three-dimensional case with α= 1. One of the first results was given\nin [38], where the authors constructed global small solutions with initial d ata\n(v0,E0,B0)∈˙B1\n2\n2,1×˙H1\n2×˙H1\n2.\nFor large initial data in some ℓ1weighted space in Fourier side, the authors in [ 40] have been provided the local\nin time existence of mild solutions. Moreover, by using the fact that t he damped-wave operator does not have\nany smoothing effect, they also showed these local solutions lost re gularity in some finite time. Later on, the\nabove result in [ 38] was extended in [ 32] in which either local large initial data solutions or global small intial\ndata ones was provided for initial data in the following space\n(v0,E0,B0)∈˙H1\n2×˙H1\n2×˙H1\n2.\nRecently, the existence of weak solutions was built in [ 2] for small initial data (the smallness assumption is\nrelated to only L2and˙Hsnorms) with\n(v0,E0,B0)∈L2×Hs×Hsfors∈/bracketleftbigg1\n2,3\n2/parenrightbigg\n.\nWe also note that time-periodic small solutions and their asymptotic s tability were investiagted in [ 39]. For\nfurther results to ( NSM) (and also to ( NSM-SO )) such as GWP for small data and LWP for possibly large data,\nloss of regularity, asymptotic behaviors, existence of global weak solutions with small data, global regularity\ncriteria, time periodic solutions and so on, we prefer the reader to [ 2,3,32,38,39,40,43,45,61,71,77].\n1.3 Main results\nFor the reader’s convenience, before going to the detailed statem ents, let us first summarize the main results in\nthe present paper as follows:\n1. The GWP of ( NSM) forν >0,α=d\n2withd= 2 and d= 3 in Theorems 1.1and1.2;\n2. The LWP for ν= 0 and the inviscid limit of ( NSM) ford∈ {2,3}in Theorem 1.3;\n3. The stability near a magnetohydrostatic equilibrium with a constan t (or equivalently bounded) magnetic\nfield of ( NSM) and (H-MHD) forα= 0,ν >0,κ≥0 andd∈ {2,3}in Theorems 1.4and1.5;\nwhich will be precisely presented in the following subsubsections, res pectively.\n1.3.1 Global well-posedness\nOur first result is aiming to obtain higher regular solutions to ( NSM) in two dimensions compared to those of\nin [2,59] with a direct and sightly different proof, which is stated as follows.\nTheorem 1.1 (Higher regular solutions in two dimensions) .Letd= 2,α= 1,c,ν,σ > 0and(v0,E0,B0)∈\n(Hδ×Hs×Hs)(R2)withdivv0= divB0= 0andδ,s∈[0,∞).\n(i). If(δ,s)satisfies one of the following assumptions\n(a)0< δ≤s≤1;\n(b)0< s <1ands≤δ≤2s;\n(c)s= 1and1≤δ <2;\n(d)s >1ands≤δ≤s+1;\n(e)s >1ands−1≤δ < s;\n(f)s∈(0,1)andδ= 0;\nthen there exists a unique global solution (v,E,B)to(NSM)satisfying for any T∈(0,∞)\nv∈L∞(0,T;Hδ)∩L2(0,T;Hδ+1)∩L2(0,T;L∞)and(E,B)∈L∞(0,T;Hs),\nand fort∈(0,T)\n/ba∇dblv(t)/ba∇dbl2\nHδ+/ba∇dbl(E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblv/ba∇dbl2\nHδ+1+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblj/ba∇dbl2\nHsdτ≤C(T,δ,ν,σ,s,v 0,E0,B0).\n5(ii). Ifδ= 0ands∈(0,1)then for any t∈(0,T), in addition to Part (i)-(f), there holds\nv∈L∞(t,T;Hδ′)∩L2(t,T;Hδ′+1)forδ′∈/braceleftigg\n[0,s]ifs∈(0,1),\n[0,1]ifs∈[1\n2,1).\n(iii). Ifδ= 0ands= 1then we have the same properties as Part (i)-(f) and for any t∈(0,T)andδ′∈[0,1]\nv∈L∞(t,T;Hδ′)∩L2(t,T;Hδ′+1)and(E,B)∈L∞(t,T;H1).\nFurthermore, v∈C([0,T];Hδ)and(E,B)∈Cweak([0,T];Hs).\nRemark 1.1. We add some comments to Theorem 1.1:\n1. Strategy of proof: The proof mainly based on the usual energy method with using the Brezis-Gallouet-\nWainger inequalty ( 2.2), a logarithmic Gronwall inequality in Lemma 7.2, some well-known commutator\nestimates and a carefully treated in each case. In addition, in the ca se of (i)−(f), we also borrow\nthe idea from [ 2] with a slightly different velocity decomposition. Furthermore, in ord er to obtain the\nuniqueness, we use the idea in [ 59] with a slightly different proof. The idea here will also be applied to\nthe three-dimensional case in Theorem 1.2.\n2. The range for initial data in Theorem 1.1consists of the dark and darker regions (the region A). We also\nprovide a new proof for the region A∩B. The results obtained in [ 59] are the segment from (0 ,0) to\n(1,0) without the end points (also in [ 2]) and the darker and darkest areas (the region B) without the\nlineδ=s−2 and without the end point (2 ,0) as well (and it seems also without the segments from (1 ,0)\nto (1,1) and from (1 ,0) to (2,0) excluding the end points, as mentioned previously).\nsδ\n1234512345\n0AA∩Bδ=s\nδ=s−2Bδ=s+1\nAA\nFigure 1: The relation between sandδin Theorem 1.1and [2,59].\n3. As mentioned previously in the introduction, the results in Theore ms1.1,1.2,1.3and1.4can be easily\nobtained to ( NSM-SO ) by using mainly the divergence-free condition of j.\n4. As it will be seen later that the estimates in Theorem 1.1arec-independent, from [ 2, Corollary 1.3])\nwe can prove that ( NSM) converges to ( H-MHD) withα= 1 and κ= 0 asc→ ∞in the sence of\ndistributions, see also the proof of Theorem 1.2-(ii). In addition, Theorem 1.1also holds in the case that\n−ν∆vis replaced by ( ν2∂22v1,ν1∂11v2,ν3∆v3) in (NSM) for any positive constants ν1,ν2andν3, by using\nthe divergence-free condition of v, for example see [ 46].\n5. It seems to be not clear to us how to obtain a priori estimates for initial data in the following cases: a)\nthe triangle (0 ,0)−(1,2)−(0,1) without the segment from (0 ,0) to (1,2) including the end points; b) the\nsegment from (1 ,0) to (2,0) including the end points; c) the line δ=s−2; and d) the domain is either\nabove the line δ=s+1 or under the one δ=s−2.\n6. Note that in Part ( iii), we are not able to close the estimate of ( v,E,B) in the whole time interval (0 ,T),\nbut only in ( t,T) for any t∈(0,T). Furthermore, higher regularity for ( v,E,B) after the initial time as\nParts (ii) and (iii) can be obtained to the cases from Part ( i)−(a) to Part ( i)−(e).\nOur second result focuses on the three-dimensional case, where we obtain the GWP of ( NSM) for possibly\ncritical exponent fractional Laplacian. More precisely, it is given as follows.\n6Theorem 1.2 (Possibly critical exponent in three dimensions) .Letd= 3,α=3\n2and(v0,E0,B0)∈(Hδ×\nHs×Hs)(R3)withδ,s∈[0,∞).\n(i)(Global well-posedness) If(δ,s)satisfies one of the following conditions\n(a)δ= 0ands∈(0,3\n2);\n(b)0< δ≤s≤3\n2;\n(c)0< s <3\n2ands≤δ≤2s;\n(d)s=3\n2and3\n2≤δ <3;\n(e)s >3\n2ands−3\n2≤δ≤s+3\n2;\nthen there exists a unique global solution (v,E,B)to(NSM)satisfying for any T∈(0,∞)\nv∈L∞(0,T;Hδ)∩L2(0,T;Hδ+3\n2)∩L2(0,T;L∞)and(E,B)∈L∞(0,T;Hs),\nand fort∈(0,T)\n/ba∇dblv(t)/ba∇dbl2\nHδ+/ba∇dbl(E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblv/ba∇dbl2\nHδ+3\n2+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblj/ba∇dbl2\nHsdτ≤C(T,δ,ν,σ,s,v 0,E0,B0).\nIn addition, v∈C([0,T];Hδ)and(E,B)∈Cweak([0,T];Hs).\n(ii)(The limit as c→ ∞)Letc >0and(vc\n0,Ec\n0,Bc\n0)∈L2×Hs×Hswiths∈(0,3\n2)satisfying divvc\n0=\ndivBc\n0= 0and asc→ ∞\n(vc\n0,Ec\n0,Bc\n0)⇀(¯v0,¯E0,¯B0)inL2×Hs×Hs\nfor some (¯v0,¯E0,¯B0)withdiv¯v0= div¯B0= 0. Then, there exists a sequence of global solutions (vc,Ec,Bc)\nto(NSM)withα=3\n2and(vc,Ec,Bc)|t=0= (vc\n0,Ec\n0,Bc\n0)given as in Part (i). In addition, up to an\nextraction of a subsequence, (vc,Bc)converges to (v,B)in the sense of distributions as c→ ∞, where\n(v,B)satisfies (H-MHD)withα=3\n2,κ= 0and(v,B)|t=0= (¯v0,¯B0). The same conclusion can be\nobtained for the initial data given by one of the parts from (i)−(b)to(i)−(e).\nRemark 1.2. We add some comments to Theorem 1.2:\n1. As mentioned previously, strategy of proof is similar to that of Th eorem1.1with using in addition some\nhomogeneous Besov-type maximal regularity estimate for the fra ctional heat equation, see Proposition 7.1\nin Appendix D in Section 7. Similar to Theorem 1.1, for the reader’s convenience, we will summarize the\nconditions of ( δ,s) as follows:\nsδ\n1234512345\n0δ=s\nδ=s−3/2δ=s+3/2\n3\n2\n3\n2\nFigure 2: The relation between sandδin Theorem 1.2.\nIn addition, as it can be seen from the proof below, similar results as in Parts (i) and (ii) also hold in\nthe case α >3\n2with a modified range of the initial data, for example in the case of Par ts (i)−(a), which\nshould be replaced by δ= 0 and s∈(0,α). Similar notes as Remark 1.1-4 and 5 are also applied here.\n72. We first explain why should we choose α=3\n2as a critical case. It is well-known that the fractional Navier-\nStokes (formally setting E=B= 0 in (NSM)) and (MHD) (in the case α=βandν=µ) equations have\nthe following scaling property\n(vλ,Bλ,πλ)(x,t)/ma√sto→(λ2α−1v,λ2α−1B,λ2(2α−1)π)(λx,λ2αt) for λ >0.\nIn addition, it can be seen formally that from the energy inequality\nE(v,B) := esssup\nt∈(0,∞)/ba∇dbl(v,B)(t)/ba∇dbl2\nL2(Rd)+2ν/ba∇dbl(Λαv,ΛαB)/ba∇dbl2\nL2(0,∞;L2(Rd))≤ /ba∇dbl(v,B)(0)/ba∇dbl2\nL2(Rd),\nwhich yields E(vλ,Bλ) =λ4α−2−dE(v,B). That is why it suggests to take α=d+2\n4withα= 1 (see [ 52])\nandα=5\n4(see [56,72]) in the cases of two and three dimentions, repsectively, to obtain t he existence\nand uniqueness of global in time weak solutions. A similar observation a lso holds for the Hall equations\n(H), where E(Bλ) =λ4α−4−dE(B) by using the scaling invariance Bλ(x,t)/ma√sto→λ2α−2B(λx,λ2αt), and\nthus it is suggested to take −(−∆)αBinstead of ∆ Bwithα=3\n2(see Proposition 1.1) andα=7\n4(see\n[68] and also Proposition 1.1) in the cases of d= 2 and d= 3, respectively. Unfortunately, it does not\nseem to be the case to ( NSM), where a similar scaling as above seems does not exist mainly due to th e\nappearance of the electric field. It seems to us that the most difficu lt term in ( NSM) is the Lorentz force\nonej×B=σcE×B+σ(v×B)×B, whichdrivesthe fluid. Comparedto the usualfractionalNavier-S tokes\nsystem, we have two new terms σcE×Bandσ(v×B)×B. While the latter one satisfies the usual scaling\nproperty, we have not known any similar thing for the former one, s ince no scaling information of Ehas\nbeen provided yet. To have a better understanding the situation, it is natural to focus more carefully on\nthe (E,B) system, i.e., the Maxwell equations9\n\n\n1\nc∂tE−∇×B=−j=−σ(cE+v×B),\n1\nc∂tB+∇×E= 0,divB= 0.(M)\nSimilar to ( NSM), we do not have any scaling property to ( M) even in the case v= 0. However, if we\nformally drop out the electric current field, i.e., the term on the right -hand side of the first equation (it\ncan be done formally either by taking σ= 0 or by setting v= 0 and ignoring the electric damping term\n−σcE) then in these cases ( M) is rewritten by\n\n\n1\nc∂tE−∇×B= 0,\n1\nc∂tB+∇×E= 0,divB= 0,(M’)\nwhich is invariant under the scaling ( Eλ,Bλ)/ma√sto→λβ(E,B)(λx,λt) for any real number β. Furthermore,\nby defining10\nE(E,B) := esssup\nt∈(0,∞)/ba∇dbl(E,B)(t)/ba∇dbl2\nL2(Rd)≤ /ba∇dbl(E,B)(0)/ba∇dbl2\nL2(Rd),\nit follows that E(Eλ,Bλ) =λ2β−dE(E,B), which suggests us to choose β=d\n2withβ=3\n2in the three-\ndimensional case. Coming back to the Lorentz force j×B, if we scale vλ(t,x)/ma√sto→λγv(λx,λt) for some\nreal number γto be determined later and use the scaling property of ( M’) for (E,B) then this force is\ninvariant under choosing γ= 0. Thus, in order to control the Lorentz force term by using the fractional\nLaplacianonewiththescaling( vλ,Eλ,Bλ)(x,t)/ma√sto→(v,λβE,λβB)(λx,λt), itsuggestsustotake α≥β=d\n2.\nTherefore, in two dimensions this also explains the GWP result given in T heorem1.1in the case α= 1\nwith a slightly stronger assumption on the initial data, i.e., ( v0,E0,B0)∈L2×Hs×Hsfor anys∈(0,1)\nand probably raisesa difficult problem in the case α∈(0,1). We should mention here that the critical case\nα=3\n2to (NSM) can be compared to the results in [ 73] in the three-dimensional case, where the author\nproved the GWP of ( MHD) forα≥5\n4,β >0 andα+β≥5\n2(in fact the author provided general results\ninddimensions, for similar partial fractional dissipation results, see als o [75]), so if we choose β= 1 then\nwe should take α≥3\n2. Moreover, the case ( α,β) = (3\n2,1) can be obtained by taking the limit as c→ ∞\nin which ( NSM) withα=3\n2converges to ( H-MHD) withα=3\n2andκ= 0, in the sense of distributions,\nsee Theorem 1.2-(ii).\n3. In the case δ=s >0, we should remark that a more general result has been obtained in [71]. More\nprecisely, theGWP of ( NSM)is providedwith replacing −(−∆)αvby−L2v, whereforsomenondecreasing\n9Under suitable assumptions on v, the existence and uniqueness of L2weak solutions ( E,B) to (M) can be established, see\nLemma 7.5.\n10The existence and uniqueness of L2weak solutions ( E,B) to (M’) can be found in Lemma 7.5.\n8sysmetric function g≥1, the operator Lis defined via Fourier transform as follows\nF(Lu)(ξ) :=|ξ|d\n2\ng(ξ)F(u)(ξ) with/integraldisplay∞\ne1\nslog(s)g2(s)ds=∞.\nThe above stronger conditions are inspirited by the similar weaker on es for the supercritical hyperdissi-\npative Navier-Stokes equations given in [ 6,66], where the first paper did not need the above logarithmic\nterm and improved the result in the second one, which also did not ass ume the logarithmic term but\nrequiring g4instead of g2. As mentioned previously, the critical case for the usual fraction al Navier-Stokes\nequations is α=d+2\n4. By choosing g= 1,δ=s >0 and either d= 2 ord= 3, the result in [ 71] reduces\nto Theorem 1.1or Theorem 1.2. However, they have not been explained about the choice of the ex ponent\nd\n2in the definition of Land have not been considered lower regularity data cases, for inst anceδ= 0 and\ns∈(0,d\n2). In addition, it is possible to obtain (at least) the existence of globa l weak solutions to ( NSM)\nforα∈(1,3\n2) and for small data by adapting the technique provided in [ 2, Theorem 1.1]. Furthermore, it\nis also natural to ask the two following questions: 1) Can the above lo garithmic term be dropped out as\nin [6,66]; and 2) Can the regularity of v0be reduced, namely, ( v0,E0,B0)∈L2×Hs×Hsfors∈(0,d\n2)\nford∈ {2,3}.\n4. Theorem 1.2-(ii) also says that the hyperbolicity of ( NSM) (due to the Maxwell equations) is weakly\ntransferred into the parabolicity of ( H-MHD) withκ= 0 asc→ ∞. See also Lemma 7.5, where under\nsuitable assumptions on the velocity, a similar result is obtained for th e Maxwell equations ( M), even for\nL2initial data. For more general estimates on ( M), we prefer to [ 2,32,33,38,39].\nAs mentioned previously in Remark 1.2, for the sake of completeness we will summarize the GWP of the\nHall system (i.e., ( H-MHD) without the fluid equations) in the two and three-dimensional case s as follows. This\nsystem is also known as the electron MHD equations.\nProposition 1.1. Letd∈ {2,3},B0∈Hs(Rd)withs∈[0,∞),κ,σ∈(0,∞)andT∈(0,∞). Assume that\nα≥3\n2ifd= 2andα≥7\n4ifd= 3. Then the Hall system\n∂tB=−1\nσ(−∆)αB−κ\nσ∇×((∇×B)×B)anddivB= 0, (H)\nhas a unique global solution B∈L∞(0,T;Hs)∩L2(0,T;Hs+α)satisfying for t∈(0,T)\n/ba∇dblB(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblB/ba∇dbl2\nHs+αdτ≤C(T,α,κ,σ,s,B 0).\nRemark 1.3. We add some remarks to Proposition 1.1as follows: It can be seen from the proof given in\nAppendix F in Section 7that similar results as Proposition 1.1(i.e., for initial data ( v0,B0)∈Hs(Rd) with\ns≥0) can be obtained when we couple ( H) together with the fractional Navier-Stokes equations for the v elocity\nfractional Laplacian −(−∆)αvforα≥d+2\n4(as [68] ford= 3 and for s >5\n2). It seems to us the technique\nin the proof of Proposition 1.1, which can also be adapted to obtain the GWP of ( MHD) with initial data\n(v0,B0)∈Hs(Rd) fors≥0 in the case either α≥d+2\n4,β >0 andα+β≥d+2\n2(as [73] fors >d\n2+ 1),\norα≥d+2\n4andβ≥d+2\n4(as [72] fors≥max{2α,2β}). Furthermore, we do not investigate the large-time\nbehavior here, but it can be easily obtained by adapting the Fourier s plitting method provided in [ 19,62,63],\nsee also the proof of Theorem 1.5. Finally, we should also remark that the local existence of strong so lutions to\nthe inviscid ( H-MHD) (i.e.,ν= 0) has been provided in [ 20] when replacing ∆ Bby−(−∆)αBforα >1\n2.\n1.3.2 Local well-posedness\nOur next result is concerned the LWP of ( NSM) in the inviscid case. That will allow us to study further either\nthe inviscid limit as ν→0 or the limit as c→ ∞in suitable frameworks. More precisely, the statement is given\nas follows.\nTheorem 1.3 (Local well-posedness, inviscid limit and the limit as c→ ∞).Letd∈ {2,3},c,σ >0and\n(v0,E0,B0)∈Hs(Rd)withdivv0= divB0= 0ands∈R,s >d\n2+1.\n(i)(Localwell-posedness) There exists aunique local solution (v,E,B)to(NSM)withν= 0and(v,E,B)|t=0=\n(v0,E0,B0)in(0,T0)for some T0=T0(σ,s,v0,E0,B0)>0such that (v,E,B)∈L∞(0,T0;Hs)satisfying\nfort∈(0,T0)\n/ba∇dbl(v,E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblj/ba∇dbl2\nHsdτ≤C(T0,σ,s,v 0,E0,B0).\n9(ii)(Inviscid limit) Letα= 1andν >0. Then there exists a sequence of solutions (vν,Eν,Bν)to(NSM)\nwith(vν,Eν,Bν)|t=0= (v0,E0,B0)given globally as in Theorem 1.1ifd= 2and locally as in Part (i) if\nd= 3, respectively. Moreover, for t∈(0,T0)and fors′∈[0,s)\n/ba∇dbl(vν−v,Eν−E,Bν−B)(t)/ba∇dblHs′≤νs−s′\nsC(T0,σ,s,v 0,E0,B0),\nwhere(v,E,B)is the unique local solution to (NSM)withν= 0and(v,E,B)|t=0= (v0,E0,B0)given as\nin Part (i).\n(iii)(The limit as c→ ∞)Letc >0and(vc\n0,Ec\n0,Bc\n0)∈Hssatisfying divvc\n0= divBc\n0= 0and asc→ ∞\n(vc\n0,Ec\n0,Bc\n0)⇀(¯v0,¯E0,¯B0)inHs\nfor some (¯v0,¯E0,¯B0)withdiv¯v0= div¯B0= 0. Then there exists a sequence of solutions (vc,Ec,Bc)to\n(NSM)withν= 0and(vc,Ec,Bc)|t=0= (vc\n0,Ec\n0,Bc\n0)given as in Part (i) in (0,T0)for some T0>0. In\naddition, up to an extraction of a subsequence, (vc,Bc)converges to (v,B)in the sense of distributions\nasc→ ∞, where(v,B)satisfies (H-MHD)withν=κ= 0and(v,B)|t=0= (¯v0,¯B0).\nRemark 1.4. We add some comments to Theorem 1.3as follows: The proofs of Parts ( i) and (ii) share the\nsame ideas as those of the LWP of Euler equations and the invicid limit fr om the Navier-Stokes equations to\nthe Euler system. The proof of Part ( iii) follows the ideas from [ 2] and Theorem 1.2-(ii).\n1.3.3 Stability and large-time behavior\nA. The case of (NSM).Let us now focus on the stability issue of ( NSM) around its stationary states. In this\ncase, if we look for a zero-velocity steady solution, i.e., ( v∗= 0,E∗,B∗,π∗) then it should satisfy\n∇π∗=σcE∗×B∗,∇×B∗=j∗=σcE∗,∇×E∗= 0 and div B∗= 0. (S-NSM)\nIndeed, by using the following well-known identity\nj∗×B∗= (∇×B∗)×B∗=B∗·∇B∗−1\n2∇|B∗|2,\nit follows that B∗also satisfies the following stationary Euler-type equations, which is also known as the\nmagnetohydrostatic system\nB∗·∇B∗+∇p∗= 0 and div B∗= 0 where p∗:=−1\n2|B∗|2−π∗. (MHS)\nIn three dimensions, solutions to ( MHS) either in bounded domains or on the torus are recently construct ed in\n[24] as infinite time limits of Voigt approximations11of viscous and non-resistive ( MHD) (i.e., with α= 1 and\nµ= 0). It is also believed that ( MHS) plays an important role in connection to the design of nuclear fusion\ndevices such as tokamaks and stellarators. There are several ex amples of ( v∗,E∗,B∗,π∗) to either ( S-NSM) or\n(MHS) such as for x∈Rd\nv∗=E∗= 0, B∗(x) = constant vector in R3andπ∗= constant;\nv∗=E∗= 0, B∗(x)∈ {(−x1,x2,0),(x2,x1,0),(0,x3,x2),...}andπ∗= constant .\nBy setting\n¯v:=v+v∗=v,¯E:=E+E∗,¯B:=B+B∗and ¯π:=π+π∗,\nit can be seen from ( NSM) for (¯v,¯E,¯B,¯π,¯j) and (S-NSM) that the perturbation ( v,E,B,π ) satisfies\n\n\n∂tv+v·∇v+∇(π+π∗) =−ν(−∆)αv+(j+j∗)×(B+B∗),\n1\nc∂tE−∇×(B+B∗) =−(j+j∗),\n1\nc∂tB+∇×E= 0,\nσ(c(E+E∗)+v×(B+B∗)) =j+j∗=¯j,\nσ(cE+v×B) =j,\nσ(cE∗+v×B∗) =j∗,\ndivv= divB= 0,(NSM*)\nwith the initial data is denoted by ( v,E,B)|t=0= (v0,E0,B0). We are now going to the statement, which is\ngiven as follows.\n11That means ( ∂tv,∂tB) is replaced by ( ∂t(−∆)α0v,∂t(−∆)β0B) for some α0,β0>0.\n10Theorem 1.4 (Velocity damping effect on the stability near a constant magnetic fie ldB∗).Letd∈ {2,3},\nα= 0andc,ν,σ > 0. Assume that (v0,E0,B0)∈Hs(Rd)withdivv0= divB0= 0ands∈R,s >d\n2+ 1.\nSuppose that B∗is a constant vector in R3withǫ∗:=/ba∇dblB∗/ba∇dblL∞. Then the following properties hold.\n(i)(Stability around a constant magnetic field B∗)There exists a constant ǫ0=ǫ0(ν,σ,s)>0such that if\n/ba∇dbl(v0,E0,B0)/ba∇dbl2\nHs≤ǫ2\n0then there is a unique global solution (v,E,B)to(NSM*)satisfying for t >0\n/ba∇dbl(v,E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblv/ba∇dbl2\nHs+1\nσ/ba∇dbl¯j/ba∇dbl2\nHsdτ≤2ǫ2\n0.\nIn addition, for any s′∈[0,s),s′′∈[1,s)and for some constant b0∈[0,ǫ0)ast→ ∞\n/ba∇dbl(v,E,¯j,j)(t)/ba∇dblHs′,/ba∇dblB(t)/ba∇dblL∞,/ba∇dblB(t)/ba∇dbl˙Hs′′,/ba∇dblB(t)/ba∇dblL2\nloc→0and/ba∇dblB(t)/ba∇dblL2→b0.\n(ii)(The limit as c→ ∞)Letc >0and(vc\n0,Ec\n0,Bc\n0)∈Hssatisfying divvc\n0= divBc\n0= 0,\n/ba∇dbl(vc\n0,Ec\n0,Bc\n0)/ba∇dbl2\nHs≤ǫ2\n1and(vc\n0,Ec\n0,Bc\n0)⇀(¯v0,¯E0,¯B0)inHsasc→ ∞\nfor some small ǫ1=ǫ1(ν,σ,s)>0and for some (¯v0,¯E0,¯B0)∈Hswithdiv¯v0= div¯B0= 0. Then, there\nexists a sequence of global solutions (vc,Ec,Bc)to(NSM*)withα= 0and(vc,Ec,Bc)|t=0= (vc\n0,Ec\n0,Bc\n0)\ngiven as in Part (i). In addition, up to an extraction of a subs equence, (vc,Bc)converges to (v,B)in the\nsense of distributions as c→ ∞, where(v,B)satisfies (H-MHD* )withκ= 0and(v,B)|t=0= (¯v0,¯B0).\nRemark 1.5. We add some comments to Theorem 1.4:\n1. Strategy of proof: The proof is based on the energy method wit h using some nice cancellation properties,\nwhich related to the constant vector B∗, which allow us to define a suitable energy form. We then obtain\na bound for this energy form locally in time in which by using the smallness of the initial data and a\nbootstrap argument, the global in time estimate can be established . Then, the large-time behavior can\nbe ontained by using the damping structure of the system.\n2. It seems to us that Theorem 1.4is the first stability result to ( NSM). The case α= 0 means that we have\na velocity damping term. Moreover, it can be seen from the last thre e relations in ( S-NSM) that ∆B∗= 0,\nand furthermore by Liouville’s theorem (see [ 29]), ifB∗is bounded then B∗is a constant vector. Thus,\nthe boundedness of B∗is equivalent to the constant one. Note that if B∗is a constant vector in R3then\nE∗=∇π∗= 0. If we choose ǫ0even smaller then the upper bound on the right-hand side of the main\ninequality in Part ( i) can be replaced by ǫ2\n0.\n3. The case that B∗is not a constant (unbounded) vector given in the previous example s, which is much\nmore complicated and will be considered in a forthcoming work. Similar t hings happen in the case α= 1\neven in the case that B∗is a constant vector in R3. The main difficulty in these cases is the control of\neither the weighted term j×B∗or/ba∇dblv/ba∇dblL2\ntL∞xin which at the moment it seems not clear to us.\n4. How to obtain an explicit rate of convergence as t→ ∞is not clear to us in this case, which is different\nto the case of ( H-MHD) in which under additional assumptions on the initial data, a logarithm ic rate is\nobtained, see Theorem 1.5below.\n5. As it can be seen from the proofofTheorem 1.4that we alsoobtain a similar bound in Part( i) as replacing\n¯jbyjwith a slight different unper bound such as C(c,σ,ǫ∗)ǫ2\n0instead of 2 ǫ2\n0. In addition, for r∈[0,s−1)\n/ba∇dbl(∂tv,∂tE,∂tB)(t)/ba∇dblHr→0 ast→ ∞.\nMoreover, if ∂tBdecays sufficiently fast in Hr(with an explicit rate of convergence, for example t−γfor\nsomeγ >1) then we can conclude by using the fundamental theorem of calcu lus in time that B→b\nstrongly in Hrast→ ∞for some b, see [7].\nB. The case of (H-MHD).Next, we will study the stability of ( H-MHD) around its zero-velocity stationary\nsolutions with a constant magnetic field B∗. In addition, we also provide the large-time behavior of the\ncorresponding perturbation ( v,B) inL2norm under suitable assumptions on the intial data. It is inspired\nby Theorem 1.4and also by the so-called magnetic relaxation phenomena of the non- resistive ( MHD) system\n(i.e., with µ= 0). Indeed, it is given formally in [ 60] as follows: If(v,B)is a smooth solution to (H-MHD)\nwithout magnetic diffusion and with κ= 0then/ba∇dblv(t)/ba∇dblL2→0andBconverges to a stationary Euler flow\nast→ ∞. Recent related results in this direction are obtained either on d-dimensional torus or in bounded\ndomains in [ 7,24]. It can be compared to ( NSM) in Theorem 1.4, where the time limit of the perturbation B\ninL2norm as t→ ∞is given by a constant b0∈[0,ǫ0). It is not clear to us, even in addition ( v0,E0,B0)∈L1,\nthat whether or not b0= 0. However, the L∞norm of Bconverges to zero at infinite time, but without an\n11explicit rate of decaying. Therefore, it seems to us that there is a g ap between the ”magnetic relaxation” of\n(NSM) and that of ( H-MHD) in the case α= 0, even in the latter case we should assume an extra condition\n(v0,B0)∈L1, but with an explicit asymptotic behavior. If ( v∗,B∗,p∗) is a stationary solution to ( H-MHD)\nwithv∗= 0 then for j∗:=∇×B∗\n∇π∗=j∗×B∗,1\nσ∆B∗=κ\nσ∇×(j∗×B∗) and div B∗= 0. (S-H-MHD)\nAs mentioned previously, if B∗is a solution to ( S-H-MHD ) thenB∗also satisfies ( MHS). Note that the\nexamples in Case A also satisfy ( S-H-MHD ). Moreover, it follows from ( H-MHD) for (¯v,¯B,¯π,¯j) withα= 1 and\n(S-H-MHD ) that the perturbation ( v:= ¯v−v∗,B:=¯B−B∗,π:= ¯π−π∗,j:=¯j−j∗) withj:=∇×Bsatisfies\n\n\n∂tv+v·∇v+∇π=−νv+j×(B+B∗)+j∗×B,\n∂tB−∇×(v×(B+B∗)) =1\nσ∆B−κ\nσ∇×(j×(B+B∗))−κ\nσ∇×(j∗×B),\ndivv= divB= 0,(H-MHD*)\nin which the initial data is given by ( v,B)|t=0= (v0,B0). In Theorem 1.4-(ii), we prove that ( NSM*) converges\nto (H-MHD* ) withk= 0 asc→ ∞in the sense of distributions. However, we are not able to prove the\nconvergence of Bto zero in L2norm, but in L∞one, and the rate of decaying in time of ( v,B) is implicit.\nThe next result shows that we can obtain an explicit rate of converg ence ast→ ∞for (v,B), which satisfies\n(H-MHD* ), under an additional assumption of initial data, even in the case κ≥0.\nTheorem 1.5 (A counterpart of Theorem 1.4).Letd∈ {2,3},α= 0,κ≥0,ν,σ >0and(v0,B0)∈Hs(Rd)\nwiths∈R,s >d\n2+1. Assume that B∗is a constant verctor in R3withǫ∗:=/ba∇dblB∗/ba∇dblL∞andπ∗is a constant\nfunction in Rd. There exists a constant ǫ0=ǫ0(κ,ν,σ,s)>0such that if /ba∇dbl(v0,B0)/ba∇dbl2\nHs≤ǫ2\n0then there is a\nunique global solution (v,B)to(H-MHD* )such that for t >0\n/ba∇dbl(v,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblv/ba∇dbl2\nHs+1\nσ/ba∇dbl∇B/ba∇dbl2\nHsdτ≤2ǫ2\n0,\nand fors′∈[0,s),s′′∈[1,s)ands′′′∈[0,s−2)\n/ba∇dblv(t)/ba∇dblHs′,/ba∇dblB(t)/ba∇dblL∞,/ba∇dblB(t)/ba∇dbl˙Hs′′,/ba∇dbl(∂tv,∂tB)(t)/ba∇dblHs′′′→0ast→ ∞.\nIn addition, if (v0,B0)∈L1then for t >0,s′∈[0,s)and for each m∈Nwithm≥2\n/ba∇dbl(v,B)(t)/ba∇dblHs′≤C(ǫ0,ǫ∗,κ,ν,m,σ,s,v 0,B0)log−(m−1)(s−s′)\n2s(e+m−1νt).\nRemark 1.6. We add some comments to Theorem 1.5:\n1. Strategy of proof: The proof is similar to that of Theorem 1.4. In addition, to obtain explicit rate of\ndecaying in time, we can apply the Fourier splitting method provided in [ 19,62,63] with an additional\nassumption on the initial data. However, there are new terms reta led toB∗, which should be controlled\nin a different way.\n2. In two dimensions, it is well-known that for either ( MHD) withα= 0 and β= 1 or ( H-MHD* ) with\nB∗= 0 and κ= 0, the GWP for large initial data has not been established yet. For la rge initial data and\nν= 0, the authors in [ 15] have been provided the existence and uniqueness of global solutio ns inHs(R2)\nfors∈R,s >2, to (MHD) forβ >1. Their idea can be adapted to ( H-MHD* ) in the case B∗= 0,\nν=κ= 0,d= 2, and with replacing ∆ Bby−(−∆)βBforβ >1.\n3. There are also several stability results to ( MHD) in two dimentions. In this case, the authors in [ 55,74]\nstudied the stability without the magetic diffusion term, and with eithe r Laplacian or damping velocity.\nIn these two papers, the authors considered the system of ( v,φ) instead of ( v,B), where B= (∂2φ,−∂1φ),\nand they investigated the stability of ( v,φ) around ( v∗,φ∗) = (0,x2) or equivalently of ( v,B) near (v∗=\n0,B∗= (1,0)). Recently, the authors in [ 44] improved the result in [ 74] by considering lower regular data.\nMore precisely, they proved the stability near B∗= (1,0) for the initial data in a rougher space than\nH4∩L1and large-time behavior in L2norm with an optimal decay rate for H5∩W2,1initial data.\n4. Note that the authors in [ 19] have been provided the temporal decay in time of energy solutions and also\nof higher regular ones to ( H-MHD) withα= 1 and d= 3. Here, since we do not focus on obtaining an\noptimal decay then the rate of decay can be improved in one way or a nother. In addition, in Remark 6.1\nwe point out that it is difficult to obtain polynomial decay rate when usin g the Fourier splitting method\nto (H-MHD* ) in three dimensions.\nThe rest of the paper is organized as follows: The proofs of Theore ms1.1,1.2,1.3,1.4and1.5, and\nProposition 1.1will be provided in Sections 2,3,4,5,6and7, respectively. Some technical tools are also\nrecalled and proved in the appendix given in Section 7.\n122 Proof of Theorem 1.1\nIn this section, we will provide a quite simple proof of Theorem 1.1, which is mainly relied on the standard\nenergy method with using the usual Brezis-Gallouet-Wainger inequa lity in the case δ >0 to bound the norm\n/ba∇dblv/ba∇dblL2\ntL∞x. We will also revisit the case δ= 0, i.e., ( v0,E0,B0)∈L2×Hs×Hsfors∈(0,1) by taking the idea\nfrom [2] with using a slightly different decomposition of the velocity to obtain a bound on /ba∇dblv/ba∇dblL2\ntL∞x.\nProof of Theorem 1.1-(i).The proof consists of three parts with several substeps in each p art as follows.\nPart I: Approximate system and local existence. Let us fix n∈N. Assume that ( v0,E0,B0)∈\nHδ×Hs×Hswith div v0= divB0= 0 and δ,s≥0. An approximate system of ( NSM) is taken by the\nfollowing form\n\nd\ndt/parenleftbigg\nvn,En\nc,Bn\nc/parenrightbigg\n= (Fn\n1,Fn\n2,Fn\n3)(vn,En,Bn),\ndivvn= divBn= 0,\n(vn,En,Bn)|t=0=Tn(v0,E0,B0),(2.1)\nwhere for jn=σ(cEn+Tn(vn×Bn)) andi∈ {1,2,3},Fn\niare given by\nFn\n1:=ν∆vn−Tn(P(vn·∇vn))+Tn(P(jn×Bn)), Fn\n2:=∇×Bn−jnandFn\n3:=−∇×En.\nHere,TnandPare the usual Fourier truncation operator and Leray projection12, respectively. For δ,s∈R\nwithδ,s≥0, we define the following functional spaces\nHs\nn:={h∈Hs: supp(F(h))⊆Bn},\nVs\nn:={h∈Hs\nn: divh= 0},\nand the mapping\nFn:Vδ\nn×Hs\nn×Vs\nn→Vδ\nn×Hs\nn×Vs\nn\n(vn,En,Bn)/ma√sto→Fn(vn,En,Bn) := (Fn\n1,Fn\n2,Fn\n3).\nThe space Vδ\nn×Hs\nn×Vs\nnis equipped with the following norm13\n/ba∇dbl(vn,En,Bn)/ba∇dbl2\nδ,s:=/ba∇dblvn/ba∇dbl2\nHδ+/ba∇dbl(En,Bn)/ba∇dbl2\nHs.\nIt can be checked that Fnis well-defined and is locally Lipschitz continuous as well. Then the Picard theorem\n(see [58, Theorem 3.1]) implies that there exists a unique solution ( vn,En,Bn)∈C1([0,Tn\n∗),Vδ\nn×Hs\nn×Vs\nn) to\n(2.1) for some Tn\n∗>0. In addition, if Tn\n∗<∞then (see [ 58, Theorem 3.3])\nlim\nt→Tn∗/parenleftbig\n/ba∇dblvn(t)/ba∇dbl2\nHδ+/ba∇dbl(En,Bn)(t)/ba∇dbl2\nHs/parenrightbig\n=∞.\nPart II: Global existence and uniform bound. In the following steps (from Step 1 to Step 13), in\norder to verify that Tn\n∗=∞, we will assume that Tn\n∗<∞and prove the following inequality\nesssup\nt∈(0,Tn∗)/parenleftbig\n/ba∇dblvn(t)/ba∇dbl2\nHδ+/ba∇dbl(En,Bn)(t)/ba∇dbl2\nHs/parenrightbig\n<∞,\nwhich leads to a contradiction with the analysis in Part I.\nStep 1: The case 0< δ≤s <1.It can be checked that if ( vn,En,Bn)∈Vδ\nn×Hs\nn×Vs\nnthen\nTn(vn,En,Bn,jn) = (vn,En,Bn,jn) in theL2sense. In the sequel, we will write only ( v,E,B,j ) instead of\n(vn,En,Bn,jn) for simplicity. The standard energy inequality to ( 2.1) is given by\n1\n2d\ndt/ba∇dbl(v,E,B)/ba∇dbl2\nL2+ν/ba∇dbl∇v/ba∇dbl2\nL2+1\nσ/ba∇dblj/ba∇dbl2\nL2= 0,\nwhich yields for t∈(0,Tn\n∗)\n/ba∇dbl(v,E,B)(t)/ba∇dbl2\nL2+2/integraldisplayt\n0ν/ba∇dbl∇v/ba∇dbl2\nL2+1\nσ/ba∇dblj/ba∇dbl2\nL2dτ≤ /ba∇dblTn(v0,E0,B0)/ba∇dbl2\nL2≤ /ba∇dbl(v0,E0,B0)/ba∇dbl2\nL2=:E2\n0.\n12As usual, the operators TnandPare defined by\nF(Tn(f))(ξ) := /BDBn(ξ)F(f)(ξ) forn∈R,n >0,ξ∈Rd,\nP(f) :=f+∇(−∆)−1divf.\nHere, /BDBnis the characteristic function of Bn, whereBnis the ball of radius ncentered at the origin.\n13Fors∈Randξ∈Rd,F(Js(f))(ξ) := (1 + |ξ|2)s\n2F(f)(ξ) and/ba∇⌈blf/ba∇⌈blHs:=/ba∇⌈blJsf/ba∇⌈blL2withH0≡L2.\n13Moreover, the ˙Hδ-˙Hsestimate is given by14\n1\n2d\ndt/parenleftbig\n/ba∇dblv/ba∇dbl2\n˙Hδ+/ba∇dbl(E,B)/ba∇dbl2\n˙Hs/parenrightbig\n+ν/ba∇dblv/ba∇dbl2\n˙Hδ+1+1\nσ/ba∇dblj/ba∇dbl2\n˙Hs=:3/summationdisplay\nk=1Ik,\nwhere for some ǫ∈(0,1), since s,δ∈(0,1) withδ≤s\nI1=/integraldisplay\nR2(j×B)·Λ2δvdx≤ǫν\n2/ba∇dblv/ba∇dbl2\n˙H2δ+1−s+C(ǫ,ν,s)/ba∇dblj/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\n˙Hs;\nI2=−/integraldisplay\nR2v·∇v·Λ2δvdx≤ǫν\n2/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblv/ba∇dbl2\n˙Hδ;\nI3=/integraldisplay\nR2Λsj·Λs(v×B)dx\n≤C(s)/ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dbl˙Hs(/ba∇dbl∇v/ba∇dblL2+/ba∇dblv/ba∇dblL∞)\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblB/ba∇dbl2\n˙Hs,\nwhere we used the well-known inequalities (see [ 5])\n/ba∇dblf/ba∇dblLp0≤C(p0,s0)/ba∇dblf/ba∇dbl˙Hs0 fors0∈[0,1),p0=2\n1−s0,\n/ba∇dblf/ba∇dbl˙Hs1≤C(s1,s2)/ba∇dblf/ba∇dblα0\nL2/ba∇dblf/ba∇dbl1−α0\n˙Hs2fors1,s2∈(0,∞),s1< s2,α0= 1−s1\ns2,\nand the following homogeneous Kato-Ponce type inequality (see [ 36]) for 1< pi,qi≤ ∞,i∈ {1,2},s0>0 and\n1\npi+1\nqi=1\n2\n/ba∇dblΛs0(fg)/ba∇dblL2≤C(s0,pi,qi)(/ba∇dblΛs0f/ba∇dblLp1/ba∇dblg/ba∇dblLq1+/ba∇dblf/ba∇dblLp2/ba∇dblΛs0g/ba∇dblLq2).\nTherefore, since 2 δ+1−s≤δ+1 and by choosing ǫ=1\n2\nd\ndt/parenleftbig\n/ba∇dblv/ba∇dbl2\n˙Hδ+/ba∇dbl(E,B)/ba∇dbl2\n˙Hs/parenrightbig\n+ν/ba∇dblv/ba∇dbl2\n˙Hδ+1+1\nσ/ba∇dblj/ba∇dbl2\n˙Hs≤C(σ,s)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblj/ba∇dbl2\nL2/parenrightbig\n/ba∇dblB/ba∇dbl2\n˙Hs\n+ν/ba∇dblv/ba∇dbl2\nL2+C(δ,ν)/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblv/ba∇dbl2\n˙Hδ.\nStep 2: The case δ=s= 1.Similar to the previous case, we obtain\n1\n2d\ndt/ba∇dbl(v,E,B)/ba∇dbl2\n˙H1+ν/ba∇dblv/ba∇dbl2\n˙H2+1\nσ/ba∇dblj/ba∇dbl2\n˙H1=:3/summationdisplay\nk=1Ik,\nwhereI2= 0,I1andI3are estimated as follows15\nI1=−/integraldisplay\nR2(j×B)·∆vdx\n≤C/ba∇dblj/ba∇dbl1\n2\nL2/ba∇dbl∇j/ba∇dbl1\n2\nL2/ba∇dblB/ba∇dbl1\n2\nL2/ba∇dbl∇B/ba∇dbl1\n2\nL2/ba∇dbl∆v/ba∇dblL2\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\n˙H1+ǫν/ba∇dblv/ba∇dbl2\n˙H2+C(ǫ,ν,σ)/ba∇dblj/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\nL2/ba∇dbl∇B/ba∇dbl2\nL2;\nI3=/integraldisplay\nR2∇j:∇(v×B)dx\n≤C/ba∇dbl∇j/ba∇dblL2/parenleftig\n/ba∇dbl∇v/ba∇dbl1\n2\nL2/ba∇dbl∆v/ba∇dbl1\n2\nL2/ba∇dblB/ba∇dbl1\n2\nL2/ba∇dbl∇B/ba∇dbl1\n2\nL2+/ba∇dblv/ba∇dblL∞/ba∇dbl∇B/ba∇dblL2/parenrightig\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\n˙H1+ǫν/ba∇dblv/ba∇dbl2\n˙H2+C(ǫ,ν,σ)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblB/ba∇dbl2\n˙H1,\nwhich yields by choosing ǫ=1\n4\nd\ndt/ba∇dbl(v,E,B)/ba∇dbl2\n˙H1+ν/ba∇dblv/ba∇dbl2\n˙H2+1\nσ/ba∇dblj/ba∇dbl2\n˙H1≤C(ν,σ)/parenleftbig\n1+/ba∇dblB/ba∇dbl2\nL2/parenrightbig/parenleftbig\n/ba∇dblj/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblB/ba∇dbl2\n˙H1.\nStep 3: Conclusion of Steps 1 and 2. Since 0< δ≤s≤1, by using the energy inequality, we collect\nthe main estimates in the two previous steps as follows\nd\ndtYδ,s+ν/ba∇dblv/ba∇dbl2\nHδ+1+1\nσ/ba∇dblj/ba∇dbl2\nHs≤C1GYδ,s+/bracketleftbigg1\n2C2/ba∇dblB/ba∇dblHs/ba∇dblv/ba∇dblH1/parenleftbigg\n1+log1\n2/parenleftbigg/ba∇dblv/ba∇dblHδ+1\n/ba∇dblv/ba∇dblH1/parenrightbigg/parenrightbigg/bracketrightbigg2\n,\n14Theusualfractionalderivative operator isgiven by F(Λs(f))(ξ) :=|ξ|sF(f)(ξ)forξ∈R2,s∈R. Recallthat /ba∇⌈blf/ba∇⌈bl˙Hs:=/ba∇⌈blΛsf/ba∇⌈blL2\nand/ba∇⌈blf/ba∇⌈bl2\nHs≈ /ba∇⌈blΛsf/ba∇⌈bl2\nL2+/ba∇⌈blf/ba∇⌈bl2\nL2.\n15Here,A:B:=/summationtext\n1≤i,j≤3aijbijfor two matrices A=aijandB=bij.\n14whereC1(δ,ν,σ,s),C2(δ,σ,s)>0, and for t∈(0,Tn\n∗)\nYδ,s(t) :=/ba∇dblv(t)/ba∇dbl2\nHδ+/ba∇dbl(E,B)(t)/ba∇dbl2\nHsandG(t) :=/parenleftbig\n1+/ba∇dblB(t)/ba∇dbl2\nL2/parenrightbig/parenleftbig\n1+/ba∇dblj(t)/ba∇dbl2\nL2+/ba∇dbl∇v(t)/ba∇dbl2\nL2/parenrightbig\n.\nHere, in order to to bound the norm /ba∇dblv/ba∇dblL∞, we also used the well-known Brezis-Gallouet-Wainger inequality\nin the following form (for example, see [ 12] fors0= 1 and d= 2; see [ 14] fors0>max{d\n2−1,0}andd≥1, and\nsee [37] fors0∈(0,1) andd= 2) with s0=δandd= 2\n/ba∇dblf/ba∇dblL∞(Rd)≤C(s0)/ba∇dblf/ba∇dblHd\n2(Rd)/parenleftigg\n1+log1\n2/parenleftigg\n1+/ba∇dblf/ba∇dblHs0+1(Rd)\n/ba∇dblf/ba∇dblHd\n2(Rd)/parenrightigg/parenrightigg\nf/\\e}atio\\slash= 0. (2.2)\nBy applying the following inequality (see [ 37]) forα,β,γ > 0 and log+(a) := max {log(a),0},a >0\nβ(1+log+(γ))1\n2≤αγ+β/parenleftbigg\n1+log/parenleftbigg\n1+β\nα/parenrightbigg/parenrightbigg1\n2\nto the case where\nα=√ν\n2/ba∇dblv/ba∇dblH1, β=C2/ba∇dblB/ba∇dblHs/ba∇dblv/ba∇dblH1andγ=/ba∇dblv/ba∇dblHδ+1\n/ba∇dblv/ba∇dblH1,\nwe find that\nR:=C2\n2/ba∇dblB/ba∇dblHs/ba∇dblv/ba∇dblH1/parenleftbigg\n1+log1\n2/parenleftbigg/ba∇dblv/ba∇dblHδ+1\n/ba∇dblv/ba∇dblH1/parenrightbigg/parenrightbigg\n≤C2/ba∇dblB/ba∇dblHs/ba∇dblv/ba∇dblH1/parenleftbigg\n1+log/parenleftbigg/ba∇dblv/ba∇dblHδ+1\n/ba∇dblv/ba∇dblH1/parenrightbigg/parenrightbigg1\n2\n≤√ν\n2/ba∇dblv/ba∇dblHδ+1+C2/ba∇dblB/ba∇dblHs/ba∇dblv/ba∇dblH1/parenleftbigg\n1+log/parenleftbigg\n1+2C2√ν/ba∇dblB/ba∇dblHs/parenrightbigg/parenrightbigg1\n2\n,\nwhich yields\nR2≤ν\n2/ba∇dblv/ba∇dbl2\nHδ+1+2C2\n2/ba∇dblB/ba∇dbl2\nHs/ba∇dblv/ba∇dbl2\nH1/parenleftbigg\n1+log/parenleftbigg\n1+C2ν√\n2/ba∇dblB/ba∇dblHs/parenrightbigg/parenrightbigg\nand\nd\ndtYδ,s+ν\n2/ba∇dblv/ba∇dbl2\nHδ+1+1\nσ/ba∇dblj/ba∇dbl2\nHs≤C1GYδ,s+C(δ,ν,σ,s)/ba∇dblv/ba∇dbl2\nH1(1+log(1+ Yδ,s))Yδ,s.\nTherefore, it follows from Lemma 7.2that\nYδ,s(t)≤exp{(log(e+Yδ,s(0))+(1+ Tn\n∗)C(E0,ν,σ,s))exp{(1+Tn\n∗)C(E0,δ,ν,σ,s)}},\nwhich gives us the conclusion in Steps 1 and 2. In addition, since v∈L2\ntHδ+1\nxforδ >0, it implies that\nv∈L2\ntL∞\nx. We should remark here that in Steps 1 and 2, we obtain the double ex ponential bound in time, i.e.,\nin the form of Cexp{CTn\n∗exp{CTn\n∗}}for some constant Cdepending on the parameters and the intial data.\nHowever, if we use directly Step 14 below in these two steps then the bound can be given in the form of either\nCexp{CTn\n∗}orC(Tn\n∗)C.\nStep 4: The case δ∈(0,1)ands= 1.By applying Step 1 (the case δ=s), we are able to close the Hδ\nestimate of ( v,E,B), in particular\n/ba∇dblv/ba∇dblL2\ntHδ+1\nx≤C(Tn\n∗,δ,ν,σ,v 0,E0,B0).\nIt remains to obtain the H1estimate of ( E,B). It can be seen that\n1\n2d\ndt/ba∇dbl(E,B)/ba∇dbl2\nH1+1\nσ/ba∇dblj/ba∇dbl2\nH1=:I31+I32,\nwhere for some ǫ∈(0,1), since δ∈(0,1)\nI31=/integraldisplay\nR2j·(v×B)dx≤ǫ\nσ/ba∇dblj/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dblB/ba∇dbl2\nL2/ba∇dbl∇B/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dblv/ba∇dbl2\nL2/ba∇dbl∇v/ba∇dbl2\nL2;\nI32=/integraldisplay\nR2∇j:∇(v×B)dx\n15≤C(δ)/ba∇dbl∇j/ba∇dblL2/parenleftig\n/ba∇dbl∇v/ba∇dbl\nL2\n1−δ/ba∇dblB/ba∇dblL2\nδ+/ba∇dblv/ba∇dblL∞/ba∇dbl∇B/ba∇dblL2/parenrightig\n≤C(δ)/ba∇dbl∇j/ba∇dblL2/parenleftbig\n/ba∇dblΛδ+1v/ba∇dblL2/ba∇dblΛ1−δB/ba∇dblL2+/ba∇dblv/ba∇dblHδ+1/ba∇dbl∇B/ba∇dblL2/parenrightbig\n≤ǫ\nσ/ba∇dbl∇j/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dblv/ba∇dbl2\nHδ+1/ba∇dblB/ba∇dbl2\nH1.\nBy choosing ǫ=1\n2, it follows that\nd\ndt/ba∇dbl(E,B)/ba∇dbl2\nH1+1\nσ/ba∇dblj/ba∇dbl2\nH1≤C(σ)/ba∇dblv/ba∇dbl2\nL2/ba∇dbl∇v/ba∇dbl2\nL2+C(σ)/parenleftbig\n/ba∇dblB/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nHδ+1/parenrightbig\n/ba∇dblB/ba∇dbl2\nH1,\nwhich is closable. Thus, the conclusion follows.\nStep 5: The case1\n2≤s <1ands < δ≤1.We first focus on obtaining the Hsestimate for ( E,B). Since\nδ > s, as in Step 1 (for the case δ=s) we are able to bound the norms\n/ba∇dblv/ba∇dblL∞\ntHs\nx∩L2\ntHs+1\nx,/ba∇dbl(E,B)/ba∇dblL∞\ntHsxand/ba∇dblj/ba∇dblL2\ntHsx.\nIt remains to bound the norm /ba∇dblv/ba∇dblL∞\ntHδx∩L2\ntHδ+1\nx. It can be seen that\n1\n2d\ndt/ba∇dblv/ba∇dbl2\n˙Hδ+ν/ba∇dblv/ba∇dbl2\n˙Hδ+1=:I1+I2,\nwhereI2is bounded as in Step 1 (for δ∈(0,1)) andI2= 0 (for δ= 1), and since s∈[1\n2,1) andδ≤1, for some\nǫ∈(0,1)\nI1=/integraldisplay\nR2(j×B)·Λ2δvdx\n≤C(s)/ba∇dblj/ba∇dbl˙H1−s/ba∇dblB/ba∇dbl˙Hs/ba∇dblΛ2δv/ba∇dblL2\n≤C(ǫ,ν,s)/ba∇dblj/ba∇dbl2\nH1−s/ba∇dblB/ba∇dbl2\nHs+ǫν/parenleftbig\n/ba∇dblv/ba∇dbl2\n˙Hδ+1+/ba∇dblv/ba∇dbl2\nL2/parenrightbig\n≤C(ǫ,ν,s)/ba∇dblj/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs+ǫν/parenleftbig\n/ba∇dblv/ba∇dbl2\n˙Hδ+1+/ba∇dblv/ba∇dbl2\nL2/parenrightbig\n.\nIt implies the closable of the Hδestimate of vby choosing ǫ=1\n4.\nStep 6a: The case1\n2< s <1and1< δ≤2s.In this case, we can estimate ( E,B) exactly as in Step 5.\nWe now focus on the estimates of I1andI2. Firstly, since s∈(1\n2,1) andδ∈(1,2s]\nI1=/integraldisplay\nR2Λδ−1(j×B)·Λδ+1vdx\n≤C(δ)(/ba∇dblj/ba∇dbl˙Hδ−s/ba∇dblB/ba∇dbl˙Hs+/ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dbl˙Hδ−s)/ba∇dblv/ba∇dbl˙Hδ+1\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/ba∇dblj/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs.\nSecondly, for σ′:=δ−1∈(0,1) we find that\nI2=−/integraldisplay\nR2Λσ′(v·∇v)·Λδ+1vdx\n≤C(δ)/ba∇dblΛδ+1v/ba∇dblL2×/braceleftigg\n/ba∇dblΛσ′v/ba∇dblL4/ba∇dbl∇v/ba∇dblL4+/ba∇dblv/ba∇dbl\nL4\n1−2σ′/ba∇dblΛσ′∇v/ba∇dbl\nL4\n1+2σ′ifσ′∈(0,1\n2),\n/ba∇dblΛσ′v/ba∇dblL4/ba∇dbl∇v/ba∇dblL4+/ba∇dblv/ba∇dblL6/ba∇dblΛσ′∇v/ba∇dblL3 ifσ′∈[1\n2,1).\nMoreover,\n/ba∇dblΛσ′v/ba∇dblL4,/ba∇dblv/ba∇dbl\nL4\n1−2σ′≤C(δ)/ba∇dblΛσ′+1\n2v/ba∇dblL2≤C(δ)/ba∇dblv/ba∇dbl3\n2(δ+1)\nL2/ba∇dblΛδ+1v/ba∇dbl2s−1\n2(δ+1)\nL2,\n/ba∇dbl∇v/ba∇dblL4,/ba∇dblΛσ′∇v/ba∇dbl\nL4\n1+2σ′≤C(δ)/ba∇dblΛ1\n2∇v/ba∇dblL2≤C(δ)/ba∇dbl∇v/ba∇dbl2δ−1\n2δ\nL2/ba∇dblΛδ+1v/ba∇dbl1\n2δ\nL2,\n/ba∇dblv/ba∇dblL6≤C/ba∇dblv/ba∇dbl1\n3\nL2/ba∇dbl∇v/ba∇dbl2\n3\nL2,\n/ba∇dblΛσ′∇v/ba∇dblL3≤C(δ)/ba∇dblΛσ′+1\n3∇v/ba∇dblL2≤C(δ)/ba∇dbl∇v/ba∇dbl2\n3δ\nL2/ba∇dblΛδ+1v/ba∇dbl3δ−2\n3δ\nL2,\nwhich yields\nI2≤C(δ)/ba∇dblv/ba∇dbl3\n2(δ+1)\nL2/ba∇dbl∇v/ba∇dbl2δ−1\n2δ\nL2/ba∇dblΛδ+1v/ba∇dbl4δ2+2δ+1\n2(δ+1)δ\nL2+C(δ)/ba∇dblv/ba∇dbl1\n3\nL2/ba∇dbl∇v/ba∇dbl2δ+2\n3δ\nL2/ba∇dblΛδ+1v/ba∇dbl6δ−2\n3δ\nL2.\nIn addition, since 1 < δthen an application of Step 5 (with δ= 1) gives us the bound on /ba∇dbl∇v/ba∇dblL∞\ntL2x. It can be\nseen that since δ >1\n4δ2+2δ+1\n2(δ+1)δ,6δ−2\n3δ<2,\n16which implies the closable of the Hδestimate as in Step 5 by using Young inequalty with ǫ=1\n6.\nStep 6b: The case 0< s <1\n2ands < δ≤2s.Similar to the previous case, we only need to focus on the\nestimate of v. Indeed, I2can be bounded as in Step 1 since δ∈(0,1). In addition, for some ǫ∈(0,1), since\ns∈(0,1),s < δ≤2sand 1−(δ−s),δ−s\n2∈(0,1)\nI1=/integraldisplay\nR2[(Λs(j×B)−Λsj×B−j×ΛsB)+Λsj×B+j×ΛsB]·Λ2δ−svdx=:3/summationdisplay\nk=1I1k,\nI11≤ /ba∇dblΛs(j×B)−Λsj×B−j×ΛsB/ba∇dbl\nL2\n2+s−δ/ba∇dblΛ2δ−sv/ba∇dbl\nL2\nδ−s\n≤C(δ,s)/ba∇dblΛ4s−δ\n4j/ba∇dbl\nL4\n2+s−δ/ba∇dblΛδ\n4B/ba∇dbl\nL4\n2+s−δ/ba∇dblv/ba∇dbl˙Hδ+1\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν,s)/ba∇dblΛ2s+δ\n4j/ba∇dbl2\nL2/ba∇dblΛ3δ−2s\n4B/ba∇dbl2\nL2\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν,s)/ba∇dblj/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs,\nI12≤ /ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dbl\nL2\n1−(δ−s)/ba∇dblΛ2δ−sv/ba∇dbl\nL2\nδ−s\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,s)/ba∇dblj/ba∇dbl2\n˙Hs/ba∇dblB/ba∇dbl2\n˙Hs,\nI13≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,s)/ba∇dblj/ba∇dbl2\n˙Hs/ba∇dblB/ba∇dbl2\n˙Hs,\nwhere we used the well-known Kenig-Ponce-Vega commutator estim ate (see [ 48])\n/ba∇dblΛs0(fg)−gΛs0f−fΛs0g/ba∇dblLp0(Rd)≤C(d,p0,p01,p02,s0,s01,s02)/ba∇dblΛs01f/ba∇dblLp01(Rd)/ba∇dblΛs02g/ba∇dblLp02(Rd),(2.3)\nfor 0< s0,s01,s02<1 andp0,p01,p02∈(1,∞) satisfying s0=s01+s02and1\np0=1\np01+1\np02.\nStep 7: The case s= 1and1< δ <2.In this case, we only need to bound I1, other terms can be done\nexactly as in Step 5 (the estimate of ( E,B)) and Step 6a (the estimate of I2). Indeed, similar to Step 6a, since\nδ∈(1,2) withδ−1∈(0,1) for some ǫ∈(0,1)\nI1=/integraldisplay\nR2Λδ−1(j×B)·Λδ+1vdx\n≤C(δ)/parenleftig\n/ba∇dblΛδ−1j/ba∇dbl\nL2\nδ−1/ba∇dblB/ba∇dbl\nL2\n2−δ+/ba∇dblj/ba∇dbl\nL2\nδ−1/ba∇dblΛδ−1B/ba∇dbl\nL2\n2−δ/parenrightig\n/ba∇dblv/ba∇dbl˙Hδ+1\n≤C(δ)(/ba∇dblj/ba∇dbl˙H1/ba∇dblB/ba∇dbl˙Hδ−1+/ba∇dblj/ba∇dbl˙Hδ−1/ba∇dblB/ba∇dbl˙H1)/ba∇dblv/ba∇dbl˙Hδ+1\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/ba∇dblj/ba∇dbl2\nH1/ba∇dblB/ba∇dbl2\nH1.\nStep 8: The case δ=s >1(revisited). This case has been treated in [ 59] with a different proof, but\nto make the present work self-contained, we revisit this case with p roviding our simple proof. The proof of this\ncase is also useful for later use, for instance in Steps 9, 10, 12 and 13 below, which are also included in the main\naims of this paper. It can be seen that\n1\n2d\ndt/ba∇dbl(v,E,B)/ba∇dbl2\n˙Hs+ν/ba∇dblv/ba∇dbl2\n˙Hs+1+1\nσ/ba∇dblj/ba∇dbl2\n˙Hs=:3/summationdisplay\nk=1Ik,\nwhere for some ǫ∈(0,1), since s >1\nI1=/integraldisplay\nR2Λs(j×B)·Λsvdx\n≤C(s)(/ba∇dblj/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs+/ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dblL∞)/ba∇dblv/ba∇dbl˙Hs\n≤C(s)/parenleftig\n/ba∇dblj/ba∇dbls−1\ns\nL2/ba∇dblj/ba∇dbl1\ns\n˙Hs/ba∇dblB/ba∇dbl˙Hs+/ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dbls−1\ns\nL2/ba∇dblB/ba∇dbl1\ns\n˙Hs/parenrightig\n/ba∇dbl∇v/ba∇dbl1\ns\nL2/ba∇dblv/ba∇dbls−1\ns\n˙Hs+1\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\nHs+2ǫν\n3/ba∇dblv/ba∇dbl2\n˙Hs+1+C(ǫ,ν,σ,s)/parenleftig\n/ba∇dblj/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblB/ba∇dbl2(s−1)\nL2/ba∇dbl∇v/ba∇dbl2\nL2/parenrightig\n/ba∇dblB/ba∇dbl2\nHs;\nI2=−/integraldisplay\nR2Λs(v·∇v)·Λsvdx\n≤C(s)/parenleftbig\n/ba∇dblΛsv/ba∇dblL4/ba∇dbl∇v/ba∇dblL4+/ba∇dblv/ba∇dblL∞/ba∇dblΛs+1v/ba∇dblL2/parenrightbig\n/ba∇dblΛsv/ba∇dblL2\n≤C(s)/parenleftig\n/ba∇dblΛs−1\n2∇v/ba∇dblL2/ba∇dblΛ1\n2∇v/ba∇dblL2+/ba∇dblv/ba∇dblL∞/ba∇dblΛs+1v/ba∇dblL2/parenrightig\n/ba∇dblΛsv/ba∇dblL2\n≤C(s)/parenleftbig\n/ba∇dblΛs+1v/ba∇dblL2/ba∇dbl∇v/ba∇dblL2+/ba∇dblv/ba∇dblL∞/ba∇dblΛs+1v/ba∇dblL2/parenrightbig\n/ba∇dblΛsv/ba∇dblL2\n≤2ǫν\n3/ba∇dblv/ba∇dbl2\n˙Hs+1+C(ǫ,s,ν)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblv/ba∇dbl2\n˙Hs;\nI3=/integraldisplay\nR2Λsj·Λs(v×B)dx\n17≤C(s)/ba∇dblj/ba∇dbl˙Hs(/ba∇dblB/ba∇dbl˙Hs/ba∇dblv/ba∇dblL∞+/ba∇dblB/ba∇dblL∞/ba∇dblv/ba∇dbl˙Hs)\n≤C(s)/ba∇dblj/ba∇dbl˙Hs/ba∇dblv/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs+C(s)/ba∇dblj/ba∇dbl˙Hs/ba∇dblB/ba∇dbls−1\ns\nL2/ba∇dblB/ba∇dbl1\ns\n˙Hs/ba∇dbl∇v/ba∇dbl1\ns\nL2/ba∇dblv/ba∇dbls−1\ns\n˙Hs+1\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\nHs+2ǫν\n3/ba∇dblv/ba∇dbl2\n˙Hs+1+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nL∞/ba∇dblB/ba∇dbl2\nHs+C(ǫ,σ,s)/ba∇dblB/ba∇dbl2(s−1)\nL2/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\nHs,\nhere we used the following Agmon-type inequality (its simple proof can be found in Appendix B in Section 7)\n/ba∇dblf/ba∇dblL∞≤C(s0)/ba∇dblf/ba∇dbls0−1\ns0\nL2/ba∇dblf/ba∇dbl1\ns0\n˙Hs0fors0>1. (2.4)\nBy choosing ǫ=1\n4and using the energy estimate, it follows that\nd\ndtYs+ν/ba∇dblv/ba∇dbl2\nHs+1+1\nσ/ba∇dblj/ba∇dbl2\nHs≤C(ν,σ,s)GsYs+C(ν,σ,s)/ba∇dblv/ba∇dbl2\nH1/parenleftbigg\n1+log/parenleftbigg\n1+Ys\n/ba∇dblv/ba∇dbl2\nH1/parenrightbigg/parenrightbigg\nYs,\nwhere for t∈(0,Tn\n∗)\nYs(t) :=/ba∇dbl(v,E,B)(t)/ba∇dbl2\nHsandGs(t) :=/parenleftig\n1+/ba∇dblB(t)/ba∇dbl2(s−1)\nL2/parenrightig/parenleftbig\n1+/ba∇dblj(t)/ba∇dbl2\nL2+/ba∇dbl∇v(t)/ba∇dbl2\nL2/parenrightbig\n,\nand here in order to bound /ba∇dblv/ba∇dblL∞, we also used ( 2.2) withs0=s−1>0 andd= 2. It can be seen that the\nabove estimate of Ysthat\nd\ndtYs≤C(ν,σ,s)GsYs+C(ν,σ,s)/ba∇dblv/ba∇dbl2\nH1Ys/parenleftbig\n1+log(/ba∇dblv/ba∇dbl2\nH1+Ys)−log(/ba∇dblv/ba∇dbl2\nH1)/parenrightbig\n≤C(ν,σ,s)GsYs+C(ν,σ,s)/ba∇dblv/ba∇dbl2\nH1Ys(1+log(1+ Ys)),\nwhere we used the fact that |xlog(x)| ≤exp{−1}forx∈(0,1). Therefore, for t∈(0,Tn\n∗) Lemma 7.2gives us\nYs(t)≤exp{(log(e+Ys(0))+(1+ Tn\n∗)C(E0,ν,σ,s))exp{(1+Tn\n∗)C(E0,ν,σ,s)}}.\nStep 9: The case s >1ands < δ < s +1.Sinceδ > s, we are able to close the Hsestimate of ( v,E,B)\nas in Step 8. It remains to focus on the Hδestimate of v. Moreover, it can be seen that I2can be bounded\nexactly as the previous step with replacing sbyδ, i.e.,\nI2=−/integraldisplay\nR2Λδ(v·∇v)·Λδvdx≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblv/ba∇dbl2\n˙Hδ.\nWe continue with the bound I1as follows. Since δ∈(s,s+1), we can write δ=s+ǫ0forǫ0∈(0,1) and find\nthat since δ >1\nI1=/integraldisplay\nR2Λδ−1(j×B)·Λδ+1vdx\n≤C(ǫ0)/parenleftig\n/ba∇dblΛδ−1j/ba∇dbl\nL2\nǫ0/ba∇dblB/ba∇dbl\nL2\n1−ǫ0+/ba∇dblj/ba∇dbl\nL2\n1−ǫ0/ba∇dblΛδ−1B/ba∇dbl\nL2\nǫ0/parenrightig\n/ba∇dblΛδ+1v/ba∇dblL2\n≤C(ǫ0)/parenleftbig\n/ba∇dblΛδ−ǫ0j/ba∇dblL2/ba∇dblΛǫ0B/ba∇dblL2+/ba∇dblΛǫ0j/ba∇dblL2/ba∇dblΛδ−ǫ0B/ba∇dblL2/parenrightbig\n/ba∇dblΛδ+1v/ba∇dblL2\n≤C(ǫ0)(/ba∇dblj/ba∇dbl˙Hδ−ǫ0/ba∇dblB/ba∇dbl˙Hǫ0+/ba∇dblj/ba∇dbl˙Hǫ0/ba∇dblB/ba∇dbl˙Hδ−ǫ0)/ba∇dblv/ba∇dbl˙Hδ+1\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ0,ǫ,ν)/ba∇dblj/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs.\nTherefore, by choosing ǫ=1\n4as in Step 5, the conclusion follows.\nStep 10: The case s >1andδ=s+1.This case is very similar to Step 9. We only need to bound I1\nas follows\nI1=/integraldisplay\nR2Λs(j×B)·Λδ+1vdx\n≤C(s)(/ba∇dblΛsj/ba∇dblL2/ba∇dblB/ba∇dblL∞+/ba∇dblj/ba∇dblL∞/ba∇dblΛsB/ba∇dblL2)/ba∇dblΛδ+1v/ba∇dblL2\n≤ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,ν,s)/ba∇dblj/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs.\nThus, the conclusion follows.\nIn the next few steps, we will consider a domain for ( δ,s) withs >1 ands−1≤δ < s, which has been\nprovided in [ 59] with a different proof. We aim to revisit this domain of initial data with a new proof.\nStep 11: The case s∈(1,2]ands−1≤δ≤1(revisited). Similar to the previous case, we find that\n1\n2d\ndt/parenleftbig\n/ba∇dblv/ba∇dbl2\n˙Hδ+/ba∇dbl(E,B)/ba∇dbl2\n˙Hs/parenrightbig\n+ν/ba∇dblv/ba∇dbl2\n˙Hδ+1+1\nσ/ba∇dblj/ba∇dbl2\n˙Hs=:3/summationdisplay\nk=1Ik,\n18where for some ǫ∈(0,1), since δ∈[s−1,1] ands∈(1,2]\nI1=/integraldisplay\nR2(j×B)·Λ2δvdx\n≤C(δ)/braceleftigg\n/ba∇dblj/ba∇dblL2/ba∇dblB/ba∇dbl˙Hδ/ba∇dblΛ2δv/ba∇dbl\nL2\n1−δifδ∈[s−1,1),\n/ba∇dblj/ba∇dbl1\n2\nL2/ba∇dbl∇j/ba∇dbl1\n2\nL2/ba∇dblB/ba∇dbl1\n2\nL2/ba∇dbl∇B/ba∇dbl1\n2\nL2/ba∇dbl∆v/ba∇dblL2ifδ= 1,\n≤/braceleftigg\nǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/ba∇dblj/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\n˙Hδ ifδ∈[s−1,1),\nǫ\nσ/ba∇dblj/ba∇dbl2\n˙H1+ǫν/ba∇dblv/ba∇dbl2\n˙H2+C(ǫ,ν,σ)/ba∇dblj/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\nL2/ba∇dblB/ba∇dbl2\n˙H1ifδ= 1;\nI2=−/integraldisplay\nR2(v·∇v)·Λ2δvdx≤/braceleftigg\nǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,δ,ν)/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblv/ba∇dbl2\n˙Hδifδ∈[s−1,1),\n0 if δ= 1;\nI3=/integraldisplay\nR2Λsj·Λs(v×B)dx\n≤/braceleftigg\nC(s)/ba∇dblj/ba∇dbl˙Hs/parenleftig\n/ba∇dblΛsv/ba∇dbl\nL2\n1−(δ+1−s)/ba∇dblB/ba∇dbl\nL2\nδ+1−s+/ba∇dblv/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs/parenrightig\nifδ > s−1,\nC(s)/ba∇dblj/ba∇dbl˙Hs/parenleftbig\n/ba∇dblΛδ+1v/ba∇dblL2/ba∇dblB/ba∇dblL∞+/ba∇dblv/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs/parenrightbig\nifδ=s−1,\n≤/braceleftigg\nC(s)/ba∇dblj/ba∇dbl˙Hs/parenleftbig\n/ba∇dblΛδ+1v/ba∇dblL2/ba∇dblΛs−δB/ba∇dblL2+/ba∇dblv/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs/parenrightbig\nifδ > s−1,\nC(s)/ba∇dblj/ba∇dbl˙Hs/parenleftbig\n/ba∇dblΛδ+1v/ba∇dblL2/ba∇dblB/ba∇dblL∞+/ba∇dblv/ba∇dblL∞/ba∇dblB/ba∇dbl˙Hs/parenrightbig\nifδ=s−1,\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\n˙Hs+ǫν/ba∇dblv/ba∇dbl2\n˙Hδ+1+C(ǫ,ν,s)/ba∇dblj/ba∇dbl2\n˙Hs/parenleftbig\n/ba∇dblB/ba∇dbl2\n˙Hs−δ+/ba∇dblB/ba∇dbl2\nL∞/parenrightbig\n+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nL∞/ba∇dblB/ba∇dbl2\n˙Hs.\nTherefore, an application of Step 3 gives us the conclusion.\nStep 12: The case s∈(1,2]and1< δ < s (revisited). In this case, we apply Step 8 to obtain\n(v,E,B)∈L∞\ntHδ\nx∩L2\ntHδ+1\nx×L∞\ntHδ\nx×L∞\ntHδ\nx. It remains to obtain the Hsestimate of ( E,B). We only need\nto bound I3. Sinces∈(1,2] and 1 < δ < swithδ+1−s∈(0,1),I3is given and bounded as the previous step.\nStep 13: The case s >2ands−1≤δ < s(revisited). Similar to the previous step, an application of\nStep 8 gives us v∈L∞\ntHδ\nx∩L2\ntHδ+1\nx. We now focus on I3by using s≤δ+1\nI3=/integraldisplay\nR2Λsj·Λs(v×B)dx\n≤C(s)/ba∇dblj/ba∇dblHs/ba∇dblv/ba∇dblHδ+1/ba∇dblB/ba∇dblHs\n≤ǫ\nσ/ba∇dblj/ba∇dbl2\nHs+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nHδ+1/ba∇dblB/ba∇dbl2\nHs.\nIn addition, from Step 8 to Step 13, it can be seen that since δ,s >1, fort∈(0,Tn\n∗)\n/ba∇dblj(t)/ba∇dblHmin{δ,s}≤C(c,σ)(/ba∇dblE(t)/ba∇dblHs+/ba∇dblv(t)/ba∇dblHδ/ba∇dblB(t)/ba∇dblHs).\nStep 14: The bound of /ba∇dblv/ba∇dblL2\ntL∞x(revisited). We revisit this case with a sightly different decomposition\nas it has been considered in [ 2,59] before. In addition, the idea here will be applied to the proof of The orem\n1.2below. Assume that v0∈L2and (E0,B0)∈Hsfor some s∈(0,1). In the previous steps, to close the main\nestimates, we need to assume that v0∈Hδfor some δ >0. Thus, the same argument might not work in the\ncaseδ= 0 with v∈L∞\ntL2\nx∩L2\ntH1\nx, which is not enough to bound the norm /ba∇dblv/ba∇dblL2\ntL∞x. In the two-dimensional\ncase, there is a way to overcome this difficulty in which the idea comes f rom the recent result in [ 2], where the\nauthors have introduced a suitable decomposition of the velocity, w hich is useful for obtaining first an estimate\nof/ba∇dblv/ba∇dblL2\ntL∞xin terms of /ba∇dbl(E,B)/ba∇dblL∞\nt˙Hsxfors∈(0,1) and then closing the estimate of /ba∇dbl(E,B)/ba∇dblL∞\nt˙Hsx. As a\nconsequence, they are able to bound the norm /ba∇dblv/ba∇dblL2\ntL∞x. More precisely, they decomposed the velocity and\npressure by v= ¯v1+¯v2+¯v3andπ= ¯π1+ ¯π2, where ¯v1, ¯v2and ¯v3are solutions of the following heat equation\nand Stokes systems\n∂t¯v1−ν∆¯v1= 0, div¯v1= 0, ¯v1\n|t=0=v0,\n∂t¯v2−ν∆¯v2+∇¯π1=−v·∇v, div¯v2= 0, ¯v2\n|t=0= 0,\n∂t¯v3−ν∆¯v3+∇¯π2=j×B, div¯v3= 0, ¯v3\n|t=0= 0.\nThat allows them to study each part of the decomposition separate ly, where the most difficult part is dealing\nwith ¯v3in which they overcomed this issue by introducing a suitable iteration. In fact, it is possible to combine\n¯v1and ¯v2parts together by decomposing vn=vn,1+vn,2in such a way that vn,i∈L2\nnfori∈ {1,2}. In the\nsequel, we will write ( v1,v2) instead of ( vn,1,vn,2) for simplicity. Indeed, we first define v1be a divergence-free\nvector with supp( F(v1))⊆Bnand be a solution of the first equation below. It can be seen that fro m the\n19properties of vthat such a v1∈L2exists (see the estimate below). Then, we set v2=v−v1, which leads to\nv2∈L2, supp(F(v2))⊆Bnand divv2= 0. It follows from ( 2.1) that\n∂tv1−ν∆v1=−P(Tn(v·∇v)), divv1= 0, v1\n|t=0=Tn(v0),\n∂tv2−ν∆v2=P(Tn(j×B)), divv2= 0, v2\n|t=0= 0.\nIn the sequel, we firstly explain how /ba∇dblv/ba∇dblL2\ntL∞xcan be controlled only in terms of /ba∇dblv2/ba∇dblL2\ntL∞xand secondly use the\ntechnique in [ 2] to bound /ba∇dblv2/ba∇dblL2\ntL∞x. We first focus on obtaining estimates of v1. Its energy estimate is given\nby\nd\ndt/ba∇dblv1/ba∇dbl2\nL2+ν/ba∇dbl∇v1/ba∇dbl2\nL2≤C(ν)/ba∇dblv/ba∇dbl2\nL2/ba∇dbl∇v/ba∇dbl2\nL2,\nwhich implies for t∈(0,Tn\n∗) by using the energy estimate of v\n/ba∇dblv1(t)/ba∇dbl2\nL2+ν/integraldisplayt\n0/ba∇dbl∇v1/ba∇dbl2\nL2dτ≤ /ba∇dblv1(0)/ba∇dbl2\nL2+C(ν)/ba∇dblv/ba∇dbl2\nL∞\ntL2x/integraldisplayt\n0/ba∇dbl∇v/ba∇dbl2\nL2dτ≤C(E0,ν).\nMoreover, it follows that for t∈(0,Tn\n∗) andq∈Z\n1\n2d\ndt/ba∇dbl∆qv1/ba∇dbl2\nL2+ν/ba∇dbl∆q∇v1/ba∇dbl2\nL2≤ /ba∇dbl∆qv1/ba∇dblL2/ba∇dbl∆q(v·∇v)/ba∇dblL2.\nIt can be seen from the definition of nonhomogeneous dyadic blocks (see Appendix A in Section 7) that for\nq∈Zwithq≥0\n/ba∇dbl∆q∇v1/ba∇dbl2\nL2≥C22q/ba∇dbl∆qv1/ba∇dbl2\nL2,\nwhich yields\nesssup\nt∈(0,Tn∗)/ba∇dbl∆qv1(t)/ba∇dbl2\nL2+C(ν)/parenleftigg/integraldisplayTn\n∗\n022q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n≤R2\nqifq≥0,\nesssup\nt∈(0,Tn∗)/ba∇dbl∆qv1(t)/ba∇dbl2\nL2≤R2\nqifq≥ −1,\nwhere\nRq:=/integraldisplayTn\n∗\n0/ba∇dbl∆q(v·∇v)/ba∇dblL2dτ+/ba∇dbl∆qv(0)/ba∇dblL2.\nFurthermore,\n/summationdisplay\nq≥−1esssup\nt∈(0,Tn∗)/ba∇dbl∆qv1(t)/ba∇dbl2\nL2+C(ν)/summationdisplay\nq≥0/parenleftigg/integraldisplayTn\n∗\n022q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n≤2/summationdisplay\nq≥−1R2\nq.\nWe now estimate the right-hand side as follows\n/summationdisplay\nq≥−1R2\nq≤C/summationdisplay\nq≥−1/parenleftigg/integraldisplayTn\n∗\n0/ba∇dbl∆q(v·∇v)/ba∇dblL2dτ/parenrightigg2\n+C/ba∇dblv0/ba∇dbl2\nL2\n≤C\n/integraldisplayTn\n∗\n0\n/summationdisplay\nq≥−1/ba∇dbl∆q(v·∇v)/ba∇dbl2\nL2\n1\n2\ndτ\n2\n+C/ba∇dblv0/ba∇dbl2\nL2\n≤C/parenleftig\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x/parenrightig\n/ba∇dbl∇v/ba∇dbl2\nL2\ntL2x+C/ba∇dblv0/ba∇dbl2\nL2,\nwhere we used the fact that L1(0,Tn\n∗;B0\n2,2(R2))⊂˜L1(0,Tn\n∗;B0\n2,2(R2)), see the definition of this functional space\nin Appendix A in Section 7. We first focus on obtaining a bound on the norm /ba∇dblv1/ba∇dblL2\ntL∞x. The Littlewood–Paley\ndecomposition gives us\n/integraldisplayTn\n∗\n0/ba∇dblv1/ba∇dbl2\nL∞dτ≤C/integraldisplayTn\n∗\n0/summationdisplay\nq≥−12q/ba∇dbl∆qv1/ba∇dblL2/summationdisplay\nk≥−12k/ba∇dbl∆kv1/ba∇dblL2dτ,\n≤C/integraldisplayTn\n∗\n0\n/summationdisplay\n|q−k|≤N+/summationdisplay\n|q−k|>N\n2q/ba∇dbl∆qv1/ba∇dblL22k/ba∇dbl∆kv1/ba∇dblL2dτ=:¯R1+¯R2,\n20whereN∈Nto be determined later and we used the following Bernstein-type est imate (see [ 5]) for 1≤q0≤\np0≤ ∞\n/ba∇dblf/ba∇dblLp0(Rd)≤C(p0,q0,d)λd/parenleftBig\n1\nq0−1\np0/parenrightBig\n0 /ba∇dblf/ba∇dblLq0(Rd)if supp( F(f))⊂/braceleftbig\nξ∈Rd:|ξ| ≤λ0/bracerightbig\n.\nThe terms on the right-hand side can be bounded as follows\n¯R1=C/integraldisplayTn\n∗\n0/summationdisplay\nq≥−12q/ba∇dbl∆qv1/ba∇dblL2/summationdisplay\nq−N≤k≤q+N2k/ba∇dbl∆kv1/ba∇dblL2dτ\n≤C/integraldisplayTn\n∗\n0/summationdisplay\nq≥−1\n22q/ba∇dbl∆qv1/ba∇dbl2\nL2+/summationdisplay\nq−N≤k≤q+N22k/ba∇dbl∆kv1/ba∇dbl2\nL2\ndτ\n≤CN/parenleftig\n/ba∇dbl∇v1/ba∇dbl2\nL2\ntL2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x/parenrightig\n,\nand by using Young inequality for sequences\n¯R2=C21−N∞/summationdisplay\nq=N/integraldisplayTn\n∗\n022q/ba∇dbl∆qv1/ba∇dblL2q−N−1/summationdisplay\nk=−12k−(q−N)/ba∇dbl∆kv1/ba∇dblL2dτ\n≤C21−N\n∞/summationdisplay\nq=N/parenleftigg/integraldisplayTn\n∗\n022q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n1\n2\n∞/summationdisplay\nk0=0/parenleftiggk0−1/summationdisplay\nk=−12−(k0−k)esssup\nt∈(0,Tn∗)/ba∇dbl∆kv1/ba∇dblL2/parenrightigg2\n1\n2\n≤C21−N\n∞/summationdisplay\nq=0/parenleftigg/integraldisplayTn\n∗\n022q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n1\n2/parenleftigg∞/summationdisplay\nk=−1esssup\nt∈(0,Tn\n∗)/ba∇dbl∆kv1/ba∇dbl2\nL2/parenrightigg1\n2∞/summationdisplay\nk=−12−k.\nTherefore,\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x≤C(ν)2−N/parenleftig\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x/parenrightig\n/ba∇dbl∇v/ba∇dbl2\nL2\ntL2x+CN/parenleftig\n/ba∇dbl∇v1/ba∇dbl2\nL2\ntL2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x/parenrightig\n+C/ba∇dblv0/ba∇dbl2\nL2,\nwhich by choosing16\nN=/ceilingleftig\nlog2/parenleftig\n4+2C(ν)/ba∇dbl∇v/ba∇dbl2\nL2\ntL2x/parenrightig/ceilingrightig\nand using the energy estimates of v1andvyields\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x≤C(E0,ν)/parenleftig\n(/ba∇dbl∇v1/ba∇dbl2\nL2\ntL2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x)(1+/ba∇dbl∇v/ba∇dbl2\nL2\ntL2x)+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x+1/parenrightig\n≤C(E0,ν)(1+Tn\n∗)/parenleftig\n/ba∇dblv2/ba∇dbl2\nL2\ntL∞x+1/parenrightig\n.\nIn addition,\n/ba∇dblv/ba∇dbl2\nL2\ntL∞x≤C/parenleftig\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x/parenrightig\n≤C(E0,ν)(1+Tn\n∗)/parenleftig\n/ba∇dblv2/ba∇dbl2\nL2\ntL∞x+1/parenrightig\n.\nIt remains to bound the term /ba∇dblv2/ba∇dblL2\ntL∞x. It can be seen that\n/ba∇dblv2/ba∇dbl2\nL2\nt˙H1x≤C/parenleftig\n/ba∇dblv/ba∇dbl2\nL2\nt˙H1x+/ba∇dblv1/ba∇dbl2\nL2\nt˙H1x/parenrightig\n≤C(E0,ν).\nLet us now summarize how the authors in [ 2] obtained a bound on /ba∇dblv2/ba∇dblL2\ntL∞xin terms of /ba∇dbl(E,B)/ba∇dblL2\nt˙Hsxand use\nthis relation to close the estimate of /ba∇dbl(E,B)/ba∇dblL∞\nt˙Hs\nxitself. By decomposing v2into high and low frequencies in\n[2, Lemmas 7.3 and 7.4], respectively, they proved that for 0 ≤t0< t < Tn\n∗\n/ba∇dblv2/ba∇dbl2\nL2(t0,t;L∞)≤C(E0,ν)log(e+t−t0)+C(σ,s)/ba∇dblv2/ba∇dbl2\nL2(t0,t;˙H1)log/parenleftigg\ne+E2\n0/ba∇dblB/ba∇dbl2\nL∞(t0,t;˙Hs)\n/ba∇dblv2/ba∇dbl2\nL2(t0,t;˙H1)/parenrightigg\n.\nMoreover, as the estimate of I3in Step 1 (consider only the equations of ( E,B)), we find that17\n1\n2d\ndt/ba∇dbl(E,B)/ba∇dbl2\n˙Hs+1\nσ/ba∇dblj/ba∇dbl2\n˙Hs≤ǫ\nσ/ba∇dblj/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/parenleftbig\n/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblB/ba∇dbl2\n˙Hs\n16Here⌈·⌉denotes the usual ceiling function.\n17There is a slightly different here, where the constant C(σ,s) does not depend on c. However, it seems not to be the case as\nin [2], where the authors used the relation j=σ(cE+v×B) and considered σcEas a damping term on the left-hand side in the\nequation of E.\n21and then by choosing ǫ=1\n2for 0≤t0< t < Tn\n∗\n/ba∇dbl(E,B)(t)/ba∇dbl2\n˙Hs≤ /ba∇dbl(E,B)(t0)/ba∇dbl2\n˙Hsexp/braceleftbigg\nC(σ,s)/integraldisplayt\nt0/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dblv/ba∇dbl2\nL∞dτ/bracerightbigg\n,\nand by using the bound on /ba∇dblv/ba∇dblL2\ntL∞\nxin terms of /ba∇dblv2/ba∇dblL2(t0,t;L∞)and those of /ba∇dbl(v,v2)/ba∇dblL2\nt˙H1x, an iteration as in\n[2, The proof of Theorem 1.2] can be applied to the above bound of ( E,B), which gives us18\nE2\n0/ba∇dbl(E,B)/ba∇dbl2\nL∞(0,t;˙Hs)≤/parenleftbig\ne+E2\n0/ba∇dbl(E0,B0)/ba∇dbl2\n˙Hs+t/parenrightbigC(Tn\n∗,E0,ν,σ,s)fort∈(0,Tn\n∗),\nwhich by using the increasing of the function z/ma√sto→zlog(e+C\nz) forz >0, implies that\n/ba∇dblv2/ba∇dbl2\nL2(0,t;L∞)+/ba∇dblv/ba∇dbl2\nL2(0,t;L∞)≤C(Tn\n∗,E0,ν,σ,s)/parenleftbig\nE2\n0+/parenleftbig\ne+E2\n0/ba∇dbl(E0,B0)/ba∇dbl2\n˙Hs+t/parenrightbig/parenrightbigC(Tn\n∗,E0,ν,σ,s),\nwhereC(Tn\n∗,E0,ν,σ,s) = (1+ Tn\n∗)C(E0,ν,σ,s).\nStep 15: Conclusion of Part II. Collecting the main estimates from Step 1 to Step 14, we find that\nTn\n∗=∞. Moreover, by replacing Tn\n∗by any given (does not depend on n)T∈(0,∞) and repeating the above\ncalculations, it follows that for t∈(0,T) and for the same range of ( s,δ) from Step 1 to Step 14\n/ba∇dblvn(t)/ba∇dbl2\nHδ+/ba∇dbl(En,Bn)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblvn/ba∇dbl2\nHδ+1+/ba∇dblvn/ba∇dbl2\nL∞+/ba∇dbljn/ba∇dbl2\nHsdτ≤C(T,δ,ν,σ,s,v 0,E0,B0).\nPart III: Pass to the limit and uniqueness. Although this part is quite standard, it has not been given\nin details in [ 2,59]. Our aim in this part is to fulfill this gap for the sake of completeness a nd also for later\nuse in the proofs of next theorems. Firstly, by using the ideas in [ 31,46,58], we prove that ( vn,En,Bn) and\n(∇vn,jn) are Cauchy sequences in L∞(0,T;L2(R2)) andL2(0,T;L2(R2)), respectively, for any T∈(0,∞) and\nδ,s >1, which allows us to pass to the limit from ( 2.1) in a stronger sense than the usual one (the sense of\ndistributions) in the case either δ∈[0,1] ors∈(0,1]. Secondly, the uniqueness can be obtained by a carefully\nanalysis.\nStep 16: Cauchy sequence. Assume that ( vn,En,Bn) and (vm,Em,Bm) form,n∈Nwithm > nare\ntwo solutions to ( 2.1) with the same initial data. It follows that\n1\n2d\ndt/ba∇dbl(vn−vm,En−Em,Bn−Bm)/ba∇dbl2\nL2+ν/ba∇dbl∇(vn−vm)/ba∇dbl2\nL2+1\nσ/ba∇dbljn−jm/ba∇dbl2\nL2=:8/summationdisplay\nk=4Ik,\nwhere for some ǫ∈(0,1), since δ,s >1\nI4=/integraldisplay\nR2P(−Tn(vn·∇vn)+Tm(vm·∇vm))·(vn−vm)dx=:3/summationdisplay\nk=1I4k,\nI41=−/integraldisplay\nR2(Tn−Tm)(vn·∇vn)·(vn−vm)dx≤C(δ)n−(δ−1)/ba∇dblvn/ba∇dbl2\nHδ/ba∇dblvn−vm/ba∇dblL2,\nI42=−/integraldisplay\nR2Tm((vn−vm)·∇vn)·(vn−vm)dx≤C(ǫ,ν)/ba∇dbl∇vn/ba∇dbl2\nL2/ba∇dblvn−vm/ba∇dbl2\nL2+ǫν/ba∇dbl∇(vn−vm)/ba∇dbl2\nL2,\nI43=−/integraldisplay\nR2Tm(vm·∇(vn−vm))·(vn−vm)dx= 0;\nI5=/integraldisplay\nR2P(Tn(jn×Bn)−Tm(jm×Bm))·(vn−vm)dx=:3/summationdisplay\nk=1I5k,\nI51=/integraldisplay\nR2(Tn−Tm)(jn×Bn)·(vn−vm)dx≤C(s)n−2s/ba∇dbljn/ba∇dbl2\nHs+/ba∇dblBn/ba∇dbl2\nHs/ba∇dblvn−vm/ba∇dbl2\nL2,\nI52=/integraldisplay\nR2Tm((jn−jm)×Bn)·(vn−vm)dx,\nI53=/integraldisplay\nR2Tm(jm×(Bn−Bm))·(vn−vm)dx\n≤C(ǫ,ν)/ba∇dbljm/ba∇dbl2\nHs/ba∇dblBn−Bm/ba∇dbl2\nL2+ǫν/parenleftbig\n/ba∇dblvn−vm/ba∇dbl2\nL2+/ba∇dbl∇(vn−vm)/ba∇dbl2\nL2/parenrightbig\n;\n18Here, the authors in [ 2] used a suitable time decomposition of the whole time interv al (0,∞) based on the fact that /ba∇⌈blv2/ba∇⌈blL2\nt˙H1x<\n∞, which allowed them to set up an iteration and obtain the boun d of/ba∇⌈bl(E,B)/ba∇⌈blL∞\nt˙Hsxon each smalltime interval, then they obtained\nthe bound on the whole time interval by using the continuous i n time of regularized solutions. Here, we only need to change (0,∞)\ninto (0,Tn\n∗).\n22I6=−/integraldisplay\nR2(jn−jm)·(−Tn(vn×Bn)+Tm(vm×Bm))dx=:3/summationdisplay\nk=1I6k,\nI61=/integraldisplay\nR2(jn−jm)·(Tn−Tm)(vn×Bn)dx≤ǫ\nσ/ba∇dbljn−jm/ba∇dbl2\nL2+C(ǫ,δ,σ,s)n−2min{δ,s}/ba∇dblvn/ba∇dbl2\nHδ/ba∇dblBn/ba∇dbl2\nHs,\nI62=/integraldisplay\nR2(jn−jm)·Tm((vn−vm)×Bn)dx=−I52,\nI63=/integraldisplay\nR2(jn−jm)·Tm(vm×(Bn−Bm))dx≤ǫ\nσ/ba∇dbljn−jm/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dblvn/ba∇dbl2\nHδ/ba∇dblBn−Bm/ba∇dbl2\nL2;\nI7=/integraldisplay\nR2∇×(Bn−Bm)·(c(En−Em))dx;\nI8=−/integraldisplay\nR2∇×(En−Em)·(c(Bn−Bm))dx=−I7;\nhere we used the fact that Hs0(R2) is an algebra for s0>1 and the following inequality (see [ 31])\n/ba∇dblTn(f)−f/ba∇dblHs1≤n−s2/ba∇dblf/ba∇dblHs1+s2fors1,s2∈R,s2≥0.\nTherefore, by choosing ǫ=1\n4\nd\ndtEnm+ν/ba∇dbl∇(vn−vm)/ba∇dbl2\nL2+1\nσ/ba∇dbljn−jm/ba∇dbl2\nL2≤C(ν,σ)/parenleftbig\n1+/ba∇dblvn/ba∇dbl2\nHδ+/ba∇dbljm/ba∇dbl2\nHs+/ba∇dblBn/ba∇dbl2\nHs/parenrightbig\nEnm\n+C(δ,σ,s)/parenleftig\nn−(δ−1)/ba∇dblvn/ba∇dbl2\nHδ/ba∇dblvn−vm/ba∇dblL2+n−2s/ba∇dbljn/ba∇dbl2\nHs+n−2min{δ,s}/ba∇dblvn/ba∇dbl2\nHδ/ba∇dblBn/ba∇dbl2\nHs/parenrightig\n.\nBy denoting for t∈(0,T)\nEnm(t) :=/ba∇dbl(vn−vm,En−Em,Bn−Bm)(t)/ba∇dbl2\nL2\nand using Enm(0) = 0 and Step 15, it follows that for C=C(T,δ,ν,σ,s,v 0,E0,B0)\nEnm(t)+/integraldisplayt\n0/ba∇dbl∇(vn−vm)/ba∇dbl2\nL2+/ba∇dbljn−jm/ba∇dbl2\nL2dτ≤Cmax/braceleftig\nn−(δ−1),n−2s,n−2min{δ,s}/bracerightig\n,\nwhich ends the proof by letting n→ ∞.\nStep 17: Pass to the limit. There are two substeps in this step as follows.\nStep 17a: The case δ,s >1.We use the notation →,⇀and∗⇀to denote the usual strong, weak and\nweak-star convergences, respectively. From the previous step , there exists ( v,E,B,j ) such that as n→ ∞\n(vn,En,Bn)→(v,E,B) in L∞(0,T;L2(R2)),\n(∇vn,jn)→(∇v,j) in L2(0,T;L2(R2)),\nwhich implies by using interpolation inequalities and Step 15 that for all s′∈(1,min{δ,s}) asn→ ∞\n(vn,En,Bn)→(v,E,B) in L∞(0,T;Hs′(R2)),\n(∇vn,jn)→(∇v,j) in L2(0,T;Hs′(R2)),\n(∆vn,∇×En,∇×Bn)→(∆v,∇×E,∇×B) in L2(0,T;Hs′−1(R2)).\nMoreover, for the nonlinear terms as n→ ∞\nTn(vn·∇vn,vn×Bn)→(v·∇v,v×B) in L∞(0,T;Hs′−1(R2)×Hs′(R2)),\nTn(jn×Bn)→j×B inL2(0,T;Hs′(R2)),\nsince\nN1:=/ba∇dblTn(vn·∇vn)−v·∇v/ba∇dblHs′−1≤ns′−δ/ba∇dblvn/ba∇dbl2\nHδ+/ba∇dblvn−v/ba∇dblHs′(/ba∇dblvn/ba∇dblHδ+/ba∇dblv/ba∇dblHδ);\nN2:=/ba∇dblTn(vn×Bn)−v×B/ba∇dblHs′\n≤ns′−min{δ,s}/ba∇dblvn/ba∇dblHδ/ba∇dblBn/ba∇dblHs+/ba∇dblvn−v/ba∇dblHs′/ba∇dblBn/ba∇dblHs+/ba∇dblv/ba∇dblHδ/ba∇dblBn−B/ba∇dblHs′;\nN3:=/integraldisplayT\n0/ba∇dblTn(jn×Bn)−j×B/ba∇dbl2\nHs′dt\n≤C/integraldisplayT\n0ns′−s/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs+/ba∇dbljn−j/ba∇dbl2\nHs′/ba∇dblBn/ba∇dbl2\nHs+/ba∇dblBn−B/ba∇dbl2\nHs′/ba∇dblj/ba∇dbl2\nHsdt.\n23In addition, ( 2.1) gives us for t∈(0,T),σ∈ {s′−1,δ−1}andσ′∈ {s′−1,s−1}\n/integraldisplayt\n0/ba∇dbl∂tvn/ba∇dbl2\nHσdτ≤C/integraldisplayt\n0/ba∇dbl(vn·∇vn,ν∆vn,jn×Bn)/ba∇dbl2\nHσdτ,\n/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nc(∂tEn,∂tBn)/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nHσ′dτ≤C/integraldisplayt\n0/ba∇dbl(∇×En,∇×Bn,jn)/ba∇dbl2\nHσ′dτ,\nwhich together with Step 15 and the above strong convergences le ads to there exists a subsequence denoted by\n(vnk,Enk,Bnk) such that as nk→ ∞\n(∂tvnk,1\nc∂tEnk,1\nc∂tBnk)⇀(∂tv,1\nc∂tE,1\nc∂tB) inL2(0,T;Hδ−1(R2)×Hs−1(R2)×Hs−1(R2)),\n(∂tvnk,1\nc∂tEnk,1\nc∂tBnk)→(∂tv,1\nc∂tE,1\nc∂tB) inL2(0,T;Hs′−1(R2)).\nIn addition, the above strong convergences and ( 2.1) imply that in L2(0,T;Hs′−1(R2))\n\n\n∂tv+P(v·∇v) =ν∆v+P(j×B),\n1\nc∂tE−∇×B=−j\n1\nc∂tB+∇×E= 0\nσ(cE+v×B) =j,\ndivv= divB= 0.(2.5)\nIndeed, it can be checked that as n→ ∞\ndivvn→divv inL2(0,T;Hs′(R2)),\ndivBn→divB inL2(0,T;Hs′−1(R2)),\n(vn,En,Bn)|t=0=Tn(v0,E0,B0)→(v0,E0,B0) in Hδ(R2)×Hs(R2)×Hs(R2),\nwhich leads to div v= divB= 0 and ( v,E,B)|t=0= (v0,E0,B0). Then, the theorem de Rham (see [ 67]) ensures\nthe existence of a scalar function πsuch that ( v,E,B,π ) satisfies ( NSM) at least in the sense of distributions.\nFrom the uniform bound in Step 15, we also have as nk→ ∞\n(vnk,Enk,Bnk)∗⇀(v,E,B) in L∞(0,T;Hδ(R2)×Hs(R2)×Hs(R2)),\n(∇vnk,jnk)⇀(∇v,j) in L2(0,T;Hδ(R2)×Hs(R2)),\nwhich implies that for δ,s >1,\n(v,E,B)∈L∞(0,T;Hδ×Hs×Hs) and ( v,j)∈L2(0,T;Hδ+1×Hs)\nsatisfying for t∈(0,T)\n/ba∇dblv(t)/ba∇dbl2\nHδ+/ba∇dbl(E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblv/ba∇dbl2\nHδ+1+/ba∇dblj/ba∇dbl2\nHsdτ≤C(T,δ,ν,σ,s,v 0,E0,B0).\nIn fact, after possibly being redefined on a set of measure zero, v∈C([0,T];Hδ(R2)) (see [29,67]) since\nv∈L2(0,T;Hδ+1(R2)) and∂tv∈L2(0,T;Hδ−1(R2)). Furthermore, we find that from the uniform bound in\nStep 15, ( E,B) is weak continuous in time with values in Hs(R2).\nStep 17b: The case either δ∈[0,1]ors∈(0,1].It is enough to focus on the case where δ= 0 and\ns∈(0,1), which is considered in Step 14 above. Other cases follow as conse quences. It follows from the uniform\nbound in Step 15 that\n(vn,En,Bn) is uniformly bounded in L∞(0,T;L2(R2)×Hs(R2)×Hs(R2))\nsatisfying for t∈(0,T)\n/ba∇dblvn(t)/ba∇dbl2\nL2+/ba∇dbl(En,Bn)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dblvn/ba∇dbl2\nH1+/ba∇dbljn/ba∇dbl2\nHsdτ≤C(T,ν,σ,s,v 0,E0,B0).\nIn addition, for φ∈H1(R2;R3) with/ba∇dblφ/ba∇dblH1≤1, it yields for τ∈(0,T)19\n/integraldisplayτ\n0|(∂tvn,φ)|2dt≤C(ν)/integraldisplayτ\n0/parenleftbig\n1+/ba∇dblvn/ba∇dbl2\nL2/parenrightbig\n/ba∇dbl∇vn/ba∇dbl2\nL2+/ba∇dbljn/ba∇dbl2\nL2/ba∇dblBn/ba∇dbl2\nHsdt≤C(T,ν,σ,s,v 0,E0,B0),\n19Here, (·,·) is the standard L2inner product.\n241\nc/integraldisplayτ\n0|(∂tEn,φ)|2+|(∂tBn,φ)|2dt≤/integraldisplayτ\n0/ba∇dblEn/ba∇dbl2\nL2+/ba∇dblBn/ba∇dbl2\nL2+/ba∇dbljn/ba∇dbl2\nL2dt≤C(T,ν,σ,v 0,E0,B0),\nwhich implies that20\n(∂tvn,∂tEn,∂tBn) is uniformly bounded in L2(0,T;H−1(R2)).\nTherefore, there exists a subsequence (still denoted by) ( vn,En,Bn,jn) and (v,E,B,j ) such that as n→ ∞\n(vn,En,Bn)∗⇀(v,E,B) in L∞(0,T;L2(R2)×Hs(R2)×Hs(R2)),\n(vn,jn)⇀(v,j) in L2(0,T;H1(R2)×Hs(R2)),\n(∂tvn,∂tEn,∂tBn)⇀(∂tv,∂tE,∂tB) in L2(0,T;H−1(R2)).\nRecall that the injections H1֒→L2֒→H−1andHs֒→L2֒→H−1fors∈(0,1) are locally compact by using\nthe Rellich–Kondrachov and Schauder theorems (see [ 11,51]) then an application of the Aubin-Lions lemma\n(see [10]) implies that as n→ ∞\n(vn,En,Bn)→(v,E,B) in L2(0,T;L2\nloc(R2)).\nFurthermore, it can be seen from ( 2.1) that (vn,En,Bn) satisfies\na)/integraldisplayT\n0/integraldisplay\nR2vn·∂tφ−P(Tn(vn·∇vn))·φ−ν∇vn:∇φ+P(Tn(jn×Bn))·φdxdt=−/integraldisplay\nR2vn(0)·φ(0)dx,\nb)/integraldisplayT\n0/integraldisplay\nR21\ncEn·∂tϕ+Bn·(∇×ϕ)−jn·ϕdxdt=−/integraldisplay\nR21\ncEn(0)·ϕ(0)dx,\nc)/integraldisplayT\n0/integraldisplay\nR21\ncBn·∂tϕ−En·(∇×ϕ)dxdt=−/integraldisplay\nR21\ncBn(0)·ϕ(0)dx,\nwhereφ,ϕ∈C∞\n0([0,T)×R2;R3) with div φ= 0. By using the above weak and strong convergences as n→ ∞,\nwe can pass to the limit for the linear terms easily. It remains to check the convergence of the nonlinear terms.\nMoreover, we find that21\nNL1:=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2P(Tn(vn·∇vn)−v·∇v)·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(Tn(vn⊗vn)−vn⊗vn) :∇φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2((vn−v)⊗vn) :∇φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(v⊗(vn−v)) :∇φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblTn(vn⊗vn)−vn⊗vn/ba∇dblL2\ntH−1\nx/ba∇dbl∇φ/ba∇dblL2\ntH1x+/ba∇dblvn−v/ba∇dblL2\nt,x(supp(φ))/ba∇dblvn/ba∇dblL∞\ntL2x/ba∇dbl∇φ/ba∇dblL2\ntL∞x\n+/ba∇dblvn−v/ba∇dblL2\nt,x(supp(φ))/ba∇dblv/ba∇dblL∞\ntL2x/ba∇dbl∇φ/ba∇dblL2\ntL∞x\n→0 asn→ ∞\nby using the strong convergence of vn, and Steps 14 and 15 with\n/ba∇dblTn(vn⊗vn)−vn⊗vn/ba∇dblL2\ntH−1\nx≤1\nn/ba∇dblvn⊗vn/ba∇dblL2\ntL2\nx≤1\nn/ba∇dblvn/ba∇dblL∞\ntL2x/ba∇dblvn/ba∇dblL2\ntL∞\nx.\nSimilarly,\nNL2:=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2P(Tn(jn×Bn)−j×B)·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(Tn(jn×Bn)−jn×Bn)·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(j×(Bn−B))·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2((jn−j)×Bn)·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblTn(jn×Bn)−jn×Bn/ba∇dblL2\ntH−1\nx/ba∇dblφ/ba∇dblL2\ntH1x+/ba∇dblBn−B/ba∇dblL2\nt,x(supp(φ))/ba∇dblj/ba∇dblL2\nt,x/ba∇dblφ/ba∇dblL∞\nt,x\n20As usual, for s∈Rwiths >0, the space H−s(R2) can be considered as the dual space of Hs(R2), see [5].\n21Here,v⊗u:= (viuj)1≤i,j≤3forv= (v1,v2,v3) andu= (u1,u2,u3).\n25+/ba∇dblBn−B/ba∇dblL2\nt,x(supp(φ))/ba∇dbljn−j/ba∇dblL2\nt,x/ba∇dblφ/ba∇dblL∞\nt,x+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2((jn−j)×B)·φdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n→0 asn→ ∞\nby using the strong and weak convergences of Bnandjn, Steps 14 and 15, and ( 7.2) with\n/ba∇dblTn(jn×Bn)−jn×Bn/ba∇dblL2\ntH−1\nx≤1\nn2s/ba∇dbljn×Bn/ba∇dblL2\ntH2s−1\nx≤1\nn2s/ba∇dblBn/ba∇dblL∞\ntHsx/ba∇dbljn/ba∇dblL2\ntHsx.\nFurthermore, we also use jn=σ(cEn+Tn(vn×Bn)) with\nNL3:=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(Tn(vn×Bn)−v×B)·ϕdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(Tn(vn×Bn)−vn×Bn)·ϕdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2((vn−v)×Bn)·ϕdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT\n0/integraldisplay\nR2(v×(Bn−B))·ϕdxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblTn(vn×Bn)−vn×Bn/ba∇dblL2\ntH−1\nx/ba∇dblϕ/ba∇dblL2\ntH1x+/ba∇dblvn−v/ba∇dblL2\nt,x(supp(ϕ))/ba∇dblBn/ba∇dblL∞\ntL2x/ba∇dblϕ/ba∇dblL2\ntL∞x\n+/ba∇dblBn−B/ba∇dblL2\nt,x(supp(ϕ))/ba∇dblv/ba∇dblL∞\ntL2x/ba∇dblϕ/ba∇dblL2\ntL∞x\n→0 asn→ ∞.\nIt can be seen that div v= divB= 0 in the sense of distributions and ( v,E,B)|t=0= (v0,E0,B0). It shows\nthat (v,E,B,j ) satisfies ( 2.5) in the sense of distributions (similar to those of a),b) andc) without n) with\n(v,E,B)|t=0= (v0,E0,B0). In addition, v∈C([0,T];L2), (E,B) is weak continuous in time with values in Hs,\nand (v,E,B) shares the same bounds in the case either δ∈[0,1] ors∈(0,1] as that of ( vn,En,Bn) given in\nStep 15. As mentioned in the previous case, a scalar pressure πcan be recovered such that ( v,E,B,π ) satisfies\n(NSM) in the sense of distributions.\nStep 18: Uniqueness. Although the uniqueness has been considered in [ 59] with a different functional\nspace and has not been mentioned in [ 2], we adapt the idea in [ 59] by providing a slightly different proof,\nwhich will take the advantage of the bound /ba∇dblv/ba∇dblL2\ntL∞xgiven in Step 14 compared to [ 59]. We note that our\nmodified proof can also be useful in the three-dimensional case in Th eorem1.2. Assume that ( v,E,B,j,π ) and\n(¯v,¯E,¯B,¯j,¯π) are two solutions to ( NSM) with the same initial data ( v0,E0,B0)∈L2×Hs×Hsfors∈(0,1).\nIt is worth mentioning that we can not use the usual energy method here due to the lack of smoothness of\n(E,B). It can be seen that the difference satisfies\n\n\n∂t(v−¯v)+(v−¯v)·∇v+ ¯v·∇(v−¯v)+∇(π−¯π) =ν∆(v−¯v)+(j−¯j)×B+¯j×(B−¯B),\n1\nc∂t(E−¯E)−∇×(B−¯B) =−(j−¯j),\n1\nc∂t(B−¯B)+∇×(E−¯E) = 0,\nσ(cE+v×B) =j,\nσ(c¯E+ ¯vׯB) =¯j,\n1\nσ(j−¯j)−(v×B)+(¯vׯB) =c(E−¯E),\ndivv= divB= div¯v= div¯B= 0,\nwhich gives us by using the continuity in time of both vand ¯vthat\n1\n2d\ndt/ba∇dblv−¯v/ba∇dbl2\nL2+ν/ba∇dblv−¯v/ba∇dbl2\n˙H1=:3/summationdisplay\nk=1¯Ik,\nwhere for some ǫ∈(0,1) and for any s′∈(0,s]\n¯I1=−/integraldisplay\nR2(v−¯v)·∇v·(v−¯v)dx≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H1+C(ǫ,ν)/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblv−¯v/ba∇dbl2\nL2;\n¯I2=/integraldisplay\nR2(j−¯j)×B·(v−¯v)dx≤C(s′)/ba∇dblj−¯j/ba∇dblL2/ba∇dblB/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbl˙H1−s′\n≤C(s′)/ba∇dblj−¯j/ba∇dblL2/ba∇dblB/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbls′\nL2/ba∇dblv−¯v/ba∇dbl1−s′\n˙H1\n26≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H1+C(ǫ,ν,s′)/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblj−¯j/ba∇dbl2\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2;\n¯I3=/integraldisplay\nR2¯j×(B−¯B)·(v−¯v)dx≤C(s′)/ba∇dbl¯j/ba∇dblL2/ba∇dblB−¯B/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbl˙H1−s′\n≤C(s′)/ba∇dbl¯j/ba∇dblL2/ba∇dblB−¯B/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbls′\nL2/ba∇dblv−¯v/ba∇dbl1−s′\n˙H1\n≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H1+C(ǫ,ν,s′)/ba∇dbl¯j/ba∇dbl2\ns′+1\nL2/parenleftbig\n/ba∇dblB−¯B/ba∇dbl2\n˙Hs′+/ba∇dblv−¯v/ba∇dbl2\nL2/parenrightbig\n.\nTherefore, by choosing ǫ=1\n6and taking T∗∈(0,T]\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/integraldisplayT∗\n0/ba∇dblv−¯v/ba∇dbl2\n˙H1dτ dτ≤3/summationdisplay\nk=1¯Jk,\nwhere\n¯J1:=C(ν)/integraldisplayT∗\n0/ba∇dbl∇v/ba∇dbl2\nL2/ba∇dblv−¯v/ba∇dbl2\nL2dτ≤C(ν)/ba∇dblv/ba∇dbl2\nL2(0,T∗;˙H1)/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2);\n¯J2:=C(ν,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblj−¯j/ba∇dbl2\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/parenleftbigg\n/ba∇dblE−¯E/ba∇dbl2\ns′+1\nL2+/ba∇dbl(v−¯v)×B/ba∇dbl2\ns′+1\nL2+/ba∇dbl¯v×(B−¯B)/ba∇dbl2\ns′+1\nL2/parenrightbigg\n/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n=:3/summationdisplay\nk=1¯J2k,\n¯J21=C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblE−¯E/ba∇dbl2\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)T∗/ba∇dblB/ba∇dbl2\ns′+1\nL∞(0,T∗;˙Hs′)/parenleftig\n/ba∇dblE−¯E/ba∇dbl2\nL∞(0,T∗;L2)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n,\n¯J22=C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dbl(v−¯v)×B/ba∇dbl2\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblv−¯v/ba∇dbl2\ns′+1\nL2\ns′/ba∇dblB/ba∇dbl2\ns′+1\nL2\n1−s′/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblv−¯v/ba∇dbl2\ns′+1\n˙H1−s′/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl4\ns′+1\n˙Hs′/ba∇dblv−¯v/ba∇dbl4s′\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2(1−s′)\ns′+1\n˙H1dτ\n≤C(c,ν,σ,s′)T2s′\ns′+1\n∗/ba∇dblB/ba∇dbl4\ns′+1\nL∞(0,T∗;˙Hs′)/parenleftig\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗˙H1)/parenrightig\n,\n¯J23=C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dbl¯v×(B−¯B)/ba∇dbl2\ns′+1\nL2/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl2\ns′+1\n˙Hs′/ba∇dbl¯v/ba∇dbl2s′\ns′+1\nL2/ba∇dbl¯v/ba∇dbl2(1−s′)\ns′+1\n˙H1/ba∇dblB−¯B/ba∇dbl2\ns′+1\n˙Hs′/ba∇dblv−¯v/ba∇dbl2s′\ns′+1\nL2dτ\n≤C(c,ν,σ,s′)T2s′\ns′+1\n∗/ba∇dblB/ba∇dbl2\ns′+1\nL∞(0,T∗;˙Hs′)/ba∇dbl¯v/ba∇dbl2s′\ns′+1\nL∞(0,T∗;L2)/ba∇dbl¯v/ba∇dbl2(1−s′)\ns′+1\nL2(0,T∗;˙H1)\n×/parenleftig\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n;\n¯J3:=C(ν,s′)/integraldisplayT∗\n0/ba∇dbl¯j/ba∇dbl2\ns′+1\nL2/parenleftbig\n/ba∇dblB−¯B/ba∇dbl2\n˙Hs′+/ba∇dblv−¯v/ba∇dbl2\nL2/parenrightbig\ndτ\n≤C(ν,s′)Ts′\ns′+1\n∗/ba∇dbl¯j/ba∇dbl2\ns′+1\nL2(0,T∗;L2)/parenleftig\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n.\nIn addition, by using Lemma 7.4, it follows that\n/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′)≤C(c)/ba∇dblj−¯j/ba∇dbl2\nL1(0,T∗;Hs′)=:6/summationdisplay\nk=4¯Jk,\nwhere for any s′∈(0,s)\n¯J4=C(c,σ)/ba∇dblE−¯E/ba∇dbl2\nL1(0,T∗;Hs′)≤C(c,σ)T2\n∗/ba∇dblE−¯E/ba∇dbl2\nL∞(0,T∗;Hs′);\n27¯J5=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;Hs′)≤¯J51+¯J52,\n¯J51:=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;L2)\n≤C(c,σ,s′)Ts′+1\n∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)/ba∇dblv−¯v/ba∇dbl2s′\nL∞(0,T∗;L2)/ba∇dblv−¯v/ba∇dbl2(1−s′)\nL2(0,T∗;˙H1),\n≤C(c,ν,σ,s′)Ts′+1\n∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)/parenleftig\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H1)/parenrightig\n,\n¯J52:=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;˙Hs′)\n≤C(c,ν,σ,s′)T∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H1)\n+C(c,σ,s,s′)Ts−s′+1\n∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs)/ba∇dblv−¯v/ba∇dbl2(s−s′)\nL∞(0,T∗;L2)/ba∇dblv−¯v/ba∇dbl2(1−(s−s′))\nL2(0,T∗;˙H1)\n≤C(c,ν,σ,s′)T∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H1)\n+C(c,ν,σ,s,s′)Ts−s′+1\n∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs)/parenleftig\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H1)/parenrightig\n;\n¯J6=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;Hs′)≤¯J61+¯J62,\n¯J61:=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;L2)\n≤C(c,σ,s′)Ts′+1\n∗/ba∇dbl¯v/ba∇dbl2s′\nL∞(0,T∗;L2)/ba∇dbl¯v/ba∇dbl2(1−s′)\nL2(0,T∗;˙H1)/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′),\n¯J62:=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;˙Hs′)\n≤C(c,σ,s′)T∗/parenleftig\n/ba∇dbl¯v/ba∇dbl2\nL2(0,T∗;˙H1)+/ba∇dbl¯v/ba∇dbl2\nL2(0,T∗;L∞)/parenrightig\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;Hs′).\nCombining all the above estimates and using Step 18, we find that for sufficiently small T∗(depending on the\nparameters and initial data)\nA(v−¯v,E−¯E,B−¯B) :=/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H1)+/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′)\n≤1\n2A(v−¯v,E−¯E,B−¯B),\nwhich yields v= ¯v,E=¯EandB=¯Bin (0,T∗). By repeating this process, we obtain the conclusion in the\nwhole time interval (0 ,T). Finally, we note that only the estimate of ¯J52needss′< sand other ones hold for\ns′=sas well.\nProof of Theorem 1.1-(ii) and (iii). In this part, by applying the previous one, we obtain more regularity for\n(v,E,B).\nStep 19: Higher regularity. In the case δ= 0 and s∈(0,1), an application of Step 14, which allows us\nto bound /ba∇dblvn/ba∇dblL2\ntL∞\nxand/ba∇dbl(En,Bn)/ba∇dblL∞\ntHsx. In addition, it follows from the energy estimate that vn(t′)∈H1for\na.et′∈(0,T) and for any T∈(0,∞). Thus, for any t′∈(0,T) there exists t∗∈(0,t′) such that vn(t∗)∈H1.\nBy fixing t∗, we define for t∈[0,T−t∗),un(t) :=vn(t∗+t) withun\n|t=0:=Tn(vn(t∗))∈Hδ′forδ′∈(0,1] and\nconsider ( 2.1) inR2×(0,T−t∗) by replacing unbyvnwith the initial data ( un,En,Bn)|t=0∈Hδ′×Hs×Hs\nforδ′∈(0,1] ands∈(0,1). An application of Steps 1, 5 and 17b in the proof of Part ( i), which allows us to\nbound/ba∇dblun/ba∇dblL∞\ntHδ′\nx∩L2\ntHδ′+1\nxand pass to the limit as n→ ∞. Furthermore, a similar argument can be applied\nto the case where δ= 0 and s= 1 by using in addition Steps 2, 4 and 17b in the proof of Part ( i), we skip\nfurther details. We note that this step can be applied to other case s to gain more regularity for ( v,E,B) after\nthe initial time, but we will not investigate here. Thus, the proof of t his part is complete.\n3 Proof of Theorem 1.2\nIn this section, we will provide a standard proof of Theorem 1.2, which follows the idea as that of Theorem 1.1.\nProof of Theorem 1.2-(i).The proof is divided into several steps as follows.\nStep 1: Local existence. We will use exactly the approximate system ( 2.1) with replacing ν∆vnby\n−ν(−∆)3\n2vn. Then there exists a unique solution ( vn,En,Bn)∈C1([0,Tn\n∗);Vδ\nn×Hs\nn×Vs\nn) for some Tn\n∗>0.\nIn what follows, we will assume that Tn\n∗<∞. In the sequel, it is sufficient to focus on the case α=3\n2. The\ncaseα >3\n2can be done similarly for more general initial data.\nStep 2: The case δ= 0ands∈(0,3\n2).Similar to the two-dimensional case, the energy estimate is given\nfort∈(0,Tn\n∗) by\n/ba∇dbl(vn,En,Bn)(t)/ba∇dbl2\nL2+/integraldisplayt\n0ν/ba∇dblvn/ba∇dbl2\n˙H3\n2+1\nσ/ba∇dbljn/ba∇dbl2\nL2dτ≤ /ba∇dbl(v0,E0,B0)/ba∇dbl2\nL2=:E2\n0.\n28In addition, the ˙Hsestimate of ( En,Bn) reads\n1\n2d\ndt/ba∇dbl(En,Bn)/ba∇dbl2\n˙Hs+1\nσ/ba∇dbljn/ba∇dbl2\n˙Hs=/integraldisplay\nR3Λs(vn×Bn)·Λsjndx,\nwhere the right-hand side can be bounded as follows\nRHS≤C(s)/parenleftig\n/ba∇dblΛsvn/ba∇dblL6\n2s/ba∇dblBn/ba∇dbl\nL6\n3−2s+/ba∇dblvn/ba∇dblL∞/ba∇dblΛsBn/ba∇dblL2/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/parenleftig\n/ba∇dblvn/ba∇dbl2\n˙H3\n2+/ba∇dblvn/ba∇dbl2\nL∞/parenrightig\n/ba∇dblBn/ba∇dbl2\n˙Hs,\nhere we used the following homogeneous Sobolev inequality (see [ 5])\n/ba∇dblf/ba∇dblLp0≤C(s0,p0)/ba∇dblf/ba∇dbl˙Hs0fors0∈/bracketleftbigg\n0,3\n2/parenrightbigg\n,p0=6\n3−2s0.\nTherefore, by choosing ǫ=1\n2, we obtain for 0 ≤t0< t≤Tn\n∗\n/ba∇dbl(En,Bn)(t)/ba∇dbl2\n˙Hs≤ /ba∇dbl(En,Bn)(t0)/ba∇dbl2\n˙Hsexp/braceleftbigg\nC(σ,s)/integraldisplayt\nt0/ba∇dblvn/ba∇dbl2\n˙H3\n2+/ba∇dblvn/ba∇dbl2\nL∞dτ/bracerightbigg\n.\nIt remains to control /ba∇dblvn/ba∇dblL2\ntL∞x. Defining v1andv2(we only write ( v1,v2) instead of ( vn,1,vn,2) for simplicity)\nas Step 14 in the proof of Theorem 1.1with\n∂tv1+ν(−∆)3\n2v1=−P(Tn(vn·∇vn)),divv1= 0, v1\n|t=0=Tn(v0),\n∂tv2+ν(−∆)3\n2v2=P(Tn(jn×Bn)), divv2= 0, v2\n|t=0= 0.\nIt follows that for t∈(0,Tn\n∗)\n/ba∇dblv1(t)/ba∇dbl2\nL2+ν/integraldisplayt\n0/ba∇dblv1/ba∇dbl2\n˙H3\n2dτ≤ /ba∇dblv1(0)/ba∇dbl2\nL2+C(ν)t1\n3/ba∇dblvn/ba∇dbl8\n3\nL∞(0,t;L2)/ba∇dblvn/ba∇dbl4\n3\nL2(0,t;˙H3\n2)≤C(E0,ν)/parenleftig\n1+(Tn\n∗)1\n3/parenrightig\n.\nMoreover, for t∈(0,Tn\n∗) andq∈Z\n1\n2d\ndt/ba∇dbl∆qv1/ba∇dbl2\nL2+ν/ba∇dbl∆q(−∆)3\n4v1/ba∇dbl2\nL2≤ /ba∇dbl∆qv1/ba∇dblL2/ba∇dbl∆q(vn·∇vn)/ba∇dblL2,\nand by using for q≥0\n/ba∇dbl∆q(−∆)3\n4v1/ba∇dbl2\nL2≥C23q/ba∇dbl∆qv1/ba∇dbl2\nL2,\nwhich yields (we use the same notation as the two-dimensional case)\n/summationdisplay\nq≥−1esssup\nt∈(0,Tn∗)/ba��dbl∆qv1(t)/ba∇dbl2\nL2+C(ν)/summationdisplay\nq≥0/parenleftigg/integraldisplayTn\n∗\n023q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n≤2/summationdisplay\nq≥−1R2\nq,\nand\n/summationdisplay\nq≥−1R2\nq≤C/parenleftig\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x/parenrightig\n/ba∇dbl∇v/ba∇dbl2\nL2\ntL2x+C/ba∇dblv0/ba∇dbl2\nL2.\nThe Littlewood–Paley decomposition and Bernstein-type estimate g ive us\n/integraldisplayTn\n∗\n0/ba∇dblv1/ba∇dbl2\nL∞dτ≤C/integraldisplayTn\n∗\n0\n/summationdisplay\n|q−k|≤N+/summationdisplay\n|q−k|>N\n23q\n2/ba∇dbl∆qv1/ba∇dblL223k\n2/ba∇dbl∆kv1/ba∇dblL2dτ=:¯R1+¯R2.\nThe terms on the right-hand side can be bounded as follows\n¯R1=C/integraldisplayTn\n∗\n0/summationdisplay\nq≥−123q\n2/ba∇dbl∆qv1/ba∇dblL2/summationdisplay\nq−N≤k≤q+N23k\n2/ba∇dbl∆kv1/ba∇dblL2dτ\n≤C/integraldisplayTn\n∗\n0/summationdisplay\nq≥−1\n23q/ba∇dbl∆qv1/ba∇dbl2\nL2+/summationdisplay\nq−N≤k≤q+N23k/ba∇dbl∆kv1/ba∇dbl2\nL2\ndτ\n≤CN/parenleftbigg\n/ba∇dblv1/ba∇dbl2\nL2\nt˙H3\n2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x/parenrightbigg\n;\n29¯R2=C21−3\n2N∞/summationdisplay\nq=N/integraldisplayTn\n∗\n023q/ba∇dbl∆qv1/ba∇dblL2q−N−1/summationdisplay\nk=−123\n2(k−(q−N))/ba∇dbl∆kv1/ba∇dblL2dτ\n≤C21−3\n2N\n∞/summationdisplay\nq=N/parenleftigg/integraldisplayTn\n∗\n023q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n1\n2\n∞/summationdisplay\nk0=0/parenleftiggk0−1/summationdisplay\nk=−12−3\n2(k0−k)esssup\nt∈(0,Tn\n∗)/ba∇dbl∆kv1/ba∇dblL2/parenrightigg2\n1\n2\n≤C21−3\n2N\n∞/summationdisplay\nq=0/parenleftigg/integraldisplayTn\n∗\n023q/ba∇dbl∆qv1/ba∇dblL2dτ/parenrightigg2\n1\n2/parenleftigg∞/summationdisplay\nk=−1esssup\nt∈(0,Tn∗)/ba∇dbl∆kv1/ba∇dbl2\nL2/parenrightigg1\n2∞/summationdisplay\nk=−12−3\n2k.\nTherefore,\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x≤C(ν)2−N/parenleftig\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x/parenrightig\n/ba∇dbl∇vn/ba∇dbl2\nL2\ntL2x+CN/parenleftbigg\n/ba∇dblv1/ba∇dbl2\nL2\nt˙H3\n2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x/parenrightbigg\n+C/ba∇dblv0/ba∇dbl2\nL2,\nwhich by choosing\nN=/ceilingleftig\nlog2/parenleftig\n4+2C(ν)/ba∇dbl∇vn/ba∇dbl2\nL2\ntL2\nx/parenrightig/ceilingrightig\nand using the energy estimates of v1andvnyields\n/ba∇dblv1/ba∇dbl2\nL2\ntL∞x≤C(ν)/parenleftbigg/parenleftbigg\n/ba∇dblv1/ba∇dbl2\nL2\nt˙H3\n2x+/ba∇dblv1/ba∇dbl2\nL2\ntL2x/parenrightbigg\n(1+/ba∇dbl∇vn/ba∇dbl2\nL2\ntL2x)+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x+E2\n0/parenrightbigg\n≤C(E0,ν)/parenleftig\n1+(Tn\n∗)1\n3+(Tn\n∗)5\n3/parenrightig\n+C(ν)/ba∇dblv2/ba∇dbl2\nL2\ntL∞\nx.\nIn addition,\n/ba∇dblv/ba∇dbl2\nL2\ntL∞x≤ /ba∇dblv1/ba∇dbl2\nL2\ntL∞x+/ba∇dblv2/ba∇dbl2\nL2\ntL∞x≤C(E0,ν)/parenleftig\n1+(Tn\n∗)1\n3+(Tn\n∗)5\n3/parenrightig\n+C(ν)/ba∇dblv2/ba∇dbl2\nL2\ntL∞x.\nIt remains to bound the term /ba∇dblv2/ba∇dblL2\ntL∞\nx. It can be seen that for 0 ≤t0< t≤Tn\n∗\n/ba∇dblv2/ba∇dbl2\nL∞(t0,t;L2)+/ba∇dblv2/ba∇dbl2\nL2(t0,t;˙H3\n2)≤ /ba∇dbl(vn,v1)/ba∇dbl2\nL∞(t0,t;L2)+/ba∇dbl(vn,v1)/ba∇dbl2\nL2(t0,t;˙H3\n2)≤C(E0,ν)/parenleftig\n1+(Tn\n∗)1\n3/parenrightig\n.\nWe will use the following Besov-type maximal regularity estimate for t he forced fractional heat equation of v2\nin which its proof will be provided in Appendix D (see Section 7) fort∈(0,Tn\n∗]\n/ba∇dblv2/ba∇dbl\nL2(0,t;˙Bs+3\n2\n2,1)≤C(s)/ba∇dbljn×Bn/ba∇dbl\nL2(0,t;˙Bs−3\n2\n2,1)≤C(s)/ba∇dbljn/ba∇dblL2(0,t;L2)/ba∇dblBn/ba∇dblL∞(0,t;˙Hs), (3.1)\nwhere we used the paraproduct rule ( 7.1) in the second inequality. An application of [ 2, Lemmas 7.3 and 7.4]\nyields for s′>3\n2and 0≤t0< t≤Tn\n∗\n/ba∇dbl(Id−˙S0)v2/ba∇dblL2(t0,t;L∞)≤C(s)/ba∇dblv2/ba∇dblL2(t0,t;˙H3\n2)log1\n2/parenleftigg\ne+/ba∇dblv2/ba∇dblL2(t0,t;˙Bs′\n2,1)\n/ba∇dblv2/ba∇dblL2(t0,t;˙H3\n2)/parenrightigg\n,\n/ba∇dbl˙S0v2/ba∇dblL2(t0,t;L∞)≤C(s)/ba∇dblv2/ba∇dblL2(t0,t;˙H3\n2)log1\n2/parenleftigg\ne+/ba∇dblv2/ba∇dblL2(t0,t;L2)\n/ba∇dblv2/ba∇dblL2(t0,t;˙H3\n2)/parenrightigg\n.\nBy choosing s′=s+3\n2, using (3.1), the estimate of v2and the increasing of the function z/ma√sto→zlog(e+C\nz) for\nz >0, we find that\n/ba∇dblv2/ba∇dbl2\nL2(t0,t;L∞)≤C(E0,ν,s)/parenleftig\n1+(Tn\n∗)1\n3/parenrightig\nlog(e+t−t0)+C(σ,s)/ba∇dblv2/ba∇dbl2\nL2(t0,t;˙H3\n2)log\ne+E2\n0/ba∇dblBn/ba∇dbl2\nL2(t0,t;˙Hs)\n/ba∇dblv2/ba∇dbl2\nL2(t0,t;˙H3\n2)\n.\nBy using an upper bound on /ba∇dblv/ba∇dblL2\ntL∞xin terms of /ba∇dblv2/ba∇dblL2\ntL∞xand that of /ba∇dblv2/ba∇dbl\nL2\nt˙H3\n2x, an iteration as [ 2, The proof\nof Theorem 1.2] can be applied (replacing /ba∇dblue/ba∇dblL2\nt˙H1xby/ba∇dblv2/ba∇dbl\nL2\nt˙H3\n2x) to the ˙Hsestimate of ( En,Bn), which yields\nfort∈(0,Tn\n∗]\nE2\n0/ba∇dbl(En,Bn)/ba∇dbl2\nL∞(0,t;˙Hs)≤/parenleftbig\ne+E2\n0/ba∇dbl(E0,B0)/ba∇dbl2\n˙Hs+t/parenrightbigC(E0,ν,σ,s)/parenleftBig\n1+(Tn\n∗)1\n3+(Tn\n∗)5\n3/parenrightBig\n,\n30and for some C(E0,ν,σ,s)>1\n/ba∇dblvn/ba∇dbl2\nL2(0,t;L∞)≤C(E0,ν,σ,s)/parenleftig\n1+(Tn\n∗)1\n3+(Tn\n∗)5\n3/parenrightig/parenleftbig\ne+E2\n0/ba∇dbl(E0,B0)/ba∇dbl2\n˙Hs+t/parenrightbigC(E0,ν,σ,s)/parenleftBig\n1+(Tn\n∗)1\n3+(Tn\n∗)5\n3/parenrightBig\n.\nStep 3: The case δ=s.This step includes three substeps as follows.\nStep 3a: The case δ=s∈(0,3\n2).Similarly, it can be seen that\n1\n2d\ndt/parenleftbig\n/ba∇dblvn/ba∇dbl2\n˙Hδ+/ba∇dbl(En,Bn)/ba∇dbl2\n˙Hs/parenrightbig\n+ν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+1\nσ/ba∇dbljn/ba∇dbl2\n˙Hs=:3/summationdisplay\nk=1Ik,\nwhere for some ǫ∈(0,1), since δ=s\nI1=/integraldisplay\nR3Λδ(jn×Bn)·Λδvndx\n≤ /ba∇dbljn/ba∇dblL2/ba∇dblBn/ba∇dbl\nL6\n3−2δ/ba∇dblΛ2δvn/ba∇dblL6\n2δ\n≤ǫν\n2/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nL2/ba∇dblBn/ba∇dbl2\n˙Hδ;\nI2=−/integraldisplay\nR3Λδ(v·∇vn)·Λδvndx\n≤ /ba∇dblvn/ba∇dbl\nL6\n3−2δ/ba∇dbl∇vn/ba∇dblL2/ba∇dblΛ2δvn/ba∇dblL6\n2δ\n≤ǫν\n2/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/parenleftig\n/ba∇dblvn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙H3\n2/parenrightig\n/ba∇dblvn/ba∇dbl2\n˙Hδ;\nI3=/integraldisplay\nR3Λs(v×Bn)·Λsjndx\n≤C(s)/parenleftig\n/ba∇dblΛsvn/ba∇dblL6\n2s/ba∇dblBn/ba∇dbl\nL6\n3−2s+/ba∇dblvn/ba∇dblL∞/ba∇dblΛsBn/ba∇dblL2/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤C(s)/parenleftig\n/ba∇dblvn/ba∇dbl˙H3\n2/ba∇dblBn/ba∇dbl˙Hs+/ba∇dblvn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hs/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\n˙H3\n2/ba∇dblBn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nL∞/ba∇dblBn/ba∇dbl2\n˙Hs.\nTherefore, by choosing ǫ=1\n2and using ( 2.2) withd= 3 and s0=s+1\n2\nd\ndt/parenleftbig\n/ba∇dblvn/ba∇dbl2\nHδ+/ba∇dbl(En,Bn)/ba∇dbl2\nHs/parenrightbig\n+ν/ba∇dblvn/ba∇dbl2\nHδ+3\n2+1\nσ/ba∇dbljn/ba∇dbl2\nHs≤C(δ,ν,σ)G/ba∇dbl(vn,Bn)/ba∇dbl2\nHs+ν/ba∇dblvn/ba∇dbl2\nL2\n+/bracketleftigg\n1\n2C(σ,s)/ba∇dblBn/ba∇dblHs/ba∇dblvn/ba∇dblH3\n2/parenleftigg\n1+log1\n2/parenleftigg\n/ba∇dblvn/ba∇dblHs+3\n2\n/ba∇dblvn/ba∇dblH3\n2/parenrightigg/parenrightigg/bracketrightigg2\n,\nwhere\nG(t) :=/ba∇dblvn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙H3\n2+/ba∇dbljn/ba∇dbl2\nL2,\nwhich yields the conclusion as that of Step 3 in the proof of Theorem 1.1.\nStep 3b: The case δ=s=3\n2.In this case, we find that for some ǫ0∈(0,3\n2)\nI1≤ /ba∇dbljn/ba∇dblL6/ba∇dblBn/ba∇dblL3/ba∇dblΛ2δvn/ba∇dblL2≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nH1/ba∇dblBn/ba∇dbl2\nH1;\nI2≤ /ba∇dblvn/ba∇dblL6/ba∇dbl∇vn/ba∇dblL3/ba∇dblΛ2δvn/ba∇dblL2≤ǫν/ba∇dblvn/ba∇dbl2\n˙H3\n2+C(ǫ,ν,δ)/parenleftig\n/ba∇dblvn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙H3\n2/parenrightig\n/ba∇dblvn/ba∇dbl2\n˙H3\n2;\nI3≤C(s)/parenleftig\n/ba∇dblΛsvn/ba∇dbl\nL6\n2ǫ0/ba∇dblBn/ba∇dbl\nL6\n3−2ǫ0+/ba∇dblvn/ba∇dblL∞/ba∇dblΛsBn/ba∇dblL2/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\n˙Hs−ǫ0+3\n2/ba∇dblBn/ba∇dbl2\n˙Hǫ0+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nL∞/ba∇dblBn/ba∇dbl2\n˙Hs,\nwhich together with Step 3a (to bound /ba∇dblvn/ba∇dbl\nL2\ntHs−ǫ0+3\n2x,/ba∇dblvn/ba∇dblL2\ntL∞\nxand/ba∇dbljn/ba∇dblL2\ntH1\nx) closes the Hsestimate.\nStep 3c: The case δ=s >3\n2.In this case, it follows that for some ǫ∈(0,1)\nI1≤C(δ)(/ba∇dbljn/ba∇dbl˙Hδ/ba∇dblBn/ba∇dblL∞+/ba∇dbljn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hδ)/ba∇dblvn/ba∇dbl˙Hδ=:I11+I12,\nI11≤C(δ)/ba∇dbljn/ba∇dbl˙Hδ/ba∇dblBn/ba∇dbl2δ−3\n2δ\nL2/ba∇dblBn/ba∇dbl3\n2δ\n˙Hδ/ba∇dblvn/ba∇dbl3\n2δ\n˙H3\n2/ba∇dblvn/ba∇dbl2δ−3\n2δ\n˙Hδ+3\n2\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+C(ǫ,δ,σ)/ba∇dblBn/ba∇dbl2δ−3\nδ\nL2/ba∇dblBn/ba∇dbl3\nδ\n˙Hδ/ba∇dblvn/ba∇dbl3\nδ\n˙H3\n2/ba∇dblvn/ba∇dbl2δ−3\nδ\n˙Hδ+3\n2\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,σ)/ba∇dblBn/ba∇dbl2(2δ−3)\n3\nL2/ba∇dblBn/ba∇dbl2\n˙Hδ/ba∇dblvn/ba∇dbl2\n˙H3\n2,\n31I12≤C(δ)/ba∇dbljn/ba∇dbl2δ−3\n2δ\nL2/ba∇dbljn/ba∇dbl3\n2δ\n˙Hδ/ba∇dblBn/ba∇dbl˙Hδ/ba∇dblvn/ba∇dbl3\n2δ\n˙H3\n2/ba∇dblvn/ba∇dbl2δ−3\n2δ\n˙Hδ+3\n2\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+C(ǫ,δ,σ)/ba∇dbljn/ba∇dbl2(2δ−3)\n4δ−3\nL2/ba∇dblBn/ba∇dbl4δ\n4δ−3\n˙Hδ/ba∇dblvn/ba∇dbl6\n4δ−3\n˙H3\n2/ba∇dblvn/ba∇dbl4δ−6\n4δ−3\n˙Hδ+3\n2\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,σ)/ba∇dbljn/ba∇dbl2δ−3\nδ\nL2/ba∇dblvn/ba∇dbl3\nδ\n˙H3\n2/ba∇dblBn/ba∇dbl2\n˙Hδ\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,σ)/parenleftig\n/ba∇dbljn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙H3\n2/parenrightig\n/ba∇dblBn/ba∇dbl2\n˙Hδ;\nI2=/integraldisplay\nR3Λδ−1\n2(vn⊗vn)·Λδ+3\n2vndx\n≤C(δ)/ba∇dblΛδ−1\n2vn/ba∇dblL3/ba∇dblvn/ba∇dblL6/ba∇dblΛδ+3\n2vn/ba∇dblL2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/parenleftig\n/ba∇dblvn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙H3\n2/parenrightig\n/ba∇dblvn/ba∇dbl2\n˙Hδ;\nI3≤C(δ)(/ba∇dblvn/ba∇dbl˙Hδ/ba∇dblBn/ba∇dblL∞+/ba∇dblvn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hδ)/ba∇dbljn/ba∇dbl˙Hδ=:I31+I32,\nI31≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,σ)/ba∇dblBn/ba∇dbl2(2δ−3)\n3\nL2/ba∇dblBn/ba∇dbl2\n˙Hδ/ba∇dblvn/ba∇dbl2\n˙H3\n2,\nI32≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hδ+C(ǫ,δ,σ)/ba∇dblvn/ba∇dbl2\nL∞/ba∇dblBn/ba∇dbl2\n˙Hδ,\nwhere we used Lemma 7.1. Thus, we can use Step 3a to bound /ba∇dblvn/ba∇dblL2\ntL∞x, which closes the main estimate.\nStep 4: The case 0< δ≤s <3\n2.In this case, by using Step 3a, we need to estimate only ( En,Bn), with\nI3is bounded as follows\nI3≤C(s)/parenleftig\n/ba∇dblΛsvn/ba∇dblL6\n2s/ba∇dblBn/ba∇dbl\nL6\n3−2s+/ba∇dblvn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hs/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/parenleftig\n/ba∇dblvn/ba∇dbl2\n˙H3\n2+/ba∇dblvn/ba∇dbl2\nHδ+3\n2/parenrightig\n/ba∇dblBn/ba∇dbl2\n˙Hs.\nStep 5: The case 0< δ≤s=3\n2.Similar to the previous step,\nI3≤C(s)/parenleftig\n/ba∇dblΛsvn/ba∇dbl\nL6\n3−2δ/ba∇dblBn/ba∇dblL6\n2δ+/ba∇dblvn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hs/parenrightig\n/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,δ,σ,s)/ba∇dblvn/ba∇dbl2\n˙Hs+δ/ba∇dblBn/ba∇dbl˙Hδ+C(ǫ,δ,σ,s)/ba∇dblvn/ba∇dbl2\nHδ+3\n2/ba∇dblBn/ba∇dbl2\n˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,δ,σ,s)/ba∇dblvn/ba∇dbl2\nHδ+3\n2/ba∇dblBn/ba∇dbl2\nHs.\nStep 6: The case s >3\n2ands−3\n2≤δ < s.In this case, we find that\nI3≤C(s)(/ba∇dblvn/ba∇dbl˙Hs/ba∇dblBn/ba∇dblL∞+/ba∇dblvn/ba∇dblL∞/ba∇dblBn/ba∇dbl˙Hs)/ba∇dbljn/ba∇dbl˙Hs\n≤ǫ\nσ/ba∇dbljn/ba∇dbl2\n˙Hs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nHδ+3\n2/ba∇dblBn/ba∇dbl2\nHs.\nStep 7: The case s >3\n2ands < δ < s +3\n2.In this case, by using Step 3c, we need to estimate only vn.\nWe write δ=s+ǫ0for some ǫ0∈(0,3\n2), and bound I2as in Step 3c and I1as follows\nI1=/integraldisplay\nR3Λδ−3\n2(jn×Bn)·Λδ+3\n2vndx\n≤C(δ)/parenleftig\n/ba∇dblΛδ−3\n2jn/ba∇dbl\nL6\n2ǫ0/ba∇dblBn/ba∇dbl\nL6\n3−2ǫ0+/ba∇dbljn/ba∇dbl\nL6\n3−2ǫ0/ba∇dblBn/ba∇dbl\nL6\n2ǫ0/parenrightig\n/ba∇dblvn/ba∇dbl˙Hδ+3\n2\n≤C(δ)(/ba∇dbljn/ba∇dbl˙Hs/ba∇dblBn/ba∇dbl˙Hǫ0+/ba∇dbljn/ba∇dbl˙Hǫ0/ba∇dblBn/ba∇dbl˙Hs)/ba∇dblvn/ba∇dbl˙Hδ+3\n2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 8: The case s >3\n2andδ=s+3\n2.Similarly, I2is bounded as in Step 3c and\nI1=/integraldisplay\nR3Λs(jn×Bn)·Λδ+3\n2vndx≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν,s)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 9: The case s=3\n2and3\n2< δ <3.Sinceδ−3\n2∈(0,3\n2), we bound I2as in Step 3c and\nI1=/integraldisplay\nR3Λδ−3\n2(jn×Bn)·Λδ+3\n2vndx\n≤C(δ)/parenleftigg\n/ba∇dblΛδ−3\n2jn/ba∇dbl\nL6\n2(δ−3\n2)/ba∇dblBn/ba∇dbl\nL6\n3−2(δ−3\n2)+/ba∇dbljn/ba∇dbl\nL6\n3−2(δ−3\n2)/ba∇dblΛδ−3\n2Bn/ba∇dbl\nL6\n2(δ−3\n2)/parenrightigg\n/ba∇dblvn/ba∇dbl˙Hδ+3\n2\n32≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\n˙H3\n2/ba∇dblBn/ba∇dbl2\n˙Hδ−3\n2+C(ǫ,δ,ν)/ba∇dblBn/ba∇dbl2\n˙H3\n2/ba∇dbljn/ba∇dbl2\n˙Hδ−3\n2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 10: The case3\n4< s <3\n2and3\n2< δ≤2s.Similar to the previous case, it can be seen that\nI1≤C(s)/parenleftig\n/ba∇dblΛδ−3\n2jn/ba∇dblL6\n2s/ba∇dblBn/ba∇dbl\nL6\n3−2s+/ba∇dbljn/ba∇dbl\nL6\n3−2s/ba∇dblΛδ−3\n2Bn/ba∇dblL6\n2s/parenrightig\n/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2\n≤C(s)(/ba∇dbljn/ba∇dbl˙Hδ−s/ba∇dblBn/ba∇dbl˙Hs+/ba∇dbljn/ba∇dbl˙Hs/ba∇dblBn/ba∇dbl˙Hδ−s)/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 11: The case3\n4≤s <3\n2ands < δ≤3\n2.In this case, I2can bounded as in Steps 3a and 3b. Since\n3\n2−s≤sand 2δ≤δ+3\n2\nI1=/integraldisplay\nR3(jn×Bn)·Λ2δvndx\n≤C(s)/parenleftig\n/ba∇dbljn/ba∇dblL6\n2s/ba∇dblBn/ba∇dbl\nL6\n3−2s+/ba∇dbljn/ba∇dbl\nL6\n3−2s/ba∇dblBn/ba∇dblL6\n2s/parenrightig\n/ba∇dblvn/ba∇dbl˙H2δ\n≤ǫν/parenleftig\n/ba∇dblvn/ba∇dbl2\nL2+/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2/parenrightig\n+C(ǫ,δ,ν)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 12: The case 0< s <3\n4ands < δ≤2s.In this case, I2is estimated as in Step 3a. We\nonly need to focus on the estimate of I1. In addition, for some ǫ∈(0,1), since s∈(0,3\n4),s < δ≤2sand\n3\n2−(δ−s),δ−s\n2∈(0,3\n2), by using ( 2.3)\nI1=/integraldisplay\nR3[(Λs(jn×Bn)−Λsjn×Bn−jn×ΛsBn)+Λsjn×Bn+jn×ΛsBn]·Λ2δ−svndx=:3/summationdisplay\nk=1I1k,\nI11≤ /ba∇dblΛs(jn×Bn)−Λsjn×Bn−jn×ΛsBn/ba∇dbl\nL6\n3+2(3\n2−(δ−s))/ba∇dblΛ2δ−svn/ba∇dbl\nL6\n3−2(3\n2−(δ−s))\n≤C(δ,s)/ba∇dblΛ6s−2δ\n4jn/ba∇dbl\nL12\n3+2(3\n2−(δ−s))/ba∇dblΛδ−s\n2Bn/ba∇dbl\nL12\n3+2(3\n2−(δ−s))/ba∇dblvn/ba∇dbl˙Hδ+3\n2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν,s)/ba∇dblΛsjn/ba∇dbl2\nL2/ba∇dblΛδ−sBn/ba∇dbl2\nL2\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν,s)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs,\nI12≤ /ba∇dbljn/ba∇dbl˙Hs/ba∇dblBn/ba∇dbl\nL6\n3−2(δ−s)/ba∇dblΛ2δ−svn/ba∇dbl\nL6\n2(δ−s)\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,s)/ba∇dbljn/ba∇dbl2\n˙Hs/ba∇dblBn/ba∇dbl2\n˙Hδ−s,\n≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν,s)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs,\nI13≤ǫν/ba∇dblvn/ba∇dbl2\n˙Hδ+3\n2+C(ǫ,δ,ν,s)/ba∇dbljn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 13: Conclusion from Step 2 to Step 12. From Step 2 to Step 12, we can close the Hδ−Hs\nestimate of ( vn,En,Bn), which yields Tn\n∗=∞and uniform bounds in terms of nwith replacing Tn\n∗by any\nT∈(0,∞).\nStep 14: Pass to the limit. This step can be done by applying Steps 16 and 17a for s,δ >3\n2and Step\n17b for either s∈(0,3\n2] orδ∈(0,3\n2], in the proof of Theorem 1.1. We only mention that in the case δ= 0 and\ns∈(0,3\n2), we have ∂tvnis uniformly bounded in L2\ntH−3\n2x. Thus, we will use the injections H1֒→L2֒→H−3\n2\nforvninstead of the previous one in two dimensions. Therefore, we can pa ss to the limit in the same way. We\nomit further details.\nStep 15: Uniqueness. It is enough to prove the uniqueness in the case δ= 0 and s∈(0,3\n2). Similar to\nStep 18 in the proof of Theorem 1.1, the usual energy method does not work here for s∈(0,1). Indeed,\n1\n2d\ndt/ba∇dblv−¯v/ba∇dbl2\nL2+ν/ba∇dblv−¯v/ba∇dbl2\n˙H3\n2=:3/summationdisplay\nk=1¯Ik,\nwhere for some ǫ∈(0,1) and for any s′∈(0,s]\n¯I1=−/integraldisplay\nR3(v−¯v)·∇v·(v−¯v)dx\n≤C/ba∇dblv/ba∇dbl˙H3\n2/ba∇dblv−¯v/ba∇dblL2/ba∇dblv−¯v/ba∇dbl˙H1\n≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H3\n2+C(ǫ,ν)/ba∇dblv/ba∇dbl3\n2\n˙H3\n2/ba∇dblv−¯v/ba∇dbl2\nL2;\n33¯I2=/integraldisplay\nR3(j−¯j)×B·(v−¯v))dx\n≤C(s′)/ba∇dblj−¯j/ba∇dblL2/ba∇dblB/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbl˙H3\n2−s′\n≤C(s′)/ba∇dblj−¯j/ba∇dblL2/ba∇dblB/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbl2s′\n3\nL2/ba∇dblv−¯v/ba∇dbl3−2s′\n3\n˙H3\n2\n≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H3\n2+C(ǫ,ν,s′)/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dblj−¯j/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2;\n¯I3=/integraldisplay\nR3¯j×(B−¯B)·(v−¯v)dx\n≤C(s′)/ba∇dbl¯j/ba∇dblL2/ba∇dblB−¯B/ba∇dbl˙Hs′/ba∇dblv−¯v/ba∇dbl˙H3\n2−s′\n≤ǫν/ba∇dblv−¯v/ba∇dbl2\n˙H3\n2+C(ǫ,ν,s′)/ba∇dblB−¯B/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dbl¯j/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2.\nTherefore, by choosing ǫ=1\n6and taking T∗∈(0,T]\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/integraldisplayT∗\n0/ba∇dblv−¯v/ba∇dbl2\n˙H3\n2dτ≤3/summationdisplay\nk=1¯Jk,\nwhere\n¯J1:=C(ν,s′)/integraldisplayT∗\n0/ba∇dblv/ba∇dbl3\n2\n˙H3\n2/ba∇dblv−¯v/ba∇dbl2\nL2dτ≤C(ν)T1\n4∗/ba∇dblv/ba∇dbl3\n2\nL2(0,T∗;˙H3\n2)/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2);\n¯J2:=C(ν,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dblj−¯j/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ≤3/summationdisplay\nk=1¯J2k,\n¯J21:=C(c,ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dblE−¯E/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(c,ν,σ,s′)T∗/ba∇dblB/ba∇dbl6\n2s′+3\nL∞(0,T∗;˙Hs′)/parenleftig\n/ba∇dblE−¯E/ba∇dbl2\nL∞(0,T∗;L2)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n,\n¯J22:=C(ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dbl(v−¯v)×B/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dblv−¯v/ba∇dbl6\n2s′+3\n˙H3\n2−s/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl12\n2s′+3\n˙Hs′/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl2(3−2s′)\n2s′+3\n˙H3\n2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,σ,s′)T4s′\n2s′+3\n∗/ba∇dblB/ba∇dbl12\n2s′+3\nL∞(0,T∗;˙Hs′)/parenleftbigg\n/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)/parenrightbigg\n,\n¯J23:=C(ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dbl¯v×(B−¯B)/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,σ,s′)/integraldisplayT∗\n0/ba∇dblB/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dbl¯v/ba∇dbl4s′\n2s′+3\nL2/ba∇dbl¯v/ba∇dbl2(3−2s′)\n2s′+3\n˙H3\n2/ba∇dblB−¯B/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,σ,s′)T4s′\n2s′+3\n∗/ba∇dblB/ba∇dbl6\n2s′+3\nL∞(0,T∗;˙Hs′)/ba∇dbl¯v/ba∇dbl4s′\n2s′+3\nL2(0,T∗;L2)/ba∇dbl¯v/ba∇dbl2(3−2s′)\n2s′+3\nL2(0,T∗;˙H3\n2)\n×/parenleftig\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n;\n¯J3:=C(ν,s′)/integraldisplayT∗\n0/ba∇dblB−¯B/ba∇dbl6\n2s′+3\n˙Hs′/ba∇dbl¯j/ba∇dbl6\n2s′+3\nL2/ba∇dblv−¯v/ba∇dbl4s′\n2s′+3\nL2dτ\n≤C(ν,s′)T2s′\n2s′+3\n∗/ba∇dbl¯j/ba∇dbl6\n2s′+3\nL2(0,T∗;L2)/parenleftig\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′)+/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)/parenrightig\n.\nIn addition, by using Lemma 7.4, it follows that\n/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′)≤C(c)/ba∇dblj−¯j/ba∇dbl2\nL1(0,T∗;Hs′)=:6/summationdisplay\nk=4¯Jk,\nwhere for any s′∈(0,s)\n¯J4=C(c,σ)/ba∇dblE−¯E/ba∇dbl2\nL1(0,T∗;Hs′)≤C(c,σ)T2\n∗/ba∇dblE−¯E/ba∇dbl2\nL∞(0,T∗;Hs′);\n34¯J5=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;Hs′)≤¯J51+¯J52,\n¯J51:=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;L2)\n≤C(c,σ,s′)T3+2s′\n3∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)/ba∇dblv−¯v/ba∇dbl4s′\n3\nL∞(0,T∗;L2)/ba∇dblv−¯v/ba∇dbl2(3−2s′)\n3\nL2(0,T∗;˙H3\n2),\n≤C(c,ν,σ,s′)T3+2s′\n3∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)/parenleftbigg\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)/parenrightbigg\n,\n¯J52:=C(c,σ)/ba∇dbl(v−¯v)×B/ba∇dbl2\nL1(0,T∗;˙Hs′)\n≤C(ν,c,σ,s′)T∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)\n+C(c,σ,s,s′)T3+2(s−s′)\n∗ /ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs)/ba∇dblv−¯v/ba∇dbl4(s−s′)\n3\nL∞(0,T∗;L2)/ba∇dblv−¯v/ba∇dbl2(3−2(s−s′))\n3\nL2(0,T∗;˙H3\n2)\n≤C(c,ν,σ,s′)T∗/ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs′)ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)\n+C(c,ν,σ,s,s′)T3+2(s−s′)\n∗ /ba∇dblB/ba∇dbl2\nL∞(0,T∗;˙Hs)/parenleftbigg\n/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)/parenrightbigg\n;\n¯J6=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;Hs′)≤¯J61+¯J62,\n¯J61:=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;L2)\n≤C(c,σ,s′)T3+2s′\n3∗/ba∇dbl¯v/ba∇dbl4s′\n3\nL∞(0,T∗;L2)/ba∇dbl¯v/ba∇dbl2(3−2s′)\n3\nL2(0,T∗;˙H3\n2)/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;˙Hs′),\n¯J62:=C(c,σ)/ba∇dbl¯v×(B−¯B)/ba∇dbl2\nL1(0,T∗;˙Hs′)\n≤C(c,σ)T∗/parenleftbigg\n/ba∇dbl¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)+/ba∇dbl¯v/ba∇dbl2\nL2(0,T∗;L∞)/parenrightbigg\n/ba∇dblB−¯B/ba∇dbl2\nL∞(0,T∗;Hs′).\nCombining all the above estimates and using Step 1, we find that for s ufficiently small T∗\nA(v−¯v,E−¯E,B−¯B) :=/ba∇dblv−¯v/ba∇dbl2\nL∞(0,T∗;L2)+ν/ba∇dblv−¯v/ba∇dbl2\nL2(0,T∗;˙H3\n2)+/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′)\n≤1\n2A(v−¯v,E−¯E,B−¯B),\nwhich yields v= ¯v,E=¯EandB=¯Bin (0,T∗). By repeating this process, we obtain the conclusion in the\nwhole time interval (0 ,T). Finally, we note that only the estimate of ¯J52needss′< sand other ones hold for\ns′=sas well.\nProof of Theorem 1.2-(ii).Wewillfollowtheideain theproofof[ 2, Corollary1.3],wheretheauthorsconsidered\nthe case ν >0,α= 1 and d= 2, and proved that up to an extraction of a subsequence ( vc,Bc)→(v,B) as\nc→ ∞in the sense of distributions. We aim to apply the same idea to the case ν >0,α=3\n2andd= 3. It\nsuffices to focus on the case δ= 0 and s∈(0,3\n2). It can be seen from ( NSM) withα=3\n2that\n\n\n∂tvc+vc·∇vc+∇πc=−ν(−∆)3\n2vc+jc×Bc,\n1\nc∂tEc−∇×Bc=−jc,\n∂tBc−∇×(vc×Bc) =−1\nσ∇×jc,\ndivvc= divBc= 0.(3.2)\nBy applying Step 1 in Part ( i), we know that ( vc,Ec,Bc,jc) is uniformly bounded in terms of cfor any\nT∈(0,∞) in the following spaces\nvc∈L∞(0,T;L2)∩L2(0,T;H3\n2),(Ec,Bc)∈L∞(0,T;Hs) and jc∈L2(0,T;Hs),\nwhichimpliesthatthereexists( v,E,B,j ) suchthatuptoanextractionofasubsequence(usethe sameno tation)\nasc→ ∞\n(vc,Ec,Bc)∗⇀(v,E,B) in L∞\nt(L2\nx×Hs\nx×Hs\nx),\n(vc,jc)⇀(v,j) in L2\nt(H3\n2x×Hs\nx).\nIn addition, we find from ( 3.2) that\n(∂tvc,∂tBc) is uniformly bounded in L2\nt(H−3\n2x×H−1\nx)\n35and by using the Aubin-Lions lemma as c→ ∞\n(vc,Bc)→(v,B) (locally in space) in L2\nt,x.\nAs in Step 17b in the proof of Theorem 1.1, forφ,ϕ∈C∞\n0([0,T)×R3;R3) with div φ= 0, the weak form of\n(3.2) is given by (similar to those of a),b) andc))\na′)/integraldisplayT\n0/integraldisplay\nR3vc·∂tφ+(vc⊗vc) :∇φ−νvc·(−∆)3\n2φ+(jc×Bc)·φdxdt=−/integraldisplay\nR3vc(0)·φ(0)dx,\nb′)/integraldisplayT\n0/integraldisplay\nR31\ncEc·∂tϕ+Bc·(∇×ϕ)−jc·ϕdxdt=−/integraldisplay\nR31\ncEc(0)·ϕ(0)dx,\nc′)/integraldisplayT\n0/integraldisplay\nR3Bc·∂tϕ+[(vc×Bc)−1\nσjc]·(∇×ϕ)dxdt=−/integraldisplay\nR3Bc(0)·ϕ(0)dx.\nTherefore, we can pass to the limit by using ( vc\n0,Ec\n0,Bc\n0)⇀(¯v0,¯E0,¯B0) inL2×Hs×Hsand the above strong\nconvergences as c→ ∞to obtain that ( 3.2) converges in the sense of distributions to\n\n\n∂tv+v·∇v+∇π=−ν(−∆)3\n2v+j×B,\n∂tB−∇×(v×B) =−1\nσ∇×j,\ndivv= divB= 0,\nwherej=∇×Band (v,B)|t=0= (¯v0,¯B0). Thus, the proof is finished since ∇×(∇×B) =−∆B.\n4 Proof of Theorem 1.3\nIn this subsection, we focus on giving the standard proof of Theor em1.3, which shares similar ideas as those of\nTheorems 1.1and1.2.\nProof of Theorem 1.3.The proof consists of several steps as follows.\nStep 1: Local existence, Hsestimate and uniform bound. As the proofs of Theorems 1.1and1.2,\nwe will use ( 2.1) withν= 0 as an approximate system. It can be seen from ( 2.1) withν= 0 that\n1\n2d\ndt/ba∇dbl(vn,En,Bn)/ba∇dbl2\nHs+1\nσ/ba∇dbljn/ba∇dbl2\nHs=:3/summationdisplay\nk=1Jk,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nJ1+J3=/integraldisplay\nRdJs(jn×Bn)·Jsvn+Jsjn·Js(vn×Bn)dx≤ǫ\nσ/ba∇dbljn/ba∇dbl2\nHs+C(ǫ,σ,s)/parenleftbig\n/ba∇dbl(vn,Bn)/ba∇dbl2\nHs/parenrightbig2;\nJ2=−/integraldisplay\nRd[Js(vn·∇vn)−vn·∇Jsvn]·Jsvndx≤C(s)/ba∇dblvn/ba∇dbl3\nHs,\nhere we used the following well-known Kato-Ponce commutator estim ate (see [ 47])\n/ba∇dblJr(fg)−fJrg/ba∇dblL2≤C(r)/parenleftbig\n/ba∇dblJrf/ba∇dblL2/ba∇dblg/ba∇dblL∞+/ba∇dbl∇f/ba∇dblL∞/ba∇dblJr−1g/ba∇dblL2/parenrightbig\n∀r >0.\nBy choosing ǫ=1\n2, we find that\nd\ndtYn,s+1\nσ/ba∇dbljn/ba∇dbl2\nHs≤C(σ,s)Y2\nn,s,\nwhereYn,s(t) :=/ba∇dbl(vn,En,Bn)(t)/ba∇dbl2\nHs+ 1 fort∈(0,Tn\n∗). It can be seen that the above estimate implies an\nuniform bound in terms of nforYn,sin (0,T0) for some T0=T0(σ,s,v0,E0,B0)>0 (does not depend on n)\nand fort∈(0,T0)\n/ba∇dbl(vn,En,Bn)(t)/ba∇dbl2\nHs+/integraldisplayt\n0/ba∇dbljn/ba∇dbl2\nHsdτ≤C(T0,σ,s,v 0,E0,B0).\nStep 2: Pass to the limit. In this case, since ν= 0 and δ=s >d\n2+1, we need to modify the estimates\nofI42andI53in Step 16 in the proof of Theorem 1.1in the following way (for I41,I51,I61andI63, we replace\nR2byRdwith using the same estimates)\nI42=−/integraldisplay\nRdTm((vn−vm)·∇vn)·(vn−vm)dx≤ /ba∇dbl∇vn/ba∇dblL∞/ba∇dblvn−vm/ba∇dbl2\nL2;\n36I53=/integraldisplay\nRdTm(jm×(Bn−Bm))·(vn−vm)dx≤ /ba∇dbljm/ba∇dblL∞/ba∇dbl(vn−vm,Bn−Bm)/ba∇dbl2\nL2,\nwhich showsthat ( vn,En,Bn) andjnareCauchy sequencesin L∞(0,T0;L2(Rd)) andL2(0,T0;L2(Rd)) by using\nStep 1. Therefore, we can pass to the limit as in Step 17a in the proof of Theorem 1.1by replacing R2byRd\nwith receiving the limit system ( 2.5) forν= 0. We skip further details.\nStep 3: Uniqueness. Assume that ( v,E,B,π ) and (¯v,¯E,¯B,¯π) are two solutions to ( NSM) withν= 0\nand the same initial data. As in Step 18 in the proof of Theorem 1.1, it follows that\n1\n2d\ndt/ba∇dbl(v−¯v,E−¯E,B−¯B)/ba∇dbl2\nL2+1\nσ/ba∇dblj−¯j/ba∇dbl2\nL2=:3/summationdisplay\nk=1¯Ik,\nwhere for some ǫ∈(0,1)\n¯I1=−/integraldisplay\nRd(v−¯v)·∇v·(v−¯v)dx≤ /ba∇dbl∇v/ba∇dblL∞/ba∇dblv−¯v/ba∇dbl2\nL2;\n¯I2=/integraldisplay\nRd(¯j×(B−¯B))·(v−¯v)dx≤ /ba∇dbl¯j/ba∇dblL∞/ba∇dbl(v−¯v,B−¯B)/ba∇dbl2\nL2;\n¯I3=/integraldisplay\nRd(j−¯j)·(¯v×(B−¯B))dx≤ǫ\nσ/ba∇dblj−¯j/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dbl¯v/ba∇dbl2\nL∞/ba∇dblB−¯B/ba∇dbl2\nL2,\nwhich yields for ǫ=1\n2\nd\ndt/ba∇dbl(v−¯v,E−¯E,B−¯B)/ba∇dbl2\nL2+1\nσ/ba∇dblj−¯j/ba∇dbl2\nL2≤C(σ)/parenleftbig\n/ba∇dbl(∇v,¯j)/ba∇dblL∞+/ba∇dbl¯v/ba∇dbl2\nL∞/parenrightbig\n/ba∇dbl(v−¯v,B−¯B)/ba∇dbl2\nL2.\nTherefore, the uniqueness follows by using Step 1.\nStep 4: Inviscid limit. We should remark here in the two-dimensional case that given T0>0 then for\nanyν >0, there exists a unique solution ( vν,Eν,Bν) to (NSM) given in Theorem 1.1in (0,T0). Ifd= 3\nthen an application of Steps 1, 2 and 3 above gives us the local existe nce and uniqueness of ( vν,Eν,Bν) to\n(NSM) withν >0 andα= 1 in the same time interval (0 ,T0). Therefore, T0does not depends on ν. Let\n(vν,Eν,Bν,πν) and (v,E,B,π ) be the corresponding solutions to ( NSM) withν >0 andν= 0 satisfying\n(vν,Eν,Bν)|t=0= (v,E,B)|t=0= (v0,E0,B0). Similar to the proof of uniqueness in the previous step with\nreplacing ( v,E,B,π,j ) and (¯v,¯E,¯B,¯π,¯j) by (vν,Eν,Bν,πν,jν) and (v,E,B,π,j ), respectively, there are two\nadditional terms, one on the left-hand side of the energy equality r elated to the viscosity and the other one on\nthe right-hand side denoted by ¯I4. We will bound ¯I4and also need to modify the estimate of ¯I1as follows\n¯I1=−/integraldisplay\nRd(vν−v)·∇vν·(vν−v)dx=−/integraldisplay\nRd(vν−v)·∇v·(vν−v)dx≤ /ba∇dbl∇v/ba∇dblL∞/ba∇dblvν−v/ba∇dbl2\nL2;\n¯I4:=ν/integraldisplay\nRd∆v·(vν−v)dx≤ν2/ba∇dbl∆v/ba∇dbl2\nL2+/ba∇dblvν−v/ba∇dbl2\nL2,\nin which we find that for Yν(t) :=/ba∇dbl(vν−v,Eν−E,Bν−B)(t)/ba∇dbl2\nL2witht∈(0,T0)\nd\ndtYν+ν/ba∇dbl∇(vν−v)/ba∇dbl2\nL2+1\nσ/ba∇dbljν−j/ba∇dbl2\nL2≤C(σ)/parenleftbig\n/ba∇dbl(∇v,|v|2,j)/ba∇dblL∞+1/parenrightbig\nYν+ν2/ba∇dbl∆v/ba∇dbl2\nL2,\nwhich yields\nYν(t)≤ν2/integraldisplayt\n0/ba∇dbl∆v/ba∇dbl2\nL2dτexp/braceleftbigg\nC(σ)/integraldisplayt\n0/ba∇dbl(∇v,|v|2,j)/ba∇dblL∞+1dτ/bracerightbigg\n.\nBy using Step 1, for s′∈[0,s)\n/ba∇dbl(vν−v)(t)/ba∇dblHs′≤ /ba∇dbl(vν−v)(t)/ba∇dbls−s′\ns\nL2/ba∇dbl(vν−v)(t)/ba∇dbls′\ns\nHs≤νs−s′\nsC(T0,σ,s,v 0,E0,B0),\nwhich is similarly for ( Eν−E,Bν−B) and gives us the conclusion. In addition, the bound conthe right-h and\nside does not depend on νsince during the proof we do not use any bounds on ( vν,Eν,Bν,jν), but only ones\non (v,E,B,j ).\nStep 5: The limit c→ ∞.It can be seen from ( NSM) withν= 0 that\n\n\n∂tvc+vc·∇vc+∇πc=jc×Bc,\n1\nc∂tEc−∇×Bc=−jc,\n∂tBc−∇×(vc×Bc) =−1\nσ∇×jc,\ndivvc= divBc= 0.(4.1)\n37By applying Step 1, we know that the local solution ( vc,Ec,Bc,jc) is uniformly bounded in terms of cin the\nfollowing spaces\n(vc,Ec,Bc)∈L∞(0,T0;Hs) and jc∈L2(0,T0;Hs),\nwhichimpliesthatthereexists( v,E,B,j ) suchthatuptoanextractionofasubsequence(usethe sameno tation)\nasc→ ∞\n(vc,Ec,Bc)∗⇀(v,E,B) in L∞\ntHs\nx,\njc⇀ j inL2\ntHs\nx.\nIn addition, we find from ( 4.1) that\n(∂tvc,∂tBc) is uniformly bounded in L2\ntHs−1\nx\nand by using the Aubin-Lions lemma as c→ ∞\n(vc,Bc)→(v,B) (locally in space) in L2\nt,x.\nAs in Step 17b in the proof of Theorem 1.1, forφ,ϕ∈C∞\n0([0,T0)×Rd;R3) with div φ= 0, the weak form of\n(4.1) is given by (similar to those of a),b) andc))\na′′)/integraldisplayT0\n0/integraldisplay\nRdvc·∂tφ+(vc⊗vc) :∇φ+(jc×Bc)·φdxdt=−/integraldisplay\nRdvc(0)·φ(0)dx,\nb′′)/integraldisplayT0\n0/integraldisplay\nRd1\ncEc·∂tϕ+Bc·(∇×ϕ)−jc·ϕdxdt=−/integraldisplay\nRd1\ncEc(0)·ϕ(0)dx,\nc′′)/integraldisplayT0\n0/integraldisplay\nRdBc·∂tϕ+[(vc×Bc)−1\nσjc]·(∇×ϕ)dxdt=−/integraldisplay\nRdBc(0)·ϕ(0)dx.\nTherefore, we can pass to the limit by using the weak convergence o f (vc\n0,Ec\n0,Bc\n0) to (¯v0,¯E0,¯B0) inHsand the\nabove strong convergences as c→ ∞to obtain that ( 4.1) converges in the sense of distributions to\n\n\n∂tv+v·∇v+∇π=j×B,\n∂tB−∇×(v×B) =−1\nσ∇×j,\ndivv= divB= 0,\nwherej=∇×Band (v,B)|t=0= (¯v0,¯B0). Thus, the proof is finished since ∇×(∇×B) =−∆B.\n5 Proof of Theorem 1.4\nIn this section, we will provide a proof of Theorem 1.4. The proof shares similar ideas to those of the previous\nsections. However, some modifications are needed due to the appe arance of new terms, which are related to the\nconstant magnetic vector B∗.\nProof of Theorem 1.4-(i).The proof contains several steps as follows.\nStep 1: Local existence. As mentioned previously, since B∗is a constant vector in R3then∇×B∗=\nE∗=∇π∗= 0 in (NSM*). We will use an approximate system of ( NSM*), which is a slightly modification of\n(2.1), where jnis replaced by ¯jnandFn\n1is redefined as follows\njn\n∗=σTn(vn×B∗),\n¯jn=jn+jn\n∗=σ(cEn+Tn(vn×(Bn+B∗))),\nFn\n1=−νvn−P(Tn(vn·∇vn))+P(Tn(¯jn×(Bn+B∗))).\nTherefore, similar to the proof of Theorem 1.1, there exists a unique solution ( vn,En,Bn)∈C1([0,Tn\n∗),Vs\nn×\nHs\nn×Vs\nn) for some Tn\n∗>0 satisfying the following property: if Tn\n∗<∞then\nlim\nt→Tn∗/ba∇dbl(vn,En,Bn)(t)/ba∇dbl2\nHs=∞.\nStep 2: Hsestimate. Assume that Tn\n∗<∞. The energy balance is given by\n1\n2d\ndt/ba∇dbl(vn,En,Bn)/ba∇dbl2\nL2+ν/ba∇dblvn/ba∇dbl2\nL2+1\nσ/ba∇dbl¯jn/ba∇dbl2\nL2= 0.\n38In addition, the Hsestimate is\n1\n2d\ndt/ba∇dbl(vn,En,Bn)/ba∇dbl2\nHs+ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl¯jn/ba∇dbl2\nHs=:5/summationdisplay\ni=1Ji,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nJ1=/integraldisplay\nRdJs(¯jn×Bn)·Jsvndx+/integraldisplay\nRdJs(¯jn×B∗)·Jsvndx=:J11+J12,\nJ11≤ǫ\nσ/ba∇dbl¯jn/ba∇dbl2\nHs+C(ǫ,σ,s)/ba∇dblBn/ba∇dbl2\nHs/ba∇dblvn/ba∇dbl2\nHs,\nJ12=/integraldisplay\nRd(Js¯jn×B∗)·Jsvndx;\nJ2=−/integraldisplay\nRd[Js(vn·∇vn)−vn·∇Jsvn]·Jsvndx≤C(s)/ba∇dblvn/ba∇dbl3\nHs;\nJ3=/integraldisplay\nRdJs¯jn·Js(vn×Bn)dx+/integraldisplay\nRdJs¯jn·Js(vn×B∗)dx=:J31+J32,\nJ31≤ǫ\nσ/ba∇dbl¯jn/ba∇dbl2\nHs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs,\nJ32=/integraldisplay\nRdJs¯jn·(Jsvn×B∗)dx=−J12;\nJ4=/integraldisplay\nRdJs(∇×Bn)·Js(cEn)dx;\nJ5=−/integraldisplay\nRdJs(∇×En)·Js(cBn)dx=−J4.\nTherefore, by choosing ǫ=1\n4\nd\ndt/ba∇dbl(vn,En,Bn)/ba∇dbl2\nHs+ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl¯jn/ba∇dbl2\nHs≤C(s)/ba∇dblvn/ba∇dbl3\nHs+C(s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs.\nStep 3: Bootstrap argument. By defining the following energy form for t≥0\nEn(t) := esssup\nτ∈[0,t]/ba∇dbl(vn,En,Bn)(τ)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl¯jn/ba∇dbl2\nHsdτ,\nit follows that for some fixed positive constants C1=C1(ν,s) andC2=C2(ν,σ,s), and for t∈(0,Tn\n∗)\nEn(t)≤En(0)+C1E3\n2n(t)+C2E2\nn(t). (5.1)\nTo the end of this step, we aim to prove the following property. Claim:LetSn:={t∈(0,Tn\n∗) :En(t)≤2ǫ2\n0}.\nThenSn= (0,Tn\n∗) andTn\n∗=∞.\n3a) Hypothesis implies conclusion. Assume that for some t∈(0,Tn\n∗)\nEn(t)≤min/braceleftbigg1\n16C2\n1,1\n4C2/bracerightbigg\n=:1\nC2\n0. (5.2)\nTherefore, by choosing ǫ0>0 such that 2 C0ǫ0≤1, it follows from ( 5.1) and (5.2) that\nEn(t)≤2En(0)≤2ǫ2\n0≤1\n2C2\n0. (5.3)\n3b) Conclusion is stronger than hypothesis. Assume that ( 5.3) holds for some t0∈(0,Tn\n∗). For a given\nδ0>0, by the continuity in time of ( vn,En,Bn) inHs, there exists a small tδ0such that\nEn(t)< En(t0)+δ0≤1\n2C2\n0+δ0∀t∈(t0−tδ0,t0+tδ0),\nwhich yields ( 5.2) if we choose δ0≤1\n2C2\n0.\n3c) Conclusion is closed. Lettmandtin (0,Tn\n∗) such that tm→tasm→ ∞. IfEn(tm)≤2ǫ2\n0for all\nm∈Nthen by the continuity in time of ( vn,En,Bn) inHs,En(t)≤2ǫ2\n0as well.\n3d) Base case. By the continuity in time of ( vn,En,Bn) inHs, we can find some Tn\n∗∗∈(0,Tn\n∗)\nEn(t)≤2En(0)≤2ǫ2\n0≤1\n2C2\n0∀t∈(0,Tn\n∗∗).\n39This implies that Snis a non-empty set. We then apply the abstract bootstrap principle (see [65, Proposition\n1.21]) to obtain the first part of the claim, while the second part follow s immediately by using Step 1. Moreover,\nfort≥0\n/ba∇dbl(vn,En,Bn)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl¯jn/ba∇dbl2\nHsdτ≤2ǫ2\n0. (5.4)\nStep 4: Cauchy sequence, pass to the limit and uniqueness. Assume that ( vn,En,Bn) and\n(vm,Em,Bm) form,n∈Rwithm > n > 0 are two solutions to the approximate system with the same\ninitial data. Therefore, it follows that\n1\n2d\ndt/ba∇dbl(vn−vm,En−Em,Bn−Bm)/ba∇dbl2\nL2+ν/ba∇dblvn−vm/ba∇dbl2\nL2+1\nσ/ba∇dbl¯jn−¯jm/ba∇dbl2\nL2=:6/summationdisplay\nk=4Ik,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nI4=/integraldisplay\nRd(−Tn(vn·∇vn)+Tm(vm·∇vm))·(vn−vm)dx=:3/summationdisplay\nk=1I4k,\nI41=−/integraldisplay\nRd(Tn−Tm)(vn·∇vn)·(vn−vm)dx≤C(s)n−(s−1)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblvn−vm/ba∇dblL2,\nI42=−/integraldisplay\nRdTm((vn−vm)·∇vn)·(vn−vm)dx≤4ǫν/ba∇dblvn−vm/ba∇dbl2\nL2+C(ǫ,ν)/ba∇dbl∇vn/ba∇dbl2\nL∞/ba∇dblvn−vm/ba∇dbl2\nL2,\nI43=−/integraldisplay\nRdTm(vm·∇(vn−vm))·(vn−vm)dx= 0;\nI5=/integraldisplay\nRd(Tn(¯jn×(Bn+B∗))−Tm(¯jm×(Bm+B∗)))·(vn−vm)dx=:4/summationdisplay\nk=1I5k,\nI51=/integraldisplay\nRd(Tn−Tm)(¯jn×Bn)·(vn−vm)dx≤C(s)n−s/ba∇dblBn/ba∇dblHs/parenleftbig\n/ba∇dblvn−vm/ba∇dbl2\nL2+/ba∇dbl¯jn/ba∇dbl2\nHs/parenrightbig\n,\nI52=/integraldisplay\nRdTm((¯jn−¯jm)×Bn)·(vn−vm)dx,\nI53=/integraldisplay\nRdTm(¯jm×(Bn−Bm))·(vn−vm)dx≤4ǫν/ba∇dblvn−vm/ba∇dbl2\nL2+C(ǫ,ν)/ba∇dbl¯jm/ba∇dbl2\nL∞/ba∇dblBn−Bm/ba∇dbl2\nL2,\nI54=/integraldisplay\nRd(Tn(¯jn×B∗)−Tm(¯jm×B∗))·(vn−vm)dx=:I541+I542,\nI541=/integraldisplay\nRd(Tn−Tm)(¯jn×B∗)·(vn−vm)dx≤C(s)n−s/ba∇dblB∗/ba∇dblL∞/parenleftbig\n/ba∇dbl¯jn/ba∇dbl2\nHs+/ba∇dblvn−vm/ba∇dbl2\nL2/parenrightbig\n,\nI542=/integraldisplay\nRdTm((¯jn−¯jm)×B∗)·(vn−vm)dx;\nI6=−/integraldisplay\nRd(¯jn−¯jm)·(−Tn(vn×(Bn+B∗))+Tm(vm×(Bm+B∗)))dx=:4/summationdisplay\nk=1I6k,\nI61=/integraldisplay\nRd(¯jn−¯jm)·(Tn−Tm)(vn×Bn)dx≤ǫ\nσ/ba∇dbl¯jn−¯jm/ba∇dbl2\nL2+C(ǫ,σ,s)n−2s/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs,\nI62=/integraldisplay\nRd(¯jn−¯jm)·Tm((vn−vm)×Bn)dx=−I52,\nI63=/integraldisplay\nRd(¯jn−¯jm)·Tm(vm×(Bn−Bm))dx≤ǫ\nσ/ba∇dbl¯jn−¯jm/ba∇dbl2\nL2+C(ǫ,σ)/ba∇dblvm/ba∇dbl2\nHs/ba∇dblBn−Bm/ba∇dbl2\nL2,\nI64=−/integraldisplay\nRd(¯jn−¯jm)·(−Tn(vn×B∗)+Tm(vm×B∗))dx=:I641+I642,\nI641=−/integraldisplay\nRd(¯jn−¯jm)·(Tm−Tn)(vn×B∗)dx≤C(s)n−s/ba∇dblB∗/ba∇dblL∞/parenleftbig\n/ba∇dbl¯jn−¯jm/ba∇dbl2\nL2+/ba∇dblvm/ba∇dbl2\nHs/parenrightbig\n,\nI642=/integraldisplay\nRd(¯jn−¯jm)·Tm((vn−vm)×B∗)dx=−I542.\nAs Step 16 in the proof of Theorem 1.1, by choosing ǫ=1\n8, it follows that ( vn,En,Bn) and (vn,¯jn) are Cauchy\nsequencesin L∞(0,∞;L2(Rd)) andL2(0,∞;L2(Rd)), respectively. Therefore, we canpassto the limit asin Step\n17a in the proof of Theorem 1.1to obtain a limiting system, which is similar to ( 2.5) with replacing P(j×B)\nandjbyP(¯j×(B+B∗)) and¯j, respectively, where ¯j=σ(cE+v×(B+B∗)). Moreover, the limiting solution\n40(v,E,B) satisfies for t >0\n/ba∇dbl(v,E,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblv/ba∇dbl2\nHs+1\nσ/ba∇dbl¯j/ba∇dbl2\nHsdτ≤2ǫ2\n0. (5.5)\nWe can also prove the uniqueness as in Step 3 in the proof of Theorem 1.3. We omit further details.\nStep 5: Large-time behavior. It can be seen from the Ohm’s law that for some ǫ∈(0,1)\nσc/integraldisplay∞\n0/ba∇dblE/ba∇dbl2\nL2dτ=/integraldisplay∞\n0/integraldisplay\nRd(¯j−σ(v×(B+B∗)))·Edxdτ\n≤1\ncC(ǫ,σ)/integraldisplay∞\n0(1+/ba∇dbl(B,B∗)/ba∇dbl2\nL∞)/ba∇dbl(v,¯j)/ba∇dbl2\nL2dτ+3ǫσc/integraldisplay∞\n0/ba∇dblE/ba∇dbl2\nL2dτ,\nwhich yields by using ( 5.5) and choosing ǫ=1\n6\n/integraldisplay∞\n0/ba∇dbl(v,E)/ba∇dbl2\nL2dτ≤1\nc2C(ǫ∗,ν,σ,s)(ǫ2\n0+ǫ4\n0)+C(ν)ǫ2\n0. (5.6)\nIn addition, we observe that ( v,E)∈C([0,T];L2)22since (∂tv,∂tE)∈L2(0,T;H−1) for any T∈(0,∞) (see\n[29,67]), which implies by using ( 5.5) that\n1\n2d\ndt/ba∇dbl(v,E)/ba∇dbl2\nL2+ν/ba∇dblv/ba∇dbl2\nL2+1\nσ/ba∇dbl¯j/ba∇dbl2\nL2=/integraldisplay\nRd(∇×B)·cEdx≤2cǫ2\n0.\nTherefore, for 0 ≤t′< t <∞\n/ba∇dbl(v,E)(t)/ba∇dbl2\nL2−/ba∇dbl(v,E)(t′)/ba∇dbl2\nL2≤4cǫ2\n0(t−t′). (5.7)\nBy using ( 5.6)-(5.7), it follows that /ba∇dbl(v,E)(t)/ba∇dblL2→0 ast→ ∞(see [50, Lemma 2.3]). As a consequence, we\nfind that\n/ba∇dbl¯j(t)/ba∇dbl2\nL2=σ/integraldisplay\nRd¯j(t)·(cE+v×(B+B∗))(t)dx\n≤1\n2/ba∇dbl¯j(t)/ba∇dbl2\nL2+σ2c2/ba∇dblE(t)/ba∇dbl2\nL2+C(s)σ2/ba∇dblv(t)/ba∇dbl2\nL2/ba∇dblB(t)/ba∇dbl2\nHs+Cσ2/ba∇dblB∗/ba∇dbl2\nL∞/ba∇dblv(t)/ba∇dbl2\nL2,\nwhich yields /ba∇dbl¯j(t)/ba∇dblL2→0 ast→ ∞. By using again ( 5.5) and the above L2decay properties, for s′∈[0,s)\nandf∈ {v,E,¯j}\n/ba∇dblf(t)/ba∇dblHs′≤C(s)/ba∇dblf(t)/ba∇dbls−s′\ns\nL2/ba∇dblf(t)/ba∇dbls′\ns\nHs→0 ast→ ∞,\nwhere if f≡¯jthen we also used the following estimate\n/ba∇dbl¯j(t)/ba∇dblHs≤C(c,σ,s)(/ba∇dblE(t)/ba∇dblHs+/ba∇dblv(t)/ba∇dblHs/ba∇dblB(t)/ba∇dblHs+/ba∇dblB∗/ba∇dblL∞/ba∇dblv(t)/ba∇dblHs)≤C(c,ǫ0,ǫ∗,σ,s).\nIn addition, for s′∈[0,s)\n/ba∇dblj(t)/ba∇dblHs′≤ /ba∇dbl¯j(t)/ba∇dblHs′+/ba∇dblj∗(t)/ba∇dblHs′→0 ast→ ∞.\nAs a consequence, for f∈ {E,B,¯j}the following quantities for s′∈[0,s)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdJs′(v(t))·Js′(f(t))dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblv(t)/ba∇dblHs′/ba∇dblf(t)/ba∇dblHs′→0 ast→ ∞.\nFurthermore, similar to the case of ( v,E) above, we also have B∈C([0,T],L2) for any T∈(0,∞), it follows\nthat1\n2d\ndt/ba∇dbl(v,E,B)/ba∇dbl2\nL2+ν/ba∇dblv/ba∇dbl2\nL2+1\nσ/ba∇dbl¯j/ba∇dbl2\nL2= 0,\nwhich implies that 0 < f(t) :=/ba∇dbl(v,E,B)(t)/ba∇dbl2\nL2< ǫ2\n0andf(t) is a strictly decreasing function. By using the L2\ndecay in time property of ( v,E), we find that /ba∇dblB(t)/ba∇dblL2→b0ast→ ∞for some constant b0∈[0,ǫ0). Since\n∂tB=−c∇×Ethen/ba∇dbl∂tB(t)/ba∇dblHr−1=c/ba∇dbl∇×E(t)/ba∇dblHr−1→0 ast→ ∞forr∈[1,s). In the sequel, we aim to\nprove that\n/ba∇dbl∂tE(t)/ba∇dblL2→0 ast→ ∞. (5.8)\n22It is after possibly being redefined on a set of measure zero.\n41Indeed, for any T∈(0,∞) we find from ( 5.5) that\n\n\n¯j=σ(cE+v×(B+B∗)) ∈L2(0,∞;Hs),\n∂tv=−P(v·∇v)−νv+P(¯j×(B+B∗)) ∈L2(0,∞;Hs−1),\n1\nc∂tE=∇×B−¯j ∈L2(0,T;Hs−1),\n1\nc∂tB=−∇×E ∈L2(0,T;Hs−1),\n1\nc∂ttE=∇×∂tB−cσ∂tE−σ(∂tv×(B+B∗)+v×∂tB)∈L2(0,T;Hs−2).\nFurthermore, since B∈L2(0,T;Hs) and∂tB∈L2(0,T;Hs−1) withs−1>d\n2≥1 for any T∈(0,∞), it follows\nfrom [67, Lemma 1.2, Chapter 3] that B∈C([0,T];H1) and since v,E,B∈C([0,T];L2)\n∂tE(t) =c∇×B(t)−c2σE(t)−cσv(t)×(B(t)+B∗)∈C([0,T];L2), (5.9)\nwhich gives us the meaning for the value of ∂tEatt= 0 and suggests us to take\n∂tE|t=0=/parenleftbig\nc∇×B−c2σE−cσv×(B+B∗)/parenrightbig\n|t=0. (5.10)\nIn addition, it can be seen that\n1\n2d\ndt/ba∇dbl∂tE/ba∇dbl2\nL2+c2σ/ba∇dbl∂tE/ba∇dbl2\nL2=:4/summationdisplay\nk=1Rk,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nR1=c/integraldisplay\nRd∇×∂tB·∂tEdx=−c/integraldisplay\nRd∇×(∇×cE)·∂tEdx\n=−c/integraldisplay\nRd∇×(∇×(1\nσ¯j−v×(B+B∗)))·∂tEdx\n≤3ǫc2σ/ba∇dbl∂tE/ba∇dbl2\nL2+C(ǫ,σ,s)/parenleftbig\n/ba∇dbl¯j/ba∇dbl2\nHs+/ba∇dblv/ba∇dbl2\nHs(/ba∇dblB/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\n;\nR2=−cσ/integraldisplay\nRd(∂tv×B)·∂tEdx\n≤ǫc2σ/ba∇dbl∂tE/ba∇dbl2\nL2+C(ǫ,σ,s)/ba∇dblB/ba∇dbl2\nHs/ba∇dbl∂tv/ba∇dbl2\nL2\n≤ǫc2σ/ba∇dbl∂tE/ba∇dbl2\nL2+C(ǫ,ν,σ,s)/ba∇dblB/ba∇dbl2\nHs/parenleftbig\n/ba∇dblv/ba∇dbl4\nHs+/ba∇dblv/ba∇dbl2\nHs+/ba∇dbl¯j/ba∇dbl2\nHs(/ba∇dblB/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\n;\nR3=−cσ/integraldisplay\nRd(∂tv×B∗)·∂tEdx\n≤ǫc2σ/ba∇dbl∂tE/ba∇dbl2\nL2+C(ǫ,ν,σ,s)/ba∇dblB∗/ba∇dbl2\nL∞/parenleftbig\n/ba∇dblv/ba∇dbl4\nHs+/ba∇dblv/ba∇dbl2\nHs+/ba∇dbl¯j/ba∇dbl2\nHs(/ba∇dblB/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\n;\nR4=−cσ/integraldisplay\nRd(v×∂tB)·∂tEdx=cσ/integraldisplay\nRd(v×(∇×cE))·∂tEdx\n=cσ/integraldisplay\nRd(v×(∇×(1\nσ¯j−v×(B+B∗))))·∂tEdx\n≤3ǫc2σ/ba∇dbl∂tE/ba∇dbl2\nL2+C(ǫ,σ,s)/parenleftbig\n/ba∇dblv/ba∇dbl2\nHs/ba∇dbl¯j/ba∇dbl2\nHs+/ba∇dblv/ba∇dbl4\nHs(/ba∇dblB/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\n,\nhere in the estimates of R2andR3, we employed the following fact\n/ba∇dbl∂tv(t)/ba∇dbl2\nHs−1≤C(ν)/parenleftbig\n/ba∇dblv(t)/ba∇dbl4\nHs+/ba∇dblv(t)/ba∇dbl2\nHs+/ba∇dbl¯j(t)/ba∇dbl2\nHs(/ba∇dblB(t)/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\nfort >0.\nTherefore, by choosing ǫ=1\n16and using ( 5.5), (5.9)-(5.10), it follows that for 0 ≤t′< t <∞\n/ba∇dbl∂tE(t)/ba∇dbl2\nL2−/ba∇dbl∂tE(t′)/ba∇dbl2\nL2≤C(c,ǫ0,ǫ∗,σ,s)(t−t′),\n/integraldisplayt\n0/ba∇dbl∂tE/ba∇dbl2\nL2dτ≤C(c,ǫ0,ǫ∗,σ,s),\nwhere we also used the following estimate\n/ba∇dbl¯j(t)/ba∇dbl2\nHs≤C(c,σ,s)/parenleftbig\n/ba∇dblE(t)/ba∇dbl2\nHs+/ba∇dblv(t)/ba∇dbl2\nHs(/ba∇dblB(t)/ba∇dbl2\nHs+/ba∇dblB∗/ba∇dbl2\nL∞)/parenrightbig\n≤C(c,ǫ∗,σ,s)(ǫ2\n0+ǫ4\n0).\nThus, (5.8) follows by using [ 50, Lemma 2.3] again. Combining ( 5.8) and the decay in time of /ba∇dbl¯j/ba∇dblL2, we find\nthat/ba∇dbl∇×B(t)/ba∇dblL2=/ba∇dbl∇B(t)/ba∇dblL2→0 ast→ ∞. Therefore, for any s′∈[0,s−1) and for some suitable s′′>d\n2,\nby using ( 5.5), interpolation inequalities and Lemma 7.1, it follows that as t→ ∞\n/ba∇dblB(t)/ba∇dbl˙Hs′+1≤C(s′)/ba∇dbl∇B(t)/ba∇dbls−1−s′\ns−1\nL2/ba∇dbl∇B(t)/ba∇dbls′\ns−1\n˙Hs−1→0,\n42/ba∇dblB(t)/ba∇dblL∞≤C(d,s′′)/ba∇dblB(t)/ba∇dbl1−d\n2s′′\nL2/ba∇dblB(t)/ba∇dbld\n2s′′\n˙Hs′′→0.\nAs a consequence, /ba∇dblB(t)/ba∇dblL2\nloc→0 ast→ ∞. Finally, the convergences in time of ( ∂tv,∂tE) follows from those\nof (v,E,B,¯j), (5.5) and the following estimates for r∈[0,s−1)\n/ba∇dbl∂tv(t)/ba∇dblHr≤ /ba∇dblv(t)·∇v(t)/ba∇dblHr+ν/ba∇dblv(t)/ba∇dblHr+/ba∇dbl¯j(t)×(B(t)+B∗)/ba∇dblHr,\n/ba∇dbl∂tE(t)/ba∇dblHr≤ /ba∇dbl∇×B(t)/ba∇dblHr+/ba∇dbl¯j(t)/ba∇dblHr.\nProof of Theorem 1.4-(ii).We now focus on the limit as c→ ∞. It can be seen from ( NSM*) withα= 0 that\n\n\n∂tvc+vc·∇vc+∇πc=−νvc+¯jc×(Bc+B∗),\n1\nc∂tEc−∇×Bc=−¯jc,\n∂tBc−∇×(vc×(Bc+B∗)) =−1\nσ∇ׯjc,\ndivvc= divBc= 0.(5.11)\nBy using the smallness condition on the initial data ( vc\n0,Ec\n0,Bc\n0), an application of Part ( i) gives us the existence\nof a sequence of global solutions ( vc,Ec,Bc) to (5.11) with some uniform bounds in terms of c. Therefore, this\nstep can be done as Step 5 in the proof of Theorem 1.3, which by using the weak convergence of ( vc\n0,Ec\n0,Bc\n0) in\nHsand some mentioned vector identities implies that ( 5.11) converges to ( H-MHD* ) withκ= 0 in the sense\nof distributions as c→ ∞. Here, we note in addition that ∇×(B+B∗) =∇×B=¯j, whereBand¯jare the\ncorresponding limits of Bcand¯jcasc→ ∞, respectively. We skip further details and end the proof.\n6 Proof of Theorem 1.5\nIn this subsection, we will provide a simple proof of Theorem 1.5, which shares a similar idea as that of Theorem\n1.4.\nProof of Theorem 1.5.The proof is quite similar to that of Theorem 1.4, which contains several steps as follows.\nIn this case, since B∗is a constant vector in R3, we have j∗=∇×B∗= 0. Thus, ( H-MHD* ) is reduced to the\nfollowing system\n\n∂tv+v·∇v+∇π=−νv+j×(B+B∗),\n∂tB−∇×(v×(B+B∗)) =1\nσ∆B−κ\nσ∇×(j×(B+B∗)),\ndivv= divB= 0,\nwhich can be further equivalently rewritten as follows for p∗:=π+1\n2|B+B∗|2\n\n\n∂tv+v·∇v+∇p∗=−νv+B·∇B+B∗·∇B,\n∂tB+v·∇B=1\nσ∆B+B·∇v+B∗·∇v+κ\nσ(j·∇B−B·∇j−B∗·∇j),\ndivv= divB= 0,(6.1)\nby using the following vector indentities for f∈ {v,j}\n(∇×(B+B∗))×(B+B∗) = (B+B∗)·∇(B+B∗)−1\n2∇|B+B∗|2,\n∇×(f×(B+B∗)) =−f·∇(B+B∗)+(B+B∗)·∇f.\nStep 1: Local existence. We will consider the following approximate system to ( 6.1)\nd\ndt(vn,Bn) = (Fn\n1,Fn\n2)(vn,Bn),divvn= divBn= 0,(vn,Bn)|t=0=Tn(v0,B0), (6.2)\nwhere\nFn\n1=−Tn(P(vn·∇vn))−νvn+Tn(P(Bn·∇Bn))+Tn(P(B∗·∇Bn)),\nFn\n2=−Tn(vn·∇Bn)+Tn(Bn·∇vn)+Tn(B∗·∇vn)+1\nσ∆Bn+k\nσTn(jn·∇Bn−Bn·∇jn−B∗·∇jn).\n43By defining Fn:Vs\nn×Vs\nn→Vs\nn×Hs\nnwithFn:= (Fn\n1,Fn\n2), the existence of a unique local solultion ( vn,Bn)∈\nC1([0,Tn\n∗);Vs\nn×Vs\nn) to (6.2) follows as previously.\nStep 2: Hsestimate and global uniform bound. It can be seen from ( 6.2) that\n1\n2d\ndt/ba∇dbl(vn,Bn)/ba∇dbl2\nHs+ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl∇Bn/ba∇dbl2\nHs=:9/summationdisplay\nk=1Ik,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nI1=−/integraldisplay\nRd[Js(P(Tn(vn·∇vn))−vn·∇Jsvn]·Jsvndx≤C(s)/ba∇dblvn/ba∇dbl3\nHs;\nI2=/integraldisplay\nRdJs(P(Tn(Bn·∇Bn))·Jsvndx≤ǫ\nσ/ba∇dbl∇Bn/ba∇dblHs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs;\nI3=/integraldisplay\nRdJs(P(Tn(B∗·∇Bn))·Jsvndx=/integraldisplay\nRdB∗·∇JsBn·Jsvndx;\nI4=−/integraldisplay\nRdJs(Tn(vn·∇Bn))·JsBndx≤ǫ\nσ/ba∇dbl∇Bn/ba∇dblHs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs;\nI5=/integraldisplay\nRdJs(Tn(Bn·∇vn))·JsBndx≤ǫ\nσ/ba∇dbl∇Bn/ba∇dblHs+C(ǫ,σ,s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs;\nI6=/integraldisplay\nRdJs(Tn(B∗·∇vn))·JsBndx=/integraldisplay\nRdB∗·∇Jsvn·JsBndx=−I3;\nI7=κ\nσ/integraldisplay\nRdJs(Tn(jn·∇Bn))·JsBndx≤κ\nσC(s)/ba∇dbl∇Bn/ba∇dbl2\nHs/ba∇dblBn/ba∇dblHs;\nI8=−κ\nσ/integraldisplay\nRdJs(Tn(Bn·∇jn))·JsBndx≤κ\nσC(s)/ba∇dbl∇Bn/ba∇dbl2\nHs/ba∇dblBn/ba∇dblHs;\nI9=−κ\nσ/integraldisplay\nRdJs(Tn(B∗·∇jn))·JsBndx=−κ\nσ/integraldisplay\nRdJs(−jn·∇B∗+B∗·∇jn)·JsBndx\n=−κ\nσ/integraldisplay\nRdJs(∇×(j×B∗))·JsBndx=−κ\nσ/integraldisplay\nRdJs(jn×B∗)·Jsjndx= 0.\nBy chossing ǫ=1\n6\nd\ndt/ba∇dbl(vn,Bn)/ba∇dbl2\nHs+ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl∇Bn/ba∇dbl2\nHs≤C(s)/ba∇dblvn/ba∇dbl3\nHs+C(σ,s)/ba∇dblvn/ba∇dbl2\nHs/ba∇dblBn/ba∇dbl2\nHs+C(κ,σ,s)/ba∇dblBn/ba∇dblHs/ba∇dbl∇Bn/ba∇dbl2\nHs.\nSimilar to the previous parts, under the smallness assumption on initia l data, it follows that for t≥0\n/ba∇dbl(vn,Bn)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblvn/ba∇dbl2\nHs+1\nσ/ba∇dbl∇Bn/ba∇dbl2\nHsdτ≤2ǫ2\n0.\nStep 3: Cauchy sequence, pass to the limit and uniqueness. Thanks to the uniform bound obtained\nin the prevous step, we then can prove that ( vn,Bn) and (vn,∇Bn) are Cauchy sequences in L∞(0,∞;L2(Rd))\nandL2(0,∞;L2(Rd)), respectively. That allows us to pass to the limit and to obtain\n/ba∇dbl(v,B)(t)/ba∇dbl2\nHs+/integraldisplayt\n0ν/ba∇dblv/ba∇dbl2\nHs+1\nσ/ba∇dbl∇B/ba∇dbl2\nHsdτ≤2ǫ2\n0. (6.3)\nThe proof of the uniqueness is standard. This step is similar to the pr evious parts then we omit further details.\nStep 4a: Large-time behavior: implicit rate. It can be seen from ( 6.1) that\n1\n2d\ndt/ba∇dblv/ba∇dbl2\nL2+ν/ba∇dblv/ba∇dbl2\nL2=:R5,\n1\n2d\ndt/ba∇dblB/ba∇dbl2\n˙H1+1\nσ/ba∇dblB/ba∇dbl2\n˙H2=:11/summationdisplay\nk=6Rk,\nwhere for some ǫ∈(0,1), since s >d\n2+1\nR5=/integraldisplay\nRd(B+B∗)·∇B·vdx≤C(s)(/ba∇dblB/ba∇dblHs+/ba∇dblB∗/ba∇dblL∞)/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dblB/ba∇dbl2\n˙H1/parenrightbig\n;\nR6=/integraldisplay\nRdv·∇B·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\n˙H1;\n44R7=−/integraldisplay\nRdB·∇v·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\nHs;\nR8=−/integraldisplay\nRdB∗·∇v·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblv/ba∇dbl2\nHs/ba∇dblB∗/ba∇dbl2\nL∞;\nR9=−κ\nσ/integraldisplay\nRdj·∇B·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblB/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\n˙H1;\nR10=κ\nσ/integraldisplay\nRdB·∇j·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblB/ba∇dbl2\nHs/ba∇dblB/ba∇dbl2\n˙H2;\nR11=κ\nσ/integraldisplay\nRdB∗·∇j·∆Bdx≤ǫ\nσ/ba∇dblB/ba∇dbl2\n˙H2+C(ǫ,σ,s)/ba∇dblB∗/ba∇dbl2\nL∞/ba∇dblB/ba∇dbl2\n˙H2.\nTherefore, similar to Step 5 in the proof of Theorem 1.4, by choosing ǫ=1\n12, and using the energy estimate\nand (6.3), we have /ba∇dbl(v,∇B)(t)/ba∇dblL2→0 ast→ ∞. In addition, for s′∈[0,s) ands′′∈[1,s), interpolation\ninequalities and Lemma 7.1yield/ba∇dblv(t)/ba∇dblHs′→0,/ba∇dblB(t)/ba∇dblL∞and/ba∇dblB(t)/ba∇dbl˙Hs′′→0 ast→ ∞. The convergences\nin time of ( ∂tv,∂tB) follows from those of ( v,B) and the following estimates for r∈[0,s−2)\n/ba∇dbl∂tv(t)/ba∇dblHr≤ /ba∇dblv(t)·∇v(t)/ba∇dblHr+ν/ba∇dblv(t)/ba∇dblHr+/ba∇dblB(t)·∇B(t)/ba∇dblHr+/ba∇dblB∗·∇B(t)/ba∇dblHr,\n/ba∇dbl∂tB(t)/ba∇dblHr≤ /ba∇dblv(t)·∇B(t)/ba∇dblHr+1\nσ/ba∇dbl∆B(t)/ba∇dblHr+/ba∇dblB(t)·∇v(t)/ba∇dblHr+/ba∇dblB∗·∇v(t)/ba∇dblHr\n+κ\nσ(/ba∇dblj(t)·∇B(t)/ba∇dblHr+/ba∇dblB(t)·∇j(t)/ba∇dblHr+/ba∇dblB∗·∇j(t)/ba∇dblHr).\nWe only show how to deal with the most difficult term as follows, other o nes can be done similarly. Indeed,\nsinces >d\n2+1 then for some suitable s′∈(d\n2+1,s) andr≥0 withr+1≤s′−1\n/ba∇dblj(t)·∇B(t)/ba∇dblHr≤C(s)/ba∇dblj(t)⊗B(t)/ba∇dblHr+1≤C(s)/ba∇dblj(t)⊗B(t)/ba∇dblHs′−1\n≤C(s)/ba∇dblj(t)/ba∇dblHs′−1/ba∇dblB(t)/ba∇dblHs′−1\n≤C(s)(/ba∇dbl∇B(t)/ba∇dblL2+/ba∇dblB(t)/ba∇dbl˙Hs′)/ba∇dblB(t)/ba∇dblHs→0 ast→ ∞.\nStep 4b: Large-time behavior: explicit rate. If in addition ( v0,B0)∈L1then an explicit rate of\nconvergence in suitable norms can be established. More precisely, w e can follow closely the ideas in [ 62,63] (for\nthe Navier-Stokes equations) by applying the Fourier-splitting met hod to obtain the L2decay in time of ( v,B).\nIndeed, there is a new difficulty, which is related to the perturbation terms (see ( S3,S6,S9) below) since at\nsome point we need to control L∞norm of such a bab term F(B∗⊗B) and it seems leading to the L1estimate\nofB, which has not been obtained yet. Thus, the techniques in [ 62,63] can not be applied directly and new\nideas should be suggested. To overcome this new issue, we will estima te more carefully the bad term, especially\nusing the velocity damping kernel, which allows us to gain more good fac tors. For fixed ν >0, by defining for\n(x,t)∈Rd×(0,∞) and for some m∈Nwill be chosen later\n(vν,Bν)(x,t) :=m\nν(v,B)/parenleftbigg\nx,mt\nν/parenrightbigg\n, pν(x,t) :=m2\nν2p/parenleftbigg\nx,mt\nν/parenrightbigg\n, B∗\nν:=m\nνB∗andjν:=∇×Bν,\nwe reduce ( 6.1) to\n\n\n∂tvν+vν·∇vν=−mvν+Bν·∇Bν+B∗\nν·∇Bν+∇pν,\n∂tBν+vν·∇Bν=Bν·∇vν+B∗\nν·∇vν+m\nνσ∆Bν+κ\nσ(jν·∇Bν−Bν·∇jν−B∗\nν·∇jν),\ndivvν= divBν= 0,(6.4)\nwith the initial data is given by ( vν,Bν)|t=0=mν−1(v0,B0). From the previous step, we know the existence and\nuniqueness of solutions ( vν,Bν) to (6.4) satisfying ( vν,Bν)∈L∞(0,∞;Hs(Rd)), (vν,∇Bν)∈L2(0,∞;Hs(Rd))\nfors >d\n2+1 and the estimate ( 6.3) withC(ν,m)ǫ2\n0instead of 2 ǫ2\n0. It can be seen from the energy balance of\n(6.4) that\nd\ndt/parenleftbigg\nhν(t) :=/integraldisplay\nRd|F(vν)(t)|2+|F(Bν)(t)|2dξ/parenrightbigg\n=−2m/integraldisplay\nRd|F(vν)(t)|2dξ−2m\nνσ/integraldisplay\nRd|ξ|2|F(Bν)(t)|2dξ.\nAs in [63], for some β >0 to be determined later, we define for t >0\nS(t) :={ξ∈Rd:|ξ| ≤g(t)}and ˜g(t) := exp/braceleftbigg\nβ/integraldisplayt\n0g2dτ/bracerightbigg\nwithg2(t) :=m\nβ(e+t)log(e+t).\nForSc(t) :=Rd\\S(t), by choosing β=2m\nνσand using the fact that βg2≤m, it follows that\nd\ndt(˜g(t)hν(t)) =d\ndt(˜g(t))hν(t)+ ˜g(t)d\ndthν(t)\n45≤βg2(t)˜g(t)hν(t)+ ˜g(t)/parenleftigg\n−2/integraldisplay\nSc(t)|F(vν)(t)|2dξ−2\nνσg2(t)/integraldisplay\nSc(t)|F(Bν)(t)|2dξ/parenrightigg\n≤βg2(t)˜g(t)/parenleftigg/integraldisplay\nS(t)|F(vν)(t)|2+|F(Bν)(t)|2dξ=:Svν(t)+SBν(t)/parenrightigg\n.\nIt remains to control the integral on the right-hand side. It can b e seen that\n∂tF(vν)+mF(vν) =F(P(−vν·∇vν+Bν·∇Bν+B∗\nν·∇Bν)),\n∂tF(Bν)+m\nνσ|ξ2|F(Bν) =F(−vν·∇Bν+Bν·∇vν+B∗\nν·∇vν+κ\nσ(jν·∇Bν−Bν·∇jν−B∗\nν·∇jν)),\nit implies that for t >0\nSvν(t)≤Cgd(t)/ba∇dblv0/ba∇dbl2\nL1+/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp{−m(t−τ)}|F(P(−vν·∇vν+Bν·∇Bν+B∗\nν·∇Bν))|dτ/parenrightbigg2\ndξ\n≤Cgd(t)/ba∇dblv0/ba∇dbl2\nL1+C3/summationdisplay\nk=1Sk.\nWe are going to estimate each term on the right-hand side by using ( 6.3) as follows\nS1:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp{−m(t−τ)}|F(P(vν·∇vν))|dτ/parenrightbigg2\ndξ\n≤/integraldisplay\nS(t)/integraldisplayt\n0exp{−2m(t−τ)}dτ/integraldisplayt\n0|F(vν·∇vν)|2dτdξ\n≤C(m)g2(t)/integraldisplayt\n0/integraldisplay\nS(t)|F(vν⊗vν)|2dξdτ\n≤C(m)gd+2(t)/integraldisplayt\n0/ba∇dblF(vν⊗vν)/ba∇dbl2\nL∞dτ\n≤C(m)gd+2(t)/integraldisplayt\n0/ba∇dblvν⊗vν/ba∇dbl2\nL1dτ\n≤C(ǫ0,ν,m)gd+2(t);\nS2:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp{−m(t−τ)}|F(P(Bν·∇Bν))|dτ/parenrightbigg2\ndξ≤C(ǫ0,ν,m)tgd+2(t);\nS3:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp{−m(t−τ)}|F(P(B∗\nν·∇Bν))|dτ/parenrightbigg2\ndξ\n≤/integraldisplay\nS(t)/integraldisplayt\n0exp{−2m(t−τ)}dτ/integraldisplayt\n0|F(B∗\nν·∇Bν)|2dτdξ\n≤C(m)/ba∇dblB∗/ba∇dbl2\nL∞g2(t)/integraldisplayt\n0/integraldisplay\nRd|F(Bν)|2dξdτ\n≤C(ǫ0,ǫ∗,ν,m)tg2(t).\nSimilarly, we find that\nSBν(t)≤Cgd(t)/ba∇dblB0/ba∇dbl2\nL1+C/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(−vν·∇Bν+Bν·∇vν+B∗\nν·∇vν)|dτ/parenrightbigg2\ndξ\n+C(κ,σ)/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(jν·∇Bν−Bν·∇jν−B∗\nν·∇jν)|dτ/parenrightbigg2\ndξ\n≤Cgd(t)/ba∇dblB0/ba∇dbl2\nL1+C(κ,σ)9/summationdisplay\nk=4Sk,\nwhere each term on the right-hand side is bounded by\nS4:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(vν·∇Bν)|dτ/parenrightbigg2\ndξ≤C(ǫ0,ν,m)tgd+2(t);\n46S5:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(Bν·∇vν)|dτ/parenrightbigg2\ndξ≤C(ǫ0,ν,m)tgd+2(t);\nS6:=/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(B∗\nν·∇vν)|dτ/parenrightbigg2\ndξ≤C(ǫ0,ǫ∗,ν,m)tg2(t);\nS7:=C(κ,σ)/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(jν·∇Bν)|dτ/parenrightbigg2\ndξ\n≤C(κ,σ,m)tg2+d(t)/integraldisplayt\n0/ba∇dbljν⊗Bν/ba∇dbl2\nL1dτ\n≤C(κ,σ,m)tg2+d(t)/integraldisplayt\n0/ba∇dbl∇B/ba∇dbl2\nL2/ba∇dblBν/ba∇dbl2\nL2dτ\n≤C(ǫ0,κ,ν,m,σ )tgd+2(t);\nS8:=C(κ,σ)/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(Bν·∇jν)|dτ/parenrightbigg2\ndξ≤C(ǫ0,κ,ν,m,σ )tgd+2(t);\nS9:=C(κ,σ)/integraldisplay\nS(t)/parenleftbigg/integraldisplayt\n0exp/braceleftbigg\n−m|ξ|2\nνσ(t−τ)/bracerightbigg\n|F(B∗\nν·∇jν)|dτ/parenrightbigg2\ndξ≤C(ǫ0,ǫ∗,κ,ν,m,σ )tg2(t).\nTherefore,\nd\ndt(˜g(t)hν(t))≤C(ǫ0,ǫ∗,κ,ν,m,σ,v 0,B0)˜g(t)(e+t)g4(t). (6.5)\nMoreover, it can be easily checked that\n˜g(t) = exp/braceleftbigg\nβ/integraldisplayt\n0g2dτ/bracerightbigg\n= logm(e+t),\nwhich by choosing m= 2 yields for s >0\nG:=/integraldisplays\n0˜g(t)(e+t)g4(t)dt=/integraldisplays\n01\n(e+t)log2−m(e+t)dt≤log(e+s).\nThus, (6.5) implies that for t >0\n/ba∇dbl(vν,Bν)(t)/ba∇dbl2\nL2≤C(ǫ0,ǫ∗,κ,ν,m,σ,v 0,B0)log−1(e+t). (6.6)\nBy using ( 6.6), we observe that for C=C(ǫ0,ǫ∗,κ,ν,m,σ,v 0,B0)\n/integraldisplayt\n0/ba∇dbl(vν,Bν)/ba∇dbl2\nL2dτ≤C/integraldisplayt\n0log−1(e+τ)dτ=C/integraldisplaylog(e+t)\n1euu−1du≤C(e+t)log−1(e+t),\nwhich will improve the estimates of S3,S6andS9as follows\nS3,S6,S9≤C(ǫ0,ǫ∗,κ,ν,m,σ )(e+t)g2(t)log−1(e+t),\nthat leads to a multiplication by the factor log−1(e+t) to the right-hand side of ( 6.5). Therefore, by changing\nthe estimate of Gwith choosing again m= 3 as follows\n˜G:=/integraldisplays\n0˜g(t)(e+t)g4(t)log−1(e+t)dt=/integraldisplays\n01\n(e+t)log3−m(e+t)dt≤log(e+s),\nthe estimate ( 6.6) can be replaced by\n/ba∇dbl(vν,Bν)(t)/ba∇dbl2\nL2≤C(ǫ0,ǫ∗,κ,ν,m,σ,v 0,B0)log−2(e+t). (6.7)\nBy repeating this iteration, it can be seen that ( 6.7) can be improved for each m∈N,m≥3\n/ba∇dbl(vν,Bν)(t)/ba∇dbl2\nL2≤C(ǫ0,ǫ∗,κ,ν,m,σ,v 0,B0)log−(m−1)(e+t).\nThat finishes the proof by combining the above inequality, ( 6.6), a change of variables from ( vν,Bν) to (v,B)\nand interpolation inequalities.\nRemark 6.1. (The case d= 3) In Step 4 above, we condsidered both cases d= 2 and d= 3 at the same\ntime. However, in the three-dimensional case, it would be expected to obtain a faster L2decay rate such as\n(t+1)−3\n4, which is known in the case of either the Navier-Stokes [ 62,63] or the Hall-MHD equations [ 19]. We\n47now give a remark on this case, where it seems to be difficult to obtain p olynomial decay in time compared to\nthe case of the Navier-Stokes and Hall-MHD equations (see [ 19,62,63]), unless new estimates of S3,S6andS9\nare provided. Similarly, for fixed ν >0, by defining for ( x,t)∈R3×(0,∞)\n(vν,Bν)(x,t) :=3\nν(v,B)/parenleftbigg\nx,3t\nν/parenrightbigg\n, pν(x,t) :=9\nν2p/parenleftbigg\nx,3t\n2ν/parenrightbigg\n, B∗\nν:=3\nνB∗andjν:=∇×Bν,\nwe reduce ( 6.1) to\n\n\n∂tvν+vν·∇vν=−3vν+Bν·∇Bν+B∗\nν·∇Bν+∇pν,\n∂tBν+vν·∇Bν=Bν·∇vν+B∗\nν·∇vν+3\nνσ∆Bν+κ\nσ(jν·∇Bν−Bν·∇jν−B∗\nν·∇jν),\ndivvν= divBν= 0,(6.8)\nwith the initial data is given by ( vν,Bν)|t=0= 3ν−1(v0,B0). From Step 3 above, we know that ( vν,Bν) to (6.4)\nsatisfying ( vν,Bν)∈L∞(0,∞;Hs(R3)), (vν,∇Bν)∈L2(0,∞;Hs(R3)) fors >5\n2and the estimate ( 6.3) with\nC(ν)ǫ2\n0instead of 2 ǫ2\n0. Therefore, the energy balance of ( 6.8) is given by\nd\ndt/parenleftbigg\nhν(t) :=/integraldisplay\nR3|F(vν)(t)|2+|F(Bν)(t)|2dξ/parenrightbigg\n=−6/integraldisplay\nR3|F(vν)(t)|2dξ−6\nνσ/integraldisplay\nR3|ξ|2|F(Bν)(t)|2dξ=:F1+F2.\nFor some β >0 to be determined later, by defining\nS(t) :={ξ∈R3:|ξ| ≤g(t)}withg2(t) :=3\nβ(t+1)andSc(t) :=R3\\S(t),\nand using βg2≤3, we find that\nF1≤ −3/integraldisplay\nS(t)|F(vν)(t)|2dξ−3/integraldisplay\nSc(t)|F(vν)(t)|2dξ\n≤ −3/integraldisplay\nS(t)|F(vν)(t)|2dξ−3\nt+1/integraldisplay\nSc(t)|F(vν)(t)|2dξ±3\nt+1/integraldisplay\nS(t)|F(vν)(t)|2dξ\n≤ −3\nt+1/integraldisplay\nR3|F(vν)(t)|2dξ+3\nt+1/integraldisplay\nS(t)|F(vν)(t)|2dξ,\nand by choosing β:=3\nνσ, we obtain\nF2≤ −3\nνσ/integraldisplay\nS(t)|ξ|2|F(Bν)(t)|2dξ−3\nνσ/integraldisplay\nSc(t)|ξ|2|F(Bν)(t)|2dξ\n≤ −3\nνσ/integraldisplay\nS(t)|ξ|2|F(Bν)(t)|2dξ−3\nt+1/integraldisplay\nSc(t)|F(Bν)|2dξ±3\nt+1/integraldisplay\nS(t)|F(Bν)(t)|2dξ\n≤ −3\nt+1/integraldisplay\nR3|F(Bν)(t)|2dξ+3\nt+1/integraldisplay\nS(t)|F(Bν)(t)|2dξ.\nTherefore,\nd\ndthν(t)+3\nt+1hν(t)≤3\nt+1/integraldisplay\nS(t)|F(vν)(t)|2+|F(Bν)(t)|2dξ\nand by multiplying ( t+1)3both sides\nd\ndt(hν(t)(t+1)3) = (t+1)3d\ndthν(t)+3(t+1)2hν(t)≤3(t+1)2/integraldisplay\nS(t)|F(vν)(t)|2+|F(Bν)(t)|2dξ.\nIt remains to bound the integral on the right-hand side. Similar to th e previous case, it follows that\n∂tF(vν)+3F(vν) =F(P(−vν·∇vν+Bν·∇Bν+B∗\nν·∇Bν)),\n∂tF(Bν)+3\nνσ|ξ2|F(Bν) =F(−vν·∇Bν+Bν·∇vν+B∗\nν·∇vν+κ\nσ(jν·∇Bν−Bν·∇jν−B∗\nν·∇jν)),\nwhich by using the same notation for Siwithi∈ {1,...,9}with a small modification in the exponential factors\nand assuming ( v0,B0)∈L1yields\n/integraldisplay\nS(t)|F(vν)(t)|2+|F(Bν)(t)|2dξ≤Cg3(t)/ba∇dbl(v0,B0)/ba∇dbl2\nL1+C(κ,σ)9/summationdisplay\nk=1Sk,\n48where the estimates of Sifori /∈ {3,6,9}can be given as in Step 4 above, for the remaining terms, if we use\nagain those estimates, i.e., S3,S6,S9≤Ctg2≤Cfor some positive constant Cdepending on the parameters,\nthen we will find thatd\ndt(hν(t)(t+1)3)≤C(t+1)2,\nwhich unfortunately does not provide us a decay in time after integr ating in time. So as mentioned previously,\nnew ideas should be suggested to overcome this issue. For instance ,\nS3≤Cg2+2γ0(t)/integraldisplayt\n0/integraldisplay\nS(t)|F(B∗\nν⊗Bν)|2|ξ|−2γ0dξdτ for some γ0∈/parenleftbigg\n0,3\n2/parenrightbigg\n≤C/ba∇dblB∗/ba∇dbl2\nL∞g2+2γ0(t)/integraldisplayt\n0/integraldisplay\nR3|F(Bν)|2|ξ|−2γ0dξdτ\n=C/ba∇dblB∗/ba∇dbl2\nL∞g2+2γ0(t)/integraldisplayt\n0/ba∇dblBν/ba∇dbl2\n˙H−γ0dτ\n≤C(ǫ∗,γ0,ν)g2+2γ0(t)/integraldisplayt\n0/ba∇dblBν/ba∇dbl2\nL6\n3+2γ0dτ,\nwhich leads to the study of Lpestimate of Bνfor some p∈(1,2). We hope that the above time integral can be\ncontrolled nicely, for example, S3≤Ctα0g2+2γ0for some constant α0such that α0<1+γ0. It seems hardly to\nbe the case since the standard dinemsional analysis shows that this integral has 3+2 γ0dimensions. However,\nif it is the case then that will lead to a polynomial decay rate as ( t+1)−(1+γ0−α0)at the first level. We leave\nit as an open question for the interested reader.\nAcknowledgements\nK.Kang’sworkissupportedbyNRF-2019R1A2C1084685. J.Lee’sw orkissupportedbyNRF-2021R1A2C1092830.\nD. D. Nguyen’s work is supported by NRF-2019R1A2C1084685 and N RF-2021R1A2C1092830.\n7 Appendix\n7.1 Appendix A: Besov spaces\nLet us quickly recall the definitions of the standard nonhomogeneo us and homogeneous Besov spaces, see more\ndetails in [ 5]. There exist two smooth radial functions χ,ϕ:Rd→[0,1] ford≥1 such that\nsupp(χ)⊂/braceleftbigg\nξ∈Rd:|ξ| ≤4\n3/bracerightbigg\n, supp(ϕ)⊂/braceleftbigg\nξ∈Rd:3\n4≤ |ξ| ≤8\n3/bracerightbigg\n,\nχ(ξ)+/summationdisplay\n0≤j∈Zϕ(2−jξ) = 1∀ξ∈Rd,/summationdisplay\nj∈Zϕ(2−jξ) = 1∀ξ∈Rd\\{0}.\nDefining ˜h:=F−1(χ) andh:=F−1(ϕ), where F−1denotes the usual inverse Fourier transform. The nonho-\nmogeneous and homogeneous dyadic blocks are defined by\n∆jf:=\n\n0 if j≤ −2,\n˜h∗f ifj=−1,\n2jdh(2j·)∗fifj≥0,and ˙∆jf:= 2jdh(2j·)∗f∀j∈Z,\nwhere∗stands for the usual convolution operator. Then the nonhomoge neous and homogeneous low-frequency\ncut-off operators are set for any k∈Zby\nSkf:=/summationdisplay\n−1≤j≤k−1∆jfand ˙Sk:= 2kd˜h(2k·)∗f=/summationdisplay\nj≤k−1,j∈Z˙∆jf.\nFors∈Randp,q∈[1,∞], the nonhomogeneous and homogeneous Besov spaces are estab lished as follows\nBs\np,q(Rd) :=/braceleftig\nf∈ S′(Rd) :/ba∇dblf/ba∇dblBsp,q(Rd):=/ba∇dbl2sj/ba∇dbl∆jf/ba∇dblLp(Rd)/ba∇dblℓq(Z)<∞/bracerightig\n,\n˙Bs\np,q(Rd) :=/braceleftig\nf∈ S′\nh(Rd) :/ba∇dblf/ba∇dbl˙Bsp,q(Rd):=/ba∇dbl2sj/ba∇dbl˙∆jf/ba∇dblLp(Rd)/ba∇dblℓq(Z)<∞/bracerightig\n,\nwhereS′(Rd) denotes the dual space of the usual Schwartz class S(Rd), the so-called the space of tempered\ndistributions and\nS′\nh(Rd) :=/braceleftbigg\nf∈ S′(Rd) : lim\nλ→∞/ba∇dblg(λD)f/ba∇dblL∞= 0∀g∈C∞\n0(Rd)/bracerightbigg\n,\n49hereforanymeasurablefunction gonRdwithatmostpolynomialgrowthatinfinity, g(D)f:=F−1(g(ξ)F(f)(ξ)).\nIt is also convenient to use the identities Bs\n2,2(Rd)≈Hs(Rd) and˙Bs\n2,2(Rd)≈˙Hs(Rd) fors∈R. In addition,\nthe Littlewood–Paley decompositions are given by\nf=/summationdisplay\n−1≤j∈Z∆jfinS′(Rd) and f=/summationdisplay\nj∈Z˙∆jfinS′(Rd)∀f∈ S′\nh(Rd).\nWe also recall a product rule in homogeneous Besov spaces (see [ 5, Corollary 2.55]) for s1,s2∈(−d\n2,d\n2) and\ns1+s2>0\n/ba∇dblfg/ba∇dbl˙Bs1+s2−d\n2\n2,1(Rd)≤C(d,s1,s2)/ba∇dblf/ba∇dbl˙Hs1(Rd)/ba∇dblg/ba∇dbl˙Hs2(Rd). (7.1)\nAn application of ( 7.1) is the following Sobolev product estimate for s1,s2∈(0,d\n2)\n/ba∇dblfg/ba∇dblHs1+s2−d\n2(Rd)≤C(d,s1,s2)/ba∇dblf/ba∇dblHs1(Rd)/ba∇dblg/ba∇dblHs2(Rd). (7.2)\nIndeed, if s1+s2∈(0,d\n2) then for h∈Hd\n2−(s1+s2)\n/integraldisplay\nRdfghdx≤ /ba∇dblf/ba∇dbl\nL2d\nd−2s1/ba∇dblg/ba∇dbl\nL2d\nd−2s2/ba∇dblh/ba∇dbl\nL2d\nd−2(d\n2−(s1+s2))≤C(d,s1,s2)/ba∇dblf/ba∇dblHs1(Rd)/ba∇dblg/ba∇dblHs2(Rd)/ba∇dblh/ba∇dblHd\n2−(s1+s2).\nOn the other hand, if s1+s2≥d\n2then (7.1) yields\n/ba∇dblfg/ba∇dblL2≤ /ba∇dblf/ba∇dbl\nL2d\nd−2s1/ba∇dblg/ba∇dbl\nL2d\nd−2(d\n2−s1)≤C(d,s1)/ba∇dblf/ba∇dbl˙Hs1/ba∇dblg/ba∇dbl˙Hd\n2−s1≤C(d,s1)/ba∇dblf/ba∇dblHs1/ba∇dblg/ba∇dblHs2,\n/ba∇dblfg/ba∇dbl˙Hs1+s2−d\n2≤C(d,s1,s2)/ba∇dblfg/ba∇dbl˙Bs1+s2−d\n2\n2,2≤C(d,s1,s2)/ba∇dblfg/ba∇dbl˙Bs1+s2−d\n2\n2,1≤C(d,s1,s2)/ba∇dblf/ba∇dblHs1/ba∇dblg/ba∇dblHs2.\nIt is also convenient to recall the time-space Besov spaces. For T >0,s∈R,r0,p0,q0∈[1,∞], the Chemin-\nLerner spaces ˜Lr0(0,T;˙Bs\np0,q0(Rd)) and˜Lr0(0,T;Bs\np0,q0(Rd)) were introduced in [ 23] (see [5] for more details)\nand are given as follows\n˜Lr0(0,T;˙Bs\np0,q0(Rd)) :=/braceleftig\nf∈ S′\n0(Rd) :/ba∇dblf/ba∇dbl˜Lr0(0,T;˙Bsp0,q0(Rd)):=/ba∇dbl2sq/ba∇dbl˙∆qf/ba∇dblLr0(0,T;Lp0(Rd))/ba∇dblℓq0(Z)<∞/bracerightig\n,\nS′\n0(Rd) :=/braceleftbigg\nf∈ S′(Rd) : lim\nk→−∞/ba∇dbl˙Skf/ba∇dblLr0(0,T;Lp0(Rd))= 0/bracerightbigg\n,\n˜Lr0(0,T;Bs\np0,q0(Rd)) :=/braceleftig\nf∈ S′(Rd) :/ba∇dblf/ba∇dbl˜Lr0(0,T;Bsp0,q0(Rd)):=/ba∇dbl2sq/ba∇dbl∆qf/ba∇dblLr0(0,T;Lp0(Rd))/ba∇dblℓq0(Z)<∞/bracerightig\n.\nBy using Minkowski inequality for integrals, the following relations hold\n˜Lr0(0,T;˙Bs\np0,q0(Rd))⊂Lr0(0,T;˙Bs\np0,q0(Rd)) and ˜Lr0(0,T;Bs\np0,q0(Rd))⊂Lr0(0,T;Bs\np0,q0(Rd)) ifr0≥q0,\nLr0(0,T;˙Bs\np0,q0(Rd))⊂˜Lr0(0,T;˙Bs\np0,q0(Rd)) and Lr0(0,T;Bs\np0,q0(Rd))⊂˜Lr0(0,T;Bs\np0,q0(Rd)) ifr0≤q0.\n7.2 Appendix B: Homogeneous Sobolev inequalities and proof of(2.4)\nThere is a proof of ( 2.4) in [49] in a more general Lpframework. In Hilbert spaces, the proof is much more\nsimpler. However, we do not find a specific reference for the proof of the three-dimensional case, especially for\nthe homogeneous Sobolev norm version, so for the sake of complet eness, we provide a standard proof of ( 2.4)\nand its three-dimensional version as follows. Since we used both ver sions in the previous proofs. We note that\nfor the nonhomogeneous Sobolev norm version, it is a consequence of a result in [ 13].\nLemma 7.1. Assume that f∈Hs(Rd)withs >d\n2andd∈ {2,3}then\n/ba∇dblf/ba∇dblL∞≤C(s)\n\n/ba∇dblf/ba∇dbls−1\ns\nL2/ba∇dblf/ba∇dbl1\ns\n˙Hsifd= 2,\n/ba∇dblf/ba∇dbl2s−3\n2s\nL2/ba∇dblf/ba∇dbl3\n2s\n˙Hsifd= 3.(7.3)\nProof of Lemma 7.1.It can be seen that for x∈Rd\nf(x) =/integraldisplay\n|ξ|≤Mexp{ix·ξ}F(f)(ξ)dξ+/integraldisplay\n|ξ|>Mexp{ix·ξ}F(f)(ξ)dξ=:F1+F2,\nwhereMis a positive constant to be determined later and\n|f(x)| ≤ |F1|+|F2| ≤/braceleftigg\nCM/ba∇dblf/ba∇dblL2+C(s)M1−s/ba∇dblf/ba∇dbl˙Hsifd= 2,\nCM3\n2/ba∇dblf/ba∇dblL2+C(s)M3\n2−s/ba∇dblf/ba∇dbl˙Hsifd= 3.\nThus, (7.3) follows by choosing M=/ba∇dblf/ba∇dbl−1\ns\nL2/ba∇dblf/ba∇dbl1\ns\n˙Hs.\n507.3 Appendix C: A logarithmic Gronwall inequality\nIn this subsection, we provide a simple proof of a lograrithmic Gronwa ll inequality, which is used several times\nbefore.\nLemma 7.2. Assume that h1,h2,y≥0satisfying h1,h2∈L1\nloc(0,∞),y(0)≥0and forα≥1\nd\ndty(t)≤h1(t)y(t)+h2(t)log(α+y(t))y(t)fort >0, (7.4)\nthen\ny(t2)≤exp/braceleftbigg/parenleftbigg\nlog(α+y(t1))+/integraldisplayt2\nt1h1dτ/parenrightbigg\nexp/braceleftbigg/integraldisplayt2\nt1h2dτ/bracerightbigg/bracerightbigg\nfor0≤t1≤t2<∞.(7.5)\nProof of Lemma 7.2.By setting v(t) := log(α+y(t)), it can be seen from ( 7.4) that\nd\ndtv(t)≤h1(t)+h2(t)v(t).\nTherefore, for 0 ≤t1≤t2<∞\nv(t2)exp/braceleftbigg\n−/integraldisplayt2\n0h2dτ/bracerightbigg\n≤v(t1)exp/braceleftbigg\n−/integraldisplayt1\n0h2dτ/bracerightbigg\n+/integraldisplayt2\nt1h1exp/braceleftbigg\n−/integraldisplayτ\n0h1ds/bracerightbigg\ndτ,\nwhich implies that\nlog(α+y(t2))≤/parenleftbigg\nlog(α+y(t1))+/integraldisplayt2\nt1h1dτ/parenrightbigg\nexp/braceleftbigg/integraldisplayt2\nt1h2dτ/bracerightbigg\nand (7.5) follows.\n7.4 Appendix D: Fractional heat equation and proof of (3.1)\nFor the sake of completeness, we will provide here a proof of ( 3.1), which is a special case of a more general\nsituation below. Let us consider a fractional heat equation given in t he following form for suitable force f,\ninitial data w0,α∈[0,∞),ν∈(0,∞),T∈(0,∞] andd≥1\n∂tw+ν(−∆)αw=fin (0,T)×Rdandw|t=0=w0. (7.6)\nIt is well-known that the solution to ( 7.6) can be represented by the following Duhamel formula\nw(t) = exp{tν(−∆)α}w0+/integraldisplayt\n0exp{(t−τ)ν(−∆)α}f(τ)dτfort∈(0,T),\nwhere we have been used the notation\nexp{tν(−∆)α}f:=F−1(exp{−νt|ξ|2α}F(f)(ξ)).\nIn the sequel, we aim to prove the following result, which mostly follows the ideas in [ 2, Proposition 3.1], where\nthe authors focused on the case α= 1. We note that for a similar result in form of Chemin-Lerner spaces , see\n[17, Proposition 2].\nProposition 7.1. Letd≥1andwbe a solution to (7.6)withw|t=0=w0,α∈[0,∞), andν∈(0,∞).\nAssume that δ0∈R,p∈[1,∞],1< r≤m <∞,1≤q≤m,T∈(0,∞],w0∈˙Bδ0+2α\np,q(Rd))and\nf∈Lr(0,T;˙Bδ0+2\nrp,q(Rd). Then there are some positive constants C1=C1(α,d,δ0,m,ν,p,q,r )andC2=\nC2(α,d,δ0,m,ν,p,q )such that\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nmp,q (Rd))≤C1/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nrp,q(Rd))+C2/ba∇dblw0/ba∇dbl˙Bδ0+2α\np,q(Rd). (7.7)\nOnce the above proposition is established, ( 3.1) follows by choosing α=3\n2,δ0=s−3,m=r=p= 2,q= 1\nandw=v2withw0=v2\n0= 0. Before going to the proof of Proposition 7.1, we need to establish the following\ntechnical lemma, which follows the ideas in [ 5, Lemma 2.4], [ 22, Lemma 2.1], where the authors considered the\ncaseα= 1. See also [ 17, Lemma 1] with a similar proof in the case d= 3 and α≥0.\nLemma 7.3. Letd≥1andC(c1,c2)be an annulus with the smaller radius c1>0and the bigger radius c2>0.\nThere exist positive constants C3=C3(α,c1,c2,d)andC4=C4(α,c1,d)such that for any α∈[0,∞),p∈[1,∞]\nand any pair (t,λ)of positive real numbers the following property holds. If supp(F(u))⊂λCthen\n/ba∇dblexp{tν(−∆)α}u/ba∇dblLp(Rd)≤C4exp{−C3νtλ2α}/ba∇dblu/ba∇dblLp(Rd). (7.8)\n51Proof of Lemma 7.3.It can be seen that in the case p= 2, (7.8) follows immediately by using the Plancherel’s\nidentity. For p∈[1,∞], we will closely follow the idea in [ 5, Lemma 2.4] by focusing mainly on the case λ= 1.\nIndeed, the case λ/\\e}atio\\slash= 1 can be transformed to the case λ= 1 as follows. Assume that ( 7.8) holds in the case\nλ= 1 for any t′∈(0,∞) and for any fsatisfying supp( F(f))⊂ C, i.e.,\n/ba∇dblexp{t′ν(−∆)α}f/ba∇dblLp≤C3exp{−C4νt′}/ba∇dblf/ba∇dblLp. (7.9)\nWe now fix ( t,λ)∈(0,∞). Letube a function such that supp( F(u))⊂λC. We define for x∈Rd\nv(x) :=1\nλdu/parenleftigx\nλ/parenrightig\nwith F(v)(ξ) =F(u)(λξ)∀ξ∈Rd,\nwhich yields supp( F(v))⊂ C. An application of ( 7.9) to the case f=vandt′=tλ2αgives us\n/ba∇dblexp{tλ2ν(−∆)α}v/ba∇dblLp≤C3exp{−C4νtλ2}/ba∇dblv/ba∇dblLp=λd\np−dC3exp{−C4νtλ2α}/ba∇dblu/ba∇dblLp.(7.10)\nFurthermore, it can be verified that\nλd−d\np/ba∇dblexp{tλ2ν(−∆)α}v/ba∇dblLp=λd−d\np/ba∇dblF−1(exp{−νt|λξ|2α}F(v)(ξ))(·)/ba∇dblLp\n=/ba∇dblλd−d\npF−1(exp{−νt|λξ|2α}F(u)(λξ))(·)/ba∇dblLp\n=/ba∇dblλ−d\npF−1(exp{−νt|ξ|2α}F(u)(ξ))(λ−1·)/ba∇dblLp\n=/ba∇dblF−1(exp{−νt|ξ|2α}F(u)(ξ))(·)/ba∇dblLp=/ba∇dblexp{tν(−∆)α}u/ba∇dblLp,\nwhich combines with ( 7.10) leading to ( 7.8). Therefore, it remains to check ( 7.8) in the case λ= 1. By choosing\nφ∈C∞\n0(Rd\\{0}) with 0 ≤φ≤1,φ= 1 inC(c1,c2) andφ= 0 outside of C(1\n2c1,3\n2c2). Since supp( F(u))⊂ C\nandφ= 1 inC(c1,c2) then by using Young inequality for convolution\n/ba∇dblexp{tν(−∆)α}u/ba∇dblLp=/ba∇dblF−1(φ(ξ)exp{−νt|ξ|2α}F(u)(ξ))/ba∇dblLp≤ /ba∇dblG(t,·)/ba∇dblL1/ba∇dblu/ba∇dblLp,\nwhere for x∈Rd\nG(t,x) := (2π)−d/integraldisplay\nRdexp{ix·ξ}φ(ξ)exp{−νt|ξ|2α}dξ.\nIt remains to bound /ba∇dblG(t,·)/ba∇dblL1. By using integration by parts, Gcan be rewritten by\nG(t,x) = (2π)−d(1+|x|2)−d/integraldisplay\nRdexp{ix·ξ}(Id−∆ξ)d/parenleftbig\nφ(ξ)exp{−νt|ξ|2α}/parenrightbig\ndξ.\nWe need to control the second term inside of the above integral. It can be checked that\n(Id−∆ξ)d(φ(ξ)exp{−νt|ξ|2α}) =/summationdisplay\n0≤j≤d/summationdisplay\n0≤|α0|≤2jC(α0,j)∂|α0|(φ(ξ))∂2j−|α0|(exp{−νt|ξ|2α}).\nIn addition, since supp( φ)⊂ C(1\n2c1,3\n2c2) then for ξ∈supp(φ), we find that |∂|α0|(φ(ξ))| ≤C(c1,c2,d) and\n|∂2j−|α0|exp{−νt|ξ|2α}| ≤C(α,d)/summationdisplay\n0≤i≤2j−|α0|(νt|ξ|2α)i|ξ|−2j+|α0|exp{−νt|ξ|2α}\n≤C(α,d)|ξ|−2j+|α0|(1+νt|ξ|2α)2dexp{−νt|ξ|2α}\n≤C(α,c1,c2,d)exp{−C(α,c1,d)νt},\nwhere we also used another fact that sexp{−s} ≤exp{1}exp{−1\n2s}for anys∈R,s≥0, which leads to\n(1+s)2dexp{−s} ≤C(d)exp{−c(d)s}as well. Therefore,\n|G(t,x)| ≤C(α,c1,c2,d)(1+|x|2)−dexp{−C(α,c1,d)νt},\nwhich implies that\n/ba∇dblG(t,·)/ba∇dblL1≤C(α,c1,c2,d)exp{−C(α,c1,d)νt}.\nThus, the proof is complete.\nProof of Proposition 7.1.The proof consists of the following steps.\nStep 1: Parabolic regularity estimate. We aim to obtain the following standard estimate\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nmp,q (Rd))≤C1/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nrp,q(Rd))+C2/ba∇dblw0/ba∇dbl˙Bδ0+2α\np,q(Rd). (7.11)\n52for the same range of parameters α,δ0,m,p,rand similar constants C1,C2as (7.7), but only for r≤q≤m. In\norder to prove ( 7.11), as the usual case α= 1, we decompose ˙∆kw:=˙∆kw1+˙∆kw2for each k∈Z, where\n∂t˙∆kw1+ν(−∆)α˙∆kw1= 0, ˙∆kw1(t= 0) =˙∆kw0,\n∂t˙∆kw2+ν(−∆)α˙∆kw2=˙∆kf, ˙∆kw2(t= 0) = 0 ,\nand estimate w1andw2in the desired norm. We begin with the bound of w1by using the fact ˙∆kw1=\nexp{tν(−∆)α(˙∆kw0)}in which Lemma 7.3yields\n/ba∇dbl˙∆kw1/ba∇dblLp≤C3exp{−C4νt22αk}/ba∇dbl˙∆kw0/ba∇dblLp,\nand a direct calculation, which implies that for m,p,q∈[1,∞]\n/ba∇dblw1/ba∇dbl˜Lm(0,T;˙Bδ0+2α+2α\nmp,q )≤C(α,d,ν,m )/parenleftigg/summationdisplay\nk∈Z2(δ0+2α)kq/ba∇dbl˙∆kw0/ba∇dblq\nLp/parenrightigg1\nq\n=C(α,d,ν,m )/ba∇dblw0/ba∇dbl˙Bδ0+2α\np,q.\nSimilarly, since\n˙∆kw2(t) =/integraldisplayt\n0exp{(t−τ)ν(−∆)α˙∆kf dτ,\nby using Lemma 7.3and Minkowski inequality for integrals with r,m,p∈[1,∞] andr≤m\n/ba∇dbl˙∆kw2/ba∇dblLm(0,T;Lp)≤C3/parenleftigg/integraldisplayT\n0/parenleftbigg/integraldisplayt\n0exp{−C4ν(t−τ)22αk}/ba∇dbl˙∆kf/ba∇dblLpdτ/parenrightbiggm\ndt/parenrightigg1\nm\n=C3/parenleftigg/integraldisplayT\n0/parenleftbigg/integraldisplayt\n0exp{−C4ν(t−τ)22αk} /BDt≥τ(t)/ba∇dbl˙∆kf(τ)/ba∇dblLpdτ/parenrightbiggm\ndt/parenrightigg1\nm\n≤C3/integraldisplayT\n0/parenleftigg/integraldisplayT\nτexp{−C4mν(t−τ)22αk}dt/parenrightigg1\nm\n/ba∇dbl˙∆kf(τ)/ba∇dblLpdτ\n≤C(C3,C4,m,ν)/integraldisplayT\n02−2αk\nmexp{−C4ντ22αk}/ba∇dbl˙∆kf(τ)/ba∇dblLpdτ\n≤C(C3,C4,m,ν)2−2αk\nm/parenleftigg/integraldisplayT\n0exp/braceleftbigg\n−C4ντ22αkr\nr−1/bracerightbigg\ndτ/parenrightiggr−1\nr\n/ba∇dbl˙∆kf/ba∇dblLr(0,T;Lp)\n≤C(C3,C4,m,ν,r)2−2αk\nm2−2αkr−1\nr/ba∇dbl˙∆kf/ba∇dblLr(0,T;Lp),\nwhich leads to for 1 ≤r≤mand form,p,q∈[1,∞]\n/ba∇dblw2/ba∇dbl˜Lm(0,T;˙Bδ0+2α+2α\nmp,q )≤C(α,d,ν,m,r )/ba∇dblf/ba∇dbl˜Lr(0,T;˙Bδ0+2α\nrp,q).\nTherefore, for m,r,p,q ∈[1,∞] with 1≤r≤m\n/ba∇dblw/ba∇dbl˜Lm(0,T;˙Bδ0+2α+2α\nmp,q )≤C1/ba∇dblf/ba∇dbl˜Lr(0,T;˙Bδ0+2α\nrp,q)+C2/ba∇dblw0/ba∇dbl˙Bδ0+2α\np,q.\nFurthermore, by using the properties of Chemin-Lerner spaces g iven in Appendix A, we find that\nLr(0,T;˙Bδ0+2α\nrp,q)⊂˜Lr(0,T;˙Bδ0+2α\nrp,q) if r≤q,\n˜Lm(0,T;˙Bδ0+2α+2α\nmp,q )⊂Lm(0,T;˙Bδ0+2α+2α\nmp,q ) if q≤m.\nThus, (7.11) follows by the previous estimate.\nStep 2: The case w0= 0,m=randq= 1.Similar to [ 2, Proposition 3.1] by using the duality argument,\nfor allg∈Lr′(0,T) with1\nr+1\nr′= 1 and for C=C(α,d,δ0,ν,p,r), it is enough to prove that\nI:=/summationdisplay\nk∈Z/integraldisplayT\n0g(t)2k(δ0+2α+2α\nr)/ba∇dbl˙∆kw(t)/ba∇dblLpdt=/integraldisplayT\n0g(t)/ba∇dblw(t)/ba∇dbl˙Bδ0+2α+2α\nr\np,1dt≤C/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nr\np,1)/ba∇dblg/ba∇dblLr′(0,T).\nIt can be seen from the representation formula and Lemma 7.3that fort∈(0,T) andk∈Z\n/ba∇dbl˙∆kw(t)/ba∇dblLp≤/integraldisplayt\n0/ba∇dblexp{(t−τ)ν(−∆)α}˙∆kf(τ)/ba∇dblLpdτ≤C(α,d)/integraldisplayt\n0exp{−C(d)ν(t−τ)22αk}/ba∇dbl˙∆kf(τ)/ba∇dblLpdτ,\n53where we also used the property supp( F(˙∆kw))⊂ C(3\n42k,8\n32k), the annulus with the smaller radius3\n42kand\nthe bigger radius8\n32k, which yields\nI≤C(α,d)/summationdisplay\nk∈Z/integraldisplayT\n0/integraldisplayt\n022αkexp{−C(α,d)ν(t−τ)22αk}|g(t)|2(δ0+2α\nr)k/ba∇dbl˙∆kf(τ)/ba∇dblLpdτdt\n=C(α,d,ν)/summationdisplay\nk∈Z/integraldisplayT\n0/integraldisplayT\nτν22αkexp{−C(α,d)ν(t−τ)22αk}|g(t)|2(δ0+2α\nr)k/ba∇dbl˙∆kf(τ)/ba∇dblLpdtdτ\n=C(α,d,ν)/summationdisplay\nk∈Z/integraldisplayT\n0/integraldisplayT\n0ν22αk/BDt≥τ(t)exp{−C(α,d)ν(t−τ)22αk}|g(t)|2(δ0+2α\nr)k/ba∇dbl˙∆kf(τ)/ba∇dblLpdtdτ\n≤C(α,d,ν)/summationdisplay\nk∈Z/integraldisplayT\n0Mg(τ)2(δ0+2α\nr)k/ba∇dbl˙∆kf(τ)/ba∇dblLpdτ\n=C(α,d,ν)/integraldisplayT\n0Mg(τ)/ba∇dblf(τ)/ba∇dbl˙Bδ0+2α\nr\np,1dτ\n≤C(α,d,ν)/ba∇dblMg/ba∇dblLr′(0,T)/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nr\np,1)\n≤C(α,d,ν,r′)/ba∇dblg/ba∇dblLr′(0,T)/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nr\np,1),\nwhere\nMg(τ) :=/braceleftigg\nsupρ>0/integraltextT\n0ρ /BDt≥τ(t)exp{−C(α,d)(t−τ)ρ}|g(t)|dtifτ∈(0,T),\n0 if τ /∈(0,T).\nIt remains to check the last inequality in the previous estimate. Inde ed, it can be seen that if τ∈(0,T) then\nMg(τ) can be rewritten by\nMg(τ) = sup\n1\nρ>0(K1\nρ∗˜g)(τ) =/integraldisplay\nRK1\nρ(τ−t)˜g(t)dt,\nwhere for t∈R\nK(t) := exp{−C(α,d)|t|}, K 1\nρ(t) :=ρK(tρ) and ˜ g(t) := /BDt≥τ(t) /BD(0,T)(t)|g(t)|.\nIn addition, we can verify that Ksatisfies all conditions in [ 35, Theorem 2.1.10], which yields\nMg(τ) = sup\n1\nρ>0(K1\nρ∗˜g)(τ)≤ /ba∇dblK/ba∇dblL1(R)M(˜g)(τ)≤C(α,d)M(˜g)(τ),\nwhere the centered Hardy–Littlewood maximal function of ˜ gis defined by\nM(˜g)(τ) := sup\nr>01\n2r/integraldisplay\nB(τ,r)|˜g(t)|dtwith B(τ,r) :={s∈R:|s−τ|< r}.\nFinally, an application of [ 35, Theorem 2.1.6], which is on the boundedness of the maximal operato rMfrom\nLp0toLp0forp0∈(1,∞), implies that\n/ba∇dblMg/ba∇dblLr′(R)≤C(α,d)/ba∇dblM˜g/ba∇dblLr′(R)≤C(α,d,r′)/ba∇dbl˜g/ba∇dblLr′(R)=C(α,d,r′)/ba∇dblg/ba∇dblLr′(0,T).\nStep 3: The case w0= 0,1< r≤m <∞and1≤q≤m.We use exactly the argument in [ 2]. More\nprecisely, it follows from ( 7.7) form=r,q= 1 and from ( 7.11) forq=r= 1, respectively, that\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nm\np,1)≤C1/ba∇dblf/ba∇dbl\nLm(0,T;˙Bδ0+2α\nm\np,1),\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nm\np,1)≤C1/ba∇dblf/ba∇dblL1(0,T;˙Bδ0+2α\np,1),\nwhich combines with interpolation theory in [ 8, Theorems 5.1.2 and 6.4.5] yielding for 1 < r≤m <∞\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nm\np,1)≤C1/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nr\np,1).\nIn addition, ( 7.11) withq=m, which gives us\n/ba∇dblw/ba∇dbl\nLm(0,T;˙Bδ0+2α+2α\nmp,m )≤C1/ba∇dblf/ba∇dbl\nLr(0,T;˙Bδ0+2α\nrp,m).\nThus, combining the two previous estimates finishes the proof of th is step. Therefore, ( 7.7) withw0= 0 follows.\nStep 4: The case w0/\\e}atio\\slash= 0,1< r≤m <∞and1≤q≤m.In this step, in order to prove ( 7.7) in the case\nw0/\\e}atio\\slash= 0, we can repeat Step 1 (use the same estimate of w1) with using ( 7.7) forw0= 0 (in the estimate of w2)\nto obtain the desired result. We omit further details. Thus, the pro of of the proposition now is finished.\n547.5 Appendix E: Remarks on the Maxwell equations\nIn this subsection, show that under suitable assumptions on the ve locity, the existence and uniqueness of L2\nweak solutions to ( M) can be provided. We first recall the following result.\nLemma 7.4. ([33, Lemma 2.2], [ 59, Lemma 1.10]) If(E0,B0)∈Hs(Rd)withs∈Randj∈L1\nloc(0,∞;Hs(Rd))\nthen any solution (E,B)to(M)satisfying for any T∈(0,∞)\n/ba∇dbl(E,B)/ba∇dbl2\nL∞(0,T;Hs)≤2/ba∇dbl(E0,B0)/ba∇dbl2\nHs+2c2/ba∇dblj/ba∇dbl2\nL1(0,T;Hs).\nLemma 7.5. Letd∈ {2,3}andE,B,v:Rd×(0,∞)→R3satisfying (M)with(E,B)|t=0= (E0,B0).\n(i)(Global well-posedness) Ifv∈L∞\nloc(0,∞;L2(Rd))∩L2\nloc(0,∞;Hd\n2(Rd)∩L∞(Rd))and(E0,B0)∈Hs(Rd)\nwiths∈[0,d\n2)then there exists aunique global weak solution (E,B)to(M)satisfying (E,B)∈L∞(0,T;Hs)\nandE∈L2(0,T;Hs)for anyT∈(0,∞).\n(ii)(The limit as c→ ∞)Letvbe given as in Part (i). Letc >0and(Ec\n0,Bc\n0)∈Hs(Rd)withs∈[0,d\n2)\nsatisfying divBc\n0= 0and asc→ ∞\n(Ec\n0,Bc\n0)⇀(¯E0,¯B0)inHs\nfor some (¯E0,¯B0)withdiv¯B0= 0. Then there exists a sequence of global solutions (Ec,Bc)to(M)with\n(Ec,Bc)|t=0= (Ec\n0,Bc\n0)given as Part (i). In addition, up to an extraction of a subseq uence,Bcconverges\ntoBin the sense of distributions as c→ ∞, whereBsatisfies\n∂tB−∇×(v×B) =1\nσ∆B,divB= 0andB|t=0=¯B0.\nProof of Lemma 7.5.The proof is very simple, which shares the ideas as those of Theorem s1.1and1.2and can\nbe done as follows.\nStep 1: The existence of Part (i).We first consider an approximate system to ( M) by\n1\ncd\ndt(En,Bn) =Fn(En,Bn),divBn= 0 and ( En,Bn)|t=0=Tn(E0,B0),\nwhereFn= (Fn\n1,Fn\n2) withFn\n1=∇×Bn−jn,jn=σ(cEn+Tn(v×Bn)) andFn\n2=−∇×En. Furthermore,\nfors∈[0,d\n2),Fn:Hs\nn×Vs\nn→Hs\nn×Vs\nnis well-defined and is a locally Lipschitz function as well. Therefore,\nthere exists a unique solution ( En,Bn)∈C1([0,Tn\n∗);Hs\nn×Vs\nn) for some Tn\n∗∈(0,∞] satisfying in addition if\nTn∗<∞then\nlim\nt→Tn∗/ba∇dbl(En,Bn)/ba∇dbl2\nHs=∞.\nAssume that Tn\n∗<∞then the energy balance\nd\ndt/ba∇dbl(En,Bn)/ba∇dbl2\nL2+1\nσ/ba∇dbljn/ba∇dbl2\nL2=/integraldisplay\nRdTn(v×Bn)·jndx,\nwhich implies for t∈(0,Tn\n∗)\n/ba∇dbl(En,Bn)(t)/ba∇dbl2\nL2≤ /ba∇dbl(E0,B0)/ba∇dbl2\nL2exp/braceleftigg\nC(σ)/integraldisplayTn\n∗\n0/ba∇dblv/ba∇dbl2\nL∞dτ/bracerightigg\n.\nSimilarly, for s∈(0,d\n2)\n/ba∇dbl(En,Bn)(t)/ba∇dbl2\n˙Hs≤ /ba∇dbl(E0,B0)/ba∇dbl2\n˙Hsexp/braceleftigg\nC(σ,s)/integraldisplayTn\n∗\n0/ba∇dblv/ba∇dbl2\n˙Hd\n2+/ba∇dblv/ba∇dbl2\nL∞dτ/bracerightigg\n.\nThe above estimates give us a contradiction to the assumption Tn\n∗<∞and yield Tn\n∗=∞. Replacing Tn\n∗by\nanyT∈(0,∞), we obtain uniform bounds (in terms of n) of (En,Bn) inL∞(0,T;Hs) andEninL2(0,T;Hs).\nThat leads to the existence of ( E,B) such that up to an extraction of a subsequence\n(En,Bn)∗⇀(E,B) in L∞(0,T;Hs(Rd))\nEn⇀ E inL2(0,T;Hs(Rd)).\nMoreover, as Step 17b in the proof of Theorem 1.1, by using the following weak formulation for ϕ∈C∞\n0([0,T)×\nRd;R3)\n/integraldisplayT\n0/integraldisplay\nRd1\ncEn·∂tϕ+Bn·(∇×ϕ)−σ(cEn+Tn(v×Bn))·ϕdxdt=−/integraldisplay\nRd1\ncEn(0)·ϕ(0)dx,\n55/integraldisplayT\n0/integraldisplay\nRd1\ncBn·∂tϕ−En·(∇×ϕ)dxdt=−/integraldisplay\nRd1\ncBn(0)·ϕ(0)dx,\nwe can pass to the limit as n→ ∞easily by using the assumptions of v.\nStep 2: The uniqueness of Part (i).For two solutions ( E,B) and (¯E,¯B) to (M) with the same initial\ndata (E0,B0) and for s′∈[0,s), it follows from Lemma 7.4that\n/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T;Hs′)≤c2/ba∇dblj−¯j/ba∇dbl2\nL1(0,T;Hs′)≤6/summationdisplay\nk=4¯Jk.\nBy repeating either Step 18 in the proof of Theorem 1.1ford= 2 or Step 15 in the proof of Theorem 1.2for\nd= 3, we find that for v= ¯vand for sufficiently small T∗∈(0,T)\n/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′)≤1\n2/ba∇dbl(E−¯E,B−¯B)/ba∇dbl2\nL∞(0,T∗;Hs′),\nwhich yields E=¯EandB=¯Bin (0,T∗). By repeating this process, we obtain the conclusion in the whole\ntime interval (0 ,T).\nStep 3: Proof of Part (ii).Since the main estimates in Step 1 are independent on c, then the proof\nfollows as that of Part ( ii) in Theorem 1.2by using the above weak formulation ( nis replaced by cand without\nTn), the fact jc=σ(cEc+v×Bc) and the weak convergence of ( Ec\n0,Bc\n0). Thus, the proof now is complete.\n7.6 Appendix F: Proof of Proposition 1.1\nFor the sake of completeness, we will give a proof of Proposition 1.1below.\nProof of Proposition 1.1-The case d= 2.The proof consits of the following steps.\nStep 1: Local and global existence. As previous parts, an approximate system to ( H) is given by\n∂tBn=−1\nσ(−∆)αBn−κ\nσ∇×Tn(jn×Bn),divBn= 0, Bn\n|t=0=Tn(B0), jn:=∇×Bn,(7.12)\nand there exists a unique solution Bnto (7.12) withBn∈C1([0,Tn\n∗);Vs\nn) for some Tn\n∗>0. It is sufficient to\nfocus on the case α=3\n2. The case α >3\n2can be done in the same way in which we will omit the details. It can\nbe seen from ( 7.12) that for s >0\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\nL2+1\nσ/ba∇dblBn/ba∇dbl2\n˙H3\n2= 0,\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\n˙Hs+1\nσ/ba∇dblBn/ba∇dbl2\n˙Hs+3\n2=−κ\nσ/integraldisplay\nR2Λs(jn×Bn)·Λsjndx=:H.\n•Ifs∈(0,1\n2) then for some ǫ∈(0,1) the˙Hsestimate is closable as follows\nH=−κ\nσ/integraldisplay\nR2(jn×Bn)·Λ2sjndx≤κ\nσC(s)/ba∇dbljn/ba∇dblL4/ba∇dblBn/ba∇dbl\nL2\n1−s/ba∇dblΛ2sjn/ba∇dbl\nL2\n1−(3\n2−(s+1))\n≤ǫ\nσ/ba∇dblBn/ba∇dbl2\n˙Hs+3\n2+C(ǫ,κ,σ,s)/ba∇dblBn/ba∇dbl2\n˙H3\n2/ba∇dblBn/ba∇dbl2\n˙Hs,\n•Ifs=1\n2then 2s+1 =s+3\n2= 2 and\nH≤κ\nσ/ba∇dbljn/ba∇dblL4/ba∇dblBn/ba∇dblL4/ba∇dblΛ2sjn/ba∇dblL2≤ǫ\nσ/ba∇dblBn/ba∇dbl2\n˙H2+C(ǫ,κ,σ)/ba∇dblBn/ba∇dbl2\n˙H3\n2/ba∇dblBn/ba∇dbl2\n˙H1\n2.\n•Ifs >1\n2then\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\nHs+1\nσ/ba∇dblΛ3\n2Bn/ba∇dbl2\nHs=−κ\nσ/integraldisplay\nR2[Js(jn×Bn)−Jsjn×Bn]·Jsjndx=:H,\nwhere for some ǫ∈(0,1), the Kato-Ponce commutator estimate gives\nH≤C(s)κ\nσ(/ba∇dbljn/ba∇dblHs−1/ba∇dbl∇Bn/ba∇dblL∞+/ba∇dbljn/ba∇dblL∞/ba∇dblBn/ba∇dblHs)/ba∇dblBn/ba∇dblHs+1\n≤ǫ\nσ/ba∇dblBn/ba∇dbl2\nHs+3\n2+C(ǫ,κ,σ,s)/ba∇dbl∇Bn/ba∇dbl2\nL∞/ba∇dblBn/ba∇dbl2\nHs,\nwhich by choosing ǫ=1\n2and using ( 2.2) withd= 2,f=∇Bnands0=s−1\n2>0 implies that\nd\ndt/ba∇dblBn/ba∇dbl2\nHs+1\nσ/ba∇dblBn/ba∇dbl2\nHs+3\n2≤1\nσ/ba∇dblBn/ba∇dbl2\nL2+/bracketleftigg\n1\n2C(κ,σ,s)/ba∇dblBn/ba∇dblHs/ba∇dbl∇Bn/ba∇dblH1/parenleftigg\n1+log1\n2/parenleftigg\n/ba∇dblBn/ba∇dblHs+3\n2\n/ba∇dbl∇Bn/ba∇dblH1/parenrightigg/parenrightigg/bracketrightigg2\n.\n56By using the previous case to bound /ba∇dblBn/ba∇dblL2\ntH2\nx, the conclusion follows as Step 3 in the proof of Theorem 1.1.\nStep 2: Pass to the limit. This step can be done as either Step 16a for s >1 or Step 16b for s∈[0,1] in\nthe proof of Theorem 1.1. We omit further details.\nStep 3: Uniqueness. It is enough to consider the case s= 0. Let Bbe the limit in Step 2. It can be seen\nthatB∈L2(0,T;H3\n2) and∂tB∈L2(0,T;H−3\n2), which implies that B∈C([0,T];L2) (see [67]) after possibly\nbeing redefined on a set of measure zero. Assume that B1andB2are two solutions to ( H) with the same initial\ndataB0∈L2andji=∇×Bifori∈ {1,2}. Thus, we find that\n1\n2d\ndt/ba∇dblB1−B2/ba∇dbl2\nL2+1\nσ/ba∇dblB1−B2/ba∇dbl2\n˙H3\n2=−/integraldisplay\nR2(j1×B1−j2×B2)·(j1−j2)dx=:H1\nwhere for some ǫ∈(0,1)\nH1≤ /ba∇dblj1/ba∇dblL4/ba∇dblB1−B2/ba∇dblL2/ba∇dblj1−j2/ba∇dblL4≤ǫ\nσ/ba∇dblB1−B2/ba∇dbl2\n˙H3\n2+C(ǫ,σ)/ba∇dblB1/ba∇dbl2\n˙H3\n2/ba∇dblB1−B2/ba∇dbl2\nL2,\nwhich yields B1=B2and ends the proof.\nProof of Proposition 1.1-The case d= 3.The proof is divided into several steps as follows.\nStep 1: Local and global existence. Similar to the previous case, we will focus on ( 7.12) withα=7\n4.\nIn addition, for s >0\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\nL2+1\nσ/ba∇dblBn/ba∇dbl2\n˙H7\n4= 0,\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\n˙Hs+1\nσ/ba∇dblBn/ba∇dbl2\n˙Hs+7\n4=−κ\nσ/integraldisplay\nR3Λs(jn×Bn)·Λsjndx=:H.\n•Ifs∈(0,3\n4) then\nH≤κ\nσC(s)/ba∇dbljn/ba∇dblL4/ba∇dblBn/ba∇dbl\nL6\n3−2s/ba∇dblΛ2sjn/ba∇dbl\nL6\n3−2(3\n2−(s+3\n4))\n≤ǫ\nσ/ba∇dblBn/ba∇dbl2\n˙Hs+7\n4+C(ǫ,κ,σ,s)/ba∇dblBn/ba∇dbl2\n˙H7\n4/ba∇dblBn/ba∇dbl2\n˙Hs.\n•Ifs=3\n4then 2s+1 =s+7\n4and\nH≤ /ba���dbljn/ba∇dblL4/ba∇dblBn/ba∇dblL4/ba∇dblΛ2sjn/ba∇dblL2≤ǫ\nσ/ba∇dblBn/ba∇dbl2\n˙Hs+7\n4+C(ǫ,κ,σ,s)/ba∇dblBn/ba∇dbl2\n˙H7\n4/ba∇dblBn/ba∇dbl2\n˙H3\n4.\n•Ifs >3\n4then\n1\n2d\ndt/ba∇dblBn/ba∇dbl2\nHs+1\nσ/ba∇dblΛ7\n4Bn/ba∇dbl2\nHs=−κ\nσ/integraldisplay\nR3[Js(jn×Bn)−Jsjn×Bn]·Jsjndx=:H,\nwhere for some ǫ∈(0,1), the Kato-Ponce commutator estimate gives\nH≤C(s)κ\nσ(/ba∇dbljn/ba∇dblHs−1/ba∇dbl∇Bn/ba∇dblL∞+/ba∇dbljn/ba∇dblL∞/ba∇dblBn/ba∇dblHs)/ba∇dblBn/ba∇dblHs+1\n≤ǫ\nσ/ba∇dblBn/ba∇dbl2\nHs+7\n4+C(ǫ,κ,σ,s)/ba∇dbl∇Bn/ba∇dbl2\nL∞/ba∇dblBn/ba∇dbl2\nHs,\nwhich by choosing ǫ=1\n2and using ( 2.2) withd= 3,f=∇Bnands0=s−1\n4>1\n2implies that\nd\ndt/ba∇dblBn/ba∇dbl2\nHs+1\nσ/ba∇dblBn/ba∇dbl2\nHs+7\n4≤1\nσ/ba∇dblBn/ba∇dbl2\nL2+/bracketleftigg\n1\n2C(κ,σ,s)/ba∇dblBn/ba∇dblHs/ba∇dbl∇Bn/ba∇dblH3\n2/parenleftigg\n1+log1\n2/parenleftigg\n/ba∇dblBn/ba∇dblHs+7\n4\n/ba∇dbl∇Bn/ba∇dblH3\n2/parenrightigg/parenrightigg/bracketrightigg2\n.\nTherefore, the conclusion follows.\nStep 2: Pass to the limit and uniqueness. This step follows as that of in the previous case. Indeed,\nthe uniqueness in the case s= 0 is proceeded with\nH1≤ /ba∇dblj1/ba∇dblL4/ba∇dblB1−B2/ba∇dblL2/ba∇dblj1−j2/ba∇dblL4≤ǫ\nσ/ba∇dblB1−B2/ba∇dbl2\n˙H7\n4+C(ǫ,σ)/ba∇dblB1/ba∇dbl2\n˙H7\n4/ba∇dblB1−B2/ba∇dbl2\nL2,\nwhich finishes the proof.\n57References\n[1] Marion Acheritogaray, Pierre Degond, Amic Frouvelle, and Jian-G uo Liu. Kinetic formulation and global\nexistence for the Hall-Magneto-hydrodynamics system. Kinet. Relat. Models , 4(4):901–918, 2011.\n[2] Diogo Ars´ enio and Isabelle Gallagher. Solutions of Navier-Stokes -Maxwell systems in large energy spaces.\nTrans. Amer. Math. Soc. , 373(6):3853–3884, 2020.\n[3] Diogo Ars´ enio, Slim Ibrahim, and Nader Masmoudi. A derivation of t he magnetohydrodynamic system\nfrom Navier-Stokes-Maxwell systems. Arch. Ration. Mech. Anal. , 216(3):767–812, 2015.\n[4] Hantaek Bae and Kyungkeun Kang. On the existence and tempor al asymptotics of solutions for the two\nand half dimensional Hall MHD. J. Math. Fluid Mech. , 25(2):Paper No. 24, 30, 2023.\n[5] HajerBahouri,Jean-YvesChemin, andRapha¨ elDanchin. Fourier analysis and nonlinear partial differential\nequations , volume 343 of Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of\nMathematical Sciences] . Springer, Heidelberg, 2011.\n[6] David Barbato, Francesco Morandin, and Marco Romito. Global r egularity for a slightly supercritical\nhyperdissipative Navier-Stokes system. Anal. PDE , 7(8):2009–2027, 2014.\n[7] Rajendra Beekie, Susan Friedlander, and Vlad Vicol. On Moffatt’s m agnetic relaxation equations. Comm.\nMath. Phys. , 390(3):1311–1339, 2022.\n[8] J¨ oran Bergh and J¨ orgen L¨ ofstr¨ om. Interpolation spaces. An introduction , volume No. 223 of Grundlehren\nder Mathematischen Wissenschaften . Springer-Verlag, Berlin-New York, 1976.\n[9] DieterBiskamp. Nonlinear magnetohydrodynamics , volume1of Cambridge Monographs on Plasma Physics .\nCambridge University Press, Cambridge, 1993.\n[10] Franck Boyer and Pierre Fabrie. Mathematical tools for the study of the incompressible Navi er-Stokes\nequations and related models , volume 183 of Applied Mathematical Sciences . Springer, New York, 2013.\n[11] Haim Brezis. Functional analysis, Sobolev spaces and partial differenti al equations . Universitext. Springer,\nNew York, 2011.\n[12] Haim Brezis and Thierry Gallouet. Nonlinear Schr¨ odinger evolutio n equations. Nonlinear Anal. , 4(4):677–\n681, 1980.\n[13] Haim Brezis and Petru Mironescu. Gagliardo-Nirenberg inequalitie s and non-inequalities: the full story.\nAnn. Inst. H. Poincar´ e C Anal. Non Lin´ eaire , 35(5):1355–1376, 2018.\n[14] Haim Brezis and Stephen Wainger. A note on limiting cases of Sobole v embeddings and convolution\ninequalities. Comm. Partial Differential Equations , 5(7):773–789, 1980.\n[15] Chongsheng Cao, Jiahong Wu, and Baoquan Yuan. The 2D incomp ressible magnetohydrodynamics equa-\ntions with only magnetic diffusion. SIAM J. Math. Anal. , 46(1):588–602, 2014.\n[16] Dongho Chae, Pierre Degond, and Jian-Guo Liu. Well-posedness for Hall-magnetohydrodynamics. Ann.\nInst. H. Poincar´ e C Anal. Non Lin´ eaire , 31(3):555–565, 2014.\n[17] Dongho Chae and Jihoon Lee. On the global well-posedness and s tability of the Navier-Stokes and the\nrelated equations. In Contributions to current challenges in mathematical fluid m echanics, Adv. Math.\nFluid Mech., pages 31–51. Birkh¨ auser, Basel, 2004.\n[18] Dongho Chae and Jihoon Lee. On the blow-up criterion and small d ata global existence for the Hall-\nmagnetohydrodynamics. J. Differential Equations , 256(11):3835–3858, 2014.\n[19] DonghoChaeandMariaE.Schonbek. Onthetemporaldecayfo rtheHall-magnetohydrodynamicequations.\nJ. Differential Equations , 255(11):3971–3982, 2013.\n[20] Dongho Chae, Renhui Wan, and Jiahong Wu. Local well-posedne ss for the Hall-MHD equations with\nfractional magnetic diffusion. J. Math. Fluid Mech. , 17(4):627–638, 2015.\n[21] Dongho Chae and J¨ org Wolf. Regularity of the 3 Dstationary hall magnetohydrodynamic equations on the\nplane.Comm. Math. Phys. , 354(1):213–230, 2017.\n[22] Jean-Yves Chemin. Th´ eor` emesd’unicit´ e pour le syst` eme de Navier-Stokestridimensionnel. J. Anal. Math. ,\n77:27–50, 1999.\n58[23] Jean-Yves Chemin and Nicolas Lerner. Flot de champs de vecteu rs non lipschitziens et ´ equations de\nNavier-Stokes. J. Differential Equations , 121(2):314–328, 1995.\n[24] Peter Constantin and Federico Pasqualotto. Magnetic relaxat ion of a Voigt-MHD system. Comm. Math.\nPhys., 402(2):1931–1952, 2023.\n[25] Mimi Dai. Nonunique weak solutions in Leray-Hopf class for the th ree-dimensional Hall-MHD system.\nSIAM J. Math. Anal. , 53(5):5979–6016, 2021.\n[26] Rapha¨ el Danchin and Jin Tan. On the well-posedness of the Hall- magnetohydrodynamics system in critical\nspaces.Comm. Partial Differential Equations , 46(1):31–65, 2021.\n[27] Rapha¨ el Danchin and Jin Tan. The global solvability of the Hall-ma gnetohydrodynamics system in critical\nSobolev spaces. Commun. Contemp. Math. , 24(10):Paper No. 2150099, 33, 2022.\n[28] Peter A. Davidson. Introduction to magnetohydrodynamics . Cambridge Texts in Applied Mathematics.\nCambridge University Press, Cambridge, second edition, 2017.\n[29] LawrenceC.Evans. Partial differential equations , volume19of Graduate Studies in Mathematics . American\nMathematical Society, Providence, RI, second edition, 2010.\n[30] Jishan Fan and Tohru Ozawa. Global solutions to the Maxwell-Nav ier-Stokes system in a bounded domain\nin 2D.Z. Angew. Math. Phys. , 71(4):Paper No. 136, 6, 2020.\n[31] Charles L. Fefferman, David S. McCormick, James C. Robinson, a nd Jose L. Rodrigo. Higher order\ncommutator estimates and local existence for the non-resistive M HD equations and related models. J.\nFunct. Anal. , 267(4):1035–1056, 2014.\n[32] Pierre Germain, Slim Ibrahim, and Nader Masmoudi. Well-posednes s of the Navier-Stokes-Maxwell equa-\ntions.Proc. Roy. Soc. Edinburgh Sect. A , 144(1):71–86, 2014.\n[33] Yoshikazu Giga, Slim Ibrahim, Shengyi Shen, and Tsuyoshi Yone da. Global well posedness for a two-fluid\nmodel.Differential Integral Equations , 31(3-4):187–214, 2018.\n[34] Yoshikazu Giga and Zensho Yoshida. On the equations of the two -component theory in magnetohydrody-\nnamics.Comm. Partial Differential Equations , 9(6):503–522, 1984.\n[35] Loukas Grafakos. Classical Fourier analysis , volume 249 of Graduate Texts in Mathematics . Springer, New\nYork, third edition, 2014.\n[36] Loukas Grafakos and Seungly Oh. The Kato-Ponce inequality. Comm. Partial Differential Equations ,\n39(6):1128–1157, 2014.\n[37] Weiwei Hu, Igor Kukavica, and Mohammed Ziane. Persistence of regularity for the viscous Boussinesq\nequations with zero diffusivity. Asymptot. Anal. , 91(2):111–124, 2015.\n[38] Slim Ibrahim and Sahbi Keraani. Global small solutions for the Nav ier-Stokes-Maxwell system. SIAM J.\nMath. Anal. , 43(5):2275–2295, 2011.\n[39] Slim Ibrahim, Pierre-Gilles Lemari´ e-Rieusset, and Nader Masmo udi. Time-periodic forcing and asymptotic\nstability for the Navier-Stokes-Maxwell equations. Comm. Pure Appl. Math. , 71(1):51–89, 2018.\n[40] Slim Ibrahim and Tsuyoshi Yoneda. Local solvability and loss of sm oothness of the Navier-Stokes-Maxwell\nequations with large initial data. J. Math. Anal. Appl. , 396(2):555–561, 2012.\n[41] Juhi Jang and Nader Masmoudi. Derivation of Ohm’s law from the k inetic equations. SIAM J. Math.\nAnal., 44(5):3649–3669, 2012.\n[42] In-Jee Jeong and Sung-Jin Oh. On the Cauchy problem for the H all and electron magnetohydrodynamic\nequations without resistivity I: Illposedness near degenerate sta tionary solutions. Ann. PDE , 8(2):Paper\nNo. 15, 106, 2022.\n[43] Ning Jiang, Yi-Long Luo, and Shaojun Tang. Convergence from two-fluid incompressible Navier-Stokes-\nMaxwell system with Ohm’s law to solenoidal Ohm’s law: classical solution s.J. Differential Equations ,\n269(1):349–376, 2020.\n[44] Min JunJo, Junha Kim, andJihoonLee. Globalwell-posednessoft he partiallydamped 2DMHD equations\nvia a direct normal mode method for the anisotropic linear operator , arXiv:2210.10283, 2022.\n59[45] Ensil Kang and Jihoon Lee. Notes on the global well-posedness f or the Maxwell-Navier-Stokes system.\nAbstr. Appl. Anal. , pages Art. ID 402793, 6, 2013.\n[46] Kyungkeun Kang, Jihoon Lee, and Dinh Duong Nguyen. Global we ll-posedness and stability of the 2D\nBoussinesq equations with partial dissipation near a hydrostatic eq uilibrium. J. Differential Equations ,\n393:1–57, 2024.\n[47] TosioKato and GustavoPonce. Commutatorestimates and the Euler and Navier-Stokesequations. Comm.\nPure Appl. Math. , 41(7):891–907, 1988.\n[48] Carlos E. Kenig, Gustavo Ponce, and Luis Vega. Well-posedness and scattering results for the generalized\nKorteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. , 46(4):527–620, 1993.\n[49] Igor Kukavica, Fei Wang, and Mohammed Ziane. Persistence of regularity for solutions of the Boussinesq\nequations in Sobolev spaces. Adv. Differential Equations , 21(1-2):85–108, 2016.\n[50] Suhua Lai, Jiahong Wu, and Yueyuan Zhong. Stability and large-t ime behavior of the 2D Boussinesq\nequations with partial dissipation. J. Differential Equations , 271:764–796, 2021.\n[51] GiovanniLeoni. A first course in fractional Sobolev spaces , volume229of Graduate Studies in Mathematics .\nAmerican Mathematical Society, Providence, RI, 2023.\n[52] Jean Leray. ´Etude de diverses ´ equations int´ egrales non lin´ eaires et de quelqu es probl` emes que pose\nl’hydrodynamique. J. Math. Pures Appl. , 9(12):1–82, 1933.\n[53] Jean Leray. Sur le mouvement d’un liquide visqueux emplissant l’esp ace.Acta Math. , 63(1):193–248, 1934.\n[54] Michael J. Lighthill. Studies on magneto-hydrodynamic waves an d other anisotropic wave motions. Philos.\nTrans. Roy. Soc. London Ser. A , 252:397–430, 1960.\n[55] Fanghua Lin, Li Xu, and Ping Zhang. Global small solutions of 2-D incompressible MHD system. J.\nDifferential Equations , 259(10):5440–5485, 2015.\n[56] Jacques L. Lions. Quelques m´ ethodes de r´ esolution des probl` emes aux limit es non lin´ eaires . Dunod, Paris;\nGauthier-Villars, Paris, 1969.\n[57] LvqiaoLiu and Jin Tan. Global well-posednessfor the Hall-magnet ohydrodynamicssystem in largercritical\nBesov spaces. J. Differential Equations , 274:382–413, 2021.\n[58] Andrew J. Majda and Andrea L. Bertozzi. Vorticity and incompressible flow , volume 27 of Cambridge\nTexts in Applied Mathematics . Cambridge University Press, Cambridge, 2002.\n[59] Nader Masmoudi. Global well posedness for the Maxwell-Navier- Stokes system in 2D. J. Math. Pures\nAppl. (9) , 93(6):559–571, 2010.\n[60] Henry K. Moffatt. Magnetostatic equilibria and analogous Euler fl ows of arbitrarily complex topology. I.\nFundamentals. J. Fluid Mech. , 159:359–378, 1985.\n[61] Yi Peng, Huaqiao Wang, and Qiuju Xu. Derivation of the Hall-MHD e quations from the Navier-Stokes-\nMaxwell equations. J. Nonlinear Sci. , 32(6):Paper No. 90, 27, 2022.\n[62] Maria E. Schonbek. L2decay for weak solutions of the Navier-Stokes equations. Arch. Rational Mech.\nAnal., 88(3):209–222, 1985.\n[63] Maria E. Schonbek. Large time behaviour of solutions to the Nav ier-Stokes equations. Comm. Partial\nDifferential Equations , 11(7):733–763, 1986.\n[64] Jin Tan. New energy functionals for the incompressible Hall-MHD s ystem. Preprint hal-03855638, 2022.\n[65] Terence Tao. Nonlinear dispersive equations , volume 106 of CBMS Regional Conference Series in Math-\nematics. Published for the Conference Board of the Mathematical Science s, Washington, DC; by the\nAmerican Mathematical Society, Providence, RI, 2006. Local and global analysis.\n[66] Terence Tao. Global regularity for a logarithmically supercritica l hyperdissipative Navier-Stokes equation.\nAnal. PDE , 2(3):361–366, 2009.\n[67] Roger Temam. Navier-Stokes equations . AMS Chelsea Publishing, Providence, RI, 2001. Theory and\nnumerical analysis, Reprint of the 1984 edition.\n60[68] Renhui Wan. Global regularity for generalized Hall magneto-hy drodynamics systems. Electron. J. Differ-\nential Equations , (179):1–18, 2015.\n[69] Renhui Wan and Yong Zhou. Global well-posedness, BKM blow-up criteria and zero hlimit for the 3D\nincompressible Hall-MHD equations. J. Differential Equations , 267(6):3724–3747, 2019.\n[70] Renhui Wan and Yong Zhou. Global well-posedness for the 3D inc ompressible Hall-magnetohydrodynamic\nequations with Fujita-Kato type initial data. J. Math. Fluid Mech. , 21(1):Paper No. 5, 16, 2019.\n[71] Zhihong Wen and Zhuan Ye. Regularity results for the Navier-St okes-Maxwell system. Commun. Math.\nSci., 18(2):339–358, 2020.\n[72] Jiahong Wu. Generalized MHD equations. J. Differential Equations , 195(2):284–312, 2003.\n[73] Jiahong Wu. Global regularity for a class of generalized magneto hydrodynamic equations. J. Math. Fluid\nMech., 13(2):295–305, 2011.\n[74] Jiahong Wu, Yifei Wu, and Xiaojing Xu. Global small solution to the 2D MHD system with a velocity\ndamping term. SIAM J. Math. Anal. , 47(4):2630–2656, 2015.\n[75] Wanrong Yang, Quansen Jiu, and Jiahong Wu. The 3D incompress ible magnetohydrodynamic equations\nwith fractional partial dissipation. J. Differential Equations , 266(1):630–652, 2019.\n[76] ZhuanYe. Well-posednessresults forthe 3DincompressibleHall- MHD equations. J. Differential Equations ,\n321:130–216, 2022.\n[77] Gaocheng Yue and Chengkui Zhong. On the global well-posedne ss to the 3-D Navier-Stokes-Maxwell\nsystem.Discrete Contin. Dyn. Syst. , 36(10):5817–5835, 2016.\n61" }, { "title": "2401.14854v1.Analysis_of_a___boldsymbol_P__1_oplus__boldsymbol_RT__0__finite_element_method_for_linear_elasticity_with_Dirichlet_and_mixed_boundary_conditions.pdf", "content": "arXiv:2401.14854v1 [math.NA] 26 Jan 2024Analysis of aP1⊕RT 0finite element method for\nlinear elasticity with Dirichlet and mixed\nboundary conditions\nHongpeng Li1, Xu Li2,1, Hongxing Rui1*\n1School of Mathematics, Shandong University, Jinan, 250100 , China.\n2Eastern Institute for Advanced Study, Eastern Institute of Technology,\nNingbo, 315200, China.\n*Corresponding author(s). E-mail(s): hxrui@sdu.edu.cn ;\nContributing authors: lihongpeng sd@163.com ;xulisdu@126.com ;\nAbstract\nIn this paper, we investigate a low-order robust numerical m ethod for the linear\nelasticity problem. The method is based on a Bernardi–Rauge l-likeH(div) -\nconforming method proposed first for the Stokes flows in [Li an d Rui, IMA\nJ. Numer. Anal. 42(2022) 3711–3734]. Therein the lowest-order H(div) -\nconforming Raviart–Thomas space ( RT 0) was added to the classical conforming\nP1×P0pair to meet the inf-sup condition, while preserving the div ergence\nconstraint and some important features of conforming metho ds. Due to the inf-\nsup stability of the P1⊕RT 0×P0pair, a locking-free elasticity discretization\nwith respect to the Lam´ e constant λcan be naturally obtained. Moreover, our\nscheme is gradient-robust for the pure and homogeneous disp lacement boundary\nproblem, that is, the discrete H1-norm of the displacement is O(λ−1)when the\nexternal body force is a gradient field. We also consider the m ixed displacement\nand stress boundary problem, whose P1⊕RT 0discretization should be carefully\ndesigned due to a consistency error arising from the RT 0part. We propose both\nsymmetric and nonsymmetric schemes to approximate the mixe d boundary case.\nThe optimal error estimates are derived for the energy norm a nd/orL2-norm.\nNumerical experiments demonstrate the accuracy and robust ness of our schemes.\nKeywords: linear elasticity, divergence-free element, gradient-ro bust, locking-free,\nmixed boundary conditions\n11 Introduction\nThis paper is concerned with a low-order finite element metho d for the linear elasticity\nproblem. Assume that Ω ⊂Rd(d= 2,3) is a bounded domain with polyhedral and\nLipschitz-continuous boundary ∂Ω. The symmetric d×dstress tensor is defined as\nσ(u) := (2µǫ(u) +λ(∇ ·u)I) withǫ(u) := ( ∇u+∇u⊤)/2,\nwhereuis the displacement of the elastic material and I∈Rd×dis the identity\nmatrix.λandµare two Lam´ e parameters which satisfy 0 < λ 0< λ < ∞and\n0<µ 1<µ<µ 2. Then the linear elasticity problem with homogeneous displ acement\nboundary condition is as follows:\n−∇ ·σ(u) =fin Ω,\nu=0on∂Ω,(1)\nwithf∈[L2(Ω)]dbeing an external body force. Using the Green formulation we\nobtain a variational formulation of ( 1): Findu∈V:= [H1\n0(Ω)]dsuch that\n2µa(u,v) +λ(∇ ·u,∇ ·v) = (f,v)∀v∈V, (2)\nwhere ( ·,·) denotes the L2inner products, a(u,v) := (ǫ(u),ǫ(v)) andH1\n0(Ω) consists\nof the functions with vanishing trace in H1(Ω). Two kinds of robustness are considered\nin this contribution: locking-free property and gradient- robustness. The former means\nthat the error estimates do not blow up as the Lam´ e constant λ→ ∞ , while the latter\nmeans the dominant gradient fields in the governing equation do not lead to spurious\ndisplacement. To be more precise, if fis a gradient field, it was proven in [ 16] that\n/ba∇dbl∇u/ba∇dbl=O(λ−1) (i.e., asλ→ ∞ , the true solution ushould tend to zero). Then a\ngradient-robust method should preserve this property.\nThe locking phenomenon in elasticity problems is usually ca lled “volume locking”\nor “Poisson locking” . When λis very large, the material is nearly incompressible (i.e.,\n∇ ·u≈0). The standard finite element method, such as the continuou s piecewise\nlinear element, can behave very badly [ 9,34]. Babuˇ ska and Suri [ 5] found that any\npolynomial of degree k≥1 cannot avoid locking on quadrilateral mesh. Volume\nlocking has been dealt with in many different discretization approaches. We divide the\ndiscretization approaches into three large classes. The fir st class is based on the primal\ndisplacement equation ( 1). A variety of finite element methods have been implemented\nfor this, such as the nonconforming Crouzeix–Raviart (CR) e lement [ 20], the enriched\nGalerkin method [ 41], the weak Galerkin method [ 40] and the discontinuous Galerkin\nmethod [ 38], to name just a few. The second class is to introduce the “sol id pressure”\np=λ∇ ·uas an independent unknown. Then the primal formulation ( 1) can be\nreformulated as a generalized Stokes problem\n−2µ∇ ·ǫ(u)− ∇p=f,∇ ·u−λ−1p= 0 in Ω. (3)\n2Any inf-sup stable mixed element method which is appropriat e for the Stokes problem,\nwould provide a locking-free formulation for the linear ela sticity problem, cf. [ 8,11,\n25,36]. Note that an inf-sup stable mixed formulation can usually be transformed into\na primal formulation by static condesation of the pressure u nknowns if the discrete\npressure is discontinuous. The last class transforms the li near elasticity equations ( 1)\ninto the Hellinger-Reissner formulation based on the Helli nger-Reissner variational\nprinciple [ 2]. And this method produces direct approximations to both st ress and\ndisplacement. The most popular methods include mixed finite element methods [ 2,3,\n21,23,28], dual-mixed methods [ 17–19], and hybrid discontinuous Galerkin methods\n[12,35].\nCompared to volume locking, gradient robustness is a new con cept. The definition\nof gradient-robustness is given in Section 2. A related concept of gradient-robustness\nhas been introduced first for the steady compressible isothe rmal Stokes equations\nin [1]. For the incompressible Stokes problem, gradient-robust ness means pressure-\nrobustness [ 26], that is, when the external force in the momentum equation i s a\ngradient field, it is only balanced by the pressure gradient. Fu et al. [ 16] proposed\nand analyzed an H(div)-conforming HDG scheme for linear elasticity ( 1), and the\nscheme is both locking-free and gradient-robust. Basava an d Wollner [ 6] applied the\npressure-robust reconstruction methods for the Stokes pro blem [ 30,32] to elasticity\ndiscretizations to get a gradient-robust method. Numerica l schemes of linear elastic-\nity may perform well when the body force is divergence-free, but may fail when the\nbody force is a gradient field. The concept of gradient robust ness gives us a new per-\nspective to analyze the effectiveness of numerical schemes. Our goal is to construct\nalgorithms that maintain parameter robustness about λ, and are also accurate when\nthe body force in the momentum balance equation is dominated by a gradient field.\nWe also consider mixed boundary conditions in this paper. As sume that the bound-\nary∂Ω consists of two parts: Γ D⊂∂Ω, with |ΓD|>0, and Γ N:=∂Ω\\ΓD. The\nelasticity problem with mixed boundary conditions becomes the primal formulation\nin (1) with\nu=0on Γ D,σn=gon Γ N. (4)\nWe defineVΓD:={v∈[H1(Ω)]d:v|ΓD=0}and the traction g∈[H1/2\n00(ΓN)]d,\nwhere [H1/2\n00(ΓN)]d:={v|ΓN:v∈VΓD}. The associated duality pairing with respect\nto the [L2(ΓN)]dis denoted by /a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htΓN. The variational formulation of ( 4) is that: Find\nu∈VΓDsuch that\n2µa(u,v) +λ(∇ ·u,∇ ·v) = (f,v) +/a\\}b∇acketle{tg,v/a\\}b∇acket∇i}htΓN∀v∈VΓD. (5)\nUnder the assumption |ΓD|>0, the Korn’s inequality holds [ 9], i.e., there exists a\npositive constant such that\n/ba∇dblv/ba∇dbl1≤Ckorn/ba∇dblǫ(u)/ba∇dbl ∀u∈VΓD, (6)\nwhere /ba∇dbl · /ba∇dbl 1and/ba∇dbl · /ba∇dbl denote the usual H1norms and L2norms, respectively. Thus\nthe unique solvability of ( 2) and ( 5) holds.\n3The starting point of this paper is a kind of low-order inf-su p stable mixed ele-\nment method for the Stokes problem proposed in [ 29], where the velocity space is\nobtained by enriching the space of conforming piecewise lin ear polynomials ( P1)\nwith theH(div)-conforming lowest-order Raviart–Thomas space, and the pressure\nspace consists of piecewise constants with zero mean ( P0). The resulting scheme is\ndivergence-free and pressure-robust, and the discrete for mulation consists of volume\nintegrals only, which is different from the usual H(div)-conforming methods with dis-\ncontinuous Galerkin (DG) formulation [ 13,16,26,37]. Moreover, since the pressure is\ndiscontinuous, as it is mentioned before, this method natur ally leads to a locking-free\nprimal discretization of the elasticity problem. The main c ontribution of this paper\nis twofold: on the one hand, for the pure Dirichlet problem, w e prove that the result-\ning method is gradient-robust, on the other hand, for the mix ed boundary problem,\nwe point out that the extension from pure Dirichlet boundary case is not straight-\nforward and then propose some strategies to deal with the Neu mann part, which is\nnot involved in [ 29]. A priori error estimate shows that all schemes achieve opt imal\nconvergence order in the energy norm and L2-norm. Numerical experiments show that\nan inappropriate treatment to mixed boundary conditions ca n lead to reduction of\nthe convergence rates of the discrete solution. And our prop osedP1⊕RT 0schemes\nare numerically accurate and robust for both Dirichlet and m ixed boundary condi-\ntions. For mixed boundary problems, in contrast to the Crouz eix–Raviart element\nmethod using interior jump stabilization [ 20], our formulation consists of standard\nvolume integrals and face integrals over the Neumann bounda ry. Note that interior\nface integrals like jump stabilization can dramatically ch ange the sparsity of the coef-\nficient matrix. Although this issue of Crouzeix–Raviart ele ment can be bypassed by\nreplacing one component with conforming linear elements [ 27], its three-dimensional\nextension has to apply higher-order elements for one compon ent [22,43].\nThe rest of the paper is organized as follows. In Section 2, we present the fun-\ndamental results about gradient-robustness. We discuss so me finite element schemes\nfrom the perspective of locking-free and gradient-robust p roperties. In Section 3we\npropose theP1⊕RT 0finite element schemes and analyze the well-posedness. The\nuniform convergence analysis about λand the gradient-robustness are analyzed in\nSection 4. The case of mixed boundary conditions is considered in Sect ion5. Finally\nwe do some numerical studies in Section 6.\n2 Gradient-robustness and some finite element\nschemes\nIn this section and next two sections, the elasticity proble m with homogeneous dis-\nplacement boundary conditions is considered. With convent ion the boundary value\nproblem with pure displacement (resp. traction) boundary c onditions is called a pure\ndisplacement (resp. traction) problem. We introduce some f undamental results about\nHelmholtz decomposition and Helmholtz projector [ 30]. Let us define\nH(div; Ω) := {v∈[L2(Ω)]d:∇ ·v∈L2(Ω)}.\n4Every vector field f∈[L2(Ω)]dhas a unique decomposition into a irrotational field ∇φ\nwithφ∈H1(Ω)/Rand a divergence-free component f0∈H(div; Ω), i.e.,f=∇φ+f0.\nMoreover,f0isL2-orthogonal to ∇ψfor allψ∈H1(Ω). The Helmholtz projector\nP(f) :=f0preserves the divergence-free part. For any ψ∈H1(Ω), it holds P(∇ψ) = 0.\nWe define the divergence-free subspace Vdivand its orthogonal complement V⊥\ndivby\nVdiv:={u∈V: (∇ ·u,q) = 0 ∀q∈W}={u∈V:∇ ·u= 0},\nV⊥\ndiv:={u∈V:a(u,v) = 0 ∀v∈Vdiv},\nrespectively, where W:=L2\n0(Ω) = {q∈L2(Ω) :/integraltext\nΩqdx= 0}is the space of L2\nfunctions with zero mean. Now any function u∈Vcan be uniquely decomposed as\nu=u0+u⊥∈Vdiv⊕V⊥\ndiv. The following boundedness of uin theH1-norm can be\nderived.\nLemma 1. Letube the solution of ( 2). It satisfies the following stability estimate:\n/ba∇dblu/ba∇dbl1≤1\n2µ/C2\nkorn+λβ2/ba∇dblf/ba∇dbl−1+C2\nkorn\n2µ/ba∇dblP(f)/ba∇dbl−1, (7)\nwhere /ba∇dbl · /ba∇dbl −1denotes theH−1norm.\nProof. The inf-sup condition [ 25]\nsup\nv∈V\\{0}(∇ ·v,q)\n/ba∇dblv/ba∇dbl1≥β/ba∇dblq/ba∇dbl (8)\nimplies the divergence operator is bijective from V⊥\ndivtoW, and\n/ba∇dblu⊥/ba∇dbl1≤1\nβ/ba∇dbl∇ ·u⊥/ba∇dbl. (9)\nWe setv=u⊥in the variational formulation ( 2), and use the a(·,·)-orthogonality to\nget\n2µa(u⊥,u⊥) +λ(∇ ·u⊥,∇ ·u⊥) = (f,u⊥).\nBy Korn’s inequality ( 6), (9) and the Cauchy-Schwarz inequality, we get\n2µ\nC2\nkorn/ba∇dblu⊥/ba∇dbl2\n1+λβ2/ba∇dblu⊥/ba∇dbl2\n1≤ /ba∇dblf/ba∇dbl−1/ba∇dblu⊥/ba∇dbl1, (10)\nThus\n/ba∇dblu⊥/ba∇dbl1≤1\n2µ/C2\nkorn+λβ2/ba∇dblf/ba∇dbl−1. (11)\nNext, testing with an arbitrary divergence-free function v0∈Vdivin (2) gives\n2µa(u0,v0) = (f,v0) = (P(f),v0).\n5Settingv0=u0, by Korn’s inequality ( 6) it yields\n2µ\nC2\nkorn/ba∇dblu0/ba∇dbl2\n1≤ /ba∇dblP(f)/ba∇dbl−1/ba∇dblu0/ba∇dbl1,\nwhich implies\n/ba∇dblu0/ba∇dbl1≤C2\nkorn\n2µ/ba∇dblP(f)/ba∇dbl−1. (12)\nThen inequality ( 7) follows immediately from ( 11) and ( 12).\nFollowing the proof of Lemma 1, we can easily prove the following lemma, which\nis introduced in [ 16]. It characterizes an important feature of the exact soluti on of\nnearly incompressible linear elasticity.\nLemma 2. Iffin (1) is a gradient field, i.e., f=∇φ,φ∈H1(Ω)/R, the solution\nu=u0+u⊥of (2) satisfies,\nu0=0,/ba∇dblu/ba∇dbl1=/ba∇dblu⊥/ba∇dbl1≤1\n2µ/C2\nkorn+λβ2/ba∇dblφ/ba∇dbl. (13)\nProof. The proof of Lemma 1impliesu0=0. For the inequality in ( 13), one has\n2µ\nC2\nkorn/ba∇dblu⊥/ba∇dbl2\n1+λβ2/ba∇dblu⊥/ba∇dbl2\n1≤(∇φ,u⊥)≤(φ,∇ ·u⊥)≤ /ba∇dblφ/ba∇dbl/ba∇dblu⊥/ba∇dbl1.\nThen the inequality can be obtained analogously to ( 11).\nDefinition 1 (Gradient-robustness) .A discretization of the linear elasticity problem\n(1) is called gradient-robust, if on an arbitrary but fixed grid , the discrete displacement\nsolutionuhsatisfies\n/ba∇dbluh/ba∇dbl1,h=O(λ−1)\nin casefis a gradient field, where /ba∇dbl · /ba∇dbl 1,his a discrete H1norm defined in finite\nelement spaces.\nIt is worth noting that in this paper the gradient-robustnes s is only considered\nunder homogeneous displacement boundary conditions. In [ 42] the role of some differ-\nent boundary conditions on pressure-robustness for the inc ompressible linear elasticity\nproblem is discussed, such as the normal or tangential compo nents of displacement\nboundary conditions.\nAccording to ( 3), whenλ→ ∞ , the elasticity problem tends to a Stokes problem\n−2ν∇ ·ǫ(u)− ∇p=f,∇ ·u= 0 in Ω. (14)\nConsider the case f=∇φwithφ∈H1(Ω)/R. For ( 14) it holds (u,p) = (0,−φ)\n(cf. [26]), while for ( 1) (or ( 3)) the displacement solution of a gradient-robust method\ntends to zero from Definition 1whenλ→ ∞ , which coincides with the solution of\nits Stokes limit. In this sense, we say a gradient-robust dis cretization is asymptotic\n6preserving (AP) [ 24]. From [ 26] the discrete velocity solution of a pressure-robust\nmethod for ( 14) is also zero. An elasticity discretization should be gradi ent-robust if\nit corresponds to a pressure-robust scheme for the Stokes pr oblem [ 16].\nNext we list some finite element schemes, and take some experi ments to verify their\nproperties about locking-free and gradient-robustness. W e omit the details of these\nelements, which can be found in other references. Pk(T) (k≥0) denotes the space of\npolynomials of degree no more than kon an element T. We assume the region is unit\nsquare, i.e., Ω = (0 ,1)2. We use the uniform triangular partition Th(see Fig. 4(left)),\nwhere the spatial steps hrange from 1 /8 to 1/128. Example 1is designed to satisfy\n∇ ·u→0 when the Lam´ e constant λ→ ∞ . It is taken to verify locking-free property.\nThe right-hand term fis determined by Equation ( 1). Example 2is taken from [ 16],\nthe right-hand term is designed to be a gradient field to verif y the gradient-robustness\nproperty. We use homogeneous Dirichlet boundary condition s for both examples. The\nLam´ e constants equal λ= 1,102,104,106, andµ= 1.\nExample 1. The exact solutions are chosen as follows\n\n\nu1=sin(2πy)(−1 +cos(2πx)) +1\nµ+λsin(πx)sin(πy),\nu2=sin(2πx)(1−cos(2πy)) +1\nµ+λsin(πx)sin(πy).\nExample 2. We takef=∇ψwithψ=x6+y6.\nP1scheme. LetVh={v∈V:v|T∈[P1(T)]2∀T∈ T h}be the piecewise linear\ncontinuous finite element space. The finite element scheme fo r (1) is that: Find uh∈\nVhsuch that\n2µa(uh,vh) +λ(∇ ·uh,∇ ·vh) = (f,vh)∀vh∈Vh. (15)\nIt is well-known that the continuous piecewise linear eleme nt would result in a poor\nconvergence rate of the displacement. We take Wh={q∈W:q|T∈P0(T)∀T∈ T h}\nsuch that ∇ ·Vh⊂Wh. As shown in Figure 1(left), when λ= 1 we obtain the\noptimal convergence rate of the displacement. As λbecomes large, the convergence\nrate deteriorates on the chosen meshes. Figure 1(right) shows that /ba∇dbl∇uh/ba∇dbl=O(λ−1).\nThe scheme ( 15) is gradient-robust, but it is not free of volumetric lockin g.\nBR scheme. The scheme is motivated by the discretization for the poroel asticity\nproblem [ 39]. The Bernardi–Raugel (BR) pair [ 7] is the linear space enriched by edge\nbubble functions. We choose Vh⊂Vto be the BR element space. The finite element\nscheme for ( 1) is that: Find uh∈Vhsuch that\n2µa(uh,vh) +λ(Ph∇ ·uh,Ph∇ ·vh) = (f,vh)∀vh∈Vh. (16)\nPhis the orthogonal L2projection defined in Section 3. In ( 16) we implement the\ntechnique of reduced integration [ 33] to obtain the uniform convergence with respect\ntoλ. In the poroelasticity problem, the linear elasticity equa tion is used to describe\nthe displacement of the solid medium. The proof of the unifor m convergence of ( 16)\nwith respect to λcan be derived from [ 39]. As shown in Figure 2(left), the lines for\nλ= 1,102,104,106are coincident. The difference for different λis very small. So the\n71/8 1/16 1/32 1/64 1/128\nh10-1100101|| u- uh||\n--O(h)=1\n=102\n=104\n=106\n1/8 1/16 1/32 1/64 1/128\nh10-710-610-510-410-310-210-1100|| uh||=1\n=102\n=104\n=106\nFig. 1 :P1element for Example 1(left) and Example 2(right).\nscheme ( 16) is uniformly convergent about λ. However, the Bernardi–Raugel element\nincludes the edge bubble functions, which is piecewise quad ratic polynomials. It is\nwell-known that the classical Stokes discretization with t his pair is not divergence-free\nor pressure-robust [ 26], and hence ( 16) is not gradient-robust.\nOne should also note that, although a classical discretizat ion with the Bernardi–\nRaugel element is not gradient-robust, it has been proven in [6] that the lack of\ngradient-robustness can be overcome by a reconstruction st rategy from the Stokes\ndiscretizations such as [ 31,32], which is the so-called pressure-robust reconstruction.\nFor the Crouzeix–Raviart element method below, such a strat egy is also considered\nand employed.\n1/8 1/16 1/32 1/64 1/128\nh10-1100|| u- uh||\n--O(h)=1\n=102\n=104\n=106\n1/8 1/16 1/32 1/64 1/128\nh10-310-210-1|| uh||=1\n=102\n=104\n=106\nFig. 2 : BR element for Example 1(left) and Example 2(right).\nCR scheme. Inspired by the reconstruction method [ 6,30], we try the follow-\ning gradient-robust reconstruction scheme. We use the first order nonconforming\nCrouzeix–Raviart (CR) element [ 15] to approximate the linear elasticity ( 1). When\n8the boundary condition is homogeneous displacement condit ion, the equation ( 1) can\nbe rewritten as\n−µ∆u−(µ+λ)∇(∇ ·u) =f. (17)\nLetVhbe the CR element space. Its locking-free property is derive d by Brenner and\nSung [ 10]. The piecewise gradient and piecewise divergence operato rs are defined as\n(∇hu,∇hv) :=/summationdisplay\nT∈Th(∇u,∇v)T,(∇h·u,∇h·v) :=/summationdisplay\nT∈Th(∇ ·u,∇ ·v)T,\nrespectively. To obtain the gradient-robustness property , we apply a reconstruction\noperator to the test function. Then the finite element scheme is that: Find uh∈Vh\nsuch that\nµ(∇huh,∇hvh) + (µ+λ)(∇h·uh,∇h·vh) = (f,ΠR\nhvh)∀vh∈Vh.(18)\nHere ΠR\nh:Vh→VR\nhis the lowest order Raviart–Thomas interpolation defined by (23),\nwhereVR\nhis the lowest-order Raviart–Thomas space (see ( 22)). This reconstruction\nmethod is first introduced in [ 30]. As shown in Figure 3, the scheme ( 18) is both\nlocking-free and gradient-robust. Note that the variation al form from ( 17) is only\nvalid for pure displacement problems. If Γ N/\\e}atio\\slash=∅, a variational form from ( 1) can be\nemployed. In this case a stabilized version is essential to g uarantee the discrete Korn’s\ninequality for CR elements [ 20], where the jump stabilization need to be added to\ninterior faces.\nProposition 1. The finite element scheme ( 18) is gradient-robust in the sense of\nDefinition 1.\nProof. Like in the continuous case, we define the discretely diverge nce-free space V0\nh\nand its orthogonal complement V⊥\nhas\nV0\nh:={vh∈Vh:∇h·vh= 0},\nV⊥\nh:={uh∈Vh: (∇huh,∇hvh) = 0 ∀vh∈V0\nh}.\nNote that ΠR\nhvh∈H(div; Ω). Testing the equation ( 18) with arbitrary vh∈V0\nhand\nf=∇φyields\n(∇φ,ΠR\nhvh) =−(φ,∇ ·ΠR\nhvh) =−(φ,∇h·vh),\nµ(∇huh,∇hvh) =−(µ+λ)(∇h·uh,∇h·vh)−(φ,∇h·vh) = 0,\nthusuh∈V⊥\nh. Considervh=uh, integration by parts for the right hand side gives\nµ(∇huh,∇huh) + (µ+λ)(∇h·uh,∇h·uh) =−(φ,∇ ·ΠR\nhuh). (19)\nUsing the fact that ∇ ·ΠR\nhuh=∇h·uh(see ( 27)), Eq. ( 19) implies\nµ/ba∇dbl∇huh/ba∇dbl2+ (µ+λ)/ba∇dbl∇h·uh/ba∇dbl2≤ /ba∇dblφ/ba∇dbl/ba∇dbl∇ h·uh/ba∇dbl. (20)\n9Similarly to [ 25, Lemma 3.58], it holds\n/ba∇dbl∇huh/ba∇dbl ≤1\nβcr/ba∇dbl∇h·uh/ba∇dbl,\nwithβcrbeing the discrete inf-sup constant for the CR element. Thus , applying the\nfact that /ba∇dbl∇h·uh/ba∇dbl ≤√\nd/ba∇dbl∇huh/ba∇dbl, formula ( 20) can be rewritten as\nµ/ba∇dbl∇huh/ba∇dbl2+ (µ+λ)β2\ncr/ba∇dbl∇huh/ba∇dbl2≤√\nd/ba∇dblφ/ba∇dbl/ba∇dbl∇ huh/ba∇dbl.\nDividing by /ba∇dbl∇huh/ba∇dbl, one gets\n/ba∇dbl∇huh/ba∇dbl ≤√\nd\n(1 +β2cr)µ+λβ2cr/ba∇dblφ/ba∇dbl,\nwhich demonstrates that /ba∇dbl∇huh/ba∇dbl=O(λ−1).\n1/8 1/16 1/32 1/64 1/128\nh100||h u-h uh||\n--O(h)=1\n=102\n=104\n=106\n1/8 1/16 1/32 1/64 1/128\nh10-710-610-510-410-310-210-1100||h uh||=1\n=102\n=104\n=106\nFig. 3 : CR element for Example 1(left) and Example 2(right).\n3 TheP1⊕RT 0finite element schemes\nIn this section we propose an H(div)-conforming finite element method for the linear\nelasticity problem with homogeneous displacement boundar y condition. The P1⊕RT 0\nfinite element is proposed by Li and Rui [ 29] for Stokes flow, and it is the continu-\nous vector-valued piecewise linear polynomial space ( P1) enriched by the lowest-order\nRaviart-Thomas space ( RT 0). Let {Th}be a family of triangluations of Ω. Let hTand\nhedenote the diameters of elements Tand facese, respectively, and h:= max T∈ThhT.\nThe set of interior faces and boundary faces of Thare denoted by E0andE∂, respec-\ntively, and E:=E0∪E∂. An unit normal vector to the face eis denoted by ne. The\n10family of meshes {Th}is assumed to be shape-regular, that is, there exists a const ant\nγ, independent of h, such that\nhT\nρT≤γ∀T∈ T h, (21)\nwhereρTdenotes the diameter of the largest ball contained in T. We define the\nspaceH0(div; Ω) := {v∈H(div; Ω) :v·n= 0 on∂Ω}. For easier understanding of\nnotations, the piecewise linear polynomial space is rename d as\nV1\nh:=/braceleftBig\nv∈V:v|T∈[P1(T)]d∀T∈ T h/bracerightBig\n.\nThe lowest-order Raviart–Thomas finite element space [ 8] is denoted by\nVR\nh:=/braceleftBig\nv∈H0(div; Ω) :v|T∈[P0(T)]d⊕xP0(T)∀T∈ T h/bracerightBig\n. (22)\nThe space of piecewise constants reads\nWh:={q∈W:q|T∈P0(T)∀T∈ T h}.\nThe nodal interpolation is denoted by Π1\nh:V∩C0(Ω)→V1\nh. Moreover, we define\nthe Raviart-Thomas interpolation ΠR\nh:V→VR\nhand the orthogonal L2projection\nPh:W→Whby\n((v−ΠR\nhv)·ne,1)e= 0 ∀e∈E, (23)\nand\n(r−Phr,w h) = 0 ∀wh∈Wh, (24)\nrespectively. The following approximation and commutativ e properties can be found\nin [8,9]:\n/ba∇dblv−Π1\nhv/ba∇dblT+hT|v−Π1\nhv|1,T≤Ch2\nT|v|2,T ∀T∈ T h, (25)\n/ba∇dblv−ΠR\nhv/ba∇dbl0,T≤ChT|v|1,T ∀T∈ T h,\n/ba∇dblr−Phr/ba∇dbl0,T≤Chs\nT|r|s,T ∀T∈ T h, s = 0,1, (26)\n∇ ·ΠR\nhv=Ph∇ ·v. (27)\nBy [29, Lemma 2.1], we know V1\nh∩VR\nh={0}. Then theP1⊕RT 0finite element\nspaceVhis a direct sum of these two spaces, i.e., Vh:=V1\nh⊕VR\nh. For anyuh∈Vh, it\ncan be uniquely decomposed into u1\nh+uR\nh, whereu1\nh∈V1\nhanduR\nh∈VR\nh. We know\nthatVhis anH(div)-conforming space and Vh×Whis a divergence-free pair in the\nsense of [ 26], i.e., ∇ ·Vh=Wh. Moreover, we define\nV0\nh: ={vh∈Vh: (∇ ·vh,qh) = 0 ∀qh∈Wh}\n={vh∈Vh:∇ ·vh= 0}.\n11P1⊕RT 0scheme 1. Based on this element, we propose the following finite elemen t\nscheme of ( 1): Finduh∈Vhsuch that\nah1(uh,vh) :=2µah(uh,vh) +λ(∇ ·uh,∇ ·vh) = (f,vh)∀vh∈Vh, (S1)\nwhereah(uh,vh) =a(u1\nh,v1\nh)+aR(uR\nh,vR\nh). Letψedenote the Raviart-Thomas basis\nfunction for the face esuch thatuR\nh∈VR\nhcan be rewritten as uR\nh=/summationtext\ne∈E0ueψe.\nFollowing [ 29],aR(·,·) has three choices as follows\naR(uR\nh,vR\nh) =a0(uR\nh,vR\nh) :=/summationdisplay\nT∈ThαTh−2\nT(uR\nh,vR\nh)T,\naR(uR\nh,vR\nh) =aD(uR\nh,vR\nh) :=/summationdisplay\nT∈Th/summationdisplay\ne∈∂T∩E0αTh−2\nTueve(ψe,ψe)T,\naR(uR\nh,vR\nh) =adiv(uR\nh,vR\nh) :=/summationdisplay\nT∈Th/summationdisplay\ne∈∂T∩E0αTueve(∇ ·ψe,∇ ·ψe)T.\nThe three forms are spectrally equivalent (see [ 29, Lemma 3.2]) and the parameters\nαTare positive constants. Relevant proof can be found in [ 29]. For brevity, we choose\naR=a0in analysis.\nAnalogously to the continuous setting, we can define the orth ogonal complement\nofV0\nhwith respect to the bilinear form ah(·,·)\nV⊥\nh:=/braceleftbig\nuh∈Vh:ah(uh,vh) = 0 ∀vh∈V0\nh/bracerightbig\n.\nRemark 1. The construction of the P1⊕RT 0element is similar to the Bernardi–\nRaugel element. Both of them are based on continuous linear p olynomials and\nsupplemented with some stable functions to satisfy the inf- sup condition. The differ-\nence is that the P1⊕RT 0element uses lowest-order Raviart–Thomas edge functions,\nwhile the BR element uses quadratic bubble functions. Thus t heP1⊕RT 0×Whpair is\na divergence-free and pressure-robust pair. And compared t oBR scheme (16), in our\nscheme ( S1) the term with λdoes not require L2projection. Moreover, the proposed\nscheme is easy to implement, it does not involve any face inte grals.\nWe define a larger space V(h) :=V⊕VR\nhfor analysis. For all v∈V(h) we\nsimilarly have the unique decomposition v=v1+vR, wherev1∈VandvR∈VR\nh.\nWe define the following norms or seminorms on V(h):\n/ba∇dblv/ba∇dbl2\nR:=aR(vR,vR),/ba∇dblv/ba∇dbl2\nh:=ah(v,v),/ba∇dblv/ba∇dbl2\nh1:=ah1(v,v).\nBecause of Korn’s inequality ( 6),/ba∇dbl · /ba∇dbl hand/ba∇dbl · /ba∇dbl h1are two norms. Moreover, we define\nthe interpolation Π h:V∩C0(¯Ω)→Vhas\nΠhv:= Π1\nhv+ ΠR\nh(v−Π1\nhv). (28)\n12Let/ba∇dbl·/ba∇dblh,Tand/ba∇dbl·/ba∇dblh1,Tbe the elementwise counterparts of /ba∇dbl·/ba∇dblhand/ba∇dbl·/ba∇dblh1, respectively,\nsuch that /ba∇dbl·/ba∇dbl2\nh=/summationtext\nT∈Th/ba∇dbl·/ba∇dbl2\nh,Tand/ba∇dbl·/ba∇dbl2\nh1=/summationtext\nT∈Th/ba∇dbl·/ba∇dbl2\nh1,T. It was proven in [ 29] that\n∇ ·Πhv=Ph∇ ·v ∀v∈V∩C0(¯Ω), (29)\n/ba∇dblv−Πhv/ba∇dblT+hT/ba∇dblv−Πhv/ba∇dblh,T≤Ch2\nT|v|2,T ∀v∈[H2(T)]d,T∈ T h.(30)\nThus we obtain\n/ba∇dblv−Πhv/ba∇dblh1,T≤(2µ)1\n2/ba∇dblv−Πhv/ba∇dblh,T+λ1\n2/ba∇dbl∇ ·(v−Πhv)/ba∇dbl0,T\n≤(2µ)1\n2/ba∇dblv−Πhv/ba∇dblh,T+λ1\n2/ba∇dbl∇ ·v−Ph∇ ·v/ba∇dbl0,T\n≤ChT((2µ)1\n2|v|2,T+λ1\n2|∇ ·v|1,T)∀v∈[H2(Ω)]d.(31)\nLemma 3 (Inf-Sup Stability) .There exists a positive constant βis, dependent on β,\nγandαT,T∈ T h, but independent of h, satisfying the inf-sup condition\nsup\nvh∈Vh\\{0}(∇ ·vh,qh)\n/ba∇dblv/ba∇dblh≥βis/ba∇dblqh/ba∇dbl ∀qh∈Wh, (32)\nwhereβandγare the constants in (8)and (21), respectively. In addition, for all\nqh∈Whthere exists a unique u⊥\nh∈V⊥\nhsuch that\n∇ ·u⊥\nh=qh,/ba∇dblu⊥\nh/ba∇dblh≤β−1\nis/ba∇dblqh/ba∇dbl. (33)\nProof. A similar inf-sup condition has already been proven in [ 29]:\nsup\nvh∈Vh\\{0}(∇ ·vh,qh)\n|||vh|||≥βis/ba∇dblqh/ba∇dbl ∀qh∈Wh,\nwhere |||v|||2:=/ba∇dbl∇v1/ba∇dbl2+aR(vR,vR) is a norm defined on V(h). Then ( 32) follows\nfrom the fact that /ba∇dblǫ(v1)/ba∇dbl ≤ /ba∇dbl∇v1/ba∇dblfor anyv1∈V. Further, since ∇ ·Vh=Wh, the\nstatement concerning ( 33) is a direct consequence of [ 25, Lemma 3.12]. This completes\nthe proof.\nLemma 4. The numerical scheme ( S1) has unique solution uh∈Vh.\nProof. It is trivial to prove\nah1(vh,vh) =/ba∇dblvh/ba∇dbl2\nh1 ∀vh∈Vh, (34)\nah1(uh,vh)≤ /ba∇dbluh/ba∇dblh1/ba∇dblvh/ba∇dblh1 ∀uh,vh∈Vh. (35)\nSo the unique solvability of ( S1) is established.\n13Theorem 1. The finite element scheme ( S1) is gradient-robust in the sense of Defi-\nnition 1. If the right-hand side equals f=∇φfor someφ∈H1(Ω), then the solution\nuh=u0\nh+u⊥\nh∈V0\nh⊕V⊥\nhof (S1) satisfies\nu0\nh=0,/ba∇dblu⊥\nh/ba∇dblh≤√\n2 max {1,Cinv/√α}\n2µ+λβ2\nis/ba∇dblφ/ba∇dbl, (36)\nwhereα:= min T∈ThαTandCinvis a constant such that the inverse estimate /ba∇dbl∇ ·\nvR/ba∇dblT≤Cinvh−1\nT/ba∇dblvR/ba∇dblTholds for any vR∈VR\nhandT∈ T h.\nProof. Consideringvh=u0\nhin (S1), by theah(·,·)-orthogonality, and integrating by\nparts for the right hand side we get\n2µah(u0\nh,u0\nh) = ( ∇φ,u0\nh) =−(φ,∇ ·u0\nh) = 0,\nwhich implies u0\nh=0. Considering vh=u⊥\nhin (S1), it follows\n2µah(u⊥\nh,u⊥\nh) +λ(∇ ·u⊥\nh,∇ ·u⊥\nh) = ( ∇φ,u⊥\nh) =−(φ,∇ ·u⊥\nh)≤ /ba∇dblφ/ba∇dbl/ba∇dbl∇ ·u⊥\nh/ba∇dbl.\nBy an inverse estimate and a triangle inequality, it holds fo r anyv∈V(h) that\n/ba∇dbl∇ ·v/ba∇dbl ≤ /ba∇dbl∇ ·v1/ba∇dbl+/ba∇dbl∇ ·vR/ba∇dbl ≤ /ba∇dblǫ(v1)/ba∇dbl+ (Cinv/√α)/ba∇dblv/ba∇dblR\n≤max{1,Cinv/√α}(/ba∇dblǫ(v1)/ba∇dbl+/ba∇dblv/ba∇dblR)≤√\n2 max {1,Cinv/√α}/ba∇dblv/ba∇dblh.\nLemma 3implies that\n/ba∇dblu⊥\nh/ba∇dblh≤β−1\nis/ba∇dbl∇ ·u⊥\nh/ba∇dbl,\nwhich, together with the above two estimates, yields\n(2µ+λβ2\nis)/ba∇dblu⊥\nh/ba∇dbl2\nh≤√\n2 max {1,Cinv/√α}/ba∇dblφ/ba∇dbl/ba∇dblu⊥\nh/ba∇dblh.\nThen ( 36) follows. This completes the proof.\nRemark 2. The scheme ( S1) satisfies both locking-free and gradient-robust properti es.\nThe reconstruction operator in CRscheme ( 18) is used to map discretely divergence-\nfree functions to divergence-free functions, while the sch eme ( S1) does not require\nreconstructing.\nNext we analyze the consistency error caused by this H(div)-conforming P1⊕RT 0\nelement. Let ube the true solution of ( 1). The consistency error is denoted by\nδh1(u,v) := (f,v)−ah1(u,v).\nNote that ah1(u,v) = 2µa(u,v1) +λ(∇ ·u,∇ ·v) and (f,v) =\n(−∇ · (2µǫ(u) +λ(∇ ·u)I),v) for allv∈V(h). Integrating by parts for the\n142µa(u,v1) term, one obtains the consistency error:\n|δh1(u,v)|=/vextendsingle/vextendsingle(−2µ∇ ·ǫ(u),vR)/vextendsingle/vextendsingle≤2µ/summationdisplay\nT∈ThhT|u|2,Th−1\nT/ba∇dblvR/ba∇dbl0,T\n≤2µ/parenleftBigg/summationdisplay\nT∈Thh2\nT|u|2\n2,T/parenrightBigg1/2/parenleftBigg/summationdisplay\nT∈Thh−2\nT/ba∇dblvR/ba∇dbl2\n0,T/parenrightBigg1/2\n≤(2µ/√α)h|u|2/ba∇dblv/ba∇dblR≤(2µ/√α)h|u|2/ba∇dblv/ba∇dblh.(37)\nHereαis the same as in Theorem 1.\n4 Error estimates\nTheorem 2. Letube the solution of ( 2) anduhbe the solution of ( S1). Then\nassumingu∈[H2(Ω)]dandf∈[L2(Ω)]d, we have the following error estimates:\n/ba∇dblu−uh/ba∇dblh≤Ch|u|2, (38)\n/ba∇dblu−uh/ba∇dblh1≤Ch((2µ)1\n2|u|2+λ1\n2|∇ ·u|1). (39)\nIf additionally Ωis convex and d= 2, one further has\n/ba∇dblu−uh/ba∇dblh1≤Ch((2µ)1\n2+λ−1\n2)/ba∇dblf/ba∇dbl.\nIn all estimates the constants Care independent of λ,µandh.\nProof. First, we split the error into\nu−uh=u−Πhu−(uh−Πhu) :=ηu−ξu.\nSubtracting ( S1) from ( 2), we get the following error equation\n2µah(u−uh,vh) +λ(∇ ·(u−uh),∇ ·vh) =−δh1(u,vh)∀vh∈Vh. (40)\nSettingvh=ξu, we get\n2µah(ξu,ξu) +λ(∇ ·ξu,∇ ·ξu) = 2µah(ηu,ξu) +λ(∇ ·ηu,∇ ·ξu)−δh1(u,ξu).(41)\nA combination of ( 24), (29) and the fact that ∇ ·ξu∈Whimplies that ( ∇ ·ηu,∇ ·\nξu) = 0. Then it follows from the coercivity and the boundedness ofah(·,·), and the\nconsistency error ( 37) that\n2µ/ba∇dblξu/ba∇dbl2\nh≤2µ/ba∇dblηu/ba∇dblh/ba∇dblξu/ba∇dblh+ (2µ/√α)h|u|2/ba∇dblξu/ba∇dblh,\nwhich implies\n/ba∇dblξu/ba∇dblh≤ /ba∇dblηu/ba∇dblh+ (√α)−1h|u|2.\n15The estimate ( 38) follows from a combination of the above inequality, the tri angle\ninequality and the interpolation error with respect to /ba∇dbl · /ba∇dbl h.\nLet us prove ( 39). Note that (2 µ)1\n2/ba∇dbl · /ba∇dbl h≤ /ba∇dbl · /ba∇dbl h1. By Equation ( 41), the coercivity\nofah1(·,·), the boundedness of ah(·,·), and the consistency error ( 37), we obtain\n/ba∇dblξu/ba∇dbl2\nh1≤2µ/ba∇dblηu/ba∇dblh/ba∇dblξu/ba∇dblh+ (2µ/√α)h|u|2/ba∇dblξu/ba∇dblh≤ /ba∇dblηu/ba∇dblh1/ba∇dblξu/ba∇dblh1+ (2µ/α)1\n2h|u|2/ba∇dblξu/ba∇dblh1,\nwhich implies\n/ba∇dblξu/ba∇dblh1≤ /ba∇dblηu/ba∇dblh1+ (2µ/α)1\n2h|u|2.\nThe above estimate gives, together with the approximation p roperties ( 31) and the\ntriangle inequality,\n/ba∇dblu−uh/ba∇dblh1≤ /ba∇dblu−Πhu/ba∇dblh1+/ba∇dblΠhu−uh/ba∇dblh1\n≤2/ba∇dblu−Πhu/ba∇dblh1+ (2µ/α)1\n2h|u|2\n≤Ch((2µ)1\n2|u|2+λ1\n2|∇ ·u|1).(42)\nThen the last inequality in Theorem 2follows immediately from the H2-regularity\nestimate [ 10],\n/ba∇dblu/ba∇dbl2+λ/ba∇dbl∇ ·u/ba∇dbl1≤C/ba∇dblf/ba∇dbl, (43)\nwhich holds true in case Ω is a convex polygon in two dimension s. Thus we complete\nthe proof.\nUsing duality argument it is not hard to obtain the error esti mate forL2-norm\n(cf. [10,29])\n/ba∇dblu−uh/ba∇dbl ≤Ch2/ba∇dblf/ba∇dbl,\nwhen ( 43) holds true. Here we propose a strategy to get a sharper L2estimate for our\nmethod like ( 38), which is based on a specifically designed projection Πβ\nh:V→Vh,\ndefined by\nΠβ\nhv:= Πe\nhv+ ΠR\nh(v−Πe\nhv),\nwhere Πe\nh:V→V1\nhis an elliptic projection defined by\na(Πe\nhv,wh) :=a(v,wh)∀wh∈V1\nh.\nFrom known theory of elliptic type projection one has\n/ba∇dblv−Πe\nhv/ba∇dbl+h/ba∇dbl∇(v−Πe\nhv)/ba∇dbl ≤Cinf\nwh∈V1\nh/ba∇dbl∇(v−wh)/ba∇dbl.\n16Analogously, following the same strategy one can prove that\n∇ ·Πβ\nhv=Ph∇ ·v, (44)\n/ba∇dblv−Πβ\nhv/ba∇dbl+h/ba∇dblv−Πβ\nhv/ba∇dblh≤Ch2|v|2, (45)\nwhich is similar to ( 29) and ( 30), but an elementwise estimate for Πβ\nhis not available.\nTheorem 3. Letube the solution of variational formulation ( 2) anduhbe the\nsolution of ( S1). Under the assumption that (43)holds, one has\n/ba∇dblu−uh/ba∇dbl ≤C(1 + 2µ)h2|u|2, (46)\nwhereCis a positive constant independent of λ,µandh.\nProof. First we introduce the following duality problem:\n−∇ · (2µǫ(φ) +λ(∇ ·φ)I) = Πβ\nhu−uhin Ω,\nφ=0 on∂Ω.\nSince Πβ\nhu−uh∈[L2(Ω)]d, from ( 43) we have the following regularity\n/ba∇dblφ/ba∇dbl2+λ/ba∇dbl∇ ·φ/ba∇dbl1≤C/ba∇dblΠβ\nhu−uh/ba∇dbl. (47)\nMultiplyingv∈V(h), and integrating by parts, we can get\nah1(v,φ) = (Πβ\nhu−uh,v)−δh1(φ,v)∀v∈V(h).\nTakingv= Πβ\nhu−uhgives\n/ba∇dblΠβ\nhu−uh/ba∇dbl2=ah1(Πβ\nhu−uh,φ) +δh1(φ,Πβ\nhu−uh)\n=ah1(Πβ\nhu−uh,φ−Πhφ) +ah1(Πβ\nhu−uh,Πhφ) +δh1(φ,Πβ\nhu−uh)\n(48)\nSimilarly, note that ∇ ·(Πβ\nhu−uh)∈Wh, from ( 29) one has\n|ah1(Πβ\nhu−uh,φ−Πhφ)|=|2µah(Πβ\nhu−uh,φ−Πhφ)|\n≤2µ/ba∇dblΠβ\nhu−uh/ba∇dblh/ba∇dblφ−Πhφ/ba∇dblh≤Ch(2µ)/ba∇dblΠβ\nhu−uh/ba∇dblh|φ|2.\nOn the other hand, a combination of the definition of ah1, Πe\nhandδh1, together with\n(44) and the fact ∇ ·Πhφ∈Wh, implies\nah1(Πβ\nhu−uh,Πhφ) = 2 µ/parenleftBig\na(Πe\nhu−u1\nh,(Πhφ)1) +aR((Πβ\nhu)R−uR\nh,(Πhφ)R)/parenrightBig\n+λ(∇ ·(Πβ\nhu−uh),∇ ·Πhφ)\n= 2µ/parenleftBig\na(u−u1\nh,(Πhφ)1) +aR((Πβ\nhu)R−uR\nh,(Πhφ)R)/parenrightBig\n17+λ(∇ ·(u−uh),∇ ·Πhφ)\n=ah1(u−uh,Πhφ) + 2 µaR/parenleftBig\n(Πβ\nhu)R,(Πhφ)R/parenrightBig\n=−δh1(u,Πhφ) + 2 µaR/parenleftBig\n(Πβ\nhu)R,(Πhφ)R/parenrightBig\n.\nFrom the definition of /ba∇dbl · /ba∇dbl Rand/ba∇dbl · /ba∇dbl hone has\n/ba∇dblv/ba∇dblR≤ /ba∇dblv−w/ba∇dblh∀w∈V. (49)\nFrom ( 37) and ( 49), we have\n|δh1(u,Πhφ)| ≤Ch(2µ)|u|2/ba∇dblΠhφ/ba∇dblR\n≤Ch(2µ)|u|2/ba∇dblφ−Πhφ/ba∇dblh≤Ch2(2µ)|u|2|φ|2,\n|aR/parenleftBig\n(Πβ\nhu)R,(Πhφ)R/parenrightBig\n| ≤ /ba∇dbl Πβ\nhu/ba∇dblR/ba∇dblΠhφ/ba∇dblR≤C/ba∇dblu−Πβ\nhu/ba∇dblh/ba∇dblφ−Πhφ/ba∇dblh\n≤Ch/ba∇dblu−Πβ\nhu/ba∇dblh|φ|2,\n|δh1(φ,Πβ\nhu−uh)| ≤Ch(2µ)|φ|2/ba∇dblΠβ\nhu−uh/ba∇dblR\n≤Ch(2µ)(/ba∇dblu−Πβ\nhu/ba∇dblh+/ba∇dblu−uh/ba∇dblh)|φ|2.\nSubstituting above estimates into ( 48) gives\n/ba∇dblΠβ\nhu−uh/ba∇dbl2≤C(h/ba∇dblu−uh/ba∇dblh+h/ba∇dblu−Πβ\nhu/ba∇dblh+h2|u|2)(2µ)|φ|2.\nBy the regularity assumption ( 47), the error estimates ( 38) and ( 45), and the triangle\ninequality we get\n/ba∇dblu−uh/ba∇dbl ≤C(1 + 2µ)h2|u|2.\nThus we complete the proof.\n5 Mixed boundary conditions\nWe define the space compatible with mixed boundary condition s (4). In case no\nambiguity occurs, we use the same notations in Section 4. We define\nHΓD(div; Ω) := {v∈H(div; Ω) :v·n|ΓD= 0},\nVR\nh:=/braceleftbig\nvh∈HΓD(div; Ω) :vh|T∈[P0(T)]d⊕xP0(T)∀T∈ T h/bracerightbig\n,\nV1\nh:=/braceleftbig\nvh∈VΓD:vh|T∈[P1(T)]d∀T∈ T h/bracerightbig\n,\nVh:=V1\nh⊕VR\nh,V(h) :=VΓD⊕VR\nh.\nRemark 3. The scheme for mixed boundary conditions should be designed carefully\nforP1⊕RT 0element to obtain an optimally convergent consistency erro r. To be more\nprecise, denote by am\nh(·,·)andF(·)two generic forms representing the left-hand side\n18and right-hand side of a discretization, respectively. Let ube the true solution related\nto(4). The principle to designing Fandam\nhis that we hope they satisfy\nF(v)−am\nh(u,v) = ( −2µ∇ ·ǫ(u),vR)for allv∈V(h), (50)\nlikeδh1in Section 3. In this way the consistency error is still optimally conver gent.\nA trivial extension from the pure Dirichlet problem to the mi xed boundary problem\nmight read\n2µah(uh,vh) +λ(∇ ·uh,∇ ·vh) = (f,vh) +/integraldisplay\nΓNg·vhds. (51)\nHowever, one can check that this does not satisfy the princip le: some additional consis-\ntency error arises from the Neumann boundary part because th e discretization related\ntoRT 0part inahis not obtained from integration by parts. The numerical exp eri-\nments later also demonstrate that the above scheme is not opt imal. To overcome this\nissue, we should modify the discretization in the case of mix ed boundary conditions.\nSeveral schemes which satisfy the principle are listed belo w.\nP1⊕RT 0scheme 2. The nonsymmetric finite element scheme to deal with mixed\nboundary conditions ( 4) reads\naNS(uh,vh) := 2µah(uh,vh) +λ(∇ ·uh,∇ ·vh) +/integraldisplay\nΓN2µǫ(u1\nh)n·vR\nhds\n−/integraldisplay\nΓN2µǫ(v1\nh)n·uR\nhds= (f,vh) +/integraldisplay\nΓNg·vhds.(S2)\nP1⊕RT 0scheme 3. The symmetric finite element scheme to deal with mixed\nboundary conditions ( 4) reads\naS(uh,vh) := 2µah(uh,vh) +λ(∇ ·uh,∇ ·vh) +/integraldisplay\nΓN2µǫ(u1\nh)n·vR\nhds\n+/integraldisplay\nΓN2µǫ(v1\nh)n·uR\nhds= (f,vh) +/integraldisplay\nΓNg·vhds.(S3)\nP1⊕RT 0scheme 4. A modified version of ( S2) or ( S3) reads\naM(uh,vh) := 2µah(uh,vh) +λ(∇ ·uh,∇ ·vh) +/integraldisplay\nΓN(2µǫ(u1\nh)n·n)(vR\nh·n)ds\n±/integraldisplay\nΓN(2µǫ(v1\nh)n·n)(uR\nh·n)ds= (f,vh) +/integraldisplay\nΓN/bracketleftbig\ng·v1\nh+ (g·n)(vR\nh·n)/bracketrightbig\nds.\n(S4)\nThe main feature of ( S4) is that it only involves the normal component of vR\nhon the\nstress boundary, which matches the degrees of freedom of RT 0well and, hence, makes\nthe scheme easier to implement. It can be verified that all the three schemes satisfy\n19the designing principle ( 50) in Remark 3by integration by parts. For example, the\nconsistency error of ( S2) is\nδNS(u,v) := (f,v) +/integraldisplay\nΓNg·vds−aNS(u,v)\n= (−∇ · (2µǫ(u) +λ∇ ·uI),v) +/integraldisplay\nΓN[2µǫ(u)n·v+λ∇ ·u(v·n)]ds−aNS(u,v)\n= 2µa(u,v1) +λ(∇ ·u,∇ ·v) +/parenleftbig\n−∇ · (2µǫ(u)),vR/parenrightbig\n+/integraldisplay\nΓN2µǫ(u)n·vRds−aNS(u,v)\n=/parenleftbig\n−∇ · (2µǫ(u)),vR/parenrightbig\n.\n(52)\nRemark 4. Compared to (51), the third term in the left-hand side of schemes\n(S2)–(S4)is introduced to satisfy the designing principle, while the fourth term is a\nconsistent term to guarantee that a scheme is symmetric or no nsymmetric but stable\nas long asαT,T∈ T h, are positive. This strategy is very similar to the disconti nuous\nGalerkin (DG) methods for the elliptic problem [ 4]. However, there is a fundamental\ndifference between our method and DG methods. In contrast to t he DG methods, the\nproposed schemes here do not involve any interior jump stabi lization or face integral\nover interior faces, which are simpler to implement and do no t change the sparsity\npattern of the coefficient matrix.\nThe analysis of these schemes is indeed very similar to the pu re Dirichlet boundary\ncase. For brevity, we only analyze ( S2) and ( S3) below. We redefine norm /ba∇dbl · /ba∇dbl hon\nV(h):\n/ba∇dblv/ba∇dbl2\nh:=a(v1,v1) +aR(vR,vR) +/summationdisplay\ne∈ΓNhe/ba∇dblǫ(v1)/ba∇dbl2\ne, (53)\nwherev1∈VΓD,vR∈VR\nhand/ba∇dblǫ(v1)/ba∇dbl2\ne:=/integraltext\neǫ(v1) :ǫ(v1)ds. Then we define a norm\n/ba∇dblv/ba∇dbl2\nh2:= 2µ/ba∇dblv/ba∇dbl2\nh+λ(∇ ·v,∇ ·v).\nAssume that T∈ T his an element with eas one edge. By the trace inequality and\nthe inverse inequality we can get\nhe/ba∇dblǫ(v1\nh)/ba∇dbl2\ne≤C/ba∇dbl∇v1\nh/ba∇dbl2\nT∀vh∈Vh.\nAs a result, the two norms /ba∇dbl · /ba∇dbl h1and/ba∇dbl · /ba∇dbl h2are equivalent in Vhspace, i.e., /ba∇dblvh/ba∇dblh2≤\nC/ba∇dblvh/ba∇dblh1. For anyv∈[H2(T)]dande⊂∂T, we have\nh1\n2e/ba∇dblǫ(v−Π1\nhv)/ba∇dble≤C(|v−Π1\nhv|1,T+h|v−Π1\nhv|2,T) =C(|v−Π1\nhv|1,T+h|v|2,T),\nwhich means the interpolation estimate ( 30) still holds for new /ba∇dbl · /ba∇dbl hnorm ( 53), and\ntogether with ( 31), implies\n/ba∇dblv−Πhv/ba∇dblh2≤Ch((2µ)1\n2|v|2+λ1\n2|∇ ·v|1)∀v∈[H2(Ω)]d. (54)\n20Define Th(ΓN) := {T∈ T h:|∂T∩ΓN| /\\e}atio\\slash= 0}. The following lemma is concerned\nwith the unique solvability of ( S2) and ( S3).\nLemma 5. The numerical scheme ( S2) has a unique solution uh∈Vh, and the\nnumerical scheme ( S3) has a unique solution uh∈Vhif allαT,T∈ T h(ΓN),are large\nenough.\nProof. For anyvh∈Vh, we have\naNS(vh,vh) = 2µah(vh,vh) +λ(∇ ·vh,∇ ·vh)\n≥ /ba∇dblvh/ba∇dbl2\nh1≥C/ba∇dblvh/ba∇dbl2\nh2.\nThus the coercivity of aNS(·,·) holds. It follows from Cauchy-Schwarz inequality and\ntrace inequality that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\ne∈ΓN/integraldisplay\ne2µǫ(u1)n·vRds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2µ/parenleftBigg/summationdisplay\ne∈ΓNhe/ba∇dblǫ(u1)/ba∇dbl2\ne/parenrightBigg1\n2/parenleftBigg/summationdisplay\ne∈ΓNh−1\ne/ba∇dblvR/ba∇dbl2\ne/parenrightBigg1\n2\n≤2µC/parenleftBigg/summationdisplay\ne∈ΓNhe/ba∇dblǫ(u1)/ba∇dbl2\ne/parenrightBigg1\n2\n/summationdisplay\nT∈Th(ΓN)h−2\nT/ba∇dblvR/ba∇dbl2\nT\n1\n2\n≤2µC/ba∇dblu/ba∇dblh/ba∇dblv/ba∇dblR∀u,v∈V(h).\n(55)\nThe term/integraltext\nΓN2µǫ(v1\nh)n·uR\nhdscan be bounded in the same way. Using the above\ninequality, we can get the boundedness of aNS(·,·). By Lax–Milgram Theorem, we\nget the unique solvability of ( S2).\nNext we analyze the unique solvability of ( S3). From the bound of the edge integrals\n(55) and Holder’s inequality, we can get\naS(vh,vh)≥2µ(ǫ(v1\nh),ǫ(v1\nh)) + 2µ/summationdisplay\nT∈ThαTh−2\nT(vR\nh,vR\nh)T+λ(∇ ·vh,∇ ·vh)\n−/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay\nΓN2µǫ(v1\nh)n·vR\nhds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥2µ(ǫ(v1\nh),ǫ(v1\nh)) + 2µ/summationdisplay\nT∈ThαTh−2\nT(vR\nh,vR\nh)T+λ(∇ ·vh,∇ ·vh)\n−\n2µε/summationdisplay\nT∈Th(ΓN)/ba∇dbl∇v1\nh/ba∇dbl2\nT+ 2µC/summationdisplay\nT∈Th(ΓN)h−2\nT/ba∇dblvR\nh/ba∇dbl2\nT\n\n= 2µ\n(ǫ(v1\nh),ǫ(v1\nh))−ε/summationdisplay\nT∈Th(ΓN)/ba∇dbl∇v1\nh/ba∇dbl2\nT\n\n+ 2µ\n/summationdisplay\nT∈ThαTh−2\nT(vR\nh,vR\nh)T−C/summationdisplay\nT∈Th(ΓN)h−2\nT/ba∇dblvR\nh/ba∇dbl2\nT\n+λ(∇ ·vh,∇ ·vh).\n21The constant Cdepends on the constants of the trace inequality and Holder’ s\ninequality. The parameters αTover Th(ΓN) should be chosen greater than C. The\nboundedness of aS(·,·) is very similar to aNS(·,·). Thus we complete the proof.\nBased on the coercivity and boundedness analyzed in Lemma 5, the consistency\nerror such as ( 52), the estimates of interpolation ( 54) andH2-regularity in two\ndimensions, one can similarly obtain the following estimat es.\nTheorem 4. Letube the solution of ( 5) anduhbe the solution of ( S2) or ( S3).\nThen assuming u∈[H2(Ω)]d, we have the following error estimates:\n/ba∇dblu−uh/ba∇dblh≤Ch|u|2,\n/ba∇dblu−uh/ba∇dblh2≤Ch((2µ)1\n2|u|2+λ1\n2|∇ ·u|1),\nwhereCis a positive constant independent of λ,µandh.\n6 Numerical experiments\nIn this section, we divide into two subsections to verify the theoretical results in\nTheorem 1, Theorem 2, Theorem 3and Theorem 4. The numerical experiments in\nthe first subsection follows the examples in Section 2. The second subsection is the\nCook’s Membrane problem, which is used to show the robustnes s of our novel scheme\nfor nearly incompressible elasticity. We set αT=α= 1 for all T∈ T h. We choose\naR(·,·) =adiv(·,·), because it is related to a diagonal block, and all the terms of the\nmatrix can be calculated using the barycentric quadrature r ule.\n6.1 Parameter-robustness test\nThe examples to show the locking-free and gradient-robust p roperties are the same as\nin Section 2. We always use the primal formulation ( 1) to take numerical experiments.\nWhen we show the accuracy of our novel schemes, the boundary c onditions are divided\ninto two cases. One is homogeneous Dirichlet boundary condi tionu=0on∂Ω, and\nthe other is mixed boundary conditions ( 4)\nu=0on Γ D,σn=gon Γ N.\nThe Neumann boundary Γ Nis posed on the right boundary ( x= 1) of the domain,\nwhile on the other three sides the Dirichlet condition is use d.\nTable 1 : Numerical results of Example 1, using scheme ( S1) with homogeneous\nDirichlet boundary condition when λ= 1.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.43E-1 – 2.65 – 186 1.08E-1 – 2.98 –\n1378 4.16E-2 1.78 1.32 1.00 690 3.22E-2 1.75 1.54 0.95\n5314 1.08E-2 1.93 6.58E-1 1.00 2658 8.39E-3 1.94 7.75E-1 0.99\n20866 2.75E-3 1.98 3.28E-1 1.00 10434 2.11E-3 1.98 3.87E-1 0.99\n82690 6.90E-4 1.99 1.64E-1 1.00 41346 5.30E-4 1.99 1.94E-1 0.99\n220 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91\nG10 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91\nG2\nFig. 4 : Structured grid (G1) and unstructured grid (G2).\nTable 2 : Numerical results of Example 1, using scheme ( S1) with homogeneous\nDirichlet boundary condition when λ= 106.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.42E-1 2.63 186 1.15E-1 2.95\n1378 4.16E-2 1.77 1.31 1.01 690 3.46E-2 1.74 1.51 0.96\n5314 1.09E-2 1.93 6.44E-1 1.01 2658 8.96E-3 1.95 7.61E-1 0.96\n20866 2.76E-3 1.98 3.21E-1 1.00 10434 2.24E-3 1.99 3.80E-1 1.00\n82690 6.92E-4 1.99 1.60E-1 1.00 41346 5.59E-4 2.00 1.90E-1 1.00\nNote that the discrete H1-seminorm is bounded by the energy norm, we measure\nthe error in the discrete H1-seminorm /ba∇dbl∇h(u−uh)/ba∇dblto validate theoretical analysis.\nThe term ‘ndof’ denotes the number of degrees of freedom, and it is approximately\nequal to twice the number of vertices plus the number of edges in the triangular\npartitions Th. We focus on the errors and convergence rates on structured m esh grid\nand unstructured mesh grid (see Figure 4). The scheme ( S1) is used to handle the\nDirichlet boundary condition, while the schemes ( S2), (S3) and ( S4) are used to handle\nthe mixed boundary conditions. From Table 1-Table 2, we find that scheme ( S1) for\nhomogeneous displacement boundary condition has optimal c onvergence rates, and\nscheme ( S1) is parameter-robust about λ. Table 3shows the numerical results of\nEquation ( 51), where the convergence rate of L2-norm decreases. So the modification\nof the left-hand side is necessary.\nTable 3 : Numerical results of Example 1, using the scheme ( 51) with mixed bound-\nary condition when λ= 1.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.57E-1 2.68 186 1.05E-1 2.98\n1378 4.83E-2 1.70 1.34 1.00 690 3.26E-2 1.68 1.54 0.94\n5314 1.37E-2 1.81 6.70E-1 1.00 2658 1.02E-2 1.67 7.78E-1 0.99\n20866 4.06E-3 1.76 3.39E-1 0.98 10434 3.44E-3 1.57 3.90E-1 0.99\n82690 1.32E-3 1.62 1.74E-1 0.98 41346 1.22E-3 1.48 1.96E-1 0.98\n23From Table 4-Table 7, all the schemes for mixed boundary conditions have the\noptimal convergence rates. Especially when λ= 106, all the schemes are stable and\nlocking-free for nearly incompressible situations. Note t hat the difference between the\nschemes ( S2), (S3) and ( S4) does not affect the uniform convergence of λ. For ( S3)\nand ( S4), we only take numerical experiments when λ= 106. The errors of L2-norm\nandH1-seminorm vary little when λtakes different values.\nTable 4 : Numerical results of Example 1, using the nonsymmetric scheme ( S2)\nwith mixed boundary condition when λ= 1.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.53E-1 2.66 186 1.11E-1 2.99\n1378 4.53E-2 1.75 1.32 1.00 690 3.23E-2 1.79 1.54 0.95\n5314 1.19E-2 1.92 6.59E-1 1.01 2658 8.37E-3 1.94 7.75E-1 0.99\n20866 3.03E-3 1.97 3.28E-1 1.00 10434 2.11E-3 1.98 3.88E-1 0.99\n82690 7.62E-4 1.99 1.64E-1 1.00 41346 5.28E-4 1.99 1.94E-1 1.00\nTable 5 : Numerical results of Example 1, using the nonsymmetric scheme ( S2)\nwith mixed boundary condition when λ= 106.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.49E-1 2.64 186 1.21E-1 2.96\n1378 4.47E-2 1.73 1.30 1.02 690 3.48E-2 1.80 1.52 0.96\n5314 1.18E-2 1.91 6.45E-1 1.01 2658 9.03E-3 1.94 7.61E-1 0.99\n20866 3.01E-3 1.97 3.21E-1 1.00 10434 2.28E-3 1.98 3.80E-1 1.00\n82690 7.52E-4 2.00 1.60E-1 1.00 41346 5.71E-4 1.99 1.90E-1 1.00\nTable 6 : Numerical results of Example 1, using the symmetric scheme ( S3) with\nmixed boundary condition when λ= 106.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.48E-1 2.65 186 1.20E-1 2.97\n1378 4.40E-2 1.75 1.30 1.01 690 3.38E-2 1.83 1.52 0.96\n5314 1.16E-2 1.91 6.46E-1 1.01 2658 8.78E-3 1.94 7.63E-1 1.00\n20866 2.96E-3 1.97 3.21E-1 1.00 10434 2.21E-3 1.98 3.81E-1 1.00\n82690 7.45E-4 1.99 1.60E-1 1.00 41346 5.55E-4 1.99 1.90E-1 1.00\nTable 8and Table 9are used to show the gradient-robustness of the scheme ( S1)\nwith homogeneous displacement boundary condition. From Th eorem 1, we have the\nbound\n/ba∇dbluh/ba∇dblh≤c\nλ+µ/ba∇dblφ/ba∇dbl\n24Table 7 : Numerical results of Example 1, using the nonsymmetric form of scheme\n(S4), with mixed boundary condition when λ= 106.\nndof(G1) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate ndof(G2) /ba∇dblu−uh/ba∇dblRate /ba∇dbl∇h(u−uh)/ba∇dblRate\n370 1.50E-1 2.64 186 1.14E-1 2.96\n1378 4.51E-2 1.74 1.30 1.01 690 3.41E-2 1.74 1.52 0.96\n5314 1.19E-2 1.91 6.45E-1 1.01 2658 8.89E-3 1.94 7.61E-1 0.99\n20866 3.03E-3 1.97 3.21E-1 1.00 10434 2.23E-3 1.99 3.80E-1 1.00\n82690 7.62E-4 1.99 1.60E-1 1.00 41346 5.57E-4 2.00 1.90E-1 1.00\nfor the gradient-robust discretization. As a comparison, f or non-gradient-robust\nmethods we have the following bound from [ 6]\n/ba∇dbluh/ba∇dbl1,h≤c\nµ/parenleftbigg1\nλ+ 1/parenrightbigg\n/ba∇dblφ/ba∇dbl.\nBy analyzing Table 8horizontally, we can find that /ba∇dbl∇huh/ba∇dblis independent of the dis-\ncretizations. And the vertical direction of the table indic ates that /ba∇dbl∇huh/ba∇dbl=O(λ−1).\nForλ= 104andµ∈(0,1],1\nλ+µ≈c(constant). Table 9shows that for different scaled\nµ, the quantity /ba∇dbl∇huh/ba∇dblonly varies very little, which verifies Theorem 1.\nTable 8 : Norm /ba∇dbl∇huh/ba∇dblof Example 2withµ= 1, different λand\ndifferent discretizations on structured mesh grid (G1).\n/ba∇dbl∇huh/ba∇dblndof=370 ndof=1378 ndof=5314 ndof=20866 ndof=82690\nλ= 1 1.089E-1 1.124E-1 1.136E-1 1.139E-1 1.140E-1\nλ= 10 3.389E-2 3.631E-2 3.721E-2 3.750E-2 3.759E-2\nλ= 1024.443E-3 4.839E-3 4.994E-3 5.046E-3 5.063E-3\nλ= 1044.608E-5 5.032E-5 5.198E-5 5.256E-5 5.274E-5\nλ= 1064.609E-7 5.034E-7 5.200E-7 5.258E-7 5.276E-7\nTable 9 : Norm /ba∇dbl∇huh/ba∇dblof Example 2withλ= 104, differentµand\ndifferent discretizations on structured mesh grid (G1).\n/ba∇dbl∇huh/ba∇dbl ndof=370 ndof=1378 ndof=5314 ndof=20866 ndof=82690\nµ= 10−64.6099E-5 5.0344E-5 5.2009E-5 5.2585E-5 5.2768E-5\nµ= 10−44.6099E-5 5.0344E-5 5.2009E-5 5.2585E-5 5.2768E-5\nµ= 10−24.6099E-5 5.0344E-5 5.2009E-5 5.2584E-5 5.2768E-5\nµ= 10−14.6097E-5 5.0342E-5 5.2007E-5 5.2582E-5 5.2766E-5\nµ= 1 4.6082E-5 5.0324E-5 5.1987E-5 5.2563E-5 5.2746E-5\n6.2 Cook’s Membrane Problem\nThis is a popular benchmark problem [ 14] for linear elasticity. As shown in Figure\n5, the domain Ω is a convex region formed by connecting four ver tices (0,0), (48,44),\n(48,60) and (0,44). The displacement boundary condition u=0is imposed on the\n25Fig. 5 : Cook’s membrance problem.\nleft side of the domain. A uniform vertical traction is impos ed on the right side, that\nis to say, the boundary condition on the right side is g= (0,1\n16)⊤. The rest of the\nboundary has no traction force. The body force f=0, the elasticity modulus E= 1,\nand the Lam´ e constants are given by\nλ=Eν\n(1 +ν)(1−2ν), µ =E\n2(1 +ν).\nAsν→1\n2andλ→ ∞ , the material becomes nearly incompressible. We choose the\nPossion’s ratio as 0 .33 and 0.4999, while ν= 0.33 denotes copper and ν= 0.4999\ndenotes rubber. There is no analytical solution to this prob lem. We solving this prob-\nlem using both the classical lagrangian element P1and theP1⊕RT 0element on\nunstructured triangulation mesh. Figure 6and Figure 7show the numerical dilation\n∇·uhusing theP1andP1⊕RT 0element, respectively. When ν= 0.33, both numer-\nical methods have good approximation results. The area’s to p-left corner is squeezed\nand the dilation ∇ ·uhis negative. The bottom of the area is stretched and ∇ ·uh\nis positive. When ν= 0.4999, the material is nearly incompressible. The classical\nGalerkin method exhibits locking phenomenon, the dilation oscillation occurs. The\nP1⊕RT 0scheme ( S2) yields a good numerical dilation approximation. Due to the\nnearly incompressible feature of the material, the dilatio n is numerically much smaller\nthan that of the compressible material.\n26Fig. 6 : Numerical dilation by P1element on unstructured mesh. ν= 0.33 (left);\nν= 0.4999 (right).\nFig. 7 : Numerical dilation by P1⊕RT 0element on unstructured mesh. ν= 0.33\n(left);ν= 0.4999 (right).\nFunding This work was supported by the National Natural Science Foun dation of\nChina (Grant 12131014).\nData Availability All data generated or analysed during this study are include d in\nthis manuscript.\nDeclarations\nConflict of Interest The authors declare that they have no conflict of interest.\n27References\n[1] Akbas M, Gallou¨ et T, Gaßmann A, et al (2020) A gradient-r obust well-balanced\nscheme for the compressible isothermal Stokes problem. Com put Methods Appl\nMech Engrg 367:113069\n[2] Arnold D, Falk R, Winther R (2007) Mixed finite element met hods for linear\nelasticity with weakly imposed symmetry. Math Comp 76(260) :1699–1723\n[3] Arnold DN, Winther R (2002) Mixed finite elements for elas ticity. Numer Math\n92:401–419\n[4] Arnold DN, Brezzi F, Cockburn B, et al (2002) Unified analy sis of discontinuous\nGalerkin methods for elliptic problems. SIAM J Numer Anal 39 (5):1749–1779\n[5] Babuˇ ska I, Suri M (1992) Locking effects in the finite elem ent approximation of\nelasticity problems. Numer Math 62(1):439–463\n[6] Basava SR, Wollner W (2023) Gradient robust mixed method s for nearly\nincompressible elasticity. J Sci Comput 95:93\n[7] Bernardi C, Raugel G (1985) Analysis of some finite elemen ts for the Stokes\nproblem. Math Comp 44(169):71–79\n[8] Boffi D, Brezzi F, Fortin M (2013) Mixed finite element metho ds and applications,\nSpringer Series in Computational Mathematics, vol 44. Spri nger, Heidelberg\n[9] Brenner SC, Scott LR (2008) The mathematical theory of fin ite element methods,\nTexts in Applied Mathematics, vol 15, 3rd edn. Springer, New York\n[10] Brenner SC, Sung LY (1992) Linear finite element methods for planar linear\nelasticity. Math Comp 59(200):321–338\n[11] Chiumenti M, Valverde Q, De Saracibar CA, et al (2002) A s tabilized for-\nmulation for incompressible elasticity using linear displ acement and pressure\ninterpolations. Comput Methods Appl Mech Engrg 191(46):52 53–5264\n[12] Cockburn B, Shi K (2013) Superconvergent HDG methods fo r linear elasticity\nwith weakly symmetric stresses. IMA J Numer Anal 33(3):747– 770\n[13] Cockburn B, Kanschat G, Sch¨ otzau D (2007) A note on disc ontinuous Galerkin\ndivergence-free solutions of the Navier–Stokes equations . J Sci Comput 31(1-\n2):61–73\n[14] Cook RD (1974) Improved two-dimensional finite element . J Structural Division\n100(9):1851–1863\n28[15] Crouzeix M, Raviart PA (1973) Conforming and nonconfor ming finite ele-\nment methods for solving the stationary Stokes equations i. Revue fran¸ caise\nd’automatique informatique recherche op´ erationnelle Ma th´ ematique 7(R3):33–75\n[16] Fu G, Lehrenfeld C, Linke A, et al (2021) Locking-free an d gradient-robust\nH(div)-conforming HDG methods for linear elasticity. J Sci Comput 86(3):1–30\n[17] Gatica GN (2006) Analysis of a new augmented mixed finite element method for\nlinear elasticity allowing RT0−P1−P0approximations. ESAIM: Math Model\nNumer Anal 40(1):1–28\n[18] Gatica GN, Gatica LF, Stephan EP (2007) A dual-mixed fini te element method\nfor nonlinear incompressible elasticity with mixed bounda ry conditions. Comput\nMethods Appl Mech Engrg 196(35-36):3348–3369\n[19] Gatica GN, M´ arquez A, Meddahi S (2009) An augmented mix ed finite element\nmethod for 3D linear elasticity problems. J Comput Appl Math 231(2):526–540\n[20] Hansbo P, Larson MG (2003) Discontinuous Galerkin and t he Crouzeix–Raviart\nelement: Application to elasticity. ESAIM: Math Model Nume r Anal 37(1):63–72\n[21] Hu J (2015) Finite element approximations of symmetric tensors on simplicial\ngrids inRn: The higher order case. J Comput Math pp 283–296\n[22] Hu J, Schedensack M (2018) Two low-order nonconforming finite element meth-\nods for the Stokes flow in three dimensions. IMA J Numer Anal 39 (3):1447–1470\n[23] Hu J, Zhang S (2016) Finite element approximations of sy mmetric tensors on\nsimplicial grids in Rn: The lower order case. Math Model Method Appl Sci\n26(09):1649–1669\n[24] Jin S (1999) Efficient asymptotic-preserving (ap) schem es for some multiscale\nkinetic equations. SIAM J Sci Comput 21(2):441–454\n[25] John V (2016) Finite element methods for incompressibl e flow problems, Springer\nSeries in Computational Mathematics, vol 51. Springer, Cha m\n[26] John V, Linke A, Merdon C, et al (2017) On the divergence c onstraint in mixed\nfinite element methods for incompressible flows. SIAM Rev 59( 3):492–544\n[27] Kouhia R, Stenberg R (1995) A linear nonconforming finit e element method for\nnearly incompressible elasticity and Stokes flow. Comput Me thods Appl Mech\nEngrg 124(3):195–212\n[28] Lamichhane BP (2009) Inf–sup stable finite-element pai rs based on dual meshes\nand bases for nearly incompressible elasticity. IMA J Numer Anal 29(2):404–420\n29[29] Li X, Rui H (2022) A low-order divergence-free H(div)-c onforming finite element\nmethod for Stokes flows. IMA J Numer Anal 42(4):3711–3734\n[30] Linke A (2014) On the role of the Helmholtz decompositio n in mixed methods for\nincompressible flows and a new variational crime. Comput Met hods Appl Mech\nEngrg 268:782–800\n[31] Linke A, Merdon C (2016) Pressure-robustness and discr ete Helmholtz projectors\nin mixed finite element methods for the incompressible Navie r–Stokes equations.\nComput Methods Appl Mech Engrg 311:304–326. https://doi.org/10.1016/j.cma.\n2016.08.018\n[32] Linke A, Matthies G, Tobiska L (2016) Robust arbitrary o rder mixed finite ele-\nment methods for the incompressible Stokes equations with p ressure independent\nvelocity errors. ESAIM: Math Model Numer Anal 50(1):289–30 9\n[33] Malkus DS, Hughes TJ (1978) Mixed finite element methods –reduced and selec-\ntive integration techniques: a unification of concepts. Com put Methods Appl\nMech Engrg 15(1):63–81\n[34] Phillips PJ, Wheeler MF (2009) Overcoming the problem o f locking in linear\nelasticity and poroelasticity: an heuristic approach. Com put Geosci 13:5–12\n[35] Qiu W, Shen J, Shi K (2018) An HDG method for linear elasti city with strong\nsymmetric stresses. Math Comp 87(309):69–93\n[36] Rui H, Sun M (2018) A locking-free finite difference metho d on staggered grids\nfor linear elasticity problems. Comput Math Appl 76(6):130 1–1320\n[37] Wang J, Ye X (2007) New finite element methods in computat ional fluid dynamics\nby H(div) elements. SIAM J Numer Anal 45(3):1269–1286\n[38] Wihler T (2006) Locking-free adaptive discontinuous G alerkin FEM for linear\nelasticity problems. Math Comp 75(255):1087–1102\n[39] Yi SY (2017) A study of two modes of locking in poroelasti city. SIAM J Numer\nAnal 55(4):1915–1936\n[40] Yi SY (2019) A lowest-order weak Galerkin method for lin ear elasticity. J Comput\nAppl Math 350:286–298\n[41] Yi SY, Lee S, Zikatanov L (2022) Locking-free enriched G alerkin method for\nlinear elasticity. SIAM J Numer Anal 60(1):52–75\n[42] Zdunek A, Neunteufel M, Rachowicz W (2023) On pressure r obustness and inde-\npendent determination of displacement and pressure in inco mpressible linear\nelasticity. Comput Methods Appl Mech Engrg 403:115714\n30[43] Zhang M, Zhang S (2017) A 3D conforming-nonconforming m ixed finite element\nfor solving symmetric stress Stokes equations. Int J Numer A nal Model 14(4-\n5):730–743\n31" }, { "title": "2401.14932v1.Super_exponential_quantum_advantage_for_finding_the_center_of_a_sphere.pdf", "content": "Super-exponential quantum advantage for finding the center of a sphere\nGuanzhong Li and Lvzhou Li∗\nInstitute of Quantum Computing and Software, School of Computer Science and Engineering,\nSun Yat-sen University, Guangzhou 510006, China\nThis article considers the geometric problem of finding the center of a sphere in vector space\nover finite fields, given samples of random points on the sphere. We propose a quantum algorithm\nbased on continuous-time quantum walks that needs only a constant number of samples to find\nthe center. We also prove that any classical algorithm for the same task requires approximately as\nmany samples as the dimension of the vector space, by a reduction to an old and basic algebraic\nresult—Warning’s second theorem. Thus, a super-exponential quantum advantage is revealed for\nthe first time for a natural and intuitive geometric problem.\nIntroduction. — A primary goal of quantum computing\nis to identify problems for which quantum algorithms can\noffer speedups over their classical counterpart. Grover’s\nalgorithm [1] offers a quadratic speedup for unstructured\ndatabase search problem. Shor’s algorithm [2] is expo-\nnentially faster than the best classical algorithm in fac-\ntoring integers. Recent studies have also found exponen-\ntial speedups in specific problems such as simulating the\ncoupled oscillators [3], traversing a decorated welded-tree\ngraph using adiabatic quantum computation [4], a graph\nproperty testing problem in the adjacency list model [5], a\nclassification problem for quantum kernel methods based\non discrete logarithm problem [6], a specific NP search\nproblem based on random black-box function [7], and\nso on. However, to the best of our knowledge, no one\nhas ever found a geometric problem that exhibits sharp\nquantum speedups.\nIn this letter, we consider a natural geometric problem\nof finding the center of a sphere given samples of random\npoints on the sphere. To illustrate this problem, we begin\nwith an intuitive example: if we want to pin down a\ncircle on a paper, we will need 3 points ( A, B, C ) which\nare not on the same line, as depicted in Fig. 1 (a); and\nif we want to fix an inflatable balloon, we will need 4\npoints ( A, B, C, D ) which are not on the same plane, as\ndepicted in Fig. 1 (b). As such, n+ 1 points are required\nto determine a sphere in Rn.\nAB\nC(a)\n(b)\nA\nB CD\nFIG. 1. Spheres in the R2andR3.\nTo accommodate the discrete nature of qubit-based\nquantum computer, we will replace the continuous vec-tor space RnwithFn\nq, the vector space over a finite field\nFqwith qbeing a prime number. Define the length of a\nvector x= (x1,···, xn)∈Fn\nqbyl(x) :=Pn\nj=1x2\nj. Then\na sphere with radius r∈Fq\\ {0}and center t∈Fn\nqis\ndenoted by Sr+t={x+t|x∈Sr}, where\nSr:={x|x∈Fn\nq, l(x) =r}. (1)\nThe problem is then formally described as follows.\nProblem 1. Find the unknown center t∈Fn\nqof the\nsphere Sr+t(r > 0 is given) with as few samples of\npoints on Sr+tas possible.\nIn analogy to sampling a point uniformly random\nfrom Sr+t, each usage of the state |Sr+t⟩:=\n1√srP\nv∈Sr+t|v⟩, where sr:=|Sr|, is regarded as a quan-\ntum sample. In this letter, we obtain the following theo-\nrem.\nTheorem 1. There is a quantum algorithm that solves\nProblem 1 with bounded error, and uses O(logq) [8] sam-\nples. However, any classical algorithm that solves Prob-\nlem 1 with bounded error requires Ω( n) samples.\nNote that when qis fixed to be a constant, the quan-\ntum algorithm needs constant O(1) samples, while any\nclassical algorithm requires Ω( n) samples. This is ac-\ntually a super-exponential speedup because the classical\nΩ(n) v.s. quantum O(logn) separation is already an ex-\nponential speedup.\nTo prove the classical lower bound Ω( n), we find un-\nexpectedly that it can be reduced to a basic algebraic\ntheorem proved around 1935 by Warning [9], which gives\na lower bound on the number of zeros of multi-variable\npolynomials over finite fields. We then design the quan-\ntum algorithm based on a continuous-time quantum walk\n(CTQW) on the Euclidean graph on Fn\nq, where two points\nx, yare connected if and only if l(x−y) =r.\nQuantum walks are an analogy to classical random\nwalks, and have become a widely adopted paradigm to\ndesign quantum algorithms for various problems, such as\nspatial search [10–14], element distinctness [15], matrix\nproduct verification [16], triangle finding [17, 18], group\ncommutativity [19], the welded-tree problem [20–22], andarXiv:2401.14932v1 [quant-ph] 26 Jan 20242\nso on. There are two types of quantum walks: the\nCTQW and the discrete-time quantum walk (DTQW).\nCTQW is relatively simple, and mainly involves simulat-\ning a Hamiltonian Hthat encodes the structure of the\ngraph. DTQW is more diverse, ranging from the earli-\nest and simplest coined quantum walk [23, 24] to various\nMarkov chain based frameworks [25–29]. The quantum\nalgorithm proposed in this letter is based on CTQW and\nis different from previous ones with sharp speedups fea-\nturing quantum Fourier transform, and thus our result\nmay inspire the discovery of new quantum algorithms\nwith sharp speedups.\nClassical lower bound. — In the introduction section\nwe have intuitively shown that n+ 1 points are required\nto determine a sphere in Rn. Here we will rigorously\nprove that any classical algorithm requires Ω( n) samples\nof points on the sphere Sr+tto determine the center\ntwith high probability. To prove this classical lower\nbound, we will use the following lemma, also known as\nWarning’s second theorem [30–32], attributed to Ewald\nWarning [9].\nLemma 1 (Warning’s second theorem) .Suppose\nf1, f2,···, fs∈Fq[x1,···, xn] are multivariate polyno-\nmials over Fqwith di= deg( fi). Let d:=d1+···+ds\nbe the total degree and V:={x= (x1,···, xn)∈Fn\nq|\nfi(x) = 0 ,∀1≤i≤s}be the set of common zeros of fi.\nAssume n > d andV̸=∅. Then |V| ≥qn−d.\nWe can now obtain a relationship between the number\nof different points that a classical algorithm has sampled\nonSr+tand the upper bound of its success probability.\nLemma 2. Suppose cis a constant such that 0 < c <\n1/2. If a classical algorithm has sampled less than cn\ndifferent points on Sr+t, then it has at most 1 /q(1−2c)n\nprobability of obtaining the correct center t.\nProof. Denoted by X+t, where X:={(xi1,···, xin)|\ni= 1, . . . , s } ⊆Sr, the s≤cndifferent points that the\nclassical algorithm has obtained on Sr+t. We claim that\nthere are at least q(1−2c)npossible centers {t′} ⊆Fn\nqsuch\nthatX+t⊆Sr+t′. In other words, all of these possible\nspheres {Sr+t′}can lead to the same set of samples\nX+t. Having obtained only X+t, the algorithm cannot\ndistinguish between the possible centers {t′}, where only\none of them is the correct center t. As|{t′}| ≥q(1−2c)n,\nthe possibility of obtaining t∈ {t′}is at most 1 /q(1−2c)n.\nTo prove our claim, note that by letting ¯t:=t′−t, the\ninclusion X+t⊆Sr+t′becomes X⊆Sr+¯t, which\nresults in the following system of polynomial equations\nover finite fields about variables ¯t= (¯t1,···,¯tn):\nfi:=nX\nj=1(xij−¯tj)2−r= 0,1≤i≤s. (2)\nSince di= deg( fi) = 2 for all i, and ¯t=⃗0 is a solution\n(recall that X⊆Sr), the above equations have at leastqn−2s≥q(1−2c)ncommon roots ¯tby Lemma 1. Thus\nthere are at least q(1−2c)npossible centers t′=¯t+tsuch\nthat X+t⊆Sr+t′.\nIf we let c= 1/3, then a classical algorithm that has\nsampled less than n/3 different points on Sr+tcan only\nsucceed with probability less than 1 /qn/3, which is expo-\nnentially small. Therefore, if a classical algorithm wants\nto solve Problem 1 with high probability, Ω( n) samples\nare needed.\nQuantum algorithm. — The quantum algorithm for\nProblem 1 is concise and consists of only four steps as\nshown below. The main idea is to use CTQW on the\nEuclidean graph GonFn\nqto move amplitude from the\nsphere to its center. The Euclidean graph Ghas vertex\nsetFn\nq, and two vertices x, x′∈Fn\nqare connected by an\nundirected edge if and only if l(x−x′) =r.\nThe Hamiltonian ¯Aof the CTQW on the Euclidean\ngraph Gis approximately the adjacency matrix Aof\nthe Euclidean graph G, but with A’s largest eigenvalue\nλ0=srreplaced by 0. Specifically, ¯A:=A−λ0|e0⟩⟨e0|,\nwhere |˜0⟩is the corresponding eigenvector of eigenvalue\nλ0and it is the equal superposition of all points in Fn\nq\n(see [33, Proposition 2] for the spectral decomposition of\nA). Note that Ais symmetric since the graph Gis undi-\nrected, so ¯Ais a valid Hamiltonian. We will see later from\nnumerical simulation that Aitself as the Hamiltonian is\ngood enough.\n1. Prepare the quantum sample |Sr+t⟩=\n1√srP\nv∈Sr+t|v⟩.\n2. Apply a CTQW ei¯At0to|Sr+t⟩for time t0=\n1/p\nqn−1logq, and then measure in the computa-\ntional basis obtaining a point in Fn\nq.\n3. Repeat the above two steps for O(logq) times.\n4. Output the point with the highest frequency.\nThe following lemma lower bounds the success proba-\nbility of step 2. It can be seen as a fine-grained version\nof [34, Lemma 4].\nLemma 3. The final state |ψt0⟩:=ei¯At0|Sr+t⟩of\nCTQW on the Euclidean graph Ghas the following prop-\nerties:\n|⟨x|ψt0⟩|(\n≤O(q−(n−1)/2) for x̸=t,\n≥h(q,n)√logqforx=t,(3)\nwhere the function h(q, n) is monotonically increasing in\nboth qandn, and h(127,8)≥0.0016.\nFrom Lemma 3, the probability to obtain the center\ntis Ω(1 /logq) in step 2, while the probability to obtain\nany point other than tis exponentially small. Step 3 then\nguarantees the center tto be found with high probability.\nOverall, the quantum algorithm needs O(logq) samples.3\nProof of Lemma 3. As shown in Appendix, the CTQW\nei¯At0=P∞\nk=0(it0)k\nk!¯Akhas a ( q+1)-dimensional invariant\nsubspace H0spanned by the following orthonormal basis:\nB0={|t⟩,|S0+t⟩,|S1+t⟩,···,|Sq−1+t⟩},(4)\nwhere S0:={v∈Fn\nq:l(v) = 0} \\ {⃗0}(see Eq. (1) for\nSrwith r̸= 0). The following equation [33, Theorem 1]\nshows that sr:=|Sr| ≈qn−1for all r∈Fq.\nsr=\n\nqn−1+χ((−1)n−1\n2r)qn−1\n2 nodd, r̸= 0,\nqn−1−χ((−1)n\n2)qn−2\n2 neven, r̸= 0,\nqn−1−1 nodd, r= 0,\nqn−1+χ((−1)n\n2)(q−1)qn−2\n2−1neven, r= 0,\n(5)\nwhere χ(a)∈ {0,1,−1}indicates whether ais zero, or\nthe square of some element in Fq, or otherwise.\nWe first consider x̸=t. Since |ψt0⟩ ∈spanB0, we\nhave:\n|⟨x|ψt0⟩| ≤max\nr′∈[q]|⟨x|Sr′+t⟩| (6)\n= max\nr′∈[q]1/√sr′=O(q−(n−1)/2). (7)\nWe then consider the lower bound of |⟨t|ψt0⟩|as follows:\n|⟨t|ψt0⟩|=\f\f\f⟨t|ei¯At0|Sr+t⟩\f\f\f=\f\f\f\f\f⟨t|∞X\nk=0(it0)k\nk!¯Ak|Sr+t⟩\f\f\f\f\f\n(8)\n≥t0⟨t|\u0000\nA−λ0|e0⟩⟨e0|\u0001\n|Sr+t⟩ −∞X\nk=2tk\n0\nk!\f\f⟨t|¯Ak|Sr+t⟩\f\f\n(9)\n≥t0\u0012√sr−sr√sr√qn1√qn\u0013\n−∞X\nk=2(2t0p\nqn−1)k\nk!(10)\n=q\nsrt2\n0−sr\nqnq\nsrt2\n0−(e2t0√\nqn−1−1−2t0p\nqn−1).\n(11)\nWe have used ⟨t|Sr+t⟩= 0 and the triangle inequality\n|x+y| ≥ |x| − |y|and the fact that ¯A=A−λ0|e0⟩⟨e0|\nis a real matrix in Formula (9). Formula (10) is because\nthe adjacency matrix Amaps |t⟩to√sr|Sr+t⟩, and\n|⟨t|¯Ak|Sr+t⟩| ≤ ∥ ¯Ak∥ ≤ (2p\nqn−1)k, where the sec-\nond upper bound follows from the fact that the spectral\nradius of ¯Ais less than 2p\nqn−1[33, Theorem 3].\nFrom Eq. (5) we know qn−1−q(n−1)/2≤sr≤qn−1+\nq(n−1)/2. Recall that t0= 1/p\nqn−1logq. Thus we\nhave1\nlogq(1−1\nq(n−1)/2)≤srt2\n0≤1\nlogq(1 +1\nq(n−1)/2),\nandsr/qn≤1\nq(1 +1\nq(n−1)/2). Using basic inequalities√a−ε≥√a−√εand√a+ε≤√a+√ε, we can con-tinue to calculate the lower bound of |⟨t|ψt0⟩|as follows:\n|⟨t|ψt0⟩| ≥1√logq(1−1\nq(n−1)/4)\n−1√logq(1 +1\nq(n−1)/4)(1\nq+1\nq(n+1)/2)\n−(e2/√logq−1−2���logq) (12)\n≥1√logq(1−1\nq−4\nq(n−1)/4−g(p\nlogq)) (13)\n:=1√logqh(q, n) (14)\nIn Formula (13), g(x) :=xe2/x−x−2 is monotonically\ndecreasing when x >0, and g(√logq)<1 when q≥127.\nThus h(q, n) is monotonically increasing in both qandn,\nand it can be verified that h(127,8)≥0.0016.\nNumerical simulation. — To illustrate that Ais good\nenough as the Hamiltonian, we consider the simplest\ncases where the size of the finite field is q= 3 and the\nradius of the sphere is r= 1. The invariant subspace\nH0= span B0of CTQW eiAt 0is then 4-dimensional:\nB0={|t⟩,|S0+t⟩,|S1+t⟩,|S2+t⟩}.\nWe first consider the case where n= 5. Using Lemma 4\nin Appendix, we have the following matrix expression of\nAon the basis B0.\nMA=\n0 0 3√\n10 0\n0 27 24√\n2 9√\n10\n3√\n10 24√\n2 33 12√\n5\n0 9√\n10 12√\n5 30\n. (15)\nDenote by p(t1) :=\f\f⟨t|eiAt 1·t0|Sr+t⟩\f\f2the success prob-\nability of reaching the target state after time t1·t0, then\np(t1) =\f\f⟨0|eiMAt1·t0|2⟩\f\f2for the case where q= 3, r=\n1, n= 5. By numerical calculation, we obtain p(t1) with\nt1∈[0,10] as shown in Fig. 2. It shows that the success\nprobability of the CTQW oscillates periodically with its\nevolution time, and the earliest time to achieve the max-\nimum success probability is around t1= 1.5.\nWe then consider the cases where n= 3∼12. Fig. 3\nshows that as the dimension nvaries, the optimal evolu-\ntion time tmax:= arg max\nt1∈[0,3]p(t1) lies in [1 ,2], and the cor-\nresponding maximum success probability p(tmax)≥0.4.\nIn summary, we have found a super-exponential quan-\ntum advantage for a natural and geometric problem of\nfinding the center of a sphere in Fn\nqgiven samples on\nthe sphere. While any classical bounded-error algorithm\nrequires Ω( n) samples on the sphere, the quantum algo-\nrithm based on CTQW on the Euclidean graph on Fn\nq\nneeds only O(1) sample.\nThis work is supported by the National Natural Sci-\nence Foundation of China Grant No. 62272492, and the\nGuangdong Basic and Applied Basic Research Founda-\ntion Grant No. 2020B1515020050.4\n0 1 2 3 4 5 6 7 8 9 10\nTime t100.050.10.150.20.250.30.350.40.450.5Success probability p(t1)\nFIG. 2. The success probability p(t1) for the case where\nq= 3, r= 1, n= 5.\n3 4 5 6 7 8 9 10 11 12\ndimension n0.40.60.811.21.41.61.82success probability / optimal walk timetmax\np(tmax)\nFIG. 3. Optimal CTQW time tmaxand corresponding suc-\ncess probability p(tmax). Here, q= 3, r= 1, and n= 3∼12.\n∗lilvzh@mail.sysu.edu.cn\n[1] L. K. Grover, A fast quantum mechanical algorithm for\ndatabase search, in Proceedings of the Twenty-Eighth An-\nnual ACM Symposium on Theory of Computing , STOC\n’96 (Association for Computing Machinery, New York,\nNY, USA, 1996) p. 212–219.\n[2] P. W. Shor, Polynomial-time algorithms for prime factor-\nization and discrete logarithms on a quantum computer,\nSIAM Journal on Computing 26, 1484 (1997).\n[3] R. Babbush, D. W. Berry, R. Kothari, R. D. Somma,\nand N. Wiebe, Exponential quantum speedup in simulat-\ning coupled classical oscillators, Phys. Rev. X 13, 041041\n(2023).\n[4] A. Gily´ en, M. B. Hastings, and U. Vazirani,\n(sub)exponential advantage of adiabatic quantum\ncomputation with no sign problem, in Proceedings ofthe 53rd Annual ACM SIGACT Symposium on Theory\nof Computing , STOC 2021 (Association for Computing\nMachinery, New York, NY, USA, 2021) p. 1357–1369.\n[5] S. Ben-David, A. M. Childs, A. Gily´ en, W. Kretschmer,\nS. Podder, and D. Wang, Symmetries, graph properties,\nand quantum speedups, in 2020 IEEE 61st Annual Sym-\nposium on Foundations of Computer Science (FOCS)\n(2020) pp. 649–660.\n[6] Y. Liu, S. Arunachalam, and K. Temme, A rigorous and\nrobust quantum speed-up in supervised machine learn-\ning, Nature Physics 17, 1013 (2021).\n[7] T. Yamakawa and M. Zhandry, Verifiable quantum ad-\nvantage without structure, in 2022 IEEE 63rd An-\nnual Symposium on Foundations of Computer Science\n(FOCS) (2022) pp. 69–74.\n[8] The standard asymptotic notations O(·) and Ω( ·) are\nused. We say the complexity is O(f(n)) (resp. Ω( f(n))),\nif for large enough n, it is at most (resp. at least) cf(n)\nfor some constant c.\n[9] H. Chevalley and E. Warning, Bemerkung zur vorste-\nhenden arbeit, Abhandlungen aus dem Mathematischen\nSeminar der Universit¨ at Hamburg 11, 76 (1935).\n[10] A. M. Childs and J. Goldstone, Spatial search by quan-\ntum walk, Phys. Rev. A 70, 022314 (2004).\n[11] A. Ambainis, J. Kempe, and A. Rivosh, Coins make\nquantum walks faster, in Proceedings of the Sixteenth\nAnnual ACM-SIAM Symposium on Discrete Algorithms ,\nSODA ’05 (Society for Industrial and Applied Mathe-\nmatics, USA, 2005) p. 1099–1108.\n[12] S. Chakraborty, L. Novo, A. Ambainis, and Y. Omar,\nSpatial search by quantum walk is optimal for almost all\ngraphs, Phys. Rev. Lett. 116, 100501 (2016).\n[13] D. Qu, S. Marsh, K. Wang, L. Xiao, J. Wang, and P. Xue,\nDeterministic search on star graphs via quantum walks,\nPhys. Rev. Lett. 128, 050501 (2022).\n[14] Y. Xu, D. Zhang, and L. Li, Robust quantum walk search\nwithout knowing the number of marked vertices, Phys.\nRev. A 106, 052207 (2022).\n[15] A. Ambainis, Quantum walk algorithm for element dis-\ntinctness, SIAM J. Comput. 37, 210 (2007).\n[16] H. Buhrman and R. Spalek, Quantum verification of ma-\ntrix products, in Proceedings of the Seventeenth ACM-\nSIAM Symposium on Discrete Algorithms (2006) pp.\n880–889.\n[17] F. Magniez, M. Santha, and M. Szegedy, Quantum algo-\nrithms for the triangle problem, SIAM J. Comput. 37,\n413 (2007).\n[18] G. Li and L. Li, Derandomization of quantum algorithm\nfor triangle finding (2023), arXiv:2309.13268 [quant-ph].\n[19] F. Magniez and A. Nayak, Quantum complexity of test-\ning group commutativity, Algorithmica 48, 221 (2007).\n[20] A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann,\nand D. A. Spielman, Exponential algorithmic speedup by\na quantum walk, in Proceedings of the Thirty-Fifth An-\nnual ACM Symposium on Theory of Computing , STOC\n’03 (Association for Computing Machinery, New York,\nNY, USA, 2003) p. 59–68.\n[21] S. Jeffery and S. Zur, Multidimensional quantum walks,\ninProceedings of the 55th Annual ACM Symposium\non Theory of Computing , STOC 2023 (Association for\nComputing Machinery, New York, NY, USA, 2023) p.\n1125–1130.\n[22] G. Li, L. Li, and J. Luo, Recovering the original sim-\nplicity: succinct and deterministic quantum algorithm5\nfor the welded tree problem, in Proceedings of the 2024\nAnnual ACM-SIAM Symposium on Discrete Algorithms\n(SODA) , pp. 2454–2480.\n[23] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and\nJ. Watrous, One-dimensional quantum walks, in Proceed-\nings of the 33rd ACM Symposium on Theory of Comput-\ning(2001) pp. 37–49.\n[24] D. Aharonov, A. Ambainis, J. Kempe, and U. V. Vazi-\nrani, Quantum walks on graphs, in Proceedings of the\n33rd ACM Symposium on Theory of Computing (2001)\npp. 50–59.\n[25] M. Szegedy, Quantum speed-up of markov chain based\nalgorithms, in 45th Annual IEEE Symposium on Foun-\ndations of Computer Science (2004) pp. 32–41.\n[26] F. Magniez, A. Nayak, J. Roland, and M. Santha, Search\nvia quantum walk, SIAM J. Comput. 40, 142 (2011).\n[27] A. Belovs, Quantum walks and electric networks (2013),\narXiv:1302.3143 [quant-ph].\n[28] H. Krovi, F. Magniez, M. Ozols, and J. Roland, Quantum\nwalks can find a marked element on any graph, Algorith-\nmica 74, 851 (2016).\n[29] S. Apers, A. Gily´ en, and S. Jeffery, A Unified Frame-\nwork of Quantum Walk Search, in 38th International\nSymposium on Theoretical Aspects of Computer Science\n(STACS 2021) , Leibniz International Proceedings in In-\nformatics (LIPIcs), Vol. 187, edited by M. Bl¨ aser and\nB. Monmege (Schloss Dagstuhl – Leibniz-Zentrum f¨ ur\nInformatik, Dagstuhl, Germany, 2021) pp. 6:1–6:13.\n[30] S. Asgarli, A new proof of warning’s second theorem, The\nAmerican Mathematical Monthly 125, 549 (2018).\n[31] P. L. Clark, A. Forrow, and J. R. Schmitt, Warning’s\nsecond theorem with restricted variables, Combinatorica\n37, 397 (2017).\n[32] P. L. Clark, Warning’s second theorem with relaxed out-\nputs, Journal of Algebraic Combinatorics 48, 325 (2018).\n[33] A. Medrano, P. Myers, H. Stark, and A. Terras, Finite\nanalogues of euclidean space, Journal of Computational\nand Applied Mathematics 68, 221 (1996).\n[34] A. M. Childs, L. J. Schulman, and U. V. Vazirani, Quan-\ntum algorithms for hidden nonlinear structures, in 48th\nAnnual IEEE Symposium on Foundations of Computer\nScience (FOCS’07) (2007) pp. 395–404.\n[35] M. Aschbacher, Finite Group Theory , 2nd ed., Cam-\nbridge Studies in Advanced Mathematics (Cambridge\nUniversity Press, 2000).\nAppendix —We will prove the following Lemma 4, used\nin the proof of Lemma 3 and also in obtaining the reduced\nmatrix MAin Eq. (15). Lemma 4 is extracted from [34,\nLemma 4], but with a more detailed proof for the con-\nvenience of the reader. Lemma 4 implies that H0is an\ninvariant subspace of the CTQW ei¯At0=P∞\nk=0(it0)k\nk!¯Ak,\nsince ¯A=A−λ0|e0⟩⟨e0|, and√qn|˜0⟩=P\nx∈Fnq|x⟩=\n|t⟩+Pq−1\nr′=0√sr′|Sr′+t⟩ ∈ H 0.\nLemma 4. The adjacency matrix Ahas a ( q+ 1)-\ndimensional invariant subspace H0spanned by the fol-\nlowing orthonormal basis:\nB0={|t⟩,|S0+t⟩,|S1+t⟩,···,|Sq−1+t⟩}.(16)Specifically, we have\nA|t⟩=√sr|Sr+t⟩, (17)\nA|Sr′+t⟩=cr′|t⟩+q−1X\nr′′=0c(r′′, r′)|Sr′′+t⟩,(18)\nwhere c(r′′, r′) :=qsr′′\nsr′|Sr∩(Sr′+v′′\n0)|for arbitrary\nv′′\n0∈Sr′′, and cr′:=δr′,r√sr.\nProof. Eq. (17) follows from the definition that Amaps\nanyx∈Fn\nqto{y∈Fn\nq|l(y−x) =r}=Sr+x. To\nprove Eq. (18), we consider any point v′′+tonSr′′+t,\nwhere v′′∈Sr′′. We calculate\n⟨v′′+t|A|Sr′+t⟩ (19)\n=1√sr′X\nv∈SrX\nv′∈Sr′⟨v′′+t|v−v′+t⟩ (20)\n=1√sr′|Sr∩(Sr′+v′′)|. (21)\nWe used |Sr′+t⟩=1√sr′P\nv′∈Sr′|−v′+t⟩in Eq. (20),\nand Eq. (21) follows from the fact that the condition\nv′′+t=v−v′+tis equivalent to v=v′+v′′, where\nv∈Srandv′∈Sr′.\nWe will later show that |Sr���(Sr′+v′′)|is the same\nfor any v′′∈Sr′′. Thus we have\nA|Sr′+t⟩ (22)\n=cr′|t⟩+q−1X\nr′′=0X\nv′′∈Sr′′⟨v′′+t|A|Sr′+t⟩|v′′+t⟩(23)\n=cr′|t⟩+q−1X\nr′′=01√sr′|Sr∩(Sr′+v′′\n0)|X\nv′′∈Sr′′|v′′+t⟩\n(24)\n=cr′|t⟩+q−1X\nr′′=0c(r′′, r′)|S′′\nr+t⟩. (25)\nWe used ⟨t|A|Sr′+t⟩=⟨Sr′+t|A|t⟩=δr′,r√s1in\nEq. (23), where the first equality follows from Abeing\nsymmetric. In Eq. (24), v′′\n0∈Sr′′is arbitrary.\nWe now show that |Sr∩(Sr′+v′′)|is the same for\nanyv′′∈Sr′′, or equivalently, |Sr∩(Sr′+x)|=|Sr∩\n(Sr′+z)|for any x∈Sr′′andz∈Sr′′. We will later\nconstruct an isometry τofFn\nqsuch that z=τ(x). As τis\na distance-preserving bijection, we have τ(Sr) =Srand\nτ(Sr′) =Sr′, and thus τ(Sr∩(Sr′+x)) =Sr∩(Sr′+z),\nwhich implies |Sr∩(Sr′+x)|=|Sr∩(Sr′+z)|. The\nisometry τcan be constructed by extending the isometry\nτ′:ax7→azthat maps subspace U={ax:a∈Fq}\nto subspace W={az:a∈Fq}(note that UandW\nare both 1-dimensional subspace, since xandzare both\nnonzero points), to an isometry τof the whole space Fn\nq\nusing Witt’s Lemma [35, Section 20]." }, { "title": "2401.14937v1.Scanning_Tunneling_Microscopy_for_Molecules__Effects_of_Electron_Propagation_into_Vacuum.pdf", "content": "Scanning Tunnelling Microscopy for Molecules: Effects of Electron\nPropagation into Vacuum\nAbhishek Grewal,1Christopher C. Leon,1Klaus Kuhnke,1,∗Klaus Kern,1, 2and Olle Gunnarsson1,†\n1Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany\n2Institut de Physique, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland\n(Dated: December 7, 2023)\nUsing scanning tunneling microscopy (STM), we experimentally and theoretically investigate isolated plat-\ninum phthalocyanine (PtPc) molecules adsorbed on atomically thin NaCl(100) vapor deposited on Au(111).\nWe obtain good agreement between theory and constant-height STM topography. We examine why strong\ndistortions of STM images occur as a function of distance between molecule and STM tip. The images of the\nhighest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) exhibit, for\nincreasing distance, significant radial expansion due to electron propagation in the vacuum. Additionally, the\nimaged angular dependence is substantially distorted. The LUMO image has substantial intensity along the\nmolecular diagonals where PtPc has no atoms. In the electronic transport gap the image differs drastically\nfrom HOMO and LUMO, even at energies very close to these orbitals. As the tunneling becomes increasingly\noff-resonant, the eight angular lobes of the HOMO or of the degenerate LUMOs diminish and reveal four lobes\nwith maxima along the molecular axes, where both, HOMO and LUMO have little or no weight. These images\nare strongly influenced by low-lying PtPc orbitals that have simple angular structures.\nINTRODUCTION\nScanning tunneling microscopy (STM) was developed\nacross several papers by Binnig et al. [1–4] The technique\nhas been extensively discussed in reviews [5–9], as well as\nin early theoretical works [10–25]. Significant theoretical\nprogress was achieved by Tersoff and Hamann [10, 11].\nThey assumed that the electrons tunnel to or from an 𝑠-\norbital on the tip. Using the Bardeen theory [26], they\ndemonstrated that the tunneling current is determined by\nthe hypothetical value of the wave function for the tun-\nneling electron at the center of the 𝑠-orbital. Substantial\neffort has been dedicated to improving this assumption\n[13, 17–25, 27–32]. However, these refinements necessi-\ntate a thorough understanding of the electronic structure\nof the tip. Given the limited information available, we\nadopt the assumption of Tersoff and Hamann. Repp et al.\n[33] introduced a NaCl buffer when studying a molecule\nto reduce the influence of the substrate. They observed\nthat STM shows spatially expanded images of molecular\norbitals (MO) of a pentacene molecule [33].\nAb initio calculations of STM images have been per-\nformed for clean surfaces and for very small molecules\nadsorbed directly on metal surfaces [15, 16, 34–42]. More\nclosely related to the present work are studies of single\ncopper phthalocyanine (CuPc) molecules using a model\nthat includes the highest occupied MO (HOMO), the low-\nest unoccupied MO (LUMO) and one 𝜎-orbital, located\njust below the energy of the CuPc HOMO [43–45]. CuPc\nhas been widely studied due to its demonstration of neg-\native differential resistance [46], while the related com-\npound H2Pc has garnered significant interest due to ob-\nservations of up-conversion electroluminescence at tun-\nneling energies in the H2Pc transport gap [47–49]. These\nexperiments raise important and fundamental questions\n∗k.kuhnke@fkf.mpg.de\n†o.gunnarsson@fkf.mpg.deabout electrons tunneling through the transport gap.\nHere, we study a similar organic luminescent system,\nnamely PtPc adsorbed on a few layers of a NaCl(100) film\non an Au substrate, as illustrated in Fig. 1. In particular\nour study serves as a means to better understand the very\nnature of tunneling through vacuum in STM studies. We\nshow how images of the HOMO and the LUMO substan-\ntially expand radially and explain their origin. The LUMO\nimage also distorts angularly and has significant weight\neven at spatial positions where the underlying molecule\nhas no atoms, for reasons to be discussed below. The im-\nages of electrons tunneling through the transport gap are\nshown to differ drastically from the HOMO or LUMO im-\nages, even for energies very close to these states. We show\nthat in an expansion in terms of MOs this is due to con-\ntributions from MOs with few nodal surfaces at substan-\ntially lower energy than the HOMO. This effect is very\nimportant for understanding experiments manipulating\nelectrons (tunneling through the gap) and photons.\nHere we treat a single PtPc molecule adsorbed flat on\nthe surface. We expand the wave function in vacuum in\ncylindrical coordinates. In planes parallel to the surface\nwe use trigonometric functions, cos(𝑚𝜙)and sin(𝑚𝜙),\nto describe the angular behavior and integer Bessel func-\ntions,𝐽𝑚(𝑘𝑚𝑖𝜌)with𝑖−1radial nodes in the region stud-\nied, to describe the radial behavior. Perpendicular to the\nsurface the wave function is described by exponentially\ndecaying functions. For the Au-NaCl-PtPc system we em-\nploy a tight-binding (TB) formalism [50, 51].\nWe investigate electrons arriving at the tip with a given\ntotal kinetic energy, comprising positive contributions\nfrom the angular and radial variations parallel to the sur-\nface and negative contributions from the exponentially\ndecaying part perpendicular to the surface. We study\nthese effects for the HOMO, the LUMO, and states in the\ntransport gap.\nBasis states with a) radial functions with many nodes\n(large𝑘𝑚𝑖), or b) angular functions with many nodes\n(large𝑚) have large (positive) kinetic energies in planesarXiv:2401.14937v1 [cond-mat.mes-hall] 26 Jan 20242\nFigure 1. Scheme of the system investigated experimentally and\nsimulated theoretically: PtPc on a thin NaCl(100) film on Au(111)\nin the STM tunnel junction. The four “arms” of the molecule are\nthe isoindole units, shown aligned with the cardinal directions.\nparallel to the surface. These basis states are combined\nwith exponential functions perpendicular to the surface,\nwhich have large negative kinetic energies, to obtain\nstates with the total kinetic energy corresponding to the\nbias. These states decay rapidly in the perpendicular di-\nrection and play a small role for the topography at the tip.\nThis effect exponentially favors components with a) small\n𝑘𝑚𝑖and b) small 𝑚, i.e., components with few nodal sur-\nfaces in the angular and radial parts of the wave function.\nFactor a) results in a significant expansion of the STM\nimage in the radial direction when the smallest significant\nvalue of𝑚is not too small. For PtPc, this phenomenon is\napplicable to both the HOMO and LUMO. To observe the\neffects of factor b), we notice that PtPc has four “arms”\nalong the𝑥- and𝑦-axis. The HOMO exhibits a total of\neight lobes, with a pair of lobes tending to align along\neach of the molecular arms. Due to factor b) this ten-\ndency diminishes during electron propagation into vac-\nuum. The two-fold degenerate LUMOs collectively form\na set of eight lobes. Due to factor b) its image markedly\naccumulates intensity between the molecular arms in the\n𝑥±𝑦-directions, even though there are no atoms in the\nunderlying molecule in these directions. While factor b)\ntends to position the eight lobes of the HOMO at equal\nangles, the LUMO has a tendency to overshoot.\nPropagation through the transport gap becomes impor-\ntant for electron and photon manipulations. Factors a)\nand b) mentioned above exponentially favor contributions\nfrom basis functions with few angular and radial nodes.\nIn Ref. 51 we emphasized how propagation through the\nAu-NaCl-PtPc system also favors this type of basis states.\nConsequently, even for energies very close to the HOMO\nor LUMO, the in-gap image differs qualitatively from the\nHOMO and LUMO images. While the HOMO, as well as\nthe two degenerate LUMOs together, have eight lobes, the\nin-gap image has only four lobes. Despite the HOMO hav-\ning no intensity right on the 𝑥- or𝑦-axes, and the LUMO\nhaving little intensity as well, the image in the gap has its\nfour lobes centered on these axes. Specifically, in an ex-\npansion in terms of MOs, we find that the lowest 𝜋orbital,\nwith no nodal planes perpendicular to the surface, has a\nlarge weight due to its small kinetic energy parallel to the\nsurface. More detailed information about the model canbe found in the Supplementary Information (SI). The next\nsection provides a detailed description of the theoretical\nformalism. In the following section, we present the exper-\nimental and theoretical results.\nTheoretical formalism\nWe study a PtPc molecule adsorbed on a NaCl(100) film\nof three atomic layers on a Au(111) substrate. As in earlier\nwork [50, 51], we use a TB formalism to describe the Au-\nNaCl-PtPc system, essentially following prescriptions of\nHarrison [52] (see SI). The resulting Hamiltonian is\n𝐻=∑︁\n𝑖𝜎𝜀Au\n𝑖𝑛Au\n𝑖𝜎+∑︁\n𝑖𝜎𝜀NaCl\n𝑖𝑛NaCl\n𝑖𝜎+182∑︁\n𝑖=1∑︁\n𝜎𝜀PtPc\n𝑖𝑛PtPc\n𝑖𝜎\n+∑︁\n𝑖𝑗𝜎[𝑉Au−NaCl\n𝑖𝑗(𝑐Au\n𝑖𝜎)†𝑐NaCl\n𝑗𝜎+ℎ.𝑐.] (1)\n+∑︁\n𝑖𝑗𝜎[𝑉NaCl−PtPc\n𝑖𝑗(𝑐NaCl\n𝑖𝜎)†𝑐PtPc\n𝑗𝜎+ℎ.𝑐.]\nThe first three terms encode the energies of the Au states,\nthe NaCl states, and the PtPc states, respectively. The last\ntwo terms encode the coupling between Au and NaCl, and\nbetween NaCl and PtPc, respectively. This Hamiltonian\nis used for describing the Au-NaCl-PtPc system inside a\nmatching plane at 𝑧0=1Å outside the nuclei of the PtPc\nmolecule. The calculations account for the Cl character of\nthe states in the NaCl band gap due to the Cl character of\nthe NaCl valence and conduction band [50, 53–55].\nBeyond the matching plane, we introduce cylindrical\ncoordinates with the radial coordinate, 𝜌, the azimuthal\nangle,𝜙, and a coordinate perpendicular to the surface, 𝑧.\nWe follow Tersoff and Hamann[10, 11] and assume that\nthe important orbital on the tip is an 𝑠orbital. The tip has\na local radius of curvature 𝑅. The center of the curvature\nis located a distance 𝑧from the surface, which is also the\ncenter of the tip 𝑠orbital. The tip apex is then at a distance\n𝑧−𝑅. In the following, we do not specify the value of 𝑅,\nand present our results as a function of 𝑧, corresponding\nto a tip distance of 𝑧−𝑅. Furthermore, we neglect changes\nof the potential in the NaCl-PtPc-vacuum system induced\nby the tip.\nUnder these assumptions, the tip does not explicitly fac-\ntor into the calculations. Coulomb interactions are also\nnot explicitly taken into account. However, the HOMO\nand LUMO positions are adjusted to the measured val-\nues determined from scanning tunneling spectroscopy\n(STS). As long as the important PtPc states are the neu-\ntral ground-state and states with one extra electron or one\nhole, Coulomb effects and image potential effects are im-\nplicitly accounted for by using level positions determined\nby STS. This formalism, however, is incapable of describ-\ning exciton effects. The TB solutions for substrate-barrier-\nmolecule complex are matched continuously to the vac-\nuum solution outlined below.\nFor𝑧>𝑧0, we assume that the potential 𝑉(𝜌,𝜙,𝑧)takes\nthe work function value 𝑉0=4.3eV [56] inside a cylinder3\nradius𝜌0=12Å, and positive infinity outside. Then, the\npotential𝑉(𝜌,𝜙,𝑧)can be expressed as follows:\n𝑉(𝜌,𝜙,𝑧)=/b√︂aceleftigg\n𝑉0,if𝜌≤𝜌0and𝑧≥𝑧0\n∞,if𝜌>𝜌0and𝑧≥𝑧0(2)\nThe energy zero is set at the Fermi energy of the substrate.The cylinder radius (12 Å) significantly exceeds the dis-\ntance from the cylinder axis to the outermost H atoms (7.6\nÅ) or the outermost C atoms (6.6 Å). Consequently, it is\nthen a reasonable assumption that the true wave function\nof the tunneling electrons is localized within the cylin-\nder. We introduce the Schrödinger equation for an elec-\ntron with the energy 𝜀(<𝑉0):\n/b√︂acketleftbigg\n−/√︁a√︂enleftbigg𝜕2\n𝜕𝑧2+1\n𝜌𝜕\n𝜕𝜌+𝜕2\n𝜕𝜌2/√︁a√︂en√︂ightbigg\n+1\n𝜌2𝜕2\n𝜕𝜙2+𝑉(𝜌,𝜙,𝑧)/b√︂acket√︂ightbigg\n𝜓(𝜌,𝜙,𝑧)=𝜀𝜓(𝜌,𝜙,𝑧) (3)\nHere we have expressed energies in Rydberg units (Ryd =\n13.6 eV) and lengths are given in Bohr radii ( 𝑎0=0.529Å). A solution in vacuum outside the PtPc molecule can\nbe written as follows:\n𝜓(𝜌,𝜙,𝑧,𝜀)=∑︁\n𝑚𝑖/b√︂acketleftig\n𝑐(s)\n𝑚𝑖sin(𝑚𝜙)+𝑐(c)\n𝑚𝑖cos(𝑚𝜙)/b√︂acket√︂ightig\n𝐽𝑚[𝑘𝑚𝑖𝜌]𝑒−𝜅𝑚𝑖(𝜀)(𝑧−𝑧0)(4)\nwhere𝑚is a non-negative integer, and 𝐽𝑚is an integer\nBessel function. The coefficients 𝑘𝑚𝑖are defined such that\n𝐽𝑚[𝑘𝑚𝑖𝜌0]=0, ensuring that the wave function is zero for\n𝜌=𝜌0. To obtain the correct energy 𝜀, we require\n[𝜅𝑚𝑖(𝜀)]2=[𝑘𝑚𝑖]2−(𝜀−𝑉0). (5)\nFor𝑚≥1,sin(𝑚𝜙)and for𝑚≥0𝑐𝑜𝑠(𝑚𝜙)describe the\nazimuthal angle 𝜙dependence, while the Bessel functions\n𝐽𝑚[𝑘𝑚𝑖𝜌],𝑖=1,2,... describe the radial behavior for a\ngiven𝑚value.\nThe factor exp[−2𝜅𝑚𝑖(𝜀)𝑧]describes the exponential\ndecay of the square of the wave function in the 𝑧-\ndirection. This factor introduces a strong energy depen-\ndence via the energy dependence of 𝜅𝑚𝑖(𝜀)in Eq. (5). The\nleading contribution results from exp[−2𝜅01(𝜀)(𝑧−𝑧0)].\nThis factor varies over two orders of magnitude as the en-\nergy ranges from the bottom of the electronic transport\ngap, at−1.3eV, to the top, at 1.7eV. Additional relative\nvariations among the different components are induced\nbyexp{−2[𝜅𝑚𝑖(𝜀)−𝜅01(𝜀)](𝑧−𝑧0)}, as discussed in Ta-\nble I below. These considerations are rooted in the as-\nsumption that the potential reaches its vacuum value di-\nrectly outside the molecule. In this approach we have ne-\nglected the potential from the tip. This potential is sub-\nstantially higher in the study of the LUMO compared to\nthat for the HOMO. If this potential were included in the\ncalculation, the pronounced enhancement of the LUMO\nversus the HOMO would be substantially smaller. Fur-\nthermore, the potential is lowered outside the molecule,\nwhich we have neglected. The image potential also con-\ntributes beyond what is implicitly included in the HOMO\nand LUMO positions. While these factors should alleviate\nthe strong energy dependence, they are neglected here.\nThese dependencies exist alongside the effects that occur\nduring the propagation through the NaCl buffer and the\nPtPc molecule.RESULTS AND DISCUSSION\nSTM measurements\nThe experiments were performed in a home-built low-\ntemperature ( 𝑇∼5K) STM operated under ultra-high\nvacuum at a base pressure of ≤1×10−11mbar [57].\nThe preparation of single molecules of PtPc adsorbed on 3\nmonolayer NaCl(100) on Au(111) follows a procedure de-\nscribed previously [49, 51]. All experimental data shown\nhere are measured in constant-height mode. The data in\nFig. 2 is presented for comparison with the theoretical cal-\nculations discussed below. PtPc molecules are sensitive to\nthe local inhomogeneity that results from the incommen-\nsurability of NaCl(100) with the herringbone reconstruc-\ntion of Au(111). Consequently, some molecules are more\nstably adsorbed and may thus be preferentially selected\nby the experimentalist. Additionally, the substrate causes\nsome molecules to adsorb with the two-fold degeneracy of\nthe LUMO lifted, enabling the imaging of one of the LU-\nMOs in isolation. We chose tip-sample distances that en-\nable imaging molecules with a good signal-to-noise ratio\nwhile maintaining stable scanning conditions. For exces-\nsively high currents and excessively short distances from\nthe tip apex, molecules tend to move laterally, creating\nstreaks in the image or even hop irreversibly to the tip.\nThe onset of this instability typically occurs at a tip-\nmolecule distance of roughly 3 Å, accompanied by a maxi-\nmum tunnel current of 200 pA, with substantial variability\nin both values. These values strongly depend on the ap-\nplied voltage. In the experiments presented here, the con-\ncern is not only to minimize the distance to the molecule\nbut also to maximize the image contrast of the radial com-\nponents of the molecular features. The experimental im-\nages are intended to elucidate the effects of vacuum prop-4\nFigure 2. Typical differential conductance (d 𝐼/d𝑉) spectrum\non PtPc indicating the energies of the molecular frontier or-\nbitals with broad local maxima at −1.35V (HOMO) and+2.05\nV (LUMO). Bottom row: Typical constant-height STM images of\nthe frontier orbitals and of the feature appearing in the trans-\nport gap of the molecule. The color scale is valid for all figures\nwith either measured (current) or calculated (DOS = density of\nstates) images and spans the range from the minimum to the\nmaximum of the absolute tunnel current. Here, the maxima of\nabsolute currents are 66 pA (HOMO), 5.3 pA (gap feature), and\n187 pA (LUMO). All images show an area of 20×20Å2.\nagation in the electron tunneling process.\nAs we will demonstrate below, the constant height im-\nages (Fig. 2) obtained using the described method match\nvery well with the calculations for molecule-tip distances\nranging from 4-8 Å. It is important to recall that we define\nthe apex of the tip to be at 𝑧−𝑅, where𝑅is curvature of\nthe tip. Note also that the color bar used in Fig. 2 is com-\nmon to all 2D plots in this paper. Further details on its\nconstruction are given in SI.\nTheoretical results\nWe now present calculated images of a PtPc molecule.\nFig. 3 shows the results for the HOMO ( 𝜀=−1.3eV)\nfor different values of 𝑧−𝑧0. The molecule has four\n“arms” along the 𝑥- and𝑦-axes, with the HOMO featuring\nnodes along these axes, resulting in eight lobe-like fea-\ntures as a function of azimuthal angle. These lobes are\nparticularly visible at larger 𝑧values. The HOMO image\nrapidly expands with increasing 𝑧and undergoes notice-\nable changes in its shape. As discussed in the experimen-\ntal section above, some 𝑧values are too small to be ex-\nperimentally accessible, yet observing the evolution with\nvarying𝑧yields valuable insights.\nFig. 4 display the results for the two degenerate LU-\nMOs (𝜀=1.7eV) separately, while Fig. 5 presents their\nsum, corresponding to the typical topographical image\nobtained by STM. One LUMO has most of its weight close\nto the𝑥-axis, with a node along the 𝑥-axis. The other\nLUMO has its weight distributed analogously along the\n𝑦-axis. The sum of the images results in eight features as\na function of angle. Interestingly, for 𝑧−𝑧0=0, the image\nbears a striking resemblance to the HOMO image at thesame distance. As 𝑧increases, the LUMO image expands\nradially in a manner similar to the HOMO image.\nThe behavior of the angular features is quite different.\nNote the behavior near the high symmetry axes. For the\nHOMO, the eight angular features group into four pairs\nthat straddle the positive and negative 𝑥- and𝑦-axes. As𝑧\nis increased, the features shift somewhat away from the 𝑥-\nand𝑦axis, with the maxima moving towards, e.g., ±22.5◦\nand minima at, e.g., ±45◦. On the other hand, for the\nLUMO, this angular shift is much more important. Sub-\nstantial weight is built up at, e.g., ±45◦. Strikingly, this\nimplies that weight accumulates in directions where the\nunderlying molecule has no atoms.\nOne might have expected that for energies between the\nHOMO and LUMO the image would show similarities to\nthese MOs. It is remarkable, however, that even for en-\nergies very close to either the HOMO or LUMO, the im-\nages look qualitatively distinct from their corresponding\norbitals. In contrast to the eight angular features of the\nHOMO and LUMO, only four features appear as a func-\ntion of angle at voltages in the electronic transport gap.\nThe HOMO image has no weight, and the image compris-\ning the two degenerate LUMOs has little weight along the\n𝑥- and𝑦-axes. In stark contrast, in-gap images have sub-\nstantial weight built up along these axes. Fig. 6 shows that\nthere is no clear radial expansion of the images with in-\ncreasing𝑧. This behavior stands in stark contrast to the\nclear expansion observed for the HOMO and LUMO.\nSimilar results have been obtained in experiments for\nCuPc by Uhlmann et al. [58] for the HOMO, LUMO, and\nthe energies in between. Our theoretical results for the\nHOMO and LUMO at 𝑧−𝑧0=4Å(𝑧=5Å)closely\nresemble the theoretical results obtained by Siegert et al.\n[45] for CuPc at the same tip-sample distance. However,\ntheoretical results for energies lying between the HOMO\nand LUMO for CuPc, as reported in Ref. 44, significantly\ndeviate from both our theoretical and experimental re-\nsults for PtPc. These results also deviate from experimen-\ntal findings by Uhlmann et al. for CuPc [58]. The reason\nis probably that the calculations in Ref. 44 did not include\nthe lower-energy 𝜋-orbitals, which we demonstrate in the\nfollowing to play a crucial role for topography images ob-\ntained at energies in the transport gap of the molecule.\nDetailed analysis of electron propagation effects\nWe now turn our attention to the propagation of\nelectrons through the vacuum region between the PtPc\nmolecule and the tip. This is done by examining the in-\ntricate details of the coefficients that describe the electron\nwave function. In a forthcoming study, we will address\nthe role of the propagation through the buffer (NaCl).\nAngular distortions and radial expansion\nWe first discuss the exponential factor exp[−𝜅𝑚𝑖(𝜀)𝑧],\nwhich describes the 𝑧-dependence in Eq. (4) and Eq. (5) of5\nFigure 3. Calculated images at the energy of the HOMO ( 𝜀=−1.3eV) for different distances from the molecular plane, 𝑧−𝑧0= 0 Å\n(A), 4 Å (B), 8 Å (C) and 12 Å (D). All panels show an area of 20×20Å2. In panel D the molecular structure is superposed.\nFigure 4. Calculated images of the two individual LUMO degeneracy lifted orbitals ( 𝜀=1.7eV) at the distances 𝑧−𝑧0= 0 Å (panels\nA and C) and 12 Å (panels B and D). All panels show an area of 20×20Å2. In panel D the molecular structure is superposed. Panel\nE shows a constant height STM image at a bias of only 1.35 V for a PtPc molecule for which the LUMO degeneracy happened to be\nlifted, probably due to a nearby substrate defect.\nthe wave function amplitude. Table I presents the values\nof𝜅𝑚𝑖(𝜀). In parentheses we show the relative damping\nof the intensity of components 𝑚𝑖\nexp{−2[𝜅𝑚𝑖(𝜀)−𝜅01(𝜀)](𝑧−𝑧0)} (6)\nrelative to the 𝑚=0and𝑖=1component for(𝑧−𝑧0)=8Å\nand𝜀=0, close to the middle of the gap and well below\nthe vacuum level at 𝑉0=4.3eV. The table illustrates that\ni) components for small 𝑚values are less damped and are\nthus strongly favored. For each 𝑚value, ii) components\nwith small values of 𝑖are strongly favored. As discussed\nin the introduction, these effects emerge because a larger\nvalue of𝑚or𝑖leads to a larger kinetic energy in a plane\nparallel to the surface. Since we study electrons of a given\nenergy𝜀, the kinetic energy perpendicular to the surface\nis then correspondingly more negative and the decay in\nthe𝑧-direction exponentially more rapid.\nWe now turn to the Bessel functions describing the ra-\ndial behavior that is the dependence on 𝜌. Fig. 7 shows\nthese functions for 𝑚= 0, 1 and 4. With exception of\nsmall values of 𝑚, for e.g.,𝑚=0and𝑚=1, the function\n𝐽𝑚(𝑘𝑚1𝜌)primarily characterizes the outer regions of the\nTable I.𝜅𝑚𝑖(𝜀)(Å−1) determining the decay of the 𝑧-dependent\nfunctions [Eq. (5)]. The numbers in parenthesis show the rela-\ntive damping exp{−2[𝜅𝑚𝑖(𝜀)−𝜅01(𝜀)](𝑧−𝑧0)}of the intensity\nof a component relative to the 𝑚=0and𝑖=1component for\n𝑧−𝑧0=8Å and𝜀=0.\n𝑖 𝑚 =0𝑚=1𝑚=2𝑚=4𝑚=8\n1 1.08 (1.00) 1.11 (.637) 1.15 (.358) 1.24 (.084) 1.47 (.002)\n2 1.16 (.294) 1.21 (.122) 1.27 (.046) 1.41 (.005) 1.71 (.000)\n3 1.28 (.039) 1.36 (.012) 1.44 (.003) 1.60 (.000) 1.95 (.000)\n4 1.45 (.003) 1.54 (.001) 1.63 (.000) 1.81 (.000) 2.19 (.000)molecule, encompassing the outermost C atoms and the\nsurrounding space outside these atoms. Conversely, the\ninner regions are predominantly characterized by 𝐽𝑚(𝑘𝑚𝑖)\nfor𝑖>1.\nBelow we demonstrate that the two effects previously\ndiscussed, i) and ii), along with the the corresponding be-\nhavior of the Bessel functions, are key for understand-\ning the substantial distortion occuring in vacuum for the\nHOMO, LUMO and the in-gap images.\nHOMO\nWe first discuss the HOMO in more detail. Given its\neight features as a function of angle, it is described by an-\ngular functions with 𝑚=4𝜈,𝜈=1,2,.... The relative\nweight,\n𝑆𝑚=∑︁\n𝑖[(𝑐(s)\n𝑚𝑖)2+(𝑐(c)\n𝑚𝑖)2]𝑒−2[𝜅𝑚𝑖(𝜀)−𝜅01(𝜀)](𝑧−𝑧0),(7)\nis shown in Fig. 8 as a function of 𝑚for𝜀=𝜀HOMO , high-\nlighting the dominance of the 𝑚=4terms over higher\nvalues of𝑚across all𝑧values.\nWe first focus on 𝑚=4. The numbers in parenthe-\nsis in Table I show that the 𝑖=1component is signifi-\ncantly less suppressed than the components with 𝑖>1as\n𝑧increases. This leads to a noticeable radial expansion of\nthe image due to relatively higher weight of 𝐽4(𝑘4𝑖𝜌)for\n𝑖=1for large𝜌than the𝑖>1components (see Fig. 7).\nConsequently, the centers of the eight HOMO lobes are\npositioned around 𝑅=7.5Å (0.63𝜌0) for𝑧−𝑧0=8Å,\nextending beyond the centers of the outermost C atoms6\nFigure 5. Calculated images of the sum of both LUMOs ( 𝜀=1.7eV) for distances, 𝑧−𝑧0of 0 Å (A), 4 Å (B), 8 Å (C), and 12 Å (D). All\npanels show an area of 20×20Å2. In panel D the molecular structure is superposed.\nFigure 6. Calculated images of the states at the substrate Fermi energy ( 𝜀=0.0eV), that is in the molecular transport gap, for\ndistances,𝑧−𝑧0of 0 Å (A), 4 Å (B), 8 Å (C), and 12 Å (D). All panels show an area of 20×20Å2. In panel D the molecular structure\nis superposed.\nat𝑅=6.6Å (0.55𝜌0)1. The maximum of the image\nis at a somewhat smaller value of 𝜌than the maximum\n(𝜌/𝜌0=0.70) of𝐽4(𝑘41𝜌/𝜌0)due to the influence of mixed\nterms such as ones with 𝑖=1and2, even for𝑧−𝑧0=8Å.\nThe𝑚=4component results in eight intense features,\nprominently positioned at angles ±22.5◦and separated by\nan angle of 45◦. The higher order components, e.g., 𝑚=8,\nmove these features pairwise together towards the 𝑥- and\n𝑦-axes, as is illustrated for the case of 𝑧=1Å. With in-\ncreasing𝑧, the influence of the 𝑚>4components di-\nminishes rapidly, causing the angular placement of the\nfeatures in each quadrant to approach ±22.5◦. However,\neven for𝑧−𝑧0=8Å, these features are weakly displaced\ntowards the 𝑥- and𝑦-axes due to the non-negligible con-\ntributions from higher 𝑚-values.\nLUMO\nWe now focus on the doubly degenerate LUMO. Fig. 5\npresents the sum of the images of both states. As men-\ntioned earlier, HOMO and LUMO appear very similar for\n𝑧−𝑧0=0, both undergoing significant expansion in the\n𝜌-direction with increasing 𝑧. However, their angular be-\nhavior diverge considerably. Notably, the summation of\nthe LUMOs builds up intensity along the angles ±45◦and\n±135◦for large values of 𝑧.\nTo understand this build up, we first notice that the\nLUMOs are described by terms sin(𝑚𝜙)andcos(𝑚𝜙)=\n1These contributions are not due to the H atoms. Suppressing the con-\ntributions from H atoms made no visible change in the image. Actu-\nally, the MOs with 𝜋character do not couple to the 1𝑠H orbital, and\n𝜎orbitals, of importance in this context, have little 1𝑠H character.sin(𝑚𝜙+90◦)for odd values of 𝑚. Let us consider the one\nLUMO with prominent features close to the 𝑥-axis, de-\nscribed by sin(𝑚𝜙)and shown in Fig. 4 A, B. The weights\nof its components with different 𝑚-values are shown in\nFig. 8. For 𝑧−𝑧0=0, the𝑚=5component has the\nmost weight. This component has its maximum weight\nat the angle 90◦/5=18◦. As a result, this intensity ap-\npears approximately at ±18◦for this LUMO, and natu-\nrally at 90◦±18◦for the other LUMO (Fig. 4A, B). The\n𝑚=3components are also relatively important, adding\nintensity that peaks at 90���/3=30◦, thereby slightly shift-\ning the overall intensity distribution away from the axes.\nThe two arms of one LUMO (at ≈±18◦) are then about\n36◦apart, while the two arms of two different LUMOs are\nabout 90◦−36◦=54◦apart. This leads to the features in\nFig. 5, displaying both LUMOs, being relatively close to\nthe𝑥- and𝑦-axes for𝑧−𝑧0=0Å.\nWe can now turn our attention to 𝑧−𝑧0=8Å. As\ndemonstrated in Fig. 8, the 𝑚=1and𝑚=3components\nare the largest. For the LUMO with arms along the 𝑥-axis,\nthese arms are shifted away from the axis, since sin(3𝜙)\nhas its maximum for a larger 𝜙compared to, e.g., sin(5𝜙),\nas evident in the right section of Fig. 4 (for 𝑧−𝑧0=12Å).\nSimilarly, for the other LUMO, the arms move away from\nthe𝑦-axis. The result is the build-up of an appreciable\nweight around, e.g., 𝜙=45◦, even though there are no\natoms in the underlying molecule in this direction.\nGap states\nFinally, we consider states in the electronic transport\ngap. These states were previously investigated in our ear-\nlier publication,[51] wherein we concentrated on electron\npropagation from the substrate out to the PtPc molecule.7\nFigure 7. Bessel functions 𝐽𝑚(𝑘𝜌/𝜌0)for𝑚= 0, 1 and 4 (from left to right) and for 𝑘=𝑘𝑚𝑖,𝑖= 1, 2, 3 and 4. The arrow in the right\nfigure marks the positions of the outermost C atoms.\nFigure 8. Relative weights of different 𝑚components in images\nof the HOMO ( 𝜀=−1.3eV) and LUMO ( 𝜀=1.7eV) at distances\n𝑧−𝑧0=0and8Å. For each value of 𝑧, the results are normalized\nto the maximum value for that 𝑧. For the HOMO this is for 𝑚=\n4, for both values of 𝑧, while for the LUMO this is 𝑚=5for\n𝑧−𝑧0=0and𝑚=1for𝑧−𝑧0=8Å.\nHowever, in the current work we focus on the propaga-\ntion through the vacuum. A striking result in Ref. 51 was\nthe rather simple image with four lobes along the 𝑥- and\n𝑦-axes with no resemblance to the HOMO or LUMO, even\nthough the measurement was performed at for energies\nrelatively close to the HOMO or LUMO. For the example\nof the state at the substrate Fermi energy (0.0 eV), Fig. 6 il-\nlustrates that the weight of states within the transport gap\nof the molecule does not experience significant outward\nexpansion. This is in contrast to the behavior observed in\nthe case of the HOMO and the LUMO.\nAs discussed in Ref. 51, inside the PtPc molecule the\nstates within the molecule’s gap are to a good approxima-\ntion linear combinations of bound PtPc states. Very close\nin energy to the HOMO, the HOMO dominates the gap\nstate, producing a HOMO-like image (not shown here).\nHowever, moving up to energies above the HOMO, bound\nPtPc states below the HOMO and with small values of 𝑚\nrapidly begin to contribute significantly to the image for\n𝑧=𝑧0. The smallest 𝑚value contributing to the HOMO\nis𝑚=4, as shown in Table I, which efficiently under-\ngoes exponential suppression as 𝑧is increased. This sup-\npression then causes states with 𝑚=0, 1 or 2 to gain\nrelative prominence. The exponential suppression effect\nremains influential even for energies a few tenths of an eV\nabove the HOMO, rendering components with smaller 𝑚values predominant in the image. Despite the correspond-\ning molecular states being considerably lower in energy\nthan the HOMO, these smaller 𝑚components dominate\ndue to this strong suppression. Analogous phenomena oc-\ncur at energies below the LUMO. This is illustrated by the\nsharp and reversible transition from imaging the gap fea-\nture and the LUMO within a few tens of meV as shown\nin the STM measurements in Fig.9 B, C, D. The theoret-\nical result (panel A) corroborates that this transition in-\ndeed happens very rapidly as a function of tunneling en-\nergy. Given the great importance of orbitals below the\nHOMO and LUMO for images within the transport gap, it\nis not surprising that these images could not be described\nin Ref. 44, where these orbitals were not included.\nWe observe that the discussed effects render MOs be-\nlow the HOMO very important for electron propagation\nat energies within the gap. However, there is an additional\nphysical mechanism at play. Most of the MOs above the\nLUMO have many nodal surfaces, since they have to be or-\nthogonal to lower states. Consequently, these MOs then\ntend to have large kinetic energies parallel to the surface.\nFor a given bias voltage, they tend to have a very negative\nkinetic energy perpendicular to the surfaces, and there-\nfore decay rapidly with increasing 𝑧values. This intro-\nduces a fundamental asymmetry between MOs well below\nversus most of those well above the gap.\nFor the HOMO and LUMO images, we found that the\nimage strongly expands radially as 𝑧−𝑧0increases from\n0 to 12 Å. By comparison with Fig.6 it becomes apparent\nthat a similar strong expansion does not occur for the gap\nstates. For the HOMO, the expansion was attributed to\nthe increasing dominance of the extended 𝑖=1states as\n��increases. A similar predominance of small values of 𝑖\nhappens also for gap states. However, for these states, the\n𝑚=0and𝑚=1components are particularly important,\nand within this range of 𝑚values, the𝑖=1and the𝑖=2\nstates have a comparable radial extension (see Fig. 7)2.\n2If a much larger cylinder radius 𝜌0is used, these arguments become\nmore complicated, since the basis states for small values of 𝑖are then\nvery extended. The numerical results for the images, however, are\nessentially unchanged.8\nFigure 9. Constant height STM images (B, C, D) illustrating the sharp transition between the gap feature and the LUMO. The bias\nvoltages are indicated in each panel. Panel A shows theoretical results for the bias 1.5 V. All panels show an area of 20×20Å2.\nVacuum propagation strongly favors certain MOs\nTo gain a more comprehensive understanding of the\norigin of images for energies within the PtPc transport\ngap, we continue the PtPc MOs at specific energies 𝜀into\nthe vacuum region. The continuation, Ψ𝜈(𝜌,𝜙,𝑧,𝜀), for\nMO𝜈, is formulated in terms of the basis functions em-\nployed in Eq. (4). The function Ψ𝜈matches the MO 𝜈\ncontinuously at 𝑧=𝑧0. Next, we consider a solution ˜Ψ𝛼\nof the combined system comprising the substrate-buffer-\nPtPc and the vacuum at an energy 𝜀𝛼. Within the vacuum\nwe write the solution as\n˜Ψ𝛼(𝜌,𝜙,𝑧)=182∑︁\n𝜈=1𝑎(𝛼)\n𝜈Ψ𝜈(𝜌,𝜙,𝑧,𝜀𝛼) (8)\nWe then calculate the overlap integral, 𝑆𝜈,𝜇(𝑧)of two func-\ntionsΨ𝜈andΨ𝜇, over a plane parallel to the surface and\nat a distance 𝑧from the molecule. Finally, we introduce a\ndensity matrix\n𝑃𝜈,𝜇=𝐶∑︁\n−0.8≤𝜀𝛼≤1.2[𝑎𝛼\n𝜈]∗𝑎𝛼\n𝜇𝑆𝜈,𝜇 (9)\nwhere we sum over all states within the energy range\nfrom−0.8𝑒𝑉to1.2𝑒𝑉in the transport gap. To ensure\nnormalization, the constant 𝐶is chosen so that\n182∑︁\n𝜈,𝜇=1𝑃𝜈𝜇(𝑧)=1. (10)\nIt is notable that some of the off-diagonal terms might\nbe negative. This matrix illustrates how the propagation\nthrough the different MOs contributes to the image.\nFig. 10 shows the 𝑧-dependence of important elements\nof𝑃𝜈𝜇. Some of the corresponding orbitals are shown in\nFig. 11. Remarkably, even the lowest ( 𝜎1) state, at -26 eV,\ncontributes more significantly than the HOMO (-1.3 eV)\nand the LUMO (1.7 eV). As previously discussed, this phe-\nnomenon stems from to the lowest 𝜎orbital having only a\nsmall positive kinetic energy in the plane of the molecule,\ndue to the absence of nodes in this plane. For a given total\nenergy𝜀, the kinetic energy perpendicular to the molecule\nis then not very negative, and the exponential decay in\nvacuum not so rapid.\nSimilar effects happen for the lowest 𝜋orbital,𝜋1, al-\nthough its smaller energy difference to 𝜀𝐹=0results\nin a significantly larger contribution. The 𝜋2orbital hasa radial node (as seen in Fig. 11), resulting in a some-\nwhat larger kinetic energy parallel to the surface, a more\nnegative kinetic energy perpendicular to the surface, and\nconsequently, a more rapid decay in the vacuum. Panel\nD shows the calculated image at a value just above the\nHOMO energy. It indeed shows great similarities with a\nlinear combination of the orbitals in panels B and C.\nThe HOMO has components with 𝑚=4or higher𝑚\nvalues. Table I shows that the 𝑚=4components decay\nrapidly. As shown in the SI, most of the HOMO weight\nis furthermore in components with fairly large values of\n𝑖∼4. This configuration results in the kinetic energy par-\nallel to the surface being very large, and a corresponding\nrapid decay outside the molecule. We have also summed\nthe diagonal contributions from 7 orbitals (“7 orbital”),\nwithin a calculated energy range of approximately 0.1eV,\nand around 1.8eV below the HOMO. These orbitals have\nmuch more weight than the HOMO and the LUMO, ren-\ndering them potentially important in some specific con-\ntexts. The SI shows that the leading components of the\nLUMO are also characterized by large values of 𝑚and\n𝑖, and the corresponding rapid decay in vacuum. How-\nFigure 10. Theory predicted weights 𝑃𝜈,𝜇(𝑧)describing contri-\nbutions to the image due to products of tails of important MOs.\nThe product was integrated over a plane at 𝑧−𝑧0and over the\nbias range (-0.8 eV ≤𝜀≤1.2eV). We show diagonal elements,\n𝑃𝜈𝜈, for the lowest 𝜋-orbital (𝜋1,𝜀−𝜀𝐹=−8eV), a𝜋-orbital\nwith an approximately cylindrical nodal surface ( 𝜋2,𝜀−𝜀𝐹=−5\neV), the diagonal terms summed over 7 orbitals about 1.8 eV be-\nlow the HOMO as well as the HOMO and LUMO. The latter are\nplotted twice, each with the indicated enhancement factor. We\nalso show two off-diagonal contributions involving the lowest 𝜎\norbital (𝜎1,𝜀−𝜀𝐹=−26eV) and𝜋1-orbital as well as the 𝜋1and\n𝜋2orbitals.9\nFigure 11. Important MOs for images in the gap. The panels show the absolute value of (A) the lowest 𝜎MO (𝜎1in Fig. 10), (B) the\nlowest𝜋MO (𝜋1), and (C) a 𝜋MO with one “radial” node ( 𝜋2). The second MO ( 𝜋1) is positive everywhere (for 𝑧>0), while the\nfirst MO (𝜎1) is slightly negative at parts very close to the Pt nucleus ( ≤0.7Å) due to the Pt 3𝑧2−𝑟2d-orbital. We emphasize the\nsimilarity of the 𝜎1and𝜋1orbitals. The inner parts of 𝜋2are negative and the outer parts are positive. In constructing the images from\nFig. 3 to Fig. 6 the coefficient of the 𝜎orbital has the opposite sign to the coefficients of the two 𝜋-orbitals which leads to an overall\ncancellation of weight in the inner parts of the image. The orbitals were calculated at 𝑧−𝑧0=0Å. Panel D shows the calculated\nimage at -1.0 V (a value just above the HOMO energy at -1.3 V) for 𝑧−𝑧0=6Å. All panels show an area of 20×20Å2.\never, the LUMO also has components with small values\nof𝑚and𝑖. Despite these components’ small amplitudes\nat the matching surface ( 𝑧−𝑧0=1Å), their relatively\nslow decay in vacuum results in their significant contri-\nbution. Furthermore, much of the contribution from the\nLUMO comes from the upper part of the studied energy\ninterval, where the wave functions generally decay more\nslowly. Consequently, the LUMO contribution ( 4×10−4\nfrom both LUMOs) is not as drastically reduced during the\npropagation through vacuum as observed forthe HOMO\ncontribution ( 2×10−5). The observation that the most im-\nportant contributions stem from orbitals lacking angular\nnodes (as shown in Fig. 10) elucidates the shapes of the\nimages presented in Fig. 6.\nSUMMARY\nWe have analysed the transport of electrons through\nthe vacuum region in STM studies of PtPc adsorbed on\na thin NaCl film atop an Au substrate. For propagation\nthrough vacuum, basis functions with few nodes in the\nangular and radial directions parallel to the surface are\nfavored. Such functions exhibit small kinetic energy in\nthese directions, subsequently leading to a less negative\nkinetic energy perpendicular to the surface at a given bias\nvoltage. This results in their slower exponential decay\nin the vacuum region. These effects are directly influ-\nenced by the thickness of the vacuum layer, i.e., the dis-\ntance to the tip. When considering the HOMO or LUMO,\nboth experimental and theoretical analyses reveal a sub-\nstantial radial expansion of the image. This expansion is\nattributable to the dominance of vacuum basis functions\nthat lack radial nodes. In the case of the LUMO, there is\nalso a substantial angular distortion, due to the additional\nemphasis of basis functions with few angular nodes. This\nleads to a substantial weight along the diagonals 𝑥=𝑦and\n𝑥=−𝑦, despite the underlying molecule having no atoms\nat these positions. The analysis moreover demonstrates\nthat the low resolution observed for orbitals with many\nnodes has its physical origin in the electron propagation\ninto vacuum and is not exclusively imposed by the limited\nspatial resolution of the STM tip. For propagation withinthe transport gap, the image is due to a linear combina-\ntion of waves originating from many MOs. Notably, the\nsignificance of HOMO and LUMO quickly reduces when\nentering the transport gap, already at energies just a few\ntenths of an eV above the HOMO or below the LUMO en-\nergy, leading to a radical change in the resulting STM to-\npography image. Waves characterized by few angular and\nradial nodes then make a large contribution. Particularly,\nthis accentuates the role of 𝜋-MOs located energetically\nwell below the HOMO. These intricate details provide a\nconsistent and rational basis for understanding the inves-\ntigative capabilities of the STM.\n[1] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Applied\nPhysics Letters 40, 178 (1982).\n[2] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Physical\nReview Letters 49, 57 (1982).\n[3] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Surface\nScience Letters 131, L379 (1983).\n[4] G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, Physical\nReview Letters 50, 120 (1983).\n[5] G. Binnig and H. Rohrer, Reviews of Modern Physics 59,\n615 (1987).\n[6] D. Drakova, Reports on Progress in Physics 64, 205 (2001).\n[7] W. A. Hofer, A. S. Foster, and A. L. Shluger, Reviews of Mod-\nern Physics 75, 1287 (2003).\n[8] F. Evers, R. Korytár, S. Tewari, and J. M. van Ruitenbeek,\nReviews of Modern Physics 92, 035001 (2020).\n[9] C. J. Chen, Introduction to Scanning Tunneling Microscopy:\nThird Edition (Oxford University Press, Oxford, New York,\n2021).\n[10] J. Tersoff and D. R. Hamann, Physical Review Letters 50,\n1998 (1983).\n[11] J. Tersoff and D. R. Hamann, Physical Review B 31, 805\n(1985).\n[12] N. García, C. Ocal, and F. Flores, Physical Review Letters\n50, 2002 (1983).\n[13] A. Baratoff, Physica B+C Proceedings of the 4th General\nConference of the Condensed Matter Division of the EPS,\n127, 143 (1984).\n[14] E. Stoll, Surface Science 143, L411 (1984).\n[15] N. D. Lang, Physical Review Letters 55, 230 (1985).\n[16] N. D. Lang, Physical Review Letters 56, 1164 (1986).10\n[17] M. S. Chung, T. E. Feuchtwang, and P. H. Cutler, Surface\nScience 187, 559 (1987).\n[18] W. Sacks, S. Gauthier, S. Rousset, J. Klein, and M. A. Esrick,\nPhysical Review B 36, 961 (1987).\n[19] W. Sacks, S. Gauthier, S. Rousset, and J. Klein, Physical Re-\nview B 37, 4489 (1988).\n[20] C. J. Chen, Journal of Vacuum Science & Technology A 6,\n319 (1988).\n[21] C. J. Chen, Physical Review Letters 65, 448 (1990).\n[22] C. J. Chen, Physical Review B 42, 8841 (1990).\n[23] W. Sacks and C. Noguera, Physical Review B 43, 11612\n(1991).\n[24] W. Sacks and C. Noguera, Journal of Vacuum Science &\nTechnology B: Microelectronics and Nanometer Structures\nProcessing, Measurement, and Phenomena 9, 488 (1991).\n[25] D. Šestović and M. Šunjić, Physical Review B 51, 13760\n(1995).\n[26] J. Bardeen, Physical Review Letters 9, 147 (1962).\n[27] M. Bode, R. Pascal, and R. Wiesendanger, Zeitschrift für\nPhysik B Condensed Matter 101, 103 (1996).\n[28] A. D. Gottlieb and L. Wesoloski, Nanotechnology 17, R57\n(2006).\n[29] A. N. Chaika, S. S. Nazin, V. N. Semenov, S. I. Bozhko,\nO. Lübben, S. A. Krasnikov, K. Radican, and I. V. Shvets,\nEurophysics Letters 92, 46003 (2010).\n[30] K. Palotás, G. Mándi, and L. Szunyogh, Physical Review B\n86, 235415 (2012).\n[31] V. Grushko, O. Lübben, A. N. Chaika, N. Novikov,\nE. Mitskevich, A. Chepugov, O. Lysenko, B. E. Murphy, S. A.\nKrasnikov, and I. V. Shvets, Nanotechnology 25, 025706\n(2013).\n[32] G. Mándi and K. Palotás, Physical Review B 91, 165406\n(2015).\n[33] J. Repp, G. Meyer, S. M. Stojković, A. Gourdon, and\nC. Joachim, Physcal Review Letters 94, 026803 (2005).\n[34] Ó. Paz and J. M. Soler, physica status solidi (b) 243, 1080\n(2006).\n[35] G. Teobaldi, M. Peñalba, A. Arnau, N. Lorente, and W. A.\nHofer, Physical Review B 76, 235407 (2007).\n[36] E. T. R. Rossen, C. F. J. Flipse, and J. I. Cerdá, Physical Re-\nview B 87, 235412 (2013).\n[37] R. Zhang, Z. Hu, B. Li, and J. Yang, The Journal of Physical\nChemistry A 118, 8953 (2014).\n[38] A. Gustafsson, Theoretical Modeling of Scanning Tunneling\nMicroscopy , Ph.D. thesis, Linnéuniversitetet, Växjö, Sweden\n(2017).\n[39] A. Gustafsson and M. Paulsson, Physical Review B 93,\n115434 (2016).[40] N. Okabayashi, A. Gustafsson, A. Peronio, M. Paulsson,\nT. Arai, and F. J. Giessibl, Physical Review B 93, 165415\n(2016).\n[41] A. Gustafsson, N. Okabayashi, A. Peronio, F. J. Giessibl, and\nM. Paulsson, Physical Review B 96, 085415 (2017).\n[42] A. Gustafsson and M. Paulsson, Journal of Physics: Con-\ndensed Matter 29, 505301 (2017).\n[43] B. Siegert, A. Donarini, and M. Grifoni, Beilstein Journal of\nNanotechnology 6, 2452 (2015).\n[44] A. Donarini, B. Siegert, S. Sobczyk, and M. Grifoni, Physical\nReview B 86, 155451 (2012).\n[45] B. Siegert, A. Donarini, and M. Grifoni, Physical Review B\n93, 121406 (2016).\n[46] X. W. Tu, G. Mikaelian, and W. Ho, Physical Review Letters\n100, 126807 (2008).\n[47] G. Chen, Y. Luo, H. Gao, J. Jiang, Y. Yu, L. Zhang, Y. Zhang,\nX. Li, Z. Zhang, and Z. Dong, Physical Review Letters 122,\n177401 (2019).\n[48] A. Farrukh, X.-j. Tian, F.-f. Kong, Y.-j. Yu, S.-h. Jing,\nG. Chen, Y. Zhang, Y. Liao, Y. Zhang, and Z.-c. Dong, Chi-\nnese Journal of Chemical Physics 34, 87 (2021).\n[49] A. Grewal, Single molecule fluorescence and phosphorescence\nstudies using a scanning tunneling microscope , Ph.D. the-\nsis, École polytechnique fédérale de Lausanne, Lausanne\n(2022).\n[50] C. C. Leon, A. Grewal, K. Kuhnke, K. Kern, and O. Gunnars-\nson, Nature Communications 13, 981 (2022).\n[51] A. Grewal, C. C. Leon, K. Kuhnke, K. Kern, and O. Gunnars-\nson, ACS Nano 17, 13176 (2023).\n[52] W. Harrison, Elementary Electronic Structure (World Scien-\ntific Publishing, Singapore, 1999).\n[53] T. D. Clark and K. L. Kliever, Phys. Lett. A 27, 167 (1968).\n[54] P. K. de Boer and R. A. de Groot, American Journal of\nPhysics 67, 443 (1999).\n[55] F. E. Olson, M. Persson, J. Repp, and G. Meyer, Physical Re-\nview B 71, 075419 (2005).\n[56] Z. Li, H.-Y. T. Chen, K. Schouteden, E. Janssens, C. V. Hae-\nsendonck, P. Lievens, and G. Pacchioni, Nanoscale 7, 2366\n(2015).\n[57] K. Kuhnke, A. Kabakchiev, W. Stiepany, F. Zinser, R. Vo-\ngelgesang, and K. Kern, Review of Scientific Instruments\n81, 10.1063/1.3480548 (2010).\n[58] C. Uhlmann, I. Swart, and J. Repp, Nano Letters 13, 777\n(2013).Supplementary Information for\nScanning Tunnelling Microscopy for Molecules: Effects of Electron\nPropagation into Vacuum\nTight-binding model – NaCl(100) on Au(111)\nWe use the same model for NaCl on Au as in our earlier\nwork [1]. This model consists of a NaCl(100) three-layer\nfilm on a Au(111) substrate. The NaCl film contains 9×9×4\natoms per layer. We use three different clusters represent-\ning the Au substrate and average the results. The three Au\nclusters have four, six, or eight layers with 1020, 780, and\n572 Au atoms per layer, respectively. In total there are\nthen 486 Na, 486 Cl, and 4080, 4680, or 4576 Au atoms.\nWe impose periodic boundary conditions parallel to the\nsurface for both, the NaCl slab and the Au slabs. Hopping\nintegrals are constructed according to the nearest neigh-\nbour hopping rules rules of Harrison [2, 3], including 𝑠−𝑑\n[3] and𝑝−𝑑hopping. The NaCl and Au surfaces are non\ncommensurate. We place the central Na atom on top of a\ncentral Au atom in the neighbouring NaCl and Au layers.\nTo describe the Au substrate, we use the lattice param-\neter𝑎Au=4.07Å [4]. We use the Harrison level energies\n𝜀6𝑠=−6.98eV and𝜀5𝑑=−17.78eV as a reference and add\na6𝑝level at 5 eV above the 4𝑠-level. We then shift the 5𝑑\nlevel relative to the 6𝑠and6𝑝levels so that the top of the\n5𝑑band is placed at 1.7 eV below the Fermi energy [5]. Fi-\nnally all energies are shifted so that the Fermi energy is at\nzero. The resulting parameters are summarised in Table I.\nTo describe the NaCl film, we essentially follow our ear-\nlier work [1, 6] and choose parameters such that the con-\nduction band has mainly Cl 4𝑠character [6, 7]. To achieve\nthis, we replace the Cl 3𝑠level by a 4𝑠level, which has\nbeen strongly lowered by the Madelung potential, while\nthe Na levels are strongly shifted upwards. We fine-tune\nthe Harrison parameters to replicate the experimentally\nobserved bulk NaCl band gap (8.5 eV) [8]. Image poten-\ntial effects were neglected. According to calculations [9]\nusing the GW method [10], the top of the valence band is\n5 eV below the Fermi energy. We then shift all the NaCl\nenergies relative to the Au energies correspondingly. The\nresulting parameters are shown in Table I. The calcula-\ntions were performed for lattice parameter 𝑎NaCl=5.54\nÅ [11].\nWe use the calculated separation 𝑑Au−NaCl=3.12Å be-\ntween the Au surface and the NaCl film [11]. Since the\nNaCl film and the substrate are incommensurate, several\nAu atoms can have similar distances to a given NaCl atom,\nand the nearest neighbours are poorly defined. We then\nuse a smooth distance dependent cut-off of the Harrison\nprescription for the hopping between the substrate and\nthe film. Thus, the Harrison prescription for these hop-\nping integrals is multiplied by a factor\nexp/√︁a√︂enleftigg\n−(𝑑−𝑑Au−NaCl)2\n𝜆2\nSB/√︁a√︂en√︂ightigg\n, (1)\nwhere𝑑is the distance between an Au atom and a NaClElement 𝑠 𝑝 𝑑\nAu (6s, 6p, 5d) 4.1 9.1 −3.7\nNa (3s, 3p) 12.8 16.8 −\nCl (4s, 3p) 10.2 −5.0−\nC (2s, 2p)−19.38−11.07−\nN (2s, 2p)−26.22−13.84−\nPt (6s, 5d)−6.85− − 16.47\nH (1s)−13.61− −\nTable I. Level energies used for PtPc, NaCl, and Au after shifts\ndescribed in the text. The Fermi energy is put at zero.\natom at the Au-NaCl interface. Here 𝜆SBis chosen such\nthat summing these factors over all the Au neighbours of\na NaCl atom and averaged over the NaCl atoms in the in-\nnermost layer adds up to four. Then the innermost NaCl\natoms effectively couple to four Au atoms.\nTight-binding model – PtPc\nWe study the absorbed molecule platinum phthalocya-\nnine (PtPc). The coordinates are obtained from a density\nfunctional calculation. The tight-binding parameters are\nprimarily obtained from Harrison [2]. For the H atoms we\ninclude the 1𝑠level at the energy −13.6eV (not given by\nHarrison). Guided by Miwa et al. we use the separation 3.4\nÅ between the the PtPc molecule and the NaCl film. PtPc\nis absorbed atop a Na atom [12]. For PtPc the four “arms”\nof the molecule are along the NaCl (100) directions. For\nthe hopping between the molecule and NaCl, we also use\nthe rules of Harrison [2], but modified as in Eq. (1) above.\nAgain a𝜆𝐵𝑀is chosen so that, on average, from each atom\nin the molecule there is effectively hopping to four atoms\nin the NaCl buffer. The Au slab breaks the four-fold sym-\nmetry of PtPc which has been reintroduced in the plots.\nThese parameters incorrectly puts a 𝜎-orbital below the\nHOMO. We therefore shift this orbital by 3.2 eV upwards,\nslightly above the LUMO. We also adjust the parameters\nso that the experimental gap is obtained, including image\neffects. Finally we align the levels with the Fermi energy\n(𝐸𝐹=0) of the system, so that the HOMO is located at −1.3\neV and LUMO at 1.7 eV, in agreement with experiment.\nThe resulting parameters are shown in Table I.\nPropagation in vacuum\nIn Eq. (3) in the main text, we presented the expansion\nof the MOs of PtPc in basis function appropriate for vac-\nuum propagation\n∑︁\n𝑚𝑖/b√︂acketleftig\n𝑐(s)\n𝑚𝑖sin(𝑚𝜙)+𝑐(c)\n𝑚𝑖cos(𝑚𝜙)/b√︂acket√︂ightig\n𝐽𝑚[𝑘𝑚𝑖𝜌]𝑒−𝜅𝑚𝑖𝑧,(2)arXiv:2401.14937v1 [cond-mat.mes-hall] 26 Jan 20242\nFigure 1. Constant height STM images illustrating the sharp transition between the gap feature and the LUMO. The bias voltages are\nindicated in each panel. All panels show an area of 20×20Å2.\nLowest𝜎Lowest𝜋Higher𝜋 HOMO LUMO\n𝑚 𝑖 𝑐(c)\n𝑚𝑖𝑚 𝑖 𝑐(c)\n𝑚𝑖𝑚 𝑖 𝑐(𝑐)\n𝑚𝑖𝑚 𝑖 𝑐(s)\n𝑚𝑖𝑚 𝑖 𝑐(c)\n𝑚𝑖\n0 1 1.00 0 1 1.00 0 1 0.81 4 1 −0.231 1−0.03\n0 2 0.78 0 2 0.73 0 3 −0.924 3 0.41 1 2 0.13\n0 4−0.320 4−0.360 5 0.47 4 4 1.00 3 4 −0.69\n0 5−0.460 5−0.404 1 0.62 4 5 0.90 5 4 1.00\n0 6−0.360 6−0.254 2 1.00 4 6 0.42 5 5 0.97\n0 7−0.164 5 0.17 4 3 0.47 8 3 −0.385 6 0.56\nTable II. Largest expansion coefficients for a few MOs [Eq. (2)].\nIn addition, we show a few coefficients for small 𝑖and𝑚values.\nThe table shows results for the lowest 𝜎MO, the lowest 𝜋MO, a\n𝜋MO with one radial node as well as the HOMO and one LUMO.\nFor a given orbital, the coefficients have been renormalized so\nthat the largest coefficient is unity.\nwhere𝑚(≥0)is an integer and 𝐽𝑚is an integer Bessel\nfunction. The values of 𝜅𝑚𝑖, determining the exponential\ndecay with 𝑧, were given in Table I in the main text. As\ndiscussed in that context, the different components 𝑚𝑖de-\ncay at very different rates in vacuum, with the 𝑚=0and\n𝑖=1component decaying most slowly. Table II shows the\nexpansion coefficients for some important MOs in terms\nof these vacuum functions. The largest component for the\nlowest𝜎and lowest 𝜋orbitals is the 𝑚=0𝑖=1com-\nponent, which, therefore, decay relatively slowly with 𝑧.\nThe𝜋orbital with one radial node has somewhat larger\namplitudes for higher components, and decays somewhat\nfaster. The HOMO only has components with 𝑚=4or\nhigher𝑚values, which decay rapidly with 𝑧. In partic-\nular, most of these components, in addition, have rather\nhigh values of 𝑖, making the decay even faster. The LUMO\nhas the largest amplitude for an 𝑚=5component with a\nhigh𝑖value. This component decays very rapidly. How-\never, there are also components with 𝑚=1and𝑚=3,\nwhich have small amplitudes, but are still important be-\ncause of their slower decay. Overall, the LUMO therefore\ndoes not decay as rapidly as the HOMO.\nTransition from gap image to LUMO\nAs an example of how abrupt the transition from the or-\nbital image to the transport gap feature occurs, we show inFig. 1 images that were recorded with voltage differences\nof only 50 mV. We remark that a similarly sharp transi-\ntion was observed also for the transition from HOMO to\nthe gap feature at negative voltages.\nColour bar generation\nLet𝛾=1.25and the range of 𝑥be[0,1]. The colour\nbars of the main text are parameterised by the following\nequations for each primary colour component.\nred(𝑥)=𝑥𝛾\n2\ngreen(𝑥)=𝑥3𝛾\nblue(𝑥)=/b√︂aceleftbiggsin(2𝜋𝑥𝛾),0≤𝑥𝛾≤0.5\n0, 0.5<𝑥𝛾≤1\n[1] A. Grewal, C. C. Leon, K. Kuhnke, K. Kern, and O. Gunnars-\nson, ACS Nano 17, 13176 (2023).\n[2] W. Harrison, Elementary Electronic Structure (World Scien-\ntific Publishing, Singapore, 1999).\n[3] W. Harrison, Electronic Structure and the Properties of Solids:\nThe Physics of the Chemical Bond (Freeman, 1980).\n[4] W. P. Davey, Physical Review 25, 753 (1925).\n[5] P. M. Sheverdyaeva, R. Requist, P. Moras, S. K. Mahatha,\nM. Papagno, L. Ferrari, E. Tosatti, and C. Carbone, Physical\nReview B 93, 035113 (2016).\n[6] C. C. Leon, A. Grewal, K. Kuhnke, K. Kern, and O. Gunnars-\nson, Nature Communications 13, 981 (2022).\n[7] P. K. de Boer and R. A. de Groot, American Journal of\nPhysics 67, 443 (1999).\n[8] R. T. Poole, J. G. Jenkin, J. Liesegang, and R. C. G. Leckey,\nPhysical Review B 11, 5179 (1975).\n[9] S. Wang, N. Kharche, E. Costa Girão, X. Feng, K. Müllen,\nV. Meunier, R. Fasel, and P. Ruffieux, Nano Letters 17, 4277\n(2017).\n[10] L. Hedin, Physical Review 139, A796 (1965).\n[11] H.-Y. T. Chen and G. Pacchioni, Physical Chemistry Chem-\nical Physics 16, 21838 (2014).\n[12] K. Miwa, H. Imada, S. Kawahara, and Y. Kim, Physical Re-\nview B 93, 165419 (2016)." }, { "title": "2401.14954v1.pyMBE__the_Python_based_Molecule_Builder_for_ESPResSo.pdf", "content": "pyMBE: the Python-based Molecule Builder for ESPResSo\nDavid Beyer,1,a)Paola B. Torres,2,a)Sebastian P. Pineda,3Claudio F. Narambuena,2Jean-Noël Grad,1Peter\nKošovan,3and Pablo M. Blanco3, 4, 5\n1)Institute for Computational Physics, University of Stuttgart, Allmandring 3, 70569 Stuttgart, Germany\n2)Grupo de Bionanotecnologia y Sistemas Complejos. Infap-CONICET & Facultad Regional San Rafael, Universidad Tecnológica\nNacional, 5600 San Rafael, Argentina\n3)Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, Hlavova 8, 128 40 Prague 2,\nCzech Republic\n4)Department of Material Science and Physical Chemistry, Research Institute of Theoretical and Computational Chemistry\n(IQTCUB), University of Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain\n5)Department of Physics, NTNU - Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(*Electronic mail: pablb@ntnu.no)\n(Dated: January 29, 2024)\nWe present the Python-based Molecule Builder for ESPResSo (pyMBE), an open source software designed to build\ncoarse-grained models of polyelectrolytes, peptides and globular proteins of arbitrary topology into the Extensible\nSimulation Package for Research on Soft Matter (ESPResSo). ESPResSo features the constant pH (cpH) and grand-\nreaction (G-RxMC) methods, which are powerful tools to study macromolecular systems with many reactive groups,\npermitting to efficiently sample systems with multiple coupled chemical equilibria. However, setting up these methods\nfor macromolecules with many different reactive groups is a non-trivial and error-prone task, especially for beginners.\npyMBE enables the automatic setup of cpH and G-RxMC simulations in ESPResSo, lowering the barrier for newcomers\nand opening the door to investigate complex systems not yet studied with these methods. To demonstrate some of the\napplications of pyMBE, we showcase several study cases where pyMBE successfully reproduces previous simulations\nin the literature done with ESPResSo and other software, including various simulations of different peptides in bulk\nsolution, simulations of weak polyelectrolytes in dialysis and simulations of globular proteins in bulk solution. pyMBE\nis publicly available as a GitLab repository (https://gitlab.com/blancoapa/pyMBE) which includes its source code and\nvarious sample and test scripts, including the ones that we used to generated the data presented in this article.\nKeywords: molecule builder, coarse-grained molecular modelling, constant pH simulation, Reaction ensemble Monte\nCarlo, charge regulation, pKa, protein, peptide\nI. INTRODUCTION\nComputer simulations using molecular models are a valu-\nable tool to rationalize experimental observations, validate\nthe applicability of simplified theories and provide numeri-\ncal results beyond the capabilities of analytical calculations.\nThe most popular molecular models are atomistic models,\nin which each atom is represented by an explicit particle.\nWhile atomistic models provide a detailed description of a\nsystem, they are often limited to nanoscopic scales due to\nthe current computing capabilities. Coarse-graining tech-\nniques circumvent this problem by reducing the number of\ndegrees of freedom of a system, at the cost of losing infor-\nmation about the atomistic details of the system. Coarse-\ngrained (CG) models are often preferred to investigate macro-\nmolecular systems, since they permit to reach longer time and\nlength scales than atomistic models. Despite the popularity\nof CG models to simulate macromolecular systems, most of\nthe current software packages for molecular modelling are de-\nsigned for atomistic models. Only a few software packages\nnatively support CG models, such as the Extensible Simu-\nlation Package for Research on Soft Matter (ESPResSo),1,2\nESPResSo++,3HOOMD-blue,4the modular molecular sim-\na)These two authors contributed equally.\nFigure 1. Logo of the Python-based Molecule Builder for ESPResSo\n(pyMBE). The logo has been edited from an image originally gener-\nated using Artificial Intelligence powered tools from Canva.7\nulation (MOLSIM)5and the Large-scale Atomic/Molecular\nMassively Parallel Simulator (LAMMPS).6\nCG models are especially well-suited for Monte Carlo\nmethods, which can be used to sample reversible chemi-\ncal reactions in reactive systems. Reversible chemical reac-\ntions can have a significant impact on the physical properties\nof macromolecular systems. For example, the coupling be-\ntween the ionization states and the conformation of flexible\npolyelectrolytes and short peptides determines their physical\nproperties.8,9Constant pH Monte Carlo10(cpH) simulations\nare a popular tool to study pH-sensitive molecules in bulkarXiv:2401.14954v1 [cond-mat.soft] 26 Jan 20242\nsolution, including weak polyelectrolyte chains,11,12flexible\npolypeptides,13–15proteins,16–19microgels20–22and charge\nregulating nanoparticles.23,24More advanced techniques, such\nas the grand-reaction25,26(G-RxMC) method, are required to\nstudy two-phase systems. In these two-phase systems, poly-\nelectrolytes are confined in one phase, creating an asymmetric\ndistribution of ions, leading to the so-called Donnan effect.25\nSuch systems naturally appear in many applications, including\nweak polyelectrolyte hydrogels,27,28coacervates,29brushes30\nand the dialysis of polyelectrolytes25or proteins.31Both\nof these Monte Carlo methods are available in ESPResSo,\nhowever, setting them up for polyelectrolytes and biomacro-\nmolecules with many different reactive groups is non-trivial.\nThe preliminary step before performing a computer simu-\nlation is to build the molecular model into the software. Many\nmolecule builders such as Avogadro,32the TopoTools plugin\nfor Visual Molecular Dynamics33(VMD), the Builder tool in\nPyMOL,34the LEaP molecular builder35from the Amber-\nTools suite,36the mBuild37,38tool in MoSDeF,39or Atlas40\nare available to facilitate the setup of atomistic models into\nsoftware for molecular simulation. While some of these all-\natom molecule builders can be repurposed for CG modeling\nby defining custom molecular residue types, this procedure\nis time-consuming and requires detailed knowledge of the\ninternal representation of all-atom residues in the software.\nMolecule builders for CG models are scarce, because CG\nmodels are usually highly problem-specific. Therefore, many\nusers need to design their own scripts to build CG models\ninto simulation software such as LAMMPS and ESPResSo,\nwhich can consume a significant amount of their working\ntime. ESPResSo offers a flexible Python scripting interface\ncapable of building custom CG models, resulting in a conve-\nnient software to perform computer simulations of CG mod-\nels. However, users of ESPResSo still need to build the CG\nmodel by hand, which in practice has prevented its users from\nusing the software to simulate models with complex architec-\ntures or force-fields.\nIn this paper, we present the Python-based Molecule\nBuilder for ESPResSo (pyMBE), an open-source software\ndeveloped as a collaborative project between various re-\nsearch groups in the field of charge regulation of weak poly-\nelectrolytes and biomacromolecules. pyMBE is a molecule\nbuilder designed to build CG models of macromolecules with\ndifferent complex architectures, ranging from models of flex-\nible polyelectrolytes and peptides to globular proteins with a\nrigid structure. The software facilitates the setup of the cpH\nand G-RxMC methods in ESPResSo, lowering the barrier to\nentry for new users of these methods. In Fig. 1, we present the\nlogo of pyMBE with its mascot: a (fictional) python snake that\nloves drinking espresso while coding. To demonstrate the ap-\nplications of pyMBE, we showcase examples where pyMBE\nsuccessfully reproduces reference data from previous works\ndone with ESPResSo and other simulation software. Alto-\ngether, pyMBE aims to aid researchers in the field, both en-\nsuring data reproducibility and reusability of the software pro-\nduced.Table I: Features of pyMBE\n•Molecule builder designed for ESPResSo.\n•Build custom polymer molecules using simple blocks.\n•Support for coarse-grained models of peptides and proteins.\n•Automated setup of cpH and G-RxMC simulations.\n•Bookkeeping of the topology of the molecules.\nII. PYMBE: THE PYTHON-BASED MOLECULE BUILDER\nFOR ESPRESSO\nWe outline in Table I the main features of pyMBE, which\nwe will explain throughout this section. pyMBE is currently\nunder active development and new functionalities are sporadi-\ncally implemented by the development team. Every new func-\ntionality is reviewed by at least one member of the devel-\noping team and tested to reproduce previous reference data\nfrom our groups before being merged into the stable version\nof pyMBE. A stable version of pyMBE is publicly available\nas a GitLab repository (https://gitlab.com/blancoapa/pyMBE),\nincluding the source code, an application programming inter-\nface (API) documentation, a Jupyter notebook tutorial for be-\nginners and several sample scripts and test scripts. The API\ndocumentation is automatically generated from the source\ncode of the library using the Python library pdoc.41A list of\nthe dependencies of pyMBE can be found in the Appendix A.\nA. Basics of pyMBE\npyMBE is imported as a Python library,\nimport pyMBE\n# Create an instance of pyMBE\npmb = pyMBE . pymbe_library ()\nwhich allows to use it with standard ESPResSo scripts. It em-\nploys the Python library Pint42for unit manipulation to define\nand operate on quantities. When the user creates an instance\nof pyMBE, pyMBE creates an instance of a UnitRegistry ob-\nject from Pint, which is stored as an attribute of pyMBE. The\nuser can access this attribute to perform operations with units,\nas conventionally done when using Pint:\nKb = 1.38e -23* pmb . units .J/pmb. units .K\nT = 298.15* pmb. units .K\nthermal_energy = Kb*T\nThe use of Pint permits an easy and reliable transformation\nfrom the international system of units to the reduced system\nof units used in ESPResSo. By default, the set of reduced\nunits is chosen using typical values for coarse-grained models\nof polymers as follows: the unit of reduced length is 0 .355nm,\nthe unit of reduced charge is the elementary charge e=1.602·\n10−19C and the unit of reduced energy is the thermal energy\nat room temperature, kBT, with temperature T=298.15K and\nthe Boltzmann constant kB=1.381·10−23J/K. The user can3\nParticlesIBA111111111111111 111111111111111central\nbeadsside\nchains\nRes1Res2\nResidues\nMolecules\nFigure 2. Schematic representation of the essential building blocks\nused in pyMBE: particles, residues and molecules. A particle is the\nsmallest building block in pyMBE, representing an atom or a coarse-\ngrained group of atoms. Residues consist of a central particle, re-\nferred as central bead, with one or multiple side chains bonded to it.\nMolecules are built as a linear sequence of residues with an arbitrary\nuser-defined composition.\nset a different set of reduced units in pyMBE and consult the\nactive set of reduced units in the library:\n# Set a different set of reduced units\npmb . set_reduced_units (\nunit_length = 0.5* pmb. units .nm ,\nunit_charge = 2* pmb. units .e,\ntemperature = 300* pmb. units .K)\n# Print the set of reduced units\npmb . print_reduced_units ()\nBy default, pyMBE does not define a unit of reduced mass be-\ncause this choice does not influence the equilibrium properties\nof the system for the Monte Carlo methods that pyMBE has\nbeen designed for.\nAs schematically shown in Fig. 2, pyMBE uses a hierarchi-\ncal object-oriented structure consisting of three fundamental\nbuilding blocks: particles, residues and molecules. This hier-\narchical approach is, in part, inspired by the architecture used\nin the modular molecular simulation (MOLSIM) software.5A\nparticle is the smallest building block in pyMBE, represent-\ning an atom or a coarse-grained group of atoms. Users can\nuse pyMBE to define the properties of each different particle\ntype, including its acid-base properties:\n# Inert particle\npmb . define_particle (name = \"I\",\nacidity = \" inert \",\nq = 0,\ndiameter = 0.3* pmb. units .nm ,\nepsilon = thermal_energy )\n# Acidic particle\npmb . define_particle (\nname = \"A\",\nacidity = \" acidic \",\npka = 4,\ndiameter = 0.4* pmb. units .nm ,\nepsilon = thermal_energy )\n# Basic particle\npmb . define_particle (\nname = \"B\",\nacidity = \" basic \",\npka = 9,\ndiameter = 0.5* pmb. units .nm ,\nepsilon = thermal_energy )\npyMBE uses the argument name as an identifier for each type\nof particle. The argument acidity determines if pyMBE\nconsiders the particle an acid, a base or a chemically inert par-\nticle that does not exhibit charge regulation ( e.g. a molecule\nbackbone or a strong salt ion). For acidic and basic particles,\nthe optional argument pka sets the value of the thermody-\nnamic equilibrium acidity constant. When setting up the con-\nstant pH or the grand-reaction method, pyMBE uses acidity\nandpkato set the up the chemical reactions for the different\ncharge regulating particles, as explained in Section II E. For\ninert particles, the argument qsets the charge of a particle.\nFor acidic and basic particles, whose charges fluctuate during\nthe simulation, the initial charge is set to match that of the pro-\ntonated form, i.e.0 for an acid and +1efor a base. diameter\nandepsilon are used as input parameters during the setup\nof pair-wise Lennard-Jones interactions between particles, as\nexplained in Section II C.\nIn pyMBE, residues are monomeric units used to build\nmolecules. A residue is defined as a central particle with a\nset of objects bonded to it:\npmb . define_residue (\nname = \" Res_1 \",\ncentral_bead = \"I\",\nside_chains = [\"A\",\"B\"])\npmb . define_residue (\nname = \" Res_2 \",\ncentral_bead = \"I\",\nside_chains = [\" Res_1 \"])\nHere, name is the identifier used by pyMBE to identify the\nresidue. The particle name in central_bead defines what\nparticle pyMBE places in the center of the residue. The\nnames listed in side_chains determine which objects will be\nbonded to central_bead , and they must correspond to either\nparticle or a residue objects. If the user provides the name of a\nparticle object, pyMBE creates a particle of such type bonded4\ntocentral_bead . If the name of a residue object is provided\ninstead, pyMBE creates a residue of such type, whose central\nbead is bonded to the central_bead of the defined residue.\nMolecules are built as a linear sequence of residues with an\narbitrary user-defined composition:\npmb . define_molecule (\nname = \" A_molecule \",\nresidue_list = [\" Res_1 \",\" Res_2 \",\n\" Res_1 \",\" Res_1 \",\n\" Res_2 \"])\nname is the identifier of the molecule within pyMBE. pyMBE\nwill create in sequential order one residue of the correspond-\ning name per item in the list residue_list . The logic pre-\nsented in this section permits a flexible setup of molecules\nwith several architectures, but still requires the user to define\neach type of particle and residue in the molecule. Further-\nmore, pyMBE also provides specific functions to build coarse-\ngrained models for peptides and proteins which define the\namino acid particles and residues automatically, circumvent-\ning manual setup by the user. pyMBE internally bookkeeps all\nthe information of the particles, residues and molecules pro-\nvided by the user, allowing to easily sort all the information\nabout the system.\nB. Information storage in pyMBE\npyMBE uses the Python library Pandas43,44to bookkeep all\nthe information regarding the particle objects into a Pandas\ndataframe which can be accessed in the following way:\n# Print the full pyMBE dataframe\nprint (pmb .df)\nThe pyMBE dataframe can be operated on as a standard Pan-\ndas dataframe and allows the user to easily store and sort\nthe topology of the system and all the parameters defined\nin pyMBE. Users can filter the dataframe by specifying the\npyMBE object of interest in the argument pmb_type :\n# Filter the dataframe by particle\npmb . filter_df ( pmb_type = \" particle \")\npmb_type supports pyMBE objects of the following types:\n\"particle\" ,\"residue\" ,\"molecule\" and\"bond\" . In Ta-\nble II, we show an example of the output of this function when\nsorting by particles, following the setup done in Section II A.\nIn this example, each column corresponds to a parameter of a\npyMBE particle object and each row contains the information\nof a different particle in the system. Additional examples of\nsubsets of the pyMBE dataframe can be found in Appendix B.\nThere are two different types of information stored in the\ndataframe: the input parameters provided by the user when\ndefining a particle object and identifiers internally generated\nby pyMBE. Examples of input parameters are name ,acidity\n,pka,diameter andepsilon . For acidic and basic particles,\ntheir pkavalue is also stored in the dataframe. In the case of\ninert particles, the pkais set to NaN. The pyMBE dataframe\nstores diameter andepsilon with their corresponding Pintunits. pyMBE uses the values of diameter andepsilon in\nthe dataframe during the setup of Lennard-Jones interactions\nbetween particles in ESPResSo, as explained in Section II C.\nExamples of identifiers internally generated by pyMBE\nare:pmb_type ,particle_id ,residue_id ,molecule_id\n,state_one andstate_two . pyMBE assigns a numer-\nical identifier to each individual particle ( particle_id ),\nresidue ( residue_id ) or molecule ( molecule_id ) object\nthat pyMBE creates into the ESPResSo system. These identi-\nfiers are stored in the pyMBE dataframe in which pyMBE cre-\nates one row per each individual object. They permit to map\nwhich specific particles belong to which specific residues and\nmolecules, allowing to back-trace the exact topology of the\nsystem. state_one andstate_two summarize the informa-\ntion about the possible ionization states of the particles. Each\nstate has a secondary multi-index header with the following\nindexes: label ,es_type andcharge .label is a string-\nlike identifier of a particular ionization state of the particle.\nes_type is the numerical identifier that pyMBE will assign to\nparticles in that state when creating them into the ESPResSo\nsystem or setting up the Monte Carlo methods described in\nSection II E. charge is the valency of the particle in the corre-\nsponding ionization state. pyMBE considers inert particles to\nhave a unique state ( state_one ) whose label matches with its\nname . For acidic and basic particles, \"state_one\" corresponds\nto the protonated state and \"state_two\" the deprotonated state.\npyMBE labels state_one by adding the character ’H’ at the\nend of the name of the particle. state_two is labelled using\nthe same label as in the name of the particle. Additionally,\npyMBE assigns the charge of each state using the acidity\nof the particle. For an acidic particle, charge is set to 0 in\nstate_one (corresponding to a protonated monoprotic acid)\nandcharge is set to to −1einstate_two (corresponding to a\ndeprotonated monoprotic acid). For a basic particle, charge\nis set to +1einstate_one (corresponding to a protonated\nmonoprotic base) and charge is set to 0 in state_two (cor-\nresponding to a deprotonated monoprotic base).\nThe user can store the pyMBE dataframe into a local file in\nCSV format using the Pandas native function\npmb .df. to_csv (\"df.csv\")\nDue to the multi-index organization of the pyMBE dataframe,\nthe user cannot directly use the Pandas native function to load\nfiles in CSV files. However, pyMBE provides a helper func-\ntion to read the information from a pyMBE dataframe stored\nin a local CSV file\npmb . read_pmb_df ( filename = \"df.csv\")\nwhere filename is the path to the local CSV file with the\npyMBE dataframe.\nC. How to set up coarse-grained models of flexible\nmolecules in pyMBE\nAfter reviewing the basic features of pyMBE, let us show-\ncase some specific examples how pyMBE can be used to build\ncustom CG models of flexible molecules, such as polyelec-\ntrolytes and peptides.5\nTable II. Example of a subset of the pyMBE dataframe, filtered by pmb_type = \"particle\" . The cells under the columns name ,acidity ,\npKa,acidity ,diameter andepsilon store the parameters provided by the user when defining each particle. The values of the parameters\nmatch the example proposed in Section II A. pmb_type ,particle_id ,residue_id ,molecule_id ,state_one andstate_two correspond\nto identifiers internally assigned by pyMBE when it creates the corresponding objects into ESPResSo.\nname pmb_type particle_id residue_id molecule_id acidity pKa diameter epsilonstate_one state_two···label es_type charge label es_type charge\nI particle 0 0 0 inert NaN 0.3 nm 1 reduced_energy I 0 0 NaN NaN NaN ···\nA particle 1 0 0 acidic 4.0 0.4 nm 1 reduced_energy AH 1 0 A 2 -1 ···\nB particle 2 0 0 basic 9.0 0.5 nm 1 reduced_energy BH 3 +1 B 4 0 ···\n................................................\nFirst, let us define a set of example particles and residues in\nthe same fashion as in the previous section\n# Particles\n## Backbone particle\npmb . define_particle (\nname = \"I\",\nacidity = \" inert \",\nq = 0,\ndiameter = 0.35* pmb. units .nm ,\nepsilon = thermal_energy )\n## Acidic particle\npmb . define_particle (\nname = \"A\",\nacidity = \" acidic \",\npka = 4.5 ,\ndiameter = 0.35* pmb. units .nm ,\nepsilon = thermal_energy )\n## Basic particle\npmb . define_particle (\nname = \"B\",\nacidity = \" basic \",\npka = 10.5 ,\ndiameter = 0.35* pmb. units .nm ,\nepsilon = thermal_energy )\n# Residues\n## Acidic residue\npmb . define_residue (\nname = \"IA\",\ncentral_bead = \"I\",\nside_chains = [\"A\"])\n## Basic residue\npmb . define_residue (\nname = \"IB\",\ncentral_bead = \"I\",\nside_chains = [\"B\"])\nwhere we have defined the residues using a two-bead model\nwith an inert particle ( \"I\") in the center and an acidic ( \"A\")\nor a basic particle ( \"B\") as side chain. pyMBE allows to build\nlinear flexible polymers with arbitrary composition by lettingthe user define a custom list of residues ( residue_list ).\nLet us showcase this feature by defining a set of example\nmolecules with common polymer architectures\n## Molecules\n# Polyacid\npmb . define_molecule (\nname = \" polyacid \",\nresidue_list = [\"IA\" ]*10)\n# Alternating polyampholyte\npmb . define_molecule (\nname = \" alternating_polymer \",\nresidue_list = [\"IA\",\"IB\" ]*5)\n# Diblock polyampholyte\npmb . define_molecule (\nname = \" diblock_polymer \",\nresidue_list = [\"IA\" ]*5+[ \"IB\" ]*5)\nUsing the logic presented, the user can define CG models of\nflexible linear polymers with any arbitrary composition. Flex-\nible peptides are an example of such a molecule with a well-\ndefined number of possible residues, the natural amino acids.\nFor convenience, pyMBE has an additional method to define\nmodels of flexible peptides\npmb . define_peptide (\nname = \" Cys2His3 \",\nsequence = \" cCCHHHn \",\nmodel = \"2 beadAA \")\nwhere name is the string-like identifier of the peptide\nmolecule. sequence is a string with the amino acid sequence\nof the peptide. Currently, pyMBE supports a sequence us-\ning either the one letter or the three letter standard codes\nfor protein sequences. For example, valid sequence inputs\nfor the peptide Cys2His 3are\"CCHHH\" and\"CYS-CYS-HIS-\nHIS-HIS\" . The method define_peptide defines internally\none residue object into the pyMBE dataframe for each type\nof amino acid in sequence . pyMBE will define additional\nsingle-bead residues for the carboxyl and amino ends if the\nlabels \"c\"and\"n\"are provided at the ends of the sequence\n, respectively. The argument model defines the structure\nof these residues. Currently, pyMBE supports CG models\nwith one bead per amino acid ( model = \"1beadAA\" ) and CG6\nmodels with two beads per amino acid ( model = \"2beadAA\n\"). Independently of the model of choice, only one bead will\nbe added per each \"c\"and\"n\"end.\npyMBE can be used to build the designed coarse-grained\nmodels into ESPResSo, but prior to that the user needs to de-\nfine how each pair of particle types should be bonded:\n# Bonds\n## I - I bond\npmb . define_bond (\nbond_object = II_bond ,\nbond_type = \" harmonic \",\nparticle_name1 = \"I\",\nparticle_name2 = \"I\")\n## I - A bond\npmb . define_bond (\nbond_object = IA_bond ,\nbond_type = \" harmonic \",\nparticle_name1 = \"I\",\nparticle_name2 = \"A\")\n## I - B bond\npmb . define_bond (\nbond_object = IB_bond ,\nbond_type = \" harmonic \",\nparticle_name1 = \"I\",\nparticle_name2 = \"B\")\nHere, bond_object is an instance of a bond object of the\nESPResSo library and bond_type is a string what kind of\nbond that object is. Currently, pyMBE supports the har-\nmonic potential ( bond_type = \"harmonic\" ) and the Fi-\nnite Extension Non-linear Elastic potential ( bond_type =\n\"FENE\" ) as model interactions for bonding the particles.1,2\nparticle_name1 andparticle_name2 are the name identi-\nfiers of the two types of particles to be bonded by pyMBE. All\nbonds defined in pyMBE are stored in the pyMBE dataframe,\nas explained in Appendix B. The user can add all bonds de-\nfined in the pyMBE dataframe into the ESPResSo system us-\ning the helper function\npmb . add_bonds_to_espresso (\nespresso_system = es_system )\nHere, espresso_system is an instance of an ESPResSo sys-\ntem from the ESPResSo library.\npyMBE provides a set of helper functions to aid the setup\nof a simulation with the defined molecules in ESPResSo. The\nuser can create any pyMBE object into ESPResSo with the\nhelper function\npmb . create_pmb_object_in_espresso (\nname = \" alternating_polymer \",\nespresso_system = es_system ,\nnumber_of_objects = 2)\nname is the pyMBE string identifier of the object to be created\ninto ESPResSo. Currently, this function supports the follow-\ning types of pyMBE objects: particles, residues, molecules\nand peptides. pyMBE features special-purpose functionsto create models of globular proteins into ESPResSo, as\nexplained in the next section. number_of_objects is\nthe number of the objects to be created into ESPResSo.\nIn the above code example, pyMBE would create 2\nmolecules in ESPResSo with the features defined by the\nuser in the pyMBE dataframe for molecules with name =\n\"alternating_polymer\" . pyMBE would also update the\npyMBE dataframe with information about each individual\nparticle, residue and molecule created, as explained in Ap-\npendix B. By default, the pyMBE object will be created in a\nrandom position in the simulation box but this function takes\nan optional argument position which can be used to control\nin which position pyMBE will create the object.\npyMBE also features a helper function to load the topology\nfrom a local file into the pyMBE dataframe as an alternative\nto defining the molecule topology by hand in the script\npmb . load_interaction_parameters (\nfilename = \" path_to_file \")\nwith the path to the local file filename . An example\nof a pyMBE topology file can be found in the folder\n‘reference_parameters/interaction_parameters ’,\nwhich includes the parameters used to build the CG model\nfor oligopeptides used in Ref.45. If the user calls this function\nto define bonds in pyMBE, we note those bonds still need to\nbe added to the ESPResSo system either manually or using\npmb.add_bonds_to_espresso .\nFor convenience, pyMBE also provides a helper function to\nload a set of p KAvalues\npmb . load_pka_set (\nfilename = pka_set_file )\nwhere filename is the path to the local file where the p KAset\nis stored. For molecules with many acidic and basic groups,\nsuch as peptides or proteins, one may find it more convenient\nto load a set of p KAvalues instead of setting them up manually\nfor each type of particle. In the pyMBE repository, we provide\nexamples of files with a format suitable to be processed by\npyMBE in the folder ‘ reference_parameters/pka_sets ’.\nThese files store reference p KAsets from the following refer-\nences: Bienkiewicz and Lumb,46CRC Handbook of Chem-\nistry and Physics,47Dobrev et al. ,48,49Hass and Mulder,50\nNozaki and Tanford,51Platzer et al.52and Thurlkill et al.53.\nFinally, pyMBE has another helper function that automati-\ncally sets up Lennard-Jones (LJ) interactions between the par-\nticles in ESPResSo that have been defined using pyMBE:\npmb . setup_lj_interactions_in_espresso (\nespresso_system = es_system )\nThe LJ potential between each pair of particle types is set up\nby combining the diameter andepsilon of both particle\ntypes using the Lorentz-Berthelot combining rules. pyMBE\nsets the value of the σparameter of the LJ potential to\nbe equal to the unit of reduced length in the system of re-\nduced units defined by the user. The size of the parti-\ncles is then included in the LJ potential by adding an off-\nsetroff= (di+dj)/2−σ, where dianddkare the diame-\nters of the particles with arbitrary types iandj, respectively.7\nsetup_lj_interactions_in_espresso takes the optional\nargument cutoff which defines a cutoff distance rcutfor the\nLJ potential. By default, pyMBE sets rcut=6√\n2σ+roff, cor-\nresponding to a purely repulsive LJ potential also known as\nthe Weeks-Chandler-Andersen potential.\nD. How to set up coarse-grained models of globular proteins\nin pyMBE\npyMBE aids the setup of coarse-grained models of globu-\nlar proteins in ESPResSo using a rigid body representation.\nCurrently, pyMBE supports models of globular proteins con-\nsisting of either 1 bead or 2 beads per amino acid. Two ad-\nditional beads are added to represent carboxylic and amino\nend groups. In Fig. 3, we depict an example of a coarse-\ngrained model of α-lactalbumin (PDB code 1F6S54) built with\npyMBE, consisting of a rigid object with 2 beads per amino\nacid.\nFigure 3. Snapshot of the coarse-grained representation of α-\nlactalbumin (PDB code 1F6S54) using a coarse-grained model with\ntwo beads per amino acid. The beads of each amino acid are colored\nwith the following color code: acidic bead (red), basic bead (blue),\ninert side chain bead (green) and inert backbone bead (grey). The\nsize of the beads has been scaled by an arbitrary value for visualiza-\ntion purposes.\nThe first step when setting up a globular protein with\npyMBE is to load its topology into the pyMBE dataframe.\nThe standard way of loading the protein into pyMBE is\nby providing a VTF file with the coordinates of the beads\nand the information about the protein topology. To aid the\nsetup of such VTF file, we provide a supporting script in the\npyMBE repository, handy_scripts/create_cg_from_pdb\n.py. This script creates a VTF file with the coarse-grained\nmodel of a protein from its corresponding PDB file\npython3 create_cg_from_pdb .py\n-- filename FILENAME\n-- download_pdb PDB_CODE\n--model MODEL\n-- chain_id CHAIN_IDThe script supports two options to provide the PDB file of the\nprotein: providing a path to a local copy of the PDB file using\nthe argparse argument --filename or providing the PDB\ncode using the argparse argument --download_pdb . The\nuser can select a specific chain of the protein by providing\nthe id of the chain to the argparse argument --chain_id .\nFrom the PDB coordinates, the script creates a coarse-grained\nmodel consisting of either one bead per amino acid ( --model\n1beadAA ) or two beads per amino acid ( --model 2beadAA ).\nAs mentioned before, the two bead model represents each\namino acid using two beads, one representing the backbone\nand the other one for the side chain (Fig. 3). The script places\nthe backbone bead at the position of the α-carbon of the\namino acid in PDB files. The second bead is placed at the\ncenter of mass of the side chain, which is calculated including\nall the atoms in the side chain of the amino acid. The radius\nof the beads in the side chains is estimated as the radius of\ngyration of the atoms in the side chain of each amino acid.\nTwo additional beads are added for the carboxyl and amino\nend groups: the bead representing the carboxyl group is\nplaced in the position of the carbon atom of the group and\nthe bead of the amino group is placed in the position of the\nnitrogen atom of the group, using the input PDB coordinates.\nFor the one bead model, only the beads in the side chains and\nthe ends are kept and the beads in the backbone of the protein\nare disregarded. The script returns a VTF file with all the nec-\nessary information to use pyMBE to build the coarse-grained\nmodel of the globular protein into ESPResSo. We provide\nexamples of VTF files in the pyMBE repository in the folder\n‘/reference_parameters/coarse_grained_structures/ ’,\nwhich have a format suitable to be processed by pyMBE.\npyMBE includes a function to read such files:\ntopology = pmb. read_protein_vtf_in_df (\nfilename = protein_filename )\nThe function returns a dictionary ( topology ) with all the in-\nformation about the protein topology. pyMBE uses this dictio-\nnary to define the protein object and store all the information\nin the pyMBE dataframe:\npmb . define_protein (\nname = protein_name ,\ntopology_dict = topology ,\nmodel = \"2 beadAA \")\nwhere name is the identifier of the protein within pyMBE.\nmodel supports the keywords 1beadAA and2beadAA , corre-\nsponding to the two types of CG models for globular proteins\nsupported in pyMBE. Internally, pyMBE defines particles and\nresidues objects for each type of amino acid, using the same\nlogic as described in Sec. II A. Once a protein is defined in the\npyMBE dataframe, users can employ pyMBE to create pro-\nteins into the ESPResSo system\npmb . create_protein (\nname = protein_name ,\nnumber_of_proteins = 1,\nespresso_system = es_system ,\ntopology_dict = topology )8\nwhere name is the pyMBE identifier of the protein,\nespresso_system is an instance of an ESPResSo system\nfrom the ESPResSo library and number_of_proteins is the\nnumber of proteins of that type to be created into es_system .\nBy default, pyMBE fixes the position of particles belonging\nto proteins when creating them into ESPResSo since they are\nmodelled as a rigid object. This setup is convenient for simu-\nlations with a single protein in the simulation box, for which\npyMBE features the option to center the protein in the simu-\nlation box:\npmb . center_molecule_in_simulation_box (\nmolecule_id = protein_id ,\nespresso_system = es_system )\nmolecule_id is the numeric identifier within the pyMBE\ndataframe of the protein to be centered in the simulation box.\nAlthough this function was designed for this particular appli-\ncation, its implementation is general and it can be used with\nany molecule defined in pyMBE. To set up simulations hav-\ning multiple proteins in the simulation box, pyMBE permits\nto active the motion of the rigid object as a whole through the\nsimulation box\npmb . activate_motion_of_rigid_object (\nespresso_system = es_system ,\nname = protein_name )\nwhere name is the pyMBE identifier of the type of proteins\nwhose motion should be activated.\nE. Monte Carlo methods supported in pyMBE\npyMBE supports the automated configuration of two dis-\ntinct methods Monte Carlo methods suitable to study acid-\nbase equilibria: the constant-pH method for the simulation of\nacid-base equilibria10in a single phase and the grand-reaction\nmethod,25,26specifically designed for two-phase systems. In\nthe following, a short introduction to these methods and their\nimplementation and use in pyMBE is given.\n1. The constant-pH method\nThe constant-pH (cpH) method was originally developed\nby Reed and Reed10to simulate the pH-response of weak\npolyelectrolyte chains at a given pH in a buffer solution, as\nshown schematically in Figure 4 (a). In the cpH method, the\npH of the buffer solution is only considered implicitly and it\nis a direct input parameter of the method. The method uses\nMonte Carlo moves to sample different ionization states of\nweak acid and/or base groups according to the chemical equa-\ntions\nHA− −⇀↽− −A−+X+(1)\nHB+− −⇀↽− −B+X+, (2)\nwith the protonated state of an acid (HA) and a base (HB+)\nand the corresponding deprotonated states A−and B. X+\nSystem Reservoir\nHA A−H+Na+OH−Cl−HA A−X+Na+Cl−(a) cpH method\n(b) G-RxMC methodFigure 4. Schematic representation of weak polyelectrolyte systems\n(here represented by weak polyacids with monomers HA) in differ-\nent statistical ensembles. (a): In the cpH-ensemble, a single-phase\nsystem at a given buffer pH and salt concentration is considered. (b):\nIn the grand-reaction method, a two-phase system at a fixed reservoir\ncomposition is considered. The polyelectrolyte chains cannot leave\nthe \"system\" phase, while small ions can be partitioned between the\n\"system\" and the \"reservoir\", leading to a Donnan equilibrium.\nstands for a generic neutralizing counterion rather than H+,\nbecause the pH is decoupled from the activity of H+-ions and\nonly considered implicitly. This feature of the algorithm limits\nthe applicability of the cpH method to the regime of interme-\ndiate pH-values, where the ionic screening is dominated by\nthe salt (e.g. NaCl) rather than by H+or OH−.8,55In the cpH\nmethod, trial moves consist of changing the ionization state of\na charge-regulating particle and randomly inserting/deleting a\nneutralizing counterion according to Equation 1 or Equation 2.\nThese trial moves are accepted with a probability of\nPcpH=min[1,exp(−β∆U+ξln(10)(pH−pKA))].(3)\nHere, β=1/kBTis the inverse thermal energy, ξthe ex-\ntent of reaction ( ξ= + 1 for a deprotonation and ξ=−1\nfor a protonation), ∆Uis the change in potential energy be-9\ntween the old state and the proposed new state. KA=10−pKA\nis the acidic equilibrium constant, specific to the considered\ngroup, for which tabulated values (at the standard concen-\ntration c⊖=1mol /kg) can be used.56As an output of cpH-\nsimulations, one obtains, among other quantities, the degree\nof ionization of the various titrable groups. Thus, by compar-\ning to titration measurements, a direct comparison with exper-\niments is possible.14,15,45\nTo facilitate setting up constant-pH reactions in ESPResSo,\npyMBE implements the function setup_cpH , which is called\nin the following fashion:\nRE , rx_labels = pmb. setup_cpH (\ncounter_ion = cation_name ,\nconstant_pH = pH ,\nSEED = REACTION_SEED )\nAs inputs, one has to provide the name pyMBE identifier\nof the generic neutralizing counterion X+(counter_ion ),\nthe pH at which the system is simulated ( constant_pH )\nand the seed for the pseudorandom number generator ( SEED\n). The function automatically sets up the cpH-reactions\nfor all particles that have the property acidic orbasic\nwith the corresponding p KA-values specified in the pyMBE\ndataframe. It returns an instance of the reaction_methods\n.ConstantpHEnsemble object of the ESPResSo library ( RE)\nand a list containing the labels of the particles for whom reac-\ntions have been set up ( rx_labels ).\nBelow, the use of the cpH method in pyMBE is demon-\nstrated in the case studies on charge-regulating peptides (sec-\ntion III A) and globular proteins (section III C).\n2. The Grand-Reaction method\nIn contrast to the cpH method, the grand-reaction method\n(G-RxMC) of Landsgesell et al.25,26was conceived to study\nthe behaviour of two-phase systems, as shown schematically\nin Figure 4 (b). In these systems, the charge-regulating macro-\nmolecules cannot leave the \"system\" phase, e.g. due to co-\nvalent crosslinking (hydrogels) or the presence of a semi-\npermeable membrane (dialysis). While the macromolecules\nare thus confined to one phase, small ions (H+, Na+, OH−,\nCl−) can freely partition between the polyelectrolyte phase\nconstituting the \"system\" and the \"reservoir\", which contains\nonly small ions. The macroscopic constraint that both phases\nhave to be electroneutral leads to an uneven partitioning of\nsmall ions between the system and the reservoir, the so-called\nDonnan equilibrium.25\nThe G-RxMC method was designed to mimic the setting\nthat is typically present in experimental setups of the above-\nmentioned two-phase systems: the reservoir is of effectively\ninfinite size ( i.e.much larger than the system) and is thus not\naffected by the partitioning of ions. This allows one to fix the\nreservoir composition ( i.e. pHresand the salt concentration\ncres\nNaCl) and to study the properties of the system, e.g.the par-\ntitioning of ions into the system or the ionization response of\nthe ionizable residues, for the specified reservoir conditions.\nIn the G-RxMC, this setting is achieved by grand-canonicallycoupling the system to a virtual reservoir. In the method, both\nthe chemical reactions inside the system, i.e.\nHA− −⇀↽− −A−+H+(4)\nHB+− −⇀↽− −B+H+, (5)\nand the exchange of ion pairs with the reservoir, represented\nby a set of \"virtual\" reactions,\n/ 0− −⇀↽− −H++OH−(6)\n/ 0− −⇀↽− −Na++Cl−(7)\n/ 0− −⇀↽− −Na++OH−(8)\n/ 0− −⇀↽− −H++Cl−, (9)\nare implemented using the Reaction ensemble Monte Carlo\n(RxMC) method.57,58The RxMC method is a general frame-\nwork for sampling chemical equilibria in CG simulations,\nwhich uses MC moves with an acceptance probability of\nPRxMC =min\"\n1,exp(−β∆U)Kξ\nc(VNA)¯νξ∏\niNi!\n(Ni+νiξ)!#\n.\n(10)\nIn this equation, Kcis the concentration-based equilibrium\nconstant of the considered reaction, Vdenotes the box vol-\nume, NAis the Avogadro number, Nithe number of particles\nof type i,νithe stoichiometric coefficient corresponding to\ntype Iand¯ν=∑iνi. The equilibrium constants of the differ-\nent reactions are determined by the p KA-values of the acidic\nand basic residues and by the reservoir composition (pHres\nandcres\nNaCl). In general, the mapping between reservoir com-\nposition and equilibrium constants is non-trivial and requires\nauxiliary simulations of the reservoir.25,59Instead of distin-\nguishing between different ion types of the same charge, one\ncan also formulate the G-RxMC method in terms of unified\nion types.60For the sake of completeness, pyMBE contains\nimplementations of both the original G-RxMC method as well\nas the reformulation in therms of unified ion types.\npyMBE facilitates setting up simulations using the G-\nRxMC method via the function setup_grxmc :\nRE , rx_labels , I_res = pmb. setup_grxmc (\npH_res = 2,\nc_salt_res = salt_conc_res ,\nproton_name = name_proton ,\nhydroxide_name = name_hydroxide ,\nsodium_name = name_sodium ,\nchloride_name = name_chloride ,\nSEED = REACTION_SEED ,\nexcess_chemical_potential = mu_ex )\nThis function takes as required arguments the reservoir com-\nposition ( i.e. the desired pH value ( pH_res ) and salt con-\ncentration ( c_salt_res ), the names of the various small ions\n(proton_name , etc.), and a seed for the pseudorandom num-\nber generator ( SEED ). Furthermore, one can provide a function\nin the optional argument excess_chemical_potential\nthat calculates the excess chemical potential of an ion pair in\na solution of ions at a given concentration. If the user does not10\nprovide an excess_chemical_potential , pyMBE will set\nup the G-RxMC methos considering an ideal reservoir, i.e.a\nreservoir of ions without any interactions. Internally, when\nsetting up the G-RxMC method, pyMBE self-consistently\nsolves a coupled system of nonlinear equations involving\npH_res ,c_salt_res andexcess_chemical_potential ,\nallowing one to map the reservoir composition to the required\nequilibrium constants.25The required conversion of the ob-\ntained concentration-based equilibrium constants to the re-\nduced unit system is automatically taken care of using the Pint\nlibrary. To set up the G-RxMC method with unified ion types,\nthe function setup_grxmc_unified is employed in an anal-\nogous fashion.\nWe showcase the use of the G-RxMC method in pyMBE\nbelow for the dialysis of weak poylelectrolyte chains (section\nIII B).\nIII. CASE STUDIES\nA. Peptides in bulk solution\nAs a first scenario of a usecase of pyMBE, we showcase\nin Fig. 5 the titration curves, i.e. the pH-dependent charge,\nof various peptides: LYS 5ASP 5(Fig. 5 (a)), GLU 5HIS 5\n(Fig. 5 (b)) and histatin-5 (Fig. 5 (c)). The two first\npeptides are synthetic peptides that were used by Lunkad\net al.45to validate their coarse-grained (CG) model against\ntitration experiments using potentiometry and Nuclear Mag-\nnetic Resonance techniques. Lunkad et al.45designed a CG\nmodel consisting of two beads per amino acid whose prop-\nerties they computed with constant pH (cpH) simulations us-\ning ESPResSo. The third peptide, histatin-5, is a natural pep-\ntide present in human saliva with antifungal and antibacterial\nproperties.61–63Blanco et al.13used a simple coarse-grained\nmodel of histatin-5, consisting of one bead per amino acid,\nand investigated the ionization properties under different con-\nditions of macromolecular crowding using cpH simulations\nperformed in MOLSIM. As explained in Section II C, pyMBE\nallows to easily set up all these CG models of peptides in\nESPResSo, permitting to quantitatively reproduce the results\nproduced in this previous research as shown in Fig. 5.\nTo calculate the charge on a molecule, pyMBE includes\nthe function calculate_net_charge , which enables the\ncalculation of the net charge of a given molecule type\n(molecule_name ), averaged over all specimen:\ncharge_dict = pmb. calculate_net_charge (\nespresso_system = es_system ,\nobject_name = molecule_name )\nThe returned dictionary contains the mean net charge per\nmolecule, as well as a dictionary containing the mean net\ncharge of the individual residues.\nA common benchmark when studying the ionization\nproperties of macromolecules is the so-called Henderson–\nHasselbalch (HH) equation, which is the analytical solution of\nthe acid-base chemical equilibrium (Eqs. 1 and 2) assuming\na non-interacting system, i.e.under ideal conditions. Under\n−505Net charge Z/e\nKKKKKDDDDD(a)HH\nLunkad et al.pyMBE\nBlanco et al.\n−505Net charge Z/e\nEEEEEHHHHH(b)\n2 4 6 8 10 12\npH in the solution010Net charge Z/e\nnDSHAKRHHGYKRKFHEKHHSHRGYc(c)Figure 5. Net charge Zof various peptides as a function of the\npH. Panels (a) and (b): titration curve of the synthetic peptides\nLYS 5ASP 5(a) and GLU 5HIS 5(b) obtained with constant pH Monte\nCarlo (cpH) simulation using ESPResSo by Lunkad et al. (blue\nmarkers).45Panel (c): titration curve of histatin-5 computed by cpH\nsimulation using MOLSIM by Blanco et al. (green markers).13The\ndata from these references can be easily reproduced using pyMBE\nto set up these systems into ESPResSo (orange markers). The black\nlines follow the Henderson–Hasselbalch (HH) equation (Eqs. 11 and\n12) using different sets of p KA-values: the Chemical Rubber Com-\npany (CRC) Handbook of Chemistry and Physics47(a and b) and\nNozaki and Tanford51(c).\nthis assumption, the degree of ionization αiof a monoprotic\nacid or base iis\nαiideal=1\n1+10zi(pKi\nA−pH), (11)\nwhere Ki\nA=10−pKi\nAis the acidic equilibrium constant of iand11\npH=−logaH+is defined in terms of the proton activity aH+.\nziis the charge of the group iwhen it is fully ionized; i.e.\nzi= +1efor a monoprotic base and zi=−1efor a monopro-\ntic acid. The net charge Zof a macromolecule with Nioniz-\nable groups can be calculated by summing over all ionizable\ngroups:\nZ=N\n∑\ni=1ziαi. (12)\npyMBE provides a function that facilitates to calculate the\nideal net charge of any molecule defined with a given\nmolecule_name :\nZ_HH = pmb. calculate_HH (\nobject_name = molecule_name ,\npH_list = pH_range )\nHere, pH_list is the list of pH-values at which the\nHenderson–Hasselbalch equation is evaluated. This function\nis particularly handy for molecules with many different acid\nand basic groups, for example in peptides and proteins. The\ndeviations from the ideal theory that are observed in simu-\nlations (e.g. Fig. 5) are due to the electrostatic interactions\nbetween the various ionizable groups, a phenomenon that has\nbeen termed the \"polyelectrolyte effect\".25Comparison with\nthe ideal theory thus serves as an important metric for quanti-\nfying the effects of interactions, particularly electrostatics, on\nthe ionization behaviour of macromolecules.\nB. Dialysis of weak polyelectrolyte chains\nAs our next case study, we consider one of the simplest\nnon-trivial systems that requires the G-RxMC method in or-\nder to correctly model the interplay of charge regulation and\nion partitioning: a solution of weak polyacid chains which is\nseparated from an aqueous solution of small ions via a semi-\npermeable membrane. Such a setup corresponds to a dialysis\nof the polyelectrolyte, where the role of the semi-permeable\nmembrane is played by the dialysis bag. While this exam-\nple is simpler than the other case studies from the point of\nview of molecular architecture (there is only one type of titra-\nble group), setting up this kind of simulation in ESPResSo\nis still non-trivial, as it requires the setup of multiple, mu-\ntually coupled chemical reactions. Landsgesell et al.25were\nthe first to investigate this system using the G-RxMC method\nand here we show that we can reproduce the earlier results us-\ning pyMBE. We perform simulations for a simple bead-spring\npolymer model derived from the Kremer–Grest model.64In\nour simulations, we consider a system of 16 chains, each of\nlength N=50, with monomers that have p KA=4.0. The\nsimulations are performed at different reservoir compositions\n(salt concentration and pH) and concentrations of the poly-\nelectrolyte within the system. A more detailed description of\nall parameters can be found in Ref.25\nIn Fig. 6, we show a plot of the degree of ionization αof\nthe polyacid chains vs. the pH in the reservoir, obtained for\na salt concentration of cres\nNaCl=10mM in the reservoir and a\n2 4 6 8 10 12\npH in the reservoir0.000.250.500.751.00Degree of ionization a\nHH\nHH+DonLandsgesell et al.\npyMBEFigure 6. Degree of ionization αof a weak polyacid solution\ncoupled to a reservoir vs. the pH in the reservoir, obtained for a salt\nconcentration of cres\nNaCl=10mM in the reservoir and a monomer\nconcentration of cmon=435mM in the solution. The reference data\nby Landsgesell et al.25was also obtained using ESPResSo. \"HH\"\ncorresponds to the result calculated using the ideal Henderson–\nHasselbalch equation (Eq. 11), while \"HH+Don\" results from a\ncoupled systems of equations involving the Henderson–Hasselbalch\nequation and the ideal Donnan theory.\nmonomer concentration of cmon=435mM inside the system.\nWe note that the results obtained using pyMBE are in quanti-\ntative agreement with the results of Landsgesell et al. . Com-\nparing the simulation results to the Henderson–Hasselbalch\nequation (Eq. 11), it can be observed that the ionization ob-\ntained from the simulation is suppressed as compared to the\nideal result. As discussed earlier in more detail,25,27we can\nexplain this shift of the ionization curve as a combination of\ntwo effects: the “polyelectrolyte effect” and the “Donnan ef-\nfect”. The polyelectrolyte effect describes the suppression of\nthe ionization due to the repulsive electrostatic interaction be-\ntween ionized monomers as we already mentioned for the case\nof the peptides in the previous section. In contrast, the Donnan\neffect is a property of two-phase systems. It is a consequence\nof the emergence of an electrostatic potential difference be-\ntween the phases due to the uneven partitioning of the small\nions, the so-called Donnan potential ψDon. Theoretically, it\ncan be shown that the Donnan potential leads to a difference in\npH between the system (pHsys) and the reservoir (pHres),25,27\npHsys−pHres=βeψDon\nln10, (13)\nwhich can be identified with a shift of the ionization curve of\na weak polyelectrolyte.\nThe distinctiveness of the polyelectrolyte effect and the\nDonnan effect can be demonstrated by comparing the simu-\nlation results in Fig. 6 with a theoretical calculation taking12\ninto account the Donnan effect but not the polyelectrolyte ef-\nfect (shown as a dashed line in the plot). This calculation\nrequires one to solve a coupled system of equations involv-\ning the partition coefficients of small ions according to the\nideal Donnan theory and the Henderson–Hasselbalch equa-\ntions for the various ionizable residues in the system. In order\nto perform this computation, pyMBE implements the function\ncalculate_HH_Donnan :\nZ_HH = pmb. calculate_HH_Donnan (\nespresso_system = es_system ,\nobject_names = [pep1 , pep2 ],\nc_salt = salt_conc_res ,\npH_list = pH_range )\nNote that in contrast to the function calculate_HH , one\nneeds to provide a list of allcharged molecule types (here\nshown for a system containing two different peptides) as well\nas the salt concentration in the reservoir. This additional in-\nformation is required to correctly calculate the Donnan parti-\ntioning, which depends on the ionic strength of the reservoir\nand the concentration of impermeable charges in the system.\nWhile the above example is a rather simple system that\ncould also be easily set up without pyMBE, combining com-\nplex molecular architectures with the G-RxMC method is now\npossible in a straightforward manner. For example, using the\nnew implementation, one can study the dialysis of complex\ncharge-regulating polypeptides and globular proteins, a pro-\ncess that is important from the point of view of chemical en-\ngineering and pharmaceuticals.\nC. Globular proteins in bulk solution\nAs our last example of a use case of pyMBE, we study the\npH-response of the two globular proteins most abundant in\nwhey: α−lactalbumin and β−lactoglobulin. Motivated by\nthe multiple industrial applications of milk whey, Torres et\nal.18,19investigated these proteins using coarse-grained (CG)\nmodels built from the crystallographic structure available in\nthe Protein Data Bank (PDB). Owing to the fairly rigid struc-\nture of globular proteins, they used a CG model consisting\nof a rigid body with two beads per amino acid as described\nin Section II D. Torres et al. measured the titration curves of\nα−lactalbumin and β−lactoglobulin using an in-house cpH\nMonte Carlo software developed in their group, observing that\nthe isoelectric point of both proteins was reasonably in line\nwith other computational models.65\nSetting up a cpH simulation of a globular protein is a non-\ntrivial task, even for such a simple CG model, since one needs\nto correctly build the protein structure into the software and to\nproperly set up the cpH method for all the different reactive\ngroups in the protein. As explained in Section II D, pyMBE\nprovides tools to easily set up CG models of globular proteins\nfrom their PDB structures. In Fig. 7, we compare the net\ncharge Zofα−lactalbumin (panel a) and β−lactoglobulin\n(panel b) at various pH values in dilute solution measured\nwith pyMBE with the corresponding ones reported by Torres\net al. .18,19In addition, we plot the solution of the Henderson–\n−1001020Net charge Z/e\na-lactalbumin (1F6S)(a)HH\nTorres et al.pyMBE\n2 3 4 5 6 7\npH in the solution−1001020Net charge Z/e\nb-lactoglobulin (1BEB)(b)Figure 7. Net charge Zofα−lactalbumin (panel a) and\nβ−lactoglobulin (panel b) as a function of the pH. pyMBE has been\nused to set up a coarse-grained model of each protein from their\nPDB crystallographic data: α−lactalbumin (PDB code: 1F6S54) and\nβ−lactoglobulin (PDB code: 1BEB66). The net charge measured\nwith pyMBE (orange circles) matches the reference data reported by\nTorres et al.18,19(blue squares) within the estimated error. For ref-\nerence, the analytical solution of the Henderson–Hasselbalch (HH)\nequation Eqs. 11-12 for each protein is plotted as a black line.\nHasselbalch (HH) equation (Eqs. 11 and 12) as a benchmark\ncorresponding to expected net charge of each protein under\nideal, non-interacting, conditions. As a general trend, the ab-\nsolute value of the net charge measured by cpH simulation is\nbelow to that of the HH equation due to the “polyelectrolyte\neffect” described in Section III C. In general, pyMBE reason-\nably reproduces the data reported by Torres et al.18,19and we\nobserve a quantitative agreement between both sets of data\nwithin the reported error bars.\nIn summary, pyMBE provides tools to easily set up CG\nmodels of globular proteins in ESPResSo which, in com-\nbination with the above-mentioned Monte Carlo reaction\nmethods, opens the door to many potential applications that\ncould by studied by molecular simulation. Globular proteins\nare ubiquitous in nature and they have many important bi-\nological functions such as binding67,68, catalysis,69regula-\ntion, transport,69immunity, cellular signaling and more.67,70\nMolecular simulations using CG models could bring new in-\nsights on the underlying mechanisms controlling these bio-13\nlogical functions for example by modelling system consist-\ning on proteins in dilute and concentrated solution71, protein-\npolyelectrolyte complexes72,73and two-phase systems of pro-\nteins in dialysis conditions.31\nIV. CONCLUSIONS AND OUTLOOKS\nWe presented the Python-based Molecule Builder for\nESPResSo (pyMBE). pyMBE is an open-source software that\nfacilitates the setup of coarse-grained (CG) models of com-\nplex molecules, such as peptides and proteins, in ESPResSo.\nWe showed that pyMBE reproduces data from cpH and G-\nRxMC simulation produced by different research groups and\nsoftware, which served as both a validation of the library and\na future set of test cases for further development of pyMBE.\nThe development of a common tool to do these simulations\nrepresents also a step forwards for data reproducibility in the\ncommunity. The use of pyMBE lowers the barrier to set up\nconstant pH and grand-reaction Monte Carlo (G-RxMC) sim-\nulations, opening the door to simulate systems not studied us-\ning these approaches yet, such as solutions of globular pro-\nteins in dialysis,31peptide hydrogels74or dendrimer polyelec-\ntrolytes with peptidic tails.75In particular, simulations using\nthe G-RxMC methods are still scarce in the literature despite\nits many potential applications to study two-phase systems, as\nwe will describe in an upcoming perspective article.\nThere is a growing number of tools to automate the coarse-\ngraining of existing all-atom models (martinize.py76,\nMartinize2,77Auto_Martini,78MAD,79PDB→UNF\nConverter,80GENESIS-CG-tool81), as well as special-\npurpose CG molecular builders (e.g. for DNA origami,82,83\nand lipid membranes84). However, general-purpose CG\nmolecule builders capable of generating arbitrary polymer\ntopologies are still very scarce. To our knowledge, cur-\nrently the only molecule builders with such capabilities\nare Moltemplate85and MoSDeF-GOMC.86Moltemplate is\ndesigned to facilitate the setup of input files for LAMMPS\nand, although it can also be used to set up simulations with\nESPResSo, its compatibility is limited to a deprecated version\nof the software (ESPResSo v3.3.1). MoSDeF-GOMC is a\nPython interface for GOMC,87designed to set up Monte\nCarlo simulations in various statistical ensembles of CG\nmodels of small molecules and crystals with applications in\nMaterial Science. pyMBE has been designed specifically for\nESPResSo, with a special focus on supporting the setup of\nMonte Carlo methods to model reactive Soft Matter systems.\npyMBE is currently under active development as a collab-\norative project between researchers on the field of charge reg-\nulation of polyelectrolytes and biomacromolecules. We plan\nto continue developing the software by extending pyMBE to\nbuild CG models of other molecules, such as hydrogels, den-\ndrimers and nanoparticles. We also plan to develop pyMBE\nto facilitate the setting up of established force fields for CG\nmodels such as MARTINI88into ESPResSo.ACKNOWLEDGMENTS\nWe thank the early users of pyMBE who have contributed\nto its development by providing valuable feedback on their\nexperience when using the library: Corinna Dannert, Rita S.\nDias, Sergio Madurga, Alberto Martinez-Serra, Francesc Mas,\nMagdaléna Nejedlá and Raju Lunkad.\nD.B. acknowledges the German Research Foundation\n(DFG) for funding within the Research Unit FOR2811 “Adap-\ntive Polymer Gels with Model Network Structure” under\ngrant 423791428 along with grant 397384169 (TP7). P.B.T.\nacknowledges Ph.D. fellowship from CONICET. S.P. and\nP.K. acknowledge financial support of the Czech Science\nfoundation, grant 21-31978J. C.F.N. acknowledges the finan-\ncial support from AGENCIA I+D+i (FONCyT), PICT-2021-\nGRFTI-00090 and from Universidad Tecnologica Nacional\n(PIDs PATCASR0008459 and PATCASR0008463). J.-N.G.\nacknowledges the DFG for funding under grant 528726435\n(PI: Holm). P.M.B. acknowledges the financial support\nfrom the Spanish Ministry of Universities (Margarita Salas\nGrant MS98), from the Generalitat de Catalunya (Grant\n2021SGR00350) and from the European Union’s Horizon Eu-\nrope research and innovation programme under the Marie\nSklodowska-Curie grant agreement No 101062456 (ModE-\nMUS).\nDATA AVAILABILITY STATEMENT\nThe original data showcased in this article in Figs. 5-7\nis available in the pyMBE repository in the folder ’/refer-\nence_data’ with the permission of its original authors. These\ndata sets were originally published in Refs.13,18,19,25,45. We\nalso provide the set of scripts that we used to reproduce the\ndata of these articles with pyMBE in the repository of the soft-\nware in the folder ’/tests’.\nCONFLICT OF INTEREST\nThe authors have no conflicts to disclose.\nAUTHOR CONTRIBUTIONS\nD.B.: Software, Validation, Visualization, Writing – origi-\nnal draft, Writing – review & editing. P.B.T.: Software, Val-\nidation, Visualization, Writing – original draft. S.P.P.: Soft-\nware, Visualization, Writing – original draft. C.F.N.: Funding\nacquisition, Supervision, Writing – review & editing. J.-N.G.:\nSoftware, Writing – review & editing. P.K.: Conceptualiza-\ntion, Funding acquisition, Supervision, Writing – review &\nediting. P.M.B: Conceptualization, Funding acquisition, Soft-\nware, Supervision, Validation, Visualization, Writing – origi-\nnal draft, Writing – review & editing.14\nAppendix A: Dependencies of pyMBE\npyMBE has dependencies in the following libraries:\nESPResSo (v4.2.1),1NumPy(v1.23),89Pandas(v1.5.3),43,44\nPint(v0.20.01),42Pint-Pandas (v0.5)90and, only for develop-\ners, pdoc (v14.3).41\nAppendix B: Additional examples of the pyMBE Dataframe\nTo extend the description of the pyMBE dataframe that\nwe provided in Section II B, let us consider one of the ex-\nample cases presented in Section II C. For simplicity, let us\nconsider a user that uses pyMBE to create two molecules\nof the type name = \"alternating_polymer\" with a small\nnumber of monomeric units, for example 2. When creat-\ning objects into ESPResSo, pyMBE adds one row per ob-\nject into the pyMBE dataframe to bookkeep the information\nof every individual object created. For example, if the user\nfilters the pyMBE dataframe by pmb_type = \"residue\" ,\npyMBE returns the dataframe depicted in Table III. The first\nfour rows correspond to residues belonging to a molecule\nwith molecule_id = 0 while the last four correspond to a\nsecond molecule with a molecule_id = 1 . The user can\nalso filter by pmb_type = \"bond\" to check how pyMBE has\nbonded the particle to create the two molecules into the sys-\ntem as shown in Table IV. In this case, the subset of the\npyMBE dataframe displays 14 rows, corresponding to the\nnumber of bonds needed to build the two example molecules.\nparticle_id1 andparticle_id2 store the numerical iden-\ntifiers of the two bonded particles within the ESPResSo sys-\ntem. bond_object stores the instance of the bond object\nfrom the ESPResSo library used to set up the bonding poten-\ntial between the particle, in which the parameters of the bond-\ning potential can be consulted. If the user filters the pyMBE\ndataframe by the pmb_type = \"molecule\" , pyMBE returns\nthe dataframe depicted in Table V. Each row of the dataframe\ncorresponds to a different molecule in the ESPResSo sys-\ntem, where the user can observe that the molecules with\nmolecule_id of 0 and 1 correspond of molecules of the type\ngiven by name = \"alternating_polymer\" .\nREFERENCES\n1F. Weik, R. Weeber, K. Szuttor, K. Breitsprecher, J. de Graaf, M. Kuron,\nJ. Landsgesell, H. Menke, D. Sean, and C. Holm, “ESPResSo 4.0 – an\nextensible software package for simulating soft matter systems,” European\nPhysical Journal Special Topics 227, 1789–1816 (2019).\n2R. Weeber, J.-N. Grad, D. Beyer, P. M. Blanco, P. Kreissl, A. Reinauer,\nI. Tischler, P. Košovan, and C. Holm, “ESPResSo, a versatile open-source\nsoftware package for simulating soft matter systems,” in Comprehensive\nComputational Chemistry , edited by M. Yáñez and R. J. Boyd (Elsevier,\nOxford, 2024) 1st ed., pp. 578–601.\n3H. V . Guzman, N. Tretyakov, H. Kobayashi, A. C. Fogarty, K. Kreis, J. Kra-\njniak, C. Junghans, K. Kremer, and T. Stuehn, “ESPResSo++ 2.0: Ad-\nvanced methods for multiscale molecular simulation,” Computer Physics\nCommunications 238, 66–76 (2019), 1806.10841.\n4J. A. Anderson, J. Glaser, and S. C. Glotzer, “HOOMD-blue: A Python\npackage for high-performance molecular dynamics and hard particle Monte\nCarlo simulations,” Computational Materials Science 173, 109363 (2020).Table III. Example of a subset of the pyMBE dataframe, filtered\nbypmb_type = \"residue\" . The cells under the columns name\n,central_bead ,side_chains store the parameters provided by\nthe user when defining each residue. pmb_type ,residue_id\nandmolecule_id correspond to identifiers internally assigned by\npyMBE. The values correspond to the molecule with name = \"\nalternating_polymer\" , showcased in Section II C.\nname pmb_type residue_id molecule_id central_bead side_chains ···\nIA residue 0 0 I [I,A] ···\nIB residue 1 0 I [I,B] ···\nIA residue 2 0 I [I,A] ···\nIB residue 3 0 I [I,B] ···\nIA residue 4 1 I [I,A] ···\nIB residue 5 1 I [I,A] ···\nIA residue 6 1 I [I,B] ···\nIB residue 7 1 I [I,A] ···\n.....................\nTable IV . Example of a subset of the pyMBE dataframe, filtered by\npmb_type = \"bond\" . The values correspond to the molecule with\nname = \"alternating_polymer\" , showcased in Section II C.\nname pmb_type particle_id particle_id2 bond_object ···\nI-A bond 0 1 HarmonicBond() ···\nI-I bond 0 2 HarmonicBond() ···\nI-B bond 2 3 HarmonicBond() ···\nI-I bond 2 4 HarmonicBond() ···\nI-A bond 4 5 HarmonicBond() ···\nI-I bond 4 6 HarmonicBond() ···\nI-B bond 6 7 HarmonicBond() ···\nI-A bond 8 9 HarmonicBond() ···\nI-I bond 8 10 HarmonicBond() ···\nI-B bond 10 11 HarmonicBond() ···\nI-I bond 10 12 HarmonicBond() ···\nI-A bond 12 13 HarmonicBond() ···\nI-I bond 12 14 HarmonicBond() ···\nI-B bond 14 15 HarmonicBond() ···\n..................\n5R. Jurij and L. Per, “MOLSIM: A modular molecular simulation software,”\nJournal of Computational Chemistry 36, 1259–1274 (2015).\n6A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M.\nBrown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D.\nNguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton,\n“LAMMPS - a flexible simulation tool for particle-based materials model-\ning at the atomic, meso, and continuum scales,” Computer Physics Com-\nmunications 271, 108171 (2022).\n7The Canva development team, “Canva: Visual suite for everyone,” (2023).\n8J. Landsgesell, L. Nová, O. Rud, F. Uhlík, D. Sean, P. Hebbeker, C. Holm,\nand P. Košovan, “Simulations of ionization equilibria in weak polyelec-\ntrolyte solutions and gels,” Soft Matter 15, 1155–1185 (2019).\n9P. M. Blanco, C. F. Narambuena, S. Madurga, F. Mas, and J. L. Garcés,\n“Unusual aspects of charge regulation in flexible weak polyelectrolytes,”15\nTable V . Example of a subset of the pyMBE dataframe, filtered\nbypmb_type = \"molecule\" . The cells under the columns name\n,residue_list store the parameters provided by the user when\ndefining each molecule. pmb_type andmolecule_id correspond to\nidentifiers internally assigned by pyMBE. The values correspond to\nthe molecule with name = \"alternating_polymer\" , showcased\nin Section II C.\nname pmb_type molecule_id residue_list ···\nalternating_polymer molecule 0 [IA,IB,IA,IB] ···\nalternating_polymer molecule 1 [IA,IB,IA,IB] ···\n...............\nPolymers 15, 2680 (2023).\n10C. E. Reed and W. F. Reed, “Monte Carlo study of titration of linear poly-\nelectrolytes,” The Journal of Chemical Physics 96, 1609–1620 (1992).\n11M. Ullner, B. Jönsson, B. Söderberg, and C. Peterson, “A Monte Carlo\nstudy of titrating polyelectrolytes,” The Journal of Chemical Physics 104,\n3048–3057 (1996).\n12M. Ullner and B. Jönsson, “A Monte Carlo study of titrating polyelec-\ntrolytes in the presence of salt,” Macromolecules 29, 6645–6655 (1996).\n13P. M. Blanco, S. Madurga, J. L. Garcés, F. Mas, and R. S. Dias, “Influ-\nence of macromolecular crowding on the charge regulation of intrinsically\ndisordered proteins,” Soft Matter 17, 655–669 (2021).\n14R. Lunkad, A. Murmiliuk, Z. Tošner, M. Št ˇepánek, and P. Košovan,\n“Role of p KAin charge regulation and conformation of various peptide\nsequences,” Polymers 13, 214 (2021).\n15R. Lunkad, P. Biehl, A. Murmiliuk, P. M. Blanco, P. Mons, M. Št ˇepánek,\nF. H. Schacher, and P. Košovan, “Simulations and potentiometric titrations\nenable reliable determination of effective p Kavalues of various polyzwitte-\nrions,” Macromolecules 55, 7775–7784 (2022).\n16M. Lund and B. Jönsson, “On the charge regulation of proteins,” Biochem-\nistry44, 5722–5727 (2005).\n17F. L. B. da Silva and B. Jönsson, “Polyelectrolyte–protein complexation\ndriven by charge regulation,” Soft Matter 5, 2862–2868 (2009).\n18P. B. Torres, E. Quiroga, A. J. Ramirez-Pastor, V . Boeris, and C. F.\nNarambuena, “Interaction between β-lactoglobuline and weak polyelec-\ntrolyte chains: A study using Monte Carlo simulation,” The Journal of\nPhysical Chemistry B 123, 8617–8627 (2019).\n19P. B. Torres, P. M. Blanco, J. L. Garcés, and C. F. Narambuena, “The\nelectrostatic potential inside and around α-lactalbumin: Fluctuations and\nmean-field models,” The Journal of Chemical Physics 157, 205101 (2022).\n20C. Hofzumahaus, P. Hebbeker, and S. Schneider, “Monte Carlo simulations\nof weak polyelectrolyte microgels: pH-dependence of conformation and\nionization,” Soft Matter 14, 4087–4100 (2018).\n21C. Hofzumahaus, C. Strauch, and S. Schneider, “Monte Carlo simulations\nof weak polyampholyte microgels: pH-dependence of conformation and\nionization,” Soft Matter 17, 6029–6043 (2021).\n22C. Strauch and S. Schneider, “Ionisation and swelling behaviour of weak\npolyampholyte core–shell networks – a Monte Carlo study,” Soft Matter\n19, 938–950 (2023).\n23A. Clavier, M. Seijo, F. Carnal, and S. Stoll, “Surface charging behav-\nior of nanoparticles by considering site distribution and density, dielectric\nconstant and pH changes–a Monte Carlo approach,” Physical Chemistry\nChemical Physics 17, 4346–4353 (2015).\n24M. Stornes, P. M. Blanco, and R. S. Dias, “Polyelectrolyte-nanoparticle\nmutual charge regulation and its influence on their complexation,” Colloids\nand Surfaces A: Physicochemical and Engineering Aspects 628, 127258\n(2021).\n25J. Landsgesell, P. Hebbeker, O. Rud, R. Lunkad, P. Košovan, and C. Holm,\n“Grand-reaction method for simulations of ionization equilibria coupled to\nion partitioning,” Macromolecules 53, 3007–3020 (2020).\n26D. Beyer, J. Landsgesell, P. Hebbeker, O. Rud, R. Lunkad, P. Košovan, and\nC. Holm, “Correction to “Grand-reaction method for simulations of ion-\nization equilibria coupled to ion partitioning”,” Macromolecules 55, 1088(2022).\n27J. Landsgesell, D. Beyer, P. Hebbeker, P. Košovan, and C. Holm, “The pH-\ndependent swelling of weak polyelectrolyte hydrogels modeled at different\nlevels of resolution,” Macromolecules 55, 3176–3188 (2022).\n28D. Beyer, P. Košovan, and C. Holm, “Simulations explain the swelling\nbehavior of hydrogels with alternating neutral and weakly acidic blocks,”\nMacromolecules 55, 10751–10760 (2022).\n29R. Sta ˇno, J. van Lente, S. Lindhoud, and P. Košovan, “Sequestration of\nsmall ions, weak acids and bases by polyelectrolyte complex studied by\nsimulation and experiment,” ChemRxiv (2023), 10.26434/chemrxiv-2023-\n6q159.\n30D. Beyer, P. Košovan, and C. Holm, “Explaining giant apparent p Kashifts\nin weak polyelectrolyte brushes,” Physical Review Letters 131, 168101\n(2023).\n31T. Briskot, N. Hillebrandt, S. Kluters, G. Wang, J. Studts, T. Hahn, T. Huuk,\nand J. Hubbuch, “Modeling the Gibbs–Donnan effect during ultrafiltration\nand diafiltration processes using the Poisson–Boltzmann theory in com-\nbination with a basic Stern model,” Journal of Membrane Science 648,\n120333 (2022).\n32M. D. Hanwell, D. E. Curtis, D. C. Lonie, T. Vandermeersch, E. Zurek,\nand G. R. Hutchison, “Avogadro: an advanced semantic chemical editor,\nvisualization, and analysis platform,” Journal of Cheminformatics 4(2012),\n10.1186/1758-2946-4-17.\n33W. Humphrey, A. Dalke, and K. Schulten, “VMD: visual molecular dy-\nnamics,” Journal of molecular graphics 14, 33–38 (1996).\n34Schrödinger, LLC, “The PyMOL molecular graphics system, ver-\nsion 2.5.7,” (2023).\n35R. Salomon-Ferrer, D. A. Case, and R. C. Walker, “An overview of the Am-\nber biomolecular simulation package,” Wiley Interdisciplinary Reviews:\nComputational Molecular Science 3, 198–210 (2012).\n36D. A. Case, H. M. Aktulga, K. Belfon, D. S. Cerutti, G. A. Cisneros,\nV . W. D. Cruzeiro, N. Forouzesh, T. J. Giese, A. W. Götz, H. Gohlke,\nS. Izadi, K. Kasavajhala, M. C. Kaymak, E. King, T. Kurtzman, T.-S. Lee,\nP. Li, J. Liu, T. Luchko, R. Luo, M. Manathunga, M. R. Machado, H. M.\nNguyen, K. A. O’Hearn, A. V . Onufriev, F. Pan, S. Pantano, R. Qi, A. Rah-\nnamoun, A. Risheh, S. Schott-Verdugo, A. Shajan, J. Swails, J. Wang,\nH. Wei, X. Wu, Y . Wu, S. Zhang, S. Zhao, Q. Zhu, T. E. Cheatham, D. R.\nRoe, A. Roitberg, C. Simmerling, D. M. York, M. C. Nagan, and K. M.\nMerz, “AmberTools,” Journal of Chemical Information and Modeling 63,\n6183–6191 (2023).\n37J. Sallai, G. Varga, S. Toth, C. Iacovella, C. Klein, C. McCabe, A. Ledeczi,\nand P. T. Cummings, “Web- and cloud-based software infrastructure for\nmaterials design,” Procedia Computer Science 29, 2034–2044 (2014).\n38C. Klein, J. Sallai, T. J. Jones, C. R. Iacovella, C. McCabe, and P. T. Cum-\nmings, “A hierarchical, component based approach to screening properties\nof soft matter,” in Molecular Modeling and Simulation (Springer Singapore,\n2016) pp. 79–92.\n39P. T. Cummings, C. MCabe, C. R. Iacovella, A. Ledeczi, E. Jankowski,\nA. Jayaraman, J. C. Palmer, E. J. Maginn, S. C. Glotzer, J. A. Anderson,\nJ. Ilja Siepmann, J. Potoff, R. A. Matsumoto, J. B. Gilmer, R. S. DeFever,\nR. Singh, and B. Crawford, “Open-source molecular modeling software in\nchemical engineering focusing on the Molecular Simulation Design Frame-\nwork,” AIChE Journal 67(2021), 10.1002/aic.17206.\n40Molydyn Ltd., “Atlas: Supporting your simulations,” (2023).\n41A. Gallant and M. Hils, “pdoc – generate API documentation for Python\nprojects,” GitHub (2023).\n42H. E. Grecco and J. Chéron, “pint: makes units easy,” GitHub (2023).\n43W. McKinney, “Data structures for statistical computing in Python,” in Pro-\nceedings of the 9thPython in Science Conference , edited by S. van der Walt\nand J. Millman (2010) pp. 56–61.\n44The pandas development team, “pandas-dev/pandas: Pandas, ver-\nsion 1.5.3,” Zenodo (2023).\n45R. Lunkad, A. Murmiliuk, P. Hebbeker, M. Boublík, Z. Tošner,\nM. Št ˇepánek, and P. Košovan, “Quantitative prediction of charge regulation\nin oligopeptides,” Molecular Systems Design and Engineering 6, 122–131\n(2021).\n46E. A. Bienkiewicz and K. J. Lumb, “Random-coil chemical shifts of\nphosphorylated amino acids,” Journal of Biomolecular NMR 15, 203–206\n(1999).16\n47D. R. Lide, ed., CRC Handbook of Chemistry and Physics , 72nd ed. (CRC\nPress, Boca Raton, Florida, USA, 1991).\n48P. Dobrev, S. P. B. Vemulapalli, N. Nath, C. Griesinger, and H. Grubmüller,\n“Probing the accuracy of explicit solvent constant pH molecular dynamics\nsimulations for peptides,” Journal of Chemical Theory and Computation 16,\n2561–2569 (2020).\n49P. Dobrev, S. P. B. Vemulapalli, N. Nath, C. Griesinger, G. Groenhof,\nand H. Grubmüller, “Correction to “Probing the accuracy of explicit sol-\nvent constant pH molecular dynamics simulations for peptides”,” Journal\nof Chemical Theory and Computation 16, 4753–4753 (2020).\n50M. A. S. Hass and F. A. A. Mulder, “Contemporary NMR studies of protein\nelectrostatics,” Annual Review of Biophysics 44, 53–75 (2015).\n51Y . Nozaki and C. Tanford, “Examination of titration behavior,” Enzyme\nStructure , , 715–734 (1967).\n52G. Platzer, M. Okon, and L. P. McIntosh, “pH-dependent random coil1H,\n13C, and15N chemical shifts of the ionizable amino acids: a guide for\nprotein p Kameasurements,” Journal of Biomolecular NMR 60, 109–129\n(2014).\n53R. L. Thurlkill, G. R. Grimsley, J. M. Scholtz, and C. N. Pace, “pK values\nof the ionizable groups of proteins,” Protein Science 15, 1214–1218 (2006).\n54E. D. Chrysina, K. Brew, and K. R. Acharya, “Crystal structures of apo- and\nholo-bovine α-lactalbumin at 2.2-Å resolution reveal an effect of calcium\non inter-lobe interactions,” Journal of Biological Chemistry 275, 37021–\n37029 (2000).\n55P. Košovan, J. Landsgesell, L. Nová, F. Uhlík, D. Beyer, P. M. Blanco,\nR. Sta ˇno, and C. Holm, “Reply to the ‘Comment on “Simulations of ion-\nization equilibria in weak polyelectrolyte solutions and gels”’ by J. Lands-\ngesell, L. Nová, O. Rud, F. Uhlík, D. Sean, P. Hebbeker, C. Holm and P.\nKošovan, Soft Matter , 2019, 15, 1155–1185,” Soft Matter 19, 3522–3525\n(2023).\n56A. D. McNaught and A. Wilkinson, IUPAC. Compendium of Chemical\nTerminology (Blackwell Scientific Publications, Oxford, United Kingdom,\n1997).\n57W. R. Smith and B. Triska, “The reaction ensemble method for the com-\nputer simulation of chemical and phase equilibria. I. Theory and basic ex-\namples,” The Journal of Chemical Physics 100, 3019–3027 (1994).\n58J. K. Johnson, A. Z. Panagiotopoulos, and K. E. Gubbins, “Reactive canon-\nical Monte Carlo: A new simulation technique for reacting or associating\nfluids,” Molecular Physics 81, 717–733 (1994).\n59D. Beyer and C. Holm, “A generalized grand-reaction method for mod-\neling the exchange of weak (polyprotic) acids between a solution and a\nweak polyelectrolyte phase,” The Journal of Chemical Physics 159, 014905\n(2023).\n60T. Curk, J. Yuan, and E. Luijten, “Accelerated simulation method for charge\nregulation effects,” The Journal of Chemical Physics 156, 044122 (2022).\n61K. Hyltegren, T. Nylander, M. Lund, and M. Skepö, “Adsorption of the\nintrinsically disordered saliva protein histatin 5 to silica surfaces. a Monte\nCarlo simulation and ellipsometry study,” Journal of Colloid and Interface\nScience 467, 280–290 (2016).\n62B. J. MacKay, L. Denepitiya, V . J. Iacono, S. B. Krost, and J. J. Pol-\nlock, “Growth-inhibitory and bactericidal effects of human parotid salivary\nhistidine-rich polypeptides on streptococcus mutans,” Infection and Immu-\nnity44, 695–701 (1984).\n63S. Puri and M. Edgerton, “How does it kill?: Understanding the candidaci-\ndal mechanism of salivary histatin 5,” Eukaryotic Cell 13, 958–964 (2014).\n64G. S. Grest and K. Kremer, “Molecular dynamics simulation for polymers\nin the presence of a heat bath,” Physical Review A 33, 3628 (1986).\n65A. A. R. Teixeira, M. Lund, and F. L. Barroso da Silva, “Fast proton titra-\ntion scheme for multiscale modeling of protein solutions,” Journal of Chem-\nical Theory and Computation 6, 3259–3266 (2010).\n66S. Brownlow, J. H. Morais Cabral, R. Cooper, D. R. Flower, S. J. Yewdall,\nI. Polikarpov, A. C. T. North, and L. Sawyer, “Bovine β-lactoglobulin\nat 1.8 Å resolution — still an enigmatic lipocalin,” Structure 5, 481–495\n(1997).\n67D. J. McClements, “Modulation of globular protein functionality by weakly\ninteracting cosolvents,” Critical Reviews in Food Science and Nutrition 42,\n417–471 (2002).\n68L. H. Lucas and C. K. Larive, “Measuring ligand-protein binding using\nNMR diffusion experiments,” Concepts in Magnetic Resonance Part A\n20A, 24–41 (2004).69T. E. Creighton, Proteins: Structures and Molecular Properties , 2nd ed.\n(W. H. Freeman and Company, 1993).\n70E. Dickinson and D. J. McClements, Advances in Food Colloids (Springer\nScience & Business Media, 1995).\n71J. K. Cheung and T. M. Truskett, “Coarse-grained strategy for modeling\nprotein stability in concentrated solutions,” Biophysical Journal 89, 2372–\n2384 (2005).\n72F. Liu, D. J. McClements, C. Ma, and X. Liu, “Novel colloidal food ingre-\ndients: Protein complexes and conjugates,” Annual Review of Food Science\nand Technology 14, 35–61 (2023).\n73F. L. Barroso da Silva, M. Lund, B. Jönsson, and T. Åkesson, “On the com-\nplexation of proteins and polyelectrolytes,” The Journal of Physical Chem-\nistry B 110, 4459–4464 (2006).\n74P. Katyal, F. Mahmoudinobar, and J. K. Montclare, “Recent trends in pep-\ntide and protein-based hydrogels,” Current Opinion in Structural Biology\n63, 97–105 (2020).\n75C. Dannert, I. Mardal, R. Lale, B. T. Stokke, and R. S. Dias, “DNA con-\ndensation by peptide-conjugated PAMAM dendrimers. influence of peptide\ncharge,” ACS Omega 8, 44624–44636 (2023).\n76D. H. de Jong, G. Singh, W. F. D. Bennett, C. Arnarez, T. A. Wassenaar,\nL. V . Schäfer, X. Periole, D. P. Tieleman, and S. J. Marrink, “Improved\nparameters for the martini coarse-grained protein force field,” Journal of\nChemical Theory and Computation 9, 687–697 (2012).\n77P. C. Kroon, F. Grünewald, J. Barnoud, M. v. Tilburg, P. C. T. Souza,\nT. A. Wassenaar, and S.-J. Marrink, “Martinize2 and Vermouth: Unified\nframework for topology generation,” (2022), 10.48550/arXiv.2212.01191,\n2212.01191v2.\n78T. Bereau and K. Kremer, “Automated parametrization of the coarse-\ngrained Martini force field for small organic molecules,” Journal of Chem-\nical Theory and Computation 11, 2783–2791 (2015).\n79C. Hilpert, L. Beranger, P. C. T. Souza, P. A. Vainikka, V . Nieto, S. J.\nMarrink, L. Monticelli, and G. Launay, “Facilitating CG simulations with\nMAD: The MArtini Database server,” Journal of Chemical Information and\nModeling 63, 702–710 (2023).\n80D. Kut’ák, E. Poppleton, H. Miao, P. Šulc, and I. Bariši ´c, “Unified nan-\notechnology format: One way to store them all,” Molecules 27, 63 (2021).\n81C. Tan, J. Jung, C. Kobayashi, D. U. L. Torre, S. Takada, and Y . Sugita,\n“Implementation of residue-level coarse-grained models in GENESIS for\nlarge-scale molecular dynamics simulations,” PLOS Computational Biol-\nogy18, e1009578 (2022).\n82J. P. K. Doye, T. E. Ouldridge, A. A. Louis, F. Romano, P. Šulc, C. Matek,\nB. E. K. Snodin, L. Rovigatti, J. S. Schreck, R. M. Harrison, and W. P. J.\nSmith, “Coarse-graining DNA for simulations of DNA nanotechnology,”\nPhysical Chemistry Chemical Physics 15, 20395–20414 (2013).\n83J. M. Majikes and J. A. Liddle, “DNA origami design: A how-to tutorial,”\nJournal of Research of the National Institute of Standards and Technology\n126(2021), 10.6028/jres.126.001.\n84T. A. Wassenaar, H. I. Ingólfsson, R. A. Böckmann, D. P. Tieleman, and\nS. J. Marrink, “Computational lipidomics with insane : A versatile tool\nfor generating custom membranes for molecular simulations,” Journal of\nChemical Theory and Computation 11, 2144–2155 (2015).\n85A. I. Jewett, D. Stelter, J. Lambert, S. M. Saladi, O. M. Roscioni, M. Ricci,\nL. Autin, M. Maritan, S. M. Bashusqeh, T. Keyes, R. T. Dame, J.-E. Shea,\nG. J. Jensen, and D. S. Goodsell, “Moltemplate: A tool for coarse-grained\nmodeling of complex biological matter and soft condensed matter physics,”\nJournal of Molecular Biology 433, 166841 (2021).\n86B. Crawford, U. Timalsina, C. D. Quach, N. C. Craven, J. B. Gilmer, C. Mc-\nCabe, P. T. Cummings, and J. J. Potoff, “MoSDeF-GOMC: Python soft-\nware for the creation of scientific workflows for the Monte Carlo simula-\ntion engine GOMC,” Journal of Chemical Information and Modeling 63,\n1218–1228 (2023).\n87Y . Nejahi, M. S. Barhaghi, G. Schwing, L. Schwiebert, and J. Potoff, “Up-\ndate 2.70 to “GOMC: GPU optimized Monte Carlo for the simulation of\nphase equilibria and physical properties of complex fluids”,” SoftwareX 13,\n100627 (2021).\n88P. C. T. Souza, R. Alessandri, J. Barnoud, S. Thallmair, I. Faustino,\nF. Grünewald, I. Patmanidis, H. Abdizadeh, B. M. H. Bruininks, T. A.\nWassenaar, P. C. Kroon, J. Melcr, V . Nieto, V . Corradi, H. M. Khan, J. Do-\nma´nski, M. Javanainen, H. Martinez-Seara, N. Reuter, R. B. Best, I. Vat-\ntulainen, L. Monticelli, X. Periole, D. P. Tieleman, A. H. de Vries, and17\nS. J. Marrink, “Martini 3: a general purpose force field for coarse-grained\nmolecular dynamics,” Nature Methods 18, 382–388 (2021).\n89C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virta-\nnen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern,\nM. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. delRío, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy,\nW. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array program-\nming with NumPy,” Nature 585, 357–362 (2020).\n90H. E. Grecco, “Pint-pandas: Extend pandas dataframe with physical quan-\ntities module,” GitHub (2023)." }, { "title": "2401.14987v1.Null_controllability_for_the_beam_equation_with_structural_damping__Part_1__Distributed_control.pdf", "content": "arXiv:2401.14987v1 [math.OC] 26 Jan 2024Null-controllability for the beam equation with\nstructural damping.\nPart 1. Distributed control\nSergei Avdonin, Julian Edward, and Sergei Ivanov\nJanuary 20, 2024\nAbstract\nLet ∆ be the Dirichlet Laplacian on the interval (0 ,π). The null controllability\nproperties of the equation\nutt+∆2u+ρ(∆)αut=F(x,t)\nare studied. Let T >0, and assume initial conditions ( u0,u1)∈Dom(∆)×L2(0,π).\nWe first prove finite dimensional null control results: suppo seF(x,t) =f1(t)h1(x) +\nf2(t)h2(x) withh1,h2given functions. For α∈[0,3/2), we prove that there exist\nh1,h2∈L2(0,π) such that for any ( u0,u1), there exist L2null controls ( f1,f2).For\nα <1 andρ <2, we prove null controllability with f2= 0 and h1belonging to a\nlarge class of functions. For α∈[3/2,2), we prove spectral and null controllability\nboth generally fail, but two dimensional weak controllabil ity holds. Our second set\nof results pertains to F(x,t) =χΩ(x)f(x,t), with Ω any open subset of (0 ,π). For\nanyα∈[0,3/2),we prove there exists a null control f∈L2(Ω×(0,T)) To prove our\nmain results, we use the Fourier method to rewrite the contro l problems as moment\nproblems. These are then solved by constructing biorthogon al sets to the associated\nexponential families. These constructions seem to be non-s tandard and may be of\nindependent interest.\n1 Introduction\nLet ∆ be the Laplacian: ∆ = −∂2\nx,with operator domain H2(0,π)∩H1\n0(0,π). It is well\nknown that this operator is self-adjoint with positive spectrum, an d hence ∆αis defined by\nthe Spectral Theorem for all α. We will study control problems for the equation\nutt+∆2u+ρ(∆)αut=Bf, x∈(0,π), t >0,\nwith a positive constant ρ,α∈[0,2],and where Bis the control operator.\nThis system is actuated through a control mechanism prescribed b y the operator B\n(possibly unbounded to take into account trace operators presc ribing the boundary value\nof distributed states). Throughout this paper, controllability will a lways mean the ability\n1of steering any initial state ( u(x,0),ut(x,0)) to zero over a finite time by some appropriate\ninput function f(i.e. exact controllability to zero or null controllability).\nThe term (∆)αutmodels a specific dissipative effect, known as structural damping, w hen\nα∈(0,2). To the best of our knowledge, this was introduced in [5] assuming α= 1: “The\nbasic property of structural damping, which is said to be consisten t with empirical studies,\nis that the amplitudes of the normal modes of vibration are attenua ted at rates which are\nproportional to the oscillation frequencies.” This model was also stu died under the name\n“proportional damping” (cf. [3]). The quite different case α= 2 is known as “Kelvin–Voigt”\ndamping. When Bis the identity and α∈(0,2], this is the first class of parabolic-like control\nmodels considered in [10],[15], see also [1].\nThe paper consists of two parts. In this first part we consider inte rior controllability, and\nin the second part boundary controllability.\nFirst, we consider two-dimensional interior control:\nutt+∆2u+ρ(∆)αut=f1(t)h1(x)+f2(t)h2(x),\nx∈(0,π), t >0, (1.1)\nu(0,t) =uxx(0,t) =uxx(π,t) =u(π,t) = 0, (1.2)\nu(x,0) =u0(x), ut(x,0) =u1(x). (1.3)\nHereh1,h2will be fixed functions (profiles), and f1,f2would serve as controls.\nThroughout this paper, we will denote Xp=Dom(∆p/2), soX0=L2(0,π), etc. In the\ntheorem below,\nQ(T)≤/braceleftbiggC′/T, α ≤1,\nC′/T1/(3α−2), α∈(1,3/2).(1.4)\nwhere the constant C′depends on αandρonly.\nTheorem 1 Considerthe system(1.1), with boundaryconditions(1.2) a ndinitial conditions\n(1.3). Suppose (u0,u1)∈X2×X0andT >0.\nA) (One dimensional control.) Suppose α= 0, orα∈(0,1]andρ≤2. Seth2= 0.\nSuppose the Fourier coefficients {h1\nn}ofh1∈L2(0,π)satisfy|h1\nn| ≍1/npfor some constant\np >1/2. Then there exists f1∈L2(0,T)such that the solution uto the system above solves\nu(x,T) =ut(x,T) = 0,\nwith\n/bardblf/bardblL2(0,T)≤CeQ(T)(/bardblu0\nxx/bardblL2(0,π)+/bardblu1/bardblL2(0,π)).\nHere the constant Cis depends only on α,ρ.\nB) (Two dimensional control.) Suppose α∈(0,1]andρ >2, orα∈(1,3/2).Then there\nexisth1,h2∈L2(0,1)such that for any pair (u0,u1), there exist f1,f2∈L2(0,T)such that\nthe solution uto the system above solves\nu(x,T) =ut(x,T) = 0,\nwith\n/bardblf1/bardblL2(0,T)+/bardblf2/bardblL2(0,T)≤CeQ(T)(/bardblu0\nxx/bardblL2(0,π)+/bardblu1/bardblL2(0,π)).\nHere the constant Cis depends only on α,ρ, andh1,h2.\n2C) Ifα≥3/2, then the system is neither null controllable nor spectrall y controllable with\ntwo dimensional (or with any finite dimensional) control.\nD) (Two dimensional weak controllability.) Suppose α∈[3/2,2). Then there exist\nh1,h2∈L2(0,1)such that for any pair (q0,q1)∈X2×X0and any ǫ >0, there exist\nf1,f2∈L2(0,T)such that the solution uto the system above satisfies\n/bardbluxx(x,T)/bardblL2(0,π)+/bardblut(x,T)/bardblL2(0,π)< ǫ.\nRemark 1 Recall that spectral null-controllability is equivalent t o the fact that jgfor all the\ninitial data (u0,u1), whereu0andu1are eigenfunctions or zero, the system can be steered\nto rest and equilibrium.\nTo prove this result, we first apply the Fourier method, i.e. find the s olution as\n/summationdisplay\nak(t)eiλktϕ|k|(x),\nwherekruns over K:=Z\\0, the sequence Λ consists of the frequencies λkarising in this\nmethod and ϕkare the eigenfunctions of the Laplace operator. Then we rewrite t he associ-\nated control problem as a moment problem.\nFor part A, we need to distinguish the case ρ <2 from all other cases. In this case,\nthe frequencies are separated. Hence the associated exponent ial family, {h1\n|k|eitλk,k∈K},\nis minimal on L2(0,T), and hence the moment problem has a (formal) solution via the\nbiorthogonal elements. The separation condition, together with t he asymptotics of {λk},\nallow us to use a result in [13] to conclude the biorthogonal function s satisfy an exponential\nestimate that implies the formal solution converges in L2(0,T).\nFor the case ρ= 2 and α= 1 the family has the form E1={eitλn,teitλn:n∈N}.\nWe show that the family E1is also minimal, and the elements of the biorthogonal family\nsatisfy the necessary estimates to prove the theorem. The cons truction of the biorthogonal\nfamily, an adaptation of the argument of [13], is formulated in Propos ition 1 and might be\nof independent interest. In this context, we note that the const ruction of, and estimates on,\nsets of functions biorthogonal to {eitλk}have been a subject of considerable research, see for\ninstance [4] and references therein.\nForρ= 2 and α <1, we have a single double frequency. This can be treated similarly\ntoρ= 2 and α= 1.\nFor the part B, the frequency set {λn}no longer necessarily satisfies the separation\ncondition. Moreover it is possible for some ρthat two elements (no more!) can coincide and\none dimensional control fails. Associated to the moment problem in t his case is the ‘vector’\nexponential family\nE=/braceleftBigg/parenleftbiggh1\n|k|\nh2\n|k|/parenrightbigg\neiλkt,k∈K/bracerightBigg\n, (1.5)\nwherehj\nkare the Fourier coefficients of the profiles hj. This family Ecan be made minimal\nby carefully choosing the Fourier coefficients of h1,h2. Roughly speaking we split our vector\nexponential family into two orthogonal families, and thus the origina l moment problem is\nsplit into two solvable moment problems, one for f1and one for f2.\n3The proof of part C follows from the theory of moment problems, be cause the frequency\nset does not satisfy the Blaschke condition. The proof of part D us es an adaptation of the\nsplitting argument from part B, together with the following sufficient condition for weak\nlinear independence of a family of exponentials {eλnt;n∈N}onL2(0,T), where we assume\nλn>0:\nlim\nn→∞ln(n)\nλn= 0.\nThe next control problem is the following. Let Ω be an open subset of (0,π).Suppose\nwe have the initial boundary value problem (IBVP)\nutt+∆2u+ρ(∆)αut=χΩf(t,x), x∈(0,π), t >0,(1.6)\nu(0,t) =uxx(0,t) =uxx(π,t) =u(π,t) = 0 (1.7)\nu(x,0) =u0(x), ut(x,0) =u1(x). (1.8)\nHereχΩis the characteristic function of Ω.\nTheorem 2 Letα∈[0,3/2).Given(u0,u1)∈X2×X0andT >0, there exists f∈\nL2(Ω×(0,T))such that the solution uto the system (1.6)-(1.8) solves\nu(x,T) =ut(x,T) = 0,\nwith\n/bardblf/bardblL2(Ω×(0,T))≤CeQ(T)(/bardblu0\nxx/bardblL2(0,π)+/bardblu1/bardblL2(0,π)).\nHereQ(T)is as in Theorem 1, and the constant Cdepends on αandρ.\nFor the proof, we again reduce the control problem to the moment one, now with respect\nto the exponential family\nE2={eiλktϕ|k|(x)/vextendsingle/vextendsingle\nΩ,k∈K}.\nFor the case of non-separated spectrum we split the the family usin g the fact that the angle\nbetween the eigenfunctions ϕnandϕminL2(a,b) is separated from zero.\nIn the Part 2 of the paper, to be published separately, we will deal w ith boundary\ncontrollability. We will study the dynamical system of the same form a nd discuss several\napproaches to treat non-homogeneous boundary conditions\nu(0,t) =uxx(0,t) = 0, (1.9)\nu(π,t) =f(t), (1.10)\nuxx(π,t) =g(t). (1.11)\nWe will then prove null controllability for this system.\nThis paper is organized as follows. In the next subsection, we compa re our results with\nthe literature. In Section 2.1, we discuss the spectral solution of t he uncontrolled system,\ndiscussing how α,ρdetermine the separation properties of the frequencies {λn}. In Section\n2.2, we adapt an argument from [13] to prove a existence of a biorth ogonal to {eλ+\nnt, teλ+\nnt}\nfamily of functions, with the norms satisfying an exponential estima te, that will be used to\nsolve the moment problems associated to Theorems 2 and 1. Theore m 1 is proven in Section\n3 and Theorem 2 is proven in Section 4 .\n41.1 Literature review\nWe first compare our Theorem 2 to the relevant literature. Lasieck a and Triggiani, in [10],\nconsidered the case an abstract system which, for dimension 1, ca n be reduced to the beam\nequation with control distributed throughout the interval, and α∈[1,2). Excepting the case\nα= 1,ρ= 2, they prove null controllability. An important ingredient in their ca lculations\nis Parseval’s Formula, which requires Ω = (0 ,π).The well-posedness and regularity of the\nequation, and also for plate equations in higher dimensions, is discuss ed in [14] using the\ntheory of analytic semigroups, but assuming αis an integer.\nMiller [11] considered (1.6) in a bounded domain of Rn,with distributed controls sup-\nported on a subset of the interior, which for n= 1 could a arbitrary open subset Ω ⊂(0,π).\nMiller proves null-controllability for α∈(1/2,3/2) when Ω is a proper subset, and for any\nα <1 if Ω = (0 ,π).In place of Parseval’s Formula, he uses the inequality (which follows\nfrom a Carleman estimate due to Lebeau and Robiano)\n/integraldisplay\nΩ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj≤ωjcjφj(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx≥C1e−C2ωj/summationdisplay\nj≤ωj|cj|2,\nalong with the natural damping properties of the system. Here {ωj,φj}are the spectrum\nand corresponding orthonormal eigenfunctions for the Laplacian . Edward [6] considered also\ninterior control for α <1/2, and for α= 1/2 with small ρ, and proved null controllability.\nMore recently, Mitra [12] considered the case α= 1 on the interval with periodic boundary\nconditions. Using a Carleman estimate, null controllability is proven fo r controls supported\non an open subset of (0 ,1).\nSince our Theorem 2 covers α∈[0,3/2) for all ρ, this theorem can be viewed as comple-\nmentary to the results of Miller, Edward and Mitra. And compared wit h Edward’s result,\nour Theorem 2 has the advantage of the estimate on the cost blowu p rate.\nAs a final remark about Theorem 2, the case α≥3/2, for Ω a proper subset of (0 ,π),\nremains open. If Ω = (0 ,π), null controllabilty was proven in [10].\nRegarding our Theorem 1, we are unaware of any related papers fo r the structurally\ndamped beam equation with finite dimensional interior control. Well-po sedness and regu-\nlarity for one dimensional control is discussed in [14].\nWe conclude this section with a brief discussion on the function Q(T), which appears in\nourupper bound, CeQ(T), onthecost ofcontrol. The rateat which thecost ofa control blo ws\nup asT→0+has been the subject of interest, motivated by problems in non-line ar partial\ndifferential equations and stochastic differential equations. Reca ll we showed the estimate\n(1.4)for some constant C′, which follows from the construction of Qgiven in [13]. Miller\nalso estimates Q(T), and comparing his Corollary 1 with our Theorem 2, one sees that his\nresult for interior control is sharper for the power of T. The problem was also studied in\n[1]. The results there are given in an abstract setting, but restrict ing to the 1-dimensional\nbeam equation, the control function is assumed to be distributed t hroughout the interior,\nand under this strong hypothesis the control cost has the much s maller upper bound C/Tβ,\nwhereβis determined by ρ,α.\n52 Frequency set and biorthogonal functions\n2.1 Frequency set\nConsider the eigenvalue problem\n∆ϕ=λϕ, ϕ(0) =ϕ(π) = 0.\nClearly an orthonormal basis of eigenfunctions is {ϕn;n∈N},ϕn(x) =/radicalBig\n2\nπsin(nx), with\ncorresponding eigenvalues n2.Obviously the (∆)αhas the same eigenfunctions, with corre-\nsponding eigenvalues n2α.\nConsider the IBVP , whose solution we will refer to as the “free” wav e,\nwtt+∆2w+ρ(∆)αwt= 0, (2.12)\nw(0,t) =w(π,t) =wxx(0,t) =wxx(π,t) = 0, (2.13)\nw(x,0) =w0(x), (2.14)\nwt(x,0) =w1(x). (2.15)\nSetw=/summationtext∞\nn=1an(t)ϕn(x). Then (2.12) implies\n/summationdisplay\n(a′′\nn+ρa′\nnn2α+ann4)ϕn(x) = 0,\nhence\na′′\nn+ρn2αa′\nn+n4an= 0,∀n∈N. (2.16)\nSolvingλ2+ρn2αλ+n4= 0, we get\nλ=−ρn2α±/radicalbig\nρ2n4α−4n4\n2=:λ±\nn.\nThus, ifλ+\nn/\\e}atio\\slash=λ−\nn,\nw(x,t) =∞/summationdisplay\n1(c+\nneλ+\nnt+c−\nneλ−\nnt)ϕn(x), (2.17)\nwith coefficients c±\nndetermined by the initial conditions:\nc+\nn+c−\nn=w0\nn,\nλ+\nnc+\nn+λ−\nnc−\nn=w1\nn,\nwherew0\nnandw1\nnaretheFouriercoefficientsof w0andw1. Thisgivesthefollowingexpression\nc+\nn= (w1\nn−λ−\nnw0\nn)/qn, (2.18)\nc−\nn= (−w1\nn+λ+\nnw0\nn)/qn, (2.19)\nwhere\nqn=λ+\nn−λ−\nn=/radicalbig\nρ2n4α−4n4. (2.20)\nIf for some nwe haveλ+\nn=λ−\nn, then we change the corresponding term in wto\n(c+\nneλ+\nnt+c−\nnteλ+\nnt)ϕn(x),\n6with\nc+\nn=w0\nn,\nc−\nn= (w1\nn−λ+\nnw0\nn)/(λ+\nn+1).\nWe now examine the gap properties of {λ±\nn}. In what follows we will use the frequency\nset\nΛ ={λk}k∈K,K=Z\\{0}, λk=/braceleftbigg\n−iλ+\nk, k >0,\n−iλ−\nk, k <0.(2.21)\nThus it is easy to see Λ ⊂C+. We need (following [13]) to introduce a function ν: [0,∞)/mapsto→\n[0,∞) which describes the density of Λ\n#{λn∈Λ\\λk:|λn−λk|< r} ≤ν(r),∀k.\nWe will require that Λ satisfies\nν(r) = 0, r < R 0, (2.22)\nfor a positive R. This assumption is equivalent to\ninf\nk/\\egatio\\slash=n|λk−λn|>0,\nwhich we well refer to as separability of Λ .\nExample 1. It is not hard to see that for λn=sgn(n)|n|p, withp >1 andn∈N, we\nhave\nν(r)≍r1/p, (2.23)\nfor large r, see also [13].\nThus, if one assumes α∈[0,3/2), then the asymptotics below will show that both\nsequences {λk:k <0}and{λk:k >0}satisfy (2.22), (2.23) with p= 2 forα≤1 and\np= 2αforα >1.\nLemma 1\n(1) Forα∈[0,1]andρ <2,the frequency set is separable,\n(2) For(α,ρ) = (1,2), we have\nλ+\nn=λ−\nn\nfor alln. Also, for ρ= 2and any α≥0, we have λ+\n1=λ−\n1.\n(3) For (i) α= 1andρ >2, or (ii) α >1, there are an infinite number of ρsuch that\nthe sequence Λcontains two equal elements, i.e. for some mandnwithm/\\e}atio\\slash=nwe have\nλ+\nm=λ−\nn. (2.24)\n(4) Forα≥3/2, the frequency set does not satisfy the Blashke condition [9 ], in other\nwords,/summationdisplay/vextendsingle/vextendsingle/vextendsingle/vextendsingleℑ1\niλ+\nn/vextendsingle/vextendsingle/vextendsingle/vextendsingle=∞\n(5) Forα= 2, we have λ+\nn=O(1).\n7Proof:(1) (i) For α <1 we have the following asymptotics:\nλ±\nn=−1\n2ρn2α(1+o(1))±in2(1+o(1)).\nFrom here we see that the frequency set is separable for large n. Also for ρ <2 the set Λ\nhas no coinciding elements because the real parts of the branches λ±\nnare strictly increasing\ninnwhile two branches have opposite signs of the imaginary parts.\n(2) This is easily verified.\n(3) (i) Let α= 1, and ρ >2. Then\nλ+\nn=n2(−ρ/2+/radicalbig\nρ2/4−1),λ−\nn=n2(−ρ/2−/radicalbig\nρ2/4−1).\nSettingr= (ρ+/radicalbig\nρ2−4)/2, and we have\nλ+\nm=−2\nrm2, λ−\nn=−r\n2n2. (2.25)\nTakem < n. Then (2.25) implies that we can find ρsatisfying (2.24).\n(ii) Letα∈(1,3/2). Then\nλ+\nn=−n4−2α\nρ/parenleftBig\n1+o(1)/parenrightBig\nandλ−\nn=−ρn2α/parenleftBig\n1+o(1)/parenrightBig\n. (2.26)\nThe situation is similar to 3(i). Fix mandn. Ifρruns (0,∞) the main terms of the\nbranches in (2.26) change from −∞to 0 and from 0 to −∞, This means that we can find ρ\nsatisfying (2.24).\n(4) The statement can be checked directly.\n(5) Follows immediately from (2.26). ✷\nRemark 2 The case α= 2is known as Kelvin-Voight damping, and because of part (5) of\nthe lemma, the methods of this paper mostly cannot be used. In the case of controls distributed\non(0,π), this case is discussed in [10], also see [2].\nFinally, we discuss the properties of the solution to the IBVP (2.12)– (2.15). By (2.17)\nand the asymptotics of {λ±\nn}, we have\nLemma 2 The mapping (w0,w1)/mapsto→wis a continuous map\nX2×X0/mapsto→C(0,T;X2×X0))∩C1(0,T;X0).\nProof:Assume ( w0,w1)∈X2×X0×L2(0,π), so that\n/summationdisplay\nn2(w0\nn)2<∞,/summationdisplay\n(w1\nn)2<∞.\nNoww(·,t)∈X2×X0iff/summationdisplay\nn2|c+\nneλ+\nnt+c−\nneλ−\nnt|2<∞.\nBecause all λ±\nnhave a negative real part, it is enough to check that\n/summationdisplay\nn2(|c+\nn|2+|c−\nn|2)<∞.\n8By (2.18),(2.19), and (2.20), it suffices to show that\n/summationdisplay\nn2(|(w1\nn)2+(|λ+\nn|2+|λ−\nn|2)|w0\nn|2)/q2\nn<∞. (2.27)\nIt is easy to see that for α∈[0,2], n/qnandλ±\nn/qnare bounded and (2.27) is correct.\nBecause the series converges uniformly in t, we have w∈C(0,T;X2×X0).\nThe rest of the lemma can be proved similarly. ✷\n2.2 Biorthogonal Functions\nAn important part of our proof is to construct suitable sets of bior thogonal functions asso-\nciated to {eλ±\nnt:n∈N}.\nFor completeness, we begin by providing results from [13]. Let Lbe a subset of Z.\nTheorem 3 ([13]) Suppose Λ ={µl:l∈L} ⊂C+. LetR0>0.\nA- Suppose there exists a function ν(r)such that for all l∈L,Λsatisfies\n#{µl∈Λ\\µm:|µl−µm|< r} ≤ν(r)\nwithν(r) = 0forr < R 0, andν(r)/r2integrable. Then for any δ >0,T0>0, there exists a\nconstant ˜C=˜C(δ,T0)such that for any sequence {al}, we have\nΣl|aleiµlδ|2≤˜C/integraldisplayT0\n0|Σlaleiµlt|2dt.\nB- Suppose T=δ=T0. Ifν(r)≍r1/pfor large r, then the constant ˜Csatisfies\n˜C≤C1exp(C2/T1/(p−1)),\nwith constants C1,C2independent of T.\nIn what follows, we will often have {µl;l∈L}={λk;k∈K}, withλkgiven by (2.21).\nAssuming α <3/2, the following estimates follow from Section 2. There exist positive\nconstants depending on ρ,αsuch that one can choose ν(r) satisfying\nC0rκ≤ν(r)≤C1rκ, (2.28)\nwhereκ= 1/2 forα <1 orα= 1,ρ≤2,κ= 1/2αforα∈(1,3/2).\nProposition 1, below, is a generalization of a result proven in [13], whic h in turn gener-\nalizes a result in [7]. Recall\n/a\\}bracketle{tf,g/a\\}bracketri}ht=/integraldisplayT\n0f(t)g(t)dt,\nwhere the bar denotes complex conjugation.\nProposition 1 LetT >0. Suppose there exists a function ν(r)satisfying the estimates\n(2.28) (2.22). Then there exists a family of functions {gm,j(t);m∈L, j= 1,2}inL2(0,T)\nsatisfying\n/a\\}bracketle{tgm,1,teiλnt/a\\}bracketri}ht= 0,/a\\}bracketle{tgm,1,eiλnt/a\\}bracketri}ht=δm,n,/a\\}bracketle{tgm,2,teiλnt/a\\}bracketri}ht=δm,n,/a\\}bracketle{tgm,2,eiλnt/a\\}bracketri}ht= 0.\n9Furthermore, there exist positive constants C2,C3depending only on R0,T,C0,C1such for\nj= 1,2,\n/bardblgm,j/bardblL2(0,T)≤C2eC3(ℑ(λm))κ,\nwhere all constants are defined in (2.28) or Theorem 3.\nAn immediate consequence of the proposition is:\nCorollary 1 Assume the hypothesesof Proposition1. Then there existsa f amilyoffunctions\n{gj(t) :j∈K}inL2(0,T)satisfying\n/a\\}bracketle{tgj,eiλnt/a\\}bracketri}ht=δjn.\nFurthermore, there exist positive constants C2,C3depending only on R0,T,C0,C1such\n/bardblgj/bardblL2(0,T)≤C2eC3(ℑ(λj))κ. (2.29)\nProof of Proposition 1: We adapt the construction used in [13]. We define\nFj,1(z) =/parenleftBigg/productdisplay\nk∈Z,k/\\egatio\\slash=j/parenleftBigg\n1−/parenleftbiggz−λj\nλk−λj/parenrightbigg2/parenrightBigg/parenrightBigg2\n,j∈N.\nThen\nFj,1(λk) =δj,k,F′\nj,1(λk) = 0, j,k∈N,\nand by (2.28) and [13, Lemma 3], Fj,1(z) is entire of exponential type zero with\n|Fj,1(λj+z)| ≤e2θ(|z|)forz∈C,\nwhere\nθ(s) = 2/integraldisplay∞\n0ν(r)\nrs2\ns2+r2dr.\nThusθis a positive increasing function with\nθ(s)≍sκ, s→ ∞. (2.30)\nWe now define\nFj,2(z) = (z−λj)Fj,1(z).\nThen for all j,k∈N,we have\nFj,2(λk) = 0,F′\nj,2(λk) =δj,k.\nFurthermore, by increasing θslightly, we have\n|Fj,2(λj+z)| ≤e2θ(|z|), z∈C, j∈N. (2.31)\nIndeed, we can replace θ(s) byθ(s)+log(s+1)−logmin|λj|. In what follows, we will employ\nthis slightly larger θ. In what follows, it will be convenient to set αj=ℜ(λj),βj=ℑ(λj),\nsoβj≥0.\n10By [13, Theorem 2] there exists an entire function Phaving the following properties\n(i)|P(z)| ≤1 forz∈C+,andP(0) = 1,\n(ii)P(is) is real and positive for s≥0, and there exists a positive constant C4with\nP(is)≥e−C4sκ, s >0, (2.32)\n(iii)\n|P(s)| ≤eQ(T)e−3θ(|s|), s∈R, (2.33)\nwithQ(T) a constant,\n(iv)P(z)e−izT/2is of exponential type T/2.\nFurthermore, we have\nLemma 3 Forr≥0, there exists CP>0such that\n|P′(ir)|< CP.\nProof:We recall some facts from [13].\nP(z) =∞/productdisplay\nn=01\n2(1+e2ianz),\nwith{an}a positive sequence satisfying/summationtext\nnan=δ/2 for some δ >0.SinceP′(z) =\nP(z)(logP(z))′and|P(z)| ≤1 in the upper half plane,\n|P′(ir)| ≤ |(logP(ir))′|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBigg/summationdisplay\nnlog/parenleftbigg1\n2+1\n2e−anr/parenrightbigg/bracketrightBigg′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn−aje−anr\n1+e−anr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C1/summationdisplay\nnan≤CP.✷\nWe now continue with the proof of the proposition. Define, for n= 1,2\nGj,n(z) =Fj,n(z)P(z−αj)\nP(iβj), j∈N. (2.34)\nBy(2.31)and(2.33), Gj,n(s)∈L2(−∞,∞). Furthermore, forall j,k∈N,wehave Gj,1(λk) =\nδjk, andGj,2(λk) = 0.By (2.34), we have\nG′\nj,1(λk) =F′\nj,1(λk)P(λk−αj)+Fj,1(λk)P′(λk−αj)\nP(iβj)=δjkP′(iβj)\nP(iβj)\nand\nG′\nj,2(λk) =F′\nj,2(λk)P(λk−αj)+Fj,2(λk)P′(λk−αj)\nP(iβj)=δjk.\nDefine\ngj,2(t) =1\n2π/integraldisplay\nRGj,2(s)eistds\n11and\ngj,1(t) =1\n2π/integraldisplay\nR(Gj,1(s)−P′(iβj)\nP(iβj)Gj,2(s))eistds\nwithj∈N.\nThen,\n/a\\}bracketle{tgj,1(t),eiλkt/a\\}bracketri}htL2(0,T)=δj,k,/a\\}bracketle{tgj,2(t),eiλkt/a\\}bracketri}htL2(0,T)= 0,∀k∈N.\nAlso,\n/a\\}bracketle{tgj,1(t),teiλkt/a\\}bracketri}ht=−id\ndλ/a\\}bracketle{tgj,1(t),eiλt/a\\}bracketri}ht|λ=λk=−i(G′\nj,1(λk)−P′(iβj)\nP(iβj)G′\nj,2(λk)) = 0, n= 1,\n/a\\}bracketle{tgj,2(t),teiλkt/a\\}bracketri}ht=−id\ndλ/a\\}bracketle{tgj,2(t),eiλt/a\\}bracketri}ht|λ=λk=−id\ndλGj,n(λ)|λ=λk=δjk,n= 2.\nFurthermore, Fj,n(z) is entire of exponential type zero, and e−izT/2P(z−βj) is entire\nof exponential type T/2 in both halfspaces, and so by the Paley-Wiener Theorem, gj,n∈\nL2(0,T).\nWe see{gj,n, j∈N, n= 1,2}is a biorthogonal set to {eiλkt,teiλkt;k∈N}.We now\nestimate the elements of this set. By (2.31),(2.30), (2.32), and (2.3 3), we have for s∈R\n|Gj,2(s+αj)|=|Fj,2(λj+s−iβj)P(s)/P(iβj)|\n≤e2θ(|s−iβj|)eQ(T)−3θ(|s|)/e−C4(βj)κ\n≤eQ(T)−C5|s|κ+(1+C4)(βj)κ.\nSince the Fourier transform is unitary,\n/bardblgj,2/bardbl ≤Ce(1+C4)(βj)κ,\nwith with Cdepending only on T,R0,ǫ,Cj,j= 1−5.Similarly, we estimate gj,1where we\nmust use Lemma 3:\n/bardblgj,1/bardbl=/bardbl(Gj,1(s)−P′(iβj)\nP(iβj)Gj,2(s))/bardbl ≤Ce(1+C4)βκ\nj+CPCe(1+2C4)βκ\nj.\n✷\n3 Proof of Theorem 1\nFixh1,h2∈L2(0,π). We consider the following initial boundary value problem on (0 ,π)×\n(0,T)\nutt+∆2u+ρ(∆)αut=h1(x)f1(t)+h2(x)f2(t),(3.35)\nu(0,t) =u(π,t) =uxx(0,t) =uxx(π,t) = 0, (3.36)\nu(x,0) =u0(x), ut(x,0) =u1(x). (3.37)\nHere (u0,u1)∈X2×X0. We wish to prove null-controllability.\n12We can represent the solution to (3.35)-(3.37) as a sum of a “free” wave, corresponding\ntof1=f2= 0, and a “controlled” wave, corresponding to u0=u1= 0. Let us express the\nfree wave, ufree, as a Fourier series. Suppose for j= 0,1, the initial conditions have Fourier\ncoefficients {u0\nn},{u1\nn}respectively. Then, similarly to (2.17)-(2.20), if we assume λ+\nn/\\e}atio\\slash=λ−\nn\nfor alln,\nufree(x,t) =/summationdisplay\n(c+\nneλ+\nnt+c−\nneλ−\nnt)ϕn(x),\nwith\nc+\nn=λ−\nnu0\nn−u1\nn\nλ−\nn−λ+\nn, c−\nn=λ+\nnu0\nn−u1\nn\nλ+\nn−λ−\nn.\nIn the calculations below, we will assume λ+\nn/\\e}atio\\slash=λ−\nn. In the cases where λ+\nn=λ−\nn, the\ncalculations below can be adapted by replacing eλ+\nnt,eλ−\nntbyeλ+\nnt,teλ+\nnt. The details of the\nadaptation are left to the reader, but also see the paragraph at t he end of the proof of part\nA below. Thus\nufree(x,T) =/summationdisplay\nγ1\nnϕn(x) :=/summationdisplay\n(c+\nneλ+\nnT+c−\nneλ−\nnT)ϕn(x), (3.38)\nufree\nt(x,T) =/summationdisplay\nγ2\nnϕn(x) :=/summationdisplay\n(λ+\nnc+\nneλ+\nnT+λ−\nnc−\nneλ−\nnT)ϕn(x). (3.39)\nWe now derive a formula for the controlled wave, denoted ufwithf= (f1,f2), and setting\nu0=u1= 0. Let hj\nnare the Fourier coefficients of hj,hj\nn=/a\\}bracketle{thj,φn/a\\}bracketri}ht, and let uf(x,t) =/summationtextan(t)ϕn(x). Putting this into (3.35), we get the following family of ODE:\na′′\nn+ρn2αa′\nn+n4an=f1(s)h1\nn+f2(s)h2\nn, an(0) =a′\nn(t) = 0,∀n∈N.\nThen the solution to the ODE above is\nan(t) =−1\nqn/integraldisplayt\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig/parenleftBig\neλ+\nn(t−s)−eλ−\nn(t−s)/parenrightBig\nds, n∈N.\nComparing this with (3.39), we see that null controllability in time Tis equivalent to\nγ1\nn=1\nqn/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig/parenleftBig\neλ+\nn(T−s)−eλ−\nn(T−s)/parenrightBig\nds,\nγ2\nn=1\nqn/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig/parenleftBig\nλ+\nneλ+\nn(T−s)−λ−\nneλ−\nn(T−s)/parenrightBig\nds,\nor, equivalently,\nζ1\nn=:qnγ1\nn−γ2\nn\nλ+\nn\n(−1+λ−\nn\nλ+\nn)=/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig\neλ−\nn(T−s)ds, n∈N,(3.40)\nζ2\nn=:qnγ1\nn−γ2\nn\nλ−\nn\n(1−λ+\nn\nλ−\nn)=/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig\neλ+\nn(T−s)ds, n∈N. (3.41)\n13We setζk=ζ1\nkfork >0,ζk=ζ2\n−kfork <0. Forj= 1,2, we extend hj\nntoKbyhj\nk:=hj\n|k|.\nRecall\nλk=/braceleftbigg−iλ+\nk, k >0,\n−iλ−\nk, k <0.\nHence system (3.40),(3.41) can be rewritten in terms of iλn:\nζk=/integraldisplayT\n0/parenleftbig\nf1(s)h1\nk+f2(s)h2\nk/parenrightbig\neiλk(T−s)ds, k∈K. (3.42)\nRemark 3 Relating the above formula for our discussion in the introdu ction leading to\n(1.5), we have obtained the moment problem in L2/parenleftbig\n(0,T)×C2/parenrightbig\nwith respect to the solution\n˜F(t) =/parenleftbigf1(T−t)\nf2(T−t)/parenrightbig\nand the exponential family complex conjugate to (1.5).\nProof of part A .\nIn this case, set f2= 0.Assume for the moment 0 < α <1,ρ <2. The other cases will\nbe addressed at the end of the paragraph. We have\nλ−\nn=−in2(1+o(1))−ρn2α\n2(1+o(1)), λ+\nn=in2(1+o(1))−ρn2α\n2(1+o(1)).\nWe see that the set Λ satisfies the hypotheses of Corollary 1, and h ence{eiλkt,k∈K}admits\na biorthogonal family {gk,k∈K}satisfying (2.29). It follows from (3.42) that\nf1(t) :=/summationdisplay\nj∈Kζj\nh1\nj¯gj(T−t)\nformally satisfies the moment problem. Recall the hypothesis that |h1\nk| ≍ |k|−pfor some\npositive constant p. Combining this with (2.29) (with κ= 1/2), (3.38), (3.39), (3.40),\n(3.41), we get f1∈L2(0,T).\nWe now discuss the case α= 1. Ifρ <2 the frequency set satisfies the hypotheses of\nCorollary 1, and we can argue the same as the case α <1,ρ <2.Ifρ= 2, then we have\nλ+\nn=λ−\nnfor alln∈N. In this case, we can represent the control problem as a moment\nproblem using the family {etλ+\nn,tetλ+\nn:n∈N}. Then by Proposition 1, there exists a\nbiorthogonal family of functions which can be used to solve the mome nt problem. The\ndetails are left to the reader.\nFinally, suppose α= 0. A simple calculation shows multiple frequencies only arise when\nρ= 2n. Forρ/\\e}atio\\slash= 2n, we can argue as in the case α∈(0,1),ρ <2 to prove the theorem. For\nρ= 2n0for some n0∈N, we have λ+\nn=λ−\nmif and only if m=n=n0, and in this case we\ncan still apply Proposition 1 to obtain a biorthogonal family of functio ns which can be used\nto solve the moment problem.\nRemark 4 Ifρ >2andα≤1, the frequency set will have multiplicities for various val ues\nofρandα, in which case Proposition 1 won’t apply, so we are in the situ ation Part B.\nIndeed, the sets {λk}k<0and{λk}k>0separately satisfy the Proposition 1, and the clusters\n(if any!) consist of two points.\n14Proof of part B\nWe will present the proof for α∈(1,3/2); it will be easy to see that the case α≤1,ρ >2\ncan be covered by the same argument.\nRecall that, for α >1,\nλ−\nn≍ −ρn2α, λ+\nn≍ −1\nρn4−2α.\nThe difficulty in solving the moment problem, (3.42), is that we do not kn ow whether the\nsequence {λk:k >0}is separated from {λk:k <0}.We address this as follows.\nForǫ >0, we will refer to the pair ( λ+\nn,λ−\nl) as anǫcluster if |λ+\nn−λ−\nl|< ǫ. Let\nǫ >0 be sufficiently small that any ǫclusters involving element of {λ+\nn},{λ−\nl}will involve\nonly two elements. Let ιbe the bijection within the set of ǫclusters that maps λ+\nnto its\ncluster-counterpart λ−\nl. Let\nN+={n∈N:∃l=ι(n) such that |λ+\nn−λ−\nl|< ǫ}= domain( ι),andN−= range( ι).\n(3.43)\nIt is worth noting that if N+∩N−=∅, then the construction of {h1\nn,h2\nn}is easy: it suffices\nto define\nh1\nn= 1/n,h2\nn= 0 ifn∈N\\N−,andh1\nn= 0,h2\nn= 1/nifn∈ N−.(3.44)\nIndeed, the moment equalities (3.40) and (3.41) then take the form of the moment equalities\nwith respect to two separated sets of exponentials\nξ1\nn= (1\nneλ−\nnt,/tildewidef1), n∈N\\N+(3.45)\nξ2\nn= (1\nneλ+\nnt,/tildewidef2), n∈ N+. (3.46)\nHere/tildewidefj(t) =¯fj(T−t), and the set {ξj\nn}is the renumbered set {ζj\nn}\nLet us consider the general case: N±:=N+∩N−is not empty. The goal is to obtain\nalso in this case two moment problems with respect to separated set s of exponentials. The\nfirst step is the first step in (3.44):\nh1\nn= 1/n,h2\nn= 0 ifn∈N\\N+.\nAt this moment we have the moment equalities for (3.45) or for a part of (3.40). Evidently,\nthe set{λ−\nn}n∈N\\N+is separated.\nThe second step is close to the second step in (3.44):\nh1\nn= 0,h2\nn= 1/nifn∈ N+\\N±.\nThe corresponding moment equalities are the part N+\\N±of (3.46). The set {λ+\nn}n∈N+\\N±\nis separated.\nBecause of the asymptotics of λ±\nn, there exists M >0 such that n > Mimplies\nn > ι(n). (3.47)\nIn the calculations that follow, we will assume (3.47) holds for all n, leaving the simple\nadaptations for the general case to the reader.\n15The third step will involve an induction in which we define hj\nnforn∈ N±. Letmthe\nsmallest element in N±, and let l=ι(m).Becausel < m, we have\nl∈ N−\\N±⊂N\\N+.\nHenceh1\nl= 0,h2\nl= 1/l. Thus we choose\nh1\nm= 1/m,h2\nm= 0.\nForn∈ N±forn > m, we carry out the following inductive step:\nifh2\nι(n)= 0 then set h1\nn= 0,h2\nn= 1/n, (3.48)\nand ifh1\nι(n)= 0 then set h1\nn= 1/n,h2\nn= 0. (3.49)\nIn the first case, the corresponding moment equality has the same form as in (3.46), and in\nthe second case, the moment equality has the same form as in (3.45) . The key point is that\nwe never have λ+\nn,λ−\nι(n)both appearing in one of (3.45),(3.46). In the other words, similarly\nto (3.45), (3.46) we obtain moment problem with respect to f1andf2with two sets of\n”scalar ” exponentials. Moreover, each set is a separated subset of all (λ+\nn,λ−\nl) and satisfy\nthe Proposition 1.\nProof of part C\nHere, we will use the theory of the moment problem and its application to control prob-\nlems, ([2, Ch. I, V]). If α≥3/2, then by Lemma 1 the sequence {λ+\nn}fails the Blaschke\ncondition, and hence the exponential family is not minimal. Thus both n ull controllability\nand spectral controllability will fail.\nProof of part D\nWe now prove the weak controllability for α∈[3/2,2).Recall\nλ+\nn=−n4−2α\nρ/parenleftBig\n1+o(1)/parenrightBig\nandλ−\nn=−ρn2α/parenleftBig\n1+o(1)/parenrightBig\n.\nIt is easy to see that the sets {λ+\nn}and{λ−\nn}are each simple, and\nlim\nn→∞ln(n)\nλ±\nn= 0.\nHence, by ([2], Theorem II.6.3 and ), the families {eλ+\nnt},{eλ−\nnt}are each weakly linearly\nindependent. But their union might not be, due to multiple frequencie s. For this reason, we\nneed to use two dimensional control.\nWe will present the proof for ρ >2,in which case λ+\nn> λ−\nnfor alln. The adaptations\nfor cases ρ= 2 and ρ <2 will be indicated at the end of the proof.\nWe express the controllability problem using moment problems (3.40),( 3.41):\nζ1\nn=/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig\neλ−\nn(T−s)ds, n∈N,\n16ζ2\nn=/integraldisplayT\n0/parenleftbig\nf1(s)h1\nn+f2(s)h2\nn/parenrightbig\neλ+\nn(T−s)ds, n∈N.\nWe now adapt the construction of h1,h2from part B. Recall that any multiple frequency\nwill have multiplicity at most 2. Let ιbe the bijection within the set of double frequencies\nthat maps λ+\nnto its counterpart λ−\nl. Because of the asymptotics of λ±\nn, there exists M >0\nsuch that n > Mimplies\nn > ι(n). (3.50)\nIn the calculations that follow, we will assume (3.50) holds for all n, leaving the simple\nadaptations for the general case to the reader. Let\nN+={n∈N:∃l=ι(n) such that λ+\nn=λ−\nl}= domain( ι),andN−= range( ι).\nWe can now argue exactly as in the proof for two dimensional null-con trollability for α∈\n(1,3/2) to construct, from N±, the functions h1,h2.As in part B, the moment problem turns\ninto two distinct moment problems, one each for f1andf2, and in each the frequency set\nis simple, so the associated exponential families are weakly linearly inde pendent. By ([2],\nThm. III.3.3), weak controllability follows.\nFinally, if ρ≤2, than it is possible that there exists msuch that λ+\nm=λ−\nm, and also\npossibly a finite number of complex frequencies. Thus we must exten d our exponential\nfamily to {eλ+\nnt,teλ+\nmt,n∈N},with a finite number of distinct non-real frequencies. A\ncareful reading of the proof of ([2], Theorem II.6.3) shows that th is extended family remains\nweakly linearly independent. The remaining adaptations of the proof of above are left to the\nreader.\nOur proof of weak controllability is complete. ✷\n4 Interior control of structurally damped beam. The\nproof of Theorem 2\nLet Ω⊂(0,π) be a proper open subset. We consider:\nutt+∆2u+ρ(∆)αut=χΩ(x)f(x,t), x∈(0,π), t >0,(4.51)\nu(0,t) =u(π,t) =uxx(0,t) =uxx(π,t) = 0, (4.52)\nu(x,0) =u0(x), ut(x,0) =u1(x). (4.53)\nHereu0∈H2∩H1\n0, u1∈L2.\nFirst, we represent the solution to (4.51)-(4.53) using the calculat ions and notation from\nthe previous section, see (3.38),(3.39). In this section, we will assu meλ+\nn/\\e}atio\\slash=λ−\nnfor alln,\nleaving the simple adaptations in the other case to the reader. Thus att=T, we have the\nfree wave satisfying\nufree(x,T) =/summationdisplay\nγ1\nnϕn(x) :=/summationdisplay\n(c+\nneλ+\nnT+c−\nneλ−\nnT)ϕn(x),\nufree\nt(x,T) =/summationdisplay\nγ2\nnϕn(x) :=/summationdisplay\n(λ+\nnc+\nneλ+\nnT+λ−\nnc−\nneλ−\nnT)ϕn(x). (4.54)\n17We now adapt the argument of the previous section to derive a form ula for the controlled\nwave, denoted uf. Letfn(t) are the Fourier coefficients of χΩ(x)f(x,t), so\nfn(t) =/integraldisplayπ\n0χΩ(x)f(x,t)φn(x)dx=/integraldisplay\nΩf(x,t)φn(x)dx,\nand letuf(x,t) =/summationtextan(t)ϕn(x). We have the set of ODEs\na′′\nn+ρn2αa′\nn+n4an=fn(t), an(0) =a′\nn(t) = 0,∀n∈N.\nHence\nan(t) =−1\nqn/integraldisplayt\n0fn(s)/parenleftBig\neλ+\nn(t−s)−eλ−\nn(t−s)/parenrightBig\nds, n∈N.\nComparing this with (4.54), we see that null controllability in time Tis equivalent to\nγ1\nn=1\nqn/integraldisplayT\n0fn(s)/parenleftBig\neλ+\nn(T−s)−eλ−\nn(T−s)/parenrightBig\nds,\nγ2\nn=1\nqn/integraldisplayT\n0fn(s)/parenleftBig\nλ+\nneλ+\nn(T−s)−λ−\nneλ−\nn(T−s)/parenrightBig\nds,\nor, equivalently,\nζ1\nn=:qnγ1\nn−γ2\nn\nλ+\nn\n(−1+λ−\nn\nλ+\nn)=/integraldisplayT\n0fn(s)eλ−\nn(T−s)ds, n∈N, (4.55)\nζ2\nn=:qnγ1\nn−γ2\nn\nλ−\nn\n(1−λ+\nn\nλ−\nn)=/integraldisplayT\n0fn(s)eλ+\nn(T−s)ds, n∈N. (4.56)\nWe setζk=ζ1\nkfork >0,ζk=ζ2\n−kfork <0.Recall we have λk=−iλ+\nkfork >0, and\nλk=−iλ−\n|k|fork <0.We extend fntoKbyfk:=f|k|, and similarly ϕk(x) =ϕ|k|.Then\nsystem (4.55), (4.56) can be rewritten\nζk=/integraldisplayT\n0/integraldisplay\nΩf(x,t)ϕk(x)eiλk(T−s)dxds, k∈K. (4.57)\nThe remainder of this section will be devoted to solving this moment pr oblem on L2(Ω×\n(0,T)) by constructing a suitably bounded biorthogonal set to {ϕk(x)eiλk(T−s),k∈K}. We\nassume first that for all n,λ+\nn/\\e}atio\\slash=λ−\nn, which is equivalent to ρ/2/\\e}atio\\slash=n2−2α. At the section’s\nend, we briefly discuss the adaptations necessary in the other cas e.\nLemma 4 The infimum of the angles between ϕnandϕminL2(a,b)is positive.\nThe elementary proof of this lemma is deferred to the appendix.\nLetφnbe the restrictions of the eigenfunctions ϕnto (a,b), normalized in the space\nL2(a,b).\n18Proposition 2 LetT >0.Suppose α∈[0,3/2). Assume λ+\nn/\\e}atio\\slash=λ−\nnfor alln.Then there\nexists a set {hk(x,t) :k∈K}biorthogonal to {φk(x)exp(iλkt),k∈K}inL2((a,b)×(0,T)).\nFurthermore, there exist positive constants C2,C3, depending, on a,b,T, such that\n/integraldisplayT\n0/integraldisplayb\na|hk(x,t)|2dxdt≤C2exp(C3(Imλk)κ). (4.58)\nHereκ= 1/2forα <1orα= 1,ρ≤2, andκ= 3−2αforα∈(1,3/2).\nProof:First, if we assume α≤1 andρ <2, then the frequency set is separated, so by\nProposition 1 we can use hk(x,t) =gk(t)φk(x).Next, note ρ= 2 is ruled out because we\nassumeλ+\n1/\\e}atio\\slash=λ−\n1. In the remainder of the proof, we consider the harder cases α∈(1,3/2),\norα≤1 andρ >2, so that the union {λ+\nn}∪{λ−\nn}is not necessarily separated.\nRecall that the cardinality of any cluster of frequencies can be at m ost two. We use the\nnotation introduced in the previous section, in the proof of Theore m 1. In particular, there\nexistsǫ >0 such that the set Nǫofǫ-close frequencies can be parametrized as\nNǫ={(ln,mn)}n∈N.\nThus (ln,mn)∈ Nǫmeans|λln−λmn|< ǫ. We introduce now two reduced sets of the\nindices:Kǫ1:=K\\{mn},where{mn}is the set of the second indices of pairs from Nǫ,and\nKǫ:=Kǫ1\\{ln},where{ln}istheset ofthefirstindices ofpairsfrom Nǫ.Then{λk:k∈Kǫ1}\nis separated, and so by Proposition 1, there exists {θk,k∈Kǫ1},a family biorthogonal to\n{expiλkt, k∈Kǫ1},in the space L2(0,T),and the following estimate holds\n/bardblθk/bardblL2(0,T)≤C5exp(C6(ℑλk)κ), (4.59)\nwithC5,C6positive constants that depend only on T,α,ρ. Now we construct the family\n{hk(x,t),k∈K}in the following way. For k∈Kǫwe set\nhk(x,t) =φk(x)θk(t).\nFork /∈Kǫ, there exist n,ln,mnsuch that ( ln,mn)∈ Nǫand either ln=kormn=k.\nAssume the latter; the argument in the other case is similar. By Lemm a 4, there exist a pair\nηln(x),ηmn(x) of functions biorthogonal to φln(x),φmn(x) onL2(a,b), and furthermore there\nexists a positive constant C, independent of n, such that\n/integraldisplayb\na|ηln(x)|2+|ηmn(x)|2≤C. (4.60)\nLet\nhln(x,t) =ηln(x)θln(t), hmn(x,t) =ηmn(x)θln(t).\nIt is then easy to check that\n/integraldisplayT\n0/integraldisplayb\nahj(x,t)exp(iλkt)φk(x)dxdt=δjk,∀j,k∈K.\nFinally, by (4.59) and (4.60), the estimate (4.58) follows. ✷\n19We now complete the proof of Theorem 2. Assume for the moment λ+\nn/\\e}atio\\slash=λ−\nnfor anyn.\nRecall for α <1,ρ <2 we have ℑλk≍ |k|2α, while for α∈[1,3/2), we have\nℑλk=|k|4−2α\nρ/parenleftBig\n1+o(1)/parenrightBig\n, k >0,andℑλk=ρ|k|2α/parenleftBig\n1+o(1)/parenrightBig\n, k <0.\nThe moment problem (4.57) is formally solved by\n/summationdisplay\nk∈Kζkhk(x,t).\nIt suffices to prove convergence of this series. By (4.55),(4.56),(4 .54), there exist positive\nconstants C3,C4such that\n|ζk| ≤C3e−C4(ℑλk),∀k∈K.\nSinceκ<1,by (4.58) and the asymptotics of {ℑλk}, the series converges in L2((a,b)×\n(0,T)).\nFinally, suppose λ+\nn=λ−\nnfor some n. It is easy to see this nwill be unique. Here,\nwe need to replace the pair ( eλ+\nnt,eλ−\nnt) by the pair ( eλ+\nnt,teλ+\nnt). The construction of the\nbiorthogonal set {θk(t)}can now proceed same as in the previous section, and then the\nconstruction of {hk(x,t)}can now proceed as above in this section. The details are left to\nthe reader. This finishes the proof of Theorem 2. ✷\n5 Conclusion\nOur results on finite dimensional control (Theorem 1) are in some se nse definitive. One\npossible extension would be to consider perturbations of the Laplac ian, replacing uxxby\n(r(x)ux)x+q(x)u(x). Another possible extension would be to replace Dirichlet bound-\nary conditions by Neumann or Robin boundary conditions. In all thes e cases, provided\nthe Sturm-Liouville problem is regular, the spectrum would remain simp le, with the same\nasymptotics as in this paper. Thus the frequency set will have multip licity at most two,\nand two dimensional null-controllabilty will always be possible for α <3/2, and will fail for\nα≥3/2.\nIn the case of controls distributed on an open proper subset of (0 ,π), Theorem 2, the\ncaseα≥3/2 remains open. In case the associated Sturm-Liouville problem is per turbed\nregularly, as in the previous problem, the methods of this paper will a pply. For α∈(1,3/2),\none would need to generalize Lemma 4.\nAcknowledgements. The research of Sergei Avdonin was supported in part by the\nNational Science Foundation, grants DMS 1909869 and 2308377.\n6 Appendix\nProof of Lemma 4: Denoteby Φ( n,m) theanglebetween ϕnandϕminL2(a,b),Φ(n,m)∈\n[0,π/2].Evidently, for m/\\e}atio\\slash=n, the functions sin mxand sinnxare linearly independent on\n20(a,b), which implies Φ( n,m)>0. Therefore we can restrict to the large m,n. In what\nfollows we suppose m > n. By the definition\ncosΦ(n,m) =|(ϕn,ϕm)L2(a,b)|\n/bardblϕn/bardblL2(a,b)/bardblϕm/bardblL2(a,b).\nFurther,/integraldisplayb\nasinmxsinnxdx=1\n2/bracketleftbiggsin(n−m)x\nn−m−sin(n+m)x\nn+m/bracketrightbiggb\na.\nThis gives the asymptotic relation\n(ϕn,ϕm)L2(a,b)=2\nπ(m−n)sin/bracketleftbigg1\n2(m−n)(b−a)/bracketrightbigg\ncos/bracketleftbigg1\n2(m−n)(b+a)/bracketrightbigg\n+O(1/(m+n)).\nSimilarly\n/bardblϕm/bardbl2\nL2(a,b)=2\nπ/integraldisplayb\nasin2mxdx=b−a\nπ+O(1/m).\nThen\ncosΦ(n,m) =2\n(b−a)(m−n)sin/bracketleftbigg1\n2(m−n)(b−a)/bracketrightbigg\ncos/bracketleftbigg1\n2(m−n)(b+a)/bracketrightbigg\n+O(1/n)≤\n≤2\n(b−a)(m−n)sin/bracketleftbigg1\n2(m−n)(b−a)/bracketrightbigg\n+O(1/n)\nThe function\nf(x) =2sin(x/2)\nx,\ndefined on a semiaxis ( ǫ,∞) with a positive ǫsatisfies\nsupf <1.\nIndeed,f(x)<1 and this functions goes to zero as xgoes to infinity,\nThus,\nsup\nm/\\egatio\\slash=ncosΦ(n,m)<1.\n✷\nReferences\n[1] G. Avalos and I. Lasiecka, ”Optimal blowup rates for the minimal e nergy null control\nof the strongly damped abstract wave equation”. Ann. Sc. Norm. Super. Pisa Cl. Sci.\n(5) 2 (2003), no. 3, 601–616.\n[2] S. A. Avdonin and S. A. Ivanov, Families ofExponentials. TheMethodofMoments\ninControllability Problems forDistributed Parameter Systems, Cambridge University\nPress, New York, London, Melbourne, 1995.\n21[3] A.V. Balakrishnan, Damping operators in continuum models of flexib le structures: Ex-\nplicit models for proportional damping in beam bending with end-bodies , Appl. Math.\nOptim. 21 (3) (1990) 315–334.\n[4] M. Gonzalez-Burgos and L Ouaili, SHARP ESTIMATES FOR BIORTHOG ONAL\nFAMILIES TO EXPONENTIAL FUNCTIONS ASSOCIATED TO COMPLEX SE -\nQUENCES WITHOUT GAP CONDITIONS, Evolution Equations and Contr ol Theory\nVol. 13, No. 1, February 2024, pp. 215-279 doi:10.3934/eect.2023 044.\n[5] G. Chen and D.L. Russell, ”A mathematical model for linear elastic s ystems with struc-\ntural damping”, Quart. Appl. Math., 39, (1982), 433-454.\n[6] Edward, J. Complex Ingham type inequalities and applications to co ntrol theory, Jour-\nnal of Mathematical Analysis and Applications, 324 (2006)\n[7] S.W. Hansen, ”Bounds on functions biorthogonal to sets of com plex exponentials; con-\ntrol of elastic damped systems”, J. Math. Anal. Appl. 158 (1991), 487-508\n[8] A.E. Ingham, ”Some trigonometrical inequalities with applications t o the theory of\nseries”, Mathematicsche Zeitschrift, 41 (1936), 367-379.\n[9] Koosis, P. Introduction toHpSpaces (2nd ed., Cambridge Tracts in Mathematics).\nCambridge: Cambridge University Press. 1999.doi:10.1017/CBO9780 511470950\n[10] I. Lasieka and R. Triggiani, ”Exact null-controllability of structu rally damped and\nthermo-elastic parabolic models”, Rend. Mat. Acc. Lincet, s.9, v.9 ( 1998), p.43-69.\n[11] Miller, LucNon-structural controllability of linear elastic system s withstructural damp-\ning. J. Funct. Anal. 236 (2006), no. 2, 592–608.\n[12] Mitra, Sourav Carleman estimate for an adjoint of a damped bea m equation and an\napplication to null controllability. J. Math. Anal. Appl. 484 (2020), no . 1, 123718, 29\npp.\n[13] Seidman, T. I., Avdonin, S. A.; Ivanov, S. A., The ”window problem ” for series of\ncomplex exponentials. J. Fourier Anal. Appl. 6 (2000), no. 3, 233–2 54.\n[14] R. Triggiani,JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICAT IONS\n161, 2999331 (1991) Regularity of Some Structurally Damped Prob lems with Point\nControl and with Boundary Control\n[15] R. Triggiani, ”Optimal estimates of norms of fast controls in exa ct null controllability\nof two non-classical abstract parabolic systems”. Adv. Different ial Equations 8 (2003),\nno. 2, 189–229.\n22" }, { "title": "2401.15001v1.Accelerated_relaxation_enhancing_flows_cause_total_dissipation.pdf", "content": "arXiv:2401.15001v1 [math.AP] 26 Jan 2024ACCELERATED RELAXATION ENHANCING FLOWS CAUSE\nTOTAL DISSIPATION\nKEEFER ROWAN\nAbstract. We show that by “accelerating” relaxation enhancing flows, o ne\ncan construct a flow that is smooth on [0 ,1)×Tdbut highly singular at t= 1\nso that for any positive diffusivity, the advection-diffusio n equation associated\nto the accelerated flow totally dissipates solutions, takin g arbitrary initial data\nto the constant function at t= 1.\n1.Introduction\nWe consider the evolution of advection-diffusion equations on the to rusTdwith\nincompressible advecting flow. That is we study solutions to the equa tion\n∂tθ−ν∆θ+u·∇θ= 0,\nwhere∇ ·u= 0 and ν≥0.As this equation preserves the mean of θ, we will\nwithout loss of generality suppose/integraltext\nTdθ(x)dx= 0 throughout. The presence of a\npositive diffusivity ν >0 ensures that /ba∇dblθ/ba∇dblL2(Td)(t) is strictly decreasing in time.\nA flow is relaxation enhancement if it causes this dissipation of the L2norm of\nθto happen faster than if no flow were present. To make this precise , let us\nintroduce some definitions from the literature [5, 8, 11]. These ideas originate\nin [5], which originally defined relaxation enhancing. We however use a diff erent—\nbut equivalent—definition of relaxation enhancing.\nDefinition 1.1. We denote\nL2\n0(Td) :={θ∈L2(Td) :/integraldisplay\nθdx= 0}.\nDefinition 1.2. For a flow u∈L∞\nloc([0,∞)×Td) such that ∇·u= 0, a diffusivity\nν >0, and times 0 ≤s≤t <∞, let Φu,ν\ns,t:L2\n0(Td)→L2\n0(Td) denote the solution\noperator to the PDE\n(1.1) ∂tθ−ν∆θ+u·∇θ= 0\nso thatθ(t) := Φu,ν\ns,tθ0solves (1.1) on [ s,∞)×Tdwith initial data θ0.\nDefinition 1.3. Define the dissipation time of a flow as the maximum time it take\nfor half of the L2norm of a solution to (1.1) to diffuse, that is\nτu(ν) := sup\ns≥0/parenleftBig\ninf{t−s:t≥s,/ba∇dblΦu,ν\nt,s/ba∇dblL2\n0→L2\n0≤1\n2}/parenrightBig\n.\nDefinition 1.4. Say a flow u∈L∞\nloc([0,∞)×Td) such that ∇·u= 0 isrelaxation\nenhancing if\nlim\nν→0ντu(ν) = 0.\n2010Mathematics Subject Classification. Primary 35Q35, 76R99.\n12 KEEFER ROWAN\nA very wide variety of flows are relaxation enhancing, including all mixin g flows,\nas will be discussed below. Before stating the main result, we need to introduce\nthe notion of accelerating a flow.\nDefinition 1.5. For a flow u∈L∞\nloc([0,∞)×Td) and an increasing diffeomorphism\nσ: [0,1)→[0,∞), we define the accelerated flow uσ∈L∞\nloc([0,1)×Td) by\nuσ(t,x) :=σ′(t)u(σ(t),x).\nThis definition is given so that if θsolves the transport equation associated to\nu, thenθ(σ(t),x) solves the transport equation associated to uσ. We can now state\nthe main result.\nTheorem 1.6. Suppose that u∈L∞\nloc([0,∞)×Td)such that ∇·u= 0is relaxation\nenhancing. Then there exists some acceleration of uthat is totally diffusive on\n[0,1]for all diffusivities ν >0, that is there exists some smooth diffeomorphism\nσ: [0,1)→[0,∞)such that for all ν >0,s∈[0,1),θ0∈L2\n0(Td),\nΦuσ,ν\ns,1θ0= 0.\nA natural question is what space uσbelongs to. Let Xbe some Banach space\nof functions Td→Rd. For the sake of clarity, we consider the case that the flow\nuis constant-in-time (a perfectly valid possibility), though this discus sion is easily\nadapted to a variety of assumptions on a time-dependent flow, mos t simply that\nu∈Cb([0,∞),X).\nWe first note that uσ/ne}ationslash∈L1([0,1],X), as\n/integraldisplay1\n0/ba∇dbluσ/ba∇dblX(t)dt=/ba∇dblu/ba∇dblX/integraldisplay1\n0σ′(t)dt=/ba∇dblu/ba∇dblXσ(1) =∞.\nOn the other hand, we have that uσ∈C∞([0,1),X).Thus we see that uσis regular\naway from t= 1, but so highly singular at t= 1 so as not to live in any Lebesgue\nspace. On the other hand, uσlives quite naturally in weighted-in-time spaces, in\nparticular we trivially have\n1\nσ′uσ∈L∞([0,1],X),\nwhere we note that lim t→11\nσ′(t)= 0. Then to determine which weighted spaces uσ\nbelongs to, we need to get an upper bound for σ′. The bound on σ′in turn depends\non how relaxation enhancing the flow uis—that is how fast ντu(ν)ν→0→0. The\nnext result follows from a general bound on σ′and gives the general weighted space\nthatuσbelongs to. Additionally, we give a more concrete bound in the particu larly\nrelevant case of exponential mixing.\nTheorem 1.7. LetXbe some Banach space of functions Td→Rd. Then the uσ\nin Theorem 1.6 can be taken so that\n(1.2)/vextenddouble/vextenddouble/vextenddoublef−1/parenleftBig\n1−t\n4(|log2(1−t)|+1)/parenrightBig\n2(|log2(1−t)|+1)uσ(t)/vextenddouble/vextenddouble/vextenddouble\nL∞([0,1],X)≤ /ba∇dblu/ba∇dblL∞([0,∞),X),\nwhere\nf(a) := sup\n0≤b≤abτu(b).ACCELERATED RELAXATION ENHANCING FLOWS CAUSE TOTAL DISSIP ATION 3\nIn particular, if u∈L∞([0,∞),C∞(Td))is exponentially mixing, that is if for every\nθ0∈L2\n0(Td), we have for some K,p >0,the estimate\n(1.3) /ba∇dblΦu,0\n0,tθ0/ba∇dbl˙H−1(Td)≤Ke−tp/K/ba∇dblθ0/ba∇dblH1(Td),\nthen we can take uσin Theorem 1.6 so that\n(1.4)1−t\n|log2(1−t)|2+2/p+1uσ(t)∈L∞([0,1],C∞(Td)).\n2.Discussion\nRecently, there has been substantial interest in the phenomenon of anomalous\ndissipation for the passive scalaradvection-diffusion equation [6, 4, 1, 7, 3]. In these\nworks, an incompressible flow u∈L∞([0,1]×Td) is constructed1such that for some\ninitial data θ0∈L2(Td),2\nliminf\nν→0/ba∇dblΦu,ν\n0,1θ0/ba∇dblL2(Td)0 if the flow u∈L∞\nt,x, by the\nunique continuation result of [10].3\nLastly, wenotethat[8,11]showthatanymixingflowisrelaxationenh ancing.4In\nparticular, [8, 11] show quantitative relations between mixing rates and dissipation\ntimes. This implies the broad class of mixing flows are relaxation enhanc ing and\nfurther gives a quantitative bound on the rate of relaxation enhan cement (the rate\nthatντu(ν)→0) in terms of the rate of mixing. Thus through Theorem 1.7, quan-\ntitatively mixing flows allow us to construct uσwhich belong to specific weighted\nL∞spaces. As given in Theorem 1.7, we compute the weighted space uσbelongs\nto when uis exponentially mixing, but this computation can be repeated for any\ngiven rate of mixing, using [8] to convert the mixing rate to an enhan ced dissipa-\ntion rate and then using Theorem 1.7 to determine the appropriate w eighted space.\nThe existence of an exponential mixer u∈L∞([0,∞),C∞(Td)) withp= 1 is given\nby [2], so there are flows ufor which we can apply the estimate (1.4).\n1In each work, ubelongs to much stronger space than just L∞\nt,x, but for our sake this is the\nrelevant fact.\n2In most of these works, a substantially stronger result is sh own than anomalous dissipation\nfor justsomeinitial data. In particular, in [6], anomalous dissipation is shown for all initial data\nthat is sufficiently close to eigenfunctions of the Laplacian . In [1], anomalous dissipation is shown\nfor allθ0∈H1(Td). In [7], anomalous dissipation is shown for all θ0∈W1,∞(Td)∩H1+s(Td)\nfor some s >2/5.In [3], the construction of [1] is modified so that the flow usolves the Euler\nequation.\n3Note however that this doesn’t prevent one from having asymp totically total dissipation in\nthe limit as ν→0, i.e. that lim ν→0Φu,ν\n0,1θ0= 0, for a flow u∈L∞\nt,x.\n4For precise definitions of the relevant sense of mixing, see [ 8, 11].4 KEEFER ROWAN\n3.Proofs\nWe now provide the straightforward and short proofs of Theorem s 1.6 and 1.7.\nProofs of Theorems 1.6 and 1.7. We define the diffeomorphism σas a regulariza-\ntion of a piecewise linear flow. Let us first specify the piecewise linear fl ow. Fix\nsome strictly increasing sequence of time Tjsuch that T0= 0; the remaining Tj\nwill be specified later. Then let\n˜σ(1−2−j) =Tj,\nwith ˜σ(t) taken to be linear on [1 −2−j,1−2−(j+1)].We then define σas a strictly\nincreasing regularization of ˜ σso thatσ∈C∞(0,1) and so that σ′≤2˜σ′and for\nanyj∈Nand and any t∈[1−2−(j−1),1−2−(j−1)+2−(j+1)]\nσ(t) = ˜σ(t).\nThenσ: [0,1)→[0,∞) is a smooth diffeomorphism. We now fix ν >0. Our goal\nnow is to choose Tjindepedently of νso that for jsufficiently large\n/ba∇dblΦuσ,ν\n1−2−(j−1),1−2−(j−1)+2−(j+1)/ba∇dblL2\n0(Td)→L2\n0(Td)≤1\n2.\nNote that this clearly implies the total dissipation of Theorem 1.6.\nBy a simple change of variables\nΦuσ,ν\n1−2−(j−1),1−2−(j−1)+2−(j+1)= Φu,2−jν\nTj−Tj−1\nTj−1,Tj−1+Tj\n2,\nthus by the definition of τu, it suffices to choose Tjso that for jsufficiently large\nτu/parenleftBig2−jν\nTj−Tj−1/parenrightBig\n≤Tj−Tj−1\n2.\nThis in turn is implied by\n2−(j+1)ν≥f/parenleftBig2−jν\nTj−Tj−1/parenrightBig\n,\nwherefis defined as in Theorem 1.7. Note that fis increasing and for any ν >0,\nwe have that eventually j−1≤ν≤j, so it suffices to choose Tjso that\n(3.1)2−(j+1)\nj=f/parenleftBig2−jj\nTj−Tj−1/parenrightBig\n.\nThus we take Tjso that\nTj−Tj−1:=2−jj\nf−1/parenleftBig\n2−(j+1)\nj/parenrightBig.\nChoosing Tjin this way concludes the construction of σand by the arguments\nabove concludes the proof of Theorem 1.6.\nWhat remains is to prove the bounds of Theorem 1.7. Note that on th e interval\nt∈(1−2−(j−1),1−2−j), we have\nσ′(t)≤2˜σ′(t) = 2j+1(Tj−Tj−1) =2j\nf−1/parenleftBig\n2−(j+1)\nj/parenrightBig.\nNote then that on this interval\n2−j≤1−t≤2−(j−1),ACCELERATED RELAXATION ENHANCING FLOWS CAUSE TOTAL DISSIP ATION 5\nthus using that f−1is increasing\nσ′(t)≤2(|log2(1−t)|+1)\nf−1/parenleftBig\n1−t\n4(|log2(1−t)|+1)/parenrightBig.\nPlugging this into the definition of uσ, we get (1.2).\nIn order to conclude, we just need to specialize (1.2) to the case of exponential\nmixing. Suppose now that usatisfies the mixing estimate (1.3). From [11], (1.3)\nimplies that there exists some C(K,p)<∞such that for ν≤C−1\nτu(ν)≤C|logν|2/p.\nThus for a≤C−1,\nf(a)≤Ca|loga|2/p.\nIt’s somewhat unwieldy to compute a good bound directly from (1.2), so let us\ninstead return to (3.1) to give for jsufficiently large,\n2−(j+1)\nj=f/parenleftBig2−jj\nTj−Tj−1/parenrightBig\n≤C2−jj\nTj−Tj−1/vextendsingle/vextendsingle/vextendsingle/vextendsinglelog2−jj\nTj−Tj−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/p\n≤C2−jj\nTj−Tj−1/parenleftBig\nj2/p+|log(Tj−Tj−1)|2/p/parenrightBig\n.\nThus\nTj−Tj−1≤Cj2/parenleftBig\nj2/p+|log(Tj−Tj−1)|2/p/parenrightBig\n.\nOne can verify that this inequality implies that\nTj−Tj−1≤Cj2+2/p.\nThen, as above, we have that on the interval t∈(1−2−(j−1),1−2−j),\nσ′(t)≤2˜σ′(t) = 2j+1(Tj−Tj−1)≤C2j+1j2+2/p≤C|log2(1−t)|2+2/p+1\n1−t.\nThis estimate then gives the result. /square\nAcknowledgements. I would like to thank Scott Armstrong and Vlad Vicol for\nstimulating discussion. The author was partially supported by NSF gr ants DMS-\n1954357 and DMS-2000200 as well as a Simons Foundation grant.\nReferences\n1. Scott Armstrong and Vlad Vicol, Anomalous diffusion by fractal homogenization , May 2023,\narXiv:2305.05048 [math-ph, physics:physics].\n2. AlexBlumenthal, MicheleCoti Zelati, and RishabhS.Gval ani,Exponential mixing for random\ndynamical systems and an example of Pierrehumbert , The Annals of Probability 51(2023),\nno. 4, 1559–1601.\n3. Jan Burczak, L´ aszl´ o Sz´ ekelyhidi Jr., and Bian Wu, Anomalous dissipation and Euler flows ,\nOctober 2023, arXiv:2310.02934 [math].\n4. Maria Colombo, Gianluca Crippa, and Massimo Sorella, Anomalous Dissipation and Lack\nof Selection in the Obukhov–Corrsin Theory of Scalar Turbul ence, Annals of PDE 9(2023),\nno. 2, 21 (en).\n5. Peter Constantin, Alexander Kiselev, Lenya Ryzhik, and A ndrej Zlatoˇ s, Diffusion and Mixing\nin Fluid Flow , Annals of Mathematics 168(2008), no. 2, 643–674.6 KEEFER ROWAN\n6. Theodore D.Drivas,Tarek M.Elgindi,Gautam Iyer, and In- Jee Jeong, Anomalous Dissipation\nin Passive Scalar Transport , Archive for Rational Mechanics and Analysis 243(2022), no. 3,\n1151–1180 (en).\n7. Tarek M. Elgindi and Kyle Liss, Norm Growth, Non-uniqueness, and Anomalous Dissipation\nin Passive Scalars , September 2023, arXiv:2309.08576 [physics].\n8. Yuanyuan Feng and Gautam Iyer, Dissipation enhancement by mixing , Nonlinearity 32\n(2019), no. 5, 1810 (en).\n9. Martina Hofmanov´ a, Umberto Pappalettera, Rongchan Zhu , and Xiangchan Zhu, Anomalous\nand total dissipation due to advection by solutions of rando mly forced Navier-Stokes equations ,\nMay 2023, arXiv:2305.08090 [math].\n10. Chi-Cheung Poon, Unique continuation for parabolic equations , Communications in Partial\nDifferential Equations 21(1996), no. 3-4, 521–539 (English).\n11. Michele Coti Zelati, Matias G. Delgadino, and Tarek M. El gindi,On the Relation between\nEnhanced Dissipation Timescales and Mixing Rates , Communications on Pure and Applied\nMathematics 73(2020), no. 6, 1205–1244 (en).\nCourant Institute of Mathematical Sciences, New York Universi ty, New York 10012\nEmail address :keefer.rowan@cims.nyu.edu" }, { "title": "2401.15009v1.Double_pulse_all_optical_coherent_control_of_ultrafast_spin_reorientation_in_antiferromagnetic_rare_earth_orthoferrite.pdf", "content": "Double pulse all-optical coherent control of ultrafast spin-reorientation in\nantiferromagnetic rare-earth orthoferrite\nN. E. Khokhlov,1A. E. Dolgikh,1B. A. Ivanov,1, 2and A. V. Kimel1\n1)Radboud University Nijmegen, Institute for Molecules and Materials, 6525 AJ Nijmegen,\nThe Netherlands\n2)Institute of Magnetism, NAS and MES of Ukraine, 36b Vernadsky Blvd., Kiev 03142,\nUkraine\n(*nikolai.khokhlov@ru.nl)\n(Dated: 29 January 2024)\nA pair of circularly polarized laser pulses of opposite helicities are shown to control the route of spin reorientation\nphase transition in rare-earth antiferromagnetic orthoferrite SmTbFeO 3. The route can be efficiently controlled by\nthe delay between the pulses and the sample temperature. Simulations employing earlier published models of laser-\ninduced spin dynamics in orthoferrites failed to reproduce the experimental results. It is suggested that the failure is due\nto neglected temperature dependence of the antiferromagnetic resonance damping in the material. Taking into account\nthe experimentally deduced temperature dependence of the damping, we have been able to obtain a good agreement\nbetween the simulations and the experimental results.\nAntiferromagnets are the largest, but probably the least ex-\nplored, class of magnetically ordered materials discovered\nonly in the 20thcentury1,2. The magnetic order in antifer-\nromagnets is characterized by mutually antiparallel alignment\nof neighboring spins, such that their net magnetic moment is\neither zero or vanishingly small. In the simplest case of a two-\nsublattice antiferromagnet, the order can be modeled as two\nferromagnets with mutually antiparallel magnetizations of the\nsub-lattices – M1andM2– so that the whole material is de-\nscribed by the antiferromagnetic Néel vector L=M1−M2.\nDue to the high frequencies of intrinsic spin resonances,\noften reaching the landmark of 1 THz, the antiferromagnets\nare seen as the materials that may facilitate the fastest and\nleast-dissipative mechanisms for writing magnetic bits in fu-\nture data storage3. Understanding how to control spins in\nantiferromagnets and revealing the characteristic time scales,\nwhich define the fundamental limits on the speed of such a\ncontrol, are thus among the most heavily debated questions in\ncontemporary magnetism4.\nRare-earth orthoferrites have been long offering a very\nfruitful playground for this research. First, because of the very\nstrong temperature dependence of magnetic anisotropy, these\nmaterials possess a heat-induced spin-reorientation phase\ntransition (SRT). Using the femtosecond laser pulse as an ul-\ntrafast heater, it is possible to launch spin dynamics and study\nspin reorientation in antiferromagnets at an unprecedentedly\nfast timescale5. Second, due to strong opto-magnetic effects,\ncircularly polarized femtosecond laser pulses can act on spins\nin these materials as equally short pulses of effective magnetic\nfield with the polarity defined by the helicity of light6.\nA combination of these two mechanisms of launching the\nspin dynamics led, in particular, to the discovery of spin in-\nertia in antiferromagnets7and to the routes of coherent con-\ntrol of SRT8. Although intuitively heat-induced SRT can pro-\nceed along two energetically equivalent routes with the ma-\nterial eventually ending up in a multidomain state, ultrashort\npulses of opto-magnetic fields were suggested to dynamically\nbreak the degeneracy and steer the medium to a state defined\nby the helicity of the light pulse8. Later, the same principle ofdynamical degeneracy breaking was employed to demonstrate\ncoherent control of SRT in orthoferrites with the help of a pair\nof pulses – a properly timed femtosecond laser heat pulse and\na nearly single-cycle pulse of the THz magnetic field9.\nHere, we further explore the coherent control of SRT with a\npair of pump pulses. Employing two circularly polarized op-\ntical pulses acting as both ultrafast heater and opto-magnetic\nfield, we reveal a strong and previously ignored effect of heav-\nily increased damping of spin precession near the phase transi-\ntion. The damping significantly affects the result of the action\nof the pair of pulses. We showed that if the first pulse heats\nthe orthoferrite to a temperature near the SRT, the material be-\ncomes practically insensitive to the second pulse in the subse-\nquent time window of 5-20 ps. According to the simulations\nbased on the models employed before, such an insensitivity\ncan indeed be observed, but in this case it must be observed\nperiodically at later time delays as well. We propose an up-\ngrade for the existing model that accounts for the increased\ndamping and enables a match of the modeling with the exper-\nimental results.\nThe idea of double-pulse coherent control employs SRT in\nrare-earth orthoferrite. Here we employ (Sm 0.55Tb0.45)FeO 3\nas a sample. The crystal was grown using the floating zone\ntechnique10. For the study, the bulk crystal is cut in the\nform of 158- µm-thin plane-parallel plate with normal along\nthecaxis. The magnetic structure of the crystal can be\nmodeled as a two-sublattice antiferromagnet with magneti-\nzations M1andM2, respectively. The exchange interaction\nfavors their mutually antiparallel orientations, but due to the\nDzyaloshinskii-Moriya interaction M1andM2are slightly\ncanted of about 1 degree, resulting in non-zero net magne-\ntization M=M1+M2̸=0. Magnetization Mand the anti-\nferromagnetic Néel vector Lare orthogonal to each other [in-\nsets on Fig. 1(c)]. At temperatures T<215 K, the spins and\nLare aligned along the ccrystallographic axis, while Mis\nalong the aaxis ( Γ2phase). Due to the strong temperature\ndependence of magneto-crystalline anisotropy, in the range\n215 K 250 K, the spins are along the aarXiv:2401.15009v1 [cond-mat.str-el] 26 Jan 20242\nFIG. 1. (a) Schematic illustration of the coherent optical control of\nfinal magnetization at SRT. (b) Domain patterns of the sample at dif-\nferent temperatures. (c) Temperature dependence of magneto-optical\ncontrast between opposite domains (symbols). Line is the theoret-\nical dependence found using the model from11. Insets depict the\norientations of antiferromagnetic vector Land net magnetization M\nof the sample in the low-temperature ( Γ2), angular ( Γ24), and high-\ntemperature ( Γ4) phases.\naxis, while Mis along the caxis (Γ4 phase). Measuring the\nFaraday effect for light propagating along the caxis and thus\nsensitive to the magnetization component along the caxis, we\ncan experimentally confirm the presence of SRT in the studied\nsample. It is seen from the images that a temperature increase\npromotes spin reorientation from phase Γ2 to phase Γ4 along\none of the two equivalent routes leading either to a state with\nthe magnetization \"up\" or to a state with the magnetization\n\"down\". Hence, even if in the Γ2 phase the sample was in a\nsingle-domain state, upon a temperature increase the sample\nsplits into multiple magnetic domains with mutually opposite\nmagnetizations.\nThe idea of our experiment is to control the route with a\npair of laser pulses and reveal how the final state depends on\nthe time delay between the pulses in a pair. An expected sce-\nnario that we aim to verify is shown in Fig. 1(a). The first\npulse acts as an ultrafast heater and a pulse of opto-magnetic\nfield. Hence, it launches a low-amplitude spin precession and,\nsimultaneously, causes transient changes of thermodynamic\nequilibrium. Using a properly timed second pulse, which also\nacts as an ultrafast heater and an ultrashort pulse of the opto-\nmagnetic field, one can push the spin system either to the state\nwith the magnetization \"up\" or to the state with the magneti-\nzation \"down\".\nTo study the magnetization reorientation induced by such\na double-pulse excitation, we used a time-resolved magneto-\noptical pump-probe technique combined with magneto-\noptical imaging12. The sample is pumped with two 50 fs\ncircularly polarized laser pulses with a central wavelength of\n800 nm, generated by Ti:sapphire amplifier at 1 kHz repeti-\ntion rate. The time delay between these two pumps τis me-\nchanically controlled in a range from -150 to +150 ps. The\npumps follow the same path and approach the sample at an\nincidence angle of 11◦. They are focused in spots with full\nwidth at a half maximum of 100 µm. Pump-induced changes\nin the sample are probed with a linearly polarized pulse with\na wavelength converted from 800 to 650 nm using an optical\nparametric amplifier. The probe is unfocused to cover an area\nof approximately 1 ×1 mm2on the sample with a fluence four\nFIG. 2. (a) Magneto-optical images of the reorientation process\natt=1.5 ns for different pump-pump delays τobserved with CCD\ncamera at T=190 K. (b) Experimental diagram of magnetization’s\nfinal state in coordinates of pump-pump delay τand initial temper-\nature of the sample T. Fluences of both pumps are 67 mJ/cm2. (c)\nThe diagram of the final state calculated with Eq.(1). The color code\nshows the value cos θ, as the experimental scheme is sensitive to out-\nof-plane component of magnetization.\norders of magnitude lower than that of a pump pulse. The time\ndelay between the first-arrived pump and probe is mechani-\ncally controlled from -0.5 to +1.5 ns. Two complementary\nsets of experiments are performed. The first one used a CCD\ncamera as a detector to obtain magneto-optical images of the\nsample13. In the second set of experiments, a diaphragm is\nset in the probe beam, selecting only the pumped area. After\nthis spatial filtering, the probe is detected with a balanced de-\ntector and lock-in amplifier, synchronized with a mechanical\nchopper placed on the pump path. In both cases, the measure-\nments are sensitive to the out-of-plane component of magneti-\nzation parallel to the caxis. The sample is placed in a cold fin-\nger cryostat to control its initial temperature T. Experiments\nare done without an external magnetic field. The helicities of\nthe two pumps were set to the right-handed σ+and the left-\nhanded σ−, respectively, with quarter-wave plates.\nFor double-pump experiments, we set the fluences of the\npump pulses to 67 mJ/cm2to such that any of the pulses could\nnot launch the SRT alone, but two pump pulses were able to\ninitiate and steer the phase transition. Figure 2(a) shows that\nthe sign of magnetization in the final state, measured at 1.5\nns after the laser excitation, depends dramatically on the time\ndelay between the pump pulses τ.\nTo obtain a better understanding of the observed dynamics,\nwe performed double-pump experiments with the balanced\nphotodetector as a function of temperature Tand time de-\nlayτ. In Fig. 2(b) we plot experimentally defined diagram\nshowing the magneto-optical contrast of the pumped domain\nat 1.5 ns after the laser excitation as functions of Tandτ.\nAt temperatures just below the SRT, i.e. 190 τc, the magneti-\nzation orientation is defined by the helicity of the latest pulse.\nAt lower temperatures of T<170 K, the optical pump pulses\ndo not sufficiently heat the system to trigger SRT.\nTo simulate laser-induced spin dynamics, we solved the\nequation of motion for the antiferromagnetic vector Néel L\nderived using the principles of Lagrangian mechanics14,15.\nThe resulting dynamics of Lwithin the acplane is described\nby the angle θbetween Landaaxis as7,16:\nd2θ\ndt2+2ζdθ\ndt+γHexdWa(θ)\ndθ=γ2HDHp(t)sinθ,(1)\nwhere the total length of Lis assumed to be conserved; ζis\na damping parameter in the units of frequency; Hp(t)is the\npulse effective opto-magnetic field with duration of 50 fs and\naligned either parallel ( σ+) or antiparallel ( σ−) with respect\nto the caxis; HDis the effective field of the Dzyaloshinskii-\nMoriya interaction; Hexrepresents the exchange field of the\nantiferromagnet; γis the gyromagnetic ratio; tis time after the\npump. The function Wa(θ)is the potential energy described\nby the magnetic anisotropy of the antiferromagnet. In our\nmodel, Wa(θ)is a function of temperature in accordance with\nthe conventionally accepted model11.To mimic laser-induced\nheating, we assume that the temperature is a function of the\npump-probe time delay, similarly to the model from Ref.17.\nIn particular, we take into account that the time dependence\nof magnetic anisotropy is due to temperature-induced repop-\nulation of the electronic states in highly anisotropic Sm3+\nand Tb3+ions. This repopulation occurs on a time scale of\nelectron-phonon interaction for rare-earth ions, which could\nbe estimated to be around 15 ps17. Furthermore, we suggest\nan increase in temperature of 25 K after one pump, since the\naction of both pumps is enough to induce SRT at T=170 K.\nThe results of the modeling are similar to those from Ref.9\n(see Appendix A), but they are clearly different with respect\nto the experimental observations. In the modeling, we indeed\nobserve a triangle centered around zero pump-pump delay,\nsimilar to the experimental diagram in Fig. 2(b). This trian-\ngle reproduces the insensitivity of spins in the antiferromagnet\nto the second pump pulse observed experimentally. However,\ncontrary to the experiment, this insensitivity also appears pe-\nriodically at longer times τin the simulations (Fig.A1), but\nis clearly absent in the experiment. Nevertheless, we note\nthat none of the models suggested before took into account\nthe fact that the damping parameter ζin the equation of mo-\ntion (1) must also have a strong temperature dependence. The\nopto-magnetic pulse triggers spin oscillations at the frequency\nof the quasi-ferromagnet mode of the antiferromagnetic reso-\nnance in the orthoferrite6. This mode is known to \"soften\"\ndown to zero frequency at the temperatures of SRT from Γ2to\nΓ24, as well as from Γ24toΓ4. It is a well-known experimental\nfact that softening of magnetic resonances is accompanied by\na dramatic increase in damping. This is also the case in our ex-\nperiment. Measuring the frequency and damping of the spin\nFIG. 3. Two scenarios of the coherent magnetization control under\ndouble-pulse fluence. The final state is defined by pumps’ helicities\nand pump-pump delay τ. Detail description is in the text.\noscillations as a function of temperature, we found that the\ndamping peaks at T1indeed. If we add this experimentally de-\nfined temperature dependence of damping to the simulations,\nthe results of the simulations appear to be in good qualitative\nagreement with the experiment (Fig.2b,c).\nOn the basis of the experimental observations and model-\ning, we suggest the next path of the reorientation process un-\nder double-pump excitation. In phase Γ2(Tθ0(Fig. 3, top frame). Subsequently, there are\ntwo scenarios for the subsequent dynamics determined by τ.\nThe first scenario is realized if the second pump arrives too\nearly, that is, τis smaller than a critical value τcr(Fig. 3, left\nframes). In this scenario, the second pulse cannot reverse the\nforward motion of the system or accelerate it sufficiently in the\nbackward motion to overcome the potential barrier in phase\nΓ24before the barrier appears. The second pump will further\nheat the sample and thus help further establish the state deter-\nmined by the helicity of the first pump if τ<τcr. The second\nscenario occurs if τ>τcr, that is, the second pump arrives\nwhen the system has reversed its momentum and passed a crit-\nical coordinate θcrin backward motion (Fig. 3, right frames).\nIn this scenario, the torque of the second pulse is sufficient to\ntransfer the system to θ<θ0before the potential barrier ap-\npears in Γ24. We notice that the same scenarios work for the\ninitial combination of σ+andθ0=−π/2, since the torque\ninduced by IFE does not change sign with the sign of θ018,19.\nThus, helicity σ+works in the same way for both initial ori-\nentations of the domains in the sample. The change in helic-\nity sign flips the initial torque and corresponding final state,\nresulting in the symmetrical T−τdiagram with respect to\nτ=0.\nIn conclusion, we experimentally and numerically studied\ncoherent control of the ultrafast phase transition in antiferro-\nmagnetic rare-earth orthoferrite using double-pulse excitation.\nWe show that the final state, formed at 1.5 ns after pump ex-\ncitation, depends on the time delay between the pump pulses.4\nAt a temperature close to the phase transition, the final state\nis fully defined by the helicity of the earliest pump pulse. At\nlower temperature, we distinguish two regions. In particular,\nwe show that at delays larger than a critical time, the state\nis defined by the helicity of the latest pulse, while at shorter\ndelays it is the earliest pulse in the pair that defines the fi-\nnal magnetization. We show that earlier published models are\nunable to reproduce the experimental results and suggest that\nthe reason for the discrepancy is the neglected temperature de-\npendence of the damping. Finally, we note that the peak-like\nbehavior of damping at SRT temperature is more general and\ninherent to other kinds of phase transitions, where the soft-\nening of the corresponding mode appears: damping peak of\nmagnons at antiferromagnetic-paramagnetic phase transition\nat Néel temperature20,21; easy-axis to easy-plane Morin tran-\nsition in antiferromagnets22; damping peak of the mode with\nout-of-phase magnetizations precession in synthetic antifer-\nromagnets at spin-flop transition23; decrement of spin-lattice\nrelaxation time in nuclear quadrupole resonance studies of\nstructural phase transitions in cubic antifluorite and cubic per-\novskite structures24; peak of the relative absorption coefficient\nof dynamics in multiferroic crystals at magnetic and ferroelec-\ntric phase transitions25.\nACKNOWLEDGMENTS\nThe work is supported by the European Research Council\nERC Grant Agreement No. 101054664 (SPARTACUS).\nCONFLICT OF INTEREST STATEMENT\nThe authors have no conflicts to disclose.\nAUTHOR CONTRIBUTIONS\nNikolai E. Khokhlov : Investigation (equal); Methodology\n(experiment); Visualization (equal); Data Curation, Writing -–\noriginal draft (equal); Writing — review and editing (equal).\nAlexander E. Dolgikh : Investigation (equal); Visualization\n(equal); Software, Writing -– original draft (equal). Boris A.\nIvanov : Methodology (theoretical model); Writing — origi-\nnal draft (equal). Alexei V . Kimel : Conceptualization; Super-\nvision; Writing — review and editing (equal); Project Admin-\nistration.\nDATA AVAILABILITY STATEMENT\nSource data for figures are publicly available at https:\n//doi.org/10.34973/5nxv-2q76 . All other data support-\ning the findings of this article are available from the corre-\nsponding author upon request.APPENDIX\nA. Simulations with low damping parameter\nFIG. A1. Diagram of the final state of magnetization in the coor-\ndinates of the pump-pump delay τand the initial temperature of the\nsample T, calculated with Eq.(1) at dimensionless damping parame-\nterζ/(2πf) =0.1.\nB. Single-pump experiments at low pump fluence\nFigure A2 represents experimental data on single-\npump excitation with low fluence. The oscilla-\ntions on Fig. A2(a) are fitted with the function\nF(t) =Aexp(−2πfλt)sin(2πft+ϕ) +y(t), where Fis\nFaraday signal, λ=ζ/(2πf)is the dimensionless decay\nrate; f,A,ϕare the frequency, amplitude, and initial phase of\noscillations, respectively; y(t)is a slow varying offset.\nFIG. A2. (a) Magnetization dynamics in single-pump experiments\nat different initial temperatures. The difference of two pump-probe\nsignals at opposite pump helicities σ−andσ+is shown. The fluence\nof the pump is 25 mJ/cm2. Symbols -– experiment; lines — fits with\ndamping sine function. The data sets are shifted along the vertical\naxis for the convenience. (b) Temperature variation of the frequency\nfand the dimensionless decay rate λ=ζ/(2πf), estimated from\ndata on panel (a). Solid lines are guides to the eye.5\nREFERENCES\n1L. D. Landau, “A possible explanation of the field dependence of the sus-\nceptibility at low temperatures,” Phys. Z. Sowjet 4, 675 (1933).\n2Néel, M. Louis, “Propriétés magnétiques des ferrites ; ferrimagnétisme et\nantiferromagnétisme,” Ann. Phys. 12, 137–198 (1948).\n3T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, “Antiferromagnetic\nspintronics,” Nature nanotechnology 11, 231–241 (2016).\n4C. Song, Y . You, X. Chen, X. Zhou, Y . Wang, and F. Pan, “How to ma-\nnipulate magnetic states of antiferromagnets,” Nanotechnology 29, 112001\n(2018).\n5A. Kimel, A. Kirilyuk, A. Tsvetkov, R. Pisarev, and T. Rasing, “Laser-\ninduced ultrafast spin reorientation in the antiferromagnet TmFeO 3,” Na-\nture429, 850–853 (2004).\n6A. Kimel, A. Kirilyuk, P. Usachev, R. Pisarev, A. Balbashov, and T. Ras-\ning, “Ultrafast non-thermal control of magnetization by instantaneous pho-\ntomagnetic pulses,” Nature 435, 655–657 (2005).\n7A. Kimel, B. Ivanov, R. Pisarev, P. Usachev, A. Kirilyuk, and T. Rasing,\n“Inertia-driven spin switching in antiferromagnets,” Nature Physics 5, 727–\n731 (2009).\n8J. A. de Jong, I. Razdolski, A. M. Kalashnikova, R. V . Pisarev, A. M. Bal-\nbashov, A. Kirilyuk, T. Rasing, and A. V . Kimel, “Coherent control of\nthe route of an ultrafast magnetic phase transition via low-amplitude spin\nprecession,” Phys. Rev. Lett. 108, 157601 (2012).\n9T. Kurihara, H. Watanabe, M. Nakajima, S. Karube, K. Oto, Y . Otani, and\nT. Suemoto, “Macroscopic magnetization control by symmetry breaking of\nphotoinduced spin reorientation with intense terahertz magnetic near field,”\nPhys. Rev. Lett. 120, 107202 (2018).\n10A. Balbashov and S. Egorov, “Apparatus for growth of single crystals of\noxide compounds by floating zone melting with radiation heating,” Journal\nof Crystal Growth 52, 498–504 (1981).\n11H. Horner and C. M. Varma, “Nature of spin-reorientation transitions,”\nPhys. Rev. Lett. 20, 845–846 (1968).\n12K. Vahaplar, A. M. Kalashnikova, A. V . Kimel, D. Hinzke, U. Nowak,\nR. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, “Ultrafast\npath for optical magnetization reversal via a strongly nonequilibrium state,”\nPhys. Rev. Lett. 103, 117201 (2009).\n13Y . Hashimoto, A. R. Khorsand, M. Savoini, B. Koene, D. Bossini,\nA. Tsukamoto, A. Itoh, Y . Ohtsuka, K. Aoshima, A. V . Kimel, A. Kiri-\nlyuk, and T. Rasing, “Ultrafast time-resolved magneto-optical imaging of\nall-optical switching in gdfeco with femtosecond time-resolution and a µm\nspatial-resolution,” Review of Scientific Instruments 85, 063702 (2014).14A. K. Zvezdin, “Dynamics of domain walls in weak ferromagnets,” JETP\nLetters 29, 605 (1979).\n15V . G. Bar’yakhtar, B. A. Ivanov, and M. V . Chetkin, “Dynamics of domain\nwalls in weak ferromagnets,” Uspekhi Fizicheskikh Nauk , 417–458 (1985),\ntranslated as: Soviet Physics Uspekhi, 28, 563 (1985).\n16B. A. Ivanov, “Spin dynamics of antiferromagnets under action of femtosec-\nond laser pulses (review article),” Low Temperature Physics 40, 91–105\n(2014).\n17J. A. de Jong, A. V . Kimel, R. V . Pisarev, A. Kirilyuk, and T. Ras-\ning, “Laser-induced ultrafast spin dynamics in ErFeO 3,” Phys. Rev. B 84,\n104421 (2011).\n18F. Hansteen, A. Kimel, A. Kirilyuk, and T. Rasing, “Nonthermal ultra-\nfast optical control of the magnetization in garnet films,” Phys. Rev. B 73,\n014421 (2006).\n19L. A. Shelukhin, V . V . Pavlov, P. A. Usachev, P. Y . Shamray, R. V . Pisarev,\nand A. M. Kalashnikova, “Ultrafast laser-induced changes of the magnetic\nanisotropy in a low-symmetry iron garnet film,” Phys. Rev. B 97, 014422\n(2018).\n20D. Bossini, S. Dal Conte, Y . Hashimoto, A. Secchi, R. V . Pisarev, T. Ras-\ning, G. Cerullo, and A. V . Kimel, “Macrospin dynamics in antiferromag-\nnets triggered by sub-20 femtosecond injection of nanomagnons,” Nature\ncommunications 7, 10645 (2016).\n21D. Afanasiev, J. R. Hortensius, M. Matthiesen, S. Mañas-Valero,\nM. Šiškins, M. Lee, E. Lesne, H. S. J. van der Zant, P. G. Steeneken, B. A.\nIvanov, E. Coronado, and A. D. Caviglia, “Controlling the anisotropy of a\nvan der Waals antiferromagnet with light,” Science Advances 7, eabf3096\n(2021).\n22R. Lebrun, A. Ross, O. Gomonay, V . Baltz, U. Ebels, A.-L. Barra,\nA. Qaiumzadeh, A. Brataas, J. Sinova, and M. Kläui, “Long-distance spin-\ntransport across the morin phase transition up to room temperature in ultra-\nlow damping single crystals of the antiferromagnet α-Fe2O3,” Nature com-\nmunications 11, 6332 (2020).\n23S. Sorokin, R. A. Gallardo, C. Fowley, K. Lenz, A. Titova, G. Y . P. Atch-\neson, G. Dennehy, K. Rode, J. Fassbender, J. Lindner, and A. M. Deac,\n“Magnetization dynamics in synthetic antiferromagnets: Role of dynami-\ncal energy and mutual spin pumping,” Phys. Rev. B 101, 144410 (2020).\n24R. L. Armstrong, “Pure nuclear quadrupole resonance studies of structural\nphase transitions,” Journal of Magnetic Resonance (1969) 20, 214–231\n(1975).\n25E. Golovenchits and V . Sanina, “Magnetic and magnetoelectric dynamics\nin RMn 2O5(R= Gd and Eu),” Journal of Physics: Condensed Matter 16,\n4325 (2004)." }, { "title": "2401.15017v1.Protocol_for_certifying_entanglement_in_surface_spin_systems_using_a_scanning_tunneling_microscope.pdf", "content": "Protocol for certifying entanglement in surface spin systems using a scanning\ntunneling microscope\nRik Broekhoven,1Curie Lee,2, 3Soo-hyon Phark,2, 4Sander Otte,1and Christoph Wolf2, 4,∗\n1Department of Quantum Nanoscience, Kavli Institute of Nanoscience,\nDelft University of Technology, 2628 CJ Delft, the Netherlands.\n2Center for Quantum Nanoscience, Institute for Basic Science (IBS), Seoul 03760, Korea\n3Department of Physics, Ewha Womans University, Seoul 03760, Korea\n4Ewha Womans University, Seoul 03760, Korea\n(Dated: January 29, 2024)\nI. ABSTRACT\nCertifying quantum entanglement is a critical step towards realizing quantum-coherent applications of surface spin\nsystems. In this work, we show that entanglement can be unambiguously shown in a scanning tunneling microscope\n(STM) with electron spin resonance by exploiting the fact that entangled states undergo a free time evolution with\na distinct characteristic time constant that clearly distinguishes it from any other time evolution in the system.\nBy implementing a suitable phase control scheme, the phase of this time evolution can be mapped back onto the\npopulation of one entangled spin in a pair, which can then be read out reliably using a weakly coupled sensor spin in\nthe junction of the scanning tunneling microscope. We demonstrate through open quantum system simulations with\nrealistic spin systems, which are currently available with spin coherence times of T2≈300 ns, that a signal directly\ncorrelated with the degree of entanglement can be measured at a temperature range of 100 −400 mK accessible in\nsub-Kelvin cryogenic STM systems.\nII. INTRODUCTION\nRecent advances in quantum control of surface spin systems have shown that this platform can be used to design\nquantum-coherent systems by tailoring the interaction of individual spins using the scanning tunneling microscope\n(STM) and atom manipulation.[1–3] In such a system, quantum coherent control of single and multiple spins was\nachieved by electron spin resonance (ESR), which in the STM is facilitated by resonant electric fields.[4–9] When\ncombining the atomic manipulation aspect and quantum coherent control, one can envision that this platform can be\nused to implement a re-configurable quantum simulator in hardware using only a few atoms and an ESR-STM. The\nnext logical step is to certify entanglement in such as system, which is a strong prerequisite to the study of quantum-\ncoherent phenomena beyond single spin quantum gate operations.[10, 11] This, however, is not as straightforward in\nthe ESR-STM since it only allows for time-averaged single spin read-out with long measurement time (ms or kHz),[12]\ncompared to the typical time-scale for coherence time ( T2) of only several hundred nanoseconds.[5, 13] Previous works\n[14] have suggested to use the magnetic susceptibility as entanglement witness, however no experimental realization of\nthis idea has yet been shown. Alternatively, one could exploit the fact that entangled states are no longer eigenstates\nin the Zeeman basis of the constituent spins, and therefore will undergo a time evolution that is distinctively different\nfrom the evolution of any non-entangled state. This approach, also called phase reversal tomography,[15] has been\npreviously shown in phosphorous donor semiconductor qubits.[16] Here, we present a protocol to adapt and optimize\nthis method for ESR-STM, by using the fact that it can be used to probe the free time evolution of spins [3] and has\nhighly sensitive population read out.[7]\nIII. CREATING AND MEASURING ENTANGLEMENT\nIt has been established that ESR-STM provides a universal gate set based on single-spin (or qubit) phase control [17]\nand controlled-NOT gates.[18] In the following, we will discuss how to create entanglement in a surface spin system\nand subsequently measure it. We start from two weakly interacting spins, which can be realized in the experiment\nby using two Ti atoms.[2, 7] We require that the interaction between these spins is sufficiently weak so that their\ncombined eigenstates can be written in good approximation as Zeeman product states, e.g. |↑⟩A⊗ |↑⟩B=|↑↑⟩, where\n↑(↓) denotes the ground (excited state) of each spin and the subscripts A, B label the two spins and ⊗denotes\nthe tensor product. As shown in Fig. 1(a), two spins |↑↑⟩can be entangled by a Hadamard gate H followed by aarXiv:2401.15017v1 [cond-mat.mes-hall] 26 Jan 20242\nnegative controlled-NOT gate (CNOT= |↑⟩⟨↑| ⊗ 1+|↓⟩⟨↓| ⊗ σx), resulting in an entangled state |↑↓⟩+|↓↑⟩. In order\nto detect this entanglement, we can now exploit the fact that the entangled state is not an eigenstate of the Zeeman\nproduct basis and thus undergoes a free evolution.[3] During this evolution the state picks up a phase at a rate that\nis proportional to the energy splitting between |↑↓⟩and|↓↑⟩(Fig. 1(b)). The accumulated phase is distinct from\nthe free evolution of any other non-entangled state and thus allows to uniquely witness the state as being entangled.\nIn particular, maximally correlated states have no accumulated phase. We can measure the phase through a Bell\nstate disentanglement measurement, realized by a CNOT followed by a Hadamard, which projects the phase onto\none of the two spins followed by read-out of that spin. To be more precise the |↑↓⟩+|↓↑⟩is projected upon |↑↓⟩\nwhereas |↑↓⟩ − |↓↑⟩ , which has a phase of π, is projected upon |↑↑⟩. The full protocol is illustrated in Fig. 1(a). By\nrepeating the scheme with increasing delay times between the entanglement and disentanglement sequences we can\nprobe the full phase accumulation during free evolution. For the maximally entangled state it shows up as a slow\n(relative to the Larmor frequencies of the individual spins) variation of ⟨Sz⟩as shown in Fig. 1(c). This variation can\nbe read-out through the sensor-spin, where ⟨Sz⟩ ∝∆IESR, i.e. the change in the tunneling current at spin resonance\nin the ESR-STM experiment.[7] In contrast, the maximally correlated state will result in a flat signal (Fig. 1(d)). To\ndirectly and unambiguously evidence entanglement, one has to ensure that the measurement gives an oscillation of\nspinAwhereas spin Bstays constant.\nIn a practical implementation, probing free evolution might be slightly disadvantageous when the evolution time\nis either very short and approaches typical rise and fall times of the signal generator, or very long and rivals the\ncoherence times T2. Fortunately, the effect of free evolution can also be captured by adjusting the phase ϕon the\nsecond CNOT gate such that ϕ=τ/∆E. In the following, we will use such as phase-sweep instead of a delay-time\nsweep.\nH H\nDelay\nτSB\nSAX X\nΔE\na b\nc dDetection Creation\nFIG. 1: Sequence of quantum logic gates to demonstrate entanglement in ESR-STM. (a) shows the pulse\nscheme using a quantum gate notation. From left to right this scheme applies a Hadamard gate to spin A, a\nnegative CNOT gate, a pulse delay (or phase sweep) gate, followed by an disentangling gate scheme. Finally, both\nstates can be measured to determine their respective populations. (b) the Bloch sphere shows the time-evolution of\nthe entangled state on the equator. (c) expected measurement signal for spin Aand spin Bwhen entangled and in\ncontrast according to the density matrix in the inset (d) the same measurement for two spins that are not entangled.3\nIV. IMPLEMENTATION\nWe will now discuss some details of the implementation. All simulations were carried out using the QuTiP package in\nthe Lindblad formalism using collapse operators parameterized by T1andTϕfor energy relaxation and pure dephasing\nof each spin, respectively (see methods). [19] The total system consists of three spin 1/2 (labeled A,B,Rin Fig. 2\n(a)), which are exchanged coupled to one another sufficiently weakly so that the state diagram can be written in good\napproximations as Zeeman product states (details of the system can be found in the methods section). We emphasize\nthat only spins AandBwill be the target of this entanglement scheme whilst spin Racts as sensor. The Fe atoms\nare added in the experiment to provide the local field gradients for driving ESR of the remote spins ( AandB).[7, 18]\nTo achieve the desired gate sequence for entanglement, we first combine two rotations (labeled as Xπ\n2, Yπ, where\nX, Y denotes the rotation axis and the subscript the rotation angle) to perform a Hadamard gate and then a single-\nfrequency pulse Xπto perform a CNOT (Fig. 2(b)). Note that in general in this system a single driving frequency\nalways performs a conditional operation whilst an unconditional NOT gate requires multi-frequency driving.[18] We\nfound that at low enough temperatures single-frequency driving can be used for all gates due to negligible population\nin the excited states (Fig. 2(b)). This no longer holds true at elevated temperatures, where excited states can have\nnon-negligible populations. In such a case, the Hadamard gate can result in an admixture of entangled states reflecting\nthe excited state population. To avoid this, we also use single frequency driving for the Hadamard gate, which ensures\nthat only the targeted fraction of population will be entangled, at the loss of overall signal amplitude. We have\nconfirmed that this maximizes the readout of the sensor spin (denoted by WRin the following) in our scheme and\ndoes not influence the outcome of the entanglement.\nWe drive all spins on resonance using a control field of the form Ω cos( ωRFt+ϕ) ˆσx, with Ω the Rabi rate, ωRF\nthe angular radio-frequency resonant with a desired transition, ϕan adjustable phase and ˆ σxthe Pauli matrix. It\nwas previously shown that this approach leads to efficient ESR in excellent agreement with the experiment.[7]. For\ndisentanglement we use the same gate sequence but in opposite order whilst matching the initial phase of each\nsubsequent pulse to the phase of the previous pulse. The top 3 plots of Fig. 2 (c) show the expectation values for the\nspin operator ⟨S⟩under these driving fields. We note that the appearance of filled areas is due to crosstalk of the\ndriving frequencies of the pulses and the very fast Larmor precession (10-20 GHz) of each individual spin (see inset),\ndue to the choice that we implemented the simulation in a lab frame of reference. ϕof the second CNOT was chosen\nsuch that it mimics half a free evolution in the entangled state resulting in a spin flip of Aat the end of the scheme\nwhereas spin Bremains unchanged. At the point where the spin should be entangled the expectation value of ⟨Sz⟩for\nAandBare 0, indicating that the spins lie at the equator of their Bloch spheres. To further confirm entanglement we\nalso plot the concurrence C, which is bounded by 0 for non-entangled and 1 for maximally entangled states.[20] For a\nbipartite qubit density matrix ρAB,Cis straightforward to calculate and at the point of entanglement the concurrence\napproaches 1 for the chosen parameter set.\nRead-out of the final target spin states is achieved by a long RF pulse on Rconditioned on the spin state to be read\nout. In this part of the sequence quantum properties like the phase of the pulse play a lesser role, as the coherence of\nRis known to be limited by the conduction electrons.[7] Fig. 2(b) shows the transition that is driven for read-out of\nspinA. In the STM-ESR experiment a long DC voltage pulse could be used to measure the resulting oscillation as a\nchange in the tunneling current ∆ IESR. We set a fast decay time ( T1= 20 ns) for Rmimicking this DC pulse and\nmake sure the pulse is relatively long (100 ns) such that Rquickly reaches the steady state and the signal becomes\nonly dependent on the spin which is read out. Note that we consider this relaxation only during the read-out since for\nthe other parts of the scheme the DC pulse would not be present. The bottom plot of Fig. 2 (c) shows the evolution of\nthe sensor spin during read-out of spin A. Since oscillation of Aas a function of ϕserves as a witness of entanglement\nand in Fig. 2 (c) ϕis such that this oscillation is at its maximum, we refer to the maximum variation of the sensor\nspin as the measurement contrast WR. Due to the nature of the steady state it is at most half the amplitude of ⟨Sz⟩\nofA. We see that in this case Rapproaches this value showing that here where the concurrence is 1 the read-out\nscheme gives a correct output for WR.\nV. RESULTS\nTo demonstrate the concept, we will first discuss results without any relaxation of spins AandBand at a very\nlow temperature of 10 mK. Larmor frequencies, exchange couplings, and Rabi rates must be chosen such that we\nstay in the weakly coupled regime while limiting crosstalk, i.e. unwanted driving of other transitions depending on\nthe realistic resonance line widths in the experiment. In addition, we want the Larmor frequencies to be as high as\npossible to ensure most of the population is in the ground state. Here, we limited the frequency range to 10 −204\nR\nAB17\n20140.2\n0.4\n0.4a\nbc\nΔE\nR\nAB\nFeFe\nFe\nFe\nFIG. 2: Two-spin entanglement scheme using sensor spin read-out (a) Two relatively long-lived spins (A,\nB) are entangled whilst a third, short-lived sensor spin ( R) is used for the read-out. Each pair of titanium and iron\n(Fe) atom serves as a logical qubit in the ESR-STM experiment. (b) energy level diagram showing CNOT (red),\nHadamard (blue) spin control and read-out (purple). (c) actual pulse scheme as implemented in the simulations as\nwell as expectation values along x, y, z for each spin involved. The top panel shows the implemented pulse scheme\nwhere XandYrepresent the rotation axis and the subscript the rotation angle. The next two panels show the\ntime-evolution of spins AandBunder driving, followed by the concurrence which serves as direct measure of\nentanglement in the simulation. The last panel shows the time-evolution of the sensor spin when reading out spin A.\nIdealized parameters were used for clarity: T= 10 mK, Ω = 0.04 GHz, TR\n1= 20 ns, no relaxation for AandB, and\nLarmor frequencies and exchange couplings are in GHz as indicated in panel (a)\nGHz which is routinely achieved for single Ti spins on magnesium oxide (MgO) surfaces ( S= 1/2) in ESR-STM\nsetups.[5, 21, 22] The system parameters are shown in Fig. 2(a), which lists the Larmor frequencies and exchange\ncoupling strengths. In Fig. 3(c-f) we show the results in two ways: first, the variation of each target spin as directly\nobtained from the density matrix, which serves as evidence of the entanglement but is not accessible with the ESR-\nSTM. Second, we show the expected readout signal WR, which is a direct observable of the experiment since for the\nsensor spin ⟨Sz⟩ ∝∆IESR. Contrasting both shows that whilst WRis reduced, clearly the signal on the sensor spin\ndirectly reflects the spin dynamics of the measurement scheme. We note that in this scheme the phase of the second\nHadamard is swept by the equal amount of the free time-evolution, such that ϕ=ωτ, where ωis the angular frequency\nassociated with the entangled state. Whilst the pulse sequence in Fig. 3(a) can be practically implemented, a real\nESR-STM measurement also requires an empty cycle (’B-cycle’), which can be implemented as shown in Fig. 3(b).\nIn this cycle, the background current of the experiment can be measured by simply not entangling the states, which\nis achieved by removing the Hadamard gate during the entanglement step. Finally in Fig. 3 (e) we show that the\nmethod is not limited to the ( ↑↓,↓↑) subspace. Here, we initialise the system in ↓↑such that the Hand CNOT gate\nbring the overall target state to ↑↑+↓↓. Note that here we drive ↓↑to↑↑forH. The major difference in Fig. 3 (e)\nwhen compared to Fig. 3 (c) is that now the oscillation appears in the read-out of |↓A↑B⟩instead of |↑A↓B⟩. This in\nturn allows to identify all the different Bell states in this system.\nWe now turn to the effect of finite lifetime and elevated temperatures relevant to typical ESR-STM experiments.[6,\n7, 18] Previous works have shown that the coherence of Ti spins on two monolayers of MgO deposited on Ag(001)\nsingle crystals seem to be lifetime limited such that T2= 2T1, which allows us to discuss the first results without\nconsidering additional pure dephasing.[7, 17, 18]. As can be seen in Fig. 4(a) the T2time of the two entangled spins\n(taken here to be identical whilst for the sensor spin T1= 20 ns) has a rather modest influence in the experimentally\nrelevant range of T2>300 ns. This is illustrated as well by Fig. 4 (b), which shows a slice at T= 0.1 K. Such5\nHSB\nSAX X(φ)\nHSB\nSAX X(φ)\nH\n|↓↑⟩+|↑↓⟩|↑↑⟩\n|↑↑⟩+|↓↓⟩|↑↓⟩a b\nc d\ne fA- Cycle B- Cycle\nFIG. 3: Simulations of two-qubit entanglement in ESR-STM showing expected measurement outcomes,\nwhere we compare the expectation values ⟨Sz⟩on each spin (inaccessible in the experiment) as well as the indirect\nreadout WRof these values through the sensor spin. From top to bottom (panels a,c,e) we compare the entangled\nsubspace for different initial states ( ↑↑,↑↓). The panels b, d, f on the right side shows simulations for the same\nsystem but for not entangled states (achieved by removing the Hadamard gate), which could serve as empty cycle\nfor the lock-in detection in the ESR-STM experiment. The system parameters are as shown in Fig. 2(a), T= 10\nmK, no relaxation for spins A,B, and T1= 20 ns for the sensor spin during read-out\nlow temperatures are typically achieved by using a dilution refrigerator equipped ESR-STM which can reach base-\ntemperatures close to 20 mK.[23] Clearly, in all cases a T2of around 300 ns allows for efficient entanglement detection.\nTemperature is a more critical parameter as becomes apparent in Fig. 4 (c). Here, we show another slice of Fig. 4\n(a) but now for T2= 300 ns. Above 300 mK the concurrence as well as WRdrastically drop and the concurrences\nreaches 0 at 700 mK. In the intermediate temperature regime of 400 mK, which can be achieved in a3He-cooled\nSTM system, a small degree of entanglement is achievable, with C ≈ 0.2 and WR≈0.1. The strong temperature\ndependence is a consequence of reduced population contrast in our system, which is initialized purely by temperature.\nThis means, in turn, that alternative systems where the initialization is achieved by active pumping, might not be\nas severely limited by temperature. Note that in contrast to Cabove 700 mK WRis still non-zero meaning that\nhere it is no valid witness of entanglement anymore. We further investigate this in Fig. 4 (d) where we plot WR\nagainst CforTsweeps at various T2. Clearly the observation stands that above 700 mK WRis not a valid witness.\nFortunately, for temperatures below 700 mK WRscales with Cmaking a reliable entanglement witness. This relation\ncan best be fitted with a single exponential including an offset, which reflects how at higher temperatures exponentially\nmore population is in unwanted excited states, hereby increasing the effect of crosstalk during the read-out and thus\ndecreasing WRcompared to its ideal value based on the concurrence. The solid lines in Fig. 4 (d) are these fits of the\nformWR= (c+bC)eaC(details of the fitting results can be found in the methods section, Tab.I). Finally, in Fig. 4 (e)\nwe plot WRagainst CforT2sweeps at various temperatures. Again, we see that the dependence can best be fitted\nexponentially, which here reflects that for lower T2there is exponentially more decay of the read-out. Solid lines in\nFig. 4 (e) represent fits of the form WR=bCeaC(details in Tab.II).6\na b c\nd e\nFIG. 4: Influence of finite lifetime and temperature on the entanglement (a) shows WRas function of\ntemperature and decoherence time T2(where T2= 2·T1) (b) slice of (a) showing WRtogether with the concurrence\nCforT= 0.1 K as achievable by dilution refrigerators. (c) slice of (a) showing WRtogether with CforT2= 300 ns.\nClearly, temperature is a critical factor and the concurrence drops drastically above 0.3 K. (d) relation of CandWR\nfor three different T2with an offset exponential fit (e) relation of concurrence and WRfor four different\ntemperatures. Solid lines are exponential fits. The system parameters are as shown in Fig. 2(a)\nFinally, we address systems where coherence is not lifetime limited. In such systems, the coherence of the system\nis reduced by additional pure dephasing processes, such that an effective coherence time can be defined as 1 /T∗\n2=\n1/T2+ 1/Tϕ, with Tϕbeing the time constant of the pure dephasing process. In the following, T2= 2T1= 300 ns as\ntypical for the experiments [18]. As show in Fig. 5 (a) and (b) even fast dephasing processes with a dephasing time\naround T∗\n2= 75 ns still allow for sufficient concurrence and WR. It is not surprising that longer T∗\n2times are desirable\nas this is generally the case in quantum coherent systems, but it is encouraging that in the typical experimental range\nofT∗\n2≈300 ns [7, 17, 18] concurrence and WRare still relatively high.\nVI. CONCLUSION\nIn this work we have shown by open quantum systems simulations that two exchange coupled relatively long-lived\nspins can be entangled and that the entanglement can be directly measured using a third, weakly coupled sensor\nspin. Our simulations indicate that temperature is critical to achieve high entanglement and WR, due to the fact\nthat the populations are initialized into thermal equilibrium. Systems that can be initialized more independently\nfrom temperature as usually done in optical qubits in trapped ion systems for example, could overcome the strict\ntemperature requirement. For physical spins on surface systems available today, such as the widely studied Ti on\nMgO/Ag(001), entanglement should be achievable and measurable with T2= 300 ns for the quantum spins and\nT1≈20 ns for the sensor spin. High degrees of entanglement C>0.8 and corresponding read-out can be reached\nwhen using a dilution refrigerator at T <100 mK.7\na b\nFIG. 5: Influence of pure dephasing on the entanglement (a) shows the achievable concurrence and WRas\nfunction of pure dephasing time Tϕ. (b) shows CandWRas function of T2for a fixed T2= 300 ns for the two spins\nandT= 0.1 K\nVII. METHODS\nAll calculations were performed by time-evolution for an appropriate amount of time for the entire pulse\nscheme and the read-out using a converged time step smaller than 8 ps. Following previous works,[7] we mod-\neled each spin as an on-site energy term 2 πfL,iSz,iwith fL,ithei-th Larmor frequency, and pairwise isotropic\nexchange coupling terms Ji,j⃗Si⃗Sj. ESR driving is achieved by applying the necessary single frequency driving terms\nΩkcos(ωk(t−tstart\nk) +ϕk)σx,i(tstart\nk < t < tend\nk), with ωkthe frequency the pulse is send at matching the desired\nenergy transition, tstart\nk andtend\nkthe start and end times of the pulse and ϕkis an adjustable phase. Ω kis the\non-resonance Rabi rate, k= 1. . . N the index of driving frequency terms. The maximum number of driving terms in\nour simulation was N≤7. The total system Hamiltonian can be written as follows:\nSystem Hamiltonian:\nHtot=3X\ni=12πfL,iSz,i+3X\ni=13X\nj>iJi,j⃗Si⃗Sj+X\nk3X\ni=1Ωkcos (ωk(t−tstart\nk) +ϕk)σx,i(tstart\nk < t < tend\nk) (1)\nLindblad equation : We solved a Lindblad equation for the reduced density matrix ρof the following form\ndρ\ndt=−i\n¯h[Htot, ρ] +X\nl\u0012\nLlρL†\nl−1\n2L†\nlLlρ−1\n2ρL†\nlLl\u0013\n(2)\nThe last term on the right hand side are the collapse operators for our system. We used two sets of collapse\noperators LKondo+Lϕto model spin energy relaxation as well as pure dephasing.\nCollapse operators: The first set of collapse operators was defined acting on the coupled 3 spin system in order\nto model Kondo spin relaxation, known to be the main source of decoherence for these system.[5, 24, 25]. We arrive\nat these terms by writing the known rate equation (see for example Eq. 4 of supplementary of [26]) in Lindblad form.\nThe operator acting between energy level mandnof the system is\nLKondo\nm,n =sX\nlJlX\nsi,sf⟨m, s i|⃗ s⊗⃗Sl|n, sf⟩ϵmn\n1−eϵmn/kBT|m⟩⟨n|. (3)8\nT2(ns) a b c\n300 0.68±0.07 0.15±0.014 0.029 ±0.027\n600 0.58±0.05 0.19±0.011 0.040 ±0.025\n900 0.56±0.04 0.21±0.011 0.047 ±0.024\nTABLE I: Fitting results of WR= (c+bC)eaCfor\nthe data shown in Fig. 4 (d). Uncertainties\nrepresent the 2 σconfidence intervalT(K) a b\n0.1 3.0±0.06 (2.7±0.14)×10−2\n0.2 2.5±0.07 (5.4±0.32)×10−2\n0.3 2.2±0.04 (9.9±0.27)×10−2\n0.4 1.9±0.36 (1.8±2.5)×10��1\nTABLE II: Fitting results of WR=bCeaCfor the\ndata shown in Fig. 4 (e). Uncertainties represent\nthe 2 σconfidence interval\nHere the first sum is over the ldifferent atomic spins and the second sum is over the initial ( si) and final ( sf)\nstate of the itinerant electron spin interacting with these spins. ⃗Sand⃗ sare the respective spin operators. ϵmnis the\nenergy difference between mandnof the three spin system. Finally, Jlis the strength of the interaction with each\natomic spin. In low temperature approximation it relates to the isolated l-th spin relaxation time T1,land energy of\nits Larmor frequency ϵlas (see Eq. 69 of [25])\nJl=s\n1\nϵlT1,l. (4)\nThe second set of operators is for pure dephasing. Here, the standard operators are used relating the pure dephasing\nrate to the pure dephasing time Tϕ,lvia the Pauli-z matrix for the l-th spin, i.e. σz,l=1⊗11. . .1⊗lσz⊗l+11. . .\nLϕ\nl=s\n1\n2Tϕ,lσz,l. (5)\nRead out: For read-out long pulses were sent resonant with transitions of SR. The expectation value of SRwas\naveraged in 16000 time steps for a time of 100 ns. In order to have a converged expectation value a Rabi strength\nwas used double the other strengths used in the scheme.\nFitting results: The relation between WRandCin Fig. 4 (d) were best fit using an exponential function of the\nformWR= (c+bC)eaC. The fitting results are reported in Tab. I. The relation between WRandCin Fig. 4 (e) were\nbest fit using afunction of the form WR=bCeaC. The fitting results are reported in Tab. II.\nConcurrence: For concurrence calculation first the partial trace over SRwas taken leaving the reduced matrix in\nthe target spin basis. Then for each entanglement scheme the maximum was reported.\nCode availability: The underlying code for this study is available and can be accessed via this link 10.5281/zen-\nodo.10528113.\nAUTHOR CONTRIBUTIONS\nRB, SP, and CW conceived the paper, RB and CL performed numerical simulations, all authors contributed to the\ndiscussion and the writing of the manuscript.\nACKNOWLEDGMENTS\nThe authors thank N. Lorente for his insight related to entanglement in surface spin systems. Further we thank G.\nGiedke, F. Donati and H. Stemp for discussions.\nThis work was supported by the Institute for Basic Science (IBS-R027-D1). R. B. and S. O. acknowledge sup-\nport from the Netherlands Organisation for Scientific Research (NWO Vici Grant VI.C.182.016).\nCOMPETING INTERESTS\nAll authors declare no financial or non-financial competing interests.9\nDATA AVAILABILITY\nAll data generated or analysed during this study are included in 10.5281/zenodo.10530510\nREFERENCES\n∗wolf.christoph@qns.science\n[1] Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal surface. Science\n262, 218–220 (1993).\n[2] Yang, K. et al. Engineering the Eigenstates of Coupled Spin- 1/2 Atoms on a Surface. Physical Review Letters 119, 1–8\n(2017).\n[3] Veldman, L. M. et al. Free coherent evolution of a coupled atomic spin system initialized by electron scattering. Science\n372, 964–968 (2021).\n[4] Baumann, S. et al. Electron paramagnetic resonance of individual atoms on a surface. Science 350, 417–420 (2015).\n[5] Yang, K. et al. Coherent spin manipulation of individual atoms on a surface. Science 366, 509–512 (2019).\n[6] Phark, S. h. et al. Electric-Field-Driven Spin Resonance by On-Surface Exchange Coupling to a Single-Atom Magnet.\nAdvanced Science 10, 1–8 (2023).\n[7] Phark, S. H. et al. Double-Resonance Spectroscopy of Coupled Electron Spins on a Surface. ACS Nano (2023).\n[8] Reina-G´ alvez, J., Lorente, N., Delgado, F. & Arrachea, L. All-electric electron spin resonance studied by means of Floquet\nquantum master equations. Physical Review B 104, 1–16 (2021).\n[9] Reina-G´ alvez, J., Wolf, C. & Lorente, N. Many-body nonequilibrium effects in all-electric electron spin resonance. Physical\nReview B 107, 235404 (2023). URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.235404 .\n[10] Jozsa, R. & Linden, N. On the role of entanglement in quantum-computational speed-up. Proceedings of the Royal\nSociety of London. Series A: Mathematical, Physical and Engineering Sciences 459, 2011–2032 (2003). URL https:\n//royalsocietypublishing.org/doi/abs/10.1098/rspa.2002.1097 .\n[11] Rieffel, E. G. & Polak, W. H. Quantum Computing: A Gentle Introduction . Scientific and Engineering Computation (MIT\nPress, 2011). URL https://books.google.co.kr/books?id=iYX6AQAAQBAJ .\n[12] Bastiaans, K. M. et al. Amplifier for scanning tunneling microscopy at MHz frequencies. Review of Scientific Instruments\n89, 93709 (2018). URL https://doi.org/10.1063/1.5043267 .\n[13] Willke, P. et al. Coherent Spin Control of Single Molecules on a Surface. ACS Nano 15, 17959–17965 (2021).\n[14] Del Castillo, Y. & Fern´ andez-Rossier, J. Certifying entanglement of spins on surfaces using ESR-STM. Physical Review B\n108, 1–5 (2023).\n[15] Wei, K. X. et al. Verifying multipartite entangled Greenberger-Horne-Zeilinger states via multiple quantum coherences.\nPhysical Review A 101, 1–10 (2020).\n[16] Stemp, H. G. et al. Tomography of entangling two-qubit logic operations in exchange-coupled donor electron spin qubits\n1–9 (2023). URL http://arxiv.org/abs/2309.15463 .\n[17] Wang, Y. et al. Universal quantum control of an atomic spin qubit on a surface. npj Quantum Information 9, 3–8 (2023).\n[18] Wang, Y. et al. An atomic-scale multi-qubit platform. Science 382, 87–92 (2023).\n[19] Johansson, J. R., Nation, P. D. & Nori, F. QuTiP: An open-source Python framework for the dynamics of open quantum\nsystems. Computer Physics Communications 183, 1760–1772 (2012). URL http://dx.doi.org/10.1016/j.cpc.2012.02.\n021.\n[20] Hill, S. & Wootters, W. K. Entanglement of a Pair of Quantum Bits. Physical Review Letters 78, 5022–5025 (1997).\n[21] Reale, S. et al. Electrically Driven Spin Resonance of 4f Electrons in a Single Atom on a Surface (2023). URL http:\n//arxiv.org/abs/2309.02348 .\n[22] Seifert, T. S. et al. Single-atom electron paramagnetic resonance in a scanning tunneling microscope driven by a radio-\nfrequency antenna at 4 K. Physical Review Research 2(2020).\n[23] Kim, J. et al. Spin resonance amplitude and frequency of a single atom on a surface in a vector magnetic field. Physical\nReview B 104(2021).\n[24] Ternes, M. Spin excitations and correlations in scanning tunneling spectroscopy. New Journal of Physics 17, 63016 (2015).\nURL http://dx.doi.org/10.1088/1367-2630/17/6/063016 .\n[25] Delgado, F. & Fern´ andez-Rossier, J. Spin decoherence of magnetic atoms on surfaces. Progress in Surface Science 92,\n40–82 (2017). URL http://dx.doi.org/10.1016/j.progsurf.2016.12.001 .\n[26] Loth, S., Lutz, C. P. & Heinrich, A. J. Spin-polarized spin excitation spectroscopy. New Jour-\nnal of Physics 12, 125021 (2010). URL http://stacks.iop.org/1367-2630/12/i=12/a=125021?key=crossref.\n22720a4de1b70f37508646ce6db51c68 ." }, { "title": "2401.15021v1.Volterra_equations_with_affine_drift__looking_for_stationarity.pdf", "content": "Volterra equations with affine drift: looking for\nstationarity\nGilles Pag `es∗†\nJanuary 29, 2024\nAbstract\nWe investigate the properties of the solutions of scaled Volterra equations (i.e. with an\naffine mean-reverting drift) in terms of stationarity at both a finite horizon and on the long\nrun. In particular we prove that such an equation never has a stationary regime, except if\nthe kernel is constant (i.e. the equation is a standard Brownian diffusion) or in some fully\ndegenerate pathological settings. We introduce a deterministic stabilizer ςassociated to the\nkernel which produces a fake stationary regime in the sense that all the marginals share the\nsame expectation and variance. We also show that the marginals of such a process starting from\nwhen starting various initial values are confluent in L2as time goes to infinity. We establish\nthat for some classes of diffusion coefficients (square root of positive quadratic polynomials)\nthe time shifted solutions of such Volterra equations weakly functionally converges toward a\nfamily of L2-stationary processes sharing the same covariance function. We apply these results\nto (stabilized) rough volatility models (when the kernel K(t) =tH−1\n2, 0< H <1\n2) which leads\nto produce a fake stationary quadratic rough Heston model.\nIntroduction\nWe investigate in this paper conditions under which stochastic Volterra equations with affine\ndrifts and convolutive kernels of the form\nXt=X0+Zt\n0K(t−s)(µ(s)−λXs)ds+Zt\n0K(t−s)σ(Xs)dWs\nhave a stationarity regime in the classical sense, i.e. the distribution of the process is invariant by\ntime shift (see [15]) or in a weaker sense – that we called fake stationary regime (of type I) – in\nwhich the solution only has constant mean and variance at every instant t. In the Gaussian case\n(typically pseudo-Ornstein-Uhlenbeck process), it implies that the process has the same 1-marginal\ndistribution at very time t(which could be called a fake stationary regime (of type II )). However\nsuch a property if it holds does not imply stationarity nor classical weak L2-stationarity (based on\ncovariance).\n∗Laboratoire de Probabilit´ es, Statistique et Mod´ elisation, UMR 8001, Sorbonne Universit´ e, case 158, 4, pl. Jussieu,\nF-75252 Paris Cedex 5, France. E-mail: gilles.pages@sorbonne-universite.fr\n†This research benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque.\n1arXiv:2401.15021v1 [math.PR] 26 Jan 2024In terms of application this may have consequences on rough volatility models (see [2, 8, 11, 12])\nwhich met a great success in the financial community during these last ten years, in particular\nbecause it provides a consistent modeling of the dynamics of financial markets (mostly equity) at\nvarious scales from the order book to the pricing of derivative products in spite of its non-Markovian\nfeature. In the original version of these rough stochastic volatility models, mostly derived from the\nregular Heston model, the mean-reverting CIR dynamics of the volatility is replaced by a Volterra\nversion of this CIR process in which the convolution kernel is singular and of fractional form\nK(t) =tH−1\n2for some H∈(0,1/2) (Hurst coefficients). In the late 1990’s, attempts to model\nthe long memory effect on continuous time stochastic volatility models (see [6, 7]) also relied on\nsimilar stochastic Volterra equations using the same family of kernels but with α∈(1/2,1). In\nboth cases the aim as to mimic the behaviour of a stochastic differential equations (SDE) driven\nby a fractional Brownian motion without having to face the technicalities of such equations which\nrely on rough path theory and which can be extreme when Hgets close to 0.\nIf we come back for a while to the standard Heston model equipped with its CIR driven stochas-\ntic volatility process, it has been observed for long by practitioners that this model was difficult\nto calibrate (on vanilla products) to produce good prices for derivatives with short maturities,\nespecially path-dependent ones. This is most likely due to the fact that the CIR process is usu-\nally initialized by a deterministic value at the origin (often the long run mean). So this volatility\ndynamics actually has two regimes, first, one for short maturities when the starting value is promi-\nnent and the variance is very small, then it deploys until attaining its its stationary regime – a\nγ-distribution –. A solution to get rid of this drawback is to consider directly a stationary Heston\nin which the volatility is taken under its stationary regime (see e.g. [21]) which also has the ad-\nvantage to reduce by one the number of parameters. A quantization based approach is provided\nfor calibration and pricing of path-dependent derivatives (at least under the Feller condition that\nensures positivity of the volatility) in a recent paper [18].\nIf, as it is likely, a similar problem occurs for rough volatility models based on the rough Volterra\nversion of the CIR model, the search for some stationarity properties of such a rough CIR model\ncan be of interest to make possible a calibration consistent for both short and long maturities. This\ndoes not provide a plug-and-play solution to this problem but explores in an elementary way what\nare or could be the connexions between stationarity and Volterra equations with “mean-reverting”\naffine drift with a mitigate conclusion. This could be compared to the recent paper [15] about\nergodicity of Volterra processes.\nThe paper is organized as follows. In Section 1 we recall the main properties of stochastic\nVolterra equations with convolutive drifts: existence, moment control, etc, with a focus on those\nof these processes having an affine drift. In that case some specific tools are available like the\nResolvent and the solution Wiener-Hopf equation which corresponds when K(t) =eλt(Markovian\nframework) to apply Itˆ o’s lemma to eλtXt. In Section 2 we elucidate when a Volterra equation\nwith affine has an invariant distribution (distribution invariant by time shift). We show that this\nsituation is essentially degenerate in the sense that it implies that the kernel Kis constant i.e. that\nthe Volterra equation is a standard Brownian equation. In Section 3 we define some notions of\nfakestationarity like having constant mean and variance (type I), or constant 1-marginal distribu-\ntion (type II). To this end we are led to introduce a multiplicative deterministic function ςin the\nBrownian convolution, called stabilizer to “control” the process (through its volatility coefficient).\nThis stabilizer is solution to an intrinsic convolution equation related to the kernel of the equation.\n2In Section 4 we provide examples of such“fake stationary” stabilized processes with some explicit\ncomputations when\nσ(x) =q\nκ0+κ1(x−µ0\nλ) +κ2(x−µ0\nλ)2with κi≥0, i= 1,2, κ2\n1≤4κ0κ2.\nThis form of the diffusion function appears e.g. in [12] when modeling the volatility in the quadratic\nrough Heston model. We also investigate for such stabilized processes the functional weak asymp-\ntotics of the time shifted process ( Xt+s)s≥0ast→+∞which turns out to be a weak L2-stationary\nprocess. Finally in Section 5 we apply our results to the case case of fractional kernels Kα(t) =tα−1,\nα∈(0,1) covering “rough” models (where α=1\n2+h,H∈(0,1/2), the Hurst coefficient). In par-\nticular we prove the existence of the stabilizer with a semi-closed form as (fractional) power series\nwhose coefficients can be computed by induction. We prove in an appendix that this series has an\ninfinite convergence radius.\nNotations. •Leb1denotes the Lebesgue measure on RorR+.\n•Forf, g∈ L1\nR+,loc(R+,Leb1), we define their convolution by f∗g(t) =Rt\n0f(t−s)g(s)ds,t≥0.\n•Forf, g∈ L2\nR+,loc(R+,Leb) and Wa Brownian motion, we define their stochastic convolution by\nfW∗g=Zt\n0f(t−s)g(s)dWs, t≥0.\n•[X] stands for the distribution of the random variable/vector/process X.\n•X⊥ ⊥Ywill stand for the independence of the two random objects XandY.\n1 Background on stochastic Volterra equations with convolutive\nkernels\n1.1 Volterra processes with convolutive kernels\nWe are interested in the stochastic Volterra equation\nXt=X0+Zt\n0K(t−s)b(s, X s)ds+Zt\n0K(t−s)σ(s, X s)dWs, t≥0, (1.1)\nwhere b: [0, T]×R→R,σ: [0, T]×R→Rare Borel measurable, K∈ L2\nloc,R+(Leb 1) is a convolutive\nkernel and ( Wt)t≥0is a standard Brownian motion independent from the R-valued random variable\nX0both defined on a probability space (Ω ,A,P). Let ( Ft)t≥0be a filtration (satisfying the usual\nconditions) such that X0isF0-measurable and Wis an ( Ft)-Brownian motion independent of X0.\nAssume that the kernel Ksatisfies\n∃β >1 such that K∈L2β\nloc(Leb 1) (1.2)\nand that, for every T >0,\n(Kcont\nT,θ)∃κT<+∞,∃θT>0,∀δ∈(0, T),sup\nt∈[0,T]\u0014Zt\n0|K(\u0000\ns+δ)∧T\u0001\n−K(s))|2ds\u00151\n2\n≤κTδθT.\n(1.3)\n3Assume bandσsatisfy the following Lipschitz-linear growth assumption uniform in time\n(i)∀t∈[0, T],∀x, y∈R,|b(tx)−b(t, y) +|σ(t, x)−σ(t, y)| ≤Cb,σ,T|x−y|,\n(ii) sup\nt∈[0,T]|b(t,0)|+|σ(t,0)|<+∞.\nIt has been established in [17] that, if X0∈Lp(P) for some p >0, then Equation (1.1) admits a\nunique pathwise continuous solution on R+starting from X0satisfying (among other properties),\n∀T >0,∃Cp,T>0,\r\r\rsup\nt∈[0,T]|Xt|\r\r\r\np≤Cp,T(1 +∥X0∥p). (1.4)\nThis result appears as a generalization of a strong existence-uniqueness result of pathwise continuous\nsolutions is established in [25] provided the starting value X0has finite polynomial moments at any\norder (The framework is more general).\n1.2 Laplace Transform\nLet us first introduce the Laplace transform of a Borel function f:R+→R+by\n∀t≥0, L f(t) =Z+∞\n0e−tuf(u)du∈[0,∞].\nThis Laplace transform is non-increasing and if Lf(t0)<+∞for some t0≥0, then Lf(t)→0 as\nt→+∞.\nOne can define the Laplace transform of a Borel function f:R+→Ron (0 ,+∞) as soon as\nL|f|(t)<+∞for every t >0 by the above formula. The Laplace transform can be extended to R+\nas anR-valued function if f∈ L1\nR+(Leb 1).\n1.3 Resolvent of a convolutive kernel\nFor every λ∈R, the resolvent Rλassociated to Kandλis defined as the unique solution – if\nit exists – to the deterministic Volterra equation\n∀t≥0, R λ(t) +λZt\n0K(t−s)Rλ(s)ds= 1 (1.5)\nor, equivalently, written in terms of convolution,\nRλ+λK∗Rλ= 1. (1.6)\nThe solution always satisfies Rλ(0) = 1 and formally reads\nRλ=X\nk≥0(−1)kλk(1∗K∗k) (1.7)\nwhere, by convention in this formula, K∗0=δ0(Dirac mass at 0).\nFrom now on we will assume that the kernel Ksatisfies\n(LK)∀t >0, L K(t)<+∞. (1.8)\n4•Ifλ >0, (1.7) defines an absolutely convergent series for every t∈(tK,+∞) where tK= inf{t:\nLK(t)≤1/λ}<+∞.\n•If the (non-negative) kernel Ksatisfies\n0≤K(t)≤Cebtta−1for some a, b, C > 0∈R+ (1.9)\nthen one easily checks by induction that\n1∗K∗n(t)≤CnebtΓ(a)n\nΓ(an+ 1)tan, (1.10)\nwhere Γ( a) =R+∞\n0e−uua−1du. Then, one shows using Stirling formula that, for such kernels, the\nabove series (1.7) is absolutely converging for every t >0 so that the function Rλis well-defined\non (0 ,+∞).\n•IfKis regular enough (say continuous) the resolvent Rλis differentiable and one checks that\nfλ=−R′\nλsatisfies, for every t >0,\n−fλ(t) +λ\u0000\nRλ(0)K(t)−K∗fλ(t)\u0001\n= 0\nthat is fλis solution to the equation\nfλ+λK∗fλ=λK. (1.11)\nIn particular, if Rλturns out to be non-increasing, then fλis non-negative and satisfies 0 ≤fλ≤\nλK. In that case one also has thatR+∞\n0fλ(t)dt= 1−Rλ(+∞), so that fλis a probability density\nif and only if lim\nt→+∞Rλ(t) = 0.\nExamples of kernels .\n1.Trivial kernel (Markov) . Let K(t) =1R+(t). It obviously satisfies (1.2), (1.3) and (1.9).\nThen Rλ(t) =e−λt.\n2.Exponential kernel . Let K(t) = e−u. It clearly satisfies (1.2), (1.3) and (1.9). Then\nRλ(t) =(\nt+ 1 if λ=−1\n1+λe−(λ+1)t\nλ+1ifλ̸=−1.\n3.Fractional integration kernel . Let\nK(t) =Kα(t) =uα−1\nΓ(α)1R+(t), α > 0. (1.12)\nThese kernels satisfy (1.2) and (1.3) for α >1/2 (with θT= (α−1\n2)∧1, see [24]) among many\nothers and trivially (1.9). This family of kernels corresponds to the fractional integrations\nof order α >0.\nIt follows from the easy identity Kα∗Kα′=Kα+α′that\nRα,λ(t) =X\nk≥0(−1)kλktαk\nΓ(αk+ 1)=Eα(−λtα)t≥0, (1.13)\n5where Eαdenotes the standard Mittag-Leffler function\nEα(t) =X\nk≥0tk\nΓ(αk+ 1), t∈R.\nOne shows (see [13] and Section 5 further on) that Eαis increasing and differentiable on the\nreal line with lim\nt→+∞Eα(t) = +∞andEα(0) = 1. In particular, Eαis an homeomorphism\nfrom ( −∞,0] to (0 ,1]. Hence, if λ >0, the function fα,λdefined on (0 ,+∞) by\nfα,λ(t) =−R′\nα,λ(t) =αλtα−1E′\nα(−λtα) =λtα−1X\nk≥0(−1)kλk tαk\nΓ(α(k+ 1))(1.14)\nis a probability density – called Mittag-Leffler density – since fα,λ>0 andZ+∞\n0fα,λ(t)dt=\nRα,λ(0)−Rα,λ(+∞) = 1. Note that when α= 1 (i.e. K=K1=1),E1(t) =et,R1,λ(t) =\ne−λtandf1,λ(t) =λe−λt.\n4.Exponential-Fractional integration kernel . Let K(t) =Kα,ρ(t) =e−ρtuα−1\nΓ(α)1R+(t) with α, ρ >\n0>0. One checks that these kernels also satisfy (1.2) and (1.3) for α > 1/2 (with θT=\n(α−1\n2)∧1) and trivially (1.9). Moreover Kα,ρ∗Kα′,ρ=Kα+α′,ρso that the resolvent reads\nRα,ρ,λ(t) =e−ρtRα,0,λ(t).\nThis last example is important for applications, especially in Finance for the wide class of rough\nstochastic volatility models (see [16, 11, 8, 12]) since in that case α=H+1\n2where H∈(0,1) is\nkind of Hurst coefficient representative or the roughness of the model ( H <1\n2) or its long memory\n(1\n2< H < 1).\n1.4 Application to the Wiener-Hopf equation\nWe come now to the main result of these preliminaries.\nProposition 1.1 (Wiener-Hopf equation) Letg:R+→Rbe a locally bounded Borel func-\ntion, let K∈L1\nloc(LebR+)and let λ∈R. Assume that Rλis differentiable on (0,+∞)with a\nderivative R′\nλ∈L1\nloc(LebR+), that both RλandR′\nλadmit a finite Laplace transform on R+and\nlim\nu→+∞e−tuRλ(u) = 0 for every t >0. Then, the Wiener-Hopf equation\n∀t≥0, x(t) =g(t)−λZt\n0K(t−s)x(s)ds (1.15)\n(also reading x=g−λK∗x) has a solution given by\n∀t≥0, x(t) =g(t) +Zt\n0R′\nλ(t−s)g(s)ds (1.16)\nor, equivalently,\nx=g−fλ∗g,\nwhere fλ=−R′\nλ. This solution is uniquely defined on R+up to dt-a.e.equality.\n6We provide a proof of this classical result for the reader’s convenience (see e.g. [12]).\nProof. First note that, by an integration by parts, that for every t >0,\ntLRλ(t) = 1 + LR′\nλ(t).\nOn the other hand it follows from (1.5) that tLRλ(t)(1 + λLK(t)) = 1, t >0. Consequently,\nLx(t) =Lg(t)\n1 +λLK(t)=tLgLRλ(t) =Lg(t)\u0000\n1 +LR′\nλ(t)\u0001\n=Lg+R′\nλ∗g(t)\nwhich completes the proof since Laplace transform is injective. 2\n2 Genuine and fake stationarity of a scaled Volterra equation\nFrom now we focus on the special case of a scaled Volterra equation associated to a convolutive\nkernel K:R+→R+satisfying ( K), (1.2), (1.3) and (1.9):\nXt=X0+Zt\n0K(t−s)(µ(s)−λXs)ds+Zt\n0K(t−s)σ(s, X s)dWs, X 0⊥ ⊥W, (2.17)\nwhere λ >0,µ:R+→Ris a bounded Borel function (hence having a well-defined finite Laplace\ntransform on (0 ,+∞)) and σ:R+×R→Ris Lipschitz continuous in x, locally uniformly in\nt∈[0, T], for every T >0. Note that the drift b(t, x) =µ(t)−λxis clearly Lipschitz continuous in\nx, uniformly in t∈R.\nThen, Equation (2.17) has a unique ( FX0,W\nt )-adapted solution starting from X0∈L0(P) (Ap-\nply [17, Theorem 1.1] on every time interval [0 , T],T∈Nand paste the solutions taking advantage\nof the uniform in time linear growth of the drift and of σ).\nHave in mind that under our assumptions, if p >0 and E|X0|p<+∞, then Esupt∈[0,T]|Xt|p<\nCT(1+∥X0∥p)+∞for every T >0 (see [17, Theorem 1.1]). Combined with the fact that |σ(t, x)| ≤\nC′\nT(1 +|x|),t∈[0, T], this implies that supt∈[0,T]∥σ(t, Xt)∥p< C(1 +∥X0∥p)<+∞for every\nT > 0. This allows us to use in what follows without restriction both regular and stochastic\nFubini’s theorems.\nThroughout what follows we assume that the λ-resolvent Rλof the kernel Ksatisfies for every\nλ >0\n(K)\u001a(i)Rλdecreases to 0 as t→+∞andfλ=−R′\nλ>0 on (0 ,+∞).\n(ii)fλ∈ L2\nloc(Leb 1)(2.18)\nIn particular, under this assumption, the function fλis a probability density.\nThis assumption is satisfied by our examples 1-3-4 of kernels.\nWe may read the above equation (2.17) “pathwise” as a Wiener-Hopf equation with x(t) =Xt(ω)\nand\ng(t) =X0(ω) + (µ∗K)t+\u0000\nKW∗σ(., X.(ω))\u0001\nt.\n7Then\nXt=X0+ (µ∗K)t+\u0000\nKW∗σ(., X.)\u0001\nt+Zt\n0R′\nλ(t−s)h\nX0+ (µ∗K)s+\u0000\nKW∗σ(., X.)\u0001\nsi\nds\n=X0+ (µ∗K)t+\u0000\nKW∗σ(., X.)\u0001\nt+X0Zt\n0R′\nλ(t−s)ds\n+Zt\n0R′\nλ(t−s)(µ∗K)sds\n| {z }\n=:(a)+Zt\n0R′\nλ(t−s)\u0000\nKW∗σ(., X.)\u0001\nsds\n| {z }\n=:(b).\nBy commutativity and associativity (which relies on regular Fubini’s theorem) of regular con-\nvolution, one has\n(a) =−fλ∗(µ∗K)t=−\u0000\n(fλ∗K)∗µ\u0001\nt. (2.19)\nwhere fλ=−R′\nλ. On the other hand by differentiating Equation (1.5) we know that that fλ\nsatisfies Equation (1.11)\n−fλ∗K=1\nλfλ−K.\nConsequently, plugging this identity in (2.19) yields\n(a) =1\nλ(fλ∗µ)t−(K∗µ)t. (2.20)\nUsing stochastic Fubini’s theorem and, again, Equation (1.11) recalled above satisfied by fλ, we\nderive that\n(b) =1\nλ\u0000\nfλW∗σ(·, X·)\u0001\nt−\u0000\nKW∗σ(·, X·)\u0001\nt. (2.21)\nPlugging (2.20) and (2.21) into (2.17) finally yields\nXt=X0Rλ(t) +1\nλ(fλ∗µ)t+1\nλ(fλW∗σ(·, X·)t\ni.e.\nXt=X0Rλ(t) +1\nλZt\n0fλ(t−s)µ(s)ds+1\nλZt\n0fλ(t−s)σ(s, X s)dWs. (2.22)\nRemark. When K(t) =1, the above computations correspond to the application of Itˆ o ’s Lemma\ntoeλtXt.\n2.1 Stationarity of the mean\nBefore tackling the problem of the stationary regime of the “scaled” Volterra equation (2.17),\na first step can be to determine when this equation has a constant mean if X0∈L2(P) (1) i.e.\nEXt=EX0for every t≥0. As\nE\u0012Zt\n0fλ(t−s)σ(s, X s)dWs\u00132\n=Zt\n0f2\nλ(t−s)σ2(s, X s)ds≤C(1 +∥X0∥2\n2)Zt\n0f2\nλ(u)du < +∞,\n1.X0∈Lp(P) for some p >1 would be sufficient by standard argument based on BDG Inequality in this section.\n8we have\nEZt\n0fλ(t−s)σ(s, X s)dWs= 0\nso that\nEXt=Rλ(t)EX0+1\nλ(fλ∗µ)t.\nUsing that Rλ(t) = 1−Rt\n0fλ(t−s)ds, we derive that, for every t≥0,\nEXt=EX0+Zt\n0fλ(t−s)\u0010µ(s)\nλ−EX0\u0011\nds.\nHence EXtis constant if and only if\n∀t≥0,Zt\n0fλ(t−s)\u0010µ(s)\nλ−EX0\u0011\nds= 0.\nThis also reads in terms of Laplace transform\nLfλ·Lµ(·)\nλ−EX0= 0.\nAsLfλ(t)>0 on (0 ,+∞) owing to Assumption ( K)(ii) (see (2.18)), this implies thatµ(t)\nλ−EX0= 0\ndt-a.e.i.e.\nµ(t) =µ0dt-a.e. and EX0=µ0\nλ.\n2.2 Stationarity of the variance, fake stationary regime\nIf we assume that X0∈L2(P), under our assumptions, Xis well-defined on the whole positive\nreal line and Xt∈L2(P) for every t≥0. In fact (see (1.4)), ∥supt∈[0,T]|Xt|∥2< C T(1 +∥X0∥2)<\n+∞for every T >0 where CTis a (non-exploding) positive real constant. Then the non-negative\nfunction defined by\nt7−→¯σ2(t) :=Eσ2(t, Xt), t≥0. (2.23)\nis locally bounded on R+since σhas at most linear growth in space, locally uniformly in t≥0.\n2.2.1 Autonomous volatility coefficient σ\nWe assume in this section the case of an autonomous volatility coefficient defined by\n∀(t, x)∈R+×R, σ (t, x) =σ(x).\nNote that if the solution ( Xt)t≥0is a stationary (2) and X0∈L2(P) then the mean of Xand the\nvariance of Xtare constant as functions of tas well as the expectations of any function of Xtwith\nat most quadratic growth. Typically such is the case of x7→x,x7→x2andσ2. As a consequence,\n(see (1.4))\n∀t≥0,EXt=µ0\nλ,Var(Xt) =v0≥0 and ¯ σ2:=Eσ2(Xt)≥0.\nThe theorem below shows what are the consequences of these three constraints\n2. in the sense that the shifted processes ( Xt+u)u≥0and ( Xu)u≥0have the same distribution as processes on the\ncanonical space C(R+,R).\n9Theorem 2.1 (Autonomous σand µ=µ0is constant) Setµ(t) = µ0andσ(t, x) = σ(x)\nin(2.17) . Assume X0∈L2(P)withEX0=µ0\nλ. If\n∀t≥0,Var(Xt) =v0≥0and ¯σ2(t) = ¯σ2≥0,\nonly two situations can occur:\n(i)If¯σ >0the kernel Kis constant so that (2.17) is a standard Brownian SDE and (Xt)t≥0is a\n(Markov) Brownian diffusion process.\n(ii)If¯σ= 0, then σ\u0000µ0\nλ\u0001\n= 0andXt=µ0\nλP-a.s.\nProof. First note that as µ(t) =µ0and = R′\nλ−fλ, Equation (2.22) reads\nXt=\u0010\nX0−µ0\nλ\u0011\nRλ(t) +1\nλZt\n0fλ(t−s)σ(Xs)dWs.\nBy Itˆ o’s isomorphism and Fubini’s Theorem\nVar\u0010Zt\n0K(t−s)σ(Xs)dWs\u0011\n=E\u0010Zt\n0K(t−s)σ(Xs)dWs\u00112\n=Zt\n0fλ(t−s)2¯σ(s)2ds= (f2\nλ∗¯σ2)(t).\nThen, it follows from (2.22) that\n∀t≥0, v 0= Var( Xt) =v0R2\nλ(t) +¯σ2\nλ2Zt\n0f2\nλ(s)ds (2.24)\nor, equivalently\nv0\u0000\n1−R2\nλ(t)\u0001\n=¯σ2\nλ2Zt\n0f2\nλ(s)ds. (2.25)\n(i) If ¯σ2̸= 0, differentiating this equality implies, since R′\nλ=−fλ,Rλandfλin never null on\n(0,+∞) which is a by-product of Assumption ( K) (see (2.18))\n∀t >0, R′\nλ(t) =−κ Rλ(t)\nwith κ=2λ2v0\n¯σ2so that\n∀t≥0, R λ(t) =e−κt\n(since Rλ(t) = 1).\nNote this also reads equivalently in terms of in terms of Laplace transform, LRλ(t) =1\nκ+t,u >0.\nThus Equation (1.6), rewritten in terms of Laplace transform, reads\nLK(t) =1\nλ\u00101\ntLRλ(t)−1\u0011\n=κ\nλ1\nt, u > 0\nwhich implies\nK(t) =κ\nλ, t > 0.\n10Hence the kernel Kis necessary constant which yields the announced conclusion.\n(ii) If ¯σ2= 0, then v0= 0 since Rλ(t)>1 (at least for tlarge enough). Consequently Var( Xt) = 0\nfor every t≥0. As EXt=EX0=µ0\nλowing to what was done in section 2.1, it follows that\nXt=µ0\nλP-a.s.. But then σ2\u0000µ0\nλ\u0001\n= ¯σ2= 0. 2\nExamples. \u0003The case σ(t, x) =σ >0andµ(t) =µ0.Asσ >0, the existence of a stationary\nregime implies that the kernel Kis constant. We can normalize it at K=1. The process X\nis then a Gaussian process and, more precisely, a regular a mean-reverting Ornstein-Uhlenbeck\nprocess whose stationary regime does exist, is unique, with N\u0010\nµ0\nλ,σ2\n2λ\u0011\nas a 1-marginal distribution\n(a.k.a. an invariant distribution in such a Markovian setting). This regime is of course obtained\nby considering X0∼ N\u0010\nµ0\nλ,σ2\n2λ\u0011\n, independent of W.\n\u0003Volterra CIR. This is the class of processes of the form\nXt=X0+Zt\n0K(t−s)(µ0−λXs)ds+Zt\n0K(t−s)ϑp\nXsdWs, X 0≥0\nwithEX0=µ0\nλ, where ϑ, λ > 0 and µ0≥0. These processes are extensively used in rough volatility\nmodeling with a kernel Kgiven by K(u) =KH+1\n2(u) =uH−1\n2,H∈(0,1\n2)(see [16, 8, 11, 12, 15]),\nprovided the equation has at least a non-negative weak solution as established e.g. in [16]). However,\nthis specific choice of Kplays no role here. For this dynamics,\n¯σ(t)2=ϑ2EXt=ϑ2µ0\nλ\nis constant whatever the choice of the kernel is since EXtis constant by assumption.\n– If ¯σ2>0, then we are in the case where the kernel Kis constant (say K=1w.l.g.) i.e. the\nregular CIR model whose 1-marginal invariant distribution is known to be the gamma distribution\nγ\u00002µ\nϑ2,2λ\nϑ2\u0001\nwith our notations.\n– If ¯σ2= 0, it follows that ¯ σ2=µ0\nλso that µ0= 0. In turn (2.25) implies v0= 0 since Rλ(t)̸= 1\nfortlarge enough. If such is the case, then the trivial process Xt= 0,t≥0 is the only stationary\nsolution.\n3 Toward fake stationarity for scaled Volterra equations\nIn this section we investigate the case where\nµ(t) =µ0and σ(t, x) =ς(t)σ(x), ς(t), σ(x)>0.\nNote that if σisLipschitz continuous andςis abounded Borel function , then Equation (2.17)\nhas a unique pathwise solution (see [17, Theorem 1.1]).\nNotational warning ! Also note that from now on we will still denote\n¯σ2(t) =Eσ2(Xt).\n11We saw that a necessary condition for the existence of a stationary regime is that EXtand\nVar(Xt) are constant but also Eσ2(Xt),t≥0.\nHowever the converse is clearly false as illustrated in the previous section). The existence of\nsuch regimes remain interesting in practice to produce somewhat stable models.\nThis leads us to introduce two notions of fake stationary regimes :\n▶Fake stationary regime of type I : The mean EXt, the variance Var( Xt) and ¯ σ2=Eσ2(Xt) are\nall constant as functions of t.\nAsking Eσ2(Xt) to constant may appear as a more technical assumption that can be discussed.\nHowever it an also be seen as an additional constraint which is one more step toward stationarity.\nWe will see in Proposition 3.2 further on that imposing that ¯ σ2(t) is constant turns out to be\nsuperfluous as it can be obtained as a by-product of the constance of the first two quantities.\n▶Fake stationary regime of type II : The solution ( Xt)t≥0has the same marginal distribution i.e.\nXtd=X0for every t≥0.\nBy mimicking the proof of Theorem 2.1 where the autonomous σ(x) is replaced by ς(t)σ(x)\nyields a necessary condition for a fake stationary regime of type I to exist, namely the triplet\u0000\nv0,¯σ2, ς2(t)\u0001\nwhere v0= Var( X0), ¯σ2=Eσ(X0)2satisfies the equation\n∀t >0, cλ2\u0000\n1−R2\nλ(t)\u0001\n= (f2\nλ∗ς2)tandv0\n¯σ2\n0=c >0. (3.26)\nDefinition 3.1 (Corrector/Stabilizer) The function ς=ςλ,csolution to the above equation (if\nany) can be called a corrector or a stabilizer of the scaled Volterra equation.\nNote that there is a degree of freedom for the specification of v0and ¯σ2which corresponds to\nthe balance of the two terms in the product ς(t)σ(x).\nThis equation can be rewritten formally in terms of Laplace transform\ncλ2L1−R2\nλ=Lf2\nλLς2.\nIn order to get rid of the Laplace transform of 1 −R2\nλ, we perform an integration by parts\nL1−R2(t) =L1(t)−LR2\nλ(t) =1\nt−\u0010R2\nλ(0)\nt−2\ntLRλfλ(t)\u0011\n=2\ntLRλfλ(t)\nso that the Laplace counterpart of (3.26) finally reads\n∀t >0, t Lf2\nλ(t).Lς2(t) = 2 cλ2LRλfλ(t). (3.27)\nNote that, c being fixed, the solution ς2\nλ,cof (3.26), if any, is unique. Indeed, as fλis a\nprobability density owing to Assumption ( K)(ii) (see (2.18)), Lf2\nλ>0. Then Lς2is uniquely\ndetermined by (3.27) which in turn implies uniqueness of ς2(at least dt-a.e.).\n12To carry on our investigations of the fake stationary regimes we make the following existence\nassumption on Equation (3.26), in particular in terms of positivity: let λ,c >0\n(Eλ,c) Equation (3.26) has a unique positive bounded Borel solution ςλ,con (0 ,+∞).\nUniqueness is not a difficult point in practice (see e.g. the proof of the proposition below). The\nmain condition in ( Eλ,c) is clearly positiveness.\nRemarks. •IfK= 1, then Rλ(t) =e−λtandfλ(t) =λe−λtso that ς2\nλ,c= 2λc.\n•Formally one can write the solution of Equation (3.26) as\nςλ,c= 2ec Rλfλ∗X\nk≥0(−1)k\u0000ef2\nλ(u)du−δ0\u0001∗k,\nwhereec=λ2R\nR+f2\nλ(u)duandef2\nλ=f2\nλ R\nR+f2\nλ(u)duwithout presuming of the convergence of the series in\nthe right hand side of the equation, nor its sign.\nNumerical aspects of this equation in the “rough setting”, that is K=Kα,α=H+1\n2with\nH∈(0,1\n2), are investigated in Section 5.1 where (partial) theoretical results are established like the\nsquare integrability of fλand the existence of a solution ς2\nλ,c(but not its positiveness).\nIf Assumptions ( Eλ,c), (1.2) and (1.3) made on the kernel Kare in force and if µ(t) =µ0\nandσ(t, x) =ςλ,c(t)σ(x) with σ:R→Ris Lipschitz continuous with coefficient [ σ]Lip, then the\nscaled Volterra equation (2.17) has a unique ( FX0,W\nt )t≥0-adapted strong solution starting from\nX0∈L2(Ω,A,P).\nBefore exploring the main two situations for σ, let us come back to the notion of fake stationary\nregime of type I. The following proposition emphasizes that there is in fact an equivalence between\nhaving a constant variance and a constant value for Eσ2(Xt).\nProposition 3.2 (More on fake stationary regime of type I) Assume f2\nλhas a finite non\nidentically 0Laplace transform on (0,+∞)andEX0=µ0\nλandc=v0\n¯σ2\n0∈(0,+∞). Then, the\nfollowing claims are equivalent for a solution of the scaled Volterra equation (2.17) :\n(i)∀t≥0,Var(Xt) = Var( X0),\n(ii)∀t≥0,Eσ2(Xt) =Eσ2(X0).\nProof. (ii)⇒(i)Then ¯ σ2= ¯σ2\nt=Eσ2(Xt),t≥0, so that it follows from Equation (3.26)\nVar(Xt) = Var( X0)R2\nλ((t) +¯σ2\nλ2f2\nλ∗ς2\nλ,c\n= Var( X0)R2\nλ(t) +1\ncλ2f2\nλ∗ς2\nλ,c\n=v0R2\nλ(t) +\u0000\n1−R2\nλ(t)\u0001\nv0=v0.\n(i)⇒(ii)Assume Var( Xt) = Var( X0) =v0for every t≥0. If v0= 0, then Xt=µ0\nλa.s.for every\nt≥0. Consequently Eσ2(Xt) =Eσ2\u0000µ0\nλ\u0001\n.\n13Ifv0>0,Eσ2(X0)>0 since cis finite. As consequence one checks that the function\ny(t) =ς2\nλ,c\u0010Eσ2(Xt)\nEσ2(X0)−1\u0011\nis solution to f2\nλ∗y= 0,y(0) = 0. Moreover by the linear growth assumption on σ, we know that\nEσ2(Xt) is bounded since EX2\ntis. Consequently yhas a Laplace transform as well as its positive\nand negative parts y±. One has Lf2\nλ·Ly+=Lf2\nλ·Ly−. IfLf2\nλis not identically 0, then it is positive\non (0 ,+∞) so that Ly+=Ly−which implies y+=y−, hence y= 0. 2\n\u0003The case σ(x) =σ. Then ¯ σ2=σ2. In that case ( Xt)t≥0is a Gaussian process with a constant\nmeanµ0\nλ, constant variance v0. As a consequence Xt∼ N\u0000µ0\nλ, v0\u0001\nfor every t≥0 i.e. has a\nfake stationary regime of type II. But it cannot be a true stationary regime if the kernel Kis not\nconstant.\n\u0003The general case σ(x).\nWe first recall a result on the constant of an Lp-Burkholder-Davis-Gundy (BDG) inequality.\nLemma 3.3 (Best constant in a BDG inequality (see Remark 2 in [5])) LetMbe a con-\ntinuous local martingale null at t= 0. Then, for every p≥1\n∥Mt∥p≤2√p∥⟨M⟩1\n2∥p.\nProposition 3.4 (Moment control) (a) Quadratic moments . Assume fλ∈L2(Leb 1),σ(t, x) :=\nς(t)σ(x)where ς=ςλ,cis a non-negative and bounded solution to (2.24) for some fixed λ, c > 0.\nLet(Xt)t≥0be the solutions to the Volterra equation (2.17) starting from X0∈L2(P). Assume σis\nLipschitz and c∈\u0000\n0,1\n[σ]2\nLip\u0001\n. Then, setting ρ:=c[σ]2\nLip∈(0,1), one has\nsup\nt≥0\r\r\rXt−µ0\nλ\r\r\r\n2≤\"√c\nρ1\n4(1−√ρ)\f\fσ(µ0\nλ)\f\f#\n∨\r\r\rX0−µ0\nλ\r\r\r\n2<+∞.\n(b)Lp-moments . Let p∈(2,+∞)and let Cp=CBDG\np = 2√p. Ifcis such that ρp:=c C2\np[σ]2\nLip<1\nandEX0=µ0\nλ, then\nsup\nt≥0\r\r\rXt−µ0\nλ\r\r\r\np≤ inf\nϵ∈(0,1\nρp−1)\"√c Cp(1 +ϵ)1\n2\nρp(1 +ϵ)1\n4(1−p\nρp(1 +ϵ))\f\fσ(µ0\nλ)\f\f#\n∨\u0014\n(1 + 1 /ϵ)1\n2\r\r\rX0−µ0\nλ\r\r\r\np\u0015\n<+∞\nProof. (a) Let η∈(0,1−ρ) be a free parameter such that ρ+η∈(0,1). One has for every x∈R,\nσ2(x)≤\u0000\f\fσ(µ0\nλ)\f\f+ [σ]Lip\f\fx−µ0\nλ|\u00012≤κ1+κ2\f\fx−µ0\nλ|2\nwith real constant κidepending on ηand reading\nκ1= (1 + ρη−1)|σ(µ0\nλ)|2and κ2= [σ]2\nLip(1 +ηρ−1)\n14so that cκ2=ρ+η <1. Using that f2\nλ∗ς2=cλ2(1−R2\nλ) elementary computations show that for\nevery t≥0\nE\u0010\nXt−µ0\nλ\u00112\n≤E\u0010\nX0−µ0\nλ\u00112\nR2\nλ(t)+κ1\u0000\n1−R2\nλ(t)\u0001\n+κ2\nλ2Zt\n0f2\nλ(t−s)ς2(s)E\u0010\nXs−µ0\nλ\u00112\nds(3.28)\nNow let A > ¯Aη:=κ1c\n1−κ2c∨E\u0010\nX0−µ0\nλ\u00112\n,δ >0 and\ntδ= infn\nt:E\u0010\nXt−µ0\nλ\u00112\n≥A+δo\n.\nAst7→E\u0010\nXt−µ0\nλ\u00112\nis continuous and A >E\u0010\nX0−µ0\nλ\u00112\nit follows from the above inequality and\nthe identity satisfied by ςthat, if tδ<+∞,\nA+δ=E\u0010\nXtδ−µ0\nλ\u00112\n< AR2\nλ(tδ) +\u0000\nκ1c+κ2c(A+δ)\u0001\u0000\n1−R2\nλ(tδ)\u0001\n.\nNow, we have κ1c+κ2cA < A by construction of A, hence\nA+δ=E\u0010\nXtδ−µ0\nλ\u00112\n< AR2\nλ(tδ) +A(1−R2\nλ(tδ)) +κ2cδ\u0000\n(1−R2\nλ(tδ)\u0001\n< A+δ.\nwhich yields a contradiction. Consequently, tδ= +∞which implies that E\u0010\nXt−µ0\nλ\u00112\n≤A+δfor\nevery t≥0. Letting δ→0 and A→¯Aηsuccessively, yields\nsup\nt≥0E\u0010\nXt−µ0\nλ\u00112\n≤¯Aη.\nThen one checks that η7→¯Aηis minimal over (0 ,1−ρ) atη=√ρ−ρwhich finally yields the\nannounced result.\n(b) Let p≥2. Using BDG inequality to the (a priori) local martingale Ms=Rs\n0fλ(t−u)ζ(u)σ(Xu)dWu,\n0≤s≤t, (see [23, Proposition 4.3]) and the generalized Minkowski inequality, we get\n\r\r\rXt−µ0\nλ\r\r\r\np≤\r\r\rX0−µ0\nλ\r\r\r\npRλ(t) +Cp\nλ\r\r\rZt\n0f2\nλ(t−s)ς2(s)σ(Xs)2ds\r\r\r1\n2\np\n2\n≤\r\r\rX0−µ0\nλ\r\r\r\npRλ(t) +Cp\nλ\u0010Zt\n0f2\nλ(t−s)ς2(s)∥σ(Xs)2∥p\n2\u00111\n2\n(with an equality when p= 2 in the first line). Now let ϵ∈(0,1/ρp−1). It follows from the\nelementary inequality ( a+b)2≤(1 + 1 /ϵ)a2+ (1 + ϵ)b2that\n\r\r\rXt−µ0\nλ\r\r\r2\np≤\r\r\rX0−µ0\nλ\r\r\r2\npR2\nλ(t)(1 + 1 /ϵ) +C2\np\nλ2(1 +ϵ)Zt\n0f2\nλ(t−s)ς2(s)∥σ(Xs)2∥p\n2ds\nLet ˜ρp=ρp(1 +ϵ)∈(0,1) and for η∈(0,1−˜ρp), set now κidepending on ηand reading\nκ1= (1 + ˜ ρpη−1)|σ(µ0\nλ)|2and κ2= [σ]2\nLip(1 +η˜ρ−1\np)\n15so that cC2\np(1 +ϵ)κ2= ˜ρp+η <1. Asp\n2≥1,\n∥σ(Xs)2∥p\n2≤κ1+κ2\r\r\rXs−µ0\nλ\r\r\r2\np\nwhich entails, combined with the identity f2\nλ∗ς2=cλ2(1−R2\nλ), that, for every t≥0,\n\r\r\rXt−µ0\nλ\r\r\r2\np≤\r\r\rX0−µ0\nλ\r\r\r2\npR2\nλ(t)(1 + 1 /ϵ) +C2\np(1 +ϵ)\u0010\nκ1c\u0000\n1−R2\nλ(t)\u0001\n+κ2\nλ2Zt\n0f2\nλ(t−s)ς2(s)\r\r\rXs−µ0\nλ\r\r\r2\npds\u0011\n(3.29)\nNow let A > ¯Aη,ϵ:=κ1cC2\np(1+ϵ)\n1−κ2cC2p(1+ϵ)∨h\n(1 + 1 /ϵ)\r\r\rX0−µ0\nλ\r\r\r2\npi\n,δ >0 and\ntδ= infn\nt:\r\r\rXt−µ0\nλ\r\r\r2\np≥A+δo\n.\nIftδ<+∞, it follows on the one hand from the continuity of t7→\r\r\rXt−µ0\nλ\r\r\r2\npthatA+δ=\r\r\rXtδ−µ0\nλ\r\r\r2\np\nand, on the other hand, from Equation (3.26) satisfied by ς, that\nZt\n0f2\nλ(t−s)ς2(s)\r\r\rXs−µ0\nλ\r\r\r2\npds≤A(1−R2\nλ(t)).\nMoreover as A >\r\r\rX0−µ0\nλ\r\r\r2\np(1 + 1 /ϵ), we deduce from (3.29)\nA+δ=\r\r\rXtδ−µ0\nλ\r\r\r2\np< AR2\nλ(tδ) +C2\np(1 +ϵ)\u0000\nκ1c+κ2c(A+δ)\u0001\u0000\n1−R2\nλ(tδ)\u0001\n.\nNow, we have C2\np(1 +ϵ)c(κ1+κ2A)< Aby definition of A. Hence\nA+δ=\r\r\rXtδ−µ0\nλ\r\r\r2\np< AR2\nλ(tδ) +A(1−R2\nλ(t)) +C2\np(1 +ϵ)cκ2δ\u0000\n(1−R2\nλ(tδ)\u0001\n< A+δ.\nsince C2\np(1 + ϵ)cκ2<1. This yields a contradiction. Consequently, tδ= +∞which implies that\r\r\rXtδ−µ0\nλ\r\r\r2\np≤A+δfor every t≥0. Letting δ→0 and A→¯Aη,ϵsuccessively, yields\nsup\nt≥0\r\r\rXt−µ0\nλ\r\r\r\np≤¯A1\n2η,ϵ<+∞.\nThen one checks that η7→¯Aηis minimal over (0 ,1−˜ρp) atη=p˜ρp−˜ρpwhich finally yields the\nannounced result. 2\nThe following result can be compared to the confluence property satisfied by he mean-reverting\nOrnstein-Uhlenbeck process (without rate of convergence however).\n16Proposition 3.5 ( L2-confluence) Assume that the above assumptions are in force and that fλ∈\nL2(Leb 1),σ(t, x) := ςλ,c(t)σ(x)where ςλ,cis solution to (3.26) for some λ > 0,c∈\u0000\n0,1\n[σ]2\nLip\u0001\nsatisfying (Eλ,c).\nForX0, X′\n0∈L2(P), we consider the solutions to Volterra equation (2.17) denoted (Xt)t≥0and\n(X′\nt)t≥0starting from X0andX′\n0∈L2(P)respectively. Set\n∆t=Xt−X′\nt∈ L2(Leb 1), t≥0.\nThen, there exists a continuous non negative function φ∞:R+→[0,1]such that φ∞(0) = 1 ,\nlim\nt→+∞φ∞(t) = 0 such that\n∀t≥0,E∆2\nt≤φ∞(t)E∆2\n0.\nProof . We still denote ρ=c[σ]2\nLip∈(0,1). It follows from the reduced form (2.22), Itˆ o’s isometry\nand the Lipschitz property of σthat\nE∆2\nt≤R2\nλ(t)E∆2\n0+[σ]2\nLip\nλ2Zt\n0f2\nλ(t−s)ς2(s)E∆2\nsds. (3.30)\nLet¯δt=∥∆t∥2to alleviate notations. One checks that, under our assumptions, the function\nt7→¯δtis continuous (see [17]).\nLetη >0 such that ρ(1 +η)2<1. We define τη:= inf{t:¯δt>(1 +η)¯δ0}. Ifτη<+∞, then\n¯δs≤(1 +η)¯δ0for every s∈(0, τη) and by continuity ¯δ2\nτη= (1 + η)2¯δ2\n0. Plugging this in the above\ninequality at time τηyields\n¯δ2\nτη≤¯δ2\n0\u0002\nR2\nλ(τη) + (1 −R2\nλ(τη))ρ(1 +η)2\u0003\n< ρ(1 +η)2¯δ2\n0\nwhich yields a contradiction so that ¯δs≤(1 + η)¯δ0for every s >0. This holds for every η >0\nwhich in turn implies that ¯δt≤¯δ0for every t≥0. Plugging this again in (3.30) implies that, for\nevery t >0,\nδ2\nt≤δ2\n0φ1(t) with φ1(t) :=R2\nλ(t) + (1 −R2\nλ(t))ρ.\nNote that φ1(t) =ρ+R2\nλ(t)(1−ρ) satisfies\nφ1(0) := 1 , φ1(t)∈(0,1), t > 0 and φ1is non-decreasing and continuous.\nPlugging this upper-bounds of δ2\ntinto (3.30) straightforwardly yields\nδ2\nt≤¯δ0φ2(t) with φ2(t) :=R2\nλ(t) +ρZt\n0f2\nλ(t−s)ς2(s)φ1(s)ds\nλ2c.\nOne checks using identity (3.26) satisfied by ς2and the definition of φ1that\nφ2(t) :=φ1(t)−ρZt\n0f2\nλ(t−s)ς2(s)\u0000\n1−φ1(s)\u0001ds\nλ2c\nso that 0 ≤φ2< φ 1<1 on (0 ,+∞). By induction one shows that\nδ2\nt≤δ2\n0φk(t)\n17with\nφk(t) =R2\nλ(t) +ρZt\n0f2\nλ(t−s)ς2(s)φk−1(s)ds\nλ2c\n=φ1(t)−ρZt\n0f2\nλ(t−s)ς2(s)\u0000\n1−φk−1(s)\u0001ds\nλ2c.\nwhere we used again (3.26) satisfied by ς2and the definition of φ1.\nConsequently, starting from 0 ≤φ2< φ 1<1 on (0 ,+∞), we show by induction that 0 ≤φk<\nφk−1<1 on (0 ,+∞) for every k≥2. One checks again by induction that φkis continuous since\nby change of variable\nφk(t) =φ1(t)−ρZt\n0f2\nλ(s)ς2(t−s)\u0000\n1−φk−1(t−s)\u0001ds\nλ2c.\n(since ς2is bounded and continuous owing to Assumption Eλ,c).\nBy the first Dini Lemma, it follows that φk↓φ∞∈ C(R+,R) uniformly on compact intervals\nwith φ∞(0) = 1. In particular φ∞satisfies the functional equation\nφ∞(t) =R2\nλ(t) +ρZt\n0f2\nλ(t−s)ς2(s)φ∞(s)ds\nλ2c\nLetℓ∞:= lim sup\nt→+∞φ∞(t)∈[0,1]. For every ε >0 there exists Aε>0 such that for t≥Aε,\nφ∞(t)≤ℓ∞+ε. Then\nZt\n0f2\nλ(t−s)ς2(s)φ∞(s)ds\nλ2c≤Zt\nAεf2\nλ(t−s)ς2(s)(ℓ+ε) +∥ς∥∞\ncλ2Zt\nt−Aεf2\nλ(u)du.\nConsequently, as fλ∈L2(Leb 1) and lim t→+∞R2\nλ(t) = 0,\nlim sup\nt→+∞φ∞(t)≤ρ(ℓ∞+ε)\nso that ℓ∞≤ρ ℓ∞which in turn implies ℓ∞= 0 since ρ∈[0,1). 2\nRemark. If [σ]2\nLip<λ2\n∥ς2\nλ,c∥∞R+∞\n0f2\nλ(u)du<1 and Rλ∈ L2(Leb 1), then one derives from Fubini-\nTonelli’s theorem that\nZ+∞\n0φ2\n∞(s)ds≤λ2\nλ2−[σ]2\nLip∥ς2∥∞R+∞\n0f2\nλ(u)duZ+∞\n0R2\nλ(t)dt < +∞.\n4 Example of fake stationary regimes of type I and II\nIn this section we specify a family of scaled models where b(x) =µ0−λ xand\nσ(x) =q\nκ0+κ1(x−µ0\nλ) +κ2(x−µ0\nλ)2with κi≥0, i= 1,2, κ2\n1≤4κ0κ2. (4.31)\nMoreover we will always set κ1= 0 whenever κ2= 0. This is this type of vol-vol term that appears\nin the quadratic rough volatility model introduced in [12] to solve the problem of the the joint S&P\n500/VIX smile calibration problem.\n184.1 Fake stationary regimes\nProposition 4.1 (Fake stationary regimes (types I and II) and first asymptotics) . Con-\nsider the diffusion coefficient σgiven by (4.31) and let λ >0,c∈(0,1\nκ2)such that (Eλ,c)is satisfied.\nThen, let X0∈L2(P)with\nEX0=µ0\nλand v0= Var( X0) =cκ0\n1−cκ2.\n(a)Ifκ2>0, then (Xt)t≥0solution to (2.17) has a fake stationary regime of type I in the sense\nthat\n∀t≥0,EXt=µ0\nλ,Var(Xt) =v0=cκ0\n1−cκ2and Eσ2(Xt) = ¯σ2\n0=κ0\n1−cκ2.\nMoreover, for any starting value X0∈L2(P),\nEXt→µ0\nλand Var(Xt)→cκ0\n1−cκ2ast→+∞.\n(b)Ifκ2=κ1= 0, then σ(x) =√κ0is constant (the choice of cbecomes free) and if X0∼ν∗:=\nN\u0000µ0\nλ,cκ0\n1−cκ2\u0001\n, then (Xt)t≥0is a Gaussian process with a fake stationary regime of type II with ν∗\nas1-marginal distribution .\nPractitioner’s corner .•The above proposition covers at least partially the rough volatility\ndynamics recently introduced in [12] (the volatility process being defined as Vt=σ2(Xt) with our\nnotations). In this model, the traded asset and its volatility are driven by t he same Brownian\nmotion and its purpose is to propose a joint calibration of the VIX and the SP500 in order to take\ninto account the so-called Zumbach effect which connects the evolution of the asset (here an index)\nand its volatility.\n•Note that in practice, when κ1>0, if we fix the value of v0, then c=v0\nκ0+v0κ2so that, σbeing√κ2-Lipschitz continuous, one has κ2[σ]2\nLip=v0κ2\nκ0+v0κ2<1 which ensures the L2-confluence of the\npaths of the solution.\nMoreover the presence of the stabilizer ςλ,callows a better control of the behaviour of the\nequation since it induces an L2-confluence and a stability of first two moments if needed.\nRemark. If we consider a dynamics more inspired by the original CIR model in which µ(t) =µ0>\n0,σ(t, x) =ςλ,c(t)σ(x) with σ(x) :=q\nκ0+κ1(x−µ0\nλ),κ1>0,κ0> κ 1µ0\nλ, the resulting Volterra\nequation (2.17) may have (at least) a non-negative weak solution e.g. as the C-weak limit of a\nsystem of Hawkes processes (as it has been done in [16] without the presence of the corrector ςλ,c,\nsee also [14], [1]). If such a weak solution on the whole non-negative real line does exists starting\nfrom X0with meanµ0\nλthen\nVar(Xt) = Var( X0)R2\nλ(t) +κ0\u0000\n1−R2\nλ(t)\u0001\n.\nA fake stationary regime of type I then should have constant mean EXt=µ0\nλand a variance given\nby Var( Xt) =cκ0respectively and one easily checks that such is the case. Note that if κ0= 0 – like\nin the Volterra CIR like model – one retrieves the degenerate situation where Xt=µ0\nλP-a.s.\n19Proof. (a) Note that, if κ2>0, [σ]Lip=√κ2since\nσ(x) =s\nκ2(x−µ0\nλ−κ1\n2κ2)2+κ0−κ2\n1\n4κ2\nso that the condition 0 < c < 1/[σ]2\nLipis satisfied. We know that\nEXt=µ0\nλR2\nλ(t) +µ0\nλ(1−R2\nλ(t)) =µ0\nλ\nfor every t≥0. As for the variance, we have\nVar(Xt) = Var( X0)R2\nλ(t) +1\nλ2f2\nλ∗\u0000\nς2Eσ2(X·)\u0001\nt\n= Var( X0)R2\nλ(t) +κ0\nλ2(f2\nλ∗ς2)t+κ1×0 +κ2\nλ2f2\nλ∗\u0000\nς2Var(X·)\u0001\nt. (4.32)\nIf we assume that Var( Xt) is constant i.e. Var( Xt) =v0for every t≥0 and take advantage of the\nidentity (3.26) satisfied by ς=ςλ,c, the above equation reads\nv0(1−R2\nλ(t)) = ( cκ0+cκ2v0)(1−R2\nλ(t)), t≥0\ni.e.\nv0=cκ0\n1−cκ2>0.\nwhich is clearly solution to the equation.\nConversely one checks that this constant value for the variance solves the above equation. Let\nus prove that it is the only one. By the linearity of Equation (4.32), it suffices to show that the\nequation in x∈ C(R+,R)\nx(t) =κ2\nλ2\u0000\nf2\nλ∗(ς2. x)\u0001\nt, x(0) = 0\nonly has the null function as solution. If xsolves the above equation, then\n|x(t)| ≤κ2\nλ2(f2\nλ∗ς2)tsup\n0≤s≤t|x(s)|=cκ2sup\n0≤s≤t|x(s)|.\nIfx≡/0, there exist ε >0 such that τε= inf{t:|x(t)|> ε}<+∞. By continuity of xit is clear\nthat τε>0 and |x(τε)|= sup0:≤s≤t|x(s)|=εwhich is impossible since κ2c <1. Consequently\nx≡0.\nHence ( Xt)t≥0is a fake stationary regime of type II with the above mean and variance. The\nlast claim is a straightforward consequence of Proposition 3.5.\n(b) is obvious once noted that ( Xt)t≥0is a Gaussian process (and [ σ]Lip= 0). 2\n4.2 Long run behaviour\nTheorem 4.2 Letλ >0. Assume the kernel Kand its λ-resolvent Rλsatisfy\nZ+∞\n0fλ(t)2β(t)dt < +∞ for some β >1\n20and assume that there exists ϑ∈(0,1]such that\nZ+∞\n0(fλ(t+δ)−fλ(t))2dt≤Cδ2ϑ.\nAssume that ς=ςλ,csatisfies ς(0) = 0 ,ς2is differentiable with a derivative having locally finite\nvariation and that σandX0satisfy one of the conditions below:\n(SSL σ,a):σstrictly sublinear . Assume that c <1\n[σ]2\nLipand that there exists a∈h\n0,\u0000\n2(ϑ∧2β\nβ−1)\u0001\n∧1\u0011\nsuch that X0∈L2\na(P)and\n∀x∈R,|σ(x)| ≤κ\u0000\n1 +|x|2a\u00011/2for some κ≥0\nor\n(SLσ):σsublinear . Assume c <1\n4p[σ]2\nLipandX0∈Lp(P)for some p >2and\n∀x∈R,|σ(x)| ≤κ\u0000\n1 +|x|2\u00011/2for some κ≥0.\n(a)Assume ¯σ2=Eσ2(Xt),t≥0, is constant. Then, for every t1, t2≥0,t1≤t2,\nCov(Xt+t1, Xt+t2)t→+∞−→ Cfλ(t1, t2) :=v0R+∞\n0f2\nλ(s)dsZ+∞\n0fλ(t2−t1+u)fλ(u)du. (4.33)\nFurthermore, the family of shifted processes (Xt+u)u≥0isC-tight and uniformly square integrable\nast→+∞. Thus, for any limiting distribution Pon the canonical space Ω0:=C(R+,R), the\ncanonical process Yt(ω) =ω(t),ω∈Ω0, has, for any small enough η >0, a\u0000\n(ϑ∧2β\nβ−1)−a\n2−η\u0001\n-\nH¨ older pathwise continuous P-modification. Moreover, under P,Yis an L2-stationary processes\nwith meanµ0\nλand covariance function Cfλ(s, t),s, t≥0.\nWhen a= 0(i.e. σbounded) the above results hold true provided X0∈ ∩p>0Lp(P).\n(b)When σ(x) =σ >0is constant and X0has a normal distribution, then (Xt)t≥0satisfies\nXt+·C−→Ξ(fλ)ast→+∞,\nwhere Ξ(fλ)is the stationary Gaussian process with covariance function CfλandC→stands for\nfunctional weak convergence on C(R+,R)equipped with the topology of uniform convergence on\ncompact sets.\nTo be compared to the more precise result from [10] for the Volterra CIR Volterra process.\nProof. Step 1. One may assume w.l.g. that\n∀x∈R,|σ(x)| ≤κ\u0000\n1 +\f\fx−µ0\nλ\f\f2a\u00011/2for some κ≥0.\nMoreover, Young’s inequality implies that for every r >0\n|σ(x)| ≤\u0010\nκ+ (1−a)rκ|{z}\n=:˜κ1+a\u0000κ\nr\u00011/a\n|{z}\n=:˜κ2\f\f\fx−µ0\nλ\f\f\f2\u00111/2\n21so that for rlarge enough, c·˜κ2<1. By mimicking the computation in the beginning of the proof\nof Proposition 3.4( a) we derive that\nsup\nt≥0E\u0010\nXt−µ0\nλ\u00112\n<+∞.\nStep 2. Now it follows from BDG inequality that, if X0∈Lp(P) for a p >2, then\n\r\r\rXt−µ0\nλ\r\r\r\np≤\r\r\rX0−µ0\nλ\r\r\npRλ(t) +CBDG\np\u00121\nλ2Zt\n0f2\nλ(t−s)ς(s)2∥σ2(Xs)∥p\n2ds\u00131/2\nand, if p∈[1,2),\n∥σ2(Xs)∥p\n2≤κ(1 +∥X−µ0\nλ∥2a\nap).\nSetting p=2\na>1 yields\nsup\ns≥0∥σ2(Xs)∥p\n2≤κ\u0000\n1 + sup\ns≥0∥Xs−µ0\nλ∥2a\n2)<+∞.\nAs a consequence, still relying on the identity f2\nλ∗ς2=λ2c(1−R2\nλ), we derive\n∀t≥0,\r\r\rXt−µ0\nλ\r\r\r\np≤\r\r\rX0−µ0\nλ\r\r\r\npRλ(t) +CBDG\np sup\ns≥0∥σ2(Xs)∥1/2\np\n2√c\u0000\n1−Rλ(t)2\u00011/2\n≤\r\r\rX0−µ0\nλ\r\r\r\np+CBDG\np sup\ns≥0∥σ2(Xs)∥1/2\np\n2√c <+∞.\nStep 3 (Kolmogorov criterion). One writes for s, t≥0,s≤t\nXt−Xs=\u0000\nRλ(t)−Rλ(s)\u0001\u0010\nX0−µ0\nλ\u0011\n+Zt\n0fλ(t−u)ς(u)σ(Xu)dWu−Zs\n0fλ(s−u)ς(u)σ(Xu)dWu.\nLet us denote by ( A) and ( B) the two terms of the sum on the right hand side of the above equality.\nFirst note that if p=2\nathen\n∥A∥p=\r\r\rX0−µ0\nλ\r\r\r\npZt\nsfλ(u)du\n≤\r\r\rX0−µ0\nλ\r\r\r\np\u0012Z+∞\n0f2β\nλ(u)du\u00131\n2β\n|t−s|1−1\n2β\n=CX0,β,fλ|t−s|1−1\n2β.\nOn the other hand\n∥B∥p≤\r\r\r\rZt\nsfλ(t−u)ς(u)σ(Xu)dWu\r\r\r\r\np+\r\r\r\rZs\n0\u0000\nfλ(t−u)−fλ(s−u)\u0001\nς(u)σ(Xu)dWu\r\r\r\r\np.\n22Let us denote by ( B.1) and ( B.2) the two terms on the right hand side. Combining the Lp-BDG\nand the generalized Minkowski inequality yields for ( B.1)\n∥(B.1)∥p≤CBDG\np\u0010Zt\nsf2\nλ(t−u)ς2(u)∥σ(Xu)∥2\npdu\u00111/2\n≤CBDG\np sup\nu≥0∥σ(Xu)∥p∥ς∥∞\u0010Zt\nsf2\nλ(t−u)du\u00111/2\n≤CBDG\np sup\nu≥0∥σ(Xu)∥p∥ς∥∞\u0010Z+∞\n0f2β\nλ(u)du\u00111/2\n|t−s|1\n2(1−1\nβ)\n≤Cp,σ,f λ|t−s|β−1\n2β.\nAs for ( B.2)\n∥(B.2)∥p≤Cp,σ,f λ\u0012Z+∞\n0\u0000\nfλ(t−s+v)−fλ(v)\u00012du\u00131/2\n≤Cp,σ,f λ|t−s|ϑ.\nFinally, after noticing thatβ−1\n2β≤1−1\n2βwhat precedes proves the existence of a real constant\nCp,λ>0 such that\nE|Xt−Xs|p≤Cp,λ|t−s|p(ϑ∧β−1\n2β)\nhaving in mind that p=2\naso that p(ϑ∧β−1\n2β)>1.\nThen it follows from Kolmogorov’s C-tightness criterion (see [23, Theorem 2.1]), that the family\nof shifted processes Xt+·,t≥0, isC-tight.\nStep 4. (SLσ)setting. In that setting we know from Proposition 3.4( b) that ( ∥Xt−µ0\nλ∥p)t≥0is\nbounded. Then then the computations carried out in Step 3 are still valid and we have\nE|Xt−Xs|p≤Cp,λ|t−s|p(ϑ∧β−1\n2β).\nRemark. We would need ∥supt≥0|Xt|∥p<+∞for some p >2 to derive relative compactness for\n(functional) W2-compactness (quadratic Wasserstein distance).\nStep 5 . Let XandX′be two solutions of Equation (2.17) starting from X0andX′\n0respectively,\nboth square integrable. Using Proposition 3.5, we derive that for every 0 ≤t1< t2<···< tN<\n+∞\nW2\u0000\n[(Xt+t1,···, Xt+tN)],[(X′\nt+t1,···, X′\nt+tN)])→0 ast→+∞\nAs a consequence, the weak limiting distributions of [ Xt+·] and [ X′\nt+·] are the same in the sense\nthat, if [ Xtn+·](C)−→Pfor some subsequence tn→+∞(where Pis a probability measure on\nC(R+,R) equipped with the Borel σ-field induced by the sup-norm topology), then [ X′\ntn+·](C)w−→P\nand conversely.\nStep 6 .L2-stationarity\nBy an integration by part and using that ς(0) = 0 and ςis bounded, one derives that\nL(ς2)′(t) =t Lς2(t) =2cλ2LRλfλ\nLf2\nλ(t)\n23where the second equality follows from (3.27). As f2\nλ∈ L2(Leb 1) and 2 LRλfλ(0) =−R+∞\n0(R2\nλ)′(u)du=\n1−0 = 1, one has\nL(ς2)′(t)0∼cλ2\nR+∞\n0f2\nλ(s)ds.\nBy the Hardy-Littlewood Tauberian Theorem (see [9]), we get that\nς2(t) =Zt\n0(ς2)′(s)ds+∞∼cλ2\nR+∞\n0f2\nλ(s)ds.\nNow let us consider the asymptotic covariance between Xt+t1andXt+t2, 0< t1< t2when Xt\nstarts for X0with meanµ0\nλ,v0and ¯σ2=Eσ(Xt)2,t≥0.\nCov(Xt+t1, Xt+t2) = Var( X0)R2\nλ(t) +1\nλ2EZt+t1\n0fλ(t+t2−s)fλ(t+t1−s)ς2(s)Eσ2(Xs)ds\n= Var( X0)R2\nλ(t) +¯σ2\nλ2EZt+t1\n0fλ(t2−t1+u)fλ(u)ς2(t+t1−u)du\nAsfλ(t2−t1+·)fλ∈ L2(Leb 1) since fλ∈ L2(Leb 1) and 1{0≤u≤t+t1}ς2(t+t1−u)→cλ2\nR+∞\n0f2\nλ(s)dsfor\nevery u∈R+ast→+∞,\nCov(Xt+t1, Xt+t2)t→+∞−→v0R+∞\n0f2\nλ(s)dsZ+∞\n0fλ(t2−t1+u)fλ(u)du\nwhere we also used that R2\nλ(t)→0 ast→+∞.\nRemark. Be ware that at this stage we do no t have uniqueness of the limit distributions since\nthey are not characterized by their lean and covariance functions, except in Gaussian setting ( Xis\nGaussian process).\nExamples. •When σ(x) =σ >0 and X0has a Gaussian distribution, the process Xis Gaussian\nand this proves (at least for finite dimensional weak convergence i.e. weak convergence of all\nmarginals at any order).\n(Xt+·)Df−→X(∞)ast→+∞\nwhere X(∞)is a Gaussian process with meanµ0\nλand covariance function given by\nCov(X(∞)\nt1, X(∞)\nt2) =v0Z+∞\n0fλ(t2−t1+u)fλ(u)duR+∞\n0f2\nλ(v)dv.\n•When σis given by σ(x) =q\nκ1+κ2(x−µ0\nλ)2like in Proposition 4.1, it has linear growth so the\nabove long run behaviour can be established when there exists p >2 such that X0∈Lp(P) with\nEX0=µ0\nλ, Var( X0) =v0>0 andv0κ2\nκ1+v0κ2<1/(2√p). This is always possible if\nX0∈L2+(P) =∩p>2Lp(P) and v0<κ1\nκ2(2√\n2−1).\n245 Back to α-fractional kernels, 0< α < 1\nThis case is of special interest since it corresponds to the recent introduction of fractional\nVolterra SDEs to devise “rough models” of stochastic volatility dynamics in Finance (see [16, 2,\n11, 8]) or long memory volatility models when H∈(1/2,1) (see [6, 7]). In these models Volterra\nequations with fractional kernels Kα,α=H+1\n2with H∈(0,1\n2) appear as more tractable dynamics\nthan solutions of SDEs involving stochastic integrals with respect to true H-fractional Brownian\nmotions ( Hbeing the Hurst coefficient) which would require to call upon “high order” rough path\ntheory.\nOur aim in this section is to prove that for such kernels Kαthe resolvent Rα,λsatisfies the\nmonotonicity assumption (2.18) for every ( λ >0), that fα,λ=−Rα,λexists and is square integrable\nw.r.t. the Lebesgue measure on R+, at least when1\n2< α < 1. Thus the above results established\nin Sections 3 and 4 in the case σ(t, x) =σ(t) (Gaussian setting) and σ(t, x) =ς(t)σ(x) apply.\nTo this end, it is enough to study Rα,1(λ= 1) given by its expansion – denoted eαin the\nliterature and known as Mittag-Leffler function – given by\neα(t) =X\nk≥0(−1)ktαk\nΓ(αk+ 1), t≥0.\nIn fact this function is completely monotone (CM) in the real line in the sense that ( −1)ne(n)\nα(t)≥0\nat every order n≥0. This follows from the fact that eαis the Laplace transform\neα(t) =LHα(t) =Z+∞\n0e−tuHα(u)du\nof a non-negative Lebesgue integrable function Hα:R+→R+. This representation was first\nestablished in [22]. More recently, a synthetic formula was found for Hαin [20] (see (F.22) p.31,\nsee also [19]). One starts from the Laplace transform of eαwhich writes on C,\nLeα(z) =zα−1\nzα+ 1, z∈C,ℜ(z)>1\nwhere zα:=|z|αeiαarg(z),−π < arg(z)< π. The above identity is based on the inverse Laplace\ntransform (Bromwich-Mellin formula)\neα(t) =1\n2iπZs=1+i·∞\ns=1−i·∞estsα−1\nsα+ 1ds\nwhich finally yields a closed form for Hαgiven by\n∀u∈R+, Hα(u) =−1\n2π·2ℑm\u0010zα−1\nzα+ 1\u0011\n|z=ueiπ=sin(απ)\nπuα−1\nu2α+ 2uαcos(πα) + 1>0 (5.34)\nsince 0 < α < 1 implies sin( απ)>0 and\nu2α+ 2uαcos(πα) + 1≥1−cos2(απ) = sin2(απ)>0 (5.35)\nMoreover as Hαis continuous on (0 ,+∞),\nHα(u)0∼uα−1sin(πα)\nπand Hα(u)+∞∼sin(πα)\nπuα+1,\n25it is clear that Hα∈ L1\nR+(Leb 1) and that both functions\nu7→uHα(u) and u7→uα+1Hα(u) are bounded on R+.\nThus, for every ε >0,R+∞\n0e−εuuHα(u)du < +∞so that eαis differentiable on (0 ,+∞) with\ne′\nα(t) =−Z+∞\n0e−tuuHα(u)du < 0, t > 0. (5.36)\nProposition 5.1 Letλ >0and let α∈(0,1).\n(a)Theλ-resolvent Rα,λsatisfies Rα,1=eαandRα,λ=Rα,1(λ1/α·). The function Rα,λis com-\npletely monotonic (hence infinitely differentiable on (0,+∞)). Moreover Rα,λ(0) = 1 ,Rα,λde-\ncreases to 0,Rα,λ∈ Lr(Leb 1)for every r >1\nαandfα,λ=−R′\nα,λis a completely monotonic\nfunction (hence convex), decreasing to 0and satisfying\n∀t >0, f α,λ(t) :=−R′\nα,λ(t) =λ1\nαZ+∞\n0e−λ1\nαtuuHα(u)du > 0.\n(b)Moreover, if α∈(1\n2,1),fα,λ∈ L2β(Leb 1)for every β∈\u0000\n0,1\n2(1−α)\u0001\nfor every ϑ∈\u0000\n0, α−1\n2\u0001\n, there\nexists a real constant Cϑ,λ>0such that\n∀δ >0,\u0014Z+∞\n0\u0000\nfα,λ(t+δ)−fα,λ(t)\u00012\u00151/2\n≤Cϑ,λδϑ.\nProof. (a) follows from the fact that Rα,λ=eα(λ1/α·) =Rα,1(λ1/α·), hence completely monotonic,\ndecreasing to 0 and differentiable on (0 ,+∞) with fα,λ>0. It follows from (5.34) and (5.35) that\nHα(u)≤uα−1sin(πα)\nπsin2(πα)=uα−1\nπsin(πα). (5.37)\nHence, for every t≥0,\nRα,1(t) =eα(t)≤1\nπsin(πα)Z+∞\n0e−tuuα−1du=Γ(α)\nπsin(πα)t−α\nso that Rα,1∈ Lγ(Leb 1) for every γ >1\nα. This extends to Rλ,αby scaling.\n(b) Let us prove the L2β-integrability of fα,λ. Once noted that fα,λ=λ1/αfα,1(λ1/α·) so thatR+∞\n0f2β\nα,λ(t)dt=λ2β−1\nαR+∞\n0f2β\nα,1(t)dt, it is clear that it is enough to prove that fα,1isL2β-integrable.\nIt follows from (5.37) that for every t >0\nfα,1(t) =−e′\nα(t)≤1\nπsin(πα)Z+∞\n0e−tuuαdu=Γ(α+ 1)\ntα+1πsin(πα).\nOn the other hand\nfα,1(t)0∼tα−1\nΓ(α)\n26owing (3) to (1.14). Combing these two remarks implies the existence of a real constant Cα>0\nsuch that for t∈(0,1],\nfα,1(t)≤Cα\u00101\nt1−α∧1\ntα+1\u0011\nAst7→\u0010\n1\nt1−α∧1\ntα+1\u0011\n∈ L2β(Leb 1) for any β∈\u0000\n1,1\n2(1−α)\u0001\nsince α∈(1\n2,1),fα,1∈ L2β(Leb 1).\nAnother consequence is that, for every t≥1,\nRα,1(t) =eα(t) =Z+∞\ntfα,1(s)ds≤C′\nαtα\nso that Rα,1∈ L2(Leb 1).\nAs for the L2(R+)-ϑ-H¨ older continuity of fα,λ, one may again assume w.l.g. that λ= 1. Let\nδ >0. One has\nfα,1(t+δ)−fα,1(t) =Z+∞\n0e−tu(1−e−δu)uHα(u)du.\nAs 0≤1−e−v≤(1−e−v)ϑ, for every v≥0, since ϑ∈(0,1)\nZ+∞\n0\u0000\nfα,1(t+δ)−fα,1(t)\u00012dt≤Z\n(0,+∞)2(uv)1+ϑHα(u)Hα(v)Zt\n0e−t(u+v)dt du dv δ2ϑ\n=Z\n(0,+∞)2(uv)1+ϑ\nu+vHα(u)Hα(v)du dv δ2ϑ\n≤1\n2Z\n(0,+∞)2(uv)1\n2+ϑHα(u)Hα(v)du dv δ2ϑ\n=1\n2\"Z\n(0,+∞)u1\n2+ϑHα(u)du#\nδ2ϑ,\nwhere we used Fubini-Tonelli’s theorem in the first line to interwind the order of integration and\nthe elementary inequality√uv≤1\n2(u+v) when u, v≥0 in the penultimate line. Now, we derive\nform (5.34) that Now note that\nHα(u)0∼sin(πα)\nπuα−1and Hα(u)+∞∼sin(πα)\nπu−(α+1),\nConsequently\nu1\n2+ϑHα(u)0∼sin(πα)\nπuα−1\n2+ϑand u1\n2+ϑHα(u)+∞∼sin(πα)\nπu−(1\n2+α−ϑ),\nwhich implies that Z\n(0,+∞)u1\n2+ϑHα(u)du < +∞ iff ϑ < α −1\n2.\n2\n3. Another argument is to note that uHα(u)+∞∼sin(πα)\nπu−αso that it follows from Karamata’s theorem (see [3])\nthat\nfα,1(t)0∼Γ(1−α)sin(πα)\nπ.\nThe two formulas coincide since Γ( α)Γ(1−α) =π\nsin(πα).\n27Corollary 5.2 (a)Ifα∈\u00001\n2,1\u0001\nandσ,X0andcsatisfy (SSL)σ,aor(SL)σthen the family of\nshifted processes Xt+·,t≥0isC-tight as t→+∞and its (functional) limiting distributions are\nall of L2-stationary processes with covariance function C∞given by (4.33) which do not depend on\nthe initial distribution [X0]of the process XinL2(P).\n(b)For rough volatility models with Hurst constant H∈(0,1\n2), the condition on ain(SSL)σ,areads\na∈(0,2H), excluding the diffusion coefficients with linear growth. For diffusion coefficients with\nlinear growth one should check (SL)σ, in particular the condition c <1/(2[σ]2\nLip√p)forX0∈Lp,\np >2.\nProof. We check that2\na(\u0000β−1\n2β)∧ϑ\u0001\n>1 if 0 < ϑ < α −1\n2and 1 < β <1\n2(1−α), then\nlim\nϑ→α−1\n2,β→1\n2(1−α)2(\u0000β−1\n2β)∧ϑ\u0001\n= 2α−1. For claim ( b) juste note that 2 α−1 = 2 H. 2\n5.1 Computing the function ς2\nα,λ,csolution to Equation (3.27) when α∈(1\n2,1)\nIn this section we want de compute ςλ,cas a power series in tkα. To this end we rely on the\nLaplace version (3.27) of the equation satisfied by ς2\nλ,c, namely\n∀t >0, t Lf2\nλ(t).Lς2(t) = 2 cλ2LRλfλ(t).\nGiven the form of the kernel Kα(u) =uα−1\nΓ(α)and the expansion of the resolvents Rλ(we drop\nthe dependence in αfor simplicity) and it derivative −fλ, we check that\nRλfλ(t)0∼λtα−1\nΓ(α)and f2\nλ(t)0∼λ2t2(α−1)\nΓ(α)2\nso that – at least heuristically (4) –\nLRλfλ(t)+∞∼λt−αand Lf2\nλ(t)+∞∼λ2Γ(2α−1)t−(2α−1)\nΓ(α)2.\nThis implies that\nLς2(t)+∞∼2λ cΓ(α)2\nΓ(2α−1)t−(2−α)\nowing to Equation (3.27). This in turn suggests that\nς2(t)0∼2λcΓ(α)2\nΓ(2α−1)Γ(2−α)t1−α(so that ς(0) = 0 since α <1).\nThis suggests to search ς2(t) of the form\nς2(t) =ς2\nα,λ,c(t) := 2 λ c t1−αX\nk≥0(−1)kckλktαk. (5.38)\n4. We use here heuristically a dual version of the Hardy-Littlewood Tauberian theorem for Laplace transform,\nnamely ς2(t)0∼Ctγ,γ >−1, iff Lς2(t)+∞∼Ct−(γ+1)Γ(γ+ 1).\n28with c0=Γ(α)2\nΓ(2α−1)Γ(2−α).\nIt is important to note at this stage that, αbeing fixed, all functions ς2\nα,λ,cfrom (5.38) are\ngenerated by the same function\nς2\nα(t) := 2 t1−αX\nk≥0(−1)kcktk(5.39)\n(with in mind that the ckdepend on α) by the formula\nς2\nα,λ,c(t) =cλας2\nα(λtα). (5.40)\nLet us establish a a recurrence formula satisfied by the coefficients ckwhich make possible the\ncomputation of the functions ςα,λ,c. To this end, it is convenient (but not mandatory) to switch to\nLaplace transforms.\nFirst we can rewrite the expansions (1.13) and (1.14) that define Rλandfλas\nRλ(t) =X\nk≥0(−1)kakλktαkwith ak=1\nΓ(αk+ 1), k≥0,\nand\nfλ(t) =λ tα−1X\nk≥0(−1)kbkλktαkwith bk=1\nΓ(α(k+ 1)), k≥0.\nThen, using that Luγ(t) =t−(γ+1)Γ(γ+ 1),\nLRλfλ(t) =λt−αX\nk≥0(−1)k(a∗b)kλkt−αkΓ(α(k+ 1))\nand\nLf2\nλ(t) =λ2t−2α+1X\nk≥0(−1)k(b∗2)kλkt−αkΓ(α(k+ 2)−1)\nwhere, for two sequences of real numbers ( uk)k≥0and ( vk)k≥0, (u∗v)k=Pk\nℓ=0uℓvk−ℓ. Set\nebk= (b∗2)kΓ(α(k+ 2)−1) and eck=ckΓ(α(k−1) + 2) , k≥0.\nThen if ς2(t) has the expected form (5.38), one has\nLς2(t) = 2 λ ctα−2X\nk≥0(−1)keckλkt−αk\nso that Equation (3.27) reads, by identification of the coefficients of the expansions on both sides\nof the equation,\n∀k≥0,(eb∗ec)k= (a∗b)kΓ(α(k+ 1)) .\nNote that this equation does not depend on λnorcso that the coefficients ckare intrinsic in the\nsense that they only depend on α(orH). Elementary computations confirm that\nc0=Γ(α)2\nΓ(2α−1)Γ(2−α)\n29and show that, for every k≥1,\nck=Γ(α)2\nΓ(α(k−1) + 2)Γ(2 α−1)\"\nΓ(α(k+ 1))( a∗b)k−kX\nℓ=1Γ(α(ℓ+ 2)−1)Γ(α(k−ℓ−1) + 2)( b∗2)ℓck−ℓ#\n.\n(5.41)\nUsing the classical identities Γ( a)Γ(b) = Γ( a+b)B(a, b),a, b > 0 where B(a, b) =R1\n0ua−1(1−\nu)b−1duand Γ( a+ 1) = aΓ(a), one gets the alternative formulation for the ckwhich is much more\nappropriate for numerical computations: for every k≥1,\nck=Γ(α)2Γ(α(k+ 1))\nΓ(2α−1)Γ(αk+ 2−α)\"\n(a∗b)k−α(k+ 1)kX\nℓ=1B\u0000\nα(ℓ+ 2)−1, α(k−ℓ−1) + 2\u0001\n(b∗2)ℓck−ℓ#\n.\n(5.42)\nFollowing classical techniques already used in [4] among many others, one shows that the con-\nvergence radius ρ=\u0000\nlim inf n\u0000\n(cn)1/n/λ\u0001\u0001−1/αof the power series defined by the ckis infinite.\nThis is done in Proposition A.1 in Appendix A. This does not prove yet the existence of a solution\nsince he question of the positiveness of the resulting alternate series involved in the definition of the\nfunction(s) ς2on the whole positive real line remains open (beyond a right neighbourhood of 0).\nExtensive Numerical tests (illustrated here by Figures 5.1 and 5.1 suggests that such is the case: ς\nis a positive function on the whole positive real line. Proving non-negativity rigorously is clearly\nthe next step of this work.\nLast but not least, we need to check that the resulting candidate to be ς2is non-negative which\nis usually not easy given the fact the function ς2is defined by a series whose terms have alternate\nsigns. Typical examples of graphs is displayed in Figures below with parameters H= 0.4 (α= 0.9,\nFigure 1) and H= 0.1 (α= 0.6, Figure 2), λ= 0.2,v0= 0.3,T= 10 (with 100 steps per unit of\ntime). These examples and many others strongly suggest that ς2is non-negative in our framework\n(boundedness follows from the continuity of ςand its finite limit at infinity).\nFigure 1 – Graph of t7→ςα,λ,c(t)over time interval [0, T],T= 10,H= 0.4,c= 0.3,λ= 0.2.\n30Figure 2 – Graph of t7→ςα,λ,c(t)over time interval [0, T],T= 10,H= 0.1,c= 0.3,λ= 0.2.\n5.2 A numerical illustration\nWe consider the scaled Volterra equation (2.17) with kernel K=Kα, mean-reverting coefficient\nλ >0, long run meanµ0\nλand diffusion coefficient σ(t, x) given – as in Proposition 4.1 – by\nσ(t, x) =ς(t)r\nκ1+κ2\u0000\nx−µ0\nλ\u00012, κ 1, κ2>0,\nwhere X0satisfies EX0=µ0\nλ, Var( X0) =v0>0 and the stabilizer ςis solution to (3.26) with\nc=v0\nκ1+v0κ2. Then σis√κ2-Lipschitz and cκ2=v0κ2\nκ1+v0κ2<1 since κ1>0.\nWe want to illustrate the fact that the variance of the resulting Volterra process has constant\nvariance.\nWe introduce an Euler scheme with stepT\nnof the semi-integrated form (2.22) of the equation\n(with µ(s) =µ0and noting thatRt\n0fλ(s)ds= 1−Rλ(t)) namely, for every k= 0, . . . , n ,\n¯Xtk+1=µ0\nλ+\u0000\nX0−µ0\nλ\u0001\nRλ(tk) +1\nλkX\nℓ=1fλ(tk−tℓ−1)\u0010\nκ1+κ2\u0000¯Xtℓ−1−µ0\nλ\u00012\u00111/2\n(Wtℓ−Wtℓ−1)\nwhere tk=kT\nn,k= 0, . . . , n .\nWe set numerical values of the parameters of the Volterra equation to\nH= 0.4, µ0= 2, v 0= 0.09, λ = 1.2, κ 1= 0.25, κ 2= 0.384\nso that c≃0.3163 and cκ2≃0.1215. Finally we specified X0asX0∼ N\u0000µ0\nλ;v0\u0001\n.\nWe set the number of time steps at n= 1000 and the horizon time at T= 1. We performed a\nMonte Carlo simulation of successive size M= 100 000.\nWe depicted in Figure 3 the empirical variances of the process at each discretization instant,\nnamely\ntk7−→Var(tk, M)\n31of this simulation where\nvar(tk, M) =1\nMMX\nm=1\u0000¯X(m)\ntk−µ0\nλ\u00112\n, k = 0, . . . , n.\nwhere X(m),m= 1, . . . , M are i.i.d. copies of the Euler scheme. (We are aware that this is not the\nusual unbiased estimation of the variance in a Monte Carlo simulation).\nFigure 3 – Graph of tk7−→p\nvar(tk, M)over the time interval [0, T],T= 1,H= 0.4,µ0= 0.2,\nv0= 0.09,λ= 1.2,κ1= 0.25,κ2= 0.384. Number of steps: n= 1 000 , Simulation size:\nM= 100 000 —.\nNote that, in beyond the clear stabilization effect, one observes a significant numerical instability\nin spite of the fact that we considered H= 0.4 (and not H= 0.1) in this simulation.\nAcknowledgement: I thank N. Fournier, A. Pannier and M. Rosenbaum for helpful discussions.\nReferences\n[1] Aur ¨Alien Alfonsi. Nonnegativity preserving convolution kernels. application to stochastic volterra equa-\ntions in closed convex domains and their approximation, 2023.\n[2] Christian Bayer, Peter Friz, and Jim Gatheral. Pricing under rough volatility. Quant. Finance ,\n16(6):887–904, 2016.\n[3] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation , volume 27 of Encyclopedia of\nMathematics and its Applications . Cambridge University Press, Cambridge, 1989.\n[4] Giorgia Callegaro, Martino Grasselli, and Gilles Pag` es. Fast hybrid schemes for fractional Riccati\nequations (rough is not so tough). Math. Oper. Res. , 46(1):221–254, 2021.\n[5] Eric Carlen and Paul Kr´ ee. Lpestimates on iterated stochastic integrals. Ann. Probab. , 19(1):354–368,\n1991.\n[6] F. Comte and E. Renault. Long memory continuous time models. J. Econometrics , 73(1):101–149,\n1996.\n[7] Fabienne Comte and Eric Renault. Long memory in continuous-time stochastic volatility models. Math.\nFinance , 8(4):291–323, 1998.\n32[8] Omar El Euch and Mathieu Rosenbaum. Perfect hedging in rough Heston models. Ann. Appl. Probab. ,\n28(6):3813–3856, 2018.\n[9] William Feller. An introduction to probability theory and its applications. Vol. II . John Wiley & Sons,\nInc., New York-London-Sydney, second edition, 1971.\n[10] Martin Friesen and Peng Jin. Volterra square-root process: Stationarity and regularity of the law, 2022.\n[11] Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. Volatility is rough. Quant. Finance ,\n18(6):933–949, 2018.\n[12] Jim Gatheral, Paul Jusselin, and Mathieu Rosenbaum. The quadratic rough Heston model and the\njoint S&P 500/VIX smile calibration problem. arXiv e-prints , page arXiv:2001.01789, January 2020.\n[13] Rudolf Gorenflo and Francesco Mainardi. Fractional calculus: integral and differential equations of\nfractional order. In Fractals and fractional calculus in continuum mechanics (Udine, 1996) , volume 378\nofCISM Courses and Lect. , pages 223–276. Springer, Vienna, 1997.\n[14] Eduardo Abi Jaber, Martin Larsson, and Sergio Pulido. Affine volterra processes. The Annals of Applied\nProbability , 2017.\n[15] Antoine Jacquier, Alexandre Pannier, and Konstantinos Spiliopoulos. On the ergodic behaviour of affine\nVolterra processes. arXiv e-prints , page arXiv:2204.05270, April 2022.\n[16] Thibault Jaisson and Mathieu Rosenbaum. Rough fractional diffusions as scaling limits of nearly\nunstable heavy tailed Hawkes processes. Ann. Appl. Probab. , 26(5):2860–2882, 2016.\n[17] Benjamin Jourdain and Gilles Pag` es. Convex ordering for stochastic Volterra equations and their Euler\nschemes. arXiv e-prints , pages arXiv:2211.10186, to appear in Fin. & Stoch. , November 2022.\n[18] Vincent Lemaire, Thibaut Montes, and Gilles Pag` es. Stationary Heston model: calibration and pricing\nof exotics using product recursive quantization. Quant. Finance , 22(4):611–629, 2022.\n[19] Francesco Mainardi. On some properties of the Mittag-Leffler function Eα(−tα), completely monotone\nfort >0 with 0 < α < 1.Discrete Contin. Dyn. Syst. Ser. B , 19(7):2267–2278, 2014.\n[20] Francesco Mainardi and Rudolf Gorenflo. Fractional calculus: special functions and applications. In\nAdvanced special functions and applications (Melfi, 1999) , volume 1 of Proc. Melfi Sch. Adv. Top. Math.\nPhys. , pages 165–188. Aracne, Rome, 2000.\n[21] Gilles Pag` es and Fabien Panloup. Approximation of the distribution of a stationary Markov process\nwith application to option pricing. Bernoulli , 15(1):146–177, 2009.\n[22] Harry Pollard. The completely monotonic character of the Mittag-Leffler function Ea(−x).Bull. Amer.\nMath. Soc. , 54:1115–1116, 1948.\n[23] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion , volume 293 of Grundlehren\nder mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-\nVerlag, Berlin, third edition, 1999.\n[24] Alexandre Richard, Xiaolu Tan, and Fan Yang. Discrete-time simulation of stochastic Volterra equa-\ntions. Stochastic Process. Appl. , 141:109–138, 2021.\n[25] Xicheng Zhang. Stochastic Volterra equations in Banach spaces and stochastic partial differential\nequation. J. Funct. Anal. , 258(4):1361–1425, 2010.\nA More about ζ2\nα\nThe aim of this additional section is to prove that the power series involved in the defini-\ntion (5.39) of the function ζ2\nα(t) = 2 t1−αP\nk≥0cktkhas an infinite convergence radius. We entirely\n33rely on the definition by induction (5.42) of the coefficients ckand the definitions for the sequences\nak=1\nΓ(αk+1)andbk=1\nΓ(α(k+1)),k≥0. By the triangle inequality we get the bound\n|ck| ≤Γ(α)2Γ(α(k+ 1))\nΓ(αk+ 2−α)Γ(2α−1)\"\n(a∗b)k+α(k+ 1)kX\nℓ=1B\u0000\nα(ℓ+ 2)−1, α(k−ℓ−1) + 2\u0001\n(b∗2)ℓ|ck−ℓ|#\n.\n(A.43)\nWe will extensively use the standard identities, for a,b >0:\nΓ(a+ 1) = aΓ(a), B (a, b) :=Γ(a)Γ(b)\nΓ(a+b)\nProposition A.1 For every α∈(1/2,1), the convergence radius of the power seriesP\nk≥0cktkis\ninfinite. To be more precise, there exists K≥1\nΓ(2−α)andA≥2αsuch that\n∀k≥0,|ck| ≤KAk\nΓ(αk+ 2−α).\nAs a consequence the expansion (5.38) holds for every t∈R+(in fact R).\nLemma A.2 Letα∈(1/2,1). For every k≥1,\n—(a∗b)k≤2αk\nΓ(α(k+ 1))\u0000\n1 + (k+ 1)(1 + log k)\u0001\n.\n—b∗2\nk≤α2αk\nΓ(α(k+ 2))(k+ 1)2.\nProof . Using the identity linking the Beta function and the Γ function, we have for every k≥1\n(a∗b)k=kX\nℓ=01\nΓ(αℓ)Γ(α(k−ℓ+ 1))\n=1\nΓ(α(k+ 1))\u0012\n1 +kX\nℓ=11\nℓ1\nB(αℓ, α(k−ℓ+ 1))\u0013\n.\nIfℓ= 1 or k,B(αℓ, α(k−ℓ+ 1)) = B(α, αk ) =R1\n0uα−1(1−u)αk−1du≥1\nαk.\nIf 2≤ℓ≤k−1\nB(αℓ, α(k−ℓ+ 1)) =Z1/2\n0uαℓ−1(1−u)α(k−ℓ+1)du=Z1/2\n0. . . du +Z1\n1/2. . . du\n≥1\n2α(k+1)−1\u00101\nαℓ+1\nα(k−ℓ+ 1)\u0011\n≥1\nα2α(k+1)8\nk+ 1\n≥1\nα2α(k+1)4\nk+ 1=1\nα2α(k+1)−21\nk+ 1.\n34The last downgraded inequality is also satisfied by when ℓ= 1, k. Hence\n(a∗b)k≤1\nΓ(α(k+ 1))\u0012\n1 +α2α(k+1)−2(k+ 1)kX\nℓ=11\nℓ\u0013\n≤1\nΓ(α(k+ 1))\u0012\n1 +α2α(k+1)−2(k+ 1)kX\nℓ=11\nℓ\u0013\n≤2αk\nΓ(α(k+ 1))\u0000\n1 + (k+ 1)(1 + log k)\u0001\nsince α2α−1<1 when α∈(0,1).\nFork= 0,c0=Γ(α)2Γ(2−α)\nΓ(2α−1)≤K\nΓ(2−α). Now let k≥1.\nLet us deal now with b∗2\nkfollowing the same principle.\nb∗2\nk=1\nΓ(α(k+ 2))kX\nℓ=01\nB(α(ℓ+ 1), α(k−ℓ+ 1).\nIfℓ= 0, k,\nB(α(ℓ+ 1), α(k−ℓ+ 1) = B(α, α(k+ 1)) =Z1\n0uα−1(1−u)α(k+)−1du≥1\nα(k+ 1)\nand, if 1 ≤ℓ≤k−1,\nB(α(ℓ+ 1), α(k−ℓ+ 1)≥k+ 2\nα2α(k+2)−1(ℓ+ 1)( k−ℓ+ 1)≥4\nα2α(k+2)−3(k+ 2).\nHence, for every k≥1,\nb∗2\nk≤1\nΓ(α(k+ 2))\u0012\n2α(k+ 1 + α2α−3k−1X\nℓ=12αk(k+ 2)\u0013\n=α2αk(k+ 1)2\nΓ(α(k+ 2))\u00122\n(k+ 1)2αk+ 2α−3(k+ 1)( k+ 2)\n(k+ 1)2\n| {z }\n=:φα(k)\u0013\n.\nOne checks that supk≥1φα(k)<2−1\n2+ 2−2<1 for every α∈(1/2,1) which finally entails that, for\nevery k≥1,\nb∗2\nk≤α2αk\nΓ(α(k+ 2))(k+1)2. 2\nProof of Proposition A.1 . Let us prove now by induction that there exists A > 2α,K >\n|c0|Γ(2−α) such that\n∀k≥0,|ck| ≤KAk\nΓ(αk+ 2−α).\n35Assume cℓsatisfies this inequality or every ℓ= 0, . . . , k −1. Then, for every ℓ= 1, . . . , k ,\nb∗2\nℓ|ck−ℓ| ≤Kα2αℓ(ℓ+ 1)2Ak−ℓ\nΓ(α(ℓ+ 2))Γ( α(k−ℓ) + 2−α)\nso that, noting that Γ( α(k−ℓ) + 2−α) = Γ( α(k−ℓ−1) + 2),\nB(α(ℓ+ 2)−1, α(k−ℓ−1) + 2)) b∗2\nℓ|ck−ℓ| ≤Kα2αℓ(ℓ+ 1)2Ak−ℓ\nα(ℓ+ 2)Γ( α(k+ 1) + 1)\n≤K2αℓ(ℓ+ 1)Ak−ℓ\n(k+ 1)Γ( α(k+ 1)).\nPlugging this bound into (A.43) yields\n|ck| ≤Γ(α)2\nΓ(2α−1)Γ(αk+ 2−α)\u0012(a∗b)k\nα(k+ 1)+K AkkX\nℓ=1\u00102α\nA\u0011ℓ\n(ℓ+ 1)\u0013\n.\nUsing the elementary inequality\n∀ρ∈(0,1),X\nn≥1nρn−1≤1\n(1−ρ)2\nwith ρ=ρ(A) := 2α/A, dividing the above inequality by Akand using the upper bound for ( a∗b)k\nfrom Lemma A.2 we get\n|ck|\nAk≤Γ(α)2\nΓ(2α−1)Γ(αk+ 2−α)\u0012\nρkα(1 + log k) +K1\n(1−ρ)2\u0013\n.\nLetε >0 and let A=Aα,εbe large enough so that\nsup\nk≥1ρk(k+ 1)(1 + log k)<ε\nαand1\n(1−ρ)2<1 +ε\nthen\n|ck|\nAk≤Γ(α)2\nΓ(2α−1)Γ(αk+ 2−α)\u0012\nε+K(1 +ε)\u0013\n=K\nΓ(αk+ 2−α)Γ(α)2\nΓ(2α−1)·\u0012ε\nK+ 1 + ε\u0013\n.\nTo establish the heredity inequality we need to prove that\nΓ(α)2\nΓ(2α−1)<1\nsince then it is possible to choose εsmall enough and Klarge enough so thatΓ(α)2\nΓ(2α−1)·\u0000ε\nK+1+ε\u0001\n<1.\nTaking advantage of the log-convexity of the Γ-function we have\nlog Γ( α)≤1\n2\u0000\nlog Γ(2 α−1) + log Γ(1)\u0001\n=1\n2log Γ(2 α−1)\nwhich implies the above inequality.\nFinally note that\nlim\nk|ck|1\nk≤lim\nk\u0012A\nΓ(αk+ 2−α)\u00131/k\n= 0\nsince Γ( αk+ 2−α)1/k∼e−α(αk+ 2−α)αask→+∞owing to Stirling’s formula. 2\n36" }, { "title": "2401.15034v1.Explicit_Subcodes_of_Reed_Solomon_Codes_that_Efficiently_Achieve_List_Decoding_Capacity.pdf", "content": "arXiv:2401.15034v1 [cs.IT] 26 Jan 2024Explicit Subcodes of Reed–Solomon Codes that\nEfficiently Achieve List Decoding Capacity\nAmit Berman1, Yaron Shany1, and Itzhak Tamo2,1\n1Samsung Semiconductor Israel R&D Center, 146 Derech Menach em\nBegin St., Tel Aviv 6492103, Israel. Emails: {amit.berman,\nyaron.shany }@samsung.com\n2Department of Electrical Engineering-Systems, Tel Aviv Un iversity, Tel\nAviv 6997801, Israel. Email: zactamo@gmail.com\nJanuary 29, 2024\nAbstract\nIn this paper, we introduce a novel explicit family of subcod es of Reed-Solomon\n(RS) codes that efficiently achieve list decoding capacity wi th a constant output\nlist size. Our approach builds upon the idea of large linear s ubcodes of RS codes\nevaluated on a subfield, similar to the method employed by Gur uswami and Xing\n(STOC 2013). However, our approach diverges by leveraging t he idea of permuted\nproduct codes , thereby simplifyingthe construction by avoiding the need ofsubspace\ndesigns.\nSpecifically, the codes are constructed by initially formin g the tensor product\nof two RS codes with carefully selected evaluation sets, fol lowed by specific cyclic\nshifts to the codeword rows. This process results in each cod eword column being\ntreated as an individual coordinate, reminiscent of prior c apacity-achieving codes,\nsuch as folded RS codes and univariate multiplicity codes. T his construction is\neasily shown to be a subcode of an interleaved RS code, equiva lently, an RS code\nevaluated on a subfield.\nAlternatively, the codes can be constructed by the evaluati on of bivariate poly-\nnomials over orbits generated by twoaffine transformations with coprime orders,\nextendingtheearlier useof a singleaffine transformation in folded RScodes andthe\nrecent affinefolded RScodesintroducedbyBhandari et al.(IEEET-IT,Feb. 2024).\nWhileourcodesrequirelarge, yet constant characteristic , thetwo affinetransforma-\ntions facilitate achieving code length equal to the field siz e, without the restriction\nof the field being prime, contrasting with univariate multip licity codes.\n1 Introduction\nError-correcting codes are used for reliably transmitting data ov er noisy communication\nchannels. To achieve this goal, a code C⊆Σn(for some alphabet Σ andlengthn) is\n1typically a proper subset of Σn, consisting only of |Σ|kcodewords for some k < n. We\nthen say that the code has rateR:=k/n, and one of the fundamental goals of coding\ntheory is to maximize the minimum (Hamming) distance between codewo rds for a given\nrateR.\nFor a code of minimum distance dandnormalized distance δ:=d/n, the transmitted\ncodewordiscompletelydeterminedfromthenoisychanneloutputif thelatterhaserrorsin\nlessthanafractionof δ/2coordinates, whileforalargerfractionoferrors, thetransmitt ed\ncodeword is in general not uniquely determined. Since the Singleton bound implies that\nδ≤1−R, the maximum possible guaranteed unique decoding radius of a code of rate R\nis therefore (1 −R)/2, and this is achieved by Reed–Solomon (RS) codes.\nAs originally suggested by Elias [Eli57] and Wozencraft [Woz58], to go be yond the\nunique decoding radius, the decoder must be allowed to output a listof potential trans-\nmitted codewords. We say that a code Cis (ρ,L)-list decodable if for any received word\nw, there are at most Lcodewords c∈Csuch that cdisagrees with win at most a fraction\nρof the coordinates.\nIn a breakthrough work, Sudan [Sud97] presented the first polyn omial-complexity\nalgorithm for list-decoding RS codes beyond (1 −R)/2 for low R. This was later consid-\nerably improved by Guruswami and Sudan [GS99], who presented an e fficient algorithm\nfor decoding RS codes up to the Johnson radius 1−√\nRwith a polynomial list size. How-\never, it is well-known that the maximum list-decoding radius with a guar anteed constant\nlist size is much higher than the Johnson radius; a random coding argu ment (see, e.g.,\n[GRS19]) shows that a ( ρ,L)-list decodable code with constant Lmay be achieved for\nanyρ <1−R(for a large enough alphabet), while it is clear that for ρ >1−R, the list\nsizeLmust be exponential. Hence, ρ= 1−Ris called the list decoding capacity .\nGuruswami and Rudra [GR08] presented the construction of folded RS codes (FRS),\nthe first explicit family of codes that achieve list decoding capacity. F RScodes are closely\nrelated to RS codes; they are obtained by “folding” cyclic RS codewo rds into the shape\nof matrices, where each column is considered as a single coordinate. In fact, not only that\nFRS codes achieve list decoding capacity, they achieve capacity efficiently , i.e., there is a\ndeterministic algorithm that can decode up to a fraction of 1 −R−εerrors for any ε >0,\nwith list size and complexity polynomial in the code length. In a later wor k, Guruswami\nand Wang [GW13] presented a simple linear-algebraic decoding algorith m for FRS codes,\nand proved that the output list is contained in a subspace of dimensio nO(1/ε). Hence,\ndespite the considerable simplification of the decoding algorithm, the list size remained\npolynomial in the code length. To achieve a constant output list size, Guruswami and\nWang proposed in [GW13] the use of pre-encoding with a combinatoria l stucture they\ncalledsubspace evasive sets . They also constructed these using a probabilistic argument.\nSubsequently, Dvir and Lovett [DL12], gave an explicit construction of subspace evasive\nsets.\nRecently, Kopparty et al.[KRZSW23] revealed that the list size of FRS codes them-\nselves, as well as that of other capacity achieving codes, such as univariate multiplicity\ncodes(originally introduced by Rosenbloom and Tsfasman [RT97]; see also [G W13] and\n[KRZSW23]) is in fact constant, even without using a pre-encoding st ep. Furthermore,\nit was shown in [KRZSW23] that the constant list size can be achieved e fficiently with a\nrandomized algorithm. Lastly, the list-size bounds of [KRZSW23] wer e further tightened\nin [Tam23].\n2We comment that while folded RS codes univariate and multiplicity codes have a\npolynomial alphabet size, there are several papers, culminating in [G R22], that use con-\nstructions based on algebraic-geometry (AG) codes to obtain cap acity-achieving codes\nwith both constant list size andconstant alphabet size; see [GR22] and the references\ntherein. Since this is outside the main scope of the current paper, w e will not further\nelaborate on this subject; the interested reader is referred to t he introduction of [GR22]\nfor a comprehensive account.\nAlthough folded RS codes are closely related to RS codes, they are in general not RS\ncodes themselves, neither (informally) large subcodes of RS codes . Moreover, although\nit is known that for an appropriate choice of the evaluation set , RS codes achieve list de-\ncoding capacity combinatorially [BGM23], [GZ23], [AGL23] (withexponential, quadratic,\nand linear finite-field size, resp.), the evaluation sets in these works are not explicit, and\nthere is no known efficient algorithm for decoding these RS codes up t o capacity. It is\ntherefore natural to ask:\nQ1What is an explicit evaluation set, if exists, for which an RS code can be efficiently\nlist decoded up to capacity?\nAsQ1appearstobeahardquestion, itisalsoofinterest toconsider thefollowingsimpler,\nyet non-trivial, question:\nQ2What is an explicit evaluation set for which an (informally) large linear su bcode of\nan RS code can be efficiently decoded up to the list-decoding capacity ?\nAn answer given in [GX13] to Q2 is as follows. Start with an evaluation s et that is a\nsubfield of the finite field over which the RS code is defined. While the re sulting RS code\nitself has an exponential list size (albeit with a smaller exponent than t he trivial one), it\nturns out that the list of coefficients of potential information polyn omials is a so-called\n“periodic subspace”. This fact is then used in [GX13] to show that pr e-encoding each\ncolumn of the information matrix with a different subspace from a subspace design (also\ndefined in [GX13]1) results in an efficient capacity achieving code. Also, since the above\nsubspaces have a small (informally) co-dimension, this indeed result s in a large linear\nsubcode of an RS code.\n1.1 Results and methods\nOur main contribution is a new and simple answer to Q2. In detail, similarly to [GX13],\nwe construct a large subcode of an RS code evaluated on a subfield, and that efficiently\nachieves list decoding capacity. However, our construction does n ot require the rather\ninvolved concepts of periodic subspaces and subspace designs. In stead, it is based on the\nwell-known and simple construction of (tensor) product codes.\nInformally, we start with the product of two RS codes over Fq,2and coprime lengths\nmandn, where the row code of length nis cyclic. We then apply cyclic shifts to the rows\n1Besides defining subspace designs, Guruswami and Xing [GX13] also g ave a randomized construction\nof subspace design. An explicit construction was later found by Gur uswami and Kopparty [GK16].\n2Fqis the finite field of qelements, whose characteristic is large enough. See Section 4.3.\n3of the resulting codewords, where row iis cyclically shifted to the left by a·icoordinates,\nwitha≡m−1mod (n). We refer to the resulting code as a permuted product code .\nThe permuted product code is obviously a subcode of the interleave d RS code defined\nby requiring only that all rows are in the cyclic row code. Moreover, it is well-known that\nthelatter interleaved codeis anRScodeover Fqmwith anevaluationset in Fq, where each\nentry of a codeword is replaced by the column vector of coefficients in its representation\naccording to some basis of Fqm/Fq. Hence showing that, indeed, the permuted product\ncode is a subcode of an RS code. Our main result is the following theore m.\nTheorem (Informal, see Theorem 4.12 below) .ForR∈(0,1), small enough ε >0,\nand all powers qof a prime p=O(1/ε3), there are instances of the permuted product\ncode over Fqwith alphabet size qO(1/ε3), rateRand block length q−1, that are efficiently\nlist-decodable from error fraction 1−R−ε, with an output list of size (1/ε)O(1/ε2).\nAlternatively, the permuted product code can be constructed by evaluating bivariate\npolynomials on orbits of two elements under the action of twoaffine transformations of\ncoprimeorders. This extends theprevious usageofasingle affinetr ansformationinfolded\nRS codes, additive folded RS codes, and, more generally, in the rece nt affine folded RS\ncodes of Bhandari et al.[BHKS24]. While folded RS codes cannot reach a length that\nis as large as the size of the underlying finite field Fq(as the code length is shorter by\na factor of the folding length), the current construction can rea ch a length of q−1. We\nnote that this is also possible with univariate multiplicity codes. Howeve r, the usage of\ntwo affine transformations enables reaching a length that is as large as the finite-field size\nfor all rates (albeit with high, yet constant, characteristic), with a finite field that needs\nnot be prime.3\nTable 1 below compares the parameters of the permuted product c odes of this work\nwith those of some other polynomial-based capacity achieving codes . We note that while\nthe parameters of the construction of [GX13] are better, the ma in contribution of the\ncurrent construction is not in its parameters, but rather that it p rovides a simpleanswer\nto Q2, constructing capacity achieving subcodes of RScodes, usin g basic coding-theoretic\nconcepts.\nTo summarize, our main contributions are as follows:\n•We construct a new and simple large subcode of an RS code that efficie ntly achieves\nlist-decoding capacity.\n•We show that up to some cyclic shifts, the product of RS codes can b e used to\nachieve efficient capacity-achieving codes (where each column of a c odeword is re-\ngarded as a coordinate). This provides a new method for construc ting capacity\nachieving codes.\n•We show how twoaffine transformations can be used to define capacity-achieving\ncodes whose length is as large as the underlying finite field.\n3Note that univariate multiplicity codes have an exponential list size if t he dimension is sufficiently\nlarger than the characteristic.\n4Code Field size, q logq(|Σ|)List size bound\nFolded RS codes q=O(n/ε2) O(1/ε2) (1/ε)4/ε\nUnivariate\nmultiplicity codes\nof dimension d+1q=pr(r∈N∗)\np≥Ω(1/ε) prime\nq≥nO(1/ε2)(1/ε)4\nε/parenleftig\n1+d\np/parenrightig\n[GX13]\nInterleaved RS\ncodes +\nsubspace designsq≥Ω(1/ε2)\nq≥nO(1/ε2)(1/ε)O(1/ε2)\nPermuted\nproduct codes\n(this work)q=pr(r∈N∗)\np≥Ω(1/ε3) prime\nq > nO(1/ε3)(1/ε)O(1/ε2)\nTable 1: Some capacity achieving list-decodable codes of rate R, lengthn, and alphabet\nΣ that are list decodable from 1 −R−εfraction of errors. The list size bounds are\nobtained via [Tam23]. Specifically, the list size bound for FRS codes is fr om [Tam23,\nCorollary 3.6], the bound for univariate multiplicity codes is from [Tam23 , Theorem 3.8],\nwhile the other two bounds on the list size are obtained by using [Tam23 , Lemma 3.1]\nwith [GK16, Theorem 23] and Theorem 4.12 below. Note that by the list size bound for\nunivariate multiplicity codes, the rate is positive and the list size bound is constant only\nif the code length is O(p).\n1.2 Organization\nIn Section 2 we provide some required definitions and notations. In S ection 3 we define\nthe permuted product codes and prove that they are indeed subc odes of an RS code\nwith an appropriate evaluation set. Then, a linear-algebraic list deco ding algorithm up\nto capacity is presented in Section 4. Finally, Section 5 includes some o pen questions\nfor further research. The paper is supplemented by an appendix, in which we study the\nproperties of the “unfolded” code as a cyclic code.\n2 Preliminaries\nThis section includes some definitions and notation that will be used th roughout the\npaper.\nWe write Fqfor the finite field of qelements, where qis a prime power. Throughout,\nwe fix a prime pandqa power of p. Unless otherwise noted, all vectors are row vectors.\nAlso, (·)Tstands for matrix transposition.\n2.1 Reed–Solomon codes and their tensor products\nFor integers 0 ≤k≤n≤qand a set A={a0,...,a n−1} ⊆Fq, theReed–Solomon (RS)\ncodeRSFq(k,A) with evaluation set Aand dimension k, is defined as\nRSFq(k,A) :=/braceleftbig/parenleftbig\nf(a0),...,f(an−1)/parenrightbig/vextendsingle/vextendsinglef∈Fq[x],deg(f)< k/bracerightbig\n⊆Fn\nq.\nFor simplicity, when the underlying finite field is clear from the context , we will write\nsimplyRS(k,A) forRSFq(k,A).\n5Next, it will be useful to recall the definition of the tensor product of two RS codes.\nThetensor product C2⊗C1of linear codes C1,C2⊆Fn\nqof dimensions k1,k2(resp.) is the\nspace of matrices whose columns are in C2and whose rows are in C1. It can be verified\nthatC2⊗C1is indeed a tensor product: it is generated as an Fq-space by the outer\nproducts c2⊗c1:=cT\n2c1forci∈Ci,i= 1,2. In particular, it can be verified that if\n{b′\n1,...,b′\nk2},{b1,...,bk1}are bases for C2andC1(resp.), then {b′\ni⊗bj}i,jis a basis for\nC2⊗C1. It follows that dim( C2⊗C1) =k1k2, and it is easily verified that if the minimum\ndistances of C1andC2ared1,d2(resp.), then the minimum distance of C2⊗C1isd1d2.\nIt also follows that for two sets A={ai},B={bj} ⊆Fq, and non-negative integers\ns≤ |A|,t≤ |B|,\nRS(s,A)⊗RS(t,B) =/braceleftig\n{f(ai,bj)}i∈{0,...,|A|−1}\nj∈{0,...,|B|−1}∈F|A|×|B|\nq/vextendsingle/vextendsingle/vextendsinglef∈Fq[x,y],degx(f)< s,degy(f)< t/bracerightig\n.\n2.2 The affine group\nTheaffine group GA(q) is the group whose underlying set is {ax+b|(a,b)∈F∗\nq×Fq} ⊂\nFq[x], while the group operation is polynomial composition: for ℓi:=aix+bi∈GA(q)\n(i= 1,2),ℓ2◦ℓ1:=ℓ2(ℓ1(x)) =a2a1x+a2b1+b2. It can be verified that this is indeed a\ngroup, with identity element x, and inverse ( ax+b)−1=a−1x−a−1b. Forℓ(x)∈GA(q)\nandi∈N∗, we letℓi:=ℓ◦···◦ℓ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ni, andℓ0:=x. Theorderofℓ, ord(ℓ), is defined as usual\nas the smallest i∈N∗such that ℓi=x.\nIfℓ(x) =ax+bwitha/ne}ationslash= 1, then ℓi(x) =aix+bai−1\na−1, from which it is clear that\nord(ℓ) = ord(a), where ord( a) is the order of ainF∗\nq. In addition, if a= 1 and b/ne}ationslash= 0,\nℓi(x) =x+ib, so that ord( ℓ) =p. To conclude,\nord(ax+b) =\n\nord(a)a/ne}ationslash= 1\np a = 1,b/ne}ationslash= 0\n1a= 1,b= 0.(1)\nWe let GA( q) act on Fqin the obvious way, by setting ℓ·ζ:=ℓ(ζ) forℓ∈GA(q)\nandζ∈Fq. It is easily verified that if ζis not a fixed point of ℓ(i.e.ℓ(ζ)/ne}ationslash=ζ),\nthen the stabilizer of ζin the cyclic subgroup /an}⌊ra⌋ketle{tℓ/an}⌊ra⌋ketri}htgenerated by ℓis trivial, so that the\norbit{ℓ(ζ),···,ℓord(ζ)(ζ)}has ord(ℓ) distinct elements. This fact will be used frequently\nwithout further mention throughout the paper.\n2.3 The splitting field of xq−ax−b\nThe following properties of the splitting field of xq−ℓ(x) forℓ∈GA(q) will be useful\nahead.\nProposition 2.1. Letℓ(x) =ax+b∈GA(q). LetLbe the splitting field of h(x) :=\nxq−ℓ(x). Then[L:Fq] = ord(ℓ). Moreover, either (a,b) = (1,0)andL=Fq, orL/supersetnoteql Fq,\nandL=Fq(ζ)for any root ζofhoutsideFq.\n6Proof.First, ord( ℓ) = 1 if and only if a= 1 and b= 0, in which case L=Fq. Suppose,\ntherefore, that ( a,b)/ne}ationslash= (1,0). Since his separable,4monic, has degree q, and is not equal\ntoxq−x, its splitting field is not Fq. Letζ∈L/integerdivideFqbe a root of h.\nLetoℓ:= ord(ℓ). We claim that pζ(x) :=/producttextoℓ−1\ni=0/parenleftbig\nx−ℓi(ζ)/parenrightbig\nis the minimal polynomial\nofζoverFq. Clearly pζ(ζ) = 0. In addition, since ζ /∈Fq,ℓ(ζ)/ne}ationslash=ζ(for otherwise ζq=ζ,\nasζis a root of h), and therefore the roots {ℓi(ζ) =ζqi}oℓ−1\ni=0ofpζare distinct elements5\nin the orbit of ζunder the action of Gal( L/Fq). Actually, the roots of pζare anentire\norbit, as ζqoℓ\n=ℓoℓ(ζ) =ζ. This proves our claim.\nHence, for any root ζoutsideFq, [Fq(ζ) :Fq] =oℓ, so all these roots lie in the same\nfieldL=Fqoℓ, as required.\n3 Code construction\nIn this section, we first define the permuted product code as an ev aluation code. It then\nfollows almost immediately that the code is indeed a permuted product code, and that\nit is a subcode of an RS code.\nLetℓ1(x),ℓ2(x)∈GA(q) be two affine polynomials of coprime orders m,n, respec-\ntively, and let α,β∈Fqbe such that ℓ1(α)/ne}ationslash=α,ℓ2(β)/ne}ationslash=β, i.e.,α,βare not fixed points\nofℓ1andℓ2, respectively.\nForf∈Fq[x,y] andj∈ {0,...,n−1}, let\nevj(f) :=\nf(ℓjm\n1(α),ℓjm\n2(β))\nf(ℓjm+1\n1(α),ℓjm+1\n2(β))\n...\nf(ℓjm+m−1\n1(α),ℓjm+m−1\n2(β))\n∈Fm\nq,\nand let ev: Fq[x,y]→(Fm\nq)nbe the function that maps fto the vector whose j-th entry\nis the column vector ev j(f),j∈ {0,...,n−1}. Explicitly, ev( f) equals\n\n\nf(ℓ0\n1(α),ℓ0\n2(β))\nf(ℓ1(α),ℓ2(β))\n...\nf(ℓm−1\n1(α),ℓm−1\n2(β))\n,\nf(ℓm\n1(α),ℓm\n2(β))\nf(ℓm+1\n1(α),ℓm+1\n2(β))\n...\nf(ℓ2m−1\n1(α),ℓ2m−1\n2(β))\n, ...,\nf(ℓ(n−1)m\n1(α),ℓ(n−1)m\n2(β))\nf(ℓ(n−1)m+1\n1(α),ℓ(n−1)m+1\n2(β))\n...\nf(ℓnm−1\n1(α),ℓnm−1\n2(β))\n\n.\n(2)\nLets≤mandt≤nbepositiveintegers. Writing Fs,t\nq[x,y] :={f∈Fq[x,y]|degx(f)<\ns,degy(f)< t}, the permuted product code CPPC(s,t)⊆(Fm\nq)nis defined as\nCPPC(s,t) :=/braceleftbig\nev(f)|f∈Fs,t\nq[x,y]/bracerightbig\n.\nTo simplify notation, we will sometimes identify ( Fm\nq)nwithFm×n\nq, so that codewords\nofCPPC(s,t) will be regarded either as vectors of column vectors, or as m×nmatrices\nin the obvious way.\n4Ash′(x) =−a/ne}ationslash= 0 is coprime to h.\n5Recall that since ζis not a fixed point, its orbit under the action of /an}⌊ra⌋ketle{tℓ/an}⌊ra⌋ketri}hthasoℓelements.\n7Next, we would like to show that CPPC(s,t) is indeed a permuted product code.\nToward this end, let l2:=ℓm\n2, and note that ord( l2) isn, sincem,nare coprime. Also, β\nis not a fixed point of l2.6LetA:={ℓi\n1(α)}i∈{0,...,m−1},B:={lj\n2(β)}j∈{0,...,n−1}. Finally,\nlet [m]−1be a representative for the inverse of minZ/nZ. Then we have the following\nproposition.\nProposition 3.1. It holds that\nCPPC=π/parenleftbig\nRS(s,A)⊗RS(t,B)/parenrightbig\n,\nwhereπis the permutation that shifts row i∈ {0,...,m−1}to the left by i·[m]−1.\nProof.Letν∈ {0,...,mn −1}be a “folded running index” in an m×nmatrix, where\nfor a coordinate index ( i,j) (i∈ {0,...,m−1},j∈ {0,...,n−1}), we let ν:=mj+i.\nThe (i,j)-th entry of the codeword corresponding to f∈Fs,t\nq[x,y] is\nf/parenleftbig\nℓν\n1(α),ℓν\n2(β)/parenrightbig\n=f/parenleftbig\nℓi\n1(α),ℓm(j+[m]−1i)\n2 (β)/parenrightbig\n=f/parenleftbig\nℓi\n1(α),lj+[m]−1i\n2(β)/parenrightbig\n. (3)\nOntheotherhand, the( i,j)-thentryofthecodewordof RS(s,A)⊗RS(t,B)corresponding\ntof∈Fs,t\nq[x,y] isf/parenleftbig\nℓi\n1(α),lj\n2(β)/parenrightbig\n, and it is clear that (3) corresponds to the stated cyclic\nshifts of the rows.\nThe following corollary gives the basic parameters of the permuted p roduct code.\nCorollary 3.2. The code CPPC(s,t)is anFq-linear code of length n, rateR=st\nmnand\nminimum distance at least n−t+1.\nProof.All assertions followimmediately fromProposition3.1. Forexample, fo ranonzero\nc∈CPPC(s,t), each non-zero row has weight at least n−t+1 as it is a nonzero codeword\nofRS(t,B), therefore the number of non-zero columns is certainly at least n−t+1.\nNote that the code is close to being MDS if sis close to m.\nRemark 3.3. Some remarks are in place:\n1. The construction of folded RS codes [GR08] involves a single affine p olynomial, γx,\nfor a primitive γ∈F∗\nq. This results in a code whose length is smaller than q−1 by a\nfactorofthe folding parameter . Asimilar assertion isalsotruefor additive folded RS\ncodes[GR08], [BHKS24], and for the more general affine folded RS codes [BHKS24],\nwhich again use a single affine polynomial. The idea of using twoaffine polynomials\nof coprime orders is a generalization that enables to construct a ca pacity-achieving\ncode of length q−1, as will be shown below.\n2. The product structure can be interpreted as follows. If s=m, then the vertical\ncode is just Fm\nq, and the product is an interleaved RS code [SSB09]. While the\ninterleaved code itself does not guarantee a small list [GX13], moving froms=m\ntosslightly smaller than m(informally) results in a guaranteed small list, as will\nbe shown below. While in [GX13], the non-trivial concept of subspace designs was\n6Sincem,nare coprime, ℓm\n2(x)/ne}ationslash=xand therefore a fixed point of ℓm\n2is a fixed point of ℓ2.\n8required for assuring a small list, here, the simpler construction of tensor product\nwith shifts is used.7\nAt this point, it is fairly clear that CPPC(s,t) can be viewed as a linear subcode of\nRSFqm(t,B), which is an interleaved code, as B⊆Fq. We record this property in the\nfollowing proposition.\nProposition 3.4. CPPC(s,t)can be viewed as a linear subcode of RSFqm(t,B).\nProof.Let/tildewiderRSFqm(t,B)⊂(Fm\nq)nbe the code obtained by replacing each entry of each\ncodeword of RSFqm(t,B) by the column vector of its coefficients in the decomposition\naccording to a fixed basis for Fqm/Fq. Then, it is sufficient to show that CPPC(s,t)⊆\n/tildewiderRSFqm(t,B). Since the evaluation set Bis a subset of Fq,/tildewiderRSFqm(t,B) is the interleaved\ncode whose codewords are obtained by choosing freely mrows from RSFq(t,B), regardless\nof the basis choice. Note that by the definition of B,RSFq(t,B) is cyclic, as cyclically\nshifting the evaluation vector of a polynomial f(x) onBresults in the evaluation vector\noff(l2(x)) onB. SinceCPPC=π/parenleftbig\nRSFq(s,A)⊗RSFq(t,B)/parenrightbig\nby Proposition 3.1, each row\nofCPPCis a cyclic shift of a codeword of RSFq(t,B), and therefore again a codeword of\nRSFq(t,B), and the result follows.\n4 List decoding the permuted product code\nLet the received, possibly corrupted, version of the codeword be\nr=\nr0,0r0,1... r 0,n−1\nr1,0r1,1... r 1,n−1\n............\nrm−1,0rm−1,1... rm−1,n−1\n∈Fm×n\nq. (4)\nThe goal is to recover all polynomials f(x,y)∈Fs,t\nq[x,y] whose encoding (2) agrees\nwithron at least γof the columns, for some agreement parameter γ. For large enough γ,\nsay, at least half of the minimum distance bound given in Corollary 3.2, t he polynomial\nf, if exists, is unique. We would like to decode beyond the unique decodin g regime, i.e.,\nfor a much smaller agreement parameter γ, by sacrificing the uniqueness and instead\noutputting a list of possible codewords. To this end, we adapt the kn own algebraic\ntechnique to list-decode folded RS codes and their variants.\n4.1 Polynomial interpolation\nIn what follows, we assume that s < m. For a positive integer w≤m−s, consider\npolynomials of the form\nw−1/summationdisplay\ni=0pi(x,y)zi, where degx(pi)≤m−s−wand degy(pi)≤D−t,for alli,(5)\n7We also note that the construction of [GX13] has some resemblanc e to a product code: instead of\nusing a free matrix of information symbols, each column is constraine d to be in a different subspace from\na subspace design.\n9inFq[x,y,z0,...,z w−1], for some integer Dto be determined later.\nThe goal in the interpolation step is to interpolate a nonzero polynom ialQof the\nform (5) such that for each 0 ≤j≤n−1,\nQ(ℓjm+i\n1(α),ℓjm+i\n2(β),ri,j,...,r i+w−1,j) = 0,for 0≤i≤m−w. (6)\nNote that for each j, the constraints (6) are a collection of m−w+1 homogeneous linear\nconstraints on the coefficients of the polynomial Q, and in total there are n(m−w+1)\nsuch constraints. The following lemma shows that a nonzero interpo lation polynomial Q\nexists and can be found efficiently.\nLemma 4.1. With hindsight, for\nD:=/floorleftignm\nw(m−s−w+1)/floorrightig\n+t, (7)\na nonzero polynomial Qof the form (5)which satisfies the interpolation constraints (6)\nexists and can be found in O((nm)3)field operationsover Fq. Furthermore, we can assume\nthatQand the polynomial yq−ℓ2(y)are coprime.\nProof.The total number of free variables in Qis\nw(D−t+1)(m−s−w+1) =w/parenleftig/floorleftignm\nw(m−s−w+1)/floorrightig\n+1/parenrightig\n(m−s−w+1)\n> wnm\nw(m−s−w+1)·(m−s−w+1) =mn > n(m−w+1),\nwhere the right-hand side is the number of homogeneous linear equa tions for all inter-\npolation constraints. This proves that a non-zero Qsatisfying all constraints does exist,\nand the system of equations (which has at most nmconstraints) has a nontrivial solution\nthat can be found efficiently.\nLastly, we can assume that Qandyq−ℓ2(y) are coprime, since otherwise let g(y) =\ngcd(Q,yq−ℓ2(y)) and write yq−ℓ2(y) =g(y)h(y). We claim that g(y) has no roots in\nthe orbit of βunder the action of /an}⌊ra⌋ketle{tℓ2/an}⌊ra⌋ketri}ht, and therefore the polynomial Q/gsatisfies too the\nconstraints (6). Indeed, recall that βwas chosen to be a non-fixed point of ℓ2(y), hence\nalso any other element β′in the orbit of βunder the action of /an}⌊ra⌋ketle{tℓ2/an}⌊ra⌋ketri}htis too a non-fixed\npoint. Therefore,\n0/ne}ationslash=β′−ℓ2(β′) = (β′)q−ℓ2(β′) =g(β′)h(β′),\nand the result follows.\nNote that given a polynomial Qof the form (5) that satisfies the interpolation con-\nstraints (6), it is straightforward to modify Qto be coprime to yq−ℓ2(y) while still satis-\nfying the constraints. This can be achieved by dividing Qby any power of an irreducible\nfactor of yq−ℓ2(y) that divides it. Importantly, there is no need for general factor ization\nalgorithms in this process, as we focus in the sequel on the case whe reℓ2(x) =γxfor\na primitive γ. In such a scenario, yq−ℓ2(y) =y(yq−1−γ) is the decomposition into\nirreducible factors [GR08, Lemma 3.5].8\n8In the somewhat more general case where ℓ2(x) =γx+bwith non-zero b, it follows from the proof\nof Proposition 2.1 that xq−ℓ2(x) factors as ( x−δ)h(x) withh(x) irreducible of degree q−1, andδ, the\nonly root of xq−ℓ2(x) inFq, can be easily found by linear algebra methods. We omit the details.\n10Now, dividing out the largest powers of the irreducible factors that divide it in\npolynomial time is straightforward: For example, for dividing out the largest power of\nv(y) :=yq−1−γ, it is possible to iteratively divide all the pi(x,y) byv(y), until the first\ntime at least one of the pi’s is not divisible by it anymore.\nTo continue, we will need the following definition.\nDefinition 4.2. For a polynomial Qof the form (5), and for f∈Fq[x,y], we associate\nthe bivariate polynomial\nˆQf(x,y) :=Q/parenleftbig\nx,y,f(x,y),f(ℓ1(x),ℓ2(y)),...,f(ℓw−1\n1(x),ℓw−1\n2(y))/parenrightbig\n=w−1/summationdisplay\ni=0pi(x,y)f/parenleftbig\nℓi\n1(x),ℓi\n2(y)/parenrightbig\n.\nConsequently, if f∈Fs,t\nq[x,y],\ndegx(ˆQf)≤m−s−w+degx(f(x,y))< m−w, (8)\ndegy(ˆQf)≤D−t+degy(f(x,y))< D (9)\nThe following lemma shows the usefulness of the interpolation step fo r list decoding.\nLemma 4.3. LetQbe a polynomial of the form (5)that satisfies the interpolation con-\nstraints(6). Assume that the received word (4)agrees with the encoding of f(x,y)at the\nj-th coordinate for some j∈ {0,...,n−1}, i.e.,evj(f)equals the j-th column of r. Then\nˆQf(ℓjm+i\n1(α),ℓjm+i\n2(β)) = 0fori= 0,...,m−w.\nProof.For simplicity, assume that j= 0, and note that the general case follows similarly.\nThe following is easy to verify.\nˆQf(ℓi\n1(α),ℓi\n2(β)) =\nQ(ℓi\n1(α),ℓi\n2(β),f(ℓi\n1(α),ℓi\n2(β)),...,f(ℓi+w−1\n1(α),ℓi+w−1\n2(β))) =\nQ(ℓi\n1(α),ℓi\n2(β),ri,0,...,r i+w−1,0) = 0,\nwhere the last equality follows by (6).\n4.2 Outputting the list\nIn this section, we present a method that uses the interpolation po lynomial in order to\noutput the list of all polynomials f∈Fs,t\nq[x,y] whose encoding is close enough to the\nreceived word r. Before we proceed, we will need the following simple lemma.\nLemma 4.4. Letf(x,y)∈Fs,t\nq[x,y]be a polynomial. Assume that there exists a set\nS⊆Fqof sizesand a set Tα⊆Fqof sizetfor anyα∈S, such that\nf(α,β) = 0for anyα∈Sandβ∈Tα.\nThen necessarily f≡0.\n11Proof.Letf(x,y) =/summationtextdegy(f)\ni=0fi(x)yiand letα∈S. The univariate polynomial f(α,y) is\nof degree less than t, however it vanishes on at least tpoints, for each β∈Tα, therefore\nf(α,y)≡0, equivalently fi(α) = 0 for any i.However, fi(x) is a univariate polynomial of\ndegree less than sthat vanishes on at least spoints, for each α∈S, therefore fi(x)≡0\nfor anyi, and the result follows.\nAssume that we have a polynomial Qsatisfying the interpolation constraints. Next,\nwe would like to show that for a codeword that is close enough to the r eceived word (4),\nthe corresponding polynomial which generated thecodeword isa ro otofQ. The following\nlemma shows exactly this.\nLemma 4.5. Letf∈Fs,t\nq[x,y]be a polynomial whose encoding agrees with the received\nword on at least\nn/parenleftigm\nw(m−s−w+1)+t\nn/parenrightig\ncoordinates. Then ˆQf(x,y)is the zero polynomial.\nProof.As before, let ν∈ {0,...,nm−1}bea running index inthe codeword array, where\nfor row index i∈ {0,...,m−1}and column index j∈ {0,...,n−1},ν(i,j) :=mj+i.\nFor convenience, we will write i(ν) :=νmodmandj(ν) :=⌊ν/m⌋.\nWhenνruns on an entire column except for the last w−1 coordinates (explicitly,\nν∈ {jm,jm+1,...,jm+m−w}for some j∈ {0,...,n−1}),ℓν\n1(α) runs on the same set\nS:={ℓi\n1(α)|i∈ {0,...,m−w}}ofm−w+1 elements, regardless of the column j. Fixing\nα′∈S, the total number of choices of νsuch that: 1. j(ν) is an agreement column, and\n2.ℓν\n1(α) =α′(and therefore i(ν)≤m−w), is exactly the number of agreement columns,\nthat is, at least\nn/parenleftigm\nw(m−s−w+1)+t\nn/parenrightig\n=mn\nw(m−s−w+1)+t≥D. (10)\nMoreover, running on these choices of ν,ℓν\n2(β) runs on distinct values,9and hence on a\nsetTα′of size at least D.\nSinceˆQf(α′,β′) = 0 for all α′∈Sand allβ′∈Tα′by Lemma 4.3, |S|=m−w+1>\ndegx(ˆQf), and|Tα′| ≥D >degy(ˆQf) for allα′(using (8), (9)), it follows from Lemma\n4.4 that ˆQf= 0.\nBytheabovelemma, weconcludethatanypolynomial fthatgeneratesaclose-enough\ncodeword to the received word, satisfies ˆQf= 0. Therefore, the list decoding problem\nboils down to efficiently finding all such polynomials ffor which ˆQf= 0. To this end,\nwe consider below a related univariate polynomial over a large extens ion field KofFq.\nBefore proceeding, it is important to note that as opposed to [GR0 8], where it is\neventually required to solve a polynomial equation over an extension field, here Kis\nused mainly as a tool for analyzing the list size, and for easily deriving linear-algebraic\ndecoding over Fq-itself, as in [GW13]. See more on this in Remark 4.11 below.\n9Any two distinct such choices of ν, sayν2> ν1, satisfy m|(ν2−ν1). Since 0 < ν2−ν1< mnand\ngcd(m,n) = 1, we must have ℓν1\n2(β)/ne}ationslash=ℓν2\n2(β).\n12Proposition 4.6. Suppose that both m,n/ne}ationslash= 1. Lethi(x) :=xq−ℓi(x), andζibe a root\nofhioutsideFq,i= 1,2. Let also K=Fq(ζ1,ζ2)be the splitting field of h1h2. Then\n[K:Fq(ζ1)] =n, and{ζj\n2}n−1\nj=0is a basis for K/Fq(ζ1). Hence{ζi\n1ζj\n2}0≤i≤m−1\n0≤j≤n−1is a basis for\nK/Fq.\nProof.By Proposition 2.1, Fq(ζi) is the splitting field of hi,i= 1,2, and we have the\nfollowing diagram of field extensions and extension degrees:\nK=Fq(ζ1,ζ2)\nd2\n♣♣♣♣♣♣♣♣♣♣♣d1\n◆◆◆◆◆◆◆◆◆◆◆\nFq(ζ1)\nm❖❖❖❖❖❖❖❖❖❖❖❖❖Fq(ζ2)\nn\n♦♦♦♦♦♦♦♦♦♦♦♦♦\nFq\nSinced2m=d1n,d1≤m,d2≤n(as, e.g., the minimal polynomial of ζ2overFq(ζ1)\ndivides that over Fq), and gcd( m,n) = 1, it must hold that d1=mandd2=n. Hence,\n[K:Fq(ζ1)] =n,{ζj\n2}n−1\nj=0is a basis for K/Fq(ζ1), and{ζi\n1ζj\n2}0≤i≤m−1\n0≤j≤n−1is a basis for\nK/Fq.\nDefinition 4.7. Using the terminology of Proposition 4.6, let\nA(z) :=w−1/summationdisplay\ni=0pi(ζ1,ζ2)zqi∈K[z].\nLemma 4.9 below shows that the decoding problem reduces to the pro blem of finding\nthe roots of the linearized polynomial A(z).10In the lemma, we will use the following\nobservation, whose omitted proof is by straightforward induction on they-degree.\nObservation 4.8. LetKbe a field, let f(x,y)∈K[x,y]and letg(y)∈K[y]be a non-\nzero polynomial. Then there exist q(x,y),r(x,y)∈K[x,y]such that: 1. f=q(x,y)g(y)+\nr(x,y), 2.degy(r)0.\nTakem=p≈1\nε3,w≈1\n2·1\nε2, andssuch that m−s≈1\nε2(which is indeed ≥w). Recall\nthat the normalized number of required agreement columns is at leas tm\nw(m−s−w+1)+t\nn.\nLet us consider each summand separately. First,\nm\nw(m���s−w+1)0, and all powers qof a prime\np=O(1/ε3), there are instances of the permuted product code over Fqwith alphabet size\nqO(1/ε3), rateRand block length q−1, that are list-decodable from error fraction 1−R−ε,\nwith an output list of size L= (1/ε)O(1/ε2)by a randomized algorithm that outputs the list\nwith probability at least 1−αin timepoly(log/parenleftbig1\n1−α/parenrightbig\n,1/ε,q,L).\nBefore we proceed with the proof of the theorem we will need the fo llowing result of\n[Tam23] specialized to our case of list decoding.\nLemma 4.13. [Tam23, Lemma 3.1] Let C ⊆(Fm\nq)nbe a linear code with relative mini-\nmum distance δ >0that is(δ−ε,L)-list decodable. Assume further that the output list\nsize is contained in subspace V⊆ Cof dimension at most r, then the output list size\nL≤1\nεr. (13)\n15Moreover, there is a randomized algorithm that, given a basi s forV, with probability at\nleast1−αlist decodes Cwith the above parameters in time poly(logq,log/parenleftbig1\n1−α/parenrightbig\n,m,n,L).\nProof of Theorem 4.12. The only part that still requires proof is the assertion regarding\nthelist sizeandtherunning timeoftheoverall algorithm. ByTheorem 4.10thealgorithm\noutputs an Fq-subspace of the permuted product code of dimension at most O/parenleftbig\n1/ε2/parenrightbig\nand\ntherefore, by Lemma 4.13 it follows that the list is of size at most (1 /ε)O(1/ε2). The\nrunning time follows by the running times of the deterministic algorithm in Theorem\n4.10 and the randomized algorithm in Lemma 4.13.\n5 Open questions\nWe conclude the paper with some open questions for future resear ch.\n1.More than two affine polynomials. In this work, we considered codes achieving\nlist-decoding capacity constructed by two affine polynomials ℓ1,ℓ2∈GA(q) of co-\nprime orders. This raises the question of potential benefits from e mploying a larger\nnumber of affine polynomials. In particular, is it possible to construct capacity\nachieving codes of longer length than q−1, using more than two affine polynomials?\nNote that if r≥2 affine polynomials ℓ1,...,ℓ rare used, the number of distinct vec-\ntors (ℓi\n1,...,ℓi\nr)∈GA(q)r, asivaries over N, is lcm(ord( ℓ1),...,ord(ℓr))≤p(q−1).\nThis inequality is a consequence of (1), indicating that the maximum nu mber of\nevaluation points does not increase beyond that achievable with two affine polyno-\nmials, as shown in this paper.\nHowever, thisdoesnot rule outthe possibility of having a longer code . Forexample,\nifq−1 =abfor coprime integers a,b >1 witha < p, we may take ℓ1,ℓ2,ℓ3∈GA(q)\nwith orders a,p,b(resp.) and construct codewords with column length a, similarly\nto (2). The resulting code length will therefore be pb > q−1.\n2.Using AG codes. Similarly to [GX13], is it possible to extend the current results\nto the setup of AG codes in order to reduce the alphabet size?\n3.Efficient encoding. Can the product structure of the construction be used for\nefficient encoding? In particular, since the horizontal code is define d over the entire\nmultiplicative group of Fq, can this be used for some fast evaluation algorithm?\nReferences\n[AGL23] Omar Alrabiah, Venkatesan Guruswami, and Ray Li. Randomly punctured\nReed–Solomon codes achieve list-decoding capacity over linear-size d fields.\narXiv preprint arXiv:2304.09445 , 2023.\n[BGM23] Joshua Brakensiek, Sivakanth Gopi, and Visu Makam. Gener ic Reed–\nSolomon codes achieve list-decoding capacity. In Barna Saha and Ro cco A.\nServedio, editors, Proceedings of the 55th Annual ACM Symposium on The-\nory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 20 23, pages\n1488–1501. ACM, 2023.\n16[BHKS24] Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, and Madhu Sudan.\nIdeal-theoretic explanation of capacity-achieving decoding. IEEE Transac-\ntions on Information Theory , 70(2):1107–1123, 2024.\n[CMSvS91] G. Castagnoli, J.L. Massey, P.A. Schoeller, and N. von See mann. On\nrepeated-root cyclic codes. IEEE Transactions on Information Theory ,\n37(2):337–342, 1991.\n[DKSS13] Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, and Mad hu Sudan. Exten-\nsions to the method of multiplicities, with applications to kakeya sets a nd\nmergers. SIAM Journal on Computing , 42(6):2305–2328, 2013.\n[DL12] Zeev Dvir and Shachar Lovett. Subspace evasive sets. In Proceedings of\nthe 44th Symposium on Theory of Computing Conference (STOC) , pages\n351–358. ACM Press, 2012.\n[Eli57] Peter Elias. List decoding for noisy channels. Wescon Convention Record,\nPart 2, Institute of Radio Engineers , pages 99–104, 1957.\n[GK16] Venkatesan Guruswami and Swastik Kopparty. Explicit subs pace designs.\nComb., 36(2):161–185, 2016.\n[GR08] Venkatesan Guruswami and Atri Rudra. Explicit codes achie ving list decod-\ning capacity: Error-correctionwithoptimal redundancy. IEEE Transactions\non Information Theory , 54(1):135–150, 2008.\n[GR22] Zeyu Guo and Noga Ron-Zewi. Efficient list-decoding with const ant alpha-\nbet and list sizes. IEEE Trans. Inf. Theory , 68(3):1663–1682, 2022.\n[GRS19] Venkatesan Guruswami, Atri Rudra, and Madhu Su-\ndan. Essential coding theory. Draft available at\nhttp://cse.buffalo.edu/faculty/atri/courses/coding- theory/book/ ,\n2019.\n[GS99] Venkatesan Guruswami and Madhu Sudan. Improved decod ing of Reed–\nSolomon and algebraic-geometry codes. IEEE Transactions on Information\nTheory, 45(6):1757–1767, 1999.\n[GW13] Venkatesan Guruswami and Carol Wang. Linear-algebraic lis t decoding\nfor variants of Reed–Solomon codes. IEEE Transactions on Information\nTheory, 59(6):3257–3268, 2013.\n[GX13] Venkatesan Guruswami and Chaoping Xing. List decoding Ree d–Solomon,\nalgebraic-geometric, and Gabidulin subcodes up to the singleton bou nd. In\nDan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium\non Theory of Computing Conference, STOC’13, Palo Alto, CA, U SA, June\n1-4, 2013 , pages 843–852. ACM, 2013.\n17[GZ23] Zeyu Guo and Zihan Zhang. Randomly punctured Reed–Solomo n codes\nachieve the list decoding capacity over polynomial-size alphabets. In 64th\nIEEE Annual Symposium on Foundations of Computer Science, F OCS\n2023, Santa Cruz, CA, USA, November 6-9, 2023 , pages 164–176. IEEE,\n2023.\n[KRZSW23] Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, a nd Mary Wootters.\nImproved list decoding of folded reed-solomon and multiplicity codes. SIAM\nJournal on Computing , 52(3):794–840, 2023.\n[MS78] F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting\nCodes. North-holland Publishing Company, 2nd edition, 1978.\n[Rot06] Ron Roth. Introduction to Coding Theory . Cambridge University Press,\nUSA, 2006.\n[RT97] M. Yu. Rosenbloom and M. A. Tsfasman. Codes for them-met ric.Problemy\nPeredachi Informatsii , 33(1):55–63, 1997.\n[SSB09] Georg Schmidt, Vladimir Sidorenko, and Martin Bossert. Colla borative\ndecodingofinterleaved Reed–Solomoncodesandconcatenatedco dedesigns.\nIEEE Trans. Inf. Theory , 55(7):2991–3012, 2009.\n[Sud97] MadhuSudan. DecodingofReed-Solomoncodesbeyondthe error-correction\nbound.Journal of Complexity , 13(1):180–193, 1997.\n[Tam23] Itzhak Tamo. Tighter List-Size Bounds for List-Decoding a nd Recovery of\nFolded Reed-Solomon and Multiplicity Codes, 2023. arXiv:2312.17097.\n[Woz58] John M. Wozencraft. List decoding. Quarterly progress r eport, Research\nLaboratory of Electronics, MIT, 1958.\nA Properties of the unfolded code\nIn this appendix, we consider the properties of the “unfolded” cod eCunfolded\nPPC(s,t)⊆Fmn\nq,\nwhose codewords are the evaluation vectors\n/parenleftbig\nf(ℓ0\n1(α),ℓ0\n2(β)),f(ℓ1\n1(α),ℓ1\n2(β)),...,f(ℓmn−1\n1(α),ℓmn−1\n2(β))/parenrightbig\n,\nfor allf∈Fs,t\nq[x,y]. In particular, we show that the code is cyclic, and we find its\ngenerator polynomial.\nIt is interesting to note that while for folded RS codes, the unfolded code is an\nMDS code, Cunfolded\nPPC(s,t) is far from MDS, as its minimum distance equals ( m−s+\n1)(n−t+ 1) by Proposition 3.1. On the other hand, similarly to the case for fo lded\nRS codes, Cunfolded\nPPC(s,t) is cyclic, since cyclically shifting the codeword corresponding to\nf(x,y)∈Fs,t\nq[x,y] results in the codeword corresponding to f(ℓ1(x),ℓ2(y))∈Fs,t\nq[x,y]. It\n18is therefore natural to ask what is the generator polynomial ofCunfolded\nPPC(s,t) as a cyclic\ncode.11\nFor the choice of ℓ1(x) =x+1,ℓ2(x) =γx,α= 0,β= 1 from Section 4.3, we answer\nthis question in Proposition A.1 below. Note that the length N:=mn=p(q−1) is not\ncoprime to q, and in general the code is a repeated-root cyclic code , see e.g., [CMSvS91].\nIt will be useful to note that with the above choice of ℓ1,ℓ2,α,β, it holds that for all\nf∈Fq[x,y] andν∈ {0,...,N−1},\nf(ℓν\n1(α),ℓν\n2(β)) =f(ν,γν). (14)\nProposition A.1. The generator polynomial of Cunfolded\nPPC(s,t)is\ng(X) :=/parenleftbig\nxq−1−1/parenrightbigp\n/producttextq−1\nj=q−t(x−γj)s.\nThe proof relies on the following lemma.\nLemma A.2. A polynomial c(x) =c0+c1x+···+cN−1xN−1∈Fq[x]has a root β∈Fq\nof multiplicity at least r≤piff its vector of coefficients c:= (c0,c1,...,c N−1)satisfies\nHcT=0, where\nH:={jiβj}i∈{0,...,r−1},j∈{0,...,N−1}.\nProof.By the definition of the Hasse derivative (e.g., in [DKSS13]), βis a root of mul-\ntiplicity at least rofc(x) iff the Hasse derivatives of order 0 ,...,r−1 ofc(x) vanish at\nβ. As observed in [CMSvS91], this means that having βas a root of multiplicity ris\nequivalent to H0·cT=0, where\nH0:=/braceleftbigg/parenleftbiggj\ni/parenrightbigg\nβj/bracerightbigg\ni∈{0,...,r−1},j∈{0,...,N−1}.\nNote that the row index isatisfiesi≤r−1≤p−1. Now, for i < p,i! is invertible in\nFp⊆Fq, and/parenleftbiggj\ni/parenrightbigg\n= (i!)−1·j(j−1)···(j−i+1)\n(note that this holds also for j < i, where/parenleftbigj\ni/parenrightbig\n= 0). In Fp[x], writea0+a1x+···+aixi:=\n(i!)−1·x(x−1)···(x−i+1), where ai= (i!)−1/ne}ationslash= 0, and set a:= (a0,...,a i). Then/parenleftbigj\ni/parenrightbig\nis thea-linear combination of 1 ,j,...,ji(sameafor allj), and therefore row iofH0is\nthea-linear combination of rows 0 ,...,iofH. Hence, H0is obtained by multiplying H\nfrom the left by an invertible lower triangular matrix, and it follows tha t both matrices\nhave the same row space.\n11We will assume some background on cyclic codes, as appearing, e.g., in [MS78, Ch. 7], or [Rot06,\nCh. 8]. Recall that considering cyclic codes of length N′as ideals in Fq[x]/(xN′−1) is valid for any\nN′, not necessarily coprime to the characteristic. By corresponden ce of ideals of Fq[x] and those of\nF[x]/(xN′−1), any ideal of the quotient , i.e., any cyclic code of length N′, is the image of an ideal of\nFq[x] containing ( xN′−1), that is, the image of ( g(x)) for some g(x) dividing xN′−1. The unique monic\nsuchg(x) is called the generatorpolynomial of the code. The check-polynomial ish(x) := (xN′−1)/g(x).\nIt is shown, e.g., [MS78, Theorem. 7.5.4, p. 196], that the generator polynomial of the dual code is the\n“reversed h,” that is, h(0)−1·xdeg(h)h(x−1), and the proof remains valid when N′is not coprime to q.\n19With the lemma, the proof of Proposition A.1 is now straightforward:\nProof of Proposition A.1. A generator matrix of Cunfolded\nPPC(s,t) can be obtained by eval-\nuating the monomials xiyj,i∈ {0,...,s−1},j∈ {0,...,t−1}, as defined in (14), for\nν∈ {0,...,N−1}. The resulting matrix in Fst×N\nqhas rows\n/braceleftbig\nνi(γj)ν/bracerightbig\nν∈{0,...,N−1}\nfor alli∈ {0,...,s−1},j∈ {0,...,t−1}.Noting that s≤p, it follows from Lemma\nA.2 that the generator polynomial of the dual code is g⊥(x) :=/producttextt−1\nj=0(x−γj)s, so that\nthe check polynomial of Cunfolded\nPPC(s,t) is\nh(x) :=t−1/productdisplay\nj=0(x−γ−j)s=q−1/productdisplay\nj=q−t(x−γj)s.\n20" }, { "title": "2401.15051v1.The_Norm_Functor_over_Schemes.pdf", "content": "arXiv:2401.15051v1 [math.AG] 26 Jan 2024THE NORM FUNCTOR OVER SCHEMES\nPHILIPPE GILLE, ERHARD NEHER, AND CAMERON RUETHER\nAbstract: We construct a globalization of Ferrand’s norm functor over rings\nwhich generalizes it to the setting of a finite locally free mo rphism of schemes\nT→Sof constant rank. It sends quasi-coherent modules over Tto quasi-\ncoherent modules over S. These functors restrict to the category of quasi-\ncoherent algebras. We also assemble these functors into a no rm morphism\nfrom the stack of quasi-coherent modules over a finite locall y free of constant\nrank extension of the base scheme into the stack of quasi-coh erent modules.\nThis morphism also restricts to the analogous stacks of alge bras. Restricting\nour attention to finite ´ etale covers, we give a cohomologica l description of the\nnorm morphism in terms of the Segre embedding. Using this coh omological\ndescription, we show that the norm gives an equivalence of st acks of algebras\nA2\n1≡D2, akin to the result shown in The Book of Involutions.\nKeywords: Algebraic Groups, Azumaya algebras, Exceptiona l Iso-\nmorphism, Norm functor MSC 2020: 16H05, 14F20, 20G10, 20G35\nContents\nIntroduction 2\n1. Preliminaries 8\n1.1. Flat Sites 8\n1.4. Ringed Sites 9\n1.5. Quadratic Pairs 11\n1.6. Algebraic Groups 11\n1.7. Contracted Products 13\n1.8. Stacks 13\n2. Globalizing Ferrand’s Norm Functor 22\n2.1. Ferrand’s Norm Functor over Rings 22\n2.8. The Norm of Modules 33\nDate : January 26, 2024.\nThe first author was supported by the project “Group schemes, root systems, and\nrelated representations” founded by the European Union - Ne xtGenerationEU through\nRomania’s National Recovery and Resilience Plan (PNRR) cal l no. PNRR-III-C9-2023-\nI8, Project CF159/31.07.2023, and coordinated by the Minis try of Research, Innovation\nand Digitalization (MCID) of Romania. The research of the se cond author was partially\nsupported by an NSERC grant. The research of the third author was partially supported\nby the NSERC grants of the second author, Kirill Zainoulline , Mikhail Kotchetov, and\nYorck Sommerh¨ auser.\n12 P. GILLE, E. NEHER, AND C. RUETHER\n2.15. The Norm of Algebras 41\n3. Cohomological Description 48\n3.1. Automorphism Sheaves 48\n3.4. Cohomology Maps 51\n3.9. The Norm and the Brauer Group 54\n4. An Equivalence A2\n1≡D2 61\n4.1. A Quadratic Triple over Z 61\n4.3. Restricting the Segre Homomorphism 62\n4.6. Twisting Quadratic Triples 64\nAppendix A. Twisted Sheaves and Weil Restriction 68\nAppendix B. Cohomology of Semi-direct Products with Permut ation\nGroups 70\nAppendix C. Quasi-coherent sheaves on SchS 74\nC.8. Stack Morphism 80\nReferences 88\nIntroduction\nOne of the coincidences in the theory of algebraic groups is t he excep-\ntional isomorphism between the Dynkin diagrams of type A1+A1and of\ntypeD2. One way this manifests is as an isomorphism between split si m-\nply connected groups SL2×SL2∼=Spin4, or between split adjoint groups\nPGL 2×PGL 2∼=PSO 4. However, due to the relationship between alge-\nbraic groups and algebras with involution, this also manife sts as the following\nequivalence of groupoids shown in [ KMRT , 15.B]. Let Fbe an arbitrary field\nand\n(i) letA2\n1be the groupoid of Azumaya algebras of degree 2 over a qua-\ndratic ´ etale extension of FwithF–algebra isomorphisms as arrows,\nand\n(ii) letD2be the groupoid of central simple F–algebras of degree 4\nequipped with quadratic pairs (see [ KMRT , §5]) with F–algebra iso-\nmorphisms respecting the quadratic pair as arrows.\nThen, there is an equivalence of categories A2\n1≡D2. In particular, they\nshow in [ KMRT , 15.7] that a norm functor N:A2\n1→D2provides this\nequivalence.\nThe norm functor used in [ KMRT ] is with respect to finite ´ etale extensions\nof the base field. It is a generalization of the corestriction with respect to\na finite separable field extension K/Fintroduced by Riehm in [ R]. Riehm’s\ncorestriction sends a central simple K–algebraAof degreerto a central\nsimpleF–algebra cor K/F(A) of degree r[K:F]in such a way that the induced\nmap on Brauer groups\nBr(K) =H2(Gal(Fsep,K),F×\nsep)→H2(Gal(Fsep,F),F×\nsep) = Br(F)\n[A]/maps⊔o→[corK/F(A)]THE NORM FUNCTOR 3\nagrees with the usual corestriction in Galois cohomology. T his was general-\nized by Knus and Ojanguren, who defined the norm functor of a fin ite ´ etale\nextensions of rings in [ KO], and theirs is the version used in [ KMRT ]. The\nnorm functor was then extended further still by Ferrand in [ Fer] to include\nthe case of a finite locally free extension of rings. By [ Fer, §5], his new\nconstruction agrees with the previous notions of norm funct or. We review\nFerrand’s construction in Section 2.1.\nIn this paper we continue the “tradition” of extending the no rm functor\nto new settings. We fix a base scheme Sand work on the big fppf ringed\nsite (SchS,O) of schemes over Swith the global sections functor O. In fact,\nwe extend the norm functor to a morphism of stacks over SchSbetween\ncertain stacks of quasi-coherent sheaves. We are able to do s o because of\nthe following key property of Ferrand’s norm functor. For a fi nite locally\nfree ring extension R→R′, letNR′/R:ModR′→ModRdenote Ferrand’s\nnorm functor. If R→Qis any other ring homomorphism, thus making\nQ→R′⊗RQa finite locally free extension as well, then for any R′–module\nM′there is an isomorphism\nNR′/R(M′)⊗RQ∼=N(R′⊗RQ)/Q(M′⊗RQ)\nand this is functorial in M′. This compatibility with tensor products is ex-\nactly what allows the norm to be generalized to quasi-cohere nt sheaves as\nthey are characterized by a similar condition, see Lemma C.1. Our con-\nstruction is a general one which takes\n(i) a family Iof affine morphisms in SchSwhich is closed under arbi-\ntrary pullbacks and which allows descent (precisely, Ishould be a\nsubstack of the stack of affine morphisms AffMor as in Appendix\nC.8),\n(ii) for each h:U′→UinIwithUandU′affine, a functor\nFh:Mod O(U′)→Mod O(U),\n(iii) and for every fiber product diagram in SchS\nV′U′\nV Uh′h\nwhereU,U′,V,V′are all affine, a natural isomorphism of functors\nFh()⊗O(U)O(V)∼− → Fh′(⊗O(U′)O(V′))\nand assembles them into a morphism of stacks F:QCohI→QCoh . Here,\nQCohIis the stack with objects ( T′→T,M′) consisting of a morphism\nT′→TinIand a quasi-coherent O|T′–module M′, whileQCoh is the stack\nwith objects ( X,M) consisting of a scheme X∈SchSand a quasi-coherent\nO|X–module M. For a review of quasi-coherent modules on ( SchS,O) and\nthe details of this construction, see Appendix C.4 P. GILLE, E. NEHER, AND C. RUETHER\nUnsurprisingly, Ferrand’s norm functors, ranging over the family of fi-\nnite locally free morphisms of a fixed degree d, satisfy the necessary con-\nditions of the constructions in Appendix C. WhenIis the family of finite\nlocally free morphisms of degree d, we write QCohI=QCohd\nflf. There-\nfore, we obtain a stack morphism N:QCohd\nflf→QCoh , as well as a func-\ntorNT/S:QCoh (T)→QCoh (S) between categories of quasi-coherent O|T–\nmodules and quasi-coherent O–modules for any finite locally free morphism\nT→Sof degreed. We verify this in Section 2.8. The functor NT/S\nof course has additional specific properties analogous to Fe rrand’s functor,\nmost of which are in regards to polynomial laws. Since f:T→Sis finite\nlocally free, by [ St, Tag 0BD2] there is a functor\nnorm:f∗(O|T)→ O\nwhich arises from the determinant of left multiplication by elements of\nf∗(O|T) on itself. Given a quasi-coherent O|T–module M, we define a\nnormic polynomial law to be a natural transformation φ:f∗(M)→ N ,\nwhere Nis a quasi-coherent O–module, such that\nφ(tm) = norm(t)φ(m)\nholds for all sections t∈f∗(O|T)(X) andm∈f∗(M)(X) and for all X∈\nSchS. We show that the norm functor NT/Shas the following properties.\nA.Theorem. Letf:T→Sbe a finite locally free morphism of schemes\nand letNT/S:QCoh (T)→QCoh (S)be the norm functor.\n(i)For every quasi-coherent MoverTthere exists a normic polynomial\nlawνM:f∗(M)→NT/S(M)such that the pair (NT/S(M),νM)is\nuniversal in the following sense; if ν′:f∗(M)→ N′is any other\nnormic polynomial law into a quasi-coherent O–module, then there\nis a unique O–module morphism ϕ:NT/S(M)→ N′such thatν′=\nϕ◦νM.\n(ii)The universal property determines the image of NT/Son morphisms.\nIfϕ:M1→ M 2is a morphism of quasi-coherent modules over T,\nthenνM2◦f∗(ϕ)is a normic polynomial law and NT/S(ϕ)is the\nunique O–module morphism making the diagram\nf∗(M1)NT/S(M1)\nf∗(M2)NT/S(M2)νM1\nf∗(ϕ) NT/S(ϕ)\nνM2\ncommute.THE NORM FUNCTOR 5\n(iii)The norm functor and universal normic polynomial law respec ts base\nchange. If we have a fiber product diagram in SchS\nT′T\nS′Sg′\nf′f\ng\nthen there is an isomorphism of functors NT′/S′◦g′∗∼− →g∗◦NT/S\nand for any M ∈QCoh (T)the diagram\nf′\n∗(M|T′)NT′/S′(M|T′)\nf∗(M)|S′NT/S(M)|S′ν(M|T′)\ncan\n∼\n(νM)|S′\ncommutes.\n(iv)NT/S(O|T) =OandνO|T= norm:f∗(O|T)→ O .\n(v)IfBis a quasi-coherent O|T–algebra, then NT/S(B)is naturally an\nO–algebra and νBis multiplicative. The norm also preserves algebra\nhomomorphisms and thus restricts to the categories of quasi -coherent\nalgebras.\nFurther, if T→Sis finite ´ etale of degree dwe have the following.\n(vi)NT/Ssends (finite) locally free modules to (finite) locally free m od-\nules.\n(vii)NT/Ssends Azumaya algebras to Azumaya algebras.\n(viii)NT/Ssends locally free O|T–modules of constant rank rto locally\nfree modules of constant rank rdand it sends Azumaya algebras of\nconstant degree rto Azumaya algebras of constant degree rd.\nProperties (i)and(ii)are shown in Proposition 2.10and Corollary 2.11re-\nspectively. The isomorphism of functors in Property (iii)comes from Lemma\nC.9and the statement about the universal normic laws is Corolla ry2.12.\nProperty (iv)follows from Example 2.14after considering a sufficient local-\nization. Property (v)is shown in Lemma 2.17. Finally, when T→Sis\n´ etale, property (vi)follows from Example 2.14, property (vii) follows from\nproperty (viii) , and property (viii) is Lemma 2.13 and Lemma 2.20.\nIn Section 3we give a description of maps on cohomology induced by the\nnorm morphism. In particular, we restrict the norm morphism of stacks to\nvarious substacks, in fact subgerbes, which are equivalent to the gerbes of\ntorsors for some semi-direct products of groups. As is discu ssed in Appendix\nB, the cohomology set H1(S,(GLr)d⋊ Sd), where Sdis the permutation\ngroup, classifies isomorphism classes of objects in the fiber overSof the\ngerbe whose objects are pairs ( T′→T,M′) whereT′→Tis a degreed´ etale\ncover and M′is a locally free O|T′–module of constant rank r. Similarly,\nisomorphism classes in the fiber over Sof the gerbe whose objects are pairs6 P. GILLE, E. NEHER, AND C. RUETHER\n(T′→T,A′), where now A′is an O|T′–Azumaya algebra of degree r, are\nclassified by H1(S,(PGLr)d⋊Sd). Details on the definitions of these gerbes\nare given at the beginning of Section 3. We know by Theorem A(viii) that\nthe norm will send such objects to O–modules of rank rdor Azumaya O–\nalgebras of degree rdrespectively. Isomorphism classes of these modules\nare classified by H1(S,GLrd) and isomorphisms of those Azumaya algebras\nare classified by H1(S,PGLrd). A general fact about stack morphisms,\n[Gir, III.2.5.3], states that the resulting maps on isomorphism classes will be\ninduced from group homomorphisms ( GLr)d⋊Sd→GLrdand ( PGLr)d⋊\nSd→PGLrd. We show that these are the Segre embeddings. On the\nGLlevel, it sends elements ( A1,...,Ad)∈(GLr)dtoA1⊗...⊗Adand\nelements of Sdto the corresponding permutation of the tensor factors of\nO(rd)∼=(Or)⊗d. The map on the PGL level is defined similarly.\nB.Theorem. The map on cohomology sets induced by the norm functor,\nnamely\n/tildewideN:H1(S,(GLr)d⋊Sd)→H1(S,GLrd)\n[(T→S,M)]/maps⊔o→[NT/S(M)],\nagrees with the map induced by the Segre embedding (GLr)d⋊Sd→GLrd.\nFurthermore, the behaviour of the norm functor on Azumaya al gebras in-\nduces a map of cohomology sets\n/tildewidestNalg:H1(S,(PGLr)d⋊Sd)→H1(S,PGLrd)\n[(T→S,A)]/maps⊔o→[NT/S(A)].\nwhich likewise agrees with the map induced by the Segre embed ding (PGLr)d⋊\nSd→PGLrd.\nOur cohomological description of the norm functor also exte nds beyond\nfirst cohomology. Over a scheme, Brauer classes of Azumaya al gebras lie\nin the Brauer-Grothendieck group Br(S) =H2(S,Gm), where Gmis the\nmultiplicative group. Since we are assuming f:T→Sis finite ´ etale, we\nknow that the norm functor preserves Azumaya algebras and th us maps\nclasses of Azumaya algebras in Br( T) to classes in Br( S). We show that this\ninduced action is compatible with the trace morphism tr: Br( T)→Br(S)\ninduced by the trace morphism f∗(Gm|T)→Gmof [SGA4 , IX.5.1.2]. The\nfollowing is Proposition 3.14.\nC.Proposition. LetT→Sbe a degree d´ etale cover. Let Bbe an Azu-\nmaya O|T–algebra of constant degree and let Abe an Azumaya O–algebra\nof constant degree. Denoting Brauer classes in square brack ets, we have\n(i) [NT/S(B)] = tr([ B])∈Br(S), and\n(ii) [NT/S(A|T)] =d[A]∈Br(S).THE NORM FUNCTOR 7\nOur goal in Section 4is to show that the exceptional isomorphism dis-\ncussed at the beginning of this introduction still occurs at the level of Azu-\nmaya algebras over schemes. In fact, we show that it holds as a n equivalence\nof stacks. We consider the gerbes\n(i)A2\n1of quaternion algebras over a degree 2 ´ etale extension, and\n(ii)D2of degree 4 Azumaya algebras with a quadratic pair.\nQuadratic pairs are a characteristic agnostic analogue of o rthogonal involu-\ntions and thus are appropriate for discussing groups of type Dand related\nobjects over a general scheme. We include background on quad ratic pairs\nin Section 1.5. We begin Section 4by constructing a quadratic pair over Z\nwhich, after restriction, acts as the split object in Dn, the gerbe of Azumaya\nO–algebras of degree 2 nwith quadratic pair. By restricting the Segre ho-\nmomorphism to symplectic and orthogonal groups, we then sho w how the\nnorm gives a morphism from the stack of Azumaya algebras of de gree 2r\nwith symplectic involution over an ´ etale extension of degr eedinto the stack\nD2d−1rd. By noting that when r= 1 andd= 2 we get an equivalence of\ncategories, we obtain the following generalization of [ KMRT , 15.7] to our\nsetting.\nD.Theorem. There is an equivalence of stacks\nN:A2\n1→D2\n(T′→T,B)/maps⊔o→/parenleftbigT,(NT′/T(B),σNT′/T(B),fNT′/T(B))/parenrightbig\ngiven by the norm functor.\nThis is Theorem 4.9. If we focus on the fiber over Sof this morphism, we\nget an equivalence of categories between the groupoid of qua ternion algebras\nover a degree 2 ´ etale extension of Sinto the groupoid of degree 4 Azumaya\nO–algebras with quadratic pair, which is a more direct analog ue of [ KMRT ,\n15.7].\nThe organization of the paper is essentially outlined above . Section 1re-\ncalls the common objects and some basic results we use throug hout the pa-\nper. The important details of Ferrand’s construction over r ings are in Section\n2which also contains the construction of our new norm morphis m/functor\nand the proofs of the many parts of Theorem A. Section 3develops the co-\nhomological interpretation of the newly constructed norm f unctor. Section\n4uses these tools to show Theorem D.\nWe also include three Appendices containing general techni cal lemmas.\nAppendix Aconsiders a degree d´ etale cover f:T→Sand then for any\nsheaf FonSchSdescribesf∗(F|T) in terms of twisting sheaves with torsors\nor in terms of Weil restrictions in the case Fis representable. Appendix B\ngives a cohomological description of stacks related to semi -direct products of\ngroups which appear frequently in Sections 3and4. Finally, as mentioned\nabove, Appendix Ccontains a review of the properties of quasi-coherent\nmodule on ( SchS,O) as well as the details of the general construction we\nuse to define the norm morphism.8 P. GILLE, E. NEHER, AND C. RUETHER\n1.Preliminaries\n1.1.Flat Sites. Following the style of [ CF] and [ GNR ], most of our objects\nwill be sheaves on a category of schemes equipped with the fpp f topology.\nWe now review this setting, highlighting definitions of obje cts that are most\nimportant to us.\nFor a scheme Xwe denote by SchXthe big fppf site of Xas in [ SGA3 , Ex-\npos´ e IV]. The objects of SchXare schemes with a fixed structure morphism\nY→X, morphisms are scheme morphisms which respect the structur e mor-\nphisms, and coverings in the site consist of families of the f orm{Yi→Y}i∈I\nwhich are jointly surjective and where each Yi→Yis flat and locally of\nfinite presentation. When given a cover, we denote Yij=Yi×YYj, as well\nasYijk=Yi×YYj×YYk, etc. Affine schemes will commonly be denoted\nwithUorVand affine covers by {Ui→Y}i∈Ior likewise with V. SinceY\nneed not be a separated scheme, an affine cover need not have its Uij’s be\naffine.\n1.2.Remark. In [St, Tag 021S], the Stacks project defines “a” big fppf site\nofX, instead of “the” . This distinction, due to set theoretic co nsiderations,\nis avoided in [ SGA3 ] through the use of universes. We also simply use “the”\nbig fppf site and similarly use “the” big affine fppf site intro duced below.\nWe denote by AffXthe big affine fppf site of Xas in [ St, Tag 021S (2)].\nIt is the full subcategory of SchXconsisting of affine schemes over Xand\nthe covers are fppf covers of the form {Ui→U}m\ni=1where each UiandU\nare affine schemes. We will frequently use the following lemma to discuss\nsheaves on all of SchXby instead working with sheaves on AffX.\n1.3.Lemma. There is an equivalence of categories between sheaves on SchX\nand sheaves on AffXgiven by restricting a sheaf F:SchX→Sets to the\nobjects of AffX. Under this equivalence, intrinsic properties, such as be-\ning finite locally free or being quasi-coherent as recalled i n Section 1.4, are\npreserved.\nProof. The equivalence of categories (more precisely, equivalenc e of topoi)\nis [St, Tag 021V]. The properties of Section 1.4are intrinsic properties by\n[St, Tag 03DM]. By the definition of intrinsic property at the beg inning of\n[St, Tag 03DG], they are preserved under equivalences of topoi. /square\nIfY∈SchX, thenSchYis naturally a subcategory of SchXby composing\nthe structure morphisms Y′→Ywith the structure morphism Y→X. For\na sheaf FonSchX, we denote by F|Ythe restriction of the sheaf to SchY.\nIfY′→Yis a morphism of X–schemes, then borrowing notation from [ St],\nwe uset|Y′to denote the image under F(Y)→ F(Y′) of a section t∈ F(Y).\nThis will also be referred to as the restriction of ttoY′. It will be clear from\ncontext which notion of restriction, for sheaves or for sect ions, is intended.\nBy a slightly further abuse of notation, we may talk about a se ctiont∈ F,THE NORM FUNCTOR 9\nby which we mean a section t∈ F(Y) for some Y∈SchS, i.e.,tmay be\nany section over any Y.\nGiven two schemes Y1,Y2∈SchX, their presheaf of homomorphisms is\nthe functor\nHom(Y1,Y2):SchX→Sets\nZ/maps⊔o→HomZ(Y1×XZ,Y 2×XZ)\nsendingZto the set of Z–scheme morphisms between the two fiber products.\nIt is a sheaf by [ St, Tag 040L]. The subsheaf of isomorphisms, denoted\nIsom(Y1,Y2), is then also a sheaf.\n1.4.Ringed Sites. We now fix a base scheme S. Unless otherwise stated,\nwe assume that a ring is unital, commutative, and associativ e. The global\nsections functor\nO:SchS→Rings\nX/maps⊔o→ OX(X)\nwhereRings is the category of commutative rings, is a sheaf with respect\nto the fppf topology by [ St, Tag 03DU]. It makes ( SchS,O) into a ringed\nsite as in [ St, Tag 03AD] and we call Othestructure sheaf . IfX∈SchSis\nanother scheme, then the structure sheaf of SchXisO|X.\nFrom [ St, Tag 03CW], an O–module is a sheaf M:SchS→Abof abelian\ngroups with a map of sheaves\nO × M → M\nsuch that for each X∈SchS, the map O(X)× M (X)→ M (X) gives\nM(X) the structure of an O(X)–module. A morphism of O–modules is a\nmorphism of sheaves such that the map on Xpoints is O(X)–linear for all\nX∈SchS. The notion of O|X–module on SchXis analogous and likewise\nfor the properties discussed below.\nTheinternal homomorphism functor of two O–modules MandNis\nHomO(M,N):SchS→Ab\nT/maps⊔o→Hom O|T(M|T,N |T).\nIt is another O–module by [ St, 03EM]. The internal endomorphisms of an\nO–module Mare denoted by EndO(M) =HomO(M,M), and the subsheaf\nof automorphisms is denoted AutO(M).\nIfg:X→Sis a morphism of schemes, then we have a direct image (or\npushforward) functor g∗and a pull-back functor g∗as in [ St, Tag 03D6].\nThese form an adjoint pair where g∗is left adjoint to g∗, [St, Tag 03D7].\nThat is, for each O|X–module Eand each O–module F, we have a natural\nisomorphism\nHom O|X−Mod(g∗F,E)∼− →Hom O−Mod(F,g∗E).10 P. GILLE, E. NEHER, AND C. RUETHER\nIn particular for F=Owe have\nHom O|X−Mod(O|X,E)∼− →Hom O−Mod(O,g∗E)\nor in other words E(X)∼− →(g∗E)(S).\n1.4.1. Local Types of O–modules. We refer to [ St, Tags 03DE, 03DL] for\ndefinitions of various properties of O–modules. Since S∈SchSis a final\nobject, [ St, Tag 03DN] applies and it suffices for us to define local conditi ons\nfor an fppf-covering of S. Quasi-coherent modules are reviewed in Appendix\nC. Here we briefly review finite locally free modules.\nWe call an O–module Mfinite locally free orlocally free of finite type\nif for allX∈SchS, there is a covering {Xi→X}i∈Isuch that for each\ni∈I, the restriction M|Xiis a free O|Xi–module of finite rank. Explicitly,\nM|Xi∼=O|ni\nXifor some non-negative ni∈Z. Ifni>0 for alli∈Iwe say\nthat Mis of finite positive rank . If allni=nfor some integer nthen we\nsayMhasconstant rank n. Neither of these notions depend on the cover.\nIfMis finite locally free, then so is EndO(M).\n1.4.2. O–Algebras. AnO–algebra is an abelian sheaf B:SchS→Abto-\ngether with sheaf morphisms\nO → B andB ⊗ OB → B\nwhich makes B(X) into a O(X)–algebra for all X∈SchS. It is unital,\nassociative, commutative, etc., if each B(X) has that property. For an O–\nmodule M, the sheaf EndO(M) is naturally an O–algebra with multiplication\ncoming from composition as usual.\nAnO-algebra Ais an Azumaya O–algebra if it is finite locally free and it\nsatisfies the following equivalent conditions.\n(i) The enveloping morphism\nA ⊗ OAop→ EndO(A)\na⊗b/maps⊔o→(x/maps⊔o→axb)\nis an isomorphism.\n(ii) For any U∈AffS, we have that A(U) is an Azumaya O(U)–algebra\nin the sense over rings such as in [ Fo] or [Knu, III §5]. In particular,\nonAffSan Azumaya O–algebra is a sheaf of Azumaya algebras.\n(iii) There exists a cover {Xi→S}i∈Isuch that for each i∈I,A|Xi∼=\nEndO|Xi(Mi) for a locally free O|Xi–module Mof finite positive rank.\n(iv) There exists a cover {Xi→S}i∈Isuch that for each i∈I,A|Xi∼=\nMni(O|Xi) for some 0 0 (4)\nwhere\nζ=V−νβ−mνs\nb⟨J⟩. (5)\nTo include radiation damping, we could evaluate Eq. 1 for\nIm(V) = +1 /τxwhere τxis the radiation damping time,\nalthough the integral would then have to be evaluated\nnumerically, slowing down the calculation significantly.\nIn practice, the radiation damping does not make a large\ndifference in the case of the TMCI so it is neglected.\nThe inverse of the dispersion relation is subtracted\nfrom the diagonal elements of Chin’s scaled coupling ma-\ntrixνsMmk\nnlfor head-tail and mode-coupling instabilities\nas given by Eq. 2.44 in [6]. Solutions are determined nu-\nmerically by equating the determinant of the resulting\nmatrix to zero:\ndet(I−1\nmδmlδnk−νsMmk\nnl) = 0 (6)\nwhere δijis the Kronecker delta. In practice, there are\ntwo unknowns left to determine: the tune spread at ⟨J⟩\n(∆ν(⟨J⟩) =b⟨J⟩) and the coherent frequency of oscilla-\ntion Re( V).\nFigure 1. Simulated coherent beam spectrum of the bunch at\nzero chromaticity and zero ADTS, showing the tune shift with\nincreasing bunch current and the resulting mode coupling at\nthe TMCI threshold at 3.5 mA.\nII. EXPERIMENTAL SETUP\nThe measurements presented in this paper were per-\nformed with the machine parameters given in Table I if\nnot stated otherwise. Only a single bunch was stored in\nthe machine to be able to use diagnostics that are not\nbunch-resolved such as beam position monitors (BPM)\nand the synchrotron light monitor (SLM). To ensure that\nonly a pure single bunch is filled, great care has been\ntaken to clean out residual charge from the other buck-\nets. The total acceleration voltage in the main cavities\nwas set to a fixed value for better comparability of dif-\nferent measurements. Due to the usage of a single bunch\nand therefore a low absolute beam current, the passive\nLandau cavities are not elongating the bunch. Likewise,\nall insertion device gaps were opened for the sake of com-\nparability. The reference orbit for the orbit correction\nwas set to zero-orbit without any beam line bumps laid\nin, to use an optic close to the design optics used in the\nsimulations. During the measurements itself the orbit\ncorrection was only in use while changing to new settings,\ne.g. changing chromaticity or ADTS and was afterwards\nswitched off. When changing settings, it was also checked\nthat the tunes were at the standard working point. The\nTable I. Beam Parameters during Measurements\nParameter Value\nBeam energy / GeV 3.0\nCircumference / m 528\nRF frequency / MHz 99.931\nHarmonic number 176\nRF voltage / kV 864\nSynchrotron freq. / Hz 830\nSynchrotron tune 0.00146\nVertical tune 16.275\nHorizontal tune 42.23\n0 10000 20000 30000 40000 50000 60000\nTurns / 10.15\n0.10\n0.05\n0.000.050.100.15Center-of-mass position / mm\nFigure 2. BPM signal showing the amplitude modulation\ncaused by the TMCI on the vertical center-of-mass position\nas function of turns.\nvertical chromaticity was reduced via sextupole magnets\nin the vertical plane while the horizontal chromaticity re-\nmained at the value for standard operation. For adjust-\nments of the ADTS the three octupole magnet families\ncan be used [7].\nA. ADTS - measurement and control\nThe center-of-mass (COM) motion is measured via\nthe turn-by-turn data from the beam position monitors\n(BPM). The position at each turn can be written out\nfor 216consecutive turns. As the measurements where\ntaken in single-bunch operation, this gives the COM po-\nsition of the bunch at each turn at every BPM. Addition-\nally, the data observed by the bunch-by-bunch system is\nsaved with a turn-by-turn resolution as well. The coher-\nent tunes can therefore be calculated from the Fourier\ntransform of this data.\nThe amplitude-dependent tune shift (ADTS) was mea-\nsured by kicking the bunch in each transverse plane in-\ndividually with increasing amplitude while detecting the\ncenter-of-mass movement on the turn-by-turn BPM data\nfor both transverse planes. The tune was calculated\nvia the Numerical Analysis of Fundamental Frequencies\n(NAFF) algorithm [8, 9] based on the first 100 turns af-\nter each kick. The resulting tunes for the different kick\namplitudes show the tune shift as a function of the center-\nof-mass displacement ˆ xat each BPM position. For the\nconversion between the measured maximal amplitude ˆ x\nat each BPM and the action J, it is assumed that at\nmaximum displacement ˆ xthe action Jcan be calculated\nvia the corresponding value of the beta-function βsat\nthe position of measurement s, in this case the position\nof each BPM:\nJ=ˆx2\ns\nβs. (7)\nFigure. 3 shows the near linear dependence of the tune\n0 1 2 3 4\nAction J / m\n 1e7\n0.2710.2720.2730.2740.275Tune / 1\nb=(11095.9±15.3)/m\nMeasurement2\nFigure 3. Measurement of the betatron tune as function of the\naction ⟨J⟩in the vertical plane, showing the linear dependence\ngiving the amplitude-dependent tune shift coefficient b.\nshift on the action Jin an example measurement.\nTo ease the operation during the experimental scans\nof the ADTS value, a response matrix Mwas measured\nfor the resulting change of the ADTS caused by changes\nto two of the octupole families. This matrix allows fast\ncalculation of the necessary change ∆ Ioct,uin octupole\ncurrent for a requested change of ∆ buin ADTS coeffi-\ncient, with urepresenting the planes xandy, without\nthe need for intermediate measurements.\n\u0012\n∆Ioct,x\n∆Ioct,y\u0013\n=M−1∗\u0012\n∆bx\n∆by\u0013\n(8)\nNevertheless, after arriving at a new ADTS value and\nchecking, and if necessary correcting, the chromaticity\nand the tunes, and before conducting dedicated measure-\nments, the ADTS value was measured to ensure accuracy.\nB. bunch size\nThe bunch size is measured at the two diagnostic beam\nlines [10] via synchrotron light monitors (SLM) with in-\nterferometric source point imaging. Synchrotron radi-\nation in the visible wavelength range is detected with\nCMOS cameras after passing though a double-slit for the\nhorizontal plane and a diffraction obstacle in the vertical\nplane. The beam sizes can be calculated from the in-\nterferometric visibility in the resulting interference pat-\ntern. During the instability the bunch size is blown up\nto such a degree that the interference pattern is not vis-\nible anymore (Fig. 4b) and the distribution is fitted by\na Gaussian. The required exposure time of the cameras\nto gather enough intensity, does not allow for turn-by-\nturn detection. Nevertheless, the exposure time is short\nenough ( ≈1 ms) to be able to resolve the time structure\nof the characteristic amplitude modulation observed on\nthe center-of-mass position (see Fig. 2) caused by the\ndynamics above the instability threshold. In this case,\nit has to be taken into account that during the exposure4\n(a)\n (b)\nFigure 4. Synchrotron light spot of a vertically unstable\nbunch measured at a diagnostic beam line for interferometric\nbunch size measurement. a) The additional, residual low-\ncurrent bunches are stable and show the typical interferomet-\nric pattern on top of the unstable main bunch. b) After clean-\ning the residual bunches, only the vertically unstable single\nbunch is visible.\ntime the camera integrates over the observed center-of-\nmass oscillations providing a superposition of the center-\nof-mass motion and the interference pattern containing\nthe beam size information. Furthermore, it has do be as-\nserted that no residual bunches are present during these\nmeasurements. As visible in Fig. 4, even a small amount\nof charge in additional residual bunches around the main\nsingle bunch, in this case approximately 5% of the charge,\ncan significantly influence the observed spot profile on\nthe SLMs. As these low-charge residual bunches are be-\nlow the instability threshold and therefore stable, their\nlight shows the classical interference pattern. The inten-\nsity of the focused pattern is therefore overshadowing the\nsmeared out spot profile from the unstable main bunch.\nC. synchronised measurements\nThe readout of the cameras can be triggered so that\nsynchronised images can be taken with respect to the\nturn-by-turn center-of-mass motion measured with the\nBPMs. The synchronization was aligned with triggered\nkicks to the beam which can be observed in both sys-\ntems (BPMs and SLMs). The timing between the cam-\nera acquisition and BPM acquisition is chosen such that\nthe camera’s exposure time window lies roughly at three-\nquarters of the BPM measurement window of 216turns\n(≈115 ms). By this, the center-of-mass movement is\nknown for some time before and after the bunch size\nmeasurement. The alignment accuracy depends on the\n0.00065 0.00070 0.00075 0.00080 0.00085\nEnergy spread / 12.502.753.003.253.503.754.00Threshold current / mAFigure 5. Simulated TMCI threshold at a chromaticity of\n0.05,b= 1000 and different energy spreads of 80%, 90%,\n100% and 110% showing the expected increase in threshold\nfor increased energy spread.\ncamera exposure time used and is in the presented mea-\nsurements better than 1 ms.\nIn case of multiple such measurement sets being taken\nduring the instability with the characteristic amplitude\nmodulation on the center-of-mass movement (see Fig. 2),\nthe repetitive behaviour seen on the BPMs can be used\nto overlay multiple measurement sets aligned by this pat-\ntern. This will provide a “sampled” image of the changes\nin the light spot observed on the SLM cameras. In other\nwords, due to the measurement trigger not being synchro-\nnised to the instability dynamics, different phases of the\namplitude modulation are sampled with every measure-\nment set taken and the repetitiveness of the amplitude\nmodulation can be used to reconstruct a time resolved\nimage. The spot size is the result of the superposition of\nthe blown up bunch size and the center-of-mass oscilla-\ntion within the exposure time window.\nIII. SIMULATION TOOL\nTo simulate the beam dynamics observed, especially\nabove the threshold current, particle tracking with the\nmbtrack2 [11] python code was performed.\nA broadband resonator was used for the vertical\nimpedance with a shunt impedance of 200 kOhm/m at\nthe resonant frequency of 11.5 GHz and a quality Qof\n1 [12]. The mbtrack2 simulations also included a longi-\ntudinal impedance (732 Ohm at 6 GHz with Q= 1 [13])\nto account for bunch lengthening with increasing bunch\ncurrent. mbtrack2 allows for the optics parameter to be\nread-in from an AT lattice file using pyAT. The RF volt-\nage was set to the same value as in the measurements (see\nTable I). The tune shift contribution by the ADTS is cal-\nculated in mbtrack2 based on the action J(see Sec. I A\nEq. 3). Intra-Beam Scattering (IBS) is not yet imple-\nmented in mbtrack2. The shown simulations were con-\nducted with 50000 macro-particles and were run on the\nCOSMOS cluster of LUNARC at Lund University.5\n10000\n 5000\n 0 5000 10000 15000\nADTS coefficient / 1/m1.01.52.02.53.03.5Threshold current / mA\nChroma\n0.00±0.01\n0.05±0.02\n0.15±0.10\n15000\n 10000\n 5000\n 0 5000 10000 15000\nADTS coefficient/ 1/m1.01.52.02.53.03.5Threshold current / mAChroma\n0.00\n0.05\n-0.15\nFigure 6. left: Single bunch threshold currents during injection shown as a function of ADTS coefficient measured at chro-\nmaticities of 0, 0.05 and -0.15. right: Single bunch threshold currents simulated in mbtrack2 (including bunch lengthening by\naddition longitudinal impedance) for chromaticities of 0, 0.05, -0.15.\nIV. RESULTS\nIn the following the measurement and simulation re-\nsults will be presented side by side and grouped by the\ndifferent beam properties affected by the instability. The\nmeasurements were conducted in the vertical plane. Be-\nsides the threshold current, the beam loss at threshold,\nthe bunch position and size as well as the betatron tune\nshift with current below and above the instability thresh-\nold are discussed. Additionally, theoretical calculations\non the Landau damping in combination with the trans-\nverse mode-coupling instability will be discussed in con-\ntext of the observed asymmetry with respect to the sign\nof the amplitude-dependent tune shift.\nA. Instability Threshold\nWhen studying an instability, the threshold current is\na very important parameter as it is the limit up to what\ncurrent stable operation is possible.\nDuring the experimental investigations of the TMCI it\nwas observed, that the threshold current changes depend-\ning on the beam conditions while reaching the thresh-\nold. For example, the observed threshold current was\nlower when the beam current slowly decayed while the\nbeam was unstable, compared to the threshold current\nobserved when charge was injected into a stable beam.\nAdditionally, within a certain bunch current range, it was\npossible to stabilize an unstable beam with the bunch-\nby-bunch feedback system and the beam remained stable\nafter switching off the feedback. Furthermore, the insta-\nbility could be triggered by excitations or kicks to the\nbeam even below the injection threshold, but not below\nthe decaying threshold. In summary, a hysteresis effect\nwas observed for the experimental TMCI threshold cur-\nrent, where a stable beam shows a higher threshold cur-rent than an already unstable or excited beam.\nA possible explanation for the observed hysteresis in\nthreshold is found when considering the effects of Intra-\nBeam Scattering (IBS). For beams with small transverse\nemittances, IBS can lead, amongst other things, to an\nincrease in energy spread. This can be mitigated by in-\ncreasing the vertical emittance either via coupling or with\nvertical excitations of the beam. With respect to the\nTMCI, IBS would have the following effect. For a sta-\nble beam the vertical emittance is small and the energy\nspread is increased by IBS. An increased energy spread\nresults in an increase of the theoretical TMCI threshold,\nas the current-dependent tune shift is inversely propor-\ntional to the bunch length which again is proportional\nto the energy spread [14]. As soon as the beam becomes\nunstable, either by crossing the (higher) threshold or by\nexcitation, the vertical emittance increases and the ef-\nfect of the IBS is reduced leading to a reduction in en-\nergy spread. The lower energy spread finally results in\na lower TMCI threshold current. This results in a hys-\nteresis of the instability threshold depending on whether\nthe threshold is measured starting with a stable or an\nunstable beam.\nAs the mbtrack2 simulations do not include IBS this\nhysteresis can not be directly simulated. Nevertheless,\nsimulations with the energy spread manually set to differ-\nent values show the expected dependence of the threshold\ncurrent on the energy spread (see Fig. 5).\nFor the following studies the threshold during injec-\ntion was selected as it can be quickly measured reliably\nand accurately compared to the other thresholds. Fur-\nthermore, the disturbance to the stored beam caused by\nthe used Multipole Injection Kicker (MIK) is known to\nbe very small [15], so the observed threshold during in-\njection should be very close to the theoretical threshold\n(including IBS) if the charge in a stable beam is slowly\nincreased.6\nThe left hand side of Fig. 6 shows the thresholds mea-\nsured during injection for different ADTS coefficients.\nThese thresholds where determined by injecting (using\nthe MIK) into a single bunch and observing the center-\nof-mass movement on the BPMs. As soon as the center-\nof-mass movement movement grew unstable the injection\nwas stopped and all charge in residual bunches from a\nnon-perfect single bunch injection was cleaned. The re-\nsulting threshold currents differ as expected depending\non the chromaticity. To separate the TMCI from the\nhead-tail instability [16], the measurements where con-\nducted either at a vertical chromaticity of zero or nearly\nzero chromaticity (0.05) in contrast to a chromaticity of\n≈1.1 during standard operation. The thresholds for\nboth chromaticity values are very similar and lie around\n2.8 mA. Additionally, measurements were conducted at a\nslightly negative vertical chromaticity of −0.15. As ex-\npected during operation with a positive momentum com-\npaction factor and a negative chromaticity (e.g. [14]),\nthey show a much lower threshold current of around\n1 mA. The same is visible in the simulated thresholds\nshown on the right hand side of Fig. 6. The simulated\nthresholds for a chromaticity of zero and 0.05 lie both at\naround 3.45 mA and are higher than seen in the measure-\nments by about 0.5 mA. At the same time, the simulated\nthreshold for the slightly negative chromaticity matches\nthe measurements at around 1 mA.\nThe measurements and the simulations were conducted\nfor a range of positive and negative ADTS coefficients.\nNo significant correlation between threshold currents and\nthe value of the ADTS coefficient is observed in either\nmeasurement or simulations. This is not unexpected as\nthe experimental ADTS coefficients reached only result in\na very small tune shift for the center-of-mass oscillation\nand the bunch size observed in a stable beam. A typical\nmeasured ADTS coefficient of b= 5000 /m leads with\na stable bunch size of below 10 µm or a center-of-mass\nmovement with a maximal amplitude of 10 µm to a tune\nshift in the order of only ∆ ν≈10−7. Consequently,\nan ADTS coefficient in this order of magnitude is not\nrelevant until the instability starts to “blow up” the beam\nleading to a bigger contribution of the ADTS due to the\nthen drastically increased center-of-mass oscillation and\nbunch size. As is shown in Fig. 2, during the instability\nthe center-of-mass amplitudes reach values of the order of\nhundreds of micrometers and, as will be shown later, the\nbunch size blows up to similar sizes. Then the tune shift\nby ADTS is in the order of ∆ ν≈0.001 which corresponds\nalready to two-thirds of the synchrotron tune. Therefore,\nit is then, above the instability threshold, that the ADTS\nis expected to influence the dynamics.\nB. Beam Losses at Thresholds\nA significant influence of the value and sign of the\nADTS coefficients can be observed in the amount of\ncharge lost when the instability threshold is crossed dur-\n10000\n 5000\n 0 5000 10000 15000\nADTS coefficient / 1/m020406080Beam loss crossing threshold / %\nChroma\n0.00±0.01\n0.05±0.02\n0.15±0.10\nFigure 7. Current loss in percent at the TMCI threshold as\nfunction of the amplitude-dependent tune shift at chromatic-\nities of 0, 0.05 and -0.15.\ning injection. Figure 7 shows the beam loss in percent for\ndifferent values of the ADTS coefficient with a chromatic-\nity close to zero or with slightly negative chromaticity\nvalues. For negative ADTS coefficients up to nearly zero\n(≈ −500/m ) no beam loss is encountered at all when\ncrossing the instability threshold during injection. This\nis already noteworthy as it shows, that the instability\nis not destructive even though it leads to strong center-\nof-mass oscillations and an increase in bunch size. On\nthe other side, for positive ADTS coefficients a partial\nbeam loss is observed when crossing the threshold. For\nvalues from zero up to 6000 /m more than 40 and up to\n90 percent of the beam current is lost. For higher pos-\nitive ADTS coefficients the loss goes down close to zero\nagain. So, for ADTS coefficients up to 6000 /m there is\na difference in the observed behaviour for a positive and\nnegative sign of the ADTS coefficient. While at negative\ncoefficients the instability is self-containing, for positive\ncoefficients a partial beam loss is observed until the beam\nstabilizes again.\nTo investigate this difference in behaviour above the\nthreshold the time domain signal of the center-of-mass\noscillation and the bunch size was studied in measure-\nment and simulation.\nC. Bunch Position and Size\nFor a negative ADTS coefficient the dynamic above\nthreshold shows clear, regular, sawtooth like bursts in\nbunch size and as amplitude modulation of the center-\nof-mass oscillations. For measurements this is visible in\nthe BPM trace directly (Fig. 2) as well as in the syn-\nchronous measurement of bunch position and bunch size\nin Fig. 8. The contribution of the bunch size can be seen\nin the fact that the spot size goes down slower than the\ncenter-of-mass oscillations seen by the BPMs. Addition-7\nFigure 8. Synchronized measurement of vertical spot profile\nand center-of-mass amplitude as function of time. The im-\nage shows the vertical light spot measured at different points\nin the sawtooth like dynamic. In red the envelope of the\ncenter-of-mass motion is displayed with a point for each spot\nprofile measurement. The yellow dots indicate the spot size\ngained from a Gaussian fit. Measured at ADTS coefficient\nb=−10000 and current of 2.6 mA.\nFigure 9. Simulated center-of-mass oscillation and bunch size\nat an ADTS coefficient b=−15000 and a current of 4.8 mA.\nally, the calculated spot size of up to 1.7 mm (determined\nby Gaussian fit) in Fig. 8 compared to the maximal de-\ntected center-of-mass oscillation amplitude of 0.25 mm\n(= 0.5 mm peak-peak) indicates that the bunch size has\na non-negligible contribution. Furthermore, the center-\nof-mass oscillation goes back to nearly zero for some hun-\ndred turns between the increases in oscillation amplitude.\nDuring this time the observed spot size slowly damps\ndown indicating that this corresponds to the bunch size\ndamping down, reaching minimal values in the order of\n0.3 mm before the next peak. The same behaviour is\npresent in the simulation shown in Fig. 9.\nThis dynamic indicates a stabilizing mechanism which\nleads to a containment of the instability instead of a con-\ntinuous growth until charge is lost. The observed be-\nhavior in the amplitude of the center-of-mass oscillationsand the bunch size show that at some point a temporary\nstabilization occurs which leads to a damping down of\nthe oscillation to below the noise limit of the measure-\nment. The bunch size is also damped down during this\nstable period but it does not reach the expected stable\nbunch size before the instability is triggered again lead-\ning to a fast blow up of the bunch size and the onset of\nstrong center-of-mass oscillations. A possible mechanism\ncould be via an increased tune spread due to the blown up\nbunch size and the ADTS leading to an increased Landau\ndamping effect. While the IBS could be the cause of the\nhysteresis observed in the Instability threshold, it is not a\ncandidate for explaining the self-containing dynamics, as\nit operates in the wrong sense, ie. a blown-up beam has\na lower threshold current and is therefore more unstable\nand does not contribute to a self-containing effect, where\nthe threshold would need to increase to temporarily sta-\nbilize the unstable beam.\nFor positive ADTS, the dynamics above the threshold\ncan only be observed in measurements at high ADTS co-\nefficients where no instantaneous charge loss occurs. For\nhigher bunch currents the dynamics in the center-of-mass\noscillation and the bunch size have a similar sawtooth like\npattern as observed for negative ADTS coefficients (see\nleft side of Fig. 10). The pattern changes for lower bunch\ncurrents as shown on the right side in Fig. 10. Here, the\ncenter-of-mass oscillation amplitude is more constant and\nlightly modulated in the measurements, and nearly con-\nstant in the simulations.\nComparing the simulated maximal oscillation ampli-\ntudes and bunch sizes reached in, for example, Fig. 9 and\nthe lower left plot of Fig. 10, shows a stronger blow up\nof the beam for the positive sign of the ADTS. The same\ndifference is observed in measurements when comparing\nthe maximal amplitude of the center-of-mass oscillation\nin Fig. 2 and the upper left plot in Fig. 10.\nThis indicates, that the level of “blow up” at which\nthe instability is contained and finds some kind of equi-\nlibrium, pseudo-stable state is different for negative and\npositive ADTS. The asymmetry is visible clearly in\nFig. 11, where for a bunch current slightly above thresh-\nold, the maximal bunch size and the maximal oscillation\namplitude of the center-of-mass is given as a function of\nADTS coefficient for simulations1. The range in ADTS\ncoefficient where partial current loss would occur proba-\nbly depends on the combination of the center-of-mass os-\ncillations and the total bunch size which, above a certain\nvalue, would lead to parts of the charge being “scraped”\nby the beam pipe. Independent of the exact value, it can\nbe seen from Fig. 11 that the affected ADTS range would\nnot be symmetric around zero but rather shifted to pos-\nitive ADTS coefficients. In measurements, the same de-\npendence of the maximal center-of-mass oscillation am-\n1To be more robust against outliers the 95th percentile of the\nbunch size and the center-of-mass oscillation amplitude are\ntaken.8\n0 10000 20000 30000 40000 50000 60000\nTurns / 10.3\n0.2\n0.1\n0.00.10.20.3Center-of-mass position / mm\n0 10000 20000 30000 40000 50000 60000\nTurns / 10.3\n0.2\n0.1\n0.00.10.20.3Center-of-mass position / mm\nFigure 10. Top: Measured center-of-mass oscillation at 3.07 mA (left) and 2.79 mA (right) and an ADTS coefficient b= 13720/m.\nThe measurement at high current show a similar sawtooth pattern to measurements at negative ADTS coefficient (Fig. 2). The\nmeasurement at low current is more constant (noise level would be +-0.02 mm). Bottom: Simulated center-of-mass oscillation\nand bunch size at b= 15000/m and bunch currents of 4.8 mA (left) and 4.6 mA (right).\nplitude2as a function of the ADTS coefficient is observed\n(Fig. 12) for negative ADTS. While it is not measur-\nable at the lower positive ADTS coefficients, due to the\npartial beam losses, the measured values at higher posi-\ntive ADTS coefficients are higher than the corresponding\nvalues at negative ADTS showing the same asymmetry\nas in the simulations in Fig. 11. So to reach the same\nlevel of suppression of the instability, meaning low values\nin maximal bunch size and center-of-mass oscillations, a\nhigher positive than negative ADTS coefficient would be\nneeded. This asymmetry observed in both measurement\nand simulations supports the explanation for the asym-\nmetry observed in the beam loss at the threshold (see\nFig. 7). Figure 11 also shows the value of the aver-\nage action of the particle ensemble ⟨J⟩at the times of\nminimal bunch size and center-of-mass oscillations3as a\nfunction of ADTS coefficient, again with the asymmetry\nfor the different signs of the ADTS coefficient visible. The\nminimal value ⟨J⟩reaches can be connected to the point\nwere the instability is no longer damped and the beam\n2Again, the 95th percentile of center-of-mass oscillation is taken.\n3The 5th percentile is taken as value for the minimal action ⟨J⟩.\n15000\n 10000\n 5000\n 05000 10000 15000\nADTS coefficient / 1/m0.000.250.500.751.001.251.501.75COM oscillation amplitude 95%ile / mm\n Bunch size 95%ile / mmCOM 95%ile\nSize 95%ile\nJ 5%ile\n0.000.250.500.751.001.251.501.752.00\nAction J 95%ile / m\n1e6\nFigure 11. Simulated maximal (95percentile) bunch size\nand center-of-mass oscillation amplitude and minimal (5per-\ncentile) action ⟨J⟩as a function of the ADTS coefficient for\ncurrents slightly above threshold. The lines highlight the\n1/x1/2(bunch size and COM) respective 1 /x(⟨J⟩dependency\nwith the dotted line being the mirror of the solid line at neg-\native ADTS.9\n15000\n 10000\n 5000\n 0 5000 10000 15000\nADTS coefficient / 1/m0.20.30.40.50.60.7COM oscillation amplitude 95%ile / mm\nFigure 12. Measurements of the maximal center-of-mass os-\ncillation amplitude (95percentile) as a function of the ADTS\ncoefficient. The Measurements were taken at bunch currents\nclose to the threshold and with a chromaticity of 0.05. The\nlines highlight the 1 /x1/2dependency with the dotted line\nbeing the mirror of the solid line at negative ADTS. The er-\nrors show the standard deviation between multiple consecu-\ntive measurements per point.\n15000\n 10000\n 5000\n 0 5000 10000 15000\nADTS coefficient / 1/m0.0004\n0.0002\n0.00000.00020.00040.00060.00080.0010Tune shift / 1calculated from:\nJ 95%ile\nJ 5%ile\nFigure 13. Tune shift calculated from the maximal (95per-\ncentile) and the minimal (5percentile) action ⟨J⟩as function\nof ADTS coefficient.\nbecomes unstable again, as will be described further in\nthe next section (Sec. IV D).\nBoth, the maximal bunch size and center-of-mass oscil-\nlation amplitudes as well as the minimal action ⟨J⟩show\na characteristic dependence on the ADTS coefficient. For\n⟨J⟩it follows a 1 /xdependency and the bunch size as well\nas the center-of-mass oscillation amplitude has a 1 /√x\ndependency. This already hints at a connection with the\ntune shift via Equ. 3. Going one step further and cal-\nculating the tune shift due to ADTS from the simulated\nvalues of the minimal action ⟨J⟩for each ADTS coeffi-\ncient, shows a constant but different level of tune shiftfor each sign of the ADTS (Fig. 13). While for negative\nADTS the calculated tune shift of ≈0.00012 is approxi-\nmately 8% of the synchrotron tune, the shift for positive\nADTS is with ≈0.00094 already 65% of the synchrotron\ntune. This significant difference gives rise to the hypoth-\nesis, that for the positive and negative ADTS a different\nlevel of tune shift is required to contain the instability.\nThe tune shift calculated from the maximal ⟨J⟩shows a\nvery similar behavior, where the small difference between\nmaximal and minimal values on the positive side of the\nADTS is explained by the fact that nearly no sawtooth\nis observed for positive ADTS coefficients and currents\nclose to the threshold (compare Fig. 10). Overall, it can\nbe concluded that the tune shift stays between these two\nlevels, with the higher one being the point were the beam\nstabilises and starts to damp and the lower one indicates\nwhen the stabilizing effect stops and the beam goes un-\nstable and blows up again. This is supported in the next\nsection by theoretical calculations evaluating the insta-\nbility threshold of the transverse mode coupling taking\ninto account the Landau damping.\nD. Theoretical Calculations\nFigure 14 shows stability diagrams calculated from\nEq. 6 for the cases of positive and negative ADTS. Con-\ntours are drawn by plotting the imaginary part of the in-\nverse dispersion relation I−1\nmagainst the real part. These\ncontours map out a teardrop shape in complex frequency\nspace pointing towards negative coherent tune shifts in\nthe case of negative ADTS. Changing the sign of the\nADTS to positive reflects the contour about the line of\nzero coherent frequency shift. Also shown on the fig-\nure are the eigenvalues of the coupling matrix without\nLandau damping where two modes are included: the az-\nimuthal head tail modes m= 0 and m=−1. For a head-\ntail mode to be stable in isolation, its complex coherent\nfrequency shift would have to be within the Landau con-\ntour. The condition in the presence of mode coupling is\nslightly different, given by the zero determinant in Eq. 6,\nbut the images are still illustrative none the less. It can\nbe seen that, in order to influence the stability, a posi-\ntive ADTS coefficient must be much larger in magnitude\nthan a negative one, matching the observations in mea-\nsurement and simulation shown in Fig. 11 and 12. This\nis intuitive as a negative ADTS coefficient means that\nthe tune spread is towards negative tune shifts and the\ncurrent-dependent tune shift of the m= 0 mode is also\nnegative for most broadband impedances.\nOne feature of storage rings used for fourth-generation\nsynchrotron light sources, particularly those using low-\nRF frequencies (such as the 100 MHz of the 3 GeV ring\nat MAX IV), that is beneficial in this regard is the low\nincoherent synchrotron frequency. This means that the\ncoupling frequency of the m= 0 and m=−1 head-tail\nmodes is not so far out of the spread in betatron tune of\nthe electron bunches. Nevertheless, both in simulation10\nFigure 14. Stability diagrams for negative ADTS coefficient (left) and positive ADTS coefficient (right). The numbers that\nlabel the Landau contours indicate the magnitude of the tune shift at ⟨J⟩ ×103. The colored points are the eigenvalues of the\nscaled coupling matrix νsMmk\nnlwhere the color represents the bunch current.\nFigure 15. Predicted equilibrium actions ⟨J⟩and those deter-\nmined from simulations in Mbtrack2 for an ADTS coefficient\nb= 1000 tm−1.\nand measurement at the 3 GeV ring at MAX IV, the\nmagnitude and sign of the ADTS coefficient does not im-\npact the threshold current of the TMCI. As discussed in\nSec. IV A, this is because the point of the mode-coupling\nis outside the tune spread of the bunch when it is stable.\nWhen the bunch goes unstable and increases in size, how-\never, the tune spread increases until the Landau damping\nkicks in. Figure 15 shows the saturation equilibrium aver-\nage particle actions ⟨J⟩in simulation and from the results\nof Eq. 6 for the two azimuthal head-tail modes and for a\ngiven ADTS coefficient b. Radiation damping is included\nin the simulations so they show the same sawtooth be-\nhavior seen in the measurements. The saturation action\n⟨J⟩is then taken as the minimum in this sawtooth pat-tern, corresponding to the point where the damping stops\nand the bunch blows up again, because during the rest\nof the sawtooth period, the beam is either being damped\n(and is therefore stable) or has some centroid motion,\nwhich has a negative effect on the Landau damping (this\nis a potential root cause of the sawtooth behavior). The\nagreement is not perfect, particularly in the value of the\nthreshold current. This may change with the number\nof head-tail modes included in the theoretical prediction,\nalthough including more significantly complicates the nu-\nmerical optimisation.\nE. Betatron Tune Shift with Current\nObserving the vertical betatron tune as function of\ncurrent directly shows the expected current-dependent\ntune shift due to the transverse impedance from the zero-\ncurrent tune of ν0= 0.275 towards the -1 mode at the\nfirst synchrotron frequency side band ( ν0−0.00146). In\nsimulations of the coherent beam spectrum the threshold\nof the TMCI is clearly visible as the current at which the\ntune couples to the -1 mode (top row in Fig. 16). This is\nthe same for both signs of the ADTS. The difference for\nnegative and positive ADTS starts above the threshold\nwhere for negative ADTS the tune seems to continue its\nshift towards a lower tune with a similar slope as below\nthe threshold (Fig. 16a). For the positive ADTS, the be-\nhaviour looks very different. While a slight shift in the\nopposite direction to higher tunes would not be unex-\npected, due to the positive sign of the ADTS, the tune\njumps within a very small current range above the thresh-\nold from the -1 mode back to the 0 mode and then shows\na continuous shift to higher tunes from there (Fig. 16b).\nThis drastic difference in behaviour can also be seen in\nmeasurements. Figures 16c and 16d show the measured11\n(a) Simulation at negative ADTS\n (b) Simulation at positive ADTS\n(c) Measurement at negative ADTS\n (d) Measurement at positive ADTS\nFigure 16. Coherent motion spectrum showing the current dependent betatron tune shift below and above the instability\nthreshold. Simulation (top): Fourier transform of the center-of-mass oscillation plotted as a function of the bunch current for\nnegative (left) and positive (right) ADTS coefficient b= 15000/m. Measurement(bottom): Fourier transform of the center-of-\nmass oscillation as a function of bunch current for an ADTS coefficient of b=−10000/m (left) and b= 13720/m. During the\nmeasurement the instability was “switched” on and off (see text).\ntune spectra at different bunch currents and for nega-\ntive and positive ADTS respectively. The measurements\nwere conducted in such a way that the previously de-\nscribed hysteresis of the instability threshold (Sec. IV A)\nwas used to get comparative measurements for the tune\nof a stable and an unstable beam. To this end, the mea-\nsurement was started at high bunch currents and the tune\nspectrum was recorded alternately for a beam stabilized\nby the BBB feedback system4and for an unstable beam,\nwhere the instability was triggered by a short excitation5.\nFor the stable beam, the tune continues its current-\ndependent shift towards lower values. For both signs of\n4After initial stabilization the feedback is switched off during the\nmeasurement.\n5The excitation is switched off as well before the measurement is\ntaken.the ADTS, it is clearly visible that the presence of the\ninstability shifts the tune compared to the tune of the\nstable beam. For negative ADTS (Fig. 16c), the shift is\nsmall and towards slightly lower tune values. For posi-\ntive ADTS (Fig. 16d), the tune is shifted back towards\nthe zero-current tune (0 mode) and shows a very small\ncurrent-dependent shift towards higher tune. Except for\nthe difference in threshold and the threshold hysteresis\nobserved in the measurements, the simulation and the\nmeasurements agree very well with respect to the tune\nshifts below and above threshold.\nAt higher bunch currents, additional features appear.\nIn the measurement at negative ADTS, an upper and\nlower sideband shows up, moving with the tune as func-\ntion of current. In the case of the positive ADTS the\ntune peak is broadened greatly and nearly spans from\nthe -1 mode to the +1 mode. Comparing with the cal-\nculated tune shift of ≈0.001 resulting from the maximal12\naction Jsimulated in case of positive ADTS (Fig. 13),\nshows that the jump of the coherent betatron tune by\none synchrotron tune ( νs= 0.00146) towards the 0 mode\nis only slightly bigger. Overall, from the measurement\nand tracking simulations it is not apparent whether this\ndifference in the behaviour of the coherent tune above\nthreshold is the cause or a consequence of the observed\nasymmetry in the level of beam “blow up” for negative\nversus positive ADTS.\nV. SUMMARY AND CONCLUSION\nLandau damping has been investigated in the past\nas a possible source as mitigation mechanism of mode-\ncoupling instabilities, also in connection with the\namplitude-dependent tune shift as the source of the re-\nquired tune spread.\nWhile during standard operations the bunch current\nat the 3 GeV ring at the MAX IV Laboratory is below\nthe TMCI threshold, an asymmetric dependence on the\nsign of the amplitude-dependent tune shift (ADTS) has\nbeen previously observed in dedicated experiments. Sys-\ntematic studies were now conducted to investigate this\nobserved asymmetry in dedicated single bunch experi-\nments. It was observed, that for some ADTS coeffi-\ncients the beam was lost when crossing the threshold\nwhile at others a saw-tooth shaped amplitude modula-\ntion was observed on the center-of-mass oscillation as\nwell as on the bunch size leading to a self-contained in-\nstability. The presented simulations with the tracking\ntool mbtrack2 and the conducted measurements are in\ngood agreement. Both show that the observed threshold\ncurrent is independent of the ADTS coefficient and an\nobserved hysteresis in the measured threshold can be at-\ntributed to intra-beam scattering effects. For the dynam-\nics above the threshold, both measurements and simula-\ntion, show that for positive ADTS coefficients the maxi-\nmal center-of-mass oscillation amplitude and bunch size\nthat is reached before the instability stabilizes is system-\natically higher than for negative ADTS coefficients, indi-\ncating that this could be the cause of the observed partial\nbeam current losses. The same asymmetry is also visible\nin the tune shift at the minimal ⟨J⟩required for damp-\ning. The shift is constant and the value is only dependent\non the sign of the ADTS, with the tune shift calculated\nfor positive ADTS coefficients already being at ≈65% of\nthe synchrotron tune. Stability diagrams with Landau\ncontours, calculated to include the amplitude-dependent\ntune shift, show as well that higher positive ADTS co-\nefficients compared to negative ones are required for the\ninstability to be Landau damped. Furthermore, simula-tions and measurements of the coherent tunes as a func-\ntion of the bunch current, show a strong difference in the\ntunes development above threshold. For negative ADTS\ncoefficients the tune is slightly shifting to lower values\nstarting from the -1 mode at the threshold. In contrast,\nfor positive ADTS coefficients, the coherent tune jumps\nback to the 0 mode and only then shows with increasing\ncurrent a slight shift to higher tune values, as might be\nexpected for positive ADTS coefficients.\nOverall, it can be concluded that, for the presented\nmeasurements, the sign of the ADTS leads to an asymme-\ntry in how strongly the vertical TMCI is self-containing\nand in which tune shift with current is observed above\nthe threshold. As expected, a higher absolute value of the\nADTS coefficient leads to a lower maximal center-of-mass\noscillation and bunch size blow up. But when comparing\nsigns, a higher positive ADTS coefficient is required for\nthe same amount of suppression of the instability than\nfor a negative ADTS.\nCompared to previous studies on using Landau damp-\ning together with ADTS to mitigate the TMCI, two dif-\nferences were found for the presented investigations for\nthe parameters at the 3 GeV ring at MAX IV. It is clear\nthat, the Landau damping comes in as a stabilization\nmechanism only after the beam begins to go unstable,\nas the threshold was shown to not be dependent on the\nADTS coefficient, leading to the observed saw-tooth pat-\ntern. The second result and rather interesting finding is\nthat the maximal level that the center-of-mass motion\nand bunch size reaches before being contained by Lan-\ndau damping is higher for a positive than a negative sign\nof the ADTS coefficient. This could not only be shown\nin the measurement but also in tracking simulations and\nin theoretical stability considerations including Landau\ndamping and the ADTS as well.\nA key contribution to making this asymmetry visible,\nmight by the rather low synchrotron frequency at the\n3 GeV ring due to the low momentum compaction fac-\ntor, typical in fourth-generation light-source storage rings\ncombined with the low RF frequency of 100 MHz. Both of\nthese aspects lead to a synchrotron frequency that starts\nto be within the betatron tune spread of the bunch when\nit is blown up by the instability. This would indicate\nthat the presented findings might become more relevant\nin new fourth-generation light-source storage rings with\neven more extreme parameters than MAX IV.\nACKNOWLEDGMENTS\nThe computations were enabled by resources provided\nby LUNARC.\n[1] H. G. Hereward, Landau damping by non-linearity , Tech.\nRep. (CERN, Geneva, 1969).[2] N. Mounet, Landau damping in the transverse plane,\nCERN Yellow Reports: Conference Proceedings Vol.13\n9, 45 Pages (2020), artwork Size: 45 Pages Publisher:\nCERN Yellow Reports: Conference Proceedings.\n[3] L. Carver, X. Buffat, K. Li, E. M´ etral, and M. Schenk,\nTransverse beam instabilities in the presence of linear\ncoupling in the Large Hadron Collider, Physical Review\nAccelerators and Beams 21, 044401 (2018).\n[4] P. F. Tavares, E. Al-Dmour, ˚A. Andersson, F. Cullinan,\nB. N. Jensen, D. Olsson, D. K. Olsson, M. Sj¨ ostr¨ om,\nH. Tarawneh, S. Thorin, and A. Vorozhtsov, Commis-\nsioning and first-year operational results of the MAXIV\n3GeV ring, Journal of Synchrotron Radiation 25, 1291\n(2018).\n[5] F. J. Cullinan, Collective effects in MAX IV (Presented\nat the 7th Low Emittance Rings Workshop, 2018).\n[6] Y. H. Chin, Hamiltonian Formulation for Transverse\nBunched Beam Instabilities in the presence of Betatron\nTune Spread, CERN SPS/85-9 (1985).\n[7] P. F. Tavares, S. C. Leemann, M. Sj¨ ostr¨ om, and ˚A. An-\ndersson, The MAXIV storage ring project, Journal of\nSynchrotron Radiation 21, 862 (2014).\n[8] S. Kostoglou, N. Karastathis, Y. Papaphilippou, D. Pel-\nlegrini, and P. Zisopoulos, Development of Computa-\ntional Tools for Noise Studies in the LHC, in Proc. of\nInternational Particle Accelerator Conference (IPAC 17),\nCopenhagen, Denmark, 14 to 19 May, 2017 .\n[9] Konstantinos Paraschou and Sofia Kostoglou and Dario\nPellegrini,.\n[10] J. Breunlin and A. Andersson, Emittance Diagnostics at\nthe Max Iv 3 Gev Storage Ring, in Proc. of International\nParticle Accelerator Conference (IPAC’16), Busan, Ko-\nrea, May 8-13, 2016 , International Particle Accelera-\ntor Conference No. 7 (JACoW, Geneva, Switzerland,\n2016) pp. 2908–2910, doi:10.18429/JACoW-IPAC2016-WEPOW034.\n[11] A. Gamelin, W. Foosang, and R. Nagaoka, mb-\ntrack2, a Collective Effect Library in Python, in Proc.\nIPAC’21 , International Particle Accelerator Confer-\nence No. 12 (JACoW Publishing, Geneva, Switzerland,\n2021) pp. 282–285, https://doi.org/10.18429/JACoW-\nIPAC2021-MOPAB070.\n[12] G. Skripka, R. Nagaoka, M. Klein, F. Cullinan, and P. F.\nTavares, Simultaneous computation of intrabunch and in-\nterbunch collective beam motions in storage rings, Nu-\nclear Instruments and Methods in Physics Research Sec-\ntion A: Accelerators, Spectrometers, Detectors and As-\nsociated Equipment 806, 221 (2016).\n[13] G. Skripka, A. Andersson, F. Cullinan, R. Na-\ngaoka, and P. Tavares, Impedance Characterization\nand Collective Effects in the MAX IV 3 GeV\nRing, in Proc. of North American Particle Accelera-\ntor Conference (NAPAC’16), Chicago, IL, USA, Oc-\ntober 9-14, 2016 , North American Particle Accelera-\ntor Conference No. 3 (JACoW, Geneva, Switzerland,\n2017) pp. 843–846, https://doi.org/10.18429/JACoW-\nNAPAC2016-WEA3CO04.\n[14] A. W. Chao, Physics of collective beam instabilities in\nhigh-energy accelerators (1993).\n[15] P. Alexandre, R. B. E. Fekih, A. Letr´ esor, S. Thoraud,\nJ. da Silva Castro, F. Bouvet, J. Breunlin, ˚Ake Ander-\nsson, and P. Fernandes Tavares, Transparent top-up in-\njection into a fourth-generation storage ring, Nuclear In-\nstruments and Methods in Physics Research Section A:\nAccelerators, Spectrometers, Detectors and Associated\nEquipment 986, 164739 (2021).\n[16] K. Y. Ng, Physics of intensity dependent beam instabil-\nities (World Scientific, Hoboken, NJ, 2006) publication\nTitle: Hackensack, USA: World Scientific (2006) 776 p." }, { "title": "2401.15104v1.On_the_supposed_mass_of_entropy_and_that_of_information.pdf", "content": "On the supposed mass of entropy and that of information\nD. Lairez∗\nLaboratoire des solides irradi´ es, ´Ecole polytechnique, CEA, CNRS, IPP, 91128 Palaiseau, France\n(Dated: January 30, 2024)\nIn the theory of special relativity, energy can be found in two forms: kinetic energy and rest\nmass. Potential energy of a body is actually stored under the form of rest mass, interaction energy\ntoo, temperature is not. Information acquired about a dynamical system can be potentially used\nto extract useful work from it. Hence the “mass-energy-information equivalence principle” that has\nbeen recently proposed. In this paper, it is first recalled that for a thermodynamic system made of\nnon interacting entities at constant temperature, the internal energy is also constant. So that, the\nenergy involved in a variation of entropy ( T∆S) differs from a change in potential energy stored\nor released and cannot be associated to a corresponding variation of mass of the system, even if it\nis expressed in term of quantity of information. This debate gives us the opportunity to deepen\nthe notion of entropy seen as a quantity of information, to highlight the difference between logical\nirreversibility (a state dependent property) and thermodynamical irreversibility (a path dependent\nproperty) and to return to the nature of the link between energy and information that is dynamical.\nINTRODUCTION\nThe link between information and energy finds its ori-\ngin in Maxwell’s demon who, by acquiring information\nabout a thermodynamic system, is able to act on it and\nproduce work in return [1]. Later, Shannon [2] formalized\nmathematically this link by considering the quite differ-\nent problem of information processing. He demonstrated\nthat the minimum average number Hof bit to encode a\nrandom variable emitted by a source, let say the current\nmicrostate of a dynamical system, is equal to a factor ln 2\nto the Gibbs entropy S, that is itself equal to the Clau-\nsius entropy of the system: S=Hln 2 (in this paper Sis\ndimensionless and temperature Tis in Joule). Hence the\nlink: from the second law of thermodynamics, acquiring\none bit of information about a dynamical system has a\nminimum energy cost equal to Tln 2 that can in return\nbe potentially used to extract at best the same quantity\nof energy from the system.\nLandauer followed by Bennett [3–5] tackled the prob-\nlem in a quite different way. In their spirit, the logical\nstates 0 or 1 of one bit of information correspond neces-\nsarily to two different thermodynamic states. Even more,\nany irreversible logical operation, such as erasing one bit,\ncorresponds to an irreversible non-quasistatic thermody-\nnamic process that consequently has a non-zero mini-\nmum energy cost when performed cyclically. This is the\nso called “Landauer principle”. In this way, it is believed\nthat “Information is physical” [6] in a much convincing\nmanner than with Shannon’s information theory.\nBased on the Landauer-Bennett idea, a new step (in\na wrong direction) has recently been done. Information\nstored under the form of physical bits is considered a kind\nof potential energy to which, in the framework of spe-\ncial relativity, it can be assigned a mass [7–10]. This is\nthe “mass-energy-information equivalence principle” that\n∗didier.lairez@polytechnique.edu“states that information is a form of matter, it is phys-\nical, and it can be identified by a specific mass per bit\nwhile it stores information... It is shown that the mass\nof a bit of information at room temperature (300K) is\n3.19×10−38Kg.” [7]. This idea has been already crit-\nicized [11] at an epistemological and ontological level:\nwhat exactly does “physical” mean in “Information is\nphysical” ? [6]. The aim of this paper is to show that this\nidea is also false for at least three reasons, which this\ntime are at a more prosaic level.\nIn the first section, the “mass-information equivalence\nprinciple”, is addressed from the thermodynamic side as a\n“mass-entropy equivalence principle”. It is a recall of the\nbasic difference between potential energy and entropy:\nthe elastic energy of a spring is fundamentally different\nfrom that of a rubber or from that of a compressed vol-\nume of gas. For a spring it originates from a microscopic\ninteraction potential, whereas it is emergent for a rubber\nor a gas. It will be shown that a monothermal variation\nof entropy ( T∆S) of a body is not accompanied by any\nvariation of its mass.\nIn the second section, the “mass-energy-information\nequivalence principle” is addressed at its root, that is to\nsay the Landauer principle. In a previous paper [12], it\nhas been shown that the Landauer-Bennet idea cannot be\na general principle but is only true in a particular case. It\nfollows that any derivative of this “principle” is logically\nruled out. Here, new examples will be given to illus-\ntrate that logical and thermodynamical irreversibilities\nare uncoupled. In fact, as defined by Landauer himself,\nthe logical irreversibility of an operation is intrinsic to\nits initial and final states and is independent of the path\nused to achieve the operation, contrary to the thermody-\nnamic irreversibility that is a property of the path.\nIn the third section, the last argument against the\n“mass-energy-information equivalence principle” is given:\nthe link between information and energy is valid for fresh\ninformation about a dynamical system. Old information,\nor information detached from its subject matter, is no\nlonger information and has no value.arXiv:2401.15104v1 [physics.gen-ph] 25 Jan 20242\nI. POTENTIAL VERSUS ENTROPIC FORCES\nAfter Shannon [2], we know that the thermodynamic\nstate quantity S, named entropy and introduced by Clau-\nsius [13] to account for exchanges of heat during a process,\nis to a factor ln 2 mathematically equal to the minimum\naverage number Hof bits necessary to encode in which\nmicrostate the dynamical system is currently, namely the\nquantity of information emitted by the system.\nS=Hln 2 (1)\nEven if Shannon’s information theory is not used by Lan-\ndauer and Bennett, they do not question its correctness.\nIt follows that the hypothetical “mass-information equiv-\nalence” is nothing other than a “mass-entropy equiva-\nlence” that can be addressed in a pure thermodynamic\nframework in the context of special relativity. This is the\naim of this section.\nIn the theory of special relativity, the energy of a body\ntakes two forms: kinetic energy and mass (or rest mass,\nor rest energy). Mass is the energy that is stored by the\nbody when it is at rest. For any monothermal transfor-\nmation, the product of temperature T(in Joule) to the\nvariation of entropy ∆ S(dimensionless) has the dimen-\nsion of an energy. Behind the idea of a “mass-entropy\nequivalence” is that T∆Sis a sort of potential energy\nstored (or released) somewhere in the system at the end\nof the transformation, to which it can be attributed an\nequivalent mass difference T∆S/c2(where cis the celer-\nity of light), in virtue of the Einstein famous equation.\nBefore to address this analogy, let us first deal with the\ncase of potential energy in mechanics.\nA. Mechanical potential energy\nWhen a force is applied to a body over a given dis-\ntance, mechanical work is done, that is to say energy is\ntransferred from one body (the one applying the force)\nto another (the one we are interested in). But an impor-\ntant point is that “Work is a process; once done it no\nlonger exists. It is something that cannot be stored; what\nis stored is energy” [14].\nWhen a stone is transported from the ground to a ta-\nble, mechanical work is done against the force of gravity.\nThe energy transferred to the stone is recovered under the\nform of kinetic energy when the stone falls back down. If\nenergy is conserved, where is it between these two pro-\ncesses? We usually say that it is stored under the form\nof potential energy in the earth-stone system. But as\nnoted by Hecht [15], kinetic energy can be measured, as\nwell as the work that has been done, without affecting\ntheir integrity, but the potential energy of the stone on\nthe table cannot. When we measure it, it is no longer\npotential energy, it is kinetic energy. Potential energy\nis a concept that was introduced to ensure the conser-\nvation of energy, the energy is actually stored under theform of mass, a physical quantity that can in principle\nbe measured without affecting its integrity. For instance,\nthe mass equivalence of potential energy can be measured\nfor nuclear fission: the mass of a nucleus is smaller than\nthe sum of those of its nucleons taken independently. The\ndifference is due to the attractive strong interactions be-\ntween nucleons and is divided between the different parts\nwhen they separate. Even if it is not measurable for a\nstone on a table, for the consistency of the theory we are\nobliged to assume the same effect. The stone has more\nmass on the table than on the ground.\nThe elastic potential energy of a constrained metal\nspring is of the same nature (see Fig.1). “When work is\ndone on the spring, the spring’s rest energy increases in\nthe form of ∆m”[15]. Compared to the gravitational po-\ntential energy of a stone that has an arbitrary zero at the\nground level, the spring can be stretched or compressed\nwith an identical restoring force (up to the sign) towards\nthe equilibrium position that unambiguously defines the\nzero of potential energy of the spring. This equilibrium\nposition originates from the microscopic net interaction\npotential between the atoms of the crystal: each atom is\nin the minimum of a potential well made by the presence\nof others. The work necessary to constrain the spring is\nthat needed to deviate atoms from this minimum. The\npotential energy of the spring is the sum of those of its\natoms.\nFIG. 1. When a weight (in brown) compresses a vertical\nspring (in blue) it undergoes a decrease of mass (less poten-\ntial energy), while that of the spring increases (more elastic\npotential energy).\nNot only metal springs are elastic. So are pieces of\nrubber. But contrary to what it was suggested in ref. [15]\n(but this point is marginal in the paper) the origin of this\nelasticity is different. It has the same origin as that of a\nvolume of gas in thermal equilibrium with its surround-\nings but at a different pressure. It is entropic [16].\nB. Entropic forces\nConsider a volume of gas in a container equipped with\na piston allowing its contents to be compressed or ex-3\npanded. Like the spring, the piston has an equilib-\nrium position that corresponds to equal forces applied\non it. When deviating it from this position by pushing\nor pulling, we feel an elastic restoring force that is ap-\nparently comparable to what it would be if the gas were\nreplaced by a spring. So that it is legitimate to state that\nwhen the piston deviates from its equilibrium position\nthe overall system stores an amount of elastic potential\nenergy. But at a microscopic level, for a perfect gas there\nis no interaction potential between molecules. Even for\nreal gas, for which pair-interactions can be modelized by\na Lennard-Jones potential, interactions can be neglected\nas soon as the particles are not in contact (between two\ncollisions). Gas particles do not interact at distance and\ndo not have equilibrium position.\nFIG. 2. If the spring of Fig.1 is replaced by a gas compressed\nwith a piston at constant temperature, the gas has less en-\ntropy but the same internal energy and its mass is unchanged.\nThe elastic force we feel on the piston is due to the\nbalance between the many collisions it experiences with\nthe gas molecules on both sides (each collision involving\nthe kinetic energy of one given particle) and the force\napplied by the surroundings on the piston.\nFirst, note that the internal energy of a system made\nofNnon-interacting independent entities is its tempera-\ntureT(in Joule). For the sake of simplicity, let us assume\nthat this is the case for the gas inside the container and\nfor the atmosphere outside (that of the surroundings),\nwhich is reasonable in the case where both are air close\nto atmospheric pressure. Neglecting interactions means\nin particular neglecting hydrodynamic interactions, fric-\ntion and viscosity and thus the time delay to reach the\nequilibrium after any perturbation. In terms of thermo-\ndynamics it means that the overall system (gas plus sur-\nroundings) is always at equilibrium and that the trans-\nformation is reversible. The existence of such a reversible\nprocess allowing to pass from one state to another is the\nonly way in phenomenological thermodynamics to mea-\nsure (and thus to define) entropy that is given by its exact\ndifferential:\ndS=dQr\nT(2)where Qris the heat exchanged for a reversible process.\nNote that the notion of instantaneous equilibrium, and\nconsequently that of reversible process, appears to be in-\ncompatible with special relativity because in principle\nnothing (and in particular the propagation of a pertur-\nbation) can go faster than light. The same issue exists\nfor mechanical potential energy [17, 18]. But this is not a\nproblem as far as we are concerned by the initial and final\nstates of a process and not by the process in itself (e.g.\nas far as we consider monothermal and not isothermal\nprocesses). The notion of reversible process in thermo-\ndynamics is equivalent to the classical mechanics limit\nof special relativity, which is conditioned on the two as-\nsumptions: velocities of particles are small compared to\nthat of light; characteristic distances in the system are\nsmall, so are delays in the propagation of signals.\nTo go further into our problem two cases for the gas\ncontainer are worth considering:\n1. the container is adiabatic, i.e. it prevents heat ex-\nchanges with the surroundings;\n2. the container is diathermal, i.e. it allows heat ex-\nchanges with the surrounding.\nConsider the gas in an adiabatic container. Compress-\ning the gas by pushing the piston, we produce work and\nprovide to the gas an equivalent amount of energy. Do-\ning so, as the gas cannot dissipate heat, its internal en-\nergy necessarily increases, and so its temperature. From\nEq.2, it follows that if there is no heat exchanged, there is\nno variation of entropy. Clearly, the reversible adiabatic\ncase is not the matter of the “mass-entropy equivalence”\nthat envisages differences of entropy between two states\nat the same temperature, but it is worth considering to\nwhat follows. Can the increase in internal energy (tem-\nperature) be assimilated to the elastic potential energy?\nNo, it cannot. Because when pulling (instead of pushing)\nthe piston, this time the gas decreases in internal energy,\nbut there is still a restoring force and a positive potential\nenergy stored somewhere. The elastic potential energy is\nin fact stored in the solid container, and in the external\nmechanical part of the device that drives the piston, un-\nder a form quite comparable to that it is for an elastic\nspring, but it is not stored in the gas.\nConsider the ideal diathermal container. Heat ex-\nchanges ensure that for any transformation the initial\nand final states of the gas are both in thermal equilibrium\nwith the surroundings. If in addition, the surroundings\nis so large that its temperature can be considered con-\nstant, the transformation is monothermal: The temper-\natures of the initial and final states are the same. Then\nby definition, the internal energy of the gas is also un-\nchanged. Whatever the work provided to it, the gas does\nnot store additional energy compared to what it initially\ncontained. The system can receive work W, but it dis-\nsipates an equal amount of energy to the surroundings\nunder the form of heat Q, or do the reverse. For any4\nmonothermal transformation:\nW+Q= 0 (3)\nThe entropy change is given by considering a process\nslow enough to ensure a constant temperature at all times\n(isothermal transformation). By integrating Eq.2, one\ngets\nT∆S=Qr=−Wr (4)\nThe differential of the work is d Wr=−PdV, with Pthe\npressure and Vthe volume. As P=NTV−1, one has\ndW=−NTV−1dV, integration from V0toVgives\nT∆S=TNln(V/V 0) (5)\nSo that in the monothermal case, this time under the ac-\ntion of the piston the gas undergoes a change in entropy,\nbut the corresponding elastic potential energy is still not\nstored by the gas itself. It is stored by the container, the\nmechanical part that drives the piston and the surround-\nings. The gas itself does not store more or less potential\n(or internal) energy which could correspond to any vari-\nation in its mass. In this, it differs from the spring (see\nFig.1 and 2).\nThe case of a piece of rubber is even more illustrative\nbecause there is no need of a container. Rubber is made\nof cross linked linear polymer chains which form a three\ndimensional network. Let Nbe the number of indepen-\ndent chain segments of the chain portion between two\nfirst neighbor crosslinks at distance R0when the rubber\nis unconstrained. In this state, the chain conformation\nis random with R0andNlinked by the scaling relation\nN∝R2\n0. Stretching the rubber, causes the distance be-\ntween crosslinks to increase in the same direction, forcing\nthe chain to be less random. The corresponding variation\nof entropy is such as:\n∆S∝ −(R/R 0)2(6)\nBut as for the gas, at constant temperature the internal\nenergy is constant ([16] p.31). It follows that the elastic\npotential energy, even if it originates from the rubber, is\nnot stored inside the rubber but by the mechanical part\nof the surrounding that is responsible for its stretching.\nThis result for a gas or a piece of rubber is actually\ngeneral. The internal energy of a set of independent en-\ntities, such as a set of bits, is its temperature. It follows\nthat any monothermal variation of entropy interpreted in\nterms of potential energy stored under the form of rest\nmass cannot be localized in such a system, but only in\nits surroundings.\nII. LOGICAL VERSUS THERMODYNAMICAL\nIRREVERSIBILITIES\nThe second law of thermodynamics is two folds. The\nfirst is the definition of entropy as a state quantity de-\nfined by its exact differential given by Eq.2 valid for areversible process. The second accounts for the general\ncase. It is the Clausius inequality that writes at constant\ntemperature:\n−Q≥ −T∆S (7)\nThis means that the quantity −Qof heat dissipated\n(and received by the surroundings) is always higher than\n−T∆S. For a system made of independent entities at\nconstant temperature, the internal energy is also con-\nstant (Eq.3) so that the heat dissipation is compensated\nby the same amount of work ( W=−Q) provided to the\nsystem. Generally, heat is unwanted and work is more\nvalued and can be viewed as an energy cost. With Eq.1\nthe Clausius inequality rewrites in this case:\nW≥ −T∆Hln 2 (8)\nConsider the process of reducing the volume of the phase\nspace of the dynamical system (∆ S≤0). The uncer-\ntainty about the current microstate of the system, or the\nquantity of information it emits, is reduced by ∆ S/ln 2.\nThe total amount of information we lack to describe\nthe system is reduced accordingly, as if we had acquired\n∆H= ∆S/ln 2 bits of data about the system. So that\nEq.8 can be expressed per bit (∆ H=−1) of acquired\ndata:\nWacq/bit ≥Tln 2 (9)\nwhich expresses that Tln 2 is the minimum energy cost\nto acquire 1 bit of data about the dynamical system un-\nder consideration. This statement is nothing more than\na reformulation of the Clausius inequality in terms of\nquantity of information.\nEquation 9 suffices in itself to understand the func-\ntioning of demonic engines like that of Maxwell [1] or\nthe simplified version of Szilard [19] (see Fig.3) and their\nphysical implementations under the form of ratchet-pawl\nmechanisms [20, 21]. Each bit of information needed for\nthe engine to work has a minimum energy cost of Tln 2.\nThis is a direct application of the second law that pro-\nvides the link between information and energy.\nFIG. 3. Szilard demon detects when the particle (in blue) is\nin the suitable compartment. In doing so, it acquires 1 bit of\ninformation for an energy cost −W1≥Tln 2 (Eq.9). Then it\ncan install a piston for free ( W2= 0), allowing the device to\nsubsequently produce work ( W3≤Tln 2). The overall energy\nbalance is W3+W1+W2≤0.\nThe Landauer principle [3–5] reaches the same result\nwithout information theory but with an indirect applica-5\ntion of the second law. It is at the heart of the “mass-\nenergy-information equivalence principle” [7, 9]. Let us\nreport some quotes:\n“The M/E/I principle [the mass-energy-information\nequivalence principle] states that information is a form\nof matter, it is physical, and it can be identified by a spe-\ncific mass per bit while it stores information or by an\nenergy dissipation following the irreversible information\nerasure operation, as dictated by the Landauer principle.”\n(Vopson [9])\n“We demonstrated that the Landauer principle supplies\nthe estimation of the minimal mass of the particle allow-\ning the recording/erasure of information within the sur-\nrounding medium at temperature T”(Bormashenko [22]).\nThe purpose of this section is not to discuss the rea-\nsonings leading to these conclusions, but rather to show\nthat the root of them, i.e. the Landauer principle, is not\ncorrect because it results: 1) from considering a doubly\nparticular case; 2) from a confusion between logical and\nthermodynamical irreversibilities.\nA. Landauer “principle” is a doubly particular case\nAt the basis of the Landauer principle allowing Eq.9\nto be found without any reference to the Shannon’s en-\ncoding problem are the two assumptions below:\n1. For a cyclic process (such as that of the Szilard\nengine), the recording or acquisition of a data bit\nis supposed to first require the erasure of that bit.\n2. The erasure of a data bit is supposed to necessar-\nily involve an irreversible non-quasistatic stage (i.e.\nuncontrollable and similar to the free expansion of\na gas), so that when performed cyclically it has a\nminimum energy cost of Tln 2.\nThe first supposed requirement will be discussed in the\nsecond part of this section. Here, we only focus on the\nsecond.\nLandauer and Bennett first imagine a one-to-one cor-\nrespondence between logic and thermodynamic states.\nThey imagine a particle in a bi-stable potential well al-\nlowing two different positions (labeled 0 and 1, respec-\ntively) to be equally stable. The ERASE operation con-\nsists is putting the particle in position 0 (SET TO 0). It\nis done in three elementary stages or operations.\nLandauer’s ERASE procedure:\n1. SET TO S (standard state): set the particle in an\nundetermined position by lowering the energy bar-\nrier between the two positions.\n2. BIAS TO 0: apply a bias favoring the zero position.\n3. STABILIZE: raise the energy barrier and stop the\nbias.The point is that the path chosen by Landauer to achieve\nthe first stage, throws the probability density of the par-\nticle position out of control and causes it to leak from one\npotential well to the other. It is similar to the free ex-\npansion of a gas, initially confined in one half-volume of\na box (with label 0 or 1) and suddenly allowed to occupy\nthe entire volume. Whereas the last two stages can be\ndone in a quasistatic manner equivalent to the isothermal\ncompression the gas in the correct half-volume of the box\n(with label 0). During the first stage, neither heat nor\nwork are exchanged with the surroundings, whereas the\nlast two have an energy cost at least equal to Tln 2. The\nnet energy balance of the total operation is thus:\nWerase 1 bit ≥Tln 2 (10)\nConjointly with the necessity to erase prior to acquire\n1 bit of data, this last equation allows us to found Eq.9.\nThe non-quasistatic irreversibility of the first stage is\nsupposed by Landauer and Bennett to be unavoidable. In\na previous paper [12], it has been shown that the best way\nto avoid any probability-density leakage between the two\npotential-wells is to have only one, but still two logical\nstates. This is subject to the possibility of establishing\na two-to-one correspondence between logic and thermo-\ndynamic states. An example of such correspondence has\nbeen given in ref. [12] which ruins the generality of the\nLandauer principle. For the purpose of this paper, let us\ngive another counter-example.\nIn Fig.4, we imagine a cam that can compress two\nsprings (which can be replaced if we want by two vol-\numes of gas compressed by pistons). The angle of the\ncam defines the bit state, whereas the state of the springs\ndefines the thermodynamic (or mechanical) state. The\ncam has a smooth shape that continuously passes from\nan elliptical section on one side (front) to a circular one\non the other (rear), both being centered on the axis of\nrotation provided with a steering wheel allowing it to be\ndriven. When the steering wheel is pushed (left and right\nin Fig.4), the bit is stabilized in position 0 or 1 by the two\nsprings. In both positions, the constraint they undergo\nare the same and it is not possible to know the bit-state\nby simply observing the state of the springs. There is\nactually only one thermodynamic state and thus a two-\nto-one correspondence between logic and thermodynamic\nstates. The ERASE operation can be done by pulling the\nsteering wheel, so that the springs are in contact with\nthe circular section of the cam and the energy barrier\nbetween the two logical states is zero, the bit is then in\nthe undetermined S-state. Then, turn the steering wheel\nin order to align the red mark of the cam with position 0\n(apply a bias), finally push the steering wheel to restore\nthe energy barrier. The entire operation only involves\nfriction, so that the heat dissipation tends to zero as the\nvelocity of the steering wheel manipulation tends to zero.\nIt is quasistatic. Note that if in addition, the small ra-\ndius of the ellipse is equal to the radius of the circle (as in\nFig.4), the state of constraint of the springs is the same\nwhether the bit is in position 0, D or 1. So that the entire6\nERASE operation can be done at constant elastic poten-\ntial energy of the springs. Note also that in the two logic\nstates (0 and 1), as the elastic potential energies of the\nsprings are the same, their mass in the framework of spe-\ncial relativity is also the same. A set of independent bits,\nbuilt with this physical implementation, can be erased in\na quasistatic manner and with no variation of rest mass.\nFIG. 4. Two-to-one correspondence between logic and ther-\nmodynamic states. The bit-state (in red) is represented by\nthe angle of a cam centered on a rotating axis controlled by a\nsteering wheel. The cam-profile is elliptical in front and cir-\ncular in back and can constrain two vertical springs (in blue)\nthat define the thermodynamic (or mechanical) state. When\nthe steering wheel is pushed (left and right) the bit has two\nstable logical states 0 or 1. By slowly pulling the steering\nwheel, the energy barrier between the two states vanishes in\na quasistatic manner and drives the bit (middle) in an unde-\ntermined S-state (following Landauer-Bennet terminology).\nA two-to-one correspondence could be viewed as a very\nparticular case of physical implementation of logic and\nthe one-to-one correspondence perceived as the general\ncase. Interestingly, the example just given above can be\nslightly modified in order to obtain a one-to-one corre-\nspondence similar to that of Landauer but with no leak.\nFor this, it is enough to decentre the elliptical face of the\ncam (while the circular face remains centered) as it is\nshown in Fig.5. Then, the logical states 0 or 1 are still\nstable and well separated by an energy barrier, but the\nstate of the springs are not the same in both cases. It\nis now possible to know the bit-state by only observing\nin which state are the springs (compressed up or down).\nThere is a one-to-one correspondence between logic and\nthermodynamic states. The procedure to change the bit\nstate (e.g. SET TO 0 or ERASE) is the same as in\nthe previous case and is quasistatic. During this oper-\nation, the potential energy of the springs is first released\n(when pulling the steering wheel), then the same amount\nis stored again (when pushing the steering wheel). The\nsum of the potential energy of the springs is the same\nwhatever the bit state so that it can be set to 0 with no\nFIG. 5. One-to-one correspondence between logic and ther-\nmodynamic states. The mechanism is similar to that of Fig.4\nbut this time the cam is not centered in its elliptical part\n(but is still centered in its circular part). So that depend-\ning on the bit state (in red), the two springs are not equally\nconstrained allowing to identify two different thermodynamic\n(or mechanical) states (in blue) each one corresponding to a\ndifferent logical state. Just like for the two-to-one implemen-\ntation (Fig.4) the bit can be set to 0 (erased) in a quasistatic\nmanner that only involves friction and avoids any leakage be-\ntween the two states.\nchange in rest mass.\nFrom these two counter-examples, it appears clearly\nthat logical and thermodynamical irreversibilities are not\nlinked. The reason for this is explained just below.\nB. Irreversibilities\nThe logical irreversibility of an operation is defined by\nLandauer: “We shall call a device [an operation] logically\nirreversible if the output of a device does not uniquely\ndefine the inputs.” (Landauer [3]). The ERASE opera-\ntion is logically irreversible: two possible initial states 0\nor 1 (the input of the operation) lead to only one final\nstate 0 (the output). Further in the same paper, Lan-\ndauer writes: “Logical irreversibility, we believe, in turn\nimplies physical [thermodynamical] irreversibility” . This\nlast point is discussed in this section.\nAs soon as we deal with the physical implementation\nof a logical operation on a bit, this operation becomes a\nthermodynamical transformation (or a process). With\nthe above definition, it is clear that the logical irre-\nversibility of an operation is defined only by the prop-\nerties of the bit before and after the transformation. The\nlogical irreversibility is a property of the initial and final\nstates of the bit. It is not a property of the path that has\nbeen used to perform the transformation.\nIn other words, a transformation (say a transport) from\nAtoBis logically irreversible if once in Bthe infor-7\nmation from where the system started has been lost, so\nthat it is not possible to return to the starting point\nA. The same transformation is thermodynamically irre-\nversible if it is not possible to return to Aby using the\nsame path backward. Thermodynamical irreversibility is\na path property.\nDue to this fundamental difference, logical and thermo-\ndynamical irreversibilities cannot be linked by a material\nimplication.\nIn thermodynamics, when we deal with the path al-\nlowing a system to change from an initial to a final state,\nwe first wonder whether or not it can be decomposed\nin a succession of infinitely small changes, i.e. in a suc-\ncession of quasi-equilibrium states. In other words, con-\nsider the variation of entropy versus the extent of the\nchange, we wonder whether or not this variation is dif-\nferentiable. If so, the path is said quasistatic and can\nbe potentially reversible if slow down enough. Other-\nwise, the very point where the discontinuity occurs is\nan inherently irreversible step. At this point, the process\nevolves spontaneously in an uncontrollable manner. This\noccurs when a system suddenly finds itself far from equi-\nlibrium when an internal constraint has been released (a\ntypical example is that of the free expansion of a gas).\nConsider such an irreducible step as a process in itself.\nNo work can be extracted from it ( W= 0). At a given\ntemperature, according to Eq.3 and 7, one gets:\n∆S >0 (11)\nAn increase of entropy is in fact a necessary condition\nfor an inherently irreversible (non-quasistatic and irre-\nducible) process to occur. But it is not at all a sufficient\ncondition. The same change in entropy can occur using\na reversible path, otherwise it would not be defined in\nthermodynamics because it could not be associated with\nany measurable quantity.\nConsider the ERASE operation on the thermostatis-\ntics side and let Γ be the phase space. The two initial\npossibilities Γ = {0,1}result in only one Γ = {0}once\nthe bit has been erased. So that\n∆SERASE <0 (12)\nThis means that ERASE is not inherently irreversible,\nbut that any thermodynamic path leading to this oper-\nation can be decomposed into elementary steps (just as\nLandauer did). In fact, in Landauer’s physical imple-\nmentation, only the first step (the one which brings the\nsystem into an indeterminate state S) can possibly be in-\ntrinsically irreversible, because Γ initially {0,1}changes\ninto [0 ,1], so that\n∆SSET TO D >0 (13)\nFor this step, another path than that of Landauer can be\nchosen, quasistatic this time, as shown in section II A.\n“[Logical irreversible] operations are quite numerous in\ncomputer programs as ordinary written: besides erasure,they include overwriting of data by other data” (Ben-\nnett [4]). On the thermostatics side, the status of OVER-\nWRITE depends on which new data replaces the old\nones. For cyclic recording of data, as that needed when\nimplementing a Szilard engine, if the system is in a sta-\ntionary regime, the probability distribution of the data\nis unchanged from one cycle to the next. So that over-\nwriting old data by new ones leaves the phase space un-\nchanged, thus:\n∆SOVERWRITE = 0 (14)\nIt follows that OVERWRITE is also not an inherent ir-\nreversible process. It can be decomposed in quasistatic\nsteps. To “elucidate” the functioning of the Szilard en-\ngine, Landauer-Bennett chose to break down OVER-\nWRITE into ERASE then WRITE, which brings us back\nto the previous discussion about ERASE. But OVER-\nWRITE can be done in a direct manner without ERASE,\njust like an old magnetic tape can be reused without to\nbe reset in a virgin state (erased).\nLogical irreversibility is not linked to thermodynamic\nirreversibility. There is no conceptual impediment for\na logical irreversible operation to be quasistatic and for\nheat dissipation to vanish as the operation slow down.\nIII. INFORMATION IS DYNAMICAL\n“To test the hypothesis [the mass-energy-information\nequivalence principle] we propose here an experiment,\npredicting that the mass of a data storage device would\nincrease by a small amount when is full of digital infor-\nmation relative to its mass in erased state. For 1Tb de-\nvice the estimated mass change is 2.5×10−25Kg.” (Vop-\nson [7]). Beyond the already recognized difficulty of car-\nrying out such a measurement [9], we will show here that\nthis idea is nonsense and inconsistent with everything we\nknow about thermodynamics (and physics).\nThe first argument directly comes from the fundamen-\ntal difference between logical and thermodynamical irre-\nversibilities that has been exposed in the previous section.\nConsider the above hard drive full of data and the three\nexperiments below:\n1. Directly erase the hard drive. This operation is\nlogical irreversible (it is impossible to retrieve the\ndata once they have been erased).\n2. Make a copy of the hard drive then erase the orig-\ninal one. This operation is logical reversible (with\nthe copy it is possible to restore the original hard\ndrive to its original state).\n3. Make a copy, erase one hard drive then the sec-\nond. The erasure of the first hard drive is logical\nreversible whereas that of the second is irreversible.\nFor the copy to be of any use in preserving data integrity,\nit must be physically independent from the original (we8\ncan imagine moving it to the other side of the earth). It\nfollows that the mass defect (if there is one) that would\nbe measured for the above four erase operations would\nhave exactly the same value. If there is a mass defect,\nit has nothing to do with logical irreversibility nor with\ninformation that would be lost or not.\nThe independence of two hard drive also holds for two\nbits. This is implicit in the mass-energy-information\n“equivalence” when it is expressed per bit: “Using the\nmass-energy-information equivalence principle, the rest\nmass of a digital bit of information at room temperature\nismbit= 3.19×10−38kg.” (Vopson [8]). But sometime\nit is explicit: “Essentially, a bit of information could be\nseen as an abstract information particle, with no charge,\nno spin, and rest mass” (Vopson [8]). It follows that\nthe above three experiments performed with a hard drive\ncould be done with a bit with the same conclusion.\nData are stored with bits set either to 0 or 1, the two\nvalues equally participate to the storage of information.\nIf a bit of information has a rest mass, the latter is in-\ndependent of its value 0 or 1. The ERASE operation is\nusually presented as a SET TO 0 operation, but this is a\nconvention and it could be SET TO 1 (as Landauer did\nin his first paper [3]). This suggest another experiment\nin two stages:\n1. Erase (SET TO 0) one given bit of information\n(with value either 0 or 1).\n2. Erase it a second time.\nAccording to the mass-energy-information equivalence\nprinciple, a mass defect should be observed in the first\nstage, while it should not be observed in the second.\nThe only explanation for this difference should be the\nchange in entropy caused by the ERASE: in the first\nstage Γ : {0,1} → { 0}, whereas for the second stage\nΓ :{0} → { 0}. This brings us back to the first section\nof this paper: at constant temperature no change in rest\nmass accompanies a change in entropy.\nIn fact, the data stored in the hard drives or the bits\nabove are not information in the sense given to that word\nby Shannon. The physical support of these data can be\nconsidered as a thermodynamic system in its own right.\nBut for this, it must be read and emit outcomes just like\nother dynamical systems do. To consider these data as\ninformation, they must not be detached from their sub-\nject matter, i.e. from the dynamical system that emits\nthis information. Once detached from this dynamical\nsystem, the information becomes frozen and outdated, it\nhas no value and no link with energy. Let us detail this\npoint.\nIn the word “thermodynamic” there is “dynamic”.\nThis is a truism but apparently worth remembering: ∆ S\nmust be multiplied by temperature Tto become an en-\nergy. In other words, the link between energy and infor-\nmation only exists when the renewal of the latter obeys\nto the same dynamics as that of the system it concerns.\nThis is particularly clear with the Szilard engine. Imag-\nine that the position of the particle is recorded on a harddrive for a given time interval. Once this is done, these\nold data cannot be of any utility to extract energy from\nthe current system in return to that spent on their ac-\nquisition and recording.\nCONCLUSION\nEntropy (and quantity of information) is a concept.\nJust like potential energy is. There are actually many\ncommon points between them. For instance, just like po-\ntential energy, it is not possible to measure entropy with-\nout changing it into something else (i.e. changing poten-\ntial energy into kinetic energy and changing entropy into\nheat). Also, zeros for both quantities may appear arbi-\ntrary and not intrinsic to the nature of things. Neverthe-\nless, these concepts have some fundamental differences.\nPotential energy was introduced to satisfy a conservation\nprinciple for energy (first law of thermodynamics), while\nentropy was introduced to account for the irreversibil-\nity of changes in the form of energy (second law), that\nis to say a change in quality but not in quantity. Basi-\ncally, this difference rules out the idea of a mass-entropy\nequivalence (section I).\nThis idea of a mass-entropy equivalence (or mass-\ninformation equivalence) is actually the last attempt to\nmaterialize the link between information and energy, that\nis to say to make it more “physical”, more tangible, less\nelusive. It originates from the Landauer principle. The\nlatter is actually due to a confusion between logical ir-\nreversibility (that is a state dependent property) and\nthermodynamical irreversibility (that is a path depen-\ndent property). Once this has been clarified, it appears\nthat there is no finite limit of heat dissipation to erase\na bit. In other words, a bit does not store more energy\nwhether it is set to a given data value or erased (section\nII).\nThis brings us to the last confusion at the origin of\nthe mass-information equivalence: stored data is not in-\nformation in the Shannon sense. Stored data are frozen,\ninformation is dynamical. Stored data are actually out-\ncomes of a dynamical system that have been acquired\n(thus brought to our knowledge) then copied somewhere\n(stored). But information is very special, as soon it has\nbeen given (acquired), it is no longer information, it is\noutdated. Information cannot be given twice. The link\nbetween energy and information is that of a dynamical\nsystem as a source of information in the spirit of Shan-\nnon (section III). Just like the link between energy and\nentropy.\nIn 1948, when Shannon [2] identified the minimum\nnumber of bits (which he called quantity of information)\nto encode the behavior of a dynamic system as its statis-\ntical entropy, this was a great advance: entropy became\ninformation. Although this alone was of great impor-\ntance for computer scientists or for communications en-\ngineers, for physicists the major breakthrough did not\nlie in this identification, which may appear to them as a9\nsimple question of vocabulary. The major breakthrough\nwas in the second part of work of Shannon who identified\nalso this quantity of information with a measure of the\nuncertainty about the system. The resulting principle of\nmaximum entropy [23–25] made it possible to legitimizea priori probabilities and resolve many inconsistencies in\nstatistical mechanics (for a review on this topic see [26]).\nEntropy is information, as fascinating as that may be,\nwe must not forget that the relationship also holds in the\nother direction: information is entropy and is just that.\n[1] J. C. Maxwell, Theory of heat , 3rd ed. (Longmans, Green\nand Co., London, 1872).\n[2] C. E. Shannon, A mathematical theory of communica-\ntion, The Bell System Technical Journal 27, 379 (1948).\n[3] R. Landauer, Irreversibility and heat generation in the\ncomputing process, IBM Journal of Research and Devel-\nopment 5, 183 (1961).\n[4] C. H. Bennett, The thermodynamics of computation - a\nreview, International Journal of Theoretical Physics 21,\n905 (1982).\n[5] C. H. Bennett, Notes on Landauer 's principle, reversible\ncomputation, and Maxwell 's demon, Studies in History\nand Philosophy of Science Part B: Studies in History and\nPhilosophy of Modern Physics 34, 501 (2003).\n[6] R. Landauer, Information is physical, Physics Today 44,\n23 (1991).\n[7] M. M. Vopson, The mass-energy-information equivalence\nprinciple, AIP Advances 9, 095206 (2019).\n[8] M. M. Vopson, The information catastrophe, AIP Ad-\nvances 10, 10.1063/5.0019941 (2020).\n[9] M. M. Vopson, Experimental protocol for testing the\nmass-energy-information equivalence principle, AIP Ad-\nvances 12, 035311 (2022).\n[10] E. Dˇ zaferovi´ c-Maˇ si´ c, Missing information in the universe\nas a dark matter candidate based on the mass-energy-\ninformation equivalence principle, Journal of Physics:\nConference Series 1814 , 012006 (2021).\n[11] M. Burgin and R. Mikkilineni, Is information physical\nand does it have mass?, Information 13, 540 (2022).\n[12] D. Lairez, Thermodynamical versus logical irreversibility:\nA concrete objection to Landauer’s principle, Entropy\n25, 1155 (2023).\n[13] R. Clausius, The mechanical theory of heat (Macmillan\n& Co, London, UK, 1879).[14] E. Hecht, Understanding energy as a subtle concept: A\nmodel for teaching and learning energy, American Jour-\nnal of Physics 87, 495 (2019).\n[15] E. Hecht, Relativity, potential energy, and mass, Euro-\npean Journal of Physics 37, 065804 (2016).\n[16] P.-G. de Gennes, Scaling concepts in polymer physics\n(Cornell Univ. Press, 1979).\n[17] L. Brillouin, The actual mass of potential energy, a cor-\nrection to classical relativity, Proceedings of the National\nAcademy of Sciences 53, 475 (1965).\n[18] L. Brillouin, The actual mass of potential energy ii, Pro-\nceedings of the National Academy of Sciences 53, 1280\n(1965).\n[19] L. Szilard, On the decrease of entropy in a thermody-\nnamic system by the intervention of intelligent beings,\nBehavioral Science 9, 301 (1964).\n[20] R. P. Feynman, R. B. Leighton, and M. Sands, The Feyn-\nman lectures on physics (Addison-Wesley, Reading, MA,\n1966) Chap. 46, pp. 1–9.\n[21] L. Brillouin, Can the rectifier become a thermodynamical\ndemon?, Physical Review 78, 627 (1950).\n[22] E. Bormashenko, The Landauer principle: Re-\nformulation of the second thermodynamics law or a step\nto great unification?, Entropy 21, 918 (2019).\n[23] E. T. Jaynes, Information theory and statistical mechan-\nics, Phys. Rev. 106, 620 (1957).\n[24] E. T. Jaynes, Prior probabilities, IEEE Transactions on\nSystems Science and Cybernetics 4, 227 (1968).\n[25] J. Shore and R. Johnson, Axiomatic derivation of the\nprinciple of maximum entropy and the principle of min-\nimum cross-entropy, IEEE Transactions on Information\nTheory 26, 26 (1980).\n[26] D. Lairez, Thermostatistics, information, subjectivity,\nwhy is this association so disturbing? (2023)." }, { "title": "2401.15156v1.Quasibound_state_reminiscent_in_de_Sitter_black_holes__Quasinormal_modes_and_the_decay_of_massive_fields.pdf", "content": "Quasibound state reminiscent in de-Sitter black holes: Quasinormal modes and the decay of\nmassive fields\nMateus M. Corrêa,1,∗Caio F. B. Macedo,1, 2,†and João Luís Rosa3, 4,‡\n1Programa de Pós-Graduação em Física, Universidade Federal do Pará, 66075-110, Belém, PA, Brazil\n2Faculdade de Física, Campus Salinópolis, Universidade Federal do Pará, 68721-000, Salinópolis, Pará, Brazil\n3Institute of Theoretical Physics and Astrophysics, University of Gda´ nsk, Jana Ba˙ zy´ nskiego 8, 80-309 Gda´ nsk, Poland\n4Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia\n(Dated: January 30, 2024)\nMassive perturbations in asymptotic flat black holes leave a distinct signature in their late-time evolution ‘tail’:\nan oscillatory behavior modulated by the Compton wavelength of the field, which can be associated with the so-\ncalled quasibound state spectrum. In asymptotically de-Sitter spacetimes, however, the massive perturbations\nalways leak to the cosmological horizon, which indicates the absence of a quasibound part of the spectrum.\nIn this work, we show that an additional mode exists in asymptotically de-Sitter black holes that produces an\nimprint similar to that of the quasibound states in the late-time behavior of massive scalar perturbations. If the\nCompton wavelength is larger than a certain critical value (which depends on the cosmological constant), the\noscillatory behavior of the tail turns into an exponential decay due to the fact that the de-Sitter mode is purely\nimaginary. Even for black holes with typical length scales small in comparison to the size of the cosmological\nhorizon, the late-time tail behavior of the massive perturbations is modified as compared to the usual t−5/6for\nSchwarzschild black holes, thus being a distinctive feature induced by the presence of a cosmological constant.\nI. INTRODUCTION\nBlack holes (BHs) are the standard objects with strong\ngravitational fields. Their signature comes from many astro-\nphysical channels, such as electromagnetic and gravitational\nwave observations [1–8]. The dynamics of BHs in General\nRelativity (GR) are fairly understood, covering perturbative\npost-Newtonian methods to full-blown numerical relativity.\nIn particular, the quasinormal spectrum of BHs is crucial to\nunderstanding how these objects relax to a final state of equi-\nlibrium [9–11].\nPerturbations of BHs and the study of quasinormal modes\n(QNM) have been the subject of intensive study in the litera-\nture. The seminal studies of the stability of the Schwarzschild\nspacetime [12–14] paved the way to analyze the mode sta-\nbility in di fferent scenarios, Reissner-Nordström [15, 16] and\nKerr [17] being the natural extension, but also testing di fferent\nobjects within and beyond GR [18–24]. The standard picture\nof the evolution of initial perturbations is i)an initial prompt\nthat depends on the initial conditions; ii)the ringdown phase,\nin which the real part of the modes dictate the oscillatory pat-\ntern while the imaginary part the decaying of the signal; and\niii)finally in the late-time, where we have no oscillations and\ndecay of the field, usually described by a power law behav-\nior. These features are illustrated in Fig. 1 where we show the\nevolution of a scalar Gaussian wave packet in Schwarzschild\nspacetime.\nThe standard evolution profile depicted above depends on\nthe type of perturbation induced. For instance, scalar fields\nin asymptotic flat black hole spacetimes evolve di fferently\nwhether they are massive or charged [25–28]. Some of these\n∗malato.mateus@gmail.com\n†caiomacedo@ufpa.br\n‡joaoluis92@gmail.com\n50 100 150 200 25010-910-50.1FIG. 1. Time evolution of a massless scalar field in Schwarzschild\nspacetime, highlighting each phase: i) the initial prompt, ii) the ring-\ndown phase, and iii) the late-time.\nproperties can e fficiently confine the perturbations, decreasing\ntheir leaking to infinity. As a result, we have a modification\nof the late-time behavior due to long-lived modes [25, 29–\n31]. These modes may be perceived as quasi-resonant modes\nor quasibound states (QBS), akin to the hydrogen spectrum,\nbut with an imaginary part due to the dissipative nature of the\nevent horizon. These types of modes have been analyzed in\ndifferent spacetimes [30, 32–37]. Additionally, whenever su-\nperradiance is present, this trapping can trigger instabilities in\nthe spacetime [35, 37–39], and can be considered as a natural\n“mirror” for the black hole bomb phenomena [40, 41]. As we\nshow in what follows, this e ffect is non-existent when asymp-\ntotic de-Sitter spacetimes are considered, due to the existence\nof a cosmological horizon.\nAsymptotically de-Sitter spacetimes are solutions with a\npositive cosmological constant. Studies on massless scalar\nperturbations in de-Sitter spacetimes, without a black hole,\nfound purely imaginary quasinormal frequencies (QNF) [42,arXiv:2401.15156v1 [gr-qc] 26 Jan 20242\n43]. Considering a massive field instead, one can modify the\nevolution of the perturbation leading to the presence of an os-\ncillation, i.e., it can modify the quasinormal spectrum by in-\ntroducing a real part to the frequencies [43, 44].\nThe Schwarzschild-de-Sitter (SdS) spacetime inherits some\nproperties from both the black hole and cosmological sides. It\nhas been shown that QNMs in SdS exist in two families, one\nrelated to the black hole and the other to the de-Sitter asymp-\ntotic behavior, which are known as photon sphere (PS) modes\nand de-Sitter (dS) modes, respectively [43, 45–47]. The PS\nmodes reduce to the usual Schwarzshild BH spectrum when\nwe take the limit of the vanishing cosmological constant and,\nin the evolution of initial perturbations, they dictate the ring-\ndown. In contrast, the de-Sitter modes are purely imaginary\nand retrieve the de-Sitter spectrum when the BH mass goes to\nzero, and are better seen in the late time tail [47].\nThe QNF spectrum has been calculated for massive scalar\nfields in SdS spacetime in [45, 48]. In [45] the authors found\nthat the mass can remove the anomalous decay for the PS\nmodes (where modes with larger angular momentum num-\nbers are more stable than the lower modes), and introduce a\nreal part in the dS modes, inducing oscillations. Increasing\nthe mass can also lead to a smaller imaginary part, similar to\nthe asymptotically flat case [29, 48]. Additionally, it has been\nshown that massive charged fields in Reissner-Nordström-de-\nSitter may present some long-lived ringing, and for massless\nperturbations the dS modes can be extracted from the late time\ntails [47, 49]. From these, we expect to relate the late time tails\nand the dS modes for di fferent values for the mass of the field.\nAs previously stated, the cosmological horizon prevents\nQBS to form in SdS BHs. Nonetheless, since many of the\nfeatures of the spacetime are shared with their asymptoti-\ncally flat counterparts, we should expect that at some point we\ncan recover the same signatures as the cosmological constant\ntends to zero. Moreover, since the e ffective potential changes\nasymptotically, the universal late-time behavior for massive\nfields, t−5/6, should change when the cosmological constant is\nconsidered. These issues are adressed in what follows.\nIn this work we analyze the spectrum and the time-\nevolution of perturbations of the massive scalar field in the\nSdS spacetime. In Sec. II we lay out the scalar field equation\nand its decomposition. In Sec. III we outline the methods em-\nployed to compute the modes spectra. In Sec. IV we provide\nour numerical results, confirming that QBS are non-existent\neven in the limit Λ→0 (rc→∞ ), with Λbeing the cosmo-\nlogical constant ( rcthe cosmological horizon). We also inves-\ntigate the influence of the mass in the late-time tail and de-\ntermine whether the oscillations due to the real part in the dS\nmodes can be seen as reminiscent quasibound states from the\nSchwarzschild case by extracting the frequencies through the\nProny method. Finally, in Sec. V we present our final remarks.\nThroughout this work we use the metric signature ( −,+,+,+)\nand a geometrized unit system for which G=c=1.II. SCALAR FIELD IN SCHWARZSCHILD-DE SITTER\nSPACETIME\nThe Schwarzschild-de Sitter (SdS) spacetime is a static and\nspherically symmetric BH solution with a cosmological hori-\nzon. The line element that describes such a spacetime in the\nstandard Schwarzschild-like coordinates ( t,r,θ,ϕ) is\nds2=−f(r)dt2+1\nf(r)dr2+r2(dθ2+sin2θdϕ2),(1)\nwhere we have defined\nf(r)=1−2M\nr−Λ\n3r2, (2)\nwith Mbeing the mass of the BH and Λthe cosmological con-\nstant. The radii of the event horizon rhand the cosmological\nhorizon rccan be extracted by analyzing the roots of f(r), i.e.,\nf(rh)=f(rc)=0. This analysis allows one to rewrite the\nmass Mand the cosmological constant Λin terms of the radii\nof the horizons as:\nM=rcrh(rc+rh)\n2(r2c+rcrh+r2\nh),Λ =3\nr2c+rcrh+r2\nh. (3)\nFollowing the results above, one can rewrite the lapse function\nf(r) in a more suitable way as\nf(r)=Λ\n3(rc−r)(r−rh)(r+rc+rh)\nr, (4)\nwhich we shall use to parameterize the computations that fol-\nlow. Note that rh≤rc, and also that rc→∞ andrh→2M\nin the limit Λ→0, thus recovering the Schwarzschild space-\ntime.\nConsider now a massive scalar field Ψ. The equation of mo-\ntion for such a scalar field is given by the Klein-Gordon equa-\ntion\u0010\n□−µ2\u0011\nΨ = 0, where □=gαβ∇α∇βis the d’Alembert\noperator, with∇αdenoting the covariant derivatives written in\nterms of the metric gαβ, andµis the mass of the scalar field.\nThis equation can be written in the form\n1\n(−g)1/2∂αh\n(−g)1/2gαβ∂βΨi\n−µ2Ψ = 0, (5)\nwhere gis the metric determinant, and gαβthe contravariant\nmetric. Given the spherical symmetry of the spacetime under\nanalysis, it is convenient to introduce the following decompo-\nsition of the scalar field as\nΨ(t,r,θ,ϕ)=ψ(t,r)\nrYlm(θ,ϕ) (6)\nwhere Ylm(θ,ϕ) denote the spherical harmonics, with landm\nthe angular momentum and magnetic numbers, respectively,\nandψ(t,r)is an angle-independent wave function. Further-\nmore, introducing a redefinition of the radial coordinate into\nthe so-called tortoise coordinate r∗defined by\ndr∗=dr\nf(r), (7)3\nthe Klein-Gordon equation given in Eq. (5) takes the form of\na Schrödinger wavelike equation as\n∂2ψ(t,r∗)\n∂r2∗−∂2ψ(t,r∗)\n∂t2−V(r)ψ(t,r∗)=0, (8)\nwhere the e ffective potential Vis defined as\nV(r)=f(r)\nr2h\nl(l+1)+r2µ2+r f′(r)i\n. (9)\nWe can further decompose the time-dependence in terms of\nthe field angular frequency ωasψ(t,r∗)≡ψ(r∗)e−iωt, the so-\ncalled harmonic ansatz, leading to\nd2ψ(r∗)\ndr2∗+h\nω2−V(r)i\nψ(r∗)=0. (10)\nIn Fig. 2 we plot the behavior of the e ffective potential Vfor\ndifferent values of the mass of the field µand cosmological\nhorizon rc. Two distinctive features induced by the cosmo-\nlogical constant are i)the existence of a region for which the\npotential is negative (which happens solely for l=0), and ii)\nthe vanishing of the potential as r∗→∞ , even for the massive\ncase. As such, instead of having an exponential suppression\nof the field, the solutions behave asymptotically as waves, in\nboth boundaries. As the cosmological constant increases for\na constant mass of the field, the maximum value of the po-\ntential decreases, the region in which the e ffective potential\nis negative broadens, and the minimum value of the potential\ndecreases. On the other hand, the mass of the field induces\na different behavior in the potential. As the mass increases\nfor a constant value of the cosmological constant, the whole\neffective potential increases in the region r∗>0, eventually\ncompensating for the negative contribution of the cosmologi-\ncal constant. In the region r∗<0, the potential resembles the\nSchwarzschild BH case.\nIII. QUASINORMAL MODES AND EVOLUTION\nQuasinormal mode frequencies are found by requiring the\nsolutions of the wave equation to satisfy a set of physically\nmotivated boundary conditions. Usually, in BH spacetimes,\nthis implies that the scalar field must behave asymptotically\nas follows\nψ(r∗)∼eG(ω)r∗,forr∗→−∞\neF(ω)r∗,forr∗→∞,(11)\nwhere the functions ( F,G) depend on the asymptotic forms of\nthe spacetime, i.e., the behavior of the e ffective potential at\nthe boundaries. For massive fields in Schwarzschild BHs, for\ninstance, we have that G(ω)=−iωandF(ω)=±ip\nω2−µ2.\nNote that, in this case, when Re[p\nω2−µ2]>0 we have a\nwave-like behavior in both boundaries, corresponding to the\nQNM, and when Re[p\nω2−µ2]<0 the solution is exponen-\ntially damped at infinity, corresponding to the QBS [35]. Both\nquasinormal and quasibound spectra co-exist, having di fferentimprints in the evolution of the fields. For instance, the qua-\nsibound spectrum is relevant in computing the superradiant\ninstabilities in BH spacetimes (see Ref. [41] and the refer-\nences therein). In SdS spacetime, since the e ffective potential\nvanishes at both boundaries, one can write G(ω)=−iωand\nF(ω)=iω, and therefore we have a wave-like behavior in\nboth boundaries independently of the mass of the field, i.e.,\nthe quasibound spectrum is absent. One thus obtains\nψ(r∗)∼e−iωr∗,forr∗→−∞ (r→rh)\neiωr∗,forr∗→∞ (r→rc).(12)\nIn what follows, we apply three di fferent methods to compute\nthe QNF of SdS BHs.\nA. Continued fraction method\nIn this section, we implement the continued fraction\nmethod, also known as the Leaver’s method, to extract the\nQNFs of massive scalar fields in SdS and Schwarzschild\nspacetimes, separately. Such a distinction is necessary since\nthe results for the Schwarzschild spacetime can not be directly\nobtained as a limit of those in SdS spacetimes when Λ→0\ndue to the fundamental di fference in the boundary conditions\nstated previously, as we clarify in what follows.\n1. Schwarzschild-de Sitter Spacetime\nLet us now expose how we can implement the boundary\nconditions in Eq. (12) to find the quasinormal spectrum of\nmassive scalar fields in SdS. First, we note that for r→rhwe\nhave\ne−iωr∗= 1\nr−1\nrh!ρh 1\nrc−1\nr!−ρc 1\nr+1\nrc+rh!ρh−ρc\n,(13)\nwith\nρh=iω\n2M\u00101\nrc−1\nrh\u0011\u00101\nrh+1\nrh+rc\u0011,\nρc=iω\n2M\u00101\nrc−1\nrh\u0011\u00101\nrc+1\nrh+rc\u0011.(14)\nWith the aid of the above result, we can write the solution to\nthe wave equation in the form of a Frobenius series, akin to\nLeaver’s method [50], to search for the quasinormal frequen-\ncies. This solution takes the form\nψ(r)= 1\nrc−1\nr!ρc 1\nr−1\nrh!ρh 1\nr+1\nrc+rh!ρh−ρc\n×∞X\nn=0an1\nr−1\nrh\n1\nrc−1\nrhn\n.(15)4\n-20 -10 0 10 20 300.000.020.040.060.080.10\n-10 0 10 20 30 40 500.000.020.040.060.080.100.12\nFIG. 2. The e ffective potential Vas a function of r∗with l=0, for a constant µ=0 and di fferent values of rc(left panel) and for a constant\nrc=20rhand di fferent values of the field mass µ(right panel).\nSubstituting Eq. (15) into Eq. (10), we find a 5-term recur-\nrence relation for the ancoefficients\nα0a1+β0a0=0,\nα1a2+β1a1+γ1a0=0,\nα2a3+β2a2+γ2a1+δ2a0=0,\nαnan+1+βnan+γnan−1+δnan−2+σnan−3=0,n≥3,(16)\nwhere the forms of αn,βn,γn,δnandσnare presented in Ap-\npendix A. Note that the expansion in Eq. (15) satisfies the\nboundary conditions at both horizons, provided that the re-\ncurrence relation (16) is satisfied.\nTo solve the recurrence relations, we can use a two-step\nGaussian elimination procedure to reach a three-term recur-\nrence relation. Details on this procedure can be found in Ap-\npendix A. Once a three-term recurrence relation is reached,\none can proceed with the standard Leaver’s method to obtain\nthe QNF as the roots of the algebraic equation [16, 50]:\n0=β′′\n0\nα′′\n0−γ′′\n1\nβ′′\n1−α′′\n1γ′′\n2\nβ′′\n2−α′′\n2γ′′\n3\nβ′′\n3−... (17)\nwhere the definitions of α′′\nn,β′′\nnandγ′′\nncan be found in Ap-\npendix A. The roots of this equation are the quasinormal\nfrequencies and they ensure the convergence of the series\nEq. (15).\n2. Schwarzschild Spacetime\nIn the Schwarzschild spacetime we follow similar steps to\nobtain both QNM and QBS [35, 51]. Since both spacetimes\npresent spherical symmetry, the equations in Sec. I have the\nsame forms, with di fferences arising only at the level of the\nlapse function f(r), since Λ→0 in Eq. (2). The bound-\nary condition at infinity for the case of a massive field in\nSchwarzschild spacetime needs to take into consideration the\nsub-dominant term at infinity due to the presence of mass. Theappropriate Frobenius series is thus\nψ(r)= r\nrh−1!−iωrh\neF(ω)r r\nrh!\u0012\nF(ω)+iω−µ2\n2F(ω)\u0013\nrh\n×∞X\nn=0an\u0012r−rh\nr\u0013n(18)\nwhere F(ω)=±ip\nω2−µ2. As mentioned after Eq. (11) the\nchoice of signal for Re[ F(ω)] determines the mode behavior\nat infinity. Indeed, if Re[( F(ω)]>0 it corresponds to quasi-\nnormal modes, whereas if Re[( F(ω)]<0 it corresponds to\nquadibound states. Substituting Eq. (18) into Eq. (10) yields\na 3-term recurrence relation for the ancoefficients\nα0a1+β0a0=0,\nαnan+1+βnan+γnan−1=0,n≥1,(19)\nwhereαn,βnandγnare given in Appendix A. Proceeding with\nthe standard Leaver’s method leads to an equation of the same\nform of Eq. (17), without the primes, and solving numeri-\ncally we find the QNF for Re[ F(ω)]>0 or the or QBS for\nRe[F(ω)]<0, respectively.\nAt this point, it is interesting to highlight two properties of\nthe equations until now. If we consider Eqs. (12)–(15), and\ntake the limit rc→∞ , we recover the correspondent equa-\ntions for the massless scalar field in Schwarzschild spacetime,\nalthough there is no condition imposed on the mass of the\nfield. This is a direct consequence of the potential behavior\nclose to the cosmological horizon, as already mentioned af-\nter the Eq. (11). Further, since there are no assumptions on\nthe mass of the field, the coe fficients from the recurrence rela-\ntion do not recover all the correspondent coe fficients obtained\nfrom the asymptotic flat case.\nB. Direct integration method\nAnother way to impose that the solutions satisfy the bound-\nary conditions in Eq. (12) at the boundaries is to numerically5\nintegrate the equations under these conditions. This method is\nusually known as the direct integration (DI) method [52, 53],\nbut it consists essentially of a shooting method for a two-point\nboundary value problem. We outline the procedure in what\nfollows.\nWe start by numerically integrating the radial equation out-\nwards, starting from the event horizon, with the following\nboundary condition\nψ(r≈rh)=NX\nn=0bn(r−rh)ne−iωr∗, (20)\nwhere the coe fficients bnare found by expanding the equations\nnear the event horizon and solving iteratively order by order\nin (r−rh). The minimum order of the expansion at the event\nhorizon is chosen such that the computation of the mode fre-\nquency is numerically stable, which in our case corresponds\ntypically to N=5. Therefore, We construct a numerically\nintegrated solution, say ψ−, from the event horizon, which\nsatisfies the boundary condition at the event horizon but not\nnecessarily at the cosmological horizon.\nThe procedure outlined above can then be repeated for an\ninwards integration starting from the cosmological horizon,\nunder the boundary condition\nψ(r≈rc)=NX\nn=0cn(r−rc)neiωr∗, (21)\nwhere the cncoefficients are found by expanding the equations\nat the cosmological horizon and solving iteratively. The order\nof the expansions is typically set at N=5 as well, and check-\ning convergence with other results in the literature. With this,\nwe construct a second numerical solution, say ψ+, that satis-\nfies the boundary condition at the cosmological horizon.\nQuasinormal mode solutions are obtained by requiring that\nthe proper boundary conditions, namely Eqs. (20) and (21),\nare satisfied simultaneously. This is achieved by searching for\nthe frequency for which the Wronskian Wofψ−andψ+van-\nishes, which implies that the solutions ψ−andψ+are linearly\ndependent. The solutions can thus be extracted at an interme-\ndiate matching point r=rm, at which one can find the roots ω\nof\nW(ψ−,ψ+)=ψ−(rm)ψ′\n+(rm)−ψ′\n−(rm)ψ+(rm)=0. (22)\nWe can verify if the mode is stable numerically by relaxing the\nnumber of coe fficients considered in the expansions, as well\nas varying the matching point and values considered for the\nnumerical horizons.\nC. Time-evolution of initial data\nTo study the time evolution of the scalar field and the influ-\nence of the mass in the time domain profile, we rewrite Eq. (8)\nin terms of the null coordinates, du=dt−dr∗anddv=dt+dr∗,\nwhich leads to the following partial di fferential equation\n4∂2ψ(u,v)\n∂u∂v+V(r)ψ(u,v)=0. (23)The integration is to be held in a u-vgrid, where we use the\nGundlach-Price-Pulling discretization procedure [54]\nψ(N)=ψ(W)+ψ(E)−ψ(S)−h2V(S)ψ(W)+ψ(E)\n8+O(h4),\n(24)\nwhere his the length of the edge of the grid, and the letters\nN,W,E, and Srepresent di fferent points on the grid, namely\nN=(u+h,v+h),W=(u+h,v),E=(u,v+h), and S=\n(u,v) [11]. The initial data is assumed to be a Gaussian wave\npacket in the v=v0null surface, and constant in the u=\nu0. Modifying the initial condition does not change the time\ndomain profile of the perturbation in the ringdown phase nor\nin the late-time. To obtain the profile, we extract the field at\nr∗=10rh(for other values the behavior remained consistent\nwith our results). We use h=5×10−1for the grid, verifying\nthat smaller values do not change considerably the results, and\nalso testing convergence in some particular points. Finally, we\nuse a Prony method to find the dominant QNF that appears\nin the ringdown phase and in the late-time (see, for instance,\nRef. [11]), fitting the signal with the series expansion\nψ=NX\ni=0Cie−iωit, (25)\nand finding the (complex-valuated) coe fficients ( Ci,ωi).\nIV . RESULTS\n0.0 0.5 1.0 1.5-0.20.00.20.40.60.81.01.2\nFIG. 3. Real and imaginary part of the fundamental PS and dS quasi-\nnormal modes of massive scalar fields in SdS BHs as a function of\nthe mass of the field. The PS modes are represented by dashed lines\nand the dS modes by solid lines. We show the cases for l={0,1,2}\nandrc=20rh, but the results qualitatively hold for other values of\nrc.\nIn SdS BHs there are two families of modes. The first fam-\nily is the PS modes, whose frequencies are denoted by ωPS,\nand it asymptotically approaches the usual Schwarzschild\nQNMs as rc→∞ . These are related to the unstable null circu-\nlar geodesics and can also be determined by the WKB method\n[48]. On the other hand, the dS modes, ωdS, result from the6\n100 200 300 400 500 60010-1510-1210-910-60.0011\nFIG. 4. Time-evolution of scalar wave packets for µrh=0.01 and\nl=2, and di fferent values of the cosmological horizon ( rc), includ-\ning the Schwarzschild case ( Λ = 0). A nonzero cosmological con-\nstant removes the quasibound states part of the spectrum and, in these\ncases, the dS modes are purely imaginary.\nde-Sitter behavior of the system and can be purely imaginary,\ndepending on the mass of the field. Using the methods de-\nscribed in Sec. III, we compute the QNFs and compare our\nresults with the ones presented in the literature for massless\nscalar fields [45]. Our results are presented in detail in Ap-\npendix B. The massive case shares some features with the\nmassless case, but there are some crucial di fferences intro-\nduced in the massive case, as expected given the modifications\ninduced in the e ffective potential (see Fig. 2).\nIn Fig. 3, we plot the QNFs as functions of the mass of the\nfield for a constant value of the cosmological constant. The\nbehavior of the modes can be qualitatively divided into three\nparts. For µ < µ c, for some critical value µcthe dS mode\nfrequencies are purely imaginary and, consequently, after the\nringdown we have an exponential “tail” corresponding to this\nmode. The PS modes vary only slightly in this region. The\nvalue ofµcdepends on the cosmological constant, and for the\nde-Sitter case (with M=0) they correspond to\nµ2\nc=3\n4Λ. (26)\nAlthough such an explicit form is not attainable for the\nSdS case, the result for the de-Sitter spacetime presented in\nRef. [44] and [45] provides a good estimate for our purposes.\nIn this region, the imaginary part of the dS modes increases in\nabsolute value, implying that as µ→µcthe decay timescale\ndecreases.\nForµrh∼ωPS\nRrh> µrh> µ crh, as we show in what fol-\nlows, the time domain profile can be separated between a clear\n\"photon sphere\" ringdown and a further decay dominated by\nthe dS modes, with oscillations that resemble that of a massive\nfield in Schwarzshild spacetime. In this regime, the dS modes\nhave both a real and imaginary parts. At intermediate times,\nthe signal presents some interference due to the influence of\nboth PS and dS modes. The decay time of the dS modes still\ndecreases slightly for µ=µc+δfor a small value of δ, before\nincreasing with µrh. For instance, we have that δ+µc=0.08forrc=20rh(see Fig. 3).\nThe third region occurs for µrh>ωPS\nRrhandωPS\nRrh>µ crh.\nIn this region, the PS modes have a significant change. We\nsee that the real part of both PS and dS modes are bounded by\nthe mass of the field, i.e. ωPS,dS\nRrh< µrh, and they increase\napproximately linearly with µ, converging to similar values\nin the large µlimit. The imaginary part of the dS modes\napproaches a constant value regardless the multipole l. For\ninstance, for rc=20rh,ωdS\nI∼ −0.03633 for large µ. The\nmain distinction between the PS and dS modes in this regime\ncomes from the imaginary part. The time domain profile of\nthe field ceases to present an interference profile and the dS\nmodes dominate the signal as we can see through the Prony\nmethod.\nNote that the transitions between the above regions occur\nfor considerably large values of the mass of the field. For\ninstance, we can see in Fig. 3 that for l=0 and rc=20rhthe\nfirst transition happens at µrh∼0.073, and the second one at\nµrh∼ωRrh∼0.22.\nTo illustrate the features explained above in the time-\nevolution of the signals, in Fig. 4 we present the time domain\nprofile of the massive scalar field with the angular number\nl=2. In particular, we consider a massive scalar field with\nµrh=0.01 in the SdS with the cosmological horizon located\natrc=20rh(µcrh≈0.073) and 50 rh(µcrh≈0.03) and for the\nasymptotic flatness case ( Λ = 0). Notice that in the cases with\ncosmological constant we have µ < µ cand, therefore, the dS\nmodes are purely imaginary. We see that there is a late-time\noscillation introduced by the mass of the field in the asymp-\ntotically flat case, as usually is the case for massive fields in\nSchwarzschild. For the SdS cases, as µ < µ c, the exponential\nlate-time decay is dictated by the purely imaginary dS mode.\nWe highlight that the ringdown phase of the signals presented\nin Fig. 4 are essentially the same, which consistent with the\nfact that the PS mode frequencies are approximately the same\nforµ<µ c.\nIn Fig. 5 we show the time domain profile of the per-\nturbation for fixed cosmological horizon rc=20rh(µcrh≈\n0.073). In the left panel (right panel) we plot the l=0\n(l=1) mode, for di fferent values of the scalar field mass\nµrh={0.07,0.08,0.15}. In both cases, we see the dS modes\nacquiring an oscillatory pattern for µ>µ c, indicating that the\nmodes acquire a real part. We also see an increasing decay-\ning rate, in agreement with the behavior previously discussed.\nThe corresponding QNM frequencies for the configurations\nin Fig. 5 are presented in Tab. I. We show the numbers found\nthrough both the Leaver’s and Prony methods, with the direct\nintegration method computations in a close agreement with\nthose of the Leaver’s method. The inaccuracy of the Prony\nmethod for l=0 is caused by the small number of oscillations\npresent in the ringdown phase.\nIt is relevant to investigate whether there is a limit at which\nthe signal of SdS BHs coincides with their asymptotically\nflat counterparts. To access that, in Fig. 6 we compare the\ntime evolution of massive scalar fields in Schwarzschild with\nthe SdS for a choice of cosmological horizon radius large in\ncomparison with the BH length scale ( rc=200rh). We also\nconsider a large value of the scalar field mass, µrh=0.5,7\n100 200 300 400 500 60010-2410-1910-1410-910-4\n100 200 300 400 500 60010-3910-2910-1910-9\nFIG. 5. Time domain profile of the scalar field for l=0 (left panel) and l=1 (right panel), for di fferent scalar field masses. The field’s masses\nare chosen such that each value highlights the di fferent behaviors of the tail as we cross the limit for the purely imaginary modes. The shaded\nregion indicates the interval in which the frequencies are extracted using the Prony method.\nTABLE I. Fundamental QNF for the configurations of Fig. 5, using the Leaver and Prony method.\nlµrh ωdS\nLeaverrh ωdS\nPronyrh ωPS\nLeaverrh ωPS\nPronyrh\n0.07−0.0523507 i −0.052711 i 0.219859−0.208169 i 0.24547−0.22047 i\n0 0.08 0.034485−0.0715712 i 0.034621−0.071687 i 0.220026−0.20755 i 0.25084−0.21524 i\n0.15 0.130808−0.0643422 i 0.13113−0.06447 i 0.222553−0.201355 i 0.30933−0.19301 i\n0.07−0.100924 i −0.10115 i 0.582771−0.194107 i 0.59178−0.19401 i\n1 0.08 0.0336226−0.120839 i 0.033702−0.121088 i 0.583106−0.193909 i 0.59532−0.19470 i\n0.15 0.130563−0.117177 i 0.13071−0.11733 i 0.586701−0.191773 i 0.59104−0.19245 i\n200 400 600 800 1000 1200 140010-1510-1110-70.001\n50 100 150 200 250 30010-510-40.0010.0100.100\n200 400 600 800 1000 1200 140010-1510-1110-70.001\n50 100 150 200 250 30010-610-510-40.0010.0100.100\nFIG. 6. Time evolution of a massive scalar field with a mass µrh=0.5, in Schwarzschild and Schwarzscild-de Sitter with rc=200rh, for l=0\n(left panel), and l=1 (right panel). We see that the time evolution is comparable in early stages, but the decay in SdS is quicker, while in the\nSchwarzshild case it behaves as t−5/6at late times.\nto analyze whether quasibound features appear in SdS BHs.\nFor completeness, we show the cases l=0 (left panel) and\nl=1 (right panel). Initially, the profiles are almost identical,\neven for moderate time intervals (see also the inset in Fig. 6).\nHowever, at late times the universal behavior of massive fields\nin Schwarzschild BHs ( t−5/6) changes when the cosmological\nconstant is present. For SdS BHs the late time is described by\nthe dS modes.\nIt is important to mention that for small values of the cos-\nmological constant (or rc≫rh) the contributions of PS and\ndS modes are more clearly distinguishable. As rc→rh, thecritical mass is µcrh≈0.87, and therefore the range of mass\nof the field in which the dS modes have real part is relatively\nlarge. For such larger masses, the time domain profile does\nnot present clear distinguishable features from the PS and dS\nmodes.\nThe frequencies of the dS modes have a direct relation with\nthe tail present in the time evolution and also with the mass\nof the field. When µ > µ c, we haveωdS\nR,0, and oscillations\nappear in the tail. Furthermore, the condition ωdS\nR/µ< 1 is al-\nways verified. These are properties similar to the case of qua-8\nTABLE II. Quasinormal frequencies of de Sitter modes as rc→∞ and the quasibound states in Schwarzschild spacetime (QBS) for di fferent\nvalues of angular number land forµrH=0.5.\nl ωdS\nLeaver/µ(rc=2000 rh) ωQBS/µ ωdS\nLeaver/µ(rc→∞ )\n0 0.960624−0.0154134 i 0.990915−0.0025461 i 0.96064−0.0153784 i\n1 0.984614−0.00390827 i 0.991565−2.97451×10−6i 0.984633−0.00382118 i\n2 0.987621−0.0174992 i 0.998−4.82897×10−12i 0.98763−0.0173872 i\n5 10 50 100 500 10000.010.050.100.50\n50 100 500 100010-510-40.0010.0100.1001\nFIG. 7. The quasinormal frequencies of dS modes (solid lines)\nand the quasibound frequencies in Schwarzschild spacetime (dashed\nlines) forµrh=0.5, different values of l=0, 1 and 2, and increasing\nthe cosmological horizon coordinate.\nsibound states of a massive scalar field in the Schwarzschild\nspacetime [25, 35]. From these similarities, we might inter-\npret the dS modes as some sort of reminiscent of quasibound\nstates from the asymptotic flat case, being more and more\nsimilar in the limit rc→ ∞ . To investigate the validity of\nthis statement, we track the dS modes as we approach the\nasymptotic flat case for a fixed value of the mass µrh=0.5,\nand the result can be seen in Fig. 7. These results indicate\nthat there is always a gap between the quasibound and the\ndS modes, which is smaller for larger values of l. Particu-\nlarly, we have calculated ωdSthrough two di fferent methods,\nnamely by considering the dS modes for rc=2000 rHand by\nexpanding the coe fficients in Appendix A for rc→∞ . The\nresults can be seen in Tab. II together with the fundamental\nquasibound states in Schwarzschild spacetime. We highlight\nthat, although the spectrum of dS modes does not coincide\nwith the spectrum of quasibound states in the rc→∞ case,\nthe time evolution shares similar profiles, especially at early\ntimes, with the dS modes appearing in the late-time evolution.\nV . CONCLUSION\nIn the present paper, we have studied the evolution of a\nmassive scalar field in the background of a Schwarzschild-\nde-Sitter black hole. We determined the quasinormal frequen-\ncies using the Leaver, Chandrasekhar, and Prony methods. We\nfound two branches of frequencies, the photons sphere modes,\nwhich are related to the circular null geodesics, and the de-Sitter modes which seem to be related to the asymptotic po-\ntential behavior.\nThe photon sphere modes for large values of the cosmolog-\nical horizon dominate the ringdown phase of the time evolu-\ntion. Increasing the mass of the scalar field mainly increases\nits oscillation behavior and increases its decay time. For small\nvalues of the cosmological horizon, these modes dominate the\nwhole time evolution of the perturbation, overshadowing any\ncontribution from the de-Sitter modes.\nThe de Sitter modes increase in magnitude for small values\nofrc, but their main influence is better observed for greater\nvalues of the cosmological horizon. For a massless scalar\nfield, the dS modes are purely imaginary and can be extracted\nthrough the Prony method from the exponential tail. Increas-\ning the mass of the scalar field, these modes can acquire a real\npart, which changes the tail, introducing some oscillations.\nAs we increase the mass of the field, we initially see these\nmodes decreasing the time decay. Still, once a threshold is\ncrossed, the imaginary part of the dS modes decreases, giving\nrise to modes that are similar to the long lived modes in the\nSchwarzschild asymptotic flat case.\nTo verify whether the de-Sitter modes are a reminiscent\nof the quasibound states, we track them as we increase the\ncosmological horizon to recover the asymptotic flatness. We\nfound that, for the range of parameters considered, the de Sit-\nter modes share similar properties with the quasibound states,\nalthough their spectrum does not retrieve the quasibound state\nspectrum in the limit rc→∞ .\nWe also note here the similarities between massive fields\nin de Sitter black holes and the e ffective mass of the photon\nintroduced by astrophysical plamas [55, 56]. Since the plasma\nis localized, the e ffective mass of the field does not introduce\na quasi-bound spectrum. Nonetheless, dynamical evolution\nshows a similar time-domain profile to the true massive case at\nearly stages, akin to what we presented in this paper. It would\nbe interesting to analyze common features in both scenarios.\nFinally, we note that the natural extension of our results\nwould be to consider spinning black holes. In the rotating\ncase, the mass of the field is connected with superradiant in-\nstabilities [41] and the existence of a cosmological horizon\ncould quench such instabilities since the field is not confined\nanymore. Therefore, the existence of a cosmological horizon\nwould potentially relax the bounds on the rotation of black\nholes as well as on the mass of ultralight bosons [57, 58]. We\nleave this for future work.9\nACKNOWLEDGMENTS\nWe thank Hector O. Silva for the discussions. CFBM and\nMMC acknowledge Fundação Amazônia de Amparo a Es-\ntudos e Pesquisas (FAPESPA), Conselho Nacional de De-\nsenvolvimento Científico e Tecnológico (CNPq) and Coor-\ndenação de Aperfeiçoamento de Pessoal de Nível Superior\n(CAPES) – Finance Code 001, from Brazil, for partial fi-\nnancial support. JLR acknowledges the European Regional\nDevelopment Fund and the programme Mobilitas Pluss for\nfinancial support through Project No. MOBJD647, project\nNo. 2021 /43/P/ST2/02141 co-funded by the Polish National\nScience Centre and the European Union Framework Pro-\ngramme for Research and Innovation Horizon 2020 under theMarie Sklodowska-Curie grant agreement No. 94533, Fun-\ndação para a Ciência e Tecnologia through project number\nPTDC /FIS-AST /7002/2020, and Ministerio de Ciencia, Inno-\nvación y Universidades (Spain), through grant No. PID2022-\n138607NB-I00.\nAppendix A: Recurrence Relations and Gaussian Elimination\na. Schwarzschild Case\nIn subsection III A, we found a 3-term recurrence relation\nfor the Schwarzchild case, see Eq. (19). The coe fficientsαn,\nβnandγnare given explicitly by\nαn=(1+n)(1+n−2i rhω), (A1)\nβn=(2n+1)rh\u0010\n3µ2+4iω(F(ω)+iω)\u0011\n2F(ω)+ iω3\nF(ω)−2iωF(ω)−F(ω)(F(ω)+iω)+3ω2!\nr2\nh−l(l+1)−2n(n+1)−1,(A2)\nγn=−nrh\u0010\nµ2+2iω(F(ω)+iω)\u0011\nF(ω)+r2\nh\u0010\n4iµ2ω(F(ω)+2iω)+8ω3(ω−iF(ω))+µ4\u0011\n4F(ω)2+n2, (A3)\nwhere F(ω)=±ip\nω2−µ2. The ratio between successive series coe fficients anleads to an infinite continued fraction\n0=β0−α0γ1\nβ1−α1γ2\nβ2−α2γ3\nβ3−..., (A4)\nwhich can be rewritten as\n0=β0\nα0−γ1\nβ1−α1γ2\nβ2−α2γ3\nβ3−... (A5)\nWe solve this numerically to find the QNF or QBS as the roots of the Eq. (A5).\nb. Schwarzschild-de Sitter case\nIn the SdS case, we found a 5-term recurrence relation given by Eq. (16), the full form of the coe fficientsαn,βn,γn,δn, and\nσnare\nαn=2M(1+n)(rc+2rh)\nr3\nh(rc+rh)(1+n+2ρh), (A6)\nβn=−µ2−l(l+1)\nr2\nh+2M\nr3\nhrc(rc+rh)\u0010\n(r2\nc+rcrh)(1+2n(1+2n)+4ρh(1+3n+2ρh))\u0011\n+\n−2M\nr3\nhrc(rc+rh)r2\nh(2+n(1+5n)+2ρh(1+6n+2ρh)),(A7)\nγn=2l(l+1)(rc−rh)\nrcr2\nh+2M(rc−rh)\u0010\nrcrh+r2\nc\u0011\u0010\n22nρh+20ρ2\nh−10ρh+6(n−1)n+3\u0011\nr2cr3\nh(rc+rh)+\n−4M(rc−rh)r2\nh(2ρh+n−1) (2ρh+2n−1)\nr2cr3\nh(rc+rh),(A8)10\nδn=−l(l+1)(rc−rh)2\nr2cr2\nh−2M(rc−rh)2\u0010\nrcrh+r2\nc\u0011\u0010\n2n(8ρh−5)+4ρh(4ρh−5)+4n2+7\u0011\nr3cr3\nh(rc+rh)+\n+2M(rc−rh)2(2ρh+n−2) (2ρh+n−1)\nr3crh(rc+rh),(A9)\nσn=2M(rc−rh)3\nr3cr3\nh(2ρh+n−2)2. (A10)\nTo calculate the quasinormal modes we need to perform a two-\nstep Gaussian elimination to reach a three-term recurrence re-\nlation [11, 16]. The first step is to define new coe fficients\ngiven by\nα′\nn≡αn, β′\nn≡βn, γ′\nn≡γn, δ′\nn≡δn,forn=0,1,(A11)\nand\nα′\nn≡αn, β′\nn≡βn−α′\nn−1\nδ′\nn−1σn,\nγ′\nn≡γn−β′\nn−1\nδ′\nn−1σn, δ′\nn≡δn−γ′\nn−1\nδ′\nn−1σn, σ′\nn≡0 for n≥2.\n(A12)\nAfter this first step, we obtain a similar recurrence relation,\nbut now with 4 terms, α′\nn,β′\nn,γ′\nn, andδ′\nn. We perform a second\nGaussian elimination, defining the coe fficients\nα′′\nn≡α′\nn, β′′\nn≡β′\nn, γ′′\nn≡γ′\nn,forn=0,1, (A13)and\nα′′\nn≡α′\nn, β′′\nn≡β′\nn−α′′\nn−1\nγ′′\nn−1δ′\nn,\nγ′′\nn≡γ′\nn−β′′\nn−1\nγ′′\nn−1δ′\nn, δ′′\nn≡0,forn≥2.(A14)\nThese new coe fficients obey a 3-term recurrence relation\nα′′\n0a1+β′′\n0a0=0,\nα′′\nnan+1+β′′\nnan+γ′′\nnan−1=0,n≥1.(A15)\nNow with a 3-term recurrence relation we can proceed in the\nsame way as the Schwarzschild case. Equation (15) is a con-\nvergent series for the characteristic frequency ωthat solves\nthe equation the Eq. (A5) for this case.\nAppendix B: Massless scalar modes\nFor consistency check, we compare the quasinormal fre-\nquencies, both PS (Tab. III) and dS (Tab. IV) modes, ob-\ntained through the Leaver’s method and the direct integration\n(DI) with previous results in the literature for lower overtones\n[45, 51]. The Leaver’s method is also consistent with the high\novertones presented in [59] but, since this work aims at ana-\nlyzing the dominant modes and possible long-lived modes, we\ndo not include a table for comparison.\n[1] B. P. Abbot and et al. (LIGO Scientific Collaboration and Virgo\nCollaboration), Observation of Gravitational Waves from a Bi-\nnary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).\n[2] B. P. Abbot and et al. (LIGO Scientific Collaboration and\nVirgo Collaboration), GW170817: Observation of Gravita-\ntional Waves from a Binary Neutron Star Inspiral, Phys. Rev.\nLett. 119, 161101 (2017).\n[3] The EHT Collaboration et al., First M87 Event Horizon Tele-\nscope Results. I. The Shadow of the Supermassive Black Hole,\nAstr. J. Lett. 875, L1 (2019).\n[4] K. Akiyama et al. (Event Horizon Telescope), First M87 Event\nHorizon Telescope Results. VII. Polarization of the Ring, As-\ntrophys. J. Lett. 910, L12 (2021), arXiv:2105.01169 [astro-\nph.HE].[5] The EHT Collaboration et al., First Sagittarius A* Event Hori-\nzon Telescope Results. I. The Shadow of the Supermassive\nBlack Hole in the Center of the Milky Way, Astr. J. Lett. 930,\nL12 (2022).\n[6] M. Bauböck et al. (GRA VITY), Modeling the orbital motion\nof Sgr A*’s near-infrared flares, Astron. Astrophys. 635, A143\n(2020), arXiv:2002.08374 [astro-ph.HE].\n[7] R. Abuter et al. (GRA VITY), Detection of the Schwarzschild\nprecession in the orbit of the star S2 near the Galactic cen-\ntre massive black hole, Astron. Astrophys. 636, L5 (2020),\narXiv:2004.07187 [astro-ph.GA].\n[8] R. Abuter et al. (GRA VITY), Polarimetry and astrometry\nof NIR flares as event horizon scale, dynamical probes for\nthe mass of Sgr A*, Astron. Astrophys. 677, L10 (2023),\narXiv:2307.11821 [astro-ph.GA].11\nTABLE III. Comparing the quasinormal frequencies for the photon sphere modes ( nPS=0) for massless scalar field with l=1, from the Ref.\n[45] (ωPS), with our results obtained through Leaver ( ωLeaver ) and direct integration ( ωDI).\nΛM2ωPSM[45, 51] ωLeaver M ωDIM\n0.02 0.2603−0.0911 i 0.26028785−0.091002536 i 0.260288−0.0910025 i\n0.04 0.2247−0.0821 i 0.22468492−0.082051288 i 0.224685−0.0820513 i\n0.06 0.1854−0.0701 i 0.18536931−0.070060118 i 0.185369−0.0700601 i\n0.08 0.1404−0.0542 i 0.14040900−0.054023195 i 0.140409−0.0540232 i\n0.09 0.11392−0.04397 i 0.11399675−0.043882695 i 0.113997−0.0438827 i\n0.1 0.08156−0.03121 i 0.081589747−0.031233161 i 0.0815901−0.0312332 i\n0.11 0.02549−0.00965 i 0.025490450−0.0096494266 I 0.0255404−0.00972708 i\nTABLE IV . Comparing the quasinormal frequencies for the de Sitter modes ( ndS=0) for massless scalar field with l=1, from the Ref.[45]\n(ωdS), with our results obtained through Leaver ( ωLeaver ) and direct integration ( ωDI).\nΛM2ωdSM[45] ωLeaver M ωDIM\n0.02 −0.081565496 i −0.081565496 i −0.0815655 i\n0.04 −0.11524810 i −0.11524810 i −0.115248 i\n0.06 −0.14100253 i −0.14100253 i −0.141003 i\n0.08 −0.16268011 i −0.16268004 i −0.16268 i\n0.09 −0.17249210 i −0.17249210 i −0.172492 i\n0.1 −0.18177480 i −0.18177480 i −0.181775 i\n0.11 −0.19057630 i −0.1905 i\n[9] H.-P. Nollert, Quasinormal modes: the characteristic ‘sound’ of\nblack holes and neutron stars, Class. Quantum Grav. 16, R159\n(1999).\n[10] E. Berti, V . Cardoso, and A. O. Starinets, Quasinormal modes of\nblack holes and black branes, Class. Quantum Grav. 26, 163001\n(2009).\n[11] R. A. Konoplya and A. Zhidenko, Quasinormal modes of black\nholes: From astrophysics to string theory, Rev. Mod. Phys. 83,\n793 (2011).\n[12] T. Regge and J. A. Wheeler, Stability of a schwarzschild singu-\nlarity, Phys. Rev. 108, 1063 (1957).\n[13] F. J. Zerilli, Gravitational field of a particle falling in a\nschwarzschild geometry analyzed in tensor harmonics, Phys.\nRev. D 2, 2141 (1970).\n[14] C. V . Vishveshwara, Scattering of gravitational radiation by a\nschwarzschild black-hole, Nature 227, 936 (1970).\n[15] K. D. Kokkotas and B. F. Schutz, Black-hole normal modes: A\nwkb approach. iii. the reissner-nordström black hole, Phys. Rev.\nD37, 3378 (1988).\n[16] E. W. Leaver, Quasinormal modes of reissner-nordström black\nholes, Phys. Rev. D 41, 10 (1990).\n[17] S. Teukolsky, Rotating black holes: Separable wave equations\nfor gravitational and electromagnetic perturbations, Phys. Rev.\nLett. 29, 1114 (1972).\n[18] C. Molina, P. Pani, V . Cardoso, and L. Gualtieri, Gravi-\ntational signature of Schwarzschild black holes in dynami-\ncal Chern-Simons gravity, Phys. Rev. D 81, 124021 (2010),\narXiv:1004.4007 [gr-qc].\n[19] J. L. Blázquez-Salcedo, C. F. B. Macedo, V . Cardoso, V . Fer-\nrari, L. Gualtieri, F. S. Khoo, J. Kunz, and P. Pani, Perturbed\nblack holes in Einstein-dilaton-Gauss-Bonnet gravity: Stabil-\nity, ringdown, and gravitational-wave emission, Phys. Rev. D\n94, 104024 (2016), arXiv:1609.01286 [gr-qc].\n[20] C. F. B. Macedo, Dynamical signatures of black holes in mas-\nsive Chern-Simons gravity: Quasibound modes and time evo-\nlution, Phys. Rev. D 98, 084054 (2018), arXiv:1809.08691 [gr-qc].\n[21] E. Berti, K. Yagi, H. Yang, and N. Yunes, Extreme Grav-\nity Tests with Gravitational Waves from Compact Binary Co-\nalescences: (II) Ringdown, Gen. Rel. Grav. 50, 49 (2018),\narXiv:1801.03587 [gr-qc].\n[22] V . Cardoso, M. Kimura, A. Maselli, E. Berti, C. F. B. Macedo,\nand R. McManus, Parametrized black hole quasinormal ring-\ndown: Decoupled equations for nonrotating black holes, Phys.\nRev. D 99, 104077 (2019), arXiv:1901.01265 [gr-qc].\n[23] R. McManus, E. Berti, C. F. B. Macedo, M. Kimura, A. Maselli,\nand V . Cardoso, Parametrized black hole quasinormal ring-\ndown. II. Coupled equations and quadratic corrections for\nnonrotating black holes, Phys. Rev. D 100, 044061 (2019),\narXiv:1906.05155 [gr-qc].\n[24] Z. S. Moreira, H. C. D. Lima Junior, L. C. B. Crispino, and\nC. A. R. Herdeiro, Quasinormal modes of a holonomy corrected\nschwarzschild black hole, Phys. Rev. D 107, 104016 (2023).\n[25] L. M. Burko and G. Khanna, Universality of massive scalar\nfield late-time tails in black-hole spacetimes, Phys. Rev. D 70,\n044018 (2004).\n[26] H. Koyama and A. Tomimatsu, Asymptotic power law tails of\nmassive scalar fields in Reissner-Nordstrom background, Phys.\nRev. D 63, 064032 (2001), arXiv:gr-qc /0012022.\n[27] H. Koyama and A. Tomimatsu, Asymptotic tails of massive\nscalar fields in Schwarzschild background, Phys. Rev. D 64,\n044014 (2001), arXiv:gr-qc /0103086.\n[28] H. Koyama and A. Tomimatsu, Slowly decaying tails of mas-\nsive scalar fields in spherically symmetric space-times, Phys.\nRev. D 65, 084031 (2002), arXiv:gr-qc /0112075.\n[29] R. A. Konoplya and A. Zhidenko, Decay of massive scalar field\nin a schwarzschild background, Phys. Lett. B. 609, 377 (2005).\n[30] A. Ohashi and M. Sakagami, Massive quasi-normal mode,\nClass. Quantum Grav. 21, 3973 (2004).\n[31] S. Hod, Slowly decaying resonances of charged massive scalar\nfields in the reissner-nordström black-hole spacetime, Phys.\nLett. B 761, 53 (2016).12\n[32] E. Berti, R. Brito, C. F. B. Macedo, G. Raposo, and J. L. Rosa,\nUltralight boson cloud depletion in binary systems, Phys. Rev.\nD99, 104039 (2019), arXiv:1904.03131 [gr-qc].\n[33] V . Cardoso, R. Brito, and J. L. Rosa, Superradiance in stars,\nPhys. Rev. D 91, 124026 (2015), arXiv:1505.05509 [gr-qc].\n[34] R. A. Konoplya, Massive charged scalar field in a reiss-\nner–nordstrom black hole background: quasinormal ringing,\nPhys. Lett. B 550, 117 (2002).\n[35] S. R. Dolan, Instability of the massive klein-gordon field on the\nkerr spacetime, Phys. Rev. D 76, 084001 (2007).\n[36] S. Detweiler, Klein-gordon equation and rotating black holes,\nPhys. Rev. D 22, 2323 (1980).\n[37] T. Damour, N. Deruelle and R. Ru ffini, On quantum resonances\nin stationary geometries., Lett. Nuovo. Cim. 15, 257 (1970).\n[38] V . Cardoso, O. J. C. Días, J. P. S. Lemos, and S. Yoshida,\nBlack-hole bomb and superradiant instabilities, Phys. Rev. D\n70, 044039 (2004).\n[39] J. L. Rosa, J. P. S. Lemos, and F. S. N. Lobo, Stability of Kerr\nblack holes in generalized hybrid metric-Palatini gravity, Phys.\nRev. D 101, 044055 (2020), arXiv:2003.00090 [gr-qc].\n[40] W. H. Press and S. A. Teulkolski, Floating orbits, supperadiant\nscattering and the black hole bomb, Nature 238, 211 (1972).\n[41] R. Brito, V . Cardoso, and P. Pani, Superradiance (Springer\nCham, 2020).\n[42] A. Lopez-Ortega, Absorption and quasinormal modes of clas-\nsical fields propagating on 3a .nd 4d .e sitter spacetime, Gen. Rel.\nGrav. 38, 743 (2006).\n[43] A. Lopez-Ortega, On the quasinormal modes of the de sitter\nspacetime, Gen. Rel. Grav. 44, 2387 (2012).\n[44] A. Lopez-Ortega, Quasinormal modes of d-dimensional de sit-\nter spacetime, Gen. Rel. Grav. 38, 1565 (2006).\n[45] P. A. Aragón, Almendraand González, E. Papantonopoulos, and\nY . Vásquez, Anomalous decay rate of quasinormal modes in\nschwarzschild-ds and schwarzschild-ads black holes, J. High\nEnerg. Phys. 120(2020).\n[46] V . Cardoso, J. L. Costa, K. Destounis, P. Hintz, and A. Jansen,\nQuasinormal modes and strong cosmic censorship, Phys. Rev.\nLett. 120, 031103 (2018).\n[47] R. A. Konoplya and A. Zhidenko, Nonoscillatory gravitational\nquasinormal modes and telling tails for schwarzschild-de sitter\nblack holes, Phys. Rev. D 106, 124004 (2022).[48] B. Toshmatov and Z. Stuchlík, Slowly decaying resonances\nof massive scalar fields around schwarzschild-de sitter black\nholes, Eur. Phys. J. Plus. 132, 324 (2017).\n[49] Z. Zhu, S.-J. Zhang, C. E. Pellicer, B. Wang, and E. Abdalla,\nStability of Reissner-Nordström black hole in de Sitter back-\nground under charged scalar perturbation, Phys. Rev. D 90,\n044042 (2014).\n[50] E. W. Leaver, An analytic representation for the quasinormal\nmodes of kerr black holes, Phys. R. Soc. Lond. A. 402, 285\n(1985).\n[51] R. A. Konoplya and A. Zhidenko, Quasinormal modes of\nschwarzschild-de sitter black holes, Class. Quant. Grav. 21, 273\n(2004).\n[52] S. Chandrasekhar and S. Detweiler, The quasi-normal modes of\nthe schwarzschild black hole, Proceedings of the Royal Society\nof London. Series A, Mathematical and Physical Sciences 344,\n441 (1975).\n[53] C. F. B. Macedo, V . Cardoso, L. C. B. Crispino, and P. Pani,\nQuasinormal modes of relativistic stars and interacting fields,\nPhys. Rev. D 93, 064053 (2016), arXiv:1603.02095 [gr-qc].\n[54] C. Gundlach, R. H. Price, and J. Pullin, Late-time behavior\nof stellar collapse and explosions: I. linearized perturbationss,\nPhys. Lett. D. 49, 883 (1994).\n[55] V . Cardoso, W.-D. Guo, C. F. B. Macedo, and P. Pani, The\ntune of the Universe: the role of plasma in tests of strong-\nfield gravity, Mon. Not. Roy. Astron. Soc. 503, 563 (2021),\narXiv:2009.07287 [gr-qc].\n[56] Z. Wang, T. Helfer, K. Clough, and E. Berti, Superradiance in\nmassive vector fields with spatially varying mass, Phys. Rev. D\n105, 104055 (2022), arXiv:2201.08305 [gr-qc].\n[57] P. Pani, V . Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi,\nBlack hole bombs and photon mass bounds, Phys. Rev. Lett.\n109, 131102 (2012), arXiv:1209.0465 [gr-qc].\n[58] R. Brito, V . Cardoso, and P. Pani, Massive spin-2 fields on black\nhole spacetimes: Instability of the Schwarzschild and Kerr solu-\ntions and bounds on the graviton mass, Phys. Rev. D 88, 023514\n(2013), arXiv:1304.6725 [gr-qc].\n[59] A. Jansen, Overdamped modes in Schwarzschild-de Sitter and a\nMathematica package for the numerical Computation of quasi-\nnormal modes, The European Physical Journal Plus 132, 22\n(2017), arXiv:1709.09178v2 [gr-qc]." }, { "title": "2401.15195v1.Bounded_degree_Low_Rank_Parity_Check_Codes.pdf", "content": "arXiv:2401.15195v1 [cs.IT] 26 Jan 2024Bounded-degree\nLow Rank Parity Check Codes\nErmes Franch, Chunlei Li\nEmails:{ermes.franch, chunlei.li }@uib.no\nDepartment of Informatics, University of Bergen,\n5020, Norway\nAbstract\nLow rank parity check (LRPC) codes are the rank-metric analo gue\nof low density parity check codes. In this paper we investiga te a sub-\nfamily of LRPCcodes, whichhave aparity check matrix defined over a\nsubspace Vα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFq/subsetnoteql Fqm, whereFqmis the finite field\nofqmelements and dis significantly smaller than m. These codes are\nnamed bounded-degree LRPC (BD-LRPC) codes and are the same\nas the standard LRPC codes of density 2 when the degree d= 2,\nwhile BD-LRPC codes of degree d >2 constitute a proper subset of\nLRPC codes of density d. Exploiting the particular structure of their\nparity check matrix, we show that the BD-LRPC codes of degree d\ncan uniquely correct errors of rank weight rwhenn−k≥r+ufor\ncertainu≥1, in contrast to the condition n−k≥drrequired for\nthe standard LRPC codes, where d≥n/(n−k). This underscores\nthe superior decoding capability of the proposed BD-LRPC co des. As\nthe code length napproaches infinity, when n/m→0, it is shown\nthatucan be chosen as a certain constant, which indicates that the\nBD-LRPC codes with a code rate of Rcan be, with a high probability,\nuniquely decodable with the decoding radius ρ=r/napproaching the\nSingleton bound 1 ���Rforn→ ∞; and when b=n/mis a constant,\nthe BD-LRPC codes can have unique decoding radius ρ= 1−R−ǫ\nfor a small ǫ, which can easily lead to ρ >(1−R)/2 with properly\nchosen parameters.\n11 Introduction\nRank metric codes, a type of error-correction codes equipped wit h rank met-\nric, have undergone significant advancements over the past deca des and have\na wide range of applications, including network coding [20], criss-cro ss error\ncorrection [19], and cryptography [9]. Many important results for H amming\nmetric codes have been analogously extended to their counterpar ts in rank\nmetric [3,7], such as the Singleton bound, the MacWilliams identities and\nefficient unique decoding algorithms from Reed-Solomon codes to Gab idulin\ncodes. In 2013 Gaborit, Murat, Ruatta, and Z´ emor proposed low rank parity\ncheck codes [17], which can be deemed as an equivalent of the low den sity\nparity check (LDPC) codes [13] for Hamming metric. LRPC codes allo w\nfor unique decoding through a probabilistic, polynomial-time algorithm and\nexhibit sounding randomness that is highly valued in cryptographic ap plica-\ntions. Since then they have been adopted in several post-quantu m crypto-\ngraphic primitives, including RankSign [12], identity-based encryption [10],\nROLLO [16], the signature scheme Durandal [1], etc.\nDecoding capability is a crucial aspect of rank metric codes for their\napplications. The majority of efficient unique decoding algorithms for Reed-\nSolomon codes have been extended to the linearized setting for Gab idulin\ncodes, allowing for uniquely decoding errors of ranks rwithin half of the\nminimum rank distance, that is r≤d−1\n2=n−k\n2[3,7,8]. However, when an\nerror has rank beyondn−k\n2, it is challenging to efficiently decode it for rank\nmetric codes, even when one considers list decoding. A rank metric c ode is\nsaid to be ( ρ,L)-list decodable when errors ofrank r=ρncanbe successfully\ndecoded with a list of Lcodewords. List decodability of Gabidulin codes has\nbeen intensively investigated. Wachter-Zeh in [22] pointed out tha t square\nGabidulin codes cannot be decoded with a polynomial-size list when the\n(relative) decoding radius ρ=r\nnis beyond the Johnson radius 1 −√\nR,\nwhereR=k\nnis the code rate. Later Raviv and Wachter-Zeh [18] further\nshowed that Gabidulin codes with rate Rcannot be list decoded beyond1−R\n2\nfor a certain family of parameters. The list decodability of random ra nk\nmetric codes was settled [5]. More specifically, Ding [5] showed that a r ank\nmetric code in Fn\nqmwith a constant column-to-row ratio b=n\nmand rate Ris\n(ρ,L) list-decodable with L=O(poly(mn)) must satisfy R≤(1−ρ)(1−bρ);\nand she also showed that for ǫ∈(0,1), any rank metric code in Fn\nqmof rateR\n2cannot be (1 −R−ǫ,poly(n))-list decodable when the ratio b≥2ǫ\n(1−R−ǫ)(R+ǫ).\nAlthough random rank metric codes with R= (1−ρ)(1−bρ)−ǫfor a\nsmall 0< ǫ <1 can be ( ρ,O(1\nǫ))-list decodable with a high probability [5],\nthere has been little progress on explicit constructions of rank met ric codes\nthat approach the theoretic limit while allowing for efficient decoding. S ilva,\nKschischang, andKotterin[20]introducedaclassofsimplematrixco desover\nFq, which can be efficiently decoded with the decoding radius ρapproaching\nthe Singleton bound 1 −Rwhen the ratio b→0. Guruswami, Wang and\nXing later proposed a family of subcodes of Gabidulin codes through p oint\nevaluations in a subfield [14], where they showed that those subcode s can be\nlist decoded with code rate (1 −ǫ)Rand decoding radius ρ=s\ns+1(1−R) for\nanyǫ∈(0,1) and positive integers s, where the ratio bis in order of O(ǫ2)\nand the list Lis of exponential size. It indicates that as ǫand 1/sapproaches\nzero, these codes can approach the Singleton bound ρ= 1−Ras well. For\nrank metric codes in Fn\nqmwith a constant ratio b=n\nm, it appears more\nchallenging to explicitly construct codes that are ( ρ,L)-list decodable with\na radius approaching the theoretic limit established in [5]. The interleav ed\nGabidulin codes [23] and folded Gabidulin codes [4] were shown to have lis t\ndecoding radius beyond1\n2(1−R) with exponential-size lists. Recently Xing\nand Yuan [24] raised an open problem on the construction of rank me tric\ncodes with list decoding radius beyond1\n2(1−R), a polynomial-size list, and\nan efficient decoding algorithm. They also proposed a class of rank-m etric\ncodes with a constant bup to1\n2and list decoding radius beyond the unique\ndecoding radius1−R\n2, where the list size after randomization can be reduced\ntoO(1\nǫ).\nCompared to the Gabidulin codes, LRPC codes have more random-like\nparity check matrices, which not only are favored in cryptographic applica-\ntions but also appear to allow for better decoding capability. Recent ly the\ndecoding capability of LRPC codes was further investigated in [2], whe re for\ndensityd= 2 and m≥3rd−2, LRPC codes are shown to be uniquely\ndecodable with decoding radius ρ=2\n3(1−R). In [6] we proposed improved\ndecodings for LRPC codes when n−k < rd. Motivated by the improved\ndecoding capability of LRPC codes, in this paper we proposed a subfa mily of\nLRPC codes, which allow for a parity check matrix Hwhose entries are ran-\ndomly drawn from a subspace Vα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFq/subsetnoteql Fqmfor a positive\n3integerdsignificantly smaller than m. We term these codes bounded-degree\nLRPC (BD-LRPC) codes according to the structure of Vα,d. This particular\nstructure allows for efficient decoding of errors even when S/subsetnoteqlVα,d.E, where\nE,Sare theFq-subspaces generated by the components of the error e∈Fn\nqm\nand the syndrome s=eH⊺for a parity check matrix HoverVα,d, respec-\ntively, and Vα,d.Eis the product of two subspaces. We observe that, when\nmultiplied with a subspace Vα,t, the subspaces SandVα,d.Ewill be expanded\nin different paces, leading to Vα,t.S=Vα,t.Vα,d.E=Vα,d+t−1.Efor certain\npositive integer t. This equality is shown to hold with a high probability\nwith technical calculations in Sections 4 and 5. More concretely, in Se ction\n4 we transform the equality Vα,t.S=Vα,d+t−1.Eto a condition requiring a\nmatrixM(t)\nt(as defined in Eq. (24) derived from the equality) to have rank\nr(t+d−1), which is equivalent to requiring its r×rsub-matrix Y1to be\ninvertible and its ut×r(d−1) sub-matrix Mt(Z(1),A) to have rank r(d−1);\nin Section 5 we investigate the probability of Rank( Mt(Z(1),A)) =r(d−1)\nwith the help of Ferrers diagrams. Similarly to the decoding for stand ard\nLRPC codes, from the equality Vα,t.S=Vα,d+t−1.Ewe can recover the error\nsupportEand then correct the error ewith the syndrome equations given\nbys=eH⊺. Asymptotically, when napproaches infinity with the ratio\nb=n\nm→0, the BD-LRPC code can be uniquely decoded with a code rate R\nand decoding radius ρ= 1−R−µ, whereµ=u\nncan be arbitrarily small. In\naddition, for a constant b=n\nm, the BD-LRPC code can be uniquely decoded\nwith decoding radius ρ≥1\n2(1−R) when relevant parameters are properly\nchosen. This suggests that the BD-LRPC codes serve as another solution to\nthe open problem proposed in [24].\nThe remainder of this paper is organized as follows. Section 2 recalls\nbasic notations, auxiliary results on rank metric codes and Ferrers diagrams.\nSection 3 introduces the BD-LRPC codes and shows that the decod ing ca-\npability of these codes approaches the Singleton bound. The main re sult is\ngiven in Theorem 2, for which important theoretical arguments and technical\ncalculations are given in Proposition 1 in Section 4 and in Theorems 3 and\n4 in Section 5, respectively. Section 6 concludes our works in this pap er.\n42 Preliminaries\nLetqbe a prime power. It is well known that the finite field Fqmwithqm\nelements can be seen as an m-dimensional vector space over the base field Fq.\nWe will denote Fq-linear subspaces of Fqmby calligraphic letters. A vector of\nFn\nqmwill be denoted by a lower-case letter in bold, and its components will b e\ndenoted by the same letter in normal form, e.g., a vector v= (v1,...,v n)∈\nFn\nqm. For two positive integers l≤n, the notation [ l...n] denotes the set\n{l,...,n} ⊆Z, and we will use the shorthand notation [ n] forl= 1.\nGiven a set S⊆Fqm, therank support ofS, denoted by /a\\}⌊ra⌋k⌉tl⌉{tS/a\\}⌊ra⌋k⌉tri}htFq, is\ndefined as the Fq-linear span of all the elements in S. This notion can be\nnaturally applied to vectors and matrices over Fqm, namely, the support of a\nvectorv∈Fn\nqmis given by /a\\}⌊ra⌋k⌉tl⌉{tv/a\\}⌊ra⌋k⌉tri}htFq=/a\\}⌊ra⌋k⌉tl⌉{tv1,...,v n/a\\}⌊ra⌋k⌉tri}htFqand the support of a matrix\nA∈Fr×t\nqmis given by /a\\}⌊ra⌋k⌉tl⌉{tA/a\\}⌊ra⌋k⌉tri}htFq=/a\\}⌊ra⌋k⌉tl⌉{tai,j|i∈[r],j∈[t]/a\\}⌊ra⌋k⌉tri}htFq. For an element α∈Fqm\nwe denote αS={αs|s∈S}. GivenV=/a\\}⌊ra⌋k⌉tl⌉{tv1,...,v n/a\\}⌊ra⌋k⌉tri}htFq⊆FqmanFq-linear\nsubspace of Fqmof dimension n, it is clear that αV=/a\\}⌊ra⌋k⌉tl⌉{tαv1,...,αv n/a\\}⌊ra⌋k⌉tri}htFqfor\nanyα/\\⌉}atio\\slash= 0 is still an Fq-linear subspace of dimension n. We will denote by\nFn×k\nqthe set of all n×kmatrices over Fq.\nLetv= (v1,...,v d) andu= (u1,...,u r) be two vectors over Fqmcon-\nsisting of Fq-linearly independent components. We define the product of\nthe two subspaces V=/a\\}⌊ra⌋k⌉tl⌉{tv/a\\}⌊ra⌋k⌉tri}htFqandU=/a\\}⌊ra⌋k⌉tl⌉{tu/a\\}⌊ra⌋k⌉tri}htFqasV.U=/a\\}⌊ra⌋k⌉tl⌉{tv⊗u/a\\}⌊ra⌋k⌉tri}htFq,where\nv⊗u= (v1u,...,v du)∈Fdr\nqm.It is readily seen that the product subspace\nV.U ⊆FqmhasFq-dimension at most dr.\n2.1 Rank metric codes and their list decodability\nRank metric codes have both matrix and vectorial representation s, for which\nthere exists a one-to-one correspondence with respect to a bas is ofFqmover\nFq. Below we introduce some basics of rank metric codes in vectorial re pre-\nsentation.\nArankmetric code Cissimply a subset of Fn\nqmequipped withrankmetric,\nnamely, its minimum rank distance is given by\nd(C) := min{dR(x,y)|x,y∈ C,x/\\⌉}atio\\slash=y}\nwithdR(x,y) = dim( /a\\}⌊ra⌋k⌉tl⌉{tx−y/a\\}⌊ra⌋k⌉tri}htFq).Similarly to block codes with Hamming\nmetric, there exist the Singelton bound and Gilbert-Varshamov bou nd on\n5rank metric codes. Given a rank metric code C ⊆Fn\nqmwith minimum rank\ndistance d, its size satisfies\nlogq|C| ≤min{n(m−d+1),m(n−d+1)}.\nWithout loss of generality, we may assume that n≤msince the minimum\nrank distance and size of a rank metric code are preserved under t ranspose.\nLetR=logq|C|\nmnbe the code rate and δ=d(C)−1\nnbe the relative distance. The\nasymptotic version of the Singleton bound, as n→ ∞, is given by R≤1−δ.\nIt is well known that, given a rank metric code C, a received word y\ncontaining an error eof rank weight r≤ ⌊d(C)−1\n2⌋can be uniquely decoded,\nindicating that there exists a unique codeword c∈ Csatisfying dR(c,y) =r.\nOn the other hand, when ycontains an error with rank weight r >⌊d(C)−1\n2⌋,\nthere might be a list of codewords having rank distance rfromyand one\nneeds to turn to the list decoding. However, list decoding for rank m etric\ncodes appears more challenging than its counterpart in Hamming met ric\n[5,22,24]. Below we recall some results of list decoding for rank metr ic\ncodes.\nA rank metric code C ⊆Fn\nqmis said to be ( ρ,L)-list decodable if for every\nx∈Fn\nqm, we have\n|C ∩B(x,ρn)| ≤L,\nwhereB(x,r) :={y∈Fn\nqm|dR(x,y)≤r}is the rank metric ball of center\nxwith radius r. It is clear that for relative decoding radius ρ≤ ⌊d(C)−1\n2⌋,\nany rank metric code is ( ρ,1) list-decodable, which corresponds to the case\nof unique decoding radius. Ding in [5] studied the theoretic limit of the lis t\ndecodability of rank metric codes.\nTheorem 1. [5, Th. 1] Let m,n,Lbe positive integers such that b=n/m\nis a constant and L=O(poly(mn)). Then, for any R∈(0,1),ρ∈(0,1), a\n(ρ,L)-list decodable rank metric code C ⊆Fn\nqmof rateRmust satisfy\nR≤(1−ρ)(1−bρ).\nIt is also shown [5] that for any ǫ∈(0,1), a random Fq-linear rank metric\ncode is (ρ,O(exp(1\nǫ)))-list decodable with rate R= (1−ρ)(1−bρ)−ǫ, which\nis considered as the Gilbert–Varshamov bound for the list decoding o f rank\nmetric codes. However, there have been only a few efficient list deco dable\nrank metric codes in the literature [14,21,24].\n6The rank metric code in [21] was proposed to achieve the capacity of an\nadditive matrix channel and was given in matrix form\nC=/parenleftigg\n0v×v0v×(n−v)\n0(m−v)×vU/parenrightigg\n,\nwhereU∈F(m−v)×(n−v)\nq is a data matrix1. The authors in [21] showed that\ntakingr=v, this simple codehas code rate R=(n−r)(m−r)\nmninq-aryunits and\nthat for any ǫ∈(0,1),r= (ρ+ǫ)nand a constant b=n/m, asmincreases,\nthis simple code can be uniquely decoded with overall complexity O(nmr)\noperations in Fq, the failure rate decreasing exponentially with n, and the\ncode rate satisfying R≥(1−ρ)(1−bρ)−(1 +b)ǫ.A modified version of\nthis code was also discussed in the context of code-based cryptog raphy [11].\nIn [14] an Fq-linear subcode ofGabidulin codewas shown to belist-decodable\nwith code rate (1 −ǫ)R, decoding radius ρ=s\ns+1(1−R), list size O(qs2/ǫ2)\nfor anyǫ∈(0,1) and integer s≥1. Note that as sincreases, the decoding\nradiusρapproaches the theoretic limit 1 −R. Nevertheless, the column-to-\nrow ratio b=n\nmfor the code is required to approach zero. Xing and Yuan\nrecently in [24] reviewed the progress of list decoding of rank metric codes\nand proposed the following open problem:\nOpen Problem. For a given constant ratio b=n\nm, explicitly construct\nrank-metric codes of rate RinFm×n\nqwith list decoding radius ρ >1\n2(1−R)\nand efficiently decode them.\nIn [24] they provided an explicit construction of rank metric codes, for\nwhich they provided both deterministic and probabilistic algorithms fo r de-\ncoding the codes. They showed that the constructed code for a c onstantbup\nto 1/2 and any ǫ∈(0,1) is (ρ,exp(O(1/ǫ2)))-list decodable with ρbeyond\n1\n2(1−R) and complexity poly( n,exp(1/ǫ)).\nIn subsequent sections, we will study a subfamily of Fqm-linear low-rank\nmetric codes [17] and show that the code can be, with a high probabilit y,\nuniquely decoded with radius ρ= 1−R−u\nnfor a constant b=n\nmand\na positive integer u. The proposed codes therefore serve as a probabilistic\nsolution to the above open problem.\n1Here we transpose the code matrix in [21] for the consistency of no tation in this paper.\n72.2 Counting Tools and Ferrers diagrams\nThe error probability of decoding the proposed BD-LRPC codes will b e de-\nrived from technical calculations with the help of Ferrers diagrams. Below\nwe introduce some useful counting tools used in Section 5.\nThe number of possible k-tuples of linearly independent vectors in Fn\nq\ncan be obtained as follows. Consider klinearly independent vectors g1,...gk\ninFn\nq. The vector g1must be not null and can then be chosen in ( qn−1)\nways. The second vector can be chosen in ( qn−q) ways among the vectors in\nFn\nq\\/a\\}⌊ra⌋k⌉tl⌉{tg1/a\\}⌊ra⌋k⌉tri}htFq. Following thisargument wesee that gk∈Fn\nq\\/a\\}⌊ra⌋k⌉tl⌉{tg1,...,gk−1/a\\}⌊ra⌋k⌉tri}htFqcan\nbe chosen in ( qn−qk−1) ways. Hence the number of klinearly independent\nvectors in Fn\nqis given by Aq(n,k) =/producttextk−1\ni=0(qn−qi).It is natural to involve\nAq(n,k)inthecounting of k-dimensional linearsubspaces in Fn\nq. The number\nof distinct subspaces of dimension kcan be computed as the number of pos-\nsible distinct choices of klinearly independent vectors up to a change of base,\nindicating that |{V ⊆Fn\nq|dim(V) =k}|=Aq(n,k)/Aq(k,k). This number\nis also known as the Gaussian binomial coefficient/bracketleftbign\nk/bracketrightbig\nq=Aq(n,k)/Aq(k,k).\nIn addition, it can be shown that the number of matrices in Fn×m\nqwith rank\nk≤min{n,m}is given by Aq(n,k)Aq(m,k)/slashbig\nAq(k,k). This immediately\nimplies that the number of invertible matrices in Fk×k\nqis given by Aq(k,k).\nAn alternative way to express Aq(n,k) is to use the quantity Hq(n) =/producttextn\ni=1(1−q−i),which corresponds to the fraction of invertible matrices among\nall the matrices in Fn×n\nqsinceqn2Hq(n) =/producttextn\ni=1(qn−qn−i) =Aq(n,n).\nIt can be proved that the limit of Hq(n) forn→ ∞converges to Hq=\nlimn→∞Hq(n)≤1 and lim q→∞Hq= 1.Notice that Hq(n)≤Hq(n−k) and\n0.28≈H2≤Hq< Hq(n)<1. Using some simple algebraic manipulation\none can show that\nAq(n,k) =qnkHq(n)/Hq(n−k). (1)\nThis expression reflects the asymptotic value of Aq(n,k). When we want to\nestimate the proportion of the matrices of rank kamong all the matrices in\nFk×n\nq,we have\nAq(n,k)/qnk=Hq(n)/Hq(n−k) =n/productdisplay\ni=n−k+1(1−q−i),\n8which can be approximated as\n1−q−nk−1/summationdisplay\ni=0qi= 1−q−nqk−1\nq−1≥1−q−(n−k)\nq−1.\nAn alternative way to count the subspaces of dimension kinFn\nq,is to\ncount the number of full rank matrices in Fn×k\nqin column reduced echelon\nform. Recall that a matrix A∈Fn×k\nqof rankkis in column reduced echelon\nform if:\n•The topmost nonzero entry of each column is a 1 (called a leading one) .\n•The other entries of a row where there is a leading one are all zeroes .\n•If we denote by nithe position of the leading one of the i-th column,\nthenn1< n2<···< nk≤n.\nAs an example the following matrix is in column reduced echelon form\nA=\n0 0 0\n1 0 0\n0 1 0\n• •0\n• •0\n0 0 1\n• • •\n,\nwhere each •represents a free value in Fq. The number of matrices following\nthe same pattern of A∈F7×3\nqas above is q7where 7 is the number of •inA.\nLet us now consider the alternative counting. Let V ⊆Fn\nqbe a subspace\nof dimension kand letG∈Fn×k\nqbe a matrix such that its columns form a\nbasis ofV.The column reduced echelon form of Gis a basis of Vand it is\nunique. All vector spaces of dimension kcorrespond to a unique generator\nin column reduced echelon form.\nRemoving all the rows containing a leading one from a reduced column\nechelon form matrix in Fn×k\nqgives a Ferrers diagram. If we consider only\nmatrices offull rank k,there will be exactly kleading ones, then theresulting\nFerrers diagram will resemble to an ( n−k)×kmatrix. In our example, the\nFerrers diagram associated to Ais the 4×3 diagram\nF=/parenleftbigg0 0 0\n• •0\n• •0\n• • •/parenrightbigg\n. (2)\nEachn×kFerrersdiagram Fhasanaturalrepresentationasanon-increasing\nsequence [ f1,...,f k] of length kof integers 0 ≤fi≤n.The sequence repre-\nsentation of (2) is [3 ,3,1].\n9Consider an n×kFerrers diagram Fas a matrix. Denote by ai,jthe\nelement at the intersection of the i-th row with the j-th column. Whenever\ni/\\⌉}atio\\slash=nandj/\\⌉}atio\\slash= 1,ifai,j=•,thenai+1,j=•andai,j−1=•.IfFis an×k\nFerrers diagram, its weight |F|correspond to the number of free variables •.\nIt is easy to see that, for each ( n−k)×kFerrers diagram F,there are\nexactlyq|F|possible column reduced echelon matrices A∈Fk×r\nqthat have F\nas the associated Ferrers diagram. Let Fn−k,kbe the set of all ( n−k)×k\nFerrers diagrams. Knut [15] showed the following relation between Ferrers\ndiagram and the Gaussian binomial coefficients:\n/summationdisplay\nF∈Fn−k,kq|F|=/bracketleftbiggn\nk/bracketrightbigg\nq=Aq(n,k)/Aq(k,k) (3)\nsince each column reduced echelon matrix corresponds to a k-dimensional\nsubspace in Fn\nq. We will see that this formula is important to attack Problem\n1 in Section 5.\nConsidering the set Fn,kofn×kFerrers diagrams as non-decreasing se-\nquences, it is natural to define a partial order relation over this se t. the\nsetFn,kofn×kFerrers diagrams. Let F= [f1,...,f k]∈ Fn,kandF′=\n[f′\n1,...,f′\nk]∈ Fn,kbe two Ferrers diagrams, we define the partial order rela-\ntionF≤F′iffi≤f′\nifor 1≤i≤k. We will denote by /hatwideF={F′∈ Fn,k|\nF≤F′},the set of all the Ferrers diagrams bigger than F.We can combine\n(3) and the partial order on Ferrers diagrams to determine the nu mber of a\ncertain class of matrices.\nLemma 1. LetF= [f1,...,f k]be au×kFerrers diagram such that fi=u\nfori≤k−tandfi=u−sotherwise. Then we have\n/summationdisplay\nF′∈/hatwideFq|F′|=/bracketleftbiggs+t\ns/bracketrightbigg\nqquk−st(4)\nwhere/hatwideF={F′∈ Fu,k|F≤F′}.\nProof.ForF′∈ˆF,denoteF′=F+Lwhere the addition between FandL\nis the component wise sum.\n10FL1u\nkst\nWewrite Lastheconcatenationoftwo sequences L0||L1, whereL0haslength\nk−tandL1has length t. It is clear that L0= [0,...,0] andL1is a non\nincreasing sequence of integers upper bounded by s. HenceL1describes an\ns×tFerrers diagram. From (3) we see that\n/summationdisplay\nL1∈Fs,tq|L1|=/bracketleftbiggs+t\nt/bracketrightbigg\nq.\nCombining the fact q|F|=quk−st, we obtain the desired result.\n3 Bounded degree LRPC codes\nIn this section we will consider a particular family of subspaces in Fqm. The\nproperties of these subspaces motivate us to study the BD-LRPC codes.\nDefinition 1. Letα∈Fqmanddbe a positive integer. The bounded degree\nsubspace generated by αof degree dis described as\nVα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFq.\nAssumeFq[x]/(xd)is the set of all the polynomials in xoverFqof degree\nstrictly less than d.The subspace Vα,dcan be equivalently given by\nVα,d=/braceleftbig\np(α) :p∈Fq[x]/(xd)/bracerightbig\n.\nFrom the definition, if α∈Fqmdoes not belong to any proper subfield of\nFqm, we immediately have Vα,m=Fqm. The product of two bounded degree\nsubspaces Vα,iandVα,jhas the following properties.\nLemma 2. Forα∈Fqmandi, j∈[m], ifαdoes not belong to any proper\nsubfield of Fqm,we havedim(Vα,i) =iand\nVα,i.Vα,j=Vα,i+j−1, (5)\n11in particular, for j= 2we have\nVα,i+1=Vα,i.Vα,2=Vα,i+αVα,i. (6)\nProof.For any i,j∈[m], a basis of Vα,iis given by (1 ,α,...,αi−1) and a\nbasis ofVα,jis given by (1 ,α,...,αj−1).Furthermore, by the definition of\nVα,i.Vα,jwe see that (1 ,α,...,αi+j−2) is a basis of Vα,i.Vα,j, indicating that\nVα,i.Vα,j=Vα,i+j−1.By taking j= 2, one has Vα,i.Vα,2=Vα,i+1, which has\na basis (1 ,α,...,αi).SinceVα,i+αVα,ihas the same basis (1 ,α,...,αi), the\nstatement follows.\nRecall from [17] that an LRPC code has its parity check matrix H∈\nF(n−k)×n\nqmsatisfydim( /a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq) =d≪m, orequivalently, itsparitycheck matrix\nHis taken from V(n−k)×nwhereV/subsetnoteql Fqmhas dimension d≪m. The main\nidea behind the decoding of LRPC codes is as follows. Assume e∈Fn\nqmis\nan error of rank weight r. Let/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq=/a\\}⌊ra⌋k⌉tl⌉{th1,...,h d/a\\}⌊ra⌋k⌉tri}htFq, andEbe the support\nof error given by /a\\}⌊ra⌋k⌉tl⌉{te/a\\}⌊ra⌋k⌉tri}htFq=/a\\}⌊ra⌋k⌉tl⌉{tε1,...,ε r/a\\}⌊ra⌋k⌉tri}htFq. Then the syndrome s=eH⊺has\nsupportS=/a\\}⌊ra⌋k⌉tl⌉{ts/a\\}⌊ra⌋k⌉tri}htFqcontained in the product space /a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq.E.With properly\nchosen parameters, one can assume that S=/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq.Eholds with a high\nprobability, and one can recover EfromSand/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFqunder the assumption\nE=d/intersectiondisplay\ni=1h−1\niS=d/intersectiondisplay\ni=1h−1\ni(/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq.E).\nHowever, this approach does not work for the cases where S/subsetnoteql/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq.E. This\nmotivates us to consider the following subfamily of LRPC codes.\nDefinition 2 (BD-LRPC code) .LetVα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFqbe a bounded-\ndegree subspace and H∈ V(n−k)×n\nα,dhave full rank n−k. A code C/subsetnoteql Fn\nqm\nhavingHas parity check matrix is said to be a bounded degree low-rank\nparity check (BD-LRPC) code when dis significantly smaller than m.\nRemark 1. Observe that LRPC codes of density 2correspond exactly to BD-\nLRPC code of degree 2.Let/a\\}⌊ra⌋k⌉tl⌉{ta1,a2/a\\}⌊ra⌋k⌉tri}htFqbe the support of the parity check matrix\nH∈F(n−k)×n\nqmof an LRPC code of density 2. Then the matrix H′= (a−1\n1)H\nwill be a parity check matrix of the same code and\n/a\\}⌊ra⌋k⌉tl⌉{tH′/a\\}⌊ra⌋k⌉tri}htFq=a−1\n1/a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq=/a\\}⌊ra⌋k⌉tl⌉{t1,a−1\n1a2/a\\}⌊ra⌋k⌉tri}htFq=Vα,2,\nwhereα=a−1\n1a2.\n12Fqm-linear codes\nLRPC-d\nd= 3\nd= 2\nBD-LRPC\nFigure 1: BD-LRPC inside the set of LRPC codes\nIn Figure 1 we depict the situation of BD-LRPC of degree dbeing a\nsubset of LRPC codes of density dford >2 while being the same set for\nd= 2.\nForBD-LRPCcodeswith /a\\}⌊ra⌋k⌉tl⌉{tH/a\\}⌊ra⌋k⌉tri}htFq=Vα,d,asadirectconsequenceofLemma\n2, we have the following result regarding the product space of Vα,dandE.\nLemma 3. Letα∈Fqm,Vα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFq⊆Fqmand letE ⊆Fqmbe\nanFq-linear subspace of Fqmof dimension r. Then, for any positive integer\ni≥1, we have\nVα,i+1.E=Vα,2.Vα,i.E=Vα,i.E+αVα,i.E. (7)\nThe dimension of Vα,i.Eis in most cases equal to irfor integers i≤m/r\nandit is always upper boundedby ir. Consider S=/a\\}⌊ra⌋k⌉tl⌉{ts1,...,s n−k/a\\}⌊ra⌋k⌉tri}htFq/subsetnoteqlVα,d.E\nand suppose its dimension is dim( S) = (n−k) t.Finally, if dim(Ω 0(Z,A))≥1,then dim(Ω t′(Z,A))≥\n25t′+1orthereexist0 ≤t < t′suchthatΩ t′(Z,A) = Ωt(Z,A)anddim(Ω t(Z,A))≥\nt+1.Since Ω t(Z,A)⊆Fr\nq,its dimension is upper bounded by r.Then from\nr≥dim(Ω t(Z,A))≥t+1 we obtain that t≤r−1.For the particular case\nof Ω0(Z,A) ={0},we haveZ= 0 and Ω t′(Z,A) ={0}for allt′>0.\nAs a direct consequence of (iii), we see that the expansion of Ω i(Z,A)\nstops at certain integer t≤r−1. That is to say, Ω t(Z,A) = Ωr−1(Z,A) for\nanyt≥r−1 and it suffices to consider Ω r−1(Z,A). Meanwhile, it is clear\nthat Ω r−1(Z,A) is the smallest subspace Ω ⊆Fr\nqsuch that RowSpan( Z)⊆Ω\nand ΩA={xA|x∈Ω} ⊆Ω.For simplicity, we denote Ω r−1(Z,A) as Ω(Z,A)\nin the sequel and denote\nC(u,r)\nk=C(u,r,r−1)\nk={(Z,A)∈Fu×r\nq×Fr×r\nq|dim(Ω(Z,A)) =k}.(32)\nThe following property of the set C(u,r,t)\nkwill facilitate our subsequent\ncalculations.\nLemma 6. LetC(u,r,t)\nkbe defined as in (30)andBbe an invertible matrix\ninFr×r\nq.For any(Z,A)∈C(u,r,t)\nk, we have (ZB,B−1AB)∈C(u,r,t)\nk.\nProof.For (Z,A)∈C(u,r,t)\nk, we can express Ω t(Z,A) as the row span of the\nmatrixMt(Z,A)\nΩt(Z,A) = RowSpan\nZ\nZA\n...\nZAt\n. (33)\nNotice that ( B−1AB)(B−1AB) =B−1A2B, implying ( B−1AB)t=B−1AtB.\nThis gives ZB(B−1AB)t=ZAtB.Hence, for the pair ( ZB,B−1AB) we\nobtain\nΩt(ZB,B−1AB) = RowSpan\nZB\nZAB\n...\nZAtB\n= Ωt(Z,A)B.\nSinceBis invertible, it holds dim(Ω t(Z,A)B) = Ωt(Z,A) =k,implying that\n(ZB,B−1AB)∈C(u,r,t)\nk.\n26Algorithm 1: Generation of a basis of Ω t(Z,A) for (Z,A),t\nInput:A matrix Z∈Fu×r\nq, a matrix A∈Fr×r\nqand an integer t.\nOutput: A matrix Gsuch that RowSpan( G) = Ωt(Z,A).\n// Initialize Ωas the zero subspace and Gas an empty list.\n1Ω ={0};\n2G= [] ;\n3forj∈[0...t]do\n// Add all the new linearly independent rows obtained from\nZAjusing the convention A0=Ir.\n4G(j)= [];\n5fori∈[u]do\n6 if ziAj/∈Ωthen\n7 G(j).append(ziAj);\n8 Ω = Ω+ /a\\}⌊ra⌋k⌉tl⌉{tziAj/a\\}⌊ra⌋k⌉tri}htFq;\n9 end\n10end\n11G.append (G(j));\n12end\n13ReturnG;\nTo solve Problem 1, we need to consider linearly independent rows in\nthe matrix Mt(Z,A) as it expands. This is a process of extracting those\nlinearly independent row vectors in ZAjforj= 0,1,...,tin sequential order\nfor a matrix pair ( Z,A). This process is summarized in Algorithm 1. The\nfollowinglemmashowsthatthematrix G,obtainedasoutputfromAlgorithm\n1, generates the subspace Ω t(Z,A) of dimension kon input ( Z,A),twhen\n(Z,A) is taken from C(u,r,t)\nk.\nLemma 7. For(Z,A)inC(u,r,t)\nk, the rows of the output matrix\nG=\nG(0)\nG(1)\n...\nG(t)\n\nof Algorithm 1 form a basis of Ωt(Z,A)⊆Fr\nq. In particular, when t=r−1,\nthe rows of Gform a basis of Ω(Z,A).\n27Proof.InAlgorithm 1, thevector space Ω is initialized as Ω = {0}.Using the\nconvention A0=Ir, in the initial execution of the inner loop, we add to G(0)\nall the linearly independent rows of ZA0=Z.Each time a new row is added\ntoG(0)the subspace Ω is enlarged to encompass that row. By the end of th e\nfirst inner loop we will have Ω = RowSpan( G(0)) = RowSpan( Z) = Ω0(Z,A).\nOn the second iteration j= 1, all the rows of ZAthat do not belong to Ω\nwill be added to G(1). For each row we add to G(1), the subspace Ω is\nextended to encompass that row as well. At the end of this iteration , the\nsubspace Ω correspond to RowSpan( G) forG=/bracketleftbigG(0)\nG(1)/bracketrightbig\nwhich is equal to\nΩ1(Z,A).Continuing in this way, after the execution of the j-th inner loop\nwe have Ω = RowSpan( G) = Ωj(Z,A) forG=/parenleftigg\nG(0)\n...\nG(j)/parenrightigg\n.The statement thus\nfollows.\nBelow we provide an example to show the steps in Algorithm 1.\nExample 1. Letu= 3,r= 4,t= 3andk= 3. Assume\nZ=\n1 0 0 0\n0 1 0 0\n1 1 0 0\n, A=\n0 0 1 0\n1 1 1 0\n0 0 0 1\n1 1 1 1\n.\nThen, from Algorithm 1 one obtains\nG=\nG(0)\nG(1)\nG2\n=\n1 0 0 0\n0 1 0 0\n0 0 1 0\n0 0 0 1\n.\nNote that in Algorithm 1 G(3)in iteration j= 3is an empty matrix.\nWe now take a closer look at the steps of Algorithm 1. Denote Z(0)=Z\nand letG(0)be the matrix obtained after the iteration j= 0. Notice that\nG(0)are sequentially formed by rows in Z(0)of which each row is linearly\nindependent of rows in Z(0)before it; and that any row of Z(0)is a linear\ncombination of rows in G(0).For instance, as in Example 1 we have\nZ(0)=\n1 0 0 0\n0 1 0 0\n1 1 0 0\n=⇒G(0)=/parenleftigg\n1 0 0 0\n0 1 0 0/parenrightigg\n28It is clear that, all the rows in matrix Z(0)Acan be by linearly expressed\nby rows in Z(1)=G(0)A. Similarly, denote by G(1)the matrix sequentially\nformed by rows in Z(1), which cannot be linearly expressed by rows in G(0)\nand preceding rows in Z(1). For instance,\nZ(1)=G(0)A=/parenleftigg\n0 0 1 0\n1 1 1 0/parenrightigg\n=⇒G(1)=/parenleftig\n0 0 1 0/parenrightig\n.\nWe continue this process as in Algorithm 1 and denote Z(j)=G(j−1)Afor\n1≤j < r−1 andG(j)the matrix that sequentially extracts rows from Z(j)\nwhich cannot be linearly expressed by rows in G(0),...,G(j−1)and preceding\nrows inZ(j). Then from Algorithm 1 we have\n\nZ(0)\nZ(1)\n...\nZ(r−1)\n=⇒G=\nG(0)\nG(1)\n...\nG(r−1)\n, (34)\nwhere for jlarger than certain value t, the matrices G(j)could be empty\nmatrices.\nFrom the above process, it is easily seen that different pairs of ( Z,A) can\nproduce the same output G∈Fk×r\nqby Algorithm 1. The number of pairs\nproducing the same output G∈Fk×r\nqturns out to be independent from the\nchoice of Gamong the set of all the k×rfull rank matrices over Fq. This fact\nwill be shown in the following proposition, which enables us to concentr ate\non the special output matrix Ek∈Fk×r\nq, formed by the first krows of the\nidentity matrix Ir∈Fr×r\nq, in our calculations.\nLemma 8. LetG∈Fk×r\nqbe a matrix of rank k≤rand denote\nC(u,r,t)\nG={(Z,A)∈Fu×r\nq×Fr×r\nq:Alg(Z,A,t) =G}.\nLetEkbe the submatrix formed by the first krows ofIroverFq. Then we\nhave\n|C(u,r,t)\nG|=|C(u,r,t)\nEk|.\nProof.To prove this result we will create a bijection between C(u,r,t)\nGand\nC(u,r,t)\nEk.Sincek≤r, there exists an invertible matrix B∈Fr×r\nqsuch that\n29GB=Ek.Consider the map B: (Z,A)→(ZB,B−1AB),sinceBis in-\nvertible this is a bijection of Fu×r\nq×Fr×r\nqto itself. We will prove that, for\neach (Z,A) such that Alg(Z,A,t) =G,we have that Alg(ZB,B−1AB,t) =\nGB=Ek.\nWe compare the procedure of Algorithm 1 on two inputs ( Z,A,t) and\n(ZB,B−1AB,t),showing that, when the first yields Gas output, the sec-\nond will yield Ek.Letzibe a row of Zthat is linearly independent from\nz1,...,zi−1,alsoziBis independent from z1B,...,zi−1B.This means that,\nevery time we add a row ziin the first execution if we add ziBin the second.\nIn this way, at each step, G′B=Ek′,whereG′andEk′are the matri-\nces obtained so far in the algorithm. At the end of the first inner loop , if\nRank(Z) =k0≤u,we will have G(0)B=Ek0where both G(0),Ek0∈Fk0×r\nq.\nIn thej-th inner loop, the algorithm will do the following. On initial\ninput (Z,A), it will compute ZAjthen, for each row vector ziAj, it will\ncheck if it is already in Ω = /a\\}⌊ra⌋k⌉tl⌉{tg1,...,gk′/a\\}⌊ra⌋k⌉tri}htFq, if not, it will add it as the last\nrow ofG′and update Ω to /a\\}⌊ra⌋k⌉tl⌉{tg1,...,gk′+1/a\\}⌊ra⌋k⌉tri}htFq. On initial input ( ZB,B−1AB),\nit will compute ZB(B−1AB)jthen, for each row vector ziB(B−1AB)j, it will\ncheck if it is already in Ω B=/a\\}⌊ra⌋k⌉tl⌉{te1,...,ek′/a\\}⌊ra⌋k⌉tri}htFq, if not, it will add it as the last\nrow ofEk′and update Ω to /a\\}⌊ra⌋k⌉tl⌉{te1,...,ek′+1/a\\}⌊ra⌋k⌉tri}htFqIn the proof of Lemma 6 we\nhave already seen that ziB(B−1AB)j=ziAjB.\nWe will show that, at each step, G′B=Ek′.LetGj−1andEj−1=Gj−1B\nbe the matrices obtained after we complete the inner loop for the ( j−1)-th\ntime. It is easy to see that\nziAj∈RowSpan( Gj−1)+/a\\}⌊ra⌋k⌉tl⌉{tz1,...,zi−1/a\\}⌊ra⌋k⌉tri}htFqAj,\nif and only if\nziAjB∈RowSpan( Ej−1)+/a\\}⌊ra⌋k⌉tl⌉{tz1,...,zi−1/a\\}⌊ra⌋k⌉tri}htFqAjB.\nwhere we used the fact that RowSpan( Ej−1) = RowSpan( Gj−1)B.\nBased on Lemma 8 we have the following corollary, which significantly\nsimplifies our calculations.\nProposition 2. LetC(u,r,t)\nkbe given as in (30)and\nC(u,r,t)\nEk={(Z,A)∈Fu×r\nq×Fr×r\nq:Alg(Z,A,t) =Ek}.\n30Then we have\n|C(u,r,t)\nk|=Aq(r,k)|C(u,r,t)\nEk|.\nProof.A pair (Z,A)∈C(u,r,t)\nkiff its output Alg(Z,A,t) through Algorithm\n1 is a full rank matrix G∈Fk×r\nq. There are Aq(k,r) distinct matrices in\nFk×r\nqof rankk. For each distinct matrix G∈Fk×r\nqof rankk, Lemma 8\nimplies that |C(u,r,t)\nG|=|C(u,r,t)\nEk|. Hence there are |C(u,r,t)\nEk|distinct tuples\n(A,Z) that generate each full-rank matrix in Fk×r\nq. The desired statement\nthus follows.\nBy Proposition 2, to solve Problem 1, it suffices to find the value of\n|C(u,r,t)\nEk|.We shall investigate |C(u,r,t)\nEk|forsome integers tthat areparticularly\ninteresting in the decoding of BD-LRPC codes.\n5.2 The cardinality of C(u,r,t)\nkfort≥r−1\nThis subsection presents an explicit formula to compute the cardina lity of\nC(u,r)\nkdefined in (32) and the detailed calculations.\nTheorem 3. The setC(u,r)\nkdefined in (32)has cardinality\n/vextendsingle/vextendsingle/vextendsingleC(u,r)\nk/vextendsingle/vextendsingle/vextendsingle=Aq(r,k)/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nqqr(r−k)+k=/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nqHq(r)\nHq(r−k)qr(r−k)+k.\nThe calculation of the formula in Theorem 3 involves several paramet ers.\nBelow we shall first discuss the calculation of |C(1,r)\nEk|, which is easier to follow\nbut exhibits the key idea for the general case u≥2. Then we will present\nthe calculation of C(u,r)\nEkwith the help of Ferrers diagrams.\n5.2.1 The case u= 1.\nForu= 1,the matrix Z∈Fu×r\nqis just a row vector z∈Fr\nq.Letyi=zAi−1\nand consider the matrix Y∈Fr×r\nqwhose rows are the vectors yi. Following\nProposition 2, we focus on the case G=Ek, where Gis the output of\nAlgorithm 1 on input ( z,A),r−1. We can characterize the structure of z\n31andAthat gives this output. We have\nY=\ny1\ny2\n...\nyr\n=\nz\nzA\n...\nzAr−1\n=⇒G=Ek=\ne1\ne2\n...\nek\n.\nAt first, itisclear that z=e1, otherwise onecannotobtain G=Ek, implying\nz=y1=e1. Then the second row y2=e1Ais the first row of A. Note\nthat the first row of Acannot be e1. Otherwise y2,y3,...,ykwill be always\ne1, contradicting G=Ek. Thus, in order to obtain G=Ek, the first row of\nAmust be e2. As the iterations proceed, with a similar argument, the first\n(k−1) rows of Amust be e2,...,ek, respectively, in order to yield G=Ek.\nThat is to say,\nYk=\nz\nzA\n...\nzAk−1\n=\nz\na1\n...\nak−1\n=\ne1\ne2\n...\nek\n,\nwhereaiis thei-th row of the matrix A. This relation fixes zand the first\n(k−1) rows of A. Consider yk+1=zAk=ekA=ak. SinceG=Ekhas only\nkrows,yk+1=akmust belong to /a\\}⌊ra⌋k⌉tl⌉{te1,...,ek/a\\}⌊ra⌋k⌉tri}htFq.This means that there are qk\nchoices of the k-th rowakofA. There are no extra restrictions on A. That\nis to say, the vector ztogether with matrix Ahas the following structure:\n/parenleftigg\nz\nA/parenrightigg\n=\nIk\na0\nˆA\n, (35)\nwhereacontains the first kcoordinates of akwhich can be freely chosen\nfromFk\nq,0is the zero matrix in F(k+1)×(r−k)\nq , andˆAis an arbitrary matrix\ninF(r−k)×r\nq. Therefore, there are in total q(r−k)r+kpossible pairs ( z,A) that\ngenerate G=Ekas output from Algorithm 1. This implies that\n|C(1,r)\nk|=Aq(r,k)|C(1,r)\nEk|=Aq(r,k)qr(r−k)+k. (36)\nThis is consistent with the statement in Theorem 3 for u= 1.\n325.2.2 The cases u≥2.\nSimilarly to the case of u= 1, we proceed to count the cardinality of C(u,r)\nEk\nby describing the pattern which the pairs of matrices ( Z,A) in this set must\nfollow. This counting will involve the use of Ferrers diagrams introduc ed in\nSection 2.2.\nProposition 3. The cardinality of the set\nC(u,r)\nEk=/braceleftbig\n(Z,A)∈Fu×r\nq×Fr×r\nq|Alg(Z,A,r−1) =Ek/bracerightbig\n,\nis given by\n|C(u,r)\nEk|=/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nqqr(r−k)+k. (37)\nProof.Let (Z,A) be a pair of matrices in C(u,r)\nEk.This implies that Z=\n(ˆZ|0)∈Fu×r\nqwhereˆZ∈Fu×k\nqand0∈Fu×(r−k)\nqis the zero matrix. The\nmatrixAshould satisfy the condition vA∈ /a\\}⌊ra⌋k⌉tl⌉{tEk/a\\}⌊ra⌋k⌉tri}htFqfor allv∈ /a\\}⌊ra⌋k⌉tl⌉{tEk/a\\}⌊ra⌋k⌉tri}htFq.This\nmeans that Acan be written as\nA=/parenleftigg\nY0\nˆA/parenrightigg\n, (38)\nwhereY∈Fk×k\nq,0is the zero matrix in Fk×(r−k)\nqandˆA∈F(r−k)×k\nq.Hence\nwe have/parenleftigg\nZ\nA/parenrightigg\n=\nˆZ\nY0\nˆA\n=/parenleftigg\nW\nˆA/parenrightigg\n, (39)\nwhere0is thenull matrixin F(u+k)×(r−k)\nq , W=/parenleftiggˆZ\nY0/parenrightigg\n∈F(u+k)×r\nq.Notice\nthat thesubmatrix ˆA∈F(r−k)×r\nqhasnoimpact ontheexecution ofAlgorithm\n1, so it can be chosen in qr(r−k)distinct ways. In the sequel we analyse the\nrequired properties of Win detail by following the steps in Algorithm 1.\nRecall from the discussion after Example 1 that Z(0)=Z, whileG(0)is\nobtained by sequentially extracting all the linearly independent row v ectors\ninZ(0).Forj= 1,...,r−1,Z(j)=G(j−1)A,andG(j)is formed by all the\nrows inZ(j)which cannot be linearly expressed by rows in G(0),...,G(j−1)\nand preceding rows in Z(j).It is clear that Z(j)is a sub-matrix of Z(j−1)A,\n33and is thus a sub-matrix of Z(0)Aj=ZAj.Suppose that in (34) the matrices\nG(0),...,G(r−1)havek0,...,k r−1rows, respectively. Then k0≥k1≥...≥\nkr−1andk0+k1+···+kr−1=k.Below we analyze the properties of ( Z,A)\nthat gives as output the matrix G=Ek.\nAs indicated in (34), if G=EkthenG(0)=Ek0for some k0≤min(u,k).\nThis implies that the matrix Z(1)=G(0)Acorresponds to the first k0rows of\nthe matrix A.That is to say, Z(1)is formed by the first k0rows of the matrix\nA, and it corresponds to the rows ranging from u+1 tou+k0in matrix W.\nMore generally, for j= 1,...,r−2, the matrix G(j)is sequentially formed\nby the row vectors eiwithk0+···+kj−1< i≤k0+...+kj. This implies\nthat, the matrix Z(j+1)=G(j)A,is formed by the rows of Ain the same\nrange, which corresponds to the rows ranging from u+(k0+···+kj−1)+1\ntou+(k0+···+kj) in the matrix W.Hence we can summarize the relation\nbetween W,Z(j)andG(j)as follows:\nW=\nZ(0)\nZ(1)\n...\nZ(r−1)\n=⇒\nG(0)\nG(1)\n...\nG(r−1)\n=Ek, (40)\nwhere matrices G(j)might be an empty matrix for jlarger than certain\nintegertifekalready occurs in G(t).\nDenote by withei-th row of Wfor 1≤i≤u+k, and denote by\nwi1,...,wiktherowsin Wcorresponding tothefirst occurrenceof e1,...,ek.\nFrom the above discussion we see that, for any iwithij≤i < ij+1, where\n1≤j < k, the row wican be linearly expressed by e1,...,ej, namely, wican\nbe any of the qjvectors of /a\\}⌊ra⌋k⌉tl⌉{te1,...,ej/a\\}⌊ra⌋k⌉tri}htFq; and for those i=ijfor 1≤j≤k,\nthe rowwimust be the vector ej. This indicates that there are u“free” rows\ninWin addition to the krowse1,...,ek. Ifik′+l < ik′+1,the rowwik′+lcan\nbe expressed as a linear combination of e1,...,ek′, indicating that wik′+lcan\nbe substituted by any vector such that its first k′coordinates are randomly\nchosen from Fqand the remaining ( r−k′) coordinates are set to zero. Since\nwe know that the output must be Ek,it must be that either ekbelongs to\nZor thatek=eiAfor some i < k.Then, similarly to the case of u= 1, the\nlast row of W, is always free to be any of the qkelements of Ek.\nHence, we only need to consider the ( u−1) “free” rows in the matrix W.\n34Below we provide a toy example to the above discussion before the re maining\nanalysis. E.g., in the case of k= 4,u= 4, the following matrix\nW=\n1 0 0 0\n•0 0 0\n0 1 0 0\n• •0 0\n0 0 1 0\n• • •0\n0 0 0 1\n• • • •\n=\nZ(0)\nZ(1)\nZ(2)\nZ(3)\n=⇒E4=\nG(0)\nG(1)\nG(2)\nG(3)\n=/parenleftbigg1 0 0 0\n0 1 0 0\n0 0 1 0\n0 0 0 1/parenrightbigg\n,\nwhereG(3)is an empty matrix, and each •stands for a free variablein Fq.We\ncan consider the submatrix W′made by the first u+k−1 rows of W. The\nmatrix obtained from the free rows of this matrix resembles to a ( u−1)×k\nFerrers diagram FW. Continuing with the example above we would obtain\nFW′=/parenleftig•0 0 0\n• •0 0\n• • •0/parenrightig\n.\nEach ofthepossiblediagramsarecounted as q|FW′|where|FW′|isthenumber\nof free variables in FW′.Recall from (4) that the sum of all n×kFerrers\ndiagram counted by qelevated to their weights is equal to/bracketleftbign+k\nk/bracketrightbig\nq. Applying\n(4) to our case, there are/bracketleftbigk+u−1\nu−1/bracketrightbig\nqpossible distinct matrices W′.\nTo sum up, we have\n/parenleftigg\nZ\nA/parenrightigg\n=\nW′\nwu+k0\nˆA\n,\nwhere the matrix ˆAcanbechosen in qr(r−k)distinct ways among thematrices\ninF(r−k)×r\nq, the vector wk+ucan be chosen in qkways among the vectors in\nEkandW′can be chosen in/bracketleftbigk+u−1\nu−1/bracketrightbig\nqdistinct ways. This gives the cardinality\n|C(u,k)\nEk|=/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nqqr(r−k)+k.\nThe desired statement thus follows.\nFrom the results in Propositions 2 and 3, the desired statement in Th eo-\nrem 3 directly follows.\nBefore ending this section, below we calculate the probability that a u ni-\nformly distributed random choice of ( Z,A)∈Fu×r\nq×Fr×r\nqexpands the row\nspace ofZto the whole space Fr\nq.\n35Corollary 1. The probability that a random matrix Y= (Z\nA)uniformly\ndrawn from F(r+u)×r\nqproduces a k-dimensional subspace Ω(Z,A)is given by\n|C(u,r)\nk|\nqr(r+u)=Hq(r)Hq(k+u−1)\nHq(r−k)Hq(k)Hq(u−1)q−u(r−k). (41)\nIn particular, for k=rthe probability is given by\n|C(u,r)\nr|\nqr(r+u)=Hq(r+u−1)\nHq(u−1)≥1−q−u+1\nq−1. (42)\nProof.There are qr(r+u)matrices Y. It follows from Theorem 3 that the\nprobability is given by\n|C(u,r)\nk|\nqr(r+u)=Aq(r,k)/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nqqr(r−k)+k−r(r+u).\nWe can substitute/bracketleftigg\nk+u−1\nu−1/bracketrightigg\nq=Aq(k+u−1,u−1)/Aq(u−1,u−1) and\nuse (1) for each of the expression in Aq.In this way we obtain\n|C(u,r)\nk|\nqr(r+u)=Hq(r)Hq(k+u−1)\nHq(r−k)Hq(k)Hq(u−1)q−u(r−k).\nUsing the convention Hq(0) = 1 andsubstituting k=r, we obtainthe second\nexpression in (42). The approximation is obtained from\nHq(a)/Hq(a−b)≈1−q−aqb���1\nq−1.\nIn our case we have a=r+u−1 andb=r, which concludes the proof.\n5.3 Calculation of C(u,r,t)\nr\nIn the previous subsection we calculate the probability that dim(Ω( Z,A)) =\nk.From its application in Section 4, we are particularly interested in the\ncasek=r.To derive this probability we counted the pairs ( Z,A) such that\nΩ(Z,A) = Ωr−1(Z,A) =Fr\nq.From experimental results we observed that, in\nmostofthecases, ifdim(Ω( Z,A)) =r,thendim(Ω t(Z,A)) =rfort=⌈r\nu⌉+1.\nThis means that, in most of the cases, Algorithm 1 outputs the same result\n36on (Z,A),twithtmuch smaller than r−1. In Section 3, we have seen that\nm≥(2(d+t)−3)rwheretrepresents the number of expansions (i.e. the\nlast input of Algorithm 1). In general ⌈r\nu⌉+1 is much smaller than r−1.It\nis clear that using such a small value tallows for a much smaller value of m.\nIn particular, asymptotically we can still correct an error with a pro bability\narbitrarily close to 1 and a value of msuch that lim n→∞n\nm=bfor some\nb >0.\nSuppose that udividesr, fort=r\nu,it is already possible that Ω t(Z,A) =\nΩ(Z,A) =Fr\nq. Each of the matrices Z,ZA,...,ZAthaveurows. Combining\nall their rows we get tu=rrow vectors of Fr\nq,if all these vectors are linearly\nindependent, then Ω r/u(Z,A) =Fr\nq.\nWe will show how, for large values of uandr,the probability that\nΩt(Z,A) =Fr\nq,conditioned on Ω( Z,A) =Fr\nqis lower bounded by 1 −\nq−u/2/(q−1) fort=r\nu+ 1.The desired probability is obtained as the\nratio between the cardinality of C(u,r)\nr={(Z,A)|Ω(Z,A) =Fr\nq}and\nC(u,r,t)\nr={(Z,A)|Ωt(Z,A) =Fr\nq}noticing that C(u,r,t)\nr⊆C(u,r)\nr.In The-\norem 3 we already calculated |C(u,r)\nr|.We will use a similar process to count\n|C(u,r,t)\nr|.The proof will be structured in the following way.\n•ByProposition1, wehave |C(u,r,t)\nk|=Aq(k,r)|C(u,r,t)\nEk|forcertaininteger\nt. We are interested in the case that r=k, indicating |C(u,r,t)\nr|=\nAq(r,r)|C(u,r,t)\nIr|, whereEkfork=rbecomes the identity matrix Irin\nFr×r\nq.\n•As in the proof of Proposition 3, we consider the matrices W= (Z\nA)\nthat have output Er(i.e. ther×ridentity matrix) through Algorithm\n1. We partition these matrices according to their associated Ferre rs\ndiagram. Then, we establish a relation between the number of inner\ncycles needed to obtain the output Erfrom a certain matrix W and its\nassociated Ferrers diagram.\n•Using this relation, we identify a particular subset of Ferrers diagra ms\nsuch that, all the matrices matching those diagrams, need at most\nt=⌈r\nu⌉+1 inner cycles to obtain the desired output Er.\n•Although this set is not representing all the Ferrers diagrams with t hat\nproperty, whencountedwithmultiplicity, itgivesusagoodlowerboun d\nfor|C(u,r,t)\nEr|as we will see in Theorem 4.\n37We describe now more in detail the lower bound we give to |C(u,r,t)\nEr|.\nTheorem 4. For an integer t=⌈r\nu⌉+1, one has\n|C(u,r,t)\nEr| ≥/bracketleftigu+u\n2u\n2/bracketrightig\nqqur−u2\n2. (43)\nProof.InProposition3weobtained |C(u,r)\nEk|=|C(u,r,r−1)\nEk|=qr(r−k)+k/bracketleftbigk+u−1\nu−1/bracketrightbig\nq.\nWe are interested in the case k=r. The formula of |C(u,r)\nEk|in Proposition 3\nwas obtained by counting the possible matrices\nW=/parenleftigg\nZ\nA/parenrightigg\n=\nZ(0)\nZ(1)\n...\nZ(r−1)\n, (44)\nsuch that the each row is either eior a combination of the previous rows and\nthe rows corresponding to eiappear in the order from e1toek.\nWe managed to count all these matrices associating them with ( u−1)×r\nFerrers diagrams matching their structure. From the Ferrers dia gram as-\nsociated to W, it is possible to count the number of steps tneeded such\nthat\nEr=/parenleftigg\nG(0)\n...\nG(t)/parenrightigg\n.\nAs we have seen in Subsection 2.2, a ( u−1)×rFerrers diagram Fcan be\ndescribed by a sequence of rintegers [ f1,...,f r] such that u−1≥fi≥\nfi+1≥0.Starting form this sequence it is possible to reconstruct a set of\nmatrices W⊆F(u+k)×r\nqhavingFas their associated Ferrers diagram. We\nstart from f1and consider f0=u−1,fr+1= 0.Iff1< f0then the first\nf1−f0rows ofWwill all be zero rows. Otherwise, we consider the first\nindexi1such that fi1=f0andfi1+1< fi1.We add the rows e1,...,ei1to\nWfollowed by fi1−fi1+1rows with i1initial•followed by r−i1zeroes. We\nproceed considering fjforj≥i1+1.As long as fj=fi1+1, we continue to\naddei1+1,...,ejtoWuntil we find the next index i2such that fi2+1< fi2.\nAt this point we add fi2−fi2+1identical rows with i2initial•followed by\nr−i2zeroes. We continue in this way until we reach fr.Iffr>0,then we\naddfr+1rows entirely filled by •,otherwise we addjust onerow ofthis type.\n(In Example 2 we show this procedure on three different Ferrers dia grams.)\n38Recall that Z(j)is a submatrix of Z(0)Ajand is obtained from G(j−1)A.\nWe can divide the matrix in blocks as follows. The first block is always\nZ=Z(0)and corresponds to the first urows. The second block is given by\nZ(1)=G(0)A,whereG(0)is the matrix obtained removing all the free rows\nofZ(0). Suppose that Z(0)hasl0free rows, then G(0)will have u−l0rows\nandZ(1)=G(0)Awill have the same number of rows which will correspond\nto the rows of Whaving index between u+1 and 2 u−l0.\nIfl0,...,ljare the number of free rows in Z(0),...,Z(j), then the block\nZ(j+1)=G(j)Awill be constituted by the u−(l0+···+lj) rows following\nthe block Z(j).In Example 2, we provide some examples of how to pass from\na Ferrers diagram to its respective matrix form divided into the block sZ(j).\nWe need to find a way to characterize the Ferrers diagrams such th at\nEr=/parenleftigg\nG(0)\n...\nG(t)/parenrightigg\n(45)\nfort≤ ⌈r\nu⌉+1.Wewillnotcompletelycharacterizethesediagramsbutwewill\nshow a very large subset of Ferrers diagram with the desired prope rty. The\nmatrices matching this subset, when counted with multiplicities, repr esents\nthe vast majority of the possible cases. This leads to a lower bound f or\n|C(r,u,t)\nEr|.\nOne simple way to count how many iterations are needed to generate the\nwhole space Fr\nqstarting from a matrix Whaving an associated ( u−1)×r\nFerrers diagram FWis the following. First we add the row ( •...•) on the\nbottom of FW.Then we can represent the Ferrers diagram as a u×rgrid of\nsquares and draw a line from the top-left corner to the bottom-rig ht corner.\nThis line will separate the cells containing a •from the cells containing a\nzero. The line is always constituted of precisely rhorizontal segments and\nuvertical segments. Beginning from the top-left corner, we move a long the\nline byusegments and mark the point we reach. In subsequent steps, we\nstart from the last marked point, we continue to move along the line b y a\nnumber of segments corresponding to the number of horizontal s egments we\ncovered in the previous step and we mark the new point that we reac h.\nThis procedure will divide the line into at most rchunks. We repeat\nthis procedure until all the horizontal segments are covered. Th e number of\nchunks correspond to the number of blocks Z(j)inSneeded to generate Fr\nq.\n39In Example 3 we consider some Ferrers diagram and show their respe ctive\nline partition according to the procedure described above.\nWe can identify a subset of Ferrers diagrams covered by a small num ber\nof steps.\nFort=⌈r\nu⌉+1 we have t−1≥r\nuwith equality only when udividesr.\nThen assuming that udividesrrepresents the most challenging scenario and\ncan then be used to set a lower bound.\nLetFbe theu×rFerrers diagram described by the sequence of non-\ndecreasing positive integers, i.e., F= [f1,...,f r],wherefi=uifi≤r−u\nandfi=u\n2otherwise. Assume /hatwideF={F′∈ Fu,r|F′≥F}is the set of all\nthe possible Ferrers diagrams bigger than Fas in Lemma 1. If we depict the\nu×rFerrers diagrams as paths over a u×rgrid,/hatwideFis the set of all paths\nthat are always on top or equal to F.\nA1 At−2\nF u\nr= (t−2)uu\n2\nIn dark gray, we represented the Ferrers diagram F.All the Ferrers diagrams\nin/hatwideFwill cover that area. In light gray, we represented the part that c an\nchange between two different elements of /hatwideF.\nAssumeF′≥F.Thenthecorrespondingpathwillgothrough A1,...,A t−2\nint−2 steps. The first part of the path consists of t−2 horizontal steps each\nof length ucovering a total of r−uhorizontal segments. In the following\nsteps, we can move by at mostu\n2vertical segments, then we will always move\nby at leastu\n2horizontal segments until we reach the right border. It takes\nat most two more steps to reach the right border from At−2.In total we will\nthen need at most tsteps.\nLetTbe the set of all the matrices Wthat match with a Ferrers diagram\nin/hatwideF,we will have\n|C(u,r,t)\nEr| ≥ |T|=/summationdisplay\nF′∈/hatwideFq|F′|. (46)\n40Finally, we can apply Lemma 1 to /hatwideFto obtain\n|C(u,r,t)\nEr| ≥/bracketleftigu+u\n2u\n2/bracketrightig\nqqur−u2\n2. (47)\nTo help the reader we provide some examples of Ferrers diagrams, t heir\nrelative matrix conversion and division into the blocks Z0,...,Z(t).\nExample 2. Letu= 4andr= 8. The Ferrers diagram below can be\nconverted into a matrix Wwith the following structure\n/parenleftig• • • • 0 0 0 0\n• • • • 0 0 0 0\n• • • • • • • •/parenrightig\n→W=\n1 0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n• • • • 0 0 0 0\n• • • • 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n• • • • • • • •\n. (48)\nWe divided the resulting matrix in blocks. These blocks corr esponding to\nZ0,...,Z(t)while, if we remove the free rows, the blocks correspond to th e\nmatrices G0,...,G(t).\nAn extreme example is provided by the following Ferrers diag ram\n/parenleftig• • • • • • • •\n• • • • • • • •\n• • • • • • • •/parenrightig\n→W=\n1 0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n• • �� • • • • •\n• • • • • • • •\n• • • • • • • •\n. (49)\nIn this case G(0)andG(1)coincide with Z(0)andZ(1),most important, the\nrows in the first two blocks already generates Fr\nq, while in (48) we needed to\nconsider the first 3blocks to span Fr\nq.\n41An opposite example can be given by the following\n/parenleftig•0 0 0 0 0 0 0\n•0 0 0 0 0 0 0\n• • • • 0 0 0 0/parenrightig\n→W=\n1 0 0 0 0 0 0 0\n•0 0 0 0 0 0 0\n•0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n• • • • 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n. (50)\nIn this case we need to consider Z(0),...,Z(5)to spanFr\nq.This means that\nFr\nq=5/summationdisplay\ni=0RowSpan( ZAi),\nwhich is close to the theoretical upper bound of r−1 = 7.\nThe first thing to notice from the Ferrers diagrams in Example 2 is tha t,\nif (Z,A) form a matrix that match a Ferrers diagram with an high weight\n(i.e. a big number of •), then after few iteration Z,ZA,...,ZAjwill span\nthe whole space Fr\nq.The second thing to notice is that, when counted with\nmultiplicities, these matrices are disproportionately more than the m atrices\nwith a low weight Ferrers diagram.\nStarting form the first row, each free row represents an interru ption to\nthe expansion. If an interruption happens early, all the following st eps will\nbe shorter by one, so it will take more steps to reach Fr\nq.For example, if in\nZthere are only u−1 out ofurows which are linearly independent, we will\nneed at leastr\nu−1steps to cover Fr\nqinstead ofr\nuas we could expect from a\nfull rank matrix.\nExample 3. We illustrate the line procedure we used in the proof of Theor em\n4 on the Ferrers diagrams we introduced in the previous examp le.\n42→\n1 0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n• • • • 0 0 0 0\n• • • • 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n• • • • • • • •\n.\nAn example illustrating (48)\n→\n1 0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n• • • • • • • •\n• • • • • • • •\n• • • • • • • •\n.\nAn example illustrating (49)\n→\n1 0 0 0 0 0 0 0\n•0 0 0 0 0 0 0\n•0 0 0 0 0 0 0\n0 1 0 0 0 0 0 0\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n• • • •0 0 0 0\n0 0 0 0 1 0 0 0\n0 0 0 0 0 1 0 0\n0 0 0 0 0 0 1 0\n0 0 0 0 0 0 0 1\n• • • • • • • •\n\nAn example illustrating (50)\n43We provide another example to clarify the last part of the proof of T he-\norem 4.\nExample 4. Letu= 4andr= 8, consider the following set of examples\n/parenleftig• • • • • • • •\n• • • • • • • •\n• • • • • • • •/parenrightig\n, (51)\nwhere•can either be a 0or•.\nConsider Sa matrix with an associated Ferrers diagram matching this\npattern. We claim that Fr\nq=/summationtext2\ni=0RowSpan( Z(i)).\nTo see that we can consider the associated grid\nA\n.\nAny path that stays above the path we draw will be such that the first marked\npoint is A.In the subsequent steps we can move by at most two vertical\nsegments, then we will always move down by at least two horizo ntal segments\nuntil we reach the right edge. Since we already covered 4horizontal segments\nin the first step, with two more steps of length min(2,8−c)wherecis the\nnumber of horizontal steps we did so far, we will reach the the right edge in\nat most3steps.\nLet us call Tthe set of all the matrices Shaving a Ferrers diagram that\nmatch one of the possible Ferrers diagram in (51). The possib le different\npaths we can consider correspond to all the possible Ferrers diagram we can\ndraw inside the top-left rectangle. This can be counted as al l the possible 2×4\nFerrers diagrams. If we count the set Twith the multiplicity we will get\n|T|=q24[6\n2]q. (52)\nIf we compare it to the number of all the possible matrices Sthat gives us\noutputErwe obtain\n|T|\n|C(4,8)\nE8|=[6\n2]q\n[11\n3]qq16=Hq(6)Hq(8)Hq(3)\nHq(4)Hq(2)Hq(11).\nWhich is lower bounded byH(6)\nH(2)≈1−q−2\nq−1.\n44This example shows us how to prove the probability in the following\ncorollary.\nCorollary 2. LetZ∈Fu×r\nqandA∈Fr×r\nqbe two matrices uniformly inde-\npendently chosen in their respective domains. For t≥ ⌈r\nu⌉+1,the probability\nProb/parenleftbig\nΩt(Z,A) =Fr\nq|Ω(Z,A) =Fr\nq/parenrightbig\n≥1−q−u/2\nq−1.\nProof.Observe that C(u,r,t)\nr⊆C(u,r)\nrwe have\nP(Ωt(Z,A) =Fr\nq|Ω(Z,A) =Fr\nq) =|C(u,r,t)\nr|\n|C(u,r)\nr|=|C(u,r,t)\nEr|\n|C(u,r)\nEr|.(53)\nTheorem 4 provides us with the lower bound\n|C(u,r,t)\nEr| ≥/bracketleftigu+u\n2u\n2/bracketrightig\nqqur−u2\n2. (54)\nSubstituting it in (53) we get\n|C(u,r,t)\nEr|\n|C(u,r)\nEr|≥/bracketleftigu+u\n2u\n2/bracketrightig\nq/bracketleftbigr+u−1\nu−1/bracketrightbig\nqq(u−1)r−u2\n2=Hq(u+u\n2)Hq(u−1)Hq(r)\nHq(u\n2)Hq(u)Hq(r+u−1),(55)\nwhich can be further lower bounded as\n|C(u,r,t)\nEr|\n|C(u,r)\nEr|≥Hq(u+u\n2)\nHq(u\n2)=u/productdisplay\ni=u\n2+1(1−q−i)≈1−q−u\n2\nq−1.(56)\n6 Conclusion\nThis paper presents a novel family of bounded degree low rank parit y check\n(BD-LRPC) codes, which exhibits remarkable error correction cap abilities.\nThe proposed BD-LRPC codes are constructed by leveraging the in herent\nproperties of the subspace Vα,d=/a\\}⌊ra⌋k⌉tl⌉{t1,α,...,αd−1/a\\}⌊ra⌋k⌉tri}htFq, from which the decoding\nradius can approach the Singleton bound.\nOur contributions inthis paper aretwofold. First, we introduce this novel\nfamily of BD-LRPC codes, demonstrating their superior error corr ection ca-\npability compared to LRPC codes andtheir ability to approach the Sing leton\n45bound. Second, we formulate an intriguing counting problem (Proble m 1)\nthat arises from the study of these codes. We provide a sophistica ted ap-\nproach to tackling this problem using Ferrers diagrams for the case s that\nt=r−1 andt=⌈r\nu⌉+ 1. The counting problem seems to have its own\ninterest in combinatorics and may have applications in other domains. We\ncordially invite interested readers to find a tighter bound or an exac t solution\nfor any⌈r\nu⌉ ≤t < r−1.\nReferences\n[1] N. Aragon, O. Blazy, P. Gaborit, A. Hauteville, and G. Z´ emor. Duran-\ndal: A Rank Metric Based Signature Scheme , pages 728–758. Springer,\n04 2019.\n[2] N. Aragon, P. Gaborit, A. Hauteville, O. Ruatta, and G. Z´ emor. Low\nrank parity check codes: New decoding algorithms and applications t o\ncryptography. IEEE Transactions on Information Theory , 65(12):7697–\n7717, 2019.\n[3] H. Bartz, L. Holzbaur, Liu, S. Puchinger, Renner, and A. Wacht er-Zeh.\nRank-Metric Codes and Their Applications , volume19(3), page390–546.\n2022.\n[4] H.BartzandV.Sidorenko. Algebraicdecodingoffoldedgabidulinc odes.\nDesigns, Codes and Cryptography , 82(1-2):449–467, 2016.\n[5] Y. Ding. On list-decodability of random rank metric codes and subs pace\ncodes.IEEE Transactions on Information Theory , 61(1):51–59, 2015.\n[6] E. Franch and C. Li. Two new algorithms for error support recov ery of\nlow rank parity check codes. In 2023 IEEE International Symposium on\nInformation Theory (ISIT) , pages 2368–2373, 2023.\n[7] E. Gabidulin. Rank Codes . TUM.University Press, 2021.\n[8] E. M. Gabidulin. Theory of codes with maximum rank distance. Prob-\nlemy Peredachi Informatsii , 21(1):3–16, 1985.\n46[9] E. M. Gabidulin, A. V. Paramonov, and O. V. Tretjakov. Ideals ov er\na non-commutative ring and their application in cryptology. In D. W.\nDavies, editor, Advances in Cryptology – EUROCRYPT’91 , pages 482–\n489. Springer, 1991.\n[10] P. Gaborit, A. Hauteville, D. H. Phan, and J.-P. Tillich. Identity-b ased\nencryption fromcodes with rankmetric. InJ. KatzandH. Shacham , ed-\nitors,Advances in Cryptology – CRYPTO 2017 , pages194–224.Springer\nInternational Publishing, 2017.\n[11] P. Gaborit, A. Hauteville, D. H. Phan, and J.-P. Tillich. Identity-b ased\nencryption from codes with rank metric. In J. Katz and H. Shacham ,\neditors,AdvancesinCryptology– CRYPTO2017 ,pages194–224,Cham,\n2017. Springer International Publishing.\n[12] P. Gaborit, O. Ruatta, J. Schrek, and G. Z´ emor. Ranksign: a n effi-\ncient signature algorithm based on the rank metric. In M. Mosca, ed i-\ntor,Post-Quantum Cryptography , pages 88–107. Springer International\nPublishing, 2014.\n[13] R. Gallager. Low-density parity-check codes. IRE Transactions on In-\nformation Theory , 8(1):21–28, 1962.\n[14] V. Guruswami, C. Wang, and C. P. Xing. Explicit list-decodable ran k-\nmetric and subspace codes via subspace designs. IEEE Transactions on\nInformation Theory , 62(5):2707–2718, 2016.\n[15] D. E. Knuth. Subspaces, subsets, and partitions. Journal of Combina-\ntorial Theory, Series A , 10(2):178–180, 1971.\n[16] C. A. Melchor, N. Aragon, M. Bardet, S. Bettaieb, L. Bidoux, O . Blazy,\nJ.-C. Deneuville, P. Gaborit, A. Hauteville, A. Otmani, O. Ruatta, J.-\nP. Tillich, and G. Z´ emor. ROLLO (merger of Rank-Ouroboros, LAKE\nand LOCKER). In Second round submission to the NIST post-quantum\ncryptography call , April, 2020.\n[17] Philippe Gaborit, Ga´ etan Murat, Olivier Ruatta, and Gilles\nZ´ emor. Low rank parity check codes and their applica-\ntion to cryptography. in proceedings of the workshop on coding\n47and cryptography WCC’2013 Bergen Norway 2013. available on\nwww.selmer.uib.no/wcc2013/pdfs/gaborit.pdf.\n[18] N. Raviv and A. Wachter-Zeh. Some Gabidulin codes cannot be list\ndecoded efficiently at any radius. IEEE Transactions on Information\nTheory, 62(4):1605–1615, 2016.\n[19] R. M. Roth. Maximum-rank array codes and their application to\ncrisscross error correction. IEEE Transactions on Information Theory ,\n37(2):328–336, 1991.\n[20] D. Silva, F. R. Kschischang, and R. Koetter. A rank-metric app roach\nto error control in random network coding. IEEE Transactions on In-\nformation Theory , 54(9):3951–3967, Sept 2008.\n[21] D. Silva, F. R. Kschischang, and R. Kotter. Communication over\nfinite-field matrix channels. IEEE Transactions on Information The-\nory, 56(3):1296–1305, mar 2010.\n[22] A. Wachter-Zeh. Bounds on list decoding of rank-metric codes .IEEE\nTransactions on Information Theory , 59(11):7268–7277, 2013.\n[23] A. Wachter-Zeh and A. Zeh. List and unique error-erasure de coding\nof interleaved gabidulin codes with interpolation techniques. Designs,\nCodes and Cryptography , 73(2):547–570, 2014.\n[24] C. P. Xing and C. Yuan. A new class of rank-metric codes and the ir\nlist decoding beyond the unique decoding radius. IEEE Transactions on\nInformation Theory , 64(5):3394–3402, 2018.\n48" }, { "title": "2401.15220v1.From_equilibrium_to_non_equilibrium_statistical_mechanics_of_liquids.pdf", "content": "arXiv:2401.15220v1 [cond-mat.stat-mech] 26 Jan 2024From equilibrium to non-equilibrium\nstatistical mechanics of liquids.\nO. Joaqu´ ın-Jaime1, R. Peredo-Ortiz1,2, M. Medina-Noyola1and L.F. Elizondo-Aguilera4.\n1Instituto de F´ ısica, Universidad Aut´ onoma de San Luis Pot os´ ı,\n´Alvaro Obreg´ on 64, 78000 San Luis Potos´ ı, SLP, M´ exico\n2Facultad de Ciencias F´ ısico-Matem´ aticas,\nBenem´ erita Universidad Aut´ onoma de Puebla,\nApartado Postal 1152, CP 72570, Puebla, PUE, M´ exico and\n4Instituto de F´ ısica, Benem´ erita Universidad Aut´ onoma d e Puebla,\nApartado Postal J-48, 72570 Puebla, Mexico.\n(Dated: January 30, 2024)\n1Abstract\nRelevant and fundamental concepts of the statistical mecha nical theory of classical liquids are\nordinarily introduced in the context of the description of t hermodynamic equilibrium states. This\nmakes explicit reference to probability distribution func tions ofequilibrium statistical ensembles\n(canonical, microcanonical, ...) in the derivation of gene ral and fundamental relations between\ninter-particle interactions and measurable macroscopic p roperties of a given system. This includes,\nfor instance, expressing the internal energy and the pressu re as functionals of the radial distribu-\ntion function, or writing transport coefficients (diffusion co nstant, linear viscosity, ...) in terms of\nintegral relations involving both, static and dynamic auto -correlation functions (density-density,\nstress-stress, ...). Most commonly, however, matter is not in thermodynamic equilibrium, and this\ncalls for the extension of these relations to out-of-equili brium conditions with the aim of under-\nstanding, for example, the time-dependent transient state s during the process of equilibration, or\nthe aging of glass- and gel-forming liquids during the forma tion of non-equilibrium amorphous\nsolid states. In this work we address this issue from both, a g eneral perspective and an illustra-\ntive concrete application focused on the first principles de scription of rheological and viscoelastic\nproperties of glass- and gel-forming liquids.\nPACS numbers:\n2I. INTRODUCTION\nThe thermodynamic and statistical mechanical description of equilib rium liquids rests\non firm and well-established fundamental basis [1]. Thermodynamic co ncepts such as equa-\ntions of state, equilibrium phases and phase diagrams [2], as well as st atistical mechanical\nconcepts such as pair correlation functions, the Ornstein-Zernik e equation, or free energy\ndensity functionals, are nowadays well-understood textbook mat erial [3–6]. One reason for\nthe beauty and simplicity of these concepts is that their ordinary de finition and application\nonly refers to macroscopic states contained in the universal set ( or “catalog”) of thermody-\nnamic equilibrium states of matter, which are understood in terms of the maximum-entropy\nprinciple [2] together with Boltzmann’s expression S=kBlnWfor the entropy Sin terms\nof the number Wof microscopic states [4]. In general, when gases, liquids and cryst alline\nsolids reach a thermodynamic equilibrium state, their properties are stationary, indepen-\ndent of the preparation protocol, and determined by the solution o f the equation dS[A] = 0,\nwhereSis the total entropy (including reservoirs) and the components of the vector Aare\nthe extensive thermodynamic variables.\nIn contrast, it is not clear how Boltzmann’s principle operates in gene ral to describe, for\nexample, the formation of very common non-equilibrium amorphous materials (glasses, gels,\netc.), whose properties mayexhibit agingandmight depend ontheir p reparationprotocol[7,\n8]. Although the amorphous solidification of glass- and gel-forming liqu ids is an ubiquitous\nnon-equilibrium process of enormous relevance in physics, chemistr y, biology, and materials\nscience andengineering [9], their fundamental understanding issom etimes referred to as “the\ndeepest and most interesting unsolved problem in solid stat e theory” [10]. The macroscopic\nstates in which all the known (and unknown) non-equilibrium amorpho us materials are\nfound, constitute a second universal catalog of states of matte r, additional and disjoint to\nthe catalog of stationary equilibrium states that solve the equation dS[A] = 0. Thus, one\nmight claim that the referred fundamental problem cannot be decla red as “solved” until\na general and fundamental physical principle is identified, from whic h one can derive the\nequation whose solutions, in principle, describe and predict the prop erties of materials in\nthis second universal catalog of states of matter.\nThis fundamental problem, however, is only one of the many concer ns of non-equilibrium\nstatistical mechanics, which comprises many different theoretical tools, such as Boltz-\n3mann kinetic equation, time correlation function formalism, project ion operator techniques,\nstochastic equations, the mode-coupling theory, and the dynamic density functional the-\nory, amply described in authoritative textbooks and reviews [11–16 ], and in the references\ntherein. Somehow, however, andinspiteofthelong-standingscien tific interest andextensive\nresearch efforts to extend statistical mechanical methods to no n-equilibrium conditions, the\nprogress in its application to the fundamental theoretical unders tanding of the amorphous\nsolidification of supercooled liquids has been rather modest.\nLet us recall as a reference that almost two centuries ago, Clapey ron summarized the\nexperimental results of Boyle, Charles and Avogadro into the empir ical ideal gas equation\nof state. The need to explain this experimental phenomenology in mo lecular terms, in turn,\nled Clausius, Maxwell and Boltzmann to elaborate the kinetic molecular theory of ideal\ngases, thus inaugurating the theoretical methods of statistical mechanics [17]. We may say\nthat the current understanding of non-equilibrium amorphous solid s is still in the stage of\ngathering empirical experimental information, with increasingly gre ater (even microscopic)\ndetail, particularly when complemented with molecular simulation metho ds. Although no\nanalogous simple phenomenological synthesis has emerged from the overwhelmingly varied\naccumulation of experimental data describing all the features tha t characterize the real\nphysical behavior of glass- and gel-forming liquids [7, 8], insightful ph enomenological models\nexist that describe relevant features of glass behavior. This is illust rated, for example, by\ntheTool-Narayanaswamy-Moynihan [18–20] andtheKovacs-Aklo nis-Hutchinson-Ramos [21]\nmodels, commonly used in industry to predict aging effects [22], and w hose development\ninvolved a rich discussion of many relevant issues [23]. These phenome nological models\nintelligently compile many previous partial discoveries, just like Clapey ron compiled the\nempirical data represented by the ideal gas equation of state pV=nRT.\nContinuing with the previous analogy, the notorious missing piece is th e first-principles\ntheoretical description of the experimental phenomenology of gla ss-forming liquids during\nthe process of amorphous solidification. In spite of a rich and well-do cumented theoretical\ndiscussion of relevant aspects of the behavior of viscous liquids [24– 27], we are still missing\nthe non-equilibrium analog for glasses, of the molecular statistical m echanical theory de-\nveloped by Clausius, Maxwell, and Boltzmann to understand ideal gas es, which was later\nextended to equilibrium non-ideal gases by van der Waals [28] and, e ventually, to liquids by\nOrnstein and Zernike [29], Widom [30], and many others (see [3–6]). Build ing the analog of\n4these developments in the context of non-equilibrium glass- and gel- forming liquids, poses a\nrelevant challenge to the “beautiful and profound subject” [11] of non-equilibrium statistical\nmechanics, which thus has the opportunity to become the theoret ical counterpart of experi-\nments and simulations, in the search for the fundamental underst anding of non-equilibrium\nstates of matter. Thus, a relevant initiative is now to focus this rich theoretical infras-\ntructure on the specific theoretical challenge of understanding a morphous materials from\nfirst principles, including the behavior of liquids during the irreversible transient process\nof dynamic-arrest (or “aging”), occurring in highly viscous liquids du ring their amorphous\nsolidification into glassy and gelled states.\nThis was precisely the main aim of the recently-developed statistical physics formalism\nreferredtoasthe non-equilibrium self-consistentgeneralizedLangevin eq uation (NE-SCGLE)\ntheory [31–35], whose essence is a set of time-evolution equations f or the structural and\ndynamical properties of a non-equilibrium liquid, namely, Eqs. (4.1)-( 4.7) of Ref. [32].\nThe NE-SCGLE theory originated, somewhat off the beaten path, f rom the assumption\nthat the manner in which Boltzmann’s postulate S=kBlnWexplains non-equilibrium\nstates, isprovidedbyaspatiallynon-localandtemporallynon-Mark ovianandnon-stationary\ngeneralization [31] of Onsager’s linear irreversible thermodynamics [3 6, 37] and the Onsager-\nMachlup theory of thermal fluctuations [38, 39]. We can say that th e resulting NE-SCGLE\nequations, and the remarkable predicted scenario they have unve iled, constitute a highly\nrelevant contribution to the foundations of the non-equilibrium ext ension of the statistical\nmechanical theory of equilibrium liquids.\nA brief and updated account of the fundamental origins of the NE- SCGLE theory is\nprovided by Ref. [40] and, hence, here we do not dwell on this subje ct. Similarly, here we\nshall not review the applications of the NE-SCGLE theory, i.e., the so lution of Eqs. (4.1)-\n(4.7) of Ref. [32] for a variety of model systems, which illustrate th e competition between\nthe kinetic processes of thermodynamic equilibration, and the ultra -slow kinetic processes of\nformation of non-equilibrium amorphous solids; these contributions will be the subject of a\nforthcoming publication [41]. Instead, the general aim of the prese nt work is to illustrate in\ndetail the possible strategies to identify the non-equilibrium analog o f some well-established\nelementary concepts of the statistical mechanics of equilibrium liquid s. For this we first\ndiscuss – with the support of the NE-SCGLE theory – the non-equilib rium role of two well-\nestablished notions of the (equilibrium) liquid state theory, namely, t he Ornstein-Zernike\n5equation and the Wertheim-Lovett relation [3–5].\nWe then follow a similar route to provide an approximate expression fo r a highly relevant\ndynamic property, namely, the frequency-dependent dynamic sh ear viscosity η(ω), written\nin terms of the structure factor and intermediate scattering fun ctions. Such an expression\nwas first derived by Geszti for atomic fluids [42] and by N¨ agele and B ergenholtz [43, 44]\nfor colloids, albeit only for thermodynamic equilibrium conditions. Here , instead, we shall\nzoom in on the detailed theoretical arguments leading to a general e xpression for the non-\nequilibrium dynamic shear viscosity η(ω;t) of a liquid at a (waiting) time tafter being\nsuddenly quenched to arbitrary final temperature and density. T his derivation follows a\nsimple strategy, consisting of inspecting the derivation of the equilib rium counterpart, to\nsee if in reality the assumption of equilibrium conditions was really essen tial, for example,\nif at some point an explicit and indispensable use was made of any equilibr ium statistical\nensemble.\nWe start this work in Section II by illustrating this strategy with a simp le example,\nnamely, the extension to non-equilibrium of the so called energy equa tion, which is shown\nto be identical to the usual expression for the internal energy in t erms of the radial distri-\nbution function g(r), except that g(r) is replaced by the t-dependent non-equilibrium radial\ndistribution function g(r;t). This poses the crucial questions of how to determine g(r;t). At\nequilibrium, g(r;t) is independent of t, and is related with the so-called direct correlation\nfunction c(r) by means of the Ornstein-Zernike equation, whose validity or exte nsion at\nnon-equilibrium conditions is also a natural and intriguing question. In Section II we also\nshow that the NE-SCGLE theory actually provides a straightforwa rd response to both of\nthese issues.\nSectionIIIexploresapossibleroutetoextendtono-equilibriumcon ditionsanotherfunda-\nmental relation derived and employed in the statistical thermodyna mic theory of inhomoge-\nneous fluids at equilibrium, namely, the so called Wertheim-Lovett rela tion [3]. This relation\nwrites the gradient of the equilibrium local particle number density n(r) as a convolution\nof the two-particle correlation function and the pairwise force bet ween particles. From the\nconventional arguments employed in its equilibrium derivation, it would be understood to\nbe valid only at equilibrium. In Section III, however, we demonstrate that a particular case\nof the Wertheim-Lovett relation derives from symmetry considera tions that are completely\ntransportable to non-equilibrium conditions.\n6Section IV then focuses on the most ambitious objective of this con tribution, namely,the\nderivation of the non-equilibrium expression for the dynamic shear v iscosityη(ω;t), which\nturns out to be almost identical to its equilibrium counterpart. Our p resent derivation,\nhowever, does not really follow in detail the arguments and steps em ployed in the orig-\ninal derivations, constrained to thermodynamic equilibrium [42–44]. However, it is not\nfundamentally different, except for the fact that our derivation a ssumes the condition of\nstationarity, rather than the condition of thermodynamic equilibriu m. To the best of our\nknowledge, such a closed equation for η(ω;t) has never been proposed before. Finally, in\nSection V we provide a brief discussion of perspectives and a summar y of conclusions.\nII. NON-EQUILIBRIUM VERSION OF ELEMENTARY EQUILIBRIUM CON-\nCEPTS\nThe statistical mechanical theory of classical fluids was the subje ct of active development\nin the second half of the last century. This development mostly focu sed on the description\nof the properties of systems in thermodynamic equilibrium states, a s recorded in influential\nmonographs and textbooks [1–6]. One of the main aims, for example, was to relate inter-\nparticleinteractionswithmeasurablemacroscopicpropertiesofag ivensystem, asillustrated,\nfor instance, by the so-called energy equation [4, 5]\nU\nN=3\n2kBT+n\n2/integraldisplay\nu(r)g(r)d3r, (2.1)\nwhich expresses the internal energy Uof a fluid of Nspherical particles in a volume Vat\ntemperature Tand number density n=N/V, in terms of the pair potential u(r) and of the\nradial distribution function g(r). Similar expressions were derived for other thermodynamic\nproperties (e.g. pressure, isothermal compressibility) [4, 5]. In ad dition, transport coeffi-\ncients (e.g. diffusion constant, linear viscosity) were written in term s of integral relations\ninvolving both, structural and dynamical auto-correlation funct ions (density-density, stress-\nstress, etc) [6], which approximate theories [45–48] were able to wr ite in terms of u(r) and\ng(r). This, in fact, is another reason why much of the early liquid state t heory was centered\non the determination of g(r).\nThe standard derivation of general equilibrium relations, such as th e energy equation\nabove, makes explicit use of probability distribution functions of equilibrium (canonical,\n7microcanonical, ...) statistical ensembles [3–6]. We may thus be conditio ned to believe that\ntheir validity is restricted to systems in thermodynamic equilibrium. Th ere are, however,\nmany reasons to revise these relations, concepts, and derivation s, with the aim of extending\nthem to more general out-of-equilibrium conditions, and here we st art precisely with the\nenergy equation.\nA. Macroscopic properties and statistical ensembles of non-equilibrium liquids\nLet us start by recalling that the microscopic dynamics of a many-bo dy system is gov-\nerned by the fundamental dynamical (Newton’s or Hamilton’s) equa tions describing the\nmotion of each of the Nparticles comprising the system. Thus, if ri(t) denotes the po-\nsition of the ith particle at time tandpi(t) its momentum, then the time-evolution of\nany dynamical variable ˆA(t)≡ˆA(rN(t),pN(t)), with rN(t)≡(r1(t),r2(t),...,rN(t)) and\npN(t)≡(p1(t),p2(t),...,pN(t)), will be rooted in these microscopic equations of motion, as\ndescribed in any reference textbook of classical mechanics [49]. Th e fundamental postulate\nof statistical mechanics [4, 5] isthat any measurable observable A(t) ofa macroscopic system\ncorresponds to the average value of a specific dynamic variable ˆA≡ˆA(rN,pN), i.e.,\nA(t) =/angb∇acketleftˆA(t)/angb∇acket∇ight ≡/integraldisplay\nˆA(rN,pN)PN(rN,pN;t)drNdpN, (2.2)\nwhere the brackets /angb∇acketleft···/angb∇acket∇ightindicate average over a statistical ensemble, written here in terms\nof theN-particle probability distribution function (PDF) PN(rN,pN;t) that represents the\nconditions imposed on the system.\nRestricting ourselves to thermodynamic equilibrium states (which ar e strictly stationary)\nA(t) =A=/angb∇acketleftˆA/angb∇acket∇ighteq, where the label “ eq” indicates any of the conventional equilibrium sta-\ntistical ensembles (canonical, microcanonical, etc.). For instance, in the canonical ensemble\nwe may write Aas\nA=/angb∇acketleftˆA/angb∇acket∇ighteq≡/integraldisplay\nˆA(rN,pN)Peq\nN(rN,pN)drNdpN, (2.3)\nwherePeq\nN(rN,pN) is the equilibrium N-particle PDF, given by\nPeq\nN(rN,pN)≡1\nh3NN!e−βH(rN,pN)\nQN, (2.4)\n8whereβ−1=kBT, and with H(rN,pN) being the Hamiltonian of the system and QNthe\ncanonical partition function\nQN≡1\nh3NN!/integraldisplay\ne−βH(rN,pN)drNdpN. (2.5)\nEqs. (2.3)-(2.5) provide the fundamental basis for the conventio nal statistical mechanical\nderivation of general expressions for the thermodynamic observ ables. For example, Refs.\n[4, 5] describe in detail the steps and arguments that lead, from th ese equations, to the\nexpression for the internal energy in Eq. (2.1). To start with a simp le illustrative example,\nlet us now discuss to what extent those arguments and steps can b e extended to non-\nequilibrium conditions.\nB. The non-equilibrium energy equation.\nUnder general non-equilibrium conditions, the macroscopic state o f a system may be de-\nscribedbyastatisticalensemble, nowrepresentedbythetime-de pendentPDF PN(rN,pN;t).\nThe measurable observable A(t) is then the mean value of ˆA≡ˆA(rN,pN) according to Eq.\n(2.2). For example, let the Nparticles of our system interact only through pairwise forces,\nwhose interaction potential between two particles at positions randr′is denoted by u(r,r′),\nand which are also subjected to an external field such that the pot ential energy of one parti-\ncle at position ris Ψ(r). Then the total mechanical energy is ˆU(rN,pN) =ˆK(pN)+ˆV(rN),\nwith\nˆK(pN) =/summationdisplay\n1≤i≤Np2\ni/2M (2.6)\nbeing the kinetic energy and with\nˆV(rN)≡/summationdisplay\n1≤i0 to new stationary conditions. During this relaxation transient, th e non-\nstationary total shear stress relaxation function η(t,t′) can be written as η(t,t′) =η(t−t′;t),\ni.e., asη(τ;t), withτ≡t−t′. This is the non-equilibrium shear stress relaxation function\n27referred to in the introduction, whose Fourier-Laplace transfor mη(ω;t) is the dynamic shear\nviscosity, related with the dynamic shear modulus G(ω;t) byG(ω;t) =iωη(ω;t), whose real\nand imaginary parts are the elastic and loss moduli G′(ω;t) andG′′(ω;t).\nThe total shear stress relaxation function, η(τ;t), can be written as η(τ;t) = 2δ(τ)η0+\n∆η(τ;t), withη0being the “short-time” (or “infinite-frequency”) viscosity, relat ed with the\n“short-time” (or “free”) self-diffusion coefficient D0by the Stokes-Einstein relation η0=\nkBT/3πσD0. The function ∆ η(τ;t) is the contribution to η(τ;t) due to the inter-particle\nforces. In the absence of hydrodynamic interactions, η0is the viscosity of the pure solvent,\nbut under some circumstances, such as for concentrated hard- sphere suspensions, the effects\nof hydrodynamic interactions act virtually instantaneously, simply r enormalizing the value\nofη0andD0, but otherwise behaving as if hydrodynamic interactions were abse nt [69, 70].\nThis will be a general assumption in what follows.\nOurmainpurposenowistoobtainanapproximatebutgeneral expre ssion forthefunction\n∆η(τ;t) in terms of both, the non-equilibrium structure factor S(k;t) and the t-evolving and\nτ-dependent intermediate scattering function F(k,τ;t). For this, our starting point is the\nGreen-Kubo relation [5, 6] that can be obtained from the fluctuatio n-dissipation relation,\nnamely\n∆η(τ;t) = (β/V)/angb∇acketleftσxy(t+τ)σxy(t)/angb∇acket∇ight, (4.1)\nwhereσxy(t) is the microscopic expression for the configurational component of the stress\ntensor, given by [6, 43]\nσxy(t) =−ˆx·/bracketleftBiggN/summationdisplay\ni=1ri(t)Fi(t)/bracketrightBigg\n·ˆy (4.2)\nwhereri(t) andFi(t) are the position and total force on the i-th colloidal particle, re-\nspectively. As before, in Eq. (4.1), the brackets /angb∇acketleft.../angb∇acket∇ightindicate a general (not necessarily\nequilibrium ) ensemble average.\nAlso, in the same equation, β≡1/kBT, whereTis the molecular temperature T(t)\ndefined in Eq. (2.17), assumed to coincide with the ���nal temperatur e of the quench. This,\nhowever, entails another drastic simplification that must be made ex plicit here. We refer to\nthe assumption that the system is in contact with a thermal reserv oir at temperature TR(t),\nand that heat is conducted instantaneously through the surface and bulk of the fluid, so\nthat the local time-dependent molecular temperature T(t) defined in Eq. (2.17) as T(t)≡\n/angb∇acketleftp2(t)/3MkB/angb∇acket∇ight, is uniform and equal to TR(t). For the particular case of an instantaneous\n28temperature quench at t= 0 from an initial temperature Tito a final temperature Tthat\nremains constant for t >0, the temperatures T(t) =TR(t) will remain constant, or T(t) =\nTR(t) =Tfort >0.\nIn the absence of external fields, we may rewrite equation (4.2) as\nσxy(t) =−N/summationdisplay\ni=1Rx\ni(t)Fy\ni(t) =1\n2N/summationdisplay\ni,j=1xijdu(Rij)\ndyij, (4.3)\nwhereRx\ni(t)≡ˆx·ri(t) =xi(t),Fy\ni(t)≡Fi(t)·ˆy,xij(t) =xi(t)−xj(t), anddu(Rij)/dyij≡\n(∇iju(Rij))·ˆy. In terms of the local density of particles, one can rewrite Eq. (4.3 ) as\nσxy(t) =1\n2/integraldisplay\ndr/integraldisplay\ndr′(x−x′)du(|r−r′|)\nd(y−y′)N/summationdisplay\ni=1δ(r−ri(t))N/summationdisplay\nj=1δ(r′−rj(t))\n=1\n2/integraldisplay\ndr/integraldisplay\ndr′(x−x′)du(|r−r′|)\nd(y−y′)ˆn(r,t)ˆn(r′,t), (4.4)\nwhich, inserted in Eq. (4.1), leads to\n∆η(τ;t) =β\n4V/integraldisplay\ndr1dr2dr3dr4(x1−x2)du(|r1−r2|)\nd(y1−y2)(x3−x4)du(|r3−r4|)\nd(y3−y4)\n×/angbracketleftBig\nˆn(r1,t+τ)ˆn(r2,t+τ)ˆn(r3,t)ˆn(r4,t)/angbracketrightBig\n. (4.5)\nUsing the Fourier transform u(k) =/integraltext\ndre−ik·ru(r) of the pair potential u(r), it is straight-\nforward to show that −r∇u(r) = (1/(2π)3)/integraltext\ndkeik·r∇k[ku(k)], whose xycomponent (writ-\ningr=r1−r2) is\n(x1−x2)du(|r1−r2|)\nd(y1−y2)=−1\n(2π)3/integraldisplay\ndkeik·(r1−r2)ˆx·∇k[k·ˆyu(k)] =−1\n(2π)3/integraldisplay\ndkeik·(r1−r2)/parenleftbigg∂[kyu(k)]\n∂kx/parenrightbigg\n.\n(4.6)\nThis result allows us to rewrite Eq. (4.5) as\n∆η(τ;t) =(β/V)\n4(2π)6/integraldisplay\ndk/integraldisplay\ndk′/parenleftbigg∂[kyu(k)]\n∂kx/parenrightbigg/parenleftBigg\n∂/bracketleftbig\nk′\nyu(k′)/bracketrightbig\n∂k′x/parenrightBigg/integraldisplay\ndr1dr2dr3dr4eik·(r1−r2)eik′·(r3−r4)\n×/angbracketleftBig\nˆn(r1,t+τ)ˆn(r2,t+τ)ˆn(r3,t)ˆn(r4,t)/angbracketrightBig\n, (4.7)\nwhich can also be written as\n∆η(τ;t) =(β/V)\n4(2π)6/integraldisplay\ndk/integraldisplay\ndk′/parenleftbigg∂[kyu(k)]\n∂kx/parenrightbigg/parenleftBigg\n∂/bracketleftbig\nk′\nyu(k′)/bracketrightbig\n∂k′x/parenrightBigg/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\n,\n(4.8)\nwheren(k,t) is the Fourier transform n(k,t)≡/integraltext\ndreik·rn(r,t) ofn(r,t).\n29B. Expression for η(τ;t)in terms of S(k;t)andF(k,τ;t)\nEq. (4.8) writes ∆ η(τ;t) in terms of the Fourier-transform /angb∇acketleftn(k,t+τ)n(−k,t+\nτ)n(k′,t)n(−k′,t)/angb∇acket∇ightof the four-point correlation function /angb∇acketleftˆn(r1,t+τ)ˆn(r2,t+\nτ) ˆn(r3,t)ˆn(r4,t)/angb∇acket∇ight, whose calculation is probably impossible without some form of\nsimplifying approximation. For this, here we adopt its Gaussian facto rization, which\napproximates/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\nby a sum of products of two-point\nand one-point correlation functions. In Appendix A we demonstrat e that, restricting\nourselves to the case of a homogeneous and isotropic liquid, such fa ctorization allows us to\nwrite Eq. (4.8) as\n∆η(τ;t) =βn2\n2(2π)3/integraldisplay\ndk/integraldisplay\ndk′/parenleftbigg∂[kyu(k)]\n∂kx/parenrightbigg/parenleftBigg\n∂/bracketleftbig\nk′\nyu(k′)/bracketrightbig\n∂k′x/parenrightBigg\nF(k,τ;t)F(k′,τ;t)δ(k+k′).\n(4.9)\nAt this point, let us substitute in this equation, the expression for [ kzu(k)] provided\nby the WL relation derived in the previous section, Eq. (3.13), namely , [kyu(k)] =\n−kBTS−1(k;t)[kyh(k;t)]. This allows us to rewrite Eq. (4.9) as\n∆η(τ;t) =βn2\n2(2π)3/integraldisplay\ndk/parenleftbigg∂[kyu(k)]\n∂kx/parenrightbigg2\n[F(k,τ;t)]2\n=kBTn2\n2(2π)3/integraldisplay\ndk/parenleftbigg∂[kyS−1(k;t)h(k;t)]\n∂kx/parenrightbigg2\n[F(k,τ;t)]2\n=kBT\n2(2π)3/integraldisplay\ndkk2\ny/parenleftbigg∂[1−S−1(k;t)]\n∂kx/parenrightbigg2\n[F(k,τ;t)]2, (4.10)\nwhich can be more conveniently rewritten as\n∆η(τ;t) =kBT\n2(2π)3/integraldisplay\ndk/parenleftbiggkxkz\nk/parenrightbigg2/bracketleftbigg1\nS(k;t)/parenleftbiggdS(k;t)\ndk/parenrightbigg/bracketrightbigg2/bracketleftbiggF(k,τ;t)\nS(k;t)/bracketrightbigg2\n.(4.11)\nUpon angular integration/integraltext\ndk/parenleftbigkxkz\nk/parenrightbig2f(k) =/parenleftbig4π\n15/parenrightbig/integraltext∞\n0k4dkf(k), this expression reads\n∆η(τ;t) =kBT\n60π2/integraldisplay∞\n0dkk4/bracketleftbigg1\nS(k;t)/parenleftbiggdS(k;t)\ndk/parenrightbigg/bracketrightbigg2/bracketleftbiggF(k,τ;t)\nS(k;t)/bracketrightbigg2\n,(4.12)\nwhich is the expression for η(τ;t) in terms of S(k;t) andF(k,τ;t) that we set out to derive.\nLet us finally highlight that the structure of this equation is identical to that derived for\nthermodynamic equilibrium conditions by Geszti for atomic fluids [42] a nd by N¨ agele and\nBergenholtz [43, 44] using mode coupling theory.\n30V. DISCUSSION AND SUMMARY\nThis work was aimed to contribute to the discussion of the relevant a nd general chal-\nlenge of extending fundamental concepts of the statistical mech anical theory of classical\nequilibrium liquids, to out-of-equilibrium conditions. In addressing this challenge our spe-\ncific motivation and perspective derived from the development of th e statistical physical\nformalism referred to as the non-equilibrium self-consistent generalized Langevin eq uation\ntheory of irreversible processes of liquids [31–35]. Given its success ful first-principles de-\nscription of aging and other essential fingerprints of the amorpho us solidification of liquids,\nwe deemed important to highlight some methodological aspects implicit in the derivation\nof the NE-SCGLE equations, since they will continue to be employed in further extensions\nand applications of this non-equilibrium theoretical approach.\nThe simplest of these methodological aspects consisted in revising t he derivation of a\ngiven set of theoretical results of the equilibrium theory of liquids, t o see if they really\nemploy the actual condition of thermodynamic equilibrium, through t he use, for example,\nof an equilibrium (canonical, microcanonical, ...) probability distribution f unction. As it\nhappens, many steps in these equilibrium derivations actually employ o nly the condition of\nstationarity (but not of thermodynamic equilibrium), as well as othe r temporal or spatial\nsymmetries (spatial homogeneity and/or isotropy, time-reversa l, etc.). Our method was\nfirst illustrated in Section II with a simple exercise, namely, the deriva tion of the non-\nequilibrium energy equation, followed in Section III by a second examp le, the derivation of\nthe non-equilibrium extension of the Wertheim-Lovett relation.\nThese two specific illustrative examples clearly prepared the stage f or the main specific\ncontribution of this work, namely, the derivation of the non-equilibr ium extension of an\nexpression – first derived by Geszti [42] and by N¨ agele and Berge nholtz [43, 44]– for the\nrheological and viscoelastic properties of liquids in terms of the stru ctural and dynamical\nproperties of the system. This extension, carried out in Section IV , led us to an approximate\nbut general expression, Eq. (4.12), that connects the non-equ ilibrium shear stress relaxation\nfunction η(τ;t) of a non-equilibrium liquid with the kinetics of the structural relaxat ion, en-\ncoded inthe t-evolution of thenon-equilibrium structure factor S(k;t), andthedynamic cor-\nrelations represented by the (collective and self) intermediate sca ttering functions F(k,τ;t)\nandFS(k,τ;t). These are, according to Eq. (4.12), the main microscopic element s that\n31determine the value of η(τ;t) and of the instantaneous viscosity η(t)≡/integraltext∞\n0dτη(τ;t), thus\ndirectly relating the viscoleastic response of a glass- or gel- forming system with explicit mi-\ncroscopic details, such as the potential of interaction between th e constitutive particles, and\nthe protocol of fabrication (here simplified by considering only an ins tantaneous quench).\nTo the best of our knowledge, such a connection had never been es tablished before.\nAs we shall demonstrate in a separate work, Eq. (4.12), together with the NE-SCGLE\nequations Eqs. (2.31)-(2.35), constitute a proposal of a canonic al theoretical protocol to\ndetermine the viscoelasticity of non-equilibrium liquids from first-prin ciples. The resulting\napproachisnowreadyforitssystematicapplicationtothecharact erizationoftheviscoelastic\nresponse ofa diversity of qualitatively different glass andgel formin g systems, such as liquids\nwith Lennard Jones-like interactions [71, 72] or systems with comp eting interactions (short-\nrangedattractionpluslong-rangedrepulsion)[73]. Inthesesyste ms, theinterferencebetween\nthermodynamical instabilities (spinodal line, λ-line) and dynamical arrest mechanisms leads\nto the possibility of qualitatively different glassy states, ranging fro m porous glasses, gels\nand Wigner glasses [74, 75].\nAlthough rather secondary to the main line of arguments just desc ribed, in Section II\nwe also addressed the natural question of the non-equilibrium exte nsion of the Ornstein-\nZernike equation Seq(k) = 1/nE(k;n,T). There, we concluded that this equation is actually\naconditionforthermodynamicequilibrium, andthatthedeviations[ S(k;t)−1/nE(k;n,T)],\naccording to Eq. (2.31), drive the rate of change of the structur e of the liquid (represented\nbyS(k;t)).\nLet us finally notice that there are no fundamental barriers that p revent the extension\nof the arguments and equations presented here, to much more co mplex conditions, involv-\ning glass and gel forming systems with multiple relaxation channels. Th is is the case, for\nexample, of colloidal suspensions comprised by dipolar particles (fer rofluids), in which the\ndecoupling of the orientational and translational dynamics allows to investigate partially\narrested states, and also, of mixtures with disparate size ratios, which allow for the devel-\nopment of glassy states with qualitatively different structural and dynamical characteristics\nupon tuning the molar distribution and total concentration. The dis cussion of the non-\nequilibrium viscoelastic response of these more complex materials is an additional example\nof areas of opportunity left for subsequent work.\n32VI. ACKNOWLEDGMENTS\nACKNOWLEDGMENTS: This work was supported by the Consejo Nacio nal de Cien-\ncia y Tecnolog´ ıa (CONACYT, Mexico) through Postdoctoral Fellows hips Grants No.\nI1200/224/2021 and I1200/320/2022; and trough grants 3209 83, CB A1-S-22362, and LAN-\nIMFE 314881.\nAppendix A: Gaussian approximation for the four-point correlation function and\nFourier transforms\nThisAppendix discusses theGaussianfactorizationforthefour-p ointcorrelationfunction/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\nunder more general conditions [76, 77] than em-\nployed intheMCT descriptionofequilibrium viscoelasticity [43]. Forclarit y, let usintroduce\nthe following notation for the four microscopic densities involved, na mely,n(k,t+τ)≡n1,\nn(−k,t+τ)≡n2,n(k′,t)≡n3, andn(−k′,t)≡n4; and for their averages and fluctuations,\n/angb∇acketleftni/angb∇acket∇ight ≡niandδn(ri;t)≡δni(withi= 1,2,3, and 4). Then,\n/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\n=/angb∇acketleftn1n2n3n4/angb∇acket∇ight=/angb∇acketleft(¯n1+δn1)(¯n2+δn2)(¯n3+δn3)(¯n4+δn4)/angb∇acket∇ight\n=/angb∇acketleftδn1δn2δn3δn4/angb∇acket∇ight+/angb∇acketleftδn1δn2δn3/angb∇acket∇ight¯n4+/angb∇acketleftδn1δn2δn4/angb∇acket∇ight¯n3+/angb∇acketleftδn1δn3δn4/angb∇acket∇ight¯n2\n+/angb∇acketleftδn2δn3δn4/angb∇acket∇ight¯n1+/angb∇acketleftδn1δn3/angb∇acket∇ight¯n2¯n4+/angb∇acketleftδn1δn2/angb∇acket∇ight¯n3¯n4+/angb∇acketleftδn2δn3/angb∇acket∇ight¯n1¯n4\n+/angb∇acketleftδn3δn4/angb∇acket∇ight¯n1¯n2+/angb∇acketleftδn1δn4/angb∇acket∇ight¯n2¯n3+/angb∇acketleftδn2δn4/angb∇acket∇ight¯n1¯n3+/angb∇acketleftδn1/angb∇acket∇ight¯n2¯n3¯n4\n+/angb∇acketleftδn2/angb∇acket∇ight¯n1¯n3¯n4+/angb∇acketleftδn3/angb∇acket∇ight¯n1¯n2¯n4+/angb∇acketleftδn4/angb∇acket∇ight¯n1¯n2¯n3+ ¯n1¯n2¯n3¯n4. (A1)\nIf each of the variables niabove represented a stationary Gaussian stochastic process,\nthen from Isserlis-Wick’s theorem [76, 77] it would follow that\n/angb∇acketleftδni/angb∇acket∇ight=/angb∇acketleftδniδnjδnk/angb∇acket∇ight= 0, (A2)\n/angb∇acketleftδn1δn2δn3δn4/angb∇acket∇ight=/angb∇acketleftδn1δn2/angb∇acket∇ight/angb∇acketleftδn3δn4/angb∇acket∇ight+/angb∇acketleftδn1δn3/angb∇acket∇ight/angb∇acketleftδn2δn4/angb∇acket∇ight+/angb∇acketleftδn1δn4/angb∇acket∇ight/angb∇acketleftδn2δn3/angb∇acket∇ight,(A3)\n33The stochastic process represented by the variables niabove is not strictly stationary. How-\never, it will be assumed to be piece-wise stationary [32]. It is also not n ecessarily Gaussian.\nNevertheless, we adopt this factorization as an approximation. As a result, and after some\nstraightforward algebraic steps, one gets\n/angb∇acketleftn1n2n3n4/angb∇acket∇ight ≈(¯n1¯n2+/angb∇acketleftδn1δn2/angb∇acket∇ight)(¯n3¯n4+/angb∇acketleftδn3δn4/angb∇acket∇ight)+(¯n1¯n3+/angb∇acketleftδn1δn3/angb∇acket∇ight)(¯n2¯n4+/angb∇acketleftδn2δn4/angb∇acket∇ight)\n+(¯n1¯n4+/angb∇acketleftδn1δn4/angb∇acket∇ight)(¯n2¯n3+/angb∇acketleftδn2δn3/angb∇acket∇ight)−2¯n1¯n2¯n3¯n4\n=/angb∇acketleftn1n2/angb∇acket∇ight/angb∇acketleftn3n4/angb∇acket∇ight+/angb∇acketleftn1n3/angb∇acket∇ight/angb∇acketleftn2n4/angb∇acket∇ight+/angb∇acketleftn1n4/angb∇acket∇ight/angb∇acketleftn2n3/angb∇acket∇ight−2¯n1¯n2¯n3¯n4. (A4)\nGoing back to the original notation, this equation reads\n/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\n≈/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)/angbracketrightBig/angbracketleftBig\nn(k′,t)n(−k′,t)/angbracketrightBig\n+/angbracketleftBig\nn(k,t+τ)n(k′,t)/angbracketrightBig/angbracketleftBig\nn(−k,t+τ)n(−k′,t)/angbracketrightBig\n+/angbracketleftBig\nn(k,t+τ)n(−k′,t)/angbracketrightBig/angbracketleftBig\nn(k′,t)n(−k,t+τ)/angbracketrightBig\n−2/angb∇acketleftn(k,t+τ)/angb∇acket∇ight/angb∇acketleftn(−k,t+τ)/angb∇acket∇ight/angb∇acketleftn(k′,t)/angb∇acket∇ight/angb∇acketleftn(−k′,t)/angb∇acket∇ight\n(A5)\nBytranslationalinvariance, thevanHovefunction G(r,t;r′,t′)≡ /angb∇acketleftn(r;t)n(r′;t′)/angb∇acket∇ightcanonly\ndepend on the difference r−r′and by spatial isotropy, G(r−r′;t,t′) can only depend on the\nmagnitude |r−r′|. Similarly, as a consequence of translational invariance, the corre lation\n/angb∇acketleftn(k,t)n(k′,t′)/angb∇acket∇ightis non-zero only if k′=−kand by rotational invariance, F(k;t,t′) can only\ndependonthemagnitudeof k, i.e.,N−1/angb∇acketleftn(k,t)n(k′,t′)/angb∇acket∇ight=F(k,t;k′,t′) =F(k;t,t′)δ(k+k′).\nOn the other hand, the mean values /angb∇acketleftn(r,t)/angb∇acket∇ightand/angb∇acketleftn(k,t)/angb∇acket∇ightdepend in general on t, but\nin the present application they are constrained for simplicity to be un iform and constant,\n/angb∇acketleftn(r,t)/angb∇acket∇ight=n≡N/V, sothat/angb∇acketleftn(k,t)/angb∇acket∇ight= (N/V)(2π)3δ(k). Thetwo-timecorrelationfunction\nF(k;t,t′), which under stationary conditions only depends on the time differe nceτ≡t−t′,\ndepends in general on both times, tandt′(or, equivalently, on tandτ≡t−t′), so that we\nshall actually write N−1/angb∇acketleftn(k,t)n(k′,t′)/angb∇acket∇ight=F(k,τ;t)δ(k+k′). Finally, let us notice that\nthe equal-time intermediate scattering function F(k,τ= 0;t) is just the time-dependent\nstructure factor S(k;t), i.e.,F(k,τ= 0;t) =S(k;t).\n34With these previsions, Eq. (A5) becomes\n/angbracketleftBig\nn(k,t+τ)n(−k,t+τ)n(k′,t)n(−k′,t)/angbracketrightBig\n≈S(k;t+τ)S(k′;t)\n+N2F(k,τ;t)F(k′,τ;t)(2π)3V−1δ(k+k′)\n+N2F(k,τ;t)F(k′,τ;t)(2π)3V−1δ(k−k′)\n−2(n/V)2(2π)6δ(k)δ(k′),\n(A6)\nwhich allows us to approximate ∆ η(τ;t) in Eq. (4.8) by\n∆η(τ;t) =(β/V)\n4(2π)6/integraldisplay\ndk/integraldisplay\ndk′/parenleftbigg∂[kzu(k)]\n∂kx/parenrightbigg/parenleftbigg∂[k′\nzu(k′)]\n∂k′x/parenrightbigg\n×/braceleftBig\nS(k;t+τ)S(k′t)+N2F(k,τ;t)F(k′,τ;t)(2π)3V−1δ(k+k′)\n+N2F(k,τ;t)F(k′,τ;t)(2π)3V−1δ(k−k′)−2/parenleftBign\nV/parenrightBig2\n(2π)6δ(k)δ(k′)/bracerightBig\n.(A7)\nThus, ∆η(τ;t) is a sum of four terms. The first of them becomes a product of two factors,\neach vanishing because the integrand is an odd function of kz. The last term also vanishes\nbecause it also factorizes, with each factor being proportional to/integraltext\ndkkz(∂u(k)/∂kx)δ(k) =\n[kz(∂u(k)/∂kx)]k=0= 0. Thus, the only surviving terms are the second and the third, wh ich\nare clearly identical. These considerations allow us to rewrite Eq. (A7 ) as\n∆η(τ;t) =βn2\n4(2π)3/integraldisplay\ndk/integraldisplay\ndk′/parenleftbigg∂[kzu(k)]\n∂kx/parenrightbigg/parenleftbigg∂[k′\nzu(k′)]\n∂k′\nx/parenrightbigg\n×/braceleftBig\n2F(k,τ;t)F(k′,τ;t)δ(k+k′)/bracerightBig\n, (A8)\nwhich is Eq. (4.9).\n35[1]“The equilibrium theory of classical fluids: a lecture note an d reprint volume” , Editors: H. L.\nFrisch and J. L. Lebowitz, W.A. Benjamin, New York (1964).\n[2] H. Callen. Thermodynamics . John Wiley (1960).\n[3] R. Evans Adv. Phys. 28143-200 (1979).\n[4] D. A. McQuarrie Statistical Mechanics , Harper & Row (New York, 1976).\n[5] J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids (Elsevier-Academic Press, 2006).\n[6] J. L. Boon and S. Yip, Molecular Hydrodynamics (Dover Publications Inc. N. Y., 1980).\n[7] Angell C. A., Ngai K. L., McKenna G. B., McMillan P. F. and M artin S. F., J. Appl. Phys.\n883113 (2000).\n[8] K. L. Ngai et al. J. Phys.: Cond. Mat. 20, 244125 (2008).\n[9] K. A. Dawson, Current Opinion in Colloid & Interface Scie nce7, 218 (2002).\n[10] P. Anderson. Science 267, 1615 (1995).\n[11] B. Bagchi, Nonequilibrium Statistical Mechanics. An Introduction wit h Applications , Chapman\n& Hall (2023).\n[12] G. Nicolis and I. Prigogine, Self-Organization in non-equilibrium Systems: From dissi pative\nstructures to order through fluctuations . Wiley-Interscience, New York (1977).\n[13] R. Zwanzig, Nonequilibrium Statistical Mechanics , Oxford University Press, USA (2001).\n[14] G. Lebon, D. Jou, and J. Casas-V´ azquez, Understanding Non-equilibrium Thermodynamics\nFoundations, Applications, Frontiers , Springer-Verlag Berlin Heidelberg (2008).\n[15] H. C. ¨Ottinger, Beyond Equilibrium thermodynamics (Wiley, N.Y. 2006).\n[16] M. te Vrugt, H. L¨ owen and R. Wittkowski, Advances in Phy sics 69, 121 (2020).\n[17] K. S. Schmitz, Physical Chemistry Concepts and Theory, Elsevier (2016).\n[18] A. Q. Tool, J. Am. Ceram. Soc. 29, 240 (1946).\n[19] O. S. Narayanaswamy, J. Am. Ceram. Soc. 54, 491 (1971).\n[20] C. T. Moynihan, A. J. Easteal, D. C. Tran, J. A. Wilder, E. P. Donovan, J. Am. Ceram. Soc.\n59, 137 (1976).\n[21] A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson, and A. R. Ra mos, J. Polym. Sci. Polym. Phys.\n17, 1097 (1979).\n[22] L. Hornboll, T. Knusen, Y. Yue, and X. Guo, Chem. Phys. Le tt.494, 37 (2010).\n36[23] I. M. Hodge, J. Non-Cryst. Solids 169, 211 (1994).\n[24] J. C. Mauro, D. C. Allan, and M. Potuzak, Phys. Rev. B 80, 0 94204 (2009).\n[25] L. Berthier and G. Biroli, Rev. Mod. Phys. 83(2011).\n[26] Berthier L, Biroli G, Bouchaud JP, Cipelletti L, van Saa rloos W, editors. Dynamical Hetero-\ngeneities in Glasses, Colloids, and Granular Media. Oxford: Oxford University Press (2011).\n[27] P. Charbonneau, Y. Jin, G. Parisi, and F. Zamponi, PNAS 1 11, 15025 (2014).\n[28]J.D. van der Waals: On the Continuity of the Gaseous and Liquid S tates, StudiesinStatistical\nMechanics XIV, ed. J.S. Rowlinson (North-Holland, 1988).\n[29] L. S. Ornstein and F. Zernike, Proc. K. Ned. Akad. Wet. 17, 793 (1914).\n[30] B. Widom, Physica A 263, 500 (1999).\n[31] P. E. Ram´ ırez-Gonz´ alez and M. Medina-Noyola, Phys. Rev. E 82, 061503 (2010).\n[32] P. E. Ram´ ırez-Gonz´ alez and M. Medina-Noyola, Phys. R ev. E82, 061503 (2010).\n[33] P. E. Ram´ ırez-Gonz´ alez and M. Medina-Noyola, Phys. R ev. E82, 061504 (2010).\n[34] L. E. S´ anchez-D´ ıaz, P. E. Ram´ ırez-Gonz´ alez, and M. Medina-Noyola, Phys. Rev. E 87, 052306\n(2013).\n[35] J. M. Olais-Govea, L. L´ opez-Flores, and M. Medina-Noy ola, J. Chem Phys. 143, 174505\n(2015).\n[36] L. Onsager, Phys. Rev. 37, 405 (1931).\n[37] L. Onsager, Phys. Rev. 38, 2265 (1931).\n[38] L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953).\n[39] S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953).\n[40] R. Peredo-Ortiz, L. F. Elizondo-Aguilera, P. Ram´ ırez -Gonz´ alez, Edilio L´ azaro-L´ azaro, P.\nMendoza-M´ endez, and M. Medina-Noyola, (2023) Non-equilibrium Onsager-Machlup theory ,\nMolecular Physics, DOI: 10.1080/00268976.2023.2297991.\n[41] O. Joaqu´ ın-Jaime, P. Ojeda-Mart´ ınez, A. G. Carretas -Talamante, L. F. Elizondo-Aguilera\n(Manuscript in preparation, 2024).\n[42] T. Geszti, J. Phys. C 16, 5805 (1983).\n[43] G. N¨ agele and J. Bergenholtz, J. Chem. Phys. 1089893 (1998).\n[44] A.J. Banchio, G.N¨ agele and J. Bergenholtz J. Chem. Phys. 111, 8721–8740 (1999)\n[45] W. G¨ otze and L. Sj¨ ogren, Rep. Prog. Phys. 55, 241 (1992).\n[46] W. G¨ otze, E. Leutheusser and S. Yip, Phys. Rev. A 23, 2634 (1981).\n37[47] W. G¨ otze, Complex Dynamics of Glass-Forming Liquids (Oxford University Press, 2009).\n[48] W. Hess and R. Klein, Adv. Phys. 32, 173 (1983).\n[49] Herbert Goldstein, Classical Mechanics (2nd ed.) Addison-Wesley (1980).\n[50] J.M. Brader and M. Schmidt, J. Chem. Phys. 139, 104108 (2013).\n[51] P. Pusey in Liquids, Freezing and Glass Transition , Ed. J. P. Hansen, D. Levesque, and J.\nZinn-Justin (Elsevier, Amsterdam, 1991).\n[52] M. Sperl, Phys. Rev. E 69, 011401 (2004).\n[53] P. E. Ram´ ırez-Gonz´ alez, A. Vizcarra-Rend´ on, F. de J . Guevara-Rodr´ ıguez, and M. Medina-\nNoyola, J. Phys.: Cond. Matter, 20, 20510 (2008).\n[54] L.F. Elizondo-Aguilera and Th. Voigtmann, Phys. Rev. E 100042601 (2019).\n[55] R. Peredo-Ortiz, M. Medina-Noyola, Th. Voigtmann and L .F. Elizondo-Aguilera, J. Chem.\nPhys. 156 244506 (2022).\n[56] L. Yeomans-Reyna, M. A. Ch´ avez-Rojo, P. E. Ram´ ırez-G onz´ alez, R. Ju´ arez-Maldonado, M.\nCh´ avez-P´ aez, and M. Medina-Noyola, Phys. Rev. E 76, 041504 (2007)\n[57] L. Yeomans-Reyna and M. Medina-Noyola, Phys. Rev. E 623382 (2000).\n[58] L. Yeomans-Reyna, H. Acu˜ na-Campa and M. Medina-Noyol a, Phys. Rev. E 623395 (2000).\n[59] J.B. Zepeda-L´ opez and M. Medina-Noyola, J. Chem. Phys. 154, 174901 (2021).\n[60] G. F. Vander Voort. Atlas of Time Temperature Diagrams for Irons and Steels. ASM IN-\nTERNATIONAL, Materials Park, OH (1991).\n[61] K. Nakashima, K. Noda, and K. Mori, J. Am. Ceram. Soc., 80, 1101 (1997).\n[62] M. Medina-Noyola, Faraday Discuss. Chem. Soc. 83, 21 (1987).\n[63] M. Medina-Noyola and J. L. del R´ ıo-Correa, Physica 146A , 483 (1987).\n[64] L.F. Elizondo-Aguilera, A. Carrillo-R´ ıos, G. Rodr´ ı guez Li˜ n´ an F. L´ opez Gonz´ alez, F. Donado\nand F. Pacheco V´ azquez, manuscript submitted to Physica A\n[65] Brownian motion in non-equilibrium systems and the Orn stein-Uhlenbeck stochastic process,\nF. Donado, R.E. Moctezuma, L. L´ opez-Flores, M. Medina-Noy ola and J. L. Arauz-Lara,\nScientific Reports, 7, 12614 (2017). Doi: 10.1038/s41598-0 17-12737-1.\n[66] R. Kubo, Rep. Prog. Phys., 29255 (1966).\n[67] H. Mori, Prog. Theor. Phys. 33, 423 (1965).\n[68] S. Ciarella, R. A. Biezemans, and L. M. C. Janssen, “Unde rstanding, predicting, and tuning\nthe fragility of vitrimeric polymers”. Proc Natl Acad Sci U S A. 116(50):25013-25022 (2019).\n38[69] M. Medina-Noyola, Phys. Rev. Lett. 60, 2705 (1988).\n[70] P. Mazur and U. Geigenm¨ uller, Physica A 146, 657 (1987).\n[71] J.M. Olais-Govea, L. L´ opez-Flores and M. Medina-Noyo la, “Non-equilibrium theory of ar-\nrested spinodal decomposition” J. Chem. Phys. 143, 174505 (2015)\n[72] J.M. Olais-Govea, L. L´ opez-Flores, J.B. Zepeda-L´ op ez and M. Medina-Noyola. “Interference\nbetween the glass, gel, and gas-liquid transitions”. Sci Rep9, 16445 (2019).\n[73] A.G. Carretas-Talamante, J.B. Zepeda-L´ opez, E. L´ az aro-L´ azaro, et al J. Chem. Phys .158,\n064506 (2023).\n[74] L.E. S´ anchez-D´ ıaz, A. Vizcarra REnd´ on and R. Ju´ are z-Maldonado, Phys. Rev. Lett. 103,\n179902 (2009) / Erratum Phys. Rev. Lett. 103, 179902 (2009).\n[75] N. Benitez-Camacho, J.M. Olais-Govea, L. L´ opez-Flor es and H. Ruiz-Estrada J. Chem. Phys.\n159, 044906 2023.\n[76] L. Isserlis Biometrika 12134-139 (1918).\n[77] J.K.G. Dhont, An introduction to dynamics of colloids: Volume 2 Editors: D. M¨ obius and R.\nMiller, Elsevier Science (1996).\n39" }, { "title": "2401.15226v1.Existence_of_spiral_waves_in_oscillatory_media_with_nonlocal_coupling.pdf", "content": "EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH\nNONLOCAL COUPLING\nGABRIELA JARAMILLO\nUniversity of Houston, Department of Mathematics, 3551 Cullen Blvd., Houston, TX 77204-3008,\nUSA\nAbstract. We prove existence of spiral waves in oscillatory media with nonlocal coupling. Our\nstarting point is a nonlocal complex Ginzburg-Landau (cGL) equation, rigorously derived as an\namplitude equation for integro-differential equations undergoing a Hopf bifurcation. Because this\nreduced equation includes higher order terms that are usually ignored in a formal derivation of the\ncGL, the solutions we find also correspond to solutions of the original nonlocal system. To prove\nexistence of these patterns we use perturbation methods together with the implicit function theorem.\nWithin appropriate parameter regions, we find that spiral wave patterns have wavenumbers, κ, with\nexpansion κ∼Ce−a/ε, where ais a positive constant, εis the small bifurcation parameter, and the\npositive constant Cdepends on the strength and spread of the nonlocal coupling. The main difficulty\nwe face comes from the linear operators appearing in our system of equations. Due to the symmetries\npresent in the system, and because the equations are posed on the plane, these maps have a zero\neigenvalue embedded in their essential spectrum. Therefore, they are not invertible when defined\nbetween standard Sobolev spaces and a straightforward application of the implicit function theorem\nis not possible. We surpass this difficulty by redefining the domain of these operators using doubly\nweighted Sobolev spaces. These spaces encode algebraic decay/growth properties of functions, near\nthe origin and in the far field, and allow us to recover Fredholm properties for these maps.\nRunning head: Existence of Spiral Waves\nKeywords: pattern formation, nonlocal diffusion, integro-differential equations, Fredholm operators.\nAMS subject classification: 45K05, 45G15, 46N20, 35Q56, 35Q92\n1.Introduction\nThe term oscillatory media describes systems which combine self-sustained time oscillations with\nmechanisms that allow for spatial interactions, or coupling. Examples include electrochemical sys-\ntems [31, 32], oscillating chemical reactions [11, 48], colonies of aggregating slime mold [16], and\nunder certain assumptions, even heart [53] and brain tissue [46]. Interest in these systems stems in\npart from their ability to generate beautiful spatio-temporal structures like target patterns, traveling\nwaves, and spiral waves. While properties of these patterns have been extensively studied in the\ncase of oscillatory media with local coupling, not many results address the case of systems involving\nThis work is supported by NSF DMS-1911742.\nAMS subject classification: 45K05, 45G15, 46N20, 35Q56, 35Q92.\n1arXiv:2401.15226v1 [math.AP] 26 Jan 2024EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 2\nFigure 1. Spiral chimeras.\nlong-range interactions. In this paper we take on this challenge, focusing on existence of spiral waves\nin planar spatially extended oscillatory media with nonlocal coupling.\nOur interest in spiral waves comes from numerical experiments done by Kuramoto and coauthors,\nwhich focused on an abstract FitzHugh-Nagumo system that incorporates a convolution term in place\nof the standard Laplacian, [44]. Their simulations show that, depending on system parameters, the\nnonlocal coupling described by this convolution operator can give rise to a new type of pattern\nknown as a spiral chimera. As the name suggest, this novel structure looks very much like a spiral\nwave in the far field, but has a core which does not act in synchrony with the rest of the pattern,\n(see Figure 1). As a first step towards understanding the formation of these new structures, here\nwe address how nonlocal forms of coupling affect the formation and shape of ‘regular’ spiral waves.\nTo model nonlocal coupling, we use convolution operators of diffusive type. These operators are\ndescribed by convolution kernels whose Fourier symbols are radially symmetric, uniformly bounded,\nanalytic, and have a quadratic tangency near the origin. This choice of coupling then leads to model\nequations that are nonlocal and that take the form,\nUt=DL ∗U+F(U;µ)U∈R2, x∈R2, µ ∈R. (1)\nHere Dis a matrix of diffusion coefficients, while Lrepresents our choice of convolution operator.\nThe symbol Fthen describes reaction terms that undergo a Hopf bifurcation as the parameter µ\ncrosses the origin.\nWhile similar in structure to reaction-diffusion equations modeling oscillatory media with local\ncoupling, integro-differential equations like (1) are in general more challenging to analyze. On the\none hand, the convolution operator is easier to handle if the equations are posed on the plane, since in\nthis case one does not have to worry about imposing boundary constraints that are compatible with\nthe operator, and which in addition respect desired modeling assumptions. But this simplification\ncomes at a price. When posed on R2, the linearization of equation (1) about a steady state is now\nan operator with real essential spectrum touching the origin. Moreover, due to the translational\nsymmetry of the equation this operator has a non-trivial nullspace, or equivalently, a zero-eigenvalue\nembedded in the essential spectrum. Consequently, the linearization is not an invertible, nor a\nFredholm operator, when viewed as a map between standard Sobolev spaces. As a result, one cannot\nimmediately use perturbation methods together with the implicit function theorem, or Lyapunov-\nSchmidt reduction, to prove existence of solutions which bifurcate from a steady state.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 3\nA similar difficulty is encountered when considering reaction-diffusion equations posed on un-\nbounded domains. While in this case it is possible to prove existence of solutions by reformulating\nthe problem as an ordinary differential equation and employing methods from spatial dynamics\n[29, 30, 42], this approach is not readily applicable for integro-differential equations (unless one as-\nsumes the convolution kernel has a particular form that allows one to write the equations as pde’s,\nsee [?]). An exception is the construction of a center manifold, which can be done without reference\nto a phase space using fixed point methods, see [12, 8]. However, this approach implicitly assumes\nthat the system is in a regime where the nonlocal coupling is well approximated by local interactions,\nas evidenced by the fact that the resulting equations describing the ’flow’ on the center manifold are\ndifferential equations. In contrast, here we are interested in the opposite regime, where interactions\nbetween oscillating elements are truly nonlocal.\nTherefore, our starting point will be a nonlocal complex Ginzburg-Landau equation, rigorously\nderived in [21] as an amplitude equation for rotating wave solutions of integro-differential equations\nof the form (1). To prove existence of spiral waves we use perturbation methods together with the\nimplicit function theorem. To overcome the course of the zero-eigenvalue, we follow the approach\ntaken in [25, 24, 22], where it is shown that one can recover Fredholm properties (closed range,\nfinite dimensional kernel and cokernel) for convolution operators of diffusive type, and related maps,\nusing algebraically weighted Sobolev spaces. In the rest of this introduction, we briefly describe the\nderivation of the nonlocal amplitude equation, state our main theorem, and give a short outline for\nthe paper. We finish this introduction with a discussion of our results.\n1.1.The Nonlocal Amplitude Equation. Close to the onset of oscillations and under the as-\nsumption of weak local coupling, oscillatory media may be described by the complex Ginzburg-\nLandau (cGL) equation. This reduced equation describes variation in the amplitude of oscillations\nwhich occur over long spatial and time scales, and can thus be formally derived using a multiple-\nscale analysis, see [34, 51]. This method can also be extended to account for other forms of coupling,\nincluding global and nonlocal coupling [47, 14], and to incorporate feedback mechanisms and forcing\nterms [15]. Since this is a formal approach, it is then necessary to justify the validity of the equation.\nThat is, one must prove that the approximate solutions obtained using the cGL are close to the ac-\ntual solutions of the corresponding system in an appropriate metric. See for instance [28, 33, 43, 50]\nfor works that address this question.\nInstead, the work presented in [21] takes a different approach. There, the method of multiple-\nscales is given a rigorous treatment in order to derive, and validate, an amplitude equation for\nrotating wave solutions of (1). The result is the following nonlocal complex Ginzburg-Landau equa-\ntion,\n0 =K∗w+ (1 + i λ)w−(1 + i β)|w|2w+N(w;ε), x = (r, θ)∈R2, (2)\nwhere the symbol Krepresents a scaled version of the original operator describing the nonlocal\ncoupling, the unknown w=w(r) is a radial complex-valued function, λandβare real parameters,\nandεis a small number measuring how close the system is to the Hopf instability. The term N(w, ε)\nthen summarizes nonlinear higher order correction terms of size O( ε) which, as explained below, are\nneeded in order to rigorously prove the existence of spiral waves.\nAs in the formal derivation of the local cGL, the method used to arrive at equation (2) focuses\non small amplitude oscillations that emerge close to the Hopf bifurcation. In the formal derivation\nof the amplitude equation this difference in scales then allows one to expand solutions to equationEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 4\n(1) in powers of ε, i.e\nU=εU1+ε2U2+ε3U3+···.\nGathering similar terms one then obtains a sequence of equations. Using an appropriate ansatz on a\nco-rotating frame that moves with the rotational speed, c, of the wave (i.e. U(r, θ, t ) =U(r, θ−ct)),\none then finds that the order εandε2equations can easily be solved, while the cGL equation then\nappears as a solvability condition at order ε3. As mentioned above, because all higher order terms\nare then ignored it is then necessary to justify the validity of the equation.\nIn contrast, to place the multiple-scales method in a more rigorous setting, the approach in [21]\nassumes solutions can be written using a finite expansion, i.e. U=εU1+ε2U2+ε3U3.Again, one\nfinds that the order εandε2equation can readily be solved, while all remaining terms of order O( ε3)\nare now gathered into one main equation. The results from [21] show that this main equation can be\nsplit into an invertible system and a reduced equation. Using the implicit function theorem one can\nthen solve the invertible system, thus obtaining a family of solutions parametrized by the first order\ncorrection term, U1. That is, one finds Ui= Ψ i(U1;ε) with i= 2,3, for some C1functions, Ψ i. After\ninserting this family of solutions into the reduced equation and projecting onto the angular Fourier\nmodes, e±iθ, one arrives at the nonlocal cGL equation (2), where one now sees that the symbol\nN(w;ε) encodes all remaining terms of order O( ε4). As a result, solving the amplitude equation\n(2) is equivalent to solving the original integro-differential equation (1). Thus, to rigorously prove\nexistence of solution to (1) representing spiral waves, it is enough to prove this result for the nonlocal\ncGL equation given by (2).\n1.2.Main Result. Before stating our main result, let us point out a couple of properties of the\namplitude equation (2) and of the solutions we seek.\nFirst, because the reduce equation (2) comes from using a suitable projection onto an angular\nmode, and because it is based on an ansatz that moves in a co-rotating frame, the unknown w\ndepends only on the radial variable r. While it is assumed that the solution, U, to the original\nintegro-differential equation is rotating with speed c, the value of this parameter is unknown. In\nthe reduced equation (2), the speed of the wave is captured by the parameter λvia the relation\nc=ω+ε2λ, where ωrepresents the frequency of the time oscillations emerging from the Hopf\nbifurcation, see [21]. Thus, using the gauge symmetry of the equation, we have that an equivalent\nformulation for our problem is to find solutions to\n˜wt=K∗˜w+ ˜w−(1 + i β)|˜w|2˜w+N( ˜w;ε), x = (r, θ)∈R2, (3)\nwhere ˜ w(r, t) =w(r)e−iλt.\nSecond, because in the co-rotating frame spiral waves look like target patterns, our goal is to\nfind constants κandA, as well as solutions to (3) of the form ˜ w(r, t) =ρ(r)ei(ϕ(r)−λt), such that\nϕ(r)→κandρ(r)→A, asrgoes to infinity. This is in essence the content of Theorem 1.\nTo complete the formulation of our problem, we now state our main assumptions regarding the\nthe nonlinear terms N(w;ε) and convolution operator K.\nHypothesis (H1).The nonlinear function N(w;ε)is order O(ε|w|4w), and every term in this\nexpression is of the form c|w|2nw, with c∈C, and n∈ {1,2,3,···}.\nA justification for Hypothesis [H1] is provided in Appendix A.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 5\nHypothesis (H2).The convolution operator K∗is defined by a radially symmetric kernel given by\nK(|x|) =η\nε2D\u0010\nK1(|x|/√\nε2D)−1\u0011\nwhere η, D are positive constants, 0< ε << 1, and K1(ξ)is the order one modified Bessel function\nof the second kind.\nWith these considerations in this paper we prove the following theorem.\nTheorem 1. Letε, βbe real numbers, with β < 0. Let Kdenote a convolution operator of the\nform described by Hypothesis (H2) and take N(w;ε)to be higher order terms of the form stated\nin Hypothesis (H1). Then, there exists a small positive number, ε0,and a family of solutions,\nw(r;ε) =ρ(r;ε)ei(ϕ(r;ε)−λ(ε)t), to the nonlocal complex Ginzburg-Landau equation (3), which is valid\nandC1in the ε- neighborhood (−ε0, ε0). Moreover, with δ∼ε1/4, this family has expansions,\nλ(ε) =β+δ2Ω(β)\nρ(r;ε) =ρ0(˜r) +δ2(R0(δ˜r) +δR1(δ˜r))\nϕ(r;ε) =ϕ0(δ˜r) +δϕ1(δ˜r),\n˜r=r/p\nη−ε2D,\nwhere ηandDare coefficients appearing in the definition of the operator K, and\nΩ(β) = 4 ˜C(β) exp(−M/β2),\nfor some constant M > 0and a C1(R)function ˜C(β). In addition, the lower order correction terms\nsatisfy,\n•ϕ0(δ˜r)∼1\nβlog(K0(Λδ˜r)) +c1asS=δ˜r→ ∞ , for some c1∈R,\n•R0(δ˜r) =−1\n2ρ0(˜r)(∂Sϕ0(δ˜r))2,\nwhile\n•ρ(r;ε)−→1 +δ2c2,for some c2∈R, and\n•∂rϕ(r;ε)−→κ=−Λ\nβδp\n��−ε2Dwith Λ =p\n−βΩ(β),\nasr→ ∞ .\nBefore continuing let us highlight how the results from Theorem 1 establish a relation between\nproperties of the operator Kand the shape of the spiral wave. From Hypothesis (H2) we see that\nthe parameters ηandDin the definition of the operator Kcontrol the strength and the spread of\nthe convolution kernel, respectively. These parameters then appear in the far-field approximation\nof the spiral’s wavenumber,\nκ=−Λ\nβδp\nη−ε2D.\nThe above expression shows that as the strength, η, of the convolution kernel increases, the spiral’s\nwavenumber, κ, decreases. On the other hand, if the spread, D, of the operator increases past a\ncertain threshold, the approximation for κbreaks down. These results are in good agreement with\nour simulations, see Figure 2 and Figure 3. In particular, notice how when the parameter Dis too\nlarge, we no longer obtain spiral waves but rather spiral chimeras.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 6\nFigure 2. Simulation of FitzHugh-Nagumo system appearing in [44] on a square\ndomain of length L= 100, using a cosine spectral method and an implicit Euler time\nstepping scheme with N= 1024 nodes and a time step h= 0.05. Convolution kernel\nused has Fourier symbol ˆK(ξ) =−η|ξ|2\n1+ε2D|ξ|2with fixed ε2D= 0.1 and varying η. From\nleft to right we have η= 0.5, η= 1.0, η= 1.5,andη= 2.0.\n1.3.Outline/ Sketch of Proof for Theorem 1. Although both formulations for our problem,\ni.e. equation (2) and equation (3), are equivalent, to prove Theorem 1 we use the first equation. We\nnow give a short summary of how we arrive at our main result.\nFirst, notice that the form of the convolution operator Kstated in Hypothesis [H2] comes from\nprojecting the map\nL=η(1−ε2D∆)−1∆\nonto the angular Fourier modes e±iθ. As a result we can formally write Kas\nK∗w=η(1−ε2D∆1)−1∆1w, (4)\nwhere ∆ 1=∂rr+1\nr∂r−1\nr2. Notice also that the operator Lhas a radially symmetric Fourier\nsymbol ˆL(ξ) =−η|ξ|2/(1 + ε2D|ξ|2). Because the Fourier Transform commutes with orthogonal\ntransformations, we have that the Fourier symbol of K∗ ·+ Id is also radially symmetric [45], and\nis thus given by\nF(K∗w+w) =\u0014−η|ξ|2\n1 +ε2D|ξ|2+ 1\u0015\nˆw=\u0014\n−(η−ε2D)|ξ|2+ηε2D|ξ|4\n1 +ε2D|ξ|2\u0015\nˆw.\nTherefore, equation (2) can also be written as\n0 = ( η−ε2D)∆1w+ (1 + i λ)w−(1 + i β)|w|2w+ε2J∗w+N(w;ε), (5)\nwith ˆJ(ξ) =ηD|ξ|4\n1+ε2D|ξ|2.\nAt the same time, we can take advantage of the formal definition of the operator Kgiven in (4)\nand precondition equation (2) by (1 −ε2D∆1) to arrive at\n0 = ( η−ε2D)∆1w+ (1 + i λ)w+ (1−ε2D∆1)[−(1 + i β)|w|2w+N(w;ε)]. (6)\nSince both formulations, equations (5) and (6), share similar first order terms, the first order\napproximations to both systems will be the same. Thus, we choose to work with the second formu-\nlation for ease of exposition. However, we point out that the ideas used to arrive at equation (5) can\nbe extended to more general convolution operators. We plan to tackle this more general framework\nusing this approach in a future paper.\nContinuing with the sketch of the proof of Theorem 1, to show existence of solutions to equation\n(6), we use polar coordinates to represent the unknown variable as w(r) =ρ(r)eiϕ(r), and thenEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 7\nFigure 3. Simulation of FitzHugh-Nagumo system appearing in [44] on a square\ndomain of length L= 100, using a cosine spectral method and an implicit Euler time\nstepping scheme with N= 1024 nodes and a time step h= 0.05. Convolution kernel\nused has Fourier symbol ˆK(ξ) =−η|ξ|2\n1+ε2D|ξ|2with fixed η= 1 and varying ˜D=ε2D.\nFrom left to right we have ˜D= 0.5,˜D= 1.0,˜D= 1.5,and˜D= 2.0. First row depicts\nfull spiral, second row zooms in into core of spiral.\nseparate the equation into its real and imaginary parts. Following the results from [10], in Section 3\nwe pick appropriate scalings and, using a regular expansion for both ρandϕ, we obtain a hierarchy\nof equations at different powers of the small parameter δ∼ε1/4. Our goal is to then show that this\nsequence of equations has solutions representing spiral waves. This is done in Section 4.\nIn Subsection 4.1 we work with the order O(1) equation and show that the first order correction\nto the complex amplitude, ρ0, converges quickly to a constant. Thus, the spiral wave pattern is\ndescribed by the complex phase, ϕ. In Subsection 4.3 we show that, not surprisingly, the first order\ncorrection term for the phase variable, ϕ0, satisfies the following viscous eikonal equation,\n−Ω =∂rrϕ0+1\nr∂rϕ0+β(∂rϕ0)2+βg(r). (7)\nwhich is a known phase dynamics approximation for target patterns in oscillatory media. While in\nthe case of target patterns the inhomogeneity grepresents the impurity, or defect, that gives rise\nthese structures, in the case of spiral waves this perturbation comes from the first order correction to\nthe amplitude, ρ0. We show that gdecays algebraically in the far field and satisfies g(r)∼O(1/r2)\nasr→ ∞ . We are then able to use the results from [22], where it is shown that equation (7) admits\nsolutions representing target patterns. That is, solutions satisfying ∇ϕ0→κasrgoes to infinity.\nWe show that the constant κ >0, indicating that the impurity gacts as a pacemaker, producing\ntraveling waves that move away from the core. Because we are working in a co-rotating frame, we\nconclude that these same solutions correspond to spiral waves patterns in our setting.\nTo complete our proof, we gather all terms of order O( δ3) into one system of equations, which\nwe refer to as the closing equations . Following a similar approach as in [21], in Section 5 we use\nthe implicit function theorem to prove existence of solutions. As mentioned before, the difficulty\nwith using this strategy comes from the fact that the linear operators appearing in our equationsEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 8\nare not invertible when viewed as maps between standard Sobolev spaces. We overcome this diffi-\nculty by establishing Fredholm properties for these, and related operators, using carefully selected\nalgebraically weighted spaces. This is done in Section 2, where we also give a precise definition for\nthe spaces we will be working with. A bordering lemma then allows us to recover the invertibility\nof the operator.\n1.4.Discussion. Spiral wave patterns have been extensively studied experimentally [39, 40, 52, 3],\nanalytically [30, 17, 18, 20, 42, 34, 41, 5, 38, 49], and through simulations [4, 36, 27, 41] since the\n1970s, and have been shown to exist in both excitable and oscillatory media. In this paper we focus\non the latter case, where one can assume that the intrinsic dynamics of the system allows for the\nformation of a limit cycle via a Hopf bifurcation. While most analysis of spiral waves concentrate\non systems where these intrinsic ‘oscillators’ interact via local forms coupling, here we assume that\nthese connections are long-ranged and can be modeled using a convolution operator. Since the\nlength scale of these patterns is small compared to the experimental set up, we pose the model\nequations on the whole plane.\nWithin this context, meaning spatially extended oscillatory media, past results on existence of\nspiral waves assume that coupling is well represented by the Laplace operator, even in the case of\nnonlocal coupling, see [30, 17, 18, 20, 42]. As mentioned above, this assumption then allows one to use\ntools from spatial dynamics to analyze these problems. While these techniques have been extended\nto study nonlocal neural field models [37, 9, 7], these results apply only to nonlocal operators that\nhave fractional Fourier symbols. The key idea is that in this case, the model equations can be\ntransformed into partial differential equations by preconditioning the system with an appropriate\ndifferential operator. Because this assumption is very restrictive, our goal for this paper has been to\ndevelop an alternative method based on functional analysis, which can be adapted to more general\nconvolution operators.\nOur efforts, summarized in Theorem 1, together with the work presented in reference [21], pro-\nvide the first rigorous proof for the existence of small amplitude spiral waves in oscillatory media\nwith nonlocal coupling. Theorem 1 also provides first order approximations for the amplitude and\nwavenumber of the pattern, and is the first work to rigorously establish a connection between the\nwavenumber, κ, and properties of the nonlocal coupling. In addition, notice that our expansion for\nκmatches the results obtained by Hagan [20] and Aguareles et al [2] in the limiting case when the\nnonlocal operator reduces to the Laplacian. More precisely, we find that the wavenumber is small\nbeyond all orders of the bifurcation parameter, a result that, due to the connection between the\nwavenumer and speed of the wave, also applies to the parameter Ω.\nRegarding our first order approximations, notice that because we are working in the weak coupling\nregime, spiral waves solutions to our nonlocal cGL equation (2) are characterized mainly by the\ncomplex phase ϕ. Thus, the viscous eikonal equation (7), which describes first order corrections\nfor this variable, plays a central role in our proof. Because the parameter Ω is an unknown, this\nequation can be interpreted as a nonlinear eigenvalue problem. The difficulty in solving this equation\nthen comes from the fact that Ω, as pointed out above, is small beyond all orders of the bifurcation\nparameter. As a result one cannot use regular perturbation techniques to find solutions, and instead\none has to find approximations to the pattern, both near the core and in the far field. This analysis\nwas done in [22], where it is shown that one can find the value of the parameter Ω, and thus solve\nthe nonlinear eigenvalue problem, by matching these two approximations.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 9\nThis split between near and far field approximations is also necessary at the level of the closing\nequations (i.e. the order O( δ3) system analyzed in Section 5). However, instead of finding and\nthen matching these approximations, as was the strategy in references [2] and [22], here we use a\nspecial class of doubly weighted Sobolev spaces, which are able to capture the behavior of solutions\nin these two regimes. In particular, we use spaces that encode algebraic decay or growth properties\nof functions, both at infinity and near the origin. With these spaces we are then able to 1) describe\nthe higher order correction terms to our spiral wave solution, 2) show that the closing equations\ndefine a bounded map, and 3) prove that the corresponding linear operators are invertible. As a\nresult, we are then able to prove existence of solutions using the implicit function theorem.\nAlthough the main goal of this paper was to show existence of spiral waves, a large portion of this\narticle is devoted to establishing Fredholm properties for elliptic operators using weighted Sobolev\nspaces. These results are new and interesting in and on themselves and, as shown in this article and\nin references [23, 24, 25, 22], they are the key ingredient to proving existence of patterned solutions\nin spatially extended systems via perturbation methods.\nFinally, let us point out that even though our assumptions on Knarrow down the type of nonlocal\ncoupling covered by our proof, this choice of convolution map does fall under the broader family of\noperators that are of diffusive type. We concentrate on kernels like those defined by Hypothesis [H2]\nfor simplicity of exposition. Using the approach given in the derivation of equation (6), the methods\ndeveloped here can be adapted to more general convolution operators, with Fourier symbols that\nare uniformly bounded, analytic, and that have a quadratic (or even higher order) tangency near\nthe origin. These operators naturally appear in neural field models, so it is possible to modify the\ntechniques presented here to these systems, provided one can write the modeling equations in the\nform (1). In particular this means assuming that the firing rate function is smooth (i.e. sigmoidal)\nand that the space-clamped system admits a Hopf bifurcation, see for instance [27] for an example\nof such a model. However, notice that although these assumptions are reasonable mathematically,\nit is not clear if the resulting system is physically relevant.\n2.Preliminaries\nIn this section we first introduce weighted and doubly-weighted Sobolev spaces. We then consider\nthe main linear operators appearing in our proofs of existence (Section 5) and establish their Fred-\nholm properties. In particular, in Subsection 2.2 we show the connection between their Fredholm\nindex and the type of weighted Sobolev spaces used to define their domain and range. On a first\nreading, one may skip Subsection 2.2 and refer back to it when needed.\n2.1.Sobolev Spaces. Throughout this section the letters dandsrepresent non-negative integers,\nwhile γandσare real numbers. To define the norm of our weighted Sobolev spaces we use the\nsymbols\n⟨x⟩= (1 + |x|2)1/2and m(|x|) =|x|(1−χ(x)),\nwhere χ(x)∈C∞(Rd) denotes a smooth radial cut-off function satisfying χ(x) = 0 for |x|<1 and\nχ(x) = 1 for |x|>2.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 10\nDefinition 2.1. We define the weighted space Hs\nγ(Rd)as the completion of C∞\n0(Rd,C)with respect\nto the norm\n∥u∥Hsγ(Rd)=X\n|α|≤s∥Dαu(x)⟨x⟩γ∥L2(Rd).\nIt is clear from this definition that Hs\nγ(Rd) is a Hilbert space. Its inner product is given by\n⟨f, g⟩=X\n|α|−1) at infinity, see Figure 4. Moreover, the embedding Hs\nγ1(Rd)⊂Hs\nγ2(Rd) holds,\nprovided γ1> γ 2, while Hs\nγ(Rd)⊂Hk\nγ(Rd) is valid whenever s > k . In terms of notation, when\ns= 0 we write L2\nγ(Rd) instead of H0\nγ(Rd).\nDefinition 2.2. We define the doubly-weighted space Hs\nγ,σ(Rd)as the completion of C∞\n0(Rd,C)with\nrespect to the norm\n∥u(x)∥Hsγ,σ(Rd)=X\n|α| γ2ands > k , respectively, while Hs\nγ,σ1(Rd)⊂Hs\nγ,σ2(Rd)\nis valid whenever σ1< σ2.\nAs before, functions in these doubly-weighted spaces have a level of growth or decay at infinity\nthat is controlled by the weight γ. However, in contrast to Hs\n��(Rd), functions in Hs\nγ,σ(Rd) are also\nallowed to grow near the origin, see Figure 4. In particular, elements in L2\nγ,σ(Rd) are allowed to\nhave a singularity at the origin of order O(1 /|x|α), with α∈(0, σ+d/2), see Lemma 2.6.\nIn addition to the above weighted Sobolev spaces, in this article we also use their restriction to\nradially symmetric functions. This is summarized in the following definition.\nDefinition 2.3. We denote by Hs\nr,γ(Rd)andHs\nr,γ,σ(Rd)the subspaces of Hs\nγ(Rd)andHs\nγ,σ(Rd),\nrespectively, consisting of radially symmetric functions.\nRemark 2.4. To avoid confusion, throughout the paper we always use γto denote the strength of\nthe weight ⟨x⟩, while the symbol σwill always represent the strength of the weight m(|x|). As a\nresult, we will use γto encode growth or decay properties of functions at infinity, while the value of\nσwill indicate the behavior of functions near the origin. In addition, in what follows we will often\napproximate m(|x|)∼ |x|=when defining the norm ∥ · ∥ L2γ,σ(B1). Here B1represents the unit ball\ninR2.\nNext, we present four lemmas summarizing properties of functions in Hk\nσ(R2) and in Hk\nγ,σ(R2).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 11\n0123-1-2-3AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Rj04jGCeUCyhN7JbDJmZnaZmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqeDG+v63t7K6tr6xWdgqbu/s7u2XDg4bJsk0ZXWaiES3IjRMcMXqllvBWqlmKCPBmtHwduo3n5g2PFEPdpSyUGJf8ZhTtE5qdPooJXZLZb/iz0CWSZCTMuSodUtfnV5CM8mUpQKNaQd+asMxasupYJNiJzMsRTrEPms7qlAyE45n107IqVN6JE60K2XJTP09MUZpzEhGrlOiHZhFbyr+57UzG1+HY67SzDJF54viTBCbkOnrpMc1o1aMHEGqubuV0AFqpNYFVHQhBIsvL5PGeSW4rAT3F+XqTR5HAY7hBM4ggCuowh3UoA4UHuEZXuHNS7wX7937mLeuePnMEfyB9/kDiOOPGw==\u0000\nAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzU8Prlilt15yCrxMtJBXLU++Wv3iBmaYTSMEG17npuYvyMKsOZwGmpl2pMKBvTIXYtlTRC7WfzQ6fkzCoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDGz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m5INwVt+eZW0LqreVdVrXFZqt3kcRTiBUzgHD66hBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBfJWMuw==1AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6rHoxWMF+wFtKJvtpl262Q27E6GE/AwvHhTx6q/x5r9x2+agrQ8GHu/NMDMvTAQ36HnfTmltfWNzq7xd2dnd2z+oHh61jUo1ZS2qhNLdkBgmuGQt5ChYN9GMxKFgnXByN/M7T0wbruQjThMWxGQkecQpQSv1+pEmNPPzTOeDas2re3O4q8QvSA0KNAfVr/5Q0TRmEqkgxvR8L8EgIxo5FSyv9FPDEkInZMR6lkoSMxNk85Nz98wqQzdS2pZEd67+nshIbMw0Dm1nTHBslr2Z+J/XSzG6CTIukxSZpItFUSpcVO7sf3fINaMoppYQqrm91aVjYlNAm1LFhuAvv7xK2hd1/6ruP1zWGrdFHGU4gVM4Bx+uoQH30IQWUFDwDK/w5qDz4rw7H4vWklPMHMMfOJ8/nHiReQ==1rAAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9kUUY9FLx4r2A9o15JNs21oNrsm2UJZ9nd48aCIV3+MN/+NabsHbX0w8Hhvhpl5fiy4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS0eJoqxJIxGpjk80E1yypuFGsE6sGAl9wdr++HbmtydMaR7JBzONmReSoeQBp8RYyesFitAUZ6l6rGX9csWtunOgVYJzUoEcjX75qzeIaBIyaaggWnexGxsvJcpwKlhW6iWaxYSOyZB1LZUkZNpL50dn6MwqAxREypY0aK7+nkhJqPU09G1nSMxIL3sz8T+vm5jg2ku5jBPDJF0sChKBTIRmCaABV4waMbWEUMXtrYiOiM3B2JxKNgS8/PIqadWq+LKK7y8q9Zs8jiKcwCmcA4YrqMMdNKAJFJ7gGV7hzZk4L86787FoLTj5zDH8gfP5A8nDkh0=1r2\nAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUUP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvquo1Liu12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A3xmM/A==rAAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9moqMeiF48V7Ae0a8mm2TY0m12TbKEs+zu8eFDEqz/Gm//GtN2Dtj4YeLw3w8w8PxZcG9f9dgorq2vrG8XN0tb2zu5eef+gqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dDv1W2OmNI/kg5nEzAvJQPKAU2Ks5HUDRWiKs1Q9nme9csWtujOgZYJzUoEc9V75q9uPaBIyaaggWnewGxsvJcpwKlhW6iaaxYSOyIB1LJUkZNpLZ0dn6MQqfRREypY0aKb+nkhJqPUk9G1nSMxQL3pT8T+vk5jg2ku5jBPDJJ0vChKBTISmCaA+V4waMbGEUMXtrYgOic3B2JxKNgS8+PIyaZ5V8WUV319Uajd5HEU4gmM4BQxXUIM7qEMDKDzBM7zCmzN2Xpx352PeWnDymUP4A+fzB8tIkh4=1r3AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9lIUY9FLx4r2A9o15JNs21oNrsm2UJZ9nd48aCIV3+MN/+NabsHbX0w8Hhvhpl5fiy4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS0eJoqxJIxGpjk80E1yypuFGsE6sGAl9wdr++HbmtydMaR7JBzONmReSoeQBp8RYyesFitAUZ6l6rGX9csWtunOgVYJzUoEcjX75qzeIaBIyaaggWnexGxsvJcpwKlhW6iWaxYSOyZB1LZUkZNpL50dn6MwqAxREypY0aK7+nkhJqPU09G1nSMxIL3sz8T+vm5jg2ku5jBPDJF0sChKBTIRmCaABV4waMbWEUMXtrYiOiM3B2JxKNgS8/PIqaV1U8WUV39cq9Zs8jiKcwCmcA4YrqMMdNKAJFJ7gGV7hzZk4L86787FoLTj5zDH8gfP5A8zNkh8=1r40123-1-2-3AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzU8Prlilt15yCrxMtJBXLU++Wv3iBmaYTSMEG17npuYvyMKsOZwGmpl2pMKBvTIXYtlTRC7WfzQ6fkzCoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDGz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m5INwVt+eZW0LqreVdVrXFZqt3kcRTiBUzgHD66hBvdQhyYwQHiGV3hzHp0X5935WLQWnHzmGP7A+fwBfJWMuw==1AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi6rHoxWMF+wFtKJvtpl262Q27E6GE/AwvHhTx6q/x5r9x2+agrQ8GHu/NMDMvTAQ36HnfTmltfWNzq7xd2dnd2z+oHh61jUo1ZS2qhNLdkBgmuGQt5ChYN9GMxKFgnXByN/M7T0wbruQjThMWxGQkecQpQSv1+pEmNPPzTOeDas2re3O4q8QvSA0KNAfVr/5Q0TRmEqkgxvR8L8EgIxo5FSyv9FPDEkInZMR6lkoSMxNk85Nz98wqQzdS2pZEd67+nshIbMw0Dm1nTHBslr2Z+J/XSzG6CTIukxSZpItFUSpcVO7sf3fINaMoppYQqrm91aVjYlNAm1LFhuAvv7xK2hd1/6ruP1zWGrdFHGU4gVM4Bx+uoQH30IQWUFDwDK/w5qDz4rw7H4vWklPMHMMfOJ8/nHiReQ==1rAAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9kUUY9FLx4r2A9o15JNs21oNrsm2UJZ9nd48aCIV3+MN/+NabsHbX0w8Hhvhpl5fiy4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS0eJoqxJIxGpjk80E1yypuFGsE6sGAl9wdr++HbmtydMaR7JBzONmReSoeQBp8RYyesFitAUZ6l6rGX9csWtunOgVYJzUoEcjX75qzeIaBIyaaggWnexGxsvJcpwKlhW6iWaxYSOyZB1LZUkZNpL50dn6MwqAxREypY0aK7+nkhJqPU09G1nSMxIL3sz8T+vm5jg2ku5jBPDJF0sChKBTIRmCaABV4waMbWEUMXtrYiOiM3B2JxKNgS8/PIqadWq+LKK7y8q9Zs8jiKcwCmcA4YrqMMdNKAJFJ7gGV7hzZk4L86787FoLTj5zDH8gfP5A8nDkh0=1r2\nAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHssHM0nQj+hQ8pAzaqzUUP1yxa26c5BV4uWkAjnq/fJXbxCzNEJpmKBadz03MX5GleFM4LTUSzUmlI3pELuWShqh9rP5oVNyZpUBCWNlSxoyV39PZDTSehIFtjOiZqSXvZn4n9dNTXjjZ1wmqUHJFovCVBATk9nXZMAVMiMmllCmuL2VsBFVlBmbTcmG4C2/vEpaF1Xvquo1Liu12zyOIpzAKZyDB9dQg3uoQxMYIDzDK7w5j86L8+58LFoLTj5zDH/gfP4A3xmM/A==rAAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9moqMeiF48V7Ae0a8mm2TY0m12TbKEs+zu8eFDEqz/Gm//GtN2Dtj4YeLw3w8w8PxZcG9f9dgorq2vrG8XN0tb2zu5eef+gqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dDv1W2OmNI/kg5nEzAvJQPKAU2Ks5HUDRWiKs1Q9nme9csWtujOgZYJzUoEc9V75q9uPaBIyaaggWnewGxsvJcpwKlhW6iaaxYSOyIB1LJUkZNpLZ0dn6MQqfRREypY0aKb+nkhJqPUk9G1nSMxQL3pT8T+vk5jg2ku5jBPDJJ0vChKBTISmCaA+V4waMbGEUMXtrYgOic3B2JxKNgS8+PIyaZ5V8WUV319Uajd5HEU4gmM4BQxXUIM7qEMDKDzBM7zCmzN2Xpx352PeWnDymUP4A+fzB8tIkh4=1r3\nAAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKRI9BLx4jmAckS5idzCZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNaJMornQnwoZyJmnTMstpJ9EUi4jTdjS+nfntJ6oNU/LBThIaCjyULGYEWye1eoYNBe6XK37VnwOtkiAnFcjR6Je/egNFUkGlJRwb0w38xIYZ1pYRTqelXmpogskYD2nXUYkFNWE2v3aKzpwyQLHSrqRFc/X3RIaFMRMRuU6B7cgsezPxP6+b2vg6zJhMUkslWSyKU46sQrPX0YBpSiyfOIKJZu5WREZYY2JdQCUXQrD88ippXVSDWjW4v6zUb/I4inACp3AOAVxBHe6gAU0g8AjP8ApvnvJevHfvY9Fa8PKZY/gD7/MHnlePKQ==\u0000\nAAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkR9Vj04rGi/YA2ls120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKwecJxwP6IDJULBKFrpXj9We6WyW3FnIMvEy0kZctR7pa9uP2ZpxBUySY3peG6CfkY1Cib5pNhNDU8oG9EB71iqaMSNn81OnZBTq/RJGGtbCslM/T2R0ciYcRTYzoji0Cx6U/E/r5NieOVnQiUpcsXmi8JUEozJ9G/SF5ozlGNLKNPC3krYkGrK0KZTtCF4iy8vk2a14l1UvLvzcu06j6MAx3ACZ+DBJdTgFurQAAYDeIZXeHOk8+K8Ox/z1hUnnzmCP3A+fwAEg42gr2\nFigure 4. Examples of algebraic decay/growth for weighted, L2\nγ(R2), and doubly-\nweighted, L2\nγ,σ(R2), Sobolev spaces. Parameter γencodes decay/growth properties\nof functions at infinity, while parameter σencodes decay/growth rates of functions\nnear the origin.\nLemma 2.5. Letγ∈Randd∈N. A function fis in L2\nγ(Rd)if and only if there is a number\nα <−γ−d/2and a positive constant C, such that for a.e. x∈Rd\n|f(x)| ≤C|x|αasx→ ∞ .\nProof. Letf:Rd−→C, and suppose there is a constant C, such that |f(x)|< C|x|α, with αas\nstated in the Lemma. We may then write\n∥f∥L2γ(Rd)=Z\nRd|f(x)|2⟨x⟩2γdx≤CZ∞\n1r2αr2γrd−1dr.\nSince this last integral is finite when α <−γ−d/2, it then follows that f∈L2\nγ(R2).\nTo prove the second implication we use its contrapositive. To that end, suppose now that f(x)\nis such that |f(x)|>|x|βwith β=−γ−d/2 +ε, and that this holds for all ε >0. Then, there is a\nconstant c >0 such that\n∞=Z∞\n1r2βr2γrd−1dr < cZ\nRd|f(x)|2⟨x⟩2γdx=c∥f∥L2γ(Rd),\nConsequently, fis not in L2\nγ(Rd), and the results of the lemma then follow. □\nA similar argument as in the proof of the last lemma gives us decay properties near the origin for\nelements in the space L2\nγ,σ(R2). This result is then summarized in Lemma 2.6. Examples illustrating\nthe results of Lemma 2.5 and Lemma 2.6, for the case of functions defined in R2, are also summarized\nin Figure 4.\nLemma 2.6. Letγ, σ∈R, and d∈N. A function fis in L2\nγ,σ(R2)if and only if there is a number\nα >−σ−d/2and a positive constant C, such that for a.e. x∈Rd,\n|f(x)| ≤C|x|αas|x| →0.\nThe next lemma gives assumptions under which the space Hk\nγ,σ(R2) is a Banach algebra.\nLemma 2.7. Letkbe an integer, and take γ, σ∈Rwithγ >−1andσ <−1. Then Hk\nγ,σ(R2)is a\nBanach algebra.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 12\nProof. Suppose that fandgare in Hk\nγ,σ(R2). Since γ >−1, from Lemma 2.5 we know that functions\nin this space decay at infinity. Therefore, the inequality |fg|< C|f|holds for large values of |x|\nand some C > 0. As a result, the product fg∈L2\nγ,σ(R2\\B1), where B1⊂R2represents the unit\nball. A similar argument then shows that Dα(fg) is also in L2\nγ,σ+|α|(R2\\B1), for all indices αwith\n|α| ≤k.\nOn the other hand, to show that expression fgsatisfies the desired properties near the origin, we\nlook at the derivative Ds(fg). Taking s∈Z∩[0, k] and letting Dsfany partial derivative of order\ns, we obtain\nDs(fg) =sX\nℓ=0\u0012s\nℓ\u0013\nDℓfDs−ℓg.\nFrom the definition of the spaces Hk\nγ,σ(R2) and Lemma 2.6 we know that near the origin,\n|Dℓf| ≤C|x|α1α1>−(σ+ℓ+ 1),\n|Ds−ℓg| ≤C|x|α2α2>−(σ+s−ℓ+ 1),\nand as a result,\n∥DℓfDs−ℓg∥L2\nγ,σ+s(B1)≤Z1\n0r2(α1+α2)r2(σ+s)r dr.\nSince σ <−1, the inequality 2( α1+α2) + 2( σ+s) + 2 >0 is satisfied. Consequently, this last\nintegral is finite and we obtain Ds(fg)∈L2\nγ,σ+s(R2), as desired. The results of the Lemma then\nfollow. □\nLemma 2.8. Letk, s∈N∪ {0}satisfying s≤k, take γ1, γ2>−1and pick σ1, σ2>−1. Suppose\nthatf∈Hk\nγ1,σ1(R2)andg∈Hs\nγ2,σ2(R2), then the product\nfg∈Hs\nγ,σ(R2)\nwhere γ= min( γ1, γ2)andσ=σ1+σ2+ 1.\nProof. As in the previous lemma, since for i∈ {1,2}we have that γi<−1, both fandgdecay at\ninfinity. If, without loss of generality, we assume γ1< γ2, then the inequality |fg|<|f|holds, and\nwe conclude that the product fg∈L2\nγ1,σ(R2\\B1), where σis arbitrary for now. A similar reasoning\nalso shows that Dα(fg)∈L2\nγ1,σ+|α|(R2\\B1).\nOn the other hand, if we now take p∈Z∩[0, s] and let Dpfdenote any partial derivative of\norder p, we may write\nDp(fg) =pX\nℓ=0\u0012p\nℓ\u0013\nDℓfDp−ℓg.\nUsing Lemma 2.6 we again find that for |x| ∼0,\n|Dℓf| ≤C|x|α1α1>−(σ1+ℓ+ 1),\n|Ds−ℓg| ≤C|x|α2α2>−(σ2+s−ℓ+ 1).\nLetting σ=σ1+σ2+ 1, the above expressions, along with a similar reasoning as in the proof of\nLemma 2.5, then imply that the norm |Ds(fg)∥L2\nγ,σ+s(B1)is finite. This follows since the inequality\n2(α1+α2) + 2( σ+s) + 2 >0 is satisfied. □EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 13\n2.2.Fredholm Operators. In this subsection we prove Fredholm properties for the various dif-\nferential operators that appear in this article. Recall that a linear operator, L, is Fredholm if it\nhas closed range, finite dimensional kernel, Ker( L), and finite dimensional co-kernel, coKer( L). Its\nindex is then given by i= dim (Ker( L))−dim (coKer( L)).\nWe start with the operator ∆ −Id, which is known to be invertible from Hs(R2) into Hs−2(R2).\nLemma 2.9, shown below, extends this result to the weighted Sobolev spaces defined in Subsection\n2.1. A proof of this result can be found in [25], but for convenience we reproduce it in Appendix B.\nLemma 2.9. Lets≥2and suppose γ∈R. Then, the operator\n∆−Id :Hs\nγ(R2)−→Hs−2\nγ(R2)\nis invertible.\nIn the following corollary we view the Hilbert spaces Hs\nγ(R2) as direct sums given by,\nHs\nγ(R2) =⊕nhs\nn,γ,\nwhere n∈Zand the spaces hs\nn,γare defined as\nhs\nn,γ={u∈Hs\nγ(R2) :u(r, θ) =un(r)einθ, u n(r)∈Hs\nr,γ(R2)},\nHeuristically, the above decomposition comes from a change of coordinates into polar coordinates\nand a Fourier series decomposition of uin the θvariable. For a complete description of this for-\nmulation, see [45]. Recall also that we have assumed our weighted Sobolev spaces are composed of\ncomplex-valued functions.\nCorollary 2.10. Letn∈N∪ {0}, and take γ∈R. Then, the operator\n∆n−Id :Hs\nr,γ(R2)−→Hs−2\nr,γ(R2)\nwith ∆n=∂rr+1\nr∂r−n2\nr2is invertible.\nProof. The result follows from the fact that the Laplacian commutes with rotations, so that if we\nview Hs\nγ(R2) =⊕nhs\nn,γ, then the invertible operator ∆ −Id :Hs\nγ(R2)−→Hs−2\nγ(R2) is a diagonal\noperator in this last space. That is,\n(∆−Id)u=f\nX\nn(∆n−Id)uneinθ=X\nnfneinθ,\nwhere by the definition of hs\nn,γ, the functions un(r)∈Hs\nr,γ(R2) and fn(r)∈Hs−2\nr,γ(R2). It then\nfollows that each operator ∆ n−Id :Hs\nr,γ(R2)−→Hs−2\nr,γ(R2) is invertible. □\nIn the next two lemmas we establish Fredholm properties for the radial operator L=∂r+1\nr−λ\nwith λ >0. At this stage it is convenient to restrict attention to real-valued functions. So in the\nrest of this section we assume that Hs\nr,γ(R2) =Hs\nr,γ(R2,R) and Hs\nr,γ,σ(R2) =Hs\nr,γ,σ(R2,R).\nLemma 2.11. Letγbe a real number, san integer satisfying s≥1, and take λ >0. Then, the\noperator L:Hs\nr,γ(R2)−→Hs−1\nr,γ(R2), defined by\nLu=∂ru+1\nru−λu,\nis Fredholm with index i=−1and cokernel spanned by e−λr.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 14\nProof. We first show the results of the lemma for the case when s= 1.\nWe start by proving that the operator L:H1\nr,γ(R2)−→L2\nr,γ(R2) has a trivial nullspace. Indeed,\none can check that the function u0=1\nreλris the only function that satisfies Lu0= 0. However,\nbecause of its singularity at the origin and its exponential growth, this function does not belong to\nthe space H1\nr,γ(R2) for any weight γ∈R.\nA short calculation using integration by parts and the pairing between elements in L2\nr,γ(R2) and\nits dual, L2\nr,−γ(R2), shows that the adjoint of Lis the operator L∗:L2\nr,−γ(R2)−→H−1\nr,−γ(R2), given\nbyL∗u=−(∂ru+λu). The kernel of this operator is then spanned by e−λr, which is an element of\nthe space L2\nr,−γ(R2) for all γ∈R. As a result the cokernel of Lis spanned by e−λrand therefore\nhas dimension one.\nTo prove the results of the lemma, we are left with showing that the range of the operator\nL:H1\nr,γ(R2)−→L2\nr,γ(R2) is given by\nR={u∈L2\nr,γ(R2)|Z∞\n0u(r)e−λrr dr= 0}.\nNotice that because the function e−λris an element of L2\nr,−γ(R2), it defines a bounded linear func-\ntional, ℓ:L2\nr,γ(R2)−→R, via the integral\nℓ(u) =Z∞\n0u(r)e−λrr dr.\nSince the space Rcorresponds to the nullspace of ℓ, we immediately obtain that it is a closed\nsubspace of L2\nr,γ(R2).\nTo prove the above claim, we show that the inverse of L, defined by the operator\nR−→ H1\nr,γ(R2)\nf(r)7→ u(r) =1\nrRr\n∞e−λ(s−r)f(s)s ds\nis bounded. We start by proving the bound ∥u∥L2r,γ(R2)≤ ∥f∥L2r,γ(R2)using the inequality\n∥u∥L2r,γ(R2)≤ ∥u∥L2r,γ(B1)+∥u∥L2r,γ(R2\\B1),\nwhere B1is the unit ball in R2. First, notice that\n∥u∥L2r,γ(R2\\B1)=\"Z∞\n1\f\f\f\fZr\n∞f(s)e−λ(s−r)s\nrds\f\f\f\f2\n⟨r⟩2γr dr#1/2\n≤2|γ−1|\"Z∞\n1\f\f\f\fZr\n∞f(s)⟨s⟩γ−1se−λ(s−r)⟨s−r⟩|γ−1|r1/2ds\f\f\f\f2\ndr#1/2\n,\n≤2|γ−1|\"Z∞\n1\f\f\f\fZ0\n∞|f(z+r)|⟨z+r⟩γ−1(z+r) e−λz⟨z⟩|γ−1|r1/2dz\f\f\f\f2\ndr#1/2\n≤2|γ−1|Z∞\n0e−λz⟨z⟩|γ−1|\u0014Z∞\n1|f(z+r)|2⟨z+r⟩2γr dr\u00151/2\ndz\n≤C1(λ, γ)∥f∥L2r,γ(R2),\nwhere the second line follows from the relation ⟨s⟩−η⟨r⟩η≤2|η|⟨s−r⟩|η|, and the approximation\n⟨r⟩2γ/r2∼ ⟨r⟩2(γ−1), which is valid given that r >1. The inequality on the third line comes from theEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 15\nchange of variables z=s−r, while the fourth line follows from Minkowski’s inequality for integrals\n[13, Theorem 6.19]. In the last line, we let C1(γ, λ) = 2|γ−1|R∞\n0e−λz⟨z⟩|γ−1|dz.\nNext, we use the solvability conditionR∞\n0f(r)e−λrr dr= 0 to write the inverse as\nu(r) =1\nrZr\n0e−λ(s−r)f(s)s ds\nand bound the norm in B1as follows:\n∥u∥L2r,γ(B1)=\"Z1\n0\f\f\f\f1\nrZr\n0e−λ(s−r)f(s)s ds\f\f\f\f2\n⟨r⟩2γr dr#1/2\n≤\"Z1\n0\f\f\f\f1\nrZr\n0eλrf(s)s ds\f\f\f\f2\n⟨r⟩2γr dr#1/2\n≤Z1\n0\u0014Z1\ns|f(s)|2s2e2λr⟨r⟩2γ1\nrdr\u00151/2\nds\n≤Z1\n0eλ2γ\u0014Z1\ns1\nrdr\u00151/2\n|f(s)|s ds\n≤2γeλZ1\n0[−log(s)]1/2|f(s)|s ds\n≤2γeλ∥f∥L2r(B1)\u0012Z1\n0|log(s)|s ds\u00131/2\n≤C2(λ, γ)∥f∥L2r,γ(B1),\nIn this case, the second line comes from the inequality 0 <−(s−r)< r, the third line follows from\napplying Minkowski’s inequality for integrals, and the second to last line from H¨ older’s inequality.\nIn the last line, we also let C2(λ, γ) = 2γ−1eλ. The result is,\n∥u∥L2r,γ(R2)≤(C1(λ, γ) +C2(λ, γ))∥f∥L2r,γ(R2).\nOur next step is to show that ∥∂ru∥L2r,γ(R2)≤ ∥f∥L2r,γ(R2)using again the inequality\n∥∂ru∥L2r,γ(R2)≤ ∥∂ru∥L2r,γ(B1)+∥∂ru∥L2r,γ(R2\\B1).\nFrom the relation ∂ru+1\nru−λu=f, it is easy to see that the second term on the right hand side\nof the inequality satisfies,\n∥∂ru∥L2r,γ(R2\\B1)≤[(1 + λ)(C1(λ, γ) +C2(λ, γ) + 1]∥f∥L2r,γ(R2).\nTo bound the term ∥∂ru∥L2r,γ(B1), we first observe that\n|u(r)|\nr≤1\nr2Zr\n0|f(s)|s ds≤1\nr2Zr\n0|f(s)−f(y)|s ds+1\n2|f(y)|,\nwhere y∈[0, r]. Letting B(y,2r) denote the ball of radius 2 rcentered at y, and defining ˜B(y,2r) =\nB(y,2r)∩B(0, r)⊂R2, the above inequality can also be written as,\n|u(r)|\nr≤|˜B(y,2r)|\n2πr2 \n1\n|˜B(y,2r)|Z\n˜B(y,2r)|f(s)−f(y)|s ds dθ!\n+1\n2|f(y)|. (8)EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 16\nwith|˜B(y,2r)|representing the measure of the set ˜B(y,2r). Because of the embedding L2(B(0, M))⊂\nL1(B(0, M)), which holds for any bounded ball B(0, M)⊂R2, the function fis in L1\nloc(B(0, M)).\nAs a result, by the Lebesgue Differentiation Theorem [13, Theorem 3.21], the expression in paren-\nthesis approaches zero as r→0, while the fraction in front remains bounded, since |˜B(y,2r)| ≥\n2πr2. Thus, near the origin, the function |u(r)|/ris bounded above by |f(r)|. From the equation\n∂ru=f−λu−u/r, it then follows that\n∥∂ru∥L2r,γ(B1)≤(2 +λC1(λ, γ))∥f∥L2r,γ(R2).\nConsequently, for f∈R, the solution to ( ∂r+1\nr−λ)u=fsatisfies,\n∥∂ru∥L2r,γ(R2)≤(3 + 2(1 + λ)(C1(γ, λ) +C2(γ)))∥f∥L2r,γ(R2).\nTo prove that the general operator ∂r+1\nr−λ:Hs\nr,γ(R2)−→Hs−1\nr,γ(R2) is Fredholm index i=−1,\none can proceed by induction: Assuming that fanduare in Hs−1\nr,γ(R2) one shows that ∂s\nruis in\nL2\nr,γ(R2) using the relation ∂s\nru=∂s−1\nr(f+λu−u\nr). The fact that ∂s−1\nr\u0010u\nr\u0011\nis in the correct space\nfollows by a similar argument as the one done above to prove u/r∈L2\nr,γ(R2). □\nLemma 2.12. Letγ, σbe real numbers, sand integer satisfying s≥1, and take λ >0. Then, the\noperator L:Hs\nr,γ,σ(R2)−→Hs−1\nr,γ,σ +1(R2), defined by\nLu=∂ru+1\nru−λu,\nis Fredholm. Moreover,\ni) ifσ <0, it has index i=−1. It is injective and its cokernel is spanned by e−λr;\nii) if σ >0, it has index i= 0and it is invertible.\nProof. We first concentrate in the case when s= 1. As in Lemma 2.11, the kernel and cokernel\nof the operator are spanned by eλr/rande−λr, respectively. Because the function eλr/rgrows\nexponentially, it does not belong to the domain of the operator no matter what the values of σand\nγare. Therefore the kernel of the operator is trivial. On the other hand, the function e−λrbelongs\ntoL2\nr,−γ,−(σ+1)(R2) and thus, it is in the co-kernel of the operator provided σ <0, see Lemma 2.6\nand Figure 4.\nIn case i) the results of this lemma follow a similar argument as the ones used to prove Lemma\n2.11 above. The main difference comes from bounding the solution unear the origin. Therefore, we\nconcentrate only on proving this result.\nSuppose then that f∈L2\nr,γ,σ +1(R2) with σ <0. We first want to establish the bound ∥u∥L2r,γ,σ(B1)≤\n∥f∥L2\nr,γ,σ +1(R2). Using the solvability condition, we may write the inverse as\nu(r) =1\nrZr\n0e−λ(s−r)f(s)s ds.\nThe expression in (8) then shows that the function |u(r)|/ris bounded above by |f(r)|provided\nf(r)∈L1(B1). One can then show that these last condition is satisfied if σ <0.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 17\nNext, we show that the derivative, ∂ru, satisfies the bound ∥∂ru∥L2\nr,γ,σ +1(B1)≤ ∥f∥L2\nr,γ,σ +1(R2).\nUsing the equation we may write ∂ru=f−1\nru+λu, to obtain,\n|∂ru| ≤ |f|+1\nr|u|+λ|u|\n∥∂ru∥L2\nr,γ,σ +1(B1)≤ ∥f∥L2\nr,γ,σ +1(B1)+∥u∥L2r,γ,σ(B1)+∥u∥L2\nr,γ,σ +1(B1)\n≤(2 +λ)∥f∥L2\nr,γ,σ +1(R2),\nwhere in the last line we used the embedding L2\nr,γ,σ(B1)⊂L2\nr,γ,σ +1(B1) and the previous bound for\nthe norm of u.\nIn case ii) we no longer have a solvability condition, and the inverse is given by\nu(r) =1\nrZr\n∞e−λ(s−r)f(s)s ds,\nAs in the previous case, the same arguments as in Lemma 2.11 give us the bounds for ∥u∥L2r,γ,σ(R2\\B1).\nTo bound the solution near the origin we take a different approach. Letting g(r) = e−λsf(s)sand\nwriting ∂rv=g, our goal is to show that ∥v∥L2\nr,γ,σ−1(B1)≤ ∥g∥L2r,γ,σ(R2). Then,\n∥u∥L2r,γ,σ(B1)≤ ∥v/r∥L2r,γ,σ(B1)=∥v∥L2\nr,γ,σ−1(B1)≤ ∥g∥L2r,γ,σ(R2)≤ ∥f∥L2\nr,γ,σ +1(R2),\nas desired.\nConsider then the change of variables τ= lnrforr∈(0,1) and define\nw(τ) =v(eτ)e−βτh(τ) =g(eτ)e−βτeτ,\nwith β=−σ < 0. Then, ∂τ(w(τ)eβτ) =h(τ)eβτ, and a straightforward computation using the\nabove change of variables shows that ∥v∥L2\nr,γ,σ−1(B1)=∥w∥L2(−∞,0). In addition,\nw(τ) =Zτ\n∞h(s)eβ(s−τ)ds.\nTo bound the L2norm of w, we can then write\n∥w∥2\nL2(−∞,0)=Z0\n−∞\u0014Zτ\n∞h(s)eβ(s−τ)ds\u00152\ndτ\n≤Z0\n−∞\u0014Z0\n∞h(z+τ)eβzdz\u00152\ndτ\n∥w∥L2(−∞,0)≤Z∞\n0\u0014Z0\n−∞|h(z+τ)|2e2βzdτ\u00151/2\ndz\n≤Z∞\n0eβz∥h∥L2(R2)dz\n≤C(β)∥h∥L2(R2).\nHere, the second line comes from the change of coordinates z=s+τ, the third line follows from\nMinkowski’s inequality for integrals [13, Theorem 6.19], and the fourth line is a consequence β=\n−σ <0.\nTo bound ∥h∥L2(R)notice that\n∥h∥L2(R)=Z∞\n−∞|g(eτ)|2e−2βτe2τdτ≤Z∞\n0e−2λr|f(r)|2r2σ+2r dr≤C(λ, γ)∥f∥L2\nr,γ,σ +1(R2),EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 18\nz→0 z→ ∞\nKν(z) ∼1\n2Γ(ν)\u0002z\n2\u0003−ν∼pπ\n2ze−z\nIν(z) ∼1\nΓ(ν+1)\u0002z\n2\u0003ν∼q\n1\n2zπez\nd\ndzZν(z) = Zν+1(z) +ν\nzZν(z)\nTable 1. Asymptotic behavior for the first-order Modified Bessel functions of the\nfirst and second kind, and their derivatives, taken from [1, (9.6.8), (9.6.9), (9.7.2)].\nHere, the integer ν≥1, Γ( ν) is the Euler-Gamma function, and Zν=\n{Iν(z), eiπνKν(z)}.\nand we may indeed conclude that ∥u∥L2r,γ,σ(B1)≤ ∥f∥L2\nr,γ,σ +1(R2). To bound the derivative ∂ru, one\nfollows the approach used in case i).\nFinally, as in Lemma 2.11, to prove that the general operator ∂r+1\nr−λ:Hs\nr,γ,σ(R2)−→\nHs−1\nr,γ,σ +1(R2) is Fredholm index i=−1, one can proceed by induction. □\nIn the Section 3 we find that the normal form describing the evolution of one-armed spiral waves\ninvolves the linear radial operator (∆ 1−Id)u=∂rru+1\nr∂ru−1\nr2u−u. Corollary 2.10 showed that the\noperator is invertible when defined over the weighted spaces Hs\nr,γ(R2). The following proposition\nshows that this operator is also invertible over the double weighted Sobolev spaces Hs\nr,γ,σ(R2),\nprovided the weights σare chosen appropriately.\nProposition 2.13. Letγ∈R,σ∈(−2,0)and take s≥2to be an integer. Then, the operator\n∆1−Id :Hs\nr,γ,σ(R2)−→Hs−2\nr,γ,σ +2(R2), defined by\n(∆1−Id)u=∂rru+1\nr∂ru−1\nr2u−u,\nis invertible.\nProof. The proof of this proposition is carried out in a number of steps. In Step 1 we first determine\nthe elements in the kernel and cokernel of the operator. In Step 2 we find the Green’s function\nfor the operator. Finally, in Steps 3 and 4 we show that for values of σ∈(−2,0), and for s= 2,\nthe operator has a bounded inverse from Hs−2\nr,γ,σ(R2) into Hs\nr,γ,σ(R2). In Step 5 we use an induction\nargument to prove the results for integer values s >2.\nStep 1: Notice first that the equation (∆ 1−Id)u= 0, represents the modified Bessel equation\nof order one. It has as solutions the Modified Bessel functions K1(r), I1(r), which satisfy the de-\ncay/growth properties summarized in Table 1. Since I1(r) grows exponentially, this function is not\nin any of the spaces H2\nr,γ,σ(R2). On the other hand, since K1(r)∼O(1/r) near the origin, a short\ncalculations shows that this function belongs to H2\nr,γ,σ(R2) provided σ >0. See also Lemma 2.6 and\nFigure 4.\nUsing integration by parts, one can check that the adjoint of the operator is given by\n∆1−Id :L2\nr,−γ,−(σ+2)(R2)−→ H−2\nr,−γ,−σ(R2)\nIt then follows that only K1(r) is an element of the co-kernel, and this holds for values of σ <−2.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 19\nStep 2: The Green’s function, G(r, ρ), for the operator ∆ 1−Id satisfying G(r, ρ)→0 asr→ ∞ ,\nandG(0, ρ) = 0 on the interval [0 ,∞) is\nG(r, ρ) =(I1(r)K1(ρ)\nW(r)for 0 < r < ρ,\nK1(r)I1(ρ)\nW(r)forρ < r < ∞.\nHere, the function Wrepresents the Wronskian W(r) =I′\n1(r)K1(r)−I1(r)K′\n1(r) =1\nr. It then\nfollows that the formal inverse of the operator ∆ 1−Id is given by\nL2\nr,γ,(σ+2)(R2)−→ H2\nr,γ,σ(R2)\nf(r) 7→ u(r) =R∞\n0G(r;ρ)f(ρ)dρ.\nMore precisely,\nu(r) =I1(r)Z∞\nrK1(ρ)f(ρ)ρ dρ+K1(r)Zr\n0I1(ρ)f(ρ)ρ dρ. (9)\nIn Steps 3 and 4, we use expression (9) to show that the bound ∥u∥H2r,γ,σ(R2)≤ ∥f∥L2\nr,γ,σ +2(R2)\nholds for −2< σ < 0. Throughout, we use the inequality\n∥ · ∥ L2γ,σ(R2)≤ ∥ · ∥ L2γ,σ(B1)+∥ · ∥ L2γ,σ(R2\\B1).\nwhere B1represents the unit ball centered at the origin. In this case, ∥·∥L2γ,σ(R2\\B1)=∥·∥L2γ(R2\\B1).\nStep 3: We start by bounding the L2\nr,γ,σ(R2\\B1) norm of u, ∂ru,and∂rru. Using expression\n(9) we have that\n∥u∥2\nL2γ,σ(R2\\B1)≤A+B\nwhere\nA=Z∞\n1\u0014\nI1(r)Z∞\nrK1(ρ)f(ρ)ρ dρ\u00152\n⟨r⟩2γrdr\nB=Z∞\n1\u0014\nK1(r)Zr\n0I1(ρ)f(ρ)ρ dρ\u00152\n⟨r⟩2γrdr\nSince 1 < r < ρ , we can use the decay/growth properties of I1(r) and K1(ρ), asr, ρ→ ∞ , and write\nA≤CZ∞\n1\"Z∞\nre−(ρ−r)\n√r√ρf(ρ)ρ dρ#2\n⟨r⟩2γrdr\n≤CZ∞\n1\u0014Z∞\nre−(ρ−r)f(ρ)⟨r−ρ⟩|γ|⟨ρ⟩γρ1/2dρ\u00152\ndr\n≤CZ∞\n1\u0014Z∞\n0e−zf(z+r)⟨z⟩|γ|⟨z+r⟩γ(z+r)1/2dz\u00152\ndr,\nwhere the second line follows from the inequality ⟨r⟩γ⟨ρ⟩−γ≤ ⟨r−ρ⟩|γ|, and the third line comes\nfrom the change of coordinates z=ρ−r. Applying Minkowski’s inequality for integrals, we then\nobtain\nA1/2≤CZ∞\n0\u0014Z∞\n1e−2z|f(z+r)|2⟨z⟩2|γ|⟨z+r⟩2γ(z+r)dr\u00151/2\ndz\n≤CZ∞\n0e−z⟨z⟩|z|∥f∥L2r,γ(R2\\B1)dz\n≤C(γ)∥f∥L2\nr,γ,σ +2(R2).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 20\nA similar analysis also shows that B1/2≤C(γ)∥f∥L2\nr,γ,σ +2(R2).It therefore follows that\n∥u∥L2r,γ,σ(R2\\B1)≤C(γ)∥f∥L2\nr,γ,σ +2(R2).\nTo bound ∥∂ru∥L2\nr,γ,σ +1(R2\\B1), we first compute this derivative using expression (9):\n∂ru=I′\n1(r)Z∞\nrK1(ρ)f(ρ)ρ dρ+K′\n1(r)Zr\n0I1(ρ)f(ρ)ρ dρ. (10)\nSince I′\n1(r) and K′\n1(r) have the same growth/decay properties as I1(r) and K1(r), respectively, the\nsame computations as above show that\n∥∂ru∥L2r,γ,σ(R2\\B1)≤C(γ)∥f∥L2\nr,γ,σ +2(R2).\nTo bound the second derivative ∂rruin the same norm, we use the equation and write\n∂rru=f−1\nr∂ru+1\nr2u+u,\narriving at the desired result,\n∥∂rru∥L2r,γ,σ(R2\\B1)≤4C(γ)∥f∥L2\nr,γ,σ +2(R2).\nStep 4: To complete the proof of the proposition, we need to now bound the norms ∥u∥L2r,γ,σ(B1),\n���∂ru∥L2\nr,γ,σ +1(B1), and∥∂rru∥L2\nr,γ,σ +2(B1).\nWe start with the bound ∥u∥L2r,γ,σ(B1)≤ ∥f∥L2\nr,γ,σ +2(R2). Recalling the expression for u, along\nwith the decay properties for I1(r) and K1(r) near the origin, one can approximate the first integral\nin (9) by\nA(r) =I1(r)Z∞\nrK1(ρ)f(ρ)ρ dρ≤CrZ∞\nrK1(ρ)f(ρ)ρ dρ.\nWith g(ρ) =K1(ρ)f(ρ)ρ, in what follows we show that v(r) =R∞\nrg(ρ)dρsatisfies ∥v∥L2\nr,γ,σ +1(B1)≤\nC∥f∥L2\nr,γ,σ +2(R2)for some generic positive constant C. As a result\n∥A∥L2r,γ,σ(B1)≤C∥rv(r)∥L2r,γ,σ(B1)≤C∥v(r)∥L2\nr,γ,σ +1(B1)≤C∥f(r)∥L2\nr,γ,σ +2(R2). (11)\nRecalling that ∂rv=g, letting τ= lnrwith r∈(0,1), and defining\nw(τ) =v(eτ)e−βτh(τ) =g(eτ)e−βτeτ\nwe find that ∂τ(w(τ)eβτ) =h(τ)eβτ. Taking β=−(σ+ 2)<0, a short calculation shows that\nw(τ) =Zτ\n∞h(s)eβ(s−τ)ds\nand that ∥w∥L2(−∞,0)=∥v∥L2\nr,γ,σ +1(B1). We can then bound\n∥w∥2\nL2(−∞,0)=Z0\n−∞\u0014Zτ\n∞h(s)eβ(s−τ)ds\u00152\ndτ\n≤Z0\n−∞\u0014\n−Z∞\n0h(z+τ)eβzdz\u00152\ndτ\n∥w∥L2(−∞,0)≤Z∞\n0eβz\u0014Z0\n−∞|h(z+τ)|2dτ\u00151/2\ndz\n≤C(β)∥h∥L2(R)EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 21\nwhere the second inequality comes from the change of coordinates z=s−τ, the third line is a\nconsequence of Minkowski inequality for integrals [ ?], and the last line follows from our assumption\nβ=−(σ+ 2)<0.\nLastly, notice that\n∥h∥L2(R)=Z∞\n−∞|h(τ)|2dτ\n=Z∞\n−∞|g(eτ)|2e−2βτe2τdτ\n≤Z1\n0|f(r)|2r2(σ+2)r dr+Z∞\n1|g(r)|2⟨r⟩2(σ+2)r dr\n≤C∥f∥L2\nr,γ,σ +2(R2).\nHere the third line follows from the change of variables r=eτand the fact that g(r)∼O(f(r)) for\nr∼0, while the last line follows from the definition of g(r) and the decay properties of the Bessel\nfunction K1(r) asr→ ∞ .\nOn the other hand, for the second integral on expression (9) we may write\nB=K1(r)Zr\n0I1(ρ)f(ρ)ρ dρ≤C1\nrZr\n0f(ρ)ρ2dρ. (12)\nAs in the proof of Lemma 2.11 we have that for s∈[0, r]\n|B|/r≤1\nr2Zr\n0|f(ρ)ρ−f(s)s|ρ dρ+1\n2|f(s)s|.\nLetting B(s,2r) denote the ball of radius 2 rcentered at s, and writing ˜B(s,2r) =B(s,2r)∩B(0, r),\nwe then obtain\n|B|/r≤|˜B(s,2r)|\n2πr2 \n1\n|˜B(s,2r)|Z\n˜B(s,2r)|f(ρ)ρ−f(s)s|ρ dρ!\n+1\n2|f(s)s|. (13)\nSince f∈L2\nr,γ,σ +2(R2) with σ <0, it follows f(ρ)∈L1(B1). We may therefore apply the Lebesgue\nDifferentiation Theorem to conclude that as rgoes to the origin the term inside the parenthesis\ngoes to zero, while the fraction in front of it remains bounded. Consequently |B(r)|< r2f(r) and\nit follows that ∥B∥L2r,γ,σ(B1)≤C∥f(r)∥L2\nr,γ,σ +2(R2).This bound together with (11) leads to\n∥u∥L2r,γ,σ(B1)≤ ∥f∥L2\nr,γ,σ +2(R2).\nTo obtain the bound ∥∂ru∥L2r,γ,σ(B1)≤ ∥f∥L2\nr,γ,σ +2(R2),recall first the expression for ∂ruwritten\nin (10). Given that\nd\ndzZν(z) =Zν+1(z) +ν\nzZν(z)\nwhere Zν(z) represents either Bessel function, Iν(z) orKν(z), it follows from the decay properties\nsummarized in Table 1 that I′\n1(r)∼O(1) and K′\n1(r)∼O(1/r2) near the origin. Therefore, we may\nbound\n|∂ru| ≤CZ∞\nrg(ρ)dρ+1\nr2Zr\n0f(ρ)ρ2dρ,\nwhere again g(ρ) =K1(ρ)f(ρ)ρ.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 22\nTo complete our argument, we define\nE(r) =CZ∞\nrg(ρ)dρ, F (r) =1\nr2Zr\n0f(ρ)ρ2dρ.\nTo bound ∥E∥L2\nr,γ,σ +1(R2)one can follow the same approach as was done in the case of A(r), see\ninequality (11). While the bound for ∥F∥L2\nr,γ,σ +1(R2)comes from a similar approach as inthe case of\nB(r), see inequalities (12) and (13). It then follows that\n∥∂ru∥L2\nr,γ,σ +1(B1)≤C∥f∥L2\nr,γ,σ +2(R2).\nFinally, to derive the bounds for the second derivative ∂rru, we use the equation to write\n|∂rru| ≤ |f|+1\nr|∂ru|+1\nr2|u|+|u|\n∥∂rru∥L2\nr,γ,σ +2(B1)≤ ∥f∥L2\nr,γ,σ +2(B1)+∥∂ru∥L2\nr,γ,σ +1(B1)+∥u∥L2r,γ,σ(B1)+∥u∥L2\nr,γ,σ +2(B1)\n≤4C∥f∥L2\nr,γ,σ +2(R2).\nHere the last inequality follows from our previous bounds for uand∂ru, and the embedding\nL2\nr,γ,σ(B1)⊂L2\nr,γ,σ +2(B1).\nStep 5: Lastly, to prove that the general operator (∆ 1−Id) : Hs\nr,γ,σ(R2)−→Hs−2\nr,γ,σ +2(R2) is\ninvertible one can use induction: Suppose that f∈Hs−2\nr,γ,σ +2(R2) and that u∈Hs−1\nr,γ,σ(R2). Then,\nthe result that ∂s\nru∈L2\nr,γ,σ +s(R2), follows from the equation\n∂s\nru=∂s−2\nr(f+u+1\nr2u−1\nr∂ru),\nand the embedding Hs−1\nr,γ,σ(R2)⊂Hs−1\nr,γ,σ +k(R2), which hold for k >0. □\n3.Normal Form\nAs mentioned in the introduction, the following nonlocal version of the complex Ginzburg-Landau\nequation was rigorously derived in reference [21] as an amplitude equation for spiral waves patterns\nin oscillatory media with nonlocal coupling:\n0 =K∗w+ (1 + i λ)w−(1 + i β)|w|2w+N(w;ε), x = (r, θ)∈R2(14)\nNotice that when compared to the original complex Ginzburg-Landau (cGL) equation, the above\nexpression uses a convolution operator, K∗, in place of the Laplacian operator. In addition, although\nthe equation is posed on the plane, R2, the unknown function, w, depends only on the radial\nvariable r, that is w=w(r). Another difference with the standard cGL equation is the term\nN(w, ε)∼O(ε|w|2w), which captures all higher order terms that are usually ignored in a formal\nderivation of this equation, see Appendix A. As shown in [21], these terms need to be taken into\naccount in order to conclude that our results are valid approximations to the solutions of the original\nsystem.\nIn this Section we prepare our normal form so that the analysis can go more smoothly. First,\nrecalling Hypothesis (H2) and the discussion from Subsection 1.3, we write the convolution K∗as\nK∗u=η(1−ε2D∆1)−1∆1u,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 23\nwhere, as already stated, we use the symbol ∆ nto denote the map\n∆nu=∂rru+1\nr∂ru−n\nr2u.\nWe may then precondition equation (14) with the operator (1 −ε2D∆1). The result is the following\nexpression,\n0 =d∆1w+ (1 + i λ)w−(1 + i β)|w|2w+˜N(w;ε),\nwhere the parameter d∈Cis given by\nd=dR+ idI= (η−ε2D)−iε2Dλ.\nNext, we rescale the radial variable r=√dRy, letα=dI/dR, and write\n0 = (1 + iα)∆1,yw+ (1 + i λ)w−(1 + i β)|w|2w+˜N(w;ε),\nwhere the nonlinearities are now given by\n˜N(w;ε) =ε2D\ndR(1 + i β)∆1,y|w|2w+ (1−ε2D\ndR∆1,y)N(w;ε).\nFinally, we write the complex valued function, w, using polar coordinates w=ρ(y)eiϕ(y), and\nseparate the equation into its real and imaginary parts. This gives the following system of equations,\nwhere for convenience we revert back to the original radial variable, r,\n0 =\u0002\n∆1ρ−(∂rϕ)2ρ\u0003\n−α[ρ∆0ϕ+ 2∂rϕ∂rρ] +ρ−ρ3+ Reh\n˜N(w;ε)e−iϕi\n(15)\n0 = [ ρ∆0ϕ+ 2∂rϕ∂rρ] +α\u0002\n∆1ρ−(∂rϕ)2ρ\u0003\n+λρ−βρ3+ Imh\n˜N(w;ε)e−iϕi\n. (16)\nTo prove the existence of solutions to the above system, we proceed via a perturbation analysis.\nWe rescale the variable rby defining S=δr, where δis assumed to be a small positive parameter\nsuch that ε∼δ4. We also use the following expressions for the unknown functions:\nρ=ρ0+δ2(R0+δR1), ρ 0=ρ0(r), R i=Ri(δr)i= 0,1\nϕ=ϕ0+δϕ1, ϕ i=ϕi(δr)i= 0,1.\nNotice that these are the same type of scalings used in [10] to derive a phase dynamics approximation\nfor the local cGL equation. For the free parameter, λ, representing the rotational speed of the wave,\nwe choose λ=β+δ2Ω,with βas above and Ω to be determined.\nThe result is a hierarchy of equations at different powers of δthat we present next. In terms of\nnotation, we use the subscript Sto distinguish operators that are applied to functions that depend\non this variable, i.e. ∆ 0,S. The absence of this subscript, indicates that the operator is applied to a\nfunction of the original variable r.\nRecalling that α∼O(ε2), we find the following system of equations.\n•At order O(1):\n0 =∆ 1ρ0+ρ0−ρ3\n0,\n0 =βρ0−βρ3\n0.\n•At order O( δ):\n0 =−2∂rρ0∂Sϕ0.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 24\n•At order O( δ2):\n0 =−ρ0(∂Sϕ0)2+R0(1−3ρ2\n0),\n0 =ρ0∆0,Sϕ0−2∂rρ0∂sϕ1+βR0(1−3ρ2\n0) + Ω ρ0.\n•Remaining higher order terms:\n0 =δ2∆1,SR1+R1(1−3ρ2\n0)−2ρ0∂Sϕ0∂Sϕ1+N1(R1, ϕ1;δ, ε),\n0 =ρ0∆0,Sϕ1+βR1(1−3ρ2\n0) +N2(R1, ϕ1;δ, ε),\nwhere\nN1(R1, ϕ1;δ, ε) = δ\u0010\n∆1,SR0+ (R0+δR1)(∂Sϕ0+δ∂Sϕ1)2−ρ0(∂Sϕ1)2\u0011\n−δ\u0010\n3ρ0(R0+δR1)2+δ2(R0+δR1)3\u0011\n−δ2α\u0010\nρ0+δ2(R0+δR1\u0011\n(∆0,Sϕ0+δ∆0,Sϕ1)\n−2δα\u0010\n∂rρ0+δ3(∂SR0+δ∂SR1\u0011\n(∂Sϕ0+δ∂Sϕ1)\n+ Reh\n˜N(w;ε)e−iϕi\n,(17)\nN2(R1, ϕ1;δ, ε) = δ\u0010\n(R0+δR1)(∆0,Sϕ0+δ∆0,Sϕ1)−2(∂SR0+δ∂SR1)(∂Sϕ0+δ∂Sϕ1)\u0011\n+δ\u0010\nΩ(R0+δR1)−3βρ0(R0+δR1)2−βδ2(R0+δR1)3\u0011\n+α\u0010\n∆1ρ0+δ4(∆1,SR0+δ∆1,SR1)\u0011\n−δ2α\u0010\nρ0+δ2(R0+δR1)\u0011\n(∂Sϕ0+δ∂Sϕ1)2\n+ Imh\n˜N(w;ε)e−iϕi\n.(18)\nIn the next section we solve the O(1) and O( δ2) equations explicitly. We then use these results,\ntogether with all remaining higher order terms, to define operators Fi(R1, ϕ1;δ) with i= 1,2, noting\nthat the zeros of these operators will then correspond to the solutions we seek. In Section 5 we show\nthat these maps satisfy the conditions of the implicit function theorem, and thus prove the existence\nof solutions to the nonlocal CGL equation representing spiral waves.\n4.The approximations\nIn this section we look in more detail at the O(1), O( δ), and O( δ2) equations. In addition,\nwe elaborate on the connection between our approximations and well known results regarding the\nexistence of spiral waves in reaction diffusion equations, [30, 17, 18].\n4.1.The order one equation. We study the system,\n0 =∆ 1ρ0+ρ0−ρ3\n0,\n0 =βρ0−βρ3\n0,(19)\nand to start we concentrate on the first expression, which we view as a boundary value problem:\n0 = ∆ 1ρ+ (1−ρ2)ρ, ρ →1 as r→ ∞ . (20)EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 25\nHere, we prove the existence of solutions to this equation and show that they converge to 1 at a rate\nof O(1 /r2), a result which will be important for us in later sections. We then use this information\nto show that the second equation in the above system is really of order O( δ2). These results are\nsummarized in the following proposition.\nProposition 4.1. Consider the system (19) defined by the order O(1) terms. There exists an\napproximation, ρ0(r), which solves the first equation and that satisfies:\ni.)ρ0(r)∈C2((0,∞])∩C1([0,∞])is positive and increasing for r >0.\nii.)ρ0(r)∼O(r)asr→0, while ρ0(r)→1asr→ ∞ .\niii.)(1−ρ0(r)2)∈H3\nr,γ(R2)withγ∈(0,1)and satisfies (1−ρ0(r)2)∼O(1/r2)asr→ ∞ .\nTo prove Proposition 4.1 we begin by recalling previous work by Kopell and Howard [30], which\nlooks at a more general version of the above boundary value problem, equation (20), but does not\nprovide the rate at which the solutions converge to one. The precise equation and results from [30]\nare summarized next.\nProposition 4.2. [ Theorem 3.1 in [30]] Let m∈Nand consider the function f(ρ), with f(1) = 0\nandf′<0. Then, the boundary value problem,\n∆mρ+f(ρ)ρ= 0, ρ (r)∼brmasr−→0,\nhas a unique solution, ρm(r), satisfying\ni)0< ρm(r)<1for all r >0,\nii)ρ′\nm(r)>0for all r >0, and\niii)ρm(r)→1asr→ ∞ .\nThe above results by Kopell and Howard prove the existence of solutions to our boundary value\nproblem and also give us items i) and ii) in Proposition 4.1. Notice that similar results can be found\nin Greenberg’s papers [17, 18], where it is assumed that the function f(ρ) in the above proposition\nisf(ρ) = 1−ρ.\nIn what follows we present an alternative proof for the existence of solutions to the boundary\nvalue problem (20). Our results not only give us existence, but also the level of decay with which\nthese solutions approach 1 at infinity, proving item iii) in Proposition 4.1. More precisely, we show\nthe following result.\nLemma 4.3. The boundary value problem\n0 = ∆ 1ρ+ (1−ρ2)ρ, ρ →1asr→ ∞ ,\nhas a unique solution. Moreover, the difference (1−ρ(r))is of order O(1/r2), and\n(1−ρ(r)2)∼2\nr2+ o(1 /r2)\nasrgoes to infinity.\nProof. First, to simplify the analysis, we consider the ansatz ρ(ξ) = 1 + u(r). The corresponding\nequation for u, is then\n∆1u−2u=1\nr2+ (3u2+u3).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 26\nLetu=un+uℓ, and notice that this is a solution to the above equation provided uℓsolves the\nlinear problem,\n∆1uℓ−2uℓ−1\nr2= 0, u ℓ→0 as r→ ∞ ,\nandunsolves the nonlinear equation\n∆1un−2un−\u0010\n3(un+uℓ)2+ (un+uℓ)3\u0011\n= 0. (21)\nWe will show that the term 1 + uℓcaptures the main behavior of the solution, ρ(r), while un\ncorresponds to a small correction.\nFor ease of exposition we prove the existence of solutions, uℓ, to the linear problem in Lemma\n4.4, shown below. Lemma 4.4 also shows that the solution uℓdecays at O(1 /r2) asrapproaches\ninfinity.\nNext, having obtained the asymptotic decay of uℓ, we move on to proving the existence of solutions\nto equation (21) which bifurcate from zero using the implicit function theorem. We assume that un\nhas a regular expansion un=εu1+ε2u2+···, and consider the left hand side of equation (21) as\nan operator F:Hs\nr,γ(R2)×R−→Hs−2\nr,γ(R2).\nFrom Corollary 2.10 we know that the linearization about the origin DuF(0; 0) : Hs\nr,γ(R2)−→\nHs−2\nr,γ(R2), given by the expression\nDuF(0; 0) = ∆ 1−2,\ndefines an invertible operator for all values of γ. To complete our argument, we need to pick the\ncorrect value of the weight γthat guarantees that the operator Fis well defined. In particular, we\nare concerned with showing that the nonlinear terms,\n\u0000\n3(un+uℓ)2+ (un+uℓ)3\u0001\n= 0.\nare in Hs\nr,γ(R2).\nNotice that if we let un∈H2\nr,γ(R2) with γ >0, then by Sobolev embeddings unis a bounded\nand continuous function. As a result any product up\nnis in the space L2\nr,γ(R2). Similarly, because\nuℓis bounded near the origin and decays like 1 /r2at infinity, any term of the form up\nnuq\nℓis also in\nL2\nr,γ(R2), for γ >0. Therefore, the level of algebraic localization of the nonlinear terms is controlled\nby the term u2\nℓ. Since we have that u2\nℓ∼O(1/r4) in the far field, then the nonlinearities are in the\nspace L2\nr,γ∗for values of γ∗<3.\nWe may conclude then that the solution unis in the space H2\nr,γ∗(R2) and that it has the same\nlevel of decay at infinity as u2\nℓ. Consequently, the solution, ρ, to the original boundary value problem\n(20) satisfies ρ(r)−1∼O(uℓ) = O(1 /r2).\nFinally, a short computation, together with our result that un∼O(u2\nℓ), shows that (1 −ρ2(r)) =\n−2uℓ(r) + o(1 /r2). It then follows from Lemma 4.4 that\n(1−ρ2(r)) =2\nr2+ o(1 /r2).\n□\nThe following Lemma captures the asymptotic behavior of the solution to the linear inhomoge-\nneous problem mentioned in the above analysis.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 27\nLemma 4.4. There exists solutions to the ordinary differential equation,\n∂ξξu+1\nξ∂ξu−1\nξ2u−u=1\nξ2\nsatisfying u∼O(1/ξ2)asξ→ ∞ , while u∼O(1) asξ→0.\nProof. The proof of this lemma relies on Proposition 2.13 where it is shown that the operator\n(∆1−Id) : Hs\nr,γ,σ(R2)−→Hs−2\nr,γ,σ +2(R2) is invertible for values of σ∈(−2,0) and γ∈R. A short\ncalculation then shows that f= 1/ξ2is in the space Hs−2\nr,γ,σ(R2), for any integer s≥2, and for values\nofγ∈(0,1) and σ+ 2∈(1,2). It then follows that the solution uto\n(∆1−Id)u=∂ξξu+1\nξ∂ξu−1\nξ2u−u=1\nξ2\nis in the space Hs\nr,γ,σ(R2). In particular, uhas the same level of decay at infinity as f, sou∼O(1/ξ2)\nasξ→ ∞ .\nTo obtain the growth rate of unear the origin, we use the Green’s function for the operator\n(∆1−Id) and write\nu(ξ) =I1(ξ)Z∞\nξK1(ρ)\nρdρ+K1(ξ)Zξ\n0I1(ρ)\nρdρ.\nUsing the asymptotic approximation for I1(ξ) near the origin, we may bound the first term by\nA(ξ) =I1(ξ)Z∞\nξK1(ρ)\nρdρ≤CξZ∞\nξK1(ρ)\nρdρ.\nBecause K1(ρ)/ρdecays at infinity, there is a constant Msuch that\nA(ξ)≤Cξ\u0014Z1\nξK1(ρ)\nρdρ+Z∞\n1K1(ρ)\nρdρ\u0015\n≤Cξ\u0014Z1\nξ1\nρ2dρ+M\u0015\n,\nand it then follows that A(ξ)∼O(1) as ξ→0.\nSimilarly, using the asymptotic expansions for K1andI1for values of ξ∼0, we may write\nB(ξ) =K1(ξ)Zξ\n0I1(ρ)\nρdρ≤C\nξZξ\n01dρ < C\nIt then follows that B(ξ)∼O(1) as ξ→0. The results of the lemma then follow. □\nRemark 4.5. Notice that because ρ0solves the second order ode (20), it is an element of C2((0,∞))∩\nC1([0,∞)). The decay rates for (1−ρ(r)2)presented in Lemma 4.3 then imply that this function is\ninC2([0,∞))⊂H3\nr,γ(R2)with 0< γ < 1, see also Figure 4. This completes the proof of item iii) in\nProposition 4.1.\nWe close this section by showing that the expression defining the second equation in our order\nO(1) approximation, that is\nβρ0−βρ3\n0=βρ0(1 +ρ0)(1−ρ0),\nis of order O( δ2). To do this we use the results of the previous lemma.\nIndeed, because ρ0solves equation (20), and the function ρ0satisfies ( ρ0−1)∼O(1/r2) in the\nfar field, one sees that the above expression is also of order O(1 /r2). Letting S=δrand recalling\nthatρ0is bounded near the origin, one finds that, in terms of the variable S,\nβ(1−ρ0)(1 + ρ0)ρ0∼O(δ2).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 28\nThis ‘left over’ term will be included in the O( δ2) equation in the next subsection. For notational\nconvenience we will also use the following definition.\nDefinition 4.6. We let ˜ρ0=ρ0(S/δ)and define\ng(S) =1\nδ2(1−˜ρ2\n0),\nwhich satisfies g(S)∼O(1/S2)asS→ ∞ .\n4.2.The order δequation. In this short subsection we look at the terms appearing in the order\nO(δ) equation,\n−2∂rρ0∂Sϕ0.\nand show that these are really of order O( δ4). This result follows from this next lemma.\nLemma 4.7. Letρ0satisfy the boundary value problem\n0 = ∆ 1ρ0+ (1−ρ2\n0)ρ0, ρ 0→1asr→ ∞ .\nThen, given f(r, S) =∂rρ0∂Sϕ0and the rescaling S=δr, the term\n˜f(S) =f(S/δ, S )∼O(δ3)asS→ ∞ .\nProof. From Lemma 4.3 we know that the solution, ρ0, to the given boundary value problem satisfies\n(ρ0−1)∼O(1/r2) asr→ ∞ . As a result, we may conclude that ∂rρ0∼O(1/r3) asr→ ∞ . Using\nthe rescaling, r=S/δwe arrive at the desired result. □\n4.3.The order δ2equation. We now move on to the next set of equations\n0 =−ρ0(∂Sϕ0)2+R0(1−3ρ2\n0)\n0 =ρ0∆0,Sϕ0−2∂rρ0∂Sϕ1+βR0(1−3ρ2\n0) + Ω ρ0+ρ0βg(S)\nwhere the function g(S) is given as in Definition 4.6.\nTo simplify the analysis of the above system, we use the information presented in Proposition\n4.1. In particular:\n•A similar proof as in Lemma 4.7, shows that the expression −2∂rρ0∂Sϕ1is of order O( δ3).\nTherefore, this term can be moved to the next set of higher order equations.\n•Similarly, because 1 −ρ2\n0∼O(1/r2) when ris large, we can write\n1−3ρ2\n0=−2 + 3(1 −ρ2\n0) =−2 +h(S)\nand conclude that the function h(S) = 3(1 −ρ2\n0) is of oder O( δ2). Thus, we also move these\nterms to the next set of equations.\nAs a result, in this section we concentrate on solving the system\n0 =−ρ0(∂Sϕ0)2−2R0 (22)\n0 =ρ0∆0,Sϕ0−2βR0+ Ωρ0+ρ0βg(S). (23)\nMore precisely, we prove the following proposition, which also states smoothness and decay properties\nof the solutions R0andϕ0. In the proposition, we use χD∈C∞to denote a radial cut-off function,\nwithχD(|x|) = 0 for |x|< D, for some positive number D, and χD(|x|) = 1 for |x|>2D. The symbol\nγeis also used here to represent the Euler-Mascheroni constant, while K0denotes the modified Bessel\nfunction of the second kind.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 29\nProposition 4.8. Letg(S)be as in Definition 4.6, pick D > 0and assume β < 0. Then, the\nsolutions to the equations (22) and(23), are given by,\nϕ0(S) =1\nβχ(ΛS) log( K0(ΛS)) +ac\n| {z }\n˜ϕ1+˜ϕ2(S), Λ2=−βΩ(a), c∈R,\nR0(S) =−1\n2ρ0(∂Sϕ0)2,\nwhere Ω(a) = 4 C(a)e−2γeexp(2 /a), with C(z)aC1(R)function in zand\na=−β2Z∞\n0gc(S)S dS; gc= (1−χD)g.\nMoreover, we have that\ni.)∂S˜ϕ2∈H3\nr,γ(R2)forγ∈(0,1),\nii.)∂Sϕ0→κandR0→1\n2κ2asS→ ∞ , with κ=−Λ\nβ>0, and\niii.)(∂Sϕ0−κ),(R0−1\n2κ2)∈H3\nr,¯γfor¯γ∈(−1,0)\nBefore proving Proposition 4.8, notice first that we can easily solve for the unknown R0using\nequation (22),\nR0=−1\n2ρ0(∂Sϕ0)2.\nInserting this result into the expression (23) then leads to the viscous eikonal equation,\n0 = ∆ 0,Sϕ0+β(∂Sϕ0)2+ Ω + βg(S).\nFor convenience, in the analysis that follows we let b=−β >0 (see Hypothesis (H1) ) and define\n¯g(S) =bg(S)/ϵ, for some arbitrarily small number ϵ >0. With this notation, the above viscous\neikonal equation then reads\n0 = ∆ 0,Sϕ0−b(∂Sϕ0)2+ Ω−ϵ¯g(S). (24)\nNotice that, not surprisingly, we have arrived at the same phase dynamics approximation that\ncan be formally derived for systems undergoing a Hopf bifurcation, see for example [29, 25]. Our\ngoal in this subsection is to solve this nonlinear equation. It is important to note that this equation\nconstitutes a nonlinear eigenvalue problem, since we need to find the solution, ϕ0, together with the\ncorresponding value of Ω. This is because Ω is related to the unknown parameter λ, which represents\nthe rotational speed of the spiral pattern, through the expression λ=β+δ2Ω. In particular, we are\ninterested in solutions, ϕ0(S), which in the far field behave like planar waves, since as mentioned\nin the introdcution these solutions would then represent spiral waves. That is, we require that\n∂Sϕ0→kasSgoes to infinity, for some constant k. We also ask that the group velocity of these\nsolutions be positive, i.e. ∂Sϕ0·S > 0, indicating that these planar waves move outwards, away\nfrom the spiral’s core.\nTo prove the existence of solutions to equation (24), we use the results from reference [22], where\nthe same viscous eikonal equation is used to model target patterns in 2-d oscillatory media. In\nthis context, the function g(S) that appears in equation (24) represents a defect, or impurity, that\nis present in the medium. This defect acts as a pacemaker entraining the rest of the system and\ngenerating a series of concentric waves that propagate away from it. n contrast to previous resultsEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 30\nz→0 z→ ∞\nK0(z) −log(z/2)−γe+ O(z2)pπ\n2ze−z\u0010\n1 + O(1 /z)\u0011\nK1(z)1\nz+ O(z)pπ\n2ze−z\u0010\n1 + O(1 /z)\u0011\nTable 2. Asymptotic behavior for the Modified Bessel functions of the second kind\nof zero-th and first-order, taken from [1, (9.6.8), (9.6.9), (9.7.2)]\nthat also deal with the existence of target pattern solutions (see for example [35, 30, 19, 29, 23, 25]),\nin reference [22] the function g(S) is assumed to represent a large defect, in the sense that it does not\nhave finite mass. More precisely, it is assumed that g(S)∼O(1/Sm) asS→ ∞ , with 1 < m≤2.\nSince the function g(S) considered here is of order O(1 /S2), and because in the far field our spiral\nwave solutions, as well as target patterns, satisfy ∇ϕ·x\n|x|→kas|x| → ∞ , the results from [22] are\nprecisely the ones we need in this section. We therefore summarize them in the next theorem using\nthe notation already introduced. We start by stating the hypothesis placed on gin reference [22].\nHypothesis 4.9. The inhomogeneity, g, lives in Hk\nγ(R2), with k≥2andγ∈(0,1), is radially\nsymmetric, and positive. In addition, this function can be split into the sum of two positive functions,\ngc, gf, satisfying\n•The function gfis in Hk\nγ(R2)for0< γ < 1. In particular, gf∼O(1/rm)asr→ ∞ , with\n1< m≤2, while near the origin gf(|x|) = 0 for|x|<1.\n•The function gcis in Hk\n˜γ(R2)for˜γ >1. In particular, gc∼O(1/rd)withd >2asr→ ∞ .\nTheorem 2. Letk≥2andγ∈(0,1)and consider a function g∈Hk\nr,γ(R2)satisfying Hypothesis\n4.9. Then, there exists a constant ϵ0>0and a C1([0, ϵ0))family of eigenfunctions ϕ=ϕ(S;ϵ)and\neigenvalues Ω = Ω( ϵ)that bifurcate from zero and solve the equation\n∆0ϕ−b(∂Sϕ)2+ Ω−ϵg(S) = 0 S=|x| ∈[0,∞), b > 0.\nMoreover, this family has the form\nϕ(S;ϵ) =−1\nbχ(ΛS) log( K0(ΛS)) +ϕ2(S;ϵ) +ϵc, Λ2=bΩ(ϵ) (25)\nwhere\ni)cis a constant that depends on the initial conditions of the problem,\nii)K0(z)represents the zeroth-order Modified Bessel function of the second kind,\niii)∂Sϕ2∈Hk\nr,γ(R2), and\niv)Ω = Ω( ϵ) =C(ϵ)4e−2γϵexp[2 /a], with\na=−ϵbZ∞\n0gc(S)S dS,\nandC(ϵ)aC1constant that depends on ϵ.\nAs already mentioned, a complete proof of the above theorem can be found in [22]. In the\ntheorem, the function χis a radial C∞cut-off function satisfying χ(x) = 0 for |x|<1 and χ(x) = 1\nfor|x|>2.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 31\nProof of Proposition 4.8\nProof. We first confirm that the function ¯ g(S) =bg(S)/ϵin (24) satisfies Hypothesis 4.9. From\nthe Definition 4.6 and Proposition 4.1, it follows that ¯ g(S)∼(1−ρ2\n0)∼O(1/S2) asSapproaches\ninfinity. Similarly, from Proposition 4.2 we know that the term (1 −ρ2\n0) is always positive, so that\n¯g(S) is a positive function provided b=−β >0. In addition, Definition 4.6 specifies that gis in\nH3\nr,γ(R2) with γ∈(0,1), as desired. Recalling that χD∈C∞denotes a radial cut-off function, with\nχD(|x|) = 0 for |x|< D, for some positive number D, and χD(|x|) = 1 for |x|>2D, we can then\ndefine\n¯gc= (1−χD)¯g ¯gf=χD¯g, (26)\nwith both functions, ¯ gfand ¯gcsatisfying the rest of Hypothesis 4.9.\nWe can therefore invoke Theorem 2 so that for ϵ∈(0, ϵ0), target-pattern solutions, ϕ(S;ϵ), given\nas in expression (25) to equation (24), exist. The value of ϵis then related to the constant aby\na=−ϵbZ∞\n0¯gc(S)S dS =−β2Z∞\n0gc(S)S dS,\nwhich is fixed once the value of Dis picked. As a result, we arrive at the form of the solutions ϕ0\nandR0stated in Proposition 4.8.\nWe are left with showing the decay properties of these functions, which we restate here for\nconvenience:\ni.)∂S˜ϕ2∈H3\nr,γ(R2) for γ∈(0,1),\nii.)∂Sϕ0→κandR0→1\n2κ2asS→ ∞ , with κ=−Λ\nβ, and\niii.) ( ∂Sϕ0−κ),(R0−1\n2κ2)∈H3\nr,¯γfor ¯γ∈(0,1).\nItem i) follows from Theorem 2. To prove items ii) and iii) for ϕ0, we recall the form of this solution\nshown in Proposition 4.8, and calculate its derivative,\n∂Sϕ0(S) =∂S˜ϕ1+∂S˜ϕ2=Λ\nbK1(ΛS)\nK0(ΛS)+∂S˜ϕ2.\nSince ∂S˜ϕ2∈H3\nr,γwith γ∈(0,1), it then follows from Lemma 2.5 that as Sapproaches infinity,\nthe function ∂S˜ϕ2∼O(Sα) with α <−γ−1. Similarly, using the decay properties of the Modified\nBessel functions (or Mathematica), we find that\nK1(ξ)\nK0(ξ)∼1 +1\n2ξ−1\n8ξ2+1\n8ξ3+ O(1 /ξ4),asξ→ ∞ . (27)\nConsequently, ∂Sϕ0→κasSapproaches infinity, with κ=Λ\nb=−Λ\nβ. In addition, the term\n(∂Sϕ0−κ)∼O(1/S) in the far field, and we may conclude that this function is in H3\nr,¯γ(R2) with\n¯γ∈(−1,0), see Lemma 2.5 and Figure 4.\nNext, recall that R0(S) =1\n2ρ0(∂Sϕ0)2. Expanding the quadratic term, we can write\n(∂Sϕ0)2= (∂S˜ϕ1)2+ 2∂S˜ϕ1∂S˜ϕ2+ (∂S˜ϕ2)2.\nSince elements in H3\nr,γ(R2), with γ∈(0,1), are bounded it then follows that ( ∂S˜ϕ2)2∈H3\nr,γ(R2). In\naddition, because the cut-off function χremoves the singularity at the origin, the function ∂S˜ϕ1is\nsmooth and uniformly bounded. As a result, the product ∂S˜ϕ1∂S˜ϕ2is also in the space H3\nr,γ(R2).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 32\nThus, the far field behavior of R0is determined by ( ∂S˜ϕ1)2. Items ii) and iii) for R0then follow\nfrom the definition of ∂S˜ϕ1, the expansion (27), and the fact that ( R0−1\n2κ2)∼O(1/S) for large\nS. □\nWe end this section with some remarks regarding the value of Ω and the behavior of ∂SΦ0near\nthe origin.\nRemark 4.10. Notice that the results of Theorem 2, and consequently Proposition 4.8, imply that\nthe frequency, Ω, of our spiral waves only depends on gc, the core of the function g. Unfortunately,\nthe theorem does not provide us with a way of determining the bounds for what is consider the core.\nThis is why one can pick any cut-off function χD, as defined in the proof above, to determine gf\nandgc. This choice affects the accuracy of our approximation for Ωandκ, but Theorem 1 makes\nit clear that no matter what choice we make for χD, the form of the phase solution, ϕ0, does not\nchange.\nRemark 4.11. Notice that we do not have precise details regarding the asymptotic behavior for\n∂Sϕ0near the origin. Nonetheless, Theorem 2 and Proposition 4.8 allow us to conclude at leas that\n∂Sϕ0∼O(1) near the origin. This follows from the fact that ˜ϕ1involves the cut off function χ,\nwhich removes the singularity near the origin for this logarithmic term, while the fact that ∂S˜ϕ2∈\nH3\nr,γ(R2)⊂C(R)implies that this function is bounded near the origin, therefore at least of order\nO(1).\n5.Existence of spiral waves\nIn this section we gather all terms of order O( δ3) and higher, and write them as a system of\nequations. Rearranging and separating linear and nonlinear terms we arrive at,\n0 =(δ2∆1,S−2)R1−2∂Sϕ0∂Sϕ1+˜N1(R1, ∂Sϕ1;δ), (28)\n0 =−2βR1+ ∆ 0,Sϕ1+˜N2(R1, ∂Sϕ1;δ). (29)\nThis is the same system appearing in Section 3, except that now the nonlinear terms are given by\n˜N1(R1, ∂Sϕ1;δ) =N1(R1, ϕ1;δ, δ4) + (1 −ρ0)[2∂Sϕ0∂Sϕ1] + 3(1 −ρ2\n0)(R0+δR1)/δ\n˜N2(R1, ∂Sϕ1;δ) =N2(R1, ϕ1;δ, δ4)−(1−ρ0)[∆0,Sϕ1] + 3β(1−ρ2\n0)(R0+δR1)/δ,\nwhere N1andN2are as in expressions (17) and (18), respectively, with ε=δ4, and the last terms\ncome from the order O( δ2) equations after rewriting\n(1−3ρ2\n0) =−2 + 3(1 −ρ2\n0),\n(see Subsection 4.3). Because the nonlinearities in the expressions ˜Ni,i= 1,2, depend on ϕ1through\nits derivative, ∂Sϕ1, we write them as functions of this last variable. Also, notice from the scaling\nS=δrand Lemma 4.3 that the terms (1 −ρ0) and (1 −ρ2\n0) are order O( δ2).\nOur goal is to prove the existence of solutions to the above system which bifurcate from zero.\nUsing the results from Section 4 we assume the functions ρ0andR0, ϕ0are given by the expressions\nstated in Propositions 4.1 and 4.8, respectively. Because ρ0, R0, ϕ0then represent the first order\napproximations to spiral wave solutions, this immediately implies the existence of solutions to the\nnonlocal complex Ginzburg-Landau equation (14) representing these patterns. Thus, Theorem 3,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 33\nshown below, together with Propositions 4.1 and 4.8, then imply the main results of this paper,\nTheorem 1.\nTheorem 3. There exists a real number δ0>0, such that equations (28) and(29) have a family\nof solutions (R1(S;δ), ∂Sϕ1(S;δ))valid for δ∈[0, δ0). Moreover, this family is C1in the parameter\nδand satisfies,\n•R1(S; 0) = 0 and∂Sϕ1(S; 0) = 0 , as well as having\n•both∂SΦ1(S;δ)andR1(S;δ)of order O(1) asS→ ∞ ,\nTo prove Theorem 3 we work with the equivalent system,\n0 =(δ2∆1,S−2)R1−2∂Sϕ0∂Sϕ1+˜N1(R1, ∂Sϕ1;δ), (30)\n0 =−2βδ2∆1,SR1+ ∆ 0,Sϕ1+ 2β∂Sϕ0∂Sϕ1+˜N2(R1, ∂Sϕ1;δ)−β˜N1(R1, ∂Sϕ1;δ). (31)\nwhich can be obtained by adding a −βmultiple of equation (28) to equation (29) to arrive at the\nsecond equation, (31). We carry out the proof as a series of steps. In Step 1, we use Fredholm\nproperties of the operator Lψ= (∂S+1\nS−k)ψ(Lemma 2.12), a result that is known as a Bordering\nLemma (Lemma 5.6), and the implicit function theorem to obtain a family of solutions ∂Sϕ1=\n∂Sϕ1(R1, δ) to equation (31). This result is then summarized in Proposition 5.3. In Step 2, we use\nProposition 2.13, which implies that the operator ( δ2∆1,S−2) is invertible, together with our family\nof solutions ∂Sϕ1and the implicit function theorem, to prove existence of solutions, R1=R1(δ),\nto equation (30). This in turn proves Theorem 3. In the course of the analysis we find that it is\nnecessary to show that the nonlinear terms ˜N1,˜N2define bounded operators between appropriate\nweighted Sobolev spaces, and that they depend continuously on the variables R1, ∂Sϕ1and the\nparameter δ. This is shown in Step 3.\nRemark 5.1. Notice that all terms appearing in equations (30)and(31)involve the derivative ∂Sϕ1\nand not variable ϕ1. We therefore view ∂Sϕ1, and not ϕ1, as our unknown.\nBefore presenting our result, let us properly define the spaces that we will be using. Here, and\nin the rest of this section, we let γ1∈(−1,0),σ1∈(−2,−1) and consider\nX=H2\nr,γ1,σ1(R2)⊕ {ζ(S)} ⊕R\nY=H1\nr,γ1,σ1+1(R2)⊕ {1\nS} ⊕R\nZ=L2\nr,γ1,σ1+2(R2)⊕\b1\nS2\t\n⊕R(32)\nwhere {u}denotes the span of the function uover the reals, and the function ζ(S) satisfies\n(δ2∆1,S−2)ζ=−1.\nThese spaces are then endowed with the norms\n∥x∥X=∥R∥H2r,γ1,σ1(R2)+|x1|+|x2| ∀ x:= (R, x 1, x2)∈X\n∥y∥Y=∥ϕ∥H1\nr,γ1,σ1+1(R2)+|y1|+|y2| ∀y:= (ϕ, y1, y2)∈Y\n∥z∥Z=∥N∥L2\nr,γ1,σ1+2(R2)+|z1|+|z2| ∀z:= (N, z 1, z2)∈Z.\nThe decay rates for ∂Sϕ1andR1follow from this choice of Banach spaces.\nIn this section we will also use the following result, which establishes decay properties for the\nfunction ζ(S).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 34\nLemma 5.2. Letζbe the solution to the equation\n(δ2∆1,S−2)ζ=−1\nThen, ζ(S) =ζ1(S) +ζfwhere ζ1∈H3\nr,γ(R2), with γ∈(0,1), and ζf∈R.\nProof. Notice that with the change of variables ξ=√\n2S/δwe are able to write the equation\n(δ2∆1,S−2)ζ=−1 as\n∂ξξζ+1\nξ∂ξζ−1\nξ2ζ−ζ=−1/2\nLetting ζf= 1/2 so that ζ=ζ1(ξ) + 1/2, we find that ζ1(ξ) solves\n∂ξξζ1+1\nξ∂ξζ1−1\nξ2ζ1−ζ1=1\n21\nξ2\nThe proof of Lemma 4.4 then shows that solutions to the above equation satisfy ζ1∼O(1/ξ2) asξ\ngoes to infinity. It then follows from Lemma 2.5 that ζ1∈L2\nr,γ(R2), with γ∈(0,1), see also Figure\n4. Lemma 4.4 also shows that ζ1∼O(1) as ξ→0. Because, ζ1satisfies a second order ODE and\nit is bounded near the origin, it follows that ζ1∈C2([0,∞))∩C1([0,∞)) and is consequently an\nelement in H3\nr,γ(R2).\n□\n5.1.Step 1. In this section we use the implicit function theorem to find solutions to equation (31)\nthat bifurcate from zero and are of the form ∂Sϕ1=∂Sϕ1(R1, δ). To do this, we first look at the\nnonlinear terms in this equation,\nN(R1, ∂Sϕ1;δ) =−2βδ2∆1,SR1+˜N2(R1, ϕ1;δ)−β˜N1(R1, ϕ1;δ),\n=−2βδ2∆1,SR1−(1−ρ0)[∆0,Sϕ1+ 2β���Sϕ0∂Sϕ1)\n+N2(R1, ϕ1;δ, δ4)−βN1(R1, ϕ1;δ, δ4),\nwhere N1andN2are given as in expressions (17) and (18), respectively, with ε=δ4. In Step 3, we\nprove that these nonlinear terms give us a well defined map N:X×Y×R2−→Z. This result is\nsummarized in Proposition 5.14. We can therefore write\nN(R1, ∂Sϕ1;δ) =C(R1)(∂Sϕ1)2+D(R1, ∂Sϕ1)1\nS2+M(R1, ∂Sϕ1;δ), (33)\nwhere, using the notation R1=Rn+ζ(S) +Rf∈Xand∂Sϕ1=∂Sϕn+a\nS+∂Sϕf∈Y, we have\ndefined\n•C(R1)∈Rwith\nC(R1) =βδ−βδ3(1\n2k2+δζf+δRf)−δ4α(1 +δ2(1\n2k2+δζf+δRf)) + O( ε=δ4) as δ→0.(34)\n•D(R1, ∂Sϕ1)∈Rwith,\nD(R1, ∂Sϕ1) = (−βδ−βδ2a2+αδ2−δ6αa2)[δRf+δζf+1\n2k2] + O( ε=δ4) as δ→0.(35)\n•M:X×Y×R−→˜Z=L2\nr,γ1,σ1+2(R2)⊕R,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 35\nEssentially, equation (33) comes from extracting all constant multiples of ( ∂Sϕ1)2and1\nS2from the\nnonlinearities N. This calculation is done in Appendix C. The notation C(R1) and D(R1, ∂Sϕ1)\nis meant to indicate that these numbers depend on the constant elements, ζf, Rfanda, ϕf, which\nappear in the definition of the unknowns, R1and∂Sϕ1, respectively. Here and in the rest of the\npaper, we also define the space ˜Z=L2\nr,γ1,σ1+2(R2)⊕R, with γ1∈(−1,0),σ1∈(−2,−1), and norm\n∥z∥˜Z=∥N∥L2\nr,γ1,σ1+2(R2)+|z1| ∀z:= (N, z 1)∈˜Z. (36)\nNext, to prove the results of Step 1, we let ∂Sϕ1=∂S˜ϕ1+qwith\n∂S˜ϕ1:=∂Sϕn+a\nS+∂Sϕf∈Y a, ∂ Sϕf∈R,\nandqsatisfying the equation,\n∂Sq+1\nSq+ 2a1\nSq+ 2β∂Sϕ0q+Cq2+ (a2C+D)1\nS2= 0. (37)\nHere, the constants CandDare as in (51) and (52), respectively. Inserting the anstaz ∂Sϕ1=\n∂S˜ϕ1+qinto expression (31) we obtain\n0 =∆ 0,S˜ϕ1+ 2β∂Sϕ0∂S˜ϕ1+M(R1, ∂Sϕ1;δ)\n+C(R1)h\n2(q+a\nS)(∂Sϕn+∂Sϕf) + (∂Sϕn+∂Sϕf)2i\n.(38)\nThe right hand side of this equation then defines an operator F:Y×X×R−→ ˜Z. Our goal\nis to show that the map Fsatisfies the assumptions of the implicit function theorem. The result is\nthe following proposition.\nProposition 5.3. LetXandYbe Banach spaces defined as in (32), let ˜Zbe defined as in (36),\nand consider the map F:Y×X×R−→ ˜Zgiven by (38). Then, there exist a constant δ0>0,\na neighborhood X0⊂Xcontaining 0, and a C1function ∂Sϕ:X0×(−δ0, δ0)−→Ysatisfying:\n∂Sϕ(0,0) = 0 andF(∂Sϕ(R, δ);R, δ) = 0 whenever (R, δ)∈X0×(−δ0, δ0). Moreover, if qsatisfies\nequation (37), then the vector (q+∂Sϕ(R, δ), R, δ)solves equation (31) in the same neighborhood of\nzero, i.e. (R, δ)∈X0×(−δ0, δ0).\nProof. We need to check that the operator F:Y×X×R−→˜Z,\n1) is well defined and C1with respect to all its variables,\n2) that F(0; 0,0) = 0, and\n3) that its derivative D∂Sϕ1F|(0;0,0):Y−→˜Zis invertible.\nBecause both M(R1, ∂Sϕ;δ) and C(R1) are order O( δ) asδ→0, it is easy to check that item 2)\nholds. Given that the derivative map is\nD∂Sϕ1F|(0;0,0)u=Lϕu=∂Su+1\nSu+ 2β∂Sϕ0u,\nitem 3) then follows from Proposition 5.7 given at the end of this subection, which shows that the\noperator Lϕ:Y−→ ˜Zhas a bounded inverse. We are left with showing that Fis well defined\nand continuously differentiable with respect to all its variables. We begin by showing that Fis well\ndefined.\nFirst, Proposition 5.7, shows that the linear part of the operator, Lϕ, maps to ˜Z. Next, from\nthe definition of the map Mand Proposition 5.14 we may conclude that these nonlinearities areEXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 36\nwell defined in ˜Z, provided qis also an element in Y. This last point is shown in Lemma 5.4 given\nbelow. Thus, we only need to check that the remaining terms,\nC(R1)h\n2(q+a\nS)(∂Sϕn+∂Sϕf) + (∂Sϕn+∂Sϕf)2i\nare elements in the space ˜Z.\nNotice that because the terms in C(R1) belong to R, we can concentrate on showing that the\nelements in the brackets are in ˜Z. We start by noticing that the expression ( q+a\nS) is an element in\nY. This follows from the definition of this space and Lemma 5.4, which shows qis also in Y. As a\nresult, we may write\n(q+a\nS)(∂Sϕn+∂Sϕf) =(h+˜a\nS+c)(∂Sϕn+∂Sϕf)\n=h∂Sϕn+˜a\nS∂Sϕn+c∂Sϕn+h∂Sϕf+˜a\nS∂Sϕf+c∂Sϕf,\nwhere h, ∂Sϕn∈H1\nr,γ1,σ1+1(R2) while c,˜a, ∂Sϕfare in R. This observation then allows to look at\neach term individually, and conclude that\nc∂Sϕn+h∂Sϕf∈H1\nr,γ1,σ1+1(R2)⊂L2\nr,γ1,σ1+2(R2)\n˜a\nS∂Sϕf∈L2\nr,γ1,σ1+2(R2)\n˜a\nS∂Sϕn∈H1\nr,γ1,σ1+2(R2)\nc∂Sϕf∈R\nh∂Sϕn∈L2\nr,γ1,2σ1+3(R2)⊂L2\nr,γ1,σ1+2(R2)\nwhere the first embedding of the last line follows from Lemma 2.8, while the second embedding is\na consequence of Definition 2.2 and our assumption that σ∈(−2,−1). More precisely, this last\ncondition shows that 2 σ1+ 3< σ 1+ 2 so that the embedding, L2\nr,γ1,2σ1+3(R2)⊂L2\nr,γ1,σ1+2(R2),\nholds. It then follows that the term ( q+a\nS)(∂Sϕn+∂Sϕf) is in ˜Z.\nA similar reasoning allows us to show that ( ∂Sϕn+∂Sϕf)2∈˜Z, so we omit the details.\nBecause all terms in the definition of the operator Finvolve linear or polynomial functions of\nthe variables R, δ, ε, and∂sϕ1, and because ∂Sϕ1=∂S˜ϕ+qwith qaC2function of ε, δ, the map F\nis continuously differentiable. This concludes the proof of this proposition. □\nOur next task is to show that the function qsatisfying equation (37) lives in the space Y. This\nfollows from Lemma 5.4 shown below, which establishes properties for solutions to this type of\nequations. One can check that the coefficients in (37) satisfy the assumptions presented in this\nlemma.\nLemma 5.4. Leta, b∞, c, d be positive constants and take b(x)to be a function in C1([0,∞),R)\nsuch that b(x)∼ −b∞+ O(1 /x)asx→ ∞ . Then, there exists a solution, q(x), to the ordinary\ndifferential equation,\nq′+a\nxq+b(x)q−cq2+d\nx2= 0 x≥0\nsatisfying,\nq(x)∼O(1/x)asx→0, q (x)∼q∞+ O(1 /x)asx→ ∞EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 37\nfor some q∞∈R. Moreover, there is a constant νand a function q2∈H1\nr,γ1,σ1+1(R2), with γ1∈\n(−1,0)andσ1∈(−2,−1), such that\nq(x) =q2(x) +ν\nx+q∞∈Y.\nThe proof of this lemma is presented in Appendix D. The idea is to use the Hopf-Cole transform,\nq=αy′\ny, with α=−1/c, to arrive at the linear ordinary differential equation,\ny′′+a\nxy′+b(x)y′−cd\nx2y= 0,\nOne can then use the Frobenious method to determine the asymptotic behavior of solutions, F,\nto this linear equation as x→0 and as x→ ∞ . The results of Lemma 5.4 then follow from the\nrelationship between Fandqand the definition of the space H1\nr,γ1,σ1+1(R2).\nTo conclude this section, we show that the linear operator Lϕ:Y−→ ˜Zis invertible. This is\ndone in Proposition 5.7, which relies in Lemma 5.5 and Lemma 5.6, shown next. We start with the\nproof of Lemma 5.5, which shows that the operator Lϕ=∂S+1\nS−λ∂Sϕ0is Fredholm when its\ndomain is defined appropriately.\nLemma 5.5. Letγandσbe real numbers, san integer satisfying s≥1, and take λ >0. If the\nfunction ϕ0(r)∈L2\nr,˜γ−1(R2), with ˜γ∈(0,1), satisfies:\n•∂rϕ0→κasr→ ∞ , with κ >0,\n•(∂rϕ0−κ)∼O(1/r)asr→ ∞ , and\n•(∂rϕ0−κ)∈H3\nr,˜γ(R2),\nthen the operator Lϕ:Hs\nr,γ,σ(R2)−→Hs−1\nr,γ,σ +1(R2), given by\nLϕu=∂ru+1\nru−λ∂rϕ0u\nis Fredholm. If moreover, ϕ0(r)→cwithc∈R, asr→0, then\ni) if σ <0, the operator has index i=−1, it is injective and its cokernel is spanned y e−λϕ0(r);\nand\nii) if σ >0, it has index i= 0and it is invertible.\nProof. Recall first the results from Lemma 2.12, which shows that for ˜λ > 0, the operator L:\nHs\nr,γ,σ(R2)−→Hs−1\nr,γ,σ +1(R2), defined by L=∂r+1\nr−˜λ, is Fredholm. In particular, we have that\nforσ <0 the operator Lhas index i=−1, while for σ >0 it is invertible. In what follows we show\nthatLϕis a compact perturbation of L. As a result, Lϕis also Fredholm with the same index as L.\nTo this end, we let ˜λ=λκand write\nLϕ=∂r+1\nr−˜λ+λ(κ−∂rϕ0) =L+T,\nwhere Tis the multiplication operator T=λ(κ−∂rϕ0). Since ( κ−∂rϕ0)∈O(1/r) asr→ ∞ and\nL2\nr,γ,σ(R2)⊂L2\nr,γ,σ +1(R2), it follows that T:H1\nr,γ,σ(R2)−→L2\nr,γ,σ +1(R2) is bounded. To prove that\nTis compact, we show that it can be approximated by a sequence of compact operators given by\nTN=λχN(κ−∂rϕ0). Here, χNis a radial function in C∞(R2) satisfying χN(r) = 1 for r < N and\nχN(r) = 0 for r > N + 1.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 38\nFirst, it is clear from the following inequalities that TN→Tin the operator norm\n∥(T−TN)u∥2\nL2\nr,γ,σ +1(R2)≤λCZ∞\n0|1−χN|2\f\f\f\f1\nr\f\f\f\f2\n|u|2m(r)2(σ+1)⟨r⟩2γr dr\n≤λCZ∞\nN\f\f\f\f1\nr\f\f\f\f2\n|u|2⟨r⟩2γr dr\n≤λC\nN2∥u∥2\nL2r,γ,σ(R2).\nNext, to show that the multiplication operators TNare compact notice that the image, TNu, is\na function with compact support inside the ball B2Nof radius 2 N. In particular, given a sequence\n{un} ⊂H1\nr,γ,σ(R2)⊂H1\nr,γ,σ +1(R2), we have that TNun(r)⟨r⟩γm(r)σ+1∈H1(B2N)⊂⊂L2(B2N),\nwhere this last embedding is compact by the Rellich-Kondrachov theorem. Thus, there is a subse-\nquence unksuch that TNunk→¯vwith ¯ v(r)⟨r⟩γm(r)σ+1∈L2(R2), as desired.\nTo complete the proof of this lemma, we need to show that given any σ∈R, the operator has\ntrivial kernel, and that its cokernel is spanned by e−λϕ0only if σ < 0. First, it is clear that the\nonly function satisfying Lϕu= 0 is u(r) =eλϕ0(r)\nr. Since ϕ0(r)→κrwith κ >0, this function grows\nexponentially and thus, it is not in the domain of the operator no matter what the values of γand\nσare. On the other hand, the adjoint of the operator is L∗\nϕ:L2\nr,−γ+1,−(σ+1)(R2)−→H−1\nr,−γ,−σ(R2),\nis given by\nL∗\nϕ=−(∂r+λ∂ϕ 0).\nThe kernel of this map is then spanned by e−λϕ0(r). Since e−λϕ0(r)���e−λcasrgoes to zero, it then\nfollows that this function is in the domain of L∗\nϕprovided σ <0. The results of the lemma then\nfollow. □\nThis next result, which is known as a Bordering Lemma, gives us conditions under which a\nFredholm operator becomes invertible.\nLemma 5.6. [Bordering Lemma] Let XandYbe Banach spaces, and consider the operator\nS=\u0014A B\nC D\u0015\n:X×Rp−→Y×Rq,\nwith bounded linear operators A:X−→Y,B:Rp−→Y,C:X−→Rq,D:Rp−→Rq. IfAis\nFredholm of index i, then Sis Fredholm of index i+p−q.\nProof. One can write Sas the sum of a block diagonal operator with the indicated index, i+p−q,\nand a compact operator consisting of the off-diagonal elements. Since compact perturbations do not\nalter the index of a Fredholm operator, the result then follows. □\nThis next proposition shows that the linear operator in equation (31), when considered as a map\nbetween Yand ˜Z, is invertible.\nProposition 5.7. Letϕ0be as in Lemma 5.5, take β <0. Then, the map\nLϕ:Y−→ ˜Z\nu7−→ ∂Su+1\nSu+ 2β∂Sϕ0u\nwithY=H1\nr,γ1,σ1+1(R2)⊕ {1\nS} ⊕Rand ˜Z=L2\nr,γ1,σ1+2(R2)⊕R, is invertible.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 39\nProof. We first take a closer look at the operator\n˜Lϕ:H1\nr,γ1,σ1+1(R2)−→ L2\nr,γ1,σ1+2(R2)\nu 7−→ ∂Su+1\nSu+ 2β∂Sϕ0u.\nThe following calculations show that the function Lϕ1\nS= (∂S+1\nS+ 2β∂Sϕ0)1\nSspans the cokernel of\nthe operator ˜Lϕ:H1\nr,γ1,σ1+1−→L2\nr,γ1,σ1+2, which is given by e2βϕ0(S). Since these Hilbert spaces\ncome endowed with the inner product\n⟨u, v⟩=Z∞\n0u(S)v(S)S dS,\nit suffices to show that ⟨Lϕ1\nS,e2βϕ0(S)⟩ ̸= 0.This last result follows easily from an integration by\nparts:\n⟨Lϕ1\nS,e2βϕ0(S)⟩=1\nSe2βϕ0(S)S\f\f\f∞\n0+Z∞\n01\nSL∗\nϕ(e2βϕ0(S))S dS\n=−e2βϕ0(0)\n̸= 0.\nUsing the Borderling Lemma we can therefore conclude that the extended operator\n˜Lϕ:H1\nr,γ1,σ1+1(R2)× {1\nS} −→ L2\nr,γ1,σ1+2,\nis Fredholm and has index i= 0. One can now check that the only element in the kernel of this\noperator is the trivial solution. Consequently, the operator is invertible. Indeed, letting u0=u1+a\nS\nwith a∈R, we find that the equation ˜Lϕu0= 0 can be written as\nLϕu1=−2β∂Sϕ0a\nS.\nSince ∂Sϕ0converges to a constant, both near the origin and in the far field, it follows that the right\nhand side is in L2\nr,γ,σ +2with γ∈(−1,0) and σ+2∈(0,1), see Lemma 2.5 and Figure 4. The results\nof Lemma 5.5 then imply that u1∈H1\nr,γ,σ +1, but with σ∈(−2,−1). Therefore, this solution u0is\nnot in the space Y, which uses σ1∈(−1,0) (see definition of Yin (32)).\nTo complete the proof of this proposition, we need to show that the following operators are\nbounded,\nLϕ:R−→ ˜Z\nL−1\nϕ:R−→ Y(39)\nHere again, Lϕu=∂Su+1\nSu+ 2β∂Sϕ0u. Letting c∈R, we may write\nLϕc=c\nS+ 2βc(∂Sϕ0−k) + 2βck.\nSince\u0000c\nS+ 2βc(∂Sϕ0−k)\u0001\n∈L2\nr,γ1,σ1+2, it follows that the right hand side in the above expression\nis in ˜Z. As a result, the first operator defined in (39) is bounded. On the other hand, notice that\nproving that the second operator appearing in (39) is bounded, is equivalent to solving the equation\nLϕu= 1 and showing that u∈Y. To that end, let u=u1+c, with c= 1/2βkand notice that u1\nthen has to solve:\nLϕu1=−c\nS−2βc(∂Sϕ0−k).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 40\nSince the right hand side is an element in L2\nr,γ1,σ1+2(R2), and since we just showed that the operator\n˜Lϕ:H1\nr,γ1,σ1+1(R2)× {1\nS} −→ L2\nr,γ1,σ1+2\nis invertible, we then have that u1∈H1\nr,γ1,σ1+1(R2)×{1\nS}. Consequently u=u1+c∈Y, as desired.\n□\n5.2.Step 2. From Proposition 5.3 we know that there is a function ∂Sϕ1(R1, δ) representing solu-\ntions to (31) that bifurcate from zero. We can now insert this function into equation (30) with the\ngoal of proving existence of solutions to the resulting expression. To do this we precondition equation\n(30) by L−1(δ) = (δ2∆1,S−2)−1. The right hand side then defines an operator F:X×R−→X,\ngiven by\nF(R1;δ) =R1+L−1(δ)h\n−2∂Sϕ0∂Sϕ1(R1, δ) +˜N1(R1, ∂Sϕ1(R1, δ);δ)i\n. (40)\nIn what follows we show that Fsatisfies the conditions of the implicit function theorem, thus proving\nthe results of this next proposition.\nProposition 5.8. LetXbe the Banach space defined as in (32), and consider the operator F:X×\nR−→Xdefined in (40). Then there exist a constant δ0>0, and a C1function, R: [0, δ0)−→X,\nsuch that R(0) = 0 . Moreover, the function R(δ)is a zero of the operator F, and thus it is also a\nsolution to equation (30), whenever δ∈[0, δ0)and∂Sϕis as in Proposition 5.3.\nThe results of Proposition 5.8 rely on:\n•Lemma 5.9, which shows that the operator L−1(δ) :Z−→Xis bounded, and C1with\nrespect to the parameter δ, forδ >0.\n•Proposition 5.14, which shows that the nonlinear terms define bounded operators of the form\n˜Ni:X×Y×R2−→Z(this also includes the term 2 ∂Sϕ0∂Sϕ1).\nFor ease of exposition at this stage we only summarize these results. The proof of Lemma 5.9 is\ngiven at the end of this subsection, while the proof of Proposition 5.14 appears in Subsection 5.3.\nProof. The results from Lemma 5.9 and Proposition 5.14 show that the operator Fis well defined.\nTo show that F(0; 0,0) = 0 notice first that the nonlinear map ˜N1is order O( δ). At the same time,\nProposition 5.3 shows that the function ∂Sϕ1(R1,0) solves equation 38 with δ= 0, i.e.\n0 =∆ 0,S˜ϕ1+ 2β∂Sϕ0∂S˜ϕ1+M(R1, ∂Sϕ1;δ)\n+C(R1)h\n2(q+a\nS)(∂Sϕn+∂Sϕf) + (∂Sϕn+∂Sϕf)2i\n.\nSince both, M(R1, ∂Sϕ1;δ) and C(R1) are order O( δ), and because by Proposition 5.7 the operator\nLϕ:Y−→˜Z⊂Zdefined by\nLϕ∂S˜ϕ1= (∂S+1\nS+ 2β∂Sϕ0)∂S˜ϕ1= ∆ 0,S˜ϕ1+ 2β∂Sϕ0∂S˜ϕ1,\nis invertible, if follows that ∂Sϕ1(R1,0) = 0. Hence F(0; 0,0) = 0. Because the norm ∥L−1(δ)∥is\nbounded, the above remarks also show that the derivative map, DXF|(0;0,0):X−→X, is just the\nidentity and is therefore invertible.\nTo show that Fsatisfies the conditions of the implicit function theorem, we are left with checking\nthat the operator defines a C1map with respect to all its variables. This easily follows for R1∈X,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 41\nand∂Sϕ=∂Sϕ(R1, δ)∈Yas in Proposition 5.3, since all nonlinear terms in ˜N1(R1, ∂Sϕ1;δ) involve\nlinear or polynomial functions of these variables and because, by Proposition 5.3, the function ∂Sϕ\nisC1in all its variables.\nTo check that the operator FisC1with respect to δ, notice first that from Lemma 5.9 we have\nthat the linear map L−1(δ) isC1in the parameter δfor values of δ >0. Because the nonlinear\nterms in ˜N1are smooth with respect to δ, it then follows that Fis also C1with respect to this\nparameter when δ >0.\nTo establish this same result for δ= 0, we first check that Fis continuous at this point. This\ncan be done by showing there is a constant C >0 such that\n∥F(R, h)−F(R,0)∥X≤C|h|.\nTo this end, we determine first that\nF(R, h)−F(R,0) =\u0010\nR+L−1(h)[−2∂Sϕ0∂Sϕ1(R1, h) +˜N1(R, ∂ Sϕ(R, h);h)]\u0011\n−\u0010\nR+L−1(0)[−2∂Sϕ0∂Sϕ1(R1,0) + ˜N1(R, ∂ Sϕ(R,0); 0)\u0011\n=L−1(h)[−2∂Sϕ0∂Sϕ1(R1, h) +˜N1(R, ∂ Sϕ(R, h);h)].\nThen\n∥F(R, h)−F(R,0)∥X≤h∥L−1(h)∥h\n∥2∂Sϕ0∂Sϕ1(R1, h)/h∥Z+∥˜N1(R, ϕ(R, h), h)/h∥Zi\n∥F(R, h)−F(R,0)∥X≤hC(γ),\nwhere the last inequality holds since both, ˜Nand∂Sϕ1(R, δ), are order O( δ) asδ→0. From the\nabove calculation we also deduce that DδF|(R,0)is bounded. As a result the map FisC1for\nδ∈R+, and the results of the proposition then follow.\n□\nWe now proceed with the proof of Lemma 5.9 which shows that the operator L(δ) :X−→Zis\ninvertible.\nLemma 5.9. Letδ >0, and take γ1∈Randσ1∈(−2,0)in the definition of the spaces XandZ\ngiven in (32) . Then, the operator\nL−1(δ) : Z −→ X\n(f+a/S2+b)7−→ (δ2∆1,S−2)−1(f+a/S2+b)\nis bounded and C1with respect to the parameter δ.\nProof. We split the proof of this lemma into two steps. In Step A, we show that the operator is\nbounded, and in Step B we show that it is C1with respect to δ.\nStep A: Suppose\n(f+a/S2+b)∈Z=L2\nr,γ1,σ1+2(R2)⊕\u001a1\nS2\u001b\n⊕R.\nIn what follows we show that\n(δ2∆1,S−2)−1(f+a/S2+b)∈X=H2\nr,γ1,σ1(R2)⊕ {ζ(S)} ⊕R.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 42\nFirst, since δ >0 by Proposition 2.13 we know that the operator (∆ 1−Id) : H2\nr,γ1,σ1(R2)−→\nL2\nr,γ1,σ1+2(R2) has a bounded inverse provided σ1∈(−2,0). Therefore, by a suitable re-scaling, this\nsame lemma implies that\n(δ2∆1,S−2)−1f∈H2\nr,γ1,σ1(R2).\nNext, taking\n(a/S2+b) = (a/S2+ 2a/δ2)|{z }\nf2+ (b−2a/δ2)|{z}\nf3\nwe show that ( δ2∆1,S−2)−1f2∈Rand ( δ2∆1,S−2)−1f3∈ {ζ(S)}.\nFirst, notice that the first term\nf2:=a\nδ2(δ2/S2+ 2)∈L2\nr,¯γ,¯σ+2(R2)\nfor ¯γ∈(−2,−1) and ¯ σ+ 2∈(1,2). By Proposition 2.13, it again follows that the function\n(δ2∆1,S−2)−1f2=:Rf∈H2\nr,¯γ,¯σ(R2)\nexists and is unique. Since\n(δ2∆1,S−2)1 = −(δ2\nS2+ 2) = −δ2\naf2\nwe then obtain that Rf=−a/δ2∈R.\nFinally, because f3:= (b−2a/δ2)∈R, then\n(δ2∆1,S−2)−1f3=f3(δ2∆1,S−2)−11∈ {ζ(S)}\nwhere, we recall that the function ζis defined as satisfying −ζ= (δ2∆1,S−2)−11, see (32).\nStep B: We start by showing the operator L−1(δ) is continuous with respect to δ. This is\nequivalent to showing that there exists a constant C >0 such that\nsup\n∥f∥Z=1∥\u0000\nL−1(δ+h)− L−1(δ)\u0001\nf∥X≤ |h|C.\nSuppose then that f∈Zwith∥f∥Z= 1, and let u(δ) =L−1(δ)f. Notice that\n\u0000\nL−1(δ+h)− L−1(δ)\u0001\nf=−(u(δ)−u(δ+h))\n=−L−1(δ) [L(δ+h)− L(δ)]u(δ+h)\n=−(2δh−h2)L−1(δ)∆1,SL−1(δ+h)f.\nIf we show that the operator ∆ 1,S:X−→Zis bounded, we then obtain\n∥\u0000\nL−1(δ+h)− L−1(δ)\u0001\nf∥X≤ |2hδ+h2|∥L−1(δ)∥∥∆1,S∥∥L−1(δ+h)∥∥f∥Z (41)\n≤ |h|C,\nas desired.\nTo check that the map ∆ 1,S:X−→Zis indeed well defined, let f=f1+aζ+b∈X. Because\nf1∈H2\nr,γ1,σ1(R2), it then follows from the definition of this space that ∆ 1,Sf1∈L2\nr,γ1,σ1+2(R2). From\nLemma 5.2 we know that the function ζcan be decomposed as ζ=ζ1+ζf, where ζ1∈H2\nr,γ(R2)\nwith γ∈(0,1), and ζf∈R. The definition of this last weighted space then implies that the term\n∆1,Sζ1is in L2\nr,γ(R2)⊂L2\nr,γ1,σ1+2(R2). This last embedding holds since γ > γ 1andσ1+ 2>0 .\nLastly, combining ζf+b∈Rwe obtain that ∆ 1,S(ζf+b) =−(ζf+b)/S2, showing that ∆ 1,Sf∈Z.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 43\nTo complete the proof of this Lemma, notice that from inequality (41) we are able to deduce that\nthe derivative of the operator L−1(δ) with respect to δis given by\n−2δL−1(δ)∆1,SL−1(δ) :Z−→X.\nSince this map is the composition of bounded operators, it is well defined. We can therefore conclude\nthat the operator L−1(δ) :Z−→XisC1with respect to δ, forδ >0. □\n5.3.Step 3. In this section we consider the nonlinear terms,\n˜N1(R1, ∂Sϕ1;δ) =N1(R1, ϕ1;δ, δ4) + (1 −ρ0)[2∂Sϕ0∂Sϕ1] + 3(1 −ρ2\n0)(R0+δR1)/δ\n˜N2(R1, ∂Sϕ1;δ) =N2(R1, ϕ1;δ, δ4)−(1−ρ0)[∆0,Sϕ1] + 3β(1−ρ2\n0)(R0+δR1)/δ,\nwhere N1andN2are as in expressions (17) and (18), respectively, with ε=δ4. We show that these\nnonlinearities define bounded operators between appropriate weighted Sobolev spaces. We start\nwith two preliminary results followed by two lemmas regarding terms of the form ( ∂Sϕ0+δ∂Sϕ1)\nand ( R0+δR1). We then move on to prove the main result of this subsection in Proposition 5.14.\nNotation: Throughout this section ϕ0andR0are as in Proposition 4.8. We also let\n∂Sϕ1(S) =∂Sϕn(S) +a\nS+∂Sϕf∈Y, (42)\nR1(S) =Rn(S) +aζ(S) +Rf∈X, (43)\n(44)\nwhere the spaces X=H2\nr,γ1,σ1(R2)⊕ {ζ(S)} ⊕RandY=H1\nr,γ1,σ1+1(R2)⊕ {1\nS} ⊕Rare defined in\n(32). (Recall that we assume γ∈(−1,0), while σ∈(−2,−1)).\nLemma 5.10. Letkbe an integer satisfying k≤3, and take γ≥γ1>−1, and σ >−1. Then, the\nembedding\nH3\nr,γ(R2)⊂Hk\nr,γ1,σ(R2)\nholds.\nProof. Letf∈H3\nr,γ(R2). We first study the behavior of this function in the far field R2\\B1, where\nB1is the unit ball. Since γ1≤γ, we have that the inequality ⟨x⟩γ1<⟨x⟩γholds. It then follows\nthat for all multi-index αwith 0 ≤ |α| ≤k, the function Dαf∈L2\nr,γ1,σ+|α|(R2\\B1).\nTo show that fis inHk\nr,γ1,σ(B1), notice first that we have the following relations,\nH3\nr,γ(B1)≡H3(B1)⊂C1(B1),\nHere, the first equivalence is a consequence of the weight ⟨x⟩being bounded near the origin, and\nthe last inclusion follows from standard Sobolev embeddings. As a result, we can bound the norm\n∥f∥L2r,γ1,σ(B1)≤ ∥f∥2\nL∞(B1)Z1\n0r2σr dr < ∞,\nwhere the last inequality follows from our assumption that σ >−1. On the other hand, for any\nderivative of order 1 ≤s≤k, which for simplicity we represent as Dsf, we have that the norm\n∥Dsf∥L2\nr,γ1,σ+s(B1)≤CZ1\n0|Dsf|2r2(σ+s)r dr≤CZ1\n0|Dsf|2r dr < ∞.\nIn this case, the second inequality is a consequence of σ+s >0, while the last inequality follows\nfrom our assumption that f∈H3\nr,γ(R2). The results of the Lemma then follow. □EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 44\nLemma 5.11. Letσ∈R, and take γ1, γ2>−1. Then, given f∈H3\nr,γ1(R2)andg∈H2\nr,γ2,σ(R2),\ntheir product\nfg∈H2\nr,γ,σ(R2),\nwhere γ= min {γ1, γ2}.\nProof. Because the value of γ1, γ2>−1 functions in the spaces H3\nr,γ1(R2) and H2\nr,γ2,σ(R2), as well as\ntheir derivatives, decay at infinity. As a result, given an index αwith 0 ≤ |α| ≤k, the expressions\nDα(fg) are bounded by an L2\nr,γ(R2\\B1) function, where γ= min {γ1, γ2}. We may thus conclude\nthat the product fgis inHk\nr,γ,σ(R2\\B1).\nTo show that the function fghas the desired properties near the origin we use the following\ninclusions,\nf∈H3\nr,γ1(B1)⊂H3(B1)⊂C1(B1)\ngrσ+2∈Hk(B1)⊂Ck−2(B1).\nThese follow from Sobolev embeddings and the inequality |g|rσ+s<|g|rσ, which is valid for values\nofs >0 and r <1. A short computation then shows that any function of the form Dα(fg)rσ+|α|,\nwhere 0 ≤ |α| ≤k, can be written as the product of a C(B1) function and an element of L2(B1).\nFor example, letting D2(fg) denote a derivative of order 2, we can use the product rule to write\nD3(fg)rσ+2= (D2f)grσ+2+ (Df)(Dgrσ+2) +f(D2grσ+2).\nIn this case, the functions grσ+2,(Df) and fare bounded, while the remaining terms are in L2(B1).\nThe results of the Lemma then immediately follow. □\nWith the help of the above Lemmas, and properties of the spaces Hk\nr,γ,σ(R2) summarized in\nSubsection 2.1, we now bound the terms ( ∂Sϕ0+δ∂Sϕ1)2and ( R0+δR1)min appropriate weighted\nspaces. These lemmas will then be used to prove the main result of this subsection, Proposition 5.14.\nWhen looking at the proofs of these results, it is worth keeping in mind Figure 4, which summarizes\ndecay properties of the spaces, Hk\nr,γ,σ(R2) and Hk\nr,γ(R2).\nLemma 5.12. Letγ1∈(−1,0),σ1∈(−2,−1), define Y=H1\nr,γ1,σ1+1(R2)⊕\b1\nS\t\n⊕R, and take\n∂Sϕ1=∂Sϕn+a\nS+∂Sϕf∈Y.\nThen, the expression\n(∂Sϕ0+δ∂Sϕ1)2∈L2\nr,γ1,σ1+2(R2)⊕\u001a1\nS2\u001b\n⊕R.\nProof. From Proposition 4.8 we know that the function ∂Sϕ0→κasS→ ∞ , with κ∈R; while the\nfunction ( ∂Sϕ0−κ)∈H3\nr,γ1(R2). We can then expand\n(∂Sϕ0+δ∂Sϕ1) =(∂Sϕ0−κ) +δ∂Sϕn+δa\nS+ (κ+δ∂Sϕf).\nBecause σ1+ 1>−1, Lemma 5.10 shows that H3\nr,γ1(R2)⊂H1\nr,γ1,σ1+1(R2). We can therefore write\n(∂Sϕ0+δ∂Sϕ1)2= (h+δa\nS+c)2\n= [h2+ 2δha\nS+ 2ch+ 2δca\nS] +δ2a2\nS2+c2EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 45\nwhere h= (∂Sϕ0−κ) +δ∂Sϕn∈H1\nr,γ1,σ1+1(R2) and c= (κ+δ∂Sϕf)∈R. To finish the proof of\nthis lemma we must show that the terms in the brackets are in L2\nr,γ1,σ1+2(R2).\nNotice first thata\nS∈L2\nr,γ1,σ1+2(R2), while the embedding H1\nr,γ1,σ1+1(R2)⊂L2\nr,γ1,σ1+2(R2), which\nholds since σ1+1< σ1+2, shows that his the correct space. Recalling the definition of our weighted\nSobolev spaces, we also find thata\nSh∈L2\nr,γ1,σ1+2(R2). Finally, Lemma 2.8 shows that the product\nh2is in the space H1\nr,γ1,2σ1+3(R2)⊂L2\nr,γ1,σ1+2(R2). Notice that this last embedding follows from the\ninequality 2 σ1+ 3< σ1+ 2, which holds thanks to the assumption that σ1∈(−2,−1). □\nIn this next lemma, the function ζ(S) is as in (32).\nLemma 5.13. Letγ1∈(−1,0),σ1∈(−2,−1), define X=H2\nr,γ1,σ1(R2)⊕ {ζ(S)} ⊕R, and take\nR1=Rn+aζ(S) +Rf∈X.\nThen, for any m∈N∪ {0}, the expression\n(R0+δR1)m∈H3\nr,γ1(R2)⊕H2\nr,γ1,σ1(R2)⊕R.\nProof. From Proposition 4.8 we know that the function R0→ −1\n2κ2asS→ ∞ , with κ∈R; while\n(R0−1\n2κ2)∈H3\nr,γ1(R2). Thus, we may expand,\n(R0+δR1) = (R0−1\n2κ2) +δRn+δaζ(S) + (1\n2κ2+δRf).\nThen from Lemma 5.2 we know that ζ(S) =ζ1+ζfwhere ζ1∈H3\nr,γ(R2) with γ∈(0,1), and ζfis\na constant. We can therefore write\n(R0+δR1) =f+g+d\nwith f= (R0−1\n2κ2) +ζ1∈H3\nr,γ1(R2),g=δRn∈H2\nr,γ1,σ1(R2), and d= (1\n2κ2+δRf)∈R.\nNow, looking at the product,\n(f+g)m=mX\nℓ=0\u0012m\nℓ\u0013\nfℓgm−ℓ,\nand using Lemma 5.11, one can show that all terms of the form fℓgm−ℓare in H2\nr,γ1,σ1(R2), whenever\n1< ℓ, m −ℓ < m . Indeed, since H3\nr,γ1(R2)⊂C1(R2), it follows that this space is a Banach algebra\nand therefore fp∈H3\nr,γ1(R2) for any integer p. Similarly, because σ1<−1, it follows from Lemma\n2.7 that the space H2\nr,γ1,σ1(R2) is also closed under multiplication, so that gpis in H2\nr,γ1,σ1(R2),\nagain for any integer p. Therefore, fℓgm−ℓis always the product of a bounded function, fℓ, times\na function in gm−ℓ∈H2\nr,γ1,σ1(R2). If we now include terms with ℓ= 0, m, we may conclude that\n(f+g)m∈H3\nr,γ1(R2)⊕H2\nr,γ1,σ1(R2). If we then write\n(f+g+c)m=mX\nℓ=0\u0012m\nℓ\u0013\n(f+g)ℓcm−ℓ\nwe immediately see that the results of the lemma hold.\n□\nWe now state and prove the main result of this section.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 46\nProposition 5.14. LetX, Y, Z be Banach spaces defined as in (32). Then, the maps ˜Ni:X×Y×\nR−→Z, for i= 1,2, defined by\n˜N1(R1, ∂Sϕ1;δ) =N1(R1, ϕ1;δ, δ4) + (1 −ρ0)[2∂Sϕ0∂Sϕ1] + 3(1 −ρ2\n0)(R0+δR1)/δ\n˜N2(R1, ∂Sϕ1;δ) =N2(R1, ϕ1;δ, δ4)−(1−ρ0)[∆0,Sϕ1] + 3β(1−ρ2\n0)(R0+δR1)/δ,\nwhere N1andN2are as in expressions (17) and(18), respectively, are bounded.\nProof. To prove the proposition one establishes decay properties at infinity and near the origin for\neach the nonlinear terms in ˜Ni,i= 1,2. This analysis is very similar to the one done in the proofs\nof Lemmas 5.12 and 5.13. We therefore omit most of the details and just state the results. We will\nonly show that terms ˜N(w;ε) appearing in the expressions Ni,i= 1,2 are in the space Z. Here\nε=δ4, but for convenience we omit this detail in the proof and just write ε.\n•Elements in L2\nr,γ1,σ1+2(R2):\n❖(1−ρ0)(∆0,Sϕ1+ 2∂Sϕ0∂Sϕ1)\n❖(ρ0+R0+δR1)(∆0,Sϕ0+δ∆0,Sϕ1)\n❖(∂SR0+δ∂SR1)(∂Sϕ0+δ∂Sϕ1)\n•Elements in L2\nr,γ1,σ1+2(R2)⊕\b1\nS2\t\n⊕R:\n❖(R0+δR1)(∂Sϕ0+δ∂Sϕ1)2\n•Elements in L2\nr,γ1,σ1+1(R2)⊕R:\n❖ρ0(R0+δR1)2\n❖(R0+δR1)3\n•Elements in H2\nr,γ1,σ1+1(R2):\n❖(1−ρ0)(R0+δR1)\n•Elements in L2\nr,γ1,σ1+2(R2)⊕\b1\nS2\t\n:\n❖∆1,S(R0+δR1)\n•Elements in L2\nr,γ1,σ1+2(R2)⊕R:\n❖ρ0(∂Sϕ0+δ∂Sϕ1)2\n•Since ρ0∈H3\nr,γ(R2) we then have that ∂rρ0∈H2\nr,γ(R2) and ∆ 1ρ0∈H1\nr,γ(R2) and ∂rρ∂Sϕ0∈\nH2\nr,γ(R2).\nThe statement of the proposition then follows from the above results, the embeddings Hs\nr,γ1,σ1(R2)⊂\nHk\nr,γ1,ν(R2), which hold whenever σ1< νands > k , and the inclusion H3\nr,γ(R2)⊂H2\nr,γ1,σ1+2(R2)\nwhich follows from Lemma 5.10.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 47\nWe conclude by showing that ˜N(w;ε)e−iϕ∈Z.\nFirst, recall that\n˜N(w;ε) =ε2D\ndR(1 + i β)∆1|w|2w+ (1−ε2D\ndR∆1)N(w;ε).\nwhere N(w;ε) is given as in Hypothesis (H1), and is thus composed of elements of the form |w|2nw\nwith n∈ {1,2,3,···}. We show that ˜N(w;ε)e−iϕ∈Zin three steps.\nStep A. We consider first the term (∆ 1|w|2w)e−iϕ. Writing |w|2w=ρ3eiϕand letting W=ρ3,\nwe can then expand\n(∆1Weiϕ)e−iϕ= ∆ 1W+ iW∆0ϕ−2i∂rϕ∂rW−W(∂rϕ)2. (45)\nIn what follows we check that each element is in the space Z=L2\nr,γ1,σ1+2(R2)⊕\b1\nS2\t\n⊕R, keeping\nin mind that S=δr.\nWe start by noticing that\nW(r) :=ρ3= (ρ0(r) +δ2(R0+δR1))3∈H3\nr,γ(R2)⊕H2\nr,γ1,σ1(R2)⊕R. (46)\nThis follows from the fact that ( ρ0−1)∈H3\nr,γ(R2) and the results of Lemma 5.13, which show that\nthe space H3\nr,γ(R2)⊕H2\nr,γ1,σ1(R2)⊕Ris a Banach algebra. Similarly, in the proof of Lemma 5.12 it\nis shown that\n∂Sϕ=∂Sϕ0+∂Sϕ1∈H1\nr,γ1,σ1+1(R2)⊕\u001a1\nS\u001b\n⊕R. (47)\nWe will use these simplifications in the analysis for each of the terms appearing in expression (45).\nTo prove that ∆ 1W∈Z, we use (46) and write\n∆1W= ∆ 1(h1+h2+c) = ∆ 1h1+ ∆ 1h2+c\nr2\nwhere c∈Rand the functions, h1andh2are in H3\nr,γ(R2) and H2\nr,γ1,σ1(R2), respectively. The result\nthen follows from the definition of these spaces, and the embedding H1\nr,γ(R2)⊂L2\nr,γ1,σ1+2(R2), given\nby Lemma 5.10.\nNext, because the function Wis uniformly bounded, to show W∆0ϕ∈Z, we only need to show\nthat ∆ 0ϕ∈Z. To do this, we use (47) and write\n∆0ϕ=∂r(h+a\nr+c) +1\nr(h+a\nr+c) =∂rh+1\nrh+c\nr\nwhere h∈H1\nr,γ1,σ1+1(R2) and a, c∈R. The result again follows from the definition of these spaces,\nand the fact that the function1\nr∈L2\nr,γ1,σ1+2(R2) (see Figure 4 in Section 2).\nTo study the term ∂rW∂rϕ, we consider (47) and expand,\n∂rW∂rϕ=∂rW(h+a\nr+c) =∂rWh+a\nr∂rW+c∂rW\nwith h∈H1\nr,γ1,σ1+1(R2) and a, c∈R. Because\n∂rW∈H2\nr,γ(R2)⊕H1\nr,γ1,σ1+1(R2)⊂H1\nr,γ1,σ1+1(R2)\n(see Lemma 5.10), thena\nr∂rW∈L2\nr,γ1,σ1+2(R2). Similarly, we have that ∂rW∈H1\nr,γ1,σ1+1(R2)⊂\nL2\nr,γ1,σ1+2(R2), which follows from the definition of these doubly-weighted Sobolev spaces. Lastly, us-\ning Lemma 2.8 we see that term ∂rWh∈L2\nr,γ1,2σ1+3(R2)⊂L2\nr,γ1,σ1+2(R2), where this last embedding\nfollows from the inequality 2 σ1+ 3≤σ1+ 2.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 48\nFinally, because Wis uniformly bounded, the results from Lemma 5.12 allow us to conclude that,\nW(∂rϕ)2∈L2\nr,γ1,σ1+2(R2)⊕\b1\nS2\t\n⊂Z.\nStep B. We now look at the nonlinear terms included in the expression N(w;ε)e−iϕ, which as\nmentioned above, are of the form |w|2nwe−iϕwith n∈ {1,2,3,···}. Given that w=ρeiϕ, we may\nwrite,\n|w|2nwe−iϕ=ρ2n+1.\nThus, it suffices to show that ρp∈Zfor all odd integers p≥3. Notice that this easily follows from\nthe definition of ρ,\nρ=ρ0(r) +δ2(R0+δR1))3∈H3\nr,γ(R2)⊕H2\nr,γ1,σ1(R2)⊕R,\nthe fact that this last space is closed under multiplication, and the embeddings\nH3\nr,γ(R2)⊕H2\nr,γ1,σ1(R2)⊂H2\nr,γ1,σ1(R2)⊂L2\nr,γ1,σ1+2(R2),\nwhich follow from Lemma 5.10 and the definition of these spaces.\nStep C. To show [∆ 1N(w;ε)]e−iϕ, or equivalently [∆ 1ρpeiϕ]e−iϕ, is in the space Z, we use the\nresults from Step 2 to write ρpeiϕ=V(r)eiϕ, for some function V(r)∈H2\nr,γ1,σ1(R2)⊕R. Then,\n[∆1V(r)eiϕ]e−iϕ= [∆ 1(Veisϕ)]e−iϕ= ∆ 1V+iV∆0ϕ+ 2i∂rϕ∂rV−V(∂rϕ)2.\nWe can then use a similar argument as in Step A, to show that these terms are in the space Z.\n□\nAppendix A.\nIn this appendix we justify hypothesis (H1), reproduced below.\nHypothesis (H1).The nonlinear term N(w;ε)is of order O(ε|w|4w), and every term in this\nexpression is of the form c|w|2nw, with c∈Candn∈ {1,2,3,···}.\nTo justify Hypothesis (H1) we ‘briefly’ review the results from [21], where the nonlocal complex\nGinzburg-Landau equation (14) is derived as an amplitude equation for systems that undergo a\nHopf bifurcation and that include nonlocal diffusion.\nAs in the case of reaction diffusion equations, the approach from [21] takes advantage of the\nseparation in scales between the frequency of time oscillations that emerge from the Hopf bifurcation,\nt, and the long spatial, R=ε|x|, and time scales, T=ε2t, that dictate changes in the amplitude\nof these oscillation. To derive a reduce equation for spiral waves it is then assumed that these\npatterns bifurcate from the zero solution and have an amplitude that is proportional to the small\nparameter ε. In polar coordinates ( r, ϑ), these patterned solutions are also assumed to take the form\nU(r, θ, R ;��, µ) =U(r, R;ϑ+ωt+ε2µt), where ωandµare constants related to the rotational speed\nof the spiral wave and where θ=ϑ+ωt+ε2µt.\nWith these two assumptions one can then write the solution as a regular perturbation in ε, i.e.\nU(r, θ, R ;ε, µ) =εU1(θ, R;ε, µ) +ε2U2(θ, R;ε, µ) +ε3U3(r, θ;ε, µ).\nInserting this ansatz into the original nonlocal system and separating terms of equal powers in ε\none obtains a sequence of equations. Assuming that the first order correction term is of the form\nU1(θ, R;ε, µ) =W1w(R;ε, µ)eiθ+¯W1¯w(R;ε, µ)e−iθ,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 49\nwith (i ω, W 1) the eigenvalue-eigenvector pair associated with the Hopf bifurcation, one can then\nshow that U1solves the O( ε) equation, with w(R) an arbitrary complex valued function. Moreover,\nit is then possible to prove that the O( ε2) equation has a solution, U2∼O(U2\n1), which depends\nsmoothly on U1.\nFinally, the nonlocal Ginzburg-Landau equation is derived after gathering all terms of order O( ε3)\nand higher into a single equation, which for ease of exposition in this summary we call the ‘main\nequation’. The results from [21] then show that there exist a Banach space, X, and projections,\nP:X−→X∥and ( I−P) :X−→X⊥, such that the linear part of this main equation, L, can be\nsplit into an invertible operator L⊥:X⊥−→X⊥and a bounded operator L∥:X∥−→X∥. Using\nthese projections and Lyapunov-Schmidt reduction, the main equation can then be split into an\ninvertible equation for the variable U3, and a reduced equation for the unknown amplitude w(r).\nThe invertible equation can then be written as\n0 =G(U3;U1, ε, µ) (48)\nwith G:X∥×X⊥×R2−→X⊥a smooth well defined map with a derivative operator DU3G=L⊥:\nX⊥−→X⊥that is invertible. The implicit function theorem then proves the existence of a smooth\nmap Ψ : U ⊂X∥× V ⊂ R2−→X⊥, such that U3= Ψ( U1, ε, µ) solves equation (48), and satisfies\nΨ(0, ε, µ) = 0 and DU1Ψ(0;ε, µ) for ( ε, µ)∈ V. Inserting this map Ψ into the reduced equation and\nprojecting onto the space X∥then results in the nonlocal complex Ginzburg-Landau equation (14).\nA couple of remarks are in order:\n•Notice that unlike a formal multiple scales approach, where this reduced equation comes\nfrom applying a solvability condition to the O( ε3) equation, the nonlocal Ginzburg-Landau\nequation (14) derived in [21] includes terms of order O( ε3) and higher. In equation (14) the\nhigher order terms are then summarized in the expression N(w;ε), which therefore includes\npowers of the function U3= Ψ( U3, ε, µ)∼O(U2\n1).\n•The projection P:X−→X∥takes the form\nPu=Z2π\n0⟨u, W 1⟩eiθdθ+Z2π\n0⟨u,¯W1⟩e−iθdθ\nbut because we are interested in real solutions, it suffices to consider only the projection\nonto the span of {eiθ}.\n•The terms in the expression N(w, ε) are generated from the nonlinearities that appear in\nthe original nonlocal equation. We assume that these terms are polynomial functions of the\nunknown variable.\nThe above discussion then implies that elements in N(w, ε) come from terms of the form U2\n2∼U4\n1\norUp\n1Uq\n3with p, q≥1. Looking first at U2and using the definition of U1we find\nU2∼U4\n1=4X\ns=0\u00124\ns\u0013\nws¯w4−seiθ(4−2s). (49)\nOn the other hand, since U3= Ψ( U1, ε, µ)∼O(U2\n1) is a smooth function, we can Taylor expand it\nabout the point U1= 0. As a result, we may write\nUp\n1Uq\n3=Up\n1Ψq=∞X\ns=0asU2q+p+s\n1 ,EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 50\nwhere, again from the definition of U1, we have that\nUm\n1=mX\nk=0\u0012m\nk\u0013\nwk¯wm−keiθ(2k−m)(50)\nform≥2q+p≥3. Notice then that projecting terms of the form (49) or (50) into the span of\n{eiθ}, gives\nPUm\n1=mX\nk=0\u0012m\nk\u0013\nPwk¯wm−keiθ(2k−m)\n=mX\nk≥(1+m)/2,\n2k−m=1\u0012m\nk\u0013\nwk¯wk−1eiθ\n=mX\nk≥(1+m)/2,\n2k−m=1\u0012m\nk\u0013\n|w|2(k−1)weiθ.\nBecause we have the restrictions 2 k−m= 1,k≥(1+m)/2, and m≥3, we may conclude k−1≥1.\nTherefore, elements in N(w;ε) are given by |w|2nwwith n={1,2,3,···}.\nAppendix B.\nTo prove Lemma 2.9 will use a result by Kato [26], which we state next.\nProposition B.1 (Kato, p.370) .LetT(γ)be a family of compact operators in a Banach space X\nwhich are holomorphic for all γ∈C. Call γa singular point if 1 is an eigenvalue of T(γ). Then\neither all γ∈Dare singular points or there are only finitely many singular points in each compact\nsubset of D.\nWith the above result in hand, we prove that ∆ −Id is invertible when posed over weighted and\ndoubly-weighted Sobolev spaces.\nLemma. 2.9 Let s≥2and suppose γ∈R. Then, the operator\n∆−Id :Hs\nγ(R2)−→Hs−2\nγ(R2)\nis invertible.\nProof. We work with the commutative diagram,\nHs\nγ(Rd) Hs−2\nγ(Rd)\nHs(Rd) Hs−2(Rd)(∆−Id)\n⟨x⟩γ⟨x⟩γ\nL(γ)\nwhere the operator Lis given by L(γ)u= (∆−Id)u+T(γ)uwith\nT(γ)u= 2γ⟨x⟩−2∇u·x+ (γ(γ−1)|x|2⟨x⟩−4+γ⟨x⟩−2)u.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 51\nEach term in the map T(γ) involves a multiplication operator that can be approximated by a\nfunction with compact support. It then follows, by Sobolev embeddings, that T(γ) is a compact\nperturbation of the invertible operator (∆ −Id) : Hs(Rd)→Hs−2(Rd). Consequently, (∆ −Id) :\nHs\nγ(Rd)→Hs−2\nγ(Rd) is Fredholm of index zero.\nTo check that the operator has a trivial kernel, we proceed by contradiction. Suppose for the\nmoment that there is a number γ∗∈Rand a function u∈Hs\nγ∗(Rd) such that L(γ∗)u= 0. Then, the\ncommutative diagram and the embedding Hs\nγ∗(Rd)⊂Hs\nγ(Rd), with γ < γ∗, imply that L(σ)u= 0\nfor all γ < γ∗.\nOn the other hand, the operator\nHs(Rd)−→ Hs−2(Rd)\nu 7−→ (∆−Id)−1T(γ)u,\nwhich is compact and analytic for all γ∈C, has λ= 1 as an eigenvalue if and only if L(γ) has\na non trivial kernel. It then follows, by our previous argument, that λ= 1 is an eigenvalue of\n(Id−∆)−1T(γ) for all γ < γ∗. Using Proposition B.1 we may then conclude that λ= 1 is an\neigenvalue for all γ∈C. In particular, this result holds for γ= 0, which is a contradiction since\n(∆−Id)−1T(0) = (Id −∆)−1is invertible. As a result, ker L(γ) ={0}for all γ∈Rand the map\n(∆−Id) :Hs\nγ(Rd)→Hs−2\nγ(Rd) is therefore an isomorphism.\n□\nAppendix C.\nIn this section we work on re-grouping the nonlinear terms\nN(R1, ∂Sϕ1;δ) =−2βδ2∆1,SR1+˜N2(R1, ϕ1;δ)−β˜N1(R1, ϕ1;δ),\n=−2βδ2∆1,SR1−(1−ρ0)[∆0,Sϕ1+ 2β∂Sϕ0∂Sϕ1)\n+N2(R1, ϕ1;δ, δ4)−βN1(R1, ϕ1;δ, δ4),\nto arrive at equation (33),\nN(R1, ∂Sϕ1;δ) =C(R1)(∂Sϕ1)2+D(R1, ∂Sϕ1)1\nS2+M(R1, ∂Sϕ1;δ).\nWe recall that here, N1andN2are given as in expressions (17) and (18), respectively, with ε=δ4,\nand\n•C(R1)∈Rwith\nC(R1) =βδ−βδ3(1\n2k2+δζf+δRf)−δ4α(1 +δ2(1\n2k2+δζf+δRf)) + O( ε=δ4) as δ→0.(51)\n•D(R1, ∂Sϕ1)∈Rwith,\nD(R1, ∂Sϕ1) = (−βδ−βδ2a2+αδ2−δ6αa2)[δRf+δζf+1\n2k2] + O( ε=δ4) as δ→0.(52)\n•M:X×Y×R−→˜Z=L2\nr,γ1,σ1+2(R2)⊕R.\nAs mentioned in Section 5, this is the result of extracting multiples of ( ∂Sϕ1)2and1\nS2fromN.\nWe start with the constant C(R1).EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 52\nUsing the expressions for N1, given in (17), and N2, given in (18), we see that there are three\nterms in N2−βN1, which involve the function ( ∂Sϕ1)2(ignoring the order O( ε=δ3) terms, since\nthese are already higher order). These are,\n• −δ2α(ρ0+δ2(R0+δR1))(δ∂Sϕ1)2,\n• −βδ(R0+δR1)(δ∂Sϕ1)2\n•βδρ0(∂Sϕ1)2\nSince ρ0andR0are given as in Propositions 4.1 and 4.8, we may re-arrange,\n(ρ0+δ2(R0+δR1)) =( ρ0−1) + 1 + δ2[(R0−1\n2κ2) +δ(Rn+ζ(S) +Rf) +1\n2κ2]\n=(ρ0−1) +δ2(R0−1\n2κ2) +δ3(ζ1(S) +Rn) +\u0014\n1 +δ2\u00121\n2κ2+δζf+δRf\u0013\u0015\nwhere κ∈Ris given in Proposition 4.8, and we have also used the notation R1=Rn+ζ(S)+Rf∈X.\nIn addition, ζ(S) =ζ1(S) +ζfis as in Lemma 5.2. As a result\n−δ2α(ρ0+δ2(R0+δR1)) =h−δ2α\u0014\n1 +δ2\u00121\n2κ2+δζf+δRf\u0013\u0015\n,\nwith h∈˜Zand the remaining terms in R.\nA similar analysis then shows that\n• −βδ(R0+δR1) =h−βδ(1\n2κ+δ(ζf+Rf))\n•βδρ0=h+βδ\nwhere again we use has a generic function in ˜Z. The results for C(R1) then follow.\nNext, we gather all terms that are multiples of1\nS2. These are,\n•(αδ2−βδ)[∆1,SR0+δ∆1,SR1]\n• −(αδ4+βδ)[(R0+δR1)(∂Sϕ0+δ∂Sϕ1)2]\nTo see why, using Proposition 4.8 and Lemma 5.2, we first expand\n(R0+δR1) =(R0−1\n2κ2) +δ(Rn+ζ1(S) +ζf+Rf) +1\n2κ2\n= (R0−1\n2κ2) +δζ1(S) +δRn\n| {z }\nh+\u00141\n2κ2+δ(ζf+Rf)\u0015\nwith h∈H3\nr,γ(R2)⊕H2\nr,γ1,σ1(R2),γ∈(0,1), γ1∈(−1,0) and σ1∈(−2,−1). It then follows that\n∆1,S(R0+δR1) = ∆ 1,Sh−\u00141\n2κ2+δ(ζf+Rf)\u00151\nS2\nwhere, from the definition of these spaces, we may conclude that ∆ 1,Sh∈H1\nr,γ(R2)⊕L2\nr,γ1,σ1+2(R2)⊂\n˜Z.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 53\nTo analyze the second expression, [( R0+δR1)(∂Sϕ0+δ∂Sϕ1)2], we first write\n(R0+δR1) =h1+\u00141\n2κ2+δ(ζf+Rf)\u0015\n| {z }\nd\nwith h1=has above and d∈R. Similarly, using the notation ∂Sϕ1=∂Sϕn+a\nS+∂Sϕf∈Y\nintroduced in Section 5, together with Lemma 5.12 stated in Subsection 5.3, we may write\n(∂Sϕ0+δ∂Sϕ1)2=h2+δ2a2\nS2+c\nwith h2∈L2\nr,γ1,σ1+2(R2) and a, c∈R. We then have\n(R0+δR1)(∂Sϕ0+δ∂Sϕ1)2=(h1+d)(h2δ2a2\nS2+c)\n= (h1+d)h2+ (h1+d)c+δ2a2\nS2h1+δ2da2\nS2\nwhere, thanks to Proposition 5.14 proved in Subsection 5.3, we may conclude that the first three\nterms are in ˜Z. The results for D(R1, ∂Sϕ1) then follow.\nAppendix D.\nLemma. Leta, b∞, c, d be positive constants and take b(x)to be a function in C1([0,∞),R)such\nthatb(x)∼ −b∞+O(1 /x)asx→ ∞ . Then, there exists a solution, q(x), to the ordinary differential\nequation,\nq′+a\nxq+b(x)q−cq2+d\nx2= 0 x≥0\nsatisfying,\nq(x)∼O(1/x)asx→0, q (x)∼q∞+ O(1 /x)asx→ ∞\nfor some q∞∈R. Moreover, there is a constant νand a function q2∈H1\nr,γ1,σ1+1(R2), with γ1∈\n(−1,0)andσ1∈(−2,−1), such that\nq(x) =q2(x) +ν\nx+q∞∈Y.\nProof. We start by using the Hopf-Cole Transform, q=αy′\ny, with α= 1/c, to arrive at the linear\nequation\ny′′+a\nxy′+b(x)y′−dc\nx2y= 0.\nWe first use the Frobenius Method [6] to determine the behavior of solutions near the origin.\nIt is straight forward to check that x= 0 is a regular singular point of the equation. We can\ntherefore look for solutions of the form y(x) =xrP∞\nk=0akxk.Multiplying the equation by x2and\ninserting this guess, we obtain,\n0 =∞X\nk=0ak(r+k)(r+k−1)xr+k+aka(r+k)xr+k+akb(x)(r+k)xr+k+1−akcdxk+1,\nand one finds that the indicial equation is\nr2+ (a−1)r−cd= 0.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 54\nSince the discriminant ( a−1)2+ 4cd > 0, it follows that the roots, r1, r2, of this equation are real.\nWe then have two cases, r1−r2is or is not an integer. If the difference between the roots is not an\ninteger, we then have the two linearly independent solutions of the form\ny1(x) =xr1∞X\nk=0akxk, y 2(x) =xr2∞X\nk=0bkxk.\nIf, on the other hand, r1−r2is an integer then the two linearly independent solutions are\ny1(x) =xr1∞X\nk=0akxk, y 2(x) =xr2∞X\nk=0bkxk+Clog(x)y1(x),\nfor some constant C. In either case, we see that q(x) =αy′\ny∼O(1/x) as x→0.\nNext, to determine the behavior of solutions at infinity, we use the change of coordinates ξ= 1/x,\nwhich leads to the equation\nd2y\ndξ2+(2−a)\nξdy\ndξ−˜b(ξ)\nξ2dy\ndξ−dc\nξ2y= 0. (53)\nThe point ξ= 0 is now an irregular singular point, so we first look for solutions of the form\ny(ξ) = eA/ξY(ξ) with Y(ξ) =ξr∞X\nk=0akξk.\nThis leads to the following equation for Y(ξ)\nY′′+ \n(2−a)\nξ−(˜b(ξ) + 2A)\nξ2!\nY′+ \nA(˜b(ξ) +A)\nξ4+aA\nξ3−cd\nξ2!\nY= 0.\nMultiplying the equation by ξ2and inserting the power series for Y(ξ), we obtain\n0 =∞X\nk=0ak(r+k)(r+k−1)ξr+k+ak(2−a)(r+k)ξr+k−ak(˜b(ξ) + 2A)(r+k)ξr+k−1\n+akA(˜b(ξ) +A)\nξξr+k−1+akaAξr+k−1−akcdxr+k.\nLetting A=−b∞, we find that the termA(˜b(ξ)+A)\nξ∼O(1) as ξ→0. Therefore, the indicial equation,\nwhich is determined by the expression\n−(˜b(ξ) + 2A)r+A(˜b(ξ) +A)\nξ+aA= 0,\ngives us r=a+ O(1) >0.\nAlthough, the sum describing Y(ξ) might not converge, it still provides us with an approximation\nfory(ξ) when ξ∼0. Keeping in mind that ξ= 1/x, we conclude that\ny(x)∼e−b∞xxr∞X\nk=0ak\nxkx→ ∞ .\nIt then follows that q(x) =αy′\ny∼ −αb∞+ O(1 /x) asx→ ∞ .EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 55\nLastly, to find a second solution to equation (53) that is linearly independent from the solution\nfound above, we assume y(ξ) =P∞\nk=0bkξr+k. Multiplying the equation by ξ2and inserting this\nguess, gives us\n∞X\nk=0bk(r+k)(r+k−1)ξr+k+bk(2−a)(r+k)ξr+k−bk˜b(ξ)ξr+k−1−bkcdξr+k.\nThe lowest term in the above expression must satisfy ˜b(ξ)r= 0, which shows that r= 0. We\nconclude that this second solution to the equation satisfies\ny(x)∼∞X\nk=0bk1\nxkasx→ ∞\nConsequently, q(x) =αy′\ny∼O(1/x) asx→ ∞ .\nBecause the function q(x) solves a first order differential equation, it is a function in C1((0,∞),R).\nThe asymptotic expansions found above, also show that it is possible to find constants q∞, ν∈R\nsuch that\nq(x) =q2(x) +ν\nx+q∞\nwith q2(x)∼O(1) near the origin and q2(x)∼O(1/x) at infinity. It then follows that q2∈\nH1\nr,γ1,σ1+1(R2) with γ1∈(−1,0) and σ1∈(−2,0).\n□\nE-mail: gabriela@math.uh.edu\nReferences\n[1] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions with formulas, graphs, and mathe-\nmatical tables , volume 55. US Government printing office, 1948.\n[2] Maria Aguareles, Inmaculada Baldom´ a, and Tere M Seara. A rigorous derivation of the asymptotic wavenumber\nof spiral wave solutions of the complex Ginzburg-Landau equation. arXiv preprint arXiv:2306.02191 , 2023.\n[3] Andrew L. Belmonte, Qi Ouyang, and Jean-Marc Flesselles. Experimental survey of spiral dynamics in the\nbelousov-zhabotinsky reaction. J. Phys. II France , 7(10):1425–1468, 1997.\n[4] Dwight Barkley. A model for fast computer simulation of waves in excitable media. Physica D: Nonlinear Phe-\nnomena , 49(1):61–70, 1991.\n[5] Andrew J. Bernoff. Spiral wave solutions for reaction-diffusion equations in a fast reaction/slow diffusion limit.\nPhysica D: Nonlinear Phenomena , 53(1):125–150, 1991.\n[6] William E Boyce, Richard C DiPrima, and Douglas B Meade. Elementary differential equations and boundary\nvalue problems . John Wiley & Sons, 2021.\n[7] Paul C Bressloff. Spatiotemporal dynamics of continuum neural fields. Journal of Physics A: Mathematical and\nTheoretical , 45(3):033001, dec 2011.\n[8] Olivia Cannon and Arnd Scheel. Coherent structures in nonlocal systems: Functional analytic tools. Journal of\nDynamics and Differential Equations , 2023.\n[9] S. Coombes. Waves, bumps, and patterns in neural field theories. Biological Cybernetics , 93(2):91–108, 2005.\n[10] Arjen Doelman, Bj¨ orn Sandstede, Arnd Scheel, and Guido Schneider. The dynamics of modulated wave trains .\nCWI. Modelling, Analysis and Simulation [MAS], 2005.\n[11] Irving R Epstein, Kenneth Kustin, Patrick De Kepper, and Mikl´ os Orb´ an. Oscillating chemical reactions. Scientific\nAmerican , 248(3):112–123, 1983.\n[12] Gr´ egory Faye and Arnd Scheel. Center manifolds without a phase space. Transactions of the American Mathe-\nmatical Society , 370(8):5843–5885, 2018.\n[13] Gerald B Folland. Real analysis: modern techniques and their applications , volume 40. John Wiley & Sons, 1999.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 56\n[14] Vladimir Garc´ ıa-Morales and Katharina Krischer. Nonlocal complex Ginzburg–Landau equation for electrochem-\nical systems. Phys. Rev. Lett. , 100:054101, Feb 2008.\n[15] Vladimir Garcia-Morales and Katharina Krischer. The complex Ginzburg–Landau equation: an introduction.\nContemporary Physics , 53(2):79–95, 2012.\n[16] Albert Goldbeter. Oscillations and waves of cyclic amp in dictyostelium: A prototype for spatio-temporal orga-\nnization and pulsatile intercellular communication. Bulletin of Mathematical Biology , 68(5):1095–1109, 2006.\n[17] J. M. Greenberg. Spiral waves for λ-ωsystems. SIAM Journal on Applied Mathematics , 39(2):301–309, 1980.\n[18] J.M Greenberg. Spiral waves for λ-ωsystems, II. Advances in Applied Mathematics , 2(4):450–455, 1981.\n[19] Patrick S Hagan. Target patterns in reaction-diffusion systems. Advances in Applied Mathematics , 2(4):400–416,\n1981.\n[20] Patrick S. Hagan. Spiral waves in reaction-diffusion equations. SIAM Journal on Applied Mathematics , 42(4):762–\n786, 1982.\n[21] Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete\nand Continuous Dynamical Systems-S , 15(9):2513–2551, 2022.\n[22] Gabriela Jaramillo. Can large inhomogeneities generate target patterns? Zeitschrift f¨ ur angewandte Mathematik\nund Physik , 74(4):134, 2023.\n[23] Gabriela Jaramillo and Arnd Scheel. Pacemakers in large arrays of oscillators with nonlocal coupling. Journal of\nDifferential Equations , 260(3):2060–2090, 2016.\n[24] Gabriela Jaramillo, Arnd Scheel, and Qiliang Wu. The effect of impurities on striped phases. Proceedings of the\nRoyal Society of Edinburgh: Section A Mathematics , 149(1):131– 168, 2019.\n[25] Gabriela Jaramillo and Shankar C Venkataramani. Target patterns in a 2d array of oscillators with nonlocal\ncoupling. Nonlinearity , 31(9):4162–4201, jul 2018.\n[26] Tosio Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013.\n[27] Zachary P. Kilpatrick and Paul C. Bressloff. Spatially structured oscillations in a two-dimensional excitatory\nneuronal network with synaptic depression. Journal of Computational Neuroscience , 28(2):193–209, 2010.\n[28] Pius Kirrmann, Guido Schneider, and Alexander Mielke. The validity of modulation equations for extended\nsystems with cubic nonlinearities. Proceedings of the Royal Society of Edinburgh: Section A Mathematics , 122(1-\n2):85–91, 1992.\n[29] Richard Koll´ ar and Arnd Scheel. Coherent structures generated by inhomogeneities in oscillatory media. SIAM\nJournal on Applied Dynamical Systems , 6(1):236–262, 2007.\n[30] N Kopell and L.N Howard. Target pattern and spiral solutions to reaction-diffusion equations with more than\none space dimension. Advances in Applied Mathematics , 2(4):417 – 449, 1981.\n[31] Marc TM Koper. Non-linear phenomena in electrochemical systems. Journal of the Chemical Society, Faraday\nTransactions , 94(10):1369–1378, 1998.\n[32] Katharina Krischer. Principles of Temporal and Spatial Pattern Formation in Electrochemical Systems , pages\n1–142. Springer US, Boston, MA, 2002.\n[33] Christian Kuehn and Sebastian Throm. Validity of amplitude equations for nonlocal nonlinearities. Journal of\nMathematical Physics , 59(7):071510, 2018.\n[34] Yoshiki Kuramoto. Chemical oscillations, waves, and turbulence . Courier Corporation, 2003.\n[35] Yoshiki Kuramoto and Tomoji Yamada. Pattern formation in oscillatory chemical reactions. Progress of Theoret-\nical Physics , 56(3):724–740, 09 1976.\n[36] Carlo R. Laing. Spiral waves in nonlocal equations. SIAM Journal on Applied Dynamical Systems , 4(3):588–606,\n2005.\n[37] Carlo R. Laing and William C. Troy. PDE methods for nonlocal models. SIAM Journal on Applied Dynamical\nSystems , 2(3):487–516, 2003.\n[38] Daniel Margerit and Dwight Barkley. Cookbook asymptotics for spiral and scroll waves in excitable media. Chaos:\nAn Interdisciplinary Journal of Nonlinear Science , 12(3):636–649, 08 2002.\n[39] Stefan C. M¨ uller, Theo Plesser, and Benno Hess. Two-dimensional spectrophotometry of spiral wave propagation\nin the Belousov-Zhabotinskii reaction: I. experiments and digital data representation. Physica D: Nonlinear\nPhenomena , 24(1):71–86, 1987.\n[40] Z. Nagy-Ungvarai, John J. Tyson, S. C. Mueller, L. T. Watson, and B. Hess. Experimental study of spiral waves\nin the cerium-catalyzed Belousov-Zhabotinskii reaction. The Journal of Physical Chemistry , 94(24):8677–8682,\n11 1990.EXISTENCE OF SPIRAL WAVES IN OSCILLATORY MEDIA WITH NONLOCAL COUPLING 57\n[41] Joseph Paullet, Bard Ermentrout, and William Troy. The existence of spiral waves in an oscillatory reaction-\ndiffusion system. SIAM Journal on Applied Mathematics , 54(5):1386–1401, 1994.\n[42] Arnd Scheel. Bifurcation to spiral waves in reaction-diffusion systems. SIAM Journal on Mathematical Analysis ,\n29(6):1399–1418, 1998.\n[43] Guido Schneider. The validity of generalized Ginzburg–Landau equations. Mathematical Methods in the Applied\nSciences , 19(9):717–736, 1996.\n[44] Shin-ichiro Shima and Yoshiki Kuramoto. Rotating spiral waves with phase-randomized core in nonlocally coupled\noscillators. Phys. Rev. E , 69:036213, Mar 2004.\n[45] Elias M Stein and Guido Weiss. Introduction to Fourier analysis on Euclidean spaces, volume 32 of. Princeton\nMathematical Series , 1971.\n[46] Klaus M. Stiefel and G. Bard Ermentrout. Neurons as oscillators. Journal of Neurophysiology , 116(6):2950–2960,\n2016. PMID: 27683887.\n[47] Dan Tanaka and Yoshiki Kuramoto. Complex Ginzburg–Landau equation with nonlocal coupling. Phys. Rev. E ,\n68:026219, Aug 2003.\n[48] Annette F. Taylor. Mechanism and phenomenology of an oscillating chemical reaction. Progress in Reaction\nKinetics and Mechanism , 27(4):247–326, 2002.\n[49] John J. Tyson and James P. Keener. Singular perturbation theory of traveling waves in excitable media (a review).\nPhysica D: Nonlinear Phenomena , 32(3):327–361, 1988.\n[50] A. van Harten. On the validity of the Ginzburg–Landau equation. Journal of Nonlinear Science , 1(4):397–422,\n1991.\n[51] Wim Van Saarloos. The complex Ginzburg–Landau equation for beginners. In Santa Fe Institute Studies in the\nSciences of Complexity–Proceedings Volume- , volume 21, pages 19–19. Addison–Wesley Publishing CO, 1995.\n[52] Arthur T. Winfree. Rotating chemical reactions. Scientific American , 230(6):82–95, 1974.\n[53] J. J. Zebrowski, K. Grudzinski, T. Buchner, P. Kuklik, J. Gac, G. Gielerak, P. Sanders, and R. Baranowski.\nNonlinear oscillator model reproducing various phenomena in the dynamics of the conduction system of the\nheart. Chaos: An Interdisciplinary Journal of Nonlinear Science , 17(1):015121, 03 2007." }, { "title": "2401.15281v1.Improved_confidence_intervals_for_nonlinear_mixed_effects_and_nonparametric_regression_models.pdf", "content": "Improved confidence intervals for nonlinear mixed-effects and\nnonparametric regression models\nNan Zhenga, Noel Cadiganb∗\naDepartment of Mathematics and Statistics,\nMemorial University of Newfoundland, St. John’s, NL, Canada. A1C 5S7\nbCentre for Fisheries Ecosystems Research,\nFisheries and Marine Institute of Memorial University of Newfoundland, St. John’s, NL, Canada. A1C\n5R3\n1. Introduction\nThe joint density function of a mixed-effects model (linear and nonlinear) can be written,\nwith wide generality, as f(D,Ψ|θ) =f(D|Ψ, θ)f(Ψ|θ), where D,Ψ and θare respectively\ndata, random effects (REs) and model parameters (FEs, fixed effects), and f(·) represents\nthe corresponding probability density/mass function (pdf/pmf). The logarithm of the joint\nlikelihood of data and REs is\nlj(Ψ, θ) = log f(D,Ψ|θ) = log f(D|Ψ, θ) + log f(Ψ|θ) =lc+lr, (1)\nwhere lc= log f(D|Ψ, θ) and lr= log f(Ψ|θ) denote respectively the log-likelihood condi-\ntional on REs and the log-likelihood of REs. We are addressing a general class of models\nwith broad applicability in the ecology and evolution field. These models, while not limited\nto, include integrated state-space fishery assessment models that incorporate various data\nsources depending on the fish stock. We do not introduce notation for specific models and\ndata, but see Aeberhard et al (2018) for an example. We do provide simpler examples of\nrandom walk state-space models and a generalized additive model.\nIn many applications of mixed-effects models, the objective of statistical inference is for\na single set of REs, which are considered to be fixed and frequentist inference is based on\nrepeated sampling of the data given the REs. Basically, the REs are conceptually treated\n∗Corresponding author. E-mail address: noel.cadigan@mi.mun.ca\nPreprint submitted to Elsevier January 30, 2024arXiv:2401.15281v1 [stat.ME] 27 Jan 2024as FEs, possibly with high dimension, and they are modelled as REs with marginal distri-\nbution f(Ψ|θ) for pragmatic purposes. This is opposed to basing statistical inference on\nrepeated sampling of the REs and the data, which is the common statistical basis assumed\nfor mixed-effects models. For example, in a fisheries stock assessment model, even though\nthe population abundances for consecutive years may be modeled as REs following a yearly\nAR(1) (auto-regressive model of order 1) model, the population abundance for a specific\nyear remains fixed for sampling events in different locations and dates of the year. In this\ncontext, statistical inference should be conditional on the true but unknown REs, instead\nof integrating out REs over their marginal distribution f(Ψ|θ). We refer to the former\ninferential setting as “conditional”, and the latter as “marginal”.\nFrequently, a substantial part of the domain of f(Ψ|θ)≫0 involves values of Ψ that can\nproduce unrealistic data via f(D|Ψ, θ), such as the extinction of a fish stock, in contradiction\nto the observed data. Nevertheless, in marginal inference, the properties of RE predictors\nˆΨ(ˆθ) and parameter estimators ˆθ, such as the mean E {ˆΨ(ˆθ)}and prediction error covariance\nCov{ˆΨ(ˆθ)−Ψ}, are evaluated by integrating over all values of the REs (realistic and other-\nwise) and the resulting sample space, leading to biased conclusion. For instance, if ˆΨ(ˆθ) is\nthe posterior mode and the posterior distribution of REs is approximately symmetric, then\nthe marginal bias E {ˆΨ(ˆθ)−Ψ}=O(1/T) (Zheng and Cadigan, 2021), while the conditional\nbias E {ˆΨ(ˆθ)−Ψ|Ψ}=O(1) (Zheng and Cadigan, 2023). In this paper Tdenotes the to-\ntal number of observational units (e.g., Kass and Steffey, 1989; Flores-Agreda and Cantoni,\n2019). For example, in a yearly time-series setting, Tindicates the number of years. The\nsample size for each observational unit substantially affects the accuracy and precision of RE\npredictions. However, such unit-wise sample sizes can vary from zero to large numbers. In\norder to accommodate all these cases, we do not impose restriction on the unit-wise sample\nsizes and hence we do not develop notations for them.\nWith parametric empirical Bayes inference (Kass and Steffey, 1989), the posterior mean\nand variance of Ψ given the observed data Dare used respectively as the measures of central\ntendency and variability of the RE predictions, and the results of Kass and Steffey (1989) are\nwidely adopted by modeling packages for nonlinear mixed-effects models, such as Template\nModel Builder (TMB, Kristensen et al, 2015) and AD Model Builder (ADMB, Fournier et al,\n22012), to construct prediction intervals for REs and nonlinear functions of REs. However, the\nposterior mean and variance in Bayesian inference are equal respectively to the frequentist\nmarginal E {ˆΨ(ˆθ)}and Cov {ˆΨ(ˆθ)−Ψ}to the first order approximation (proof in Zheng\nand Cadigan, 2021); that is, inference given the data still results in marginal inference, and\nfundamentally this cannot provide accurate measures of variability of RE predictions when\nREs behave like high-dimensional FEs (namely, the conditional inferential setting). Note\nthat the covariance of the prediction error, Cov {ˆΨ(ˆθ)−Ψ}, is equal to the marginal mean\nsquared error (MSE) of ˆΨ(ˆθ) to a first order approximation (Zheng and Cadigan, 2021). The\nsimulation studies in Zheng and Cadigan (2023) demonstrated that under the conditional\nsetting, confidence intervals (CIs) constructed using marginal root MSEs can lead to coverage\nprobabilities substantially different from the nominal levels.\nThese marginal inference methods have also been used with nonparametric regression\nmodels, such as using Bayesian posterior variance, or equivalently, the frequentist marginal\nMSE, to construct CIs in generalized additive models (GAMs, e.g., Marra and Wood, 2012;\nWood, 2020). Here nonparametric regression models refer generally to the statistical models\nthat have parametric and nonparametric components, and both components can be of inter-\nest. When marginal inference is applied in the conditional inferential setting, component-wise\nCI coverage probabilities may be unreliable, and the “across-the-function” coverage prop-\nerty has been used to validate the inferential performance (Marra and Wood, 2012; Wood,\n2020). If we still use Ψ to denote the high-dimensional parameters, the component-wise CI\ncoverage probability is for each component Ψ iof Ψ, and the across-the-function coverage\nprobability is averaged across all the Ψ i’s. Zheng and Cadigan (2023) found that if marginal\nMSEs are used for conditional inference, in some cases even the across-the-function coverage\nprobability can deviate substantially from the nominal one.\nThe penalized log-likelihood for a GAM model can be written as (Wood et al, 2016)\nL(βββ) =l(βββ)−1\n2MX\nj=1λjβββ⊤SSSjβββ, (2)\nwhere the vector βββincludes both the basis coefficients and other model parameters θ,l(βββ) is\nthe likelihood, βββ⊤SSSjβββare the smoothing penalties with known sparse matrices SSSj, and λjare\nthe smoothing parameters. Wood et al (2016) suggested to estimate the model coefficients βββ\n3with the modes of L(βββ), and the smoothing parameters λjby maximizing the approximate\nmarginal likelihood where βββare integrated out using the Laplace approximation. Similar\ntoλj, the θparameters can also be estimated by maximizing the Laplace approximate\nmarginal likelihood (LAML), where the basis coefficients are integrated out using the Laplace\napproximation. This is equivalent to estimating both λjandθusing REML (restricted\nmaximum likelihood) in the sense of Laird and Ware (1982). For this approach, we can\ndenote the basis coefficients as Ψ, and re-write the penalized log-likelihood (2) as\nL(Ψ, θ) =l(Ψ, θ)−1\n2MX\nj=1λjΨ⊤SSSjΨ\n=l(Ψ, θ)−1\n2Ψ⊤(MX\nj=1λjSSSj)\nΨ,(3)\nwhere the likelihood l(Ψ, θ) and the penalty term correspond respectively to lcandlrin (1).\nThe GAM penalized log-likelihood (3) and the mixed-effects model log-joint likelihood\n(1) can be optimized in the same way with maximum LAML for θandλj, and the posterior\nmode for Ψ. The way the parameters θare estimated is the only difference between GAM\npenalized log-likelihoods (2) and (3). The correlation structure of lrin (1) can involve some\nmodel parameters such as the autocorrelation parameter in an AR(1) model, while that of (3)\ndoes not. In this sense, GAM models can be regarded as special cases of mixed-effects models\nin the conditional setting (see Wand, 2003; Wood et al, 2013). Maximum LAML estimation\nof the model parameters results in a bias of O(T−1/2) in the conditional setting (Zheng\nand Cadigan, 2023). Such bias generally appears in semi-parametric inference (e.g., He and\nSeverini, 2016; Cheng et al, 2018), where in some cases the bias can be larger (see, e.g., Eq.\n79 in Zheng and Sutradhar, 2018). Despite the biases, the variances of the maximum LAML\nestimators in the conditional setting are in general smaller than the corresponding values\nfor the marginal setting due to the restriction on randomization when Ψ is fixed (Zheng and\nCadigan, 2023). Overall, the MSEs of the maximum LAML estimators in the conditional\nsetting are equal to the marginal MSEs with an approximation order of o(1/T) (Zheng\nand Cadigan, 2023). In the marginal setting, the MSEs are the same as the variances of\nmaximum marginal likelihood estimators (MMLEs), because the biases are negligible when\naveraged over the marginal distribution of Ψ. Due to the consistency and efficiency of\n4maximum likelihood estimators (MLEs), using maximum LAML estimators for θandλj\nin the conditional setting is well justified. In this work, we will extend the methodology of\nconditional inference for mixed-effects models to GAMs based on the penalized log-likelihood\n(3). The resulting confidence intervals (CIs) of Ψ’s depend on the conditional MSEs of θ\nestimates, which are equal to the corresponding marginal MSEs, namely, the variance of\nMMLEs, as referred above.\nThe objective of this work is to develop interval estimates for REs in the conditional\nsetting that have accurate coverage probabilities. We refer to these RE interval estimates as\nconditional CIs instead of prediction intervals (PIs), which is the conventional terminology\nfor REs, because the REs are actually fixed in the conditional inference setting. In Sec. 2,\nwe will first identify the reason why the commonly used conditional CIs based on the MSEs\nof the posterior mode RE estimates do not perform well, and then propose improved interval\nestimators accordingly. In Sec. 3, we will use simulation studies to examine the performance\nof the new interval estimators for mixed-effects models and GAMs, and compare them with\nthe interval estimators based on marginal variances that are typically used. Discussions and\nconcluding remarks are provided in Sec. 4.\n2. Methods\n2.1. Notations and background\nLet Ψ denote the vector of REs in the nonlinear mixed effects model described by (1).\nΨ can also represent the basis coefficients in a GAM, because of the equivalence between\nmixed-effects models in the conditional setting and GAMs, as reflected by (1) and (3). The\nmodel parameters are denoted as θ. For GAMs, θalso includes the smoothing parameters.\nθare estimated by maximizing the marginal likelihood, which is the marginal distribution\nof data Dobtained by integrating out REs Ψ from the joint distribution of Ψ and D. This\nMMLE is denoted as ˆθ. Ψ are estimated by maximizing the log-joint likelihood (1), or\nequivalently the log-posterior distribution of Ψ, with ˆθused for θ. This log-joint likelihood\nformally includes the GAM penalized log-likelihood (3). The posterior mode estimators of\nΨ are denoted as ˆΨ(ˆθ).\nIf the distribution of Ψ conditional on observed data is approximately multivariate normal\n5(MVN), the conditional bias and covariance of ˆΨ(ˆθ) are given by (Zheng and Cadigan, 2023)\nE{ˆΨ(ˆθ)|Ψ} −Ψ =−¨lj(Ψ, θo)−1¨lr(Ψ, θo)[Ψ−E{Ψ}] +O(T−1/2),\nCov{ˆΨ(ˆθ)|Ψ} ≈ − ¨lj(Ψ, θo)−1+¨lj(Ψ, θo)−1¨lr(Ψ, θo)¨lj(Ψ, θo)−1\n+∂ˆΨ(θo)\n∂θ⊤\noCov( ˆθ|Ψ)∂ˆΨ⊤(θo)\n∂θo,(4)\nwhere the double dots above ljandlrdenote second order derivatives with respect to Ψ.\nCov( ˆθ|Ψ) is of O(T−1) (Zheng and Cadigan, 2023), but its explicit form is not relevant to\nthis paper.\nWhen deriving the conditional MSE of ˆΨ(ˆθ) given Ψ, the bias-squared term ¨lj(Ψ, θo)−1¨lr(Ψ, θo)[Ψ−\nE{Ψ}][Ψ⊤−E{Ψ}⊤]¨lr(Ψ, θo)¨lj(Ψ, θo)−1cannot be evaluated correctly because Ψ is unknown\nand can only be estimated with bias of O(1) according to (4). Let Ω = (Ψ⊤, θ⊤\no)⊤and\nˆΩ = ( ˆΨ(ˆθ)⊤,ˆθ⊤)⊤. Zheng and Cadigan (2023, Appendix D) showed that if the bias-squared\nterm is approximated by its marginal expectation, then the conditional MSE is approxi-\nmately\nMSE( ˆΩ|Ψ) = Cov( ˆΩ|Ψ) + E( ˆΩ−Ω|Ψ)E( ˆΩ−Ω|Ψ)⊤\n≈Cov( ˆΩ|Ψ) + En\nE(ˆΩ−Ω|Ψ) E( ˆΩ−Ω|Ψ)⊤o\n=\n−¨l−1\nj+∂ˆΨ(ˆθ)\n∂ˆθ⊤Cov( ˆθ)∂ˆΨ⊤(ˆθ)\n∂ˆθ∂ˆΨ(ˆθ)\n∂ˆθ⊤Cov( ˆθ)\nCov( ˆθ)∂ˆΨ⊤(ˆθ)\n∂ˆθCov( ˆθ)\n+o(1/T).(5)\nThe last matrix in (5) is the MSE in the marginal setting (Zheng and Cadigan, 2021) and\nis also the posterior variability in empirical Bayes (Kass and Steffey, 1989). Note that\nin this conditional setting, ˆθhas a bias of order O(T−1/2). The conditional MSE of ˆθis\napproximately the marginal covariance Cov( ˆθ) =−¨l−1\nm(ˆθ), where lmis the logarithm of the\nmarginal likelihood obtained by integrating the joint likelihood across REs Ψ, and ¨lm(θ)\ndenotes its second order derivatives with respect to θ.\nThe approximation made in (5) indicates that when applying the marginal MSE (or\nequivalently the Bayesian posterior variance) to construct CIs in the conditional setting, we\nactually approximate the bias-squared term by integrating it across the marginal distribution\nof REs, which introduces biases of order O(1) to the conditional MSE. As a result, the\n6corresponding component-wise CI coverage probabilities can be unsatisfactory. When the\nCI coverage probabilities are averaged over all the REs in the model, conditional MSEs, and\nhence bias-squared terms, are somewhat averaged over REs approximately, which in turn is\nsimilar to an average over the marginal distribution of REs based on the notion of ergodicity,\ni.e., the ensemble average equals the time average (Feller, 2008). Therefore, the coverage\nprobability averaged across the REs can be substantially closer to the nominal level than\nthe component-wise CI coverage probabilities, as discussed for smoothing splines in Nychka\n(1988).\n2.2. Conditional CI based on bias-corrected RE estimation\nThe bias-squared term in the conditional MSE cannot be estimated well and hence ad-\nversely affects the component-wise CI coverage. Using a bias corrected RE estimator can\navoid this term in the conditional MSE and hence improve CI coverage probabilities. To\nsimplify notation we write the bias formula in (4) as\nE{ˆΨ(ˆθ)|Ψ} −¨lj(Ψ, θo)−1¨lr(Ψ, θo)E{Ψ}=BΨ +O(T−1/2), (6)\nwhere B= III−¨lj(Ψ, θo)−1¨lr(Ψ, θo) and I II is an identity matrix. The O(T−1/2) term comes\nfrom the expectation of ∂ˆΨ(θo)/∂θ⊤\no(ˆθ−θo). If some components of ∂ˆΨ(θo)/∂θ⊤\no(ˆθ−θo) are\nlarge, i.e., close to 1 in absolute value, this term can become large. Therefore, we make\nthe assumption in our subsequent development that ∂ˆΨ(θo)/∂θ⊤\nois not large. Reparameter-\nization can often resolve the issue of large components of ∂ˆΨ(θo)/∂θ⊤\noin cases where they\noccur. For instance, if µ+ Ψ follows an AR(1) model with mean µ, then the components\nof∂ˆΨ/∂µare close to 1. However, if we redefine Ψ to follow an AR(1) model with mean µ,\nthen the components of ∂ˆΨ/∂µbecome small.\nIf all the REs are well supported by data and hence Bis nonsingular, multiplying both\nsides of (6) by B−1does not change the approximation order O(T−1/2), and\nˆΨBC(ˆθ) =B−1h\nˆΨ(ˆθ)−¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}i\n(7)\nis an unbiased estimator of Ψ to the order O(T−1/2). Here the subscript “BC” denotes “bias\ncorrection”, and E {Ψ}as a function of θis evaluated at ˆθ. In (7) ¨lj(Ψ, θo) and ¨lr(Ψ, θo)\nare all evaluated at ˆΨ(ˆθ) and ˆθ, which does not change the approximation order O(T−1/2)\n7because ˆθ−θoisOp(T−1/2) as the MSE of ˆθisO(T−1), and ljandlrare approximately MVN\nrespecting Ψ. In Sec. 2.3 we extend (7) when some REs are not well supported by the data\nso that Bis singular or close to singular.\nIfˆΨBC(ˆθ) is an unbiased estimator of Ψ then its conditional MSE is equal to its conditional\ncovariance,\nMSE( ˆΨBC(ˆθ)|Ψ) = Cov( ˆΨBC(ˆθ)|Ψ) + O(T−1)\n=B−1h\n−¨lj(Ψ, θo)−1+¨lj(Ψ, θo)−1¨lr(Ψ, θo)¨lj(Ψ, θo)−1\n+Υ Cov( ˆθ) Υ⊤i\b\nB−1\t⊤+O(T−1),\nΥ =∂ˆΨ(θo)\n∂θ⊤\no−¨lj(Ψ, θo)−1¨lr(Ψ, θo)∂E{Ψ}\n∂θ⊤\no.(8)\nHere Cov( ˆθ|Ψ) in (4) is replaced by Cov( ˆθ) because conditional on Ψ ˆθis a biased estimator\nofθoand hence its overall variability should be measured by its MSE Cov( ˆθ) (see Eq. 5).\nWhen applying other estimators of θ(e.g., Eq. 2, the penalized likelihood maximization in\nGAM), Cov( ˆθ) in (8) should be replaced by the corresponding MSEs. A 100(1 −α)% CI for\ntheith component of Ψ, Ψ i, can be constructed as\nˆΨBC,i(ˆθ)±zα/2ˆσΨ,i, (9)\nwhere ˆ σΨ,i=q\nMSE( ˆΨBC(ˆθ)|Ψ)ii,zα/2is the z-value corresponding to an area α/2 in the\nupper tail of a standard normal distribution, and the subscript idenotes the ith element.\nA CI for a vector-valued function of Ψ, g(Ψ), can be constructed with the generalized-\ndelta method (see, e.g., Kristensen et al, 2015), where g(Ψ) is regarded as a MVN random\nvariable with mean ˆ µg=g(ˆΨBC(ˆθ)) and covariance ˆΣg={∂g(Ψ)/∂Ψ⊤}MSE( ˆΨBC(ˆθ)|Ψ){∂g⊤(Ψ)/∂Ψ}.\nWhen g(·) is a nonlinear function, a better estimator of its mean ˆ µgcan be obtained through\nthe bias correction proposed in Thorson and Kristensen (2016). A 100(1 −α)% CI for the\nith component of g(Ψ) is\nˆµg,i±zα/2ˆσg,i, (10)\nwhere ˆ σg,i=q\nˆΣg,ii. The generalized-delta method is similarly implemented with the other\nΨ estimators proposed in this paper.\n82.3. Conditional CI when some REs are not associated with data\nLet 000 denote a matrix or vector of zeros with the appropriate dimension. If there is no\ndata, namely ¨lc(Ψ, θo) =¨lj(Ψ, θo)−¨lr(Ψ, θo) = 000,¨lj(Ψ, θo)−1¨lr(Ψ, θo) = III in (6) and hence\nB= 000. In this case, ˆΨ(ˆθ) is zero. When we have no information about REs then we have\nto use the marginal MSE to provide CIs if we have enough knowledge about the model\nparameters. Even if only a small number of REs are not supported by data, which will\ncommonly occur if there are missing data or other types of irregular sampling, then Bwill\nbe singular.\nWe need to invert Bin (6) for our unbiased estimator of the REs (i.e., Eq. 7). How-\never, when REs associated with few or no data cause singularity in B, it is not possible to\nimplement (7) directly. In this case, a straightforward remedy for those REs not supported\nby data is to use the corresponding components of ˆΨ(ˆθ) for ˆΨBC(ˆθ), with uncertainty given\nby the marginal MSE (5). We consider this further in the Discussion Section. Therefore,\nwe first devise a method to identify whether the REs are supported by data sufficiently or\nnot, and then for those REs with sufficient data we use (7) and (8) to construct CIs, and for\nthose REs with insufficient data we use ˆΨ(ˆθ) and (5).\nWe use the singular value decomposition, B=UΓV⊤, where UandVare orthogonal\nmatrices, and Γ is a diagonal matrix of non-negative singular values. Multiplying both sides\nof (6) with U⊤gives\nE{U⊤[ˆΨ(ˆθ)−¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}]|Ψ}= ΓV⊤Ψ +O(T−1/2).\nHere we applied U⊤O(T−1/2) =O(T−1/2).V⊤Ψ is an alternative set of REs to the original\nΨ, and is estimated by U⊤[ˆΨ(ˆθ)−¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}]. Component-wise, E {[U⊤[ˆΨ(ˆθ)−\n¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}]]i|Ψ}= Γ ii[V⊤Ψ]i+O(T−1/2), where the subscript idenotes the\nith element. If multiplying both sides with Γ−1\niidoes not change the approximation order\nO(T−1/2), namely, if Γ ii> γ c, say, then Γ−1\nii[U⊤[ˆΨ(ˆθ)−¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}]]iis approx-\nimately an unbiased estimator of [ V⊤Ψ]i; otherwise, [ V⊤Ψ]iis estimated by [ V⊤ˆΨ(ˆθ)]iwith\na bias −[V⊤¨l−1\nj¨lr(Ψ−E{Ψ})]i, according to (4). This component-by-component treatment\nleads to an estimator of V⊤Ψ, [Γ−1\ngU⊤+ΓcV⊤]ˆΨ(ˆθ)−Γ−1\ngU⊤¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}, with a\nbias−ΓcV⊤¨l−1\nj¨lr(Ψ−E{Ψ}), where Γcis a diagonal matrix with value 1 replacing the zero or\n9very small singular values ( ≤γc) in Γ, but 0 otherwise, and Γ−1\ngis similar to the generalized\ninverse (Ben-Israel and Greville, 2003) of Γ, namely, setting very small singular values ( ≤γc)\nto zero, but inverting the other diagonal values. This estimator of V⊤Ψ is then transformed\nto an estimator of Ψ, ˆΨSD(ˆθ) =V[(Γ−1\ngU⊤+ ΓcV⊤)ˆΨ(ˆθ)−Γ−1\ngU⊤¨lj(Ψ, θo)−1¨lr(Ψ, θo) E{Ψ}],\nwith\nE{ˆΨSD(ˆθ)|Ψ} −Ψ =−VΓcV⊤¨l−1\nj¨lr(Ψ−E{Ψ}) +O(T−1/2). (11)\nHere the subscript “SD” denotes “singular value decomposition”.\nThe MSE of ˆΨSD(ˆθ) is\nMSE( ˆΨSD(ˆθ)|Ψ) = V\u0002\nΓ−1\ngU⊤Cov 11UΓ−1\ng+ Γ−1\ngU⊤Cov 12VΓc\n+ΓcV⊤Cov⊤\n12UΓ−1\ng+ ΓcV⊤MSE( ˆΨ(ˆθ)|Ψ)VΓci\nV⊤,(12)\nwhere using Υ defined in (8),\nCov 11=−¨lj(Ψ, θo)−1+¨lj(Ψ, θo)−1¨lr(Ψ, θo)¨lj(Ψ, θo)−1+ Υ Cov( ˆθ) Υ⊤,\nCov 12=−¨lj(Ψ, θo)−1+¨lj(Ψ, θo)−1¨lr(Ψ, θo)¨lj(Ψ, θo)−1+ Υ Cov( ˆθ)∂ˆΨ⊤(θo)\n∂θo.(13)\nEq. (12) says that the conditional MSE for those elements of Ψ with few data is the marginal\nMSE, and for the other REs we use the conditional MSE for bias-corrected estimators (8).\nThe 100(1 −α)% CI for Ψ ican be constructed as ˆΨSD,i(ˆθ)±zα/2q\nMSE( ˆΨSD(ˆθ)|Ψ)ii. When\nγcgoes to zero, this CI converges to the conditional CI based on the bias-corrected estimator\n(9). When γcbecomes large, this CI converges to the CI based on the marginal MSE. The\nREs are usually defined for observational units (Kass and Steffey, 1989; Flores-Agreda and\nCantoni, 2019). To determine γc, we can count the number ncof observational units that\nhave data, and γcis specified so that exactly ncsingular values exceed γc. If different types\nof data are involved, e.g., survey catches and fisheries catches in a fisheries stock assessment\nmodel, then ncshould be the summation of the observational units with data for the various\ntypes. ncis also equal to the number of nonzero singular values from the singular value\ndecomposition of ¨lc.\nWe use ˆΩSD= (ˆΨSD(ˆθ)⊤,ˆθ⊤)⊤to estimate Ω = (Ψ⊤, θ⊤\no)⊤for the case with data insuffi-\n10ciency. The conditional MSE of ˆΩSDis given by\nMSE( ˆΩSD|Ψ) = MSE\n\nˆΨSD(ˆθ)\nˆθ\n\f\f\f\f\f\fΨ\n\n=\nMSE( ˆΨSD(ˆθ)|Ψ)GCov( ˆθ)\nCov( ˆθ)G⊤Cov( ˆθ)\n,(14)\nwhere G=V[(Γ−1\ngU⊤+ ΓcV⊤)∂ˆΨ(θo)/∂θ⊤\no−Γ−1\ngU⊤¨lj(Ψ, θo)−1¨lr(Ψ, θo)∂E{Ψ}/∂θ⊤\no].\n3. Simulation studies\nIn this section we use a random walk example and one based on the global temperature\nanomaly data in Wood (2020, Fig. 3) to examine the performance of the conditional CIs\nderived from the bias-corrected RE estimation in Sec. 2.2. We also examine the impact of\nmissing data on CI performance.\n3.1. Random walk example\nThe random walk is Ψ t|Ψt−1indep∼N(Ψt−1, σ2\nΨ) for t= 2, ..., T , and Ψ 1indep∼N(0, σ2\nΨ).\nHere N(µ, σ2) denotes the normal distribution with mean µand variance σ2. At each time-\nstep there are nindependent observations of the process, Yt,i|Ψtindep∼N(Ψt, σ2\nϵ), i= 1, ..., n\nandt= 1, ..., T . The parameters are θ= (σΨ, σϵ)⊤and the REs are Ψ = (Ψ 1, ...,ΨT)⊤which\nis aT×1 vector. We randomly generated one set of random-walk Ψ t’s, and then generate\n1000 simulations of the data yconditional on the Ψ t’s, with σΨ= 1, σϵ= 0.5, and two\nchoices each for n= 2,5 and T= 50 ,200. We estimated the parameters and Ψ t’s using\nTMB (Kristensen et al, 2015) and the nlminb procedure in R (R Core Team, 2022). Biases\nand CI coverage depend on the values of Ψ t’s, and to generalize results we also repeated the\nsimulations for 500 different random walk Ψ’s, and then averaged results over these random\nΨ’s.\nThe squared bias, (E {ˆΨBC|Ψ} −Ψ)2, was much smaller than for ˆΨ, when averaged over\nthe 500 simulation Ψ’s (Fig. 1). This indicates that the bias correction is effective. In\nFig. 2 the “Random” results are the simulated coverage probabilities of 95% CIs based on\nˆΨi±zα/2×RMMSE( ˆΨi), where RMMSE is the root marginal mean squared error (5). The\npoints are the mean probabilities for the 500 Ψ’s, and the green shaded regions indicate the\n11lower 5thand upper 95thpercentiles. These coverages are accurate on average (with respect\nto Ψ), but for specific values of Ψ the coverages can deviate from 95%, which is indicated\nby the width of the shaded regions. The blue points and shaded regions indicate results\nfor the bias-corrected conditional CI (9). These coverages are also accurate on average, but\nmore accurate for specific values of Ψ compared to the random intervals. The red points\nindicate intervals based on ˆΨiand the conditional variance (4). These are biased because of\nthe conditional bias in ˆΨi.\nSimulated coverages for one set of random walk Ψ’s are shown in Fig. 3. This figure\ndemonstrates that the conditional bias-corrected CIs have coverages that are close to 95%,\ncompared to the biased conditional intervals or intervals based on RMMSE. The latter are the\nintervals that are commonly used for inferences about random effects in mixed-effects models,\nand they can be substantially inaccurate and potentially misleading when the random effects\nare not actually random.\n50 2002 5\n010203040500501001502000.000.050.100.15\n0.000.050.100.15\nTimeAverage Squared BiasΨ^Ψ^\nBC\nFigure 1: Simulated squared bias for estimators of the random walk Ψ’s, averaged over 500 simulated Ψ’s. For\neach set of Ψ’s, the squared bias is based on 1000 simulated data sets Yt,i|Ψ with i= 1, ..., n andt= 1, ..., T .\nPanel rows indicate choices for nand columns indicate T. The colors correspond to the “posterior mode”\nestimator ( ˆΨ) and the bias-corrected estimator ( ˆΨBC).\n1250 2002 5\n01020304050050100150200859095\n859095\nTimeConfidence Interval Coverage (%)Conditional Random Conditional BCFigure 2: Simulated coverage of 95% CIs for the random walk Ψ’s. Points indicate average CI coverages,\nand shaded regions indicate 5% and 95% quantiles, from the 500 simulated Ψ’s. Panels are described in Fig.\n1.\n13859095\n0 10 20 30 40 50\nTimeConfidence Interval Coverage (%)Conditional Random Conditional BCFigure 3: Simulated coverage of 95% CIs for an illustrative example of random walk Ψ’s. Red lines indicate\nCIs based on the ”posterior mode” estimator ( ˆΨ) and conditional-Ψ standard errors (SEs), green lines indicate\nCIs based on ˆΨ and random-Ψ SEs, and blue lines indicate CIs based on the bias-corrected estimator ( ˆΨBC)\nand its conditional-Ψ SEs.\n143.2. Random walk example with missing data\nWe remove the yobservations for the last three random walk time points, and repeated the\nsimulation study described in Section 3.1. We used the bias-correction procedure outlined\nin Section 2.3 and γc= 0.1. The simulated CI coverages (Fig. 4) for time points with\ndata are very similar to those in Fig. 2. Note that the scale of the y-axis is different in\nthese two figures. However, for the last three time points with no data, the coverages can\nbe substantially different from 95% for specific Ψ’s, and they are only reasonably accurate\nwhen averaged over the 500 Ψ’s in our simulations. Note that the random and conditional\nintervals in Fig. 4 are identical for the times with no data.\n50 2002 5\n01020304050050100150200708090100\n708090100\nTimeConfidence Interval Coverage (%)Random Conditional BC\nFigure 4: Simulated coverage of 95% CIs for the random walk Ψ’s. yobservations for the last three time\npoints are missing. Points indicate average CI coverages, and shaded regions indicate 5% and 95% quantiles,\nfrom the 500 simulated Ψ’s. Panels are described in Fig. 1.\n3.3. Global temperature anomaly example\nWe compare the performances of CIs constructed from ˆΨBC(ˆθ) and ˆΨ(ˆθ) for a GAM\napplied to the global annual mean temperature anomalies in Fig. 3 of Wood (2020). We\nobtained the global temperature anomalies from Rohde and Hausfather (2020) and used\n15the data during 1850 to 2010, following Fig. 3 of Wood (2020). We fit the data using a\ncubic spline with 50 basis coefficients. The smoothing parameter was estimated by marginal\nlikelihood (REML) maximization. The annual temperature anomalies were assumed to have\nindependent normal distributions with means given by an intercept plus the cubic spline,\nand a common standard deviation. We then assumed that the estimated mean and model\nparameters were the true values, and used the model to simulate 10000 sets of temperature\nanomalies. We fit the simulated data with the same model, and constructed CIs using ˆΨBC(ˆθ)\nandˆΨ(ˆθ), where the former is based on (10) and the latter is based on the marginal MSE.\nThe estimates of the smoothing parameter λin (3) were effectively infinite (at a magnitude\nofe11), and hence the usual theory about the variability of MLEs based on the regularity\nconditions (e.g., the interiority of a parameter to the parameter space) may not apply to\nthe estimation of log( λ) here. When evaluating the variance of the estimator of log( λ) by\nthe inverse of the Hessian of the negative log marginal likelihood with respect to log( λ), too\nwide CIs based on (8) resulted in our simulation studies. Therefore, we follow the convention\nof neglecting the uncertainty in log( λ) estimator (Wood et al, 2016). We set up the spline\nstructure using the mgcv package (Wood, 2011) in R, and implemented the GAM using\nTMB.\nThe simulated 95% CI coverage rates and squared biases are presented in Fig. 5. The\ncomponent-wise coverage rates of the CIs based on ˆΨBC(ˆθ) and (10) are fairly close to the\nnominal value, while those based on ˆΨ(ˆθ) and marginal MSEs deviate substantially from the\nnominal value in almost every year. However, the average CI coverage rate across the years\n(i.e., across-the-function) for the ˆΨ(ˆθ) and marginal MSEs approach is 0.946, close to the\nnominal value and agreeing with Marra and Wood (2012) and Wood (2020). The across-the-\nfunction CI coverage rate for the ˆΨBC(ˆθ) and (10) approach is 0.949. Similarly, the squared\nbiases for ˆΨBC(ˆθ) are virtually 0, while the squared biases of ˆΨ(ˆθ) are substantially larger.\nFig. 5 demonstrates that when the bias is large the corresponding CI coverage rate is less\nthan the nominal value due to an underestimation of the bias-squared term in MSE with the\nmean squared bias in (5). When the squared bias is small, the corresponding CI coverage\nrate is larger than the nominal value due to an overestimation of the bias-squared term in\nMSE. Hence, the lack of accuracy in the estimation of the bias-squared term in MSE is\n1685 90 95 100\n1850:2010100 * CI_cov/sim_totalΨ^\nΨ^\nBCCI coverage (%)\n1850 1900 1950 20000.0000 0.0002 0.0004\n1850:2010abs(colMeans(muSpline_mat) − muSpline_save)^2 Squared Bias\nYearFigure 5: Simulated 95% CI coverage rates and squared biases for estimating the yearly mean global temper-\nature anomalies obtained by fitting the global annual temperature anomalies (Rohde and Hausfather, 2020)\nusing a GAM. Upper panel: the black lines are the coverage rates for the true mean annual temperature\nanomalies of the 95% CIs constructed using ˆΨ(ˆθ) and marginal MSEs, and the red lines are those based on\nˆΨBC(ˆθ) and (10); the blue dashed reference line is at 95%. Lower panel: the black and red lines are the\nsimulated squared biases for estimating the true mean annual temperature anomalies based on ��Ψ(ˆθ) and\nˆΨBC(ˆθ) respectively, and the blue dashed reference line is at 0.\nan important reason for the unsatisfactory component-wise coverage of the CIs constructed\nwith ˆΨ(ˆθ) and marginal MSE.\n4. Discussion\nWe investigated statistical inference for nonlinear mixed-effects models for the case when\nrandom effects (REs) were used as a pragmatic approach to estimate high dimensional fixed\neffect parameters (FEs). This case includes GAMs and other nonparametric regression meth-\nods. If the FEs possess some intrinsic correlation, i.e., FEs that are close together spatially\nand/or temporally tend to have similar values, then modeling them as REs with multivariate\ncorrelation structure can be a suitable estimation procedure, and the conditional inference\ndiscussed in this paper is appropriate. This is also applicable to some implementations of\n17mixed-effects models where the REs actually remain fixed during repeated samplings, and\nhence conceptually should be classified as FEs. We proposed a novel procedure for less biased\nestimation of the REs, and improved confidence intervals. We demonstrated using simulation\nstudies that our new methods provide substantially improved statistical inferences.\nFor conditional confidence interval (CI) when there are REs not associated with data, we\ncan have an outlook alternative to that in Sec. 2.3. Let Ψ Abe the subset of REs Ψ associated\nwith data D, and Ψ Bbe the subset not associated with data, which can be represented by\na conditional independence statement\nf(D|ΨA,ΨB, θ) =f(D|ΨA, θ). (15)\nTherefore, all the conditional quantities such as mean and variance given Ψ are actually con-\nditioning on Ψ A. The estimates of θand Ψ Aare based on the joint density f(D|ΨA, θ)f(ΨA|θ),\nnamely, θare estimated by ˆθmaximizing the marginal likelihood f(D|θ) integrating out Ψ A,\nand Ψ Aare estimated using the posterior mode, or equivalently, the mode of the joint density\nf(D|ΨA,ˆθ)f(ΨA|ˆθ) respecting Ψ A. Since all the elements of Ψ Aare associated with data,\nits bias-corrected estimator and CI can be constructed with (7)–(9). This whole inferential\nprocedure does not involve Ψ B; that is, the estimation of Ψ Bbelongs to prediction of “fu-\nture”, even though Ψ Bcan spatiotemporally interweave with Ψ Aover the range of the latter.\nMathematically, the predictor ˆΨBof Ψ Bshould be a function of the estimator ˆΨAof Ψ A.\nIf Ψ = (Ψ⊤\nA,Ψ⊤\nB)⊤follows a multivariate normal (MVN) distribution, this function has an\nanalytical form obtained as follows. Let the precision matrix of the RE density f(ΨA,ΨB|θ)\nbe\nΣ−1=\nΛAΛAB\nΛBA ΛB\n,\nwhich has been partitioned according to the dimensions of Ψ Aand Ψ B. Prediction of Ψ B\nis obtained by maximizing the log-joint likelihood log f(D|ˆΨA,ˆθ) + log f(ˆΨA,ΨB|ˆθ) with\nrespect to Ψ B, which gives the normal equation Λ BAˆΨA+ Λ BˆΨB= 0, and hence\nˆΨB=−Λ−1\nBΛBAˆΨA. (16)\nDue to the conditional independence property (15), there is no inference conditional on Ψ B.\nAs a result, a bias-corrected estimator cannot be constructed for Ψ B, and marginal inference\n18has to be adopted for Ψ B; that is, the marginal MSE (5) is applied to set up CI for Ψ B.\nThe covariance between ˆΨBand the bias-corrected estimator for Ψ Aconditional on Ψ Acan\nbe derived using (13) and (16). In order to partition Ψ into Ψ Aand Ψ B, we suggest to use\nthe singular value decomposition of the Hessian matrix of log f(D|ΨA, θ) with respect to Ψ.\nThe components of Ψ with singular values virtually equal to zero are Ψ Band the others\nare Ψ A. In the random-walk example of Sec. 3.2, where the last three time points had no\ndata, the prediction inference for new REs were basically applied. The above analysis can be\ndeveloped into a method to deal with REs without data. Our simulations indicated that this\nmethod performed close to the method proposed in Sec. 2.3; hence, we only fully formulate\nthe latter method in this paper.\nReferences\nAeberhard WH, Mills Flemming J, Nielsen A (2018) Review of state-space models for fisheries\nscience. Annual Review of Statistics and Its Application 5:215–235\nBen-Israel A, Greville TN (2003) Generalized inverses: theory and applications, vol 15.\nSpringer Science & Business Media\nCheng MY, Huang T, Liu P, Peng H (2018) Bias reduction for nonparametric and semipara-\nmetric regression models. Statistica Sinica 28(4):2749–2770\nFeller W (2008) An introduction to probability theory and its applications, vol 2. John Wiley\n& Sons\nFlores-Agreda D, Cantoni E (2019) Bootstrap estimation of uncertainty in prediction for\ngeneralized linear mixed models. Computational Statistics & Data Analysis 130:1–17\nFournier DA, Skaug HJ, Ancheta J, Ianelli J, Magnusson A, Maunder MN, Nielsen A, Sibert J\n(2012) Ad model builder: using automatic differentiation for statistical inference of highly\nparameterized complex nonlinear models. Optimization Methods and Software 27(2):233–\n249\n19He H, Severini TA (2016) A flexible approach to inference in semiparametric regression\nmodels with correlated errors using gaussian processes. Computational Statistics & Data\nAnalysis 103:316–329\nKass RE, Steffey D (1989) Approximate bayesian inference in conditionally independent hi-\nerarchical models (parametric empirical bayes models). Journal of the American Statistical\nAssociation 84(407):717–726\nKristensen K, Nielsen A, Berg CW, Skaug H, Bell B (2015) Tmb: automatic differentiation\nand laplace approximation. arXiv preprint arXiv:150900660\nLaird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics pp\n963–974\nMarra G, Wood SN (2012) Coverage properties of confidence intervals for generalized additive\nmodel components. Scandinavian Journal of Statistics 39(1):53–74\nNychka D (1988) Bayesian confidence intervals for smoothing splines. Journal of the Ameri-\ncan Statistical Association 83(404):1134–1143\nR Core Team (2022) R: A Language and Environment for Statistical Computing. R Foun-\ndation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/\nRohde RA, Hausfather Z (2020) The berkeley earth land/ocean temperature record. Earth\nSystem Science Data 12(4):3469–3479\nThorson JT, Kristensen K (2016) Implementing a generic method for bias correction in\nstatistical models using random effects, with spatial and population dynamics examples.\nFisheries Research 175:66–74\nWand MP (2003) Smoothing and mixed models. Computational statistics 18(2):223–249\nWood SN (2011) Fast stable restricted maximum likelihood and marginal likelihood estima-\ntion of semiparametric generalized linear models. Journal of the Royal Statistical Society:\nSeries B (Statistical Methodology) 73(1):3–36\n20Wood SN (2020) Inference and computation with generalized additive models and their\nextensions. Test 29(2):307–339\nWood SN, Scheipl F, Faraway JJ (2013) Straightforward intermediate rank tensor product\nsmoothing in mixed models. Statistics and Computing 23(3):341–360\nWood SN, Pya N, S¨ afken B (2016) Smoothing parameter and model selection for general\nsmooth models. Journal of the American Statistical Association 111(516):1548–1563\nZheng N, Cadigan N (2021) Frequentist delta-variance approximations with mixed-effects\nmodels and tmb. Computational Statistics & Data Analysis 160:107,227\nZheng N, Cadigan N (2023) Frequentist conditional variance for nonlinear mixed-effects\nmodels. Journal of Statistical Theory and Practice 17(1):1–30\nZheng N, Sutradhar BC (2018) Inferences in semi-parametric dynamic mixed models for\nlongitudinal count data. Annals of the Institute of Statistical Mathematics 70(1):215–247\n21" }, { "title": "2401.15291v1.Improved_Construction_of_Robust_Gray_Code.pdf", "content": "arXiv:2401.15291v1 [cs.IT] 27 Jan 2024Improved Construction of Robust Gray Codes\nDorsa Fathollahi\nDepartment of Electrical Engineering\nStanford University\nStanford, CA, USA\nEmail: dorsafth@stanford.eduMary Wootters\nDepartments of Computer Science and Electrical Engineerin g\nStanford University\nStanford, CA, USA\nEmail: marykw@stanford.edu\nAbstract\nArobust Gray code , formally introduced by (Lolck and Pagh, SODA 2024), is a Gra y code that additionally\nhas the property that, given a noisy version of the encoding o f an integer j, it is possible to reconstruct ˆjso that\n|j−ˆj|is small with high probability. That work presented a transf ormation that transforms a binary code Cof\nrateRto a robust Gray code with rate Ω(R), where the constant in the Ω(·)can be at most 1/4. We improve\nupon their construction by presenting a transformation fro m a (linear) binary code Cto a robust Gray code with\nsimilar robustness guarantees, but with rate that can appro achR/2.\nI. I NTRODUCTION\nIn [1], Lolck and Pagh introduce the notion of a robust Gray code . Informally, a robust Gray code G ⊆\n{0,1}dhas an encoding map EncG:{0,...,N−1} → {0,1}dthat maps integers to bitstrings, with the\nfollowing desiderata.\n•Gshould be a Gray code.1That is, for any j∈ {0,...,N−2},|EncG(j)−EncG(j+1)|= 1.\n•Gshould be “noise robust.” Informally, this means that we should be able to approximate ly recover\nan integer j∈ {0,...,N−1}given a noisy version of EncG(j). Slightly more formally, Gshould\nhave a decoding map DecG:{0,1}d→ {0,...,N−1}, so that when η∼Ber(p)n, the estimate\nˆj= Dec G(EncG(j)⊕η)should be close to jwith high probability.\n•Gshould have high rate. The rated\nlogNofGshould be as close to 1as possible.\n•Gshould have efficient algorithms. BothEncGandDecGshould have running time polynomial (ideally,\nnear-linear) in d.\nRobust Gray codes have applications in differential privac y; see [1]–[4] for more details on the connection. It\nis worth mentioning that there exist non-binary codes based on the Chinese Remainder Theorem [5], [6] that\nhave nontrivial sensitivity, but in our work, we focus on bin ary codes.\nOur Contributions. In this paper, we improve upon the construction of [1] by givi ng a construction of a robust\nGray code with the same robustness guarantees, but better ra te.\nMore precisely, for p∈(0,1/2), [1] give a general recipe for turning a binary error-correc ting code Cwith\nrateRinto a robust Gray code Gwith rate Ω(R), and with the following robustness guarantee:\nPr[|j−DecG(EncG(j)+η)| ≥t≤exp(−Ω(t))+exp( −Ω(d))+O(Pfail(C)), (1)\nwhere the probability is over the noise vector η∼Ber(p)d, andPfail(C)is the failure probability of the code\nCon the binary symmetric channel with parameter p.\nOur main result is a similar transformation that turns a (lin ear) binary code Cwith good performance on\nthe binary symmetric channel into a robust Gray code G. We obtain a similar robustness guarantee as (1) (see\nTheorem 1 for the precise statement), but with better rate. C oncretely, if the original code Chas rateR∈(0,1),\nthe rate of the robust Gray code from [1] is proven to be Ω(R), where the constant inside the Ωapproaches 1/4\nwhenChas sublinear distance; this comes from the fact that the a co deword in their final construction involves\nfour codewords from C. In contrast, under the same conditions, our robust Gray cod eGhas rate approaching\nR/2; this is because our construction involves only twocodewords from C. (See Observation 2 for the formal\nstatement). Moreover, if the encoding and decoding algorit hms forCare efficient, then so are the encoding\n1The paper [1] also gives a more general definition, where the c ode should have low sensitivity , meaning that |EncG(j)−EncG(j+1)|\nis small; however, both their code and our code is a Gray code, so we specialize to that case (in which the sensitivity is 1).and decoding algorithms for our construction G; concretely, the overhead on top of the encoding and decodin g\nalgorithms for CisO(d)(see Lemma 7 for the formal statement).\nAs a result, when instantiated with, say, a constant-rate Re ed-Muller code or a polar code (both of which have\nsublinear distance and good performance on the BSC( p) (see, e.g., [7]–[11])), our construction gives efficient\nrobust Gray codes with a rate about two times larger than than previous work, approaching 1/2.\nMain Idea. The idea of our transformation is quite simple, and follows t he same high-level structure as [1].\nWe begin with our base code C, and use it to construct an intermediate code W(with an appropriate ordering).\nThen we add new codewords to Wto complete it to a Gray code. For example, if wi,wi+1are two consecutive\ncodewords in W, then we will insert ∆(wi,wi+1)−1codewords in between them, iteratively flipping bits to\nmove from witowi+1.\nThe main difference between our construction and that of pre vious work is how we build and order W. First,\nwe use a standard Gray code to construct an ordering of the codewords in C. Then, we build Was follows.\nLetcibe thei’th codeword in C. Then the i’th codeword in Wis given by\nwi=si◦ci◦si◦ci◦si,\nwheresiis a short string that is all zeros if iis even and all ones otherwise, and ◦denotes concatenation.\nThen we form Gby interpolating as described above.\nOur decoding algorithm ends up being rather complicated, bu t the idea is simple. Suppose that for a codeword\ng∈ G, we see a corrupted version g+η∈Fd\n2, whereηis a noise vector. As described above, gis made up\nof a prefix from wi+1and a suffix from wi, for some i. Leth∈[d]be the index where g“crosses over” from\nwi+1towi. Notice that, as this crossover point can only be in one place , at least one of the two codewords\nofCappearing in gwill be complete, and equal to either ciorci+1. Thus, if we could identify where the\ncrossover point hwas, then we could use C’s decoder to decode whichever the complete C-codeword was to\nidentifyi; and then use our knowledge of where his to complete the decoding. The simple observation behind\nour construction is that, because the strings si(which are either all zeros or all ones) flip with the parity of\ni, we cantell (approximately) where hwas! Indeed, these strings will be all zeros before hand all ones after\nh, or vice versa. Of course, some noise will be added, but provi ded that the length of the strings siare long\nenough, we will still be able to approximately locate hwith high probability.\nHowever, there are several challenges to implementing this simple idea. For example, given iandh, how do\nwe efficiently compute j? (Here is where the fact that we ordered Ccarefully comes in; it’s not trivial because\nthe number of codewords of Ginserted between wiandwi+1depends on i, so naively adding up the number\nof codewords of Gthat come before wiand then adding hwould take exponential time.) Or, what happens\nwhen the crossover point his very close to the end of gj+η? (Here, it might be the case that we mis-identify\ni; but we show that this does not matter, with high probability , because our final estimate will still be close to\njwith high probability). In the rest of the paper, we show how t o deal with these and other challenges.\nII. P RELIMINARIES\nWe begin by setting notation. Throughout, we work with linea r codes over F2, so all arithmetic between\ncodewords is modulo 2. For x,y∈Fℓ\n2, let∆(x,y)denote the Hamming distance between xandy. We use\n/⌊ard⌊lx/⌊ard⌊lto denote the Hamming weight of a vector x∈Fℓ\n2. For a code C ⊆Fn\n2, the minimum distance of the code\nis given by D(C) := max c/ne}ationslash=c′∈C∆(c,c′).\nFor two strings s1ands2, we use s1◦s2to denote the concatenation of s1ands2. For a string s∈Fℓ\n2\nand fori≤ℓ, we use prefi(s)∈Fi\n2to denote the prefix of the string sending at (and including) index i.\nAnalogously, we use suff i(s)∈Fℓ−i\n2be defined as the suffix of sstarting at (and including) index i. For an\nintegerℓ, we use [ℓ]to denote the set {1,...,ℓ}.\nForℓ∈Z, letMajℓ:Fℓ\n2→F2be majority function on ℓbits. (In the case that ℓis even and a string y∈Fℓ\n2\nhas an equal number of zeros and ones, Majℓ(y)is defined to be a randomized function that outputs 0or1\neach with probability 1/2.) We use Ber(p)to denote the Bernoulli- pdistribution on F2, so ifX∼Ber(p),\nthenXis1with probability pand0with probability 1−p.\nNext we define Binary Reflected Codes, a classical Gray code ( [ 12]; see also, e.g., [13]); we will use these\nto define our ordering on C.Definition 1 (Binary Reflected Code, [12]) .Letkbe a positive integer. The Binary Reflected Code (BRC)\nis a map Rk:{0,...,2k−1} →Fk\n2defined recursively as follows.\n1) Fork= 1,R1(0) = 0 andR1(1) = 1 .\n2) Fork >1, for any i∈ {0,...,2k−1},\n•Ifi <2k−1, thenRk(i) = 0◦Rk−1(i)\n•Ifi≥2k−1, thenRk(i) = 1◦Rk−1(2k−1−(i−2k−1)) = 1◦Rk−1(2k−i).\nIt is not hard to see that for any two successive integers iandi+1, the encoded values Rk(i)andRk(i+1)\ndiffer in exactly one bit. We will need one more building-blo ck, the Unary code.\nDefinition 2 (Unary code) .The Unary code U ⊆Fℓ\n2is defined as the image of the encoding map EncU:\n{0,...,ℓ} →Fℓ\n2given by EncU(v) := 1v◦0ℓ−v.The decoding map DecU:Fℓ\n2→ {0,...,ℓ}is given by\nDecU(x) = argminv∈{0,...,ℓ}∆(x,EncU(v)).\nNext, we define the failure probability of a code C.\nDefinition 3. LetC ⊆Fn\n2be a code with message length kand encoding and decoding maps DecCandEncC\nrespectively. The probability of failure of Cis\nPfail(C) = max\nv∈Fk\n2P[DecC(EncC(v)+ηp)/ne}ationslash=v],\nwhere the probability is over a noise vector ηp∈Fn\n2withηp∼Ber(p)n.\nIII. C ONSTRUCTION\nWe recall the high-level overview of our construction from t he introduction: To construct Gwe will start\nwith a base code CwhereC ⊆Fn\n2, which we will order in a particular way (Definition 4). Then w e construct\nan intermediate code W={w0,...,w 2k−1} ⊆Fd\n2by transforming the codewords of C(Definition 5); the\ncodewords of Winherit an order of C. Finally, we create final code G ⊆Fd\n2by adding new codewords that\n“interpolate” between the codewords of Wso that it satisfies the Gray code condition (Definition 6). We discuss\neach of these steps in the subsequent subsections.\nA. Base Code C\nGiven a base code C ⊂Fn\n2, we define an ordering on the elements of Cas follows.\nDefinition 4. [Ordering on C] LetC ⊆Fn\n2be a linear code with block length nand dimension k. LetAC∈Fk×n\n2\nbe a generator matrix for C, and let aidenote the i-th row of AC. Giveni∈ {0,...,2k−1}, definezito be\nthe unique integer so that Rk(i)[zi]/ne}ationslash=Rk(i+1)[zi].2Letc0= 0n. Then, for all i∈ {1,...,2k−1}, thei-th\ncodeword of Cis defined by\nci=ci−1+azi.\nOur next lemma establishes that indeed this ordering hits al l of the codewords.\nLemma 1. LetCbe a linear code, and consider the ordering defined in Definiti on 4. For every c∈ C, there is\na unique index i∈ {0,...,2k−1}such that ci=c.\nProof. Observe that, by construction, we have\nci=Rk(i)TAC.\nSinceRk:{0,...,2k−1} →Fk\n2is a bijection and ACis full rank, this implies that each codeword in Cis\nuniquely represented as some cifori∈ {0,...,2k−1}.\n2As noted after Definition 1, Rk(i)andRk(i+1) differ in only one bit, so ziis well-defined.B. Intermediate Code W\nNext, we describe how to generate our intermediate code W.\nDefinition 5. LetC ⊆Fn\n2be a linear code of dimension k. Let(c0,...,c 2k−1)denote the ordering of codewords\ninCas per Definition 4. Let d= 2n+3D(C). The intermediate code W, along with its ordering, is defined\nas follows. For each i∈ {0,...,2k−1}, definewi∈Fd\n2by the equation\nwi=/braceleftBigg\n0D(C)◦ci◦0D(C)◦ci◦0D(C)ifiis even\n1D(C)◦ci◦1D(C)◦ci◦1D(C)ifiis odd(2)\nThen,Wis the subset of Fd\n2defined by W={wi:i∈ {1,...,2k−1}}, where the code Wis ordered as\n(w0,...,w 2k−1).\nC. Final Code G\nTo create our robust Gray code G, given any two consecutive codewords in W, we inject extra codewords\nbetween them to create G, as follows.\nDefinition 6 (Definition of our robust Gray code G; and the parameters ri,hi,j).LetW ⊆Fd\n2be a code\ndefined as in Definition 5. For each i∈[2k], defineri=/summationtexti\nℓ=1∆(wℓ−1,wℓ), and let N=r2k. Fori∈[2k]\nand1≤j <∆(wi,wi+1), lethi,j∈ {0,...,d−1}be thej-th index where codewords wiandwi+1differ.\nDefine the zero’th codeword of Gasg0=w0. Fixj∈ {1,...,N−1}. Ifj=rifor some i, we define\ngj∈Fd\n2bygj=wi. On the other hand, if j∈(ri,ri+1)for some i, then we define gj∈Fd\n2as\ngj=prefhi,j−ri(wi+1)◦suffhi,j−ri+1(wi). (3)\nFinally, define G ⊆Fd\n2byG={gi:i∈ {0,...,N−1}},along with the encoding map EncG:{0,...,N−\n1} →Fd\n2given by EncG(i) =gi.\nWe will also define hi= (hi,1,hi,2,...,h i,∆(wi,wi+1)−1)∈ {0,...,d−1}∆(wi,wi+1)−1to be the vector of\nall indices in which wiandwi+1differ, in order, except for the last one.3\nIt will frequently be useful to be able to locate j∈[N]within a block [ri,ri+1). To that end, we introduce\nthe following notation.\nDefinition 7. Letj∈ {0,...,N−1}. Leti∈ {0,...,2k−1}be such that j∈[ri,ri+1). Then we will use\nthe notation ¯jto denote j−ri. That is, ¯jis the index of jin the block [ri,ri+1).\nNote that, in this notation, when j∈[ri,ri+1), the last bit that has been flipped to arrive at gjin the ordering\nofG(that is, the “crossover point” alluded to in the introducti on) ishi,¯j.We make a few useful observations\nabout Definition 6. The first two follow immediately from the d efinition.\nObservation 1. Gis a Gray code. That is, For any j∈ {0,...,N−1}, we have that ∆(gj,gj+1) = 1 .\nObservation 2. Suppose that C ⊆Fn\n2has rateR= log2|C|/nand distance o(n). Then the code Gconstructed\nas in Definition 6 has rate that approaches R/2asN→ ∞ .\nObservation 3. Letgj∈ G, and suppose that j∈(ri,ri+1)for some i∈ {0,...,2k−1}. Then\n(gj+wi)[hi] = Enc U(¯j),\nwhereU ⊂F∆(wi,wi+1)\n2 is the unary code of length ∆(wi,wi+1). Above, (gj+wi)[hi]denotes the restriction\nof the vector gi+wi∈Fd\n2to the indices that appear in the vector hi.\nProof. By definition, hicontains the indices on which wiandwi+1differ, and also by definition, by the time\nwe have reached gj, the first j−ri=¯jof these indices have been flipped from agreeing with wito agreeing\nwithwi+1. Thus, if we add gjandwi(mod 2), we will get 1on the first j−riindices and 0on the on the\nrest.\n3The reason we don’t include the last one is because once the la st differing bit has been flipped, gjwill lie in [wi+1,wi+2), not\n[wi,wi+1).Before we move on, we show Definition 6 actually defines an inje ctive map.\nLemma 2. LetGbe a code with encoding map EncGbe as defined in Definition 6. Then EncGis injective.\nProof. Assume, for the sake of contradiction, that there are two dis tinctj,j′∈ {0,...,N−1}such that\ngj=gj′. Without loss of generality assume that j′> j. There are three scenarios possible.\n1)Case 1: Bothjandj′are in the interval [ri,ri+1). Then we claim that gj[hi,¯j′]/ne}ationslash=gj′[hi,¯j′]. The reason\nis thatgj[hi,¯j′] =wi[hi,¯j′]andgj′[hi,¯j′] =wi+1[hi,¯j′]; but by definition of hi,¯j′,wi[hi,¯j′]/ne}ationslash=wi+1[hi,¯j′].\nThus,gj/ne}ationslash=gj′.\n2)Case 2: j∈[ri−1,ri)andj′∈[ri,ri+1). Thengjis an interpolation of wi−1andwi, andgj′is an\ninterpolation of wiandwi+1. Notice that gj[d−1] =wi−1[d−1]andgj′[d−1] =wi[d−1], as the\nlast index of the codewords gjandgj′has not been flipped yet. However, as iandi−1have different\nparity,wi[d−1]/ne}ationslash=wi−1[d−1], which implies gj/ne}ationslash=gj′.\n3)Case 3:j∈[ri,ri+1)andj′∈[ri′,ri′+1)where|i−i′|>1. In this scenario, {ci,ci+1}∩{ci′,ci′+1}=∅.\nSuppose that neither hi,¯jnorhi′,¯j′are in[D(C),D(C) +n). Then˜c1(gj)∈ {ci,ci+1}and˜c1(gj′)∈\n{ci′,ci′+1}leading to ˜c1(gj)/ne}ationslash= ˜c1(gj′)hencegj/ne}ationslash=gj′. The same holds if neither are in [2D(C) +\nn,2D(C) + 2n), repeating the argument with ˜c2. The final sub-case is that hi,¯j∈[D(C),D(C) +n)\nandhi′,¯j′∈[2D(C)+n,2D(C)+2n)or vice versa. If this occurs (suppose without loss of genera lity,\nit is the first one, not the “vice versa” case), then according to Lemma 4, s1(gj)/ne}ationslash=s2(gj), however\nforgj′,s1(gj′) =s1(wi′+1) =s2(wi′+1) =s2(gj′). This implies that either s1(gj)/ne}ationslash=s1(gj′)or\ns2(gj)/ne}ationslash=s2(gj′), which implies that gj/ne}ationslash=gj′, as desired.\nIV. D ECODING ALGORITHM AND ANALYSIS\nIn this section, we define our decoding algorithm and analyze it. We begin with some notation for the different\nparts of the codewords gj∈ G. For a string x, we use x[i:i′]to denote the substring (xi,xi+1,...,x i′−1).\nWith this notation, for any x∈Fd\n2, define the following substrings:\n•s1(x) =x[0 :D(C)]\n•˜c1(x) =x[D(C) :D(C)+n]\n•s2(x) =x[D(C)+n: 2D(C)+n]\n•˜c2(x) =x[2D(C)+n,2D(C)+2n]\n•s3(x) =x[2D(C)+2n: 3D(C)+2n].\nNotice that if x∈ G, then˜c1and˜c2are in locations corresponding to the codewords of Cthat appear in\ncodewords of W, whiles1,s2, ands3are in locations corresponding to the 0D(C)and1D(C)strings.\nBefore we formally state the algorithm (Algorithm 2 below), we prove a few lemmas to motivate its structure.\nOur first lemma formalizes the intuition in the introduction that at most one of the “chunks” in each codeword\ngjis broken up by the crossover point hi,¯j.\nLemma 3. Fixj∈ {0,...,N−1}. Suppose that i∈ {0,...,2k−1}is such that j∈[ri,ri+1), sogj∈ G can\nbe written as gj=s1◦˜c1◦s2◦˜c2◦s3as above. Then at most one of the substrings in S={s1,s2,s3,˜c1,˜c2}\nthat is not equal to the corresponding substring in wiorwi+1.\nProof. First, suppose that j=ri. Then in that case gj=wiand all of the substrings in Sare equal to their\ncorresponding substring. Otherwise, j∈(ri,ri+1). In that case, ¯j∈(0,ri+1−ri) = (0,∆(wi,wi+1)). This\nmeans that hi,¯j(the “crossover point” for gj) is defined, and indexes a position in gj, and in particular in\none of the sub-strings in S. Then other substrings strictly to the left of hi,¯jare equal to their corresponding\nsubstring in wi+1; and the ones strictly to the right are equal to the correspon ding substring in wi.\nUsing the language of Lemma 3, we say that a substring in Sthat is equal to its corresponding substring\ninwiis afull chunk . Thus, Lemma 3 implies that there are at least four full chunk s in anygj∈ G. Notice\nthat it is possible that a substring ˜cℓis inCbut is not a full chunk. We say that all full chunks are decoded\ncorrectly if, for full chunk of x, when we run the corresponding decoder, we get the right answ er. That is, if\n˜c1(x)is a full chunk, then if we were to run DecCon˜c1(x)we would obtain ˜c1(gj), and similarly for ˜c2; and\nifs1(x)is a full chunk, and we were to run MajD(C)ons1(x), we would obtain s1(gj), and similarly for s2\nands3.Next, we show that if the “crossover point” hi,¯jdoes not point to one of chunks s1,s2, ors3, then there are\nat least two of them that differ.\nLemma 4. LetGbe a code defined as in Definition 6. Fix any gj∈ G and letribe such that j∈[ri,ri+1).\nSuppose that hi,¯j∈[D(C),D(C)+n)∪[2D(C)+n,2D(C)+2n); that is, hi,¯jindexes a position in ˜ci′(gj)\nfor some i′∈ {1,2}. Thensi′(gj)/ne}ationslash=si′+1(gj).\nProof. Without loss of generality, suppose that i′= 1. By definition, we have\ngj= prefhi,¯j(wi+1)◦suffhi,¯j+1(wi).\nIn particular, since the “cut-off” hi,¯jpoints to a position within ˜c1(gj), we have that both s1(gj)ands2(gj)\nare full chunks, and further s1(gj)agrees with wi+1, whiles2(gj)agrees with wi. Sinceiandi+ 1 have\ndifferent parities, either s1(gj) = 0D(C)ands2(gj) = 1D(C), or the other way around; in either case, they are\ndifferent. The same argument holds when i′= 2.\nFinally, we break things up into three cases, which will be re flected in our algorithm. In each case, we\ncan use the pattern of the chunks s1(x),s2(x),s3(x)to get an estimate for ciorci+1, and bound where the\ncrossover point hi,¯jwill be.\nLemma 5. Letgj∈ Gand letibe such that j∈[ri,ri+1). Letη∼Ber(p)dwherep∈(0,1/2). Letx=gj+η\nbe a received input. Then define ˆvi′= Dec C(˜ci′(x))fori′∈ {1,2}andbi′= MajD(C)(si′(x))fori′∈ {1,2,3}.\nAssume that all full chunks are decoded correctly by their co rresponding decoder. Then the following hold.\n1) If(b1,b2,b3)∈ {(1,1,0),(0,0,1)}, thenEncC(ˆv1) =ci+1andhi,¯j≥n+D(C).\n2) If(b1,b2,b3)∈ {(0,1,1),(1,0,0)}, thenEncC(ˆv2) =ciandhi,¯j≤n+2D(C).\n3) If(b1,b2,b3)∈ {(0,0,0),(1,1,1)}, thenEncC(ˆv1) = Enc C(ˆv2)∈ {ci,ci+1}andhi,¯j∈[0,D(C))∪[d−\nD(C),d). Moreover, if hi,¯j∈[0,D(C)), then˜c1(gj) = ˜c2(gj) =ci; and otherwise they are equal to ci+1.\nProof. We address each case individually.\n1) If(b1,b2,b3)∈ {(1,1,0),(0,0,1)}then we claim that hi,¯j≥n+D(C). Assume otherwise. If hi,¯j∈\n[0,D(C)), thengj[D(C) :] =wi[D(C) :]. This means that s2(gj) =s3(gj), and are full chunks. Given the\nassumption that all the full chunks are decoded correctly, b2= Maj(s2(x)) = Maj( s3(x)) =b3but that\ncontradicts our assumption for this case; so we conclude tha thi,¯j/ne}ationslash∈[0,D(C)). Thus,hi,¯j∈[D(C),n+\nD(C)). But then then s1(gj)ands2(gj)are full chunks, and according to Lemma 4, s1(gj)/ne}ationslash=s2(gj).\nAgain using the assumption of correct decoding of all full ch unks, this implies that b1/ne}ationslash=b2, which again\ncontradicts our assumption for this case. This establishes the claim that hi,¯j≥n+D(C).\nFinally, the fact that hi,¯j≥n+D(C)implies that ˜c1(gj) = ˜c1(wi+1) =ci+1, and˜c1(gj)is a full chunk.\nUsing the assumption of correct decoding of all full chunks, we get that EncC(ˆv1) =ci+1\n2) If(b1,b2,b3)∈ {(0,1,1),(1,0,0)}, then the conclusion follows by an argument nearly identica l to case\n1.\n3) If(b1,b2,b3)∈ {(0,0,0),(1,1,1)}, then we claim that hi,¯j∈[0,D(C))∪[d−D(C),d). Assume otherwise,\nthens1(gj) =s1(wi+1)ands3(gj) =s3(wi)and they are full chunks. Now as iandi+1do not have the\nsame parity, s1(gj)/ne}ationslash=s3(gj). As a result, if all full chunks are decoded correctly, we hav e thatb1/ne}ationslash=b3,\nwhich contradicts our assumption in this case. This proves o ur claim that hi,¯j∈[0,D(C))∪[d−D(C),d).\nIfhi,¯j∈[0,D(C))then��c1(gj) = ˜c2(gj) =ci; ifhi,¯j∈[d−D(C),d)then˜c1(gj) = ˜c2(gj) =ci+1; and\nin either case both are full chunks. Using the assumption tha t all full chunks are decoded correctly, we\nsee thatEncC(ˆv1) = Enc C(ˆv2)∈ {ci,ci+1}, as desired.\nA. Decoding Algorithm\nBefore we state our main algorithm (Algorithm 2 below), we in clude a helper algorithm, compute-r\n(Algorithm 1). This algorithm takes an index i∈ {0,...,2k−1}and returns ri. Note that this is not trivial to\ndo efficiently: If we wanted to compute ridirectly from the definition, that would require computing o r storing\n∆(wℓ,wℓ+1)for allℓ≤iand adding them up, which may take time Ω(2k). Instead, we do something much\nfaster.Algorithm 1 compute-r\nInput:i∈ {0,...,2k−1}\nˆri= 0\nforz∈ {0,...,k−1}do\nˆri= ˆri+2⌊max(i−2z,0)\n2z+1⌋·/⌊ard⌊laz/⌊ard⌊l+3D(C) ⊲ azis thez’th row of the generator matrix of C.\nend for\nReturn: ˆri\nLemma 6. The Algorithm compute-r (Algorithm 1) correctly computes ri.\nProof. Recall that ri=/summationtexti−1\nℓ=0∆(wℓ,wℓ+1). Consider a fixed difference ∆(wℓ,wℓ+1). This is precisely\n∆(wℓ,wℓ+1) = 2/⌊ard⌊lazℓ/⌊ard⌊l+3D(C), (4)\nwherezℓis the unique index so that Rk(ℓ)[zℓ]/ne}ationslash=Rk(ℓ+1)[zℓ]: indeed, by Definition 4, ∆(cℓ,cℓ+1) =/⌊ard⌊lazℓ/⌊ard⌊l,\nand from that (4) follows from the definition of W(Definition 5). Thus, in order to compute\nri=i−1/summationdisplay\nℓ=0(2/⌊ard⌊lazℓ/⌊ard⌊l+3D(C)),\nit suffices to count how often each index z∈ {0,...,k−1}shows up as some zℓin that sum. This is precisely /floorleftBig\nmax(i−2z,0)\n2z+1/floorrightBig\n,by the definition of Rk.\nOur final algorithm is given in Algorithm 2. It is organized in to the three cases of Lemma 5. To help\nthe reader, we have included comments saying what each estim ate “should” be. Here, “should” is under the\nassumption that each full chunk is decoded correctly.\nB. Analysis\nNext, we analyze the correctness and running time of Algorit hm 2. We begin with the running time.\nLemma 7. LetC ⊆Fn\n2be a constant rate code. Suppose that DecCruns in time TDecC(n), andEncCruns in time\nTEncC(n), and that D(C) =o(n). LetACbe the generator matrix of C, with rows azforz∈ {0,...,2k−1}.\nSuppose that /⌊ard⌊laz/⌊ard⌊lcan be computed in time O(1). Then the running time is of DecGis\nO(TDecC(d)+TEncC(d)+d),\nand the running time of EncGisO(TEncC(d)).\nRemark 1 (Time to compute /⌊ard⌊laz/⌊ard⌊l).We note that if Cis, say, a Reed-Muller code RM(r,m), then indeed, given\nz,/⌊ard⌊laz/⌊ard⌊lcan be computed in time O(1): if the binary expansion of zhas weight t≤r, then the corresponding\nrow has weight 2m−t. For codes that may not have closed-form expressions for the ir generator matrices, we\ncan pre-compute each /⌊ard⌊laz/⌊ard⌊l(in total time O(d2)) and store them to allow for O(1)lookup time.4\nProof of Lemma 7. As we are assuming that D(C) =o(n), finding the bℓalso takes time o(n)and is negligible.\nAmong the other steps, the only non-tivial ones are running t he encoding and decoding maps for C(each of\nwhich happens O(1)times); running compute-r (which takes time O(k) =O(d)if/⌊ard⌊laz/⌊ard⌊lcan be computed\nin timeO(1)); and running RkandR−1\nk, which can be done in time O(k) =O(d).\nNext, we move on to the analysis of the correctness and failur e probability of Algorithm 2. The final statement\nis Theorem 1 below, but we will need several lemmas first. We be gin by showing that, if all full chunks are\ndecoded correctly, then gˆjis equal to gjon the portion of indices where the crossover point hi,¯jis guaranteed\nnot to be.\n4If a lookup table is not desirable and the /bardblaz/bardblcannot otherwise be computed on the fly, then our algorithm st ill works, and DecG\nruns in time at most O(TDecC(d)+TEncC(d)+d2),where we recall that d= Θ(n)andO(logN).Algorithm 2 DecG: Decoding algorithm for G\n1:Input:x=gj+η∈Fd\n2\n2:Output: ˆj∈[N]\n3:forℓ∈ {1,2}do\n4:ˆvℓ= Dec C(˜cℓ(x)) ⊲Decode˜c1(x)and˜c2(x)individually to obtain ˆv1,ˆv2∈ {0,...,2k−1}.\n5:end for\n6:forℓ∈ {1,2,3}do\n7:bℓ= MajD(C)(sℓ(x)) ⊲Decode each sℓ(x)to obtain bℓ∈ {0,1}.\n8:end for\n9:⊲Below, in the comments we note what each value “should” be. Th is is what these values will be under\nthe assumption that each full chunk is decoded correctly.\n10:if(b1,b2,b3)∈ {(1,1,0),(0,0,1)}then ⊲Case 1:EncC(ˆv1)should be ci+1\n11:ˆι=R−1\nk(ˆv1) ⊲ˆιshould be i+1\n12:ˆv=Rk(ˆι−1) ⊲ˆvshould be the BRC corresponding to i\n13:ˆc1= Enc C(ˆv) ⊲ˆc1should be ci\n14:ˆc2= Enc C(ˆv1) ⊲ˆc2should be ci+1\n15:ˆw1= Enc W(ˆc1) ⊲ˆw1should be wi\n16:ˆw2= Enc W(ˆc2) ⊲ˆw2should be wi+1\n17:H′={ℓ∈hˆι−1:ℓ≥n+D(C)}⊲Should be the set of indices that appear in hithat are at least\nn+D(C).\n18:u= Dec U(x[H′]+ ˆw1[H′]) ⊲uis an estimate of hi,¯j−∆(ci,ci+1)−D(C)\n19:ˆj=u+D(C)+∆(ˆc1,ˆc2)+compute-r (ˆι−1)\n20:else if(b1,b2,b3)∈ {(0,1,1),(1,0,0)}then ⊲Case 2:EncC(ˆv2)should be ci\n21:ˆι=R−1\nk(ˆv2) ⊲ˆιshould be i\n22:ˆv=Rk(ˆι+1) ⊲ˆvshould be the BRC corresponding to i+1\n23:ˆc1= Enc C(ˆv2) ⊲ˆc1should be ci\n24:ˆc2= Enc C(ˆv) ⊲ˆc2should be ci+1\n25:ˆw1= Enc W(ˆc1) ⊲ˆw1should be wi\n26:ˆw2= Enc W(ˆc2) ⊲ˆw2should be wi+1\n27:H′={ℓ∈hˆι−1:ℓ < n+2D(C)}\n28: ⊲H′should be the set of indices that appear in hithat are less than n+2D(C).\n29:u= Dec U(x[H′]+ ˆw1[H′]) ⊲uis an estimate of hi,¯j≤2D(C)+n\n30:ˆj=u+compute-r (ˆι)\n31:else if(b1,b2,b3)∈ {(0,0,0),(1,1,1)}then\n32: ⊲Case 3:EncC(ˆv1)andEncC(ˆv2)should be equal to each other, and to either ciorci+1, but we\nneed to figure out which one.\n33:ˆι=R−1\nk(ˆv1) ⊲ˆιshould be either iori+1\n34:ˆv=Rk(ˆι−1) ⊲ˆvshould be BRC encoding of iori−1depending on ˆι\n35:ˆc1= Enc C(ˆv) ⊲ˆc1=ciifˆι=i+1andˆc1=ci−1ifˆι=i\n36:ˆc2= Enc C(ˆv1) ⊲ˆc2=ci+1ifˆι=i+1andˆc1=ciifˆι=i\n37:ˆw= Enc W(ˆc2) ⊲ˆw=wi+1ifˆι=i+1andˆw=wiifˆι=i\n38:u1= Dec U(x[< D(C)]+bD(C)\n1) ⊲u1is an estimate of hi,¯j< D(C)assuming ˆw=wi\n39:u2= Dec U(x[>2D(C)+2n]+¯bD(C)\n1)\n40: ⊲u2is an estimate of hi,¯j−2D(C)−2∆(ci,ci+1)assuming ˆw=wi+1\n41:ˆj1=u1+compute-r (ˆι) ⊲Estimate for jassuming ˆw=wi\n42:ˆj2=u2+2D(C)+2∆(ˆc1,ˆc2)+compute-r (ˆι−1)) ⊲Estimate for jassuming ˆw=wi+1\n43:ˆg1= Enc G(ˆj1)\n44:ˆg2= Enc G(ˆj2)\n45:ˆj= minj′∈{ˆj1,ˆj2}∆(x,ˆgj′)\n46:end ifLemma 8. Letx=gj+ηbe the noisy input to DecG, wheregj∈ G andη∼Ber(p)nand letˆj= Dec G(x).\nAssume that all full chunks are decoded correctly in Lines 3 t o 8. Define I ⊂ {0,...,d−1}as the set of indices\nthathi,¯jcan be equal to depending on the pattern of (b1,b2,b3)according to Lemma 5.5Thengˆj[¯I] =gj[¯I].\nProof. First notice that the indices in ¯Iare indices corresponding to a subset S⊆ {s1,s2,s3,˜c1,˜c2}. As\nhi,j/ne}ationslash=¯Ithen all chunks in Sare full chunks. Given that full chunks are decoded correctl y, then we know that\nfor˜ci∈S,EncC(ˆvi) = ˜ci(gj)and forsi∈Swe have that bD(C)\ni=si(gj). As a result the decoder fixes the\nvalues of these indices and only estimates the values of the r est of the bits in lines 18, 29, 38, and 39. Thus,\ngj[¯I] =gˆj[¯I].\nLemma 9. For ai∈ {0,...,2k−2}letjbe such that j∈[ri,ri+1)andhi,¯j∈[D(C),d−D(C)). Let\nx=gj+ηbe the noisy input for DecGandˆjbe the estimate given by DecG. Assuming that all full chunks\nare decoded correctly, then ˆj∈[ri,ri+1), Moreover, |j−ˆj|= ∆(gj,gˆj).\nProof. We first claim that either Case 1 or Case 2 of Lemma 5 has occurre d. Indeed, the fact that hi,¯j∈\n[D(C),d−D(C))implies that s1(gj)ands3(gj)are both full chunks, and our assumption that each full chunk\nis correctly decoded implies that s1(gj) =s1(wi+1)whiles3(gj) =s3(wi). Asiandi+ 1 have different\nparities,s1(gj)/ne}ationslash=s3(gj), which implies that we are in Case 1 or Case 2 of Lemma 5.\nNext, we establish that the estimate ˆjreturned by the algorithm in Cases 1 or 2 satisfies ˆj∈[ri,ri+1).\nSuppose without loss of generality that we are in Case 1, so (b1,b2,b3) = (0,0,1)or(1,1,0)(Case 2 is\nsymmetric). We first go through Case 1 of Algorithm 2, which st arts at Line 10. Since we are in Case 1 of\nLemma 5, that lemma implies that EncC(ˆv1) =ci+1and that hi,¯j≥n+D(C). Thus, in the first case in\nAlgorithm 2, under the assumption that all full chunks are co rrectly decoded, we have ˆι−1 =i,ˆc1=ci,\nˆc2=ci+1,ˆw1=wi, andˆw2=wi+1. At the end of this case, the final estimate ˆjis set to be\nˆj=u+D(C)+∆(ˆc1,ˆc2)+compute-r (ˆι−1).\nBy the above, we have ˆι−1 =i, so by Lemma 6, compute-r (ˆι−1) =ri. Note also that ∆(C) =\n∆(s1(wi),s1(wi+1)). Plugging in to our expression for ˆjand subtracting rifrom both sides, we have\nˆj−ri=u+∆(s1(wi),s1(wi+1))+∆(˜c1(wi),˜c1(wi+1)). (5)\nNow, recall that in Algorithm 2, we have u= Dec U(x[H′]+ ˆw1[H′]), whereH′is the set of ℓappearing\ninhiso thatℓ≥n+D(C). It thus follows from the definition that\nu≤ |H′|<∆(wi[n+D(C) :],wi+1[n+D(C) :]),\nfrom the definition of H′. Plugging this into (5), we see that ˆj−ri<∆(wi,wi+1), which implies that\nˆj∈[ri,ri+1), as desired.\nFinally, we argue that |ˆj−j|= ∆(gj,gˆj). Indeed, we may write\ngj+gˆj= (gj+wi)+(gˆj+wi) = (gj+wi)[hi]+(gˆj+wi)[hi].\nBy Observation 3, (gj+wi)[hi] = Enc U(j); and by that observation along with the fact that ˆj∈[ri,ri+1),\nwe also have (gˆj+wi)[hi] = Enc U(ˆj).Thus,\n/⌊ard⌊lgj+gˆj/⌊ard⌊l=/⌊ard⌊l(gj+wi)[hi]+(gˆj+wi)[hi]/⌊ard⌊l=/⌊ard⌊lEncUj+Enc Uˆj/⌊ard⌊l=|j−ˆj|,\nwhich finishes the proof of the lemma.\nLemma 10. Fori∈ {0,...,2k−2}, letjbe such that j∈[ri,ri+D(C))∪[ri+1−D(C),ri+1). Further let\nx=gj+ηbe the noisy input and ˆjbe the estimate obtained from DecG. Assuming that all full chunks are\ndecoded correctly, then either\n1)ˆj∈[ri,ri+1)and∆(gj,gˆj) =|j−ˆj|; or\n2)ˆj∈[ri+1,ri+2)∪[ri−1,ri)and|j−ˆj| ≤2D(C)and∆(gj,gˆj) =|j−ˆj|.\n5That is, if (b1,b2,b3) = (0,0,1)or(1,1,0), thenI= [n+D(C),2n+3D(C)), and so on.Proof. Unlike in Lemma 9, it is now the case that any of the three cases in Lemma 5 could occur. We first\nconsider Cases 1 and 2, so (b1,b2,b3)/∈ {(0,0,0),(1,1,1)}in line 7. The proof is quite similar to that of\nLemma 9, so we just sketch it here. Suppose that we are in Case 1 of Lemma 5 (Case 2 is symmetric). In\nCase 1, we claim that ˆj∈[ri,ri+1). Indeed, since all full chunks are decoded correctly, as we a rgued in the\nproof of Lemma 9, the values ˆι,ˆv,ˆcℓ,ˆwℓare computed correctly, meaning that they match the values t hat the\ncomments in Algorithm 2 say they should be. In particular, as before we have\nˆj=u+D(C)+∆(˜c1,˜c2)+compute-r (ˆι−1)\n=u+D(C)+∆(ci,ci+1)+ri\n<∆(wi,wi+1)+ri\n< ri+1.\nThis establishes that ˆj∈[ri,ri+1), which proves the claim; the rest of this case follows identi cally to the proof\nof Lemma 9.\nNext we consider Case 3, so (b1,b2,b3)∈ {(0,0,0),(1,1,1)}. In this case, Lemma 5 (and the assumption\nthat all full chunks are correctly decoded) says that EncC(ˆv1) = Enc C(ˆv2)∈ {ci,ci+1}, which implies that\nthe value of ˆwcomputed in line 37 is either wiorwi+1. Algorithm 2 then computes two estimates ˆj1and\nˆj2, which are meant to be an estimate of jin the two cases the ˆw=wiandˆw=wi+1; eventually it picks\nwhichever ˆjℓproduces a codeword closer to the received word x.\nThere are two main sub-cases. In the first, the algorithm gues ses correctly, meaning that either ˆj=ˆj1and\nˆw=wi; or that ˆj=ˆj2andˆw=wi+1. In the other case, the algorithm guesses incorrectly, mean ing that\nˆj=ˆj1andˆw=wi+1, orˆj=ˆj2andˆw=wi. We consider each of these in turn.\nFirst suppose that the algorithm guesses correctly. This ca se is again quite similar to that of Lemma 9, and\nwe sketch the proof. Suppose that ˆj=ˆj1andˆw=wi; the other way of “guessing correctly” leads to a similar\nargument. Now, we claim that ˆj∈[ri,ri+1). To see this, notice that in this case, we have\nˆj=ˆj1=u1+compute-r (ˆι)\nin Line 41. Since ˆw=wi, given our assumption that all full chunks are correctly dec oded, it is not hard to\nsee thatˆι=i. Lemma 6 then implies that compute-r (ˆι) =ri, so\nˆj=u1+ri≤D(C)+ri<∆(wi,wi+1)+ri=wi+1.\nThis shows that ˆj∈[wi,wi+1). Once we have this, the rest of the argument follows as in Lemm a 9.\nNow we move onto the second sub-case of Case 3, when the algori thm guesses incorrectly. Unlike the\nprevious cases we have considered, this is different than Le mma 9, because ˆjmay end up outside of [ri,ri+1).\nWithout loss of generality, suppose that ˆw=wibut thatˆjhas been set to ˆj2. (The other case is similar).\nClaim 1. In this sub-case, the following hold.\nA.ˆj∈[ri−1,ri)\nB.j < ri+D(C)\nC.ˆj≥ri−1+2∆(ci,ci−1)+2D(C) =r−D(C).\nProof. We begin with B. First, since ˆw=wi, this implies that ˜c1(gj) = ˜c2(gj) =wi, so Lemma 5 (Case 3)\nimplies that hi,¯j∈[0,D(C)). Then since ¯j≤hi,¯j≤D(C), we have\nj=¯j+ri≤ri+D(C).\nThis proved B.\nNext we prove C. This follows from the computation of ˆj2in Algorithm 2, along with the assumption that\nall full chunks are decoded correctly. Indeed, we have\nˆj=ˆj2=u2+2D(C)+2∆(ˆc1,ˆc2)+compute-r (ˆι−1).\nSinceˆw=wi, we are in the case where ˆι=i,ˆc1=ci−1,ˆc2=ci, and the above implies that\nˆj=u2+2D(C)+2∆(ci−1,ci)+ri−1.The fact that u2≥0proves inequality in part C. The equality in part C follows si nce by the definition of W,\nwe have\n2∆(ci,ci−1)+2D(C) = ∆(wi,wi−1)−D(C).\nFinally, we move onto A. The fact that ˆj≥ri−1follows immediately from C. The fact that ˆj < rifollows\nfrom the fact that, by a computation similar to that above, we have\nˆj=ˆj2=u2+ri−D(C),\nwhich is less than riasu2< D(C).\nGiven the claim, we can finish the proof of the lemma in this sub -case. First, we observe that |j−ˆj| ≤D(C);\nindeed this follows directly from B and C in the claim.\nFinally, we show that ∆(gj,gˆj) =|j−ˆj|. To see this, we first write\n/⌊ard⌊lgj+gˆj/⌊ard⌊l=/⌊ard⌊l(gj+wi)+(gˆj+wi)/⌊ard⌊l.\nWe claim that gjandwidiffer on only the indices in [0,D(C)). This follows from the fact that hi,¯j∈[0,D(C)),\nwhich we saw in the proof of Claim 1 (part B). Next, we claim tha tgˆjandwidiffer only on the indices in\n[d−D(C),d). Indeed, part C of Claim 1 implies that ri−ˆj≤D(C). Sinceˆj∈[ri−1,ri)(part A of Claim 1),\nthis means that ˆjis in the last chunk (the s3chunk) of [ri−1,ri), which proves the claim. Thus, we have that\n/⌊ard⌊l(gj+wi)+(gˆj+wi)/⌊ard⌊l=/⌊ard⌊lgj+wi/⌊ard⌊l+/⌊ard⌊lgˆj+wi/⌊ard⌊l,\nas the two parts differ on disjoint sets. Moreover, since s1(wi) =s1(wi+1), we have /⌊ard⌊lgj+wi/⌊ard⌊l= (j−ri),\nsincewi+1andwidiffer on allof the first D(C)bits, sogjandwidiffer on all of the first j−ribits. Similarly,\n/⌊ard⌊lgˆj+wi/⌊ard⌊l=ri−ˆj. Putting everything together, we conclude that\n∆(gj,gˆj) = (j−ri)+(ri−ˆj) =j−ˆj.\nSinceˆj < j in this case (as ˆj∈[wi−1,wi), whilej∈[wi,wi+1), this proves the last component of the lemma.\nThe following lemma is included in [1]. We include a short pro of for completeness.\nLemma 11 ( [1]) .LetC ⊆Fn\n2be a linear code with message length kand minimum distance D(C). Further\nletp∈(0,1/2)andηp∼Ber(p)D(C). Then\nPr/bracketleftbigg\n/⌊ard⌊lηp/⌊ard⌊l>D(C)\n2/bracketrightbigg\n+1\n2Pr/bracketleftbigg\n/⌊ard⌊lηp/⌊ard⌊l=D(C)\n2/bracketrightbigg\n≤Pfail(C). (6)\nProof. LetDecMLD be the maximum likelihood decoder for C. That is, given x∈Fn\n2,DecMLD(x) =\nargminc∈C∆(x,c). If there are multiple codewords c∈Cthat attain the minimum above, then DecMLD\nchooses randomly among them. Then\nPfail(C)≥max\nv∈Fk\n2Pr[Dec MLD(EncC(v)+ηp)/ne}ationslash=v] =:pMLD.\nFix a message v∈Fk\n2, and let cv= Enc C(v). Letv′∈Fk\n2be such that ∆(cv,cv′) =D(C), wherecv′=\nEncC(v′).LetHv,v′={i∈[n] : (cv)i/ne}ationslash= (cv′)i}be the set on which cvandcv′disagree. Let η′\np∼Ber(p)nbe\na noise vector, and define ηp=η′\np[Hv,v′], the restriction of η′\npto the positions indexed by Hv,v′. Observe that\nηp∼Ber(p)D(C)as in the lemma statement. Suppose that /⌊ard⌊lηp/⌊ard⌊l> D(C)/2. Thenv/ne}ationslash= Dec MLD(EncC(v)).On\nthe other hand, if /⌊ard⌊lηp/⌊ard⌊l=D(C)/2, then with probability at least 1/2,v/ne}ationslash= Dec MLD(EncC(v)).\nTogether, we conclude that\nPfail(C)≥pMLD≥Pr[/⌊ard⌊lηp/⌊ard⌊l> D(C)/2]+1\n2Pr[/⌊ard⌊lηp/⌊ard⌊l=D(C)/2].\nLemma 12. Letη∼Ber(p)D(C)be a vector in FD(C)\n2, forp∈(0,1/2). LetYbe the repetition code of length\nD(C), so thatEncY:{0,1} → {0D(C),1D(C)}. Then for any b∈ {0,1},Pr[MajD(C)(EncY(b)+η)/ne}ationslash=b]≤Pfail(C),\nwhere we recall that MajD(C)denotes the majority function. Above, the randomness is ove r both the choice\nofηand any randomness that MajD(C)uses to break ties.\nProof. Fixb∈ {0,1}. Suppose that MajD(C)(EncY(b) +η)/ne}ationslash=b. Then either /⌊ard⌊lη/⌊ard⌊l> D(C)/2, or else /⌊ard⌊lη/⌊ard⌊l=\nD(C)/2and the random choice of MajD(C)was incorrect, which happens with probability 1/2. Thus,\nPr[MajD(C)(EncY(b)+η)/ne}ationslash=b]≤Pr/bracketleftbigg\n/⌊ard⌊lη/⌊ard⌊l>D(C)\n2/bracketrightbigg\n+1\n2Pr/bracketleftbigg\n/⌊ard⌊lη/⌊ard⌊l=D(C)\n2/bracketrightbigg\n,\nwhich by Lemma 11 is at most Pfail(C).\nBefore we prove our main theorem establishing correctness o fDecGwith high probability (Theorem 1), we\nneed one more concentration bound. We use the following from [1].\nLemma 13 ( [1]) .Letx1,x2∈Fd\n2. Letη∼Ber(p)dforp∈(0,1/2). Then\nPr[∆(x2,x1+η)≤∆(x1,x1+η)]≤exp/parenleftbigg\n−(1−2p)2\n4p+2∆(x1,x2)/parenrightbigg\n(7)\nLemma 14. Letgj∈ G, and letˆj= Dec G(gj+ηp). Suppose that all full chunks are decoded correctly. Then\n∆(gj+ηp,gˆj)≤∆(gj+ηp,gj).\nProof. By the analysis in the proof of Lemmas 9 and 10, if all full chun ks are decoded correctly, then all the\nquantities computed in Algorithm 2 beforeu(in Cases 1 and 2), or before u1,u2(in Case 3) are what they\n“should” be. That is, in Cases 1 and 2, all of the quantities co mputed before Lines 18 and 29, respectively are\nwhat the comments claim they should be. In Case 3, all of the co mments computed before Line 39) are what\nthe comments claim they should be. Thus, any error in ˆjcomes from the estimates of u(in Cases 1 or 2) or\nu1andu2(in Case 3).\nWe first work out Case 1. First, we observe that it suffices to lo ok only on the set H′defined as in Algorithm 2.\nThat is, it suffices to show that\n∆((gj+ηp)[H′],gˆj[H′])≤∆(gj+ηp)[H′],gj[H′]).\nIndeed,gjandgˆjdiffer only on H′. Next, recall from Observation 3 that (gj+wi)[hi] = Enc U(¯j); that is,\nrestricted to the elements in hi,gj+wihas¯jones followed by all zeros. Since ¯j≥n+D(C)in Case 1 (as\nshown in the proof of Lemma 9), (gj+wi)[H′]is¯j−(∆(ci,ci+1)+D(C))ones followed by zeros. Thus,\nx[H′]+ ˆw1[H′] =gj[H′]+η[H′]+wi[H′]\nis a vector of ¯j−(∆(ci,ci+1)+D(C))ones followed by zeros, plus the noise from η[H′]. Therefore,\nu= Dec U(x[H′]+ ˆw1[H′])\nis the decoding of EncU(¯j−(∆(ci,ci+1)+D(C)))+ηp[H′]. For notational convenience, in the following we\nwill introduce the notation X=D(C) + ∆(ci,ci+1). With this notation (and the fact that DecUreturns the\nclosest codeword in U), we conclude that\n∆(Enc U(u),EncU(¯j−X)+ηp[H′])≤∆(Enc U(¯j−X),EncU(¯j−X)+ηp[H′]). (8)\nWe claim that in fact the first of the terms in (8) is equal to ∆(gˆj[H′],x[H′])and the second is equal to\n∆(gj[H′],x[H′]), which will immediately prove the statement in Case 1. To see the first part of the claim,\nfirst notice that in Algorithm 2 (using the fact that all the es timates are what they “should” be, as above), we\nhaveˆj=u+X+ri, sou=¯ˆj−X.Then\n∆(gˆj[H′],x[H′]) =/⌊ard⌊lgˆj[H′]+gj[H′]+ηp[H′]/⌊ard⌊l\n=/⌊ard⌊lEncU(¯ˆj−X)+Enc U(¯j−X)+ηp[H′]/⌊ard⌊l\n=/⌊ard⌊lEncU(u)+Enc U(¯j−X)+ηp[H′]/⌊ard⌊l\n= ∆(Enc U(u),EncU(¯j−X)+ηp[H′])Above, we used the fact that on H′,gˆjandgjdiffer precisely on the indices between¯ˆj−Xandˆj−X. This\nproves the first part of the claim. For the next part, we have th at\n∆(gj[H′],x[H′]) =/⌊ard⌊lη[H′]/⌊ard⌊l,\nwhich is the right hand side of (8). This finishes proving the c laim, and the lemma in this case.\nCase 2 is similar to Case 1 and we omit the analysis. In Case 3, w e have two candidates, u1andu2. By an\nanalysis similar to the above, at least one of the following s tatements hold:\n•ˆw1=wiandu1= Dec U(EncU(j)+η′\np),or;\n•ˆw2=wi+1andu2= Dec U(EncU(j−2D(C)−2n)+η′\np),\nwhere in both cases η′\nphas i.i.d. Ber(p)coordinates. Thus, if the first is the case, we have that\n∆(gj+ηj,gˆj1)≤∆(gj+ηj,gj)\nand in the second case we have (again with an analysis similar to that above) that\n∆(gj+ηj,gˆj2)≤∆(gj+ηj,gj).\nThus, by the definition of ˆjin this case (Line 45), we have\n∆(gj+ηp,gˆj) = min{∆(gj+ηj,gˆj1),∆(gj+ηj,gˆj2)} ≤∆(gj+ηj,gj),\nas desired.\nTheorem 1. Fixp∈(0,1/2). LetC ⊆Fn\n2be a linear code. Let Gbe defined as in Definition 6 from C. Let\nηp∈Fd\n2∼Ber(p)dThen\nPr[|j−DecG(EncG(j)+ηp)|> t]≤γexp(−αt)+5Pfail(C) (9)\nwherePfail(C)is the block error probability of C, andαandγare constants given by α=−(1−2p)2\n4p+2and\nγ=2\n1−exp(−α).\nProof. LetSbe the event that at least one of the full chunks in gj= Enc G(j)is decoded incorrectly. Let\nˆj= Dec G(EncG(j)+ηp). Then\nPr[|j−ˆj|> t] = Pr/bracketleftBig\n|j−ˆj|> t|¯S/bracketrightBig\n·Pr[¯S]+Pr/bracketleftBig\n|j−ˆj|> t|S/bracketrightBig\n·Pr[S]\n≤Pr/bracketleftBig\n|j−ˆj|> t|¯S/bracketrightBig\n+Pr[S]\nLeti∈ {0,...,2k−1}be such that j∈[ri,ri+1). We will bound each of the two terms above individually.\nWe begin with Pr/bracketleftBig\n|j−ˆj|> t|¯S/bracketrightBig\n. There are two scenarios, depending on whether jis safely in the middle\nof the interval [ri,ri+1)(that is, in the middle three chunks), or if it is near the edge s (the outermost chunks).\nIn either case, if all full chunks are decoded correctly, the n∆(Enc G(j),EncG(ˆj)) =|j−ˆj|. Indeed, in the\nfirst case this follows from Lemma 9, while in the second case i t follows from Lemma 10. Thus, we see that\nin either case, the probability DecGof returning a particular ˆjis\nPr/bracketleftBig\nˆj|¯S/bracketrightBig\n≤Pr/bracketleftBig\n∆(Enc G(j)+ηp,EncG(ˆj))≤∆(Enc G(j)+ηp,EncG(j))/bracketrightBig\n≤exp(−α∆(Enc G(j),EncG(ˆj)))\n= exp(−α|j−ˆj|),\nAbove, the first line follows from Lemma 14. The second line fo llows from Lemma 13, while the last line\nfollows from the fact that ∆(Enc Gj,EncG(ˆj)) =|j−ˆj|as noted above. Thus, for any integer zthat might be\nequal to|j−ˆj|, we have\nPr[|j−ˆj|=z,¯S]≤2exp(−αz),where the factor of two comes from the fact that ˆjmight be either j+zorj−z. Thus,\nP[|j−ˆj| ≥t|¯S]≤∞/summationdisplay\nz=t2exp(−αz) =2exp(−αt)\n1−exp(−α).\nNow we turn our attention to the second term, Pr[S], the probability that at least one of the full chunks is\ndecoded incorrectly. The full chunks ˜cℓ(gj)forℓ∈ {1,2}are codewords in Cand are decoding using DecC.\nThus, the probability that either of these chunks (assuming they are full chunks) are decoded incorrectly is at\nmost twice the failure probability is Pfail(C).\nOn the other hand, the full chunks sℓforℓ∈ {1,2,3}are repetition codes of length ∆(C). Then Lemma 12\nthat the probability that any of these (assuming it is a full c hunk) is at most Pfail(C), so the probability that any\nof these fail is at most 3Pfail(C). Altogether, the probability that at least one full chunk is decoded incorrectly\nis at most Pr[S]≤5Pfail(C).Summing both terms up we see that\nPr/bracketleftBig\n|j−ˆj| ≥t/bracketrightBig\n≤2exp(−αt)\n1−exp(−α)+5Pfail(C),\nwhich completes the proof of the theorem.\nACKNOWLEDGMENT\nMW and DF are partially supported by NSF Grants CCF-2231157 a nd CCF-2133154. The first author thanks\nRasmus Pagh for bringing our attention to this problem.\nREFERENCES\n[1] D. R. Lolck and R. Pagh, “Shannon meets gray: Noise-robus t, low-sensitivity codes with applications in differentia l privacy,” in\nProceedings of the 2024 Annual ACM-SIAM Symposium on Discre te Algorithms (SODA) . SIAM, 2024, pp. 1050–1066.\n[2] M. Aumüller, C. J. Lebeda, and R. Pagh, “Differentially p rivate sparse vectors with low error, optimal space, and fas t access,” in\nProceedings of the 2021 ACM SIGSAC Conference on Computer an d Communications Security , 2021, pp. 1223–1236.\n[3] J. Acharya, Y . Liu, and Z. Sun, “Discrete distribution es timation under user-level local differential privacy,” in International Conference\non Artificial Intelligence and Statistics . PMLR, 2023, pp. 8561–8585.\n[4] J. Acharya, C. Canonne, Y . Liu, Z. Sun, and H. Tyagi, “Dist ributed estimation with multiple samples per user: Sharp ra tes and phase\ntransition,” Advances in neural information processing systems , vol. 34, pp. 18 920–18 931, 2021.\n[5] L. Xiao, X.-G. Xia, and Y .-P. Wang, “Exact and robust reco nstructions of integer vectors based on multidimensional c hinese remainder\ntheorem (md-crt),” IEEE Transactions on Signal Processing , vol. 68, pp. 5349–5364, 2020.\n[6] W. Wang and X.-G. Xia, “A closed-form robust chinese rema inder theorem and its performance analysis,” IEEE Transactions on\nSignal Processing , vol. 58, no. 11, pp. 5655–5666, 2010.\n[7] G. Reeves and H. D. Pfister, “Reed–muller codes on bms chan nels achieve vanishing bit-error probability for all rates below capacity,”\nIEEE Transactions on Information Theory , 2023.\n[8] E. Arikan, “A performance comparison of polar codes and r eed-muller codes,” IEEE Communications Letters , vol. 12, no. 6, pp.\n447–449, 2008.\n[9] V . Guruswami and P. Xia, “Polar codes: Speed of polarizat ion and polynomial gap to capacity,” IEEE Transactions on Information\nTheory , vol. 61, no. 1, pp. 3–16, 2014.\n[10] M. Mondelli, S. H. Hassani, and R. L. Urbanke, “Unified sc aling of polar codes: Error exponent, scaling exponent, mod erate\ndeviations, and error floors,” IEEE Transactions on Information Theory , vol. 62, no. 12, pp. 6698–6712, 2016.\n[11] H.-P. Wang, T.-C. Lin, A. Vardy, and R. Gabrys, “Sub-4.7 scaling exponent of polar codes,” IEEE Transactions on Information\nTheory , 2023.\n[12] F. Gray, “Pulse code communication,” Mar. 17 1953, uS Pa tent 2,632,058.\n[13] D. E. Knuth, The art of computer programming, volume 4A: combinatorial a lgorithms, part 1 . Pearson Education India, 2011." }, { "title": "2401.15326v1.Energetic_particle_modes_in_burning_plasmas.pdf", "content": "Energetic particle modes in burning plasmas\nYang Li1∗\n1. Southwestern Institute of Physics, PO Box 432, Chengdu 610041, People’s Republic of China\nIn this work, we present a novel kind of energetic particle modes with frequencies in the range\nof thermal ion diamagnetic and/or transit frequencies in burning plasmas. It is shown that the\ncontinuum structure can be broken by the energetic particles. In addition, the mode destabilization\ndoes not require the energetic particle drive to overcome the continuum damping, which indicates\nthat they are much easier to be excited and thus manifest themselves as wide-band modes. Based on\nthe analysis, a switch or even an oscillation between phases of discrete eigenmodes and wide-band\nmodes can be predicted.\nEnergetic particles (EPs), as the product of fusion reaction and auxiliary heating, play a key role in magnetically\nconfined fusion devices. Such EPs are thermalized by Coulomb collision, which provides the energy source for a\nsustainable igniting fusion reactor. However, EPs can also resonantly excite collective modes such as shear Alfv´ en\nwaves (SAWs) which lead to anormalous EP loss[1]. The EP induced SAWs have been investigated extensively[2]. In\nprevious studies, the density of EPs is considered to be low (about O\u0000\nβ3/2\u0001\nof the thermal ion density), especially\nin the analysis of inertial layer[3], where βis the ratio of plasma energy to magnetic field energy. Such condition\nis usually valid when only auxiliary heating is taken into account. In future tokamaks such as ITER or even fusion\nreactors, however, the product of D-T reaction together with power injection will probably lead to a higher proportion\nof EPs. Thus, it is necessary to consider the effect of higher EP proportion on the SAW excitation. In this letter, a\nnovel mechanism for the excitation of SAWs by high proportion of EPs in burning plasmas is presented.\nThe SAWs usually exhibit themselves as discrete eigenmodes, such as Beta-induced Alfv´ en Eigenmode (BAE) [ ? ?\n]etc., due to magnetic field geometry and non-uniformities. These discrete modes can only exist in the gaps of SAW\ncontinuum spectrum since the modes within the continuum are suppressed by the continuum damping mechanism[6].\nIn theoretical study, the continuum spectrum is determined by the property of the so-called inertia, which is calculated\nfrom the asymptotic behavior of the solution of the vorticity equation in inertial layer. If the resonant drive from EPs\nexceeds the threshold of the damping of continuum spectrum, energetic particle modes (EPMs) are then excited[7].\nFor low frequency SAWs, when the EP effect is neglected in the inertial layer due to low density, the vorticity\nequation in inertial layer is then periodic. Here low frequency modes are at the frequency range of beta-induced\nAlfv´ en eigenmode (BAE) and kinetic balooning mode (KBM)[8, 9]. However, if the density of EPs increases from\nO\u0000\nβ3/2\u0001\ntoO(β) of the thermal ions, the EP effect should be taken into account in the inertial layer analysis. In this\nletter, it will be demonstrated that the EP effect, in inertial layer, breaks the periodicity of the vorticity equation and\nthe structure of continuum spectrum, which makes the EPMs much more unstable. It is found that there will not be\nany threshold for EP drive to excite EPMs.\nIn this work, the standard gyrokinetic model is applied and the perturbed distribution function can be expressed\nas[10, 11]\nδfs=\u0010e\nm\u0011\ns\u0014∂F0\n∂Eδϕ−QF 0\nωeiLkJ0(k⊥ρs)δψ\u0015\ns\n+\u0010e\nm\u0011\ns∂F0s\nB∂µ\u0014\u0000\n1−J0(k⊥ρs)eiLk\u0001\u0010\nδϕ+iv∥\nωb· ∇ψ\u0011\n−v⊥\nk⊥cJ1(k⊥ρs)eiLkδB∥\u0015\ns\n+eiLksδKs, (1)\nwhere δKssatisfies\n[ωtr∂ϑ−i(ω−ωd)]sδKs=i\u0010e\nm\u0011\nsQF 0s[J0(k⊥ρs) (δϕ−δψ)] (2)\n+i\u0010e\nm\u0011\nsQF 0s\u0014\u0010ωd\nω\u0011\nsJ0(k⊥ρs)δψ+v⊥\nk⊥cJ1(k⊥ρs)δB∥\u0015\n,\nwhere δϕandδB∥are, respectively, the perturbed electrostatic potential and the parallel magnetic field, δψis related\nto the parallel perturbed vector potential δA∥byδA∥=−i(c/ω)b·∇δψ,ωis mode frequency, b=B/B, the subscript\nsindicates particle species except for core electrons which are assumed to be adiabatic; i.e., s=E, ifor EPs and\nthermal ions respectively, esandmsare the electric charge and mass for the species, F0sis the equilibrium distribution\n∗leeyang˙swip@outlook.comarXiv:2401.15326v1 [physics.plasm-ph] 27 Jan 20242\nfunction, E=v2/2 the energy per unit mass, µ=v2\n⊥/(2B) the magnetic moment, QF 0s= (ω∂E+ ˆω∗)sF0s, ˆω∗sF0s=\nω−1\ncs(k×b)· ∇F0s,ωcs=esB/(msc) is the cyclotron frequency, k=−i∇is the wave vector, J0andJ1are the\n0th and 1st order Bessel functions of the first kind, k⊥the perpendicular to bwave vector, ρLs=mscv⊥/esB\nthe Larmor radius, Lks=ω−1\ncs(k×b)·vis the generator of coordinate transformation from the guiding center to\nparticle variables, ωtr=v∥/qR 0the transit frequency with qbeing the safety factor and R0being the major radius,\nωds=ω−1\ncs(k×b)·\u0010\nµ∇B+v2\n∥κ\u0011\nis the magnetic drift frequency and κ=b·∇bis the magnetic field curvature. Here\nthe ballooning representation [12] is already considered and ϑis the extended poloidal angle. With the s−αmodel[12],\nit can be expressed that ωds=g(ϑ)kϑc\u0010\nv2\n⊥/2 +v2\n∥\u0011\n/qsBR 0,g(ϑ) = cos ϑ+(sϑ−αsinϑ) sinϑandα=−R0q2β′with\na prime denoting derivation with respect to r.\nFinite βtokamak plasmas are considered and ε∼β1/2is used as small parameter for formal orderings. And\nε∼Lp/R0, where Lpis the radial inhomogeneity scale length. A higher density of EPs is considered as nE∼ε2ni\nwith temperature TE∼ε−2Ti. The frequency orderings are given as ω∗piε−1∼ωtiε−1∼ωε−1∼ωA∼ωtE∼εω∗pE,\nwhere ω∗ps=ω∗ns+ω∗Ts,ω∗ns= (cTs/esB) (k×b)· ∇ns/ns,ω∗Ts= (c/esB) (k×b)· ∇Ts,ωts=p\n2Ts/ms/qR 0,\nωA=vA/qR 0is the Alfv´ en frequency and vA=B/√4πnimiis the Alfv´ en velocity. It should be noted that the desity\nof EPs is still much smaller than that of the thermal ions and the Alfv´ en velocity is determined by the thermal ions.\nThe wavelength orderings can be written as kϑρE∼εandkϑρi∼ε2. And ωds∼qk⊥ρsωts, noting again s=E, i.\nIn order to study the SAW dispersion relation, the problem can be determined by a close set of governing equations\nincluding the quasineutrality equation, the perpendicular Amp` ere’s law and the vorticity equation. By assuming that\nthe electron response is adiabatic, the quasineutrality equation can be given as\n\u0012eini\nTi+eni\nTe\u0013\n(δϕ−δψ) +eni\nTi\u0010\n1−ω∗pi\nω\u0011\nbiδψ=*X\ns=i,eJ0δKs+\n, (3)\nwhere ⟨···⟩ = 2πP\nv∥/|v∥|´\n(···)BdµdE/\f\fv∥\f\f,bs=k2\n⊥c2msTs/e2B2and the density of electron is approximately\nequal to the thermal ion density since nE/ni∼ε2. The perpendicular Amp` ere’s law can be cast as\nδB∥=4cπδψ\nωB2(k×b)· ∇P−X\ns4πms\u001c2J1\nk⊥ρsµδK s+eE\nmEQF 0s\nω\u0012\n1−2J1J0\nk⊥ρs\u0013\nµδψ\u001d\n. (4)\nHere argument of J0andJ1isk⊥ρs. The equilibrium distribution function of EPs is regarded as isotropic since the\nEPs are mainly generated by fusion reaction. Pis the total pressure including both thermal particles and EPs. By\nintegrating Eq.(2) with multiper 4 πeiωJ 0/k2\nϑc2and following the procedures in Ref.[10], the vorticity equation can be\nobtained as\nBb· ∇k2\n⊥\nBk2\nϑb· ∇δψ+ω2\nv2\nA\u0010\n1−ω∗pi\nω\u0011k2\n⊥\nk2\nϑδϕ+αg(ϑ)\nq2R2\n0δψ+ωα\n2ckϑq2R0δB∥=*X\ns=i,E4πes\nk2\nϑc2J0ωωdsδKs+\n+4πω2\nk2\nϑc2\u001c\neEµQF 0E\nω\u0012\n1−2J1J0\nk⊥ρE\u0013\nδB∥\u001d\n+\u001c4π\nk2\nϑc2\u0000\n1−J2\n0\u0001\nωωdEe2\nE\nmEQF 0E\nω\u001d\nδψ+4πω2\nk2\nϑc2\u001ce2\nE\nmEQF 0E\nω\u0000\n1−J2\n0\u0001\u001d\nδϕ.\n(5)\nNote that here the parallel Amp` ere’s law is applied,\nk2\n⊥\nk2\nϑb· ∇δψ=4π\nk2\nϑc2iω*X\nsesv∥δfs+\n. (6)\nTo gain further insights of Eqs. (3)-(5), the two-scale approach can be applied. The problem is seperated into\nthe large |ϑ|region (or inertial layer) and the moderate |ϑ|region (or ideal region). In the inertial layer, the mode\nstructure varies on the large scale |ϑ1| ∼ε−1and the small scale |ϑ0| ∼1. The equation set above can be solved order\nby order. In inertial layer, Larmor radius parameters order as k⊥ρi∼εandk⊥ρE∼1. By redefining the perturbed\nfield as δΦ = k⊥δϕ/k ϑ,δΨ = k⊥δψ/k ϑandδˆB∥=k⊥δB∥/kϑ, the solutions of perturbed fields and distribution\nfunction can be described for instance,\nδΨ =δΨ(0)+εδΨ(1)+ε2δΨ(2)+···, (7)\nδKs=δK(0)\ns+εδK(1)\ns+ε2δK(2)\ns+···. (8)3\nThe 0-th order solutions of the governing equations can be obtained as δK(0)\ni= 0,δΦ(0)=δΨ(0),∂2\nϑ0δΨ(0)= 0,\nδK(0)\nE≃eE\nωmEQF 0Ekϑ\nk⊥J0δΨ(0), (9)\nδˆB(0)\n∥≃ckϑ\nωq2R0αc\n2δΨ(0), (10)\nwhere α=αE+αcincludes both thermal particles ( αc) and EPs components ( αE). Note that δΨ(0)is indepen-\ndent of ϑ0. The solution of δB(1)\n∥will not be necessary in later analysis and the related 1st order perpendicu-\nlar Amp` ere’s law is not presented here. By decomposing\u0002\nδΨ(1), δΦ(1)\u0003\n= [δΨs, δΦs] sinϑ0+ [δΨc, δΦc] cosϑ0and\nmatching the Fourier component in ϑ0, the solution of 1st order of the governing equations can be obtained as[3]\nδΨs≃ −sϑ1αEδΨ(0)/2|sϑ1|2,δΨc≃δΨs/(sϑ1),δΦs=sϑ1δΦc,\nδΦc−δΨc\nδΨ(0)=−2cTi\neB0kϑ\nωR 0N(qR0ω/vti)\nD(qR0ω/vti), (11)\nδK(1)\ni=ikϑ\nk⊥(e/m)iQF 0i\nω2−ω2\nti\u0014\n(iω+sϑ1ωti)kϑΩdi\nωδΨ(0)+iω(δΦc−δΨc) +ωti(δΦs−δΨs)\u0015\nsinϑ0\n+ikϑ\nk⊥(e/m)iQF 0i\nω2−ω2\nti\u0014\n(isϑ 1ω−ωti)kϑΩdi\nωδΨ(0)+iω(δΦs−δΨs)−ωti(δΦc−δΨc)\u0015\ncosϑ0, (12)\nwhere N(x) =\u0000\n1−ω∗ni\nω\u0001\u0002\nx+\u00001\n2+x2\u0001\nZ(x)\u0003\n−ω∗Ti\nω\u0002\nx\u00001\n2+x2\u0001\n+\u00001\n4+x4\u0001\nZ(x)\u0003\n,D(x) =\u00001\nx\u0001\u0000\n1 +1\nτ\u0001\n+\u0000\n1−ω∗ni\nω\u0001\nZ(x)−\nω∗Ti\nω\u0002\nx+\u0000\nx2−1\n2\u0001\nZ(x)\u0003\n,τ=Te/TiandZ(x) is the plasma dispersion function. The average of the 1st equation of\nδK(1)\nEinϑ0is already enough for the analysis in this work and can be given as\nωdEJ0δK(1)\nE=\u001ceE\nmEQF 0Ekϑ\nk⊥J2\n0δΨ(0)\u001d\n+eE\nmEQF 0kϑ\nk⊥ωdE\nωJ2\n0δΨ(1)+\u001c\nQF 0ikϑ\nk⊥µ\nk⊥ρE2J0J1δˆB(0)\n∥\u001d\n(13)\nwhere (···) =1\n2π¸\n(···)dϑ0. The average of the 2nd vorticity equation in ϑ0can be cast as\n∂2\n∂ϑ2\n1δΨ(0)+ω2\nω2\nA\u0010\n1−ω∗p\nω\u0011\nδΨ(0)+ωkϑαR 0\n2ck2\n⊥δˆB(0)\n∥=4πω\nk⊥kϑc2q2R2\n0D\neiωdiδK(1)\ni+eEJ0ωdEδK(1)\nEE\n+4π\nk2\n⊥c2q2R2\n0\u001c\n(1−J2\n0)ωωdEe2\nE\nmEQF 0E\nω\u001d\nδΨ(1)+4πω2q2R2\n0\nk2\n⊥c2\u001ce2\nE\nmEQF 0E\nω\u0000\n1−J2\n0\u0001\u001d\nδΦ(0)\n+4πω2q2R2\n0\nk2\n⊥c2\u001c\neEµQF 0E\nω\u0012\n1−2J1J0\nk⊥ρE\u0013\nδˆB(0)\n∥\u001d\n. (14)\nBy substituting δB(0)\n∥,δK(1)\ni,δK(1)\nE,δΨ(1)and Eq.(13) into Eq.(14), the 2nd order vorticity equation can be re-written\nas\n∂2\n∂ϑ2\n1δΨ(0)+QδΨ(0)+L\nϑ2\n1δΨ(0)= 0, (15)\nwhere Q(ω) =ω2\nω2\nA\u0000\n1−ω∗pi\nω\u0001\n+q2ωωti\nω2\nAh\u0000\n1−ω∗ni\nω\u0001\nF\u0010\nω\nωti\u0011\n−ω∗Ti\nωG\u0010\nω\nωti\u0011\n−N2(ω/ωti)\nD(ω/ωti)i\n,F(x) =x\u0000\nx2+3\n2\u0001\n+\u0000\nx4+x2+1\n2\u0001\nZ(x),\nG(x) =x\u0000\nx4+x2+ 2\u0001\n+\u0010\nx6+x4\n2+x2+3\n4\u0011\nZ(x), and\nL(ω) =−α2\nc\n4q2s2−α2\nE\ns2+ωω∗nE\ns2ω2\nAnEmE\nk2\nϑρ2\ntEnimi, (16)\nwhere ρ2\ntE=mEc2TE/e2B2. Note that Eq. (15) is not periodic anymore and the Floquet theory cannot be applied\nto analyze the asymptotic behavior of its solution as before[2]. As shown, for L= 0, Eq.(15) goes back to the original4\nvorticity equation in Ref. [3]. So the cases that L̸= 0 are mainly concerned in the rest of this letter. For Q= 0, the\nmodes correspond to the accumulation points in the previous study[3]. For Q̸= 0,the solution to Eq.(15) is\nδΨ(0)=(p\n|ϑ1|Id(hp\nϑ2\n1), for ℜh <0p\n|ϑ1|Kd(hp\nϑ2\n1), for ℜh >0(17)\nwhere IdandKdis d-th order the modified Bessel function of first and second kind, h2=−Qandd=±p\n1/4−L. As\n|ϑ1| → ∞ ,p\n|ϑ1|Id∼O\u0000\neh|ϑ1|\u0001\nandp\n|ϑ1|Kd∼O\u0000\ne−h|ϑ1|\u0001\n. Since the solution should be bounded as |ϑ1| → ∞ and\nalso square integrable to avoid infinite mode energy near resonant surface, there are two types of physical solutions\nin Eq. (17). The quantity Λ = −iδΨ(0)∗∂ϑδΨ(0)\f\f\f\f0+\n0−is of more interest than the explicit solution of δΨ(0).\nFor the ideal region,the following quadratic forms are used for matching the inertial layer and ideal region,\nδWk=1\n2ˆ\nδΨ∗\nI4πωe\nk⊥kϑc2q2R2\n0⟨ωdEδKE,t⟩dϑ (18)\nδWf=1\n2ˆ∞\n−∞dϑ\"\f\f\f\fdδΨI\ndϑ\f\f\f\f2\n+\"\nk4\nϑ(s−αcosϑ)2\nk4\n⊥−k2\nϑαcosϑ\nk2\n⊥#\n|δΨI|2#\n. (19)\nBy matching the solution of the inertial layer and the ideal region in integral form, the dispersion relation of concerned\nmodes can be given as[3]\niΛ = 2 δWk+ 2δWf. (20)\nFrom the equation above, the condition that |Λ|=∞is impossible to the satisfied. Thus the only physical cases are\nthose with finite Λ, which can always be satisfied for Q̸= 0, since Qis a complex number in general and either of the\ntwo branches in Eq. (17) is the physical solution. A spectial case is that ℜd= 0. And this alows a band of oscillating\nmodes with real frequency ω <−s2ω2\nAk2\nϑρ2\nEnimi\u0000\n1/4 +α2\nc/4q2s2+α2\nE/s2\u0001\n/nEmEω∗nE. However, for complex ω,\nthe condition ℜd̸= 0 can always be satisfied. Thus, this work is mainly focused on the most unstable cases that\nℜd̸= 0. If ℜd >0, the solutionp\n|ϑ1|Id(hp\nϑ2\n1) is stuitable and behaves like O\u0010\n|ϑ1||ℜd|+1/2\u0011\nas|ϑ1| →0. Otherwise,\np\n|ϑ1|Kd(hp\nϑ2\n1) is physical solution and also behaves like O\u0010\n|ϑ1||ℜd|+1/2\u0011\nas|ϑ1| →0. Thus, Λ = 0 as long as it is\nfinite and ℜd̸= 0. For marginal stable EPMs, the dispersion relation for real frequency and the growth rate can be\ngiven as\nℜδWK(ωr) +δWf≃0, γ≃ −ℑδWk\n∂ℜδWk/∂ωr. (21)\nIn low frequency regime, ℑδWkis from the resonance at precession frequency of trapped EPs. And the real frequency\nof mode is determined by non-resonant EPs. Thus, there is no threshold for EP drive and the equations of non-resonant\nparticles in ideal region are symmetric for space-time reversing[7, 13], either the ( ωr, kϑ) branch or the ( −ωr,−kϑ)\nbranch satisfies the condition that −∂ℜδWk,u/∂ωr>0 and becomes unstable. For instance, a trial function can be\nassumed as δΨI=κ⊥/\u0000\n1 +s2ϑ2\u0001\n, where κ⊥=k⊥/kϑ. This trial function and its derivative at ϑ→ ±∞ can match\nwith δΨ(0)atϑ→0. A slowing down distribution of EPs is assumed to be\nF0E=3P0E\n4πmEEFH(EF− E)\n(2E)3/2+ (2Ec)3/2, (22)\nwhere His the Heaviside step function, Ec≪ E FandEFis the energy of alpha particles in fusion reaction. The drive\npartℑδWk,tcan be obtained as\nℑδWk,t=ℑ1\n2ˆ4πωe\nk⊥kϑc2q2R2\n0δΨ∗\nI⟨ωdEδKE,t⟩dϑ=3π2ϵ1/2αEω\n16√\n2|s|ωdF, (23)\nwhere ωdF=−kϑEF/ΩcERis the precession frequency evaluated at EF. Compared to discrete eigenmodes, the\nEPMs of this kind are a wider band of modes and more unstable. From the analysis above, a prediction of physical5\nFIG. 1. The illustration of the oscillation mechanism.\nphenomenon can be made when the proportion of EPs is near O(β). As the density of EPs increases, the continuum\nstructure is regulated and the wide-band EPMs are excited. Thus, a switch from discrete eigenmodes to wide-band\nEPMs will appear. However, the collective EPMs are very unstable and the drive is strong. These modes cause\nenhanced loss of EPs. If this happens in the igniting scenario and the EPs production rate is not adequate to\novercome the enhanced loss, the EPs proportion will decrease again. The continuum structure is then restored and\nthe SAWs again become discrete eigenmodes. With the fusion reaction and auxiliary heating continuing, the density\nof EPs will increase again. Thus, as illustrated in Figure 1, the switch of phases between wide-band EPMs and discrete\nSAWs can become a quasi-periodic oscillation, which can possibly be detected in experiments and simulations. This\nprocess may also be affected by the parametric decay, nonlinear transport and zonal structure[14], which is beyond\nthe scope of this letter.\nIn summary, we have presented a linear model for low frequency EPM in burning tokamak plasmas. It is shown\nthat the continuum structure is regulated by EPs and a wide-band of EPMs can be excited if the proportion of EPs\nis about O(β). Compared with the analysis before[7], the EP drive can excite the EPMs without overcoming the\ncontinuum damping. And the EPMs become more unstable than discrete eigenmodes. Moreover, a switch or even\nan oscillation between discrete eigenmodes and wide-band EPMs can possibly be detected in future experiments and\nsimulations.\nThis work currently receives no support of fundings.\n[1] Chen L, Zonca F. Theory of Alfv´ en waves and energetic particle physics in burning plasmas. Nuclear Fusion.\n2007;47(10):S727-34.\n[2] Chen L, Zonca F. Physics of Alfven waves and energetic particles in burning plasmas. Reviews of Modern Physics.\n2016;88(1):015008.6\n[3] Zonca F, Chen L, Santoro RA. Kinetic theory of low-frequency Alfv´ en modes in tokamaks. Plasma Physics and Controlled\nFusion. 1996;38:2011-28.\n[4] Heidbrink WW, Strait EJ, Chu MS, Turnbull AD. Observation of beta-induced Alfv´ en eigenmodes in the DIII-D tokamak.\nPhys Rev Lett. 1993;71(6):855-8.\n[5] Turnbull A, Strait E, Heidbrink W, Chu M, Duong H, Greene J, et al. Global Alfv´ en modes: Theory and experiment.\nPhysics of Fluids B: Plasma Physics. 1993;5(7):2546-53.\n[6] Chen L. Plasma heating by spatial resonance of Alfv´ en wave. Physics of Fluids. 1974;17(7):1399.\n[7] Chen L. Theory of magnetohydrodynamic instabilities excited by energetic particles in tokamaks. Physics of Plasmas.\n1994 May;1(5):1519-22.\n[8] Tang WM, Connor JW, Hastie RJ. Kinetic-ballooning-mode theory in general geometry. Nuclear Fusion. 1980;20(11):1439.\n[9] Cheng CZ. Kinetic theory of collisionless ballooning modes. Physics of Fluids. 1982;25(6):1020.\n[10] Chen L, Hasegawa A. Kinetic Theory of Geomagnetic Pulsations Internal Excitations by Energetic Particles. Journal of\nGeophysical Research. 1991;96(A2):1503-12.\n[11] Antonsen TM, Lane B. Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Physics of Fluids.\n1980;23(6):1205.\n[12] Connor JW, Hastie RJ, Taylor JB. Shear, Periodicity, and Plasma Ballooning Modes. Phys Rev Lett. 1978;40(6):394.\n[13] Tsai ST, Chen L. Theory of kinetic ballooning modes excited by energetic particles in tokamaks. Physics of Fluids B:\nPlasma Physics. 1993;5(9):3284.\n[14] Chen L, Lin Z, White RB, Zonca F. Non-linear zonal dynamics of drift and drift-Alfv´ en turbulence in tokamak plasmas.\nNuclear Fusion. 2001;41(6)." }, { "title": "2401.15334v1.Equatorial_source_of_oblique_electromagnetic_ion_cyclotron_waves__peculiarities_in_the_ion_distribution_function.pdf", "content": "manuscript submitted to JGR: Space Physics\nEquatorial source of oblique electromagnetic ion\ncyclotron waves: peculiarities in the ion distribution\nfunction\nDavid S. Tonoian1, Xiao-Jia Zhang1,2, Anton Artemyev2, Xin An2\n1Department of Physics, University of Texas at Dallas, Richardson, TX, USA\n2Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, USA\nKey Points:\n•We report observations of very oblique EMIC waves around their equatorial source\nregion\n•Oblique EMIC wave generation is associated with field-aligned thermal ion streams\nand hot, transversely anisotropic ions\n•The presence of field-aligned thermal ion streams attributes oblique EMIC wave\ngeneration to its possible ionospheric source\nCorresponding author: David S. Tonoian, david.tonoian@utdallas.edu\n–1–arXiv:2401.15334v1 [physics.space-ph] 27 Jan 2024manuscript submitted to JGR: Space Physics\nAbstract\nElectromagnetic ion cyclotron (EMIC) waves are important for Earth’s inner magneto-\nsphere as they can effectively drive relativistic electron losses to the atmosphere and en-\nergetic (ring current) ion scattering and isotropization. EMIC waves are generated by\ntransversely anisotropic ion populations around the equatorial source region, and for typ-\nical magnetospheric conditions this almost always produces field-aligned waves. For many\nspecific occasions, however, oblique EMIC waves are observed, and such obliquity has\nbeen commonly attributed to the wave off-equatorial propagation in curved dipole mag-\nnetic fields. In this study, we report that very oblique EMIC waves can be directly gen-\nerated at the equatorial source region. Using THEMIS spacecraft observations at the dawn\nflank, we show that such oblique wave generation is possible in the presence of a field-\naligned thermal ion population, likely of ionospheric origin, which can reduce Landau\ndamping of oblique EMIC waves and cyclotron generation of field-aligned waves. This\ngeneration mechanism underlines the importance of magnetosphere-ionosphere coupling\nprocesses in controlling wave characteristics in the inner magnetosphere.\n1 Introduction\nElectromagnetic ion cyclotron (EMIC) waves is a natural emission generated by\ntransversely anisotropic ion (proton) populations in the Earth’s inner magnetosphere (Cornwall\net al., 1970; Cornwall & Schulz, 1971; Min et al., 2015; Yue et al., 2019). These waves\nare responsible for resonant scattering of relativistic electrons and energetic (ring cur-\nrent) ions (see review Usanova et al., 2016, and references therein), which makes this wave\nmode principally important for the inner magnetosphere dynamics.\nTransversely anisotropic ion populations, either injected from the plasma sheet or\nformed due to the magnetosphere compression by solar wind transients (Jun et al., 2019,\n2021; Kim et al., 2021), usually generate field-aligned EMIC waves (Chen et al., 2010,\n2011), whereas generation of oblique waves is often suppressed by wave Landau damp-\ning due to suprathermal ions (see discussions in de Soria-Santacruz et al., 2013). Thus,\noblique EMIC waves have been mostly detected off the equator (Liu et al., 2013), be-\ncause wave propagation in highly inhomogeneous magnetic field and plasma will result\nin wavevector deviation from the field-aligned direction (Rauch & Roux, 1982; Thorne\n& Horne, 1997; Fraser & Nguyen, 2001). It has been shown that oblique EMIC waves\ncan scatter particles via several resonant mechanisms not available to field-aligned waves\n(see review by Usanova, 2021). First, wave obliquity modifies the efficiency of relativis-\ntic electron scattering (see Khazanov & Gamayunov, 2007; Lee et al., 2018; Hanzelka et\nal., 2023a). Second, the Landau resonance between oblique EMIC waves and cold ions\nallows EMIC waves to serve as a transmitter of energy between hot and cold ion pop-\nulations (Omura et al., 1985; Kitamura et al., 2018; Ma et al., 2019). Third, oblique EMIC\nwaves can resonate with cold plasmasphere electrons (in linear and nonlinear regimes,\nsee B. Wang et al., 2019) and can accelerate and/or precipitate them to form the sta-\nble red aurora arcs (Cornwall et al., 1971; Thorne & Horne, 1992). Fourth, oblique EMIC\nwaves may scatter energetic ( ∼100keV) electrons via the bounce resonance (Q. Wang\net al., 2018; Blum et al., 2019). All these mechanisms imply the importance of under-\nstanding possible sources of oblique EMIC waves: can such waves be generated around\nthe equator or only off-equator as the field-aligned waves propagate away from their equa-\ntorial source?\nThe equatorial generation of oblique EMIC waves requires a viable mechanism to\nsuppress the Landau damping, i.e., reduction of the field-aligned gradient in the ion phase\nspace density around the Landau resonant energies (hundreds of eV). For the electron-\nscale whistler-mode waves, it has been shown that oblique wave generation can be ex-\nplained by field-aligned electron streams generated by ionosphere outflow (see discus-\nsions in Artemyev & Mourenas, 2020), which largely suppress Landau damping (see, e.g.,\n–2–manuscript submitted to JGR: Space Physics\nMourenas et al., 2015; Li et al., 2016). Such ionosphere outflows also include field-aligned\nionospheric ion populations, which are often detected in the inner magnetosphere and\nnear-Earth plasma sheet (Yue et al., 2017; Artemyev et al., 2018). Therefore, oblique EMIC\nwave generation may be explained by a combination of near-equatorial, transversely anisotropic\nhot ions and field-aligned thermal ion streams.\nIn this study, we investigate near-equatorial observations of very oblique EMIC waves\nby Time History of Events and Macroscale Interactions during Substorms (THEMIS)\nspacecraft (Angelopoulos, 2008). Combining measurements of ion distribution functions\n(McFadden et al., 2008) and linear dispersion solver (Astfalk & Jenko, 2017), we reveal\nthe conditions of such wave generation and demonstrate that very oblique EMIC waves\nare associated with field-aligned ion streams. The rest of this paper is structured as fol-\nlows: Section 2 describes THEMIS instrumentation, plasma and wave measurements, meth-\nods of data analysis, and details of the linear dispersion solver. Section 3 examines the\ngeneration of multiple very oblique EMIC wave events by analyzing the observed ion dis-\ntribution function and comparing this distribution with that during field-aligned EMIC\nwaves. Section 4 summarizes our results and discusses their implication for modelling\nof particle dynamics in the inner magnetosphere.\n2 Dataset and instruments\nWe combine near-equatorial THEMIS measurements of EMIC waves (of 16Hz sam-\npling rate from the flux-gate magnetometer, see Auster et al., 2008), plasma sheet ion\ndistributions (at 3s resolution from the electrostatic analyzer (ESA), covering <25keV\nenergy range and full pitch-angle range, see McFadden et al., 2008), total plasma den-\nsity inferred from the spacecraft potential (see Bonnell et al., 2008; Nishimura et al., 2013),\nand the numerical dispersion solver for electromagnetic waves – Linear Electromagnetic\nOscillations in Plasmas with Arbitrary Rotationally-symmetric Distributions (LEOPARD,\nsee Astfalk & Jenko, 2017). We use the single-spacecraft maximum variance analysis tech-\nnique (Means, 1972) to estimate the wave normal angle, which comes with an ambigu-\nity of parallel versus anti-parallel directions: the wave normal angle from this technique\nranges from 0◦to 90◦, rather than 0◦to 180◦.\nAn important advantage of LEOPARD model is it can accommodate arbitrary gy-\nrotropic distribution functions in a uniform grid of velocities parallel and perpendicu-\nlar to the background magnetic field, ( v∥, v⊥). Compared to dispersion solvers that re-\nquire fitting the measurements to prescribed particle distributions (such as Maxwellians),\nLEOPARD significantly reduces the uncertainties due to fitting and hence can give a more\nreliable dispersion solution. Therefore, we use ESA ion distributions in 23 logarithmi-\ncally spaced energy channels between 5 and 25 keV and 8 linearly spaced pitch-angle chan-\nnels and interpolate them to a denser ( v∥, v⊥) grid, which is then passed to LEOPARD\nto evaluate the observed EMIC wave dispersion and growth rate. We will combine LEOP-\nARD and THEMIS measurements to reveal properties of specific ion distributions that\nare responsible for the generation of very oblique EMIC waves, quite atypical type of EMIC\nemission. Note that although THEMIS ESA provides plasma sheet ( <30keV) electron\ndistributions, this electron population usually does not resonate with EMIC waves or al-\nter the wave dispersion. Thus, we only treat electrons as the cold background plasma\n(with electron β= 10−2) and do not examine their contribution to EMIC wave growth.\n3 Oblique EMIC events\nWe now analyze in details multiple events with very oblique EMIC waves. These\nare typical events of this wave population and hence their characteristics will be repre-\nsentative of the entire population.\n–3–manuscript submitted to JGR: Space Physics\n3.1 Detailed analysis of the first event\nThe first event shows very oblique EMIC waves observed by THEMIS-A spacecraft\non February 17, 2020, between 17:50-17:58 UT (Figure 1). During this event, hydrogen-\nband EMIC waves (of frequencies within 0 .3−0.9 of the proton cyclotron frequency fcH+)\nwere detected in the dawn flank of the outer edge of the inner magnetosphere (MLT ∼\n07,L∼10) with the presence of hot, plasma sheet ions. The EMIC wave normal an-\ngle can reach 80◦during this event. The ion beta is β∼1.7, typical for the inner plasma\nsheet edge/outer edge of the inner magnetosphere (Yue et al., 2017; Artemyev et al., 2018).\nFigure 1 shows the field-aligned population of thermal ( ≲1 keV) ions and hot ( >\n2keV), transversely anisotropic ion population. Both the flux and anisotropy of hot ions\nare large right around the moment of intense, very oblique EMIC waves. Such transversely\nanisotropic ions are likely responsible for EMIC wave generation (Chen et al., 2010, 2011;\nYue et al., 2019), but instead of more typical field-aligned waves we observe very oblique\nwaves. Therefore, certain features in the ion distribution significantly alter the gener-\nation mechanism. Most likely the thermal field-aligned ion population affects wave gen-\neration and moves the positive growth rate to high wave normal angles. To verify this\nassumption, we will combine the measured ion distribution and linear dispersion solver.\nBoth the field-aligned thermal ion population and hot, transversely anisotropic pop-\nulation can be well seen in Figure 2, where we plot the 1-min averaged (around the time\nof the most intense wave spectrum) ion distribution in ( v∥, v⊥) plane. The velocities are\nnormalized to the Alfv´ en velocity vA=B/p4πnm p, where nis the plasma density cal-\nculated from spacecraft potential, mpis proton mass. Compared with the isotropic dis-\ntribution (shown in black dashed curves), observations clearly show a strong transverse\nanisotropy at v⊥>0.5vAand field-aligned anisotropy at v⊥<0.25vA. This ( v∥, v⊥)\ndistribution is then passed to the LEOPARD solver (Astfalk & Jenko, 2017) to calcu-\nlate the wave dispersion relation, where we can determine the frequency–wave normal\nangle region of positive growth rate and compare this with observations of very oblique\nEMIC waves. Figure 3 shows results of LEOPARD calculations: the positive growth rate\nis indeed confined to very oblique wave normal angles, peaking at 74◦, and to wave fre-\nquencies around 0 .35−0.55fcH+(note throughout the paper we also use Ω ci= 2πfcH+).\nThese ranges of wave normal angles and frequencies are quite close to those observed by\nTHEMIS (see Fig.1), which confirms that the ion distribution from Fig. 2 can reliably\nreproduce the main wave properties. Note that there is no positive growth rate for field-\naligned (or small wave normal angle) waves (not shown).\nTo reveal the ion distribution that is responsible for the generation of oblique EMIC\nwaves, we decompose the observed ( v∥, v⊥) distribution into three ion populations, each\nfitted by a bi-maxwellian distribution. Figure 4 shows (a) the observed distribution, (b)\nthe cold ion population with β∼10−2(this population does not contribute to EMIC\nwave generation, but is needed to realistically match the total ion density with obser-\nvations), (c) sum of cold and thermal, with β⊥/β∥∼0.45, ion populations (the latter\none is parallel anisotropic and used to represent the <1keV field-aligned ion streams),\n(d) sum of cold, thermal, and hot, with β⊥/β∥∼3.8, ion populations (the latter one\nis transversely anisotropic and provides free energy for EMIC generation). Comparison\nof panels (a) and (d) in Fig. 4 shows that this three-component fitting agrees well with\nthe observed ion distribution. Using this fitted distribution, we then evaluate the con-\ntribution of each ion population to the EMIC wave dispersion and generation. Figure\n5 shows the dispersion relations of EMIC waves with 74◦propagation angle from the ob-\nserved ion distribution and from the fitted distributions in Figs. 4 (b-d). For compar-\nison, we also show the cold plasma dispersion for field-aligned waves (Stix, 1962). So-\nlutions of the wave dispersion for cold-only (blue trace) and cold + thermal (cyan trace)\npopulations are quite similar, apart from the stronger damping in the latter. Introduc-\ning the hot transversely anisotropic population (orange trace) results in positive growth\nrate, which is larger than the growth rate from the observed ion distribution, but occu-\n–4–manuscript submitted to JGR: Space Physics\n0.000.250.500.751.00Frequency, HzfcH+\nfcHe+a)TH-A, L=10.1, MLT=07:36\n0.000.250.500.751.00Frequency, Hzb)\n04590135180 0.1 - 1.0 keV\nPitch angle, ◦c)\n04590135180 1.0 - 2.0 keV\nPitch angle, ◦d)\n04590135180 2.0 - 5.0 keV\nPitch angle, ◦e)\n04590135180 5.0 - 10.0 keV\nPitch angle, ◦f)\n17:50 17:51 17:52 17:53 17:54 17:55 17:56 17:57 17:58\nTime 2020-Feb-1704590135180 10.0 - 25.0 keV\nPitch angle, ◦g)\n103\n101\n101\n(nT)^2/Hz\n255075\nWNA\n10.0\n9.5\n9.0\n10.6\n10.4\n10.2\n10.6\n10.4\nlog(PSD)\n11.25\n11.00\n10.75\n10.50\n12.0\n11.5\nFigure 1. Observation of oblique, hydrogen-band EMIC waves by THEMIS-A spacecraft on\nFebruary 17, 2020. a) Wave magnetic field power spectrum, b) wave normal angle; top and bot-\ntom lines represent hydrogen ion H+and helium ion He+cyclotron frequencies, respectively;\nvertical dashed lines mark the time interval that is used to average the ion phase space density\nfor subsequent investigations of wave dispersion properties. c-g) Ion pitch-angle distributions for\ndifferent energy ranges as a function of time.\n–5–manuscript submitted to JGR: Space Physics\n2000\n 1000\n 0 1000 2000\nv, km/s\n4\n 2\n 0 2 4\nv/vA\n024v/vA\nTH-A 2020-02-17\n17:53:30 - 17:54:30\nFigure 2. Snapshot of the ion distribution observed by THEMIS-A during the event on\nFebruary 17, 2020 from Fig. 1; ESA measurements are averaged over 17:53:30-17:54:30 UT,\nwhich are used to plot the contours of the ion phase space density (blue). For reference, the black\ndashed traces show contours of the particle energy.\n70 75 80\nWNA, ◦0.300.350.400.450.500.550.60ω/ΩciTH-A 2020-02-17\n17:53:30 - 17:54:30\n20\n10\n01020\nγ/Ωci×103\nFigure 3. The estimated EMIC wave growth rate in the plane of wave normal angle and wave\nfrequency. The growth rate is evaluated for hydrogen-electron plasma with the ion distribution\nfrom Fig. 2 .\n–6–manuscript submitted to JGR: Space Physics\n4\n 2\n 0 2 4\nv/vA\n024v/vA\na)THEMIS A\n2020-02-17\n4\n 2\n 0 2 4\nv/vA\nb)β/β=1.565\n4\n 2\n 0 2 4\nv/vA\nc)β/β=0.454\n4\n 2\n 0 2 4\nv/vA\nd)β/β=3.84\n103\n102\n101\n100101\nPSD\nFigure 4. Observed ion distribution during the event on February 17, 2020 (a) and its fitting\nto a sum of three bi-maxwellian distributions (b-d): panel (b) shows a single cold ion population\nwith β < 10−2, panel (c) shows a summation of cold and thermal ion populations, panel (d)\nshows a summation of cold, thermal, and hot ion populations.\n0.0 0.5 1.0 1.5 2.0\nkdi0.20.40.60.81.0ω/Ωci\nWNA = 74.0◦, bi-max fit\ncold \n1 maxw.\n2 maxw.\n3 maxw.\nTHEMIS distr.\n0.0 0.2 0.4 0.6 0.8 1.0\nω/Ωci0.04\n0.03\n0.02\n0.01\n0.000.010.020.030.04γ/Ωci\nFigure 5. Solutions of the wave dispersion (left) and wave growth rate (right) for EMIC\nwaves with 74◦wave normal angle: results for the observed ion distribution are shown in red,\nresults for the three bi-maxwellian distributions from Figure 4 are shown in blue (cold population\nonly), cyan (cold and thermal populations), and orange (cold, thermal, and hot populations),\nrespectively. Left panel shows wave dispersion relation: Ω ciis the proton cyclotron frequency,\ndi= c/ω piis proton inertial length. Black line shows the analytical solution for field-aligned\nEMIC waves in cold proton-electron plasma with background parameters from observations dur-\ning the event (Stix, 1962). Black dashed traces in the right panel show the absolute value of the\ngrowth rate, |γ|=ω.\npies a similar frequency range. The main role of the thermal, field-aligned anisotropic\npopulation is to provide strong cyclotron damping of low wave normal angle waves: note\nthat the energy of the first cyclotron resonance decreases with wave normal angle decrease\nand reaches 1keV (around the energy of the thermal population) for field-aligned waves.\nA secondary, yet important, role of the thermal population is in reducing the total ion\nanisotropy: although the anisotropy of the entire ion distribution, β⊥/β∥= 1.8, does\nnot exceed much the threshold for field-aligned EMIC generation (Yue et al., 2019), the\nanisotropy of the hot population is sufficiently large ( β⊥/β∥≈4) to produce very oblique\nwaves (see the analogical mechanism of oblique whistler-mode wave generation by the\nhighly anisotropic electron component in Gary et al., 2011)\nWe also investigate the contribution of different resonances to the oblique EMIC\nwave dispersion and growth rate. Figure 6 shows the result for the first cyclotron res-\n–7–manuscript submitted to JGR: Space Physics\n0.0 0.5 1.0 1.5 2.0\nkdi0.20.40.60.81.0ω/Ωci\nWNA = 74◦\ncold \n|n|1\n|n|=1\n|n|10\n0.0 0.2 0.4 0.6 0.8 1.0\nω/Ωci0.04\n0.02\n0.000.020.04γ/Ωci\nFigure 6. Solutions of the wave dispersion (left) and wave growth rate (right) for 74◦wave\nnormal angle, while keeping different orders of resonances in the dielectric tensor calculation. The\nresult from the observed distribution for all resonances (in red, same as the red trace in Figure 5)\nincludes up to |n|= 10 resonances.\nonance only, n= 1 (orange), first cyclotron and Landau resonances |n| ≤ 1 (cyan),\nand for all resonances up to |n|= 10 (red). There is barely any difference between wave\ndispersions for these three cases, i.e., the wave dispersion is mainly provided by the ion\npopulation contributing to the first cyclotron resonance. The comparison of wave growth\nrates shows that the same ion population is responsible for wave generation, whereas Lan-\ndau damping reduces the magnitude of the growth rate, especially at small wave frequen-\ncies.\nFigure 7 further confirms the principal role of the first cyclotron resonance in gen-\nerating the observed oblique EMIC waves, by combining contours of constant phase phase\ndensity (from Fig. 2) and resonant conditions for two approximations. The grey shaded\nregion in the left panel shows the cyclotron resonant velocities calculated from the dis-\npersion relation for 74◦wave normal angle and wave frequencies of positive growth rate,\nω/Ωci∈[0.38,0.55]. Comparison of constant phase phase density contours (blue traces)\nand resonance curves (contours of constant energy in the wave rest frame, shown in red)\ndemonstrates that within the grey shaded region the gradient along the resonance curves\n(seen from the comparison of constant energy curves, shown in dashed black, and phase\nspace density contours, shown in blue) corresponds to an increase in the phase space den-\nsity, which will drive EMIC wave growth (Lyons & Williams, 1984). Similarly, the same\nconclusion can be drawn for the resonant velocity range for monochromatic waves, with\na frequency corresponding to the peak wave intensity in observations and variations of\nthe resonance condition along magnetic latitudes (right panel): the positive gradient of\nthe ion phase space density along resonance curves within the purple shaded region will\namplify the waves as they propagate away from the generation region near the equator.\n3.2 Comparison of ion distributions during field-aligned and oblique EMIC\nevents\nFigure 8(b-e) shows four more examples of THEMIS observed very oblique EMIC\nwaves in the dawn flank. All these events share similar properties of the event discussed\nin Section 3.1:\n–8–manuscript submitted to JGR: Space Physics\n4\n 2\n 0 2 4\nv/vA\n024v/vA\nWNA = 74◦\n4\n 2\n 0 2 4\nv/vA\n01234v/vA\nWNA = 74◦, ω=0.47Ωci,eq\nFigure 7. Contours of constant phase space density from Figure 2 (blue), with resonant ve-\nlocities for the first cyclotron resonance (shaded region), and resonant curves corresponding to\ncontours of constant ion energy in the wave rest frame (red). Orange vertical line marks the\nLandau resonant energy. Left panel shows the range of resonant velocities at the equator for the\nfrequency range with positive growth rate (grey shaded area), right panel shows the range of res-\nonant velocities for a fixed equatorial frequency 0 .47Ω ciand a latitudinal range of 0◦≤λ≤25◦\n(purple shaded region).\n•Waves are proton band EMIC waves with clear maximum of the wave intensity\naround [0 .3,0.5] of local proton cyclotron frequency and with wave normal angles\ntypically exceeding 60◦(left panels).\n•Ion distribution functions (middle panels) observed around the oblique EMIC wave\nburst include a very anisotropic hot (a few keVs) ion population ( β⊥/β∥>3)\nand a field-aligned anisotropic thermal ( ≤1keV) ion population. The latter can\nlead to strong cyclotron damping of field-aligned EMIC waves.\n•Combining the measured ion distribution and the linear dispersion solver, LEOP-\nARD, we show positive growth rates for very oblique EMIC waves (right panels),\nas well as damping for the field-aligned waves (not shown). Therefore, the most\nimportant condition for very oblique EMIC wave generation is likely the combi-\nnation of transversely anisotropic hot ions (providing wave growth via cyclotron\nresonance) and field-aligned thermal ions (providing cyclotron damping of small\nwave normal angle waves).\nTo verify the importance of the thermal field-aligned population for very oblique\nEMIC wave growth, we further selected a typical field-aligned EMIC wave event observed\nby THEMIS in the dusk flank. As shown in Figure 8(f), in the absence of field-aligned\nthermal ions, the field-aligned waves can be generated via cyclotron resonance with trans-\nversely anisotropic hot ions. Moreover, in the absence of the field-aligned population, the\nanisotropy of the entire ion distribution is mainly determined by the anisotropy of hot\nions, which does not need to be as large as in events with oblique waves in order to gen-\nerate field-aligned waves. As a result, in the event from Fig. 8(f) the hot ion anisotropy,\nβ⊥/β∥= 1.3, is sufficient to generate field-aligned EMIC waves, but insufficient to gen-\nerate oblique waves (the growth rate of oblique waves is negative for this event; not shown).\n4 Discussion and conclusions\nIn this study we analyze several events with THEMIS observations of very oblique\nEMIC waves around the equator. During these events, the observed ion distributions con-\nsist of a highly transversely anisotropic ( β⊥/β∥∼4,>2keV) hot ion population and\na field-aligned anisotropic thermal ion population ( β⊥/β∥∼0.3,<1keV). It is this\n–9–manuscript submitted to JGR: Space Physics\na) Oblique EMIC 2020-02-17 TH-A, L=10.1, MLT=07:36, ωpe/Ωce=14.9\n17:50 17:51 17:52 17:53 17:54 17:55 17:56 17:57 17:58\nTime, HH:MM 2020-Feb-170.00.20.40.60.81.0Frequency, Hz\n103\n102\n101\n100101\n(nT)^2/Hz\n70 80\nWNA, ◦0.30.40.50.6ω/Ωci\n2000\n 0 2000\nv, km/s\n2.5\n 0.0 2.5\nv/vA\n024v/vA\n20\n10\n01020\nγ/Ωci×103\nb) Oblique EMIC 2020-02-14 TH-E, L=11.0, MLT=05:29, ωpe/Ωce=12.8\n0.00.20.40.60.81.0Frequency, Hz0.30.40.5\n05:36 05:37 05:38 05:39 05:40 05:41 05:42 05:43 05:44\nTime, HH:MM0.00.2\n103\n102\n101\n100101\n(nT)^2/Hz\n75 80\nWNA, ◦0.300.350.400.450.50ω/Ωci\n2000\n 0 2000\nv, km/s\n2\n 0 2\nv/vA\n02v/vA\n4\n2\n024\nγ/Ωci×103\nc) Oblique EMIC 2020-04-06 TH-D, L=11.4, MLT=04:40, ωpe/Ωce=17.0\n15:30 15:35 15:40 15:45 15:50 15:55 16:00\nTime, HH:MM 2020-Apr-060.00.10.20.30.4Frequency, Hz\n103\n102\n101\n100101\n(nT)^2/Hz\n100 105 110\nWNA, ◦0.250.300.350.400.45ω/Ωci\n2000\n 0 2000\nv, km/s\n2.5\n 0.0 2.5\nv/vA\n024v/vA\n20\n10\n01020\nγ/Ωci×103\nd) Oblique EMIC 2021-05-15 TH-E, L=9.1, MLT=02:27, ωpe/Ωce=8.8\n05:00 05:05 05:10 05:15\nTime, HH:MM 2021-May-150.00.10.20.30.40.5Frequency, Hz\n103\n102\n101\n100101\n(nT)^2/Hz\n75 80\nWNA, ◦0.400.450.500.550.60ω/Ωci\n2000\n 0 2000\nv, km/s\n2\n 0 2\nv/vA\n012v/vA\n20\n020\nγ/Ωci×103\ne) Oblique EMIC 2021-05-27 TH-E, L=7.4, MLT=05:30, ωpe/Ωce=8.0\n09:30 09:35 09:40\nTime, HH:MM 2021-May-270.00.20.40.60.8Frequency, Hz\n103\n102\n101\n100101\n(nT)^2/Hz\n95 100 105\nWNA, ◦0.300.350.400.45ω/Ωci\n2000\n 0 2000\nv, km/s\n2\n 0 2\nv/vA\n012v/vA\n10\n5\n0510\nγ/Ωci×103\nf) Parallel EMIC 2020-10-01 TH-E, L=8.6, MLT=15:45, ωpe/Ωce=11.0\n18:25 18:30 18:35\nTime, HH:MM 2020-Oct-010.00.20.40.60.81.0Frequency, Hz\n103\n102\n101\n100101\n(nT)^2/Hz\n0 10 20\nWNA, ◦0.30.40.5ω/Ωci\n2000\n 0 2000\nv, km/s\n2\n 0 2\nv/vA\n02v/vA\n1.0\n0.5\n0.00.51.0\nγ/Ωci×103\nFigure 8. Wave magnetic field power spectrum (left column), ion distribution (center col-\numn), and wave growth rate (right column) for oblique (a-e) and field-aligned (f) EMIC wave\nevents. In the wave power spectra: for oblique EMIC waves, red contours show the domain where\nwave normal angle > 60◦, whereas for field-aligned EMIC wave event orange contours show the\ndomain where wave normal angle < 20◦; the two horizontal white lines mark hydrogen and he-\nlium ion gyrofrequencies from top to bottom. During the February 14, 2020 event (b), part of the\nwave spectrum has been cut to remove the spin tones that obscure the waves.\n–10–manuscript submitted to JGR: Space Physics\n0.3 0.4 0.5 0.6 0.7\nω/Ωcp0246810Emin,H+[keV]\ncold H+:+1\ncold H+:Landau\nobs. H+:+1\nobs. H+:Landau\n104\n103\n102\n101\n100101102103104\nEmin,e[keV]WNA =74◦, ωpe/Ωce=14.9\nFigure 9. Minimum resonant energies for protons (blue, linear scale) and electrons (red, log-\narithmic scale) in Landau resonance and first cyclotron resonance (” + 1” for protons, ” −1” for\nelectrons). Dashed and dotted lines are calculated using generic bi-maxwellian distributions (with\nβ= 10−2), whereas solid and dot-dashed lines are based on dispersion results for the first event\n(Figure 8 a).\nfield-aligned thermal ion population that prohibits the generation of field-aligned EMIC\nwaves: the cyclotron resonant energy of waves with small wave normal angles is ∼1keV,\nwhere the transverse anisotropy is insufficient to drive these waves. Therefore, such ther-\nmal ( <1 keV) field-aligned ion population play a key role in producing very oblique EMIC\nwaves, which are often observed on the dawn flank. The energy range and field-aligned\nanisotropy of this population suggest that these are likely ionospheric outflow ions (see\nstatistics of ion pitch-angle distributions in Artemyev et al., 2018; Yue et al., 2017). Over-\nlapping of such outflow, probably enhanced at the dawn flank due to strong plasma sheet\nelectron precipitation driven by whistler-mode waves (Thorne et al., 2010; Ni et al., 2016),\nand the hot ion population, likely drifted from the dusk flank after being injected from\nthe plasmasheet (e.g., Birn et al., 1997; Gabrielse et al., 2014; Ukhorskiy et al., 2018),\ncreates favorable conditions for the generation of very oblique EMIC waves. This fur-\nther implies that very oblique waves are likely a result of magnetosphere-ionosphere cou-\npling, in contrast to the more typical field-aligned waves generated in the dusk flank due\nto plasmasheet injections or on the day side due to magnetosphere compression by the\nsolar wind (Yue et al., 2019; Jun et al., 2019, 2021).\nIn addition to the possible resonant interactions between oblique EMIC waves and\nmagnetospheric particles, as discussed in the introduction, a new resonant mechanism\nhas been proposed recently that makes oblique EMIC waves potentially more important.\nIt has been shown by Hanzelka et al. (2023b) that very oblique and sufficiently intense\nEMIC waves may resonate with energetic electrons via the so-called fractional (or sub-\nharmonic) resonances (Lewak & Chen, 1969; Smirnov & Frank-Kamenestki ˇi, 1968). In\ncontrast to cyclotron resonances with integer resonant numbers, the factional resonance\nis a purely nonlinear effect providing electron scattering in resonances with fractional num-\n–11–manuscript submitted to JGR: Space Physics\nbers (Terasawa & Matsukiyo, 2012), which reduces the electron energy in resonance with\nEMIC waves to sub-relativistic values (Hanzelka et al., 2023b). THEMIS observations\nof very oblique EMIC waves and the proposed formation mechanism of these waves sug-\ngest that sub-relativistic electron precipitation on the dawn side (see statistics of such\nprecipitation in Tsai et al., 2023) may be partly driven by EMIC waves, not exclusively\nby whistler-mode waves.\nFigure 9 shows the energies of electrons and ions in resonance with very oblique\nEMIC waves (we use a wave normal angle of 74◦and plot resonant energies as a func-\ntion of wave frequency for typical plasma frequencies as from Figs. 8). Note that the elec-\ntron resonant energies for fractional resonances fall between the regions bounded by the\nfirst cyclotron resonance, n=−1, and Landau resonance, n= 0 (Hanzelka et al., 2023b).\nComparing the results for the cold plasma dispersion ( β= 10−2) and for the observed\nhot plasma with β≈1.7, we can see a higher resonant energy for the hot plasma case\ndue to a decrease of the wave number at a fixed frequency (Silin et al., 2011). Cyclotron\nand Landau resonant energies for protons are within [1 ,10]keV for ω/Ωcp<0.5, where\nmost of wave power is observed (see Fig. 8 (a)). This energy range allows oblique EMIC\nwaves to heat thermal ( ∼1keV) ions and scatter ring current ( ∼10keV) ions into the\nloss cone (note that these resonant energies are calculated at the equator, which will in-\ncrease in the off-equatorial region). The range of resonant energies for electrons, on the\nother hand, is much wider: from ∼1eV in Landau resonance to 1MeV in cyclotron res-\nonance. The fractional resonances with a resonance number ∈[0,1] will fill this gap and\nallow EMIC waves to also scatter hundreds of keV electrons. This is likely the most in-\nteresting and potentially important implication of very oblique waves.\nTo conclude, using THEMIS observations in the inner magnetosphere, we have in-\nvestigated the generation mechanism of very oblique EMIC waves. Six typical events of\nsuch waves are observed at the dawn flank, in contrast to the more common field-aligned\nwaves observed at the dusk and noon sectors. Very oblique EMIC waves are usually ac-\ncompanied by ion distributions consisting of two main populations (except for the cold\nplasma population contributing to the total density). 1) A thermal ion population, at\n≤1 keV, with the field-line anisotropy providing the cyclotron damping of field-aligned\nEMIC waves and potentially reducing the Landau damping of oblique EMIC waves. This\npopulation shares the same properties of ionospheric outflow as reported previously for\nthe inner magnetosphere (e.g., Yue et al., 2017). 2) The hot ion population, at >5keV,\nwith a strong transverse anisotropy ( β⊥/β∥∼4) providing the cyclotron resonant growth\nof oblique EMIC waves with wave normal angles exceeding 60◦. These observations un-\nderline the importance of magnetosphere-ionosphere coupling in producing very oblique\nEMIC waves.\nAcknowledgments\nWe acknowledge the support of NASA contract NAS5-02099 for the use of data from the\nTHEMIS Mission, specifically K. H.Glassmeier, U. Auster and W. Baumjohann for the\nuse of FGM data (provided under the lead of the Technical University of Braunschweig\nand with financial support through the German Ministry for Economy and Technology\nand the German Center for Aviation and Space (DLR) under contract 50 OC 0302). Work\nof D.S.T., X.-J. Z., and A.V.A. are supported by NSF grant #2329897, and NASA grants\n#80NSSC20K1270, #80NSSC23K0403, #80NSSC23K0108 .\nOpen Research\nTHEMIS data is available at http://themis.ssl.berkeley.edu. Data access and process-\ning was performed using SPEDAS V4.1 and its Python-based implementation, see Angelopoulos\net al. (2019) and (Grimes et al., 2022).\n–12–manuscript submitted to JGR: Space Physics\nReferences\nAngelopoulos, V. (2008, December). The THEMIS Mission. Space Sci. Rev. ,141, 5-\n34. doi: 10.1007/s11214-008-9336-1\nAngelopoulos, V., Cruce, P., Drozdov, A., Grimes, E. W., Hatzigeorgiu, N., King,\nD. A., . . . Schroeder, P. (2019, January). The Space Physics Environ-\nment Data Analysis System (SPEDAS). Space Sci. Rev. ,215, 9. doi:\n10.1007/s11214-018-0576-4\nArtemyev, A. V., & Mourenas, D. (2020, March). On Whistler Mode Wave Rela-\ntion to Electron Field-Aligned Plateau Populations. Journal of Geophysical Re-\nsearch (Space Physics) ,125(3), e27735. doi: 10.1029/2019JA027735\nArtemyev, A. V., Zhang, X. J., Angelopoulos, V., Runov, A., Spence, H. E., &\nLarsen, B. A. (2018, Jul). Plasma Anisotropies and Currents in the Near-\nEarth Plasma Sheet and Inner Magnetosphere. Journal of Geophysical Re-\nsearch (Space Physics) ,123(7), 5625-5639. doi: 10.1029/2018JA025232\nAstfalk, P., & Jenko, F. (2017, January). LEOPARD: A grid-based dispersion re-\nlation solver for arbitrary gyrotropic distributions. Journal of Geophysical Re-\nsearch (Space Physics) ,122(1), 89-101. doi: 10.1002/2016JA023522\nAuster, H. U., Glassmeier, K. H., Magnes, W., Aydogar, O., Baumjohann,\nW., Constantinescu, D., . . . Wiedemann, M. (2008, December). The\nTHEMIS Fluxgate Magnetometer. Space Sci. Rev. ,141, 235-264. doi:\n10.1007/s11214-008-9365-9\nBirn, J., Thomsen, M. F., Borovsky, J. E., Reeves, G. D., McComas, D. J., Belian,\nR. D., & Hesse, M. (1997, February). Substorm ion injections: Geosynchronous\nobservations and test particle orbits in three-dimensional dynamic MHD fields.\nJ. Geophys. Res. ,102, 2325-2342. doi: 10.1029/96JA03032\nBlum, L. W., Artemyev, A., Agapitov, O., Mourenas, D., Boardsen, S., & Schiller,\nQ. (2019, Apr). EMIC Wave-Driven Bounce Resonance Scattering of Energetic\nElectrons in the Inner Magnetosphere. Journal of Geophysical Research (Space\nPhysics) ,124(4), 2484-2496. doi: 10.1029/2018JA026427\nBonnell, J. W., Mozer, F. S., Delory, G. T., Hull, A. J., Ergun, R. E., Cully, C. M.,\n. . . Harvey, P. R. (2008, December). The Electric Field Instrument (EFI) for\nTHEMIS. Space Sci. Rev. ,141, 303-341. doi: 10.1007/s11214-008-9469-2\nChen, L., Thorne, R. M., & Bortnik, J. (2011, August). The controlling effect of ion\ntemperature on EMIC wave excitation and scattering. Geophys. Res. Lett. ,\n38(16), L16109. doi: 10.1029/2011GL048653\nChen, L., Thorne, R. M., Jordanova, V. K., Wang, C.-P., Gkioulidou, M., Lyons,\nL., & Horne, R. B. (2010, Jul). Global simulation of EMIC wave excitation\nduring the 21 April 2001 storm from coupled RCM-RAM-HOTRAY model-\ning. Journal of Geophysical Research (Space Physics) ,115(A7), A07209. doi:\n10.1029/2009JA015075\nCornwall, J. M., Coroniti, F. V., & Thorne, R. M. (1970). Turbulent loss of ring cur-\nrent protons. J. Geophys. Res. ,75, 4699. doi: 10.1029/JA075i025p04699\nCornwall, J. M., Coroniti, F. V., & Thorne, R. M. (1971, January). Unified theory of\nSAR arc formation at the plasmapause. J. Geophys. Res. ,76(19), 4428. doi:\n10.1029/JA076i019p04428\nCornwall, J. M., & Schulz, M. (1971, November). Electromagnetic ion-cyclotron\ninstabilities in multicomponent magnetospheric plasmas. J. Geophys. Res. ,\n76(31), 7791-7796. doi: 10.1029/JA076i031p07791\nde Soria-Santacruz, M., Spasojevic, M., & Chen, L. (2013, May). EMIC waves\ngrowth and guiding in the presence of cold plasma density irregularities. Geo-\nphys. Res. Lett. ,40(10), 1940-1944. doi: 10.1002/grl.50484\nFraser, B. J., & Nguyen, T. S. (2001, July). Is the plasmapause a preferred source\nregion of electromagnetic ion cyclotron waves in the magnetosphere? Journal\nof Atmospheric and Solar-Terrestrial Physics ,63(11), 1225-1247. doi: 10.1016/\nS1364-6826(00)00225-X\n–13–manuscript submitted to JGR: Space Physics\nGabrielse, C., Angelopoulos, V., Runov, A., & Turner, D. L. (2014, April). Statisti-\ncal characteristics of particle injections throughout the equatorial magnetotail.\nJ. Geophys. Res. ,119, 2512-2535. doi: 10.1002/2013JA019638\nGary, S. P., Liu, K., & Winske, D. (2011, August). Whistler anisotropy instability at\nlow electron β: Particle-in-cell simulations. Physics of Plasmas ,18(8), 082902.\ndoi: 10.1063/1.3610378\nGrimes, E. W., Harter, B., Hatzigeorgiu, N., Drozdov, A., Lewis, J. W., Angelopou-\nlos, V., . . . others (2022). The space physics environment data analysis system\nin python. Frontiers in Astronomy and Space Sciences ,9, 1020815.\nHanzelka, M., Li, W., & Ma, Q. (2023a, April). Parametric analysis of pitch an-\ngle scattering and losses of relativistic electrons by oblique EMIC waves. Fron-\ntiers in Astronomy and Space Sciences ,10, 1163515. doi: 10.3389/fspas.2023\n.1163515\nHanzelka, M., Li, W., & Ma, Q. (2023b, April). Parametric analysis of pitch an-\ngle scattering and losses of relativistic electrons by oblique EMIC waves. Fron-\ntiers in Astronomy and Space Sciences ,10, 1163515. doi: 10.3389/fspas.2023\n.1163515\nJun, C.-W., Miyoshi, Y., Kurita, S., Yue, C., Bortnik, J., Lyons, L., . . . Shinohara,\nI. (2021, June). The Characteristics of EMIC Waves in the Magnetosphere\nBased on the Van Allen Probes and Arase Observations. Journal of Geophysi-\ncal Research (Space Physics) ,126(6), e29001. doi: 10.1029/2020JA029001\nJun, C. W., Yue, C., Bortnik, J., Lyons, L. R., Nishimura, Y., & Kletzing, C. (2019,\nMar). EMIC Wave Properties Associated With and Without Injections in\nThe Inner Magnetosphere. Journal of Geophysical Research (Space Physics) ,\n124(3), 2029-2045. doi: 10.1029/2018JA026279\nKhazanov, G. V., & Gamayunov, K. V. (2007, October). Effect of electromagnetic\nion cyclotron wave normal angle distribution on relativistic electron scatter-\ning in outer radiation belt. Journal of Geophysical Research (Space Physics) ,\n112(A10), A10209. doi: 10.1029/2007JA012282\nKim, H., Schiller, Q., Engebretson, M. J., Noh, S., Kuzichev, I., Lanzerotti, L. J.,\n. . . Fromm, T. (2021, February). Observations of Particle Loss due to Injec-\ntion Associated Electromagnetic Ion Cyclotron Waves. Journal of Geophysical\nResearch (Space Physics) ,126(2), e28503. doi: 10.1029/2020JA028503\nKitamura, N., Kitahara, M., Shoji, M., Miyoshi, Y., Hasegawa, H., Nakamura, S., . . .\nBurch, J. L. (2018, September). Direct measurements of two-way wave-particle\nenergy transfer in a collisionless space plasma. Science ,361(6406), 1000-1003.\ndoi: 10.1126/science.aap8730\nLee, D.-Y., Shin, D.-K., & Choi, C.-R. (2018, June). Effects of Oblique Wave\nNormal Angle and Noncircular Polarization of Electromagnetic Ion Cy-\nclotron Waves on the Pitch Angle Scattering of Relativistic Electrons.\nJournal of Geophysical Research (Space Physics) ,123(6), 4556-4573. doi:\n10.1029/2018JA025342\nLewak, G. J., & Chen, C. S. (1969, September). Higher order resonances\nin a plasma. Journal of Plasma Physics ,3(3), 481-497. doi: 10.1017/\nS0022377800004554\nLi, W., Mourenas, D., Artemyev, A. V., Bortnik, J., Thorne, R. M., Kletzing, C. A.,\n. . . Spence, H. E. (2016, September). Unraveling the excitation mechanisms of\nhighly oblique lower band chorus waves. Geophys. Res. Lett. ,43, 8867-8875.\ndoi: 10.1002/2016GL070386\nLiu, Y. H., Fraser, B. J., & Menk, F. W. (2013, September). EMIC waves observed\nby Cluster near the plasmapause. Journal of Geophysical Research (Space\nPhysics) ,118(9), 5603-5615. doi: 10.1002/jgra.50486\nLyons, L. R., & Williams, D. J. (1984). Quantitative aspects of magnetospheric\nphysics. (Lyons, L. R. & Williams, D. J., Ed.).\nMa, Q., Li, W., Yue, C., Thorne, R. M., Bortnik, J., Kletzing, C. A., . . . Spence,\n–14–manuscript submitted to JGR: Space Physics\nH. E. (2019, June). Ion Heating by Electromagnetic Ion Cyclotron Waves and\nMagnetosonic Waves in the Earth’s Inner Magnetosphere. Geophys. Res. Lett. ,\n46(12), 6258-6267. doi: 10.1029/2019GL083513\nMcFadden, J. P., Carlson, C. W., Larson, D., Ludlam, M., Abiad, R., Elliott, B.,\n. . . Angelopoulos, V. (2008, December). The THEMIS ESA Plasma In-\nstrument and In-flight Calibration. Space Sci. Rev. ,141, 277-302. doi:\n10.1007/s11214-008-9440-2\nMeans, J. D. (1972). Use of the three-dimensional covariance matrix in analyzing\nthe polarization properties of plane waves. J. Geophys. Res. ,77, 5551. doi: 10\n.1029/JA077i028p05551\nMin, K., Liu, K., Bonnell, J. W., Breneman, A. W., Denton, R. E., Funsten, H. O.,\n. . . Wygant, J. R. (2015, April). Study of EMIC wave excitation using direct\nion measurements. Journal of Geophysical Research (Space Physics) ,120(4),\n2702-2719. doi: 10.1002/2014JA020717\nMourenas, D., Artemyev, A. V., Agapitov, O. V., Krasnoselskikh, V., & Mozer, F. S.\n(2015). Very oblique whistler generation by low-energy electron streams. J.\nGeophys. Res. ,120, 3665–3683. doi: 10.1002/2015JA021135\nNi, B., Thorne, R. M., Zhang, X., Bortnik, J., Pu, Z., Xie, L., . . . Gu, X. (2016,\nApril). Origins of the Earth’s Diffuse Auroral Precipitation. Space Sci. Rev. ,\n200, 205-259. doi: 10.1007/s11214-016-0234-7\nNishimura, Y., Bortnik, J., Li, W., Thorne, R. M., Ni, B., Lyons, L. R., . . . Auster,\nU. (2013, Feb). Structures of dayside whistler-mode waves deduced from\nconjugate diffuse aurora. Journal of Geophysical Research (Space Physics) ,\n118(2), 664-673. doi: 10.1029/2012JA018242\nOmura, Y., Ashour-Abdalla, M., Gendrin, R., & Quest, K. (1985, September). Heat-\ning of thermal helium in the equatorial magnetosphere: A simulation study. J.\nGeophys. Res. ,90(A9), 8281-8292. doi: 10.1029/JA090iA09p08281\nRauch, J. L., & Roux, A. (1982, October). Ray tracing of ULF waves in a multi-\ncomponent magnetospheric plasma: Consequences for the generation mech-\nanism of ion cyclotron waves. J. Geophys. Res. ,87(A10), 8191-8198. doi:\n10.1029/JA087iA10p08191\nSilin, I., Mann, I. R., Sydora, R. D., Summers, D., & Mace, R. L. (2011, May).\nWarm plasma effects on electromagnetic ion cyclotron wave MeV electron\ninteractions in the magnetosphere. Journal of Geophysical Research (Space\nPhysics) ,116(A5), A05215. doi: 10.1029/2010JA016398\nSmirnov, Y. N., & Frank-Kamenestki ˇi, D. A. (1968, March). Nonlinearity and Para-\nmetric Resonance in a Plasma. Soviet Journal of Experimental and Theoretical\nPhysics ,26, 627.\nStix, T. H. (1962). The Theory of Plasma Waves .\nTerasawa, T., & Matsukiyo, S. (2012, November). Cyclotron Resonant Interactions\nin Cosmic Particle Accelerators. Space Sci. Rev. ,173, 623-640. doi: 10.1007/\ns11214-012-9878-0\nThorne, R. M., & Horne, R. B. (1992, February). The contribution of ion-cyclotron\nwaves to electron heating and SAR-arc excitation near the storm-time plasma-\npause. Geophys. Res. Lett. ,19(4), 417-420. doi: 10.1029/92GL00089\nThorne, R. M., & Horne, R. B. (1997, July). Modulation of electromagnetic ion\ncyclotron instability due to interaction with ring current O+during magnetic\nstorms. J. Geophys. Res. ,102(A7), 14155-14164. doi: 10.1029/96JA04019\nThorne, R. M., Ni, B., Tao, X., Horne, R. B., & Meredith, N. P. (2010, October).\nScattering by chorus waves as the dominant cause of diffuse auroral precipita-\ntion. Nature ,467, 943-946. doi: 10.1038/nature09467\nTsai, E., Artemyev, A., Angelopoulos, V., & Zhang, X.-J. (2023, August). In-\nvestigating Whistler-Mode Wave Intensity Along Field Lines Using Electron\nPrecipitation Measurements. Journal of Geophysical Research (Space Physics) ,\n128(8), e2023JA031578. doi: 10.1029/2023JA031578\n–15–manuscript submitted to JGR: Space Physics\nUkhorskiy, A. Y., Sorathia, K. A., Merkin, V. G., Sitnov, M. I., Mitchell, D. G.,\n& Gkioulidou, M. (2018, Jul). Ion Trapping and Acceleration at Dipo-\nlarization Fronts: High-Resolution MHD and Test-Particle Simulations.\nJournal of Geophysical Research (Space Physics) ,123(7), 5580-5589. doi:\n10.1029/2018JA025370\nUsanova, M. E. (2021, September). Energy Exchange Between Electromag-\nnetic Ion Cyclotron (EMIC) Waves and Thermal Plasma: From Theory to\nObservations. Frontiers in Astronomy and Space Sciences ,8, 150. doi:\n10.3389/fspas.2021.744344\nUsanova, M. E., Mann, I. R., & Darrouzet, F. (2016, February). EMIC Waves in the\nInner Magnetosphere. Washington DC American Geophysical Union Geophysi-\ncal Monograph Series ,216, 65-78. doi: 10.1002/9781119055006.ch5\nWang, B., Li, P., Huang, J., & Zhang, B. (2019, April). Nonlinear Landau resonance\nbetween EMIC waves and cold electrons in the inner magnetosphere. Physics\nof Plasmas ,26(4), 042903. doi: 10.1063/1.5088374\nWang, Q., Fu, S., Ni, B., Cao, X., Gu, X., & Huang, H. (2018, Aug). Bounce reso-\nnance scattering of ring current electrons by H+band EMIC waves. Physics of\nPlasmas ,25(8), 082903. doi: 10.1063/1.5043522\nYue, C., Bortnik, J., Thorne, R. M., Ma, Q., An, X., Chappell, C. R., . . . Kletzing,\nC. A. (2017). The characteristic pitch angle distributions of 1 ev to 600 kev\nprotons near the equator based on van allen probes observations. J. Geo-\nphys. Res. . Retrieved from http://dx.doi.org/10.1002/2017JA024421 doi:\n10.1002/2017JA024421\nYue, C., Jun, C.-W., Bortnik, J., An, X., Ma, Q., Reeves, G. D., . . . Kletzing, C. A.\n(2019, Apr). The Relationship Between EMIC Wave Properties and Proton\nDistributions Based on Van Allen Probes Observations. Geophys. Res. Lett. ,\n46(8), 4070-4078. doi: 10.1029/2019GL082633\n–16–" }, { "title": "2401.15382v1.Inference_on_an_heteroscedastic_Gompertz_tumor_growth_model.pdf", "content": "Inference on an heteroscedastic Gompertz tumor\ngrowth model\nG. Albanoa,∗, V. Giornob, P. Rom´ an-Rom´ anc,d, S. Rom´ an-Rom´ ane, J.J.\nSerrano-P´ erezc, F. Torres-Ruizc,d\naDipartimento di Studi Politici e Sociali, Universit` a di Salerno, Italy\nbDipartimento di Informatica, Universit` a di Salerno, Italy\ncDepartamento de Estad´ ıstica e Investigaci´ on Operativa, Universidad de Granada, Spain\ndInstituto de Matem´ aticas (IEMath-GR), Universidad de Granada, Spain\neD´ ep. de Recherche Translationnelle, Institut Curie, France\nAbstract\nWe consider a non homogeneous Gompertz diffusion process whose parame-\nters are modified by generally time-dependent exogenous factors included in the\ninfinitesimal moments. The proposed model is able to describe tumor dynam-\nics under the effect of anti-proliferative and/or cell death-induced therapies.\nWe assume that such therapies can modify also the infinitesimal variance of\nthe diffusion process. An estimation procedure, based on a control group and\ntwo treated groups, is proposed to infer the model by estimating the constant\nparameters and the time-dependent terms. Moreover, several concatenated hy-\npothesis tests are considered in order to confirm or reject the need to include\ntime-dependent functions in the infinitesimal moments. Simulations are pro-\nvided to evaluate the efficiency of the suggested procedures and to validate the\ntesting hypothesis. Finally, an application to real data is considered.\nKeywords: Tumor growth, anti-proliferative and cell death-induced therapies,\nmodified Gompertz diffusion process, inference in diffusion processes,\nbootstrap tests.\n∗Corresponding author\nEmail addresses: pialbano@unisa.it (G. Albano), giorno@unisa.it (V. Giorno),\nproman@ugr.es (P. Rom´ an-Rom´ an), sergio.Roman-Roman@curie.fr (S. Rom´ an-Rom´ an),\njjserra@ugr.es (J.J. Serrano-P´ erez), fdeasis@ugr.es (F. Torres-Ruiz)\nPreprint submitted to Mathematical Biosciences January 31, 2024arXiv:2401.15382v1 [stat.ME] 27 Jan 20241. Introduction\nDiffusion processes are widely used in the literature to describe phenomena\nin a lot of fields, ranging from economics [1, 2] to biology [3, 4]. Concerning the\ntumor growth modeling, many efforts have been devoted in the last years since\nnowadays the cancer represents one of the main causes of death in our society.\nFurther, the availability of modern diagnostic and prognostic methodologies al-\nlows to build ever more faithful models, giving useful insights into the dynamics\nof such disease [5, 6, 7]. On the other hand, the mathematical tractability of\nthe model must be taken into account because it allows to better handle explicit\nsolutions of the involved dynamics. In this context, Gompertz growth model\nseems successfully overcome the “trade off” between these two aspects. Indeed,\nit is widely accepted that such model is able to capture dynamics of solid tu-\nmors and several models based on this growth have been proposed by looking\nat deterministic and stochastic behaviors [8, 9, 10, 11, 12, 13].\nThe Gompertz curve belongs to the Richards family of sigmoidal growth\nmodels, along with familiar models such as the negative exponential, the logis-\ntic, and the Bertalanffy [14]. These curves, although born in deterministic con-\ntexts, have been generalized to include stochastic effects aimed at bridging the\ngaps that often exist between experimental data and theoretical results. Con-\ncerning the stochastic version of the Gompertz growth, in the literature various\ncontributions can be found concerning both theoretical probabilistic properties\nand statistical characteristics [15, 16, 17].\nRecently, the attention has been focused on non-homogeneous Gompertz dif-\nfusion process describing the effect of some exogenous generally time-dependent\nfactors. In preclinical tumor growth studies it is useful to understand as experi-\nmental therapies can modify the natural cancer cells’ growth rates. A modified\nGompertz equation is considered in Cabrales et al. [18] to describe tumor re-\nsponses to electrochemical treatments and the possible decay of solutions is\ninvestigated both from theoretical and numerical points of view. In [19] a con-\ntrol approach to predict an optimal drug dosage shrinking the cancer tumor-cell\n2population was proposed. The predictive control problem is hence based on the\ndifference between the probability density function and the desired probability\ndensity function calculated at each time instant.\nIn the mathematical framework, an untreated tumor volume can be modeled\nby an homogeneous Gompertz stochastic process. In order to include the effect\nof an anti-angiogenic therapy in [15, 20, 21] the infinitesimal moments of the\nhomogeneous process were modified by introducing suitable continuous time-\ndependent functions modeling the exogenous factors; in such a way, the therapy\npresence leads to a time non homogeneous stochastic process. A statistical\napproach was also proposed in [22, 23, 24] to fit the modifications in the natural\ngrowth rates due to several therapies.\n1.1. Motivations and plan of the paper\nIn the present paper we provide a natural generalization of the results pro-\nvide in [23] and [24]. Specifically, we assume that the two applied therapies (one\nof anti-proliferative and the other inducing the cancer cells death) are generally\nable to modify both the drift and the infinitesimal variance of the process mak-\ning it time-dependent. We propose a statistical methodology to estimate the\nnatural tumor rates and to fit the exogenous term including also the infinites-\nimal variance of the resulting process. The procedure uses a control group G\ndescribed by a homogeneous Gompertz diffusion process and two treated groups\nG1andG2described by two time non-homogeneous processes. The group G1is\nassumed to be treated with a therapy (anti-proliferative or cell death-induced)\nwhile the group G2is treated the same therapy of G1together with an other ther-\napy of the other type. Further, we consider the case in which the two groups are\ncharacterized by two generally different time-dependent infinitesimal variance.\nThe procedure works as follows. In a first step, the control group Gis\nused to estimate the constant parameters included in the homogeneous process,\nwhereas in the subsequent steps the treated groups, G1andG2, are used to fit\nthe unknown time-dependent functions describing the effect of the therapies.\nPrecisely, via suitable mathematical relations, the group G1is used to fit the\n3function related to the single therapy applied in it and the effect that this\ntherapy has on the infinitesimal variance. Then, the group G2is used to fit the\nsecond therapy and its effect on the infinitesimal variance.\nMoreover, a bootstrap testing procedure able to evaluate the time depen-\ndence of the exogenous factors is provided. Essentially, it is directed to establish\nif the real effect of the various applied therapies has a known functional form.\nIn particular the proposed test is then used to establish if the therapy effect is\nnull or constant.\nWe point out that both the estimation procedure and the testing hypothesis\non the exogenous factors related to the stochastic diffusion processes are of\ninterest in various applicative and theoretical contexts [25, 26, 27].\nThe plan of the paper is the following. In Section 2 the model is introduced\nand its probability distribution and some statistical characteristics are derived.\nIn Section 3 the procedure to estimate the parameters and to fit the unknown\nfunctions is proposed. Various simulated-based examples are given to validate\nthe fitting procedure. In Section 4 the hypothesis test procedure is provided\nand several cases of particular interest in biological context are considered. In\nparticular, some concatenated tests are performed to evaluate the constant/null\neffect of the therapy on the rates and on the infinitesimal variance. Finally, in\nSection 5 an application to real data is provided to study the combined effect\nof Carboplatin and Taxol in ovarian cancer.\n2. The model\nIn the mathematical framework, an untreated tumor volume can be modeled\nby an homogeneous Gompertz stochastic process defined in R+with infinitesi-\nmal moments\nA1(x) =αx−βxlogx,\nA2(x) =σ2x2, (1)\nwhere α, βandσare positive constants. The parameters αandβdescribe the\ncell’s growth and death rates, respectively, σis related to more or less intense\n4environmental fluctuations introduced to justify discrepancies between clinical\ndata and theoretical predictions that quite often are detected.\nOur approach for including the effect of an anti-angiogenic therapy consists\nto modify the infinitesimal moments (1) by introducing suitable continuous time-\ndependent functions modeling the exogenous factors; in such a way, the therapy\npresence leads to a time non homogeneous stochastic process. Precisely, let\n{X(t) :t≥t0}with t0≥0 be a stochastic process in R+and satisfying the\nfollowing stochastic differential equation (SDE)\ndX(t) ={(α−C(t))−(β−D(t)) lnX(t)}X(t)dt+σp\nV(t)X(t)dW(t),\nX(t0) =X0. (2)\nHere, as in (1), α, βandσare positive constants, while C(t), D(t) and V(t)\nare functions in C1[t0,+∞) with V(t)>0 for all t≥t0,X0is a random variable\ndescribing the initial state of the process, and W(t) is a standard Wiener process\nindependent from X0fort≥t0. In the model setting, C(t) represents tumor\nregression rate due to the therapy and has the same dimension as parameter α,\nwhile the function D(t) modifies the death rate βof the process (1) in β−D(t).\nFrom a biological point of view, the function C(t) describes the effect of an\nanti-proliferative therapy, that is, able to modify the natural birth rate of cancer\ncells, while D(t) describes the effect of cell death-induced therapy (see [15, 23]).\nClearly C(t) successfully applied when it assumes positive values, while D(t) is\neffective when it is a negative function. Further it would be desirable to have\nsmall values of the function V(t), describing fluctuations in the tumor volume.\nAnyway, in experimental studies the effectiveness of a therapy has to be tested,\nso we assume that the functions C(t),D(t) have real values and V(t)>0.\nThe aim of this paper is to model the combined effect of two therapies, one\nanti-proliferative and the other that induces the death of cancer cells. In this\nsense, model (2) can be viewed as a modification of model (1) after transforming\nits infinitesimal moments by introducing the functions C(t),D(t) and V(t).\n5By considering\nZ(t) =f(X(t), t) =k(t) lnX(t), (3)\nwhere\nk(t) = exp\u0012Zt\n[β−D(s)]ds\u0013\n,\nand by applying Itˆ o’s Lemma, we can transform process X(t) into a non-\nhomogeneous Wiener process Z(t) described by the following SDE:\ndZ(t) =a(t)dt+b(t)dW(t), Z (t0) =Z0, (4)\nwith\na(t) =k(t)\u0014\nα−C(t)−σ2V(t)\n2\u0015\n, b (t) =σp\nV(t)k(t),\nwhose solution is\nZ(t) =Z0+Zt\nt0a(s)ds+Zt\nt0b(s)dW(s).\nFinally, undoing the change (3), we obtain\nX(t) = exp(\n1\nk(t)\"\nk(t0) lnX0+Zt\nt0a(s)ds+Zt\nt0b(s)dW(s)#)\n.\n2.1. Distribution of the process\nFrom (4), and if Z0is a degenerate random variable, i.e. P(Z0=z0) = 1,\nwith z0∈Ror normally distributed, i.e. Z0∼N1[µ0, σ2\n0], then Z(t) is a\nGaussian process, so, ∀n∈Nandt1<···< tn, vector ( Z(t1), . . . , Z (tn))Thas\nan-dimensional normal distribution Nn[ε,Σ], where the components of vector\nεand matrix Σare\nεi=E[Z0] +Zti\nt0k(s)\u0014\nα−C(s)−σ2V(s)\n2\u0015\nds, i = 1, . . . , n\nand\nσij=V ar[Z0] +σ2Zmin(ti,tj)\nt0k2(s)V(s)ds, i, j = 1, . . . , n,\nrespectively.\n6Therefore, by (3) all the finite-dimensional distributions of the process X(t)\nare lognormal; specifically, ∀n∈N,\n(X(t1), . . . , X (tn))T∼Λn[ξ,∆], (5)\nwhere ξi=εi\nk(ti)andδij=σij\nk(ti)k(tj),i, j= 1. . . , n , are the components of ξ\nand∆, respectively.\nIn particular, by considering X0∼Λ1[µ0;σ2\n0], we have\nX(t)∼Λ1\u0002\nM∗(t|µ0, t0);V∗(t|σ2\n0, t0)\u0003\n,\nwhere, for τ < t ,\nM∗(t|u, τ) =u¯k(t|τ) +Zt\nτ\u0012\nα−C(s)−σ2V(s)\n2\u0013\n¯k(t|s)ds,\nand\nV∗(t|u, τ) =u¯k2(t|τ) +Zt\nτσ2V(s)¯k2(t|s)ds,\nwith ¯k(t|τ) =k(τ)/k(t).\nIn the following we will assume that X0is a degenerate random variable in\nx0. This assumption is quite common in the context of tumor growth since the\nvariable of interest is usually the relative volume of the tumor, and x0= 1 is\nthe relative volume at the detection of the tumor. So, we will assume P[X0=\nx0] = 1. In this case\nX(t)∼Λ1[m1(t);u(t)],\nwith\nm1(t) =M∗(t|lnx0, t0) (6)\nand\nu(t) =V∗(t|0, t0). (7)\n7So, the mean and the variance functions of X(t) are:\nE[X(t)] = exp\u0012\nm1(t) +1\n2u(t)\u0013\n,\n(8)\nV ar[X(t)] = exp (2 m1(t) +u(t))×[exp ( u(t))−1],\nrespectively.\n3. Estimation of the model\nIn this section we propose a procedure to estimate the parameters α, β, σ ,\nand to approximate the functions C(t),D(t) and V(t) in [t0, T]. To this end, in\npractice it is necessary to have data from three experimental groups of individ-\nuals. Concretely:\n•an untreated (control) group, say G,\n•a first group, G1, treated with a single therapy that affects only one of the\ntwo rates that model the untreated tumor volume,\n•a second group, G2, treated with two therapies. One of them must be the\nsame therapy applied in G1, whereas the other one affects the rate not\nmodified in G1.\nThe control group is associated to the stochastic process X(t) described by\nthe SDE\ndX(t) = [α−βlnX(t)]X(t)dt+σX(t)dW(t) X(t0) =x0. (9)\nMoreover, group G1is modeled by a stochastic process X1(t) for which two cases\ncan be considered:\n•G1is treated with an anti-proliferative therapy, i.e. mainly affecting cell\ngrowth. In this case, X1(t) follows the SDE\ndX1(t) ={[α−C(t)]−βlnX1(t)}X1(t)dt+σp\nV1(t)X1(t)dW(t), X1(t0) =x0.\n(10)\n8•G1is treated with a therapy that induces, or mainly induces, the death of\ncancer cells. Now the SDE followed by X1(t) is\ndX1(t) ={α−[β−D(t)] lnX1(t)}X1(t)dt+σp\nV1(t)X1(t)dW(t), X1(t0) =x0.\n(11)\nFinally, group G2is described by a stochastic process X2(t) solution of\ndX2(t) ={[α−C(t)]−[β−D(t)] lnX2(t)}X2(t)dt+σp\nV2(t)X2(t)dW(t), X2(t0) =x0.\n(12)\nThe basic idea is to use data from the control group to estimate the param-\neters α, βandσ2, whereas the treated groups are used to fit the functions C(t),\nD(t),V1(t) and V2(t).\n3.1. Some basic expressions\nIn this subsection we introduce some expressions that are the basis of the\nestimation procedure developed in the next one.\nFrom (8), we define\nm2(t) = ln E[X(t)] =m1(t) +1\n2u(t),\nand by considering (6) and (7), after some algebra, the following relationships\nare obtained:\nC(t) =α−(β−D(t))(m1(t) +u(t))−m′\n1(t)−1\n2u′(t)\n=α−(β−D(t))(2m2(t)−m1(t))−m′\n2(t) (13)\nD(t) =β+m′\n1(t) +1\n2u′(t)−α+C(t)\nm1(t) +u(t)=β+m′\n2(t)−α+C(t)\n2m2(t)−m1(t)(14)\nV(t) =1\nσ2(u′(t) + 2( β−D(t))u(t))\n=2\nσ2[(m′\n2(t)−m′\n1(t)) + 2( β−D(t))(m2(t)−m1(t))]. (15)\n9We point out that the two expressions obtained for C(t), D(t) and V(t) in\n(13), (14) and (15) respectively, can be alternatively used to fit the functions\ndepending on the behavior of the sampling versions of these functions in real\napplications.\n3.2. The estimation procedure\nLet us consider dsample-paths from the control group, observed at the\nsame time instants tj,j= 0, . . . , n −1, in the interval [ t0, T]. Let {xij, i=\n1, . . . , d ;j= 0, . . . , n −1}be the observed values of the sample paths. Moreover,\nlet{x(k)\nij, i= 1, . . . , d k;j= 0, . . . , n −1}be the values of dksample paths from\nthe treated group Gk,k= 1,2, observed at the same previous time instants.\nMaking use of Equations (13)-(15), and denoting m(k)\n1(t) = E[lnXk(t)],\nm(k)\n2(t) = ln E[Xk(t)] and u(k)(t) =V ar[lnXk(t)],k= 1,2, we can estimate\nthe three models from data provided by the control and the two treated groups.\nTo this end we provide the following stepwise procedure:\n•Obtain the maximum likelihood (ML) estimates of α,βandσ2by solving\nthe likelihood equation system (18)-(20) in Appendix for C(t) =D(t) = 0\nandV(t) = 1, from the data of group G. Denote by bα,bβandbσ2such\nestimates.\n•Calculate at each time instant tj,j= 0, . . . , n −1, the values\nbm(k)\n1(tj) = ¯yk\nj,bm(k)\n2(tj) = ln(¯ x(k)\nj),bu(k)(tj) =s2\nj(k), k= 1,2\nwhere\n- ¯yk\njis the sample mean of the logarithms of the values of the sample\npaths of the group Gk(k= 1,2) at tj,\n- ¯x(k)\njis the sample mean of the values of the sample paths of Gk\n(k= 1,2) at tj,\n-s2\nj(k)is the unbiased sample variance of the logarithms of the values\nof the sample paths of Gk(k= 1,2) at tj.\n10•Fork= 1,2, approximate the derivatives of m(k)\n1(t),m(k)\n2(t) and u(k)(t),\nattjfrom the values obtained in the previous step. Denote by bm(k)′\n1(tj),\nbm(k)′\n2(tj) andbu(k)′(tj) the obtained values.\n•Estimating C(t),D(t),V1(t) and V2(t) as follows:\n–IfG1is modeled by (10), i.e. it is treated with an anti-proliferative\ntherapy, obtain an initial estimate of C(tj) and V1(tj) by applying\nthe observed data of this group to expressions (13) and (15), with\nD(t) = 0. This leads to\nbCj=bα−bβ\u0010\nbm(1)\n1(tj) +bu(1)(tj)\u0011\n−bm(1)′\n1(tj)−1\n2bu(1)′(tj)\n=bα−bβ\u0010\n2bm(1)\n2(tj)−bm(1)\n1(tj)\u0011\n−bm(1)′\n2(tj)\nand\nbV1,j=1\nbσ2\u0010\nbu(1)′(tj) + 2bβbu(1)(tj)\u0011\n=2\nbσ2\u0010\nbm(1)′\n2(tj)−bm(1)′\n1(tj) + 2bβ\u0010\nbm(1)\n2(tj)−bm(1)\n1(tj)\u0011\u0011\n.\nNext, for each tj, calculate initial estimates of D(tj) and V2(tj) for\nprocess X2(t), by considering (14) and (15) for the data of group\nG2and the previous bCjvalues. In this way the following values are\nobtained:\nbDj=bβ+bm(2)′\n1(tj) +1\n2bu(2)′(tj)−bα+bCj\nbm(2)\n1(tj) +bu(2)(tj)\n=bβ+bm(2)′\n2(tj)−bα+bCj\n2bm(2)\n2(tj)−bm(2)\n1(tj)\nand\nbV2,j=1\nbσ2\u0010\nbu(2)′(tj) + 2(bβ−bDj)bu(2)(tj)\u0011\n=2\nbσ2\u0010\nbm(2)′\n2(tj)−bm(2)′\n1(tj) + 2(bβ−bDj)\u0010\nbm(2)\n2(tj)−bm(2)\n1(tj)\u0011\u0011\n.\n11–IfG1is treated with a therapy that induces the death of cancer cells,\ni.e. the model (11) is now considered, determine values bDjandbV1,j\n(initial estimates of D(tj) and V1(tj),j= 0, . . . , n −1) from (14) and\n(15) by considering C(t) = 0 and the data of G1, thus obtaining\nbDj=bβ+bm(1)′\n1(tj) +1\n2bu(1)′(tj)−bα\nbm(1)\n1(tj) +bu(1)(tj)\n=bβ+bm(1)′\n2(tj)−bα\n2bm(1)\n2(tj)−bm(1)\n1(tj)\nand\nbV1,j=1\nbσ2\u0010\nbu(1)′(tj) + 2(bβ−bDj)bu(1)(tj)\u0011\n=2\nbσ2\u0010\nbm(1)′\n2(tj)−bm(1)′\n1(tj) + 2(bβ−bDj)\u0010\nbm(1)\n2(tj)−bm(1)\n1(tj)\u0011\u0011\n.\nThen, for process X2(t), compute initial estimates of C(tj) and V2(tj),\ntj,j= 0, . . . , n −1, from (13) and (15) by taking the data of group\nG2and the values bDjpreviously estimated. This leads to\nbCj=bα−(bβ−bDj)\u0010\nbm(2)\n1(tj) +bu(2)(tj)\u0011\n−bm(2)′\n1(tj)−1\n2bu(2)′(tj)\n=bα−(bβ−bDj)\u0010\n2bm(2)\n2(tj)−bm(2)\n1(tj)\u0011\n−bm(2)′\n2(tj)\nand\nbV2,j=1\nbσ2\u0010\nbu(2)′(tj) + 2(bβ−bDj)bu(2)(tj)\u0011\n=2\nbσ2\u0010\nbm(2)′\n2(tj)−bm(2)′\n1(tj) + 2(bβ−bDj)\u0010\nbm(2)\n2(tj)−bm(2)\n1(tj)\u0011\u0011\n.\n•Obtain bC(t),bD(t),bV1(t) andbV2(t) as follows:\n–Calculate the final estimated values bC(tj),bD(tj),bV1(tj) andbV2(tj)\nby using local regression of bCj,bDj,bV1,jandbV2,jontj, respectively.\n–Interpolate, by means of spline functions1, the data points ( tj,bC(tj)),\n1Since the functions C(t),D(t) and V(t) are sufficiently smooth (they are C1-class), we\nuse the natural cubic spline interpolation.\n12(tj,bD(tj)), (tj,bV1(tj)) and ( tj,bV2(tj)), respectively.\n3.3. Simulation-based applications\nIn order to validate the proposed estimation procedure, we have developed\ntwo applications based on simulated data:\n•In the former, we consider an untreated group ( G), a first group ( G1)\ntreated with an anti-proliferative therapy (so, it is modeled by (10)), and\na second group ( G2) that is treated with the same therapy as the first\ngroup together with another one inducing the death of cancer cells. This\ngroup is modeled by (12).\n•In the second case, in addition to the control group ( G), we consider a\ngroup G1treated with a therapy that induces the death of cancer cells\n(modeled by (11)), whereas G2is treated with the same therapy as the\nfirst group together with an anti-proliferative therapy, so this group is\nmodeled by (12).\nIn the two applications the untreated group is modeled by (9) with the same\nparameters. Table 1 summarizes the parameters and functions considered. The\nchoice of α,βandσvalues has been made so that the simulated paths present\nvalues similar to real situations. On the other hand, the therapeutical functions\nin our simulation experiment are in line with [15, 23]. In Application 1 we\nconsider the case in which the group G1is treated with an anti-proliferative\nlinear therapy, while the group G2is treated with a cell death-induced therapy\nhaving a “bump effect” when it is applied and asymptotically reduces of 12%\nthe natural death rate of the tumor. In Application 2 the two therapies are\nreversed. The infinitesimal variances V1(t) and V2(t) involve two lognormal\nprobability density functions since we expect that the variability of the process\nis greatly influenced by the therapies when they are applied, then they restore\nto natural values. This assumption is close to what is observed in real situations\nlike the one presented in Section 5.\n13Table 1: Parameters and functions considered in each example (being Λ 1(t, µ, σ2) the density\nfunction of a lognormal distribution Λ 1(µ, σ2)).\nGroup Application 1 Application 2\nG α= 0.5,β= 0.2,σ= 0.01\nG1C(t) = 0 .005t\nV1(t) = (0 .7+10 Λ 1(t,3,0.5))2D(t) =−0.12t2/(50 + t(t−10))\nV1(t) = (0 .7+10 Λ 1(t,3,0.5))2\nG2D(t) =−0.12t2/(50 + t(t−10))\nV2(t) = (0 .7+15 Λ 1(t,3,0.5))2C(t) = 0 .005t\nV2(t) = (0 .7+10 Λ 1(t,3,0.5))2\nThe estimation procedure has been replicated 100 times in both the exam-\nples. In each replication, 25 sample paths have been simulated by considering\n51 time instants equally spaced in the interval [0 ,50]. The sample paths have\nbeen simulated using the snssde1d function from the R package Sim.DiffProc\n[28]. This function allows to simulate the solution of a stochastic differential\nequation from its discretization by using differents numerical schema as Euler-\nMaruyama and Milstein among others (see Iacus [29] for details). Further, a\ndegenerate initial distribution at x0= 1 has been considered. This choice has\nbeen made because in real studies on the evolution of tumors (such as the one\nshown in section 5) the data provided are relative volumes of the tumors.\nThe estimates in the control group were bα= 0.496477, bβ= 0.198469 and\nbσ= 0.010043. In Figure 1 the fit of the functions C(t), V1(t), D(t) and V2(t) in\nthe models (10) and (12) in Application 1 are plotted on the top. The mean\nand variances of the processes X1(t) and X2(t) along with their fitted versions\nare also shown on the bottom. The absolute difference functions between the\nsimulated and fitted function are also represented in green. Results related to\nApplication 2 are shown in Figure 2. In both the applications, the procedure\nprovides estimated functions (red lines) very close to the theoretical ones (black\nlines).\n14010203040500.000.050.100.150.200.25C(t) and its estimate\nt●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.60.81.01.2V1(t) and its estimate\nt●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●\n01020304050−0.25−0.20−0.15−0.10−0.050.00D(t) and its estimate\nt●●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.51.01.5V2(t) and its estimate\nt●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●\n010203040502468E[X1(t)] and its estimate\nt\n010203040500.0000.0050.0100.0150.020Var[X1(t)] and its estimate\nt\n010203040501.01.52.02.53.03.54.0E[X2(t)] and its estimate\nt\n010203040500.00000.00050.00100.0015Var[X2(t)] and its estimate\nt\nFigure 1: Fit of the functions C(t), V1(t), D(t) and V2(t) in the models (10) and (12) in Application 1 are plotted on the top. The mean and variances\nof the processes X1(t) and X2(t) along with their fitted versions (in red) are also shown on the bottom. The absolute difference functions between\nthe simulated and fitted function are represented in green.\n1501020304050−0.25−0.20−0.15−0.10−0.050.00D(t) and its estimate\nt●●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.60.81.01.2V1(t) and its estimate\nt●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●●●\n010203040500.000.050.100.150.200.25C(t) and its estimate\nt●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.60.81.01.2V2(t) and its estimate\nt●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●\n●●●●●●●●●●●●●●●●●\n010203040501.01.52.02.53.03.54.0E[X1(t)] and its estimate\nt\n010203040500.00000.00050.00100.0015Var[X1(t)] and its estimate\nt\n010203040501.01.52.02.53.03.54.0E[X2(t)] and its estimate\nt\n010203040500.00000.00020.00040.00060.00080.00100.0012Var[X2(t)] and its estimate\nt\nFigure 2: Fit of the functions D(t), V1(t), C(t) and V2(t) in the models (11) and (12) in Application 2 are plotted on the top. The mean and variances\nof the processes X1(t) and X2(t) along with their fitted versions (in red) are also shown on the bottom. The absolute difference functions between\nthe simulated and fitted function are represented in green.\n16Moreover, in order to have a measure of goodness of fit of the obtained\nestimates, we have considered the following mean squared error\nMSE (H) =1\nnn−1X\nj=0\u0010\nbH(tj)−H(tj)\u00112\n,\nwhere Hrepresents one of the C, D, V 1, V2functions. Table 2 includes the\nvalues obtained for these errors, confirming the closeness between theoretical\nand estimated functions.\nTable 2: Mean squared errors of estimated functions in groups G1andG2for Applications 1\nand 2\nApplication 1\nGroup G1 Group G2\nFunction MSE Function MSE\nC(t) 5 .779772 e−07 D(t) 2 .637550 e−06\nV1(t) 3 .006884 e−04 V2(t) 4 .192476 e−04\nE(X1(t)) 1 .684190 e−04 E(X2(t)) 2 .448863 e−05\nV ar(X1(t)) 7 .953060 e−09 V ar(X2(t)) 5 .742807 e−10\nApplication 2\nGroup G1 Group G2\nFunction MSE Function MSE\nD(t) 1 .942332 e−06 C(t) 1 .111295 e−06\nV1(t) 2 .428458 e−04 V2(t) 2 .102767 e−04\nE(X1(t)) 3 .193397 e−05 E(X2(t)) 2 .437206 e−05\nV ar(X1(t)) 4 .368352 e−10 V ar(X2(t)) 2 .315756 e−10\nWe point out that the fit functions for Applications 1 and 2 are obtained by\nconsidering as data-generating process the model X1(t) and X2(t) in (10) and\n(12) for Application 1 and Eqs (11) and (12) for Application 2.\n17In the next section we consider the problem of testing if the influence over\ntime of a therapy follows a given functional scheme and, in particular, if its\napplication results in a constant modification of the natural parameters of the\nprocess.\n4. Testing hypothesis about functions C(t),D(t) and V(t)\nIn the tumor context outlined in this paper, in order to evaluate the effec-\ntiveness of experimental therapies, the following questions are of special interest:\n•Is the real effect on the growth rate of an anti-proliferative therapy null?\n•Is the real effect on the death rate of a therapy that induces the death of\ncancer cells null?\n•Does the therapy, or combination of therapies, affect the infinitesimal\nvariance?\n•Do the effects of a therapy or combination of therapies depend on time?\n•Do the functions that model the effect of the therapy or combination of\ntherapies have a specific form?\nSince the functions included in model (2) represent different effects of a\ntherapy, or combination of therapies on tumor growth, to answer these questions\nwe propose to perform hypothesis testing about the functions C(t),D(t) and\nV(t) in model (2) (note that models (10), (11) and (12) are particular cases of\nthis).\nThe null hypothesis can be formulated in a unified way as\nH0:H(t) =h(t),\nwith H(t) any of the functions in model (2) and h(t) a given function.\nTo test the null hypothesis we propose to use a bootstrap test (b-Test) based\non the statistic D=n−1X\nj=0|bH(tj)−h(tj)|.Calculation of values of this statistic is\n18based on a bootstrap procedure following the line proposed in Rom´ an-Rom´ an\net al. [24, 30]. Concretely, the schema is the following:\n•Generate mbootstrap samples of the considered model. Each bootstrap\nsample consists of dksample paths (depending of the treated group Gk\nbeing considered) simulated in the same way as the one previously exposed\nby taking function h(t) inH0and the estimates of the parameters and the\nrest of functions via the procedure proposed in the previous section.\n•Estimate H(t) from the sample paths of each bootstrap sample, and cal-\nculate a value Dl,l= 1, . . . , m , of the statistic D.\n•Calculate the p-value as the proportion of values Dlgreater than or equal\ntoD.\nThe case h(t) =his of special interest because it means that the effect of\ntherapy represented by H(t) does not depend on time. Even more, C(t) = 0\nmeans that the therapy has no anti-proliferative effect; D(t) = 0 signifies that\nthe therapy does not induce the death of cancer cells, whereas V(t) = 1 leads\nto the non-influence of the therapy on the infinitesimal variance of the process.\nIn such case, the constant hto be included in H0has to be chosen. If it is\nnot known a priori, as usual in applications, we propose to choose has the value\nobtained from the ML estimation of h(t) =hin model (2). Concretely,\n•Testing C(t) constant. In this case, H0:C(t) =c. The value of cis\nobtained from the ML estimate of the growth rate in model (2) by solving\n(18) in Appendix taking C(t) = 0, D(t) =bD(t) (ifD(t)̸= 0), V(t) =bV(t)\n(ifV(t)̸= 1), and considering β=bβandσ=bσ, the ML estimates of β\nandσfor group G. In this way we obtain [α−c, from which c=bα−[α−c.\n•Testing D(t) constant. Now, H0:D(t) =d, where the value dis obtained\nas in the previous case, by changing C(t) with D(t) and αwith β. Note\nthat in this case the equation (19) in Appendix must be solved.\n19•Testing V(t) constant. This case is performed considering H0:V(t) =v,\nwhere the constant vis obtained from v=dσ2v/cσ2, weredσ2vmatches the\nML estimate of σ2in the model (2) by solving (20) in Appendix taking\nV(t) = 1, α=bα,β=bβ,C(t) =bC(t) (ifC(t)̸= 0) and D(t) =bD(t) (if\nD(t)̸= 0).\n4.1. Simulation study\nIn order to show the behavior of the proposed bootstrap tests we have per-\nformed a simulation study considering a control group Gand two treated groups\nG1andG2, modeled by (9), (10) and (12), respectively. The present study is\nlimited to the case in which G1is treated with an anti-proliferative therapy. The\nstudy in the case in which G1is treated with a therapy that induces the death\nof cancer cells would be carried out in a similar way.\nOnce the models are estimated following the corresponding procedure in\nSection 3.1, we test hypotheses about all the functions included in (10) and (12)\nas follows.\n1. For group G1,H0:V1(t) =v1is tested, where v1is proposed following\nthe comments mentioned above. Then, we test H0:C(t) =c, taking into\naccount that:\n•IfH0:V1(t) =v1is not rejected, the value of cis determined con-\nsidering V1(t) =v1.\n•IfH0:V1(t) =v1is rejected, cis determined considering V1(t) =\nbV1(t).\n2. For group G2,H0:V2(t) =v2is tested. To this end, v2is determined\nmaking use of C(t) =c, ifH0:C(t) =cis not rejected, or C(t) =bC(t)\notherwise, and D(t) =bD(t). Then we test H0:D(t) =d, noting in this\ncase that:\n•IfH0:V2(t) =v2is not rejected, dis determined making use of\nC(t) =bC(t), ifH0:C(t) =cis rejected, or C(t) =con the contrary,\nandV2(t) =v2.\n20•IfH0:V2(t) =v2is rejected, the value of dis determined considering\nC(t) =bC(t) , if H0:C(t) =cis rejected, or C(t) =cotherwise, and\nV2(t) =bV2(t).\nIn the simulation study we have considered models (9), (10) and (12) with\nα= 0.5,β= 0.2 and σ= 0.01 for all possible combinations of the functions\nin Table 3. Precisely, we consider 3 choices for functions C(t),D(t) and V1(t)\nand four cases for V2(t), obtaining 9 cases for group G1and 108 for group G2.\nThese functions have been selected in order to simulate the tumor growth in\nthe groups treated with therapies of diverse effects, ranging from therapies that\ndo not produce any improvement, until therapies that produce a significant\nreduction both in the mean relative volume of the tumor and in its variability.\nFor each model, 25 sample paths have been simulated over 51 equally time\ninstants in [0 ,50]. The estimates obtained for model (9), from the data of\nthe control group G, were bα= 0.4972273, bβ= 0.1987757 and bσ= 0.0100692.\nFurther, in each case for groups G1andG2, the number of bootstrap samples\nused to perform each b-Test was m= 1500.\nThe complete simulation study is presented in schematic form in Supple-\nmentary Material. In each case, for groups G1andG2, we show:\n•the sample paths simulated for each model,\n•the estimates of the functions included in each model,\n•the results of the hypothesis tests listed above (concatenated b-Tests)\n•the sample mean and variance functions,\n•the theoretical mean and variance functions, together with their estimated\nversions, before and after the concatenated b-Tests are performed.\nIn this section we focus on two cases that have a specific meaning. These\ntwo cases show how the proposed tests allow obtaining a better estimation of\nthe mean functions and variances of the simulated processes.\n21Table 3: Different values of functions C(t),V1(t),D(t) and V2(t) for the simulation study,\nbeing Λ 1(t, µ, σ2) the density function of a lognormal distribution Λ 1(µ, σ2).\nC(t) V1(t) D(t) V2(t)\n0 1 0 1\n0.025 0 .49 −0.05 0 .49\n0.005t(0.7+10 Λ 1(t,3,0.5))2−0.12t2\n50 + t(t−10)(0.7+10 Λ 1(t,3,0.5))2\n(0.7+15 Λ 1(t,3,0.5))2\n•Case 1 .C(t) = 0, V1(t) = 1, D(t) = 0 and V2(t) = 1.\nIn this case, in the group G1, the anti-proliferative effect of the first ther-\napy is null, and this therapy also does not affect the infinitesimal variance\nof process X1(t). Moreover, in group G2, the effect of therapy that in-\nduces the death of cancer cells is null, and the combined effect of the two\ntherapies does not affect the infinitesimal variance of the process X2(t).\nFigure 3 shows the simulated sample paths of models (9), (10) and (12).\n0102030405024681012Simulated sample paths\nt\nFigure 3: Simulated sample paths of models (9), (10) and (12), in red, blue and green,\nrespectively, with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0, V1(t) = 1, D(t) = 0 and V2(t) = 1.\nModel (10) has been adjusted from the data of treated group G1with\nα=bα,β=bβandσ=bσ, obtaining the estimates bC(t) andbV1(t). These\nestimates are shown in Figure 4 as well as the estimated mean and variance\nfunctions of the process X1(t). Specifically, in Figures 4(a) and 4(b), the\n22red points correspond to the estimated values bCiandbV1,i, the red solid\nlines represent the estimated functions bC(t) and bV1(t), while the black\nsolid lines indicate the theoretical functions. In Figures 4(c) and 4(d),\nthe red points correspond to the sample mean and variance functions,\nrespectively; the red solid lines represent the estimated mean and variance\nfunctions of the process X1(t) whereas the black solid lines indicate the\ntheoretical mean and variance functions of process X1(t). In all the cases,\nthe absolute difference functions between the simulated and fitted function\nare also represented in green.\n01020304050−0.0050.0000.0050.010C(t) and its estimate\nt●●\n●\n●●●\n●●●\n●\n●●\n●●●\n●●●\n●●●●\n●\n●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●\n●●\n010203040500.40.60.81.01.21.41.6V1(t) and its estimate\nt●●●●\n●●●\n●\n●●\n●●●\n●●\n●●●\n●●●●\n●\n●\n●●●●●●●●\n●\n●●●●\n●\n●\n●\n●●●●\n●\n●●\n●\n●●●\n(a) (b)\n0102030405024681012Sample mean, E[X1(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.000.010.020.030.040.05Sample variance, Var[X1(t)] and its estimate\n●●●●●●●●●●\n●●●●●●●●\n●\n●●●●\n●●\n●●●●●●●\n●\n●\n●●●●●\n●\n●\n●●●●\n●●\n●\n●●●\nt\n(c) (d)\nFigure 4: Fit of simulated data of model (10) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 and\nV1(t) = 1. The absolute difference functions between the simulated and fitted function are\nrepresented in green.\nFigure 4 seems to indicate that the estimated values bCiandbV1,ivary\naround values close to 0 and 1, respectively, and it makes sense to test if\nthe functions C(t) and V1(t) are constant.\n23First, we test if V1(t) is constant. The constant is chosen via the ML\nestimation as previously indicated. In this case, v1= 0.9710093. The value\nof the D-statistics is D= 2.5203844 and the associated p-value is 0 .974,\nso there is no evidence to reject that the effect of the anti-proliferative\ntherapy on the infinitesimal variance of the process that models tumor\ngrowth does not depend on time. Moreover, v1≈1 suggests that the\nanti-proliferative therapy hardly affects such infinitesimal variance.\nThen, under the assumption V1(t) = 0 .9710093, we test if C(t) is constant,\ni.e.H0:C(t) =cwhere c= 0.0002401 has been determined as described\nbefore. The value of the D-statistics is D= 0.0158729 and the associated\np-value is 0 .755, so there is no evidence to reject that the effect of the\nanti-proliferative therapy on the rate of growth does not depend on time.\nIn fact, since c≈0, we can conclude that the supposed anti-proliferative\neffect of the therapy on the growth rate has been almost null.\nFigure 5, similarly to Figures 4(c) and 4(d), shows the estimated mean\nand variance functions in the group G1withbα,bβ,bσ,bV1(t) = 0 .9710093\nandbC(t) = 0 .0002401, together with the sample and theoretical mean and\nvariance functions of process X1(t). Observe how now the estimated mean\nand variance functions better reproduce the theoretical ones.\n0102030405024681012Sample mean, E[X1(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.000.010.020.030.040.05Sample variance, Var[X1(t)] and its estimate\n●●●●●●●●●●\n●●●●●●●●\n●\n●●●●\n●●\n●●●●●●●\n●\n●\n●●●●●\n●\n●\n●●●●\n●●\n●\n●●●\nt\nFigure 5: Fit of simulated data of model (10) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0\nandV1(t) = 1, assuming that it is accepted first that V1(t) is constant, and then, that C(t)\nis constant. The absolute difference functions between the simulated and fitted function are\nrepresented in green.\n24Next, from data of the treated group G2, model (12) have been adjusted\nby using bα,bβ,bσandbC(t) = 0 .0002401. As in Figure 4, the estimated\nfunctions bD(t) andbV2(t) in Figure 6 are plotted as well as the estimated\nmean and variance functions of the process X2(t), showing how the values\nbDiandbV2,iare close to 0 and 1, respectively, which leads us to test whether\nthe functions D(t) and V2(t) are constant.\n01020304050−0.006−0.004−0.0020.0000.0020.004D(t) and its estimate\nt●\n●●\n●\n●●●●\n●●●●\n●\n●●●●\n●●●●●\n●●●●●●\n●\n●●●●●●●\n●\n●●●●\n●●●●●●●●●\n●\n010203040500.40.60.81.01.21.41.6V2(t) and its estimate\nt●●●●\n●\n●\n●\n●●●●\n●●\n●●●\n●●●\n●\n●\n●●\n●\n●●●●●●●\n●\n●\n●●●●●●●●\n●\n●●●●●●\n●\n●\n●\n(a) (b)\n0102030405024681012Sample mean, E[X2(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.000.010.020.030.040.05Sample variance, Var[X2(t)] and its estimate\n●●●●●●●●●●●\n●●\n●●●\n●●●●●\n●●\n●\n●●●●●●●\n●●\n●\n●\n●●●\n●●●\n●\n●●\n●●\n●●●\n●\n●\nt\n(c) (d)\nFigure 6: Fit of simulated data of model (12) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0,\nD(t) = 0 and V2(t) = 1. The absolute difference functions between the simulated and fitted\nfunction are represented in green.\nFirst, we test the hypothesis H0:V2(t) =v2, where now v2= 0.9778234.\nThe bootstrap test results in D= 2.9449791 and the associated p-value=\n0.931 and, therefore, there is no evidence to reject that the effect of the\ncombination of therapies on the infinitesimal variance of the process X2(t)\ndoes not depend on the time. In addition, as v2≈1, the combination of\ntherapies has a negligible effect on such infinitesimal variance.\n25Thus, assuming V2(t) = 0 .9778234, we test H0:D(t) =d, being now\nd= 0.0002078. The value of the D-statistics is 0 .0084735 and the p-value\nis 0.897, so that there is no evidence to reject that the effect of the therapy\ninducing the death of cancer cells does not depend on time. Furthermore,\nsince d≈0, we can conclude that the therapy has hardly induced the\ndeath of cancer cells.\nFigure 7 shows the estimated mean and variance functions in the group\nG2forbα= 0.4972273, bβ= 0.1987757, bσ= 0.0100692, bC(t) = 0 .0002401,\nbV2(t) = 0 .9778234 and bD(t) = 0 .0002078, together with the sample and\nthe theoretical mean and variance functions of process X2(t). Comparing\nthis figure with Figures 6(c) and 6(d), we can see that the estimated mean\nand variance functions better reproduce the theoretical ones.\n0102030405024681012Sample mean, E[X2(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.000.010.020.030.040.05Sample variance, Var[X2(t)] and its estimate\n●●●●●●●●●●●\n●●\n●●●\n●●●●●\n●●\n●\n●●●●●●●\n●●\n●\n●\n●●●\n●●●\n●\n●●\n●●\n●●●\n●\n●\nt\nFigure 7: Fit of simulated data of model (12) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0,\nD(t) = 0 and V2(t) = 1, knowing that C(t) is constant and assuming that it is accepted at\nfirst that V2(t) is constant, and then, that D(t) is constant\nIn Figure 8 the Gaussian kernel density estimations of the D-statistics\nfor the tests just discussed are plotted based on m= 1500 runs. The\nbandwidth is chosen by using pilot estimation of derivatives as indicated\nin [31]. To summarize the results of the tests the values of the D-statistics\n(green points) and the critical region with significance 0 .05 (red) are also\nshown.\n•Case 2. C(t) = 0 .025,V1(t) = 0 .49,D(t) =−0.05 and V2(t) = 0 .49\n26051015200.000.050.100.15\n●H0:V1(t)=0.9710093Density\nD\nm = 1500 Bandwidth = 0.7\n0.00 0.02 0.04 0.06010203040\n●DensityH0:C(t)=0.0002401\nD\nm = 1500 Bandwidth = 0.0024\n051015200.000.050.100.15\n●H0:V2(t)=0.9778234Density\nD\nm = 1500 Bandwidth = 0.54\n0.000.010.020.030.040102030405060\n●H0:D(t)=0.0002078Density\nD\nm = 1500 Bandwidth = 0.0018Figure 8: Gaussian kernel density estimation of D-statistics for the tests associated to Case\n1. Green points are the values of the D-statistics in our simulation experiment. In red the\ncritical region with significance 0 .05 is shown.\nIn this case, in group G1, an anti-proliferative therapy that affects the\ninfinitesimal variance of the process X1(t) is considered, although its ef-\nfects do not depend on time. In group G2, the effect of the therapy that\ninduces the death of cancer cells does not depend on time and such ther-\napy does not affect the infinitesimal variance previously modified by the\nanti-proliferative therapy.\nFigure 9 shows the simulated sample paths of models (9), (10) and (12)\nin red, blue and green, respectively.\nFigure 10 shows the estimated functions bC(t) andbV1(t) in model (10) as\nwell as the estimated mean and variance functions of process X1(t). We\ncan see that the estimated values bCiandbV1,ivary around values close to\n0.025 and 0 .49, respectively, which suggests that the functions C(t) and\nV1(t) could be constant.\n270102030405024681012Simulated sample paths\ntFigure 9: Simulated sample paths of models (9), (10) and (12), in red, blue and green,\nrespectively, with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 .025,V1(t) = 0 .49,D(t) =−0.05 and\nV2(t) = 0 .49.\n010203040500.000.010.020.030.040.050.060.07C(t) and its estimate\nt●●\n●\n●\n●\n●●●●\n●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.20.30.40.50.60.7V1(t) and its estimate\nt●●●●●\n●●\n●●\n●●\n●●●\n●●\n●\n●●●●\n●●\n●●\n●●●●●●●●\n●\n●●●\n●●\n●\n●●●\n●\n●●●\n●●●●\n01020304050246810Sample mean, E[X1(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.0000.0050.0100.0150.020Sample variance, Var[X1(t)] and its estimate\n●●●●●●●●●●●\n●●●\n●●\n●●●●●\n●●\n●●\n●●\n●●●●●●\n●\n●●●\n●●\n●\n●●●\n●\n●●●\n●●●●\nt\nFigure 10: Fit of simulated data of model (10) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 .025\nandV1(t) = 0 .49. The absolute difference functions between the simulated and fitted function\nare represented in green.\nFirst we test the hypothesis H0:V1(t) =v1where v1= 0.4761118. The\nvalue of the D-statistics is D= 1.0633344 and the p-value is 0 .996, so there\nis no evidence to reject that the effect of the anti-proliferative therapy on\nthe infinitesimal variance of the process X1(t) does not depend on time.\n28Then, under the assumption V1(t) = 0 .4761118, we test H0:C(t) =c\nwith c= 0.0248451. The bootstrap test provides D= 0.0234247 and a\np-value of 0 .334, so there is no evidence to reject that the effect of the\nanti-proliferative therapy on the growth rate does not depend on time.\nFigure 11 shows the estimated mean and variance functions in the group G1\nwithbα= 0.4972273, bβ= 0.1987757, bσ= 0.0100692, bC(t) = 0 .0248451 and\nbV1(t) = 0 .4761118, together with the sample and the theoretical mean and\nvariance functions of the process X1(t). It is clear that now the estimated\nmean and variance functions reproduce more appropriately the theoretical\nones.\n01020304050246810Sample mean, E[X1(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.0000.0050.0100.0150.020Sample variance, Var[X1(t)] and its estimate\n●●●●●●●●●●●\n●●●\n●●\n●●●●●\n●●\n●●\n●●\n●●●●●●\n●\n●●●\n●●\n●\n●●●\n●\n●●●\n●●●●\nt\nFigure 11: Fit of simulated data of model (10) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 .025\nandV1(t) = 0 .49, assuming that it is accepted first that V1(t) is constant, and then, that C(t)\nis constant. The absolute difference functions between the simulated and fitted function are\nrepresented in green.\nFigure 12 shows the estimated functions bD(t) andbV2(t) in model (12) by\nusingbα,bβ,bσandbC(t) = 0 .0248451, as well as the estimated mean and\nvariance functions of process X2(t). The estimated values bDiandbV2,i\nin this figure vary around values close to −0.05 and 0 .49, respectively,\nand it seems reasonable to test whether the functions D(t) and V2(t) are\nconstant.\nFirst we test H0:V2(t) =v2where v2= 0.4842422. The value of the\nD-statistics is D= 0.9347132 and the p-value is 0 .981, so that there is no\n2901020304050−0.10−0.08−0.06−0.04−0.020.00D(t) and its estimate\nt●\n●●●\n●\n●●\n●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n010203040500.20.40.60.8V2(t) and its estimate\nt●●●\n●●●●\n●\n●●●\n●\n●●●\n●\n●\n●●●●\n●\n●●●●\n●\n●\n●●\n●●●\n●\n●●●\n●\n●●●\n●●●\n●\n●●●●●●\n01020304050123456Sample mean, E[X2(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.0000.0010.0020.0030.0040.0050.006Sample variance, Var[X2(t)] and its estimate\n●●●●●●●●●\n●●●\n●●●●\n●\n●●\n●●●\n●●●●●●\n●●\n●●●\n●\n●●●\n●●\n●●\n●●●●\n●\n●●●●●\ntFigure 12: Fit of simulated data of model (12) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 .025,\nD(t) =−0.05 and V2(t) = 0 .49.\nevidence to reject that the effect of the combination of therapies on the\ninfinitesimal variance of the process X2(t) is independent on time.\nHence, under the assumption V2(t) = 0 .4842422, we test H0:D(t) =d\nwhere the proposed value for dis−0.0497542. The bootstrap test results\ninD= 0.0359978 and p-value=0 .321 and consequently there is no evidence\nto reject that the therapy that induces the death of cancer cells does not\ndepend on time.\nFigure 13 shows the estimated mean and variance functions in group G2\nwithbα= 0.4972273, bβ= 0.1987757, bσ= 0.0100692, bC(t) = 0 .0248451,\nbV2(t) = 0 .4842422 and bD(t) =−0.0497542, together with the theoret-\nical and the sample mean and variance functions of the process X2(t).\nAgain, the estimated mean and variance functions better reproduce the\ntheoretical ones.\nAs in Figure 8, the Gaussian kernel density estimations of the D-statistics\n3001020304050123456Sample mean, E[X2(t)] and its estimate\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\nt\n010203040500.0000.0010.0020.0030.0040.0050.006Sample variance, Var[X2(t)] and its estimate\n●●●●●●●●●\n●●●\n●●●●\n●\n●●\n●●●\n●●●●●●\n●●\n●●●\n●\n●●●\n●●\n●●\n●●●●\n●\n●●●●●\ntFigure 13: Fit of simulated data of model (12) with α= 0.5,β= 0.2,σ= 0.01,C(t) = 0 .025,\nD(t) =−0.05 and V2(t) = 0 .49 knowing that C(t) is constant and assuming that it is accepted\nfirst that V2(t) is constant, and then that D(t) is constant.\nfor the tests just discussed are plotted in Figure 14.\n5. Application to real data of tumor growth\nIn this section we apply the stochastic process introduced in this paper to\nmodel experimental data, obtained in mice, in order to study the effect of two\ntreatments on ovarian cancer.\nIn particular, we analyze the effects of Carboplatin and Paclitaxel treatments\non the growth of OVA014HENp9 tumor from data of three experimental groups\nof 9, 8 and 8 mice. These data has been provided by the Laboratory of Preclin-\nical Investigation (LIP) that belongs to the Translational Research Department\nof the Institute Curie, Paris. Carboplatin plus paclitaxel regimen remains the\nstandard chemotherapy for the initial treatment of ovarian cancer and it is less\ntoxic and easier to administrate compared to other drug combinations (Ozols\net al. [32]).\nThe first group, G, was a control (untreated); the second group, G1, was\ntreated with Carboplatin (66mg/kg/day the days 1 and 22); and the third group,\nG2, received Carboplatin(idem)+Paclitaxel (12mg/kg/week over a period of six\nweeks). The relative volume of tumor was measured at days 1, 4, 11, 16, 19, 31,\n34, 38, 41, 53, 59 and 66.\nFigures 15 and 16 show the sample paths and the sample mean and variance,\n31024680.000.050.100.150.200.250.30\n●H0:V1(t)=0.4761118Density\nD\nm = 1500 Bandwidth = 0.4\n0.000.010.020.030.040.050102030405060\n●H0:C(t)=0.0248451Density\nD\nm = 1500 Bandwidth = 0.0028\n024680.000.050.100.150.200.250.300.35\n●H0:V2(t)=0.4842422Density\nD\nm = 1500 Bandwidth = 0.26\n0.010.020.030.040.050.06010203040506070\n●H0:D(t)=−0.0497542Density\nD\nm = 1500 Bandwidth = 0.0018Figure 14: Gaussian kernel density estimation of D-statistics for the tests associated to Case\n2. Green points are the values of the D-statistics in our simulation experiment. In red the\ncritical region with significance 0 .05 is shown.\nrespectively, of the relative tumor volume for the three experimental groups as\na function of the days after starting the treatment.\nThe ML estimation of the parameters in control group provide bα= 0.06964254,\nbβ= 0.01238329, bσ= 0.08964128.\nSince the therapy with Carboplatin induces the death of cancer cells, we\nhave adjusted the model (11) to the data of treated group G1. Figure 17 shows\nthe estimates of the D(t) and V1(t) functions as well as the fit of the sample\nmeans and variances of data by using E(bX1(t)) and V ar(bX1(t)), respectively.\nIn the same way, since the therapy with Paclitaxel is anti-proliferative, we\nhave adjusted the model (12) to the data of the treated group G2. Figure 18\nshows the estimates of the C(t) and V2(t) functions as well as the fit of the sample\nmeans and variances of data by using E(bX2(t)) and V ar(bX2(t)), respectively.\nThe results of the fitting function D(t) (Figure 17) show that the Carbo-\n32010203040506005101520253035\n●●●●●●●●●●●●\n●●● ●●●● ●●● ●●\n●●● ●●●● ●●●●●\n●●●●●●●●●●●●\n●●● ●●●●●●●●●\n●●● ●●●● ●●● ●●\n●●●●●●● ●●●●●\n●●●●●●●●●●●●\n●●● ●●●● ●●●●●\nDays after start of treatmentRelative volume\n010203040506005101520253035\n●● ● ●●●●●●●●●\n●●●●●●● ●●●●●\n●●● ●●●● ●●●●●\n●●\n●●●●●●●●● ●\n●● ●●●●● ●●● ●●\n●● ●●●●● ●●●●●\n●● ●●●●● ●●●●●\n●● ● ●●●● ●●●● ●\nDays after start of treatmentRelative volume\n010203040506005101520253035\n●●●●● ●●●●● ●●\n●●●●● ●●●●●●●\n●● ●●●●● ●●●●●\n●●●●●●● ●●●●●\n●● ●●●●●●●●●●\n●● ● ●●●\n● ●●●●●\n●●● ●●●● ●●●●●\n●●●●●●● ●●●●●\nDays after start of treatmentRelative volume(a) (b) (c)\nFigure 15: Sample paths of relative volume of tumor in control group (a), and Carboplatin\n(b) and Carboplatin+Paclitaxel (c) treated groups.\n01020304050605101520\n●●●●●●●●●●●●\n●● ●●●●●●●●●●\n●●●●●●●●●●●●\nDays after start of treatmentSample mean of relative tumor volume●\n●\n●Control group\nCarboplatin treated group\nCarboplatin+Taxol treated group\n0102030405060020406080\n●●● ●●●●●\n●●●●\n●● ● ●●●●●●●●●\n●● ●●● ●●●●●●●\nDays after start of treatmentSample variance of relative tumor volume●\n●\n●Control group\nCarboplatin treated group\nCarboplatin+Taxol treated group\nFigure 16: Sample mean and variance of the relative tumor volume in control and treated\ngroups.\nplatin treatment is effective in the first 15-20 days in which it present a negative\npeak, then it becomes ineffective. The infinitesimal variance V1(t) seems to be\ngreatly influenced by the therapy when it is effective, after that the variability\nof the process restore to natural constant values. Concerning the treated Carbo-\nplatin+Paclitaxel group G2(Figure 18), we observe that the therapy is effective\nin the first days of the treatment, then its effectivenes declines corresponding to\na negative bump, followed from values close to zero. The function V2(t) presents\na similar behaviour with respect to V1(t), although it shows a lower peak.\nThe estimated models in both treated groups provide a good fit of the sample\nmeans and variances of the relative volume of tumor. Table 4 presents the mean\nsquared errors between the sample mean and variance functions of the simulated\n330102030405060−1.0−0.50.0Estimate of D(t)\nDays after start of treatment\n010203040506001020304050Estimate of V1(t)\nDays after start of treatment\n010203040506051015Sample mean and its estimate\n●●●●●●●●●●●●\nDays after start of treatment\n0102030405060020406080Sample variance and its estimate\n●●●●● ●●●●●●●\nDays after start of treatmentFigure 17: Fit of model (11) in Carboplatin treated group (estimates in solid line).\nprocess and the estimated ones, that is\nMeanMSE =1\nnnX\nj=1(mj−ˆmj)2, V arMSE =1\nnnX\nj=1(σ2\nj−ˆσ2\nj)2\nwhere ( mj, σ2\nj) are the values of the sample mean and variance functions at tj,\nj= 1, . . . , n whereas ( ˆ mj,ˆσ2\nj) are the estimated ones.\nTable 4: Mean squared errors for the fits of the sample means and variances in the treated\ngroups.\nMSEs Carboplatin Carboplatin+Paclitaxel\nMeanMSEs 0 .0445 0 .0595\nVarMSEs 5 .3507 2 .5464\nTaking into account the estimates of functions D(t),V1(t),C(t) and V2(t),\nit seems reasonable to test only if C(t) is constant. However, we have tested,\nin a concatenated form, if each of the functions could be constant, as in the\n340102030405060−0.20−0.15−0.10−0.050.000.050.10Estimate of C(t)\nDays after start of treatment\n0102030405060051015202530Estimate of V2(t)\nDays after start of treatment\n010203040506024681012Sample mean and its estimate\n●●●●●●●●●●●●\nDays after start of treatment\n010203040506005101520Sample variance and its estimate\n●●●●●●●●●●●●\nDays after start of treatmentFigure 18: Fit of model (12) in Carboplatin+Paclitaxel treated group (estimates in solid line).\nsimulation study in Section 4.1. The corresponding constants to be included in\nthe null hypothesis are estimated by ML as described in Section 4.\nFor group G1,H0:V1(t) = 9 .612252 was tested first. The associated p-\nvalue was 0 .04 and therefore we reject that V1(t) be constant. Then, we have\ntested H0:D(t) =d, where d= 0.001296 is determined by ML considering\nV1(t) =bV1(t). The test produced a p-value of 0 .03 and we also reject that D(t)\nbe constant.\nNext, for group G2,H0:V2(t) = 8 .097848 was tested (making use of D(t) =\nbD(t)). This hypothesis is rejected with a p-value of 0 .01. Finally, we have\ntested H0:C(t) =c, where c= 0.023195 is determined by ML considering\nV2(t) =bV2(t). The resulting p-value was 0 .03 and we must also reject that C(t)\nbe constant.\nThus, we can conclude that, in group G1, the effect of Carboplatin on the\ndeath of cancer cells and the infinitesimal variability of relative volume of tumor\nis time-dependent. In a similar way, in group G2, the same comment can be\n35done about the combined effect of Carboplatin and Paclitaxel on the growth\nand death of cancer cells, as well as on the infinitesimal variability of relative\nvolume of tumor. Therefore, based on the tests carried out, we can conclude\nthat the therapies applied are time dependent, so the models (11) and (12) seem\nto be appropriated to describe the combined effect of the two therapies.\nAppendix A. Maximum likelihood estimates of the parameters of the\nprocess\nThe objective of this appendix is to provide the ML estimation of the pa-\nrameters of the process in model (2) for known C(t),D(t) and V(t) functions.\nIn addition we provide the ML estimation of each parameter for known val-\nues of the rest ones, which will be useful when establishing the null hypotheses\nH0:H(t) =h, with H(t) any of the functions in model (2).\nFrom (5) in Section 2.1, the transition pdf of the process can be obtained,\nresulting in\nX(t)|X(s) =y∼Λ1\u0002¯k(t|s) lny+θ(t|s), σ2Ω(t|s)\u0003\n(16)\nwhere\nθ(t|τ) =Zt\nτ\u0012\nα−C(s)−σ2\n2V(s)\u0013\n¯k(t|s)ds,\nΩ(t|τ) =Zt\nτ¯k2(t|s)V(s)ds\nLet us consider a discrete sampling {xij, i= 1, . . . , d ;j= 0, . . . , n i−1}of the\nprocess based on dsample paths at times tij, (i= 1, . . . , d, j = 0, . . . , n i−1)\nwith ti0=t0andxi0=x0,i= 1, . . . , d . Denote by X=\u0000\nXT\n1|···|XT\nd\u0001T, where\nXi= (Xi0, . . . , X i,ni−1)T,i= 1, . . . , d , with Xij=X(tij),j= 0, . . . , n i−1.\nBy taking X(t0) a degenerate random variable, i.e. P[X(t0) =x0] = 1, from\n(16), the probability density function of Xis\nfX(x) =dY\ni=1ni−1Y\nj=1exp \n−\u0002\nδij\nβ−θij\nξ\u00032\n2σ2Ωij\nβ!\nxijσq\n2πΩij\nβ\n36where δij\nβ= ln xij−¯kij\nβlnxi,j−1,θij\nξ=θ(tij|ti,j−1) and Ωij\nβ= Ω( tij|ti,j−1),\nwith ¯kij\nβ=¯k(tij|ti,j−1) = exp\u0010\n−Rtij\nti,j−1(β−D(s))ds\u0011\n,i= 1,2, . . . , d ,j=\n1, . . . , n i−1, and ξ= (α, β, σ2)T.\nThen, for a fixed value xof the sample and known C(t),D(t) and V(t)\nfunctions, the log-likelihood function is\nLx(ξ) =−nln(2π)\n2−nlnσ2\n2−Zβ+ Φ ξ−2Γξ\n2σ2−1\n2Υβ−dX\ni=1ni−1X\nj=1ln xij(17)\nwhere n=dX\ni=1(ni−1),Zβ=dX\ni=1ni−1X\nj=1(δij\nβ)2\nΩij\nβ, Φ ξ=dX\ni=1ni−1X\nj=1(θij\nξ)2\nΩij\nβ,\nΓξ=dX\ni=1ni−1X\nj=1δij\nβθij\nξ\nΩij\nβand Υ β=dX\ni=1ni−1X\nj=1lnΩij\nβ.\nIn order to obtain the ML estimate of α,βandσ2we denote\nΨl,m,p,q\nβ,ij=Ztij\nti,j−1(tij−s)l(C(s))m(V(s))p\u0000¯k(tij|s)\u0001qds,\nfrom which we deduce:\n∂Ψl,m,p,q\nβ,ij\n∂β=−qΨl+1,m,p,q\nβ,ij\nθij\nξ=αΨ0,0,0,1\nβ,ij−Ψ0,1,0,1\nβ,ij−σ2\n2Ψ0,0,1,1\nβ,ij\nΩij\nβ= Ψ0,0,1,2\nβ,ij\nThe likelihood equations are:\n∂Lx\n∂α=−1\n2σ2\u0012∂Φξ\n∂α−2∂Γξ\n∂α\u0013\n= 0\n∂Lx\n∂β=−1\n2σ2\u0012∂Zβ\n∂β+∂Φξ\n∂β−2∂Γξ\n∂β\u0013\n+1\n2∂Υβ\n∂β= 0\n∂Lx\n∂σ2=−1\n2σ2\u0012\nn−1\nσ2(Zβ+ Φ ξ−2Γξ) +∂Φξ\n∂σ2−2∂Γξ\n∂σ2\u0013\n= 0\nor equivalently, after calculus,\n2α Xβ\n1−2Xβ\n2−σ2Xβ\n3−2Xβ\n4= 0 (18)\nXβ\n5−Xβ\n6−Xβ\n7−Xβ\n8−Xβ\n9+ 2Xβ\n10+Xβ\n11+σ2\n2Xβ\n12= 0 (19)\nσ4\n4Xβ\n13+nσ2−(Zβ+α2Xβ\n1−2α Xβ\n2−2α Xβ\n4+Xβ\n14+ 2Xβ\n15) = 0 (20)\n37where\nXβ\n1=dX\ni=1ni−1X\nj=1\u0010\nΨ0,0,0,1\nβ,ij\u00112\nΨ0,0,1,2\nβ,ij, Xβ\n2=dX\ni=1ni−1X\nj=1Ψ0,1,0,1\nβ,ijΨ0,0,0,1\nβ,ij\nΨ0,0,1,2\nβ,ij,\nXβ\n3=dX\ni=1ni−1X\nj=1Ψ0,0,1,1\nβ,ijΨ0,0,0,1\nβ,ij\nΨ0,0,1,2\nβ,ij, Xβ\n4=dX\ni=1ni−1X\nj=1δij\nβΨ0,0,0,1\nβ,ij\nΨ0,0,1,2\nβ,ij,\nXβ\n5=dX\ni=1ni−1X\nj=1¯kij\nβ∆ijθij\nξlnxij\nΨ0,0,1,2\nβ,ij, Xβ\n6=dX\ni=1ni−1X\nj=1θij\nξφij\nξ\nΨ0,0,1,2\nβ,ij,\nXβ\n7=dX\ni=1ni−1X\nj=1\u0010\nθij\nξ\u00112\nΨ1,0,1,2\nβ,ij\n\u0010\nΨ0,0,1,2\nβ,ij\u00112, Xβ\n8=dX\ni=1ni−1X\nj=1\u0010\nδij\nβ\u00112\nΨ1,0,1,2\nβ,ij\n\u0010\nΨ0,0,1,2\nβ,ij\u00112,\nXβ\n9=dX\ni=1ni−1X\nj=1δij\nβ¯kij\nβ∆ijlnxij\nΨ0,0,1,2\nβ,ij, Xβ\n10=dX\ni=1ni−1X\nj=1δij\nβΨ1,0,1,2\nβ,ijθij\nξ\u0010\nΨ0,0,1,2\nβ,ij\u00112,\nXβ\n11=dX\ni=1ni−1X\nj=1δij\nβφij\nξ\nΨ0,0,1,2\nβ,ij, Xβ\n12=dX\ni=1ni−1X\nj=1Ψ1,0,1,2\nβ,ij\nΨ0,0,1,2\nβ,ij,\nXβ\n13=dX\ni=1ni−1X\nj=1\u0010\nΨ0,0,1,1\nβ,ij\u00112\nΨ0,0,1,2\nβ,ij, Xβ\n14=dX\ni=1ni−1X\nj=1\u0010\nΨ0,1,0,1\nβ,ij\u00112\nΨ0,0,1,2\nβ,ij,\nXβ\n15=dX\ni=1ni−1X\nj=1δij\nβΨ0,1,0,1\nβ,ij\nΨ0,0,1,2\nβ,ij,\nbeing ∆ ij=ti,j+1−tijandφij\nξ=−αΨ1,0,0,1\nβ,ij+ Ψ1,1,0,1\nβ,ij+σ2\n2Ψ1,0,1,1\nβ,ij.\n6. Conclusions\nMathematical modeling of tumor growth can help investigators to improve\nthe design of preclinical or clinical trials and to better predict treatment out-\ncome. Actually a comprehensive description of tumor dynamics during therapy\n38in preclinical setting allows to accurately compare different schemes of drug\nadministration. Ultimately, preclinical correlations between tumor growth dy-\nnamics during treatment and the efficacy of drug(s) could help to tailor the\nschedule to be proposed to patients in clinical trials.\nIn this paper a modified Gompertz diffusion process including exogenous\nfactors in its infinitesimal moments has been considered in order to model both\nthe effect of anti-proliferative and/or cell death-induced therapies. A procedure\nto estimate the parameters and time functions included in the model has been\nproposed, and our simulation studies show how said procedure adequately re-\nproduces both the parameters and the form of the functions involved, as well as\nthe mean and variance of the simulated data. In addition, from the estimated\nmodel, we have provided bootstrap tests about the form of the true functions\nin the model. The estimated functions C(t),D(t) and V(t) that finally result\nallow to understand how therapies affect tumor growth. Thus, our model could\nconstitute a valuable tool to adjust the drug administration scheme in the pre-\nclinical setting, in order to improve the efficacy of treatment, and to optimize\nthe schedule to be proposed to patients in clinical trials.\nAcknowledgments\nThe authors are very grateful to Dr. Didier Decaudin (Institute Curie, Paris)\nfor providing the data used in this research. They acknowledge the constructive\ncriticism of anonymous reviewers on the earlier version of this paper.\nThis work was supported in part by the Ministerio de Econom´ ıa, Industria y\nCompetitividad, Spain, under Grant MTM2017-85568-P and by the Consejer´ ıa\nde Econom´ ıa y Conocimiento de la Junta de Andaluc´ ıa, Spain under Grant\nA-FQM-456-UGR18 and by MIUR - PRIN 2017, Project “Stochastic Models\nfor Complex Systems” no. 2017JFFHSH. The authors Albano and Giorno are\nmembers of the research group GNCS-INdAM.\n39References\n[1] J. M. Steele, Stochastic Calculus and Financial Applications, Vol. 45 of\nStochastic Modelling and Applied Probability, Springer Science and Busi-\nness Media, 2012.\n[2] S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models,\nSpringer Science and Business Media, 2004.\n[3] L. J. S. Allen, An Introduction to Stochastic Processes with Applications\nto Biology, 2nd Edition, CRC Press, 2010.\n[4] D. J. Wilkinson, Stochastic modelling for quantitative description of het-\nerogeneous biological systems, Nature Reviews Genetics 10 (2009) 122–133.\ndoi:10.1038/nrg2509 .\n[5] H. Ahn, H. Moon, S. Kim, K. R. L., A newton-based approach for attribut-\ning tumor lethality in animal carcinogenicity studies, Computational Statis-\ntics and Data Analysis 38 (2002) 263–283. doi:10.1016/S0167-9473(01)\n00041-X .\n[6] A. Ochab-Marcinek, E. Gudowska-Nowak, Population growth and control\nin stochastic models of cancer development, Physica A: Statistical Mechan-\nics and its Applications 343 (15) (2004) 557–572. doi:10.1016/j.physa.\n2004.06.071 .\n[7] A. Talkington, R. Durrett, Estimating tumor growth rates in vivo, Bul-\nletin of Mathematical Biology 77 (2015) 1934–1954. doi:10.1007/\ns11538-015-0110-8 .\n[8] H. P. de Vladar, J. A. Gonzales, Dynamic response of cancer under the\ninfluence of immunological activity and therapy, Journal of Theoretical\nBiology 227 (2004) 335–348. doi:10.1016/j.jtbi.2003.11.012 .\n[9] L. Ferrante, S. Bompadre, L. Leone, M. P. Montanari, A stochastic formu-\nlation of the gompertzian growth model for in vitro bactericidal kinetics:\n40parameter estimation and extinction probability, Biometrical Journal 47 (3)\n(2005) 309–318. doi:10.1002/bimj.200410125 .\n[10] C. F. Lo, Stochastic gompertz model of tumor cell growth, Journal of The-\noretical Biology 248 (2) (2007) 317–321. doi:10.1016/j.jtbi.2007.04.\n024.\n[11] C. F. Lo, A modified stochastic gompertz model for tumor cell growth,\nComputational and Mathematical Methods in Medicine 11 (1) (2008) 3–\n11.doi:10.1080/17486700802545543 .\n[12] K. M. C. Tjørve, E. Tjørve, The use of gompertz models in growth anal-\nyses, and new gompertz-model approach: An addition to the unified-\nrichards family, PLoS ONE 12 (6) (2017) 1–17. doi:10.1371/journal.\npone.0178691 .\n[13] D. Yang, P. Gao, C. Tian, Y. Sheng, Gompertz tracking of the growth tra-\njectories of the human-liver-cancer xenograft-tumors in nude mice, Com-\nputer Methods and Programs in Biomedicine 191 (2020) 1055412. doi:\n10.1016/j.cmpb.2020.105412 .\n[14] E. Tjørve, K. M. C. Tjørve, A unified approach to the richards-model family\nfor use in growth analyses: Why we need only two model forms, Journal of\nTheoretical Biology 267 (3) (2010) 417–25. doi:10.1016/j.jtbi.2010.\n09.008 .\n[15] G. Albano, V. Giorno, P. Rom´ an-Rom´ an, F. Torres-Ruiz, On the effect of\na therapy able to modify both the growth rates in a gompertz stochastic\nmodel, Mathematical Biosciences 245 (1) (2013) 12–21. doi:10.1016/j.\nmbs.2013.01.001 .\n[16] V. Giorno, A. G. Nobile, Restricted gompertz-type diffusion processes with\nperiodic regulation functions, Mathematics 7 (6) (2019) 555. doi:10.3390/\nmath7060555 .\n41[17] G. Ascione, E. Pirozzi, On the construction of some fractional stochas-\ntic gompertz models, Mathematics 8 (1) (2020) 60. doi:10.3390/\nmath8010060 .\n[18] L. E. B. Cabrales, J. I. Montijano, M. Schonbek, A. R. S. Casta˜ neda, A\nviscous modified gompertz model for the analysis of the kinetics of tumors\nunder electrochemical therapy, Mathematics and Computers in Simulation\n151 (2018) 96–110. doi:10.1016/j.matcom.2018.03.005 .\n[19] E. Shakeri, G. Latif-Shabgahi, A. E. Abharian, Predictive drug dosage con-\ntrol through a fokker-planck observer, Computational and Applied Mathe-\nmatics 37 (3) (2018) 3813–3831. doi:10.1007/s40314-017-0542-x .\n[20] G. Albano, V. Giorno, A stochastic model in tumor growth, Journal of\nTheoretical Biology 242 (2) (2006) 329–336. doi:10.1016/j.jtbi.2006.\n03.001 .\n[21] G. Albano, V. Giorno, P. Rom´ an-Rom´ an, F. Torres-Ruiz, Inference on a\nstochastic two-compartment model in tumor growth, Computational Statis-\ntics and Data Analysis 56 (2012) 1723–1736. doi:10.1016/j.csda.2011.\n10.016 .\n[22] G. Albano, V. Giorno, P. Rom´ an-Rom´ an, F. Torres-Ruiz, Inferring the ef-\nfect of therapy on tumors showing stochastic Gompertzian growth, Journal\nof Theoretical Biology 276 (1) (2011) 67–77. doi:10.1016/j.jtbi.2011.\n01.040 .\n[23] G. Albano, V. Giorno, P. Rom´ an-Rom´ an, S. Rom´ an-Rom´ an, F. Torres-\nRuiz, Estimating and determining the effect of a therapy on tumor dy-\nnamics by a modified gompertz diffusion process, Journal of Theoretical\nBiology 364 (7) (2015) 206–219. doi:10.1016/j.jtbi.2014.09.014 .\n[24] P. Rom´ an-Rom´ an, S. Rom´ an-Rom´ an, J. J. Serrano-P´ erez, F. Torres-Ruiz,\nModeling tumor growth in the presence of a therapy with an effect on rate\ngrowth and variability by means of a modified gompertz diffusion process,\n42Journal of Theoretical Biology 407 (1) (2016) 1–17. doi:10.1016/j.jtbi.\n2016.07.023 .\n[25] L. J. H¨ o¨ ok, E. Lindstr¨ omb, Efficient computation of the quasi likelihood\nfunction for discretely observed diffusion processes, Computational Statis-\ntics and Data Analysis 103 (2016) 426–437. doi:10.1016/j.csda.2016.\n05.014 .\n[26] K. Ignatieva, E. Platen, Estimating the diffusion coefficient function for a\ndiversified world stock index, Computational Statistics and Data Analysis\n56 (2012) 1333–1349. doi:10.1016/j.csda.2011.10.004 .\n[27] A. Mandal, W. T. Huang, S. K. Bhandari, A. Basu, Goodness-of-fit testing\nin growth curve models: A general approach based on finite differences,\nComputational Statistics and Data Analysis 55 (2011) 1086–1098. doi:\n10.1016/j.csda.2010.09.003 .\n[28] A. C. Guidoum, K. Boukhetala, Sim.diffproc: Simulation of\ndiffusion processes. r package version 4.6, https://CRAN.R-\nproject.org/package=Sim.DiffProc (2020).\n[29] S. M. Iacus, Simulation and inference for stochastic differential equations,\nSpringer, 2008.\n[30] P. Rom´ an-Rom´ an, S. Rom´ an-Rom´ an, J. J. Serrano-P´ erez, F. Torres-\nRuiz, Fitting real data by means of non-homogeneous lognormal diffu-\nsion processes, Statistics and Its Interface 10 (4) (2017) 585–600. doi:\n10.4310/SII.2017.v10.n4.a5 .\n[31] S. Sheather, M. Jones, A reliable data-based bandwidth selection method\nfor kernel density estimation, Journal of the Royal Statistical Society series\nB 53 (1991) 683–690. doi:10.1111/j.2517-6161.1991.tb01857.x .\n[32] R. Ozols, B. Bundy, B. G. et al., Phase iii trial of carboplatin and paclitaxel\ncompared with cisplatin and paclitaxel in patients with optimally resected\n43stage iii ovarian cancer: A gynecologic oncology group study, Journal of\nClinical Oncology 21 (17) (2003) 3194–3200. doi:10.1200/JCO.2003.02.\n153.\n44" }, { "title": "2401.15388v1.On_sets_where_lip_f__is_infinite_for_monotone_continuous_functions.pdf", "content": "ON SETS WHERE lipfIS INFINITE\nFOR MONOTONE CONTINUOUS FUNCTIONS\nMARTIN RMOUTIL AND THOMAS ZÜRCHER\nAs a teacher and coauthor, our beloved colleague Jan Malý helped to shape our\nmathematical and world view. He is sorely missed. With admiration, we dedicate\nthis paper to him.\nAbstract. For a function f:R→R, we let\nlipf(x) = lim inf\nr→0+sup\ny∈[x−r,x+r]|f(y)−f(x)|\nr.\nGivenany FσδsetA⊆RofLebesguemeasurezero, weexplainhow\nto obtain a nondecreasing absolutely continuous function g:R→\n[0,1]such that g′(x) =∞for every x∈A, and lipg(x)<∞for\nevery x /∈A.\n1.Introduction\nIn [Rad19], H. Rademacher proved that Lipschitz functions between\nEuclideanspacesaredifferentiablealmosteverywhere, seeSatzI.Aswe\nlook at functions on the real line, we mention that the one-dimensional\ncase (actually for the larger class of functions of bounded variation)\nis due to Lebesgue, see page 128 in [Leb04], republished in [Leb09].\nA strengthening of Rademacher’s result is by W. Stepanov in [Ste23],\nbut before detailing it, let us introduce some notation. Although our\nsetting is the real line, the following definition is of a metric flavour,\nand hence we give it for metric spaces.\nDefinition 1.1. Let(X, d X)and(Y, d Y)be metric spaces and f:X→\nYbe a mapping. Then we define\nlipf(x):= lim inf\nr→0+sup\ny∈B(x,r)dY(f(y), f(x))\nr,\nLipf(x):= lim sup\nr→0+sup\ny∈B(x,r)dY(f(y), f(x))\nr.\nIn case we have X, Y⊆R, we sometimes replace B(x, r)in the above\nformulae by (x−r, x]and[x, x+r). We indicate this by using Lip(x−)\nandLip(x+), respectively and the same for lip.\nStepanov proved that functions f:Rm→Rnare differentiable at\nalmost every point in the set where Lipfis finite. We also would\nDate: January 30, 2024.\n1arXiv:2401.15388v1 [math.CA] 27 Jan 20242 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nlike to mention the slick proof of this statement by J. Malý in [Mal99].\nZ. Balogh and M. Csörnyei showed in [BC06] that there are functions f\nthat fail to be differentiable at almost every point in the set where\nlipfis finite, see Theorems 1.3 and 1.4 in their paper highlighting\ntwo different issues. However, they also showed in Theorem 1.2 that\nStepanov-type theorems still hold if the integrability of lipand the\nsize of the set where lipis infinite satisfy certain carefully balanced\nrestrictions.\nWhile these results shed light on the size of the set where lipis\ninfinite and its connection to differentiability, in the current paper, we\nare interested to know more about the structure of this set. This is in\nthe tradition of studying the structure of the set where the derivative of\na function is infinite. Note that lipf(x) =∞whenever f′(x) =∞. A\ntestament to the inquiry of the sets where the derivative is infinite are\nfor example the two papers (written in German) by V. Jarník, [Jar33],\nandbyZ.Zahorski, [Zah41]. JarníkprovedinSatz3thatgivena Gδset\nG⊂Rof measure zero, there is a nondecreasing continuous function\nf:R→Rsuch that f′(x) =∞forx∈Gand all Dini derivatives are\nfinite for x /∈G. Zahorski found such a function that has an infinite\nderivative at each point in Gand a finite derivative at each point in\nthe complement of G.\nThe analysis for Lipis much easier to do than the one for lip. For\nexample, Theorem 3.35 and Lemma 2.4 in [BHRZ19] show that, given\na set A⊆R, there is a continuous function f:R→Rsatisfying A=\n{x∈R: Lip f(x) =∞}if and only if Ais aGδset. The just mentioned\nLemma 2.4 also states that the set where lipis infinite for a continuous\nfunction is an Fσδset. We conjecture that whenever Ais an Fσδset\nin the real line, there exists a continuous function f:R→Rsuch that\nA={x∈R: lipf(x) =∞}. In [BHRZ19], such functions are given in\ncaseAis an Fσset. In this paper, we replace the assumption that Abe\nanFσsetby Abeingan Fσδsetofmeasurezero. Moreover, thefactthat\nthe set has vanishing measure enables us to find appropriate functions\nthat are not only continuous but actually absolutely continuous and\nnondecreasing.\nTo close the introduction, we look at some further papers connected\nto our research. One part of B. Hanson’s Theorem 1.3 in [Han21]\ntells us that if E⊂Ris aGδset with measure zero, then there ex-\nists a continuous, monotonic function f:R→Rsuch that E={x∈\nR: Lip f(x) =∞}and{x∈R: lipf(x) =∞}=∅. Moreover fmay\nbe constructed so that lipf(x) = 0for all x∈E. Theorem 1.2 deals\nwith the case where fis not monotonic.\nA similar topic as the one in our paper is the study of sets Esuch\nthat there is a continuous function fsatisfying lipf(x) =χE(x)(or\nLipf(x) =χE). Werefertheinterestedreadertothepapers[BHMV20a,\nBHMV20b, BHMV21], where characterizations of such sets are found.ON SETS WHERE lipfIS INFINITE 3\n2.The statement of the main result and an explanation\nof the strategy of its proof\nTheorem 2.1 (main result) .Assume that A⊂Ris an Fσδset of\nmeasure zero, then there exists a nondecreasing absolutely continuous\nfunction g:R→Rsuch that g′(x) = lip g(x) =∞for every x∈Aand\nlipg(x)<∞for every x /∈A.\nWe already mentioned the results by Jarník and Zahorski in the\nintroduction. Our proof actually makes use of the function that Jarník\nfound. Moreover, our overall proof strategy shows similarities to the\none employed by Jarník. Neither Jarník nor Zahorski state that the\nconstructed function is absolutely continuous.\nLet us give a brief description of Jarník’s construction with an inter-\nwovenargumentwhytheconstructedfunctionisabsolutelycontinuous.\nHestartswithanarbitrary GδsetGwrittenascountableintersection\nof open sets On. In the next step, he replaces the open sets by better\nsuited open sets Un. Following this, he defines fk(x) =L1((−∞, x)∩\nUk). Finally he adds up all the functions fkto obtain a function f\nhaving the claimed properties. That f′(x) =∞forx∈Gfollows since\nf′\nk(x) = 1for all such x. Note that f′\nk(x) = 0forxoutside the closure\nofUk. The sets Ukare chosen so small thatP∞\nk=1∥fk∥1<∞. This\nguarantees that fis absolutely continuous, see Lemma 3.1.\nThat the Dini derivatives are finite outside Gneeds the cleverly\nchosen sets Ukand some careful computations and estimates.\nBehind the choice of fklies the fact that given a set A, setting\nf(x) =Rx\n0χAguarantees that f′(x) = 1, whenever xis a density point\nofA.\nIn Jarník’s case, the main reason for the modification of the sets On\nto the sets Unis to make sure that the points outside the intersection of\nall these open sets are such that the function has finite derivative there;\nin our case we also need to focus on the points in the intersection. In\nsome sense, we need to transform these points into density points.\nHaving reviewed Jarník’s proof strategy, it is now time to talk about\nthe ideas behind the results in our paper.\nThe main part of our main result is covered by Lemma 4.5, which\nis almost our main result Theorem 2.1, but contains the additional\nassumption that the FσδsetAbe meagre. A brief outline and an\nexplanation of the proof follow.\nThedesiredfunction gwillbeconstructedintheformP∞\nk=1gk,where\neach gkwill be carefully crafted to have the desired properties. We\nrecord the creation of these functions in Lemma 4.4 and its proof al-\nlowing for particular choices of the parameters of the lemma, especially\nthe sets E, F, H.4 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nFirst we express the set Ain a more convenient form, namely we find\nclosed sets Fk(k∈N) such that Ais exactly the set of points belonging\ntoinfinitelymany Fk’s. Moreover, thesets Fkarechosensothatforany\nindices k, l∈Nwith k < l, ifFk∩Fl̸=∅, then Fl⊆Fk. Our method\nof proof relies heavily on these properties. It is useful to note that\nsuch an arrangement of closed sets is essentially a level-disjoint Suslin\nscheme. The disjointness and nestedness are essential for our method,\nand they can be achieved thanks to the assumption that Abe meagre;\nwithout it, we can run into problems: For example, let A= (0,1)and\nA=TLnwith Lnof the type Fσfor every n, and Ln⊆Lmwhenever\nn > m. Then for some n0and any n⩾n0we have 0,1/∈Ln, and\nby intersecting with [0,1], we may assume also that Ln⊆[0,1], so\nLn= (0,1). But then, by a classical result of W. Sierpiński [Sie18], Ln\ncannot be expressed as the union of countably many pairwise disjoint\nclosed sets.\nTo resume our thoughts about the Suslin scheme, we note that in\nparticular, the sets Fkare naturally arranged (by inclusion) in a tree.\nAny vertex of this tree (i.e. any of the sets Fk) can be seen as the root of\na subtree, which we (for the purposes of this explanatory remark) call a\nfamily of (all descendants of) the set Fk. Clearly, if we pick two indices\nk, lsuch that Fk∩Fl=∅, then the corresponding subtrees (families)\nare also disjoint, and any member of one subtree is disjoint from any\nset of the other. The two families, i.e. the one started by Fkand the\none started by Fl, can thus be seen as “unrelated”. For the purposes of\nthis remark, we shall use the words “descendant” and “ancestor” in the\nobvious sense of the tree order, while the words “previous” or “past”,\nand “later” or “future” will refer simply to the order of indices; that is,\nifj < k, then Fjis previous to FkandFkis future to Fj.\nNext we use the fact that Ais Lebesgue null to find pairwise disjoint\nmeasurable sets Mkcontained in the complement AcofA, each with\npositive measure in every nonempty open interval in R. These are\nused in Lemma 4.3 to obtain the compact sets Hksatisfying Fk⊆\nHk⊆Fk∪Mk; the set Hkis where the function gkis later allowed\nto grow (see Lemma 4.4 (i)). Requirement (ii) in Lemma 4.4 on the\nfunction makes it clear that the set Fkitself is not sufficient for this\npurpose and must be enlarged.\nNowwemaketheimportantsteptodefinetheclosedsets# »Hk=SHj\nwhere the union is over all jwith Fj⊆Fk, i.e. over all the family of Fk.\nIt can be seen from the definition that the ordering by inclusion of the\nsets# »Hkis the same as that of the sets Fkin the sense that if Fk⊆Fl,\nthen also# »Hk⊆# »Hl. However,# »Hkand# »Hjneed not be disjoint, even if\nFkandFjare, a fact that is a source of some complications. The set# »Hkcontains Fk(as well as all of its descendants, of course) as its “core”\nand it also covers the whole space in which any of the corresponding\ndescendant functions (i.e. all gjforjsuch that Fj⊆Fk) will be allowedON SETS WHERE lipfIS INFINITE 5\nto grow. The notation tries to convey that# »Hktakes responsibility for\nthe whole future of the family.\nWe also define the closed sets Ek=S# »Hjwhere the union is over\nallj < k(so the union is already clear to be over finitely many sets)\nwith Fj∩Fk=∅, that is, over “ previous families unrelated to that of\nFk”. The union is finite, so only finitely many families are involved; on\nthe other hand, each of them is involved as a whole because we use the\nsets# »Hj(and not just Hj) in the definition of Ek.\nAtthispointwearefinallyreadytoapply, foreach k∈N, Lemma4.4\nwith E, F, Hreplaced by Ek, Fk, Hk, and obtain the functions gk. We\nmay adopt the “family terminology” also for the functions, e.g. gjis a\ndescendant of gkifFj⊆Fk.\nTheroleofthesets EkiskeyandislargelyrevealedbyLemma4.4(iv):\nloosely speaking, gkis forced to “behave nicely” in the vicinity of Ek.\nButEkis the set where the previous unrelated functions (i.e. with\nsmaller indices from different families), as well as all their descendants,\nare allowed to grow. This means that gkis chosen so carefully that it\ndoes not “provide unsolicited support to the achievements of previous\nunrelated functions”; simply put, the growths of all the functions gkdo\nnot add up too much where they should not, and this allows us later\nto prove that for g:=P∞\nk=1gkwe have lipg(x)<∞whenever x /∈A.\nAnother way to put this is as follows: For any two unrelated functions,\nsaygjandgkwith j < k, we have Hj⊆Ek(even# »Hj⊆Ek) and so gk\nmust “behave nicely” close to where gjgrows. That is, the later of the\ntwo unrelated functions is taking responsibility for the control we need.\nOf course, this is just a very rough general idea. The remainder of the\nproof is to show precisely that gindeed enjoys the desired properties.\nIt is easy to show that g′(x) =∞at each point x∈A, asxbelongs\ntoFkfor infinitely many indices k, and for such kwe have g′\nk(x) = 1by\nLemma 4.4 (iii); a notable condition in (iii) is that x /∈Ek, and it must\ntherefore be shown that this is the case whenever x∈A∩Fk. This is\nnot as trivial as it might seem at a first glance because – as mentioned\nabove – the sets# »Hkand# »Hjare not necessarily disjoint even if Fkand\nFjare. But it follows from the construction that these intersections are\ncontained in the sets Mk, which are all disjoint from A; since x∈A,\nwe indeed get that x /∈Ek.\nAssume, now, that x /∈A. Showing that this implies lipg(x)<∞is\nmore involved, and it seems to require the careful preparation above.\nSince x∈Fkfor finitely many k, one can easily show the same also\nfor the sets# »Hk, so let lbe the largest index with x∈# »Hl. Since\nLipgk(x)<∞for every k, we do not have to care about finitely many\nsummands gk. Hence, we only look at indices k > l, in particular,\nindices ksuch that x /∈# »Hk. So let k > l. Ifx∈Ek, then any interval6 MARTIN RMOUTIL AND THOMAS ZÜRCHER\ncontaining xmeets Ek, andthefunction gksatisfiesthestrongestimates\nLemma 4.4 (iv); this takes care of all gkforksuch that x∈Ek.\nThe core of our argument deals with the set of indices J:={k >\nl:x /∈Ek}.We set h=P\nj∈Jgjand want to prove that liph(x)<∞.\nSo we wish to find a suitable decreasing sequence of radii (rp)∞\np=1, one\nthat witnesses that the lower limit in the definition of liph(x)is finite.\nWe define the radii as follows: let j1∈ Jbe minimal such that# »Hj1\nmeets the interval (x−1, x+ 1), and r1= dist( x,# »Hj1). Next, let\nj2∈ Jbe minimal such that# »Hj2meets (x−r1, x+r1), and set r2=\ndist(x,# »Hj2). We continue this process; if it stops after finitely many\nsteps, it means that his constant on an open neighbourhood of x.\nSimilarly, if R:= lim p→∞rp>0we easily reach the same conclusion:\nhis constant on (x−R, x+R).\nThe main case is when limp→∞rp= 0. To treat it, we fix an arbitrary\nj∈ Jand aim to estimate the oscillation of gjon every interval Ip:=\n(x−rp, x+rp)forp∈N. So we fix a p∈N. IfIp∩Hj=∅, then\ngjis constant on Ip, so assume Ip∩Hj̸=∅. Then we obtain that\nj > j p, and it also follows that gjbelongs to an unrelated family, i.e.\nFj∩Fjp=∅. Thus, by the definition of Ej, we have that Hjp(even# »Hjp) is contained in Ej. (Indeed, Ej“looks at past unrelated families”;\nbutjpis the “past” as j > j p.) But the content of Lemma 4.4 (iv) is to\nprovide an oscillation estimate for gjon any interval that meets Ej, in\nparticular on the interval Ipthat clearly meets Ej, even# »Hjp. Summing\noverj∈ Jwe find that also h“behaves nicely” on Ip, for any p. Thus\nwe obtain that liph(x)<∞.\nTheprecedingparagraphdescribesthecentralargumentoftheproof.\nIt reveals the reason for using the sets# »Hk, especially in the definition\nofEk, instead of just Hk: without this trick we would not be able to\nprove that j > j p. Therefore we would not necessarily have that the\nappropriate set (in this case that would be Hjp) is contained in Ej, and\nin turn, gjwould not be guaranteed to “behave nicely” in Ip.\n3.Preliminaries\nHere we list some of the notation and conventions that we use, be-\nsides the notions of lipandLipintroduced in Definition 1.1. Through-\nout the paper, we work with the real line R, functions from RtoR, the\nLebesgue measure on Retc. We follow the convention 0/∈N, and we\nuse the term “countable” for “at most countable”. We use the standard\nnotation (a, b)for open intervals, and [a, b]for closed intervals in R.\nWhen we talk about intervals, then we tacitly assume that they are\nnontrivial, i.e. a < b. Given x∈Randr > 0, by B(x, r)we mean\nthe usual open ball (with respect to the usual metric on R), i.e. the\nopen interval (x−r, x+r); of course, any bounded open interval is an\nopen ball, a fact we will occasionally use without further explanation.ON SETS WHERE lipfIS INFINITE 7\nForx∈RandA⊆R, we write dist(x, A):= inf{|x−y|:y∈A},\nthe distance of xfrom A. Given any set A⊆R, we denote its comple-\nmentR\\AbyAc, its closure by Aand its boundary A∩Acby∂A; since\nwe only use the notion of boundary for intervals, we could equivalently\nsay that, for an interval I,∂Iis the set of its (at most two) endpoints.\nFor a Lebesgue measurable (or just “measurable”), A⊆Rwe use the\nsymbol |A|to denote its Lebesgue measure, or |A|=L1(A). A set in R\nisnowhere dense if its closure has empty interior; a set A⊆Ris said to\nbemeagreif it can be written as the union of countably many nowhere\ndense sets.\nFor any function f:R→Rand a set U⊆R, we denote by osc(f, U)\ntheoscillation offoverU, i.e. osc(f, U):= sup{|f(x)−f(y)|:x, y∈\nU}. However, we use this notation exclusively for nondecreasing func-\ntions fand intervals U, soosc(f, U)is just the increment of foverU.\nThe support of fis the set supp( f):={x∈R:f(x)̸= 0}. We de-\nnote the right (resp. left) derivative of fatxbyf′\n+(x)(resp. f′\n−(x)).\nWe write ∥f∥1=R\nR|f|for the L1-norm of f; the supremum norm is\n∥f∥∞= supx∈R|f(x)|. Given a set A⊆R, the symbol 1Adenotes the\ncharacteristic function of A.\nWeshallbeusingsomewell-knownfactsaboutabsolutelycontinuous\nfunctions, some of which are arranged into the following lemma.\nWe do not give the detail of its proof; it can be proved by a combina-\ntion of Fubini’s theorem about interchanging the order of summation\nand differentiation (see for example Theorem 1.4.1 in [KK96]) and the\nmonotone convergence theorem.\nLemma3.1. Letfk:R→Rbe locally absolutely continuous and mono-\ntone increasing such that f:=P∞\nk=1fkexists. Then the function fis\nlocally absolutely continuous.\nIf moreoverP∞\nk=1∥f′\nk∥1<∞, then fis absolutely continuous. A\nspecial case is if φ∈L1, and g:R→Ris defined by g(x) =Rx\n−∞φ(t) dt,\nthen gis absolutely continuous.\nWe shall also need the following simple lemma. We shall be using\nthe term disjoint Kσ(which we abbreviate DK σ) for any set that can\nbe expressed as the union of countably many pairwise disjoint compact\nsets.\nLemma 3.2. Any meagre Fσ-set in RisDK σ.\nProof.Letusfirstmaketwosimpleobservations, providingaproofonly\nfor the second one:\n•A countable union of pairwise disjoint DK σ-sets is itself DK σ.\n•LetF⊆Rbe a nowhere dense closed set, and I= (a, b)⊆R\n(a, b∈R∪{−∞ ,∞}) be an open interval. Then F∩IisDK σ.\nThe first observation is obvious. To prove the second one, we use the\nnowhere denseness of Fto find an increasing sequence (xn)∞\nn=−∞in8 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nI\\Fsuch that limn→−∞ xn=aandlimn→∞xn=b. Then we have the\nfollowing expression of F∩I, which makes it apparent that F∩Iis\nDK σ:\nF∩I=F∩(a, b) =∞[\nn=−∞[xn, xn+1]∩F.\nHaving taken care of the two observations, we take an arbitrary\nmeagre set B⊆Rof the type Fσ. Then Bcan be written asS∞\nn=1Fn\nwhere all Fnare compact. Moreover, each Fnis also nowhere dense:\nIndeed, if Fnwerenotnowheredense, thenitwouldcontainanontrivial\nopen interval, which would make Bnonmeagre by the Baire category\ntheorem.\nWe can express Bas the following disjoint union\nB=F1∪(F2\\F1)∪(F3\\(F1∪F2))∪ ··· =∞[\nn=1\u0012\nFn\\n−1[\nk=1Fk\u0013\n,\nso (by the first observation) it suffices to show, given a natural number\nn⩾2, that Fn\\Sn−1\nk=1FkisDK σ. To that end, define Ito be the set\nof all components of R\\Sn−1\nk=1Fk; then the elements of Iare pairwise\ndisjoint open intervals. Now we have\nFn\\n−1[\nk=1Fk=[\nI∈I(Fn∩I),\nwhere the union on the right-hand side is clearly disjoint, and each of\nthe sets Fn∩IisDK σby the second observation. Hence, the first\nobservation implies Fn\\Sn−1\nk=1Fkto be DK σas required. The proof is\ncomplete. □\n4.Proofs\nIn this section, we gradually build towards a proof of Theorem 2.1.\nAlthough the longest proof is that of Lemma 4.4 as we need to prove\nthe function gconstructed therein has many particular properties, the\nmain ideas are contained in the proof of Lemma 4.5, which is essentially\nthe same as the main theorem but contains the extra assumption that\nAbe meagre. Getting rid of meagreness is then a simple task.\nNotation 4.1. Given a closed set F⊆R, we denote\n(F)ε={x∈R: dist( x, F)< ε}and\nbF=F∪∞[\nn=1n\nx∈R: dist( x, F) =1\nno\n.\nIn the next lemma, we say that a subset of the real line has every-\nwherepositivemeasure(EPM)ifititsintersectionwitheverynonempty\nopen interval has positive Lebesgue measure.ON SETS WHERE lipfIS INFINITE 9\nLemma 4.2. LetM⊆Rhave EPM. Then there are disjoint subsets\nM1, M 2⊆M, both having EPM.\nProof.Let(In)∞\nn=1be a basis of open sets in Rconsisting of open in-\ntervals. By the regularity of the Lebesgue measure, we may choose\ndisjoint compact sets K1, L1⊆I1∩Mof positive Lebesgue measure;\nwe can also assume them to be nowhere dense: indeed, if e.g. K1were\nnot, then it would contain a nontrivial interval Jas it is closed. We\nwould then replace K1by a “fat Cantor set” contained in J.\nNow, assume that the nowhere dense compacta K1, L1, . . . , K n, Ln\nhave already been constructed. Then K:=Sn\ni=1(Ki∪Li)is nowhere\ndense, and so In+1\\Kcontains a nonempty open interval, say eIn+1.\nAgain, choosedisjointnowheredensecompactsets Kn+1, Ln+1⊆eIn+1∩\nMof positive measure.\nSetting M1=S∞\nn=1KnandM2=S∞\nn=1Ln, it is easy to observe that\nM1, M 2⊆M, we have M1∩M2=∅, and that both sets have EPM. □\nLemma 4.3. Suppose F⊆Ris closed and M⊆Ris a measurable set\nthat meets every interval in a set of positive measure. Then for every\nε >0there is a closed set Hsuch that\n(1)F⊆H⊆(F∪M)∩(F)ε;\n(2)Hmeets the middle third of every component of bFc∩(F)εin a\nset of positive measure.\nProof.Assume F̸=∅; the statement is trivial otherwise. The set bFis\neasily seen to be closed, so if Jis an arbitrary component of bFc∩(F)ε,\nit is an open interval. Given any such J=B(c, r), the regularity of\nthe Lebesgue measure permits us to choose a compact set HJwith\n|HJ|>0and\n(4.1) HJ⊆B\u0010\nc,r\n3\u0011\n∩M⊆J∩M;\nin particular, HJis contained in the middle third of J. We define\n(4.2) H:=F∪[\nJHJ,\nwhere the union is over all components JofbFc∩(F)ε; then (1) and (2)\nare obviously satisfied (by the choice of HI).\nWe are left to show that His closed. Pick an arbitrary x /∈H.\nThen x /∈F, so there exists a component (a, b)ofFc(with a, b∈\nR∪{−∞ ,∞})containing x. Weset δ0:=1\n2dist(x, F), whichispositive\nasFis closed, and we have\na < a +δ0⩽x−δ0< x < x +δ0⩽b−δ0< b.10 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nLetJbe the family of all the components of bFc∩(F)εthat meet the\ninterval (a+δ0, b−δ0). Then clearly\n(4.3) (a+δ0, b−δ0)∩\u0012\nF∪[\nJ /∈JHJ\u0013\n=∅.\nIt is easy to see that Jis finite (including the cases when aorbis\ninfinite). Hence,\nC:=[\nJ∈JHJ\nis closed. Since x /∈HandC⊆H, we have x /∈Cimplying that\nδ1:= dist( x, C)>0. We set δ:= min {δ0, δ1}; then B(x, δ)∩C=∅.\nTogether with (4.3), this shows that B(x, δ)∩H=∅, concluding the\nproof. □\nLemma 4.4. Suppose E, F⊆Rare closed, A, M⊆Rare measurable,\n|A|= 0andMmeets every interval in a set of positive measure. Let\nε >0andHbe the closed set from Lemma 4.3:\n(1)F⊆H⊆(F∪M)∩(F)ε;\n(2)Hmeets the middle third of every component of bFc∩(F)εin a\nset of positive measure.\nThen there is a nondecreasing absolutely continuous function g:R→\n[0, ε]such that\n(i)supp( g′)⊆H;\n(ii)Lipg(x)<∞for every x∈R;\n(iii)g′(x) = 1for every x∈A∩F∩Ec;\n(iv)osc(g, U)⩽ε|U|whenever Uis an interval meeting E;\n(v)∥g′∥1< ε.\nProof.Without loss of generality we may assume that all components\nofEcare bounded: Indeed, since Ahas measure zero, we may find\na strictly increasing sequence (xk)∞\nk=−∞without accumulation points\nsuch that none of the xk’s lies in Aand limk→−∞ xk=−∞and\nlimk→∞xk=∞. It is now clear from the statement of the lemma, that\nif we prove it with Ereplaced by the (obviously closed) set E∪{xk:k∈\nZ}, it will also be proved for E. Hence, there is no loss in generality in\nthe assumption, which we shall adopt, that the complement of Eis a\ncountable union of bounded open intervals.\nLetIbe the collection of all components IofEc. Denote eA=\nA∩F∩Ec, and let G0⊆Rbe an open set such that eA⊆G0⊆(F)ε\nand|G0|< ε.\nPick any I∈ I; then I= (a, b)is a bounded open interval. Let us\nchoose a sequence (Gn\nI)∞\nn=1of open subsets of Isatisfying, for every\nn∈N, the following conditions (as |eA|= 0, we can find such sets):\n•eA∩I⊆Gn\nI⊆G0∩I;ON SETS WHERE lipfIS INFINITE 11\n•(a+|I|\nn, b−|I|\nn)⊆Gn\nI;\n•\f\f\fGn\nI∩(a, a+|I|\nn)\f\f\f<ε\n4·|I|\nn+1;\n•\f\f\fGn\nI∩(b−|I|\nn, b)\f\f\f<ε\n4·|I|\nn+1.\nSetGI:=T∞\nn=1Gn\nI; we show that GIis open: Indeed, given any x∈\nGI⊆I= (a, b), there clearly exist δ1>0andn0∈Nsuch that\nfor all n⩾n0we have B(x, δ 1)⊆(a+|I|\nn, b−|I|\nn)⊆Gn\nI. Of course,\nthe sets Gn\nI,n∈ {1, . . . , n 0−1}, are all open and contain x, so there\nis some δ2>0such that B(x, δ 2)⊆Tn0−1\nn=1Gn\nI, and it follows that\nB(x,min{δ1, δ2})⊆GI.\nClaim:For any interval U⊆I= (a, b)with ∂U∩∂I̸=∅we have\n(4.4) |GI∩U|<ε\n4|U|.\nFor the proof of the claim, suppose U= (a, c)for some c∈(a, b]; the\nother case can be dealt with using a symmetric argument. Take the\nunique n∈Nwith\na+|I|\nn+ 1< c⩽a+|I|\nn.\nThe following estimates prove the claim:\n|GI∩U|⩽|Gn\nI∩U|⩽\f\f\f\fGn\nI∩\u0010\na, a+|I|\nn\u0011\f\f\f\f<ε\n4·|I|\nn+ 1<ε\n4(c−a) =ε\n4|U|.\nHaving finished the above construction for every I∈ I, we now set\nG:=S\nI∈IGI. Note that\n(4.5) eA⊆G⊆Ec∩G0⊆(F)ε\nand|G|< ε; indeed, eA∩I⊆GIfor all I∈ I,eA⊆Ec, and Ec=SI;\nmoreover, |G0|< εandG0⊆(F)ε.\nNext we set (following Notation 4.1)\n(4.6) bF:=F∪∞[\nn=1n\nx∈R: dist( x, F) =1\nno\n.\nClearly, bFis closed as Fis closed. Further, we define\n(4.7) Jto be the set of all components Jof(bF)cwith J⊆G;\nobserve that the set bF∪SJcontains eAin its interior. Further, each\nJ∈ Jis contained in some I∈ IasJis connected and J⊆G⊆Ec,\nso it is contained in some component of Ec. Now, for every interval\nJ=B(c, r)∈ J, choose a bounded measurable function φJ⩾0\nsupported in H∩B(c,r\n3)such thatR\nJφJ=|J|; this is possible thanks\nto the fact that Hmeets the middle third of every J∈ Jin a set\nof positive measure, as follows from Lemma 4.3 (2) (note that every12 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nJ∈ Jsatisfies J⊆G⊆(F)ε). Remembering that the sets GIare\npairwise disjoint and that their union is by definition G, we define\n(4.8)\nφ:=X\nJ∈JφJ+X\nI∈I1F∩GI=X\nJ∈JφJ+1F∩Gand g(t):=Zt\n−∞φ.\nWe show ghas the desired properties, starting with (v). First we\nobserve that φis integrable; indeed, all the summands in the definition\nofφare nonnegative, so the monotone convergence theorem yields\nZ\nRφ=X\nJ∈JZ\nRφJ+Z\nR1F∩G=X\nJ∈J|J|+|F∩G|⩽|G|,\nwhere the last inequality follows from the fact that the intervals J∈\nJ, together with F∩G, are pairwise disjoint subsets of G. As φis\nnonnegative, gisnondecreasing. Since |G|< ε,wehave g:R→[0, ε],as\nrequired. Moreover, Lemma 3.1 implies that gis absolutely continuous.\nTherefore, g′=φalmost everywhere, whence ∥g′∥1=R\nR|g′|=R\nR|φ|=R\nRφ⩽|G|< ε, implying (v).\nWe show (i). By their definitions, all of the summands in (4.8) are\nsupported in the closed set H. Therefore, given any x /∈H, there exists\nδ >0such that φ= 0inB(x, δ); that obviously makes gconstant on\nB(x, δ), whence x /∈supp( g′). Thus is proved (i).\nTo prove (ii), we pick an arbitrary x∈R. If x /∈H, then gis\nconstant on a neighbourhood of xby the above, so Lipg(x) = 0; hence,\nwe assume x∈H. We divide the task into two parts by setting\ng1(t):=Zt\n−∞X\nJ∈JφJ,\ng2(t):=Zt\n−∞1F∩G;\nwe then have g=g1+g2and since Lipg(x)⩽Lipg1(x) + Lip g2(x), it\nsuffices to prove that Lipg1(x)andLipg2(x)are both finite.\nWe start with g1; recall that we are in the case where x∈H. If\nx∈Jfor some J∈ J, then for any y∈Jwe have\n|g1(y)−g1(x)|=\f\f\f\fZy\nxφJ\f\f\f\f⩽|y−x|∥φJ∥∞.\nAsφJis bounded, this shows that Lipg1(x)⩽∥φJ∥∞<∞.\nIf, on the other hand, x /∈Jfor any J∈ J, then we observe what\nhappens on either side (i.e. left or right) of x. If there is a δ >0such\nthat (x, x+δ)∩SJ=∅, then we obviously get Lipg1(x+) = 0. Hence,\nlet us assume xis approximated from the right by intervals in J, and\ntake an arbitrary y > x.ON SETS WHERE lipfIS INFINITE 13\nAssume first that y /∈SJ. Taking the sum over all intervals J∈ J\nthat are contained in (x, y), we have\n(4.9) |g1(y)−g1(x)|=X\n|J|⩽|y−x|.\nIf, on the other hand, there is J∈ Jwith y∈J, then there are two\ncases: (a) yis in the open left third of J, or (b) not. Let z= inf J;\nby the assumption on x, we have x⩽z.\nAssume (a) is the case. As φJis supported in the middle third of J,\nwe see that g1(z) =g1(y). Using this and the estimate (4.9) with y\nreplaced by z, we obtain\n|g1(y)−g1(x)|=|g1(z)−g1(x)|⩽|z−x|<|y−x|.\nIfthecasethatoccursis(b),then |y−z|⩾|J|\n3,andclearly |g1(y)−g1(z)|⩽\n|J|by the choice of φJ; hence\n|g1(y)−g1(z)|⩽3|y−z|.\nMoreover, (again by (4.9)) we have |g1(z)−g1(x)|⩽|z−x|. It follows\nthat\n|g1(y)−g1(x)|⩽|g1(y)−g1(z)|+|g1(z)−g1(x)|\n⩽3|y−z|+|z−x|⩽3|y−z|+ 3|z−x|= 3|y−x|.\nThus we conclude that if x /∈SJ, then Lipg1(x+)⩽3. Similarly, we\nobtain Lipg1(x−)⩽3, and so, again, Lipg1(x)<∞.\nAs for g2, by its definition we have, for all x, y∈R,\n|g2(y)−g2(x)|=\f\f\f\fZy\nx1F∩G\f\f\f\f⩽|y−x|,\nsog2is1-Lipschitz. Thus we obtain Lipg(x) = Lip( g1+g2)(x)⩽\nLipg1(x) + Lip g2(x)<∞, and (ii) is proved.\nInstead of proving directly the full version of (iii), we only aim to\nprove that the right derivative equals 1at any point of eA=A∩F∩Ec,\nwhich easily follows from Claim 3 below. The left derivative can be\nhandled analogously. Our strategy shall be to prove Claim 3 below; we\nthen apply this result, almost immediately obtaining g′\n+(x) = 1for any\nx∈eA. We shall need to distinguish several cases based on the choice\nofy. In Claim 1, we start with the simplest one, y∈bF, which will later\nbe useful multiple times. We then move on to Claim 2, proving the\nestimate under the assumption (x, y)∩F=∅. The proof of Claim 3\nconsists in applying both Claims 1 and 2 to treat the remaining case\n(x, y)∩F̸=∅. We keep in mind, in particular, the definitions of bF,J\nandg(see (4.6), (4.7), and (4.8)).\nClaim 1: For any x∈F∩G, and any y∈bF∩(x,∞)such that\n[x, y]⊆G∩(bF∪SJ), we have g(y)−g(x) =y−x.14 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nTo prove the claim, choose arbitrary x, yas in the statement. Let us\nnote that bF\\Fis countable ( Fchas countably many components and\nbFis countable in each of them), and\n(4.10) (x, y)⊆F∪(bF\\F)∪[\n(∗)J\nwhere by (∗)we mean all J∈ Jwith J⊆(x, y)(recall that y∈\nbF). Explaining the individual equalities just below, we perform the\nfollowing computation:\n(4.11)g(y)−g(x) =Zy\nx1F∩G+Zy\nxX\nJ∈JφJ\n=Zy\nx1F+X\n(∗)Zy\nxφJ\n=|F∩(x, y)|+X\n(∗)|J|\n=|(x, y)|=y−x.\nThe first equality is from the definition of g, the second equality holds\nas(x, y)⊆Gand by the monotone convergence theorem, and the\nfourth equality is by (4.10). Claim 1 is proved.\nClaim 2: For any x∈G∩F, any y > xwith [x, y]⊆G∩(bF∪SJ)\nand(x, y)∩F=∅, and any N⩾2with y < x + 1/N, we have\n(4.12)g(y)−g(x)\ny−x∈\u0012N−1\nN+ 1,N+ 1\nN−1\u0013\n.\nIn order to prove the claim, we pick arbitrary x, y, Nas in the state-\nment. We may further assume y /∈bFas Claim 1 clearly covers the\nopposite possibility. Let J= (u, v)∈ Jbe the one element sat-\nisfying y∈J; it exists as y∈SJ. Then Jis bounded because\ny∈(x, x+1/N)⊆(x, x+1/2). Asu∈bF, we have g(u)−g(x) =u−x\nby Claim 1 (where we replace ybyu). Therefore we obtain\ng(y)−g(x) =g(y)−g(u) +g(u)−g(x) =g(y)−g(u) +u−x.\nSince g(y)−g(u) =Ry\nuφJ∈[0,|J|]by the choice of φJ, it follows that\ng(y)−g(x)∈h\nu−x, u−x+|J|i\n= [u−x, v−x],\nfrom where we infer, using that y∈J= (u, v),\n(4.13)g(y)−g(x)\ny−x∈\u0012u−x\nv−x,v−x\nu−x\u0013\n.\nNow, if yis in an unbounded component of Fc(i.e. (x,∞)∩F=\n∅), then the interval J∈ Jcontaining yis of the form (u, v) =ON SETS WHERE lipfIS INFINITE 15\n\u0000\nx+1\nn+1, x+1\nn\u0001\nfor some natural number n; clearly, n⩾Nasy <\nx+ 1/N. But this implies\n(4.14)\u0012u−x\nv−x,v−x\nu−x\u0013\n=\u0012n\nn+ 1,n+ 1\nn\u0013\n⊆\u0012N\nN+ 1,N+ 1\nN\u0013\n.\nOf course, (4.13) and (4.14) yield\n(4.15)g(y)−g(x)\ny−x∈\u0012N\nN+ 1,N+ 1\nN\u0013\n.\nOn the other hand, if yis in a bounded component of Fc, say (x, b),\nthen the distribution of bFin(x, b)is symmetric, and the I∈ Jwith\nI⊆(x, b)are of three types:\n(a)Iis contained in the left half of (x, b);\n(b)Iis contained in the right half of (x, b);\n(c)Icontains the centre of (x, b).\nIfJis of type (a), then J= (u, v) =\u0000\nx+1\nn+1, x+1\nn\u0001\n, and we ob-\ntain (4.14), and subsequently (4.15), in the same way as above. If we\nhave (b), by the symmetry of bFin(x, b), the interval J= (u, v)has\na symmetric counterpart J′= (u′, v′)in the left half of (x, b). Set\nd:=u−u′>0(then d=v−v′). Now we have\n\u0012u−x\nv−x,v−x\nu−x\u0013\n=\u0012u′−x+d\nv′−x+d,v′−x+d\nu′−x+d\u0013\n⊆\u0012u′−x\nv′−x,v′−x\nu′−x\u0013\n⊆\u0012N\nN+ 1,N+ 1\nN\u0013\nwhere the last inclusion follows from case (a): Recall that J′= (u′, v′)\nis in the left half of (x, b), so (4.14) is satisfied with u, vreplaced by\nu′, v′, respectively. By virtue of (4.13) we, again, obtain (4.15).\nFinally, if (c) occurs, we have J=\u0000\nx+1\nn, b−1\nn\u0001\nfor some n∈N.\nThen n > Nas1/n < y −x <1/N. Let us now observe that x+1\nn−1\nis not in the left half of (x, b)as otherwise we would have x+1\nn−1∈bF\nandx+1\nn−1∈\u0000\nx+1\nn, b−1\nn\u0001\n=J, soJwould not be a component of\nbFc, a contradiction. Since Jshares its centre with (x, b), it follows that\nx+1\nn−1is not in the left half of J=\u0000\nx+1\nn, b−1\nn\u0001\neither, and thus we\nobtain\n|J|\n2⩽\u0012\nx+1\nn−1\u0013\n−\u0012\nx+1\nn\u0013\n=1\nn(n−1).16 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nDenoting J= (u, v), we therefore have u−x=1\nnandv−x=u−x+\n|J|=1\nn+|J|⩽1\nn+2\nn(n−1), whence\nu−x\nv−x⩾1\nn\n1\nn+2\nn(n−1)=1\n1 +2\nn−1=n−1\nn+ 1.\nUsing this and (4.13), we immediately derive (4.12):\n(4.16)g(y)−g(x)\ny−x∈\u0012n−1\nn+ 1,n+ 1\nn−1\u0013\n⊆\u0012N−1\nN+ 1,N+ 1\nN−1\u0013\n.\nWe summarize our findings by stating that in every possible case we\neither have (4.11), or (4.15), or (4.16), and that each of these proposi-\ntions implies (4.12). The claim is proved.\nClaim 3: For any x∈G∩F, any y > xwith [x, y]⊆G∩(bF∪SJ),\nand any N⩾2with y < x + 1/N, we have (4.12).\nFor the proof of the claim, we again pick any x, y, Nsatisfying the\nassumptions of our claim, and by the preceding claims we may further\nassume that y /∈Fand(x, y)∩F̸=∅.\nSetz= max(( x, y)∩F)and let us check the assumptions of Claim 2\nfor the interval [z, y], i.e. with xreplaced by z: By the choice of z, we\nhave z∈Fand(z, y)∩F=∅. Moreover, [z, y]⊆[x, y]⊆G∩(bF∪SJ),\nthe second inclusion being one of the assumptions of the present claim,\nand we also have y < x + 1/N < z + 1/N.\nBy Claim 2 (with xreplaced by z) we thus have\ng(y)−g(z)\ny−z∈\u0012N−1\nN+ 1,N+ 1\nN−1\u0013\n.\nand by Claim 1 (with yreplaced by z, and noting that z∈F⊆bF),\ng(z)−g(x)\nz−x= 1.\nSince the average slope of gover [x, y](i.e., the quotient (g(y)−\ng(x))/(y−x)) is a convex combination of the average slopes over [x, z]\nand[z, y], we immediately see that (4.12) holds also in the present case,\nconcluding the discussion of all possible cases and thus also the proof\nof the claim.\nTo conclude the proof of (iii), pick any x∈eA=A∩F∩Ecand any\nη >0; by (4.5), x∈G∩F. Recall that Jis the set of all components\nofbFccontained in G, which easily implies that bF∪SJcontains eAin\nits interior. Therefore there is N⩾2such that\u0014\nx, x+1\nN\u0015\n⊆G∩\u0010\nbF∪[\nJ\u0011\nand\n\u0012N−1\nN+ 1,N+ 1\nN−1\u0013\n⊆(1−η,1 +η).ON SETS WHERE lipfIS INFINITE 17\nObviously, for any y∈[x, x+1/N]we now have [x, y]⊆G∩(bF∪SJ),\nwhence, by virtue of Claim 3,\ng(y)−g(x)\ny−x∈\u0012N−1\nN+ 1,N+ 1\nN−1\u0013\n⊆(1−η,1 +η).\nThe proof of (iii) is finished.\nFinally, we prove (iv), which states that, for any interval U⊆Rwith\nU∩E̸=∅, we have\n(4.17) osc(g, U)⩽ε|U|.\nRecall that ε >0was fixed already in the statement of the lemma, Eis\nclosed, and Iis the collection of all components of Ec. Recalling (4.7),\nwe see that the elements of Jare pairwise disjoint open intervals, each\ncontained in some I∈ I.\nWe start by choosing an arbitrary I= (a, b)∈ Iand any open\ninterval U⊆Iwith ∂U∩∂I̸=∅; our first aim shall be to prove (4.17)\nin this case. (Note that osc(g, U) = osc( g,U)and that, unlike U, the\nclosed interval Umeets E.) There is no loss of generality in assuming\nthatUshares with Iits left endpoint, i.e. U= (a, c)for some c∈(a, b];\nthe other case can be dealt with analogously.\nKeeping in mind the definitions of φandg(cf. (4.8)), in particular\nthe fact that φ⩾0sogis monotone, we compute (explanations of the\nlastthreestepsandthecorrespondingnotationsfollowjustafterwards):\n(4.18)osc(g, U) =g(supU)−g(infU) =Z\nUφ\n=Z\nU\u0012X\nJ∈JφJ+1G∩F\u0013\n=|G∩F∩U|+X\nJ∈JZ\nUφJ\n=|GI∩F∩U|+X\n(∗)|J|+X\n(∗∗)Z\nUφJ\n⩽|GI∩U|+X\n(∗∗)Z\nUφJ\n<ε\n4|U|+X\n(∗∗)Z\nUφJ.\nHere by (∗)we mean all the intervals J∈ Jwith J⊆U. By (∗∗)we\nmean all the intervals J∈ Jwith J∩U̸=∅andJ⊈U. The fifth\nequality is a consequence of the following facts: U⊆IandG∩I=GI,\nwhence G∩U=GI∩U; for any J∈ J,R\nRφJ=R\nJφJ=|J|. The next\ninequality follows from the fact that all J∈ Jare pairwise disjoint,18 MARTIN RMOUTIL AND THOMAS ZÜRCHER\ndisjoint from F, and the ones pertaining to (∗)are contained in GI∩U.\nThe final inequality is from (4.4).\nSince each J∈ Jis contained in some I∈ Iand we are looking at\nsuch an Isharing its left endpoint with the one of U, it is easy to see\nthat (∗∗)represents at most one interval, so the sum has at most one\nsummand. If the sumP\n(∗∗)in the above computation is empty, then\nwe are done. So let us assume there is K∈ Jsuch that K∩U̸=∅\nandK⊈U. Then, clearly, K= (u, v)⊆Iwith u < c < v . There are\ntwo cases: |K|⩽3|U|or|K|>3|U|.\nWe consider the case |K|>3|U|, first. We have intervals U, K⊆I\nsuch that Kis more than three times the length of UandUshares its\nleft endpoint with I. From this it is obvious that U∩Kis contained in\nthe (open) left-hand side third of the interval K, and so (by the choice\nofφK) we haveR\nUφK= 0. That is, the sumP\n(∗∗)consists of one\nsummand whose value is 0, and we are, again, done by (4.18).\nSuppose |K|⩽3|U|and set U1:=U∪K= (a, v), and replace Uby\nU1in computation (4.18) and adapt the meaning of (∗)and(∗∗); then\nthe sumP\n(∗∗)becomes empty. We now obtain the desired estimate as\nfollows:\nosc(g, U)⩽osc(g, U 1)<ε\n4|U1|+ 0\n⩽ε\n4(|K|+|U|)⩽ε\n4(3|U|+|U|) =ε|U|.\nWe have proved (4.17) for any I∈ Iand any open interval U⊆I\nwith ∂U∩∂I̸=∅. Remembering that Eis closed and Ec=SIreveals\nthat we have actually proved (4.17) for any interval U⊆Ecsatisfying\n∂U∩E̸=∅.\nNow, let Ube any interval meeting E. Remembering that Gis\ncontained in Ec, it follows immediately from the definitions that φ= 0\nonE, and obviously\nU= (U∩E)∪(U∩Ec) = (U∩E)∪[\n(+)(U∩I),\nwhere (+)represents all I∈ Iwith U∩I̸=∅. Using these observations\nwe get\nosc(g, U) =Z\nUφ=Z\nU∩Eφ+X\n(+)Z\nU∩Iφ=X\n(+)Z\nU∩Iφ.\nFor every I∈ I, we set UI:=U∩I; then each nonempty UIis an\ninterval. Let us observe that for any I∈ Iwith (+)(i.e.UI̸=∅) we\nhave ∂UI∩∂I̸=∅: Indeed, we assumed U∩E̸=∅, i.e. U⊈Ec, and\nin particular, no I∈ Icontains U. Hence, for any I∈ Imeeting U,\nthe set Ucontains a point outside of I, and by its convexity, it also\ncontains a point of ∂I.ON SETS WHERE lipfIS INFINITE 19\nWe have just checked that each nonempty UIshares an endpoint\nwith I, sobythefirstpartoftheproofof(iv), weget osc(g, U I)⩽ε|UI|.\nUsing these observations, we may resume the above computation as\nfollows:\nosc(g, U) =X\n(+)Z\nUIφ=X\n(+)osc(g, U I)⩽εX\n(+)|UI|⩽ε|U|.\nThis shows (4.17) for any interval Umeeting E, and thus it concludes\nthe proof of (iv), and the lemma. □\nLemma 4.5. Suppose A⊆RisFσδ, meagre, and Lebesgue null. Then\nthere is a nondecreasing absolutely continuous function g:R→Rsuch\nthatg′(x) =∞for every x∈Aandlipg(x)<∞for every x /∈A.\nProof.As the zero function takes care of the case where A=∅, we\nassume that A̸=∅. We claim that there are nonempty compact sets\nFk⊆R(k∈N) such that\n(a)F0=R;\n(b) for all k, l∈N, ifk > landFk∩Fl̸=∅, then Fk⊆Fl;\n(c)A=T\nN∈NS\nk⩾NFk, that is, Ais precisely the set of points\nx∈Rbelonging to infinitely many Fk’s.\nIndeed, the meagreness of Aimplies the existence of an Fσmeagre\nsetBcontaining A. Since AisFσδ, there exist FσsetsBnsuch that\nA=T∞\nn=1Bn. By intersecting each Bnwith B, we may, and will,\nassume that Bnis meagre for every n. Moreover, we may assume that\nthe sets Bnare nested as the intersection of any two Fσ-sets is Fσ;\nthat is, we have Bn⊇Bn+1for every n∈N. By Lemma 3.2, each Bn\nis disjoint Kσ, i.e. can be expressed as the union of countably many\npairwise disjoint compact sets. It follows that Acan be expressed in\nthe form A=T∞\nn=1Bn=T∞\nn=1S∞\nm=1Fm\nnwhere each Fm\nnis compact\nand nowhere dense, and for any n∈Nand any distinct m1, m 2∈Nwe\nhave Fm1n∩Fm2n=∅.\nWe shall now recursively construct (for all n∈N) the systems\n(Dm\nn)∞\nm=1of pairwise disjoint compact sets withS∞\nm=1Dm\nn=Bnsuch\nthat\n(4.19) for any n⩾2and any mthere is ksuch that Dm\nn⊆Dk\nn−1.\n(Ifweconsiderthelowerindex nas“level”, thenthisrequirementmeans\nthat each set of level nis contained in some set of level n−1.) We\nstart with level n= 1where for each m∈Nwe set Dm\n1:=Fm\n1.\nAssuming we have already performed the construction up to level\nn−1for some n⩾2, we now take all the sets of the form Fi\nn∩Dj\nn−1,\ni, j∈N, discard the ones that are empty and order the remaining ones\ninto a sequence (Dm\nn)∞\nm=1.20 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nHaving finished this construction for every n, we need to check that\nthe above conditions on the sets Dm\nnare all met. The pairwise disjoint-\nness, compactness, and condition (4.19) are obvious; by induction we\nshall prove, for every n∈N, the equalityS∞\nm=1Dm\nn=Bn.\nForn= 1the statement is obvious, so assume we have already\nprovedS∞\nm=1Dm\nn−1=Bn−1for some n⩾2. Take any x∈Bn; then\nx∈Bn−1by the nestedness of Bn’s, so there exists j∈Nsuch that\nx∈Dj\nn−1. Moreover, as x∈Bn=S∞\nm=1Fm\nn, there also exists isuch\nthat x∈Fi\nn. Thus x∈Fi\nn∩Dj\nn−1, so this last intersection is one of\nthe nonempty sets that get arranged into the sequence (Dm\nn)∞\nm=1in the\nn-th step of the above recursive construction. Thus x∈S∞\nm=1Dm\nn,\nand we get Bn⊆S∞\nm=1Dm\nn; the opposite inclusion follows immediately\nfrom the factS∞\nm=1Dm\nn⊆S∞\nm=1Fm\nn=Bn.\nIt is now easy to order all the sets Dm\nn(n, m∈N) into a sequence\n(Fn)∞\nn=1satisfyingconditions(b),and(c), andweset F0=R, satisfying\n(a).\nSince Ais Lebesgue null, it easily follows from Lemma 4.2 that we\ncan find pairwise disjoint measurable sets Mk(k∈N) such that for\nevery k, we have A∩Mk=∅and|I∩Mk|>0for every kand every\ninterval I. Now, for each k⩾1, we use Lemma 4.3 to obtain a compact\nsetHksuch that (cf. Notation 4.1)\n•Fk⊆Hk⊆(Fk∪Mk)∩(Fk)2−k;\n•Hkmeets the middle third of every component of bFc\nk∩(Fk)2−k\nin a set of positive measure.\nNext, we define H0=R,# »H0=R,E0=∅, and for k⩾1we set\n(4.20)# »Hk:=[\nFj⊆FkHjand Ek:=[\nj0and\nwe can pick j0> ksuch that 2−j0< α/ 2. It follows that for any j⩾j0\nwith Fj⊆Fk,d(x, H j)⩾α/2. Thus\nd(x,# »Hk)⩾min({d(x, H j):Fj⊆Fkandj < j 0} ∪ {α/2})>0,\nand we see that# »Hkis closed.\nFinally, given k⩾1, we recall that we obtained in Lemma 4.4 a non-\ndecreasing absolutely continuous function gk:R→[0,2−k]satisfying\nthe following properties:\n(i)supp( g′\nk)⊆Hk;\n(ii)Lipgk(x)<∞for every x∈R;\n(iii)g′\nk(x) = 1for every x∈A∩Fk∩Ec\nk;\n(iv)osc(gk, I)<2−k|I|whenever Iis an interval meeting Ek;\n(v)∥g′\nk∥1<2−k.ON SETS WHERE lipfIS INFINITE 21\nWe will show that the statement holds with g=P∞\nk=1gk. That g\nis nondecreasing is clear, and the absolute continuity is a consequence\nof Lemma 3.1.\nFirst, we prove that g′(x) =∞for every x∈A. We do this by\nusing (iii). As x∈A, there are infinitely many indices ksuch that\nx∈Fk. We fix such an index k, and hence have x∈A∩Fk. In view\nof (iii), we want to show that x /∈Ek=S\nj lwith x∈# »Hp=S\nFj⊆FpHj, then there would exist j\nsuch that Fj⊆Fp(whence j⩾pby (b)) and x∈Hj. Since j⩾p > l,\nthis would be in contradiction with our choice of l.\nTo prove lipg(x)<∞, we divide g=P∞\nj=1gjinto three summands\nas we now describe. First, we introduce\nJ={j > l :x /∈Ej},\nK={j > l :x∈Ej}\nand then write\ng=X\nj∈Jgj+X\nj∈Kgj+lX\nj=1gj.\nWe will show that the first summand has finite lipatx, while the other\ntwo even have finite Lipat this point. We define\nh=X\nj∈Jgj,22 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nand want to show that liph(x)<∞. If there is r > 0such that\n(x−r, x+r)∩S\nj∈J# »Hj=∅, then (i) implies hto be constant on\n(x−r, x+r), soliph(x) = 0, and we are done.\nHence we assume\n(4.21) (x−r, x+r)∩[\nj∈J# »Hj̸=∅for every r >0.\nIt suffices to find a sequence (rp)∞\np=1of positive radii converging to 0,\nand for which\nlim\np→∞sup\ny∈B(x,rp)|h(y)−h(x)|\nrp<∞.\nLetr0= 1, and for p= 1,2, . . ., we define recursively jp∈ Jand\nrp>0by letting\njp= min {j∈ J: (x−rp−1, x+rp−1)∩# »Hj̸=∅},\nrp= dist( x,# »Hjp),\nIp= (x−rp, x+rp).\nBy (4.21), jpandrpare well-defined, and it is easy to see that j1<\nj2<···, and r0> r 1>···. We now argue by contradiction that\nR:= lim p→∞rp= 0; so assume not, i.e. R > 0. Take k∈ Jsuch that\n(x−R, x+R)∩# »Hk̸=∅. Since limp→∞jp=∞, we may pick p∈Nwith\nk < j p. As rp−1> R, we have (x−rp−1, x+rp−1)∩# »Hk̸=∅; therefore,\nthe fact that k < j pis in contradiction with the definition of jp.\nNext we note that Fjp∩Fjq=∅whenever p̸=q: Assume, for a\ncontradiction, p < qandFjp∩Fjq̸=∅. Then jp< jq, and (b) yields\nFjp⊇Fjq, whence# »Hjp⊇# »Hjq, implying rp⩽rq, a contradiction.\nFix arbitrary p∈Nandj∈ J; we shall estimate the oscillation of\ngjonIp. IfIp∩Hj=∅, then gjis constant on Ip; hence, we shall\nassume Ip∩Hj̸=∅. Then Ip∩# »Hj̸=∅, and rp>dist(x,# »Hj). By the\nconstruction of jpandrp, this yields jp< jandjp+1⩽j.\nWe verify next that Fj∩Fjp=∅. Indeed, by (b) and as j > j p, the\nonly alternative is Fj⊆Fjp; but in that case we would also have# »Hj⊆# »Hjp, which is impossible as# »Hjmeets Ipand# »Hjp(by the definition of\nIp) does not.\nTo summarize, we have fixed arbitrary p∈N,j∈ J, and assumed\nIp∩Hj̸=∅, obtaining jp< jp+1⩽jandFj∩Fjp=∅. Thus\nEj=[\nk lsuch that x∈Ej(i.e.j /∈ J); then any interval\ncontaining xmeets Ej. Thus, for any r >0,osc(gj,(x−r, x+r))<\n2−j·2r= 21−j·r. Setting f=P\nj>l, j /∈Jgj, we obtain\nLipf(x) = lim sup\ny→x|f(y)−f(x)|\n|y−x|⩽lim sup\nr→0+osc(f,(x−r, x+r))\nr\n= lim sup\nr→0+P\nj>l, j /∈Josc(gj,(x−r, x+r))\nr\n⩽lim sup\nr→0+21−jr\nr⩽∞X\nj=221−j= 1.\nFinally, as g=h+f+Pl\nj=1gj, we obtain the following estimate:\nlipg(x)⩽liph(x)+Lip f(x)+lX\nj=1Lipgj(x)⩽2+1+lX\nj=1Lipgj(x)<∞.\nThis concludes the proof. □\nFinally, we are ready for the proof of our main result.\nProof of Theorem 2.1. Since all Borel sets have the Baire property (see\ne.g.[Oxt80,Theorem4.3]),by[Oxt80,Theorem4.4],thereareameagre\nFσδsetA0and a GδsetGsuch that A0∩G=∅andA0∪G=A(the\ndisjointness is not clearly stated in [Oxt80], but can easily be seen from\nthe proof).\nJarník constructed a nondecreasing absolutely continuous function\nf:R→Rsuch that f′(x) =∞for every x∈GandLipf(x)<∞\nfor every x /∈G. Note that this also shows that Lipf(x) =∞for\nx∈G. From the meagre case, Lemma 4.5, we know that there is\na nondecreasing absolutely continuous function g:R→Rsuch that\ng′(x) =∞for every x∈A0andlipg(x)<∞forx /∈A0. Again, we\nnote that Lipg(x) =∞for every x∈A0.24 MARTIN RMOUTIL AND THOMAS ZÜRCHER\nWe define h:=f+g. It is clear that his nondecreasing and abso-\nlutely continuous. If f′(x) =∞org′(x) =∞, then also h′(x) =∞.\nThis shows that h′(x) =∞forx∈A, and hence that liph(x) =∞.\nNow, we assume that x /∈A. Hence, Lipf(x)<∞andlipg(x)<∞.\nThis tells us that liph(x)<∞forx /∈A.\nWe have found a function with all the required properties. □\nAcknowledgments This project started while we were at the Uni-\nversity of Warwick as members of a research group led by David Preiss.\nWewanttogiveourheartfeltthanksforhismentoringandforthemany\ndiscussions about numerous aspects of life and mathematics, and in\nparticular about questions concerning lip. We are grateful to him for\nsuggesting the idea of our proof.\nThe research leading to these results has received funding from the\nEuropeanResearchCouncilundertheEuropeanUnion’sSeventhFrame-\nworkProgramme(FP/2007–2013)/ERCGrantAgreementno.291497.\nFurthermore, the second author was supported by the University of\nSilesia Mathematics Department (Iterative Functional Equations and\nReal Analysis program).\nReferences\n[BC06] Zoltán M. Balogh and Marianna Csörnyei, Scaled-oscillation and\nregularity , Proc. Amer. Math. Soc. 134(2006), no. 9, 2667–2675.\nMR 2213746\n[BHMV20a] Zoltán Buczolich, Bruce Hanson, Balázs Maga, and Gáspár Vértesy,\nCharacterization of lip sets , J. Math. Anal. Appl. 489(2020), no. 2,\n124175, 11. MR 4093796\n[BHMV20b] ,Lipschitz one sets modulo sets of measure zero , Math. Slovaca\n70(2020), no. 3, 567–584. MR 4104346\n[BHMV21] ,Big and little Lipschitz one sets , Eur. J. Math. 7(2021), no. 2,\n464–488. MR 4256959\n[BHRZ19] Zoltán Buczolich, Bruce Hanson, Martin Rmoutil, and Thomas\nZürcher, On sets where lipfis finite, Studia Math. 249(2019), no. 1,\n33–58. MR 3984283\n[Han21] Bruce Hanson, Sets where Lipfis infinite and lipfis finite, J. Math.\nAnal. Appl. 499(2021), no. 2, Paper No. 125071, 11. MR 4216971\n[Jar33] Vojtěch Jarník, Über die Menge der Punkte, in welchen die Ableitung\nunendlich ist , Tôhoku Math. J. 37(1933), 248–253 (German).\n[KK96] R. Kannan and Carole King Krueger, Advanced analysis on the real\nline, Universitext, Springer-Verlag, New York, 1996. MR 1390758\n[Leb04] H. Lebesgue, Leçons sur l’intégration et la recherche de fonctions\nprimitives. , 1904 (French).\n[Leb09] Henri Leon Lebesgue, Cambridge library collection - mathematics:\nLecons sur l’integration et la recherche des fonctions primitives pro-\nfessees au college de france , Cambridge University Press, Cambridge,\nEngland, July 2009 (French).\n[Mal99] J. Malý, A simple proof of the Stepanov theorem on differentiabil-\nity almost everywhere , Exposition. Math. 17(1999), no. 1, 59–61.\nMR 1687460ON SETS WHERE lipfIS INFINITE 25\n[Oxt80] John C. Oxtoby, Measure and category. A survey of the analogies\nbetween topological and measure spaces , second ed., Graduate Texts\nin Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980.\nMR 584443\n[Rad19] Hans Rademacher, Über partielle und totale Differenzierbarkeit von\nFunktionen mehrerer Variabeln und über die Transformation der Dop-\npelintegrale , Math. Ann. 79(1919), no. 4, 340–359. MR 1511935\n[Sie18] W. Sierpiński, Un théorème sur les continus. , Tôhoku Math. J. 13\n(1918), 300–303 (French).\n[Ste23] Wiatscheslaw Stepanoff, Über totale Differenzierbarkeit , Math. Ann.\n90(1923), no. 3-4, 318–320 (German). MR 1512177\n[Zah41] Zygmunt Zahorski, Über die Menge der Punkte in welchen die\nAbleitung unendlich ist , Tôhoku Math. J. 48(1941), 321–330 (Ger-\nman). MR 27825\nDepartment of Mathematics Education, Faculty of Mathematics and\nPhysics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech\nRepublic\nEmail address :rmoutil@karlin.mff.cuni.cz\nInstytut Matematyki, Uniwersytet Śląski, Bankowa 14, 40-007 Ka-\ntowice, Poland\nEmail address :thomas.zurcher@us.edu.pl" }, { "title": "2401.15408v2.The_appearance_of_de_Sitter_black_holes_and_strong_cosmic_censorship.pdf", "content": "ICTS-USTC/PCFT-24-05\nThe appearance of de Sitter black holes and strong cosmic censorship\nLi-Ming Caoa ,b∗, Long-Yue Lib†, Xia-Yuan Liub‡, and Yu-Sen Zhoub§\naPeng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China and\nbInterdisciplinary Center for Theoretical Study and Department of Modern Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n(Dated: March 5, 2024)\nWe study the optical appearance of Schwarzschild-de Sitter and Reissner-Nordstr ¨om-de Sitter black holes\nviewed by distant observers inside cosmological horizons. Unlike their asymptotically flat counterparts, due\nto the positive cosmological constant, there are outermost stable circular orbits in the spacetimes, resulting\nin significant outer edges in the images. Besides, when the Reissner-Nordstr ¨om-de Sitter black hole has a\nstable Cauchy horizon, the photons from the preceding companion universe can be received by the observer in\nour universe. These rays create a multi-ring structure in the image. Since the stable Cauchy horizon violates\nthe strong cosmic censorship conjecture, this novel image shed some light on the test of the conjecture by\nastronomical observations.\nI. INTRODUCTION\nDespite its incredible success, general relativity (GR) indicates its own breakdown at singularities. This was demonstrated by\nHawking and Penrose [1, 2], who showed that within the framework of GR, gravitational collapse inevitably leads to singularities.\nThese singularities mark the catastrophic edge where determinism breaks down in our current understanding of physics. To\nremedy this issue, two conjectures were proposed, the weak cosmic censorship conjecture (WCCC) [3] and the strong cosmic\ncensorship conjecture (SCCC) [4]. The WCCC primarily concerns on the visibility of singularities by far away observers,\npostulating that singularities are always concealed behind a horizon. For instance, a Kerr-Newman (KN) black hole can not\nbe over charged or over spun to become a naked KN singularity by absorbing matter [5, 6]. On the other hand, the SCCC is\ndevoted to rescue the determinism of GR. It states that a physical spacetime is always globally hyperbolic [4, 7, 8]. However,\nThe extension of spacetime beyond the Cauchy horizon processed in Reissner-Nordstr ¨om (RN) black hole seems to violate the\nSCCC. Fortunately, the Cauchy horizon of the RN black hole has been shown to be unstable [9–15]. Several methods have been\nproposed to examine the validity of the SCCC within a given spacetime. For example, the perturbation of RN black hole grows\nunbounded near the Cauchy horizon [9–12]. And the method involved the backreation by considering ingoing null flux [13].\nFurthermore, Poisson and Israel [14] found that the mass inflation, i.e., the divergence of the Hawking mass or renormalized\nHawking mass, can be triggered by the presence of an outgoing null flux. Therefore, the Cauchy horizon of RN black hole will\nconvert into a null singularity after a small perturbation, which prevents the observer to pass through it. The Cauchy horizons of\nmany other black holes, whose Penrose diagrams are consist of repeated universes as RN black hole, are also unstable [16–19].\nIt was commonly believed that the SCCC holds true and may only be violated in some peculiar modified gravity or when\nextraordinary matter is present. However, even in the context of GR, which is the most well-established theory of gravity, coun-\nterexamples exist. One such counterexample is the RN-de Sitter (dS) black hole, which is distinguished by a single additional\nparameter compared to the RN black hole. It is one of the most typical solutions with a non-vanishing cosmological constant.\nUnder certain parameters, the RN-dS black hole can exhibit a stable Cauchy horizon [20–24]. Consequently, the Penrose dia-\ngram of the RN-dS black hole can be extended to be consisted of countless repeated identical universes. Therefore, observers\ncould travel through a black-white hole bridge and then enter another separate universe, which violates the SCCC. More pre-\ncisely, the Cauchy horizon remains stable when the surface gravity of the Cauchy horizon is less than that of the cosmological\nhorizon [25, 26]. The mass inflation in more sophisticate Einstein-Maxwell-scalar field model has been thoroughly studied in\nthe trilogy [27–30], in particular, a detailed range of parameters for mass inflation has been achieved.\nIt is widely believed in astronomy that massive stars will eventually collapse to form rotating black holes, i.e., Kerr black\nholes. There are many similarities between RN black holes and Kerr one. For example, they both have singularities and Cauchy\nhorizons. The Penrose diagram of the equatorial plane of Kerr black hole is similar to that of RN black hole. This implies that\nthe global structure of Kerr black hole is similar to that of RN one to some extent. Just as the cosmological constant stabilizes\nthe Cauchy horizon of the RN black hole, similar scenarios occur for the Kerr black hole. In other words, the Cauchy horizon\nof the Kerr black hole is also unstable, which respects SCCC [31]. However, when the cosmological constant is taken into\naccount, the Cauchy horizon of the Kerr-dS black hole can be stable for certain parameters, which violates SCCC [32]. On the\nother hand, a positive cosmological constant can describe the inflation of the early universe and plays a crucial role in explaining\nthe universe’s accelerating expansion. Multiple observations now also favor a positive cosmological constant. Therefore, the\n∗e-mail address: caolm@ustc.edu.cn\n†e-mail address: lily26@mail.ustc.edu.cn\n‡e-mail address: liuxiayuan@mail.ustc.edu.cn\n§e-mail address: zhou ys@mail.ustc.edu.cnarXiv:2401.15408v2 [gr-qc] 2 Mar 20242\nKerr-dS black hole has garnered considerable interest. In short, the similarities mentioned above even appear when they are\nimmersed in de Sitter space. Based on the similarities, the study of SCCC in RN-dS black hole has reference significance for\nthe study of Kerr-dS one and could potentially serve as a natural experimental site for testing the SCCC.\nSeveral years ago, the images of the supermassive objects at the center of M87 [33] and Milky Way galaxies [34] was captured\nby the Event Horizon Telescope (EHT). This provides us a novel method to detect the compact objects and the black holes. In\nthese black hole pictures, a bright ring surrounds a dark shadow, formed by light emitted directly from the accretion disk.\nAccording to calculations for the null geodesics, light rays emitted from the vicinity of the photon sphere, where the effective\npotential reaches its maximum, can circulate around the black hole numerous times before being received by the observer. A\nmore general definition of the photon sphere (surface) can be found in [35]. However, despite anticipating them, this type of light\nremained undetected on these images. Utilizing ray-tracing method [36], we can trace light rays backwards from the observer’s\nimage plane. The trajectory of light rays intersects with the accretion disk, subsequently converging to a bright ring in the\nimage plane. These rings is called a “lensed ring” for two intersections, and “photon ring” for more intersections. However,\nthe Lyapunov exponent being positive indicates that the photon sphere is chaotic. This means that the separation between two\ninitially close trajectories will diverge exponentially over time. Consequently, the photon ring is so close to the lensed ring that\nit can not be distinguished apart, and is much narrower that its contribution to the overall intensity is negligible. The lensed rings\nare also covered by the directly emitted light due to the current limitations of astronomical observation precision. As the impact\nparameter reaches its critical limit, the rays rotate at the photon sphere for infinite times. Since the accretion disk is distributed\nin a limited range, the rays emitted from the edge of the source will form a sharp edge on the image. The position of the lensed\nring or photon ring reveals the information about the background geometry, while the total appearance of a compact object is\nheavily influenced by the position and profile of the light source.\nRecently the images of many compact objects were studied by the ray-tracing method proposed in [36], such as Kazakov-\nSolodukhin black hole, the regular black hole with de Sitter core, the quantum-corrected black hole, the wormholes and so\non [37–44]. All of these works have exhibited the reliability of the way to get the optical appearance of the black holes or\nother compact objects. Therefore, it is valuable to apply the method to study the image of the black hole when the SCCC is\ndestroyed, and find some specific and indicative features which might provide some enlightenment on the astronomical obser-\nvation. Actually, in [45], we have studied the image of a regular black hole with a stable Cauchy horizon, which is inconsistent\nwith SCCC. The rays from the preceding companion universe can be received by the observer in our universe due to the stable\nCauchy horizon. This produces many new rings inside the shadow area in the image. This novel multi-rings structure may be\ndetected astronomically. The fly in the ointment is that a precise physical process that leads to the formation of such black hole\nstill remains a mystery. On the other hand, it is clear that the RN-dS black hole with a stable Cauchy horizon has a reasonable\nexplanation in the framework of Einstein gravity theory. It is natural to exploit its image in the same way. However, unlike\nthe asymptotically flat black holes, some difficulties arise when the positive cosmological constant is presented. Firstly, for the\nRN-dS black hole, the observers have to be located inside the cosmological horizon rather than the null infinity. But the image is\ninfluenced greatly by the location of the observer while there is no specific location is given privileged status. The image has an\nouter edge if the observer is close to the black hole. And the image formed by the rays emitted directly is more narrow compared\nwith that of Schwarzschild one. Secondly, unlike the asymptotically flat black holes, there are outermost stable circular orbits\n(OSCO) in the spacetimes of the dS or charged dS black hole [46–48]. A physically reasonable thin accretion disk is distributed\nbetween innermost stable circular orbit (ISCO) and OSCO. And this leads to a significant outer edge in the image. Thirdly, dur-\ning the propagation of light, the redshift or blueshift factor in RN-dS black hole is quite different from the one in asymptotically\nflat case, for example, it tends to infinity near the cosmological horizon. A very large intensity of light will be received by the\nobservers near the cosmological horizon. All of these will be discussed in detail in section III.\nAs the regular black hole with a stable Cauchy horizon, the stable Cauchy horizon in RN-dS black hole also gives rise to the\nmulti-rings structure. Hence it may be a potential method to test SCCC in astronomical observations. However, the multi-rings\nstructure also occurs in the compact object without horizons and the wormholes [49–52]. And the similar multi images structure\ncan be found in the image of stars caused by gravitational lensing of a massive object [53, 54]. The difference between these\nobjects and the black hole violating SCCC is whether they have horizons. In [55], a general relativistic formalism is proposed to\nget the parameter of the RN black hole (such as the mass and the charge) from some directly observable quantities (such as the\ntotal frequency shift, aperture angle of the telescope, and redshift rapidity). And a method to get the parameter of the Kerr-dS\nblack hole is proposed in [56]. Maybe we can distinguish the compact object, wormhole and the RN-dS black hole with the\nassistance of multiple methods, especially the experiment involving a significant redshift in the presence of an event horizon.\nThis paper is organized as follows. In section II, we introduce the ISCO and OSCO of the RN-dS black hole and get the range\nwhen there is a stable Cauchy horizon, an ISCO and OSCO in RN-dS spacetime. In section III, we draw up the image of the\nSchwarzschild-dS black hole and analyse the effect of observer distance on the image. Then the image of the RN-dS black hole\nis investigated in IV. Finally, we give conclusions and discussions in section V.\nII. THE ISCO AND OSCO IN RN-DS BLACK HOLE\nThe existence of an ISCO for massive particles in the Schwarzschild black hole at r= 6mis a well-known fact. However,\nin addition to the ISCO, there is an OSCO in the spacetime of the Schwarzschild-dS black hole or more general RN-dS black3\nhole. Physically acceptable substances that can emit light are generally massive. As a result, we adapt the model of the accretion\ndisk, which is regarded as the source of light emission, to consist of the stable orbits of these massive substances. Consequently,\nthe accretion disk in an RN-dS black hole resembles a band distributed between the ISCO and OSCO. The accretion disk of a\nKerr-dS spacetime is studied in [57]. In this section, we will give a brief introduction to the ISCO and OSCO of the RN-dS black\nhole.\nIn static coordinates, the metric of RN-dS black hole can be written as\nds2=−f(r)dt2+1\nf(r)dr2+r2dθ2+r2sin2θdϕ2, (2.1)\nwhere\nf(r) = 1−2M\nr+Q2\nr2−Λ\n3r2. (2.2)\nHere, Mis mass parameter, Qis electric charge parameter, and Λ>0is the cosmological constant. We are interested in case\nwith three horizons, i.e., the Cauchy horizon, the event horizon, and the cosmological horizon. The radii of these horizons are\ndenoted as r−,r+, and rc, respectively, which are three real roots of f(r) = 0 . With the help of the Killing vector fields ∂/∂t\nand∂/∂ϕ , we can get the conserved quantities associated with the geodesic, which can be used to simplify the equations of\nmotion of the particle. The motion of a particle with energy Eand angular momentum lis governed by the equation\n˙r2+V(r) =1\nb2, (2.3)\nwhere b=l/Eis the impact parameter, and\nV(r) =\u0012l2\nr2+ϵ\u0013\nf(r) (2.4)\nis the effective potential. Here, ϵ= 0,1for massless and massive particle respectively, and “·”denotes the derivative with\nrespect to the parameter of the geodesic. Of course, this parameter is the so-called affine parameter when the particle is massless.\nFor the circular orbits, we have Vr= 0, where Vrdenotes the derivative of Vwith respect to r. Similarly, in the following\ndiscussion, VrrandVrrrrepresent the second and third derivative of Vwith respect to rrespectively. Furthermore, Vrr<0\nfor the unstable circular orbits and Vrr>0for the stable circular orbits. Solving Vr= 0 for the massless particles, we can\nget a stable photon sphere between the event horizon and the Cauchy horizon, and an unstable photon sphere outside the event\nhorizon. However, in the case of massive particles, the situation is complicated. Fig.1 represents the parameter space of circular\norbits, where each point corresponds to a unique circular orbit.1The curves depicted in the figure is composed of points that\nsatisfy Vr= 0, representing physically allowed circular orbits. The curves consist of two parts. The lower part, appearing almost\nhorizontal, is concealed behind the event horizon, thus is not of interest to us. However, the stability of a point on the curve\nrequires further calculations on Vrr. As the radius rapproaches positive infinity or 0, the derivative of effective potential Vr\ntends towards negative infinity. The Vrwithin the region between the two branches of the curve is positive. Intuitively, consider\nan auxiliary vertical line with fixed l. The intersections of the line with the curve represent the zeros of Vr. As we gradually\nincrease the value of rfrom bottom to top, crossing the intersections of the line and the curve, if Vrtransits from negative to\npositive, then Vrr≥0. Conversely, if Vrtransits from positive to negative, then Vrr≤0. A more rigorous discussion appears in\nthe following paragraphs.\nWe have two special circular orbits represented by point AandBwhere dl/dr= 0. Actually, the circular orbit associated\nwithAwhere d2l/dr2>0is nothing but the ISCO. Accordingly, the orbit presented by the point Bwhere d2l/dr2<0is\nthe OSCO mentioned in the previous section. It should be noted that although neither point Anor point Brepresent stable\ncircular orbits since Vr=Vrr= 0,Vrrr̸= 0at these points, stable orbits can indeed be found within any small vicinity around\nthem. We will prove this in the next paragraph. Besides the situation depicted in Fig.1 or Fig.2(a), there is a critical case where\nISCO and OSCO are coincided, which has been shown in Fig.2(b). It can be seen that d2l/dr2is vanished on the black spot of\nFig.2(b). Unlike the usual Schwarzschild black hole, for some range of parameters, RN-dS spacetime does not possess stable\ncircular orbits outside the event horizon, which can be understood from Fig.2(c). Obviously, as l→ ∞ , the effect potential Vof\nthe massive particles tends to that of the massless particles. Meanwhile, the two curves of r−lin Figs. 2(a), or 2(b) and 2(c),\ntend to the photon spheres as l→ ∞ .\nThe ISCO and OSCO can be obtained by solving the equations Vr= 0 andVrr= 0. This is actually the usual definition of\nthe stable circular orbits in literature [46–48]. The r−lcurve in Fig.1 is the solution of Vr= 0. Denote the parameter of the\nupper branch of the curve as σ. Consider the Vas a function of randl. Since each runiquely corresponds to a point on the\n1Here and below, unless otherwise stated, all numerical values of Q, r, andlrepresent quantities in units of M. Similarly, Λis given in units of M−2.4\n2.6 2.7 2.8 2.9 3.0 3.1 3.2 l24681012r\nABVrr<0\nVrr>0\nVrr<0\nFIG. 1: The r−ldiagram. The RN-dS black hole with Q= 0.995,Λ = 0 .001.lis the angular momentum of a massive particle and ris\nthe corresponding radius of the circular orbital radius. Two horizontal black dashed lines are photon spheres, while the two solid curves in\nthe figure are derived from Vr= 0. The below one (blue), which is inside the event horizon, nearly overlaps with the black dashed line. The\ncurve above is divided into three parts by points AandB. The part AB(blue) represents the stable circular orbit because Vrr>0and the red\ncurves represent the unstable circular orbit because Vrr<0. Therefore Ais the ISCO and Bis the OSCO. The two orange vertical dashed\nline, whose abscissas are lAandlB, represent the angular momentum lfor point AandB.\n(a)\n (b)\n (c)\nFIG. 2: Some possible r−ldiagram. The RN-dS black hole have an ISCO and an OSCO in figure (a). The ISCO and OSCO is coincide in\nfigure (b). And there is no ISCO or OSCO in figure (c).\ncurve, we can take the σvaries monotonically with r, which further implies that dr/dσ̸= 0. We have\n0 =dVr\u0000\nr(σ), l(σ)\u0001\ndσ=∂Vr\n∂rdr\ndσ+∂Vr\n∂ldl\ndσ=Vrrdr\ndσ+∂Vr\n∂ldl\ndσ(2.5)\non the curve. Thus\nVrr=−∂Vr\n∂ldl\ndr= 0, (2.6)\nwhere we have used dl/dr= 0at points AandB. Therefore, the turning points AandBin Fig.1 are satisfied by Vr=Vrr= 0,\nwhich means they are actually the ISCO and OSCO.\nThe boundary of the range of parameters in which ISCO and OSCO exist is the critical case as Fig.2(b), where ISCO and\nOSCO coincide. In this case Vrrr= 0 and this condition is equivalent to d2l/dr2= 0. The proof is as follows. At the point\ncoincided by AandB,Vrr= 0anddl/dr= 0. If we chose dr/dσis positive and finite, then\ndl\ndσ=dl\ndrdr\ndσ= 0. (2.7)5\nFrom\n0 =d2Vr\ndσ2=Vrrr\u0012dr\ndσ\u00132\n+ 2∂Vrr\n∂ldl\ndσdr\ndσ+∂Vr\n∂ld2l\ndσ2+∂2Vr\n∂l2\u0012dl\ndσ\u00132\n+Vrrd2r\ndσ2, (2.8)\nwe get\nVrrr=−2∂Vrr\n∂ldl\ndr−∂Vr\n∂ld2l\ndσ2\u0012dr\ndσ\u0013−2\n−∂2Vr\n∂l2\u0012dl\ndr\u00132\n−Vrrd2r\ndσ2\u0012dr\ndσ\u0013−2\n=−∂Vr\n∂ld2l\ndσ2\u0012dr\ndσ\u0013−2\n, (2.9)\nwhere we have used dl/dr= 0 andVrr= 0 for ISCO and OSCO, and Vr= 0 for the entire curve in Fig.2(b). Hence when\nVrrr= 0and∂Vr/∂l̸= 0, we have d2l/dσ2= 0. And then\nd2l\ndr2=d2l\ndσ2\u0012dr\ndσ\u0013−2\n−dl\ndσ\u0012dr\ndσ\u0013−3d2r\ndσ2= 0. (2.10)\nTherefore, when Vrrr=Vrr= 0anddl/dr= 0, we have d2l/dr2= 0. To summarize, the orbit radii of ISCO and OSCO are\nsatisfied by Vr=Vrr= 0. And the critical case where ISCO and OSCO coincide, which is also the critical case that there is no\nISCO and OSCO, is satisfied by Vr=Vrr=Vrrr= 0.\nIf there is no photon sphere, i.e., the spacetime is the RN-dS naked singularity, the r−ldiagrams will be different, and are\nshown in Fig.3.\n(a)\n (b)\n (c)\nFIG. 3: Some possible r−ldiagrams without photon sphere. The spacetime have two ISCOs and two OSCOs in figure (a) (the three turning\npoints and an ISCO with l= 0). An ISCO and OSCO is coincide in figure (b). And there is an ISCO ( l= 0) and an OSCO (the turning point)\nin this case. There is an ISCO ( l= 0) and an OSCO (the turning point) in figure (c).\nIn order to see the various scenarios clearly, the relation of the existence of ISCO and OSCO and Qis shown in Fig.4. As we\ncan see, the unstable photon sphere is always outside the event horizon. The dependence on the charge Qis given as follows\n(i). For Q∈(0, Q2), there are three horizons and two photon spheres. There is no ISCO and OSCO for Q∈(0, Q1), as\nFig.2(c), and an ISCO and an OSCO for Q∈(Q1, Q2), as Fig.2(a). When Q=Q1, the ISCO and OSCO coincide, as\nFig.2(b).\n(ii). For Q∈(Q2, Q3), it is a RN-dS naked singularity with two photon spheres, and there is an ISCO and an OSCO as\nFig.2(a), too.\n(iii). For Q∈(Q3, Q4), the RN-dS naked singularity has no photon sphere and the blue curve has three turning points. Stable\ncircular orbits exist for two distinct intervals of rin this case, as Fig.3(a).\n(iv). When Q⩾Q4, there is an ISCO and an OSCO, and the corresponding r−ldiagrams are Fig.3(b) for Q=Q4and\nFig.3(c) for Q > Q 4.\nThe stable Cauchy horizon emerges when κ−< κc[25, 26], where κ−andκcare the surface gravities of the Cauchy horizon and\ncosmological horizon respectively. Once a stable Cauchy horizon exists, the SCCC will be broken down and the predictability\nof classical theory is threatened. Thus it aroused great interest in the community of general relativity and gravity theory. We\nwill look for the range of the parameters of the black hole which has a stable Cauchy horizon as well as stable circular orbits\n(i.e., ISCO and OSCO). Firstly, the range of the parameters for the black hole with three horizons has been drawn in Fig.5(a).\nA part of the boundary of this range is given by r+=r−andr+=rc. The parameter range for Kerr-dS black holes can be6\n0.6 0.7 0.8 0.9 1.0 1.1 1.2Q 02468r\nFIG. 4: The blue curve is solution of Vr=Vrr= 0, i.e., the ISCO or OSCO. The orange curve is the Cauchy horizon and the event horizon\n(The radius of the cosmological horizon is too large to be shown in the figure, so we omit it). The green curve is the photon spheres. The\nparameter of the RN-dS spacetime is Λ = 0 .001. The four blue dashed line passing the turning points are located at Q1= 0.654242 , Q2=\n1.00017 , Q3= 1.06066 , Q4= 1.11927 .\nfound in [56]. Although their graph closely resembles ours, it is important to note that their abscissa represents the angular\nmomentum parameter α, while our abscissa corresponds to the charge parameter Q. This observation highlights the similarity\nbetween angular momentum and charge, illustrating that the RN-dS black hole can serve as a valuable reference for further study\non Kerr-dS black holes. Secondly, the range of the parameters of the RN-dS black hole with a stable Cauchy horizon is shown\nas the small crescent in Fig.5(b). The left boundary of this area is κ−=κcand the right boundary is κ−= 0, i.e., r−=r+.\n(a)\n (b)\nFIG. 5: Figure (a) is the range of the parameter of the RN-dS black hole with three horizons. The boundary of the range is r−=r+and\nr+=rc. Figure (b) is the range of the parameter of the RN-dS black hole with κ−< κ c. The boundary is κ−=κcandκ−= 0.\nFinally, the range of parameters for the presence of stable circular orbits, i.e., ISCO and OSCO, has been found from the\ndefinition the result is put in Fig.6. In summary, the final range is the gray area in the right figure of Fig.6.\nFIG. 6: The left figure is the range where there are three horizons, one of which is a stable Cauchy horizon, and an ISCO and OSCO. The\nmiddle figure and the right figure are the partial enlarged figure the left one. The range is marked as a gray area in the right figure.7\nIII. THE APPEARANCE OF THE SCHWARZSCHILD-DS BLACK HOLE VIEWED BY OBSERVERS WITH DIFFERENT\nDISTANCE\nIn this section we will study the effect of the position of the observer on the image of the Schwarzschild-dS black hole. The\npositions of the black hole, accretion disk, observer and the trajectory of the photons are shown in Fig.7. We use the stereographic\nprojection to get the image, and thus the abscissa of the image yp[58] is\nyp= 2 tanθ\n2= 2 \n1√fr\nb−s\nr2\nfb2−1!\f\f\f\f\f\nr=robs, (3.1)\nwhere bis the impact parameter and robscan be understood as the distance between the observer and the black hole.\nFIG. 7: The schematic picture showing the trajectory of the photons and the stereographic projection. The black hole is located at Oand the\nobserver is at A. The black ring is the event horizon of the black hole. The yellow line represents the accretion disk and the red curve is the ray\nfrom the accretion disk to the observer. θis the angle of incidence. The green triangle is a schematic diagram of the stereographic projection.\nWe use the ray-tracing method [36] to draw the image of the black holes. The normalized number of orbits n=ϕ/(2π)\nrelates to the number of intersections with the equatorial plane of a particular light ray, where ϕis the azimuthal angle. Light\nbends greatly around the massive object, some even trace a circular path at the peak of the effective potential. This circular path\nalso known as the photon sphere or “critical curve”. At the critical impact parameter, the number of rotations of the photons\nincreases infinitely. This results in a bright ring in the image, and the size of the ring only depends on the background geometry\nof spacetime. Images produced by rays intersecting the accretion disk once, twice, or more than twice are referred to as “direct\nemission”, “lensed ring”, and “photon ring”, respectively. However, in most cases, the photon ring is so close to the lensed ring\nthat it can not be distinguished, and its contribution to the overall intensity is negligible. It means we can see only one bright\nring in the image. At each intersection, newly emitted photons from the accretion disk join in the journey towards the screen or\nthe celestial sphere of the observer. Each of intersections of light with the accretion disk contributes to the intensity received by\nthe observer. Besides, considering the effect of gravitational redshift on the intensity of the emission, the intensity of the light\nreceived by the observer, Iobs, is [36, 38, 59]\nIobs=X\nnIem(r)f2(r)\nf2(robs)\f\f\f\f\nr=rn, (3.2)\nwhere rnis the position of n-th intersection with the accretion disk.\nA phenomenon of discontinuity arises when the location of the observer is finite, and the intensity received by the observer\n“jumps” to null at certain impact parameter. In order to see the “jump” of the Iobscaused by the position of the observer, we\nchoose a Schwarzschild-dS black hole with Λ = 0 .001M−2, which has no ISCO and OSCO. The luminous intensity of the\naccretion disk is\nIem(r) =\n\nπ/2−arctan( r−2)\nπ/2−arctan( r+−2),ifr+< r < r c.\n0, ifr⩽r+orr⩾rc.(3.3)8\nThis kind of jump is different from the jump caused by the OSCO, which will be discussed later. The image viewed by observers\nat different distance is given by Fig.8.\n0.0 0.5 1.0 1.5 2.00.00.20.40.60.8\n(a)\n (b)\n0.0 0.5 1.0 1.5 2.00.00.10.20.30.40.50.6 (c)\n (d)\n0.0 0.5 1.0 1.5 2.00.00.10.20.30.40.50.6\n(e)\n (f)\n0.0 0.5 1.0 1.5 2.00.00.10.20.30.40.50.6 (g)\n (h)\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.10.20.30.40.5\n(i)\n (j)\n0.0 0.1 0.2 0.3 0.4 0.501234567 (k)\n (l)\nFIG. 8: The appearance of the Schwarzschild-dS black hole with Λ = 0 .001 viewed by observers with different distance. The observed\nintensity Iobsis normalized to the maximum value I0of the emitted intensity. We choose six different observation positions, i.e., robs=\n4,5,6,10,0.5rcand0.9rc. For each given observation position, we draw the corresponding observed intensity and shadow image separately.\nEach item corresponds to an observer located at a specific position in a sequential manner. (e.g., Figure (a) and (b) is the intensity and the\nimage corresponds to observer at robs= 4, Figure (c) and (d) corresponds to observer at robs= 5, Figure (e) and (f) corresponds to observer\natrobs= 6, etc.)\nAs we can see, there is a significant discontinuous jump in Iobsand an edge in the image when the observer is near the black\nhole. This is because some rays with large bcan not be received by the observer located too close to the black hole. To explain\nthis better, we draw the trajectories of photons in Fig.9. The rays with large b, like the blue curve, can not be received by the\nobserver because its perihelion is farther than the position of the observer. Therefore, there is a jump in the observed intensity.\nAnd the image of the black hole has a significant edge if the observer is close enough to the black hole. In order to avoid this\nkind of jump, the observer has to be located far way from the black hole. In this case, only light rays with a significantly large\nvalue of bcannot be received by the observer. Such light can only be emitted from the remote regions of the accretion disk.\nConsequently, its contribution to the total intensity is negligible.\nWe also find that observer at 0.9rcreceive a much stronger intensity Iobscompared to those at other locations in Fig.8 since\nthey have different redshift factor. As shown in schematic diagram 10, although the two rays are emitted at the same position and\nthey have same f(rem)andIem(rem), they have different f(robs). The two observers are located at AandBwithfAobs= 0.633\nandfBobs= 0.190, while fem(rem) = 0 .743. Therefore, from (3.2), the observed intensity IobsforAandBvaries by a factor of\nf2\nAobs/f2\nBobs≈11. In conclusion, for observers located near the cosmological horizon, the intensity they received will become\nbrighter as they get closer to the cosmological horizon. In fact, this intensity can reach infinity. As the observer moves away\nfrom the black hole, the metric function f(robs)increases and then decreases. Due to the same reason, the intensity (3.2) is large\nat small robsand near the cosmological horizon, while diminishes a lot in the middle of these two horizons.\nThe image viewed by the observer far away from the Schwarzschild-dS black hole without ISCO or OSCO has the similar\nshape to the usual Schwarzschild black hole. So we draw up the image of the Schwarzschild-dS black hole with an ISCO and\nOSCO with Λ = 0 .0005 in Fig.11. In this case, the accretion disk is distributed between ISCO and OSCO. And the luminous9\n-40 -20 0 20 40-60-40-200204060\nFIG. 9: The trajectory of photons. The parameter of black hole is Λ = 0 .001. The blue ring is the event horizon and the black ring is the\ncosmological horizon. The observer is located at A. The rays with small bare received by the observer, which are shown as red curves. But\nsome rays with large b, whose radius of the perihelion is larger than robs, can not pass through the observer. This kind of the rays is shown as\nthe blue curve. And the critical case, when the perihelion of the ray is the position of the observer, is shown as the green\ncurve.\n-10 10 20 30 40 50-101020304050\nFIG. 10: Two trajectories of rays from the accretion disk to the observers AatrA= 30 andBatrB= 48 . The three rings are the Cauchy\nhorizon, event horizon and cosmological horizon of the Schwarzschild-dS black hole with Λ = 0 .001in turn.\nintensity of the accretion disk is\nIem(r) =\n\n1\n(r−rISCO + 1)2,ifrISCO⩽r⩽rOSCO\n0, ifr < r ISCO orr > r OSCO. (3.4)\nTo avoid the undesirable discontinuity due to the position of the observer, we set the observers as far as possible. Actually, in\nfollowing discussion, robs= 0.9rccan reach the requirement.\nAs in Fig.8, the image of Schwarzschild-dS black hole with an ISCO and OSCO in Fig.11 also has a edge. Particularly, when\nthe observer is too close to the black hole, e.g., robs= 4, only the rays emitted from a little range of the accretion disk can be\nreceived by the observer. In this case, despite being direct emission, the image contract to a ring. However, the edges of the\nSchwarzschild-dS black hole with and without an ISCO and OSCO have different causes. Besides the influence of the observed\ndistant, the edge in the image of black hole with ISCO and OSCO is also resulting for that there is no light source outside the\nOSCO, i.e., the light source has a cutoff at OSCO. Therefore, the image of the black hole with cosmological constant will be a\nlittle bit different from that of the usual asymptotically flat Schwarzschild black hole. The presence of the cosmological constant\nmay lead to the emergence of OSCO, and this produce an edge in the image of the black hole.10\n0.0 0.5 1.0 1.5 2.00.00.51.01.52.0\n(a)\n (b)\n0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.4 (c)\n (d)\n0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.4\n(e)\n (f)\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.01.21.4 (g)\n (h)\n0.0 0.1 0.2 0.3 0.40.00.20.40.60.81.01.21.4\n(i)\n (j)\n0.00 0.02 0.04 0.06 0.08 0.1002468101214 (k)\n (l)\nFIG. 11: The appearance of the Schwarzschild-dS black hole with Λ = 0 .0005 viewed by observers with different distance. It has an\nISCO and an OSCO. As the Schwarzschild-dS black hole without an ISCO and OSCO, we choose six different observation positions, i.e.,\nrobs= 4,5,6,10,0.5rcand0.9rc. (e.g., Figure (a) and (b) corresponds to observer at robs= 4, Figure (c) and (d) corresponds to observer at\nrobs= 5, Figure (e) and (f) corresponds to observer at robs= 6, etc.\nIV . THE APPEARANCE OF THE RN-DS BLACK HOLE\nThe position of the observer will affect the image of the black hole, especially the ray with large bwill not be received by\nthe observer. Therefore, we set the observer far from the black hole and near the cosmological horizon in this section, i.e.,\nrobs= 0.9rc. A part of the Penrose diagram of the RN-dS black hole is given by Fig.12. If the Cauchy horizon is stable, the\ninfinite repetition of the spacetime allow the ray emitted from the accretion disk in the previous universe Ato fall into the black\nhole, fly out from the white hole and be received by the observer in universe Bfinally. This kind of trajectory of the photon is\nshown as the red curve in Fig.12. Besides, the photons can be emitted from the accretion disk and received by the observer both\nin universe B, as the green curve in Fig.12. The images caused by this two kinds of ray are quite different. This can be found in\nthe following discussions.\nA. The appearance of the RN-dS black hole with ISCO and OSCO\nThe appearance of the RN-dS black hole with Q= 1.0001,Λ = 0 .001is given by Fig.13 and the observer is located at\nrobs= 48.378.robs/rc= 0.9The light source distributes between ISCO and OSCO and the luminous intensity of the accretion\ndisk is described by Eq.(3.4).\nCompare to that of the Schwarzschild black hole, the image of the RN-dS black hole has a significant edge. This is because\nthere is no light source outside the OSCO. This has already been encountered in Schwarzschild-dS black hole with an ISCO and\nOSCO. Besides, the lensed ring or the photon ring of the RN-dS black hole is smaller than that of the Schwarzschild black hole,\nalthough their photon spheres are the same size. The intensity of the Schwarzschild black hole is weaker than the RN-dS black\nhole, which is caused by redshift factor. More precisely, when the observer is located far from the black hole, the metric function11\nFIG. 12: The Penrose diagram of the RN-dS black hole. The blue curves are the accretion disks in universe AandB, respectively. The\nobserver is located near the cosmological horizon. The green curve is the ray from the accretion disks in universe Band received by the\nobserver. And the red curve is the ray from the accretion disks in universe A.\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.00.10.20.30.40.5\n(a)\n (b)\n0.00 0.02 0.04 0.06 0.08 0.100246810 (c)\n (d)\n0.00 0.01 0.02 0.03 0.040246810\n(e)\n (f)\n0.00 0.02 0.04 0.06 0.08 0.100246810 (g)\n (h)\nFIG. 13: The intensity and the image of the Schwarzschild black hole with the photon sphere at rsp= 1.9996 (M= 0.6665 ) and the RN-dS\nblack hole with Q= 1.0001,Λ = 0 .001. Figure (a) and (b) are the observed intensity and the image of the Schwarzschild black hole with\nphoton sphere whose radius is same as that of the RN-dS black hole. Figure (c) and (d) are the observed intensity and the image of the RN-dS\nblack hole when the photons are emitted from the accretion disk in universe B. Figure (e) and (f) are the observed intensity and the image of\nthe RN-dS black hole when the photons are emitted from the accretion disk in universe A. Figure (g) and (h) are the observed intensity and\nthe image of the RN-dS black hole when the photons are emitted from the accretion disk in both universe AandB.\nfSch(robs)tends to 1, while fRNdS(robs)tends to 0. The cosmological constant chosen by us is Λ = 0 .001. This means the\nmagnitude of ΛM2is10−3, in other words, it is a supermassive black hole since the astronomical observations have revealed\na tiny value for the cosmological constant. From Eq.(3.2), the intensity is greatly amplified when the observer is very close to\nthe cosmological horizon because of the nearly divergent redshift factor. Therefore, if the observer is close to the cosmological\nhorizon enough, the intensity emitted near the OSCO is magnified so much that an edge can be seen clearly in the image. In\nshort, the images of the supermassive RN-dS black holes may have significant edge.\nIf the accretion disk is located at the universe A, then the multi-ring structure occurs. The rings insides the photon ring are\nseparated obviously and we can distinguish them easily. The reasons for the formation of the multi-rings structure have been\ndescribed in detail in [45]. The key is that the stable Cauchy horizon allows photons to cross it and the infinitely repetitive\nspacetime allows photons to travel through different universes. As we had mentioned, this breaks the SCCC. In some special\ncases, if the accretion disks are located at both universe AandB, then the image is shown in Fig.13(h). Many rings occur inside\nthe area where is traditionally thought to be the shadow, and the closer to the photon ring, the denser the rings become. And it\nalso has a significant edge. Therefore, if the SCCC is broken down, then this novel phenomenon may be observed. However, the12\nmulti-ring structure not only occurs when there is a stable Cauchy, but also occurs in case of compact objects and wormholes\n[49–52]. In fact, when the effective potential V(r)of the spacetime have a local maximum, the photon sphere (also called critical\ncurve) occurs and there is a bright ring in the image. Furthermore, if there is an another higher maximum inside the outer one,\norVis diverging at somewhere inside the photon sphere, then the multi-ring structure appears [51]. To distinguish compact\nobjects, wormholes and the black hole with a stable Cauchy horizon, we require additional observation.\nB. The appearance of the RN-dS black hole without ISCO and OSCO\nWe have already discussed the situation of RN-dS black hole with an ISCO and an OSCO. However, there are also RN-dS\nblack holes that have no ISCO or OSCO. The accretion disk around these black holes will have no significant edge. In this\nsubsection, we will study the image of these black holes. The luminous intensity of the accretion disk is (3.3). The image of the\nRN-dS black hole with Q= 1.0002,Λ = 0 .003is given by Fig.14. The observer is located at robs= 27.530 = 0 .9rc.\n0.00 0.05 0.10 0.15 0.20 0.25 0.3002468\n(a)\n (b)\n0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07012345\n(c)\n (d)\n0 2 4 6 8 10 120246810\n(e)\n (f)\nFIG. 14: The intensity and the image of the RN-dS black hole with Q= 1.0002,Λ = 0 .003. Figure (a) and (b) are the observed intensity and\nthe image of the RN-dS black hole when the photons are emitted from the accretion disk in universe B. Figure (c) and (d) are the observed\nintensity and the image of the RN-dS black hole when the photons are emitted from the accretion disk in universe A. Figure (e) and (f) are the\nobserved intensity and the image of the RN-dS black hole when the photons are emitted from the accretion disk in both universe AandB.\nFor the RN-dS black hole without ISCO and OSCO, the shape of the image is similar to that of the Schwarzschild one if the\naccretion disk is located at universe B. There is no outer edge in the image. However, for the case of the accretion disk located\nat universe A, or at both AandB, the multi-rings structure appears. Like the image in Fig.13(h), there are many rings inside the13\nshadow. And, similarly, the closer to the photon ring (or lensed ring), the denser the rings become. Compare to Fig.13(f) and\n13(h), the rings in Fig.14(d) and 14(f) are more wider. Besides, there is a little bit of overlap between some of the neighboring\nrings. In [51], it had also mentioned this similar phenomenon that, there are wider rings in the image if the edge of accretion\ndisk is closer to the center of object. Here, to explain this phenomenon, the trajectory of rays has been drawn in Fig.15.\n-10 -5 0 5 10-10-50510\n1 2 3\n(a)\n-10 -5 0 5 10-10-505104 5 6 (b)\nFIG. 15: The trajectory of rays around the RN-dS black hole. The black ring is the event horizon. The black line is the the accretion disk\nbetween ISCO and OSCO, and the dashed black line is the accretion disk between r+andrc. The rays are for n= 7/4(the three solid rays\nlabeled 1 ∼3 in (a)) and n= 9/4(the three dashed rays labeled 4 ∼6 in (b) )\n.\nThe red, green, and blue solid rays are for the third peak (ring) with the normalized number n= 7/4in Fig.14(c). And the\nred, green, and blue dashed rays are for the fourth one with n= 9/4. If the accretion disk is between ISCO and OSCO (black\nline), then the third ring is from b2tob3, where biis the impact parameter of the ray with corresponding number labeled in Fig.\n15. And the fourth ring is from b5tob6. However, if the accretion disk is distribute outside the event horizon (black dashed\nline), then the third ring begins at b1and the fourth ring begins at b4. Therefore, the rings caused by the accretion disk outside\nall the event horizon is wider than the rings caused by the the accretion disk between ISCO and OSCO. Take the green dashed\nray (labeled 5) for an example, it intersects with the accretion disk below twice and the accretion disk above twice. It contributes\nto the third ring and the fourth ring, too. Thus, all six rays intersect with the accretion disk below for n= 7/4, and only the red,\ngreen, and blue dashed rays intersect with accretion disk above for n= 9/4. This means the fourth ring begins when the third\nring is not finished. Therefore, the third and the fourth ring in Fig.14(c) have a little bit of overlap.\nV . CONCLUSIONS AND DISCUSSION\nSCCC is an important subject in general relativity. It guarantees the predictability of the gravitation theory. However, al-\nthough some potential observations violating WCCC is proposed in the RN singularity [60], testing SCCC from astronomical\nobservations has been a challenge. Hence it is significant to study the observational effect of SCCC. The Cauchy horizon is the\nboundary of the region where the initial data can predict. When the Cauchy horizon is stable, one can pass it and arrive at a area\nthat is not determined by the initial data which violates the SCCC. RN-dS black hole is one of the most famous black hole with\na stable Cauchy horizon.\nUnlike Schwarzschild black hole and RN black hole, there may exist an ISCO and an OSCO in RN-dS black hole. The range\nof parameter when there is an ISCO and an OSCO in the RN-dS black hole with an stable Cauchy horizon is shown in Fig.6.\nThe accretion disk distributed between the ISCO and OSCO results in significant inner and outer edges in the image. Taking\ninto account the positive cosmological constant, static observers only exist in the region inside the cosmological horizon. So we\nhave to place the observer between the event horizon and the cosmological horizon and use stereographic projection to obtain\nthe image. To analyse the influence of the observed distant on the image, we first draw up the image of Schwarzschild-dS black\nhole without ISCO and OSCO viewed by observers at different distance.\nWhen the observer is at a finite distance, the observed intensity is truncated, and the corresponding image is clearly bounded.\nThis effect is particularly noticeable when the observer is close to the black hole, but it becomes progressively less distinguishable\nas the distance increases. This is because, for an observer at a finite distance, there is always light with extremely large impact\nparameter that cannot be received. As the observed distance increases, the intensity decreases and then increases, which resulting\nfrom that the denominator of the redshift factor in Eq.(3.2) is nearly zero when the observer is too close to the event horizon and\nthe cosmological horizon. If the observer is far enough away from the black hole, e.g., 0.9rc, the edge may disappear. We also\ndraw up the image of Schwarzschild-dS black hole with an ISCO and an OSCO in Fig.11. The image has an outer edge, too.\nHowever, the cause of this edge is different. Besides the influence of the observed distance, the OSCO would result in an outer14\nedge in the image, too. The reason is that there is no light source outside the OSCO, and this edge always exists no matter how\nfar the observer is.\nIn order to avoid the influence of the observed distance, we draw up the image of RN-dS black hole with an ISCO, an OSCO\nand a stable Cauchy horizon viewed by a far away observer in Fig.13. The observer is located at universe B, while there are\ntwo kinds of light sources. When the photons is emitted from the accretion disk in universe B, the image’s shape is similar\nto that of the Schwarzschild one, except that there is an outer edge in the image of RN-dS black hole, which is resulted from\nthe OSCO. Besides, the photon ring of the RN-dS black hole is smaller than the Schwarzschild one, even though they have the\nphoton spheres of the same size. And the intensity of the RN-dS one is much larger than the Schwarzschild one, due to the\nredshift factor. If the observer is close to the cosmological horizon enough, the redshift factor tends to infinity and the edge of\nthe image is bright enough to be viewed clearly. However, when the accretion disk is located in universe A, the photons can be\nreceived by the observer in universe Bby the black-white hole channel. These photons produce many extra bright rings in the\nimage. When the accretion disk is located in both universe AandB, the image has a outer edge, and the multi-rings structure\ninside the traditional shadow area. The image of the RN-dS black hole without ISCO and OSCO is also shown in Fig.14. The\naccretion disk is distributed between the event horizon and the cosmological horizon, and sharply peaked at event horizon. The\nmulti-rings structure also occurs when there is only one accretion disk in universe A. As a result, the widely distributed accretion\ndisk leads to the wider rings inside the shadow. Some of the neighboring rings even have a little overlap. In conclusion, the\nimage of RN-dS black hole with a stable Cauchy horizon is much different from the Schwarzschild one.\nThe RN-dS black hole with a stable Cauchy horizon is one of the simplest examples inconsistent with SCCC. Based on our\nanalysis, the biggest differences between its image and the Schwarzschild’s are the outer edge and the multi-rings structure.\nAnother example which violating SCCC is the regular black hole with a stable Cauchy horizon. There is also a multi-rings\nstructure in its image [45]. Therefore, if SCCC is broken down, this multi-rings structure may be observed. This provides us a\nnew way to test SCCC in astronomical observation. It is worth noting that such a multi-ring structure can also appear in case of\ncompact objects and wormholes, which have no horizon. And we need other methods to distinguish them.\nWithin classical theory, some black holes have a stable Cauchy horizons. However, certain studies suggest that in such sce-\nnarios, quantum effects will play an important role and eventually render these Cauchy horizons unstable [61–63]. Nevertheless,\nwhether quantum effects will overturn established classical results remains an unresolved issue. Probing the multi-ring structure\ncould help examine the SCCC, and further assessing the significance of quantum effects.\nOur future research will focus on two intriguing directions. Firstly, we aim to study the images of rotating black holes with\nstable Cauchy horizons and further explore the presence of multi-rings structures which is similar to those observed in RN-dS\nimages. Secondly, we plan to investigate other types of black holes that violate the SCCC. The images of these more complex\nblack holes may reveal new types of structures in black hole images beyond what is currently known. We hope that these\ninvestigations will provide new methods for testing the SCCC in astronomical observations.\nAcknowledgement\nThis work was supported in part by the National Natural Science Foundation of China with grants No.12075232 and\nNo.12247103. It is also supported by the National Key R&D Program of China Grant No.2022YFC2204603.\n[1] R. Penrose, Phys. Rev. Lett. 14, 57-59 (1965) doi:10.1103/PhysRevLett.14.57\n[2] S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A 314, 529-548 (1970) doi:10.1098/rspa.1970.0021\n[3] R. Penrose, Riv. Nuovo Cim. 1, 252-276 (1969) doi:10.1023/A:1016578408204\n[4] R. Penrose, Gravitational collapse. In C. Dewitt-Morette, editor, Gravitational Radiation and Gravitational Collapse, volume 64 of IAU\nSymposium, pages 82–91. Springer, 1974.\n[5] R. Wald, Annals Phys. 82, no.2, 548-556 (1974) doi:10.1016/0003-4916(74)90125-0\n[6] J. Sorce and R. M. Wald, Phys. Rev. D 96, no.10, 104014 (2017) doi:10.1103/PhysRevD.96.104014 [arXiv:1707.05862 [gr-qc]].\n[7] R. Penrose, Singularities of Spacetime, in Theoretical Principles in Astrophysics and Relativity, edited by N.R. Lebovitz, W. H. Reid,\nP.O.Vandervoort, 1978\n[8] R. Penrose, “Singularities and time-asymmetry,” in General relativity, and Einstein centenary survey, edited by S.W.Hawking and W.\nIsrael(1979).\n[9] M. Simpson and R. Penrose, Int. J. Theor. Phys. 7, 183-197 (1973) doi:10.1007/BF00792069\n[10] Mcnamara, John. (1977). Royal Society of London Proceedings Series A. 358. 499-517. 10.1098/rspa.1978.0024.\n[11] McNamara John Michael 1978 Proc. R. Soc. Lond. A364121-134\n[12] S. Chandrasekhar, and J. B. Hartle, : 1982, Proceedings of the Royal Society of London Series A 384, 301. doi:10.1098/rspa.1982.0160.\n[13] W. A. Hiscock, Evolution of the interior of a charge dblack hole, Phys. Lett. A, 83(1991).\n[14] E. Poisson and W. Israel, Phys. Rev. D 41, 1796-1809 (1990) doi:10.1103/PhysRevD.41.1796\n[15] A. Ori, Phys. Rev. Lett. 67, 789-792 (1991) doi:10.1103/PhysRevLett.67.789\n[16] L. M. Cao, L. Y . Li, L. B. Wu and Y . S. Zhou, [arXiv:2308.10746 [gr-qc]].\n[17] E. G. Brown, R. B. Mann and L. Modesto, Phys. Rev. D 84, 104041 (2011) doi:10.1103/PhysRevD.84.104041 [arXiv:1104.3126 [gr-qc]].15\n[18] M. Z. Iofa, J. Exp. Theor. Phys. 135, no.5, 647-654 (2022) doi:10.1134/S1063776122110048 [arXiv:2206.09460 [gr-qc]].\n[19] R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio and M. Visser, JHEP 05, 132 (2021) doi:10.1007/JHEP05(2021)132\n[arXiv:2101.05006 [gr-qc]].\n[20] P. R. Brady, D. Nunez and S. Sinha, Phys. Rev. D 47, 4239-4243 (1993) doi:10.1103/PhysRevD.47.4239 [arXiv:gr-qc/9211026 [gr-qc]].\n[21] F. Mellor and I. Moss, Phys. Rev. D 41, 403 (1990) doi:10.1103/PhysRevD.41.403\n[22] V . Cardoso, J. L. Costa, K. Destounis, P. Hintz and A. Jansen, Phys. Rev. Lett. 120, no.3, 031103 (2018)\ndoi:10.1103/PhysRevLett.120.031103 [arXiv:1711.10502 [gr-qc]].\n[23] J. Costa, P. M. Gir ˜ao, J. Nat ´ario and J. D. Silva, J. Phys. Conf. Ser. 600, no.1, 012031 (2015) doi:10.1088/1742-6596/600/1/012031\n[24] J. L. Costa and A. T. Franzen, Annales Henri Poincare 18, no.10, 3371-3398 (2017) doi:10.1007/s00023-017-0592-z [arXiv:1607.01018\n[gr-qc]].\n[25] P. R. Brady and E. Poisson, Class. Quantum Grav. 9121 doi:10.1088/0264-9381/9/1/011\n[26] R. G. Cai and R. K. Su, Phys. Rev. D 52, 666-671 (1995) doi:10.1103/PhysRevD.52.666\n[27] J. L. Costa, P. M. Gir ˜ao, J. Nat ´ario and J. D. Silva, Class. Quant. Grav. 32, no.1, 015017 (2015) doi:10.1088/0264-9381/32/1/015017\n[arXiv:1406.7245 [gr-qc]].\n[28] J. L. Costa, P. M. Gir ˜ao, J. Nat ´ario and J. D. Silva, Commun. Math. Phys. 339, no.3, 903-947 (2015) doi:10.1007/s00220-015-2433-6\n[arXiv:1406.7253 [gr-qc]].\n[29] J. L. Costa, P. M. Gir ˜ao, J. Nat ´ario and J. D. Silva, doi:10.1007/s40818-017-0028-6 [arXiv:1406.7261 [gr-qc]].\n[30] F. Rossetti, [arXiv:2309.14420 [gr-qc]].\n[31] A. Ori, Phys. Rev. Lett. 68, 2117-2120 (1992) doi:10.1103/PhysRevLett.68.2117\n[32] C. M. Chambers, Annals Israel Phys. Soc. 13, 33 (1997) [arXiv:gr-qc/9709025 [gr-qc]].\n[33] K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019) doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238\n[astro-ph.GA]].\n[34] K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L12 (2022) doi:10.3847/2041-8213/ac6674\n[arXiv:2311.08680 [astro-ph.HE]].\n[35] C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, J. Math. Phys. 42, 818-838 (2001) doi:10.1063/1.1308507 [arXiv:gr-qc/0005050\n[gr-qc]].\n[36] S. E. Gralla, D. E. Holz and R. M. Wald, Phys. Rev. D 100, no.2, 024018 (2019) doi:10.1103/PhysRevD.100.024018 [arXiv:1906.00873\n[astro-ph.HE]].\n[37] J. Peng, M. Guo and X. H. Feng, Chin. Phys. C 45, no.8, 085103 (2021) doi:10.1088/1674-1137/ac06bb [arXiv:2008.00657 [gr-qc]].\n[38] I. De Martino, R. Della Monica and D. Rubiera-Garcia, Phys. Rev. D 108, no.12, 124054 (2023) doi:10.1103/PhysRevD.108.124054\n[arXiv:2310.11039 [gr-qc]].\n[39] Z. L. Wang, Eur. Phys. J. Plus 138, no.12, 1131 (2023) doi:10.1140/epjp/s13360-023-04756-x [arXiv:2307.12361 [gr-qc]].\n[40] C. Zhang, Y . Ma and J. Yang, Phys. Rev. D 108, no.10, 104004 (2023) doi:10.1103/PhysRevD.108.104004 [arXiv:2302.02800 [gr-qc]].\n[41] H. Huang, J. Kunz, J. Yang and C. Zhang, Phys. Rev. D 107, no.10, 104060 (2023) doi:10.1103/PhysRevD.107.104060\n[arXiv:2303.11885 [gr-qc]].\n[42] Z. Luo, H. Yu and J. Li, [arXiv:2312.07018 [gr-qc]].\n[43] J. Peng, M. Guo and X. H. Feng, Phys. Rev. D 104, no.12, 124010 (2021) doi:10.1103/PhysRevD.104.124010 [arXiv:2102.05488 [gr-\nqc]].\n[44] S. Guo, G. R. Li and E. W. Liang, Eur. Phys. J. C 83, no.7, 663 (2023) doi:10.1140/epjc/s10052-023-11842-y [arXiv:2210.03010 [gr-qc]].\n[45] L. M. Cao, L. Y . Li, X. Y . Liu and Y . S. Zhou, [arXiv:2312.04301 [gr-qc]].\n[46] T. Berry, A. Simpson and M. Visser, Universe 7, no.1, 2 (2020) doi:10.3390/universe7010002 [arXiv:2008.13308 [gr-qc]].\n[47] P. Boonserm, T. Ngampitipan, A. Simpson and M. Visser, Phys. Rev. D 101, no.2, 024050 (2020) doi:10.1103/PhysRevD.101.024050\n[arXiv:1909.06755 [gr-qc]].\n[48] Y . Song, Eur. Phys. J. C 81, no.10, 875 (2021) doi:10.1140/epjc/s10052-021-09623-6 [arXiv:2108.00696 [gr-qc]].\n[49] G. J. Olmo, D. Rubiera-Garcia and D. S. C. G ´omez, Phys. Lett. B 829, 137045 (2022) doi:10.1016/j.physletb.2022.137045\n[arXiv:2110.10002 [gr-qc]].\n[50] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia and D. G ´omez S ´aez-Chill ´on, Phys. Rev. D 105, no.8, 084057 (2022)\ndoi:10.1103/PhysRevD.105.084057 [arXiv:2202.03809 [gr-qc]].\n[51] M. Guerrero, G. J. Olmo, D. Rubiera-Garcia and D. S ´aez-Chill ´on G ´omez, Phys. Rev. D 106, no.4, 044070 (2022)\ndoi:10.1103/PhysRevD.106.044070 [arXiv:2205.12147 [gr-qc]].\n[52] G. J. Olmo, J. L. Rosa, D. Rubiera-Garcia and D. Saez-Chillon Gomez, Class. Quant. Grav. 40, no.17, 174002 (2023) doi:10.1088/1361-\n6382/aceacd [arXiv:2302.12064 [gr-qc]].\n[53] N. Tsukamoto, Phys. Rev. D 104, no.6, 064022 (2021) doi:10.1103/PhysRevD.104.064022 [arXiv:2105.14336 [gr-qc]].\n[54] N. Tsukamoto, Phys. Rev. D 105, no.8, 084036 (2022) doi:10.1103/PhysRevD.105.084036 [arXiv:2202.09641 [gr-qc]].\n[55] G. Morales-Herrera, P. Ortega-Ruiz, M. Momennia and A. Herrera-Aguilar, [arXiv:2401.07112 [gr-qc]].\n[56] M. Momennia, A. Herrera-Aguilar and U. Nucamendi, Phys. Rev. D 107, no.10, 104041 (2023) doi:10.1103/PhysRevD.107.104041\n[arXiv:2302.11547 [gr-qc]].\n[57] Z. Stuchl ´ık, M. Kolo ˇs, J. Kov ´aˇr, P. Slan ´y and A. Tursunov, Universe 6, no.2, 26 (2020) doi:10.3390/universe6020026\n[58] GRENZEBACH A. The shadow of black holes[M/OL]. Cham: Springer International Publishing, 2016: 55-79.\nhttps://doi.org/10.1007/978-3-319-30066-5 4.\n[59] G. B. Rybicki, Wiley-VCH, 2004, ISBN 978-0-471-82759-7, 978-3-527-61817-0 doi:10.1002/9783527618170\n[60] J. Wagner, [arXiv:2301.13210 [astro-ph.CO]].\n[61] R. G. Cai, Phys. Rev. D 53, 5698-5704 (1996) doi:10.1103/PhysRevD.53.5698\n[62] R. G. Cai, Phys. Rev. D 59, 104004 (1999) doi:10.1103/PhysRevD.59.104004 [arXiv:gr-qc/9810071 [gr-qc]].\n[63] S. Hollands, R. M. Wald and J. Zahn, Class. Quant. Grav. 37, no.11, 115009 (2020) doi:10.1088/1361-6382/ab8052 [arXiv:1912.06047\n[gr-qc]]." }, { "title": "2401.15450v1.Characterization_of_frames_for_source_recovery_from_dynamical_samples.pdf", "content": "arXiv:2401.15450v1 [math.DS] 27 Jan 2024CHARACTERIZATION OF FRAMES FOR SOURCE RECOVERY\nFROM DYNAMICAL SAMPLES\nA. ALDROUBI, R. DIAZ MARTIN, L. GONG, J. MASHREGHI, I. MEDRI\nAbstract. In this paper, we address the problem of recovering constant sou rce\nterms in a discrete dynamical system represented by xn+1=Axn+w, wherexnis\nthen-th state in a Hilbert space H,Ais a bounded linear operator in B(H), andwis\na source term within a closed subspace WofH. Our focus is on the stable recovery\nofwusing time-space sample measurements formed by inner products w ith vectors\nfrom a Bessel system G ⊂ H. We establish the necessary and sufficient conditions for\nthe recovery of wfrom these measurements, independent of the unknown initial sta te\nx0and for any w∈W. This research is particularly relevant to applications such as\nenvironmental monitoring, where precise source identification is cr itical.\n1.Introduction\n1.1.Dynamical sampling for source recovery. The problem we study in this\npaper is the recovery of constant source terms driving a discrete dynamical system,\nusing time-space samples of an evolving physical quantity. Specifica lly, we consider the\nfollowing discrete-time dynamical system:\n(1.1) xn+1=Axn+w, n∈N, w∈W,\nwherexn∈ His then-th state of the system , andHis a separable Hilbert space. The\noperator A∈ B(H) is called the dynamic operator ,w∈W⊆ His thesourceorforcing\nterm, andWis a closed subspace of H. The term x0∈ His called the initial state .\n(Akram Aldroubi) Department of Mathematics, Vanderbilt Un iversity, Nashville,\nTennessee 37240-0001 USA\n(Roc´ıo Mart ´ın D´ıaz) Department of Mathematics, Vanderbilt University, Na shville,\nTennessee 37240-0001 USA\n(Le Gong) Department of Mathematics, Vanderbilt Universit y, Nashville, Ten-\nnessee 37240-0001 USA\n(Javad Mashreghi) Laval University, Qu ´ebec, QC, G1V 0A6, Canada\n(Ivan Medri) Tennessee State University, Department of Com puter Science,\nNashville, TN 37209, USA\nE-mail address :aldroubi@math.vanderbilt.edu, rocio.p.diaz.martin@va nderbilt.edu,\nle.gong@vanderbilt.edu, javad.mashreghi@mat.ulaval.c a, imedri@tnstate.edu .\n2010Mathematics Subject Classification. 46N99, 42C15, 94O20.\nKey words and phrases. Sampling Theory, Forcing, Frames, Reconstruction, Continuous S ampling.\nThe research of Akram Aldroubi and Le Gong is supported in part by grant DMS-2208030. Javad\nMashreghi was supported by grants from the Fulbright Foundatio n and the Canada Research Chair\nprogram.\n12 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREGH I, IVAN MEDRI\nTime-space sample measurements\n(1.2) D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n,j\nare obtained by inner products /an}bracketle{txn,gj/an}bracketri}htwith vectors of a Bessel system G={gj}j≥1⊂\nH, referred to as the set of spatial sampling vectors , and organized in the matrix\nD(x0,w). The data matrix is called the dataof the system (also called the set of time-\nspace samples ,measurements , orobservations ). The problem we will analyze is to find\nnecessary and sufficient conditions for recovering wfrom the data ( 1.2) in astable way ,\nindependent of the unknown x0and for any w∈W. The concept of stability will be\nmade precise in Section 4.\nForexample, if H=ℓ2, thevalue xn(j)represents thevalueofthestateattime nand\nspatial position j. Given an orthonormal basis {bk}k∈KforW, the vector w=/summationtext\nkckbk\ncan be viewed as a weighted sum of source terms bk, each located at positions k∈K,\nwith magnitude ck. A set of spatial sampling vectors G={gj}j≥1, consisting of spatial\nsampling vectors (not necessarily in W), can be used to obtain the space time-samples\n(1.2).\nThe mathematical problem described above is inspired by environmen tal monitoring\napplicationsforidentifying thelocationsandmagnitudeofpollutionso urces. Typically,\nthis necessitates the strategic placement of sensors across diffe rent locations to collect\nrelevant data. For instance, in the topic of air pollution highlighted in [ 1] and related\nworks, thegoal isto determine the emission levels froma specific num ber ofsmokestack\nsources by employing a limited set of sensors.\nThe recent work [ 2], provides necessary and sufficient conditions for recovering a\nstationary source in a finite-dimensional dynamical system. In this article, the under-\nlying space is an infinite-dimensional separable Hilbert space H, and the techniques\nand some of the results are vastly different.\n1.2.Context of this work. The area of Dynamical Sampling has been extensively\nstudied in the literature [ 3–16]. Dynamical sampling problems are connected to several\nareas of mathematics, including control theory, frame theory, f unctional analysis, and\nharmonic analysis [ 17–33], as well as having numerous applications in science and\nengineering [ 34–37]. The three main problems in the area of dynamical sampling are\nas follows.\n(i)System identification : Recovering the dynamic operator Aby knowledge of the\nset of measurements D(x0,w). See, e.g., [ 38–41].\n(ii)Initial state recovery : Assumingthe Aisknownand ω≡0orknown, theobjective\nis to determine the necessary and sufficient conditions on the spatia l sampling set\nof vectors G={gj}j≥1and on the operator Afor recovering the initial condition\nx0in a stable way from the data D(x0,w). This problem is also known as the\n“time-space trade-off in sampling for recovering the initial state” s ince the goal\nis to exploit time to achieve reconstruction even when Gis finite. See, e.g., [ 4,5,\n11,12,16,22,26,27,42].\n(iii)Source recovery : The focus is on identifying specific types of source terms that\ndrive the dynamical system. See, e.g., [ 2,43,44].DYNAMICAL SAMPLING FOR SOURCE RECOVERY 3\nInthiswork, we investigate thepreviously mentioned sourcerecov ery problemwithin\nthe dynamical sampling framework. The key distinction, compared t o those in [ 2,43,\n44], lies in establishing the necessary and sufficient conditions under whic h the set\nG={gj}j≥1enables stable recovery of the sources. Conversely, the cited wo rks [2,\n43,44] prescribe a set Gand provide algorithms for approximating specific types of\ntime-dependent source terms.\n1.3.Organization of the paper. The organization of the paper is as follows. In\nSection2we describe the intrinsic mathematical structure of the dynamical system\nthatisour mainconcerninthiswork. Section 3contains allthemainresults, consisting\noffive major theorems . This part is followed by three consecutive Sections 4,5, and6\nwhich contain the proofs of Main Theorems. In fact, Section 4contains some technical\nlemmas that are interesting in their own right, as well as a detailed des cription of\nB(ℓ2,ℓ∞) andBs(ℓ2,ℓ∞) spaces. The latter is crucial since it is used as the ambient\nspace of measurements in the dynamical system and is exploited in th e stable recovery\nof source file. Finally, Section 7contains a delicate descriptive example showing that\neven if thesource wbelongs toa one-dimensional subspace WwithinH, therecovery of\nwrequires an infinite number of time samples when His an infinite-dimensional space.\nThis example reveals the sharpness of our main results. Section 8constitutes the final\nsegment of the paper, encompassing concluding remarks, potent ial generalizations, and\navenues for future research.\n2.The mathematical description of problem\nAs part of the main problem, given a dynamical system ( 1.1), we wish to recover the\nsource term win a stable way from the data provided in measurements D(x0,w). To\ndescribe the notion of stable reconstruction we need to describe some ambient spaces\nBin which the data sits together with an appropriate norm /bardbl·/bardblBin each case. This\nsetting allows us to describe the reconstruction operator Ras a continuous linear\nmapping from the data space Bto the Hilbert space Hcontaining the source term w.\n2.1.The measurement space. There are two cases of dynamical systems that we\nwill wish to consider. Briefly speaking, they are as follows.\n(i) In the first case, the data matrix D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1is obtained from\nfinitely many iterations, where [ N] ={0,1,2,...,N−1},N≥1.\n(ii) In the second setting, the data matrix D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n≥0,j≥1stems from\ninfinitely many time iterations.\nIn the first case, all data measurements sit in the space B(ℓ2,CN), which can be\ndescribed as the family of all infinite matrices D= [dij] with (finitely many) Nrows\nr1,...,r N, where each row ri= (di1,di2,...)∈ℓ2. This space B(ℓ2,CN) is endowed\nwith the norm\n(2.1) /bardblD/bardblℓ2→CN=N/summationdisplay\ni=1/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n,forD∈ B(ℓ2,CN).4 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREGH I, IVAN MEDRI\nForthesecondcaseofinfinitelymanytimeiterations, wewill usethes paceBs(ℓ2,ℓ∞)\nwhich is a closed subspace of B(ℓ2,ℓ∞). The latter is the family of all infinite matrices\nfor which the norm\n(2.2) /bardblD/bardblℓ2→ℓ∞= sup\ni≥1/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n,forD∈ B(ℓ2,ℓ∞),\nis finite. The former space Bs(ℓ2,ℓ∞) is the closed subspace consisting of matrices\nwhose rows form a Cauchy sequence in ℓ2. More explicitly, we provide the following\ndefinition.\nDefinition 2.3. The space Bs(ℓ2,ℓ∞)is the set of matrices {D= [di,j] :i≥1,j≥1}\nsuch that each row riofDbelongs to ℓ2, and there exists a t∈ℓ2such that limi→∞/bardblri−\nt/bardblℓ2= 0. The norm /bardblD/bardblℓ2→ℓ∞is defined as supi≥1/bardblri/bardblℓ2.\nNotethatduetotheequivalence ofnormsin CN,we mayreplaceN/summationtext\ni=1by sup\n1≤i≤Nin(2.1),\nand soBs(ℓ2,CN) =B(ℓ2,CN). A detailed description of these spaces, in particular, an\nequivalent description of Bs(ℓ2,ℓ∞), is available in Section 4. Throughout the general\ndescription of the spaces B(ℓ2,CN) andB(ℓ2,ℓ∞), we use the index i, commencing from\nthe initial value 1, for the rows of matrices involved in the discussion. However, when\nanalyzing dynamical systems, we adopt a different indexing scheme, mostly denoted\nbynand starting at 0.\n2.2.Generalized source recovery problem. We also treat dynamical systems that\nare a generalized version of ( 1.1). In this general setting, we assume that the states\nxn,n≥1, are obtained via a recursive equation\n(2.4) xn=Fn(x0,...,x n−1,w), n≥1,\nwithwbelonging to a closed subspace WofH. In particular, Fncan be a nonlinear\nfunctional of its arguments. Another prototype example of such a general system is\nxn=An,0x0+···+An,n−1xn−1+Bnw, n ≥1,\nwhereAs andBs are bounded linear operators on H.\nTo present some solid results in the setting ( 2.4), we assume the system satisfies the\nfollowing properties.\n(i) Foreach w∈W, thereisa corresponding unique stationary state . Moreexplicitly,\ngiven any w∈W, there is an initial state x0(w) such that\nxn=x0(w), n≥1.\n(ii) The correspondence between wand its unique stationary state x0(w) is bounded.\nThat is, the mapping S:W→ Hdefined by S(w) :=x0(w) is a bounded linear\noperator, and we call Sthestationary mapping operator .\n(iii) For any source term w∈Wand any arbitrary initial state x0∈ H, we have\nlim\nn→∞xn=S(w),\nwhere the above limit is in /bardbl·/bardblH.DYNAMICAL SAMPLING FOR SOURCE RECOVERY 5\nAs an illustrative example, when considering a dynamical system of th e form (1.1),\nunder the hypothesis ρ(A)<1, we will see that x0(w) = (I−A)−1wis the unique\nstationary state corresponding to a given w. Besides, notice that if Fnis linear and S\nis well-defined, i.e., (i) holds, then Sis necessarily a linear mapping.\nDefinition 2.5. A dynamical system (2.4)satisfying the above properties (i)–(iii) will\nbe referred to by the quadruple (H, W,F,S).\n2.3.Stable recovery. Consider a dynamical system of the form ( 1.1) or of the form\n(H,W,F,S), starting at an arbitrary initial state x0∈ Hwith measurements D(x0,w)\ngiven by sampling through a Bessel sequence G={gj}j≥1inHas in (1.2).\n(i) If there arefinitely many time iterations, we say that the source termw∈W⊆ H\ncan be recovered from the data D(x0,w) in a stable way if there exists a bounded\nlinear operator R:B(ℓ2,CN)→ Hsuch that\nR/parenleftbig\nD(x0,w)/parenrightbig\n=w\nfor allx0∈ Hand allw∈W.\n(ii) Ifwehaveinfinitelymanytimeiterations, wesaythatthesourcet ermw∈W⊆ H\ncan be recovered from the data D(x0,w) in a stable way if there exists a bounded\nlinear operator R:Bs(ℓ2,ℓ∞)→ Hsuch that\nR/parenleftbig\nD(x0,w)/parenrightbig\n=w\nfor allx0∈ Hand allw∈W.\nThe differences between the measurement spaces B(ℓ2,CN) andBs(ℓ2,ℓ∞), and con-\nsequently the emerging reconstruction operators R, are profound and is discussed in\ndepth in the following sections.\n3.Main Results\nIn this section, we gather the main results accompanied by concise d escriptions. In\nlater sections, we provide the proofs.\n3.1.The reconstruction. Our first results reveal the main property of the Bs(ℓ2,ℓ∞)\nspace and its role in the stable reconstruction process.\nTheorem 3.1. LetHbe a separable Hilbert space, and let G={gj}j≥1be any Besselse-\nquence in Hwith optimal Bessel bound CG>0. Then, for each D= [dij]∈ Bs(ℓ2,ℓ∞),\nthe limit\nlimDG:= lim\ni→∞∞/summationdisplay\nj=1dijgj\nexists in Hand, moreover, the mapping\nRG:Bs(ℓ2,ℓ∞)−→ H\nD/ma√sto−→limDG\nis a well-defined bounded operator whose norm is precisely√CG.6 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREGH I, IVAN MEDRI\n3.2.Finite time iterations. In this part, the reconstruction of the source term is\ndone by using a finite number of time iterations of the dynamical syst em (1.1).\nIn our first result, the source wcan be any point of the ambient space H. From\nthe practical point of view, this case is not as interesting as the upc oming restricted\ncase to closed subspaces, since in general, the sources are not loc ated at every spatial\nlocation. However, from the mathematical point of view, it has an ele gant description\nof the solution to the source recovery problem.\nTheorem 3.2. LetHbe a separable Hilbert space, and let G={gj}j≥1be a Bessel\nsequence in H. Considerthe dynamicalsystem (1.1), with an arbitrary initial state x0∈\nH. Then the source term w∈ Hcan be recovered from the measurements D(x0,w) =\n[/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1in a stable way for some 1≤N <∞if and only if G={gj}j≥1is a\nframe for H.\nIn the next result, we restrict the source term to be in a closed sub spaceWofH.\nFrom a practical point of view, despite being mathematically more cha llenging, this is\nthe most interesting case. As a matter of fact, in applications such as environmental\nmonitoring, one has prior knowledge of where the main pollution sourc es are located\nwhich is translated into considering closed subspaces of the ambient spaceH. However,\nthe mathematical description of a solution turns out to be more sub tle.\nTheorem 3.3. LetHbe a separable Hilbert space, and let G={gj}j≥1be a Bessel\nsequence in H. LetWbe a closed subspace of H, letPW:H → Hbe the orthogonal pro-\njection onto W, and assume that source term wbelongs to W. Consider the dynamical\nsystem(1.1), with an arbitrary initial state x0∈ H, and with 1/∈σ(A). If the source\ntermw∈Wcan be recovered from the measurements D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1in\na stable way for some 1≤N <∞, then{PW(I−A∗)−1gj}j≥1is a frame for W.\nThere is an example in Section 7, that shows that in the case where W/subsetnoteqlH, the\nsourcewcannot be recovered from finitely many iterations N. Thus, on the one hand,\nthis example reveals the sharpness of Theorem 3.3. On the other hand, in the next\nsubsection, we consider the case where W/subsetnoteqlHand the data D(x0,w) consists of\ninfinitely many time samples.\n3.3.Infinite time iterations. Dynamical system ( 1.1) is a special of the more gen-\neral dynamical system ( 2.4). In fact, the Model ( 2.4) can be even nonlinear. The\nfollowing theorem provides a full characterization of stable recons truction from data\nmeasurements D(x0,w) for the system ( 2.4).\nTheorem 3.4. LetHbe a separable Hilbert space, let Wbe a closed subspace of\nH, and let G={gj}j≥1be a Bessel sequence in H. Consider the dynamical system\n(H, W,F,S)(see Definition 2.5) with any initial state x0∈ H, and assume that F\nis linear. Then, each source term w∈Wcan be recovered from the measurements\nD(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n≥0,j≥1in a stable way if and only if {S∗gj}j≥1is a frame for W.\nAs a special case of Theorem 3.4, we characterize stable reconstruction for the dy-\nnamical system ( 1.1) created with an operator Awith the spectral radius ρ(A)<1.DYNAMICAL SAMPLING FOR SOURCE RECOVERY 7\nTheorem 3.5. LetHbe a separable Hilbert space, let Wbe a closed subspace of H, and\nletG={gj}j≥1be a Bessel sequence in H. Consider the dynamical system (1.1)with\nan arbitrary initial state x0∈ H, and with ρ(A)<1. Then each source term w∈W\nof the system can be recovered from the measurements D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n≥0,j≥1in a\nstable way if and only if {PW(I−A∗)−1gj}j≥1is a frame for W.\n4.Stable reconstruction and the proof of Theorem 3.1\nBefore presenting the proof of Main Theorems, we need to develop the required\nmathematical background. In particular, our main goal is to provid e a clear foundation\nfor the notion of stable reconstruction which was exploited in the announcement of\ntheorems. This important concept needs a detailed discussion, whic h is fulfilled in this\nsection.\nLemma 4.1. LetDbe a bounded operator from the sequence Hilbert space ℓ2to the\nsequence Banach algebra ℓ∞. Let[dij]be the matrix representation of Dwith respect\nto the canonical basis (en)n≥1, i.e.,\nDej=∞/summationdisplay\ni=1dijei, j≥1.\nThen\n/bardblD/bardblℓ2→ℓ∞= sup\ni≥1/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n.\nProof.Letx= (xj)j≥1∈ℓ2. Then, by Cauchy–Schwartz inequality,\n/bardblDx/bardblℓ∞= sup\ni≥1|(Dx)i|= sup\ni≥1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\nj=1dijxj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\ni≥1/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n/bardblx/bardblℓ2.\nHence,\n/bardblD/bardblℓ2→ℓ∞≤sup\ni≥1/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n.\nFor the reverse inequality, fix a row iinD. Define the vector x= (xj)j≥1by\nxj:=\n\n|dij|2/dijifdij/ne}ationslash= 0,\n0 if dij= 0.\nThese coefficients are designed to have\n(Dx)i=∞/summationdisplay\nj=1dijxj=∞/summationdisplay\nj=1|dij|2,\nand, at the same time,\n/bardblx/bardblℓ2=/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n.8 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREGH I, IVAN MEDRI\nThus,\n/bardblDx/bardblℓ∞\n/bardblx/bardblℓ2≥|(Dx)i|\n/bardblx/bardblℓ2=/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n,\nwhich gives\n/bardblD/bardblℓ2→ℓ∞≥/parenleftBigg∞/summationdisplay\nj=1|dij|2/parenrightBigg1/2\n.\n/square\nWe denote the family of all bounded linear operators from ℓ2toℓ∞byB(ℓ2,ℓ∞).\nLemma4.1characterizes such operators as those whose rows, in the canon ical matrix\nrepresentation, are uniformly bounded in ℓ2. We define a closed subspace of B(ℓ2,ℓ∞)\nand use it to tackle the question of stable reconstruction. The pre cise definition is as\nfollows.\nLetHbe a separable Hilbert space. Then Bs(ℓ2,ℓ∞) (where sstands for strong)\nconsists of all operators D∈ B(ℓ2,ℓ∞) such that the limit\n(4.2) lim\ni→∞∞/summationdisplay\nj=1dijgj\nexists in the norm of Hfor any Bessel sequence G={gj}j≥1⊂ H. By the unitary\nequivalence of separable Hilbert spaces, the definition does not dep end onH. Also\nnote that since Gis a Bessel sequence and each row of Dis inℓ2, the sum in ( 4.2) is\nwell-defined and represents an element of H. In the following, we will write lim DGfor\nthe limit in ( 4.2). More explicitly, whenever D∈ Bs(ℓ2,ℓ∞) andGis a Bessel sequence\ninH, we will write\n(4.3) lim DG:= lim\ni→∞∞/summationdisplay\nj=1dijgj\nLemma 4.4. LetD= [dij]∈ B(ℓ2,ℓ∞). ThenD∈ Bs(ℓ2,ℓ∞)if and only if the rows\nofDare norm convergent in ℓ2, i.e., there is a vector t∈ℓ2such that\nlim\ni→∞/bardblri−t/bardblℓ2= 0,\nwhereri:= (di1,di2,...)denotes the i-th row of D.\nProof.Assume that D∈ Bs(ℓ2,ℓ∞). If we consider even a single sequence Gwhich is\nan orthonormal basis in H, then the series in ( 4.2) is unitarily equivalent to the vector\nri:= (di1,di2,...)∈ℓ2. Thus, the assumption on the existence of a limit precisely\nmeans that the rows are convergent in ℓ2.\nConversely, let D∈ B(ℓ2,ℓ∞), assume that its rows are convergent in ℓ2norm to a\nvectort= (t1,t2,...)∈ℓ2, and fix an arbitrary Bessel sequence G={gj}j≥1⊂ H.\nThen,\nh:=∞/summationdisplay\nj=1tjgjDYNAMICAL SAMPLING FOR SOURCE RECOVERY 9\nis a well-defined element of H. Moreover,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh−∞/summationdisplay\nj=1dijgj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1tjgj−∞/summationdisplay\nj=1dijgj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH\n=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1(tj−dij)gj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH\n≤/radicalbig\nCG/bardblt−ri/bardblℓ2,\nwhereCGis the optimal bound of the Bessel sequence G(see, for e.g. [ 45, Thm. 3.2.3]).\nBy assumption, /bardblt−ri/bardblℓ2→0, asi→ ∞. Hence,\nlim\ni→∞∞/summationdisplay\nj=1dijgj=h,\nwhich means that D∈ Bs(ℓ2,ℓ∞). /square\nCorollary 4.5. Bs(ℓ2,ℓ∞)is a closed subspace of B(ℓ2,ℓ∞).\nProof.LetD(n),n≥1, be a sequence in Bs(ℓ2,ℓ∞) which converges to D∈ B(ℓ2,ℓ∞)\nin the topology of B(ℓ2,ℓ∞). Denote the i-th rows of D(n)andDrespectively by r(n)\ni\nandri. Write\nri−ri′= (ri−r(n)\ni)+(r(n)\ni′−ri′)+(r(n)\ni−r(n)\ni′).\nFor the first two terms on the right side we have\n/bardblri−r(n)\ni/bardblℓ2≤ /bardblD−D(n)/bardblℓ2→ℓ∞,\nand\n/bardblri′−r(n)\ni′/bardblℓ2≤ /bardblD−D(n)/bardblℓ2→ℓ∞,\nTherefore, given ε >0, we take nso large that /bardblD−D(n)/bardblℓ2→ℓ∞≤ε. Hence,\n/bardblri−ri′/bardblℓ2≤2ε+/bardblr(n)\ni−r(n)\ni′/bardblℓ2.\nNow, by Lemma 4.4applied to D(n), there is and index i0such that\n/bardblr(n)\ni−r(n)\ni′/bardblℓ2< ε, i,i′> i0.\nTherefore, for all i,i′> i0, we will have\n/bardblri−ri′/bardblℓ2≤3ε.\nThis means that the rows of Dare norm Cauchy, and thus norm convergent. Hence,\none again using Lemma 4.4, we conclude that D∈ Bs(ℓ2,ℓ∞). /square\nAll the previous results were designed to arrive at Theorem 3.1which is a funda-\nmental result. We are now able to prove this theorem.10 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\n4.1.Proof of Theorem 3.1.Fix anyD∈ Bs(ℓ2,ℓ∞). Letri:= (di1,di2,...) denotes\nthei-th row of D. Define\n(DG)i:=∞/summationdisplay\nj=1dijgj, i≥1.\nThen,\n/bardbl(DG)i/bardblH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1dijgj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH≤/radicalbig\nCG/bardblri/bardblℓ2≤/radicalbig\nCG/bardblD/bardblℓ2→ℓ∞, i≥1.\nLeti→ ∞to obtain\n/bardblRG(D)/bardblH=/bardbllimDG/bardblH≤/radicalbig\nCG/bardblD/bardblℓ2→ℓ∞.\nTherefore, the operator RGis bounded and\n/bardblRG/bardblBs(ℓ2,ℓ∞)→H≤/radicalbig\nCG.\nTo prove the reverse inequality, let Dbe the operator in Bs(ℓ2,ℓ∞) whose rows are\nall equal to a fixed vector\nd= (d1,d2,...)∈ℓ2.\nThen clearly\n/bardblD/bardblℓ2→ℓ∞=/bardbld/bardblℓ2,\nand\nRG(D) = limDG=∞/summationdisplay\nj=1djgj.\nHence, the inequality\n/bardblRG(D)/bardblH≤ /bardblRG/bardblBs(ℓ2,ℓ∞)→H/bardblD/bardblℓ2→ℓ∞\ntransforms to/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1djgj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH≤ /bardblRG/bardblBs(ℓ2,ℓ∞)→H/bardbld/bardblℓ2, d∈ℓ2.\nBut, since d∈ℓ2is arbitrary, this estimation implies\n/radicalbig\nCG≤ /bardblRG/bardblBs(ℓ2,ℓ∞)→H.\n/square\n5.Proofs of theorems for finite time iterations\nLemma 5.1. Consider the dynamical system (1.1)with any initial state x0∈ H.\nGiven a Bessel sequence G={gj}j≥1⊂ Hand1≤N <∞, then the data matrix\nD(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1belongs to B(ℓ2,CN).DYNAMICAL SAMPLING FOR SOURCE RECOVERY 11\nProof.The proof holds since G={gj}j≥1⊂ His a Bessel sequence. Indeed, if CGis\nthe optimal Bessel bound, then\n/bardblD(x0,w)/bardblℓ2→CN=N−1/summationdisplay\nn=0∞/summationdisplay\nj=1|/an}bracketle{txn,gj/an}bracketri}ht|2≤CGN−1/summationdisplay\nn=0/bardblxn/bardbl2\nH<∞.\n/square\nProposition 5.2. Consider the dynamical system (1.1), and let 1≤N <∞. Given\na frameG={gj}j≥1forH, consider the data operator Ddefined by\nD:H×H −→ B (ℓ2,CN)\nD(x0,w) := [/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1.\nThen the image of Dis a closed subspace in B(ℓ2,CN).\nProof.Considerasequence {(xk\n0,wk)}k∈NinH×Hsuchthatthesequence {D(xk\n0,wk)}k∈N\nconverges in B(ℓ2,CN). Letr∈ B(ℓ2,CN) be such limit\nlim\nk→∞D(xk\n0,wk) =r.\nWe recall that, by definition, ris of the form r:= (r0,r1,...,r N−1) where, for each\n0≤n≤N−1,rn:= (rn1,rn2,...)∈ℓ2.\nSinceG={gj}j≥1is a frame for H, the image of the analysis operator\nAG:H −→ℓ2\nAG(h) := (/an}bracketle{th,gj/an}bracketri}ht)j≥1\nis a closed linear subspace in ℓ2. Notice that\n(5.3) r0= lim\nk→∞AG(xk\n0)\nwhere the above limit is in ℓ2-norm. Since the image of AGis closed, there exists\nx0∈ Hsuch that\nr0=AG(x0) =/parenleftbig\n/an}bracketle{tx0,g1/an}bracketri}ht,/an}bracketle{tx0,g2/an}bracketri}ht,.../parenrightbig\n.\nSinceGis a frame with optimal lower and upper bounds cGandCG, we obtain that\n/bardblxk\n0−x0/bardbl2\nH≤1\ncG∞/summationdisplay\nj=1|/an}bracketle{txk\n0−x0,gj/an}bracketri}ht|2=1\ncG/bardblAG(xk\n0)−AG(x0)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nr0/bardbl2\nℓ2,\nand thus, by ( 5.3),\n(5.4) lim\nk→∞/bardblxk\n0−x0/bardbl2\nH≤lim\nk→∞1\ncG/bardblAG(xk\n0)−r0/bardbl2\nℓ2= 0.\nThis means that {xk\n0}k∈Nconverges to x0inH.\nSimilarly,\nr1= lim\nk→∞AG(Axk\n0+wk) =AG(Ax0)+ lim\nk→∞AG(wk).12 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\nIn particular {AG(wk)}k∈Nconverges in ℓ2, and since the image of AGis closed in ℓ2,\nthere exists w∈ Hsuch that\nlim\nk→∞/bardblAG(wk)−AG(w)/bardbl2\nℓ2= 0.\nBy repeating the argument given in ( 5.4) we see that {wk}k∈Nconverges to winH.\nThus,\nr1=AG(Ax0+w).\nFinally, as the n-th state of the dynamical system ( 1.1) with initial state xk\n0∈ Hand\nsourcewk∈ Hcan be written as\nxk\nn=Anxk\n0+(I+A+···+An−1)wk, n≥1,\nand, analogously, the n-th state of the dynamical system ( 1.1) with initial state x0∈ H\nand source w∈ Hcan be written as\nxn=Anx0+(I+A+···+An−1)w, n ≥1,\nwe have\nrn= lim\nk→∞AG(xk\nn) = lim\nk→∞AG/parenleftbig\nAnxk\n0+(I+A+···+An−1)wk/parenrightbig\n=AG/parenleftbig\nAnx0+(I+A+···+An−1)w/parenrightbig\n=AG(xn).\nThis implies that\nlim\nk→∞D(xk\n0,wk) =r=D(x0,w),\nthat is,rbelongs to the image of the data operator D. /square\n5.1.Proof of Theorem 3.2.Consider the dynamical system ( 1.1), and suppose that\na stable recovery is possible in Ntime of iterations. More explicitly, for any x0∈ H\nand anyw∈ H, the source wcan be recovered by applying a bounded linear operator\nR:B(ℓ2,CN)→ Hto the measurements D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n∈[N],j≥1, whereG=\n{gj}j≥1is a Bessel sequence in H. Hence, there exists a positive constant Csuch that\n(5.5) /bardblR(D)/bardbl2\nH≤CN−1/summationdisplay\nn=0∞/summationdisplay\nj=1|dnj|2, D∈ B(ℓ2,ℓ∞),\nand\n(5.6) R(D(x0,w)) =w, x 0,w∈ H.\nOur objective is to show that G={gj}j≥1is a frame for H.\nAccording to ( 5.5) and (5.6), we have\n/bardblw/bardbl2\nH≤C/parenleftBigg∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2+···+∞/summationdisplay\nj=1|/an}bracketle{txN−1,gj/an}bracketri}ht|2/parenrightBigg\n. (5.7)\nHence, by ( 1.1), we obtain\n(5.8) /bardblw/bardbl2\nH≤C\n∞/summationdisplay\nj=1N−1/summationdisplay\nn=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg\nAnx0+n−1/summationdisplay\nk=0Akw,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,DYNAMICAL SAMPLING FOR SOURCE RECOVERY 13\nwhere ifn= 0, we understand Anx0+/summationtextn−1\nk=0Aksimply as x0.\nNext, for x0∈ H, we choose w= (I−A)x0, and we substitute it into ( 5.8). After\nsome simplifications, the above relation is rewritten as\n/bardbl(I−A)x0/bardbl2\nH≤NC∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2, x 0∈ H.\nRepeated use of this estimation implies\n/bardbl(I−A)kx0/bardbl2\nH≤NC∞/summationdisplay\nj=1|/an}bracketle{t(I−A)k−1x0,gj/an}bracketri}ht|2\n≤NCCG/bardbl(I−A)k−1x0/bardbl2\nH\n≤N2C2CG∞/summationdisplay\nj=1|/an}bracketle{t(I−A)k−2x0,gj/an}bracketri}ht|2\n...\n≤Ck∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2, x 0∈ H, (5.9)\nwhere we have used the fact that G={gj}j≥1is a Bessel sequence in Hwith optimal\nboundCG, and where the constant Ck=NkCkCk−1\nGdepends on constant C, on the\nnumber of time iterations of the dynamical system N, and on the power k≥1.\nWe go back to ( 5.8) again, but this time we consider x0= 0. If we write Aas\nA=I−(I−A), then the expressionn−1/summationtext\nk=0Akwcan be rewritten as\nn−1/summationdisplay\nk=0Akw=n−1/summationdisplay\nk=0αn,k(I−A)kw,1≤n≤N,\nfor some complex coefficients αn,kwhose precise values are not relevant here. Thus, we\nget\n/bardblw/bardbl2\nH≤C\n∞/summationdisplay\nj=1N−1/summationdisplay\nn=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg\nAnx0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n0+n−1/summationdisplay\nk=0Akw,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n=C\n∞/summationdisplay\nj=1N−1/summationdisplay\nn=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBiggn−1/summationdisplay\nk=0Akw,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n=C\n∞/summationdisplay\nj=1N−1/summationdisplay\nn=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBiggn−1/summationdisplay\nk=0αn,k(I−A)kw,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n≤C(sup|αn,k|)2/parenleftBigg∞/summationdisplay\nj=1N−1/summationdisplay\nn=1n−1/summationdisplay\nk=0/vextendsingle/vextendsingle/angbracketleftbig\n(I−A)kw,gj/angbracketrightbig/vextendsingle/vextendsingle2/parenrightBigg14 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\n≤C(sup|αn,k|)2(N−1)/parenleftBigg∞/summationdisplay\nj=1N−2/summationdisplay\nk=0/vextendsingle/vextendsingle/angbracketleftbig\n(I−A)kw,gj/angbracketrightbig/vextendsingle/vextendsingle2/parenrightBigg\n=C′/parenleftBigg∞/summationdisplay\nj=1N−2/summationdisplay\nk=0|/an}bracketle{t(I−A)kw,gj/an}bracketri}ht|2/parenrightBigg\n≤C′CGN−2/summationdisplay\nk=1/bardbl(I−A)kw/bardbl2\nH+C′/parenleftBigg∞/summationdisplay\nj=1|/an}bracketle{tw,gj/an}bracketri}ht|2/parenrightBigg\n≤C′′∞/summationdisplay\nj=1|/an}bracketle{tw,gj/an}bracketri}ht|2,\nwhereC′=C(N−1)(sup{|αn,k|: 0≤k≤n−1,1≤n≤N−1})2, the third\ninequality again stems from the fact that G={gj}j≥1is a Bessel sequence in H, and\nthe last inequality holds because of ( 5.9) withC′′=C′(CGN−2/summationtext\nk=1Ck+1). Therefore,\n1\nC′′/bardblw/bardbl2\nH≤∞/summationdisplay\nj=1|/an}bracketle{tw,gj/an}bracketri}ht|2≤CG/bardblw/bardbl2\nH, w∈ H\nand we finally conclude that G={gj}j≥1is a frame for H.\nNowweprovetheconversestatement: wcanberecoveredinastableway G={gj}j≥1\nis a frame for H.\nIfG={gj}j≥1is a frame for H, there exists constants cG,CG>0 such that\n(5.10) cG/bardblh/bardbl2\nH≤∞/summationdisplay\nj=1|/an}bracketle{th,gj/an}bracketri}ht|2≤CG/bardblh/bardbl2\nH, h∈ H.\nAdditionally, let /tildewideG={/tildewidegi}i≥1be the canonical dual frame of G={gj}j≥1. Then, for\neachj≥1,\n(5.11) A∗gj=∞/summationdisplay\ni=1aijgi,where aij=/an}bracketle{tA∗gj,/tildewidegi/an}bracketri}ht.\nGiven the dynamical system ( 1.1), we have\n/an}bracketle{tw,gj/an}bracketri}ht=/an}bracketle{txn+1,gj/an}bracketri}ht−/an}bracketle{tAxn,gj/an}bracketri}ht\n=/an}bracketle{txn+1,gj/an}bracketri}ht−/an}bracketle{txn,A∗gj/an}bracketri}ht\n=/an}bracketle{txn+1,gj/an}bracketri}ht−/angbracketleftBigg\nxn,∞/summationdisplay\ni=1aijgi/angbracketrightBigg\n=/an}bracketle{txn+1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{txn,gi/an}bracketri}ht. (5.12)\nNotice that ( 5.12) holds for any two consecutive states xnandxn+1.DYNAMICAL SAMPLING FOR SOURCE RECOVERY 15\nSinceGis a frame for H,w∈ Hcan be written as\nw=∞/summationdisplay\nj=1/an}bracketle{tw,gj/an}bracketri}ht/tildewidegj.\nBy (5.12), every coefficient /an}bracketle{tw,gj/an}bracketri}htcan be written in terms of the measurements\n{/an}bracketle{txn,gj/an}bracketri}ht,/an}bracketle{txn+1,gj/an}bracketri}ht}j≥1,\nfor all values of n≥0. Therefore, we have the following reconstruction expression fo r\nwin terms of the measurements of the dynamical system ( 1.1):\n(5.13) w=∞/summationdisplay\nj=1/parenleftBigg\n/an}bracketle{txn+1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{txn,gi/an}bracketri}ht/parenrightBigg\n/tildewidegj.\nIn particular, one can consider the space samples of the first two s tates of the system\nx0andx1and get\n(5.14) w=∞/summationdisplay\nj=1/parenleftBigg\n/an}bracketle{tx1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/parenrightBigg\n/tildewidegj.\nMoreover, from ( 5.10) and (5.12), we have\n∞/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=∞/summationdisplay\nj=1|/an}bracketle{tAx0,gj/an}bracketri}ht|2\n≤CG/bardblAx0/bardbl2\nH≤CG/bardblA/bardbl2\nH→H/bardblx0/bardbl2\nH\n≤CG\ncG/bardblA/bardbl2\nH→H∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2, (5.15)\nand thus\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1/parenleftBigg\n/an}bracketle{tx1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/parenrightBigg\n/tildewidegj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH≤1\ncG∞/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}bracketle{tx1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤2\ncG∞/summationdisplay\nj=1\n|/an}bracketle{tx1,gj/an}bracketri}ht|2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n≤2\ncG∞/summationdisplay\nj=1|/an}bracketle{tx1,gj/an}bracketri}ht|2+2CG\nc2\nG/bardblA/bardbl∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2\n≤/tildewideC/parenleftBigg∞/summationdisplay\nj=1|/an}bracketle{tx1,gj/an}bracketri}ht|2+∞/summationdisplay\nj=1|/an}bracketle{tx0,gj/an}bracketri}ht|2/parenrightBigg\n, (5.16)\nwhere/tildewideC= 2max{1\ncG,CG\nc2\nG/bardblA/bardbl2\nH→H}. These estimations imply the boundedness of the\nreconstruction formula ( 5.14).16 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\nWe are now able to define the operator R:B(ℓ2,C2)−→ Hby\nR(D) =∞/summationdisplay\nj=1/parenleftBigg\nd1j−∞/summationdisplay\ni=1aij/angbracketleftBigg∞/summationdisplay\nk=1d0k/tildewidegk,gi/angbracketrightBigg/parenrightBigg\n/tildewidegj,\nwhere we recall that [2] = {0,1}andD= [dnj]n∈[2],j≥1. Equivalently, by ( 5.11), the\nabove operator can be re-written as\nR(D) =∞/summationdisplay\nj=1/parenleftBigg\nd1j−/angbracketleftBigg∞/summationdisplay\nk=1d0k/tildewidegk,A∗gj/angbracketrightBigg/parenrightBigg\n/tildewidegj\n=∞/summationdisplay\nj=1/parenleftBigg\nd1j−/angbracketleftBigg\nA/parenleftBigg∞/summationdisplay\nk=1d0k/tildewidegk/parenrightBigg\n,gj/angbracketrightBigg/parenrightBigg\n/tildewidegj. (5.17)\nBy using ( 5.17) and the same bounds as in ( 5.15) and (5.16) we can show that Ris a\nwell-defined bounded operator. Indeed,\n/bardblR(D)/bardbl2\nH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1/parenleftBigg\nd1j−/angbracketleftBigg\nA/parenleftBigg∞/summationdisplay\nk=1d0k/tildewidegk/parenrightBigg\n,gj/angbracketrightBigg/parenrightBigg\n/tildewidegj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH\n≤/tildewideC\n∞/summationdisplay\nj=1|d1j|2+∞/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg∞/summationdisplay\nk=1d0k/tildewidegk,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n≤/tildewideC/parenleftBigg∞/summationdisplay\nj=1|d1j|2+∞/summationdisplay\nj=1|d0j|2/parenrightBigg\n=/tildewideC/bardblD/bardblB(ℓ2,C2).\nTo justify the last inequality, let u:=/summationtext∞\nk=1d0k/tildewidegk∈ H. Since{gj}j≥1and{/tildewidegj}j≥1are\na pair of canonical dual frames, we also have that u=/summationtext∞\nj=1/an}bracketle{tu,gj/an}bracketri}ht/tildewidegj. By [46, Lemma\nVIII]), the coefficients ( /an}bracketle{tu,gj/an}bracketri}ht)j≥1have the least ℓ2-norm. In particular,\n∞/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg∞/summationdisplay\nk=1d0k/tildewidegk\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nu,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=∞/summationdisplay\nj=1|/an}bracketle{tu,gj/an}bracketri}ht|2≤∞/summationdisplay\nj=1|d0j|2.\nFinally, the operator Ris linear, and it is a reconstruction operator since by ( 5.14) we\nhave\nR/parenleftbig\n[/an}bracketle{txn,gj/an}bracketri}ht]n∈[2],j≥1/parenrightbig\n=∞/summationdisplay\nj=1\n/an}bracketle{tx1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/angbracketleftBigg∞/summationdisplay\nk=1/an}bracketle{tx0,gk/an}bracketri}ht/tildewidegk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nx0,gi/angbracketrightBigg\n/tildewidegj.\n=∞/summationdisplay\nj=1/parenleftBigg\n/an}bracketle{tx1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{tx0,gi/an}bracketri}ht/parenrightBigg\n/tildewidegj=w,\nwhich concludes the proof.\n/squareDYNAMICAL SAMPLING FOR SOURCE RECOVERY 17\n5.2.Proof of Theorem 3.3.Consider the dynamical system ( 1.1) and suppose that\nfor eachx0∈ Handw∈Wwe achieve stable recovery of the source term win finite\ntimeNby measuring the states of the system with a Bessel sequence G={gj}j≥1⊂ H.\nPrecisely, the source w∈Wcan be recovered by applying a bounded linear operator\nR:B(ℓ2,CN)→ H. That is, Rsatisfies ( 5.5) and\n(5.18) R(D(x0,w)) =w, x 0∈ H, w∈W.\nOur objective is to show that, {PW(I−A∗)−1gj}j≥1is a frame for W.\nAccording to ( 5.5) and (5.18), we have\n/bardblw/bardbl2\nH≤CN−1/summationdisplay\nn=0∞/summationdisplay\nj=1|/an}bracketle{txn,gj/an}bracketri}ht|2\n=C\n∞/summationdisplay\nj=1N−1/summationdisplay\nn=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftBigg\nAnx0+n−1/summationdisplay\ni=0Aiw,gj/angbracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n\n=C/parenleftBigg∞/summationdisplay\nj=1N−1/summationdisplay\nn=0|/an}bracketle{tAnx0+(I−An)(I−A)−1w,gj/an}bracketri}ht|2/parenrightBigg\n, w∈W,\nwhere we have used that the states of ( 1.1) can be expressed as\n(5.19) xn=Anx0+(I+A+···+An−1)w, n ≥1.\nGiven any w∈W, by considering x0= (I−A)−1wand substituting it into the above\ninequality we get\n/bardblw/bardbl2\nH≤NC∞/summationdisplay\nj=1|/an}bracketle{t(I−A)−1w,gj/an}bracketri}ht|2≤C′/bardbl(I−A)−1w/bardbl2≤C′′/bardblw/bardbl2, (5.20)\nwhere in the second inequality follows since G={gj}j≥1is a Bessel sequence in H,\nand the last inequality holds because of ( I−A)−1is a bounded linear operator on\nH. Note that, if CGdenotes the optimal Bessel bound for G, thenC′=NCCGand\nC′′=C′/bardbl(I−A)−1/bardblH→H.\nSince\n∞/summationdisplay\nj=1|/an}bracketle{t(I−A)−1w,gj/an}bracketri}ht|2=∞/summationdisplay\nj=1|/an}bracketle{tw,(I−A∗)−1gj/an}bracketri}ht|2=∞/summationdisplay\nj=1|/an}bracketle{tw,PW(I−A∗)−1gj/an}bracketri}ht|2,\nfrom (5.20) we have\n1\nNC/bardblw/bardbl2\nH≤∞/summationdisplay\nj=1|/an}bracketle{tw,PW(I−A∗)−1gj/an}bracketri}ht|2≤C′′/bardblw/bardbl2, w∈W,\nand so{PW(I−A∗)−1gj}j≥0is a frame for the subspace W. /square18 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\n6.Proofs of theorems for infinite time iterations\nLemma 6.1. Consider the dynamical system (H, W,F,S)(see Definition 2.5) with\nany initial state x0∈ H. IfG={gj}j≥1⊂ His a Bessel sequence, then the data matrix\nD(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n≥0,j≥1belongs to Bs(ℓ2,ℓ∞).\nProof.By assumption, the states {xn}of the dynamical system ( 2.4) satisfy\n(6.2) /bardblxn−S(w)/bardblH→0, n→ ∞.\nThen-th row of D(x0,w) is\nrn:=/parenleftbig\n/an}bracketle{txn,g1/an}bracketri}ht,/an}bracketle{txn,g2/an}bracketri}ht, .../parenrightbig\n.\nDefine\nr:=/parenleftbig\n/an}bracketle{tS(w),g1/an}bracketri}ht,/an}bracketle{tS(w),g2/an}bracketri}ht, .../parenrightbig\n.\nSinceG={gj}j≥1is a Bessel sequence in H(with optimal Bessel bound CG), by (6.2),\nwe have\n/bardblrn−r/bardbl2\nℓ2=∞/summationdisplay\nj=1|/an}bracketle{txn,gj/an}bracketri}ht−/an}bracketle{tS(w),gj/an}bracketri}ht|2\n=∞/summationdisplay\nj=1|/an}bracketle{t(xn−S(w)),gj/an}bracketri}ht|2\n≤CG/bardblxn−S(w)/bardbl2\nH→0, n→ ∞. (6.3)\nHence, by Lemma 4.4,D(x0,w)∈ Bs(ℓ2,ℓ∞). /square\n6.1.Proof of Theorem 3.4.Fix any x0∈ Hand any source w∈W, and put\nD(x0,w) :=/bracketleftbig\n/an}bracketle{txi,gj/an}bracketri}ht/bracketrightbig\nn≥0,j≥1. By Lemma 6.1,D(x0,w)∈ Bs(ℓ2,ℓ∞).\nSuppose that there is a bounded linear operator R:Bs(ℓ2,ℓ∞)→ Hsuch that\nR/parenleftbig\nD(x0,w)/parenrightbig\n=w∀x0∈ H, w∈W.\nHence,\n(6.4) /bardblw/bardblH≤ /bardblR/bardbl Bs(ℓ2,ℓ∞)→Hsup\nn≥0/parenleftBigg∞/summationdisplay\nj=1|/an}bracketle{txn,gj/an}bracketri}ht|2/parenrightBigg1/2\n, x 0∈ H, w∈W.\nSincex0∈ His arbitrary, we can take x0=S(w). Hence, xn=S(w) for alln≥0,\nand (6.4) becomes\n/bardblw/bardblH≤ /bardblR/bardbl Bs(ℓ2,ℓ∞)→H/parenleftBigg∞/summationdisplay\nj=1|/an}bracketle{tS(w),gj/an}bracketri}ht|2/parenrightBigg1/2\n, w∈W,\nwhich we rewrite as\nm/bardblw/bardbl2\nH≤∞/summationdisplay\nj=1|/an}bracketle{tw,S∗gj/an}bracketri}ht|2, w∈WDYNAMICAL SAMPLING FOR SOURCE RECOVERY 19\nwithm=/bardblR/bardbl−2\nBs(ℓ2,ℓ∞)→H. Finally, since Gis a Bessel sequence (with optimal Bessel\nboundCG), we also have\n∞/summationdisplay\nj=1|/an}bracketle{tw,S∗gj/an}bracketri}ht|2=∞/summationdisplay\nj=1|/an}bracketle{tS(w), gj/an}bracketri}ht|2≤CG/bardblS(w)/bardbl2\nH\n≤CG/bardblS/bardbl2\nW→H/bardblw/bardbl2\nH=M/bardblw/bardbl2\nH,∀w∈W.\nwithM=CG/bardblS/bardbl2\nW→H. Therefore, we conclude that {S∗gj}j≥1is a frame for W.\nFor the converse implication, suppose that {S∗gj}j≥1is a frame for W, and let\n/tildewideG={/tildewidegj}j≥1⊂Wbe a dual frame of {S∗gj}j≥1. By definition of D(x0,w),\n(D(x0,w)/tildewideG)n=∞/summationdisplay\nj=1/an}bracketle{txn,gj/an}bracketri}ht/tildewidegj, n≥0.\nClearly, ( D(x0,w)/tildewideG)n∈W. Since{/tildewidegj}j≥1is a dual frame of {S∗gj}j≥1, we can also\nwrite\nw=∞/summationdisplay\nj=1/an}bracketle{tw,S∗gj/an}bracketri}ht/tildewidegj.\nHence, by ( 6.3), we have\n/bardbl(D/tildewideG)n−w/bardbl2\nH=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nj=1(/an}bracketle{txn,gj/an}bracketri}ht−/an}bracketle{tw,S∗gj/an}bracketri}ht)/tildewidegj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH\n≤C/tildewideG∞/summationdisplay\nj=1/vextendsingle/vextendsingle/an}bracketle{txn,gj/an}bracketri}ht−/an}bracketle{tw,S∗gj/an}bracketri}ht/vextendsingle/vextendsingle2\n=C/tildewideG∞/summationdisplay\nj=1/vextendsingle/vextendsingle/an}bracketle{txn−S(w),gj/an}bracketri}ht/vextendsingle/vextendsingle2\n=C/tildewideGCG/bardblxn−S(w)/bardbl2\nH→0.\nThis means that R:Bs(ℓ2,ℓ∞)→ Hdefined by R(D) = limD/tildewideGsatisfies\nR/parenleftbig\nD(x0,w)/parenrightbig\n=w.\nIn other words, by Theorem 3.1, we have a stable reconstruction. /square\n6.2.Proof of Theorem 3.5.It is enough to show that the conditions of Theorem\n3.4are fulfilled. To do so, note that the equation ( 1.1) simplifies to\n(6.5) xn=Anx0+(I−An)(I−A)−1w, n ≥1.\nWe rewrite this identity as\nxn−(I−A)−1w=An/parenleftbig\nx0−(I−A)−1w/parenrightbig\n, n≥1.\nHence,\n/bardblxn−(I−A)−1w/bardblH=/bardblx0−(I−A)−1w/bardblH/bardblAn/bardblH→H, n≥1.20 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\nSince we assumed that ρ(A)<1, we have\n/bardblAn/bardblH→H→0, n→ ∞.\nTherefore,\n(6.6) /bardblxn−(I−A)−1w/bardblH→0, n→ ∞.\nThis suggests that the stationary point of the dynamical system is uniquely given\nS(w) = (I−A)−1w.\nIn fact, it is easy to see that if we take x0= (I−A)−1w, thenxn=x0for all\nn≥1. Moreover, the operator S:= (I−A)−1|Wis bounded and invertible. Hence,\nthe conclusion follows from Theorem 3.4. Note that the adjoint of Sis given by\nS∗=PW(I−A∗)−1. /square\n7.A descriptive Example\nWe present an example demonstrating that the recovery of w∈W⊂ Hrequires an\ninfinite number of time samples when His an infinite-dimensional Hilbert space. It is\ninteresting to note that the subspace Win this example is one-dimensional.\nLetH=ℓ2, let\n(7.1) w= (1/2,1/4,1/8, ...)\nwithWbeing the one-dimensional subspace of ℓ2generated by w. Let\nA=\nλ10 0···\n0λ20···\n0 0λ3···\n............\n,\nwhere 0 < λi<1,i≥1, andλi/ne}ationslash=λjifi/ne}ationslash=j. Hence, Aacts as a bounded linear\noperator on ℓ2.\nLetg= (I−A)w, which gives PW(I−A∗)−1g=w. Therefore, {PW(I−A∗)−1g}is\na frame for W. Moreover, by ( 7.1) and that Ais diagonal, we have\n(7.2) g= (g1,g2,g3,...) =/parenleftbig\n(1−λ1)/2,(1−λ2)/4,(1−λ3)/8, .../parenrightbig\n.\nNote that each coordinate giis nonzero.\nConsider the dynamical system\nxn=Axn−1+cw,\nfor some c/ne}ationslash= 0. Then\n(7.3) /an}bracketle{txn,g/an}bracketri}ht=/an}bracketle{tx0,Ang/an}bracketri}ht+c/an}bracketle{tw,Λng/an}bracketri}ht, n≥1,\nwhere Λ n=I+A+···+An−1. Let\n(7.4) x0= (a1,a2,...,a N,0,0,...).DYNAMICAL SAMPLING FOR SOURCE RECOVERY 21\nUsing (7.1) and (7.2), in terms of the coordinates of vectors involved, the system ( 7.3)\nforn= 0,1,...,N−1 can be written as\n\n/an}bracketle{tx0,g/an}bracketri}ht\n/an}bracketle{tx1,g/an}bracketri}ht\n/an}bracketle{tx2,g/an}bracketri}ht\n...\n/an}bracketle{txN−1,g/an}bracketri}ht\n=\ng1g2···gN0\nλ1g1λ2g2···λNgNb1\nλ1g1λ2g2···λNgNb1...............\nλN−1\n1g1λN−1\n2g2···λN−1\nNgNbN−1\n\na1\na2\na3\n...\nc\n,\nwherebi=/an}bracketle{tw,Λig/an}bracketri}ht. Note that the matrix is of dimension N×(N+ 1). Moreover,\nsincegi/ne}ationslash= 0,i≥1, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg1g2···gN\nλ1g1λ2g2···λNgN\nλ1g1λ2g2···λNgN............\nλN−1\n1g1λN−1\n2g2···λN−1\nNgN/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ne}ationslash= 0.\nHence, the columns in the above N×Nmatrix are linearly independent. Thus, for\nanyc/ne}ationslash= 0, there exists a unique x0of the form ( 7.4) such that\n\ng1g2···gN0\nλ1g1λ2g2···λNgNb1\nλ1g1λ2g2···λNgNb1...............\nλN−1\n1g1λN−1\n2g2···λN−1\nNgNbN−1\n\na1\na2\na3\n...\nc\n=\n0\n0\n0\n...\n0\n.\nTherefore, with this choice of x0, we necessarily have\n/an}bracketle{tx0,g/an}bracketri}ht=/an}bracketle{tx1,g/an}bracketri}ht=···=/an}bracketle{txN−1,g/an}bracketri}ht= 0,\nand thus it is impossible to recover the source term cw, if we only have the first N\nsamples of measurements.\n8.Concluding remarks\nWe conclude this article by first, underlying the frame condition in the four last\nmain theorems to guarantee stable source recovery, and second ly, by stating two future\ngeneralizations of the source recovery problem.\n8.1.Unstable recovery. In Theorems 3.2,3.3,3.4, and3.5the frame condition is\na meeting point for stable recovery. In the following example, we sho w that recon-\nstruction of the source term is possible under weak assumptions on the sampling set\nof vectors G={gj}j≥1, but we lack stability.\nLetH=ℓ2and consider the dynamical system ( 1.1) withA=I(the identity\noperator) and measurements given by projecting onto the vecto rsgj:=1\njejforj≥1\nwhere{ej}j≥1form the standard orthonormal basis for ℓ2. In this case, G={1\njej}j∈N22 AKRAM ALDROUBI, ROCIO DIAZ MARTIN, LE GONG, JAVAD MASHREG HI, IVAN MEDRI\nis a Bessel sequence for ℓ2but it is not a frame for ℓ2. Consider R:B(ℓ2,C2)→ℓ2\ndefined by\nR(D) =∞/summationdisplay\nj=1j(d1j−d0j)ej, D= [dnj]n∈[2],j≥1∈ B(ℓ2,C2).\nThen, given the data matrix D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n∈[2],j≥1, we have\nR(D(x0,w)) =∞/summationdisplay\nj=1j/an}bracketle{tx1−x0,1\njej/an}bracketri}htej=∞/summationdisplay\nj=1/an}bracketle{tw,ej/an}bracketri}htej=w.\nIn conclusion, although Ris a linear map and allows source reconstruction, it is an\nunbounded operator.\n8.2.Generalizations of the source recovery problem.\n8.2.1.Time-dependent source term. If the source term depends on time, that is, if the\ndynamical system ( 1.1) is replaced by one of the form\nxn+1=Axn+wn, n∈N,\nwhereforeach n∈N,wn∈ H, thereconstructionformulaprovidedbyTheorem 3.2can\nbe adopted in this framework. Indeed, assume that the spatial sa mpling vectors G=\n{gj}j≥1form a frame for H. Then, for recovering each source term wn, two iterations\nof the dynamical system are needed: Given the measurements {/an}bracketle{txn,gj/an}bracketri}ht,/an}bracketle{txn+1,gj/an}bracketri}ht}j≥1,\nthe source term wncan be linearly recovered by\nwn=∞/summationdisplay\nj=1/parenleftBigg\n/an}bracketle{txn+1,gj/an}bracketri}ht−∞/summationdisplay\ni=1aij/an}bracketle{txn,gi/an}bracketri}ht/parenrightBigg\n/tildewidegj (8.1)\nwhere{/tildewidegj}j≥1is a dual frame for Gand the coefficients aijare given according to\nformula ( 5.11) (i.e.,aij=/an}bracketle{tA∗gj,/tildewidegi/an}bracketri}ht) (cf. (5.13)). Notice that if the reconstruction\nformulas ( 8.1) are used, then we need all the measurements D(x0,w) = [/an}bracketle{txn,gj/an}bracketri}ht]n≥0,j≥1\nfor recovering the source terms {wn}n∈N. The exploration of natural conditions on the\nmeasurements D(x0,w), as well as necessary and sufficient on Gto guarantee stable\nrecovery in this case, are part of future work.\n8.2.2.Continuous-time dynamical systems. As part of future work, we aim to explore\na continuous version of the source recovery problem and address some similarities with\nthe discrete version. Precisely, consider the continuous-time dyn amical system on a\nseparable Hilbert space H\n(8.2)/braceleftBigg\n˙x(t) =Ax(t)+w, t ∈R≥0,\nx(0) =x0\nwhereAis the infinitesimal generator of a strongly continuous semigroup T:R≥0→\nB(H),x0∈ H, andthesourcetermisanunkwown vector winH. IntheparticularcaseREFERENCES 23\nwhenA∈ B(H), thenT(t) =etA, whereeA:=∞/summationtext\nn=01\nn!An. As before, let G={gj}j≥1be\na Bessel sequence for H, and let us denote by\nDc= [/an}bracketle{tx(t),gj/an}bracketri}ht]\nthe continuous time-space samples of the dynamical system (which depend on x0and\nw). In this setting, Dccan be viewed as a curve of observations\nDc: [0,t0)→ℓ2\nDc(t) := (d1(t),d2(t),...) := (/an}bracketle{tx(t),g1/an}bracketri}ht,/an}bracketle{tx(t),g2/an}bracketri}ht,...). (8.3)\nIfG={gj}j≥1is a frame for H, then the ideas from Theorem 3.2can be adapted\nso that one can reconstruct wby considering the continuous time-space samples Dc=\n[/an}bracketle{tx(t),gj/an}bracketri}ht]t∈[0,t0),j≥1for some 0 < t0≤ ∞. Indeed, by having access to such a curve\n(8.3), we know in particular its value at t= 0, i.e.\nDc(0) = (d1(0),d2(0),...) = (/an}bracketle{tx0,g1/an}bracketri}ht,/an}bracketle{tx0,g2/an}bracketri}ht,...),\nand to its derivative, which using ( 8.2) turns out to be\nD′(t) = (d′\n1(t),d′\n2(t),...) = (/an}bracketle{tAx(t),g1/an}bracketri}ht+/an}bracketle{tw,g1/an}bracketri}ht,/an}bracketle{tAx(t),g2/an}bracketri}ht+/an}bracketle{tw,g2/an}bracketri}ht,...)\nIn particular,\nD′(0) = (d′\n1(0),d′\n2(0),...) = (/an}bracketle{tx0,A∗g1/an}bracketri}ht+/an}bracketle{tw,g1/an}bracketri}ht,/an}bracketle{tx0,A∗g2/an}bracketri}ht+/an}bracketle{tw,g2/an}bracketri}ht,...).\nTherefore,\nw=∞/summationdisplay\nj=1/parenleftBigg\nd′\nj(0)−∞/summationdisplay\ni=1aijdi(0)/parenrightBigg\n/tildewidegj.\nwhere{/tildewidegj}j≥1is a dual frame for G, and the coefficients aijare given by ( 5.11). Notice\nthat the values d′\nj(0) are in place of d1j=/an}bracketle{tx1,gj/an}bracketri}htin (5.14).\nThe exploration of natural spaces for the measurements Dc, as well as necessary and\nsufficient on Gto guarantee stable recovery in this case, are part of future wor k.\nReferences\n[1] Juri Ranieri et al. “Sampling and reconstruction of time-varying atmospheric\nemissions”.In: 2012IEEE International Conferenceon Acoustics, Speech an d Sig-\nnalProcessing(ICASSP) .2012,pp.3673–3676. doi:10.1109/ICASSP.2012.6288713 .\n[2] Akram Aldroubi, Rocio Diaz Martin, and Ivan Medri. “Dynamical Sam pling for\nthe Recovery of Spatially Constant Source Terms in Dynamical Syst ems”. In:\narXiv preprint arXiv:2308.01462 (2023).\n[3] Roza Aceska, Armenak Petrosyan, andSui Tang. “Multidimensio nal signal recov-\nery in discrete evolution systems via spatiotemporal trade off”. In :Sampl. Theory\nSignal Image Process. 14.2 (2015), pp. 153–169. issn: 1530-6429.\n[4] A. Aldroubi et al. “Iterative actions of normal operators”. In :J. Funct. Anal.\n272.3(2017),pp.1121–1146. issn:0022-1236. doi:10.1016/j.jfa.2016.10.027 .\nurl:http://dx.doi.org/10.1016/j.jfa.2016.10.027 .24 REFERENCES\n[5] Akram Aldroubi et al. “Dynamical sampling”. In: Applied and Computational\nHarmonic Analysis 42.3 (2017). doi: 10.1016/j.acha.2015.08.014, pp. 378–401.\nissn: 1063-5203. doi:http://dx.doi.org/10.1016/j.acha.2015.08.014 .url:\nhttp://www.sciencedirect.com/science/article/pii/S1 063520315001177 .\n[6] AkramAldroubi,JacquelineDavis,andIlyaKrishtal.“Dynamicalsa mpling:time-\nspacetrade-off”.In: Appl. Comput. Harmon. Anal. 34.3(2013),pp.495–503. issn:\n1063-5203. doi:10.1016/j.acha.2012.09.002 .url:http://dx.doi.org/10.1016/j.acha.2012. \n[7] A. Aldroubi, J. Davis, and I. Krishtal. “Exact Reconstruction of Signals in Evo-\nlutionary Systems Via Spatiotemporal Trade-off”. English. In: Journal of Fourier\nAnalysisandApplications 21.1(2015),pp.11–31. issn:1069-5869. doi:10.1007/s00041-014-9359- \nurl:http://dx.doi.org/10.1007/s00041-014-9359-9 .\n[8] A. Aguilera et al. “Frames by orbits of two operators that commu te”. In:Appl.\nComput. Harmon. Anal. 66 (2023), pp. 46–61. issn: 1063-5203,1096-603X. doi:\n10.1016/j.acha.2023.04.006 .url:https://doi.org/10.1016/j.acha.2023.04.006 .\n[9] Akram Aldroubi et al. “Sampling the Flow of a Bandlimited Function”. In:The\nJournal ofGeometric Analysis 31.9(2021),pp.9241–9275. doi:10.1007/s12220-021-00617-0 .\nurl:https://doi.org/10.1007/s12220-021-00617-0 .\n[10] Akram Aldroubi and Armenak Petrosyan. “Dynamical sampling a nd systems\nfrom iterative actions of operators”. In: Frames and other bases in abstract and\nfunction spaces . Appl. Numer. Harmon. Anal. Birkh¨ auser/Springer, Cham, 2017,\npp. 15–26.\n[11] C. Cabrelli et al. “Dynamical Sampling on Finite Index Sets”. In: ArXiv e-prints\n(Feb. 2017). arXiv: 1702.03384 [math.FA] .\n[12] Carlos Cabrelli et al. “Dynamical sampling on finite index sets”. In :J. Anal.\nMath.140.2(2020),pp.637–667. issn:0021-7670. doi:10.1007/s11854-020-0099-2 .\nurl:https://doi-org.proxy.library.vanderbilt.edu/10.100 7/s11854-020-0099-2 .\n[13] Cheng Cheng, Yingchun Jiang, and Qiyu Sun. “Spatially Distribute d Sampling\nand Reconstruction”. In: CoRRabs/1511.08541 (2015). arXiv: 1511.08541 .url:\nhttp://arxiv.org/abs/1511.08541 .\n[14] Nabin Kumar Sahu, Shalini Chauhan, and Ram N. Mohapatra. “Re presenta-\ntions of frames via iterative actions of operators in tensor produc t spaces”. In:\nJ. Pseudo-Differ. Oper. Appl. 14.4 (2023), p. 72. issn: 1662-9981,1662-999X. doi:\n10.1007/s11868-023-00566-y .url:https://doi.org/10.1007/s11868-023-00566-y .\n[15] Alexander Ulanovskii and Ilya Zlotnikov. “Reconstruction of ba ndlimited func-\ntions from space-time samples”. In: J. Funct. Anal. 280.9 (2021), Paper No.\n108962,14. issn:0022-1236. doi:10.1016/j.jfa.2021.108962 .url:https://doi-org.proxy.libr \n[16] Qingyue Zhang, Bei Liu, and Rui Li. “Dynamical sampling in multiply g enerated\nshift-invariant spaces”. In: Applicable Analysis 96.5 (2017), pp. 760–770. doi:\n10.1080/00036811.2016.1157586 .eprint:https://doi.org/10.1080/00036811.2016.1157586 .\nurl:https://doi.org/10.1080/00036811.2016.1157586 .\n[17] Roza Aceska and Yeon H. Kim. “Scalability of Frames Generated b y Dynamical\nOperators”. In: Frontiers in Applied Mathematics and Statistics 3 (2017), p. 22.\nissn:2297-4687. doi:10.3389/fams.2017.00022 .url:https://www.frontiersin.org/article/ REFERENCES 25\n[18] A. Aguilera et al. “Dynamical sampling for shift-preserving oper ators”. In: Appl.\nComput. Harmon. Anal. 51 (2021), pp. 258–274. issn: 1063-5203,1096-603X. doi:\n10.1016/j.acha.2020.11.004 .url:https://doi.org/10.1016/j.acha.2020.11.004 .\n[19] Jonathan Ashbrock and Alexander M Powell. “Dynamical dual fr ames with an\napplication to quantization”. In: Linear Algebra and its Applications 658 (2023),\npp. 151–185.\n[20] Robert Beinert and Marzieh Hasannasab. “Phase retrieval an d system identifica-\ntion in dynamical sampling via Prony’s method”. In: Adv. Comput. Math. 49.4\n(2023),PaperNo.56. issn:1019-7168,1572-9044. doi:10.1007/s10444-023-10059-7 .\nurl:https://doi.org/10.1007/s10444-023-10059-7 .\n[21] F. Bozkurt and K. Kornelson. “Norm retrieval from few spatio -temporal sam-\nples”. In: J. Math. Anal. Appl. 519.2 (2023), Paper No. 126804, 17. issn: 0022-\n247X,1096-0813. doi:10.1016/j.jmaa.2022.126804 .url:https://doi.org/10.1016/j.jmaa.2 \n[22] Carlos Cabrelli, Ursula Molter, and Daniel Su´ arez. “Multi-orbita l frames through\nmodel spaces”. In: Complex Analysis and Operator Theory 15 (2021), pp. 1–22.\n[23] Carlos Cabrelli, Ursula Molter, and Daniel Su´ arez. “Frames of it erations and\nvector-valued model spaces”. In: arXiv preprint arXiv:2203.01301 (2022).\n[24] Ole Christensen and Marzieh Hasannasab. “Frame properties o f systems aris-\ning via iterated actions of operators”. In: Appl. Comput. Harmon. Anal. 46.3\n(2019), pp. 664–673. issn: 1063-5203. doi:10.1016/j.acha.2018.04.002 .url:\nhttps://doi-org.proxy.library.vanderbilt.edu/10.101 6/j.acha.2018.04.002 .\n[25] Ole Christensen and Marzieh Hasannasab. “Frames and genera lized operator or-\nbits”. In: Sampling Theory, Signal Processing, and Data Analysis 21.2 (2023),\np.22.doi:10.1007/s43670-023-00061-x .url:https://doi.org/10.1007/s43670-023-00061- \n[26] Roc´ ıo D´ ıaz Mart´ ın, Ivan Medri, and Ursula Molter. “Dynamical sampling: a view\nfrom control theory”. In: Excursions in harmonic analysis. Vol. 6 . Appl. Numer.\nHarmon. Anal. Birkh¨ auser/Springer, Cham, [2021] ©2021, pp. 269–295.\n[27] Roc´ ıo D´ ıaz Mart´ ın, Ivan Medri, and Ursula Molter. “Continuou s and discrete dy-\nnamical sampling”. In: J. Math. Anal. Appl. 499.2 (2021), Paper No. 125060, 19.\nissn:0022-247X. doi:10.1016/j.jmaa.2021.125060 .url:https://doi-org.proxy.library.vande \n[28] Rocio Diaz Martin, Ivan Medri, and Juliana Osorio. “Error analysis on the initial\nstate reconstruction problem”. In: Sampling Theory, Signal Processing, and Data\nAnalysis 21.1 (2023), p. 18.\n[29] Daniel Freeman and Darrin Speegle. “The discretization problem for continuous\nframes”. In: Adv. Math. 345 (2019), pp. 784–813. issn: 0001-8708,1090-2082. doi:\n10.1016/j.aim.2019.01.006 .url:https://doi.org/10.1016/j.aim.2019.01.006 .\n[30] Karlheinz Gr¨ ochenig et al. “On minimal trajectories for mobile sa mpling of ban-\ndlimited fields”. In: Appl. Comput. Harmon. Anal. 39.3(2015),pp. 487–510. issn:\n1063-5203. doi:10.1016/j.acha.2014.11.002 .url:https://doi.org/10.1016/j.acha.2014.11 \n[31] Zaur A. Kasumov and Aydin Sh. Shukurov. “On frame propertie s of iterates of\na multiplication operator”. In: Results Math. 74.2 (2019), Paper No. 84, 8. issn:\n1422-6383. doi:10.1007/s00025-019-1009-8 .url:https://doi-org.proxy.library.vanderbilt 26 REFERENCES\n[32] Tesfa Mengestie. “Closed range weighted composition operato rs and dynamical\nsampling”. In: J. Math. Anal. Appl. 515.1 (2022), Paper No. 126387, 11. issn:\n0022-247X,1096-0813. doi:10.1016/j.jmaa.2022.126387 .url:https://doi.org/10.1016/j.jmaa \n[33] Tesfa Mengestie. “Closed range Volterra-type integral oper ators and dynamical\nsampling”. In: Monatsh. Math. 202.1 (2023), pp. 161–170. issn: 0026-9255,1436-\n5081.doi:10.1007/s00605-022-01768-0 .url:https://doi.org/10.1007/s00605-022-01768- \n[34] RoxanaAlexandru,ThierryBlu,andPierLuigiDragotti.“Diffusio nSLAM:local-\nizing diffusion sources from samples taken by location-unaware mobile sensors”.\nIn:IEEE Trans. Signal Process. 69 (2021), pp. 5539–5554. issn: 1053-587X,1941-\n0476.doi:10.1109/TSP.2021.3113789 .url:https://doi.org/10.1109/TSP.2021.3113789 .\n[35] RoxanaAlexandruandPierLuigiDragotti.“Reconstructingcla ssesofnon-bandlimited\nsignals from time encoded information”. In: IEEE Trans. Signal Process. 68\n(2020), pp. 747–763. issn: 1053-587X. doi:10.1109/TSP.2019.2961301 .url:\nhttps://doi.org/10.1109/TSP.2019.2961301 .\n[36] J. Murray-Bruce and P. L. Dragotti. “Estimating Localized Sou rces of Diffu-\nsion Fields Using Spatiotemporal Sensor Measurements”. In: IEEE Transac-\ntions on Signal Processing 63.12 (2015), pp. 3018–3031. issn: 1053-587X. doi:\n10.1109/TSP.2015.2419187 .\n[37] John Murray-Bruce and Pier Luigi Dragotti. “A sampling framew ork for solving\nphysics-driven inverse source problems”. In: IEEE Trans. Signal Process. 65.24\n(2017),pp.6365–6380. issn:1053-587X,1941-0476. doi:10.1109/TSP.2017.2742983 .\nurl:https://doi.org/10.1109/TSP.2017.2742983 .\n[38] Akram Aldroubi et al. “Dynamical Sampling with Additive Random No ise”. In:\nSampl. Theory Signal Image Process. 17.2 (2018), pp. 153–182.\n[39] Akram Aldroubi and Ilya Krishtal. “Krylov subspace methods in d ynamical sam-\npling”. In: Sampl. Theory Signal Image Process. 15 (2016), pp. 9–20.\n[40] Jiahui Cheng and Sui Tang. “Estimate the spectrum of affine dy namical systems\nfrom partial observations of a single trajectory data”. In: Inverse Problems 38.1\n(2022),PaperNo.015004,42. issn:0266-5611,1361-6420. doi:10.1088/1361-6420/ac37fb .\nurl:https://doi.org/10.1088/1361-6420/ac37fb .\n[41] SuiTang.“Systemidentificationindynamical sampling”.In: Adv. Comput. Math.\n43.3(2017),pp.555–580. issn:1019-7168. url:https://doi-org.proxy.library.vanderbilt.ed \n[42] IlyaZlotnikov.“OnplanarsamplingwithGaussiankernelinspaces ofbandlimited\nfunctions”. In: J. Fourier Anal. Appl. 28.3 (2022), Paper No. 55, 24. issn: 1069-\n5869,1531-5851. doi:10.1007/s00041-022-09948-0 .url:https://doi.org/10.1007/s00041-022- \n[43] Akram Aldroubi, Le Gong, and Ilya Krishtal. “Recovery of rapidly decaying\nsource terms from dynamical samples in evolution equations”. In: Sampling The-\nory, SignalProcessing,andDataAnalysis 21.1(2023),p.15. doi:10.1007/s43670-023-00054-w .\nurl:https://doi.org/10.1007/s43670-023-00054-w .\n[44] Akram Aldroubi et al. “Predictive algorithms in dynamical sampling for burst-\nlike forcing terms”. In: Applied and Computational Harmonic Analysis 65 (2023),\npp.322–347. issn:1063-5203. doi:https://doi.org/10.1016/j.acha.2023.03.003 .\nurl:https://www.sciencedirect.com/science/article/pii/S 1063520323000246 .REFERENCES 27\n[45] Ole Christensen et al. An introduction to frames and Riesz bases .Vol. 7. Springer,\n2003.\n[46] R. J. Duffin and A. C. Schaeffer. “A class of nonharmonic Fourier series”. In:\nTrans. Amer. Math. Soc. 72 (1952), pp. 341–366. issn: 0002-9947." }, { "title": "2401.15451v1.Constant_roll_inflation_and_primordial_black_holes_with_Barrow_holographic_dark_energy.pdf", "content": "Constant-roll inflation and primordial black holes with Barrow\nholographic dark energy\nQihong Huang1a, He Huang2, Bing Xu3and Kaituo Zhang4\n1School of Physics and Electronic Science,\nZunyi Normal University, Zunyi, Guizhou 563006, China\n2Institute of Applied Mechanics, Zhejiang University, Hangzhou, Zhejiang 310058, China\n3School of Electrical and Electronic Engineering,\nAnhui Science and Technology University, Bengbu, Anhui 233030, China\n4Department of Physics, Anhui Normal University, Wuhu, Anhui 241000, China\nAbstract\nWe investigate the constant-roll inflation and the evolution of primordial black holes (PBHs) with\nBarrow holographic dark energy (BHDE). Using the modified Friedmann equation and the constant-\nroll condition in BHDE model, we calculate the constant-roll parameters, the scalar spectral index\nparameter and the tensor-to-scalar ratio with the chaotic potential V0ϕn. Then, we show that a\nsuitable value of the power exponent is n= 1 by using the Planck 2018 data. Considering the\naccretion process and the evaporation due to Hawking radiation, we discuss the evolution of PBHs\nin BHDE model and obtain that the PBHs mass is in the mass window of PBHs.\naCorresponding author: huangqihongzynu@163.com\n1arXiv:2401.15451v1 [gr-qc] 27 Jan 2024I. INTRODUCTION\nInflation, a short period of an exponential accelerating expansion before the radiation\ndominated era, is the currently widely accepted paradigm of modern cosmology [1, 2]. The\ninflation not only addresses the puzzles of the standard hot Big Bang cosmology but also\nprovides an explanation for the quantum origin of the Cosmic Microwave Background tem-\nperature anisotropies and the Large-Scale Structure [3–5]. In general, a scalar field named\nas the inflaton field drives the exponential accelerating expansion of the universe during the\ninflationary epoch. The mechanism of inflation is based on the generation of small quantum\nfluctuations in the inflaton field. The small quantum fluctuations are amplified in physical\nscale during the inflationary epoch and lead to a Gaussian, scale-invariant and adiabatic\nprimordial density perturbations [6]. This information is encoded into the primordial scalar\npower spectrum described by the scalar spectral index ns, which is constrained by the Planck\n2018 results [105]. In addition, inflation also predicts the generation of tensor perturbations\nas primordial gravitational waves described by the tensor-to-scalar ratio r[8]. Usually, the\ndynamics of inflation is based on the slow-roll approximation, in which the scalar poten-\ntial is chosen to be nearly flat so that the scalar field can slowly roll down this potential.\nOnce the minimum value of this potential is reached, inflation ends. This scenario is called\nslow-roll inflation [2, 6, 9], which can be measured by the parameters ϵ1=−˙H/H2and\nϵ2=¨ϕ/H ˙ϕ. For the slow-roll inflation, the small values of the parameters ϵ1≪1 and ϵ2≪1\nare required. If the scalar potential is assumed to be extremely flat, the second condition\nbecomes ϵ2=−3 which corresponds to the ultra slow-roll condition [10–12]. In the ultra\nslow-roll inflation, the curvature perturbations are not frozen at the super Hubble scales thus\nleading to non-Gaussianities. Furthermore, when the second condition ϵ2is generalized to\nbe constant, a more generalized scenario named as constant-roll inflation was proposed [13–\n19], in which the rate of acceleration and velocity of the inflaton field are constants. The\nconstant-roll inflation has novel dynamical features enriching physics, which is different from\nthe slow-roll inflation. When the constant-roll inflation was proposed, it drawed widespread\nattention and was widely studied in lots of theories, such as f(R) gravity [20, 21], f(R, ϕ)\ngravity [22], f(Q, T) gravity [23], scalar-tensor gravity [24], brane gravity [25], Dirac-Born-\n2Infeld theory [26], Galilean model [27], holographic dark energy [28], Tsallis holographic dark\nenergy [29], and so on.\nAfter inflation ended, the primordial inhomogeneities of the primordial power spectrum\non small scales re-entered the Hubble horizon in the radiation dominated era, significant\namount of primordial black holes (PBHs) can be formed as a result of the gravitational\ncollapse [30, 31] if the amplitude of the primordial power spectrum produced during inflation\nis strong enough. After PBHs were proposed, they were found that they could be a candidate\nof dark matter [32] and were reconsidered [33, 34] after black holes mergers were detected by\nLIGO [35]. PBHs can offer us an opportunity to explore the physics of the early universe and\nmay play some important roles in cosmology. Thus, PBHs were widely discussed in some\nmodified gravity theories, such as scalar-tensor gravity [36], f(T) gravity [37], Brans-Dicke\ngravity [38], f(Q) gravity [39], teleparallel gravity [40], f(Q, T) gravity [23], and studied in\nsome inflationary models including slow-roll inflation [41–47], ultra-slow-roll inflation [48–52]\nand constant-roll inflation [23, 53, 54]. PBHs also have attracted lots of attention since they\nmay constitute part or all of dark matter [55–67], they may play a role in the synthesis of\nheavy elements [68, 69] and could be responsible for some astrophysical phenomena, such\nas, seeding supermassive black holes [55, 60, 62, 70], seeding galaxies [55, 60], explaining\nthe gravitational wave signals observed by the LIGO detectors [62, 71]. The existence of\nPBHs formed in the early universe still remains an open question. PBHs with mass smaller\nthan 5 ×1011kgwould have been evaporated by Hawking radiation now [72]. When PBHs\nare considered as dark matter, the currently allowed mass window for PBHs was shown\naround 1020g[58], 10−10−10−8M⊙[73], 5 ×10−16M⊙, 2×10−14M⊙and 25 −100M⊙[74],\n1016−1017g, 1020−1024gand 1−103M⊙[57], 104−106M⊙[66], 106−1012M⊙[75], so the\nallowed mass window for PBHs is seemingly broad and a more accurate mass window needs\nfurther investigation.\nThe holographic dark energy is one competitive candidates of dark energy [76–78], which\nis used to explain the late time acceleration of the universe, and is proposed based on the\nholographic principle stating that the entropy of a system is scaled on its surface area [79, 80].\nThe cornerstone of holographic dark energy is the horizon entropy, and different horizon\n3entropy will result in different holographic dark energy models. Recently, combining the\nholographic principle and Barrow entropy [81], Barrow holographic dark energy(BHDE) with\ndifferent IR cutoff was proposed [82–86] and subsequently widely studied in theories [87–97]\nand observations [98–101].\nRecently, the constant-roll inflation was studied in holographic dark energy model [28] and\nTsallis holographic dark energy model [29], the results show that the constant-roll inflation\ncan be realized in these models under some conditions. So, whether the constant-roll inflation\ncan also be realized in BHDE model. This is the main goal of this paper. Then, the evolution\nof PBHs, which is formed after inflation ended, is analyzed. The paper is organized as follows.\nIn section II, we briefly review the modified Friedmann equation in BHDE model. In section\nIII, we study the constant-roll inflation in BHDE model. In section IV, we analyze the\nevolution of PBHs in BHDE model. Finally, our main conclusions are shown in Section V.\nII. MODIFIED FRIEDMANN EQUATIONS\nIn a homogeneous and isotropic Friedmann-Robertson-Walker universe described by the\nFriedmann-Lema ˆitre-Robertson-Walker metric\nds2=−dt2+a(t)2γijdxidxj, (1)\nwhere a(t) denotes the scale factor, and γijrepresents the metric on the three-sphere\nγijdxidxj=dr2\n1−Kr2+r2(dθ2+ sin2θdϕ2). (2)\nHere, Kis the spatial curvature. Considering the apparent horizon as the IR cutoff, the\nFriedmann equation in BHDE model can be modified as [85]\n\u0010\nH2+K\na2\u00111−δ\n2=8πGeff\n3ρϕ, (3)\nwhere δsatisfies the relation 0 ≤δ≤1 and stands for the amount of the quantum-\ngravitational deformation effects [102], δ= 0 represents the standard holographic dark energy\nandδ= 1 denotes the most intricate and fractal structure of the horizon, and Geffrepresents\n4the effective Newtonian gravitational constant as\nGeff=A0\n4\u00102−δ\n2 +δ\u0011\u0010A0\n4π\u0011δ\n2. (4)\nIn the case δ→0, the area law of entropy is recovered, and we have A0→4G. As a result,\nGeff→Gand the standard Friedmann equation is obtained.\nWe consider the matter contents of the early universe as a scalar field, and the energy\ndensity and the pressure take the form\nρϕ=1\n2˙ϕ2+V(ϕ), p ϕ=1\n2˙ϕ2−V(ϕ), (5)\nwhich satisfies the continuity equation\n˙ρϕ+ 3H(ρϕ+pϕ) = 0 . (6)\nIt can be written as the Klein-Gordon equation\n¨ϕ+ 3H˙ϕ+Vϕ= 0. (7)\nCombing Eqs. (3) and (6), the second Friedmann equation can be derived as [85]\n(2−δ)\u0010\n1 +˙H\nH2\u0011\u0010\nH2+k\na2\u0011−δ\n2H2+ (1 + δ)\u0010\nH2+k\na2\u00111−δ\n2=−8πGeffpϕ. (8)\nIII. CONSTANT-ROLL INFLATION\nIn this section, we will discuss the constant-roll condition on the Friedmann equations\ngiven by Eqs. (3) and (8) in a flat universe. During inflation, the universe is characterized\nby the scalar spectral index parameter nsand the tensor-to-scalar ratio rwhich are given\nby [103, 104]\nns≃1−4ϵ1−2ϵ2, r = 16ϵ1, (9)\nwith\nϵ1=−˙H\nH2, ϵ 2=¨ϕ\nH˙ϕ. (10)\n5For the slow-roll inflation, ϵ1≪1 and ϵ2≪1 are required. While in the constant-roll\ninflation, imposing˙ϕ2\n2<1, only ϵ1≪1 is required to occur in the inflation, and the other\nconstant-roll condition is given by\n¨ϕ=γH˙ϕ. (11)\nHere, γis a dimensionless real parameter. For γ=−3, the ultra slow-roll condition, which\nhas a flat potential Vϕ= 0, is recovered. And the slow-roll condition is obtained for γ= 0.\nIn the flat universe, considering˙ϕ2\n2<1, Eqs. (3), (8) and (7) can be written as\nH2≃\u00108πGeff\n3\u0011 2\n2−δV2\n2−δ, (12)\n˙H≃ −3˙ϕ2\n2(2−δ)\u00108πGeff\n3\u0011 2\n2−δVδ\n2−δ, (13)\n˙ϕ=−Vϕ\n(γ+ 3)H. (14)\nThen, the constant-roll parameters become\nϵ1=3˙ϕ2\n2(2−δ)V, ϵ 2=γ. (15)\nSimilar to the slow-roll inflation occuring at the horizon crossing point, we also consider in-\nflation begins at the horizon crossing point. Thus, the e-folding number N, which determines\nthe amount of inflation, is given as\nN=Ztend\nt∗Hdt, (16)\nwhere t∗andtendrepresent the horizon crossing time and the end of inflation time, respec-\ntively. Rewriting the e-folding number Naccording to the scalar field, we obtain\nN=Zϕend\nϕ∗H\n˙ϕdϕ. (17)\nTo analyze the e-folding number, we consider the potential Vwith the form\nV=V0ϕn, (18)\nwhich are the chaotic potentials. Here, both V0andnare positive parameters. At the end\nof inflation, the first constant parameter ϵ1satisfies ϵ1(ϕend)≃1 which gives\nϕend=\" \n8πGeff\n3! 2\n2−δ2(2−δ)(γ+ 3)2\n3n2V−δ\nδ−2\n0# δ−2\n4−2δ+nδ\n. (19)\n6Here, Eqs. (12), (14) and (15) are taken into consideration. Then, the e-folding number\ncan be written according ϕendandϕ∗\nN=\u00108πGeff\n3\u0011 2\n2−δγ+ 3\nnVδ\n2−δ\n0δ−2\n4−2δ+nδ\u0010\nϕ−4−2δ+nδ\nδ−2\nend −ϕ−4−2δ+nδ\nδ−2\n∗\u0011\n. (20)\nSolving the above equation, we obtain the value of the scalar field at the horizon crossing ϕ∗\nϕ∗=\"\u00108πGeff\n3\u0011−2\n2−δV−δ\n2−δ\n0\u00103n2\n2(2−δ)(γ+ 3)2−Nn\nγ+ 34−2δ+nδ\nδ−2\u0011#−δ−2\n4−2δ+nδ\n. (21)\nUsing this result, the first constant-roll parameter ϵ1can be written as\nϵ1=3n\n3n+ 2(γ+ 3)[4 + ( n−2)δ]N. (22)\nThen, the scalar spectral index parameter nsand the tensor-to-scalar ratio rbecome\nns= 1−12n\n3n+ 2(γ+ 3)[4 + ( n−2)δ]N−2γ, (23)\nr=48n\n3n+ 2(γ+ 3)[4 + ( n−2)δ]N, (24)\nwhich show that the parameters nsandrdepend on the power-term nof the chaotic poten-\ntial, the constant-roll parameter γ, the parameter of the quantum-gravitational deformation\neffects δand the e-folding number N.\nTo determine the value of the power-term nin the chaotic potential V=V0ϕn, we choose\nr0.002<0.058 and ns= 0.9668±0.0037 which was constrained by the Planck TT,TE,EE +\nlowE + lensing + BK15 + BAO [105]. Then, using Eqs. (23) and (24), we have plotted\nthe prediction regions in γ−nplane with the number of e-folding number Nfrom 50 to 70\nin Fig. (1) and the prediction regions in ( δ, γ, n ) parameter space in Fig. (2), respectively.\nThese figures show that the value of γis very small, and nis determined by the e-folding\nnumber N. Ifδtakes a small value, n≤1.4 is required, while n≤1.1 is required for δ∼1.\nTo match our results with the observations, we consider 0 .004< γ < 0.018 and δtakes a\nsmall value according to the result of observation [98–101]. With the fixed values of nand\nδ, we depict the predictions of the chaotic potential (18) in r0.002−nsplane in Fig. (3) in\nwhich we overlap our analytical results with Planck 2018 data. These figures show that the\n70.0 0.5 1.0 1.5 2.0 2.50.0050.0100.0150.0200.025\nnγN=50\nN=55\nN=60\nN=65\nN=70\n0.0 0.5 1.0 1.5 2.0 2.50.0050.0100.0150.0200.025\nnγN=50\nN=55\nN=60\nN=65\nN=70FIG. 1. Prediction regions in γ−nplane. The left panel is plotted for δ= 10−4, while the right\none is for δ= 0.094.\nFIG. 2. Prediction regions in ( δ, γ, n ) parameter space.\n8value of rdecreases as Nvaries from 50 to 70, and an increasing γleads to a smaller value of\nns. It is obvious that the constant-roll parameter γhas an apparent influence on ( r0.002, ns)\nbehavior. With this comparison with the observation, the results show a good consistency\nfor a specific range of the constant-roll parameter γwith Planck 2018 data, and the results\nprefer to the case δ= 10−4,γ= 0.012 and n= 1. It is interesting to note that n= 1 is also\nobtained in Tsallis holographic dark energy [29]. It is easy to see that an increased value of\nnwill lead to mismatching with the observation.\nIV. PRIMORDIAL BLACK HOLES\nIn previous section, we have analyzed the constant-roll inflation in BHDE model. And\nthen, in this section, we will analyze the evolution of PBHs, which were supposed to formed\nin the radiation dominated era, in BHDE model. The black holes (BHs) can evolve with an\nincreasing mass by absorbing other matter, stars and BHs. During this process, BHs can\nemit particles. And the quantum properties of BHs show that the possibility of emitting\nparticles with a thermal spectrum is related to BHs surface gravity [106]. During the process\nof emitting particles, BHs may lose mass. In the following, we will analyze the thermal\nproperties of evaporating PBHs. The corresponding PBHs temperatures given as [106, 107]\nTBH=ℏc3\n8πGM BHk∼1.06\u00101010\nMBH\u0011\nGeV, (25)\nwhere MBHis the total mass of PBHs with the form MBH=Meva+Maccr, in which Meva\nandMaccrrepresent the evaporation mass and the accretion mass respectively.\nTo discuss the evaporation mass of PBHs, we consider the main process to decrease the\nmass of PBHs is Hawking evaporation which is defined as [108, 109]\n\u0010dM\ndt\u0011\neva=−ℏc4\nG2αs\nM2, (26)\nwhere αsis the spin parameter of the emitting particle. Integrating Eq. (26), the evaporation\nmass of PBHs is obtained\nMeva=Mi\u0010\n1−t\nteva\u00111\n3, (27)\n9γ=0.004\nγ=0.006\nγ=0.008\nγ=0.010\nγ=0.012\nγ=0.014\nγ=0.016\nγ=0.018\n0.94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.200.25\nnsr0.002N=50\nN=70Planck TT,TE,EE+lowE\nPlanck TT,TE,EE+lowE+lensing\nPlanck TT,TE,EE+lowE+lensing+BK15+BAO\nγ=0.004\nγ=0.006\nγ=0.008\nγ=0.010\nγ=0.012\nγ=0.014\nγ=0.016\nγ=0.018\n0.94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.200.25\nnsr0.002\nN=50\nN=70Planck TT,TE,EE+lowE\nPlanck TT,TE,EE+lowE+lensing\nPlanck TT,TE,EE+lowE+lensing+BK15+BAO\nγ=0.004\nγ=0.006\nγ=0.008\nγ=0.010\nγ=0.012\nγ=0.014\nγ=0.016\nγ=0.018\n0.94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.200.25\nnsr0.002\nN=50\nN=70Planck TT,TE,EE+lowE\nPlanck TT,TE,EE+lowE+lensing\nPlanck TT,TE,EE+lowE+lensing+BK15+BAO\nγ=0.004\nγ=0.006\nγ=0.008\nγ=0.010\nγ=0.012\nγ=0.014\nγ=0.016\nγ=0.018\n0.94 0.95 0.96 0.97 0.98 0.99 1.000.000.050.100.150.200.25\nnsr0.002\nN=50\nN=70Planck TT,TE,EE+lowE\nPlanck TT,TE,EE+lowE+lensing\nPlanck TT,TE,EE+lowE+lensing+BK15+BAOFIG. 3. Predictions of the chaotic potential (18) in r0.002−nsplane. These figures are depicted\nfor: (i) n= 3, δ= 10−4; (ii) n= 2, δ= 10−4; (iii) n= 1, δ= 10−4; (iv) n= 1, δ= 0.094.\nwith\nteva=G2\nℏc4M3\ni\n3αs, (28)\nwhere Miis the initial mass of PBHs and tevadenotes the Hawking evaporation time scale.\nThe initial mass of PBHs is given as the order of the particle horizon mass when it was\nformed [110]\nMi≈c3tf\nG≈1012tf\n10−23. (29)\n10Here, tfis the time of its formation. So, the PBHs formed in the late time of the universe\nmust have more mass than that formed in the early time. For the case of PHBs formed at\nPlanck time 10−43s, it had the mass 10−8kg. For tf= 2×10−5s, one has Mi= 2×1030kg\nwhich is the mass of sun M⊙.\nEq. (27) shows the evolution of the evaporation mass of PBHs with respect to t/teva.\nThe evaporation mass decreasing as tapproaches to tevaand becomes 0 for t=teva, which\nindicates PBHs evaporate completely. The evolution of Meva/Mias the functions of t/teva\nis plotted in the left panel of Fig. (4), which is also given in Ref. [23]. So, the Hawking\nevaporation time scale tevadetermines the evaporation time of PBHs. According to Eq. (28),\nwe can find that tevaincreases with the increase of Miand decreases with the increase of\nαs. In the right panel of Fig. (4), by considering the evaporation time tevaas a function of\nthe initial mass of PBHs Mi, we have plotted the relation between tevaandMi. The red\ndashed line in this figure represents the current age of the universe. This figure shows that\nthe PBHs need more time to achieve a complete evaporation with the increase of the initial\nmass, and it needs a evaporation time longer than the age of the universe when the initial\nmass is larger than 5 ×1011kg, which is shown in Ref. [72]. For αs∼10−4[108, 110], we can\nwrite tevaas\nteva∼10−17M3\ni. (30)\nDuring the process of evaporation, the accretion of fluid surrounding PBHs will prolong\nthe evaporation of PBHs. So, we require to consider the process of the mass accretion rate\nfor PBHs with fluid, which is given as [110, 111]\n\u0010dM\ndt\u0011\naccr=16πG2\nc3M2(ρeff+peff). (31)\nHere, ρeffandpeffrepresent the effective energy density and pressure, respectively. Using\nEqs. (3) and (8), we can obtain\nρeff=3\n8πGH2, ω eff=peff\nρeff=2ωϕ+δ\n2−δ, (32)\nin which ωϕ=pϕ/ρϕ. When δtakes a small value, one can obtain ωeff≈ωϕ. Considering\n110.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.01.2\nt\ntevaMeva\nMi\n0 2×10114×10116×10118×101101×10172×10173×10174×10175×10176×1017\nMiteva\nαs=2.0×10-4\nαs=3.5×10-4\nαs=5.0×10-4FIG. 4. In the left panel, plotted is the evolution of Meva/Mias the function of t/teva. In the\nright panel, shown is relation between the initial mass of PBHs Miand the evaporation time teva.\nThe red dashed line denotes the age of the universe.\na∼t2\n3(1+ωeff)andH=2\n3(1+ωefft), we can write ρeff+peffas\nρeff+peff=1\n6πG(1 +ωeff)t2. (33)\nHere, we consider that ρeff+peffevolves with tinstead of a constant in Ref. [23]. Then,\nEq. (31) can be integrated as\nMaccr=Mi\n1−β\u0010\n1−ti\nt\u0011 (34)\nwith\nβ=8G\n3c2(1 +ωeff)Mi\nti, (35)\nin which tiis the time that PBHs begins to accrete, βdenotes the product of the accretion\nefficiency and the fraction of the horizon mass [112]. Assuming PBHs begins to accrete at\nthe time when it formed, we obtain ti=tf. Then, using Eq. (29), βandti/tcan be written\nas\nβ≈0.6588\n1 +ωeff, (36)\n12which indicates βis only determined by ωeffand decreases with the increase of ωeff, and\nti\nt=10−35Mi\nt, (37)\nwhich equals to 1 at the time PBHs begins to accrete and decays very fast. So, according to\nEq. (34), we can obtain Maccr≃Miat the initial time ti. The evolution curve of βandti/t\nare plotted in the left and right panel of Fig. (5) respectively. In the left panel of Fig. (6), we\nhave plotted the evolutionary curves for Maccr/Mias the function of t. This figure shows that\nMaccr/Miincreases with the decrease of ωeff, and it increases rapidly in a short time and\nthen keeps as a constant. The right panel of Fig. (6) shows the relation between Maccr/Mi\nandωeff. From these figures, we can see that Maccr/Miincreases to approach 102when ωeff\ndecreases from 1 /3 to−1/3.\n-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30.40.50.60.70.80.91.0\nωeffβ\n10-40.001 0.010 0.100 110-510-40.0010.0100.1001\ntti\nt\nFIG. 5. In the left panel, the evolution of βas the function of ωeffis plotted. In the right\npanel, the evolution of ti/tas the function of tis plotted. The initial mass of PBHs takes the value\nMi=M⊙.\nThe left panel of Fig. (7) shows the evolutionary curves of PBHs mass MBHas the function\noft. From this figure, we can see that MBHincreases rapidly in a short time and then keeps\nas a constant, and MBHincreases with the decrease of ωeff. In the right panel of Fig. (7), we\n1310-40.001 0.010 0.100 1151050100\ntMaccr\nMi\nωeff=1/3ωeff=0ωeff=-1/3\n-0.4 -0.2 0.0 0.2 0.420406080100\nωeffMaccr\nMiFIG. 6. Evolutionary curves of Maccr/Mi. The left panel sets tas the variable, while the right\npanel sets ωeff. The initial mass of PBHs takes the value Mi=M⊙.\nhave plotted the evolutionary curves of PBHs temperature TBHas the function of t. Since\nTBHis inversely proportional to MBH, which is given in Eq. (25), TBHdecreases rapidly in a\nvery short time and then becomes a constant, and TBHdecreases once ωeffbecomes small.\nWhen ωeffevolves from 1 /3 to−1/3, the PBHs temperature TBHdecreases from 10−21GeV\nto 10−23GeV, i.e. TBHdecreases from 10−8Kto 10−10K.\nAfter inflation ended and PHBs formed, the universe evolves from the radiation dominated\nepoch to the pressureless matter dominated epoch, and then enters into the dark energy\ndominated epoch, the effective state of equation parameter evolves from 1 /3 to less than\n−1/3, the accretion mass Maccrincreases to approach 102Mi, the PBHs temperature TBH\ndecreases with the increase of MBH.\nV. CONCLUSION\nBased on the holographic principle and the Barrow entropy, a new BHDE model with the\napparent horizon as IR cutoff has been proposed and the corresponding Friedmann equation\n1410-40.001 0.010 0.100 11×10305×10301×10315×10311×10325×1032\ntMBH\nωeff=1/3ωeff=0ωeff=-1/3\n10-40.001 0.010 0.100 15.×10-231.×10-225.×10-221.×10-215.×10-21\ntTBH\nωeff=1/3ωeff=0ωeff=-1/3FIG. 7. Evolutionary curves of PBH mass MBHand PBH temperature TBH. The initial mass of\nPBHs takes the value Mi=M⊙.\nhas been modified. In this paper, using the modified Friedmann equation and the constant-\nroll condition, we calculate the constant-roll parameters ϵ1andϵ2, the scalar spectral index\nparameter nsand the tensor-to-scalar ratio rwith the chaotic potential V0ϕn. Then, using\nthe Planck 2018 data, we plot the γ−nplane, the ( δ, γ, n ) parameter space and the r0.002−ns\nplane and then obtain a suitable value of the power exponent n= 1, the parameter of BHDE\nδand the second constant-roll parameter γ.\nThen, we discuss the evolution of PBHs, which were formed in the radiation dominated\nera after inflation ended, in BHDE model. By analyzing the evolution of the evaporation\nmass Mevaand the accretion mass Maccrof PBHs with the initial mass Mi=M⊙when\nthe effective state of equation parameter evolves from 1 /3 to less than −1/3, we find the\naccretion mass Maccrincreases to approach 102M⊙which is in the mass window of PBHs,\nthe PBHs mass MBHincreases to 1032kgand the temperature TBHdecreases to 10−10K.\n15ACKNOWLEDGMENTS\nThis work was supported by the National Natural Science Foundation of China under\nGrants Nos. 12265019, 12305056, 11865018, the University Scientific Research Project of\nAnhui Province of China under Grants No. 2022AH051634.\n[1] A. Guth, Phys. Rev. D 23, 347 (1981).\n[2] A. Linde, Phys. Lett. B 108, 389 (1982).\n[3] V. Mukhanov, G. Chibisov, JETP Lett. 33, 532 (1981).\n[4] A. Lewis, A. Challinor, A. Lasenby, Astrophys. J. 538, 473 (2000).\n[5] F. Bernardeau, S. Colombi, E. Gaztanaga, R. Scoccimarro, Phys. Rep. 367, 1 (2002).\n[6] S. Weinberg, Cosmology, Oxford Univ. Press (2008).\n[7] Planck Collaboration, Astron. Astrophys. 641, A10 (2020).\n[8] M. Maggiore, Gravitational Waves, Oxford Univ. Press (2018).\n[9] H. Noh and J. Hwang, Phys. Lett. B 515, 231 (2001).\n[10] J. Martin, H. Motohashi, T. Suyama, Phys. Rev. D 87, 023514 (2013).\n[11] K. Dimopoulos, Ultra slow-roll inflation demystified. Phys. Lett. B 775, 262 (2017).\n[12] C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, JCAP 08, 048 (2018).\n[13] H. Motohashi, A. Starobinsky, J. Yokoyama, JCAP 09, 018 (2015).\n[14] Q. Gao, Sci. China Phys. Mech. Astron. 60, 090411 (2017).\n[15] Q. Gao, Sci. China Phys. Mech. Astron. 61, 70411 (2018).\n[16] Z. Yi, Y. Gong, JCAP 03, 052 (2018).\n[17] L. Anguelova, P. Suranyi, L.C.R. Wijewardhana, JCAP 02, 004 (2018).\n[18] F. Cicciarella, J. Mabillard, M. Pieroni, JCAP 01, 024 (2018).\n[19] M. Guerrero, D. Rubiera-Garcia, D. Gomez, Phys. Rev. D 102, 123528 (2020).\n[20] S. Nojiri, S. Odintsov, and V. Oikonomou, Class. Quantum Grav. 34, 245012 (2017).\n[21] H. Motohashi, A. Starobinsky, Eur. Phys. J. C 77, 538 (2017).\n16[22] S. Panda, A. Rana, and R. Thakur, Eur. Phys. J. C 83, 297 (2023).\n[23] K. Bourakadi, M. Koussour, G. Otalora, M. Bennai, and T. Ouali, Physics of the Dark\nUniverse 41, 101246 (2023).\n[24] H. Motohashi and A. Starobinsky, JCAP 11, 025 (2019).\n[25] A. Mohammadi, T. Golanbari, S. Nasri, and K. Saaidi, Phys. Rev. D 101, 123537 (2020).\n[26] S. Lahiri, Mode. Phys. Lett. A 37, 2250003 (2022).\n[27] R. Herrera, M. Shokri, and J. Sadeghi, Physics of the Dark Universe 41, 101232 (2023).\n[28] A. Mohammadi, Physics of the Dark Universe 36, 101055 (2022).\n[29] A. Keskin and K. Kurt, Eur. Phys. J. C 83, 72 (2023).\n[30] S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 (1971).\n[31] B. Carr and S. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399 (1974).\n[32] G. Chapline, Nature 253, 251 (1975).\n[33] S. Bird, I. Cholis, J. Munoz, Y. Ali-Haimoud, M. Kamionkowski, E. Kovetz, A.Raccanelli,\nand A. Riess, Phys. Rev. Lett. 116, 201301 (2016).\n[34] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Phys. Rev. Lett. 117, 061101 (2016).\n[35] B. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. D 93,\n122003 (2016).\n[36] J. Barrow and B. Carr, Phys. Rev. D 54, 3920 (1996).\n[37] J. Bhadra and U. Debnath, Int. J. Theor. Phys. 53, 645 (2013).\n[38] G. Aliferis and V. Zarikas, Phys. Rev. D 103, 023509 (2021).\n[39] A. Chanda and B. Paul, Eur. Phys. J. C 82, 616 (2022).\n[40] K. Bourakadi, B. Asfour, Z. Sakhi, M. Bennai, and T. Ouali, Eur. Phys. J. C 82, 792 (2022).\n[41] B. Carr and J. Lidsey, Phys. Rev. D 48, 543 (1993).\n[42] P. Ivanov, P. Naselsky, and I. Novikov, Phys. Rev. D 50, 7173 (1994).\n[43] J. Yokoyama, Phys. Rev. D 58, 083510 (1998).\n[44] J. Garcia-Bellido and E. Ruiz Morales, Physics of the Dark Universe 18, 47 (2017).\n[45] M. Biagetti, G. Franciolini, A. Kehagias, and A. Riotto, JCAP 07, 032(2018).\n[46] C. Fu, P. Wu, and H. Yu, Phys. Rev. D 102, 043527 (2020).\n17[47] M. Davies, P Carrilho, and D. Mulryne, JCAP 06, 019(2022).\n[48] H. Di and Y. Gong, JCAP 07, 007 (2018).\n[49] C, Fu, P, Wu, and H, Yu, Phys. Rev. D 100, 063532 (2019).\n[50] G. Ballesteros, J. Rey, M. Taoso, and A. Urbano, JCAP 08, 043 (2020).\n[51] Y. Liu, Q. Wang, B. Su, and N. Li, Physics of the Dark Universe 34, 100905 (2021).\n[52] D. Figueroa, S. Raatikainen, S. Rasanen, and E. Tomberg, JCAP 05, 027 (2022).\n[53] H. Motohashi, S. Mukohyama, and M. Oliosi, JCAP 03, 002 (2020).\n[54] E. Tomberg, Phys. Rev. D 108, 043502 (2023).\n[55] S. Clesse and J. Garcia-Bellido, Phys. Rev. D 92, 023524 (2015).\n[56] M. Kawasaki, A. Kusenko, Y. Tada, and T. T. Yanagida, Phys. Rev. D 94, 083523 (2016).\n[57] B. Carr, F. Kuhnel, and M. Sandstad, Phys. Rev. D 94,083504 (2016).\n[58] K. Inomata, M. Kawasaki, K. Mukaida, Y. Tada, and T. Yanagida, Phys. Rev. D 96, 043504\n(2017).\n[59] K. Inomata, M. Kawasaki, K. Mukaida, and T. Yanagida, Phys. Rev. D 97, 043514 (2018).\n[60] B. Carr and J. Silk, Mon. Not. Roy. Astron. Soc. 478, 3756 (2018).\n[61] C. Germani and I. Musco, Phys. Rev. Lett. 122, 141302 (2019).\n[62] A. Kusenko, M. Sasaki, S. Sugiyama, M. Takada, V. Takhistov, and E. Vitagliano, Phys.\nRev. Lett. 125, 181304 (2020).\n[63] R. Calabrese, D. Fiorillo, G. Miele, S. Morisi, and A. Palazzo, Phys. Lett. B 829, 137050\n(2022).\n[64] J. de Freitas Pacheco, E. Kiritsis, M. Lucca, and J. Silk, Phys. Rev. D 107, 123525 (2023).\n[65] M. Flores and A. Kusenko, JCAP 05, 013 (2023).\n[66] V. Dike, D. Gilman, T. Treu, Mon. Not. Roy. Astron. Soc. 522, 5434 (2023).\n[67] J. de Freitas Pacheco and J. Silk, Phys. Rev. D 101, 083022 (2020).\n[68] G. Fuller, A. Kusenko, and V. Takhistov, Phys. Rev. Lett. 119, 061101 (2017).\n[69] C. Keith, D. Hooper, N. Blinov, and S. McDermott, Phys. Rev. D 102, 103512 (2020).\n[70] M. Kawasaki, A. Kusenko, and T. Yanagida, Phys. Lett. B 711, 1 (2012).\n[71] K. Inomata, M. Kawasaki, K. Mukaida, Y. Tada, and T. Yanagida, Phys. Rev. D 95, 123510\n18(2017).\n[72] D. Page, Phys. Rev. D 13, 198 (1976).\n[73] F. Kuhnel and K. Freese, Phys. Rev. D 95, 083508 (2017).\n[74] B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen, and H. Veermae, Phys. Rev. D 96, 023514\n(2017).\n[75] R. DAgostino, R. Giambo, and O. Luongo, Phys. Rev. D 107, 043032 (2023).\n[76] S. Hsu, Phys. Lett. B 59413 (2004).\n[77] R. Horvat, Phys. Rev. D 70087301 (2004).\n[78] M. Li, Phys. Lett. B 6031 (2004).\n[79] E. Witten, Adv. Theor. Math. Phys. 2253 (1998).\n[80] R. Bousso, Rev. Modern Phys. 74825 (2002).\n[81] J. D. Barrow, Phys. Lett. B 808135643 (2020).\n[82] E. N. Saridakis, Phys. Rev. D 102123525 (2020).\n[83] F. K. Anagnostopoulos, S. Basilakos, and E. N. Saridakis, Eur. Phys. J. C 80826 (2020).\n[84] S. Srivastava and U. K. Sharma, Int. J. Geom. Methods Mode. Phys. 182150014 (2021).\n[85] A. Sheykhi, Phys. Rev. D 103, 123503 (2021).\n[86] A. Oliveros, M. Sabogal, and M. Acero, Eur. Phys. J. Plus 137, 783 (2022).\n[87] Q. Huang, H. Huang, B. Xu, F. Tu, and J. Chen, Eur. Phys. J. C 81, 686 (2021).\n[88] P. Adhikary, S. Das, S. Basilakos, and E. Saridakis, Phys. Rev. D 104, 123519 (2021).\n[89] A. Mamon, A. Paliathanasis, and S. Saha, Eur. Phys. J. Plus 136, 134 (2021).\n[90] S. Rani and N. Azhar, Universe 7, 268 (2021).\n[91] G. Luciano, Phys. Rev. D 106, 083530 (2022).\n[92] S. Nojiri, S. Odintsov, and T. Paul, Phys. Lett. B 825, 136844 (2022).\n[93] B. Paul, B. Roy, and A. Saha, Eur. Phys. J. C 82, 76 (2022).\n[94] G. Luciano, Physics of the Dark Universe 41, 101237 (2023).\n[95] N. Boulkaboul, Physics of the Dark Universe 40, 101205 (2023).\n[96] R. Pankaj, U. Sharma, and N. Ali, Astrophys Space Sci. 368, 15 (2023).\n[97] S. Ghaffari, G. Luciano, and S. Capozziello, Eur. Phys. J. Plus 138, 82 (2023).\n19[98] M. Asghari and A. Sheykhi, Eur. Phys. J. C 82, 388 (2022).\n[99] K. Jusufi, M. Azreg-Ainou, M. Jamil, and E. Saridakis, Universe 8, 102 (2022).\n[100] J. Barrow, S. Basilakos, and E. Saridakis, Phys. Lett. B 815, 136134 (2021).\n[101] F. Anagnostopoulos, S. Basilakos, and E. Saridakis, Eur. Phys. J. C 80, 826 (2020).\n[102] J. Barrow, Phys. Lett. B 808, 135643 (2020).\n[103] S. Nojiri, S. Odintsov, and V. Oikonomou, Phys. Rep. 692, 1 (2017).\n[104] J. Hwang and H. Noh, Phys. Rev. D 71, 063536 (2005).\n[105] Planck Collaboration, A&A 641, A10 (2020).\n[106] S. Hawking, Nature 248, 30 (1974).\n[107] A. Coogan, L. Morrison, and S. Profumo, Phys. Rev. Lett. 126, 171101 (2021).\n[108] D. Page, Phys. Rev. D 13, 198 (1976).\n[109] B. Nayak and L. Singh, Pramana 76, 173 (2011).\n[110] M. Jamil and A. Qadir, Gen. Relativ. Gravit. 43, 1069 (2011).\n[111] E. Babichev, V. Dokuchaev, and Y. Eroshenko, Phys. Rev. Lett. 93, 021102 (2004).\n[112] B. Nayak and L. Singh, Phys. Rev. D 82, 127301 (2010).\n20" }, { "title": "2401.15456v5.Product_Mixing_in_Compact_Lie_Groups.pdf", "content": "arXiv:2401.15456v5 [math.CO] 18 Mar 2024Product Mixing in Compact Lie Groups\nDavid Ellis∗Guy Kindler†Noam Lifshitz‡Dor Minzer§\nAbstract\nIfGis a group, we say a subset SofGisproduct-free if the equation xy=zhas\nno solutions with x,y,z∈S. ForD∈N, a groupGis said to be D-quasirandom\nif the minimal dimension of a nontrivial complex irreducible representa tion ofGis at\nleastD. Gowers showed that in a D-quasirandom finite group G, the maximal size of a\nproduct-free set is at most |G|/D1/3. This disproved a longstanding conjecture of Babai\nand S´ os from 1985.\nFor the special unitary group, G= SU(n), Gowers observed that his argument\nyields an upper bound of n−1/3on the measure of a measurable product-free subset.\nIn this paper, we improve Gowers’ upper bound to exp( −cn1/3), wherec >0 is an\nabsolute constant. In fact, we establish something stronger, na mely,product-mixing for\nmeasurablesubsetsofSU( n) with measureat leastexp( −cn1/3); forthis product-mixing\nresult, then1/3in the exponent is sharp.\nOur approach involves introducing novel hypercontractive inequa lities, which imply\nthat the non-Abelian Fourier spectrum of the indicator function of a small set concen-\ntrates on high-dimensional irreducible representations. Our hype rcontractive inequali-\nties are obtained via methods from representation theory, harmo nic analysis, random\nmatrix theory and differential geometry. We generalize our hyperc ontractive inequali-\nties from SU( n) to an arbitrary D-quasirandom compact connected Lie group for Dat\nleast an absolute constant, thereby extending our results on pro duct-free sets to such\ngroups.\nWe also demonstrate various other applications of our inequalities to geometry\n(viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, a nd the theory\nof growth in compact Lie groups. A subsequent work due to Arunac halam, Girish and\nLifshitz uses our inequalities to establish new separation results bet ween classical and\nquantum communication complexity.\n∗Department of Mathematics, University of Bristol.\n†Department of Computer Science, Hebrew University of Jerus alem.\n‡Einstein Institute of Mathematics, Hebrew University of Je rusalem. Supported by the Israel Science\nFoundation (grant no. 1980/22).\n§Department of Mathematics, Massachusetts Institute of Tec hnology. Supported by a Sloan Research\nFellowship, NSF CCF award 2227876 and NSF CAREER award 22391 60.\n1Contents\n1 Introduction 3\n1.1 Quasirandomness for groups, and mixing. . . . . . . . . . . . . . . . . . . . 4\n1.2 Ideas and techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n2 Preliminaries: Quasirandomness and min-rank 16\n3 Good groups and fine groups 18\n3.1 Graded groups and ‘good’ groups. . . . . . . . . . . . . . . . . . . . . . . . 18\n3.2 Fine groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n3.3 Goodness and fineness are preserved under taking product s and quotients. . 22\n4 Growth in good groups 25\n4.1 Level dinequalities and the eigenvalues of convolution operators . . . . . . 26\n4.2 Upper bounds on the measures of product-free sets. . . . . . . . . . . . . . 30\n4.3 Product mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0\n4.4 Equidistribution of convolutions . . . . . . . . . . . . . . . . . . . . . . . . . 32\n5 Growth in fine groups 33\n5.1 Product mixing in fine groups . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n5.2 Non-Abelian Brunn-Minkowski type inequalities for fine groups . . . . . . . 35\n5.3 Diameter bounds for fine groups . . . . . . . . . . . . . . . . . . . . . . . . 37\n6 The strong quasirandomness of the simply connected compact Lie groups 37\n6.1 The Peter-Weyl theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\n6.2 Weyl’s construction for SO( n), and its applications . . . . . . . . . . . . . . 38\n6.3 Obtaining strong quasirandomness for SO( n). . . . . . . . . . . . . . . . . . 42\n6.4 Obtaining strong quasirandomness for Spin( n). . . . . . . . . . . . . . . . . 42\n6.5 Two descriptions of the compact symplectic group Sp (n). . . . . . . . . . 43\n6.6 Weyl’s construction in Sp( n), and its applications . . . . . . . . . . . . . . . 44\n6.7 Obtaining strong quasirandomness for Sp( n). . . . . . . . . . . . . . . . . . 47\n6.8 Weyl’s construction for SU( n), and its applications. . . . . . . . . . . . . . . 47\n6.9 Obtaining strong quasirandomness for SU( n). . . . . . . . . . . . . . . . . . 53\n7 Simply connected compact Lie groups are fine 54\n8 Showing that Sp( n),Spin(n) and SU( n) are good 56\n8.1 The Gaussian noise operator, a.k.a. the Ornstein–Uhlen beck operator . . . 57\n8.2 Constructing the noise operator T ρ. . . . . . . . . . . . . . . . . . . . . . . 57\n8.3 The noise operator T ρis hypercontractive . . . . . . . . . . . . . . . . . . . 60\n29 Comfortable polynomials on SO (n), and their properties. 62\n9.1 Comfortable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62\n9.2 Reducing Theorem 8.8 to a statement about low-degree tru ncations of T col. 66\n10 Low-degree truncations of Tcol: Proof of Lemma 9.9. 66\n10.1 Comparing L2(µ) andL2(γ). . . . . . . . . . . . . . . . . . . . . . . . . . . 66\n10.2 The main argument for proving Lemma 9.9. . . . . . . . . . . . . . . . . . . 67\n10.3L2(µ) is dominated by L2(γ) on column or row comfortable juntas: Proof of Lemma 10.1. 71\n10.4L2(γ) is dominated by L2(µ) on comfortable juntas: Proof of Lemma 10.2. . 73\nA Bounding the dimensions of high level representations 81\nA.1 The special orthogonal group SO( n) . . . . . . . . . . . . . . . . . . . . . . 81\nA.2 The spin group Spin( n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82\nA.3 The special unitary group SU( n) . . . . . . . . . . . . . . . . . . . . . . . . 83\nB The required adaptations for showing that Sp (n)and SU(n)are good 85\nC Bounding the eigenvalues of the Laplace-Beltrami operator 96\nC.1 Bounding the eigenvalues of the Laplace-Beltrami opera tor in SO(n) . . . . 96\nC.2 Bounding the eigenvalues of the Laplace–Beltrami opera tor in SU(n) . . . . 99\nC.3 Bounding the eigenvalues of the Laplace-Beltrami opera tor in Sp(n) . . . . 101\n1 Introduction\nA subsetAof a group Gis said to be product-free ifgh /∈Afor allg,h∈A. The study\nof product-free subsets of groups has attracted significant attention over the past three\ndecades. In 1985, Babai and S´ os [3] considered the problem o f determining the largest\nsize of a product-free set in a finite group G. They conjectured that exists an absolute\npositive constant c0>0 such that any finite group Ghas a product-free set of size at least\nc0|G|. In the Abelian case, this is quite easy to see, and had previo usly been observed by\nErd˝ os, in an unpublished communication to Babai and S´ os. ( In the cyclic case ( Zn,+),\none can take a ‘middle-third’ construction, viz., {x∈Zn:n/3< x/lessorequalslant2n/3}, as a large\nproduct-free set, and one can reduce to the cyclic case by obs erving that any finite Abelian\ngroup has a nontrivial cyclic quotient, and that the preimag e of a product-free set under\na quotient map is also product-free and of the same measure.) The exact answer in the\nAbelian case was given by Green and Ruzsa [15] in 2003: the lar gest product-freesubsetof a\nfinite Abelian group Ghas sizec|G|, where the function c=c(G)∈[2/7,1/2] was explicitly\ndetermined by Green and Ruzsa. The general Babai-S´ os conje cture was disproved in 2008\nby Gowers [14], who showed that if Gis a finite group such that the minimal dimension of\na nontrivial irreducible complex representation of Gis equal to D, then any product-free\nsubset ofGhas size at most D−1/3|G|. It remains to observe that the quantity D=D(G)\nis unbounded over finite non-Abelian groups G. For example, for the projective special\nlinear group PSL 2(Fq) (forqan odd prime power), we have D(PSL2(Fq)) = (q−1)/2, so\n3the measure of a product-free subset of PSL 2(Fq) is at most O(q−1/3), which tends to zero\nasqtends to infinity.\nGowers observed that his argument also implies that if Gis an (infinite) compact group\nfor which the minimal dimension of a nontrivial irreducible complex continuous representa-\ntion is equal to D, then the maximal Haar measure of a measurable, product-fre e set inG\nis at mostD−1/3. For SU(n) we haveD(SU(n)) =n, implying an upper bound of n−1/3on\nthe measure of a measurable product-free subset of SU( n). However, Gowers conjectured\nthat for SU( n), the true answer is exponentially small in n. Indeed, as Gowers states, it\nseems difficult to come up with an example better than the follo wing. Recall that group\nSU(n) acts on the complex unit sphere {v∈Cn:/⌊ar⌈⌊lv/⌊ar⌈⌊l2= 1}, and takeAto be the set of\nall matrices A∈SU(n) such that the real part of /a\\}⌊ra⌋k⌉tl⌉{tAe1,e1/a\\}⌊ra⌋k⌉tri}htis less than−1/2. As noted by\nGowers, it follows from the triangle inequality that this se t is product-free, and it is easy\nto check that the measure of Ais 2−Ω(n).\nIn this work, we make progress towards proving Gowers’ conje cture. Specifically, we\nimprove Gowers’ upper bound by a stretched exponential fact or, viz., from n−1/3toe−cn1/3.\nTheorem 1.1. There exists an absolute constant c >0such that the following holds. Let\nn∈Nand letA⊂SU(n)be Haar-measurable and product-free. Then µ(A)/lessorequalslantexp(−cn1/3).\n1.1 Quasirandomness for groups, and mixing.\nGowers’ bound for product-free sets relies on a relationshi p between spectral gaps and\ndimensionsofirreduciblerepresentations, arelationshi pwhichwasfirstdiscoveredbySarnak\nand Xue [40]. In fact, Gowers’ proof uses a beautiful connect ion between the problem and a\npurely representation-theoretic notion that Gowers calle dquasirandomness (due to a rough\nequivalence with the graph-quasirandomnessof certain Cay ley graphs, an equivalence which\nwe shall explain below). For a group Gwe denote by D(G) the minimal dimension of a\nnon-trivial complex irreducible continuous representati on ofG. (Henceforth, for brevity, we\nwill use the term representation to mean continuous representation.) For d∈N, we say that\na groupGisd-quasirandom ifD(G)/greaterorequalslantd.1Denoting by α(G) the largest possible density|A|\n|G|\nof a product-free set A⊆G(ifGis a finite group), Gowers showed that for any finite group\nG,α(G)/lessorequalslantD(G)−1/3. SinceD(G) can be arbitrarily large (as is the case for the alternating\ngroups, which have D(An) =n−1 for alln/greaterorequalslant7, and the groups PSL 2(Fq) as mentioned\nabove, and for many other natural infinite families of finite g roups), this disproved the\nconjecture of Babai and S´ os.\nFor finite groups, the quasirandomness parameter gives an al most complete description\nof the maximal size of a product free set. Pyber (see [14]) use d the Classification of Finite\nSimple Groups, together with a construction of Kedlaya, sho wing thatα(G)/greaterorequalslantD(G)−C\nwhereC >0 is an absolute constant. Nikolov and Pyber [34] later impro ved this to\nα(G)/greaterorequalslant1\nCD(G). This established a remarkable fact, namely that the purely representation\n1To avoid confusion with the quasirandomness parameter for g raphs, it might have been less ambiguous\nto call this notion ‘ d-group-quasirandomness’, but as the latter is rather cumbe rsome we have opted for the\nabove shorter formulation; we hope that this will not cause t he reader confusion, in the sequel.\n4theoretic quasirandomness parameter D(G) is polynomially related to the thecombinatorial\nquantityα(G).\n(CD(G))−1/lessorequalslantα(G)/lessorequalslantD(G)−1/3(1)\nFor compact connected Lie groups we obtain the following gen eral variant of Theo-\nrem 1.1, which gives an upper bound on the size of a product-fr ee set in the group.\nTheorem 1.2. There exists an absolute constant c >0such that the following holds. Let\nGbe a compact connected Lie group, and let ˜Gbe its universal cover. Let A ⊂Gbe\nHaar-measurable and product-free. Let µdenote the Haar probability measure on G. Then\nµ(A)/lessorequalslantexp(−cD(˜G)1/3).\nAn elegant argument of Gowers [14] (proof of Theorem 4.6, the rein) for finite groups,\nwhich generalises very easily to the case of compact groups, shows that if Gis a compact\ngroup then it has a measurable product-free subset of measur e at least exp(−Ω(D(G)). In\nSection 2 we show that D(G) =O(D(˜G)2). These two facts combine with Theorem 1.2 to\ngive the following analogue of (1) for compact connected Lie groups.\nCorollary 1.3. There exists an absolute constant c>0such that the following holds. For\nevery compact connected Lie group G,\ncD(G)1/6/lessorequalslantlog(1/α(G))/lessorequalslant1\ncD(G).\n(We remark that our logs will always be taken with respect to t he natural basis.) Corollary\n1.3 says that, as with finite groups, the maximal measure of a m easurable product-free set\nin a compact connected Lie group is controlled by the quasira ndomness parameter, but this\ntime the control moves to the exponent.\nQuasirandomness (for groups) was a crucial ingredient in th e ‘Bourgain–Gamburd ex-\npansion machine’, which is a three-step method for obtainin g spectral gaps for Cayley\ngraphs (see e.g. Tao [41], for an exposition). Briefly, this ‘ machine’ proceeds as follows: one\nfirst shows that the graph has high girth, then one shows that t here are no ‘approximate\nsubgroups’ in which a random walk could be entrapped, and the n quasirandomness is used\n(together with with the trace method) to finally obtain a spec tral gap. Quasirandomness\n(for groups) has many other applications, such as in boundin g the diameters of Cayley\ngraphs (see e.g. the survery of Helfgott [19]).\nThe term ‘quasirandomness’ was used (for groups) by Gowers, due to the following\nconnection with the (now classical) notion of quasirandomn ess for graphs. (There are, of\ncourse, now notions of quasirandomness for a huge variety of combinatorial and algebraic\nstructures; roughly speaking, these say the structure beha ves in a random-like way, in\nan appropriate sense.) We now need some more terminology. Th enormalized adjacency\nmatrixAH∈RV×Vof ad-regular graph H= (V,E) has (i,j)-th entry equal to 1 /dif\n{i,j}∈E, and equal to zero otherwise. The graph His said to be ε-quasirandom if all\nthe nontrivial eigenvalues of AHare at most εin absolute value (here, ‘nontrivial’ means\nhaving an eigenvector orthogonal to the constant functions ).\n5One of the striking consequences of d-quasirandomness for a finite group G, is that it\nimplies that Cayley graphs of the form Cay(G,S) are (1/poly(d))-quasirandom, whenever\nSis a dense subset of G. The fact that this only relies on density considerations an d does\nnot require any assumption on the structure of S, makes the notion of quasirandomness for\ngroups rather powerful.\nMore generally, applications of quasirandomness for a grou pGcan often be (re)phrased\nas follows. Suppose that Gisd-quasirandom, and that we have a linear operator T:\nL2(G)→L2(G) whose nontrivial eigenvalues we want to bound (in absolute value) from\nabove; suppose further that Tcommutes with either the left or the right action of Gon\nL2(G). (In Gowers’ proof, slightly rephrased, the operator Tcould be viewed as B∗B,\nwhereBis the bipartite adjacency matrix of the bipartite Cayley gr aph with vertex-classes\nconsisting of two disjoint copies of G, and where the edges are all pairs of the form ( g,sg)\nforg∈Gands∈S,Sbeing a product-free set in G.) Then by the commuting property,\neach eigenspace of Tis a nontrivial representation of G, and therefore has dimension at\nleastd; it follows that each nontrivial eigenvalue of Ghas multiplicity at least d. But the\nsum of the squares of the eigenvalues of Tis equal to Trace( T2), and this yields the bound\nd|λ|2/lessorequalslantTrace(T2) for all nontrivial eigenvalues λofT. This is often called the Sarnak-Xue\ntrick, as it was first employed in [40]\nBourgain and Gamburd used their ‘expansion machine’ (allud ed to above) to show that\ntaking two uniformly random elements a,b∈SL2(Fp) is sufficient for the Cayley graph\nCay/parenleftbig\nSL2(Fp,{a,b,a−1,b−1}/parenrightbig\nto be an expander with high probability, ptending to infinity.\nIt is a major open problem in the theory of Cayley graphs to obt ain a similar result in\nthe unbounded-rank case, for example for SL n(Fp) wherepis fixed and ntends to infinity.\nOne of the properties that breaks down when one attempts to us e the Bourgain–Gamburd\nexpansionmachineinthecaseof unboundedrank, isthedepen denceofthequasirandomness\nparameter on the cardinality of the group. Specifically, in o rder for the Bourgain–Gamburd\nexpansion machine to work effectively for a group G, the quasirandomness parameter D(G)\nneeds to be polynomial in the cardinality of G. In the unbounded rank case, this no longer\nholds. For example, D(SLn(Fp))/lessorequalslantpn(consider the representation of dimension pninduced\nby the natural action of SLn(Fp) onFn\np). The situation is even worse for the alternating\ngroupAn, asD(An) =n−1 forn/greaterorequalslant7, andn−1 is less than logarithmic in the cardinality\nof the group.\n1.2 Ideas and techniques\nTo improve on the upper bound of Gowers, we need to find methods for ‘dealing with’\nthe low-dimensional irreducible representations (more pr ecisely, for dealing with the corre-\nsponding parts of the Fourier transform). In this paper, we d evelop some new techniques\nfor this in the case of compact connected Lie groups. These te chniques turn out also to be\nuseful for finite groups; for example, in [28], analogues of s ome of our methods are devel-\noped for the alternating group An(where the idea of mixing is replaced by a refined notion,\nreferred to therein as a ‘mixing property for global sets’).\nBelow we give indications of the new techniques that are used to obtain our improved\n6bounds, and the various areas of mathematics from which they originate.\nLeveldinequalities and hypercontractivity\nOne of our key ideas is motivated by the (now well-developed) theory of the analysis of\nBoolean functions. A function\nf:{−1,1}n→R\nhas aFourier expansion f=/summationtext\nS⊆[n]ˆf(S)χS, whereχS:{−1,1}n→{−1,1}is defined by\nχS(x) :=/producttext\ni∈Sxifor eachx∈{−1,1}nandS⊆[n]. The functions χS, known as the\nFourier-Walsh functions orcharacters , are orthonormal (with respect to the natural inner\nproduct on R[{−1,1}n] induced by the uniform measure). The Fourier expansion giv es rise\nto a coarser orthogonal decomposition, f=/summationtextn\nd=0f=d,where\nf=d:=/summationdisplay\n|S|=dˆf(S)χS.\nThis is known as the degree decomposition (as each function f=dis a homogeneous polyno-\nmial of total degree din thexi’s).\nTheleveldinequality for the Boolean cube (essentially due to Kahn–Kalai–Linial [27]\nand Benjamini–Kalai–Schramm [6]) states that there exists an absolute constant C >0,\nsuch that for a set A⊆{−1,1}nof density|A|\n2n=α, ifd/lessorequalslantlog(1/α) then the characteristic\nfunctionf= 1Asatisfies/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslantα2/parenleftBig\nClog(1/α)\nd/parenrightBigd\n. Roughlyspeaking, thelevel dinequality\nsays that indicators of small sets are very much uncorrelate d with low degree polynomials.\nOne of our key ideas in this paper is to generalize the level dinequality from the Boolean\ncube to the setting of compact connected Lie groups.\nThe main tool in the proof of the Boolean level dinequality is the Bonami–Gross–\nBeckner hypercontactivity theorem. It states that the noise operator Tρf:=/summationtextn\nd=0ρdf=d\nisacontraction asanoperatorfrom LqtoLp, forallq>p/greaterorequalslant1provided0 /lessorequalslantρ/lessorequalslant/radicalBig\nq−1\np−1. This\nimmediately implies that /⌊ar⌈⌊lf=d/⌊ar⌈⌊lq/lessorequalslantρ−d/⌊ar⌈⌊lf=d/⌊ar⌈⌊lpfor any function f. Roughly speaking, this\nlast inequality says that Lp-norm of a low-degree function does not change too drastical ly\nwithp. This is in stark contrast with the behaviour of indicator fu nctions of small sets,\nf= 1A. These satisfy/⌊ar⌈⌊lf/⌊ar⌈⌊lp=α1/p, which does change rapidly with p. This difference in\nbehaviours can be used to prove the level d-inequality, stating that indicators of small sets\nare essentially orthogonal to the low degree functions.\nThe same proof-concept works hand in hand with the represent ation theory of compact\nsimple Lie groups. For simplicity, let us restrict our atten tion (at first) to the group G=\nSO(n). For each d∈N∪{0}, we letV/lessorequalslantd⊆L2(G) denote the subspace of L2(G) spanned\nby the polynomials of degree at most din the matrix entries of X∈G= SO(n); so, for\nexample,X11X22∈V/lessorequalslant2. We also let V=d:=V/lessorequalslantd∩(V/lessorequalslantd−1)⊥, for eachd∈N. Given\nf∈L2(G), we letf/lessorequalslantddenote the orthogonal projection of fontoV/lessorequalslantd, and we let f=d\ndenote the orthogonal projection of fontoV=d, so thatf=d=f/lessorequalslantd−f/lessorequalslantd−1. The subspaces\n7V/lessorequalslantdandV=dare two-sided ideals of L2(G) (i.e., they are closed under both left and right\nactions ofGonL2(G)). Now, if Jis a two-sided ideal of L2(G) andT:L2(G)→L2(G) is\na linear operator that commutes with either the left or the ri ght action of G(as will be the\ncase with all the operators we will work with), it follows fro m the classical representation\ntheory of compact groups (viz., the Peter-Weyl theorem and S chur’s lemma) that ThasJ\nas an invariant subspace. Hence, such an operator Thas eachV=das an invariant subspace,\nso each eigenspace of Tcan be taken to be within one of the V=d’s. It therefore makes\nsense to consider quasirandomness relative to the degree de composition. For each d∈N,\nwe letDdbe the smallest dimension of a subrepresentation of the G-representation V=d.\nThe obvious adaptation of the Sarnak-Xue trick, described a bove, then yields that for any\neigenvalueλofTwith eigenspace within V=d, we haveDd|λ|2/lessorequalslantTrace(T2). It turns out\nthatDdgrows very fast with d, yielding very strong upper bounds on the corresponding |λ|\nfor larged.\nOn the other hand, an ideal level dinequality would imply that if Ais an indicator of\na small set, then most of its mass lies on the high degrees. Thi s combines with the fast\ngrowth ofDd(withd) to give a much more powerful form of quasirandomness, one th at\ntakes into account the fact that fis{0,1}-valued, and gives much better bounds.\nWe remark that the above degree decomposition can be easily e xtended to all compact\nlinear Lie groups G/lessorequalslantGLn(C) by letting V/lessorequalslantdbe the space of degree /lessorequalslantdpolynomials in the\nreal and imaginary parts of the matrix entries of X∈G. (In fact, this notion generalizes\nfairly easily to arbitrary compact simple Lie groups, even w hen they are not linear.) As in\nthe SO(n) case, we let f/lessorequalslantddenote the orthogonal projection of fontoV/lessorequalslantd.\nWe obtain the following level dinequality.\nTheorem 1.4. There exists absolute constants c,C >0such that the following holds. Let\nGbe a simple compact Lie group equipped with its Haar probabili ty measure µ. Suppose\nthatD(G)/greaterorequalslantC. LetA⊆Gbe a measurable set with α:=µ(A)/greaterorequalslante−cD(G). Then for each\nd∈N∪{0}withd/lessorequalslantlog(1/α), we have/⌊ar⌈⌊lf/lessorequalslantd/⌊ar⌈⌊l2\n2/lessorequalslantα2/parenleftBig\n2log(1/α)\nd/parenrightBigCd\n.\nWhenGis simply connected and d/lessorequalslantc√nwe are able to obtain an even stronger level\ndinequality, which is similar to the one on the Boolean cube wi thout the extra Cfactor in\nthe exponent. This leads to the following.\nTheorem 1.5. There exists absolute constants C,c>0such that the following holds. Let\nGbe a compact connected Lie group, let ˜Gdenote its universal cover, and write n=D(˜G).\nSuppose that n/greaterorequalslantC. LetA⊆Gbe a measurable set with α:=µ(A)/greaterorequalslantexp(−cn1/2). Then\nfor eachd∈N∪{0}withd/lessorequalslantlog(1/α), we have/⌊ar⌈⌊lf/lessorequalslantd/⌊ar⌈⌊l2\n2/lessorequalslantα2/parenleftBig\nClog(1/α)\nd/parenrightBigd\n.\nIt is this second level d-inequality that is responsible for the 1 /3 in the exponent of\nTheorem 1.2. Unfortunately, one would not be able to improve our 1/3 in the exponent\nto the (conjectural) right one, merely by strengthening thi s leveld-inequality. Indeed, our\nsecond level dinequality can be easily seen to be sharp up to the value of the absolute\n8constantC, by considering sets of the form {A∈SO(n) :/a\\}⌊ra⌋k⌉tl⌉{tAe1,e1/a\\}⌊ra⌋k⌉tri}ht>1−t}for appropriate\nvalues oft, whenG= SO(n), for example.\nBoth of our level dinequalities are inspired by the same ideas from the Boolean setting,\ntogether with an extra representation theoretic ideas. Nam ely, in order to show a level d\ninequality, we upper-bound q-norms of low degree polynomials in terms of their 2-norms,\nand then use H¨ older’s inequality. In the Boolean cube, such upper bounds follow from two\nfacts. The first is that the noise operator T ρis hypercontractive. The second is that all the\neigenvalues of the restriction of T ρtoV/lessorequalslantdare bounded from below by ρd. Our approach is\nto construct operators on L2(G) that satisfy the same two properties.\nDifferential geometry and Markov diffusion processes\nOur leveldinequalities stem from two techniques for obtaining hyperc ontractivity. Our\nfirst leveldinequality, Theorem 1.4, is obtained via the following meth od. First, we observe\nthat we may assume without loss of generality that our group Gis simply connected.\n(This is because every compact simple Lie group is a quotient of its universal cover by a\ndiscrete subgroup of its centre.) We then make use of classic al lower bounds on the Ricci\ncurvature of our (simply connected) compact simple Lie grou p. The Bakry-Emery criterion\n[4] translates such lower bounds on the Ricci curvature into log-Sobolev inequalities for\nthe Laplace-Beltrami operator L. We then apply an inequality of Gross [16] to deduce\na hypercontractive inequality for the operator e−tLfrom the log-Sobolev inequality. This\ninequality then allows us to prove our first level dinequality. The operator e−tLis the one\ncorresponding to Brownian motion on G. In order to deduce our level d-inequality we rely\non a formula for the eigenvalues of the Laplacian in terms of a step vector corresponding\nto each eigenspace. This formula is well-known in the theory of Lie groups; it is given for\nexample in Berti and Procesi [7].\nRandom walks on bipartite graphs\nThere are two mutually adjoint linear operators that corres pond to a random walk on\nad-regular bipartite graph B⊆L×R. We denote those by T:L2(L)→L2(R) and\nT∗:L2(R)→L2(L) and they are given by taking expectations over a random neig hbour;\nexplicitly, ( Tf)(x) =Ey∼xf(y) forf∈L2(L) andx∈R, and (T∗g)(y) =Ex∼yg(x) for\ng∈L2(R) andy∈L. It is easy to see that both operators are contractions with r espect\nto any norm. It turns out that given such a bipartite graph and given a hypercontractive\noperatorSonRone gets for free that the operator T∗STis hypercontractive. Filmus et al\n[11] used this idea to obtain a ‘non-Abelian’ hypercontract ive estimate for ‘global’ functions\non the symmetric group, from an ‘Abelian’ hypercontractive result for ‘global’ functions on\n(Zn)n. (Informally, a ‘global’ function is one where one cannot in crease the expectation\nvery much by restricting the values of a small number of coord inates.)\nInthiswork, weextendthisideatothecontinuousdomain, by replacingabipartitegraph\nby acouplingof two probability distributions. Specificall y, weconsider theprobability space\n(Rn×n,γ) ofnbynGaussian matrices (i.e., Rn×nwith each entry being an independent\n9standardGaussian), andtheHaar measureonO( n). For (Rn×n,γ), theOrnstein–Uhlenbeck\noperatorUρis a hypercontractive analogue of the noise operator from th e Boolean case. We\ncouple (Rn×n,γ) with SO(n) by applying the Gram–Schmidt operation on the columns of\na given Gaussian matrix (flipping the sign of the last column, if necessary, so as to ensure\nthat the determinant is equal to one). We note that essential ly the same coupling has been\nused before, e.g. by Jiang [23]; however, it has not been used before (to our knowledge) to\nanalyse the distribution of high-degree polynomials in the matrix-entries, which is crucial\nin our work.\nThis coupling gives rise to operators T coland T∗\ncol, similar to the ones in the discrete\ncase. The hypercontractive inequality for the Ornstein–Uh lenbeck operator Uρ, together\nwith our coupling implies a hypercontractive inequality fo r the operator T′\nρ:= T∗\ncolUρTcol.\nWe then use a symmetrization trick to obtain an operator T ρ:=EB∼µR∗\nBT′\nρRB, where\nRBcorresponds to right multiplication by B. The symmetrization does not change the\nhypercontractive properties, which are the same as for Uρ(see Theorem 8.8), but it has the\nadvantage of allowing us to analyse more easily the eigenval ues of the operator.\nRepresentation theory\nThe hypercontractive inequality for the operator T ρis useful due to the fact that it imme-\ndiately gives bounds on the norms of eigenfunctions of T ρ. Because of the symmetrization,\nTρcommutes with the action of Gfrom both sides. Therefore, the Peter-Weyl theorem\nimplies that every isotypical2component of L2(G) is contained in an eigenspace of T ρ.\nWe eventually show that the eigenvalues of the restriction o f TρtoVdare at least ( cρ)d,\nfor some absolute constant c >0. This implies that T ρis indeed a good analogue of\nthe noise operator on the Boolean cube, and of the Ornstein–U hlenbeck operator Uρ, i.e.\nthe noise operator on Gaussian space. We obtain this lower bo und by showing that each\nisotypical component contains certain functions that are n ice to deal with, functions we call\nthecomfortable juntas .\nThe latter are defined as follows. We define a d-juntato be a function in the matrix\nentries ofX∈SO(n) that depends only upon the upper-left dbydminor ofX. Such ad-\njunta is said to be comfortable d-junta if it is contained in the linear span of the monomials\n{mσ:σ∈Sd}, wheremσ: SO(n)→Ris defined by mσ(X) =/producttextd\ni=1Xi,σ(i)for each\nX∈SO(n), for each permutation σ∈Sd.\nRandom matrix theory\nOne of the main discoveries of random matrix theory is that th e entries of a random or-\nthogonal matrix behave (in an appropriate sense) like indep endent Gaussians of the same\nexpectation and variance: at least, when one restricts mino rs of the matrix that are not\ntoo large. (In fact, this holds for many different models of ran dom matrices, not just the\northogonal ensemble.) The power of this discovery is of cour se that a Gaussian random\n2Ifρis an irreducible representation of GandVis aG-module, the ρ-isotypical component ofVis the\nsum of all subrepresentations of Vthat are isomorphic to ρ.\n10matrix is a priori much easier to analyse than e.g. the random matrix given by th e Haar\nmeasure on a group.\nOne way to test that two distributions are similar is to apply a continuous ‘test function’\nand take expectations. Usually, for applications in random matrix theory, the test function\ncan be taken to be an arbitrary fixed polynomial.\nWhen computing the eigenvalues of our operator T ρwe need to show a similarity in\ndistribution between the upper d×d-minor of O( n) and thed×dminor of a random\nGaussian matix. For us, however, it is not sufficient to look at a single polynomial of\nfixed degree. Instead, we need to show a similarity in the dist ribution with respect to our\ncomfortable d-juntas (where dmay be as large as√n, rather than an absolute constant).\nHence, while the philosophy is similar to that of random matr ix theory, we require new\ntechniques enabling us to deal with the distributions of pol ynomials whose degrees may be\na function of n, indeed up to√n.\n1.3 Applications\nIn this section we list several applications of our hypercon tractive theory: to some problems\nin group theory, in geometry, and in probability.\nTo state some of our results, we need some more terminology. I fGis a compact\nconnected Lie group, we define n(G) :=D(˜G), where ˜Gdenotes the universal cover of\nG. It is well-known that, for each m∈N, we haveD(SU(m)) =D(Spin(m)) =mand\nD(Sp(m)) = 2m(and all these groups are simply connected except for Spin(2 )); we also\nhaveD(SO(m)) =m. Since Spin( m) is the universal cover of SO( m) for allm>2, we have\nn(SO(m)) =mfor allm >2. As we will see in the next section, any compact connected\nsemisimple Lie group GwithD(G) at least an absolute constant, can be written in the\nform (/producttextr\ni=1Ki)/Fwhere each Kiis one of SU( ni),Spin(ni) or Sp(ni) for someni/greaterorequalslant3, and\nFis a finite subgroup of the centre of/producttextr\ni=1Ki; the universal cover of such is/producttextr\ni=1Ki,\nandD(/producttextr\ni=1Ki) = Θ(min ini). Hence, the quantity n(G) has a very explicit description in\nterms of the structure of the Lie group G.\nGrowth in groups: the diameter problem\nThe theory of growth in groups has been a very active area of st udy in recent decades, and\nan important class of problem in this area is to determine the diameter of a metric space\ndefined by a group (e.g., the diameter of a Cayley graph of the g roup). For a compact group\nGequipped with its Haar probability measure, and a measurabl e generating setA⊆Gof\nmeasureµ, it is natural to consider the metric space on Gwhere the distance between xand\nyis defined to bethe minimal length of a word in the elements of Aand their inverses which\nis equal to xy−1. The diameters of such metric spaces in the case where Gis finite have\nbecome a focus of intense study in the last two decades: see e. g. the works of Liebeck and\nShalev [31], Helfgott [18], Helfgott and Seress [20], Pyber and Szabo [36] and Breuillard,\nGreen and Tao [8].\n11For a subsetAof a groupGandt∈N, we define\nAt:={a1·a2···at|a1,a2,...,at∈A}.\nThediameter problem for Gwith respect toAasks for the smallest positive integer tfor\nwhichAt=G. For a compact group Gand a real number 0 <α/lessorequalslant1, thediameter problem\nfor sets of measure αinGasks for the minimum possible diameter of a measurable set in\nGof measure α.\nIn the case where Gis a compact and connected group, we note that the diameter\nofGwith respect to any subset Aof positive measure is finite. This follows (almost)\nimmediately from Kemperman’s theorem [29], which states th at for any compact connected\ngroupG(equipped with its Haar probability measure µ) and any measurable A,B⊂G, we\nhaveµ(AB)/greaterorequalslantmin{µ(A)+µ(B),1}.\nWe make the following conjecture, concerning the diameter o f large sets.\nConjecture 1.6. LetGbe one of SU(n),SO(n),Spin(n)orSp(n), and letA⊆Gbe a\nmeasurable subset of measure ν. Then the diameter of Gwith respect toAisO(ν−1/(ℓn)),\nwhereℓ= 1in the case of SO(n)andSpin(n),ℓ= 2in the case of SU(n), andℓ= 4in\nthe case of Sp(n). In particular, if ν/greaterorequalslante−cn, then the diameter of Gwith respect toAis at\nmostOc(1).\nWe note that if true, the conjecture is essentially tight, as can be seen for SO( n) by\nconsidering the set\nSε:={X∈SO(n): the angle between Xe1ande1is at mostε},\nForε/lessorequalslant1/2, we have µ(Sε) = (Θ(ε))n, and the diameter of SO( n) with respect to S��is\nΘ(1/ε). Ifπ: Spin(n)→SO(n) is the usual (double) covering homomorphism, then the\nliftπ−1(Sε) is a subset of Spin( n) of the same measure as Sε(using, of course, the Haar\nprobability measure on both groups), and the diameter of Spi n(n) with respect to π−1(Sε)\nis the same the diameter of SO( n) with respect to Sε, sinceπ(At) = (π(A))tfor any subset\nA⊂Spin(n) and anyt∈N. Hence,π−1(Sε) demonstrates tightness for Spin( n). The group\nSU(n) acts transitively on the unit sphere in Cn, which can be identified with S2n−1, and\nthe group Sp( n) acts transitively on the unit sphere in Hn, which can be identified with\nS4n−1; both actions are angle-preserving (in S2n−1andS4n−1respectively). So our above\nconstruction for SO( n) (which comes from the action of SO( n) onSn−1) has the obvious\nanalogues for SU( n) and Sp(n), which we conjecture are sharp for those groups.\nWe show that for a compact connected Lie group Gwithn(G) =n, for allδ>0 and all\nmeasurable subsets AofGwith measure at least 2−cn1−δ, the diameter of Gwith respect\ntoAis at mostOδ(1).\nTheorem 1.7. For eachδ>0there existn0,k>0such that the following holds. Let n>n0\nand letGa compact connected Lie group with n(G) =n. IfA⊂Gis a Haar-measurable\nset, andµ(A)/greaterorequalslant2−n1−δ, thenAk=G.\n12Doubling inequalities for groups\nTheorem 1.7 follows from a new lower boundon µ(A2), whereA⊆Gis a measurable subset\nof the compact connected Lie group G. We prove the following ‘doubling inequality’.\nTheorem 1.8. There exists absolute constants C,c>0such that the following holds. Let\nGbe a compact connected Lie group with n(G) =n/greaterorequalslantC. LetA⊆Gbe a measurable set\nwithµ(A)/greaterorequalslante−cn. Thenµ(A2)/greaterorequalslantµ(A)0.1.\nThe problem of giving a lower bound on µ(A2) in terms of µ(A), forAa measurable\nsubset of a compact group G, dates back to the work of Henstock and Macbeath [21]\nfrom 1953, the aforemenentioned bound of Kemperman [29] fro m 1964, and the work of\nJenkins [22] from 1973. Several recent works of Jing, Tran an d Zhang have introduced some\npowerful new methods into the field. For instance, in [26], Ji ng, Tran and Zhang generalized\nthe Brunn-Minkowskii inequality from Rnto an arbitrary connected Lie group, using the\nIwasawa decomposition to facilitate an inductive approach ; their result is essentially sharp\nfor helix-free Lie groups. In [25], they used techniques fro mO-minimal geometry to show\nthat thatµ(A2)/greaterorequalslant3.99µ(A) for all measurable subsets A⊆SO(3) of sufficiently small\nmeasure. In a forthcoming paper [24] they prove that there ex ists a function δ=δ(n) and\nan absolute constant c>0, such that ifA⊆SO(n) is a measurable set of measure at most\nδ(n),thenµ(A2)/greaterorequalslant2cn1/10µ(A); the function δ(n) satisfiesδ(n)/lessorequalslant2−n1+c′\nwherec′>0\nis an absolute constant. (For comparison, we note that Theor em 1.8, in conjunction with\nTheorem 1.10 below, imply the existence of an absolute const antc>0 such that µ(A2)/greaterorequalslant\nmin{2cn1/2µ(A),0.99}for all measurable subsets A⊂SO(n) of measure at least 2−cn, so our\nresult and that of Jing, Tran and Zhang leave a ‘gap’ between t hem.) It remains an open\nproblem to determine whether µ(A2)/greaterorequalslantmin{2n/10µ(A),0.99}for all subsetsA⊂SO(n).\nSpectral gaps\nWe also give the following upper bound on the spectral gaps of the operator corresponding\nto convolution by1A\nµ(A). IfGis a compact group equipped with its (unique) Haar proba-\nbility measure µ, for a measurable set A⊂Gwe writex∼Ato mean that xis chosen\n(conditionally) according to the Haar measure µ, conditional on the event that x∈A.\nTheorem 1.9. There exist absolute constants c,C >0such that the following holds. Let G\nbe compact connected Lie group and suppose n:=n(G)/greaterorequalslantC. LetA=A−1be a symmetric,\nmeasurable set in Gand suppose that µ(A)/greaterorequalslante−cn1/2. Then the nontrivial spectrum of the\noperatorTdefined byTf(x) =Ea∼A[f(ax)]is contained in the interval\n/bracketleftBigg\n−/radicalbigg\nClog1/α\nn,/radicalbigg\nClog1/α\nn/bracketrightBigg\n.\nMixing times\nLetGbe a compact group, equipped with its (unique) Haar probabil ity measure; then\nevery measurable subset A⊆Gof positive Haar measure corresponds to a random walk\n13onG. Indeed, we may define a (discrete-time) random walk RA= (Xt)t∈N∪{0}onG,\nby lettingX0= Id, and for each t∈N, ifXt−1=xthenXt=ax, whereais chosen\nuniformly at random from A. In the case where Gis finite andAis closed under taking\ninverses, this is the usual random walk associated to the Cay ley graph Cay( G,A). One of\nthe fundamental problems associated to such random walks is to determine their mixing\ntime. (Following Larsen and Shalev [30], we say that the mixing time of a Markov chain\n(Xt)t∈N∪{0}is the minimal non-negative integer Tsuch that the total variation distance\nbetween the distribution of XTand the uniform distribution, is at most 1 /e. We note that\n1/ecould be replaced by any other absolute constant c∈(0,1), without materially altering\nthe definition; Larsen and Shalev use the constant 1 /eas it makes the statement of certain\nresults concerning SnandAnmore elegant.)\nLarsen and Shalev [30] considered the case where Ais a normal set, i.e. a set closed\nunder conjugation, and Gis the alternating group An. They showed that for each ε >0,\nifA⊆Anof density2|A|\nn!/greaterorequalslantexp/parenleftbig\n−n1/2−ε/parenrightbig\n, then the mixing time of RAis 2, provided that\nn/greaterorequalslantn0(ε) is sufficiently large depending on ε. Their proof was based upon a heavy use\nof character theory. Their result is almost sharp, in the sen se the number 1 /2 cannot be\nreplaced by any number smaller than 1 /2. We show that a similar phenomenon holds for\ncompact connected Lie groups, even when Ais not a normal set.\nTheorem 1.10. There exist absolute constants c,n0>0, such that the following holds. Let\nGbe a compact connected Lie group with n:=n(G)>n0. LetA⊆Gbe a measurable set\nwith Haar measure at least e−cn1/2. Then the mixing time of the random walk RAis 2.\nThis result is essentially best possible. For instance, tak ingG= SO(n), we may take\nA={X∈SO(n) :X11>10/n1/4}. It is easy to see that the mixing time of RAis 3, while\nµ(A) = exp(−Θ(n1/2)).\nProduct mixing\nGowers’ proof of his upper bound on the sizes of product-free sets actually establishes a\nstronger phenomenon, known as product mixing . We say that a compact group G(equipped\nwith its Haar probability measure µ) is an (α,ε)-mixerif for all sets A,B,C⊆Gof Haar\nprobability measures /greaterorequalslantα, when choosing independent uniformly random elements a∼A\nandb∼B, the probability that ab∈Clies in the interval ( µ(C)(1−ε),µ(C)(1+ε)).\nGowers’ proof actually yields the following statement: the re exists an absolute constant\nC >0, such that if Gis aD-quasirandom compact group, then it is a ( CD−1/3,0.01)-\nmixer. (The proof is given only for finite groups, but it gener alises easily to all compact\ngroups.) For finite groups, Gowers’ product-mixing result i s sharp up to the value of the\nconstantC. Here, we obtain an analogous result for compact connected L ie groups, where\nthen−1/3moves to the exponent.\nTheorem 1.11. For anyε>0, there exist c,n0>0such that the following holds. Let n>\nn0and letGbe a compact connected Lie group with n:=n(G)>n0. Setα= exp(−cn−1/3).\nThenGis an(α,ε)-mixer.\n14This result is sharp up to the dependence of the constants c=c(ε) andn0=n0(ε) upon\nε. Indeed, we may take G= SO(n) and letA=B={X∈SO(n) :X11>10/n1/3}and\nC={X∈SO(n) :X11<−10/n1/3}, to obtain a triple of sets each of measure e−Θ(n1/3),\nsuch that when choosing a∼Aandb∼Bindependently, the probability that ab∈Cis\nsmaller than1\n2µ(C).\nHomogeneous dynamics and equidistribution\nSuppose that a compact Lie group Gacts on a topological space X. The space Xis said to\nbeG-homogeneous ifGacts transitively and continuously on X(the latter meaning that\nthe action map from G×XtoXis continuous); in this case, Xhas a unique G-invariant\nprobability measure, which is called the Haar probability m easure. We obtain the following\nequidistribution result for homogeneous spaces.\nTheorem 1.12. For eachε>0there existc,n0>0such that the following holds. Let Gbe\na compact connected Lie group with n(G) =:n>n0. LetXbe aG-homogeneous topological\nspace, and let µXdenotes its unique G-invariant (Haar) probability measure. Suppose that\nA⊆GandB⊆Xare both measurable sets of Haar probability measures /greaterorequalslante−cn1/2. Let\nνbe the probability measure on Xwhich is given by the distribution of a(b), for a uniform\nrandoma∼Aand an (independent) uniform random b∼B. Then the total variation\ndistance between µandνis less than ε.\nLq-norms of low degree polynomials\nWe obtain the following upper bounds on the q-norms of low degree polynomials (we state\nthe result for SO( n), for simplicity).\nTheorem 1.13. There exist absolute constants c,C >0such that the following holds. Let\nq>2and letf∈L2(SO(n))be a polynomial of degree din the matrix entries of X∈SO(n).\nIfd/lessorequalslantcn, then\n/⌊ar⌈⌊lf/⌊ar⌈⌊lLq(µ)/lessorequalslantqCd/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\nIf moreover, d/lessorequalslantc√n, then\n/⌊ar⌈⌊lf/⌊ar⌈⌊lLq(µ)/lessorequalslant(C√q)d/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\nSeparation of quantum and classical communication complexity\nStarting with their introduction to computer science in the seminal paper of Kahn Kalai\nand Linial [27], hypercontractive inequalities have found a huge number of applications in\nvarious branches of computer science and related fields (see e.g. [13, 32, 10, 37], to name\nbut a few). While these applications have generally require d hypercontractive inequalities\nfor functions on discrete sets, some applications require c ontinuous domains. For example,\nin the paper of Klartag and Regev [38], a hypercontractive in equality for functions on\nthen-sphere is used to obtain a lower bound on the number of (class ical) communication\n15bits required for two parties to jointly compute a certain fu nction. While with quantum\ncommunication, the value of that function can be computed by one party transmitting\nonlyO(logn) quantum-bits to the other, it was shown in [38] that classic al communication\nrequires at least Ω( n1/3) bits of communication to besent, even if parties areallowe d to send\nbits both ways, thereby showing an exponential separation b etween the power of classical\ncommunication and that of one-way quantum communication.\nIn the field of quantum communication, establishing a signifi cant separation between\nclassical communication and practically implemented mode s of quantum communication re-\nmains a major open problem. In a forthcoming work, Arunachal am, Girish, and Lifshitz [2]\napply our hypercontractive inequality for SU( n) to make a substantial step towards this\ngoal. They used it to obtain an exponential separation betwe en classical communication\nand a more realistic version of quantum communication, name ly the one-clean-qubit model.\n2 Preliminaries: Quasirandomness and min-rank\nIn this section we show that, for Dat least an absolute constant, the universal cover ˜Gof a\nD-quasirandom compact connected Lie group Gis a product of ‘classical’ (compact, simple,\nsimply connected) Lie groups of the form Spin( n),SU(n),Sp(n). We then make use of this\nto determine D(˜G), and we show that D(˜G)/lessorequalslant4D(G)2.\nIn what follows, as usual, a compact group Gis a Hausdorff topological group for which\nthegroupoperations(orequivalently, themap( g,h)/ma√sto→gh−1)arecontinuous. Werecall that\na compact group has a unique left-multiplication-invarian t probability measure (called the\nHaar measure), which is also the unique right-multiplicati on-invariant probability measure.\nAs usual, if Gis a compact group, we let\nL2(G) ={f:G→C:Eµ[|f|2]<∞}/∼,\nwhere the expectation is with respect to the Haar probabilit y measure µonGand the\nequivalence relation ∼is defined by f∼gifff=g µ-almost-everywhere, and we view\nL2(G) as a Hilbert space, with the natural inner product,\n/a\\}⌊ra⌋k⌉tl⌉{tf,g/a\\}⌊ra⌋k⌉tri}ht:=Eµ[fg].\nWe make use of the following fact, appearing for example in [3 5] Chapter 10, Section\n7.2, Theorem 4, page 380. (Note that the word ‘Lie’ is missing from the statement of this\ntheorem; this omission is clearly just a typographical erro r.)\nFact 2.1. Every compact connected Lie group is Lie-isomorphic to a group of the form\n(/producttextr\ni=1Ki×T)/F,where each Kiis a simply connected simple compact Lie group (equiva-\nlently,Kiis one of Sp(ni)for someni/greaterorequalslant1,Spin(ni)for someni/greaterorequalslant3,SU(ni)for some\nni/greaterorequalslant2, or the compact form of one of the five exceptional Lie groups, f or eachi∈[r]),T\nis a finite-dimensional torus (i.e. T= (R/Z)mfor some integer m), andFis a finite group\ncontained in the center of/producttextr\ni=1Ki×T, withF∩T={1}.\n16Lemma 2.2. Suppose that D >1and thatGis aD-quasirandom compact connected Lie\ngroup. Then Gis semisimple.\nProof.WriteG= (/producttextr\ni=1Ki×T)/F, where each Kiis a simply connected simple compact\nLie group,Tis a finite-dimensional torus (i.e. T= (R/Z)mfor some integer m), andFis\na finite group contained in the center of/producttextr\ni=1Ki×T, withF∩T={1}, as in the above\nfact. Semisimplicity of Gis equivalent to T={1}. Suppose on the contrary that T/\\⌉}atio\\slash={1}.\nLetπbe the projection map from/producttextr\ni=1Ki×Tonto theTcomponent. Since F∩T={1},\nwe haveπ(F) ={1}, so the projection πinduces a (surjective) group homomorphism ˜ π\nfromGtoT, and therefore Ghas a quotient isomorphic to ( R/Z)mfor some integer m/greaterorequalslant1;\nany nontrivial complex one-dimension irreducible represe ntation of the latter quotient lifts\nto one ofG, contradicting the D-quasirandomness of G(for anyD >1) and proving the\nlemma.\nWe also recall the following standard fact.\nFact 2.3. Every compact semisimple Lie group has finite centre.\nWe now show that if Gis sufficiently quasirandom, then the exceptional groups do n ot\nmake an appearance as some Kiwhen writing G= (/producttextr\ni=1Ki)/F.\nLemma 2.4. SetD0= 248. LetGbe a compact connected Lie group, and suppose that\nit isD-quasirandom for some D > D 0. ThenG= (/producttextr\ni=1Ki)/F, where each Kiis one of\nSp(ni),Spin(ni),SU(ni)for someni/greaterorequalslant√\nD/2andFis a subgroup of the (finite) centre of/producttextr\ni=1Ki.\nProof.By the previous lemma, provided D0>1,Gis semisimple. Hence, we may write\nG= (/producttextr\ni=1Ki)/F, where each Kiis one of Sp( ni), Spin(ni), SU(ni) or the compact form of\none of the five exceptional Lie groups, for each i, andFis a finite group contained in the\n(finite) center of/producttextr\ni=1Ki. As the quotient of a D-quasirandom group is D-quasirandom,\nwe may project to any one of the components and still obtain a D-quasirandom group\nKi/F′. (In detail, let πidenote projection of/producttextr\nj=1Kjonto theKifactor;πiinduces a\nsurjective homomorphism from GontoKi/πi(F), and since Fis a subgroup of the centre\nof/producttextr\nj=1Kj,Fiis a subgroup of the centre of Ki. The group Ki/πi(F) is therefore a\nquotient of G, and so inherits its D-quasirandomness.) It is therefore sufficient to consider\nthe case where G=K1/F′. We now note that the adjoint representation of K1factors\nthroughK1/F′(sinceF′is contained in the centre of K1), so it can also be viewed as a\nrepresentation of K1/F′. As the adjoint representation of K1is not a sum of copies of\nthe trivial representation (this follows from the fact that K1is non-Abelian), its dimension\n(which is the same as the dimension of the Lie group K1) is at least D. The five exceptional\nLie groups, E6,E7,E8,F4andG2, have dimensions 78, 133, 248, 52 and 14 respectively, so\nK1cannot equal any of these (since D>D 0= 248). Hence, K1is one of Sp( n1), Spin(n1)\nor SU(n1). The dimensions of these Lie groups are n1(2n1+ 1),n1(n1−1)/2 andn2\n1−1\nrespectively, so we obtain n1(2n1+1)/greaterorequalslantD, which implies that n1/greaterorequalslant√\nD/2. This completes\nthe proof of the lemma.\n17The following (non-standard) definition will be convenient for us.\nDefinition 2.5. LetGbe a compact, connected, semisimple Lie group and write G=\n(/producttextr\ni=1Ki)/F, where, as above, Kiis one of Sp(ni),Spin(ni)orSU(ni)for eachi, and\nFis a finite subgroup of the (finite) centre of G. We define the min-rank ofGto be\nmin{n1,...,n r}.\nUsing this terminology, the above lemma can be restated by sa ying that if a compact\nconnected Lie group GisD-quasirandom for large enough D, then it has min-rank at least√\nD/2.\nWe remark that the rankof a Lie group is defined to be the dimension of any one of its\nCartan subgroups, so the ranks of Sp( ni), Spin(ni) and SU(ni) are respectively ni,⌊ni/2⌋\nandni−1, so in particular are all Θ( ni); hence, while the min-rank of Gis not exactly the\nminimum of the ranks of the Ki’s (where the Ki’s are as above), it is within an absolute\nconstant factor thereof. (We hope this slight abuse of termi nology will not cause confusion.)\nTo establish that D(˜G)/lessorequalslant4D(G)2we also need the following.\nLemma 2.6. LetGbe a compact, connected, semisimple Lie group of min-rank m. Then\nits universal cover ˜GsatisfiesD(˜G)∈{m,2m}.\nProof.WriteG=/producttextr\ni=1Ki/F. As the projection map from/producttextr\ni=1KitoGis a cover map,\nand since/producttextr\ni=1Kiis simply connected, we obtain that ˜G=/producttextr\ni=1Ki. The lemma now\nfollows from the fact that the complex irreducible represen tations of a product/producttextr\ni=1Kiof\nfinitely many compact groups are tensor products of complex i rreducible representations,\nof the form ρ1⊗...⊗ρrwhereρiis an complex irreducible representation of ρi, for eachi\ntogether with the fact that D(SU(n)) =D(SO(n)) =n,D(Sp(n)) = 2n.\nLemma 2.6 shows that, when proving the theorems in the introd uction, we may replace\nD(˜G) with the min-rank of G.\n3 Good groups and fine groups\n3.1 Graded groups and ‘good’ groups.\nIn this section we define some basic properties of compact con nected groups, which we later\nuse to prove various growth properties. We define gradedandstrongly quasirandom groups,\nandhypercontractive groups; we say that groups satisfying all these properties a regood. We\nalso define a somewhat weaker (or, technically, incomparabl e) notion of a finegroup.\nThe compact, simple, simply connected real Lie groups of lar ge enough rank, i.e SU( n),\nSp(n) and Spin( n), are indeed good (this is proved in Section 8). We show that g ood-\nness is preserved when taking products and quotients (quoti ents, that is, by closed normal\nsubgroups, as usual), thereby showing that every D-quasirandom group is a good graded\ngroup, provided Dis sufficiently large.\n18Definition 3.1 (Graded groups) .Forn∈N, we say a compact connected group Gis\nn-gradedif there exists an orthogonal direct sum,\nL2(G) =⌈n/2⌉−1/circleplusdisplay\nd=0V=d⊕V/greaterorequalslantn/2,\nsuch that the spaces V=dare invariant under the action of Gfrom both sides, and where V=0\ncontains only the constant functions. For an n-graded group and an integer 0/lessorequalslantd0d0the direct sum V>d0:=⌈n/2⌉−1/circleplusdisplay\nd=d0+1V=d⊕V/greaterorequalslantn/2. (Note that we will sometimes\nwriteVG\n=din place of V=d, when we want to stress that the group in question is G, e.g. if\nthere are several groups involved in our argument.)\nRemark 3.2. Note that it follows from the definition of an n-grading that V/greaterorequalslantn/2is also\ninvariant under the action of Gfrom both sides.\nGrading for the compact simply connected simple Lie groups (of large enough\nrank). We note in this section that SO( n), SU(n), Sp(n) and Spin( n) are alln-graded,\nforn/greaterorequalslant3. For the group SO( n), for each integer 0 /lessorequalslantd < n/2 we define V/lessorequalslantdto be\nthe subspace of L2(SO(n)) spanned by degree /lessorequalslantdmultivariate polynomials in the matrix\nentries ofX∈SO(n) (so, forexample, V/lessorequalslant2contains thepolynomial X11X12). For notational\nconvenience, for 0 0.\nDefinition 3.3 (Strongly quasirandom graded group) .We say an n-graded compact group\nGis/parenleftBig\n(Qd)⌈n/2⌉−1\nd=0,Q/parenrightBig\n-strongly-quasirandom if the minimal dimension of a subrepresenta-\ntion ofV=d(as a leftG-module) is /greaterorequalslantQdfor all integers 0/lessorequalslantd/lessorequalslant⌈n/2⌉−1, and is/greaterorequalslantQfor\nV/greaterorequalslantn/2.\nForc >0we say that the n-graded compact group Gisc-strongly-quasirandom if it is\n((Qd)⌈n/2⌉−1\nd=1,Q)-strongly-quasirandom when we set Qd:=/parenleftbigcn\nd/parenrightbigdford < cn/(1 +c), and\nQ,Qd:= (1+c)cn/(1+c)ford/greaterorequalslantcn/(1+c).\nIn Section 3.4, we show that all the (infinite families of) com pact simply connected\nsimple Lie groups are c-strongly quasirandom for some absolute constant c>0.\nTheorem 3.4. Then-graded compact groups SU(n),Sp(n),Spin(n)(forn/greaterorequalslant3) are all\nc-strongly quasirandom for some absolute constant c >0, when equipped with our chosen\nn-grading.\nDefinition 3.5 (Beckner operator for graded groups) .LetGbe ann-graded compact group,\nletrbe an integer with 0/lessorequalslantr < n/2, and let 0/lessorequalslantδ/lessorequalslant1. We define the Beckner operator\nTδ,r:L2(G)→L2(G)byTδ,r(f) :=/summationtextr\ni=0δif=i, for allf∈L2(G).\nDefinition 3.6 (Hypercontractive group) .LetC >0and letrbe an integer with 0/lessorequalslantr<\nn/2. We say that an n-graded group compact Gis(r,C)-hypercontractive if for every q/greaterorequalslant2\nand every 0/lessorequalslantδ/lessorequalslant1/(C√q), we have/⌊ar⌈⌊lTδ,r/⌊ar⌈⌊l2→q/lessorequalslant1.\nThe following is an easy consequence of hypercontractivity .\nLemma 3.7. LetGbe ann-graded(r,C)-hypercontractive group, where 0/lessorequalslantr1and1/lessorequalslantr < n/2. Ann-\ngraded compact group Gis(r,C)-weakly hypercontractive if for every function f∈L2(G)\nand everyq/greaterorequalslant2and0/lessorequalslantδ/lessorequalslant1/qCwe have\n/⌊ar⌈⌊lTδ,rf/⌊ar⌈⌊lq/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2.\nThe following lemma follows similarly to Lemma 3.7.\nLemma 3.12. LetGbe ann-graded(r,C)-weakly hypercontractive compact group, where\nC >1and1/lessorequalslantr < n/2. Then for any integer d/lessorequalslantr, everyq/greaterorequalslant2, and any function\nf∈V=d, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊lq/lessorequalslantqCd·/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nDefinition 3.13 (Fine groups) .Ifc>0andC >1, ann-graded compact group Gis said\nto be(C,c)-fineif it is both (cn,C)-weakly hypercontractive, and c-strongly-quasirandom.\nIn Section 7 we show that the (infinite families of) compact si mply connected simple\ngroups, viz. Sp( n),SU(n), and Spin( n), are all fine, as n-graded groups equipped with our\nchosen gradings.\nTheorem 3.14. Forn/greaterorequalslant3, then-graded groups Sp(n),SU(n), andSpin(n)are(C,c)-fine,\nfor some absolute constants C >1andc>0, when equipped with our chosen n-gradings.\n213.3 Goodness and fineness are preserved under taking product s and quo-\ntients.\nIn this subsection, we show that goodness and fineness are pre served under taking products,\nand quotients (quotients, that is, by closed normal subgrou ps) — more precisely, that suit-\nable gradings can be defined on products and quotients. This, together with Theorems 3.10\nand 3.14, implies that all compact Lie groups of large-enoug h min-rank are both good and\nfine.\nLemma 3.15. Assuming Theorem 3.10, the following holds. There exist posit ive constants\nc,Candn0such that if n>n0, then every compact connected Lie group of min-rank ncan\nbe equipped with an n-grading that makes it (C,c)-good, as an n-graded group.\nLemma 3.16. Assuming Theorem 3.14, the following holds. There exist posit ive constants\nc,Candn0such that if n>n0, then every compact connected Lie group of min-rank ncan\nbe equipped with an n-grading that makes it (C,c)-fine, as an n-graded group.\nGrading for group products. LetGandHbe compact groups, and let fandgbe\nfunctions on GandHrespectively. We write f⊗gfor the function on G×Hgiven by\n(x,y)/ma√sto→f(x)g(y). IfUis aclosed linear subspaceof L2(G) andVis aclosed linear subspace\nofL2(H), then we denote by U⊗Vbe the Hilbert-space tensor product of UandV, i.e.\nthe linear subspace of L2(G×H) consisting of the closure of the linear span of the set of\nfunctions{f⊗g:f∈U, g∈V}. We recall that if {ui}∞\ni=1is a Hilbert-space basis for U\nand{vi}∞\ni=1is a Hilbert-space basis for V, then{ui⊗vj}∞\ni,j=1is a Hilbert-space basis for\nU⊗V.\nDefinition 3.17. Letn/greaterorequalslantm, letGbe ann-graded compact group and Hbe anm-graded\ncompact group. We give G×Hthe structure of an m-graded compact group by setting\nVG×H\n=d=/circleplusdisplay\nd1+d2=dVG\n=d1⊗VH\n=d2\nfordn/2) constitute a grading of L2(G/H). The fact that i∗◦iis the\nidentity also imiplies that VG/H\n=d⊆i∗(VG\n=d) for eachd risi◦i∗-invariant.\nWe can therefore use fact that the operator Tannihilates VG\n>rto deduce that the operator\nT◦i◦i∗agrees with TonVG\n>r. Our claim will follow once we show that i◦i∗is the identity\nonVG\n/lessorequalslantr. To accomplish that we note that iis injective, and by the hypothesis the restriction\nofito the corresponding V=dspaces is also surjective, and thus so is its restriction to VG\n/lessorequalslantr.\nAs the operator i∗◦iis the identity we obtain that the restriction of i◦i∗toVG\n/lessorequalslantris the\nidentity as well.\nWe can therefore write T=i◦T′◦i∗, and using the fact that iis anLq-isometric\nembedding and that i∗contracts 2-norms we obtain:\n/⌊ar⌈⌊lT/⌊ar⌈⌊l2→q=/⌊ar⌈⌊lT′◦i∗/⌊ar⌈⌊l2→q/lessorequalslant/⌊ar⌈⌊lT′/⌊ar⌈⌊l2→q/⌊ar⌈⌊li∗/⌊ar⌈⌊l2→2/lessorequalslant1.\nConclusion. We have shown that (weak) hypercontractivity is preserved u nder taking\nproducts and quotients. It is easy to check that c-strong-quasirandomness is also preserved\nunder taking products or quotients (using the gradings abov e), so we immediately obtain\nLemma 3.15 and Lemma 3.16.\n4 Growth in good groups\nIn this section we prove Theorems 1.2, 1.5, 1.9, 1.10 1.11 and 1.12. For now, the reader\nmay consider the objective of proving Theorem 1.2 as motivat ion for what follows.\nLetGbe a compact group, and let µbe the Haar probability measure on G. We would\nlike to bound µ(A) for a setA⊆Gthat is product free. We first note that the property of\nbeing product free can be stated in terms of convolutions.\nDefinition 4.1. For two functions f,g∈L2(G), we define their convolution f∗g∈L2(G)\nby\nf∗g(x) :=/integraldisplay\nf(xy−1)g(y)dµ(y).\nForf∈L2(G), we write Tffor the linear operator from L2(G) to itself defined by\ng/ma√sto→g∗f. Observe that if A⊂Gis a product-free set of density µ(A) =α, andf=1A\nα,\nthen/a\\}⌊ra⌋k⌉tl⌉{tTf1A,1A/a\\}⌊ra⌋k⌉tri}ht= 0. IfGis ann-graded group, we can decompose g:= 1Ainto its\northogonal projections onto the V=d’s, and write g=/summationtext⌈n/2⌉−1\nd=0g=d+g/greaterorequalslantn/2, whereg=dis\nthe orthogonal projection of gontoV=d. Noting that g=0≡α, this allows us to expand\n/a\\}⌊ra⌋k⌉tl⌉{tTf1A,1A/a\\}⌊ra⌋k⌉tri}ht=/a\\}⌊ra⌋k⌉tl⌉{tTfg,g/a\\}⌊ra⌋k⌉tri}ht\nas a sum of a main term, α2, and other terms of the form /a\\}⌊ra⌋k⌉tl⌉{tTfg=d,g=d/a\\}⌊ra⌋k⌉tri}htor/a\\}⌊ra⌋k⌉tl⌉{tTfg/greaterorequalslantn/2,g/greaterorequalslantn/2/a\\}⌊ra⌋k⌉tri}ht.\nWe upper-bound each term using Cauchy-Schwarz:\n|/a\\}⌊ra⌋k⌉tl⌉{tTfg=d,g=d/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/⌊ar⌈⌊lg=d/⌊ar⌈⌊l2\n2,\n25where for a closed subspace M/lessorequalslantL2(G) and a linear operator T:L2(G)→L2(G) we write\n/⌊ar⌈⌊lT/⌊ar⌈⌊lMfor the supremum of/⌊ar⌈⌊lTv/⌊ar⌈⌊l2\n/⌊ar⌈⌊lv/⌊ar⌈⌊l2over all nonzero v∈M.\nOur goal will be to show that these other term make a negligibl e contribution to the sum\ncompared to the main term α2. We accomplish that by observing that the space V=disTf-\ninvariant. This shows that the operator T∗\nfTfcan be diagonalized inside V=d.It also implies\nthat/⌊ar⌈⌊lTf/⌊ar⌈⌊l2\nV=dis equal to the maximal eigenvalue of T∗\nfTfinsideV=d. We then upper bound\nthe maximal eigenvalue of T∗\nfTfinsideV=d, showing that these eigenvalues get smaller and\nsmaller as the degree gets larger. Finally, we combine our up per bound on the eigenvalues\nofTfwith a level d-inequality which shows that the L2-mass ofgis concentrated on the\nhigh degrees. Together, we obtain that the sum of terms /a\\}⌊ra⌋k⌉tl⌉{tTfg=d,g=d/a\\}⌊ra⌋k⌉tri}htis indeed negligible.\nOur upper bound on the ‘degree d’ eigenvalues of T∗\nfTffollows by combining a level d\ninequality with a lower bound on the dimension of each eigens pace ofT∗\nfTf.We use the fact\nthat each such eigenspace is a subrepresentation of V=dand therefore by strong quasiran-\ndomness must have dimension /greaterorequalslantQd,for the appropriate quasirandomness parameter Qd.\nWe upper bound |/a\\}⌊ra⌋k⌉tl⌉{tTfg/greaterorequalslantn/2,g/greaterorequalslantn/2/a\\}⌊ra⌋k⌉tri}ht|in a similar fashion.\n4.1 Level dinequalities and the eigenvalues of convolution operators\nRecall that the Hilbert-Schmidt norm of a linear operator Ton a separable Hilbert space\nHis defined by\n/⌊ar⌈⌊lT/⌊ar⌈⌊l2\nHS:=∞/summationdisplay\ni=1/⌊ar��⌊lT(ei)/⌊ar⌈⌊l2\n2,\nwhere{ei}∞\ni=1is any Hilbert-space basis for H; ifTis a compact operator, then /⌊ar⌈⌊lT/⌊ar⌈⌊lHS\nis the square root of the sum of the eigenvalues of T∗T(counted with multiplicity). One\nstandard fact (see e.g. [9], page 267) is the following.\nFact 4.2. Letf∈L2(G)and letTfbe the linear operator from L2(G)to itself defined by\ng/ma√sto→g∗f.ThenTfis a compact operator, and the Hilbert–Schmidt norm of Tfis equal to\nthe 2-norm of f:\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lHS=/⌊ar⌈⌊lf/⌊ar⌈⌊l2.\nWe recall our notation for the norm of an operator on a subspac e ofL2(G).\nDefinition 4.3. LetM/lessorequalslantL2(G)be a closed subspace. Let T:L2(G)→L2(G)be a linear\noperator; then we write /⌊ar⌈⌊lT/⌊ar⌈⌊lM:= sup{/⌊ar⌈⌊lTf/⌊ar⌈⌊l2//⌊ar⌈⌊lf/⌊ar⌈⌊l2:f∈M\\{0}}.\nWe now give our upper bound on the level deigenvalues of Tf.\nLemma 4.4. LetGbe ann-graded((Qd)n/2\nd=1,Q)-quasirandom group. Let f∈L2(G). Then\nthe spacesV=d,V/greaterorequalslantn/2areTf-invariant. Moreover, /⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/lessorequalslant/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2√Qdand/⌊ar⌈⌊lTf/⌊ar⌈⌊lV/greaterorequalslantn/2/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2√Q.\nProof.We first claim that the subspaces V=d,V/greaterorequalslantn/2are allTfinvariant. To see this, observe\nthat ifU/lessorequalslantL2(G) is a closed subspace that is invariant under the right-acti on ofG, then\n26for everyg∈U, we haveg∗f∈Uas well. (Indeed, let h∈U⊥; then\n/a\\}⌊ra⌋k⌉tl⌉{tg∗f,h/a\\}⌊ra⌋k⌉tri}ht=/integraldisplay /integraldisplay\ng(xy−1)f(y)h(x)dµ(y)dµ(x) =/integraldisplay /integraldisplay\ng(xy−1)f(y)h(x)dµ(x)dµ(y) = 0,\nusing Fubini and the fact that for each fixed y∈Gthe function x/ma√sto→g(xy−1)f(y) lies in\nU. Hence,g∗f∈(U⊥)⊥=U.) Applying this with U=V=d, which is a closed subspace\ninvariant under the right action of G, we see that the spaces V=dare indeed Tf-invariant.\nSimilarly, applying it with U=V/greaterorequalslantn/2(which is also a closed subspace invariant under the\nright action of G), we see that V/greaterorequalslantn/2is alsoTf-invariant.\nFixd0there exist c′,n0>0such that the following holds. Let\nn>n0and letGbe ann-graded(C,c)-good group. Let A⊆Gbe measurable, and suppose\nthat\nα:=µG(A)∈(e−c′√n,c′).\nWritef=1A\nαandt=log(1/α)\n2. Then for all 1/lessorequalslantd/lessorequalslantt, we have\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/lessorequalslant/parenleftbiggC′log(1/α)\nn/parenrightbiggd/2\n,\nwhereC′:=10C2\nc. Moreover,\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV>t/lessorequalslant/parenleftBigα\nn/parenrightBig10\n.\nProof.The lemma follows by applying Lemma 4.4 with the values of QdandQwhich are\npromised by the goodness of G, and then bounding/vextenddouble/vextenddoublef=d/vextenddouble/vextenddouble\n2using Theorem 4.5.\nLet us begin with the range d/lessorequalslantt. Since we know that Gisc-strongly-quasirandom,\nhe have that Gis ((Qd)⌈n/2⌉−1\nd=1,Q) graded where Qd/greaterorequalslant/parenleftbigcn\nd/parenrightbigdfor alld/lessorequalslantt. Hence, by\nLemma 4.4, we have\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/lessorequalslant/vextenddouble/vextenddoublef=d/vextenddouble/vextenddouble\n2√Qd\n/lessorequalslant/vextenddouble/vextenddouble/vextenddoublef=d/vextenddouble/vextenddouble/vextenddouble\n2·/parenleftBigcn\nd/parenrightBig−d/2\n=/vextenddouble/vextenddoubleα·f=d/vextenddouble/vextenddouble\n2\nα·/parenleftBigcn\nd/parenrightBig−d/2\n/lessorequalslant/parenleftbiggd\ncn/parenrightbiggd/2\n·/parenleftbigg10C2\nd/parenrightbiggd/2\nlog(1/α)d/2\n(using Lemma 4.5)\n/lessorequalslant/parenleftbigg10C2log(1/α)\ncn/parenrightbiggd/2\n,\nwhich is the desired bound.\nNext, consider the range d/greaterorequalslantt. Fordin this range, and provided that n0is sufficiently\nlarge, we have from c-strong-quasirandomness that Qd/greaterorequalslant(cn\nt)t. By Lemma 4.4 we therefore\nhave, by a similar computation to before,\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV>t/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2/parenleftBigcn\nt/parenrightBig−t/2\n/lessorequalslantn−t/4e−21tα−1/2/lessorequalslant/parenleftBigα\nn/parenrightBig10\n,\n28where we usedcn\nt/greaterorequalslante42n1/2,\nwhich holds provided c′is sufficiently small.\nLemma 4.7. Letc,C,c′>0. LetGbe ann-gradedc-strongly-quasirandom group. Let\nA⊆Gbe measurable, and suppose that\nα:=µG(A)/greaterorequalslantc′.\nWritef=1A\nα. Then for 1/lessorequalslantd0there exist c′,C′>0such that the following holds. Let\nGbe ann-graded, (c,C)-good group, let f:G→{0,1}be measurable, and suppose that\nα:=E[f]/greaterorequalslante−c′√n. Then for all d∈N, we have\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslant(C′)dα2logd(e/α). (2)\nProof.ProvidedC′is sufficiently large depending on c′, we may (and shall) assume that\nα/lessorequalslantc′.Indeed, for α/greaterorequalslantc′the right-hand side is greater than one, and we always have th e\ntrivial upper bound /⌊ar⌈⌊lf/lessorequalslantd/⌊ar⌈⌊l2\n2/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\n2=α. Letr=c√n, and note that in the case where\nd>r, the claimed upper bound is trivial, as in this case the right -hand side of (2) is (again)\ngreater than one.\nLetq= log(1/α).Letq′be the H¨ older conjugate of q. Ford/lessorequalslantr, we have by ( r,C)-\nhypercontractivity and Lemma 3.7 that\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2=/a\\}⌊ra⌋k⌉tl⌉{tf,f=d/a\\}⌊ra⌋k⌉tri}ht/lessorequalslant/⌊ar⌈⌊lf=d/⌊ar⌈⌊lq/⌊ar⌈⌊lf/⌊ar⌈⌊lq′/lessorequalslant(C2q)d/2/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2α1−1/q.\nAfter rearranging we obtain\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslante2(C2q)dα2=e2C2dα2logd(1/α).\nThis gives the claimed upper bound, provided C′is sufficiently large depending on C.\n294.2 Upper bounds on the measures of product-free sets.\nLet us now show how Lemma 4.6 implies an upper bound on the meas ure of a product-free\nset in a good group.\nTheorem 4.9. For anyc,C >0, there exist c′,n0>0such that the following holds. Let\nn>n0and letGbe a(c,C)-goodn-graded group. Then every measurable product-free set\ninGhas Haar measure at most e−c′n1/3.\nBefore proving Theorem 4.9, let us note that together with Th eorem 3.10 it implies\nTheorem1.2. Thisfollows since, byLemma3.15, forall n>n0every compact connected Lie\ngroup of min-rank nis a (c,C)-goodn-graded group for some absolute constants c,C >0.\nProof.LetA⊆Gbe product-free and measurable; write α=E[1A] andt=log(1/α)\n2.\nAssume w.l.o.g. that α < c′, wherec′is to be chosen later (if not, then replace Aby a\nsmaller product-free set). Let f=1A\nα. Suppose for a contradiction that α/greaterorequalslante−c′n1/3. We\nhave\n0 =/a\\}⌊ra⌋k⌉tl⌉{tTff,f/a\\}���ra⌋k⌉tri}ht=E3[f]+⌊t⌋/summationdisplay\nd=1/a\\}⌊ra⌋k⌉tl⌉{tTff=d,f=d/a\\}⌊ra⌋k⌉tri}ht+/a\\}⌊ra⌋k⌉tl⌉{tTff>t,f>t/a\\}⌊ra⌋k⌉tri}ht. (3)\nProvidedc′is sufficiently small, we may now apply Lemma 4.6 and Theorem 4. 5 with\n1A=α·fto obtain, for C′sufficiently large depending on c,Cand all 1 /lessorequalslantd/lessorequalslantt,\n|/a\\}⌊ra⌋k⌉tl⌉{tTff=d,f=d/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslant/parenleftbiggC′log3(1/α)\nnd2/parenrightbiggd/2\n/lessorequalslant100−d,\nwhere the last inequality holds provided c′is sufficiently small depending on C′. We may\nalso apply Lemma 4.6 to obtain\n|/a\\}⌊ra⌋k⌉tl⌉{tTff>t,f>t/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTf/⌊ar⌈⌊lV>t/⌊ar⌈⌊lf/⌊ar⌈⌊l2\n2/lessorequalslant/parenleftBigα\nn/parenrightBig10\nα−1.\nAsE[f] = 1, these two upper bounds contradict (3).\n4.3 Product mixing\nThe proof of Theorem 4.9 in fact gives the following stronger statement, which implies\nTheorem 1.11.\nTheorem 4.10. For anyε,c,C > 0, there exist c′,n0>0such that the following holds.\nLetn>n0and letGbe a(c,C)-goodn-graded group. Let A,B,Cbe measurable subsets of\nG, each with Haar measure at least e−c′n1/3. Letf=1A\nµ(A),g=1B\nµ(B),h=1C\nµ(C). Then\n|/a\\}⌊ra⌋k⌉tl⌉{tf∗g,h/a\\}⌊ra⌋k⌉tri}ht−1|<ε.\n30Proof.Assume without loss of generality that Bhas the smallest measure of the three sets.\n(Note that, while the trilinear form T(f,g,h) :=/a\\}⌊ra⌋k⌉tl⌉{tf∗g,h/a\\}⌊ra⌋k⌉tri}htis not quite symmetric with\nrespect to permuting f,gandh, we may swap the positions of fandgor ofgandhif we\nreplace some of A,BandCby their inverses, meaning A−1:={x−1:x∈A}etc, which\nhave the same measures. So there is indeed no loss of generali ty in assuming the above.)\nWriteµ(A) =α,µ(B) =β,µ(C) =γandC′=10C2\nc.\nNote that/a\\}⌊ra⌋k⌉tl⌉{tf∗g,h/a\\}⌊ra⌋k⌉tri}ht=/a\\}⌊ra⌋k⌉tl⌉{tTgf,h/a\\}⌊ra⌋k⌉tri}ht. First we quickly handle the case where β/greaterorequalslantc′. Here\nwe may apply Lemma 4.7 to obtain that /⌊ar⌈⌊lTg−I0/⌊ar⌈⌊l2→2< εc′, whereI0denotes operator\nF/ma√sto→E[F] whichsendsafunction totheconstant function ofthesame e xpectation, provided\nthatn0is sufficiently large. Using the fact that E[g] = 1 we then have\n|/a\\}⌊ra⌋k⌉tl⌉{tTgf,h/a\\}⌊ra⌋k⌉tri}ht−1|=|/a\\}⌊ra⌋k⌉tl⌉{tTgf,h/a\\}⌊ra⌋k⌉tri}ht−/a\\}⌊ra⌋k⌉tl⌉{tI0f,h/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTg−I0/⌊ar⌈⌊l2→2·/⌊ar⌈⌊lf/⌊ar⌈⌊l2·/⌊ar⌈⌊lh/⌊ar⌈⌊l2<εc′\nα1/2γ1/2/lessorequalslantε,\nyielding the conclusion of the theorem.\nThe proof of the case min {α,β,γ}=β t,h>t/a\\}⌊ra⌋k⌉tri}ht.(4)\nUsing Lemma 4.6, we obtain the upper bound\n|/a\\}⌊ra⌋k⌉tl⌉{tTgf>t,h>t/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTg/⌊ar⌈⌊lV>t/⌊ar⌈⌊lf/⌊ar⌈⌊l2/⌊ar⌈⌊lh/⌊ar⌈⌊l2/lessorequalslantβ10\nα1/2·γ1/2·n10<ε/2, (5)\nprovided that n0is sufficiently large. For 1 /lessorequalslantd/lessorequalslantt, we use the upper bound\n|/a\\}⌊ra⌋k⌉tl⌉{tTgf=d,h=d/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lTg/⌊ar⌈⌊lV=d/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2/⌊ar⌈⌊lh=d/⌊ar⌈⌊l2. (6)\nBy Lemma 4.6, we have /⌊ar⌈⌊lTg/⌊ar⌈⌊lV=d/lessorequalslant/parenleftBig\nC′log(1/α)\nn/parenrightBigd/2\n, whereC′:= 10C2/c. By Theorem 4.8,\nwe have the upper bound\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslantC′dlogd(e/α),\nand similarly for h,\n/⌊ar⌈⌊lh=d/⌊ar⌈⌊l2\n2/lessorequalslantC′dlogd(e/γ).\nSubstituting the last three bounds into (6), we obtain\n|/a\\}⌊ra⌋k⌉tl⌉{tTgf=d,h=d/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/parenleftbiggC′3log(e/α)log(e/β)log(e/γ)\nn/parenrightbiggd/2\n/lessorequalslantε4−d, (7)\nprovided that c′is sufficiently small depending on C,candε.\nThe sum of the contribution from (5) and of those from (7) for 1 /lessorequalslantd/lessorequalslantt, to the\nright-hand side of (4), is clearly less than ε, yielding|/a\\}⌊ra⌋k⌉tl⌉{tTgf,h/a\\}⌊ra⌋k⌉tri}ht−1|<ε, as required.\n314.4 Equidistribution of convolutions\nLetA⊆Gbe a positive-measure subset of a good Lie group, and suppose thatXis a\nG-homogeneous topological space (equipped with its G-invariant Haar probability measure\nµX), and that B⊆Xis a positive-measure subset. The next theorem states that a s long\nas the measures of Aand ofBare not too small, applying a uniformly random element of\nAto a uniformly random element of Byields an almost uniformly random element of X\n(meaning, a random element with respect to the Haar probabil ity measure). Note that\nTheorem 4.11 below, together with Lemma 3.15, imply Theorem s 1.10 and 1.12.\nTheorem 4.11. For eachC,c,ε > 0there exists c′,n0>0, such that the following holds.\nLetn > n 0, letGbe ann-graded(C,c)-good compact connected Lie group, and let X\nbe aG-homogeneous topological space (equipped with the G-action(g,x)/ma√sto→gx), and let\nµXdenote the G-invariant Haar probability measure on X. Suppose that A⊆Gand\nB⊆Xare measurable sets of Haar probability measures /greaterorequalslante−c′√n. LetµAdenote the\nHaar probability measure on G, conditioned on the event A, and letµBdenote the Haar\nprobability measure on Xconditioned on the event B, i.e.µB(Y) =µX(B∩Y)/µX(B)for\na measurable set Y⊆X, andµA(Z) =µG(A∩Z)/µG(A)for a measurable set Z⊆G.\nThen the total variation distance between µXand the distribution of abwherea∼µAand\nb∼µBindependently, is less than ε.\nProof.Considerfirstthecasewhere X=GandGacts onitself byleftmultiplication; inthis\ncase, since the distribution of abisµA∗µB, we need to show that /⌊ar⌈⌊lµA∗µB−µG/⌊ar⌈⌊lTV<ε.\nWe associate µAwith the function fA=1A\nµ(A)∈L2(G) and similarly, we associate µBwith\nthe function fB= 1B/µ(B)∈L2(G); it follows easily from the Cauchy-Schwarz inequality\nthat/⌊ar⌈⌊lµA∗µB−µG/⌊ar⌈⌊lTV/lessorequalslant/⌊ar⌈⌊lfA∗fB−1/⌊ar⌈⌊l2. So our aim is now to prove that\n/⌊ar⌈⌊lfA∗fB−1/⌊ar⌈⌊l2<ε. (8)\nIn proving (8), we argue that we may assume, without loss of ge nerality, that µ(B)/lessorequalslantc′.\nIndeed, if this does not hold, then write B=∪i∈IBias a finite, disjoint union of sets Bi\nsuch thatc′/2/lessorequalslantµ(Bi)/lessorequalslantc′. Once we have proved (8) for sets of measure at most c′, we\nobtain the desired bound for Bby convexity, noting that fBis a convex combination of the\nfunctionsfBi.\nSett=log(1/µ(B))\n2. We now have\n/⌊ar⌈⌊lfA∗fB−1/⌊ar⌈⌊l2\n2=t/summationdisplay\nd=1/⌊ar⌈⌊lTfBf=d\nA/⌊ar⌈⌊l2\n2+/⌊ar⌈⌊lTfBf>t\nA/⌊ar⌈⌊l2\n2.\nNow\n/⌊ar⌈⌊lTfBf=d\nA/⌊ar⌈⌊l2/lessorequalslant/⌊ar⌈⌊lTfB/⌊ar⌈⌊lV=d/⌊ar⌈⌊lf=d\nA/⌊ar⌈⌊l2.\nand\n/⌊ar⌈⌊lTfBf/greaterorequalslantt\nA/⌊ar⌈⌊l2/lessorequalslant/⌊ar⌈⌊lTfB/⌊ar⌈⌊lV>t/⌊ar⌈⌊lf>t\nA/⌊ar⌈⌊l2.\n32The bound (8) now easily follows from Theorem 4.8 and Lemma 4. 6, similarly to in the\nproof of Theorem 4.10. Indeed, writing C′:= 10C2/c, these yield\n/⌊ar⌈⌊lTfB/⌊ar⌈⌊lV=d/⌊ar⌈⌊lf=d\nA/⌊ar⌈⌊l2/lessorequalslant/parenleftbiggC′2log(1/µ(A))log(1/µ(B))\nn/parenrightbiggd/2\n/lessorequalslantε4−d\nfor all 1/lessorequalslantd/lessorequalslantt, and\n/⌊ar⌈⌊lTfB/⌊ar⌈⌊lV>t/⌊ar⌈⌊lf>t\nA/⌊ar⌈⌊l2/lessorequalslant/parenleftbiggβ\nn/parenrightbigg10\nα−1/2/lessorequalslantε/2,\nprovided that n0is sufficiently large and c′sufficiently small depending on c,Candε.\nTo prove the general case, note that we may choose an arbitrar yx0∈Xand set\n˜B={b∈G:bx0∈B}. If˜b∼µ˜Bthen˜bx0∼µB, and ifg∼µGthengx0∼µX; it follows\nthat/⌊ar⌈⌊lµab−µX/⌊ar⌈⌊lTV/lessorequalslant/⌊ar⌈⌊lµA∗µ˜B−µG/⌊ar⌈⌊lTV, whereµabdenotes the distribution of abfora∼µA\nandb∼µB(independently). The result for the pair ( A,˜B) therefore implies the result for\n(A,B).\n5 Growth in fine groups\nIn this section we show that Theorems 1.4, 1.7 and 1.8 follow f rom Theorem 3.14.\nFirst, thetheorem below is a variation on Theorem4.5, with h ypercontractivity replaced\nby weak hypercontractivity. Together with Lemma 3.16 it sho ws that Theorem 3.14 implies\nTheorem 1.4.\nTheorem 5.1. Letc >0,C >1and letGbe an(r,C)-weakly hypercontractive group.\nLetf:G→{0,1}be measurable, and write α:=E[f].Letd∈Nbe such that 0< d/lessorequalslant\nmin/parenleftBig\nlog(1/α)\n2,r/parenrightBig\n. Then\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2/lessorequalslant/parenleftbiggelog(1/α)\nd/parenrightbigg2Cd\nα2.\nProof.Letq= log(1/α)/d, and letq′be the H¨ older conjugate of q. Then by H¨ older, weak\nhypercontractivity and Lemma 3.12, we have\n/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2\n2=/a\\}⌊ra⌋k⌉tl⌉{tf,f=d/a\\}⌊ra⌋k⌉tri}ht/lessorequalslant/⌊ar⌈⌊lf=d/⌊ar⌈⌊lq/⌊ar⌈⌊lf/⌊ar⌈⌊lq′/lessorequalslantqCdα1−1/q/⌊ar⌈⌊lf=d/⌊ar⌈⌊l2.\nThe theorem follows by rearranging and substituting α−1/q=ed. (Note that C >1, by the\ndefinition of weak hypercontractivity.)\nThe following lemma is a variant of Lemma 4.6 for fine groups – t he proof is the same\nas for Lemma 4.6, only using Theorem 5.1 instead of Theorem 4. 5. We will make use of it\nwhen proving our diameter bounds for fine groups.\n33Lemma 5.2. For eachc >0andC >1there exist c′,C′,n0>0such that the following\nholds. Let n > n0and letGbe ann-graded(C,c)-fine group. Let A⊆Gbe measurable,\nand suppose that\nα:=µG(A)∈(e−c′n,c′).\nWritef=1A\nαandt=log(1/α)\n2. Then for 1/lessorequalslantd/lessorequalslanttwe have\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV=d/lessorequalslant/parenleftbiggelog(1/α)\nd/parenrightbiggCd\n·/parenleftbiggd\ncn/parenrightbiggd/2\n.\nWe also have\n/⌊ar⌈⌊lTf/⌊ar⌈⌊lV>t/lessorequalslantα10\nn10.\nWe now show that if fhas small expectation, then most of the Fourier mass of flies\non the high degrees.\nLemma 5.3. For eachc>0andC >1there existc′,n0>0such that the following holds.\nLetn > n 0, letGbe a(C,c)-finen-graded group, and let f:G→{0,1}be measurable.\nSuppose that α:=E[f]/greaterorequalslante−c′n, and let 0/lessorequalslantt/lessorequalslantlog(1/α)\n106C2. Then/⌊ar⌈⌊lf/lessorequalslantt/⌊ar⌈⌊l2\n2/lessorequalslant2·2α1.99.\nProof.Inserting a factor of 2 into the bound from Theorem 5.1, we hav e for any 0 0andC >1, there exist c′,n0>0such that the following\nholds. Letn>n0, letGbe a(C,c)-finen-graded group, and let f:G→{0,1}be measurable.\nSuppose that α:=E[f]/greaterorequalslante−c′nand letd∈Nwith00andC >1, there exists δ>0such that the following holds.\nLetGbe a(C,c)-fine,n-graded group, let A,B,C⊆Gbe measurable sets of measures at\nleaste−nδ, and letf=1A\nµ(A),g=1B\nµ(B),h=1C\nµ(C). Then|/a\\}⌊ra⌋k⌉tl⌉{tf∗g,h/a\\}⌊ra⌋k⌉tri}ht−1|<ε.\nWe remark that Lemma 5.5 is only weaker than the correspondin g Theorem 4.10 for\ngood groups. We include it even though the groups of interest to us are both good and\nfine. We decided to include the lemma mainly for aesthetic rea sons. Our diameter bounds\nrely on Lemma 5.5 and we preferred to show that our diameter bo unds hold for fine groups\nrather than groups that are both good and fine.\nBelow we use a trick of Nikolov and Pyber [34] (who observed th at product mixing\nimplies an upper bound on the diameter), to bound the diamete r in fine groups.\nCorollary 5.6. For eachc>0andC >1, there exists δ>0such that if Gis a(C,c)-fine\ngroup andA⊆Gis a measurable set of measure at least e−nδ, thenµ(A2)>1−e−nδ, and\nA3=G.\nProof.The claim about µ(A2) follows by applying Lemma 5.5 while taking A=B=A,\nC=G\\A2andε= 1/2 (in fact, any value of εless than one, will do). As for the claim\naboutA3, suppose for a contradiction that A3/\\⌉}atio\\slash=G. Letx∈G\\A3.ThenA2∩xA−1=∅.\nThis contradicts Lemma 5.5 when setting A=B=AandC=xA−1.\n5.2 Non-Abelian Brunn-Minkowski type inequalities for fine groups\nThe following theorem is a restatement of Theorem 1.8.\nLemma 5.7. There exist absolute constants c′,n0>0such that the following holds. Let G\nbe a compact connected Lie group with n:=n(G)>n0, and letA⊆Gbe a measurable set\nof measure at least e−c′n. Thenµ(A2)/greaterorequalslantµ(A)1/10.\n35Proof.First note that µ(A)>1\n2implies that A2=G, as ifx∈G\\A2, thenAandxA−1\nare disjoint sets each of measure greater than 1 /2, a contradiction. By Corollary 5.6, we\nmay also assume that µ(A)/lessorequalslante−nc′\n, providedc′is sufficiently small.\nLetf=1A\nµ(A)andg= 1A2. Then we have /a\\}⌊ra⌋k⌉tl⌉{tf∗f,g/a\\}⌊ra⌋k⌉tri}ht= 1.On the other hand, by\nCauchy–Schwarz, we have\n|/a\\}⌊ra⌋k⌉tl⌉{tf∗f,g/a\\}⌊ra⌋k⌉tri}ht|/lessorequalslant/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2/⌊ar⌈⌊lg/⌊ar⌈⌊l2=/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2/radicalbig\nµ(A2).\nThisyields µ(A2)/greaterorequalslant1\n/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2\n2.Lett=log(1/µ(A))\n106C2.Wehave/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2\n2=/⌊ar⌈⌊lf/lessorequalslantt∗f/lessorequalslantt/⌊ar⌈⌊l2\n2+/⌊ar⌈⌊lf/greaterorequalslantt∗f/greaterorequalslantt/⌊ar⌈⌊l2\n2.\nBy applying Lemma 5.3 to 1 A, we obtain\n/⌊ar⌈⌊lf/lessorequalslantt∗f/lessorequalslantt/⌊ar⌈⌊l2\n2/lessorequalslant/⌊ar⌈⌊lf/lessorequalslantt/⌊ar⌈⌊l4\n2/lessorequalslantα−0.02.\nBy applying Lemma 5.2 to 1 A, we obtain\n/⌊ar⌈⌊lf>t∗f>t/⌊ar⌈⌊l2\n2/lessorequalslant/⌊ar⌈⌊lTf/⌊ar⌈⌊l2\nV>t/⌊ar⌈⌊lf/⌊ar⌈⌊l2\n2/lessorequalslant/parenleftBigα\nn/parenrightBig20\nα−1/lessorequalslant1.\nCombining these two bounds completes the proof.\nLemma 5.8. For eachε,c >0andC >1there exist δ,n0>0such that the following\nholds. Letn>n0and letGbe a(C,c)-finen-graded group. If A⊂Gis a measurable set\nwith\nµ(A) :=e−nζ∈/parenleftBig\ne−n1−ε,e−nε/parenrightBig\n,\nthenµ(A2)/greaterorequalslante−nζ−δ.\nProof.Wemayandshallassume,throughouttheproof,that δissufficientlysmalldepending\nonε,Candc, and that n0is sufficiently large depending on δ. Letf=1A\nµ(A). As in the\nproof of the previous lemma, we have\nµ(A2)/greaterorequalslant1\n/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2\n2. (9)\nLett1=log(1/µ(A))\nn4δandt2=log(1/µ(A))\n106C2.We bound/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2\n2from above by decomposing it\nas follows:\n/⌊ar⌈⌊lf∗f/⌊ar⌈⌊l2\n2=/⌊ar⌈⌊lft2∗f>t2/⌊ar⌈⌊l2\n2. (10)\nBy applying Lemma 5.2, we obtain\n/⌊ar⌈⌊lf>t2∗f>t2/⌊ar⌈⌊l2/lessorequalslant/⌊ar⌈⌊lTf/⌊ar⌈⌊lV>t2/⌊ar⌈⌊lf/⌊ar⌈⌊l2/lessorequalslantα10\nn10α−1/2/lessorequalslant1.\nApplying Lemma 5.4 (with αfin place off, andεtaken to be 4 δ), we have\n/⌊ar⌈⌊lf0andC >1, there exist m,n0>0, such that the following\nholds. Let n > n 0and suppose that Gis a(C,c)-finen-graded group. Let A⊆Gbe a\nmeasurable set with µ(A)>e−n1−ε. ThenAm=G.\nProof.Note that we may assume εis as small as we please (depending on candC). First\napply Lemma 5.8 repeatedly ( Ntimes, say), until A2Nhas measure /greaterorequalslante−nε. In other words,\nµ(Am1)/greaterorequalslante−nε, wherem1depends upon εalone. We can now apply Corollary 5.6 to obtain\nA3m1=G, provided that εis sufficiently small depending on candC.\n6 The strong quasirandomness of the simply connected com-\npact Lie groups\nIn this section we describe the degree decomposition of the n-graded simply connected\nsimple compact Lie groups in terms of their irreducible subr epresentations. We also show\nthat all of them are c-strongly-quasirandom, for some absolute constant c>0. In addition,\nwe introduce the notion of comfortable d-juntas. These will be important in our proofs.\n37One of the goals of this section is to show that each Peter-Wey l idealWρ⊆V=dcontains\na comfortable d-junta. This is useful because any linear operator on L2(G) that commutes\nwith the action of Gfrom both sides, has each Wρas an eigenspace. The operators we use\nin Section 8 will have this commuting property, and so when co mputing their eigenvalues\nwe can simply consider the action of the relevant operator on a comfortable d-junta.\n6.1 The Peter-Weyl theorem\nWe now recall some classical facts from the representation t heory of compact groups. The\nPeter-Weyl theorem states that if Gis a compact group, equipped with its Haar probability\nmeasure, then L2(G) has the following decomposition as an orthogonal direct su m:\nL2(G) =/circleplusdisplay\nρ∈ˆGWρ,\nwhereˆGdenotes a complete set of complex irreducible unitary repre sentations of G(here,\ncomplete means having one irreducible representation from each equi valence class of irre-\nducible representations), and Wρis the subspace of L2(G) spanned by functions of the form\ng/ma√sto→utρ(g)v, foru,v∈V, whereVis the vector space on which ρacts. The latter functions\nare known as the matrix coefficients or thematrix entries ofρ. The subspaces Wρare\ntwo-sided ideals (meaning, they are closed under both left a nd right actions of G), and they\nare also topologically closed; in fact, they are precisely t he minimal non-zero topologically\nclosed two-sided ideals of L2(G), and they are therefore irreducible as G×G-modules (the\nG×Gaction beingdefinedin theobvious way, with the firstfactor a cting onL2(G) from the\nleft and the second from the right). We call them the Peter-Weyl ideals ofL2(G), though\nthis terminology is non-standard. The space Wρcan be decomposed as a direct sum of\ndim(ρ) irreducible left-representations.\nSince the Peter-Weyl ideals Wρare precisely the minimal closed two-sided ideals of\nL2(G), every closed two-sided ideal of L2(G) can be decomposed as a direct sum of some\nof theWρ. Letd∈N∪{0}; sinceV=dis a closed, two-sided ideal of L2(G), there exists a\nsetLdof irreducible representations of Gsuch that\nV=d=/circleplusdisplay\nρ∈LdWρ.\nIfρ∈Ldfor some integer 0 /lessorequalslantd0, for each d < n/2. We will also show that the other\nirreducible representations of SO( n) all have dimension at least exponential in n, i.e. at\nleast exp(c′n) for some absolute constant c′>0.\n38We briefly recall Weyl’s construction of the irreducible rep resentations of SO( n). For\nmore detail on Weyl’s construction, the reader is referred f or example to the book [12]\nof Fulton and Harris. (We note that, though the description i n [12] is of SO( n,C), the\nirreducible representations of SO( n) := SO(n,R) are in a dimension-preserving one-to-\none correspondence with those of its complexification SO( n,C).) We start by describing\nthe irreducible representations of O( n) :=O(n,R). LetV=Rndenote the standard\nrepresentation of O( n), defined by ρV(g)(v) =g·v— meaning, multiplication of the matrix\ngwith the column-vector v. (We note, for later, that the restriction of this represent ation to\nSO(n) is known as ‘the standard representation of SO( n)’.) For a partition λ= (λ1,...,λ ℓ)\nof some non-negative integer, let d=/summationtextℓ\ni=1λi. Consider the group algebra of the symmetric\ngroup ondelements, R[Sd], with the standard basis {eg:g∈Sd}, and with multiplication\ndefined byegeh=eghforg,h∈Sd. (Where there is no risk of confusion, we will sometimes\nwritegin place of eg, as an element of R[Sd], as is usual practice.) Let Tbe the standard\nYoung tableau of shape λwith the numbers 1 ,2,...,λ 1(in order) in the first row, the\nnumbersλ1+1,λ1+2,...,λ 1+λ2(in order) in the second row, and so on. Also, let Pbe\nthe subgroup of Sdstabilising each of the rows of T(as sets), let Qbe the subgroup of Sd\nstabilising each of the columns of T(as sets), and let\ncλ=\n/summationdisplay\ng∈Peg\n\n/summationdisplay\ng∈Qsign(g)eg\n\nbe theYoung symmetrizer ofλcorresponding to T. The group Sdacts onV⊗dfrom the\nright, permuting the factors:\n(v1⊗v2⊗...⊗vd)g=vg(1)⊗vg(2)⊗...⊗vg(d),\nand, extending linearly, so does R[Sd].\nWe define the Weyl module Sλ(V) :=V⊗dcλ. Clearly, Sλ(V) is a left O( n)-submodule\nofV⊗d. It is reducible in general. However, we can obtain an irredu cible left O( n)-module\nby considering S[λ](V) :=V[d]cλ, whereV[d]is defined to be the intersection of the kernels\nof all/parenleftbigd\n2/parenrightbig\nlinear maps on V⊗dof the form\nv1⊗v2⊗...⊗vd/ma√sto→/a\\}⌊ra⌋k⌉tl⌉{tvi,vj/a\\}⌊ra⌋k⌉tri}htv1⊗v2⊗...⊗vi−1⊗vi+1⊗...⊗vj−1⊗vj+1⊗...⊗vd.\nSuch linear maps are called contractions . It turns out that when the sum of the lengths of\nthe first two columns of the Young diagram of λis greater than n, we haveS[λ](V) ={0}.\nThe other modules S[λ](V) (corresponding to those partitions λsuch that the sum of the\nfirst two columns of the Young diagram of λis at mostn) form a complete set of pairwise\ninequivalent irreducible complex representations of O( n).\nWeyl’s construction for SO( n) requires only one additional ingredient. We say two\npartitionsλandµareassociated if the sum of the lengths of the first column of λand the\nfirstcolumn of µis equal ton, andtheith columnof λhasthesamelength as the ithcolumn\nofµfor eachi >1. Ifλandµare a pair of distinct associated partitions, then S[λ](V)\n39andS[µ](V) restrict to isomorphic representations of SO( n). Ifλis self-associated (which\nhappens iff nis even and the first column of λhas lengthn/2), then S[λ](V) restricts to a\ndirect sum of two isomorphic irreducible representations o f SO(n); ifλis not self-associated,\nthenS[λ](V) restricts to an irreducible representation of SO( n), and ifλ′is the partition\nassociated to λ, thenS[λ′](V) restricts to the same irreducible representation of SO( n). In\nthe latter case, it is customary to choose (as the representa tive of its equivalence class), the\npartition with first column of length less than n/2. Note that, importantly for us, for any\npartitionλwith/summationtext\niλi0. Since the spaces Wρare pairwise orthogonal, the induction hypothesis\nreduces our task to showing that\nV/lessorequalslantd=d/circleplusdisplay\ni=0/circleplusdisplay\nρ∈˜LdWρ.\nLet us write ˜V/lessorequalslantd:=/circleplustextd\ni=0/circleplustext\nρ∈˜LdWρ. We now use Lemma 6.1, namely that all the irre-\nducible subrepresentations of V⊗dare elements of ˜Lifor somei/lessorequalslantd. The matrix coefficients\nof the representation V⊗dinclude the entries of the matrix X⊗d, whereX∈SO(n) is the\ninput matrix. These are exactly the degree- dmonomials in the entries of X. Decomposing\nV⊗dinto irreducible representations, we see that all the homog eneous degree- dpolynomials\nbelong to ˜V/lessorequalslantd; using the induction hypothesis again, all polynomials of d egree at most d−1\nbelong to ˜V/lessorequalslantd−1⊂˜V/lessorequalslantd, and therefore V/lessorequalslantd⊆˜V/lessorequalslantd. The reverse inclusion ( ˜V/lessorequalslantd⊆V/lessorequalslantd) follows\nimmediately from the fact that if/summationtext\niλi=d, then the matrix coefficients of S[λ](V) are\nhomogeneous degree- dpolynomials in the entries of the input matrix.\nWe can now extend the notion of level to all the representatio ns of SO(n).\nDefinition 6.3. Letρbe an irreducible representation of SO(n)and letλbe the corre-\nsponding partition, whose Young diagram has first column of l ength at most n/2. Then we\ndefine the levelofρto be/summationtext\niλi.\nComfortable d-juntas\nWe now digress a little and show that Weyl’s construction imp lies that each Peter-Weyl\nidealWρcontains a certain ‘nice’ function. This will be used later, in Section 8.\nOur ‘nice’ functions are as follows.\nDefinition 6.4. Thecomfortable d-juntasonSO(n)are the functions on SO(n)of the form\nX/ma√sto→/summationdisplay\nσ∈Sdaσx1,σ(1)···xd,σ(d)\nforaσ∈R.\nWe remark that we use the term ‘junta’ here, by analogy with ju ntas in the theory\nof Boolean functions (on {0,1}n), because functions of the above form depend only upon\nthe upperd×dminor, though we stress that we will be interested in the case wheredis\npolynomial in n(e.g.d∼√n), rather than just the case of dfixed andnlarge.\nLettinge1,...,enbe the standard orthonormal basis of Rn, since/a\\}⌊ra⌋k⌉tl⌉{tei,ej/a\\}⌊ra⌋k⌉tri}ht= 0 for all\ni,j∈[d] we havee1⊗...⊗ed∈V[d]. Therefore ( e1⊗...⊗ed)cλ∈S[λ](V),and thus the\nfunctionPλinL2(SO(n)) defined by\nPλ(X) :=/a\\}⌊ra⌋k⌉tl⌉{tX((e1⊗...⊗ed)cλ),e1⊗...⊗ed/a\\}⌊ra⌋k⌉tri}ht\n41is a matrix coefficient of S[λ](V). Moreover, the function Pλis clearly a comfortable\nd-junta: writing cλ=/summationtext\nσ∈Sdεσσ, whereεσ∈{−1,0,1}for eachσ∈Sd, we have\nPλ(X) =/summationdisplay\nσ∈Sdεσd/productdisplay\ni=1xi,σ(i).\nMoreover, we clearly have Pλ(Id) = 1, so Pλis a non-zero element of L2(SO(n)). We obtain\nthe following conclusion, upon which we rely crucially in th e sequel.\nFact 6.5. Let0/lessorequalslantd < n/2. For each irreducible representation ρ∈LdofSO(n), the\nPeter-Weyl ideal Wρcontains a nonzero comfortable d-junta.\n6.3 Obtaining strong quasirandomness for SO (n).\nThefollowinglowerboundonthedimensionofanirreducible representationofSO( n)follows\nimmediately from the analysis in [39] of Weyl’s original dim ension formulae [42]. (We note\nthatourcomfortable d-juntamachinerycouldbeusedtoeasilyobtainaslightly we akerlower\nbound of/parenleftbig⌊n/2⌋\nd/parenrightbig\n. We use such an argument later, when showing strong quasiran domness\nfor SU(n).)\nLemma 6.6. Ifρis an irreducible representation of SO(n)of leveld/lessorequalslantn, then\ndim(ρ)/greaterorequalslant(n−d)d\nd!.\nWe also need the following lower bound, whose proof is deferr ed to the Appendix.\nLemma 6.7. Letn/greaterorequalslant5. Ifρis an irreducible representation of SO(n)of leveld/greaterorequalslantn/2,\nthen\ndim(ρ)/greaterorequalslantexp(n/32).\nLemmas 6.6 and 6.7 immediately give strong quasirandomness .\nTheorem 6.8. For eachn/greaterorequalslant2, then-graded group SO(n)isc-strongly-quasirandom, for\nsome absolute constant c>0.\n6.4 Obtaining strong quasirandomness for Spin (n).\nThe strong quasirandomness of the group Spin( n) follows from the fact that it is a double\ncover of SO( n).\nTheorem 6.9. For eachn/greaterorequalslant3, then-graded group Spin(n)isc-strongly-quasirandom, for\nsome absolute constant c>0.\n42Proof.Recall that the spin group Spin( n) is the double-cover of SO( n) for alln/greaterorequalslant2. It is\na simply-connected real Lie group for all n/greaterorequalslant3, so its complex irreducible representations\nare in an explicit (and dimension-preserving) one-to-one c orrespondence with those of its\nLie algebra. It has the same Lie algebra as SO( n); this Lie algebra so(n,R) is simple for\nalln/greaterorequalslant5, and its complexification so(n,C) is likewise simple (for all n/greaterorequalslant5), so by e.g. [12]\n(26.14), for all n/greaterorequalslant5 the complex irreducible representations of so(n,R) are restrictions\nof unique complex irreducible representations of so(n,C). The dimensions of the complex\nirreduciblerepresentations of Spin( n) (for alln/greaterorequalslant5) are thereforegiven by equations (24.29)\nand (24.41) in [12] (pages 408 and 410). For all n= 2k+1/greaterorequalslant5 odd, we have\ndim(ρλ) =/productdisplay\n1/lessorequalslanti0.\n6.8 Weyl’s construction for SU (n), and its applications.\nThe final group to consider is the special unitary group. We st art by relating our degree\ndecomposition of L2(SU(n)) to the decomposition of L2(SU(n)) into Peter-Weyl ideals.\n47Degree decomposition\nEarlier, we defined V/lessorequalslantdto consist of the polynomials of (total) degree at most dpolynomials\nin the real parts and the imaginary parts of the entries of the input matrix X∈SU(n).\nEquivalently, V/lessorequalslantdconsists of the polynomials of (total) degree at most din the entries of\nthe input matrix and their complex conjugates.\nWeyl’s construction for SU(n)\nWe now recall Weyl’s construction of the irreducible repres entations of SU( n), and deduce\nfrom it the consequences we need. (As before, for more detail on Weyl’s construction, the\nreader is referred to Fulton and Harris [12], noting that the complex irreducible represen-\ntations of SU( n) are the same as those of SL( n,C), since SU( n) is a maximal compact\nsubgroup of SL( n,C).) LetV=Cndenote the standard representation of SU( n), defined\nbyρV(g)(v) =g·v. For a partition λwith at most n−1 parts, let d=d(λ) =/summationtext\niλi.\nLetTbe the standard Young tableau of shape λ(defined in Section 6.2) and let cλbe the\ncorresponding Young symmetrizer (also defined in Section 6. 2). We define the correspond-\ningWeyl module bySλ(V) :=V⊗dcλ. Clearly, Sλ(V) is a left SU( n)-submodule of V⊗d.\nThe modules Sλ(V), asλranges over all partitions with at most n−1 parts, constitute a\ncomplete set of pairwise inequivalent complex irreducible representations of SU( n). (Unlike\nin the cases of SO( n) and Sp(n), we do not need to pass to a subrepresentation of the Weyl\nmodule; the latter is already irreducible as a left SU( n)-module.)\nUnlike in the case of Sp( n), however, the complex conjugates of the entries of the inpu t\nmatrix are no longer matrix coefficients of the standard repre sentation. Instead, they are\nmatrix coefficients of the dual of the standard representatio n, i.e. they are the entries of\nthe matrix ( A−1)t=¯A. (Recall that the dual of a representation ρis the representation ρ∗\ndefined byρ∗(g) = (ρ(g−1))t.) We note that ( Sλ(V))∗∼=Sλ(V∗) = (V∗)⊗dcλ.\nWe have three notions of level for a representation ρof SU(n), and our goal is to show\nthat they agree when the level is lexλ′iffλi> λ′\niwherei= min{j:λj/\\⌉}atio\\slash=λ′\nj}. For a (not necessarily irreducible)\nrepresentation ρof SU(n), its step vector is defined to be the lexicographically larg est step\nvector of an irreducible subrepresentation of ρ.\nThe step vector is better-behaved than the corresponding pa rtition, with respect to\ntaking duals and tensors. The dual of ρhas the reversed step vector ( an−1,...,a 1), cor-\nresponding to the partition ( a1+···+an−1,···,a1+a2,a1). Moreover, if ρ1andρ2are\ntwo representations whose step vectors are ( a1,...,an−1) and (b1,...,bn−1), then the step\nvector of their tensor product is ( a1+b1,...,an−1+bn−1).\nWe say that an irreducible representation ρ=Sλ(V)of SU(n) isefficient ifλ⌊n\n2⌋= 0.\nEquivalently, ρis efficient if its step vector w= (a1,...,a n−1) has the property that its\nsecond half w′′:= (a⌈n/2⌉,...,a n−1) consists of zeros. We say that ρisdually-efficient if\nitsfirst halfw′:= (a1,...,...,a ⌈n/2⌉−1) consists of zeroes. Alternatively, in terms of the\ncorresponding partition λ, the irreducible representation ρis dually-efficient if λ1=λ2=\n···=λ⌈n/2⌉−1. (Note that the dual to each efficient representation is duall y-efficient, and if\nnis odd, the converse also holds. For neven, our definition leads to a somewhat arbitrary\nchoice of how to handle the middle part of the step vector, but this does not matter when\nthe level is smaller than n/2.) We call the partition αwith step vector w′theefficient\ntruncation ofλand the partition βwith step vector w′thedually-efficient truncation ofλ.\nDefinition 6.18. We define the total level of a representation ρ=Sλ(V)with step vector\n(a1,...,a n−1)to be\nn−1/summationdisplay\ni=1aimin{i,n−i}.\nFor eachd∈N∪{0}, let˜Lddenote set of irreducible representations of SU( n) with total\nleveld. Our next aim is to show that Ld=˜Ldfor all 0 /lessorequalslantd0such that if d/greaterorequalslantn/4andρis an\nirreducible representation of SU(n)of leveld, thendim(ρ)/greaterorequalslant2cn.\nLemmas 6.25 and 6.26 immediately yield the strong quasirand omness of SU( n).\nLemma 6.27. For eachn/greaterorequalslant2, the group SU(n)isc-strongly-quasirandom as an n-graded\ngroup, where c>0is an absolute constant.\n7 Simply connected compact Lie groups are fine\nIn this section, we prove Theorem 3.14. The proof has two part s. In the first part, we\nidentify a natural noise operator UδonL2(G) for the groups G= SU(n),Sp(n),Spin(n)\nwhich is guaranteed to satisfy a certain hypercontractive i nequality, thanks to the fact\nthat the Ricci curvature of these groups is bounded from belo w. (We note that this noise\noperatorUδis not quite the same as the Beckner operator Tδ,rthat we defined earlier.) The\nsecond part consists of inferring the weak hypercontractiv ity of the operator Tδ,rfrom the\nhypercontractive inequality for Uδ. We accomplish that by analyzing the eigenvalues of Uδ\nand showing that they are all larger than the eigenvalues of t he operator TδC,rfor some\nabsolute constant C >1. This will allow us to write TδC,r=UδSfor a linear operator Son\nL2(G) satisfying/⌊ar⌈⌊lS/⌊ar⌈⌊l2→2/lessorequalslant1.We will thus have\n/⌊ar⌈⌊lTδC,r/⌊ar⌈⌊l2→q/lessorequalslant/⌊ar⌈⌊lS/⌊ar⌈⌊l2→2·/⌊ar⌈⌊lUδ/⌊ar⌈⌊l2→q/lessorequalslant1,\nas needed.\nThe hypercontractive inequality\nHere we rely on concepts from differential geometry, such as a R iemannian metric, the\nLaplace–Beltrami operator, and the Ricci curvature/tenso r. We use the notation of Ander-\nson, Guionnet and Zeitouni [1, Sections E and F], and we refer the reader to that work for\nmore details.\nThe compact Lie groups are equipped with the structure of a bi -invariant Riemannian\nmanifold (M,g); this is uniqueupto normalization if the compact Lie group is simple. Once\na normalization is set, and denoting by ∆ the Laplace–Beltra mi operator, it is also known\nthat the Hilbert space L2(M) has an orthonormal basis of eigenvectors of ∆, that ∆ is self -\nadjoint and negative semidefinite, and that 0 is a simple eige nvalue of ∆ (with the constant\nfunctions as corresponding eigenvectors). For a given comp act Lie group Gwith a Riemann\nmanifold structure, we let u0,u1,u2,...be such a basis, with 0 = λ0>λ1/greaterorequalslantλ2/greaterorequalslant...being\nthe corresponding eigenvalues, so that λi<0 for alli/greaterorequalslant1, and with u0being the constant\nfunction with value 1. For any f∈L2(M), we may write funiquely in the form\nf=∞/summationdisplay\ni=0ciui,\n54whereci∈Rfor eachi/greaterorequalslant0 (we have ci=/a\\}⌊ra⌋k⌉tl⌉{tf,ui/a\\}⌊ra⌋k⌉tri}htfor eachi/greaterorequalslant0). Forδ >0, we define the\nnoise operator Uδby\nUδ:L2(M)→L2(M);Uδ(f) =∞/summationdisplay\ni=0ciδ−λiui,\nforf=/summationtext∞\ni=0ciui. We note that Ue−tis, in fact, the heat kernel corresponding to ∆,\nwhich is the averaging operator with respect to the Brownian motion on the corresponding\nmanifold.\nFor a Riemaniann manifold ( M,g) andC >0, we say that ( M,g) hasRicci curvature\nbounded from below by Cif for all points p∈M, the Ricci tensor Ric p(·,·) atpsatisfies\nRicp(X,X)/greaterorequalslantCgp(X,X)\nfor all tangent vectors Xatp.\nThe hypercontractive inequality we need is the following.\nTheorem 7.1. LetC >0and let(M,g)be a compact, connected Riemann manifold whose\nRicci curvature is bounded from below by C, let2/lessorequalslantp/lessorequalslantqand letf∈Lp(M). Then\n/⌊ar⌈⌊lUδ(f)/⌊ar⌈⌊lq/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lp∀0/lessorequalslantδ/lessorequalslant/parenleftbiggp−1\nq−1/parenrightbigg1/C\n.\nAs explained in Klartag and Regev [38] the Bakry-Emery crite rion yields a log-Sobolev\ninequality (as given for example in [1, Corollary 4.4.25] ap plied with Φ = 0), which implies\na hypercontractive inequality by a theorem of Gross [16, The orem 6]).\nUtilizing the Riemannian structure. As mentioned above, the compact simple Lie\ngroups have a unique (up to normalization) bi-invariant Rie mannian manifold structure.\nIn order to set it up one needs to assign an inner-product on th e tangent space at the\nidentity Id of G, i.e., on the Lie algebra of G. (The inner product on all the other tangent\nspaces is then determined by using a push-forward with respe ct to left multiplication by\nthe appropriate group element.)\nThe tangent space of Spin( n) at the identity is the Lie algebra of SO( n). As in [1], we\nequip it with the usual Euclidean / Hilbert-Schmidt norm, i. e. the norm of a matrix is\nthe sum of the squares of its entries. This norm gives rise to a bi-invariant metric when\napplying push-forward maps to define the norm on all the other tangent spaces. The norm\non the Lie algebras of SU( n) and Sp(n) is defined similarly by taking the sum of squares of\nthe components of each entry.\nIt is well-known (see [1, 4.4.30]), that the Ricci curvature of the simply connected com-\npact Lie groups Spin( n),SU(n),Sp(n) is bounded from below by ( n−2)/4,n\n2,n+1 respec-\ntively.\nUsing Theorem 7.1, we can prove Theorem 3.14, assuming the fo llowing lemma.\n55Lemma 7.2. There exists an absolute constant C, such that the following holds. Let Gbe\neitherSU(n),Sp(n)orSO(n). Letd∈Nwith0/lessorequalslantd0.\n568.1 The Gaussian noise operator, a.k.a. the Ornstein–Uhlen beck operator\nInthis section, werecall thedefinitionof theGaussian nois eoperator andseveral of its prop-\nerties. For simplicity of notation, we present this theory f or functions in L2(Rn,γ), however\neverything applies more generally to functions in L2(Rn×m,γ). (Here and elsewhere, we\nabuse notation slightly and denote by γa Gaussian distribution, where the domain is clear\nfrom context.)\nDefinition 8.1. Forρ∈[0,1], we define Uρ:L2(Rn,γ)→L2(Rn,γ)by\nUρf(X) =EY∼γ[f(ρX+/radicalbig\n1−ρ2)Y].\nIt is a well known fact that Uρis hypercontractive [33]:\nTheorem 8.2. Letf:Rn→Rand let0/lessorequalslantρ/lessorequalslant1√q−1. Then/⌊ar⌈⌊lUρf/⌊ar⌈⌊lLq(γ)/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nBelow we use Uρto construct an operator T ρoverL2(√nSO(n)) which is hypercontrac-\ntive, and on which we have lower bounds on the eigenvalues cor responding to low-degrees,\nthereby showing that the group SO( n) is good.\n8.2 Constructing the noise operator Tρ\nIn this section, we design our noise operator T ρonL2(SO(n)). En route, we define auxiliary\noperators that act on both L2(Rn×n,γn×n) andL2(√nSO(n),µ).\nLeft and right multiplication by matrices from SO (n)\nDefinition 8.3. For a matrix U∈SO(n), we define the operator LUacting both on\nL2(Rn×n,γn×n)andL2(√nSO(n),µ), as follows. For a function f:Rn×n→R, the function\nLUf:Rn×n→Ris defined by\nLUf(X) =f(UX).\nFor a function f:√nSO(n)→R, we similarly define LUf(X) =f(UX).\nWe similarly define the operator RVcorresponding to right multiplication.\nDefinition 8.4. For a matrix V∈SO(n), we define the operator RVacting both on\nL2(Rn×n,γ)andL2(√nSO(n),µ), as follows. For a function f:Rn×n→R, the function\nRVf:Rn×n→Ris defined by\nRVf(X) =f(XV).\nFor a function f:√nSO(n)→R, we similarly define RVf(X) =f(XV).\n57The Gram–Schmidt operators.\nNext, we define the operators T row,Tcolthat capture our coupling and map L2(SO(n),µ)\ntoL2(Rn×n,γ), as well as their adjoint operators that go in the reverse di rection. To do so,\nwe use the Gram-Schmidt process.\nFix a matrix X∈Rn×nand letc1,...,cnbe its columns. Provided det( X)/\\⌉}atio\\slash= 0 (which\nfor a Gaussian matrix happens with probability one), we may a pply the Gram-Schmidt\nprocess on ( c1,...,cn) to get an orthonormal set of vectors ˜ c1,...,˜cn. Abusing notation\nslightly, we define the matrix GScol(X)∈√nSO(n) as the matrix whose ithcolumn is√n˜ci\nfor alli0) or−√n˜cn(if det(X)<0);\nthis is of course a (column-) dilation of the (column-) Gram- Schmidt matrix corresponding\ntoX. Since the Gram-Schmidt process preserves the sign of the de terminant, this matrix\nGScol(X) is indeed in√nSO(n).\nSimilarly, letting r1,...,rnbe the rows of X, we let ˜r1,...,˜rnbe the resulting set\nof vectors by applying the Gram-Schmidt process on ( r1,...,rn) and define the matrix\nGSrow(X) as the matrix whose ithrow is√n˜rifor alli < n, and whose nth row is either√n˜rn(if det(X)>0) or−√n˜rn(if det(X)<0).\nThedilatedGram–Schmidtprocessesabovedefinecouplings( X,GScol(X))and(X,GSrow(X))\nbetweenγandµ, and we use these to define the operators T rowand Tcol:\nDefinition 8.5. We define Trow:L2(√nSO(n),µ)→L2(Rn×n,γ)as follows. For a func-\ntionf:√nSO(n)→R, we define Trowf:Rn×n→Rby\nTrowf(X) =f(GSrow(X)).\nDefinition 8.6. We define Tcol:L2(√nSO(n),µ)→L2(Rn×n,γ)as follows. For a function\nf:√nSO(n)→R, we define Tcolf:Rn×n→Rby\nTcolf(X) =f(GScol(X)).\nThe operator Tρ.\nThe operators T coland T rowallow us to move from L2(µ) toL2(γ). We can also go in\nthe reverse direction, using their adjoints. It is easy to se e, using Jensen’s inequality, that\nT∗\ncolUρTcolhas the same hypercontractive properties as Uρ. Thus, it is natural to consider\nthe operator T∗\ncolUρTcolas an analogue of the Gaussian noise operator, for√nSO(n). We\ndo not know, however, how to bound from below the eigenvalues of T∗\ncolUρTcolso as to\ndeduce Theorem 1.13. The reason is that to bound its eigenval ues, we (naturally) need\nsome information about the eigenvectors corresponding to t hem, and we only know how to\nobtain such information from classical representation fac ts about SO( n). To use these facts\n(so as to ensure the eigenspaces are ‘nice’, and easy to analy se), it is necessary that our\noperator commutes with the action of SO( n)from both sides . For the operator T∗\ncolUρTcol\nabove, one can show that it commutes with multiplication fro m the left, i.e. with the\noperatorsLU, but unfortunately, it does not commute with multiplicatio n from the right.\n58To overcome this, we obtain commutation with the action of SO (n) from the right with an\naveraging trick, which is an analogue of the famous Weyl unit ary trick.\nDefinition 8.7. We setTρ=EV∼SO(n)[R∗\nVT∗\ncolUρTcolRV].\nThe following result asserts that T ρis hypercontractive, and it also gives lower bounds\non its eigenvalues (which are required for deducing our leve ldinequalities).\nTheorem 8.8. For eachρ∈(0,1), the operator Tρis self adjoint on L2(√nSO(n),µ)and\nhas the following properties:\n1.Tρcommutes with both left and right multiplication by matrice s fromSO(n).\n2. Ifρ/lessorequalslant1√q−1, andf∈L2(√nSO(n),µ), then/⌊ar⌈⌊lTρf/⌊ar⌈⌊lLq(µ)/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\n3. There exist absolute constants δ>0,C >1, such that if d/lessorequalslantδn1/2andf∈Vd, then\n/⌊ar⌈⌊lTρf/⌊ar⌈⌊lL2(µ)/greaterorequalslantC−dρd/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\nLet use show how Theorem 8.8 immediately implies that SO( n) and Spin( n) are good.\nTheorem 8.9. There exist absolute constants c,C >0such that the n-graded groups SO(n)\nandSpin(n)are(C,c)-good.\nProof, given Theorem 8.8. The strong quasirandomness of Spin( n) and SO(n) was already\nestablished in Section 6. It remains to show that they are ( cn1/2,C)-hypercontractive for\nabsolute constants C,c>0. LetCbe the constant in the statement of Theorem 8.8 and let\nc=δwhereδis the constant in the statement of Theorem 8.8. By Lemma 3.24 , it suffices\nto show that SO( n) is (cn1/2,C)-hypercontractive. Let T=T1/(C√q), cn1/2be the Beckner\noperator. By the monotonicity (in δ) of/⌊ar⌈⌊lTδ,r/⌊ar⌈⌊l2→q, it suffices to show that /⌊ar⌈⌊lT/⌊ar⌈⌊l2→q/lessorequalslant1.\nLet us show (using Theorem 8.8) that we may write T=T1/√q−1◦S, where/⌊ar⌈⌊lS/⌊ar⌈⌊l2→2/lessorequalslant1.\nIndeed, asT1/√q−1is self-adjoint and commutes with the action of SO( n) from both sides, it\nhas the Peter-Weyl ideals Wρas its eigenspaces. By Part 3 of Theorem 8.8, the eigenvalues\nofT1/√q−1corresponding to a representations ρof leveldare at least ( C√q)−d, and thus\nare at least the corresponding eigenvalue of TonWρ. This shows that the desired operator\nSexists. We may now apply Part 2 of Theorem 8.8 to obtain:\n/⌊ar⌈⌊lT/⌊ar⌈⌊l2→q/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoubleT1√q−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2→q/⌊ar⌈⌊lS/⌊ar⌈⌊l2→2/lessorequalslant1.\nThe noise operator Tρcommutes with the action of SO (n)from both sides\nIn this section, we establish Part 1 of Theorem 8.8. We phrase it as a lemma.\nLemma 8.10. The operator Tρcommutes with the action of SO(n)from both sides.\nProof.We show the commuting from the left and the right separately.\n59Commuting from the left. For this, it suffices to show that for each U,V∈SO(n), the\noperatorsLUandR∗\nVT∗\ncolTρTcolRVcommute. It is easy to see that LUandRVcommute.\nHence, it suffices to show that LUand T∗\ncolTρTcolcommute.\nFirst note that if X∈Rn×nis a matrix, it holds that GScol(UX) =UGScol(X). It\nfollows that\nLUT∗\ncolf(X) = T∗\ncolf(UX) = E\nY:GScol(Y)=UX[f(Y)] = E\nZ:GScol(Z)=X[f(UZ)] = T∗\ncolLUf(X),\nsoLUcommutes with T∗\ncol. The adjointness immediately implies that LUalso commutes\nwith the operator T col.It is easy to see directly that LUcommutes with the operator Uρ,\nand so it commutes with the composition T∗\ncolTρTcol.\nCommuting from the right. FixV′∈SO(n), then\nRV′Tρ=E\nV∼SO(n)[RV′R∗\nVT∗\ncolTρTcolRV] =E\nV∼SO(n)[R∗\nVV′tT∗\ncolTρTcolRV]\nMaking the change of variables V←VV′t, we obtain\nRV′Tρ=E\nV∼SO(n)[R∗\nVT∗\ncolTρTcolRVV′] =E\nV∼SO(n)[R∗\nVT∗\ncolTρTcolRV]RV′= TρRV′.\n8.3 The noise operator Tρis hypercontractive\nIn this section, we prove part 2 of Theorem 8.8. To do so, we firs t adopt a different point of\nview of the couplings defined by T rowand Tcolthat will often be easier for us to work with.\nThe ‘Gaussian maker distribution’\nRather than going from Y∼γtoX∼µby applying the Gram–Schmidt process on its\ncolumns and dilating by√n(and flipping the sign of the last column if necessary), we can\ngo the other way and construct YfromX. This is accomplished as follows. We define a pair\nof independent random variables ( X,G) such that XGis distributed according to γand\nX∼µ. We call the distribution of GtheGaussian maker distribution and we abbreviate\nit to GMD.\nDefinition 8.11. We define the Gaussian maker distribution to be the distribution of the\nupper-triangular matrix G= (gij)constructed as follows. First, independently choose one-\ndimensional Gaussians gij∼N(0,1\nn)of expectation zero and variance 1/n, for eachi0such\nthat:\n1. IfHSis column comfortable (i.e. the jkare distinct), then T∗\ncolHS=λSHS.\n2. IfHSis row comfortable, then T∗\nrowHSt=λSHSt.\n3.C−d/lessorequalslantλS/lessorequalslant1for allS.\n4. IfSis supported on [n]×[d], then|λS−1|=O(d2/n).\nProof.It suffices to prove the first, third and fourth items, as the sec ond follows from the\nfirst by taking transposes. By Lemma 8.12 we have (T∗\ncolHS)(X) =EG∼GMD[HS(XG)].\nUsing the fact that the entries of XGcorresponding to different columns are independent\nand that each jkappears at most once, we obtain\n(T∗\ncolHS)(X) =E\nG∼GMD[HS(XG)]\n=E\nG∼GMD/bracketleftBiggd/productdisplay\nk=1(XG)ik,jk/bracketrightBigg\n=d/productdisplay\nk=1E\nG∼GMD[(XG)ik,jk]\n=d/productdisplay\nk=1/parenleftbigg\nXE\nG∼GMD[G]/parenrightbigg\nik,jk.\nObserve that EG∼GMD[G] is a diagonal matrix with ( E[G])j,jbeing equal to 1 /√ntimes\nthe expectation of the length (= Euclidean norm) of an ( n−j+1)-dimensional standard\nGaussian random vector, for each j 0andδ>0such that the following holds\nfor alld/lessorequalslantδn1/2. For all comfortable d-juntasf:Rn×n→R, we have\nE\nV∼SO(n)/bracketleftbigg/vextenddouble/vextenddouble/vextenddouble(TcolRVf)/lessorequalslantd/vextenddouble/vextenddouble/vextenddouble2\nL2(γ)/bracketrightbigg\n/greaterorequalslantC−d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nWe now show how to complete the proof of Theorem 8.8, assuming Lemma 9.9.\nProof of Theorem 8.8 (assuming Lemma 9.9). Lemmas8.10and8.14givethefirsttwoitems,\nso it remains to prove the third item. Namely, letting f∈Vdwe want to show that\n/⌊ar⌈⌊lTρf/⌊ar⌈⌊lL2(µ)/greaterorequalslant(cρ)d/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ). By Claim 9.1 we may assume that fis a comfortable d-junta.\nBy Cauchy–Schwarz, it is enough to show that /a\\}⌊ra⌋k⌉tl⌉{tTρf,f/a\\}⌊ra⌋k⌉tri}htL2(µ)/greaterorequalslant(cρ)d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\n2, and we next show\nthat the last assertion follows by Lemma 9.9.\nUsing the fact that Uρ=U∗√ρU√ρ, we have\n/a\\}⌊ra⌋k⌉tl⌉{tTρf,f/a\\}⌊ra⌋k⌉tri}htL2(µ)=E\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊lU√ρTcolRVf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n/greaterorequalslantE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊l(U√ρTcolRVf)/lessorequalslantd/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n/greaterorequalslantρdE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊l(TcolRVf)/lessorequalslantd/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n/greaterorequalslantρdC−d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ),\nwhere we used Lemma 9.9 and the fact that for each function gof degree /lessorequalslantdit holds that\n/⌊ar⌈⌊lU√ρg/⌊ar⌈⌊lL2(γ)/greaterorequalslantρd/2/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(γ).\n10 Low-degree truncations of Tcol: Proof of Lemma 9.9.\nIn this section, we prove Lemma 9.9. Our main idea is to show th at one can approximate\ntheL2-norms of (row) comfortable d-juntas with respect to (√nSO(n),µ) by theirL2-norms\nwith respect to ( Rn×n,γ).\n10.1 Comparing L2(µ)andL2(γ).\nThe following lemmas assert that the 2-norm in L2(√nSO(n)) of a row comfortable d-junta\nis roughly bounded by its 2-norm in Gaussian space. We defer t he proofs of the lemmas to\nSections 10.3 and 10.4.\n66Lemma 10.1. For allε >0, there exists δ >0such that if d/lessorequalslantδn1/2, then for any\nfunctionf:Rn×n→Rthat is either (i) a column comfortable, d-column-junta or (ii) a row\ncomfortable, d-row-junta, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/lessorequalslant(1+ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nIn the case where the function fis comfortable, we show that in fact the same inequality\nwith the measures swapped.\nLemma 10.2. For allε >0, there exists δ >0such that if d/lessorequalslantδn1/2andf:Rn×n→R\nis a comfortable d-junta, then\n/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)/lessorequalslant(1+ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\n10.2 The main argument for proving Lemma 9.9.\nWe now give the proof of Lemma 9.9, assuming Lemmas 10.1 and 10 .2.\nSwapping TrowandTcol.\nOur first step is to show that on the left-hand side of Lemma 9.9 , we can replace T colby\nTrow. The benefit of this exchange is that T rowandRVcommute.\nLemma 10.3. There exist absolute constants C >0,δ >0such that the following holds.\nLetd<δnand letf:Rn×n→Rbe a comfortable d-junta. Then\nE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊l(TcolRVf)/lessorequalslantd/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n/greaterorequalslantC−dE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊lΠcomf,dTrowRVf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n.\nProof.Applying Claim 9.7, we have\n/⌊ar⌈⌊l(TcolRVf)/lessorequalslantd/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant/summationdisplay\nHScomfortable of degree d/a\\}⌊ra⌋k⌉tl⌉{tTcolRVf,HS/a\\}⌊ra⌋k⌉tri}ht2\nL2(γ)\n=/summationdisplay\nHScomfortable of degree d/a\\}⌊ra⌋k⌉tl⌉{tRVf,T∗\ncolHS/a\\}⌊ra⌋k⌉tri}ht2\nL2(µ)\n=λ2\nS/summationdisplay\nHScomfortable of degree d/a\\}⌊ra⌋k⌉tl⌉{tRVf,HS/a\\}⌊ra⌋k⌉tri}ht2\nL2(µ)\n=λ2\nS\nλ2\nSt/summationdisplay\nHScomfortable of degree d/a\\}⌊ra⌋k⌉tl⌉{tRVf,T∗\nrowHS/a\\}⌊ra⌋k⌉tri}ht2\nL2(µ)\n/greaterorequalslantC−d/summationdisplay\nHScomfortable of degree d/a\\}⌊ra⌋k⌉tl⌉{tTrowRVf,HS/a\\}⌊ra⌋k⌉tri}ht2\nL2(γ).\nThe lemma follows by plugging in the definition of Π comf,d.\n67Trowfis close to f.\nWe have now reduced our task to understanding the average of t he square of the 2-norm\nof Πcomf,dTrowRVf= Πcomf,dRVTrowf(this equality holds because T rowandRVcommute).\nThe following claim further simplifies our task and shows tha t Trowfis close tof, thereby\neffectively reducing our task to estimating the 2-norm of Π comf,dRVf. (Though some care is\nrequired to make this precise, as we are applying a projectio n operator on top, which may\ndecrease norms considerably.)\nClaim 10.4. For anyε>0, there exists δ>0such that the following holds. Let d/lessorequalslantδn1/2,\nand letf:Rn×n→Rbe a comfortable d-junta. Then\n/⌊ar⌈⌊lTrowf−f/⌊ar⌈⌊l2\nL2(γ)/lessorequalslantε/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ).\nProof.Letgbe a comfortable d-junta satisfying T∗\nrowg=f, i.e. writing f=/summationtext\nSαSHS, we\ntakeg=/summationtextλ−1\nStαSHS, whereλSis as in Claim 9.7. Then by Parseval, we have\n/⌊ar⌈⌊lg−f/⌊ar⌈⌊lL2(γ)/lessorequalslantε/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ),\nprovidedδis sufficiently small. Hence, using Cauchy–Schwarz, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)=/a\\}⌊ra⌋k⌉tl⌉{tf,T∗\nrowg/a\\}⌊ra⌋k⌉tri}ht=/a\\}⌊ra⌋k⌉tl⌉{tTrowf,g/a\\}⌊ra⌋k⌉tri}ht=/a\\}⌊ra⌋k⌉tl⌉{tTrowf,f/a\\}⌊ra⌋k⌉tri}ht+/a\\}⌊ra⌋k⌉tl⌉{tTrowf,g−f/a\\}⌊ra⌋k⌉tri}ht\n/lessorequalslant/a\\}⌊ra⌋k⌉tl⌉{tTrowf,f/a\\}⌊ra⌋k⌉tri}ht+/⌊ar⌈⌊lTrowf/⌊ar⌈⌊lL2(γ)/⌊ar⌈⌊lg−f/⌊ar⌈⌊lL2(γ),\nimplying that\n/a\\}⌊ra⌋k⌉tl⌉{tTrowf,f/a\\}⌊ra⌋k⌉tri}ht/greaterorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)−ε/⌊ar⌈⌊lTrowf/⌊ar⌈⌊lL2(γ)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)=/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)−ε/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nThus, we obtain\n/⌊ar⌈⌊lTrowf−f/⌊ar⌈⌊l2\nL2(γ)=/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/⌊ar⌈⌊lTrowf/⌊ar⌈⌊l2\nL2(γ)−2/a\\}⌊ra⌋k⌉tl⌉{tTrowf,f/a\\}⌊ra⌋k⌉tri}ht\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)−/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)+2ε/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)\n/lessorequalslant3ε/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ),\nusing Lemmas 10.1 and 10.2 (again provided δis sufficiently small depending on ε). This\ncompletes the proof.\nThe projection of RVTrowfonto the subspace of comfortable polynomials\nAt this point, it would seem that to finish the proof of Lemma 9. 9 it suffices to estimate the\ntypical 2-norm of Π comf,dRVf. While this is indeed the case, some care is needed, as one\ncannot switch particularly smoothly from T rowftofin the previous statement. To address\nthis, we must be able to estimate the 2-norm of Π comf,dRVfunder the weaker hypothesis\nthatfis arow-comfortable d-junta. This is the content of the following claim.\n68Claim 10.5. Letd/lessorequalslantn/2and letf:Rn×n→Rbe a row-comfortable d-row-junta. Then\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,dRVf/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\n2dnd((n/2)−d)!/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nProof.We first note that if HαandHβare row-comfortable monomials such that the\nset of rows appearing in αis different from the set of rows appearing in β, then we have\n/a\\}⌊ra⌋k⌉tl⌉{tHα,Hβ/a\\}⌊ra⌋k⌉tri}htL2(γ)=/a\\}⌊ra⌋k⌉tl⌉{tHα,Hβ/a\\}⌊ra⌋k⌉tri}htL2(µ)= 0. Indeed,thefirstequalityisimmediatefromthepairwise\northogonality of the Hermite polynomials with respect to L2(γ), and the second equality\nfollows from the fact that, if Hαcontains a variable from the ith row and Hβdoes not, then\nflipping the sign of both the ith row and the ( ⌊n/2⌋+ 1)th row of a matrix X∈SO(n)\nyields a (Haar-)measure-preserving map on SO( n) that sends Hα(X)Hβ(X) to its negative,\nand therefore/a\\}⌊ra⌋k⌉tl⌉{tHα,Hβ/a\\}⌊ra⌋k⌉tri}htL2(µ)=EX∼µ[Hα(X)Hβ(X)] = 0.\nForS⊂[n], letWSdenote the linear span of the row-comfortable homogeneous m ono-\nmials of degree|S|which depend only upon variables from the rows in S(and therefore\ndepend upon precisely one variable from each of the rows in S). It is clear that the WS\nare pairwise orthogonal with respect to L2(γ), and by the above observation they are also\npairwise orthogonal with respect to L2(µ).\nBy the above remarks, for any row-comfortable d-row junta g:Rn×n→R, we may\nwritegas an orthogonal direct sum,\ng=/summationdisplay\nS⊂[d]g(=S),\nwhereg(=S)denotes the orthogonal projection of gontoWS(with respect to L2(γ)). It is\nclear that (RVg)(=S)=RV(g(=S)) for anyV∈SO(n) and any row-comfortable d-row-junta\ng.\nNow letf:Rn×n→Rbe a row-comfortable d-row-junta. Since the f(=S)are pairwise\northogonal with respect to L2(µ) as well as with respect to L2(γ), we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)=/summationdisplay\nS⊂[d]/⌊ar⌈⌊lf(=S)/⌊ar⌈⌊l2\nL2(µ),\nand therefore by averaging, there exists S⊂[d] such that\n/⌊ar⌈⌊lf(=S)/⌊ar⌈⌊l2\nL2(µ)/greaterorequalslant1\n2d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ). (11)\nFor brevity, write h=f(=S),d′:=|S|andS={i1,...,id′}. Our next aim is to show\nthat\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,=SRVh/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\nnd′((n/2)−d′)!/⌊ar⌈⌊lh/⌊ar⌈⌊l2\nL2(µ). (12)\nInproving(15), wemay andshall assume(without loss of gene rality) that S={1,2,...,d′}.\nWe first assert that, in this case,\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,=SRVh/⌊ar⌈⌊l2\nL2(γ)=(n/2)!\n((n/2)−d′)!EV/a\\}⌊ra⌋k⌉tl⌉{tRVh,H{(1,1),...,(d′,d′)}/a\\}⌊ra⌋k⌉tri}ht2\nL2(γ).\n69Indeed, this follows from the fact that we may write Vas the product of a random SO( n)\nmatrixV′and a random permutation matrix Pσ,V=PσV′say. Now the Sn-orbit{RPσm:\nσ∈Sn}of a monomial m=m(X) of the form/producttextd′\ni=1Xi,viwith theviall distinct, consists\nof all the monomials of the form/producttextd′\ni=1Xi,wi, with thewiall distinct.\nNowwriteh=/summationtext\nα=((1,i1),...,(d′,id′))ˆh(α)Hα. Thenforeach αoftheform{(1,j1),...,(d′,jd′)}\n(forj1,...,jd′all distinct), we have\nRVHα(X) =Hα(XV) =d/productdisplay\nk=1[XV]k,jk=/summationdisplay\nr1,...,rd′d′/productdisplay\nk=1Xk,rkVrk,jk.\nTherefore,\n/a\\}⌊ra⌋k⌉tl⌉{tRVHα(X),H{(1,1),...,(d′,d′)}/a\\}⌊ra⌋k⌉tri}htL2(γ)=Hα(V) =n−d′/2Hα(√nV).\nExpanding h=/summationtext\nαˆh(α)Hαand taking 2-norms with respect to√nV∼µ, we see that\nindeed,\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,=SRVh/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\nnd′((n/2)−d′)!/⌊ar⌈⌊lh/⌊ar⌈⌊l2\nL2(µ),\nproving (15).\nCombining this with (11) and using the fact that d′/lessorequalslantd, yields\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,=SRVf(=S)/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\n2dnd((n/2)−d)!/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nFinally, since\nΠcomf,dRVf= Πcomf,dRV\n/summationdisplay\nS⊂[d]f(=S)\n\n= Πcomf,d\n/summationdisplay\nS⊂[d](RVf)(=S)\n\n=/summationdisplay\nS⊂[d]Πcomf,=S/parenleftbig\n(RVf)(=S)/parenrightbig\n,\nand since the last sum is an orthogonal direct sum (again usin g the pairwise orthogonality\nof theWS’s with respect to L2(γ)), we have\nEV∼SO(n)/⌊ar⌈⌊lΠcomf,dRVf/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslantEV∼SO(n)/⌊ar⌈⌊lΠcomf,=S/parenleftbig\n(RVf)(=S)/parenrightbig\n/⌊ar⌈⌊l2\nL2(γ)\n=EV∼SO(n)/⌊ar⌈⌊lΠcomf,=SRV(f(=S))/⌊ar⌈⌊l2\nL2(γ)\n/greaterorequalslant(n/2)!\n2dnd((n/2)−d)!/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ),\nproving the claim.\n70We now show how to lower-bound the right-hand side of the expr ession in the statement\nof Lemma 10.3.\nLemma 10.6. There exist δ >0andC >0such that the following holds. Let d/lessorequalslantδn1/2.\nThen for all comfortable d-juntasf:Rn×n→R, we have\nE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊lΠcomf,dRVTrowf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n/greaterorequalslantC−d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nProof.We write Π comf,d= Πcomf,dΠcomf,row,d, and setg= Πcomf,row,dTrowf. Note that gis\na row-comfortable d-row-junta. We have\nΠcomf,dRVTrowf= Πcomf,dRVg,\nwhere we used the fact that Π comf,row,dcommutes with RV. Taking the squares of the\n2-norms and expectations over Vwe may apply Claim 10.5 to g, obtaining\nE\nV/bracketleftBig\n/⌊ar⌈⌊lΠcomf,dTrowRVf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n=(n/2)!\nnd((n/2)−d)!/⌊ar⌈⌊lg/⌊ar⌈⌊l2\nL2(µ)/greaterorequalslantC−d/⌊ar⌈⌊lg/⌊ar⌈⌊l2\nL2(µ).(13)\nBy Lemma 10.1, as g−fis a row comfortable, d-row-junta, we have\n/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(µ)/greaterorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)−/⌊ar⌈⌊lg−f/⌊ar⌈⌊lL2(µ)/greaterorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)−2/⌊ar⌈⌊lg−f/⌊ar⌈⌊lL2(γ)\nAsfis comfortable, we can apply Claim 10.4 to it to obtain\n/⌊ar⌈⌊lg−f/⌊ar⌈⌊lL2(γ)=/⌊ar⌈⌊lΠcomf,row,d(Trowf−f)/⌊ar⌈⌊lL2(γ)/lessorequalslant/⌊ar⌈⌊lTrowf−f/⌊ar⌈⌊lL2(γ)/lessorequalslantε/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nOn the other hand Lemma 10.2 shows that /⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)/lessorequalslant2/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ). Putting these three facts\ntogether, we obtain /⌊ar⌈⌊lg/⌊ar⌈⌊lL2(µ)>(1−2ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ). Plugging this into (13), and adjusting the\nvalue ofC, completes the proof.\nFinishing the proof of Lemma 9.9.\nUsing Lemma 10.3, the left-hand side of Lemma 9.9 is at least\nC−dE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊lΠcomf,dTrowRVf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n=C−dE\nV∼SO(n)/bracketleftBig\n/⌊ar⌈⌊lΠcomf,dRVTrowf/⌊ar⌈⌊l2\nL2(γ)/bracketrightBig\n,\nand using Lemma 10.6 the last quantity is at least C′−d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ), as required.\n10.3L2(µ)is dominated by L2(γ)on column or row comfortable juntas:\nProof of Lemma 10.1.\nOur remaining tasks are to prove Lemmas 10.1 and 10.2. First, we show that the operator\nT∗\ncolis close to the identity on column comfortable d-column-juntas.\n71Claim 10.7. For allε >0there exists δ >0such that if d/lessorequalslantδn1/2andfis a column\ncomfortable d-column-junta, then\n/⌊ar⌈⌊lT∗\ncolf−f/⌊ar⌈⌊lL2(γ)/lessorequalslantε/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nProof.The lemma follows immediately from Parseval and Claim 9.7 as |λS−1|<εfor each\nSas above, provided δis sufficiently small (similarly to in the proof of Claim 10.4) .\nClaim 10.7 is particularly useful as it implies that T∗\ncolis invertible on the space spanned\nbycolumn-comfortable d-column-juntas, andthusgivesusanaturalway ofgoingfrom L2(µ)\ntoL2(γ). We are now ready to prove Lemma 10.1, restated below.\nLemma 10.1 (Restated). For allε >0, there exists δ >0such that if d/lessorequalslantδn1/2, then\nfor any function f:Rn×n→Rthat is either (i) a column comfortable, d-column-junta or\n(ii) a row comfortable, d-row-junta, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/lessorequalslant(1+ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nProof.Letδ=δ(ε)>0 be as in the statement of Claim 10.7. Without loss of general ity,\nwe assume that fis a column comfortable d-column-junta, i.e., is contained in the subspace\nW:= Span{HS:HSis column comfortable and Sis supported on [ n]×[d]},\notherwise we may consider f′(X) =f(Xt). Letg:Rn×n→Rbe a function such that\nT∗\ncolg=f.Then as T∗\ncolis a contraction, we have /⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/lessorequalslant/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(γ).In order to complete\nthe proof, we note (using Claim 10.7) that T∗\ncol(W) =W, T∗\ncolis invertible on the subspace\nW, and that\n/⌊ar⌈⌊l(T∗\ncol|W)−1/⌊ar⌈⌊lL2(γ)→L2(γ)/lessorequalslant1+2ε. (14)\nIndeed, the facts that T∗\ncol(W) =Wand that T∗\ncolis invertible on the subspace W, follow\nimmediately fromClaim9.7. Moreover, if(14)fails, thenth ereexistsh∈Wwith/⌊ar⌈⌊lh/⌊ar⌈⌊lL2(γ)<\n1/(1 +2ε) and/vextenddouble/vextenddouble(T∗\ncol|W)−1h/vextenddouble/vextenddouble\nL2(γ)= 1, and then for h′:= (T∗\ncol|W)−1h(which lies in W),\nwe obtain\n/vextenddouble/vextenddoubleT∗\ncolh′−h′/vextenddouble/vextenddouble\nL2(γ)=/vextenddouble/vextenddoubleh−(T∗\ncol|W)−1h/vextenddouble/vextenddouble\nL2(γ)\n/greaterorequalslant/vextenddouble/vextenddouble(T∗\ncol|W)−1h/vextenddouble/vextenddouble\nL2(γ)−/⌊ar⌈⌊lh/⌊ar⌈⌊lL2(γ)\n>1−1/(1+2ε)\n/greaterorequalslantε\n=ε/⌊ar⌈⌊lh′/⌊ar⌈⌊lL2(γ)\nprovidedε/lessorequalslant1/2, contradicting Claim 10.7.\nWe haveg= (T∗\ncol|W)−1f, and so\n/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ)/lessorequalslant/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(γ)=/vextenddouble/vextenddouble(T∗\ncol|W)−1f/vextenddouble/vextenddouble\nL2(γ)/lessorequalslant(1+2ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ),\ncompleting the proof.\n7210.4L2(γ)is dominated by L2(µ)on comfortable juntas: Proof of Lemma\n10.2.\nTo prove Lemma 10.2, we define an auxiliary distribution on Rn×n, which we refer to as the\n‘over-Gaussian’ distribution.\nDefinition 10.8. LetG∼GMD, and choose Y∼γindependently. We define the distri-\nbutionνto be the distribution of YG, and call it the over-Gaussian distribution .\nWe refer to νby this name since it can be produced taking X∼µ(i.e., at random\naccording to the Haar measure on SO( n)), and then multiplying Xby two independent\ncopies of GMD, thereby ‘overshooting’ the Gaussian distrib ution.\nIn the following section, we show that the distribution νis close toγin the sense that\nthe expectation of a certain kind of test function is roughly the same under both measures.\nIn fact, the relevant test functions are the squares of the co mfortabled-juntas, and their\nexpectations areroughlythesameeven ifweallow thedegree dtobeas largeasΘ(√n). The\nfollowing lemma asserts that if fis a comfortable d-junta, then its over-Gaussian 2-norm\ncannot be much larger than its Gaussian 2-norm.\nLemma 10.9. For allε>0there exists δ>0suchthat if d/lessorequalslantδn1/2, then for all comfortable\nd-juntasf:Rn×n→R, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(ν)/lessorequalslant(1+ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ).\nTo prove Lemma 10.9, we need some more notation, and two techn ical claims. For a\npermutation σ∈Sd, we writexσ:=x1,σ(1)···xd,σ(d); this is a function on Rd×d. Fix a com-\nfortabled-juntaf:Rd×d→Randwritef=/summationtext\nI=(i1,...,id)aIxIwherexI:=x1,i1x2,i2···xd,id,\nthe sum ranging over all Isuch thati1,...,id∈[d] are distinct. (As usual, we write ( S)d\nfor the set of ordered d-tuples of distinct elements of the set S; using this notation we may\nwriteI∈([d])d.) We have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)=/summationdisplay\nIa2\nI/⌊ar⌈⌊lxI/⌊ar⌈⌊l2\nL2(ν)+/summationdisplay\nI/n⌉}ationslash=JaIaJ/a\\}⌊ra⌋k⌉tl⌉{txI,xJ/a\\}⌊ra⌋k⌉tri}htν.\nWe now need two technical claims, which handle respectively the diagonal terms and the\noff-diagonal terms in the above expansion of /⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν). To handle the diagonal terms, we\nwill use the following claim.\nClaim 10.10. Ifi1,...,id∈[n]are distinct, then xI:=x1,i1x2,i2···xd,idsatisfies\n/⌊ar⌈⌊lxI/⌊ar⌈⌊l2\nL2(ν)= 1.\nForI,J∈[n]dwe letd(I,J) :=|{r|ir/\\⌉}atio\\slash=jr}|denote the Hamming distance from Ito\nJ. To handle the off-diagonal terms, we need the following claim .\nClaim 10.11. For anyI,Jsuch thatd(I,J) =ℓ, we have|/a\\}⌊ra⌋k⌉tl⌉{txI,xJ/a\\}⌊ra⌋k⌉tri}ht|L2(ν)/lessorequalslantεℓ, where\nεℓ:= 2ℓ+4n−ℓ/22dℓ/√n.\n73Claims 10.10 and 10.11 imply Lemma 10.9\nWe first show how to deduce Lemma 10.9 from Claims 10.10 and 10. 11. Writing f=/summationtext\nIaIxI, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)=/summationdisplay\nIa2\nI/⌊ar⌈⌊lxI/⌊ar⌈⌊l2\nL2(ν)+/summationdisplay\nI/n⌉}ationslash=JaIaJ/a\\}⌊ra⌋k⌉tl⌉{txI,xJ/a\\}⌊ra⌋k⌉tri}htν\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/summationdisplay\nI/n⌉}ationslash=Ja2\nI+a2\nJ\n2|/a\\}⌊ra⌋k⌉tl⌉{txI,xJ/a\\}⌊ra⌋k⌉tri}htν|\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+d/summationdisplay\nℓ=1/summationdisplay\nIa2\nI|{J:d(J,I) =ℓ}|·εℓ\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)d/summationdisplay\nℓ=1εℓ/parenleftbiggd\nℓ/parenrightbigg\nℓ!\n=/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)/parenleftBigg\n1+d/summationdisplay\nℓ=1εℓdℓ/parenrightBigg\n.\nUsing the upper bound on εℓ, we obtain\nd/summationdisplay\nℓ=1εℓdℓ/lessorequalslantd/summationdisplay\nℓ=12ℓ+4n−ℓ/22dℓ/√ndℓ/lessorequalslant16∞/summationdisplay\nℓ=1/parenleftBigg\n2d·2d/√n\n√n/parenrightBiggℓ\n/lessorequalslantε\n2,\nwhere we used d/lessorequalslantδn1/2and the fact that δis sufficiently small compared to ε.\nProof of Claims 10.10 and 10.11.\nTo prove the two claims, we need the following simple fact abo ut the Gaussian maker\ndistribution.\nClaim 10.12. LetI= (i1,...,id)∈[n]dandJ= (j1,...,jd)∈[n]dbe such that d(I,J) =ℓ,\nand such that in the product\nGi1j1Gi2j2···Gidjd,\nno matrix entry of Gappears more than twice. Then\n|EG∼GMD[Gi1j1Gi2j2···Gjdid]|/lessorequalslant/parenleftbigg1\nn/parenrightbiggℓ/2\n.\nProof.If, in the product\nGi1j1Gi2j2···Gidjd,\nsome off-diagonal matrix entry of Gappears exactly once, then the expectation of the\nproduct is zero. We may therefore assume that every off-diagon al matrix entry of Gappears\n74either exactly twice, or not at all, in the above product. If t here are exactly ℓvalues ofr\nsuch thatir/\\⌉}atio\\slash=jr, then the above expectation factorises into a product of the expectations\nof the squares of ℓ/2 off-diagonal and of the squares of ( d−ℓ)/2 diagonal entries:\n/productdisplay\nk∈DE[G2\nk,k]/productdisplay\n(i,j)∈EE[G2\ni,j],\nwhereE⊂[n]2\\{(k,k) :k∈[n]},|D|= (d−ℓ)/2 and|E|=ℓ/2. We have E[G2\ni,j] = 1/n\nfor all (i,j)∈EandE[G2\nk,k] = (n−k+1)/n/lessorequalslant1 for allk∈D, proving the claim.\nWe are now ready to prove Claim 10.10.\nProof of Claim 10.10. LetxI=x1,i1x2,i2···xd,id, wherei1,...,id∈[n] are distinct. We\nhave\n/⌊ar⌈⌊lxI/⌊ar⌈⌊l2\nL2(ν)=EG∼GMDEY∼γ[(YG)2\n1,i1(YG)2\n2,i2···(YG)2\nd,id].\nSince for each h∈[d], (YG)h,ih=/summationtextih\nk=1Yh,kGk,ihinvolves only entries of Yin rowhand\nentries ofGin columnih(and theihare distinct), the random variables (( YG)2\nh,ih:h∈[d])\nform a system of independent random variables, and therefor e\n/⌊ar⌈⌊lxI/⌊ar⌈⌊l2\nL2(ν)=EG∼GMDEY∼γ[(YG)2\n1,i1]EG∼GMDEY∼γ[(YG)2\n2,i2]···EG∼GMDEY∼γ[(YG)2\nd,id].\nFor eachh∈[d], we have\nEG∼GMDEY∼γ[(YG)2\nh,ih] =EG∼GMDEY∼γ\n/parenleftBiggih/summationdisplay\nk=1Yh,kGk,ih/parenrightBigg2\n\n= 2/summationdisplay\n1/lessorequalslantk0, there exists δ >0such that if d/lessorequalslantδn1/2and\nf:Rn×n→Ris a comfortable d-junta, then/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)/lessorequalslant(1+ε)/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(µ).\nProof.We may assume that ε/lessorequalslant1/2. By the triangle inequality, it suffices to show that\n/⌊ar⌈⌊lf(X)−f(XG)/⌊ar⌈⌊lL2(µ,GMD)<ε\n2/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ), whereX∼µandG∼GMD. Since for any fixed,\nupper triangular G∈Rn×n, the function gGdefined by\ngG:X/ma√sto→f(X)−f(XG)\nis a row-comfortable d-row-junta, by Lemma 10.1 we have\n/⌊ar⌈⌊lf(X)−f(XG)/⌊ar⌈⌊l2\nL2(µ,GMD)=EG∼GMDEX∼µ[(f(X)−f(XG))2]\n=EG∼GMDEX∼µ[gG(X)2]\n=EG∼GMD[/⌊ar⌈⌊lgG/⌊ar⌈⌊l2\nL2(µ)]\n/lessorequalslant(1+ε)2EG∼GMD/⌊ar⌈⌊lgG/⌊ar⌈⌊l2\nL2(γ)\n= (1+ε)2EG∼GMDEY∼γ[(f(Y)−f(YG))2]\n= (1+ε)2/⌊ar⌈⌊lf(Y)−f(YG)/⌊ar⌈⌊l2\nL2(γ,GMD),\nand therefore it suffices to show that\n/⌊ar⌈⌊lf(Y)−f(YG)/⌊ar⌈⌊lL2(γ,GMD)<ε\n4/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ),\n77whereY∼γandG∼GMD is independent of Y. Expanding, we note that\n/⌊ar⌈⌊lf(Y)−f(YG)/⌊ar⌈⌊l2\nL2(γ,GMD)=/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)−2/a\\}⌊ra⌋k⌉tl⌉{tf,T∗\ncolf/a\\}⌊ra⌋k⌉tri}htL2(γ).\nTo handle the cross term, we note that by Cauchy-Schwarz and C laim 10.7, we have\n/vextendsingle/vextendsingle/a\\}⌊ra⌋k⌉tl⌉{tf,T∗\ncolf−f/a\\}⌊ra⌋k⌉tri}htL2(γ)/vextendsingle/vextendsingle/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊lL2(γ)/⌊ar⌈⌊lT∗\ncolf−f/⌊ar⌈⌊lL2(γ)/lessorequalslantε\n10/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ),\nso/a\\}⌊ra⌋k⌉tl⌉{tf,T∗\ncolf/a\\}⌊ra⌋k⌉tri}htL2(γ)/greaterorequalslant(1−ε/10)/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ). Using Lemma 10.9, we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)/lessorequalslant(1+ε/20)/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)\nprovidedδis sufficiently small, and so\n/⌊ar⌈⌊lf(Y)−f(YG)/⌊ar⌈⌊l2\nL2(γ,GMD)/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)−2(1−ε/10)/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)/lessorequalslant(ε/4)/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ),\ncompleting the proof.\nReferences\n[1] G. W. Anderson, A. Guionnet, and O. Zeitouni. An Introduction to Random Matrices .\nCambridge University Press, Cambridge, 2010.\n[2] S. Aruanachalam, U. Girish, and N. Lifshitz. One clean qu bit suffices for quantum\ncommunication advantage. Preprint. arXiv:2310.02406, 20 23.\n[3] L. Babai and V.T. S´ os. Sidon sets in groups and induced su bgraphs of Cayley graphs.\nEurop. J. Combin. , 6:101–114, 1985.\n[4] D.BakryandM. ´Emery. Diffusionshypercontractives. InJ.Az´ emaandM.Yor, editors,\nS´ eminaire de Probabilit´ es XIX 1983/84 , pages 177–206. Springer, Berlin, Heidelberg,\n1985.\n[5] W. Beckner. Inequalities in Fourier Analysis. Ann. Math. , 102:159–182, 1975.\n[6] I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivit y of Boolean functions and\napplications to percolation. Publ. Math. Inst. Hautes Etudes Sci. , 90:5–43, 1999.\n[7] M. Berti and C. Procesi. Nonlinear wave and Schr¨ odinger equations on compact Lie\ngroups and homogeneous spaces. Duke Math. J. , 159:479–538, 2011.\n[8] E. Breuillard, B. Green, and T. Tao. Approximate subgrou ps of linear groups. Geom.\nFunct. Anal. , 21:774–819, 2011.\n[9] J. B. H. Conway. A Course in Functional Analysis . Springer, New York, secondedition,\n1985.\n78[10] I. Dinur and S. Safra. On the hardness of approximating m inimum vertex cover. Ann.\nMath., 162:439–485, 2005.\n[11] Y.Filmus, G.Kindler,N.Lifshitz, andD.Minzer. Hyper contractivity onthesymmetric\ngroup.Forum Math. Sigma. , 12:e6, 2024.\n[12] W. Fulton and J. Harris. Representation Theory: A First Course . Springer, New York,\n2013.\n[13] D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz, and R. De Wol f. Exponential separations\nfor one-way quantum communication complexity, with applic ations to cryptography.\nInProceedings of the 39th Annual ACM Symposium on Theory of Comput ing, pages\n516–525, 2007.\n[14] W. T. Gowers. Quasirandom Groups. Combin. Probab. Comput. , 17:363–387, 2008.\n[15] B. Green and I. Z. Ruzsa. Sum-free sets in Abelian groups .Israel J. Math. , 147:157–\n188, 2005.\n[16] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. , 97:1061–1083, 1975.\n[17] Brian C. Hall. Lie Groups, Lie Algebras and Representations: An Elementary In tro-\nduction. Springer, Cham, 2015.\n[18] H. A. Helfgott. Growth and generation in SL2(Z/pZ).Ann. Math. , 167:601–623, 2008.\n[19] H. A. Helfgott. Growth in groups: ideas and perspective s.Bull. Amer. Math. Soc. ,\n52(3):357–413, 2015.\n[20] H. A. Helfgott and ´A. Seress. On the diameter of permutation groups. Ann. Math. ,\n179:611–658, 2014.\n[21] R. Henstock and A. M. Macbeath. On the Measure of Sum-Set s. (I) the Theorems of\nBrunn, Minkowski and Lusternik. Proc. Lond. Math. Soc. , 3:182–194, 1953.\n[22] J. W. Jenkins. Growth of connected locally compact grou ps.J. Funct. Anal. , 12:113–\n127, 1973.\n[23] T. Jiang. How many entries of a typical orthogonal matri x can be approximated by\nindependent normals? Ann. Probab. , 34:1497–1529, 2006.\n[24] Y. Jing, C.-M. Tran, and R. Zhang. Effective measure growt h for small sets in compact\nLie groups. In preparation.\n[25] Y.Jing, C.-M.Tran, andR.Zhang. Measuredoublingofsm allsetsinSO 3(R). Preprint.\narXiv:2304.09619, 2023.\n[26] Y.Jing, C.-M. Tran, andR.Zhang. Anon-AbelianBrunn-M inkowskiinequality. Geom.\nFunct. Anal. , 33:1048–1100, 2023.\n79[27] J. Kahn, G. Kalai, and N. Linial. The influences of variab les on Boolean functions.\nProceedings of the 29th Annual Symposium on Foundations of Co mputer Science , pages\n68–80, 1988.\n[28] P. Keevash and N. Lifshitz. Sharp hypercontractivity f or symmetric groups and its\napplications. Preprint. arXiv:2307.15030, 2023.\n[29] J. H. B. Kemperman. On products of sets in a locally compa ct group. Fund. Math. ,\n56:51–68, 1964.\n[30] M. Larsen and A. Shalev. Characters of symmetric groups : sharp bounds and appli-\ncations. Invent. Math. , 174:645–687, 2008.\n[31] M. Liebeck and A. Shalev. Diameters of finite simple grou ps: sharp bounds and\napplications. Ann. Math. , 154:383–406, 2001.\n[32] E. Mossel. A quantitative Arrow theorem. Probab. Theory Relat. Fields , 154:49–88,\n2012.\n[33] E. Nelson. Construction of quantum fields from Markoff fie lds.J. Funct. Anal. , 12:97–\n112, 1973.\n[34] N.Nikolov andL.Pyber. Productdecompositionsofquas irandomgroups, andaJordan\ntype theorem. Europ. J. Combin. , 13:1063–1077, 2011.\n[35] C. Procesi. Lie groups: an approach through invariants and representati ons. Springer,\nNew York, 2007.\n[36] L. Pyber and E. Szab´ o. Growth in finite simple groups of L ie type. J. Amer. Math.\nSoc., 29:95–146, 2016.\n[37] P. Raghavendra. Optimal algorithms and inapproximabi lity results for every CSP?\nInProceedings of the 40th Annual ACM Symposium on Theory of Comput ing, pages\n245–254, 2008.\n[38] O. Regev and B. Klartag. Quantum one-way communication can be exponentially\nstronger than classical communication. In Proceedings of the 43rd Annual ACM Sym-\nposium on Theory of Computing , pages 31–40, 2011.\n[39] N. El Samra and R.C. King. Dimensions of irreducible rep resentations of the classical\nLie groups. Journal of Physics A: Mathematical and General , 12:2317, 1979.\n[40] P. Sarnak and X. Xue. Bounds for multiplicities of autom orphic representations. Duke\nMath. J. , 64(1):207–227, 1991.\n[41] T. Tao. Expansion in finite simple groups of Lie type . American Mathematical Society,\nProvidence, Rhode Island, 2015.\n80[42] H.Weyl. TheoriederDarstellungkontinuierlicher hal b-einfacherGruppendurchlineare\ntransformationen. II. Math. Z. , 24:328–376, 1926.\nA Bounding the dimensions of high level representations\nA.1 The special orthogonal group SO (n)\nLemma 6.7 (Restated). Letn/greaterorequalslant5. Ifρis an irreducible representation of SO(n)of level\nd/greaterorequalslantn/2, then\ndim(ρ)/greaterorequalslantexp(n/32).\nProof.This follows from Weyl’s original dimension formulae, toge ther with a short compu-\ntation. First suppose that n= 2k+1 is odd. As mentioned above, the equivalence classes\nof irreducible representations of SO(2 k+1,R) are in an explicit one-to-one correspondence\nwith the partitions λ(of non-negative integers) whose Young diagrams have at mos tkrows.\nWeyl’s dimension formula states that for any such partition λ, the correspondingirreducible\nrepresentation ρλof SO(2k+1,R) has\ndim(ρλ) =/productdisplay\n1/lessorequalslanti n/2, then considering all the terms in the\nabove product corresponding to j=nandi/lessorequalslant⌊n/2⌋+1, we obtain\n/productdisplay\n1/lessorequalslantij,gi,jis 1/√ntimes aQNrandom variable, the\nentries ofGare independent, and independent of all the entries of X. This (distribution\nover) quaternionic upper-triangular matrices Gis our Gaussian Maker Distribution (or\nGMD, for short) in the Sp( n) case. To generalise the SO( n) proof, the only (important)\nfacts we need are the fact that Sp( n) acts transitively (from either the left or the right) on\nthe setS={v∈Hn:/a\\}⌊ra⌋k⌉tl⌉{tv,v/a\\}⌊ra⌋k⌉tri}ht= 1}of quaternionic vectors of unit norm, and that ( Hn×n,γ)\nis invariant under both left and right actions of Sp( n).\nWe can therefore write\nT∗\ncolf(X) =EG∼GMDf(XG)∀X∈√nSp(n).\nSimilarly, we can write\nT∗\nrowf(X) =EG∼GMDf(GTX)∀X∈√nSp(n).\nOur ‘nice’ functions on L2(Hn×n,γ) are polynomials (with complex coefficients) where\neach variable is a real-part, an i-part, aj-part or a k-part of one of the matrix entries. We\nsay amonomial inthesevariables is comfortable if notwo of itsvariables comefromthesame\nrow, notwo of its variables come from thesamecolumn, andall variables come from thefirst\n⌊n/2⌋rows and the first ⌊n/2⌋columns. Similarly, we say it is row comfortable (respectively\ncolumn comfortable if no two of its variables come from the same row (respectivel y column),\nand all of its variables come from the first ⌊n/2⌋rows (respectively columns). A comfortable\npolynomial (respectively row-comfortable polynomial or c olumn-comfortable polynomial)\nis a complex linear combination of comfortable (respective ly row-comfortable or column-\ncomfortable) monomials. Such a polynomial is said to be a d-juntaif it is homogeneous of\ndegreedand depends only upon the top-left dbydminor, it is said to be a d-row-junta if\nit depends only upon the first drows, and it is said to be a d-column-junta if it depends\nonly upon the first dcolumns.\nAs in the SO( n)-case, we define Π comf:L2(Hn×n,γ)→L2(Hn×n,γ) to be orthogonal\nprojection onto the linear subspace of comfortable polynom ials. Similarly, we define the\n88operators Π comf,d, Πcomf,col,dand Πcomf,row,dto be the orthogonal projections onto the space\nof comfortable, row comfortable and column comfortable pol ynomials of degree at most d,\nrespectively. Finally, if S⊂[n] is a set of rows, we define Π comf,=Sto be the projection\nonto the subspace spanned by the comfortable homogeneous mo nomials of degree |S|which\ndepend only upon variables from the rows in S.\nSince the real-part, the i-part, the j-part and the k-part of each matrix entry of X∼\n(Hn×n,γ) isN(0,1/4)-distributed rather than N(0,1)-distributed, to guarantee orthonor-\nmality we must multiply by a factor of 2d, soa ‘generic’ row-comfortable monomial of degree\ndis of the form\n2d(Xi1,j1)q1-part·(Xi2,j2)q2-part·...·(Xid,jd)qd-part,\nwhereqk∈{real,i,j,k}for allk∈[d] andi1,...,idare distinct integers between 1 and n/2;\nfor brevity we denote this by Hαwhereα={(i1,j1;q1),(i2,j2;q2),...,(id,jd;qd)}.\nThe proof (and statement) of Claim 9.7 is readily adapted. If\nS={(i1,j1;q1),(i2,j2;q2),...,(id,jd;qd)}\nis such that the jkare all distinct, then we have\nT∗\ncolf(X) =EG∼GMDHS(XG)\n= 2dEG∼GMD/bracketleftBiggd/productdisplay\nk=1((XG)ik,jk)qk-part/bracketrightBigg\n= 2dEG∼GMD/bracketleftBiggd/productdisplay\nk=1/parenleftBiggjk/summationdisplay\nℓ=1(Xik,ℓGℓ,jk)qk-part/parenrightBigg/bracketrightBigg\n= 2dd/productdisplay\nk=1EG∼GMD/bracketleftBiggjk/summationdisplay\nℓ=1(Xik,ℓGℓ,jk)qk-part/bracketrightBigg\n= 2dd/productdisplay\nk=1/parenleftBiggjk/summationdisplay\nℓ=1(Xik,ℓEG[Gℓ,jk])qk-part/parenrightBigg\n= 2dd/productdisplay\nk=1(Xik,jkEG[Gjk,jk])qk-part\n= 2dd/productdisplay\nk=1(Xik,jk)qk-partEG[Gjk,jk]\n=λSHS(X),\nwhere\nλS:=d/productdisplay\nk=1EG[Gjk,jk].\n(Note that we used the fact that EG[Gℓ,jk] = 0 for all ℓ/\\⌉}atio\\slash=jk, and that EG[Gjk,jk]∈R.) The\nrest of the proof is almost exactly the same as before.\n89The analogue of Claim 10.5 is that, for any row-comfortable d-row-juntaf:Hn×n→C,\nwe have\nEV∼Sp(n)/⌊ar⌈⌊lΠcomf,dRVf/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\n2dnd((n/2)−d)!/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nThe proof of Claim 10.5 is readily adapted. We first note that, as before, if HαandHβ\nare row-comfortable monomials such that the set of rows appe aring inαis different from\nthe set of rows appearing in β, then we have/a\\}⌊ra⌋k⌉tl⌉{tHα,Hβ/a\\}⌊ra⌋k⌉tri}htL2(γ)=/a\\}⌊ra⌋k⌉tl⌉{tHα,Hβ/a\\}⌊ra⌋k⌉tri}htL2(µ)= 0; the same\nproof works as before.\nForS⊂[n], letWSdenote the (complex) linear span of the row-comfortable mon omials\nof degree exactly |S|which depend only upon variables from the rows in S. As before, the\nWSare pairwise orthogonal with respect to both L2(γ) andL2(µ).\nAgain as before, for any row-comfortable d-row juntag:Hn×n→C, we may write gas\nan orthogonal direct sum,\ng=/summationdisplay\nS⊂[d]g(=S),\nwhereg(=S)denotes the orthogonal projection of gontoWS(with respect to L2(γ)). It is\nclear that ( RVg)(=S)=RV(g(=S)) for anyV∈Sp(n) and any row-comfortable d-row-junta\ng.\nNow letf:Hn×n→Cbe a row-comfortable d-row-junta. Since the f(=S)are pairwise\northogonal with respect to L2(µ) as well as with respect to L2(γ), we have\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ)=/summationdisplay\nS⊂[d]/⌊ar⌈⌊lf(=S)/⌊ar⌈⌊l2\nL2(µ),\nso there exists S⊂[d] such that/⌊ar⌈⌊lf(=S)/⌊ar⌈⌊l2\nL2(µ)/greaterorequalslant1\n2d/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(µ).\nFor brevity, write h=f(=S),d′:=|S|andS={i1,...,id′}. We need to show that\nEV∼Sp(n)/⌊ar⌈⌊lΠcomf,=SRVh/⌊ar⌈⌊l2\nL2(γ)/greaterorequalslant(n/2)!\nnd′((n/2)−d′)!/⌊ar⌈⌊lh/⌊ar⌈⌊l2\nL2(µ). (15)\nIn proving this, we may assume without loss of generality tha tS={1,2,...,d′}. In this\ncase, we may write\nh=/summationdisplay\nα={(1,j1;q1),...,(d,jd;qd)}ˆf(α)Hα. (16)\nFor brevity, write Hα0:=H{(1,1;R),(2,2;R),...,(d′,d′;R)}, i.e.,Hα0denotes thedegree- d′monomial\nX/ma√sto→2d(X1,1)real-part·(X2,2)real-part·...·(Xd′,d′)real-part.\nIfα={(1,j1;q1),(2,j2;q2),...,(d′,jd′;qd′)}is such that the jkare all distinct, then\nHα=RVαHα0, whereVα= ΣD, Σ is the permutation matrix corresponding to some\npermutation σ∈Snsatisfyingσ−1(i) =jifor alli∈[d′] (Σi,j:=δ{j=σ(i)}for alli,j∈[n]),\n90andDis a diagonal matrix with Di,i=qifor alli∈[d′]. It follows that Hα0=RV−1\nαHα.\nSinceVα∈Sp(n) for any such Vα, and since VαV∼Sp(n) forV∼Sp(n), we have\nEV∼Sp(n)/⌊ar⌈⌊lΠcomf,dRVf/⌊ar⌈⌊l2\nL2(γ)=/summationdisplay\nα:HαcomfortableEV∼Sp(n)|/a\\}⌊ra⌋k⌉tl⌉{tRVf,Hα/a\\}⌊ra⌋k⌉tri}ht|2\n=/summationdisplay\nα:HαcomfortableEV∼Sp(n)|/a\\}⌊ra⌋k⌉tl⌉{tRVαVf,Hα/a\\}⌊ra⌋k⌉tri}ht|2\n=/summationdisplay\nα:HαcomfortableEV∼Sp(n)|/a\\}⌊ra⌋k⌉tl⌉{tRVαRVf,Hα/a\\}⌊ra⌋k⌉tri}ht|2\n=/summationdisplay\nα:HαcomfortableEVαV∼Sp(n)|/a\\}⌊ra⌋k⌉tl⌉{tRVf,RV−1\nαHα/a\\}⌊ra⌋k⌉tri}ht|2\n= 4d′(n/2)!\n((n/2)−d′)!EV∼Sp(n)|/a\\}⌊ra⌋k⌉tl⌉{tRVf,Hα0/a\\}⌊ra⌋k⌉tri}ht|2.\nNow observe that\nRVHα(X) =Hα(XV) = 2d′d′/productdisplay\nk=1((XV)k,jk)qk-part= 2d′d′/productdisplay\nk=1/parenleftBiggn/summationdisplay\ni=1Xk,iVi,jk/parenrightBigg\nqk-part,\nand therefore\n/a\\}⌊ra⌋k⌉tl⌉{tRVHα(X),Hα0(X)/a\\}⌊ra⌋k⌉tri}ht=d′/productdisplay\nk=1(Vk,jk)qk-part= 2−d′Hα(V).\nUsing the expansion (16) of h, it follows that\n/a\\}⌊ra⌋k⌉tl⌉{tRVh,Hα0/a\\}⌊ra⌋k⌉tri}ht= 2−d′/summationdisplay\nαˆf(α)Hα(V) = 2−d′f(V),\nand therefore\nEV∼Sp(n)/⌊ar⌈⌊lΠcomf,=SRVh/⌊ar⌈⌊l2\nL2(γ)= 4d′(n/2)!\n((n/2)−d′)!EV∼Sp(n)4−d′|h(V)|2\n=(n/2)!\nnd′((n/2)−d′)!/⌊ar⌈⌊lh/⌊ar⌈⌊l2\nL2(µ).\nThe rest of the proof is exactly as before.\nWe now need analogues of some of the results of Section 10. (We omit those whose\nstatements and proofs are trivial to adapt.) As in the SO( n) case, we define the ‘over-\nGaussian’ distribution νto be the distribution of YG, whereY∼γandG∼GMD. Fix a\ncomfortable d-juntafand write\nf=/summationdisplay\nSaI(2dxS),\n91where forS={(1,i1;q1),...,(d,id;qd)}(withi1,...,id∈[n] distinct, and q1,...,qd∈\n{real,i,j,k}:=R), we write\nxS:=d/productdisplay\nh=1(xh,ih)qh-part.\nNote that{2dxS}Sforms an orthonormal set of vectors in L2(γ), whereSranges over tuples\nof the above form.\nWe need the following analogue of Claim 10.10.\nClaim B.1. LetS={(1,i1;q1),...,(d,id;qd)}wherei1,...,id∈[n]are distinct and\nq1,...,qd∈{real,i,j,k}:=R, and letxS:=/producttextd\nh=1(xh,ih)qh-part. Then\n/⌊ar⌈⌊l2dxS/⌊ar⌈⌊l2\nL2(ν)= 1.\nProof.In what follows, for q∈{real,i,j,k}andh∈H, we define ( h)−q-part:=−(h)q-part,\nfor notational convenience. Observe that\n/⌊ar⌈⌊l2dxS/⌊ar⌈⌊l2\nL2(ν)= 4dEG∼GMDEY∼γ[(((YG)1,i1)q1-part)2(((YG)2,i2)q2-part)2···(((YG)d,id))2\nqd-part].\nSince for each h∈[d], ((YG)h,ih)qh-part=/summationtextih\nk=1/summationtext\nr∈R(Yh,k)r-part(Gk,ih)r−1qh-partinvolves\nonly entries of Yin rowhand entries of Gin columnih(and theihare distinct), therandom\nvariables{(((YG)h,ih)qh-part)2:h∈[d]}form a system of independent random variables,\nand therefore\n/⌊ar⌈⌊l2dxS/⌊ar⌈⌊l2\nL2(ν)= 4dd/productdisplay\nh=1EG∼GMDEY∼γ[(((YG)h,ih)qh-part)2].\nFor eachh∈[d], we have\nEG∼GMDEY∼γ[(((YG)h,ih)qh-part)2]\n=EGEY\n/parenleftBiggih/summationdisplay\nk=1/summationdisplay\nr∈R(Yh,k)r-part(Gk,ih)r−1qh-part/parenrightBigg2\n\n=/summationdisplay\n(k,r)/n⌉}ationslash=(k′,r′)EGEY[(Yh,k)r-part(Yh,k′)r′-part(Gk,ih)r−1qh-part(Gk′,ih)(r′)−1qh-part]\n+ih/summationdisplay\nk=1EG∼GMDEY∼γ[Y2\nh,kG2\nk,ih]\n= 0+ih/summationdisplay\nk=1/summationdisplay\nr∈REG∼GMD[((Gk,ih)r−1qh-part)2]EY∼γ[((Yh,k)r-part)2]\n=ih/summationdisplay\nk=1/summationdisplay\nr∈REG∼GMD[((Gk,ih)r−1qh-part)2]·1\n4\n=1\n4(4(ih−1)(1/(4n)) +(n−ih+1)/n)\n= 1/4.\n92(Here, for the third equality we use the independence of\n(Yh,k)r-part,(Yh,k′)r′-part,(Gk,ih)r−1qh-part,(Gk′,ih)(r′)−1qh-part\nandthefactthat( Yh,k)r-partand(Yh,k′)r′-partbothhavezeroexpectation.) Hence, /⌊ar⌈⌊l2dxS/⌊ar⌈⌊l2\nL2(ν)=\n1, as required.\nSimilarly, weneedthefollowinganalogueofClaim10.11. Fo rS={(1,i1;q1),...,(d,id;qd)}\nandT={(1,j1;p1),...,(d,jd;pd)}we set\nd(S,T) :=|{h∈[d] :ih/\\⌉}atio\\slash=jhorqh/\\⌉}atio\\slash=ph}|.\nClaim B.2. For anyS,Tsuch thatd(S,T) =ℓ, we have/vextendsingle/vextendsingle/angbracketleftbig\n2dxS,2dxT/angbracketrightbig/vextendsingle/vextendsingle\nL2(ν)/lessorequalslantεℓ, where\nεℓ:= 2ℓ+4n−ℓ/22dℓ/√n.\nTo prove this we first need the following simple analogue of Cl aim 10.12.\nClaim B.3. Let(i1,...,id)∈[n]dand(j1,...,jd)∈[n]dbe such that|{h∈[d] :ih/\\⌉}atio\\slash=\njh}|=ℓand such that in the product\n(Gi1j1)r1-part(Gi2j2)r2-part···(Gidjd)rd-part,\nno matrix entry of Gappears more than twice. Then\n|EG∼GMD[(Gi1j1)r1-part(Gi2j2)r2-part···(Gidjd)rd-part]|/lessorequalslant/parenleftbigg1\n4n/parenrightbiggℓ/2\n.\nProof.If, in the product\n(Gi1j1)r1-part(Gi2j2)r2-part···(Gidjd)rd-part,\nsome off-diagonal matrix entry of Gappears exactly once, then the expectation of the\nproduct is zero. We may therefore assume that every matrix en try ofGappears either\nexactly twice, or not at all, in the above product. If there ar e exactlyℓvalues ofhsuch\nthatih/\\⌉}atio\\slash=jh, then the above expectation factorises into a product of the expectations of\nthe squares of ℓ/2 off-diagonal and of the squares of ( d−ℓ)/2 diagonal entries:\n/productdisplay\nk∈DE[((Gk,k)qk-part)2]/productdisplay\n(i,j)∈EE[((Gi,j)ri-part)2],\nwhereE⊂[n]2\\{(k,k) :k∈[n]},|D|= (d−ℓ)/2,|E|=ℓ/2 andqk,ri∈Rfor alliand\nk. We have E[((Gi,j)ri-part)2] = 1/(4n) for all (i,j)∈EandE[((Gk,k)qk-part)2]/lessorequalslant1 for all\nk∈D, proving the claim.\n93Proof.Letℓ/greaterorequalslant1 and fixS={(1,i1;q1),...,(d,id;qd)}andT={(1,j1;p1),...,(d,jd;pd)}\nsuch thatd(S,T) =ℓ/greaterorequalslant1. SinceGis upper-triangular and ih,jh/lessorequalslantdfor allh∈[d], we\nhave\n((YG)h,ih)qh-part=ih/summationdisplay\nk=1/summationdisplay\nr∈R(Yh,k)r-part(Gk,ih)r−1qh-part=d/summationdisplay\nk=1/summationdisplay\nr∈R(Yh,k)r-part(Gk,ih)r−1qh-part\nand\n((YG)h,jh)ph-part=jh/summationdisplay\nk=1/summationdisplay\nr∈R(Yh,k)r-part(Gk,jh)r−1ph-part=d/summationdisplay\nk=1/summationdisplay\nr∈R(Yh,k)r-part(Gk,jh)r−1ph-part\nfor allh∈[d]. Hence,\nxS(YG) =d/productdisplay\nh=1((YG)h,ih)qh-part\n=/summationdisplay\nK=(k1,...,kd)∈[d]d,\nR=(r1,...,rd)∈Rd(Y1,k1)r1-part···(Yd,kd)rd-part(Gk1,i1)r−1\n1q1-part···(Gkd,id)r−1\ndqd-part\nand\nxT(YG) =/summationdisplay\nK=(k1,...,kd)∈[d]d,\nR=(r1,...,rd)∈Rd(Y1,k1)r1-part···(Yd,kd)rd-part(Gk1,j1)r−1\n1p1-part···(Gkd,jd)r−1\ndpd-part,\nso, using the fact that, under ν, the ((Yi,j)r-part:i,j∈[n], r∈R) are independent and of\nexpectation zero (and are independent of the Gi,j), we obtain\n/angbracketleftBig\n2dxS,2dxT/angbracketrightBig\nν\n=/summationdisplay\nK∈[d]d,\nR∈RdEG∼GMD/bracketleftBig\n(Gk1i1)r−1\n1q1-part(Gk1j1)r−1\n1p1-part···(Gkdid)r−1\ndqd-part(Gkdjd)r−1\ndpd-part/bracketrightBig\n.\n(17)\nFor ad-tuple (K;R) = (k1;r1...,kd;rd)∈([d]×R)d, we writem1=m1(K) =m1(K;R) :=\n|{h∈[d] :jh=ih, kh/\\⌉}atio\\slash=ih}|,m2=m2(K) =m2(K;R) :=|{h∈[d] :jh/\\⌉}atio\\slash=ih, kh/∈\n{ih,jh}}|andm3=m3(K;R) :=|{h∈[d] :jh/\\⌉}atio\\slash=ih, kh∈{ih,jh}}|; note that these\nquantities dependonly on Kand not on R. We letK(m1,m2,m3) denote the set of d-tuples\n(K;R) with parameters m1,m2andm3. For (K;R)∈K(m1,m2,m3), by Claim 10.12 we\nhave\n/vextendsingle/vextendsingle/vextendsingleEG/bracketleftBig\n(Gk1i1)r−1\n1q1-part(Gk1j1)r−1\n1p1-part···(Gkdid)r−1\ndqd-part(Gkdjd)r−1\ndpd-part/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/lessorequalslant(4n)−2m1+2m2+m3\n2.\nWe further note that, for ( K;R)∈K(m1,m2,m3), the expectation in the above inequality\nis zero unless the following four conditions hold:\n94•Wheneverih=jhandkh/\\⌉}atio\\slash=ih, we haveph=qh.\n•Wheneverih=jh=khwe haveph=qhandr−1\nhph= real.\n•Wheneverih/\\⌉}atio\\slash=jhandkh=ihwe haver−1\nhqh= real.\n•Wheneverih/\\⌉}atio\\slash=jjandkh=jhwe haver−1\nhph= real.\nIn view of this we let K∗(m1,m2,m3) be the set of all d-tuples (K;R)∈K(m1,m2,m3)\nsuchthattheabovefourconditionshold. For( K;R)∈K∗(m1,m2,m3)wehavem2+m3=ℓ.\nSumming over all K, we see that|/a\\}��ra⌋k⌉tl⌉{txS,xT/a\\}⌊ra⌋k⌉tri}ht|is at most\n/summationdisplay\nm1,m2,m3/summationdisplay\nK∈K∗(m1,m2,m3)(4n)−2m1+2m2+m3\n2\n/lessorequalslant/summationdisplay\nm1,m2,m3(4n)−2m1+2m2+m3\n2|K∗(m1,m2,m3)|.\nNow\n|K∗(m1,m2,m3)|/lessorequalslant/parenleftbiggd\nm1/parenrightbigg\ndm1/parenleftbiggℓ\nm2/parenrightbigg\ndm22m3·4m1+m2/lessorequalslantd2m1+m2ℓm22m3\nm1!m2!·4m1+m2;\nnote that the only difference with the corresponding expressi on in the proof of Claim 10.12\nis the extra factor of 4m1+m2, which comes from the fact that rhcan vary freely over R\n(and still satisfy the above conditions) when ih/\\⌉}atio\\slash=jhandkh/∈{ih,jh}, or whenih=jhand\nkh/\\⌉}atio\\slash=ih, but in no other cases.\nSumming over all m1,m2,m3withm2+m3=ℓcompletes the proof, just as in the proof\nof Claim 10.11; the extra factor of 4−2m1+2m2+m3\n2cancels out (or more than cancels out) the\nextra factor of 4m1+m2.\nThe analogue of Lemma 10.9 is proven from the above claims in a very similar way.\nWriting\nf=/summationdisplay\nSαS(2dxS),\n95we obtain\n/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(ν)/lessorequalslant/summationdisplay\nS|αS|2/⌊ar⌈⌊l2dxS/⌊ar⌈⌊l2\nL2(ν)+/summationdisplay\nS/n⌉}ationslash=T|αS||αT|/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n2dxS,2dxT/angbracketrightBig\nν/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/summationdisplay\nS/n⌉}ationslash=T|aS|2+|aT|2\n2/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n2dxS,2dxT/angbracketrightBig\nν/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+d/summationdisplay\nℓ=1/summationdisplay\nS|aS|2|{T:d(T,S) =ℓ}|·εℓ\n/lessorequalslant/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)+/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)d/summationdisplay\nℓ=1εℓ/parenleftbiggd\nℓ/parenrightbigg\nℓ!4ℓ\n=/⌊ar⌈⌊lf/⌊ar⌈⌊l2\nL2(γ)/parenleftBigg\n1+d/summationdisplay\nℓ=1εℓ(4d)ℓ/parenrightBigg\n,\nand therest of theproof is essentially unchanged, upto redu cingthevalue of δby aconstant\nfactor.\nC Bounding the eigenvalues of the Laplace-Beltrami opera-\ntor\nC.1 Bounding the eigenvalues of the Laplace-Beltrami opera tor in SO (n)\nIn this section, we obtain our desired bound on the eigenvalu es of the Laplace-Beltrami\noperator in SO n, which we need in order to prove fineness.\nLemma C.1. Letρ∈ˆSO(n)be of level D, then the corresponding eigenvalue λρof the\nLaplacian ∆satisfies\nλρ/greaterorequalslant−C(nD+D2).\nTo prove this, we will use [7, Theorems 2.3, 2.4]. It is well-k nown that the matrix\ncoefficients of the irreducible representations of SO( n) are eigenvectors of the Laplace–\nBeltrami operator. The aforementiond theorems in [7] give a formula for the eigenvalues, in\nterms of the fundamental weights of the irreducible represe ntations. (The reader is referred\nto [17, p. 219] for a relatively concise definition of the fund amental weights.) The following\nstatement (which suffices for our purposes) combines Theorem 2.3 and 2.4 from [7].\nTheorem C.2. LetGbe a simply connected Lie group of rank k. Then the eigenvalues of\nthe Laplace-Beltrami operator ∆ :L2(G)→L2(G)are in one-to-one correspondence with\nthe equivalence classes of complex irreducible representa tions ofG. Furthermore, there is a\nset of vectors{w1,...,w k}in a finite-dimensional inner-product space, known as a systemof\nfundamental weights for G, such that there is an explicit one-to-one correspondence b etween\n96the equivalence classes of irreducible representations of G, and the elements of the discrete\ncone /braceleftBiggk/summationdisplay\ni=1riwi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingler1,...,rk∈N/bracerightBigg\n.\nFurthermore, defining ρ:=k/summationtext\ni=1wi, the eigenvalue λvof∆corresponding to v=k/summationtext\ni=1riwi\nsatisfies\nλv=−/⌊ar⌈⌊lv+ρ/⌊ar⌈⌊l2\n2+/⌊ar⌈⌊lρ/⌊ar⌈⌊l2\n2=−2/a\\}⌊ra⌋k⌉tl⌉{tv,ρ/a\\}⌊ra⌋k⌉tri}ht−/⌊ar⌈⌊lv/⌊ar⌈⌊l2\n2.\nWe will choose a known system of fundamental weights for SO( n) as in [35, Section 5.1].\nWe denote by Ei,j∈Rn×nthenbynmatrix with a one in the ( i,j)-th entry and zeros\nelsewhere. (We omit nfrom notation, as it will always be clear from context.) The s ystem\nof fundamental weights depends on whether nis even or odd, and we deal with the two\ncases separately.\nThe case of odd n\nLetn= 2k+1. In this case, the rank of SO( n) isk, and a system of fundamental weights\n{w1,w2,...,w k}is given by wi=i/summationtext\nj=1ujfor 1/lessorequalslanti/lessorequalslantk−1 andwk=1\n2k/summationtext\nj=1uj, where\nuj:=Ej,k+j−Ek+j,jfor each 1 /lessorequalslantj/lessorequalslantk.\nThe equivalence class of representations corresponding to v=k/summationtext\ni=1riwiis indexed by\nYoung diagrams λ= (λ1,...,λ k), whereri=λi−λi+1for each 1 /lessorequalslanti/lessorequalslantk−1 andrk= 2λk.\nThe corresponding degree is therefore\nD=k/summationdisplay\ni=1λi=k−1/summationdisplay\ni=1\n/summationdisplay\ni/lessorequalslantj/lessorequalslantk−1rj+rk\n2\n+rk\n2=/summationdisplay\n1/lessorequalslantj/lessorequalslantk−1jrj+k\n2rk. (18)\nWe willusethisequality toestimate theeigenvalue given in TheoremC.2. Usingtheformula\ntherein, the corresponding eigenvalue is\nλv=−/⌊ar⌈⌊lv+ρ/⌊ar⌈⌊l2\n2+/⌊ar⌈⌊lρ/⌊ar⌈⌊l2\n2,\nwhere here and henceforth we view the matrices vandρas vectors of length n2, using the\nthe natural flattening, and we use the unnormalized inner pro duct on Rn2(equivalently,\n/a\\}⌊ra⌋k⌉tl⌉{tA,B/a\\}⌊ra⌋k⌉tri}ht=Tr(AtB)). Then\nλv=−2/a\\}⌊ra⌋k⌉tl⌉{tv,ρ/a\\}⌊ra⌋k⌉tri}ht−/⌊ar⌈⌊lv/⌊ar⌈⌊l2\n2.\nIt remains to estimate the norm of vand the inner product of vandρ. To estimate the\nnorm ofv, we write\nv=k/summationdisplay\ni=1wi=k−1/summationdisplay\ni=1rii/summationdisplay\nj=1uj+rk\n2k/summationdisplay\nj=1uj=k/summationdisplay\nj=1\nrk\n2+k−1/summationdisplay\ni=jri\nuj,\n97and since the uj’s are mutually orthogonal and each has /⌊ar⌈⌊luj/⌊ar⌈⌊l2\n2= 2, we obtain\n/⌊ar⌈⌊lv/⌊ar⌈⌊l2\n2= 2k/summationdisplay\nj=1\nrk\n2+k−1/summationdisplay\ni=jri\n2\n/lessorequalslant2\nk/summationdisplay\nj=1\nrk\n2+k−1/summationdisplay\ni=jri\n\n2\n= 2\nk/summationdisplay\nj=1krk\n2+k−1/summationdisplay\ni=1iri\n2\n,\nwhich is at most 2 D2by (18).\nTo bound the inner product of vandρ, note that ρis the vector vin which we take all\ntheri’s to be 1, hence by the computation above, we have\nρ=k/summationdisplay\ni=1wi=k/summationdisplay\nj=1/parenleftbigg\nk−j+1\n2/parenrightbigg\nuj,\nand so\n/a\\}⌊ra⌋k⌉tl⌉{tρ,v/a\\}⌊ra⌋k⌉tri}ht= 2k/summationdisplay\nj=1/parenleftbigg\nk−j+1\n2/parenrightbigg\nrk\n2+k−1/summationdisplay\ni=jri\n/lessorequalslant2kk/summationdisplay\nj=1\nrk\n2+k−1/summationdisplay\ni=jri\n/lessorequalslant2kD.\nwhere in the last inequality we used (18). Overall, we see tha t the eigenvalue λvsatisfies\nλv/greaterorequalslant−2D2−nD, as required.\nThe case of even n\nLetn= 2k. In this case, the rank of SO( n) isk, and a system of fundamental weights\n{w1,...,w k}is given by wi=i/summationtext\nj=1ujfor 1/lessorequalslanti/lessorequalslantk−2,wk−1=1\n2k/summationtext\nj=1ujandwk=wk−1−uk,\nwhere again we define uj:=Ej,k+j−Ek+j,jfor 1/lessorequalslantj/lessorequalslantk.\nConsider the the equivalence class of representations corr esponding to v=k/summationtext\ni=1riwi. We\nnow need to inspect the corresponding Young diagram to relat e theri’s to the degree of\nthe representation, and we recall that the corresponding Yo ung diagram λ= (λ1,...,λ k)\nmaybe either have knon-zero rows or at most k−1 non-zero rows.\nYoung diagrams with at most k−1rows.In this case we have ri=λi−λi+1for each\n1/lessorequalslanti/lessorequalslantk, and so the degree is\nD=k/summationdisplay\ni=1λi=k/summationdisplay\ni=1k−1/summationdisplay\nj=irj=k−1/summationdisplay\nj=1jrj.\nUsing the same estimates as before, we obtain\nv=k−2/summationdisplay\ni=1rii/summationdisplay\nj=1uj+k/summationdisplay\nj=1rk−1\n2uj=k/summationdisplay\nj=1\nk−2/summationdisplay\ni=jri+rk−1\n2\nuj,\n98so\n/⌊ar⌈⌊lv/⌊ar⌈⌊l2\n2/lessorequalslant2k/summationdisplay\nj=1\nk−2/summationdisplay\ni=jri+rk−1\n2\n2\n/lessorequalslant2\nk−2/summationdisplay\ni=jiri+krk−1\n2\n2\n/lessorequalslant2D2.\nWe also obtain ρ=k−2/summationtext\nj=1(k−1\n2−j)uj+1\n2uk−1+1\n2uk, and so\n/a\\}⌊ra⌋k⌉tl⌉{tv,ρ/a\\}⌊ra⌋k⌉tri}ht= 2k−2/summationdisplay\nj=1/parenleftbigg\nk−1\n2−j/parenrightbigg\nk−2/summationdisplay\ni=jri+rk−1\n2\n+rk−1/lessorequalslant2kk−2/summationdisplay\nj=1\nk−2/summationdisplay\ni=jri+rk−1\n2\n+rk−1/lessorequalslant2kD+D,\nwhich is at most 2 nD, as desired.\nYoung diagrams with krows.Inthis case, we have ri=λi−λi+1foreach 1 /lessorequalslanti/lessorequalslantk−2,\nand (rk−1,rk) is either ( λk−1−λk,λk−1+λk) or (λk−1+λk,λk−1−λk). The computation\nin both cases is similar and goes along the same lines as the co mputations so far, hence we\ndeal only with the case where ( rk−1,rk) = (λk−1−λk,λk−1+λk).\nThe degree here is\nD=k/summationdisplay\ni=1λi=k−2/summationdisplay\ni=1\nk−2/summationdisplay\nj=irj+1\n2(rk−1+rk)\n=k−2/summationdisplay\nj=1jrj+k−2\n2rk−1+k−2\n2rk.\nWe have the same formulae for vandρas before, and so\n/⌊ar⌈⌊lv/⌊ar⌈⌊l2\n2/lessorequalslant2k/summationdisplay\nj=1\nk−2/summationdisplay\ni=jri+rk−1\n2\n2\n/lessorequalslant2\nk−2/summationdisplay\ni=jiri+krk−1\n2\n2\n/lessorequalslant2D2,\nand\n/a\\}⌊ra⌋k⌉tl⌉{tv,ρ/a\\}⌊ra⌋k⌉tri}ht= 2k−2/summationdisplay\nj=1/parenleftbigg\nk−1\n2−j/parenrightbigg\nk−2/summationdisplay\ni=jri+rk−1\n2\n+rk−1/lessorequalslant2kk−2/summationdisplay\nj=1\nk−2/summationdisplay\ni=jri+rk−1\n2\n+rk−1/lessorequalslant2kD+D,\nwhich is at most 2 nD, as desired.\nC.2 Bounding the eigenvalues of the Laplace–Beltrami opera tor in SU (n)\nThe following lemma, analogous to Lemma C.1, gives our desir ed bound on the eigenvalues\nof the Laplace–Beltrami operator in SU( n).\nLemma C.3. Forρ∈/hatwiderSU(n)of levelD, the corresponding eigenvalue λρof∆satisfies\nλρ/greaterorequalslant−C(nD+D2),\nwhereCis an absolute constant.\n99Proof.The proof proceeds by a similar computation to the proof of Le mma C.1. We apply\nTheorem C.2 in the case of G= SU(n), which has rank k=n−1, and use a system of\nfundamental weights from [7]. Thesystem of fundamental wei ghts{w1,...,w n−1}is defined\nby\nwi=i/summationdisplay\nj=1ej−i\nnn/summationdisplay\nj=1ej(1/lessorequalslanti/lessorequalslantn−1),\nwhere{ei}n\ni=1is the standard basis of Rn; here,eicorresponds to\niEi,i, (19)\nwherei=√−1 andEi,jis the matrix with a one in the ( i,j)th-entry and zeros elsewhere,\nas before. For each 1 /lessorequalslantk/lessorequalslantl/lessorequalslantn−1, we have\n/a\\}⌊ra⌋k⌉tl⌉{twk,wl/a\\}⌊ra⌋k⌉tri}ht=k(n−l)/n.\nSetσ:=/summationtextn−1\ni=1wi. For a partition λwhose Young diagram has less than nrows, the\ncorresponding weight vector is\nv=n−1/summationdisplay\ni=1aiwi,\nwhereai=λi−λi+1for alli∈[n−1] andλn:= 0; the level Dof the corresponding\nrepresentation is given by\nD=n−1/summationdisplay\ni=1aimin{i,n−i}.\nIt follows that, if v=/summationtextn−1\nk=1akwk, then\n/a\\}⌊ra⌋k⌉tl⌉{tv,σ/a\\}⌊ra⌋k⌉tri}ht=/angbracketleftBiggn−1/summationdisplay\nk=1akwk,n−1/summationdisplay\nk=1wk/angbracketrightBigg\n=/summationdisplay\n1/lessorequalslantk/lessorequalslantl/lessorequalslantn−1akk(n−l)/n+/summationdisplay\n1/lessorequalslantl1,\nit is not obvious what the suitable definition of log-concavi ty should be. We discuss this in Section\n1.2. Moreover, we prove a d-dimensional analogue (Theorem 11) of the following result , due to Bobkov,\nMarsiglietti and Melbourne [4], who also studied discrete v ersions of the EPI (up to multiplicative\nconstants) for R´ enyi entropies of log-concave distributi ons and which is an important tool in the one-\ndimensional case:\n2Theorem 2. [4, Theorem 1.1] If a random variable Xfollows a discrete log-concave p.m.f. fonZ, then\nmax\nk∈Zf(k)≤1√\n1+4σ2, (5)\nwhereσ2= Var(X).\n1.2 Notations and definitions\nBig- and small- Onotation. Letfbe a real-valued function and ganother strictly positive function.\nWe write f=O(g) if there exist positive absolute constants N,Csuch that |f(x)| ≤Cg(x) for every\nx≥N. IfN,Care absolute up to a parameter d, we write f=Od(g). Analogously, we write f= Ω(g),\nif|f(x)| ≥Cg(x) for every x≥N. Iff= Ω(g) andf=O(g), we write f= Θ(g) (with the analogous\ndefinitions for Ω dand Θ d). When it is more convenient, we will write f/lessorsimilardgforf=Od(g) andf≃dg\nforf= Θ(g). We write f(x) =o(g(x)) if lim x→∞f(x)\ng(x)= 0.\nConvex bodies. Aconvex body is a convex set that is compact and has a non-empty interior. F or a\nconvex body K∈Rd, one denotes by |K|its volume. Let A,Bconvex bodies in Rd, we write A≈Bif\nthere are constants C1(d),C2(d) that depend on the dimension, such that C1(d)B⊂A⊂C2(d)B.\nLog-concavity and convexity in Zd.Ford >1 there are more than one definitions of log-concavity\nthat have been used in different contexts. We will use a quite g eneral definition that implies several other\nnotions of log-concavity, the one of extensible log-concave functions . The reader is referred to Murota\n[21] for an extensive discussion of discrete convexity in hi gher dimensions.\nA function f:Zd→R∪ {+∞}is said to be convex-extensible if there exists a convex function\n¯f:Rd→R∪{+∞}such that\n¯f(z) =f(z) (∀z∈Zd).\nWe refer to ¯fas theextension off.\nBased on the above, we define\nDefinition 3 (Log-concave extensiblefunctions) .Afunction f:Zd→R∪{+∞}issaid tobelog-concave\nextensible if\nf(z) =e−V(z)\nwhereV:Zd→R∪ {+∞}is convex-extensible. Throughout when we say that fis log-concave, we\nmean that fis log-concave extensible. We say that a random vector X, taking values in Zdand p.m.f.\np:Zd→[0,1] is log-concave, if pis log-concave.\nSimilarly, we define a Zd-convex set (commonly referred in the literature as hole-free set [21]):\nDefinition 4 (Zd-convexity) .A setA⊂Zdis said to be Zd-convex if and only if its indicator function\nIAdefined as IA(x) = 0 ifx∈Aand +∞otherwise, is convex-extensible or equivalently if\nA= conv(A)∩Zd,\nwhere conv denotes the convex hull ofA, that is, the smallest convex set (in the usual sense) contai ning\nthe setA.\nRemark 5. Ford= 1our definition of log-concavity is equivalent to the usual de finitionp(k)2≥\np(k−1)p(k+ 1),k∈Z, which is preserved under convolution (e.g. [14]). However , ford >1log-\nconcavity may not be preserved in general, as pointed out by M urota [21, Example 3.15] by considering\ntwo log-concave distributions supported on S1={(0,0),(1,1)}andS2={(0,1),(1,0)}respectively. This\nis because the Minkowski sum S1+S2is notZ2-convex. On the other hand, in the i.i.d. case, we shall\nsee in our Proposition 35 below that/summationtextn\ni=1AiisZd-convex for any Zd-convex set A. We do not know\nwhether our definition of log-concavity is preserved under s elf-convolution.\nNotion of isotropicity . The isotropic constant of a function f:Rd→R+is defined by\nLf:=/parenleftbiggmaxRdf/integraltext\nRdf/parenrightbigg1\nd\ndet(Cov( f))1\n2d, (6)\n3where Cov( f) is theinertiaorcovariance matrix defined, for 1 ≤i,j≤d, by\n[Cov(f)]ij:=/integraltext\nRdxixjf(x)dx/integraltext\nRdf(x)dx−/integraltext\nRdxif(x)dx/integraltext\nRdxjf(x)dx\n/parenleftBig/integraltext\nRdf(x)dx/parenrightBig2.\nWe say that fisisotropic if Cov(f) =σ2Id, for some σ >0,where I dis thed×didentity matrix. Let\nKbe a convex body in Rd. Then its covariance matrix is Cov( K) := Cov( 1K) and its isotropic constant\nisLK:=L1K. The convex body Kis called isotropic if1Kis isotropic.\nSimilarly, in the discrete case p:Zd→R+, we define the covariance matrix Cov( p) by\n[Cov(p)]ij:=/summationtext\nk∈Zdkikjp(k)/summationtext\nkp(k)−/summationtext\nk∈Zdkip(k)/summationtext\nk∈Zdkjp(k)\n/parenleftBig/summationtext\nk∈Zdp(k)/parenrightBig2.\nDefinition 6. A family {fσ}σ∈R+of non-negative functions on Rdisalmost isotropic if, asσ→ ∞,\nCov(fσ)i,j=σ2+O(σ) fori=j,\n=O(σ) fori/ne}ationslash=j.\nRemark 7. It is straightforward to check that the family {fσ}σ∈R+is almost isotropic if and only if\n||Cov(fσ)−σ2Id||op=O(σ),\nwhere||A||opis the operator norm.\nWe are interested in log-concave densities f, for which detCov( f)→ ∞. Thus, when we write that f\nis almost isotropic, it is meant that the underlying paramet er isσ:= detCov( f)1\n2d. When its not clear\nfrom the context, we write Cov R(f) and Cov Z(f) to distinguish between the continuous and discrete\ncovariance matrix of f.\n1.3 Main results and paper outline\nOur first main result is the following one:\nTheorem 8. LetX1,...,X nbe i.i.d. random vectors on Zdsuch that the sums X1+···+Xnand\nX1+···+Xn+1are log-concave with almost isotropic extension. Then\nH(X1+···+Xn+1)≥H(X1+···+Xn)+d\n2log/parenleftBign+1\nn/parenrightBig\n−o(1)\nasH(X1)→ ∞. In fact, we have the rate of convergence o(1) =Od,n/parenleftBig\nH(X1)e−1\ndH(X1)/parenrightBig\n.\nTheorem 8 readily follows from the next theorem together wit h the generalised EPI in Rd.\nTheorem 9. LetX1,...,X nbe i.i.d. random vectors on Zdsuch that their sum X1+···+Xnis\nlog-concave with almost isotropic extension. Then\nh(X1+···+Xn+U1+···+Un) =H(X1+···+Xn)+o(1), (7)\nwhereo(1)→0asH(X1)→ ∞, where hstands for the differential entropy, Hdenotes the discrete\nShannon entropy and U1,...,U nare independent continuous uniforms on [0,1]d. In fact, we have the\nrate of convergence o(1) =Od,n/parenleftBig\nH(X1)e−1\ndH(X1)/parenrightBig\n.\nRemark 10. LetX1,...,X nbe i.i.d. random vectors on Zdsuch that their sum is log-concave with\nalmost isotropic extension. Then, since Cov/parenleftBig/summationtextn\ni=1Xi/parenrightBig\n=nCov(X1), X1is also almost isotropic. Denote\nthe covariance matrix of X1byC,σ2:= det(C)1\ndand letU1a uniform on the open box [0,1]d. Then\nH(X1) =h(X1+U1)≤d\n2log/parenleftbigg\ndet(C+1\n12Id)1\nd2πe/parenrightbigg\n≃d\n2log/parenleftbigg\nσ22πe/parenrightbigg\n, (8)\nby the fact that the Gaussian maximizes the entropy under fixe d covariance matrix, which implies that\nfor any random variable XonRdone has\nh(X)≤d\n2log(2πe(detCov( X))1/d). (9)\nThus,H(X1)→ ∞impliesσ→ ∞and therefore it suffices to prove (7)witho(1)→0asσ→ ∞.\n4Analogously to the one-dimensional case, we will prove in ou r Theorem 11 below, a generalisation to\ndimensions d >1 of Theorem 2, that is an upper bound of the maximum probabili ty in terms of the\ninverse of the determinant of the discrete covariance matri x. For the proof of Theorem 9 we will also\nneed a different generalisation of Theorem 2, namely Lemma 16 (see also Lemma 31), which reads\n/summationdisplay\nl∈Zd−1max\nk∈Zf(k,l) =O/parenleftBig1\nσ/parenrightBig\n.\nWe will use this bound to control the error term arising from t he approximation of the density of/summationtext\niXi+Uiby the p.m.f. of/summationtext\niXi. Combined with the Taylor-type estimate of Lemma 34, which i s\nanalogous to an estimate used in [25], this approximation is the main technical step in the proof of\nTheorem 9. Theorem 11 below will ensure that the conditions o f that lemma are satisfied.\nTheorem 11 may also be seen as a discrete analogue of a dimensi onal upper bound on the isotropic\nconstant Lf:\nThe following lower bound on Lfcan be deduced from (9) and the fact that e−h(X)≤max(f):\nLf=/parenleftbigg\nmax\nRdf/parenrightbigg1\nd\ndet(Cov( f))1\n2d≥e−h(X)\nddet(Cov( f))1\n2d≥1√\n2πe.\nIt is well known that a dimensional upper bound also exists an d we denote by Ldthe maximum of the\nisotropic constants Lfamong log-concave functions finRd. The latter is related to the famous hyper-\nplane conjecture (or slicing problem) from convex geometry , one of the central questions in the field. The\nhyperplane conjecture is equivalent to the isotropic const ant conjecture [5, Theorem 3.1.2].\nHyperplane conjecture: there exists a universal constant c >0 such that, for any dimension d\nand for any convex body Kin isotropic position and of volume 1 in Rdand any direction θ∈Sd−1, one\nhas|K∩θ⊥|d−1≥c.\nIsotropic constant conjecture: there exists a universal constant Csuch that for any dimension\nd, one has Ld≤C.\nA recent result of Klartag [16] gives the best known constant depending on the dimension dfor the\nslicing problem and thus also the best upper bound known for Ld:\nLd≤C/radicalbig\nlogd , (10)\nfor some absolute constant C.\nOur discrete analogue of this (and dimensional analogue of ( 5)) reads:\nTheorem 11. Suppose pis a log-concave p.m.f. on Zdwith almost isotropic extension and covariance\nmatrixCov(p). Then there exists a constant C′\nddepending on the dimension only, such that\nmax\nk∈Zdp(k)≤C′\nd\ndet/parenleftBig\nCovZd(p)/parenrightBig1\n2\nprovided that det(CovZd(p))is large enough depending on d.\nOur method for proving Theorem 11 is to use the corresponding continuous result. To this end, we\nobtain the approximations\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdf−/summationdisplay\nZdf/vextendsingle/vextendsingle/vextendsingle=od(1),asσ→ ∞,\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdxf−/summationdisplay\nk∈Zdkf(k)/vextendsingle/vextendsingle/vextendsingle=Od(1),asσ→ ∞,\n/vextendsingle/vextendsingle/vextendsingledet/parenleftBig\nCovZd(f)/parenrightBig\n−det/parenleftBig\nCovRd(f)/parenrightBig/vextendsingle/vextendsingle/vextendsingle=Od(σ2d−1),asσ→ ∞,\nfor any isotropic log-concave density f. This is done in Section 2. Although our results hold under th e\nmore general almost isotropicity assumption, for better il lustration of the ideas, we assume first that the\ncontinuous extension fis isotropic. In Section 3 we show how to relax this assumptio n (see Remark 28).\nThen, in Section 4 we prove Theorems 8 and 9. Finally, in Secti on 5 we conclude with a brief discussion\nof our assumptions as well as open questions.\n52 Proof of Theorem 11 for isotropic f\nIn this section we prove Theorem 11 under the assumption that the continuous extension fis isotropic.\nIn what follows, the inradius r(K) of a convex body K∈Rdwith 0 in its interior is defined as the\nlargestr >0 for which rBd\n2⊂Kand thecircumradius R(K) ofKis defined as the smallest R >0 such\nthatK⊂RBd\n2, whereBd\n2is thed-dimensional Euclidean ball. Let ωdbe the volume of the ball Bd\n2and\nlethC:Rd→Rbe thesupport function of a convex body Cdefined as hC(x) = max y∈C/an}⌊ra⌋ketle{tx,y/an}⌊ra⌋ketri}ht.\n2.1 Ball’s bodies\nFor most results in this section we will assume that f:Rd→Ris a centered, isotropic, log-concave\ndensity, that is/integraltext\nRdf= 1,/integraltext\nRdxf= 0 and/integraltext\nRdxTxf=σ2Id.To this function f, we attach its Ball’s\nbodies. This important family of bodies was introduced by Ba ll [2]. We refer to the book [5] for the\nproperties of these.\nDefinition 12. Letf:Rd→[0,∞) be an integrable, centered log-concave function. For any p >0, the\nsetKp(f) is defined as follows\nKp(f) :=/braceleftbigg\nx∈Rd:/integraldisplay∞\n0prp−1f(rx)dr≥f(0)/bracerightbigg\n.\nBall [2] established that the set Kp(f) is a convex body. Moreover, its radial function is\nρKp(f)(x) =/parenleftBig1\nf(0)/integraldisplay∞\n0prp−1f(rx)dr/parenrightBig1\npforx/ne}ationslash= 0.\nFrom integration in polar coordinates (see [5, Proposition 2.5.3]), it follows that Kd+1(f) is centered and,\nfor anyp≥0 andu∈Sd−1,\n/integraldisplay\nKd+p(f)|/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht|pdx=1\nf(0)/integraldisplay\nRd|/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht|pf(x)dx. (11)\nFinally, the following inclusion relations between Kp(f) andKq(f) hold [5, Proposition 2.5.7], for p < q,\nΓ(p+1)1\np\nΓ(q+1)1\nqKq(f)⊂Kp(f)⊂ed\np−d\nqKq(f).\nApplying these inclusions first for p=dandq=d+1 then for p=d+1 andq=d+2 and using classical\ninequalities on Gamma functions, it follows that there exis t universal constants 0 < c1< c2such that\nfor any dimension d≥1, one has\nc1Kd+1(f)⊂Kd(f)⊂c2Kd+1(f) and c1Kd+2(f)⊂Kd+1(f)⊂c2Kd+2(f). (12)\nWeshall apply to Kd+1(f) thefollowing theorem duetoKannan, Lov´ asz andSimonovit s [15, Theorem\n4.1] to which we give a new and simple proof.\nTheorem 13. LetKbe a centered convex body in Rdandu∈Sd−1. Then\nhK(u)2\nd(d+2)≤1\n|K|/integraldisplay\nK/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht2dx≤d\nd+2hK(u)2. (13)\nProof.Letf(t) =|{x∈K;/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht=t}. Denote by [ −b,a] its support. Then one has a=hK(u),\n/integraldisplay\nRf(t)dt=|K|,/integraldisplay\nRtf(t)dt=/integraldisplay\nK/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}htdxand/integraldisplay\nRt2f(t)dt=/integraldisplay\nK/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht2dx.\nNow we define g:R→R+by\ng(t) =α/parenleftbigg\n1+t\nda/parenrightbiggd−1\n1[−da,a](t),whereα=1\na/parenleftbig\n1+1\nd/parenrightbigd/integraldisplay\nRf(t)dt.\nAnd leth:R→R+be defined by h(t) =dh(−dt). Then, it is not difficult to see that\n/integraldisplay\nRg(t)dt=/integraldisplay\nRh(t)dt=/integraldisplay\nRf(t)dtand/integraldisplay\nRtg(t)dt=/integraldisplay\nRth(t)dt=/integraldisplay\nRtf(t)dt= 0.(14)\n6We shall prove that/integraldisplay\nRt2g(t)dt≤/integraldisplay\nRt2f(t)dt≤/integraldisplay\nRt2h(t)dt,\nfrom which the inequality (13) follows immediately by a simp le calculation. If f/ne}ationslash=g, it follows from\n(14) that there exist at least two points t1(g)< t2(g) such that g−fchanges sign at these points and,\nin the same way, if f/ne}ationslash=hthere exists at least two points t1(h)< t2(h) such that f−hchanges sign at\nthese points. Since the function f1/(d−1)is concave on its support [ −b,a] and the functions g1/(d−1)and\nh1/(d−1)are affine on their respective supports [ −da,a] and [−a\nd,a], it follows that the functions g−fand\nf−hhave exactly two sign changes and that they are non positive o n [t1,t2] and non negative outside\n[t1,t2]. Let us define vto be either g−forf−h. Then, for any t∈R, one has ( t−t1)(t−t2)v(t)≥0.\nIntegrating this inequality, we deduce that/integraltext\nt2v(t)dt≥0, which is the result.\nWe shall deduce from the preceding theorem the following imp ortant technical lemma.\nLemma 14. Letd≥1be an integer. There exist two constants 0< C′\nd< Cdsuch that for any\nf:Rd→R+centered, isotropic, log-concave density and for every θ∈Sd−1,\nC′\nd≤/parenleftbigg/integraldisplay∞\n0drd−1f(rθ)dr/parenrightbigg1\nd\n≤Cd, (15)\nwhereCdandC′\ndare constants depending only on the dimension d. In fact, we may take\nC′\nd=cd+2\n1√\n2πe3\n2andCd= (d+1)cd+2\n2Ld, (16)\nwherec1andc2are the absolute constants appearing in (12).\nProof.The function fbeing isotropic, we have Cov( f) =σ2Id, for some σ >0 and/integraltext\nf= 1, thus\nLf= max(f)1\ndσ. Note that proving (15) is equivalent to proving that\nC′\ndBd\n2⊂f(0)1\ndKd(f)⊂CdBd\n2.\nApplying Theorem 13 to the centered body Kd+1(f), we get\nhKd+1(f)(u)2\nd(d+2)≤1\n|Kd+1(f)|/integraldisplay\nKd+1(f)/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht2dx≤d\nd+2hKd+1(f)(u)2. (17)\nUsing the inclusion relations (12), we get upper and lower bo und of the integral over Kd+1(f) in the\nmiddle term by an integral over Kd+2(f). Then, we use (11) for p= 2 to get that\n/integraldisplay\nKd+2(f)|/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht|2dx=1\nf(0)/integraldisplay\nRd|/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht|2f(x)dx=σ2\nf(0).\nThis gives\ncd+2\n1σ2\nf(0)≤/integraldisplay\nKd+1(f)/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht2dx≤cd+2\n2σ2\nf(0).\nUsing again the inclusion relations (12), we get upper and lo wer bound of |Kd+1(f)|by|Kd(f)|= 1/f(0),\nby (11) for p= 0. Thus, altogether we have\nc2d+2\n1σ2≤1\n|Kd+1(f)|/integraldisplay\nKd+1(f)/an}⌊ra⌋ketle{tx,u/an}⌊ra⌋ketri}ht2dx≤c2d+2\n2σ2.\nUsing these inequalities, (17) and the inclusion relations (12), we get\n/radicalbigg\nd+2\ndcd+2\n1σBd\n2⊂Kd(f)⊂/radicalbig\nd(d+2)cd+2\n2σBd\n2.\nNow, from [6, Theorem 4], we have f(0)≥e−dmax(f) hence\nLf\ne≤f(0)1\ndσ=/parenleftbiggf(0)\nmax(f)/parenrightbigg1\nd\nLf≤Ld.\nUsing that Lf≥1/√\n2πe, we conclude.\n72.2 A concentration lemma\nThe following concentration lemma will be required through out the work.\nLemma 15 (Concentration Lemma) .Letcd:= 31\ndCd, whereCdis the constant from (16). Then, for\nevery log-concave, isotropic, centered density function fand for every x∈Rdsuch that /⌊ard⌊lx/⌊ard⌊l2≥cd/f(0)1\nd,\nf(x)≤f(0)2−/⌊ard⌊lx/⌊ard⌊l2f(0)1\nd\ncd. (18)\nProof.Letxmax∈Rdbe a value where the maximum of fis attained and for every θ∈Sd−1, let\nrmax(θ)∈Rdbe a value where the maximum of r/ma√sto→f(rθ) is attained. Since fis log-concave, for every\nθ∈Sd−1, the function r/ma√sto→f(rθ) is non-increasing for r≥rmax(θ). Therefore, for every r0(θ)≥rmax(θ),\n/integraldisplay∞\n0drd−1f(rθ)dr≥/integraldisplayr0(θ)\nrmax(θ)drd−1f(rθ)dr≥f(r0(θ))(r0(θ)d−rmax(θ)d). (19)\nWe choose r0=r0(θ) as\nr0(θ) =/parenleftBig\n2d/integraldisplay∞\n0rd−1f(rθ)\nf(0)dr+rd\nmax/parenrightBig1\nd, (20)\nso that, by (19), one has f(r0(θ))≤f(0)\n2. Sincer/ma√sto→f(rθ) is non-decreasing on [0 ,rmax(θ)], one has\n/integraldisplay+∞\n0drd−1f(rθ)dr≥/integraldisplayrmax(θ)\n0drd−1f(0)dr=f(0)rmax(θ)d,\nand therefore, by Lemma 14,\nrmax(θ)≤/parenleftbig\nd/integraltext+∞\n0rd−1f(rθ)dr/parenrightbig1\nd\nf(0)1\nd≤Cd\nf(0)1\nd,\nwhereCdis given by (16). Thus, by (20) and the definition of cd,\nr0(θ)≤/parenleftbigg3\nf(0)/parenrightbigg1\nd\nCd=cd\nf(0)1\nd.\nSince for every r≥r0(θ), one has r0(θ) =r0(θ)\nr·r+(1−r0(θ)\nr)·0, by log-concavity, we deduce\nf(r)≤f(0)/parenleftbiggf(r0(θ))\nf(0)/parenrightbiggr\nr0(θ)\n≤f(0)2−r\nr0(θ)≤f(0)2−rf(0)1\nd\ncd.\nHence, if /⌊ard⌊lx/⌊ard⌊l2≥cd/f(0)1\nd, then\nf(x)≤f(0)2−/⌊ard⌊lx/⌊ard⌊l2f(0)1\nd\ncd.\n2.3 Sum of maxima of isotropic log-concave functions\nThe following lemma bounds the sum of the maxima of a log-conc ave density using our previous concen-\ntration result.\nLemma 16. Letfbe a centered, isotropic, log-concave density on Rdwith covariance σ2Id. Let0≤i≤2\nand0≤j≤d−1. Then, as σ→ ∞,\n/summationdisplay\nl∈Zd−j|l1|imax\nk∈Zjf(k1,...,k j,l1,...,l d−j) =Od/parenleftBig1\nσj−i/parenrightBig\n. (21)\nProof.Setλ=cdf(0)−1/d, withcdis given in Lemma 15. Then f(x)≤f(0)2−/⌊ard⌊lx/⌊ard⌊l2/λ, for/⌊ard⌊lx/⌊ard⌊l2≥λ.\nMoreover, from the definition of Lf, its bounds and the inequality f(0)≥e−dmax(f), we have\ncdσ\nLd≤λ=cdf(0)−1/d≤cdeσ\nLf≤cd√\n2πe3σ.\n8Note that\n/summationdisplay\nl∈Zd−j|l1|imax\nk∈Zjf(k,l) =/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞≤λ|l1|imax\nk∈Zjf(k,l)+/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λ|l1|imax\nk∈Zjf(k,l),\nThe first sum is easily upper bounded by Od(1\nσj−i). Indeed,\n/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞≤λ|l1|imax\nk∈Zjf(k,l)≤(maxf)(2λ)d+i−j=Ld\nf\nσd(2λ)d+i−j≤Ld\nd(2cd√\n2πe3)d+i−j\nσj−i=Od/parenleftbigg1\nσj−i/parenrightbigg\n.\nUsing the tails estimates of Lemma 15 and the fact that for l∈Zd−j, one has /⌊ard⌊ll/⌊ard⌊l2≥ /⌊ard⌊ll/⌊ard⌊l∞and\n/⌊ard⌊ll/⌊ard⌊l2≥/⌊ard⌊ll/⌊ard⌊l1√d−j≥/⌊ard⌊ll/⌊ard⌊l1√\ndthe second sum can be expressed as follows\n/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λ|l1|imax\nk∈Zjf(k,l)≤f(0)/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λ|l1|i2−/bardbll/bardbl1\nλ√\nd\n=f(0)\n/summationdisplay\n|l1|>λ|l1|i2−|l1|\nλ√\nd\n\n/summationdisplay\n|n|>λ2−|n|\nλ√\nd\nd−j−1\n.\nPuttingx= 2−1/λ√\ndand assuming that λ√\nd≥1, we have x∈[1\n2,1) and1\n2≤λ√\nd(1−x)≤ln(2). Thus\n/summationdisplay\n|n|>λ2−|n|\nλ√\nd= 2/summationdisplay\nn>λxn≤2\n1−x≤4λ√\nd.\nHence, for i= 0, replacing λby its value, for σ≥Ld√\ndcd, we have λ≥1/√\ndand thus\n/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λmax\nk∈Zjf(k,l)≤f(0)/parenleftBig\n4λ√\nd/parenrightBigd−j\n=/parenleftBig\n4√\ndcd/parenrightBigd−j\nf(0)j\nd≤(4√\ndcd)d−jLj\nd\nσj=Od/parenleftbigg1\nσj/parenrightbigg\n.\nAnd, for i= 1, it is not difficult to see that, for λ≥1/√\nd, we have\n/summationdisplay\n|l1|>λ|l1|2−|l1|\nλ√\nd= 2/summationdisplay\nn>λnxn= 2x/parenleftBigg/summationdisplay\nn>λxn/parenrightBigg′\n≤6\n(1−x)2≤24dλ2.\nThus, again for σ≥Ld√\ndcd, which ensures that λ≥1/√\nd, we deduce\n/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λ|l1|max\nk∈Zjf(k,l)≤f(0)×24dλ2/parenleftBig\n4λ√\nd/parenrightBigd−j−1\n≤2f(0)/parenleftBig\n4λ√\nd/parenrightBigd−j+1\n=Od/parenleftbigg1\nσj−1/parenrightbigg\n.\nIn the same way, for i= 2, we easily get that, for λ≥1/√\nd,\n/summationdisplay\n|l1|>λl2\n12−|l1|\nλ√\nd= 2/summationdisplay\nn>λn2xn= 2x2/parenleftBigg/summationdisplay\nn>λxn/parenrightBigg′′\n+2/summationdisplay\nn>λnxn≤14\n(1−x)3+12\n(1−x)2≤26(2λ√\nd)3.\nThus, again for σ≥Ld√\ndcd, which ensures that λ≥1/√\nd, we deduce\n/summationdisplay\nl∈Zd−j,/⌊ard⌊ll/⌊ard⌊l∞>λl2\n1max\nk∈Zjf(k,l)≤f(0)×26(2λ√\nd)3/parenleftBig\n4λ√\nd/parenrightBigd−j−1\n≤4f(0)/parenleftBig\n4λ√\nd/parenrightBigd−j+2\n=Od/parenleftbigg1\nσj−2/parenrightbigg\n.\n92.4 Approximating the integral, mean and covariance discre tely\nIn this section, we will approximate the isotropic constant of an isotropic, log-concave density f∈Rdin\na discrete way. To do so, we will first approximate the integra l as a sum and subsequently approximate\nthe continuous covariance of fby its discrete covariance.\nThe following proposition allows us to approximate the inte gral of an isotropic log-concave function\nby its sum.\nProposition 17. Letf:Rd→Rbe a log-concave isotropic density function with covarianc e matrix of\nthe form σ2Id. Then/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdfdx−/summationdisplay\nk∈Zdf(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=Od/parenleftBig1\nσ/parenrightBig\n.\nProof.Let us proceed by induction. Let us first consider the one dime nsional base case. Let f:R→R\nbe a log-concave function and let suppose that the maximum of fis attained at x0∈R. Letk0∈Zsuch\nthatk0≤x0< k0+1. Then, one has\n/integraldisplay\nRfdx=/summationdisplay\nk∈Z/integraldisplayk+1\nkf(x)dx\n≥/summationdisplay\nk0.\nSince for every y∈Zd−1, the function x/ma√sto→f(x,y) is quasi-concave, one may use (22) to get that the\nsecond term is upper bounded by/summationdisplay\ny∈Zd−1max\nx∈Zf(x,y).\nWe conclude by Lemma 16.\nA remark from convex geometry: the case of isotropic convex b odies. Applying the\nprevious proposition to f=1K/|K|d, whereKis an isotropic convex body in Rdand using that, in this\ncase,σ=LK|K|1/d, we deduce the following proposition.\nProposition 18. For any isotropic convex body KinRd,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle#(K∩Zd)\n|K|d−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Md\n|K|1\nd,\nfor some constant Md>0, depending only on the dimension, where #Kdenotes the cardinality of a\ndiscrete set K.\n10Remark 19. The hypothesis of isotropicity is necessary, since in Zd, ford≥2, it is easy to construct\nconvex bodies that do not contain integer points, but whose v olumes are increasingly large.\nProposition 20. Letf:Rd→Rbe a centered, isotropic, log-concave density. Then, for 1≤i≤d,\n/summationdisplay\nk∈Zdkif(k) =/integraldisplay\nRdxif(x)dx+Od(1) =Od(1).\nIn particular, for d= 1, the following finite bound holds for any log-concave center ed integrable function\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRxf(x)dx−/summationdisplay\nk∈Zkf(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(e+1)/summationdisplay\nk∈Zf(k).\nProof.The proof is by induction again. Let us make a one dimensional remark that will be used on the\nbase case of the induction. Let f:R→Rbe a centered log-concave function and let suppose that the\nmaximum of fis attained at x0∈Rand let us suppose without loss of generality that x0>0. Then\nsincefis log-concave and by the inequality [6, Theorem 4] one has\n/integraldisplay\nRf(x)dx≥/integraldisplayx0\n0f(x)dx≥f(0)x0≥maxf\nex0. (23)\nHence, we get x0maxf≤e/integraltext\nRf(x)dx. The case of the summation is analogous. Let k0∈Zbe such that\nk0≤x0< k0+1 . Then, we deduce\n/summationdisplay\nk∈Zf(k)≥k0/summationdisplay\nk=0f(k)≥(k0+1)f(0)≥maxf\ne(k0+1). (24)\nLet us start the induction. Let fbe a centered, isotropic, log-concave density and let suppo se that the\nmaximum of fis attained at x0. We decompose the integral/integraltext\nRxf(x)dxas follows\n/integraldisplay\nRxf(x)dx=/summationdisplay\nk<0/integraldisplayk+1\nkxf(x)dx+k0−1/summationdisplay\nk=0/integraldisplayk+1\nkxf(x)dx+/integraldisplayk0+1\nk0xf(x)dx+/summationdisplay\nk≥k0+1/integraldisplayk+1\nkxf(x)dx\n≥/summationdisplay\nk<0kf(k+1)+k0−1/summationdisplay\nk=0kf(k)+k0min{f(k0),f(k0+1)}+/summationdisplay\nk≥k0+1kf(k+1)\n≥/summationdisplay\nk∈Zkf(k)−k0f(k0)−k0f(k0+1)+k0min{f(k0),f(k0+1)}−/summationdisplay\nk∈Zf(k).\nTherefore,\n/integraldisplay\nRxf(x)dx≥/summationdisplay\nk∈Zkf(k)−/summationdisplay\nk∈Zf(k)−k0maxf\n≥/summationdisplay\nk∈Zkf(k)−/summationdisplay\nk∈Zf(k)−e/summationdisplay\nk∈Zf(k) =/summationdisplay\nk∈Zkf(k)−(e+1)/summationdisplay\nk∈Zf(k),\nwhere in the last line we have used the inequality (24). Bound ing from the above/integraltext\nRxf(x)dxwe obtain\na similar inequality. Thus, we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRxf(x)dx−/summationdisplay\nk∈Zkf(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤(e+1)/summationdisplay\nk∈Zf(k),\nand the one-dimensional case follows by Proposition 17.\nFor the inductive step, let F:Rd−1→Rbe defined, for x∈R,y∈Rd−1, by\nF(y) :=/integraldisplay\nRf(x,y)dx.\nThenFis log-concave, centered and isotropic as the marginal of a l og-concave, centered and isotropic\ndensity and, for i= 1,...,d−1\n/integraldisplay\nRdyif(x,y)dxdy=/integraldisplay\nRd−1yiF(y)dy.\n11We have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdyif(x,y)dxdy−/summationdisplay\nk1∈Z,k∈Zd−1kif(k1,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdyif(x,y)dxdy−/summationdisplay\nk∈Zd−1kiF(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈Zd−1kiF(k)−/summationdisplay\nk∈Zdkif(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nBut the first term is Od(1) by inductive hypothesis and for the second we note that f(·,k) is log-concave\nfor every k∈Zd−1, so we may apply (22) to get, for any k∈Zd−1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF(k)−/summationdisplay\nk1∈Zf(k1,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRf(x,k)dx−/summationdisplay\nk1∈Zf(k1,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤max\nx∈Rf(x,k),\nwhich gives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdyif(x,y)dxdy−/summationdisplay\nk1∈Z,k∈Zd−1kif(k1,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Od(1)+/summationdisplay\nk∈Zd−1kimax\nx∈Rf(x,k) =Od/parenleftbig\n1/parenrightbig\n,\nwhere the last equality follows by Lemma 16.\nNow we wish to show that a continuous isotropic, log-concave densityfinRdis almost isotropic in\nthe discrete sense, meaning that its discrete covariance ma trix Cov Z(f) hasO(σ) off-diagonal elements\nandσ2+O(σ) diagonal elements.\nProposition 21. Letf:Rd→Rbe a centered, isotropic, log-concave density with Cov(f) =σ2Id. Then\n/summationdisplay\nk∈Zdf(k)k2\ni=σ2+Od/parenleftbig\nσ/parenrightbig\n,for every 1≤i≤d.\nProof.We proceed by induction on the dimension. For the one-dimens ional case let\nx0= inf{x∈R:f(x) = max f}. Then\n/integraldisplay\nRf(x)x2=/integraldisplay\nx≤x0f(x)x2dx+/integraldisplay\nx>x0f(x)x2dx.\nNoting that fis non-decreasing on ( −∞,x0] and non-increasing on [ x0,+∞) by log-concavity, we get\n/integraldisplay\nRf(x)x2dx=/summationdisplay\nk<⌊x0⌋/integraldisplay\n[k,k+1)f(x)x2dx+/summationdisplay\nk>⌊x0⌋/integraldisplay\n[k,k+1)f(x)x2dx+/integraldisplay⌊x0⌋+1\n⌊x0⌋f(x)x2dx\n≤/summationdisplay\nk<⌊x0⌋f(k+1)(k2+k+1\n3)+/summationdisplay\nk>⌊x0⌋f(k)(k2+k+1\n3)+/integraldisplay⌊x0⌋+1\n⌊x0⌋f(x)x2dx\n≤/summationdisplay\nk∈Zf(k)k2−/summationdisplay\nk<⌊x0⌋f(k+1)k−/summationdisplay\nk>⌊x0⌋f(k)k+f(x0)/parenleftBig\n⌊x0⌋2+⌊x0⌋+1\n3/parenrightBig\n≤/summationdisplay\nk∈Zf(k)k2−/summationdisplay\nk∈Zf(k)k+f(x0)/parenleftBig\n⌊x0⌋2+⌊x0⌋+1\n3/parenrightBig\n(25)\nNow we observe that f(x0) = max f≤1\nσand, by (23), x0=O(σ). Hence, the last term in (25) is O(σ).\nFurthermore, by Proposition 17, the second term in (25) is O(1). Thus,\n/integraldisplay\nRf(x)x2dx≤/summationdisplay\nk∈Zf(k)k2+O(σ).\nSimilarly, using the piecewise monotonicity of fin the reverse way, we obtain\n/integraldisplay\nRf(x)x2dx≥/summationdisplay\nk∈Zf(k)k2+O(σ).\nThis proves the base case.\n12Letx∈R,y∈Rd−1and letF(y) :=/integraltext\nRf(x,y)dx. Then, by the inductive hypothesis applied to F,\nfori= 1,...,d−1,\n/integraldisplay\nRdf(x,y)y2\nidxdy=/integraldisplay\nRd−1F(y)y2\nidy=/summationdisplay\nk∈Zd−1F(k)k2\ni+Od(σ). (26)\nSince, for every k∈Zd−1, the function f(·,k) is log-concave in R, we have, by (22),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleF(k)−/summationdisplay\nk′∈Zf(k′,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRf(x,k)dx−/summationdisplay\nk′∈Zf(k′,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤max\nxf(x,k). (27)\nSo by (26) and (27) we get, for i= 1,...,d−1\n/integraldisplay\nRdf(x,y)y2\nidxdy=/summationdisplay\nk′∈Z,k∈Zd−1f(k′,k)k2\ni+/summationdisplay\nk∈Zd−1O(max\nxf(x,k))k2\ni+Od(σ)\n=/summationdisplay\nk′∈Z,k∈Zd−1f(k′,k)k2\ni+Od(σ),\nby Lemma 16, and the result follows.\nWe will need the following lemma for the base case of the induc tion of Proposition 23.\nLemma 22. Letfbe a centered, isotropic, log-concave density in R2, withCov(f) =σ2I2. Forx∈R,\nlet\nNmax(x) := inf{y∈R:f(x,y) = max\nyf(x,y)}.\nThen /integraldisplay\nRxNmax(x)max\nyf(x,y)dx=O(σ) (28)\nand /summationdisplay\nk∈ZkNmax(k)max\nyf(k,y) =O(σ). (29)\nProof.Letcbe an absolute constant to be determined later and denote the setAσ:={x∈R:\n|x|,|Nmax(x)| ≤cσ}.We have\n/integraldisplay\nRxNmax(x)max\nyf(x,y)dx=/integraldisplay\nAσxNmax(x)max\nyf(x,y)dx+/integraldisplay\nACσxNmax(x)max\nyf(x,y)dx\nThe first term can be trivially bounded as\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nAσxNmax(x)max\nyf(x,y)dx/vextendsingle/vextendsingle/vextendsingle≤O(σ3)maxf=O(σ).\nFor the second term we have, for some absolute constant C′\n2and forclarge enough,\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nACσ|x|Nmax(x)max\nyf(x,y)dx/vextendsingle/vextendsingle/vextendsingle≤max(f)/integraldisplay\nACσxNmax(x)e−|x|+Nmax(x)\nσC′\n2dx\n≤O/parenleftBig1\nσ2/parenrightBig\nσ/integraldisplay\nx>cσxe−x\nσC′\n2dx (30)\n=O(σ). (31)\nHere (30) follows by using the fact that ye−y≤1. This gives (28), (29) follows in a similar way.\nProposition 23. Letfbe a centered, continuous, isotropic, log-concave density inRd, withCov(f) =\nσ2Id. Then, for all i/ne}ationslash=j/summationdisplay\nk∈Zdf(k)kikj=Od(σ).\n13Proof.We proceed by induction on d. Consider first the base case d= 2. Then, letting Nmax(x) :=\ninf{y∈R:f(x,y) = max yf(x,y)}andMmax(y) := inf{x∈R:f(x,y) = max xf(x,y)}and assuming\nwithout loss of generality that the infima are achieved,\n/integraldisplay\nR2\n+f(x,y)xydx=/integraldisplay\nR+x/integraldisplay\n0≤y≤Nmax(x)f(x,y)ydy+/integraldisplay\ny>Nmax(x)f(x,y)ydydx.\nNow/integraldisplay\n0≤y≤Nmax(x)f(x,y)ydy≤/summationdisplay\n0≤k≤Nmax(x)/integraldisplay\n[k,k+1)f(x,y)ydy\n≤/summationdisplay\n0≤k≤Nmax(x)−1(k+1)f(x,k+1)+max\nyf(x,y)(Nmax(x)+1) (32)\nand/integraldisplay\ny>Nmax(x)f(x,y)ydy≤/summationdisplay\nk≥Nmax(x)(k+1)f(x,k). (33)\nBy (32) and (33)\n/integraldisplay\nR2\n+f(x,y)xydx≤/integraldisplay\nRx/parenleftBig/summationdisplay\nk≥0kf(x,k)+/summationdisplay\nk≥Nmax(x)f(x,k)+max\nyf(x,y)(Nmax(x)+1)/parenrightBig\ndx (34)\nFor the first term in (34) we have\n/integraldisplay\nRx/summationdisplay\nk≥0kf(x,k)dx≤/summationdisplay\nk≥0/summationdisplay\n0≤m≤Mmax(k)−1k(m+1)f(m+1,k)+/summationdisplay\nk≥0/summationdisplay\nm>M max(k)k(m+1)f(m,k)\n+/summationdisplay\nk≥0k(Mmax(k)+1)max\nxf(x,k)\n≤/summationdisplay\nk≥0/summationdisplay\nm≥0kmf(m,k)+/summationdisplay\nk≥0/summationdisplay\nm≥0kf(m,k)+/summationdisplay\nk≥0k(Mmax(k)+1)max\nxf(x,k) (35)\n≤/summationdisplay\nk≥0/summationdisplay\nm≥0kmf(m,k)+/summationdisplay\nk≥0/summationdisplay\nm≥0kf(m,k)+O(σ), (36)\nwhere the last inequality follows by Lemma 22. On the other ha nd, for the second term in (34) we have\n/integraldisplay\nR/summationdisplay\nk≥Nmaxxf(x,k)dx≤/parenleftBig/integraldisplay\nR/summationdisplay\nk∈Zx2f(x,k)dx/parenrightBig1\n2=O(σ) (37)\nby the Cauchy-Schwarz inequality and Proposition 21. Final ly, the third term in (34) is\n/integraldisplay\nRmax\nyf(x,y)(Nmax(x)+1)dx=O(σ) (38)\nby Lemma 22. Plugging (36), (37) and (38) into (34) we obtain\n/integraldisplay\nR2\n+f(x,y)xydx≤/summationdisplay\nk≥0/summationdisplay\nm≥0kmf(m,k)+/summationdisplay\nk≥0/summationdisplay\nm≥0kf(m,k)+O(σ). (39)\nUsing the reverse bounds to those in (32) and (33) obtained by the monotonicity of f, we get the lower\nbound /integraldisplay\nR2\n+f(x,y)xydx≥/summationdisplay\nk≥0/summationdisplay\nm≥0kmf(m,k)+/summationdisplay\nk≥0/summationdisplay\nm≥0kf(m,k)+O(σ).\nIn a completely analogous way, we may bound the integral on th e rest of the quadrants and obtain the\napproximations\n/integraldisplay\nR2\n−f(x,y)xydx=/summationdisplay\nk≤0/summationdisplay\nm≤0kmf(m,k)+/summationdisplay\nk≤0/summationdisplay\nm≤0kf(m,k)+O(σ), (40)\n/integraldisplay\nR−×R+f(x,y)xydx=/summationdisplay\nk≥0/summationdisplay\nm≤0kmf(m,k)+/summationdisplay\nk≥0/summationdisplay\nm≤0kf(m,k)+O(σ) and (41)\n/integraldisplay\nR+×R−f(x,y)xydx=/summationdisplay\nk≤0/summationdisplay\nm≥0kmf(m,k)+/summationdisplay\nk≤0/summationdisplay\nm≥0kf(m,k)+O(σ). (42)\n14Since, by Proposition 20,/summationtext\nk/summationtext\nmkf(m,k) =O(1), by adding (39)–(42) we obtain\n0 =/integraldisplay\nR2f(x,y)xydx=/summationdisplay\nk∈Z/summationdisplay\nm∈Zkmf(m,k)+O(σ).\nThis completes the base case d= 2.\nThe inductive step is similar to that in the proof of Proposit ion 21. Let x∈R,y= (y1,...,y d−1)∈\nRd−1and letF(y) :=/integraltext\nRf(x,y)dx. Then, by the inductive hypothesis, for 1 ≤i,j≤d−1,\n/integraldisplay\nRdf(x,y)yiyjdy=/integraldisplay\nRd−1F(y)yiyjdy=/summationdisplay\nk∈Zd−1F(k)kikj+Od(σ). (43)\nSince for every k= (k1,...,k d−1)∈Zd−1, the function f(·,k) is log-concave R, we have by Proposition\n17\nF(k)−/summationdisplay\nk′∈Zf(k′,k) =/integraldisplay\nRf(x,k)dx−/summationdisplay\nk′∈Zf(k′,k) =Od/parenleftbig\nmax\nxf(x,k)/parenrightbig\n. (44)\nSo by (43) and (44) we get, for 1 ≤i,j≤d−1\n/integraldisplay\nRdf(x,y)yiyjdxdy=/summationdisplay\nk′∈Z,k∈Zd−1f(k′,k)kikj+/summationdisplay\nk∈Zd−1O/parenleftBig\nmax\nxf(x,k)/parenrightBig\nkikj+Od(σ).(45)\nBut by the Cauchy-Schwarz inequality and Lemma 16,/summationtext\nk∈Zd−1O/parenleftBig\nmaxxf(x,k)/parenrightBig\nkikj=Od(σ) and the\nresult follows.\nCorollary 24. Letfbe a centered, isotropic, log-concave density in Rd, withCov(f) =σ2Id.Then\ndet/parenleftBig\nCovZd(f)/parenrightBig\n=σ2d+Od(σ2d−1).\nProof.By Proposition 23\n[CovZd(f)]ij=Od(σ),fori/ne}ationslash=j\nand\n[CovZd(f)]ij=σ2+Od(σ),fori=j.\nThus,\ndet/parenleftBig\nCovZd(f)/parenrightBig\n=/summationdisplay\nτsgn(τ)d/productdisplay\ni=1CovZd(f)iτ(i)\n=σ2d+O(σ2d−1),\nsince for τbeing the identity permutation sgn( τ)/producttextd\ni=1CovZd(f)iτ(i)=/parenleftbig\nσ2+O(σ)/parenrightbigd=σ2d+O(σ2d−1)\nand for any other permutation τ′\nd/productdisplay\ni=1CovZd(f)iτ′(i)=O((σ2)d−2σ2) =Od(��2d−2).\nFinally, Corollary 25 below is a version of Theorem 11 with th e assumption that the continuous\nfunction fis isotropic. It can be observed from the proof that the depen dence of of C′\ndondcan be taken\nto be the same as in the continuous case (cf. (10)). It can also be seen that due to use of Corollary 24,\nσneeds to be taken at least Ω( d!). We do not know whether this is the best (lowest) rate.\nCorollary 25 (Theorem 11 for isotropic f).Letpbe log-concave p.m.f. on Zd, whose continuous\nlog-concave extension, say f, is isotropic. Then there exists a constant C′\ndthat depends on the dimension\nonly, such that\nmax\nk∈Zdp(k)≤C′\nd\ndet/parenleftBig\nCov(p)/parenrightBig1\n2\nprovided that σ:= det/parenleftBig\nCovR(f)/parenrightBig\nis large enough depending on d.\n15Proof.Sincepis extensible log-concave, there exists a continuous log-c oncave function f(not necessarily\na density) such that f(k) =p(k) for allk∈Zdand by assumption fis isotropic. Thus\nmax\nk∈Zdp(k)≤max\nx∈Rdf(x) =Ld\nf/integraltext\nRdf\nσd≤Cd/integraltext\nRdf\nσd, (46)\nwhereCdis an upper bound of Ld\nf. Noting that by definition Cov( Zp) = Cov( p) for any normalising\nconstant Z∈R+and applying Proposition 17 and Corollary 24 tof(·)/integraltext\nf, we obtain\nmax\nk∈Zdp(k)≤Cd(1+Od(1\nσ))\n/parenleftBig\ndet/parenleftBig\nCov(p)/parenrightBig\n+Od(σ2d−1)/parenrightBig1/2≤2Cd\ndet/parenleftBig\nCov(p)/parenrightBig1\n2\nprovided that σis large enough depending on d, using that Od(σ2d−1)≥ −1\n2det/parenleftBig\nCov(p)/parenrightBig\n≃ −1\n2σ2das\nσ→ ∞.\n3 Relaxing the isotropicity assumption on f\nIn this section, we relax the assumptions of Lemma 14, so that we obtain conclusions of the same\nkind under some weaker assumptions that relate to the eigenv alues of the continuous covariance matrix\nCovR(f). Specifically, we will prove:\nProposition 26. For every centered and log-concave density f:Rd→Rand for every θ∈Sd−1one\nhas\nf(0)/parenleftbig/radicalbig\nλmin(Cov(f))/parenrightbigd/lessorsimilard/integraldisplay∞\n0rd−1f(rθ)dr/lessorsimilardf(0)/parenleftbig/radicalbig\nλmax(Cov(f))/parenrightbigd.\nCorollary 27. Letfbe a a centered, almost isotropic log-concave density. Then for every θ∈Sd−1\n/integraldisplay∞\n0rd−1f(rθ)dr= Θd(1),\nasσ:= det(Cov( f))→ ∞.\nProof.Sincefis almost isotropic, we have\nλmin/parenleftbig\nCov(f)/parenrightbig\n≃dλmax/parenleftbig\nCov(f)/parenrightbig\n≃dσ2\nasσ→ ∞. The result follows from Proposition 26 since f(0)≃dmaxf≃d1\nσd.\nRemark 28. The isotropicity assumption was only used in Lemma 14, which in turn allowed us to prove\nthe concentration Lemma 15 which was repeatedly evoked in Se ction 2 to bound the sums of maxima of\nlog-concave functions and thus also the error terms. By Coro llary 27, a version of Lemma 15 holds\ntrue for almost isotropic densities, although with possibl y different constants and for large enough σ.\nTherefore, all the results of Section 2 hold true and thus als o Theorem 11.\nRemark 29. Note that if the function fis in addition isotropic i.e. Cov(f) =σ2Id, one has that\nf(0)≃d1\nσdand so we recover the result of Lemma 14.\nFor the above-mentioned proposition, the following lemma w hich gives bounds for the inradius and\ncircumradius of convex bodies in not necessarily isotropic position will be essential .\nLemma 30 (Inradius and circumradius of convex bodies) .LetKbe a convex body of volume one in Rd.\nThen\nR(K)≤(d+1)||Cov(K)||1\n2op= (d+1)λmax(Cov(K))1\n2and\nr(K)≥/radicalbigg\nd+2\ndλmin(Cov(K))1\n2.\nwhereCov(K) = Cov R(f)withf=1K, i.e.Cov(K) = [/integraltext\nKyiyj]n\ni,j=1.\nThe proof is an adaptation of the proof of [15, Theorem 4.1].\n16Proof.Letx∈K. One defines h:Sd−1→Rby\nh(u) = max{t≥0 :x+tu∈K}.\nThe volume of Kcan be expressed as follows\n1 =|K|=dωd/integraldisplay\nSd−1/integraldisplayh(u)\n0td−1dtdσ(u) =wd/integraldisplay\nSd−1hd(u)dσ(u).\nFor allθ∈Sd−1,\n/integraldisplay\nK/an}⌊ra⌋ketle{ty,θ/an}⌊ra⌋ketri}ht2dy=dwd/integraldisplay\nSd−1/integraldisplayh(u)\n0td−1/an}⌊ra⌋ketle{tx+tu,θ/an}⌊ra⌋ketri}ht2dtdσ(u)\n=dwd/integraldisplay\nSd−1/integraldisplayh(u)\n0(td−1/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht2+2td/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tu,θ/an}⌊ra⌋ketri}ht+td+1/an}⌊ra⌋ketle{tu,θ/an}⌊ra⌋ketri}ht2)dtdσ(u)\n=dwd/integraldisplay\nSd−1(hd(u)\nd/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht2+2hd+1(u)\nd+1/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tu,θ/an}⌊ra⌋ketri}ht+hd+2(u)\nd+2/an}⌊ra⌋ketle{tu,θ/an}⌊ra⌋ketri}ht2)dσ(u)\n=dwd/integraldisplay\nSd−1/parenleftBigg\nhd(u)\nd(d+1)2/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht2+hd(u)/parenleftBigh(u)/an}⌊ra⌋ketle{tu,θ/an}⌊ra⌋ketri}ht√\nd+2+√\nd+2/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht\nd+1/parenrightBig2/parenrightBigg\ndσ(u)\n≥/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht2\n(d+1)2wd/integraldisplay\nSd−1hd(u)dσ(u) =/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht2\n(d+1)2.\nThus, for every x∈Kand every θ∈Sd−1,\n|/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht| ≤(d+1)/parenleftbig/integraldisplay\nK/an}⌊ra⌋ketle{ty,θ/an}⌊ra⌋ketri}ht2dy/parenrightbig1\n2.\nLet us note that\n/an}⌊ra⌋ketle{tCov(K)θ,θ/an}⌊ra⌋ketri}ht=/summationdisplay\nijCov(K)ijθiθj=/integraldisplay\nK/summationdisplay\ni,jyiyjθiθjdy=/integraldisplay\nK/an}⌊ra⌋ketle{ty,θ/an}⌊ra⌋ketri}ht2dy.\nTherefore, taking the maximum over θ∈Sd−1one gets\n||x||2= max\nθ∈Sd−1|/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht| ≤(d+1)||Cov(K)||1\n2op,\nand that concludes the first part of the proof.\nOn the other hand, by Proposition 8 of the Phd’s thesis of Frad elizi, we have the following inequality\n/radicalbigg\nd+2\nd/an}⌊ra⌋ketle{tCov(K)θ,θ/an}⌊ra⌋ketri}ht1\n2=/radicalbigg\nd+2\nd/parenleftbig/integraldisplay\nK/an}⌊ra⌋ketle{ty,θ/an}⌊ra⌋ketri}ht2dy/parenrightbig1\n2≤hK(θ).\nMinimizing over θ∈Sd−1on the left-hand side of the equation one gets\nh\n(/radicalBig\nd+2\ndλmin(Cov(K))1\n2)Bd\n2(θ) =/radicalbigg\nd+2\ndλmin(Cov(K))1\n2≤hK(θ),\nand the result follows.\nProof of Proposition 26. Letf:Rd→Rbe a centered log-concave function with density one. Since\n/tildewideKd+2(f) =Kd+2(f)\n|Kd+2(f)|1\ndis a convex body of volume 1, we can apply the previous Lemma. B y the inradius\nand circumradius bounds on convex bodies of volume one, the f ollowing inclusion holds\n/radicalbigg\nd+2\ndλmin(Cov(/tildewideKd+2(f)))1\n2Bd\n2⊂/tildewideKd+2(f) =Kd+2(f)\n|Kd+2(f)|1\nd⊂(d+1)λmax(Cov(/tildewideKd+2(f))1\n2Bd\n2.\nThus,\nKd+2(f)⊆(d+1)|Kd+2(f)|1\nd/radicalBig\nλmax(Cov(/tildewideKd+2(f))Bd\n2.\n17Note that the inequality/integraltext\nKd+2(f)|/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht|2dx=1\nf(0)/integraltext\nRn|/an}⌊ra⌋ketle{tx,θ/an}⌊ra⌋ketri}ht|2f(x)dx(see [5, Proposition 2.5.3 (ii)])\ncan be reformulated as Cov( /tildewideKd+2(f))≃df(0)2\ndCov(f).\nIn that way, using the previous equivalence and the fact that |Kd+2(f)|1\nd≃d|Kd(f)|1\nd=1\nf(0)1\ndwe\nfinally get\n/radicalbigg\nd+2\nd/radicalbig\nλmin(Cov(f)Bd\n2⊆Kd+2(f)⊆(d+1)/radicalbig\nλmax(Cov(f))Bd\n2.\nThe latter translates in terms of the radial function as/radicalbig\nλmin(Cov(f)/lessorsimilardρKd+2(f)(θ)/lessorsimilard/radicalbig\nλmax(Cov(f))\nfor allθ∈Sd−1. Since,ρKd(f)≃dρKd+2(f)one finally gets\nf(0)/parenleftbig/radicalbig\nλmin(Cov(f))/parenrightbigd/lessorsimilard/integraldisplay∞\n0drd−1f(rθ)dr/lessorsimilardf(0)/parenleftbig/radicalbig\nλmax(Cov(f))/parenrightbigd,\nand the conclusion follows.\n4 Proof of the monotonicity of discrete entropy in Zd\nFirst we need two technical lemmas.\nLemma 31. Fixd≥1and letpbe a log-concave p.m.f. on Zdwith almost isotropic extension. Then,\nfor every 1≤i≤d,\n/summationdisplay\nk∈Zd|p(k)−p(k−ei)|=Od/parenleftBigg\n1\ndet(Cov( p))1\n2d/parenrightBigg\n,\nwhereei∈Zdis defined as the vector with the i-th coordinate 1and all the other coordinates 0.\nProof.By log-concavity, for any one-dimensional quasi-concave n on-negative real sequence ( an)n∈Z, one\nhas/summationtext\nn∈Z|an−an−1| ≤2max n∈Zan. Thus, denoting ki=/summationtext\nj/ne}ationslash=ikjej∈Zd−1, one has\n/summationdisplay\nk∈Zd|p(k)−p(k−ei)| ≤2/summationdisplay\nki∈Zd−1max\nki∈Zp(k1,...,k i−1,ki,ki+1,...,k d),\nHence, as in (21), we have\n/summationdisplay\nki∈Zd−1max\nki∈Zp(k1,...,k i−1,ki,ki+1,...,k d) =Od/parenleftBig1\ndet/parenleftbig\nCov(p)/parenrightbig1\n2d/parenrightBig\nand the result follows.\nThe following result is a dimensional generalisation of [8, Lemma 2].\nLemma 32. Fixn,d≥1and letSbe a log-concave random vector on Zdwith almost isotropic extension.\nLetU1,...,U nbe i.i.d. continuous uniforms on the unit cube. Let fndenote the density of S+/summationtextn\ni=1Ui\nandpSthe p.m.f. of S. Then, for each k∈Zd,x∈k+[0,1)dif we define\ng(k,x) =fn(x)−pS(k),then/summationdisplay\nk∈Zdsup\nx∈k+[0,1)dg(k,x) =Od/parenleftBig1\ndet/parenleftbig\nCov(pS)/parenrightbig1\n2d/parenrightBig\n.\nProof.We proceed by induction on n. Recall that the density of U1+U2is, forz∈[0,2)d, fU1+U2(z) =/producttextd\ni=1fi(zi),wherefi(u) =u,foru∈[0,1) andfi(u) = 2−u,foru∈[1,2).\nThus, we have for k∈Zd,x∈k+[0,1)d\nf2(x) =/summationdisplay\ns∈{0,1}dfU1+U2(s+x−k)pS(k−s)\n=/summationdisplay\ns∈{0,1}d/productdisplay\ni:si=0(xi−ki)/productdisplay\nj:sj=1(1−xj+kj)pS(k−s). (47)\n18Now we claim that (47) is\npS(k)+Od/parenleftBiggd/summationdisplay\ni=1/summationdisplay\ns∈{0,1}d,si=0|pS(k−s)−pS(k−s−ei)|/parenrightBigg\n, (48)\nwhere, as before, ei∈Zdis the vector with the i-th coordinate 1 and the other coordinates 0.\nTo justify (48) we argue by induction on d: Ford= 1,we have\n(x1−k1)pS(k)+(1−x1+k1)pS(k−1) =pS(k)+(1−x1+k1)(pS(k−1)−pS(k)),\nwhere (1 −x1+k1)∈[0,1]. Assuming the claim (48) for d−1 we have\n/summationdisplay\ns∈{0,1}d/productdisplay\ni:si=0(xi−ki)/productdisplay\nj:sj=1(1−xj+kj)pS(k−s)\n=/summationdisplay\ns′∈{0,1}d−1/productdisplay\ni:s′\ni=0(xi−ki)/productdisplay\nj:s′\nj=1(1−xj+kj)/parenleftBig\n(x1−k1)pS(k1,s′)+(1−x1+k1)pS(k1−1,s′)/parenrightBig\n=/summationdisplay\ns′∈{0,1}d−1/productdisplay\ni:s′\ni=0(xi−ki)/productdisplay\nj:s′\nj=1(1−xj+kj)/parenleftBig\npS(k1,s′)+(1−x1+k1)(pS(k1−1,s′)−pS(k1,s′))/parenrightBig\nand the claim (48) follows by the ( d−1) case and the fact that ( xi−ki),(1−xi+ki)∈[0,1].\nNow, using (48) we have\nf2(x) =pS(k)+g(k,x)\nwith\n/summationdisplay\nk′∈Zsup\nu∈k′+[0,1)d|g(k′,u)|=Od/parenleftBig\n2dd/summationdisplay\ni=1/summationdisplay\nk′∈Zd|pS(k′)−pS(k′−ei)|/parenrightBig\n(49)\nand the case n= 2 follows from Lemma 31.\nNext, we have\nfS+U(n+1)(x) =/integraldisplay\n[0,1)dfS+U(n)(x−u)du\n=/integraldisplay\n[0,1)d∩x−k−[0,1)dfS+U(n)(x−u)du+/integraldisplay\n[0,1)d∩x−k+[0,1)dfS+U(n)(x−u)du. (50)\nUsing the inductive hypothesis, denoting 1d∈Zdthe vector with all coordinates equal to 1, (50) is equal\nto\nλpS(k)+(1−λ)pS(k−1d)+/integraldisplay\n[0,1)d∩x−k−[0,1)dgn(k,x−u)du+/integraldisplay\n[0,1)d∩x−k+[0,1)dgn(k−1d,x−u)du\nfor some λ∈(0,1), andgn(k,u) satisfying (49). Thus, we can write\nfS+U(n+1)(x) =pS(k)+(1−λ)(pS(k−1d)−pS(k))\n+/integraldisplay\n[0,1)d∩x−k−[0,1)dgn(k,x−u)du+/integraldisplay\n[0,1)d∩x−k+[0,1)dgn(k−1d,x−u)du\n=pS(k)+gn+1(k,x), (51)\nwhere\n/summationdisplay\nk′∈Zsup\nu∈k′+[0,1)d|gn+1(k′,x)| ≤/summationdisplay\nk′∈Zd|pS(k′−1d)−pS(k′)|+2Od/parenleftBigd/summationdisplay\ni=1/summationdisplay\nk′∈Zd|pS(k′)−pS(k′−ei)|/parenrightBig\n.(52)\nBut\n/summationdisplay\nk′∈Zd|pS(k′−1d)−pS(k′)|=/summationdisplay\nk′∈Zd|pS(k′\n1−1,...,k′\nd−1)−pS(k′\n1,...,k′\nd)|\n≤/summationdisplay\nk′∈Zd|pS(k′\n1,...,k′\nd)−pS(k′\n1−1,k′\n2,...,k′\nd)|+/summationdisplay\nk′∈Zd|pS(k′\n1−1,k′\n2,...,k′\nd)−pS(k′\n1−1,...,k′\nd−1)|\n19and repeating dtimes we get\n/summationdisplay\nk′∈Zd|pS(k′−1d)−pS(k′)| ≤Od/parenleftBigd/summationdisplay\ni=1/summationdisplay\nk′∈Zd|pS(k′)−pS(k′−ei)|/parenrightBig\n. (53)\nThe inductive step now follows by (52), (53) and Lemma 31. The proof is complete.\nProof of Theorem 9. Assume without loss of generality that X1has zero mean. Let F(x) =xlog1\nx,x >0\nand note that F(x) is increasing for x≤1/e. Denote Sn=/summationtextn\ni=1Xi, U(n)=/summationtextn\ni=1Ui, and let fSn+U(n)\nbe the density of Sn+U(n)on theRdandpSnthe p.m.f. of Sn. We have\nh(X1+···+Xn+U1+···+Un)\n=/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx+/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2>σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx. (54)\nFirst we bound the second term. Note that for σlarge enough fSn+U(n)≤1/eso thatFis increasing.\nWe observe that from the proof of Lemma 32\nfSn+U(n)(x)≤C′\nd,npSn(k−Kd,n) (55)\nfor some constants C′\nd,n>0,Kd,n∈Zd\n+. Using Lemma 15 and the fact that the ℓ2andℓ∞norms are\ncomparable, we have\n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2>σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx\n/lessorsimilard,n/summationdisplay\nk:/⌊ard⌊lk−Kd,n/⌊ard⌊l2>σ2/integraldisplay\nk+[0,1)dF/parenleftBig\npSn(0)2−/⌊ard⌊lk−Kd,n/⌊ard⌊l2pSn(0)1\nd\nCd/parenrightBig\ndx (56)\n/lessorsimilard,n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l∞>σ2+K′\nd,n1\n(nσ)d2−C′′\nd/bardblk/bardbl∞\nnσ/⌊ard⌊lk/⌊ard⌊l∞\nnσ+/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l∞>σ2+K′\nd,n1\n(nσ)d2−C′′\nd/bardblk/bardbl∞\nnσlogσ\n/lessorsimilard,n1\nσd+1/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l∞>σ2+K′\nd,n2−C′′\nd,n/bardblk/bardbl∞\nσ/⌊ard⌊lk/⌊ard⌊l∞\n/lessorsimilard,n1\nσd+1σ2+d\n2C′′′\nd,nσ\n/lessorsimilard,nσ\n2C′′′\nd,nσ\nby a similar calculation to the one in the proof of Lemma 16. He reC′′\nd,n,C′′′\nd,n>0,K′\nd,n∈Zd\n+are again\nconstants that depend only on dandn.\nNow we will show that the first term in (54) is approximately H(Sn). Denoting by Bd\n2the standard\nEuclidean ball in Rd,\n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx≃d,nlogσ2/integraldisplay\nσ2Bd\n2fSn+U(n)(x)dx\n+/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)d(F(fSn+U(n)(x))−fSn+U(n)(x)logσ2)dx (57)\n= logσ2P/parenleftbig\nSn+U(n)∈σ2Bd\n2/parenrightbig\n+/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)d/parenleftbig\nF(fSn+U(n)(x))−fSn+U(n)(x)logσ2/parenrightbig\ndx. (58)\nTo justify (57), we note that/integraltext\nσ2Bd\n2fSn+U(n)(x)dx=P/parenleftbig\nSn+U(n)∈σ2Bd\n2/parenrightbig\n= 1−Od,n(1\nσ2) (see (65)\nbelow) and, using the elementary estimate /⌊ard⌊lx+1d/⌊ard⌊l2\n2=/⌊ard⌊lx/⌊ard⌊l2\n2+Od(/⌊ard⌊lx/⌊ard⌊l2),x∈Rd,we have/summationtext\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraltext\nk+[0,1)dfSn+U(n)(x)dx=P/parenleftbig\nSn+U(n)∈Od(σ2)Bd\n2/parenrightbig\n= 1−Od,n(1\nσ2) by the same reasoning.\n20Now we will apply the estimate of Lemma 34 to the integrand of t he second term in (58) with\nG(x) =F(x)−xlog(σ2), µ= Θd,n(1\nσ),D= Θd,n(σd),M=σ2d,a=fSn+U(n)(x) andb=pSn(k).The\nassumption of the lemma is satisfied since by (55) and Theorem 11,a,b/lessorsimilard,nmaxpSn(k)/lessorsimilard,n1\nσd.Thus,\nsince there are at most Θ( σ2d) elements in the set {k∈Zd:/⌊ard⌊lk/⌊ard⌊l2≤σ2},\n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nk+[0,1)dG(fSn+U(n)(x))dx−G(pSn(k))/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilard,nσ2dlogσ\nσ2d+1+logσ/summationdisplay\nk∈Zd/integraldisplay\nk+[0,1)d|fSn+U(n)(x)−pSn(k)|dx (59)\n/lessorsimilard,nlogσ\nσ+logσ/summationdisplay\nk∈Zdsup\nx∈k+[0,1)d|gn(k,x)| (60)\n/lessorsimilard,nlogσ\nσ, (61)\nwheregn(k,x) is given by Lemma 32 applied to the log-concave random varia bleSnand therefore\n/summationtext\nksupx∈k+[0,1)d|gn(k,x)|/lessorsimilardet/parenleftBig\nCov(Sn)/parenrightBig−1\n2d≃d,n1\nσ.Therefore, by (58) and (61),\n/vextendsingle/vextendsingle/vextendsingleH(Sn)−/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilard,n/vextendsingle/vextendsingle/vextendsingleH(Sn)−/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2G(pSn(k))−P/parenleftbig\nSn+U(n)∈σ2Bd\n2/parenrightbig/vextendsingle/vextendsingle/vextendsingle+logσ\nσ\n/lessorsimilard,n/vextendsingle/vextendsingle/vextendsingleH(Sn)−/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2F(pSn(k))/vextendsingle/vextendsingle/vextendsingle+logσ2/vextendsingle/vextendsingle/vextendsingleP/parenleftbig\nSn+U(n)∈σ2Bd\n2/parenrightbig\n−P/parenleftbig\nSn∈σ2Bd\n2/parenrightbig/vextendsingle/vextendsingle/vextendsingle+logσ\nσ\n/lessorsimilard,n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2F(pSn(k))+logσ2/vextendsingle/vextendsingle/vextendsingleP/parenleftbig\nSn+U(n)∈σ2Bd\n2/parenrightbig\n−P/parenleftbig\nSn∈σ2Bd\n2/parenrightbig/vextendsingle/vextendsingle/vextendsingle+logσ\nσ. (62)\nBut, in view of (55) and (56), we can bound the discrete tails i n the same way:\n/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2>σ2F(pSn(k))/lessorsimilard,nσ\n2Θd,n(σ). (63)\nFinally, we note that by Markov’s inequality and the almost i sotropicity assumption\n0≤P/parenleftbig\nSn+U(n)/∈σ2Bd\n2/parenrightbig\n=P/parenleftbig\n/⌊ard⌊lSn+U(n)/⌊ard⌊l2\n2> σ4/parenrightbig\n(64)\n≤E/parenleftbig\n/⌊ard⌊lSn+U(n)/⌊ard⌊l2\n2/parenrightbig\nσ4\n=trCov(Sn+U(n))+dn2\n4\nσ4\n=σ2+Od,n(σ)\nσ4=Od,n(1\nσ2) (65)\nand the same upper bound applies to P/parenleftbig\nSn/∈σ2Bd\n2/parenrightbig\n. Since both probabilities inside the absolute value\nin (62) are also upper bounded by 1, replacing the bounds (63) and (65) into (62), we get\n/vextendsingle/vextendsingle/vextendsingleH(Sn)−/summationdisplay\nk:/⌊ard⌊lk/⌊ard⌊l2≤σ2/integraldisplay\nk+[0,1)dF(fSn+U(n)(x))dx/vextendsingle/vextendsingle/vextendsingle\n≤Od,n/parenleftBigσ\n2Θd,n(σ)/parenrightBig\n+Od,n/parenleftBiglogσ\nσ/parenrightBig\n+Od,n/parenleftBiglogσ\nσ2/parenrightBig\n=Od,n/parenleftBiglogσ\nσ/parenrightBig\n.\nFinally, by (54) and the exponential bounds on the entropy ta ils (61), (63), we obtain\n/vextendsingle/vextendsingleh(Sn+U(n))−H(Sn)/vextendsingle/vextendsingle=Od,n/parenleftBiglogσ\nσ/parenrightBig\n+Od,n/parenleftBigσ\n2Θd,n(σ)/parenrightBig\n(66)\n=Od,n/parenleftBiglogσ\nσ/parenrightBig\n21completing the proof.\nProof of Theorem 8. By the continuous EPI (1),\nh/parenleftBign+1/summationdisplay\ni=1Xi+Ui/parenrightBig\n≥h/parenleftBign/summationdisplay\ni=1Xi+Ui/parenrightBig\n+d\n2log/parenleftBign+1\nn/parenrightBig\nand by Theorem 9 applied to the differential entropies on both sides\nH/parenleftBign+1/summationdisplay\ni=1Xi/parenrightBig\n+Od,n/parenleftBiglogσ\nσ/parenrightBig\n≥H/parenleftBign/summationdisplay\ni=1Xi/parenrightBig\n+Od,n/parenleftBiglogσ\nσ/parenrightBig\n+d\n2log/parenleftBign+1\nn/parenrightBig\n.\nBut by (8), Od,n/parenleftBig\nlogσ\nσ/parenrightBig\n=Od,n/parenleftBig\nH(X1)e−1\ndH(X1)/parenrightBig\nand the result follows.\nRemark 33. The rate can be improved to superpolynomial by splitting the sum according to σ1+ǫand\noptimising over ǫ.\nLemma 34 ([8]).LetD,M≥1and, for x >0,consider G(x) =F(x)−xlogM, whereF(x) =−xlogx.\nThen, for 0≤a,b≤D\nMand any 0< µ <1\ne,we have the estimate\n|G(b)−G(a)| ≤2µ\nMlog1\nµ+|b−a|logeD\nµ.\nProof.See [8].\n5 Concluding Remarks and Open Questions\n1. As noted in the Introduction, for d= 1, our definition of log-concavity is equivalent to the usua l\ndefinition p(k)2≥p(k−1)p(k+1),k∈Z, which is preserved under convolution (e.g. [14]). However ,\nford >1 log-concavity may not be preserved in general as we have alr eady observed in Remark 5.\nOn the other hand in the i.i.d. case, Proposition 35 below sho ws that/summationtextn\ni=1Aiis convex for any\ndiscrete convex set A, meaning that conv( A)∩Zd=A. We do not know whether our definition of\nlog-concavity is preserved under self-convolution.\nProposition 35. LetA⊂ZdbeZdconvex. Then, for any integer n≥2,An:=/summationtextn\ni=1AisZd-\nconvex.\nProof.It is clear that\nAn⊂conv(An)∩Zd.\nFor the inverse inclusion, we first note that conv( An) =nconv(A). Thus, if a∈conv(An)∩Zd,\nthen there are a1,a2∈Aandλ∈[0,1] such that a=n/parenleftbig\n(1−λ)a1+λa2/parenrightbig\n∈Zd.\nWe distinguish ncases depending on the value of λ. Ifk\nn≤λ≤k+1\nnforkinteger such that\n0≤k≤n−1, we express aconveniently as follows\na= (nλ−k)a2+/parenleftbig\n1−(nλ−k)/parenrightbig\na1+(n−k−1)a1+ka2.\nThus, we get that a∈conv(A)∩Zd+An−k−1+Ak. ByZd-convexity, conv( A)∩Zd=Aand\nAn−k−1+Ak=An−1. Therefore, a∈A+An−1=An.\nIt is therefore natural to ask the following question:\nQuestion 36. LetX1,X2be i.i.d. log-concave random vectors on Zd. IsX1+X2log-concave?\nFurthermore, is/summationtextn\ni=1Xilog-concave for every n?\nIf the answer to Question 36 is positive, a sufficient conditio n for the assumptions of Theorems 8\nand 9 would be that X1is log-concave. Answering Question 36 even for quantised mu ltivariate\nGaussians seems to be non-trivial. Nevertheless, the follo wing simple example shows that the\nanswer is positive for quantised isotropic multivariate Ga ussians, yielding an example that satisfies\nthe assumptions of Theorems 8 and 9.\n22Example 37. LetX1be a random variable with p.m.f. on Zdproportional to e−|k|2\n2σ2; that is X1\nis a multivariate centered isotropic Gaussian quantised on Zd. Then X1has the distribution of\n(Z1,...,Z d), whereZiare i.i.d. one-dimensional quantised centered Gaussians w ith variance σ2.\nNote that if pis a p.m.f. on Zdwhich is a product of log-concave p.m.f.s {pi}d\ni=1onZ\np(k) =d/productdisplay\ni=1p(i)(ki), k= (k1,...,k n)∈Zd,\nthenpis clearly log-concave on Zd.\nTherefore, since log-concavity is preserved under convolu tion in dimension one [4] and the coordi-\nnates of/summationtextn\ni=1Xiare independent, it follows that/summationtextn\ni=1Xiis log-concave for every n.\n2. For Theorem 11 and therefore for Theorems 8 and 9 as well, we have assumed that there exists\na continuous extension f, which is almost isotropic. We have then shown that fis also almost\nisotropic in the discrete sense. It would be more natural to s tart with the assumption that the\ndiscrete p.m.f. is isotropic. However, in this case we would not be able to use the continuous\ntoolkit to prove our concentration lemma. Nevertheless, we suspect that if the discrete p.m.f. is\nisotropic, then there exists a continuous extension almost isotropic and therefore our results would\nstill hold under this assumption:\nQuestion 38. LetXbe a log-concave random vector with p.m.f. ponZd. Assume that pis\nisotropic (respectively almost isotropic). Is there a cont inuous log-concave extension of ponRd\nwhich is isotropic (respectively almost isotropic)?\nReferences\n[1] S. Artstein, K. Ball, F. Barthe, and A. Naor. Solution of S hannon’s problem on the monotonicity\nof entropy. Journal of the American Mathematical Society , 17(4):975–982, 2004.\n[2] K. Ball. Logarithmically concave functions and section s of convex sets in Rn. Studia Math , 88(1):69–\n84, 1988.\n[3] A. R. Barron. Entropy and the central limit theorem. The Annals of Probability , pages 336–342,\n1986.\n[4] S. G. Bobkov, A. Marsiglietti, and J. Melbourne. Concent ration functions and entropy bounds for\ndiscrete log-concave distributions. Combinatorics, Probability and Computing , 31(1):54–72, 2022.\n[5] S. Brazitikos, A. Giannopoulos, P. Valettas, and B.-H. V ritsiou.Geometry of isotropic convex bodies ,\nvolume 196. American Mathematical Soc., 2014.\n[6] M. Fradelizi. Sections of convex bodies through their ce ntroid.Archiv der Mathematik , 69(6):515–\n522, 1997.\n[7] M. Fradelizi. Hyperplane sections of convex bodies in is otropic position. Beitr¨ age Algebra Geom. ,\n40(1):163–183, 1999.\n[8] L. Gavalakis. Approximate discrete entropy monotonici ty for log-concave sums. Combinatorics,\nProbability and Computing , page 1–14, 2023.\n[9] L. Gavalakis. Discrete generalised entropy power inequ alities for log-concave random variables. In\n2023 IEEE International Symposium on Information Theory (I SIT), pages 42–47, 2023.\n[10] L. Gavalakis and I. Kontoyiannis. Entropy and the discr ete central limit theorem. Stochastic\nProcesses and their Applications , page 104294, 2024.\n[11] S. Haghighatshoar, E. Abbe, and E. Telatar. Adaptive se nsing using deterministic partial hadamard\nmatrices. In 2012 IEEE International Symposium on Information Theory Pr oceedings , pages 1842–\n1846. Ieee, 2012.\n[12] S. Haghighatshoar, E. Abbe, and I. E. Telatar. A new entr opy power inequality for integer-valued\nrandom variables. IEEE transactions on information theory , 60(7):3787–3796, 2014.\n[13] P. Harremo´ es, C. Vignat, et al. An entropy power inequa lity for the binomial family. JIPAM. J.\nInequal. Pure Appl. Math , 4(5):93, 2003.\n[14] S. Hoggar. Chromatic polynomials and logarithmic conc avity.Journal of Combinatorial Theory,\nSeries B , 16(3):248–254, 1974.\n23[15] R. Kannan, L. Lov´ asz, and M. Simonovits. Isoperimetri c problems for convex bodies and a local-\nization lemma. Discrete & Computational Geometry , 13:541–559, 1995.\n[16] B. Klartag. Logarithmic bounds for isoperimetry and sl ices of convex sets. arXiv preprint\narXiv:2303.14938 , 2023.\n[17] M. Madiman and A. Barron. Generalized entropy power ine qualities and monotonicity properties\nof information. IEEE Transactions on Information Theory , 53(7):2317–2329, 2007.\n[18] M. Madiman, J. Melbourne, and C. Roberto. Bernoulli sum s and R´ enyi entropy inequalities.\nBernoulli , 29(2):1578–1599, 2023.\n[19] M. Madiman, L. Wang, and J. O. Woo. Majorization and R´ en yi entropy inequalities via Sperner\ntheory.Discrete Mathematics , 342(10):2911–2923, 2019.\n[20] M. Madiman, L. Wang, and J. O. Woo. Entropy inequalities for sums in prime cyclic groups. SIAM\nJournal on Discrete Mathematics , 35(3):1628–1649, 2021.\n[21] K. Murota. Discrete convex analysis: monographs on dis crete mathematics and applications. Com-\nputing Reviews , 45(6):339, 2004.\n[22] C. E. Shannon. A mathematical theory of communication. The Bell system technical journal ,\n27(3):379–423, 1948.\n[23] D. Shlyakhtenko. A free analogue of Shannon’s problem o n monotonicity of entropy. Adv. Math. ,\n208(2):824–833, 2007.\n[24] A. J. Stam. Some inequalities satisfied by the quantitie s of information of Fisher and Shannon.\nInformation and Control , 2(2):101–112, 1959.\n[25] T. Tao. Sumset and inverse sumset theory for Shannon ent ropy.Combinatorics, Probability and\nComputing , 19:603–639, 07 2010.\n[26] A. M. Tulino and S. Verd´ u. Monotonic decrease of the non -Gaussianness of the sum of independent\nrandom variables: a simple proof. IEEE Trans. Inform. Theory , 52(9):4295–4297, 2006.\n[27] J. O. Woo and M. Madiman. A discrete entropy power inequa lity for uniform distributions. In 2015\nIEEE International Symposium on Information Theory (ISIT) , pages 1625–1629. IEEE, 2015.\n24" }, { "title": "2401.15568v1.Intriguing_Equivalence_Structures_of_the_Embedding_Space_of_Vision_Transformers.pdf", "content": "Intriguing Equivalence Structures of the Embedding Space of Vision Transformers\nShaeke Salman1, Md Montasir Bin Shams1, Xiuwen Liu1\n1Department of Computer Science, Florida State University, FL 32306, USA\n{salman, liux }@cs.fsu.edu, mshams@fsu.edu,\nAbstract\nPre-trained large foundation models play a central role in\nthe recent surge of artificial intelligence, resulting in fine-\ntuned models with remarkable abilities when measured on\nbenchmark datasets, standard exams, and applications. Due\nto their inherent complexity, these models are not well un-\nderstood. While small adversarial inputs to such models are\nwell known, the structures of the representation space are not\nwell characterized despite their fundamental importance. In\nthis paper, using the vision transformers as an example due to\nthe continuous nature of their input space, we show via anal-\nyses and systematic experiments that the representation space\nconsists of large piecewise linear subspaces where there ex-\nist very different inputs sharing the same representations, and\nat the same time, local normal spaces where there are visu-\nally indistinguishable inputs having very different representa-\ntions. The empirical results are further verified using the local\ndirectional estimations of the Lipschitz constants of the un-\nderlying models. Consequently, the resulting representations\nchange the results of downstream models, and such models\nare subject to overgeneralization and with limited semanti-\ncally meaningful generalization capability.\nIntroduction\nBuilt on large pre-trained foundation models (Bommasani\net al. 2022), applications have exhibited unprecedented ca-\npabilities for a wide range of tasks, setting new state-of-\nthe-art on benchmark datasets, acing standard exams, and\npassing professional exams (OpenAI 2023; Brandes et al.\n2022; Kung et al. 2023; Mainuddin, Duan, and Dong 2021;\nIslam et al. 2023b; Emdad et al. 2023; Choi et al. 2023).\nLoosely speaking, applications have a relatively (very) small\napplication-specific component, which is fine-tuned on top\nof the shared foundation models. Therefore we focus on the\nfoundation models and the outputs of such models, referred\nto as representations and also embeddings. Transformers\nhave become a hallmark component in models for many ap-\nplications and have led to significant improvements in per-\nformance (Vaswani et al. 2023; Dosovitskiy et al. 2021; De-\nvlin et al. 2018; Mainuddin et al. 2022; Islam et al. 2023a;\nFeng et al. 2023), but there is no systematic study of the\nunderlying embeddings in terms of fundamental character-\nistics. Given a representation of a model, to understand the\ngeneralization and overgeneralization, one must know the\nequivalence classes of inputs that share the same represen-tation as the downstream applications will treat them the\nsame. Similarly, knowing the characteristics of resulting em-\nbeddings of semantically equivalent inputs is also crucial:\nif these inputs can have very different representations, the\nmodels underlying all applications will have limited consis-\ntent generalization.\nIt is well known that neural networks as classifiers exhibit\nan intriguing property in that they are subject to adversar-\nial attacks: some small changes to an input could result in\nsubstantial changes in the classifier’s outputs (Goodfellow,\nShlens, and Szegedy 2015; Szegedy et al. 2014; Chakraborty\net al. 2018; Madry et al. 2019). Conceptually speaking, those\ninputs are the ones that are close to the decision bound-\naries but near the given input; finding them leads to an op-\ntimization problem tied to the classifier and heuristic meth-\nods such as the fast gradient sign method and related varia-\ntions, are often effective (Goodfellow, Shlens, and Szegedy\n2015; Kurakin, Goodfellow, and Bengio 2017; Chen et al.\n2017; Moosavi-Dezfooli, Fawzi, and Frossard 2016). How-\never, these methods cannot be applied to studying the equiv-\nalences of the underlying representations given by the mod-\nels.\nIn this paper, using gradient-descent-based optimization\nprocedures, we show empirically that perturbing an input to\na deployed model in unnoticeable ways can alter the result-\ning representation to match that of any chosen one. Further-\nmore, we show that the resulting inputs will result in dra-\nmatic changes in classification results with no modifications\nto the classifiers. To highlight the key results of our frame-\nwork, we use the ImageBind model as an example (Girdhar\net al. 2023). Fig. 1 shows several images along with their\nrepresentations and the classification results. The three vi-\nsually indistinguishable pairs in Fig. 1, (a) and (e), (b) and\n(f), and (c) and (g), respectively (see Fig. 2 for pixel dif-\nferences) have very different representations, as shown by\ntheir low-dimensional projections. On the other hand, the\nimages in (e), (f), and (c) have very similar representations\neven though they are semantically very different; the images\nin (a) and (g) show another set. When we pass these images\nto the unmodified multimodal ImageBind model, the images\nwith similar embeddings are classified into the same class,\nregardless of their semantic similarity, as shown in Fig. 1 (d)\nand (h). These and additional results shown in the Experi-arXiv:2401.15568v1 [cs.CV] 28 Jan 2024agama lizard\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(c)embedding projection\n(d)Image\n(a)\n(b)\n(c)lizard peacock wombat\n1.0\n1.1187e-07\n2.2586e-125.8658e-10\n1.0\n8.6227e-121.6041e-12\n8.2994e-12\n1.0vision x text\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(e)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(f)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component6\n3\n0369projected value\n(g)embedding projection\n(h)Image\n(e)\n(f)\n(g)lizard peacock wombat\n4.2853e-11\n3.7758e-12\n1.02.7194e-11\n2.7308e-11\n1.0218e-091.0\n1.0\n9.7523e-12vision x textFigure 1: Typical examples from ImageNet obtained using the proposed framework. Three pairs of visually indistinguishable\nimages (a and e, b and f, c and g) have different representations from each other as shown in their low-dimensional projections.\nIn contrast, very similar representations are seen for the images in (e), (f), and (c), despite their substantial semantic differences;\nsimilar goes with images in (a) and (g). Note that the arrow in the title ( original →target ) signifies a derived image from the\noriginal one by aligning the embedding of the original image with the target image using our method. The matrices (d) and (h)\nshow the classification outcomes from the multimodal ImageBind pre-trained model used directly with no modifications.\n+.02×\n =\n+.02×\n =\n+.02×\n =\nFigure 2: Pixel differences between the two images in each\nof the three pairs in Fig. 1; they are multiplied by 50 for\nvisualization.\nmental Results section, along with the fact we have obtained\nthe same findings on all the images we have used, demon-\nstrate convincingly that there are visually indistinguishable\ninputs having very different embeddings and yet that there\nare very different images having almost identical embed-\ndings. Through the estimation of lower bounds on the local\ndirectional Lipschitz constants and the structures of the Ja-\ncobian matrices, we show such models are inherently vul-\nnerable to adversarial attacks. Note that our method pro-\nduces adversarial inputs as a by-product. By analyzing the\nequivalence classes of the embeddings of foundational mod-\nels, the problem we solve is very different from the opti-\nmization problem for finding an adversarial input, and con-\nsequently, our results are more general and do not depend on\napplication-specific classifiers.Our main contributions are as follows:\n• We clearly demonstrate the algebraic and geometric\nstructure of the embedding space of vision transform-\ners. More specifically, we show that the input space con-\nsists of large piecewise linear subspaces where different\nimages share the same representation and local normal\nspaces where visually indistinguishable images can have\nvery different representations.\n• We have proposed efficient computational procedures for\nfinding equivalence structures of the embedding space\nand demonstrated their effectiveness in deployed models.\nAs an additional outcome, we are able to identify adver-\nsarial examples to the representations which will affect\nall downstream applications.\n• We show how to estimate the local directional Lipschitz\nconstants robustly by understanding and overcoming the\nnumerical issues of large models.\nRelated Work\nWith the availability of large datasets for challenging tasks\nsuch as natural language processing and computer vi-\nsion, large foundational models have dominated the top-\nperforming models and methods. In this new paradigm, such\nlarge models are trained on large datasets with huge com-\nputation, and then applications built on top with relatively\nsmall application-specific components to be further tuned\nusing much smaller datasets. The trend has been further ac-\ncelerated by the recent prompting-based models and multi-\nmodal models. The joint multimodal models have demon-\nstrated significant benefits by employing a shared embed-\nding space across various modalities. One of them is Im-\nageBind (Girdhar et al. 2023), which aims to learn a single\nshared representation space by leveraging multiple types of\nimage-paired data. This model aligns the embedding of eachmodality to image embeddings, resulting in an emergent\nalignment across all modalities via paired modeling based\non the CLIP (the underlying vision and text model of Image-\nBind) (Radford et al. 2021). The ImageBind results suggest\nsemantic representation in the embedding space by match-\ning embeddings of different modalities. Our results show\nthat images with the same visual content can have highly\ndissimilar embeddings, whereas images with notable differ-\nences can have embeddings that are nearly identical.\nWhile recent works have improved the model performance\non benchmark datasets and tasks, the fundamental issues of\nunderstanding how such models generalize, overgeneralize\nand memorize remain an open challenge (Zhang et al. 2016,\n2017; Neyshabur et al. 2017).\nSeveral researchers focus on the “Activation Regions” con-\ncept and their potential role in understanding neural net-\nworks (Crabb ´e and van der Schaar 2022). Activation regions\nrefer to specific regions in the input space that lead to cer-\ntain activation patterns in the hidden layers of neural net-\nworks. Rectified Linear Units (ReLUs) are a common choice\nof activation functions in deep learning models. For neural\nnetworks where ReLU is used, they lead to piecewise lin-\near regions (Montufar et al. 2014). In their work, Hanin and\nRolnick (2019) demonstrate that, despite the vast number of\npossible input patterns, deep neural networks with ReLU ac-\ntivations exhibit surprisingly few distinct activation patterns\nin their hidden layers. The linear approximation works well\nand we do not need to change an input much in order to\nmatch the embedding of another input.\nAnother line of research trying to understand the model is\nby probing the models to identify new properties. The most\nwell-studied problem is adversarial attacks, where unnotice-\nable changes to the input can cause the models, mostly clas-\nsifiers, to change their predictions. Bhojanapalli et al. (2021)\nand Shao et al. (2022) investigate the robustness of ViTs\nagainst attacks where the attacker has access to the model’s\ninternal structure. Their findings indicate that ViTs gener-\nally exhibit higher resilience than CNNs. Qin et al. (2023)\nand Salman et al. (2021) examine the robustness of Vision\nTransformers (ViTs) by focusing on the architectural struc-\nture based on patches. Herrmann et al. (2022) additionally\ndevelop a pyramid adversarial training approach incorporat-\ning augmentation techniques to enhance both the sanity and\nrobust performance of Vision Transformers (ViTs). A recent\nwork (Carlini et al. 2023) explores the interplay between\nalignment techniques and adversarial attacks in neural net-\nworks, highlighting the potential vulnerabilities of aligned\nmodels. Though most adversarial examples have been ap-\nplied to image classification tasks (Szegedy et al. 2014), the\navailability of multimodal models facilitates the application\nto text and other domains. In our work, based on a gradient-\ndescent-based optimization procedure, we are able to find\nadversarial attacks to the embedding of any given image.\nA common explanation of the existence of adversarial at-\ntacks is that the Lipschitz constants for deep neural net-\nworks are large, and therefore, models are sensitive to small\nchanges (Fazlyab et al. 2023; Szegedy et al. 2014; Good-fellow, Shlens, and Szegedy 2015). Several papers focus on\nestimating the Lipschitz constants, both global and local.\nPrior research works have shown that Lipschitz properties\nreveal intriguing behaviors of neural networks, such as ro-\nbustness and generalization (Szegedy et al. 2014). In recent\ntimes, numerous studies have delved into the exploration\nof optimization-based methods for bounding or approximat-\ning the Lipschitz constant of neural networks (Scaman and\nVirmaux 2019; Latorre, Rolland, and Cevher 2020; Fazlyab\net al. 2023). Avant and Morgansen (2021) determines guar-\nanteed upper bounds on the local Lipschitz constant of larger\nneural networks with ReLU activations. The LipsFormer ar-\nchitecture by Qi et al. (2023), attempts to address the issue of\ntraining instability in transformers, a challenge particularly\npronounced during the initial training phases. They derive\ntheoretical upper limits for the Lipschitz constants, provid-\ning valuable insights into this aspect. Our results are com-\nplementary in nature; we show the distributions of the local\ndirectional Lipschitz constants of real trained large models\nand are able to estimate them accurately.\nPreliminaries\nUnderstanding the large foundational models requires an un-\nderstanding of all the components. However, such models\nare very complex due to the number of parameters used. To\novercome the challenges, we roughly divide a model into\ntwo stages: a large foundational stage that is common to\ndifferent applications and then an application-specific stage,\nconsisting of classifiers and other application-specific com-\nponents. To simplify the analyses, we assume the founda-\ntional model stage is fixed. As we focus on vision transform-\ners, here we first describe the transformers mathematically\nand describe the vision transformers.\nTransformers can be described mathematically succinctly,\nconsisting of a stack of transformer blocks. A transformer\nblock is a parameterized function class fθ:Rn×d→Rn×d.\nIfx∈Rn×dthen fθ(x) = zwhere Q(h)(xi) =\nWT\nh,qxi, K(h)(xi) = WT\nh,kxi, V(h)(xi) =\nWT\nh,vxi, W h,q, Wh,k, Wh,v∈Rd×k. The key multi-\nhead self-attention is a softmax function applying row-wise\non the inner products.1\nα(h)\ni,j=softmax j \nQ(h)(xi), K(h)(xj)\u000b\n√\nk!\n(1)\nThe outputs from the softmax are used as weights to com-\npute new features, emphasizing the ones with higher weights\ngiven by\nu′\ni=HX\nh=1WT\nc,hnX\nj=1αi,jV(h)(xj), W c,h∈Rk×d.(2)\nThe new features then pass through a layer normalization,\nfollowed by a ReLU layer, and then another layer normaliza-\ntion. Typically transformer layers are stacked to form deep\n1Note that there are other ways to compute the attention\nweights.models. Such models are used for natural language process-\ning tasks, including various language models and machine\ntranslation.\nRecently the transformer architectures are adapted to vision\ntasks by using image blocks on the basic units, and spa-\ntial relationships among the units are captured via the self-\nattention mechanism. Since images can change smoothly\nand continuously, they make the analyses of the embedding\nspace amendable to mathematical analyses. For example, vi-\nsion transformers transform image patches into an embed-\nding using a multi-layer perception applied on the output\nfrom the transformers (Dosovitskiy et al. 2021).\nWhile the proposed method applies to all transformer-based\nmodels with continuous inputs, we focus on the CLIP\nmodel (Radford et al. 2021), which jointly models images\nand text using the same shared embedding space used in the\nImageBind model.\nProposed Framework\nHere we describe the framework that enables us to explore\nthe embedding space, analyze their properties, and verify\nthem in large models. Generally, we model the represen-\ntation given by a (deep) neural network (including a trans-\nformer) as a function f:Rm→Rn. The fundamental ques-\ntion is to have a computationally efficient and effective way\nto explore the embeddings of inputs in the representation\nspace by finding the inputs whose representation will match\nthe one given by f(xtg), where xtgis an input whose em-\nbedding we like to match. Informally, given an image of a\nlizard in Fig. 1 as an example, all the images that share its\nrepresentation given by a model will be treated as a lizard. In\naddition, we like to know the local algebraic and geometric\nstructures of a representation; as adversarial examples are\nknown to exist in neural network models as classifiers, we\nwould like to know whether adversarial examples exist for\nrepresentations. More importantly, we like to know how lo-\ncal spaces are connected.\nA Simple and Effective Procedure\nNote that it is much more challenging to find inputs that\nwould match the representation of a target input. Since we\nneed to match two vectors, we define the loss for finding an\ninput matching a given representation as\nL(x) =L(x0+ ∆x) =1\n2∥f(x0+ ∆x)−f(xtg)∥2,(3)\nwhere x0is an initial input and f(xtg)specifies the target\nembedding. The gradient is given by\n∂L\n∂x≈\u0012∂f\n∂x\f\f\f\nx=x0\u0013T\n(f(x0+ ∆x)−f(xtg)).(4)\nEq. 4 shows how the gradient of the mean square loss func-\ntion is related to the Jacobian of the representation func-\ntion at x=x0. While optimal solutions can be obtained\nby solving a quadratic programming problem or linear pro-\ngramming problem, depending on the norm to be used when\nminimizing ∆x, the gradient function works effectively forall the cases we have tested due to the Jacobian of the trans-\nformer.\nOne of the practical issues using the gradient descent-based\nprocedure is how to determine the learning rate. In the case\nof the transformers, the model can be approximated by a\nlinear model when it moves within one activation region;\nnote that it is approximate due to the nonlinearity by the\nsoftmax, whose gradient is known. This property allows the\ngradient method to be very effective. We call the procedure\nembedding matching procedure.\nLocal Algebraic and Geometric Structures\nGiven an input x0, the local structures decide how the model\nbehaves in the local neighborhood; for transformer-based\nmodels, note that the local neighborhood can be large spa-\ntially in the input space. Since we know the nonlinearity of\nthe transformers is due to the ReLU function being used and\nthe softmax function, the linear approximation of the func-\ntion in a local neighborhood should be effective, given by\nf(x0+ ∆x)≈f(x0) +∂f\n∂x\f\f\f\nx=x0×∆x. (5)\nas in Eq. 4,∂f\n∂xis the Jacobian matrix of the function at\nx=x0. As a result, for deployed models, where m > n ,\nthere is a null space where the embeddings do not change\nas the input changes; it can be obtained via a reduced singu-\nlar decomposition of the Jacobian. There is a normal space\nin the space perpendicular to the null space, where the em-\nbeddings can change quickly. To quantify how sensitive a\nrepresentation is to local perturbations in the input space,\nwe compute an accurate estimate of the extended local Lip-\nschitz constant, given by the smallest L, such that\n||f(x0+ ∆y)−f(x0+ ∆x)|| ≤L||∆y−∆x||,(6)\nwhere ∆xand∆yspecify the accepted neighborhood of x0.\nSince the derivative of the ReLU function is not defined at 0,\nthe definition avoids the issue. Lcan be estimated accurately\nusing the largest singular value of the Jacobian, and we also\nverify numerically.\nAs the model is high dimensional in nature, its behavior de-\npends on the directions as well. To quantify that, we also\ndefine and estimate the local directional Lipschitz constant\n(LLDLC ), along a given direction. The estimate is helpful to\ncharacterize how fast the model changes along the direction.\nSince x0could be near or even on the boundary between dif-\nferent activation regions in terms of the ReLU network, the\nLLDLC is defined as the smallest number that satisfies\n||f(x0+β∆x0)−f(x0+α∆x0)|| ≤LLDLC|β−α|,(7)\nwhere 0≤ |α|,|β| ≤ϵLD,∆x0is a unit length vector,\nspecifying the direction, and ϵLDis a parameter specifying\nthe range of αandβ. Estimated LLDLC values and their\ndistributions allow us to quantify the changes in the normal\nspace and the null space.\nManifold Structures of the Embedding Space and\nTheir Implications\nPutting all together, it is clear that the embedding space con-\nsists of subspaces where the representations do not changelocally and are therefore invariant to all the changes in the\nspace; invariance to nuance changes is desirable, and results\nin generalization but invariance to other changes will lead to\nharmful overgeneralization. These subspaces together form\na manifold in the space. Since ReLU is piecewise linear and\nreduces to a linear function within one activation region, the\nmanifold is piecewise linear in nature, corresponding to the\nactivation regions. The manifold is locally a subspace, and\ntherefore the connection with the Grassmannian manifold\ncan be exploited to characterize them formally (Gallivan\net al. 2003). In this paper, we adopt a numerical approach\nand leave the formal exploration as future work. There are\nalso normal directions where the small changes in the input\ncan lead to large changes in the representation, causing the\nmodel not to generalize well and be subject to adversarial\nattacks. The rate of change is bounded by the largest singu-\nlar value of the Jacobian matrix and can be studied formally\nand numerically.\nWhile the description is high level, we instantiate it using the\nCLIP model (Radford et al. 2021), a commonly deployed vi-\nsion transformer. In addition, as the algebraic and geometric\nstructures do not depend on the specifications of a model,\nwe expect the results should be similar with other vision\ntransformers and other models where the Jacobian can be\nestimated. We have validated this and provided detailed in-\nsights in the Experimental Results section and Appendix.\nExperiments\nIn this section, we begin by providing the specifics of our\nexperimental settings and implementation details. Our pro-\nposed framework is systematically applied across various\ndatasets and multiple vision transformer models; in the sub-\nsequent subsections, we present both the experimental out-\ncomes and quantitative results.\n0 200 400 600 800 1000\nnumber020406080100singular values\n0 200 400 600 800 1000\nnumber020406080100ldlc values\nFigure 3: Local structures of the embedding space. (top) The\nsingular values of the Jacobian Matrix for Fig. 1(a); (bot-\ntom) The estimated local directional Lipschitz constant val-\nues along the directions given by the right singular vectors,\nwhich are consistent with the singular values.Our findings showcase the capability to align any image\nwith another image through imperceptible adversarial at-\ntacks within a vision transformer model. More importantly,\nwe show that our framework exhibits versatility, being ag-\nnostic to both the model architecture and dataset character-\nistics.\nDatasets and Settings\nDatasets. We conduct extensive experiments to evaluate our\nproposed framework on widely recognized vision datasets,\nnamely ImageNet (Deng et al. 2009), MS-COCO (Lin et al.\n2015) and Google Open Images (Kuznetsova et al. 2020).\nImplementation Details. To demonstrate the feasibility of\nthe proposed method on large models, we have used the pre-\ntrained model publicly available by ImageBind2, which in\nturn uses a CLIP model3. More specifically, ImageBind uti-\nlizes the pre-trained vision (ViT-H 630M params) and text\nencoders (302M params) from the OpenCLIP (Ilharco et al.\n2021; Girdhar et al. 2023). The input size is 224×224×3,\nand the dimension of the embedding is 1024. As a result, the\nJacobian matrix is of size 1,024×150,528.\nExperimental Results\nWe have tested the embedding matching procedure using\nmany image pairs. Fig. 5 shows a typical example, where\nthe left one shows the evolution of loss when matching a\nspecified target embedding. We use a small step size to make\nsure it converges. The right one shows that cosine similarity\nincreases steadily. We also show the average pixel value dif-\nference between the new input and the original image at each\nstep; one can see the values remain very small even though\nthey increase as well. The algorithm is not sensitive to the\nlearning rate and works effectively across a broad range of\nvalues, spanning from 0.001 to 0.09. For instance, with a\nlearning rate of 0.001, convergence is achieved in around\n25,000 iterations, while 0.09 requires around 3,000 itera-\ntions. The visual differences in the resulting images are not\nnoticeable. Eqn. 4 and 5 provide an explanation, as the gra-\ndient for our loss is insensitive to the learning rate. We will\nprovide source code for all our experiments in GitHub4.\nQuantitative evaluation. We use reduced singular value de-\ncomposition to write the Jacobian as UΣVT=P1023\ni=0si×\nU(:, i)×V(:, i)T, where Tdenotes the matrix transpose op-\nerator. The top plot in Fig. 3 shows the singular values of the\nJacobian matrix in Fig. 1(a). The distribution of the singular\nvalues shows that the Jacobian has several dominating direc-\ntions, reflecting the training set and the training algorithm\nbeing used. Note that the largest singular value gives us an\nestimation of the LLDLC at the input image. It shows that\nthe model is sensitive to small changes along those direc-\ntions. We also empirically estimate the LLDLC values along\nthe directions; the results are shown in the bottom plot of the\nfigure. One can see that the values match well and indicate\nthat the linear model provides a good approximation locally.\n2https://github.com/facebookresearch/ImageBind\n3https://github.com/mlfoundations/open clip\n4https://github.com/programminglove08/EquivalenceStruct0.2 0.4 0.6 0.8 1.0\nldlc values0.00.51.01.52.02.53.03.5Density\n0.0020 0.0025 0.0030 0.0035 0.0040\nldlc values0200400600800100012001400Density\n0 1 2 3 4\ndirection01020304050ldlc valuesFigure 4: The distribution of the estimated local directional\nLipschitz constant values along the directions given by ran-\ndom Gaussian vectors (top left), random Gaussian vectors in\nthe null space of singular vectors (top right), and the gradi-\nent optimization procedure (bottom).\n0 500 1000 1500 2000\nsteps0.00.10.20.30.4loss\n0 500 1000 1500 2000\nsteps0.60.70.80.91.0cosine similarity\ncosine_similarity\navg_diff0.0000.0050.0100.0150.020\naverage pixel differance\nFigure 5: The evolution of loss while matching a target em-\nbedding. (left) the loss w.r.t. steps. (right) the cosine similar-\nity between the embeddings of the new input and the target\nw.r.t. the steps, along with the average pixel value difference\nbetween the new input and the original image.\nFig. 4 (top right) shows distributions of the estimated\nLLDLC values for 10,000 such randomly generated direc-\ntions in the null space, and the values are consistently small.\nIn comparison, Fig. 4 (top left) shows the same but for ran-\ndom directions. Note that when a random direction is used,\nthe resulting direction is a mixture of the null subspace and\nnormal space. As expected, their values are much larger than\nthose in the null space. Fig. 4 (bottom) shows the estimated\nLLDLC values along the directions given by our gradient op-\ntimization procedure. Those values are two orders of mag-\nnitude larger than the random directions and four orders of\nmagnitude larger than the values in the null space, showing\nthe effectiveness of the procedure.\nAs a by-product, understanding the algebraic and geomet-\nric structures of the embedding space allow us to explore\nthe space effectively. For example, we can find adversarial\nattacks to the embedding of any given image using the pro-\nposed gradient procedure. Fig. 1 shows three examples. To\ndemonstrate the universal applicability of the procedure and\nthe adversarial examples that exist almost everywhere, Fig.\n8 shows more examples from different categories from the\nImageNet dataset. See the Appendix for additional exam-\nples.\nQualitative evaluation. Our key result is that the seman-\ntic meanings of the embeddings given by transformer mod-\n0.0 0.2 0.4 0.6 0.8 1.0\n0.50.60.70.80.91.0cosine similarityinterpolated images similarity with original\n0.0 0.2 0.4 0.6 0.8 1.0\n0.50.60.70.80.91.0cosine similarityinterpolated images similarity with finalFigure 6: The change of the embeddings as the input changes\nlinearly for the image pair in Fig. 1(a). (left) cosine similar-\nity between embeddings of the interpolated and the original.\n(right) same as left but for the final image.\n0.0 0.2 0.4 0.6 0.8 1.0\n0.60.70.80.91.0cosine similarityinterpolated image -> first image\nwith first\nwith last\n0.0 0.2 0.4 0.6 0.8 1.0\n0.50.60.70.80.91.0cosine similarityinterpolated image -> last image\nwith first\nwith last\nFigure 7: (Left) The cosine similarity along the path that\nmatches the embedding of the first image. (Right) Same as\nthe left except along the path for that of the last image.\nels are fundamentally limited as different inputs share simi-\nlar embeddings while visually indistinguishable inputs have\nvery different embeddings. As the techniques are model\nand dataset-agnostic, they should be effective on differ-\nent transformer models and datasets, including ones for\nother modalities. We have conducted experiments with vari-\nous vision transformer models, including MAE-like models\nfrom HuggingFace5, such as BEiT, DEiT, Swin, ViTMAE,\nViTMSN (Bao et al. 2022; Touvron et al. 2021; Liu et al.\n2021; He et al. 2021; Assran et al. 2022) and two examples\nare given in Fig. 9. Please refer to the Appendix for addi-\ntional results with other models and datasets.\nIn general, as shown in the examples, the proposed tech-\nnique works well with any randomly chosen image from a\ndifferent target class. Additionally, Fig. 3 displays the sin-\ngular values of the Jacobian matrix, revealing notable differ-\nences in the singular value distributions between the original\nand manipulated images.\nSo far, we have shown the structures of the embedding space\nof a particular point using concrete examples. Our frame-\nwork allows us to explore the paths and the space more\nbroadly. Fig. 6 shows how the embeddings change as the in-\nput changes from one image to the one that matches a spec-\nified target but remains visually indistinguishable. The plot\nshows the changes roughly linearly.\nCompared to existing adversarial attack methods, one dis-\ntinctive feature of our proposed framework is that we can\nexploit how different subspaces are connected. Fig. 7 shows\none such path example. By applying the match-finding pro-\n5https://huggingface.co/docs/transformers/model doc/beitcat -> dog\n cat -> tiger\n cat -> racing car\n cat -> panda\n cat -> zebra\n0.99966 7.534e-07 3.0398e-08 0.00033748 8.9922e-07\n1.7605e-08 0.99996 3.4396e-13 6.2466e-07 3.5846e-05\n5.0006e-06 4.8747e-09 0.99999 6.0516e-06 1.6799e-09\n2.3207e-08 9.9777e-08 1.6882e-12 1.0 6.6274e-08\n1.6442e-07 3.1709e-07 2.2325e-11 1.9986e-07 1.0vision x text\nchurch bulding -> boy with tench \n golf carts -> boy with tench\n flower pot -> boy with tench\n cauliflower -> boy with tench\n strawberry -> boy with tench\n0.00024826 3.0604e-10 6.9593e-09 1.9486e-10 9.5458e-09 0.99975\n1.2976e-15 6.9238e-13 6.4869e-14 8.8301e-15 8.2113e-13 1.0\n2.0479e-15 2.1143e-15 6.3306e-13 1.0105e-13 1.025e-12 1.0\n6.4728e-17 7.3682e-17 3.984e-14 1.5059e-13 3.4323e-12 1.0\n6.1611e-17 1.4762e-16 1.0718e-13 1.3643e-13 5.7408e-11 1.0vision x textFigure 8: (top) More examples where visually indistinguishable images have very different embeddings and consequently are\nclassified to other classes as in Fig. 1. Cat images are classified as a dog, a tiger, a racing car, a panda, and a zebra. (bottom)\nVisually very different images (e.g., a church building, some golf carts, flower pot, cauliflower, strawberry) have very similar\nembeddings and are classified as a boy with tench. For more details on the additional images in the first row and the second\nrow, please see the Appendix.\nagama lizard\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component75\n50\n25\n0255075projected value\n(f)embedding projection\nagama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(f)embedding projection\nFigure 9: Examples obtained while the proposed framework\nis applied on different vision transformer models, such as\n(top two rows) BEiT, and (the next two rows) Swin Trans-\nformer. The results are given in the same format as depicted\nin Fig. 1. Additional plots for other models are also consis-\ntent and added to the Appendix. The example demonstrates\nthat the method is model-agnostic.\ncedure, we are able to construct and connect different sub-\nspaces.\nThe results show that the embeddings are inherently lim-\nited semantically when analyzed systematically. Locally, the\nmodel is sensitive to small changes along the directions in\nthe normal space. In the null space, the embeddings remain\nconstant while the input changes substantially. By connect-\ning local null spaces, we have connected spaces where em-\nbeddings are similar, but inputs can be very different. As the\nembedding space is high dimensional in nature, testing us-\ning datasets is inherently limited. The systematic analysesare essential.\nDiscussion\nBy using computational procedures with mathematical anal-\nyses, we characterize the embedding space of a vision trans-\nformer both locally and globally. Note that the proposed\nframework can be applied to characterizing any model di-\nrectly as long as the input varies continuously so that the\nJacobian can be estimated properly. With multimodal mod-\nels, the framework can also be used to study other models\nwith discrete inputs indirectly via other joint embeddings.\nIt may be attempting to categorize our framework as an ad-\nversarial attack technique. Our primary focus is on analyzing\nthe embedding space; we utilize the ImageBind solely as a\nclassifier to validate our findings and is not used otherwise.\nWhile our embedding matching procedure can be used to\ngenerate effective adversarial examples, it is fundamentally\ndifferent. Our technique is classifier agnostic and does not\nexploit features specific to classifiers. Consequently, our ex-\namples with matched embeddings will appear to be the same\nto any classifier or downstream model that builds on embed-\ndings. On the other hand, traditional adversarial attacks are\nspecific to classifiers and applications, focusing on altering\ntheir outputs by changing the input.\nThe plausible root cause of such adversarial examples and\nalso semantically different images with identical embed-\ndings is that transformers do not require the inputs to be\naligned to have similar embeddings. By adding alignment-\nsensitive components to the embedding could mitigate the\nproblem, which is being investigated further. Additionally,\nbased on the singular values of the Jacobian matrix, it ap-\npears possible to evaluate the robustness of the models,\nwhich is being investigated.The results shown in this paper seem not to be consistent\nwith the impressive results demonstrated by such models.\nNote that almost all existing results are measured on bench-\nmark datasets. Due to the high dimensionality of the embed-\nding space and the input space, even the largest dataset will\ncover the spaces very sparsely. We believe that systematic\nevaluations such as ours are necessary if one likes to evalu-\nate models to be able to predict their behaviors in the entire\nspace rather than on samples.\nNote that the problem of how to estimate the global and lo-\ncal Lipschitz constants of neural networks and transformers\nhas been studied mathematically. In particular, LipsFormer\n(Qi et al. 2023) shows that degenerated cases can cause the\nLipschitz constant to be unbounded. However, none of these\ntechniques have been scaled to the large models that are be-\ning deployed, including the ones we have used. Our results\nare also complementary in nature; we show the distributions\nof the local directional Lipschitz constants of real trained\nlarge models and are able to estimate them accurately us-\ning the Jacobian matrix. For applications, the Lipschitz con-\nstants themselves provide an upper bound of the rate of the\nchange and may not be sufficient to understand their behav-\nior for typical inputs.\nConclusion\nIn this paper, we show the structures of the embedding\nspaces using algorithms and mathematical analyses. It is at-\ntempting to conclude that recent pre-trained models can be\nused to build any effective applications based on their perfor-\nmance on benchmark datasets. While such models give im-\npressive performance, their inherent generalization abilities\nare limited by the properties of the underlying embedding\nspaces. Before this fundamental limitation can be addressed,\nsuch models should not be used for critical applications.\nReferences\nAssran, M.; Caron, M.; Misra, I.; Bojanowski, P.; Bor-\ndes, F.; Vincent, P.; Joulin, A.; Rabbat, M.; and Ballas, N.\n2022. Masked Siamese Networks for Label-Efficient Learning.\narXiv:2204.07141.\nAvant, T.; and Morgansen, K. A. 2021. Analytical bounds on the\nlocal Lipschitz constants of ReLU networks. arXiv:2104.14672.\nBao, H.; Dong, L.; Piao, S.; and Wei, F. 2022. BEiT: BERT Pre-\nTraining of Image Transformers. arXiv:2106.08254.\nBhojanapalli, S.; Chakrabarti, A.; Glasner, D.; Li, D.; Unterthiner,\nT.; and Veit, A. 2021. Understanding Robustness of Transformers\nfor Image Classification. arXiv:2103.14586.\nBommasani, R.; Hudson, D. A.; Adeli, E.; Others; and et al.\n2022. On the Opportunities and Risks of Foundation Models.\narXiv:2108.07258.\nBrandes, N.; Ofer, D.; Peleg, Y .; Rappoport, N.; and Linial, M.\n2022. ProteinBERT: a universal deep-learning model of protein\nsequence and function. Bioinformatics , 38(8): 2102–2110.\nCarlini, N.; Nasr, M.; Choquette-Choo, C. A.; Jagielski, M.; Gao,\nI.; Awadalla, A.; Koh, P. W.; Ippolito, D.; Lee, K.; Tramer, F.;and Schmidt, L. 2023. Are aligned neural networks adversarially\naligned? arXiv:2306.15447.\nChakraborty, A.; Alam, M.; Dey, V .; Chattopadhyay, A.; and\nMukhopadhyay, D. 2018. Adversarial Attacks and Defences: A\nSurvey. arXiv:1810.00069.\nChen, P.-Y .; Zhang, H.; Sharma, Y .; Yi, J.; and Hsieh, C.-J. 2017.\nZOO. In Proceedings of the 10th ACM Workshop on Artificial In-\ntelligence and Security .\nChoi, J. H.; Hickman, K. E.; Monahan, A.; and Schwarcz, D. B.\n2023. ChatGPT Goes to Law School. Journal of Legal Education\n(Forthcoming) .\nCrabb ´e, J.; and van der Schaar, M. 2022. Concept Activation\nRegions: A Generalized Framework For Concept-Based Explana-\ntions. arXiv:2209.11222.\nDeng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L.\n2009. ImageNet: A large-scale hierarchical image database. In\n2009 IEEE Conference on Computer Vision and Pattern Recogni-\ntion, 248–255.\nDevlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2018. BERT:\nPre-training of Deep Bidirectional Transformers for Language Un-\nderstanding. arXiv:1810.04805.\nDosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai,\nX.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.;\nGelly, S.; Uszkoreit, J.; and Houlsby, N. 2021. An Image is\nWorth 16x16 Words: Transformers for Image Recognition at Scale.\narXiv:2010.11929.\nEmdad, F. B.; Tian, S.; Nandy, E.; Hanna, K.; and He, Z. 2023. To-\nwards Interpretable Multimodal Predictive Models for Early Mor-\ntality Prediction of Hemorrhagic Stroke Patients. AMIA Summits\non Translational Science Proceedings .\nFazlyab, M.; Robey, A.; Hassani, H.; Morari, M.; and Pappas, G. J.\n2023. Efficient and Accurate Estimation of Lipschitz Constants for\nDeep Neural Networks. arXiv:1906.04893.\nFeng, Q.; Yuan, J.; Emdad, F. B.; Hanna, K.; Hu, X.; and He,\nZ. 2023. Can Attention Be Used to Explain EHR-Based Mor-\ntality Prediction Tasks: A Case Study on Hemorrhagic Stroke.\narXiv:2308.05110.\nGallivan, K.; Srivastava, A.; Liu, X.; and Van Dooren, P. 2003. Ef-\nficient algorithms for inferences on Grassmann manifolds. In IEEE\nWorkshop on Statistical Signal Processing, 2003 , 315–318.\nGirdhar, R.; El-Nouby, A.; Liu, Z.; Singh, M.; Alwala, K. V .;\nJoulin, A.; and Misra, I. 2023. ImageBind: One Embedding Space\nTo Bind Them All. arXiv:2305.05665.\nGoodfellow, I. J.; Shlens, J.; and Szegedy, C. 2015. Explaining and\nHarnessing Adversarial Examples. arXiv:1412.6572.\nHanin, B.; and Rolnick, D. 2019. Deep ReLU Networks Have Sur-\nprisingly Few Activation Patterns. arXiv:1906.00904.\nHe, K.; Chen, X.; Xie, S.; Li, Y .; Doll ´ar, P.; and Girshick,\nR. 2021. Masked Autoencoders Are Scalable Vision Learners.\narXiv:2111.06377.\nHerrmann, C.; Sargent, K.; Jiang, L.; Zabih, R.; Chang, H.; Liu,\nC.; Krishnan, D.; and Sun, D. 2022. Pyramid Adversarial Training\nImproves ViT Performance. arXiv:2111.15121.Ilharco, G.; Wortsman, M.; Wightman, R.; Gordon, C.; Carlini, N.;\nTaori, R.; Dave, A.; Shankar, V .; Namkoong, H.; Miller, J.; Ha-\njishirzi, H.; Farhadi, A.; and Schmidt, L. 2021. OpenCLIP.\nIslam, M.; Shuvo, S. S.; Ahmed Shohan, J.; and Faruque, O. 2023a.\nForecasting of PV Plant Output Using Interpretable Temporal Fu-\nsion Transformer Model. In 2023 North American Power Sympo-\nsium (NAPS) .\nIslam, M. M.; Faruque, M. O.; Butterfield, J.; Singh, G.; and Cooke,\nT. A. 2023b. Unsupervised clustering of disturbances in power\nsystems via deep convolutional autoencoders. arXiv:2306.06124.\nKung, T. H.; Cheatham, M.; Medenilla, A.; and et al., O. 2023. Per-\nformance of ChatGPT on USMLE: Potential for AI-assisted medi-\ncal education using large language models. PLOS Digital Health .\nKurakin, A.; Goodfellow, I.; and Bengio, S. 2017. Adversarial Ma-\nchine Learning at Scale. arXiv:1611.01236.\nKuznetsova, A.; Rom, H.; Alldrin, N.; Uijlings, J.; Krasin, I.; Pont-\nTuset, J.; Kamali, S.; Popov, S.; Malloci, M.; Kolesnikov, A.;\nDuerig, T.; and Ferrari, V . 2020. The Open Images Dataset V4:\nUnified image classification, object detection, and visual relation-\nship detection at scale. IJCV .\nLatorre, F.; Rolland, P.; and Cevher, V . 2020. Lipschitz constant\nestimation of Neural Networks via sparse polynomial optimization.\narXiv:2004.08688.\nLin, T.-Y .; Maire, M.; Belongie, S.; Bourdev, L.; Girshick, R.;\nHays, J.; Perona, P.; Ramanan, D.; Zitnick, C. L.; and Doll ´ar,\nP. 2015. Microsoft COCO: Common Objects in Context.\narXiv:1405.0312.\nLiu, Z.; Lin, Y .; Cao, Y .; Hu, H.; Wei, Y .; Zhang, Z.; Lin, S.; and\nGuo, B. 2021. Swin Transformer: Hierarchical Vision Transformer\nusing Shifted Windows. arXiv:2103.14030.\nMadry, A.; Makelov, A.; Schmidt, L.; Tsipras, D.; and Vladu, A.\n2019. Towards Deep Learning Models Resistant to Adversarial\nAttacks. arXiv:1706.06083.\nMainuddin, M.; Duan, Z.; and Dong, Y . 2021. Network Traffic\nCharacteristics of IoT Devices in Smart Homes. arXiv:2109.01855.\nMainuddin, M.; Duan, Z.; Dong, Y .; Salman, S.; and Taami, T.\n2022. IoT Device Identification Based on Network Traffic Char-\nacteristics. In GLOBECOM 2022 - 2022 IEEE Global Communi-\ncations Conference .\nMontufar, G. F.; Pascanu, R.; Cho, K.; and Bengio, Y . 2014. On the\nNumber of Linear Regions of Deep Neural Networks. In Advances\nin Neural Information Processing Systems , volume 27. Curran As-\nsociates, Inc.\nMoosavi-Dezfooli, S.-M.; Fawzi, A.; and Frossard, P. 2016. Deep-\nFool: a simple and accurate method to fool deep neural networks.\narXiv:1511.04599.\nNeyshabur, B.; Bhojanapalli, S.; McAllester, D.; and Srebro, N.\n2017. Exploring Generalization in Deep Learning. In NIPS .\nOpenAI. 2023. GPT-4 Technical Report. arXiv:2303.08774.\nQi, X.; Wang, J.; Chen, Y .; Shi, Y .; and Zhang, L. 2023. Lips-\nFormer: Introducing Lipschitz Continuity to Vision Transformers.\narXiv:2304.09856.\nQin, Y .; Zhang, C.; Chen, T.; Lakshminarayanan, B.; Beutel, A.;\nand Wang, X. 2023. Understanding and Improving Robustness ofVision Transformers through Patch-based Negative Augmentation.\narXiv:2110.07858.\nRadford, A.; Kim, J. W.; Hallacy, C.; Others; and et al. 2021.\nLearning Transferable Visual Models From Natural Language Su-\npervision. arXiv:2103.00020.\nSalman, H.; Jain, S.; Wong, E.; and Madry, A. 2021. Certified Patch\nRobustness via Smoothed Vision Transformers. arXiv:2110.07719.\nScaman, K.; and Virmaux, A. 2019. Lipschitz regularity\nof deep neural networks: analysis and efficient estimation.\narXiv:1805.10965.\nShao, R.; Shi, Z.; Yi, J.; Chen, P.-Y .; and Hsieh, C.-J.\n2022. On the Adversarial Robustness of Vision Transformers.\narXiv:2103.15670.\nSzegedy, C.; Zaremba, W.; Sutskever, I.; Bruna, J.; Erhan, D.;\nGoodfellow, I.; and Fergus, R. 2014. Intriguing properties of neural\nnetworks. arXiv:1312.6199.\nTouvron, H.; Cord, M.; Douze, M.; Massa, F.; Sablayrolles, A.;\nand J ´egou, H. 2021. Training data-efficient image transformers &\ndistillation through attention. arXiv:2012.12877.\nVaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.;\nGomez, A. N.; Kaiser, L.; and Polosukhin, I. 2023. Attention Is\nAll You Need. arXiv:1706.03762.\nXu, P.; Zhu, X.; and Clifton, D. A. 2023. Multimodal learning with\ntransformers: A survey. IEEE Transactions on Pattern Analysis\nand Machine Intelligence .\nZhang, C.; Bengio, S.; Hardt, M.; Recht, B.; and Vinyals, O. 2016.\nUnderstanding deep learning requires rethinking generalization.\nCoRR , abs/1611.03530.\nZhang, C.; Bengio, S.; Hardt, M.; Recht, B.; and Vinyals, O. 2017.\nUnderstanding deep learning requires rethinking generalization.\narXiv:1611.03530.\nZhu, D.; Chen, J.; Shen, X.; Li, X.; and Elhoseiny, M. 2023.\nMiniGPT-4: Enhancing Vision-Language Understanding with Ad-\nvanced Large Language Models. arXiv preprint arXiv:2304.10592 .Appendix\nMore on Vision Transformers\nVery recently, several multi-modal models have been intro-\nduced (Xu, Zhu, and Clifton 2023; Zhu et al. 2023; OpenAI\n2023; Girdhar et al. 2023). By using a shared embedding\nspace among different modalities, such joint models have\nshown to have advantages. Vision transformers have been\nsuccessful in various vision tasks due to their ability to treat\nan image as a sequence of patches and utilize self-attention\nmechanisms.\nTransformer \nEncoder \nBlock\nLinear \nProjection \nof \nflattened \npatches\npositional \nencoding\nMLP \nHead\nTransformer \nEncoder \nBlock\nClass\n \nDog\nCat \n \nBird \n....\nImage \nPatches\nn \nblocks\nFigure 10: Vision Transformer (ViT) architecture (Dosovit-\nskiy et al. 2021).\nA collection of transformer blocks make up the Vision\nTransformer Architecture. Each transformer block com-\nprises two sub-layers: a multi-headed self-attention layer\nand a feed-forward layer. The self-attention layer computes\nattention weights for each pixel in the image based on its\nrelationship with all other pixels, while the feed-forward\nlayer applies a non-linear transformation to the self-attentionlayer’s output. The patch embedding layer separates the im-\nage into fixed-size patches before mapping each patch to a\nhigh-dimensional vector representation. These patch embed-\ndings are then supplied into the transformer blocks to be pro-\ncessed further (Dosovitskiy et al. 2021).\nAdditional Results\nHere we provide more details and additional information\nabout the results we have included in the main text.\nFig. 11 and Fig. 12 show the interpolated images along the\npath.\nTo further demonstrate the effectiveness of the gradient pro-\ncedure to match embeddings, we have applied the procedure\nto numerous images from different sources. As random im-\nages are typical in the input image space, we have applied\nthe procedure to match a specified embedding from ran-\ndomly generated images. Fig. 13 shows that we can match\nthe embeddings of images from a random image; These re-\nsults, along with outcomes from other datasets, demonstrate\nthe efficacy of our technique across all the images we have\nutilized.\nIn the main paper, the results are generated using the pre-\ntrained ImageBind (Girdhar et al. 2023) model, which uti-\nlizes a pre-trained CLIP model (ViT-H-14). As the frame-\nwork does not rely on the specifics of the ImageBind, it is\neffective for other models and datasets as well. To demon-\nstrate that our framework works equally well with other vari-\nants, Fig. 14 shows the results on several different variants of\nthe original vision transformer models6. To further showcase\nthe model-agnostic nature of our techniques, we conduct ex-\nperiments with diverse vision transformer models, including\nDEiT, ViTMAE, and ViTMSN. Please refer to Fig. 9 in the\nmain paper and Fig. 15 for detailed results.\nFig. 16 provides more examples on ImageNet, where vi-\nsually indistinguishable images have very different embed-\ndings and consequently are classified into other classes. In\ncontrast, visually very different images have very similar\nembeddings, aligned to the embedding of a particular image\nand classified into the corresponding class. Additionally, in\nFig. 17 and Fig. 18, we present further examples applying\nour proposed framework to the MS-COCO and Open Im-\nages datasets, affirming the dataset-agnostic nature of our\napproach.\nFig. 19 provides the original images from ImageNet used\nin Fig. 8 and Fig. 16. Similarly, Fig. 20 shows the original\nimages from MS-COCO and Open Images dataset used to\ngenerate the Fig. 17 and Fig. 18.\n6https://github.com/openai/CLIP: 0.0\n : 0.1\n : 0.2\n : 0.3\n : 0.4\n : 0.5\n : 0.6\n : 0.7\n : 0.8\n : 0.9\n : 1.0\nFigure 11: Interpolated Images for Fig. 6.\n: 0.0\n : 0.1\n : 0.2\n : 0.3\n : 0.4\n : 0.5\n : 0.6\n : 0.7\n : 0.8\n : 0.9\n : 1.0\nFigure 12: Interpolated Images for Fig. 7.\nrandom image\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n random image -> agama lizard\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n agama lizard\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\nrandom image\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n random image -> peacock\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n peacock\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\nrandom image\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n random image -> wombat\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\n wombat\n0 1 2 3 4 5\nprincipal component12\n9\n6\n3\n036912projected valueembedding projection\nFigure 13: Example of random image (left) that matches a target embedding (right), with the final image shown in the middle.agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(f)embedding projection(a)\nagama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(f)embedding projection\n(b)\nagama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(f)embedding projection\n(c)\nFigure 14: More examples from ImageNet obtained using the proposed framework with different variants of the original vision\ntransformer, such as (top) ViT-B-16, which has the embedding dimension of 512, (center) ViT-B-32, which has the embedding\ndimension of 512, (bottom) ViT-L-14 which has the embedding dimension of 768.agama lizard\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(f)embedding projection\nagama lizard\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component6\n4\n2\n0246projected value\n(f)embedding projection\nagama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(a)embedding projection\n peacock\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(b)embedding projection\n wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(c)embedding projection\nagama lizard -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(d)embedding projection\n peacock -> wombat\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(e)embedding projection\n wombat -> agama lizard\n0 1 2 3 4 5\nprincipal component9\n6\n3\n0369projected value\n(f)embedding projectionFigure 15: Same as Fig. 1 and Fig. 9, in support of demonstrating that the proposed framework is model-agnostic; shown for\ndifferent other vision transformer models, such as (top two rows) DEiT, (middle two rows) ViTMAE and (the next two rows)\nViTMSN.dog -> cat\n dog -> tiger\n dog -> racing car\n dog -> panda\n dog -> zebra\n0.9975 0.0024792 4.7461e-11 2.3377e-07 1.6917e-05\n8.6353e-05 0.99984 1.7668e-12 2.2632e-06 7.041e-05\n5.9585e-05 5.0451e-10 0.99994 1.422e-06 3.8814e-10\n5.4175e-07 2.6145e-08 2.0719e-12 1.0 3.3574e-08\n1.7533e-06 1.8503e-07 3.2159e-11 3.458e-07 1.0vision x text\nflamingo -> blue heron\n flamingo -> hummingbird\n flamingo -> goldfish\n flamingo -> jellyfish\n flamingo -> mushroom\n0.99998 1.5891e-05 2.7918e-08 3.5024e-07 3.9912e-08\n6.5692e-07 1.0 1.7838e-10 2.7508e-09 1.5289e-09\n3.483e-11 4.3782e-09 1.0 6.7194e-09 8.9677e-11\n1.6602e-09 2.5472e-10 2.5002e-07 1.0 1.4489e-07\n3.1548e-09 6.6183e-10 1.0718e-08 0.00029618 0.9997vision x text\ncanoes -> sunglass\n ladybug -> sunglass\n balloons -> sunglass\n volleyballs -> sunglass\n bell peppers -> sunglass\n1.6779e-09 6.2218e-11 1.1666e-10 1.3876e-10 3.7205e-11 1.0\n1.3182e-11 1.0752e-08 2.4351e-11 1.7684e-11 5.6193e-11 1.0\n6.4583e-11 6.2146e-11 1.3227e-09 5.2623e-10 8.0156e-11 1.0\n1.5072e-11 3.6325e-11 1.3151e-10 1.2676e-10 1.9663e-11 1.0\n1.1775e-10 2.7743e-10 1.8898e-10 9.0745e-11 9.5447e-10 1.0vision x text\numbrellas -> fountain\n purse -> fountain\n teddy bear -> fountain\n vases -> sunglass\n corns -> fountain\n6.9849e-06 9.6605e-09 2.1942e-11 8.6076e-07 8.5166e-09 0.99999\n5.5887e-08 1.0471e-07 1.7128e-10 9.5419e-07 2.6608e-09 1.0\n5.197e-10 4.0554e-09 2.2333e-09 9.0886e-08 7.0402e-10 1.0\n5.8297e-10 2.4471e-09 3.7085e-12 4.5997e-07 2.7914e-10 1.0\n1.3212e-08 9.7316e-09 1.4019e-10 4.2713e-07 1.8929e-08 1.0vision x textFigure 16: (first row) Additional examples where visually indistinguishable images have very different embeddings and conse-\nquently are classified to other classes as in Fig. 1 and Fig. 8. Dog images are classified as a cat, a tiger, a racing car, a panda,\nand a zebra. (second row) Similar as first row, flamingo images are classified as a heron, hummingbird, goldfish, jellyfish, and\nmushroom. (third row) Visually very different images (e.g., some canoes, a ladybug, some balloons, some volleyballs, some bell\npeppers) have very similar embeddings and are classified as sunglass. (fourth row) Similar as third row, different images (e.g.,\nsome umbrellas, a purse, a teddy bear, some vases, some corns) are classified as fountain. The examples are strictly randomly\nchosen. There is no postselection involved.apple -> banana\n apple -> orange\n apple -> vase\n apple -> donut\n apple -> teddy bear\n1.0 7.2258e-09 1.9618e-10 3.6651e-08 1.6196e-09\n0.0010576 0.99808 4.1193e-07 0.00085294 6.9156e-06\n2.8226e-08 2.0859e-06 1.0 1.8908e-06 2.2715e-07\n1.1689e-10 3.4756e-09 1.0704e-12 1.0 1.2119e-09\n6.067e-12 2.1414e-12 6.0786e-10 2.243e-07 1.0vision x text\nbus -> sandwich \n stop sign -> sandwich\n elephants -> sandwich\n traffic light -> sandwich\n motorbike -> sandwich\n0.00014094 0.00022636 2.1401e-09 2.4687e-05 1.8011e-06 0.99961\n6.3003e-06 0.0066666 5.5741e-10 2.4037e-05 1.1805e-07 0.9933\n2.272e-06 8.1179e-05 7.2943e-09 6.0124e-06 2.7624e-07 0.99991\n4.1042e-05 0.0013815 4.0774e-09 0.00016947 4.1124e-06 0.9984\n0.00013757 0.00033407 2.9864e-09 3.7515e-05 0.0001003 0.99939vision x textFigure 17: More examples involving MS-COCO dataset. (top) Visually indistinguishable images have very different represen-\ntations via embedding alignment with the corresponding images and therefore very different classification outcomes. (bottom)\nVisually very different images have very similar embeddings, aligned to the embedding of a specific image and classified into\nthe corresponding class. Again the samples are randomly chosen.\nbee -> dragonfly\n bee -> fish\n bee -> leopard\n bee -> mushroom\n bee -> sheep\n1.0 2.0434e-06 2.0546e-11 1.5034e-11 1.076e-10\n1.4596e-05 0.99999 3.5442e-09 1.1857e-10 5.2593e-09\n3.8112e-07 1.7095e-05 0.99998 3.2473e-10 3.6246e-07\n1.2075e-07 2.3502e-08 1.3974e-10 1.0 1.6441e-08\n2.8852e-08 1.8774e-07 8.1137e-10 1.228e-08 1.0vision x text\nlily -> popcorn \n grape -> popcorn\n high heels -> popcorn\n pumpkin -> popcorn\n alpaca -> popcorn\n1.767e-07 3.3131e-08 4.0687e-12 1.5213e-09 2.0049e-10 1.0\n3.7177e-08 4.8941e-07 2.3625e-12 3.2305e-10 3.7273e-10 1.0\n2.2895e-09 2.0374e-08 7.5533e-11 1.0546e-10 1.2803e-09 1.0\n2.9533e-09 3.7373e-08 2.2788e-12 5.4501e-09 2.5034e-10 1.0\n4.6421e-09 2.2973e-08 5.6506e-12 2.0513e-10 7.9991e-09 1.0vision x text\nFigure 18: More examples involving Open Images dataset having high-resolution images. (top) Visually indistinguishable\nimages have very different representations via embedding alignment with the corresponding images and therefore very different\nclassification outcomes. (bottom) Visually very different images have very similar embeddings, aligned to the embedding of a\nspecific image and classified into the corresponding class. The samples are randomly chosen.Figure 19: The original images from ImageNet corresponds to Fig. 8 and Fig 16.\nFigure 20: The original images correspond to Fig. 17 and Fig 18. (first two rows) MS-COCO, and (the next two rows) Google\nOpen Images." }, { "title": "2401.15570v1.Estimation_of_domain_truncation_error_for_a_system_of_PDEs_arising_in_option_pricing.pdf", "content": "ESTIMATION OF DOMAIN TRUNCATION ERROR FOR A SYSTEM OF\nPDES ARISING IN OPTION PRICING\nANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nAbstract. In this paper, a multidimensional system of parabolic partial differential equations\narising in European option pricing under a regime-switching market model is studied in details.\nFor solving that numerically, one must truncate the domain and impose an artificial boundary\ndata. By deriving an estimate of the domain truncation error at all the points in the truncated\ndomain, we extend some results in the literature those deal with option pricing equation under\nconstant regime case only. We differ from the existing approach to obtain the error estimate that\nis sharper in certain region of the domain. Hence, the minimum of proposed and existing gives\na strictly sharper estimate. A comprehensive comparison with the existing literature is carried\nout by considering some numerical examples. Those examples confirm that the improvement in\nthe error estimates is significant.\nKeywords: regime switching market model, existence and uniqueness of solution, theory of\nsystem of PDEs, far field boundary error estimates, near field error estimates\n1.Introduction\nA multidimensional system of parabolic partial differential equations (PDEs), arising in Eu-\nropean option pricing in a regime-switching market model, is considered in this paper. Its one-\ndimensional version appears in [1, 5, 6] and in many other related works. The system of PDEs\nunder consideration is given by\n\u0012∂ϕ\n∂t+Lϕ\u0013\n(t, s, i) = 0 , (1)\nϕ(T, s, i ) =K(s), (2)\nfor all t∈(0, T),s∈(0,∞)d, and i∈ X:={1,2, ..., k}where\nLϕ(t, s, i) := \nr(i)dX\nl=1sl∂ϕ\n∂sl+1\n2dX\nl=1dX\nl′=1all′(i)slsl′∂2ϕ\n∂sl∂sl′−r(i)ϕ!\n(t, s, i) +kX\nj=1λijϕ(t, s, j ).\n(3)\nHere a(i) = (all′(i))d×dis the diffusion matrix defined as all′(i) :=Pd\nj=1σl,j(i)σl′,j(i), where the\nd×dmatrix σ, having ( l, l′) entry σl,l′:X →R, is called the volatility matrix. The matrix\nΛ := ( λij)k×kis the instantaneous transition rate of a Markov chain on X, and the terminal data\nK(·) is non-negative and Lipschitz continuous.\nWe recall that the existence and uniqueness results of the classical solution of a general class of\nsystem of parabolic PDEs appear in [9, Theorem 9.3, page 256], and [9, Theorem 9.6, page 260]\nrespectively. It is evident that the second order operator in (1) is not strictly elliptic, which is\nnecessary for applying the results of [9]. However, after a suitable transformation (as in [7]), the\nsystem of PDEs (1)-(3) can be written as a constant coefficient problem. Finally, the existence and\nuniqueness of the resulting constant coefficient system follow from a general result [9, Theorem\n2.10]. The analytical solution for the special cases of (1)-(3) have been discussed in [3, 18, 23].\nHowever, all these literature are limited to two state regime-switching economy, and the analytical\n* Corresponding author.\nAuthors acknowledge the support for the financial support from Department of Science & Technology under the\ngrant no. DST/INT/DAAD/P-12/2020 and from National Board for Higher Mathematics under the grant number\n02011-32-2023-R&D-II-13347.\n1arXiv:2401.15570v1 [q-fin.CP] 28 Jan 20242 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nsolutions are expressed in term of cumbersome integrals. Therefore, the development of numerical\ntechniques is unavoidable for solving (1)-(3). For numerical computation of the solution, one must\ntruncate the domain and impose an artificial boundary condition. Thus for estimating as well\nas reducing the truncation error, the growth analysis of the solution on the unbounded domain\nbecomes essential. We prove the positivity and at most linear growth properties of the solution of\n(1)-(3) by using the growth of the terminal data. These results do not follow from [9]. Moreover,\nas an offshoot, we provide a self-contained proof for the existence and uniqueness results for (1)-\n(3) via the semigroup approach. This does not require a transformation to a constant coefficient\nproblem. To the best of our knowledge, the proposed approach is novel.\nVarious numerical methods have been developed in [1, 2, 10, 11, 15, 14] to solve the system\n(1)-(3). However, an analysis of truncation error is absent in these works. A detailed analysis\nof truncation error for a multidimensional PDE, i.e., when Xis a singleton, appears in [12]. We\nextend that analysis in the settings of the system of PDEs in this paper. More precisely, using\nthe derived growth estimate, we have estimated the error at the the boundary caused due to the\nartificial boundary data. Furthermore, an abstract error estimate is obtained at an interior point\nof the domain using a parameterized function. Subsequently, the point-wise estimate is expressed\nin terms of the model parameters and the maximum error on the far boundary.\nOur analysis helps to improve the error estimate given in [12, Theorem 4] in three different\nways. First of all, the error bound has been obtained for a system of multi-dimensional PDEs\ninstead of a single multidimensional PDE. Secondly, the proposed error bound is sharper in certain\nregion of the domain. Hence, the minimum of both gives a strictly sharper estimate. Finally, the\nproposed expression of the error estimate is valid for all the points in the domain unlike the\nexpression given in [12, Theorem 4] which works only on a strictly smaller subdomain. We have\nalso included a comparison with the existing literature by considering some numerical examples\nwith realistic parameter values. Those illustrations confirm that the improvement in the error\nestimates is significant.\nThe present paper is structured as follows: The regime switching market dynamics is briefly\npresented in Sec. 2. The system of PDEs is studied in Sec. 3, and the existence and uniqueness of\nthe solution is proved. The problem on truncated domain is considered in Sec. 4. In this section,\nfar boundary error estimates and near field estimates are derived. Some numerical examples are\npresented to verify the theoretical claims. Sec. 5 includes the concluding remarks with some future\nresearch directions. Appendices A and B are supplemented to provide the proofs of Lemmas, which\nare used to prove the main results.\n2.Regime switching market dynamics\nAssume that r:X → [0,∞),µl:X →R, and σl:X →Rdare given positive functions for each\nl= 1, . . . , d . We consider a friction-less market consisting of one locally risk-free asset with price\nS0anddrisky assets which may be referred to as stocks with prices S1, . . . , Sd. Consider the\nfiltered probability space (Ω ,F,{Ft}t≥0, P) and assume that X:={Xt}t≥0andW:={Wt}t≥0\nareXvalued Markov chain and a d-dimensional standard Brownian motion respectively which are\nindependent to each other and adapted to {Ft}t≥0. We further assume that the Markov chain X\non the statespace Xhas instantaneous transition rate Λ := ( λij)k×k. Let r(Xt) be the floating\ninterest rate of the ideal bank at time t. Therefore, S0at time t, given by S0(t) solves\ndS0(t) =r(Xt)S0(t)dt, S 0(0) = 1 .\nConsider the following stochastic differential equation (SDE)\ndSl(t) =Sl(t)\nµl(Xt)dt+dX\nj=1σl,j(Xt)dWj\nt\n, (4)\nSl(0) = sl, s l>0,\nwhere Wj:={Wj\nt}t≥0is the jthcomponent of Wfor each j= 1, . . . , d . Here µlandσl=\n(σl,1, . . . , σ l,d) represent the growth rate and volatility coefficient of lthasset respectively. WeESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 3\nfurther assume that for each i∈ X,σ(i) is invertible. Since each coefficient is adapted, linear\nin space variable, and bounded in time variable, (4) has an almost sure unique strong solution.\nThe price of the lthstock at time tis modelled by Sl(t). We denote ( S1(t), . . . , S d(t)) by S(t)\nand{S(t)}t≥0byS. Since σ(i) is invertible for each i∈ X,Wis also an adapted d-dimensional\nstandard Brownian motion w.r.t. the completion of filtration generated by ( S, X). Without loss of\ngenerality, we assume this filtration {Ft}t≥0to be right continuous. We consider K(S(T)) as the\npayoff at terminal time T. Let the locally risk-minimizing price of the above payoff in European\nstyle at time tbe denoted as ϕ(t, s, i) where at tthe stock prices are s= (s1, s2, ..., s d), and Markov\nchain is at state i. Then it has been shown in [1, 5, 6] that ϕsolves (1)-(3) classically.\n3.Existence and Uniqueness\n3.1.Existence. In order to show existence of classical solution to (1)-(3), we consider the follow-\ning integral equation (IE)\nϕ(t, s, i) =e−λi(T−t)ηi(t, s) +ZT−t\n0e−{λi+r(i)}vX\nj̸=iλijZ\nx∈(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv,\n(5)\nwhere λi=|λii|, and\n\n\nα(x, s, i, v ):=exph\n−1\n2Pd\nl=1Pd\nl′=1(S−1\nll′)(zl−˜zl)(zl′−˜zl′)i\np\n(2π)d|S|x1.x2...xd,\nS(i) =va(i), zl= ln\u0012xl\nsl\u0013\n,˜zl=\u0012\nr(i)−1\n2all′(i)\u0013\nv.(6)\nHere, |S|denotes the determinant of S. We also recall that a(i) =σ(i)σ(i)∗, where transpose\nof the matrix σis denoted as σ∗. By setting z= (z1, z2, ..., z d), ˜z= (˜z1,˜z2, ...,˜zd), and ln( s) =\n(lns1, . . . , lnsd), we introduce Z∼N(ln(s)+ ˜z,S(i)), ad-dimensional normal random variable for\nevery i∈ X, s∈(0,∞)d, and v >0. Clearly, αis the density of the multi-dimensional lognormal\nrandom vector ( eZ1, eZ2, ..., eZd). Let DandC(D) denote (0 , T)×(0,∞)d×Xand the space of all\nreal-valued component-wise continuous functions on D, respectively. By setting ∥s∥1=Pd\n1|si|,\nwe define\nV:=\u001a\nϕ∈C(D)\f\f\f∥ϕ∥V:= sup\ns,t,i\f\f\f\fϕ(t, s, i)\n1 +∥s∥1\f\f\f\f<∞\u001b\n. (7)\nThe existence and uniqueness results for the integral equation (5) are obtained in Theorem\n3.6, whereas the smoothness of the solution is established in Theorems 3.8 and 3.9. Finally,\nTheorem 3.10 states that the solution to the IE indeed solves the system of PDEs (1)-(3). Below\nsome important properties of α(x, s, i, v ) given in (6) are stated which are crucial for subsequent\nanalysis, and the proofs of those are given in Appendix A.\nLemma 3.1. For every s∈(0,∞)d, v > 0,andi∈ X, we have\nZ\n(0,∞)d \n1 +dX\nl=1xl!\nα(x, s, i, v )dx= 1 +dX\nl=1sler(i)v.\nLemma 3.2. For a fixed s∈(0,∞)d, v > 0,andi∈ X, we have\f\f\f\f1\nα∂α\n∂v\f\f\f\f=O(ln∥x∥1)2,as∥x∥1→ ∞ .\nLemma 3.3. For a fixed s∈(0,∞)d, v > 0, i∈ X, and 1≤l0≤n, we have\n∂α\n∂sl0=α\nsl0O(ln∥x∥),as∥x∥1→ ∞ .\nFollowing lemma includes a tail behaviour of the density function α. Indeed, the present family\nof density functions with family parameter vcoming from a bounded set can be asymptotically\ndominated by a single density function with a larger value of the parameter v.4 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nLemma 3.4. Given s, s′∈(0,∞)d, and v′, ϵ > 0there exists a sufficiently large Rsuch that\nlnα(x, s′, i, v′)>lnα(x, s, i, v )for all x /∈(0,R)dand for all v≤v′−ϵ.\nFor each i∈ X, we consider the following initial value problem (IVP)\n \n∂ηi\n∂t+r(i)dX\nl=1sl∂ηi\n∂sl+1\n2dX\nl=1dX\nl′=1all′(i)slsl′∂2ηi\n∂sl∂sl′!\n(t, s) =r(i)ηi(t, s), (8)\non (0 , T)×(0,∞)dwith terminal condition ηi(T, s) =K(s). For fixed i, (8) is known as the\nBlack-Scholes-Merton PDE for European option price with payoff K(S(T)). Due to the linear\ngrowth property of the payoff K(·),ηi(t, s) also has at most linear growth in svariable for every i.\nIndeed, a log transformation of variables, followed by applications of [9, Theorem 9.3, page 256]\nand Feynman-Kac formula [13, Theorem 4.4.2, page 268] give\nηi(t, s) =Eh\ne−r(i)(T−t)K(Y(T))|Y(t) =si\n, (9)\nwhere Ysatisfies the SDE: dYl(t) =Yl(t)h\nr(i)dt+Pd\nj=1σl,j(i)dWj\nti\n.Hence, x7→α(x, y, i, t )\ngives the density of Y(t) for each t, having Y(0) = y. Consequently, at most linear growth\nproperty of ηi(t,·) follows from the direct application of Lemma 3.1. Moreover, from (9) the\nnon-negativity of ηis evident as K ≥0.\nRemark 3.5. Throughout this article, the various constant notations appear in several subsequent\nproofs. They carry the same meaning inside a single proof but may have different meanings when\nthey appear in a different proof.\nTheorem 3.6. Let\nAϕ(t, s, i) :=e−λi(T−t)ηi(t, s) +ZT−t\n0e−{λi+r(i)}vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv,\n(10)\nfor every ϕ∈V. Then for every ϕ∈V,\n(i)Aϕ∈C(D),\n(ii)Aϕ∈VandA:V→Vis a contraction,\n(iii) IE (5) has a unique solution in the Banach space (V,∥·∥V).\nProof. (i) To show that Aϕis inV, its continuity in tandsvariables will be shown first. As the\nfirst term on the right side of (10) is continuous, it is enough to show the continuity of the integral\nterm with respect to tandsvariables.\nContinuity in tvariable: We need to show that\n\nZT−t−ϵ\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v+ϵ, x, j )α(x, s, i, v )dxdv\n−ZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv\n→0 asϵ→0.\nNote that if ϵ > 0 we can split the second integral in two parts, where first part will be an\nintegration from 0 to ϵ, and second part will be an integration from ϵtoT−t. On the other hand,\nifϵ <0, we can split the first integral in two parts where the first part will be an integration from\n0 toT−tand second part will be an integration from T−ttoT−t−ϵ. Since the analyses for\nϵ >0 and ϵ <0 are very similar, we present only for the ϵ >0 case. Using a suitable substitutionESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 5\nof variables in the first integral term, we get\nZT−t\nϵX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\u0010\ne−(λi+r(i))(v−ϵ)α(x, s, i, v −ϵ)−e−(λi+r(i))v\nα(x, s, i, v ))dxdv−Zϵ\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv,\n=term 1 +term 2 (say).\nAsϕis in V, the direct use of Lemma 3.1 implies that the integral with respect to xvariable\nappearing in term 2 is a bounded function in v. Hence term 2 is an integral of a bounded function\ninvover the interval (0 , ϵ). Therefore as ϵ→0,term 2→0.\nWe denote the factor\u0002\ne−(λi+r(i))(v−ϵ)α(x, s, i, v −ϵ)−e−(λi+r(i))vα(x, s, i, v )\u0003\nof integrand of\nterm 1 asβϵ(x, s, i, v ). Since αande−(λi+r(i))vare continuously differentiable, we can write using\nmean value theorem (MVT):\nβϵ(x, s, i, v ) =−ϵ∂\n∂vh\ne−(λi+r(i))(v−ϵ1)α(x, s, i, v −ϵ1)i\n,\nfor some 0 < ϵ1< ϵ. From Lemma 3.2, there exist C1(v) and C2(v) such that\n∂α\n∂v≤\u0000\nC1+C2ln2∥x∥1\u0001\nα∀x /∈B, (11)\nwhere B= (0,R)d, for a sufficiently large R. For each v >0, inner integral of term 1 is equal to\nϵZ\n(0,∞)dϕ(t+v, x, j )e−(λi+r(i))(v−ϵ1)\u0010\n(λi+r(i))α(x, s, i, v −ϵ1)−∂α\n∂v(x, s, i, v −ϵ1)\u0011\ndx. (12)\nWe know that if 0 < ϵ≪v, sup0<ϵ1<ϵα(x, s, i, v −ϵ1) is bounded on B. We write the integral\n(12) as sum of two integrals by decomposing the domain (0 ,∞)das union of BandBc. For the\nintegral over Bwith a finite domain and uniformly bounded integrand, the convergence is obvious\ndue to the dominated convergence theorem (DCT). Next from (11), we note that the integrand\noverBcis dominated by ( C3+C4∥x∥1)(C5+C2ln2∥x∥1)α(x, s, i, v −ϵ1) for some C2, C3, C4, and\nC5chosen independent of ϵ1. On the other hand, using Lemma 3.4 we get some R>0 such that\nsup\n0<ϵ1<ϵα(x, s, i, v −ϵ1)≤α(x, s, i, v + 2ϵ)∀x /∈B.\nUsing above inequality it is evident that integrand in (12) is dominated by ( C6+C7∥x∥2\n1)α(x, s, i, v +\n2ϵ) onBc. Now we have the following claim:\nZ\n(C6+C7∥x∥2\n1)α(x, s, i, v + 2ϵ)dxconverges to− − − − − − − − →Z\n(C6+C7∥x∥2\n1)α(x, s, i, v )dx <∞. (13)\nTo prove above claim, we note that the integrand of L.H.S. of (13) is a product of a fixed quadratic\nfunction, and a lognormal density. This is uniformly integrable family of functions in xwhere\nfamily parameter, v+ 2ϵ, vary on a bounded set away from {0}. This family is also tight as a\nconsequence of tightness of Gaussian measures with parameters from a bounded set (bounded\nmean and variance). Then from generalized Vitali’s theorem ( pp.98, [21]) (13) holds. Using (13),\nand General Lebesgue Convergence Theorem (Theorem 19, Chapter 4 in [21]), we assert that as\nϵ→0, the integral in (12) converges to\nZ\n(0,∞)dϕ(t+v, x, j )\u0014\n−∂\n∂v\u0010\ne−(λi+r(i))vα(x, s, i, v )\u0011\u0015\ndx. (14)\nHowever, the expression in (12) is product of ϵand the integral. Hence, for the outer integral\nw.r.t. vvariable in term 1, the integrand converges to zero pointwise as ϵgoes to 0. For a fixed s,\nthis convergence is indeed uniform as the integrand of R.H.S. of (13) is a bounded function of v\nover the interval (0 , T−t). Thus using the DCT, term 1 converges to 0 as ϵ→0.6 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nContinuity in slvariables: Let 1 ldenote the unit vector along lth direction and 1 l(l′) denote\nthel′th component of 1 l, and\nγϵ(x, s, i, v ) := ( α(x, s+ϵ1l, i, v)−α(x, s, i, v )).\nWe need to show that for each 1 ≤l≤d\nZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )γϵ(x, s, i, v )dxdv→0, (15)\nasϵ→0 from either sides. Since αis continuously differentiable, we write using the MVT,\nγϵ(x, s, i, v ) =ϵ∂α\n∂sl(x, s+ϵ11l, i, v), for some 0 <|ϵ1|<|ϵ|. Now, inner integral of (15) is equal to\nϵZ\n(0,∞)dϕ(t+v, x, j )∂α\n∂sl(x, s+ϵ11l, i, v)dx. (16)\nWe know that sup0<|ϵ1|<|ϵ|ϕ(t+v, x, j )∂α\n∂sl(x, s+ϵ11l, i, v) is bounded on Bfor each s,i,v,l,\nj, and |ϵ|< sl. We write the integral (16) as sum of two integrals by decomposing the domain\n(0,∞)das union of BandBc. For the first integral with a finite domain and bounded integrand,\nthe convergence is obvious due to the DCT. Hence, we consider the integral on Bconly. Again,\nLemma 3.3 guarantees the existence of constants C1(v) and C2(v) such that\n∂α\n∂sl≤(C1+C2ln∥x∥1)α\nsl,∀x /∈B,\nwhere B= (0,R)dfor some R>0. Consequently, on Bc, the integrand of (16) is dominated\nby1\nsl−|ϵ|(C3+C4∥x∥1)(C1+C2ln∥x∥1)α(x, s+ϵ11l, i, v) for some C1, C2, C3andC4chosen\nindependent of ϵ1. On the other hand using Lemma 3.4 we get\nsup\n|ϵ1|<|ϵ|α(x, s+ϵ11l, i, v)≤α(x, s, i, v +|ϵ|),∀x /∈B.\nUsing this, it is evident that expression in (16) is dominated by\n1\nsl− |ϵ|(C6+C5∥x∥2\n1)α(x, s, i, v +|ϵ|).\nSince (13) holds, we also have\nZ\n(C6+C5∥x∥2\n1)α(x, s, i, v +|ϵ|)dxconverges to− − − − − − − − →Z\n(C6+C5∥x∥2\n1)α(x, s, i, v )dx <∞. (17)\nNext, the convergence of outer integral of (15) can be argued in a similar way as done for the\nouter integral of term 1 while proving continuity in tvariable. Hence Aϕ∈C(D).\n(ii) Prior to show that the range of Ais in ( V,∥·∥V), we consider\n∥Aϕ1− Aϕ2∥V= sup\nD\f\f\f\fAϕ1− Aϕ2\n1 +∥s∥1\f\f\f\f,\n= sup\nD\f\f\f\f\f\fZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)n|ϕ1−ϕ2|(x)α(x, s, i, v )\n(1 +∥s∥1)dxdv\f\f\f\f\f\f,\n= sup\nD\f\f\f\f\f\fZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)n(1 +∥x∥1)|ϕ1−ϕ2|(x)\n(1 +∥x∥1)α(x, s, i, v )\n(1 +∥s∥1)dxdv\f\f\f\f\f\f,\n≤∥ϕ1−ϕ2∥Vsup\nD\f\f\f\f\f\fZT−t\n0e−(r(i)+λ(i))vX\nj̸=iλijZ\n(0,∞)d(1 +∥x∥1)α(x, s, i, v )\n(1 +∥s∥1)dxdv\f\f\f\f\f\f.ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 7\nWe simplify the above inequality using the result in Lemma 3.1, and obtain\n∥Aϕ1− Aϕ2∥V≤ ∥ϕ1−ϕ2∥Vsup\nD\f\f\f\f\f\fZT−t\n0e−(r(i)+λ(i))vX\nj̸=iλij1 +Pd\nl=1sler(i)v\n1 +Pd\nl=1sldv\f\f\f\f\f\f,\n≤ ∥ϕ1−ϕ2∥Vsup\nDZT−t\n0e−λivX\nj̸=iλijdv,\n=∥ϕ1−ϕ2∥Vsup\nDZT−t\n0λie−λivdv=C∥ϕ1−ϕ2∥V, (18)\nfor some C <1. To show that ∥Aϕ∥Vis finite for every ϕ∈V, we take ϕ2= 0 in above inequality\n(18), and get\n∥Aϕ1∥V≤ ∥A ϕ1− A0∥V+∥A0∥V≤C∥ϕ1∥V+∥A0∥V.\nUsing (10), A0=e−λi(T−t)ηi(t, s). Hence, from the argument below Eq. (9) ∥A0∥V<∞which\nimplies ∥Aϕ1∥Vis also finite. Again since Aϕ∈C(D), we have A:V→V. Thus (18) implies\nthatAis a contraction on V.\n(iii) A direct application of Banach fixed point theorem [17, Theorem A1, page 528] gives that A\nhas a unique fixed point in V. Hence, (5) has a unique solution. □\nFor showing that the unique solution ϕ∈Vof (5) solves (1)-(3) classically, we prove in Theorems\n3.8 and 3.9 that ϕhas required smoothness. The following lemma is required in the proof of\nTheorem 3.8 and its proof is given in Appendix A.\nLemma 3.7. Letϕbe the solution of integral equation (5). Then for each t, s, j, j′, we have\nlim\nu↓0Z\n(0,∞)dϕ(t+u, x, j )α(x, s, j′, u)dx=ϕ(t, s, j ). (19)\nTheorem 3.8. Letϕ∈Vand solves (5), then it is differentiable in tvariable. Furthermore, for\ng1(x, s, i, v ) :=\u0000∂α\n∂v/α\u0001\n(x, s, i, v )\n∂ϕ\n∂t(t, s, i) =−r(i)e−λi(T−t)ηi(t, s) +e−λi(T−t)∂ηi\n∂t−X\nj̸=iλijϕ(t, s, j ) + (λi+r(i))×ϕ(t, s, i)\n−ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )e−(λi+r(i))v×g1(x, s, i, v )α(x, s, i, v )dxdv. (20)\nProof. The partial derivative∂ϕ\n∂t(t, s, i), if exists, can be written as follows:\n∂ϕ\n∂t(t, s, i) =∂\n∂t\u0010\ne−λi(T−t)ηi(t, s)\u0011\n+ lim\nϵ→01\nϵ\nZT−t−ϵ\n0e−(λi+r(i))vX\nj̸=iλij×Z\n(0,∞)dϕ(t+v+ϵ, x, j )\nα(x, s, i, v )dxdv−ZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv#\n.\nThe partial derivative, i.e. the first term on R.H.S., exists as it is the derivative of product of two\nsmooth functions. Next, we consider the second (limit) term. Using a suitable substitution, the\nlimit term is equals to\n1\nϵ\"ZT−t\nϵX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j ) \ne−(λi+r(i))(v−ϵ)α(x, s, i, v −ϵ)−e−(λi+r(i))v\nα(x, s, i, v )!\ndxdv#\n−1\nϵZϵ\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv,\n=term 1 +term 2 (say).8 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nFrom Lemma 3.7, as ϵ→0,term 2→ −P\nj̸=iλijϕ(t, s, j ). As explained in the proof of Theorem\n3.6, the inner integral of term 1 converges to (14) with an ϵmultiplied. After cancelling the ϵwith\n1\nϵ, and using the uniform boudedness of the integrand of the outer integral, we get\nterm 1→ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\u0014\n−∂\n∂v\u0010\ne−(λi+r(i))vα(x, s, i, v )\u0011\u0015\ndxdv.\nTherefore, we can write\n∂ϕ\n∂t(t, s, i)−∂\n∂t\u0010\ne−λi(T−t)ηi(t, s)\u0011\n+X\nj̸=iλijϕ(t, s, j )\n=ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\u0014\n−∂\n∂v\u0010\ne−(λi+r(i))vα(x, s, i, v )\u0011\u0015\ndxdv,\n=ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\u0014\n(λi+r(i))e−(λi+r(i))vα(x, s, i, v )−e−(λi+r(i))v∂α\n∂v\u0015\ndxdv,\n=−ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )e−(λi+r(i))v∂α\n∂vdxdv\n+ (λi+r(i))ZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )α(x, s, i, v )dxdv. (21)\nBy rewriting the last term of (21) using (5), the above becomes\n∂ϕ\n∂t(t, s, i) =∂\n∂t\u0010\ne−λi(T−t)ηi(t, s)\u0011\n−ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )e−(λi+r(i))v\n∂α\n∂v(x, s, i, v )dxdv−X\nj̸=iλijϕ(t, s, j ) + (λi+r(i))\u0010\nϕ(t, x, i )−e−λi(T−t)ηi\u0011\n.\nThe simplification of right side of above expression gives (20). □\nTheorem 3.9. Ifϕ∈Vsolves (5), then for each l≤d,ϕis twice differentiable in slvariable.\nFurthermore,\n∂ϕ\n∂sl(t, s, i) =e−λi(T−t)∂ηi\n∂sl+ZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )gl\n2(x, s, i, v )\nα(x, s, i, v )dxdv, (22)\n∂2ϕ\n∂sl∂sl′(t, s, i) =e−λi(T−t)∂2ηi\n∂sl∂sl′+ZT−t\n0e−(λi+r(i))vX\nj̸=iλij\nZ\n(0,∞)dϕ(t+v, x, j )\u0012∂\n∂sl′gl\n2+gl\n2gl′\n2\u0013\n(x, s, i, v )α(x, s, i, v )dxdv, (23)\nwhere gl\n2(x, s, i, v ) :=\u0010\n∂α\n∂sl/α\u0011\n(x, s, i, v ).\nProof. From (5), we can write\n∂ϕ\n∂sl(t, s, i) =∂\n∂sl\"\ne−λi(T−t)ηi+ZT−t\n0e−{λi+r(i)}vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )×α(x, s, i, v )dxdv#\n,\n(24)ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 9\nprovided the partial derivative exists. In other words, (24) can be rewritten as\n∂ϕ\n∂sl(t, s, i) =e−λi(T−t)∂ηi\n∂sl+ lim\nϵ→01\nϵZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\n×(α(x, s+ϵ1l, i, v)−α(x, s, i, v ))dxdv,\nif the above limit exists. Now we consider the second (limit) term only. We have already proved\nin Theorem 3.6 (continuity in svariable part) that as ϵ→0, this second (limit) term converges to\nZT−t\n0e−(λi+r(i))vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )∂α\n∂sl(x, s, i, v )dxdv,\nwhich gives the desired expression for∂ϕ\n∂sl(t, s, i) as in (22), since gl\n2α:=∂α\n∂sl. Now for the second\norder partial derivative∂2ϕ\n∂sl∂s′\nl(t, s, i), we write\n∂2ϕ\n∂sl∂sl′(t, s, i) =∂2\n∂sl∂sl′\"\ne−λi(T−t)ηi+ZT−t\n0e−{λi+r(i)}vX\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, j )\n×α(x, s, i, v )dxdv#\n,(25)\nprovided the partial derivative in (25) exists. One can show the existence of this in a similar\nline of that for the first order derivative with the only difference arising from the tail estimate\nof∂2α\n∂sl∂sl′. However in this case, similar to Lemma 3.3, we have∂2α\n∂sl∂sl′=α\nslsl′O(ln∥x∥)2, for a\nfixed s∈(0,∞)d, v > 0. Thus, we can dominate the second order partial derivative of αby the\nproduct of α, and a quadratic function in x. Note that this dominating function has also arisen\nin Theorem 3.6 (continuity in svariable part). Hence, the rest of the proof of this theorem will\nexactly be the same as the proof of Theorem 3.6 (continuity in svariable part). □\nWe have proved that unique solution ϕof IE (5) is sufficiently smooth. Now we prove that it\nalso satisfies the system of PDEs (1) in the following Theorem.\nTheorem 3.10. Letϕ(t, s, i)be the unique solution of IE (5) then\n(i)ϕ(t, s, i)also satisfies PDE (1)-(3).\n(ii)ϕ(t, s, i)is non-negative and of at-most linear growth.\nProof. Using (20) from Theorem 3.8, and (22) & (23) from Theorem 3.9, we get\n\u0012∂ϕ\n∂t+r(i)dX\nl=1sl∂ϕ\n∂sl+1\n2dX\nl=1dX\nl′=1slsl′all′(i)∂2ϕ\n∂sl∂sl′\u0013\n(t, s, i)\n=e−λi(T−t)\"\n∂ηi\n∂t−r(i)ηi+r(i)dX\nl=1sl∂ηi\n∂sl+1\n2dX\nl=1dX\nl′=1slsl′all′(i)∂2ηi\n∂sl∂sl∂sl′#\n−X\nj̸=iλijϕ(t, s, j ) + (r(i) +λi)ϕ(t, s, i) +ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, i )e−(λi+r(i))v\nα(x, s, i, v )×\"\n−g1+r(i)dX\nl=1slgl\n2+1\n2dX\nl=1dX\nl′=1slsl′all′(i)\u0012∂\n∂sl′gl\n2+gl\n2gl′\n2\u0013#\ndxdv,10 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nwhich implies\n\u0012∂ϕ\n∂t+r(i)dX\nl=1sl∂ϕ\n∂sl+1\n2dX\nl=1dX\nl′=1slsl′all′(i)∂2ϕ\n∂sl∂sl′−r(i)ϕ\u0013\n(t, s, i) +kX\nj=1λijϕ(t, s, j )\n=ZT−t\n0X\nj̸=iλijZ\n(0,∞)dϕ(t+v, x, i )e−(λi+r(i))vα(x, s, i, v )\"\n−g1+r(i)\n×dX\nl=1slgl\n2+1\n2dX\nl=1dX\nl′=1slsl′all′(i)\u0012∂\n∂sl′gl\n2+gl\n2gl′\n2\u0013#\n(x, s, i, v )dxdv.\nWe refer to Appendix A for proof of the following identity\n\"\n−g1+r(i)dX\nl=1slgl\n2+1\n2dX\nl=1dX\nl′=1slsl′all′(i)\u0012∂\n∂sl′gl\n2+gl\n2gl′\n2\u0013#\n(x, s, i, v ) = 0 . (26)\nUsing (26), the equation above reduces to\n\u0012∂ϕ\n∂t+r(i)dX\nl=1sl∂ϕ\n∂sl+1\n2dX\nl=1dX\nl′=1slsl′all′(i)∂2ϕ\n∂sl∂sl′\u0013\n(t, s, i) +kX\nj=1λijϕ(t, s, j )−r(i)ϕ(t, s, i) = 0 .\nAgain, in (5), by substituting t=T, we get ϕ(T, s, i ) =ηi(T, s) =K(s). The last equality is due\nto (8). Thus, unique solution ϕ(t, s, i) of IE (5) satisfies (2) along with system of PDEs (1).\n(ii) Since, Kis non-negative, ηin (10) is non-negative too. Therefore, the left side of (10) is\nnon-negative, provided ϕ≥0. Thus A:H→H, where His the set of all non-negative functions\ninV. Clearly, His a complete metric space with metric d(h1, h2) =∥h1−h2∥Vtoo. Moreover,\nTheorem 3.6-(ii) implies d(Aϕ1,Aϕ2)≤Cd(ϕ1, ϕ2) for some 0 < C < 1. Finally, the result follows\nfrom [4, Theorem 17 .1(a) ]. □\n3.2.Uniqueness. In this subsection, we aim to prove that the system of PDEs (1)-(3) has unique\nclassical solution in ( V,∥·∥V) via probabilistic approach. Let 1 ≤l≤dand ˜Sl:={˜Sl(t)}t≥0be\nthe strong solution of the following SDE\nd˜Sl(t) =˜Sl(t)\u0010\nr(Xt)dt+ ˆσl(Xt)dˆWl\nt\u0011\n,˜Sl(0)>0, (27)\nwhere ˆ σl(i) =\r\rσl(i)\r\r\n2, and ˆWl\nt=P\njσl,j(Xt)Wj\nt\nˆσl. Note that, ˆWl\ntis a Brownian motion using [19,\nTheorem 8 .4.2, pp. 143]. We denote ( ˜S1, . . . , ˜Sd) by ˜S, and the filtration generated by ( ˜S, X) by\n{˜Ft}t. The following lemmas, whose proofs are given in Appendix A, are crucial to achieve the\nuniqueness result.\nLemma 3.11. Thelthcomponent ˜Slof˜Sis a sub-martingale for each l.\nDue to the above lemma, and the finite second order moment of ˜Sl, we can apply the Doob’s\nmaximal inequality [22, pp.132, Theorem 7.3.2] on ˜Slto get\nE\u0012\nsup\ns≤t|˜Sl(s)|\u0013\n<∞, (28)\nfor each l= 1, . . . , d , and t≥0.\nLemma 3.12. Ifϕ(t, s, i)is a classical solution of (1) with at most linear growth, then the process\nNϕ={Nϕ\nt}t≥0given by\nNϕ\nt=e−Rt\n0r(Xu)duϕ(t,˜S(t), Xt), (29)\nis a martingale.ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 11\nFrom Theorem 3.10, we know that (1)-(3) has at least one classical solution in V. Again, the\nabove lemma implies that every such solution ϕwould produce a martingale Nϕ(as in (29)). Then\nusing Nϕ\nt=E[Nϕ\nT|˜Ft], Markovity of ( ˜S, X), and the terminal condition of (1)-(3), we get\ne−Rt\n0r(Xu)duϕ(t,˜St, Xt) =E\u0010\ne−RT\n0r(Xu)duK(˜S(T))|˜Ft\u0011\n,\nor,ϕ(t, s, i) =E\u0010\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t) =s, Xt=i\u0011\n. (30)\nSince every classical solution of (1)-(3) in Vhas the identical expression (30), all of them are\nidentical. By this we have established the following theorem.\nTheorem 3.13. The problem (1)-(3) has a unique classical solution in V.\nIt is evident from the above discussion and Theorem 3.13 that the classical solution of (1)-(3)\nhaving at most linear growth also solves the IE (5). However, above results do not indicate how\nto derive the IE from the PDE. For an independent interest, we produce the derivation of the IE\n(5) using the stochastic representation (30) of ϕ(t, s, i) in Appendix B.\n4.Truncated domain problem\nDue to the absence of analytical solution of (1)-(3), it needs to be solved numerically by truncating\nthe unbounded domain to a bounded one. Let ψ: [0, T]×Πd\nl=1[sb\nl, su\nl]× X → Rbe such that in\nthe interior\n\u0012∂ψ\n∂t+r(i)dX\nl=1sl∂ψ\n∂sl+1\n2dX\nl=1dX\nl′=1all′(i)slsl′∂2ψ\n∂sl∂sl′−r(i)ψ\u0013\n(t, s, i) +kX\nj=1λijψ(t, s, j ) = 0 ,(31)\nψ(T, s, i ) =K(s)∀s∈R, i∈ X, (32)\nψ(t, s, i) =h(t, s, i)∀(t, s, i)∈(0, T)×Γ× X, (33)\nwhere R= Πd\nl=1(sb\nl, su\nl), 0≤sb\nl< su\nl,∀l, and Γ = ∂R∩(0,∞)d. For each i,h(·,·, i) is set as a\nsufficiently smooth function on closure of (0 , T)×R. The existence and uniqueness of the classical\nsolution of (31)-(33) can be borrowed from the Theorem 3 .5 onpp.291 in [8], and also in Theorem\n10.1 onpp.616 in [16]. However, the application of these Theorems requires smoothness of Γ. For\nour case, this is achieved by a smooth approximation of R, which is explained below.\nFor any ε∈(0,1), let U0\nε:={s∈Rd|Pd\nl=1|sl|1/ε<1}and the diagonal matrix Mbe such\nthat the lth diagonal element issu\nl−sb\nl\n2. Then clearly Uε:=1\n2(sb+su) +MU0\nεis contained in R,\nand having smooth boundary. Furthermore, Uε1⊂Uε2for any 1 ≥ε1> ε2>0, and\n[\nε∈(0,1)Uε=R.\nHence, due to the smoothness of h,ψε, the classical solution of (31)-(33), where Ris replaced by\nUε, approximates ψfor sufficiently small ε >0.\n4.1.Growth estimate. Next, we derive a growth estimate of the solution of the un-truncated\nproblem depending on the growth of the terminal data. This is useful in several aspects, including\nin estimating the error due to the boundary data of the truncated domain problem. A similar\nresult appeared in [12, Theorem 2] for Black-Scholes-Merton PDE, which is extended for a system\nof PDEs here. It is worth noting that the present proof is entirely different from that given in [12].\nTheorem 4.1. Letϕ(t, s, i)be the solution of (1)-(3). In addition to the non-negativity and\nLipschitz continuity, we further assume that\n−k3+k4·s≤ K(s)≤k1+k2·s∀s∈(0,∞)d, (34)\nfor some k1, k3≥0and vectors k2, k4∈Rd, where “·”represents inner product. Then\n−k3e−(min ir(i))(T−t)+k4·s≤ϕ(t, s, i)≤k1e−(min ir(i))(T−t)+k2·s∀(t, s, i)∈D. (35)12 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nProof. From (30) and (34), we can write,\nEh\ne−RT\ntr(Xu)du(−k3+k4·˜S(T))|˜S(t), Xti\n≤ϕ(t,˜S(t), Xt)\n≤Eh\ne−RT\ntr(Xu)du(k1+k2·˜S(T))|˜S(t), Xti\n.\nUsing the Markovity of ( ˜S, X) w.r.t. {˜Ft}t, we have\nEh\ne−RT\ntr(Xu)du(−k3+k4·˜S(T))|˜Fti\n≤ϕ(t,˜S(t), Xt)\n≤Eh\ne−RT\ntr(Xu)du(k1+k2·˜S(T))|˜Fti\n.\nMultiplying e−Rt\n0r(Xu)duto each term in the above inequality, we obtain\nEh\ne−RT\n0r(Xu)du(−k3+k4·˜S(T))|˜Fti\n≤e−Rt\n0r(Xu)duϕ(t,˜S(t), Xt)\n≤Eh\ne−RT\n0r(Xu)du(k1+k2·˜S(T))|˜Fti\n,\nor\n−k3Eh\ne−RT\n0r(Xu)du|˜Fti\n+k4·Eh\ne−RT\n0r(Xu)du˜S(T)|˜Fti\n≤e−Rt\n0r(Xu)duϕ(t,˜S(t), Xt)\n≤k1Eh\ne−RT\n0r(Xu)du|˜Fti\n+k2·Eh\ne−RT\n0r(Xu)du˜S(T)|˜Fti\n.\nNote that the function φl∈Vgiven by φl(t, s, i) := slsolves (1) classically for each l. By\napplying Lemma 3.12, we get that {e−Rt\n0r(Xu)du˜Sl\nt}t≥0is martingale for each l. Hence, using\nEh\ne−RT\n0r(Xu)du˜S(T)|˜Fti\n=e−Rt\n0r(Xu)du˜S(t), the above inequality reduce to\n−k3Eh\ne−RT\n0r(Xu)du|˜Fti\n+k4·e−Rt\n0r(Xu)du˜S(t)≤e−Rt\n0r(Xu)duϕ(t,˜S(t), Xt),\n≤k1Eh\ne−RT\n0r(Xu)du|˜Fti\n+k2·e−Rt\n0r(Xu)du˜S(t).\nCancelling e−Rt\n0r(Xu)dufrom each term, we get\n−k3Eh\ne−RT\ntr(Xu)du|˜Fti\n+k4·˜S(t)≤ϕ(t,˜S(t), Xt)\n≤k1Eh\ne−RT\ntr(Xu)du|˜Fti\n+k2·˜S(t)−k3e−(min ir(i))(T−t)+k4·˜S(t)\n≤ϕ(t,˜S(t), Xt)≤k1e−(min ir(i))(T−t)+k2·˜S(t),\nalmost surely for all t∈[0, T]. Hence (35), obtained by replacing ˜S(t) =sandXt=i, follows for\nalmost every ( t, s, i)∈D, since Xis irreducible, and ˜Sis not degenerate on the positive orthant.\nIn fact, the inequality holds for all ( t, s, i)∈D, asϕis continuous, for every ( t, s, i)∈D. □\nRemark 4.2 (Far boundary estimate) .From Theorem 4.1, we can obtain the error bound ∥ϕ−ψ∥V1,\nwhere ϕandψare solutions of (1)-(3) and (31)-(33) respectively, and\nV1=\u0012\nC((0, T)×Γ× X),∥ϕ∥V1= sup\nt,s,i|ϕ(t, s, i)|\n1 +∥s∥1\u0013\n.\nTo be more precise, the maximum error on the boundary due to the imposition of artificial data\nis not more than maxn\n∥k1e−(min ir(i))(T−t)+k2·s−h(t, s, i)∥V1,∥ −k3e−(max ir(i))(T−t)+k4·s−\nh(t, s, i)∥V1o\n.In literature, this bound is often termed as far field boundary error estimate.ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 13\n4.2.Near field estimates. In this subsection, we establish a few intermediate results for de-\nveloping our final result Theorem 4.7, which is an extension of [12, Theorem 4]. The following\nlemma, which resembles to [12, Lemma 1], is proved here using a probabilistic method instead of\nan analytical approach. While [12, Lemma 1] is for Black-Scholes-Merton model and deals with\na scalar equation, the following lemma is for a parabolic system of equations originating from the\nregime-switching extension of the Black-Scholes-Merton model.\nLemma 4.3. Letf1andf2be in\nC\u0000\n[0, T]×(¯U ∩(0,∞)d)× X\u0001\n∩C1,2((0, T)× U × X ),\nwith at most linear growth in space variable, where Uis an open domain in (0,∞)d. We also\nassume that\u0012∂f1\n∂t+Lf1\u0013\n(t, s, i)≤0on(0, T)× U × X , (36)\n\u0012∂f2\n∂t+Lf2\u0013\n(t, s, i)≥0on(0, T)× U × X , (37)\nandf1≥f2att=T, and on (0, T)×(∂U ∩(0,∞)d)× X. Then f1≥f2in(0, T)× U × X .\nProof. Given any two real numbers a, b, leta∧bdenote min( a, b). We fix a ( t, s, i)∈(0, T)×U×X .\nWe define τ=T∧τ′, where τ′:= inf{t′≥t|˜S(t′)/∈ U} , gives the exit time of {˜S(t′)}t′≥tfrom\nU. We further specify that ˜Ssolves the SDE in (27) with ˜S(t) =s. The transition kernel of the\nMarkov chain {Xt′}t′≥tis as before with Xt=i. Let fjbe in C\u0000\n[0, T]×(¯U ∩(0,∞)d)× X\u0001\n∩\nC1,2((0, T)× U × X ), with\nsup\f\f\f\ffj(t′, s′, i′)\n1 +∥s′∥1\f\f\f\f<∞,(t′, s′, i′)∈[0, T]×(¯U ∩(0,∞)d)× X,\nforj= 1,2. For each jlet the processes Nfj:={Nfj\nt′}t′≥tbe defined as in (29) by\nNfj\nt′=e−Rt′∧τ′\ntr(Xu)dufj(t′∧τ′,˜S(t′∧τ′), Xt′∧τ′).\nThen as in the proof of Lemma 3.12, we obtain that\nt′7→Mfj\nt′:=Nfj\nt′−Zt′∧τ′\nte−Ru\ntr(Xu′)du′\u0012∂fj\n∂t+Lfj\u0013\n(u,˜S(u), Xu)du,\nis a local martingale for each j= 1,2. Let us introduce a sequence of stopping times {τn}n, where\nτnrepresents the exit time of ˜S(starting from sat time t) from an open neighbourhood of swhere\nthe modulus of the functions f1,f2, and their first order time derivative, all first and second order\npartial space derivatives are bounded by n. Thus in the expression of Mfj\nt′∧τn, the boundedness of\nfjand its partial derivatives may be assumed. Hence, by an argument similar to that appearing\nin the proof of Lemma 3.12, {Mfj\nt′∧τn}t′≥tis a martingale for each n. We also note that for every\nt′≥t,E[Mfj\nt′∧τn] =E[Mfj\nt∧τn]. But Mfj\nt∧τn=Mfj\nt=Nfj\nt=fj(t, s, i). So, for each j\nE(Nfj\nt′∧τn) =fj(t, s, i) +E\"Zt′∧τ′∧τn\nte−Ru\ntr(Xu′)du′\u0012∂fj\n∂t+Lfj\u0013\n(u,˜S(u), Xu)du#\n.\nThe difference of above equations for j= 1 and 2 gives\nEh\ne−Rt′∧τ′∧τn\ntr(Xu)du(f1−f2)(t′∧τ′∧τn,˜S(t′∧τ′∧τn), Xt′∧τ′∧τn)i\n= (f1−f2)(t, s, i) +E\"t′∧τ′∧τnZ\nte−Ru\ntr(Xu′)du′\u001a\u0012∂f1\n∂t+Lf1\u0013\n−\u0012∂f2\n∂t+Lf2\u0013\u001b\n(u,˜S(u), Xu)du#\n≤(f1−f2)(t, s, i), (38)14 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nasf1andf2satisfy (36) and (37) respectively. On the other hand, τn→ ∞ almost surely as\nn→ ∞ . Therefore, due to the growth constraint on fjand (28), the left side of (38) converges\nast′↑Tandn→ ∞ toEh\ne−RT∧τ′\ntr(Xu)du(f1−f2)(T∧τ′,˜S(T∧τ′), XT∧τ′)i\n,which is non-\nnegative due to the assumption that f1≥f2att=T, and on (0 , T)×∂U × X . Thus the right\nside of (38), i.e., ( f1−f2)(t, s, i) is non-negative too for every fixed ( t, s, i)∈(0, T)× U × X .□\nBy following [12], we introduce a parameterized function that satisfies a relevant partial differ-\nential inequality for certain choices of the parameters. In Lemmas 4.4 and 4.5, we specify those\nranges of parameters along with detailed proofs. For each l∈ {1,2, ..., d}, consider scalars ϵl>0,\nγl≥0, and kl≥1 and a function yl: [0, T]×R× X → R+such that\nyl(t, s, i) =1√T+ϵl−texp\"\n−γl\u0012\nlnsl\nklsu\nl\u00132\n/(T+ϵl−t)#\n, (39)\nwhich essentially depends only on time and the lth component of space variable. Since, this is\nconstant on X, the termPk\nj=1λijyl(t, s, j ), which appears in Lyl(t, s, i), is zero because the row\nsums are zero in Λ = [ λij]. Furthermore, we have\n\n\n∂yl\n∂t(t, s, i) =1\n(T+ϵl−t)2\u0014\nT+ϵl−t\n2−γl\u0010\nlnsl\nklsu\nl\u00112\u0015\nyl(t, s, i),\n∂yl\n∂sl(t, s, i) =−2γl\nsl(T+ϵl−t)h\nlnsl\nklsu\nli\nyl(t, s, i),\n∂2yl\n∂s2\nl(t, s, i) =2γl\ns2\nl(T+ϵl−t)2\u0014\u0010\nlnsl\nklsu\nl−1\u0011\n(T+ϵl−t) + 2γl\u0010\nlnsl\nklsu\nl\u00112\u0015\nyl(t, s, i).(40)\nWe recall from (3) that all(i) =Pd\nj=1σ2\nlj(i). For each l∈ {1,2, ..., d}, we set\nDl:= min {all(i)−2r(i) :i∈ X} . (41)\nLemma 4.4. Fix an l∈ {1,2, ..., d}. IfDlas in (41) is positive, we set\nγl:=1\n2 max\ni{all(i)},andkl:= exp max\ni{all(i)}\nDl!\n. (42)\nThen we have on (0, T)×R× X for any ϵl>0\n∂yl\n∂t(t, s, i) +Lyl(t, s, i)≤0. (43)\nProof. Using the expressions in (40), the LHS of (43) becomes yltimes the following term\nγl(2all(i)γl−1)\u0012\nlnsl\nklsu\nl\u00132\n+γl(all(i)−2r(i)) (T+ϵl−t)\u0012\nlnsl\nklsu\nl\u0013\n(44)\n+\u00121\n2−γlall(i)−r(i)(T+ϵl−t)\u0013\n(T+ϵl−t),\nfor all ( t, s, i)∈(0, T)×R× X. After substituting γl, the above expression becomes\n1\n2max\ni{all(i)}\n2all(i)\n2max\ni{all(i)}−1\n\u0012\nlnsl\nklsu\nl\u00132\n+(all(i)−2r(i))\n2max\ni{all(i)}(T+ϵl−t)\n\u0012\nlnsl\nklsu\nl\u0013\n+\n1\n2−all(i)\n2max\ni{all(i)}−r(i)(T+ϵl−t)\n(T+ϵl−t).ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 15\nBy substituting klin the second additive term of the above expression, we get\n1\n2max\ni{all(i)}\n2all(i)\n2max\ni{all(i)}−1\n\u0012\nlnsl\nklsu\nl\u00132\n+(all(i)−2r(i))\n2max\ni{all(i)}(T+ϵl−t)\n×(lnsl−lnsu\nl)−(all(i)−2r(i))\n2max\ni{all(i)}(T+ϵl−t)max\ni{all(i)}\nDl+1\n2(T+ϵl−t)\n−all(i)\n2max\ni{all(i)}(T+ϵl−t)−r(i)(T+ϵl−t)2.\nNote that except the fourth term, all other terms are non-positive because ( T+ϵl−t) is always\npositive. Since Dl>0, it is clear that the third term dominates the fourth term in magnitude.\nHence (44) is non-positive, and the result follows. □\nLemma 4.5. Let us fix l∈ {1, . . . , d }, and assume that Dl≤0. Fix a point (ˆt,ˆs)∈[0, T)×R\nsuch that\nlnsu\nl\nˆsl>−Dl(T−ˆt). (45)\nUsing this point, we set the following values of the parameters\nϵl=(T−ˆt) lnkl\nlnsu\nl\nˆsl, γ l0, using (45), 0 0,ϵl\nT+ϵl−t′<1 and\n−γl\nT+ϵl−t′\u0014\nlnsu\nl\ns′\nlln\u0012k2\nlsu\nl\ns′\nl\u0013\n−ln2\u0012klsu\nl\ns′\nl\u0013\u0015\n=γlln2kl\nT+ϵl−t′<γlln2kl\nϵl.\nNext, we substitute the parameters ϵl,γl, and klin (52) by not violating the constraints in Lemmas\n4.4, and 4.5 to obtain the upper bound (50) of Yl(t′, s′, i′). We first set ϵl=lnkl\nmax\ni{all(i)}+|Dl|and18 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nget\nYl(t′, s′, i′)≤exp\n−γllnsu\nl\ns′\nl\u0010\nlnsu\nl\ns′\nl+ 2 ln kl\u0011\n−(max\ni{all(i)}+|Dl|) lnkl(T−t′)\n(T−t′) +lnkl\nmax\ni{all(i)}+|Dl|)\n,\nbecause\n−γlln2\u0010\nklsu\nl\ns′\nl\u0011\nT−t′+lnkl\nmax\ni{all(i)}+|Dl|+γl(max\ni{all(i)}+|Dl|) ln2kl\nlnkl\n=−γlln2\u0010\nklsu\nl\ns′\nl\u0011\n−(max\ni{all(i)}+|Dl|) lnkl(T−t′)−ln2kl\n(T−t′) +lnkl\nmax\ni{all(i)}+|Dl|),\n=−γllnsu\nl\ns′\nlln\u0010\nk2\nlsu\nl\ns′\nl\u0011\n−(max\ni{all(i)}+|Dl|) lnkl(T−t′)\n(T−t′) +lnkl\nmax\ni{all(i)}+|Dl|).\nCase 1(Dl>0) : Using (42), i.e., ln kl=max\ni{all(i)}\nDlandγl=1\n2max\ni{all(i)}, we get\nYl(t′, s′, i′)≤exp\n−lnsu\nl\ns′\nl\u0012\nlnsu\nl\ns′\nl+ 2max\ni{all(i)}\nDl\u0013\n−(max\ni{all(i)}+Dl)max\ni{all(i)}\nDl(T−t′)\n2max\ni{all(i)}\u0012\n(T−t′) +max\ni{all(i)}\nDl(max\ni{all(i)}+Dl)\u0013\n,\n= exp\n−lnsu\nl\ns′\nl\u0012\nDl\nmax\ni{all(i)}lnsu\nl\ns′\nl+ 2\u0013\n+ (max\ni{all(i)}+Dl)(T−t′)\n2\u0012\nDl(T−t′) +max\ni{all(i)}\n(max\ni{all(i)}+Dl)\u0013\n. (53)\nCase 2(Dl≤0) : The above choice of ϵland (46) imply that 0 0. Then\nYl(t′, s′, i′)≤exp\n−ˆγllnsu\nl\ns′\nl\u0010\u0010\nlnsu\nl\ns′\nl\u0011\n/lnkl+ 2\u0011\n−\u0010\nmax\ni{all(i)} −Dl\u0011\n(T−t′)\n(T−t′)/lnkl+ 1/(max\ni{all(i)} −Dl)\n.\nSince, Lemma 4.5 holds for sufficiently large kl, letting kl→ ∞ in the above, we get Yl(t′, s′, i′)≤\nexp\u0010\n−ln\u0010\nsu\nl\ns′\nl\u0011\n+\u0010\nmax\ni{all(i)} −Dl\u0011\n(T−t′)/2\u0011\n. By combining the inequalities appearing above\nand in (53), we get for both the cases\nYl(t′, s′, i′)≤exp\n−ln(su\nl\ns′\nl)\u0012\nD+\nl\nmax\ni{all(i)}ln(su\nl\ns′\nl) + 2\u0013\n+ (max\ni{all(i)}+|Dl|)(T−t′)\n2\u0012\nD+\nl(T−t′) +max\ni{all(i)}\n(max\ni{all(i)}+D+\nl)\u0013\n.(54)\nHence, (50) follows from the above bound and (51). □\nRemark 4.8. In the preamble of Theorem 4.7, we have mentioned the possibility of deriving an\nestimate Ψl(t, s, i)ofYl, that is valid on D:={(t, s, i)∈[0, T]×R× X | lnsu\nl\nsl+Dl(T−t)≥\n0,∀l= 1, . . . , d }, by mimicking the approach of [12]. On the other hand a globally valid estimate\n¯Ψl(t, s, i)ofYlhas been obtained in (54). Thus (50)may be improved as |v(t′, s′, i′)−w(t′, s′, i′)| ≤\nsup[t′,T)×Γ×X|v−w|Pd\nl=1ˆΨl(t′, s′, i′)where, ˆΨl:= min {Ψl,¯Ψl}1D+¯Ψl1([0,T]×R×X)\\D.ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 19\n4.3.Numerical Study. A comparison of two estimates, mentioned in Remark 4.8 is presented\nbelow by considering a couple of numerical examples. The estimate of Ylin (54) for a single regime\ncase may easily be compared with the estimate presented in [12]. We present the comparison for\nthe single asset case, i.e., d= 1. It will be shown that none dominates the other. We set X={1},\nand denote σ:=σ(1),r:=r(1),D:=D1. So, the estimates in [12] and (54) are\nΨ1(t, s1,1) = exp\u0012\n−ln(su\n1/s1)(ln(su\n1/s1) + min {0, D}(T−t))\n2σ2(T−t)\u0013\n,\n¯Ψ1(t, s1,1) = exp\n−ln(su\n1/s1)\u0010\nD+\nσ2ln(su\n1/s1) + 2\u0011\n+ (σ2+|D|)(T−t)\n2\u0010\nD+(T−t) +σ2\n(σ2+D+)\u0011\n,\nrespectively. Now we set T= 1, sb\n1= 0, su\n1= 20. Figure 2a presents a contour plot of Ψ 1−¯Ψ1,\nwhere σ= 0.4, and r= 1%. Since D:=σ2−2r= 0.14>0, both the estimates are valid on\nthe full region. The contour plot in Figure 2a, where t, and s1variables are along vertical and\nhorizontal axes, shows that Ψ 1−¯Ψ1takes both positive and negative values for this example.\nHence ˆΨ1= min {Ψ1,¯Ψ1}is strictly sharper than both Ψ 1and¯Ψ1. Next by setting σ= 0.1 and\n-0.5-0.4\n-0.3-0.3\n-0.2-0.2\n-0.1-0.1-0.1 -0.1\n000\n0\n00.10.1\n0.1\n0 2 4 6 8 10 12 14 16 18 20\nstock price00.10.20.30.40.50.60.70.80.91time\n(a)Contour plot of Ψ 1−¯Ψ1where σ= 0.4, and\nr= 0.01.\n(b)Surface plot of Ψ 1,¯Ψ1, and the indicator function\nof the domain Dwhere σ= 0.1, and r= 0.2.\nFigure 2. Comparison of estimates.\nr= 20%, we get D=−0.39<0. Figure 2b includes surface plots of Ψ 1,¯Ψ1, and indicator\nfunction of the domain D:={(t, s1)|lnsu\n1\ns1+D(T−t)≥0}, against t-s1plane. Those surfaces are\ncolored in blue, green, and red respectively. We observe that Ψ 1−¯Ψ1changes its sign even inside\nD, where Ψ 1is a valid estimate for this example. Hence ˆΨ1= min {Ψ1,¯Ψ1}1D+¯Ψ11Dcgives a\nstrictly sharper estimate on the whole truncated domain.\nThe question regarding the location for the artificial boundary depending on the error tolerance\nhas been addressed in [12] by using the estimate discussed above. The analysis in [12], that\nconsiders few realistic numerical examples, convinces of the usefulness of a result like Theorem\n4.7. Therefore, in view of Remark 4.8, we have significantly improved the applicability of the\nresult presented in [12] by sharpening the estimate, extending the region of validity, and including\nthe regime switching scenario.\n5.Conclusions\nUsing the probabilistic method a self-contained proof has been developed for the existence of a\nunique solution of a system of PDEs in the class of functions having at most linear growth. The\nsystem under consideration originates from the regime-switching extension of the Black-Scholes-\nMerton model. A growth estimate has been derived for the solution depending on the growth of20 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nthe terminal data. This growth estimate has further been utilized in estimating the maximum\nerror at the boundary due to the imposed boundary data of the truncated domain problem. An\nerror estimate has also been obtained at every interior point of the domain. Finally, the estimate\nhas been expressed using the model parameters and maximum error on the far boundary. These\nresults are useful for allocating the artificial boundary depending on the error tolerance. For more\ndetails in this regard, [12] may be referred to. The error estimate obtained in that paper has been\ncompared with the estimate of the present paper under the special case where regime-switching\nis absent. By considering realistic numerical examples, we show that our results significantly\nimprove over the results presented in [12] by sharpening the estimate and extending the region of\nvalidity of the estimate. Moreover, we obtain point-wise estimates for domain truncation error of\na system of PDE that arises in the option pricing problem under the regime-switching scenario.\nAs a future direction, the proposed results may be extended to the option pricing equations with\nmore sophisticated model assumptions. For example, a similar study for the regime-switching\nstochastic volatility models is absent in the literature.\nAppendix A.Proofs of Lemmata in Section 3\nProof of Lemma 3.1. Using the expectation of log-normal random variable, we can write\nZ\n(0,∞)d \n1 +dX\nl=1xl!\nα(x, s, i, v )dx= 1 +dX\nl=1Z\n(0,∞)dxlα(x, s, i, v )dx,\n= 1 +dX\nl=1e[lnsl+(r(i)−1\n2all(i))v+1\n2all(i)v],\n= 1 +dX\nl=1sler(i)v.\n□\nProof of Lemma 3.2. From (6), we have\n∂˜zl\n∂v=r(i)−1\n2all(i),S(i) =va(i) =⇒∂S(i)\n∂v=a(i),S(i)−1=1\nva−1(i),\ntrace\u0012\nS(i)−1∂S(i)\n∂v\u0013\n=trace\u00121\nvI\u0013\n=d\nv,\n∂\n∂v(S(i)−1)ll′=∂\n∂v\u00121\nva−1(i)\u0013\nll′=−1\nv2\u0000\na−1(i)\u0001\nll′.\n\n(55)\nTaking logarithm on both the sides of (6), we get\nlnα(x, s, i, v ) =−ln \n1p\n(2π)dx1.x2...xd!\n−1\n2ln|S(i)| −1\n2dX\nl=1dX\nl′=1(S(i)−1)ll′(zl−˜zl)(zl′−˜zl′).\n(56)ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 21\nDifferentiating above expression (56) w.r.t. vand using (55) and Jacobi’s formula for derivative\nof determinant, we can write\n∂α\n∂v=\"\n−d\n2v−1\n2dX\nl=1dX\nl′=1\u0012−1\nv2(a−1(i))ll′(zl−˜zl)(zl′−˜zl′)\u0013\n+1\n2dX\nl=1dX\nl′=11\nv(a−1(i))ll′\n(zl′−˜zl′)\u0012\nr(i)−1\n2all(i)\u0013\n+1\n2dX\nl=1dX\nl′=11\nv(a−1(i))ll′\u0012\nr(i)−1\n2al′l′(i)\u0013\n(zl−˜zl)#\nα,\n=\"\n−d\n2v+1\n2dX\nl=1dX\nl′=1\u00121\nv2(a−1(i))ll′(zl−˜zl)(zl′−˜zl′)\u0013\n+dX\nl=1dX\nl′=11\nv(a−1(i))ll′(i)\n(zl′−˜zl′)\u0012\nr(i)−1\n2all(i)\u0013\u0015\nα, (57)\nusing the symmetry of a−1. From (57), we can see: Using the fact that ( i)a−1is positive definite\n(directly follows as a=σσT), and ( ii)zl= ln\u0010\nxl\nsl\u0011\n, second term has growth O(ln∥x∥1)2and is\nnon-negative. This term would be dominating term as ∥x∥1→ ∞ as the third term has growth\nofO(ln∥x∥1) and the first term does not depend on xl. This implies that for a fixed value of s,\ni, and σ, there exists some large R, and constants C1, and C2(does not depend on xbut may\ndepend on s, i, v ), such that\n\f\f\f\f1\nα∂α\n∂v\f\f\f\f≤C1+C2(ln∥x∥1)2,\nfor all x∈(0,∞)d\\(0, R)d. □\nProof of Lemma 3.3. Recall that 1 ldenotes the unit vector along lth direction and 1 l(l′) is the\nl′th component of 1 l. Differentiating (56) w.r.t. sl, and using Jacobi’s formula for derivative of\ndeterminant, we have\n1\nα∂α\n∂sl0= 0−1\n2×0−1\n2dX\nl=1dX\nl′=1\u0014\n(S−1)ll′\u0012−1\nsl\u0013\n(zl′−˜zl′)1l0(l) + (S−1)ll′(zl−˜zl)\u0012−1\nsl\u0013\n1l0(l′)\u0015\n=1\n2X\nl′̸=l0(S−1)l0l′\u00121\nsl0\u0013\n(zl′−˜zl′) +1\n2X\nl̸=l0(S−1)ll0\u00121\nsl0\u0013\n(zl−˜zl) + (S−1)l0l0\u00121\nsl0\u0013\n(zl0−˜zl0).\nSinceSis symmetric, we can write\n∂α\n∂sl0=dX\nl=1(S−1)ll0\u00121\nsl0\u0013\n(zl(s)−˜zl(v))α.\nWe have written z(s) and ˜ z(v) at the places of zand ˜zto denote their dependency on sandv\nvariables respectively. Therefore from (6), we have\n∂α\n∂sl0=α\nsl0O(ln∥x∥),as∥x∥1→ ∞ .\n□\nProof of Lemma 3.4. For notational convenience, let us write z(s) and ˜ z(v) to denote their de-\npendency on sandvvariables respectively. Furthermore, we denote ( a(i))−1byA. Moreover,\nz(s)−˜z(v) which is a difference of two vectors can be written as ln( x)−(ln(s) + ˜z(v)), where\nln of a vector is the vector of logarithm of components.. Let us write ln( s) + ˜z(v) asξ(s, v) for22 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nnotational convenience. From (6) and (56), we can write\nlnα(x, s′, i, v′)−lnα(x, s, i, v )\n=−d\n2ln\u0012v′\nv\u0013\n−1\n2v′(lnx−ξ(s′, v′))∗A(lnx−ξ(s′, v′)) +1\n2v(lnx−ξ(s, v))∗A(lnx−ξ(s, v)),\n=−d\n2ln\u0012v′\nv\u0013\n−1\n2v′h\n(lnx)∗A(lnx)−2(lnx)∗Aξ(s′, v′) +ξ(s′, v′)∗Aξ(s′, v′)\n−\u0012v′\nv\u0013\u0010\n(lnx)∗A(lnx)−2(lnx)∗Aξ(s, v) +ξ(s, v)∗Aξ(s, v)\u0011i\n,\n=−d\n2ln\u0012v′\nv\u0013\n−1\n2v′\u0014\u0012\n1−v′\nv\u0013\n(lnx)∗A(lnx)−2(lnx)∗A\u0010\nξ(s′, v′)−\u0012v′\nv\u0013\nξ(s, v)\u0011\n+ξ(s′, v′)∗Aξ(s′, v′)−\u0012v′\nv\u0013\nξ(s, v)∗Aξ(s, v)\u0015\n.\nClearly, as ∥x∥ → ∞ , the quadratic term (ln x)∗A(lnx), which appears in the above expression\ndominates. On the other hand for v′> v, the sign of that term is positive. In other words,\nlnα(x, s′, i, v′)−lnα(x, s, i, v )>0 for large ∥x∥. To be more precise, for every fixed positive scalar\nv′,ϵ, and vectors s, and s′, there is a sufficient large Rsuch that ln α(x, s′, i, v′)−lnα(x, s, i, v )>0\nfor all ∥x∥>Rand for all v≤v′−ϵ. □\nProof of Lemma 3.7. First we fix the variables t,jandj′and hence we ignore their influence on\nother variables, to be defined in this proof. Since ϕ(·,·,·)∈V, for all s∈(0,∞)d, supt,i|ϕ(t, s, i)| ≤\n∥ϕ∥V(1 +∥s∥1). Let {ul}l∈Nbe a decreasing sequence on (0 ,1) such that ul→0. Let αl(x) :=\nα(x, s, j′, ul). Since {αl}l∈Nis a family of lognormal density functions with mean and variance\nlying on a bounded set, we have the following uniform integrability\nlim\nR→∞sup\nlZ\n(0,∞)d\\(0,R)d(1 +∥x∥1)α(x, s, j′, ul)dx= 0,\nfor every fixed sandj′. Thus, for any ϵ >0, there is R >0 such that\nZ\n(0,∞)d\\(0,R)d(1 +∥x∥1)α(x, s, j′, ul)dx < ϵ for all l∈N. (58)\nNow consider\nϕn(t, x, j ) :=22n−1X\ni=0i\n2n1[i\n2n,i+1\n2n)(ϕ(t, x, j )),\nwhich is a non-negative increasing sequence converging to ϕuniformly on every compact set. Then,\ngiven ϵ >0 and R, we can find Nsuch that for all n≥N,\n0≤Z\n(0,∞)d(ϕ(t+ul, x, j)−ϕn(t+ul, x, j))α(x, s, j′, ul)dx,\n=Z\n(0,R)d(ϕ(t+ul, x, j)−ϕn(t+ul, x, j))α(x, s, j′, ul)dx\n+Z\n(0,∞)d\\(0,R)d(ϕ(t+ul, x, j)−ϕn(t+ul, x, j))α(x, s, j′, ul)dx,\n≤ϵZ\n(0,R)dα(x, s, j′, ul)dx+Z\n(0,∞)d\\(0,R)d∥ϕ∥V(1 +∥x∥1)α(x, s, j′, ul)dx,\n<(1 +∥ϕ∥V)ϵ, (59)\nfor all l, using (58). Also,\nZ\n(0,∞)dϕn(t+ul, x, j)α(x, s, j′, ul)dx=22n−1X\ni=0i\n2nZ\nA(n)\nl,iα(x, s, j′, ul)dx (60)ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 23\nwhere A(n)\nl,i:=\b\nx∈(0,∞)d|ϕ(t+ul, x, j)∈[i\n2n,i+1\n2n)\t\n. Now, if i(n, s) be such that ϕ(t, s, j )∈\n[i(n,s)\n2n,i(n,s)+1\n2n), then for every s, due to the continuity of ϕintvariable, there is a sufficiently\nlarge l′such that for all l≥l′,ϕ(t+ul, s, j)∈(i(n,s)−1\n2n,i(n,s)+1\n2n) asul↓0. Therefore, sis in the\ninterior of A(n)\nl,i(n,s)−1∪A(n)\nl,i(n,s)for all l≥l′. Again since variance of αlgoes to zero and mean\nconverges to sasul↓0,Z\nA(n)\nl,i(n,s)−1∪A(n)\nl,i(n,s)α(x, s, j′, ul)dx→1,\nbecause A(n)\nl,i(n,s)−1∪A(n)\nl,i(n,s)contains sin the interior and its Lebesgue measure does not shrink\nto zero. Hence the integral of the density function on the complementary domain converges to\nzero. Thus, for each n, using the boundedness of ϕnand (60)\ni(n, s)−1\n2n≤lim\nl→∞Z\n(0,∞)dϕn(t+ul, x, j)α(x, s, j′, ul)dx≤i(n, s)\n2n.\nTherefore, from the above inequality\ni(n, s)−1\n2n≤lim\nl→∞Z\n(0,∞)dϕn(t+ul, x, j)α(x, s, j′, ul)dx,≤lim\nl→∞Z\n(0,∞)dϕ(t+ul, x, j)α(x, s, j′, ul)dx,\n= lim\nl→∞Z\n(0,∞)d(ϕ(t+ul, x, j)−ϕn(t+ul, x, j) +ϕn(t+ul, x, j))α(x, s, j′, ul)dx,\n≤(1 +∥ϕ∥V)ϵ+i(n, s)\n2n,\nusing (59). For a given ϵ >0 consider N(>1−log2ϵ). Now, from the definition of i(n, s), for all\nn≥N,ϕ(t, s, j )−ϵ≤ϕ(t, s, j )−2\n2nwhich is less thani(n,s)−1\n2n, the left side of above inequality.\nFinally we note that the right side is less or equal to (1 + ∥ϕ∥V)ϵ+ϕ(t, s, j ). Combining these, we\nget\nϕ(t, s, j )−ϵ≤lim\nl→∞Z\n(0,∞)dϕ(t+ul, x, j)α(x, s, j′, ul)dx≤(1 +∥ϕ∥V)ϵ+ϕ(t, s, j ).\nThe result follows as ϵis an arbitrary positive number. □\nProof of Identity (26). From the second order partial derivative of αw.r.t. sl0andsl′\n0, we have\n∂\n∂sl′\n0gl0\n2=1\nsl0∂\n∂sl′\n0dX\nl=1(S−1)ll0(zl−˜zl) =1\nsl0∂\n∂sl′\n0\u0002\n(S−1)l′\n0l0(zl′\n0−˜zl′\n0)\u0003\n=−1\nsl0sl′\n0(S−1)l′\n0l0−δ(l0, l′\n0)1\nsl0gl0\n2,\nwhere δ(l, l′) is the Kronecker delta function of landl′, i.e., δ(l, l′) = 1 if and only if l=l′and is\nzero otherwise. Using the symmetry of a−1and expressions for g1andgl\n2, we simplify the L.H.S.\nof (26) as follows\n−g1+r(i)dX\nl0=1sl0gl0\n2+1\n2dX\nl0=1dX\nl′\n0=1sl0sl′\n0al0l′\n0\u0012∂\n∂sl′\n0gl0\n2+gl0\n2gl′\n0\n2\u0013\n=−\u0014−d\n2v−1\n2dX\nl=1dX\nl′=1\u0012−1\nv2a−1\nll′(zl−˜zl)(zl′−˜zl′)\u0013\n+dX\nl=1dX\nl′=1a−1\nll′\nv\u0012\nr(i)−1\n2all\u0013\n(zl′−˜zl′)\u0015\n+r(i)dX\nl0=1sl0\"dX\nl=1a−1\nll0\nv\u00121\nsl0\u0013\n(zl−˜zl)#\n+1\n2dX\nl0=1dX\nl′\n0=1sl0sl′\n0al0l′\n0\" \n−1\nsl0sl′\n0a−1\nl′\n0l0\nv−δ(l0, l′\n0)1\nsl0dX\nl=1a−1\nll0\nv1\nsl0(zl−˜zl)!\n+ dX\nl=1a−1\nll0\nv1\nsl0(zl−˜zl)dX\nl=1a−1\nll′\n0\nv1\nsl′\n0(zl−˜zl)!#\n,24 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\n=d\n2v+1\n2dX\nl=1dX\nl′=1\u0012−a−1\nll′\nv2(zl−˜zl)(zl′−˜zl′)\u0013\n−dX\nl=1dX\nl′=1a−1\nll′\nv\u0012\nr(i)−1\n2all\u0013\n(zl′−˜zl′)\n+r(i)dX\nl0=1\"dX\nl=1a−1\nll0\nv(zl−˜zl)#\n+1\n2dX\nl0=1dX\nl′\n0=1al0l′\n0\" \n−a−1\nl′\n0l0\nv!\n+(dX\nl=1dX\nl′=1a−1\nll0a−1\nl′l′\n0\nv2\n(zl−˜zl)(zl′−˜zl′))#\n+1\n2dX\nl0=1dX\nl′\n0=1sl0sl′\n0al0l′\n0 \n−δ(l0, l′\n0)1\nsl0dX\nl=1a−1\nll0\nv1\nsl0(zl−˜zl)!\n,\n=1\n2dX\nl=1dX\nl′=1\u0012−a−1\nll′\nv2(zl−˜zl)(zl′−˜zl′)\u0013\n−r(i)\nvdX\nl=1dX\nl′=1a−1\nll′(zl′−˜zl′) +r(i)\nvdX\nl=1dX\nl′=1\na−1\nll′(zl−˜zl) +dX\nl=1dX\nl′=1alla−1\nll′\n2v(zl′−˜zl′) +1\n2dX\nl0=1dX\nl′\n0=1al0l′\n0(dX\nl=1dX\nl′=1a−1\nll0a−1\nl′l′\n0\nv2(zl−˜zl)\n(zl′−˜zl′))\n−dX\nl0=1dX\nl=1al0l0a−1\nll0\n2v(zl−˜zl),\n=−1\n2v2dX\nl=1dX\nl′=1a−1\nll′(zl−˜zl)(zl′−˜zl′) +dX\nl=1dX\nl′=1alla−1\nll′\n2v(zl′−˜zl′)−dX\nl0=1dX\nl=1al0l0a−1\nl0l\n2v\n(zl−˜zl) +1\n2v2dX\nl′\n0=1dX\nl=1\"dX\nl0=1al′\n0l0a−1\nl0l#dX\nl′=1a−1\nl′l′\n0(zl−˜zl)(zl′−˜zl′),\n=−1\n2v2dX\nl=1dX\nl′=1a−1\nll′(zl−˜zl)(zl′−˜zl′) +1\n2v2dX\nl′\n0=1dX\nl=1dX\nl′=1a−1\nl′l′\n0(zl−˜zl)(zl′−˜zl′)δ(l′\n0, l),\n=−1\n2v2dX\nl=1dX\nl′=1a−1\nll′(zl−˜zl)(zl′−˜zl′) +1\n2v2dX\nl=1dX\nl′=1a−1\nl′l(zl−˜zl)(zl′−˜zl′) = 0 .\n□\nProof of Lemma 3.11. From the closed form expression of strong solution of SDE, we can write\nfort′> t\nE(˜Sl(t′)|˜Ft) =E\u0010\n˜Sl(t)e(Rt′\nt(ru−1\n2ˆσl(Xu)2)du+Rt′\ntˆσl(Xu)dˆWl\nu)|˜Ft\u0011\n,\n=˜Sl(t)E\u0010\ne(Rt′\nt(ru−1\n2ˆσl(Xu)2)du+Rt′\ntˆσl(Xu)dˆWl\nu)|˜Ft\u0011\n,\nwhere ru=r(Xu). The conditional distribution of the term inside expectation is log-normal given\n˜Ft∨ FX\nt′with parameters\u0010Rt′\nt(ru−ˆσl(Xu)2\n2)du,Rt′\ntˆσl(Xu)2du\u0011\n. Therefore, we can write\nE(˜Sl(t′)|˜Ft) =˜Sl(t)E[e\u0010Rt′\nt(ru−1\n2ˆσl(Xu)2)du+1\n2Rt′\ntˆσl(Xu)2du\u0011\n|˜Ft] =˜Sl(t)E[eRt′\ntrudu|Xt],\n>˜Sl(t),\nusing ru>0 for all u≥0. Therefore, for each l,˜Slis a sub-martingale. □\nProof of Lemma 3.12. Using the infinitesimal generator of the Markov chain X, we can write\nϕ(t,˜S(t), Xt)−ϕ(t,˜S(t), Xt−) =X\nj̸=Xt−λXt−,j\u0010\nϕ(t,˜S(t), j)−ϕ(t,˜S(t), Xt−)\u0011\ndt+dM(t),ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 25\nfor some local martingale M. Now, using Itˆ o’s formula in (29), we get\ndNϕ\nt=−r(Xt)Nϕ\ntdt+e−Rt\n0r(Xu)dudϕ(t,˜S(t), Xt),\n=−r(Xt)Nϕ\ntdt+e−Rt\n0r(Xu)du\"\n∂ϕ(t,˜S(t), Xt)\n∂tdt+dX\nl=1∂ϕ(t,˜S(t), Xt)\n∂sld˜Sl(t)\n+1\n2dX\nl=1dX\nl′=1all′˜Sl(t)˜Sl′(t)∂2ϕ(t,˜S(t), Xt)\n∂sl∂sl′dt+ϕ(t,˜S(t), Xt)−ϕ(t,˜S(t), Xt−)#\n,\n=−r(Xt)Nϕ\ntdt+e−Rt\n0r(Xu)du\" \n∂ϕ\n∂t+r(Xt)dX\nl=1˜Sl(t)∂ϕ\n∂sl+1\n2dX\nl=1dX\nl′=1all′˜Sl(t)˜Sl′(t)\n∂2ϕ\n∂sl∂sl′!\n(t,˜S(t), Xt)dt+X\nj̸=Xt−λXt−,j\u0010\nϕ(t,˜S(t), j)−ϕ(t,˜S(t), Xt−)\u0011\ndt\n+dX\nl=1∂ϕ\n∂sl(t,˜S(t), Xt)˜Sl(t)ˆσl(Xt)dˆWl\nt+dM(t)#\n,\n=e−Rt\n0r(Xu)du\" \n−r(Xt)ϕ(t,˜S(t), Xt) +∂ϕ\n∂t(t,˜S(t), Xt) +r(Xt)dX\nl=1˜Sl(t)\n∂ϕ\n∂sl(t,˜S(t), Xt) +1\n2dX\nl=1dX\nl′=1all′˜Sl(t)˜Sl′(t)∂2ϕ\n∂sl∂sl′(t,˜S(t), Xt) +X\nj̸=Xt−λXt−,jdt\n\u0010\nϕ(t,˜S(t), j)−ϕ(t,˜S(t), Xt−)\u0011!\n+∂ϕ\n∂sl(t,˜S(t), Xt)˜Sl(t)ˆσl(Xt)dˆWl\nt+dM(t)#\n.\nIt is clear that the coefficient of dtis zero from (1)-(3). Thus Nϕis a local martingale. On the\nother hand since ϕ∈V,|Nϕ\nt| ≤ ∥ϕ∥V+∥ϕ∥VP\nl˜Sl(t) for each t. Therefore, E\u0000\nsups≤t|Nϕ\ns|\u0001\n<∞\nusing (28). Thus, Nϕis a martingale, follows from [20, Theorem 51, Chapter 1, pp. 38]. □\nAppendix B.Derivation of IE (5)\nLetn(t) denote the number of transitions during (0 , t] and Tndenote the nthtransition time\ninstant. We rewrite the right side of (30) by conditioning on the next transition time Tn(t)+1\nϕ(t,˜S(t), Xt) =Eh\nE\u0010\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1\u0011\n|˜S(t), Xti\n. (61)\nLemma B.1. Tn(t)+1−tis exponentially distributed random variable given FX\nt.\nProof. The conditional cumulative distribution function (CDF) of Tn(t)+1given Xt=iis\nFi(v) :=P(Tn(t)+1≤v|Xt=i) =P(Tn(t)+1−Tn(t)≤v−Tn(t)|Xt=i),\nforv > t and zero for v≤tasTn(t)+1> talmost surely. For the same reason, Tn(t)+1−Tn(t)≥\nt−Tn(t)almost surely. Hence, we can write\nFi(v) =P(Tn(t)+1−Tn(t)≤v−Tn(t)|Xt=i),\n=P(Tn(t)+1−Tn(t)≤v−Tn(t)|Xt=i, Tn(t)+1−Tn(t)≥t−Tn(t)).\nUsing Bay’s theorem the above is equal to\nP(t−Tn(t)≤Tn(t)+1−Tn(t)≤v−Tn(t)|Xt=i)\nP(Tn(t)+1−Tn(t)≥t−Tn(t)|Xt=i).26 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\nBy additional conditioning by Tn(t), the above is rewritten as\nE[P(t−Tn(t)≤Tn(t)+1−Tn(t)≤v−Tn(t)|Xt, Tn(t))|Xt=i]\nE[P(Tn(t)+1−Tn(t)≥t−Tn(t)|Xt, Tn(t))|Xt=i].\nUsing the exponential distribution of inter transition time of the Markov chain X, the above is\nE[(1−e−λXt(v−Tn(t)))−(1−e−λXt(t−Tn(t)))|Xt=i]\nE[e−λXt(t−Tn(t))|Xt=i]=E[e−λXt(t−Tn(t))(1−e−λXt(v−t))|Xt=i]\nE[e−λXt(t−Tn(t))|Xt=i],\n= 1−e−λi(v−t).\nTherefore, by substituting v=v′+t, we get the probability density function of Tn(t)+1−tas\nd\ndv′Fi(v′+t) =λie−λiv′onv′>0 and result follows. □\nProposition B.2. Letϕbe a classical solution of (1)-(3), then ϕalso solves integral equation (5).\nProof. First we note that Tn(t)+1−tis conditionally independent to ˜S(t) given Xt. Let us consider\n(61) and using formula for expectation and Lemma B.1, we get\nϕ(t,˜S(t), Xt) =Z∞\n0Eh\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1−t=v′i\nλXte−λXtv′dv′,\n=ZT−t\n0Eh\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1−t=v′i\nλXte−λXtv′dv′\n+Z∞\nT−tEh\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1−t=v′i\nλXte−λXtv′dv′.(62)\nNote that Tn(t)+1−t=v′∈[T−t,∞) implies {Tn(t)+1≥T}. Again, under Tn(t)+1≥T, no\ntransition takes place during [ t, T]. Thus XTis identical to Xt. Hence, the conditional distribution\nof each component of ˜S(T) given the σalgebra generated by {Tn(t)+1> T}and ˜Ftis log normal.\nThat is given this σalgebra, with X(t) =ithe conditional joint distribution of {˜Sl(T)/˜Sl(t) :l=\n1,2, . . . , d }is identical to the joint distribution of {Yl(T)/Yl(t) :l= 1,2, . . . , d }, which corresponds\nto the B-S-M model having constant parameters r(i) and σ(i). Thus the conditional expectation\nEh\ne−r(i)(T−t)K(˜S(T))|˜S(t) =s, Xu=i∀u∈[t, T]i\nis identical to the R.H.S. of (9). Therefore,\n(62) can be written as\nϕ(t,˜S(t), Xt) =ZT−t\n0Eh\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1−t=v′i\nλXte−λXtv′dv′\n+ηXt(t, s)Z∞\nT−tλXte−λXtv′dv′, (63)\n=ZT−t\n0Eh\ne−RT\ntr(Xu)duK(˜S(T))|˜S(t), Xt, Tn(t)+1−t=v′i\nλXte−λXtv′dv′+ηXt(t, s)e−λXt(T−t).\nFurther, given Tn(t)+1=t+v′< T, it is clear that during [ Tn(t), t+v′),Xhas no transition.\nMoreover, at time t+v′,Xtransits to another state jwith conditional probability PXt,jgiven\nXt. Given ˜Sl(t) =sl, Xt=i, and Tn(t)+1=t+v′, we know that\n˜Sl(t+v′) =slexp\n(r(i)−1\n2all′(i))v′+dX\nj=1σl,j(i)(Wt+v′−Wt)\n,ESTIMATION OF DOMAIN TRUNCATION ERROR FOR PDES 27\nis log-normal for each l= 1,2, . . . , d . The joint conditional PDF of ˜S(t+v′) is given by αas in\n(6). In (63) by fixing ˜S(t) =s,Xt=i, and using additional conditioning w.r.t. ˜Ft+v′, we get\nϕ(t, s, i) =ZT−t\n0Eh\nE\u0010\ne−(r(i)v′+RT\nt+v′r(Xu)du)K(˜S(T))|˜S(t+v′), Xt+v′,˜S(t), Xt,\nTn(t)+1−t=v′\u0011\n|˜S(t) =s, Xt=i, Tn(t)+1−t=v′i\nλie−λiv′dv′+e−λi(T−t)ηi(t, s),\n=ZT−t\n0kX\nj=1pijhZ\n(0,∞)dE\u0010\ne−(r(i)v′+RT\nt+v′r(Xu)du)K(˜S(T))|˜S(t+v′) =x, X t+v′=j\u0011\nα(x, s, i, v′)dxi\nλie−λiv′dv′+e−λi(T−t)ηi(t, s),\n=ZT−t\n0e−(λi+r(i))v′kX\nj=1λijZ\n(0,∞)dE\u0010\ne−RT\nt+v′r(Xu)duK(˜S(T))|˜S(t+v′) =x,\nXt+v′=j\u0011\nα(x, s, i, v′)dxdv′+e−λi(T−t)ηi(t, s).\nUsing (30), we get\nϕ(t, s, i) =e−λi(T−t)ηi(t, s) +ZT−t\n0e−(λi+ri)v′kX\nj=1λijZ\n(0,∞)dϕ(t+v′, x, j)α(x, s, i, v′)dxdv′.\nThus ϕalso solves the integral equation (5). This completes the proof. □\nWe recall that uniqueness of (1)-(3) has been established in Theorem 3.13. Here, we present an\nalternative argument for the same. Let us assume that ϕ1andϕ2are two classical solutions of\n(1)-(3) in V. Then using Proposition B.2, we know that both also solve IE (5). But from Theorem\n3.6, there is only one such solution in V. Hence ϕ1=ϕ2.\nAcknowledgments\nThe corresponding author acknowledge the support from IISER Pune and IIIT Naya Raipur\nfor providing the support to conduct some part of this research work.\nReferences\n[1] G. K. Basak, M. K. Ghosh, and A. Goswami. Risk minimizing option pricing for a class of exotic options in a\nmarkov-modulated market. Stochastic Analysis and Applications , 29:259–281, 2011.\n[2] P. Boyle and T. Draviam. Pricing exotic options under regime switching. Insurance: Mathematics and Eco-\nnomics , 40:267–282, 2007.\n[3] J. Buffington and R. J. Elliott. American options with regime switching. Int. J. Theor. Appl. Finan. ,\n05(05):497–514, 2002.\n[4] Klaus Deimling. Nonlinear Functional Analysis . Springer Verlog, Berlin, 1985.\n[5] A. Deshpande and M. K. Ghosh. Risk minimizing option pricing in a regime switching market. Stoch. Anal.\nAppl. , 26(2):313–324, 2008.\n[6] G. B. Di Masi, Yu. M. Kabanov, and W. J. Runggaldier. Mean-variance hedging of options on stocks with\nmarkov volatilities. Theory of Probability & Its Applications , 39(1):172–182, 1995.\n[7] B. During and C. Heuer. High-order compact schemes for parabolic problems with mixed derivatives in multiple\nspace dimensions. SIAM J. Numer. Anal. , 53:2113–2124, 2015.\n[8] Yu. V. Egorov and M. A. Shubin. Partial differential equations VI: Elliptic and Parabolic Operators . Springer-\nVerlag, Berlin, 1994.\n[9] A. Friedman. Partial Differential Equations of Parabolic Type . Robert E. Krieger Publishing Co., Huntington,\nNY, 1983.\n[10] Anindya Goswami, Kedar Nath Mukherjee, Irvine Homi Patalwala, and Nadahalli Satish Sanjay. Regime re-\ncovery using implied volatility in Markov modulated market model. Appl. Stoch. Models Bus. Ind. , 38(6):1127–\n1143, 2022.\n[11] Y. Huang, P. A. Forsyth, and G. Labahn. Methods for pricing American options under regime switching. SIAM\nJ Sci Comput , 33(5):2144–2168, 2011.\n[12] R. Kangro and R. Nicoladies. Far field boundary conditions for Black-Scholes equations. SIAM J. Numer.\nAnal. , 38:1357–1368, 2000.\n[13] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus . Springer Verlog, New York, 1998.28 ANINDYA GOSWAMI AND KULDIP SINGH PATEL*\n[14] A. Q. M. Khaliq and R. H. Liu. New numerical scheme for pricing American option with regime-switching.\nInt. J. Theor. Appl. Finan. , 12(03):319–340, 2009.\n[15] A.Q.M. Khaliq, B. Kleefeld, and R.H. Liu. Solving complex pde systems for pricing American options with\nregime-switching by efficient exponential time differencing schemes. Numer Methods Partial Differ Equ ,\n29(1):320–336, 2013.\n[16] O. A. Ladyzenskaya, V. A. Solonnikov, and N. N. Uralceva. Linear and quasi linear equation of parabolic type .\nTranslation of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, 1967.\n[17] B. V. Limaye. Functional Analysis . New Age International, 1996.\n[18] V. Naik. Option valuation and hedging strategies with jumps in the volatility of asset returns. Journal of\nFinance , 48:1969–1984, 1993.\n[19] B. Oksendal. Stochastic Differential Equations: An Introduction with Applications . Springer Verlog, 2000.\n[20] P. E. Protter. Stochastic integration and differential equations . Springer, 1990.\n[21] H. L. Royden and P. M. Fitzpatric. Real Analysis . China Machine Press, 2010.\n[22] A. N. Shiryaev. Probability—2. Graduate Texts in Mathematics . Springer,, 2019.\n[23] S. P. Zhu, A. Badran, and X. Lu. A new exact solution for pricing European options in a two-state regime-\nswitching economy. Comput. Math. Appl , 64(8):2744–2755, 2012.\nIISER Pune, India\nEmail address :anindya@iiserpune.ac.in\nIIT Patna, India\nEmail address :kspatel@iitp.ac.in" }, { "title": "2401.15580v1.Homemade_kit_for_demonstrating_Barkhausen_Effect.pdf", "content": "Homemade kit for demonstrating Barkhausen Effect\nShantanu Shakya1and Navinder Singh2\n1Department of Physics, Indian Institute of Space Science and Technology(IIST), Thiruvananthapuram, Kerala - 695547\n2Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380009\nThis paper presents an innovative and cost-effective approach to understanding the Barkhausen\neffect through the design and implementation of an educational kit. The Barkhausen effect, char-\nacterized by Barkhausen noise (BN) during magnetization changes in soft magnetic materials, is\nexplored for its application in probing hysteresis properties and magnetization dynamics. The\nstudy investigates scaling properties, categorizing ferromagnetic materials based on scaling expo-\nnents. The primary contribution is the introduction of a practical and accessible kit for hands-on\nBarkhausen Effect demonstrations, revolutionizing the educational experience. This kit enables stu-\ndents to not only comprehend the intricacies of BN but also calculate the scaling constant ( τ) for\nSoft Iron samples. The paper demonstrates the successful construction of the kit, its signal amplifi-\ncation capabilities, and data collection accuracy, showcasing its potential for widespread educational\nuse.\nI. INTRODUCTION\nBarkhausen in 1919 noticed that changes in the\nmagnetization of an iron sample induced a special form\nof time dependent electric potential in a pickup coil\nwound over the sample. Upon amplification, this led to\nthe generation of audible ”noise” as depicted in Figure\n1. This phenomenon is widely known as ”Barkhausen\nnoise” [1]. Barkhausen suggested its association with\nirreversible alterations in magnetization. When exposed\nFIG. 1. Barkhuasen Noise in magnetization curve\nto a magnetic field undergoing a gradual temporal shift,\nsuch as the slow movement of a manually-driven mag-\nnetic yoke, a usual magnetic material like iron produces\nirregular signal in a pick-up coil. This irregular acoustic\nmanifestation contrasts sharply with the systematic\npattern of the applied magnetic field and is detectable\nthrough a microphone. This marked the initial indirect\nconfirmation of the presence of magnetic domains, a\nconcept theorized a few years earlier by Weiss [2].\nSoon after its discovery, it became apparent that\nBarkhausen noise (BN) could serve as a valuable tool\nfor delving into and understanding the magnetization\ndynamics inherent in soft magnetic materials, provid-ing insights into specific hysteresis properties. While\nthe measurement of BN is relatively straightforward, its\ninterpretation presents a greater challenge. This com-\nplexity stems from the stochastic nature of domain wall\n(DW) motion, which progresses through sudden jumps or\navalanches during slow magnetization. These dynamics\nare notably influenced by material microstructure, along\nwith additional factors such as the demagnetizing field,\nexternal stress, and others [3].\nII. EXPERIMENTAL IDEA\nIn this section, we will delve into the details of the\nexperiment, including the instruments employed, and\nthe associated circuitry. We will explore both the\nauditory aspect of detecting Barkhausen noise (BN) and\nthe methods for its measurement.\nBN is generated by alterations in the magnetization of\na ferromagnetic material. This change in magnetization\nis often attributed to the presence of impurities and\ndefects within the material, which can act as obstacles,\npinning the movement of domain walls (as shown in\nfigure 2). When we apply an external magnetic field\nand increase it beyond a critical value denoted as Hc,\na pivotal event known as the ”depinning transition”\noccurs. During this transition, the pinned domain walls\nare suddenly released or depinned, resulting in a signif-\nicant jump in the material’s magnetization curve. This\njump can be detected using a pickup coil that operates\non the principles of ”Faraday’s law of electromagnetic\ninduction”.\nAccording to Faraday’s law, any alteration in magnetic\nflux within the coil’s vicinity induces a corresponding\nvoltage at the coil’s terminals. This induced voltagearXiv:2401.15580v1 [physics.ed-ph] 28 Jan 20242\nFIG. 2. Pinning of domain wall due to impurities\nis the key to our ability to both hear and measure the\nBarkhausen noise in our experiment.\nV=N∗A∗dB\ndT(1)\nV- voltage induced in the coil\nN- number of the turns of the pick-up coil\nA- cross section area of the coil\nB- magnetic field\nWhen deliberately and gradually adjusting the mag-\nnetic field strength, either increasing or decreasing, the\ncoil induces a prominent voltage signal characterized by\na large amplitude and low frequency. In contrast, the\nrapid Barkhausen events, characterized by much smaller\namplitudes, contribute to a signal with higher frequency.\nThis high-frequency signal is superimposed upon the\nslower and larger signal. Our aim is to filter out the\nBN noise voltage and reject the slower and larger signals.\nFigure 1 of reference 4[8] illustrates the magnetization\ncurve of a NiFe film [8]. Initially appearing smooth when\nrecorded under the influence of a slowly varying exter-\nnal magnetic field, a closer inspection with magnification\nreveals non-uniform changes in magnetization. This non-\nuniformity is marked by the presence of high-frequency,\nlow-amplitude Barkhausen noise, which is superimposed\non the main signal.\nTo extract the Barkhausen noise from the recorded data,\nwe adopt a two-step approach. Initially, a high-gain am-\nplifier is employed to boost the amplitude of the signal.\nFollowing this amplification, we implement suitable fil-\ntering techniques to isolate the Barkhausen noise. This\nis accomplished using band-pass or high-pass filters, ef-\nfectively eliminating lower-frequency components associ-\nated with slower and larger signals (refer to Figure 3).\nFrequencies below 300 Hz are filtered out.[8].\nThis approach enables us to discern and analyze the\nBarkhausen noise, a phenomenon of particular interest\nfor further investigation and understanding.\nFIG. 3. This is how we are extracting the signal[9]\nIII. EXPERIMENTAL SET-UP AND\nPROPOSAL OF KIT\nTo conduct this experiment, we devised a straight-\nforward homemade kit, from which we extracted the\nBarkhausen Noise. The kit comprises the following com-\nponents:\n•Ferromagnetic Sample\n•Pick-up coil\n•Speaker\n•Permanent Magnet\n•Audio Amplifier\nA. Ferromagnetic Sample\nIn this experiment, we are utilizing a cylindrical soft\niron core with a diameter of 19.5 mm and a length of\n71 mm. Soft iron is an easily accessible ferromagnetic\nmaterial characterized by high permeability and low\ncoercivity, meaning it does not retain its magnetization\nwhen the magnetic field is removed. This property\nallows for easy magnetization and demagnetization, and\nit exhibits minimal hysteresis loss, making it the ideal\nmaterial for our experiment, where we require repeated\nswitching of the magnetic field to accurately observe\nBarkhausen noise (BN).\nIn soft Iron core of not very high purity which is used in\nstandard applications such as in concrete structures etc.\nThese impurities are the primary cause of Barkhausen\nNoise. Non-magnetic impurities, such as carbon, sulfur,\nphosphorous, and silicon, are present, along with mag-\nnetic impurities like nickel, cobalt, and manganese. Ad-\nditionally, the material contains grain boundary defects,\ndislocations within its crystal structure (which can im-\npede the movement of domain walls), as well as inclusions\nand surface defects.3\nB. Pick-up Coil\nThe pickup coil serves the crucial function of detecting\nand converting alterations in magnetic flux into electrical\nsignals. It plays a pivotal role in various devices where\nthe conversion of magnetic fields into electrical signals is\nessential for purposes such as detection, measurement,\nor transmission. The operation of a pickup coil is based\non Faraday’s law of electromagnetic induction. The\nmagnitude of the voltage signal it produces depends on\nseveral factors, including the strength of the magnetic\nfield, the number of turns within the pickup coil, and\nthe rate at which the magnetic field changes.\nFor our experiment, we have specifically designed a\nfour-layered pickup coil using enameled copper wire with\na total of 300 turns. The copper wire employed in this\ncoil has a cross-sectional diameter of 0.5 mm. This coil\nis wound around the sample (Soft Iron core), enabling it\nto effectively capture the Barkhausen noise that we are\nstudying in our experiment.\nFIG. 4. Pickup coil wound over soft iron (sample)\nC. Speaker\nA speaker is an electroacoustic transducer designed\nto transform electrical signals into sound waves. It\nfunctions through the interaction between a permanent\nmagnet and an electromagnet that are affixed to a\ndiaphragm. This interaction results in the diaphragm’s\nmovement, which corresponds to the incoming electrical\nsignal, ultimately producing sound.\nIn our experimental setup, crafted for demonstra-tion purposes, we have incorporated an 8 Ω impedance\nspeaker with a power rating of 0.25 W. This speaker\nenables us to audibly perceive the faint yet distinct\nBarkhausen noise (BN) when one listens closely to it.\nFIG. 5. speaker used for this experiment\nD. Magnet\nCertainly, for this demonstration, an external mag-\nnetic field is essential for the sample. We are utilizing\na ”ring-shaped” permanent ferrite magnet, which we\nretrieved from old speakers. This ferrite magnet can\ngenerate a magnetic field with a range spanning from\na few milli-tesla (mT) to sub tesla (T), depending on\norientation and distance from the magnet.\nFerrite magnets are the most commonly used type\nof magnets in speakers. They are typically composed\nof a mixture of iron oxide and other materials, often\nstrontium or barium. These magnets are not only cost-\neffective but also possess favorable magnetic properties.\nWhen we bring the magnet in proximity to the sample,\nthe sample becomes magnetized, and this magnetization\nprocess is accompanied by the generation of Barkhausen\nnoise, which we can audibly perceive.\nWe are using two ferrite ring magnets for our demon-\nstration. The larger one has a diameter of 110 mm and a\nheight of 15mm, while the smaller one has a diameter of\n80 mm and a height of 14mm. These magnets are com-\nbined and positioned as illustrated in the figure 6 for our\nexperiment.\nFIG. 6. Magnets used.4\nE. Audio Amplifier\nAn audio amplifier is an electronic device designed\nto increase the amplitude of low-power electronic audio\nsignals within the frequency range of approximately 20\nto 20,000 Hz. These signals can originate from various\nsources, such as a radio receiver or an electric guitar\npickup, and the amplifier’s purpose is to elevate them to\na level suitable for driving speakers or headphones.\nTo amplify the Barkhausen noise (BN) in our experi-\nment, we have developed a cost-effective ”Audio Ampli-\nfier” utilizing the LM386 integrated circuit (IC). This IC\noffers adjustable gain and boasts low power consumption.\nWe are able to power this amplifier efficiently using a 9V\nbattery.\n1. LM386\nTheLM386 is a 8 pin IC, used for designing power\namplifier which consume very low power, have adjustable\ngain of any value from 20 to 200, which could be done\nby connecting appropriate capacitor between pin 1 and\n8 (see Figure 7) [10].\n*For more information, see reference on IC LM386[10]\nFIG. 7. LM386. For details refer to:\nwww.learningaboutelectronics.com/Articles/How-to-\nconnect-a-LM386-audio-amplifier-chip\nF. Amplifier Circuit\nFigure 8 illustrates the amplifier circuit constructed\nusing the LM386 IC. This circuit is designed to enhance\nthe Barkhausen noise received from the pickup coil, with\npin 3 serving as the input. To optimize the amplification\nprocess, a 220Ω resistor is connected in series with pin\n3. Additionally, a 10 µF capacitor is positioned between\npins 1 and 8 to achieve maximum gain. A ”bypass”\ncapacitor with a value of 100 µF is connected through\npin 7, and another 100 µF capacitor is linked to the\npower supply to ground any AC components within the\npower supply.\nFIG. 8. Circuit designed for this kit\nFIG. 9. Circuit made on general purpose PCB\nAt the output, which is pin 5, a 1000 µF capacitor is\nutilized to block the DC signal (by the motion of hands)\nfrom reaching the speaker. Furthermore, a low-pass fil-\nter is incorporated in this section to eliminate any high-\nfrequency noise (specially MHz noise), particularly in the\nmegahertz range, that may be present along with the BN\nsignal.\nComponents are mounted on a piece of plywood, and\nmade a very simple kit. ( see figure 10)\nIt turns out that the total cost to make it is only around\nRs 300/-.\nIV. RESULTS\nInitially, we successfully constructed a cost-effective\nkit (refer to Figure 10), utilizing components detailed\nin Section III. This kit, as demonstrated in a rele-\nvant video accessible as supplementary material at\nhttps://tinyurl.com/BNdata, effectively captures the\nBarkhausen Noise, producing clear auditory signals\nthrough the integrated speaker. Subsequently, we\ndirected the resulting signal to the Keithley 2450\nsourcemeter , a versatile instrument serving as a digital\nmultimeter equipped with built-in ammeter, voltmeter,\nand ohmmeter functionalities. Notably, this instrument\nboasts a remarkable resolution of 10 nV for the voltmeter5\nFIG. 10. Barkhausen Experiment kit designed by us.\nand 10 fA for the ammeter, featuring 6.5 significant\ndigits and a reading capability ranging from 1 reading\nper second to 1700 readings per second.\nThe recorded waveforms are visually presented in Fig-\nure 11. Upon bringing the magnet close to the sample,\nFIG. 11. Barkahusen Avalanche waveforms\nthe resulting signals are observed in millivolt range, as\ndepicted in the above figure. This indicates the effective\nperformance of our kit, adept at amplifying the micro-\nvolt Barkhausen signal into the millivolt range. Subse-\nquently, we conducted measurements to determine the\njump size (s) and the frequency (P(s)) of the avalanches.\nThe tail part of the measured data was plotted and fit-ted with a power-law model (refer to Figure 12) given by\nP(s) =as−τ+b, where ’a’ and ’b’ represent scaling con-\nstants. The obtained τvalue is 1.303, with a=799.139\nand b=-4.834.\nFIG. 12. Curve fitting of observed Barkhausen data.\nV. STATISTICS OF BARKHAUSEN NOISE\nIn this section, we will address how does our measured\nvalue of τ= 1.303 compare with the available data.\nIn nature, numerous systems exhibit crackling noise\nwhen driven slowly, characterized by a series of sudden\navalanche-like events with a broad range of sizes and\ndistributions. Various studies have been conducted to\ncomprehend the dynamics behind these phenomena, re-\nvealing strikingly similar behavior across diverse systems\nsuch as earthquakes, plastic deformation, microfractures,\nvortices in superconductors, dimming events in stars, and\nBarkhausen noise [8].\nExtensive theoretical and experimental investigations,\nframed within the Langevian theory of domain wall\nmotion, have contributed to the interpretation of\nBarkhausen noise statistics. These studies have revealed\na power law or scaling law distribution aligning seam-\nlessly with experimental data [4, 5].\nP(s)∼s−τ(2)\nP(T)∼s−α(3)\nwhere, sis the jump size and Tis the avalanche\nduration, Durin and Zapperi [6] conducted an investi-\ngation into the scaling properties of Barkhausen Noise\nacross several ferromagnetic materials. They measured\navalanche distributions, determined scaling constants,\nand grouped the materials into two classes with τ\nvalues of τ= 1.50 ±0.05 and τ= 1.27 ±0.03. The\nfirst category encompasses Si-Fe polycrystals and the6\npartially crystallized amorphous alloy, while the second\ncategory comprises amorphous alloys subjected to stress.\nThese values agree reasonably well with our measured\nvalue of τ.\nVI. DISCUSSION\nOur designed kit serves as a valuable resource for\nuniversities and colleges, facilitating the demonstration\nof the Barkhausen Effect. Students can easily perceive\nthe Barkhausen noise through the integrated speaker.\nNot only is the kit simple to construct, but it is also\nhighly cost-effective, requiring a modest investment of\napproximately Rs. 300 (approximately $3.61 USD). This\naccessible tool fosters a deeper understanding among\nstudents regarding domain walls and the magnetization\ndynamics of ferromagnetic materials. Additionally, it\nproves useful for non-destructive testing of materials to\nidentify impurities.\nMoreover, the kit enables the calculation of the scal-\ning constant ( τ) for the soft iron core, adding an in-\ntriguing dimension to its educational utility. The ver-\nsatility of this kit extends to potential modifications for\ndemonstrating the Barkhausen Effect in various materi-\nals. However, it is important to note that the system is\nsusceptible to errors such as mechanical vibrations, mag-netic field fluctuations, and environmental noise. While\ndata obtained from this device can serve for preliminary\ncalculations, it is acknowledged that more sophisticated\nand sensitive instruments would be required for detailed\nanalyses. Nevertheless, at the college or school level, this\nkit proves to be a valuable resource for effective demon-\nstration purposes.\nVII. CONCLUSION\nIn conclusion, the Barkhausen effect is a great edu-\ncational and research topic. To advance in this field,\na foundational understanding is crucial, and this can be\nachieved through the use of a readily designed kit. Such a\nkit provides an accessible means for students to delve into\nthe intricacies of the Barkhausen effect, offering valuable\ninsights. The kit is very cost effective and can introduced\nin colleges and universities in India and abroad. More-\nover, with the aid of this straightforward kit, one can ef-\nfectively calculate the scaling exponent τvalue of a Soft\nIron sample and other samples too and can be compared\nwith simulations and other other available data.\nA. Acknowledgments\nI would like to thank Physical Research Laboratory\n(PRL) -Ahmadabad and Indian Institute of Space Sci-\nence and Technology (IIST) - Kerala for providing me\nall the required facilities.\n[1] R. Feynman, Chapter 37: Magnetic materials, Feynman\nlectures in physics , Vol. 2, Publisher: Addison–Wesley\n(1963).\n[2] P Weiss , L’hypoth` ese du champ mol´ eculaire et la pro-\npri´ et´ e ferromagn´ etique, Vol. 6, Journal: J. Phys. Theor.\nAppl. (1907).\n[3] Gianfranco Durin and Stefano Zapperi, The Barkhausen\nEffect, Vol. 2, Journal: arXiv : cond-mat/0404512v1 ?\n(2004).\n[4] Bruno Alessandro and et. al., Domain-wall dynamics and\nBarkhausen effect in metallic ferromagnetic materials. I.\nTheory, Journal: J. Appl. Phys. 68, 2901–2907 (1990).\n[5] Bruno Alessandro and et. al., Domain-wall dynamics and\nBarkhausen effect in metallic ferromagnetic materials.\nII. Experiments, Journal: J. Appl. Phys. 68, 2908–2915\n(1990).\n[6] Gianfranco Durin and Stefano Zapperi, Scaling Expo-\nnents for Barkhausen Avalanches in Polycrystalline andAmorphous Ferromagnets, Journal: Physical Review\nLetters (2000).\n[7] Sami Kaappa and Lasse Laurson, Barkhausen noise from\nformation of 360 ◦domain walls in disordered permalloy\nthin films, Vol. 5, Journal: Physical Review Research\n(2023).\n[8] Felipe Bohn and et. al., Playing with universality classes\nof Barkhausen avalanches, Publisher: Nature- Scientific\nReports (2018).\n[9] Stainslaw and Zurek, Characterisation of Soft Magnetic\nMaterials Under Rotational Magnetisation, Publisher:\nCRC Press (2019).\n[10] Irene, LM386 Power Amplifier Circuit, Publisher:\nApogeeweb (2022)." }, { "title": "2401.15581v1.Time_harmonic_elastic_scattering_by_unbounded_deterministic_and_random_rough_surfaces_in_three_dimensions.pdf", "content": "arXiv:2401.15581v1 [math.AP] 28 Jan 2024TIME-HARMONIC ELASTIC SCATTERING BY UNBOUNDED\nDETERMINISTIC AND RANDOM ROUGH SURFACES IN THREE\nDIMENSIONS\nGUANGHUI HU, TIANJIAO WANG, XIANG XU, AND YUE ZHAO\nAbstract. In this paper, we investigatewell-posednessoftime-harmonic scat teringofelas-\ntic waves by unbounded rigid rough surfaces in three dimensions. Th e elastic scattering is\ncaused by an L2function with a compact support in the x3-direction, and both determin-\nistic and random surfaces are investigated via the variational appr oach. The rough surface\nin a deterministic setting is assumed to be Lipschitz and lie within a finite d istance of a\nflat plane, and the scattering is caused by an inhomogeneous term in the elastic wave equa-\ntion whose support lies within some finite distance of the boundary. F or the deterministic\ncase, a stability estimate of elastic scattering by rough surface is s hown at an arbitrary\nfrequency. It is noticed that all constants in a prioribounds are bounded by explicit func-\ntions of the frequency and geometry of rough surfaces. Furthe rmore, based on this explicit\ndependence on the frequency together with the measurability and P-essentially separability\nof the randomness, we obtain a similar bound for the solution of the s cattering by random\nsurfaces.\n1.Introduction\nThis paper is concerned with the mathematical analysis of the time-h armonic elastic\nscattering from unbounded deterministic and random rough surfa ces in three dimensions.\nThe phrase rough means surface is a (usually nonlocal) perturbatio n of an infinite plane\nsuch that the whole surface lies within a finite distance of the original plane. Rough surface\nscattering problems have important applications in diverse scientific areas such as remote\nsensing, geophysics, outdoor sound propagation, radar techniq ues (see e.g.,[1, 2] and the\nreferences cited therein). In linear elasticity, the existence and u niqueness of solution were\nstudied in via the boundary integral equation method [3, 4, 5]. The va riational approach was\nproposed in [7, 9] to handle well-posedenss of the scattering proble ms in periodic structures\nby using the Rayleigh expansion condition (REC) and in [8, 10] for gen eral rigid rough\nsurfaces by using the angular spectrum representation (ASR).\nRecently, in [11] a mathematical formulation of the elastic rough sur face scattering prob-\nlems was presented in three dimensions. Based on a Rellich-type ident ity, the uniqueness of\nweak solutions to the variational problem was proved if the rigid surf ace was the graph of a\nuniformly Lipschitz continuous function. The existence of solutions was also proved for the\ncase of locally perturbed scattering problems. However, the well-p osedness problem for the\nscattering by a general rough surface remains unsolved. Later, the authors in [14] further\nderived an a prioribound which was explicitly dependent on the frequency. The main goa l\nof this paper is three-fold. First, we present a variational formula tion of the elastic scattering\nin three dimensions by a Lipschitz-type rough surface and prove its well-posedness. Second,\n2010Mathematics Subject Classification. 35A15, 35P25, 74J20.\nKey words and phrases. elastic waves, rough surfaces, variational formulation, explicit a prioribounds.\n12 G. HU, T. WANG, X. XU, AND Y. ZHAO\nwe derive an a prioribound which is explicitly dependent on the frequency. Third, we utilize\nthe explicit bound to derive the well-posedness for scattering by ra ndom rough surfaces as\n[13, 14]. As discussed in [6], we expect that the variational formulatio n will be suitable for\nnumerical solution via finite element discretization. Furthermore, t he explicit bounds we\nobtain should be useful in establishing the dependence of the const ants ina priori error\nestimates for finite element schemes on the frequency and the geo metry of the domain.\nThis paper utilizes methods and results contained in [8, 11, 14]. As was pointed out in\n[8], the elastic problem is more complicated than the acoustic case due to the coexistence\nof compressional and shear waves. As a consequence, the Dirichle t-to-Neumann map for\nthe elastic wave equation is tensor-valued which does not have a defi nite real part. This\nbrings difficulties in deriving the a priori estimates of solutions via Rellich identities for\narbitrary frequencies. We prove that the variational problem is we ll-posed by the theory\nof semi-Fredholm used in [8]. To this end, we first consider the case of small frequencies\nin which the Lax–Milgram theorem can be applied. Then we establish sev erala priori\nestimates. During this process, we carefully trace the dependenc e of the coefficients of these\nbounds on the frequency. In this way, we arrive at an a prioribound for the solution to the\nvariational problem which is explicitly dependent on the frequency. A fterwords, inspired by\nthe framework for scattering by random medium in [12] and random s urfaces in [13, 14], we\ncan obtain the well-posedness for a stochastic variation problem wit h an explicit a priori\nbound.\nThe rest of this paper is outlined as follows. In Section 2 we present t he variational\nformulation for the elastic scattering problem. Section 3.3 is devote d to the well-posedness\nof the variational problem for small frequencies. In Section 4 we de rivea prioribounds and\ntrace the explicit dependence on the frequency and on the geomet ry of the domain. For\nrandom cases, a similar bound is derived in Section 5. Conclusions are p resented in Section\n6.\n2.Problem formulation\nThis section is devoted to the mathematical formulation of the thre e-dimensional elastic\nwave scattering by unbounded rigid rough surfaces. Let D⊂R3be an unbounded connected\nopen set such that, for some constants mh}, x′:= (x1,x2).\nThe space Dis supposed to be filled with a homogeneous and isotropic elastic medium\nwith unit mass density. We assume that Γ := ∂Dis an unbounded rough surface, which is\nsupposed to be the graph of a uniformly Lipschitz continuous function f.More precisely, we\nassume\nΓ ={x∈R3:x3=f(x′), x′= (x1,x2)∈R2},\nand there exists a constant L>0 such that\n|f(x′)−f(y′)| ≤L|x′−y′|for allx′,y′∈R2. (2.1)\nThroughout the paper we fix some h > M. Let Γh={x∈R3:x3=h}andSh=D\\Uh.\nDenote the unit normal vector on Γ ∪Γhbyν:= (ν1,ν2,ν3) pointing into the region of x3>h\non Γhand into the exterior of Don Γ. Assume that g∈L2(D)3is an elastic source term\nwith supp(g)⊂Sh. Consider the following Navier equation in three dimensions\n∆∗u+ω2u=ginD, (2.2)INVERSE SOURCE PROBLEM 3\nwhere ∆∗=µ∆ + (λ+µ)∇∇·,u= (u1,u2,u3)⊤is the elastic displacement and ω >0 is\nthe angular frequency. Here λandµdenote the Lam´ e constants characterizing the medium\nabove Γ satisfying µ>0,λ+2µ/3>0.Since Γ is physically rigid, there holds the Dirichlet\nboundary condition\nu= 0 on Γ. (2.3)\nAs the domain Dis unbounded, a proper radiation condition should be imposed on u\nat infinity. In this paper we utilize the elastic Upward Propagation Radiation Condition\n(UPRC) at infinity to ensure the well-posedness of the boundary va lue problem (2.2)-(2.3).\nBelow we briefly introduce this radiation condition and refer to [11, 8 ] for the details. We\nbegin with the decomposition of the wave fields into a sum of compressional and shear parts\nu=1\ni(∇ϕ+∇×ψ),∇·ψ= 0inx3>h, (2.4)\nwhere the scalar function ϕand the vector function ψsatisfy the homogeneous Helmholtz\nequations\n∆ϕ+k2\npϕ= 0,∆ψ+k2\nsψ= 0 inx3>h. (2.5)\nHere,kpandksare compressional and shear wave numbers, respectively, define d by\nkp:=ω√λ+2µ, ks:=ω√µ.\nDenote by ˆvthe Fourier transform of vinR2, i.e.,\nˆv(ξ) =Fv(ξ) :=1\n2π/integraldisplay\nR2v(x′)e−ix′·ξdx′, ξ= (ξ1,ξ2)∈R2.\nTaking the Fourier transform of (2.5)and assuming that ϕ,ψfulfill the Upward Angular\nSpectrum Representation (UASR) of the Helmholtz equation in Uh(see [6]), we obtain for\nx3≥hthat\nϕ(x′,x3) =1\n2π/integraltext\nR2ˆϕ(ξ,h)eiβ(ξ)(x3−h)eiξ·x′dξ,\nψ(x′,x3) =1\n2π/integraltext\nR2ˆψ(ξ,h)eiγ(ξ)(x3−h)eiξ·x′dξ, (2.6)\nwhere\nβ(ξ) :=/braceleftBigg\n(k2\np−|ξ|2)1/2,|ξ|kp,\nand\nγ(ξ) :=/braceleftBigg\n(k2\ns−|ξ|2)1/2,|ξ|ks.\nDenotethe Fourier transform of ϕ(x′,h) andψ(x′,h) by\nAp(ξ) = ˆϕ(ξ,h),˜As(ξ) =ˆψ(ξ,h),\nrespectively. Noting that div ψ= 0, we have ( ξ,γ(ξ))·˜As(ξ)⊤= 0. For notational conve-\nnience we omit the dependence of βandγonξin the subsequent context.\nSubstituting (2.6) into (2.4), we obtain for x3≥hthat\nu(x) =1\n2π/integraldisplay\nR2/bracketleftbig\nAp(ξ)(ξ,β)⊤eiβ(x3−h)+As(ξ)eiγ(x3−h)/bracketrightbig\neiξ·x′dξ, (2.7)4 G. HU, T. WANG, X. XU, AND Y. ZHAO\nwhereAs= (A(1)\ns,A(2)\ns,A(3)\ns)⊤(ξ) := (ξ,γ)⊤טAs(ξ). It follows from (2.7) and the orthogo-\nnality (ξ,γ)·A⊤\ns= 0 that\n/bracketleftbigg\nˆu(ξ,h)\n0/bracketrightbigg\n=\nξ11 0 0\nξ20 1 0\nβ0 0 1\n0ξ1ξ2γ\n/bracketleftBigg\nAp(ξ)\nAs(ξ)/bracketrightBigg\n:=/tildewideD(ξ)A(ξ),\nwhich gives\nA(ξ) =/bracketleftBigg\nAp\nAs/bracketrightBigg\n(ξ) =/tildewideD−1(ξ)/bracketleftbigg\nˆu(ξ,h)\n0/bracketrightbigg\n=D(ξ)ˆu(ξ,h). (2.8)\nHereDis a 4×3 matrix given by\nD(ξ) =1\nβγ+|ξ|2\nξ1ξ2γ\nβγ+ξ2\n2−ξ1ξ2−ξ1γ\n−ξ1ξ2βγ+ξ2−ξ2γ\n−ξ1β−ξ2β|ξ|2\n.\nUsing (2.7)–(2.8) yields the expression of uinUh:\nu(x) =1\n2π/integraldisplay\nR2/braceleftBig1\nβγ+|ξ|2/parenleftBig\nMp(ξ)ei(ξ·x′+β(x3−h))+Ms(ξ)ei(ξ·x′+γ(x3−h))/parenrightBig\nˆusc(ξ,h)/bracerightBig\ndξ,(2.9)\nwhere\nMp(ξ) =:\nξ2\n1ξ1ξ2ξ1γ\nξ1ξ2ξ2\n2ξ2γ\nξ1β ξ2β βγ\nandMs(ξ) =\nβγ+ξ2\n2−ξ1ξ2−γξ1\n−ξ1ξ2βγ+ξ2\n1−γξ2\n−ξ1β−ξ2β|ξ|2\n.\nThe representation (2.9) will be referred to as the upward radiation condition for rough\nsurface scattering problems in linear elasticity .\nDefinethe surface traction operator\nTu:= 2µ∂νu+λ(∇·u)ν+µν×(∇×u), (2.10)\nwhereν= (ν1,ν2,ν3) stands for the normal vector on the surface. Plugging (2.9) into (2.10)\nyields the Dirichlet-to-Neumann (DtN) operator on Γ h(cf [11])\nTu=Tu(x′) :=i\n2π/integraldisplay\nR2M(ξ)ˆu(ξ)eiξ·x′dξ, (2.11)\nwhereM(ξ) is given by\nM(ξ) =1\n|ξ|2+βγ\n×\nµ[(γ−β)ξ2\n2+k2\nsβ] −µξ1ξ2(γ−β) (2µ|ξ|2−ω2+2µβγ)ξ1\n−µξ1ξ2(γ−β)µ[(γ−β)ξ2\n1+k2\nsβ] (2µ|ξ|2−ω2+2µβγ)ξ2\n−(2µ|ξ|2−ω2+2µβγ)ξ1−(2µ|ξ|2−ω2+2µβγ)ξ2 γω2\n.\nThe boundary operator Tis non-local and is equivalent to the upward radiation condi-\ntion (2.9). It is also called the transparent boundary condition (TBC ) for time-harmonic\nscattering problems in unbounded domains.INVERSE SOURCE PROBLEM 5\nBasedontheaboveDtNoperator,thewave scatteringproblem(2 .2)-(2.3)canbereduced\ntoa boundary value problem over Sh:\nµ∆u+(λ+µ)∇∇·u+ω2u=ginSh\nu= 0 on Γ\nTu=Tuon Γh.\nTo introduce the variational formulation, we introduce the energy spaceVhforh > M\nas the closure of C∞\n0(Sh∪Γh)3in theH1norm\n/ba∇dblu/ba∇dblVh= (/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3+/ba∇dblu/ba∇dblL2(Sh)3)1/2.\nMultiplying the Navier equation in(2.2)by thecomplex conjugateof th etest function v∈Vh\nand using Betti’s formula yield\n/integraldisplay\nShE(u,¯v)−ω2u·¯vdx−/integraldisplay\nΓh¯v·Tuds=/integraldisplay\nShg·¯vdx,\nwhere the bilinear form E(·,·) is defined by\nE(u,v) := 2µ3/summationdisplay\nj,k=1∂kuj∂kvj+λ∇·u∇·v−µ∇×u·∇×v,∀u,v∈Vh.\nDefine the sesquilinear form B:Vh×Vh→Cby\nB(u,v) =/integraldisplay\nShE(u,¯v)−ω2u·¯vdx−/integraldisplay\nΓh¯v·Tuds. (2.12)\nNow we can formulate the variational problem as follows:\nVariational Problem I: findu∈Vhsuch that\nB(u,v) =−/integraldisplay\nShg·¯vdxfor allv∈Vh. (2.13)\nThe variational problem is equivalent to the boundary value problem: giveng∈L2(D)3,\nwith supp(g)⊂Shfor someh > M, findu∈H1\nloc(D)3such thatu|Sh∈Vhfor every\nh > M (implyingu= 0 on Γ), the Navier equation ( △∗+ω2)u=ginDholds in a\ndistributional sense, and the radiation condition (2.9) is satisfied wit hu|Γh∈H1/2(Γh)3by\nthe trace theorem.\nThe main theorem of this paper can now be stated as follows.\nTheorem 2.1. For anyω >0, the Variational Problem I (2.13) is uniquely solvable in Vh.\nMoreover, there exists a constant Cindependent of ω,hand the Lipschitz constant Loff\nsuch that the solution satisfies the estimate\n/ba∇dblu/ba∇dblVh≤(h−m+2)/parenleftbig\nC4(ω,h)+C5(ω,h)2+C6(ω,h,L)/parenrightbig\n/ba∇dblg/ba∇dblVh (2.14)\nwhere\nC4(ω,h) =C(h+1−m)ω, C 5=C√\n1+ω−1C3(ω,h)\nand\nC6=C(ω−1+1)C1(ω,h,L)C2(ω,h,L)2.6 G. HU, T. WANG, X. XU, AND Y. ZHAO\nHere\nC1(ω,h,L) =Cω3(1+L2)1/2(h−m+1),\nC2(ω,h,L) =C(1+L2)1/4√\nh+1−m(1+ω(h+1−m)),\nC3(ω,h) =C(h+1−m)(1+ω(h+1−m))2/ω.\nThe constants C1-C6are derived from a prioribounds of the variational solution, which\nexhibit explicit dependence on the frequency ωand the geometry of the rough surface. They\nlead to the explicit a prioribound of the solution of the elastic scattering problem in three\ndimensions.\nBy the semi-Fredholm theory in [8], the results of Theorem 2.1 follow fr om the well-\nposedness of the variational problem at small frequencies (cf The orem 3.3) and an a priori\nbound of the solution to the variational problem at an arbitrary fre quency (cf Theorem\n4.3). Thus, in the subsequent two sections we shall focus on mathe matical analysis at small\nfrequencies and a prioriestimate at an arbitrary frequency.\n3.Analysis of the variational problem for small frequency\nWe first investigate mapping properties the DtN operator in three d imensions. For a\nmatrixM(ξ)∈C3×3depending on ξ, let ReM(ξ) := (M(ξ) +M(ξ)∗)/2.We shall write\nReM(ξ)>0 if ReM(ξ) is positive definite. Here M∗(ξ) is the adjoint of Mwith respect to\nthe scalar product ( ·,·)C3×3inC3×3.\nLemma 3.1. LetM(ξ)be defined in (2.11)and leth>M.\n(1)There exists a constant Kindependent of ωsuch that Re(−iM)(ξ)>0for all|ξ|>\nKω, where\nK=λ+2µ\nµ√λ+µ>1√µ.\n(2)The DtN map Tis a bounded operator from H1/2(Γh)3toH−1/2(Γh)3.\n(3)For|ξ|Kω>k s, we haveβ= i|β|andγ= i|γ|, which implies\niM(ξ) =−1\n|ξ|2−|β||γ|\na1(ξ)b(ξ)−ic(ξ)ξ1\nb(ξ)a2(ξ)−ic(ξ)ξ2\nic(ξ)ξ1ic(ξ)ξ2a3(ξ)\n:=−1\n|ξ|2−|β||γ|M′(ξ) (3.2)\nwith\na1(ξ) =µ[ξ2\n2(|γ|−|β|)+k2\ns|β|], a2(ξ) =µ[ξ2\n1(|γ|−|β|)+k2\ns|β|], a3(ξ) =ω2|γ|,\nb(ξ) =−µξ1ξ2(|γ|−|β|), c(ξ) = 2µ|ξ|2−ω2+2µ|β||γ|.\nIt is obvious that ai(ξ),b(ξ),c(ξ)∈R. Then from (3.2) we obtain\nℜ(−iM(ξ)) =1\n|ρ|M′(ξ)\nwithρ(ξ) =|ξ|2+βγ. Hence it remains to prove M′(ξ) is positive-definite when |ξ|>Kω.\nTo this end, we should verify\ni)a1(ξ)>0,ii)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1(ξ)b(ξ)\nb(ξ)a2(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle>0,iii) detM′(ξ)>0.\ni) By direct calculation, it is obvious that\na1(ξ) =µ[(|γ|−|β|)ξ2\n2+k2\ns|β|]\n=µξ2\n2(|γ|2−|β|2)+k2\ns(|β|2+|β||γ|)\n|γ|+|β|\n=µξ1k2\ns+ξ2k2\np+k2\ns|β||γ|−k2\npk2\ns\n|γ|+|β|\n≥µ(|ξ|2−k2\ns)k2\np+k2\ns|β||γ|\n|γ|+|β|>0. (3.3)\nHere the condition |ξ|>Kω >k sis used in the last step.\nii) Denoteg(ξ) = (|γ| − |β|)|ξ|2+k2\ns|β|. Similar as (3.3) we have g(ξ)>0. Then one\narrives at\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1(ξ)b(ξ)\nb(ξ)a2(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a1a2−b2\n=µ2[(|γ|−|β|)|ξ1|2+k2\ns|β|][(|γ|−|β|)|ξ2|2+k2\ns|β|]−µ2ξ2\n1ξ2\n2(|γ|−|β|)2\n=µ2k2\ns|β|g(ξ)>0.\niii) Denote h(ξ) = 2ξ1ξ2b(ξ)−a1(ξ)ξ2\n2−a2(ξ)ξ2\n1, then it can be verified that\ndet(M′(ξ)) =a3(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea1(ξ)b(ξ)\nb(ξ)a2(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+(−a1(ξ)c(ξ)2ξ2\n2+2b(ξ)c(ξ)2ξ1ξ2−a2(ξ)c(ξ)2ξ2\n1)\n=µ2k2\ns|β||γ|g(ξ)ω2+c(ξ)2h(ξ). (3.4)8 G. HU, T. WANG, X. XU, AND Y. ZHAO\nDirect calculation implies\nh(ξ) =−2µξ2\n1ξ2\n2(|γ|−|β|)−µξ2\n1[ξ2\n1(|γ|−|β|)+k2\ns|β|]\n−µξ2\n2[ξ2\n2(|γ|−|β|)+k2\ns|β|]\n=−µ(|γ|−|β|)|ξ|4−µk2\ns|β||ξ|2=−µ|ξ|2g(ξ). (3.5)\nCombining (3.4)-(3.5) gives\ndet(M′(ξ)) =µ3g(ξ){k4\ns|β||γ|−|ξ|2[2|γ|(|γ|−|β|)+k2\ns]}\n=µ3g(ξ)/braceleftBigg\nk4\ns|β||γ|−/bracketleftbigg2|γ|(k2\np−k2\ns)\n|β|+|γ|+k2\ns/bracketrightbigg2/bracerightBigg\n=µ3g(ξ)d(ξ)\n(|γ|+|β|)2\nwith\nd(ξ) =k4\ns(|γ||β|−|ξ|2)(|γ|+|β|)2+4|γ|(k2\ns−k2\np)(|γ|k2\np+|β|k2\ns)|ξ|2.\nHence we only need to verify d(ξ)>0 for|ξ|>Kω. Taking |ξ|2=K′k2\nsimplies\nd(ξ) =k8\ns[(/radicalbig\n(K′−α)(K′−1)−K′)(√\nK′−1+√\nK′−α)2\n+4(1−α)K′√\nK′−1(√\nK′−α+α√\nK′−1)]\n>k8\ns[−(√\nK′−α+√\nK′−1)2+4(1−α)K′√\nK′−1α(√\nK′−α+√\nK′−1)]\nwithα:=k2\np/k2\ns=µ/(λ+2µ)<1. In order to show d(ξ)>0, we will verify\n2(1−α)αK′√\nK′−1>√\nK′−α,\ni.e.\nK′/radicalbigg\nK′−1\nK′−α>1\n2(1−α)α. (3.6)\nTo guarantee (3.6), let/radicalbigg\nK′−1\nK′−α>1\n2, C >1\nα(1−α),\ni.e.\nK′>max/braceleftbigg4−α\n3,1\nα(1−α)/bracerightbigg\n=1\nα(1−α)=(λ+2µ)2\nµ(λ+µ).\nHence, supposing that\n|ξ|>/radicalBigg\nK′\nµω=λ+2µ\nµ√λ+µω\nguarantees d(ξ)>0, which implies det M′(ξ)>0.\n(3) Forρ(ξ) =|ξ|2+βγ, direct calculation gives\n\n\nk2\np≤ |ρ| ≤kpks,0≤ |ξ| ≤kp;\nk2\np≤ |ρ| ≤k2\ns, kp≤ |ξ| ≤ks;\ncKω2≤ |ρ| ≤k2\ns, ks≤ |ξ| ≤Kω,(3.7)\nwith\ncK=K2−/radicalbig\n(K2−1/µ)(K2−1/(λ+2µ))>1/(λ+2µ).INVERSE SOURCE PROBLEM 9\nHere to derive the inequality for ks≤ |ξ| ≤Kωwe have used the fact that the function\nρ(ξ) =|ξ|2−/radicalBig\nk2p−|ξ|2/radicalbig\nk2s−|ξ|2\nis decreasing with respect to |ξ|for|ξ| ≥ks. We also consider γ−βwhich is\nγ−β=/radicalbig\nk2s−|ξ|2−/radicalBig\nk2p−|ξ|2=\n\n|γ|−|β|,0<|ξ| ≤kp,\n|γ|−i|β|, kp<|ξ| ≤ks,\ni(|γ|−|β|),|ξ|>ks.\nThen we immediately obtain\n/braceleftbigg|γ−β| ≤/radicalbigk2\ns−k2\np, 0<|ξ| ≤kpor|ξ|>ks,\n|γ−β|=/radicalbig\n|γ|2+|β|2=/radicalbigk2s−k2p, kp<|ξ| ≤ks.(3.8)\n(3)To prove the thirdresult, it suffices toverify the inequality Mij≤Cω, fori,j= 1,2,3\nand|ξ| ≤Kω. ForM33, by (3.7) we have\n|M33|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleγω2\nρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤\n\nω2ks/k2\np=ω(λ+2µ)/√µ, 0≤ |ξ| ≤kp,\nω2/radicalbigk2s−k2p/k2\np=ω/radicalbig\n(λ+µ)(λ+2µ)/µ, kp≤ |ξ| ≤ks,\nω/radicalbig\nK2ω2−k2s/cKω2=ω/radicalbig\nK2−1/µ/cK, ks≤ |ξ| ≤Kω.(3.9)\nSimilarly,M23andM32can be estimated using (3.7) by\n|M23|=|M32|\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2µρξ2��ω2ξ2\nρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤\n\n2µkp+ω2/kP=ω(2µ/√λ+2µ+√λ+2µ),0≤ |ξ| ≤kp,\n2µks+ωks/k2\np=ω(2√µ+(λ+2µ)/√µ), kp≤ |ξ| ≤ks,\n2µKω+Kω3/cKω2=ω(2µK+K/cK), ks≤ |ξ| ≤Kω.\n(3.10)\nIt is obvious that |M13|=|M31|can also be estimated by the right-hand side of (3.10). It\nremains to estimate M11,M22,M12andM21. For convenience, denote\n/radicalBig\nk2\ns−k2\np=ω/radicalBigg\nλ+µ\nµ(λ+2µ):=Cλ,µω.\nCombining (3.7)-(3.8) gives\n|M11| ≤µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(γ−β)ξ2\n2+k2\nsβ\nρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤\n\nµ(ωCλ,µ+k2\ns/kp) =ω(µCλ,µ+√λ+2µ),0≤ |ξ| ≤kp,\nµ(Cλ,µωk2\ns/k2\np+ωCλ,µk2\ns/k2\np) = 2ωCλ,µ(λ+2µ)/µ, kp≤ |ξ| ≤ks,\nω(µCλ,µK2/cK+/radicalbig\nK2−1/(λ+2µ)/cK), ks≤ |ξ| ≤Kω.(3.11)\nObviously, |M22|can also be estimated by the right-hand side of (3.11). For |M12|and|M21|,\nwe combine (3.7)-(3.8) to obtain\n|M12|=|M21|\n≤µ/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1ξ2(γ−β)\nρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤\n\nωµCλ,µ, 0≤ |ξ| ≤kp,\nµk2\nsωCλ,µ/k2\np=ω(λ+2µ)Cλ,µ, kp≤ |ξ| ≤ks,\nµK2Cλ,µω3/cKω2=ω(µK2Cλ,µ)/cK, ks≤ |ξ| ≤Kω.\n(3.12)10 G. HU, T. WANG, X. XU, AND Y. ZHAO\nCombining the above results (3.9)-(3.12), we have\n/ba∇dblM/ba∇dbl ≤\n\nCK,1ω,0≤ |ξ| ≤kp,\nCK,2ω, kp≤ |ξ| ≤ks,\nCK,3ω, ks≤ |ξ| ≤Kω\nwith\nCK,1= max/braceleftbiggλ+2µ√µ,2µ√λ+2µ+/radicalbig\nλ+2µ, µCλ,µ+/radicalbig\nλ+2µ, µCλ,µ/bracerightbigg\n,\nCK,2= max/braceleftbigg\n(λ+2µ)Cλ,µ,2√µ+λ+2µ√µ,2Cλ,µ(λ+2µ)\nµ/bracerightbigg\n,\nCK,3= max\n\n/radicalBig\nK2−1\nµ\ncK,2µK+K\ncK,µK2Cλ,µ\ncK,µK2Cλ,µ\ncK+/radicalBig\nK2−1\nλ+2µ\ncK\n\n.\nIt can be verified that\nCK,1≤2λ+2µ√µ, CK,2≤λ+2µ√µ+2λ+2µ\nµCλ,µ+2√µ\nand\nCK,3≤K\ncK+µK2Cλ,µ\ncK+2µK.\nRecalling that cK>1/(λ+2µ), we have\nmax{CK,1, CK,2, CK,3} ≤2(λ+4µ)K+(µ(λ+2µ)K2+2(λ+2µ)/µ)Cλ,µ\n= 2(λ+4µ)K+(µ(λ+2µ)K2+2(λ+2µ)/µ)/radicalBigg\nλ+µ\nµ(λ+2µ).\nThe proof is completed. /square\nRecall that there exists a constant C0=C0(h,L,m,M )>0 independent of ωsuch that\n/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3≥1/C0||u||2\nVh,||u||2\nH1/2(Γh)≤C0||u||2\nVh, (3.13)\nfor allu∈Vh. The well-posedness result for small frequencies is stated below.\nTheorem 3.3. LetK,CK>0be given as in Lemma 3.1. Then there exists a small frequency\nω0>0such that the variational problem admits a unique solution i nVhfor allω∈(0,ω0].INVERSE SOURCE PROBLEM 11\nProof.It is clear that /ba∇dbl∇×u/ba∇dbl2\nL2(Sh)3≤ /ba∇dbl∇u/ba∇dbl2\nL2(Sh)3.Now it follows from the definition of B\nand Lemma 3.1 that\nℜB(u,u) = 2µ/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3+λ/ba∇dbl∇·u/ba∇dbl2\nL2(Sh)3−µ/ba∇dbl∇×u/ba∇dbl2\nL2(Sh)3\n−ω2/ba∇dblu/ba∇dbl2\nL2(Sh)3−ℜ/integraldisplay\nΓh¯u·Tuds\n= 2µ/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3+λ/ba∇dbl∇·u/ba∇dbl2\nL2(Sh)3−µ/ba∇dbl∇×u/ba∇dbl2\nL2(Sh)3−ω2/ba∇dblu/ba∇dbl2\nL2(Sh)3(3.14)\n+/integraldisplay\n|ξ|≤KωRe(−iM(ξ))ˆu·¯ˆudξ+/integraldisplay\n|ξ|>KωRe(−iM(ξ))ˆu·¯ˆudξ\n≥µ/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3−ω2/ba∇dblu/ba∇dbl2\nL2(Sh)3+/integraldisplay\n|ξ|≤KωRe(−iM(ξ))ˆu·¯ˆudξ\n≥µ/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3−ω2/ba∇dblu/ba∇dbl2\nL2(Sh)3−CKC0ω/ba∇dblu/ba∇dbl2\nVh, (3.15)\nwhere the constant CK>0 is given by Lemma 3.1 (3) and the constant C0is specified in\n(3.13). By Lemma 3.4 in [6] we have the following Poincare’s inequality\n/ba∇dblu/ba∇dbl2\nL2(Sh)3≤(h−m)/ba∇dbl∂3u/ba∇dbl2\nL2(Sh)3≤(h−m)/ba∇dbl∇u/ba∇dbl2\nL2(Sh)3, u∈Vh.(3.16)\nUsing (3.14)-(3.16), we obtain the estimate\nℜB(u,u)≥/parenleftbig\nµ/C0−ωC0CK−ω2(h−m)/parenrightbig\n/ba∇dblu/ba∇dbl2\nVh\n≥/parenleftbig\nµ/C0−ω0C0CK−ω2\n0(h−m)/parenrightbig\n/ba∇dblu/ba∇dbl2\nVh\nfor allu∈Vhandω∈(0,ω0]. Chooseω0sufficiently small such that\nµ/C0−ω0C0CK−ω2\n0(h−m)>0.\nThe proof is completed by applying the Lax-Milgram theorem. /square\n4.Ana prioribound for smooth rough surfaces\nIn this section, we establish an a prioribound for a smooth rough surface at any fre-\nquency. The attractive feature is that all constants in the a prioriestimates are bounded by\nexplicit functions of ω,h,m,MandL.\nLemma 4.1. Letu∈Vhbe a variational solution to (2.13)withg∈Vh. We have\n/ba∇dbl∇·u/ba∇dbl2\nL2(Γ),/ba∇dbl∇×u/ba∇dbl2\nL2(Γ)3≤C1/ba∇dblg/ba∇dblL2(Sh)3/ba∇dbl∂3u/ba∇dblL2(Sh)3,\nwhereC1=4µ−1(1+L2)1/2(ω/√µ(h−m)+1).\nProof.By[11, Lemma 4.1](see also [9, Lemma 5] for the periodic version) we have the\nfollowing Rellich identity\n2ℜ/integraldisplay\nSh(µ∆u+(λ+µ)∇∇·u+ω2u)·∂3¯udx\n=/parenleftBig\n−/integraldisplay\nΓ+/integraldisplay\nΓh/parenrightBig/braceleftBig\n2ℜ(Tu·∂3¯u)−ν3E(u,¯u)+ω2|u|2/bracerightBig\nds, (4.1)\nand\nTu·∂3¯u=ν3E(u,¯u) =µ|∂νu|2ν3+ν3(λ+µ)|∇·u|2. (4.2)12 G. HU, T. WANG, X. XU, AND Y. ZHAO\nFrom [11, Lemma 4.2 (ii)] we also have the following two identities\n/integraldisplay\nΓh/braceleftBig\n2ℜ(Tu·∂3¯u)−E(u,¯u)+ω2|u|2/bracerightBig\nds\n= 2ω2/integraldisplay\n|ξ|h.\nSince both ϕandψsatisfy the Helmholtz equation (2.5) and the UASR (2.6), one has the\nfollowing Dirichlet-to-Neumann map on the artificial boundary Γ H:\n/tildewideTw=F−1(iηFw), w∈H1/2(ΓH), (4.10)\nwherew=ϕ,ψ, andη=β,γ, respectively. Moreover, /tildewideTis a bounded linear map of\nH1/2(ΓH) toH−1/2(ΓH) by [6, Lemma 2.4]. From Lemma 4.1 we can estimate the L2norm\nof the trace won Γ as\n/ba∇dblw/ba∇dbl2\nL2(Γ)3≤C1(ω,h,L)/ba∇dblg/ba∇dblL2(SH)3/ba∇dbl∂3u/ba∇dblL2(SH)3. (4.11)\nThe following lemma provides estimates for wonSHand the trace of won ΓH.\nLemma 4.2. Assume that wsatisfies the Helmholtz equation\n∆w+k2w=g0inSH,/tildewideTw=F−1(i/radicalbig\nk2−ξ2Fw)onΓH (4.12)\nwhereg0∈L2(SH). Then there holds the estimate\n/ba∇dblw/ba∇dblL2(ΓH)3≤ /ba∇dblw/ba∇dblL2(SH)3≤/tildewideC2(L,k,h)/ba∇dblw/ba∇dblL2(Γ)3+/tildewideC3(k,h)/ba∇dblg0/ba∇dblL2(SH)3 (4.13)\nwith\n/tildewideC2(L,k,h) =C(1+L2)1/4√\nH−m(1+k(H−m))\nand\n/tildewideC3(k,h) =C(H−m)(1+k(H−m))2/k.\nProof.Consider the boundary value problem of finding v∈H1(SH) such that\n(△+k2)v= ¯winSH, v= 0 on Γ , ∂3v=/tildewideTvon ΓH.(4.14)\nBy [6, Lemma 4.6] the boundary value problem (4.14) is well-posed with t he following esti-\nmate\n/ba∇dbl∇v/ba∇dblL2(SH)+k/ba∇dblv/ba∇dblL2(SH)≤C(1+k(H−m))2(H−m)/ba∇dblw/ba∇dblL2(SH).(4.15)\nWe first prove that /ba∇dbl∂νv/ba∇dbl2\nL2(Γ)3≤C/ba∇dblw/ba∇dbl2\nL2(SH)3for some constant C >0 depending\nexplicitly on ω,Hand the Lipschitz constant Lof Γ. The Rellich identity for the Helmholtz\nequation gives:\n2ℜ/integraldisplay\nSH∂3¯v(∆v+k2v)dx\n=/parenleftBig/integraldisplay\nΓ+/integraldisplay\nΓH/parenrightBig\n{2ℜ(∂νv∂3¯v)−ν3|∇v|2+ν3k2|v|2}ds, (4.16)14 G. HU, T. WANG, X. XU, AND Y. ZHAO\nwhich can be proved in the same way as (4.1). From the proof of [6, Le mma 4.6] it holds\nthat/integraldisplay\nΓH{2ℜ(∂νv∂3¯v−ν3|∇v|2+ν3k2|v|2)}ds≤2kℑ/integraldisplay\nΓH¯v/tildewideTvds\n≤2kℑ/integraldisplay\nSH¯v¯wdx. (4.17)\nMoreover, using the identities in (4.17) of [9] on Γ and the bound for ν3in (4.8) one has\n−/integraldisplay\nΓH{2ℜ(∂νv∂3¯v−ν3|∇v|2+ν3k2|v|2)}ds=−/integraldisplay\nΓν3|∂νv|2ds\n≥(1+L2)−1/2/ba∇dbl∂νv/ba∇dbl2\nL2(Γ). (4.18)\nPlugging (4.17) and (4.18) into (4.16) and using (4.15) yield the estimat e\n/ba∇dbl∂νv/ba∇dbl2\nL2(Γ)≤(1+L2)1/2/braceleftBig\n−2ℜ/integraldisplay\nSH¯w∂3vdx+2kℑ/integraldisplay\nSH¯w¯vdx/bracerightBig\n≤2(1+L2)1/2/ba∇dblw/ba∇dblL2(SH)(k/ba∇dblv/ba∇dblL2(SH)+/ba∇dbl∇v/ba∇dblL2(SH))\n≤C(1+L2)1/2(H−m)(1+k(H−m))2/ba∇dblw/ba∇dbl2\nL2(SH), (4.19)\nwhere the constants Cis independent of w.\nNow we prove the second inequality in (4.13). Following the approach o f [8, Lemma 7],\nwe obtain that/integraldisplay\nSH{w∆v−v∆w}dx=/integraldisplay\nΓH{w∂νv−v∂νw}ds+/integraldisplay\nΓw∂νds\n=/integraldisplay\nΓH{w/tildewideTv−v/tildewideTw}ds+/integraldisplay\nΓw∂νvds\n=/integraldisplay\nΓw∂νvds.\nNote thatv= 0 on Γ, and the Dirichlet-to-Neumann operator /tildewideTdefined in (4.10) is sym-\nmetric (see Lemma 3.2 in [6]). Thus,\n/integraldisplay\nSH|w|2dx=/integraldisplay\nSHw(∆v+k2v)dx\n=/integraldisplay\nSHv(∇w+k2w)dx+/integraldisplay\nΓw∂νvds\n=/integraldisplay\nSHvgdx+/integraldisplay\nΓw∂νvds.\nNoting (4.15) and (4.19) one has\n/ba∇dblw/ba∇dbl2\nL2(SH)≤ /ba∇dblv/ba∇dbl2\nL2(SH)/ba∇dblg/ba∇dbl2\nL2(SH)+/ba∇dblw/ba∇dbl2\nL2(Γ)/ba∇dbl∂νv/ba∇dbl2\nL2(Γ)\n≤C√\nH−m(1+L2)1/4(1+k(H−m))/ba∇dblw/ba∇dblL2(SH)/ba∇dblw/ba∇dblL2(Γ)\n+C(H−m)(1+k(H−m))2\nk/ba∇dblw/ba∇dblL2(SH)/ba∇dblg0/ba∇dblL2(SH).INVERSE SOURCE PROBLEM 15\nThen the following inequality is proved\n/ba∇dblw/ba∇dblL2(SH)≤/tildewideC2(L,k,h)/ba∇dblw/ba∇dblL2(Γ)+/tildewideC3(k,h)/ba∇dblg0/ba∇dblL2(SH). (4.20)\nTo estimate the first inequality in (4.13) we use\n/integraldisplay\nΓH|w|2ds≤/integraldisplay\nΓc|w|2ds,for allc∈(h,H],\nwhich follows from the proof of [6, Lemma 2.2]. Then we have\n(H−h)/integraldisplay\nΓH|w|2dx≤/integraldisplay\nSH\\Sh|w|2ds≤/integraldisplay\nSH|w|2ds. (4.21)\nThe estimate (4.13) is proved by combing (4.20) and (4.21).\n/square\nNext we prove the estimates of the L2norms of ∇·uand∇×uonSHand ΓH. Using\nLemma 4.2 for v=ϕandψwithg0=−(i/ω2)∇·gand (i/ω2)∇×gin (4.12) , respectively,\nand (4.11), we obtain the estimate\n/ba∇dbl∇·u/ba∇dbl2\nL2(SH)+/ba∇dbl∇×u/ba∇dbl2\nL2(SH)3\n≤C2(ω,h,L)2C1(ω,h,L)/ba∇dblg/ba∇dblVh/ba∇dbl∂3u/ba∇dblL2(SH)3+C3(ω,h)2/ba∇dblg/ba∇dbl2\nVh,(4.22)\nwhere\nC2(ω,h,L) =C(1+L2)1/4√\nH−m(1+ω(H−m))\nand\nC3(ω,h) =C(H−m)(1+ω(H−m))2/ω.\nIn a similar way, from the estimates (4.13) and (4.11) we have the bou nd\n/ba∇dbl∇·u/ba∇dbl2\nL2(ΓH)+/ba∇dbl∇×u/ba∇dbl2\nL2(ΓH)3\n≤C2(ω,h,L)2C1(ω,h,L)/ba∇dblg/ba∇dblVh/ba∇dbl∂3u/ba∇dblL2(SH)3+C3(ω,h)2/ba∇dblg/ba∇dbl2\nVh.(4.23)\nThefollowingtheoremprovidesthe a prioriboundforthesolutionto Variational Problem\nIdependent on the frequency and geometry of the rough surface .\nTheorem 4.3. Assume that Γis given by the graph of a Lipschitz function fsatisfying\n(2.1), and thatu∈Vhis a solution to the variational problem (2.13). Then there exists a\nconstantCindependent of ω,hand the Lipschitz constant Loffsuch that the following a\npriori bound holds\n/ba∇dblu/ba∇dblVh≤(h−m+2)(C4(ω,h)+C5(ω,h)+C6(ω,h,L))/ba∇dblg/ba∇dblVh,\nwhere\nC4(ω,h) =C(h+1−m)ω, C 5=C√\n1+ω−1C3(ω,h),\nC6=C(ω−1+1)C1(ω,h,L)C2(ω,h,L)2.16 G. HU, T. WANG, X. XU, AND Y. ZHAO\nProof.We first assume that fis smooth. Multiplying both sides of the Navier equation by\n(x3−m)∂3¯uand using integration by parts yields\n2ℜ/integraldisplay\nSH(△∗+ω2)u·(x3−m)∂3¯udx\n=/integraldisplay\nSH/braceleftBig\nE(u,¯u)−2ℜ/braceleftBig3/summationdisplay\nj=1E(u,(x3−m)ej)∂3¯uj/bracerightBig\n−ω2|u|2/bracerightBig\ndx\n+/parenleftBig/integraldisplay\nΓH+/integraldisplay\nΓ/parenrightBig\n[−ν3E(u,¯u)+2ℜ(Tu·∂3¯u)+ν3ω2|u|2](x3−m)ds.(4.24)\nLettingv=uin the variational formulation (2.13) gives\n/integraldisplay\nSH{E(u,¯u)−ω2|u|}dx−ℜ/integraldisplay\n|ξ|>KωM(ξ)ˆu(ξ,H)·¯ˆu(ξ,H)dξ\n=−ℜ/integraldisplay\nSHg·¯udx+ℜ/integraldisplay\n|ξ|≤KωM(ξ)ˆu(ξ,H)·¯ˆu(ξ,H)dξ. (4.25)\nTaking the real part and using Lemma 3.1 we have\n/integraldisplay\nSH{E(u,¯u)−ω2|u|}dx\n≤ −ℜ/integraldisplay\nSHg·¯udx+ℜ/integraldisplay\n|ξ|≤KωM(ξ)ˆu(ξ,H)·¯ˆu(ξ,H)dξ. (4.26)\nFrom (4.24) and using (4.26) and (4.2), we have\n/integraldisplay\nSH2ℜ/braceleftBig3/summationdisplay\nj=1E(u,(x3−m)ej)∂3¯uj/bracerightBig\ndx\n−/integraldisplay\nΓ(x3−m){µ|∂νu|2+(λ+µ)|∇×u|2}ν3ds\n=/integraldisplay\nSH{E(u,¯u)−ω2|u|2}dx−2ℜ/integraldisplay\nSH(△∗+ω2)u·(x3−m)∂3¯udx\n+(H−m)/integraldisplay\nΓH{2ℜ(Tu·∂3¯u)−E(u,¯u)+ω2|u|2}ds\n≤/integraldisplay\nSH{−g·u+2ℜ(g·∂3¯u)(x3−m)}dx+ℜ/integraldisplay\n|ξ|≤KωM(ξ) ˆuH(ξ)·¯ˆuH(ξ)dξ\n+(H−m)/integraldisplay\nΓH{2ℜ(Tu·∂3¯u)−E(u,¯u)+ω2|u|2}ds. (4.27)\nAs/ba∇dblM(ξ)/ba∇dbl ≤Cωfor all|ξ|0 to be sufficiently large, we get\n/integraldisplay\nSH2ℜ/braceleftBig3/summationdisplay\nj=1E(u,(x3−m)ej)∂3¯uj/bracerightBig\ndx+C/ba∇dbl∇·u/ba∇dbl2\nL2(SH)+C/ba∇dbl∇×u/ba∇dbl2\nL2(SH)3\n=I1+I2+I3+C/ba∇dbl∂1u2−∂2u1/ba∇dbl2\nL2(SH), (4.33)18 G. HU, T. WANG, X. XU, AND Y. ZHAO\nwhere\nI1:= [C+2(λ+2µ)]/ba∇dbl∂3u3/ba∇dbl2\nL2(SH)3+C/ba∇dbl∂1u1/ba∇dbl2\nL2(SH)3+C/ba∇dbl∂2u2/ba∇dbl2\nL2(SH)3\n+(C+2λ)/parenleftBig\nℜ/integraldisplay\nSH∂1u1∂3¯u3dx+ℜ/integraldisplay\nSH∂2u2∂3¯u3dx/parenrightBig\n+Cℜ/integraldisplay\nSH∂1u1∂2¯u2dx,\n=/integraldisplay\nSHA[∂1u1,∂2u2,∂3u3]⊤·[∂1¯u1,∂2¯u2,∂3¯u3]⊤dx,\nA:=\nC C/ 2λ+C/2\nC/2C λ +C/2\nλ+C/2λ+C/2C+2(λ+2µ)\n,\nI2:= 4µ/ba∇dbl∂3u1/ba∇dbl2\nL2(SH)3+C/ba∇dbl∂3u1−∂1u3/ba∇dbl2\nL2(SH)−2µℜ/integraldisplay\nSH(∂3u1−∂1u3)∂3¯u1dx,\nI3:= 4µ/ba∇dbl∂3u2/ba∇dbl2\nL2(SH)3)+C/ba∇dbl∂2u3−∂3u2/ba∇dbl2\nL2(SH)+2µℜ/integraldisplay\nSH(∂2u3−∂3u2)∂3¯u2dx.\nDirect calculations show that Det( A)∼C2/8 asC→ ∞. Hence the matrix A∈R3×3must\nbe strictly positive for sufficiently large C >0. This gives\nI1≥C0(/ba∇dbl∂1u1/ba∇dbl2\nL2(SH)+/ba∇dbl∂2u2/ba∇dbl2\nL2(SH)+/ba∇dbl∂3u3/ba∇dbl2\nL2(SH)), (4.34)\nwhere the constant C0>0 only depends on λandµ.By arguing in the same manner one\nhas forC >µ2/4 that\nI2≥C0(/ba∇dbl∂3u1/ba∇dbl2\nL2(SH)+/ba∇dbl∂3u1−∂1u3/ba∇dbl2\nL2(SH)), (4.35)\nI3≥C0(/ba∇dbl∂3u2/ba∇dbl2\nL2(SH)+/ba∇dbl∂3u2−∂2u3/ba∇dbl2\nL2(SH)). (4.36)\nHence, it follows from (4.33)-(4.36) that\n/integraldisplay\nSH2ℜ/braceleftBig3/summationdisplay\nj=1E(u,(x3−m)ej)∂3¯uj/bracerightBig\ndx+C/ba∇dbl∇·u/ba∇dbl2\nL2(SH)+C/ba∇dbl∇×u/ba∇dbl2\nL2(SH)3\n≥C0(/ba∇dbl∂1u1/ba∇dbl2\nL2(SH)+/ba∇dbl∂2u2/ba∇dbl2\nL2(SH)+/ba∇dbl∂3u3/ba∇dbl2\nL2(SH)+/ba∇dbl∂1u2−∂2u1/ba∇dbl2\nL2(SH))\n+C0(/ba∇dbl∂3u1/ba∇dbl2\nL2(SH)+/ba∇dbl∂1u3/ba∇dbl2\nL2(SH)+/ba∇dbl∂3u2/ba∇dbl2\nL2(SH)+/ba∇dbl∂2u3/ba∇dbl2\nL2(SH)), (4.37)\nprovidedC >0 is sufficiently large. Combining (4.22), (4.31) and (4.37) and using You ng’s\ninequality gives\nRight hand side of (4.37) ≤(C4(ω,h)2+C5(ω,h)2+C6(ω,h,L)2)/ba∇dblg/ba∇dbl2\nVh.(4.38)\nHowever, we still need to estimate /ba∇dbl∂1u2/ba∇dbl2\nL2(SH)and/ba∇dbl∂2u1/ba∇dbl2\nL2(SH). Since/ba∇dbl∂3u/ba∇dbl2\nL2(SH)3can\nalso be bounded by the right hand side of (4.38), we have (see [6, Le mma 3.4])\n/ba∇dblu/ba∇dbl2\nL2(SH)≤C0/ba∇dbl∂3u/ba∇dbl2\nL2(SH)3≤(C4(ω,h)2+C5(ω,h)2+C6(ω,h,L)2)/ba∇dblg/ba∇dbl2\nVh.(4.39)\nNow, using (4.26), (4.29) and (4.39) we arrive at\nE(u,¯u)≤(C4(ω,h)2+C5(ω,h)2+C6(ω,h,L)2)/ba∇dblg/ba∇dbl2\nVh. (4.40)\nRecalling the expression of E, we find\n2µ(∂1u2+∂2u1) =E(u,u)−λ∇·u−µ∇×u−2µ(∂1u3+∂1u1+∂2u2+∂2u3+3/summationdisplay\nj=1∂3uj).INVERSE SOURCE PROBLEM 19\nIt follows from (4.40), (4.22) and (4.38) that each term on the right hand side of the previous\nidentity canbeboundedbytherighthandsideof (4.40), leading toth esameupper boundfor\n||∂1u2+∂2u1||L2(SH). Finally, recalling the upper bound for the difference ||∂1u2−∂2u1||L2(SH)\n(see (4.37)) we obtain the estimates for ||∂1u2||2\nL2(SH),||∂2u1||2\nL2(SH)and thus also for ||∇u||.\nUsing theL2-estimate for u(see (4.39)) we obtain\n||u||2\nVh≤(C4(ω,h)2+C5(ω,h)2+C6(ω,h,L)2)/ba∇dblg/ba∇dbl2\nVh.\nNow the a prioribound forfbeing smooth has been proved. It can be extended to the\ncase of a general Lipschitz function by the method of approximatio n in [8]. This completes\nthe proof.\n/square\n5.Well-posedness for random rough surfaces\nIn this section, we investigate the well-posedness of elastic scatte ring by a random rough\nsurface. Let (Ω ,A,P) be a complete probability space. Denote by S(η) a random surface\nΓ(η) :={x∈R3:x3=f(η;x1,x′),η∈Ω,x′∈R2}.\nSimilarly,D(η) andSh(η) represent the random counterparts of DandSh, respectively.\nAssumef(η;x′) is a Lipschitz continuous function with Lipschitz constant L(η) for allη∈Ω\nand it also satisfies m < f(η;x′)< M. The random source g(η) is assumed to satisfy\ng(η)∈L2(D(η))3with its support in Sh(η). Similarly as the deterministic case, we can give\nthe following random boundary value problem.\n∆∗u(η;·)+ω2u(η;·) =g(η;·) inSh(η),\nu(η;·) = 0 on Γ( η),\nTu(η;·) =Tu(η;·) on Γ h.\nFor simplicity, let Vh(η) =Vh(Sh(η)). Define a sesquilinear form ˜BηonVh(η)×Vh(η) by\n˜Bη(u,v) =/integraldisplay\nSh(η)E(u,¯v)−ω2u·¯vdx−/integraldisplay\nΓhTu·¯vds, (5.1)\nand an antilinear functional ˜GηonVh(η) by\n˜Gη(v) :=−/integraldisplay\nSh(η)g(η)·¯vdx. (5.2)\nTo define the stochastic variation problem directly is not suitable sinc eVh(η) is dependent\nonη. We take a variable transform to give a new sesquilinear form defined onVh×Vh. Let\nf0=f(η0) andg0=g(η0) for some fixed η0∈Ω and write D=D(η0),Sh=Sh(η0) and\nVh=Vh(η0) for convenience. In addition, we assume that g(η)∈H1(D(η))3and\n/ba∇dblf(η)−f0/ba∇dbl1,∞≤M0,∀η∈Ω,\nwith some constant M0>0. Moreover, the truncated height his chosen such that\n(M−m)/γ <1, (5.3)\nwhereγ=h−sup\nx′f0(x′). This condition ensures the invertiblity of the variable transform\nHwhich will be introduced later. Since Γ his artificial, choosing sufficiently large hwill be\nenough.20 G. HU, T. WANG, X. XU, AND Y. ZHAO\nDenote by Lip(R2) the set including all Lipschitz continuous functions on R2. Then\ndefine a product topology space\nC=C1×C2,\nwhere\nC1:={v∈Lip(R2) :m0 and\nC2:=H1\n0(Sh)3.\nThe topology of C1andC2are respectively given by the norms /ba∇dbl·/ba∇dbl1,∞and/ba∇dbl·/ba∇dblH1(Sh)3.\nConsider the transform H:Sh→Sh(η) defined by\nH(y) =y+α(y3−f0(y′))(f(η;y′)−f0(y′))e3, y∈Dh,\nwheree3is the unit vector in x3direction and α(x) is a cutoff function which satisfies\nα(x) =/braceleftbigg\n0, x<δ,\n1, x>γ,\nwith sufficiently small δ. It is also required to satisfy\n|α′|<1/(γ−2δ). (5.4)\nThe Jacobi matrix of His\nJH=I3+\n0 0 0\n0 0 0\nJ1J2J3\n,\nwhere\nJi=α(y3−f0(y′))(∂if(η;y′)−∂if0(y′))−α′(y3−f0(y′))∂if0(y1)(f(η;y′)−f0(y′)), i= 1,2\nand\nJ3=α′(y3−f0(y′))(f(η;y′)−f0(y′)).\nSince matrix JHis required to be non-singular so that His invertible, according to (5.4),\nwe obtain\n|J3| λan upper\nboundb∈X+with\n/bardblb/bardbl ≤λ′sup{/bardbla/bardbl:a∈A}.\nIn this paper we start with some basic properties of Banach lattices with the\nstrong Nakano property and then focus our analysis on the case o f free Banach\nlattices. To motivate this notion, let us briefly recall the construct ion of the free\nBanach lattice generated by a Banach space: Given a Banach space E, thefree\nBanach lattice generated by Ewill be a Banach lattice FBL[E] together with a\nlinear isometric embedding δE:E→FBL[E], such that for every Banach lattice\nXand every linear and bounded operator T:E→Xthere is a unique lattice\nhomomorphism ˆTmaking the following diagram commutative:\nFBL[E]\nˆT\n/d36/d36❍❍❍❍❍\nET/d47/d47δE/d79/d79\nX\nMoreover, /bardblˆT/bardbl=/bardblT/bardbl.\nThe existence of FBL[E], for every Banach space E, was proved in [ 5] via an\nexplicit construction: For a positively homogeneous function f:E∗→Rconsider\nthe expression\n(1)/bardblf/bardblFBL[E]= sup/braceleftBigm/summationdisplay\ni=1|f(x∗\ni)|:m∈N,(x∗\ni)m\ni=1⊂E∗,sup\nx∈BEm/summationdisplay\ni=1|x∗\ni(x)| ≤1/bracerightBig\n,\nand define H1[E] to be the space of positively homogeneous functions f:E∗→R\nsuch that /bardblf/bardblFBL[E]<∞, endowed with the pointwise order and lattice operations.\nIt is easy to see that H1[E] is a Banach lattice with the norm /bardbl·/bardblFBL[E].\nObserve that for each x∈E, the functions δx:E∗→Rgiven by δx(x∗) =x∗(x)\nforx∗∈E∗, can be used to define a linear isometry x/mapsto→δE(x) =δxfromEinto\nH1[E]. [5, Theorem 2.5] shows that the closed sublattice generated by δE(E) in\nH1[E], provides an explicit description of FBL[E], representing this as a space of\nfunctions on E∗.\nItisworthnotingthattheaboveconstructiondefinesafunctorf romthecategory\nof Banach spaces with bounded linear operators into that of Banac h lattices with\nlattice homomorphisms, and has been the object of intense resear ch in the last3\nfew years (see for instance [ 3,4,6,9,11,12] for different developments including\nrelations with projectivity, duality, complex scalars and p-convex Banach lattices,\namong others). Certain properties of a Banach space can be ident ified with a\ncorresponding analogue in Banach lattices via this functor (see in pa rticular [ 18,\nSection 10] for the basis of a dictionary between Banach space and Banach lattice\nproperties).\nThe paper is organized as follows: In the next section, we begin with a system-\natic study of the strong Nakano property showing its relations to o ther classical\npropertiesofBanachlattices(suchasexistence ofstrongunits, monotonicallycom-\nplete norms, or order continuity). We also provide some results abo ut stability of\nthe strong Nakano property under taking spaces of regular oper ators or sublattices\nand some facts about projections. In addition, it is observed that projective Ba-\nnach lattices have the strong Nakano property. This is a conseque nce of the fact\nthat the free Banach lattice FBL(A) =FBL[ℓ1(A)] has the strong Nakano prop-\nerty for every set A[5, Theorem 4.11]. In particular, the latter implies that free\nBanach lattices generated by a finite dimensional space Ehave the strong Nakano\nproperty with a constant depending on the Banach-Mazur distanc e between Eand\nℓ1(dim(E)). This motivates the question on the optimality of this constant. I n\nSection3, using ideas from the local theory of Banach spaces about estimat ing\nsumming norms of operators with few vectors, we will show that the re is a univer-\nsal constant λfinsuch that for every finite-dimensional Banach space E, the space\nFBL[E] has the λfin-strong Nakano property. We also show that Banach spaces\nEfor which FBL[E] has the strong Nakano property cannot contain subspaces\nisomorphic to L1norc0. Finally, in Section 4, we provide the details to see that\nthe freep-convex Banach lattice generated by ℓ1(A),FBL(p)[ℓ1(A)] always has the\nstrong Nakano property for p≥1.\nFor background on Banach lattices we refer the reader to the mon ographs [ 2,\n14,15] and for the specifics of free Banach lattices to [ 5] and [18].\n2.Basic facts\nIt is clear that every Banach lattice with the strong Nakano proper ty, has in\nparticular the Nakano property, and this in turn implies the Fatou pr operty. A\nsimple example of a Banach lattice with the Fatou property but not th e Nakano\nproperty is exhibited in [ 22, Example 2]. On the other hand, it is easy to check\nthatc0has the Nakano, but not the λ-strong Nakano property for any λ≥1\n(this can be witnessed by the summing basis). In fact, we will see nex t that the\nonly AM-spaces with the strong Nakano property are C(K) spaces. Recall that\nan AM-space is a Banach lattice whose norm satisfies\n/bardblx∨y/bardbl= max{/bardblx/bardbl,/bardbly/bardbl},\nfor every x,yinX+. A classical result due to S. Kakutani states that AM-spaces\nare always lattice isometric to a closed sublattice of some C(K), whereas in the4 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\ncase they have strong unit (an element esatisfying /bardblx/bardbl ≤1 if and only if |x| ≤e),\nthey have to be isometric to C(K) for some compact Hausdorff space K(cf. [14,\nTheorem 1.b.6]).\nProposition 2.1. LetXbe an AM-space. Xhas the strong Nakano property if\nand only if Xhas a strong unit.\nProof.Suppose Xis an AM-space with a strong unit. Then there exists a compact\nHausdorff space Ksuch that Xis lattice isometric to the space C(K) [2, Theorem\n4.29]. Since C(K) has the strong Nakano for every compact K(with constant\nfunctions playing the role of the corresponding upper bounds) the same holds for\nX.\nConversely, suppose an AM-space Xhas the strong Nakano property. The set\nB+={x∈X+:/bardblx/bardbl ≤1}is an upward directed norm bounded subset of X+.\nHence, there is e∈X+an upper bound of B+such that\n/bardble/bardbl= sup{/bardblb/bardbl,b∈B+}= 1.\nIn particular, |x| ≤ /bardblx/bardblefor every x∈X. Therefore, eis a strong unit of X./square\nA Banach lattice Xis monotonically complete (also known in the literature as\nhaving a Levi norm) if every norm-bounded upwards direct set A⊂X+has a\nsupremum in X. Note that a monotonically complete Banach lattice is always\nDedekind complete. Also note that dual Banach lattices, and KB-sp aces (those\nBanach lattices where every increasing norm bounded sequence co nverges) are\nmonotonically complete. For convenience, we will use the following ter minology:\nGivenλ≥1, a Banach lattice Xhas theλ-Fatou property if for every upwards\ndirected subset A⊂X+with supremum b, we have\n/bardblb/bardbl ≤λsup{/bardbla/bardbl:a∈A}.\nSimilarly, we will say Xhas theλ-Nakano property if for every upwards directed\norder bounded subset A⊂X+, we have\ninf{/bardblb/bardbl:ban upper bound for A} ≤λsup{/bardbla/bardbl:a∈A}.\nProposition 2.2. Suppose Xis a monotonically complete Banach lattice. For\neveryλ≥1, the following are equivalent:\n(1)Xhas theλ-Fatou property.\n(2)Xhas theλ-Nakano property.\n(3)Xhas theλ-strong Nakano property.\nProof.(1)⇒(2) Suppose that the norm on Xhas theλ-Fatou property. Let\nA⊂X+be an upwards directed order bounded subset. As Xis monotonically\ncomplete, Ahas a supremum y0inX. Hence, we get /bardbly0/bardbl ≤λsup{/bardbla/bardbl:a∈A}.\nLetBdenote the set of all upper bounds for A. It follows that\ninf{/bardblb/bardbl:b∈B} ≤ /bardbly0/bardbl ≤λsup{/bardbla/bardbl,a∈A}.5\nAs the converse inequality always holds, this finishes the proof.\n(2)⇒(3) LetAbe an upwards directed norm bounded subset of X+. SinceX\nis monotonically complete and has the λ-Nakano property, it follows that Ahas a\nsupremum yand we have\n/bardbly/bardbl ≤inf{/bardblb/bardbl:b∈B} ≤λsup{/bardbla/bardbl:a∈A},\nwhereBis the set of all upper bounds for A.\n(3)⇒(1) always holds. /square\nAs a consequence of Proposition 2.2and [15, Proposition 2.4.19] we have a fairly\nlarge class of Banach lattices with the strong Nakano property:\nCorollary 2.3. Every dual Banach lattice has the strong Nakano property.\nIt is well known that KB-spaces are in particular order continuous [ 15, Theorem\n2.4.2]. Within the class of Banach lattices with the strong Nakano prop erty, the\nconverse also holds:\nProposition 2.4. SupposeXis a Banach lattice. The following are equivalent:\n(1)Xis a KB-space.\n(2)Xhas order continuous norm and the strong Nakano property.\nProof.(1)⇒(2): LetAbe an upwards directed norm bounded set in X+. AsXis\naKB-spacethenby[ 15, Theorem2.4.2]itisaprojectionbandin X∗∗. ByCorollary\n2.3there isyinX∗∗\n+an upper bound of Asuch that /bardbly/bardbl= sup{/bardbla/bardbl:a∈A}. IfP\ndenotes the band projection onto X, it follows that Pyis an upper bound of Ain\nXwith/bardblPy/bardbl ≤ /bardbly/bardbl. HenceXhas the strong Nakano property. Order continuity\nfollows from [ 15, Theorem 2.4.2].\n(2)⇒(1): Let ( xn)n∈Nbe a monotone sequence in the unit ball of X. Without\nloss of generality, we may assume that ( xn)n∈Nis an increasing positive sequence.\nSinceXhas the strong Nakano property there is y0inX+an upper bound of\n(xn)n∈N. It follows from order continuity and [ 15, Theorem 2.4.2(iii)] that ( xn)n∈N\nis convergent. Hence, Xis a KB-space. /square\nLooking at general directed sets for the strong Nakano propert y is not needed\nwhen thedensity character of theBanachlatticeinquestion is small. Inparticular,\nwe have the following.\nLemma 2.5. LetXbe a separable Banach lattice and λ≥1.Xhas theλ-strong\nNakano property if and only if it has the λ-strongσ-Nakano property.\nProof.Let us prove the non-trivial implication. Suppose that Xhas theλ-strong\nσ-Nakano property. Let Abe an upwards directed norm bounded subset in X+.\nAsAis also separable then it contains a countable dense subset A0={a1,a2,...}.\nLet us consider the sequence\nyn=n/logicalordisplay\ni=1ai, n∈N.6 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nNote (yn)n∈Nis an increasing norm bounded sequence in X+. By the λ-strong\nσ-Nakano property there is u∈X+, an upper bound of ( yn)n∈N, hence of A0, such\nthat\n/bardblu/bardbl ≤λsup{/bardblyn/bardbl:n∈N} ≤λsup{/bardbla/bardbl:a∈A}.\nNow, for every a∈Athere is a sequence ( xn)n∈NinA0that converges in norm\ntoa. Hence, there is a subsequence ( xnk)k∈N⊂A0which converges in order to\na. Sincexnk≤uthena≤u, souis also an upper bound of Aand we have\n/bardblu/bardbl ≤λsup{/bardbla/bardbl:a∈A},soAhas theλ-strong Nakano property. /square\nThe previous equivalence does not remain true if the Banach lattice Xis not\nseparable as the following simple example shows.\nExample 2.6. Let\nX={x∈ℓ∞(R) :supp(x) is countable },\nwheresupp(x) denotes the support of x∈ℓ∞(R). Given an increasing norm\nbounded sequence ( xn) inX+, letM= supn/bardblxn/bardbland letA=/uniontext\nn∈Nsupp(xn). Let\nx∈ℓ∞(R) take the value MonAand 0 elsewhere. Clearly, xis an upper bound\nof (xn)n∈NinXwith sup {/bardblxn/bardbl∞:n∈N}=M=/bardblx/bardbl, soXhas the strong σ-\nNakano property.\nNow for each countable set A⊂RletxA=χAdenote the indicator function of\nA. Clearly, ( xA) is upwards directed (for the inclusion) and norm bounded in X+.\nHowever, if x∈ℓ∞(R) is an upper bound of ( xA) thenxmust be at least 1 on\nevery point of R. Thus,x /∈Xand this space fails the strong Nakano property.\n2.1.Stability of the strong Nakano property. Recall that if X,YareBanach\nlattices such that Yis Dedekind complete, then the space of regular operators\nLr(X,Y) is a Dedekind complete Banach lattice with the regular norm (cf. [ 2,\nSection4.4]). WecancharacterizewhenthisspacehasthestrongN akanoproperty:\nProposition 2.7. LetYbe a Dedekind complete Banach lattice. The following\nare equivalent:\n(1)Yhas the strong Nakano property.\n(2) For every Banach lattice X, the space of regular operators Lr(X,Y)has\nthe strong Nakano property.\n(3) For some Banach lattice X/\\e}atio\\slash={0}, the space of regular operators Lr(X,Y)\nhas the strong Nakano property.\nProof.(1)⇒(2): Suppose that Yhas the strong Nakano property and let Xbe\nany Banach lattice. Let ( Tα)α∈Abe an upwards directed norm bounded set in\nLr(X,Y)+. For every x∈X+, (Tαx)αis an upwards directed norm bounded set\ninY+. Hence, by the strong Nakano property of Y, (Tαx)αhas an upper bound in\nY. By Dedekind completeness, the supremum of ( Tαx)αexists in Y+. Let us call\nTxthis supremum and note that /bardblTx/bardbl= sup\nα/bardblTαx/bardbl,for every x∈X+.7\nNow, by [ 2, Theorem 1.19], the set ( Tα)αhas a supremum TinLr(X,Y) and\nT(x) = (sup\nαTα)(x) = sup{Tα(x) :α∈A}=Tx.\nMoreover,\n/bardblT/bardbl= sup\nx∈X+,/bardblx/bardbl≤1/bardblTx/bardbl= sup\nx∈X+,/bardblx/bardbl≤1/bardblTx/bardbl= sup\nx∈X+,/bardblx/bardbl≤1sup\nα/bardblTαx/bardbl= sup\nα/bardblTα/bardbl.\nHence,Lr(X,Y) has the strong Nakano property.\nThe implication (2) ⇒(3) is trivial.\n(3)⇒(1): Suppose that for some X/\\e}atio\\slash={0}, the space Lr(X,Y) has the strong\nNakano property. Let ( yα)αbe an upwards directed norm bounded set in Y+. Let\nx0∈X+with/bardblx0/bardbl= 1. By Hahn-Banach theorem we can take x∗\n0∈X∗\n+such that\nx∗\n0(x0) = 1 and /bardblx∗\n0/bardbl= 1. For each αletTαbe the rank-one operator\nTα(x) =x∗\n0(x)yα,forx∈X.\nClearly, ( Tα)αis an upwards directed norm bounded set in Lr(X,Y)+, so by the\nstrong Nakano property of Lr(X,Y) and [2, Theorem 1.19] there is TinLr(X,Y)+\nsuch that\nT(x) = sup\nαTα(x),forx∈X+,\nwith/bardblT/bardbl= sup\nα/bardblTα/bardbl.It follows then that\nsup\nαyα= sup\nαTα(x0) =T(x0)\nand\n/bardblT(x0)/bardbl ≤ /bardblT/bardbl= sup\nα/bardblTα/bardbl ≤sup\nα/bardblyα/bardbl.\nHence,Yhas the strong Nakano property. /square\nLemma 2.8. LetXbe a Banach lattice and let Ybe an order dense sublattice\nwith the strong Nakano property. Then Xhas the strong Nakano property.\nProof.Assume that Yhas the strong Nakano property. Let ( xα)αbe an upwards\ndirected norm bounded set in X+. Let us consider the set\nA={y∈Y: 0< y≤xαfor some α}.\nClearly,Ais an upwards directed (it is closed under finite suprema) norm bound ed\nsubset in Y+. Hence, there is b∈Y+which is an upper bound of Awith/bardblb/bardbl=\nsupy∈A/bardbly/bardbl. SinceYis order dense the conclusion follows. /square\nProposition 2.9. Every regular sublattice of a Banach lattice with the Fatou p rop-\nerty also has the Fatou property.8 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nProof.Suppose that Xhas the Fatou property. Let Ybe a regular sublattice of\nXand letAbe an upwards directed order bounded subset of Y+with supremum\nb.Ais also an upwards directed order bounded subset of X+andbis also the\nsupremum of AinX+(Yis a regular sublattice of X) so using that Xhas the\nFatou property we have /bardblb/bardbl= sup{/bardbla/bardbl:a∈A}. /square\nRemark 2.10. The previous proposition is not true if we replace the Fatou prop-\nerty by the strong Nakano property. For example, one can consid ercthe space\nof convergent sequences which has the strong Nakano property (as it is a C(K)\nspace). However, c0is a regular sublattice (even an ideal) of cthat fails the strong\nNakano property.\n2.2.Projections and the strong Nakano property.\nProposition 2.11. SupposeXis a Banach lattice with the strong Nakano property\nandY⊂Xis a closed sublattice complemented by a regular projection P, thenY\nhas also the /bardblP/bardblr-strong Nakano property.\nProof.LetYbe a sublattice of Xwhich is complemented by a regular projection\nP:X→Yand letAbe an upwards directed norm bounded subset of Y+. By the\nstrong Nakano property of X, there is binX+which is an upper bound of Asuch\nthat/bardblb/bardbl= sup{/bardbla/bardbl:a∈A}. Letc=|P|(b)∈Y. Note that for every a∈Awe\nhave\na=Pa≤ |P|a≤ |P|b=c,\nhence,cis also an upper bound of A. Clearly, we have\n/bardblc/bardbl=/bardbl|P|b/bardbl ≤ /bardblP/bardblrsup{/bardbla/bardbl:a∈A}.\n/square\nAs a consequence, every projection band in a Banach lattice with th e strong\nNakano property, must also have the strong Nakano property. H owever, this is no\nlonger true for arbitrary bands: Consider in C[0,1] the band\nX={f∈C[0,1] :f|[0,1\n2]= 0}.\nForn∈Nconsider the function fnsuch that fn|[0,1\n2]= 0,fn|[1\n2+1\nn,1]= 1 and fn\naffine on [1\n2,1\n2+1\nn]. Clearly, the sequence ( fn)n∈N⊂Xis increasing and norm\nbounded but has no upper bound in X.\nRecall that a Banach lattice Zis projective if for every Banach lattice X,J⊂X\na closed ideal, every lattice homomorphism T:Z→X/Jand every ε >0,\nthere exists a lattice homomorphism ˆT:Z→Xsuch that T=ˆT◦Q(where\nQ:X→X/Jdenotes the quotient map) and /bardblˆT/bardbl ≤(1+ε)/bardblT/bardbl.\nCorollary 2.12. Every projective Banach lattice has the 1+-strong Nakano prop-\nerty.9\nProof.By[19,Theorem10.3],forevery ε >0everyprojectiveBanachlatticeis(1+\nε)-isomorphictoasublattice of FBL(A), thefreebanachlatticegeneratedbysome\nsetA, and there is a lattice homomorphism projection P:FBL(A)→FBL(A)\nonto this sublattice with /bardblP/bardbl ≤1+ε. We know that FBL(A) =FBL[ℓ1(A)] ([5,\nCorollary 2.9.iii]) and FBL[ℓ1(A)] has the strong Nakano proprety for every set A\n([5, Theorem 4.11 ]). The conclusion follows from Proposition 2.11. /square\nNote that the converse of this last result is not true in general, as t here exist\nC(K) spaces which are not projective (cf. [ 19]).\nProposition 2.13. IfYis a Banach lattice with the strong Nakano property and\nYis an ideal in X, then it is a projection band.\nProof.[19, Proposition 12.3]. /square\nProposition 2.14. Suppose for every i∈I,Eiis a Banach lattice with the λi-\nstrong Nakano property. Then the direct sum ℓp(/producttext\ni∈IEi)has the (supi∈Iλi)-strong\nNakano property for 1≤p≤ ∞.\nProof.For every k∈I, let\nπk:ℓp(/producttext\ni∈IEi)−→Ek\n(xi)i∈I/mapsto−→xk\nbethecanonicalprojectiononto Ek, whichisalatticehomomorphism. Let( xα)α∈A\nbe an upwards directed norm bounded set in ℓp(/producttext\ni∈IEi)+. For every k∈I,\n(πk(xα))α∈Ais an upwards directed norm bounded set in ( Ek)+. Hence, by the\nλk-strong Nakano, there is bk∈Ekan upper bound of ( πk(xα))α∈Awith\n/bardblbk/bardblEk≤λksup\nα∈A/bardblπk(xα)/bardblEk.\nClearly,b= (bi)i∈Iis an upper bound of ( xα)α∈Aand\n/bardblb/bardbl=/parenleftBig/summationdisplay\ni∈I/bardblbi/bardblp/parenrightBig1\np≤/parenleftBig/summationdisplay\ni∈Iλp\nisup\nα∈A/bardblπi(xα)/bardblp/parenrightBig1\np\n≤sup\ni∈Iλi/parenleftBig/summationdisplay\ni∈Isup\nα∈A/bardblπi(xα)/bardblp/parenrightBig1\np= sup\ni∈Iλisup\nα∈A/parenleftBig/summationdisplay\ni∈I/bardblπi(xα)/bardblp/parenrightBig1\np\n= sup\ni∈Iλisup\nα∈A/bardblxα/bardbl\nwhere the identity on the second line follows from the KB-property o fℓp(I) (with\nobvious changes for the case p=∞). /square\n3.Free Banach lattices and the strong Nakano property\nIt was proved in [ 5] thatFBL[ℓ1(A)] has the strong Nakano property for every\nindex set A. Our motivation for the results in this section is to understand what\nconditions on a Banach space Eguarantee that FBL[E] has the strong Nakano10 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nproperty. Similarly, suppose FBL[E] has the strong Nakano property, what can\nwe say about the underlying E?\nWe will start looking at the simpler case of free Banach lattices gener ated by\nfinite dimensional spaces. Recall in this case that FBL[E] can be identified with\nan appropriate renorming of C(SE∗) the space of continuous functions on the unit\nsphere of E∗(see [18]). However, the constant in this renorming grows without\nbound when the dimension of Eincreases. This easily implies that, as long as Eis\nfinite dimensional, FBL[E] has the λE-strong Nakano property for an appropriate\nconstant λE. We will show next that actually there is a universal constant λvalid\nfor all finite dimensional E.\nLet us introduce the following notation.\nDefinition 3.1. GivenfinH1[E]andk∈N, let\n/bardblf/bardblFBLk[E]= sup/braceleftBigk/summationdisplay\ni=1|f(x∗\ni)|:x∗\n1,...,x∗\nk∈E∗,sup\nx∈BEk/summationdisplay\ni=1|x∗\ni(x)| ≤1/bracerightBig\n.\nObviously, for every f∈H1[E], (/bardblf/bardblFBLk[E])k∈Nis a non-decreasing sequence\nand we have\n/bardblf/bardblFBL[E]= sup\nk∈N/bardblf/bardblFBLk[E].\nThe following is a consequence of the work of [ 21] for estimating the summing\nnorm of finite dimensional operators with few vectors:\nLemma 3.2. There exist a universal constant λfinand a sequence of integers\n(km)m∈Nsuch that for every finite dimensional Banach space Ewe have:\n/bardblf/bardblFBL[E]≤λfin/bardblf/bardblFBLkdim(E)[E],\nfor every finH1[E].\nProof.Letf∈H1[E],n=dim(E) and pick x∗\n1,...,x∗\nN∈E∗(with arbitrary\nN∈N) such that\nsup\nx∈BEN/summationdisplay\nk=1|x∗\nk(x)| ≤1.\nSeta=/summationtextN\nk=1|f(x∗\nk)|.By [21, Lemma 6] there exists kna positive integer which\ndepends only on n, so that the statement of [ 21, Lemma 6] applied to E∗with\nwk=|f(x∗\nk\na)|, yieldsσ⊂ {1,...,N}and scalars ( µk)k∈σsuch that:\n(1)|σ| ≤kn,\n(2) supx∈BE/summationtext\nk∈σ|µkx∗\nk(x)| ≤1/C,\n(3)/summationtext\nk∈σwkµk≥C,\nwhereCis a universal constant.11\nIt follows that\nC2≤/summationdisplay\nk∈σCwkµk=/summationdisplay\nk∈σC|f(x∗\nk\na)|µk\n≤1\na/summationdisplay\nk∈σ|f(Cx∗\nk)||µk|=1\na/summationdisplay\nk∈σ|f(Cx∗\nk|µk|)| ≤1\na/bardblf/bardblFBLkn[E].\nThus,/summationtextN\nk=1|f(x∗\nk)|C2≤ /bardblf/bardblFBLkn[E]. Since ( x∗\nk)N\nk=1are arbitrary, setting λfin=\nC−2we get\n/bardblf/bardblFBL[E]≤λfin/bardblf/bardblFBLkn[E].\n/square\nLemma 3.3. LetEbe a Banach space of finite dimension. For every k∈N, every\nε >0and every f∈H1[E]+there isg∈FBL[E]such that\n(1)f≤g,\n(2)/bardblg/bardblFBLk[E]≤(1+ε)/bardblf/bardblFBLk[E].\nProof.Letδ >0 small enough to be chosen later. As Ehas finite dimension, SE∗\nis compact, so there exist y∗\n1,...,y∗\npinSE∗such that\nSE∗=p/uniondisplay\ni=1B(y∗\ni,δ),\nwhereB(y∗\ni,δ) ={x∗∈SE∗:/bardbly∗\ni−x∗/bardbl ≤δ}.\nBy Tietze’s Theorem, for every 1 ≤i≤pwe can find gi∈C(SE∗) satisfying\ngi= 0 on SE∗\\B(y∗\ni,2δ),g∗\ni= sup\nx∗∈B(y∗\ni,2δ)f(x∗) onB(y∗\ni,δ), and with values\nbetween these two elsewhere.\nLetg=p/logicalortext\ni=1gi. Clearly, we have g∈FBL[E] and 0≤f≤g. Indeed, for x∗on\nSE∗there isisuch that x∗∈B(y∗\ni,δ) and so\nf(x∗)≤sup\ny∗∈B(y∗\ni,2δ)f(y∗) =gi(x∗)≤g(x∗).\nNow, letx∗inSE∗. Bydefinition, thereis i∈ {1,...,p}suchthat g(x∗) =gi(x∗).\nWe want to show that there exists v∗such that\n(1)g(x∗)≤f(v∗)+δ,\n(2)/bardblx∗−v∗/bardbl ≤4δ.\nIndeed, we have g(x∗) =gi(x∗)≤sup\ny∗∈B(y∗\ni,2δ)f(y∗), so one can find v∗inB(y∗\ni,2δ)\nwithg(x∗) =gi(x∗)≤f(v∗)+δ. As the case g(x∗) = 0 is trivial, we can assume\nthatg(x∗) =gi(x∗)>0, which implies x∗is inB(y∗\ni,2δ), so we get /bardblx∗−v∗/bardbl ≤4δ.\nNow, we claim that for k∈N,ε >0, choosing δ >0 small enough we have that\n/bardblg/bardblFBLk[E]≤(1+ǫ)/bardblf/bardblFBLk[E].12 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nTo check this, fix x∗\n1,...,x∗\nkinE∗satisfying sup\nx∈BEk/summationtext\ni=1|x∗\ni(x)| ≤1.For each 1 ≤i≤\nk, choose as above v∗\nisuch that /bardblx∗\ni\n/bardblx∗\ni/bardbl−v∗\ni/bardbl ≤4δandg(x∗\ni\n/bardblx∗\ni/bardbl)≤f(v∗\ni)+δ. Let\nt∗\ni=/bardblx∗\ni/bardblv∗\ni.\nWe have that\nk/summationdisplay\ni=1g(x∗\ni) =k/summationdisplay\ni=1/bardblx∗\ni/bardblg/parenleftbiggx∗\ni\n/bardblx∗\ni/bardbl/parenrightbigg\n≤k/summationdisplay\ni=1/bardblx∗\ni/bardbl(f(v∗\ni)+δ) =k/summationdisplay\ni=1(f(t∗\ni)+/bardblx∗\ni/bardblδ)\n≤k/summationdisplay\ni=1f(t∗\ni)+kδ≤ /bardblf/bardblFBLk[E]sup\nx∈BEk/summationdisplay\ni=1|t∗\ni(x)|+kδ\n≤ /bardblf/bardblFBLk[E](4kδ+1)+kδ.\nForǫ >0 andk∈Nwe can choose δsuch that /bardblg/bardblFBLk[E]≤(1+ǫ)/bardblf/bardblFBLk[E].\n/square\nTheorem 3.4. For every finite dimensional Banach space E,FBL[E]has the\n(λfin)+-strong Nakano property (where λfinis the universal constant coming from\nLemma3.2).\nProof.By Lemma 2.5it is enough to check the strong Nakano property for se-\nquences. Let ( fn)n∈Nbe an increasing norm bounded sequence in FBL[E]+with\nsupn∈N/bardblfn/bardblFBL[E]= 1. Define fonE∗byf(x∗) := supn∈N{fn(x∗)}for every x∗\ninE∗. By [5, Lemma 4.2] we have f∈H1[E] with/bardblf/bardblFBL[E]= 1.\nBy Lemma 3.3there isgis inFBL[E] such that f≤gand/bardblg/bardblFBLkdim(E)[E]≤\n(1+ε)/bardblf/bardblFBLkdim(E)[E].\nTherefore, gis an upper bound of ( fn)n∈Nand by Lemma 3.2we have\n/bardblg/bardblFBL[E]≤λfin/bardblg/bardblFBLkdim(E)[E]\n≤(1+ǫ)λfin/bardblf/bardblFBL[E]= (1+ǫ)λfin.\nThis completes the proof. /square\nThe rest of this section will be devoted to show some conditions that a Banach\nspaceEmust satisfy in case FBL[E] has the strong Nakano property.\nLet us first recall that a subspace Fof a Banach space Eis called an ideal\n(cf. [10]) if the subspace\nF⊥={x∗∈E∗:x∗(y) = 0 for y∈F}\nis the kernel of a contractive projection on E∗. This notion should not be confused\nwith that of an ideal in a Banach lattice.13\nProposition 3.5. SupposeEis a Banachspacesuch that FBL[E]has theλ-strong\nNakano property for some λ. ThenEcannot contain a subspace isomorphic to c0.\nProof.Suppose that j:c0→Eis an isomorphic embedding. By [ 18, Corollary\n4.12] the extension j:FBL[c0]→FBL[E] is a lattice isomorphic embedding.\nBy [3, Theorem 4.1], there is a lattice embedding α:c0→FBL[c0] such that\nβα=idc0, whereβ=/hatwidestidc0:FBL[c0]→c0is the unique lattice homomorphism\nextending the identity on c0,idc0.\nLetsndenote the summing basis in c0, andfn=jαsn∈FBL[E]. Clearly,\n(fn)n∈Nis an increasing sequence with supn∈N/bardblfn/bardbl ≤ /bardblα/bardbl/bardblj/bardbl. SinceFBL[E] has\ntheλ-strong Nakano property, there is f∈FBL[E] such that fn≤ffor every\nn∈Nand/bardblf/bardbl ≤λ/bardblα/bardbl/bardblj/bardbl.\nUsing the density of FVL[E] =lat(δE(E)) inFBL[E], we can find a separable\nsubspace F⊂Esuch that f∈FBL[F]. Enlarging Fif needed we can assume\nwithout loss of generality that j(c0)⊂Fand that Fis an ideal in E(the latter\ndue to [20]). In particular, by [ 18, Corollary 4.14] we have /bardblf/bardblFBL[F]=/bardblf/bardblFBL[E].\nBy Sobczyk’s theorem there is a projection P:F→j(c0). Let us consider\nx=βj−1Pf∈c0. Observe that for every n∈Nwe have\nsn=βαsn=βj−1Pjαsn=βj−1Pfn≤βj−1Pf=x.\nThis is a contradiction as xwould be larger than the constant 1 sequence, hence\nx /∈c0. /square\nIt was also shown in [ 5, Theorem 4.13] that FBL[L1[0,1]] fails the Fatou prop-\nerty, hence strong Nakano. We can improve this further as follows :\nTheorem 3.6. IfFBL[E]has theλ-Fatou property for some λ, thenEdoes not\ncontain a closed subspace isomorphic to L1[0,1].\nProof.For convenience let L1=L1[0,1]. Let us first assume Eis separable.\nSuppose that Econtains a closed subspace isomorphic to L1. Hence, let T:L1→\nEbe an operator such that T(L1) =Fis isomorphic to L1. Without loss of\ngenerality we can assume that /bardblT/bardbl= 1.\nFor each n∈N, defineIn,j:= [j−1\n2n,j\n2n] for allj∈ {1,...,2n}and setyn,j:=\nT(χIn,j).\nWe define fn=2n/summationtext\ni=1|δyn,j| ∈FBL[E]+which satisfy\n/bardblfn/bardblFBL[E]≤2n/summationdisplay\ni=1/bardblδyn,j/bardbl=2n/summationdisplay\ni=1/bardblyn,j/bardbl ≤2n/summationdisplay\ni=1/bardblχIn,j/bardblL1= 1.\nIt is easy to check (see [ 5, Lemma 4.12]) that fn≤fn+1and for every y∗inE∗\nlim\nn→∞fn(y∗) =/bardblT∗y∗/bardblL1.14 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nNow, fix any g∈FBL[E]+with/bardblg/bardblFBL[E]>1 and let ˜fn:=g∧fn. The\nsequence ˜fnis increasing , bounded above by gand satisfies\n/bardbl˜fn/bardblFBL[E]≤ /bardblfn/bardblFBL[E]≤1.\nWe will show that supn∈N˜fn=ginFBL[E].\nTo this end, let ( rj)j∈Ndenote the sequence of Rademacher functions, which\nconverges to 0 in the w∗-topology of L∗\n1=L∞. Thus, we can define S0:L1→c0\nby\nS0(f) =/parenleftbigg/integraldisplay\nrnfdµ/parenrightbigg\nn∈N.\nNoteS0◦T−1is an operator from F⊂Etoc0and, asEis separable, by Sobczyk’s\ntheorem there is an extension S:E→c0withS|F=S0T−1. Hence, we can\nconsider a sequence (˜ rj)j∈N⊂E∗such that S(f) = (/a\\}bracketle{t˜rn,f/a\\}bracketri}ht)n∈Nfor every finE.\nIn other words, (˜ rj)j∈Nisw∗-null inE∗.\nGivenhinE∗, letK=/bardblg/bardblFBL[E]/bardblh/bardbl+/bardblT∗h/bardblL1+1, and define hj=h+K˜rj.\nThe sequence ( hj) isw∗-convergent to hinE∗. Since gisw∗- continuous on\nbounded sets, it follows that g(hj)→g(h) asj→ ∞. Therefore, we can find\nj0∈Nsuch that for every j≥j0we have\n(2) g(hj)≤g(h)+1≤ /bardblg/bardblFBL[E]/bardblh/bardbl+1≤K−/bardblT∗h/bardblL1.\nWe claim that for j≥j0,g(hj)≤ /bardblT∗hj/bardblL1. Indeed, we have\n/bardblT∗hj/bardblL1=/bardblT∗h+KT∗˜rj/bardblL1≥K/bardblT∗˜rj/bardblL1−/bardblT∗h/bardblL1\nand by construction of ˜ rjwe have\n/bardblT∗˜rj/bardblL1≥ /a\\}bracketle{tT∗˜rj,rj/a\\}bracketri}ht=/a\\}bracketle{t˜rj,Trj/a\\}bracketri}ht=/a\\}bracketle{trj,T−1(Trj)/a\\}bracketri}ht=/a\\}bracketle{trj,rj/a\\}bracketri}ht= 1.\nHence, by ( 2) we get\n/bardblT∗hj/bardblL1≥K−/bardblT∗h/bardblL1≥g(hj),\nas claimed. Now let φ∈FBL[E] be any upper bound of ( ˜fn)n∈N. Forj≥j0, we\nhave\nφ(hj)≥supn∈N˜fn(hj) =g(hj)∧supn∈Nfn(hj) =g(hj)∧/bardblT∗hj/bardblL1=g(hj).\nThus, by the w∗continuity of gandφwe have g(h)≤φ(h) for every h∈E∗.\nTherefore, g= supn∈N˜fnand we get a contradiction because /bardblg/bardblFBL[E]>1 was\narbitrarily large but /bardbl˜fn/bardblFBL[E]≤1.\nTo finish the proof, let us assume now that Eis non-separable and contains\na closed subspace isomorphic to L1. By [20], there is F⊂Ea separable ideal\ncontaining a subspace isomorphic to L1. Note that by [ 18]FBL[F] is a regular\nsublattice of FBL[E], so ifFBL[E] had the λ-Fatou property, using Proposition\n2.9,FBL[F] also would have the λ-Fatou property. This is a contradiction with\nthe first part of the proof. /square15\n4.Strong Nakano of FBL(p)[ℓ1(A)]\nThe purpose of this section is to show that the proof of [ 5, Theorem 4.11] that\nFBL[ℓ1(A)] has the strong Nakano property can actually be adapted for th e free\np-convex Banach lattice FBL(p)[ℓ1(A)] with minor adjustments. We have decided\nto include details here as it will be convenient for future reference.\nAs in [12], given a Banach space E, letHp[E] denote the set of positively ho-\nmogeneous functions on E∗equipped with the norm\n(3)\n/bardblf/bardblFBL(p)[E]= sup/braceleftBig/parenleftBigm/summationdisplay\ni=1|f(x∗\ni)|p/parenrightBig1\np:m∈N,(x∗\ni)m\ni=1⊂E∗,sup\nx∈BEm/summationdisplay\ni=1|x∗\ni(x)|p≤1/bracerightBig\n,\nand the pointwise order and lattice operations. The free p-convex Banach lattice\ngenerated by Ecan be identified with the closed sublattice of Hp[E] generated\nby the evaluation functionals δx:E∗→Rgiven by δx(x∗) =x∗(x) forx∈E,\nx∗∈E∗. This is in the sense that for every p-convex Banach lattice Xand every\noperator T:E→Xthere is a unique lattice homomorphisms ˆT:FBL(p)[E]→X\nextending T(see [12] for details).\nIt should be noted that in the case p=∞,FBL(∞)[E] is always an AM-space,\nso by Proposition 2.1and [18, Proposition 9.1] we get that FBL(∞)[E] has the\nStrong Nakano property precisely when Eis finite dimensional.\nLemma 4.1. LetEbe a Banach space and let Abe an upwards directed and\npointwise bounded set in Hp[E]. Define g:E∗→R+by\ng(x∗) := sup{f(x∗) :f∈A}forx∗∈E∗.\nIt follows that g∈Hp[E]+with\n/bardblg/bardblFBL(p)[E]= sup{/bardblf/bardblFBL(p)[E]:f∈A}.\nProof.Clearly, gis positively homogeneous and /bardblg/bardblFBL(p)[E]≥ /bardblf/bardblFBL(p)[E]for\neveryf∈A. To prove the converse inequality /bardblg/bardblFBL(p)[E]≤sup{/bardblf/bardblFBL(p)[E]:\nf∈A}:=αwe can assume without loss of generality that the supremum is\nfinite. Fix ǫ >0 and take any x∗\n1,...,x∗\nn∈E∗such that/summationtextn\nk=1|x∗\nk(x)|p≤1\nfor every x∈BE. SinceAis upwards directed, we can find f∈Asuch that\ng(x∗\nk)p−ǫ\nn≤f(x∗\nk)pfor allk∈ {1...,n}, therefore\nn/summationdisplay\nk=1g(x∗\nk)p≤ǫ+n/summationdisplay\nk=1f(x∗\nk)p≤ǫ+αp.\nIt follows that /bardblg/bardblp\nFBL(p)[E]≤ǫ+αp. Asǫ >0 is arbitrary, /bardblg/bardblFBL(p)[E]≤α./square\nDefinition 4.2. LetEbe a Banach space. We say that f∈Hp[E]+ismaximal if\n/braceleftbig\ng∈Hp[E]+:g≥f,/bardblg/bardblFBL(p)[E]=/bardblf/bardblFBL(p)[E]/bracerightbig\n={f}.16 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nA standard application of Zorn’s lemma allows us to prove the following ( see [5,\nLemma 4.4]):\nLemma 4.3. LetEbe a Banach space and f∈Hp[E]+. Then there exists a\nmaximal ˜f∈Hp[E]+such that f≤˜fand/bardblf/bardblFBL(p)[E]=/bardbl˜f/bardblFBL(p)[E].\nLemma 4.4. LetAbe a non-empty set and let f∈Hp[ℓ1(A)]+be maximal.\n(i)f(x∗)≤f(y∗)whenever x∗,y∗∈ℓ∞satisfy|x∗| ≤ |y∗|.\n(ii)/parenleftbign/summationtext\nk=1(f(x∗\nk))p/parenrightbig1\np≤f/parenleftbig\n(n/summationtext\nk=1(x∗\nk)p/parenrightbig1\np)for every n∈Nandx∗\n1,...,x∗\nn∈(ℓ∞)+.\n(iii)/bardblf/bardblFBL(p)[E]=/bardblf/bardbl∞.\nProof.For anyz∗∈ℓ∞letR(z∗) :={λz∗:λ >0} ⊂ℓ∞.\n(i): We argueby contradiction. Suppose that f(x∗)> f(y∗) and define g:ℓ∞→\nR+by/braceleftBigg\ng(λy∗) :=f(λx∗) for all λ >0,\ng(z∗) :=f(z∗) for all z∗∈ℓ∞\\R(y∗).\nIt is easy to check that gis positively homogeneous. Since f(x∗)> f(y∗), we\nhavef≤gandf/\\e}atio\\slash=g. Bearing in mind that fis maximal, in order to get a\ncontradiction it suffices to check that /bardblg/bardblFBL(p)[E]=/bardblf/bardblFBL(p)[E]. To this end,\ntake any x∗\n1,...,x∗\nn∈ℓ∞such that/summationtextn\nk=1|x∗\nk(a)|p≤1 for alla∈A. LetIbe the\nset of those k∈ {1,...,n}such that x∗\nk/\\e}atio\\slash∈R(y∗) and let J:={1,...,n} \\I, so\nthat for each k∈Jwe havex∗\nk=λky∗for some λk>0. Note that for every a∈A\nwe have/summationdisplay\nk∈I|x∗\nk(a)|p+/summationdisplay\nk∈J|λkx∗(a)|p=/summationdisplay\nk∈I|x∗\nk(a)|p+|x∗(a)|p/summationdisplay\nk∈Jλp\nk\n≤/summationdisplay\nk∈I|x∗\nk(a)|p+|y∗(a)|p/summationdisplay\nk∈Jλp\nk\n=n/summationdisplay\nk=1|x∗\nk(a)|p≤1.\nHence,\nn/summationdisplay\nk=1g(x∗\nk)p=/summationdisplay\nk∈If(x∗\nk)p+/summationdisplay\nk∈Jf(λkx∗)p≤ /bardblf/bardblp\nFBL(p)[ℓ1(A)].\nThis shows that /bardblg/bardblFBL(p)[ℓ1(A)]≤ /bardblf/bardblFBL(p)[ℓ1(A)], which is a contradiction.\n(ii): Setx∗:= (/summationtextn\nk=1(x∗\nk)p)1\np. By contradiction, suppose otherwise that f(x∗)<\n(/summationtextn\nk=1(f(x∗\nk))p)1\np. Defineg:ℓ∞→R+by\n/braceleftBigg\ng(λx∗) :=λ(/summationtextn\nk=1(f(x∗\nk))p)1\npfor allλ >0,\ng(z∗) :=f(z∗) for all z∗∈ℓ∞\\R(x∗).17\nClearly,gis positively homogeneous, f≤gandf/\\e}atio\\slash=g. Again by the maximality\noff, togetacontradictionitsuffices toshowthat /bardblg/bardblFBL(p)[ℓ1(A)]=/bardblf/bardblFBL(p)[ℓ1(A)].\nTakey∗\n1,...,y∗\nm∈ℓ∞such that/summationtextm\nj=1|y∗\nj(a)|p≤1 for alla∈A. LetIdenote the\nset of all j∈ {1,...,m}for which y∗\nj/\\e}atio\\slash∈R(x∗) and let J:={1,...,m}\\I, so that\nfor eachj∈Jwe can write y∗\nj=λjx∗for some λj>0. Setµ:=/summationtext\nj∈Jλp\nj. Since\n/summationdisplay\nj∈I|y∗\nj(a)|p+n/summationdisplay\nk=1|(µ)1\npx∗\nk(a)|p=/summationdisplay\nj∈I|y∗\nj(a)|p+µ(x∗(a))p\n=/summationdisplay\nj∈I|y∗\nj(a)|+/summationdisplay\nj∈J(λjx∗(a))p\n=m/summationdisplay\nj=1|y∗\nj(a)|p≤1,\nfor every a∈A, we obtain\nm/summationdisplay\nj=1g(y∗\nj)p=/summationdisplay\nj∈If(y∗\nj)p+/summationdisplay\nj∈Jλp\nj/parenleftBign/summationdisplay\nk=1f(x∗\nk)p/parenrightBig\n=/summationdisplay\nj∈If(y∗\nj)p+n/summationdisplay\nk=1f((µ)1\npx∗\nk)≤ /bardblf/bardblp\nFBL(p)[ℓ1(A)].\nIt follows that /bardblg/bardblFBL(p)[ℓ1(A)]≤ /bardblf/bardblFBL(p)[ℓ1(A)], which is a contradiction.\n(iii): We have /bardblf/bardblFBL(p)[ℓ1(A)]≥ /bardblf/bardbl∞. To prove the equality, take finitely\nmanyx∗\n1,...,x∗\nn∈ℓ∞such that/summationtextn\nk=1|x∗\nk(a)|p≤1 for every a∈A. Thenx∗:=\n(/summationtextn\nk=1|x∗\nk|p)1\np∈Bℓ∞and\nn/summationdisplay\nk=1f(x∗\nk)p(i)\n≤n/summationdisplay\nk=1f(|x∗\nk|)p(ii)\n≤f(x∗)p≤ /bardblf/bardblp\n∞.\nThis shows that /bardblf/bardblFBL[ℓ1(A)]≤ /bardblf/bardbl∞and finishes the proof. /square\nLemma 4.5. LetAbe a non-empty set and let φ:ℓ∞→Rbe a linear functional.\nDefinegφ:ℓ∞→R+by\ngφ(x∗) :=|φ(|x∗|p)|1\npfor allx∗∈ℓ∞.\nThengφ∈Hp[ℓ1(A)]+and\n/bardblgφ/bardblFBL(p)[ℓ1(A)]= sup/braceleftbig\n|φ(x∗)|1\np:x∗∈Bℓ∞/bracerightbig\n.\nProof.Clearly,gφis positively homogeneous. Take any x∗\n1,...,x∗\nn∈ℓ∞such that\nsupa∈A/summationtextn\nk=1|x∗\nk(a)|p≤1. For each k∈ {1,...,n}, letεk∈ {−1,1}be the sign of18 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nφ(|x∗\nk|p). Then/summationtextn\nk=1εk|x∗\nk|p∈Bℓ∞and so\nn/summationdisplay\nk=1gφ(x∗\nk)p=n/summationdisplay\nk=1εkφ(|x∗\nk|p) =φ/parenleftBiggn/summationdisplay\nk=1εk|x∗\nk|p/parenrightBigg\n≤sup/braceleftbig\n|φ(x∗)|:x∗∈Bℓ∞/bracerightbig\n=:αp.\nThis immediately shows that /bardblgφ/bardblFBL(p)[ℓ1(A)]≤α.\nFor the converse, pick x∗∈Bℓ∞and write it as x∗= (x∗)+−(x∗)−, the dif-\nference of its positive and negative parts. Since |((x∗)+)1\np(a)|p+|((x∗)−)1\np(a)|p=\n|(x∗)+(a)|+|(x∗)−(a)|=|x∗(a)| ≤1 for alla∈A, then we get\n|φ((x∗))| ≤ |φ(((x∗)+)1\np)p)|+|φ(((x∗)−)1\np)p)|\n=gφ(((x∗)+)1\np)p+gφ(((x∗)−)1\np)p\n≤ /bardblgφ/bardblp\nFBL(p)[ℓ1(A)].\nThis proves that α≤ /bardblgφ/bardblFBL(p)[ℓ1(A)]. /square\nLemma 4.6. LetAbe a non-empty set and let f∈Hp[ℓ1(A)]+be maximal. Then\nthere exists φ∈ℓ∞such that f=gφ.\nProof.The case f= 0 being trivial, we can suppose without loss of generality that\n/bardblf/bardblFBL(p)[ℓ1(A)]= 1. The set\nC:={x∗∈(ℓ∞)+:f((x∗)1\np)>1}\nis convex as a consequence of Lemma 4.4(ii). Let Ube the open unit ball of ℓ∞.\nSince/bardblf/bardbl∞=/bardblf/bardblFBL(p)[ℓ1(A)]= 1 (Lemma 4.4(iii)), we have C∩U=∅. As an\napplication of the Hahn-Banach separation theorem (cf. [ 7, Proposition 2.13(ii)]),\nthere isφ∈(ℓ∞)∗such that\n(4) φ(y∗)0. Fixt >1. By Lemma 4.4(i), we\nhavef(|x∗|)≥f(x∗)>0 and so\nf/parenleftbiggt\nf(|x∗|)|x∗|/parenrightbigg\n=t >1.\nTherefore/parenleftBig\nt\nf(|x∗|)|x∗|/parenrightBigp\n∈Cand (4) yields\nφ/parenleftBig/parenleftbiggt\nf(|x∗|)|x∗|/parenrightbiggp/parenrightBig\n≥sup{|φ(y∗)|:y∗∈U}=/bardblφ/bardbl= 1.\nWe conclude that ( tφ(|x∗|))p≥f(|x∗|)pfor anyt >1, so (φ(|x∗|)p≥fp(|x∗|)≥\nf(x∗)pand then gφ(|x∗|) = (φ(|x∗|p))1\np≥f(x∗). The proof is complete. /square19\nGiven any non-empty set A, it is well-known that every φ∈ℓ∗\n∞can be written\nin a unique way as φ=φ0+φ1, where\n•φ0∈ℓ1(A) (identified as a subspace of ℓ∗\n∞),\n•φ1∈ℓ∗\n∞vanishes on all finitely supported elements of ℓ∞.\nMoreover, /bardblφ/bardbl=/bardblφ0/bardbl+/bardblφ1/bardbl.\nLemma 4.7. LetAbe a non-empty set and φ∈ℓ1(A). Thengφ∈FBLp[ℓ1(A)].\nProof.Let (ea)a∈Abe the unit vector basis of ℓ1(A). SetB=supp(φ) and note\nthat forx∗∈ℓ∞(A)\ngφ(x∗) =/vextendsingle/vextendsingle/vextendsingle/summationdisplay\na∈Bφ(a)|x∗(a)|p/vextendsingle/vextendsingle/vextendsingle1\np.\nFor every finite subset S⊂B, we have that\nfS:=/vextendsingle/vextendsingle/vextendsingle/summationdisplay\na∈Sφ(a)|δa|p/vextendsingle/vextendsingle/vextendsingle1\np∈FBLp[ℓ1(A)].\nLetx∗\n1,...,x∗\nn⊂ℓ∞such that supa∈A/summationtextn\nj=1|x∗\nj(a)|p≤1. We have that\nn/summationdisplay\nj=1|gφ(x∗\nj)−fS(x∗\nj)|p=n/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\na∈Bφ(a)|x∗\nj(a)|p/vextendsingle/vextendsingle/vextendsingle1\np−/vextendsingle/vextendsingle/vextendsingle/summationdisplay\na∈Sφ(a)|x∗\nj(a)|p/vextendsingle/vextendsingle/vextendsingle1\np/vextendsingle/vextendsingle/vextendsinglep\n≤n/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/summationdisplay\na∈B\\Sφ(a)|x∗\nj(a)|p/vextendsingle/vextendsingle/vextendsingle\n≤/summationdisplay\na∈B\\S|φ(a)|.\nHence,/bardblgφ−fS/bardblFBL(p)[ℓ1(A)]≤/parenleftBig/summationtext\na∈B\\S|φ(a)|/parenrightBig1\np. Since φ∈ℓ1(A) forε >0\nwe can take finite S⊂Bsuch that/summationtext\na∈B\\S|φ(a)|< ε. Therefore, we have\ngφ∈FBLp[ℓ1(A)]. /square\nLemma 4.8. LetAbe a non-empty set, ξ:Bℓ∞→Raw∗-continuous function\nandφ∈ℓ∗\n∞. Ifξ≤gφonBℓ∞, thenξ≤gφ0onBℓ∞as well.\nProof.Thew∗-topology on Bℓ∞= [−1,1]Aagrees with the pointwise topology.\nSince the map x∗/mapsto→ |x∗|pisw∗-continuous when restricted to Bℓ∞andφ0is\nw∗-continuous, we have that gφ0isw∗-continuous on Bℓ∞. On the other hand,\nifx∗∈Bℓ∞is finitely supported, then φ1(|x∗|p) = 0 and, therefore, we have\nξ(x∗)≤gφ(x∗) =|φ0(|x∗|p)|1\np=gφ0(x∗). Since the finitely supported elements\nofBℓ∞arew∗-dense and the functions ξandgφ0arew∗-continuous on Bℓ∞, we\nconclude that ξ≤gφ0onBℓ∞. /square\nWe arrive at the main result of this section:20 Y. AZOUZI, A. BEN RJEB, AND P. TRADACETE\nTheorem 4.9. The norm of FBLp[ℓ1(A)]has the strong Nakano property for any\nnon-empty set A.\nProof.LetA⊂FBLp[ℓ1(A)]+be an upwards directed family such that\nsup{/bardblf/bardblFBLp[ℓ1(A)]:f∈A}= 1.\nWe are going to show that Ahas an upper bound of norm 1.\nNote that Ais pointwise bounded (because /bardblf/bardbl∞:={|f(x∗)|:x∗∈BE∗/bracerightbig\n≤\n/bardblf/bardblFBL(p)[E]) and let h:ℓ∞→R+be defined as h(x∗) := sup{f(x∗) :f∈A}\nfor allx∗∈ℓ∞. Lemma 4.1ensures that h∈Hp[ℓ1(A)]+and/bardblh/bardblFBLp[ℓ1(A)]= 1.\nNow letg∈Hp[ℓ1(A)]+be maximal such that g≥hand/bardblg/bardblFBLp[ℓ1(A)]= 1 (apply\nLemma4.3). Then g=gφfor some φ∈ℓ∞with/bardblφ/bardbl= 1 (combine Lemmas 4.5\nand4.6).\nGiven any f∈A⊂FBLp[ℓ1(A)], we have f≤gφand the restriction f|Bℓ∞is\nw∗-continuous (cf. [ 18, Remark 2.5]), hence Lemma 4.8yieldsf(x∗)≤gφ0(x∗) for\neveryx∗∈ℓ∞(bear in mind that both fandgφ0are positively homogeneous).\nSincegφ0∈FBLp[ℓ1(A)] (Lemma 4.7) and\n1 =/bardblφ/bardbl ≥ /bardblφ0/bardbl ≥ /bardblgφ0/bardblp\nFBL[ℓ1(A)]\n(by Lemma 4.5), it turns out that gφ0is the upper bound of AinFBLp[ℓ1(A)]\nthat we were looking for. The proof is finished. /square\nAcknowledgements\nThisresearchhasbeenfundedbyCSICi-COOPprogramundergran tCOOPB20617.\nResearch of P. Tradacete partially supported by grants PID2020 -116398GB-I00\nand CEX2019-000904-S funded by MCIN/AEI/10.13039/5011000 11033, as well as\nby a 2022 Leonardo Grant for Researchers and Cultural Creator s, BBVA Founda-\ntion.\nReferences\n[1] Y. A. Abramovich, C. D. Aliprantis, An invitation to operator theory . Graduate Studies in\nMathematics. 50. American Mathematical Society, 2002.\n[2] C. D. Aliprantis, O. Burkinshaw, Positive Operators . Springer, Dordrecht (2006). Reprint\nof the 1985 original.\n[3] A. Avil´ es, G. Mart´ ınez-Cervantes, J. Rodr´ ıguez, and P. Tr adacete, A Godefroy-Kalton\nprinciple for free Banach lattices . Israel J. Math. 247(2022), 433–458.\n[4] A. Avil´ es, G. Mart´ ınez-Cervantes, and J. D. Rodr´ ıguez-Ab ell´ an,On projective Banach\nlattices of the form C(K)andFBL[E]. J. Math. Anal. Appl. 489(2020), no. 1, 124129,\n11 pp.\n[5] A. Avil´ es, J. Rodr´ ıguez, P. Tradacete, The free Banach lattice generated by a Banach space .\nJ. Funct. Anal. 274(2018), no. 10, 2955–2977.\n[6] S. Dantas, G. Mart´ ınez-Cervantes, J. D. Rodr´ ıguez Abell´ an , and A. Rueda Zoca, Norm-\nattaining lattice homomorphisms . Rev. Mat. Iberoam. 38(2022), no. 3, 981–1002.21\n[7] M. Fabian, P. Habala, P. H´ ajek, V. Montesinos, and V. Zizler, Banach space theory.\nThe basis for linear and nonlinear analysis , CMS Books in Mathematics/Ouvrages de\nMath´ ematiques de la SMC, Springer, New York, 2011.\n[8] E.Garc´ ıa-S´ anchez,D.deHevia, P.Tradacete, Free objects in Banach space theory . Cutting-\nedge Mathematics: Biennial Conferenceofthe SpanishRoyalMath ematicalSociety. RSME\nSpringer series 2024.\n[9] E. Garc´ ıa-S´ anchez, P. Tradacete, Free dual spaces and free Banach lattices . J. Math. Anal.\nAppl.532(2024) 127931.\n[10] G. Godefroy, N. J. Kalton, P. D. Saphar, Unconditional ideals in Banach spaces. Studia\nMath.104, 13–59 (1993).\n[11] D. de Hevia and P. Tradacete, Free complex Banach lattices . J. Funct. Anal. 284(2023),\nno. 10, Paper No. 109888, 26 pp.\n[12] H. Jard´ on-S´ anchez, N. J. Laustsen, M. A. Taylor, P. Trad acete, and V. G. Troitsky, Free\nBanach lattices under convexity conditions . Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A\nMat. RACSAM 116(2022), no. 1, Paper no. 15.\n[13] N. J. Laustsen and P. Tradacete, Banach lattices of homogeneous functions associated to\na Banach space . (Forthcoming).\n[14] J. Lindenstrauss, L. Tzafriri. Classical Banach spaces II . Springer-Verlag, Berlin, 1977.\n[15] P. Meyer-Nieberg, Banach lattices . Universitext, Springer-Verlag, Berlin, 1991.\n[16] H. Nakano, ¨Uber die Charakterisierung des allgemeinen C-Raumes . Proc. Imp. Acac.\nTokyo,17(1941) 301–307.\n[17] H. Nakano, ¨Uber normierte teilweisegeordnete Moduln . Proc. Imp. Acac. Tokyo, 17(1941)\n311–317.\n[18] T. Oikhberg, M. A Taylor, P. Tradacete and V. G. Troitsky, Free Banach lattices . J. Eur.\nMath. Soc., (in press).\n[19] B. de Pagter, A. W. Wickstead, Free and projective Banach lattices . Proc. Roy. Soc. Edin-\nburgh Sect. A 145(2015), no. 1, 105–143.\n[20] B. Sims, D. Yost, Linear Hahn-Banach extension operators . Proc. Edinburgh Math. Soc.\n(2)32(1989), no. 1, 53–57.\n[21] S.J. Szarek, Computing summing norms and type constants on few vectors . Studia Math.\n98(1991), no. 2, 147–156.\n[22] A. W. Wickstead, An isomorphic version of Nakano’s characterisation of C0(Σ). Positivity\n11(2007), no. 4, 609–615.\nResearch Laboratory of Algebra, Topology, Arithmetic, and Order, Depart-\nment of Mathematics, Faculty of Mathematical, Physical and Natural Sciences\nof Tunis, Tunis-El Manar University, 2092-El Manar, Tunisi a\nEmail address :youssef.azouzi@ipest.rnu.tn\nPreparatory institute for engineering studies Nabeul, Res earch Laboratory\nof Algebra, Topology, Arithmetic, and Order, Department of Mathematics, Fac-\nulty of Mathematical, Physical and Natural Sciences of Tuni s, Tunis-El Manar\nUniversity, 2092-El Manar, Tunisia\nEmail address :asmabenrjab1992@gmail.com\nInstituto de Ciencias Matem ´aticas (CSIC-UAM-UC3M-UCM), Consejo Superior\nde Investigaciones Cient ´ıficas, C/ Nicol ´as Cabrera, 13–15, Campus de Canto-\nblanco UAM, 28049 Madrid, Spain.\nEmail address :pedro.tradacete@icmat.es" }, { "title": "2401.15655v1.Automorphisms_of_two_dimensional_quadrics.pdf", "content": "arXiv:2401.15655v1 [math.AG] 28 Jan 2024AUTOMORPHISMS OF TWO-DIMENSIONAL QUADRICS\nA. V. ZAITSEV\nAbstract. In this paper, we find the maximum values that the Jordan const ant of\nthe automorphism group of a smooth two-dimensional rationa l quadric over a field of\ncharacteristic zero can attain, depending on the arithmeti c properties of a field.\nContents\n1. Introduction 1\n2. Jordan constants and group theory 3\n3. Semidirect products 7\n4. Jordan constants of a group Aut (P1\nK×P1\nK) 9\n5. Jordan constants of groups of type PGL2(L)⋊ Z/2Z 11\n6. Computation of M(K) 14\nReferences 17\n1.Introduction\nIt is often useful to study infinite groups at the level of thei r finite subgroups. For\nexample, one can study the Jordan property of infinite groups .\nDefinition 1.1 ([11, Definition 2.1]) .LetGbe a finite group. The Jordan constant J(G)\nofGis the smallest index of a normal abelian subgroup in G. LetΓbe an arbitrary group.\nThenΓis called Jordan if the value\nJ(Γ) = sup\nG⊆Γ,|G|<∞(J(G))\nis finite. In this case the number J(Γ)is called the Jordan constant of the group Γ.\nImportant examples of infinite Jordan groups are the complet e linear groups GL n(K)\nover a field Kof characteristic zero. The fact that these groups are Jorda n was proved\nby Camille Jordan, see [ 9, §40] or [ 3, Theorem 36.13], and their Jordan constants over\nalgebraically closed fields were computed in [ 2]. As a corollary, all linear algebraic groups\nare Jordan groups. In particular, the projective linear gro upsPGLn(K), which are auto-\nmorphism groups of projective spaces.\nThe next natural question is about the group of birational au tomorphisms of the pro-\njective plane /emdash.cyr is this group Jordan or not? The fact that this group is Jordan over\nfields of characteristic zero was proved in the paper [ 12, Theor´ em` e 3.1]. The situation\nwith it’s Jordan constants is more complicated for these gro ups than for linear groups.\nAt the moment, the exact values of the Jordan constants of the group of birational auto-\nmorphisms of the projective plane have been computed over al gebraically closed fields of\ncharacteristic zero, over fields of real and rational number s, see [ 14].\nIn this paper, we deal with one of the most important steps of a question about Jordan\nconstants of the group of birational automorphisms of the pr ojective plane over fields of\nThe study has been funded within the framework of the HSE Univ ersity Basic Research Program.\n1characteristic zero. Namely, finite subgroups in the group o f birational automorphisms of\nthe projective plane act effectively on rational del Pezzo su rfaces or on rational surfaces\nwith a conic bundle structure. Therefore, it is useful to und erstand the Jordan constants\nof the automorphism groups of these surfaces. The results ab out the rational del Pezzo\nsurface of degree 9, that is, about the projective plane, are obtained in [ 6]. In this paper\nwe compute the Jordan constants of automorphism groups of ra tional del Pezzo surfaces\nof degree 8, that is, smooth two-dimensional rational quadrics. Recal l that a smooth\ntwo-dimensional quadric is rational if and only if it contai ns a rational point.\nWe are interested in the following value:\nM(K) = max\nX(J(Aut(X)),\nwhere the maximum is taken over smooth rational quadrics in P3\nK. As a result, we prove\nthe following theorem.\nTheorem 1.2. LetKbe a field of characteristic 0.\n(1)M(K) = 7200 if and only if√\n5∈K, and−1is a sum of two squares in K;\n(2)M(K) = 120 if and only if√\n5/∈K, and−1is a sum of two squares in K(√\n5);\n(3)M(K) = 60 if and only if√\n5∈K, and−1is not a sum of two squares in K;\n(4)M(K) = 8if and only if√\n5/∈K, and−1is not a sum of two squares in K(√\n5).\nCorollary 1.3. All values of M(K)from Theorem 1.2are attained:\nM(Q) = 8, M(R) = 60, M(Q(i)) = 120, M(Q(√\n−7)) = 120 , M(C) = 7200.\nDuring the proof of Theorem 1.2, we compute the Jordan constant of a surface P1\nK×P1\nK,\nmore precisely, we prove the following theorem.\nTheorem 1.4. LetKbe a field of characteristic 0.\n(1)J(Aut(P1\nK×P1\nK)) = 7200 if and only if√\n5∈Kand−1is a sum of two squares\ninK;\n(2)J(Aut(P1\nK×P1\nK)) = 72 if and only if√\n5/\\e}atio\\slash∈K, and−1is a sum of two squares\ninK;\n(3)J(Aut(P1\nK×P1\nK)) = 8 if and only if −1is not a sum of two squares in K.\nExample 1.5. All values of Jordan constant from Theorem 1.4are attained:\n•J(Aut(P1\nC×P1\nC)) = 7200 ,\n•J(Aut(P1\nQ(i)×P1\nQ(i))) = 72 ,\n•J(Aut(P1\nR×P1\nR)) =J(Aut(P1\nQ×P1\nQ)) = 8 .\nAlso we prove the following useful proposition.\nProposition 1.6. LetKbe a field of characteristic 0. LetSbe a smooth rational quadric\ninP3\nKandS/\\e}atio\\slash≃P1\nK×P1\nK. Then\nJ(Aut(S))/lessorequalslant120.\nThe plan of the paper is as follows. In Section 2we collect some auxiliary statements\nfrom group theory. In Section 3, we compute the Jordan constants of groups of the\nform(Γ×Γ)⋊Z/2Z, which are similar to the automorphism group of the surface P1×P1. In\nSection 4we prove Theorem 1.4. In Section 5we find the matrices generating the group A5\ninsidePGL2(L), and using them we estimate the Jordan constants of automorp hism\ngroups of smooth rational two-dimensional quadrics differe nt fromP1×P1. Finally, in\nSection 6we prove Theorem 1.2, Proposition 1.6and Corollary 1.3.\nWe will use the following notation. We denote the neutral ele ment of a group by e.\nWe denote the dihedral group of order 2nbyD2n. We denote the algebraic closure of the\n2fieldKbyK. IfK⊂Lis an extension of fields, and Xis a variety over K, then we\ndenote the extension of scalars of XtoLbyXL. IfH⊂Gare groups and g,g′∈G, then\nwe denote a subgroup of Ggenerated by all elements of the subgroup Hand the element g\nby/a\\}bracketle{tH,g/a\\}bracketri}ht, and we denote a subgroup generated by elements gandg′by/a\\}bracketle{tg,g′/a\\}bracketri}ht. We denote\nthe group defined by the set of generators Sand the list of relations Rby/a\\}bracketle{tS|R/a\\}bracketri}ht.\nAcknowledgements. I would like to thank my advisor Constantin Shramov for stati ng\nthe problem, useful discussions and constant attention to t his work. I also want to thank\nAndrey Trepalin for useful discussions and especially for e legant completion of proof\nof Lemma 5.6. The work was supported by the Theoretical Physics and Mathe matics\nAdvancement Foundation “BASIS”.\n2.Jordan constants and group theory\nIn this section, we collect some auxiliary statements from g roup theory. The following\nlemma is obvious and will be used without reference to it.\nLemma 2.1. LetHbe a subgroup of a Jordan group G. Then His also Jordan\nandJ(H)/lessorequalslantJ(G).\nThe following lemma is also standard and simple.\nLemma 2.2 (see for example [ 11, Lemma 2.8]) .LetGandHbe Jordan groups. Then\ngroupG×His Jordan, and J(G×H) =J(G)·J(H).\nRecall the standard definition.\nDefinition 2.3. A subgroup Hof a group Gis called a characteristic subgroup if for\nevery automorphism ϕofG, one has ϕ(H) =H.\nThe following theorem is useful for estimating Jordan const ants of finite groups.\nTheorem 2.4 (see for example [ 8, Theorem 1.41]) .LetGbe a finite group, and Abe its\nabelian subgroup. Then there exists a characteristic abeli an subgroup NinGsuch that\n[G:N]/lessorequalslant[G:A]2.\nLet us prove an auxiliary proposition from group theory.\nProposition 2.5. LetHbe a group with a trivial center. Suppose we have a short exact\nsequence of groups\n1− →H− →G− →Z/mZ− →0,\nand there exists an element g∈Gsuch that gmaps to 1, and conjugation by ginduces\nan inner automorphism of H. ThenG≃H×Z/mZ.\nProof. Denote the homomorphism from GtoZ/mZbyp. Then we have p(g) = 1. Let\nus denote by αthe automorphism of the group Hinduced by conjugation by g. By the\ncondition, αis an inner automorphism, so there exists an element h∈Hsuch that αis\na conjugation by h.\nDenoteg′=gh−1. Firstly, note that p(g′) = 1, hencep((g′)m) = 0, that is, (g′)m∈H.\nSecondly, note that conjugation by g′induces a trivial automorphism of H, so conjugation\nby(g′)minduces a trivial automorphism of H. Therefore the element (g′)mlies in the\ncenter of H, which is trivial. It follows that (g′)m=e, hence the homomorphism phas a\nsections: 1/ma√sto→g′. Thus,G≃H⋊Z/mZwith trivial action, that is G≃H×Z/mZ./square\nThe following proposition is a direct corollary of Proposit ion2.5.\n3Proposition 2.6. LetHbe a group with a trivial center. Suppose that all automorphi sms\nofHare inner. Let Abe a finite abelian group. Then any group Gwhich includes in the\nexact sequence\n1− →H− →G− →A− →0,\nis isomorphic to the direct product H×A.\nProof. SinceAis a finite abelian group, there is an isomorphism\nA≃Z/n1Z×Z/n2Z×...×Z/nrZ.\nThen we will denote elements of Aby(m1,m2,...,m r), wheremi∈Z/niZ.\nDenote the homomorphism from GtoAbyp. Let us choose elements g1,g2,...,g r∈G\nsuch that\np(gi) = (0,...,0,1\ni,0,...,0).\nLet us act on Hby conjugation by element gi. This action induces an automorphism α\nofH, andαis inner, since all automorphisms of the group Hare inner. So we are in the\ncase of Proposition 2.5. Therefore, over each of the specified cyclic subgroups ther e is a\nsection\nsi: (0,...,0,1\ni,0,...,0)/ma√sto→g′\ni,\nand conjugation by element g′\niinduces a trivial automorphism of H.\nLet us show that obtained sections are glued into a section ov er the entire group A. To\ndo this, it is enough to show that the elements g′\niandg′\njcommute for all i,j∈ {1,...,n}.\nConsider the commutator\ncij=g′\nig′\nj(g′\ni)−1(g′\nj)−1\nof elements g′\niandg′\nj. Firstly, conjugation by this element induces a trivial aut omorphism\nofH. Secondly, this element lies in H, sincep(cij) = (0,...,0).But the center of the\ngroupHis trivial, hence cij=e, that is, the elements g′\niandg′\njcommute. Therefore, we\nget a section\ns:A− →G, s: (m1,m2,...,m r)/ma√sto→(g′\n1)m1(g′\n2)m2...(g′\nr)mr,\nandG≃H⋊A. But, as we have already mentioned, conjugation by the eleme ntg′\niinduces\na trivial automorphism of Hfor anyi∈ {1,...,r}, which means that G≃H×A./square\nWe will need standard facts about automorphisms of groups SnandAn.\nTheorem 2.7 (see for example [ 4, §4.4, Exercise 18]) .Letnbe a positive integer, n/greaterorequalslant3,\nn/\\e}atio\\slash= 6. Then\nAutSn≃Sn.\nTo prove a similar result for the group An, we need the following simple lemma.\nLemma 2.8. Letnbe a positive integer. Let Cgbe the conjugacy class of an even per-\nmutation g∈Sn. Then\n•classCgsplits into two conjugacy classes in Anif and only if the permutation g\ndecomposes into independent cycles of odd lengths, and all l engths are different\n(here a fixed point is considered as a cycle of length 1);\n•the class Cgis a conjugacy class in Anif and only if the decomposition of ginto\nindependent cycles contains a cycle of even length or two cyc les of the same odd\nlength.\nProof. A simple exercise. /square\nTheorem 2.9. Letnbe a positive integer, n/greaterorequalslant4,n/\\e}atio\\slash= 6. Then\nAutAn≃Sn.\n4Proof. Immediately note that Snis embedded in AutAnforn/greaterorequalslant4. Indeed, consider the\nhomomorphism\nρ:Sn− →AutAn, τ/ma√sto→ρτ,\nwhereρτis a conjugation by permutation τ. This homomorphism is injective because the\ncentralizer of AninSnis trivial for n/greaterorequalslant4. Let us show that for n/\\e}atio\\slash= 6the homomorphism ρ\nis also surjective.\nLetϕbe an arbitrary automorphism of An. Let us show that ϕmaps cycles of length 3\ninto cycles of length 3. Sinceϕpreserves the orders of elements, then triple cycle must\nmaps into an element of order 3, that is, into the product of kpairwise disjoint triple\ncycles, for some k∈Z>0. Note that for n/lessorequalslant5we automatically have k= 1, so it remains\nto deal with the case when n/greaterorequalslant7.\nSuppose n/greaterorequalslant7. Sinceϕis an automorphism, the conjugacy classes maps into conjuga cy\nclasses. By Lemma 2.8, all triple cycles form one conjugacy class in An. The products\nofkpairwise disjoint triple cycles form one conjugacy class in Anby the same lemma.\nEquate the number of elements in these classes:\n2/parenleftbiggn\n3/parenrightbigg\n=n!\nk!3k(n−3k)!.\nTaking into account the restriction of n/greaterorequalslant7, the obtained equality is true only for k= 1.\nThus, we proved that for n/\\e}atio\\slash= 6, the automorphism ϕmaps cycles of length 3into cycles\nof length 3.\nConsider the following set of generators of An:\nA={(123),(124),(125),...,(12n)}.\nNote that the product of any two considered permutations has the order 2, which means\nthat the same is true for the set of permutations:\nB={ϕ((123)),ϕ((124)),ϕ((125)),...,ϕ((12n))}.\nLetτ1andτ2be cycles of length 3. It is easy to see that the order of permutation τ1◦τ2\nis equal to 2if and only if these permutations have the form:\nτ1= (ijk), τ2= (ijl), k/\\e}atio\\slash=l.\nIt follows that any pair of permutations from the set Bis represented in this form.\nTherefore, the entire set is represented as:\nB={(i1i2i3),(i1i2i4),(i1i2i5),...,(i1i2in)},\nwhereir/\\e}atio\\slash=isforr/\\e}atio\\slash=s. Consider a permutation µ∈Snsuch that\nµ(j) =ij, j∈ {1,...,n}.\nThen for any permutation σ∈Athe equality ϕ(σ) =µσµ−1holds. Since the set A\ngenerates An, the automorphism ϕcoincides with the automorphism ρµ. So the homo-\nmorphism ρis surjective, and therefore is an isomorphism. /square\nFrom Proposition 2.6we obtain a corollary.\nCorollary 2.10. Letnbe a positive integer, n/greaterorequalslant3,n/\\e}atio\\slash= 6. LetAbe a finite abelian\ngroup. Then any group Gwhich includes in the exact sequence\n1− →Sn→G− →A− →0,\nis isomorphic to the direct product Sn×A.\nProof. For the specified n, the group Snhas a trivial center and all its automorphisms\nare inner according to Theorem 2.7. So we can apply Proposition 2.6. /square\n5Proposition 2.11. Letnbe a positive integer, n/greaterorequalslant4,n/\\e}atio\\slash= 6. Then the group G, which\nincludes in the exact sequence\n1− →An− →G− →Z/2Z− →0,\nis isomorphic either to An×Z/2Z, or toSn.\nProof. Denote the homomorphism from GtoZ/2Zbyp. Choose an element g∈G\nsuch that p(g) = 1 . Let us act on Anby conjugation by g. This action induces the\nautomorphism αof the group An. Now, if αis an inner automorphism, then by the\nProposition 2.5we have the isomorphism G≃An×Z/2Z.\nAssume that αis not an inner automorphism. By Theorem 2.9we have an isomor-\nphismAut(An)≃Sn, so we can choose an element h0∈An⊂Gsuch that the auto-\nmorphism β∈Aut(An), induced by conjugation by element g0=gh0, is a conjugation\nby transposition. Firstly, we have p(g0) = 1. Hence p(g2\n0) = 0, that is, g2\n0∈An. Sec-\nondly, conjugation by the element g2\n0induces a trivial automorphism of An, which means\nthatg2\n0=e. Therefore, the homomorphism phas a section s: 1/ma√sto→g0, andG≃An⋊Z/2Z,\nwhere nontrivial element of the group Z/2Zacts by conjugation by transposition. It fol-\nlows that G≃Sn. /square\nProposition 2.12. Letnandmbe positive integers, n/greaterorequalslant4,n/\\e}atio\\slash= 6. Let the group Gbe\nincluded in the exact sequence\n1− →An− →G− →Z/mZ− →0.\nIfm= 2k+1, thenGis isomorphic to An×Z/mZ. Ifm= 2k, then either Gis isomorphic\ntoAn×Z/mZ, or at least contains a normal subgroup isomorphic to An×Z/kZ.\nProof. Denote the homomorphism from GtoZ/mZbyp. Consider an element g∈G\nsuch that p(g) = 1 . Let us act on Anby conjugating by g. This action induces the\nautomorphism αofAn. By Theorem 2.9, the automorphism αis a conjugation by some\npermutation σ∈Sn.\nLetmbe odd at first. Since p(gm) = 0 , thengmlies inAn, henceαmis an inner\nautomorphism, that is, conjugation by an even permutation. Therefore, σis also an even\npermutation and αis also an inner automorphism. According to Proposition 2.5, we have\nan isomorphism G≃An×Z/mZ.\nNow letmbe even, that is, m= 2k. Ifσis an even permutation, then αis an inner\nautomorphism, and according to Proposition 2.5we have an isomorphism G≃An×Z/mZ.\nIfσis an odd permutation, then conjugation by the element g2already induces an inner\nautomorphism of An. Denote G′=/a\\}bracketle{tAn,g2/a\\}bracketri}ht, then we get into conditions of Proposition 2.5\nfor the exact sequence\n1− →An− →G′− →Z/kZ− →0.\nTherefore, G′is isomorphic to An×Z/kZ. Also,G′is normal in G, since it has index 2./square\nWe will also need the following presentation of the group A5.\nLemma 2.13. Consider a group given by generators and relations:\nG=/a\\}bracketle{tx,y|x5=y2= (xy)3=e/a\\}bracketri}ht.\nThenGis isomorphic to the group A5, and there is an isomorphism which maps xto the\npermutation (12345) , andyto the permutation (12)(34) .\nProof. In the example [ 7, Kapitel I, Beispiel 19.9], it is proved that Gis isomorphic to A5.\nNow note that permutations (12345) and(12)(34) generate the group A5and satisfy the\nconditions\n(12345)5=e,((12)(34))2=e,((12345)(12)(34))3=e.\nTherefore, the specified isomorphism exists. /square\n63.Semidirect products\nIn this section, we study the Jordan constant of groups of the form(Γ×Γ)⋊ Z/2Z,\nwhere the nontrivial element of the group Z/2Zacts by permutation of factors. The\nelement of the group (Γ×Γ)⋊Z/2Zwe write as g= (g1,g2,i), whereg1,g2∈Γ, andi= 0\n(a trivial element of Z/2Z), ori= 1(a nontrivial element of Z/2Z).\nLetg= (g1,g2,i),h= (h1,h2,j)∈(Γ×Γ)⋊Z/2Z, then the group operation looks as\nfollows:\ngh=/braceleftBigg\n(g1h1,g2h2,i+j),ifi= 0;\n(g1h2,g2h1,i+j),ifi= 1.\nThe following three technical lemmas will be needed to prove Theorem 1.4.\nLemma 3.1. LetΓbe a Jordan group. Then (Γ×Γ)⋊ Z/2Zis Jordan (a nontrivial\nelement of the group Z/2Zacts by permutation of factors) and the Jordan constant is\nreached on the group H, which is included in the exact sequence\n1− →G1×G2− →H− →Z/2Z− →0,\nwhereG1andG2are isomorphic subgroups of Γ.\nProof. Group(Γ×Γ)⋊Z/2Zis obviously Jordan. Let G⊂(Γ×Γ)⋊Z/2Zbe a finite\nsubgroup on which the Jordan constant is reached, that is\nJ(G) =J((Γ×Γ)⋊Z/2Z).\nOur goal is to find a finite subgroup Hof the required form, with J(H)/greaterorequalslantJ(G)(note\nthat this condition immediately implies the equality J(H) =J(G), sinceJ(G)equals to\nthe Jordan constant of the entire group). Let p1andp2be projections of Γ×Γonto the\nfirst and the second factors, respectively.\nLet us assume that G⊂Γ×Γ. Denote G′=p1(G)andG′′=p1(G). Then we have an\ninclusion G⊂G′×G′′which implies an inequality\nJ(G′)J(G′′) =J(G′×G′′)/greaterorequalslantJ(G).\nIt follows that either J(G′)/greaterorequalslant/radicalbig\nJ(G), orJ(G′′)/greaterorequalslant/radicalbig\nJ(G). Without loss of generality,\nwe can assume that the first case holds, then we can take Hequal to (G′×G′)⋊Z/2Z.\nIndeed, for this group we have\nJ(H)/greaterorequalslantJ(G′)2/greaterorequalslantJ(G).\nNow assume that G/\\e}atio\\slash⊂Γ×Γ. Denote by G0its intersection with Γ×Γand denote\nprojections G1=p1(G0)andG2=p2(G0). Note that G1andG2are conjugate in Γ.\nIndeed,G0⊂Gis a normal subgroup. Conjugating G0by the element\nγ= (γ1,γ2,1)∈G\\G0,\nwe obtain:\nG1=p1(G0) =p1(γG0γ−1) =γ1p2(G0)γ−1\n1=γ1G2γ−1\n1.\nSimilarly we have G2=γ2G1γ−1\n2. Then we can take Hequals to /a\\}bracketle{tG1×G2,γ/a\\}bracketri}ht. Indeed, H\nis included in the exact sequence\n1− →G1×G2− →H− →Z/2Z− →0,\nwhereG1andG2are isomorphic, since they are conjugate in Γ. AlsoGis a subgroup\nofH, therefore J(H)/greaterorequalslantJ(G). /square\nLemma 3.2. LetGbe a nontrivial finite group. Then\nJ((G×G)⋊Z/2Z) = 2J(G)2,\nwhere the nontrivial element of the group Z/2Zacts by permutation of factors.\n7Proof. Firstly, assume that J(G) =|G|. In this case, there are no nontrivial normal\nabelian subgroups in G. SinceGis a non-trivial group, the specified semidirect product\nis not a direct one, and it is easy to see that in this case the gr oup(G×G)×Z/2Zalso\ndoes not contain non-trivial normal abelian subgroups, tha t is,\nJ((G×G)×Z/2Z) = 2J(G)2.\nNow assume that J(G)/\\e}atio\\slash=|G|. LetA⊂Gbe a normal abelian subgroup such\nthat[G:A] =J(G). Then\nA×A⊂(G×G)⋊Z/2Z\nis a normal abelian subgroup of index 2J(G)2. It remains to show that there are no\nnormal abelian subgroups of smaller index.\nLetH⊂(G×G)⋊Z/2Zbe a normal abelian subgroup. If His contained in G×G,\nthen we have [G×G:H]/greaterorequalslantJ(G)2by Lemma 2.2. Therefore,\n[(G×G)⋊Z/2Z:H]/greaterorequalslant2J(G)2.\nIfHis not contained in G×G, then we denote by H0the intersection of H\nandG×G. Then H0is a normal abelian subgroup of G×GandH=/a\\}bracketle{tH0,g/a\\}bracketri}ht,\nwhereg= (g1,g2,1)∈H\\H0.Letp1andp2be projections of G×Gon the first and\nthe second factors, respectively. Denote H1=p1(H0)andH2=p2(H0). ThenH1andH2\nare normal abelian subgroups in G, thus[G:H1]/greaterorequalslantJ(G)and[G:H2]/greaterorequalslantJ(G).\nSinceHis abelian, then any element\nh= (h1,h2,0)∈H,\nwhereh1∈H1,h2∈H2, commutes with g:\n(h1,h2,0)(g1,g2,1) = (g1,g2,1)(h1,h2,0).\nAfter multiplication we get\n(h1g1,h2g2,1) = (g1h2,g2h1,1).\nTherefore h1=g1h2g−1\n1andh2=g2h1g−1\n2and the map h1/ma√sto→(h1,g2h1g−1\n2,0)defines an\nisomorphism between H1andH0. Then we have\n[(G×G)⋊Z/2Z:H] = [G×G:H0] = [G:H1]·|G|/greaterorequalslantJ(G)·|G|.\nSinceJ(G)/\\e}atio\\slash=|G|, then|G|/greaterorequalslant2J(G), and we get an estimate for the index\n[(G×G)⋊Z/2Z:H]/greaterorequalslant2J(G)2.\nAs a result, we presented a normal abelian subgroup of index 2J(G)2and showed that\nthere are no normal abelian subgroups of smaller index, ther eby the lemma is proved. /square\nNow we prove the main lemma of this section, which we will appl y in the proof of\nTheorem 1.4.\nLemma 3.3. LetΓbe a Jordan group, containing a nontrivial finite subgroup. T hen\nJ((Γ×Γ)⋊Z/2Z) = 2J(Γ)2.\nProof. For finite groups, this assertion is proved in Lemma 3.2. LetΓbe an infinite\ngroup. First, let us show that\nJ((Γ×Γ)⋊Z/2Z)/greaterorequalslant2J(Γ)2.\nTo do this, we need to find a finite subgroup of (Γ×Γ)⋊ Z/2Zwith Jordan constant\nequal to 2J(Γ)2.\n8Consider finite subgroup G⊂Γon which the Jordan constant is reached, that\nisJ(G) =J(Γ)(we can assume, that Gis nontrivial, since there exist nontrivial finite\nsubgroups in Γ). Denote\n˜G= (G×G)⋊Z/2Z⊂(Γ×Γ)⋊Z/2Z.\nWe have J(˜G) = 2J(G)2= 2J(Γ)2by Lemma 3.2.\nNow let us show, that there are no finite subgroups with larger Jordan constant. By\nLemma 3.1, it is enough to prove this for finite subgroups Hincluded in the exact sequence\n1− →G1×G2− →H− →Z/2Z− →0.\nDenote the homomorphism from HtoZ/2Zbyp. Choose an element γ∈Hsuch\nthatp(γ) = 1. Thenγcan be written as\nγ= (γ1,γ2,1).\nLet us conjugate the normal subgroup G1×G2byγ, then we obtain the equalities:\nG1=γ1G2γ−1\n1, G2=γ2G1γ−1\n2.\nLetN1be a normal abelian subgroup of G1on which the Jordan constant is reached.\nThen the Jordan constant of the group G2is reached on the subgroup N2=γ2N1γ−1\n2. In\nparticular, the following inequalities hold:\n[G1:N1] =J(G1)/lessorequalslantJ(Γ),[G2:N2] =J(G2)/lessorequalslantJ(Γ).\nThe subgroup N1×N2is obviously normal in G1×G2. Let us show that it is normal\ninH. It is enough to check that N1×N2is normalized by the element γ. Take an\nelementn= (n1,n2,0)∈N1×N2and conjugate by γ:\nγnγ−1= (γ1,γ2,1)(n1,n2,0)(γ−1\n2,γ−1\n1,1) = (γ1n2γ−1\n1,γ2n1γ−1\n2,0).\nNote that γ2n1γ−1\n2∈N2by definition of a subgroup N2. Also, by definition of a sub-\ngroupN2, there exist an element n′\n1∈N1such that n2=γ2n′\n1γ−1\n2. Therefore\nγ1n2γ−1\n1=γ1γ2n′\n1γ−1\n2γ−1\n1∈N1,\nsinceγ1γ2lies inG1, sinceγ2∈G1×G2. Hence\nγnγ−1= (γ1n2γ−1\n1,γ2n1γ−1\n2,0)∈N1×N2,\nandN1×N2is a normal abelian subgroup of Hof index\n[H:N1×N2] = 2·[G1:N1]·[G2:N2]/lessorequalslant2J(Γ)2.\nTherefore J(H)/lessorequalslant2J(Γ)2, and the lemma is proved. /square\n4.Jordan constants of a group Aut (P1\nK×P1\nK)\nThe automorphism group of the surface P1\nK×P1\nKhas an explicit description:\nAut(P1\nK×P1\nK)≃(PGL2(K)×PGL2(K))⋊Z/2Z,\nwhere the nontrivial element of the group Z/2Zacts by permutation of the factors. There-\nfore, to study the Jordan constants of the group Aut (P1\nK×P1\nK), it is necessary to under-\nstand which finite subgroups does group PGL2(K)contain.\nIf the field Kis algebraically closed, it is well known that the finite subg roups of the\ngroupPGL2(K)areZ/nZ,D2n(forn/greaterorequalslant2),A4,S4andA5. IfKis an arbitrary, all finite\nsubgroups of the group PGL2(K)occur in the above list because PGL2(K)is a subgroup\nofPGL2(K), however, no one guarantees that all groups in the list are re alized as finite\nsubgroups in PGL2(K).\n9We are working with an arbitrary field Kof characteristic zero. In this case, it is also\nwell known which finite groups are realized as subgroups of PGL2(K)depending on the\narithmetic properties of the field.\nProposition 4.1 ([1, Proposition 1.1]) .LetKbe a field of characteristic 0andξmbe a\nprimitive m-th root of unity.\n(1) PGL 2(K)contains Z/mZ,D2mif and only if Kcontains ξm+ξ−1\nm(in particular,\nPGL2(K)always contains D6,D8иD12);\n(2) PGL 2(K)contains A4,S4if and only if −1is a sum of two squares in K;\n(3) PGL 2(K)contains A5if and only if −1is a sum of two squares in K, andK\ncontains√\n5.\nRemark 4.2. We have\nJ(Z/nZ) = 1, J(D4) = 1, J(D2n) = 2 forn/greaterorequalslant3,\nJ(A4) = 3, J(S4) = 6,J(A5) = 60.\nIn particular, it can be seen from here that if the group PGL2(K)does not contain A5,\nthen the Jordan constant J(PGL2(K))does not exceed 6.\nAlso note that if the group Gis isomorphic to one of these groups, then there is a\ncharacteristic abelian subgroup A⊂Gsuch that [G:A] =J(G). That is, for any\nfinite subgroup H⊂PGL2(K), the Jordan constant of the group His reached on a\ncharacteristic subgroup.\nFrom Proposition 4.1we obtain an obvious corollary about the Jordan constants of the\ngroupPGL2(K).\nCorollary 4.3. LetKbe a field of characteristic 0.\n(1)J(PGL2(K)) = 60 if and only if√\n5∈Kand−1is a sum of two squares in K;\n(2)J(PGL2(K)) = 6 if and only if√\n5/\\e}atio\\slash∈Kand−1is a sum of two squares in K;\n(3)J(PGL2(K)) = 2 if and only if −1is not a sum of two squares in K.\nCorollary 4.4. LetKbe a field of characteristic 0. Consider the semidirect product of\ngroups(PGL2(K)×PGL2(K))⋊Z/2Z, where the nontrivial element of the group Z/2Z\nacts by permutation of factors.\n(1)J((PGL 2(K)×PGL2(K))⋊Z/2Z) = 7200 if and only if√\n5∈Kand−1is a sum\nof two squares in K;\n(2)J((PGL 2(K)×PGL2(K))⋊Z/2Z) = 72 if and only if√\n5/\\e}atio\\slash∈Kand−1is a sum\nof two squares in K;\n(3)J((PGL 2(K)×PGL2(K))⋊ Z/2Z) = 8 if and only if −1is not a sum of two\nsquares in K.\nProof. Apply Lemma 3.3to Corollary 4.3. /square\nNow we can prove Theorem 1.4.\nProof (proof of Theorem 1.4.).The automorphism group of the surface P1\nK×P1\nKis\nisomorphic to the group (PGL2(K)×PGL2(K))⋊ Z/2Z, where the nontrivial element\nof the group Z/2Zacts by permutation of factors. Therefore, applying Coroll ary4.4, we\nobtain an assertion of the theorem. /square\n105.Jordan constants of groups of type PGL2(L)⋊ Z/2Z\nIn this section, we estimate and compute the Jordan constant s of groups of the form\nPGL2(L)⋊Z/2Z,\nwhereLis a field of characteristic zero, as usual.\nAn element of the group PGL2(L)⋊ Z/2Zwe will write as g= (γ,i),\nwhereγ∈PGL2(L), andi= 0(the trivial element of the group Z/2Z), ori= 1(the\nnontrivial element of the group Z/2Z).\nLetg1= (γ1,i),g2= (γ2,j)∈PGL2(L)⋊ Z/2Z, then the group operation looks as\nfollowing:\ng1g2= (γ1(ϕi(γ2)),i+j),\nwhereϕiis an automorphism of the group PGL2(L)included in the definition of a semidi-\nrect product.\nLemma 5.1. LetLbe a field of characteristic 0. Then\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant120.\nProof. LetG⊂PGL2(L)⋊ Z/2Zbe a finite subgroup. Denote by G0the intersection\nofGwithPGL2(L). Then, according to Remark 4.2, there is an abelian characteristic\nsubgroup A⊂G0, on which the Jordan constant of the group G0is reached. Hence, Ais\na normal abelian subgroup of Gof index 2J(G0), and\nJ(G)/lessorequalslant2J(G0)/lessorequalslant2J(PGL2(L))/lessorequalslant120.\nSinceGis an arbitrary finite subgroup of PGL2(L)⋊Z/2Z, the lemma is proved. /square\nLemma 5.2. LetLbe such a field of characteristic 0that group PGL2(L)does not contain\na subgroup isomorphic to A5. Then\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant6.\nProof. SincePGL2(L)does not contain a subgroup isomorphic to A5, then we have\nthe inequality J(PGL2(L)/lessorequalslant6according to Remark 4.2. It remains to show that the\nJordan constants of the finite subgroups contained in PGL2(L)⋊Z/2Z, but not contained\ninPGL2(L), also do not exceed 6.\nLetGbe a finite subgroup of PGL2(L)⋊Z/2Znot contained in PGL2(L). ThenGis\nincluded in the exact sequence\n1− →G0− →G− →Z/2Z− →0,\nwhereG0is the intersection of the group GwithPGL2(L). SincePGL2(L)does not\ncontainA5, then, according to Proposition 4.1, the group G0is isomorphic to either a\ncyclic group, or a dihedral group, or A4, orS4.\nSuppose G0is isomorphic to either Z/nZorD2norA4. By Remark 4.2it contains\ncharacteristic abelian subgroup Aof index at most 3. ThenAis normal abelian subgroup\ninGof index at most 6, andJ(G)/lessorequalslant6.\nSuppose G0≃S4. Applying Corollary 2.10and Remark 4.2, we getJ(G)/lessorequalslant6.\nThus, we have shown that the Jordan constant does not exceed 6 for all possible finite\nsubgroups in PGL2(L)⋊Z/2Z, and the lemma is proved. /square\nThe following two propositions are useful when the base field does not contain√\n5.\nProposition 5.3. LetKbe a field of characteristic 0, andr∈Kbe an element, which\nis not a square. Assume that K(√r)contains√\n5, and there exist a,b,c,d∈K, such that\n(a+b√r)2+(c+d√r)2=−1.\n11Consider two matrices in PGL2(K(√r)):\nA=/parenleftbigg\n0 1\n−1 0/parenrightbigg\n, C=/parenleftbigg\n2c+2d√r+√\n5−3 2a+2b√r−√\n5+1\n2a+2b√r+√\n5−1−2c−2d√r+√\n5−3/parenrightbigg\n.\nThe group G′=/a\\}bracketle{tA,C/a\\}bracketri}htis isomorphic to A5, and there exists an isomorphism that maps\nmatrixAto permutation (12)(34) , and matrix Cto permutation (12345) .\nProof. Taking in account Lemma 2.13, it is enough to check that the matrices AandC\nsatisfy the relations A2=e,C5=eand(CA)3=e. Direct calculations show that this is\nthe case:\nA2=/parenleftbigg\n−1 0\n0−1/parenrightbigg\n=e;\nC5=/parenleftbigg\n−2560√\n5+5632 0\n0 −2560√\n5+5632/parenrightbigg\n=e;\n(CA)3=/parenleftbigg\n−64√\n5+128 0\n0 −64√\n5+128/parenrightbigg\n=e.\nIt follows that G′is isomorphic to the quotient of the group A5. ButA5is simple,\nandG′is nontrivial, thus G′≃A5. Existence of a specified isomorphism is guaranteed by\nLemma 2.13. /square\nLemma 5.4. LetKbe a field of characteristic 0. Assume that√\n5/∈K, and−1is the\nsum of two squares in K(√\n5). Then\nJ(PGL2(K(√\n5))⋊Z/2Z) = 120,\nwhere the nontrivial element of the group Z/2Zacts by the Galois involution of the ex-\ntensionK⊂K(√\n5).\nProof. From Lemma 5.1we have the inequality\nJ(PGL2(K(√\n5))⋊Z/2Z)/lessorequalslant120.\nIt remains to find a finite subgroup with Jordan constant equal to 120.\nBy the assumption of the lemma, there are a,b,c,d∈K, such that\n(a+b√\n5)2+(c+d√\n5)2=−1.\nConsider the matrix\nR=/parenleftbigg\na+c a−c\na−c−a−c/parenrightbigg\n∈PGL2(K(√\n5)).\nLetG′⊂PGL2(K(√\n5))be a group from Proposition 5.3forr= 5. Let us show that\n/a\\}bracketle{t(G′,0),(R,1)/a\\}bracketri}ht ≃S5,\nand that’s where the proof ends, because J(S5) = 120 .\nFirstly, note that the element (R,1)has order 2. Secondly, the subgroup (G′,0)is\ninvariant with respect to the conjugation by the element (R,1). Indeed, it is enough to\ncheck that generators remain in the group after conjugation :\n(R,1)(A,0)(R,1) = (A,0);\n(R,1)(C,0)(R,1) = ((C2A)3,0).\nTherefore\n/a\\}bracketle{t(G′,0),(R,1)/a\\}bracketri}ht ≃A5⋊Z/2Z,\n12moreover, the product is not direct, since the conjugation b y element (R,1)induces a\nnoninner automorphism of the group (G′,0). Thus, according to Proposition 2.11, we\nhave the isomorphism /a\\}bracketle{t(G′,0),(R,1)/a\\}bracketri}ht ≃S5. /square\nObserve that for an element T∈PGL2(K)the valuetr2(T)\ndet(T)is well-defined and invariant\nunder conjugation in PGL2(K). Let us prove the following auxiliary lemma.\nLemma 5.5. LetB∈PGL2(K)be an element of order 5. Then\ntr2(B)\ndet(B)=3\n2±1\n2√\n5,tr2(B2)\ndet(B2)=3\n2∓1\n2√\n5.\nProof. Over the algebraic closure K, matrix Bis diagonalizable, namely there exists an\nelementC∈PGL2(K)such that\nCBC−1=/parenleftbigg\nδ10\n0δ2/parenrightbigg\n.\nSinceBhas order 5, then we have an equality δ2=ξδ1, whereξis a5-th root of unity.\nTherefore\ntr2(B)\ndet(B)=tr2(CBC−1)\ndet(CBC−1)=(δ1+ξδ1)2\nξδ2\n1=ξ−1+2+ξ=3\n2±1\n2√\n5\nand\ntr2(B2)\ndet(B2)=tr2(CB2C−1)\ndet(CB2C−1)=(δ2\n1+ξ2δ2\n1)2\nξ2δ4\n1=ξ−2+2+ξ2=3\n2∓1\n2√\n5.\n/square\nThe last lemma of this section handles the case of such fields c ontaining√\n5, that−1\nis not a sum of two squares in this fields.\nLemma 5.6. LetKbe a field of characteristic 0such that −1is not a sum of two\nsquares in K, but√\n5lies inK. LetK⊂Lbe a quadratic extension of fields. Consider\nthe group PGL2(L)⋊Z/2Z, where the nontrivial element of the group Z/2Zacts by Galois\ninvolution of the extension K⊂L. Then\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant60.\nProof. Suppose −1is not a sum of two squares in L. Then, by Proposition 4.1, the\ngroupPGL2(L)does not contain a subgroup isomorphic to A5. Therefore, by Lemma 5.2\nwe have\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant6.\nNow suppose −1is a sum of two squares in L. Note that in this case, according to\nProposition 4.1, the group PGL2(L)contains a finite subgroup isomorphic to A5.\nLetHbe an arbitrary finite subgroup of PGL2(L)⋊Z/2Z. If at the same time His a\nsubgroup of PGL2(L), then by Corollary 4.3we immediately get J(H)/lessorequalslant60. Suppose H\nis not a subgroup of PGL2(L), then we have a short exact sequence\n1− →H0− →H− →Z/2Z− →0,\nwhereH0is the intersection of Hwith(PGL2(L),0). IfH0is not isomorphic to A5, then\nby Remark 4.2we have J(H0)/lessorequalslant6, andJ(H)/lessorequalslant12. IfH0is isomorphic to A5, then\naccording to Proposition 2.11, groupHis isomorphic to either A5×Z/2Z, orS5. Let us\nshow that Hcannot be isomorphic to S5.\nSuppose that H≃S5. Consider an element (B,0)∈H0of order 5, consider an\nelement(R,1)∈Hof order2and conjugate one by other:\n(R,1)(B,0)(R,1) = (RBR,0)∈H0.\n13Since element (R,1)has order 2, thenR=R−1and we can rewrite\n(RBR,0) = (RBR−1,0)∈H0.\nAccording to Lemma 2.8, conjugacy class of an element (B,0)inHsplits into two con-\njugacy classes in H0. It is easy to see that elements (B,0)and(B2,0)lies in different\nconjugacy classes in H0. Also it is easy to see that elements (B,0)and(RBR−1,0)lies\nin different conjugacy classes in H0. Therefore (B2,0)and(RBR−1,0)lies in the same\nconjugacy class in H0. In particular there exists an element ˜R∈PGL2(L)such that\n˜RB2˜R−1=RBR−1.\nWe obtain the following\ntr2(B2)\ndet(B2)=tr2(˜RB2˜R−1)\ndet(˜RB2˜R−1)=tr2(RBR−1)\ndet(RBR−1)=tr2(B)\ndet(B)=/parenleftbiggtr2(B)\ndet(B)/parenrightbigg\n.\nButBis an element of PGL2(L)of order5, then by Lemma 5.5we have\ntr2(B)\ndet(B)∈Q(√\n5)⊂K.\nTherefore Galois involution of extension K⊂Lacts on the scalartr2(B)\ndet(B)trivially, and we\nget an equality\ntr2(B2)\ndet(B2)=/parenleftbiggtr2(B)\ndet(B)/parenrightbigg\n=tr2(B)\ndet(B).\nBut this is a contradiction with Lemma 5.5. SoHcannot be isomorphic to S5.\nTherefore, H≃A5×Z/2Z, andJ(H) = 60 .\nThus, all possible cases of finite subgroups are considered a nd the lemma is proved. /square\n6.Computation of M(K)\nConsider a field K. As mentioned in the introduction, we want to compute the fol lowing\nvalue:\nM(K) = max\nX(J(Aut(X)),\nwhere the maximum is taken over all smooth rational quadrics inP3\nK.\nThe following proposition is standard. It will be needed in o rder to use general results\nabout del Pezzo surfaces of degree 8.\nProposition 6.1. LetXbe a smooth rational surface over a field K. The following\nconditions are equivalent.\n(1)Xis isomorphic to a smooth quadric in P3\nK;\n(2)Xis a del Pezzo surface of degree 8such that XK≃P1\nK×P1\nK.\nProof. Condition 1 obviously implies 2. Let us prove the implicatio n in the other direc-\ntion. Consider the exact sequence of groups (see [ 5, Exercise 3.3.5(iii)]):\n0− →Pic(X)− →Pic(XK)Gal(K/K)− →Br(X)− →Br(K(X)).\nSinceXis rational, there is a K-point on Xby the Lang–Nishimura theorem (see for\nexample [ 10, Theorem 3.6.11]). Therefore the last homomorphism is an em bedding and\nPic(X)≃Pic(XK)Gal(K/K).\nThis means that the class of divisors of the bidegree (1,1)is defined over K, and it defines\nan embedding of XinP3\nKas a smooth quadric. /square\n14There is an explicit description of del Pezzo surfaces of deg ree 8, which become iso-\nmorphic to the product of two projective lines when passing t o the algebraic closure.\nLetK⊂Lbe a finite extension of fields, and Ybe a variety over L. ByRL/K(Y), we\ndenote the Weil restriction of scalars (see, for example, [ 5, §8]).\nLemma 6.2 ([13, Lemma 7.3]) .(1)LetXbe a del Pezzo surface of degree 8over a\nfieldKsuch that XK≃P1\nK×P1\nK. Then either rkPic(X) = 2andXis isomorphic\nto a product C×C′of two conics over K, orrkPic(X) = 1andXis isomorphic\ntoRL/K(Q), whereL⊃Kis a quadratic separable extension and Qis a conic\noverL.\n(2)LetCbe a smooth conic over K, andL⊃Kbe a quadratic separable extension.\nThen\nAut(RL/K(CL))≃Aut(CL)⋊Z/2Z,\nwhere the nontrivial element of the group Z/2Zacts by the Galois involution of\nthe extension L⊃K.\nFrom Proposition 6.1and Lemma 6.2we get the following corollary.\nCorollary 6.3. LetKbe a field of characteristic 0.\n(1) LetXbe a smooth rational quadric in P3\nK. Then either rkPic(X) = 2 andX\nis isomorphic to P1\nK×P1\nK, orrkPic(X) = 1 andXis isomorphic to RL/K(P1\nL),\nwhereL⊃Kis a quadratic extension.\n(2) LetL⊃Kbe a quadratic extension, then\nAut(RL/K(P1\nL))≃PGL2(L)⋊Z/2Z,\nwhere the nontrivial element of the group Z/2Zacts by the Galois involution of\nthe extension L⊃K.\nProof. We should immediately note that the rationality of Ximplies the exis-\ntence of a K-point on Xaccording to the Leng–Nishimura theorem (see for exam-\nple [10, Theorem 3.6.11]).\nApplying Proposition 6.1, we conclude that Xis a del Pezzo surface of degree 8\nsuch that XK≃P1\nK×P1\nK. Then, by Lemma 6.2,Xis isomorphic to either a prod-\nuctC×C′of two conics over Kor Weil scalar restriction RL/K(Q), whereK⊂Lis\na quadratic extension and Qis a conic over L. Surface C×C′contains K-point if and\nonly ifC≃C′≃P1\nK, andRL/K(Q)contains K-point if and only if Qcontains L-point\n(see [5, Exercise 8.1.2(iv)]), that is, Q≃P1\nL. Thus, assertion 1 is proved.\nAssertion 2 follows immediately from assertion 2 of Lemma 6.2and existence of iso-\nmorphism Aut(P1\nL)≃PGL2(L). /square\nNow we can rewrite value M(K)as follows:\nM(K) = max/parenleftbig\nJ(Aut(P1\nK×P1\nK)),max\n[L:K]=2(J(PGL2(L)⋊Z/2Z))/parenrightbig\nand use the results obtained in Section 4and Section 5.\nIn the following propositions, the value M(K)is computed depending on the conditions\non a field Kintroduced in Theorem 1.2.\nProposition 6.4. LetKbe a field of characteristic 0, such that√\n5∈K, and−1is a\nsum of two squares in K. ThenM(K) = 7200 .\nProof. Since√\n5∈K, and−1is a sum of two squares in K, then by Theorem 1.4\nJ(Aut(P1\nK×P1\nK)) = 7200 .\n15Moreover, for any quadratic extension L⊃K, by Lemma 5.1, we have the inequality\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant120.\nTherefore M(K) = 7200 . /square\nProposition 6.5. LetKbe a field of characteristic 0such that√\n5/\\e}atio\\slash∈K, and−1is a\nsum of two squares in K(√\n5). ThenM(K) = 120 .\nProof. Since√\n5/\\e}atio\\slash∈K, then by the Theorem 1.4\nJ(Aut(P1\nK×P1\nK))/lessorequalslant72.\nAlso for any quadratic extension L⊃K, in accordance with Lemma 5.1, we have the\ninequality\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant120.\nTherefore, it is sufficient to present a quadratic extension L0⊃Ksuch that\nJ(PGL2(L0)⋊Z/2Z) = 120.\nBy Lemma 5.4, we can take L0equals to K(√\n5). Thus, the proposition is proved. /square\nProposition 6.6. LetKbe a field of characteristic 0such that√\n5∈K, and−1is not\na sum of two squares in K. ThenM(K) = 60 .\nProof. Since−1is not a sum of two squares in K, then by Theorem 1.4,\nJ(Aut(P1\nK×P1\nK)) = 8.\nAlso for any quadratic extension L⊃K, according to Lemma 5.6, we have the inequality\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant60.\nAt the same time, for any quadratic extension L′⊃Ksuch that −1is a sum of two\nsquares in L′, according to Proposition 4.1, groupPGL2(L′)contains a group isomorphic\ntoA5. Therefore, we have\nJ(PGL2(L′)⋊Z/2Z)/greaterorequalslantJ(PGL2(L′))/greaterorequalslantJ(A5) = 60.\nThus,M(K) = 60 , and the proposition is proved. /square\nProposition 6.7. LetKbe a field of characteristic 0such that√\n5/\\e}atio\\slash∈K, and−1is not\na sum of two squares in K(√\n5). ThenM(K) = 8.\nProof. Since−1is not a sum of two squares in K(√\n5), then, in particular, −1is not the\nsum of two squares in K, so by Theorem 1.4,\nJ(Aut(P1\nK×P1\nK)) = 8.\nAlso, since√\n5/\\e}atio\\slash∈K, and−1is not a sum of two squares in K(√\n5), then from Proposi-\ntion4.1it follows that for any quadratic extension L⊃K, the group PGL2(L)does not\ncontain a subgroup isomorphic to A5. Applying Lemma 5.2, we obtain the inequality for\nany quadratic field extension L⊃K:\nJ(PGL2(L)⋊Z/2Z)/lessorequalslant6.\nTherefore, M(K) = 8. /square\nNow everything is prepared for proving the main results.\n16Proof (Proof of Theorem 1.2.).If√\n5∈K, and−1is the a of two squares in K,\nthenM(K) = 7200 according to Proposition 6.4. If√\n5∈K, but−1is not a sum of two\nsquares in K, thenM(K) = 60 according to Proposition 6.6. If√\n5/\\e}atio\\slash∈K, and−1is a sum\nof two squares in K(√\n5), thenM(K) = 120 according to Proposition 6.5. And finally,\nif√\n5/\\e}atio\\slash∈K, and−1is not a sum of two squares in K(√\n5), thenM(K) = 8 according to\nProposition 6.7. Thus, we have considered all possible fields of characteris tic zero, and\nthe theorem is proved. /square\nProof (Proof of Proposition 1.6).From Corollary 6.3, we have the isomorphism\nS≃RL/K(P1\nL),\nwhereK⊂Lis a quadratic extension, and\nAut(S)≃PGL2(L)⋊Z/2Z.\nBy Lemma 5.1, we get an estimate J(Aut(S))/lessorequalslant120. /square\nProof (Proof of Corollary 1.3.).Theorem 1.2obviously implies that\nM(Q) = 8, M(R) = 60, M(C) = 7200.\nConsider the field Q(√−7). Firstly, note that√\n5/\\e}atio\\slash∈Q(√−7). Secondly, note that −1\nis a sum of two squares in the field Q(√−7,√\n5), for example:\n/parenleftBig√−7+√\n5\n−1+√−35/parenrightBig2\n+/parenleftBig6\n−1+√−35/parenrightBig2\n=−1.\nThus, by Theorem 1.2, we have M(Q(√−7)) = 120 .\nConsider the field Q(i). Note that√\n5does not lie in Q(i)and that −1is a sum\nof two squares in Q(i), for example −1 =i2+ 02. Therefore, by Theorem 1.2, we\nhaveM(Q(i)) = 120 . /square\nReferences\n[1] A. Beauville. Finite subgroups of PGL2(K). Contemporary Mathematics, 522(2010), 23–29.\n[2] M. J. Collins. On Jordan’s theorem for complex linear gro ups. Journal of Group Theory, 10(2007),\n411–423.\n[3] C. W. Curtis, I. Reiner. Representation theory of finite g roups and associative algebras. Pure and\nApplied Mathematics, vol. XI. Interscience Publishers, a d ivision of John Wiley & Sons, New York–\nLondon, 1962.\n[4] D. S. Dummit, R. M. Foote. Abstract Algebra, 3rd edition. Wiley, Hoboken, NJ, 2003.\n[5] S. Gorchinskiy, C. Shramov. Unramified Brauer group and i ts applications. Translations of Mathe-\nmatical Monographs, 2015.\n[6] Yi. Hu. Jordan Constant of PGL3(K). arXiv:2206.0218 (2022).\n[7] B. Huppert. Endliche gruppen I. Grundlehren der mathema tischen Wissenschaften, vol. 134.\nSpringer-Verlag, Berlin–Heidelberg–New York, 2013.\n[8] I. M. Isaacs. Finite group theory. Graduate Studies in Ma thematics, vol. 92. American Mathematical\nSociety, Providence, RI, 2008.\n[9] C. Jordan. M´ emoire sur les ´ equations diff´ erentielles lin´ eaires ` a int´ egrale alg´ ebrique. Journal f¨ ur die\nreine und angewandte Mathematik, 84(1878), 89–215.\n[10] B. Poonen. Rational points on varieties. Graduate Stud ies in Mathematics, vol. 186. American Math-\nematical Society, Providence, RI, 2017.\n[11] V. Popov. On the Makar-Limanov, Derksen invariants, an d finite automorphism groups of algebraic\nvarieties. In Affine algebraic geometry. CRM Proceedings and Lecture Notes, 54(2011), 289–311.\n[12] J.-P. Serre. Le groupe de Cremona et ses sous-groupes fin is. S´ eminaire Bourbaki, Nov. 2008, 61` eme\nann´ e,1000 (2010), 75–100.\n[13] C. Shramov, V. Vologodsky. Automorphisms of pointless surfaces. arXiv:1807.06477 (2018).\n[14] E. Yasinsky. The Jordan constant for Cremona group of ra nk 2, Bulletin of the Korean Mathematical\nSociety, 54(2017), no. 5, 1859–1871.\n17National research university ”Higher school of economics”, Lab oratory of algebraic\ngeometry, 6 Usacheva str., Moscow, 119048, Russia\nEmail address :alvlzaitsev1@gmail.com\n18" }, { "title": "2401.15684v1.Vortex_on_surfaces_and_Brownian_motion_in_higher_dimensions__special_metrics.pdf", "content": "Vortex on surfaces and Brownian-motion in higher\ndimensions: special metrics\nClodoaldo Grotta-Ragazzo∗.\nAbstract\nA single hydrodynamic vortex on a surface will in general moves\nunless its Riemannian metric is a special “Steady Vortex Metric”\n(SVM). Metrics of constant curvature are SVM only in surfaces of\ngenus zero and one.\nIn this paper:\n1. I show that K. Okikiolu’s work on the regularization of the spec-\ntral zeta function leads to the conclusion that each conformal\nclass of every compact surface with a genus of two or more pos-\nsesses at least one Steady Vortex Metric (SVM).\n2. I apply a probabilistic interpretation of the regularized zeta func-\ntion for surfaces, as developed by P. G. Doyle and J. Steiner, to\nextend the concept of SVM to higher dimensions.\nThe new special metric, which aligns with the Steady Vortex Met-\nric (SVM) in two dimensions, has been termed the “Uniform Drainage\nMetric” for the following reason: For a compact Riemannian manifold\nM, the “narrow escape time” (NET) is defined as the expected time\nfor a Brownian motion starting at a point pinM\\Bϵ(q) to remain\nwithin this region before escaping through the small ball Bϵ(q), which\nis centered at qwith radius ϵand acts as the escape window. The\nmanifold is said to possess a uniform drainage metric if, and only if,\nthe spatial average of NET, calculated across a uniformly distributedarXiv:2401.15684v1 [math.DG] 28 Jan 2024set of initial points p, remains invariant regardless of the position of\nthe escape window Bϵ(q), as ϵapproaches 0.\nKey words: point vortex, Riemann surfaces, diffusion process,\nBrownian motion, special metrics, spectral zeta function.\nAMS Classification: 76B47, 30F30, 58J65, 31C12, 60J45, 53C25\nAbbreviated title: SVM and Brownian-motion special metrics\n∗Instituto de Matem´ atica e Estat´ ıstica da Universidade de S˜ ao Paulo,\nRua do Mat˜ ao 1010, 05508-090, S˜ ao Paulo, SP, Brazil.\nPartially supported by FAPESP grant 2016/25053-8.\nemail: ragazzo@usp.br\nORCID: 0000-0002-4277-4173\n21 Introduction\nThe motion of point vortices on the plane is a classical subject in fluid me-\nchanics that goes back to Helmholtz, Kelvin, and Kirchhoff. The first to\nconsider the motion of point vortices on a curved surface, the sphere embed-\nded in R3, was Zermello in 1902. The paper [7] has a historical review on the\nearly research on hydrodynamic vortices on surfaces. An intrinsic definition\nof the motion of vortices on a surface, which is independent of the embedding\nof the surface in R3and on coordinates, started with Boatto and Koiller [5]\n(see also [6] [12] [40]) and was recently completed by Bj¨ orn Gustafsson [22]\n[23].\nA single vortex in the Euclidean plane, or in the round sphere, or in\na flat torus does not move, and this motivated the definition of “Steady\nVortex Metric” [40]: a Riemannian metric for which a single vortex does not\nmove regardless of its position. J. Koiller conjectured that a single vortex\nin a compact surface of constant curvature and of a genus greater than one\ndoes move. In [39], [24] Koiller’s conjecture was numerically verified for a\nparticular surface of constant curvature of genus two: the Bolza surface. This\nresult motivated the first main question to be answered in this work: Does\na steady vortex metric exist on any orientable compact surface of a genus\ngreater than one?\nK. Okikiolu proved that a certain functional on the space of Riemannian\nmetrics, which is an analog for closed surfaces of the ADM mass from general\nrelativity, has a minimizing metric on each conformal class. It turns out\nthat the special metrics of Okikiolu are steady vortex metrics, which gives\na positive answer to the question in the paragraph above. This raises the\nquestion about the “meaning” (or properties) of this special metric. The\nsteady vortex metric minimizes a certain functional [37] and has the property\n1in its name, but does it have any other interesting geometrical property\nbesides those? This question was the second motivation for this work.\nThe special metric found by Okikiolu is a critical point of a functional\nrelated to the regularized Green’s function of the Laplacian: the “Robin\nfunction”. P. G. Doyle and J. Steiner [11] gave a probabilistic interpretation\nto the Robin function that is related to the concept of “Narrow-Escape-\nTime”(NET) [26]. The NET is defined as the expected time for a Brownian\nmotion starting at pinM\\Bϵ(q) to remain within this region before escaping\nthrough the small ball Bϵ(q), which is centered at qwith radius ϵand acts\nas the escape window.\nThe NET is an important abstraction in science, as argued by Holcman\nand Schuss in the Introduction of [26]: “The narrow escape problem in dif-\nfusion theory, which goes back to Helmholtz (Helmholtz (1860)) and Lord\nRayleigh (Rayleigh (1945)) in the context of the theory of sound, is to cal-\nculate the mean first passage time of Brownian motion to a small absorbing\nwindow.... The renewed interest in the problem is due to the emergence of\nthe narrow escape time (NET) as a key to the determination of biological cell\nfunction from its geometrical structure. The NET is ubiquitous in molecu-\nlar and cellular biology and is manifested in stochastic models of chemical\nreactions...”\nThe average NET, with respect to a uniform distribution of initial posi-\ntions (volume measure), that a particle takes to escape from S\\Bϵ(q) through\nthe small window Bϵ(q) is proportional to −logϵ+R(q) +O(ϵ), where Ris\nthe Robin function. So, for small ϵthe Robin function indicates the drainage\ncapacity of different points qinS. The Robin function is constant if, and only\nif, the metric is a steady vortex metric (SVM). Therefore, in a surface with\naSV M the drainage capacity of different points is the same and this lead\n2to the alternative name “ uniform drainage metric ”, a property that makes\nsense in dimensions larger than two. Note: the notion of hydrodynamic point\nvortex cannot be generalized to dimensions greater than two.\nThe main contribution in this paper is the definition of uniform drainage\nmetric in dimensions greater than two and its geometric characterization in\ndimensions 3 and 4.\nFollowing the same steps given in this paper, a characterization of a\nuniform drainage metric in higher dimensions can be accomplished by means\nof certain coefficients that appear in the so-called Minakshisundaram-Plejel\nasymptotic expansion of the heat kernel. I prefer not to state any results\nin this direction because, in higher dimensions, it is necessary to compute\nmore of these coefficients, which can be expressed in terms of powers of the\nLaplacian and the distance function ℓ, and they become very complicated\n[38].\nThe existence of uniform drainage surfaces of arbitrary finite genus in\nany conformal class is guaranteed by the theorem of Okikiolu. In higher\ndimensions any compact Riemannian manifold that is a homogeneous space\nis a uniform drainage manifold.1Does there exist a closed (compact and\nboundaryless) manifold that does not admit a uniform drainage metric?\nThis paper is organized as follows.\nIn Section 2 I give a precise definition of the steady vortex metric and\npresent two fundamental theorems that stem from Okikiolu’s work. I then\nuse these theorems to compare the steady vortex metric with other natu-\n1There is a special class of Riemannian metrics on closed manifolds that are critical\nmetrics of the trace of the heat kernel under conformal variations of the metric [13]. A\nmetric in this special class is always a uniform drainage metric (a consequence of Theorem\n4.1 (ii) in [13]). The metric of any Riemannian homogeneous space is critical for the trace\nof the heat kernel.\n3ral Riemannian metrics: of constant curvature, canonical or Bergman, and\nArakelov. The proofs of the two theorems are presented in Appendix A\nin a slightly different way than those given by Okikiolu. These theorems\nplus some simple arguments imply: “No orientable surface of genus 2 and of\nconstant curvature is a Steady Vortex Surface.”\nIn Section 3, I present a regularization of the Green’s function in dimen-\nsions greater than two using the Minakshisundaram-Plejel asymptotic expan-\nsion of the heat kernel. This provides a definition of the Robin function in\nhigher dimensions. In the Appendix B I show that the Robin function can be\nwritten in terms of the analytic extension of the Minakshisundaram–Pleijel\nzeta function, and therefore uniform drainage manifolds have a special spec-\ntral property derived from this relation. The relation between the Robin\nfunction and the Minakshisundaram–Pleijel zeta function appeared in [43],\nfor surfaces, and in [4], in a more general context and in dimension greater\nthan two.\nIn Section 4 I give a characterization of uniform drainage metric in di-\nmensions 2, 3 and 4. In dimension 4 a uniform drainage metric has constant\nRobin function (a global property) and constant scalar curvature (a local\nproperty).\nIn Section 5, I present a family of non-flat tori found by Okikiolu [36],\nand which will be called Okikiolu’s tori, that are uniform drainage surfaces.\nThese tori are the only non-constant curvature uniform drainage surface that\nare explicitly known. For any a >p\nπ/2 there is an Okikiolu’s torus that is\nconformally equivalent to the flat torus R2/(aZ×a−1Z). Therefore, uniform\ndrainage metrics may not be unique in a conformal class. The curvature of\nthe Okikiolu’s tori was computed in ibid., where it was realized that in the\nlimit as a→ ∞ the curvature at almost every point of the torus tends to\n41/√\n4π. In Section 5 I embed a cylinder in R3whose quotient under the\ngroup of translations along the cylinder axis is an Okikiolu’s torus. In this\nway, I can visualize the deformation of a flat torus into a pinched torus that\nis isometric to a round sphere with two opposite points being identified. The\ndeformation is done along an interesting family of uniform drainage surfaces.\nI finally remark about a possible upshot of the relation between the Robin\nfunction and the drainage capacity of different points. The importance of the\nNET in cellular biology is partially due to diffusion processes that occur in\nmembranes towards special exit gates (escape windows). The minimum of\nthe Robin function is an equilibrium position of a single vortex [24] and\nalso a point where the drainage capacity of the surface, as defined above,\nis maximum. Equilibrium positions of systems of point vortices, an issue\nthat has been extensively studied, also have a probabilistic interpretation. If\nthe position of a vortex is related to an entrance or exit gate, depending on\nthe vortex sign, then some equilibrium configurations will certainly be more\nefficient in connecting different gates by means of diffusion than others. If\nthis idea is correct, then the the importance of equilibrium configurations\non surfaces of spheres, including those which are not round, will be greatly\nenhanced.\n2 Steady vortex metrics on orientable closed\nsurfaces.\nThe definition of a hydrodynamic vortex requires some preliminaries (see\n[40]). The fundamental equations of hydrodynamics on S, Euler’s equations,\nnecessitate that Sbe endowed with a Riemannian metric g. Here, grepre-\nsents a smooth family of inner products on the tangent spaces of S. In local\n5coordinates, the Riemannian metric is given by g=P\njkgjkdxj⊗dxk. The\nassociated volume form is µ=p\n|g|dx1∧dx2, where |g|denotes the absolute\nvalue of the determinant of the matrix gjk\nIn a neighborhood of each point of S, there exist coordinates (some-\ntimes called isothermal coordinates) in which g=λ2(x)(dx2\n1+dx2\n2) and\nµ=λ2(x)dx1∧dx2. The existence of isothermal coordinates is a mani-\nfestation of the fact that any surface is locally conformal to the Euclidean\nplane. In this paper, I will also use λ2to denote the conformal factor between\narbitrary given metrics g0andg1. This will be explicitly stated when used.\nThe one-forms θ1=λdx 1andθ2=λdx 2constitute an orthonormal mov-\ning coframe. The Hodge-star operator acts linearly on forms and is defined\nby\n∗1 =θ1∧θ2=µ, ∗θ1=θ2,∗θ2=−θ1,∗µ= 1.\nThe Laplace operator acting on functions is given by ∆ = ∗d∗d=1\nλ2(∂2\nx1+\n∂2\nx2) and the Gaussian curvature by K=−1\nλ2∆ log λ.\nLetV=R\nSµbe the total area of S. The Green’s function of ( S, g) is the\nunique solution in distribution sense to the equation\n−∆qG(q, p) =δp(q)−V−1, (2.1)\nthat has the following properties (see [2], theorem 4.13):\n•for all functions ϕ∈C2\nϕ(p) =1\nVZ\nSϕµ−Z\nSG(q, p)∆ϕ(q)µ(q), (2.2)\n•G(q, p) is C∞onS×Sminus the diagonal,\n•Gis symmetric G(q, p) =G(p, q),\n•Gis bounded from below andR\nSG(q, p)µ(q) = 0.\n6A point vortex of intensity Γ ∈Rat the point pis the fluid velocity field\ndefined on S−{p}given by q→ ∗∇ ΓG(q, p), where ∇is the gradient operator\nand∗is the operator that rotates a vector by π/2.\nThe Robin function (the regularization of G) is a C∞function on S([40]\nTheorem 5.1) defined as\nR(p) = lim\nℓ(q,p)→0\u0014\nG(q, p) +1\n2πlogℓ(q, p)\u0015\n, (2.3)\nwhere ℓ(q, p) is the Riemannian distance between pandq.\nThe motion of a single vortex depends not only on its initial position but\nalso on the initial value of a harmonic velocity field (a background flow) [23].\nIn the following statement [40] [24] the initial background flow is assumed to\nbe equal to zero:\nA vortex initially placed at any point on a surface Swith Riemannian metric\ngremains at rest if, and only if, the Robin function Rassociated with gis\nconstant. A Riemannian metric with this property is called a “Steady Vortex\nMetric”.\nThe first main result in this paper is the following.\nTheorem 2.1 (Steady Vortex Metric) .LetSbe a compact Riemann surface.\nThere exists at least one steady vortex metric gcompatible with the conformal\nstructure of S. There are examples where gis not unique.\nThe theorem effectively says that there always exist a metric for which\nthe Robin function is constant.\nThis theorem is a direct consequence of a theorem proven by K. Okikiolu\n[36] [37] and its proof is given in Appendix A.\nThe second theorem in this Section requires some definitions. A one-form\nθonSis harmonic if dθ= 0 and d∗θ= 0. Since ∗rotates one-forms by\nπ/2, harmonic forms are conformal invariants. The vector space of harmonic\n7forms on Sis finite and has dimension 2 G[9], where Gis the genus of S. Let\n{θ1, . . . θ 2G}be an arbitrary orthonormal basis of harmonic one-forms in the\nsense that\n(θj, θk) =Z\nSθj∧ ∗θk=δjk. (2.4)\nNote: this definition of orthonormality depends only on the conformal struc-\nture.\nTheorem 2.2. Let(S, g)be a compact oriented Riemannian surface and σ\nbe the two-form\nσ=2GX\nk=1θk∧ ∗θk.\nThen the Robin function Ris the only solution, up to an additive constant,\nof the equation\u0012\n∆R+K\n2π−2\nV\u0013\nµ=−σ . (2.5)\nIf the genus GofSis zero, then σ= 0. So a metric on the sphere is a\nSteady Vortex Metric if, and only if, it is of constant curvature.\nIf the genus of Sis greater than zero, then σis the area form of the\nBergman metric. The most common definition of the Bergman metric ([28]\neqs. 1.4.22 and 1.4.23) uses a basis of holomorphic differentials {ω1, . . . , ω G}\nthat satisfy the orthonormality conditionsi\n2R\nSωj∧ωk=δjk(here the overbar\ndenotes complex conjugation). A form ωjis holomorphic if, and only if,\nωj=θj+∗θjfor some harmonic differential θj([14], Theorem I.3.11). If\nwe define θj+G=∗θj,j= 1, . . . ,G, then the orthogonality condition for\nholomorphic differentials implies the orthogonality condition for harmonic\ndifferentials (2.4) and\nσ=2GX\nk=1θk∧ ∗θk=iGX\nj=1ωj∧ωj,withZ\nSσ= 2G. (2.6)\n8The Bergman metric normalized as σ/(2G) can also be defined using the\nJacobian variety associated with S(see [44], [28], or equation 1.25 in [15]).\nIn several references [44] [37] [27] the normalized Bergman metric σ/(2G) is\ncalled by the alternative name “canonical metric”.\nTheorem 2.2 appeared in the work of Okikiolu [37] (proposition 2.3) as a\n“well known” result related to the Arakelov Green’s function (in Appendix\nA I give a more self-contained proof of Theorem 2.2 than that in [37]). The\nArakelov Green’s function is used in the definition of the “Arakelov metric”\nthat is characterized by the equation (see [28] eq. 1.4.24):\nKA\n2πµA= (2−2G)σ\n2G,G ≥1, (2.7)\nwhere: µAandKAdenote the area form and the curvature of the Arakelov\nmetric.\nEquation (2.5) implies that the several “natural” metrics considered in\nthis paper satisfy the following relations:\n\u0000Ksvm\n2π−2\nV\u0001\nµsvm=−σ (SVM)\n\u0000\n∆Rcc−2G\nV\u0001\nµcc=−σ (constant curvature=CC)\n\u0000\n∆RB+KB\n2π\u0001\nσ= (2−2G)σ\n2G(Bergman)\n\u0000\n∆RA−2\nV\u0001\nµA=−σ\nG(Arakelov)(2.8)\nFrom equations (2.7) and (2.8) we obtain\nCC=SV M ⇔Bergman =CC⇔Bergman =SV M ,\nArakelov =SV M ⇔Bergman =Arakelov ⇔Arakelov =CC .(2.9)\nForG ≥1, therefore, a constant curvature metric is a Steady Vortex Metric\nif and only if the Bergman metric has constant curvature. For G= 1 this is\n9the case, since the flat metric is the Bergman metric and also the Arakelov\nmetric.\nIn any closed surface Sof genus G ≥ 2 the curvature of the Bergman\nmetric is non positive [29] (theorem 5.5.1). If the curvature of the Bergman\nmetric KBis non constant in every S, which as far as I know has not been\nproved, then constant curvature metrics will never be SVM for G ≥2. The\nlast theorem in [30] states that KB(p) = 0 if and only if Sis hyperelliptic\nandpis one of the 2 G+ 2 classical Weierstrass points on S. Therefore KB\nis not constant in hyperelliptic surfaces. Since every surface of genus 2 is\nhyperelliptic, the following theorem holds.\nTheorem 2.3. No orientable surface of genus 2 and of constant curvature\nis a Steady Vortex Surface.\nThe Gauss-Bonnet theorem implies that the average curvature of the\nBergman metric is\nKBa:=1\nVBZ\nSKBσ= 2π2−2G\n2G,where VB=Z\nSσ= 2G. (2.10)\nWe define the deviatoric part KBδofKBas :\nKBδ:=KB−KBa withZ\nSKBδσ= 0. (2.11)\nEquation (2.8) then implies that the Robin function of the Bergman metric\nsatisfies the simple relation\n−∆RB=1\n2πKBδ. (2.12)\nThis equation implies that in any conformal coordinates, {z,z},RBhas a\nsimple expression in terms of the potentials Fj(z) of the holomorphic dif-\nferentials ωj=dFj=F′\nj(z)dz=∂zFj(z)dzthat appear in the definition\nofσin equation (2.6). Indeed: iPG\nj=1ωj∧ωj=iPG\nj=1F′\nj(z)F′\nj(z)dz∧\n10dz=λ2\nBi\n2dz∧dz, with λ2\nB= 2PG\nj=1F′\nj(z)F′\nj(z), ∆ =4\nλ2\nB∂z∂z,λ2\nB=\n2∂z∂zPG\nj=1Fj(z)Fj(z), and KB=−4\nλ2\nB∂z∂zlogλB; imply\nRB(z,z) =1\n4πlog\"GX\nj=1F′\nj(z)F′\nj(z)#\n+1− G\n2GGX\nj=1Fj(z)Fj(z) +constant .\n(2.13)\nThe Riemann sphere admits a six-dimensional group of conformal trans-\nformations (the Moebius group) and a three-dimensional group of isometries.\nThe Pull-back metric g1of the round metric g0by a Moebius transformation\nthat is not an isometry satisfies g1=λ2g0with λ2̸= 1 almost everywhere.\nThe Robin function associated to g1is constant because, although differ-\nent from g0,g1is isometric to g0. This type of “nonuniqueness” of a steady\nvortex metric within a conformal class will happen whenever the group of dif-\nfeomorphisms that preserves the conformal structure is larger than the group\nof isometries. Since all spheres with constant curvature are isometric to the\nround sphere, we conclude that g0is the only steady vortex metric modulo\nisometries. The question about the uniqueness of steady vortex metrics on\ntori will be postponed to Section 5.\n3 Generalization to higher dimensions.\nThe definition of hydrodynamic point vortex is restricted to two dimensions.\nThere is an analogy between vortex and electric charges in two dimensions\n[40]. Since the theory of electrostatics can be generalized to higher dimen-\nsions, electrostatics could be the physical guide to the definition of an “elet-\nrostactic force-free metric” in dimension n. The idea although interesting\nleads to some difficulties, which will be discussed in the next paragraph, and\nit will not be pursued any further.\n11The Green’s function G(q, p), solution to equation (2.1), can be under-\nstood as the electric potential due to a positive point charge at the point p\nplus a uniform distribution of negative charges. The Robin function R(p)\ndefined in equation (2.3) is the overall potential energy G(q, p) minus the\n“singular potential of the point charge”, −(2π)−1logℓ(q, p), the difference\nbeing evaluated at p. The force upon the point charge is dR(p). The most\nnatural definition of Robin function in dimension n≥3 would be\nlim\nℓ(q,p)→0\u0002\nG(q, p)−anℓn−2(q, p)\u0003\n, (3.14)\nwhere anis some constant that depends on n. Unfortunately the Robin func-\ntion defined in this way is not a smooth function unless additional hypotheses\nare imposed on the Riemannian metric (see [25] for a discussion about this\ndefinition in the context of the conformal Laplacian). Another way to define\nthe Robin function would be first to compute the force upon a small Rieman-\nnian ball of radius ϵatpand then to take the limit as ϵ→0 to obtain dR(p).\nThis procedure may lead to quite complicated computations as nincreases.\nFrom a mathematical point of view regularity is the key property of the\nRobin function, which in two dimensions is used in the definitions of vortex\nmotion and force upon an electric charge. In order to define the Robin func-\ntion in dimension greater than two we will regularize the δ-distribution, to do\nthe computations in the regularized setting, and then to take the limit back\nto recover the δ-distribution. In order to do all these limits independently of\ncoordinates we use the heat equation. This procedure naturally associates\nthe Robin function with diffusion and Brownian motion. This association\nwill be further addressed in Section 4.\nLet ( M, g) be a compact Riemannian manifold. The heat kernel K:\n12M×M×R+→Mis the fundamental solution to the heat equation\n\u0012∂\n∂t−∆q\u0013\nK(q, p, t ) = 0 ,with K(q, p,0) = δp(q). (3.15)\nThe initial condition is understood as a distribution, namely, for any ϕ∈\nC∞(M)Z\nMK(q, p, t )ϕ(q)µ(q)→ϕ(p) as t→0+.\nThe heat kernel is a C∞symmetric, K(q, p, t ) =K(p, q, t ), function. Let 0 <\nλ1≤λ2≤λ3≤. . .be the nontrivial eigenvalues to the problem ∆ ϕ+λϕ= 0\nandϕ1, ϕ2, . . .be a corresponding L2−orthonormal basis of eigenfunctions for\nfunctions that integrate to zero over M. Then the the spectral decomposition\nof the heat kernel is\nK(q, p, t ) =1\nV+∞X\nk=1e−λktϕk(q)ϕk(p),\nwith pointwise convergence (see, for instance, [41] for basic properties of the\nheat kernel). The Green’s function G(q, p) is related to the heat kernel in\nthe following way\nG(q, p) =Z∞\n0\u0012\nK(q, p, t )−1\nV\u0013\ndt\nThis is the formula that allows for the definition of the Robin function in\ndimension n >2 by means of the regularization of the heat kernel.\nAs before, let ℓ(q, p) denote the Riemannian distance between qandp.\nThere exists ϵ > 0 and a set of functions u0(q, p), u1(q, p), . . .such that\nfor any given integer N≥0 the following estimate holds ( the so-called\nMinakshisundaram-Plejel asymptotic expansion [32]; see Equations (7)–(9)\nand the accompanying text)\n\f\fK(q, p, t )−e−ℓ2(q,p)\n4t\n(4πt)n/2NX\nk=0uk(q, p)tk\f\f≤CNtN+1−n/2, (3.16)\n13for all ( q, p) with ℓ(q, p)< ϵand all t∈(0,1), where CNis a constant that\ndepends only on N( see, for instance, [41] exercise 5 in Section 3.3). The\nfunctions ukare C∞and symmetric uk(q, p) =uk(p, q) [33]. If q=pthen\nthe above expression implies\nK(p, p, t ) =1\n(4πt)n/2\u0002\na0(p) +ta1(p) +. . .+tNaN(p)\u0003\n+EN(p, t) (3.17)\nwhere |EN(p, t)|< C NtN+1−n/2for all p∈Mandt∈(0,1). The functions\nak(p) =uk(p, p) are local heat invariants of Mthat can be expressed in terms\nof powers of the Laplacian and the distance function ℓ([38], Theorem 1.2.1).\nFor instance, a0(p) = 1 and a1(p) =s(p)/6, where s(p) is the scalar curvature\n([41], proof of Lemma 3.26 and Proposition 3.29, respectively).\nSuppose that n≥2 is even and Nin equation (3.17) is chosen asn\n2−1.\nThen for 0 < ϵ < 1 equation (3.17) implies\nZ1\nϵK(p, p, t )dt=1\n(4π)n/2\nn\n2−2X\nk=0ak(p)ϵk+1−n\n2\nn\n2−k−1−an\n2−1(p) logϵ\n+C(p, ϵ)\nwhere lim ϵ→0+C(·, ϵ) is a C∞function on M. Similarly, if n≥3 is odd and\nNin equation (3.17) is chosen asn\n2−3\n2then\nZ1\nϵK(p, p, t )dt=1\n(4π)n/2\nn\n2−3\n2X\nk=0ak(p)ϵk+1−n\n2\nn\n2−k−1\n+˜C(p, ϵ)\nwhere lim ϵ→0+˜C(·, ϵ) is a C∞function on M. These computations motivate\nthe following definition of the Robin function R(p). Ifn≥2 is even then\nR(p) = lim\nϵ→0+(Z∞\nϵ\u0012\nK(p, p, t )−1\nV\u0013\ndt\n−1\n(4π)n/2\nn\n2−2X\nk=0ak(p)ϵk+1−n\n2\nn\n2−k−1−an\n2−1(p)\u0000\nlog(4 ϵ)−γ\u0001\n)\n,\n(3.18)\n14where γ=−R∞\n0e−xlogx dx = 0.577215 . . .is the Euler’s constant. The\nconstant term an\n2−1(p)(log 4 −γ)/(4π)n/2were added to the right-hand side\nof equation (3.18) to preserve the definition of the Robin function given in\nequation (2.3).\nIfn≥3 is odd then\nR(p) = lim\nϵ→0+(Z∞\nϵ\u0012\nK(p, p, t )−1\nV\u0013\ndt\n−1\n(4π)n/2\nn\n2−3\n2X\nk=0ak(p)ϵk+1−n\n2\nn\n2−k−1\n)\n.(3.19)\nTheorem 3.1. The Robin function can be written in the following alternative\nway: for n≥3odd\nR(p) = lim\nq→p\n\nG(q, p)−1\n(4π)n/2n\n2−3\n2X\nk=0uk(q, p)\u00124\nℓ2(q, p)\u0013n\n2−k−1\nΓ(n\n2−k−1)\n\n\nand for n≥2even\nR(p) = lim\nq→p(\nG(q, p)−1\n(4π)n/2n\n2−2X\nk=0uk(q, p)\u00124\nℓ2(q, p)\u0013n\n2−k−1\nΓ(n\n2−k−1)\n+1\n(4π)n/2un\n2−1(q, p)h\nlogℓ2(q, p)i)\n.\nProof. At first we prove the statement for nodd. From equation (3.19)\nR(p) = lim\nϵ→0+lim\nq→p(Z∞\n0\u0012\nK(q, p, t )−1\nV\u0013\ndt−Zϵ\n0\u0012\nK(q, p, t )−1\nV\u0013\ndt\n−1\n(4π)n/2\nn\n2−3\n2X\nk=0uk(q, p)ϵk+1−n\n2\nn\n2−k−1\n\n\n(3.20)\nThe termRϵ\n0K(q, p, t )dtis estimated in the following way. From equation\n(3.16)\nK(q, p, t ) =e−ℓ2(q,p)\n4t\n(4πt)n/2n\n2−3\n2X\nk=0uk(q, p)tk+ Φ 1(q, p, t )\n15such that |Φ1(p, p, t )| ≤ C1t−1/2,C1>0, for t∈(0,1) (the constants\nC1, C2. . .do not depend on t,q,p, orϵ). Therefore\nZϵ\n0K(q, p, t )dt=1\n(4π)n/2n\n2−3\n2X\nk=0uk(q, p)Zϵ\n0e−ℓ2(q,p)\n4tt−n\n2+kdt+ Φ 2(q, p, ϵ )\nsuch that |Φ2(p, p, ϵ )|< C 2ϵ1/2,C2>0. With the change of variables t=ℓ2\n4s\nwe obtain\nZϵ\n0e−ℓ2\n4tt−n\n2+kdt=\u0012ℓ2\n4\u0013k+1−n\n2\"Z∞\n0e−ssn\n2−k−2ds−Zℓ2\n4ϵ\n0e−ssn\n2−k−2ds#\nwhereR∞\n0e−ssn\n2−k−2ds= Γ(n\n2−k−1) is the Gamma function. Using that\ne−s= 1−sF(s), where F(s) =R1\n0e−ηsdη, an explicit computation gives\n\u0012ℓ2\n4\u0013k+1−n\n2Zℓ2\n4ϵ\n0e−ssn\n2−k−2ds=ϵ−n\n2+k+1\nn\n2−k−1−ℓ2\n4Φ3\u0012ℓ2\n4ϵ\u0013\nwhere |Φ3\u0010\nℓ2\n4ϵ\u0011\n|<1. Therefore,\nZϵ\n0e−ℓ2\n4tt−n\n2+kdt=\u0012ℓ2\n4\u0013k+1−n\n2\nΓ(n\n2−k−1)−ϵ−n\n2+k+1\nn\n2−k−1+ℓ2\n4Φ3\u0012ℓ2\n4ϵ\u0013\nFinally, using thatR∞\n0\u0000\nK(q, p, t )−1\nV\u0001\ndt=G(q, p) and substituting all the\nprevious estimates into equation (3.20) we obtain\nR(p) = lim\nϵ→0+lim\nq→p\n\nG(q, p)−1\n(4π)n/2n\n2−3\n2X\nk=0uk(q, p)\u0012ℓ2\n4\u0013k+1−n\n2\nΓ(n\n2−k−1)\n+ϵ\nV−Φ2(q, p, ϵ )−ℓ2(q, p)\n41\n(4π)n/2n\n2−3\n2X\nk=0uk(q, p)Φ3\u0012ℓ2(q, p)\n4ϵ\u0013\n\n.\nThe limit as q→pof the second line of this equation isϵ\nV. Since for a fixed\nϵ, the limit as q→pof the expression inside brackets exists then the limit\nasq→pof the sum in the first line also exists and does not depend on ϵ.\nSo, the proof for nodd is finished.\n16The proof for neven is similar. The only difference is that it is necessary\nto estimate the additional integral\nZϵ\n0e−ℓ2\n4tt−1dt=Z∞\nℓ2\n4ϵe−ss−1ds=−log\u0012ℓ2\n4ϵ\u0013\n+Z∞\nℓ2\n4ϵe−slogs ds.\nThe last integral is equal to minus the Euler’s constant as q→p. □\nWe remark that for n >2 the term of highest order in ℓ−2is\n1\n(4π)n/2\u00124n\n2−1\nℓn−2(q, p)\u0013\nΓ(n\n2−1),\nwhere we used that u0(p, p) =a0(p) = 1, is minus the “Newtonian potential”\nthat appears in equation (3.14).\nForn= 2, theorem 3.1 states that the Robin function defined by equation\n(3.18) coincides with that given in equation (2.3).\nTheorem 3.1 can also be obtained from the Hadamard parametrix, see\n[19] section 5.3.\nThe Robin function as given in Theorem 3.1 can be written in terms of\nthe analytic extension of the Minakshisundaram–Pleijel zeta function, [43]\n(dimension two) and [4] (dimension greater than one). The relation between\nthe Robin function and the zeta-function is presented in Appendix B.\n4 The “Narrow Escape Time (NET)”.\nIn the context of a compact boundaryless manifold M, the narrow escape\nproblem can be described in the following way. Consider a Brownian motion\nonM, whose infinitesimal generator is the Laplace-Beltrami operator ∆. Let\nBϵ(q)⊂Mbe a geodesic ball of small radius ϵ >0. This ball will be the\nabsorbing set or the small window through which a particle can escape. The\namount of time that a particle initially at pis expected to spend in M\\Bϵ(q)\n(the mean sojourn time) will be denoted as vϵ(p, q). This function is the\n17“narrow escape time” (NET) since it measures the mean time it takes for a\nparticle initially at pto escape through the narrow window Bϵ(q). The NET\nis the solution to the problem (see [26], equation 3.1):\nν∆pvϵ(p, q) =−1, p∈M\\Bϵ(q),\nwith vϵ(p, q) = 0 for p∈∂Bϵ(q),(4.21)\nwhere νis a diffusion coefficient with dimensional units length2/time. The\nNET averaged against a uniform distribution of initial points in M\\Bϵ(q),\nvϵ(q) =1\nV− |Bϵ(q)|Z\nM\\Bϵ(q)vϵ(p, q)µ(p), (4.22)\ngives the expected time a particle randomly placed in the manifold remains\nin it until it scapes through Bϵ(q).\nIn dimension 2 the following theorem was proved in [11] (Lemma 4.1 and\nTheorem 4.2 part 2).\nTheorem 4.1. In dimensions 2, 3, and 4, the “Narrow Escape Time” (NET)\nis given by\nvϵ(p, q) =−V\nνG(p, q) +vϵ(q) +E(p, q, ϵ ), (4.23)\nwhere limϵ→0E(p, q, ϵ ) = 0 . The average NET, equation (4.22) , is given by\nvϵ(q) =V\nν\b\n−1\n2πlogϵ+R(q) +E2(q, ϵ)\t\nn= 2\nvϵ(q) =V\nν\b1\n4πϵ−1+R(q) +E3(q, ϵ)\t\nn= 3\nvϵ(q) =V\nν\b1\n4π2ϵ−2−1\n48π2S(q) logϵ+1\n192π2S(q) +R(q) +E4(q, ϵ)\t\nn= 4,\nwhere S(q)is the scalar curvature at qandlimϵ→0En(p, q, ϵ ) = 0 .\nRemarks:\n18•The normalizationR\nMG(p, q)µ(p) = 0 (equation (2.2)) ensures the com-\npatibility of both sides of equation (4.23).\n•The NET increases as ϵdecreases in the same way as the Newtonian poten-\ntial in Rnincreases as the distance to the singularity decreases (see e.g. [26],\nsection 3, for the same result for surfaces). This is true in all dimensions,\nnot only n= 2,3,4.\n•In dimensions 2 and 3, the divergent terms of vϵ(q) with respect to ϵdo\nnot depend on q. For n= 4 this is no longer true, since vϵ(q) contains a\nlogarithmic divergent term that is proportional to the mean curvature S(q).\nIf the mean curvature is constant on M, then the dependence of vϵ(q) onq\nasϵ→0 is determined by the Robin function, as it is in dimensions 2 and 3.\nProof. We will prove only the case n=4. The proof of the cases n=2 and\nn=3 is simpler and goes along the same lines.\nWe write vϵ(p, q) =−V\nνG(p, q) +V\nνhϵ(p, q) and from equations (2.1) and\n(4.21) we obtain\n∆phϵ(p, q) = 0 , p∈M\\Bϵ(q),with hϵ(p, q) =G(p, q) for p∈∂Bϵ(q).\n(4.24)\nLetx∈Rnbe an orthonormal coordinate system on the tangent space\nofMatq. Let x(p) := exp−1\nqpbe geodesic normal coordinates in Mdefined\nin a neighborhood of q. The metric tensor in this coordinates is given by\ngij(x) =δij−1\n3Rikjlxkxl+O(|x|3). Theorem 3.1 implies that for psufficiently\nclose to q\nG(p, q) =u0(p, q)\n(4π)2\u00124\n|x|2\u0013\n−u1(p, q)\n(4π)2log|x|2+R(q) +R1(p, q).\n19From u0(p, q) = 1 /q\ndet(expqp) ([41] equation (3.11)) we obtain\nu0(q, p) =1\n(detg)1/4= 1−1\n12Ri\nkil(q)xkxl+O(|x|3)\n= 1−1\n48S(q)|x|2−1\n12Zkl(q)xkxl+O(|x|3)\nwhere: Ri\njklis the Riemman curvature tensor and Zklis the traceless Ricci\ntensor. From [41] Proposition 3.29,\nu1(p, q) =u1(q, q) +O(|x|) =1\n6S(q) +O(|x|).\nThe expressions in the previous paragraph imply that for psufficiently\nclose to q\nG(p, q) =1\n4π2|x|2−S(q)\n192π2−S(q)\n48π2log|x|+R(q)\n| {z }\nterm a−Zkl(q)\n48π2xkxl\n|x|2\n|{z}\nterm b+R2(p, q)|{z}\nterm c,\n(4.25)\nwhere lim |x|→0R2(p, q) = 0.\nThe solution hϵto the problem in equation (4.24) can be split into three\nterms a, b, and c, according to the decomposition of the boundary conditions\nas given in equation (4.25). The term a is constant for |x|=ϵ. This term\nappears in the expression for vϵ(q) in the statement of the theorem.\nThe maximum principle dictates that the maximum of the function p7→\n|hc(p, q, ϵ )|, where hc(p, q, ϵ ) is the solution to\n∆phc,ϵ(p, q) = 0 , p∈M\\Bϵ(q),with hc,ϵ(p, q) =R2(p, q) for p∈∂Bϵ(q),\nis attained on ∂Bϵ(q). Given that hc,ϵ(p, q) =R2(p, q) for p∈∂Bϵ(q), the\nlimit ϵ→0 implies p→q, and since lim p→qR2(p, q) = 0, it follows that\nlimϵ→0|hc(p, q, ϵ )| →0 for p∈M\\Bϵ(q). Consequently, the component of\nhϵcorresponding to term (c) in equation (4.25) contributes to the function\nE(q, p, ϵ ) as stated in the theorem.\n20The part of hϵassociated the term b in equation (4.25), will be denoted\nasHϵ(p, q). It satisfies the problem\n∆pHϵ(p, q) = 0 with Hϵ(p, q) =−Zkl(q)\n192π2xkxl\nϵ2for|x|=ϵ . (4.26)\nIn order to finish the proof we must show that |Hϵ(p, q)| → 0 as ϵ→0\n|Hϵ(p, q)| →0 asϵ→0Y. The proof has several steps.\nProposition 4.1.\nZ\n|x|=ϵZkl(q)xkxl\nϵ2dσ(x) =O(ϵ5),\nwhere dσ(x)is the “area form” on the geodesic sphere ∂Bϵ(q).\nProof. The area form on ∂Bϵ(q) satisfies dσ(x) = [1−1\n6Ri\nkilxkxl+O(|x|3)]dσE(x),\nwhere dσE(x) is the Euclidean area form on the sphere |x|=ϵ. The func-\ntionZkl(q)xkxlis harmonic with respect to the Euclidean Laplacian, since\nthe trace of Zis zero, and therefore its integral over |x|=ϵwith respect to\ndσE(x) is zero. The proposition follows from the expression for dσ(x) and\n|dσE|=O(ϵ3) on|x|=ϵ. □\nThe identity ∆ z[G(z, q)−G(z, p)] =δp(z), for zandpinM\\B2ϵ(q), and\nGreen’s second identity imply that for ϵsufficiently small\nHϵ(p, q) =Z\n∂B2ϵHϵ(x, q)∇x[G(x, q)−G(x, p)]·x\n|x|dσ(x)\n−Z\n∂B2ϵ[G(x, q)−G(x, p)]∇xHϵ(x, q)·x\n|x|dσ(x).(4.27)\nWe will first estimate the integral in the second line of equation (4.27).\nEquation (4.25) implies that G(x, q)−G(x, p) with |x|= 2ϵcan be written\nas a term A1=O(ϵ2) that does not depend on xand a term A2that is\nbounded by a constant C1(q) that is independent of xandϵ. The integral\nA1R\n∂B2ϵ∇xHϵ(x, q)·x\n|x|dσ(x) = 0 becauseR\nM\\B2ϵ(q)∆pHϵ(p, q)µ(p) = 0. In\n21order to estimate the integral that contains A2we will use one of the Schauder\ninterior estimates [20] (Corollary 6.3)2\nϵmax\n|x|=2ϵ\f\f\f\f∇xHϵ(z, q)·x\n|x|\f\f\f\f≤max\np∈M\\Bϵ(q)|Hϵ(p, q)| ≤max\n|x|=ϵ|Hϵ(x, q)|=C2(q),\nwhere the second inequality follows from the maximum principle and the\nconstant C2(q) does not depend on ϵ. So\f\f\fR\n|x|=2ϵA2∇xHϵ(x, q)·x\n|x|dσ(x)\f\f\f=\nO(ϵ2) and the integral in the second line of equation (4.27) is at most of the\norder of ϵ2. It remains to estimate the integral in the first line of equation\n(4.27).\nFor a fixed p∈M\\B2ϵ(q) the function ∇x[G(x, p)]·x\n|x|restricted to |x|= 2ϵ\nis uniformly bounded with respect to ϵ. Therefore, using that |Hϵ(x, q)|<\nC2(q), we obtainR\n|x|=2ϵHϵ(x, q)∇x[G(x, p)]·x\n|x|dσ(x) =O(ϵ3) and it remains\nto estimateR\n∂B2ϵHϵ(x, q)∇x[G(x, q)]·x\n|x|dσ(x).\nThe term1\n4π2|x|2in equation (4.25) is the leading order term of a parametrix\nfor the Laplace equation (see [18], equation (5.79), or [2], theorem 4.13 equa-\ntion (17)). This implies that G(x, q)−1\n4π2|x|2, where G(x, q) is given in\nequation (4.25), can be differentiated for x̸= 0 and the derivative of R2(x, q)\nis dominated by those of the other terms, so that |∇x[G(x, q)−1\n4π2|x|2]·x\n|x||=\nO(1/|x|). This and |Hϵ(x, q)|< C 2(q) imply\nZ\n|x|=2ϵHϵ(x, q)∇xG(x, q)·x\n|x|dσ=Z\n|x|=2ϵHϵ(x, q)∇x\u00141\n4π2|x|2\u0015\n·x\n|x|dσ+O(ϵ2).\nIt remains to estimate the integral in the right-hand side of this equation.\nGreen’s second identity with ∆ xHϵ(x, q) = 0 and ∇xh\n1\n4π2|x|2i\n·x\n|x|=\n2Here is the reason for having integrated over the domain M\\B2ϵ(q) and not M\\Bϵ(q).\n22−1\n2π2|x|3, which is valid because xare normal coordinates, imply\nZ\nB2ϵ(q)\\Bϵ(q)Hϵ(x, q)∆\u00141\n4π2|x|2\u0015\ndx4=\n−1\n16π2ϵ3Z\n|x|=2ϵHϵ(x, q)dσ+1\n2π2ϵ3Z\n|x|=ϵHϵ(x, q)dσ\n−1\n16π2|ϵ|2Z\n|x|=2ϵ∇Hϵ(x, q)·x\n|x|dσ+1\n4π2|ϵ|2Z\n|x|=ϵ∇Hϵ(x, q)·x\n|x|dσ .\nThe integrals in the last line are zero becauseR\nM\\Bsϵ(q)∆pHϵ(p, q)µ(p) = 0, for\ns= 1,2. Due to equation (4.26) and Proposition 4.1,1\n2π2ϵ3R\n|x|=ϵHϵ(x, q)dσ=\nO(ϵ2). A computation using the expression for the Laplacian in geodesic\nnormal coordinates [41] (Theorem 2.63) gives ∆h\n1\n4π2|x|2i\n=O(|x|−2). This\nand|Hϵ(x, q)|< C 2(q) implyR\nB2ϵ(q)\\Bϵ(q)Hϵ(x, q)∆h\n1\n4π2|x|2i\ndx4=O(ϵ2). In\nconclusion, all these estimates imply\nZ\n|x|=2ϵHϵ(x, q)∇x\u00141\n4π2|x|2\u0015\n·x\n|x|dσ=−1\n16π2ϵ3Z\n|x|=2ϵHϵ(x, q)dσ=O(ϵ2),\nwhich finishes the proof. □\n5 Examples of non-constant curvature uni-\nform drainage surfaces: Okikiolu’s tori.\nThe flat metric g0on any two-dimensional torus is a steady vortex metric\n(SVM). Equation (2.5) implies that there exists a second SVM g1conformal\ntog0,g1=λ2g0, if and only if\n\u0012K1\n2π−2\nV1\u0013\nµ1=−2\nV0µ0 (5.28)\nNormalizing the volumes µ0andµ1such that V0=V1= 1, using −∆0logλ=\nλ2K1andµ1=λ2µ0, and defining f= log λ2we get the following equation\nforf\n∆0f= 8π−8πef(5.29)\n23To each nontrivial solution to this equation corresponds a SVM g1conformal\ntog0.\nIn the following we present a family of examples due to Okikiolu [36] of\nnon flat 2-dimensional tori that have constant Robin function, and so are\nuniform drainage surfaces. Each non flat torus in the family is conformal to\na flat torus, which is also a uniform drainage surface. The Robin function\nof the non flat tori are smaller than those of the conformally equivalent flat\ntori, and so the narrow escape time of the non flat tori are smaller than those\nof the conformally equivalent flat tori. There are two differences between our\npresentation and that of Okikiolu. We simplify the proof that the Robin\nfunctions of the non flat tori are smaller than those of the flat tori and we\nrepresent the non flat tori in R3as the quotient of an isometrically embedded\ncylinder.\nConsider the torus R2/(aZ×a−1Z),a≥1, with the conformal structure\ninduced by the flat metric g0. Ifa≤2/√π, then g0is the unique uniform\ndrainage metric [35], and if a >p\nπ/2, then g0is not unique [31]. When\na >p\nπ/2 a second natural vortex metric can be constructed in the following\nway [36]. Let ( x, y) be Cartesian coordinates on R2. We will look for a\nnontrivial solution to equation (5.29) that depends only on the variable x,\n∂yf= 0, with f(x+a) =f(x). Then fmust satisfy ¨f:=d2f\ndx2= 8π(1−ef).\nThis ordinary differential equation has a single equilibrium and a first integral\nH(f, p) =p2/2 + 8 π(ef−f−1), p =˙f . (5.30)\nThis shows that all solutions fare periodic with a period T(E), where E\nis the value of the first integral associated to the solution. The linearized\nperiod at ( f,˙f) = (0 ,0) is T(0) =p\nπ/2.\nThe period function E→T(E) of equation ¨f= 8π(1−ef) was studied\nin [8] (p. 315), where it is shown thatd\ndET(E)>0. We will additionally\n24show that lim E→∞T(E) =∞. Consider the solution associated to the initial\ncondition f(0) = 0, ˙f(0) =−√\n2Eand integrate the equation ¨f= 8π(1−ef)\non the interval [0 , β], where β >0 is the smallest value such that f(β) = 0.\nSince ˙f(β) =√\n2E, the result is\n√\n2E/(4π) =β−Zβ\n0efdt < β < T (E), (5.31)\nand therefore lim E→∞T(E) =∞. As a result, equation (5.29) has nontrivial\nsolutions for all a >p\nπ/2 such that f(x+a) = f(x) (indeed as many\ndifferent solutions as we wish provided ais sufficiently large).\nFor a given a >p\nπ/2, let g1= ef(x)(dx2+dy2) be the metric associated\nto a periodic solution to ¨f= 8π(1−ef) with minimal period a. We will\nuse lemma A.1 to show that the Robin function R1associated to g1has a\nsmaller value than the Robin function R0of the flat metric. The area form\nassociated to g1is given by µ1= efµ0=h\n1−¨f\n8πi\ndx∧dyand the equation\nthat determines the function ϕin lemma A.1 becomes\n∆0ϕ dx∧dy=µ1−µ0=−¨f\n8πdx∧dy ,Z\nSϕµ0= 0\nthat implies\nϕ(x) =−f(x)\n8π+1\n8πaZa\n0f(x)dx .\nThe constant c=−1\nVR\nSϕ(p)µ1(p) in lemma A.1 can be easily computed\nand is equal to c=1\n(8π)2aRa\n0˙f2dx. These computations and equation (A.38)\nimply\nR1−R0=1\n4πaZa\n0fdx+1\n(8π)2aZa\n0˙f2dx (5.32)\nIf we use the first integral Hin equation (5.30) to eliminate fin the right-\nhand side of this equation and then use1\naRa\n0efdx= 1, which we obtain\nintegrating ¨f= 8π(1−ef) over the interval [0 , a], then\nR1−R0=−H\n32π2+1\n32π2aZa\n0˙f2dx . (5.33)\n25The equation ¨f= 8π(1−ef) can be written in Hamiltonian form with Hamil-\ntonian function H. Using the definition of the action I(E) =1\n2πH\npd ffrom\nHamiltonian mechanics [1] we can write\n1\n32π2aZa\n0˙f2dx=1\n32π2aZa\n0p˙fdx=1\n16πa1\n2πI\npd f=I(E)\n16πa\nIn this expression ais the period of f, and therefore a=T(E) where Eis\nthe value of Hassociated to f. The Hamiltonian function can be written as\na function of the action E=H(I) with H′(I) = 2 π/T(I). All these results\nimply that equation (5.33) can be written as\nR1−R0=1\n32π2(IH′(I)−H(I)) (5.34)\nSince H′(I) = 2 π/T(I)>0 and T′(I)>0 [8] (p. 315), we conclude\nthat H′′(I) =−2πT′(I)/T2(I)<0. This fact and H(0) = 0 imply that\nR1−R0<0. In Figure 1 we present a numerical estimate of the difference\nR1−R0.\nThe torus\b\nR2/(aZ×a−1Z), g1\t\ncan be represented as the quotient of a\ncylinder that is infinite along the x-axis and periodic with period a. We will\nshow that this cylinder can be isometrically embedded in the Euclidean three-\nspace. Let X, Y, Z be Cartesian coordinates in R3. We will look for an embed-\nding of the form X=X(x),Y=F(x) sin(2 πay) and Z=F(x) cos(2 πay),\nwhere x∈R, y∈R/a−1Z. The pull-back of the Euclidean metric by the\nembedding is ( ˙X2+˙F2)dx2+ 4π2a2F2dy2. We impose that the pull-back\ncoincides with g1= ef(x)(dx2+dy2) and obtain that 4 π2a2F2= efand\n˙X2+˙F2= ef. This implies that F(x) = ef(x)/2/(2πa) and\n˙X2= ef \n1−˙f2\n16π2a2!\n(5.35)\nSince X:R→Rmust be a diffeomorphism, the right-hand side of equation\n(5.35) must be strictly positive for all x∈[0, a]. We will show this in the\nfollowing paragraph.\n261 2 3 4a\n-1.0-0.8-0.6-0.4-0.20.0R1-R0\nπ\n2\n1 2 3 4a\n-0.21-0.20-0.19-0.18-0.17-0.16-0.15R\nR0\nRS\nR1Figure 1: LEFT: Difference R1−R0as a function of a, where R1(R0)\nis the Robin function of the non flat torus {R2/(aZ×a−1Z), g1}(flat\ntorus {R2/(aZ×a−1Z), g0}). RIGHT: Graphs of R1andR0as a func-\ntion of a. The horizontal line represents the value of the Robin func-\ntion RSfor a round sphere of area 1. According to [36] (Appendix):\nR0(a) =−log(2 π)\n2π−log(|η(ia2)|4a2)\n4πandRS=−1+log π\n4π, where ηis the Dedekind\neta function.\nThe first integral (5.30) and (ef−f−1)≥0 imply that ˙f2(x)≤2Efor\nx∈[0, a], where Eis the value of Hfor the solution with period T(E) =a.\nThis and inequality (5.31) imply\n1−˙f2\n16π2a2≥1−2E\n16π2T(E)2>0.\nIn Figure 2 we show the curves x→ {X(x), Z(x)},x∈[0, a] and y= 0,\nthat when rotated about the X−axis generate the embedded cylinders. These\ncurves were obtained by the numerical integration of equations ¨f= 8π(1−ef)\nand (5.35) for: a= 1.255,a= 1.50, and a= 3.0. Only one fundamental cell\nof the periodic cylinder is shown. There are two different tori with a= 3: one\nfor which the minimal period of fis 3 and another for which the minimal\nperiod of fis 1.5, and so foscillates twice inside a fundamental cell. In\nFigure 3 we show a 3-dimensional representation of a single cell of each one\nof the cylinders whose generators are in Figure 2. It is clear from Figure\n3 that for a≫1 the cylinder becomes a collection of aligned spheres each\n27one touching its neighbors at a single point. This is in agreement with the\ninterpretation given in [36]: (the non-flat torus) “is approximately spherical\nexcept for a short wormhole joining the poles”. Note: as shown in the right\npanel of Figure 1, in the limit as a→ ∞ the tori converge to a punctured\nsphere and R1(a)→RSwhere RSis the Robin function of the round sphere.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2X0.050.100.150.20Za=1.255\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7X0.050.100.150.200.250.30Za=3.00\n0.0 0.2 0.4 0.6 0.8 1.0X0.050.100.150.200.25Za=1.50\n0.0 0.5 1.0 1.5 2.0X0.050.100.150.20Za=3.00\nFigure 2: Generating functions of four periodic cylinders (each cylinder is\nconstructed rotating the graph of X→Z(X) about the X-axis). The quo-\ntient of a cylinder by the group of periodic translations gives a torus that is\nisometric to a non-flat torus with a steady-vortex metric. The value of the\nperiod aof each torus is shown in the corresponding figure. There are two\ndifferent tori with a= 3: one for which the minimal period of fis 3 and\nanother for which the minimal period of fis 1.5, and so foscillates twice\ninside a fundamental cell.\nA Proofs of theorems 2.1 and 2.2.\nLemma A.1. Letg0andg1be two different Riemannian metrics on Sin\nthe same conformal class, g1=λ2g0. Let Gj,Rj,µj,Kj,∆j,j= 0,1, be\nthe: Green’s function, Robin function, volume form, Gaussian curvature, and\nLaplace operator, of gj. Let the conformal factor λbe normalized such that\n28Figure 3: Three-dimensional representation of the tori whose generators are\nshown in Figure 2. See the caption of Figure 2 for explanations.\n29the volumesR\nSµ0=R\nSµ1=Vare the same. Let ϕbe the unique solution of\nd∗dϕ=µ1−µ0\nVwithZ\nSϕµ0= 0,\nthat is given by\nϕ(p) =−1\nVZ\nSG0(q, p)λ2(q)µ0(q) =−1\nVZ\nSG0(q, p)µ1(q) (A.36)\nThen G0,G1,R0andR1satisfy the following relations:\nG1(q, p)−G0(q, p) =ϕ(q) +ϕ(p) +c (A.37)\nR1(p) =R0(p) +1\n2πlogλ(p) + 2ϕ(p) +c (A.38)\nwhere\nc=−1\nVZ\nSϕ(p)µ1(p) =1\nV2Z\nSZ\nSG0(q, p)µ1(q)µ1(p)\nis a constant. Equation (A.37) is in [34] (equation (8)) and Equation (A.38)\nis in [43] (Theorem 4).\nProof. Letpandqbe sufficiently close to be in a domain Uof a local\nuniformizer z. Suppose that Uis such that any two points in Uare connected\nby a single geodesic in U. In this coordinates the length elements of the\nmetrics g0andg1areλ0|dz|andλ1|dz|, respectively. Notice that λ1=λλ0.\nIfµ=dx∧dyand ∆ =∂2\n∂x2+∂2\n∂y2denote the area form and the usual Laplacian\nin the coordinates z= (x, y), respectively, then\n∆j=1\nλ2\nj∆,−∆ log λj=λ2\njKj, µ j=λ2\njµ, j = 0,1.(A.39)\nThe Dirac-delta distributions associated to the volume forms µ0andµ1sat-\nisfy\nδj,w=1\nλ2\njδw,where ψ(w) =Z\nψ(x, y)δw(x, y)dx∧dy .\n30To simplify the notation we write z(q) =zandz(p) =w. In the coordi-\nnates ( z, w) equation (2.1) becomes\n−∆zGj(z, w) =δw(z)−λ2\nj(z)\nV. (A.40)\nThe Green’s function can be written as\nGj(z, w) =−1\n2πlog|z−w|+fj(z, w) (A.41)\nwhere fj(z, w) =fj(w, z). Since ∆ zlog|z−w|= 2πδw, we obtain\n∆zfj(z, w) =λ2\nj(z)\nV. (A.42)\nLetℓj(z, w) be the length with respect to the metric gjof the unique\ngeodesic connecting ztow. It can be shown that (see for instance [40] proof\nof Theorem 5.1):\nℓj(z, w) =|w−z|q\nλj(z)λj(w)[1 +O(|z−w|)]\nTherefore\nGj(z, w) +1\n2πlogℓj(z, w) =fj(z, w) +1\n4πlog[λj(z)λj(w)] +O(|z−w|).\nTaking the limit as |z−w| →0 we obtain\nRj(z) =fj(z, z) +1\n2πlogλj(z). (A.43)\nIf we subtract equation (A.40) for j= 0 from that for j= 1 we obtain\n∆zG1(z, w)−∆zG0(z, w) =λ2\n1(z)−λ2\n0(z)\nV. (A.44)\nThis equation can be written intrinsically in terms of two-forms as\ndq∗dqG1(q, p)−dq∗dqG2(q, p) =µ1−µ0\nV=dq∗dqϕ(q)\n31where ϕis the function in the statement of the theorem. Thus G1(q, p)−\nG0(q, p) =ϕ(q)+ψ(p) that, due to the symmetry Gj(p, q) =Gj(q, p), implies\nequation (A.37). Equation dq∗dqϕ(q) =µ1−µ0\nVcan be written as ∆ 0ϕ= (λ2−\n1)/V. The representation formula (2.2) for ϕplus the relationsR\nSϕµ0= 0\nandR\nSG0(q, p)µ0(q) = 0 imply that ϕcan be written as in equation (A.36).\nIntegrating both sides of equation (A.37) with respect to µ1(q) over Swe\nobtain the expression for cin the lemma. In the z-coordinates, equation\n(A.37) implies f1(z, w)−f0(z, w) =ϕ(z) +ϕ(w) +c. This equation and\nequation (A.43) imply equation (A.38).\n□\nLemma A.2. Letg0andg1be two different Riemannian metrics on Sin\nthe same conformal class, as in Lemma A.1. Let z=x+iybe a local\nuniformizer and to simplify the notation write z(q) =wandz(p) =z. Then\n\u0012\n∆1R1+K1\n2π−2\nV\u0013\nµ1=\u0012\n∆0R0+K0\n2π−2\nV\u0013\nµ0=−˜σ , (A.45)\nwhere\n−˜σ= 8h(z)dx∧dy= 4ih(z)dz∧dz,\nwith\nh(z) =∂\n∂w∂\n∂zG0(z, w)\f\f\f\nw=z=∂\n∂w∂\n∂zG1(z, w)\f\f\f\nw=z.\nProof. In this proof we follow the notation of the proof of Lemma A.1. In\nthez-coordinates, equation (A.38) becomes\nR1(z) =R0(z) +1\n2πlogλ1(z)−1\n2πlogλ0(z) + 2ϕ(z) +c.\nTaking the Laplacian ∆ zof both sides of this equation, using ∆ zϕz=\n(λ2\n1−λ2\n0)/V, and the relations (A.39) for conformal metrics we obtain the\nfirst equality in equation (A.45). We recall that∂\n∂z=1\n2\u0010\n∂\n∂x−i∂\n∂y\u0011\n,∂\n∂z=\n321\n2\u0010\n∂\n∂x+i∂\n∂y\u0011\n, ∆z= 4∂\n∂z∂\n∂z, and dx∧dy=i\n2dz∧dz. From equation (A.43)\nwe obtain for j= 0,1\n∂\n∂z∂\n∂zRj(z) =∂\n∂z∂\n∂zfj(z, w)\f\f\f\nw=z+∂\n∂w∂\n∂wfj(z, w)\f\f\f\nw=z\n+∂\n∂z∂\n∂wfj(z, w)\f\f\f\nw=z+∂\n∂w∂\n∂zfj(z, w)\f\f\f\nw=z+1\n2π∂\n∂z∂\n∂zlogλj(z).\nFrom equation (A.42) and fj(z, w) =fj(w, z) we get\n∂\n∂z∂\n∂zfj(z, w)\f\f\f\nw=z=∂\n∂w∂\n∂wfj(z, w)\f\f\f\nw=z=1\n4∆zfj(z, w)\f\f\f\nw=z=1\n4λ2\nj(z)\nV\nFrom equation (A.41) and from the symmetry∂\n∂z∂\n∂wfj(z, w) =∂\n∂z∂\n∂wfj(w, z)\nwe get\n∂\n∂w∂\n∂zGj(z, w)\f\f\f\nw=z=∂\n∂w∂\n∂zfj(z, w)\f\f\f\nw=z=∂\n∂z∂\n∂wfj(z, w)\f\f\f\nw=z.(A.46)\nFinally, from the above equations and from equation (A.39) we obtain\n∆zRj(z) +λ2\nj(z)\n2πKj(z)−2λ2\nj(z)\nV= 8∂\n∂w∂\n∂zGj(z, w)\f\f\f\nw=z\nIf we multiply both sides of this equation by dx∧dywe obtain\n\u0012\n∆jRj+Kj\n2π−2\nV\u0013\nµj= 8∂\n∂w∂\n∂zGj(z, w)\f\f\f\nw=zdx∧dy\nforj= 0,1. Since we have already shown that the left hand side of this\nequation gives the same 2-form for j= 0 and j= 1, then the right hand side\nhas the same property. □\nThe expression∂\n∂¯w∂\n∂zGj(z, w) is formally analogous to the traditional\nBergman kernel for bounded domains in the complex plane. Indeed, equa-\ntion (A.41), which represents the decomposition of the Green’s function into\nits singular and regular parts, applies as well to the Green’s function for\nbounded domains in the plane. The distinction between the two situations\nlies in the regular part f, which is harmonic in bounded domains, whereas,\n33in this paper, the non-harmonicity of fstems from the additional term of\nconstant “background vorticity.”\nFollowing [42], let ∂be an operator defined on complex valued functions\nby∂=1\n2(d+i∗d) and ∂=1\n2(d−i∗d). In terms of a local uniformizer zwe\nhave ∂f=∂f\n∂zdzand∂f=∂f\n∂zdz.\nLemma A.3. IfG(q, p)is the Green’s function associated to a given metric\nand{θ1, . . . , θ 2G}is an orthonormal basis of harmonic forms then\n−2(∂p∂qG+∂p∂qG) =−(dpdqG+∗p∗qdpdqG) =2GX\nk=1θk(q)θk(p) (A.47)\nis the Bergman reproducing kernel for harmonic forms in S. Moreover, if q\nandpare in the domain of a local uniformizer with z(q) =wandz(p) =z\nthen\n2(∂p∂qG+∂p∂qG) = 4Re {∂p∂qG}= 4Re\u001a∂\n∂z∂\n∂wG(w, z)dwdz\u001b\n(A.48)\nProof. The equality 2( ∂p∂qG+∂p∂qG) =dpdqG+∗p∗qdpdqGand equation\n(A.48) are direct consequences of the definition of the operators ∂and∂.\nDue to equation (A.46) the function∂\n∂z∂\n∂wG(w, z) is C∞for all values of z\nandwincluding z=w. So the double one-form dpdqG+∗p∗qdpdqGis C∞\nonS×S.\nThe Bergman reproducing kernel for harmonic forms H(q, p) =P2G\nk=1θk(q)θk(p)\nis characterized by the following properties:\nFor an arbitrary function ψonS:\nZ\nSdψ(p)∧H(q, p) = 0\nZ\nS∗pdψ(p)∧H(q, p) = 0 ,\n34where the integrations are with respect to the variable p; and for any\nharmonic one-form νonS\nν(q) =Z\nSν(p)∧ ∗pH(q, p) =2GX\nk=1θk(q)Z\nS(p)ν(p)∧ ∗pθk(p).\nIn order to prove the equality dpdqG+∗p∗qdpdqG=−H(q, p) we use the\nregularity of dpdqG+∗p∗qdpdqGonS×S. So, for any function ψonS\nZ\nSdpψ(p)∧(dpdqG+∗p∗qdpdqG) =−Z\nSψ(p)∧dp(dpdqG+∗p∗qdpdqG)\n=−lim\nϵ→0Z\nS−Bϵ(q)ψ(p)∧dp(dpdqG+∗p∗qdpdqG)\nwhere Bϵ(q) is a small ball (with respect to any local uniformizer) of radius\nϵwith center at q. For poutside Bϵ(q),\ndp(dpdqG+∗p∗qdpdqG) =∗qdq(dp∗pdpG) =∗qdq\u0012µ(p)\nV\u0013\n= 0,\nsoR\nSdpψ(p)∧(dpdqG+∗p∗qdpdqG) = 0. In the same way it is possible to\nprove thatR\nS∗pdψ(p)∧(dpdqG+∗p∗qdpdqG) = 0.\nIt remains to show that ν(q) =−R\nSν(p)∧ ∗p(dpdqG+∗p∗qdpdqG) for\nany harmonic one-form νonS. This is a consequence of\nZ\nSν(p)∧ ∗p(dpdqG+∗p∗qdpdqG) = lim\nϵ→0Z\nS−Bϵ(q)ν(p)∧ ∗p(dpdqG+∗p∗qdpdqG)\n= lim\nϵ→0Z\n−∂Bϵ(q)∗pν(p)dqG+Z\n−∂Bϵ(q)ν(p)∗qdqG\nAn explicit computation using a local uniformizer gives that this last integral\nis equal to −ν(q). □\nTheorem 2.2 is a consequence of lemmas A.2 and A.3 and the following\nreasoning. Let z(p) =z=x+iyandz(q) =w=ξ+iηbe the components of\nthe local uniformizer used in lemma A.3 and θk(p) =θk1(z)dx+θk2(z)dyand\n35θk(q) =θk1(w)dξ+θk2(w)dηbe the components of θk. Lemma A.3 implies\nthat\n2GX\nk=1θk(q)θk(p) = 2GX\nk=1θk1(w)θk1(z)!\ndxdξ + 2GX\nk=1θk2(w)θk2(z)!\ndydη\n+ 2GX\nk=1θk2(w)θk1(z)!\ndxdη + 2GX\nk=1θk1(w)θk2(z)!\ndydξ\n=−4Re\u001a∂\n∂z∂\n∂wG(w, z)dwdz\u001b\nForq=panddz=dw, the right hand side of this equation becomes\n−4∂\n∂z∂\n∂wG(w, z)\f\f\f\nw=z(dx2+dy2)\nthat implies\n2GX\nk=1θ2\nk1(z) =2GX\nk=1θ2\nk2(z) =−4∂\n∂z∂\n∂wG(w, z)\f\f\f\nw=z,and2GX\nk=1θk1(z)θk2(z) = 0\nSo, the form σin theorem 2.2 can be written as\nσ(z) =2GX\nk=1θk(z)∧ ∗θk(z) =2GX\nk=1[θk1(z)dx+θk2(z)dy]∧[θk1(z)dy−θk2(z)dx]\n=2GX\nk=1[θ2\nk1(z) +θ2\nk2(z)]dx∧dy=−8∂\n∂z∂\n∂wG(w, z)\f\f\f\nw=zdx∧dy= ˜σ,\nwhere ˜ σis the form in equation (A.45). This proves that equation (2.5) holds\nand finishes the proof of theorem 2.2. □\nNow we prove theorem 2.1. The Robin function on a Riemannian mani-\nfold ( S, g) is constant whenever ( S, g) admits a transitive Lie group action of\nisometries. So, the Robin function is constant for the round sphere and for\nall flat tori. Let Sbe a sphere (torus) endowed with a Riemannian metric g0.\nThe uniformization theorem implies the existence of a diffeomorphism from\n(S, g 0) to the round sphere (a flat torus) ( S2, g1) such that the pull-back of g1\n36is conformal to g0. So, the existence of a steady vortex metric on the sphere\n(torus) is proved.\nThe proof is more complicated when Sis compact and has a genus larger\nthan one. Equation (A.38) implies:\n∆0R1(p) = ∆ 0R0(p) +1\n2π∆0logλ(p) + 2λ2−1\nV(A.49)\nImposing that R1is constant, normalizing the volume VofSto be equal to\none, and defining\nu= 4πR0+ log λ2\nwe get the following equation for u\n∆0u= 8π−8πheu(A.50)\nwhere h= e−4πR0. To each solution of this equation corresponds a Rie-\nmannian metric g1conformal to g0such that ∆ R1= 0 and therefore R1\nis constant. Equation (A.50) was very much studied for several reasons.\nIt appears in the problem of finding a Riemannian metric on the sphere\nwith a prescribed curvature hthat is conformal to the standard metric with\ncurvature 4 π(the conformal factor is eu). It also appears in the so-called\nChern-Simons-Higgs theory (see [10] for references). The following theorem\nwas taken from [10] (it is a combination of their theorem 1.2 plus their remark\n1.3).\nTheorem A.1 (Ding, Jost, Li,and Wang) .Let(S, g 0)be a compact Riemann\nsurface and let K0be its Gauss curvature. Let hbe a positive smooth function\nonS. Suppose that the function 8πR0+ 2 log hachieves its maximum at p.\nIf∆0logh(p)>−(8π−2K0(p))then equation (A.50) has a smooth solution.\nIt is remarkable that in the case we are interested in h= e−4πR0and\n8πR0+ 2 log h= 0. So, any point in Sis a point of maximum and therefore\n37to finish the proof it is sufficient to show the existence of a point pinS\nwhere the inequality 0 >−∆0logh(p)−(8π−2K0(p)) holds. The Gauss-\nBonet theorem impliesR\nSK0µ0= 2π(2−2G), where Gis the genus of S.\nSinceR\nµ0= 1, the integral of the right hand side of the inequality above is\n−8πG<0. This finishes the proof of existence of a natural vortex metric if\nG>1. □\nB The Robin function and the Minakshisun-\ndaram–Pleijel zeta function.\nThe Minakshisundaram–Pleijel zeta function, which will be referred as the\nzeta function, is defined as\nζ(q, p, s ) =∞X\nk=1ϕk(q)ϕk(p)\nλs\nk=1\nΓ(s)Z∞\n0\u0012\nK(q, p, t )−1\nV\u0013\nts−1dt , (B.51)\nwhere s∈Cand Re s > n/ 2 (the convergence is a consequence of inequality\n(3.16)).\nAccording to the theorem in Section 5 of [32], the function\nζ(p, s) :=ζ(p, p, s )\ncan be extended as a meromorphic function to the whole complex plane. If\ndimension n≥3 is odd, then the only possible poles of ζ(p, s) are located\nats=n/2, n/2−1, . . . , 3/2,1/2,−1/2, . . .. If the dimension nis even, then\nζ(p, s) has at most a finite number of poles that are possibly located at\ns=n/2, n/2−1, . . . , 2,1 and the residue at the poles can be computed [32].\nIn particular, if nis even and sis close to s= 1, then\nζ(p, s) =1\n(4π)n/2an/2−1(p)\ns−1+ convergent power series in ( s−1),(B.52)\n38where an/2−1(p) is the function that appears in equation (3.17).\nIfsis made equal to one in equation (B.51), then we obtain a formal\nexpression\nG(q, p) =Z∞\n0\u0012\nK(q, p, t )−1\nV\u0013\ndt=∞X\nk=1ϕk(q)ϕk(p)\nλk′=′ζ(q, p,1) (B.53)\nthat indicates a possible relation between the regularization of G(q, p) and\nζ(q, p, s ) asq→pands→1. Indeed, for n= 2 the following result holds\n(see, e.g. [43], Proposition 2 and the Appendix):\nR(p) = lim\nℓ(q,p)→0\u0014\nG(q, p) +1\n2πlogℓ(q, p)\u0015\n= lim\ns→1\u0014\nζ(p, s)−1\n(4π)1\ns−1\u0015\n+log 4−2γ\n4π(B.54)\nwhere γis the Euler’s constant. In the following theorem we show that\nthis result can be generalized to higher dimensions. The same result, for\nan elliptic operator that appears in the context of quantum field theory in\ncurved spacetime, was obtained by Bilal and Ferrari in [4] (Section 3). If the\nparameters mandψthat appear in their elliptic operator are set equal to\nzero, then the formulas in equations (3.45) and (3.46) of [4] are exactly ours\nin theorem (B.1).\nTheorem B.1. The Robin function can be written in terms of the analytic\nextension of the Minakshisundaram–Pleijel zeta function as\nR(p) = lim s→1h\nζ(p, s)−1\n(4π)1\ns−1i\n+log 4−2γ\n(4π)n/2ifnis even,\nR(p) = ζ(p,1) ifnis odd .(B.55)\nProof. We will prove the theorem only for neven, since the proof for nodd\nis similar. For s > n/ 2 both sides of equation (B.51) converge. The idea\nis to add terms to both sides of that equation such that the integral in the\n39right-hand side of equation (B.51) converges when s= 1. In analogy to what\nwe did to define the Robin function we rewrite equation (B.51) for s > n/ 2\nas\nζ(p, s)−n/2−1X\nk=0ak(p)\n(4π)n/2Γ(s)Z1\n0tk−n/2ts−1dt=1\nΓ(s)Z∞\n1\u0012\nK(p, p, t )−1\nV\u0013\nts−1dt\n+ lim\nϵ→0+1\nΓ(s)Z1\nϵ\nK(p, p, t )−1\nV−n/2−1X\nk=0ak(p)\n(4π)n/2tk−n/2\nts−1dt .\n(B.56)\nFors > n/ 2, the left-hand side of this equation can be written as\nζ(p, s)−an/2−1(p)\n(4π)n/2Γ(s)1\ns−1−n/2−2X\nk=0ak(p)\n(4π)n/2Γ(s)1\ns−n/2 +k. (B.57)\nDue to equations (3.16) and (3.17), the integrand in the last line of equation\n(B.56) is bounded by a constant times ts−1, and therefore the right-hand side\nof equation (B.56) is an analytic function of sfor Re s >0. This implies\nthat the analytic continuation of ζ(p, s) to Re s >0 is given by the regular\nfunction at the right-hand side of equation (B.56) plus the poles at s=\n1,2, . . . , n/ 2 explicitly given in the left-hand side of the same equation. With\nthis understanding, we can compute the regularized value of ζ(p, s) ats= 1\nas\nlim\ns→1\u0014\nζ(p, s)−an/2−1(p)\n(4π)n/2Γ(s)1\ns−1\u0015\n= lim\ns→1\u0014\nζ(p, s)−an/2−1(p)\n(4π)n/21\ns−1\u0015\n+an/2−1(p)\n(4π)n/2Γ′(1)\n=n/2−2X\nk=0ak(p)\n(4π)n/21\n1−n/2 +k+Z∞\n1\u0012\nK(p, p, t )−1\nV\u0013\ndt\n+ lim\nϵ→0+Z1\nϵ\nK(p, p, t )−1\nV−n/2−1X\nk=0ak(p)\n(4π)n/2tk−n/2\ndt ,(B.58)\n40where we used that the integrand in the last line of equation (B.56) is\nbounded by a constant times ts−1to exchange the order of the limits. Per-\nforming the integrals of the terms that are polynomials in tin the right-hand\nside of equation (B.58), using the definition of the Robin function given in\nequation (3.18), and that Γ′(1) =−γwe obtain the result in the statement\nof the theorem. □\nAcknowledgments. This paper is dedicated to Jair Koiller who introduced\nme to the subject of vortices on surfaces and presented to me the work of\nOkikiolu and Steiner. Jair has been a constant source of inspiration.\nData sharing not applicable to this article as no datasets were generated or\nanalysed during the current study.\nReferences\n[1] Vladimir Igorevich Arnol’d. Mathematical methods of classical mechan-\nics, volume 60. Springer Science & Business Media, 2013.\n[2] Thierry Aubin. Some nonlinear problems in Riemannian geometry .\nSpringer Science & Business Media, 2013.\n[3] Catherine Bandle and Martin Flucher. Harmonic radius and concentra-\ntion of energy; hyperbolic radius and liouville’s equations δu=eˆu and\nδu=uˆn+2n-2. Siam Review , 38(2):191–238, 1996.\n[4] Adel Bilal and Frank Ferrari. Multi-loop zeta function regularization\nand spectral cutoff in curved spacetime. Nuclear Physics B , 877(3):956–\n1027, 2013.\n41[5] Stefanella Boatto and Jair Koiller. Vortices on closed surfaces. arXiv\npreprint arXiv:0802.4313 , 2008.\n[6] Stefanella Boatto and Jair Koiller. Vortices on closed surfaces. In Ge-\nometry, Mechanics, and Dynamics , pages 185–237. Springer, 2015.\n[7] AV Borisov, IS Mamaev, and SM Ramodanov. Coupled motion of a\nrigid body and point vortices on a two-dimensional spherical surface.\nRegular and Chaotic Dynamics , 15(4-5):440–461, 2010.\n[8] Carmen Chicone. The monotonicity of the period function for planar\nhamiltonian vector fields. Journal of Differential equations , 69(3):310–\n321, 1987.\n[9] Georges De Rham. Differentiable manifolds: forms, currents, harmonic\nforms , volume 266. Springer Science & Business Media, 2012.\n[10] Weiyue Ding, J¨ urgen Jost, Jiayu Li, and Guofang Wang. The differential\nequation δu= 8π−8πhexpuon a compact riemann surface. In ASIAN\nJ. MATH . Citeseer, 1997.\n[11] Peter G Doyle and Jean Steiner. Spectral invariants and playing hide-\nand-seek on surfaces. arXiv preprint arXiv:1710.09857 , 2017.\n[12] David Gerard Dritschel and S Boatto. The motion of point vortices on\nclosed surfaces. In Proc. R. Soc. A , volume 471, page 20140890. The\nRoyal Society, 2015.\n[13] Ahmad El Soufi and Sa¨ ıd Ilias. Critical metrics of the trace of the\nheat kernel on a compact manifold. Journal de math´ ematiques pures et\nappliqu´ ees , 81(10):1053–1070, 2002.\n42[14] H. M. Farkas and I. Kra. Riemann surfaces. Springer, 1992.\n[15] J. Fay. Kernel functions, analytic torsion, and moduli spaces. Memoirs\nof the American Mathematical Society , v. 96,n. 464, 1992.\n[16] M Flucher and B Gustafsson. Vortex motion in two-dimensional hy-\ndromechanics. Preprint in TRITA-MAT-1997-MA-02 , 1997.\n[17] Martin Flucher. Variational problems with concentration . Springer Sci-\nence & Business Media, 1999.\n[18] Paul R Garabedian. Partial Differential Equations . Chelsea Publishing\nCompany. New York, NY, 1986.\n[19] Paul R. Garabedian. Partial Differential Equations, second edition .\nChelsea Publishing Company, New York, N.Y., 1986.\n[20] David Gilbarg and Neil S Trudinger. Elliptic partial differential equa-\ntions of second order , volume 224. springer, 2001.\n[21] Bj¨ orn Gustafsson. On the motion of a vortex in two-dimensional flow\nof an ideal fluid in simply and multiply connected domains. Research\nBulletin: TRITA-MAT-1979-7, Mathematics , 1979.\n[22] Bj¨ orn Gustafsson. Vortex motion and geometric function theory: the\nrole of connections. Philosophical Transactions of the Royal Society A ,\n377(2158):20180341, 2019.\n[23] Bj¨ orn Gustafsson. Vortex Pairs and Dipoles on Closed Surfaces. J. Non-\nlinear Sci. 32, 62 (2022). https://doi.org/10.1007/s00332-022-09822-9\n[24] C. Grotta-Ragazzo. Errata and Addenda to: “Hydrodynamic Vor-\ntex on Surfaces” and “The Motion of a Vortex on a Closed Sur-\n43face of Constant Negative Curvature”. J. Nonlinear Sci. 32 (2022).\nhttps://doi.org/10.1007/s00332-022-09817-6\n[25] Lutz Habermann, J¨ urgen Jost, et al. Green functions and conformal\ngeometry. Journal of Differential Geometry , 53(3):405–442, 1999.\n[26] David Holcman and Zeev Schuss. The narrow escape problem. siam\nREVIEW , 56(2):213–257, 2014.\n[27] J. Jorgenson and J. Kramer. Bounds on Faltings’s delta function through\ncovers. Annals of mathematics , 1-43, 2009.\n[28] J¨ urgen Jost. Geometry and physics . Springer Science & Business Media,\n2009.\n[29] J¨ urgen Jost. Compact Riemann surfaces: an introduction to contempo-\nrary mathematics . Springer Science & Business Media, 2013.\n[30] Joseph Lewittes. Differentials and metrics on riemann surfaces. Trans-\nactions of the American Mathematical Society , 139:311–318, 1969.\n[31] Chang-Shou Lin and Marcello Lucia. Uniqueness of solutions for a mean\nfield equation on torus. Journal of Differential Equations , 229(1):172–\n185, 2006.\n[32] S Minakshisundaram and A Pleijel. Some properties of the eigenfunc-\ntions of the laplace-operator on riemannian manifolds. Canadian J.\nMath , 1(242-256):9, 1949.\n[33] Valter Moretti. Proof of the symmetry of the off-diagonal heat-kernel\nand hadamard’s expansion coefficients in general c∞riemannian mani-\nfolds. Communications in Mathematical Physics , 208(2):283–308, 1999.\n44[34] Carlo Morpurgo. Zeta functions on S2.CONTEMPORARY MATHE-\nMATICS , 201:213–226, 1996.\n[35] Margherita Nolasco and Gabriella Tarantello. On a sharp sobolev-type\ninequality on two-dimensional compact manifolds. Archive for rational\nmechanics and analysis , 145(2):161–195, 1998.\n[36] Kate Okikiolu. A negative mass theorem for the 2-torus. Communica-\ntions in mathematical physics , 284(3):775–802, 2008.\n[37] Kate Okikiolu. A negative mass theorem for surfaces of positive genus.\nCommunications in Mathematical Physics , 290(3):1025–1031, 2009.\n[38] Iosif Polterovich. Heat invariants of riemannian manifolds. Israel Journal\nof Mathematics , 119(1):239–252, 2000.\n[39] C Grotta Ragazzo. The motion of a vortex on a closed surface of constant\nnegative curvature. Proceedings of the Royal Society A: Mathematical,\nPhysical and Engineering Sciences , 473(2206):20170447, 2017.\n[40] Clodoaldo Grotta Ragazzo and Humberto Henrique de Barros Viglioni.\nHydrodynamic vortex on surfaces. Journal of Nonlinear Science ,\n27(5):1609–1640, 2017.\n[41] Steven Rosenberg. The Laplacian on a Riemannian manifold: an in-\ntroduction to analysis on manifolds . Number 31. Cambridge University\nPress, 1997.\n[42] HL Royden. Function theory on compact riemann surfaces. Journal\nd’Analyse Math´ ematique , 18(1):295–327, 1967.\n[43] Jean Steiner. A geometrical mass and its extremal properties for metrics\non s2. Duke Mathematical Journal , 129(1):63–86, 2005.\n45[44] R. Wentworth. The Asymptotics of the Arakelov-Green’s Function and\nFaltings’ Delta Invariant. Commun. Math. Phys , 137: 427–459, 1991.\n46" }, { "title": "2401.15732v1.An_advance_in_the_arithmetic_of_the_Lie_groups_as_an_alternative_to_the_forms_of_the_Campbell_Baker_Hausdorff_Dynkin_theorem.pdf", "content": "arXiv:2401.15732v1 [quant-ph] 28 Jan 2024An advance in the arithmetic of the Lie groups as an alternati ve to the forms of the\nCampbell-Baker-Hausdorff-Dynkin theorem\nSunghyun Kim∗and Zhichen Liu†\nDepartment of Physics, University of Central Florida, Orla ndo, FL 32816-2385, USA\nRichard A. Klemm‡\nDepartment of Physics, University of Central Florida, Orla ndo, FL 32816-2385, USA and\nU. S. Air Force Research Laboratory, Wright-Patterson Air F orce Base, Ohio 45433-7251, USA\n(Dated: January 30, 2024)\nThe exponential of an operator or matrix is widely used in qua ntum theory, but it sometimes\ncan be a challenge to evaluate. For non-commutative operato rsXandY, according to the\nCampbell-Baker-Hausdorff-Dynkin theorem, eX+Yis not equivalent to eXeY, but is instead given\nby the well-known infinite series formula. For a Lie algebra o f a basis of three operators {X,Y,Z},\nsuch that [ X,Y] =κZfor scalar κand cyclic permutations, here it is proven that eaX+bYis\nequivalent to epZeqXe−pZfor scalar pandq. Extensions for eaX+bY+cZare also provided. This\nmethod is useful for the dynamics of atomic and molecular nuc lear and electronic spins in constant\nand oscillatory transverse magnetic and electric fields.\nINTRODUCTION\nIn1954,Rabi, RamseyandSchwingerreviewedmagneticresonance problemsin therotatingcoordinatesofanuclear\nspin [1]. The reviewed literature focused on the spin in a weak rotating magnetic field normal to a strong constant\nmagnetic field and the transition probability from a singly occupied sta te to another singly occupied state, rather\nthan upon the quantum spin wave function. A brief derivation of the spin wave function that satisfies the Schr¨ odinger\nequation at the time twas later found by Gottfried to be\nΨ(t) = eiωtJze−i[(ω−Ω)Jz−λΩJx]tΨ(0) (1)\nwhereω,Ω are scalar frequencies, λis a small dimensionless parameter, and Jx,Jzare spin operators in units of\n/planckover2pi1=h/(2π), where his Planck /acute.ts1s constant [2, see (55.19)]. As Ramsey noted, [3, see (IV.34)] the second exponential\nfactor in (1) drives the transitions from one quantum state to ano ther, but is complicated by the non-commutivity of\nthe operators.\nFrom a mathematical viewpoint, this exponential operator in (1) ca n be simplified as\ne−i[(ω−Ω)Jz−λΩJx]t→eaX+bY(2)\nwherea,bare scalars and X,Yare operators (or matrices) in a three-element subgroup of the L ie algebra approiate\nfor quantum spins. Then, the separation of eaX+bYinto a simple product of the exponentials eX,eY,eZof these three\noperators is not generally allowed by the Baker-Campbell-Hausdorff theorem.\nHere we transform this exponential factor eaX+bYinto a useful form. In Section we first remind the reader of the\nCampbell-Baker-Hausdorff-Dynkin(CBHD) theorem, in which eaX+bYis expanded into an infinite series in successive\npowers of XandY. Our transformation of this exponential factor into a more physic ally useful form is presented in\nSection .\nTHE CAMPBELL-BAKER-HAUSDORFF-DYNKIN FORMULA\nAccording to the CBHD theorem [4, 5], for any X,Y∈g, the product of two exponentials of operators or matrices\neXeYcan be rewritten as eH(X,Y), which is an infinite series of powers of XandY,\neXeY= eH(X,Y), (3)\nH(X,Y) = log(eXeY)\n=∞/summationdisplay\nk=1/summationdisplay\nm1+n1>0···/summationdisplay\nmk+nk>0(−1)k−1\nkXm1Yn1···XmkYnk\nm1!n1!···mk!nk!2\n=X+Y+1\n2[X,Y]+1\n12[X,[X,Y]]−1\n12[Y,[X,Y]]+··· (4)\n=∞/summationdisplay\nk=1/summationdisplay\nm1+n1>0···/summationdisplay\nmk+nk>0(−1)k−1\nk/summationtextk\ni=1(mi+ni)1\nm1!n1!···mk!nk!\nm1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n[X,[···,X,n1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n[Y,[···,[Y,[···mk/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n[X,[···,[X,nk/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n[Y,[···,[Y,[···]···], (5)\nwhere [X,Y] :=XY−YXis the commutator of XandYwith the understanding that [ X] :=X.\nIn this view, the exponential in (2) is not allowed to be rewritten in a pr oduct form. Instead, one may extend it in\nan infinite series, as\naX+bY=H(aX,bY)−1\n2[aX,bY]−1\n12[aX,[aX,bY]]+···, (6)\neaX+bY/ne}ationslash= eaXebY(7)\nThe operator in (2) is the exponential of a linear combination of non- commuting spin operators or matrices. To\nsolve for the probability of a transition from a most general state t o another most general state is a challenge.\nTHE RESULT\nWe desire to obtain a more useful form of the exponential eaX+bYto apply for operators representing physical\nsystems. Our approach is not from the CBHD theorem, but instead from a transformation analogous to a physical\nrotation for the relevant subgroup of the Lie algebra g. We confine our Lie algebra to physical systems with three\nbasis operators, and define the form of the unitary transformat ion needed in order to obtain our results.\nDefinition 1 A set of operators {Oµ}={X,Y,Z}is in the Lie algebra kif the following 3-cyclic relation is satisfied\n[Oµ,Oν] =κǫµνλOλ, (8)\nwhereǫµνλis the Levi-Civita symbol and summation over like Greek subs cripts is implied.\nThen, our Lie algebra kmaintains the Jacobi identity by setting [ Oµ,Oµ] = 0, or explicitly\n[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] = 0, (9)\nand the spin matrices Jx,Jy,Jzsatisfy Definition 1 with κ= i in units of /planckover2pi1.\nDefinition 2 The transformation of OνbyOµis defined as\ne−pOµOνepOµ, (10)\nwherepis a scalar.\nThe transformation is analogous to a physical rotation, but we are seeking to apply it to the exponential of two\noperators. With the above two definitions, the following theorem de monstrates the existence of a simple product form\nof the exponentials eX,eY,eZof the elements of the group.\nTheorem 1 ForX,Y,Z∈kand scalar a,b, leteU(X,Y)= eaX+bY. Then scalars pandqexist such that\neU(X,Y)= epZeqXe−pZ. (11)\nProof. ForZ,X∈k, the derivatives of the transformation of XbyZwith respect to a scalar pare in forms of\ncommutators,\nd\ndp/parenleftbig\ne−pZXepZ/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np=0=−[Z,X],\n...3\nTABLE I. Transformations of X,Y,Z\nOperation Equivalence Operation Equivalence Operation Eq uivalence\ne−pXYepXYcos(κp)−Zsin(κp) e−pXZepXZcos(κp)+Ysin(κp) e−pXXepXX\ne−pYZepYZcos(κp)−Xsin(κp) e−pYXepYXcos(κp)+Zsin(κp) e−pYYepYY\ne−pZXepZXcos(κp)−Ysin(κp) e−pZYepZYcos(κp)+Xsin(κp) e−pZZepZZ\ndm\ndpm/parenleftbig\ne−pZXepZ/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np=0= (−1)m/bracketleftbigm/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nZ,[Z,[···[Z,X]···/bracketrightbig\n. (12)\nThen, the Taylor expansion of the transformation with respect to pbecomes a series of commutators.\ne−pZXepZ=X+(−1)[Z,X]p+1\n2!(−1)2[Z,[Z,X]]p2+···\n=∞/summationdisplay\nm=0(−1)m\nm![m/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nZ,[Z,[···[Z,X]···]pm, (13)\nwhere the m= 0 term in the second line is X.\nSince the operators X,Y,Z∈ksatisfy the 3-cyclic relations in (8), the transformation in (13) bec omes a linear\nfunction of XandYwith trigonometric functions.\ne−pZXepZ=X+Y[(−1)κp]+X/bracketleftbigg1\n2!(−1)κ2p2/bracketrightbigg\n+Y/bracketleftbigg1\n3!κ3p3/bracketrightbigg\n+···\n=Xcos(κp)−Ysin(κp). (14)\nOther analogous transformations are given in Table I.\nLetU(X,Y)∈kbe a linear combination of XandY\nU={aX+bY|X,Y∈k, a,b∈R}. (15)\nThen the transformation of UbyZis\ne−pZU(X,Y)epZ= e−pZ(aX+bY)epZ\n=a/parenleftbig\ne−pZXepZ/parenrightbig\n+b/parenleftbig\ne−pZYepZ/parenrightbig\n=X/bracketleftbig\nacos(κp)+bsin(κp)/bracketrightbig\n+Y/bracketleftbig\n−asin(κp)+bcos(κp)/bracketrightbig\n. (16)\nThe choice of pin (16) is arbitrary. One could first choose pso that the coefficient of Yin (16) vanishes, leading to\np=1\nκtan−1(b/a). (17)\nFor general scalar aandb, one could write\ncos(κp) =a√\na2+b2,sin(κp) =b√\na2+b2.(18)\nThen, the transformation of Ubecomes\ne−pZU(X,Y)epZ=/radicalbig\na2+b2X≡qX. (19)\nTo transform Umanalogously, inserting the identity matrix 1= epZe−pZbetween successive factors of Uallows one\nto evaluate e−pZUmepZprecisely.\ne−pZUmepZ=/parenleftbig\ne−pZUepZ/parenrightbig/parenleftbig\ne−pZUm−1epZ/parenrightbig\n= (qX)m, (20)\nwhereqis given by Eq.(19). Therefore,\ne−pZeU(X,Y)epZ= e−pZ/parenleftBigg∞/summationdisplay\nm=0Um\nm!/parenrightBigg\nepZ4\n= eqX.\n∴eU(X,Y)= eaX+bY\n= epZeqXe−pZ./square (21)\nHowever, the selection of the scalar pis not limited to the above transformation of eU(X,Y)in terms of X. One\ncould also force the coefficient of Xin (16) to vanish, leading to p→p′given by\np′=−1\nκtan−1(a/b) (22)\neU(X,Y)= ep′ZeqYe−p′Z, (23)\nwhereqis still given by (19).\nOne may further extend the theorem for the matrix V(X,Y,Z), which is a linear combination of X,Y,andZ.\nCorollary 1.1 ForX,Y,Z∈kand scalar a,b,c, letV(X,Y,Z) =aX+bY+cZ. Then scalars p1,q1,rexist such\nthat\neV(X,Y,Z)= ep1Zeq1YerXe−q1Ye−p1Z. (24)\nProof.According to the theorem, for X,Y,Z∈k, the transformation of Vwith the operator Zis\ne−p1ZVep1Z=/radicalbig\na2+b2X+cZ, (25)\nwhere the coefficient of Yhas been set equal to zero, and p1is equal to pin (17). Then, the second transformation\nis made with respect to Y,\ne−q1Ye−p1ZVep1Zeq1Y= e−q1Y/parenleftBig/radicalbig\na2+b2X+cZ/parenrightBig\neq1Y\n=X/parenleftBig/radicalbig\na2+b2cos(κq1)−csin(κq1)/parenrightBig\n+Z/parenleftBig/radicalbig\na2+b2sin(κq1)+ccos(κq1)/parenrightBig\n. (26)\nSelecting q1so that the coefficient of Zvanishes,\nq1=−1\nκtan−1/parenleftBigc√\na2+b2/parenrightBig\n. (27)\nOne may then choose\ncos(κq1) =√\na2+b2\n√\na2+b2+c2,\nsin(κq1) =−c√\na2+b2+c2. (28)\nThen,\ne−q1Y/parenleftBig/radicalbig\na2+b2X+cZ/parenrightBig\neq1Y=/radicalbig\na2+b2+c2X≡rX. (29)\nTherefore,\ne−q1Ye−p1ZeV(X,Y,Z)ep1Zeq1Y= erX.\n∴eV(X,Y,Z)= ep1Zeq1YerXe−q1Ye−p1Z./square (30)5\nDISCUSSION\nIt is noted that CorollaryIII.1.1 is not unique. There areactually twe lvedistinct transformationorderings, resulting\nfrom the possible selections of the three basis operators {X,Y,Z}and of the three scalars {a,b,c}. As in (25), by\nfirst transforming V(X,Y,Z) with respect to Z, one can set the resulting coefficient of either XorYequal to zero.\nThen, by transforming the resultant by the operator, the coeffic ient of which had been set equal to zero, there could\nbe two additional operator forms analogous to (30). The remaining eight forms can be obtained from those four forms\nby cyclic permutations of {X,Y,Z}and{a,b,c}.\nHere we present the other three forms obtained by first transfo rmingVwith respect to Z. First, while preserving\np1in the first transformation in (25), one may choose q′\n1by forcing the coefficient of Xto vanish in the second\ntransformation, leading to\neV(X,Y,Z)= ep1Zeq′\n1YerZe−q′\n1Ye−p1Z, (31)\nq′\n1=1\nκtan−1/parenleftbig√\na2+b2\nc/parenrightbig\n.\nIn the other way, p2can be chosen for the coefficient of Xto vanish in the first transformation, similar to (22).\nThen, in the subsequent transformation with respect to X, one may force the coefficient of Zto vanish, leading to\neV(X,Y,Z)= ep2Zeq2XerYe−q2Xe−p2Z, (32)\np2=−1\nκtan−1/parenleftbiga\nb/parenrightbig\n, q2=1\nκtan−1/parenleftbigc√\na2+b2/parenrightbig\n. (33)\nThen, with p2given by (33), another way of transforming Vis to choose the coefficient of Yto vanish in the second\ntransformation. One would obtain\neV(X,Y,Z)= ep2Zeq′\n2XerZe−q′\n2Xe−p2Z, (34)\nq′\n2=−1\nκtan−1/parenleftbig√\na2+b2\nc/parenrightbig\n. (35)\nA summary of the four transformations of eVis presented in Table II, where ris given by (29).\nTABLE II. Transformations of eVby First Transformation with Respect to Z.\np q Equivalence\np11\nκtan−1/parenleftbigb\na/parenrightbig\nq1−1\nκtan−1/parenleftbigc√\na2+b2/parenrightbig\nep1Zeq1YerXe−q1Ye−p1Z\np11\nκtan−1/parenleftbigb\na/parenrightbig\nq′\n11\nκtan−1/parenleftbig√\na2+b2\nc/parenrightbig\nep1Zeq′\n1YerZe−q′\n1Ye−p1Z\np2−1\nκtan−1/parenleftbiga\nb/parenrightbig\nq21\nκtan−1/parenleftbigc√\na2+b2/parenrightbig\nep2Zeq2XerYe−q2Xe−p2Z\np2−1\nκtan−1/parenleftbiga\nb/parenrightbig\nq′\n2−1\nκtan−1/parenleftbig√\na2+b2\nc/parenrightbig\nep2Zeq′\n2XerZe−q′\n2Xe−p2Z\nObviously, one could have first transformed Vwith respect to either XorY, and in each case, there would be\nfour choices of the coefficients of the operators to force to vanis h, so that there are actually twelve expressions for\neV(X,Y,Z)containing only products of exponential factors of a scalar times o ne of the three operators, X,Y, and\nZ. The eight remaining transformation forms are obtained by cyclic pe rmutations of {X,Y,Z}and{a,b,c}, and are\npresented in Table III.\nACKNOWLEDGMENTS\nThe authors thank Prof. Joseph Brennan for helpful discussions . R. A. K. was partially supported by the U.\nS. Air Force Office of Scientific Research (AFOSR) LRIR #18RQCOR10 0, and the AFRL/SFFP Summer Faculty\nFellowship Program provided by AFRL/RQ at WPAFB.\nCONFLICTS OF INTEREST STATEMENT\nThere are no conflicts of interest.6\nTABLE III. Transformation of eVby First Transforming with Respect to XorY.\np q Equivalence\np31\nκtan−1/parenleftbigc\nb/parenrightbig\nq3−1\nκtan−1/parenleftbiga√\nb2+c2/parenrightbig\nep3Xeq3ZerYe−q3Ze−p2X\np31\nκtan−1/parenleftbigc\nb/parenrightbig\nq′\n31\nκtan−1/parenleftbig√\nb2+c2\na/parenrightbig\nep3Xeq′\n3ZerXe−q′\n3Ze−p2X\np4−1\nκtan−1/parenleftbigb\nc/parenrightbig\nq41\nκtan−1/parenleftbiga√\nb2+c2/parenrightbig\nep4Xeq4YerZe−q4Ye−p4X\np4−1\nκtan−1/parenleftbigb\nc/parenrightbig\nq′\n4−1\nκtan−1/parenleftbig√\nb2+c2\na/parenrightbig\nep4Xeq′\n4YerXe−q′\n4Ye−p4X\np51\nκtan−1/parenleftbiga\nc/parenrightbig\nq5−1\nκtan−1/parenleftbigb√\nc2+a2/parenrightbig\nep5Yeq5XerZe−q5Xe−p5Y\np51\nκtan−1/parenleftbiga\nc/parenrightbig\nq′\n51\nκtan−1/parenleftbig√\nc2+a2\nb/parenrightbig\nep5Yeq′\n5XerYe−q′\n5Xe−p5Y\np6−1\nκtan−1/parenleftbigc\na/parenrightbig\nq61\nκtan−1/parenleftbigb√\nc2+a2/parenrightbig\nep6Yeq6ZerXe−q6Ze−p6Y\np6−1\nκtan−1/parenleftbigc\na/parenrightbig\nq′\n6−1\nκtan−1/parenleftbig√\nc2+a2\nb/parenrightbig\nep6Yeq′\n6ZerYe−q′\n6Ze−p6Y\nDATA ACCESS STATEMENT\nThe manuscript is self-contained. There are no data files to access .\nETHICS STATEMENT\nNo studies on numans, animals, or plants were made.\nREFERENCES\n∗Sunghyun.Kim@ucf.edu\n†Zhichen.Liu@ucf.edu\n‡richard.klemm@ucf.edu, corresponding author\n[1] Rabi I I, Ramsey N F and Schwinger J 1954 Use of Rotating Coo rdinates in Magnetic Resonance Problems Rev. Mod. Phys.\n262\n[2] Gottfried K 1966 Quantum Mechanics Volume I: Fundamentals (CRC Press) p 431\n[3] Ramsey N F 1956 Molecular Beams (Oxford: at the Clarendon Press) p 152\n[4] M¨ uger M 2020 Notes on the Theorem of Baker-Campbell-Hausdorff-Dynkin (Radboud University) p 2–4\n[5] Bonfiglioli A and Fulci R 2012 The Theorem of Campbell, Baker, Hausdorff, and Dynkin inLecture Notes in Mathematics\n2034: Topics in Noncommutative Algebra , Morel J-M and Tessier B, eds. (Springer-Verlag, Berlin, He idelberg)." }, { "title": "2401.15761v1.A_Gaussian_Beam_Construction_of_de_Haas_vanAlfven_Resonances.pdf", "content": "arXiv:2401.15761v1 [math-ph] 28 Jan 2024A GAUSSIAN BEAM CONSTRUCTION OF DE HAAS-VAN ALFVEN\nRESONANCES\nMOUEZ DIMASSI, JEAN-CLAUDE GUILLOT AND JIM RALSTON\nAbstract. The de Haas-van Alfven Effect [ 1] arises when a metallic crystal is placed in\na constant magnetic field. One observes equally spaced peaks in its p hysical properties\nas the strength of the magnetic field is varied. Onsager [ 10] explained that the spacing\nof these peaks depended on the areas in pseudo momentum space o f the regions bounded\nby the intersections of a Fermi surface with planes perpendicular t o the magnetic field.\nHence, the dHvA effect has been quite useful in mapping Fermi surf aces. The purpose of\nthis note is to explain Onsager’s observation using gaussian beams.\nIntroduction\nThe quantum dynamics of a Bloch electron in a crystal subject to ex ternal constant\nmagnetic field A(x) is governed by the Schr¨ odinger equation\n(1) ( P−E0)u:=/bracketleftBig/planckover2pi12\n2m(i∂x+µA(ǫx))2+eV(x)−E0/bracketrightBig\nu= 0,\nwhereVis a smooth, real-valued potential, periodic with respect to a lattice Γ =⊕3\ni=1Zai\ninR3. Here (a1,a2,a3) is a basis of R3,mandeare the mass and charge of the electron,\nandµ/planckover2pi1=e. The constant ǫis the magnetic field strength. We denote T=R3/Γ.\nIn the semi-classical dynamics of Bloch electrons under slowly varyin g electric and mag-\nneticfields, recentadvanceshavebeenmade(see[ 2,3,4,15]andthereferencesgiventhere).\nSince the work of Peierls [ 12] and Slater [ 14], it is well known that, if ǫis sufficiently small,\nthe wave packets are propagating along the trajectories from th e semi-classical Hamilton-\nianH(y,p) =E(p+µA(y)),y=ǫx. HereE(k) is one of the band functions describing the\nFloquet spectrum of the unperturbed Hamiltonian −/planckover2pi12\n2m∆+eV(x).In order ǫ, the semi-\nclassical quantization conditionfor magnetic levels (well-known Onsa ger relation), contains\ntwo phases : One is the Berry’s phase, and the other is is known as th e Wilkinson-Rammal\nphase (see [ 4,15]).\nThisnoteisbasedonGuillot-Ralston-Trubowitzpaper[ 7]whereaninteresting approach,\nbased onmultiscale techniques, to electron motion inmagnetic Bloch b ands was developed.\nHowever, [ 7] contained a vague and inaccurate discussion of the role of these g eometric\nphases in the semiclassical quantization condition for cyclotron orb its, and in the de Haas-\nvan Alfven effect. Improving that was an objective of this note.\n12 M. DIMASSI, J.-C. GUILLOT AND J. RALSTON\nPreliminaries\nEquations of motion in Physical and Pseudo-momentum spaces .LetEn(k)beone\nof the band functions describing the Floquet spectrum of the unpe rturbed Hamiltonian:\nH0(k) =/planckover2pi12\n2m(−i∂x+k)2+eV(x) :L2(T)→L2(T).\nLet Φn(·,k) =e−ix·kΨn(x,k) be the corresponding normalized eigenfunction,\nH0(k)Φn(x,k) =En(k)Φn(x,k),/integraldisplay\nT|Φn(x,k)|2dx= 1,\nwhere Ψ n(·,k) is the Bloch function associated to En(k) :\nΨn(x+γ,k) =eik·γΨn(x,k),∀γ∈Γ.\nSincee−ix·γ∗H0(k)eix·γ∗=H0(k+γ∗), it follows that that\nEn(k+γ∗) =En(k),for allγ∗∈Γ∗,\nwhere Γ∗is the reciprocal lattice. Standard perturbation theory shows th at the function\nEn(k) is continuous for k∈R3and real analytic in a neighborhood of any ksuch that\nEn(k) is a simple eigenvalue, i.e.,\nEn−1(k)< En(k)< En+1(k).\nForE0∈En(T∗), we put F(E0) ={k∈T∗:En(k) =E0}1. We assume that for\neveryk= (k1,k2,k3)∈ F(E0) withk3in an open interval, En(k) is a simple eigenvalue of\nH0(k). Therefore, k/mapsto→En(k) is analytic in a neighborhood of F(E0), and we can choose\nk/mapsto→Ψn0(·,k) to be a real-analytic function near F(E0). Since we will use only one band,\nwe will suppress the index ninEn(k), Φn(·,k) and Ψ n(·,k).\nSince the magnetic field here is assumed to be constant, without loss of generality we\ncan letA(y) = (0,y1,0). Introduce the Peierls hamiltonian\nH(y,p) :=E(p+µA(y)),\nand let (( y(s),p(s)) be a trajectory of the hamiltonian flow generated by H(y,p) :\n(2) ˙ y(s) =∂H\n∂p=∂E\n∂k(p(s)+µA(y(s))),\n(3) ˙ p(s) =−∂H\n∂y=−µ(˙y2(s),0,0).\nIn the pseudo-momentum coordinate k(s) :=p(s)+µA(y(s)) one has\nE(k(s)) =E(k1(s),k2(s),k3) =E(k(0)) =E0,\n1WhenE0equals the Fermi energy EF,F(E0) is part of the Fermi surface defined by FF:={k∈\nT∗;EF∈σ(H0(k))}(see [9]). Here σ(H0(k)) denotes the spectrum of the operator H0(k).SEMICLASSICAL ASYMPTOTICS 3\nand\n(4) ˙ y(s) =∂E\n∂k(k(s)),˙k(s) =µ(−˙y2(s),˙y1(s),0).\nLetz1=p2+µy1. Since ˙p2= 0, we have\n˙z1=µ∂E\n∂k1(p1,z1,p3) and ˙p1=−µ∂E\n∂k2(p1,z1,p3).\nHence, since ˙ p3= 0, (z1(s),p1(s)) moves along a level curve for E(p1,z1,k3). We also have\n(5) ˙ y2=∂p2E(p1,p2+µy1,p3) =−˙p1/µ.\nThus there are two families of trajectories here. First there the le vel curves\nγ=γ(k3,E0) ={(k1,k2) :E(k1,k2,k3) =E0}.\nSecond there are the trajectories (with y3deleted) for the Peierls hamiltonian\nˆγ= ˆγ(k3,E0,c) ={(y1,y2,p1,p2) :E(p1,p2+µy1,k3) =E0, y2=−p1/µ+c}.\nNote that γ(k3,E0) will become ˆ γ(k3,E0,c) when one substitutes k1=p1andk2=p2+µy1\nand sets y2=−p1/µ+c. ˆγwill be used in the sections Generalized Onsager Relation and\nAppendix B.\nThroughout this note we assume that :\n(H1) (∂E\n∂k1,∂E\n∂k2)/ne}ationslash= (0,0) onγ(k3,E0),\nand\n(H2) γ(k3,E0) is a simple closed curve.\nAssumptions ( H1) and (H2) insure that {(k1,k2) :E(k1,k2,t3) =E}are simple and closed\nfor|E−E0|and|t3−k3|small enough and that it depends smoothly on ( t3,E).\nWe letS(k3) denote the area in k-space enclosed by γ(k3,E0).γ(k3,E0) is the projection\nof a helical orbit ˆΓ(k3) of the full hamiltonian in ( 2), (3).\nGaussian Beam Construction. With the change of variable y=ǫx, the operator Pis\nunitarily equivalent to\n˜P=/planckover2pi12\n2m(iǫ∂y+µA(y))2+eV(y\nǫ).\nIn [6] (see also [ 4]), we have constructed for ǫsmall enough asymptotic solutions concen-\ntrated in a tube of radius ǫ1/2around the curve Γ (formed by projecting ˆΓ(k3) onto its\ny-component) of the equation\n(6) ( ˜P−E0)u(y,ǫ)≡0.\nIn the context considered here, the asymptotic solutions had the form\n(7) u(y,ǫ) =ei[ϕ(y)\nǫ+θb+θrw+θM]a(y)Φ(x,ϕy+µA(y))+O(ǫ),4 M. DIMASSI, J.-C. GUILLOT AND J. RALSTON\nwhereϕ(y) =φ(y) +k3y3withφ(y) anda(y) are independent on y3(see Appendix A\nfor some details). Here we use ϕyfor the gradient of ϕwith respect to y. Along ˆ γ, the\ndynamical phase φsatisfies ( p1(s),p2(s)) =φy(y(s)) with\n(8) E(k(s)) =E0, k(s) = (p1(s),p2(s)+µy1(s),k3).\n•The phase θbis known as the Berry phase and θrwis the Ramanl-Wilkinson phase.\nAlong Γ, these phases are given by\n(9) ˙θb=−i/an}bracketle{tΦ(·,k(s)),˙Φ(·,k(s))/an}bracketri}ht,\nand\n(10) ˙θrw=ℑ/an}bracketle{t(H0(k(s))−E0)∂Φ\n∂k1(·,k(s)),∂Φ\n∂k2(·,k(s))/an}bracketri}ht.\nThe normalization of Φ ensures that ˙θbis real.\n•The phase θMcomes from the term ∂y·∂kE(φy+µA(y),k3) in formula ( 14). To\ndescribe it one needs to go into the details of the gaussian beam cons truction [ 11],\nand we will do that in Appendix B.\nGeneralized Onsager relation and magnetic oscillations\nGeneralized Onsager Relation. In physical space there is a motion in the y3-axis with\nvelocity ˙y3(s) =∂E\n∂k3(k(s)). Therefore, the orbits ˆΓ(k3) in physical space of the full hamil-\ntonian are helical, and do not support quasimodes, but their projec tions onto pseudo-\nmomentum produce resonances (called ”magnetic energy levels” in t he physics literature).\nOnsager’s key observation was that the magnetic energy levels det ermineS(k3) when it\nis extremal. Here we deduce that from the ”resonance condition” t hat the phase of the\nbeam must increase by an integer multiple of 2 πwhen one goes around ˆ γ. This means that\nφ(y(s))/ǫ+Θ(s) increases by a multiple of 2 π, where Θ( s) is the combined contributions\nof the Berry, Maslov and Ramal-Wilkinson phases, see ( 7).\nLet us compute the change in φaround the periodic orbit ˆ γ= ˆγ(k3,E0,c). We denote\nTits period. From ( 8),φy(y(s)) =p(s). Since ˙p2(s) = 0, it follows that\n(11)˙φ(y(s)) =p(s)·˙y(s) =p1(s)˙y1(s)+p2(s)˙y2(s) =1\nµp1(s)[˙p2(s)+µ˙y1(s)]+p2(0)˙y2(s).\nRecall that γ(k3,E0) will become ˆ γ(k3,E0,c) when one substitutes k1=p1andk2=\np2+µy1and sets y2=−p1/µ+c. Combining this with the fact that/integraltextT\n0˙y2(s)ds= 0,and\nusing Green’s theorem we obtain\nφ(y(T))−φ(y(0)) =/integraldisplayT\n0p(s)˙y(s)ds=/integraldisplayT\n01\nµp1(s)[˙p2(s)+µ˙y1(s)]ds\n=1\nµ/integraldisplay\nˆγk1dk2=S(k3)\nµ.SEMICLASSICAL ASYMPTOTICS 5\nCombining the above equality with the resonance condition and using t he fact that that\nµ/planckover2pi1=e, we obtain\n(12)/planckover2pi1\neǫS(k3)+Θ = 2 nπ,\nwhere Θ := Θ b+Θrw+ΘMnow stands for the change in Θ( s) around ˆ γ, i.e.,\nΘb=−i/integraldisplayT\n0/an}bracketle{tΦ(·,k(s)),˙Φ(·,k(s))/an}bracketri}htds=−i/integraldisplay\nˆγ/an}bracketle{tΦ(·,k),∂kΦ(·,k)/an}bracketri}htdk,\nΘrw=ℑ/integraldisplayT\n0/an}bracketle{t(H0(k(s))−E)∂Φ\n∂k1(·,k(s)),∂Φ\n∂k2(·,k(s))/an}bracketri}htds.\nWe recall that ˙Φ(·,k(s)) =˙k(s)∂Φ\n∂k(·,k(s)).\nThe phase Θ Mwill beπor 0 (see Appendix B). When Θ = 0, equation ( 12) is the\nfamous Onsager relation [ 10].\nMagnetic Oscillation. The de Haas-van Alfven effect is associated with closed curves\nγin pseudo-momentum space formed by intersecting the Fermi surf ace with planes k3=\nk0\n3, where the (constant) magnetic field is parallel to the k3-axis. Not all choices for k0\n3\ncontribute to the de Haas-van Alfven effect. Only the ”extremal” v alues ofk0\n3, i.e. those for\nwhich∂S\n∂k3= 0, contribute. For each nOnsager’s relation determines a magnetic energy\nlevel by giving a relation between ǫandS(k3). IfS(k0\n3) is extremal, as k3approaches k0\n3\nsmaller and smaller changes in ǫare needed to satisfy Onsager’s relation. Thus there is a\npeak in the density of magnetic energy levels at k0\n3.\nAppendix A\nThe paper [ 4] deals with the time-dependent Schr¨ odinger equation and the pot entials\nare time-dependent. Also instead of treating the magnetic potent ial as (0,µǫx1,0) as we\ndo here, it uses the more general form A(x) =ω×x+A(ǫx,ǫt). Once one has suppressed\nthe time dependence of the potentials and set ω= 0, only minor changes are needed to\nreduce [4] to the case considered here. Since in this note we are looking for a s olution of\nthe form\nu(y,ǫ) =ei\nǫ(φ(y)+y3k3)m(x,y;ǫ), y= (y1,y2), ǫx= (y,y3),\nwe may replace the eikonal equation (16) in [ 4] with\n(13) E(φy+µA(y),k3) = 0.\nWe will write A(y) = (0,y1) when no confusion can arise. The transport equation (29) in\n[4] is\n(14)d\nds/bracketleftBig\nf0(y(s))/bracketrightBig\n−1\n2/parenleftBig\n∂y·∂kE(φy+µA(y),k3)/parenrightBig\n|y(s)f0(y(s))+i/parenleftBig\n˙θb+˙θrw/parenrightBig\nf0(y(s)) = 0,\nwithy(s) = (y1(s),y2(s)),k= (k1,k2) andy= (y1,y2). Herek3is a parameter.6 M. DIMASSI, J.-C. GUILLOT AND J. RALSTON\nThe phases ˙θb,˙θrware given by ( 9), (10).\nAppendix B\nIn this appendix we complete the construction of the phase φand compute the Maslov\nphase Θ M. We need to construct the hessian of φas in [11] which will strengthen the\nresonance property, but φwill be independent of y3. So while it will be localized in ( y1,y2)\nnear ˆγ, it will not be the phase for a quasi-mode for the full Peierls hamiltonia n. For this\nwe need the flow of the hamiltonian\n/hatwideH(y1,y2,p1,p2) :=E(p1,p2+µy1,k3),˙y=/hatwideHp,˙p=−/hatwideHy.\nThe linearization of that flow along ˆ γ= ˆγ(k3,E0,c),\n(17)\n\n˙δy=/hatwideHppδp+/hatwideHpyδy\n˙δp=−/hatwideHypδp−/hatwideHyyδy,\nwhere the derivatives of /hatwideHare evaluated on ˆ γ, and the linearized Poincare map Ptaking\nthe data of solutions to ( 17) ats= 0 to their data at s=T.\nSince we assume that all orbits for /hatwideHnear ˆγare periodic, the algebraic eigenvalues of\nPare all 1. This is not allowed in the construction of quasi-modes in [ 11] and elsewhere.\nHowever, by following ” §2 Construction of the Phase” in [ 11] closely, one can still construct\nthe phase. For this we need two vector solutions of ( 17),v1(s) andv2(s), where v1(s) is\nthe tangent to ˆ γ, i.e.v1= (˙y1,˙y2,˙p1,˙p2(= 0)) which will satisfy ( 17) because it is the\nderivative of the flow of /hatwideHwith respect to s.\nSince we have assumed that v1(s) is never zero, v2(s) must satisfy three conditions.\nLettingσ((y,η),(w,ζ))=y·ζ−w·ηbe the symplectic two form, we need :\ni)σ(v2(s),v1(s)) = 0 for all s.\nii)σ(v2(s),v2(s)) =icwithc >0 for alls.\niii)The complex span of v1(s) andv2(s),M(s) should be periodic, i.e. M(0) =M(T).\nSinceσis constant on pairs of solutions to ( 17), will hold for all s, if they hold for s= 0.\nCondition iii) makes this construction possible in our special case. The fact that all orbits\nnear ˆγare periodic means that for any solution of ( 17)v(s),Pmapsv(0) tov(T)+av1(0)\nfor some a(see page 221 in [ 11] just before Definition 2). Hence M(T) =M(0) for any\nchoice of v2(s), and we has plenty of freedom to choose v2(0) so that i) and ii) hold2.\nNow, letting the 4 by 2 matrix with columns v1(s) andv2(s) be/parenleftbigg\nY(s)\nN(s)/parenrightbigg\n, since we\nassumed that (˙ y1(s),˙y2(s)) never vanishes (see ( 2) and (H1)),Y(s) will be invertible and\nthe hessian of φcan be chosen to be N(s)Y−1(s) aty(s).\n2For instance, we could choose ( y(0),p(0)) so that ˙ y1(0)/ne}ationslash= 0, and let v2(0) = (0,i˙y1(0),−˙y2(0),˙y1(0)).SEMICLASSICAL ASYMPTOTICS 7\nNow that Y(s) andN(s) have been constructed, a standard computation using\ntr(˙Y(s)Y−1(s)) =d\ndsln(det(Y(s))\nshows that\n(1/2)∂y·∂kE(φy+µA(y))|y(s)= (1/2)d\ndsln(det(Y(s)).\nHence, when det( Y(s)) winds around origin an even number of times as sgoes from 0\ntoT, ΘM= 0, and when it winds an odd number of times, Θ M=π.\nReferences\n[1] Ashcroft, N.W. and Mermin, N. D., Solid State Physics, Saunders College (1965).\n[2] Bellissard, J. and Rammal, R., An algebric semi-classical approach to Bloch electrons in a magnetic\nfield,J. Pysique France 51 (1990), 1803.\n[3] Buslaev, V. S., Semi-classical approximation for equations with periodic coefficients, Russ. Math.\nSurv. 42 (1987), 97–125.\n[4] Dimassi, M., Guillot, J-C and Ralston, J., Semi-Classical Asymptotics in Magnetic Bloch Bands, J.\nPhys. A: Math. Gen. 35,(2002) pp. 7597–7605.\n[5] Dimassi, M., Guillot, J.-C., and Ralston, J., On effective Hamiltonians for adiabatic perturbations of\nmagnetic Schr¨ odinger operators, J. Asymptot. Anal. 40 (2004), 137–146.\n[6] Dimassi, M., Guillot, J-C and Ralston, J., Gaussian Beam Construction for Adiabatic Perturbations.\nMathematical Physics, Analysis and Geometry (2006), pp. 187–20 1.\n[7] Guillot, J.-C,Ralston, J., TrubowitzE., Semi-classical methods in solid state physics. Commun.Math.\nPhys. (1988), 116 401–15.\n[8] B. Helffer, J. Sj¨ ostrand On diamagnetism and de Haas-van Alphen effect . Annalesde l’ I. H. P., section\nA, tome 52, no 4 (1990), p. 303–375.\n[9] I. M. Lifshitz and Moisei I Kaganov, Some Problems of the Electron Theory of Metals. I Classical a nd\nQuantum Theory of Electrons in Metals. 1960 Sov. Phyics Uspekhi 2 831\n[10] Onsager, Lars. Interpretation of the de Haas-van Alfven Effect, The Londo and Dublin Philosophical\nMag. and J. of Science, (1952), pp. 1006–1008.\n[11] J. V. Ralston, On the Construction of Quasi-modes Associated with Stable P eriodic Orbits. Commun.\nin Math. Physics, vol. 51 (1976),pp. 219-242.\n[12] Peierls, R. Zur Theorie des diamagnetimus von leitungselektronen, Z. Phys. 80 (1933), 763 Y 791.\n[13] J. Sj¨ ostrand Density of states oscillations for magnetic Schr¨ odinger o perators. Mathematics in Science\nand Engineering Volume 186, 1992, Pages 295–345.\n[14] Slater, J. C. Electrons in perturbed periodic lattices, Phys. Rev. 76 (1949), 1592–1600.\n[15] Sundaram, G. and Niu, Q. Wave packet dynamics in slowly perturbed crystals: Gradien tcorrections\nand Berry phase effects, Phys. Rev. B 59 (1999), 14915–14925.\nMouez Dimassi, IMB (UMR-CNRS 5251), UNIVERSIT ´E DE BORDEAUX, 351 COURS DE\nLA LIB ´ERATION, 33405 TALENCE CEDEX, FRANCE\nEmail address :mdimassi@u-bordeaux.fr\nJ.-C. Guillot, UNIVERSIT ´ED’AIX-MARSEILLE,CNRS, INSTITUTDE MATH ´EMATIQUES\nDE MARSEILLE, UMR 7373, 13453 MARSEILLE CEDEX 13, FRANCE\nEmail address :jcguillot@math.cnrs.fr8 M. DIMASSI, J.-C. GUILLOT AND J. RALSTON\nJ. Ralston UNIVERSITY OF CALIFORNIA, LOS ANGELES, CA 90095, USA\nEmail address :ralston@math.ucla.edu" }, { "title": "2401.15799v1.Perturbation_of_parabolic_equations_with_time_dependent_linear_operators__convergence_of_linear_processes_and_solutions.pdf", "content": "arXiv:2401.15799v1 [math.AP] 28 Jan 2024PERTURBATION OF PARABOLIC EQUATIONS WITH TIME-DEPENDENT L INEAR\nOPERATORS: CONVERGENCE OF LINEAR PROCESSES AND SOLUTIONS\nMAYKEL BELLUZI∗\nAbstract. In this work we consider parabolic equations of the form\n(uε)t+Aε(t)uε=Fε(t,uε),\nwhereεis a parameter in [0 ,ε0) and{Aε(t), t∈R}is a family of uniformly sectorial operators. As ε→0+, we\nassume that the equation converges to\nut+A0(t)u=F0(t,u).\nThe time-dependence found on the linear operators Aε(t) implies that linear process is the central object to\nobtain solutions via variation of constants formula. Under suitable conditions on the family Aε(t) and on\nits convergence to A0(t) whenε→0+, we obtain a Trotter-Kato type Approximation Theorem for th e linear\nprocessUε(t,τ) associated to Aε(t), estimating its convergence to the linear process U0(t,τ) associated to A0(t).\nThrough the variation of constants formula and assuming tha tFεconverges to F0, we analyze how this linear\nprocess convergence is transferred to the solution of the se milinear equation. We illustrate the ideas in two\nexamples. First a reaction-diffusion equation in a bounded s mooth domain Ω ⊂R3\n(uε)t−div(aε(t,x)∇uε)+uε=fε(t,uε), x∈Ω,t > τ,\nwhereaεconverges to a function a0,fεconverges to f0. We apply the abstract theory in this example, obtaining\nconvergence of the linear process and solution. As a consequ ence, we also obtain upper-semicontinuity of the\nfamily of pullback attractors associated to each problem. T he second example is a nonautonomous strongly\ndamped wave equation\nutt+(−a(t)∆D)u+2(−a(t)∆D)1\n2ut=f(t,u), x∈Ω,t > τ,\nwhere ∆ Dis the Laplacian operator with Dirichlet boundary conditio ns in a domain Ω and we analyze conver-\ngence of solution as we perturb the fractional powers of the a ssociated linear operator.\nMathematical Subject Classification 2020: 35A01, 35B40, 35B41, 35K58.\nKey words and phrases: Nonautonomous parabolic problems, time-dependent linear operators, perturbed prob-\nlems, convergence of linear process, convergence of soluti ons.\nContents\n1. Introduction 1\n2. Functional setting and main results 4\n3. Estimates and rates of convergence 8\n4. Application to reaction-diffusion equations with varying diffusion co efficients 15\n5. Application to a nonautonomous strongly damped wave equations and its fractional approximations 21\n6. Disclosures and declaration 24\nReferences 25\n1.Introduction\nIn the present paper we study singularly nonautonomous semilinear parabolic problems of the form\n(uε)t+Aε(t)uε=Fε(t,uε), t>τ,\nuε(τ) =uτ∈Y ֒→X,(1.1)\n∗Research supported by FAPESP # 2022/01439-5.\n12 M. BELLUZI\nwhereε∈[0,ε0) is a parameter, Xis a Banach space, Aε(t) :D(Aε(t))⊂X→Xis a family of sectorial\noperators (with a certain uniformity in tthat we shall specify latter), Yis a Banach Space continuously\nembedded in X, which we denote by Y ֒→X, andFε:R×Y→Xis a nonlinearity.\nThe term singularly nonautonomous expressesthe fact that this family Aε(t) is time-dependent, as a counter-\npart to the semilinear problems where Aε(t) =Aε, which we refer as nonsingular . This terminology, adopted for\ninstance in [6, 10], is not unanimous and, in the case we are considering here, does not refer to any discontinuity\nor blow-up in time, which can mean in other contexts. We shall adopt it in order to easily distinguish between\nthe case studied to the well established case where there is no time- dependence on the linear operators.\nAsε→0+, Problem (1.1) approaches to what we refer as limiting problem\n(u0)t+A0(t)u0=F0(t,u0), t>τ,\nu0(τ) =uτ∈Y,\nwhose solution is denoted by u0(t) =u0(t,τ,uτ) and referred as limiting solution .\nFor eachε∈[0,ε0) and under suitable conditions on the family {Aε(t),t∈R}and on the nonlinearity Fε,\nProblems (1.1) are well-posed. We are interested in investigating the behavior of the solution uε(t) of (1.1) as\nε→0+, comparing it to the limiting solution u0(t) and providing a rate of convergence for those solutions in\nterms ofε. In order to obtain this convergence, we must first study the ass ociated linear problem.\nThis type of analysis has already been done when the linear operator s in (1.1) do not depend on time, that is,\nAε(t)≡Aε,ε∈[0,ε0). In this case, each operator −Aεgenerates a linear semigroup, {T−Aε(t)∈ L(X),t≥0},\nthat playsanessentialroleinsolvingthe semilinearproblem. Undersu itable assumptionson Fε, thenonsingular\nproblem\n(uε)t+Aεuε=Fε(t,uε), t>τ, u ε(τ) =uτ∈Y, ε∈[0,ε0), (1.2)\nis locally solved by\nuε(t) =T−Aε(t−τ)uτ+/integraldisplayt\nτT−Aε(t−s)Fε(s,uε(s))ds (1.3)\nand we refer to the above expression as variation of constants formula .\nIn papers such as [2, 3, 4, 8, 11, 12, 14] a general routine was con ceived and applied in order to guarantee\nconvergence of solutions of Problems (1.2) as ε→0+. This routine is based in a detailed study of the behavior\nof the linear part. Precisely, the routine consists in first studying t he convergence of the linear operator A−1\nεto\nA−1\n0. This information is then used to obtain the convergence of the res olvent operator ( λ+Aε)−1to (λ+A0)−1\nin some sector and from the resolvent convergence one obtains co nvergence of the linear semigroup T−Aε(·) to\nT−A0(·). By using the variation of constants formula (1.3), one can also pr ove the convergence of the solutions\nto the limiting solution as a consequence of the linear semigroup conve rgence.\nIf the equation uε\nt+Aεuε=Fε(uε) is autonomous ( Fεdoes not depend on time), one can continue the\nanalysis and derive the upper semi-continuity of the family of global a ttractors {Aε}ε∈[0,1]and even lower\nsemi-continuity under suitable structural hypothesis on the limiting attractor A0. This is done for instance in\n[3].\nA careful analysis of those papers allows us to conclude that a huge effort goes in the direction of ensuring\nthat the linear semigroup T−Aε(·) converges to T−A0(·) in an appropriate sense. From this type of Trotter-Kato\nApproximation Theorem, the convergence of the other elements b eing studied follows.\nThe situation changes when we consider singularly nonautonomous p roblem, since the linear semigroup is\nnot the central element in obtaining the solution of the semilinear pro blem, as we discuss next. However, we\nare still able to elaborate for the singularly nonautonomous case (P roblems (1.1)) a routine similar to the one\nmentioned in the articles above to treat the matter of convergenc e for the problems. An approach like this one\nfor the singular nonautonomous case does not exist in the literatur e so far, and with the results we present in\nthis paper, we shall be able to study perturbation of singularly nona utonomous problem, incorporating several\ndifferent examples that appears in the literature.\nThe only matter that we shall not address in this paper is the lower se mi-continuity of the pullback attractor\nassociated to Problems (1.1), whenever we are able to prove that t hey exist. We do not pursuit this result due3\nto the fact that there might not exist an elliptic associated problem f or the limiting equation\n(u0)t+A0(t)(u0) =F0(t,u0)\nand we usually can not derive information on the structure of the pu llback attractor unless we require some\nsimplifying assumptions on A0(t) andF0(t,·) with respect to the time-dependence. Since this is not the purpos e\nof this article, we do not look for a result on lower semi-continuity of t he family of pullback attractors. Nev-\nertheless, upper-semicontinuity of attractors will be obtained as a consequence of the convergences established\nfor the solutions.\nThe main difference between the case where Aε(t)≡Aεto the singularly nonautonomous comes from the\nfact that instead of a linear semigroup associated to −Aεthat provides solutions for the semilinear problem\nthrough the variation of constant formula (1.3), we will have a two p arameter family of linear operators\n{Uε(t,τ)∈ L(X), t≥τ, τ∈R}\nthat will be essential in describing the solution for the semilinear prob lem. The existence of such family\nassociated to {Aε(t),t∈R}was established almost simultaneously by Sobolevski˘ ı [19] and Tana be [20, 21, 22].\nThis family Uε(t,τ) has properties similar to the ones presented by the linear semigrou p in the autonomous\ncase. In particular, there is an equivalent variation of constant fo rmula that provides solutions for (1.1) given\nby\nuε(t) =Uε(t,τ)uτ\nε+/integraldisplayt\nτUε(t,s)Fε(s,uε(s))ds. (1.4)\nTaking this into account, the outline we adopt to treat perturbatio n of singularly nonautonomous parabolic\nproblems consists in the following steps:\n(i) First we prove that, for each fixed time t∈R, the linear operator Aε(t)−1converges in an appropriate\nsense to the linear operator A0(t)−1. We also establish the rate of such convergence in terms of specific\ncharacteristics of the problem.\n(ii) We use the previous information to obtain the resolvent converg ence of (λ+Aε(t))−1to (λ+A0(t))−1\nin a sector common to all the resolvent sets of all linear operators.\n(iii) Through a well-known formulation for analytic semigroups in terms of its resolvent, we transfer the\nresolvent convergence to the linear semigroup generated by −Aε(t), for a fixed t∈R, that is, we obtain\nthe convergence (with rate) of T−Aε(t)(·) toT−A0(t)(·).\n(iv) Using the formulations of the linear process Uε(t,τ) in terms of Aε(t) andT−Aε(t)(·) (developed in [19]\nwhich we discuss in the sequel), we obtain the convergence(with rat e) ofUε(t,τ) toU0(t,τ). This result\nis presented Theorem 2.3.\n(v) Using the variation of constants formula (1.4), we obtain in Theo rem 2.5 the convergence (with rate) of\nthe solution uε(·) to the solution u0(·).\nTo attend the program proposed, this paper is structured in the f ollowing manner: In Section 2 we present\nthe assumptions required for the family of linear operators {Aε(t),t∈R}and for the nonlinearities Fεthat\nallow us to prove the results on convergence. We also enunciate in th is section the main abstract results on\nconvergence: Theorem 2.3 on the convergence of the linear proce ss and Theorem 2.5 on the convergence of the\nsolutions of (1.1) as ε→0+. Their proofs are postponed to Section 3 and they depend on follow ing steps (i) to\n(v) mentioned above. We then apply those results in two different ex amples. First, in Section 4, we consider a\nfamily of reaction-diffusion equations in a fixed bounded smooth doma in Ω⊂R3\n(uε)t−div(aε(t,x)∇uε)+uε=fε(t,uε), x∈Ω, t>τ,\n∂nuε= 0, x ∈∂Ω.\nAssuming that aεconverges to a0andfεconverges to f0, we derive in this section all the abstract conditions\nrequired in Theorems 2.3 and 2.5 that ensures convergence of the s olutionuεtou0, asε→0+. Moreover, under\nan additional dissipation assumption on the nonlinearities fε, we prove that each problem is globally well-posed,\ndefines a nonlinear dissipative process with pullback attractor {Aε(t)⊂Y, t∈R}and we prove this family4 M. BELLUZI\nof pullback attractors is upper-continuous in ε= 0. Finally, in Section 5, we apply the abstract theory to a\nnonautonomous strongly damped wave equation and its fractional approximations in the sense of [7].\nBefore we proceed to the goals proposed, we mention two points th at are important to take into account.\nThe first one concerns the linear operators Aε(t). For this paper, we shall consider a situation where the domain\nD(Aε(t)) =Dεremains fixed in tand the phase space Xwhere the linear operator is defined remains fixed in\ntandε. This assumption holds for several problems, as we shall see in applic ations. The situation where the\ndomain of the linear operator is time-dependent and the phase spac e changes with εor time shall be addressed\nin future works.\nThe second point that we want to highlight is the motivation behind con sideringsingularly nonautonomous\nproblems. In general, an evolution system in a Banach space Xcan be represented by an equation\nut=f(t,u), t>τ, u (τ) =uτ∈Y, (1.5)\nwhereY ֒→Xandf:U ⊂R×Y→X. However, the function fcan be highly nonlinear, which makes it\ndifficult to study the problem. To simplify it, we can approximate the ab ove equation around a state u0by\na linear (or semilinear) evolution equation, and then use the several tools mentioned above (and others in the\nexistent literature) to treat semilinear problems.\nThis approximation is obtained by considering the Taylor polynomial of faround the state u0(assuming\nthatfhas the necessary regularity), that is,\nf(t,u0+z) =f(t,u0)+∂f\n∂u(t,u0)z+g(t,z),\nwhereg(t,z) =o(/ba∇dblz/ba∇dblY) when/ba∇dblz/ba∇dblY→0 and∂f\n∂u(t,u0)∈ L(Y,X) is the Frechet Derivative of fwith respect to\nthe second variable. If we denote z(t) =u(t)−u0and−A(t) =∂f\n∂u(t,u0), Problem (1.5) becomes\nzt+A(t)z=f(t,u0)+g(t,z), t>τ, z (τ) = 0,\nwhich is singularly nonautonomous and is in the same format as Problem (1.1). Therefore, singularly non au-\ntonomous evolution equations seems to be a good tool to model sev eral real life phenomena and compels the\nefforts in the direction of describing its dynamics.\n2.Functional setting and main results\nIn the sequel we provide conditions on the family of linear operators {Aε(t), t∈R}that ensure existence\nof the linear process {Uε(t,τ)∈ L(X), t≥τ, τ∈R}as well as convergence of Uε(t,τ) toU0(t,τ) asε→0+.\nOnce we have convergence of the linear parts established, we prov ide conditions for the nonlinearities Fεthat\nguarantee convergence of the solutions of (1.1) as ε→0.\n2.1.A type of Trotter-Kato Approximation Theorem for the Linear processes Uε(t,τ).Consider the\nabstract singlularly nonautonomous semilinear problem (1.1) and assume that, for each ε∈[0,ε0),{Aε(t), t∈\nR}is a family of linear operators in Xsatisfying:\n(P.1)The operator Aε(t) :D(Aε(t))⊂X→Xis a closed densely defined linear operator, the domain\nDε=D(Aε(t)) is fixed in time (but it can change with ε) and there are constants C >0 andϕ∈(π\n2,π)\n(independent of ε∈[0,ε0) andt∈R) such that\nΣϕ∪{0} ⊂ρ(−Aε(t)),for allε∈[0,ε0) andt∈R,\nwhere Σ ϕ={λ∈C;|argλ| ≤ϕ}and\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(X)≤C\n|λ|+1,for allλ∈Σϕ∪{0}. (2.1)\nWe say in this case that the family Aε(t) isuniformly sectorial .5\n(P.2)The operators Aε(t) have the following regularizing property: its resolvent has its image on the Banach\nspaceY ֒→Xand there exists β∈(0,1] such that\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(Y)≤C\n|λ|+1,for allλ∈Σϕ∪{0}, (2.2)\nand\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(X,Y)≤C\n|λ|β+1,for allλ∈Σϕ∪{0}. (2.3)\n(P.3)There are constants C >0 and 0<δ≤1 (independent of ε∈[0,ε0)) such that, for any t,τ,s∈R,\n/ba∇dbl[Aε(t)−Aε(τ)]Aε(s)−1/ba∇dblL(X)≤C|t−τ|δ.\nWe say that the function R∋t/mapsto→Aε(t)Aε(s)−1∈ L(X) isδ−uniformly H¨ older continuous .\nConditions (P.1) and (P.2) state that each operator Aε(t),ε∈[0,ε0) andt∈R, is sectorial and we can\nguarantee the existence of a common sector in the spectrum of th em all as well as uniform estimate in this\nsector. Those properties can be seen as a uniform parabolicity for the family of linear operators. Moreover, to\nsay that the resolvent of A(t) has its image in Ymeans that D(A(t))⊂Y, since (λ+A(t))−1:X→D(A(t)).\nCondition (P.3) states that the H¨ older exponent for the maps t/mapsto→Aε(t)Aε(s)−1∈ L(X) can be chosen\nuniformly among all families and, as a consequence of this property,\n/ba∇dblAε(t)Aε(τ)−1/ba∇dblL(X)≤C,for all (t,τ) in a compact set and ε∈[0,ε0).\nIn this case, the graph norms defined by the operators Aε(t) andAε(τ) inDε,\n/ba∇dbl·/ba∇dblD(Aε(t))=/ba∇dblAε(t)·/ba∇dblXand/ba∇dbl·/ba∇dblD(Aε(τ))=/ba∇dblAε(τ)·/ba∇dblX,\nrespectively, are equivalent. We shall refer to both norms as /ba∇dbl·/ba∇dblX1.\nFrom conditions (P.1) to (P.3) will be able to derive uniform estimates in εfor the semigroups and linear\nprocess associated to the family {Aε(t), t∈R}. Nevertheless, in order to obtain properties of convergence as\nwe makeε→0+, we shall require further conditions on the linear operators that c onnect the different problems\nbeing studied. Those conditions are stated next:\n(P.4)There exists a continuous function ξ: [0,ε0)→R+withξ(0) = 0 such that\nsup\nt,τ∈R/ba∇dblAε(t)Aε(τ)−1−A0(t)A0(τ)−1/ba∇dblL(X)≤ξ(ε).\n(P.5)There exists a continuous function η: [0,ε0)→R+withη(0) = 0 such that\nsup\nt∈R/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(X,Y)≤η(ε).\nFor a fixed ε∈[0,ε0) andτ∈R, each operator Aε(τ) is sectorial with Σ ϕ∪{0}in the resolvent of −Aε(τ).\nHenceforth, −Aε(τ) generates an analytic semigroup which we denote by T−Aε(τ)(·) (see [18, Theorem 1.5.2])\ngiven by\nT−Aε(τ)(t) =1\n2πi/integraldisplay\nΓeλt(λ+Aε(τ))−1dλ, (2.4)\nwhere Γ is the contour of Σ ϕand it is oriented with increasing imaginary part. This linear semigroup s olves\nthe linear and homogeneous differential equation\n(uε)t+Aε(τ)uε= 0, t>0, uε(0) =u0,\nby considering uε(t,u0) =T−Aε(τ)(t)u0.\nHowever, the family {T−Aε(τ)(t)∈ L(X), t≥0}is not enough to describe the evolution of the system\nassociated to (1.1). We must obtain a two parameter family {Uε(t,τ)∈ L(X),t≥τ}of linear operators6 M. BELLUZI\nassociated to Aε(t) that plays in the singularly nonautonomous case a similar role as the s emigroup in the\nnonsingular case, that is, we should expect Uε(t,τ) to recover the solution of the homogeneous equation\n(uε)t+Aε(t)uε= 0, t>τ;uε(τ) =uτ, (2.5)\nby considering uε(t) =Uε(t,τ)uτ. In other words, we expect that ∂tUε(t,τ) =−Aε(t)Uε(t,τ). As a matter of\nfact, we search for the existence of a family of linear operators {Uε(t,τ)∈ L(X), t≥τ}with the following\nproperties:\nDefinition 2.1. LetXbe a Banach space. A family {Uε(t,τ)∈ L(X),t≥τ}of bounded linear operators is a\nlinear process associated to Aε(t) :Dε⊂X→Xif\n(1)Uε(t,t) =IandUε(t,s)Uε(s,τ) =Uε(t,τ), for allτ≤s≤t.\n(2)(t,τ,x)/mapsto→Uε(t,τ)xis continuous for t≥τand for all x∈X.\n(3) There exist C,T >0such that /ba∇dblUε(t,τ)/ba∇dblL(X)≤C, for all0≤t−τ≤T.\n(4)Uε(t,τ) :X→Dεand(τ,∞)∋t/mapsto→Uε(t,τ)x∈Xis differentiable for each x∈X.\n(5) The derivative ∂tUε(t,τ)is a bounded linear operator in X,\n∂tUε(t,τ) =−Aε(t)Uε(t,τ)\nand, forT >0, there exists C=C(T)>0such that\n/ba∇dbl∂tUε(t,τ)/ba∇dblL(X)≤C(t−τ)−1,for0≤t−τ≤T.\nConditions (P.1) to (P.3) ensure the existence of this family, as prov edin [19, Theorem 1]. We briefly mention\nthe ideas behind the construction of such family, since it depends on two auxiliary families of linear operators\nϕε(t,τ)∈ L(X) and Φ ε(t,τ)∈ L(X) that will be necessary in the sequel.\nSuppose that Uε(t,τ)∈ L(X) is a family satisfying the homogeneous differential equation given in ( 2.5), that\nis,∂tUε(t,τ) =−Aε(t)Uε(t,τ). Also, assume that there exists another family Φ ε(t,τ)∈ L(X) such that Uε(t,τ)\nis obtained trough the integral equation\nUε(t,τ) =T−Aε(τ)(t−τ)+/integraldisplayt\nτT−Aε(s)(t−s)Φε(s,τ)ds. (2.6)\nDifferentiatingin t,addingAε(t)Uε(t,τ)onbothsidesandtakingintoaccountthat ∂tUε(t,τ) =−Aε(t)Uε(t,τ),\nwe deduce\n0 = Φε(t,τ)−[Aε(τ)−Aε(t)]T−Aε(τ)(t−τ)−/integraldisplayt\nτ[Aε(s)−Aε(t)]T−Aε(s)(t−s)Φε(s,τ)ds.\nIf we set\nϕε(t,τ) = [Aε(τ)−Aε(t)]T−Aε(τ)(t−τ), (2.7)\nthen Φ ε(t,τ) would have to satisfy\nΦε(t,τ) =ϕε(t,τ)+/integraldisplayt\nτϕε(t,s)Φε(s,τ)ds (2.8)\nand it would be a fixed point of the map Sε(Ψ)(t) =ϕε(t,τ)+/integraltextt\nτϕε(t,s)Ψ(s)ds.\nIf we had a family Φ ε(t,τ) satisfying (2.8), then we could proceed in the reverse way to obta inUε(t,τ). This\nis the technique employed to construct the linear process in the par abolic case [19, 20] and the description of\nUε(t,τ) relies on this auxiliary family Φ ε(t,τ). The next proposition is proved in [18, Section 5.6] and [19]. It\nensures existence of Φ ε(t,τ) andUε(t,τ) under the conditions required previously.\nProposition 2.2. For a fixed ε∈[0,ε0), assume that {Aε(t), t∈R}satisfies (P.1), (P.2) and (P.3). Let\nδ∈(0,1]be the constant of H¨ older continuity and {ϕε(t,τ)∈ L(X),t≥τ}the family given by (2.7), then:\n(1){(t,τ)∈R2;t>τ} ∋(t,τ)/mapsto→ϕε(t,τ)∈ L(X)is continuous in the uniform topology and\n/ba∇dblϕε(t,τ)/ba∇dblL(X)≤C(t−τ)δ−1,for allt>τ,τ∈R.7\n(2) There exists a unique family {Φε(t,τ)∈ L(X),t≥τ}that satisfies (2.8)and this family is continuous\nin terms of the parameters (t,τ), that is, {(t,τ)∈R2;t>τ} ∋(t,τ)/mapsto→Φε(t,τ)∈ L(X)is continuous\nand for each T >0, there exists C=C(T)>0such that\n/ba∇dblΦε(t,τ)/ba∇dblL(X)≤C(t−τ)δ−1,for all00, independent of ε∈[0,ε0),such that\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(X)≤C(t−τ)−θeK(t−τ)ℓ(θ,ε),\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(X,Y)≤C(t−τ)−1+β(1−θ)eK(t−τ)ℓ(θ,ε),\nfor allτ∈Randt>τ, whereℓ(θ,ε) = max{[η(ε)]θ,[ξ(ε)]θ}. In particular, ℓ(θ,ε)ε→0+\n−→0.\n2.2.Rate of convergence for the solution of the semilinear probl em.In order to obtain existence\nof global solution and convergence of them as ε→0+, we need to require some properties on the family of\nnonlinearities Fε:R×Y→X. Assume that\n(NL.1) EachFε=Fε(t,u) is H¨ older continuous in t, globally Lipschitz in uand bounded. Moreover, the\nconstantsL>0 of Lipschitz and M >0 of boundedness for Fεcan be chosen uniformly in ε, that is\n/ba∇dblFε(t,u)/ba∇dblX≤M,for all (t,u)∈R×Y, ε∈[0,ε0),\n/ba∇dblFε(t,u)−Fε(t,v)/ba∇dblX≤L/ba∇dblu−v/ba∇dblY,for allε∈[0,ε0), t∈R, u,v∈Y.\n(NL.2) There exists a continuous function γ: [0,ε0)→R+withγ(0) = 0 such that\nsup\nt∈Rsup\nu∈Y/ba∇dblFε(t,u)−F0(t,u)/ba∇dblX≤γ(ε).\nRemark 2.4. Conditions required in (NL.1) are very restrictive and usua lly not found in practice. However, in\nmany situations (like the application considered in Sectio n 4) we are able to prove that dynamics of the Problems\n(1.1)eventually enters a bounded subset of Y, uniformly in ε. If that is the case, we can proceed with a cut-\noff for the nonlinearities outside this bounded set so that th e new family Fεobtained after the cut-off satisfies\nthe assumptions required in (NL.1). The parabolic problems with this new nonlinearity will differ out-side the\nbounded set, but remains the same inside it, where all the sol utions eventually go. Therefore, by restricting our\nattention to this uniform bounded absorbing set, we can assu me thatFεhave those desired properties inside it.\nIt follows from [19, Theorem 7] that Problem (1.1) is locally well-posed, that is, there exists\nuε: [τ,τ+T(ε,τ,uτ))→Ygiven byuε(t) =Uε(t,τ)uτ+/integraldisplayt\nτUε(t,s)Fε(s,uε(s))ds,\nsolution of (1.1), where T(ε,τ,uτ)>0 is the maximal interval of definition of uε(t), and it depends on the\ninitial condition and on ε. We denote the solution by uε(t,τ,uτ) if we wish to emphasize the initial condition.8 M. BELLUZI\nIn Section 3, Lemma 3.11, we shall prove that the boundedness req uired forFε, implies that /ba∇dbluε(t)/ba∇dblY\nremains bounded for tin any interval of the form [ τ,τ+T]. Therefore, the solution is globally defined in time\nand originates a nonlinear process {Sε(t,τ) :Y→Y, t≥τ, τ∈R}given by\nSε(t,τ)uτ=uε(t,τ,uτ).\nWe now present the result on convergence of the solution as ε→0+. Its proof is postponed to Section 3.\nTheorem 2.5. Assume that conditions (P.1) to (P.5) hold, as well as (NL.1) and (NL.2). Let β∈(0,1]be\nthe constant in the resolvent estimate (2.3). For anyθ∈(0,1), there exists constants C,K >0, independent of\nε,t,τsuch that, for any ε∈[0,ε0), t>τ,τ∈Randuτ∈Y, we have\n/ba∇dbluε(t,τ,uτ)−u0(t,τ,uτ)/ba∇dblY≤C(t−τ)−1+β(1−θ)eK(t−τ)[1+/ba∇dbluτ/ba∇dblY]ρ(θ,ε),\nwhere\nρ(θ,ε) = max{[η(ε)]θ,[ξ(ε)]θ,γ(ε)}.\nIn particular, the convergence of the solution is uniform fo rtin any interval of the form [τ+m,τ+M],\n00 (uniform in t∈Randε∈[0,ε0)) such that, for any λ∈Σϕ∪{0},ε∈[0,ε0) andt∈R,\n/ba∇dblAε(t)(λ+Aε(t))−1/ba∇dblL(X)≤Cand/ba∇dblAε(t)(λ+Aε(t))−1/ba∇dblL(Y)≤C. (3.1)\nSinceY ֒→X, there exists a constant C >0 such that, for each u∈Y,\n/ba∇dblu/ba∇dblX≤C/ba∇dblu/ba∇dblYand/ba∇dblI/ba∇dblL(Y,X)≤C. (3.2)\nConsequently,\n/ba∇dblAε(t)−1u−A0(t)−1u/ba∇dblX≤C/ba∇dblAε(t)−1u−A0(t)−1u/ba∇dblY≤Cη(ε)/ba∇dblu/ba∇dblX\nand\n/ba∇dblAε(t)−1u−A0(t)−1u/ba∇dblY≤η(ε)/ba∇dblu/ba∇dblX≤Cη(ε)/ba∇dblu/ba∇dblY.\nHence, we have the following estimates\n/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(X)≤Cη(ε) and /ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(Y)≤Cη(ε) (3.3)\nFrom the resolvent equality and simple algebra we can prove that the following equalities hold, for all\nλ∈Σϕ∪{0}andt∈R,\n(λ+Aε(t))−1−(λ+A0(t))−1=Aε(t)(λ+Aε(t))−1[Aε(t)−1−A0(t)−1]A0(t)(λ+A0(t))−1,(3.4)\nAε(t)(λ+Aε(t))−1−A0(t)(λ+A0(t))−1=−λ(λ+Aε(t))−1Aε(t)[Aε(t)−1−A0(t)−1]A0(t)(λ+A0(t))−1.(3.5)9\nExpression(3.4) implicates that resolventconvergenceinside the s ectorΣ ϕ∪{0}follows from the convergence\nofAε(t)−1−A0(t)−1, asε→0+, requested in (P.5), as stated in next proposition.\nProposition 3.1. There exists a constant C >0, independent of ε∈[0,ε0)ort∈R, such that, for all\nλ∈Σϕ∪{0}andt∈R,\n/ba∇dbl(λ+Aε(t))−1−(λ+A0(t))−1/ba∇dblL(X,Y)≤Cη(ε).\nProof.From the uniform estimate obtained in (3.1) for Aε(t)(λ+Aε(t))−1inL(X) andL(Y) and Equality\n(3.4), we deduce\n/ba∇dbl(λ+Aε(t))−1−(λ+A0(t))−1/ba∇dblL(X,Y)\n≤ /ba∇dblAε(t)(λ+Aε(t))−1/ba∇dblL(Y)/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(X,Y)/ba∇dblA0(t)(λ+A0(t))−1/ba∇dblL(X)\n≤Cη(ε).\n/square\nAnother estimate on the resolvent in terms of εthat will be useful in the sequel is presented next.\nLemma 3.2. There exists a constant C >0, independent of ε∈(0,ε0]andt∈R, such that, ,\n/ba∇dblAε(t)(λ+Aε(t))−1−A0(t)(λ+A0(t))−1/ba∇dblL(X)≤C|λ|η(ε),for anyλ∈Σϕ∪{0}.\nProof.It follows directly from the estimates (3.1) for Aε(t)(λ+Aε(t))−1inL(X) andL(Y), from (3.2) and\nfrom (3.5) that\n/ba∇dblAε(t)(λ+Aε(t))−1−A0(t)(λ+A0(t))−1/ba∇dblL(X)\n≤ /ba∇dblλAε(t)(λ+Aε(t))−1/ba∇dblL(Y,X)/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(X,Y)/ba∇dblA0(t)(λ+A0(t))−1/ba∇dblL(X)\n≤ |λ|/ba∇dblI/ba∇dblL(Y,X)/ba∇dblAε(t)(λ+Aε(t))−1/ba∇dblL(Y)/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(X,Y)/ba∇dblA0(t)(λ+A0(t))−1/ba∇dblL(X)\n≤C|λ|η(ε).\n/square\nLastly on the linear operator and its resolvent, we provide an estima te for a situation where we vary both ε\nand timet∈Rsimultaneously.\nLemma 3.3. Letδ∈(0,1]be the H¨ older continuity constant in (P.3). For any θ∈[0,1], there exists a constant\nC >0such that, for all t,τ∈Randε∈[0,ε0),\n/ba∇dblAε(t)Aε(τ)−1−A0(t)A0(τ)−1/ba∇dblL(X)≤C|t−τ|δ(1−θ)[ξ(ε)]θ. (3.6)\nProof.From (P.3), we deduce\n/ba∇dblAε(t)Aε(τ)−1−A0(t)A0(τ)−1/ba∇dblL(X)=/ba∇dbl[Aε(τ)−Aε(t)]Aε(τ)−1−[A0(τ)−A0(t)]A0(τ)−1/ba∇dblL(X)\n≤C|t−τ|δ. (3.7)\nNow, interpolating (3.7) and the estimate in (P.4) with an exponent θ∈[0,1], we obtain (3.6). /square\n3.2.Convergence and estimates for the semigroups. Since each operator Aε(τ) is sectorial (with an\nuniform sector and uniform resolvent estimates in terms of εandτ), classical theory on semigroups implies that\n−Aε(τ) generates an analytic semigroup, which we denote by T−Aε(τ)(·).\nIf Γ is the contour of Σ ϕ⊂ρ(−Aε(τ)), that is, Γ = {re−iϕ:r >0}∪{reiϕ:r >0}and it is oriented with\nincreasing imaginary part, then we have the following expressions\nT−Aε(τ)(t) =1\n2πi/integraldisplay\nΓeλt(λ+Aε(τ))−1dλ, (3.8)10 M. BELLUZI\nAε(τ)T−Aε(τ)(t) =1\n2πi/integraldisplay\nΓeλtAε(τ)(λ+Aε(τ))−1dλ, (3.9)\nthat can be found in [18, Section 2.5]. A direct application of estimates (2.1) and (3.1) in Expressions (3.8) and\n(3.9) implies, for any τ∈Randε∈[0,ε0),\n/ba∇dblT−Aε(τ)(t)/ba∇dblL(X)≤C,for allt≥0, (3.10)\n/ba∇dblAε(τ)T−Aε(τ)(t)/ba∇dblL(X)≤Ct−1,for allt>0. (3.11)\nUniformity of those estimates with respect to εandτfollows from (P.1) and (P.2). We can also obtain an\nestimate for this semigroup in L(X,Y), as stated next.\nLemma 3.4. Letβ∈(0,1]be the constant in (P.2). There exists a constant C >0independent of ε∈[0,ε0)\nandτ∈R, such that, for all t>0,\n/ba∇dblT−Aε(τ)(t)/ba∇dblL(X,Y)≤Ctβ−1.\nProof.Using estimate (2.3), we obtain\n/ba∇dblT−Aε(τ)(t)/ba∇dblL(X,Y)≤1\n2π/integraldisplay\nΓ|eλt|/ba∇dbl(λ+Aε(τ))−1/ba∇dblL(X,Y)|dλ| ≤C/integraldisplay∞\n0er[cosϕ]tC\n1+rβdr\n≤Ctβ−1/integraldisplay∞\n0e[cosϕ]u1\ntβ+uβdu=C(ϕ,β)tβ−1,\nwhere constant Cdepends on the angle ϕand onβ, but it is independent of ε,τandt. /square\nWe establish next a convergence of the linear semigroups relative to ε.\nLemma 3.5. Letβ∈(0,1]be the constant in (P.2). For any θ∈[0,1], there exists a constant C >0\nindependent of ε∈[0,ε0)andτ∈R, such that, for all t>0,\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X)≤Ct−θ[η(ε)]θ, (3.12)\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X,Y)≤Ct−1+β(1−θ)[η(ε)]θ. (3.13)\nProof.It follows from (3.10) that\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X)≤C. (3.14)\nConsider the curve Γ parametrized as Γ = Γ 1∨Γ−\n2where\nΓ1:={λ∈C:λ=reiϕ;r∈[0,∞)},Γ2:={λ∈C:λ=re−iϕ;r∈[0,∞)},\nand Γ−\n2stands for the reverse path. Using the symmetry of curves Γ 1and Γ2and estimate (3.3), we obtain\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X)≤1\nπ/integraldisplay\nΓ1|eλt|/ba∇dbl(λ+Aε(τ))−1−(λ+A0(τ))−1/ba∇dblL(X)|dλ|\n≤Cη(ε)/integraldisplay∞\n0er[cosϕ]tdr\n≤C(ϕ)t−1η(ε) (3.15)\nInterpolating (3.14) and (3.15) with exponents θand 1−θ, forθ∈[0,1], we obtain (3.12). In order to\nestimate (3.13), we first note from Lemma 3.4 that\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X,Y)≤Ctβ−1. (3.16)\nUsing (P.5) and the integral formulation for the semigroup (2.4), we obtain\n/ba∇dblT−Aε(τ)(t)−T−A0(τ)(t)/ba∇dblL(X,Y)≤1\nπ/integraldisplay\nΓ1|eλt|/ba∇dbl(λ+Aε(τ))−1−(λ+A0(τ))−1/ba∇dblL(X,Y)|dλ|\n≤Cϕt−1η(ε) (3.17)\nInterpolating (3.16) and (3.17) with exponents 1 −θandθ,θ∈[0,1], we obtain the desired estimate (3.13).\n/square11\nWe deduce in the sequel the last estimate on semigroup in terms of εnecessary to our future analysis.\nLemma 3.6. For anyθ∈[0,1], there exists C >0such that, for all ε∈[0,ε0), τ∈Randt>0,\n/ba∇dblAε(τ)T−Aε(τ)(t)−A0(τ)T−A0(τ)(t)/ba∇dblL(X)≤Ct−1−θ[η(ε)]θ.\nProof.Note that from (3.11), we obtain\n/ba∇dblAε(τ)T−Aε(τ)(t)−A0(τ)T−A0(τ)(t)/ba∇dblL(X)≤Ct−1. (3.18)\nOn the other hand, using the integral formulation for the semigrou p and Lemma 3.2, we deduce\n/ba∇dblAε(τ)T−Aε(τ)(t)−A0(τ)T−A0(τ)(t)/ba∇dblL(X)≤C/integraldisplay\nΓ|eλt|/ba∇dblAε(τ)(λ+Aε(τ))−1−A0(τ)(λ+A0(τ))−1/ba∇dblL(X)|dλ|\n≤Cη(ε)/integraldisplay∞\n0er[cosϕ]trdr≤C(ϕ)t−2η(ε). (3.19)\nInterpolating (3.18) and (3.19) with exponents 1 −θandθ, we obtain the desired estimate. /square\n3.3.Convergence and estimates for the families ϕε(t,τ)andΦε(t,τ).In order to achieve our final goal\nof obtaining rate of convergence for the linear process associate d to the family {Aε(t), t∈R}, we first need to\nestablish rate of convergences for the auxiliary families ϕε(t,τ) and Φ ε(t,τ). Recall that\nϕε(t,τ) = [Aε(τ)−Aε(t)]T−Aε(τ)(t−τ),\nand it follows directly from (P.3) and (3.11) that\n/ba∇dblϕε(t,τ)/ba∇dblL(X)≤C(t−τ)δ−1,for anyt>τ.\nThe rate of convergence required for the resolvent operators in Properties (P.4) and (P.5) are transfered to\nthe families ϕε(t,τ) as follows.\nLemma 3.7. Letθ∈[0,1]andδbe the constant of H¨ older continuity in (P.3). There exists a constantC >0\nsuch that, for any ε∈[0,ε0),t>τandτ∈R, we have\n/ba∇dblϕε(t,τ)−ϕ0(t,τ)/ba∇dblL(X)≤C(t−τ)−1+δ(1−θ)ℓ(θ,ε),\nwhere\nℓ(θ,ε) = max{[η(ε)]θ,[ξ(ε)]θ}. (3.20)\nIn particular, ℓ(θ,ε)ε→0−→0.\nProof.Using the previous estimates and Expression (2.7) for the family ϕε(t,τ), we obtain\n/ba∇dblϕε(t,τ)−ϕ0(t,τ)/ba∇dblL(X)\n≤ /ba∇dbl[Aε(τ)−Aε(t)]Aε(τ)−1Aε(τ)T−Aε(τ)(t−τ)−[A0(τ)−A0(t)]A0(τ)−1A0(τ)T−A0(τ)(t−τ)/ba∇dblL(X)\n≤ /ba∇dbl[Aε(τ)−Aε(t)]Aε(τ)−1/ba∇dblL(X)/ba∇dblAε(τ)T−Aε(τ)(t−τ)−A0(τ)T−A0(τ)(t−τ)/ba∇dblL(X)\n+/ba∇dblAε(t)Aε(τ)−1−A0(t)A0(τ)−1/ba∇dblL(X)/ba∇dblA0(τ)T−A0(τ)(t−τ)/ba∇dblL(X)\n≤C(t−τ)−1+δ−θ[η(ε)]θ+C(t−τ)−1+δ(1−θ)[ξ(ε)]θ\n≤C(t−τ)−1+δ(1−θ)ℓ(θ,ε),\nsince−1+δ(1−θ)<−1+δ−θ<0. /square\nAs far as estimates for the family Φ ε(t,τ), we have the following result.\nLemma 3.8. Letδ∈(0,1]be the constant of H¨ older continuity in (P.3). There exist c onstantsC,K >0such\nthat, for any ε∈[0,ε0),τ∈Randt>τ,\n/ba∇dblΦε(t,τ)/ba∇dblL(X)≤C(t−τ)δ−1eK(t−τ).12 M. BELLUZI\nProof.From previous estimates, we obtain\n/ba∇dblΦε(t,τ)/ba∇dblL(X)≤ /ba∇dblϕε(t,τ)/ba∇dblL(X)+/integraldisplayt\nτ/ba∇dblϕε(t,s)/ba∇dblL(X)/ba∇dblΦε(s,τ)/ba∇dblL(X)ds\n≤C(t−τ)δ−1+/integraldisplayt\nτC(t−s)δ−1/ba∇dblΦε(s,τ)/ba∇dblL(X)ds.\nIt follows from Gronwall’s inequality [15, p.190] that, for any t>τ,\n/ba∇dblΦε(t,τ)/ba∇dblL(X)≤C\nδ(t−τ)δ−1eK(t−τ),\nwhereK >(2CΓ(δ))1\nδ. /square\nWe obtain in the sequel a rate of convergence for the family Φ εasεconverges to zero. Unlike previous\nresults, our auxiliary θthat appears in the estimate needs to be in the open interval (0 ,1) instead of [0 ,1] in\norder to ensure convergence of integrals that feature in the est imates.\nLemma 3.9. Letθ∈(0,1)andδbe the constant of H¨ older continuity in (P.3). There exists C=C(θ,δ)>0\nandK=K(δ)>0such that, for any ε∈[0,ε0),τ∈Randt>τ, we have\n/ba∇dblΦε(t,τ)−Φ0(t,τ)/ba∇dblL(X)≤C(t−τ)−1+δ(1−θ)eK(t−τ)ℓ(θ,ε), (3.21)\nwhereℓ(θ,ε)is given in (3.20).\nProof.Using the estimates obtained earlier, we deduce\n/ba∇dblΦε(t,τ)−Φ0(t,τ)/ba∇dblL(X)≤ /ba∇dblϕε(t,τ)−ϕ0(t,τ)/ba∇dblL(X)+/integraldisplayt\nτ/ba∇dblϕε(t,s)−ϕ0(t,s)/ba∇dblL(X)/ba∇dblΦε(s,τ)/ba∇dblL(X)ds\n+/integraldisplayt\nτ/ba∇dblϕ0(t,s)/ba∇dblL(X)/ba∇dblΦε(s,τ)−Φ0(s,τ)/ba∇dblL(X)ds\n≤C(t−τ)−1+δ(1−θ)ℓ(θ,ε)+/integraldisplayt\nτC(t−s)−1+δ(1−θ)ℓ(θ,ε)(s−τ)δ−1ds\n+/integraldisplayt\nτC(t−s)δ−1/ba∇dblΦε(s,τ)−Φ0(s,τ)/ba∇dblL(X)ds\n≤C(t−τ)−1+δ(1−θ)ℓ(θ,ε)+C(t−τ)δ(1−θ)+δ−1B(δ(1−θ),δ)ℓ(θ,ε)\n+/integraldisplayt\nτC(t−s)δ−1/ba∇dblΦε(s,τ)−Φ0(s,τ)/ba∇dblL(X),\nwhereB(·,·) is the Beta function. Taking ψ(t) =/ba∇dblΦε(t,τ)−Φ0(t,τ)/ba∇dblL(X), we restate the above inequality as\nψ(t)≤C/bracketleftBig\n(t−τ)−1+δ(1−θ)+(t−τ)δ(1−θ)+δ−1/bracketrightBig\nℓ(θ,ε)+C/integraldisplayt\nτ(t−s)δ−1ψ(s)ds.\nApplying the generalized version of Gronwall’s inequality [15, p.190],\nψ(t)≤C(δ,θ)/bracketleftBig\n(t−τ)−1+δ(1−θ)+(t−τ)δ(1−θ)+δ−1/bracketrightBig\nℓ(θ,ε)eK(t−τ),\nforK >(CΓ(δ))1\nδ.Moreover, if δ(1−θ)+δ−1>0, then the growth provided by the term (t−τ)δ(1−θ)+δ−1can\nbe incorporated to the exponential term eK(t−τ), correcting the constant if necessary. If δ(1−θ)+δ−1<0,\nthen(t−τ)δ(1−θ)+δ−1≤(t−τ)δ(1−θ)−1fort−τnear zero. In both cases, Inequality (3.21) follows from the above\nestimate. /square\n3.4.Convergence and estimates for the linear process Uε(t,τ).Before we prove Theorem 2.3, we obtain\nan estimate for the linear process that will be necessary.\nLemma 3.10. Letβ∈(0,1]be the constant in (P.2). There exists C,K >0such that, for any ε∈[0,ε0),\nτ∈Randt>τ,\n/ba∇dblUε(t,τ)/ba∇dblL(X)≤CeK(t−τ),\n/ba∇dblUε(t,τ)/ba∇dblL(X,Y)≤C(t−τ)β−1eK(t−τ).13\nProof.From previous estimates and the expression for the linear process in (2.6), we obtain\n/ba∇dblUε(t,τ)/ba∇dblL(X)≤ /ba∇dblT−Aε(τ)(t−τ)/ba∇dblL(X)+/integraldisplayt\nτ/ba∇dblT−Aε(s)(t−s)/ba∇dblL(X)/ba∇dblΦε(s,τ)/ba∇dblL(X)ds\n≤C+/integraldisplayt\nτC(s−τ)δ−1eK(s−τ)ds≤C+C\nδ(t−τ)δeK(t−τ)\n≤Cmax{1,(t−τ)δ}eK(t−τ)≤CeK(t−τ),\nwhere we incorporated ( t−τ)δinto the growth presented by eK(t−τ), making adjustments in the constant C, if\nnecessary. Similarly, we have\n/ba∇dblUε(t,τ)/ba∇dblL(X,Y)≤C(t−τ)β−1+/integraldisplayt\nτC(t−s)β−1(s−τ)δ−1eK(s−τ)ds\n≤C(t−τ)β−1+C\nβ+δ(t−τ)β+δ−1B(β,δ)eK(t−τ)\n≤C(t−τ)β−1/bracketleftBig\n1+(t−τ)δeK(t−τ)/bracketrightBig\n≤C(t−τ)β−1eK(t−τ),\nalso adjusting the constant C, if necessary. /square\nWe are now able to prove Theorem 2.3.\nProof of Theorem 2.3: We first obtain the estimate in L(X) using Expression (2.6) for the linear process\nand the estimates established previously.\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(X)\n≤ /ba∇dblT−Aε(τ)(t−τ)−T−A0(τ)(t−τ)/ba∇dblL(X)+/integraldisplayt\nτ/ba∇dblT−Aε(s)(t−s)/ba∇dblL(X)/ba∇dbl[Φε(s,τ)−Φ0(s,τ)]/ba∇dblL(X)ds\n+/integraldisplayt\nτ/ba∇dbl[T−Aε(s)(t−s)−TA0(s)(t−s)]/ba∇dblL(X)/ba∇dblΦ0(s,τ)/ba∇dblL(X)ds\n≤C(t−τ)−θ[η(ε)]θ+/integraldisplayt\nτC(s−τ)−1+δ(1−θ)eK(s−τ)ℓ(θ,ε)ds+/integraldisplayt\nτC(t−s)−θ[η(ε)]θ(s−τ)δ−1eK(s−τ)ds\n≤C(t−τ)−θ[η(ε)]θ+C\nδ(1−θ)eK(t−τ)ℓ(θ,ε)(t−τ)δ(1−θ)+C[η(ε)]θ(t−τ)δ−θB(1−θ,δ)eK(t−τ)\n≤C(θ,δ)(t−τ)−θeK(t−τ)ℓ(θ,ε).\nProceeding similarly to the estimate in L(X,Y), we deduce\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(X)\n≤ /ba∇dblT−Aε(τ)(t−τ)−T−A0(τ)(t−τ)/ba∇dblL(X,Y)+/integraldisplayt\nτ/ba∇dblT−Aε(s)(t−s)/ba∇dblL(X,Y)/ba∇dbl[Φε(s,τ)−Φ0(s,τ)]/ba∇dblL(X)ds\n+/integraldisplayt\nτ/ba∇dbl[T−Aε(s)(t−s)−TA0(s)(t−s)]/ba∇dblL(X,Y)/ba∇dblΦ0(s,τ)/ba∇dblL(X)ds\n≤C(t−τ)−1+β(1−θ)[η(ε)]θ+/integraldisplayt\nτC(t−s)β−1(s−τ)−1+δ(1−θ)eK(s−τ)ℓ(θ,ε)ds\n+/integraldisplayt\nτC(t−s)−1+β(1−θ)[η(ε)]θ(s−τ)δ−1eK(s−τ)ds\n≤C(t−τ)−1+β(1−θ)[η(ε)]θ+CeK(t−τ)ℓ(θ,��)(t−τ)β+δ(1−θ)−1B(β,δ(1−θ))\n+C[η(ε)]θ(t−τ)δ+β(1−θ)−1B(β(1−θ),δ)eK(t−τ)\n≤C(θ,δ)(t−τ)−1+β(1−θ)eK(t−τ)ℓ(θ,ε).\n/square14 M. BELLUZI\n3.5.Convergence and estimates of the solution of the semilinear problem. As mentioned at Section\n2, Problem (1.1) is locally well-posed, that is, there exists\nuε(t,τ,uτ) =Uε(t,τ)uτ+/integraldisplayt\nτUε(t,s)Fε(s,uε(s))ds,\nthat solves the problem for t∈[τ,τ+T(ε,τ,uτ)). We actually have global well-posedness ( T(ε,τ,uτ) =∞) as\na consequence of the following result.\nLemma 3.11. Letβ∈(0,1]be the constant in the resolvent estimate (2.3). There exist constants C,K >0\nsuch that, for any τ∈RandT >0for whichuε(·,τ,uτ)is defined in (τ,τ+T], we have\n/ba∇dbluε(t,τ,uτ)/ba∇dblY≤C(t−τ)β−1eK(t−τ)[1+/ba∇dbluτ/ba∇dblY],for allt∈(τ,τ+T].\nIn particular, the /ba∇dbluε(t)/ba∇dblYdoes not blow-up in any finite time interval and it is globally defined.\nProof.This result follows from the expression for the solution uεand the estimates obtained previously.\n/ba∇dbluε(t,τ,uτ)/ba∇dblY≤ /ba∇dblUε(t,τ)/ba∇dblL(X,Y)/ba∇dbluτ/ba∇dblX+/integraldisplayt\nτ/ba∇dblUε(t,s)/ba∇dblL(X,Y)/ba∇dblFε(s,uε(s,τ,uτ)))/ba∇dblXds\n≤C(t−τ)β−1eK(t−τ)/ba∇dbluτ/ba∇dblY+/integraldisplayt\nτC(t−s)β−1eK(t−s)Mds\n≤C(t−τ)β−1eK(t−τ)/ba∇dbluτ/ba∇dblY+CM\nβ(t−τ)βeK(t−τ)\n≤C(t−τ)β−1eK(t−τ)[1+/ba∇dbluτ/ba∇dblY].\nTherefore, the solution is bounded in any bounded interval [ τ+m,τ+T], for 0< m < T , being globally\ndefined. /square\nWe now prove Theorem 2.5 that provides a rate at which the solutions converge.\nProof of Theorem 2.5: In the sequel we will denote the solution uε(t,τ,uτ) byuε(t). LetM >0,L>0 be\nthe boundedness and Lipschitz constant for F, respectively, and ρ(θ,ε) = max{[η(ε)]θ,[ξ(ε)]θ,γ(ε)}.Using the\nexpression for the solution and rates of convergence established earlier, we obtain\n/ba∇dbluε(t)−u0(t)/ba∇dblY≤ /ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(X,Y)/ba∇dbluτ/ba∇dblX+/integraldisplayt\nτ/ba∇dblUε(t,s)−U0(t,s)/ba∇dblL(X,Y)/ba∇dblF0(s,u0(s))/ba∇dblXds\n+/integraldisplayt\nτ/ba∇dblUε(t,s)/ba∇dblL(X,Y)[/ba∇dblFε(s,uε(s))−Fε(s,u0(s))/ba∇dblX+/ba∇dblFε(s,u0(s))−F0(s,u0(s))/ba∇dblX]ds\n≤C(t−τ)−1+β(1−θ)eK(t−τ)ℓ(θ,ε)/ba∇dbluτ/ba∇dblY+/integraldisplayt\nτC(t−s)−1+β(1−θ)eK(t−s)ℓ(θ,ε)Mds\n+/integraldisplayt\nτC(t−s)β−1eK(t−s)[L/ba∇dbluε(s)−u0(s)/ba∇dblY+γ(ε)]ds\n≤C(t−τ)−1+β(1−θ)eK(t−τ)ℓ(θ,ε)/ba∇dbluτ/ba∇dblY+CM\nβ(1−θ)(t−τ)β(1−θ)eK(t−s)ℓ(θ,ε)\n+C\nβeK(t−τ)(t−τ)βγ(ε)+CL/integraldisplayt\nτ(t−s)β−1eK(t−s)/ba∇dbluε(s)−u0(s)/ba∇dblYds\n≤C(t−τ)−1+β(1−θ)eK(t−τ)[/ba∇dbluτ/ba∇dblY+1]ρ(θ,ε)+CL/integraldisplayt\nτ(t−s)β−1eK(t−s)/ba∇dbluε(s)−u0(s)/ba∇dblYds,\nwhere we incorporated the terms ( t−τ) with a positive exponent to the exponential growth given by eK(t−τ),\nmaking adjustments in the constant C, if necessary. Multiplying both sides by e−K(t−τ)and considering\nΨ(t) =e−K(t−τ)/ba∇dbluε(t)−u0(t)/ba∇dblY, we obtain\nΨ(t)≤C(t−τ)−1+β(1−θ)[/ba∇dbluτ/ba∇dblY+1]ρ(θ,ε)+CL/integraldisplayt\nτ(t−s)β−1Ψ(s)ds.15\nWe now apply Gronwall’s inequality [15, p.190] to conclude that\nΨ(t)≤C\nβ(1−β)(t−τ)−1+β(1−θ)[/ba∇dbluτ/ba∇dblY+1]ρ(θ,ε)e˜K(t−τ),\nwhere˜K >(2CLΓ(β))1\nβ. Therefore,\n/ba∇dbluε(t)−u0(t)/ba∇dblY≤C\nβ(1−β)(t−τ)−1+β(1−θ)[/ba∇dbluτ/ba∇dblY+1]ρ(θ,ε)e(˜K+K)(t−τ)\nand\n/ba∇dbluε(t)−u0(t)/ba∇dblYε→0−→0,\nuniformly for tin compact subsets of ( τ,∞), anyτ∈Randuτin bounded sets of Y.\n/square\n4.Application to reaction-diffusion equations with varying diffusion coefficients\nAs a first application of the abstract theory developed in the previo us sections, we consider a family in\nε∈[0,1] of singularly nonautonomous reaction-diffusion equation in a boun ded smooth domain Ω ⊂R3\n(uε)t−div(aε(t,x)∇uε)+uε=fε(t,uε), x ∈Ω,t>τ,\n∂nuε= 0, x ∈∂Ω,\nuε(τ,x) =uτ(x).(4.1)\nAn autonomous version (where aεandfεdo not depend on t) was completely studied in [3] and the authors\nobtained rate of convergence of solutions and attractors in term s ofε. The nonautonomous counterpart (4.1)\nwas introduced in [6], where the authors studied global well-posedne ss and existence of pullback attractor, but\nfor a single equations rather than a family of equations parametrize d inε∈[0,1].\nWe shall apply the abstract theory developed in Section 2 in order to obtain a rate at which solutions of\n(4.1) converge as ε→0+. We assume the following conditions for the problem:\n(A.1) The functions aε:R×Ω→R+are continuously differentiable with respect to the second variable, and\naε(·,·) has its image in a closed interval [ m,M]⊂(0,∞). We also assume that the gradient function\n(inx) ofaε(t,x) is bounded, that is, ∇xaε(t,x)∈[L∞(Ω)]3.\n(A.2) Both functions aε(·,·) and∇xaε(·,·) are uniformly δ−H¨ older continuous in the first variable that is,\nthere exists δ∈(0,1] and a constant C >0 such that\n|aε(t,x)−aε(s,x)| ≤C|t−s|δ,|∇xaε(t,x)−∇xaε(s,x)| ≤C|t−s|δ,\nfor allε∈[0,1],t,s∈Randx∈Ω.\n(A.3) For each ε∈[0,1],fε∈ C1(R×R,R) and satisfies a polynomial growth condition of order ρ, that is,\nthere exists Cand 1≤ρ<3 such that\n|fε(t,ξ)−fε(t,ψ)| ≤C|ξ−ψ|(1+|ξ|ρ−1+|ψ|ρ−1),\n|fε(t,ξ)| ≤C(1+|ξ|ρ).\n(A.4) We define the quantities\n/ba∇dblaε−a0/ba∇dbl∞:= sup\nt∈R/ba∇dblaε(t,·)−a0(t,·)/ba∇dblL∞(Ω),\n/ba∇dbl∇xaε−∇xa0/ba∇dbl∞:= sup\nt∈R/ba∇dbl∇xaε(t,·)−∇xa0(t,·)/ba∇dbl[L∞(Ω)]3,\n/ba∇dblfε−f0/ba∇dbl∞:= sup\nt∈R/ba∇dblfε(t,·)−f0(t,·)/ba∇dblL∞(Ω),\nand we assume that each one of them varies continuously on ε∈[0,1]. In particular,they approach to\nzero asε→0+.16 M. BELLUZI\nThe upper bound requested for ρin (A.3) will become clear after we specify the phase space in which we p ose\nthe problem. Under conditions above, we write Problem (4.1) in its abs tract form as follows: the linear part of\nthe equation (which is time-dependent) is given by the operator Aε(t) :D(Aε(t))⊂L2(Ω)→L2(Ω) where\nD=D(Aε(t)) =/braceleftbig\nu∈H2(Ω) :∂nu= 0 in∂Ω/bracerightbig\n=:H2\nN,\nAε(t)u=−div(aε(t,x)∇u)+u,foru∈D.\nThis family of linear operators has well-known properties that we gat her in the sequel. They follow from\nclassical spectral theory (see [5, 16, 18]) and from the propertie s required upon aεin (A.1) and (A.2). To\nsimplify notation, we shall omit the domain Ω in the space norms, that is ,/ba∇dbl·/ba∇dblL2=/ba∇dbl·/ba∇dblL2(Ω).\nProposition 4.1. This family {Aε(t), t∈R}ε∈[0,1]has the following properties:\n(1)D(Aε(t))does not depend on torε. Moreover, for any fixed ε∈[0,1]andt∈R, the graph norm\n/ba∇dblAε(t)·/ba∇dblL2is equivalent to H2(Ω)−norm when restricted to D, that is, for any u∈D,\nC1/ba∇dblu/ba∇dblH2≤ /ba∇dblAε(t)u/ba∇dblL2≤C2/ba∇dblu/ba∇dblH2,\nand constants C1,C2are uniform for ε∈[0,1]andt∈R.\n(2)Aε(t)is self-adjoint and has compact resolvent.\n(3) Its spectrum consists entirely of isolated eigenvalues , all of them positive and real, with the first being 1:\nσ(Aε(t)) ={λε,i(t);i∈N∗and1 =λε,1(t)≤λε,2(t)≤...≤λε,n(t)≤...}.\n(4) For anyπ\n2<ϕ<π,Σϕ={λ∈C;|argλ| ≤ϕ} ⊂ρ(−Aε(t))and\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(L2)≤C\n|λ|+1,∀λ∈Σϕ∪{0},\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(H1)≤C\n|λ|+1,∀λ∈Σϕ∪{0},\n/ba∇dbl(λI+Aε(t))−1/ba∇dblL(L2,H1)≤C\n|λ|1\n2+1,∀λ∈Σϕ∪{0},\nwhereCdoes not depend on εort(only onϕ).\nWe restate Problem (4.1) as an abstract semilinear evolution problem :\n(uε)t+Aε(t)uε=Fε(t,uε), t>τ,\nuε(τ) =uτ∈H1(Ω),(4.2)\nwhereFεis a nonlinearity given by\nFε(t,uε)(x) =fε(t,uε(t,x)).\nSince\nH1(Ω)֒→Lr(Ω),for all 2≤r<6,\nthen the growth condition (A.3) required for fεimplies that F:R×H1(Ω)→L2(Ω), as long as 1 ≤ρ <3.\nWith the notation of Section 2, L2(Ω) will play the role of Banach space XandH1(Ω) the Banach space Y.\nMoreover, one can easily check that from (A.3) we derive, for any ε∈[0,1] andt∈R,\n/ba∇dblFε(t,u)−Fε(t,v)/ba∇dblL2≤C/ba∇dblu−v/ba∇dblH1/bracketleftBig\n1+/ba∇dblu/ba∇dblρ−1\nH1+/ba∇dblv/ba∇dblρ−1\nH1/bracketrightBig\n,\n/ba∇dblFε(t,u)/ba∇dblL2≤C/bracketleftbig\n1+/ba∇dblu/ba∇dblρ\nH1/bracketrightbig\n.\nIn order to apply the theory developed in Section 2, we first need to verify that (P.1) to(P.5) hold for (4.2).\nFrom Proposition 4.1, properties (P.1) and (P.2) already follow. Prop erty (P.3) is proved in next lemma.17\nLemma 4.2. Assume (A.1) and (A.2) hold and let δ∈(0,1]be the uniform H¨ older exponent for t/mapsto→aε(t,·)\nandt/mapsto→ ∇xaε(t,·). Then, there exists a constant C >0, independent of ε∈[0,1]orτ∈R, such that, for any\nε1,ε2∈[0,1], the function\nR∋t/mapsto→Aε1(t)Aε2(τ)−1∈ L(L2(Ω))\nis H¨ older continuous with exponent δ, that is,\n/ba∇dbl[Aε1(t)−Aε1(s)]Aε2(τ)−1/ba∇dblL(L2)≤C|t−s|δ,for allτ,s,t∈R.\nProof.For anyu∈D, we haveAε1(t)u−Aε1(s)u=−div([aε1(t,x)−aε1(s,x)]∇u) and\n/ba∇dblAε1(t)u−Aε1(s)u/ba∇dbl2\nL2=/integraldisplay\nΩ|div([aε1(t,x)−aε1(s,x)]∇u(x))|2dx\n=/integraldisplay\nΩ|∇x([aε1(t,x)−aε1(s,x)])∇u(x)+[aε1(t,x)−aε1(s,x)]∆u(x)|2dx\n≤C|t−s|2δ/integraldisplay\nΩ/braceleftbigg|∇xaε1(t,x)−∇xaε1(s,x)|\n|t−s|δ/bracerightbigg2\n|∇u(x)|2dx\n+C|t−s|2δ/integraldisplay\nΩ/braceleftbigg|aε1(t,x)−aε1(s,x)|\n|t−s|δ/bracerightbigg2\n|∆u(x)|2dx\n≤C|t−s|2δ/braceleftbig\n/ba∇dbl∇u/ba∇dbl2\nL2+/ba∇dbl∆u/ba∇dbl2\nL2/bracerightbig\n≤C|t−s|2δ/ba∇dblu/ba∇dbl2\nH2.\nTaking the square root on both sides and replacing ubyAε2(τ)−1w, then we have for any w∈L2(Ω)\n/ba∇dbl[Aε1(t)−Aε1(s)]Aε2(τ)−1w/ba∇dblL2≤C|t−s|δ/ba∇dblAε2(τ)−1w/ba∇dblH2≤C|t−s|δ/ba∇dblw/ba∇dblL2.\n/square\nIt remains to check the properties responsible to make the connec tions among the problems as εvaries in\n[0,1]. Those are conditions (P.4) and (P.5) from Section 2. We start ver ifying (P.4) and we begin by proving\nan auxiliary result.\nLemma 4.3. Letε1,ε2∈[0,1]andt,τ∈R. Then\n/ba∇dbl[Aε1(t)−A0(t)]Aε2(τ)−1/ba∇dblL(L2)≤C(/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε1−∇xa0/ba∇dbl∞).\nProof.Takeu=Aε2(τ)−1w∈D, wherewis any element in L2(Ω). We have\n/ba∇dbl[Aε1(t)−A0(t)]u/ba∇dbl2\nL2≤/integraldisplay\nΩ|div[(aε1(t,x)−a0(t,x))∇u]|2dx\n≤C/integraldisplay\nΩ[|∇x(aε1(t,x)−a0(t,x))|2|∇u|2+|aε1(t,x)−a0(t,x)|2|∆u|2]dx\n≤C(/ba∇dblaε1−a0/ba∇dbl2\n∞+/ba∇dbl∇xaε1−∇xa0/ba∇dbl2\n∞)/ba∇dblu/ba∇dbl2\nH2.\nTherefore,\n/ba∇dbl[Aε1(t)−A0(t)]Aε2(τ)−1w/ba∇dblL2≤C(/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε1−∇xa0/ba∇dbl∞)/ba∇dblAε2(τ)−1w/ba∇dblL2\n≤C(/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε1−∇xa0/ba∇dbl∞)/ba∇dblw/ba∇dblL2.\n/square\nWith the previous lemma, we are now able to prove that (P.4) holds for this problem.\nLemma 4.4. Lett,τ∈R,ε∈[0,1]. There exists C >0independent of t,τ,εsuch that\n/ba∇dblAε(t)Aε(τ)−1−A0(t)A0(τ)−1/ba∇dblL(L2)≤C(/ba∇dblaε−a0/ba∇dbl∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl∞).18 M. BELLUZI\nProof.Letw∈L2(Ω) and consider u=Aε(τ)−1w,v=A0(τ)−1w. It follows from the boundedness of aεand\nits convergence to a0that\n/ba∇dblAε(t)u−A0(t)v/ba∇dbl2\nL2≤C/integraldisplay\nΩ|div[aε(t,x)∇u]−div[a0(t,x)∇v]|2dx+C/integraldisplay\nΩ|u−v|2dx\n≤C/integraldisplay\nΩ|div[(aε(t,x)−a0(t,x))∇u]+div[a0(t,x)(∇u−∇v)]|2dx+C/integraldisplay\nΩ|u−v|2dx\n≤C/integraldisplay\nΩ{|∇xaε(t,x)−∇xa0(t,x)|2|∇u|2+|aε(t,x)−a0(t,x)|2|∆u|2\n+|∇xa0(t,x)|2|∇(u−v)|2+|a0(t,x)|2|∆(u−v)|2}dx+C/integraldisplay\nΩ|u−v|2dx\n≤C/ba∇dbl∇xaε−∇xa0/ba∇dbl2\n∞/parenleftbigg/integraldisplay\nΩ|∇u|2dx/parenrightbigg\n+C/ba∇dblaε−a0/ba∇dbl2\n∞/parenleftbigg/integraldisplay\nΩ|∆u|2/parenrightbigg\n+C/ba∇dblu−v/ba∇dbl2\nH2\n≤C(/ba∇dblaε−a0/ba∇dbl2\n∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl2\n∞)/ba∇dblu/ba∇dbl2\nH2+C/ba∇dblu−v/ba∇dbl2\nH2.\nFrom the choice of u,vand from Lemma 4.3 we obtain\n/ba∇dblAε(t)Aε(τ)−1w−A0(t)A0(τ)−1w/ba∇dbl2\nL2\n≤C(/ba∇dblaε−a0/ba∇dbl2\n∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl2\n∞)/ba∇dblAε(τ)−1w/ba∇dbl2\nH2+C/ba∇dblAε(τ)−1w−A0(τ)−1w/ba∇dbl2\nH2\n≤C(/ba∇dblaε−a0/ba∇dbl2\n∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl2\n∞)/ba∇dblw/ba∇dbl2\nH2+C/ba∇dbl[A0(τ)−Aε(t)]A0(τ)−1w/ba∇dbl2\nH2\n≤2C(/ba∇dblaε−a0/ba∇dbl2\n∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl2\n∞)/ba∇dblw/ba∇dbl2\nH2.\n/square\nTherefore, Lemma 4.4 states that (P.4) holds for\nξ(ε) =C(/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl∞).\nInspired in [3], we will use a variational formulation in order to obtain r esolvent convergence\n/ba∇dblA(t)−1\nε−A(t)−1\n0/ba∇dblL(L2,H1)ε→0+\n−→0,\nthat is, in order to prove that (P.5) holds.\nLemma 4.5. Giveng∈X=L2(Ω), a fixedt∈Randε∈[0,1], there exists a unique uε∈H2\nNsolution of\n\n\n−div(aε(t,x)∇uε)+uε=g, x∈Ω,\n∂nuε= 0, ∂Ω.(4.3)\nMoreover,\n(1) there exists C >0, independent of ε∈[0,1],g∈L2(Ω)andt∈R, such that\n/ba∇dbluε/ba∇dblH1≤C/ba∇dblg/ba∇dblL2.\n(2) There is also a constant C >0, independent of ε∈[0,1],g∈L2(Ω)andt∈R, such that\n/ba∇dbluε−u0/ba∇dblH1≤C/ba∇dblaε−a0/ba∇dbl∞/ba∇dblg/ba∇dblL2.\nProof.Existence of uεthat solves (4.3) follows from the fact that 0 ∈ρ(Aε(t)), for allt∈Randε∈[0,1]. To\nprove the first statement, we consider the weak formulation of (4 .3):\n/integraldisplay\nΩaε(t,x)∇uε∇ϕ+/integraldisplay\nΩuεϕ=/integraldisplay\nΩgϕ,forϕ∈H1(Ω).\nBy takingϕ=uε, using Young’s inequality and the fact that aε(·,·)⊂[m,M], we obtain, for any ν >0,\n/integraldisplay\nΩaε(t,x)[∇uε]2+/integraldisplay\nΩ[uε]2=/integraldisplay\nΩguε,\nm/integraldisplay\nΩ[∇uε]2+/integraldisplay\nΩ[uε]2≤/integraldisplay\nΩ|g||uε| ≤/integraldisplay\nΩ/bracketleftbigg/braceleftbigg1\nν2|g|2\n2/bracerightbigg\n+/braceleftbiggν2|uε|2\n2/bracerightbigg/bracketrightbigg\n,19\nm/integraldisplay\nΩ[∇uε]2+/parenleftbigg\n1−ν2\n2/parenrightbigg/integraldisplay\nΩ[uε]2≤1\n2ν2/integraldisplay\nΩ|g|2.\nChoosingνsmall such that 1 −ν2\n2>0 we obtain\n/ba∇dbluε/ba∇dbl2\nH1≤C/ba∇dblg/ba∇dbl2\nL2, (4.4)\nwhereC=1\n2ν2min{m,1−ν2\n2}, which does not depend on εort.\nFor the second statement we proceed similarly. Rather than taking uεas a test function, we choose uε−u0,\nobtaining\n/integraldisplay\nΩaε(t,x)∇uε(∇uε−∇u0)+/integraldisplay\nΩuε(uε−u0) =/integraldisplay\nΩg(uε−u0),\n/integraldisplay\nΩa0(t,x)∇u0(∇uε−∇u0)+/integraldisplay\nΩu0(uε−u0) =/integraldisplay\nΩg(uε−u0).\nEquality on the right side implies/integraldisplay\nΩaε(t,x)∇uε(∇uε−∇u0)+/integraldisplay\nΩuε(uε−u0) =/integraldisplay\nΩa0(t,x)∇u0(∇uε−∇u0)+/integraldisplay\nΩu0(uε−u0),\n/integraldisplay\nΩaε(t,x)∇uε(∇uε−∇u0)+/integraldisplay\nΩ(uε−u0)2=/integraldisplay\nΩa0(t,x)∇u0(∇uε−∇u0).\nWe now subtract/integraltext\nΩaε(t,x)∇u0(∇uε−∇u0) on both sides, which results\n/integraldisplay\nΩaε(t,x)(∇uε−∇u0)2+/integraldisplay\nΩ(uε−u0)2=/integraldisplay\nΩ[a0(t,x)−aε(t,x)]∇u0(∇uε−∇u0)\n≤ /ba∇dblaε−a0/ba∇dbl∞/ba∇dbl∇u0/ba∇dblL2/ba∇dbl∇uε−∇u0/ba∇dblL2\nIf1\nC= min{m,1}, we obtain from the above inequality and using (4.4),\n/ba∇dbluε−u0/ba∇dbl2\nH1=/bracketleftbigg/integraldisplay\nΩ(∇uε−∇u0)2+/integraldisplay\nΩ(uε−u0)2/bracketrightbigg\n≤C/ba∇dblaε−a0/ba∇dbl∞/ba∇dbl∇u0/ba∇dblL2/ba∇dbl∇uε−∇u0/ba∇dblL2\n≤1\n2C2/ba∇dblaε−a0/ba∇dbl2\n∞/ba∇dbl∇u0/ba∇dbl2\nL2+1\n2/ba∇dbl∇uε−∇u0/ba∇dbl2\nL2\n≤1\n2C2/ba∇dblaε−a0/ba∇dbl2\n∞/ba∇dblg/ba∇dbl2\nL2+1\n2/ba∇dbluε−u0/ba∇dbl2\nH1.\nTherefore, /ba∇dbluε−u0/ba∇dbl2\nH1≤C/ba∇dblaε−a0/ba∇dbl2\n∞/ba∇dblg/ba∇dbl2\nL2. /square\nAs an immediate consequence of the previous result, we have the fo llowing corollary.\nCorollary 4.6. The operators Aε(t)−1:L2(Ω)→H1(Ω)are uniformly bounded for t∈Randε∈[0,1]and\nthey converge to A0(t)−1in the uniform topology. More precisely, for all ε∈[0,1]andt∈R,\n/ba∇dblAε(t)−1/ba∇dblL(L2,H1)≤C, (4.5)\n/ba∇dblAε(t)−1−A0(t)−1/ba∇dblL(L2,H1)≤C/ba∇dblaε−a0/ba∇dbl∞, (4.6)\nwhereCdoes not depend on εort.\nInequality (4.6)isthestatementrequiredin(P.5), with rateofconv ergenceη(ε) givenbyη(ε) :=C/ba∇dblaε−a0/ba∇dbl∞.\nSince (P.1) to (P.5) are satisfied, we conclude that each family of linea r operators {Aε(t), t∈R}generates a\nlinear process {Uε(t,τ) :L2(Ω)→L2(Ω), t≥τ, τ∈R}and from Theorem 2.3, we obtain that there exist\nC,K >0 such that, for any ε∈[0,1],t>τandτ∈R,\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(L2)≤C(t−τ)−θeK(t−τ)ℓ(θ,ε),\n/ba∇dblUε(t,τ)−U0(t,τ)/ba∇dblL(L2,H1)≤C(t−τ)−1\n2−θ\n2eK(t−τ)ℓ(θ,ε),20 M. BELLUZI\nwhereℓ(θ,ε) = [/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl∞]θandθ∈(0,1) is arbitrary, with Cdepending on the choice\nofθ.\n4.1.Local well-posedness, global well-posedness and converge nce of the solutions. The results on\nlocal and global well-posedness that we present in the sequel can b e found in [6, Section 6]. Conditions required\nforaε(·,·) andfε(·,·) ensure that problem (4.2) admits local solution uε: [τ,τ+T(ε,τ,uτ))→H1(Ω) given by\nuε(t,τ,uτ) =Uε(t,τ)uτ+/integraldisplayt\nτUε(t,s)Fε(s,uε(s))ds,\nsuch thatuε(t)∈D=X1, for allt∈(τ,τ+T(ε,τ,uτ)), andUε(·,·) :L2(Ω)→L2(Ω) is the linear process\nassociated to {Aε(t), t∈R}.\nTo obtain global well-posedness, we assume that fεsatisfies a dissipativeness condition:\n(D)limsup|s|→∞/bracketleftbigg\nsupε∈[0,1]fε(t,s)\ns/bracketrightbigg\n<1,\nfor allt∈R. The value 1 comes from the fact that first eigenvalue of Aε(t) isλε,1(t) = 1. In next lemma, we\nrestate this dissipativeness condition in a manner suitable to applicat ions. Its proof follows directly from the\ndefinition of Limsup.\nLemma 4.7. Suppose that condition (D) holds, then there exists ω1>0such that, for each ω∈(0,ω1), there\nis a constant N >0such that\nfε(t,s)s≤(1−ω)s2+N,for alls∈R, t∈R, ε∈[0,1]. (4.7)\nMoreover,N,ωandω1are independent of ε.\nThe dissipativeness assumption allows us to obtain global well-posedn ess, as well as existence of an absorbing\nbounded set in H1(Ω), uniform in ε∈[0,1].\nTheorem 4.8. [6, Theorem 6.13] Assume that (A.1) to (A.4) and (D) hold. Let N,ωbe the constants in (4.7)\nobtained from the dissipativeness condition (D). There exi sts a constant E >0independent of ε∈[0,1]and of\nτ∈R, such that, for any bounded set B⊂H1(Ω)we can find T=T(B)>0, for which\n/ba∇dbluε(t,τ,uτ)/ba∇dblH1≤E,for anyuτ∈B, ε∈[0,1],\nas long as t−τ≥T. In particular, the solution of (4.2)is globally defined and associated to it there is a\nnonlinear process Sε(t,τ)inH1(Ω)given by\nSε(t,τ)uτ=uε(t,τ,uτ) =Uε(t,τ)uτ+/integraldisplayt\nτUε(t,s)Fε(s,uε(s))ds,for allt≥τ.\nOnce we proved that the dynamics of all the problems enter a commo n bounded set BH1[0,E] (the closed\nball inH1(Ω) centered in 0 and with radius E), we can proceed with a cut-off for the nonlinearities Fε, as\nmentioned in Remark 2.4. If that is the case, at least close to BH1[0,E], Condition (NL.1) holds. As far as\nCondition (NL.2), we have\n/ba∇dblFε(t,u)−F0(t,u)/ba∇dbl2\nL2=/integraldisplay\nΩ|fε(t,u(x))−f0(t,u(x))|2dx≤ /ba∇dblfε−f0/ba∇dbl2\n∞|Ω|.\nTherefore,\nsup\nt∈Rsup\nu∈H1/ba∇dblFε(t,u)−F0(t,u)/ba∇dblL2≤ /ba∇dblfε−f0/ba∇dbl∞|Ω|1\n2:=γ(ε),\nand Condition (NL.2) also holds. We then conclude from Theorem 2.5 th at the solutions converge as εgoes to\nzero, with a rate:\n/ba∇dbluε(t,τ,uτ)−u0(t,τ,uτ)/ba∇dblY≤C(t−τ)−1\n2−θ\n2eK(t−τ)[1+/ba∇dbluτ/ba∇dblY]ρ(θ,ε),\nwhereρ(θ,ε) = max{[/ba∇dblaε1−a0/ba∇dbl∞+/ba∇dbl∇xaε−∇xa0/ba∇dbl∞]θ,/ba∇dblfε−f0/ba∇dbl∞|Ω|1\n2},θ∈(0,1) is arbitrary and C,K >0\nare constants independent of ε∈[0,1], τ∈R, t>τoruτ∈Y, but dependent on the choice of θ.21\n4.2.Existence of pullback attractor and its upper-semicontinu ity.The existence of pullback attractor\nwas also obtained in [6, Theorem 6.14] for a single problem, rather th an a family of problems. However, in the\nproof of this theorem, the authors were able to find a compact abs orbing set in H1(Ω) depending only on the\nconstantsω,Nobtained in Lemma 4.7 and the growth ρfor the nonlinearity. Since they are all uniform for\nε∈[0,1], we can state the next theorem as a consequence of the result in [6, Theorem 6.14].\nTheorem 4.9. Assume that (A.1) to (A.4) and (D) hold. Let N,ωbe the constants in (4.7)obtained from the\ndissipativeness condition (D). Then the nonlinear process Sε(t,τ) =uε(t,τ,·)inH1(Ω)has a pullback attractor\n{Aε(t), t∈R}inH1(Ω). Moreover, there exists a compact set K⊂H1(Ω)such that\n/bracketleftBig/uniontext\nε∈[0,1]/uniontext\nt∈RAε(t)/bracketrightBig\n⊂K. (4.8)\nFrom Corollary 2.6, we obtain continuity of the family {Sε(·,·)}ε∈[0,1]and from (4.8), we conclude that\n/bracketleftBig/uniontext\nε∈[0,1]/uniontext\nt∈RAε(t)/bracketrightBig\nis relatively compact. Those are the conditions in [9, Theorem 3.6] ne cessary to ensure upper-semicontinuity of\nthe family {Aε(t),t∈R}ε∈[0,1]atε= 0.\nCorollary 4.10. Under conditions of Theorem 4.9, the family of pullback attr actos{Aε(t),t∈R}ε∈[0,1]is\nupper-semicontinuous at ε= 0.\n5.Application to a nonautonomous strongly damped wave equati ons and its fractional\napproximations\nAs a second application, we considerthe nonautonomousstronglyd amped waveequationsubjected to Dirich-\nlet boundary conditions\nutt+(−a(t)∆D)u+2(−a(t)∆D)1\n2ut=f(t,u), x ∈Ω, t>τ,\nu(t,x) = 0, x ∈∂Ω, t≥τ,\nu(τ,x) =uτ(x), ut(τ,x) =vτ(x), x ∈Ω, τ∈R,(5.1)\nwhere Ω ⊂Rn,n≥3, is a bounded smooth domain, ∆ Dis the Laplacian operator with Dirichlet boundary\ncondition and f:R×R→Ra nonlinearity. We shall assume the following additional condition:\n(B) The function a:R→R+ispositiveandhasits imagein abounded intervalofthe form[ a0,a1]⊂(0,∞).\nWe also assume that it is H¨ older continuous with an exponent δ∈(0,1], that is, there exists a constant\nC >0 such that\n|a(t)−a(s)| ≤C|t−s|δ,for allt,s∈R.\nLetE=L2(Ω) and denote by A(t) :D(A(t))⊂E→Ethe operator\nA(t)u=−a(t)∆Du,foru∈D(A(t)) =D(∆D) =H2(Ω)∩H1\n0(Ω), (5.2)\nwhereD(∆D) stands for the domain of the Laplacian with Dirichlet boundary cond itions. As expected, the\nmultiplication by a real-function a(t) does not change the domain of the Laplacian.\nTherefore, this linear operator A(t) has a time-independent domain and from the well-known properties of\nthe Laplacian operator [18] and the fact that a(t)≥a0,for allt∈R, we deduce that A(t) is a positive operator,\nself-adjoint, sectorial and −A(t) generates a compact analytic C0−semigroup in E.\nConsequently, fractional powers of A(t) in the sense of Amman [1] are well-defined. We shall denote by A(t)α\nthe power of the linear operator A(t). One can easily deduce from the expressions for fractional powe r of linear\noperators that\nA(t)α= (−a(t)∆D)α= [a(t)]α(−∆D)α,for allt∈Randα∈(0,1],22 M. BELLUZI\nand the domain of A(t)αis the same as the domain of ( −∆D)α, that is,\nD(A(t)α) =D((−∆)α),for allt∈R.\nWe then define a scale of Banach spaces given by the fractional pow ers (−∆D)α,α∈(0,1],\nEα= (−∆D)αeqquiped with the norm /ba∇dbl·/ba∇dblEα=/ba∇dbl(−∆D)α·/ba∇dblL2.\nIn particular, E0=E=L2(Ω),E1\n2=H1\n0(Ω) andE1=H2(Ω)∩H1\n0(Ω) (see [13]Theorem 3.6). From the\nboundedness of a(·), there exists 0 τ;/bracketleftBigg\nu(τ)\nv(τ)/bracketrightBigg\n=/bracketleftBigg\nu��\nvτ/bracketrightBigg\n∈E1\n2×E, (5.3)\nwhere Λ(t) :D(Λ(t))⊂E1\n2×E→E1\n2×Eis the linear operator defined in D(Λ(t)) =D=E1×E1\n2and given\nby\nΛ(t)/bracketleftBigg\nu\nv/bracketrightBigg\n=/bracketleftBigg\n0−I\nA(t) 2A(t)1\n2/bracketrightBigg/bracketleftBigg\nu\nv/bracketrightBigg\n=/bracketleftBigg\n−v\nA(t)u+2A(t)1\n2v/bracketrightBigg\n, (5.4)\nandFis the nonlinearity given by\nF/parenleftBigg\nt,/bracketleftBigg\nu\nv/bracketrightBigg/parenrightBigg\n=/bracketleftBigg\n0\nf(t,u)/bracketrightBigg\n.\nWe have the following result proved in [7, Lemma 8.1] concerning spect ral properties of Λ( t) and the calculus\nof its fractional powers.\nProposition 5.1. IfA(t)andΛ(t)are as in (5.2)and in(5.4), respectively, then the following properties hold:\n(1)Λ(t)is a positive operator with fractional powers denoted by Λ(t)α,α∈(0,1].\n(2) There exists C >0andϕ∈(π\n2,π)(independent of α) such that, for any α∈(0,1]andt∈R,\nΣϕ∪{0} ⊂ρ(−Λ(t)α),\nand the following estimates hold\n/ba∇dbl(λ+Λ(t)α)−1/ba∇dblL(E1\n2×E)≤C\n1+|λ|,for allλ∈Σϕ∪{0}.\nTherefore, each Λ(t)αis sectorial in E1\n2×Eand−Λ(t)αgenerates an analytic semigroup in L(E1\n2×E).\n(3) Given any α∈(0,1], we have the following explicitly expression for the fracti onal powers of Λ(t):\nΛ(t)α=/bracketleftBigg\n(1−α)A(t)α\n2−αA(t)−1+α\n2\nαA(t)1+α\n2(1+α)A(t)α\n2/bracketrightBigg\nandΛ(t)−α=/bracketleftBigg\n(1+α)A(t)−α\n2αA(t)−1−α\n2\n−αA(t)1−α\n2(1−α)A(t)−α\n2/bracketrightBigg\n(5.5)\nWe shall consider fractional versions of Problem (5.3), given by\nd\ndt/bracketleftBigg\nuα\nvα/bracketrightBigg\n+Λ(t)α/bracketleftBigg\nuα\nvα/bracketrightBigg\n=Fα/parenleftBigg\nt,/bracketleftBigg\nuα\nvα/bracketrightBigg/parenrightBigg\n, t>τ;/bracketleftBigg\nuα(τ)\nvα(τ)/bracketrightBigg\n=/bracketleftBigg\nuτ\nvτ/bracketrightBigg\n∈E1\n2×E, (5.6)\nwhereα∈(0,1], Λ(t)αis the fractional power of Λ( t) and\nFα/parenleftBigg\nt,/bracketleftBigg\nuα\nvα/bracketrightBigg/parenrightBigg\n=/bracketleftBigg\n0\nfα(t,uα)/bracketrightBigg\n.\nBy analyzingExpression(5.5) for the linear operatorΛ( t)α, we see that α= 1 recoversthe originalexpression\nfor Λ(t), so we might expect that as we make α→1−, the fractional problem (5.6) approaches (5.3) in a certain\nsense. We shall verify that this is the case, that is, we prove in the s equel that conditions (P.1) to (P.5) hold23\nfor this family (in α) of problems. Therefore, if Uα(t,τ) is the linear process associated to {Λ(t)α,t∈R}, then\nwe obtain its convergence to the linear process U(t,τ) associated to {Λ(t),t∈R}.\nA slightly different singularly nonautonomous wave equation and its fr actional perturbations were also con-\nsidered in [17]. However, the lack of results on the linear process con vergence prevented the authors to proceed\nwith the analysis beyond spectrum convergence of the fractional operators. They were not able to obtain\nconvergence of the associated linear processes or of the solution s, as we shall do in the sequel.\nUsing the notation developed in Section 2, we will consider Y=X=E1\n2×E. Conditions (P.1) and (P.2)\nfollows directly from Proposition 5.1. For condition (P.3), we shall nee d the following technical lemma.\nLemma 5.2. Leta(·)be the function satisfying (B). For each ω>0, the functions [a(·)]ωand[a(·)]−ωare also\nH¨ older continuous with H¨ older exponent δ, that is, for all t,s∈R,\n|[a(t)]ω−[a(s)]ω| ≤C1|t−s|δand|[a(t)]−ω−[a(s)]−ω| ≤C2|t−s|δ.\nProof.Letφ:R+→R+be given by φ(s) =sω. From the mean value theorem and the fact that a(t)∈[a0,a1],\nfor allt∈R, we obtain, for some θbetweena(s) anda(t), in particular θ∈[a0,a1], that\n|[a(t)]ω−[a(s)]ω|=|φ(a(t))−φ(a(s))| ≤ |φ′(θ)||a(t)−a(s)| ≤θω−1|a(t)−a(s)| ≤C1|t−s|δ,\nfor allt,s∈R. Moreover,\n|[a(t)]−ω−[a(s)]−ω|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle[a(s)]ω−[a(t)]ω\n[a(t)]ω[a(s)]ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|[a(s)]ω−[a(t)]ω| ≤C2|t−s|δ,for allt,s∈R.\n/square\nCondition (P.3) can now be verified.\nLemma 5.3. Assume that (B) holds and let Λ(t)be the linear operator in (5.4). IfΛ(t)αdenotes the fractional\npowers of Λ(t), thenR∋t→Λ(t)αΛ(τ)−α∈ L(E1\n2×E)isδ−H¨ older continuous, uniformly in α∈(0,1]and\nτ∈R. In other words, there exists C >0such that\n/ba∇dbl[Λ(t)α−Λ(s)α]Λ(τ)−α/ba∇dblL(E1\n2×E)≤C|t−s|δ,for allt,s,τ∈Randα∈(0,1].\nProof.Applying expression (5.5) for the fractional powers, we deduce th at\n[Λ(t)α−Λ(s)α]Λ(τ)−α=/bracketleftBigg\nΘ11Θ11\nΘ21Θ22/bracketrightBigg\n,\nwhere\nΘ11=α2/parenleftBig\n[a(t)]α−1\n2−[a(s)]α−1\n2/parenrightBig\n[a(τ)]1−α\n2+(1−α2)/parenleftbig\n[a(t)]α\n2−[a(s)]α\n2/parenrightbig\n[a(τ)]−α\n2,\nΘ12=α(1−α)(−∆D)−1\n2/braceleftBig/parenleftbig\n[a(t)]α\n2−[a(s)]α\n2/parenrightbig\n[a(τ)]−1−α\n2+/parenleftBig\n[a(s)]α−1\n2−[a(t)]α−1\n2/parenrightBig\n[a(τ)]−α\n2/bracerightBig\n,\nΘ21=−α(1+α)(−∆D)1\n2/braceleftBig/parenleftbig\n[a(t)]α\n2−[a(s)]α\n2/parenrightbig\n[a(τ)]1−α\n2+/parenleftBig\n[a(s)]α+1\n2−[a(t)]α+1\n2/parenrightBig\n[a(τ)]−α\n2/bracerightBig\n,\nΘ22=α2/parenleftBig\n[a(t)]α+1\n2−[a(s)]α+1\n2/parenrightBig\n[a(τ)]−α−1\n2+(α+1)(α−1)/parenleftbig\n[a(t)]α\n2−[a(s)]α\n2/parenrightbig\n[a(τ)]−α\n2.\nWe must obtain H¨ older continuity of each entry in its appropriate sp ace, that is, Θ 11inL(E1\n2), Θ12in\nL(E,E1\n2), Θ21inL(E1\n2,E) and Θ 22inL(E). All of them are similar and follows from Lemma 5.2. We\nillustrate how to proceed with Θ 21.\n/ba∇dblΘ21/ba∇dblL(E1\n2,E)≤ /ba∇dblΘ21(−∆D)−1\n2/ba∇dblL(E)≤α(1+α)/parenleftBig\n|[a(t)]α\n2−[a(s)]α\n2|a1−α\n2\n1+|[a(s)]α+1\n2−[a(t)]α+1\n2|a−α\n2\n0/parenrightBig\n≤C21|t−s|δ.\nThe other entries follow analogously. /square\nIn a similar way, from the Expressions (5.5) for the fractional powe rs, we can deduce property (P.4) for the\nfamily of linear operators {Λ(t)α, t∈R}α∈(0,1].24 M. BELLUZI\nLemma 5.4. Assume that (B) holds and let Λ(t)be the linear operator in (5.4)andΛ(t)αits fractional powers.\nThere exists a continuous function ξ: (0,1]→R+withξ(1) = 0such that\nsup\nt∈R/ba∇dblΛ(t)αΛ(τ)−α−Λ(t)Λ(τ)−1/ba∇dblL(E1\n2×E)≤ξ(α). (5.7)\nProof.Applying expressions (5.5) for the fractional powers, we deduce\nΛ(t)αΛ(τ)−α−Λ(t)Λ(τ)−1=/bracketleftBigg\nΓ11Γ12\nΓ21 Γ22/bracketrightBigg\n,\nwhere\nΓ11= (1−α2)[a(t)]α\n2[a(τ)]α−1\n2+(α2−1)[a(t)]α−1\n2[a(τ)]α\n2+/parenleftBig\n[a(τ)]α\n2[a(t)]α−1\n2−[a(τ)]1\n2/parenrightBig\n,\nΓ12= (1−α)α[a(t)]α−1\n2[a(τ)]−α−1\n2/parenleftBig\n[a(t)]1\n2−[a(τ)]1\n2/parenrightBig\n(−∆D)−1\n2,\nΓ21=α(1+α)(−∆D)1\n2/parenleftbigg\na(t)/braceleftbigg\n[a(t)]α−1\n2−2\nα(1+α)/bracerightbigg\n+/braceleftbigg2\nα(1+α)−[a(τ)]1−α\n2/bracerightbigg/parenrightbigg\n,\nΓ22= [a(t)]α\n2[a(τ)]−1−α\n2/parenleftBig\n(1+α2)[a(t)]1\n2−α2[a(τ)]1\n2/parenrightBig\n−[a(t)]1\n2[a(τ)]−1/parenleftBig\n2[a(t)]1\n2−[a(τ)]1\n2/parenrightBig\n.\nNote that each entry goes to zero continuously as α→1−. The presence of ( −∆D)−1\n2in Γ12or (−∆D)1\n2\nin Γ21do not represent any problem in the estimates. Actually, since Γ 12is estimated in L(E,E1\n2) and Γ 21in\nL(E1\n2,E) we have\n/ba∇dblΓ12/ba∇dblL(E,E1\n2)=/ba∇dbl(−∆D)1\n2Γ12/ba∇dblL(E)and/ba∇dblΓ21/ba∇dblL(E1\n2,E)=/ba∇dblΓ12(−∆D)−1\n2/ba∇dblL(E)\nand those powers of the Laplacian disappear when we estimate thos e terms. Therefore, there exists a continuous\nfunctionξ: (0,1]→R+withξ(1) = 0 such that (5.7) holds. /square\nEven though we are able to prove that (P.4) holds, we cannot obtain an explicit formulation for ξin (5.7),\nsince it depends on the expression of a(·). Lastly, Condition (P.5) holds following the same proof of [7, Theor em\n3.1].\nLemma 5.5. [7, Theorem 3.1] LetΛ(t)be the linear operator in (5.4)andΛ(t)α,α∈(0,1]its fractional. There\nexists a constant C >0, independent of αandt∈R, such that\n/ba∇dblΛ(t)−α−Λ(t)−1/ba∇dblL(E1\n2×E)≤C(1−α).\nSince conditions (P.1) to (P.5) hold, we have the following result, which is a restatement of Theorem 2.3.\nTheorem 5.6. Let{U(t,τ)∈ L(E1\n2×E), t≥τ}be the linear process associated to {Λ(t), t∈R}and\n{Uα(t,τ)∈ L(E1\n2×E), t≥τ}the linear process associated to {Λ(t)α, t∈R},α∈(0,1]. For anyθ∈(0,1),\nthere exists constants K,C >0, independent of α∈(0,1],τ∈Randt>τ, such that\n/ba∇dblUα(t,τ)−U(t,τ)/ba∇dblL(E1\n2×E)≤C(t−τ)−θeK(t−τ)ℓ(θ,α),\nwhereℓ(θ,α) = max{(1−α)θ,[ξ(α)]θ}.\nUnder assumptions on boundedness and Lipschitz continuity for th e family of nonlinearities fα:R×R→R,\nα∈(0,1], as well as some convergence assumption of fαtofasα→1−, we derive conditions (NL.1) and\n(NL.2) forFα:E1\n2×E→E1\n2×E, as we did in Section 4. Then we could obtain convergence of the solut ions\nuα(t,τ,[uτ,vτ]) of Problem (5.6) to the solution u(t,τ,[uτ,vτ]) of Problem (5.3) in E1\n2×E, asα→1−as a\nconsequence of Theorem 2.5.\n6.Disclosures and declaration\nThe author declares no conflicts of interest. Maykel Boldrin Belluzi has received a research grant from\nFAPESP, Brazil, process number 2022/01439-5.\nMoreover, data sharing is not applicable to this article as no new data were created or analyzed in this study.25\nReferences\n[1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I , vol. 89, Monographs in Mathematics, Birkh¨ auser Boston, I nc.,\nBoston, MA, 1995.\n[2] J.M. Arrieta, Spectral behavior and upper semicontinui ty of attractors. In International Conference on Differential Equations,\nVol. 1, 2 (Berlin, 1999) . World Sci. Publ., River Edge, NJ (2000), 615-621.\n[3] J.M. Arrieta, F.D.M. Bezerra, and A.N. Carvalho, Rate of convergence of global attractors of some perturbed reactio n-diffusion\nproblems, Topol. Methods Nonlinear Anal. 41(2) (2013), 229-253.\n[4] J.M Arrieta, and A.N. Carvalho, Spectral convergence an d nonlinear dynamics of reaction-diffusion equations under pertur-\nbations of the domain, J. Differential Equations 199(1) (2004), 143–178.\n[5] J. M. Arrieta, A. N. Carvalho, and A. Rodr´ ıguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical\nnonlinearities, J. Differential Equations 156(2) (1999), 376–406.\n[6] M. Belluzi, T. Caraballo, M.J.D Nascimento, and K. Schia bel, Smoothing effect and asymptotic dynamics of nonautonom ous\nparabolic equations with time-dependent linear operators , J. Differential Equations 314(2022), 808–849.\n[7] F.D.M. Bezerra, A.N. Carvalho, and M.J.D. Nascimento, F ractional approximations of abstract semilinear paraboli c problems,\nDiscrete Contin. Dyn. Syst. Ser. B 25(11) (2020), 4221–4255.\n[8] V.L. Carbone, A.N. Carvalho, and K. Schiabel, Continuit y of attractors for parabolic problems with localized large diffusion,\nNonlinear Anal. 68(3) (2008), 515–535.\n[9] A. N. Carvalho, J. A. Langa, and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynam ical Systems ,\nApplied Mathematical Sciences 182, Springer-Verlag, New Y ork, 2012.\n[10] A. N. Carvalho, and M. J. D. Nascimento, Singularly non- autonomous semilinear parabolic problems with critical ex ponents,\nDiscrete Contin. Dyn. Syst. Ser. S 2(3) (2009) 449-471.\n[11] A.N. Carvalho, and L. Pires, Rate of convergence of attr actors for singularly perturbed semilinear problems, J. Ma th. Anal.\nAppl.452(1) (2017), 258–296.\n[12] A.N. Carvalho, and S. Piskarev, A general approximatio n scheme for attractors of abstract parabolic problems, Num er. Funct.\nAnal. Optim. 27(7-8) (2006), 785–829.\n[13] J.W. Cholewa, and T. Dlotko, Global attractors in abstract parabolic problems , vol. 278, London Mathematical Society Lecture\nNote Series, Cambridge University Press, Cambridge, 2000.\n[14] E.A.M de Abreu, and A.N. Carvalho, Attractors for semil inear parabolic problems with Dirichlet boundary conditio ns in\nvarying domains, Mat. Contemp. 27(2004), 37–51.\n[15] D. Henry, Geometric Theory of Semilinear Parabolic Equations , vol. 840, Lecture Notes in Mathematics, Springer-Verlag,\nBerlin-New York, 1981.\n[16] T. Kato, Perturbation theory for linear operators , Classics in Mathematics, Springer-Verlag, Berlin, 1995. ,\n[17] F.D.M Bezerra, and M.J.D. Nascimento, Non-autonomous approximations governed by the fractional powers of damped wave\noperators, Electron. J. Differential Equations (2019), Pap er No. 72, 19p.\n[18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer-Verlag, New York, 1983.\n[19] P.E. Sobolevski˘ ı, Equations of parabolic type in a Ban ach space, Amer. Math. Soc. Transl. 49(1965), 1-62.\n[20] H. Tanabe, A class of the equations of evolution in a Bana ch space, Osaka Math. J. 11(1959), 121–145.\n[21] H. Tanabe, On the equations of evolution in a Banach spac e, Osaka Math. J. 12(1960), 363–376.\n[22] H. Tanabe, Remarks on the equations of evolution in a Ban ach space, Osaka Math. J. 12(1960), 145–166.\n(M. Belluzi) Universidade de S ˜ao Paulo, Instituto de Ci ˆencias Matem ´aticas e de Computac ¸ ˜ao, S˜ao Carlos SP, Brazil.\nEmail address :maykelbelluzi@icmc.usp.br" }, { "title": "2401.15808v1.Galactic_Structure_From_Binary_Pulsar_Accelerations__Beyond_Smooth_Models.pdf", "content": "Galactic Structure From Binary Pulsar Accelerations: Beyond Smooth Models\nThomas Donlon II and Sukanya Chakrabarti\nDepartment of Physics and Astronomy, University of Alabama in Huntsville,\n301 North Sparkman Drive, Huntsville, AL 35816, USA∗\nLawrence M. Widrow\nDepartment of Physics, Engineering Physics and Astronomy,\nQueen’s University, Kingston, ON K7L 3N6, Canada\nMichael T. Lam\nSETI Institute, 339 N Bernardo Ave Suite 200, Mountain View, CA 94043, USA\nSchool of Physics and Astronomy, Rochester Institute of Technology, Rochester, NY 14623, USA and\nLaboratory for Multiwavelength Astrophysics, Rochester Institute of Technology, Rochester, NY 14623, USA\nPhilip Chang\nDepartment of Physics, University of Wisconsin-Milwaukee,\n3135 N Maryland Ave, Milwaukee, WI 53211, USA\nAlice C. Quillen\nDepartment of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA\nWe measure the line-of-sight accelerations of 26 binary pulsars due to the Milky Way’s gravi-\ntational potential, and produce a 3-dimensional map of the acceleration field of the Galaxy. Ac-\nceleration measurements directly give us the change in the line-of-sight velocity at present day,\nwithout requiring any assumptions inherent to kinematic modeling. We measure the Oort limit\n(ρ0= 0.062±0.017 M ⊙/pc3) and the dark matter density in the midplane ( ρ0,DM=−0.010±0.018\nM⊙/pc3); these values are similar to, but have smaller uncertainties than previous pulsar timing\nmeasurements of these quantities. Here, we provide for the first time, values for the Oort con-\nstants and the slope of the rotation curve from direct acceleration measurements. We find that\nA= 15 .4±2.6 km/s/kpc and B=−13.1±2.6 km/s/kpc (consistent with results from Gaia ),\nand the slope of the rotation curve near the Sun is −2±5 km/s/kpc. We show that the Galactic\nacceleration field is clearly asymmetric, but due to data limitations it is not yet clear which physical\nprocesses drive this asymmetry. We provide updated models of the Galactic potential that account\nfor various sources of disequilibrium; these models are incompatible with commonly used kinematic\npotentials. This indicates that use of kinematically derived Galactic potentials in precision tests\n(e.g., in tests of general relativity with pulsar timing) may be subject to larger uncertainties than\nreported. The acceleration data indicates that the mass of the Galaxy within the Solar circle is\n2.3×1011M⊙, roughly twice as large as currently accepted models. Additionally, the residuals of\nthe acceleration data compared to existing Galactic models have a dependence on radial position;\nthis trend can be explained if the Sun has an additional acceleration away from the Galactic center.\nI. INTRODUCTION\nThe Milky Way (MW) is in dynamical disequilibrium\ndue to internal phenomena, such as the bar, spiral struc-\nture, and external perturbations from satellite galaxies\nand dark matter substructure. The timescales for these\nperturbations are comparable to the orbital periods of\nstars in the disk and stellar halo; as a result, the Galaxy\nhas not achieved relaxation in the statistical sense de-\nscribed by Lynden-Bell (1967) [1]. These disequilibria\nmanifest in the stellar Galactic disk in a variety of ways,\nsuch as corrugations, warps, and phase space spirals [2–\n4], as well as a north-south asymmetry in the density of\n∗thomas.donlon@uah.edustars [5–7], and moving groups of disk stars [8–13]. Dis-\nturbances in the HI gas [14, 15] have been interpreted as\narising from satellite perturbations [16–18], and there is\napparent bulk motion in the interstellar medium, such as\nthe Radcliffe Wave [19].\nThese disequilibrium features contain valuable infor-\nmation about the MW’s structure and history. For ex-\nample, the phase space spirals in the MW disk may have\nbeen generated by the Sagittarius Dwarf Galaxy [20] or\nthe gravitational field of the Galactic bar [21, 22]. In\ngeneral, frequent interactions with substructure leads to\na departure from equilibrium in the motions of Milky\nWay disk stars.\nHistorically, studies of the Galaxy’s structure have pri-\nmarily focused on the readily available positions and mo-\ntions of stars [23]. This method is restrictive, because\nkinematic information about the gravitational potentialarXiv:2401.15808v1 [astro-ph.GA] 29 Jan 20242\nis averaged over time and space throughout a given star’s\norbit. Kinematic techniques, such as Jeans modeling,\nrequire simplifying assumptions such as spherical or az-\nimuthal symmetry, and equilibrium [24]; these assump-\ntions restrict the ability to observe spatially complex and\ntransient features, and can therefore only produce an ap-\nproximation of the Galaxy’s true underlying gravitational\npotential field.\nHowever, the acceleration of an object is instantaneous\nin time, and produces a measure of the gravitational po-\ntential at a specific location of the Galaxy, as well as a\nnon-local measure of the density field. As a result, di-\nrect acceleration measurements allow one to probe the\nGalaxy’s structure in a fine-grained way that is free of\nkinematic assumptions. Recently, several precision tech-\nniques have been developed that enable direct line-of-\nsight acceleration measurements within a few kpc of the\nSun, including extreme precision radial velocity observa-\ntions [25], pulsar timing [26], and eclipse timing [27].\nBecause they are precise astrophysical clocks, pulsars\ncan now be used to directly measure the Galactic ac-\nceleration [26]. The spin periods of solitary pulsars are\nsubject to magnetic braking, which is not well under-\nstood; however, the orbital periods of binary pulsars can\nbe modeled precisely with general relativity [28–30] to\nextract Galactic accelerations.\nPulsar timing data has been used to test general rela-\ntivity, via the emission of gravitational waves from binary\nsystems [29, 31]. Computing this radiation term in the\npulsar timing data requires knowing the Galactic poten-\ntial at the location of the pulsar, which is currently the\nlargest source of uncertainty in these tests [30]. By ob-\ntaining an accurate model of the Galaxy’s gravitational\npotential, one could substantially lower the uncertainty\nin tests of general relativity. This can be enabled by\ndirect acceleration measurements that are independent\nof pulsar timing. Eclipse timing measurements of pre-\ncisely timed eclipsing binary stars [27], which can be em-\nployed for this purpose. Further in the future, extreme-\nprecision radial velocity measurements will also provide\ndirect measurements of the Galactic acceleration [25] that\ncan be used to determine the Galactic potential indepen-\ndent of pulsar timing.\nPreviously, Chakrabarti et al. (2021) [26, hereafter\nC21] analyzed line-of-sight acceleration data for 14 binary\npulsars; we expand this with updated data for 26 binary\npulsars, roughly doubling the size of the dataset. The\nC21 dataset spanned 2.6 kpc in Rand 1.5 kpc in Z; our\nupdated dataset extends this to 3.4 kpc in Rand 3.6 kpc\ninZ, which allows us to leverage a larger region of the\nGalaxy when constraining our models.\nIn this work, we use the expanded dataset to com-\npute updated values for several different potential mod-\nels that are fit to the observed accelerations. In addition\nto a measurement of the Oort limit and a constraint on\nthe local dark matter density, we are now able to con-\nstrain the slope of the rotation curve and the Oort Con-\nstants. We also move beyond smooth potential models,and produce a 3-dimensional, non-parametric map of the\nMW acceleration field. This map manifests an apparent\nlack of vertical and azimuthal reflection symmetry about\nthe Sun, which is evidence of local disequilibrium. We\ndemonstrate that a consideration of disequilibrium is re-\nquired in order to obtain accurate models of the MW’s\ngravitational potential from direct acceleration measure-\nments; this will continue to be important as pulsar timing\ndatasets become larger and more precise.\nII. DATA\nWe take the distance between the Sun and the Galac-\ntic center to be R⊙= 8.178 kpc [32], and the circular\nspeed at the Solar position to be VLSR = 232 .8 km/s\n[33]. Adopting other literature values of R⊙andVLSR\ndid not substantially change the results of this work.\nThis work uses a Galactocentric Cartesian coordinate\nsystem, where the Sun is located at positive x, ˆypoints\ntowards the direction of the Sun’s motion, and ˆ z=−ˆx×\nˆy.\nA. Calculating Accelerations\nWe follow the methodology in C21 in extracting the\nGalactic acceleration from measured binary pulsar accel-\nerations. The observed orbital period of a binary pulsar\nwill be Doppler shifted according to its line-of-sight ve-\nlocity. If the binary pulsar accelerates along our line of\nsight, a corresponding change in the Doppler shifted or-\nbital period will be observed. The observed line-of-sight\nacceleration is expressed as\naObs\nlos=˙PObs\nb\nPbc, (1)\nwhere Pbis the observed orbital period of the binary and\ncis the speed of light.\nThe observed time rate of change of the binary orbital\nperiod ( ˙PObs\nb) can be decomposed into a sum of different\neffects:\n˙PObs\nb=˙PShk\nb+˙PGR\nb+˙PGal\nb. (2)\nThe quantity ˙PShk\nbis known as the Shklovskii effect\n[34], and is the result of proper motion of a source leading\nto a change in the distance to that source, producing\nan apparent acceleration along our line-of-sight. It is\ncomputed as\n˙PShk\nb=Pbµ2d\nc, (3)\nwhere µis the total proper motion of the source, and d\nis the distance to that source.\nThe quantity ˙PGR\nbis the decrease in the orbital period\ndue to radiation of gravitational waves [28, 30], and is3\ncomputed as\n˙PGR\nb=−192πG5/3\n5c5\u0012Pb\n2π\u0013−5/3\u0000\n1−e2\u0001−7/2\n×\u0012\n1 +73\n24e2+37\n96e4\u0013mpmc\n(mp+mc)1/3, (4)\nwhere eis the orbital eccentricity of the binary (typically\nclose to 0), mpis the mass of the pulsar, and mcis the\nmass of its companion.\nBecause the quantity we are interested in is the ac-\nceleration due to the gravitational potential of the MW,\nwe compute the Galactic component of the line-of-sight\nacceleration as\naGal\nlos=˙PGal\nb\nPbc (5)\nusing Equations 1-4. For notational simplicity, we often\nrefer to this quantity as alosfrom this point forwards.\nB. Pulsar Data\nWe selected all available binary pulsars from the Aus-\ntralia Telescope National Facility (ATNF) pulsar catalog\n[35] that satisfied the following criteria:\n•The source had a measured Pband ˙Pb;\n•Had measured parallax πand proper motion µ;\n•Had measured orbital eccentricity e, pulsar mass\nmp, and companion mass mc; or a binary period\nPb>5 days.\n•The source cannot be a redback, black widow, etc.\nor transferring mass with its companion, which can\nchange the orbital period of the system.\n•The source cannot be in a globular cluster, as the\ninternal accelerations of globular clusters are ex-\npected to overpower the Galactic acceleration sig-\nnal.\nOur final dataset contains 26 binary pulsars; the full\nlist of the pulsars, their distances, their components of ˙P,\nand their accelerations are provided in Table I. Fourteen\nof these sources are present in the C21 dataset. Here, we\nalso include 9 sources from recent work by Moran et al.\n51. We do not use the following five sources from the\nMoran et al. (2023) dataset as they do not satisfy our\ncriteria for selection:\n•B1259-63: This pulsar has a Be star companion,\nand experiences regular mass transfer [52].\nFIG. 1. The observed pulsar data. Each source is colored ac-\ncording to the residual of its the observed line-of-sight acceler-\nation and the Gala MilkyWay2022 potential model, weighted\nby the uncertainty of each measurement. There is a global\ntrend in the data; points that are further to the left are bluer\non average than points that are further to the right. This\nsuggests that the Gala MilkyWay2022 potential model incor-\nrectly estimates the slope of the rotation curve, or that the\nSun is experiencing an additional acceleration away from the\nGalactic center.\n•J0348+0432: This source has no PTA parallax\navailable. Moran et al. 51 used a Bayesian esti-\nmate of the parallax taken from a negative Gaia\nDR3 parallax [53–55] to calculate the distance to\nthe pulsar, but we choose not to include this source\nbecause of its large distance uncertainty.\n•J0636+5128 (Listed in Moran et al. 51 as\nJ0636+5129): There is evidence for outflows\naround this source, the pulsar may have an accre-\ntion disk, and the pulsar appears to be interacting\nwith its companion in some way [56, 57].\n•J1756-2251: The proper motion in declination of\nthis source is only constrained within an upper limit\nof 20 mas; Moran et al. 51 use only the proper\nmotion in right ascension for this source (roughly 2\nmas). Additionally, the distance to this source has\nan uncertainty of 24 kpc. This makes it impossible\nto constrain ˙PShk\nb.4\nPSR Distance ˙PObs\nb˙PShk\nb˙PGR\nb aGal\nlos Reference\n(kpc) (mm s−1yr−1)\nJ0437-4715 0 .157(2) 3.728(6)E-12 3.76(5)E-12 -3.22(15)E-16 -0.8(9) R16\nJ0613-0200 1 .14(12) 3.4(7)E-14 3.15(33)E-14 -8.0(40)E-15 1.0(8) NG\nJ0737-3039A/B 1 .15(22) -1.252(17)E-12 4.4(12)E-16 -1.2484(8)E-12 -5.(18) KMS06, DBT09\nJ0740+6620 1 .0(2) 1.17(26)E-12 1.06(21)E-12 -6.07(21)E-16 2(8) NG\nJ0751+1807 1 .17(5) -3.50(5)E-14 1.19(5)E-14 -4.4(4)E-14 -1.2(17) EPTA, NSK08\nJ1012+5307 1 .0(2) 6.1(6)E-14 8.4(17)E-14 -6.0(320)E-15 -3(7) NG, EPTA\nJ1017-7156 1 .4(6) 4.(2)E-13 2.0(8)E-13 -3.9(30)E-16 3(4) PPTA\nJ1022+1001 0 .85(6) 2.18(9)E-13 7.4(20)E-13 -5.2(8)E-16 -7.3(28) EPTA\nJ1125-6014 1 .5(11) 7.(1)E-13 8.0(60)E-13 -2.24(30)E-16 -1(8) PPTA\nJ1455-3330 0 .76(6) 4.5(22)E-12 8.0(6)E-13 - 5.3(32) EPTA\nB1534+12 0 .94(7) -1.366(3)E-13 5.3(4)E-14 -1.93(6)E-13 0.8(19) FST14, DDF21\nJ1600-3053 1 .87(3) 5.(1)E-13 2.79(5)E-13 -1.2(4)E-16 1.7(8) PPTA\nJ1603-7202 2 .7(13) 1.9(4)E-13 2.2(10)E-13 - -0.4(19) PPTA, WRT22\nJ1614-2230 0 .68(4) 1.33(7)E-12 1.32(8)E-12 -4.23(3)E-16 0.1(14) NG\nJ1640+2224 1 .08(28) 9.5(19)E-12 5.3(14)E-12 -6.(3)E-18 2.6(15) EPTA\nJ1713+0747 1 .136(13) 2.64(73)E-13 6.41(7)E-13 -6.7(2)E-18 -0.61(12) EPTA\nJ1738+0333 1 .47(10) -1.70(31)E-14 8.2(6)E-15 -2.9(13)E-14 1(4) FWE12\nJ1741+1351 3 .0(9) 1.3(4)E-12 1.4(4)E-12 - -1(4) NG\nJ1909-3744 1 .12(3) 5.09(2)E-13 4.94(13)E-13 -2.90(4)E-15 1.2(10) NG\nB1913+16 4 .1(2) -2.423(1)E-12 1.6(7)E-16 -2.40263(5)E-12 -6.96(34) WH16, DWN18\nJ1933-6211 1 .6(3) 7.(1)E-13 6.6(12)E-13 -1.53(29)E-16 0.3(14) GKF23\nJ2043+1711 1 .4(1) 1.0(1)E-13 6.5(5)E-14 -2.86(16)E-15 2.8(8) NG\nJ2129-5721 3 .6(14) 1.51(9)E-12 8.9(35)E-13 - 10(6) PPTA\nJ2145-0750 0 .71(5) 1.3(2)E-13 1.76(12)E-13 - -0.7(4) PPTA\nJ2222-0137 0 .2681(12) 2.554(74)E-13 2.796(13)E-13 -7.98(12)E-15 -0.76(34) GFG21\nJ2234+0611 0 .97(4) 3.1(25)E-12 4.78(20)E-12 -2.57(12)E-17 -6(9) SFA19\nRows without ˙PGR\nbdata are expected to have negligibly small values of ˙PGR\nb.References : R16, [36]; NG (NANOGrav 15-yr\nData Release), [37]; KMS06, [38]; DBT09, [39]; EPTA (EPTA-DR2), [40]; NSK08, [41]; FST14, [42]; DDF21, [43]; PPTA\n(PPTA-DR2), [44]; WRT22, [45]; FWE12, [46]; WH16, [30]; DWN18, [47]; GKF23, [48]; GFG21, [49]; SFA19, [50].\nTABLE I. Summary of Pulsar Data.\n•J2339-0533: This source is a redback [58].\nThree of these pulsars are new sources, and were not in\neither the C21 or Moran et al. 51 datasets: J1455-3330,\nJ1640+2224, and J1933-6211; these timing solutions were\npublished after Moran et al. 51.\nC21 listed a parallax of 1.9 mas for J2129-5721 [59],\nbut more recent solutions give a parallax of 0.26 mas for\nthis source [44]. This erroneous distance measurement,\ncombined with the source’s large value of ˙PObs\nb, could\nexplain why it had the largest residual of any PSR con-\nsidered by C21.\nIn the case where e,mpandmcwere not constrained\n(and therefore ˙PGR\nbcould not be calculated), the source\nwas still included if its binary period Pb>5 days. Since\n˙PGR\nb∝P−5/3\nb, large orbital periods result in negligibly\nsmall values of ˙PGR\nb(assuming mp=mc= 1.4 M⊙,e= 0,\nandPb= 5 days, ˙PGR\nb∼10−15; for comparison, ˙PGal\nb∼\n10−12). Our dataset contains 5 sources without ˙PGR\nb,\nindicated in Table I by a dash in the ˙PGR\nbcolumn.C. Line-of-Sight Acceleration Uncertainty\nWe computed the uncertainty in alosfor each source us-\ning standard error propagation procedure. This method\ndoes not take into account any covariances between the\nuncertainties in correlated parameters, or the fact that\nuncertainty in distance contibutes to the total error by\nboth changing the position of the pulsar and also affect-\ning the calculation of ˙PShk\nb. As such, it is beneficial to\ndouble check that we are not underestimating the uncer-\ntanties in alosby estimating the uncertainties in a second\nway.\nFigure 2 shows bootstrapped uncertainties in alosfor\neach source, broken down into the contribution of the\nuncertainty of each component to the total alosuncer-\ntainty. The total uncertainty in aloswas estimated by\ncomputing the “true” alosfor each source assuming its\nparameters each individually have zero error, and then\nsampling 10,000 times with the reported errors on each\ncomponent, assuming that each uncertainty is normally\ndistributed. The contribution of each component to the\noverall alosuncertainty was estimated by assuming only\nthat component has zero uncertainty, and then sampling\nthe resulting alosuncertainty 10,000 times. The “Other”\ncomponent in the alosuncertainties is likely due to addi-5\nFIG. 2. Bootstrapped uncertainties in alosfor each source, broken down by individual component (parallax, proper motion,\n˙PGR\nb, and ˙PObs\nb). There is also a contribution to aloserror from covariances between the various components. The dashed\nblack line shows the uncertainty estimated for alosusing error propagation techniques, and is generally in agreement with the\nbootstrapped uncertainties.\ntional uncertainty contributed by covariances in the com-\nponents.\nThe dashed black line in Figure 2 shows the uncer-\ntainty that is obtained through error propagation; over-\nall this value appears to be similar to the bootstrap es-\ntimates, indicating that the uncertainties obtained via\nerror propagation are reasonable estimates.\nIn general, uncertainties in parallax and ˙PObs\nbcon-\ntribute more to the overall uncertainty in alosthan un-\ncertainties in proper motion and ˙PGR\nb. As a result, im-\nprovements in parallax and ˙PObs\nbmeasurements should\nbe prioritized in pulsar timing fits in order to obtain more\nprecise alosmeasurements in the future.\nIII. RADIAL DEPENDENCE OF\nACCELERATION RESIDUALS\nThe observed acceleration data is shown in Figure\n1 compared to a state-of-the-art theoretical potential\nmodel, MilkyWayPotential2022 from the Gala python\npackage [60]. There is a global trend in the figure; over-\nall,MilkyWayPotential2022 underestimates the acceler-\nations of the pulsar sources towards the Galactic center.\nBecause we observe a line-of-sight acceleration relative to\nthe Sun, this results in a positive residual alos(blue) for\nsources between the Sun and the Galactic center, and a\nnegative residual alos(red) for sources with R > R ⊙.\nInterestingly, we were not able to remove this trend\nby varying the total mass of the Gala MilkyWayPoten-\ntial2022 model. However, this trend can be explained\nif the Sun is accelerating relative to the pulsars in the\n6 7 8 9\nR (kpc)2\n02(aobs\nlosamodel\nlos)/obs\nalos\n100% Solar Acceleration\n6 7 8 9\nR (kpc)2\n1\n012(aobs\nlosamodel\nlos)/obs\nalos\n80% Solar AccelerationFIG. 3. The residuals of the pulsar data and the Gala Milky-\nWayPotential2022 model as a function of distance from the\nGalactic center (black). A linear fit to the data is shown as a\ndashed gray line. If the model fits the data, one would expect\nthe residuals to be clustered around 0 independent of each\npulsar’s position; however, there is a radial dependence in the\nresiduals when the Solar acceleration from the Gala model is\nused. If the Sun’s acceleration relative to the binary pulsars\nis actually only 80% of its presumed value, this trend disap-\npears (although there is still substantial scatter at some radii,\nparticularly at R∼7.5 kpc).6\ndataset.\nThe distribution of QSOs has been used to measure\nthe acceleration of the Sun relative to a cosmological ref-\nerence frame [61, 62]. These accelerations agree with pre-\ndictions of the Solar acceleration relative to the Galactic\ncenter using existing kinematic potential models of the\nMW; as a result, the measured acceleration of the Sun\nrelative to the QSO reference frame is typically treated as\nidentical to the Sun’s acceleration relative to the Galac-\ntic center. In other words, it is presumed that the ac-\nceleration of the Galaxy relative to the QSO reference\nframe is negligible. However, there is no reason that the\nGalactic center is in the same inertial reference frame as\nthe QSOs; if the Galactic center is in-fact accelerating\nrelative to the QSOs, then the acceleration of the Sun\nrelative to the QSOs is nota relative acceleration with\nrespect to the Galactic center.\nIn this scenario, the Sun could be experiencing an addi-\ntional acceleration relative to the Galactic center (which\ncannot be constrained by the QSO measurements). If\nthis is the case, then the observed relative accelerations\nof each pulsar are actually\nalos= [a(x)−a(x⊙)−apec(x⊙)]·ˆd, (6)\nwhere apec(x⊙) is a peculiar acceleration felt by the Sun\nbut not felt by each pulsar (see Equation 8 for further\ncontext) and ˆdis the line-of-sight vector from the Sun to\neach source.\nIn order to remove the observed trend, the Sun would\nneed to feel an additional acceleration of apec(x⊙) = 1 .1\nmm/s/yr of its predicted radial acceleration away from\nthe center of the Galaxy. The effect of this additional ac-\nceleration on the residual of the observed data is shown in\nFigure 3. This value was obtained by optimizing a poten-\ntial model that consisted of MilkyWayPotential2022 plus\nan additional acceleration on the Sun relative to the pul-\nsar data (see Section IV for a detailed description of the\noptimization procedure). For this analysis, we removed\nPSRs J1713+0747, B1913+16, and J2043+1711 because\nthey were outliers that substantially changed the linear\nfit to the data.\nIt is difficult to explain why the Sun would have an\nadditional acceleration relative to the Galactic center, al-\nthough a relative acceleration could potentially be caused\nby a nearby massive object, such as a nearby (dim) com-\npact object, or a dark matter subhalo. It is interesting\nto note that a Jupiter-mass object at a distance of 400\nAU from the Sun would produce a Solar acceleration of\nthe correct magnitude to explain the observed global pul-\nsar acceleration trend; constraints on the possible loca-\ntions of Planet Nine place it roughly in the direction of\nthe Galactic anticenter [63, 64], potentially making it an\nexplanation for why the Sun would experience an addi-\ntional acceleration away from the center of the Galaxy\nthat would not be felt by the pulsar sources. However,\ncurrent constraints on Planet Nine’s mass and semimajor\naxis would only produce a Solar acceleration on the order\nof∼0.05 mm/s/yr [65]; this discrepancy would need tobe resolved in order to claim that a ninth planet is the\ncause of the apparent 1.1 mm/s/yr Solar acceleration.\nZakamska & Tremaine (2005) [66] used pulsar and\nwhite dwarf timing solutions to constrain the accelera-\ntion of the Solar system, and concluded that the Solar\nsystem acceleration was consistent with zero. However,\ntheir sensitivity was limited to roughly the acceleration\nof a Jupiter-mass planet at a distance of 200 AU ( ∼4.5\nmm/s/yr), which is a few times larger than the value we\nprovide here for the Solar acceleration, and is therefore\nnot in tension with our results.\nIV. MODEL FITS\nIdeally, if one had a complete map of the gravitational\nfield, then its divergence would give a local measure of the\ndensity field; however, because we are restricted to one\ncomponent of the gravitational field with existing data,\nwe must instead rely on models to infer the local density\nand corresponding features in the Milky Way.\nGiven a model for the MW’s gravitational potential\nΦ(⃗ x), one can compute the acceleration at a given point\nfrom\na(x) =−∇Φ(x), (7)\nwhich can then be used to obtain a relative line-of-sight\nacceleration:\nalos= [a(x)−a(x⊙)]·ˆd (8)\nWe fit a number of models to the observed alosdata\nusing a maximum likelihood estimation (MLE) technique\n[68]. We first define a likelihood function, in this case\nobtained from a χ2statistic:\nχ2=NX\ni\u0000\naobs\nlos−amodel\nlos\u00012\nσ2\naobs\nlos, (9)\nwhere Nis the number of sources in the data. The cor-\nresponding log-likelihood function is ln( L) =−χ2/2.\nThe log-likelihood function was then maximized us-\ning the differential_evolution function from the\nscipy.optimize python package [69]. Differential evo-\nlution was used rather than traditional conjugate gradi-\nent descent because some of the models have “bumpy”\nlikelihood surfaces, which can cause gradient descent al-\ngorithms to get stuck in local minima.\nThe covariance matrix ( K) for the optimized parame-\nters is calculated from the weighted inverse of the Hessian\nmatrix ( H);\nKij=1\nNH−1\nij, (10)\nwhere Nis again the number of sources, and\nHij=\u0014∂2\n∂i∂jln\u0010\nˆL\u0011\u0015\n, (11)7\nLocal Model Potential Parameters AIC χ2\nEquilibrium Models\nα Φ(R, z) =V2\nLSRln\u0010\nR\nR⊙\u0011\n+1\n2αz2log10(α·Gyr2) = 3 .60±0.06 54 2.2\nα-β Φ(R, z) =V2\nLSR\n2β\u0010\nR\nR⊙\u00112β\n+1\n2αz2log10(α·Gyr2) = 3 .51±0.13 55 2.3\nβ= 0.14±0.17\nCross ( α-γ) Φ( R, z) =\u0000\nV2\nLSR−γz2\u0001\nln\u0010\nR\nR⊙\u0011\n+1\n2αz2log10(α·Gyr2) = 3 .54±0.16 55 2.3\nlog10(γ·Gyr2) = 3 .3±0.9\nMWPotential2014 Galpy Values [67] - 66 3.0\nMilkyWayPotential2022 Gala Values [60] - 70 3.5\nDamour-Taylor see Section IV A 4 - 85 3.8\nDisequilibrium Models\nAnharmonic Φ( R, z) =V2\nLSRln\u0010\nR\nR⊙\u0011\n+1\n2α1z2+1\n3α2z3log10(α1·Gyr2) = 3 .51±0.08 51 2.1\nlog10(α2·Gyr2·kpc) = 3 .4±0.2\n2α-β Φ(R, z) =V2\nLSR\n2β\u0010\nR\nR⊙\u00112β\n+1\n2α1z2+1\n3α2z3log10(α1·Gyr2) = 3 .56±0.11 53 2.2\nlog10(α2·Gyr2·kpc) = 3 .5±0.2\nβ=−0.08±0.18\nLocal Expansion see Section V A 2 log10(−∂aR/∂R·Gyr2/kpc) = 2 .6±0.3 41 1.6\nlog10(−∂aϕ/∂ϕ·Gyr2/kpc) = 3 .54±0.09\nlog10(∂az/∂z·Gyr2/kpc) = 3 .73±0.08\nSinusoidal Φ( R, z) =V2\nLSRln\u0010\nR\nR⊙\u0011\nlog10(α·Gyr2) = 3 .37±0.08 41 1.6\n+1\n2α(z+Asin (2 πR/λ +φ))2A= 0.52±0.08 kpc\nλ= 2.46±0.15 kpc\nφ= 3.5±1.2\nα-β+ 2 Point Mass Φ( R, z) =V2\nLSR\n2β\u0010\nR\nR⊙\u00112β\n+1\n2αz2+P2\niGMi\n|⃗ r−⃗ ri|log10(α·Gyr2) = 3 .65±0.09 31 0.8\nβ=−0.43±0.12\nlog10(M1/M⊙) = 8 .3±0.3\nx1= 7.4±0.2 kpc\ny1= 1.2±0.1 kpc\nz1=−0.6±0.2 kpc\nlog10(M2/M⊙) = 8 .0±0.5\nx2= 7.5±0.1 kpc\ny2= 0.5±0.1 kpc\nz2= 0.9±0.2 kpc\nTABLE II. Optimized fits to the local potential models.8\nwhere ˆLis the likelihood of the optimized parameters\ngiven the observed data. The uncertainty in the ith op-\ntimized parameter ( σi) is obtained from the diagonal of\nthe covariance matrix:\nσi= diag( Kij). (12)\nThe Akaike Information Criterion (AIC) can then be\nevaluated for each fit model using the formula\nAIC = 2 k−2 ln\u0010\nˆL\u0011\n= 2k+χ2, (13)\nwhere kis the number of free parameters in that model.\nThe AIC is a measure of significance of a likelihood score\ngiven a model; models with more free parameters typi-\ncally produce better likelihood scores, so the AIC pro-\nvides a way of evaluating whether adding additional free\nparameters to a model actually improves the quality of\nthe fit. A difference of ∆AIC= 6 between two models is\nconsidered to be strong evidence in favor of the model\nwith lower AIC [70].\nC21 used a Markov Chain Monte Carlo (MCMC) tech-\nnique to fit models to pulsar acceleration data. We chose\nto use an MLE technique instead because MCMC codes\ncan quickly become intractable for models with a large\nnumber of parameters, such as the “ α−β+ 2 Point Mass”\nmodel, which has 10 free parameters. This model is feasi-\nble in the MLE code, which is much faster and uses much\nless memory than the corresponding MCMC framework.\nThe C21 MCMC setup is practically identical to our MLE\nsetup, in that the likelihood functions are identical, and\nthe MCMC priors from C21 are (roughly) encoded in the\nuncertainties in the observed accelerations, which are de-\nrived from the usual uncertainty propagation procedure.\nWe compared several of our model fits to fits obtained\nusing C21’s MCMC approach, and the optimized param-\neters agree to well within the reported uncertainties.\nThe analytic form of each potential model can be found\nin Tables II and III, as well as the best fit parameters for\neach model and the quality of that fit.\nA. Symmetric Local Models\nOur first set of potential models are azimuthally and\nvertically symmetric models (they are invariant under\nnegation of ϕorz). These models were introduced in\nC21 as local approximations of the potential (Taylor ex-\npansions of order ≤2), which are only expected to be\naccurate within a few kpc of the Sun. For this reason,\nthese models are only fit to sources within 3 kpc of the\nSun; this removes J2129-5721 and B1913+16 from the\ndataset, and these models are only fit to 24 of the 26\nsources.\n1.αModel\nTheαmodel is the simplest model in this work. It\nis additively separable in Randz, and assumes a flatrotation curve, which is calibrated to the local standard\nof rest and the Solar position. In the vertical direction,\nthe potential is that of a harmonic oscillator, where α\ngives the angular frequency of vertical oscillations. The\nfree parameter αalso sets the vertical density profile of\nthis model, which is constant and given by 4 πGρ =α.\nOur measured value of αagrees with the value obtained\nby C21, and our updated value has smaller uncertainties.\n2.α−βModel\nTheα−βmodel is an extension of the αmodel that\nconsiders a sloped rotation curve, which is characterized\nbyβ:\nβ=\u0014R\nVcircdVcirc\ndR\u0015\nR⊙, (14)\ni.e. more positive (negative) values of βindicate a more\npositive (negative) slope of the rotation curve. Given the\nsmaller extent of the pulsars, C21 could not constrain β\nfor this model. Our larger of sample of pulsars covering\na larger radial extent enables us to constrain it to be\nslightly positive (although it is consistent with 0 within\na 1σuncertainty). Constraining βallows us to calculate\nthe Oort constants; we provide an updated estimate of\nthe Oort limit as well (see Section VI).\n3. “Cross” Model\nA potential that is additively separable in Randz\n(such as the αmodel) leads to a density that is also ad-\nditively separable through the Poisson equation, whereas\nany realistic potential will not satisfy this condition. In\nthe “cross” model, we consider a simple extension of the\nαmodel that breaks the separability assumption. The\nmagnitude of this cross term is set by γ, for which γ >0\ncorresponds to oblate isopotential contours, and a larger\nvalue of γimplies a more oblate potential.\nC21 derive typical values of γto be log10(γ/Gyr2) =\n2.93 for a log-spherical MW potential and\nlog10(γ/Gyr2) = 3 .94 for a Miyamoto-Nagai Disk\npotential [71]. Both of these values are consistent\nwith our fit value of γ, although our observed value\nofγis closer to the spheroid potential than the disk\npotential. C21 found that their value of γwas more\ndisky (log10(γ) = 4 .87) than our best-fit γvalue\n(log10(γ) = 3 .3). The difference in these values arises\nfrom the updated timing solutions for the pulsars\nrelative to what was used in C21; when fitting the cross\nmodel to only the C21 sources, but instead using the\nmore recent values for the accelerations and distances,\nwe obtain a γvalue that is consistent with our γwithin\n1σuncertainties.\nThe cross model also gives an approximation for the9\n0.00 0.25 0.50 0.75 1.00 1.25 1.50\nZ (kpc)0.50.60.70.80.91.0Vcirc/VLSR\nSharma+14\nPulsar Data, This work\nPulsar Data, C21\nExtent of Data\nFIG. 4. The ratio of VcirctoVLSRas a function of height from\nthe disk. The dashed red (this work) and blue (C21) lines\nshow the values obtained from the cross model using pulsar\nacceleration data, and the solid black line shows the values\nobtained from kinematic data by Sharma et al. (2014) [72].\nThe vertical extent of the pulsar data (which is roughly equal\nfor C21 and our local dataset) is shown as a vertical dotted\nline. The C21 pulsar data produces a more disky model than\nour updated model which uses more recent timing solutions,\nand which is now consistent with the kinematic values from\nSharma et al. 72 to the vertical extent of the data.\ncircular speed as a function of ZatR=R⊙:\nVcirc\nVLSR≈1−|γ|\n2V2\nLSRz2. (15)\nThis expression is plotted in Figure 4 for our best-fit\nvalue of γ, along with the best-fit γvalue from C21, and\na model of Vcirc/VLSRfrom Sharma et al. 72, who ana-\nlyzed kinematic data of MW stars. The Sharma et al. 72\nexpression closely agrees with canonical MW potential\nmodels, such as the model of Law & Majewski (2010)\n[73]. Overall, Vcirc/VLSRis similar for our pulsar data\nand the kinematic data; however, the C21 γvalue pro-\nduces a very disky potential model that is inconsistent\nwith the kinematic data from Sharma et al. 72. As our\nlocal dataset has the same vertical extent as the C21\ndataset, we conclude that these differences are due to\nimprovements in the reported pulsar timing values.\nIt is possible to relate γand the scale length of a\nMiyamoto-Nagai Disk potential with Eq. 15 of C21, by\nfixing the mass and scale length of the disk to the C21\nvalues, and then numerically solving for the scale height\ngiven some value of γ. This corresponds to a scale height\nof 1.0 kpc for our fit γvalue, which is somewhat larger\nthan but generally consistent with MW thick disk popu-\nlations. The C21 γmeasurement corresponds to a scale\nheight of 0.05 kpc, implying a much more condensed disk\nthan we find using the updated pulsar dataset.4. Damour-Taylor Model\nThe orbital decay via the emission of gravitational ra-\ndiation by binary pulsars can be used to test general rel-\nativity. The Damour-Taylor model [29] is an approxi-\nmation to the MW potential that is commonly used in\nthese studies [29, 74, 75]. While its functional form has\nchanged somewhat since its original definition in Damour\nand Taylor 29, the model is essentially a flat rotation\ncurve model with a roughly linear disk acceleration pro-\nfile, which is similar to our αmodel.\nWe use the version of the model from Lazaridis et al.\n75, where the acceleration profile is defined as:\nalos= aR,los+az,los\n=−V2\nLSR\nR⊙\u0012\ncosl+β\nβ2+ sin2l\u0013\ncosb (16)\n−2.27|zkpc|+ 3.68(1−e−4.31|zkpc|)\n10−9cm s−2|sinb|,\nwhere landbare the Galactic longitude and latitude co-\nordinates of the source, β= (d/R⊙) cosb−cosl, where d\nis the distance of the source from the Sun, and zkpcis the\nheight from the midplane in kpc. The trigonometry of the\nmodel is not relevant to the Galactic potential, rather it\nprovides the projection of the Galactic acceleration along\nour line-of-sight based on the source’s position.\nAs this potential is separable, we can write the vertical\nacceleration of this model as\naz=−A|z| −B\u0010\n1−e−C|z|\u0011\n, (17)\nwhich corresponds to a potential of\nΦ(z) =A\n2z2+B\u0012\n|z|+e−C|z|\nC\u0013\n, (18)\nand in the limit of small zwe recover the harmonic ver-\ntical potential\nΦ(z)≈1\n2(A+BC)z2=1\n2αz2. (19)\nPlugging in the values for A,B, and Cfrom Equation 16,\nwe can infer a value of log10(α) = 3 .77 for this potential,\nwhich is somewhat larger than our fit values of αfor the\nα,α−β, and cross potentials.\nEvaluating this model for our acceleration data pro-\nduces very large AIC and χ2values, indicating that the\nmodel is not a good fit to the observed data. This is\nworrisome, as this model is commonly used in tests of\ngeneral relativity to obtain ˙PGal\nb; if the model is not ac-\ncurate, this would lead to inconsistencies in the reported\nvalues of ˙PGR\nbin these studies. The inconsistency be-\ntween the Damour-Taylor model and the observed data\nappears to be due to the Damour-Taylor model incor-\nrectly estimating the MW disk density.\nA ∆AIC of 2 is positive evidence in favor of the model\nwith the lower AIC, while ∆AIC of 6 provides strong10\nevidence [70]. Thus, the α,α−β, and α−γmod-\nels carry similar statistical confidence, while MWPoten-\ntial2014 [76] and MilkyWayPotential2022 [60] have lower\nconfidence, with the Damour-Taylor model having signifi-\ncantly larger AIC and χ2values relative to the other kine-\nmatic models. The extremely large AIC of the Damour-\nTaylor model indicates that it should not be used to ap-\nproximate the Galactic acceleration for pulsars near the\nSun.\nB. Global Models\nThe global models, described in Table III, are combi-\nnations of a Hernquist profile [77], an NFW halo [78], and\na Miyamoto-Nagai Disk to fit the observed acceleration\ndata. These fits used all 26 sources. Note that the AIC\nof these fits cannot be directly compared to the AIC of\nthe local model fits because they use a different set of\ndata.\nThe Hernquist and NFW model fits are very similar,\nwith total MW masses of 5-8 ×1012M⊙and scale lengths\nin the low tens of kpc. These fits are similar to modern\npotential models for the MW, except that our model fits\nplace the total MW mass 2-5 times larger than kinematic\nand dynamical models, which place the total MW mass\nbetween 0.7 and ∼3×1012M⊙[33, 79–85]. These fits are\nalso an improvement over those of C21, who were not\nable to constrain a scale radius or mass for their global\nfits due to the restricted range of available data in that\nearlier work.\nWe also tested the addition of a Miyamoto-Nagai disk\nto an NFW potential, with mixed results. We were able\nto constrain the mass of the disk (although it is about\nan order of magnitude below kinematic estimates, e.g.\nBovy and Rix 86, Licquia and Newman 87, and dynam-\nical models, e.g. Newberg et al. 80), but the scale radii\nof the disk potential are not well constrained.\nThe Gala MilkyWayPotential2022 model has a very\npoor fit to the acceleration data compared to our opti-\nmized potential models. It should be noted, however,\nthat the majority of this disagreement comes from PSR\nB1913+16; if B1913+16 is excluded, MilkyWayPoten-\ntial2022 has AIC = 79 and χ2= 3.8 (although the AICs\nof the other models also drop by ∼10 when B1913+16\nis removed). The relatively poor fit of MilkyWayPoten-\ntial2022 to the observed acceleration data is in part due\nto the total mass estimate of the MW, which for Milky-\nWayPotential2022 is about 1.1 ×1012M⊙.\nAs the Milky Way is believed to be roughly spheri-\ncally symmetric over large scales, our pulsar accelera-\ntion data should be much more sensitive to the amount\nof mass within the Solar circle than outside it (via\nthe shell theorem). The mass enclosed within 8 kpc\nof the Galactic center in the NFW global model is\nMenc(8 kpc) = 2 .3×1011M⊙, which is ∼2.3 times larger\nthan the enclosed mass of the MilkyWayPotential2022\nmodel ( Menc(8 kpc) = 9 .8×1010M⊙). Thus, the pulsar\nFIG. 5. Observed and simulated asymmetry in the vertical\nacceleration profile of the disk. The fit of the anharmonic\npotential profile to the observed pulsar data is shown as the\nblack line, and 1 σuncertainties are shown as the shaded gray\nregion. The vertical acceleration profile from a simulation of\nthe MW and the Sgr dSph [18] is shown as a blue line with er-\nror bars. The shape of the two profiles are roughly the same,\nsuggesting that the observed acceleration asymmetry is con-\nsistent with the effects of orbiting dwarf galaxy interactions.\nacceleration data imply a larger MW mass than kine-\nmatic methods. If these larger masses are to be believed,\nthey imply substantial differences between orbits calcu-\nlated using potentials calibrated to modern kinematic\ndata and the potentials fit to the pulsar data. However,\nit should be noted that the current global model fits esti-\nmate an extremely large circular speed at the location of\nthe Sun (roughly 320 km/s), so it is likely that we are not\nyet producing precise constraints on the enclosed mass,\nand it is expected that the addition of future pulsar data\nwill significantly improve these fits.\nV. INFERRING DISEQUILIBRIUM FROM\nACCELERATIONS\nOne way that accelerations can be used to probe dis-\nequilibrium in the MW disk is by looking at the vertical\nacceleration profile ( azvs.z). Chakrabarti et al. 25\nshowed that simulations of a MW interacting with dwarf\ngalaxies would have a vertical acceleration profile that\nis highly asymmetric about the midplane. As a result,\nshowing that az(+z′)̸=az(−z′) for some nonzero z′, or\nalternatively that d az/dzin the midplane is nonzero,\nindicates disequilibrium in the Galactic disk.\nA. Asymmetric Local Models\nWe now move on to models that capture disequilibrium\nfeatures of the Galactic potential, i.e. these potentials are\nnot invariant under negation of ϕand/or z.\nIn general, the more flexible asymmetric models do a\nbetter job of fitting the observed acceleration data. This11\nGlobal Model Parameters AIC χ2\nHernquist log10(Mtot/M⊙) = 12 .7±0.2 78 3.1\nrs= 32±11 kpc\nNFW log10(Mvir/M⊙) = 13 .0±0.2 78 3.1\nrs= 20±7 kpc\nNFW & Miyamoto-Nagai Disk log10(Mvir/M⊙) = 12 .8±0.3 84 3.5\nrs= 16±9 kpc\nlog10(Mdisk/M⊙) = 10 ±2\na= 10±40 kpc\nb= 0.01±0.5 kpc\nMWPotential2014 Galpy Parameters [67] 223 11.0\nMilkyWayPotential2022 Gala Parameters [60] 229 12.4\nTABLE III. Optimized fits to the global potential models.\nModel A B d Vcirc/dR ρ 0 ρ0,DM\n(km/s/kpc) (km/s/kpc) (km/s/kpc) (M ⊙/pc3) (M ⊙/pc3)\nα−β 12.8±2.4 -15.7 ±2.4 4 ±5 0.060 ±0.018 -0.012 ±0.018\n2α−β 15.4±2.6 -13.1 ±2.6 -2 ±5 0.062 ±0.017 -0.010 ±0.018\nα−β+ 2PM 20.4 ±1.7 -8.1 ±1.7 -12 ±5 0.066 ±0.017 -0.006 ±0.018\nTABLE IV. Oort constants, the slope of the rotation curve, and the (dark matter) density in the midplane for different potential\nmodels.\nis strong evidence that the binary pulsar acceleration\ndata contains substantial disequilibrium effects, which\nmust be accounted for in order to obtain an accurate\nmodel of the Galactic potential.\n1. Anharmonic and 2 α-βModels\nThe first disequilibrium model is the anharmonic\nmodel, which consists of the αmodel except the verti-\ncal component of the potential is expanded one degree\nfurther in a power series expansion for a separable po-\ntential:\nΦ(z) =3X\nn=2αn−1\nnzn, (20)\nwhich provides a simple form for d az/dzin the midplane:\ndaz\ndz\f\f\f\f\nz=0=d\ndz\u0012\n−dΦ\ndz\u0013\f\f\f\f\nz=0=−α1. (21)\nIn essence, α1can be interpreted as a measure of how\nmuch the local disk potential is out of equilibrium.The additional z3term allows us to characterize the\nasymmetry in the vertical acceleration profile, which is\nshown in Figure 5. The observed value of azbelow the\ndisk has a smaller magnitude than aZabove the disk,\nwhich is consistent with simulations of the MW interact-\ning with the Sgr dSph. The methodology of the simula-\ntions is described in [16, 17], and are the same simulations\nthat are used in [18]. This is consistent with the popular\nidea that this type of disequilibrium is (at least in part)\ncaused by the disk interacting with orbiting satellites.\nThe 2 α−βmodel is identical to the anharmonic model\nexcept that it does not require a flat rotation curve; the\ninterpretation of the βparameter is the same as for the\nα−βmodel.\nWe also attempted to fit a model that included a z4\nterm, although this quartic term could not be constrained\nwith existing data. A model that includes a quartic term\nwould be useful for quantifying the behavior of the phase\nspace spiral that has been observed in the MW disk [4].12\n2. Local Expansion Model\nThe local expansion model is a 1st-order Taylor series\nexpansion of the acceleration profile centered at the posi-\ntion of the Sun assuming that the potential is additively\nseparable:\nai(x)≈ai(x⊙) +∂ai\n∂xi\f\f\f\f\n⊙(xi−xi,⊙). (22)\nWhile it is useful to characterize the slope of the rota-\ntion curve with d aR/dRand the midplane density of the\ndisk with d az/dz, these terms do not describe disequi-\nlibrium features. The nonzero d aϕ/dϕimplies disequi-\nlibrium, as it requires that the potential be a function\nof azimuth, rather than axisymmetric. The AIC of this\nmodel is a substantial improvement over the anharmonic\nand 2 α−βmodels, which indicates that the MW disk\ndensity has substantial azimuthal variation in the Solar\nneighborhood.\n3. Sinusoidal Model\nThe MW disk has been shown to contain ripples, which\nare probably related to the passage of orbiting satellites\n[3, 20, 88, 89]. We explore these features with the sinu-\nsoidal model, which follows the same idea as the density\nmodel outlined in Xu et al. 3 of varying the “midplane”\nof the potential as a sine curve, except we use a har-\nmonic oscillator disk potential instead of an exponential\ndensity profile. This model outperforms the anharmonic\nand 2 α−βmodels, suggesting that the disk ripples are\na prominent feature of the disk potential. We obtain a\nslightly smaller wavelength and a somewhat larger ampli-\ntude for the disk oscillations than was found in the stellar\ndensity distribution by Xu et al. 3, although these differ-\nences could be largely due to the different definitions of\nthe two models; we used a conservative model in order\nto minimize the number of free parameters that we had\nto fit, whereas Xu et al. 3 were able to separately fit the\ndisk density variations above and below the midplane.\n4.α-β+ 2 Point Mass Model\nThe final model we discuss here is the most flexible\nmodel, which consists of the α−βmodel with the ad-\ndition of an arbitrary number of point masses. The lo-\ncation and mass of each point mass is independently fit\nto the data simultaneously with the αandβparame-\nters. This allows for a flexible non-parametric fit to any\nvarious disequilibrium features that may be present in\nthe acceleration data. In order to determine the opti-\nmal number of point masses that should be included in\nthe model, we sequentially fit the α−βmodel with n\npoint masses, increasing nuntil the AIC no longer im-\nproved when we added another point mass to the model.The optimal number of point masses (which minimizes\nAIC) is 2. While the AIC of this model is substantially\nlower than the other models in this work, the fact that\nitsχ2<1 could indicate that it is actually overfitting\nthe data.\nIt is not immediately clear what the physical signifi-\ncance is for the point masses in this model. On the sur-\nface, the point masses are simply a flexible model that\nallows for the generation of a wide range of potential\nmodels. It is plausible that this model is producing an\napproximation of a more complex potential model, and\nthat this presently unknown more complex model is not\nwell represented using our other analytical models.\nOne possible interpretation for these point masses is\nthat each point mass corresponds to a large dark matter\nsubhalo in the Solar neighborhood. These dark matter\nsubhalos would have masses of roughly 108M⊙, which\nis large but not unreasonable for a dark matter subhalo\nin a MW-mass galaxy [e.g. 90]. If this model is actually\nidentifying real dark matter subhalos, then the number\ndensity for M∼1010M⊙subhalos must be higher than\nin Λ-CDM, which predicts that there should only be a\nfew dark matter subhalos with this mass across the en-\ntire Galaxy [91–93]. We plan to explore the sensitivity of\ndirect acceleration measurements to dark matter subha-\nlos with a future publication, as it is outside the scope of\nthis work.\nVI. FUNDAMENTAL GALACTIC\nPARAMETERS FROM PULSAR TIMING\nA. Oort Constants\nCharacterization of the Galaxy has historically been\ndone using the kinematics of stars near the Sun [23]. One\nway to do this is by empirically estimating the Oort con-\nstants [94], called A, a measure of azimuthal shear motion\nin the Solar neighborhood, and B, a measure of the ro-\ntation curve vorticity, i.e. the scale of epicyclic motion.\nThese constants are related to the rotation curve near\nthe Sun:\nA−B=\u0014Vcirc\nR\u0015\nR⊙,and (23)\nA+B=−\u0014dVcirc\ndR\u0015\nR⊙. (24)\nModern kinematic estimates from Gaia data place the\nOort constants around A= 15.1 to 15 .3 km/s/kpc, and\nB=−13.4 to−11.9 km/s/kpc [76, 95], which are con-\nsistent with the most recent estimates from Gaia DR3\n[96].\nFrom the definition of βin the Equation 14, we can\nrelate the α−βmodel to the Oort constants:\nβ=−A+B\nA−B, (25)13\nFIG. 6. Selected measurements of the Oort constants Aand\nBsince 2010. The shape of each point indicates the type of\ndata used to make the measurement, and the color of each\npoint indicates the corresponding slope of the rotation curve\ngiven by d Vcirc/dR=−(A+B). The values we infer from\nthe pulsar timing data are shown as stars; while the three\nmodels span a relatively large area on the figure, the 2 α−β\nmodel (the pale red star) is in broad agreement with many of\nthe previous measurements. The sources for this data can be\nfound in Table V.\nwhich implies a value of β≈ −0.1 from the Gaia val-\nues for the Oort constants. We can also write the Oort\nconstants as\nA=1\n2VLSR\nR⊙(1−β),and (26)\nB=−1\n2VLSR\nR⊙(1 +β). (27)\nThe values of the Oort constants for the three α−β\nmodels in this work are listed in Table IV. It is clear that\nthe different models produce substantially different val-\nues for the Oort constants, presumably because they are\nsimultaneously fitting different amounts of local disequi-\nlibrium. The 2 α−βmodel is consistent with the kine-\nmatic estimates of the Oort constants, while the other\ntwo models are inconsistent with the kinematic estimates.\nThese results suggest that it is not only important to con-\nsider disequilibrium in the acceleration models in order\nto properly extract properties of the Galaxy, but that it\nis also important how this disequilibrium is accounted\nfor.\nFigure 6 shows our inferred values of the Oort con-\nstants compared with a selection of measurements of the\nOort constants since 2010. Our values span a relatively\nlarge range, but the values we obtain for the Oort con-\nstants from the 2 α−βmodel agree well with many pre-\nvious measurements.B. The Slope of the Rotation Curve\nEquation 14 provides a way to compute the slope of\nthe rotation curve for our α-βmodels; the slopes of the\nrotation curve for each model are provided in Table IV.\nWhile we obtain a range of d Vcirc/dRvalues for our\ndifferent models, the preferred 2 α-βmodel constrains the\nslope of the rotation curve to be -2 ±5 km/s/kpc at\nthe location of the Sun. This value is consistent with\nthe slope of the rotation curve being flat. However, a\nslightly declining rotation curve at the Solar location is\nin good agreement with Gaia data, such as Figure 13 of\nGaia Collaboration et al. 97. Additionally, P˜ oder et al.\n98 contains a compilation of a few different rotation curve\nmodels, all but one of which appear to have a value of\ndVcirc/dR∼2 km/s/kpc at the Solar position.\nC. The Oort Limit and Local Dark Matter Density\nThe Oort limit, or the volumetric mass density in the\nmidplane of the disk, is another characterization of the\nlocal Galaxy that is often derived from kinematic data.\nIt can be obtained by solving the Poisson Equation in\ncylindrical coordinates for the α−βmodels, which leads\nto\n4πGρ 0=α+ 2βΩ2\n⊙, (28)\nwhere Ω = Vcirc/Ris the angular velocity of the disk at\na given radius. Here, ρ0=ρbary+ρDMis the midplane\ndensity of the baryonic matter plus the dark matter.\nC21 used this prescription to calculate an Oort limit\nofρ0= 0.08+0.05\n−0.02M⊙/pc3. When combined with the\nbaryon budget of ρ0,bary= 0.084±0.012 M ⊙/pc3from\nMcKee et al. 99, this results in a midplane dark mat-\nter density of ρ0,DM=−0.004+0.05\n−0.02M⊙/pc3. C21 state\nthat this suggests that values of the Oort limit from\nJeans analysis may be an overestimate; their value of\nρ0,DMis consistent with there being no dark matter\nin the midplane, as well as McKee et al. 99’s value of\nρ0,DM= 0.013±0.003 M ⊙/pc3.\nOur results for the Oort limit and the dark matter den-\nsity in the midplane are given in Table IV. These values\nuse a baryon budget built from a literature compilation\nof the local stellar volume mass density of ρ∗= 0.0468±\n0.0050 M ⊙/pc3[100] and the gas density from McKee\net al. 99,ρgas= 0.025±0.003 M ⊙/pc3, which gives a lo-\ncal baryon density of ρbary= 0.072±0.006 M ⊙/pc3. This\nis similar to, but slightly less than the baryon budget of\nBienaym´ e et al. 101: ρ0,bary= 0.077±0.007 M ⊙/pc3.\nThese updated values are consistent with, and have\nsmaller uncertainty than the measurement of the dark\nmatter density in the midplane from C21. These values\nare all consistent with 0 dark matter in the midplane.\nOur measurements of the local dark matter density are\nlower than recent estimates of the dark matter density\nobtained from Jeans modeling (between 0.010 and 0.01614\nFIG. 7. The effect of different choices of scale length on the GPR map. The choice of scale length increases from left to right.\nThe left-most panel has too small of a scale length, as the GPR map only estimates alosnear a source and does not inteprolate\nbetween sources. The right-most panel has too large a scale length, as it averages over complex substructure and extrapolates\nalosfar beyond where data is located. The middle two panels show that for a reasonable choice of scale length, the general\nshapes of the GPR maps are similar, and therefore the exact choice of scale length is not particularly important.\nFIG. 8. GPR map of the observed line-of-sight acceleration compared to the theoretical line-of-sight accelerations from Gala\nMilkyWayPotential2022 . The white points indicate the positions of the binary pulsars. Red (blue) indicates an apparent\nline-of-sight acceleration towards (away from) the Sun. The top row shows the R−Zplane at Y= 0, and the bottom row\nshows the X−Yplane at Z= 0. The observed line-of-sight accelerations appear to be very different from the theoretical\nacceleration maps, implying a substantial departure from an equilibrium acceleration field.15\nFIG. 9. Tomography of the observed line-of-sight acceleration GPR map. Colors are the same as in Figure 8, except the\npredicted accelerations have been divided by the corresponding uncertainty in the predicted GPR map in order to emphasize\nsignificance of the GPR map values. The top row shows slices of the R−Zplane at different azimuth, where ϕincreases in\nthe same direction as the Sun’s rotation. The bottom row shows slices of the X−Yplane at different heights below/above the\nmidplane. An equilibrium acceleration field should look identical at ±ϕand±Z, but the observed accelerations are substantially\ndifferent on either side of the Sun.\nM⊙/pc3[99–103]), which is consistent with the claim of\nC21 that these kinematic methods may be overestimating\nthe local dark matter density. Moran et al. 51 also use\npulsar timing to obtain an Oort limit of ρ0= 0.036±0.021\nM⊙/pc3, which is substantially lower than our measure-\nment by about a factor of 2, and does not appear to be\nconsistent with any available baryon budgets.\nNote that if we instead use the McKee et al. 99 baryon\nbudget, which has a much larger stellar mass density than\nthe values collated in Guo et al. 100, we get ρ0,DM=\n−0.024±0.021,−0.022±0.021, and −0.018±0.021\nM⊙/pc3for our respective models. The McKee et al. 99\nbaryon budget is similar to recent estimates from Gaia\nDR2 data, ρ0,bary∼0.09 M ⊙/pc3[104]. Either of these\nbaryon budgets produce negative values for ρ0,DMthat\nare larger than the error bars in our models, which sug-\ngests that these baryon budgets may overestimate the\nbaryon content in the midplane.\nLim et al. 105 used unsupervised machine learning in\nthe form of normalizing flows to model the phase space\ndistribution of Gaia DR3 data. They obtain a value of\nρ0= 0.0617 ±0.0020 M ⊙/pc3for the Oort limit, which\nis nearly identical to our value. Lim et al. 105 calculate\na baryon budget of only 0.0534 ±0.0042 M ⊙/pc3from\ntheGaia DR3 data; if we instead use this baryon bud-\nget, we obtain a dark matter density in the midplane ofρ0,DM= 0.008 ±0.02 M ⊙/pc3for the 2 α-βmodel. This\nvalue is still consistent with no dark matter in the Galac-\ntic midplane; however, it is non-negative, which further\nimplies that a baryon budgets of the other works could\nbe overestimated.\nSome of these differences may be due to differences\nin the effective ranges that are probed by the different\nstudies. The “local” pulsar dataset lies within ∼1 kpc of\nthe MW disk plane, but, for example, the McKee et al.\n99 data is primarily located within a few hundred pc of\nthe midplane. The differences in the vertical extents of\nthese studies may result in somewhat different Oort limits\n(since we are averaging over different vertical extents),\nwhich means that caution must be used when directly\ncomparing these values.\nVII. BEYOND SMOOTH MODELS\nAnalytical and parametric models are useful for study-\ning Galactic structure, because they enable one to calcu-\nlate and compare fundamental Galactic parameters from\ndifferent datasets. In the regime of a static, equilibrium\nGalaxy, these parametric models effectively constrain the\nbehavior of the Galaxy. However, we are now aware\nthat the MW is experiencing many disequilibrium effects,16\nwhich indicates that parametric models are approxima-\ntions of the Galaxy’s behavior. These disequilibrium ef-\nfects are difficult to quantify analytically, and lend them-\nselves to non-parametric methods with a high degree of\nflexibility. Here, we develop a non-parametric map of\nthe acceleration field of the Galaxy, and discuss ways to\nimprove this map in the future.\nA. Gaussian Process Regression Maps\nGaussian Process Regression (GPR) is a non-\nparametric supervised learning algorithm that trains on\nmeasurements of data, and produces an interpolated\nmodel of the data and its uncertainties. This interpolated\nmodel can be evaluated at any point; in our case, this al-\nlows us to build a 3-dimensional map of the line-of-sight\nacceleration field within several kpc of the Sun. This was\ndone using the GaussianProcessRegressor class from\nscikit-learn to generate the GPR model.\nGPR requires a characteristic length scale for the in-\nterpolated output, which is analogous to the scale of the\nkernel in a kernel density estimation. Typically, in GPR\nthis scale length is treated as a hyperparameter, which\nis then optimized over the data in order to produce a\nGPR model that minimizes the ordinary-least-squares\ndifference between the model and the data, along with a\npenalty for model complexity, similar to AIC [106]. How-\never, the ability of the model to successfully optimize the\nlength-scale hyperparameter is dependent on the number\nof data points available. Ideally, one would have several\ndozen datapoints per dimension that could be used to ob-\ntain a good estimate of the length scale hyperparameter\n[106], but this is not true in our case. By adding or re-\nmoving a single datapoint, we would obtain dramatically\ndifferent values for the length scale of the GPR model.\nIn order to avoid this problem, we simply set the length\nscale as 0.5 kpc rather than allow the GPR method to op-\ntimize it over the dataset. The rationale for this choice of\nlength scale is shown in Figure 7; for too-small choices of\nthe length scale, the GPR model does not interpolate be-\ntween datapoints, and for too-large choices of the length\nscale, the GPR model interpolates much further than is\nreasonable. However, for reasonable choices of length\nscale (larger than the distance between datapoints but\nsmaller than the overall extent of the data), the GPR\nmaps generally look the same regardless of the exact\nchoice of length scale.\nTheR−zandx−yplanes (2-dimensional slices) from\nthe GPR map are shown in Figure 8, along with the line-\nof-sight accelerations that are expected to be observed\nbased on the MilkyWayPotential2022 potential model. It\nis immediately clear that the general shapes of the GPR\nmap and the theoretical acceleration maps do not match,\nwhich implies that the observed acceleration data is not\nwell-modeled by an equilibrium acceleration profile of the\nGalaxy. However, note that the theoretical alosmodel\nhas structure; this is a map of the tidal field, which issqueezed in the direction of rotation and elongated along\nthe direction towards the Galactic center.\nA useful benchmark for the scale of the observed accel-\neration perturbations is to consider the radial and verti-\ncal components of the acceleration at the location of the\nSun. The radial acceleration is roughly\naR∼V2\ncirc\nR⊙= 6.8 mm/s/yr . (29)\nAssuming an infinite density sheet with a surface density\nof Σ∼50 M ⊙/pc2, the vertical acceleration is\naz∼GΣ = 0 .2 mm/s/yr . (30)\nThe observed variations are on the scale of 0.5-1\nmm/s/yr, which is sizeable compared to the vertical ac-\nceleration but a fraction of the radial acceleration scale.\nWe are only able to observe variations in alos, which\nis a projected combination of aRandaz, so the varia-\ntions in aloswe see are on the ∼10% scale of the total\nacceleration.\nFurther, we provide the tomography of the GPR map\nin Figure 9. If the Galaxy were in equilibrium, the slices\nof the map at positive and negative ϕand/or zwould be\nidentical, because the Galaxy would be azimuthally sym-\nmetric. However, it is clear from the tomography that\nthis is not the case; for example, the slice at ϕ=−5◦\nlooks substantially different than the slice at ϕ= +5◦.\nThis could indicate a large amount of disequilibrium in\nthe Galaxy’s acceleration field near the Sun; however,\nbecause all the acceleration points are within 1.5 σof the\nmodel, it’s not clear exactly how significant these varia-\ntions are, or how much of the variations are caused by\nthe large uncertainties on the individual data points.\nThis GPR map only considers the propagated uncer-\ntainty in alosfor each datapoint. However, each source\nhas its own uncertainty in distance, which will not only\nchange the source’s position on the map, but the distance\nuncertainty is correlated to the uncertainty in alosin a\nnon-linear way. In the future we plan on creating GPR\nmaps that take this type of correlated uncertainty in the\ninput and output of each datapoint into account. Despite\nthese shortcomings, the proof-of-concept GPR maps pre-\nsented here are still powerful diagnostics for disequilib-\nrium features and asymmetries in the acceleration profile\nof the Galaxy.\nVIII. CONCLUSIONS\nHere we summarize the main findings of this work:\n•We gather direct acceleration measurements for 26\nbinary pulsars, which span 3.4 kpc in R, and 3.6\nkpc in Z.\n•The residual of the pulsar accelerations and accel-\nerations from modern kinematic models appear to17\nhave a global gradient in R; this trend could not\nbe resolved by varying the mass of the theoretical\npotential model. However, the trend can be elim-\ninated if the Sun is experiencing a 1.1 mm/s/yr\nacceleration away from the Galactic center, com-\nparable to a Jupiter-mass object at a distance of\n400 AU.\n•We fit a collection of symmetric and asymmetric\npotential models to the observed acceleration data.\nThe more flexible asymmetric models that allow for\ndisequilibrium do a better job of modeling the data\ncompared to the symmetric potentials, indicating\nthat the observed accelerations are consistent with\nthe Galactic disk being in disequilibrium. These\nasymmetries are consistent with those caused by\ndwarf galaxies interacting with the MW disk, al-\nthough it is difficult to determine the exact physical\nmechanism responsible for the asymmetries given\nthe current limitations of the acceleration data.\n•The Damour-Taylor Potential [29], which is com-\nmonly used to obtain an approximation of the\nGalactic acceleration in tests of general relativ-\nity, is not a good fit to the observed acceleration\ndata. Studies that use this potential to calibrate\nthe observed acceleration of sources due to the\ngravitational field of the MW will therefore pro-\nduce systematically inaccurate results, especially\nfor sources with large heights from the Galactic\nmidplane.\n•We constrain fundamental Galactic parameters us-\ning the acceleration data. Our values of the Oort\nConstants vary substantially depending on which\nmodel we use, but a disequilibrium model that con-\nsiders the asymmetry of the vertical acceleration\nprofile produces values for the Oort constants of\nA= 15 .4±2.6 km/s/kpc and B=−13.1±2.6\nkm/s/kpc, which are consistent with literature val-\nues from kinematic estimates. This model also pro-\nduces a 3.6 σmeasurement of the Oort limit, or\nthe density in the midplane, of ρ0= 0.062±0.017\nM⊙/pc3, and a dark matter density in the midplane\nofρ0,DM=−0.010±0.018 M ⊙/pc3. However, if we\nuse the baryon budget of Lim et al. 105, we obtain\na positive dark matter density in the midplane of\nρ0,DM= 0.008±0.02 M ⊙/pc3. The inferred dark\nmatter density is consistent with there being nodark matter density in the Galactic plane.\n•We constrain the slope of the rotation curve to be\n-2±5 km/s/kpc at the location of the Sun. This\nis consistent with the slope of the rotation curve\nbeing flat, but it is also consistent with the Gaia\nfindings that the azimuthal velocity curve in the\nplane is slightly decreasing at the Solar location.\n•We measure the oblateness of the potential using\na potential model that includes a cross ( γ) term.\nOur measured value of γrelies on updating pulsar\ntiming solutions and produces a similar oblateness\nand scale height to thick disk kinematic data from\nSharma et al. 72, in contrast to the C21 dataset,\nwhich indicated a more oblate potential near the\nGalactic midplane.\n•We provide non-parametric maps of the accelera-\ntion field of the Galaxy, which exhibit a substan-\ntial amount of variation from the theoretical ac-\nceleration maps. The tomography of this map is\nconsistent with a substantial amount of disk dise-\nquilibrium, given its asymmetry in Zandϕ.\nAs binary pulsar datasets continue to improve and\ngrow, we will be able to produce progressively better con-\nstraints on the MW’s gravitational potential and its dark\nmatter content. While more work remains to determine\nthe sources of the currently unexplained variations in ac-\ncelerations compared to theoretical maps, we have now\nillustrated the need for flexible disequilibrium models in\norder to utilize the acceleration data to its full capabili-\nties.\nACKNOWLEDGMENTS\nThis work utilizes data from the Australia Telescope\nNational Facility Catalogue [35], which can be found on-\nline at http://www.atnf.csiro.au/research/pulsar/\npsrcat . MTL graciously acknowledges support received\nfrom NSF AAG award number 2009468, and NSF Physics\nFrontiers Center award number 2020265, which supports\nthe NANOGrav project. SC gratefully acknowledges sup-\nport from NSF AST 2009828.\nSoftware used: Numpy [107], Scipy [69], Sci-kit Learn\n[108], Astropy [109], Matplotlib [110], emcee [111], Gala\n[60], Galpy [67, http://github.com/jobovy/galpy ]\n[1] D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101\n(1967).\n[2] J. Binney, Annu. Rev. Astron. Astrophy. 30, 51 (1992).\n[3] Y. Xu, H. J. Newberg, J. L. Carlin, C. Liu, L. Deng,\nJ. Li, R. Sch¨ onrich, and B. Yanny, Astrophys. J. 801,\n105 (2015), arXiv:1503.00257 [astro-ph.GA].\n[4] T. Antoja, A. Helmi, M. Romero-G´ omez, D. Katz,C. Babusiaux, R. Drimmel, D. W. Evans, F. Figueras,\nE. Poggio, C. Reyl´ e, A. C. Robin, G. Seabroke,\nand C. Soubiran, Nature (London) 561, 360 (2018),\narXiv:1804.10196 [astro-ph.GA].\n[5] L. M. Widrow, S. Gardner, B. Yanny, S. Dodel-\nson, and H.-Y. Chen, Astrophys. J. 750, L41 (2012),\narXiv:1203.6861 [astro-ph.GA].18\n[6] B. Yanny and S. Gardner, Astrophys. J. 777, 91 (2013),\narXiv:1309.2300 [astro-ph.GA].\n[7] M. Bennett and J. Bovy, Mon. Not. R. Astron. Soc.\n482, 1417 (2019), arXiv:1809.03507 [astro-ph.GA].\n[8] A. C. Quillen and I. Minchev, Astron. J. 130, 576\n(2005), arXiv:astro-ph/0502205 [astro-ph].\n[9] T. Antoja, F. Figueras, D. Fern´ andez, and J. Torra,\nAstron. Astrophys. 490, 135 (2008), arXiv:0809.0511\n[astro-ph].\n[10] T. Antoja, F. Figueras, M. Romero-G´ omez,\nB. Pichardo, O. Valenzuela, and E. Moreno, Mon.\nNot. R. Astron. Soc. 418, 1423 (2011), arXiv:1106.1170\n[astro-ph.GA].\n[11] J. A. S. Hunt, J. Hong, J. Bovy, D. Kawata, and R. J. J.\nGrand, Mon. Not. R. Astron. Soc. 481, 3794 (2018),\narXiv:1806.02832 [astro-ph.GA].\n[12] P. Ramos, T. Antoja, and F. Figueras, Astron. Astro-\nphys. 619, A72 (2018), arXiv:1805.09790 [astro-ph.GA].\n[13] P. Craig, S. Chakrabarti, H. Newberg, and A. Quillen,\nMon. Not. R. Astron. Soc. 505, 2561 (2021),\narXiv:1911.01392 [astro-ph.GA].\n[14] E. S. Levine, L. Blitz, and C. Heiles, Science 312, 1773\n(2006), arXiv:astro-ph/0605728 [astro-ph].\n[15] P. M. W. Kalberla and L. Dedes, Astron. Astrophys.\n487, 951 (2008), arXiv:0804.4831 [astro-ph].\n[16] S. Chakrabarti and L. Blitz, Mon. Not. R. Astron. Soc.\n399, L118 (2009), arXiv:0812.0821 [astro-ph].\n[17] S. Chakrabarti and L. Blitz, Astrophys. J. 731, 40\n(2011), arXiv:1007.1982 [astro-ph.GA].\n[18] S. Chakrabarti, P. Chang, A. M. Price-Whelan, J. Read,\nL. Blitz, and L. Hernquist, Astrophys. J. 886, 67 (2019),\narXiv:1906.04203 [astro-ph.GA].\n[19] J. Alves, C. Zucker, A. A. Goodman, J. S. Speagle,\nS. Meingast, T. Robitaille, D. P. Finkbeiner, E. F.\nSchlafly, and G. M. Green, Nature (London) 578, 237\n(2020), arXiv:2001.08748 [astro-ph.GA].\n[20] C. F. P. Laporte, I. Minchev, K. V. Johnston, and F. A.\nG´ omez, Mon. Not. R. Astron. Soc. 485, 3134 (2019),\narXiv:1808.00451 [astro-ph.GA].\n[21] S. Khoperskov, P. Di Matteo, O. Gerhard, D. Katz,\nM. Haywood, F. Combes, P. Berczik, and A. Gomez,\nAstron. Astrophys. 622, L6 (2019), arXiv:1811.09205\n[astro-ph.GA].\n[22] J. A. S. Hunt, A. M. Price-Whelan, K. V. Johnston,\nand E. Darragh-Ford, Mon. Not. R. Astron. Soc. 516,\nL7 (2022), arXiv:2206.06125 [astro-ph.GA].\n[23] J. Bland-Hawthorn and O. Gerhard, Annu. Rev. As-\ntron. Astrophy. 54, 529 (2016), arXiv:1602.07702 [astro-\nph.GA].\n[24] J. Binney and S. Tremaine, Galactic Dynamics: Second\nEdition (2008).\n[25] S. Chakrabarti, J. Wright, P. Chang, A. Quillen,\nP. Craig, J. Territo, E. D’Onghia, K. V. Johnston, R. J.\nDe Rosa, D. Huber, K. L. Rhode, and E. Nielsen, As-\ntrophys. J. 902, L28 (2020), arXiv:2007.15097 [astro-\nph.GA].\n[26] S. Chakrabarti, P. Chang, M. T. Lam, S. J. Vige-\nland, and A. C. Quillen, Astrophys. J. 907, L26 (2021),\narXiv:2010.04018 [astro-ph.GA].\n[27] S. Chakrabarti, D. J. Stevens, J. Wright, R. R. Rafikov,\nP. Chang, T. Beatty, and D. Huber, Astrophys. J. 928,\nL17 (2022), arXiv:2112.08231 [astro-ph.GA].\n[28] P. C. Peters and J. Mathews, Physical Review 131, 435\n(1963).[29] T. Damour and J. H. Taylor, Astrophys. J. 366, 501\n(1991).\n[30] J. M. Weisberg and Y. Huang, Astrophys. J. 829, 55\n(2016), arXiv:1606.02744 [astro-ph.HE].\n[31] N. Wex and M. Kramer, Universe 6, 156 (2020).\n[32] Gravity Collaboration, R. Abuter, A. Amorim,\nM. Baub¨ ock, J. P. Berger, H. Bonnet, W. Brandner,\nY. Cl´ enet, V. Coud´ e Du Foresto, P. T. de Zeeuw, J. Dex-\nter, G. Duvert, A. Eckart, F. Eisenhauer, N. M. F¨ orster\nSchreiber, P. Garcia, F. Gao, E. Gendron, R. Gen-\nzel, O. Gerhard, S. Gillessen, M. Habibi, X. Haubois,\nT. Henning, S. Hippler, M. Horrobin, A. Jim´ enez-\nRosales, L. Jocou, P. Kervella, S. Lacour, V. Lapeyr` ere,\nJ. B. Le Bouquin, P. L´ ena, T. Ott, T. Paumard,\nK. Perraut, G. Perrin, O. Pfuhl, S. Rabien, G. Ro-\ndriguez Coira, G. Rousset, S. Scheithauer, A. Sternberg,\nO. Straub, C. Straubmeier, E. Sturm, L. J. Tacconi,\nF. Vincent, S. von Fellenberg, I. Waisberg, F. Widmann,\nE. Wieprecht, E. Wiezorrek, J. Woillez, and S. Yazici,\nAstron. Astrophys. 625, L10 (2019), arXiv:1904.05721\n[astro-ph.GA].\n[33] P. J. McMillan, Mon. Not. R. Astron. Soc. 465, 76\n(2017), arXiv:1608.00971 [astro-ph.GA].\n[34] I. S. Shklovskii, Sov. Astron. 13, 562 (1970).\n[35] R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs,\nAstron. J. 129, 1993 (2005), arXiv:astro-ph/0412641\n[astro-ph].\n[36] D. J. Reardon, G. Hobbs, W. Coles, Y. Levin, M. J.\nKeith, M. Bailes, N. D. R. Bhat, S. Burke-Spolaor,\nS. Dai, M. Kerr, P. D. Lasky, R. N. Manchester,\nS. Os lowski, V. Ravi, R. M. Shannon, W. van Straten,\nL. Toomey, J. Wang, L. Wen, X. P. You, and X. J.\nZhu, Mon. Not. R. Astron. Soc. 455, 1751 (2016),\narXiv:1510.04434 [astro-ph.HE].\n[37] G. Agazie, M. F. Alam, A. Anumarlapudi, A. M.\nArchibald, Z. Arzoumanian, P. T. Baker, L. Blecha,\nV. Bonidie, A. Brazier, P. R. Brook, S. Burke-Spolaor,\nB. B´ ecsy, C. Chapman, M. Charisi, S. Chatterjee,\nT. Cohen, J. M. Cordes, N. J. Cornish, F. Crawford,\nH. T. Cromartie, K. Crowter, M. E. Decesar, P. B. De-\nmorest, T. Dolch, B. Drachler, E. C. Ferrara, W. Fiore,\nE. Fonseca, G. E. Freedman, N. Garver-Daniels, P. A.\nGentile, J. Glaser, D. C. Good, K. G¨ ultekin, J. S.\nHazboun, R. J. Jennings, C. Jessup, A. D. Johnson,\nM. L. Jones, A. R. Kaiser, D. L. Kaplan, L. Z. Kelley,\nM. Kerr, J. S. Key, A. Kuske, N. Laal, M. T. Lam,\nW. G. Lamb, T. J. W. Lazio, N. Lewandowska, Y. Lin,\nT. Liu, D. R. Lorimer, J. Luo, R. S. Lynch, C.-P. Ma,\nD. R. Madison, K. Maraccini, A. McEwen, J. W. Mc-\nKee, M. A. McLaughlin, N. McMann, B. W. Meyers,\nC. M. F. Mingarelli, A. Mitridate, C. Ng, D. J. Nice,\nS. K. Ocker, K. D. Olum, E. Panciu, T. T. Pennucci,\nB. B. P. Perera, N. S. Pol, H. A. Radovan, S. M. Ran-\nsom, P. S. Ray, J. D. Romano, L. Salo, S. C. Sardesai,\nC. Schmiedekamp, A. Schmiedekamp, K. Schmitz, B. J.\nShapiro-Albert, X. Siemens, J. Simon, M. S. Siwek, I. H.\nStairs, D. R. Stinebring, K. Stovall, A. Susobhanan,\nJ. K. Swiggum, S. R. Taylor, J. E. Turner, C. Unal,\nM. Vallisneri, S. J. Vigeland, H. M. Wahl, Q. Wang,\nC. A. Witt, O. Young, and Nanograv Collaboration,\nAstrophys. J. 951, L9 (2023), arXiv:2306.16217 [astro-\nph.HE].\n[38] M. Kramer, I. H. Stairs, R. N. Manchester, M. A.\nMcLaughlin, A. G. Lyne, R. D. Ferdman, M. Bur-19\ngay, D. R. Lorimer, A. Possenti, N. D’Amico, J. M.\nSarkissian, G. B. Hobbs, J. E. Reynolds, P. C. C. Freire,\nand F. Camilo, Science 314, 97 (2006), arXiv:astro-\nph/0609417 [astro-ph].\n[39] A. T. Deller, M. Bailes, and S. J. Tingay, Science 323,\n1327 (2009), arXiv:0902.0996 [astro-ph.SR].\n[40] EPTA Collaboration, J. Antoniadis, S. Babak, A. S.\nBak Nielsen, C. G. Bassa, A. Berthereau, M. Bonetti,\nE. Bortolas, P. R. Brook, M. Burgay, R. N. Ca-\nballero, A. Chalumeau, D. J. Champion, S. C lianlaridis,\nS. Chen, I. Cognard, G. Desvignes, M. Falxa, R. D.\nFerdman, A. Franchini, J. R. Gair, B. Goncharov,\nE. Graikou, J. M. Grießmeier, L. Guillemot, Y. J. Guo,\nH. Hu, F. Iraci, D. Izquierdo-Villalba, J. Jang, J. Ja-\nwor, G. H. Janssen, A. Jessner, R. Karuppusamy, E. F.\nKeane, M. J. Keith, M. Kramer, M. A. Krishnakumar,\nK. Lackeos, K. J. Lee, K. Liu, Y. Liu, A. G. Lyne,\nJ. W. McKee, R. A. Main, M. B. Mickaliger, I. C.\nNit ¸u, A. Parthasarathy, B. B. P. Perera, D. Perrodin,\nA. Petiteau, N. K. Porayko, A. Possenti, H. Quelque-\njay Leclere, A. Samajdar, S. A. Sanidas, A. Sesana,\nG. Shaifullah, L. Speri, R. Spiewak, B. W. Stappers,\nS. C. Susarla, G. Theureau, C. Tiburzi, E. van der Wa-\nteren, A. Vecchio, V. Venkatraman Krishnan, J. P. W.\nVerbiest, J. Wang, L. Wang, and Z. Wu, Astron. Astro-\nphys. 678, A48 (2023), arXiv:2306.16224 [astro-ph.HE].\n[41] D. J. Nice, I. H. Stairs, and L. E. Kasian, in 40 Years\nof Pulsars: Millisecond Pulsars, Magnetars and More ,\nAmerican Institute of Physics Conference Series, Vol.\n983, edited by C. Bassa, Z. Wang, A. Cumming, and\nV. M. Kaspi (2008) pp. 453–458.\n[42] E. Fonseca, I. H. Stairs, and S. E. Thorsett, Astrophys.\nJ.787, 82 (2014), arXiv:1402.4836 [astro-ph.HE].\n[43] H. Ding, A. T. Deller, E. Fonseca, I. H. Stairs, B. Stap-\npers, and A. Lyne, Astrophys. J. 921, L19 (2021),\narXiv:2110.10590 [astro-ph.HE].\n[44] D. J. Reardon, R. M. Shannon, A. D. Cameron, B. Gon-\ncharov, G. B. Hobbs, H. Middleton, M. Shamoham-\nmadi, N. Thyagarajan, M. Bailes, N. D. R. Bhat, S. Dai,\nM. Kerr, R. N. Manchester, C. J. Russell, R. Spiewak,\nJ. B. Wang, and X. J. Zhu, Mon. Not. R. Astron. Soc.\n507, 2137 (2021), arXiv:2107.04609 [astro-ph.HE].\n[45] K. Walker, D. J. Reardon, E. Thrane, and R. Smith,\nAstrophys. J. 933, 16 (2022), arXiv:2204.11077 [astro-\nph.HE].\n[46] P. C. C. Freire, N. Wex, G. Esposito-Far` ese, J. P. W.\nVerbiest, M. Bailes, B. A. Jacoby, M. Kramer, I. H.\nStairs, J. Antoniadis, and G. H. Janssen, Mon. Not. R.\nAstron. Soc. 423, 3328 (2012), arXiv:1205.1450 [astro-\nph.GA].\n[47] A. T. Deller, J. M. Weisberg, D. J. Nice, and S. Chat-\nterjee, Astrophys. J. 862, 139 (2018), arXiv:1806.10265\n[astro-ph.SR].\n[48] M. Geyer, V. Venkatraman Krishnan, P. C. C. Freire,\nM. Kramer, J. Antoniadis, M. Bailes, M. C. i.\nBernadich, S. Buchner, A. D. Cameron, D. J. Cham-\npion, A. Karastergiou, M. J. Keith, M. E. Lower,\nS. Os lowski, A. Possenti, A. Parthasarathy, D. J. Rear-\ndon, M. Serylak, R. M. Shannon, R. Spiewak, W. van\nStraten, and J. P. W. Verbiest, Astron. Astrophys. 674,\nA169 (2023), arXiv:2304.09060 [astro-ph.HE].\n[49] Y. J. Guo, P. C. C. Freire, L. Guillemot, M. Kramer,\nW. W. Zhu, N. Wex, J. W. McKee, A. Deller, H. Ding,\nD. L. Kaplan, B. Stappers, I. Cognard, X. Miao,L. Haase, M. Keith, S. M. Ransom, and G. Theureau,\nAstron. Astrophys. 654, A16 (2021), arXiv:2107.09474\n[astro-ph.HE].\n[50] K. Stovall, P. C. C. Freire, J. Antoniadis, M. Bagchi,\nJ. S. Deneva, N. Garver-Daniels, J. G. Martinez, M. A.\nMcLaughlin, Z. Arzoumanian, H. Blumer, P. R. Brook,\nH. T. Cromartie, P. B. Demorest, M. E. DeCesar,\nT. Dolch, J. A. Ellis, R. D. Ferdman, E. C. Ferrara,\nE. Fonseca, P. A. Gentile, M. L. Jones, M. T. Lam,\nD. R. Lorimer, R. S. Lynch, C. Ng, D. J. Nice, T. T.\nPennucci, S. M. Ransom, R. Spiewak, I. H. Stairs, J. K.\nSwiggum, S. J. Vigeland, and W. W. Zhu, Astrophys.\nJ.870, 74 (2019), arXiv:1809.05064 [astro-ph.HE].\n[51] A. Moran, C. M. F. Mingarelli, K. Van Tilburg, and\nD. Good, arXiv e-prints , arXiv:2306.13137 (2023),\narXiv:2306.13137 [astro-ph.GA].\n[52] S.-X. Yi and K. S. Cheng, Astrophys. J. 844, 114 (2017),\narXiv:1706.08715 [astro-ph.HE].\n[53] Gaia Collaboration, A. Vallenari, A. G. A. Brown,\nT. Prusti, J. H. J. de Bruijne, F. Arenou, C. Babu-\nsiaux, M. Biermann, O. L. Creevey, C. Ducourant,\nD. W. Evans, L. Eyer, R. Guerra, A. Hutton, C. Jordi,\nS. A. Klioner, U. L. Lammers, L. Lindegren, X. Luri,\nF. Mignard, C. Panem, D. Pourbaix, S. Randich, P. Sar-\ntoretti, C. Soubiran, P. Tanga, N. A. Walton, C. A. L.\nBailer-Jones, U. Bastian, R. Drimmel, F. Jansen,\nD. Katz, M. G. Lattanzi, F. van Leeuwen, J. Bakker,\nC. Cacciari, J. Casta˜ neda, F. De Angeli, C. Fabri-\ncius, M. Fouesneau, Y. Fr´ emat, L. Galluccio, A. Guer-\nrier, U. Heiter, E. Masana, R. Messineo, N. Mowlavi,\nC. Nicolas, K. Nienartowicz, F. Pailler, P. Panuzzo,\nF. Riclet, W. Roux, G. M. Seabroke, R. Sordoørcit,\nF. Th´ evenin, G. Gracia-Abril, J. Portell, D. Teyssier,\nM. Altmann, R. Andrae, M. Audard, I. Bellas-\nVelidis, K. Benson, J. Berthier, R. Blomme, P. W.\nBurgess, D. Busonero, G. Busso, H. C´ anovas, B. Carry,\nA. Cellino, N. Cheek, G. Clementini, Y. Damerdji,\nM. Davidson, P. de Teodoro, M. Nu˜ nez Campos,\nL. Delchambre, A. Dell’Oro, P. Esquej, J. Fern´ andez-\nHern´ andez, E. Fraile, D. Garabato, P. Garc´ ıa-Lario,\nE. Gosset, R. Haigron, J. L. Halbwachs, N. C. Ham-\nbly, D. L. Harrison, J. Hern´ andez, D. Hestroffer,\nS. T. Hodgkin, B. Holl, K. Janßen, G. Jevardat\nde Fombelle, S. Jordan, A. Krone-Martins, A. C.\nLanzafame, W. L¨ offler, O. Marchal, P. M. Marrese,\nA. Moitinho, K. Muinonen, P. Osborne, E. Pancino,\nT. Pauwels, A. Recio-Blanco, C. Reyl´ e, M. Riello,\nL. Rimoldini, T. Roegiers, J. Rybizki, L. M. Sarro,\nC. Siopis, M. Smith, A. Sozzetti, E. Utrilla, M. van\nLeeuwen, U. Abbas, P. ´Abrah´ am, A. Abreu Aram-\nburu, C. Aerts, J. J. Aguado, M. Ajaj, F. Aldea-\nMontero, G. Altavilla, M. A. ´Alvarez, J. Alves, F. An-\nders, R. I. Anderson, E. Anglada Varela, T. Antoja,\nD. Baines, S. G. Baker, L. Balaguer-N´ u˜ nez, E. Bal-\nbinot, Z. Balog, C. Barache, D. Barbato, M. Bar-\nros, M. A. Barstow, S. Bartolom´ e, J. L. Bassilana,\nN. Bauchet, U. Becciani, M. Bellazzini, A. Berihuete,\nM. Bernet, S. Bertone, L. Bianchi, A. Binnenfeld,\nS. Blanco-Cuaresma, A. Blazere, T. Boch, A. Bom-\nbrun, D. Bossini, S. Bouquillon, A. Bragaglia, L. Bra-\nmante, E. Breedt, A. Bressan, N. Brouillet, E. Bru-\ngaletta, B. Bucciarelli, A. Burlacu, A. G. Butkevich,\nR. Buzzi, E. Caffau, R. Cancelliere, T. Cantat-Gaudin,\nR. Carballo, T. Carlucci, M. I. Carnerero, J. M. Car-20\nrasco, L. Casamiquela, M. Castellani, A. Castro-Ginard,\nL. Chaoul, P. Charlot, L. Chemin, V. Chiaramida,\nA. Chiavassa, N. Chornay, G. Comoretto, G. Con-\ntursi, W. J. Cooper, T. Cornez, S. Cowell, F. Crifo,\nM. Cropper, M. Crosta, C. Crowley, C. Dafonte,\nA. Dapergolas, M. David, P. David, P. de Laverny,\nF. De Luise, R. De March, J. De Ridder, R. de\nSouza, A. de Torres, E. F. del Peloso, E. del Pozo,\nM. Delbo, A. Delgado, J. B. Delisle, C. Demouchy,\nT. E. Dharmawardena, P. Di Matteo, S. Diakite,\nC. Diener, E. Distefano, C. Dolding, B. Edvardsson,\nH. Enke, C. Fabre, M. Fabrizio, S. Faigler, G. Fe-\ndorets, P. Fernique, A. Fienga, F. Figueras, Y. Fournier,\nC. Fouron, F. Fragkoudi, M. Gai, A. Garcia-Gutierrez,\nM. Garcia-Reinaldos, M. Garc´ ıa-Torres, A. Garo-\nfalo, A. Gavel, P. Gavras, E. Gerlach, R. Geyer,\nP. Giacobbe, G. Gilmore, S. Girona, G. Giuffrida,\nR. Gomel, A. Gomez, J. Gonz´ alez-N´ u˜ nez, I. Gonz´ alez-\nSantamar´ ıa, J. J. Gonz´ alez-Vidal, M. Granvik, P. Guill-\nout, J. Guiraud, R. Guti´ errez-S´ anchez, L. P. Guy,\nD. Hatzidimitriou, M. Hauser, M. Haywood, A. Helmer,\nA. Helmi, M. H. Sarmiento, S. L. Hidalgo, T. Hilger,\nN. H ladczuk, D. Hobbs, G. Holland, H. E. Huckle,\nK. Jardine, G. Jasniewicz, A. Jean-Antoine Piccolo,\n´O. Jim´ enez-Arranz, A. Jorissen, J. Juaristi Campillo,\nF. Julbe, L. Karbevska, P. Kervella, S. Khanna,\nM. Kontizas, G. Kordopatis, A. J. Korn, ´A. K´ osp´ al,\nZ. Kostrzewa-Rutkowska, K. Kruszy´ nska, M. Kun,\nP. Laizeau, S. Lambert, A. F. Lanza, Y. Lasne, J. F.\nLe Campion, Y. Lebreton, T. Lebzelter, S. Leccia,\nN. Leclerc, I. Lecoeur-Taibi, S. Liao, E. L. Licata,\nH. E. P. Lindstrøm, T. A. Lister, E. Livanou, A. Lobel,\nA. Lorca, C. Loup, P. Madrero Pardo, A. Magdaleno\nRomeo, S. Managau, R. G. Mann, M. Manteiga, J. M.\nMarchant, M. Marconi, J. Marcos, M. M. S. Marcos\nSantos, D. Mar´ ın Pina, S. Marinoni, F. Marocco, D. J.\nMarshall, L. M. Polo, J. M. Mart´ ın-Fleitas, G. Mar-\nton, N. Mary, A. Masip, D. Massari, A. Mastrobuono-\nBattisti, T. Mazeh, P. J. McMillan, S. Messina,\nD. Michalik, N. R. Millar, A. Mints, D. Molina, R. Moli-\nnaro, L. Moln´ ar, G. Monari, M. Mongui´ o, P. Mon-\ntegriffo, A. Montero, R. Mor, A. Mora, R. Mor-\nbidelli, T. Morel, D. Morris, T. Muraveva, C. P.\nMurphy, I. Musella, Z. Nagy, L. Noval, F. Oca˜ na,\nA. Ogden, C. Ordenovic, J. O. Osinde, C. Pagani,\nI. Pagano, L. Palaversa, P. A. Palicio, L. Pallas-\nQuintela, A. Panahi, S. Payne-Wardenaar, X. Pe˜ nalosa\nEsteller, A. Penttil¨ a, B. Pichon, A. M. Piersimoni, F. X.\nPineau, E. Plachy, G. Plum, E. Poggio, A. Prˇ sa, L. Pu-\nlone, E. Racero, S. Ragaini, M. Rainer, C. M. Rai-\nteri, N. Rambaux, P. Ramos, M. Ramos-Lerate, P. Re\nFiorentin, S. Regibo, P. J. Richards, C. Rios Diaz,\nV. Ripepi, A. Riva, H. W. Rix, G. Rixon, N. Robichon,\nA. C. Robin, C. Robin, M. Roelens, H. R. O. Rogues,\nL. Rohrbasser, M. Romero-G´ omez, N. Rowell, F. Royer,\nD. Ruz Mieres, K. A. Rybicki, G. Sadowski, A. S´ aez\nN´ u˜ nez, A. Sagrist` a Sell´ es, J. Sahlmann, E. Salguero,\nN. Samaras, V. Sanchez Gimenez, N. Sanna, R. San-\ntove˜ na, M. Sarasso, M. Schultheis, E. Sciacca, M. Segol,\nJ. C. Segovia, D. S´ egransan, D. Semeux, S. Sha-\nhaf, H. I. Siddiqui, A. Siebert, L. Siltala, A. Sil-\nvelo, E. Slezak, I. Slezak, R. L. Smart, O. N. Snaith,\nE. Solano, F. Solitro, D. Souami, J. Souchay, A. Spagna,\nL. Spina, F. Spoto, I. A. Steele, H. Steidelm¨ uller,C. A. Stephenson, M. S¨ uveges, J. Surdej, L. Szaba-\ndos, E. Szegedi-Elek, F. Taris, M. B. Taylo, R. Teix-\neira, L. Tolomei, N. Tonello, F. Torra, J. Torra, G. Tor-\nralba Elipe, M. Trabucchi, A. T. Tsounis, C. Turon,\nA. Ulla, N. Unger, M. V. Vaillant, E. van Dillen, W. van\nReeven, O. Vanel, A. Vecchiato, Y. Viala, D. Vicente,\nS. Voutsinas, M. Weiler, T. Wevers, L. Wyrzykowski,\nA. Yoldas, P. Yvard, H. Zhao, J. Zorec, S. Zucker, and\nT. Zwitter, arXiv e-prints , arXiv:2208.00211 (2022),\narXiv:2208.00211 [astro-ph.GA].\n[54] J. Antoniadis, Mon. Not. R. Astron. Soc. 501, 1116\n(2021), arXiv:2011.08075 [astro-ph.HE].\n[55] A. Moran, C. M. F. Mingarelli, M. Bedell, D. Good,\nand D. N. Spergel, Astrophys. J. 954, 89 (2023),\narXiv:2210.10816 [astro-ph.IM].\n[56] P. Draghis and R. W. Romani, Astrophys. J. 862, L6\n(2018), arXiv:1807.04249 [astro-ph.HE].\n[57] W.-C. Chen, Mon. Not. R. Astron. Soc. 501, 2327\n(2021), arXiv:2011.13923 [astro-ph.HE].\n[58] H. J. Pletsch and C. J. Clark, Astrophys. J. 807, 18\n(2015), arXiv:1504.07466 [astro-ph.HE].\n[59] D. J. Reardon, G. Hobbs, W. Coles, Y. Levin, M. J.\nKeith, M. Bailes, N. D. R. Bhat, S. Burke-Spolaor,\nS. Dai, M. Kerr, P. D. Lasky, R. N. Manchester,\nS. Os lowski, V. Ravi, R. M. Shannon, W. van Straten,\nL. Toomey, J. Wang, L. Wen, X. P. You, and X. J.\nZhu, Mon. Not. R. Astron. Soc. 455, 1751 (2016),\narXiv:1510.04434 [astro-ph.HE].\n[60] A. M. Price-Whelan, The Journal of Open Source Soft-\nware2, 10.21105/joss.00388 (2017).\n[61] Gaia Collaboration, S. A. Klioner, F. Mignard, L. Lin-\ndegren, U. Bastian, P. J. McMillan, J. Hern´ andez,\nD. Hobbs, M. Ramos-Lerate, M. Biermann, A. Bom-\nbrun, A. de Torres, E. Gerlach, R. Geyer, T. Hilger,\nU. Lammers, H. Steidelm¨ uller, C. A. Stephenson,\nA. G. A. Brown, A. Vallenari, T. Prusti, J. H. J. de\nBruijne, C. Babusiaux, O. L. Creevey, D. W. Evans,\nL. Eyer, A. Hutton, F. Jansen, C. Jordi, X. Luri,\nC. Panem, D. Pourbaix, S. Randich, P. Sartoretti,\nC. Soubiran, N. A. Walton, F. Arenou, C. A. L. Bailer-\nJones, M. Cropper, R. Drimmel, D. Katz, M. G. Lat-\ntanzi, F. van Leeuwen, J. Bakker, J. Casta˜ neda, F. De\nAngeli, C. Ducourant, C. Fabricius, M. Fouesneau,\nY. Fr´ emat, R. Guerra, A. Guerrier, J. Guiraud, A. Jean-\nAntoine Piccolo, E. Masana, R. Messineo, N. Mowlavi,\nC. Nicolas, K. Nienartowicz, F. Pailler, P. Panuzzo,\nF. Riclet, W. Roux, G. M. Seabroke, R. Sordo, P. Tanga,\nF. Th´ evenin, G. Gracia-Abril, J. Portell, D. Teyssier,\nM. Altmann, R. Andrae, I. Bellas-Velidis, K. Ben-\nson, J. Berthier, R. Blomme, E. Brugaletta, P. W.\nBurgess, G. Busso, B. Carry, A. Cellino, N. Cheek,\nG. Clementini, Y. Damerdji, M. Davidson, L. Delcham-\nbre, A. Dell’Oro, J. Fern´ andez-Hern´ andez, L. Galluc-\ncio, P. Garc´ ıa-Lario, M. Garcia-Reinaldos, J. Gonz´ alez-\nN´ u˜ nez, E. Gosset, R. Haigron, J. L. Halbwachs, N. C.\nHambly, D. L. Harrison, D. Hatzidimitriou, U. Heiter,\nD. Hestroffer, S. T. Hodgkin, B. Holl, K. Janßen,\nG. Jevardat de Fombelle, S. Jordan, A. Krone-Martins,\nA. C. Lanzafame, W. L¨ offler, A. Lorca, M. Manteiga,\nO. Marchal, P. M. Marrese, A. Moitinho, A. Mora,\nK. Muinonen, P. Osborne, E. Pancino, T. Pauwels,\nA. Recio-Blanco, P. J. Richards, M. Riello, L. Rimol-\ndini, A. C. Robin, T. Roegiers, J. Rybizki, L. M. Sarro,\nC. Siopis, M. Smith, A. Sozzetti, A. Ulla, E. Utrilla,21\nM. van Leeuwen, W. van Reeven, U. Abbas, A. Abreu\nAramburu, S. Accart, C. Aerts, J. J. Aguado, M. Ajaj,\nG. Altavilla, M. A. ´Alvarez, J. ´Alvarez Cid-Fuentes,\nJ. Alves, R. I. Anderson, E. Anglada Varela, T. An-\ntoja, M. Audard, D. Baines, S. G. Baker, L. Balaguer-\nN´ u˜ nez, E. Balbinot, Z. Balog, C. Barache, D. Bar-\nbato, M. Barros, M. A. Barstow, S. Bartolom´ e, J. L.\nBassilana, N. Bauchet, A. Baudesson-Stella, U. Bec-\nciani, M. Bellazzini, M. Bernet, S. Bertone, L. Bianchi,\nS. Blanco-Cuaresma, T. Boch, D. Bossini, S. Bouquil-\nlon, L. Bramante, E. Breedt, A. Bressan, N. Brouillet,\nB. Bucciarelli, A. Burlacu, D. Busonero, A. G. Butke-\nvich, R. Buzzi, E. Caffau, R. Cancelliere, H. C´ anovas,\nT. Cantat-Gaudin, R. Carballo, T. Carlucci, M. I.\nCarnerero, J. M. Carrasco, L. Casamiquela, M. Castel-\nlani, A. Castro-Ginard, P. Castro Sampol, L. Chaoul,\nP. Charlot, L. Chemin, A. Chiavassa, G. Comoretto,\nW. J. Cooper, T. Cornez, S. Cowell, F. Crifo, M. Crosta,\nC. Crowley, C. Dafonte, A. Dapergolas, M. David,\nP. David, P. de Laverny, F. De Luise, R. De March,\nJ. De Ridder, R. de Souza, P. de Teodoro, E. F.\ndel Peloso, E. del Pozo, A. Delgado, H. E. Delgado,\nJ. B. Delisle, P. Di Matteo, S. Diakite, C. Diener,\nE. Distefano, C. Dolding, D. Eappachen, H. Enke,\nP. Esquej, C. Fabre, M. Fabrizio, S. Faigler, G. Fe-\ndorets, P. Fernique, A. Fienga, F. Figueras, C. Fouron,\nF. Fragkoudi, E. Fraile, F. Franke, M. Gai, D. Gara-\nbato, A. Garcia-Gutierrez, M. Garc´ ıa-Torres, A. Garo-\nfalo, P. Gavras, P. Giacobbe, G. Gilmore, S. Girona,\nG. Giuffrida, A. Gomez, I. Gonzalez-Santamaria, J. J.\nGonz´ alez-Vidal, M. Granvik, R. Guti´ errez-S´ anchez,\nL. P. Guy, M. Hauser, M. Haywood, A. Helmi, S. L. Hi-\ndalgo, N. H ladczuk, G. Holland, H. E. Huckle, G. Jas-\nniewicz, P. G. Jonker, J. Juaristi Campillo, F. Julbe,\nL. Karbevska, P. Kervella, S. Khanna, A. Kochoska,\nG. Kordopatis, A. J. Korn, Z. Kostrzewa-Rutkowska,\nK. Kruszy´ nska, S. Lambert, A. F. Lanza, Y. Lasne,\nJ. F. Le Campion, Y. Le Fustec, Y. Lebreton, T. Lebzel-\nter, S. Leccia, N. Leclerc, I. Lecoeur-Taibi, S. Liao,\nE. Licata, H. E. P. Lindstrøm, T. A. Lister, E. Li-\nvanou, A. Lobel, P. Madrero Pardo, S. Managau, R. G.\nMann, J. M. Marchant, M. Marconi, M. M. S. Mar-\ncos Santos, S. Marinoni, F. Marocco, D. J. Marshall,\nL. Martin Polo, J. M. Mart´ ın-Fleitas, A. Masip, D. Mas-\nsari, A. Mastrobuono-Battisti, T. Mazeh, S. Messina,\nD. Michalik, N. R. Millar, A. Mints, D. Molina, R. Moli-\nnaro, L. Moln´ ar, P. Montegriffo, R. Mor, R. Mor-\nbidelli, T. Morel, D. Morris, A. F. Mulone, D. Munoz,\nT. Muraveva, C. P. Murphy, I. Musella, L. Noval,\nC. Ord´ enovic, G. Orr` u, J. Osinde, C. Pagani, I. Pagano,\nL. Palaversa, P. A. Palicio, A. Panahi, M. Pawlak,\nX. Pe˜ nalosa Esteller, A. Penttil¨ a, A. M. Piersimoni,\nF. X. Pineau, E. Plachy, G. Plum, E. Poggio, E. Poretti,\nE. Poujoulet, A. Prˇ sa, L. Pulone, E. Racero, S. Ra-\ngaini, M. Rainer, C. M. Raiteri, N. Rambaux, P. Ramos,\nP. Re Fiorentin, S. Regibo, C. Reyl´ e, V. Ripepi,\nA. Riva, G. Rixon, N. Robichon, C. Robin, M. Roe-\nlens, L. Rohrbasser, M. Romero-G´ omez, N. Rowell,\nF. Royer, K. A. Rybicki, G. Sadowski, A. Sagrist` a\nSell´ es, J. Sahlmann, J. Salgado, E. Salguero, N. Sama-\nras, V. Sanchez Gimenez, N. Sanna, R. Santove˜ na,\nM. Sarasso, M. Schultheis, E. Sciacca, M. Segol, J. C.\nSegovia, D. S´ egransan, D. Semeux, H. I. Siddiqui,\nA. Siebert, L. Siltala, E. Slezak, R. L. Smart, E. Solano,F. Solitro, D. Souami, J. Souchay, A. Spagna, F. Spoto,\nI. A. Steele, M. S¨ uveges, L. Szabados, E. Szegedi-\nElek, F. Taris, G. Tauran, M. B. Taylor, R. Teix-\neira, W. Thuillot, N. Tonello, F. Torra, J. Torra,\nC. Turon, N. Unger, M. Vaillant, E. van Dillen,\nO. Vanel, A. Vecchiato, Y. Viala, D. Vicente, S. Voutsi-\nnas, M. Weiler, T. Wevers, L. Wyrzykowski, A. Yoldas,\nP. Yvard, H. Zhao, J. Zorec, S. Zucker, C. Zurbach,\nand T. Zwitter, Astron. Astrophys. 649, A9 (2021),\narXiv:2012.02036 [astro-ph.GA].\n[62] O. Titov and S. Lambert, Astron. Astrophys. 559, A95\n(2013), arXiv:1310.2723 [astro-ph.IM].\n[63] C. de la Fuente Marcos and R. de la Fuente Mar-\ncos, Mon. Not. R. Astron. Soc. 462, 1972 (2016),\narXiv:1607.05633 [astro-ph.EP].\n[64] M. J. Holman and M. J. Payne, Astron. J. 152, 94\n(2016), arXiv:1604.03180 [astro-ph.EP].\n[65] K. Batygin, F. C. Adams, M. E. Brown, and J. C.\nBecker, Phys. Rep. 805, 1 (2019), arXiv:1902.10103\n[astro-ph.EP].\n[66] N. L. Zakamska and S. Tremaine, Astron. J. 130, 1939\n(2005), arXiv:astro-ph/0506548 [astro-ph].\n[67] J. Bovy, Astrophys. J. Suppl. Ser. 216, 29 (2015),\narXiv:1412.3451 [astro-ph.GA].\n[68]ˇZ. Ivezi´ c, A. J. Connolly, J. T. VanderPlas, and A. Gray,\nStatistics, Data Mining, and Machine Learning in As-\ntronomy: A Practical Python Guide for the Analysis of\nSurvey Data (2014).\n[69] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber-\nland, T. Reddy, D. Cournapeau, E. Burovski, P. Pe-\nterson, W. Weckesser, J. Bright, S. J. van der Walt,\nM. Brett, J. Wilson, K. J. Millman, N. Mayorov,\nA. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J.\nCarey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas,\nD. Laxalde, J. Perktold, R. Cimrman, I. Henriksen,\nE. A. Quintero, C. R. Harris, A. M. Archibald, A. H.\nRibeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0\nContributors, Nature Methods 17, 261 (2020).\n[70] R. E. Kass and A. E. Raftery, Journal of the\nAmerican Statistical Association 90, 773 (1995),\nhttps://www.tandfonline.com/doi/pdf/10.1080/01621459.1995.10476572.\n[71] M. Miyamoto and R. Nagai, Publ. Astron. Soc. Jap. 27,\n533 (1975).\n[72] S. Sharma, J. Bland-Hawthorn, J. Binney, K. C. Free-\nman, M. Steinmetz, C. Boeche, O. Bienaym´ e, B. K.\nGibson, G. F. Gilmore, E. K. Grebel, A. Helmi, G. Ko-\nrdopatis, U. Munari, J. F. Navarro, Q. A. Parker, W. A.\nReid, G. M. Seabroke, A. Siebert, F. Watson, M. E. K.\nWilliams, R. F. G. Wyse, and T. Zwitter, Astrophys. J.\n793, 51 (2014), arXiv:1405.7435 [astro-ph.GA].\n[73] D. R. Law and S. R. Majewski, Astrophys. J. 714, 229\n(2010), arXiv:1003.1132 [astro-ph.GA].\n[74] D. J. Nice and J. H. Taylor, Astrophys. J. 441, 429\n(1995).\n[75] K. Lazaridis, N. Wex, A. Jessner, M. Kramer, B. W.\nStappers, G. H. Janssen, G. Desvignes, M. B. Purver,\nI. Cognard, G. Theureau, A. G. Lyne, C. A. Jordan,\nand J. A. Zensus, Mon. Not. R. Astron. Soc. 400, 805\n(2009), arXiv:0908.0285 [astro-ph.GA].\n[76] J. Bovy, Mon. Not. R. Astron. Soc. 468, L63 (2017),\narXiv:1610.07610 [astro-ph.GA].\n[77] L. Hernquist, Astrophys. J. 356, 359 (1990).\n[78] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro-\nphys. J. 490, 493 (1997), arXiv:astro-ph/9611107 [astro-22\nph].\n[79] L. L. Watkins, N. W. Evans, and J. H. An, Mon. Not.\nR. Astron. Soc. 406, 264 (2010), arXiv:1002.4565 [astro-\nph.GA].\n[80] H. J. Newberg, B. A. Willett, B. Yanny, and Y. Xu,\nAstrophys. J. 711, 32 (2010), arXiv:1001.0576 [astro-\nph.GA].\n[81] P. Bhattacharjee, S. Chaudhury, and S. Kundu, Astro-\nphys. J. 785, 63 (2014), arXiv:1310.2659 [astro-ph.GA].\n[82] T. K. Fritz, G. Battaglia, M. S. Pawlowski, N. Kalli-\nvayalil, R. van der Marel, S. T. Sohn, C. Brook,\nand G. Besla, Astron. Astrophys. 619, A103 (2018),\narXiv:1805.00908 [astro-ph.GA].\n[83] A.-C. Eilers, D. W. Hogg, H.-W. Rix, and M. K. Ness,\nAstrophys. J. 871, 120 (2019), arXiv:1810.09466 [astro-\nph.GA].\n[84] S. Gardner, S. D. McDermott, and B. Yanny, Progress\nin Particle and Nuclear Physics 121, 103904 (2021),\narXiv:2106.13284 [astro-ph.GA].\n[85] P. A. Craig, S. Chakrabarti, S. Baum, and B. T. Lewis,\nMon. Not. R. Astron. Soc. 517, 1737 (2022).\n[86] J. Bovy and H.-W. Rix, Astrophys. J. 779, 115 (2013),\narXiv:1309.0809 [astro-ph.GA].\n[87] T. C. Licquia and J. A. Newman, Astrophys. J. 806, 96\n(2015), arXiv:1407.1078 [astro-ph.GA].\n[88] J. Binney and R. Sch¨ onrich, Mon. Not. R. Astron. Soc.\n481, 1501 (2018), arXiv:1807.09819 [astro-ph.GA].\n[89] P. J. McMillan, J. Petersson, T. Tepper-Garcia,\nJ. Bland-Hawthorn, T. Antoja, L. Chemin, F. Figueras,\nS. Khanna, G. Kordopatis, P. Ramos, M. Romero-\nG´ omez, and G. Seabroke, Mon. Not. R. Astron. Soc.\n516, 4988 (2022), arXiv:2206.04059 [astro-ph.GA].\n[90] B. Garc´ ıa-Conde, T. Antoja, S. Roca-F` abrega,\nF. G´ omez, P. Ramos, N. Garavito-Camargo, and\nM. G´ omez-Flechoso, arXiv e-prints , arXiv:2311.07137\n(2023), arXiv:2311.07137 [astro-ph.GA].\n[91] D. Reed, F. Governato, T. Quinn, J. Gardner, J. Stadel,\nand G. Lake, Mon. Not. R. Astron. Soc. 359, 1537\n(2005), arXiv:astro-ph/0406034 [astro-ph].\n[92] K. T. E. Chua, A. Pillepich, V. Rodriguez-Gomez,\nM. Vogelsberger, S. Bird, and L. Hernquist, Mon. Not.\nR. Astron. Soc. 472, 4343 (2017), arXiv:1611.07991\n[astro-ph.GA].\n[93] S. B. Green and F. C. van den Bosch, Mon. Not. R.\nAstron. Soc. 490, 2091 (2019), arXiv:1908.08537 [astro-\nph.GA].\n[94] J. H. Oort, Bull. Astron. Inst. Neth. 3, 275 (1927).\n[95] C. Li, G. Zhao, and C. Yang, Astrophys. J. 872, 205\n(2019).\n[96] Gaia Collaboration, O. L. Creevey, L. M. Sarro,\nA. Lobel, E. Pancino, R. Andrae, R. L. Smart,\nG. Clementini, U. Heiter, A. J. Korn, M. Fouesneau,\nY. Fr´ emat, F. De Angeli, A. Vallenari, D. L. Har-\nrison, F. Th´ evenin, C. Reyl´ e, R. Sordo, A. Garo-\nfalo, A. G. A. Brown, L. Eyer, T. Prusti, J. H. J.\nde Bruijne, F. Arenou, C. Babusiaux, M. Biermann,\nC. Ducourant, D. W. Evans, R. Guerra, A. Hut-\nton, C. Jordi, S. A. Klioner, U. L. Lammers, L. Lin-\ndegren, X. Luri, F. Mignard, C. Panem, D. Pour-\nbaix, S. Randich, P. Sartoretti, C. Soubiran, P. Tanga,\nN. A. Walton, C. A. L. Bailer-Jones, U. Bastian,\nR. Drimmel, F. Jansen, D. Katz, M. G. Lattanzi,\nF. van Leeuwen, J. Bakker, C. Cacciari, J. Casta˜ neda,\nC. Fabricius, L. Galluccio, A. Guerrier, E. Masana,R. Messineo, N. Mowlavi, C. Nicolas, K. Nienartow-\nicz, F. Pailler, P. Panuzzo, F. Riclet, W. Roux, G. M.\nSeabroke, G. Gracia-Abril, J. Portell, D. Teyssier,\nM. Altmann, M. Audard, I. Bellas-Velidis, K. Ben-\nson, J. Berthier, R. Blomme, P. W. Burgess, D. Bu-\nsonero, G. Busso, H. C´ anovas, B. Carry, A. Cellino,\nN. Cheek, Y. Damerdji, M. Davidson, P. de Teodoro,\nM. Nu˜ nez Campos, L. Delchambre, A. Dell’Oro, P. Es-\nquej, J. Fern´ andez-Hern´ andez, E. Fraile, D. Garabato,\nP. Garc´ ıa-Lario, E. Gosset, R. Haigron, J. L. Halb-\nwachs, N. C. Hambly, J. Hern´ andez, D. Hestroffer,\nS. T. Hodgkin, B. Holl, K. Janßen, G. Jevardat de\nFombelle, S. Jordan, A. Krone-Martins, A. C. Lan-\nzafame, W. L¨ offler, O. Marchal, P. M. Marrese, A. Moit-\ninho, K. Muinonen, P. Osborne, T. Pauwels, A. Recio-\nBlanco, M. Riello, L. Rimoldini, T. Roegiers, J. Ry-\nbizki, C. Siopis, M. Smith, A. Sozzetti, E. Utrilla,\nM. van Leeuwen, U. Abbas, P. ´Abrah´ am, A. Abreu\nAramburu, C. Aerts, J. J. Aguado, M. Ajaj, F. Aldea-\nMontero, G. Altavilla, M. A. ´Alvarez, J. Alves, F. An-\nders, R. I. Anderson, E. Anglada Varela, T. Antoja,\nD. Baines, S. G. Baker, L. Balaguer-N´ u˜ nez, E. Bal-\nbinot, Z. Balog, C. Barache, D. Barbato, M. Bar-\nros, M. A. Barstow, S. Bartolom´ e, J. L. Bassilana,\nN. Bauchet, U. Becciani, M. Bellazzini, A. Berihuete,\nM. Bernet, S. Bertone, L. Bianchi, A. Binnenfeld,\nS. Blanco-Cuaresma, T. Boch, A. Bombrun, D. Bossini,\nS. Bouquillon, A. Bragaglia, L. Bramante, E. Breedt,\nA. Bressan, N. Brouillet, E. Brugaletta, B. Buccia-\nrelli, A. Burlacu, A. G. Butkevich, R. Buzzi, E. Caffau,\nR. Cancelliere, T. Cantat-Gaudin, R. Carballo, T. Car-\nlucci, M. I. Carnerero, J. M. Carrasco, L. Casamiquela,\nM. Castellani, A. Castro-Ginard, L. Chaoul, P. Charlot,\nL. Chemin, V. Chiaramida, A. Chiavassa, N. Chornay,\nG. Comoretto, G. Contursi, W. J. Cooper, T. Cornez,\nS. Cowell, F. Crifo, M. Cropper, M. Crosta, C. Crow-\nley, C. Dafonte, A. Dapergolas, P. David, P. de Lav-\nerny, F. De Luise, R. De March, J. De Ridder, R. de\nSouza, A. de Torres, E. F. del Peloso, E. del Pozo,\nM. Delbo, A. Delgado, J. B. Delisle, C. Demouchy, T. E.\nDharmawardena, P. Di Matteo, S. Diakite, C. Diener,\nE. Distefano, C. Dolding, H. Enke, C. Fabre, M. Fab-\nrizio, S. Faigler, G. Fedorets, P. Fernique, F. Figueras,\nY. Fournier, C. Fouron, F. Fragkoudi, M. Gai,\nA. Garcia-Gutierrez, M. Garcia-Reinaldos, M. Garc´ ıa-\nTorres, A. Gavel, P. Gavras, E. Gerlach, R. Geyer,\nP. Giacobbe, G. Gilmore, S. Girona, G. Giuffrida,\nR. Gomel, A. Gomez, J. Gonz´ alez-N´ u˜ nez, I. Gonz´ alez-\nSantamar´ ıa, J. J. Gonz´ alez-Vidal, M. Granvik, P. Guill-\nout, J. Guiraud, R. Guti´ errez-S´ anchez, L. P. Guy,\nD. Hatzidimitriou, M. Hauser, M. Haywood, A. Helmer,\nA. Helmi, T. Hilger, M. H. Sarmiento, S. L. Hidalgo,\nN. H ladczuk, D. Hobbs, G. Holland, H. E. Huckle,\nK. Jardine, G. Jasniewicz, A. Jean-Antoine Piccolo,\n´O. Jim´ enez-Arranz, J. Juaristi Campillo, F. Julbe,\nL. Karbevska, P. Kervella, S. Khanna, G. Kordopatis,\n´A. K´ osp´ al, Z. Kostrzewa-Rutkowska, K. Kruszy´ nska,\nM. Kun, P. Laizeau, S. Lambert, A. F. Lanza, Y. Lasne,\nJ. F. Le Campion, Y. Lebreton, T. Lebzelter, S. Lec-\ncia, N. Leclerc, I. Lecoeur-Taibi, S. Liao, E. L. Li-\ncata, H. E. P. Lindstrøm, T. A. Lister, E. Livanou,\nA. Lorca, C. Loup, P. Madrero Pardo, A. Magdaleno\nRomeo, S. Managau, R. G. Mann, M. Manteiga, J. M.\nMarchant, M. Marconi, J. Marcos, M. M. S. Marcos23\nSantos, D. Mar´ ın Pina, S. Marinoni, F. Marocco, D. J.\nMarshall, L. Martin Polo, J. M. Mart´ ın-Fleitas, G. Mar-\nton, N. Mary, A. Masip, D. Massari, A. Mastrobuono-\nBattisti, T. Mazeh, P. J. McMillan, S. Messina,\nD. Michalik, N. R. Millar, A. Mints, D. Molina, R. Moli-\nnaro, L. Moln´ ar, G. Monari, M. Mongui´ o, P. Mon-\ntegriffo, A. Montero, R. Mor, A. Mora, R. Mor-\nbidelli, T. Morel, D. Morris, T. Muraveva, C. P.\nMurphy, I. Musella, Z. Nagy, L. Noval, F. Oca˜ na,\nA. Ogden, C. Ordenovic, J. O. Osinde, C. Pagani,\nI. Pagano, L. Palaversa, P. A. Palicio, L. Pallas-\nQuintela, A. Panahi, S. Payne-Wardenaar, X. Pe˜ nalosa\nEsteller, A. Penttil¨ a, B. Pichon, A. M. Piersimoni, F. X.\nPineau, E. Plachy, G. Plum, E. Poggio, A. Prˇ sa, L. Pu-\nlone, E. Racero, S. Ragaini, M. Rainer, C. M. Raiteri,\nP. Ramos, M. Ramos-Lerate, P. Re Fiorentin, S. Reg-\nibo, P. J. Richards, C. Rios Diaz, V. Ripepi, A. Riva,\nH. W. Rix, G. Rixon, N. Robichon, A. C. Robin,\nC. Robin, M. Roelens, H. R. O. Rogues, L. Rohrbasser,\nM. Romero-G´ omez, N. Rowell, F. Royer, D. Ruz\nMieres, K. A. Rybicki, G. Sadowski, A. S´ aez N´ u˜ nez,\nA. Sagrist` a Sell´ es, J. Sahlmann, E. Salguero, N. Sama-\nras, V. Sanchez Gimenez, N. Sanna, R. Santove˜ na,\nM. Sarasso, M. Schultheis, E. Sciacca, M. Segol, J. C.\nSegovia, D. S´ egransan, D. Semeux, S. Shahaf, H. I.\nSiddiqui, A. Siebert, L. Siltala, A. Silvelo, E. Slezak,\nI. Slezak, O. N. Snaith, E. Solano, F. Solitro, D. Souami,\nJ. Souchay, A. Spagna, L. Spina, F. Spoto, I. A. Steele,\nH. Steidelm¨ uller, C. A. Stephenson, M. S¨ uveges, J. Sur-\ndej, L. Szabados, E. Szegedi-Elek, F. Taris, M. B.\nTaylor, R. Teixeira, L. Tolomei, N. Tonello, F. Torra,\nJ. Torra, G. Torralba Elipe, M. Trabucchi, A. T. Tsou-\nnis, C. Turon, A. Ulla, N. Unger, M. V. Vaillant,\nE. van Dillen, W. van Reeven, O. Vanel, A. Vecchiato,\nY. Viala, D. Vicente, S. Voutsinas, M. Weiler, T. Wev-\ners, L. Wyrzykowski, A. Yoldas, P. Yvard, H. Zhao,\nJ. Zorec, S. Zucker, and T. Zwitter, Astron. Astrophys.\n674, A39 (2023), arXiv:2206.05870 [astro-ph.SR].\n[97] Gaia Collaboration, D. Katz, T. Antoja, M. Romero-\nG´ omez, R. Drimmel, C. Reyl´ e, G. M. Seabroke,\nC. Soubiran, C. Babusiaux, P. Di Matteo, F. Figueras,\nE. Poggio, A. C. Robin, D. W. Evans, A. G. A.\nBrown, A. Vallenari, T. Prusti, J. H. J. de Brui-\njne, C. A. L. Bailer-Jones, M. Biermann, L. Eyer,\nF. Jansen, C. Jordi, S. A. Klioner, U. Lammers, L. Lin-\ndegren, X. Luri, F. Mignard, C. Panem, D. Pour-\nbaix, S. Randich, P. Sartoretti, H. I. Siddiqui, F. van\nLeeuwen, N. A. Walton, F. Arenou, U. Bastian,\nM. Cropper, M. G. Lattanzi, J. Bakker, C. Cac-\nciari, J. Casta n, L. Chaoul, N. Cheek, F. De An-\ngeli, C. Fabricius, R. Guerra, B. Holl, E. Masana,\nR. Messineo, N. Mowlavi, K. Nienartowicz, P. Panuzzo,\nJ. Portell, M. Riello, P. Tanga, F. Th´ evenin, G. Gracia-\nAbril, G. Comoretto, M. Garcia-Reinaldos, D. Teyssier,\nM. Altmann, R. Andrae, M. Audard, I. Bellas-Velidis,\nK. Benson, J. Berthier, R. Blomme, P. Burgess,\nG. Busso, B. Carry, A. Cellino, G. Clementini,\nM. Clotet, O. Creevey, M. Davidson, J. De Rid-\nder, L. Delchambre, A. Dell’Oro, C. Ducourant,\nJ. Fern´ andez-Hern´ andez, M. Fouesneau, Y. Fr´ emat,\nL. Galluccio, M. Garc´ ıa-Torres, J. Gonz´ alez-N´ u˜ nez, J. J.\nGonz´ alez-Vidal, E. Gosset, L. P. Guy, J. L. Halb-\nwachs, N. C. Hambly, D. L. Harrison, J. Hern´ andez,\nD. Hestroffer, S. T. Hodgkin, A. Hutton, G. Jas-niewicz, A. Jean-Antoine-Piccolo, S. Jordan, A. J.\nKorn, A. Krone-Martins, A. C. Lanzafame, T. Lebzel-\nter, W. L¨ offler, M. Manteiga, P. M. Marrese, J. M.\nMart´ ın-Fleitas, A. Moitinho, A. Mora, K. Muinonen,\nJ. Osinde, E. Pancino, T. Pauwels, J. M. Petit, A. Recio-\nBlanco, P. J. Richards, L. Rimoldini, L. M. Sarro,\nC. Siopis, M. Smith, A. Sozzetti, M. S¨ uveges, J. Torra,\nW. van Reeven, U. Abbas, A. Abreu Aramburu, S. Ac-\ncart, C. Aerts, G. Altavilla, M. A. ´Alvarez, R. Alvarez,\nJ. Alves, R. I. Anderson, A. H. Andrei, E. Anglada\nVarela, E. Antiche, B. Arcay, T. L. Astraatmadja,\nN. Bach, S. G. Baker, L. Balaguer-N´ u˜ nez, P. Balm,\nC. Barache, C. Barata, D. Barbato, F. Barblan, P. S.\nBarklem, D. Barrado, M. Barros, M. A. Barstow,\nL. Bartholom´ e Mu˜ noz, J. L. Bassilana, U. Becciani,\nM. Bellazzini, A. Berihuete, S. Bertone, L. Bianchi,\nO. Bienaym´ e, S. Blanco-Cuaresma, T. Boch, C. Boeche,\nA. Bombrun, R. Borrachero, D. Bossini, S. Bouquil-\nlon, G. Bourda, A. Bragaglia, L. Bramante, M. A.\nBreddels, A. Bressan, N. Brouillet, T. Br¨ usemeister,\nE. Brugaletta, B. Bucciarelli, A. Burlacu, D. Busonero,\nA. G. Butkevich, R. Buzzi, E. Caffau, R. Cancelliere,\nG. Cannizzaro, T. Cantat-Gaudin, R. Carballo, T. Car-\nlucci, J. M. Carrasco, L. Casamiquela, M. Castellani,\nA. Castro-Ginard, P. Charlot, L. Chemin, A. Chiavassa,\nG. Cocozza, G. Costigan, S. Cowell, F. Crifo, M. Crosta,\nC. Crowley, J. Cuypers, C. Dafonte, Y. Damerdji,\nA. Dapergolas, P. David, M. David, P. de Laverny,\nF. De Luise, R. De March, R. de Souza, A. de Tor-\nres, J. Debosscher, E. del Pozo, M. Delbo, A. Del-\ngado, H. E. Delgado, S. Diakite, C. Diener, E. Dis-\ntefano, C. Dolding, P. Drazinos, J. Dur´ an, B. Ed-\nvardsson, H. Enke, K. Eriksson, P. Esquej, G. Ey-\nnard Bontemps, C. Fabre, M. Fabrizio, S. Faigler,\nA. J. Falc a, M. Farr` as Casas, L. Federici, G. Fe-\ndorets, P. Fernique, F. Filippi, K. Findeisen, A. Fonti,\nE. Fraile, M. Fraser, B. Fr´ ezouls, M. Gai, S. Galleti,\nD. Garabato, F. Garc´ ıa-Sedano, A. Garofalo, N. Gar-\nralda, A. Gavel, P. Gavras, J. Gerssen, R. Geyer,\nP. Giacobbe, G. Gilmore, S. Girona, G. Giuffrida,\nF. Glass, M. Gomes, M. Granvik, A. Gueguen, A. Guer-\nrier, J. Guiraud, R. Guti´ e, R. Haigron, D. Hatzidim-\nitriou, M. Hauser, M. Haywood, U. Heiter, A. Helmi,\nJ. Heu, T. Hilger, D. Hobbs, W. Hofmann, G. Hol-\nland, H. E. Huckle, A. Hypki, V. Icardi, K. Janßen,\nG. Jevardat de Fombelle, P. G. Jonker, ´A. L. Juh´ asz,\nF. Julbe, A. Karampelas, A. Kewley, J. Klar, A. Ko-\nchoska, R. Kohley, K. Kolenberg, M. Kontizas, E. Kon-\ntizas, S. E. Koposov, G. Kordopatis, Z. Kostrzewa-\nRutkowska, P. Koubsky, S. Lambert, A. F. Lanza,\nY. Lasne, J. B. Lavigne, Y. Le Fustec, C. Le Poncin-\nLafitte, Y. Lebreton, S. Leccia, N. Leclerc, I. Lecoeur-\nTaibi, H. Lenhardt, F. Leroux, S. Liao, E. Licata,\nH. E. P. Lindstrøm, T. A. Lister, E. Livanou, A. Lo-\nbel, M. L´ opez, S. Managau, R. G. Mann, G. Mantelet,\nO. Marchal, J. M. Marchant, M. Marconi, S. Marinoni,\nG. Marschalk´ o, D. J. Marshall, M. Martino, G. Mar-\nton, N. Mary, D. Massari, G. Matijeviˇ c, T. Mazeh,\nP. J. McMillan, S. Messina, D. Michalik, N. R. Mil-\nlar, D. Molina, R. Molinaro, L. Moln´ ar, P. Montegriffo,\nR. Mor, R. Morbidelli, T. Morel, D. Morris, A. F. Mu-\nlone, T. Muraveva, I. Musella, G. Nelemans, L. Nicastro,\nL. Noval, W. O’Mullane, C. Ord´ enovic, D. Ord´ o˜ nez-\nBlanco, P. Osborne, C. Pagani, I. Pagano, F. Pailler,24\nH. Palacin, L. Palaversa, A. Panahi, M. Pawlak, A. M.\nPiersimoni, F. X. Pineau, E. Plachy, G. Plum, E. Pou-\njoulet, A. Prˇ sa, L. Pulone, E. Racero, S. Ragaini,\nN. Rambaux, M. Ramos-Lerate, S. Regibo, F. Riclet,\nV. Ripepi, A. Riva, A. Rivard, G. Rixon, T. Roegiers,\nM. Roelens, N. Rowell, F. Royer, L. Ruiz-Dern, G. Sad-\nowski, T. Sagrist` a Sell´ es, J. Sahlmann, J. Salgado,\nE. Salguero, N. Sanna, T. Santana-Ros, M. Sarasso,\nH. Savietto, M. Schultheis, E. Sciacca, M. Segol,\nJ. C. Segovia, D. S´ egransan, I. C. Shih, L. Siltala,\nA. F. Silva, R. L. Smart, K. W. Smith, E. Solano,\nF. Solitro, R. Sordo, S. Soria Nieto, J. Souchay,\nA. Spagna, F. Spoto, U. Stampa, I. A. Steele, H. Stei-\ndelm¨ uller, C. A. Stephenson, H. Stoev, F. F. Suess,\nJ. Surdej, L. Szabados, E. Szegedi-Elek, D. Tapi-\nador, F. Taris, G. Tauran, M. B. Taylor, R. Teixeira,\nD. Terrett, P. Teyssandier, W. Thuillot, A. Titarenko,\nF. Torra Clotet, C. Turon, A. Ulla, E. Utrilla, S. Uzzi,\nM. Vaillant, G. Valentini, V. Valette, A. van Elteren,\nE. Van Hemelryck, M. van Leeuwen, M. Vaschetto,\nA. Vecchiato, J. Veljanoski, Y. Viala, D. Vicente,\nS. Vogt, C. von Essen, H. Voss, V. Votruba, S. Voutsi-\nnas, G. Walmsley, M. Weiler, O. Wertz, T. Wevers,\n L. Wyrzykowski, A. Yoldas, M. ˇZerjal, H. Ziaeep-\nour, J. Zorec, S. Zschocke, S. Zucker, C. Zurbach,\nand T. Zwitter, Astron. Astrophys. 616, A11 (2018),\narXiv:1804.09380 [astro-ph.GA].\n[98] S. P˜ oder, M. Benito, J. Pata, R. Kipper, H. Ram-\nler, G. H¨ utsi, I. Kolka, and G. F. Thomas, Astron.\nAstrophys. 676, A134 (2023), arXiv:2309.02895 [astro-\nph.GA].\n[99] C. F. McKee, A. Parravano, and D. J. Hollenbach,\nAstrophys. J. 814, 13 (2015), arXiv:1509.05334 [astro-\nph.GA].\n[100] R. Guo, C. Liu, S. Mao, X.-X. Xue, R. J. Long, and\nL. Zhang, Mon. Not. R. Astron. Soc. 495, 4828 (2020),\narXiv:2005.12018 [astro-ph.GA].\n[101] O. Bienaym´ e, B. Famaey, A. Siebert, K. C. Free-\nman, B. K. Gibson, G. Gilmore, E. K. Grebel,\nJ. Bland-Hawthorn, G. Kordopatis, U. Munari, J. F.\nNavarro, Q. Parker, W. Reid, G. M. Seabroke,\nA. Siviero, M. Steinmetz, F. Watson, R. F. G. Wyse,\nand T. Zwitter, Astron. Astrophys. 571, A92 (2014),\narXiv:1406.6896 [astro-ph.GA].\n[102] J. Buch, J. S. C. Leung, and J. Fan, J. Cosm.\nAstropart. Phys. 2019 , 026 (2019), arXiv:1808.05603\n[astro-ph.GA].\n[103] J.-B. Salomon, O. Bienaym´ e, C. Reyl´ e, A. C. Robin,\nand B. Famaey, Astron. Astrophys. 643, A75 (2020),\narXiv:2009.04495 [astro-ph.GA].\n[104] A. Widmark, Astron. Astrophys. 623, A30 (2019),\narXiv:1811.07911 [astro-ph.GA].\n[105] S. H. Lim, E. Putney, M. R. Buckley, and D. Shih, arXiv\ne-prints , arXiv:2305.13358 (2023), arXiv:2305.13358\n[astro-ph.GA].\n[106] C. E. Rasmussen and C. K. I. Williams, Gaussian Pro-\ncesses for Machine Learning (The MIT Press, 2005).\n[107] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom-\nmers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay-\nlor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer,\nM. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R´ ıo,\nM. Wiebe, P. Peterson, P. G´ erard-Marchant, K. Shep-\npard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke,\nand T. E. Oliphant, Nature 585, 357 (2020).[108] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel,\nB. Thirion, O. Grisel, M. Blondel, P. Prettenhofer,\nR. Weiss, V. Dubourg, J. Vanderplas, A. Passos,\nD. Cournapeau, M. Brucher, M. Perrot, and ´Edouard\nDuchesnay, Journal of Machine Learning Research 12,\n2825 (2011).\n[109] Astropy Collaboration, A. M. Price-Whelan, P. L. Lim,\nN. Earl, N. Starkman, L. Bradley, D. L. Shupe, A. A.\nPatil, L. Corrales, C. E. Brasseur, M. N¨ othe, A. Donath,\nE. Tollerud, B. M. Morris, A. Ginsburg, E. Vaher, B. A.\nWeaver, J. Tocknell, W. Jamieson, M. H. van Kerkwijk,\nT. P. Robitaille, B. Merry, M. Bachetti, H. M. G¨ unther,\nT. L. Aldcroft, J. A. Alvarado-Montes, A. M. Archibald,\nA. B´ odi, S. Bapat, G. Barentsen, J. Baz´ an, M. Biswas,\nM. Boquien, D. J. Burke, D. Cara, M. Cara, K. E.\nConroy, S. Conseil, M. W. Craig, R. M. Cross, K. L.\nCruz, F. D’Eugenio, N. Dencheva, H. A. R. Devillepoix,\nJ. P. Dietrich, A. D. Eigenbrot, T. Erben, L. Ferreira,\nD. Foreman-Mackey, R. Fox, N. Freij, S. Garg, R. Geda,\nL. Glattly, Y. Gondhalekar, K. D. Gordon, D. Grant,\nP. Greenfield, A. M. Groener, S. Guest, S. Gurovich,\nR. Handberg, A. Hart, Z. Hatfield-Dodds, D. Homeier,\nG. Hosseinzadeh, T. Jenness, C. K. Jones, P. Joseph,\nJ. B. Kalmbach, E. Karamehmetoglu, M. Ka luszy´ nski,\nM. S. P. Kelley, N. Kern, W. E. Kerzendorf, E. W.\nKoch, S. Kulumani, A. Lee, C. Ly, Z. Ma, C. MacBride,\nJ. M. Maljaars, D. Muna, N. A. Murphy, H. Nor-\nman, R. O’Steen, K. A. Oman, C. Pacifici, S. Pascual,\nJ. Pascual-Granado, R. R. Patil, G. I. Perren, T. E.\nPickering, T. Rastogi, B. R. Roulston, D. F. Ryan, E. S.\nRykoff, J. Sabater, P. Sakurikar, J. Salgado, A. Sanghi,\nN. Saunders, V. Savchenko, L. Schwardt, M. Seifert-\nEckert, A. Y. Shih, A. S. Jain, G. Shukla, J. Sick,\nC. Simpson, S. Singanamalla, L. P. Singer, J. Sing-\nhal, M. Sinha, B. M. Sip˝ ocz, L. R. Spitler, D. Stansby,\nO. Streicher, J. ˇSumak, J. D. Swinbank, D. S. Taranu,\nN. Tewary, G. R. Tremblay, M. de Val-Borro, S. J. Van\nKooten, Z. Vasovi´ c, S. Verma, J. V. de Miranda Car-\ndoso, P. K. G. Williams, T. J. Wilson, B. Winkel, W. M.\nWood-Vasey, R. Xue, P. Yoachim, C. Zhang, A. Zonca,\nand Astropy Project Contributors, Astrophys. J. 935,\n167 (2022), arXiv:2206.14220 [astro-ph.IM].\n[110] J. D. Hunter, Computing in Science & Engineering 9,\n90 (2007).\n[111] D. Foreman-Mackey, D. W. Hogg, D. Lang,\nand J. Goodman, Pub. ASP 125, 306 (2013),\narXiv:1202.3665 [astro-ph.IM].\n[112] V. V. Bobylev and A. T. Bajkova, Mon. Not. R. Astron.\nSoc.408, 1788 (2010), arXiv:1006.5152 [astro-ph.GA].\n[113] J. Branham, Richard L., Mon. Not. R. Astron. Soc. 409,\n1269 (2010).\n[114] M. Shen and H. Zhang, Chin. Astron. Astrophys. 34,\n89 (2010).\n[115] V. V. Bobylev and A. T. Bajkova, Astronomy Letters\n37, 526 (2011), arXiv:1106.5849 [astro-ph.GA].\n[116] J. Branham, R. L., Rev. Mex. Astron. Astrophys. 47,\n197 (2011).\n[117] A. Siebert, B. Famaey, I. Minchev, G. M. Seabroke,\nJ. Binney, B. Burnett, K. C. Freeman, M. Williams,\nO. Bienaym´ e, J. Bland-Hawthorn, R. Campbell, J. P.\nFulbright, B. K. Gibson, G. Gilmore, E. K. Grebel,\nA. Helmi, U. Munari, J. F. Navarro, Q. A. Parker, W. A.\nReid, A. Siviero, M. Steinmetz, F. Watson, R. F. G.\nWyse, and T. Zwitter, Mon. Not. R. Astron. Soc. 412,25\n2026 (2011), arXiv:1011.4092 [astro-ph.GA].\n[118] A. S. Stepanishchev and V. V. Bobylev, Astronomy Let-\nters37, 254 (2011).\n[119] A. S. Stepanishchev and V. V. Bobylev, Astronomy Let-\nters39, 185 (2013), arXiv:1302.1457 [astro-ph.GA].\n[120] R. L. Branham, Astrophys. Space Sci. 353, 179 (2014).\n[121] V. V. Vityazev and A. S. Tsvetkov, Mon. Not. R. As-\ntron. Soc. 442, 1249 (2014).\n[122] V. V. Bobylev and A. T. Bajkova, Astronomy Letters\n42, 90 (2016), arXiv:1512.08100 [astro-ph.GA].\n[123] V. V. Bobylev and A. T. Bajkova, Astronomy Letters\n43, 159 (2017), arXiv:1611.00794 [astro-ph.GA].\n[124] V. V. Vityazev, A. S. Tsvetkov, V. V. Bobylev,\nand A. T. Bajkova, Astrophysics 60, 462 (2017),\narXiv:1711.01476 [astro-ph.GA].\n[125] V. V. Bobylev and A. T. Bajkova, Astronomy Letters\n44, 184 (2018), arXiv:1801.07431 [astro-ph.GA].\n[126] J. J. Vickers and M. C. Smith, Astrophys. J. 860, 91\n(2018), arXiv:1805.02332 [astro-ph.GA].[127] V. V. Vityazev, A. V. Popov, A. S. Tsvetkov, S. D.\nPetrov, D. A. Trofimov, and V. I. Kiyaev, Astronomy\nLetters 44, 236 (2018).\n[128] O. I. Krisanova, V. V. Bobylev, and A. T. Bajkova,\nAstronomy Letters 46, 370 (2020), arXiv:2008.10981\n[astro-ph.GA].\n[129] M. I. Nouh and W. H. Elsanhoury, Astrophysics 63, 179\n(2020), arXiv:1905.07614 [astro-ph.SR].\n[130] F. Wang, H. W. Zhang, Y. Huang, B. Q. Chen, H. F.\nWang, and C. Wang, Mon. Not. R. Astron. Soc. 504,\n199 (2021), arXiv:2103.10232 [astro-ph.GA].\n[131] P. Nelson and L. M. Widrow, Mon. Not. R. Astron. Soc.\n516, 5429 (2022), arXiv:2206.04627 [astro-ph.GA].\n[132] S. Guo and Z. Qi, Universe 9, 252 (2023).\nAppendix A: Oort Constants Data\nIn Table V we provide the data and references used to\nmake Figure 6.26\nA B Reference Comments\n(km/s/kpc) (km/s/kpc)\n17.8±0.8 -13.2 ±1.5 [112] Masers\n14.85±7.47 -10.85 ±6.63 [113] Hipparcos F Giants\n17.42±1.17 -12.46 ±0.86 [114] Hipparcos Cepheids, 0.2 kpc < r < 3.0 kpc\n18.17±1.14 -11.91 ±0.85 Above, 0.5 kpc < r < 3.0 kpc\n17.9±0.5 -13.6 ±1.0 [115] OB III Stars\n14.05±3.28 -9.30 ±2.87 [116] Hipparcos G III Stars\n9.2±0.5 -17.4 ±0.5 [117] RAVE\n16.9±1.2 -13.5 ±1.4 [118] Masers\n16.7±0.6 -12.0 ±1.0 [119] Masers\n16.00±0.36 -14.17 ±0.28 [120] Hipparcos OB Stars\n14.2±0.2 -10.0 ±0.2 [121] UCAC4 11th Mag\n9.1±0.1 -10.9 ±0.1 Above, UCAC4 16th Mag\n12.1±0.7 -10.6 ±0.5 Above, PPMXL 11th Mag\n10.3±0.2 -12.1 ±0.2 Above, PPMXL 17th Mag\n12.2±0.4 -5.8 ±0.3 Above, XPM 11th Mag\n7.0±0.1 10.0 ±0.1 Above, XPM 17th Mag\n17.12±0.45 -11.60 ±0.77 [122] RAVE, r <250 pc\n16.53±0.52 -10.82 ±0.93 [123] Gaia DR1 OB Stars\n17.77±0.46 -13.76 ±0.71 Above, but different sample\n15.3±0.4 -11.9 ±0.4 [76] Gaia DR1, r <230 pc\n15.57±0.31 -10.72 ±0.5 [124] RAVE5 + TGAS\n15.07±0.25 -12.17 ±0.39 [125] TGAS + RAVE\n14.13 -6.7 [126] TGAS-RAVE-LAMOST, Young Stars\n18.8 -12.7 Above, Intermediate Age Stars\n17 -2.1 Above, Old Stars\n16.29±0.06 -11.9 ±0.05 [127] TGAS MS Stars, r <1.5 kpc\n13.32±0.09 -12.71 ±0.06 Above, TGAS RG Stars, r <1.5 kpc\n15.1±0.1 -13.4 ±0.1 [95] Gaia DR2, r <500 pc\n15.73±0.32 -12.67 ±0.34 [128] Gaia DR2\n15.6±1.6 -13.9 ±1.8 [129] SEGUE Halo RG Stars\n16.31±0.89 -11.99 ±0.79 [130] LAMOST A Stars, r <600 pc, |z|<100 pc\n16.2±0.2 -11.7 ±0.2 [131] Gaia DR2 “(p)”\n15.2±0.8 -12.4 ±0.9 Above, “(v)”\n15.6±1.6 -15.8 ±1.7 [132] GCNS, Gaia EDR3\nTABLE V. Selected measurements of the Oort constants AandBsince 2010." }, { "title": "2401.15809v2.Particle_dynamics_in_spherically_symmetric_electro_vacuum_instantons.pdf", "content": "PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC\nELECTRO-VACUUM INSTANTONS\nARTHUR GARNIER\nAbstract. In this paper, we study the geodesic motion in spherically symmetric electro-\nvacuum Euclidean solutions of the Einstein equation. There are two kinds of such solutions:\nthe Euclidean Reissner–Nordstr¨ om (ERN) metrics, and the Bertotti–Robinson-like (BR)\nmetrics, the latter having constant Kretschmann scalar.\nFirst, we derive the motion equations for the ERN spacetime and we generalize the\nresults of Battista–Esposito, showing that all orbits in as ERN spacetime are unbounded\nif and only if it has an event horizon. We also obtain the Weierstrass form of the polar\nradial motion, providing an efficient tool for numerical computations.\nWe then study the angular deflection of orbits in the Euclidean Schwarzschild spacetime\nwhich, in contrast to the Lorentzian background, can be either positive or negative. We\nobserve the presence of a null and a maximal deflection rings for particles with velocity at\ninfinity v >1 and we give approximate values for their size when v≳1.\nFor BR spacetimes, we obtain analytic solutions for the radial motion in proper length,\ninvolving (hyperbolic) trigonometric functions and we deduce that orbits either exponen-\ntially go to the singularity or are periodic.\nFinally, we apply the previous results and use algorithms related to Weierstrass’ elliptic\nfunctions to produce a Python code to plot orbits of the spacetimes ERN and BR, and\ndraw “shadows” of the first ones, as it was already done before for classical black holes.\nIntroduction and motivation\nInstantons (or pseudoparticles) were originally defined in [Bel+75] as solutions of the\n(classical) Yang–Mills field equations, which are non-singular on some section of a com-\nplexified spacetime. By analogy, a gravitational instanton was defined in [Haw77] to be a\nsolution of the classical Einstein field equation, which is positive-definite (i.e. Riemannian)\non some section of a complexified spacetime. Such metrics were first introduced in quantum\ngravity by Hartle and Hawking [HH76] in order to make some path integral converge, hence\ndefining the so-called Hartle–Hawking propagator . Quoting [GH79], after the development\nof instantons in Yang–Mills theory and because (super)gravity is a gauge theory, it seems\nreasonable to expect gravitational instantons to play a similar role in gravity as instantons\ndo in quantum field theory. For general discussions on gravitational instantons, see [Haw79;\nGib79; EGH80; Els84; Esp92].\nSince their introduction, gravitational instantons and their interactions with gauge instan-\ntons have been a subject of deep interest [Tek02; Nas04; MT09; OPY11]. More recently,\nthe existence and uniqueness of toric instantons have been established in [KL21]. Moreover,\npurely Euclidean instantons (i.e. the corresponding complex spacetime does not admit any\nLorentzian section) have been introduced in [CT11] and thoroughly described in [AA21].\nIt is worth mentioning that the gravitational instantons with positive cosmological constant\nwere fully described and classified in [Pag78].\nBesides quantum gravity, in the early geometric models of matter introduced in [AMS12],\nthe Fubini–Study metric on the projective plane CP2has been proposed as a (compact)\nmodel for the spacetime surrounding a neutron . Later in [AFS15], the authors rather pro-\npose the Euclidean Schwarzschild geometry as a model for the neutron. As explained in\nDate : April 3, 2024.\n2020 Mathematics Subject Classification. Primary 83C10, 83-04, 83C25; Secondary 83C22, 83C15, 83C20.\n1arXiv:2401.15809v2 [gr-qc] 2 Apr 2024ARTHUR GARNIER\n[Jan15], this has been generalized to other spin-1\n2-particles such as the proton and the elec-\ntron, for which the Taub–bolt and Taub–NUT instantons were respectively given as candi-\ndates. Moreover, interesting uniqueness results on Euclidean Schwarzschild and Taub–NUT\ninstantons were obtained in [MS99]. These proposals further motivate the investigation\nof gravitational instantons and, in particular, the study of the geodesic dynamics in such\nspaces.\nGeodesic motion in gravitational instantons has started more than thirty years ago, with\nthe pioneer work [And90], focusing on closed geodesics in compact instantons, with appli-\ncations in the determination of their injectivity radius. The general geodesic dynamics in\n(generalized) Taub–NUT instantons has been detailed in [Vis93]. More recently, the case\nof Kerr–Newman instantons is the topic of [LR18], while instantons of Eguchi–Hanson type\nwere studied in [YZ23]. Finally, we mention that the integrability of the conformal geodesic\nflow on spherically symmetric instantons motivates the work [DT22].\nIn the present paper, an instanton will designate a Riemannian solution of the Maxwell–\nEinstein equations on a 4-dimensional manifold1. We investigate the geodesic motion in\nsuch spaces, which we assume to be spherically symmetric . In particular, we will apply\nthe theory of Weierstrass elliptic functions to the radial motion, as it was already done in\nthe Lorentzian framework; see [GV12; CM22]. We also study the gravitational lensing of\ntrajectories, as done for photons and massive particles in Reissner–Nordstr¨ om (resp. Kerr–\nNewman) spacetimes in [PJ19] (resp. in [HL16]). Our methods can also be compared to the\nmore recent work [Vit+24], where tidal forces are also investigated. As explained below, one\nof the main aims of the present paper is to highlight some important dynamical differences\nbetween the Euclidean and the Lorentzian backgrounds.\nThe simplest example of a gravitational instanton is the Euclidean Schwarzschild metric\n[HH76], given in Schwarzchild coordinates ( τ, r, θ, ϕ ) onM:=R×]2M; +∞[×S2≃R2×S2\nby\n(ES) d s2=\u0012\n1−2M\nr\u0013\ndτ2+\u0012\n1−2M\nr\u0013−1\ndr2+r2(dθ2+ sin2θdϕ2),\n(M≥0 being the mass of the central body) and the link with the Lorentzian Schwarzschild\nmetric is given by setting the Euclidean time τ=it, with tbeing the (Lorentzian) coordinate\ntime. In other words, the Euclidean Schwarzschild metric is obtained from the Lorentzian\none by applying a Wick rotation . As for any metric, it is natural to study the geodesic\nmotion associated to it. Regarding the metric (ES) above, this question was addressed\nthoroughly in [BE22].\nAs suggested in [GH77, §II], one can also look for the Reissner–Nordstr¨ om analogue of\nthe Euclidean Schwarzschild solution; a metric that was used in [MM89, §4] and [MS09,\n§II.B], for instance. Assuming the central body has an electric charge Q∈R, the Euclidean\nReissner–Nordstr¨ om (ERN) metric is given by\n(ERN) d s2=\u0012\n1−2M\nr+Q2\nr2\u0013\ndτ2+\u0012\n1−2M\nr+Q2\nr2\u0013−1\ndr2+r2(dθ2+ sin2θdϕ2).\nIn this paper, we generalize the approach of [BE22] to this metric and study the motion of\na test particle in the ERN spacetime. Specifically, we prove that the method of [GV12, §3.1]\nstill applies to the ERN metric, hence obtaining analytic solutions for (non-purely radial)\ngeodesics in terms of Weierstrass’ elliptic functions. As we will see, one of the remarkable\nresults obtained in [BE22] extends to the ERN spacetime with horizon (i.e. such that\nQ2≤M2), namely the fact that the energy of an exterior geodesic is confined in the open\ninterval ] −1,1[. In particular, no elliptic-like geodesics exist in the sub-extremal case. This\nfails in the super-charged case Q2> M2, where arbitrary high energy is allowed and attained\nby a circular geodesic. As mentioned in [BE22], these facts show that these Riemannian\n1In contrast to [Haw77], we do not assume that the curvature vanishes at large distances.\n2PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nsolutions present substantial differences in their dynamics, when compared to their usual\nrelativistic avatars.\nMoreover, we will see how the polar motion equation simplifies in the Schwarzschild\ncase Q= 0 and we retrieve the results from [BE22] using only the elementary geometric\nproperties of the real elliptic curve describing the phase portrait in (affinely transformed)\nBinet variable.\nAnother remarkable dynamical distinction between Lorentzian and Euclidean Schwarzschild\nspacetimes relies in the gravitational deflection of orbits. As is well-known, given an orbit\ncoming from and to infinity, the deflection angle δϕ(in the motion plane) between its two\nasymptotic directions is always positive in Schwarzschild geometry, for photons as well as\nfor massive particles. This means that test-particles can only be attracted by the central\nbody. However, in Euclidean Schwarzschild geometry, the deflection angle can be positive\nor negative, depending on the velocity at infinity vand the perihelion rminof the orbit; in\nthis geometry, particles can be attracted or repelled by the central mass. More precisely,\nwe observe that, at fixed v < 1, we have δϕ < 0 for all values of rmin, while for v > 1,\nthe deflection δϕvanishes (resp. is maximal positive) at some perihelion rmin=ρ0(resp.\nrmin=ρmax). In (11) and (12), we give approximate values for ρ0andρmaxwhen v≳1.\nThe existence of ρmaxgives rise to a visible maximal deflection ring in the shadow of such\na spacetime. As already said, the fact the δϕ < 0 at small perihelia indicates that particles\nare repelled by the central mass and thus the horizon becomes invisible to the observer, see\nFigures 2 and 16.\nThe only other possible type of spherically symmetric instanton is given by a Bertotti–\nRobinson-like metric, whose line element has the form\n(BR) d s2=Q2\u00121−2mr+q2r2\nr2dτ2+dr2\nr2(1−2mr+q2r2)+ dθ2+ sin2θdϕ2\u0013\n,\nwhere Q̸= 0 and m, q∈Rare some constants. This is the Euclidean analogue of the\ngeneral Bertotti–Robinson electro-vacuum Lorentzian solution. As in the Lorentzian case,\nthis metric essentially differs from (ERN) is this sense that its Kretschmann invariant is\nconstant. To the knowledge of the author, the dynamics of this solution, Euclidean or\nLorentzian, doesn’t appear in the literature. This is treated in §4, where we provide a full\nanalytic solution of the geodesic equation, in terms of (hyperbolic) trigonometric functions;\nsee (23) and (25).\nFinally, the efficient algorithms available to approximate Weierstrass’ elliptic functions\n[Car95; CGL90] are used to produce a Python code2, designed to draw orbits in the two\ntypes of instantons discussed here, as well as to obtain the “shadow” of an ERN space\nby ray-tracing, as it was already done for black holes in [DA09; Vin+11; CP ¨O13; Pu+16;\nVel+22], for instance. Since there are no null geodesics in Euclidean geometry, photons\nare replaced by particles with a velocity at infinity that should be provided by the user.\nConformally to what was mentioned above concerning the deflection angle, we observe the\npresence of a maximal deflection ring when v >1 and we notice that the horizon is not\nvisible. In particular, the optical difference between the cases Q2≲M2andQ2≳M2is not\nas obvious as in the Lorentzian background. We still observe that the size of the maximal\ndeflection ring diminishes as the charge increases, see Figure 19.\nThe layout of the paper is as follows: first, we state that the metrics (ERN) and (BR)\nare the only spherically symmetric solutions of the Einstein–Maxwell field equation with\ncomplex vector potential Aµ=−iQr−1dτ. The detailed proof of this result can be found\nin the Appendix A. Then, we derive the motion equations and the motion constants for\nthe metric (ERN), and we prove that the energy Eof a geodesic satisfies E2<1 when\nQ2< M2, as mentioned above. We then obtain the Weierstrass equation from of the\npolar radial motion equation and we investigate the particular case where Q= 0. Next,\nwe study the gravitational deflection in Euclidean Schwarzschild spacetime and provide\n2available at https://github.com/arthur-garnier/euclidean_orbits_and_shadows.git\n3ARTHUR GARNIER\nthe aforementioned approximations for the deflection angles, as well as for the null and\nmaximal deflection rings. Concerning the Bertotti–Robinson family (BR), we derive the\nmotion equations and obtain analytic solutions with (hyperbolic) trigonometric functions.\nFinally, we quickly explain how the Python code is constructed and we finish with some\nfigures illustrating our results and programs.\n1.The two types of spherically symmetric electro-vacuum instantons\nIn this section, we state the unicity result for spherically symmetric electro-vacuum in-\nstantons. We systematically use Stoney units where G=c= 4πϵ0= 1. Let ( M, Q )∈R+×R\nand consider the numbers r+, r−defined by\nr±:=\u001a\nM±p\nM2−Q2ifQ2< M2,\n0 otherwise.\nLet also M:=R×]r+,+∞[×S2≃R2×S2, with coordinates xµ= (τ, r, θ, ϕ ), the pair ( θ, ϕ)\ndescribing spherical coordinates on S2. Since we work on the Euclidean section, we restrict\nour study to the open subset r > r +, just as in [Esp92]. This is a reasonable restriction, as\nwe are interested by the exterior region of the spacetime.\nWe denote by dΩ2:= dθ2+ sin2θdϕ2the usual round metric on S2and we have the\nfollowing result, the detailed proof of which can be found in Appendix A.\nLetds2=gµνdxµdxνbe a spherically symmetric solution of the Einstein–\nMaxwell equation with complex vector potential\nAµ=−iQr−1dτ,\ndefined for r≫0.\nIf the Kretschmann invariant K=RαβµνRαβµν associated to gµνis inde-\npendent of r, then there are constants m, q∈Rsuch that the metric takes the\nBertotti–Robinson form\nds2=Q2\u00141−2mr+q2r2\nr2dτ2+dr2\nr2(1−2mr+q2r2)+ dΩ2\u0015\nin which case the Kretschmann invariant is K= 8Q−4.\nOtherwise, there are coordinate transformations of the form ˜τ=Cτand\nR=r/(Ar+B)(with A∈RandB, C∈R∗) as well as constants ˜M,˜Q∈R\nsuch that gµνtakes the Reissner–Nordstr¨ om form\nds2= \n1−2˜M\nR+˜Q2\nR2!\nd˜τ2+ \n1−2˜M\nR+˜Q2\nR2!−1\ndR2+R2dΩ2,\nwhose Kretschmann invariant is K= 8R−8(6˜M2R2−12˜M˜Q2R+ 7˜Q4).\nWe make the following observations:\n•The Ricci scalar of any of the above solutions vanishes.\n•The proof shows in particular that for a vector potential Aτ=Qr−1, a spherically\nsymmetric solution of the field equation is Euclidean (resp. Lorentzian) if and only\nifQis purely imaginary (resp. is real).\n•Recalling the notation from the proof, we observe that in the Reissner–Nordstr¨ om\ncase, the new potential is\nA′\nµ=−i˜QR−1d˜τ=−iQ(r−1+αβ−1)dτ=Aµ−i∇µf,\nwhere f:=Qαβ−1τ. Therefore, the coordinate transformation ( τ, r)/map∫to/∫hortrightarrow(˜τ, R)\ninduces a gauge transformation Aµ/map∫to/∫hortrightarrowAµ−i∇µf.\n4PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n•The second metric of the statement with m=q= 0 gives the Euclidean version of\nthe original Bertotti–Robinson line element derived in [Rob59; Ber59]\nds2=Q2\nr2\u0002\ndτ2+ dr2+r2dΩ2\u0003\n.\nObserve moreover that in Binet variable u= 1/r, the general Bertotti–Robinson\nmetric has an even simpler form\nds2=Q2\u0014\n(u2−2mu+q2)dτ2+du2\nu2−2mu+q2+ dΩ2\u0015\n.\nIf now d s2=gµνdxµdxν=gµµ(dxµ)2is an asymptotically flat spherically symmetric\nelectro-vacuum instanton, then the Kretschmann scalar should vanish as r/∫hortrightarrow+∞, thus\nonly the Reissner–Nordstr¨ om form from the previous theorem is allowed, with ˜ τ=Cτand\nR=r/(Ar+B). But the asymptotic conditions\nlim\nr/∫hortrightarrow+∞(Ar+B)−2= lim\nr/∫hortrightarrow+∞gθθ\nr2= 1 = lim\nr/∫hortrightarrow+∞gττ= lim\nr/∫hortrightarrow+∞C2(1−2˜M/R +˜Q2/R2)\nimpose R=rand ˜τ=τ. Finally, the electromagnetic tensor FµνhasiFτr=˜QR−2=Qr−2,\nso that ˜Q=Qand we have obtained the following Euclidean analogue of the Birkhoff–\nHoffmann theorem:\nThe Euclidean Reissner–Nordstr¨ om metric is the only spherically symmetric,\nasymptotically (Euclidean) flat metric satisfying the electro-vacuum Einstein–\nMaxwell equation associated to the complex vector potential\nAµ:=−iQr−1dτ.\nMore precisely, if ds2=gµνdxµdxνis a such a metric, defined for r≫0, then\nthere exists a constant ˜M∈Rsuch that\nds2= \n1−2˜M\nr+Q2\nr2!\ndτ2+ \n1−2˜M\nr+Q2\nr2!−1\ndr2+r2dθ2+r2sin2θdϕ2.\nIn particular, the metric (ERN) is the only spherically symmetric solution of\nthe Einstein–Maxwell equation associated to Aµdefined on M, which reduces to\nthe Euclidean Schwarzschild metric (ES) when Q/∫hortrightarrow0.\n2.Geodesic motion in Euclidean Reissner–Nordstr ¨om instantons\n2.1.Motion equations and energy of orbits. Recall the notation from the begin-\nning of the previous section and consider a non-constant geodesic γ= (τ, r, θ, ϕ ) inM=\nR×]r+,+∞[×S2for the metric (ERN), with affine parameter λ. We will analyse the geodesic\nequation in the same fashion as in [BE22].\nBy spherical symmetry, we may assume that θ≡π/2 and letting\n∆(r) := 1 −2M\nr+Q2\nr2,\nthe relativistic Lagrangian L=1\n2gµν˙γµ˙γνreads\n2L= ∆( r) ˙τ2+ ∆( r)−1˙r2+r2˙ϕ2.\nThus, the temporal and angular Euler–Lagrange equations provide constants C, J∈Rsuch\nthat\n(1) ˙ τ=C\n∆(r),˙ϕ=J\nr2\nand thus the scalar\nH:= 2L= ∆( r)−1(C2+ ˙r2) +J2\nr2>0\n5ARTHUR GARNIER\nis conserved along γand the proper length ssatisfies d s2=Hdλ2so that we get\n(2)\u0012dϕ\nds\u00132\n=˙ϕ2\nH=L2\nr4,\u0012dr\nds\u00132\n=˙r2\nH= ∆( r)\u0012\n1−L2\nr2\u0013\n−E2.\nwhere E(resp. L) is the energy per unit mass (resp. angular momentum per unit mass ) ofγ,\ndefined by analogy with the Lorentzian framework as the proper temporal (resp. azimuthal)\nconjugate momentum\nE:=pτ(s)=1√\n|H|pτ(λ)=gτµ˙γµ\n√\nH=∆(r) ˙τ√\nH=C√\nH,\u0012\nresp. L:=1√\n|H|pϕ=J√\nH\u0013\nFor the rest of this section, we assume that γis non-purely radial and we denote the\ndifferentiation with respect to the Euclidean time τwith a dot.\nWe want to apply the Weierstrass analysis of this equation and since it has degree 4,\none first needs to choose a real root of the quartic right-hand side. Such a real root is\nguaranteed to exist as soon as E2<1, a property that we shall prove to always hold,\nprovided that the metric (ERN) presents an event horizon (that is, when Q2< M2). To do\nthis, we need expressions for the energy and angular momentum, as functions of the initial\nconditions γ(0) =: ( τ0, r0, π/2, ϕ0) and ˙ γ(0) =: (1 ,˙r0,0,˙ϕ0). By symmetry, we may assume\nthat τ0=ϕ0= 0 and if we let α:=dτ\ndλ\f\f\nλ=0, then the constant Creads C=α∆(r0) and we\nalso have J=αr2\n0˙ϕ0. Then,\nH= ∆( r0)−1 \nC2+\u0012dr\ndλ\u00132!\n+J2\nr2\n0=α2\u0010\n∆(r0) + ∆( r0)−1˙r2\n0+r2\n0˙ϕ2\n0\u0011\n,\nso that we arrive at the following expressions for the energy and angular momentum:\n(3) E=∆(r0)q\n∆(r0) + ∆( r0)−1˙r2\n0+r2\n0˙ϕ2\n0, L=r2\n0˙ϕ0q\n∆(r0) + ∆( r0)−1˙r2\n0+r2\n0˙ϕ2\n0.\nWe can now state the main result of this section, generalizing the results from [BE22] to\nthe charged case. It implies in particular that the metric (ERN) features an event horizon\nexactly when there is no bounded orbit. The proof, relying on a tedious analysis of a real\npolynomial, is given in Appendix B.\nIfQ2≤M2, then any (exterior) non-constant geodesic γfor the metric (ERN)\nhasE2<1. Otherwise, there are circular orbits with arbitrary energy.\n2.2.Reduction of the polar radial equation to Weierstrass’ form. Letγ= (τ, r, ϕ )\nbe a non-purely radial equatorial geodesic. Then J̸= 0 so that the map s/map∫to/∫hortrightarrowϕ(s) is a\ndiffeomorphism onto its image and from (2) we find the polar radial equation\n(4)\u0012dr\ndϕ\u00132\n=1−E2\nL2r4−2M\nL2r3+\u0012Q2\nL2−1\u0013\nr2+ 2Mr−Q2=:F(r)\nIn this section, the dot denotes differentiation with respect to the polar variable ϕ. We\nuse the same trick as in [GV12, §3.1] to reduce the degree of the above equation and then\nre-write it in Weierstrass form. Let r∈Cbe a root of the quartic F(which, in view of the\nresult from §2.1, is guaranteed to be real positive when Q2≤M2) and consider the shifted\nBinet variable\nu:=1\nr−r.\n6PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nThen, the equation (4) becomes\n˙u2=˙r2\n(r−r)4=u4F(r+ 1/u)\n=1\nL2\u0002\n2(2(1−E2)r3−3Mr2+ (Q2−L2)r+ML2)u3\n+(6(1 −E2)r2−6Mr+Q2−L2)u2+ 2(2(1 −E2)r−M)u+ 1−E2\u0003\nIt is now straightforward to re-write this in Weierstrass form. Indeed, we can choose a root\nr∈Cof the quartic\n1−E2\nL2r4−2M\nL2r3+\u0012Q2\nL2−1\u0013\nr2+ 2Mr−Q2= 0,\nwhich we can choose to be real positive if Q2≤M2. If we let\n(5a)\n\nδ=1−E2\nL2,\nγ= 2\u0000\n2δr−M\nL2\u0001\n,\nβ= 6r\u0000\nδr−M\nL2\u0001\n+Q2\nL2−1,\nα= 2\u0010\n2δr3−3M\nL2r2+\u0010\nQ2\nL2−1\u0011\nr+M\u0011\n,and\n\ng2:=1\n4\u0010\nβ2\n3−αγ\u0011\n,\ng3:=1\n8\u0010\nαβγ\n6−α2δ\n2−β3\n27\u0011\n,\n℘:=α\n4(r−r)+β\n12,\nthen the function ℘satisfies the Weierstrass equation\n(5b) ˙ ℘2= 4℘3−g2℘−g3.\nIn other words, the polar radial motion is given by\n(5c) r(ϕ) =r+α\n4℘(ϕ)−β/3,\nwhere ℘=℘g2,g3is the Weierstrass function associated to the pair ( g2, g3)∈C2.\nMore practically, given an initial radius r0:=r(ϕ0), we have to find some z0∈Csuch\nthat℘(z0) =α\n4(r0−r)+β\n12, a task that can be achieved using Carlson’s integrals [Car95]\n(RF) RF(x, y, z ) :=1\n2Z∞\n0dζp\n(ζ+x)(ζ+y)(ζ+z).\nThen, the equation (5c) can be recast in the following form\n(5c’) r(ϕ) =r+α\n4℘g2,g3(z0+ϕ)−β/3,with z0:=RF(℘0−z1, ℘0−z2, ℘0−z3)∈C,\nwhere z1,2,3∈Care the roots of the Weierstrass cubic 4 z3−g2z−g3and℘0:=α\n4(r0−r)+β\n12.\nThe main advantage of this formulation is that the integrals ( RF) can be approximated ef-\nficiently by the Carlson algorithm [Car95, §2] and we can approach ℘using the Coquereaux–\nGrossmann–Lautrup algorithm [CGL90, §3]. This is the method we use to approximate ERN\norbits and produce a Python code.\n2.3.The special case of the Euclidean Schwarzschild geometry. When Q= 0, we\nhave F(0) = 0 so that we may take r= 0 so that the polar equation in Binet variable\nsimplifies to\n˙u2=u4F(1/u) = 2 Mu3−u2−2M\nL2u+1−E2\nL2\nwhich is [BE22, equation (2.16)]. Therefore, the Weierstrass form\n˙℘2= 4℘3−g2℘−g3\nis obtained by letting ℘:=M/(2r)−1/12, as well as\n(gES\n12) g2:=1\n12+M2\nL2andg3:=1\n216−M2\n12L2(2−3E2).\n7ARTHUR GARNIER\nThese expression can be compared to the Lorentzian Schwarzschild case, where for a test-\nparticle of mass µ= 2L(twice the Schwarzschild Lagrangian), the constants g2andg3read\n(cf [Hag30, §4, p. 84])\n(gLS\n12) g2=1\n12+µM2\nL2andg3=1\n216−M2\n12L2(2µ+ 3E2).\nTherefore, one may view a Euclidean Schwarzschild geodesic as a space-like Lorentzian\nSchwarzschild geodesic with complex energy EEuc=iELor.Notice that this last equality is\nexpected since the energy is defined as the temporal momentum and because the Euclidean\ntime τand the Lorentzian time tare related by the relation τ=it.\nThe Weierstrass formulation also permits to derive a shorter proof of the fact that E <1\nfor every geodesic γin Euclidean Schwarzschild geometry, with initial radius r0> r+= 2M.\nAs above, we may simplify the notation by rescaling the radius and assuming that M= 1. If\nL= 0, then the second equation (2) reduces to (d r/ds)2= 1−E2−2/rso 1−E2≥2/r > 0,\nas claimed. Now if L̸= 0, then we may use Weierstrass’ form and the discriminant of the\nequation 4 z3−g2z−g3reads\n∆ := 16( g3\n2−27g2\n3) =L−6\u0002\n(1−E2)L4−(27E4−36E2+ 8)L2+ 16\u0003\n=1−E2\nL2+O\u00121\nL4\u0013\n,\nso that if, for the sake of contradiction, we assume E2>1, then ∆ <0 for L≫0, something\nwhich can be achieved by rescaling the initial azimuthal angular velocity. Therefore, we\nassume that ∆ <0 and look for an absurdity.\nConsider the Weierstrass cubic q(x) := 4 x3−g2x−g3; then the phase portrait in Weier-\nstrass variable x= 1/(2r)−1/12 describes a portion of the (real) elliptic curve\nEq:={(x, y)∈R2|y2=q(x)}.\nSince lim x/∫hortrightarrow+∞q(x) = + ∞, we have Eq∩ {x=x0} ̸=∅forx0≫0 but because q(1/6) =\n−(E/2L)2<0, the non-compact connected component of Eqlies in the open half-plane\n{x >1/6}(which corresponds in radial variable to {r <2}), so that the considered phase\nportrait cannot describe a portion of this component. However, to say that ∆ <0 amounts\nto say that Eqis connected, a contradiction.\nAt this point, we know that ∆ ≥0. If ∆ >0, then the elliptic curve Eqhas an additional\ncompact connected component and since q(−1/12) = (1 −E2)/(4L2)>0, this component\nintersects the subset {x=−1/12}. Therefore, the corresponding orbit is indeed unbounded.\nIn the case where ∆ = 0, the curve is singular but connected so that the orbit imposes\n0≤q(1/6) =−(E/2L)2≤0 sor= 2, a new contradiction. We summarize the discussion\nas follows:\nAny Euclidean Schwarzschild orbit has E2<1, and is unbounded. Moreover,\nthe phase portrait of a non-radial equatorial Euclidean Schwarzschild orbit in\nWeierstrass variable ℘=M/(2r)−1/12(with polar argument) describes a por-\ntion of the unique compact connected component of the associated (real) elliptic\ncurve E, included in E ∩ {− 1<12℘ <2}. In particular, the discriminant of E\nis positive.\n3.Gravitational deflection of Euclidean Schwarzschild orbits\nConsider an equatorial orbit γ= (τ, r, ϕ ), with energy −1< E < 1 and angular mo-\nmentum L∈R∗. Since γis unbounded, we may consider its velocity at infinity , defined by\nv2:= lim r/∫hortrightarrow∞(dr/dτ)2. Using equations (1) and (2), we find the expression\nv2= lim\nr/∫hortrightarrow∞\u0012dr\ndτ\u00132\n= lim\nr/∫hortrightarrow∞\u0012˙r\n˙τ\u00132\n= lim\nr/∫hortrightarrow∞∆(r)2\nC2\u0012\n∆(r)\u0012\nH −J2\nr2\u0013\n−C2\u0013\n=1\nE2−1\n8PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nThe equation (4) can be recast in Binet variable u= 1/rand yields\n(6)\u0012du\ndϕ\u00132\n+u2= 2Mu3−2M\nL2u+1\nb2= 2Mu\u0012\nu2−1\nL2\u0013\n+1\nb2,\nwhere the constant b:=q\nL2\n1−E2is the impact parameter ofγ, satisfying L=±bvE. Since\nlim\nr/∫hortrightarrow∞dϕ\ndτ= lim\nr/∫hortrightarrow∞J∆(r)\nCr2= 0,\nthe orbit admits asymptotic lines, and we are first interested in the deflection angle at\ninfinity δϕbetween these two asymptotic directions, as a function of the orbit’s perihelion\nr=rmin.\nb0rminϕδϕ\nδϕ/2\nFigure 1. Schematics of an orbit with angle deflection δϕ= 2ϕ−π.\n3.1.Analytic expression of the deflection angle using Carlson’s integrals. As il-\nlustrated in the Figure 1, the deflection (at infinity) δϕis given by\nδϕ= 2|ϕ(r=∞)−ϕ(r=rmin)| −π\nand using (4) again leads to the expression\nδϕ= 2Z∞\nrmindϕ−π= 2Z∞\nrmindϕ\ndrdr−π= 2Z∞\nrmin1q\n1\nb2−1\nr2\u0000\n1−2M\nr+2Mr\nL2\u0001dr\nr2−π.\nThis expression can be simplified using the Weierstrass variable ℘=Mu/ 2−1/12. Indeed,\nusing the constants g2, g3given by equation ( gES\n12), we have\nδϕ= 2\f\f\f\f\f\fZ1/rmin\n0duq\n1\nb2−u2+ 2Mu\u0000\nu2−1\nL2\u0001\f\f\f\f\f\f−π= 2Z−1\n12\n−1\n12+M\n2rmindpp\n4p3−g2p−g3−π.\nThis expression can be rewritten in terms of elliptic integrals, as in the pioneer work [Dar59].\nHowever, it is both easier and numerically more adequate to express it with the integrals\n(RF), which we numerically approximate using Carlson’s algorithm [Car95]. Observe first\nthat because r=rminis a turning point of γ, we have\n0 =d℘2\ndϕ2\f\f\f\f\nr=rmin= 4℘(r=rmin)3−g2℘(r=rmin)−g3.\n9ARTHUR GARNIER\nIn other words, the point ℘max:=M/(2rmin)−1/12 is a root of the Weierstrass cubic and\nthis leads to the factorization\n4p3−g2p−g3= (p−℘max)\u0012\n4p2+\u00122M\nrmin−1\n3\u0013\np+\u00121\n36−g2−M\n3rmin+M2\nr2\nmin\u0013\u0013\n,\nwhich allows to find the other two roots ℘±∈Cof the cubic. We then have\nZ−1\n12\n℘maxdpp\n4p3−g2p−g3=Z0\nM\n2rmindζq\n4\u0000\nζ−1\n12\u00013−g2\u0000\nζ−1\n12\u0001\n−g3\n=1\n2Z0\nM\n2rmindζq\u0000\nζ−℘max−1\n12\u0001\u0000\nζ−℘−−1\n12\u0001\u0000\nζ−℘+−1\n12\u0001\n=1\n2\"Z∞\nM\n2rmin−Z∞\n0#\ndζq\u0000\nζ−℘max−1\n12\u0001\u0000\nζ−℘−−1\n12\u0001\u0000\nζ−℘+−1\n12\u0001\n=RF\u0010\nemax+M\n2rmin, e−+M\n2rmin, e++M\n2rmin\u0011\n−RF(emax, e−, e+),\nwhere emax,±:=−℘max,±−1/12. Thus, we obtain the following expression for the deflection:\n(7) δϕ= 2\u0010\nRF\u0010\nemax+M\n2rmin, e−+M\n2rmin, e++M\n2rmin\u0011\n−RF(emax, e−, e+)\u0011\n−π\nThis is an efficient formula for numerical calculations (see below), but it is natural to ask for\nan approximation of δϕwhen rmin/∫hortrightarrow∞. This is the goal of the next subsection. Observe\nmoreover that the two terms of (7) involving the integrals ( RF) can be complex, but their\ndifference is real and non-negative.\n3.2.Approximation of the deflection angle with perturbed solution. The aim of\nthis section is to obtain an expansion of δϕin powers of the perihelion rmin, up to order\n3. This choice of order will become transparent later, when we study the null and maximal\ndeflection rings.\nTo do so, we could use for instance the well-known perturbed solution method [Str13;\nGD06; BH08; He+20]. However, we will avoid complicated calculations with a simple\nobservation. First, using (2), the turning point condition at r=rmingives the following\nrelation\n0 =\u0012\n1−2M\nr\u0013\u0012\n1−L2\nr2\u0013\n−E2,\nor, in terms of bandv,\nr2\nmin\u0012\n1−2M\nrmin\u0012\n1 +1\nv2\u0013\u0013\n=b2\u0012\n1−2M\nrmin\u0013\n,\nthat is,\n(8) b=rminvuut1−2M\nrmin\u0000\n1 +1\nv2\u0001\n1−2M\nrmin=rmins\n1−2M\nv2(rmin−2M).\nRecall also that L2=b2v2(1 +v2)−1and that the equation (4) in terms of bandLreads\n\u0012dr\ndϕ\u00132\n=r4\u00121\nb2−1\nr2\u0012\n1−2M\nr+2Mr\nL2\u0013\u0013\n.\nOn the other hand, in Lorentzian Schwarzschild geometry, the same equation\n\u0012dr\ndϕ\u00132\n=r4\u00121\nb2\nSch−1\nr2\u0012\n1−2M\nr−2Mr\nL2\nSch\u0013\u0013\n10PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\ndiffers from the above one by just a sign in the term in L−2, and we have in this case\nb2\nSch=rmins\n1 +2M\nv2(rmin−2M),\nas well as L2\nSch=b2v2(1−v2)−1. This means that the expression of the deflection angle in\nEuclidean Schwarzschild geometry with squared velocity v2is the same as the expression of\nthe Lorentzian deflection angle with same perihelion and “squared velocity” −v2. This is\nmathematically well-defined since the data only depend on the squared quantities ( b2, v2),\nand not on the pair ( b, v) itself, so no complex number is involved. Moreover, this interpre-\ntation is physically consistent with the relation τ=itbetween the Euclidean time τand\nthe Lorentzian time tand the definition of the Euclidean velocity v2= lim r/∫hortrightarrow∞(dr/dτ)2,\nwhile in the Lorentzian case, v2= lim r/∫hortrightarrow∞(dr/dt)2.\nAfter [AR02] or [Li+19] for instance, up to order 3, in Lorentzian Schwarzschild geometry\nwe have\n(9) δϕ=2M\nb\u0012\n1 +1\nv2\u0013\n+3πM2\n4b2\u0012\n1 +4\nv2\u0013\n+2M3\n3b3\u0012\n5 +45\nv2+15\nv4−1\nv6\u0013\n+O\u0012M4\nb4\u0013\n.\nTherefore, changing the sign of the terms in v−2yields the following estimate, in Euclidean\nSchwarzschild geometry,\n(δb\n3)δϕ=2M\nb\u0012\n1−1\nv2\u0013\n+3πM2\n4b2\u0012\n1−4\nv2\u0013\n+2M3\n3b3\u0012\n5−45\nv2+15\nv4+1\nv6\u0013\n+O\u0012M4\nb4\u0013\n.\nUsing now the relation (8), we arrive at the expression\nδϕ=2M\nrmin\u0012\n1−1\nv2\u0013\n+M2\nr2\nmin\u00123π\n4\u0012\n1−4\nv2\u0013\n+2\nv2\u0012\n1−1\nv2\u0013\u0013\n(δr\n3)\n+M3\nr3\nmin\u00123π\n2v2\u0012\n1−4\nv2\u0013\n+10\n3−26\nv2+9\nv4−7\n3v6\u0013\n+O\u0012M4\nr4\nmin\u0013\n.\nConcerning the metric (ERN) with non-zero charge Q̸= 0, the perturbed solution method\ngives, up to order 2 and after elementary calculations we omit,\nδϕ=2M\nb\u0012\n1−1\nv2\u0013\n+3πM2\n4b2\u0012\n1−4\nv2\u0013\n−πQ2\n4b2\u0012\n1−2\nv2\u0013\n+O(b−3),\nin agreement with [PJ19]. Expressing the impact parameter bin terms of the perihelion\nrminas\nb=rmins\n1−2Mrmin−Q2\nv2(r2\nmin−2Mrmin+Q2),\nyields the expansion\n(10)\nδϕ=2M\nrmin\u0012\n1−1\nv2\u0013\n+M2\nr2\nmin\u00123π\n4\u0012\n1−4\nv2\u0013\n+2\nv2\u0012\n1−1\nv2\u0013\u0013\n−πQ2\n4r2\nmin\u0012\n1−2\nv2\u0013\n+O(r−3\nmin).\nAs a sanity check, observe that replacing v2by−v2in the previous expression and letting\nv/∫hortrightarrow1 leads to the light deflection formula of [BH08, §III.B]\nδϕ=4M\nrmin+M2\nr2\nmin\u001215π\n4−4\u0013\n−3πQ2\n4r2\nmin+O(r−3\nmin).\n11ARTHUR GARNIER\n(a)v= 0.9\n (b)v= 1.1\n(c)v= 1.25\n(d)v= 3\nFigure 2. Orbits with different velocities and their perihelion (in units of\nM) and deflection at infinity.\n3.3.Null and maximal deflection rings. We start by observing that, at lowest order,\nthe deflection angle for a usual (Lorentzian) massive Schwarzschild orbit with velocity at\ninfinity vis given by\nδϕLor≈2M\nrmin\u0012\n1 +1\nv2\u0013\n>0,\nwhile in the Euclidean background, the estimation ( δr\n3) reads\nδϕEuc≈2M\nrmin\u0012\n1−1\nv2\u0013\n.\nThis suggests that for the Euclidean Schwarzschild solution, the deflection δϕmay vanish\nfor some values of rmin. This can also be noticed from the motion equation itself. Indeed,\nrecall from [MS98, §144] that a polar curve, parametrized in Binet variable uisconcave\nwith respect to its pole (hence, has a positive deflection) if and only if u+d2u\ndϕ2>0. Now,\nthis quantity in Schwarzschild geometry is given by\n¨u+u=M(3u2−ϵL−2),\nwhere ϵis the signature of the metric. Hence, this is always positive in the Lorentzian\nbackground ϵ=−1, while it may change of sign in the Euclidean world ϵ= 1.\nTo numerically appreciate this phenomenon, in the Figure 2 we depict pencils of orbits\nwith different velocities. Their deflection at infinity is computed using the formula (7) and\nCarlson’s algorithm. Observe that the deflection stays negative when v < 1. The more\nextreme case where v= 5 is displayed in the Figure 3, along with the graph representing\nthe deflection angle as a function of the perihelion. Observe the presence of a null deflection\natrmin≈2.16 and a maximal deflection at rmin≈2.5. The Figure 4 depicts the deflection\nas a function of the perihelion, for several velocities at infinity.\nObserve that in the Lorentzian (resp. Euclidean) Schwarzschild spacetime, the impact\nparameter bof a massive particle (resp. of any particle) satisfies bv=L/E. Moreover, a\nphoton in the usual Schwarzschild metric has b=L/E and this motivates the following\nterminology: an orbit in a Euclidean Schwarzschild spacetime with velocity at infinity vwill\nbe called of sub-photon type (resp. of sup-photon type ) ifv≲1 (resp. v≳1).\n12PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n(a)Orbits with 2 .1< r min<5.\n(b)The deflection δϕas a function of rmin.\nFigure 3. Orbits with v= 5 and corresponding deflection graph.\nFigure 4. Representations of δϕ=δϕ(rmin) for several values of v.\nWe have numerically observed that sub-photon orbits have a negative deflection (i.e. are\nrepelled by the central mass) and that the absolute value of δϕbehaves like in the usual\nSchwarzschild spacetime. This can also be noticed using the lowest order approximations\nfrom ( δr\n3) and (9) recalled at the beginning of the subsection.\nFor orbits with fixed velocity v >1, one could use numerical methods on the expression\n(7) to find an approximation of the critical value ρ0(resp. ρmax) such that the deflection\nangle of the orbit with perihelion rmin=ρ0(resp. rmin=ρmax) vanishes (resp. is maximal).\nHowever, we can also use the estimate ( δr\n3). Indeed, as it is a third order approximation of\nthe deflection, to solve δϕ= 0 at this order amounts to solve a quadratic equation. Similarly,\nwe can differentiate ( δr\n3) with respect to rminto obtain an approximation of ρmaxusing the\nquadratic formula again. This is precisely the reason why we chose this order at the first\nplace. Though the resulting expressions for ρ0andρmaxare not very enlightening, it is worth\nnoticing that they both diverge when v/∫hortrightarrow1. This suggests to expand these expressions in\npowers of v−1, yielding estimations of ρ0andρmaxfor orbits of sup-photon type. We find\n(11)ρ0\nM=9π\n16(v−1)−1+ 1−21π\n32+64\n9π+O(v−1)≈1.767\nv−1+ 1.202 + O(v−1)\nand\n(12)ρmax\nM=9π\n8(v−1)−1+ 1−21π\n16+32\n3π+O(v−1)≈3.534\nv−1+ 0.272 + O(v−1).\n13ARTHUR GARNIER\n(a)v= 1.15\n (b)v= 1.25\n(c)v= 1.35\n (d)v= 1.5\nFigure 5. Accuracy of the approximations (11) and (12) for several veloci-\nties.\nObserve that, at lowest order, we have the remarkable relation ρmax= 2ρ0. Moreover, at\nthis order, the maximal value of the deflection is given by\n(δmax) δϕmax≈16(v−1)\n9π\u0012\n1−1\nv2\u0013\n.\nHowever, this estimate badly fails when v≫1. We illustrate our approximations in the\nFigure 5.\n3.4.Deflection of orbits passing through a given point. Consider now an orbit start-\ning from a given point ( ϕ, r) = (0 , r0), for some fixed r0>2M, whose velocity vector at this\npoint makes an angle 0 < α≤π\n2with respect to the radial direction. Thinking of the point\n(ϕ, r) = (0 , r0) as an observer receiving particles coming from infinity, we are interested\nin the total deflection angle δαϕbetween the tangent line to the orbit at (0 , r0) and the\nasymptotic direction. The situation may be visualized using the Figure 6.\nWe first proceed as in the first subsection, to obtain a numerically efficient analytic formula\nforδαϕ, using Carlson’s integrals. Recall the Weierstrass variable ℘=M/(2r)−1/12,\nsatisfying the Weierstrass equation with constants g2,3=g2,3(v, rmin) given by ( gES\n12). Recall\nalso that ℘max=M/(2rmin)−1/12 is the maximal value of ℘along the geodesic. We have\nϕ=Zrmin\n∞dr\nr2q\n1\nb2−1\nr2\u0000\n1−2M\nr+2Mr\nL2\u0001=Z−1\n12\n℘maxdpp\n4p3−g2p−g3,\n14PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nbr00rmin\nα\nΦϕ\nδϕ= 2ϕ−π δαϕ\nFigure 6. Schematics of an orbit passing through the point ( ϕ, r) = (0 , r0),\nwith total deflection δαϕ= Φ + α+1\n2(δϕ−π) =α−π+ϕ+ Φ.\nand\nΦ =Zrmin\nr0dr\nr2q\n1\nb2−1\nr2\u0000\n1−2M\nr+2Mr\nL2\u0001=Z℘0\n℘maxdpp\n4p3−g2p−g3,\nwhere ℘0:=M/(2r0)−1/12, so that\nδαϕ=α−π+ϕ+ Φ = α−π+\"Z−1\n12\n℘max+Z℘0\n℘max#\ndpp\n4p3−g2p−g3.\nWe now introduce the two remaining roots ℘±̸=℘maxof the cubic 4 p3−g2p−g3, as well\nas the constants emax,±:=−℘max,±−1/12 as above and arrive at the expression\nδαϕ=RF\u0010\nemax+M\n2rmin, e−+M\n2rmin, e++M\n2rmin\u0011\n+RF\u0010\nemax+M\n2r0, e−+M\n2r0, e++M\n2r0\u0011(13)\n+α−π−2RF(emax, e−, e+).\nIt may be observed that, at fixed α∈]0;π/2], when r0≫0, we have Φ ≈π\n2−α, so\nrmin≈r0cos(Φ) ≈r0sin(α) and δαϕ≈1\n2δϕ(rmin=r0sinα). On the other hand, at fixed\nr0, when α=π\n2, we have rmin=r0and Φ = 0. We thus obtain the boundary conditions\nδαϕ≈\nr0/∫hortrightarrow∞1\n2δϕ(rmin=r0sinα) and δπ/2ϕ=1\n2δϕ(rmin=r0).\nIn particular, at lowest order,\nδαϕ≈\nr0/∫hortrightarrow∞M\nr0sinα\u0012\n1−1\nv2\u0013\n.\nWe illustrate the situation in the Figure 7.\nFor fixed values of r0andv, we now aim to find the critical angle α0(resp. αmax) at\nwhich the total deflection δαϕvanishes (resp. is maximal). First recall the following relation\n15ARTHUR GARNIER\n(a)Orbits with 0 .05< α < 0.5.\n(b)Total deflection δαϕand deflection at infinity\nδϕ, as functions of α.\nFigure 7. Orbits with r0= 10Mandv= 1.5, and the corresponding graph.\nfrom [Bel02] (which is just the Pythagorian theorem in curved space)\ntan2α=pϕ˙γϕ\npr˙γr\f\f\f\f\nr=r0=r2\u0012\n1−2M\nr\u0013\u0012dϕ\ndr\u00132\f\f\f\f\f\nr=r0=1−2M\nr0\nr2\n0\nb2−1 +2M\nr0−2Mr0\nL2\nyielding\n(14) α= arctanvuut1−2M\nr0\nr2\n0\nb2−1 +2M\nr0−2Mr0\nL2.\nThe value of the impact parameter band the angular momentum Lare obtained using the\nrelations (8) and L=bv(1 +v2)−1/2. Therefore, we obtain the value of αas a function of\nthe triple ( v, r0, rmin). With the notation of the previous subsection, we are thus interested\nin the values of αat (v, r0, ρ0) and ( v, r0, ρmax).\nFor sup-photon type orbits with high perihelion, we may inject the approximations (11)\nand (12) into (14), expand in powers of r0up to order 2, and then expand in powers of v−1\nup to order 0 and obtain the following estimates\n(15) α0=9πM\n16r0\u0012\n1 +M\nr0\u0013\n(v−1)−1+M\nr0\u001264\n9π−21π\n32\u0013\n−M2\nr2\n0\u001257π\n32−64\n9π\u0013\n+O(v−1, r−3\n0)\nand\n(16)\nαmax=9πM\n8r0\u0012\n1 +M\nr0\u0013\n(v−1)−1−M\nr0\u001221π\n16−32\n3π\u0013\n−M2\nr2\n0\u001257π\n16−32\n3π\u0013\n+O(v−1, r−3\n0).\nWe illustrate our approximations in the Figure 8.\nFinally, concerning the observable sizes R0andRmaxof the null and maximal deflection\nrings, we have R0=r0tanα0andRmax=r0tanαmax. For an orbit of sup-photon type with\nr0≫0, we may use the relation (14) and the estimates (11) and (12) to obtain\n(17) R0=9πM\n16\u0012\n1 +M\nr0\u0013\n(v−1)−1+M\u001264\n9π−21π\n32\u0013\n−M2\nr0\u001257π\n32−64\n9π\u0013\n+O(v−1, r−2\n0)\nand\n(18)\nRmax=9πM\n8\u0012\n1 +M\nr0\u0013\n(v−1)−1−M\u001221π\n16−32\n3π\u0013\n−M2\nr0\u001257π\n16−32\n3π\u0013\n+O(v−1, r−2\n0).\n16PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n(a)v= 1.15\n (b)v= 1.25\n(c)v= 1.35\n (d)v= 1.5\nFigure 8. Accuracy of the approximations (15) and (16) for several veloci-\nties and r0= 30M.\nObserve that this is consistent with the approximations (15) and (16) since, at fixed velocity,\nwe have lim r0/∫hortrightarrow∞α0= 0 and thus R0=r0tanα0∼r0α0and similarly, Rmax∼r0αmax.\nWe may proceed similarly for the charged case where Q̸= 0, but the calculations are a\nbit tougher. At the lowest order, we have\n(19) ρmax∼\nv/∫hortrightarrow1πM\n8\u0012\n9−Q2\nM2\u0013\n(v−1)−1,\nas well as\n(20) αmax∼r0/∫hortrightarrow∞\nv/∫hortrightarrow1πM\n8r0\u0012\n9−Q2\nM2\u0013\u0012\n1 +M\nr0\u0013\n(v−1)−1\nand\n(21) Rmax∼r0/∫hortrightarrow∞\nv/∫hortrightarrow1πM\n8\u0012\n9−Q2\nM2\u0013\u0012\n1 +M\nr0\u0013\n(v−1)−1.\nFurthermore, just as in the Schwarzschild case, at lowest order, we have ρmax= 2ρ0and\nsimilarly for α0andR0. Note that for these approximations to make sense, the velocity\nat infinity v=√\nE−2−1 should be real and in view of §2.1, this is ensured only when\nQ2< M2.\n17ARTHUR GARNIER\n4.Geodesic motion in Bertotti–Robinson spacetimes\nIn this section we investigate the dynamics in Euclidean and Lorentzian Bertotti–Robinson\nspacetimes and provide in particular analytic solutions for the geodesic equation. For\nm, q∈R, consider the line element in Binet radial variable\n(BRb) d s2=Q2\u0002\nϵ∆(u)dτ2+ ∆( u)−1du2+ dΩ2\u0003\n,\nwhere ∆( u) :=u2−2mu+q2. This is an electro-vacuum solution of Einstein’s equation for\nAµ=√−ϵQudτ.\nLetγ= (τ, u, θ, ϕ ) be a geodesic with respect to the metric (BRb), parametrized by an\naffine parameter λ. By spherical symmetry, we may assume that θ≡π/2, in which case the\nLagrangian reads\nL=1\n2gµµ( ˙γµ)2=Q2\n2h\nϵ∆(u) ˙τ2+ ∆( u)−1˙u2+˙ϕ2i\n.\nThus, the temporal and angular Euler–Lagrange equations immediately yield constants\nJ, C∈Rsuch that ˙ τ=C∆(u)−1and ˙ϕ=Jand so the quantity\nH:= 2L=Q2\u0002\n∆(u)−1(ϵC2+ ˙u2) +J2\u0003\nis conserved along γ. Thus, the proper length ssatisfies d s2=Hdλ2and we get\n(22)\u0012du\nds\u00132\n=˙u2\nH= ∆( u)\u00121\nQ2−L2\u0013\n−ϵE2,\nwhere we denote L:=J/√\nHandE:=C/√\nH. From now on, differentiation with respect\ntoswill be denoted by a dot. Since the right-hand side of (22) is quadratic in u, we may\nsolve it explicitly.\nFix now an exterior equatorial geodesic γ= (τ, u, ϕ ) in the spacetime (BRb), with proper\nangular momentum L=˙ϕ(0) and let u0:=u(0) and ˙ u0:= ˙u(0). The Binet component uof\nγ, as a function of the proper length s, is given by\n(23) u(s) =m+ (u(0)−m) cos( s√\nℓ) + ˙u(0)sin(s√\nℓ)√\nℓ,where ℓ:=L2−1/Q2.\nMoreover, the only circular orbit has u≡m.\nBefore establishing (23), we observe the following consequences:\n•The calculations of the previous proof allow to re-write the motion equation (22) as\n\u0012du\nds\u00132\n= (1/Q2−˙ϕ2\n0)(u−u0)(u+u0−2m) + ˙u2\n0.\nIn particular, the proper spatial dynamics does not depend on the parameter q, nor\non the signature of the metric (BRb). Moreover, the translated variable v:=u−m\nsatisfies the equation\n˙v2+ℓv2=ℓ(u0−m)2+ ˙v2\n0.\nTherefore, in the non-degenerate case ℓ̸= 0, the phase portrait in Binet variable is\neither an ellipse if ℓ >0, and a hyperbola otherwise.\n•A non-circular geodesic γ= (τ, u, θ, ϕ ) with angular momentum L=˙ϕ(0) is bounded\nif and only if Q2L2>1, in which case it has periodic radial component with proper\nperiod\nωγ=2πQp\nQ2L2−1=2π\nL+O\u00121\nQ2L3\u0013\n.\nWhen Q2L2= 1, the geodesic is affine in proper length and when Q2L2<1, we\nhave\nu(s) =\ns/∫hortrightarrow∞O\u0010\nes√\n1−Q2L2\u0011\n,\nso the Binet variable ublows-up exponentially in proper length.\n18PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n•There are exterior circular orbit only when q2> m2, in which case the only such\norbit has r= 1/m. In particular, there are no circular orbits in the original Bertotti–\nRobinson space [Ber59; Rob59].\n•Any (exterior) Bertotti–Robinson spacetime is geodesically complete in Binet vari-\nable.\nBack to the derivation of (23), observe first that, up to dilating the affine parameter, we\nmay assume that\u0000dτ\ndλ\u0001\f\f\nλ=0= 1. Then, we have\nH=Q2\"\nϵ\u0000\n∆(u0)\u0000dτ\ndλ\u0001\f\f\nλ=0\u00012+\u0000du\ndλ\u0001\f\f2\nλ=0\n∆(u0)+J2#\n=Q2\u0014\nϵ∆(u0) +H˙u2\n0+ ∆( u0)L2\n∆(u0)\u0015\n,\nimplying\nH=ϵQ2∆(u0)2\n∆(u0)(1−Q2L2)−Q2˙u2\n0=⇒E2=C2\nH=∆(u0)(1−Q2L2)−Q2˙u2\n0\nϵQ2=−ℓ∆(u0) + ˙u2\n0\nϵ\nand thus the equation (22) becomes\n(24) ˙ u2=ℓ(∆(u0)−∆(u)) + ˙u2\n0=ℓ(u0−u)(u0+u−2m) + ˙u2\n0.\nIf ˙u0= 0, then the technical lemma from Appendix C applied to y=u−m,β=u0−m\nandα=−ℓleads to the stated expression for u. Otherwise, on a neighbourhood of 0, we\nhave ˙ u̸= 0 and differentiating (22) with respect to syields\n¨u=−ℓ∆′(u)\n2=ℓ(m−u),\na linear ODE of order 2 whose solution reads\nu(s) =m+ (u0−m) cos( s√\nℓ) + ˙u0sin(s√\nℓ)√\nℓ,\nand this function indeed satisfies (24) and is globally defined.\nNow, if γis circular, then ˙ u0= 0 and applying the lemma again, we find that u(s) =\nm+ (u0−m) cos( s√\nℓ) is constant, so that ℓ= 0 or u0=m. This can also be seen by\nanalysing the potential V=√\nℓ∆. But suppose now that u̸=m, then ℓ= 0 and since\n˙u(0) = 0, we get\nℓ= 0⇐⇒L2=1\nQ2⇐⇒Q2˙ϕ2\n0\nH= 1⇐⇒∆(u0) = 0 ,\ncontradicting the fact that γis exterior.\nFrom (23) we can deduce the analytic expression for the geodesic motion in terms of affine\nparameter: if the affine parameter λis chosen so that ˙ τ0= 1, then the motion constants\nand expressions of randϕas functions of λare given as follows:\nH=Q2\u0012ϵ(1−2mr0+q2r2\n0)\nr2\n0+˙r2\n0\nr2\n0(1−2mr0+q2r2\n0)+˙ϕ2\n0\u0013\nandℓ:=˙ϕ2\n0\nH−1\nQ2, (25a)\nr(λ) = \nm+\u00121\nr0−m\u0013\ncos(λ√\nℓ)−˙r0\nr2\n0sin(λ√\nℓ)√\nℓH!−1\n,as well as ϕ(λ) =ϕ0+λ˙ϕ0. (25b)\nMoreover, the geodesic γhas energy E=1−2mr0+q2r2\n0\nr2\n0√\nHand angular momentum L=˙ϕ0√\nH.\n5.Implementation of orbits and ray-tracing\nWe have developed a package3under Python, for drawing orbits in the spacetimes (ERN)\nand (BR), as well as for ray-tracing the first ones, using a standard backward ray-tracing\n3https://github.com/arthur-garnier/euclidean_orbits_and_shadows.git\n19ARTHUR GARNIER\nmethod (see for instance [Vel+22]). As mentioned in §2.2, the Carlson and Coquereaux–\nGrossmann–Lautrup algorithms [Car95; CGL90] are used to compute geodesics in the\nReissner–Nordstr¨ om instanton.\nLet us briefly recall the backward ray-tracing procedure we use for shadowing a spacetime.\nFirst, we consider an artificial celestial hemisphere on which we project our original image,\nseeing it as a portion of its tangent plane parallel to the screen (and on the other side of the\nblack hole). As a projection, we simply choose the standard and widely used equirectangular\nprojection , which has the advantage of taking the celestial hemisphere to a square, which\nwe may rescale to fit our image.\nNext, for each pixel of the screen, we consider the geodesic for the spacetime (null in\nthe Lorentzian and normalized in the Euclidean case) starting at this point and with ve-\nlocity directed by the line from the point observer. We then solve the geodesic equations\n(backwards) and we see if the ray ends in (came from) the black hole or touches the sphere\nsomewhere. If so, the RGB value of the pixel on the screen is given by the value of the\nlanding pixel on the sphere and we carry this process on until every pixel has been worked\nout.\nTo simplify calculations, we make heavy use of the spherical symmetry of the spacetimes\nconsidered here: given an initial datum, use a linear rotation to bring the initial velocity\n(and hence the full orbit) in the plane {θ=π/2}. Then, we give values to the various\nconstants involved in the expression of the radial geodesic and, instead of computing the\nfull orbit, we simply solve the equation r=rSwhere rSis the radius of the celestial sphere,\nusing the Weierstrass function. This can be done rather easily, precisely and quickly: we\ncompute some values until we cross the sphere and the first such point is used as an initial\nvalue for the Newton method. We finally rotate the result back and find our landing pixel.\nThus, no full orbit calculation is required. For more details and illustrations, see [Gar23].\nWe should mention however that since, in contrast to the Lorentzian framework, photons\n(i.e. null geodesics) do not properly exist in the Euclidean world, we have to trace orbits\nwith some prescribed velocity at infinity v: an additional input to the program. For each\npixel (i.e each particle), we provide its initial position and the direction of its velocity vector,\nwhose norm is adjust to match the velocity v.\nWe finish our discussion by providing some figures illustrating the programs and our\nresults.\nFirst, concerning the orbits drawings, it should be mentioned that, while the functions of\nthe package are designed to draw orbits in 3D, we chose to plot planar orbits here, to make\nthe figures more readable. In each case, the mass is set to unity and we vary the charge in\nthe different plots, distinguishing between the sub-extremal and sup-extremal cases for the\ncharge. In each figure, the legend gives the values of the energy and angular momentum of\neach displayed orbit.\nThe Figure 9 depicts some orbits in the spacetime (ERN). In particular, we illustrate\nthe results from §2.1: when there is a horizon, the squared energy is smaller than unity and\nthere are circular orbits with arbitrary energy when we have a naked singularity. Notice\nalso the presence of bounded orbits in the sup-extremal case.\nIn the Figures 10 and 11, we show some Bertotti–Robinson orbits, with ϵ=±1,Q= 1/2\nandm= 1. The Figure 12 displays orbits in the original Bertotti–Robinson spaces where\nm=q= 0.\nFinally, some particular unstable orbits may be observed in the sup-extremal Euclidean\ncases. In the Figure 13 are displayed some “flower orbits” in a horizon-less ERN spacetime,\nwhile in the Figure 14, we plot some “star orbits” in a horizon-less BR space. These can be\nseen as analogues of “leaf orbits” in the usual Lorentzian solutions, see [LP08, Figs. 14, 15].\nRegarding the shadows, just as in [Gar23], we use the color grid shown in Figure 15 as our\nbase picture for the shadows. In the Figure 16, we depict the shadows of an ERN spacetime\n20PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n(a)Orbits with Q= 0.998\n (b)Orbits with Q= 1.098\n(c)Circular orbits with Q=\n1.098\nFigure 9. Some Euclidean Reissner–Nordstr¨ om orbits with M= 1.\nwith various charges. The render time of each figure is approximately 350 seconds, on an\n8-core 3.00 GHz CPU with 16 Go of RAM4.\nTo illustrate our results on deflection, we also wrote a code similar to the shadowing\nprogram, but which rather displays the total angular deflection δαϕ(here, αis the angle\nbetween the initial velocity vector and the radial direction passing through the common\nconverging point of the “light rays”. The results are depicted in the Figures 17 and 18.\nObserve that, in accordance with the Section 3, the deflection is always positive in the usual\nSchwarzschild spacetime, with a visible event horizon, while in the Euclidean world, the\nhorizon disappears and the presence of a null and maximal deflection rings is manifest.\nLastly, in the Figure 19, we give the shadows of ERN spacetimes on a celestial background.\nThe original picture is from the NASA and the render time for each shadow is about 2800\nseconds, the resolution of the picture being of 1080 pixels. While the null deflection ring is\nimpossible to pinpoint on such a picture, the maximal deflection ring is still quite noticeable.\nObserve moreover that, again in contrast to the usual Schwarzschild metric, the horizon-full\ncaseQ2< M2and the naked singularity case Q2> M2produce similar pictures, the most\nnoticeable difference being the size Rmaxof the maximal deflection ring, which decreases as\nthe charge increases. This is in accordance with the lowest order approximation of Rmax\nprovided in (21).\n4We have tested the efficiency of the shadowing program by running it on (randomly generated) images\nwith 10 ×10 to 500 ×500 pixels and we made an exponential regression on the render time. We found\ntime[s]≈e−10.6×pix2.5,\nwith a regression coefficient r >0.99.\n21ARTHUR GARNIER\n(a)q= 0\n (b)q= 0.998\n (c)q= 1.098\nFigure 10. Orbits in Lorentzian Bertotti-Robinson spacetimes with Q=\n1/2 and m= 1.\n(a)q= 0\n (b)q= 0.998\n(c)q= 1.098\nFigure 11. Orbits in Euclidean Bertotti-Robinson spacetimes with Q= 1/2\nandm= 1.\n(a)ϵ=−1\n(b)ϵ= 1\nFigure 12. Orbits in Bertotti–Robinson spaces with Q= 1 and m=q= 0.\nFigure 13. Flower orbits in ERN space with M= 1 and Q= 1.098.\n22PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nFigure 14. Star orbits in BR space with ϵ=−1,Q= 1/2,m= 1 and\nq= 1.098.\n(a)The original 600 ×600 pixels color\ngrid.\n(b) Picture obtained with the code\n(M=Q= 0,v= 1).\nFigure 15. The color grid used in the program.\n(a)Q= 0\n (b)Q= 0.998\n (c)Q= 1.098\nFigure 16. Shadows of ERN spacetimes with M= 1 and v= 1.5.\n6.Conclusions\n6.1.Summary. In this work, we study the geodesic motion in the two different kinds of\nspherically symmetric electro-vacuum Euclidean solutions of Einstein’s equation with com-\nplex vector potential, namely the Reissner–Nordstr¨ om and Bertotti–Robinson-like instan-\ntons. More precisely, after proving that these are indeed the only two possible such solutions\n(and that these two are incompatible), we derive the motion equations and give the main\nproperties of the test-particle orbits, using their motion constants such as the energy and\nangular momentum.\nWe start with the Euclidean Reissner–Nordstr¨ om solution, for which we prove that if the\nspacetime has a horizon, then all exterior (non-constant) geodesics have bounded energy\n23ARTHUR GARNIER\n(a)v= 1.5\n (b)v= 2.5\n (c)v= 4\n(d)v= 6\n (e)v= 8\n (f)v= 12\nFigure 17. Deflection maps in Euclidean Schwarzschild spacetime ( M= 1)\nwith several velocities at infinity.\n(a)v= 0.72\n (b)v= 0.998\n (c)v= 1 (photon)\nFigure 18. Deflection maps in Lorentzian Schwarzschild spacetime ( M= 1)\nwith several velocities at infinity.\nand, otherwise, there are circular orbits with arbitrary energy. In particular, the spacetime\nfeatures a horizon if and only if there are no bounded orbits. This generalizes the results\nfrom [BE22] on the Euclidean Schwarzschild solution and shows that the differences in\nthe dynamics of the Lorentzian and Euclidean solutions is deeper than the apparent mere\nsign change in the line element. Then, we show that the polar radial motion equation\nmay be put in Weierstrass form, something that we can take advantage of to numerically\nsolve the motion equation. Moreover, this approach shows that, in the special case of the\nEuclidean Schwarzschild metric, the (polar) phase portrait in Binet variable5describes a\nreal elliptic curve. This allows us to derive a new proof of the results from [BE22], using\nthe elementary geometry of the curve. Furthermore, we observe that this elliptic curve is\nalways disconnected, while in the usual Lorentzian Schwarzschild solution, it can be either\nconnected or not.\n5specifically, in an affine transform of the Binet variable\n24PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\n(a)The base picture (Arches cluster).\n (b)Q= 0\n(c)Q= 0.998\n (d)Q= 1.098\nFigure 19. Shadows of ERN spacetimes with M= 1 and v= 1.5 on a\ncelestial background (original image: https://images.nasa.gov/details/\nGSFC_20171208_Archive_e000717 ).\nWe then study the gravitational bending of geodesics in Euclidean Schwarzschild space-\ntimes. More precisely, given a geodesic with closest approach radius r=rminand velocity\nat infinity v2= lim r/∫hortrightarrow∞(dr/dτ)2, we first provide an analytic formula for the deflection\nangle δϕthat occurs between the two asymptotic directions of the geodesic, in terms of\nCarlson’s elliptic integrals, which are used to numerically compute the deflection. Then, we\ngive approximations for δϕwhen rmin/∫hortrightarrow∞using the previous results of [AR02]. We observe\nthat when v >1, there are particular values ρ0andρmaxfor which δϕ= 0 for rmin=ρ0\nandδϕis maximal for rmin=ρmax. Estimates for ρ0andρmaxare provided when v≳1.\nWe then do the same for the deflection δαϕof geodesics passing through a point at fixed\nradius r0and with varying angle 0 < α < π/ 2, modelling orbits coming from infinity to\nthe eye of an observer. We give approximations, when r0≫2Mandv≳1, for the critical\nangles α0andαmaxcorresponding to the null and maximal deflection rings, as well as for\nthe observable size of these rings. We use numerical evaluations of the analytic formulas to\ncheck the accuracy of our estimates.\nObserve that the inequality δϕ < 0 for close perihelia means that the central mass repels\nsuch test-particles, while it attracts them at bigger perihelia. In particular, at fixed initial\nradius and velocity at infinity v >1, there is a critical angle 0 < α 0< π/ 2 such that δα0ϕ= 0\n(see Figure 7b for instance), meaning that the corresponding curve in the ( r, θ, ϕ )-space is\nundistinguishable from that of a flat geodesic (i.e. a straight line).\n25ARTHUR GARNIER\nNext, we do the same for the Bertotti–Robinson solution for which, after a technical\nlemma on ordinary differential equations, we give a general analytic solution, in terms of\n(hyperbolic) trigonometric functions. In particular, orbits are either periodic or unbounded\nand there is a unique circular orbit, which is exterior exactly when the metric has no horizon.\nFinally, we provide some details on a Python code6, designed to plot orbits in the afore-\nmentioned spacetimes, as well as to draw shadows of the Euclidean Reissner–Nordstr¨ om\nfamily, by the usual backward ray-tracing method. The numerical computation takes ad-\nvantage of the Weierstrass form of the polar equation, coupled with the well-known Carlson\n[Car95] and Coquereaux–Grossmann–Lautrup [CGL90] algorithms, to produce an efficient\nand rather fast code. This method is the same as the one used by the author in [Gar23] to\nray-trace Reissner–Nordstr¨ om–(anti)de Sitter black holes. We illustrate the code and our\nresults by providing some figures.\n6.2.Discussion and perspectives. In the Figures 13 and 14, we depict some particu-\nlarly shaped unstable periodic orbits, that can be seen as analogues of the so-called “leaf\norbits” that occur in the Lorentzian framework [LP08]. However, we have no theoretical\ninterpretation for these orbits and this could be a matter of interest for future works.\nIn§2.3, we have observed that a Euclidean Schwarzschild orbit can mathematically be\ninterpreted as a Lorentzian space-like Schwarzchild orbit with purely imaginary energy. To\ndecide whether this is merely a mathematical curiosity or is a manifestation of a deeper\nphysical phenomenon is beyond the knowledge of the author and could be an interesting\nsubject to investigate as well.\nIn§3, we give approximate values for the null and maximal deflection rings of a sup-\nphoton type particle (i.e. with velocity at infinity v≳1). Moreover, we have numerically\nobserved that such particular rings occur exactly when v > 1. It would be suitable to\nprovide a rigorous proof of this fact.\nBesides the theoretical results, one aim of this work is to provide an open, transparent,\nuser-friendly and customizable Python code to plot orbits and draw shadows of spherically\nsymmetric (asymptotically flat) electro-vacuum instantons. This has a negative side though,\nwhich is that the code is under-optimal and still slow in comparison to other widely-used\nray-tracing codes, such as GYOTO ,GRay or the more recent OSIRIS [Vin+11; CP ¨O13; Vel+22].\nTherefore, it would be interesting to improve it by using a GPU parallelization, for instance.\nAs in [Gar23], the shadowing program is designed to work with a common plane back-\nground image and in order to avoid distortions, we took the compromise of projecting the\nimage on a celestial hemisphere. However, to be able to catch any possible particle that is\nray-traced, we had to take the whole celestial sphere into account and we arbitrarily chose to\nproject a mirrored version of the original image onto it. Thus, the code could be completed\nby producing a panoramic version of it.\nAcknowledgments. I am much grateful to Emmanuele Battista for bringing the subject\nto my attention and for many fascinating discussions, as well as for his careful reading of\nthe present work and his useful comments.\nI also warmly thank the referee for their useful and accurate comments and suggestions.\nFinally, a big thanks goes out to Olivier Goubet for communicating to me the nice connect-\nedness argument proving the lemma C.\nCode availability statement. This manuscript has associated code in a data repository.\nThe code generated and analysed during the current study is available in the GitHub repos-\nitory https://github.com/arthur-garnier/euclidean_orbits_and_shadows .\n6available at https://github.com/arthur-garnier/euclidean_orbits_and_shadows.git\n26PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nAppendix A.Proof of the unicity result from §1\nLet d s2=gµνdxµdxνbe a spherically symmetric solution of the field equation. As\nproved in [Hof32], a Lorentzian spherically symmetric solution of the electro-vacuum Einstein\nequations is necessarily static and the proof can be adapted verbatim to the Euclidean case.\nTherefore, the metric is diagonal: d s2=gµµ(dxµ)2and moreover, the functions gττand\ngrronly depend on r. Furthermore, since the restriction to a hypersurface with constant\ncoordinates ( τ, r) must be an SO(3)-invariant metric on S2, there is a positive smooth\nfunction ρ:]r+,+∞[−/∫hortrightarrowR∗\n+such that gθθdθ2+gϕϕdϕ2=ρ(r)r2dΩ2. Thus the metric may\nbe reduced to the form\nds2=ρ(r)\u0002\nu(r)dτ2+v(r)dr2+r2dΩ2\u0003\n.\nThe Christoffel symbols Γαµν=1\n2gαβ(gβµ,ν+gβν,µ−gµν,β) of this metric read\n\n\nΓτµν=(uρ)′\n2uρ\u00120 1 0 0\n1 0 0 0\n0 0 0 0\n0 0 0 0\u0013\n, Γrµν=1\n2vρ\n−(uρ)′0 0 0\n0 ( vρ)′0 0\n0 0 −r(rρ′+2ρ) 0\n0 0 0 −rsin2θ(rρ′+2ρ)\n,\nΓθµν=\n0 0 0 0\n0 0rρ′+2ρ\n2rρ0\n0rρ′+2ρ\n2rρ0 0\n0 0 0 −sinθcosθ\n,Γϕµν=\n0 0 0 0\n0 0 0rρ′+2ρ\n2rρ\n0 0 0 cotan θ\n0rρ′+2ρ\n2rρcotan θ 0\n,\nwhere we have dropped the dependence in the variable rfor simplicity.\nOn the other hand, the electro-magnetic field tensor associated to the potential Aµis\ngiven by\nFµν=Aµ,ν−Aν,µ=−iQ\nr2\u00120 1 0 0\n−1 0 0 0\n0 0 0 0\n0 0 0 0\u0013\nso that the Maxwell equations\n0 =Fµν;µ=Fµν,µ+ ΓµµλFλν+ ΓνµλFµλ=1√detg\u0010p\ndetgFµν\u0011\n,µ\nare satisfied for all ν̸=τand we have Fµτ;µ=Frτ,r+ΓµµrFrτ=iQ(uv)′\n(ruvρ )2, so that the Maxwell\nequations hold if and only if there is a constant k∈R∗such that v=k/u. Therefore, the\nstress-energy tensor Tµνreads\nTµν=1\nµ0\u0012\ngαβFαµFβν−1\n4gµνFαβFαβ\u0013\n=1\n4π\u0012Q2\n2kr4ρ2gµν−gααFµαFαν\u0013\n=Q2\n8πr2ρdiag\u0012−u\nkr2,−1\nur2,1\nk,sin2θ\nk\u0013\n.\nNext, the Ricci tensor Rµν= Γαµν,α−Γαµα,ν+ ΓαβαΓβµν−ΓανβΓβµαand Einstein tensor\nGµν=Rµν−1\n2Rgµνare diagonal and have\n\n\n2krρR ττ=−u(ruρ′′+rρu′′+ 2ρ′(ru′+u) + 2ρu′),\n2ruρ2Rrr=−3ruρρ′′−rρ2u′′+ 3ru(ρ′)2−2ρρ′(ru′+u)−2ρ2u′,\n2kρR θθ=−r2uρ′′−ρ′(r2u′+ 4ru) + 2ρ′(k−u−ru′),\nas well as\n\nkr2ρ2Gττ=u(r2uρρ′′−3r2u(ρ′)2/4 +rρρ′(ru′+ 4u)/2 +ρ2(k−u−ru′)),\nGrr=kGττ/u2−ρ′′/ρ+ 3/2(ρ′/ρ)2\nkρ2Gθθ=r(ruρρ′′+rρ2u′′/2−3ru(ρ′)2/4 +ρρ′(ru′+u) +ρ2u′),\n27ARTHUR GARNIER\nand we have Rϕϕ= sin2θRθθandGϕϕ= sin2θGθθ. Thus, if the Einstein equation Gµν=\n8πTµνholds, then we have\n0 =2ρ2\nu2\u0000\nu2(Grr−8πTrr)−k(Gττ−8πTττ)\u0001\n= 3(ρ′)2−2ρρ′′\nand the function ρsatisfies ( ρ/ρ′)′=−1/2, so that there are constants α, β∈Rsuch that\nρ(r) = (αr+β)−2. The metric now reads\nds2= (αr+β)−2(u(r)dτ2+ku(r)−1dr2+r2dΩ2).\nWe then distinguish two cases:\nIfβ̸= 0, then the field equations reduce to the single equation Grr= 8πTrr, which may\nbe re-written as\nβr3(αr+β)u′(r) +βr2u(r)(β−2αr) + (αr+β)2((α2Q2−k)r2+ 2Q2αβr+Q2β2) = 0 ,\nan equation which is equivalent to\n\u0012ru(r)\n(αr+β)3\u0013′\n=(k−α2Q2)r2−2Q2αβr−Q2β2\nβr2(αr+β)2\nand we integrate the right-hand side by decomposing it into simple rational fractions. For\nα̸= 0, we obtain the solution\nu(r) =(αr+β)3\nr\u0012\nγ+1\nβ\u0012Q2\nr−k\nα(αr+β)\u0013\u0013\n,\nfor some additional constant γ∈R. Ifα= 0, then we get the equation\nr3u′(r) +r2(u(r)−k) +Q2β2= 0,\nwhose solution is\nu(r) =k+ ˜α/r+Q2β2/r2\nfor some ˜ α∈R. Observe now that since β̸= 0, the map r/map∫to/∫hortrightarrowr(αr+β)−1is a diffeomorphism\nonto its image for any αand if we let\n˜τ:=τ√\nk\nβ, R:=r\nαr+β,˜Q:=Q√\nk,˜M:=\n\n1\n2\u0010\n1\nα+αQ2−β2γ\nk\u0011\nifα̸= 0,\n−˜α\n2kβotherwise,\nthen we obtain the Reissner–Nordstr¨ om form of the statement.\nAssume now that β= 0 (forcing α̸= 0), in which case the equation Grr= 8πTrr\nreduces to k=α2Q2. Moreover, the remaining non-trivial component of the field equation\nisGθθ= 8πTθθ, which yields\nr2u′′(r)−2ru′(r) + 2( u(r)−α2Q2) = 0 ,\nwhose solution reads u(r) =ˇβr2+ ˇγr+α2Q2for constants ˜β,˜γ∈R. Letting\nˇτ:=αQτ, q :=pˇβ\nαQ, m:=−ˇγ\n2α2Q2\nwe obtain the form\nds2=1\nα2\u00141−2mr+q2r2\nr2d˜τ2+dr2\nr2(1−2mr+q2r2)+ dΩ2\u0015\n.\nBut in the new coordinates (˜ τ, r, θ, ϕ ), the vector potential becomes Aµ=−iα−1r−1d˜τ, so\nthat we may assume that α= 1/Q, thus obtaining the stated form of the metric.\nTo conclude, it remains to compute the Kretschmann scalar K=RαβµνRαβµν of each\nmetric of the statement. To do this, we use the symmetry and give the non-zero components\n28PARTICLE DYNAMICS IN SPHERICALLY SYMMETRIC ELECTRO-VACUUM INSTANTONS\nRαβµν =gαλ(Γλβν,µ−Γλβµ,ν+ ΓλσµΓσβν−ΓλσνΓσβµ) of the Riemann tensor for indices\nα < β andµ < ν . First, for the Bertotti–Robinson metric we have\nRτrτr=−Q2\nr4, Rθϕθϕ=Q2sin2θ=⇒K=8\nQ4\nis indeed independent of r. For the Reissner–Nordstr¨ om form, we have\n\nRτθτθ=Rτϕτϕ\nsin2θ=(˜Q2−˜MR)(R2−2˜MR+˜Q2)\nR4 , R τrτr=2˜MR−3˜Q2\nR4,\nRrθrθ=Rrϕrϕ\nsin2θ=˜Q2−˜MR\nR2−2˜MR+˜Q2, R θϕθϕ= (2 ˜MR−˜Q2) sin2θ.\nand thus\nK=8(6˜M2R2−12˜M˜Q2R+ 7˜Q4)\nR8,\nwhich depends on Rand thus the Kretschmann scalar expressed in the original coordinates\ndepends on ras well, as claimed. □\nAppendix B.Proof of the energy constrain in the horizon-full case ( §2.1)\nAssume first that Q2≤M2. For simplicity, we may rescale the radius by 1 /Mand assume\nthatM= 1. From (3), we have\nE2≥1⇐⇒ ∆(r0)2≥∆(r0) + ∆( r0)−1˙r2\n0+r2\n0˙ϕ2\n0⇐⇒ P(r0)≤0,\nwhere P=PQ2,˙r2\n0,˙ϕ2\n0∈R8[x] is the following polynomial of degree at most 8:\nP(x) =−x6∆(x)(∆(x)2−∆(x)−˙r2\n0∆(x)−1−x2˙ϕ2\n0)\n=˙ϕ2\n0x8−2˙ϕ2\n0x7+ (Q2˙ϕ2\n0+ ˙r2\n0)x6+ 2x5−(8 +Q2)x4\n+ 8(1 + Q2)x3−2Q2(6 +Q2)x2+ 6Q2x−Q6.\nWe will inductively prove that Pis positive on the interval I=]r+,+∞[ (with r+= 1 +p\n1−Q2), hence contradicting the above inequality and establishing the result. Observe\nfirst that P(8)/8! = ˙ϕ2\n0≥0 soP(7)is non-decreasing on Iand since\n1\n7!P(7)(r+) = 2 ˙ϕ2\n0(3 + 4p\n1−Q2)≥0\nwe find that P(7)≥0 so that P(6)is non-decreasing on I. We repeat the process, with\n1\n6!P(6)(r+) = 3 ˙ϕ2\n0\u0010\n5 + 9(1 −Q2) + 14p\n1−Q2\u0011\n+ ˙r2\n0,\nwhich is positive since γis non-constant. Thus, we get P(6)>0 on IandP(5)is increasing\n(not only non-decreasing) on I. Next, we have\n1\n5!P(5)(r+) =2p\n1−Q2h\n45˙ϕ2\n0+ 25 ˙ϕ2\n0(1−Q2) + 3 ˙r2\n0i\n+ 20 ˙ϕ2\n0+ 120 ˙ϕ2\n0(1−Q2) + 2(1 + 3 ˙ r2\n0)>0,\n1\n4!P(4)(r+) =5(1 +p\n1−Q2)2h\n3(Q2˙ϕ2\n0+ ˙r2\n0) + 14 ˙ϕ2\n0(1 +p\n1−Q2)p\n1−Q2i\n+ 10p\n1−Q2+ 2−Q2>0,\n1\n3!P(3)(r+) =p\n1−Q2h\n12˙ϕ2\n0(3Q4+ 2) + 60 ˙ r2\n0+ 4 + 4(1 −Q2)(78˙ϕ2\n0+ 5 ˙r2\n0+ 1)i\n+ 20 ˙r2\n0+ 4(1−Q2)(4 + 15 ˙ r2\n0) + 6 ˙ϕ2\n0(Q2+ (1−Q2)(56−25Q2))≥0,\n29ARTHUR GARNIER\nand\n1\n2P′′(r+) =p\n1−Q2h\n60 ˙r2\n0+ 4(1−Q2)(2 + 15 ˙ r2\n0) + 2 ˙ϕ2\n0(33Q4−136Q2+ 112)i\n+ 15 ˙r2\n0Q4\n+˙ϕ2\n0(−13Q6+ 174 Q4−384Q2+ 224) + 4(1 −Q2)(30 ˙r2\n0+ (1−Q2) + 1) ≥0,\nP′(r+) =p\n1−Q2h\n6 ˙r2\n0(Q4−12Q2+ 16) + 2 ˙ϕ2\n0(−Q6+ 18Q4−48Q2+ 32)i\n+ 6 ˙r2\n0(5Q4−20Q2+ 16) + 4 ˙ϕ2\n0(−3Q6+ 19Q4−32Q2+ 16) ≥0,\nand thus, P′is increasing on Iand has P′(r+)≥0, so Pis increasing on I. We conclude\nby observing that P(r+) = ˙r2\n0r6\n+≥0.\nTo prove the second statement, assume that Q2> M2, choose e > 0 and consider a\ncircular (equatorial) geodesic γ= (τ, r, π/ 2, ϕ) with energy E=e. Introducing the potential\nV(r) :=s\n∆(r)\u0012L2\nr2−1\u0013\n,\nthe condition that γis circular reads\nV(r)2+e2= 0 =∂V\n∂r.\nThe first equation imposes\nL=±rs\n1−e2\n∆(r),\nwhile the second yields\n0 =∂V\n∂r=1\n2V\u0012\u0012L2\nr2−1\u0013∂∆\n∂r−2L2∆\nr3\u0013\n⇔(e2−1)r4+M(4−3e2)r3+ 2(Q2(e2−1)−2M2)r2+ 4MQ2r−Q4= 0.\nThis quartic has a positive root, since it evaluates to −Q4<0 atr= 0 and is positive when\nr≫0. □\nAppendix C.A technical lemma on an implicit differential equation\nLemma. For any (α, β)∈R∗×R, their is a unique smooth maximal non-constant solution\nof the incomplete initial value problem\u001a\n˙y2=α(y2−β2),\ny(0) = β.\nMoreover this solution is globally defined and given, for s∈R, by\ny(s) =βcosh( s√α).\nProof. The following argument is due to Olivier Goubet. First, up to exchanging y↔−y,\nwe may assume that β≥0. Let ybe a maximal non-constant solution of the problem,\ndefined on a (maximal, open) interval I. Consider the open subset\nU:={s∈I|y(s)> β} ⊂R∗\nand suppose that ] a;b[⊂Uis a connected component of U, of finite length. By continuity\nofy, we have β≤lims/∫hortrightarrowa+y(s) =y(a)≤β, so that y(a) =β=y(b) and by Rolle’s theorem,\nwe may choose a < t < b such that 0 = y′(t) =α(y(t)2−β2). This leads to y(t)≤ |y(t)|=β:\na contradiction.\nThis proves that Uhas no connected component of finite length, leaving four possibilities:\n•IfU=∅, then yis constant.\n30REFERENCES\n•IfU=R∗\n+, then y=βonR−and solving the well-defined ODE ˙ y=p\nα(y2−β2)\nonUleads to\ny(s) =\u001a\nβcosh( s√α) if s >0,\n1 otherwise .\n•The case U=R∗\n−is the symmetric of the previous one.\n•ifU=R∗, then we solve ˙ y=±p\nα(y2−β2) onR∗\n±and obtain the stated solution,\nthe only one being non-constant and smooth.\n□\nReferences\n[AR02] A. Accioly and S. Ragusa. “Gravitational deflection of massive particles in classi-\ncal and semiclassical general relativity”. In: Class. Quantum Grav. 19.21 (2002),\npp. 5429–5434. doi:10.1088/0264-9381/19/21/308 . Corrected in “Corrigen-\ndum”. In: Class. Quantum Grav. 20.22 (2003), pp. 4963–4964. doi:10.1088/\n0264-9381/20/22/C01 .\n[AA21] S. Aksteiner and L. Andersson. “Gravitational Instantons and special geometry”.\nIn: (Dec. 2021). arXiv: 2112.11863 [gr-qc] .\n[And90] M. T. Anderson. “Short geodesics and gravitational instantons”. In: J. Differen-\ntial Geometry 31.1 (1990), pp. 265–275. doi:10.4310/jdg/1214444097 .\n[AFS15] M. F. Atiyah, G. Franchetti, and B. J. Schroers. “Time evolution in a geometric\nmodel of a particle”. In: J. High Energ. Phys. 2015.62 (2015). doi:10.1007/\nJHEP02(2015)062 .\n[AMS12] M. F. Atiyah, N. S. Manton, and B. J. Schroers. “Geometric models of matter”.\nIn:Proc. R. Soc. A 468 (2012), pp. 1252–1279. doi:10.1098/rspa.2011.0616 .\n[BE22] E. Battista and G. Esposito. “Geodesic motion in Euclidean Schwarzschild ge-\nometry”. In: Eur. Phys. J. C 82 (1088 2022). doi:10.1140/epjc/s10052-022-\n11070-w .\n[Bel+75] A. A. Belavin et al. “Pseudoparticle solutions of the Yang–Mills equations”. In:\nPhys. Lett. B 59.1 (1975), pp. 85–87. doi:10.1016/0370-2693(75)90163-X .\n[Bel02] A. M. Beloborodov. “Gravitational bending of light near compact objects”. In:\nApJ 566.2 (2002). doi:10.1086/339511 .\n[Ber59] B. Bertotti. “Uniform electromagnetic field in the theory of general relativity”.\nIn:Phys. Rev. 116 (1959), pp. 1331–1333. doi:10.1103/PhysRev.116.1331 .\n[BH08] J. Bri¨ et and D. Hobill. Determining the dimensionality of spacetime by gravita-\ntional lensing . 2008. arXiv: 0801.3859 .\n[Car95] B. C. Carlson. “Numerical computation of real or complex elliptic integrals”. In:\nNumerical Algorithms 10.1 (1995), pp. 13–26. doi:10.1007/BF02198293 .\n[CP¨O13] C. Chan, D. Psaltis, and F. ¨Ozel. “ GRay: A massively parallel GPU-based code for\nray tracing in relativistic spacetimes”. In: ApJ 777.1 (2013). doi:10.1088/0004-\n637X/777/1/13 .\n[CT11] Y. Chen and E. Teo. “A new AF gravitational instanton”. In: Phys. Lett. B 703.3\n(2011), pp. 359–362. doi:10.1016/j.physletb.2011.07.076 .\n[CM22] A. Cie´ slik and P. Mach. “Revisiting timelike and null geodesics in the Schwarzschild\nspacetime: general expressions in terms of Weierstrass elliptic functions”. In:\nClass. Quantum Grav. 39.22 (2022). doi:10.1088/1361-6382/ac95f2 .\n[CGL90] R. Coquereaux, A. Grossmann, and B. E. Lautrup. “Iterative method for calcula-\ntion of the Weierstrass elliptic function”. In: IMA Journal of Numerical Analysis\n10 (1990), pp. 119–128.\n[Dar59] C. Darwin. “The gravity field of a particle”. In: Proc. R. Soc. Lond. A 249 (1959),\npp. 180–194. doi:10.1098/rspa.1959.0015 .\n[DA09] J. Dexter and E. Agol. “A fast new public code for computing photon orbits in a\nKerr spacetime”. In: ApJ 696.2 (2009). doi:10.1088/0004-637X/696/2/1616 .\n31REFERENCES\n[DT22] M. Dunajski and P. Tod. “Conformal geodesics on gravitational instantons”.\nIn:Math. Proc. Camb. Phil. Soc. 173.1 (2022), pp. 123–154. doi:10 . 1017 /\nS0305004121000463 .\n[EGH80] T. Eguchi, P. B. Gilkey, and A. J. Hanson. “Gravitation, gauge theories and\ndifferential geometry”. In: Phys. Rep. 66.6 (1980), pp. 213–393. doi:10.1016/\n0370-1573(80)90130-1 .\n[Els84] T. Elster. “Quantum vacuum energy near a black hole: the Maxwell field”. In:\nClass. Quantum Grav. 1 (1 1984). doi:10.1088/0264-9381/1/1/007 .\n[Esp92] G. Esposito. Quantum gravity, quantum cosmology and Lorentzian geometries .\nLecture notes in Physics Monographs. Springer, 1992. doi:10.1007/978- 3-\n662-14495-4 .\n[Gar23] A. Garnier. “Motion equations in a Kerr–Newman–de Sitter spacetime: some\nmethods of integration and application to black holes shadowing in Scilab”. In:\nClass. Quantum Grav. 40.13 (2023), p. 135011. doi:10 . 1088 / 1361 - 6382 /\naccbfe .\n[GD06] L. A. Gergely and B. Dar´ azs. “Weak gravitational lensing in brane-worlds”. In:\nPubl. Astron. Dep. Eotvos Univ. 17 (2006), pp. 213–219. doi:10.48550/arXiv.\nastro-ph/0602427 .\n[Gib79] G. W. Gibbons. “Quantum field theory in curved spacetime”. In: General rel-\nativity - An Einstein centenary survey . Ed. by S. W. Hawking and W. Israel.\nCambridge University Press, 1979.\n[GH77] G. W. Gibbons and S. W. Hawking. “Action integrals and partition functions in\nquantum gravity”. In: Phys. Rev. D 15 (10 1977). doi:10.1103/PhysRevD.15.\n2752.\n[GH79] G. W. Gibbons and S. W. Hawking. “Classification of gravitational instanton\nsymmetries”. In: Commun. Math. Phys. 66 (1979), pp. 291–310. doi:10.1007/\nBF01197189 .\n[GV12] G. W. Gibbons and M. Vyska. “The application of Weierstrass elliptic functions\nto Schwarzschild null geodesics”. In: Class. Quantum Grav. 29.6 (2012). doi:\n10.1088/0264-9381/29/6/065016 .\n[Hag30] Y. Hagihara. “Theory of the relativistic trajectories in a gravitational field of\nSchwarzschild”. In: Japanese Journal of Astronomy and Geophysics 8 (1930),\npp. 67–176.\n[HH76] J. B. Hartle and S. W. Hawking. “Path-integral derivation of black-hole radi-\nance”. In: Phys. Rev. D 13 (8 1976), pp. 2188–2203. doi:10.1103/PhysRevD.\n13.2188 .\n[Haw79] S. W. Hawking. “Euclidean Quantum Gravity”. In: Recent Developments in\nGravitation: Carg` ese 1978 . Ed. by M. L´ evy and S. Deser. Springer, 1979, pp. 145–\n173.doi:10.1007/978-1-4613-2955-8_4 .\n[Haw77] S. W. Hawking. “Gravitational instantons”. In: Phys. Lett. A 60 (2 1977), pp. 81–\n83.doi:10.1016/0375-9601(77)90386-3 .\n[HL16] G. He and W. Lin. “Gravitational deflection of light and massive particles by a\nmoving Kerr–Newman black hole”. In: Class. Quantum Grav. 33.9 (2016). doi:\n10.1088/0264-9381/33/9/095007 .\n[He+20] G. He et al. “Gravitational deflection of massive particles in Schwarzschild–de\nSitter spacetime”. In: Eur. Phys. J. C 80.835 (2020). doi:10 . 1140 / epjc /\ns10052-020-8382-z .\n[Hof32] B. Hoffmann. “On the spherically symmetric field in relativity”. In: The Quarterly\nJournal of Mathematics os-3.1 (1932), pp. 226–237. doi:10.1093/qmath/os-\n3.1.226 .\n[Jan15] R. Jante. “On the spectrum of some gravitational instantons”. PhD thesis. Heriot–\nWatt University, 2015. url:http://hdl.handle.net/10399/3083 .\n32REFERENCES\n[KL21] H. K. Kunduri and J. Lucietti. “Existence and uniqueness of asymptotically\nflat toric gravitational instantons”. In: Lett. Math. Phys. 111.133 (2021). doi:\n10.1007/s11005-021-01475-1 .\n[LP08] J. Levin and G. Perez-Giz. “A periodic table of black hole orbits”. In: Physical\nReview D 77 (10 2008). doi:10.1103/PhysRevD.77.103005 .\n[Li+19] Z. Li et al. “Gravitational deflection of massive particles by a Schwarzschild black\nhole in radiation gauge”. In: Commun. Theor. Phys. 71.10 (2019), pp. 1219–1226.\ndoi:10.1088/0253-6102/71/10/1219 .\n[LR18] A. Lindberg and S. Rayan. “Geodesics on a Kerr–Newman–(anti-)de Sitter in-\nstanton”. In: J. Geom. Phys. 132 (2018), pp. 114–130. doi:10.1016/j.geomphys.\n2018.05.018 .\n[MS99] M. Mars and W. Simon. “A proof of uniqueness of the Taub–bolt instanton”. In:\nJ. Geom. Phys. 32.2 (1999), pp. 211–226. doi:10.1016/S0393-0440(99)00023-\n6.\n[MS98] J. McMahon and V. Snyder. Elements of the differential calculus . The Cornell\nMathematical Series. American Book Company, New York, 1898.\n[MM89] F. Mellor and I. Moss. “Black holes and gravitational instantons”. In: Class.\nQuantum Grav. 6 (10 1989), pp. 1379–1385. doi:10.1088/0264-9381/6/10/008 .\n[MS09] R. Monteiro and J. E. Santos. “Negative modes and the thermodynamics of\nReissner–Nordstr¨ om black holes”. In: Phys. Rev. D 79 (6 2009). doi:10.1103/\nPhysRevD.79.064006 .\n[MT09] R. Mosna and G. Tavares. “New self-dual solutions of SU(2) Yang–Mills theory\nin Euclidean Schwarzschild space”. In: Phys. Rev. D 80 (10 2009), p. 105006.\ndoi:10.1103/PhysRevD.80.105006 .\n[Nas04] M. S. El Naschie. “How gravitational instanton could solve the mass problem\nof the standard model of high energy particle physics”. In: Chaos, Solitons &\nFractals 21.1 (2004), pp. 249–260. doi:10.1016/j.chaos.2003.12.001 .\n[OPY11] J. J. Oh, C. Park, and H. S. Yang. “Yang–Mills instantons from gravitational in-\nstantons”. In: J. High Energ. Phys. 2011.87 (2011). doi:10.1007/JHEP04(2011)\n087.\n[Pag78] D. Page. “Taub–NUT instanton with an horizon”. In: Phys. Lett. B 78.2 (1978),\npp. 249–251. doi:10.1016/0370-2693(78)90016-3 .\n[PJ19] X. Pang and J. Jia. “Gravitational lensing of massive particles in Reissner–\nNordstr¨ om black hole spacetime”. In: Class. Quantum Grav. 36.6 (2019). doi:\n10.1088/1361-6382/ab0512 .\n[Pu+16] H. Pu et al. “ Odyssey : A public GPU-based code for general-relativistic radiative\ntransfer in Kerr spacetime”. In: ApJ 820.2 (2016), pp. 105–116. doi:10.3847/\n0004-637X/820/2/105 .\n[Rob59] I. Robinson. “A solution of the Maxwell-Einstein equations”. In: Bull. Acad.\nPolon. Sci 7 (1959), pp. 351–352.\n[Str13] N. Straumann. General Relativity . Second edition. Graduate texts in Physics.\nSpringer, 2013.\n[Tek02] B. Tekin. “Yang–Mills solutions on Euclidean Schwarzschild space”. In: Phys.\nRev. D 65 (8 2002), p. 084035. doi:10.1103/PhysRevD.65.084035 .\n[Vel+22] J. M. Vel´ asquez-Cadavid et al. “ OSIRIS : a new code for ray tracing around com-\npact objects”. In: Eur. Phys. J. C 82.103 (2022). doi:10.1140/epjc/s10052-\n022-10054-0 .\n[Vin+11] F. H. Vincent et al. “ GYOTO : a new general relativistic ray-tracing code”. In:\nClass. Quantum Grav. 28.22 (2011). doi:10.1088/0264-9381/28/22/225011 .\n[Vis93] M. Visinescu. “The geodesic motion on generalized Taub–NUT gravitational in-\nstantons”. In: Z. Phys. C - Particles and Fields 60 (1993), pp. 337–341. doi:\n10.1007/BF01474631 .\n33REFERENCES\n[Vit+24] D. Viththani et al. “Particle motion and tidal force in a non-vacuum-charged\nnaked singularity”. In: (2024). arXiv: 2402.02069 [gr-qc] .\n[YZ23] Y. Yang and X. Zhang. “Geodesics on metrics of Eguchi–Hanson type”. In: Eur.\nPhys. J. C 83.574 (2023). doi:10.1140/epjc/s10052-023-11762-x .\nUniversit ´e de Picardie,\nLAMFA (UMR 7352 du CNRS),\n33 rue St Leu,\nF-80039 Amiens Cedex 1,\nFrance\nEmail address :arthur.garnier@math.cnrs.fr\n34" }, { "title": "2401.15812v1.Building_graphs_with_high_minimum_degree_on_a_budget.pdf", "content": "Building graphs with high minimum degree on a budget\nKyriakos Katsamaktsis∗Shoham Letzter†\nJanuary 30, 2024\nAbstract\nWe consider the problem of constructing a graph of minimum degree k≥1 in the following con-\ntrolled random graph process, introduced recently by Frieze, Krivelevich and Michaeli. Suppose\nthe edges of the complete graph on nvertices are permuted uniformly at random. A player,\nBuilder, sees the edges one by one, and must decide irrevocably upon seeing each edge whether\nto purchase it or not.\nSuppose Builder purchases an edge if and only if at least one endpoint has degree less than k\nin her graph. Frieze, Krivelevich and Michaeli observed that this strategy succeeds in building\na graph of minimum degree at least kbyτk, the hitting time for having minimum degree k.\nThey conjectured that any strategy using εnfewer edges, where ε >0 is any constant, fails\nwith high probability.\nIn this paper we disprove their conjecture. We show that for k≥2 Builder has a strategy which\npurchases n/9 fewer edges and succeeds with high probability in building a graph of minimum\ndegree at least kbyτk. For k= 1 we show that any strategy using εnfewer edges fails with\nprobability bounded away from 0, and exhibit such a strategy that succeeds with probability\nbounded away from 0.\n1 Introduction\nTheuniform random graph Gn,mis the random graph drawn uniformly out of all labelled n-vertex\ngraphs with medges. The series of papers by Erd˝ os and R´ enyi during 1959-1966 [6–9] started an\nin-depth study of random graphs, where they explored how a typical Gn,mvaries as mgrows from\n0 to\u0000n\n2\u0001\n. The study of random discrete structures and their applications has grown to one of the\nmost active and fruitful areas of research in combinatorics, with connections to many other fields\nof study, including theoretical computer science, number theory, analysis and statistical physics.\nLetN=\u0000n\n2\u0001\nande1, . . . , e Nbe a uniformly random permutation of the edges of the complete graph\nonV= [n]. Then we can identify Gn,mwith the first medges in the permutation. The random\ngraph process is the nested sequence ( Gn,m)N\nm=0of random graphs. One of the most striking\n∗Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK. Email:\nkyriakos.katsamaktsis.21 @ucl.ac.uk . Research supported by the Engineering and Physical Sciences Research\nCouncil [grant number EP/W523835/1].\n†Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK. Email:\ns.letzter @ucl.ac.uk . Research supported by the Royal Society.\n1arXiv:2401.15812v1 [math.CO] 29 Jan 2024discoveries of Erd˝ os and R´ enyi was that several increasing properties appear rather suddenly in\nthe random graph process. Recall that a graph property Pis a collection of graphs with vertices\ninV, and is increasing if it is closed under the addition of edges with both ends in V. For\nexample, the property of being Hamiltonian and the property of being connected are increasing\nproperties. They showed that for several increasing properties P, including being connected, there\nis an m0=m0(n) such that, when mis much larger than m0(in an appropriate sense), with high\nprobability1Gn,msatisfies P, and when mis much smaller than m0, with high probability Gn,m\ndoes not satisfy P. The hitting time for an increasing property Pis defined to be the random\nvariable τP= min {m∈[N] :Gn,msatisfies P}. A central problem in the theory of random\ngraphs asks to determine asymptotically the hitting time for various graph properties.\nThe question of determining τHC, the hitting time for Hamiltonicity, was first raised by Erd˝ os\nand R´ enyi in 1960 [7], and subsequently studied by several authors. The breakthrough is due\nto P´ osa [16] and Korshunov [14], who independently showed in 1976 that, for some constant\nC, when m≥Cnlogn, with high probability Gn,mis Hamiltonian. Their result was improved\nby various authors until Koml´ os and Szemer´ edi in 1983 [13] and independently Korshunov in\n1977 [15] showed that with high probability it is asymptotically the same as τ≥2, the hitting\ntime for having minimum degree at least 2. Erd˝ os and R´ enyi in 1961 [8] proved that with high\nprobability τ≥2= (1 + o(1))nlogn\n2, so this seems to settle the question of ‘when’ the random graph\nprocess becomes Hamiltonian. However, the precise control on the number of edges allows us to\nask more probing questions. Koml´ os and Szemer´ edi in 1983 [13] claimed and a year later Bollob´ as\nproved [3] that, in fact, with high probability, τHC=τ≥2. That is, with high probability, the\nvery edge that increases the minimum degree to 2 is the one that makes the random graph process\nHamiltonian. This type of result is the strongest possible one can hope for regarding the emergence\nof an increasing property in the random graph process.\nClearly a graph with nvertices needs only nedges to be Hamiltonian, far fewer than the approxi-\nmatelynlogn\n2at the hitting time. Is there an algorithm that finds a Hamilton cycle in the random\ngraph process by time τHCby considering only o(nlogn) edges? Motivated by this question, Frieze,\nKrivelevich and Michaeli [11] introduced the following controlled random graph process. Suppose\nthere is a player, Builder, who sees the edges in the random permutation e1, . . . , e None by one,\nand must decide irrevocably upon seeing eiwhether to purchase it or not. A ( t, b)-strategy is an\nonline algorithm (deterministic or randomised) that Builder follows in the above model where she\nsees only the first tedges in the random permutation of E(Kn) and is allowed to purchase at\nmost bof them. Builder’s objective is for her graph to satisfy a given increasing graph property\nP. For example, there is a simple ( τcon, n−1)-strategy such that Builder’s graph is connected\nat the hitting time for connectivity: Builder purchases an edge if and only if it decreases the\nnumber of connected components in her graph. Regarding Hamiltonicity, Frieze, Krivelevich and\nMichaeli [11, Theorem 3] showed that there exists a ( τHC, Cn)-strategy such that with high prob-\nability BτHCis Hamiltonian, where Cis a large constant. Anastos [1] observed that Ccannot be\nimproved to 1 + o(1). However, in the same paper he showed that if Builder is allowed to see an\nadditional ετHCedges after τHC, then she can purchase at most (1 + o(1))nedges and ensure that\nwith high probability her graph is Hamiltonian.\nThe current paper and Theorem 1 in [11] consider the problem of Builder constructing a graph\n1We say that a sequence of events ( An)n∈Nholds with high probability if lim n→∞P[An]→1.\n2of minimum degree at least a constant k≥1 at the hitting time τkfor the property of having\nminimum degree at least k. Let Bibe Builder’s graph just before the edge ei+1is revealed.\nSuppose that Builder follows the obvious greedy strategy, i.e. that she purchases eiif and only\nif at least one of its ends has degree at most k−1 in Bi−1. This strategy clearly succeeds in\nbuilding a graph with minimum degree at least kat time τk. By following this strategy, Builder’s\ngraph is distributed according to the k-th nearest neighbour random graph model. This model was\nstudied by Frieze and Cooper [5], who proved that with high probability the resulting graph Ok\nhas ( ok+o(1))nedges, for some explicit constant ok∈(k/2,3k/4] (cf. Corollary 2.10). Hence,\nthis gives a ( τk,(ok+o(1))n)-strategy for Builder that (always) succeeds in constructing a graph\nof minimum degree at least k.\nConjecture 7 in [11] asserts that for any constant ε >0, if Builder follows any ( τk,(ok−ε)n)-\nstrategy, with high probability her graph fails to have minimum degree at least k. Since o1= 3/4\nandok≥k\n2+3\n8fork≥2 (cf. Corollary 2.10), the next two theorems disprove this conjecture, for\nthe cases k≥2 and k= 1 respectively.\nTheorem 1.1. Letk≥2be an integer and δ >0be constant. Builder has a (τk,(k/2+2−k+δ)n)-\nstrategy that with high probability yields a graph with minimum degree at least k.\nTheorem 1.2. LetC > 0be constant. Builder has a\u0000\nτ1,(1/2 +C−1/2/2)n\u0001\n-strategy that with\nprobability at least (1−o(1))√\nC\neCyields a graph with minimum degree at least 1.\nHowever, as the next theorem shows, a weaker version of [11, Conjecture 7] holds for k= 1. Recall\nthato1= 3/4.\nTheorem 1.3. Letε >0be constant. Any (τ1,(3/4−ε)n)-strategy of Builder fails, with probability\nat leastε3\n100, in building a graph with minimum degree at least 1.\nNotation. Throughout nis assumed to be sufficiently large, and this is the only assumption made\nin all instances of asymptotic notation. We set V= [n] and N=\u0000n\n2\u0001\n. We use e1, . . . , e Nto denote\na uniformly random permutation of the edges of the complete graph on V, soeiis the edge at the\ni-th step of the random graph process. We denote by Gn,ithe graph at the i-th step of the random\ngraph process, i.e. the graph on Vwith edges {e1, . . . , e i}. The minimum degree of a graph Gis\nδ(G) and the maximum degree is ∆( G).\n2 Preliminaries\nThe next theorem, which estimates the number of degree dvertices in Gn,Cn, follows from a\nstraightforward application of the second moment method, see [10, Theorem 3.3]. It is also a\nsimple consequence of a more general result of Bollob´ as [2].\nTheorem 2.1. Letdbe a fixed positive integer and C, δ be positive constants. Let Xdbe the\nnumber of vertices of degree dinGn,Cn, and let µd=Cde−C\nd!·n. Then, with high probability,\n|Xd−µd| ≤δµd.\nNext we state two concentration inequalities: Chernoff’s bound and McDiarmid’s inequality.\n3Theorem 2.2 (Chernoff’s bound, see [12, eq. (2.5, 2.6) and Theorem 2.8]) .LetXbe the sum of\nmutually independent indicator random variables and write µ=E[X]. Then for any 0< t≤µ,\nP[|X−µ| ≥t]≤2e−t2\n3µ.\nTheorem 2.3 (McDiarmid’s inequality) .LetX1, . . . , X mbe independent random variables with\nXitaking values in a set Si. Let f:Q\ni∈[m]Si→Rbe a function such that for any x,x′∈Q\ni∈[m]Si\ndiffering only at the kthcoordinate we have\n\f\ff(x)−f(x′)\f\f≤ck,\nfor some ck∈R. Then, for every t >0,\nP\u0002\n|f(X1, . . . , X m)−E[f(X1, . . . , X m)]|> t\u0003\n≤2 exp\u0012\n−2t2\nPm\nk=1c2\nk\u0013\n.\nWe will use at several points the following straightforward consequence of Chernoff’s bound and\nthe union bound.\nProposition 2.4. With high probability, the maximum degree of Gn,mwithm=O(n)is at most\n10 log n.\nFor a modern treatment of the next two theorems see chapter 4 of [10]. The first one determines\nwhen, with high probability, the random graph process starts having minimum degree at least k.\nTheorem 2.5 (Erd˝ os and Renyi [8]) .Letm=n\n2(logn+ (k−1) log log n+f(n)).\nP[δ(Gn,m)≥k] =\n\n1−o(1) iff(n)→ ∞\no(1) iff(n)→ −∞ .\nThe next one shows that, with high probability, once a random graph has minimum degree at least\n1, it is also connected.\nTheorem 2.6 (Bollob´ as and Thomason [4]) .In the random graph process, with high probability\nthe hitting time for connectivity is the same as for minimum degree 1.\nFor two real random variables X, Y, we say Xstochastically dominates Yif for all t∈RP[X≥t]≥\nP[Y≥t]. The next lemma is standard, see e.g. Section 23.9 of [10] for a proof.\nLemma 2.7. LetY1, . . . , Y nbe arbitrary real random variables, and let X1, . . . , X nbe mutually\nindependent real random variables.\nSuppose that for all i∈[n]anda1, . . . , a i−1∈R,Yiconditioned on Y1=a1, . . . , Y i−1=ai−1\nstochastically dominates Xi. ThenPn\ni=1Yistochastically dominatesPn\ni=1Xi.\nSuppose instead that for all i∈[n]anda1, . . . , a i−1∈R,Yiconditioned on Y1=a1, . . . , Y i−1=ai−1\nis stochastically dominated by Xi. ThenPn\ni=1Xistochastically dominatesPn\ni=1Yi.\n4The next lemma gives a simple ( O(n),k+o(1)\n2n)-strategy that produces, with high probability, a\ngraph in which almost all vertices have degree at least k. It is due to Frieze, Krivelevich and\nMichaeli [11]. For completeness, we give a detailed proof. We remark that here and throughout\nthe paper we use the convention that Biis Builder’s graph just before the edge ei+1is revealed.\nLemma 2.8 (Lemma 2.15, [11]) .Letk≥1be an integer, ε∈(0,1)a constant and set C=kε−2.\nSuppose that for all i∈[Cn], if both ends of eihave degree less than kinBi−1, then Builder\npurchases ei. Then, with high probability, BCnhas at most εnvertices of degree at most k−1.\nProof. Let\nUi={v∈V: degBi(v)≤k−1}\nbe the set of vertices of degree at most k−1 in Builder’s graph right after she has seen eiand decided\nwhether to purchase it. Hence, if both ends of eiare in Ui−1then eiis purchased by Builder. Let\nXibe the indicator random variable for the event that either |Ui−1| ≤εn, or both ends of eiare in\nUi−1. Let a1, . . . , a i−1∈ {0,1}and write X εni\n·Ph\n|Ui−1|> εn\f\f\fX εn\f\f\fX εn\f\fX εn\f\fX0are\nconstants. Let Hbe a graph on Vwith maximum degree at most 10 log n. Suppose we draw\nuniformly at random and with repetition edges from the complete graph on Vuntil there are m\ndistinct edges. Then, with high probability, for every v∈Vsuch that at least one of the edges\ndrawn is incident to vinH, there are at least kdistinct edges incident to vwhich are not in H.\nProof. LetLv⊆E(H) be the set of edges incident to v∈V, so|Lv| ≤10 log n. Let tbe the\n(random) number of edges we draw with repetition until we see mdistinct edges.\nWe claim that with high probability t∈[m, m + (log n)3]. Indeed, let Xi, i∈[m], be the random\nvariable counting the number of edges drawn until the i-th distinct edge, and set X0= 0. Then\nXi−Xi−1is a geometric random variable with mean1\n1−(i−1)/(n\n2), and\nE[Xm] =mX\ni=1E[Xi−Xi−1] =mX\ni=11\n1−(i−1)/\u0000n\n2\u0001\n=mX\ni=1 \n1 +i−1\u0000n\n2\u0001+O\u0012i2\nn4\u0013!\n=m+ Θ\u0012m2\nn2\u0013\n+O\u0012m3\nn4\u0013\n=m+ Θ((log n)2).\nLetYbe the number of repeated edges until we see mdistinct edges. Then the above calculation\nimplies E[Y] =O((log n)2), whence from Markov’s inequality with high probability Y≤(logn)3.\nTherefore t∈[m, m + (log n)3], with high probability.\n6To complete the proof of the lemma we now show that, when drawing tedges with repetition, for\nsome t∈[m, m + (log n)3], with high probability, for every vwith at least one edge drawn from\nLv, there are at least k(distinct) incident edges not in Lv. Let ˆ e1, . . . , ˆetbe the drawn edges and\nletAv,jbe the event that ˆ ej∈Lv. Let Bv,jbe the event that the number of distinct edges in\n{ˆe1, . . . , ˆet} \\ˆejincident to vand disjoint from Lvis at most k−1. The lemma follows from the\nnext claim.\nClaim 2.12. For every t∈[m, m + (log n)3], the probability that there exist v∈V, j∈[t]such\nthatAv,j∩Bv,jholds is at most 1/n.\nProof. Fixv∈V,j∈[t]. Because ˆ e1, . . . , ˆetare drawn with repetition, the events Av,j, Bv,jare\nindependent. We have P[Av,j]≤10 log n\n(n\n2).\nFornsufficiently large,\nP[Bv,j]≤k−1X\nℓ=0\u0012t\nℓ\u0013 \nn−1\u0000n\n2\u0001!ℓ \n1−n−1−10 log n−(k−1)\u0000n\n2\u0001!t−ℓ\n≤k−1X\nℓ=0(c2nlogn+ (log n)3)ℓ2ℓn−ℓ\u0012\n1−(2−o(1))\nn\u0013(c1−o(1))nlogn\n≤k−1X\nℓ=0(2c2nlogn)ℓ2ℓn−ℓexp (−c1logn)\n≤k−1X\nℓ=0(4c2)ℓ(logn)ℓn−c1\n≤(logn)kn−c1\n≤n−c1/2.\nHence, the probability there exist v∈V, j∈[t] such that Av,j∩Bv,jholds is, by the union bound\nover the choice of v, j, at most n·t·10 log n\n(n\n2)·n−c1/2≤1/n.\nThis completes the proof of the lemma.\n3 Proof of Theorem 1.1\nGiven k≥2 and δ >0, fix 0 < ε < 1 and C >0 such that: ε <δ\n2kandC=kε−2.\nBefore delving into the details, we outline Builder’s strategy, which we give formally in Algorithm 1.\nFor the first Cnedges Builder’s strategy is as follows. For i∈[Cn], Builder purchases ei=xiyiif\nand only if either degBi−1(xi),degBi−1(yi)< kor at least one of xi, yihas degree 0 in Gn,i−1. We\ncall edges of the former kind efficient and edges of the latter kind (which are not of the former\nkind) inefficient . Builder’s strategy for i≥Cn+ 1 is to purchase eiif and only if at least one end\nofeihas degree less than kinBi−1.\nFori∈[Cn] let\nZi={v∈V: degGn,i(v) = 0}\n7Algorithm 1 Builder’s strategy for a graph with minimum degree at least kbyτk\n1:Input: Lete1, . . . , e Nbe a uniformly random permutation of E(Kn).\n2:B0← ∅\n3:fori= 1, . . . , Cn do\n4: ifboth ends of eihave degree less than kinBi−1or least one of has degree 0 then\n5: Bi←Bi−1∪ei\n6: else\n7: Bi←Bi−1\n8: end if\n9:end for\n10:fori=Cn+ 1, . . . , τ kdo\n11: ifat least one end of eihas degree less than kinBi−1then\n12: Bi←Bi−1∪ei\n13: else\n14: Bi←Bi−1\n15: end if\n16:end for\n17:return Bτk\nand\nYi={v∈V: 1≤degBi(v)≤k−1}.\nThe next claim shows that, with high probability, after the first Cnedges are exposed almost all\nvertices in Builder’s graph have degree k, yet her graph has relatively few edges.\nClaim 3.1. Reveal the first Cnedges in the random graph process and suppose Builder follows\nAlgorithm 1. Then with high probability\n1.e(BCn)≤(k/2 + 2−k+δ/2)nand\n2.|YCn∪ZCn| ≤εn.\nThe next claim implies that, with high probability, by time τkBuilder’s graph has minimum degree\nat least k.\nClaim 3.2. With high probability, for every v∈YCn, there are at least k−1edges incident to v\namong eCn+1, . . . , e τk.\nBefore proving the claims, we show how to use them to derive Theorem 1.1.\nProof of Theorem 1.1. First we argue that following Algorithm 1, with high probability, degBτk(v)≥\nkfor all v∈V. Lines 4 and 11 of Algorithm 1 imply that degBi(v) = 0 if and only if degGn,i(v) = 0.\nHence, for every v /∈YCn∪ZCn, degBCn(v)≥k. By definition of τkandZCn, every vertex in\nZCnwill see at least kincident edges among {eCn+1, . . . , e τk}, since none of the first Cnedges are\nincident to ZCn. By Claim 3.2, with high probability, every vertex in YCnwill see at least k−1\nincident edges among {eCn+1, . . . , e τk}. Therefore, in the second for-loop of Algorithm 1 Builder\nwill purchase at least kincident edges for each v∈ZCn, and at least k−degBCn(v) incident edges\nto each v∈YCn. Hence, with high probability, degBτk(v)≥kfor all v∈V.\n8It remains to show that, with high probability, e(Bτk)≤(k/2+2−k+δ)n. By Claim 3.1, with high\nprobability |YCn∪ZCn| ≤εn. Hence, the number of edges Builder purchases during the second\nfor-loop of Algorithm 1 is at most kεn. Therefore, with high probability, the total number of edges\nbought is at most\ne(BCn) +kεn≤(k/2 + 2−k+δ/2 +kε)n≤(k/2 + 2−k+δ)n,\nwhere we used the upper bound on e(BCn) from Claim 3.1.\nWe now prove the two claims.\nProof of Claim 3.1. Lemma 2.8 implies the second point of the lemma. Clearly, the number of\nefficient edges bought is at most kn/2. It remains to show that, with high probability, the number\nof inefficient edges is at most (2−k+δ/2)n.\nForr, s∈[n], r≥s,let Φ( r, s) be the collection of edges in the whole random graph process (i.e.\nall\u0000n\n2\u0001\nedges) which increase the degree of one of their ends to rand the other end to s, and set\nϕ(r, s) =|Φ(r, s)|. The inefficient edges are a subset of Gn,Cn, and by Proposition 2.4, with high\nprobability ∆( Gn,Cn)≤10 log n. Therefore, with high probability, the number of inefficient edges\nis at mostP10 log n\nr=k+1ϕ(r,1).\nWe want to show, using McDiarmid’s inequality, that ϕ(r, s) is, with high probability, concentrated\naround its mean, for s≤r≤10 log n. However, because E[ϕ(r, s)] is linear (as we shall soon see)\nand is determined by quadratically many variables, applying McDiarmid’s inequality directly to\nϕ(r, s) would not give concentration with high probability. To avoid this issue, let m=n(logn)10/2,\nlet Φ m(r, s) = Φ( r, s)∩Gn,mbe the subset of the first medges in the random graph process that\nare in Φ( r, s) and write ϕm(r, s) =|Φm(r, s)|.\nDefine\nµr,s=1\n2δ(r,s)\u0012r+s−1\ns−1\u0013\n2−r−s+1,\nwhere δ(r, s) is 1 if r=sand 0 otherwise (Kronecker delta). We will show that, with high\nprobability, ϕ(r, s) = (1 + o(1))µr,snfor all 1 ≤s≤r≤10 log n.\nTo do so, we first note that ϕ(r, s) = ϕm(r, s), with sufficiently high probability. To obtain\nconcentration, we will reveal the first msteps of the random graph process in two stages. First we\nsample the graph at step mof the random graph process, Gn,m, and show that with sufficiently\nhigh probability Gn,misalmost-regular i.e. degGn,m(v) = (1 + O((log n)−2))(log n)10for all v∈V.\nWe will then calculate the expectation of ϕm(r, s) conditioned on Gn,mbeing any given graph G,\nand show that this expectation is (1 + o(1))µr,snfor every almost-regular G. Next, conditioned\nonGn,mbeing a fixed graph G, we sample the first msteps of the random graph process by\nselecting a uniformly random permutation of E(G). Using McDiarmid’s inequality, we will show\nthat ϕm(r, s) conditioned on Gn,m=Gis, with high probability, concentrated around its mean.\nThen the required result, i.e. the concentration of ϕ(r, s) around its mean for all relevant r, s, will\nfollow easily.\nFixr, ssuch that 1 ≤s≤r≤10 log n. First, note that ϕ(r, s) =ϕm(r, s) whenever δ(Gn,m)≥\n10 log n, soϕ(r, s) =ϕm(r, s) with probability 1 −exp\u0000\n−Ω((log n)2\u0001\n.\n9Now let Gbe any graph on Vwith medges. We claim that E\u0002\nϕm(r, s)|Gn,m=G\u0003\nequals\n1\n2δ(r,s)X\nu,v∈V(G):uv∈E(G)1\ndegG(u) + degG(v)−1\u0012degG(u)−1\nr−1\u0013\u0012degG(v)−1\ns−1\u0013\n\u0012degG(u) + degG(v)−2\nr+s−2\u0013. (1)\nThis was (essentially) already noted in [5] for r, s∈[k] (kconstant), when we do not condition on\nGn,mbeing any fixed graph. To see why this is true, note that for r̸=s, the event e=uv∈Φm(r, s)\nis the union of the (disjoint) events that egives degree rtouand degree stov, and vice-versa.\nForr=sof course we have only one such event. Thus it suffices to show that the probability, over\na random permutation of E(G), that uvgives degree rtouandstovis\n1\ndegG(u) + degG(v)−1\u0012degG(u)−1\nr−1\u0013\u0012degG(v)−1\ns−1\u0013\n\u0012degG(u) + degG(v)−2\nr+s−2\u0013\nand then (1) follows by linearity of expectation. The event in question is determined by the first\nr+s−1 edges incident to {u, v}and occurs if both uvis the edge at position r+s−1 in the\npermutation of the degG(u) + degG(v)−1 edges incident to either uorv; and among the first\nr+s−2 edges incident to either uorv, exactly r−1 are incident to uands−1 are incident to v.\nIt readily follows that this event occurs with the claimed probability. Indeed, the denominator is\nthe number of ways of choosing r+s−1 edges among the degG(u) + degG(v)−1 edges touching\nuorv, with a single distinguished edge (the one in position r+s−1), and the numerator is the\nnumber of ways of choosing this many edges so that the distinguished edge is uvand the remaining\nr+s−2 edges consist of r−1 edges touching uands−1 edges touching v.\nNow suppose that Gis almost-regular, i.e. that degG(v) = (1 + O((log n)−2)(log n)10for every\nv∈V. Using this assumption, (1) and that\u0000a\nb\u0001\n=\u0010\n1 +O(b2\na)\u0011\nab\nb!, we get that for all such G,\nE\u0002\nϕm(r, s)|Gn,m=G\u0003\n= (1 + o(1))1\n2δ(r,s)\u0012r+s−1\ns−1\u0013\n2−r−s+1n. (2)\nWe now show that ϕm(r, s) conditioned on Gn,m=Gis concentrated around its mean. For\nthis we use McDiarmid’s inequality, which is applicable in the random graph process, because we\ncan generate a uniformly random permutation of E(G) in the following way2. Fix a continuous\nprobability distribution Dand for each e∈E(G) let Xe∼ D be an independent sample from D.\nOrdering the edges from smallest sampled value to largest yields a uniformly random permutation\nofE(G). To apply the inequality for ϕm(r, s), suppose we unilaterally change the value Xuvsampled\nfor the edge uv∈Gn,m, and leave the values sampled for all other edges {Xe:e∈E(G)\\uv}\nunchanged. We claim that then ϕm(r, s) changes by at most 4. To see this, observe that the\ngraph spanned by Φ m(r, s) has ∆(Φ m(r, s))≤2: each vertex can only be incident to edges whose\nposition in the random permutation increase its degree to rors. Changing Xuvcan affect only\nedges incident to uorv, hence changing ϕm(r, s) by at most 4. Thus, McDiarmid’s inequality\n2Simply permuting uniformly at random E(G) does not allow us to use McDiarmid’s inequality because the\nposition of edges are not independent from one another.\n10(Theorem 2.3) implies that\nPh\f\fϕm(r, s)−E\u0002\nϕm(r, s)|Gn,m=G\u0003\f\f≥n2/3i\n≤2 exp \n−2n4/3\n16m!\n= exp\u0000\n−Ω\u0000\n(logn)2\u0001\u0001\n.\nIn particular, using (2), with probability 1 −exp\u0000\n−Ω\u0000\n(logn)2\u0001\u0001\n, we have that ϕm(r, s) conditioned\nonGn,m=Gis (1 + o(1))µr,sn.\nTherefore, since Gn,mis almost-regular with probability 1 −exp\u0000\n−Ω\u0000\n(logn)2\u0001\u0001\n(by Chernoff’s\nbound, Theorem 2.2), because we have ϕm(r, s) =ϕ(r, s) with probability 1 −exp\u0000\n−Ω\u0000\n(logn)2\u0001\u0001\n,\nand by a union bound over s, rwith 1 ≤s≤r≤10 log n, we have that ϕ(r, s) = (1 + o(1))µr,sn,\nwith high probability.\nIn particular, with high probability, ϕ(r,1) = (1 + o(1))2−r, for every rwith k+ 1≤r≤10 log n.\nHence, with high probability, the number of inefficient edges bought is at most\n(1 +o(1)) 10 log nX\nr=k+12−rn!\n≤(2−k+δ/2)n,\nas required.\nProof of Claim 3.2. By Proposition 2.4, with high probability ∆( Gn,Cn)≤10 log n. Condition on\nthis event.\nDraw with repetition edges (ˆ ei)i≥1until t=n\n2(logn+(k−2) log log n+log log log n) distinct edges\nare drawn. Let ˆGbe the graph of these tdistinct edges and let ˆGi={ˆe1, . . . , ˆei}.\nWe couple ( ˆGi)i≥1with ( Gn,j)j≥Cn+1in the following natural way. Suppose Gn,jis the current\nstage of the random graph process and ˆ eiis the edge we have just drawn. If ˆ ei/∈Gn,j, we set\nej+1:= ˆeiand so update Gn,j+1:=Gn,j∪ {ˆei}. Otherwise, ej+1remains unrevealed, we do not\nupdate Gn,j, and keep drawing edges with repetition until we draw a new one. Clearly every edge\nnot in Gn,jhas the same probability of being added, so the coupling indeed induces the random\ngraph process.\nBy Lemma 2.11 and Theorem 2.5, with high probability, every v∈YCnhask−1 distinct incident\nedges in ˆG\\Gn,Cn: indeed, by Theorem 2.5 with high probability every vertex vhas at least\nk−1 incident edges in ˆG, and for each veither all incident edges are disjoint from Gn,Cn, or\notherwise by Lemma 2.11 at least k−1 are. The latter is applicable since t= Θ( nlogn) and\n∆(Gn,Cn)≤10 log n.\nFinally, by Theorem 2.5, with high probability, τk> t+Cn. Hence, indeed we have E(ˆG\\Gn,Cn)⊆\n{eCn+1, . . . , e τk}, with high probability.\n4 Proof of Theorems 1.2 and 1.3\nSetε=C−1/2. We assume ε <3/4, since otherwise both theorems easily hold. Theorem 1.3\ntrivially holds because then Builder’s graph is empty. Theorem 1.2 holds because then Builder can\npurchase ( o1+ 1/8)nedges by emulating O1(using o1= 3/4).\n11We will keep track of the following sets of vertices.\nXi={v∈V: degBi(v)≥1}\nYi={v∈V: degBi(v) = 0 and degGn,i(v)≥1}\nZi={v∈V: degGn,i(v) = 0}.\nFor both theorems we will use the following lemma, which we prove at the end of this section.\nNotice the first part of the lemma makes no assumption whatsoever on Builder’s strategy.\nLemma 4.1. Letm∈[Cn]and suppose Builder has gone through the first msteps of the random\ngraph process. For every strategy Builder may follow for edges ej, j≥m+ 1,Bτ1has an isolated\nvertex with probability at least(1−o(1))|Ym|\n|Ym|+|Zm|.\nSuppose that for j≥m+ 1Builder follows the strategy of purchasing ejif and only if at least one\nend of ejis isolated in Bj−1. Then with probability(1−o(1))|Zm|\n|Ym|+|Zm|,Bτ1has no isolated vertices.\nWe now prove Theorem 1.2 according to which Builder has a ( τ1,(1/2 +C−1/2/2))-strategy which\nsucceeds in building a graph with minimum degree at least 1, with probability at least (1 −o(1))√\nC\neC.\nProof of Theorem 1.2. Builder has the following strategy, given formally in Algorithm 2 (any vari-\nables inside the algorithm agree with the notation defined outside), for constructing a graph\nwith minimum degree at least 1 by time τ1. For the first Cnedges, Builder purchases an\nedge if and only if both endpoints are isolated. For the remaining edges, she purchases an\nedge if and only if at least one end is isolated. Lemma 2.8 implies that, with high probabil-\nity,|YCn∪ZCn| ≤n√\nC, and Theorem 2.1 yields that with high probability |ZCn| ≥(1−o(1))e−Cn.\nCondition on these two events. Then from Lemma 4.1, Bτ1has no isolated vertices with probability\nat least (1 −o(1))|ZCn|\n|YCn|+|ZCn|≥(1−o(1))√\nC\neC.It remains to show that e(Bτ1)≤(1/2 +C−1/2/2)n.\nInspecting Algorithm 2 (or simply the definitions), it is not hard to see that Yi∪Zi⊆Yi−1∪Zi−1:\nonly lines 15 and 28 add elements to Yi, and in both cases vertices move to Yifrom Zi−1. Also, it\nis clear that Zi⊆Zi−1, see lines 11, 16, 24 and 29 of Algorithm 2. Therefore, during the second\nfor-loop, i.e. for edges ej, j≥Cn+ 1, Builder will purchase only edges incident to YCn∪ZCn.\nMoreover, for j≥Cn+ 1,|Yj∪Zj|strictly decreases by at least 1 for each purchased edge. For\nj≤Cn+ 1, clearly |Yj∪Zj|decreases by exactly 2 (lines 7, 10 and 11 of Algorithm 2). Therefore\ne(Bτ1)≤e(BCn) +|YCn∪ZCn|\n= (n− |YCn∪ZCn|)/2 +|YCn∪ZCn| ≤(1 +C−1/2)n/2,\nas required.\nNow we prove Theorem 1.3 which asserts that any (3 /4−ε, τ1)-strategy fails in building a graph\nwith minimum degree at least 1, with probability at leastε3\n100.\nProof of Theorem 1.3. This is an immediate consequence of Lemma 4.1 and the next claim.\nClaim 4.2. If|Yi| ≤δnfor all i∈[Cn], then by time CnBuilder has purchased at least (o1−\n6δC−δ−2e−C−o(1))nedges.\n12Algorithm 2 Builder’s ( τ1,(3/4−ε)n)-strategy for a graph with minimum degree 1\n1:Input: Lete1, . . . , e Nbe a uniformly random permutation of E(Kn).\n2:B0← ∅ ▷ Biis Builder’s graph at step i\n3:X0← ∅ ▷non-isolated vertices in Biat step i\n4:Y0← ∅ ▷isolated vertices in Bi, non-isolated in random graph process at step i\n5:Z0←V ▷ isolated vertices in graph process at step i(so also in Bi)\n6:fori= 1, . . . , Cn do\n7: ifei⊆Yi−1∪Zi−1then ▷if both ends of eiare isolated in Bi−1\n8: Bi←Bi−1∪ei\n9: Xi←Xi−1∪ei\n10: Yi←Yi−1\\ei\n11: Zi←Zi−1\\ei\n12: else\n13: Bi←Bi−1\n14: Xi←Xi−1\n15: Yi←(Yi−1∪ei)\\Xi ▷add only previously isolated ends of ei\n16: Zi←Zi−1\\ei\n17: end if\n18:end for\n19:fori=Cn+ 1, . . . , τ 1do\n20: ifei∩(Yi−1∪Zi−1)̸=∅then ▷if at least one end of eiis isolated in Bi−1\n21: Bi←Bi−1∪ei ▷same updates as in first for-loop\n22: Xi←Xi−1∪ei\n23: Yi←Yi−1\\ei\n24: Zi←Zi−1\\ei\n25: else ▷same updates as in first for-loop\n26: Bi←Bi−1\n27: Xi←Xi−1\n28: Yi←(Yi−1∪ei)\\Xi ▷add only previously isolated ends of ei\n29: Zi←Zi−1\\ei\n30: end if\n31:end for\n32:return Bτ1\nSetδ=ε\n50Cand suppose Builder follows a strategy such that e(Bτ1)≤(o1−ε)n. Then e(Bτ1)<\n(o1−7δC−δ−2e−C)n(recalling ε=C−1/2), so the contrapositive of Claim 4.2 implies that for\nsome i∈[Cn],|Yi| ≥δn. Hence, by Lemma 4.1, Bτ1has an isolated vertex with probability at\nleast (1 −o(1))|Yi|\n|Yi|+|Zi|≥δ\n2=ε\n100C=ε3\n100.\nProof of Claim 4.2. Fori∈[Cn], let Iibe the indicator random variable for the event that eiis\nincident to Yi−1and|Yi−1| ≤δn. Conditioned on any choice of values for I1, . . . , I i−1, the expected\nvalue of Iiis at mostδn·(n−1)\n(n\n2)= 2δ.Hence, Lemma 2.7 implies thatPCn\ni=1Iiis stochastically\ndominated by a sum of Cnindependent indicator random variables with mean 2 δeach, for which\nwe can apply Chernoff’s bound. The latter gives that, with high probability,PCn\ni=1Ii≤3δCn.\nCondition on this being the case.\nBecause the claim assumes that |Yi| ≤δnfor all i∈[Cn],Iiis 1 if and only if eiis incident to\nYi−1. Hence,PCn\ni=1Iiis precisely the number of edges eiincident to Yi−1.\nLetX′\nCn=XCn∩\u0010SCn−1\ni=1Yi\u0011\nbe the set of vertices for which Builder did not purchase the first\n13incident edge in the random graph process, but did purchase a later edge (by step Cn). Since a\nvertex may move from Yi−1toXionly if there is an edge incident to Yi−1,|X′\nCn| ≤2PCn\ni=1Ii≤\n6δCn. Therefore, all but at most 6 δCn vertices in XCnare such that Builder bought the first\nincident edge to them. Let E1⊆E(BCn) be the collection of edges that Builder bought and were\nthe first incident edge to one of their endpoints.\nWe will compare Builder’s graph with the O1graph at this stage of the random graph process.\nRecall that O1consists of the first edge incident to each vertex in the random graph process, and\nthat O1with high probability has at least ( o1−o(1))nedges; condition on this being the case.\nLetOCn\n1be the subgraph of O1with edges E(O1)∩ {e1, . . . , e Cn}, so in particular E(OCn\n1)⊇E1.\nNotice that any other edge in E(OCn\n1) is the first edge in the random graph process incident to\nsome vertex in X′\nCn∪YCn, so\f\fE(OCn\n1)\\E1\f\f≤ |X′\nCn∪YCn|. Therefore,\n\f\fE(OCn\n1)\f\f≤ |E1|+\f\fX′\nCn\f\f+|YCn| ≤ |E1|+ 6δCn+δn. (3)\nSimilarly,\f\fE(O1)\\E(OCn\n1)\f\f≤ |ZCn|: every edge in E(O1)\\E(OCn\n1) is the first edge in the random\ngraph process incident to a vertex in ZCn. Hence, with high probability, |E(O1)| −\f\fE(OCn\n1)\f\f≤\n|ZCn| ≤2e−Cn,using Theorem 2.1 for the last inequality. Combining this with (3) we deduce that\n|E(O1)| ≤ |E1|+ 6δCn+δn+ 2e−Cn≤e(BCn) + 6δCn+δn+ 2e−Cn.\nRearranging gives e(BCn)≥(o1−6δC−δ−2e−C−o(1))n, as required.\nThis completes the proof of Theorem 1.3.\nAll that remains is to prove Lemma 4.1.\nProof of Lemma 4.1. As in Claim 3.2, draw with repetition edges (ˆ ei)i≥1from E(Kn) and let\n(ˆGi)i≥1be the corresponding graph process, i.e. ˆGiis the (simple) graph spanned by ˆ e1, . . . , ˆei.\nWe couple ( ˆGi)i≥1with ( Gn,j)j≥m+1in the obvious way; see the proof of Claim 3.2 for the details.\nLet\nˆτ1= min {i≥1 : for every v∈Ym∪Zm,deg ˆGi(v)≥1}.\nClaim 4.3. With high probability, a unique endpoint of ˆeˆτ1is isolated in ˆGˆτ1−1.\nWe prove Claim 4.3 in the end. Condition on its conclusion holding. Let v∈Ym∪Zmbe the\nendpoint of ˆ eˆτ1inYm∪Zmthat is isolated in ˆGˆτ1−1and notice that, by symmetry, vis uniformly\ndistributed in Ym∪Zm.\nSuppose v∈Ym. Then every vertex is non-isolated in the random graph process before ˆ eˆτ1is\ndrawn: vertices in Xm∪Ymare non-isolated in Gn,m, and every vertex in Zmhas an incident\nedge among {ˆe1, . . . , ˆeˆτ1−1}, so indeed Gn,τ1⊆Gn,m∪ˆGˆτ1−1. Moreover, vis isolated in Bτ1: none\nof the edges ej, j∈[m+ 1, τ1], is incident to v, sovis isolated in Bτ1, no matter what strategy\nBuilder follows for these steps, since she purchased no edge from e1, . . . , e mthat is incident to v.\nThis proves the first part of the lemma, as the probability that ˆ eˆτ1has a unique isolated vertex in\nˆGˆτ1−1, which is in Ym, is at least (1 −o(1))|Ym|\n|Ym|+|Zm|.\n14To prove the second part of the lemma, suppose v∈Zm. Then ˆ eˆτ1is the first incident edge to\nvin the random graph process, and since all other vertices in Zmhave an incident edge among\nˆe1, . . . , ˆeˆτ1−1, we deduce eτ1= ˆeˆτ1. Notice that Builder’s strategy of purchasing the first incident\nedge to each vertex ensures that degBτ1(u)≥1 for all u∈Zm, since each edge incident to\nu∈Zmis disjoint from Gn,m, and thus she can purchase it. Hence the lemma follows if, with\nhigh probability, for every u∈Ym, there is at least one incident edge in ˆGˆτ1\\Gn,m. This follows\nfrom the assumption that v∈Zˆτ1which implies that every vertex in Ymhas an incident edge in\nˆGˆτ, from Lemma 2.11, which is applicable since, with high probability, ∆( Gn,m)≤10 log nby\nProposition 2.4 and e(ˆGˆτ1) = Θ( nlogn), the latter being a direct consequence of Theorem 2.5.\nIndeed, if e(ˆGˆτ1)≥nlogn, by Theorem 2.5, with high probability ˆGˆτ1has no isolated vertices. If\ne(ˆGˆτ1)≤nlogn\n3, then ˆGˆτ1∪Gn,m⊆Gn,m′where m′=nlogn\n3+Cn, and Theorem 2.5 yields that,\nwith high probability, ˆGˆτ1∪Gn,mhas isolated vertices, which must be in Zm.\nProof of Claim 4.3. Lett=n(logn−log log n)\n2. The expected number of isolated vertices in ˆGtamong\nYm∪Zmis at most\n|Ym∪Zm| · \n1−n−1\u0000n\n2\u0001!n\n2(logn−log log n)\n≤nexp\u0012\n−2\nnn\n2(logn−log log n)\u0013\n= log n.\nTherefore, by Markov’s inequality, with high probability the number of isolated vertices in ˆGtis\nat most (log n)2. We condition on this event. We condition also on the event that ˆ τ1=O(nlogn),\nwhich holds with high probability: indeed, the argument in the paragraph immediately preceding\nthe current claim yields that, with high probability, e(ˆGˆτ1) = Θ( nlogn), and the calculations at the\nbeginning of Lemma 2.11 show that with high probability only o(nlogn) edges among Θ( nlogn)\ndrawn with replacement are repeated.\nThe claim will follow if among the remaining ˆ τ1−tedges (drawn with replacement) none has both\nends among Ym∪Zmwhich are isolated in ˆGt. As the following calculation shows, the expected\nnumber of such edges is o(1):\n(ˆτ1−t)\u0000(logn)2\n2\u0001\n\u0000n\n2\u0001≤O(nlogn)·O\u0012(logn)4\nn2\u0013\n=o(1).\nHence, from Markov’s inequality, with high probability no such edges are drawn.\nThis completes the proof of Lemma 4.1.\n5 Concluding remarks\nIn this paper we studied the problem of constructing a graph with minimum degree k≥1 in\nthe controlled random graph process introduced in [11]. Theorem 1.1 disproves [11, Conjecture 7]\nin a strong form for k≥2. Determining the optimal budget that yields a graph with minimum\n15degree at least kby time τk, with at least a given probability, remains an interesting open problem.\nSimilarly, Theorems 1.2 and 1.3 disprove [11, Conjecture 7] for k= 1. It would be interesting to\ndetermine the exact dependence between the budget and the optimal success probability.\nReferences\n[1] M. Anastos, Constructing Hamilton cycles and perfect matchings efficiently , arXiv:2209.09860,\n2022. 2\n[2] B. Bollob´ as, Vertices of given degree in a random graph , J. Graph Theory 6(1982), 147–155.\n3\n[3] ,The evolution of sparse graphs. , Proc. Cambridge Combinatorial Conf. in honour\nof Paul Erd˝ os (Bollob´ as, B., ed.), Graph Theory and Combinatorics, Academic Press, 1984,\npp. 35–57. 2\n[4] B. Bollob´ as and A. Thomason, Random graphs of small order , Annals of Discr. Math. (1985),\n47–97. 4\n[5] C. Cooper and A. Frieze, On the connectivity of random k-th nearest neighbour graphs , Comb.\nProbab. Comput. 4(1995), no. 4, 343–362. 3, 5, 10\n[6] P. Erd˝ os and A. R´ enyi, On random graphs I , Publ. Math. Debrecen 6(1959), 290–297. 1\n[7] ,On the evolution of random graphs , Publ. Math. Inst. Hungar. Acad. Sci. 5(1960),\n17–61. 1, 2\n[8] ,On the strength of connectedness of a random graph , Acta Math. Acad. Sci. Hungar.\n12(1961), 261–267. 1, 2, 4\n[9] ,On the existence of a factor of degree one of a connected random graph , Acta Math.\nAcad. Sci. Hungar. 17(1966), 359–368. 1\n[10] A. Frieze and M. Karo´ nski, Introduction to random graphs , Cambridge University Press, 2016,\nhttps://www.math.cmu.edu/ ∼af1p/BOOK.pdf. 3, 4\n[11] A. Frieze, M. Krivelevich, and P. Michaeli, Fast construction on a restricted budget ,\narXiv:2207.07251, 2022. 2, 3, 5, 6, 15, 16\n[12] S. Janson, A. Ruci´ nski, and T. Luczak, Random Graphs , John Wiley & Sons, 2011. 4\n[13] J. Koml´ os and E. Szemer´ edi, Limit distributions for the existence of Hamilton cycles in a\nrandom graph , Discrete Math. 43(1983), 55–63. 2\n[14] A. D. Korshunov, Solution of a problem of Erd˝ os and R´ enyi on Hamilton cycles in non-oriented\ngraphs , Soviet Mat. Dokl. 17(1976), 760–764. 2\n[15] ,Solution of a problem of Erd˝ os and R´ enyi about Hamilton cycles in non-oriented\ngraphs , Metody Diskr. Anal. Teoriy Upr. Syst., Sb. Trudov Novosibirsk (in Russian) 31(1977),\n17–56. 2\n[16] L. P´ osa, Hamiltonian circuits in random graphs , Discrete Math. 17(1976), 359–364. 2\n16" }, { "title": "2401.15825v2.Synaptic_motility_and_functional_stability_in_the_whisker_cortex.pdf", "content": "arXiv:2401.15825v2 [physics.bio-ph] 30 Jan 2024Synaptic motility and functional stability in the whisker c ortex\nNimrod Sherf1,2,4∗and Maoz Shamir1,2,3†\n1Department of Physics,\n2Zlotowski Center for Neuroscience and3Department of Physiology and Cell Biology,\nBen-Gurion University of the Negev,\nBeer-Sheva, Israel,4Department of Mathematics,\nUniversity of Houston, Houston, Texas, USA.\n(Dated: February 1, 2024)\nThe high motility of synaptic weights raises the question of how the brain can retain its function-\nality in the face of constant synaptic remodeling. Here we us ed the whisker system of rats and mice\nto study the interplay between synaptic plasticity (motili ty) and the transmission of sensory signals\ndownstream.\nRats and mice probe their surroundings by rhythmically movi ng their whiskers back and forth.\nThe azimuthal position of a whisker can be estimated from the activity of whisking neurons that\nrespond selectively to a preferred phase along the whisking cycle. These preferred phases are widely\ndistributed on the ring. However, simple models for the tran smission of the whisking signal down-\nstream predict a distribution of preferred phases that is an order of magnitude narrower than em-\npirically observed. Here, we suggest that synaptic plastic ity in the form of spike-timing-dependent\nplasticity (STDP) may provide a solution to this conundrum. This hypothesis is addressed in the\nframework of a modeling study that investigated the STDP dyn amics in a population of synapses\nthat propagates the whisking signal downstream.\nThe findings showed that for a wide range of parameters, STDP d ynamics do not relax to a\nfixed point. As a result, the preferred phases of downstream n eurons drift in time at a non-uniform\nvelocity which in turn, induces a non-uniform distribution of the preferred phases of the downstream\npopulation. This demonstrates how functionality, in terms of the distribution of preferred phases,\ncan be retained not simply despite, but because of the consta nt synaptic motility. Our analysis\nleads to several key empirical predictions to test this hypo thesis.\nI. INTRODUCTION\nMice and rats scan their proximal environment by mov-\ning their whiskers back and forth in a rhythmic manner\n[1–6]. Whisking provides important tactile signals that\nallow rats and mice to detect, locate, and identify nearby\nobjects [7–10]. The tactile signal from the whiskers is\ntransduced to a series of short electrical pulses (spike\ntrains) that propagate downstream the whisker system:\nfrom the trigeminal ganglion to the brain-stem, to the\nthalamus to layer 4 of the cortex downstream to layer\n2/3 [8, 10–15], Fig. 1(a).\nThroughout the whisker system, neurons that encode\nthephase along the whisking cycle have been found\n[7, 10, 11, 16]. These neurons, termed whisking neu-\nrons, fire in a selective manner to the phase along the\nwhisking cycle, Figs. 1(b)-1(c). Typically, whisking neu-\nrons are characterized by a unimodal tuning curve that\npeaks at the preferred phase of the neuron [7, 13, 16–21],\nFigs. 1(d)-1(e).\nThe preferred phases of whisking neurons are widely\ndistributed on the ring in a non-uniform manner,\nFig. 1(g). The distribution of preferred phases in spe-\ncific brain regions can be approximated by a von Mises\n∗sherfnim@post.bgu.ac.il\n†shmaoz@bgu.ac.il(circular normal) distribution,\nPr(φ) =eκcos(φ−ψ)\n2πI0(κ)(1)\nwhereI0(κ)is the modified Bessel function of order 0.\nThe parameters ψandκquantify the mean and width\n(or sharpness) of the distribution. Typically, the value\nofκis around 1, whereas the mean, ψ, depends on the\nbrain region [8, 13].\nThe distribution of preferred phases poses a challenge\nto our understanding of the propagation of sensory sig-\nnals in the central nervous system. To delve into the\nsource of this conundrum, we consider a simple model for\nthe transmission of whisking signals from layer 4 down-\nstream to layer 2/3 (L2/3) in the whisker cortex. In this\nsimplistic model, we ignore the effect of recurrent connec-\ntivity within L2/3, as well as the possible contribution of\ndirect thalamic input Fig. 1(g). Excitatory neurons in\nlayer 4 of the whisker cortex almost do not respond to\nwhisking, whereas about 30%of the inhibitory neurons\nin layer 4 (L4I) are whisking neurons [22, 23]. Thus, the\nmain source of the whisking signal to layer 2/3 originates\nfrom L4I neurons.\nWe model the activity of the downstream L2/3 neuron\nby a delayed linear response to its inputs (see [24]). This\nhigher level of abstraction (compared to the frequently\nused linear non-linear Poisson model [25–31]) is applied2\nto facilitate analysis, and is expressed as\nρL2/3(t) =Iex−1\nNN/summationdisplay\nk=1wkρk(t−d), (2)\nwhereIexdenotes an excitatory drive that is indepen-\ndent of the whisking phase, wkis the synaptic weight\nof thekth L4I neuron and d >0is the delay. Parame-\nterρkdenotes the activity of neuron kof the population\nof L4I neurons that serve as feed-forward input to the\ndownstream L2/3 neuron. We further assume that the\npreferred phases of the upstream population, {φk}N\nk=1,\nare i.i.d. according to Eq. (1) with mean ψL4Iand width\nκL4I. Thus, the preferred phase of the downstream neu-\nron is determined by a weighted average of the preferred\nphases of its inputs, weighted by their synaptic strengths.\nHowever, synaptic weights are highly volatile [33–36].\nFluctuations of about 50%in the synaptic weights were\nshown to occur over a period of several days [35, 36].\nAdditionally, activity-independent fluctuations were re-\nported to be on the same order of magnitude [36]. If\nsynaptic plasticity is purely activity-independent, the\nsynaptic weights, {wk}N\nk=1, will be random and indepen-\ndent of the preferred phases of the upstream L4I neurons.\nThus, the preferred phase of a downstream L2/3 neuron\nis determined by random pulling of the preferred phases\nof theNL4I neurons that serve as its input. As a result,\nthe width of the distribution of the preferred phases of\nL2/3 whisking neurons will decay to zero as the number\nof pooled L4I neurons, N, grows, Fig. 1(f). Even pool-\ning responses of as few as N= 20 L4I neurons results\ninκL2/3≈10, which is an order of magnitude larger\nthan typically observed [37]. Raising the question: what\nmechanism can generate this distribution?\nRecently, we studied the transmission of whisking sig-\nnals from excitatory neurons in the thalamus downstream\nto layer L4I neurons. We showed that activity-dependent\nplasticity in the form of spike-timing-dependent plastici ty\n(STDP, as detailed in section II B) can provide a mecha-\nnism that generates a non-trivial distribution of preferre d\nphases governed by the STDP rule [37]. However, this\nwork only considered the plasticity of excitatory synapses\nand capitalized on the strong positive feedback inherent\nto excitatory STDP [38]. In contrast, inhibitory STDP\nis typically characterized by a negative feedback mecha-\nnism [24, 39]. Thus, it remains unclear whether STDP\ncan account for the distribution of the preferred phases\nof L2/3 neurons.\nHere we examined the hypothesis that STDP may un-\nderlie the unexplained distribution of preferred phases\nin L2/3. We analyzed under which conditions inhibitory\nSTDP of L4I-to-L2/3 synapses can generate a non-trivial\ndistribution of preferred phases, and investigated how the\nSTDP rule shapes the resultant distribution. Below, we\nfirst define the network architecture and the response\nstatistics of L4I neurons to whisking, in section II A.\nThe STDP model is defined in section II B. Next, in sec-\ntion II C we study the STDP dynamics of a single synapsein the limit of weak coupling. This will serve as a ‘free\ntheory’ that will later shed light on the STDP dynam-\nics of a population of synapses. Then, in section II D\nwe analyze the STDP dynamics in the special case of an\nisotropic L4I population, i.e., with a uniform distribu-\ntion of preferred phases, κL4I= 0. Subsequently, we turn\nto investigate the effect of κL4I>0and study how the\nSTDP rule shapes the resultant distribution of phases\nin L2/3. Finally, we summarize our results, discuss the\nlimitations of this study, and propose several empirical\npredictions to test our hypothesis.\nII. RESULTS\nA. Network model and order parameters\nWe study STDP dynamics in a purely feed-forward ar-\nchitecture Fig. 1(g). Each downstream L2/3 neuron re-\nceives input from NL4I whisking neurons and a constant\nexcitatory input. The excitatory input is assumed to be\nnon-rhythmic and independent of the whisking phase. To\nfacilitate the analysis, the downstream neurons are as-\nsumed to be delayed linear neurons following Eq. (2).\nThe spiking activity of the L4I whisking neurons is\nmodeled by independent inhomogeneous Poisson pro-\ncesses with mean instantaneous firing rates (intensity),\ngiven by\n/an}bracketle{tρi(t)/an}bracketri}ht=D(1+γcos[θ(t)−φi]), (3)\nwhereρi(t) =/summationtext\nkδ(t−ti,k)is the spike train of neuron i\nwithti,kdenoting the time of the kth spike of the L4I neu-\nroniwith preferred phase φi. Parameter Dis the mean\nfiring rate during whisking (in the air, averaged over\none cycle), and γis the modulation depth. The phase\nalong the whisking cycle is taken to be θ(t) =νtwith\n¯ν=ν/(2π) = 1/Twdenoting the whisking frequency.\nThe angular brackets /an}bracketle{t···/an}bracketri}htdenote averaging with respect\nto the statistics of the noisy neuronal responses.\nUnder these assumptions, the firing rate of the down-\nstream neuron will also oscillate at the whisking fre-\nquency,¯ν, with\n/an}bracketle{tρpost/an}bracketri}ht=Dpost(1+γpostcos[νt−φpost]) (4)\nwhere from Eqs. (2) and (3) the mean firing rate, ampli-\ntude modulation and preferred phase of the downstream\nneuron are determined by global order parameters that\ncharacterize the synaptic weights profile:\n¯w=1\nNN/summationdisplay\nk=1wk (5)\n˜weiψ=1\nNN/summationdisplay\nk=1wkeiφk(6)3\n(a)\nTG TN VPML2/3 L4 I-20020\n0 100 200 300 400 500 600 700 800 900 1000-0\n050\n- 000.51\n(f)\n0 10 20 300246810(g)\nL2/3\nL4 I\nFIG. 1: Transmission of the whisking signal . (a) In the lemniscal pathway [10, 11, 14] the whisking signa l is\nrelayed downstream via the trigeminal ganglion (TG) to the t rigeminal nuclei (TN) of the brainstem down to the\nventral posterior medial (VPM) nucleus of the thalamus. Fro m the thalamus, the whisking signal in the lemniscal\npathway is transmitted downstream to layer 4 of the cortex (L 4I - mainly inhibitory neurons), and then down to\nlayer 2/3 (L2/3) [12–15]. The polar histograms represent th e distributions of preferred phases along the pathway,\nbased on [7, 13, 16, 18] - which are presented purely for purpo ses of illustration. Note, that the whisking signal is\nalso transmitted via the paralemniscal pathway [10, 18, 32] . (b)-(e) encoding of the whisking signal by a whisking\nneuron - illustration. (b) The angular position of a whisker ,β, is shown as a function of time. The angle is often\nmodeled as β(t) =βmid(t)+∆β(t)cos(φ(t)), whereβmid(t)and∆β(t)are the midpoint and the whisking amplitude,\nrespectively, that change slowly in time. (c) The whisking p haseφas a function of time is φ(t) = (νt)mod2π, whereν\nis the angular frequency of the whisking. (d) Raster plot. Th e phases in which a single model whisking neuron\nspiked are marked (black dots) for different whisking cycles (on the y-axis)). The preferred phase of the model\nneuron atφ=π/2. (e) Tuning curve. The normalized mean firing rate of the mode l neuron in (d) is shown as a\nfunction of the phase along the whisking cycle. (f) The expec ted width in the downstream layer, κL2/3, in the\nrandom pooling model, is depicted as a function of the number of pooled L4I neurons, N, for random pooling. The\nwidth of the distribution, κL2/3, was estimated from 10000 repetitions of drawing Npreferred phases of the\nupstream population with κL4I= 1. (g) Feed-forward architecture of the L4I-L2/3 section of t he pathway.4\nwhere¯wis the mean synaptic weight, and ˜weiψis its\n‘population vector’ with R∋˜w≥0. Thus yielding:\nDpost=Iex−D¯w,\nγpost=Dγ˜w/D post,and (7)\nφpost=π+ψ+dν.\nB. The STDP model\nSTDP can be thought of as a microscopic unsupervised\nlearning rule in which the synaptic weight is modified ac-\ncording to the temporal relation of pre- and post-spike\ntimes. A wide range of STDP rules has been observed\nempirically [40–48]. For example, Bi and Poo reported\na temporally asymmetric STPD rule in which an exci-\ntatory synapse was potentiated (strengthened) when the\nfiring was causal, i.e., the pre-synaptic neuron fired about\n20msbefore post, and depressed (weakened) when the\nfiring was acausal [41]. The STDP of inhibitory synapses\nhas also been reported. Woodin and colleagues described\na temporally symmetric rule [48], whereas Haass and col-\nleagues reported (in a different brain region) a temporally\nasymmetric STDP rule for inhibitory synapses [49].\nWe write the STDP rule as a sum of two processes,\npotentiation, and depression (see also [24, 28, 37, 50–53]:\n∆w=λ[f+(w)K+(∆t)−f−(w)K−(∆t)],(8)\nwhere∆wis the change in the synaptic weight w,λis\nthe learning rate, and ∆t=tpost−tpreis the time dif-\nference between the pre- and post-synaptic spike times.\nThe first term on the right-hand side of Eq. (8) represents\nthe potentiation ( +) and the second term the depression\n(−). For mathematical convenience, we assumed a sepa-\nration of variables, and thus write each process (potenti-\nation and depression) as a product of a weight-dependent\nfunctionf±(w)and a temporal kernel K±(∆t).\nAs in Gütig et al [50], we used the following choice for\nthe synaptic dependence for the STDP rule\nf+(w) = (1−w)µ(9)\nf−(w) =wµ, (10)\nwhereµ∈[0,1]controls the non-linearity of the learning\nrule.\nIn our numerical analysis, we used Gaussian kernels for\nthe temporal dependence of the STDP rule:\nK±(∆t) =1\nτ±√\n2πe−1\n2(∆t−T±\nτ±)2\n, (11)\nwhereτ±andT±are the widths and centers of the tem-\nporal kernels, respectively. Specifically, we focus on two\nfamilies of learning rules. One, as in Woodin et al [48], is\na temporally symmetric difference of Gaussians ‘Mexican\nhat’ learning rule, in which T+=T−= 0. We shall term\nthe upright Mexican hat rule, τ+<τ−, Hebbian, and the\ninverted Mexican hat rule, τ+>τ−, anti-Hebbian.The other, as in Haas and colleagues [54], is a tem-\nporally asymmetric rule with T±/ne}ationslash= 0. We shall term\nthe (anti-) Hebbian the case of ( T+<0ANDT−>0)\nT+>0ANDT−<0. In the limit of τ±→0the\ntemporal kernels converge to a Dirac delta function,\nK±(∆t)→δ(∆t−T±).\nIn the limit of slow learning, λ→0, STDP dynam-\nics effectively averages the noise in the neuronal activity.\nThe fluctuations in the synaptic weights become negligi-\nble and the stochastic dynamics of the synaptic weights\ncan then be replaced with deterministic dynamics for\ntheir means (see [28, 37, 50, 53, 55, 56], yielding\n˙wj(t)\nλ=I+\nj(t)−I−\nj(t) (12)\nwhere\nI±\nj(t) =f±(wj(t))/integraldisplay∞\n−∞Γj,L2/3(∆)K±(∆)d∆,(13)\nand\nΓj,L2/3(∆) =/integraldisplayTw\n0dt\nTw/an}bracketle{tρj(t)ρL2/3(t+∆)/an}bracketri}ht (14)\nis the temporal average of the cross-correlation between\nthejth L4I pre-synaptic neuron and the L2/3 post-\nsynaptic neuron, averaged over one period of the whisk-\ning cycle,Twsee section IVA.\nC. STDP dynamics of a single synapse\nWe begin by analyzing the simple case in which only\na single L4I-to-L2/3 synapse is plastic. Although highly\nartificial, this case constitutes a ‘free theory’ that will\nprove pivotal later. In the limit of large N, the activity\nof the downstream neuron will be almost unaffected by\nthe activity of the single plastic neuron. In this case, the\nfiring rate of the downstream neuron will oscillate at the\nwhisking frequency, ¯ν, with a fixed preferred phase, φpost,\ndetermined by its non-plastic inputs, following Eqs. (4)\nand (7).\nIn the limit of weak coupling, the cross-correlation\nbetween the plastic pre-synaptic neuron and the post-\nsynaptic cell can be approximated by the product of their\nmean responses, yielding for large N\nΓpre,L2/3(∆) =DpreDpost/parenleftBig\n1+γpreγpost\n2cos[φ+∆ν]/parenrightBig\n(15)\nwhereφ=φpre−φpost=φpre−(π+ψ+dν)is the\nphase difference between the pre- and post-post synaptic\nneurons. Substituting Eq. (15) into Eqs. (12) and (13)\none obtains that the synaptic weight converges to a single\nfixed point\nw∗(φ) =1\n/parenleftbig\nQ(φ)/parenrightbig1/µ+1(16)\nQ(φ) =¯K−+η˜K−cos(Ω−−φ)\n¯K++η˜K+cos(Ω+−φ)(17)5\nwhereη=γpreγpost\n2, and¯K±and˜K±eiΩ±are the Fourier\ntransforms of the STDP kernels:\n¯K±=/integraldisplay∞\n−∞K±(∆)d∆ (18)\n˜K±eiΩ±=/integraldisplay∞\n−∞K±(∆)e−iν∆d∆. (19)\nNote that in our specific choice of kernels, ¯K±= 1, by\nconstruction.\nPart of the utility of this simplified scenario is that it\nenables a complete understanding of how different pa-\nrameters affect the resultant fixed-point, Eq. (16). In\nthe limit of weak coupling, the STDP dynamics of a sin-\ngle inhibitory synapse is similar to that of an excitatory\nsynapse, which was analyzed in [28]. In brief, the synapse\nis potentiated or depressed depending on the phase dif-\nference between the pre- and the post-synaptic neurons,\nφ. The parameter µgoverns the smoothness of the tran-\nsition between potentiated and depressed synapses. In\nthe additive rule, i.e., µ= 0, the synapses are either\npotentiated to 1 or depressed to 0 with a sharp transi-\ntion. As the value of µincreases, the transition becomes\nsmoother, Figs. 2(a)-2(b).\nThe temporal structure of the STDP rule determines\nwhich phases will be potentiated and which will be de-\npressed. It is convenient to define the width of the profile\nas the range of values of φsuch thatw∗(φ)≥1/2, and\ncritical values of φby the condition Q(φc) = 1, yielding\nφc1,2=α0+ψ±π/2 (20)\nwith the definitions\n˜Keiα0=˜K−ei(Ω−+νd)−˜K+ei(Ω++νd). (21)\nThus, the width of the synaptic weights profile is always\nπ[57]. For a temporally symmetric learning rule Ω±= 0,\nyielding a symmetric profile, φpost−φc=±π/2. A Heb-\nbian learning rule will potentiate synapses with a phase\ndifference that is small in absolute values, |φ|< π/2.\nWhereas, an anti-Hebbian temporally symmetric rule will\npotentiate synapses with |φ|>π/2, Fig. 2(a).\nFor the temporally asymmetric delta rule, in the limit\nofτ±→0,˜K±= 1andΩ±=νT±, yieldingφpost−φc1=\nνT++T−\n2andφpost−φc2=νT++T−\n2+π. The resultant\nprofile for the Hebbian rule ( T+>0andT−<0) is\nshown in Fig. 2(b).\nD. STDP in an isotropic model\nWe now turn to study the STDP dynamics of a pop-\nulation ofNL4I-to-L2/3 inhibitory synapses, Fig. 1(g).\nFirst, we consider the case of an isotropic population,\nκL4I= 0, taking the preferred phases of the L4I neurons\nto be evenly spaced on the ring, φk= 2πk/N . The bio-\nlogical scenario of κL4I>0is addressed in section II E.1. The uniform fixed point\nIn the isotropic case, κL4I= 0, a uniform solution,\nwherewk=w∗,∀k∈ {1...N}always exists. In the\nuniform solution ¯w=w∗and˜w= 0. Consequently,\nγpost= 0and the whisking signal is not transmitted\ndownstream. Thus, the uniform solution prevents the\ntransmission of the whisking signal. Below we study the\nstability of the uniform solution in the limit of large N;\nsee section IVC for details. In this limit, there are two\ntypes of uniform fixed-point solutions:\nw∗\n1=1\n2(type1) (22)\nw∗\n2=Iex\nD(type2) (23)\nIntype 1 , potentiation and depression cancel each\nother out via the weight-dependence of the STDP rule,\nf+(w∗\n1) =f−(w∗\n1). Intype 2 , the mean inhibitory input\nto the downstream neuron balances the excitatory input,\nIex−Dw∗\n2= 0.\nAlong the uniform direction, the stable fixed point is\ngiven by min{w∗\n1,w∗\n2}. To further study the stability of\nthe fixed points we expand the dynamics to first order in\nfluctuations δw(φj) =w(φj)−w∗, yielding\nδ˙ w(φ)\nλ=Mδw. (24)\nwhereMis the stability matrix.\nThe stability matrix has two prominent eigenvalues.\nOne,mu, is in the uniform direction, ¯w. The other, mw,\nis in the ‘whisking’ direction, ˜w(with degeneracy 2). The\neigenvalues of the stability matrix around w∗\n1in this limit\nwere calculated and yielded\nmu=−µD2/parenleftBigIex\nD−1\n2/parenrightBig\n22−µ(25)\nmw≈mu,1+1\n22+µD2γ2˜Kcos(α0). (26)\nThus, the uniform fixed point w∗\n1is stable with respect\nto fluctuations in the uniform direction, mu<0, for\nIex> D/2. In the type 1 fixed point, muprovides a\nstabilizing term against fluctuations in the whisking di-\nrection,mw, Eq. (26). This stabilizing term can become\narbitrarily small by decreasing µ; thus enabling fluctua-\ntions in the whisking direction to develop and destabilize\nthe homogeneous fixed point. For Iex> D/2there is a\ncritical value of µ,\nµcrit=16γ2\nIex/D−1/2˜Kcos(α0) (27)\nsuch that for any µ < µ crit, the uniform solution is un-\nstable in the whisking direction. Since µ≥0, a critical\nvalue exists if and only if cos(α0)>0.\nAn analysis of the stability of type 2 (balanced) uni-\nform solution, w∗\n2, in the limit of large N, indicates that6(a)\n-\u0001 -\n\u0000\n20\n\u0000\n2\n\u000100.51\nϕ [rad]w*μ\n0.0001\n0.001\n0.01\n0.1\n0.2\n0.3(b)\n-π -π\n20π\n2π00.51\nϕ[rad]w*μ\n0.0001\n0.001\n0.01\n0.1\n0.2\n0.3\n(c)\n-π -π\n20π\n2π00.51\nϕ[rad]w*α0\nπ\n2.3\n1.8(d)\n-π -π\n20π\n2π00.51\nϕ[rad]w*ν[ H \u0002]\n5\n10\n15\n20\n253\u0003\nFIG. 2: Fixed-point solutions for STDP dynamics of a single synapse . The different panels show w∗as a\nfunction of φ, see Eq. (16), for different sets of parameters (depicted by c olors). (a) For different values of µfor the\nsymmetric kernel, Eq. (11). (b) For different values of µfor the delta kernel. (c) For different values of α0for both\nthe symmetric (solid lines) and the delta (dashed lines) rul es. The dashed horizontal purple line depicts\nw∗(φc) = 0.5. (d) For different values of νfor the symmetric rule. Unless stated otherwise, the parame ters for the\nsymmetric kernel are τ−= 20ms,τ+= 50ms,µ= 0.01,ν= 14hz andΓ1= 0.5. The parameters for the delta kernel\nareT−=−20ms,T+= 20ms,µ= 0.01, andν= 14hz. In panel (c) the symmetric kernel was plotted with\nT+≈20,30,−10ms and the delta kernel was plotted with T−≈35,15,4ms, yielding α0=π,2.3,1.8, respectively.\nthe two prominent eigenvalues are given by\nmu=D2/parenleftbig\n(w∗\n2)µ−(1−w∗\n2)µ/parenrightbig\n, (28)\nmw=1\n4D2γ2/parenleftbig\nf−(w∗\n2)˜K−cos(νd+Ω−)−\nf+(w∗\n2)˜K+cos(νd+Ω+)/parenrightbig\n. (29)\nType 2 fixed-point is stable with respect to uniform fluc-\ntuations for Iex< D/2. In contrast to the type 1 fixed-\npoint, heremudoes not provide a stabilizing term against\nfluctuations in other directions - as expected from the\nbalanced solution; see [24].\nSimilar to type 1 fixed-point, the balanced fixed-point\ntends to lose stability in the whisking direction as cos(α0)\nincreases. For both types of uniform fixed points, in the\nlimit of the additive learning rule, µ→0, the uniform\nfixed point is unstable for cos(α0)>0. Thus, the tem-\nporal structure of the learning rule governs the stability\nof the uniform fixed point via a single parameter, α0, as\ndefined in Eq. (21).\nFor the temporally symmetric difference of Gaussians\nrule (Eq. (11) with T±= 0) one obtains ˜Keiα0=/parenleftbig\ne−1\n2(ντ+)2−e−1\n2(ντ−)2/parenrightbig\neidν. In the case of a Hebbian\n(upright) Mexican hat ( τ+< τ−) one obtains α0=dν,\nwhereas for the anti-Hebbian inverted Mexican hat ( τ+>\nτ−),α0=π+dν. Thus, for short to moderate delays,\nd <1\n4Tw, the uniform fixed point will lose its stability\nfor smallµin the anti-Hebbian rule.\nFor the temporally asymmetric rule, we consider the\nlimit ofτ±→0, in which the STDP kernels, K±(t)converge to a Dirac delta. In this case ˜Kcos(α0) =\ncos[ν(d+T−)]−cos[ν(d+T+)] = 2sin/parenleftbig\n|ν|[T++T−\n2+\nd]/parenrightbig\nsin(|ν|T−−T+\n2). Thus, for example, in a Hebbian\ntemporally asymmetric rule T−<0andT+>0, for\n|T−|+|T+|< Tw,cos(α0)is positive for |T−| −2d <\nT+0,\nwhich induces a positive drift velocity, as shown by the\nblue square in Fig. 3(f). Whereas in Fig. 3(i) φ(s.c.)\npost−\nφ(ass)\npost<0, induces a negative drift velocity, as shown by\nthe green square in Fig. 3(f).\nThus, a non-uniform fixed-point (for κL4I= 0) is a spe-\ncial case in which vdrift= 0. In the generic case vdrift/ne}ationslash= 0.\nWe find that, for small µ, the drift velocity can be ap-\nproximated by (see section IVD3):\nvdrift=λ\n4D2γ2˜K(3α0sin(α0)+cos(2α0)−cos(α0)),\n(30)\ncompare the solid line, Eq. (30), and open squares (nu-\nmerical) in Fig. 3(f). Thus, the drift velocity is deter-\nmined by the STDP rule; however, only via ˜Kandα0.\nIn particular, the sign of vdriftis determined solely by α0\nin accordance with the self-consistency argument above.\nE. STDP dynamics for κL4I>0\nFigure 4(a) shows the STDP dynamics in the case of\nan un-isotropic upstream population, κL4I>0. As in\nthe isotropic case, for a wide range of parameters, the\nSTDP dynamics converge to a limit cycle. In contrast\nwith the isotropic case, the drift velocity is not con-\nstant and depends on the phase along the whisking cycle,\nFigs. 4(b)-4(c). Consequently, the time the downstream\nneuron spends at each phase is not constant and is pro-\nportional to the inverse of the drift velocity in that phase,\nFig. 4(f). Thus, STDP dynamics induces a distribution\nover time for the preferred phase of single L2/3 neurons,which in turn is translated into a distribution over the\ndownstream population, Figs. 4(d)-4(e).\nFor smallµ, a perturbation analysis for small κ(see\nsection IVD3) yields\nvdrift=v(0)\ndrift+κλD2γ2˜KF(α0)sin(ψ)(31)\nF(α0) =−3πα0sin(3α0\n2)\n16tan(α0\n2), (32)\nwherev(0)\ndriftis the drift velocity as κ= 0, Eq. (30). For\nsmallκ, the drift velocity has a single minima, either at\nπ/2or at−π/2, depending on the STDP rule via the sign\nofF(α0), see Eq. (31).\nFigure 4(h) shows the color-coded distribution of pre-\nferred phases of the downstream population, for different\nvalues ofκL4I. In the limit of an isotropic upstream pop-\nulation,κL4I= 0, the drift velocity is constant and the\ninduced distribution of preferred phases in layer 2/3 will\nbe uniform. For small κL4I, the peak of the distribution\nof the preferred phase in L2/3 will shift by either π/2+dν\nor−π/2 +dν, relative to L4I, depending on the STDP\nrule. AsκL4Igrows, the distribution of preferred phases\nin the upstream layer (L4I) converges to a delta func-\ntion, the STDP dynamics will converge to a fixed-point\nand distribution of preferred phases in L2/3 will converge\nto a delta function at dν, as shown in inset Fig . 4(h).\nIII. DISCUSSION\nSTDP can be viewed as a microscopic (unsupervised)\nlearning rule. Typically, one pictures learning as a pro-\ncess that converges to an optimum. Thus, STDP dynam-\nics is expected to relax to a fixed point or to a manifold\nattractor. However, empirical findings show that synap-\ntic weights do not converge to a fixed point. Rather,\nsynaptic weights remain volatile and vary by ∼50%over\na period of several days [35, 36]. Moreover, activity-\nindependent plasticity on a similar order of magnitude\nhas also been reported [36].\nThus, while activity-dependent plasticity (e.g. STDP)\nis considered to be an organizing force that pulls the sys-\ntem towards an attractor, activity-independent plasticit y\nis thought to introduce noise into the learning dynamics.\nThis noise will cause the synaptic weights to fluctuate\naround a point attractor or induce a random walk on\na manifold attractor. Consequently, it has been argued\nthat activity-independent plasticity can generate ‘drift -\ning representations’ [58–64]. That is, the neural rep-\nresentation of an external stimulus will not be fixed in\ntime.\nAn example of a drifting representation was described\nby Driscoll and colleagues [58]. They showed that while\nthe distribution of preferred stimuli across a population\nremained stable, the preferred stimuli of individual neu-\nrons varied over a period of several days.\nHere we showed that activity-dependent plasticity can\nalso provide a mechanism that generates drifting repre-8\n(a)\n0 50 100 15000.20.40.60.81\n 0 min\n10 min\n20 min\n30 min\n40 min(b)\n0 50 100 15000.20.40.60.81\n1280 min\n1540 min\n1800 min\n2050 min(f)\n0 20 40 60 80 100-1-0.500.51\n00.51\n0\n0+ d\n00.51\n- -/2 0 /200.51\nFIG. 3: STDP dynamics for κ= 0. (a) and (b) The synaptic weights profile, w(φ,t), is shown as a function of the\npreferred phases of the upstream neurons, φ, at different times (differentiated by color). (c) Trace depi cts synaptic\nweights are shown as a function of time. The synaptic weights are differentiated by color according to the preferred\nphase of the upstream neuron. (d) and (e) Dynamics of the orde r parameters ¯w(red) & ˜w(black) are shown in d,\nandψin e. (f) Drift velocity is shown as a function of T−. The black line follows Eq. (30), and red squares depict\nthe estimation of the drift velocity from the numerical simu lations. (g)-(i) Synaptic weights profile and the\nself-consistency argument. The black lines depict the profi le assuming zero-drift, i.e., according to Eq. (30). The\nvertical dashed grey lines (aligning with dashed black line s) showφc1,2. The input to the downstream neuron,\nψ0+π, and the phase of the downstream neuron, ψ0+π+νd, assuming zero drift (i.e., the black profile), are shown\nby the dashed red and green lines, respectively. The parabol ic approximation of the semi-phenomenological model,\nEq. (49), is depicted by the dashed magenta lines. For compar ison, the synaptic weights profile obtained by\nnumerical simulation of the STDP dynamics is shown in blue as a function of φ−φpost. The parameters used to\ngenerate the figure are as follows. In (a) - (e), the symmetric anti-Hebbian rule was used with random uniformly\ndistributed initial conditions for synaptic weights in the interval[0.3,0.7], andλ= 10−3,D= 10hz,I= 10hz,\nτ−= 20ms,τ+= 50ms,T±= 0ms,µ= 10−4,ν= 7hz,d= 5ms andγ= 1. In (f) - (i) the asymmetric Delta rule\nwas used with λ= 0.01,D= 10hz,I= 6hz,T+= 36ms,µ= 0,ν= 20hz,d= 12ms,γ= 1, andT−= 37,42,32ms\nin (g), (h), and (i), respectively. Initial conditions were w(φ,t= 0) = 0.5+0.3cos(φ). In all simulations, N= 150\nwas used.\nsentations. Moreover, this mechanism induces a distribu-\ntion over time for the preferred stimuli of single neurons.\nThis distribution is then translated into a distribution\nover the ensemble of neurons.\nOur theory leads to several empirical predictions. The\nfirst and the most natural prediction is that preferred\nphases are dynamic and have a consistent drift velocity\nthat is tightly related to the distribution of the preferred\nphases. In other words, drift velocity is expected to be\nminimal (maximal) at the most (least) common preferred\nphase.\nSecond, according to our theory, the distribution ofpreferred phases in the downstream population is gov-\nerned by the STDP rule. A wide distribution, κL2/3≈1,\nimplies that the fixed point solution of the STDP dy-\nnamics is unstable. Furthermore, the STDP rule also\ndetermines whether the mean phase of the downstream\npopulation will be advanced or delayed relative to the up-\nstream. These observations yield quantitatively testable\npredictions on the temporal structure of the STDP rule\nof L4I-to-L2/3 synapses.\nThird, according to our theory, the distribution of the\ndownstream layer is shifted and broadened due to STDP,\nwhen compared to the random pooling model, Fig. 1(f).9\n(g)\n0 0.5 1 1.5 2 --/20/2\n00.20.40.60.81\n0 0.5 1024\nFIG. 4: STDP for κ>0.(a) Synaptic weight dynamics. Each trace depicts the tempor al evolution of a single\nsynaptic weight, differentiated by color according to its pr eferred phase. (b)-(c) Dynamics of the order parameters:\n¯w(red) and ˜w(black) are shown in b, and ψis shown in c. (d)-(e) Polar histograms of the distributions of preferred\nphases of the L4I neurons (in d) with κL4I= 0.6andψL4I= 0.25πand the L2/3 neuron (in e) with κL2/3≈1.2and\nψL2/3≈2.3rad. (f) Histogram depicting the distribution of preferred phases of the downstream, L2/3, neuron over\ntime. The characteristics of the distribution are identica l to those in panel (e). The red dots show the inverse of the\ndrift velocity of the L2/3 neuron, normalized such that the n on-normalized data are multiplied by the ratio of the\nmaximum value of the histogram to the maximum value of the non -normalized data. Each red dot is the average of\n200 data points. (h) The distribution of the preferred phase s of the downstream, L2/3, neurons, ψL2/3, is shown in\ncolor code as a function of κL4I. In each column, the un-normalized distribution: Pr (ψ)/max{Pr(ψ)}, is depicted.\nThe inset shows the distribution width downstream, in layer 2/3,κL2/3, as a function of the width of the\ndistribution upstream, κL4I(blue). The identity line is shown (dashed black) for compar ison. Here we used the\nfollowing parameters: N= 150 ,¯ν=ν/(2π) = 10 hz,DL4I= 10hz,Iex= 8hz, andd= 14ms. The temporally\nsymmetric STDP rule, Eq. (11), was used with τ−= 20ms,τ+= 50ms,T±= 0ms,µ= 0.001, andλ= 10−3. Initial\nconditions were random with a uniform distribution in the in terval[0.3,0.7]. All the data presented here were\nsampled from simulations with an integer number of cycles. F or better visibility, the data presented in (a)-(c) shows\na non-integer number of cycles.10\nConsequently ‘turning off’ STDP will cause the mean\nphase at the downstream population to shift towards\nthat of the upstream, and the distribution is expected\nto become narrower. We do not yet know how to ma-\nnipulate the STDP rule. However, STDP dynamics are\ndriven by the product of the STDP rule and the cross-\ncorrelations of the neural activity. The cross-correlatio ns\ncan be modified by altering the sensory input to the sys-\ntem, e.g., by inducing artificial whisking. Such manip-\nulations have been performed in the past, illustrating\nactivity-dependent plasticity in the whisker system [65].\nThus, inducing artificial whisking at high frequencies is\nexpected to weaken the STDP; see e.g. Fig. 2(d), and\nconsequently cause the distribution to shift its center and\nbecome narrower.\nDrifting representation has implications to our under-\nstanding of the neural code. Consider a linear readout\nthat estimates the phase along the whisking cycle from\nthe responses of L2/3 neurons. A linear estimator, for\nexample, can be a weighted sum of the neural responses.\nAn optimal linear estimator is a linear estimator with a\nchoice of weights that maximizes the signal-to-noise ra-\ntio. Typically, for a linear readout the signal is embedded\nin the co-variation of the stimulus (whisking phase) and\nthe neural responses [66, 67]. However, if the neural rep-\nresentation of the stimulus is constantly changing, so is\nthe ‘signal’. As a result, the weights of the optimal linear\nestimator will have to adapt constantly.\nOne alternative to consider is that the readout may not\nbe optimal (at least not optimized to estimate the whisk-\ning phase) [67]. As the distribution of preferred phases\nis not uniform, a linear readout that pools the neural re-\nsponses with random weights will carry information on\nthe whisking phase. Even though the accuracy of a ran-\ndom pooling readout is inferior to that of the optimal\nlinear estimator, it is robust to drifting representation.\nIn our work, we made several simplifying assumptions.\nWe ignored the possible contribution of recurrent con-\nnectivity within L2/3, which may 1) affect the pre-post\ncross-correlations, and 2) be plastic itself. In addition,\nthe pre-post correlations were governed solely by rhyth-\nmic activity; however, a strong collective signal such as\ntouch may also affect the STDP dynamics. Studying\nthese effects was beyond the scope of the current work\nand will be addressed elsewhere.\nNevertheless, learning in the central nervous system\nhas been addressed empirically on two separate levels.\nOn the microscopic level, the STDP rule has been in-\nvestigated in preparations that lacked the functionality\nof the system. On the macroscopic level, functionality\nand behavior were studied in preparations that do not\nenable the investigation of the synaptic learning rules.\nThe theory developed here bridges this gap, and draws\ndirect links between these two levels that can thus serve\nas testable predictions for our hypothesis.IV. SUPPLEMENTARY MATERIALS\nA. The cross-correlation\nThe cross-correlation between the pre- and post-\nsynaptic firing is the driving force of STDP dynamics. In\nour model, the pre-post correlations are governed by the\ncorrelations within the upstream population. The cross-\ncorrelation between neurons jandkof the pre-synaptic\npopulation obey\nΓ(j,k)(∆t) =D2/parenleftBig\n1+γ2\n2cos[ν∆t+φj−φk]/parenrightBig\n+\nδjkDδ(∆t).(33)\nUsing Eq. (2), the cross-correlation between the jth\nL4I neuron and the downstream L2/3 neuron is given by\nΓj,post(∆t) =IexD−1\nNN/summationdisplay\nj=1wjΓi,j(∆t−d)\n=IexD−D\nNδ(∆t−d)wj−D2/parenleftbigg\n¯w+\nγ2\n2˜wcos[ν(∆t−d)+φj−ψ]/parenrightbigg\n,(34)\nwhere the order parameters are defined in Eqs. (5)\nand (6). In the limit of N≫1the cross-correlation\ncan be written as follows\nΓj,post(∆t) =DDpost(1+1\n2γγpostcos[φj−φpost\n+ν(∆t−d)])\n(35)\nwhereDpost,γpost,andφpostare given in Eq. (7).\nB. STDP dynamics\nSubstituting the pre-post cross-correlations in the large\nNlimit, Eq. (35), into the STDP dynamics, Eq. (12),\nyields\n˙wi\nλ=D(Iex−¯wD)(f+(wi)−f−(wi))+D2γ2\n2/parenleftBig\n˜w(˜K+(ν)\nf+(wi)cos[φi−Ω+−φpost]−˜K−(ν)f−(wi)\ncos[φi−Ω−−φpost]/parenrightBig\n,\n(36)\nwhere˜K±(ν), andΩ±(ν), are as defined in Eq. (19).\nC. Stability of the homogeneous fixed-points\nIn the homogeneous case, κ= 0, the STDP dynam-\nics in the limit of large N, Eq. (36), has two solutions,\nEqs. (22) and (23).11\nTo examine the stability of the first solution, w∗\n1, we\nconsider small perturbations around the fixed point w=\nw∗\n1+δw, writing\nδ˙w(φ)\nλ=δI(φ)+−δI(φ)−(37)\nwhere\nδI(φ)±=∂f±(w∗\n1)\n∂w(φ)D(Iex−Dw∗\n1)δw(φ)\n−f±(w∗\n1)D2/parenleftbig\nδ¯w−γ2˜K±\n2cos(φ−ψ−νd−Ω±)δ˜w/parenrightbig\n.\n(38)\nyielding\nδ˙w(φ)\nλ=−g0δw(φ) +∆fD2δ¯w+D2γ2\n2δ˜w(˜K−f−\ncos(φ−ψ−νd−Ω−)−˜K+f+cos(φ−ψ−νd−Ω+)),\n(39)\nwith the notation ∆f=f−−f+and\ng0=µD2(Iex\nw∗\n1D−1)w∗µ\n1\n1−w∗\n1=µD2(Iex\nD−1\n2)22−µ(40)\nRecalling that ∆f(w∗\n1) = 0, the eigenvalue corresponding\nto uniform fluctuations is given by\nmu,1=−g0. (41)\nNote that in this case the uniform eigenvalue does not\ndepend on the learning rule.\nThe eigenvalue in the whisking direction is\nmw,1=mu,1+1\n4D2γ2(˜K−f−(w∗\n1)cos(νd+Ω−)−\n˜K+f+(w∗\n1)cos(νd+Ω+)) =mu,1+1\n22+µD2γ2˜Kcos(α0),\n(42)\nwhere˜Kandα0were defined in Eq. (21).\nThe corresponding eigenvalues around the balanced\nfixed point, Eq. (23), obey\nmu,2=−D2/parenleftbig\n(1−w∗\n2)µ−(w∗\n2)µ/parenrightbig\n, (43)\nand\nmw,2=1\n4D2γ2/parenleftbig˜K−f−(w∗\n2)cos(νd+Ω−)−\n˜K+f+(w∗\n2)cos(νd+Ω+)/parenrightbig\n.(44)\nD. Derivation of drift velocity in the additive\nmodel\nThe phase of the synaptic weights profile, ψ, is given\nby Eq. (6) via the condition 0≤˜w∈R, namely:\n0 =/integraldisplay\nw(φ)sin(φ−ψ)Pr(φ)dφ (45)c2-c2 c1-c1\nFIG. 5: Synaptic weights profile. The\nsemi-phenomenological model for the moving profile of\nsynaptic weights as a function of φ, as described in\nEq. (49).\nwhere we take the continuum limit. Note that Pr (φ)\nis the probability density of the preferred phases in the\nupstream (L4I) layer, Eq. (1).\nTaking the temporal derivative of Eq. (45) yields the\ndynamic equation for the phase, ψ,\n˙ψ=1\n˜w/integraldisplay\n˙w(φ)sin(φ−ψ)Pr(φ)dφ. (46)\nTo utilize Eq. (46) we need a better understanding of\nthe shape of the moving profile of synaptic weights, w(φ).\nBelow, we first study the synaptic weights profile in the\nlimit of small µ, and derive a semi-phenomenological\nmodel for the moving profile. Next, we use a self-\nconsistent argument to determine the parameters that\ncharacterize this profile. Then, we use Eq. (46) to com-\npute the drift velocity.\n1. The semi-phenomenological model for small µ\nFig. 3(h) shows an example of a moving profile with a\npositive drift velocity and small µ. The synaptic weights\nprofile consists of four distinct regions. Two are saturated\nregions, in which the synapses are either fully depressed\nor fully potentiated. The other two are fronts; a preced-\ning front and a receding front.\nIn the saturated regions, w(φ) = 0 orw(φ) = 1, and\ndue to the weight dependence of the STDP rule, f±(w),\n˙w(φ) = 0. As a result, the saturated regions do not con-\ntribute to the drift velocity; see Eq. (46). In the interior\nof the fronts w(φ)∈(0,1), and in the limit of the addi-\ntive model, µ→0,f+(w) =f−(w) = 1. Consequently,\nin the fronts:\n˙w(φ) =1\n2λ˜wD2˜Kγ2cos[α0+ψ−φ], φ in fronts.(47)\nThe edges of the fronts, φc1/2, are given by the condi-\ntion of the vanishing temporal derivative, ˙w(φ) = 0, in\nEq. (47), yielding (compare to Eq. (20))\nφc1,2=α0+ψ±π/2. (48)\nTo compute the drift velocity we use the following\nsemi-phenomenological model for the synaptic weights12\nprofile (see Fig. (5)):\nw(φ) =\n\n1−/parenleftBigφ−φc1\n∆/parenrightBig2\nφ∈(φc2−∆,φc2)\n1 φ∈(φc2,φc1−∆)/parenleftBigφ−φc1\n∆/parenrightBig2\nφ∈(φc1−∆,φc1)\n0 otherwise(49)\nNote that Eq. (49) must be taken with a grain of salt,\nasφis an angular variable. Figures 3(h)-3(i) illustrates\nthe agreement between our semi-phenomenological model\nand a moving profile obtained from simulating the STDP\ndynamics. The parameter ∆denotes the width of the\nfront.\nThe synaptic weights profile, in the limit of small κ, is\nfully defined by φc1,φc2and∆. The profile determines\nψvia Eq. (6), which in turn must be consistent with φc1\nandφc2as determined by Eq. (48). This self-consistent\nconstraint determines the width of the fronts, ∆.\nBelow we determine the width of the front, ∆, and use\nit to compute the drift velocity. First, in the homoge-\nneous case, κ= 0. Then, we extend our results to a non-\nuniform distribution of preferred phases for 0< κ≪1,\nin a perturbative manner.\n2. Uniform distribution\nIn the case of a uniform distribution, κ= 0, instead of\nusing Eq. (45) to determine ψ, we demand that ψis the\n‘center of mass’, ψcom, of the synaptic weights profile:\nψ=ψcom≡1/integraltext\nw(φ)dφ/integraldisplay\nw(φ)φdφ. (50)\nUsing Eq. (49), the denominator of Eq. (50)\n/integraldisplay\nw(φ)dφ=/integraldisplay∆\n0/parenleftbiggφ\n∆/parenrightbigg2\ndφ+[φc1−∆−φc2]+\n/integraldisplay∆\n0/parenleftBig\n1−(φ/∆)2/parenrightBig\ndφ=π.(51)\nThe numerator of Eq. (50) gives\n/integraldisplay\nw(φ)φdφ=/integraldisplay∆\n0/parenleftbiggφ\n∆/parenrightbigg2\nφdφ+/integraldisplayφc1−∆\nφc2φdφ+\n/integraldisplay∆\n0/parenleftbig\n1−(φ/∆)2/parenrightbig\nφdφ=π(α0+ψ−2∆/3).\n(52)\nyielding from the self-consistency constrain, Eq. (50),\n∆ =3α0\n2. (53)Using the fact that the saturated regions do not con-\ntribute to the drift velocity, Eq. (46) yields\n˙ψ(t)\nλ=γ2D2˜K\n4π/braceleftbigg/integraldisplayφc2\nφc2−∆cos(α0+ψ−φ)sin(φ−ψ)dφ\n/integraldisplayφc1\nφc1−∆cos(α0+ψ−φ)sin(φ−ψ)dφ/bracerightbigg\n=\nγ2D2˜K\n4/parenleftbig\n2∆sin[α0]+cos[α0−2∆]−cos[α0]/parenrightbig\n,\n(54)\nsubstituting the value of ∆, Eq. (53), yields Eq. (30):\nv(0)\ndrift=λγ2D2˜K\n4/parenleftbig\ncos(2α0)−cos(α0)+3α0sin(α0)/parenrightbig\n.\n(55)\nNote thatv(0)\ndriftdoes not depend on ψ. The superscript 0\ndenotes that this is the zero-order in κ. Below we now ex-\ntend the calculation of the drift velocity in a perturbative\nmanner inκ.\n3. Nonuniform distribution in the limit of small κ\nTo study the leading non-trivial order of the drift ve-\nlocity,vdrift, inκ, we write\nvdrift(ψ) =v(0)\ndrift+κv(1)\ndrift(ψ)+O(κ2), (56)\n∆(ψ) = ∆(0)+κ∆(1)(ψ)+O(κ2), (57)\nwherev(n)\ndriftand∆(n)(n∈N∪ {0}) are independent\nofκ. The drift velocity, vdrift, can be estimated from\nEq. (46), once the synaptic weights profile w(φ)is estab-\nlished, given the width of the front, ∆.\nIn the semi-phenomenological model, Eq. (49), the\nsynaptic weights profile is fully determined by ψ(which\ndetermines φc1,2) and∆. The width of the front, ∆, can\nbe determined from Eq. (45), yielding a first order in κ\n0 =L0+L1, (58)\nwhere\nL0=/integraldisplay\nw(φ,∆(0)+κ∆(1))sin(φ−ψ)dφ (59)\nL1=κ/integraldisplay\nw(φ,∆(0))sin(φ−ψ)cos(φ)dφ. (60)\nwhere for 0< κ≪1, we approximated the probability\ndensity of preferred phases, Eq. (1), by\nPr(φ)≈1+κcos(φ)\n2π. (61)\nIn Eqs. (59) and (60) we explicitly denote the depen-\ndence of the synaptic weight profile on the width of the\nfront. Note that as Eq. (60) is O(κ), it is sufficient to\ntake only the zero order in κfor the width.13\nThe term,L0also contains zero order in κ,\nL0=L0(∆(0))+κ∆(1)∂L0\n∂∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆(0)+O(κ2). (62)\nTo compute L0(∆(0))we consider the intervals in which\nw(φ)is nonzero:\nL0=L0,0+L0,1+L0,2, (63)\nwhere\nL0,0=/integraldisplayφc1−∆\nφc2−∆sin(φ−ψ)dφ, (64)\nL0,1=−/integraldisplayφc2\nφc2−∆(φ−φc2)2\n∆2sin(φ−ψ)dφ, (65)\nL0,2=/integraldisplayφc1\nφc1−∆(φ−φc1)2\n∆2sin(φ−ψ)dφ. (66)\nThus, yielding\nL0(∆) =4\n∆2/parenleftBig\nsin(α0−∆)+∆cos( α0−∆)−sin(α0)/parenrightBig\n.\n(67)\nNote that the solution of Eq. (45) to order κ0is different\nthan the ‘center of mass’ calculation of Eq. (50). Nev-\nertheless, we find that ∆com(0)= 3α0/2yields a good\napproximation of the solution of L0(∆(0)) = 0.\nThe first order in κofL0(see Eq. (62)) obeys\nκ∆(1)∂L0\n∂∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆(0)=κ∆(1)4\n∆(0)sin(α0−∆(0)).(68)\nSubstituting ∆(0)≈3α0/2yields\nκ∆(1)∂L0\n∂∆/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆(0)=−κ∆(1)8\n3α0sin/parenleftbigα0\n2/parenrightbig\n.(69)\nThe termL1is computed using the profile for κ= 0.\nWriting\nL1=L1,0+L1,1+L1,2, (70)\nwith\nL1,0=κ/integraldisplayφc1−∆(0)\nφc2−∆(0)sin(φ−ψ)cos(φ)dφ, (71)\nL1,1=−κ/integraldisplayφc2\nφc2−∆(0)/parenleftbiggφ−φc2\n∆(0)/parenrightbigg2\nsin(φ−ψ)cos(φ)dφ,\n(72)\nL1,2=κ/integraldisplayφc1\nφc1−∆(0)/parenleftbiggφ−φc1\n∆(0)/parenrightbigg2\nsin(φ−ψ)cos(φ)dφ.\n(73)\nSubstituting ∆(0)≈3α0/2in Eq. (71) one obtains\nL1,0=−κπsin(ψ)\n2. (74)Noting that L1,1+L1,2= 0, Eq. (58) yields to first order\ninκ:\n∆(1)=−3πα0\n16sin/parenleftbigα0\n2/parenrightbigsin(ψ). (75)\nHence, up to first order in κ, the width of the front is\n∆(ψ) =3α0\n2/parenleftBigg\n1−κπ\n8sin/parenleftbigα0\n2/parenrightbigsin(ψ)/parenrightBigg\n.(76)\nExpanding the probability density to first order in κin\nEq. (46), and noting that in the limit of the additive rule,\nµ→0, only the fronts contribute to the drift velocity, we\nobtain\nvdrift=λγ2D2˜K\n4π/summationdisplay\ni=1,2/integraldisplayφci\nφci−∆(1+κcos(φ))cos(α0+ψ−φ)\nsin(φ−ψ)dφ\n(77)\nwhere to first order in κ,∆is given by Eq. (76). Eq. (77)\ncan be written as sum of three terms:\nvdrift=v(0)\ndrift+κv(1a)\ndrift+κv(1b)\ndrift, (78)\nwherev(0)\ndriftis given by Eq. (55) and\nv(1a)\ndrift=λγ2D2˜K\n4π/summationdisplay\ni=1,2/integraldisplayφci\nφci−∆φ0cos(φ)cos(α0+ψ−φ)\nsin(φ−ψ)dφ\n(79)\nv(1b)\ndrift=λ\nκγ2D2˜K\n4π/summationdisplay\ni=1,2/integraldisplayφci−∆(0)\nφci−(∆(0)+κ∆(1))cos(α0+ψ−φ)\nsin(φ−ψ)dφ\n(80)\nIntegrating the right hand side of Eq. (79) over φyields\nv(1a)\ndrift= 0, and in the limit of small κ, Eq. (80), yields\nv(1b)\ndriftλγ2D2˜K\n2πsin(3α0\n2)cos(α0\n2)∆(1). (81)\nSubstituting Eq. (75) yields\nvdrift(ψ)≈v(0)\ndrift+λκγ2D2˜K\n2πF(α0)sin(ψ).(82)\nwith\nF(α0) =−3πα0sin(3α0\n2)\n16tan(α0\n2), (83)\nas cited in Eq. (31).14\nE. Numerical simulations & pre-synaptic phase\ndistributions\nScripts of the numerical simulations were written in\nMatlab. Figure 2 was produced with Mathematica. Un-\nless stated otherwise, the numerical results presented in\nthis paper were obtained by solving Eq. (36) with the Eu-\nler method. The cftool with the Nonlinear Least Squares\nmethod and a confidence level of 95%for all parameters\nwas used for fitting the data presented in Figs. 4(d)-4(e).Unless stated otherwise, we simulated the STDP dy-\nnamics in the mean field limit without quenched disorder.\nTo this end, the preferred phase, φk, of thekth neuron\nin a population of Npre-synaptic L4I neurons was set\nby the condition/integraltextφk\n−πPr(ϕ)dϕ=k/N.\nACKNOWLEDGMENTS\nThis research was supported by the Israel Science\nFoundation (ISF) grants number 300/16 and 624/22.\n[1] Robert NS Sachdev, Takashi Sato, and Ford F Ebner.\nDivergent movement of adjacent whiskers. Journal of\nneurophysiology , 87(3):1440–1448, 2002.\n[2] Avner Wallach, David Deutsch, Tess Baker Oram, and\nEhud Ahissar. Predictive whisker kinematics reveal\ncontext-dependent sensorimotor strategies. PLoS biology ,\n18(5):e3000571, 2020.\n[3] Per Magne Knutsen, Maciej Pietr, and Ehud Ahissar.\nHaptic object localization in the vibrissal system: be-\nhavior and performance. Journal of Neuroscience ,\n26(33):8451–8464, 2006.\n[4] Samar B Mehta, Diane Whitmer, Rodolfo Figueroa,\nBen A Williams, and David Kleinfeld. Active spatial\nperception in the vibrissa scanning sensorimotor system.\nPLoS biology , 5(2):e15, 2007.\n[5] Margo Landers and H Philip Zeigler. Development of\nrodent whisking: trigeminal input and central pattern\ngeneration. Somatosensory & motor research , 23(1-2):1–\n10, 2006.\n[6] Daniel N Hill, John C Curtis, Jeffrey D Moore, and David\nKleinfeld. Primary motor cortex reports efferent con-\ntrol of vibrissa motion on multiple timescales. Neuron ,\n72(2):344–356, 2011.\n[7] Brian R Isett and Daniel E Feldman. Cortical coding of\nwhisking phase during surface whisking. Current Biology ,\n2020.\n[8] David Kleinfeld and Martin Deschênes. Neuronal basis\nfor object location in the vibrissa scanning sensorimotor\nsystem. Neuron , 72(3):455–468, 2011.\n[9] Ehud Ahissar and Per Magne Knutsen. Object localiza-\ntion with whiskers. Biological cybernetics , 98(6):449–458,\n2008.\n[10] Mathew E Diamond, Moritz Von Heimendahl,\nPer Magne Knutsen, David Kleinfeld, and Ehud\nAhissar. ’where’and’what’in the whisker sensorimotor\nsystem. Nature Reviews Neuroscience , 9(8):601–612,\n2008.\n[11] Chunxiu Yu, Dori Derdikman, Sebastian Haidarliu, and\nEhud Ahissar. Parallel thalamic pathways for whisking\nand touch signals in the rat. PLoS Biol , 4(5):e124, 2006.\n[12] David RC House, Justin Elstrott, Eileen Koh, Jason\nChung, and Daniel E Feldman. Parallel regulation of\nfeedforward inhibition and excitation during whisker map\nplasticity. Neuron , 72(5):819–831, 2011.\n[13] Jianing Yu, Diego A Gutnisky, S Andrew Hires, and\nKarel Svoboda. Layer 4 fast-spiking interneurons filter\nthalamocortical signals during active somatosensation.Nature neuroscience , 19(12):1647–1657, 2016.\n[14] Carl CH Petersen. Sensorimotor processing in the roden t\nbarrel cortex. Nature Reviews Neuroscience , 20(9):533–\n546, 2019.\n[15] Bettina Voelcker, Ravi Pancholi, and Simon Peron.\nTransformation of primary sensory cortical representa-\ntions from layer 4 to layer 2. Nature communications ,\n13(1):5484, 2022.\n[16] Kyle S Severson, Duo Xu, Margaret Van de Loo, Ling\nBai, David D Ginty, and Daniel H O’Connor. Active\ntouch and self-motion encoding by merkel cell-associated\nafferents. Neuron , 94(3):666–676, 2017.\n[17] Valérie Ego-Stengel, Julie Le Cam, and Daniel E Shulz.\nCoding of apparent motion in the thalamic nucleus of\nthe rat vibrissal somatosensory system. Journal of Neu-\nroscience , 32(10):3339–3351, 2012.\n[18] Jeffrey D Moore, Nicole Mercer Lindsay, Martin De-\nschênes, and David Kleinfeld. Vibrissa self-motion and\ntouch are reliably encoded along the same somatosensory\npathway from brainstem through thalamus. PLoS Biol ,\n13(9):e1002253, 2015.\n[19] Natalia Grion, Athena Akrami, Yangfang Zuo, Federico\nStella, and Mathew E Diamond. Coherence between rat\nsensorimotor system and hippocampus is enhanced dur-\ning tactile discrimination. PLoS biology , 14(2):e1002384,\n2016.\n[20] Jianing Yu, Hang Hu, Ariel Agmon, and Karel Svoboda.\nRecruitment of gabaergic interneurons in the barrel cor-\ntex during active tactile behavior. Neuron , 104(2):412–\n427, 2019.\n[21] Diego Adrian Gutnisky, Jianing Yu, Samuel Andrew\nHires, Minh-Son To, Michael Bale, Karel Svoboda, and\nDavid Golomb. Mechanisms underlying a thalamocorti-\ncal transformation during active tactile sensation. PLoS\ncomputational biology , 13(6):e1005576, 2017.\n[22] Simon P Peron, Jeremy Freeman, Vijay Iyer, Caiying\nGuo, and Karel Svoboda. A cellular resolution map of\nbarrel cortex activity during tactile behavior. Neuron ,\n86(3):783–799, 2015.\n[23] Simon Peron, Ravi Pancholi, Bettina Voelcker, Jason D\nWittenbach, H Freyja Ólafsdóttir, Jeremy Freeman, and\nKarel Svoboda. Recurrent interactions in local cortical\ncircuits. Nature , 579(7798):256–259, 2020.\n[24] Yotam Luz and Maoz Shamir. Balancing feed-forward\nexcitation and inhibition via hebbian inhibitory synaptic\nplasticity. PLoS computational biology , 8(1):e1002334,\n2012.15\n[25] William R Softky and Christof Koch. The highly irreg-\nular firing of cortical cells is inconsistent with temporal\nintegration of random epsps. Journal of Neuroscience ,\n13(1):334–350, 1993.\n[26] Michael N Shadlen and William T Newsome. The vari-\nable discharge of cortical neurons: implications for con-\nnectivity, computation, and information coding. Journal\nof neuroscience , 18(10):3870–3896, 1998.\n[27] Timothée Masquelier, Etienne Hugues, Gustavo Deco,\nand Simon J Thorpe. Oscillations, phase-of-firing cod-\ning, and spike timing-dependent plasticity: an efficient\nlearning scheme. Journal of Neuroscience , 29(43):13484–\n13493, 2009.\n[28] Yotam Luz and Maoz Shamir. Oscillations via spike-\ntiming dependent plasticity in a feed-forward model.\nPLoS computational biology , 12(4):e1004878, 2016.\n[29] Sen Song, Kenneth D Miller, and Larry F Abbott.\nCompetitive hebbian learning through spike-timing-\ndependent synaptic plasticity. Nature neuroscience ,\n3(9):919, 2000.\n[30] Richard Kempter, Wulfram Gerstner, and J Leo van\nHemmen. Intrinsic stabilization of output rates by\nspike-based hebbian learning. Neural computation ,\n13(12):2709–2741, 2001.\n[31] Richard Kempter, Wulfram Gerstner, and J Leo\nVan Hemmen. Hebbian learning and spiking neurons.\nPhysical Review E , 59(4):4498, 1999.\n[32] Ingrid Bureau, Francisca von Saint Paul, and Karel Svo-\nboda. Interdigitated paralemniscal and lemniscal path-\nways in the mouse barrel cortex. PLoS Biol , 4(12):e382,\n2006.\n[33] K Fox. Anatomical pathways and molecular mecha-\nnisms for plasticity in the barrel cortex. Neuroscience ,\n111(4):799–814, 2002.\n[34] Reha S Erzurumlu and Patricia Gaspar. Development\nand critical period plasticity of the barrel cortex. Euro-\npean Journal of Neuroscience , 35(10):1540–1553, 2012.\n[35] Noam E Ziv and Naama Brenner. Synaptic tenacity or\nlack thereof: spontaneous remodeling of synapses. Trends\nin neurosciences , 41(2):89–99, 2018.\n[36] Liran Hazan and Noam E Ziv. Activity dependent and in-\ndependent determinants of synaptic size diversity. Jour-\nnal of Neuroscience , 40(14):2828–2848, 2020.\n[37] Nimrod Sherf and Maoz Shamir. Stdp and the distribu-\ntion of preferred phases in the whisker system. PLOS\nComputational Biology , 17(9):1–18, 09 2021.\n[38] Maoz Shamir. Theories of rhythmogenesis. Current opin-\nion in neurobiology , 58:70–77, 2019.\n[39] Tim P Vogels, Henning Sprekeler, Friedemann Zenke,\nClaudia Clopath, and Wulfram Gerstner. Inhibitory plas-\nticity balances excitation and inhibition in sensory path-\nways and memory networks. Science , 334(6062):1569–\n1573, 2011.\n[40] Henry Markram, Joachim Lübke, Michael Frotscher,\nand Bert Sakmann. Regulation of synaptic efficacy by\ncoincidence of postsynaptic aps and epsps. Science ,\n275(5297):213–215, 1997.\n[41] Guo-qiang Bi and Mu-ming Poo. Synaptic modifications\nin cultured hippocampal neurons: dependence on spike\ntiming, synaptic strength, and postsynaptic cell type.\nJournal of neuroscience , 18(24):10464–10472, 1998.\n[42] Per Jesper Sjöström, Gina G Turrigiano, and Sacha B\nNelson. Rate, timing, and cooperativity jointly deter-\nmine cortical synaptic plasticity. Neuron , 32(6):1149–1164, 2001.\n[43] Li I Zhang, Huizhong W Tao, Christine E Holt,\nWilliam A Harris, and Mu-ming Poo. A critical win-\ndow for cooperation and competition among developing\nretinotectal synapses. Nature , 395(6697):37, 1998.\n[44] L. F. Abbott and Sacha B. Nelson. Synaptic plasticity:\ntaming the beast. Nature Neuroscience , 3(11):1178–1183,\n2000.\n[45] Robert C Froemke, Ishan A Tsay, Mohamad Raad,\nJohn D Long, and Yang Dan. Contribution of individual\nspikes in burst-induced long-term synaptic modification.\nJournal of neurophysiology , 95(3):1620–1629, 2006.\n[46] Makoto Nishiyama, Kyonsoo Hong, Katsuhiko\nMikoshiba, Mu-Ming Poo, and Kunio Kato. Cal-\ncium stores regulate the polarity and input specificity of\nsynaptic modification. Nature , 408:584–588, 2000.\n[47] Harel Z Shouval, Mark F Bear, and Leon N Cooper. A\nunified model of nmda receptor-dependent bidirectional\nsynaptic plasticity. Proceedings of the National Academy\nof Sciences , 99(16):10831–10836, 2002.\n[48] Melanie A Woodin, Karunesh Ganguly, and Mu ming\nPoo. Coincident pre- and postsynaptic activity modifies\ngabaergic synapses by postsynaptic changes in cl– trans-\nporter activity. Neuron , 39(5):807 – 820, 2003.\n[49] L. Haas. Hans berger (1873-1941), richard caton (1842-\n1926), and electroencephalography. J Neurol Neurosurg\nPsychiatry , 74(1):9–9, Jan 2003. 12486257[pmid].\n[50] Robert Gütig, Ranit Aharonov, Stefan Rotter, and Haim\nSompolinsky. Learning input correlations through non-\nlinear temporally asymmetric hebbian plasticity. Journal\nof Neuroscience , 23(9):3697–3714, 2003.\n[51] Abigail Morrison, Markus Diesmann, and Wulfram Ger-\nstner. Phenomenological models of synaptic plasticity\nbased on spike timing. Biological cybernetics , 98:459–\n478, 2008.\n[52] Gabriel Koch Ocker, Ashok Litwin-Kumar, and Brent\nDoiron. Self-organization of microcircuits in networks of\nspiking neurons with plastic synapses. PLOS Computa-\ntional Biology , 11(8):1–40, 08 2015.\n[53] Nimrod Sherf and Shamir Maoz. Multiplexing rhythmic\ninformation by spike timing dependent plasticity. PLoS\ncomputational biology , 16(6):e1008000, 2020.\n[54] Julie S Haas, Thomas Nowotny, and Henry DI Abar-\nbanel. Spike-timing-dependent plasticity of inhibitory\nsynapses in the entorhinal cortex. Journal of neurophys-\niology, 96(6):3305–3313, 2006.\n[55] Matthieu Gilson, Moritz Bürck, Anthony N Burkitt, and\nJ Leo van Hemmen. Frequency selectivity emerging from\nspike-timing-dependent plasticity. Neural computation ,\n24(9):2251–2279, 2012.\n[56] Yotam Luz and Maoz Shamir. The effect of stdp temporal\nkernel structure on the learning dynamics of single exci-\ntatory and inhibitory synapses. PloS one , 9(7):e101109,\n2014.\n[57] The width can be governed by adding a parameter\nthat controls the relative strength of depression [CITE\nLuzShamir16].\n[58] Laura N Driscoll, Noah L Pettit, Matthias Minderer, Sel -\nmaan N Chettih, and Christopher D Harvey. Dynamic\nreorganization of neuronal activity patterns in parietal\ncortex. Cell, 170(5):986–999, 2017.\n[59] Michael E Rule, Adrianna R Loback, Dhruva V Raman,\nLaura N Driscoll, Christopher D Harvey, and Timothy\nO’Leary. Stable task information from an unstable neural16\npopulation. Elife, 9:e51121, 2020.\n[60] Yaroslav Felipe Kalle Kossio, Sven Goedeke, Christian\nKlos, and Raoul-Martin Memmesheimer. Drifting as-\nsemblies for persistent memory: Neuron transitions and\nunsupervised compensation. Proceedings of the National\nAcademy of Sciences , 118(46):e2023832118, 2021.\n[61] Genki Shimizu, Kensuke Yoshida, Haruo Kasai, and Taro\nToyoizumi. Computational roles of intrinsic synaptic\ndynamics. Current Opinion in Neurobiology , 70:34–42,\n2021.\n[62] Han Chin Wang, Amy M LeMessurier, and Daniel E Feld-\nman. Tuning instability of non-columnar neurons in the\nsalt-and-pepper whisker map in somatosensory cortex.\nNature Communications , 13(1):6611, 2022.[63] Laura N Driscoll, Lea Duncker, and Christopher D Har-\nvey. Representational drift: Emerging theories for contin -\nual learning and experimental future directions. Current\nOpinion in Neurobiology , 76:102609, 2022.\n[64] Paul Manz and Raoul-Martin Memmesheimer. Purely\nstdp-based assembly dynamics: Stability, learning, over-\nlaps, drift and aging. PLOS Computational Biology ,\n19(4):e1011006, 2023.\n[65] Daniel E Feldman and Michael Brecht. Map plasticity in\nsomatosensory cortex. Science , 310(5749):810–815, 2005.\n[66] Emilio Salinas and LF Abbott. Vector reconstruction\nfrom firing rates. Journal of computational neuroscience ,\n1(1-2):89–107, 1994.\n[67] Maoz Shamir. Emerging principles of population cod-\ning: in search for the neural code. Current opinion in\nneurobiology , 25:140–148, 2014." }, { "title": "2401.15831v2.A_Liouville_type_theorem_for_the_coupled_Schrödinger_systems_and_the_uniqueness_of_the_sign_changing_radial_solutions.pdf", "content": "arXiv:2401.15831v2 [math.AP] 30 Jan 2024A Liouville-type theorem for the coupled Schr¨ odinger syst ems\nand the uniqueness of the sign-changing radial solutions\nHaoyu Li\nDepartamento de Matem´ atica,\nUniversidade Federal de S˜ ao Carlos,\nS˜ ao Carlos-SP, 13565-905, Brazil\ne-mail: hyli1994@hotmail.com\nOl´ ımpio Hiroshi Miyagaki\nDepartamento de Matem´ atica,\nUniversidade Federal de S˜ ao Carlos,\nS˜ ao Carlos-SP, 13565-905, Brazil\ne-mail: olimpio@ufscar.br\nAbstract. In this paper, we study the sign-changing radial solutions of the fo llowing coupled\nSchr¨ odinger system/braceleftBigg\n−∆uj+λjuj=µju3\nj+/summationtext\ni/negationslash=jβiju2\niujinB1,\nuj∈H1\n0,r(B1) forj= 1,···,N.\nHere,λj, µj>0 andβij=βjiare constants for i,j= 1,···,Nandi/\\e}atio\\slash=j.B1denotes the unit ball\nin the Euclidean space R3centred at the origin. For any P1,···,PN∈N, we prove the uniqueness of\nthe radial solution ( u1,···,uj) withujchanges its sign exactly Pjtimes for any j= 1,···,Nin the\nfollowing case: λj≥1 and|βij|are small for i,j= 1,···,Nandi/\\e}atio\\slash=j. New Liouville-type theorems\nand boundedness results are established for this purpose.\n2020 MSC: 35A02, 35B53, 35J47\nKey words: Coupled Schr¨ odinger systems; uniqueness; sign-changing radial solutions; Liouville-\ntype theorem.\n1Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\n1 Introduction\n1.1 Main theorems\nThispaperprovestheuniquenessoftheradialsolutionstha thaveacomponent-wiseprescribed\nnumber of nodes for the following problem:\n/braceleftBigg\n−∆uj+λjuj=µju3\nj+/summationtext\ni/ne}ationslash=jβiju2\niujinB1,\nuj∈H1\n0,r(B1) forj= 1,···,N.(1.1)\nHere,λj,µj>0, andβij=βjifori,j= 1,···,Nandi/\\e}atio\\slash=j.B1is the unit ball in R3\ncentred at the origin. H1\n0,r(B1) is the subspace consisting of radial functions of H1\n0(B1). To\nbeprecise, forany P1,···,PN∈N, weprove theuniquenessof theradial solution ( u1,···,uN)\nto Problem ( 1.1) withujchanges its sign exactly Pjtimes for any j= 1,···,N. As proved\nin [13,11], such a solution with prescribed number of nodes exists as l ong asβij< Bfor\nanyi/\\e}atio\\slash=jand for some B=B(P1,···,PN)>0. Furthermore, Li and Wang [ 11] proved\na infinitely many existence result for the above solutions. F or instance, in the case of the\ntwo-component systems, the theorem reads as\nFor Problem ( 1.1) withN= 2, assume λ1=λ2>0and−β > µ1=µ2>0. For any\nP∈N, there are infinitely many radial solutions (uk,1,uk,2)withuk,1anduk,2change\ntheir sign exactly Ptimes for any k= 1,2,···.\nIt is a natural to inquire about the boundedness or even the un iqueness of solutions within\nthe regime of existence but outside of multiplicity . In this note, we give a partial affirmative\nanswer.\nTheorem 1.1 Ifλ1,···,λN≥1, for any P1,···,PN∈N, there exists a positive number\nb=b(λ1,···,λN;µ1,···,µN;P1,···,PN)>0such that if |βij|< b, Problem ( 1.1) admits\nan unique solution (u1,···,uN)withuj(0)>0andujchanging its sign exactly Pjtimes for\nj= 1,···,N.\nThe existence part has been proved in [ 11]. We only need to prove the uniqueness part. We\nwill apply the ideas in [ 7] to the sign-changing radial solutions. A by-product is the following\nremark.\nRemark 1.2 Under the assumption of Theorem 1.1, the solution (u1,···,uN)is non-degenerate\nin(H1\n0,r(B1))N, i.e., for any (ϕ1,···,ϕN)∈H1\n0,r(B1)N, we have\n−∆ϕj+λjϕj= 3µju2\njϕj+/summationdisplay\ni/ne}ationslash=jβiju2\niϕj+2/summationdisplay\ni/ne}ationslash=jβijuiujϕi\nforj= 1,···,N.\nInordertoprovetheboundedness,thefollowingLiouville- typetheoremplaysanimportant\nrole. We are delighted to note that we benefit a lot from Profes sor Pavol Quittner, who\n2A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\nprovides the idea and various comments of the proof of Theore m1.3, cf. [17]. Consider the\nfollowing elliptic system.\n\n\n−∆uj=µju3\nj+/summationtext\ni/ne}ationslash=jβiju2\niujinR3,\nuj∈C2(R3), ujis radial,\nujis zero or changes its sign at most Pjtimes and ( u1,···,uN)/\\e}atio\\slash= 0,\n|uj|L∞(R3)≤1 forj= 1,···,N.(1.2)\nHere,µj>0 are constants for j= 1,···,N. The following Liouville-type theorem holds.\nTheorem 1.3 For any P1···,PN∈N, and any β≥0, there exists a positive number ε0=\nε0(µ1,···,µN;P1,···,PN;β)>0such that if βij∈[−ε0,β]for anyi,j= 1,···,Nandi/\\e}atio\\slash=j,\nProblem ( 1.2) admits no non-trivial solution.\nTheorem 2.7concerns the one spatial dimension case, which can be proved in a similar way.\nWe will state it in Section 2.2. As an immediate application o f Theorem 1.3and Theorem\n2.7, the following boundedness result is implied.\nTheorem 1.4 For any P1···,PN∈N, and any β≥0, there exist positive numbers ε0=\nε0(λ1,···,λN;µ1,···,µN;P1,···,PN;β)>0andC=C(λ1,···,λN;µ1,···,µN;P1,···,PN;β)>\n0such that if βij∈[−ε0,β]for anyi,j= 1,···,Nandi/\\e}atio\\slash=j, the solution Uto Problem ( 1.1)\nsatisfies |U|∞≤C.\n1.2 Some historical remarks\nIn this note, we study the uniqueness of the sign-changing ra dial solutions to Problem ( 1.1)\nwith component-wisely prescribed number of nodes. Regardi ng the uniqueness problem, the\ncurrent studies mostly focus on the positive solutions, cf. [7,19,12,21] and the references\ntherein. The studies on the uniqueness of the positive solut ion can be traced back to the\nanalogue results for the scalar field counterpart of problem (1.1), i.e. the equation −∆u+u=\n|u|p−2u. This problemhas been studiedextensively. See[ 5,10,8]. For theuniquenessof radial\nsign-changing solutions, the complete answer remains open . However, in the last decade some\npartial results have been proved. We refer to [ 1] for a slightly subcritical problem and [ 18] for\nseveral special cases. In [ 9] the authors carried out their study with the help of compute rs.\nIn this note we give a uniqueness result (cf. Theorem 1.1) for the radial sign-changing\nsolution to the coupled Schr¨ odinger systems. In [ 11] it was proved that in certain regions\nof the parameters the solutions with prescribed number of no des can be multiple. This is\nbased on a combination of the parabolic equation and the Lust ernik-Schnirelmann argument.\nHowever, there are still some cases where we can only prove ex istence but not multiplicity.\nThe aim of this note is to find some uniqueness results in the re gion of existence but outside\nthe region of multiplicity. To this end, we need new Liouvill e-type theorems for the coupled\nelliptic systems. See Theorem 1.3and Theorem 2.7. Since the seminal work [ 6], Liouville type\ntheorems have been widely applied to the semilinear ellipti c equations. There are also many\n3Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\nresults on the Liouville-type theorem for the elliptic syst ems. We quote [ 4,15,16] and the\nreferences therein for the positive solutions and [ 14] for the sign-changing radial solutions in\na very general case. It was shown by Quittner [ 14] that in addition to the number of nodes\nfor each component, the number of nodes of the difference and su m of the components, i.e.,\nthe comparisons, are also crucial. In Theorem 1.3and Theorem 2.7of this paper, we can find\na special case where the information about the comparison ca n be omitted. However, as it is\nshown in Remark 2.10, in the wider regions, the comparisons are still essential.\n1.3 The idea of the prove and the organization of this note\nOur approach is based on an implicit function theorem argume nt analogue to [ 7]. However, in\nhis work [ 7], Ikoma focused on the positive solutions. In order to solve the problem concerning\nthe sign-changing radial solutions, we should prove the cor responding a priori and the non-\ndegeneracy of the sign-changing radial solutions. Therefo re, our note is divided into the\nfollowing parts:\n•InSection2, weprovethecorrespondingLiouville-typethe oremsandthenon-degeneracy\nof the radial sign-changing solution to the scalar field coun terpart of Problem ( 1.1);\n•In Section 3, we prove Theorem 1.4and Theorem 1.1.\n2 Preliminaries\n2.1 Notations\nIn this part, we introduce some notations, in particular the number of nodes of a continuous\nradial function u: Ω→R. Here, we assume that Ω is radial, i.e., it is either a ball, an annulus,\nthe exterior domain of a ball or the whole space RN.\nDefinition 2.1 For a continuous radial function u: Ω→R, we define the number of nodes of\nthe function uto be the the largest number ksuch that there exists a sequence of real numbers\nx0,···,xksuch that 0< x0< x1<···< xkand\nu||x|=xj·u||x|=xj+1<0, j= 0,...,k−1.\nDenote the nodal number of the function ubyn(u).\nWe always assume that the functions we discuss have finite nod al numbers.\nDefinition 2.2 For a continuous radial function uwithn(u) =kandu(x0)>0, we define\nitsq-th bump for q= 1,...,k+1, by\nu1(x) =χ{u>0}·χ{|x|0}·χ{xq−2<|x|0}·χ{xq−1<|x|}·u(x).\n4A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\nFor a radial function uwithn(u) =kandu(x0)<0, we define its q-th bump q= 1,...,k+1\nby\nu1(x) =χ{u<0}·χ{|x|0}·χ{xq−1<|x|}·u(x).\nRemark 2.3 To avoid confusion, for the j-th component ujofU= (u1,...,u N), we denote\nitsq-th bump by uj,q.\nNow we present the concept of trivial solutions, semi-trivi al solutions, and non-trivial\nsolutions.\nDefinition 2.4 For a solution U= (u1,···,uN)to Problem ( 1.1),Uis said to be a trivial\nsolution if for any j= 1,···,N,uj≡0inB1.Uis semi-trivial, if it is not trivial but there is\naj0= 1,···,Nwithuj0≡0inB1.Uis non-trivial if for any j= 1,···,N,ujis non-zero.\n2.2 Liouville-type theorems and a priori estimates\nIn this part, we aim to obtain a priori estimates for the solut ions to Problem ( 1.1). Firstly,\nwe prove Theorem 1.3.\nProof of Theorem 1.3.Our argument is analogues to the one in [ 16, Proposition 21.2b].\nWe argue by contradiction via a re-scaling method. Let us den ote\nUε,β:=/braceleftbig\nU∈(C(R3))N\\{0}/vextendsingle/vextendsingleUsolves Problem ( 1.2) withε≤βij≤βfor anyi/\\e}atio\\slash=j/bracerightbig\n.\nHere,β >0 is fixed in the assumptions. Assume that for a sequence of εm→0 asm→0,\nUεm,β/\\e}atio\\slash=∅. We can find a sequence Um∈ Uεm,β. Denote Um= (um,1,···,um,N) the solution\nto \n\n−∆uj=µju3\nj+/summationtext\ni/ne}ationslash=jβm\niju2\niujinR3,\nuj∈C2(R3), ujis radial,\nujis zero or changes its sign at most Pjtimes and ( u1,···,uN)/\\e}atio\\slash= 0,\n|uj|L∞(R3)≤1 forj= 1,···,N(2.1)\nwithβm\nij∈[εm,β] fori,j= 1,···,Nandi/\\e}atio\\slash=j. Then, without loss of generality, we can\nassume that um,1/\\e}atio\\slash= 0 and supx∈R3|um,1|= supj=1,···,msupx∈R3|um,j|=:Mmholds for any\nm= 1,2,···. Up to a re-scaling, we can assume that1\n2≤supx∈R3|um,1| ≤1 for any m.\nIndeed, suppose xm∈R3such that |um,1(xm)| ≥Mm\n2. Now, let us consider the sequence\n/tildewideUm:=/parenleftBig\nM−1\nmU1,m(M−1\nm(x+Mmxm)),···,M−1\nmUN,m(M−1\nm(x+Mmxm))/parenrightBig\n=: (/tildewideUm,1,···,/tildewideUm,N).\nsatisfies1\n2≤ |/tildewideUm,1(0)| ≤supj=1,···,Nsupx∈R3|/tildewideUm,j|= supx∈R3|/tildewideUm,1| ≤1. If{Mm|xm|}mis\nbounded, as m→+∞, up to a translation in R3, there is a radial C2,αvector-valued function\n5Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\n/tildewideU∞for some α∈(0,1) such that /tildewideUm→/tildewideU∞inC2,α\nloc. Writing /tildewideU∞=/parenleftbig/tildewideU∞,1,···,/tildewideU∞,N/parenrightbig\n, it\nfollows that\n−∆/tildewideU∞,j=µj/tildewideU3\n∞,j+/summationdisplay\ni/ne}ationslash=jβ∞\nij/tildewideU2\n∞,i/tildewideU∞,jinR3(2.2)\nwith lim m→∞βm\nij=β∞\nij≥0 for any i,j= 1,···,Nandi/\\e}atio\\slash=j./tildewideU∞,jchanges its sign at most\nPjtimes for j= 1,···,N. It should be noted that /tildewideU∞,1/\\e}atio\\slash= 0 and is bounded and nodes at\nmostP1times. Then, denote O1by the interior of support of the outmost bump of /tildewideU∞,1.\nWithout loss of generality, we can assume that /tildewideU∞,1|O1>0. Then,\n−∆/tildewideU∞,1≥µ1/tildewideU3\n∞,1inO1. (2.3)\nThis contradicts with [ 2, Theorem 1.3].\nIf the sequence {Mm|xm|}mis unbounded, writing r=|x−Mmxm|and regarding /tildewideUm,j\nas one dimensional functions, then we have\n−d2\ndr2/tildewideUm,j−2\nr+Mm|xm|d\ndr/tildewideUm,j=µj/tildewideU3\nm,j+/summationdisplay\ni/ne}ationslash=jβm\nij/tildewideU2\nm,i/tildewideUm,jforr∈(−Mm|xm|,∞)\nfor anyj= 1,···,N. Then, the C2,α\nloc-limit of the sequence /tildewideUmsatisfies\n−/tildewideU′′\n∞,j=µj/tildewideU3\n∞,j+/summationdisplay\ni/ne}ationslash=jβ∞\nij/tildewideU2\n∞,i/tildewideU∞,jforr∈R (2.4)\nfor anyj= 1,···,N. And/tildewideU∞,1(0)≥1\n2and/tildewideU∞,1has at most P1zeroes. Let us denote\n−/tildewideU′′\n∞,1:=f(x,/tildewideU∞,1). Using ( 2.4), it is easy to verify that, /tildewideU∞,1f(x,/tildewideU∞,1)>0 if/tildewideU∞,1/\\e}atio\\slash= 0.\nBy [20, Theorem 1], /tildewideU∞,1oscillates on R. We have a contradiction.\nTherefore, the assumption is invalid and Theorem 1.3holds.\n✷\nTwo immediate consequences can be obtained.\nCorollary 2.5 For anyP1···,PN∈N, ifβij≥0for anyi,j= 1,···,Nandi/\\e}atio\\slash=j, Problem\n(1.2) admits no non-trivial solution.\nCorollary 2.6 For anyP1···,PN∈N, there exists a positive number ε1=ε1(µ1,···,µN;P1,···,PN)>\n0such that if |βij|< ε1, Problem ( 1.2) admits no non-trivial solution.\nA one-dimensional analogue to the above results can be prove d using a similar approach.\nTheorem 2.7 Consider the problem\n\n\n−u′′\nj=µju3\nj+/summationtext\ni/ne}ationslash=jβiju2\niujinR,\nuj∈C2(R), ujis radial,\nujis zero or changes its sign at most Pjtimes and (u1,···,uN)/\\e}atio\\slash= 0,\n|uj|L∞(R)≤1forj= 1,···,N.(2.5)\n6A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\nand the problem\n\n\n−u′′\nj=µju3\nj+/summationtext\ni/ne}ationslash=jβiju2\niujin[0,+∞),\nuj∈C2([0,+∞)), ujis radial,\nujis zero or changes its sign at most Pjtimes and (u1,···,uN)/\\e}atio\\slash= 0,\n|uj|L∞([0,+∞))≤1, uj(0) = 0forj= 1,···,N.(2.6)\nHere,µj>0are constants for j= 1,···,N. For any P1···,PN∈N, and any β≥0, there\nexists a positive number ε2=ε2(µ1,···,µN;P1,···,PN;β)>0such that if βij∈[−ε2,β]for\nanyi,j= 1,···,Nandi/\\e}atio\\slash=j, Problem ( 2.5) and Problem ( 2.6) admit no non-trivial solution.\nSimilar to the above result, we also have two immediate corol laries.\nCorollary 2.8 For anyP1···,PN∈N, ifβij≥0for anyi,j= 1,···,Nandi/\\e}atio\\slash=j, Problem\n(2.5) and Problem ( 2.6) admit no non-trivial solution.\nCorollary 2.9 For anyP1···,PN∈N, there exists a positive number ε3=ε3(µ1,···,µN;P1,···,PN)>\n0such that if |βij|< ε3, Problem ( 2.5) and Problem ( 2.6) admit no non-trivial solution.\nRemark 2.10 The novelty of these Liouville-type theorems lies in the absen ce of the need\nfor comparison between the components to conclude non-exis tence. However, comparisons are\nstill crucial in a broader range of parameters. This is becaus e the solutions to Problem ( 1.1)\nwith a component-wise prescribed number of nodes may have no a priori estimate, especially\nfor the case of βij≡β,µj≡µandβ <−µfor anyi,j= 1,···,Nandi/\\e}atio\\slash=j, cf. [11,?,14].\nIt is unreasonable to expect a Liouville-type theorem under th ese circumstance.\n2.3 [18, Theorem 1.3] and the corresponding non-degeneracy result\nConsider the scalar field counterpart of Problem ( 1.1), i.e.,\n/braceleftBigg\n−∆u+λu=µu3inBρ⊂R3,\nu∈H1\n0,r(Bρ).(2.7)\nHere,λ,µ >0 are constant. Bis a ball of radius ρwith the origin as its centre in R3. [18,\nTheorem 1.3] addresses that\nTheorem 2.11 Assume that λ=µ= 1andρ≤1. For any P= 0,1,···, Problem ( 2.7)\nadmits an unique radial solution uchanging its sign exactly Ptimes with u(0)>0.\nIn this note, we would like to go slightly further on this dire ction, i.e., to prove the following\nresult:\nTheorem 2.12 Consider Problem ( 2.7) withρ= 1. For any P∈N, we have\n7Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\n(1).ifλ≥1, for any k= 0,1,···, Problem ( 2.7) admits an unique radial solution uchanging\nits sign exactly Ptimes with u(0)>0;\n(2).the above solution is non-degenerate in the radial function al spaceH1\n0,r(B1).\nRemark 2.13 Here, by the word non-degenerate in the radial functional space , we mean that\nfor anyϕ∈H1\n0,r(B1), we have\n−∆ϕ+ϕ= 3u2ϕ.\nNow we begin to prove Theorem 1.1.\nProof of (1).This computation follows a routine process as described in [ 18]. The only\ndifference lies in the transformation of the radial functions defined on B1to one-dimensional\nfunctions, which we will restate as follows. Consider\n−d2\ndr2φ+2\nrd\ndrφ−λφ= 0\nwith\nφ(r) =√\nλcosh(r√\nλ)\nr.\nThen, we can define\nt=/integraldisplayr\n0λds/parenleftbig\ncosh(s√\nλ)/parenrightbig2\nand\ny(t) =u(r)\nφ(r).\nIt is easy to verify that\n•d\ndty(t) =r2\nλ·[d\ndruφ−ud\ndrφ];\n•d2\ndt2y+h(t)y= 0 with h(t) =λµ3·(cosh(r√\nλ))6\nr2.\nTherest partfollows aroutinesimilar tothat of [ 18, Corollary 3.4], which will not bediscussed\nfurther.\nProof of (2).The proof will be completed if we verify the following claim.\nClaim 2.14 For the radial solution uin Theorem 2.12, problem\n/braceleftBigg\n−∆v+λv= 3µu2vinB⊂R3,\nv∈H1\n0,r(B)(2.8)\nadmits only trivial solution, i.e., v= 0.\n8A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\nBy the transform\nz(t) =v(r)\nφ(r),\nwe have /braceleftBiggd2\ndt2z+3h(t)y2z= 0,\nz(0) =z(T) = 0(2.9)\nwithT=/integraltext1\n0λds/parenleftbig\ncosh(s√\nλ)/parenrightbig2. Here, since\nz(0) = lim\nr→0+r·v(r)√\nλcosh(r√\nλ),\nwe will have z(0) = 0 as long as vis a classical solution. z(T) = 0 is due to v∈H1\n0,r(B1).\nWe intend to prove that z≡0 in [0,T]. Arguing by contradiction. Assume that z/\\e}atio\\slash= 0. Since\nProblem ( 2.9) is linear in z, we can assume z′(0) = 1, without loss of generality. Consequently,\nthe problem/braceleftBiggd2\ndt2z+3h(t)y2z= 0,\nz(0) = 0, z′(0) = 1(2.10)\nadmits a non-trivial solution zwithz(T) = 0. One the other hand, notice that\n(1). T, the end point of the interval [0 ,T], is also the largest zero of the function z;\n(2).Problem ( 2.10) admits an unique C2-solution.\nThe first assertion is evident and the second is guaranteed by [18, Proposition 4.1]. Then, [ 18,\nLemma 5.2] gives us that z(T)/\\e}atio\\slash= 0. This is a impossible. Therefore, 0 is the only solution to\nProblem ( 2.9) due to the its unique solvability. This proves Theorem 2.12.\n✷\nAn immediate consequence of Theorem 2.12concerns the invertibility of the second order\nderivative of the C2functional\nI/vectorβ(u1,···,uN) =1\n2N/summationdisplay\nj=1/integraldisplay\nB1|∇uj|2+λju2\nj−1\n4N/summationdisplay\nj=1/integraldisplay\nB1µju4\nj+/summationdisplay\ni/ne}ationslash=jβiju2\niu2\nj.(2.11)\nHere,/vectorβ= (βij)i/ne}ationslash=j;i,j=1,···,N. Letting uj,Pjsolve\n−∆u+λju=µju3inB1\nwithn(uj) =Pj. Then, we get\nCorollary 2.15 It holds that the operator I′′|/vectorβ=0(u1,P1,···,uN,PN)is invertible in (H1\n0,r(B1))N.\nThis is a direct consequence of the C2-continuity of the functional I/vectorβ.\n9Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\n3 Proof of Theorem 1.1and Theorem 1.4\nThis section presents the proof for Theorem 1.1and Theorem 1.4. The former theorem is\nan immediate consequence of the a priori estimate (see Theor em1.4), the uniqueness (see\nTheorem 2.12) and non-degeneracy result (see Corollary 2.15) through an argument in [ 7].\nTo this end, we need to verify Theorem 1.4in advance.\n3.1 Proof of Theorem 1.4and its corollaries\nThe proof of Theorem 1.4is a routine and we only outline it. Arguing by contradiction ,\nsuppose for a β >0, we can find a sequence of numbers β(m)\nij∈[−ε∗,β] for any i,j= 1,···,N\nandi/\\e}atio\\slash=jwith\n•ε∗= min(ε0,ε2), the constants in Theorem 1.3and Theorem 2.7;\n•a sequence of functions ( u(m)\n1,···,u(m)\nN) with\n/braceleftBigg\n−∆u(m)\nj+λju(m)\nj=µj(u(m)\nj)3+/summationtext\ni/ne}ationslash=jβ(m)\nij(u(m)\ni)2u(m)\njinB1,\nu(m)\nj∈H1\n0,r(B1), n(u(m)\nj) =Pjforj= 1,···,N;\n•without loss of generality, |u(m)\n1|∞= max j=1,···,N|u(m)\nj|∞→ ∞asm→ ∞.\nAs in Section 2.2, denote M′\nm=|u(m)\n1|∞andx′\nm∈B1such that |u(m)\n1(x′\nm)|=|um\n1|∞for any\nm∈N. Consider the re-scaling functions\nˆUm:=/parenleftBig\n(M′\nm)−1u(m)\n1((M′\nm)−1(x+M′\nmx′\nm)),···,(M′\nm)−1u(m)\nN((M′\nm)−1(x+M′\nmx′\nm))/parenrightBig\n=: (ˆUm,1,···,ˆUm,N).\nIf the sequence {M′\nm|x′\nm|}mis bounded, the C2,α\nloclimit of ˆUmwill be a non-trivial classical\nsolution to Problem ( 1.2), which contradicts Theorem 1.3. If the sequence {M′\nm|x′\nm|}mis\nunbounded, the limit will tend to a non-trivial classical so lution to Problem ( 2.5) or Problem\n(2.6). This contradicts Theorem 2.7. Therefore, Theorem 1.4is proved.\n✷\nAs we did in Section 2.2, two corollaries follow immediately ,\nCorollary 3.1 For any P1···,PN∈N, ifβij≥0for any i,j= 1,···,Nandi/\\e}atio\\slash=j,\nthere exists a constant C=C(λ1,···,λN;µ1,···,µN;βij;P1,···,PN)>0such that for any\nsolution Uto Problem ( 1.1), we have |U|∞≤C.\nCorollary 3.2 For anyP1···,PN∈N, there exist positive numbers ε4=ε4(µ1,···,µN;P1,···,PN)>\n0andC=C(λ1,···,λN;µ1,···,µN;P1,···,PN)>0such that if |βij|< ε4, for any solution\nUto Problem ( 1.1), we have |U|∞≤C.\n10A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\n3.2 Proof of Theorem 1.1\nWe argue by contradiction. Suppose that there is a sequence ( βm\nij)i/ne}ationslash=j,i,j=1,···,N;m=1,2,···such\nthatβm\nij→0 asm→+∞for anyi/\\e}atio\\slash=jandi,j= 1,···,Nand there are two different\nsolutions U1\nmandU2\nmto the problem\n/braceleftBigg\n−∆uj+λjuj=µju3\nj+/summationtext\ni/ne}ationslash=jβm\niju2\niujinB1,\nuj∈H1\n0,r(B1), n(uj) =Pjforj= 1,···,N.\nHere,λj≥1 andµj>0 forj= 1,···,N. By Corollary 3.2, there exists a positive constant\nCindependence in msuch that |Um\n1|L∞(B1),|Um\n2|L∞(B1)≤C. Then, for any p≥2, aW2,p-\nestimate yields that /ba∇dblUm\n1/ba∇dblW2,p(B1),/ba∇dblUm\n2/ba∇dblW2,p(B1)≤C. Then, up to a subsequence, there are\nU1andU2such that\nUm\n1→U1andUm\n2→U2in (H1\n0(B1))N\nasm→ ∞. And it is evident that U1andU2solve\n/braceleftBigg\n−∆uj+λjuj=µju3\njinB1,\nuj∈H1\n0,r(B1), n(uj)≤Pjforj= 1,···,N.\nNowwewillverifythatforany j= 1,···,Nwehaven(uj) =Pj. Denote Um\ni= (um\ni,1,···,um\ni,N)\nfori= 1,2. Therefore, n(um\ni,j) =Pj. Denote by wma bump of um\ni,j. Multiplying the j-th\nequation of Problem ( 1.1) and integrating over B1, we get\n/integraldisplay\nB1|∇wm|2+λj|wm|2=µj/integraldisplay\nB1|wm|4+/summationdisplay\ni/ne}ationslash=jβij/integraldisplay\nB1u2\ni|wm|2\n≤µj|wm|4\n4+C0·sup\ni/ne}ationslash=j|βij|·|wm|2\n4.\nUsing Sobolev’s inequality,\nC1|wm|2\n4≤µj|wm|4\n4+C·sup\ni/ne}ationslash=j|βij|·|wm|2\n4.\nLetting supi/ne}ationslash=j|βij|0 independence in m. TheL4\nconvergence yields the conclusion that n(uj) =Pj. By Theorem 2.12, we have U1=U2, i.e.,\n/ba∇dblUm\n1−Um\n2/ba∇dbl →0 asm→ ∞.\nOn the other hand, due to the implicit function theorem (cf. [ 3, Theorem 1.2.1]), there is\na constant b >0 such that for any set of numbers /vectorβ= (βij)i,j=1,···,N;i/ne}ationslash=jwith|βij|< b, the\nequation\nI′\n/vectorβ(u1,···,uN) = 0\nadmits an unique non-trivial solution in {U= (u1,···,uN)|uj∈H1\n0,r(B1) and/ba∇dbluj−uj,Pj/ba∇dbl ≤\nδ}forsomesmall δ >0. Thiscontradictstheconvergenceof Um\n1andUm\n2. Theassumptionfails\nand Theorem 1.1is proved. The non-degeneracy mentioned in Remark 1.2is an immediate\nconsequence of the C2continuity.\n✷\n11Haoyu Li & Ol ´ımpio Hiroshi Miyagaki\nAcknowledgements\nBothoftheauthorswouldliketoexpresstheirsincereappre ciationtotheanonymousreviewers\nfor their helpful suggestions on improving this paper. Thei r thank also goes to Professor Pavol\nQuittner, Professor Satoshi Tanaka and Professor Zhi-Qian g Wang for their various helpful\ncommunications. In this research, HL was supported by FAPES P PROC 2022/15812-0 and\nOHM was supported by CNPq PROC 303256/2022-2 AND FAPESP PROC 2022/16407-1.\nReferences\n[1] Ao, W, Wei, J, Yao, W, Uniqueness and nondegeneracy of sig n-changing radial solutions to an almost\ncritical elliptic problem. October, 2015. Advances in Diffe rential Equations 21(12).\n[2] Bidaut-V´ eron, M.-F, Local and global behavior of solut ions of quasilinear equations of Emden-Fowler\ntype. Arch. Rational Mech. Anal. 107(1989), no.4, 293-324.\n[3] Chang, K.-C, Methods in nonlinear analysis. Springer Mo nographs in Mathematics. Berlin: Springer. ix,\n439 p. (2005).\n[4] Dancer, E. N, Wei, J, Weth, T, A priori bounds versus multi ple existence of positive solutions for a\nnonlinear Schr¨ odinger system. Annales de l’Institut Henr i Poincare (C) Non Linear Analysis, 2010, 27(3)\npp. 953-969.\n[5] Gidas, B, Ni, W. M, Nirenberg, L, Symmetry and related pro perties via the maximum principle. Com-\nmunications in Mathematical Physics, 1979, 68(3):209-243 .\n[6] Gidas, B, Spruck, J, A priori bounds for positive solutio ns of nonlinear elliptic equations. Comm. Partial\nDifferential Equations 6 (1981), no. 8, 883-901.\n[7] Ikoma, N, Uniqueness of positive solutions for a nonline ar elliptic system. NoDEA Nonlinear Differential\nEquations Appl. 16 (2009), no. 5, 555-567.\n[8] Kabeya, Y, Tanaka, K, Uniqueness of positive radial solu tions of semilinear elliptic equations in RNand\nS´ er´ e’s non-degeneracy condition. Comm. Partial Differen tial Equations24(1999), no.3-4, 563-598.\n[9] Korman, P, Li, Y, Schmidt, D. S, A computer assisted study of uniqueness of ground state solutions.\nJournal of Computational & Applied Mathematics, 2012, 236( 11):2838-2843.\n[10] Kwong, M. K, Uniqueness of positive solutions of ∆ u−u+up= 0 inRn. Archive for Rational Mechanics\nand Analysis, 1989, 105(3), 243-266.\n[11] Li, H, Wang, Z.-Q, Multiple nodal solutions having shar ed componentwise nodal numbers for coupled\nSchr¨ odinger equations. Journal of Functional Analysis 28 0(7), 2021:108872.\n[12] Liu, H, Liu, Z, Chang, J, Existence and uniqueness of pos itive solutions of nonlinear Schr¨ odinger systems.\nProceedings of the Royal Society of Edinburgh: Section A Mat hematics, 145(2), 2015:365-390.\n[13] Liu, Z, Wang, Z.-Q, Vector Solutions with Prescribed Co mponent-Wise Nodes for a Schr¨ odinger System.\nAnalysis in Theory and Applications 35(3), 2019:288-311.\n[14] Quittner, Pavol Liouville theorem and a priori estimat es of radial solutions for a non-cooperative elliptic\nsystem. Nonlinear Anal., Theory Methods Appl., Ser. A, Theo ry Methods. 222, 2022:112971.\n[15] Quittner, P, Souplet, P, Optimal Liouville-typeTheor ems for Noncooperative Elliptic Schr¨ odinger Systems\nand Applications. Communications in Mathematical Physics . 311, 2012:1-19.\n[16] Quittner, P, Souplet, P, Superlinear Parabolic Proble ms: Blow-up, Global Existence and Steady States.\nSecond edition. Birkh¨ auser Basel. 2019.\n[17] Quittner, P, Personal communications. 2023.\n12A Liouville-type theorem for the coupled Schr ¨odinger systems and ...\n[18] Tanaka, S, Uniqueness of sign-changing radial solutio ns for ∆u−u+|u|p−1u= 0 in some ball and annulus.\nJournal of Mathematical Analysis and Applications, 439(1) , 2016:154-170.\n[19] Wei, J, Yao, W, Uniqueness of positive solutions to some coupled nonlinear Schr¨ odinger equations. Com-\nmunications on Pure & Applied Analysis, 2011, 11(3):1003-1 011.\n[20] Wong, J. S. W, On second order nonlinear oscillation. Fu nkcial. Ekvac.11(1968), 207-234.\n[21] Zhou, L, Wang, Z.-Q, Uniqueness of positive solutions t o some Schr¨ odinger systems. Nonlinear Analysis,\n2020, 195(2):111750.\n13" }, { "title": "2401.15858v1.Transverse_oscillation_of_prominence_and_filament_induced_by_an_EUV_wave_from_the_farside_of_the_Sun.pdf", "content": "Draft version January 30, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nTransverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun\nYanjie Zhang ,1Qingmin Zhang ,1, 2De-chao Song,1andHaisheng Ji1\n1Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, CAS, Nanjing 210023, China\n2Yunnan Key Laboratory of Solar physics and Space Science, Kunming 650216, People’s Republic of China\nABSTRACT\nIn this paper, we report our multi-angle observations of the transverse oscillation of a prominence and a filament\ninduced by an EUV wave originating from the farside of the Sun on 2014 September 1. The prominence oscillation\nwas simultaneously observed by both Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory\n(SDO) spacecraft and Extreme-UltraViolet Imager (EUVI) onboard the Behind Solar Terrestrial Relations Observatory\n(STEREO) spacecraft. The speed of the shock travelling in the interplanetary space exceeds that of the EUV wave,\nand the coronal dimming area experiences minimal growth. This indicates that the shock wave is driven by the CME,\nwhile the EUV wave freely propagates after the lateral motion of the CME flanks has stopped. The observed oscillation\ndirection of the prominence, determined through three-dimensional reconstruction, further supports this point. Moreover,\nThe detailed investigation of the oscillations in the prominence and filament induced by the EUV wave reveals initial\namplitudes of 16.08 and 2.15 Mm, periods of 1769 and 1863 s, damping time scales of 2640 and 1259 s, and damping\nratios of 1.49 and 0.68, respectively. The radial component of magnetic field, as derived from the prominence and\nfilament oscillation measurements, was estimated to be 5.4 G and 4.1 G, respectively. In turn, utilizing the onset times\nof both the prominence and filament oscillation, the average speeds of the EUV wave are determined to be 498 km s−1\nand 451 km s−1, respectively.\nKeywords: Solar flares(1496) — Coronal mass ejections(310) — Solar prominences(1519)\n1.INTRODUCTION\nSolar prominences are often observed as bright elongated\nemission against the dark background at the solar limb in\nchromospheric and coronal lines (Tandberg-Hanssen 1995;\nMartin 1998; Labrosse et al. 2010; Chen et al. 2020). They\ncharacterise dark features, called filaments, on the disk be-\ncause they absorb intense solar atmospheric radiation. De-\npending on the location where a filament is formed, they can\ngenerally be divided into three types: active region filaments\n(AFs) (Yan et al. 2015), quiescent filament (QFs) (Heinzel\net al. 2008) and intermediate filament (IFs). The promi-\nnence and filament are the same entities and we use the term\n“prominence” throughout the paper.\nProminences are rich in dynamics, in which their large-\namplitude oscillations induced by EUV waves in association\nwith remote flaring activities has been a hot topic of research\n(Wills-Davey & Attrill 2009; Gallagher & Long 2011; Pat-\nsourakos & V ourlidas 2012; Liu & Ofman 2014; Warmuth\nCorresponding author: Yanjie Zhang\nzhangyj@pmo.ac.cn2015; Chen 2016). When the EUV wave was first discov-\nered, it was also referred to as the EIT wave, named after the\nobserving telescope Extreme-ultraviolet Imaging Telescope\n(EIT; Delaboudini `ere et al. 1995) onboard the Solar and He-\nliospheric Observatory (SOHO). They are observed as bright\nwavelike fronts propagating across most of the solar disk in\nSOHO /EIT running-di fference images if the coronal mag-\nnetic structure is simple (Thompson et al. 1998). They occur\nfollowing flare /coronal mass ejection (CME), accompanied\nby extending dimming region behind them (Thompson et al.\n2000). Initially, EUV waves were explained naturally to be\nthe coronal counterparts of chromospheric Moreton waves,\nthat is coronal fast-mode (shock) waves (Moreton 1960;\nUchida 1968, 1974; Thompson et al. 2000; Wang 2000; Wu\net al. 2001; Ofman & Thompson 2002). However, many\nEUV waves exhibit properties distinct from those of Moreton\nwaves (Eto et al. 2002; Zhang et al. 2011), giving rise to var-\nious conjectures about their physical nature (Warmuth et al.\n2001; Vr ˇsnak et al. 2002; Tripathi & Raouafi 2007; Warmuth\n2007; Wills-Davey & Attrill 2009; Gallagher & Long 2011).\nCurrently, a small fraction of events has demonstrated the\npresence of two components in EUV waves, one fast and\none slow(Chen & Wu 2011; Shen & Liu 2012a; Shen et al.arXiv:2401.15858v1 [astro-ph.SR] 29 Jan 20242 Z hang et al .\n-1000-50005001000Y (arcsec) FlareAR 12158Prominence(a) STEREO-B 195 Å 11:00:56 UT\nFlareAR 12158Prominence(b) STEREO-B 304 Å 10:56:39 UT\n-1000 -500 0 500 1000\nX (arcsec)-1000-50005001000Y (arcsec) (c) AIA 304 Å 11:00:55 UT\nS1S0\nS2\nF\n1.0\n 0.5\n 0.0 0.5 1.0\nY (HEE)1.00.50.0-0.5-1.0X (HEE)Sun\nEarthB\nA\n2014-Sep-01(d)\nFigure 1. (a and b) Images of STEREO-B 195 Å at 11:00 UT and 304 Å at 00:56 UT on 2014 September 1, respectively. The location of AR\n12158 is labelled in each panel, as well as the flare and the prominence. (c) Image of AIA 304 Å at 11:00 UT. The subfigures in the panel show\nan enlarged view of the prominence, identical to that in STEREO-B images, as well as a filament on the disk, respectively. Three slices, S1,\nS2 and S3, are utilized to plot the time-distance diagram for the prominence oscillation, while F is for the filament. (d) Position of the Earth,\nAhead (A), and Behind (B) STEREO spacecraft at 11:17 UT on 2014 September 1. An animation of the EUVI 195 Å and AIA 304 Å images,\navailable in the online Journal, illustrates the global evolution of the EUV wave from the STEREO-B perspective and provides a close-up view\nof the prominence oscillation from Earth’s perspective.\n2014a), while the majority of events have not exhibited this\nfeature.\nLarge amplitude prominence oscillations caused by EUV\nwaves are often observed (Eto et al. 2002; Okamoto et al.\n2004). Specifically, prominence oscillations can be further\nclassified into longitudinal and transverse oscillations. Lon-\ngitudinal oscillations refer to the oscillatory motion of promi-\nnence material along the guiding toroidal magnetic field, with\nmost events not exceeding 20◦(Jing et al. 2003; Li & Zhang\n2012; Luna & Karpen 2012; Zhang et al. 2012; Luna et al.\n2014; Dai et al. 2021). However, recent studies have alsoreported longitudinal oscillations exceeding 20◦(Shen et al.\n2019; Tan et al. 2023). In contrast, transverse oscillations oc-\ncur in the direction perpendicular to the spine (Hyder 1966;\nRamsey & Smith 1966; Kleczek & Kuperus 1969; Shen et al.\n2014a). Recently, the physical nature of prominence oscilla-\ntions, such as triggering mechanisms, restoring forces, and\ndamping mechanisms, has been e ffectively investigated us-\ning state-of-the-art magnetohydrodynamics (MHD) numeri-\ncal simulations (Terradas et al. 2013, 2015; Zhang et al. 2013;\nLuna et al. 2016; Zhou et al. 2017, 2018; Devi et al. 2022; Li-\nakh et al. 2023).Transverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun 3\n600 700 800 900\nX (arcsec)500600700800900Y (arcsec)\n(a) STEREO-B 304 Å 11:16:39 UTOriginal image\n600 700 800 900\nX (arcsec)Enhanced image\n(b) STEREO-B 304 Å 11:16:39 UT\nFigure 2. (a) Original image of the prominence from EUVI 304 Å\nat 11:16 UT on 2014 September 1. The colored lines are employed\nto enhance the image in the radial-filter technique, as described in\nSection 2. (b) The enhanced image after using the radial-filter tech-\nnique. The yellow line represents the contour of intensity threshold\nset at 2.8 times the average pixel intensity of the enhanced image,\nand the plus symbol represent the centroid of the prominence by us-\ning the contour.\nUntil now, oscillations in prominence and filament induced\nby EUV waves originating from the farside of the Sun have\nnot been comprehensively investigated. In this study, we ana-\nlyzed the prominence and filament oscillation associated with\na farside EUV wave on 2014 September 1. The prominence\nwas simultaneously observed by the Behind Solar Terrestrial\nRelations Observatory (STEREO-B, hereafter STB; Kaiser\net al. 2008) spacecraft and the Solar Dynamics Observatory\n(SDO; Pesnell et al. 2012), while the flare was only observed\nby STB and the filament was exclusively observed by SDO.\nWe found that the EUV wave freely propagated once the lat-\neral motion of the CME flanks stopped. In contrast, the shock\nwave in the interplanetary space is all driven by the CME, and\nits speed exceeds that of the EUV wave. The speeds of the\nEUV wave reaching the prominence and filament were cal-\nculated to be 498 km s−1and 451 km s−1, respectively. Fur-\nthermore, the unprecedented time and spatial resolution of\nmulti-wavelength EUV data recorded with SDO /AIA enable\nus to accurately determine the parameter of the prominence\nand filament oscillation, thereby the surrounding magnetic\nfield strength can be estimated.\nData analysis is described in detail in Section 2. Results\nare shown in Section 3. The discussion and summary are\ngiven in Section 4 and Section 5.\n2.DATA ANALYSIS\nThe two panels in the first row of Figure 1 show the 195\nand 304 Å images of Extreme-UltraViolet Imager (EUVI;\nWuelser et al. 2004) of the Sun Earth Connection Coro-\nnal and Heliospheric Investigation (SECCHI; Howard et al.\n2008) on board the STB on 2014 September 1, where AR\n12158 is labelled. EUVI takes full-disk images out to 1.7 R⊙\nwith a spatial resolution of 3 .′′2 in 171, 195, 284, and 304\nÅ. The 195 and 304 Å images have time cadences of 5 and\n10 minutes, respectively. The flare with obvious intensityenhancement is marked with an arrow. We have no way of\nknowing the magnitude of the flare from Soft X-ray (SXR)\nlight curves in this period recorded by the Geostationary\nOperational Environmental Satellite (GOES) spacecraft, be-\ncause it occurred on the farside of the Sun as shown by panel\n(d).\nThe red contour of the prominence in panel (a) is derived\nfrom panel (b). In our case, the intensity at the top of the\nprominence is lower than the background intensity close to\nthe solar surface in the EUVI 304 Å. Therefore, it is not fea-\nsible to obtain the profile of the prominence by simply select-\ning an intensity threshold and plotting a contour. Instead, the\nradial-filter technique is applied to enhance the emission of\nthe prominence, as demonstrated by Figure 2. Specifically,\nin the pixel coordinates, a series of concentric circles are se-\nlected and labelled 1 to 100 from lowest (close to the solar\nsurface) to highest (close to the top of the prominence) as\nshown by the colored lines in Figure 2(a). The intensity val-\nues of the EUVI 304 Å in each concentric circle are then mul-\ntiplied by the number of that concentric circle. In this way,\nthe intensity at the top of the prominence has been enhanced\nwhile that of the background close to the solar surface atten-\nuated, so the contour of the prominence can be well drawn as\ndepicted by Figure 2(b).\nThe prominence has also been recorded by the Atmo-\nspheric Imaging Assembly (AIA; Lemen et al. 2012) on\nboard the SDO. AIA took full-disk images in seven EUV\n(94, 131, 171, 193, 211, 304, and 335 Å) and two UV (1600\nand 1700 Å) wavelengths. The AIA level 1 data with a time\ncadence of 12 s and a spatial resolution of 1 .′′2 were cali-\nbrated using the standard program aiaprep.pro in the So-\nlar Software (SSW). The related CME was simultaneously\nobserved by the Large Angle and Spectrometric Coronagraph\n(LASCO; Brueckner et al. 1995) on board the SOHO space-\ncraft and the COR1 white-light (WL) coronagraph on board\nthe STB. The separation angle between the STB and Earth\nwas∼161◦as shown in Figure 1(d). The radio dynamic spec-\ntra associated with the flare and CME-driven shock was ob-\ntained from the BLENSW ground-based station belonging to\nthe e-Callisto1network.\n3.RESULTS\nThe NOAA active region (AR) 12158 is energetic as it pro-\nduced many flares during 2014 September 8 to 18 on the vis-\nible disk. The flare we studied also originated from this AR\non 2014 September 1, but on the farside of the Sun as shown\nin Figure 3. It should also be a high-energy one, because the\nflare was accompanied by a halo CME shown in Figure 5. In\naddition, the images of EUVI 195 and 304 Å during flaring\ntime su ffered from strong snow-like interference (see the on-\n1http://www.e-callisto.org4 Z hang et al .\n-50005001000Y (arcsec)Prominence(a) STB 195 Å 10:55:56-10:50:56\nProminence(b) STB 195 Å 11:00:56-10:50:56\nFlare\nProminence(c) STB 195 Å 11:05:56-10:50:56\nEUV Wave\nProminence(d) STB 195 Å 11:10:56-10:50:56\nEUV Wave\n-500 0 500 1000\nX (arcsec)-50005001000Y (arcsec)Prominence(e) STB 304 Å 11:16:39 - 10:46:49\nEUV WaveProminence\nMagnetic InterfaceFlare\n-500 0 500 1000\nX (arcsec)Prominence(f) STB 195 Å 11:20:56-10:50:56\nMagnetic Interface\n-500 0 500 1000\nX (arcsec)Prominence\nMagnetic Interface(g) STB 195 Å 11:25:56-10:50:56\n-500 0 500 1000\nX (arcsec)Prominence\nMagnetic Interface(h) STB 195 Å 11:30:56-10:50:56\nMagnetic InterfaceDimming\nFigure 3. Detailed feature of atmospheric disturbances in the EUVI 195 Å images subtracted with the image at 10:50:56 UT, except for panel\n(e) in the EUVI 304 Å image subtracted with the image at 10:46:49 UT. The dashed lines depict the wavefront in panel (c), (d) and (e). An\nexpanding dome is observed in panel (c) and is indicated by an arrow. Prominence is outlined in each image using the radial-filter technique\ndescribed in Figure 2. There is also a stationary front to the southeast of the disk as the wave front swept by, indicated by a yellow arrow. An\nanimation of the EUVI 304 Å and 195 Å subtracted images is accessible in the online journal, unveiling additional details in the evolutionary\nprocess of the EUV wave.\nline animation in Figure 1), probably caused by high-energy\nparticles that hit the image sensor at that time.\nSuccessive base-di fference images of EUVI are drawn in\nFigure 3. All of these images were subtracted from the base\nimage at about 10:50 UT, showing the detailed features of\ncoronal disturbances. The EUV wave (visible increase front)\nwas first observed at 11:05 UT in panel (c) and then propa-\ngated nearly half of the disk along all directions. However,\nit did not show an isotropic feature. Specifically, the front is\nmainly visible to the north of the flare, and a small indenta-\ntion appeared in its propagation path, which may have been\ncaused by a small AR in that area (see the online animation\nin Figure 3). In addition, a stationary front appeared in the\nsoutheast of the flare, where a magnetic interface structure\nexisted before the eruption. Therefore, this stationary bright-\nening is a result of the propagating disturbance acting on the\nstructure. However, it can still be observed that a subtle front\nwas passing through the interface in the di fference movie (see\nthe online animation in Figure 3), which demonstrates the na-\nture of the wave.\nThe dimming appeared near the AR at the same time, while\nthe EUV wave started outside the dimming area, similarly\nas reported by Eto et al. (2002); Thompson et al. (1999). It\ncan be observed that the dimming exhibited limited expan-\nsion compared to the wave front. This observation suggests\nthe scenario that the coronal dimming region maps the CMEfootprint on the solar lower layer, while the EUV wave is a\nwave driven by the associated CME (Patsourakos & V ourli-\ndas 2009; Muhr et al. 2010; Temmer et al. 2011; Shen & Liu\n2012a). We can also identify an expanding dome in panel\n(c), similar to that reported by Asai et al. (2012). The dome\nis recorded at a only single moment, indicating that its speed\nmust exceed that of the wave front. This reveals that the dome\nwas propelled by the CME, whereas in the lateral direction,\nthe wave is freely propagating as soon as the lateral expan-\nsion of the CME flanks has stopped (Veronig et al. 2010).\nSignificant propagating disturbance was also observed in\nthe EUVI 304 Å image in panel (e) and the online animation\n(Figure 3). Considering the similar kinematics to that in the\n195 Å images, we propose that the observed wave signature\nin the 304 Å passband is mainly contributed by the coronal\nSi XI line rather than the chromospheric He II lines (Shen &\nLiu 2012b).\nThe EUV wave induced oscillations in both a prominence\nand a filament. Due to the limited time resolution of the\nEUVI data and indistinct EUV Wave signals in AIA Data, it\nis inappropriate to calculate the speed by stacking the time-\ndistance diagram. Instead, the average velocity of the wave\ncan be estimated by determining the position of the flare,\nthe prominence and the filament. The position of the fil-\nament (the position of the slice F, see Figure 1(c) ) in the\nearth-based point is easily discerned (N33E9). Consider-Transverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun 5\nSTB’s viewEarth’s viewFilamentProminence\nFlare(N33E9)(N26E126)(N45E75)\n605 Mm\n1351 Mm\nFigure 4. The position of the flare, prominence and filament in the coordinate system from the earth-based vantage point, respectively. The red\ncurve is the result of the 3D reconstruction of the prominence using the AIA and EUVI 304 Å image pair at 11:16 UT.\ning the angle between STB and Earth with respect to the\nSun, the location of the flare from the Earth’s perspective\nwere calculated as (N26E126) (ignoring the solar axial tilt).\nApplying the three-dimensional (3D) reconstruction program\nsccmeasure.pro in the SSW software, the center coordi-\nnate of the prominence is obtained as (N45E75), as shown in\nFigure 4. Assuming a propagation altitude of 1.1 R⊙(the ra-\ndius of the Sun, about 695500 km) for the EUV wave, the co-\nefficient is sourced from the height of the prominence in the\n3D reconstruction. Therefore, the distances from the flare to\nthe prominence and filament can be determined as 605 Mm\nand 1351 Mm, respectively.\nThe initiation time of the flare recorded by EUVI was\n10:55:56 UT (see Figure 3), while the onset times of the\nprominence oscillation and filament oscillation observed by\nAIA were 11:16:09 UT and 11:45:57 UT, respectively (de-\ntailed discussion to ensue). Consequently, the average speeds\nof the EUV wave reaching the prominence and filament can\nbe determined as 498 km s−1and 451 km s−1, respectively,\nwith no significant velocity attenuation (assuming the EUV\nwave originated from the flare site, although their actual\nsource might be at some distance away from the flare cen-\nter, such as the edge of the dimming region (Thompson et al.\n1999; Eto et al. 2002), this seemingly has limited influence\nfor the calculations).A halo CME was also produced by the flare as shown in\nFigure 5, which was first observed by STB /COR1 at 11:05:48\nUT. It should be noted that in this moment, the prominence\nhad not yet exhibited oscillatory motion. This indicates that\nthe disturbance propagated faster in the direction of inter-\nplanetary space than along the solar surface. STB /COR1,\nLASCO /C2 and C3 all observed the shock associated with\nthe CME, as indicated by the yellow arrows in Figure 5. The\nspeed of CME is 1901 km s−1recorded by the CDAW cat-\nalog2where CMEs are recognized manually, 650 km s−1\nthe CACTus website3where CMEs are identified automati-\ncally (Yashiro et al. 2008). Figure 6 illustrates the calculation\nemployed by the two websites for the CME velocity determi-\nnation. It can be observed that the CDAW catalog involves\ntracking the fastest direction of CME propagation, whereas\nthe CACTus website averages the velocities across di fferent\npropagating directions of the CME. Simultaneously, we also\ntracked the propagation of the shock (the yellow cross sym-\nbols in Figure 5(b4), representing wavefronts in di fferent im-\nages), which exhibited a linear velocity of approximately 814\nkm s−1.\n2http://cdaw.gsfc.nasa.gov /CME list\n3http://sidc.oma.be /cactus /scan6 Z hang et al .\n(a1)\nCME\n(a2)\nShock\n(a3)\nShock\n(a4)\nShock\n(b1)\nShockCME\n(b2)\nShock\n(b3)\nShock\n(b4)\nShock\n155\n64\n(c1)\nSTB-COR1 2014/09/01 11:10:24 UTShockCME\n(c2)\nSTB-COR1 2014/09/01 11:25:24 UT\n(c3)\nSTB-COR1 2014/09/01 11:35:24 UT\n(c4)\nSTB-COR1 2014/09/01 11:50:24 UT\nFigure 5. LASCO /C2 (top), LASCO /C3 (middle) and STB /COR1 (bottom) di fference images showing the halo CME associated with the flare\nthat occurred on the farside of the Sun on 2014 September 1. Yellow arrows point to the shock associated with the CME. Panel (b2) presents\nthe two tracking angles: 64◦for the CME in Figure 6(a) and 155◦for the shock, both measured from north (the vertical line).\n(a)\n (b)\nFigure 6. (a) The time-distance diagram of the CME generated by the flare, as recorded by the CDAW catalog, with a tracking angle of 64◦\n(as shown in Figure 5(b4)). (b) The velocity distribution of the CME tracked at di fferent angles by the CACTus website.\nA flare-related type III and CME-related type II radio burst\nwas detected by the e-Callisto /BLENSW station in the fre-quency of 10∼80 MHz, which is displayed in Figure 7.Transverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun 7\n5ZQF\u0001***\u0001SBEJP\u0001CVSTU5ZQF\u0001**\u0001SBEJP\u0001CVSTU\nF\u000eDBMMJTUP\u0001#-&/48\nFigure 7. Radio dynamic spectra recorded by the e-\nCallisto /BLENSW station in the frequency of 10 ∼80 MHz. The\ntype III radio burst occurred from 11:05 UT to 11:10 UT, while the\ntype II radio burst occurred from 11:14 UT to 11:27 UT.\nThe flare was estimated to occur in 10:55 UT as shown in\nFigure 3. Subsequently, the time interval for the Type III\nradio burst was recorded between 11:05 UT and 11:10 UT,\nwhich are thought to be created by plasma emissions of flare-\naccelerated non-thermal electron beams propagating outward\nalong open field (Krucker et al. 2011; Masson et al. 2013;\nZhang et al. 2015; Wyper et al. 2018). The existence of\nopen magnetic field lines is confirmed by the Potential Field\nSource Surface (PFSS) extrapolation, as illustrated by the\npurple lines in Figure 8. The type II radio burst occurred\nimmediately after the type III burst, lasting from 11:14 UT to\n11:27 UT, which are associated with the shock waves driven\nby the fast CME (Ontiveros & V ourlidas 2009; Zucca et al.\n2018; Mancuso et al. 2019).\n(a) Viewpoint from STB (b) Viewpoint from SDO\n0QFO\u0001GJFME0QFO\u0001GJFME\n\u0013\u0011\u0012\u0015\u000e4FQ\u000e\u0011\u0012\u0001\u0011\u0017\u001b\u0011\u0015\u000165 \u0013\u0011\u0012\u0015\u000e4FQ\u000e\u0011\u0012\u0001\u0011\u0017\u001b\u0011\u0015\u000165\"3\u0001\u0012\u0013\u0012\u0016\u0019\nFigure 8. PFSS extrapolations from the STB (a) and AIA (b) per-\nspectives at 06:04 UT on 2014 September 1, displays white lines\nindicating the closed magnetic field lines, and purple lines repre-\nsenting the open magnetic field lines. The AR where the flare oc-\ncurred is marked in the figure.\nThe eruptive event on the solar backside resulted in the os-\ncillation of a prominence and a filament on the solar disk,\nsequentially. As the prominence was observed simultane-\nously by both AIA and EUVI, a 3D reconstruction was per-\nformed, as illustrated in Figure 4. Specifically, the oscilla-\ntion direction can be also determined through the reconstruc-\ntion, as depicted in Figure 9. During the period from 11:16\nUT (red segment) to 11:26 UT (green segment), a distinc-tive pattern emerges in the prominence’s motion, featuring\na simultaneous downward vertical motion (panel (a)) and a\nrightward horizontal motion (panel (b)). Subsequently, from\n11:36 UT (blue segment) to 11:46 UT (yellow segment), the\nprominence underwent simultaneous upward vertical motion\n(panel (a)) and leftward horizontal motion (panel (b)). It in-\ndicates a transverse oscillation. Moreover, the direction of\nthe oscillation can be calculated as an angle of ∼63◦to the\nlocal photosphere plane.\n\tB\n\tC\nFigure 9. 3D reconstruction of the prominence from two distinct\nvantage points (along the solar limb in panel (a) and perpendicular\nto the solar surface in panel (b)).\nTo investigate the kinematics of the prominence, we se-\nlected three artificial slices in Figure 1(c): S0, S1 and S2,\nwhich were equidistantly aligned with the direction of verti-\ncal oscillatory component. All the slices are 10′′in width.\nTime-distance diagrams are displayed in Figure 10. The os-\ncillation occurred at 11:15:33 UT and gradually ceased to\noscillate around 13:00:00 UT, lasting for about three cycles.\nThe four green points in panel (b2) represent the displace-\nment of the prominence centroid obtained from the EUVI\n304 Å images using the radial-filter technique as detailed in8 Z hang et al .\nS0A0 = 15.53 Mm\nP = 1846.10 s\n = 4.02\n = 2645.27 s\nb = 1.30 km s1\ny0 = 61.32 Mm-28.93 km s1\n(b1) AIA 304 Å\n(b2) AIA 304 Å\nS1A0 = 16.08 Mm\nP = 1807.26 s\n = 4.54\n = 2584.54 s\nb = -0.45 km s1\ny0 = 54.39 Mm-26.62 km s1\n11:20 11:40 12:00 12:20 12:40\nStart Time (2014-Sep-01 11:09:57 UT)\n(b3) AIA 304 Å\nS2A0 = 18.40 Mm\nP = 1643.82 s\n = 4.46\n = 2564.08 s\nb = -0.02 km s1\ny0 = 38.56 Mm\n-26.84 km s1\n020406080s [Mm]\n(a1) AIA 171 Å\nS0A0 = 16.78 Mm\nP = 1881.80 s\n = 5.04\n = 2544.67 s\nb = 0.93 km s1\ny0 = 62.05 Mm-27.42 km s1\n020406080s [Mm]\n(a2) AIA 171 Å\nS1A0 = 14.92 Mm\nP = 1704.12 s\n = 4.27\n = 2587.68 s\nb = -0.02 km s1\ny0 = 54.25 Mm-24.91 km s1\n11:20 11:40 12:00 12:20 12:40\nStart Time (2014-Sep-01 11:09:57 UT)020406080s [Mm]\n(a3) AIA 171 Å\nS2A0 = 14.76 Mm\nP = 1730.92 s\n = 4.36\n = 2907.49 s\nb = -2.60 km s1\ny0 = 43.73 Mm-25-15-5515\ns [arcsec]\nFigure 10. Time-distance diagrams of slit S0, S1 and S2 in Figure 1, showing the prominence oscillation. The dashed lines are fittings of the\nprominence oscillation by using Equation 1, with the white color indicating a valid fit, while the red color indicates an invalid fit. The four green\npoints in panel (b2) represent the displacement of the prominence centroid in EUVI 304 Å by using the radial-filter technique in Section 2.\n11:40 11:50 12:00 12:10 12:20\nStart Time (2014-Sep-01 11:38:48 UT)5101520s [Mm]\n11:45:57 UT\nA0 = 2.15 Mm\nP = 1862.57 s\n = 1.84\n = 1258.89 s\nb = 1.16 km s1\ny0 = 14.30 Mm\nFigure 11. Time-distance diagram of slit F, showing the oscillation\nof the filament. The onset time of the oscillation was 11:45:57 UT.\nThe dashed lines are fittings of the oscillation by using Equation 1.\nAn online animation of the AIA 171 Å data illustrates the oscillation\nprocess of the filament. This data is utilized to generate the time-\ndistance diagram using slit F, as depicted in Figure 1(c).Section 2. These instances also correspond to the four tem-\nporal points when 3D reconstruction was conducted in Fig-\nure 9..\nThe standard program mpfit.pro in the SSW is applied\nto determine the parameter of the oscillation by using the fol-\nlowing fitting function:\nA(t)=A0sin(2πt\nP+ Φ)e−t\nτ+bt+y0. (1)\nwhere A0,P,Φandτrepresent the initial amplitude,\nperiod, initial phase and damping time scale, respectively.\nbt+y0represent a linear term of the equilibrium position of\nthe prominence. These parameters obtained are listed in Ta-\nble 1. The initial amplitudes ( A0) range from 14.76 to 18.40\nMm, with an average of 16.08 Mm. The initial heights ( y0)\nfall within the range of 38.56 to 62.05 Mm, with an average\nof 52.38 Mm. The periods ( P) and damping time scales ( τ)\nspan from 1644 to 1882 seconds and 2545 to 2907 seconds,Transverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun 9\nwith mean values of 1769 seconds and 2640 seconds, respec-\ntively. The calculated damping ratios ( τ/P) range from 1.35\nto 1.68, with an average of 1.49. The initial phases varies be-\ntween 4.02 and 5.04 radians, with a mean value of 4.45 radi-\nans, suggesting oscillation in phase as a rigid body. It should\nbe noticed that the oscillation process is not well fitted by the\nfunction until after about 11:30 UT, when the prominence\nfirst reaches its height minimum. One possible reason is that\nthe prominence is not in equilibrium prior to the contact with\nthe EUV wave. Upon the onset of the downward oscillation\nof the prominence, it manifested a motion with a approxi-\nmately uniform velocity. Consequently, we performed a lin-\near fitting on this process (as indicated by the white straight\nsegment in Figure 10), yielding velocities ranging from ap-\nproximately -24.91 to -28.93 km s−1, with an average veloc-\nity of around 27 km s−1, directed towards the solar surface.\nThe filament exhibits faint transverse oscillations in the\nAIA field of view, approximately over one cycle, as depicted\nin the online animation associated with Figure 11. Due to the\nexclusive observation by AIA, 3D reconstruction is not feasi-\nble. Therefore, the selected slice F for analysis are shown in\nFigure 1(c), and the resulting time-distance diagram is shown\nin Figure 11 (We chose AIA 171 Å data because where oscil-\nlatory motion is more pronounced). Similarly, we employed\nequation 1 to fit the result, and the obtained parameters are\npresented in the Table 2.\n4.DISCUSSION\n4.1. EUV wave\nThe EUV wave investigated in this study was generated by\nCME occurring on the farside of the Sun and thus was ob-\nserved by STB. The EUV wave propagated in all directions\nbut was obstructed when encountering magnetic structures.\nSpecifically, to the southeast of the flare site, there existed an\nelongated magnetic structure. As the EUV wave interacted\nwith this structure, it generated a standing wave. In addition,\na subtle wave passing through the magnetic structure can be\nobserved in the base-di fference movie, a ffirming the funda-\nmental wave nature.\nOwing to the limited time cadence of the EUVI in this pe-\nriod (5 minutes for 195 Å and 10 minutes for 304 Å), the\nproperties of the EUV waves can not be investigated in de-\ntail. Alternatively, by determining the positions of the flare,\nprominence, and filament, we calculated the linear speed of\nthe EUV wave. The speed of the EUV wave reaching the\nprominence was 498 km s−1, while reaching the filament, it\nwas 451 km s−1. This is in accordance with the speed char-\nacteristics of EUV waves (Asai et al. 2012; Takahashi et al.\n2015). Interestingly, even after covering such a long distance\n(605 Mm to 1351 Mm), the speed of the EUV wave did not\nsignificantly decrease. In the upward direction, the CME at-\ntained a maximum speed of 1901 km s−1, with a averagespeed of 650 km s−1, and the velocity of the driven shock\nwave also reached 814 km s−1. On the other hand, the coro-\nnal dimming got darker but with only little (or no) further\nexpansion. These observational phenomena indicate that the\nupward shock is driven by the CME, whereas in the lateral\ndirection, the wave freely propagates once the lateral expan-\nsion of the CME flanks stops (Veronig et al. 2010; Shen &\nLiu 2012a). The occurrence of a type II radio burst further\nconfirms the fast-mode nature of the EUV wave.\n4.2. The oscillation of the prominence and filament\nThe prominence oscillation was simultaneously observed\nby both STB and SDO, which is unlikely to have commenced\noscillation from an equilibrium state, as its initial trajectory\ndid not align with subsequent motion that can be adequately\nfit by a single sinusoidal function. This dual perspective\nenables us to perform a 3D reconstruction to the filament.\nAs shown in Figure 9, the prominence exhibits simultane-\nous horizontal and vertical oscillations, indicative of trans-\nverse oscillations. This implies that the normal vector of the\nwave is inclined downward toward the prominence (Taka-\nhashi et al. 2015; Zheng et al. 2023; Dai et al. 2023), which\nappears to provide strong support for discussion on the prop-\nerties of EUV wave above (can also see the Figure 9 in Shen\net al. (2014b)).\nThe main restoring force of prominence oscillation is gen-\nerally attributed to magnetic tension by using a slab model\n(D´ıaz et al. 2001; Zhang & Ji 2018; Zhou et al. 2018).\nConsidering that the periods of oscillation at di fferent posi-\ntions are approximately equal (see Table 1), we believe that\nthe prominence experienced a global transverse oscillation\nof fast kink mode (Zhou et al. 2016). The magnetic field\nstrength of the prominence can be roughly estimated as fol-\nlows (Hyder 1966):\nB2\nr=πρpr2\n0(4π2P−2+τ−2). (2)\nwhere Br,ρpandr0represent the radial component of mag-\nnetic field, density, and scale height of the prominence, re-\nspectively. P and τare the period and damping time scale of\nthe oscillation. r 0is approximately 4 .3×109cm based on\nthe 3D reconstruction. By taking the average values of the\nmeasured period and decay time as 1769 s and 2640 s (see\nTable 1), respectively, and assuming ρp=4×10−14g cm−3,\nwe can estimate Brto be 5.4 G. Likewise, the radial mag-\nnetic field of the filament can also be calculated, yielding an\nestimate of 4.1 G. It is close to the values reported in previ-\nous papers (Hyder 1966; Shen et al. 2014a, 2017; Zhang &\nJi 2018; Dai et al. 2023).\n5.SUMMARY\nIn this paper, we report our multi-angle observations of the\ntransverse oscillation of a prominence and a filament sequen-10 Z hang et al .\nTable 1. Fitted Parameters of the Prominence Oscillation.\npassband A0 P Φ τ b y0 τ/P\n(Å) (Mm) (s) (rad) (s) (km s−1) (Mm)\n171 (S0) 16.78 1881.80 5.04 2544.67 0.93 62.05 1.35\n171 (S1) 14.92 1704.12 4.27 2587.68 -0.02 54.25 1.52\n171 (S2) 14.76 1730.92 4.36 2907.49 -2.60 43.73 1.68\n304 (S0) 15.53 1846.10 4.02 2645.27 1.30 61.32 1.43\n304 (S1) 16.08 1807.26 4.54 2584.54 -0.45 54.39 1.43\n304 (S2) 18.40 1643.82 4.46 2564.08 -0.02 38.56 1.56\nAverage 16.08 1769.00 4.45 2639.96 -0.14 52.38 1.49\nTable 2. Fitted Parameters of the filament Oscillation.\npassband A0 P Φ τ b y0 τ/P\n(Å) (Mm) (s) (rad) (s) (km s−1) (Mm)\n171 (F) 2.15 1862.57 1.84 1258.59 1.16 14.30 0.68\ntially, which were induced by an EUV wave on 2014 Septem-\nber 1. The main results are summarized as follows:\n1. The upward shock was propelled by the CME, whereas\nin the lateral direction, the EUV wave freely propa-\ngated upon the expansion of the CME flanks stopped,\nwhich can be supported by the coronal dimming re-\ngion with little (or no) expansion. The EUV wave\npropagated over such an extended distance (605 Mm\nto 1351 Mm) with minimal speed reduction (498 km\ns−1to 451 km s−1). Moreover, the occurrence of type\nII radio bursts further confirmed the EUV wave as a\nfast-mode shock.\n2. The EUV wave induced oscillations in both a promi-\nnence and a filament, sequentially. 3D reconstruction\nreveals that the prominence exhibits both vertical and\nhorizontal oscillations, suggesting a downward incli-nation of the normal vector of the EUV wave. The an-\ngle to the local photosphere plane can be further calcu-\nlated to be approximately 63◦. This is consistent with\nthe velocity characteristics of the EUV wave discussed\nabove. The oscillations of the prominence and filament\nalso allow for the determination of their radial mag-\nnetic field strengths, approximated at 5.4 G and 4.1 G,\nrespectively.\nSTEREO /SECCHI data are provided by a consortium of US,\nUK, Germany, Belgium, and France. SDO is a mission of\nNASA’s Living With a Star Program. AIA and HMI data are\ncourtesy of the NASA /SDO science teams. The e-Callisto\ndata are courtesy of the Institute for Data Science FHNW\nBrugg /Windisch, Switzerland. This work is supported by\nthe National Key R&D Program of China 2021YFA1600500\n(2021YFA1600502), NSFC under grant 12373065, and Yun-\nnan Key Laboratory of Solar Physics and Space Science un-\nder the number YNSPCC202206.\nREFERENCES\nAsai, A., Ishii, T. T., Isobe, H., et al. 2012, ApJL, 745, L18,\ndoi: 10.1088 /2041-8205 /745/2/L18\nBrueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995,\nSoPh, 162, 357, doi: 10.1007 /BF00733434\nChen, P. F. 2016, Washington DC American Geophysical Union\nGeophysical Monograph Series, 216, 381,\ndoi: 10.1002 /9781119055006.ch22Chen, P. F., & Wu, Y . 2011, ApJL, 732, L20,\ndoi: 10.1088 /2041-8205 /732/2/L20\nChen, P.-F., Xu, A.-A., & Ding, M.-D. 2020, Research in\nAstronomy and Astrophysics, 20, 166,\ndoi: 10.1088 /1674-4527 /20/10/166\nDai, J., Zhang, Q., Qiu, Y ., et al. 2023, ApJ, 959, 71,\ndoi: 10.3847 /1538-4357 /ad0839Transverse oscillation of prominence and filament induced by an EUV wave from the farside of the Sun 11\nDai, J., Zhang, Q., Zhang, Y ., et al. 2021, ApJ, 923, 74,\ndoi: 10.3847 /1538-4357 /ac2d97\nDelaboudini `ere, J. P., Artzner, G. E., Brunaud, J., et al. 1995, SoPh,\n162, 291, doi: 10.1007 /BF00733432\nDevi, P., Chandra, R., Joshi, R., et al. 2022, Advances in Space\nResearch, 70, 1592, doi: 10.1016 /j.asr.2022.02.053\nD´ıaz, A. J., Oliver, R., Erd ´elyi, R., & Ballester, J. L. 2001, A&A,\n379, 1083, doi: 10.1051 /0004-6361:20011351\nEto, S., Isobe, H., Narukage, N., et al. 2002, PASJ, 54, 481,\ndoi: 10.1093 /pasj/54.3.481\nGallagher, P. T., & Long, D. M. 2011, SSRv, 158, 365,\ndoi: 10.1007 /s11214-010-9710-7\nHeinzel, P., Schmieder, B., F ´arn´ık, F., et al. 2008, ApJ, 686, 1383,\ndoi: 10.1086 /591018\nHoward, R. A., Moses, J. D., V ourlidas, A., et al. 2008, SSRv, 136,\n67, doi: 10.1007 /s11214-008-9341-4\nHyder, C. L. 1966, ZA, 63, 78\nJing, J., Lee, J., Spirock, T. J., et al. 2003, ApJL, 584, L103,\ndoi: 10.1086 /373886\nKaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, SSRv, 136,\n5, doi: 10.1007 /s11214-007-9277-0\nKleczek, J., & Kuperus, M. 1969, SoPh, 6, 72,\ndoi: 10.1007 /BF00146797\nKrucker, S., Kontar, E. P., Christe, S., Glesener, L., & Lin, R. P.\n2011, ApJ, 742, 82, doi: 10.1088 /0004-637X /742/2/82\nLabrosse, N., Heinzel, P., Vial, J. C., et al. 2010, SSRv, 151, 243,\ndoi: 10.1007 /s11214-010-9630-6\nLemen, J. R., Title, A. M., Akin, D. J., et al. 2012, SoPh, 275, 17,\ndoi: 10.1007 /s11207-011-9776-8\nLi, T., & Zhang, J. 2012, ApJL, 760, L10,\ndoi: 10.1088 /2041-8205 /760/1/L10\nLiakh, V ., Luna, M., & Khomenko, E. 2023, A&A, 673, A154,\ndoi: 10.1051 /0004-6361 /202245765\nLiu, W., & Ofman, L. 2014, SoPh, 289, 3233,\ndoi: 10.1007 /s11207-014-0528-4\nLuna, M., & Karpen, J. 2012, ApJL, 750, L1,\ndoi: 10.1088 /2041-8205 /750/1/L1\nLuna, M., Knizhnik, K., Muglach, K., et al. 2014, ApJ, 785, 79,\ndoi: 10.1088 /0004-637X /785/1/79\nLuna, M., Terradas, J., Khomenko, E., Collados, M., & de Vicente,\nA. 2016, ApJ, 817, 157, doi: 10.3847 /0004-637X /817/2/157\nMancuso, S., Frassati, F., Bemporad, A., & Barghini, D. 2019,\nA&A, 624, L2, doi: 10.1051 /0004-6361 /201935157\nMartin, S. F. 1998, SoPh, 182, 107,\ndoi: 10.1023 /A:1005026814076\nMasson, S., Antiochos, S. K., & DeV ore, C. R. 2013, ApJ, 771, 82,\ndoi: 10.1088 /0004-637X /771/2/82\nMoreton, G. E. 1960, AJ, 65, 494, doi: 10.1086 /108346\nMuhr, N., Vr ˇsnak, B., Temmer, M., Veronig, A. M., & Magdaleni ´c,\nJ. 2010, ApJ, 708, 1639, doi: 10.1088 /0004-637X /708/2/1639Ofman, L., & Thompson, B. J. 2002, ApJ, 574, 440,\ndoi: 10.1086 /340924\nOkamoto, T. J., Nakai, H., Keiyama, A., et al. 2004, ApJ, 608,\n1124, doi: 10.1086 /420838\nOntiveros, V ., & V ourlidas, A. 2009, ApJ, 693, 267,\ndoi: 10.1088 /0004-637X /693/1/267\nPatsourakos, S., & V ourlidas, A. 2009, ApJL, 700, L182,\ndoi: 10.1088 /0004-637X /700/2/L182\n—. 2012, SoPh, 281, 187, doi: 10.1007 /s11207-012-9988-6\nPesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, SoPh,\n275, 3, doi: 10.1007 /s11207-011-9841-3\nRamsey, H. E., & Smith, S. F. 1966, AJ, 71, 197,\ndoi: 10.1086 /109903\nShen, Y ., Ichimoto, K., Ishii, T. T., et al. 2014a, ApJ, 786, 151,\ndoi: 10.1088 /0004-637X /786/2/151\nShen, Y ., & Liu, Y . 2012a, ApJL, 752, L23,\ndoi: 10.1088 /2041-8205 /752/2/L23\n—. 2012b, ApJ, 754, 7, doi: 10.1088 /0004-637X /754/1/7\nShen, Y ., Liu, Y ., Tian, Z., & Qu, Z. 2017, ApJ, 851, 101,\ndoi: 10.3847 /1538-4357 /aa9af0\nShen, Y ., Liu, Y . D., Chen, P. F., & Ichimoto, K. 2014b, ApJ, 795,\n130, doi: 10.1088 /0004-637X /795/2/130\nShen, Y ., Qu, Z., Yuan, D., et al. 2019, ApJ, 883, 104,\ndoi: 10.3847 /1538-4357 /ab3a4d\nTakahashi, T., Asai, A., & Shibata, K. 2015, ApJ, 801, 37,\ndoi: 10.1088 /0004-637X /801/1/37\nTan, S., Shen, Y ., Zhou, X., et al. 2023, MNRAS, 520, 3080,\ndoi: 10.1093 /mnras /stad295\nTandberg-Hanssen, E. 1995, Science, 269, 111\nTemmer, M., Veronig, A. M., Gopalswamy, N., & Yashiro, S.\n2011, SoPh, 273, 421, doi: 10.1007 /s11207-011-9746-1\nTerradas, J., Soler, R., D ´ıaz, A. J., Oliver, R., & Ballester, J. L.\n2013, ApJ, 778, 49, doi: 10.1088 /0004-637X /778/1/49\nTerradas, J., Soler, R., Luna, M., Oliver, R., & Ballester, J. L.\n2015, ApJ, 799, 94, doi: 10.1088 /0004-637X /799/1/94\nThompson, B. J., Plunkett, S. P., Gurman, J. B., et al. 1998,\nGeophys. Res. Lett., 25, 2465, doi: 10.1029 /98GL50429\nThompson, B. J., Reynolds, B., Aurass, H., et al. 2000, SoPh, 193,\n161, doi: 10.1023 /A:1005222123970\nThompson, B. J., Gurman, J. B., Neupert, W. M., et al. 1999,\nApJL, 517, L151, doi: 10.1086 /312030\nTripathi, D., & Raouafi, N. E. 2007, A&A, 473, 951,\ndoi: 10.1051 /0004-6361:20077255\nUchida, Y . 1968, SoPh, 4, 30, doi: 10.1007 /BF00146996\n—. 1974, SoPh, 39, 431, doi: 10.1007 /BF00162436\nVeronig, A. M., Muhr, N., Kienreich, I. W., Temmer, M., &\nVrˇsnak, B. 2010, ApJL, 716, L57,\ndoi: 10.1088 /2041-8205 /716/1/L57\nVrˇsnak, B., Warmuth, A., Braj ˇsa, R., & Hanslmeier, A. 2002,\nA&A, 394, 299, doi: 10.1051 /0004-6361:2002112112 Z hang et al .\nWang, Y . M. 2000, ApJL, 543, L89, doi: 10.1086 /318178\nWarmuth, A. 2007, in Lecture Notes in Physics, Berlin Springer\nVerlag, ed. K.-L. Klein & A. L. MacKinnon, V ol. 725, 107\n—. 2015, Living Reviews in Solar Physics, 12, 3,\ndoi: 10.1007 /lrsp-2015-3\nWarmuth, A., Vr ˇsnak, B., Aurass, H., & Hanslmeier, A. 2001,\nApJL, 560, L105, doi: 10.1086 /324055\nWills-Davey, M. J., & Attrill, G. D. R. 2009, SSRv, 149, 325,\ndoi: 10.1007 /s11214-009-9612-8\nWu, S. T., Zheng, H., Wang, S., et al. 2001, J. Geophys. Res., 106,\n25089, doi: 10.1029 /2000JA000447\nWuelser, J.-P., Lemen, J. R., Tarbell, T. D., et al. 2004, in Society\nof Photo-Optical Instrumentation Engineers (SPIE) Conference\nSeries, V ol. 5171, Telescopes and Instrumentation for Solar\nAstrophysics, ed. S. Fineschi & M. A. Gummin, 111–122,\ndoi: 10.1117 /12.506877\nWyper, P. F., DeV ore, C. R., & Antiochos, S. K. 2018, ApJ, 852,\n98, doi: 10.3847 /1538-4357 /aa9ffc\nYan, X. L., Xue, Z. K., Pan, G. M., et al. 2015, ApJS, 219, 17,\ndoi: 10.1088 /0067-0049 /219/2/17\nYashiro, S., Michalek, G., & Gopalswamy, N. 2008, Annales\nGeophysicae, 26, 3103, doi: 10.5194 /angeo-26-3103-2008Zhang, Q. M., Chen, P. F., Xia, C., & Keppens, R. 2012, A&A,\n542, A52, doi: 10.1051 /0004-6361 /201218786\nZhang, Q. M., Chen, P. F., Xia, C., Keppens, R., & Ji, H. S. 2013,\nA&A, 554, A124, doi: 10.1051 /0004-6361 /201220705\nZhang, Q. M., & Ji, H. S. 2018, ApJ, 860, 113,\ndoi: 10.3847 /1538-4357 /aac37e\nZhang, Q. M., Ning, Z. J., Guo, Y ., et al. 2015, ApJ, 805, 4,\ndoi: 10.1088 /0004-637X /805/1/4\nZhang, Y ., Kitai, R., Narukage, N., et al. 2011, PASJ, 63, 685,\ndoi: 10.1093 /pasj/63.3.685\nZheng, R., Liu, Y ., Liu, W., et al. 2023, ApJL, 949, L8,\ndoi: 10.3847 /2041-8213 /acd0ac\nZhou, G. P., Zhang, J., & Wang, J. X. 2016, ApJL, 823, L19,\ndoi: 10.3847 /2041-8205 /823/1/L19\nZhou, Y .-H., Xia, C., Keppens, R., Fang, C., & Chen, P. F. 2018,\nApJ, 856, 179, doi: 10.3847 /1538-4357 /aab614\nZhou, Y .-H., Zhang, L.-Y ., Ouyang, Y ., Chen, P. F., & Fang, C.\n2017, ApJ, 839, 9, doi: 10.3847 /1538-4357 /aa67de\nZucca, P., Morosan, D. E., Rouillard, A. P., et al. 2018, A&A, 615,\nA89, doi: 10.1051 /0004-6361 /201732308" }, { "title": "2401.15876v1.CMA_ES_with_Learning_Rate_Adaptation.pdf", "content": "CMA-ES with Learning Rate Adaptation\nMASAHIRO NOMURA∗,Tokyo Institute of Technology, Japan\nYOUHEI AKIMOTO, University of Tsukuba & RIKEN AIP, Japan\nISAO ONO, Tokyo Institute of Technology, Japan\nThe covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for\nsolving continuous black-box optimization problems. A practically useful aspect of the CMA-ES is that it\ncan be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable\nimpact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This\nstudy comprehensively explores the impact of learning rate on the CMA-ES performance and demonstrates\nthe necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses\nthe setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation\nmechanism for the CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate\nthe behavior of the CMA-ES with the proposed learning rate adaptation mechanism through numerical\nexperiments, and compare the results with those obtained for the CMA-ES with a fixed learning rate and with\npopulation size adaptation. The results show that the CMA-ES with the proposed learning rate adaptation\nworks well for multimodal and/or noisy problems without extremely expensive learning rate tuning.\nCCS Concepts: •Mathematics of computing →Continuous optimization .\nAdditional Key Words and Phrases: covariance matrix adaptation evolution strategy, black-box optimization,\nlearning rate adaptation\nACM Reference Format:\nMasahiro Nomura, Youhei Akimoto, and Isao Ono. 2024. CMA-ES with Learning Rate Adaptation. ACM Trans.\nEvol. Learn. 0, 0, Article 0 (January 2024), 28 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn\n1 INTRODUCTION\nThe covariance matrix adaptation evolution strategy (CMA-ES) [Hansen 2016; Hansen and Os-\ntermeier 2001] is among the most successful methods available for solving continuous black-box\noptimization problems; its effectiveness has been confirmed through various real-world appli-\ncations [Fujii et al .2018; Ha and Schmidhuber 2018; Huang et al .2022; Kikuchi et al .2021a,b;\nMaki et al .2020; Nomura et al .2021; Piergiovanni et al .2020; Purucker and Beel 2023; Tanabe\net al.2021; Tian et al .2023; Volz et al .2018]. The CMA-ES performs optimization by updating\nthe multivariate Gaussian distribution, that is, it first samples candidate solutions from the dis-\ntribution and then updates the distribution parameters (i.e., the mean vector 𝑚and covariance\nmatrix Σ=𝜎2𝐶) based on the objective function 𝑓. This update is partly based on the natural\ngradient descent [Akimoto et al .2010; Ollivier et al .2017] of the expected 𝑓, and𝑚and𝐶in the\nCMA-ES are updated to reduce the expected evaluation value. The CMA-ES is practically useful\nas it is a quasi-hyperparameter-free algorithm; practitioners can use it without hyperparameter\ntuning because default values are provided for all hyperparameters through theoretical analysis\nand extensive empirical evaluations. Specifically, the hyperparameter values are automatically\ncomputed using dimension 𝑑and population size 𝜆, where𝜆=4+⌊3 ln(𝑑)⌋by default.\n∗Corresponding author\nAuthors’ addresses: Masahiro Nomura, nomura.m.ad@m.titech.ac.jp, Tokyo Institute of Technology, 4259 Nagatsutach ¯o,\nMidori Ward, Yokohama, Kanagawa, Japan, 226-0026; Youhei Akimoto, akimoto@cs.tsukuba.ac.jp, University of Tsukuba &\nRIKEN AIP, 1-1-1 Tennodai, Tsukuba, Ibaraki, Japan, 305-8573; Isao Ono, isao@c.titech.ac.jp, Tokyo Institute of Technology,\n4259 Nagatsutach ¯o, Midori Ward, Yokohama, Kanagawa, Japan, 226-0026.\n2024. 2688-3007/2024/1-ART0 $15.00\nhttps://doi.org/10.1145/nnnnnnn.nnnnnnn\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.arXiv:2401.15876v1 [cs.NE] 29 Jan 20240:2 Nomura et al.\nAlthough the default 𝜆value works well for various unimodal problems, increasing it can\nhelp solve difficult tasks, such as solving multimodal and additive noise problems [Hansen and\nKern 2004; Nishida and Akimoto 2016, 2018]. However, in a black-box scenario, determining the\nproblem structure of 𝑓is challenging. Thus, determining the appropriate 𝜆value in advance is\nalso challenging, and online adaptation of 𝜆has been proposed to address the issue [Hellwig\nand Beyer 2016; Nguyen and Hansen 2017; Nishida and Akimoto 2016, 2018]. Population size\nadaptation (PSA)-CMA-ES [Nishida and Akimoto 2018] is a representative 𝜆adaptation mechanism\nthat has exhibited promising performance for difficult tasks, including multimodal and additive\nnoise problems.\nIt has been observed that, in the CMA-ES, increasing 𝜆has an effect similar to decreasing the\n𝑚learning rate, that is, 𝜂𝑚[Miyazawa and Akimoto 2017]1. Indeed, the 𝑚andΣlearning rates,\nthat is,𝜂, is another hyperparameter that critically affects performance. An excessively large 𝜂\nvalue results in unstable parameter updates, whereas an excessively small value degrades search\nefficiency. Miyazawa and Akimoto [Miyazawa and Akimoto 2017] reported that the CMA-ES with\neven a relatively small 𝜆(e.g.,𝜆=√\n𝑑) solves multimodal problems, through appropriate setting of\n𝜂. However, determining the appropriate 𝜂value is difficult in practice because prior knowledge is\noften limited and hyperparameter tuning entails expensive numerical investigations.\nTherefore, online adaptation of 𝜂based on the problem difficulty constitutes an important\nadvancement as it will allow practitioners to safely use the CMA-ES without requiring prior\nknowledge or expensive trial-and-error calculations. In particular, we believe that 𝜂adaptation\nis more advantageous than 𝜆adaptation from a practical perspective because the former is more\nsuitable for parallel implementations. For example, practitioners often wish to specify a certain\nnumber of workers as the value of 𝜆value to avoid wasting computational resources. However,\n𝜆adaptation may not always effectively utilize the available resources, as the values vary during\nthe optimization process. In contrast, 𝜂adaptation allows complete exploitation of the available\nresources because the value of 𝜆is fixed as the maximum number of workers. Moreover, in 𝜂\nadaptation, the parameters are regularly updated, whereas CMA-ES with 𝜆adaptation does not\nprogress until all 𝜆solutions are evaluated, making it difficult to determine the search termination\npoint.\nAlthough online 𝜂adaptation itself is not new and several studies have attempted to adapt 𝜂values\nin the CMA-ES variants, these adaptations targeted speed-up [Gissler et al .2022; Loshchilov et al .\n2014; Nomura and Ono 2022]. One notable exception is the 𝜂adaptation proposed by Krause [Krause\n2019] that aims to solve additive noise problems through new evolution strategies. However, it\nestimates the problem difficulty through resampling, that is, by repeatedly evaluating the same\nsolution; thus, it is not suitable for solving (noiseless) multimodal problems. Furthermore, as it\ninvolves significant modifications of the internal parameters of the evolution strategies, applying it\ndirectly to the CMA-ES is challlenging.\nThis study aimed to develop the CMA-ES that can solve multimodal and additive noise problems\nwithout extremely expensive 𝜂tuning or adjusting any other CMA-ES parameters except 𝜂. To\nachieve this, we first examined the impact of learning rate. Our results suggested that (i) difficult\nproblems can be realtively easily solved by decreasing the learning rate and aligning the parameter\nbehavior with the trajectory of an ordinary differential equation (ODE), and (ii) the optimal learning\nrate is approximately proportional to the signal-to-noise ratio (SNR). Based on these observations,\nwe propose an 𝜂adaptation mechanism for the CMA-ES, called the learning rate adaptation (LRA),\nthat adapts𝜂to maintain a constant SNR. The key feature of the proposed method is that it does\n1Note that, in Ref. [Miyazawa and Akimoto 2017], the rank-one update was excluded from the CMA-ES. In this study,\nhowever, we consider the CMA-ES that includes the rank-one update.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:3\nnot require specific knowledge of the internal mechanism of the distribution-parameter update\nto estimate the SNR. Consequently, the proposed method is widely applicable to various CMA-ES\nvariants, such as diagonal decoding (dd)-CMA [Akimoto and Hansen 2020], even though this study\nconsiders the most commonly used CMA-ES, which combines weighted recombination, step-size 𝜎\nadaptation, rank-one update, and rank- 𝜇update.\nThis study extends a previous work [Nomura et al .2023] as follows: In Section 3.2, we illustrate\nfrom an ODE perspective the reason why a small learning rate is essential for solving difficult\nproblems. Thereafter, we discuss the optimal learning rate in Section 3.3, which indicates that the\nproposed method adapts the learning rate to a nearly optimal value. It should be noted that Section 3\npresents entirely new information that was not included in the previous work [Nomura et al .2023].\nThereafter, the performance differences for various 𝜆values are discussed in Section 5.5. Finally,\nSection 5.6 presents a comprehensive comparison of LRA-CMA-ES and PSA-CMA-ES [Nishida and\nAkimoto 2018], a state-of-the-art 𝜆adaptation method.\nThe remainder of this paper is organized as follows: Section 2 explains the CMA-ES algorithm\nand information-geometric optimization (IGO) framework. Section 3 closely examines and explains\nthe impact of the learning rate, and presents the discussion for determining the ideal learning\nrate. Section 4 presents the proposed 𝜂adaptation mechanism based on SNR estimation. Section 5\nevaluates the performance of the proposed 𝜂adaptation for noiseless and noisy problems. Finally,\nSection 6 concludes the paper and suggests future research directions.\n2 BACKGROUND\n2.1 CMA-ES\nWe consider minimizing the objective function 𝑓:R𝑑→R. The CMA-ES employs a multivariate\nGaussian distribution to generate candidate solutions, where the distribution N(𝑚,𝜎2𝐶)is param-\neterized through three elements: mean vector 𝑚∈R𝑑, step-size𝜎∈R>0, and covariance matrix\n𝐶∈R𝑑×𝑑.\nThe CMA-ES first initializes the 𝑚(0),𝜎(0), and𝐶(0)parameters. Thereafter, the following steps\nare repeated until a pre-defined stopping criterion is met.\nStep 1. Sampling and Evaluation\nAt iteration 𝑡+1(where𝑡begins at 0),𝜆candidate solutions 𝑥𝑖(𝑖=1,2,···,𝜆)are sampled\nindependently from N(𝑚(𝑡),(𝜎(𝑡))2𝐶(𝑡)), as follows:\n𝑦𝑖=√︁\n𝐶(𝑡)𝑧𝑖, (1)\n𝑥𝑖=𝑚(𝑡)+𝜎(𝑡)𝑦𝑖, (2)\nwhere𝑧𝑖∼N( 0,𝐼)and𝐼is the identity matrix. The solutions are evaluated on 𝑓and sorted in\nascending order. Let 𝑥𝑖:𝜆be the𝑖-th best candidate solution, that is, 𝑓(𝑥1:𝜆)⩽𝑓(𝑥2:𝜆)⩽···⩽𝑓(𝑥𝜆:𝜆)\nfor minimization. In addition, we let 𝑦𝑖:𝜆and𝑧𝑖:𝜆be the intermediate vectors in Equations (1) and (2)\ncorresponding to 𝑥𝑖:𝜆.\nStep 2. Compute Evolution Paths\nThe weighted averages 𝑑𝑦=Í𝜇\n𝑖=1𝑤𝑖𝑦𝑖:𝜆and𝑑𝑧=Í𝜇\n𝑖=1𝑤𝑖𝑧𝑖:𝜆of the intermediate vectors are\ncalculated using the parent number 𝜇⩽𝜆and weight function 𝑤𝑖, whereÍ𝜇\n𝑖=1𝑤𝑖=1. The\nevolution paths are updated as follows:\n𝑝(𝑡+1)\n𝜎=(1−𝑐𝜎)𝑝(𝑡)\n𝜎+√︁\n𝑐𝜎(2−𝑐𝜎)𝜇𝑤𝑑𝑧, (3)\n𝑝(𝑡+1)\n𝑐=(1−𝑐𝑐)𝑝(𝑡)\n𝑐+ℎ(𝑡+1)\n𝜎√︁\n𝑐𝑐(2−𝑐𝑐)𝜇𝑤𝑑𝑦, (4)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:4 Nomura et al.\nwhere𝜇𝑤=1/Í𝜇\n𝑖=1𝑤2\n𝑖,𝑐𝜎, and𝑐𝑐are the cumulation factors, and ℎ(𝑡+1)\n𝜎 is the Heaviside function,\nwhich is defined as follows [Hansen and Auger 2014]:\nℎ(𝑡+1)\n𝜎=(\n1 if∥𝑝(𝑡+1)\n𝜎∥2\n1−(1−𝑐𝜎)2(𝑡+1)<\u00002+4\n𝑑+1\u0001𝑑,\n0 otherwise .(5)\nStep 3. Update Distribution Parameters\nThe distribution parameters are updated as follows [Hansen and Auger 2014]:\n𝑚(𝑡+1)=𝑚(𝑡)+𝑐𝑚𝜎(𝑡)𝑑𝑦, (6)\n𝜎(𝑡+1)=𝜎(𝑡)exp \nmin \n1,𝑐𝜎\n𝑑𝜎 \n∥𝑝(𝑡+1)\n𝜎∥\nE[∥N( 0,𝐼)∥]−1!!!\n, (7)\n𝐶(𝑡+1)=\u0010\n1+(1−ℎ(𝑡+1)\n𝜎)𝑐1𝑐𝑐(2−𝑐𝑐)\u0011\n𝐶(𝑡)\n+𝑐1\u0014\n𝑝(𝑡+1)\n𝑐\u0010\n𝑝(𝑡+1)\n𝑐\u0011⊤\n−𝐶(𝑡)\u0015\n| {z }\nrank-one update+𝑐𝜇𝜇∑︁\n𝑖=1𝑤𝑖h\n𝑦𝑖:𝜆𝑦⊤\n𝑖:𝜆−𝐶(𝑡)i\n| {z }\nrank-𝜇update,(8)\nwhere E[∥N( 0,𝐼)∥]≈√\n𝑑\u0010\n1−1\n4𝑑+1\n21𝑑2\u0011\ndenotes the expected Euclidean norm of the sample of a\nstandard normal distribution and 𝑐𝑚is the learning rate for 𝑚, which is typically set to 1.𝑐1and𝑐𝜇\nare the learning rates for the rank-one and - 𝜇updates of𝐶, respectively, and 𝑑𝜎is the damping\nfactor for the 𝜎adaptation.\n2.2 Information Geometric Optimization\nThe Information Geometric Optimization (IGO) [Ollivier et al .2017] is a unified framework for\nstochastic search methods. Given a family of probability distributions parameterized by 𝜃∈Θ, the\noriginal objective function 𝑓is transformed into a new objective function 𝐽𝜃that is defined in the\ndistribution-parameter space Θ.\nFor the family of Gaussian distributions, the IGO algorithms recover the pure rank- 𝜇-update\nCMA-ES, eliminating the 𝜎adaptation and rank-one update from the procedures in Section 2.1. To\ninvestigate the effects of learning rates on the CMA-ES, we focus on their properties within the\ncontext of the IGO framework with a family of Gaussian distributions in Section 3. This section\npresents the background of the IGO framework.\nInstead of minimizing the original objective 𝑓over the input domain R𝑑, IGO maximizes a\nnew objective 𝐽𝜃over the distribution-parameter domain Θ. Let𝑢:[0,1]→Rbe a bounded,\nnon-increasing function, and 𝑃𝜃be the Lebesgue measure on R𝑑corresponding to the probability\ndensity𝑝(𝑥;𝜃). We define the utility function 𝑊𝑓\n𝜃as\n𝑊𝑓\n𝜃(𝑥)=𝑢(𝑞𝜃(𝑥)), (9)\nwhere𝑞𝜃(𝑥)is the quantile function that is defined as 𝑞𝜃(𝑥):=𝑃𝜃[𝑦:𝑓(𝑦)⩽𝑓(𝑥)]for minimiza-\ntion. The objective of updating of 𝜃, given the current distribution parameters 𝜃(𝑡), is defined as\nthe expectation of the weighted quantile function 𝑊𝑓\n𝜃(𝑡)(𝑥)over𝑝(𝑥;𝜃):\n𝐽𝜃(𝑡)(𝜃)=E𝑥∼𝑝(𝑥;𝜃)[𝑊𝑓\n𝜃(𝑡)(𝑥)]. (10)\nThe objective 𝐽𝜃(𝑡)(𝜃)is maximized based on the natural gradient [Amari and Douglas 1998;\nAmari and Nagaoka 2000]. By using the “log-likelihood trick” under some mild conditions, the\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:5\nvanilla gradient can be calculated as\n∇𝜃𝐽𝜃(𝑡)(𝜃)=E𝑥∼𝑝(𝑥;𝜃)[𝑊𝑓\n𝜃(𝑡)(𝑥)∇𝜃ln𝑝(𝑥;𝜃)]. (11)\nThenatural gradient is obtained through the product of the inverse of the Fisher information matrix\n𝐹and the vanilla gradient as follows:\n˜∇𝜃𝐽𝜃(𝑡)(𝜃)=E𝑥∼𝑝(𝑥;𝜃)[𝑊𝑓\n𝜃(𝑡)(𝑥)˜∇𝜃ln𝑝(𝑥;𝜃)], (12)\nwhere ˜∇𝜃ln𝑝(𝑥;𝜃)=𝐹−1∇𝜃ln𝑝(𝑥;𝜃).\nIn practice, the integral cannot be calculated in a closed form and is therefore estimated using\nthe Monte-Carlo method as follows:\n˜∇𝜃𝐽𝜃(𝑡)(𝜃)≈1\n𝜆𝜆∑︁\n𝑖=1𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)˜∇𝜃ln𝑝(𝑥𝑖;𝜃), (13)\nwhere{𝑥𝑖}𝜆\n𝑖=1are𝜆i.i.d. samples obtained from probability distribution 𝑝(𝑥𝑖;𝜃). The IGO algorithms\nimplement the IGO framework using the estimated natural gradient, whose updated equation is as\nfollows:\n𝜃(𝑡+1)=𝜃(𝑡)+𝜂𝜆∑︁\n𝑖=1𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)\n𝜆˜∇𝜃ln𝑝(𝑥𝑖;𝜃(𝑡)), (14)\nwhere𝜂denotes the learning rate. In practice, 𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)is also estimated based on the ranking of\n{𝑥𝑖}𝜆\n𝑖=1.\nAs elucidated herein, the IGO framework, with a family of Gaussian distributions, recovers the\nrank-𝜇-update CMA-ES [Akimoto et al .2010; Ollivier et al .2017]. If the distribution parameter\n𝜃=(𝑚⊤,vec(𝐶)⊤)⊤, then [Akimoto et al. 2010]:\n˜∇𝜃ln𝑝(𝑥𝑖;𝜃)=\u0012𝑥−𝑚\nvec((𝑥−𝑚)(𝑥−𝑚)⊤−𝐶)\u0013\n. (15)\nThus, Eq. (14) can be rewritten as\n𝑚(𝑡+1)=𝑚(𝑡)+𝜂𝜆∑︁\n𝑖=1𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)\n𝜆(𝑥𝑖−𝑚(𝑡)), (16)\n𝐶(𝑡+1)=𝐶(𝑡)+𝜂𝜆∑︁\n𝑖=1𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)\n𝜆\u0010\n(𝑥−𝑚(𝑡))(𝑥−𝑚(𝑡))⊤−𝐶(𝑡)\u0011\n. (17)\nConsequently, by ignoring the 𝜎adaptation and rank-one update in the CMA-ES, assuming 𝑐𝑚=\n𝑐𝜇(:=𝜂), and considering that 𝑤𝑖in the CMA-ES is an approximation of 𝑊𝑓\n𝜃(𝑡)(𝑥𝑖)/𝜆in the IGO\nupdate, the𝑚and𝐶updates through the IGO algorithm (Eqs. (16) and (17), respectively) align with\nthose of the CMA-ES (Eqs. (6) and (8), respectively).\n3 LEARNING RATE IMPACT\nIn this section, we discuss the impact of the learning rate on the CMA-ES. First, Section 3.1\nsummarizes existing research on adjusting the population size, which is a common practice for\ndifficult tasks, such as multimodal problems, and the relation between the population size and\nlearning rate. In Section 3.2, we discuss the behavior from the perspective of ODEs for small learning\nrates. Consequently, we demonstrate that difficult problems can be solved by reducing the learning\nrate (i.e., closer to the solution of the ODE). However, it should be noted that an excessively small\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:6 Nomura et al.\nlearning rate can reduce the search efficiency. Therefore, Section 3.3 discusses the determination of\nthe optimal learning rate.\n3.1 Relation Between Population Size and Learning Rate\nPrevious studies generally focused on increasing the population size 𝜆to solve multimodal problems.\nHansen and Kern [2004] reported that the CMA-ES with a sufficiently large population size can often\nsolve multimodal problems with high probability. Based on this observation, Auger and Hansen\n[2005] proposed IPOP-CMA-ES, which doubles the population size with each restart. Although\nthese studies considered the CMA-ES with default learning rates, Miyazawa and Akimoto [2017]\nexperimentally evaluated the performance of the CMA-ES using small learning rates and showed\nthat multimodal problems, such as the Rastrigin function, can be solved by setting sufficiently small\nlearning rates without using a large population size. This empirical observation suggests that the\neffect of increasing the population size is similar to that of decreasing the learning rate.\nHere, we organize the relation between the population size and learning rate more formally. First,\nwe examine the relation based on the results of quality gain analysis. For the infinite-dimensional\nsphere function, the optimal value of the normalized step-size ¯𝜎∗, whose normalized step-size is\ndefined as ¯𝜎:=𝜎𝜂𝑚𝑑/∥𝑚−𝑥∗∥=O(𝜎𝜂𝑚), is¯𝜎∗=−𝜇𝑤Í𝜆\n𝑖=1𝑤𝑖E[N𝑖:𝜆]≈√︁\n2/𝜋𝜆∈O(𝜆)[Akimoto\net al.2020; Arnold 2005]. Hence, the optimal step-size is 𝜎∗∈O(𝜆/𝜂𝑚), which clearly demonstrates\nthat increasing 𝜆corresponds to decreasing 𝜂𝑚. In other words, as the population size increases or\nlearning rate decreases, the optimal step size increases. Miyazawa and Akimoto [2017] hypothesized\nthat the CMA-ES with small learning rates can solve multimodal problems owing to the effect of\nmaintaining a large step-size.\nNext, we offer another characterization of the relation between the population size and the\nlearning rate, by viewing IGO algorithms as discretizations of stochastic differential equations\n(SDEs) [Jastrzębski et al .2017]. For conciseness, we define 𝑔(𝜃):=E𝑥∼𝑝(𝑥;𝜃)[𝑊𝑓\n𝜃(𝑥)˜∇𝜃ln𝑝(𝑥;𝜃)]\nand let its Monte-Carlo estimation ˆ𝑔(𝜆)(𝜃):=(1/𝜆)Í𝜆\n𝑖=1ˆ𝑔𝑖(𝜃), where ˆ𝑔𝑖(𝜃):=𝑊𝑓\n𝜃(𝑥𝑖)˜∇𝜃ln𝑝(𝑥𝑖;𝜃),\nwhere ˆ𝑔𝑖(𝜃)is an unbiased estimator of 𝑔(𝜃). Note that, in practice, 𝑊𝑓\n𝜃must also be estimated\nusing the Monte-Carlo method; thus, ˆ𝑔𝑖(𝜃)does not necessarily provide an unbiased estimation\nof𝑔(𝜃). However, we assume the availability of 𝑊𝑓\n𝜃for this discussion. Subsequently, we denote\nthe covariance of ˆ𝑔𝑖(𝜃)as𝑆(𝜃). By using this notation, the IGO update in Eq.(14) can be written\nas𝜃(𝑡+1)=𝜃(𝑡)+𝜂ˆ𝑔(𝜆)(𝜃(𝑡)). Given a sufficiently large population size 𝜆, the following is valid\naccording to the central limit theorem:\nˆ𝑔(𝜆)(𝜃)∼N\u0012\n𝑔(𝜃),1\n𝜆𝑆(𝜃)\u0013\n. (18)\nBased on this result, we can rewrite Eq.(14) as follows:\n𝜃(𝑡+1)=𝜃(𝑡)+𝜂𝑔(𝜃(𝑡))+𝜂(ˆ𝑔(𝜆)(𝜃(𝑡))−𝑔(𝜃(𝑡))), (19)\nwhere ˆ𝑔(𝜆)(𝜃(𝑡))−𝑔(𝜃(𝑡))∼N( 0,(1/𝜆)𝑆(𝜃)). Hence, using the newly introduced random variable\n𝜖𝜃∼N( 0,𝑆(𝜃)), the IGO update can be rewritten as follows:\n𝜃(𝑡+1)=𝜃(𝑡)+𝜂𝑔(𝜃(𝑡))+𝜂√\n𝜆𝜖𝜃(𝑡). (20)\nConsequently, we consider the following SDE:\n𝑑𝜃=𝑔(𝜃)𝑑𝑡+√︂𝜂\n𝜆𝑅(𝜃)𝑑𝑊(𝑡), (21)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:7\n−3−2−1 0 1 2 3\nFig. 1. Rastrigin function.\nwhere𝑅(𝜃)𝑅(𝜃)⊤=𝑆(𝜃)and{𝑊(𝑡)}is the standard Wiener process. By discretizing the SDE using\nthe Euler–Maruyama method [Kloeden et al .1992], with the learning rate 𝜂, we obtain an equation\nidentical to Eq. (20). Therefore, from the SDE perspective, the learning rate and the population size\nappear only in the form of the ratio 𝜂/𝜆, which implies that the effect of increasing 𝜆is similar to\nthat of decreasing 𝜂.\nIn summary, although previous studies primarily adjusted the population size for solving multi-\nmodal problems, we empirically and theoretically observed that increasing the population size and\ndecreasing the learning rate have similar effects on the optimal step-size and noise.\n3.2 Effect of Decreasing the Learning Rate from an ODE Perspective\nWhen the learning rate approaches zero, the IGO algorithm is reduced to the following ODE [Aki-\nmoto et al. 2022]:\n𝑑𝜃\n𝑑𝑡=E𝑥∼𝑝(𝑥;𝜃)[𝑊𝑓\n𝜃(𝑥)˜∇𝜃ln𝑝(𝑥;𝜃)]. (22)\nTo illustrate the algorithm behavior from an ODE perspective, we consider minimizing the 1-\ndimensional Rastrigin function 𝑓Rastrigin(𝑥)=10+𝑥2−10cos(2𝜋𝑥), which is a well-structured\nmultimodal problem (Fig. 1). Assuming that 𝑊𝑓\n𝜃=−𝑓and parameterizing our Gaussian distribution\nusing𝜃=(𝑚,𝑣), where𝑚is the mean and 𝑣is the variance, the ODEs are calculated as follows:\n𝑑𝑚\n𝑑𝑡=−2𝑚𝑣−20𝜋𝑣sin(2𝜋𝑚)exp(−2𝜋2𝑣), (23)\n𝑑𝑣\n𝑑𝑡=−2𝑣2−40𝜋2𝑣2cos(2𝜋𝑚)exp(−2𝜋2𝑣). (24)\nFigure 2 shows the ODE trajectories and gradient flows of the Rastrigin function. The experiments\nwere conducted using initial distribution parameters 𝑚=3.0and𝑣∈[0.02,2.0]. It is evident that\nODEs with large initial variances exhibit trajectories converging to the optimal solution (𝑚∗,𝑣∗)=\n(0,0). Given that the algorithm behavior tends to approach the trajectory of the corresponding\nODE, as the learning rate decreases, we hypothesize that such multimodal problems can be solved\nby adequately decreasing the learning rate and employing a sufficiently large variance.\nTo verify this hypothesis, we evaluated the behavior of the distribution parameters for various\nlearning rates. For this, we employed the following discretized versions of Eq. (23) and (24) using\nthe Euler method:\n𝑚(𝑡+1)=𝑚(𝑡)−𝜂(2𝑚𝑣+20𝜋𝑣sin(2𝜋𝑚)exp(−2𝜋2𝑣)), (25)\n𝑣(𝑡+1)=𝑣(𝑡)−𝜂(2𝑣2+40𝜋2𝑣2cos(2𝜋𝑚)exp(−2𝜋2𝑣)), (26)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:8 Nomura et al.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nmean0.000.250.500.751.001.251.501.752.00varianceTrajectory and gradient flow on Rastrigin\nFig. 2. ODE trajectories and gradient flows of the\nRastrigin function. The different colors (red, orange,\nyellow-orange, and yellow) of the ODE trajectories\nindicate different attractors.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nmean0.00.51.01.52.02.5varianceTrajectory with various learning rates\nODE solution\nη= 10−5\nη= 10−4\nη= 10−3\nη= 10−2Fig. 3. Typical parameter trajectories of the Ras-\ntrigin function under various learning rates ( 𝜂=\n10−5,10−4,10−3,10−2). The ODE solution (black) is\nalso illustrated for reference.\nwhere𝜂denotes the learning rate; we used 𝜂values of 10−5,10−4,10−3, and 10−2. The initial\ndistribution parameters were set as (𝑚,𝑣)=(3.0,2.0). Figure 3 shows the typical behaviors of the\nparameter trajectories for various learning rates. It is evident that as the learning rate decreases,\nthe corresponding trajectory approaches to the ODE solution, which is also evident from the design\nof the Euler method. However, as the learning rate increases, the trajectory deviates from the ODE\ntrajectory and tends to become trapped in the local optima, failing to find the optimal solution.\nThese findings suggest the importance of setting a small learning rate for multimodal problems\nthat can be solved by moving the distribution parameters along the ODE trajectory.\nAlthough earlier discussions focused on multimodal problems, we believe that decreasing the\nlearning rate is equally important for problems with unbiased additive noise, represented as 𝑓(𝑥)+𝜖,\nwhere𝜖is an unbiased random variable, that is, E[𝜖]=0. This is because, in cases with unbiased\nnoise, the corresponding ODE remains unchanged compared with noiseless ones. That is, by\ndecreasing the learning rate and aligning the parameter updates with the corresponding ODE\ntrajectory, the distribution-parameter value can be guided closer to the optimal solution.\n3.3 Optimal Learning Rate\nAlthough setting a small learning rate can be beneficial for solving multimodal and noisy problems,\nas discussed in Section 3.2, using an excessively small value can result in slow convergence.\nTherefore, we consider in this section what the ideal value of the learning rate.\nFor simplicity, we consider the minimization of E[𝑓(𝑥)]=∫\n𝑓(𝑥)𝑝(𝑥;𝜃)𝑑𝑥=:𝐽(𝜃)and assume\nthat𝐽is twice differentiable. Additionally, we let Δbe an unbiased estimator of ˜∇𝐽(𝜃). In this case,\nthe one-step update is 𝜃−𝜂·Δ. Using the Taylor approximation, we obtain the following:\n𝐽(𝜃−𝜂·Δ)=𝐽(𝜃)−𝜂∇𝐽(𝜃)⊤Δ+1\n2𝜂2Δ⊤𝐻Δ+𝑜(𝜂2∥Δ∥2) (27)\n≈𝐽(𝜃)−𝜂∇𝐽(𝜃)⊤Δ+1\n2𝜂2Δ⊤𝐻Δ, (28)\nwhere𝐻:=∇2𝐽(𝜃). Considering the expectations over Δ, we obtain the following:\nEΔ[𝐽(𝜃−𝜂·Δ)]≈𝐽(𝜃)−𝜂∇𝐽(𝜃)⊤˜∇𝐽(𝜃)+1\n2𝜂2\u0010\n˜∇𝐽(𝜃)⊤𝐻˜∇𝐽(𝜃)+Tr(𝐻Cov[Δ])\u0011\n. (29)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:9\nThereafter, the expected improvement is approximated as follows:\n𝐽(𝜃)−EΔ[𝐽(𝜃−𝜂·Δ)]≈𝜂∇𝐽(𝜃)⊤˜∇𝐽(𝜃)−1\n2𝜂2\u0010\n˜∇𝐽(𝜃)⊤𝐻˜∇𝐽(𝜃)+Tr(𝐻Cov[Δ])\u0011\n. (30)\nBy taking the derivative wrt. 𝜂and solving it for zero, we approximate the optimal learning rate as\nfollows:\n∇𝐽(𝜃)⊤˜∇𝐽(𝜃)−𝜂\u0010\n˜∇𝐽(𝜃)⊤𝐻˜∇𝐽(𝜃)+Tr(𝐻Cov[Δ])\u0011\n=0 (31)\n∴𝜂∗≈∇𝐽(𝜃)⊤˜∇𝐽(𝜃)\n˜∇𝐽(𝜃)⊤𝐻˜∇𝐽(𝜃)+Tr(𝐻Cov[Δ])(32)\n=∥˜∇𝐽(𝜃)∥2\n𝐹\n∥˜∇𝐽(𝜃)∥2\n𝐻+Tr(𝐻Cov[Δ]), (33)\nwhere𝐹is the Fisher information matrix of the 𝜃, and∥˜∇𝐽(𝜃)∥𝑀=(˜∇𝐽(𝜃)T𝑀˜∇𝐽(𝜃))1/2is the\nnorm under 𝑀.\nTo obtain crucial insights into the determination of the optimal learning rate, we first assume\n𝐻≈𝑐𝐹for a positive constant 𝑐. This assumption is partially relevant in scenarios wherein the\ncovariance matrix of the CMA-ES successfully learns the shape of a quadratic function. This\nconcept can be illustrated as follows: For a function 𝑓(𝑥)=1\n2𝑥⊤𝐴𝑥, the Hessian 𝐻isdiag(𝐴,0).\nConsequently, given that 𝐹=diag(Σ−1,Σ−1⊗Σ−1/2), if𝐴∝Σ−1, then, to a certain extent, the\nHessian𝐻in the𝑚-part approximates 𝑐𝐹for some𝑐value; however, this does not apply to 𝐻in\ntheΣ-part. Based on this assumption, the optimal learning rate can be written as\n𝜂∗≈1\n𝑐·1\n1+SNR−1∝1\n1+SNR−1. (34)\nwhere SNR :=∥˜∇𝐽(𝜃)∥2\n𝐹\nTr(𝐹Cov[Δ]). A high SNR increases the 𝜂∗value, which aligns with our intuitive\nexpectations. In the next section, we propose a learning rate adaptation mechanism based on these\ninsights into the optimal learning rate.\n4 LEARNING RATE ADAPTATION MECHANISM\nWe consider the updating of the distribution parameters 𝜃𝑚=𝑚and𝜃Σ=vec(Σ), where vecis\nthe vectorization operator and Σ=𝜎2𝐶for the standard CMA-ES. Let Δ(𝑡)\n𝑚=𝑚(𝑡+1)−𝑚(𝑡)and\nΔ(𝑡)\nΣ=vec(Σ(𝑡+1)−Σ(𝑡))be the original updates of 𝑚andΣ, respectively. Subsequently, we introduce\nthe learning rate factors 𝜂(𝑡)\n𝑚and𝜂(𝑡)\nΣ. The modified updates are performed as 𝜃(𝑡+1)\n𝑚=𝜃(𝑡)\n𝑚+𝜂(𝑡)\n𝑚Δ(𝑡)\n𝑚\nand𝜃(𝑡+1)\nΣ=𝜃(𝑡)\nΣ+𝜂(𝑡)\nΣΔ(𝑡)\nΣ. Finally,𝜂(𝑡)\n𝑚and𝜂(𝑡)\nΣare adapted individually.\n4.1 Main Concept\nWe adapt the learning rate factor 𝜂for the component 𝜃(either𝜃𝑚=𝑚or𝜃Σ=vec(Σ)) of the\ndistribution parameters based on the SNR of the update as follows:\nSNR :=∥E[Δ]∥2\n𝐹\nTr(𝐹Cov[Δ])=∥E[Δ]∥2\n𝐹\nE[∥Δ∥2\n𝐹]−∥E[Δ]∥2\n𝐹. (35)\nThe Fisher metric is selected as it offers invariance against probability distribution parameterization.\nWe attempt to adapt 𝜂such that SNR=𝛼𝜂, where𝛼>0is a hyperparameter that determines the\ntarget SNR.\nThe following rationale is employed for selecting this concept: We assume that 𝜂is sufficiently\nsmall such that the distribution parameters do not change significantly over 𝑛iterations. Thus,\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:10 Nomura et al.\nwe assume𝜃(𝑡+𝑘)≈𝜃(𝑡)for𝑘=1,...,𝑛 . Subsequently,{Δ(𝑡+𝑘)}𝑛−1\n𝑘=0are roughly considered as i.i.d.\nHence,𝑛steps of the update are as follows:\n𝜃(𝑡+𝑛)=𝜃(𝑡)+𝜂𝑛−1∑︁\n𝑘=0Δ(𝑡+𝑘)(36a)\n≈𝜃(𝑡)+D\u0000𝑛𝜂E[Δ],𝑛𝜂2Cov[Δ]\u0001, (36b)\nwhereD(𝐴,𝐵)is a distribution with expectation 𝐴and (co)variance 𝐵. Thus, by setting a small 𝜂\nvalue and considering the results of 𝑛=1/𝜂updates, we obtain an update that is more concentrated\naround the expected behavior than that expected for an update using 𝜂=1. The expected change\nin𝜃over𝑛=1/𝜂iterations, measured using the squared Fisher norm, which approximates the\nKullback–Leibler (KL) divergence between 𝜃(𝑡)and𝜃(𝑡+𝑛), is∥E[Δ]∥2\n𝐹+𝜂Tr(𝐹Cov[Δ]), where the\nformer and latter terms come from the signal and noise, respectively. The SNR over 𝑛iterations\nis∥E[Δ]∥2\n𝐹\n𝜂Tr(𝐹Cov[Δ])=1\n𝜂SNR. Therefore, maintaining SNR=𝛼𝜂implies maintaining the SNR at 𝛼over\n𝑛=1/𝜂iterations, independent of 𝜂.\nThe rationale for using SNR can also be elucidated from the perspective of the optimal learning\nrate𝜂∗derived in Section 3.3. The results showed that 𝜂∗∝1/(1+SNR−1)approximately holds\nunder some assumptions. Additionally, we assume a relatively small SNR, for example, SNR⪅1\n(this assumption is validated in Appendix B). In this case, the approximation 1/(1+SNR−1)≈SNR\nis roughly valid. Thus, 𝜂∗∝SNR can be considered to be valid. As stated previously, we controlled 𝜂\nsuch that SNR=𝛼𝜂. Consequently, this leads to 𝜂∝SNR, which is considered to be nearly optimal.\n4.2 Signal-to-Noise Ratio Estimation\nWe estimate∥E[Δ]∥2andE[∥Δ∥2]for each component ( 𝑚andΣ) using moving averages. We let\nE(0)=0andV(0)=0, and update them as follows:\nE(𝑡+1)=(1−𝛽)E(𝑡)+𝛽˜Δ(𝑡), (37a)\nV(𝑡+1)=(1−𝛽)V(𝑡)+𝛽∥˜Δ(𝑡)∥2\n2, (37b)\nwhere𝛽is a hyperparameter; ˜Δ(𝑡)is the update at iteration 𝑡in the local coordinate at which the 𝐹at\n𝜃(𝑡)becomes the identity; ∥·∥2is theℓ2-norm. Thereafter,2−𝛽\n2−2𝛽∥E∥2\n2−𝛽\n2−2𝛽VandVare considered\nestimates of∥E[Δ]∥2\n2andE[∥Δ∥2\n2], respectively (the derivation is included in Appendix A).\nThe rationale for our estimators is as follows. Suppose that 𝜂𝑚and𝜂Σare sufficiently small for us\nto assume that the parameters 𝑚andΣdo not change significantly over 𝑛iterations. Subsequently,\nthe˜Δ(𝑡+𝑖)(𝑖=0,..,𝑛−1)are considered to be located on the same local coordinates and distributed\nindependently and identically. Then, ignoring the (1−����)𝑛terms, we obtain\nE(𝑡+𝑛)∼D\u0012\nE[˜Δ],𝛽\n2−𝛽Cov[˜Δ]\u0013\n. (38)\n(Again, the derivation is presented in Appendix A.) Thus, we have E[∥E∥2\n2]≈∥E[˜Δ]∥2\n2+𝛽\n2−𝛽Tr(Cov[˜Δ]).\nSimilarly, it is apparent that E[V]≈ E[∥˜Δ∥2\n2]=∥E[˜Δ]∥2\n2+Tr(Cov[˜Δ]).\nThe SNR is then estimated as:\nSNR :=∥E[˜Δ]∥2\nTr(Cov[˜Δ])=∥E[˜Δ]∥2\nE[∥˜Δ∥2]−∥E[˜Δ]∥2(39a)\n≈∥E∥2\n2−𝛽\n2−𝛽V\nV−∥E∥2\n2=:dSNR. (39b)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:11\n4.3 Learning Rate Factor Adaptation\nWe attempt to adapt 𝜂such that dSNR=𝛼𝜂, where𝛼>0is the hyperparameter. This adaptation is\nexpressed as follows:\n𝜂←𝜂exp \nmin(𝛾𝜂,𝛽)Π[−1,1] \ndSNR\n𝛼𝜂−1!!\n, (40)\nwhere Π[−1,1]is the projection onto [−1,1]and𝛾is a hyperparameter. If dSNR>𝛼𝜂,𝜂increases,\nand vice versa. Owing to these feedback mechanisms, dSNR/(𝛼𝜂)is expected to remain near 1. In the\nabove expression, the projection Π[−1,1]is introduced to prevent a significant change in 𝜂during an\niteration, and the damping factor min(𝛾𝜂,𝛽)is introduced because of the following reasons. First,\nthe factor𝛽is introduced to allow for the effect of the change in the previous 𝜂value to appear in\ndSNR. Second, the factor 𝛾𝜂is introduced to prevent the 𝜂value from changing more than exp(𝛾)or\nexp(−𝛾)over 1/𝜂iterations. Based on the 𝜂update through Eq. (40), the upper bound is set to 1\nusing𝜂←min(𝜂,1), to prevent unstable behavior. Although allowing 𝜂values > 1would accelerate\nthe optimization, we do not consider this because we aim to safely solve difficult problems.\n4.4 Local Coordinate-System Definition\nAlthough we estimate the SNR based on the updates Δ(·), naïvely accumulating these updates Δ(·)\nmay result in unintentional behavior, as illustrated in the following example. Consider a scenario\nwherein𝑝(𝑥;𝜃(𝑡))=N(0,100𝐼),𝑝(𝑥;𝜃(𝑡+1))=N(0,50𝐼), and𝑝(𝑥;𝜃(𝑡+2))=N(0,25𝐼). In this case,\nthe covariance matrix of the distribution decreases at a constant rate. Consequently, each of the KL\ndivergence is 𝐷KL(𝑝(𝑥;𝜃(𝑡))||𝑝(𝑥;𝜃(𝑡+1)))=𝐷KL(𝑝(𝑥;𝜃(𝑡+1))||𝑝(𝑥;𝜃(𝑡+2))). This implies that the\ndistribution is moving at a uniform pace in terms of the KL divergence. However, the updates are\nvec−1(Δ(𝑡)\nΣ)=50𝐼andvec−1(Δ(𝑡+1)\nΣ)=25𝐼, whose scales are different. Thus, accumulating these\neffects will result in unintentional behavior.\nTo address these issues, we ensure parameterization invariance by defining the local coordinate\nsystem [Nishida and Akimoto 2016, 2018] such that the Fisher information matrices, 𝐹𝑚and𝐹Σ,\ncorresponding to each component of the distribution parameters, 𝑚andΣ, respectively, are the\nidentity matrices. It is well-known that 𝐹𝑚=Σ−1and𝐹Σ=2−1Σ−1⊗Σ−1, and their square roots\nare√𝐹𝑚=√\nΣ−1and√𝐹Σ=2−1\n2√\nΣ−1⊗√\nΣ−1. Therefore, we define\n˜Δ𝑚=√\nΣ−1Δ𝑚, (41a)\n˜ΔΣ=2−1\n2vec(√\nΣ−1vec−1(ΔΣ)√\nΣ−1). (41b)\nActually, in the previous example, the local coordinate system allows us to easily verify that\nvec−1(˜Δ(𝑡)\nΣ)=vec−1(˜Δ(𝑡+1)\nΣ). This observation aligns with intuitive expectations in view of the KL\ndivergence and suggests the validity of accumulating the updates ˜Δinstead of the original Δ.\n4.5 Covariance Matrix Decomposition\nAfter updating the covariance matrix Σ(𝑡+1)=Σ(𝑡)+𝜂(𝑡)\nΣvec−1(Δ(𝑡)\nΣ), it must be split into 𝜎and𝐶.\nFor this, we adopt the following strategy:\n𝜎(𝑡+1)=det(Σ(𝑡+1))1\n2𝑑, (42a)\n𝐶(𝑡+1)=(𝜎(𝑡+1))−2Σ(𝑡+1). (42b)\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:12 Nomura et al.\n4.6 Step-size Correction\nUpdating the learning rate for the 𝑚, i.e.,𝜂𝑚, changes the appropriate 𝜎. Through a quality gain\nanalysis that analyzed the expected 𝑓value improvement in a single step, a previous study [Akimoto\net al.2020] demonstrated that the optimal 𝜎value is proportional to 1/𝜂𝑚for infinite-dimensional\nconvex quadratic functions. Therefore, to maintain the optimal 𝜎value under 𝜂𝑚variations, we\ncorrect𝜎after each𝜂𝑚update as follows:\n𝜎(𝑡+1)←𝜂(𝑡)\n𝑚\n𝜂(𝑡+1)\n𝑚𝜎(𝑡+1). (43)\n4.7 Overall Procedure\nAlgorithm 1 presents the overall LRA-CMA-ES procedure. At Line 2, the old parameters 𝑚(𝑡),𝜎(𝑡),\nand𝐶(𝑡)are input into CMA(·), which outputs new parameters 𝑚(𝑡+1),𝜎(𝑡+1),and𝐶(𝑡+1)by executing\nSteps 1–3 described in Section 2.1.\nNote that the internal parameters such as the evolution paths 𝑝𝜎and𝑝𝑐, are updated and stored\ninCMA(·). However, these values were omitted for simplicity. The subscript ·{𝑚,Σ}(e.g., as in\n𝜂{𝑚,Σ}) indicates that there are parameters for 𝑚andΣ, respectively. For example, E(𝑡+1)\n{𝑚,Σ}←\n(1−𝛽{𝑚,Σ})E(𝑡)\n{𝑚,Σ}+𝛽{𝑚,Σ}˜Δ(𝑡)\n{𝑚,Σ}is an abbreviation for the following two update equations:\nE(𝑡+1)\n𝑚←(1−𝛽𝑚)E(𝑡)\n𝑚+𝛽𝑚˜Δ(𝑡)\n𝑚andE(𝑡+1)\nΣ←(1−𝛽Σ)E(𝑡)\nΣ+𝛽Σ˜Δ(𝑡)\nΣ.\n5 EXPERIMENTS\nThis study included various experiments to investigate the following research questions (RQs):\nRQ1. Does the𝜂adaptation in the LRA-CMA-ES behave appropriately in accordance with the\nproblem structure?\nRQ2. Can the LRA-CMA-ES solve multimodal and noisy problems even though a default 𝜆value is\nused? How does its efficiency compare to that of the CMA-ES with a fixed 𝜂value?\nRQ3. How does the performance change with changes in the LRA-CMA-ES hyperparameters?\nRQ4. How does the performance depend on the population size 𝜆?\nRQ5. What are the differences in the performances of LRA-CMA-ES, which adapts the learning\nrate, and PSA-CMA-ES [Nishida and Akimoto 2018], which adapts the population size?\nThe remainder of this section is organized as follows. The experimental setups are described\nin Section 5.1. Section 5.2 demonstrates 𝜂adaptation in the LRA-CMA-ES for noiseless and noisy\nproblems ( RQ1 ). Section 5.3 compares the LRA-CMA-ES with the CMA-ES with fixed 𝜂values\n(RQ2 ). Section 5.4 investigates the effects of the LRA-CMA-ES hyperparameters ( RQ3 ). Additional\nexperimental results for the hyperparameters are presented in Appendix C. Section 5.5 evaluates\nthe performance differences under various different population sizes ( RQ4 ). Finally, Section 5.6\ncompares the LRA-CMA-ES with the PSA-CMA-ES ( RQ5 ).\n5.1 Experimental Setups\nThe benchmark problem definitions and initial distributions are presented in Table 1. In each case\n(except for the Rosenbrock function), the global optimal solution is at 𝑥=0. However, for the\nRosenbrock function, it is at 𝑥=1. Although the Rosenbrock function has local minima, in our\nstudy, it can be regarded as an almost unimodal problem. Similar to [Hansen and Kern 2004], we\nimposed additional bounds on the Ackley function. For noisy problems, we considered an additive\nGaussian noise 𝜖∼N( 0,𝜎2\n𝑛)with𝜎2\n𝑛variance.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:13\nAlgorithm 1 LRA-CMA-ES\nInput:𝑚(0)∈R𝑑,𝜎(0)∈R>0,𝜆∈N,𝛼,𝛽{𝑚,Σ},𝛾∈R\nSet:𝑡=0,𝐶(0)=𝐼,𝜂(0)\n{𝑚,Σ}=1,E(0)=0,V(0)=0\n1:while stopping criterion not met do\n2:𝑚(𝑡+1),𝜎(𝑡+1),𝐶(𝑡+1)←CMA(𝑚(𝑡),𝜎(𝑡),𝐶(𝑡))\n3: // calculate parameter one-step differences\n4: Δ(𝑡)\n𝑚←𝑚(𝑡+1)−𝑚(𝑡)\n5: Σ(𝑡+1)←\u0000𝜎(𝑡+1)\u00012𝐶(𝑡+1)\n6: Δ(𝑡)\nΣ←vec\u0000Σ(𝑡+1)−Σ(𝑡)\u0001\n7: // local coordinate\n8: ˜Δ(𝑡)\n𝑚←√\nΣ(𝑡)−1Δ(𝑡)\n𝑚\n9: ˜Δ(𝑡)\nΣ←2−1/2vec\u0010√\nΣ(𝑡)−1vec−1\u0010\nΔ(𝑡)\nΣ\u0011√\nΣ(𝑡)−1\u0011\n10: // update evolution paths and estimate SNR\n11:E(𝑡+1)\n{𝑚,Σ}←(1−𝛽{𝑚,Σ})E(𝑡)\n{𝑚,Σ}+𝛽{𝑚,Σ}˜Δ(𝑡)\n{𝑚,Σ}\n12:V(𝑡+1)\n{𝑚,Σ}←(1−𝛽{𝑚,Σ})V(𝑡)\n{𝑚,Σ}+𝛽{𝑚,Σ}∥˜Δ(𝑡)\n{𝑚,Σ}∥2\n2\n13: dSNR{𝑚,Σ}←∥E(𝑡+1)\n{𝑚,Σ}∥2\n2−𝛽{𝑚,Σ}\n2−𝛽{𝑚,Σ}V(𝑡+1)\n{𝑚,Σ}\nV(𝑡+1)\n{𝑚,Σ}−∥E(𝑡+1)\n{𝑚,Σ}∥2\n2\n14: // update learning rates\n15:𝜂(𝑡+1)\n{𝑚,Σ}←𝜂(𝑡)\n{𝑚,Σ}\n·exp\u0012\nmin(𝛾𝜂(𝑡)\n{𝑚,Σ},𝛽{𝑚,Σ})Π[−1,1]\u0012\ndSNR{𝑚,Σ}\n𝛼𝜂{𝑚,Σ}−1\u0013\u0013\n16:𝜂(𝑡+1)\n{𝑚,Σ}←min(𝜂(𝑡+1)\n{𝑚,Σ},1)\n17: // update parameters with adaptive learning rates\n18:𝑚(𝑡+1)←𝑚(𝑡)+𝜂(𝑡+1)\n𝑚Δ(𝑡)\n𝑚\n19: Σ(𝑡+1)←Σ(𝑡)+𝜂(𝑡+1)\nΣvec−1(Δ(𝑡)\nΣ)\n20: // decompose Σto𝜎and𝐶\n21:𝜎(𝑡+1)←det(Σ(𝑡+1))1\n2𝑑,𝐶(𝑡+1)←(𝜎(𝑡+1))−2Σ(𝑡+1)\n22: //𝜎correction\n23:𝜎(𝑡+1)←𝜎(𝑡+1)(𝜂(𝑡)\n𝑚/𝜂(𝑡+1)\n𝑚)\n24:𝑡←𝑡+1\n25:end while\nIn all the experiments (except for those in Section 5.5), we set the default 𝜆=4+⌊3ln𝑑⌋.\nAdditionally, we set the LRA-CMA-ES hyperparameters as 𝛼=1.4,𝛽𝑚=0.1,𝛽Σ=0.03, and\n𝛾=0.1based on preliminary experiments. As noted above, Section 5.4 presents an analysis of the\nhyperparameters sensitivity. The values of other internal parameters of the CMA-ES were set to\nthose recommended in [Hansen and Auger 2014].\n5.2 Learning Rate Behavior\nFigure 4 shows the typical LRA-CMA-ES behaviors for noiseless problems, wherein 𝜂Σmaintained\nrelatively large values for the Sphere function. However, it exhibits significantly smaller values for\nthe Ellipsoid and Rosenbrock functions. We believe that this behavior is undesirable, because the\ndefault𝜂value already works well for these unimodal problems. Although 𝜂can be increased by\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:14 Nomura et al.\nTable 1. Definitions of benchmark problems and initial distributions used in the experiments.\nDefinitions Initial Distributions\n𝑓Sphere(𝑥)=Í𝑑\n𝑖=1𝑥2\n𝑖𝑚(0)=[3,..., 3],𝜎(0)=2\n𝑓Ellipsoid(𝑥)=Í𝑑\n𝑖=1(1000𝑖−1\n𝑑−1𝑥𝑖)2𝑚(0)=[3,..., 3],𝜎(0)=2\n𝑓Rosenbrock(𝑥)=Í𝑑−1\n𝑖=1(100(𝑥𝑖+1−𝑥2\n𝑖)2+(𝑥𝑖−1)2) 𝑚(0)=[0,..., 0],𝜎(0)=0.1\n𝑓Ackley(𝑥)=20−20·exp(−0.2√︃\n1\n𝑑Í𝑑\n𝑖=1𝑥2\n𝑖)+𝑒−exp(1\n𝑑Í𝑑\n𝑖=1cos(2𝜋𝑥𝑖))𝑚(0)=[15.5,..., 15.5],𝜎(0)=14.5\n𝑓Schaffer(𝑥)=Í𝑑−1\n𝑖=1(𝑥2\n𝑖+𝑥2\n𝑖+1)0.25·[sin2(50·(𝑥2\n𝑖+𝑥2\n𝑖+1)0.1)+1] 𝑚(0)=[55,..., 55],𝜎(0)=45\n𝑓Rastrigin(𝑥)=10𝑑+Í𝑑\n𝑖=1(𝑥2\n𝑖−10 cos(2𝜋𝑥𝑖)) 𝑚(0)=[3,..., 3],𝜎(0)=2\n𝑓Bohachevsky(𝑥)=Í𝑑−1\n𝑖=1(𝑥2\n𝑖+2𝑥2\n𝑖+1−0.3 cos(3𝜋𝑥𝑖)−0.4 cos(4𝜋𝑥𝑖+1)+0.7)𝑚(0)=[8,..., 8],𝜎(0)=7\n𝑓Griewank(𝑥)=1\n4000Í𝑑\n𝑖=1𝑥2\n𝑖−Π𝑑\n𝑖=1cos(𝑥𝑖/√\n𝑖)+1 𝑚(0)=[305,..., 305],𝜎(0)=295\n10−610−2102106f(m)Sphere\n10−310−210−1100ηm\n10−210−1100ηΣ\n−303m\n10−610−3100σ(step-size)\n0 200010−610−3100√eigEllipsoid\n0 10000Rosenbrock\n0 20000Ackley\n0 7000Schaffer\n0 20000Rastrigin\n0 200000Bohachevsky\n0 4000Griewank\n0 5000\nFunction Evaluations\nFig. 4. Typical LRA-CMA-ES behaviors for 10-dimensional (10-D) noiseless problems. The coordinates of 𝑚\nand the square roots of the eigenvalues of 𝜎2𝐶(denoted by√︁\neig) are indicated with different colors.\nchanging the hyperparameters of the proposed 𝜂adaptation, this change may be detrimental for\nmultimodal problems.\nIt is evident that 𝜂𝑚is slightly smaller for multimodal problems than for unimodal problems.\nParticularly, for the Rastrigin function, 𝜂𝑚and𝜂Σclearly decrease at the beginning of the optimiza-\ntion, which reflects the difficulty of multimodal problem optimization. Subsequently, 𝜂increases as\noptimization becomes as easy as that for a unimodal problem. This behavior demonstrates that the\nLRA-CMA-ES can adapt 𝜂according to the search difficulty.\nFigure 5 shows the typical 𝜂adaptation behavior for noisy problems. The noise has a negligible\neffect in the early stages; thus, the 𝜂behavior for noisy problems is similar to that for noiseless\nproblems. However, as the optimization proceeds and the function value approaches the same scale\nas that of the noise value, the noise starts having a critical effect. In response to this, the 𝜂value\ndecreases. This adaptation ensures that the SNR remains constant. Notably, similar behavior can be\nobserved for the noisy Rastrigin function, which features both noise and multimodality.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:15\n10−2102106f(m)Noisy Sphere\n10−310−210−1100ηm\n10−310−210−1100ηΣ\n−303m\n10−610−3100σ(step-size)\n10110310510−610−3100√eigNoisy Ellipsoid\n101103105Noisy Rastrigin\n101104\nFunction Evaluations\nFig. 5. Typical LRA-CMA-ES behaviors for 10-D noisy problems. The noise variance 𝜎2𝑛was set to 1.\n5.3 Effects of Learning Rate Adaptation\nFigures 6 and 7 show the performances of the LRA-CMA-ES and that of the CMA-ES with a\nfixed learning rate (𝜂𝑚,𝜂Σ∈{100,10−1,10−2})for the noiseless problems. Note that the CMA-\nES with𝜂𝑚=1.0and𝜂Σ=1.0is the original CMA-ES with the default 𝜂value. Each trial was\nconsidered successful if 𝑓(𝑚)reached the target value 10−8before 107function evaluations or\nbefore a numerical error occurred because of an excessively small 𝜎. In addition to the success rate,\nwe employed the SP1 value [Auger and Hansen 2005], which is the average number of evaluations\namong successful trials until achieving the target value divided by the success rate. 30 trials were\nconducted for each setting.\nTo compare the performances of these strategies for the noisy problems, we employed the\nempirical cumulative density function (ECDF) of COCO, a platform for comparing continuous\noptimizers in a black-box setting [Hansen et al .2021]. Using 𝑁target target values, we recorded the\nnumber of evaluations until 𝑓(𝑚)(noiseless) reached each target value for the first time, and set\nthe maximum function evaluation to 108. We collected data by running 𝑁trialindependent trials,\nand obtained a total of 𝑁target·𝑁trialtargets for each problem. Thereafter, we set the target values\nto106−9(𝑖−1)/(𝑁target−1)for𝑖=1,...,𝑁 target, with𝑁target=30. By executing 𝑁trial=20trials, 600\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:16 Nomura et al.\n10 20 30 400.00.51.0\nSphere\n10 20 30 400.00.51.0\nEllipsoid\n10 20 30 400.00.51.0\nRosenbrock\n10 20 30 400.00.51.0\nAckley\n10 20 30 400.00.51.0\nSchaffer\n10 20 30 400.00.51.0\nRastrigin\n10 20 30 400.00.51.0\nBohachevsky\n10 20 30 400.00.51.0\nGriewank\nDimensionSuccess RateLRA\nηm= 0.01,ηΣ= 0.01ηm= 0.01,ηΣ= 0.1\nηm= 0.01,ηΣ= 1.0ηm= 0.1,ηΣ= 0.01\nηm= 0.1,ηΣ= 0.1ηm= 0.1,ηΣ= 1.0\nηm= 1.0,ηΣ= 0.01ηm= 1.0,ηΣ= 0.1\nηm= 1.0,ηΣ= 1.0\nFig. 6. Success rates according to the number of dimensions (noiseless problems).\n10 20 30 40104105\nSphere\n10 20 30 40104105106\nEllipsoid\n10 20 30 40104105106\nRosenbrock\n10 20 30 40104105106\nAckley\n10 20 30 40105106\nSchaffer\n10 20 30 40105106107\nRastrigin\n10 20 30 40104105106\nBohachevsky\n10 20 30 40104105106\nGriewank\nDimensionSP1LRA\nηm= 0.01,ηΣ= 0.01ηm= 0.01,ηΣ= 0.1\nηm= 0.01,ηΣ= 1.0ηm= 0.1,ηΣ= 0.01\nηm= 0.1,ηΣ= 0.1ηm= 0.1,ηΣ= 1.0\nηm= 1.0,ηΣ= 0.01ηm= 1.0,ηΣ= 0.1\nηm= 1.0,ηΣ= 1.0\nFig. 7. SP1 values according to the number of dimensions (noiseless problems).\ntargets were obtained for each problem. Figure 8 shows the target value percentages obtained for\neach number of evaluations.\n5.3.1 Noiseless Problems. We compared the success rates of the LRA-CMA-ES and the CMA-ES\nwith fixed𝜂values, as shown in Figure 6. For the multimodal problems, the CMA-ES with a large 𝜂\noften failed to reach the optimum. However, the CMA-ES with a small 𝜂exhibited a high success\nrate, indicating a clear dependence on 𝜂. In contrast, LRA-CMA-ES exhibited a relatively good\nsuccess rate, even though no 𝜂tuning was required. It is noteworthy that LRA-CMA-ES succeeded\nin all trials for the Rastrigin function even though the default population size (e.g., 𝜆=15for\n𝑑=40) was used and 𝜂was not tuned in advance.\nHowever, the LRA-CMA-ES performance for the Schaffer function degraded at 𝑑=40. From the\nresults indicating that the CMA-ES with an appropriately tuned, small 𝜂achieved a relatively high\nsuccess rate, the LRA-CMA-ES result may have been obtained because 𝜂was not appropriately\nadapted in that case. This will be investigated in a future work.\nFigure 7 shows the SP1 results for the LRA-CMA-ES and CMA-ES with fixed 𝜂values. The\nCMA-ES with the default 𝜂values (𝜂𝑚=1.0,𝜂Σ=1.0) outperformed the other methods for\nunimodal problems; however, the performance degraded significantly for multimodal problems\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:17\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Sphere ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Ellipsoid ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Rastrigin ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Sphere ( σ2\nn= 106)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Ellipsoid ( σ2\nn= 106)\n1011021031041051061071080.00.20.40.60.8\nNoisy Rastrigin ( σ2\nn= 106)\nFunction Evaluationsprob. of reached targetsLRA\nηm= 0.01,ηΣ= 0.01ηm= 0.01,ηΣ= 0.1\nηm= 0.01,ηΣ= 1.0ηm= 0.1,ηΣ= 0.01\nηm= 0.1,ηΣ= 0.1ηm= 0.1,ηΣ= 1.0\nηm= 1.0,ηΣ= 0.01ηm= 1.0,ηΣ= 0.1\nηm= 1.0,ηΣ= 1.0\nFig. 8. Empirical cumulative density function for 10-D noisy problems, with 𝜎2𝑛set to 1or106.\n0.20.61.01.41.82.22.63.03.43.84.20.00.51.0\nSphere\n0.20.61.01.41.82.22.63.03.43.84.20.00.51.0\nSchaffer\n0.20.61.01.41.82.22.63.03.43.84.20.00.51.0\nRastrigin\n0.20.61.01.41.82.22.63.03.43.84.2104105\n0.20.61.01.41.82.22.63.03.43.84.2105106\n0.20.61.01.41.82.22.63.03.43.84.2105106\nSP1 Success Rateαvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 9. Success rates and SP1 values with hyperparameter 𝛼for 30-D noiseless problems (30 trials).\nowing to optimization failures. In contrast, the CMA-ES with a small 𝜂sometimes exhibited\ngood performance for such multimodal problems; however, it was not efficient for unimodal and\nrelatively easy multimodal problems. Therefore, for the CMA-ES with a fixed 𝜂value, a clear\ntrade-off in efficiency exists based on the 𝜂setting. In contrast, LRA-CMA-ES exhibited stable\nand relatively good performance for unimodal and multimodal problems. Again, 𝜂was not tuned,\nwhich is significantly expensive in practice. There is scope for improvement of the LRA-CMA-ES\nperformance on unimodal problems; however, the current sub-par performance can be somewhat\nmitigated by changing the hyperparameters. The effects of the hyperparameters are discussed in\nSection 5.4.\n5.3.2 Noisy Problems. Figure 8 shows the ECDF results for the LRA-CMA-ES and for the CMA-ES\nwith fixed𝜂values. We considered two noise strengths, weak and strong, that is, 𝜎2\n𝑛=1and106,\nrespectively.\nUnder the weak-noise setting, the CMA-ES with a small 𝜂value reached all the target values. In\ncontrast, the CMA-ES with a large 𝜂value failed to approach the global optimum and yielded a\nsub-optimal solution. The LRA-CMA-ES achieved similar performance to the CMA-ES with a small\n𝜂value without tuning. However, under the strong-noise setting, even the CMA-ES with a small 𝜂\nstopped improving the 𝑓value before reaching the global optimum. By contrast, LRA-CMA-ES\ncontinued improving the 𝑓value. Notably, the results for the noisy Rastrigin function suggest that\nthe LRA-CMA-ES can simultaneously handle both noise and multimodality.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:18 Nomura et al.\n0.01 0.05 0.09 0.13 0.17 0.21 0.250.00.51.0\nSphere\n0.01 0.05 0.09 0.13 0.17 0.21 0.250.00.51.0\nSchaffer\n0.01 0.05 0.09 0.13 0.17 0.21 0.250.00.51.0\nRastrigin\n0.01 0.05 0.09 0.13 0.17 0.21 0.25104105\n0.01 0.05 0.09 0.13 0.17 0.21 0.25105106\n0.01 0.05 0.09 0.13 0.17 0.21 0.25105106\nSP1 Success RateβΣvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 10. Success rates and SP1 values with hyperparameter 𝛽Σfor 30-D noiseless problems (30 trials).\n5.4 Effects of Hyperparameters\nFigure 9 shows the success rates and SP1 values with respect to 𝛼for the 30-dimensional (30-D)\nnoiseless Sphere, Schaffer, and Rastrigin functions. For the Sphere function, a lower SP1 value could\nbe achieved with a smaller 𝛼value. However, an excessively large 𝛼results in optimization failures\nfor multimodal problems. Therefore, the current setting of 𝛼=1.4seems reasonable; however,\nfurther investigations are required.\nFigure 10 shows the success rates and SP1 values with respect to 𝛽Σ. Clearly, an excessively large\n𝛽Σcauses optimization failures in multimodal problems. However, an excessively small 𝛽Σresults\nin slow convergence for the Rastrigin function. An additional result ( 𝛽Σ∈{0.01,0.02,...,0.05}) is\npresented in Appendix C.\nWe also conducted similar experiments on the hyperparameters 𝛽𝑚and𝛾, to confirm their effects.\nThese hyperparameters mildly impacted on the overall performance compared to 𝛼and𝛽Σ(these\nresults are also presented in Appendix C).\n5.5 Effects of Population Size\nAlthough we used the default population size, 𝜆=4+⌊3log(𝑑)⌋, in all the experiments, practitioners\nmay want to employ different population sizes to fully utilize their parallel environments. In this\nsection, we describe the experiments conducted to investigate the effects of population size.\nFigure 11 shows the success rates and SP1 values with respect to 𝜆∈{14,28,42,56,70}for the\n30-D noiseless Sphere, Schaffer, and Rastrigin functions. Although the SP1 value worsens with a\nlarger𝜆for the Rastrigin function, it appears to have a mild impact for the Sphere and Schaffer\nfunctions. Figure 12 shows typical behaviors of the LRA-CMA-ES with 𝜆∈{14,42,70}on the\n30-D Sphere function. As 𝜆increases, it can be observed that the learning rates (especially 𝜂𝑚)\nalso generally increase linearly. This is because, as 𝜆increases, the SNR also increases, allowing\nfor a larger learning rate to maintain the target SNR. This phenomenon can also be theoretically\nexplained as follows: The SNR analysis for the infinite-dimensional Sphere function in Appendix B\nshows that under the assumption of the optimal step size, SNR≈𝑂(𝜆/𝑑). In this case, increasing 𝜆\ncan linearly increase the SNR; therefore, it is expected that the learning rate can be kept linearly\nlarger, which is consistent with our empirical findings. However, this analysis was conducted\nusing the (infinite-dimensional) Sphere function; thus, this discussion cannot be directly applied to\nmultimodal problems.\nAdditionally, we investigated the behavior for larger population sizes using various values\nof𝜆∈ {500,1000,1500,2000,2500}, as shown in Figure 13. Compared to the results for 𝜆∈\n{14,28,42,56,70}, the SP1 value remains almost constant for the Rastrigin function with respect to\nthe𝜆value. However, in the Sphere and Schaffer functions, the SP1 value deteriorates slightly for\nlarger𝜆values. This may be partially because the proposed method is designed to solve difficult\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:19\n14 28 42 56 700.00.51.0\nSphere\n14 28 42 56 700.00.51.0\nSchaffer\n14 28 42 56 700.00.51.0\nRastrigin\n14 28 42 56 70104105\n14 28 42 56 70105106\n14 28 42 56 70105106\nSP1 Success Rateλvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 11. Success rates and SP1 values with 𝜆∈{14,28,42,56,70}for 30-D noiseless problems (30 trials).\n0 10000 20000 30000\nFunction Evaluations0.010.040.070.10ηmλ= 14\n0 10000 20000 30000\nFunction Evaluations0.10.40.71.0ηΣ\n0.00 0.05 0.10 0.15 0.20050100hist(/hatwidestSNRm)\n0 1 2 30100hist(/hatwidestSNR Σ)0 10000 20000 30000\nFunction Evaluations0.010.040.070.10λ= 42\n0 10000 20000 30000\nFunction Evaluations0.10.40.71.0\n0.00 0.05 0.10 0.15 0.2002550\n0 1 2 30500 10000 20000 30000\nFunction Evaluations0.010.040.070.10λ= 70\n0 10000 20000 30000\nFunction Evaluations0.10.40.71.0\n0.00 0.05 0.10 0.15 0.2002040\n0 1 2 302040\nFig. 12. LRA-CMA-ES behaviors on the 30-D Sphere function with 𝜆∈{14,42,70}.𝜂𝑚,𝜂Σ, and the histograms\nof the estimated SNR wrt. 𝑚andΣ, in this order from the top. The y-axes in 𝜂𝑚and𝜂Σare shown on the\nlinear scale rather than the log scale.\n500 1000 1500 2000 25000.00.51.0\nSphere\n500 1000 1500 2000 25000.00.51.0\nSchaffer\n500 1000 1500 2000 25000.00.51.0\nRastrigin\n500 1000 1500 2000 2500105106\n500 1000 1500 2000 2500106107\n500 1000 1500 2000 2500107108\nSP1 Success Rateλvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 13. Success rates and SP1 values with 𝜆∈{500,1000,1500,2000,2500}for 30-D noiseless problems (30\ntrials).\nproblems (e.g., 𝜂𝑚,𝜂Σ⩽1). Although more aggressive learning rate updates may improve the\nperformance, such strategies were beyond the scope of this study.\n5.6 LRA-CMA-ES vs. PSA-CMA-ES\nWe compared the performance of the proposed LRA-CMA-ES with that of PSA-CMA-ES [Nishida\nand Akimoto 2018], which is a state-of-the-art population size adaptation method. For a fair\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:20 Nomura et al.\n10 20 30 400.00.51.0\nSphere\n10 20 30 400.00.51.0\nEllipsoid\n10 20 30 400.00.51.0\nRosenbrock\n10 20 30 400.00.51.0\nAckley\n10 20 30 400.00.51.0\nSchaffer\n10 20 30 400.00.51.0\nRastrigin\n10 20 30 400.00.51.0\nBohachevsky\n10 20 30 400.00.51.0\nGriewank\nDimensionSuccess RateLRA PSA\nFig. 14. Performances of LRA-CMA-ES and PSA-CMA-ES: success rates according to the number of dimensions\n(noiseless problems).\ncomparison, we employed almost the same procedure and hyperparameters for the PSA-CMA-ES\nas those for the CMA-ES described in Section 2.1. The only difference was that the PSA-CMA-ES\nrequired additional normalization factors (Eqs. (6) and (7) in [Nishida and Akimoto 2018]) to derive\nan approximate value for the parameter movement. For the step-size correction in the PSA-CMA-\nES, we used Blom’s approximation to calculate the weighted average of the expected value of\nnormal-order statistics [Akimoto et al .2020]. Additionally, we used the recommended values for\nthe PSA-CMA-ES hyperparameters [Nishida and Akimoto 2018]. The experimental settings were\nthe same as those described in Section 5.3. All LRA-CMA-ES results were obtained from Section 5.3.\nFigures 14 and 15 show the success rates and SP1 values, respectively, for the noiseless problems,\nwherein it is evident that the PSA-CMA-ES exhibits better results than the LRA-CMA-ES for most\nproblems. Figure 16 illustrates the ECDF for noisy problems. Unlike for noiseless problems, the\nperformance of the LRA-CMA-ES is better than that of the PSA-CMA-ES for most problems. For\nexample, for the Rastrigin function with a strong-noise setting (bottom right of Figure 16), the\nPSA-CMA-ES stopped improving the function value, whereas LRA-CMA-ES continued improving it.\nThese results suggest that the LRA-CMA-ES and PSA-CMA-ES are suitable for different problems.\nHowever, these performance differences can be mitigated to a certain degree by adjusting the\nhyperparameters of each method and do not necessarily suggest that there is a fundamental\nperformance difference between learning rate and population size adaptations. Although we still\nargue that learning rate adaptation is more practically useful than population size adaptation, as\ndescribed in Section 1, a detailed comparison of these methods will be an interesting direction for\nfuture work.\n6 CONCLUSION\nThis study presented the design principles and practices of LRA for the CMA-ES. We first demon-\nstrated that difficult problems can be solved relatively easily by decreasing the learning rate and\nensuring that the parameter behavior was closer to the ODE trajectory. However, decreasing it\nexcessively worsened the search efficiency. Therefore, we attempted to determine the optimal\nlearning rate for maximizing the expected improvement, which was nearly proportional to the SNR\nunder some assumptions. Based on these observations, we developed a new LRA mechanism to\nsolve multimodal and noisy problems using the CMA-ES without extremely expensive learning-rate\ntuning. The basic concept of the proposed algorithm, LRA-CMA-ES, is to adapt the learning rate\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:21\n10 20 30 40104\nSphere\n10 20 30 40104105\nEllipsoid\n10 20 30 40104105106\nRosenbrock\n10 20 30 40104105\nAckley\n10 20 30 40105106\nSchaffer\n10 20 30 40105\nRastrigin\n10 20 30 40104105\nBohachevsky\n10 20 30 40104105\nGriewank\nDimensionSP1LRA PSA\nFig. 15. Performaces of LRA-CMA-ES and PSA-CMA-ES: SP1 values according to the number of dimensions\n(noiseless problems).\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Sphere ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Ellipsoid ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Rastrigin ( σ2\nn= 1)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Sphere ( σ2\nn= 106)\n1011021031041051061071080.00.20.40.60.81.0\nNoisy Ellipsoid ( σ2\nn= 106)\n1011021031041051061071080.00.20.40.60.8\nNoisy Rastrigin ( σ2\nn= 106)\nFunction Evaluationsprob. of reached targetsLRA PSA\nFig. 16. Performances of LRA-CMA-ES and PSA-CMA-ES: Empirical cumulative density function for 10-D\nnoisy problems, with 𝜎2𝑛set to 1or106.\nsuch that the SNR can be kept constant, which is nearly optimal based on the optimal learning\nrate discussion. Experiments involving noiseless multimodal problems revealed that the proposed\nLRA-CMA-ES can adapt the learning rate appropriately depending on the search situation, and\nit works well without tuning the learning rate. Additionally, the LRA-CMA-ES provided better\nsolutions for noisy problems, even under strong-noise settings, which yielded problems that could\nnot be solved by the CMA-ES with a fixed learning rate. In conclusion, the LRA-CMA-ES effectively\nfacilitates the solving of multimodal and noisy problems to a certain extent, eliminating the need\nfor tuning the learning rate.\nHowever, the proposed LRA-CMA-ES has some limitations, which will be addressed in future\nresearch. First, it experienced several failures for the 40-D Schaffer function, although the CMA-ES\nwith an appropriately small learning rate succeeded with a high probability. We believe that a\ndetailed analysis of the SNR adaptation behavior is crucial to determine the reasons for this failure.\nOn a related note, our understanding of the appropriate hyperparameter settings in the proposed\nLRA mechanism remains limited. Our experiments revealed that the hyperparameter settings affect\nthe trade-off between stability and convergence speed. Through experiment, we identified the\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:22 Nomura et al.\nhyperparameters that perform relatively well for noiseless and noisy problems; however, better\nconfiguration methods must be developed. For example, the constant value O(1)is used for the\ncumulation factors 𝛽𝑚and𝛽Σ; however, it may be more reasonable to consider that these factors\ndepend on the parameter degrees of freedom, that is, 𝛽𝑚=O(1/𝑑)and𝛽Σ=O(1/𝑑2). In addition,\nthe method for determining the appropriate value of the target SNR 𝛼can be refined further.\nThe SNR analysis presented in Appendix B implies that 𝛼=O(𝜆/𝑑)is reasonable for an infinite-\ndimensional sphere function. On the other hand, because this analysis cannot be directly applied to\nmultimodal or noisy problems, there is still room to discuss the best method for determining 𝛼. A\ndeeper understanding of the hyperparameter effects is crucial for improving the reliability of the\nproposed LRA-CMA-ES.\nFinally, developing a more rational LRA approach is an intriguing topic for future research. While\nour discussion in Section 3.3 offers valuable insights into designing an ideal learning rate, it was\nbased on several assumptions and has the potential for improvement. A more detailed theoretical\nstudy could result in a more rational design for learning rates, which is crucial for advancing this\nline of research.\nACKNOWLEDGMENTS\nREFERENCES\nYouhei Akimoto, Anne Auger, and Nikolaus Hansen. 2020. Quality Gain Analysis of the Weighted Recombination Evolution\nStrategy on General Convex Quadratic Functions. Theoretical Computer Science 832 (2020), 42–67.\nYouhei Akimoto, Anne Auger, and Nikolaus Hansen. 2022. An ODE Method to Prove the Geometric Convergence of\nAdaptive Stochastic Algorithms. Stochastic Processes and their Applications 145 (2022), 269–307.\nYouhei Akimoto and Nikolaus Hansen. 2020. Diagonal Acceleration for Covariance Matrix Adaptation Evolution Strategies.\nEvolutionary Computation 28, 3 (2020), 405–435.\nYouhei Akimoto, Yuichi Nagata, Isao Ono, and Shigenobu Kobayashi. 2010. Bidirectional Relation between CMA Evolution\nStrategies and Natural Evolution Strategies. In International Conference on Parallel Problem Solving from Nature . 154–163.\nShun-Ichi Amari and Scott C Douglas. 1998. Why Natural Gradient?. In Proceedings of the IEEE International Conference on\nAcoustics, Speech and Signal Processing , Vol. 2. 1213–1216.\nShun-ichi Amari and Hiroshi Nagaoka. 2000. Methods of Information Geometry . Vol. 191.\nDirk V. Arnold. 2005. Optimal Weighted Recombination. In Foundations of Genetic Algorithms . 215–237.\nAnne Auger and Nikolaus Hansen. 2005. A Restart CMA Evolution Strategy with Increasing Population Size. In IEEE\nCongress on Evolutionary Computation , Vol. 2. IEEE, 1769–1776.\nGaruda Fujii, Youhei Akimoto, and Masayuki Takahashi. 2018. Exploring optimal topology of thermal cloaks by CMA-ES.\nApplied Physics Letters 112, 6 (2018).\nArmand Gissler, Anne Auger, and Nikolaus Hansen. 2022. Learning Rate Adaptation by Line Search in Evolution Strategies\nwith Recombination. In Proceedings of the Genetic and Evolutionary Computation Conference . 630–638.\nDavid Ha and Jürgen Schmidhuber. 2018. World Models. arXiv preprint arXiv:1803.10122 (2018).\nNikolaus Hansen. 2016. The CMA Evolution Strategy: A Tutorial. arXiv preprint arXiv:1604.00772 (2016).\nNikolaus Hansen and Anne Auger. 2014. Principled Design of Continuous Stochastic Search: From Theory to Practice. In\nTheory and Principled Methods for the Design of Metaheuristics . Springer, 145–180.\nNikolaus Hansen, Anne Auger, Raymond Ros, Olaf Mersmann, Tea Tušar, and Dimo Brockhoff. 2021. COCO: A Platform for\nComparing Continuous Optimizers in a Black-Box Setting. Optimization Methods and Software 36, 1 (2021), 114–144.\nNikolaus Hansen and Stefan Kern. 2004. Evaluating the CMA Evolution Strategy on Multimodal Test Functions. In\nInternational Conference on Parallel Problem Solving from Nature . 282–291.\nNikolaus Hansen and Andreas Ostermeier. 2001. Completely Derandomized Self-Adaptation in Evolution Strategies.\nEvolutionary Computation 9, 2 (2001), 159–195.\nMichael Hellwig and Hans-Georg Beyer. 2016. Evolution Under Strong Noise: A Self-Adaptive Evolution Strategy Can\nReach the Lower Performance Bound - The pcCMSA-ES. In International Conference on Parallel Problem Solving from\nNature . 26–36.\nBiwei Huang, Chaochao Lu, Liu Leqi, José Miguel Hernández-Lobato, Clark Glymour, Bernhard Schölkopf, and Kun Zhang.\n2022. Action-Sufficient State Representation Learning for Control with Structural Constraints. In International Conference\non Machine Learning . 9260–9279.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:23\nStanisław Jastrzębski, Zachary Kenton, Devansh Arpit, Nicolas Ballas, Asja Fischer, Yoshua Bengio, and Amos Storkey. 2017.\nThree Factors Influencing Minima in SGD. arXiv preprint arXiv:1711.04623 (2017).\nKotaro Kikuchi, Mayu Otani, Kota Yamaguchi, and Edgar Simo-Serra. 2021a. Modeling Visual Containment for Web Page\nLayout Optimization. In Computer Graphics Forum , Vol. 40. 33–44.\nKotaro Kikuchi, Edgar Simo-Serra, Mayu Otani, and Kota Yamaguchi. 2021b. Constrained Graphic Layout Generation via\nLatent Optimization. In Proceedings of the ACM International Conference on Multimedia . 88–96.\nPeter E Kloeden, Eckhard Platen, Peter E Kloeden, and Eckhard Platen. 1992. Stochastic Differential Equations .\nOswin Krause. 2019. Large-Scale Noise-Resilient Evolution-Strategies. In Proceedings of the Genetic and Evolutionary\nComputation Conference . 682–690.\nIlya Loshchilov, Marc Schoenauer, Michele Sebag, and Nikolaus Hansen. 2014. Maximum Likelihood-based Online Adaptation\nof Hyper-parameters in CMA-ES. In International Conference on Parallel Problem Solving from Nature . 70–79.\nAtsuo Maki, Naoki Sakamoto, Youhei Akimoto, Hiroyuki Nishikawa, and Naoya Umeda. 2020. Application of optimal\ncontrol theory based on the evolution strategy (CMA-ES) to automatic berthing. Journal of Marine Science and Technology\n25 (2020), 221–233.\nHidekazu Miyazawa and Youhei Akimoto. 2017. Effect of the Mean Vector Learning Rate in CMA-ES. In Proceedings of the\nGenetic and Evolutionary Computation Conference . 721–728.\nDuc Manh Nguyen and Nikolaus Hansen. 2017. Benchmarking CMAES-APOP on the BBOB Noiseless Testbed. In Proceedings\nof the Genetic and Evolutionary Computation Conference Companion . 1756–1763.\nKouhei Nishida and Youhei Akimoto. 2016. Population Size Adaptation for the CMA-ES Based on the Estimation Accuracy\nof the Natural Gradient. In Proceedings of the Genetic and Evolutionary Computation Conference . 237–244.\nKouhei Nishida and Youhei Akimoto. 2018. PSA-CMA-ES: CMA-ES with Population Size Adaptation. In Proceedings of the\nGenetic and Evolutionary Computation Conference . 865–872.\nMasahiro Nomura, Youhei Akimoto, and Isao Ono. 2023. CMA-ES with Learning Rate Adaptation: Can CMA-ES with Default\nPopulation Size Solve Multimodal and Noisy Problems?. In Proceedings of the Genetic and Evolutionary Computation\nConference . 839–847.\nMasahiro Nomura and Isao Ono. 2022. Towards a Principled Learning Rate Adaptation for Natural Evolution Strategies. In\nApplications of Evolutionary Computation . 721–737.\nMasahiro Nomura, Shuhei Watanabe, Youhei Akimoto, Yoshihiko Ozaki, and Masaki Onishi. 2021. Warm Starting CMA-ES\nfor Hyperparameter Optimization. In Proceedings of the AAAI Conference on Artificial Intelligence , Vol. 35. 9188–9196.\nYann Ollivier, Ludovic Arnold, Anne Auger, and Nikolaus Hansen. 2017. Information-Geometric Optimization Algorithms:\nA Unifying Picture via Invariance Principles. Journal of Machine Learning Research 18, 18 (2017), 1–65.\nAJ Piergiovanni, Anelia Angelova, and Michael S Ryoo. 2020. Evolving Losses for Unsupervised Video Representation\nLearning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition . 133–142.\nLennart Oswald Purucker and Joeran Beel. 2023. CMA-ES for Post Hoc Ensembling in AutoML: A Great Success and\nSalvageable Failure. In AutoML Conference 2023 .\nTakumi Tanabe, Kazuto Fukuchi, Jun Sakuma, and Youhei Akimoto. 2021. Level Generation for Angry Birds with Sequential\nVAE and Latent Variable Evolution. In Proceedings of the Genetic and Evolutionary Computation Conference . 1052–1060.\nStephen Tian, Yancheng Cai, Hong-Xing Yu, Sergey Zakharov, Katherine Liu, Adrien Gaidon, Yunzhu Li, and Jiajun Wu. 2023.\nMulti-Object Manipulation via Object-Centric Neural Scattering Functions. In Proceedings of the IEEE/CVF Conference on\nComputer Vision and Pattern Recognition . 9021–9031.\nVanessa Volz, Jacob Schrum, Jialin Liu, Simon M Lucas, Adam Smith, and Sebastian Risi. 2018. Evolving Mario Levels in the\nLatent Space of a Deep Convolutional Generative Adversarial Network. In Proceedings of the Genetic and Evolutionary\nComputation Conference . 221–228.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:24 Nomura et al.\nA DERIVATION FOR SECTION 4.2\nA.1 Derivations of Eq. (38)\nThis section presents the detailed derivation of Eq. (38). By ignoring (1−𝛽)𝑛,E(𝑡+𝑛)can be\napproximately calculated as follows:\nE(𝑡+𝑛)=(1−𝛽)E(𝑡+𝑛−1)+𝛽˜Δ(𝑡+𝑛−1)\n=(1−𝛽)n\n(1−𝛽)E(𝑡+𝑛−2)+𝛽˜Δ(𝑡+𝑛−2)o\n+𝛽˜Δ(𝑡+𝑛−1)\n=...\n=(1−𝛽)𝑛E(𝑡)+𝑛−1∑︁\n𝑖=0(1−𝛽)𝑖𝛽˜Δ(𝑡+𝑛−1−𝑖)\n≈𝑛−1∑︁\n𝑖=0(1−𝛽)𝑖𝛽˜Δ(𝑡+𝑛−1−𝑖).\nHere, we assume the ˜Δ(·)are uncorrelated with each other; this corresponds to the scenario where\n𝜂is sufficiently small. In this case, we can ignore the dependence of 𝑡, that is, E[˜Δ(𝑡+𝑛−1−𝑖)]=:E[˜Δ].\nThus,\nE[E(𝑡+𝑛)]=𝑛−1∑︁\n𝑖=0(1−𝛽)𝑖𝛽E[˜Δ].\nwhere\n𝑛−1∑︁\n𝑖=0(1−𝛽)𝑖=1·{1−(1−𝛽)𝑛}\n1−(1−𝛽)=1−(1−𝛽)𝑛\n𝛽.\nSubsequently, ignoring (1−𝛽)𝑛, we approximate E[E(𝑡+𝑛)]as\nE[E(𝑡+𝑛)]=[1−(1−𝛽)𝑛]E[˜Δ]≈E[˜Δ].\nNext, we consider the covariance Cov[E(𝑡+𝑛)]:\nCov[E(𝑡+𝑛)]=E[E(𝑡+𝑛)(E(𝑡+𝑛))⊤]−E[[E(𝑡+𝑛)]([E(𝑡+𝑛)])⊤.\nWe first determine the exact expression for E(𝑡+𝑛)(E(𝑡+𝑛))⊤as follows:\nE(𝑡+𝑛)(E(𝑡+𝑛))⊤=𝛽2𝑛−1∑︁\n𝑖=0(1−𝛽)2𝑖˜Δ(𝑡+𝑛−1−𝑖)(˜Δ(𝑡+𝑛−1−𝑖))⊤\n+2𝛽2𝑛−1∑︁\n𝑖,𝑗=0:𝑖≠𝑗(1−𝛽)𝑖(1−𝛽)𝑗˜Δ(𝑡+𝑛−1−𝑖)(˜Δ(𝑡+𝑛−1−𝑖))⊤.\nNote that, for 𝑖,𝑗∈ {0,···𝑛−1}(𝑖≠𝑗),E[˜Δ(𝑡+𝑛−1−𝑖)(˜Δ(𝑡+𝑛−1−𝑗))⊤]=E[˜Δ](E[˜Δ])⊤because\nwe assume that they are not correlated. For 𝑖∈ {0,···𝑛−1},E[˜Δ(𝑡+𝑛−1−𝑖)(˜Δ(𝑡+𝑛−1−𝑖))⊤]=\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:25\nE[˜Δ](E[˜Δ])⊤+Cov[˜Δ]. Thus,\nE[E(𝑡+𝑛)(E(𝑡+𝑛))⊤]\n=𝛽2𝑛−1∑︁\n𝑖=0(1−𝛽)2𝑖\u0010\nE[˜Δ](E[˜Δ])⊤+Cov[˜Δ]\u0011\n+2𝛽2𝑛−1∑︁\n𝑖,𝑗=0:𝑖≠𝑗(1−𝛽)𝑖(1−𝛽)𝑗E[˜Δ](E[˜Δ])⊤,\n=E[E(𝑡+𝑛)](E[E(𝑡+𝑛)])⊤+𝛽2𝑛−1∑︁\n𝑖=0(1−𝛽)2𝑖Cov[˜Δ].\nTherefore,\nCov[E(𝑡+𝑛)]=E[E(𝑡+𝑛)(E(𝑡+𝑛))⊤]−E[[E(𝑡+𝑛)]([E(𝑡+𝑛)])⊤\n=𝛽2𝑛−1∑︁\n𝑖=0(1−𝛽)2𝑖Cov[˜Δ].\nHere,\n𝑛−1∑︁\n𝑖=0(1−𝛽)2𝑖=1−(1−𝛽)2𝑛\n1−(1−𝛽)2=1−(1−𝛽)2𝑛\n𝛽(2−𝛽).\nThus, by ignoring(1−𝛽)2𝑛,Cov[E(𝑡+𝑛)]can be approximated as\nCov[E(𝑡+𝑛)]=[1−(1−𝛽)2𝑛]𝛽\n2−𝛽Cov[˜Δ],\n≈𝛽\n2−𝛽Cov[˜Δ].\nTherefore,E(𝑡+𝑛)approximately follows the following distribution:\nE(𝑡+𝑛)∼D\u0012\nE[˜Δ],𝛽\n2−𝛽Cov[˜Δ]\u0013\n.\nThus, the derivation of Eq. (38) is complete.\nA.2 Derivation of Estimates for ∥E[˜Δ]∥2\n2\nWe organized the relation between Eand ˜Δusing the following equation:\nE[∥E∥2\n2]=E[E]⊤𝐼E[E]+ Tr(Cov[E])\n≈∥E[˜Δ]∥2\n2+Tr\u0012𝛽\n2−𝛽Cov[˜Δ]\u0013\n=∥E[˜Δ]∥2\n2+𝛽\n2−𝛽Tr(Cov[˜Δ]).\nNow, we apply the same arguments to Vand obtain:\nE[V]=[1−(1−𝛽)𝑡+1]E[∥˜Δ∥2\n2]\n≈E[∥˜Δ∥2\n2]=∥E[˜Δ]∥2\n2+Tr(Cov[˜Δ]).\nBy reorganizing these arguments, we obtain:\n∥E[˜Δ]∥2\n2≈2−𝛽\n2−2𝛽E[∥E∥2\n2]−𝛽\n2−2𝛽E[V].\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:26 Nomura et al.\nThis provides the rationale for estimating2−𝛽\n2−2𝛽∥E∥2\n2−𝛽\n2−2𝛽Vfor∥E[˜Δ]∥2\n2.\nB THEORETICAL AND EMPIRICAL INSIGHTS INTO SNR\nIn Section 4, we assumed that the signal-to-noise ratio (SNR) is relatively small, for example,\nSNR⪅1, which validates the approximation 1/(1+SNR−1)≈SNR. In this section, we theoretically\nand empirically discuss the validity of SNR⪅1.\nTo obtain useful insights into this SNR from a theoretical perspective, we considered observing it\nin a situation wherein the objective function is the sphere function 𝑓(𝑥)=∥𝑥∥2and the covariance\nmatrix is Σ=𝜎2𝐼, where𝜎=¯𝜎∥𝑚∥\n𝑑and¯𝜎is called the normalized step-size. The quality gain analysis\n[Akimoto et al .2020; Arnold 2005] implies that for a sufficiently large 𝑑, the distribution of the 𝑖th\nranked solution among the 𝜆candidate solutions is approximated as 𝑋𝑖:𝜆=𝑚+𝜎N𝑖:𝜆𝑚\n∥𝑚∥+𝜎N⊥\n𝑖,\nwhereN𝑖:𝜆is the𝑖th order statistics among 𝜆normally distributed random variables and N⊥\n𝑖\nis an independently distributed 𝑑dimensional normal random vector with covariance matrix\n𝐼−𝑚𝑚T\n∥𝑚∥2if𝑚≠0. Using this approximation, we obtain Δ𝑚=𝜎\u0010Í𝜆\n𝑖=1𝑤𝑖N𝑖:𝜆\u0011\n𝑚\n∥𝑚∥+𝜎\u0010Í𝜆\n𝑖=1𝑤𝑖N⊥\n𝑖\u0011\n.\nLet𝒘=(𝑤1,...,𝑤𝜆),𝒏(𝜆)=(E[N1:𝜆],...,E[N𝜆:𝜆]).𝑵(𝜆)is a matrix whose (𝑖,𝑗)th element is\nE[N𝑖:𝜆N𝑗:𝜆]. Then, we obtain\nE[Δ𝑚]=𝜎(𝒘T𝒏(𝜆))𝑚\n∥𝑚∥, (44a)\nE[Δ𝑚ΔT\n𝑚]=𝜎2(𝒘T𝑵(𝜆)𝒘)𝑚𝑚T\n∥𝑚∥2+𝜎2∥𝒘∥2\u0012\n𝐼−𝑚𝑚T\n∥𝑚∥2\u0013\n. (44b)\nBecause𝐹𝑚=𝜎−2𝐼, we obtain\nSNR=𝜎−2∥E[Δ𝑚]∥2\n𝜎−2Tr(E[Δ𝑚ΔT𝑚])−𝜎−2∥E[Δ𝑚]∥2(45a)\n=(𝒘T𝒏(𝜆))2\n𝒘T𝑵(𝜆)𝒘+(𝑑−1)∥𝒘∥2−(𝒘T𝒏(𝜆))2(45b)\n≈(𝒘T𝒏(𝜆))2\n(𝑑−1)∥𝒘∥2(45c)\n=1\n𝑑−1(𝒘T𝒏(𝜆))2\n∥𝒘∥2(45d)\n≈𝜆\n𝑑−1(𝒘T𝒏(𝜆))2\n∥𝒘∥2∥𝒏(𝜆)∥2. (45e)\nHere, we used the following asymptotically true approximations for 𝜆(See Eq. (A2) provided in\n[Akimoto et al. 2020]):\n𝒘T𝑵(𝜆)𝒘\n(𝒘T𝒏(𝜆))2≈1 and∥𝒏(𝜆)∥2\n𝜆≈1. (46)\nIt should be noted that(𝒘T𝒏(𝜆))2\n∥𝒘∥2∥𝒏∥2is upper bounded by 0.25if only non-negative weights are used\nfor the𝑚-update, which aligns with our weight scheme. Therefore, we can expect that SNR⪅1\nholds if𝜆is not considerably large relative to 𝑑; for example, 𝜆⩽4(𝑑−1). Importantly, in difficult\nproblems, such as multimodal problems, the SNR tends to be smaller than that in the sphere\nfunctions. Therefore, it should be noted that the assumption of SNR≲1becomes more easily valid\nfor such difficult problems.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.CMA-ES with Learning Rate Adaptation 0:27\n−0.02 0.00 0.02 0.04 0.06 0.08 0.100100hist(/hatwidestSNRm)Sphere\n0 1 2 30100200hist(/hatwidestSNR Σ)−0.02 0.00 0.02 0.04 0.06 0.08 0.1001000Schaffer\n0 1 2 3020004000−0.02 0.00 0.02 0.04 0.06 0.08 0.1002000Rastrigin\n0 1 2 301000020000\nFig. 17. Histogram of the estimated SNR in typical trials on 30-D noiseless problems. Estimated SNR with\nrespect to (top) the mean vector 𝑚and (bottom) the covariance matrix Σ. The SNR was estimated using the\nmethod described in Section 4.2.\n0.01 0.02 0.03 0.04 0.050.00.51.0\nSphere\n0.01 0.02 0.03 0.04 0.050.00.51.0\nSchaffer\n0.01 0.02 0.03 0.04 0.050.00.51.0\nRastrigin\n0.01 0.02 0.03 0.04 0.05104105\n0.01 0.02 0.03 0.04 0.05105106\n0.01 0.02 0.03 0.04 0.05105106\nSP1 Success RateβΣvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 18. Success rate and SP1 values with hyperparameter 𝛽Σ∈{0.01,0.02,...,0.05}on 30-D noiseless problems.\nThe main limitation of the aforementioned analysis is the assumption that the dimension 𝑑and\nthe population size 𝜆are sufficiently large. To verify whether the assumption SNR⪅1works in\npractice, we conducted experiments using the LRA-CMA-ES for 30-dimensional Sphere, Schaffer,\nand Rastrigin functions using the same settings as those mentioned in Section 5.1, and 𝜆=14for\n𝑑=30. Figure 17 illustrates the typical behavior of the estimated SNR where it was estimated using\nthe method described in Section 4.2. It should be noted that this value includes estimation errors.\nAlthough the estimated SNR for the covariance in the Sphere function tends to be slightly larger,\nit often remains under 1, particularly for more difficult problems such as the Rastrigin function.\nThese results suggest that the assumption of SNR to be small, e.g., SNR⪅1, appears to be valid to\na certain degree even under finite dimensions and population sizes.\nC ADDITIONAL EXPERIMENTAL RESULTS\nFigure 18 shows the success rate and SP1 values with respect to 𝛽Σ∈{0.01,0.02,...,0.05}for the\n30-D noiseless Sphere, Schaffer, and Rastrigin functions. Clearly, the performance is not significantly\naffected by𝛽Σvalues within this range. However, as shown in Figure 10, an excessively small 𝛽Σ\nvalue decelerates the convergence for the Rastrigin function.\nFigures 19 and 20 show the success rates and SP1 values for 𝛽𝑚and𝛾, respectively. The results\nshow that the performance is relatively stable against these hyperparameters.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024.0:28 Nomura et al.\n0.05 0.10 0.15 0.20 0.250.00.51.0\nSphere\n0.05 0.10 0.15 0.20 0.250.00.51.0\nSchaffer\n0.05 0.10 0.15 0.20 0.250.00.51.0\nRastrigin\n0.05 0.10 0.15 0.20 0.25104105\n0.05 0.10 0.15 0.20 0.25105106\n0.05 0.10 0.15 0.20 0.25105106\nSP1 Success Rateβmvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 19. Success rate and SP1 values with hyperparameter 𝛽𝑚for 30-D noiseless problems.\n0.05 0.10 0.15 0.20 0.250.00.51.0\nSphere\n0.05 0.10 0.15 0.20 0.250.00.51.0\nSchaffer\n0.05 0.10 0.15 0.20 0.250.00.51.0\nRastrigin\n0.05 0.10 0.15 0.20 0.25104105\n0.05 0.10 0.15 0.20 0.25105106\n0.05 0.10 0.15 0.20 0.25105106\nSP1 Success Rateγvs. Success Rate and SP1 ( d= 30,30trials)\nFig. 20. Success rate and SP1 values with hyperparameter 𝛾for 30-D noiseless problems.\nACM Trans. Evol. Learn., Vol. 0, No. 0, Article 0. Publication date: January 2024." }, { "title": "2401.15939v1.Correcting_a_Single_Deletion_in_Reads_from_a_Nanopore_Sequencer.pdf", "content": "Correcting a Single Deletion in Reads\nfrom a Nanopore Sequencer\nAnisha Banerjee∗, Yonatan Yehezkeally∗, Antonia Wachter-Zeh∗, and Eitan Yaakobi†\n∗Institute for Communications Engineering, Technical University of Munich (TUM), Munich, Germany\n†Department of Computer Science, Technion—Israel Institute of Technology, Haifa 3200003, Israel\nEmail: anisha.banerjee@tum.de, yonatan.yehezkeally@tum.de, antonia.wachter-zeh@tum.de, yaakobi@cs.technion.ac.il\nAbstract —Owing to its several merits over other DNA se-\nquencing technologies, nanopore sequencers hold an immense\npotential to revolutionize the efficiency of DNA storage systems.\nHowever, their higher error rates necessitate further research to\ndevise practical and efficient coding schemes that would allow\naccurate retrieval of the data stored. Our work takes a step in\nthis direction by adopting a simplified model of the nanopore\nsequencer inspired by Mao et al. , which incorporates some of its\nphysical aspects. This channel model can be viewed as a sliding\nwindow of length ℓthat passes over the incoming input sequence\nand produces the L1-weight of the enclosed ℓbits, while shifting\nby one position at each time step. The resulting (ℓ+1)-ary vector,\nreferred to as the ℓ-read vector , is susceptible to deletion errors\ndue to imperfections inherent in the sequencing process. We\nestablish that at least logn−ℓbits of redundancy are needed to\ncorrect a single deletion. An error-correcting code that is optimal\nup to an additive constant, is also proposed. Furthermore, we\nfind that for ℓ⩾2, reconstruction from two distinct noisy ℓ-\nread vectors can be accomplished without any redundancy, and\nprovide a suitable reconstruction algorithm to this effect.\nI. I NTRODUCTION\nOur ever-increasing data storage requirements have\nprompted extensive research into DNA storage, as it promises\nhigh density and unmatched longevity. While there continue\nto be significant efforts to improve upon existing synthesis and\nsequencing technologies, nanopore sequencing holds particular\nappeal due to better portability, ability to read longer DNA\nstrands, and real-time analysis [1–3]. The sequencing opera-\ntion involves the transmigration of a DNA fragment through a\nmicroscopic pore in a lipid membrane, across which a voltage\ndifference exists. The nucleotides in the pore at a given time\nThis material is based upon work supported by the National Science\nFoundation under Grant No. CCF 2212437. This work has also received\nfunding from the European Research Council (ERC) under the European\nUnion’s Horizon 2020 research and innovation programme (Grant agreement\nNo. 801434). It was also funded by the European Union (ERC, DNAStorage,\n865630). Views and opinions expressed are however those of the author(s)\nonly and do not necessarily reflect those of the European Union or the\nEuropean Research Council Executive Agency. Neither the European Union\nnor the granting authority can be held responsible for them. The work\nof Yonatan Yehezkeally was supported by the Alexander von Humboldt\nFoundation under a Carl Friedrich von Siemens Post-Doctoral Research\nFellowship.\n© 2024 IEEE. Personal use of this material is permitted. Permission from\nIEEE must be obtained for all other uses, in any current or future media,\nincluding reprinting/republishing this material for advertising or promotional\npurposes, creating new collective works, for resale or redistribution to servers\nor lists, or reuse of any copyrighted component of this work in other works.instant, influence the variations in the ionic current, which are\nmeasured and fed to a basecaller that predicts the nucleotides\nin the examined DNA strand. Despite its strengths, certain\nphysical aspects of the nanopore sequencer lead to various\ndistortions in the final readout. For instance, the variations\nin the measured current are governed by multiple nucleotides\ninstead of just one due to the depth of the pore, thus hinting\nat the presence of intersymbol interference (ISI). Moreover,\nthe DNA strand often passes through the pore unevenly, i.e.,\na few nucleotides may be skipped, or some backtracking may\noccur. This naturally implies deletions and duplications in the\nfinal readout, respectively.\nPrior work in this area was largely aimed at either de-\nveloping faithful mathematical models for the sequencer or\ndesigning error-correcting codes that incorporate such models\nto correct errors in the readouts efficiently. For instance, the\nauthors of [4] introduced a channel model that incorporates\nISI, deletions, and measurement noise. Upper bounds on chan-\nnel capacity were also established. The work in [5] adopted\na more deterministic model, devised an algorithm to compute\nthe capacity of the same, and also suggested efficient coding\nschemes. A finite-state Markov channel (FSMC)-based model,\nintroduced more recently in [6], encapsulates the effects of\nISI, duplications, and noisy measurements that affect the final\nsequencing output. In [7], a specific model variation in [4] was\nconsidered, and an optimal single-substitution-correcting code\nwas presented.\nThis work endeavors to extend [7] by designing efficient\ndeletion-correcting codes for nanopore sequencers. To this end,\nwe use the channel model employed in [7]. This model was\ninspired by [4] and also resembles the transverse-read channel\n[8], which is relevant to racetrack memories. More specifically,\nnanopore sequencing is interpreted as a concatenation of three\nchannels, as illustrated in Fig. 1. The ISI component, parame-\nterized by ℓ, signifies how the measured current depends on the\nℓconsecutive nucleotides in the pore at any given time. This\nstage may be viewed as a sliding window of length ℓpassing\nover an input sequence and shifting by a single position after\neach time step, producing a sequence of ℓ-mers, i.e., strings\nofℓsymbols. Subsequently, the deletion channel accounts for\nthe effect of backtracking by corrupting the sequence of ℓ-\nmers with deletions. In the end, a discrete memoryless channel\nconverts each of the ℓ-mers in this erroneous sequence into aarXiv:2401.15939v1 [cs.IT] 29 Jan 2024Nucleotides\nxISI\nℓDeletion DMCℓ-mers\nz ˆzDiscrete\nvoltage\nlevels\ny\nFigure 1 . Simplified model of a nanopore sequencer [4]\ndiscrete voltage level based on a deterministic function, in our\nanalysis the L1-weight.\nWe now state the problem more formally. For an input\nx∈Σn\n2, letRℓ(x)represent the deletion-free channel output\n(Definition 1). Thus, we are interested in codes that correct\ntdeletions in Rℓ(x)as opposed to xitself, to guarantee\nthe unique recovery of xdespite ISI, followed by at most\ntdeletions.\nTo summarize the main contributions of this work, we\nestablish a lower bound on the redundancy required by a code\nthat corrects a single deletion in ℓ-read vectors, and suggest\nan instantiation of the same whose redundancy is optimal up\nto an additive constant. Since nanopore sequencers tend to\nproduce multiple erroneous reads for each input strand, we\nalso examine how leveraging this feature might help achieve\na lower redundancy requirement. To this end, we find that for\nanyx∈Σn\n2andℓ⩾2, two distinct noisy channel outputs that\narise from the same input x, suffice to recover xuniquely. A\nsuitable reconstruction algorithm is also stated.\nII. P RELIMINARIES\nA. Notations and Terminology\nIn the following, we let Σqindicate the q-ary alphabet\n{0,1, . . . , q −1}. Additionally, [n]is used to denote the set\n{1,2, . . . , n }. Element-wise modulo operation on a vector, say\ny∈Σn\nq, is represented as\nymod a≜\u0000\ny1mod a, y2mod a, . . . , y nmod a\u0001\n.\nFor any vector x= (x1, . . . , x n), we refer to its substring\n(xi, xi+1, . . . , x j)asxj\ni. The Hamming weight of a vector x\nis denoted by wt(x), while the number of runs in x, which are\nof length greater than or equal to some a⩾1, is represented as\nρ⩾a(x). We designate the number of all runs, i.e., when a=\n1, byρ(x). We also extensively use the Hamming distance,\nwhich is defined for any two vectors x,y∈Σn\nqas\ndH(x,y) =|{i:i∈[n], xi̸=yi}|.\nWe focus on the case of q= 2 and in this framework, the\nchannel output is defined as follows.\nDefinition 1 Theℓ-read vector of any x∈Σn\n2is of length\nn+ℓ−1overΣℓ+1, and is denoted by\nRℓ(x)≜(wt(x1\n2−ℓ),wt(x2\n3−ℓ), . . . , wt(xn+ℓ−1\nn )),\nwhere for ease of notation we simply let xi= 0 for any\ni̸∈[n](i.e., when the above definition includes indices which\nare either negative or greater than n).Additionally, the i-th element of Rℓ(x)is denoted by\nRℓ(x)i; that is, Rℓ(x)i= wt( xi\ni−ℓ+1). When clear from the\ncontext, ℓwill be removed from the preceding notations.\nExample 2 The3-read vector of x= (1,0,1,1,0,0)is given\nbyR(x) = (1 ,1,2,2,2,1,0,0). Its third element is R(x)3=\n2.\nA similar model was investigated in [8], wherein the output\nsequence, termed as the transverse-read vector, is a substring\nof the ℓ-read vector as defined here, for certain parameter\nchoices. The information limit of this transverse-read channel\nwas computed for various parameters, and several coding\nschemes enabling exact recovery were presented. Some error-\ncorrecting codes for the case of ℓ= 2 were also suggested.\nNext, to facilitate our analysis, some key properties of ℓ-\nread vectors are stated below.\nProposition 3 ([7]) P1 For any ℓ⩾1andx∈Σn\n2, it\nholds thatPn+ℓ−1\ni=1R(x)i=ℓ·wt(x), where the sum is\nperformed over integers.\nP2 For any ℓ⩾1andx∈Σn\n2, it holds that |R(x)j+1−\nR(x)j|⩽1for all j∈[n+ℓ−2].\nP3 For any x∈Σn\n2,xcan be uniquely and efficiently deter-\nmined from the first or last nelements of R(x) mod 2 .\nThe preceding definitions can be extended to the non-binary\nalphabet by replacing the notion of Hamming weights with\ncompositions, as done in [9].\nB. Error Model\nTo suitably define what constitutes an error-correcting con-\nstruction in our framework, we first let D(u)refer to the set of\nall vectors of length n−1, that can be obtained by deleting one\nsymbol from u∈Σn, for any alphabet Σ. Naturally, we are\ninterested in D(R(x))for some x∈Σn\n2. A code that corrects\na single deletion in R(x)can thus be defined as follows.\nDefinition 4 Forn⩾ℓ, a code C ⊆Σn\n2is said to be a single-\ndeletion ℓ-read code if for any two distinct x,y∈ C, it holds\nthatD(Rℓ(x))∩D(Rℓ(y)) =∅.\nIII. C ORRECTING A SINGLE DELETION\nA useful consequence of the inherent characteristics of\nℓ-read vectors, summarized in Proposition 3, is that certain\ndeletions can be corrected immediately without any redun-\ndancy, as shown in the next lemma.\nLemma 5 LetR′arise from a single deletion on R(x)for\nsome x∈Σn\n2. If there exists some i∈[n+ℓ−3]such that\n|R′\ni+1− R′\ni|>1, then R(x)(and thereby also x) can be\nreadily recovered.\nProof: From P1, we infer that the existence of an i∈\n[n+ℓ−3]such that |R′\ni+1−R′\ni|>1unambiguously reveals the\nerror location. Note that |R′\ni+1− R′\ni|⩽2. Assuming R′\ni+1−R′\ni= 2, we observe that the only xfor which R′∈D(R(x)),\nbears the following ℓ-read vector.\nR(x) = (R′\n1, . . . ,R′\ni,R′\ni+ 1,R′\ni+1, . . . ,R′\nn+ℓ−2).\nFor the case of R′\ni+1− R′\ni=−2, the argument works\nsimilarly. Once R(x)is known, xcan also uniquely recovered\nas suggested by P3.\nA. Upper Bound on the Size of Codes\nThis section seeks to obtain an upper bound on the cardinal-\nity of a single-deletion ℓ-read code. We do so by limiting our\nattention to a subset of deletion patterns on an ℓ-read vector,\nsayR(x), which bear an intriguing connection with sticky\ndeletions [10–12] on its respective binary vector, x. Here,\nwe consider a specific variant of a sticky deletion, defined\nas follows.\nDefinition 6 Anr-sticky deletion , forr⩾1, is a deletion in\na run of length at least r, inx.\nThe error ball of a single r-sticky deletion for any u∈Σn\n2\nis represented by\nDS(u;r)≜\b\n(u1u2. . . u i−1ui+1. . . u n) :i∈[n−r+ 1],\nui=···=ui+r−1\t\n.\nNaturally, DS(u; 1) = D( u)and|DS(u;r)|=ρ⩾r(u). As\nwill be established shortly, such sticky deletions translate to\nspecific deletion events on the respective ℓ-read vectors, which\nwe define formally as follows.\nDefinition 7 Ak-restricted deletion is a deletion that only\ndeletes a symbol if it equals 0ork.\nThe error ball of a k-restricted deletion for any u∈Σn\nqcan\nbe expressed as\nDR(u;k)≜{(u1u2. . . u i−1ui+1. . . u n) :i∈[n−ℓ+ 1],\nui∈ {0, k}}.\nThe upcoming lemma explains the link between ℓ-sticky\ndeletions in binary vectors and ℓ-restricted deletions in their\nrespective ℓ-read vectors.\nLemma 8 For any x∈Σn\n2with ρ⩾ℓ(0ℓ−1x0ℓ−1)⩾1,\nit holds that for each R′∈DR(R(x);ℓ), there exists a\nunique y∈Σn−1\n2 such that 0ℓ−1y0ℓ−1∈DS(0ℓ−1x0ℓ−1;ℓ)\nsatisfying R′=R(y). That is,\nDR(R(x);ℓ) ={R(y) : 0ℓ−1y0ℓ−1∈DS(0ℓ−1x0ℓ−1;ℓ)}.\nProof: Consider an R′∈DR(R(x);ℓ), that arises from\na deletion in R(x)at index i, and assume that R(x)i= 0,\nor equivalently, xi\ni−ℓ+1= 0ℓ. For the case of R(x)i=ℓ, the\nproof follows similarly.\nIfℓ+ 1⩽i⩽n, we note that for y= (xi−ℓ\n10ℓ−1xn\ni+1)∈\nDS(x;ℓ), we get R′=R(y), since xi−1\n1=yi−1\n1ensures thatR(y)j=R(x)jforj∈[i−1], while for i⩽j⩽n+ℓ−2,\nwe have R(y)j= wt( yj\nj−ℓ+1) = wt( xj+1\nj−ℓ+2) =R(x)j+1.\nNext, consider i∈[ℓ]. Since xi\n1= 0i, it follows that\nfory= (0i−1xn\ni+1),R′=R(y). Observe that (0ℓ−1y)∈\nDS(0ℓ−1x;ℓ).\nFinally when i⩾n+1, we can similarly argue that for y=\n(xi−ℓ\n10ℓ−(i−n)−1), which upholds (y0ℓ−1)∈DS(x0ℓ−1;ℓ),\nR′=R(y). In all of the aforementioned cases 0ℓ−1y0ℓ−1∈\nDS(0ℓ−1x0ℓ−1;ℓ)and since R(y)is fixed, yis clearly unique.\nThe statement of the lemma thus follows.\nExample 9 Recall x= (1,0,1,1,0,0)from Example 2, with\nthe3-read vector R(x) = (1 ,1,2,2,2,1,0,0). Consider\nR′= (1,1,2,2,2,1,0)that arises from deleting a 0inR(x),\ni.e.,R′∈DR(R(x);ℓ)where ℓ= 3 . Observe that for\ny= (1,0,1,1,0), we have 0ℓ−1y0ℓ−1∈DS(0ℓ−1x0ℓ−1;ℓ)\nandR(y) =R′.\nThus for any x∈Σn\n2with ρ⩾ℓ(0ℓ−1x0ℓ−1)⩾1,\nit holds that |DR(R(x);ℓ)|=|DS(0ℓ−1x0ℓ−1;ℓ)|=\nρ⩾ℓ(0ℓ−1x0ℓ−1)⩾ρ⩾ℓ(x).\nTo summarize, the following corollary establishes our strat-\negy for bounding the size of any single-deletion-correcting\nread code.\nCorollary 10 Any single-deletion ℓ-read code is also a single-\nℓ-sticky-deletion-correcting code.\nProof: We have noted that {R(y) :y∈DS(x;ℓ)} ⊆\nDR(R(x);ℓ)⊆D(R(x)).\nConsequently, the cardinality of a single-deletion ℓ-read\ncode is bounded from above by the size of the largest code\nthat corrects a single ℓ-sticky deletion.\nNow to establish an upper bound on the cardinality of a\ncode that corrects a single ℓ-sticky deletion. We first note that\nfor a randomly chosen x∈Σn\n2, the expected value of ρ⩾a(x)\nis given by 2−a(n−a+ 2) (see Appendix). Also note that\na change in any ximay increase or decrease ρ⩾a(x)by at\nmost 1. It then follows from McDiarmid’s inequality [13] that\nfor any ϵ >0\n2−n\f\f\f\f\u001a\nx∈Σn\n2:ρ⩾a(x) 0,(R(x)i, . . . ,R(x)i+2m−1) = ( αβ)m\nwhile (R(y)i, . . . ,R(y)i+2m−1) = (βα)m. Now the fact that\nR(x)i+1− R(x)i=R(y)i− R(y)i+1= 1 necessitates\n(xi+1, xi−ℓ+1) = (yi−ℓ+1, yi+1) = (1 ,0). However for ℓ⩾2,\nthe requirement xi−ℓ+1̸=yi−ℓ+1contradicts xi−1\n1=yi−1\n1.\nThus, xandydo not exist.\nForℓ⩾2, the previous lemma implies the following\noutcome on the redundancy needed to uniquely recover a\nbinary vector from two distinct erroneous copies of its ℓ-read\nvector.\nLemma 17 When ℓ⩾2andδ= 1, for any two distinct\nx,y∈Σn\n2,|D(R(x))∩D(R(y))|⩽1.\nProof: From [20, Corollary 1], it is known that we\nmust have |D(R(x))∩D(R(y))|⩽2since distinct binary\nvectors have distinct ℓread vectors, as suggested by P3.\nAdditionally [7, Lemma 1] asserts that for ℓ⩾2, it holds that\ndH(R(x),R(y))⩾2for distinct xandy. Upon combining\nthese facts with Proposition 15 and Lemma 16, we arrive at\nthe statement of the lemma.\nWe can thus infer the following on the redundancy required\nfor reconstruction with two noisy read vectors.\nCorollary 18 For any ℓ⩾2,x∈Σn\n2and given any two\ndistinct noisy read vectors R′,R′∈D(R(x)),R(x)can be\nuniquely recovered.Algorithm 1: Reconstruct\nInput: n,ℓ, set{R′,R′} ⊆D(R(x))for some\nx∈Σn\n2\nOutput: R(x)\ninit\nLetiandjbe the first and last indices at which\nR′andR′disagree.\nbR(x)←(R′\n1, . . . ,R′\ni−1,R′\ni,R′\ni, . . . ,R′\nn+ℓ−2);\neR(x)←(R′\n1, . . . ,R′\nj,R′\nj,R′\nj+1, . . . ,R′\nn+ℓ−2).\nifbR(x)is the ℓ-read vector1of any vector in Σn\n2then\nR(x)←bR(x).\nelse\nR(x)←eR(x).\nOne possible method to accomplish reconstruction with two\ncorrupted ℓ-read vectors is outlined in Algorithm 1 and its\ncorrectness is proved in the next lemma.\nLemma 19 For any ℓ⩾2andx∈Σn\n2such that D(R(x))⩾\n2, given any two distinct vectors in D(R(x)), Algorithm 1\nreturns R(x).\nProof: Let the two noisy reads be denoted by R′andR′\nrespectively. By virtue of Lemma 5, we deem it sufficient to\nstudy the case wherein neither of these vectors has a pair of\nconsecutive elements with an absolute difference exceeding 1.\nLetiandjdenote the first and last indices at which R′\nandR′disagree. Of course, i=jwhen dH(R′,R′) = 1 .\nAlso assume that R′andR′arise from a deletion on R(x)\nat indices aandbrespectively. Evidently, {a, b}={i, j+ 1}.\nNow consider the vectors\nbR(x) = (R′\n1, . . . ,R′\ni−1,R′\ni,R′\ni, . . . ,R′\nn+ℓ−2),\neR(x) = (R′\n1, . . . ,R′\nj,R′\nj,R′\nj+1, . . . ,R′\nn+ℓ−2).\nThese are clearly distinct and depending on whether a=i\nora=j+ 1,R(x)is either equal to bR(x)oreR(x). Next,\nobserve that bR(x)andeR(x)cannot be legitimate read vectors\naccording to [7, Proposition 1], simultaneously, as Lemma 17\nwould be otherwise contradicted. Thus, Algorithm 1 chooses\none of bR(x)andeR(x), as advised by [7, Proposition 1].\nRemark: Prior work [10, 21, 28] states that for the standard\nsingle deletion channel, i.e., ℓ= 1, the required redundancy\ndecreases gracefully from log2n+O(1)tolog2log2n−O(1)\ngiven one and two distinct erroneous copies of an ℓ-read vector\nrespectively. It is thus surprising to learn that when ℓ⩾2, the\nminimal redundancy cost remains the same for one received\nsequence, while for two noisy copies, it instantly drops to 0.\nV. C ONCLUSION\nThe aim of this work was to investigate how the inherent\nredundancy imbued by the physical aspects of a nanopore\nsequencer into its reads might help achieve more efficient\n1For more details, and a description of an efficient verification procedure,\nwe refer the reader to [7].deletion-correcting codes. To this end, we found that for\nthe simplified model of nanopore sequencing adopted here,\nthe minimal redundancy required to correct a single deletion\nreduces remarkably when the receiver is provided two distinct\nerroneous received sequences, as opposed to just one. This\nraises further questions regarding the case of multiple deletions\nand more received sequences, which we plan to explore in\nfuture work.\nREFERENCES\n[1] D. Deamer, M. Akeson, and D. Branton, “Three decades of nanopore\nsequencing,” Nature Biotechnology , vol. 34, no. 5, pp. 518–524, May\n2016.\n[2] A. H. Laszlo et al. , “Decoding long nanopore sequencing reads of\nnatural DNA,” Nature Biotechnology , vol. 32, no. 8, pp. 829–833, Aug.\n2014.\n[3] J. Kasianowicz, E. Brandin, D. Branton, and D. Deamer, “Charac-\nterization of individual polynucleotide molecules using a membrane\nchannel,” Proc. Natl. Acad. Sci. , vol. 93, no. 24, pp. 13 770–13 773,\nNov. 1996.\n[4] W. Mao, S. N. Diggavi, and S. Kannan, “Models and information-\ntheoretic bounds for nanopore sequencing,” IEEE Trans. Inf. Theory ,\nvol. 64, no. 4, pp. 3216–3236, Apr. 2018.\n[5] R. Hulett, S. Chandak, and M. Wootters, “On coding for an abstracted\nnanopore channel for DNA storage,” in IEEE Intl. Symp. on Inf. Theory\n(ISIT) , Melbourne, Australia, Jul. 2021, pp. 2465–2470.\n[6] B. McBain, E. Viterbo, and J. Saunderson, “Finite-state semi-markov\nchannels for nanopore sequencing,” in IEEE Intl. Symp. Inf. Theory\n(ISIT) , Espoo, Finland, Jun. 2022, pp. 216–221.\n[7] A. Banerjee, Y . Yehezkeally, A. Wachter-Zeh, and E. Yaakobi, “Error-\nCorrecting Codes for Nanopore Sequencing,” in IEEE International\nSymposium on Information Theory (ISIT) , Taipei, Taiwan: IEEE, Jun.\n2023, pp. 364–369.\n[8] Y . M. Chee, A. Vardy, V . K. Vu, and E. Yaakobi, “Coding for\ntransverse-reads in domain wall memories,” in IEEE Intl. Symp. Inf.\nTheory (ISIT) , Melbourne, Australia, Jul. 2021, pp. 2924–2929.\n[9] A. Banerjee, Y . Yehezkeally, A. Wachter-Zeh, and E. Yaakobi, Error-\ncorrecting codes for nanopore sequencing , 2023. arXiv: 2305.10214\n[cs.IT] .\n[10] V . Levenshtein, “Binary codes capable of correcting spurious insertions\nand deletions of ones,” Problems of Information Transmission , vol. 1,\npp. 8–17, 1965.\n[11] L. Dolecek and V . Anantharam, “Repetition Error Correcting Sets:\nExplicit Constructions and Prefixing Methods,” SIAM Journal on\nDiscrete Mathematics , vol. 23, no. 4, pp. 2120–2146, Jan. 2010.\n[12] H. Mahdavifar and A. Vardy, “Asymptotically optimal sticky-insertion-\ncorrecting codes with efficient encoding and decoding,” in 2017\nIEEE International Symposium on Information Theory (ISIT) , Aachen,\nGermany: IEEE, Jun. 2017, pp. 2683–2687.\n[13] J. L. Doob, “Regularity properties of certain families of chance\nvariables,” Transactions of the American Mathematical Society , vol. 47,\nno. 3, pp. 455–486, 1940.\n[14] A. Fazeli, A. Vardy, and E. Yaakobi, “Generalized Sphere Packing\nBound,” IEEE Transactions on Information Theory , vol. 61, no. 5,\npp. 2313–2334, May 2015.\n[15] G. M. Church, Y . Gao, and S. Kosuri, “Next-Generation Digital Infor-\nmation Storage in DNA,” Science , vol. 337, no. 6102, pp. 1628–1628,\nSep. 2012.\n[16] N. Goldman et al. , “Towards Practical, High-Capacity, Low-\nMaintenance Information Storage in Synthesized DNA,” Nature ,\nvol. 494, no. 7435, pp. 77–80, Feb. 2013.\n[17] L. Organick et al. , “Random access in large-scale DNA data storage,”\nNature Biotechnology , vol. 36, no. 3, pp. 242–248, Mar. 2018.\n[18] S. M. H. T. Yazdi, H. M. Kiah, E. Garcia-Ruiz, J. Ma, H. Zhao, and\nO. Milenkovic, “DNA-Based Storage: Trends and Methods,” IEEE\nTransactions on Molecular, Biological and Multi-Scale Communica-\ntions , vol. 1, no. 3, pp. 230–248, Sep. 2015.\n[19] V . Levenshtein, “Efficient reconstruction of sequences,” IEEE Trans-\nactions on Information Theory , vol. 47, no. 1, pp. 2–22, Jan. 2001.[20] V . I. Levenshtein, “Efficient Reconstruction of Sequences from Their\nSubsequences or Supersequences,” Journal of Combinatorial Theory,\nSeries A , vol. 93, no. 2, pp. 310–332, Feb. 2001.\n[21] J. Chrisnata, H. M. Kiah, and E. Yaakobi, “Correcting deletions with\nmultiple reads,” IEEE Trans. Inf. Theory , vol. 68, no. 11, pp. 7141–\n7158, Nov. 2022.\n[22] M. Abu-Sini and E. Yaakobi, “On Levenshtein’s Reconstruction Prob-\nlem Under Insertions, Deletions, and Substitutions,” IEEE Transactions\non Information Theory , vol. 67, no. 11, pp. 7132–7158, Nov. 2021.\n[23] T. Batu, S. Kannan, S. Khanna, and A. McGregor, “Reconstructing\nstrings from random traces,” in Proceedings of the Fifteenth Annual\nACM-SIAM Symposium on Discrete Algorithms , ser. SODA ’04, USA:\nSociety for Industrial and Applied Mathematics, Jan. 2004, pp. 910–\n918.\n[24] V . L. Phuoc Pham, K. Goyal, and H. M. Kiah, “Sequence Recon-\nstruction Problem for Deletion Channels: A Complete Asymptotic\nSolution,” in 2022 IEEE International Symposium on Information\nTheory (ISIT) , Jun. 2022, pp. 992–997.\n[25] R. Gabrys and E. Yaakobi, “Sequence Reconstruction Over the Dele-\ntion Channel,” IEEE Transactions on Information Theory , vol. 64,\nno. 4, pp. 2924–2931, Apr. 2018.\n[26] Y . Yehezkeally and M. Schwartz, “Reconstruction Codes for DNA\nSequences With Uniform Tandem-Duplication Errors,” IEEE Trans-\nactions on Information Theory , vol. 66, no. 5, pp. 2658–2668, May\n2020.\n[27] K. Cai, H. M. Kiah, T. T. Nguyen, and E. Yaakobi, “Coding for\nSequence Reconstruction for Single Edits,” IEEE Transactions on\nInformation Theory , vol. 68, no. 1, pp. 66–79, Jan. 2022.\n[28] N. J. A. Sloane, “On Single-Deletion-Correcting Codes,”\narXiv:math/0207197 , Jul. 2002. arXiv: math/0207197.\nAPPENDIX\nA. Expected number of runs exceeding a given length\nClaim 20 For any n >0anda∈[n], the expected number of\nruns with length greater than or equal to a, in a binary vector\nof length nwherein each bit is chosen uniformly, is given by\nE[ρ⩾a(x)] = 2−a(n−a+ 2).\nProof: For some x∈Σn\n2, let 1i,a(x)refer to the indicator\nfunction that evaluates to 1if a run of length greater than or\nequal to abegins exactly at the ith index of x, and 0otherwise.\nHence,\nE[ρ⩾a(x)] =n−a+1X\ni=1E[ 1i,a(x)]\n=P(xa\n1∈ {0a,1a})\n+n−a+1X\ni=2P(xi+a−1\ni−1∈ {10a,01a})\n= 2−a(n−a+ 2)." }, { "title": "2401.15947v3.MoE_LLaVA__Mixture_of_Experts_for_Large_Vision_Language_Models.pdf", "content": "MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nBin Lin1Zhenyu Tang1Yang Ye2Jiaxi Cui3Bin Zhu1Peng Jin1Jinfa Huang4Junwu Zhang1\nMunan Ning1 5Li Yuan1 5\nAbstract\nRecent advances demonstrate that scaling Large\nVision-Language Models (LVLMs) effectively im-\nproves downstream task performances. However,\nexisting scaling methods enable all model pa-\nrameters to be active for each token in the cal-\nculation, which brings massive training and in-\nferring costs. In this work, we propose a sim-\nple yet effective training strategy MoE-Tuning\nfor LVLMs. This strategy innovatively addresses\nthe common issue of performance degradation in\nmulti-modal sparsity learning, consequently con-\nstructing a sparse model with an outrageous num-\nber of parameters but a constant computational\ncost. Furthermore, we present the MoE-LLaV A ,\na MoE-based sparse LVLM architecture, which\nuniquely activates only the top- kexperts through\nrouters during deployment, keeping the remain-\ning experts inactive. Extensive experiments show\nthe significant performance of MoE-LLaV A in\na variety of visual understanding and object hal-\nlucination benchmarks. Remarkably, with only\napproximately 3Bsparsely activated parameters,\nMoE-LLaV A demonstrates performance compa-\nrable to the LLaV A-1.5-7B on various visual\nunderstanding datasets and even surpasses the\nLLaV A-1.5-13B in object hallucination bench-\nmark. Through MoE-LLaV A, we aim to establish\na baseline for sparse LVLMs and provide valuable\ninsights for future research in developing more\nefficient and effective multi-modal learning sys-\ntems. Code is released at https://github.com/PKU-\nYuanGroup/MoE-LLaV A.\n1. Introduction\nLarge Vision-Language Models (LVLMs), such as\nLLaV A (Liu et al., 2023c) and MiniGPT-4 (Zhu et al., 2023),\nhave shown promising results by leveraging an image en-\n1Peking University2Sun Yat-sen University3FarReel AI Lab\n4University of Rochester5Peng Cheng Laboratory. Correspon-\ndence to: Li Yuan , Munan Ning .\nLLaVA -Phi-2.7B\nMobileVLM -2.7BLLaMA -VID-7B\nLLaVA -1.5-7B\nmPLUG-Owl2 -7BOtterHD -8B\nLLaVA -1.5-13BLLaMA -VID-13B\nShikra -7BChat-UniVi -1.5-7BBLIP -2-13BLION -12BInternVL-Chat-19B\nInternVL-Chat-14BMoE-LLaVA -1.8B×4Hallucination Average Performance\nNumber of Activated Parameters ( Billions)Figure 1. Comparison between MoE-LLaV A-1.8B ×4 and open-\nsource LVLMs on object hallucination benchmark. We report\nthe average performance on the POPE (Li et al., 2023d) benchmark,\nwhich includes three subsets of Adversarial, Random, and Popular.\nThe red dashed line represents the linear fit to the data points of all\nmodels except MoE-LLaV A.\ncoder and several visual projection layers to enhance the\nvisual perception capabilities of the Large Language Models\n(LLMs). Typically, increasing the model size (Zhang et al.,\n2023a; Bai et al., 2023b) and dataset scale (Zhang et al.,\n2023c; Zhao et al., 2023a; Chen et al., 2023d) can improve\nmodel performance. For instance, InternVL (Chen et al.,\n2023e) has extended the image encoder to 6B parameters.\nA series of works (Li et al., 2022; Dai et al., 2023; Liu\net al., 2023b) have expanded the backend of LVLM to 13B\nparameters and achieved state-of-the-art performance on\ndownstream tasks. IDEFICS (Lauren c ¸on et al., 2023) even\ntrained an LVLM with 80B parameters. These methods have\ndemonstrated superior performance even in LLMs, which\nare typically pretrained on 34B parameters (SUSTech-IDEA,\n2023; 01-ai, 2023; FlagAI-Open, 2023) or 70B parame-\nters (Touvron et al., 2023a;b; Bai et al., 2023a; DeepSeek-\nAI, 2024; Zhang & Yang, 2023), and models surpassing\n100B parameters are common (Brown et al., 2020; Zeng\net al., 2022; Zhang et al., 2022; Scao et al., 2022; Li et al.,\n2023c; falconry, 2023) .\nIn practical applications, scaling model with high-quality\ntraining data is crucial for improving model perfor-\nmance (Lepikhin et al., 2020). However, training and de-\nploying such large models demand significant computa-\n1arXiv:2401.15947v3 [cs.CV] 17 Feb 2024MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nCopy\nweight······ ······\nWord Embedding Layer Vision Encoder & MLP\nDescribe the image?\n······ ······MoE\nLayerMoE Layer ×N\nThe image capture thebeauty and\ngrandeur ofthestatue ofliberty …\nExpert 4 Expert 1 Expert 2 Expert 3V1 T1\nV1 Vn T1 TnAttentionConvert LLM to LVLM\nVEFFN\nMLP\nWELTrainable parameter\nNon-trainable\n(b) Stage III (a) Stage I and Stage IIFrozen in Stage I \nTrained in Stage II\nFigure 2. Illustration of MoE-Tuning. The MoE-Tuning consists\nof three stages. In stage I, only the MLP is trained. In stage II,\nall parameters are trained except for the Vision Encoder (VE). In\nstage III, FFNs are used to initialize the experts in MoE, and only\nthe MoE layers are trained. For each MoE layer, only two experts\nare activated for each token, while the other experts remain silent.\ntional costs and efficient implementation on parallel devices,\nwhich can be extremely expensive. This is because each to-\nken requires computations with all model parameters, called\nthe dense model. In contrast, sparse Mixtures of Experts\n(MoE) (Jacobs et al., 1991; Eigen et al., 2013) effectively\nscale model capacity by using fixed activated parameters to\nprocess data, which has thrived in the field of NLP (Fedus\net al., 2022; Zoph et al., 2022; Komatsuzaki et al., 2022).\nRecently, Mistral LLM (Jiang et al., 2023) equipped with the\nMoE layers has gained popularity in LLMs. Mixtral-MoE-\n8×7B (Jiang et al., 2024) achieves performance comparable\nto LLaMA 2-70B with fewer computational resources.\nHowever, directly applying MoE to train sparse LVLMs is\nchallenging. We observe that simultaneously converting\nLLM to LVLM and sparsifying the model leads to signifi-\ncant performance degradation. After multiple attempts, we\nfind that proper initialization is crucial for sparsifying the\nLVLM, Therefore, we introduce a simple yet effective three-\nstage training strategy MoE-Tuning . Specifically, as shown\nin Figure 2, we first train an MLP that adapts visual tokens to\nthe LLM in stage I. Then, we pre-empower the LVLM with\na general multi-modal understanding capability by training\nthe whole LLM’s parameters in stage II. Furthermore, in\nstage III we replicate the FFN as the initialization weights\nfor the experts and only train the MoE layers. Finally, the\nsparse model gradually transitions from a general LVLM\ninitialization to sparse mixture of experts.\nIn this work, we explore a baseline for the LVLM with\nmixture of experts called MoE-LLaV A , which incorporates\nmixture of experts and learnable routers. MoE-LLaV A con-\nsists of multiple sparse paths where each token is dispatched\nto different experts through the router. The activated experts\ncollectively process the tokens, while the inactive paths re-\nmain silent. By iteratively stacking MoE encoder layers,MoE-LLaV A provides a sparse path toward a larger and\nmore powerful LVLM.\nAs a result, in Figure 1, our MoE-LLaV A with only 2.2B\nsparse activated parameters outperforms models with simi-\nlar activated parameters and LLaV A-1.5-13B, surpassing it\nby a large margin on the POPE object hallucination bench-\nmark. Additionally, MoE-LLaV A achieves comparable per-\nformance to InternVL-Chat-19B, which has approximately\n8 times the activated parameters. We further scale MoE-\nLLaV A to 3.6B sparse activated parameters, which outper-\nform LLaV A-1.5-7B by 1.9%, 0.4%, 0.9%, 30.7%, and 3.8%\nin ScienceQA, POPE, MMBench, LLaV AW, and MM-Vet,\nrespectively. Extensive experiments validate the rationality\nof our MoE-LLaV A architecture and MoE-Tuning strategy.\nWe summarize our primary contributions as follows:\n•We explore the MoE-Tuning , a novel three-stage train-\ning strategy for adapting MoE to LVLMs and prevent-\ning the model degradation caused by sparsity.\n•We propose MoE-LLaV A , a MoE-based sparse LVLM\nframework, which significantly expands the number of\nparameters while maintaining computational costs.\n•Extensive experiments demonstrate that our MoE-\nLLaV A has excellent multi-modal understanding and\nhallucination mitigation abilities. With only approx-\nimately 3Bsparse activated parameters, our method\nachieves comparable performance with SOTA 7B mod-\nels on the visual understanding benchmarks. It is worth\nnoting that MoE-LLaV A outperforms LLaV A-1.5-13B\nby 1.1% on the POPE hallucination benchmark with\n2.2B activated parameters.\n2. Related Work\n2.1. Large Vision-Language Models\nPowerful LLMs (OpenAI, 2023; Touvron et al., 2023a; Wei\net al., 2022; Touvron et al., 2023b; Zheng et al., 2023; Team,\n2023; Sun et al., 2023; Du et al., 2021; Bai et al., 2023a;\nYang et al., 2023; Penedo et al., 2023; Taori et al., 2023)\nwith strong instruction-following and generalization capa-\nbilities have been applied to LVLMs. Early works such as\nBLIP-2 (Li et al., 2023b) and FROMAGe (Koh et al., 2023)\nencoded visual signals into a sequence of visual tokens,\nsuccessfully adapting vision to LLMs through several pro-\njection layers. Subsequently, recent works have focused on\nimproving performance through methods such as expanding\nthe instruction-tuning dataset (Liu et al., 2023a;c; Zhang\net al., 2023c; Zhao et al., 2023a; Chen et al., 2023d), op-\ntimizing training strategies (Bai et al., 2023b; Chen et al.,\n2023b), increasing resolution of image (Liu et al., 2023b;\nBai et al., 2023b; Wang et al., 2023d) enhancing image en-\n2MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nFFNAdd & Norm\nSelf-AttentionAdd & Norm\nMLP\nVision\nEncoderRouterAdd & Norm\nSelf-AttentionAdd & Norm\nMLP\nVision\nEncoderMoE\nFFN 1\n(b) Stage II (c) Stage IIIImageInstruction \nRequestResponse\nImageInstruction \nRequestResponse\nFFN 3 FFN E ··· ···\nCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsCopy weight \nfor expertsFFNAdd & Norm\nSelf-AttentionAdd & Norm\nMLP\nVision\nEncoder\nImageCaption\nRequestGenerated Text Caption\n(a) Stage ICopy weightCopy weight\nCopy \nweightTrainable parameter Non-trainableCopy weight Activated forward Non-activated forward\nWord\nEmbeddingWord\nEmbedding Word\nEmbedding\nFigure 3. Training framework and strategy. MoE-LLaV A adopts a three-stage training strategy. (a) We solely train the MLP to adapt the\nLLM to visual inputs. (b) Training the LLM backend empowers multi-modal understanding capability and MoE layers are not involved.\n(c) In this stage, we replicate the weights of the FFN to initialize each expert.\ncoders (Chen et al., 2023e; Zhang et al., 2023a; Bai et al.,\n2023b), aligning the input (Lin et al., 2023) and projection\nlayers (Cha et al., 2023; Alayrac et al., 2022; Bai et al.,\n2023b; Dai et al., 2023; Ye et al., 2023; Zhao et al., 2023a).\nThese works empowered LVLMs with powerful visual un-\nderstanding capabilities by expanding the visual instruction\nfine-tuning datasets and model scales.\nCurrently, some works have endowed LVLMs with fine-\ngrained image understanding capabilities, such as region un-\nderstanding (Chen et al., 2023c; Zhao et al., 2023b; Liu et al.,\n2023e), multi-region understanding (Wang et al., 2023c;\nPi et al., 2023; Peng et al., 2023), and pixel-wise ground-\ning (Rasheed et al., 2023; Lai et al., 2023). However, the\ncost of scaling up dense visual data and models is chal-\nlenging to bear (Liu et al., 2022; Yin et al., 2023). In this\nwork, we aim to make state-of-the-art LVLMs research more\naccessible by leveraging mixture of experts.\n2.2. Mixture of Experts in Multi-modal Learning\nMixture of Experts (MoE) (Jacobs et al., 1991; Eigen et al.,\n2013) is a hybrid model consisting of multiple sub-models,\nknown as experts, which are integrated together. The key\nconcept of MoE is the use of a router to determine the token\nset that each expert handles, thereby reducing interference\nbetween different types of samples.\nHard Routers. In the hard router mode, each expert is\ntypically pre-defined as a specific pattern. This is becausemulti-modal data naturally exhibit gaps (Liang et al., 2022),\nmaking it difficult for soft routers to learn the optimal pat-\nterns for assigning tokens to different experts. A series of\nworks (Bao et al., 2022; Long et al., 2023; Satar et al., 2022;\nWang et al., 2022; Shen et al., 2023) naturally decouple ex-\nperts based on modal categories and pre-define each expert\nto handle a specific modality. An important feature of these\nhard-based routers is that they do not require learning the\nrouter. This mode is also widely applied in the task-specific\nMoE (Li et al., 2023e; Zhu et al., 2022; Ma et al., 2023;\nKudugunta et al., 2021).\nSoft Routers. Some works (Shazeer et al., 2017; Lep-\nikhin et al., 2020; Fedus et al., 2022; Zoph et al., 2022;\nKomatsuzaki et al., 2022) in natural language process have\nexplored the MoE based on soft routers. Soft routers en-\nable dynamic allocation of data among different experts,\nallowing each expert to focus on its expertise and achieve\nmodel sparsity. Therefore, our main focus is on leveraging\nsoft routers in the MoE. Small-scale (million-level) models\nbased on soft routers have also been explored in the context\nof multi-modal learning, such as EVE (Chen et al., 2023a)\nand LIMoE (Mustafa et al., 2022), which attempt a fusion\nof data by using soft routers. The work most relevant to ours\nis MoCLE (Gou et al., 2023). However, MoCLE clusters\ndifferent instruction sets and distributes them to different\nexperts, which compromises the flexibility and autonomy of\nthe experts. Differently, MoE-LLaV A relies on knowledge-\nrich routers to distribute tokens to different paths.\n3MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nTable 1. Architecture details of the MoE-LLaV A model. “FFN Factor” represents the number of linear layers in the FFN. “1.6B ×4-Top2”\nrepresents a dense foundation model with 1.6B parameters, which is equipped with a total of four experts, two of them being activated.\nName Experts Top-kMoEEmbedding Width Layers FFNFFNHeadsActivated Total\nLayers Factor Param Param\nStableLM-1.6B (Team) - - - 100352 2560 32 10240 2 32 1.6B 1.6B\nMoE-LLaV A-1.6B×4-Top2 4 2 16 100352 2560 32 10240 2 32 2.0B 2.9B\nQwen-1.8B (Bai et al., 2023a) - - - 151936 2048 24 5504 3 16 1.8B 1.8B\nMoE-LLaV A-1.8B×4-Top2 4 2 12 151936 2048 24 5504 3 16 2.2B 3.1B\nPhi2-2.7B (Microsoft, 2023) - - - 51200 2560 32 10240 2 32 2.7B 2.7B\nMoE-LLaV A-2.7B×4-Top2 4 2 16 51200 2560 32 10240 2 32 3.6B 5.3B\n3. Method\n3.1. Overview\nAs shown in Figure 3, MoE-LLaV A consists of a vision\nencoder, a visual projection layer (MLP), a word embedding\nlayer, multiple stacked LLM blocks, and MoE blocks. We\nfirst introduce the model architecture of MoE-LLaV A in\nthree stages in Section 3.2. Furthermore, in Section 3.3, we\nexplain how to train MoE-LLaV A. Finally, in Section 3.4,\nwe elaborate on the training objectives of MoE-LLaV A.\n3.2. Architecture of MoE-LLaV A\nAs shown in Table 1, we present the detailed configura-\ntion of MoE-LLaV A and more details can be found in Ap-\npendix A.1. Given a RGB image v∈RH×W×3, where\nHandWare the origin resolution. The vision encoder\nprocesses input images to obtain a visual token sequence\nZ= [z1,z2,···,zP]∈RP×C, whereP=H×W\n142repre-\nsents the sequence length of visual tokens. A visual pro-\njection layer fis used to mapZ∈RP×CtoV∈RP×D,\nwhereDrepresents the hidden size of LLM. Similarly, the\ntext undergoes a word embedding layer gand is projected to\nobtain the sequence tokens T= [t1,t2,···,tN]∈RN×D,\nwhereNrepresents the sequence length of text tokens.\nSubsequently, we concatenate the visual tokens and text\ntokens together and feed them into a large language model.\nInstead, we solely train the visual projection layer. The large\nlanguage model consists of stacked multi-head self-attention\n(MSA) and feed-forward neural networks (FFN). Layer\nnormalization (LN) and residual connections are applied\nwithin each block (Wang et al., 2019; Baevski & Auli, 2018).\nTherefore, we formulate as:\nx0= [v1,v2,···,vP,···,t1,t2,···,tN], (1)\nx′\nℓ= MSA(LN( xℓ−1)) +xℓ−1,ℓ= 1...L, (2)\nxℓ= MoE(LN( x′\nℓ)) +x′\nℓ,ℓ= 1...L, (3)\nY= LN( xL). (4)\nMoE Forward. Typically, a MoE layer consists of mul-\ntiple FFNs. As an initialization step, we replicate theFFNs from stage II to form an ensemble of experts E=\n[e1,e2,···,eE]. The router is a linear layer that predicts\nthe probability of each token being assigned to each expert.\nWe formulate as:\nP(x)i=ef(x)i\n/summationtextE\njef(x)j, (5)\nwhere the router produces weight logits f(x) =W·x,\nwhich are normalized by the softmax function. The W∈\nRD×Erepresents the lightweight training parameters and E\nrepresents the number of experts. Therefore, each token is\nprocessed by the top- kexperts with the highest probabilities,\nand the weighted sum is calculated based on the softmax\nresults of the probabilities:\nMoE( x) =k/summationdisplay\ni=1P(x)i·E(x)i. (6)\n3.3. MoE-Tuning\nStage I: In this stage, our objective is to adapt the image\ntokens to LLM, allowing the LLM to comprehend the in-\nstances in the images. To achieve this, we employ an MLP to\nproject the image tokens into the input domain of the LLM,\ntreating the image patches as pseudo-text tokens. During\nthis stage, the LLM is trained to describe the images. MoE\nlayers are not applied to the LLM during this stage.\nStage II: Tuning with multi-modal instruction data is a key\ntechnique to enhance the capabilities and controllability of\nlarge models (Zhang et al., 2023b). In this stage, LLM is\nadjusted to become an LVLM with multi-modal understand-\ning. We use more complex instructions, including tasks\nsuch as image logical reasoning and text recognition, which\nrequire the model to have a stronger multi-modal under-\nstanding. Typically, for dense models, the LVLM training is\nconsidered complete at this stage. However, we encounter\nchallenges in simultaneously transforming the LLM into an\nLVLM and sparsifying the LVLM. Therefore, MoE-LLaV A\nutilizes the weights from the second stage as initialization\nfor the third stage to alleviate the learning difficulty of the\nsparse model.\n4MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nStage III: As an initialization, we replicate the FFN multi-\nple times to initialize the experts. When image tokens and\ntext tokens are fed into the MoE layers, the router calculates\nthe matching weights between each token and the experts.\nEach token is then processed by the top- kexperts, and the\noutputs are aggregated by weighted summation based on the\nrouter’s weights. When the top- kexperts are activated, the\nremaining experts remain silent. This modeling approach\nforms the MoE-LLaV A with infinitely possible sparse path-\nways, offering a wide range of capabilities.\n3.4. Training Objectives\nTheLtotalconsists of auto-regressive loss Lregressive and auxil-\niary lossLaux, and auxiliary loss are scaled by the balancing\ncoefficientα:\nLtotal=Lregressive +α·Laux. (7)\nAuto-Regressive Loss. We optimize the output of LLM\nthrough a generative loss in an auto-regressive manner.\nGiven an image and text, MoE-LLaV A generates the output\nsequenceY= [y1,y2,···,yK]∈RK×Dby progressively\ngenerating each element, where K=P+Drepresents the\nlength of the output sequence. The formula is:\nLregressive =−N/summationdisplay\ni=1logpθ/parenleftig\nY[P+i]|V,T[:i−1]/parenrightig\n,(8)\nwhereθis a trainable parameter and we only calculate the\nloss for the generated text.\nAuxiliary Loss. Due to the presence of multiple experts,\nit is necessary to impose load balancing constraints on the\nMoE layer. We incorporate differentiable load balancing\nloss (Fedus et al., 2022) into each MoE layer to encourage\nexperts to handle tokens in a balanced manner as follows:\nLaux=E·E/summationdisplay\ni=1Fi·Gi, (9)\nwhereFrepresents the fraction of tokens processed by each\nexpertEi, andGrepresents the average routing probability\nofEi, which can be expressed by the following formulas:\nF=1\nKE/summationdisplay\ni=11{argmaxP(x) =i}, (10)\nG=1\nKK/summationdisplay\ni=1P(x)i. (11)\n4. Experiments\n4.1. Experimental Setup\nModel Settings. Following LLaV A 1.5 (Liu et al., 2023b),\nwe utilize CLIP-Large (Radford et al., 2021) as the vision en-Table 2. Composition of the data groups. For MIMIC-IT, and\nSViT datasets, we only use the LA split, and core split, respectively.\nData group Usage Source #Sample\nLLaV A-PT Stage I LLaV A 1.5-558k 558k\nHybird-FT Stage IISViT-157k, LVIS-220k964kLRV-331k, MIMIC-IT-256k\nLLaV A-FT Stage III LLaV A 1.5-mix-665k 665k\ncoder, and the MLP consists of two linear layers with GELU\nactivation function (Hendrycks & Gimpel, 2016) between\nthem. Unless otherwise specified, MoE-LLaV A employs an\nalternating replacement of FFN with MoE layers, meaning\nthat the number of MoE layers is half of the total number\nof layers. The value of balancing coefficient αis 0.01. We\nprovide additional training details in Appendix A.2.\nData Details. As shown in Table 2, we reorganize the\ncurrently available data for the three-stage training. For\nthe first stage of pretraining, we use the pretrained data\nof LLaV A 1.5-558k (Liu et al., 2023b). For the second\nstage, we collect datasets from MIMIC-IT (Li et al., 2023a),\nLRV (Liu et al., 2023a), SViT (Zhao et al., 2023a) and\nLVIS (Wang et al., 2023b) to provide a robust initialization\nfor MoE-LLaV A. For the third stage, we utilize the same\ndata pipeline as LLaV A-mix-665k (Liu et al., 2023b).\n4.2. Image Understanding Evaluation\nZero-shot Image Question Answering. As shown in Ta-\nble 3, since MoE-LLaV A is a sparse model equipped with\na soft router based on LVLM, we categorize the previous\nmodels as dense models. We evaluate the performance of\nMoE-LLaV A on five image question-answering benchmarks\nand report the number of activated parameters. Compared\nto the state-of-the-art method LLaV A 1.5, MoE-LLaV A\ndemonstrates powerful image understanding capabilities\nand performs very close to LLaV A-1.5 on five benchmarks.\nSpecifically, MoE-LLaV A-Phi-2.7B ×4 surpasses LLaV A-\n1.5-7B by 2.7% on SQAIusing 3.6B sparse activated param-\neters. Notably, MoE-LLaV A-StableLM-1.6B ×4 achieves\ncomprehensive superiority over IDEFICS-80B with only\n2.0B activated parameters. Furthermore, we observe the re-\ncent small-scale vision-language model, LLaV A-Phi. MoE-\nLLaV A-Phi-2.7B ×4 outperforms LLaV A-Phi by more than\n6.2% on VQAv2, highlighting the strong comprehension\nabilities of MoE-LLaV A in natural vision.\nEvaluation under Benchmark Toolkits. To comprehen-\nsively evaluate the multi-modal understanding capabilities\nof MoE-LLaV A, we evaluate its performance on four bench-\nmark toolkits. These benchmark toolkits typically involve\nopen-ended answers, serving as tools to verify a model’s\nability to engage in natural language questioning. In Ta-\n5MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nTable 3. Comparison among different LVLMs on image understanding benchmarks. “Res.”, “Act.”, “L”, “V”, “S”, “Q”, “P”, “M”\nand “I” respectively represent the input image resolution, activated parameters, LLaMA (Touvron et al., 2023a), Vicuna (Chiang et al.,\n2023), StableLM (Team), Qwen (Bai et al., 2023a), Phi-2 (Microsoft, 2023) MobileLLaMA (Chu et al., 2023) and IDEFICS (Lauren c ¸on\net al., 2023). Evaluation Benchmarks include VQA-v2 (Goyal et al., 2017); GQA (Hudson & Manning, 2019); VisWiz (Gurari et al.,\n2018); SQAI: ScienceQA-IMG (Lu et al., 2022); VQAT: TextVQA (Singh et al., 2019); POPE (Li et al., 2023d); MME (Fu et al., 2023);\nMMB: MMBench (Liu et al., 2023d); LLaV AW: LLaV A-Bench (in-the-Wild) (Liu et al., 2023c); MM-Vet (Yu et al., 2023).∗donates that\nthere is some overlap in the training data.†donates that the model is trained with an image resolution of 384. The best results and second\nbest results are indicated by boldface and underline , respectively.\nMethods LLM Act. Res.Image Question Answering Benchmark Toolkit\nVQAv2GQA VisWiz SQAIVQATPOPE MME MMB LLaV AWMM-Vet\nDense Model\nI-80B (Laurenc ¸on et al., 2023) L-65B 65B 224 60.0 45.2 36.0 - 30.9 - - 54.5 - -\nLLaV A-1.5 (Liu et al., 2023b) V-13B 13B 336 80.0∗63.3∗53.6 71.6 61.3 85.9 1531.3 67.7 70.7 35.4\nQwen-VL (Bai et al., 2023b) Q-7B 6.7B 448 78.8∗59.3∗35.2 67.1 63.8 - - 38.2 - -\nLLaV A-1.5 (Liu et al., 2023b) V-7B 6.7B 336 78.5∗62.0∗50.0 66.8 58.2 85.9 1510.7 64.3 63.4 30.5\nTinyGPT-V (Yuan et al., 2023) P-2.7B 2.7B 448 - 33.6∗33.4 - - - - - - -\nMobileVLM (Chu et al., 2023) M-2.7B 2.7B 336 - 59.0∗- 61.0 47.5 84.9 1288.9 59.6 - -\nLLaV A-Phi (Zhu et al., 2024) P-2.7B 2.7B 336 71.4∗- 35.9 68.4 48.6 85.0 1335.1 59.8 - 28.9\nSparse Model\nMoE-LLaV A-1.6B×4-Top2 S-1.6B 2.0B 336 76.7∗60.3∗36.2 62.6 50.1 85.7 1318.2 60.2 86.8 26.9\nMoE-LLaV A-1.8B×4-Top2 Q-1.8B 2.2B 336 76.2∗61.5∗32.6 63.1 48.0 87.0 1291.6 59.7 88.7 25.3\nMoE-LLaV A-2.7B×4-Top2 P-2.7B 3.6B 336 77.6∗61.4∗43.9 68.5 51.4 86.3 1423.0 65.2 94.1 34.3\nMoE-LLaV A-1.6B×4-Top2†S-1.6B 2.0B 384 78.6∗61.5∗40.5 63.9 54.3 85.9 1335.7 63.3 90.3 32.3\nMoE-LLaV A-2.7B×4-Top2†P-2.7B 3.6B 384 79.9∗62.6∗43.7 70.3 57.0 85.7 1431.3 68.0 97.3 35.9\nTable 4. Zero-shot object hallucination evaluation results. “Yes” indicates the proportion of positive responses to the given question.\nMethods LLM ActivatedAdersarial Popular Random\nAcc F1-Score Yes Acc F1-Score Yes Acc F1-Score Yes\nDense Model\nmPLUG-Owl (Ye et al., 2023) L-7B 6.7B 82.4 81.6 45.2 85.5 84.3 42.1 86.3 85.3 42.3\nMM-GPT (Gong et al., 2023) L-7B 6.7B 50.0 66.7 100.0 50.0 66.7 100.0 50.0 66.7 100.0\nLLaV A-1.5 (Liu et al., 2023b) V-13B 13B 85.5 84.4 43.3 87.4 86.2 41.3 88.0 87.1 41.7\nSparse Model\nMoE-LLaV A-1.6B×4-Top2 S-1.6B 2.0B 86.9 85.7 41.7 85.3 84.2 43.5 88.0 87.1 41.6\nMoE-LLaV A-1.8B×4-Top2 Q-1.8B 2.2B 86.1 85.4 44.9 88.6 87.7 42.5 88.7 88.0 43.0\nMoE-LLaV A-2.7B×4-Top2 P-2.7B 3.6B 85.9 84.9 43.2 87.5 86.4 41.8 88.5 87.7 41.8\nMoE-LLaV A-1.6B×4-Top2†S-1.6B 2.0B 86.9 85.6 41.5 85.7 84.6 43.0 88.4 87.5 41.5\nMoE-LLaV A-2.7B×4-Top2†P-2.7B 3.6B 85.5 84.2 41.9 86.7 84.4 41.7 87.9 86.9 40.6\nble 3, MoE-LLaV A-Qwen-1.8B ×4 surpasses Qwen-VL-7B\nby 21.5%, on MMBench, despite the latter utilizing higher\nimage resolutions. These results collectively demonstrate\nthat the sparse model MoE-LLaV A achieves comparable\nor even superior performance to dense models with fewer\nactivated parameters.\n4.3. Object Hallucination Evaluation\nWe adopt the evaluation pipeline of POPE (Li et al., 2023d),\na polling-based query method, to evaluate object hallucina-\ntion in MoE-LLaV A. The results are presented in Table 4,\nwhere MoE-LLaV A exhibits the best performance, indicat-\ning that MoE-LLaV A tends to generate objects consistent\nwith the given image. Specifically, MoE-LLaV A-1.8B ×4surpasses LLaV A-1.5-13B by 1.0%, 1.5%, and 0.8% in ad-\nversarial sampling, popular sampling, and random sampling,\nrespectively, with 2.2B activated parameters. Additionally,\nwe observe that the yes ratio of MoE-LLaV A remains rela-\ntively balanced, indicating that our sparse model is capable\nof providing accurate feedback based on the given questions.\n4.4. Quantitative Analysis\nRouting Distributions. In Figure 4, we present the ex-\npert loads (leftmost plot) and the modalities preferences of\ndifferent experts (four subplots on the right) through MoE-\nLLaV A-2.7B ×4-Top2 on ScienceQA. More visualization\ncan be found in Appendix B.3. To begin with, the expert\nloads in all MoE layers are totally balanced. However, as\n6MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\nFigure 4. Distribution of expert loadings. The discontinuous lines represent a perfectly balanced distribution of tokens among different\nexperts or modalities. The first figure on the left illustrates the workload among experts, while the remaining four figures depict the\npreferences of experts towards different modalities.\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\nFigure 5. Distribution of modalities across different experts.\nInterrupted lines mean a perfectly balanced distribution of tokens.\nthe model gradually becomes sparser, the expert 3 loads for\nlayers 17 to 27 suddenly increase, and they even dominate\nthe workload of almost all tokens. For the shallow layers\n(5-11), experts 2, 3, and 4 mainly collaborate. It is worth\nnoting that expert 1 only works predominantly in the first\nfew layers, and as the model becomes deeper, expert 1 grad-\nually withdraws from the workload. Therefore, the experts\nin MoE-LLaV A have learned a certain pattern that allows\nthem to divide their tasks in a specific manner.\nFurthermore, we show the distribution of modalities across\ndifferent experts in Figure 5. Similarly, experts develop\ntheir own preferences. Additionally, we find that the rout-\ning distributions for text and image are highly similar. For\nexample, when expert 3 is actively working in layers 17-27,\nthe proportions of text and image that MoE-LLaV A pro-\ncesses are similar. Each expert in MoE-LLaV A is capable of\nhandling both text tokens and image tokens simultaneously,\nwhich demonstrates that MoE-LLaV A does not exhibit a\nclear preference for any modality. This serves as evidence\nof its strong interaction in multimodal learning.\nToken Pathways. Furthermore, we examine the behavior of\nexperts at the token level. More visualization can be found\nin Appendix B.4 and Appendix B.5. We track the trajec-\ntories of all tokens on downstream tasks. For all activated\npathways, we employ PCA (Pearson, 1901) to obtain the\ntop-10 pathways, as shown in Figure 6. We found that for\na given unseen text token or image tokens, MoE-LLaV A\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthersFigure 6. Visualization of activated pathways. We highlight the\ntop-10 activated pathways on the text and image. Among them,\nthe colorful paths represent the top-2 paths for text and image,\nrespectively, while the gray paths represent the remaining 8 paths.\nTable 5. Ablation study about different training strategies. “LA”\nand “Hb” represent LLaV A-FT and Hybrid-FT in Table 2.\nMoE Stage II Stage III GQA SQAIPOPE LLaV AW\n(a)✔ - LV+Hb 58.4 58.1 81.9 88.0\n(b)✔ Hb LV 61.5 63.1 87.0 88.7\n(c)✗ LV+Hb - 60.9 60.2 86.4 86.3\n(d)✗ Hb LV 60.9 62.5 86.9 90.1\nconsistently tends to assign experts 2 and 3 to handle them\nin the deeper layers of the model. Regarding experts 1 and\n4, they tend to handle the tokens during the initialization\nphase. These findings contribute to a better understanding\nof the behavior of sparse models in multi-modal learning.\n4.5. Ablation Study\nIn this section, we first validate the necessity of the three-\nstage training strategy. We then explore the impact of differ-\nent base models and conduct ablation studies on the number\nof experts and active experts, and the MoE structure. We\nprovide additional results in Appendix B.2.\nEffect of Training Strategy. In Table 5, we conduct three\nvariant experiments to demonstrate the rationale behind us-\ning the second-stage instruction tuning as the initialization\nfor the third-stage MoE tuning. When adapting MoE to\n7MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nTable 6. Ablation study about training setting and architecture design decisions. Settings for results in Table 3 and Table 4 are\nhighlighted in blue . We report the training time on 8 V100-32G.\n(a)Tuning the parameters of different subsets.\nSubset GQA VisWiz VQATPOPE LLaV AWTime\nFFN 61.5 32.6 48.0 87.0 88.7 20h\nAll 61.3 31.9 47.6 87.0 88.1 27h(b)The number of experts.\nExperts GQA SQAIVQATPOPE LLaV AWTime\n1 60.9 60.2 48.3 86.4 86.3 13h\n2 61.2 60.8 47.0 87.5 86.5 14h\n(c)The value of top-k.\nTop-k VQAv2GQA SQAIVQATPOPE Time\n1 74.5 58.4 58.0 44.0 85.7 19h\n2 76.2 61.5 63.1 48.0 88.7 20h(d)The architectures of MoE-LLaV A.\nArchitecture VQAv2GQA SQAIVQATPOPE Time\nFirst-Half 75.9 61.3 62.4 47.0 86.9 20h\nSecond-Half 76.3 61.2 62.6 47.2 86.9 20h\nInterval 76.2 61.5 63.1 48.0 88.7 20h\nAll 74.5 61.5 62.1 47.1 87.0 32h\nLVLMs, a straightforward approach is to replace the clas-\nsic LLaV A’s FFN with a MoE layer and train it according\nto the original second-stage script, denoted as variant (a).\nHowever, variant (a) performs the worst, suggesting that\nthe current multi-modal instruction dataset is insufficient to\nsupport both the conversion from LLM to LVLM and the\nconversion from LVLM to a sparse model simultaneously.\nTherefore, we collect more data, referred to as Hybrid-FT,\nand initially convert LLM to LVLM in the second stage. Sub-\nsequently, in the third stage, LVLM is sparsified by using\nthe LLaV A-FT dataset, resulting in variant (b). Additionally,\nwe expand the data of the original LLaV A’s second stage for\nfair comparison, denoted as variant (c). The results indicate\nthat variants (b) outperformed variants (a) and (c). These\nfindings demonstrate that providing a reasonable LVLM\ninitialization allows the model to transition rapidly from\na dense model to a sparse model, validating the principle\nbehind our three-stage training strategy.\nEffect of Tuning the Parameters of Different Subsets.\nIn Table 6a, we examine the performance of fine-tuning\ndifferent parts of the parameters. “FFN” represents fine-\ntuning all FFN layers and MoE layers in the model. “All”\nindicates fine-tuning all parameters. The results indicate\ntuning the FFN is sufficient to achieve results comparable\nto full-parameter tuning, but it requires only approximately\n75% of the time. Therefore, to enhance generalization and\nreduce training costs, we only fine-tune FFN layers.\nEffect of the Number of Experts. Typically, increasing the\nnumber of experts directly leads to higher performance (Lep-\nikhin et al., 2020; Fedus et al., 2022). In Table 6b, we change\nthe number of experts while keeping the number of activated\nexperts the same, so the number of activated parameters for\nboth models remains the same. More sparse experts outper-\nform the single expert dense model by 1.1% on POPE and\n0.6% on SQAI, respectively. The results demonstrate that\nsparse experts can deliver superior performance.\nEffect of the Number of Activated Experts. To evaluateTable 7. Ablation study about the model size of MoE-LLaV A.\nModel MoE VQAv2SQAIVQATMMB LLaV AW\nStableLM✗ 74.5 62.0 48.8 58.2 83.2\n✔ 76.7 62.6 50.1 60.2 86.8\nQwen✗ 74.9 60.2 48.3 60.6 86.3\n✔ 76.2 63.1 48.0 59.7 88.7\nPhi-2✗ 75.6 67.8 50.0 65.0 91.3\n✔ 77.6 68.5 51.4 65.2 94.1\nthe effect of the number of activated experts, we compare\nthe performance of using different top- kstrategies. With\nthe number of activated experts changing from 1 to 2, it\nbrings a significant improvement with only 1h training time\nincreasing. These results show that activating more experts\ncan improve the MOE-LLaV A ability. To leverage the ad-\nvantages of the MoE scheme, we set the number of activated\nexperts to 2.\nEffect of the Architectures. In Table 6d, we explore four\nvariations of MoE architecture. Specifically, “First-Half”\nindicates that MoE layers are applied only to the first half of\nthe model while the second half retains the original dense\narchitecture. “Second-Half” means that MoE layers are\nplaced in the second half of the model while the first half\nremains dense. “Interval” represents alternating occurrences\nof MoE layers and dense layers. “All” indicates that all\nlayers are sparse MoE layers. Intuitively, it is expected that\nincorporating all MoE will enhance performance. However,\nusing “All” does not yield better results and results in longer\ntraining times compared to other architectures. Therefore,\nMoE-LLaV A alternates the insertion of MoE layers.\nEffect of the Model Size. As shown in Table 7, we compare\nthe performance of models with different parameter sizes as\nthe foundation models for MoE-LLaV A. For smaller models\nsuch as Phi2-MoE and Qwen-MoE, the performance with\nMoE surpasses that of dense models. We provide additional\nresults in Appendix B.1.\n8MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n5. Conclusion and Future Directions\nIn this work, we propose the MoE-Tuning to adapting the\nMoE architecture to LVLMs, and construct the MoE-based\nspare model MoE-LLaV A, which can find a sparse pathway\nby simultaneously handling image and text features. Our\nframework demonstrates strong ability of multi-modal un-\nderstanding and rich potential for hallucination inhibition,\nachieving comparable performance of LLaV A-1.5-7B with\nonly 3B activated parameters.\nWhile MoE-LLaV A demonstrates competitive capabilities,\nwe observe some difficulties in training stability, particularly\nwith 16-bit float precision. Furthermore, due to the presence\nof multiple experts specializing in different abilities, MoE-\nLLaV A can easily be expanded to handle additional tasks\nsuch as detection, segmentation, generation, or handling\nmore modalities such as video, depth, and thermal.\n9MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nImpact Statements\nBroader Impacts\nWhile MoE-LLaV A holds great potential and application\nvalue in multi-modal understanding, it may also have some\nnegative social impacts:\n•Information credibility: MoE-LLaV A can generate re-\nalistic texts, including false information and misleading\ncontent.\n•Bias and discrimination: The training data for MoE-\nLLaV A often comes from the internet, where various bi-\nases and discriminatory content may exist. If these un-\nequal patterns are learned and amplified by the model,\nthey may be reflected in the generated responses.\n•Social influence: People may become overly reliant on\nMoE-LLaV A for information and problem-solving, in-\nstead of actively thinking and seeking multiple sources\nof information. This can lead to increased dependency,\nreduced autonomy in thinking, and judgment skills.\nReproducibility\nIn Appendix A.2, we have provided a detailed list of all the\ntraining hyperparameters. We have open-sourced all models\nand codes. Reproducibility can be achieved by using the\ncode provided in the materials.\nCompute\nFor the main results, we conducte experiments on 8 A800-\n80G. For the ablation study, we measure the time on 8\nV100-32G.\nLicenses\nThe majority of this project is released under the Apache\n2.0 license.\n•The service is a research preview intended\nfor non-commercial use only, subject to\nthe model License of LLaMA ( https:\n//github.com/facebookresearch/llama/\nblob/main/MODEL_CARD.md ).\n•Terms of Use of the data generated by Ope-\nnAI ( https://openai.com/policies/\nterms-of-use ).\n•Privacy Practices of ShareGPT ( https:\n//chrome.google.com/webstore/\ndetail/sharegpt-share-your-chatg/\ndaiacboceoaocpibfodeljbdfacokfjb ).References\n01-ai. Building the next generation of open-source and\nbilingual llms. https://github.com/01-ai/Yi ,\n2023.\nAlayrac, J.-B., Donahue, J., Luc, P., Miech, A., Barr, I.,\nHasson, Y ., Lenc, K., Mensch, A., Millican, K., Reynolds,\nM., et al. Flamingo: a visual language model for few-shot\nlearning. Advances in Neural Information Processing\nSystems , 35:23716–23736, 2022.\nBaevski, A. and Auli, M. Adaptive input representa-\ntions for neural language modeling. arXiv preprint\narXiv:1809.10853 , 2018.\nBai, J., Bai, S., Chu, Y ., Cui, Z., Dang, K., Deng, X., Fan,\nY ., Ge, W., Han, Y ., Huang, F., et al. Qwen technical\nreport. arXiv preprint arXiv:2309.16609 , 2023a.\nBai, J., Bai, S., Yang, S., Wang, S., Tan, S., Wang, P., Lin, J.,\nZhou, C., and Zhou, J. Qwen-vl: A frontier large vision-\nlanguage model with versatile abilities. arXiv preprint\narXiv:2308.12966 , 2023b.\nBao, H., Wang, W., Dong, L., Liu, Q., Mohammed, O. K.,\nAggarwal, K., Som, S., Piao, S., and Wei, F. Vlmo:\nUnified vision-language pre-training with mixture-of-\nmodality-experts. Advances in Neural Information Pro-\ncessing Systems , 35:32897–32912, 2022.\nBrown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D.,\nDhariwal, P., Neelakantan, A., Shyam, P., Sastry, G.,\nAskell, A., et al. Language models are few-shot learners.\nAdvances in neural information processing systems , 33:\n1877–1901, 2020.\nCha, J., Kang, W., Mun, J., and Roh, B. Honeybee: Locality-\nenhanced projector for multimodal llm. arXiv preprint\narXiv:2312.06742 , 2023.\nChen, J., Guo, L., Sun, J., Shao, S., Yuan, Z., Lin, L., and\nZhang, D. Eve: Efficient vision-language pre-training\nwith masked prediction and modality-aware moe. arXiv\npreprint arXiv:2308.11971 , 2023a.\nChen, J., Zhu, D., Shen, X., Li, X., Liu, Z., Zhang, P., Krish-\nnamoorthi, R., Chandra, V ., Xiong, Y ., and Elhoseiny, M.\nMinigpt-v2: large language model as a unified interface\nfor vision-language multi-task learning. arXiv preprint\narXiv:2310.09478 , 2023b.\nChen, K., Zhang, Z., Zeng, W., Zhang, R., Zhu, F., and\nZhao, R. Shikra: Unleashing multimodal llm’s referential\ndialogue magic. arXiv preprint arXiv:2306.15195 , 2023c.\nChen, L., Li, J., Dong, X., Zhang, P., He, C., Wang, J.,\nZhao, F., and Lin, D. Sharegpt4v: Improving large\nmulti-modal models with better captions. arXiv preprint\narXiv:2311.12793 , 2023d.\n10MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nChen, Z., Wu, J., Wang, W., Su, W., Chen, G., Xing,\nS., Muyan, Z., Zhang, Q., Zhu, X., Lu, L., et al. In-\nternvl: Scaling up vision foundation models and align-\ning for generic visual-linguistic tasks. arXiv preprint\narXiv:2312.14238 , 2023e.\nChiang, W.-L., Li, Z., Lin, Z., Sheng, Y ., Wu, Z., Zhang,\nH., Zheng, L., Zhuang, S., Zhuang, Y ., Gonzalez, J. E.,\net al. Vicuna: An open-source chatbot impressing gpt-4\nwith 90%* chatgpt quality. See https://vicuna. lmsys. org\n(accessed 14 April 2023) , 2023.\nChu, X., Qiao, L., Lin, X., Xu, S., Yang, Y ., Hu, Y ., Wei,\nF., Zhang, X., Zhang, B., Wei, X., et al. Mobilevlm: A\nfast, reproducible and strong vision language assistant for\nmobile devices. arXiv preprint arXiv:2312.16886 , 2023.\nDai, W., Li, J., Li, D., Tiong, A. M. H., Zhao, J., Wang,\nW., Li, B., Fung, P., and Hoi, S. Instructblip: Towards\ngeneral-purpose vision-language models with instruction\ntuning, 2023.\nDeepSeek-AI. Deepseek llm: Scaling open-source\nlanguage models with longtermism. arXiv preprint\narXiv:2401.02954 , 2024.\nDu, Z., Qian, Y ., Liu, X., Ding, M., Qiu, J., Yang, Z.,\nand Tang, J. Glm: General language model pretrain-\ning with autoregressive blank infilling. arXiv preprint\narXiv:2103.10360 , 2021.\nEigen, D., Ranzato, M., and Sutskever, I. Learning fac-\ntored representations in a deep mixture of experts. arXiv\npreprint arXiv:1312.4314 , 2013.\nfalconry. Falcon-180b. https://falconllm.tii.\nae/, 2023.\nFedus, W., Zoph, B., and Shazeer, N. Switch transform-\ners: Scaling to trillion parameter models with simple\nand efficient sparsity. The Journal of Machine Learning\nResearch , 23(1):5232–5270, 2022.\nFlagAI-Open. Aquila2-34b. https://github.com/\nFlagAI-Open/Aquila2 , 2023.\nFu, C., Chen, P., Shen, Y ., Qin, Y ., Zhang, M., Lin, X., Yang,\nJ., Zheng, X., Li, K., Sun, X., Wu, Y ., and Ji, R. Mme:\nA comprehensive evaluation benchmark for multimodal\nlarge language models. arXiv preprint arXiv:2306.13394 ,\n2023.\nGong, T., Lyu, C., Zhang, S., Wang, Y ., Zheng, M., Zhao, Q.,\nLiu, K., Zhang, W., Luo, P., and Chen, K. Multimodal-gpt:\nA vision and language model for dialogue with humans.\narXiv preprint arXiv:2305.04790 , 2023.Gou, Y ., Liu, Z., Chen, K., Hong, L., Xu, H., Li, A., Yeung,\nD.-Y ., Kwok, J. T., and Zhang, Y . Mixture of cluster-\nconditional lora experts for vision-language instruction\ntuning. arXiv preprint arXiv:2312.12379 , 2023.\nGoyal, Y ., Khot, T., Summers-Stay, D., Batra, D., and\nParikh, D. Making the v in vqa matter: Elevating the\nrole of image understanding in visual question answer-\ning. In Proceedings of the IEEE conference on computer\nvision and pattern recognition , pp. 6904–6913, 2017.\nGurari, D., Li, Q., Stangl, A. J., Guo, A., Lin, C., Grauman,\nK., Luo, J., and Bigham, J. P. Vizwiz grand challenge:\nAnswering visual questions from blind people. In Pro-\nceedings of the IEEE conference on computer vision and\npattern recognition , pp. 3608–3617, 2018.\nHendrycks, D. and Gimpel, K. Gaussian error linear units\n(gelus). arXiv preprint arXiv:1606.08415 , 2016.\nHudson, D. A. and Manning, C. D. Gqa: A new dataset for\nreal-world visual reasoning and compositional question\nanswering. In Proceedings of the IEEE/CVF conference\non computer vision and pattern recognition , pp. 6700–\n6709, 2019.\nJacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E.\nAdaptive mixtures of local experts. Neural computation ,\n3(1):79–87, 1991.\nJiang, A. Q., Sablayrolles, A., Mensch, A., Bamford, C.,\nChaplot, D. S., de las Casas, D., Bressand, F., Lengyel,\nG., Lample, G., Saulnier, L., Lavaud, L. R., Lachaux, M.-\nA., Stock, P., Scao, T. L., Lavril, T., Wang, T., Lacroix,\nT., and Sayed, W. E. Mistral 7b, 2023.\nJiang, A. Q., Sablayrolles, A., Roux, A., Mensch, A., Savary,\nB., Bamford, C., Chaplot, D. S., de las Casas, D., Hanna,\nE. B., Bressand, F., Lengyel, G., Bour, G., Lample, G.,\nLavaud, L. R., Saulnier, L., Lachaux, M.-A., Stock, P.,\nSubramanian, S., Yang, S., Antoniak, S., Scao, T. L.,\nGervet, T., Lavril, T., Wang, T., Lacroix, T., and Sayed,\nW. E. Mixtral of experts, 2024.\nKoh, J. Y ., Salakhutdinov, R., and Fried, D. Grounding\nlanguage models to images for multimodal generation.\narXiv preprint arXiv:2301.13823 , 2023.\nKomatsuzaki, A., Puigcerver, J., Lee-Thorp, J., Ruiz,\nC. R., Mustafa, B., Ainslie, J., Tay, Y ., Dehghani, M.,\nand Houlsby, N. Sparse upcycling: Training mixture-\nof-experts from dense checkpoints. arXiv preprint\narXiv:2212.05055 , 2022.\nKudugunta, S., Huang, Y ., Bapna, A., Krikun, M., Lepikhin,\nD., Luong, M.-T., and Firat, O. Beyond distillation: Task-\nlevel mixture-of-experts for efficient inference. arXiv\npreprint arXiv:2110.03742 , 2021.\n11MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nLai, X., Tian, Z., Chen, Y ., Li, Y ., Yuan, Y ., Liu, S., and\nJia, J. Lisa: Reasoning segmentation via large language\nmodel. arXiv preprint arXiv:2308.00692 , 2023.\nLauren c ¸on, H., Saulnier, L., Tronchon, L., Bekman, S.,\nSingh, A., Lozhkov, A., Wang, T., Karamcheti, S., Rush,\nA. M., Kiela, D., Cord, M., and Sanh, V . Obelics: An\nopen web-scale filtered dataset of interleaved image-text\ndocuments, 2023.\nLepikhin, D., Lee, H., Xu, Y ., Chen, D., Firat, O., Huang, Y .,\nKrikun, M., Shazeer, N., and Chen, Z. Gshard: Scaling\ngiant models with conditional computation and automatic\nsharding. arXiv preprint arXiv:2006.16668 , 2020.\nLi, B., Zhang, Y ., Chen, L., Wang, J., Pu, F., Yang, J., Li, C.,\nand Liu, Z. Mimic-it: Multi-modal in-context instruction\ntuning. arXiv preprint arXiv:2306.05425 , 2023a.\nLi, J., Li, D., Xiong, C., and Hoi, S. Blip: Bootstrapping\nlanguage-image pre-training for unified vision-language\nunderstanding and generation. In International Confer-\nence on Machine Learning , pp. 12888–12900. PMLR,\n2022.\nLi, J., Li, D., Savarese, S., and Hoi, S. Blip-2: Boot-\nstrapping language-image pre-training with frozen im-\nage encoders and large language models. arXiv preprint\narXiv:2301.12597 , 2023b.\nLi, X., Yao, Y ., Jiang, X., Fang, X., Meng, X., Fan, S., Han,\nP., Li, J., Du, L., Qin, B., et al. Flm-101b: An open llm\nand how to train it with 100 k budget. arXiv preprint\narXiv:2309.03852 , 2023c.\nLi, Y ., Du, Y ., Zhou, K., Wang, J., Zhao, W. X., and Wen,\nJ.-R. Evaluating object hallucination in large vision-\nlanguage models. arXiv preprint arXiv:2305.10355 ,\n2023d.\nLi, Y ., Hui, B., Yin, Z., Yang, M., Huang, F., and Li, Y .\nPace: Unified multi-modal dialogue pre-training with\nprogressive and compositional experts. arXiv preprint\narXiv:2305.14839 , 2023e.\nLiang, V . W., Zhang, Y ., Kwon, Y ., Yeung, S., and Zou,\nJ. Y . Mind the gap: Understanding the modality gap\nin multi-modal contrastive representation learning. Ad-\nvances in Neural Information Processing Systems , 35:\n17612–17625, 2022.\nLin, B., Zhu, B., Ye, Y ., Ning, M., Jin, P., and Yuan,\nL. Video-llava: Learning united visual representa-\ntion by alignment before projection. arXiv preprint\narXiv:2311.10122 , 2023.\nLiu, F., Lin, K., Li, L., Wang, J., Yacoob, Y ., and Wang, L.\nAligning large multi-modal model with robust instruction\ntuning. arXiv preprint arXiv:2306.14565 , 2023a.Liu, H., Li, C., Li, Y ., and Lee, Y . J. Improved base-\nlines with visual instruction tuning. arXiv preprint\narXiv:2310.03744 , 2023b.\nLiu, H., Li, C., Wu, Q., and Lee, Y . J. Visual instruction\ntuning. arXiv preprint arXiv:2304.08485 , 2023c.\nLiu, Y ., Duan, H., Zhang, Y ., Li, B., Zhang, S., Zhao, W.,\nYuan, Y ., Wang, J., He, C., Liu, Z., et al. Mmbench: Is\nyour multi-modal model an all-around player? arXiv\npreprint arXiv:2307.06281 , 2023d.\nLiu, Z., Hu, H., Lin, Y ., Yao, Z., Xie, Z., Wei, Y ., Ning, J.,\nCao, Y ., Zhang, Z., Dong, L., et al. Swin transformer v2:\nScaling up capacity and resolution. In Proceedings of the\nIEEE/CVF conference on computer vision and pattern\nrecognition , pp. 12009–12019, 2022.\nLiu, Z., He, Y ., Wang, W., Wang, W., Wang, Y ., Chen, S.,\nZhang, Q., Lai, Z., Yang, Y ., Li, Q., et al. Interngpt:\nSolving vision-centric tasks by interacting with chatgpt\nbeyond language. arXiv preprint arXiv:2305.05662 , 3,\n2023e.\nLong, Z., Killick, G., McCreadie, R., and Camarasa, G. A.\nMultiway-adapater: Adapting large-scale multi-modal\nmodels for scalable image-text retrieval. arXiv preprint\narXiv:2309.01516 , 2023.\nLu, P., Mishra, S., Xia, T., Qiu, L., Chang, K.-W., Zhu,\nS.-C., Tafjord, O., Clark, P., and Kalyan, A. Learn to\nexplain: Multimodal reasoning via thought chains for\nscience question answering. Advances in Neural Infor-\nmation Processing Systems , 35:2507–2521, 2022.\nMa, G., Wu, X., Wang, P., and Hu, S. Cot-mote: Explor-\ning contextual masked auto-encoder pre-training with\nmixture-of-textual-experts for passage retrieval. arXiv\npreprint arXiv:2304.10195 , 2023.\nMicrosoft. Phi-2: The surprising power of\nsmall language models. https://www.\nmicrosoft.com/en-us/research/blog/\nphi-2-the-surprising-power-of-small-language-models ,\n2023.\nMustafa, B., Riquelme, C., Puigcerver, J., Jenatton, R., and\nHoulsby, N. Multimodal contrastive learning with limoe:\nthe language-image mixture of experts. Advances in\nNeural Information Processing Systems , 35:9564–9576,\n2022.\nOpenAI. Gpt-4 technical report, 2023.\nPearson, K. Liii. on lines and planes of closest fit to systems\nof points in space. The London, Edinburgh, and Dublin\nphilosophical magazine and journal of science , 2(11):\n559–572, 1901.\n12MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nPenedo, G., Malartic, Q., Hesslow, D., Cojocaru, R., Cap-\npelli, A., Alobeidli, H., Pannier, B., Almazrouei, E., and\nLaunay, J. The refinedweb dataset for falcon llm: out-\nperforming curated corpora with web data, and web data\nonly. arXiv preprint arXiv:2306.01116 , 2023.\nPeng, Z., Wang, W., Dong, L., Hao, Y ., Huang, S., Ma,\nS., and Wei, F. Kosmos-2: Grounding multimodal\nlarge language models to the world. arXiv preprint\narXiv:2306.14824 , 2023.\nPi, R., Gao, J., Diao, S., Pan, R., Dong, H., Zhang, J.,\nYao, L., Han, J., Xu, H., and Zhang, L. K. T. Detgpt:\nDetect what you need via reasoning. arXiv preprint\narXiv:2305.14167 , 2023.\nRadford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G.,\nAgarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J.,\net al. Learning transferable visual models from natural\nlanguage supervision. In International conference on\nmachine learning , pp. 8748–8763. PMLR, 2021.\nRasheed, H., Maaz, M., Shaji, S., Shaker, A., Khan, S.,\nCholakkal, H., Anwer, R. M., Xing, E., Yang, M.-H., and\nKhan, F. S. Glamm: Pixel grounding large multimodal\nmodel. arXiv preprint arXiv:2311.03356 , 2023.\nRiquelme, C., Puigcerver, J., Mustafa, B., Neumann, M.,\nJenatton, R., Susano Pinto, A., Keysers, D., and Houlsby,\nN. Scaling vision with sparse mixture of experts. Ad-\nvances in Neural Information Processing Systems , 34:\n8583–8595, 2021.\nSatar, B., Zhu, H., Zhang, H., and Lim, J. H. Rome: Role-\naware mixture-of-expert transformer for text-to-video re-\ntrieval. arXiv preprint arXiv:2206.12845 , 2022.\nScao, T. L., Fan, A., Akiki, C., Pavlick, E., Ili ´c, S., Hesslow,\nD., Castagn ´e, R., Luccioni, A. S., Yvon, F., Gall ´e, M.,\net al. Bloom: A 176b-parameter open-access multilingual\nlanguage model. arXiv preprint arXiv:2211.05100 , 2022.\nShazeer, N., Mirhoseini, A., Maziarz, K., Davis, A., Le,\nQ., Hinton, G., and Dean, J. Outrageously large neural\nnetworks: The sparsely-gated mixture-of-experts layer.\narXiv preprint arXiv:1701.06538 , 2017.\nShen, S., Yao, Z., Li, C., Darrell, T., Keutzer, K., and He, Y .\nScaling vision-language models with sparse mixture of\nexperts. arXiv preprint arXiv:2303.07226 , 2023.\nSingh, A., Natarajan, V ., Shah, M., Jiang, Y ., Chen, X.,\nBatra, D., Parikh, D., and Rohrbach, M. Towards vqa\nmodels that can read. In Proceedings of the IEEE/CVF\nconference on computer vision and pattern recognition ,\npp. 8317–8326, 2019.Sun, T., Zhang, X., He, Z., Li, P., Cheng, Q., Yan, H., Liu,\nX., Shao, Y ., Tang, Q., Zhao, X., et al. Moss: Train-\ning conversational language models from synthetic data.\narXiv preprint arXiv:2307.15020 , 7, 2023.\nSUSTech-IDEA. Sus-chat: Instruction tuning done\nright. https://github.com/SUSTech-IDEA/\nSUS-Chat , 2023.\nTaori, R., Gulrajani, I., Zhang, T., Dubois, Y ., Li, X.,\nGuestrin, C., Liang, P., and Hashimoto, T. B. Alpaca: A\nstrong, replicable instruction-following model. Stanford\nCenter for Research on Foundation Models. https://crfm.\nstanford. edu/2023/03/13/alpaca. html , 3(6):7, 2023.\nTeam, I. Internlm: A multilingual language model with\nprogressively enhanced capabilities, 2023.\nTeam, S. A. L. Stable lm 2 1.6b. URL\n[https://huggingface.co/stabilityai/\nstablelm-2-1.6b](https://huggingface.\nco/stabilityai/stablelm-2-1.6b) .\nTouvron, H., Lavril, T., Izacard, G., Martinet, X., Lachaux,\nM.-A., Lacroix, T., Rozi `ere, B., Goyal, N., Hambro, E.,\nAzhar, F., et al. Llama: Open and efficient foundation lan-\nguage models. arXiv preprint arXiv:2302.13971 , 2023a.\nTouvron, H., Martin, L., Stone, K., Albert, P., Almahairi,\nA., Babaei, Y ., Bashlykov, N., Batra, S., Bhargava, P.,\nBhosale, S., et al. Llama 2: Open foundation and fine-\ntuned chat models. arXiv preprint arXiv:2307.09288 ,\n2023b.\nWang, G., Cheng, S., Zhan, X., Li, X., Song, S., and Liu, Y .\nOpenchat: Advancing open-source language models with\nmixed-quality data. arXiv preprint arXiv:2309.11235 ,\n2023a.\nWang, J., Meng, L., Weng, Z., He, B., Wu, Z., and Jiang,\nY .-G. To see is to believe: Prompting gpt-4v for better vi-\nsual instruction tuning. arXiv preprint arXiv:2311.07574 ,\n2023b.\nWang, Q., Li, B., Xiao, T., Zhu, J., Li, C., Wong, D. F.,\nand Chao, L. S. Learning deep transformer models for\nmachine translation. arXiv preprint arXiv:1906.01787 ,\n2019.\nWang, W., Bao, H., Dong, L., Bjorck, J., Peng, Z., Liu,\nQ., Aggarwal, K., Mohammed, O. K., Singhal, S., Som,\nS., et al. Image as a foreign language: Beit pretraining\nfor all vision and vision-language tasks. arXiv preprint\narXiv:2208.10442 , 2022.\nWang, W., Chen, Z., Chen, X., Wu, J., Zhu, X., Zeng, G.,\nLuo, P., Lu, T., Zhou, J., Qiao, Y ., et al. Visionllm: Large\nlanguage model is also an open-ended decoder for vision-\ncentric tasks. arXiv preprint arXiv:2305.11175 , 2023c.\n13MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nWang, W., Lv, Q., Yu, W., Hong, W., Qi, J., Wang, Y ., Ji,\nJ., Yang, Z., Zhao, L., Song, X., et al. Cogvlm: Visual\nexpert for pretrained language models. arXiv preprint\narXiv:2311.03079 , 2023d.\nWei, J., Wang, X., Schuurmans, D., Bosma, M., Xia, F.,\nChi, E., Le, Q. V ., Zhou, D., et al. Chain-of-thought\nprompting elicits reasoning in large language models.\nAdvances in Neural Information Processing Systems , 35:\n24824–24837, 2022.\nYang, A., Xiao, B., Wang, B., Zhang, B., Bian, C., Yin,\nC., Lv, C., Pan, D., Wang, D., Yan, D., et al. Baichuan\n2: Open large-scale language models. arXiv preprint\narXiv:2309.10305 , 2023.\nYe, Q., Xu, H., Xu, G., Ye, J., Yan, M., Zhou, Y ., Wang, J.,\nHu, A., Shi, P., Shi, Y ., et al. mplug-owl: Modulariza-\ntion empowers large language models with multimodality.\narXiv preprint arXiv:2304.14178 , 2023.\nYin, S., Fu, C., Zhao, S., Li, K., Sun, X., Xu, T., and Chen,\nE. A survey on multimodal large language models. arXiv\npreprint arXiv:2306.13549 , 2023.\nYu, W., Yang, Z., Li, L., Wang, J., Lin, K., Liu, Z., Wang,\nX., and Wang, L. Mm-vet: Evaluating large multi-\nmodal models for integrated capabilities. arXiv preprint\narXiv:2308.02490 , 2023.\nYuan, Z., Li, Z., and Sun, L. Tinygpt-v: Efficient multi-\nmodal large language model via small backbones. arXiv\npreprint arXiv:2312.16862 , 2023.\nZeng, A., Liu, X., Du, Z., Wang, Z., Lai, H., Ding, M.,\nYang, Z., Xu, Y ., Zheng, W., Xia, X., et al. Glm-130b:\nAn open bilingual pre-trained model. arXiv preprint\narXiv:2210.02414 , 2022.\nZhang, P., Wang, X. D. B., Cao, Y ., Xu, C., Ouyang, L.,\nZhao, Z., Ding, S., Zhang, S., Duan, H., Yan, H., et al.\nInternlm-xcomposer: A vision-language large model for\nadvanced text-image comprehension and composition.\narXiv preprint arXiv:2309.15112 , 2023a.\nZhang, S., Roller, S., Goyal, N., Artetxe, M., Chen, M.,\nChen, S., Dewan, C., Diab, M., Li, X., Lin, X. V .,\net al. Opt: Open pre-trained transformer language models.\narXiv preprint arXiv:2205.01068 , 2022.\nZhang, S., Dong, L., Li, X., Zhang, S., Sun, X., Wang, S.,\nLi, J., Hu, R., Zhang, T., Wu, F., et al. Instruction tuning\nfor large language models: A survey. arXiv preprint\narXiv:2308.10792 , 2023b.\nZhang, X. and Yang, Q. Xuanyuan 2.0: A large chinese\nfinancial chat model with hundreds of billions parameters.InProceedings of the 32nd ACM International Confer-\nence on Information and Knowledge Management , pp.\n4435–4439, 2023.\nZhang, Y ., Zhang, R., Gu, J., Zhou, Y ., Lipka, N., Yang,\nD., and Sun, T. Llavar: Enhanced visual instruction\ntuning for text-rich image understanding. arXiv preprint\narXiv:2306.17107 , 2023c.\nZhao, B., Wu, B., and Huang, T. Svit: Scaling up vi-\nsual instruction tuning. arXiv preprint arXiv:2307.04087 ,\n2023a.\nZhao, Y ., Lin, Z., Zhou, D., Huang, Z., Feng, J., and Kang,\nB. Bubogpt: Enabling visual grounding in multi-modal\nllms. arXiv preprint arXiv:2307.08581 , 2023b.\nZheng, L., Chiang, W.-L., Sheng, Y ., Zhuang, S., Wu, Z.,\nZhuang, Y ., Lin, Z., Li, Z., Li, D., Xing, E., et al. Judging\nllm-as-a-judge with mt-bench and chatbot arena. arXiv\npreprint arXiv:2306.05685 , 2023.\nZhu, D., Chen, J., Shen, X., Li, X., and Elhoseiny, M.\nMinigpt-4: Enhancing vision-language understanding\nwith advanced large language models. arXiv preprint\narXiv:2304.10592 , 2023.\nZhu, J., Zhu, X., Wang, W., Wang, X., Li, H., Wang, X.,\nand Dai, J. Uni-perceiver-moe: Learning sparse gener-\nalist models with conditional moes. Advances in Neural\nInformation Processing Systems , 35:2664–2678, 2022.\nZhu, Y ., Zhu, M., Liu, N., Ou, Z., Mou, X., and Tang,\nJ. Llava-phi: Efficient multi-modal assistant with small\nlanguage model, 2024.\nZoph, B., Bello, I., Kumar, S., Du, N., Huang, Y ., Dean,\nJ., Shazeer, N., and Fedus, W. St-moe: Designing stable\nand transferable sparse expert models. arXiv preprint\narXiv:2202.08906 , 2022.\n14MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nAppendix for MoE-LLaV A\nA. Implementation Details\nA.1. More Model Architecture\nIn Table 8, we present additional variants of the MoE-LLaV A. We introduce how the total parameters is calculated. When\nthe number of activated experts is 2, setting Experts = 2yields the number of activated parameters.\nTotalParameters =Embedding·Width\n+Layers·(4·Width·Width +Width·FFN·FFNFactor + 2·Width )\n+Width +Width·Embedding\n+MoELayers·(Experts−1)·(Width·FFN·FFNFactor + 2·Width )\n+MoELayers·(Width·Experts )(12)\nTable 8. More architecture details of the MoE-LLaV A model. “FFN Factor“ represents the number of linear layers in the FFN. “*”\ndonates the dimension of the hidden states for the keys (k) and values (v) is 1024. “1.6B ×4-Top2” represents a dense foundation model\nwith 1.6B parameters, which will be equipped with a total of four experts, with two of them being activated. “ †” donates all layers will\nequipped with MoE layer.\nName Experts Top-kMoEEmbedding Width Layers FFNFFNHeadsActivated Total\nLayers Factor Param Param\nStableLM-1.6B (Team) - - - 100352 2560 32 10240 2 32 1.6B 1.6B\nMoE-LLaV A-1.6B×4-Top2 4 2 16 100352 2560 32 10240 2 32 2.0B 2.9B\nMoE-LLaV A-1.6B×4-Top2†4 2 32 100352 2560 32 10240 2 32 2.5B 4.1B\nQwen-1.8B (Bai et al., 2023a) - - - 151936 2048 24 5504 3 16 1.8B 1.8B\nMoE-LLaV A-1.8B×4-Top2 4 2 12 151936 2048 24 5504 3 16 2.2B 3.1B\nMoE-LLaV A-1.8B×4-Top2†4 2 24 151936 2048 24 5504 3 16 2.6B 4.3B\nPhi2-2.7B (Microsoft, 2023) - - - 51200 2560 32 10240 2 32 2.7B 2.7B\nMoE-LLaV A-2.7B×4-Top2 4 2 16 51200 2560 32 10240 2 32 3.6B 5.3B\nMoE-LLaV A-2.7B×4-Top2†4 2 32 51200 2560 32 10240 2 32 4.5B 7.8B\nOpenChat-7B (Wang et al., 2023a) - - - 32000 4096∗32 14336 3 32 6.7B 6.7B\nMoE-LLaV A-7B×4-Top2 4 2 16 32000 4096∗32 14336 3 32 9.6B 15.2B\nMoE-LLaV A-7B×4-Top2†4 2 32 32000 4096∗32 14336 3 32 12.4B 23.7B\nA.2. Training DetailsTable 9. Training hyperparameters.\nConfig Stage I Stage II Stage III\nExperts - - 4\nTop-k - - 2\nDeepspeed Zero2 Zero2 Zero2 offload\nData LLaV A-PT Hybird-PT LLaV A-FT\nImage resolution 336×336\nImage encoder CLIP-Large/336\nFeature select layer -2\nImage projector 2 Linear layers with GeLU\nEpoch 1\nLearning rate 1e-3 2e-5 2e-5\nLearning rate schdule Cosine\nWeight decay 0.0\nText max length 2048\nBatch size per GPU 32 16 16\nGPU 8 × A800-80G\nPrecision Bf16As shown in Table 9, we present the training hyperparameters\nfor all models, which are applicable to Qwen, StableLM, Phi and\nOpenChat. For the training process in all stages, we consistently\ntrain for 1 epoch, as we find that the models overfit when training\nfor 2 epochs. The batch size for the first stage is 256 and 128\nfor the second and third stages. We use an image resolution of\n336x336 for all three stages. Additionally, for smaller models\nlike Qwen-1.8B, it is feasible to train them on 8 V100-32G\nGPUs. However, during the training process, using fp16 may\nsometimes lead to loss becoming NaN. Since our models are\nsmaller than 7B, we can train them in zero2 mode. However, for\nstage 3, deepspeed temporarily does not support training MoE\narchitecture in zero3 mode. Therefore, we choose zero2 offload\nto further reduce the memory requirements and enable running\non 8 A800-80G GPUs. We enable the gradient checkpoint mode\nfor all training stage.\n15MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nB. Additional Results and Visualization\nB.1. Model ScalingTable 10. Ablation study about the model size of MoE-LLaV A.\nModel MoE VQAv2SQAIVQATMMB LLaV AW\nStableLM✗ 74.5 62.0 48.8 58.2 83.2\n✔ 76.0 62.6 47.8 59.4 85.9\nQwen✗ 74.9 60.2 48.3 60.6 86.3\n✔ 76.2 63.1 48.0 59.7 88.7\nPhi-2✗ 75.6 67.8 50.0 65.0 91.3\n✔ 77.6 68.5 51.4 65.2 94.1\nOpenChat✗ 77.9 69.0 54.7 66.9 89.7\n✔ 78.9 62.8 52.5 65.9 86.3As shown in Table 10, for models smaller than 7B,\nwe demonstrate a strong scale of law. MoE-LLaV A\nexhibits improved performance as the model size\nincreases, as exemplified by StableLM-1.6B, Qwen-\n1.8B, and Phi-2.7B. But surprisingly, the overall\nperformance of OpenChat-MoE is significantly in-\nferior to dense models. We speculate that this may\nbe due to the insufficient data for current multi-\nmodal instruction tuning to support sparse pattern\nlearning in 10B-level models, which should be ad-\ndressed in future work when scaling up to larger\nMoE-LLaV A models.\nB.2. Training Capacity\nFor MoE layers, we employ the Batch Priority Routing (BPR) strategy (Riquelme et al., 2021). This strategy utilizes the\nrouting results to determine which tokens should be dropped, ensuring a more balanced workload among the experts. During\nthe training process, the BPR strategy dynamically adjusts the routing results for each expert based on their capacity. When\nthe tokens assigned to an expert exceed its predefined capacity, the excess tokens are dropped. We conduct a ablation\nstudy on the hyperparameter capacity, as shown in Table 11. Increasing the capacity consistently improves performance for\ndifferent sizes of MoE-LLaV A.\nTable 11. Ablation study about the capacity of MoE-LLaV A. “Res.” represent the input image resolution.∗donates that there is some\noverlap in the training data.\nMethods Res. CapacityImage Question Answering Benchmark Toolkit\nVQAv2GQA VisWiz SQAIVQATPOPE MMB LLaV AWMM-Vet Avg\nMoE-LLaV A-1.6B×4-Top2 3361.5 76.7∗60.3∗36.2 62.6 50.1 85.7 60.2 86.8 26.9 60.6\n1.0 76.0∗60.4∗37.2 62.6 47.8 84.3 59.4 85.9 26.1 59.9\nMoE-LLaV A-2.7B×4-Top2 3361.5 77.6∗61.4∗43.9 68.5 51.4 86.3 65.2 94.1 34.3 64.7\n1.0 77.1∗61.1∗43.4 68.7 50.2 85.0 65.5 93.2 31.1 63.9\nMoE-LLaV A-2.7B×4-Top2 3841.5 79.9∗62.6∗43.7 70.3 57.0 85.7 68.0 97.3 35.9 66.7\n1.0 79.4∗62.7∗42.1 70.3 55.7 85.5 67.9 95.1 33.6 65.8\nB.3. Routing Distributions\nIn this section, we present the routing distributions of MoE-LLaV A-OpenChat-7B ×4-Top2, MoE-LLaV A-Phi-2.7B ×4-Top2,\nMoE-LLaV A-Qwen-1.8B ×4-Top2, and MoE-LLaV A-StableLM-1.6B ×4-Top2 on six benchmarks (ScienceQA-IMG (Lu\net al., 2022), TextVQA (Singh et al., 2019), POPE (Li et al., 2023d), MMBench (Liu et al., 2023d), VisWiz (Gurari et al.,\n2018), MM-Vet (Yu et al., 2023)). These routing distributions are based on the training up to the final checkpoint.\nFor MoE-LLaV A-OpenChat-7B ×4-Top2, it is a truly large model compared to our setting. However, as shown in Ap-\npendix B.1, its performance is not as good as expected. We provide the routing distribution of MoE-LLaV A-OpenChat\nafter sparsification in Figure 7. We can observe that even after three stages of training, the routing distributions of MoE-\nLLaV A-OpenChat and MoE-LLaV A-Phi ( Figure 8) differ significantly. MoE-LLaV A-OpenChat exhibits a relatively\nbalanced distribution overall, in terms of both expert loads and expert preferences for different modalities. On the other\nhand, MoE-LLaV A-Phi, along with other smaller models such as MoE-LLaV A-Qwen and MoE-LLaV A-StableLM, show\nsome specific patterns or, in other words, their distributions are more disordered . For example, (1) in Figure 8, MoE-\nLLaV A-Phi exhibits a prominent expert 3 in layers 17-23, which dominates the majority of the workload. (2) In Figure 9,\nMoE-LLaV A-Qwen shows a strong preference for the image modality in expert 1. (3) In Figure Figure 10, experts 2 and 3\nof MoE-LLaV A-StableLM are actively engaged in the middle layers of the model. We believe this is highly likely due to\nthe insufficient amount of current multimodal fine-tuning data (655k in our setting) to enable sparsification for 10B-level\nmodels, even starting from a well-initialized LVLM.\n16MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(a) ScienceQA-IMG\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (b) TextQA\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(c) POPE\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (d) MMBench\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(e) Viswiz\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (f) MM-Vet\nFigure 7. Distribution of expert loadings and expert preferences on MoE-LLaV A-OpenChat-7B×4-Top2 .\nIn fact, we should reflect on what behavior is expected for a sparse MoE model. Should it exhibit specific patterns for each\nexpert, like MoE-LLaV A-Phi, or should it have similar behavior among the experts, like MoE-LLaV A-OpenChat? If\nwe consider that in a sparse model, the behavior of each expert should be similar at initialization, as they are initialized from\na shared FFN and the router has not yet learned any inductive biases, then if the routing distribution continues to remain\nbalanced as the network learns, it would be similar to the initialization and may lead to confusion in the model. Therefore,\nwe speculate that the lack of sufficient data may be a reason for the poor performance of MoE-LLaV A-OpenChat.\nHowever, due to the current limitations in data and computational resources, we are unable to further explore this, and we\nhope that future work can make progress in this direction.\nAdditionally, we provide more details in Figure 11, Figure 12, Figure 13, and Figure 14.\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(a) ScienceQA-IMG\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (b) TextQA\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(c) POPE\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (d) MMBench\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image\n(e) Viswiz\n135791113151719212325272931\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layerExpert 1\nT ext Image\n135791113151719212325272931\nMoE layerExpert 2\nT ext Image\n135791113151719212325272931\nMoE layerExpert 3\nT ext Image\n135791113151719212325272931\nMoE layerExpert 4\nT ext Image (f) MM-Vet\nFigure 8. Distribution of expert loadings and expert preferences on MoE-LLaV A-Phi-2.7B×4-Top2 .\nB.4. Token Pathways\nIn Figure 11, Figure 12, Figure 13, and Figure 14, we track the paths of each token for MoE-LLaV A-OpenChat-7B ×4-Top2,\nMoE-LLaV A-Phi-2.7B ×4-Top2, MoE-LLaV A-Qwen-1.8B ×4-Top2, and MoE-LLaV A-StableLM-1.6B ×4-Top2, respectively.\nIn general, the overall trends of the token paths align with the analysis in Appendix B.3. The paths of MoE-LLaV A-\nOpenChat-7B ×4-Top2 appear more disorderly and diverse, which is attributed to a more balanced expert assignment. On the\nother hand, MoE-LLaV A-Phi-2.7B ×4-Top2, MoE-LLaV A-Qwen-1.8B ×4-Top2, and MoE-LLaV A-StableLM-1.6B ×4-Top2\neach exhibit their specific patterns.\n17MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(a) ScienceQA-IMG\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (b) TextQA\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(c) POPE\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (d) MMBench\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(e) Viswiz\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (f) MM-Vet\nFigure 9. Distribution of expert loadings and expert preferences on MoE-LLaV A-Qwen-1.8B×4-Top2 .\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(a) ScienceQA-IMG\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (b) TextQA\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(c) POPE\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (d) MMBench\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image\n(e) Viswiz\n1357911131517192123\nMoE layer0%25%50%75%100%PercentageAll experts\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layerExpert 1\nT ext Image\n1357911131517192123\nMoE layerExpert 2\nT ext Image\n1357911131517192123\nMoE layerExpert 3\nT ext Image\n1357911131517192123\nMoE layerExpert 4\nT ext Image (f) MM-Vet\nFigure 10. Distribution of expert loadings and expert preferences on MoE-LLaV A-StableLM-1.6B×4-Top2 .\nB.5. Exhibition Board\nIn Table 12, we present some classic examples using images from LLaV A (Liu et al., 2023c) and LLaV A-1.5 (Liu et al.,\n2023b). We observe that MoE-LLaV A performs comparably to them on these classic images, despite using fewer parameters.\n18MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\nTable 12. Exhibition Board of MoE-LLaV A. MoE-LLaV A demonstrates the ability to detect and answer challenging questions when\nprompted to verify them.\nVisual input example, Tricky Question and Image:\nUser If there are factual errors in the questions, point it out;\nif not, proceed answering the question.\nWhat’s happening in the desert?\nLLaV A-1.5 There are no deserts in the image. The image features\na beach with palm trees, a city skyline, and a large\nbody of water.\nMoE-LLaV A There are no deserts in the image. The scene features\na city with a beach and a large body of water, which is\nmore likely to be a coastal city.\nUser Do you know who drew this painting?\nMoE-LLaV A The painting is a portrait of the famous artist, Leonardo\nda Vinci.\nUser Do you know who drew this painting?\nMoE-LLaV A The painting is a drawing of a dog, and it is not possi-\nble to definitively determine the artist who created it\nbased on the image alone.\nUser What is the name of the man in the photo?\nMoE-LLaV A The man in the photo is named Elon Musk.\nUser What is the name of the man in the photo?\nMoE-LLaV A The man in the photo is named Elon Musk.Visual input example, Common Question:\nUser Can you explain this meme?\nGPT-4 (OpenAI, 2023) This meme is a joke that combines two unrelated\nthings: pictures of the earth from space and chicken\nnuggets. The text of the meme suggests that the image\nbelow is a beautiful picture of the earth from space.\nHowever, the image is actually of chicken nuggets ar-\nranged to vaguely resemble a map of the world. The\nhumor in this meme comes from the unexpected juxta-\nposition of the text and the image. The text sets up an\nexpectation of a majestic image of the earth, but the\nimage is actually something mundane and silly.\nMoE-LLaV A The image features a pan filled with golden-brown\nfried chicken pieces arranged in the shape of a world\nmap. There are numerous chicken pieces spread across\nthe pan, with some pieces located near the center,\nwhile others are positioned towards the edges. The\nchicken pieces are placed in a way that resembles the\ncontinents and countries, creating a visually appealing\nand creative presentation. The pan itself is placed on a\ntable, ready to be served and enjoyed.\nUser What is unusual about this image?\nMoE-LLaV A The unusual aspect of this image is that a man is stand-\ning on a clothesline that is attached to the back of a\nmoving taxi cab. This is not a typical scene, as clothes-\nlines are usually found in residential areas, patios, or\nlaundry rooms, and not on the back of a vehicle. The\nman is holding a shirt while standing on the clothes-\nline, which adds to the unusualness of the situation. It\nis likely that the man is performing some task or trying\nto dry the shirt in an unconventional manner.\nUser Describe the image and where’s the red arrow?\nMoE-LLaV A The image is a graph or chart with a red arrow pointing\nto the bottom. The graph is color-coded, with pink and\ngreen lines, and it appears to be a line graph. The red\narrow is located at the bottom of the graph, indicating\na specific point or value.\n19MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(a) ScienceQA-IMG\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (b) TextQA\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (c) POPE\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(d) MMBench\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (e) Viswiz\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (f) MM-Vet\nFigure 11. Distribution of modalities across different experts on MoE-LLaV A-OpenChat-7B×4-Top2 .\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(a) ScienceQA-IMG\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (b) TextQA\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (c) POPE\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(d) MMBench\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (e) Viswiz\n135791113151719212325272931\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n135791113151719212325272931\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (f) MM-Vet\nFigure 12. Distribution of modalities across different experts on MoE-LLaV A-Phi-2.7B×4-Top2 .\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(a) ScienceQA-IMG\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (b) TextQA\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (c) POPE\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(d) MMBench\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (e) Viswiz\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (f) MM-Vet\nFigure 13. Distribution of modalities across different experts on MoE-LLaV A-Qwen-1.8B×4-Top2 .\n20MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(a) ScienceQA-IMG\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (b) TextQA\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (c) POPE\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4\n(d) MMBench\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (e) Viswiz\n1357911131517192123\nMoE layer idx0%25%50%75%100%PercentageT ext\nExpert 1\nExpert 2Expert 3\nExpert 4\n1357911131517192123\nMoE layer idxImage\nExpert 1\nExpert 2Expert 3\nExpert 4 (f) MM-Vet\nFigure 14. Distribution of modalities across different experts on MoE-LLaV A-StableLM-1.6B×4-Top2 .\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(a) ScienceQA-IMG\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (b) TextQA\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (c) POPE\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(d) MMBench\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (e) Viswiz\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (f) MM-Vet\nFigure 15. Visualization of activated pathways on MoE-LLaV A-OpenChat-7B×4-Top2 .\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(a) ScienceQA-IMG\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (b) TextQA\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (c) POPE\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(d) MMBench\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (e) Viswiz\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (f) MM-Vet\nFigure 16. Visualization of activated pathways on MoE-LLaV A-Phi-2.7B×4-Top2 .\n21MoE-LLaV A: Mixture of Experts for Large Vision-Language Models\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(a) ScienceQA-IMG\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (b) TextQA\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (c) POPE\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(d) MMBench\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (e) Viswiz\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (f) MM-Vet\nFigure 17. Visualization of activated pathways on MoE-LLaV A-Qwen-1.8B×4-Top2 .\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(a) ScienceQA-IMG\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (b) TextQA\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (c) POPE\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers\n(d) MMBench\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (e) Viswiz\n4321Expert idx\nT ext\nT op-1\nT op-2\nOthers\n1 3 5 7 9 11 13 15 17 19 21 23\nMoE layer idx4321Expert idx\nImage\nT op-1\nT op-2\nOthers (f) MM-Vet\nFigure 18. Visualization of activated pathways on MoE-LLaV A-StableLM-1.6B×4-Top2 .\n22" }, { "title": "2401.15961v1.Genuine_entanglement_under_squeezed_generalized_amplitude_damping_channels_with_memory.pdf", "content": "arXiv:2401.15961v1 [quant-ph] 29 Jan 2024Genuine entanglement under squeezed generalized amplitud e\ndamping channels with memory\nMazhar Alia\nDepartment of Electrical Engineering, Faculty of Engineer ing,\nIslamic University Madinah, 107 Madinah, Saudi Arabia\nAbstract\nWe study genuine entanglement among 3-qubits undergoing th rough a noisy process including\ndissipation, squeezing and decoherence. We obtain a genera l solution and analyze the asymptotic\nquantum states. It turns out that most of these asymptotic st ates can be genuinely entangled\ndepending upon parameters of channel, memory parameter, an d parameters of initial states. We\nstudy Greenberger-Horne-Zeilinger (GHZ) states and W stat es, mixed with white noise and deter-\nmine the conditions for them to be genuinely entangled at infi nity. We find that for these mixtures,\nit is possible to start with bi-separable state (with specifi c mixture of white noise as described be-\nlow) and end up with genuine entangled states. However, the m emory parameter µmust be very\nhigh. We find that in contrast to two-qubit case, all three qub it asymptotic states for n→ ∞are\nnot genuinely entangled.\nPACS numbers: 03.65.Yz, 03.65.Ud, 03.67.Mn\naEmail: mazharaliawan@yahoo.com, mazhar.ali@iu.edu.sa\n1I. INTRODUCTION\nQuantum correlations among several particles, not only lead to cou nter-intuitive predic-\ntions but also have key role in future technologies [1]. There is growing interest among\nresearchers to study quantum correlations both in theory and ex periments [2–6]. Entan-\nglement is a precious resource, which is important for several applic ations. One of the big\nchallenge is to preserve quantum correlations in quantum states. Q uantum states interact\nwith their environments and it is well known that such interactions ca n degrade entangle-\nment [7–10]. Many authors have studied the effect of environments on entanglement for\nboth bipartite and multipartite systems [11–30].\nQuantum channels describe the physical situations in which a given qu antum state is\ntransformed into another quantum state as a dynamical process . These processes are re-\ngarded as maps which preserve trace and are also complete positive , which means that\na quantum state undergoing through a quantum channel is again a v alid quantum state.\nOne such channel is amplitude damping channel (ADC) which models sp ontaneous emission\nfrom excited atoms or energy dissipation at zero temperature [31 ]. This channel has been\ngeneralized to model dissipation at finite temperature [32] and calle d generalized amplitude\ndamping (GAD) channel. Another generalization of this model takes squeezing into account\nand the model is called squeezed generalized amplitude damping (SGAD ) channel [33]. The\neffect on qubit-qubit entanglement for this channel was studied [34 ], where it was found that\nthis channel can not preserve entanglement. All these channels a re considered as memory-\nless and effect of channel can be extend-able to Nqubits simply as Φ = Φ⊗N\n1. There are\nsituations where this simplification is not true and Φ /negationslash= Φ⊗N\n1[35] and channel is said to have\nmemory. The effect of memory on spin chains [36], entanglement-enh anced transmission of\nclassical information in Pauli channels [37], classical and quantum ca pacities of correlated\namplitude damping channel [38] and others [39, 40], have been studie d. Recently, qubit-\nqubit entanglement and quantum discord [41] under SGAD channel w ith memory has been\nstudied [42].\nAn array of beam-splitters can be used to model a squeezed reser voir [43]. Laser-cooled\ntrapped ions can also mimic the dynamics of an atom with a squeezed va cuum bath [44].\nAnother method to mimic coupling with squeezed bath via four level at oms driven by weak\nlaser fields [45]. Generalized amplitude damping channels are studied in e xperiments for\n2decoherence and decay of atomic states. [46, 47]\nIn this work, we study effect of memory on genuine entanglement fo r 3-qubit systems,\nwhich to our knowledge has not been studied before. We consider th ree qubits sent by three\nconsecutive use of quantum channel with memory. We determine th e asymptotic states\nfor a most general initial state of three qubits. We find that squee zing parameter mis\nnot present in the asymptotic states. This observation is in agreem ent with the fact that\nsqueezed thermal bath can suppress quantum decoherence [34] but it is unable to preserve\nquantum entanglement [48, 49]. We then analyze these asymptotic s tates for various initial\nstates and find that in most cases, they can be genuinely entangled depending on memory\nand parameter of initial states. We found that even if we start with bi-separable states then\ndepending upon thermal parameter n(must be not very large) and memory parameter µ\n(must be very large), asymptotic states can be genuinely entangle d. Although the degree\nof genuine entanglement for these asymptotis states is quite small but it is an interesting\nfeature of this dynamical process.\nThis paper is organized as follows. In section II, we briefly discuss sq ueezed generalized\namplitude damping channel for qubits and provide the general solut ion for anarbitrary state\nof 3-qubits. We briefly review the concept of genuine entanglement in section III and present\nour main findings for two important families of quantum states. Finally , we conclude our\nwork in section IV.\nII. SQUEEZEDGENERALIZED AMPLITUDE DAMPING CHANNELFORQUBITS\nQuantum theory of damping takes a two-level atom (system) as ha rmonic oscillator in-\nteracting with reservoir (or bath) which can also be treated set of harmonic oscillators. The\nHamiltonian in the interaction picture can be written as [50, 51]\nH=/planckover2pi1/summationdisplay\nkgk/bracketleftbig\nb†\nkσ−e−i(ω−νk)t+σ+bkei(ω−νk)t/bracketrightbig\n, (1)\nwhereσ−=|b/angbracketright/angbracketlefta|andσ+=|a/angbracketright/angbracketleftb|areatomiclowering andraising operators. bk(b†\nk)arebath\nannihilation (creation) operatorsfor each mode. νk=ckaredensity distributed frequencies,\nωis atomic transition frequency and gkare coupling constants. After standard quantum\noptical approximations and tracing over reservoir, we get a maste r equation for system only.\nSqueezed generalized amplitude damping channel is a noisy quantum c hannel in which a\n3qubit interacts with a bath being initially in a squeezed thermal state w ith Markov and\nBorn approximations. The master equation in interaction picture, is given as [50, 51]\ndρ\ndt=−Ω(n+1)\n2/bracketleftbigg\nσ+σ−ρ+ρσ+σ−−2σ−ρσ+/bracketrightbigg\n−Ωn\n2/bracketleftbigg\nσ−σ+ρ\n+ρσ−σ+−2σ+ρσ−/bracketrightbigg\n−Ωm/bracketleftbigg\nσ+ρσ++σ−ρσ−/bracketrightbigg\n. (2)\nnisrelatedwithnumberofthermalphotonsand misthesqueezing parameter. Thecomplete\npositivity demands that for Ω ≥0, we must have m2≤n(n+1). We note that for m= 0,\nwe have qubit interacting with thermal reservoirs, and for n=m= 0, we have (zero-\ntemperature) amplitude damping process (vacuum reservoirs).\nWecanextendthismodelformulti-qubits eitherasuncorrelatedno iseinwhichwehaveto\nsumthese 3termsforeach qubit separatelyandthensolve themas ter equationininteraction\npicture by taking Ω A= ΩB=...ΩN= Ω,nA=nB...nN=nandmA=mB...mN=m.\nThis process is the simplest model for memoryless quantum channel and the stochastic map\nΦ(ρ) can be extend-able to Φ⊗N(ρ) forN-systems or uses of quantum channel. The result is\nthat Kraus operators also have structure K=KA⊗KB⊗...KN. For a single qubit under\nsuch uncorrelated noise, the Kraus operators [52] can be written as\nK1=\nk10\n0k2\n, (3)\nK2=\n0k3\nk40\n, (4)\nK3=\n/radicalbig\ne−Ωt(n+1/2)cosh(Ωtm) 0\n0/radicalbig\ne−Ωt(n+1/2)cosh(Ωtm)\n, (5)\nK4=\n0/radicalbig\ne−Ωt(n+1/2)sinh(Ωtm)\n/radicalbig\ne−Ωt(n+1/2)sinh(Ωtm) 0\n, (6)\nwhere\nk1=/radicalig\nn\n2n+1+n+1\n2n+1e−2Ωt(n+1/2)−e−Ωt(n+1/2)cosh(Ωtm), (7)\nk2=/radicalig\nn\n2n+1e−2Ωt(n+1/2)+n+1\n2n+1−e−Ωt(n+1/2)cosh(Ωtm), (8)\nk3=/radicalig\nn\n2n+1(1−e−2Ωt(n+1/2))−e−Ωt(n+1/2)sinh(Ωtm) (9)\nk4=/radicalig\nn+1\n2n+1(1−e−2Ωt(n+1/2))−e−Ωt(n+1/2)sinh(Ωtm). (10)\n4The Kraus operators satisfy the normalization condition/summationtext\niK†\ni(t)Ki(t) =I. For three\nqubits, there are 64 such operators, that is, M1=KA\n1KB\n1KC\n1,M2=KA\n1KB\n1KC\n2,...,\nM64=KA\n4KB\n4KC\n4with/summationtext64\ni=1M†\ni(t)Mi(t) =I8. We have omitted the tensor product symbol\nbetween these operators. The stochastic map for SGAD with unco rrelated noise is\nΦu(ρ) =ρu(t) =64/summationdisplay\nj=1MjρM†\nj. (11)\nFor correlated noise Φ( ρ)/negationslash= Φ⊗N(ρ), and for three qubits, we have the correlated version\nof the master equation\ndρ\ndt=−Ω(n+1)\n2/bracketleftbigg\nσ⊗3\n+σ⊗3\n−ρ+ρσ⊗3\n+σ⊗3\n−−2σ⊗3\n−ρσ⊗3\n+/bracketrightbigg\n−Ωn\n2/bracketleftbigg\nσ⊗3\n−σ⊗3\n+ρ\n+ρσ⊗3\n−σ⊗3\n+−2σ⊗3\n+ρσ⊗3\n−/bracketrightbigg\n−Ωm/bracketleftbigg\nσ⊗3\n+ρσ⊗3\n++σ⊗3\n−ρσ⊗3\n−/bracketrightbigg\n, (12)\nwhereσ⊗3\n+=σ+⊗σ+⊗σ+. The stochastic map for the process can be written either in\nterms of Kraus operators (which are not in tensor product forma t like uncorrelated case) or\nsimply as solution of master equation (12). In any case, it can be writ ten as\nΦc(ρ) =ρc(t) =/summationdisplay\njXjρX†\nj, (13)\nwhereXj/negationslash=XA⊗XB⊗...XN. The density matrix for an arbitrary state of 3 qubits has a\nsimple solution in this case\nρc(t) =\nρ11(t)ρ12(t)ρ13(t)ρ14(t)ρ15(t)ρ16(t)ρ17(t)ρ18(t)\nρ21(t)ρ22ρ23ρ24ρ25ρ26ρ27ρ28(t)\nρ31(t)ρ32ρ33ρ34ρ35ρ36ρ37ρ38(t)\nρ41(t)ρ42ρ43ρ44ρ45ρ46ρ47ρ48(t)\nρ51(t)ρ52ρ53ρ54ρ55ρ56ρ57ρ58(t)\nρ61(t)ρ62ρ63ρ64ρ65ρ66ρ67ρ68(t)\nρ71(t)ρ72ρ73ρ74ρ75ρ76ρ77ρ78(t)\nρ81(t)ρ82(t)ρ83(t)ρ84(t)ρ85(t)ρ86(t)ρ87(t)ρ88(t)\n, (14)\n5where\nρ1s(t) =ρ1s(0)e−Ωt(n+1)\n2\nρs8(t) =ρs8(0)e−Ωtn\n2\nρ11(t) =n(ρ11+ρ88)\n2n+1+ρ11+n(ρ11−ρ88)\n2n+1e−(2n+1)Ωt\nρ18(t) =1\n2/bracketleftbig\nρ18+ρ81+e2mΩt(ρ18−ρ81)/bracketrightbig\ne−(n+m+1/2)Ωt\nρ88(t) =(1−e−(2n+1)Ωt)(1+n)ρ11\n2n+1+1+n(1+e−(2n+1)Ωt)ρ88\n2n+1, (15)\nwiths= 2,3,...,7. Finally, the stochastic map for three qubits sent by three conse cutive\nuse of channel with memory can be written as\nρ(t) =µΦc(ρ)(t)+(1−µ)Φu(ρ)(t), (16)\nwhere 0 ≤µ≤1 is degree of channel memory, which means that the noise is correla ted\nwith probability µ. We have the most general solution for the system and we can stud y the\nasymptotic statesby taking t→ ∞. Examples belowrefer tosuch states as ρ(∞). Although,\nindividual matrix elements have quite lengthy expressions, however , it is possible to study\nevolution of entanglement numerically using MATLAB as described in se ction below.\nIII. GENUINE ENTANGLEMENT UNDER SGAD CHANNELS\nIt is appropriate that we first briefly review the ideas about genuine entanglement. Let\nus consider 3-qubits to discuss the ideas with this understanding th at same arguments are\nequally valid for multipartite systems where each subsystem is a qudit . For pure states,\nwe say that a state is fully separable if |ψABC/angbracketright=|ψA/angbracketright ⊗ |ψB/angbracketright ⊗ |ψC/angbracketright. For mixed states,\na state is fully separable if ρ=/summationtext\njpjρA\nj⊗ρB\nj⊗ρC\nj. Bi-separable states can be written as\nρsep\nA|BC=/summationtext\njqj|φj\nA/angbracketright/angbracketleftφj\nA|⊗|φj\nBC/angbracketright/angbracketleftφj\nBC|, with similar construction for ρsep\nB|ACandρsep\nC|AB. Their\nmixtures are also bi-separable ρbs=p1ρsep\nA|BC+p2ρsep\nB|AC+p3ρsep\nC|AB. A multipartite state is\nsaid to be genuinely entangled if it is not fully separable and not bi-sepa rable. We should\nkeep in mind that there are quantum states which have negative par tial transpose (NPT)\nunder each partition but they are bi-separable [3].\nGenuine entanglement can be detected and quantified via positive pa rtial transpose mix-\ntures (PPT mixtures) [53–55]. PPT mixtures are characterized with semidefinite program-\nming (SDP) and this approach can be used to quantify genuine entan glement [53]. For\n6bipartite systems, this procedure is equivalent to negativity [56], so we can call it as genuine\nnegativity. For multi-qubits, it is bounded by E(ρ)≤1 with upper bound for certain pure\nstates [57]. For mixed states it is always less than 1. If this measure is positive then state is\nguaranteed to be genuine entangled, otherwise we are not sure un less some other procedure\nindicates its entanglement properties.\nIt is well known that for 3-qubits there are two inequivalent families o f quantum states,\nnamely,GHZstates andWstates, given as\n|GHZ1/angbracketright=1√\n2(|000/angbracketright+|111/angbracketright),\n|W/angbracketright=1√\n3(|001/angbracketright+|010/angbracketright+|100/angbracketright). (17)\nOther equivalent GHZ states are |GHZ2/angbracketright= 1/√\n2(|001/angbracketright+|110/angbracketright),|GHZ3/angbracketright= 1/√\n2(|010/angbracketright+\n|101/angbracketright),and|GHZ4/angbracketright= 1/√\n2(|011/angbracketright+|100/angbracketright).Wstateequivalent is |˜W/angbracketright= 1/√\n3(|011/angbracketright+|101/angbracketright+\n|110/angbracketright).GHZstateshavemaximumvalueofgenuinenegativity, thatis, E(|GHZj/angbracketright/angbracketleftGHZj|) =\n1, whereas for the Wstate,E(|W/angbracketright/angbracketleftW|) =E(|˜W/angbracketright/angbracketleft˜W|)≈0.886 [53].\nAs a first example, we consider GHZ1state mixed with white noise\nρ1=α|GHZ1/angbracketright/angbracketleftGHZ1|+1−α\n8I8, (18)\nwhere 0≤α≤1. It is known that these states are genuinely entangled for 0 .429≤α≤1\n[53]. An interesting property of SGAD channel with memory is that fo r these states, the\nonly non-zero matrix elements are only on the main diagonal and main o ff-diagonal of the\ndensity matrix whereas all other elements remain zero ( Xstructure) [58]. A result on the\ndetection of genuine entanglement states that the inequality\n|ρ18| ≤√ρ22ρ77+√ρ33ρ66+√ρ44ρ55 (19)\nis satisfied by bi-separable states and the violation implies genuine ent anglement [59]. This\ncriterion is a necessary and sufficient condition for GHZ-diagonal st ates [59]. Application of\nthis result to ρ1(∞) gives us condition that\n3/radicaligg/parenleftbiggn2(1+n)(1−µ)\n(1+2n)3+µ(1−α)/8/parenrightbigg/parenleftbiggn(1+n)2(1−µ)\n(1+2n)3+µ(1−α)/8/parenrightbigg\n≥(n(1+n))3/2α(1−µ)\n(1+2n)3.(20)\n70 0.2 0.5 0.8 10.030.1E(1()) =1\n = 0.95\n = 0.9\nFIG. 1. Genuine entanglement for states at infinity is plotte d against memory parameter µand\nn= 1. As we see that for α= 1 asymptotic states are genuine entangled for µ >0. Ifα <1, then\nsome states are genuinely entangled if we increase paramete rµ.\nThis condition is always satisfied for n→ ∞as it simplifies to µ≤(3−α)/2α. Hence\nall asymptotic states with n→ ∞are not genuinely entangled. However, it is possible\nthat condition is violated for certain values of α,nandµ. Therefore, it is possible to have\nasymptotic states which are genuinely entangled.\nIn Figure (1), we plot genuine entanglement E(ρ) for asymptotic states with n= 1.\nWe observe that as long as memory µ>0 the state ρ1(∞) withα= 1 is always genuinely\nentangled. Formixturewithwhitenoise, ifwetake α= 0.95(dashedline)or α= 0.9(dotted-\ndashed line), the asymptotic states are genuinely entangled for lar ger values of memory µ.\nWe also observe that most of genuine entanglement is lost as asympt otic states have much\nless degree of genuine entanglement. In this case, our initial state s are genuinely entangled\nand asymptotic states may be genuinely entangled as seen in Figure ( 1).\nIn contrast, if we take initial state |GHZ2/angbracketrightmixed with white noise, we expect different\nresults. As we saw in Eq.(14) |GHZ2/angbracketrightlives in decoherence free space, therefore its entan-\nglement is not changing and for µ= 1 andα= 1, we must have the same state at infinity.\nForα <1 andµ <1, we have to analyze the asymptotic states. In Figure (2), genuin e\nentanglement is plotted for asymptotic states with n= 1 for three choices of parameter α.\n80 0.2 0.4 0.6 0.8 10.20.81 E(2 ()) =1\n =0.95\n = 0.9\nFIG. 2. Multipartite entanglement is plotted for asymptoti c states against memory µ. We took\nn= 1 for this plot.\nWe can see that for µ= 1, andα= 1, the states are stationary and genuine entanglement\nis fixed at maximum value of 1. However, as memory µandαare decreased then effects of\nuncorrelated noise degrades the genuine entanglement until for a specific value of memory,\nasymptotic states are no longer genuinely entangled.\nIt turns out that condition for bi-separable states for n→ ∞gives usµ≤6−α\n13α, which is\nnot true for whole range of α. It is violated for α<0.429 and the initial states are genuinely\nentangled for 0 .429≤α≤1. As this result is for very large n, therefore, we expect that it is\npossible to start with bi-separable state and end up with genuine ent angled state, provided\nthat memory µis quite high and nis small. In Figure (3), we observe that all 3 bi-separable\nstates can end up with genuine entangled states for µ≥0.97.\nLet us now take another type of genuine entangled state, namely, Wstate mixed with\nwhite noise, as\nρW= (1−β)|W/angbracketright/angbracketleftW|+β\n8I8, (21)\nwhere 0≤β≤1. We want to mention here that 1 −β=αhas the same meanings and we\ncould have used αinstead ofβ. These states are genuinely entangled for 0 ≤β≤0.521 [53].\nFigure (4) depicts genuine entanglement for asymptotic states wit h initial states as gen-\nuine entangled for β= 0,0.2 and bi-separable for β= 0.522. We observe that even though\n90.95 0.97 0.99 10.005\nE(2()) = 0.428\n = 0.427\n = 0.426\nFIG. 3. Genuine entanglement is plotted against memory µ. This plot is for n= 0.1. We observe\nthat all 3 initially bi-separable states may become genuine entangled as a result of dynamical\nprocess.\ninitial state is bi-separable (for β= 0.522) but asymptotic state is genuinely entangled. We\nobserve that all entangled asymptotic states have very small deg ree of genuine entanglement,\nnevertheless, it is an interesting feature of this channel that it ca n convert bisepaeable states\ninto genuine entangled states.\nIV. CONCLUSIONS AND DISCUSSIONS\nWe have studied genuine entanglement under SGAD channel with mem ory for three\nqubits. We have obtained the most general solution for the system . Based on this solution,\nweareabletoanalyzetheasymptoticstates. Wehavefoundthatin thecaseof |GHZ1/angbracketrightstates\nmixed with white noise and for n→ ∞, all asymptotic states are not genuinely entangled.\nHowever, for small values of nand depending on memory µand initial states, it is possible\nto have genuine entangled asymptotic states. We have also taken in itial biseparable states\nand have found that under certain circumstances the asymptotic states may be genuine\nentangled. We found this possibility not only in GHZstates but also for Wstates mixed\nwith white noise. It is an interesting feature of this dynamical proce ss that it may bring\nbiseparable states to genuine entangled ones. However for this ph enomenon to occur, we\n100 0.2 0.4 0.6 0.8 10.20.9E(W ()) = 0\n = 0.2\n = 0.522\nFIG. 4. Genuine entanglement plotted against memory µwithn= 1 for initially Wstates mixed\nwith white noise.\nmust have very high degree of memory µin the channel. We also observed that squeezing\nparametermis absent only among asymptotic states. It is well known that expec tation\nvalues of certain physical quantities at infinity are function of aver age thermal photons\n/angbracketleftn(ω)/angbracketright[50]. This has a simple interpretation. After a long time, the oscillator in contact\nwith a heat bath gets thermalized, with the same average photon nu mber as the thermal\naverage at oscillator’s frequency. The difference between two-qu bits under SGAD and three\nqubits under similar noise can be summarized as follows. It was found t hat for two-qubits\nsinglet state with correlated noise, squeezing parameter mdoes not effect the dynamics\nof entanglement and quantum discord [42]. In addition some of the tw o-qubit asymptotic\nstates are also not dependent on squeezing parameter m. Third, it is possible that with\nsinglet states as initial states, the asymptotic states with n→ ∞can be entangled even if\ninitial states are separable. For three qubits, we get similar results with some differences.\nThe first main difference is the fact that we can only get genuine enta ngled states at infinity\nifnis not too large, because we observed that all asymptotic states w ithn→ ∞are not\ngenuinely entangled. Similar to two qubits, for correlated noise µ= 1, GHZ state living in\ndecoherence free space is invariant under dynamics. For three qu bits, we have two types of\ninequivalent genuine entangled states, whereas for two qubits, st ates are either entangled or\n11separable.\nACKNOWLEDGMENTS\nThe author is grateful to reviewers for their constructive and me ticulous comments which\nbrought much clarity in the manuscript.\n[1] Wilde M M 2017 Quantum Information Theory (Cambridge Univ. Press)\n[2] Horodecki R et al,2009 Rev. Mod. Phys. 81865\n[3] G¨ uhne O and T´ oth G 2009 Phys. Rep. 4741\n[4] Erhard M, Krenn M and Zeilinger A 2020 Nat. Rev. Phys. 2365\n[5] Friis N, Vitagliano G, Malik M and Huber M 2019 Nat. Rev. Ph ys.172\n[6] Ali M 2023 Quant. Infor. Process 22207\n[7] Yu T and Eberly J H 2002 Phys. Rev. B 66193306\n[8] Yu T and Eberly J H 2003 Phys. Rev. B 68165322\n[9] Yu T and Eberly J H 2004 Phys. Rev. Lett. 93140404\n[10] Eberly J H and Yu T 2007 Science 316555\n[11] D¨ ur W and Briegel H J 2004 Phys. Rev. Lett. 92180403\n[12] Hein M, D¨ ur W and Briegel H J 2005 Phys. Rev. A 71032350\n[13] Aolita L et al.2008 Phys. Rev. Lett. 100080501\n[14] Simon C and Kempe J 2002 Phys. Rev. A 65052327\n[15] Borras A et al.2009 Phys. Rev. A 79022108\n[16] Cavalcanti D et al.2009 Phys. Rev. Lett. 103030502\n[17] Bandyopadhyay S and Lidar D A 2005 Phys. Rev. A 72042339\n[18] Chaves R and Davidovich L 2010 Phys. Rev. A 82052308\n[19] Aolita L et al.2010 Phys. Rev. A 82032317\n[20] Carvalho A R R, Mintert F and Buchleitner A 2004 Phys. Rev . Lett.93230501\n[21] Lastra F, Romero G, Lopez C E, Fran¸ ca Santos M and Retama l J C 2007 Phys. Rev A 75\n062324\n[22] G¨ uhne O, Bodoky F and Blaauboer M 2008 Phys. Rev. A 78060301(R)\n12[23] L´ opez C E, Romero G, Lastra F, Solano E and Retamal J C 200 8 Phys. Rev. Lett. 101080503\n[24] Rau A R P, Ali M and Alber G 2008 EPL 8240002\n[25] Ali M, Alber G and Rau A R P 2009 J. Phys. B: At. Mol. Opt. Phy s.42025501\n[26] Ali M 2010 J. Phys. B: At. Mol. Opt. Phys. 43045504\n[27] Ali M 2010 Phys. Rev. A 81042303\n[28] Ali M and G¨ uhne O 2014 J. Phys. B: At. Mol. Opt. Phys. 47055503\n[29] Weinstein Y S et al.2012 Phys. Rev. A 85032324\n[30] Filippov S N, Melnikov A A and Ziman M 2013 Phys. Rev. A 88062328\n[31] Nielsen M A and Chuang I L 2010 Quantum Computation and Quantum Information (Cam-\nbridge Univ. Press)\n[32] Fujiwara A 2004 Phys. Rev. A 70012317\n[33] Srikanth R and Banerjee S 2008 Phys. Rev. A 77012318\n[34] Banerjee S and Ghosh R 2007 J. Phys. A: Math. Theor. 4013735\n[35] Banaszek K, Dragan A, Wasilewski W and Radzewics C 2004 P hys. Rev. Lett. 92257901\n[36] Plenio M B and Virmani S 2007 Phys. Rev. Lett. 99022108\n[37] Daems D 2007 Phys. Rev. A 76090501\n[38] D’Arrigo A, Benenti G, Falci G and Macchiavello C 2013 Ph ys. Rev. A 88042337\n[39] Caruso F, Giovannetti V, Lupo C and Mancini S 2014 Rev. Mo d. Phys. 861203\n[40] Guo Y N, Fang M F, Wang G Y and Zeng K 2016 Quant. Infor. Proc ess.155129\n[41] Ollivier H and Zurek W H 2001 Phys. Rev. Lett. 88017901\n[42] Jeong Y and Shin H 2019 Scientific Reports 94035\n[43] Kim M S and Imoto N 1995 Phys. Rev. A 522401\n[44] Poyatos J F, Cirac J I and Zoller P 1996 Phys. Rev. Lett. 774728\n[45] L¨ utkenhaus N, Cirac J I and Zoller P 1998 Phys. Rev. A 57548\n[46] Myatt C J, King B E, Turchette Q A, Sackett C A, Kielpinski D, Itano W M, Monroe C and\nWineland D J 2000 Nature 403269\n[47] Turchette Q A, Myatt C J, King B E, Sackett C A, Kielpinski D, Itano W M, Monroe C and\nWineland D J 2000 Phys. Rev. A 62053807\n[48] Wilson D, Lee J and Kim M S 2003 J. Mod. Opt. 501809\n[49] Banerjee S, Ravishankar V and Srikanth R 2010 Ann. Phys. 325816\n[50] Orszag M 2016 Quantum Optics (Springer)\n13[51] Scully M O and Zubairy M S 2001 Quantum Optics (Oxford University Press)\n[52] We figured out that Kraus operators provided in Ref.[42] have typos and also their number is\n6. We provide here only 4 operators satisfying the standard r estrictions on them.\n[53] Jungnitsch B, Moroder T and G¨ uhne O 2011 Phys. Rev. Lett .106190502\n[54] Novo L, Moroder T and G¨ uhne O 2013 Phys. Rev. A 88012305\n[55] Hofmann M, Moroder T and G¨ uhne O 2014 J. Phys. A: Math. Th eor.47155301\n[56] Vidal G and Werner R F 2002 Phys. Rev. A 65032314\n[57] Jungnitsch B, Moroder T and G¨ uhne O 2011 Phys. Rev. A 84032310\n[58] Rau A R P 2009 J. Phys. A: Math. Theor 42412002\n[59] G¨ uhne O and Seevinck M 2010 New. J. Phys. 12053002\n14" }, { "title": "2401.16006v1.Hyperbolic_photonic_topological_insulators.pdf", "content": "Hyperbolic photonic topologic al insulators \nLei Huang1, 2*, Lu He1, 2*, Weixuan Zhang1, 2*, #, Huizhen Zhang1, 2, Dong ning Liu3, Xue Feng3, \nFang Liu3, Kaiyu Cui3, Yidong Huang3, 4, Wei Zhang3, 4+ and Xiangdong Zhang1, 2 $ \n1Key Laboratory of advanced optoelectronic quantum architecture and measurements of Ministry of Education, \n2Beijing Key Laboratory of Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing \nInstitute of Technology, 100081 Beijing, China. \n3Frontier Science Center for Quantum Information, Beijing National Research Center for Information Science and \nTechnology (BNRist), Electronic Engineering Department, Tsinghua University, Beijing 100084, Chin a \n4Beijing Academy of Quantum Information Sciences, Beijing 100193, China. \n*These authors contributed equally to this work. $+#Author to whom any correspondence should be addressed: \nzhangxd@bit.edu.cn ; zwei@tsinghua.edu.cn ; zhang wx@bit.edu.cn \n \nAbstract \nTopological photonics provides a new degree of freedom to robustly control electromagnetic \nfields. To date, most of established topological states in photonics hav e been employed in \nEuclidean space. Motivated by unique properties of hyperbolic lattices, which are regular \ntessellations in non -Euclidean space with a constant negative curvature, the boundary -\ndominated hyperbolic topological states have been proposed. H owever, limited by highly \ncrowded boundary resonators and complicated site couplings, the hyperbolic topological \ninsulator has only been experimentally constructed in electric circuits. How to achieve \nhyperbolic photonic topological insulators is still an open question. Here, we report the \nexperimental realization of hyperbolic photonic topological insulators using coupled ring \nresonators on silicon chips. Boundary -dominated one -way edge states with pseudospin -\ndependent propagation directions have been obse rved. Furthermore, the robustness of edge \nstates in hyperbolic photonic topological insulators is also verified. Our findings have potential \napplications in the field of designing high -efficient topological photonic devices with enhanced \nboundary responses . \n \n \nIntroduction . \nEngineering photonic nano/micro -structures with non -trivial topologies has been receiving a lot of \nattention in recent years1,2, and is believed to be a key method for the realization of disorder - and \ndefect -immune photonic devices . Analogy to condensed matter physic al systems, a large number of \nfascinating topological states have been successively proposed in various photonic systems with \ndifferent characteristics. For example, topological photonic insulators and semimetals have been experimental ly realized3-22. Exotic phenomena in topological photonic quasicrystals and fractal \ninsulators have been fulfilled using co upled optical waveguide arrays23,24. In addition, the non -\nHermitian and non -linear t opological states are also widely exploded in photonics25-27, giving \nimportant platforms for the construction of functional devices with robust performances , such as \ntopological lasers28-34, topological frequency combs35 and topological source s of quantum light36. \nHowever, based on the bulk -boundary correspondence, the nth-order topological phase s are always \nfeatured by boundary states with n-dimensional lower than the bulk that hosts them. In this case , the \nvolume of topological channels is much smaller than trivial bulk domain s, limiting the utilization \nefficiency of topological structures in photonics . Hence, if we can construct topologically protected \nedge states in photonic systems with a much larger number of boundary sites than that of bulk sites, \nthe efficien cy of topological photonic device s is expected to be significantly improved. \nOn the other hand, the non -Euclidean geometry widely exists in natural and artificial systems37, \nand plays important roles in different fields . The recent ground -breaking implementations of two -\ndimensional hyperbolic lattices in circuit quantum electrodynamics37 and topolectrical circuits39 \nhave stimulated numerous advances in hyperbolic physics , including the hyperbolic band \ntheory40,41, the crystallography of hyperbolic lattices42, quantum field theories in continuous \nnegatively curved spaces43, the Breitenlohner -Freedman bound on hyperbolic tiling44, highly -\ndegenerated hyperbolic flatbands45,46, many -body hyperbolic models47 and so on48-52. Beyond those \nexotic physic al phenomena , there are many investigations on the construction of hyperbolic \ntopological states53-61. Hyperbolic topological band insulators with non -trivial first/second Chern \nnumbers and hyperbolic graphene have been theoretically creat ed and experimentally realized60,61. \nIn addition, t he robust one -way propagation of boundary -dominated hyperbolic Chern edge states \nwas also fulfille d by electric circuit networks56. It is important to note that boundary sites always \noccupy a finite portion of total site s regardless of the size for the hyperbolic topological lattice. \nHence, if we can construct such boundary -dominated hyperbolic topological state s in photonic \nsystems , the topological photonic devices with enhanced edge responses are expected to be achieved. \nHowever, limited by the requirement of crowded boundary resonators and complicated site \ncouplings, the experimental realization of hyperbolic photonic topological insulators is still lacking. \nIn this work, we give the experi mental demonstrat ion on the realization of hyperbolic photonic \ntopological insulators by coupled optical ring resonator s. We design and fabricate face -centered and vertex -centered hyperbolic topological insulators with non -trivial real -space Chern numbers using \nsilicon photonics . By measuring transmission spectr a and steady -field distributions of hyperbolic \nphotonic topological insulators , the boundary -dominated one -way edge states with pseudospin -\ndependent propagation directions have been observed. Furthermore, the robust edge propagations \nin hyperbolic photonic topological insulators with defects are also verified. Our work may have \npotential applications in designing high-efficient topological photonic devices with significantly reduced \nbulk domains. \n \nResults \nThe theory of hyperbolic photonic topological insulator s. We consider a {6, 4} hyperbolic lattice \nin the Poincaré disk , where the center of a hexagon locates at the origin , as shown in Fig. 1 (a). We \ncall this lattice model as the face -centered hyperbolic lattice. The Schläfli notation {6, 4} represents \nthe tessellation of a 2D hyperbolic plane by 6-sided regular polygons with the coordination number \nbeing 4. For clarity, the lattice site s in the first-, second -, and third -layer are represented by red, blue , \nand pink dots , respectively . The coupling patterns inside all hexagons can be divided into two \ncategories. Nearly a half number of hexagons (marked by triangle s) possess the nearest -neighbor \n(NN) hopping of 𝐽𝑒𝑖𝜑/3, the next-nearest -neighbor ( NNN ) hopping of 𝐽𝑒𝑖𝜑/6 and the next-next-\nnearest -neighbor ( NNNN ) hopping s of 𝐽, as shown in Fig. 1 (b). The remained half of hexagons only \ncontain the NN coupling of 𝐽𝑒𝑖𝜑/3, as shown in Fig. 1 (c). In this case, the hyperbolic lattice model \ncan be effectively described by a tight -binding Hamiltonian as: \n𝐻ˆ=∑𝜔0𝑎𝑖†𝑎𝑖 𝑖 +∑ 𝐽𝑒𝑖𝜑/3𝑎𝑖†𝑎𝑗 <𝑖,𝑗> +∑ 𝐽𝑒𝑖𝜑/6𝑎𝑖†𝑎𝑗 <<𝑖,𝑗>>ℎ𝑎𝑙𝑓+∑ 𝐽𝑎𝑖†𝑎𝑗 <<<𝑖,𝑗>>>ℎ𝑎𝑙𝑓+ℎ.𝑐.. (1) \nwith ai†(ai) being the creation (annihilation) operator at site i. 𝜔0 is the on -site potential of each \nlattice site. The bracket <…> indicates the summation being restricted within all NN sites of the i-th \nsite. Other two brackets <<…>>half and <<<…>>> half correspond to summation s being restricted \nwithin a half number of NNN and NNNN sites of the i-th site. \nIt is noted that the complex -valued NNN couplings can create the staggered flux into a half \nnumber of hexagons, that can break the time-reversal symmetry of the system and introduce non -\ntrivial topologies. The calculate d eigenspectra (𝜀) of the three -layer hyperbolic model with 𝜑 \nequaling to 𝜋 and 𝜋/2 are presented in two left charts of Figs. 1 (d) and 1 (e). Other parameters are set as 𝐽=1 and 𝜔0=0. The color map represents the localization strength of eigenstates on \nlattice sites at the third layer , that is quantized by 𝑉(𝜀𝑛)=∑𝑖∈𝐿=3|𝝓𝑖(𝜀𝑛)|2/∑𝑖∈𝐿=[1,3]|𝝓𝑖(𝜀𝑛)|2 \nwith 𝝓𝑖(𝜀𝑛) being the eigenmode at 𝜀=𝜀𝑛 . It is shown that there are a large number of \neigenstate s exhibit ing boundary -localiz ed spatial profiles . To further determine the topological \nproperties of these edge states, we calculate the corresponding real-space Chern number s shown in \ntwo right charts of Figs. 1(d) and 1 (e). It is shown that the non -zero platform of the real-space Chern \nnumber appear s around the eigenenergy of 𝜀=0 (𝜀=−1) with 𝜑=𝜋 (𝜑=𝜋/2 ), indicating \nthe existence of Chern -class topological edge states in our designed hyperbolic lattices . It is worth \nnoting that the calculated real-space Chern number is much closer to one than that of previous ly \nproposed {6,4} hyperbolic Haldane model with the same number of lattice site s49, showing a good \nsuperiority of our designed hyperbolic topological lattice model. In Supplementary Note 1 , we give \ndetailed numerical results on spatial profiles and robust one -way propagations of hyperbolic \ntopological edge states. The se results clearly show that non-trivial topological edge states exist in \nour designed face-centered hyperbolic topological lattice with suitably engineered staggered flux . \nIn fact, we can design the photonic nano -/micro -structures to realize the control of \nelectromagnetic fields using boundary -dominated hyperbolic topological states. For this purpose, \nwe construct the {6,4} hyperbolic topological lattice by evanescently coupled optic al resonators , \nwhere the schematic diagram of designed optical structure with two layers is shown in Fig. 1 (f). In \nthis structure, optical ring resonators can be divided into the site rings (red and blue blocks in the \nfirst and second layers ) and linking rings (the black blocks) according to their function alities . \nSpecifically , different site rings have exactly the same geometric parameters, ensuring the same \nresonant frequenc y and free spectral range (FSR) of all site rings. The suitably designed linking \nrings are used to couple different site rings to implement required site couplings. Owing to the \naperiodicity of hyperbolic lattice s, the size and coupling pattern of linking rings should be suitably \ndesigned. Here, the small -size linking ring is used to couple two boundary -site rings to simulate NN \nhoppings. In addition, six site rings are coupled by a single large -size linking ring to realize required \nNN, NNN , and NNNN couplings. In this system, we can tune the effective coupling strength J \nbetween two site rings by tuning the separation distance between site rings and the link ring. \nAdditionally, the coupling phases between NN, NNN, and NNNN site rings can be adjusted by \nengineering the phase accumulation of the wave propagating from one site ring to the other through the link ring (See Supplementary Note 2 for detail s). Geometric parameters of large coupling rings \nare illustrated in Fig. 1 (f). Figure 1(g) presents detailed parameters of r adius and width s for site \nrings and small -size linking rings, as well as the distance between linking rings and site rings. \nThrough the appropriate setting of spatial positions and coupling patterns of site rings and linking \nrings , the eigenequation of our designed evanescently coupled ring-resonator array is identical with \nthat of the topological hyperbolic lattice model (See Supplementary Note 2 for detail ed derivation s). \nIn particular , the effective site energy of each resonator is 𝜔0=193 .45 𝑇𝐻𝑧. And, t he amplitude \nand phase of effective site couplings equal to 𝐽=0.05 𝑇𝐻𝑧 and 𝜑=𝜋. In this case, our designed \nface-centered hyperbolic photonic topological insulator (FHPTI) should possess non -trivial edge \nstates around the central frequency in each FSR of site r ings. In addition, it is worth noting that all \neigenmodes of a single ring resonator can be categorized into two decoupled subspaces, namely \nclockwise modes and counterclockwise modes. The coupling strengths between clockwise and \ncounterclockwise mod es are extremely weak in our system. Consequently, we can consider \nclockwise and counterclockwise modes as a pair of decoupled pseudo -spins. In the following, we \ncall these two pseudo -spins as the clockwise pseudo -spin (CPS) and the anticlockwise pseudo -spin \n(APS). Two pseudo -spins can be suitably excited from two different ports (shown in Fig. 1 (f)). \nImportantly, it should be emphasized that the effective NN, NNN, and NNN coupling phases in \ncounterclockwise and clockwise subspaces are conjugate to each oth er. Therefore, the calculated \nChern numbers associated with counterclockwise and clockwise subspaces exhibit opposite signs. \nTo further demonstrate exotic topological effect s, we perform the full -wave simulation of wave \npropagation in FHPTI using finite element method s. It is important to note that the two -dimensional \n(2D) simulation r esults can effectively predict the performance of real 3D optical structures by \nsetting the appropriate optical parameters in the system . Here, the effective refract ive index of the \noptical waveguide (environment) is set as 2.832 (1.2) to simulate the 3D silicon waveguides \nembedding into the background of silica (See Supplementary Note 3 for details). Figure s 1(h) and \n1(i) display the simulated transmiss ion spectra by exciting the CPS and APS, respectively. Red and \nblue lines correspond to results from output ports at clockwise and counterclockwise positions with \nrespect to the input port , respectively. Around the central frequency of each FSR (highlight by dark \nregions), there is a large transmissi on platform from the clockwise (counterclockwise ) output port \nunder the CPS (APS) excitation , showing the existence of a pseudospin -dependent topological edge state. Away from the central frequency, a large number of transmissi on peaks appear and \ntransmission s of the clockwise (counterclockwise ) output port under the excitation of CPS (APS) \nare significantly decreased , indicating the excitation trivial bulk eigenmodes . Figures 1(j) and 1(k) \npresent the steady -state distribution s of electric field s by exciting the CPS and APS at 1549 .7 nm \n(equaling to the central wavelength of 2𝜋𝑐0/𝜔0). It is shown that the one-way transport of input \nsignals along the edge with pseudospin -dependent propagation directions appear s, showing key \nbehavior s of topological edge states. For compar ison, we also calculate the steady -state distribution \nof electric fields by exciting the bulk state at 1550 .3 nm, as shown in Fig. 1 (l). It is shown that the \ninput electric fields can permeate into the bulk , and t he bidirectional edge propagation also appears, \nmeaning the excitation of trivial bulk and edge states. These simulation results demonstrate the \ncorrectness on the implement ation of hyperbolic topological in sulators by coupled optical -ring \nresonators . \nExcept for the above design of FHPTIs , in the following, we show that the photonic topological \ninsulators can also be designed in hyperbolic lattices with a vertex locating at the center of the \nPoincaré disk . We call such lattice model as the vertex -centered {6, 4} hyperbolic lattice. Figure \n2(a) displays the vertex -centered hyperbolic topological lattice . Details of the corresponding \ntopological properties are provided in Supplementary Note 4. \nThen , we use evanescently coupled optical ring resonators to construct the vertex -centered \nhyperbolic photonic topological insulator (VHPTI) with three layers , as shown in Fig. 2 (d). We note \nthat the number of site rings (equaling to 113) for the three -layer VHPTI is much larger than that of \nthe two -layer face -centered counterpart (equaling to 48). In addition , the C6 symmetry of face -\ncentered structure is reduced to the C2 symmetry for VHPTI s. Hence , a greater number of large -size \nlinking rings are required to implement the VHPTI . It is worth noting that all parameters of site \nrings and small -size linking rings are identical with that used in the FHPTI . The size s of large linking \nrings are displayed in Fig. 2 (d). In this case , the effective parameters of the VHPTI are still equaling \nto 𝜔0=193 .45 𝑇𝐻𝑧, 𝐽=0.05 𝑇𝐻𝑧 and 𝜑=𝜋. \nThe calculated transmiss ion spectra of the VHPTI under CPS and APS excitations are shown \nin Figs. 2 (e) and 2 (f). Similar to the face -centered counterpart, there is a large transmissi on platform \nof the clockwise (counterclockwise) output port with the CPS (APS) being excited. And, the \ntransmission of clockwise (counterclockwise ) output port is significantly decreased in the frequency range away from the central frequency under the CPS (APS) excitation , corresponding to the \nexcitation of trivial bulk /edge states . The calculate d steady -state distributions of electric field s are \nshown in Figs. 2 (g) and 2 (h) under excitations of CPS and APS at 1549 .7 𝑛𝑚. It is found that the \nelectric field s can unidirectionally propagate along the boundary of the structure , and the \npropagation direction depend s on the input pseudospin . Additionally, during the propagation, the \nelectric field is confined to the boundary and does not penetrate into the bulk. These simulation \nresults clearly prove the existence of unidirectional boundary states in the VHPTI . In contra st, there \nare significant electric fields in the bulk region when the trivial bulk state is excited at 1550 .3 𝑛𝑚, \nas shown in Fig. 2 (i). Full wave simulations clearly prove the correctness of our design. It is \nworthwhile to note that ratios between the number of boundary optical resonators and that of optical \nresonators in the bulk are about 0.875 and 0.8496 for the two-layer FHPTI and three -layer VHPTI . \nThey are much larger than the Euclidean counterparts (0.4375 and 0.3306) wit h same numbers of \nsite rings . Hence, such an enhanced boundary response can improve efficiencies of some next -\ngeneration topological photonic device s. \n \nExperimental observation of hyperbolic photonic topological insulators by evanescently \ncoupled optical resonators. In this part, we experimentally demonstrate the realization of \nhyperbolic photonic topological insulators . The above designed optical structures are fabricated on \na 220 nm-thick Si layer coated on the SiO2 substrate using electron -beam lithography followed by \nplasma etching (See Method for details). To maintain the up -down symmetry of the structure, the \nsample is coated with a layer of SiO2. The microscopy image of the fabricated FHPTI is shown in \nFig. 3(a). The enlarged view is displayed in Fig. 3(b) . It is shown that there is a good consistence \nbetween the fabricated structure and the theoretical d esign . In experiments, to demonstrate \ntopological properties of the sample , we measure the frequency -dependent transmission spectra. \nHere, the CPS and APS can be selectively excited by injecting the optical signal from two ports \n(marked in Fig. 3(a)), respectively . In addition, we measure transmission signals from two output \nports , which are labeled by the clockwise port and the counterclockwise port, to illustrate the one-\nway edge propagation in the sample . More details on the experimental measurements can be found \nin Method . Figure 3(c) displays the measured transmittance spectra with the CPS being excited. Red \nand blue lines correspond to averaged transmittance spectra from clockwise and counterclockwise output ports, respectively. Transparent regions around data lines correspond to the fluctuation s of \nmeasured transmission signal s. It is shown that the measured transmissivity from the clockwise port \nis much larger than that from the counterclockwise port around the central frequency range in each \nFSR (highlight by dark region s), showing the excitation of one-way topological edge states. In \ncontrast, as for the case with the input frequency being away from the central frequency, the \nmeasured transmissivit ies from the clockwise output port is significantly decreased and the number \nof resonant peaks is increased , indicating the excitation trivial bulk/edge eigenmodes . Then, we \nmeasure the transmission spectra of the structure by exciting the APS, as shown in Fig. 3( d). \nContrary to the experimental results of the CPS, the measured transmission from the \ncounterclockwise port is much larger than the clockwise counterpart around the central frequency \nrange in each FSR . These measured transmissi on spectra for the FHPTI are matched to simulation \nresults in Fig. 2 . In addition, due to the large loss effect in the fabricated sample, we find that \nmeasured transmission s and topological frequency regions are lower than simulation counterpart s. \nFigures 3(e) and 3(f) present measured field distribution s of topological edge states under the \nexcitations of CPS and APS at 1550 .7 nm. It is shown that the pseudospin -dependent topological \nedge transports are observed , as highlighted by white arrows . For compar ison, we further measure \nthe field distribution by exciting trivial bulk states at 1551 .4 nm, as shown in Figs. 3(g) . A large \npart of electric fields is penetrate d into the bulk , corresponding to the excitation of trivial bulk states. \nThe above phenomen a clearly manifest the realization of pseudospin -dependent topological edge \nstates in the FHPTL . \nThe microscope image of fabricated VHPTI with three layers is shown in Fig. 4(a) . It is shown \nthat the fabricated sample is consistent with the theoretical design . Similar to the above face -\ncentered condition , we measure transmission spectra by exciting the sample under the CPS and APC , \nas shown in Figs. 4( b) and 4( c). Red and blue lines present measured transmission s from clockwise \nand counterclockwise output ports. It is clearly shown that the pseudospin -dependent one -way edge \nstates around the central frequency range in each FSR also exist in the fabricated VHPTI . \nFurthermore, we measure the field distribution s at 1549 .3 nm under the excitation s of CPS and \nAPS, as shown in Figs. 4( d) and 4( e). It is clearly shown that pseudospin -dependent edge \npropagations appear , as marked by white arrows . The measured field distribution under the \nexcitation of trivial bulk state s at 1550 .2 nm is plotted in Fig. 4(f) , where the significant bulk signals appear. These experimental result s clearly demonstrate the realization of topological edge \nstates in the three -layer VHPTI . \nFinally, we experimentally explore the robustness of topological edge states in the fabricated \nFHPTI and VHPTI . For this purpose, we introduce the defect into the boundary of FHPTI and \nVHPTI, as shown in Figs. 5(a) and 5(b). The defect we studied is caused by the missing of boundary \nring resonators, as enclosed by the red dashed box . Such a type of defect has been widely used to \ndemonstrate the robustness of topological edge state s in many topological photonic \nstructures7,30,35,62,63. We theoretically expect that there is no backscattering of hyperbolic photonic \ntopological edge states around the defect. This can be confirmed by large -valued transmissivities of \ntopological edge states in both defective and defect -free hyperbolic topological structures, along with \nanalyzing their near -field distributions. \nMeasure d transmissio n spectr a of two-layer FHPTI (three -layer VHPTI ) under the excitations \nof CPS and APS are plotted in Figs. 5(c) and 5(d) (Figs. 5(e) and 5(f)), respectively . It is clearly \nshown that the pseudospin -selected large transmission still exists in these structures with defects . In \nparticular, under the excitation of CPS (APS) on these two samples, the measured transmission s \nfrom the clockwise (counterclockwise) port are much larger than that from the counterclockwise \n(clockwise) port in the central frequency range of each FSR sustaining topological edge states (dark \nregions), being consistent with experimental results without defects. Furthermore, near-field \ndistribution s in the FHPTI (VHPTI) under excitation s of CPS and APS are also measured at the \nwavelength of 1550 .1 nm (1549 .1 nm), as shown in Figs. 5(g) and 5(h) (Figs. 5(i) and 5(j)). We \nfind that t he pseudospin -dependent one-way propagations along hyperbolic edges still exist in \nFHPTI and VHPTI with defects , showing robust boundary propagations in these two fabricated \nhyperbolic photonic topological insulators . These experimental results are in a good consistence to \nsimulation results (See Supplementary Note 5 for details) . \n \nDiscussion. In conclusion, we have given the experimen tal demonstrat ion on the realization of \nhyperbolic photonic topological insulators by coupled ring-resonator arrays. Both FHPTI and \nVHPTI have been designed and fabricated on silicon chips. The boundary -dominated optical one-\nway edge states with pseudospin -dependent propagation directions have been observed in those \nsystems. Furthermore, the robust edge propagations in hyperbolic photonic topological insulators with defects have also been verified. This provides a substantial step toward the in vestigations of \nmany other topological photonic states in non -Euclidean space, such as the higher -order topological \nstates and non -Hermitian topological states, which are expected to be realized in the future. Besides \nthe conceptual advantage, it is expected that the boundary -dominated topological edge states persist in \nour designed hyperbolic photonic structure, irrespective of local structural details. Hence, we can achieve \nincreasingly intricate layouts and shapes with boundary -dominated responses. This sho uld have potential \napplications in the field of designing high -efficient topological photonic devices, such as topological \nlasers, topological delay lines, topological quantum circuits, and topological quantum sources. \n \nMethod. \nSample fabrication . The sam ples were fabricated using standard complementary metal -oxide -\nsemiconductor processes and 248 nm deep ultraviolet (DUV) lithography processes. The substrate \nwas a silicon -on-insulator wafer with a 220 -nm-thick top Si layer. First, a thin oxide layer was \nformed on the wafer by thermal oxidation. A 150 -nm-thick polycrystalline silicon (poly -Si) layer \nwas deposited by low -pressure chemical vapor deposition for the coupling grating fabrication . The \nwafer was coated with a positive photoresist. The pattern was created by DUV lithography, and then \nthe pattern was transferred onto the poly -Si layer by double inductively coupled plasma etching \nprocesses. The etching depth for the sample is 220 nm. A special annealing process was performed \nto smooth the sidewall of the device. Subsequently, a layer of 1-μm-thick cladding oxide was \ndeposited by plasma -enhanced chemical vapor deposition. \nExperimental m easurement s. The continuous wave laser (15 00 nm-1630 nm) was employed to \nmeasure the sample in the experiment. The incident light was first coupled to the single -mode fiber \n(SMF). And then we used the polarization controllers to adjust the polarization state of the light . \nLights entered into the chip by the fiber array. The output signals were collected by another SMF of \nthe fiber array and detected by a high -speed optical power monitor. To sweep the wavelength of the \nlaser, we can obtain the transmission spectrum in the whole near -infrared band. In addition , the light \nin the silicon waveguides scatter s in the v ertical direction , thus we can observe the transport of the \nedge states. We used an optical microscopy system and an infrared InGaAs camera to directly image \nthe edge modes and bulk modes. \nIt is noted that the experimentally measured losses of the whole system for the 2 -layer and 3 -layer hyperbolic photonic topological insulators are about ~21 dB and ~26 dB, respectively. In this \ncase, we can determine the insertion loss of the hyperbolic photo nic topological insulators by \nanalyzing the individual insertion losses in each component of the experimental setup , including the \ncoupling between the fiber array and 1D gratings, the on -chip waveguide used for the selectively \nexcitation of APS and CPS mo des, and the propagation loss of fibers. To obtain the coupling loss \nbetween 1D gratings and the fiber array, we measure the transmissivity of a test structure, which \nonly consists of an input 1D coupling grating, a short connecting waveguide (130 μm), and an output \n1D coupling grating. It is noted that the propagation loss of the short waveguide (2 dB/cm ×130 μm) \ncan be ignored compared to the coupling loss between the fiber array and 1D gratings. Hence, the \ntotal coupling loss between input/output gratings and the fiber array is about ~14 dB. Moreover, the \ntotal length of the silicon waveguide, which is used to selectively excite APS or CPS mode, is about \n0.25 cm, and the associated propagation loss is about ~0.5 dB (2 dB/cm× 0.25 cm). Additionally, the \nmeasured propagation loss coming from all fibers is about ~1.5 dB. In this case, the total optical \nloss except for the hyperbolic photonic topological insulators is about ~14 dB+1.5 dB+0.5 dB=16 \ndB. After subtracting the loss of ~16 dB, the insertion losse s for 2 -layer and 3 -layer hyperbolic \nphotonic topological insulators are about ~5 dB and ~10 dB, respectively. \n \nData availability. All data are displayed in the main text and Supplementary Information . \n \nReference. \n1. Lu, L., Joannopoulos, J. D. & Soljačić . M. Topological photonics. Nat. Photon ics 8, 821 -829 \n(2014). \n2. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019). \n3. Haldane , F. D. M. & Raghu, S. Possible realization of directional optical waveguides in \nphotonic crystals with broken ti me-reversal symmetry . Phys. Rev. Lett. 100, 013904 (2008). \n4. Wang, Z., Chong, Y., Joannopoulos, J. et al. Observation of unidirectional backscattering -\nimmune topological electromagnetic states. Nature 461, 772 –775 (2009). \n5. Rechtsman, M. C. et al. Photonic floquet topological insulators . Nature 496, 196 -200 \n(2013). \n6. Khanikaev, A. B. et al. Photonic topological insulators . Nat. Mater . 12, 233 (2013). \n7. Hafezi, M. et al. Imaging topological edge states in silicon photonics . Nat. Photon ics 7, \n1001 -1005 (2013). \n8. Wu, L. H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric \nmaterial . Phys. Rev. Lett. 114, 223901 (2015). \n9. Yang, Y., Gao, Z., Xue, H. et al. Realization of a three -dimensional photonic topological insulator. Nature 565, 622 –626 (2019). \n10. Lu, L. et al. Experimental observation of Weyl points. Science 349, 622-624(2015). \n11. Chen, WJ., Xiao, M. & Chan, C. Photonic crystals possessing multiple Weyl points and the \nexperimental observation of robust surface states. Nat. Commun . 7, 13038 (2016). \n12. Yang , B. et al. Ideal Weyl points and helicoid surface states in artificial photonic crystal \nstructures. Science 359, 1013 -1016(2018). \n13. Cerjan, A., Huang, S., Wang, M. et al. Experimental realization of a Weyl exceptional ring. Nat. \nPhoton ics 13, 623 –628 (2019). \n14. Ma, T. & Shvets, G. All -Si valley -Hall photonic topological insulator. New J. Phys. 18, 025012 \n(2016). \n15. Dong, J. W., Chen, X. D., Zhu, H., Wang, Y. & Zhang, X. Valley photonic crystals for control \nof spin and topology. Nat. materials 16, 298 -302 (2017). \n16. Gao, F., Xue, H., Yang, Z. et al. Topologically protected refraction of robust kink states in \nvalley photonic crystals. Nat. Phys. 14, 140 -144(2018). \n17. Noh, J., Huang, S., Chen, K. P. & Rechtsman, M. C. Observation of photonic topological valley \nHall edge states. Phys. Rev. Lett. 120, 063902(2018). \n18. Xie, B. Y., Su, G. X., Wang, H. F. et al. Visualization of higher -order topological insulating \nphases in two -dimensional dielectric photonic crystals. Phys. Rev. Lett. 122, 233903(2019). \n19. Chen, X. D., Deng, W. M., Shi, F. L. et al. Direct observation of corner states in second -order \ntopological photonic crystal slabs. Phys. Rev. Lett. 122, 233902 (2019) . \n20. Cerjan, A ., Jü rgensen, M., Benalcazar, W. A., Mukherjee, S. & Rechtsman, M. C. Observation \nof a higher -order topological bound state in the continuum. Phys. Rev. Lett. 125, 213901 (2020) . \n21. Li, M., Zhirihin, D., Gorlach, M. et al. Higher -order topological states in ph otonic kagome \ncrystals with long -range interactions. Nat. Photonics 14, 89-94(2020). \n22. Shalaev, M. I., Walasik, W., Tsukernik, A., Xu, Y . & Litchinitser, N. M. Robust topologically \nprotected transport in photonic crystals at telecommunication wavelengths. Nat. Nanotechnol. 14, 31-\n34(2019) . \n23. Bandres, M. A., Rechtsman, M. C. & Segev , M. Topological Photonic Quasicrystals: Fractal \nTopological Spectrum and Protected Transport . Phys. Rev. X 6, 011016 (2016). \n24. Biesenthal , T. et al. Fractal photonic topological insulators. Science 376, 1114 -1119(2022). \n25. Smirnova, D., Leykam, D., Chong, Y. & Kivshar, Y. Nonlinear topological photonics . Appl. \nPhys. Rev. 7, 021306 (2020) . \n26. Ota, Y. et al. Active topological photonics. Nanophotonics 9, 547-567(2020) . \n27. Zhao , H. et al. Non-Hermitian topological light steering. Science 365, 1163 -1166(2019). \n28. Bahari, B. et al. Nonreciprocal lasing in topological c avities of arbitrary geometries. \nScience 358, 636 -640 (2017). \n29. Harari, G. et al. Topological insulator laser: theory . Science 359, eaar4003 (2018). \n30. Bandres, M. A. et al. Topological insulator laser: Experiments . Science 359, eaar4005 \n(2018). \n31. St-Jean, P. et al. Lasing in topological edge states of a one -dimensional lattice . Nat. \nPhoton ics 11, 651 -656 (2017). \n32. Shao, Z. K. et al. A high -performance topological bulk laser based on band -inversion -\ninduced reflection . Nat. Nanotech nol. 15, 67-72 (2020). 33. Zeng Y . et al. Electrically pumped topological laser with valley edge modes . Nature 578, \n246-250 (2020). \n34. Zhang, W., Xie, X., Hao, H. et al. Low-threshold topological nanolasers based on the second -\norder corner state. Light Sci . Appl . 9, 109 (2020). \n35. Mittal, S., Moille, G., Srinivasan, K. et al. Topological frequency combs and nested temporal \nsolitons. Nat. Phys. 17, 1169 –1176 (2021). \n36. Mittal, S., Goldschmidt, E.A. & Hafezi, M. A topological source of quantum light. Nature 561, \n502–506 (2018). \n37. Magnus , W. Noneuclidean tesselations and their groups (Academic Press, New York, 1974). \n38. Kollár, A. J., Fitzpatrick, M. & Houck, A. A. Hyperbolic lattices in circuit quantum \nelectrodynamics . Nature 571, 45 (2019). \n39. Lenggenhager, P. M. et al. Simulating hyperbolic space on a circuit board. Nat. Commun. 13, \n4373 (2022). \n40. Maciejko, J. & Rayan, S. Hyperbolic band theory. Sci. Adv. 7, eabe9170 (2021). \n41. Maciejko, J. & Rayan, S. Automorphic Bloch theorems for hyperbolic lattices. Proc. Natl. \nAcad. Sci. USA 119, e2116869119 (2022). \n42. Boettcher, I. et al. Crystallography of hyperbolic lattices. Phys. Rev. B 105, 125118 (2022). \n43. Boettcher, I., Bienias, P., Belyansky, R., K ollár, A. J., & Gorshkov, A. V. Quantum simulation \nof hyperbolic space with circuit quantum electrodynamics: From graphs to geometry. Phys. Rev. A \n102, 032208 (2020). \n44. Basteiro, P. et al. Breitenlohner -Freedman Bound on Hyperbolic Tilings . Phys. Rev. Lett. 130, \n091604 (2023). \n45. Bzdusek , T., & Maciejko, J. Flat bands and band -touching from real -space topology in \nhyperbolic lattices . Phys. Rev. B 106, 155146 (2022). \n46. Mosseri, R., Vogeler, R. & Vidal, J. Aharonov -Bohm cages, flat bands, and gap labeling in \nhyperbolic tilings . Phys. Rev. B 106, 155120 (2022). \n47. Zhu, X., Guo, J., Breuckmann, N. P., Guo, H. & Feng, S. Quantum phase transitions of \ninteracting bosons on hyperbolic lattices. J. Phys. Condens. Mat ter 33, 335602 (2021). \n48. Kollár, A. J., Fitzpatrick, M., Sarnak, P. & Houck, A. A. Line -graph lattices: Euclidean and \nnon-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Commun. Math. \nPhys. 376, 1909 (2020). \n49. Ikeda, K., Aoki, S. & Matsuki, Y. Hyperbolic band theory under magnetic field and Dirac cones \non a higher genus surface. J. Phys. 33, 485602 (2021). \n50. Basteiro, P., Boettcher, I., Belyansky, R., Kollár, A. J. & Gorshkov, A. V. Circuit quantum \nelectrodynamics in hyperbolic space : from photon bound states to frustrated spin models. Phys. Rev. \nLett. 128, 013601 (2022). \n51. Cheng, N. et al. Band theory and boundary modes of high dimensional representations of \ninfinite hyperbolic lattices. Phys. Rev. Lett. 129, 088002 (2022). \n52. Attar, A. & Boettcher, I. Selberg trace formula and hyperbolic band theory. Phys. Rev. E. 106, \n034114 (2022). \n53. Yu, S., Piao, X. & Park, N. Topological hyperbolic lattices. Phys. Rev. Lett. 125, 053901 (2020). \n54. Urwyler, D. M. et al. Hyperbolic Topological Band Insulator s. Phys. Rev. Lett. 129, 246402 \n(2022). 55. Liu, Z. -R., Hua, C. -B., Peng, T. & Zhou, B. Chern insulator in a hyperbolic lattice . Phys. Rev. \nB 105, 245301 (2022). \n56. Zhang, W., Yuan, H., Sun, N., Sun, H. & Zhang, X. Observation of novel topological states in \nhyperbolic lattices . Nat. Commun. 13, 2937 (2022). \n57. Stegmaier, A., Upreti, L. K., Thomale, R. & Boettcher, I. Universality of Hofstadter butterflies \non hyperbolic lattices. Phys. Rev. Lett. 128, 166402 (2022) . \n58. Liu, Z. R., Hua, C. B., Peng, T., Chen, R. & Zh ou, B. Higher -order topological insulators in \nhyperbolic lattices . Phys. Rev. B 107, 125302 (2023). \n59. Tao, Y.-L. & Xu, Y. Higher -order Topological Hyperbolic Lattices . Phys. Rev. B 107, 184201 \n(2023). \n60. Zhang, W., Di, F., Zheng, X., Sun, H. & Zhang, X. Hyperbolic band topology with non -trivial \nsecond Chern numbers . Nat. Commun. 14, 1083 (2023). \n61. Chen, A. et al. Hyperbolic matter in electrical circuits with tunable complex phases. Nat. \nCommun. 14, 622 (2023). \n62. Mittal, S., Orre, V. V., Leykam, D., Chong, Y. D. & Hafezi, M. Photonic anomalous quantum \nHall effect. Phy. Rev. Lett. 123, 043201(2019). \n63. Dai, T., Ao, Y., Bao, J. et al. Topologically protected quantum entanglement emitters. Nat. \nPhotonics 16 ,248-257 (2022). \n \nAcknowledgements. This work was supported by the National key R & D Program of China \n(2022YFA1404904 , 2018YFB2200400 ) (X. Zhang and W. Zhang) , Young Elite Scientists \nSponsorship Program by CAST (No. 2023QNRC001 ) (W. X. Zhang) , National Natural Science \nFoundation of China (No.91850205, No.12104041 ) (X. Zhang and W. X. Zhang) , and BIT Research \nand Innovation Promoting Project (No. 2022YCXY030 , No.2023YCXY020 ) (L. Huang) . \n \nAuthor contributions statement. L. Huang and W. X. Zhang finished the theoretical scheme and \ndesigned the hyperbolic photonic topological insulator. L. He finished the experiments with the help \nof H. Zhang , D. Liu, X. Feng, F. Liu, K. Cui, Y. Huang , and W. Zhang . W. X. Zhang, L. Huang and \nX. Zhang wrote the manuscript. X. Zhang initiated and designed this research project. \n \nCompeting interests statement . The authors declare no competing interests. \n \nFigure Legends/Captions . \nFigure 1. Theoretical results of hyperbolic photonic topological insulator in the face -centered {6, 4} \nhyperbolic lattice. (a). The illustration of face -centered {6, 4} hyperbolic lattice in the Poincaré disk. \nLattice sites in the first, second , and third layers are represented by red, blue , and pink dots. Here, a half \nnumber of hexagons contain the staggered flux, as marked by triangle s. (b). The coupling pattern in the \nhexagon possessing NN, NNN , and NNNN hoppings. (c). The coupling pattern in the hexagon only \npossessing NN hoppings. (d) and (e). The eigenspectra and real -space Chern numbers of the three -layer \nface-centered hyperbolic model with 𝜑 equaling to π and π/2 , respectively. (f). The schematic \ndiagram of the designed hyperbolic photoni c topological lattice with two layers. (g). The illustration of \ndetailed geometric parameters, including radius and widths of site and small -linking rings and the \ndistance between linking rings and site rings. (h) and (i). Simulated transmission spectra by exciting the \nCPS and APS, respectively. Red and blue lines correspond to results of output ports at clockwise and \ncounterclockwise positions with respect to the input port , respectively. (j) and ( k). Simulation results of \nsteady -state distributions of electric fields by exciting the CPS and APS with the wavelength being \n1549 .7 nm . (l). The steady -state distribution of electric fields with the excitation wavelength being \n1550 .3 nm. \n \n \n \nFigure 2. Theoretical results of hyperbolic photonic topological insulator in the vertex -centered {6, \n4} hyperbolic lattice. (a). The illustration of vertex -centered {6, 4} hyperbolic lattice in the Poincaré \ndisk. Lattice sites in first, second and third layers are represented by red, blue and pink dots. Here, a half \nnumber of hexagons contain the staggered flux, as marked by triangle s. (b). The schematic diagram of \nthe designed vertex -centered hyperbolic photonic topological lattice with three layers. Detailed \ngeometric parameters are presented . (c) and ( d). Simulated transmission spectra by exciting the CPS and \nAPS, respectively. (e) and ( f). Simulation results of steady -state distributions of electric fields by exciting \nthe CPS and APS with the wavelength being 1549 .7 nm. (g). The steady -state distribution of electric \nfields by exciting the trivial bulk states at 1550 .3 𝑛𝑚. \n \n \nFigure 3. Experimental results of the face-centered hyperbolic photonic topological insulator. (a) \nThe microscopy image of the fabricated face-centered hyperbolic sample. (b). The enlarged view of \nthe site optical resonator. (c) and (d). Measured transmission spectra with the CPS and APS being \nexcited. Red and blue lines correspond to transmission spectra from clockwise and counterclockwise \noutput ports, respectively. (e) and (f). Measured field distributions of topological edge states with the \nexcited pseudo -spins being CPS and APS in the face-centered hyperbolic photonic topological \ninsulator . (g). Measured field distributions of trivial bulk states in the face -centered hyperbolic \nphotonic topological insulator . \n \n \nFigure 4. Experimental results of the vortex -centered hyperbolic photonic topological insulator. (a) \nThe microscopy image of the fabricated vortex -centered hyperbolic sample. Because the microscope \nfield of view is relatively small, the whole microscope picture is taken by twice independent \nphotographing. And then we joint them together. (b) and ( c). Measured transmission spectra with the \nCPS and APS being excited. Red and blue lines corre spond to transmission spectra related to clockwise \nand counterclockwise output ports, respectively. ( d) and ( e). Measured field distributions of topological \nedge states with the excited pseudo -spins being CPS and APS in the vortex -centered hyperbolic photonic \ntopological insulator . (f). Measured field distributions of trivial bulk states in the vortex -centered \nhyperbolic photonic topological insulator . \n \n \nFigure 5. Experimental demonstrations on the robustness of hyperbolic photonic topological \ninsulators. (a) and (b). The microscopy image of the fabricated face-centered and vortex -centered \nhyperbolic sample with defects. (c) and (d). The measure transmission spectra of face -centered \nhyperbolic samples by exciting the CPS and APS, respectively. (e) and (f). The measure transmission \nspectra of vertex -centered hyperbolic samples by exciting the CPS and APS, respectively. (g) and (h). \nMeasured field distributions of topological edge states with the excited pseudo -spins being CPS and APS \nin face-centered hyperbolic photonic topological insulator s. (i) and ( j). Measured field distributions of \ntopological edge states with the excited pseudo -spins being the CPS and APS in the vortex -centered \nhyperbolic photonic topological insulator . \n \n" }, { "title": "2401.16007v1.Dissipative_effects_on_the_propagation_of_spin_modes.pdf", "content": "Dissipative effects on the propagation of spin modes\nRajeev Singh,𝑎,𝑏,∗Victor E. Ambrus𝑐,𝑑and Radoslaw Ryblewski𝑒\n𝑎Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook,\nNew York, 11794-3800, USA\n𝑏Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026,\nChina\n𝑐Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Strasse 1, D-60438\nFrankfurt am Main, Germany\n𝑑Department of Physics, West University of Timisoara, Bd. Vasile Pârvan 4, Timisoara 300223, Romania\n𝑒Institute of Nuclear Physics Polish Academy of Sciences, PL 31-342 Kraków, Poland\nE-mail: rajeevofficial24@gmail.com, victor.ambrus@e-uvt.ro,\nradoslaw.ryblewski@ifj.edu.pl\nIn relativistic hydrodynamics with spin, following de Groot–van Leeuwen–van Weert’s energy-\nmomentum and spin tensor definitions, we analyze the propagation of spin degrees of freedom.\nWe deduce an analytical formula for spin wave velocity, finding that it approaches half the speed\nof light in the ultra-relativistic limit. Only transverse degrees of freedom propagate, similar to\nelectromagnetic waves. Additionally, we explore dissipative effects and determine the damping\ncoefficients for Maxwell-Jüttner statistics.\n25th International Spin Symposium (SPIN 2023)\n24-29 September, 2023\n∗Speaker\n©Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2401.16007v1 [hep-ph] 29 Jan 2024Dissipative effects on the propagation of spin modes Rajeev Singh\n1. Introduction\nMeasurementsof Λ(¯Λ)hyperonspinpolarization[1–4]haveignitedsignificantinterestinspin\ndynamics, particularly concerning spin-orbit coupling [5–10]. The polarization is theoretically\nrootedinspin-orbitcouplingaspertheDiracequation,withVilenkin’s1980swork[11]indicating\nachiralflowalongvorticityinaDiracparticlegas[12]. Thischiralflow,linkedtotheaxialanomaly\nand affected by vorticity or electromagnetic (EM) fields, is termed ‘anomalous transport’ [12] and\nled to hydrodynamics with triangle anomalies [13]. However, for massive particles like hyperons,\naxial current conservation is explicitly broken, making the axial chemical potential model less\nsuitable [14]. Models based on spin thermodynamic equilibrium [15–18] have aligned well with\npolarization data, but differential polarization measurements remain unclear [2, 19]. This spurred\nthe integration of spin into hydrodynamics [20, 21], using energy-momentum and spin tensor def-\ninitions by de Groot, van Leeuwen, and van Weert (GLW) [22], with recent developments in this\nformalism [22–30]. This paper analyzes linear perturbations in perfect-fluid spin hydrodynam-\nics [22, 31], finding that spin degrees of freedom in the conservation equation are decoupled from\nthebackgroundfluid,leadingtoseparateanalysesforfluidandspinwavespectra[32]. Thespinten-\nsorlinearizationimpliesvalidityonlyforunpolarizedbackgrounds,yieldingageneralanalyticspin\nwave velocity expression both Maxwell-Jüttner (MJ), revealing 𝑐spin=𝑐/2in the ultra-relativistic\nlimit. Spindegreesoffreedomaredividedintoelectric( 𝐶𝜿)andmagnetic( 𝐶𝝎)parts,analogousto\nEMwaves. Ananalysisofthedissipativepartofthespintensor[24,33]fortheidealMJgasshows\nthat dissipative effects lead to the damping of both the transverse and the longitudinal components\nof the spin tensor [28]. This paper is structured as follows: 1a review of spin hydrodynamics is\npresented in Sec. 2, followed by an analysis of spin polarization perturbations and wave solutions\ninSec.3. DissipationeffectsonthepropagationofthespinwavearediscussedinSec.4. Section5\nsummarizes our conclusions.\n2. Relativistic spin hydrodynamics\nThis section summarizes the GLW-based hydrodynamic framework for spin-1\n2particles with\nmass𝑚, where spin effects are small, not influencing the conservation laws for charge, energy,\nand momentum, but stemming from angular momentum conservation. The conservation laws for\nbaryon current and energy-momentum tensor are outlined as [20, 22, 31].\n𝜕𝛼𝑁𝛼(𝑥)=0, 𝜕𝛽𝑇𝛼𝛽(𝑥)=0, (1)\nwhere𝑁𝛼and𝑇𝛼𝛽are the baryon current and the energy-momentum tensor, respectively. For a\nperfect (non-dissipative) fluid, we have [20]\n𝑁𝛼=N𝑈𝛼, 𝑇𝛼𝛽=E𝑈𝛼𝑈𝛽−PΔ𝛼𝛽. (2)\nIn this context,Nrepresents the baryon charge density, Ethe energy density, and Pthe pressure.\nThefluid’sfour-velocityisindicatedby 𝑈𝜇,andΔ𝛼𝛽=𝑔𝛼𝛽−𝑈𝛼𝑈𝛽servesastheprojectorontothe\n1WeadopttheMinkowskimetric 𝑔𝜇𝜈=diag(+1,−1,−1,−1)andnatural(Planck)units,with 𝑐=ℏ=𝑘𝐵=1(unless\nstated otherwise). The dot product of four-vectors 𝑎𝛼and𝑏𝛼is𝑎·𝑏=𝑎𝛼𝑏𝛼=𝑎0𝑏0−𝒂·𝒃, and for the Levi-Civita\ntensor𝜖𝛼𝛽𝛾𝛿, we use𝜖𝑡𝑥𝑦𝑧=+1. Antisymmetrization is denoted by square brackets, 𝑀[𝜇𝜈]=1\n2(𝑀𝜇𝜈−𝑀𝜈𝜇).\n2Dissipative effects on the propagation of spin modes Rajeev Singh\nsurfaceorthogonalto 𝑈𝜇. Thesymmetryoftheenergy-momentumtensor(2)necessitatesseparate\nconservation of spin, as per total angular momentum conservation, 𝜕𝛼𝑆𝛼,𝛽𝛾(𝑥)=0[22].\nQuantum effects like non-local collisions can lead to deviations from above conservation\nequation [34–38], potentially causing the spin polarization tensor 𝜔𝜇𝜈(5) to align with the local\nthermalvorticity. However,asthepreciserelaxationequationisstillunknown,sucheffectsarenot\nincluded in our current analysis. At leading order, the spin tensor is decomposed as [22, 27, 31].\n𝑆𝛼,𝛽𝛾=𝑆𝛼,𝛽𝛾\nph+𝑆𝛼,𝛽𝛾\nΔ, (3a)\nwhere𝑆𝛼,𝛽𝛾\nph(phenomenological) and 𝑆𝛼,𝛽𝛾\nΔ(auxiliary) contributions are [20, 27]\n𝑆𝛼,𝛽𝛾\nph=(A1+A 3)𝑈𝛼𝜔𝛽𝛾, (3b)\n𝑆𝛼,𝛽𝛾\nΔ=(2A1−A 3)𝑈𝛼𝑈𝛿𝑈[𝛽𝜔𝛾]\n𝛿+A 3\u0010\nΔ𝛼𝛿𝑈[𝛽𝜔𝛾]\n𝛿+𝑈𝛼Δ𝛿[𝛽𝜔𝛾]\n𝛿+𝑈𝛿Δ𝛼[𝛽𝜔𝛾]\n𝛿\u0011\n,(3c)\nwith the thermodynamic coefficients [28]\nA1=s2\n9\"\u0012𝜕N\n𝜕𝜉\u0013\n𝛽−2\n𝑚2\u0012𝜕E\n𝜕𝛽\u0013\n𝜉#\n,A3=2s2\n9\"\u0012𝜕N\n𝜕𝜉\u0013\n𝛽+1\n𝑚2\u0012𝜕E\n𝜕𝛽\u0013\n𝜉#\n. (4)\nWeutilizedgeneralformulasfor A1andA3thatarestatistics-independentinourkineticmodel[28].\nFor expressions specific to MJ statistics in an ideal gas, see Ref. [22]. Here, 𝜉=𝜇/𝑇denotes the\nchemicalpotentialtotemperatureratio, 𝛽istheinversetemperature,and s2=𝑠(𝑠+1),whichequals\n3/4for spin-1\n2particles, represents the square of the spin angular momentum magnitude [31].\nAdditionally, we define 𝑧=𝑚/𝑇as the mass to temperature ratio. The antisymmetric spin\npolarization tensor 𝜔𝜇𝜈is given as [20]\n𝜔𝜇𝜈=𝜅𝜇𝑈𝜈−𝜅𝜈𝑈𝜇+𝜖𝜇𝜈𝛼𝛽𝑈𝛼𝜔𝛽. (5)\nIn this structure, 𝜅𝜇and𝜔𝜇comprise six independent components. By design, these four-vectors\nare orthogonal to 𝑈𝜇, satisfying𝜅𝜇𝑈𝜇=𝜔𝜇𝑈𝜇=0,\n𝜅𝜇=𝜔𝜇𝛼𝑈𝛼, 𝜔𝜇=1\n2𝜖𝜇𝛼𝛽𝛾𝜔𝛼𝛽𝑈𝛾, (6)\nwhich, in the rest frame of the fluid, reduce to\n𝜅𝜇=(0,𝐶𝜿)=(0,𝐶𝜅𝑋,𝐶𝜅𝑌,𝐶𝜅𝑍), 𝜔𝜇=(0,𝐶𝝎)=(0,𝐶𝜔𝑋,𝐶𝜔𝑌,𝐶𝜔𝑍), (7)\nwith𝐶𝜿and𝐶𝝎being the spin polarization components [22, 31].\n3. Spin mode dispersion relation\nNow,examiningthepropagationofsmalldisturbancesinafluidwithspindegreesoffreedom,\nwe note that the background fluid’s conservation equations (1) are unaffected by polarization [31],\nyielding the familiar sound wave spectrum [32] that travels at the sound speed\n𝑐2\n𝑠=\u0012𝜕P\n𝜕E\u0013\nN+N\nE+P\u0012𝜕P\n𝜕N\u0013\nE. (8)\n3Dissipative effects on the propagation of spin modes Rajeev Singh\ncspin MJLarge zSmall z\n0.05 0.10 0.50 1 5 10 500.10.20.30.40.5\nzcspin\nνω,||\nνκ,||\nνκ,⟂ν⟂\nνω,⟂\n0.5 1 5 10 50 1000.000.050.100.150.20\nz(1/τR) * Damping coeff.\nFigure 1 – (Left panel) 𝑐spinas a function of 𝑧for MJ statistics together with small and large 𝑧limits [30].\n(Right panel) The damping coefficients’ dependence on 𝑧for longitudinal ( 𝜈𝜔,||,𝜈𝜅,||, beginning at 0.2)\nand transverse ( 𝜈𝜅,⊥,𝜈⊥, and𝜈𝜔,⊥, starting at 0.15) modes, calculated with s2=3/4as per Eqs. (25), (27),\nand (30) [28].\nTurning to excitations within the spin tensor (3a), we assume a stationary background fluid ( 𝑈𝜇=\n𝑔𝑡𝜇) and treat𝜔𝜇𝜈as a small factor, indicating an unpolarized background fluid. Under these\nconditions, Eqs. (3b) and (3c) simplify to\n𝑆𝛼,𝜇𝜈\nph=(A1+A 3)𝑔𝑡𝛼𝜔𝜇𝜈,\n𝑆𝛼,𝜇𝜈\nΔ=2(A1−2A3)𝑔𝑡𝛼𝑔𝑡[𝜇𝜔𝜈]𝑡+A 3(𝑔𝑡[𝜇𝜔𝜈]𝛼+𝑔𝛼[𝜇𝜔𝜈]𝑡−𝑔𝑡𝛼𝜔𝜇𝜈).(9)\nAssuming homogeneity in the 𝑥and𝑦directions, taking the divergence of Eq. (9) results in\n𝜕𝛼𝑆𝛼,𝜇𝜈\nph=(A1+A 3)𝜕𝑡𝜔𝜇𝜈,\n𝜕𝛼𝑆𝛼,𝜇𝜈\nΔ=(2A1−3A3)𝑔𝑡[𝜇𝜕𝑡𝜔𝜈]𝑡+A 3(𝜕[𝜇𝜔𝜈]𝑡−𝜕𝑡𝜔𝜇𝜈+𝑔𝑡[𝜇𝜕𝑧𝜔𝜈]𝑧),(10)\nwhich for𝜇=0,𝜈=𝑖and𝜇=𝑖,𝜈=𝑗, respectively, give\n𝜕𝛼𝑆𝛼,𝑡𝑖=A3\u0012\n𝜕𝑡𝜔𝑡𝑖+1\n2𝜕𝑧𝜔𝑖𝑧\u0013\n, 𝜕𝛼𝑆𝛼,𝑖𝑗=A1𝜕𝑡𝜔𝑖𝑗+A 3𝜕[𝑖𝜔𝑗]𝑡. (11)\nConsidering Eq. (7), the components of 𝜔𝜇𝜈are expressible in terms of 𝐶𝜅𝑖and𝐶𝜔𝑘as\n𝜔𝑡𝑖=−𝐶𝜅𝑖, 𝜔𝑖𝑗=−𝜖𝑡𝑖𝑗𝑘𝐶𝜔𝑘. (12)\nRequiring𝜕𝛼𝑆𝛼,𝜇𝜈=0, we get\n𝜕𝑡𝐶𝜅𝑖−1\n2𝜖𝑡𝑖𝑗𝑧𝜕𝑧𝐶𝜔𝑗=0, 𝜕𝑡𝐶𝜔𝑖−A3\n2A1𝜖𝑡𝑖𝑗𝑧𝜕𝑧𝐶𝜅𝑗=0. (13)\nThepresenceoftheLevi-Civitasymbolresultsin 𝜕𝑡𝐶𝜅𝑍=𝜕𝑡𝐶𝜔𝑍=0,meaningthatthelongitudinal\ncomponents do not propagate. Consequently, polarization propagates solely as transverse waves,\nakin to EM waves. This is derived by considering 𝑖=𝑥,𝑦in Eq. (13), yielding\n\u0012𝜕2\n𝜕𝑡2−𝑐2\nspin𝜕2\n𝜕𝑧2\u0013\nC=0, (14)\n4Dissipative effects on the propagation of spin modes Rajeev Singh\nwithC∈{𝐶𝜅𝑋,𝐶𝜅𝑌,𝐶𝜔𝑋,𝐶𝜔𝑌}and spin wave speed fulfils\n𝑐2\nspin=−1\n4A3\nA1=1\n4(𝜕E/𝜕𝑇)𝜉−𝑧2(𝜕N/𝜕𝜉)𝑇\n(𝜕E/𝜕𝑇)𝜉+𝑧2\n2(𝜕N/𝜕𝜉)𝑇. (15)\nIn𝑧→0(ultra-relativistic)limit, 𝑐spinreducesto 1/2regardlessofstatistics. Theformulafor 𝑐2\nspin\ncan be written for the ideal MJ gas as\n𝑐2\nspin=1\n4𝐾3(𝑧)\n𝐾3(𝑧)+𝑧\n2𝐾2(𝑧), (16)\nthat is independent of chemical potential 𝜉, which for𝑧≪1and for𝑧≫1(non-relativistic limit)\nreduce to\n𝑐spin\f\f\f\n𝑧≪1=1\n2\u0014\n1−𝑧2\n16+𝑂(𝑧4)\u0015\n, 𝑐 spin\f\f\f\n𝑧≫1≃1√\n2𝑧, (17)\nrespectively. Figure 1 shows that 𝑐spinmonotonically decreases with increasing 𝑧as0<𝑐 spin≤1\n2.\nThelowerlimitofthisrangecorrespondstoacold,massiveparticlegas(non-relativisticlimit),and\nthe upper limit applies to high temperatures or massless particles.\n4. Dissipative effects on spin mode propagation\nThis section examines how dissipation affects the propagation of spin modes, drawing on\nthe analysis of dissipative effects from Ref. [33]. Under the relaxation time approximation, the\ndissipative adjustments to 𝑇𝜇𝜈and𝑁𝜇are found to be independent of the spin tensor, with the\ndissipative correction to the spin component expressed as\n𝛿𝑆𝜆,𝜇𝜈=𝜏𝑅(𝐵𝜆,𝜇𝜈\nΠ𝜃+𝐵𝜅𝜆,𝜇𝜈\n𝑛∇𝜅𝜉+𝐵𝜅𝛿𝜆,𝜇𝜈\n𝜋𝜎𝜅𝛿+𝐵𝜂𝛽𝛾𝜆,𝜇𝜈\nΣ∇𝜂𝜔𝛽𝛾), (18)\nwith𝜏𝑅being the relaxation time and ∇𝜇=Δ𝜇𝜈𝜕𝜈=𝜕𝜇−𝑈𝜇𝑈𝜈𝜕𝜈.In this section, we continue\nexamining small perturbations in an unpolarized background in thermal equilibrium. Our focus is\non understanding how dissipation affects these perturbations in spin hydrodynamics and ensuring\nthe theory’s stability. We treat perturbation amplitudes, including 𝛿𝑆𝜆,𝜇𝜈∼𝜔𝜇𝜈, as infinitesimal\nbut allow for large gradient magnitudes proportional to the wave number 𝑘. This approach helps\nidentifypotentialinstabilitiesatsmallwavelengths( 𝑘→∞),thoughit’simportanttonotethatour\nanalysis may not fully capture the physics when 𝜏𝑅≫1and/or𝑘≫1. Referring to Eq. (18), the\ncoefficients𝐵𝜆,𝜇𝜈\nΠ,𝐵𝜅𝜆,𝜇𝜈\n𝑛, and𝐵𝜅𝛿𝜆,𝜇𝜈\n𝜋are linked to the spin polarization tensor 𝜔𝜇𝜈, treated as\nfirst-order relative to perturbation amplitude in an unpolarized background. These coefficients are\nmultipliedbyfirst-ordergradienttermslike 𝜃=𝜕𝜇𝑢𝜇,∇𝜅𝜉,and𝜎𝜅𝛿=1\n2(∇𝜅𝑢𝛿+∇𝛿𝑢𝜅)−1\n3𝜃Δ𝜅𝛿.\nGiven their second-order nature with respect to perturbation amplitude, the first three terms in\nEq. (18) can be disregarded, allowing us to concentrate on the last term [33]\n𝐵𝜂𝛽𝛾𝜆,𝜇𝜈\nΣ=𝐵(1)\nΣΔ𝜆𝜂𝑔𝛽[𝜇𝑔𝜈]𝛾+𝐵(2)\nΣΔ𝜆𝜂𝑢𝛾𝑢[𝜇Δ𝜈]𝛽(19)\n+𝐵(3)\nΣ(Δ𝜆𝜂Δ𝛾[𝜇𝑔𝜈]𝛽+Δ𝜆𝛾Δ𝜂[𝜇𝑔𝜈]𝛽+Δ𝛾𝜂Δ𝜆[𝜇𝑔𝜈]𝛽)+𝐵(4)\nΣΔ𝛾𝜂Δ𝜆[𝜇Δ𝜈]𝛽+𝐵(5)\nΣ𝑢𝛾Δ𝜆𝛽𝑢[𝜇Δ𝜈]𝜂,\n5Dissipative effects on the propagation of spin modes Rajeev Singh\nwhere𝐵(𝑖)\nΣare [28]\n𝐵(1)\nΣ=−4s2cosh𝜉\n3𝐼(1)\n21, 𝐵(2)\nΣ=−8s2cosh𝜉\n3𝑚2 \n𝐼(1)\n41+𝐼(1)\n41𝐼(0)\n31\n𝑚2𝐼(0)\n10−2𝐼(0)\n31!\n, 𝐵(3)\nΣ=−8s2cosh𝜉\n3𝑚2𝐼(1)\n42,\n𝐵(4)\nΣ=−8s2cosh𝜉\n3𝑚2𝐼(1)\n41𝐼(0)\n31\n𝑚2𝐼(0)\n10−(𝐼(0)\n30+𝐼(0)\n31), 𝐵(5)\nΣ=8s2cosh𝜉\n3𝑚2𝐼(1)\n41𝐼(0)\n31\n𝑚2𝐼(0)\n10−2𝐼(0)\n31, (20)\nand𝐼(𝑟)\n𝑛𝑞are thermodynamic integrals. As 𝐴𝜂∇𝜂𝜔𝛽𝛾simplifies to 𝐴𝑧𝜕𝑧𝜔𝛽𝛾, the index𝜂can be\neffectively replaced with 𝑧, hence, we proceed with the splitting\n𝜕𝜆𝛿𝑆𝜆,𝜇𝜈=𝜏𝑅∑︁\n𝑖𝐵(𝑖)\nΣT(𝑖)𝜇𝜈, (21)\nand obtain\nT(1)𝜇𝜈=−𝜕2\n𝑧𝜔𝜇𝜈,T(2)𝜇𝜈=−𝑔𝑡[𝜇𝜕2\n𝑧𝜔𝜈]𝑡,T(3)𝜇𝜈=𝜕2\n𝑧𝜔𝜇𝜈+𝑔𝑡[𝜇𝜕2\n𝑧𝜔𝜈]𝑡+2𝑔𝑧[𝜇𝜕2\n𝑧𝜔𝜈]𝑧,\nT(4)𝜇𝜈=𝑔𝑧[𝜇𝜕2\n𝑧𝜔𝜈]𝑧−𝑔𝑧[𝜇𝑔𝜈]𝑡𝜕2\n𝑧𝜔𝑡𝑧,T(5)𝜇𝜈=𝑔𝑧[𝜇𝑔𝜈]𝑡𝜕2\n𝑧𝜔𝑡𝑧. (22)\nSumming the above terms together, we have\n1\n𝜏𝑅𝜕𝜆𝛿𝑆𝜆,𝜇𝜈=−\u0010\n𝐵(1)\nΣ−𝐵(3)\nΣ\u0011\n𝜕2\n𝑧𝜔𝜇𝜈−\u0010\n𝐵(2)\nΣ−𝐵(3)\nΣ\u0011\n𝑔𝑡[𝜇𝜕2\n𝑧𝜔𝜈]𝑡+2𝐵(3)\nΣ𝑔𝑧[𝜇𝜕2\n𝑧𝜔𝜈]𝑧\n−\u0010\n𝐵(4)\nΣ−𝐵(5)\nΣ\u0011\n𝑔𝑧[𝜇𝑔𝜈]𝑡𝜕2\n𝑧𝜔𝑡𝑧+𝐵(4)\nΣ𝑔𝑧[𝜇𝜕2\n𝑧𝜔𝜈]𝑧. (23)\nWe know that 𝜔𝑡𝑖=−𝐶𝜅𝑖and𝜔𝑖𝑗=−𝜖0𝑖𝑗𝑘𝐶𝜔𝑘, thus, we get\n𝜕𝜆𝛿𝑆𝜆,𝑡𝑥=𝜈𝜅,⊥A3𝜕2\n𝑧𝐶𝜅𝑋, 𝜕 𝜆𝛿𝑆𝜆,𝑦𝑧=𝜈𝜔,⊥A1𝜕2\n𝑧𝐶𝜔𝑋,\n𝜕𝜆𝛿𝑆𝜆,𝑡𝑦=𝜈𝜅,⊥A3𝜕2\n𝑧𝐶𝜅𝑌, 𝜕 𝜆𝛿𝑆𝜆,𝑧𝑥=𝜈𝜔,⊥A1𝜕2\n𝑧𝐶𝜔𝑌,\n𝜕𝜆𝛿𝑆𝜆,𝑡𝑧=𝜈𝜅,||A3𝜕2\n𝑧𝐶𝜅𝑍, 𝜕 𝜆𝛿𝑆𝜆,𝑥𝑦=𝜈𝜔,||A1𝜕2\n𝑧𝐶𝜔𝑍, (24)\nwhereweidentified( 𝜈𝜅,||,𝜈𝜔,||)and(𝜈𝜅,⊥,𝜈𝜔,⊥)aslongitudinalandtransversekinematicviscosi-\nties, respectively,\n𝜈𝜅,||=𝜏𝑅\nA3𝐵(3)\nΣ, 𝜈𝜔,||=𝜏𝑅\nA1\u0010\n𝐵(1)\nΣ−𝐵(3)\nΣ\u0011\n,\n𝜈𝜅,⊥=𝜏𝑅\nA3\u0012\n𝐵(1)\nΣ−1\n2𝐵(2)\nΣ−1\n2𝐵(3)\nΣ\u0013\n, 𝜈𝜔,⊥=𝜏𝑅\nA1\u0012\n𝐵(1)\nΣ−2𝐵(3)\nΣ−1\n2𝐵(4)\nΣ\u0013\n.(25)\nPutting in Eq. (11) reveals that both 𝐶𝜅𝑍and𝐶𝜔𝑍demonstrate exponential decay\n𝜕𝑡𝐶𝜅𝑍−𝜈𝜅,||𝜕2\n𝑧𝐶𝜅𝑍=0, 𝜕𝑡𝐶𝜔𝑍−𝜈𝜔,||𝜕2\n𝑧𝐶𝜔𝑍=0. (26)\nAssuming𝐶𝜅/𝜔;𝑍∼𝑒−𝑖𝜔𝑡+𝑖𝑘𝑧e𝐶𝜅/𝜔;𝑍,where e𝐶𝜅/𝜔;𝑍isaconstant,weobtain 𝜔=−𝑖𝑘2𝜈𝜅/𝜔,||where\n𝜈𝜅,||=4s2𝜏𝑅\n45𝐺(𝑧)[−5𝑧+𝐺(𝑧)(3+𝑧2)−𝑧2Gi(𝑧)]≃4s2𝜏𝑅\n15\u0014\n1−𝑧2\n12+𝑂(𝑧4)\u0015\n,\n𝜈𝜔,||=4s2𝜏𝑅\n15(2𝐺(𝑧)+𝑧)[5𝑧+𝐺(𝑧)(2−𝑧2)+𝑧2Gi(𝑧)]≃4s2𝜏𝑅\n15\u0014\n1−𝑧4\n16+𝑂(𝑧5)\u0015\n,(27)\n6Dissipative effects on the propagation of spin modes Rajeev Singh\nwith𝐺(𝑧)=𝐾3(𝑧)/𝐾2(𝑧)andGi(𝑧)=Ki1(𝑧)/𝐾2(𝑧). Further employing a Fourier decomposition\nC=eC𝑒−𝑖𝜔𝑡+𝑖𝑘𝑧of the transverse modes give\n \n𝜔+𝑖𝑘2𝜈𝜅,⊥−𝑘/2\n𝑘A3\n2A1𝜔+𝑖𝑘2𝜈𝜔,⊥! \ne𝐶𝜅𝑋\ne𝐶𝜔𝑌!\n=0, \n����+𝑖𝑘2𝜈𝜅,⊥𝑘/2\n−𝑘A3\n2A1𝜔+𝑖𝑘2𝜈𝜔,⊥! \ne𝐶𝜅𝑌\ne𝐶𝜔𝑋!\n=0,(28)\nwhere the dispersion relation is\n𝜔±=−𝑖𝑘2𝜈⊥±𝑘𝑐spin, 𝜈⊥=𝜈𝜅,⊥+𝜈𝜔,⊥\n2, 𝑐2\nspin=−A3\n4A1−𝑘2\n4\u0000𝜈𝜅,⊥−𝜈𝜔,⊥\u00012.(29)\nThe damping coefficient, 𝜈⊥, is the average of the individual damping coefficients for 𝜅and𝜔\n𝜈⊥=2s2𝜏𝑅[3𝐺(𝑧)+2𝑧]\n45𝐺(𝑧)[2𝐺(𝑧)+𝑧][−5𝑧+𝐺(𝑧)(3+𝑧2)−𝑧2Gi(𝑧)]≃s2𝜏𝑅\n5\u0014\n1−𝑧2\n24+𝑂(𝑧4)\u0015\n.(30)\nThe spin wave speed undergoes a negative dissipative correction, estimated as 𝑐2\nspin=𝑐2\nspin;0(1−\n𝛿𝑐2\nspin), with𝑐2\nspin;0=−A 3/4A1>0, and\n𝛿𝑐2\nspin=𝑘2(𝜈𝜅,⊥−𝜈𝜔,⊥)2\n−A 3/A 1≃𝑘2s4𝜏2\n𝑅\n8100𝑧4[1+𝑂(𝑧6)], (31)\nwhich is suppressed for small values of 𝑧. However, for finite 𝑧, a sufficiently large wave number\ncan make𝑐2\nspinnegative, occurring when 𝑘surpasses a certain threshold value\n𝑘th=2𝑐spin;0\n|𝜈𝜅,⊥−𝜈𝜔,⊥|. (32)\nWhen𝑘exceeds𝑘th,𝑐spinturns imaginary, halting wave propagation, similar to effects seen in\nfirst-order hydrodynamics of spinless systems. An example is sound modes in ultra-relativistic\nfluids, where 𝜏𝑅𝑘th=5𝜂\n4P𝑘th=15/2[32]. In scenarios where 𝑘≫𝑘th, Eq. (29) indicates that\nstability is maintained provided, 𝜈⊥−1\n2|𝜈𝜅,⊥−𝜈𝜔,⊥|=min(𝜈𝜅,⊥,𝜈𝜔,⊥)>0, which remains true\nfor the formalism studied here. This can be checked analytically for small values of 𝑧, when\n𝜈𝜅,⊥≃s2𝜏𝑅\n5\u0014\n1−𝑧2\n72+𝑂(𝑧4)\u0015\n, 𝜈𝜔,⊥≃s2𝜏𝑅\n5\u0014\n1−5𝑧2\n72+𝑂(𝑧4)\u0015\n, (33)\nand𝜏𝑅𝑘th≃18/(5𝑧2s2). The right panel of Fig. 1 shows that both 𝜈𝜅,⊥and𝜈𝜔,⊥stay positive for\nlarge𝑧, indicating stability against linear perturbations.\nWe now assess the effect of dissipation on spin wave propagation in heavy-ion collisions,\nfocusing on the 𝑧≪1limit. In this case, the shear viscosity 𝜂relates to the relaxation time as\n𝜂=4\n5𝜏𝑅P[32]. Imposing a constant 𝜂/Sratio, whereS=(E+P−𝜇N)/𝑇≈4P/𝑇represents\nthe entropy density (assuming |𝜉|≪1), we find\n𝜏𝑅≃5\n4𝜋2𝑇×(4𝜋𝜂/S). (34)\nPutting s2=3/4, the damping time 𝑡damp;⊥=1/𝑘2𝜈⊥can be computed as\n𝑡damp;⊥≃4𝜆2𝑇/3\n4𝜋𝜂/S=\u0012𝜆\n1 fm\u00132\u0012𝑇\n600 MeV\u0013\n×4 fm/𝑐\n4𝜋𝜂/S, (35)\nwith𝜆=2𝜋/𝑘asthewavelength,indicatingthatthelifespanofspinwavesiscomparabletothatof\nthe QGP fireball.\n7Dissipative effects on the propagation of spin modes Rajeev Singh\n5. Summary\nIn this study, we explored the wave spectrum in spin hydrodynamics using the GLW pseudo-\ngauge, focusing on the antisymmetric tensor 𝜔𝜇𝜈with six independent degrees of freedom, split\ninto three electric and three magnetic components. Our findings highlight the transverse nature of\nspin waves in ideal fluids, where longitudinal components don’t propagate, but transverse ones do,\nsimilar to EM waves. The spin wave speed, 𝑐spin, varies with medium parameters (temperature\n𝑇, chemical potential 𝜇) and particle properties (mass 𝑚, statistics). In the ultra-relativistic limit\n(𝑧=𝑚/𝑇≪1),𝑐spin≃1/2, regardless of statistics. For an ideal MJ gas, 𝑐spinis unaffected by\n𝜉=𝜇/𝑇. Inthelarge 𝑧limit,wefoundthat 𝑐spin∼1/√\n2𝑧forclassical(Maxwell-Jüttner)statistics.\nDissipation effects on spin waves show that all transverse components are damped similarly ( 𝜈⊥),\nwhile longitudinal components decay at different rates ( 𝜈𝜅,||and𝜈𝜔,||). Viscous corrections affect\n𝑐spinsignificantly at high wave numbers, turning imaginary beyond a threshold wavenumber 𝑘th,\nthuspreventingwavepropagation. Thisapproach,focusingon 𝜔𝛼𝛽,doesnotencompassanomalous\ntransportphenomena. Addingvorticaltermsto 𝑁𝛼and𝑇𝛼𝛽altersthefluidsector’swavespectrum,\nintroducing excitations like the chiral magnetic wave [39], chiral vortical wave [40], or helical\nvortical wave [41]. Future research could intriguingly explore the interaction between anomalous\ntransport and spin polarization tensor dynamics.\nAcknowledgements. R.S. acknowledges the support of Polish NAWA Bekker program No.:\nBPN/BEK/2021/1/00342. V.E.A. acknowledges support through a grant of the Ministry of\nResearch, Innovation and Digitization, CNCS - UEFISCDI, project number\nPN-III-P1-1.1-TE-2021-1707, within PNCDI III. This research was also supported in part by the\nPolish National Science Centre Grant No. 2018/30/E/ST2/00432.\nReferences\n[1]STARCollaboration, L. Adamczyk et al., “GlobalΛhyperon polarization in nuclear\ncollisions: evidence for the most vortical fluid,” Nature548(2017) 62–65,\narXiv:1701.06657 [nucl-ex] .\n[2]STARCollaboration, J. Adam et al., “Polarization of Λ(¯Λ) hyperons along the beam\ndirection in Au+Au collisions at√𝑠𝑁𝑁= 200 GeV,” Phys. Rev. Lett. 123no. 13, (2019)\n132301, arXiv:1905.11917 [nucl-ex] .\n[3]ALICECollaboration, S. Acharya et al., “Evidence of Spin-Orbital Angular Momentum\nInteractions in Relativistic Heavy-Ion Collisions,” Phys. Rev. Lett. 125no. 1, (2020) 012301,\narXiv:1910.14408 [nucl-ex] .\n[4]ALICECollaboration, S. Acharya et al., “Global polarization of Λ¯Λhyperons in Pb-Pb\ncollisions at√𝑠𝑁𝑁= 2.76 and 5.02 TeV,” Phys. Rev. C 101no. 4, (2020) 044611,\narXiv:1909.01281 [nucl-ex] .\n[5] Z.-T. Liang and X.-N. Wang, “Globally polarized quark-gluon plasma in non-central A+A\ncollisions,” Phys. Rev. Lett. 94(2005) 102301, arXiv:nucl-th/0410079 [nucl-th] .\n[Erratum: Phys. Rev. Lett.96,039901(2006)].\n[6] Z.-T. Liang and X.-N. Wang, “Spin alignment of vector mesons in non-central A+A\ncollisions,” Phys. Lett. B629(2005) 20–26, arXiv:nucl-th/0411101 [nucl-th] .\n8Dissipative effects on the propagation of spin modes Rajeev Singh\n[7] J.-H. Gao, S.-W. Chen, W.-T. Deng, Z.-T. Liang, Q. Wang, and X.-N. Wang, “Global quark\npolarization in non-central A+A collisions,” Phys. Rev. C77(2008) 044902,\narXiv:0710.2943 [nucl-th] .\n[8] S.-W. Chen, J. Deng, J.-H. Gao, and Q. Wang, “A General derivation of differential\ncross-section in quark-quark scatterings at fixed impact parameter,” Front. Phys. China 4\n(2009) 509–516, arXiv:0801.2296 [hep-ph] .\n[9] D.-L. Wang and S. Pu, “Stability and causality criteria in linear mode analysis: stability\nmeans causality,” arXiv:2309.11708 [hep-th] .\n[10] X.-Q. Xie, D.-L. Wang, C. Yang, and S. Pu, “Causality and stability analysis for the minimal\ncausal spin hydrodynamics,” Phys. Rev. D 108no. 9, (2023) 094031, arXiv:2306.13880\n[hep-ph] .\n[11] A.Vilenkin,“QuantumFieldTheoryatfinitetemperatureinarotatingsystem,” Phys.Rev.D\n21(1980) 2260–2269.\n[12] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, “Chiral magnetic and vortical effects\nin high-energy nuclear collisions – status report,” Prog. Part. Nucl. Phys. 88(2016) 1–28,\narXiv:1511.04050 [hep-ph] .\n[13] D. T. Son and P. Surowka, “Hydrodynamics with Triangle Anomalies,” Phys. Rev. Lett. 103\n(2009) 191601, arXiv:0906.5044 [hep-th] .\n[14] M. N. Chernodub and V. E. Ambrus, “Phase diagram of helically imbalanced QCD matter,”\nPhys. Rev. D 103no. 9, (2021) 094015, arXiv:2005.03575 [hep-th] .\n[15] F. Becattini and I. Karpenko, “Collective Longitudinal Polarization in Relativistic Heavy-Ion\nCollisions at Very High Energy,” Phys. Rev. Lett. 120no. 1, (2018) 012302,\narXiv:1707.07984 [nucl-th] .\n[16] F. Becattini, M. Buzzegoli, and A. Palermo, “Spin-thermal shear coupling in a relativistic\nfluid,”Phys. Lett. B 820(2021) 136519, arXiv:2103.10917 .\n[17] F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, “Local Polarization\nand Isothermal Local Equilibrium in Relativistic Heavy Ion Collisions,” Phys. Rev. Lett. 127\nno. 27, (2021) 272302, arXiv:2103.14621 .\n[18] B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin, “Shear-Induced Spin Polarization in\nHeavy-Ion Collisions,” Phys. Rev. Lett. 127no. 14, (2021) 142301, arXiv:2103.10403 .\n[19]ALICECollaboration, S. Acharya et al., “Polarization of ΛandΛhyperons along the beam\ndirection in Pb-Pb collisions at√𝑠NN= 5.02 TeV,” arXiv:2107.11183 [nucl-ex] .\n[20] W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, “Relativistic fluid dynamics with\nspin,”Phys. Rev. C97no. 4, (2018) 041901, arXiv:1705.00587 [nucl-th] .\n[21] W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, and E. Speranza, “Spin-dependent\ndistribution functions for relativistic hydrodynamics of spin-1/2 particles,” Phys. Rev. D97\nno. 11, (2018) 116017, arXiv:1712.07676 [nucl-th] .\n[22] W. Florkowski, A. Kumar, and R. Ryblewski, “Thermodynamic versus kinetic approach to\npolarization-vorticity coupling,” Phys. Rev. C98no. 4, (2018) 044906, arXiv:1806.02616\n[hep-ph] .\n[23] W. Florkowski, A. Kumar, R. Ryblewski, and R. Singh, “Spin polarization evolution in a\nboost invariant hydrodynamical background,” Phys. Rev. C 99no. 4, (2019) 044910,\narXiv:1901.09655 .\n9Dissipative effects on the propagation of spin modes Rajeev Singh\n[24] S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, “Relativistic\ndissipative spin dynamics in the relaxation time approximation,” Phys. Lett. B 814(2021)\n136096, arXiv:2002.03937 .\n[25] R. Singh, G. Sophys, and R. Ryblewski, “Spin polarization dynamics in the\nGubser-expanding background,” Phys. Rev. D 103no. 7, (2021) 074024,\narXiv:2011.14907 .\n[26] R. Singh, M. Shokri, and R. Ryblewski, “Spin polarization dynamics in the\nBjorken-expanding resistive MHD background,” Phys. Rev. D 103no. 9, (2021) 094034,\narXiv:2103.02592 .\n[27] W. Florkowski, R. Ryblewski, R. Singh, and G. Sophys, “Spin polarization dynamics in the\nnon-boost-invariant background,” arXiv:2112.01856 .\n[28] V. E. Ambrus, R. Ryblewski, and R. Singh, “Spin waves in spin hydrodynamics,” Phys. Rev.\nD106no. 1, (2022) 014018, arXiv:2202.03952 .\n[29] R. Singh, M. Shokri, and S. M. A. T. Mehr, “Relativistic hydrodynamics with spin in the\npresence of electromagnetic fields,” Nucl. Phys. A 1035(2023) 122656,\narXiv:2202.11504 .\n[30] R. Singh, “Collective dynamics of polarized spin-half fermions in relativistic heavy-ion\ncollisions,” Int. J. Mod. Phys. A 38no. 20, (2023) 2330011, arXiv:2212.06569 .\n[31] W. Florkowski, R. Ryblewski, and A. Kumar, “Relativistic hydrodynamics for spin-polarized\nfluids,”Prog. Part. Nucl. Phys. 108(2019) 103709, arXiv:1811.04409 .\n[32] V. E. Ambrus, “Transport coefficients in ultrarelativistic kinetic theory,” Phys. Rev. C 97\nno. 2, (2018) 024914, arXiv:1706.05310 [physics.flu-dyn] .\n[33] S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, “Dissipative Spin\nDynamics in Relativistic Matter,” Phys. Rev. D 103no. 1, (2021) 014030,\narXiv:2008.10976 .\n[34] Y. Hidaka and D.-L. Yang, “Nonequilibrium chiral magnetic/vortical effects in viscous\nfluids,”Phys. Rev. D 98no. 1, (2018) 016012, arXiv:1801.08253 .\n[35] N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, “Generating Spin\nPolarization from Vorticity through Nonlocal Collisions,” Phys. Rev. Lett. 127no. 5, (2021)\n052301, arXiv:2005.01506 .\n[36] D.-L. Yang, K. Hattori, and Y. Hidaka, “Effective quantum kinetic theory for spin transport\nof fermions with collsional effects,” JHEP20(2020) 070, arXiv:2002.02612 .\n[37] Z. Wang, X. Guo, and P. Zhuang, “Local Equilibrium Spin Distribution From Detailed\nBalance,” arXiv:2009.10930 .\n[38] N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, “Derivation of the\nnonlocal collision term in the relativistic Boltzmann equation for massive spin-1/2 particles\nfrom quantum field theory,” Phys. Rev. D 104no. 1, (2021) 016022, arXiv:2103.04896 .\n[39] D. E. Kharzeev and H.-U. Yee, “Chiral Magnetic Wave,” Phys. Rev. D 83(2011) 085007,\narXiv:1012.6026 [hep-th] .\n[40] Y.Jiang,X.-G.Huang,andJ.Liao,“Chiralvorticalwaveandinducedflavorchargetransport\nin a rotating quark-gluon plasma,” Phys. Rev. D 92no. 7, (2015) 071501.\n[41] V. E. Ambrus and M. N. Chernodub, “Vortical effects in Dirac fluids with vector, chiral and\nhelical charges,” Eur. Phys. J. C 83no. 2, (2023) 111, arXiv:1912.11034 [hep-th] .\n10" }, { "title": "2401.16023v1.SN1987A_constraints_to_BSM_models_with_extra_neutral_bosons_near_the_trapping_regime___U_1___L_μ_L_τ___model_as_an_illustrative_example.pdf", "content": "SN1987A constraints to BSM models with extra neutral bosons\nnear the trapping regime: U(1)Lµ−Lτmodel as an illustrative\nexample\nKwang-Chang Lai,1Chun Sing Jason Leung,2and Guey-Lin Lin2\n1Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan, 333, Taiwan\n2Institute of Physics, National Yang Ming Chiao Tung University, Hsinchu, 300, Taiwan\nNew physics beyond the Standard Model (BSM) with an extra neutral boson can\nbe constrained from the observation of SN1987A, since productions of this neutral\nboson in a supernova (SN) could accelerate the SN cooling and potentially lead to a\nperiod of the neutrino burst incompatible with the observation.The constraint to the\nmodel is formulated by the condition LNB≤3×1052erg/s according to G. Raffelt\nwith LNBthe luminosity of BSM neutral boson. Computing the above luminosity in\nthe large coupling case, the so-called trapping regime, is non-trivial since the lumi-\nnosity is a competition between the large production rate and the efficient absorption\nor decay rate of the neutral boson. We illustrate such a subtlety using U(1)Lµ−Lτ\nmodel as an example where the Z′luminosity, LZ′, from the neutrinosphere is cal-\nculated.\nWe calculate Z′production, absorption, and decay rates through pair-coalescence,\nsemi-Compton, loop-bremsstrahlung from proton-neutron scattering, and their in-\nverse processes in a benchmark SN simulation with muons. We point out that, as\nthe coupling constant gZ′increases, LZ′shall be approaching to a constant plateau\nvalue for a given mZ′instead of monotonically decreasing down to zero as obtained\nin the previous literature. We demonstrate that this plateau phenomenon can be\nunderstood by physical arguments and justified by numerical calculations. With a\ndifferent result on LZ′from the previous one, we discuss impacts on the constraints to\nU(1)Lµ−Lτparameter space by SN1987A. The implication of our result to the similar\nconstraint on a generic BSM model with an extra neutral boson is also discussed.arXiv:2401.16023v1 [hep-ph] 29 Jan 20242\nI. INTRODUCTION\nDetecting neutrinos from SN1987A in the Large Magellanic Cloud confirms the paradigm\nof core-collapse supernovae and provides useful information on particle interactions and neu-\ntrino properties. The inferred energy release through neutrinos is about 3 ×1053erg, agreeing\nwell with the difference between the gravitational binding energy of the progenitor and the\nremnant. This observation is useful for constraining new physics beyond the Standard Model\n(BSM), which contains an extra neutral boson. This stems from the fact that the extra neu-\ntral boson produced in the proto-neutron star (PNS) could carry away energies, which lead\nto an accelerated cooling of PNS and consequently a shorter period of neutrino emissions.\nTo set quantitative constraints on BSM models based upon the possible accelerated cooling\nof PNS, a simple criterion provided by Raffelt [1] can be applied. The application of this\ncriterion can be divided into two separated cases: the free streaming limit and the trapping\nlimit. The free streaming constraint requires that the energy carried away by BSM parti-\ncle per unit time should not be comparable to or exceed the neutrino luminosity, which is\n3×1052erg s−1at post-bounce time tpb= 1 s. Otherwise, it would significantly reduce the\nduration of the neutrino burst and contradict what has been observed for SN1987A. In this\ncase the coupling constant between the BSM boson and SM particles should be smaller than\na critical value. On the other hand, if the above coupling constant is sufficiently large, the\nBSM boson can be reabsorbed by the surrounding matter or it can decay within PNS such\nthat its energy can be reprocessed in the star. In terms of the optical depth τof the BSM\nboson, this new boson would not be able to carry out energies provided τ≥1, with the equal\nsign understood as the trapping limit. In other words, the condition τ≥1 implies that the\nmean free path of the BSM boson is smaller than the distance between the production point\nof the BSM boson and the surface of the neutrinosphere such that the SN cooling time is\nunaffected.\nExamples of using the Raffelt criterion with these two limits are given in earlier works\n[2–7]. Nevertheless, taking τ= 1 as the trapping limit is just an approximation, and it\nunderestimates the constraint. An unified expression for the luminosity, encompassing the\nproduction and reabsorption of BSM neutral bosons, can be derived for simultaneously de-\ntermining both the free streaming and trapping constraints, as proposed by Chang et al. [8].\nIn this expression, the luminosity LNBis determined by four parameters ( Rν, Rfar, mNB, gNB),3\nwhich are the radius of the neutrinosphere, the cutoff radius for reabsorption and decay pro-\ncesses, the BSM particle mass, and the coupling strength of the BSM boson to SM particles,\nrespectively. The authors emphasize the importance of ensuring that the value of Rfaris\nlarger than Rν. For example it has been argued in [8, 9] that Rfarcan be taken as the neu-\ntrino gain radius Rgain≈100 km which is defined in the way that there are net productions\nof neutrinos between RνandRgainwhile neutrino number decreases beyond Rgain[10, 11].\nIn the case of U(1)Lµ−Lτ, asZ′boson is reabsorbed in the region between RνandRgain, its\nenergy will be inherited by nucleons absorbing Z′through n+p+Z′→n+p. Such energies\nwould then be returned to neutrinos in the neutrino production processes e−+p→n+νe\nande++n→p+ ¯νe.\nWhile Z′attenuation between RνandRgaindoes reduce LZ′as seen in [9], we stress that\nit is also necessary to consider the production of Z′in the same region for the consistency.\nIn other words, for a consistent calculation, the production and attenuation of Z′should\nbe considered within the same region, no matter it is the space within the neutrinosphere\nor a larger region up to the neutrino gain radius for instance. In this article, we shall\ndemonstrate such type of calculations using U(1)Lµ−Lτmodel as an example and consider\nthe region within the neutrinosphere for obtaining LZ′. Extending the region to the neutrino\ngain radius does not change LZ′much as the number densities of all relevant particles drop\nsignificantly outside the neutrinosphere. Furthermore, the results in [9] do include the\nRfar→Rνcase for mZ′= 0.1 eV, which is a useful point for comparing two calculations.\nWe shall see later that LZ′in the trapping regime approaches to a constant plateau value\ninstead of decreasing to zero monotonically as gZ′increases. A simple argument for this\nphenomenon will be given in later sections. We note that the result in [9] with mZ′= 0.1\neV does not exhibit this behaviour.\nTheU(1)Lµ−Lτmodel is considered a minimal extension to the Standard Model of particle\nphysics. In this model, it is possible that the new U(1) local symmetry is spontaneously\nbroken, which gives rise to a massive new gauge boson referred to as the Z′boson [12–28].\nThis neutral Z′boson couples directly to the second generation leptons and neutrinos. At\nthe one loop level, Z′can mix with photon and Zboson so that it couples indirectly to\nother fermions as well. Through these direct and indirect couplings, Z′can be produced as\nwell as reabsorbed in the PNS. As stated in [9], the mechanisms for producing Z′are pair\ncoalescence processes ν¯ν→Z′andµ+µ−→Z′, the semi-Compton process µ−γ→µ−Z′,4\nand the loop bremsstrahlung process n+p→n+p+Z′where Z′couples to proton through\nits mixing with γ. On the other hand, Z′can also decay or be reabsorbed by inverse processes\nof the above.\nThe paper is organized as follows. In Section II, we present the Lagrangian of the model,\ndescribe the Raffelt criterion, provide an overall luminosity formula, and discuss the impact\nofZ′boson emission on the energy release of SN1987A. Next, we explain the framework used\nfor calculating the Z′boson luminosity in Section III. In Sections III A and III B, we calculate\nthe emissivity and attenuation factors for each individual reaction process. Next, the overall\nluminosity of the Z′boson in the PN star is calculated by integrating the emissivity and\nattenuation factors as discussed in Section IV. The competition and cancellation between\nthe emissivity and attenuation factors will be investigated in details. Subsequently, we\npresent the model parameter space that can be excluded by SN1987A based upon our new\ncalculation of LZ′, which approaches to a constant plateau value as gZ′increases. This\nphenomenon can be seen through both numerical calculations and theoretical arguments.\nFinally, we conclude in Section V. The appendices provide comprehensive derivations of\nrelevant formulas and provide detailed explanations for the plateau phenomenon.\nII. IMPACT OF Z′BOSON ON SN1987A\nWe first discuss the U(1)Lµ−Lτextension of SM, which introduces the new boson Z′. The\nLagrangian of the model is given by [12–28]\nLZ′=LSM−1\n4Z′\nµνZ′µν−ϵ\n2Z′\nµνZµν+1\n2m2\nZ′Z′\nµZ′µ\n+gZ′Z′\nµ\u0000¯l1γµl1−¯l2γl2+ ¯µRγµµR+ ¯τRγµτR\u0001\n, (1)\nwhere gZ′isU(1)Lµ−Lτgauge coupling, l1andl2denote the electroweak doublets for the left-\nhanded leptons ( µL, νµ,L) and ( τL, ντ,L), while µRandτRare the electroweak singlets for the\nright-handed leptons. The mass of the new boson Z′can be generated through spontaneous\nU(1)Lµ−Lτsymmetry breaking. We do not include the scalar Lagrangian since it is not\nneeded in the subsequent discussions. In this model, Z′can only interact with muons,\ntaus, and their corresponding neutrinos. Therefore, to produce a considerable amount of\nZ′bosons, the core of PNS must contain a significant population of these leptons. In\nthe core of PNS, the number of electrons overwhelms that of the positron to balance the5\npositive charges of the protons. The chemical potential of these highly degenerate electrons\nis greater than the muon mass, µe> m µ, and hence can be transferred into muons through\nSM processes [32]. In addition, a significant number of neutrinos and photons are generated\nthrough thermal pair production in the core, which in turn produces a considerable amount\nof muons. While muons are produced copiously, the production of tau leptons is suppressed\ndue to the large tau lepton mass.\nAs supernova muons can be produced in large quantities within PNS, a considerable\namount of Z′bosons can be generated through Z′couplings to muons. If these Z′bosons\ncan free-stream out and carry a significant amount of energy away from the neutrinosphere,\nthe energy that drives the production of supernova neutrinos will decrease accordingly.\nTherefore, the production of Z′bosons will affect the neutrino emission of SN explosions\nand the overall PNS cooling duration. As the observations of SN1987A neutrinos are consis-\ntent with SN simulations having the neutrino emission as the only cooling mechanism, the\nZ′boson emissions cannot dominate the cooling, which then leads to a constraint on the\n(mZ′, gZ′) parameter space. Such a constraint can be formulated as an upper limit for the\nZ′luminosity, referred to as the Raffelt criterion [1] for the generic case. Raffelt’s criterion\nbased upon SN1987A observation states that\nLz′< L ν= 3×1052erg/s. (2)\nTo be consistent with observations, the energy carried away by Z′must be considerably\nlower than that carried away by neutrinos.\nWe now describe the way we calculate the Z′boson luminosity, LZ′. In the free-streaming\nregion, where no re-absorption occurs and Z′bosons can flee without attenuation, the Z′\nboson luminosity can be directly inferred from its production rate Γ prod:\nLZ′=Z\ndVZd3k\n(2π)3ωΓprod, (3)\nwhere ωis the Z′energy, and Vdenotes the volume of the neutrinosphere.\nAs the coupling strength gZ′increases, the re-absorption and decays of Z′bosons are\nnot negligible. These attenuation processes impede the free-streaming of Z′bosons and are\nconsidered by introducing the attenuation factor\nA=1\n2Z1\n−1exp\u0012\n−Zrmax\nθ\n0Γabsdrθ\u0013\ndy, (4)6\nwhere Γ absis the absorption rate of the Z′boson. The integration over y≡cosθand the\nfactor 1 /2 represent the averaging of Z′propagation direction inside the neutrinosphere. As\nshown by Fig. 1, θis the angle between the Z′propagation direction and the radial vector\n⃗ rfrom the origin to the point of Z′production. Finally, rθis the distance from where the\nZ′boson is created.\ny <0y >0\nneutrino sphere Rν ⃗ r⃗ r′⃗ rθ\nθ ��� rθ\nFIG. 1: The production and propagation of Z′boson inside the neutrinosphere. The\nmagenta line divides the region with y >0 (θ < π/ 2) and that with y <0 (θ > π/ 2).\nAdditionally, r′denotes the distance from the origin to the position of Z′in propagation.\nThe vector rθdenotes the trajectory of Z′boson after its production.\nTheZ′boson luminosity with the attenuation factor implemented is given by [8, 9]\nLZ′=ZV(Rν)\ndVZd3k\n(2π)3A(ω, r, T, R far)ωΓprod(ω, T). (5)\nWe note that Ref. [8] simplifies the angular averaging by considering only + zand−zdirec-\ntions of Z′propagation. It is interesting to see that Eq. 5 together with Raffelt’s criterion,\nEq. (2), unifies and addresses both the free-streaming and trapping constraints. Rfaris the7\nradius of the spherical volume for the attenuation processes, which would be taken as Rνin\nour study.\nIII. PRODUCTION, ABSORPTION, AND DECAY OF Z′BOSON IN\nSUPERNOVAE\nInU(1)Lµ−Lτmodel, Z′bosons can be produced at the tree level through pair coalescence\nand semi-Compton processes depicted in Fig. 2. Z′bosons can also couple to photons\nthrough an effective kinetic mixing generated by the intermediate muon or tau loops as\nshown by the third diagram of Fig. 2. Such a kinetic mixing enables the production of Z′\nboson through the neutron-proton loop bremsstrahlung process, n+p→n+p+Z′. The\nproduction of Z′bosons in supernovae is predominantly governed by these three processes,\nwhile their inverse processes then contributes to the absorption of Z′[9], which will be\ndiscussed in Sec. III A and III B respectively.\nA. The emissivity spectrum d˙ϵ/dω ofZ′in SN\n¯ν, µ+ν, µ−\nZ′\nPair Coalescenceγµ−\nµ−Z′\nSemi-Comptonµ, τ\nnp\nnpZ′\nLoop Brem\nFIG. 2: Dominant Z′emission processes in a core-collapse supernova.\nAt the energy scale of a supernova, the main Z′production processes are pair coales-\ncence, semi-Compton, and loop bremsstrahlung. Fig. 2 displays the corresponding Feynman\ndiagrams. We calculate the emissivity spectrum for each production process in this section.\nThe relation between emissivity and production rate is\nd˙ϵ=d3k\n(2π)33X\ni=1ωΓi\nprod=ω3\n2π2r\n1−m2\nZ′\nω23X\ni=1Γi\nproddω. (6)\nThe emissivity spectrum d˙ϵ/dω including all production rates Γi\nprod.in Fig. 2 for i∈ {pair-8\ncoalescence, semi-Compton, loop bremsstrahlung }is\nd˙ϵ\ndω=ω3\n2π2r\n1−m2\nZ′\nω23X\ni=1Γi\nprod, (7)\nin which the production rate of each process Γi\nprodcan be obtained by integrating the general\nphase space formula\ndΓ =1\n2ωY\nid3pif(pi)\n(2π)32EiY\nfd3pf(1±f(pf))\n(2π)32Ef(2π)4δ4(X\nipi−X\nfpf−k)|M|2, (8)\nwhere the indices iandfrefer to the initial- and final-state particles, respectively, apart\nfrom the new boson Z′. The function f(p) represents the thermal statistical distribution\nof each individual particle. The use of a plus or minus sign depends on if the particle\nadheres to Fermi-Dirac or Bose-Einstein statistics. For a fermion, a Pauli blocking factor\n(1−f(p)) should be applied. On the other hand, for a boson, a Bose enhancement factor\nof (1 + f(p)) should be used. The detailed derivation is provided in Appendices A, B and\nC for pair-coalescence, semi-Compton and loop bremsstrahlung processes shown in Fig. 2,\nrespectively.\nAfter performing the phase space integral in Eq. (8), the Z′production rate Γ µ−+µ+→Z′\nby muon pair coalescence (the first diagram in Fig. 2) is given by (see Appendix A)\nΓµ−+µ+→Z′≈g2\nZ′m2\nZ′\n4πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′×e−ω/T. (9)\nSimilarly, the Z′production rate Γ ν+¯ν→Z′via neutrino (including both νµandντ) pair\ncoalescence can be calculated in the same way. It is given by\nΓν+¯ν→Z′≈g2\nZ′m2\nZ′\n4πω×e−ω/T. (10)\nWe note that the allowed polarizations for muons and neutrinos are different. Unlike muons\nhaving two polarizations, only left-handed neutrinos can contribute to the matrix element\nsquared |M|2. The equations (9) and (10) share the same prefactor as both νµandντhave\nbeen accounted for.\nFor the semi-Compton process (the second diagram in Fig. 2), the production rate Γ sCis\ngiven by (see Appendix B)\nΓsC≈8παα Z′\n3m2\nµnµFdeg\neω/T−1r\n1−m2\nZ′\nω2, (11)9\nwhere αZ′≡gZ′/4π,nµis the muon number density, and the degeneracy factor Fdeg∈[0,1]\nis considered as a simple Pauli-blocking factor correction to the phase space of the final-state\nmuon. The definition and determination of Fdegare discussed in Appendix B.\nFinally, since the loop bremsstrahlung process (the last diagram in Fig. 2) is considerably\nmore intricate than the other processes, the emissivity spectrum is provided directly instead\nof the production rate:\nd˙ϵLB\ndω=2nnnpαϵ2\n3(πMT )3/2r\n1−\u0010mZ′\nω\u00112\u0014\n2 +\u0010mZ′\nω\u00112\u0015Z∞\nωdTcme−Tcm/TT2\ncmσ(2)\nnp(Tcm),(12)\nwhere Tcmis the kinetic energy of the initial particles in the center-of-momentum frame, M\nis the nucleon mass, nnandnpare number densities of neutrons and protons, respectively,\nϵis an effective γ−Z′mixing ϵZ′\nµνZµν/2 induced by one-loop diagrams. It is related to gZ′\nand lepton masses by\nϵ=−egZ′\n2π2Z1\n0x(1−x) ln\u0014m2\nτ−x(1−x)k2\nm2\nµ−x(1−x)k2\u0015\n, (13)\nwhere k= (ω,⃗k) is the 4-momentum of final-state Z′boson with k2=m2\nZ′. The definition\nofσ(2)\nnp(Tcm) in the integrand is\nσ(2)\nnp(Tcm)≡Z1\n−1(1−cosθcm)dσnp\ndcosθcmdcosθcm, (14)\nwhere dσnp/dcosθcmis the neutron-proton differential cross-section that can be extracted\nfrom experiments. The derivation of these formulae is given in Appendix C.\nB. The attenuation length of Z′in SN\nZ′\n¯ν, µ+ν, µ−\nZ′Decayµ−Z′\nγµ−\nInv Semi-Compµ, τ\nnpZ′\nnp\nInv Loop Brem\nFIG. 3: Dominant Z′reabsorption processes in core-collapse supernova.\nHere we discuss the attenuation length λatt, or in other words, the mean free path of\nZ′. It describes the survival distance of the Z′boson that is produced by either of the10\naforementioned production processes. The dominant decay and reabsorption processes are\nillustrated in Fig. 3. They are considered as the inverse processes of those given in Fig. 2.\nIn this section, we shall compute the inverse of the attenuation length, denoted by 1 /λatt.\nIts relation to the absorption rate Γ (in the rest frame of Z′) is given by\n1\nλatt=Γlab\nv= Γ×mZ′\nω\u001er\n1−\u0010mZ′\nω\u00112\n, (15)\nwhere Γlabandvdenote the absorption rate in the lab frame and the velocity of the new\nboson Z′, respectively, mZ′/ωis just the inverse of γfactor accounting for the time dilation\neffect in a moving frame. The last termp\n1−(mZ′/ω)2in the denominator is the speed\nfactor v. Hence the calculation of 1 /λattamounts to computing the absorption rate Γ for\neach process.\nThe absorption rate can be calculated by replacing the 4-momentum k→ − kin the\nproduction rate formula Eq. (8) and the corresponding matrix element squared for absorption\nprocesses in Fig. 3. It gives\ndΓ =1\n2ωY\nid3pif(pi)\n(2π)32EiY\nfd3pf(1±f(pf))\n(2π)32Ef(2π)4δ4(X\nipi−X\nfpf+k)|M|2,(16)\nwhere the indices iandfstand for initial and final state particles. The function f(p) is the\nthermal statistical distribution of the initial and final state particles which has already been\nintroduced in the previous section. The detail of the derivation is provided in Appendices\nA, B and C for Z′decay, semi-Compton absorption and loop bremsstrahlung absorption\nprocesses in Fig. 3, respectively.\nThe decay rate of Z′through Z′→µ−+µ+is given by Eq. (A6) (see Appendix A):\nΓlab\nZ′→µ−+µ+≈g2\nZ′m2\nZ′\n12πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′. (17)\nThe inverse of the attenuation length 1 /λcan then be obtained by dividing Eq. (17)by a\nspeed factor vaccording to the relation in Eq. (15). This gives\n1\nλZ′→µ−+µ+≈g2\nZ′m2\nZ′\n12πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′\u001er\n1−\u0010mZ′\nω\u00112\n. (18)\nThe derivation of the decay rate for Z′→ν+ ¯νis very similar to that of Z′→µ−+µ+.\nTo obtain the decay rate Γ Z′→ν+¯ν, one can simply take mµ→0 in Eq. (17), which gives\nΓlab\nZ′→ν+¯ν≈g2\nZ′m2\nZ′\n12πω. (19)11\nIt is worth noting that only left-handed neutrinos and right-handed anti-neutrinos can inter-\nact with the new gauge boson Z′. This is because, unlike muons which have two polarization\nstates, neutrinos have only one. As a result, we have Γlab\nZ′→νµ+¯νµ=1\n2Γlab\nZ′→µ−+µ+(mµ→0).\nThe pre-factor in Eq. 19 is equal to that in Eq. 17 because both Z′→νµ+¯νµandZ′→ντ+¯ντ\nhave been taken into account (for more details, see Appendix A). We obtain the inverse of\nthe attenuation length 1 /λby dividing Eq. (19) by the Z′speed v, which gives\n1\nλZ′→ν+¯ν≈g2\nZ′m2\nZ′\n12πω\u001er\n1−\u0010mZ′\nω\u00112\n. (20)\nTheZ′absorption rate through semi-Compton process (the second diagram in Fig. 3)\ncan be obtained by using the principle of detailed balance:\nΓabs\nsC=2\n3eω/T×Γprod\nsC. (21)\nSubstituting Eq. (11) into the above equation gives\nΓabs\nsC≈2\n3eω/T×8παα Z′\n3m2\nµnµFdeg\neω/T−1r\n1−m2\nZ′\nω2\n=16παα Z′\n9m2\nµ(1 +1\neω/T−1)nµFdegr\n1−m2\nZ′\nω2. (22)\nA more comprehensive derivation, with or without using the principle of detailed balance,\nis given in Appendix B. By substituting the aforementioned equation into Eq. (15), one\nobtains 1 /λsCgiven by\n1\nλsC≈16παα Z′\n9m2\nµ(1 +1\neω/T−1)nµFdeg. (23)\nFinally, the inverse of the attenuation length 1 /λfor the Z′absorption through loop-\nbremsstrahlung is significantly more complex than the others. A detailed derivation of this\ncan be found in Appendix C. The result is\n1\nλLB=8nnnpαϵ2\n9πω3\u0012πT\nM\u00133/2 \n2 + (mZ′/ω)2\np\n1−(mZ′/ω)2!\n1\n2Z∞\n0dxe−xx2σ(2)\nnp(x). (24)\nIV. RESULTS AND DISCUSSION\nTheU(1)Lµ−Lτmodel introduces a new gauge boson Z′that can be produced through in-\nteractions between standard model particles and unstable heavy leptons, specifically muons12\nand taus. This is described by the interaction Lagrangian in Eq. (1). In this section,\nwe present Z′luminosity based upon a SN simulation SFHo18.8 by Thomas Janka et\nal.) [29]. Choosing SFHo18.8 simulation enables the comparison of our calculations with\nthose in [9]. We account for the production and reabsorption processes discussed in the\nprevious sections and obtain the Z′luminosity through numerical calculation of Eq.(5)\namong different values of mZ′andgZ′. The left panel of Fig. 4 shows the Z′luminosity\nformZ′={2 eV,10 eV ,0.1 MeV ,10 MeV ,200 MeV }as a function of the coupling gZ′. The\nRaffelt bound (see Eq. (2)) is indicated by a magenta line.\n10−1110−910−710−510−310−1\ngZ′1052105410561058Luminosity L(gZ′)[erg/s]mZ′= 2 eV\nmZ′= 10 eV\nmZ′= 0.1 MeV\nmZ′= 10 MeV\nmZ′= 200 MeV\nRaffelt’s criterion\n0 25 50 75 100 125 150 175 200\nmZ′[MeV]1035103710391041104310451047104910511053L∞(mZ′)[erg/s]\nZ′Luminosity at large gZ′\n(The Plateau Value)\nRaffelt’s criterion\n(3×1052[erg/s])10−710−510−310−1101105110521053\nFIG. 4: (Left) Luminosity of the new gauge boson Z′in various mass m′\nZ. (Right) The\nplateau value L∞as a function of mZ′. The detailed behaviour of L∞formZ′below 10\nMeV is given by the inset.\nMany interesting features can be seen in Fig. 4 as pointed out by Croon et al. [9]. The peak\nvalue of LZ′goes down as mZ′decreases is a clear evidence of the Boltzmann suppression. In\nthe free streaming region, LZ′steadily increases with gZ′. On the other hand, the exponential\nsuppression of LZ′can be observed from the trapping region for a sufficiently large gZ′. The\nonly difference between the result of Ref. [9] and that of ours is that the former does not\nmanifest the plateau phenomenon when taking Rfar→Rνwith a large gZ′. To be precise,\nwe observe that each luminosity curve in the left panel of Fig. 4 approaches to a constant\nplateau value for a sufficiently large gZ′. This is however not seen from Fig. 2 of Ref. [9]\nwhere the limit Rfar→Rνis taken for LZ′with mZ′= 0.1 eV. Before we provide arguments\nfor this plateau phenomenon, we reiterates that the production and attenuation regions of\nZ′boson should be taken as identical for consistency. On the other hand, Croon et al. [9]13\npresent their results by taking Rfar≫Rν.\nThe above-mentioned plateau phenomenon can be understood by a simple argument.\nIn the large gZ′limit, only those Z′bosons produced within one interaction length from\nthe surface of the neutrinosphere can exit the sphere and contribute to LZ′. The volume\nof this spherical shell is 4 πR2\nνλwith λthe interaction length. The number of Z′bosons\nproduced in this volume is proportional to Z′production probability multiplied by 4 πR2\nνλ.\nWith the former scaled as g2\nZ′and the latter as g−2\nZ′, the overall contribution to LZ′is then\nindependent of gZ′. The details for calculating this limiting luminosity, denoted by L∞, are\ngiven in Fig. 10 and Eq. (E2). The expected plateau value L∞as a function of mZ′is given\nin the right panel of Fig. 4. For instance, in the left panel of Figure 4, a full numerical\ncalculation results in a plateau value of approximately 3 ×1052[erg/s] at mZ′= 2 eV, and\nthis plateau value is accurately predicted by the right panel of the figure. Furthermore,\nwhen the mass of Z′falls below approximately 2 eV, L∞exceeds the upper bound set by\nthe Raffelt criterion. This implies that for mZ′<2 eV, the excluded parameter region will\nno longer be bounded from above by the trapping limit. Thus, the plateau phenomenon\nsignificantly constrains the model when mZ′<2 eV. This interesting result is demonstrated\nin the complete contour plot shown in Fig. 5 [37].\n10−610−510−410−310−210−1100101102\nmZ′[MeV]10−1110−910−710−510−310−1gZ′Region excluded by the Raffelt’s criterion\n10521053105410551056105710581059\nZ′Luminosity\n10−610−510−410−310−210−1100101102\nmZ′[MeV]10−1110−910−710−510−310−1gZ′\nBorexino\nCCFRSN1987A (Ref. [9])\nSN1987A (this work)\nFIG. 5: (Left) The luminosity contour region for LZ′>1052[erg/s]. The region\ncorresponds to LZ′> L Raffelt = 3×1052[erg/s] is enclosed by the magenta line. (Right)\nThe excluded parameter region due to existing experiments and the previous work [9]. Our\nresult is shaded in red colour.\nOn the left panel of Fig. 5, we present the luminosity contour plot by scanning through14\nthe parameter space ( mZ′, gZ′) in Eq. (5). The magenta line encloses the excluded parameter\nregion where Z′luminosity LZ′> L Raffelt . AsmZ′approaches ∼2 eV from above, the upper\nmagenta curve tends towards a vertical line, which excludes all parameter space g′\nZ[37].\nPrecisely speaking, for mZ′<2 eV, the plateau value in Fig. 4 is already greater than the\nRaffelt bound, resulting in the vanishing of the trapping limit. Our work (shaded in red)\nextends the trapping limit gradually as mZ′decreases to ∼2 eV. In comparison to the result\nof Croon et al. (the magenta shaded region in the right panel of Fig. 5), our result represents\na significant extension of excluded parameter region. This difference is largely due to the\nextra attenuation region between the radii RνandRfaradopted in [9]. On the other hand,\neven in the Rfar=Rνcase, exclusion regions resulting from two calculations might still be\ndifferent as hinted by the difference in LZ′formZ′= 0.1 eV mentioned in the beginning of\nthe section.\nOur analysis shows that the extended exclusion region of the parameter space overlaps\nwith the excluded region derived from neutrino trident experiment in CCFR [31] at a 95%\nconfidence level, as well as the exclusion region derived from the Borexino solar neutrino\nmeasurement data [30]. Therefore, our findings confirm that this overlapped parameter\nregion is indeed disfavored.\nV. SUMMARY AND CONCLUSIONS\nWe pointed out that SN constraints on BSM theories involving extra neutral bosons\nnear the trapping regime should be handled with care. Specifically, we suggest that the\nattenuation boundary for the new boson should be taken to be identical to the production\nboundary, and the luminosity of the new boson will tend to a constant value instead of\ndecreasing to zero at a large coupling limit. We addressed this as the plateau phenomenon.\nIt has been argued in [9] that Rfarshould be considered a value greater than Rν. However,\nwe asserted that this is not self-consistent as the extra neutral boson can still be produced\nwithin the region between RνandRfar.\nTo demonstrate the aforementioned plateau phenomenon, we took the U(1)Lµ−Lτmodel\nas an illustrative example. We employed the luminosity formula Eq. (3) from Chang et\nal.[8] to unify the calculations of free-streaming and trapping regimes. However, we did\nnot use the approximation employed by Ref.[8] for computing the attenuation factor (see15\ncomments below Eq. (5)), and we computed it numerically with Rfar=Rν. By performing\ncomplete numerical calculations and theoretical derivations, we verified and derived the\nplateau phenomenon. Notably, for the U(1)Lµ−Lτmodel, we found that L∞(mZ′) exceeded\nLRaffelt formZ′<2 eV. Although Ref. [9] also presented the Z′boson luminosity at the limit\nRfar→RνformZ′= 0.1 eV, their result does not show the plateau phenomenon at the\nlarge gZ′limit. Before closing, we point out that our work does not consider the extra BSM\nneutral boson as the portal between visible and dark sectors. For such a scenario and the\ncorresponding SN1987A bounds to the model, see [33, 34]\nIn conclusion, we have stressed the significance of giving equal consideration to the re-\ngion of absorption processes as that of production processes. At the boundary of the neu-\ntrinosphere, the interplay between emissivity and attenuation factor results in a constant\nluminosity, denoted as L∞(m), instead of its gradual decrease to zero in the event of a large\ncoupling. Depending on the mass of the BSM particle, L∞(m) could potentially exceed the\nallowable luminosity value from SN observations and thus, strongly restrict the parameter\nspace of BSM models.\nAcknowledgements\nWe thank M.-R. Wu for discussions on strong αZ′limit as mentioned in [37]. This work\nis supported by National Science and Technology Council, Taiwan, under Grant Nos. NSTC\n111-2112-M-A49-027 and NSTC 112-2112-M-A49-017.\n[1] Georg G. Raffelt, “Stars as Laboratories for Fundamental Physics” , (1996).\n[2] J. B. Dent, F. Ferrer and L. M. Krauss, [arXiv:1201.2683 [astro-ph.CO]].\n[3] Y. Zhang, “Supernova Cooling in a Dark Matter Smog” , JCAP 1411 , no. 11, 042 (2014)\ndoi:10.1088/1475-7516/2014/11/042 [arXiv:1404.7172 [hep-ph]].\n[4] D. Kazanas, R. N. Mohapatra, S. Nussinov, V. L. Teplitz and Y. Zhang, “Supernova\nBounds on the Dark Photon Using its Electromagnetic Decay” , Nucl. Phys. B 890, 17 (2014)\ndoi:10.1016/j.nuclphysb.2014.11.009 [arXiv:1410.0221 [hep-ph]].\n[5] Ermal Rrapaj and Sanjay Reddy, “Nucleon-nucleon bremsstrahlung of dark gauge bosons and\nrevised supernova constraints” , Phys. Rev. C 94, 045805 (2016).16\n[6] Edward Hardy and Robert Lasenby, “Stellar cooling bounds on new light particles: plasma\nmixing effects” , J. High Energy Phys. 02 (2017) 033.\n[7] Cameron Mahoney, Adam K. Leibovich, and Andrew R. Zentner, “Updated Constraints on\nSelf-Interacting Dark Matter from Supernova 1987A” , Phys. Rev. D 96043018 (2017).\n[8] Jae Hyeok Chang, Rouven Essig, Samuel D. McDermott, “Revisiting Supernova 1987A Con-\nstraints on Dark Photons” , J. High Energy Phys. 01 (2017) 107.\n[9] Djuna Croon, Gilly Elor, Rebecca K. Leane and Samuel D. McDermott, “Supernova Muons:\nNew Constraints on Z′Bosons, Axions and ALPs” , J. High Energy Phys. 107(2021).\n[10] H. A. Bethe, “Supernova 1987A: An Empirical and analytic approach” , Astrophys. J. 412,\n192-202 (1993) doi:10.1086/172911\n[11] H. T. Janka, “Conditions for shock revival by neutrino heating in core collapse supernovae” ,\nAstron. Astrophys. 368, 527 (2001) doi:10.1051/0004-6361:20010012 [arXiv:astro-ph/0008432\n[astro-ph]].\n[12] X. -G. He, G. C. Joshi, H. Lew, and R. R. Volkas, “New- Z′phenomenology” , Phys. Rev. D\n43, R22(R) (1991).\n[13] Xiao-Gang He, G. C. Joshi, H. Lew, and R. R. Volkas, “Simplest Z′Model” , Phys. Rev. Lett.\n44, 2118 (1991).\n[14] E. Ma, D. P. Roy, and S. Roy, “Gauged Lµ−Lτwith large muon anomalous magnetic moment\nand the bimaximal mixing of neutrinos” , Phys. Lett. B 525, 101-106 (2002).\n[15] S. Baek, N. G. Deshpande, X.-G. He, and P. Ko, “Muon anomalous g−2and gauged Lµ−Lτ\nmodels” , Phys. Rev. D 64, 055006 (2001).\n[16] E. Salvioni, A. Strumia, G. Villadoro, and F. Zwirner, “Non-universal minimal Z′models:\npresent bounds and early LHC reach” , J. High Energy Phys. 03, 10 (2010).\n[17] J. Heeck and W. Rodejohann, “Gauged Lµ−Lτsymmetry at the electroweak scale” ,\nPhys. Rev. D 84, 075007 (2011).\n[18] K. Harigaya, T. Igari, M. M. Nojiri, M. Takeuchi, and K. Tobe, “Muon g−2and LHC\nphenomenology in the Lµ−Lτgauge symmetric model” , J. High Energy Phys. 03, 105 (2014).\n[19] M. Bauer, P. Foldenauer, and J. Jaeckel, “Hunting All the Hidden Photons” , J. High Energy\nPhys. 07, 094 (2018).\n[20] D. W. P. Amaral, D. G. Cerdeno, A. Cheek, and P. Foldenauer, “Confirming U(1)Lµ−Lτas a\nsolution for (g−2)µwith neutrinos” , Eur. Phys. J. C 81, 861 (2021).17\n[21] A. Greljo, Y. Soreq, P. Stangl, A. E. Thomsen, and J. Zupan, “Muonic force behind flavor\nanomalies” , J. High Energy Phys. 04, 151 (2022).\n[22] P. Langacker, “The Physics of Heavy Z′Gauge Bosons” , Rev. Mod. Phys. 81, 1199–1228\n(2009).\n[23] P. Langacker, “Light dark matter in a gauged U(1)Lµ−Lτmodel” , Phys. Rev. D 99, 035007\n(2019).\n[24] K. Asai, S. Okawa, and K. Tsumura, “Search for U(1)Lµ−Lτcharged dark matter with neutrino\ntelescope” , J. High Energy Phys. 03, 047 (2021).\n[25] I. Holst, D. Hooper, and G. Krnjaic, “Simplest and Most Predictive Model of Muon g−2and\nThermal Dark Matter” , Phys. Rev. Lett. 128, 141802 (2022).\n[26] M. Drees and W. Zhao, “U(1)Lµ−Lτfor light dark matter, gµ−2, the 511 keV excess and the\nHubble tension” , Phys. Lett. B 827, 136948 (2022).\n[27] T. Hapitas, D. Tuckler, and Y. Zhang, “General Kinetic Mixing in Gauged U(1)Lµ−LτModel\nfor Muon g−2and Dark Matter” , Phys. Rev. D 105, 016014 (2022).\n[28] J. Heeck and A. Thapa, “Explaining lepton-flavor non-universality and self-interacting dark\nmatter with Lµ−Lτ”, Eur. Phys. J. C 82, 485 (2022).\n[29] Robert Bollig, William DeRocco, Peter W. Graham, and Hans-Thomas Janka, “Muons in\nSupernovae: Implications for the Axion-Muon Coupling” , Phys. Rev. Lett. 125, 051104 (2020).\n[30] D.W.P. Amaral , D.G. Cerde˜ no, P. Foldenauer and E. Reid, “Solar neutrino probes of the\nmuon anomalous magnetic moment in the gauged U(1)Lµ−Lτ”, J. High Energy Phys. 155\n(2020).\n[31] Wolfgang Altmannshofer, Stefania Gori, Maxim Pospelov, and Itay Yavin, “Neutrino Trident\nProduction: A Powerful Probe of New Physics with Neutrino Beams” , Phys. Rev. Lett. 113,\n091801 (2014).\n[32] R. Bollig, H. T. Janka, A. Lohs, G. Martinez-Pinedo, C. J. Horowitz and T. Melson, Phys.\nRev. Lett. 119, no.24, 242702 (2017).\n[33] A. Sung, G. Guo and M. R. Wu, “Supernova Constraint on Self-Interacting Dark Sector\nParticles” , Phys. Rev. D 103, no.10, 103005 (2021).\n[34] C. A. Manzari, J. Martin Camalich, J. Spinner and R. Ziegler, “Supernova limits on muonic\ndark forces” , Phys. Rev. D 108, no.10, 103020 (2023).\n[35] F. E. Low, “Bremsstrahlung of Very Low-Energy Quanta in Elementary Particle Collisions”18\nPhys. Rev. 110, 974 (1958).\n[36] V.I. Zhaba, “Approximation of scattering phases for Reid93 potential” , IJARPS Vol. 5, Iss. 8\n1-6 (2018).\n[37] There is a caveat worth mentioning as αZ′approaches or even surpasses α. In such a case,\nthe interaction rate for Z′µ−→Z′µ−is comparable or even faster than that of Z′µ−→\nγµ−. Hence the former process should be considered for the propagation of Z′in addition to\nthose reabsorption processes considered in Fig. 3. Consequently the Z′trajectory inside the\nneutrinosphere is no longer a straight line. We shall not consider such a scenario in this work.\nThus one should disregard constrained regions with gZ′>0.1.19\nAppendix A: Z′decay and lepton pair coalescence\nThis section will discuss four calculation processes in detail, comprising two Z′decay\nchannels and two Z′production channels involving muon and neutrino pair coalescence. We\nwill commence by calculating the decay rates for Z′→µ++µ−andZ′→ν+ ¯ν.\nZ′\nµ+µ−\nZ′→µ++µ−mµ→0\n− − − − − − − − →\nΓZ′→µ−µ+to ΓZ′→νµ¯νµZ′\n¯νν\nZ′→ν+ ¯ν\nThe square of the amplitude for the non-polarized decay of Z′into a muon pair, Z′→\nµ++µ−, is provided by\n⟨|M|2⟩ ≡1\n3X\npol.|M|2=4\n3g2\nZ′(2m2\nµ+m2\nZ′). (A1)\nHere, the pre-factor 1 /3 incorporates the spin average of Z′. The calculation of the decay\nrate, performed in the rest frame of the initial particle Z′, follows this expression:\nΓZ′→µ−+µ+=p∗\n32π2m2\nZ′Z\n⟨|M|2⟩dΩ (A2)\n=g2\nZ′mZ′\n12π\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′, (A3)\nin which p∗=q\nm2\nZ′−4m2\nµ/2. To derive the lab frame decay rate in thermal equilibrium,\nmultiply Eq. (A3) by an inverse special relativity gamma factor, denoted as 1 /γ=mZ′/ω.\nAdditionally, consider the Pauli-blocking factor , denoted as 1 −fF(ω/2, µ) (where µdenotes\nthe chemical potential of the corresponding particle), which accounts for the presence of\nfinal state muons. The decay rate in the laboratory frame can be expressed as\nΓlab\nZ′→µ−+µ+=g2\nZ′m2\nZ′\n12πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′×(1−fF(ω/2, µµ−))(1−fF(ω/2, µµ+)),\n(A4)\nwhere ωrepresents the energy of the newly introduced gauge boson Z′, and µµ−/µµ+de-\nnotes the chemical potential of µ−/µ+. Since the chemical potential is contingent on the\nsimulation, the term related to the Pauli-blocking factor, denoted as (1 −fF(ω/2, µµ−))(1−20\nfF(ω/2, µµ+)), should not be disregarded in principle. For µ+, the temperature of SNe is\ntypically a few tenths of a MeV, significantly smaller than the muon mass, i.e., T≪mµ< ω.\nConsequently,\n1−fF(ω/2, µµ+)≡1−1\ne(ω−µµ+)/T+ 1\n= 1−1\ne(ω+µµ−)/T+ 1\n≈1.\nThe final approximation remains valid because the values of ωandµµ−in the numerator of\nthe exponential function surpass the hundred MeV scale, making them significantly larger\nthan the temperature T, which is only a few tenths of the MeV scale. In other words,\nexp[( ω+µµ−)/T]≫1. Conversely, the dependence of (1 −fF(ω/2, µµ−)) is on the simulation\ndata of the muon chemical potential µµ−. Despite being reliant on simulation data, the\nmagnitude of this quantity is constrained:\n1≥1−fF(ω/2, µµ−)≥1\n2. (A5)\nSince a factor between 1 and 1 /2 will not affect the overall order of magnitude, in this work\n(1−fF(ω/2, µµ−))≈1 has been applied for simplicity in calculations. Ultimately, the decay\nrate of Z′→µ−+µ+in the lab frame is determined as\nΓlab\nZ′→µ−+µ+≈g2\nZ′m2\nZ′\n12πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′(A6)\nForZ′→ν+ ¯ν, only left-handed neutrinos and right-handed anti-neutrinos can play a\nrole. Consequently, the squared matrix element of Z′→νµ+ ¯νµis half that of Z′→µ−+µ+.\nTaking mµ→0, we obtain the expression\n⟨|M|2⟩ ≡1\n3X\npol.|M|2=2\n3g2\nZ′m2\nZ′. (A7)\nThe decay rate of Z′→νµ+ ¯νµin the lab frame can be obtained by inserting Eq. (A7) into\nEq. (A2). Result in\nΓlab\nZ′→νµ+¯νµ=g2\nZ′m2\nZ′\n24πω. (A8)\nHence, taking into account the production of Z′from both νµandντ, the overall production\nrate amounts to\nΓlab\nZ′→ν+¯ν≈g2\nZ′m2\nZ′\n12πω. (A9)21\nIt’s important to note that neutrinos differ significantly from muons, not only in terms of\nmass but also due to the impact of helicity states on the overall prefactor. Taking mµ→0\nin Eq. (A1) leads to Eq. (A9) is merely a coincidence.\nµ+µ−\nZ′\nµ−+µ+→Z′¯νµ,¯ντνµ, ντ\nZ′\nν+ ¯ν→Z′\nFIG. 6: Feynman diagrams of muon and neutrino pair coalescence.\nAccording to the principle of detailed balance, the pair-coalescence rate Γ µ−+µ+→Z′is\nsimply related to Γlab\nZ′→µ−+µ+(Eq. (A6)) by\nΓµ−+µ+→Z′=3\n4×Γlab\nZ′→µ−+µ+×4×e−ω/T\n≈g2\nZ′m2\nZ′\n4πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′e−ω/T, (A10)\nin which the pre-factor 3 /4 and 4 account for the correct polarization averaging in the\nmatrix element squared and the degeneracy factor in reversing the process by the principle\nof detailed balance. Note that the Pauli-blocking factor is ignored for the same reason as\nEq. (A6). It gives the overall formula\nΓµ−+µ+→Z′≈g2\nZ′m2\nZ′\n4πω\u0012\n1 +2m2\nµ\nm2\nZ′\u0013s\n1−4m2\nµ\nm2\nZ′×e−ω/T. (A11)\nAccording to the principle of detailed balance, the pair-coalescence rate Γ νµ+¯νµ→Z′is\nsimply related to Γlab\nZ′→νµ+¯νµby\nΓνµ+¯νµ→Z′= 3Γlab\nZ′→νµ+¯νµe−ω/T\n≈g2\nZ′m2\nZ′\n8πωe−ω/T. (A12)\nSince the neutrino is always left-handed, the number of initial neutrino polarization combi-\nnations is 1, and we do not divide the first equality by a factor of 4. Taking into consideration\nthe production from ντ+ ¯ντ→Z′, the production rate of the new gauge boson Z′through22\nthe pair-coalescence of ν+ ¯ν→Z′is\nΓν+¯ν→Z′≈g2\nZ′m2\nZ′\n4πω×e−ω/T. (A13)\nAppendix B: Z′production and absorption in semi-Comption processes\nWe obtain a simple production rate by investigating and modifying the well-known Comp-\nton process. Since both diagrams are very similar, the semi-Compton cross-section of Z′\nproduction could be estimated by the usual Compton process.\nγµ−\nµ−γ\nComptonReplace γwith Z′\n− − − − − − − − − − − − →\nγµ−\nµ−Z′\nSemi-Compton\nBy the Klein-Nishina formula, the scattering cross-section is given by\ndσC\ndcosθ=πα2\nm2\nµ\u0012ω′\nω\u0013\u0012ω′\nω+ω\nω′−sin2θ\u0013\n, (B1)\nwhere\nω′=ω\n1 +ω\nmµ(1−cosθ), (B2)\nin which ωandω′represent the energy of initial and final state photons. They are connected\nby the energy and momentum conservation law. assuming the photon energy is much less\nthan the mass of muon, ω≪mµ, the cross-section can be written in a simpler form\ndσC\ndcosθ≈πα2\nm2\nµ\u0000\n1 + cos2θ\u0001\n, (B3)\nand\nσC≈8πα2\n3m2\nµ, (B4)\nwhere αrepresents the electromagnetic coupling constant.\nAssume the mass of the new U(1) gauge boson Z′is much smaller than the mass of muon,\nmZ′≪mµ, one can immediately obtain the cross-section of the semi-Compton process, σsC\nis simply Eq. (B3) with one of the coupling constant replaced by α→α′≡g2\nZ′/(4π), and it23\nis given by\nσsC≈8παα′\n3m2\nµ×r\n1−m2\nZ′\nω2, (B5)\nin which we have multiplied an extra speed factor of the dark boson vZ′=p\n1−m2\nZ′/ω2\nto compensate for oversimplifying the cross-section σsC. The speed factor comes from the\nphase space integration of the final state particle Z′\nZ\nd3k′= 4πZ\nk′2dk′\n= 4πZ\n(ω2−m2)dω\n=Zr\n1−m2\nZ′\nω24πω2dω. (B6)\nNote that only 4 πω2appears in the integrand if the final state particle is a photon. The\nproduction rate is therefore given by\nΓprod\nsC=nµnγvµγσsC (B7)\n=nµZd3pγ\n(2π)31\neEγ/T−1σsC. (B8)\nSince the SN neutrino energies are mostly distributed in a few tenths of MeV, we are safe\nto make an assumption that the energy of incident photons is actually much smaller than\nthe mass of muons, i.e. Eγ≪mµ. Consequently, the final state muons almost don’t recoil\nand the energy of muon doesn’t change after the scattering. Therefore, the energy of the\ninitial state photon approximately equals the energy of the final state muon Eγ≈ωdue to\nthe energy conservation law. The phase space integral in nγcan be integrated out with the\nenergy conservation delta function, and the Boltzmann distribution function in the integrand\nis moved to the later Z′phase space integral, resulting\nΓprod\nsC≈nµ1\neω/T−1σsC (B9)\n≈8παα′\n3m2\nµnµ\neω/T−1r\n1−m2\nZ′\nω2. (B10)\nFinally, we apply a simple correction to account for the Pauli-blocking factor of muons by\nmultiplying the above formula with a degeneracy factor Fdeg. Therefore,\nΓprod\nsC≈8παα′\n3m2\nµnµFdeg\neω/T−1r\n1−m2\nZ′\nω2, (B11)24\nwhere Fdegis defined as\nFdeg≡1\nnth\nµZ2d3p\n(2π)31\ne(E−µµ)/T+ 1\u0012\n1−1\ne(E−µµ)/T+ 1\u0013\n, (B12)\nin which\nnµ=Z2d3p\n(2π)3\u00121\ne(E−µµ)/T+ 1−1\ne(E+µµ)/T+ 1\u0013\n, (B13)\nand\nnth\nµ=Z2d3p\n(2π)31\ne(E−µµ)/T+ 1, (B14)\nin which the prefactor 2 in the phase space integrand represents the spin degree of freedom\nof muons. nµandnth\nµare the net muon and thermal muon number density respectively.\nThe idea of Fdegis simply inserting the Pauli-blocking factor back into the integrand of\nnth\nµ, this method is applicable only under the assumption that the final state muon does\nnot recoil. Eq. (B11) can also be derived by using its inverse process together with the\nprinciple of detail balance, the speed factor vZ′=p\n1−m2\nZ′/ω2will show up clearly during\nthe derivation also.\nInstead of applying the Klein-Nishina formula directly, a more sophisticated and complete\ncalculation can be done, with some assumptions, from the general formula of the event rate\nphase space integral. In this way, we will be able to realize where the degeneracy term Fdeg\ncomes from.\np1p2 q\np3\nγµ−\nµ−Z′\np1p2\np3q\nγµ−\nµ−Z′\nFIG. 7: Feynman diagrams of Z′production in semi-Compton process with momentum\nlabels. t channel on the left and s channel on the right.\nThe Feynman diagram with momentum labels is given in Fig. (7) for later calculation con-\nvenience. Momentum labels p1, p2, p3andqare used to represent the 4-momentum of initial\nstate particles γ, µ−and final state particles µ−, Z′respectively. The event rate formula is25\ngiven by\ndΓprod\nsC=1\n2ω|Mprod\nsC(s, t)|2(2π)4δ4(q+p3−p1−p2)\n×d3p1\n2E1(2π)3f1(E1)2d3p2\n2E2(2π)3f2(E2)2d3p3\n2E3(2π)3(1−f3(E3)), (B15)\nin which the pre-factor 2 in front of the initial and final state muon phase space d3p2and\nd3p3represent the spin degree of freedoms of Dirac particles. Multiplying the phase space\nintegral of the new gauge boson Z′(1 +fq(ω))d3q/(2π)3on both sides. We obtain\nd3q\n(2π)3(1 +fq(ω))dΓprod\nsC\n=|Mprod\nsC(s, t)|2(2π)4δ4(q+p3−p1−p2)\n×d3p1\n2E1(2π)3f1(E1)2d3p2\n2E2(2π)3f2(E2)2d3p3\n2E3(2π)3(1−f3(E3))d3q\n2ω(2π)3(1 +fq(ω))\n=|Mprod\nsC(s, t)|2(2π)4δ4(q+p3−p1−p2)\n×d3p1\n2E1(2π)3f1(E1)(1 + fq(ω))2d3p2\n2E2(2π)3f2(E2)(1−f3(E3))2d3p3\n2E3(2π)3d3q\n2ω(2π)3.\n(B16)\nSince the temperature of a SN is around a few tenths of MeV, it can assume that the energy\nof the incident photon γis much less than the mass of muon mµ≈105 MeV. In other\nwords, the muon almost doesn’t recoil after the collision E3≈E2, therefore, f3(E3)≈\nf2(E2). Furthermore, because the process is considered to be in thermal equilibrium, we\nhave fq(ω)≈f1(E1) by the energy conservation law ω≈E1. With these two assumptions,\nthe final state Pauli blocking as well as the Bose enhancement factors can be moved and\ncombined with the initial state phase space integrals to give\nd3q\n(2π)3(1 +fq(ω))dΓprod\nsC\n≈ |Mprod\nsC(s, t)|2(2π)4δ4(q+p3−p1−p2)\n×d3p1\n2E1(2π)3f1(E1)(1 + f1(E1))2d3p2\n2E2(2π)3f2(E2)(1−f2(E2))2d3p3\n2E3(2π)3d3q\n2ω(2π)3\n=d3p1\n(2π)3f1(E1)(1 + f1(E1))2d3p2\n(2π)3f2(E2)(1−f2(E2))\n×1\n2E12E2|Mprod\nsC(s, t)|2(2π)4δ4(q+p3−p1−p2)2d3p3\n2E3(2π)3d3q\n2ω(2π)3. (B17)26\nBy replacing one αtoα′in|Mprod\nsC|we get\nd3q\n(2π)3(1 +fq(ω))dΓprod\nsC\n=d3p1\n(2π)3f1(E1)(1 + f1(E1))×2d3p2\n(2π)3f2(E2)(1−f2(E2))×r\n1−m2\nZ′\nω2σC(s)\n≈d3q\n(2π)3fq(ω)(1 + fq(ω))×nµFdeg×r\n1−m2\nZ′\nω2×8παα′\n3m2\nµ, (B18)\nin which the degeneracy factor Fdegis denoted in green color. One can verify that it is just\nEq. (B12). Finally, the Z′production rate in semi-Compton process Γprod\nsCcan be obtained\nby dividing (1 + fq(ω))d3q/(2π)3on both sides. Results in\nΓprod\nsC≈8παα′\n3m2\nµnµFdeg\neω/T−1r\n1−m2\nZ′\nω2(B19)\nTherefore, the direct calculation from the general formula reveal the form of Eq. (B11) under\nthe assumptions f3(E3)≈f2(E2) and fq(ω)≈f1(E1) as expected.\nTheZ′absorption rate through semi-Compton process can be obtained by applying the\nprinciple of detailed balance on the aforementioned Z′production rate Eq. (B11), giving\nΓabs\nsC=2\n3eω/T×Γprod\nsC\n≈2\n3eω/T×8παα′\n3m2\nµnµFdeg\neω/T−1r\n1−m2\nZ′\nω2. (B20)\nSince a standard model photon γhas only two polarization while Z′has three, the pre-factor\n2/3 accounts for the correct polarization averaging of the initial Z′state in replacing the\ninitial photon state. Results in\nΓabs\nsC≈16παα′\n9m2\nµnµFdeg(1 +1\neω/T−1)r\n1−m2\nZ′\nω2, (B21)\nwhich can be verified by a direct calculation from the event rate general formula under some\nassumptions.27\np2q p3\np4\nµ−Z′\nγµ−\np2q\np4p3\nµ−Z′\nγµ−\nFIG. 8: The Feynman diagrams of Z′absorption in semi-Compton process with momentum\nlabels. t channel on the left and s channel on the right.\nThe absorption rate formula of Z′through the semi-Compton process is given by\ndΓabs\nsC=1\n2ω|Mabs\nsC(s, t)|2(2π)4δ4(q+p2−p3−p4)\n×2d3p2\n2E2(2π)3f2(E2)d3p3\n2E3(2π)3(1−f3(E3))d3p4\n2E4(2π)3(1 +f4(E4)) (B22)\nSince the temperature of SNe is around a few tenths of MeV, which is much smaller than\nthe mass of muon mµ, we assume the energy of the initial new gauge boson is much smaller\nthan the mass of muon ,ω≪mµ. In other words, the muon does not recoil under this\nassumption as the energy of the initial and final state muon does not change. Therefore,\nthe energy of the final state photon approximately equals to the energy of the initial state\nZ′, i.e., E4≈ω. The Bose enhancement factor as well as the Pauli blocking factor of the\nfinal state muon µand photon γcan then be represented by the initial state muon and Z′\nenergies. Eventually, they can be factored out from the final state phase-space integral and\ncombined with the initial state phase-space integral. The equation above becomes\ndΓabs\nsC≈1\n2ω|Mabs\nsC(s, t)|2(2π)4δ4(q+p2−p3−p4)\n×(1 +f4(ω))2d3p2\n2E2(2π)3f2(E2)(1−f2(E2))d3p3\n2E3(2π)3d3p4\n2E4(2π)3(B23)\n= (1 + f1(ω))2d3p2\n(2π)3f2(E2)(1−f2(E2))\n×1\n2ω2E2|Mabs\nsC(s, t)|2(2π)4δ4(q+p2−p3−p4)d3p3\n2E3(2π)3d3p4\n2E4(2π)3. (B24)\nSince the semi-Compton process dominates the constraint only in small mZ′, in this region,\nthe amplitude square |Mabs\nsC(s, t)|2≈(2/3)× |M C(s, t)|2. The pre-factor 2 /3 accounts for\nthe correct polarization averaging in replacing γbyZ′. Therefore,\ndΓabs\nsC≈(1 +f1(ω))×2d3p2\n(2π)3f2(E2)(1−f2(E2))×2\n3|v1−v2|σC. (B25)28\nConsequently,\nΓabs\nsC=2\n3(1 +f1(ω))×nµFdeg×r\n1−m2\nZ′\nω2σC(α2→ααZ′), (B26)\nwhere the distribution f1(ω) is the Bose-Einstein distribution and σC(α2→ααZ′)≈\n8παα Z′/(3m2\nµ). Finally, the Z′absorption rate through a semi-Compton process is\nΓabs\nsC≈16παα Z′\n9m2\nµ(1 +1\neω/T−1)nµFdegr\n1−m2\nZ′\nω2. (B27)\nAppendix C: Loop Bremsstrahlung\nFeynman diagrams for Z′production from loop Bremsstrahlung are given in Fig. 9. The\nemission rate of Z′from n−ploop Bremsstrahlung is related to the matrix element by\ndΓbrem=1\n2ωd3p1f(p1)\n(2π)32E1d3p2f(p2)\n(2π)32E2d3p3(1−f(p3))\n(2π)32E3d3p4(1−f(p4))\n(2π)32E4\n×(2π)4δ4(p1+p2−p3−p4−k)|M|2\nnpγ\n≈1\n2ωd3p1f(p1)\n(2π)32E1d3p2f(p2)\n(2π)32E2d3p3\n(2π)32E3d3p4\n(2π)32E4\n×(2π)4δ4(p1+p2−p3−p4−k)|M|2\nnpγ (C1)\nIt can be written in our shorthand notation for later calculation convenience\ndΓbrem=1\n2ωdi,2\nLIPSdf,2\nLIPS∆4\n5|M|2\nnpγ (C2)\ndΓbremdγ\nLIPS=1\n2ωdi,2\nLIPSdf,3\nLIPS∆4\n5|M|2\nnpγ\n=1\n2ωdi,2\nLIPSdσnpγF, (C3)\np2p1\np4k\np3\nnp\nnpγ\nBremsstrahlungkk+p\nk\npµ, τ γZ′\nLoop\nFIG. 9: Feynman diagrams indicating Z′production through n−ploop Bremsstrahlung.29\nwhere F= 2E1E2vrel,di,2\nLIPSanddf,2\nLIPSare the initial and final state Lorentz invariant phase\nspace integral, and ∆4\n5≡(2π)4δ4(p1+p2−p3−p4−k). With the Soft Radiation Approxi-\nmation (SRA) [5, 35], the cross section of Bremsstrahlung dσnpγcan be expressed in n−p\nelastic collision cross-section dσnpby\ndσnpγ= 4παϵ2dγ\nLIPS(ϵµJµ)2dσnp. (C4)\nExpressing dσnpγindσnp, the emission rate is\ndΓSRA\nbrem=1\n2ωdi,2\nLIPSdf,2\nLIPS(4παϵ2dγ\nLIPS(ϵµJµ)2∆4\n4|M|2\nnp) (C5)\n=4παϵ2\n2ωdi,2\nLIPSdf,2\nLIPS(ϵµJµ)2∆4\n464π2E2\ncmdσnp\ndΩcm\n=2π\nωαϵ2di,2\nLIPSdf,2\nLIPS(ϵµJµ)2∆4\n432πE2\ncmdσnp\ndθcm. (C6)\nIn which, Ecmand Ω cmdenote the total energy of the initial particles and the solid angle in\nthe center-of-momentum frame, respectively. The differential cross-sectiondσnp\ndθcmis calculated\nbased on the Reid93 nucleon-nucleon potential [36] (the result can be found at https://nn-\nonline.org). Rewriting it in the usual notation, it becomes\ndΓSRA\nbrem=2π\nωαϵ2d3p1f(p1)\n(2π)32E1d3p2f(p2)\n(2π)32E2d3p3\n(2π)32E3d3p4\n(2π)32E4\n×(ϵµJµ)2(2π)4δ4(p1+p2−p3−p4)32πE2\ncmdσnp\ndθcm,(C7)\nwhere δ4(p1+p2−p3−p4) =δ(Ecm−E3−E4)δ3(⃗ p3+⃗ p4) and the energy E=p\nM2+|⃗ p|2≈\nM+|⃗ p|2/(2M) in non-relativistic limit. The three-dimensional delta function ensures E3=\nE4. Therefore E1=E2=E3=E4=Ecm/2. The delta function can be rewritten in\nδ4(p1+p2−p3−p4) =δ(Tcm−|⃗ p3|2\nM)δ3(⃗ p3+⃗ p4)\n=M\n2|⃗ p3|δ(|⃗ p3| −p\nMT cm)δ3(⃗ p3+⃗ p4) (C8)\nWhere Tcm=Ecm−2M. By expressing Eiin terms of Ecmand substituting the 4-dimensional\ndelta function with the aforementioned result, the integration of p4can be carried out,\nleading to the determination of the emission rate as\ndΓSRA\nbrem=αϵ2\n2πωE2\ncmd3p1f(p1)\n(2π)3Ecmd3p2f(p2)\n(2π)3Ecmd3p3\n×(ϵµJµ)2M\n2|⃗ p3|δ(|⃗ p3| −p\nMT cm)32πE2\ncmdσnp\ndθcm30\n=αϵ2\n2πωd3p1f(p1)\n(2π)3Ecmd3p2f(p2)\n(2π)3Ecmp2\n3dp3\n×dΩ3(ϵµJµ)2M\n2|⃗ p3|δ(|⃗ p3| −p\nMT cm)32πdσnp\ndθcm\n=αϵ2M3/2\n4π5ωd3p1f(p1)\nEcmd3p2f(p2)\nEcmdθcm(ϵµJµ)2p\nTcmdσnp\ndθcm(C9)\n≈αϵ2\n16π5ωM1/2d3p1f(p1)d3p2f(p2)dθcm(ϵµJµ)2p\nTcmdσnp\ndθcm. (C10)\nSince we are working in a non-relativistic limit, we take Ecm≈2Min the last equality.\nNow, we take the Maxwell-Boltzmann distribution as the energy distribution of the initial\nnucleons such that\nf(p) =n\n2\u00122π\nMT\u00133/2\ne−|⃗ p|2/(2MT). (C11)\nThe emission rate can then be written as\ndΓSRA\nbrem=nnnpαϵ2\n8π2ωM7/2T3d3p1d3p2e−(|⃗ p1|2+|⃗ p2|2)/(2MT)(ϵµJµ)2p\nTcmdσnp\ndθcmdθcm. (C12)\nWe define ⃗ p+≡⃗ p1+⃗ p2and⃗ p−≡⃗ p1−⃗ p2such that one of the momenta is perpendicular\ntoTcmand can be integrated out immediately. Here |⃗ p−|2= 4MT cmand⃗ p+is irrelevant to\nTcm. We can immediately work out\nd3p1d3p2=d3p+d3p−\n8(C13)\n|⃗ p1|2+|⃗ p2|2=|⃗ p+|2+|⃗ p−|2\n2. (C14)\nSince ⃗ p+is irrelevant to Tcm,d3p+can be integrated out by using the Gaussian integral,\ngiving\nZ∞\n0|⃗ p+|2e−|⃗ p+|2/(4MT)dp+= (MT)3/2√\n4π. (C15)\nTherefore, the emission rate in the new coordinate system is\ndΓSRA\nbrem=nnnpαϵ2\n64π2ωM7/2T3d3p+d3p−e−(|⃗ p+|2+|⃗ p−|2)/(4MT)(ϵµJµ)2p\nTcmdσnp\ndθcmdθcm\n=nnnpαϵ2√\n4π\n16πωM2T3/2d3p−e−|⃗ p−|2/(4MT)(ϵµJµ)2p\nTcmdσnp\ndθcmdθcm\n=nnnpαϵ2√\n4π\n4ωM2T3/2|⃗ p−|2dp−e−|⃗ p−|2/(4MT)(ϵµJµ)2p\nTcmdσnp\ndθcmdθcm\n=nnnpαϵ2√\n4π\nωMT3/2dp−e−|⃗ p−|2/(4MT)(ϵµJµ)2(Tcm)3/2dσnp\ndθcmdθcm31\n=nnnpαϵ2√\n4π\nω√\nMT3/2dTcme−Tcm/T(ϵµJµ)2Tcmdσnp\ndθcmdθcm, (C16)\nin which the polarization sum of the massive gauge boson Z’ in ( ϵµJµ)2in given by.\n(ϵµJµ)2=X\niϵµ\niϵν\niJµJν\n=\u0012\n−gµν+kµkν\nmZ′\u0013\nJµJν. (C17)\nRecall that\nJµ=\u0012pµ\n1\np1k−pµ\n2\np2k\u0013\n. (C18)\nTherefore, the second term in the parenthesis vanishes. Such that\n(ϵµJµ)2=−gµνJµJν=−J2\n=−\u0012M2\np1k+M2\np2k−p1p2\n(p1k)(p2k)\u0013\n. (C19)\nSince we’re in non-relativistic limit ( |⃗ p| ≪M), toO(1/M2),\n(ϵµJµ)2≈1\nω2\n(⃗ p1−⃗ p3)2\nM2− \n(⃗ p1−⃗ p3)·⃗k\nMω!2\n. (C20)\nIn this simple form, the average over all Z′emission angle can be done easily\n1\n4πZ\ndΩZ′(ϵµJµ)2≈1\n4πZ\ndΩZ′1\nω2\n(⃗ p1−⃗ p3)2\nM2− \n(⃗ p1−⃗ p3)·⃗k\nMω!2\n\n=(⃗ p1−⃗ p3)2\nω2M2 \n1−|⃗k|2\n3ω2!\n=|⃗ p1|2+|⃗ p2|2−2|⃗ p1||⃗ p2|cosθcm\nω2M2 \n1−|⃗k|2\n3ω2!\n(C21)\n≈2|⃗ p1|2\nω2M2 \n1−|⃗k|2\n3ω2!\n(1−cosθcm) (C22)\n=2Tcm\nω2M \n1−|⃗k|2\n3ω2!\n(1−cosθcm). (C23)\nIn the soft limit |⃗ p1| ≈ |⃗ p2|, it has been applied to the last two equalities. Insert it into the\nemission rate dΓbremresult in\ndΓSRA\nbrem=nnnpαϵ2√\n4π\nω√\nMT3/2dTcme−Tcm/T(ϵµJµ)2Tcmdσnp\ndθcmdθcm32\n=4nnnpαϵ2√π\nω3(MT)3/2dTcme−Tcm/TT2\ncm \n1−|⃗k|2\n3ω2!\nσ(2)\nnp(Tcm), (C24)\nwhere the definition of σ(2)\nnp(Tcm) is\nσ(2)\nnp(Tcm)≡Z1\n−1(1−cosθcm)dσnp\ndcosθcmdcosθcm. (C25)\nThe emissivity can be calculated by\n˙ϵ≡Z Zd3k\n(2π)3ωdΓSRA\nbrem (C26)\n=4nnnpαϵ2√π\n(MT)3/2Z Zd3k\n(2π)31\nω2dTcme−Tcm/TT2\ncm \n1−|⃗k|2\n3ω2!\nσ(2)\nnp(Tcm) (C27)\n=2nnnpαϵ2\n(πMT )3/2Z Z\ndk|⃗k|2\nω2 \n1−|⃗k|2\n3ω2!\ndTcme−Tcm/TT2\ncmσ(2)\nnp(Tcm). (C28)\nFrom the conservation of energy, the ⃗kphase space integral should be integrated with an\nupper boundp\nT2\ncm−m2\nZ′. The integral becomes\nZ√\nT2cm−m2\nZ′\n0dk|⃗k|2\nω2 \n1−|⃗k|2\n3ω2!\n=Z√\nT2cm−m2\nZ′\n0dk|⃗k|2\nk2+m2\nZ′ \n1−|⃗k|2\n3(k2+m2\nZ′)!\n(C29)\n=Tcm\n2I(mZ′\nTcm), (C30)\nwith\nI(x)≡4\n3(1−x2\n4)√\n1−x2−xtan−1(√\n1−x2\nx). (C31)\nTherefore, the emissivity can be finally written in\n˙ϵ=nnnpαϵ2\n(πMT )3/2Z\ndTcme−Tcm/TT3\ncmσ(2)\nnp(Tcm)I(mZ′\nTcm). (C32)\nFrom the equations above, the emissivity is given by\n˙ϵ≡Z Zd3k\n(2π)3ωdΓSRA\nbrem (C33)\n=2nnnpαϵ2\n(πMT )3/2Z Z\ndk|⃗k|2\nω2 \n1−|⃗k|2\n3ω2!\ndTcme−Tcm/TT2\ncmσ(2)\nnp(Tcm). (C34)\nBy changing the variable from k→ω, the integration over momentum kcan be expressed\nin\nZ√\nT2cm−m2\nZ′\n0dk|⃗k|2\nω2 \n1−|⃗k|2\n3ω2!\n=Z√\nT2cm−m2\nZ′\n0dkω2−m2\nd\nω2\u0012\n1−ω2−m2\nd\n3ω2\u0013\n(C35)33\n=Z√\nT2cm−m2\nZ′\n0dωωp\nω2−m2\ndω2−m2\nd\nω2\u0012\n1−ω2−m2\nd\n3ω2\u0013\n(C36)\n=ZTcm\nmddωp\nω2−m2\nd\nω\u0012\n1−ω2−m2\nd\n3ω2\u0013\n. (C37)\nFurthermore, the double integral\nZ∞\nmdZTcm\nmddωdT cm...=Z∞\nmdZ∞\nωdTcmdω... (C38)\nTherefore, the loop-Bremsstrahlung emissivity spectrum d˙ϵ/dω is\nd˙ϵ\ndω=2nnnpαϵ2\n3(πMT )3/2r\n1−\u0010md\nω\u00112\u0014\n2 +\u0010md\nω\u00112\u0015Z∞\nωdTcme−Tcm/TT2\ncmσ(2)\nnp(Tcm). (C39)\nAppendix D: Inverse loop bremsstrahlung\nIn the soft limit, the only difference between the production and absorption rate in\nthe loop Bremsstrahlung process is the number of initial state polarization combinations.\ndΓSRA, abs\nbrem gain an extra factor 1 /3 during the the polarization average in the amplitude\nsquared comparing to dΓSRA\nbrem. Recall that the production rate of Z′is given by Eq. (C24),\nit is related to the absorption rate dΓSRA, abs\nbrem by\ndΓSRA, abs\nbrem =dΓSRA\nbrem\n3\n=nnnpαϵ2√\n4π\n3ω√\nMT3/2dTcme−Tcm/T(ϵµJµ)2Tcmdσnp\ndθcmdθcm\n=4nnnpαϵ2√π\n3ω3(MT)3/2dTcme−Tcm/TT2\ncm \n1−|⃗k|2\n3ω2!\nσ(2)\nnp(Tcm) (D1)\nIn which σ(2)\nnp(Tcm) is a shorthand notation defined by\nσ(2)\nnp(Tcm)≡Z1\n−1(1−cosθcm)dσnp\ndcosθcmdcosθcm (D2)\nThe mean free path λattis simply related to the absorption rate according to\n1\nλatt=ΓSRA, abs\nbrem\nv=ΓSRA, abs\nbremp\n1−(mZ′/ω)2\n=Z4nnnpαϵ2√π\n3ω3(MT)3/2dTcme−Tcm/TT2\ncm\u00122\n3+m2\nZ′\n3ω2\u0013\nσ(2)\nnp(Tcm)/v34\n=4nnnpαϵ2√π\n3ω3(MT)3/21\n3 \n2 + (mZ′/ω)2\np\n1−(mZ′/ω)2!Z\ndTcme−Tcm/TT2\ncmσ(2)\nnp(Tcm) (D3)\nIn which vis the speed of Z′. Above formula can be written in a simpler form by defining a\ndimensionless variable x≡Tcm/T. The mean free path of the inverse loop Bremsstrahlung\nprocess is\n1\nλatt=8nnnpαϵ2\n9πω3\u0012πT\nM\u00133/2 \n2 + (mZ′/ω)2\np\n1−(mZ′/ω)2!\n1\n2Z∞\n0dxe−xx2σ(2)\nnp(x) (D4)\nAppendix E: The Plateau Phenomenon\nTo understand the consequences of taking Rfar=Rνin the upper integration limit of\nboth emissivity and attenuation calculation, let us consider the case of a huge coupling\nsuch that λatt(ω)≪Rν. Since the interaction length decreases as the coupling constant\nincreases, the new gauge boson Z′produced deep inside the neutrinosphere will have less\nchance of escaping the sphere than those produced near the surface. Generally speaking,\nonly those Z′produced at a thin shell of the thickness of λatt(ω) have the chance to leave the\nneutrinosphere and contribute to the luminosity. Since Z′bosons produced within this thin\nshell can propagate either out or into the neutrinosphere, only a half of the total number\nofZ′might contribute to the luminosity. Lastly, projecting the trajectory of those particles\nto ensure that they are propagating outward, we have the luminosity ∆ L∞(mZ′) for Z′in a\nsmall energy range ∆ ω\n∆L∞(mZ′, ω) =πR2\nνλatt(mZ′, Rν, ω)d˙ϵ\ndω(mZ′, Rν, ω)∆ω (E1)\nThe overall luminosity L∞(mZ′) can be calculated by sum over all energy contribution.\nIntegrating the above equation with respect to all possible energy ωyields\nL∞(mZ′) =πR2\nνZ∞\nmZ′λatt(mZ′, Rν, ω)d˙ϵ\ndω(mZ′, Rν, ω)dω (E2)\nA pictorial demonstration for the derivation of L∞(mZ′) above is given in Fig. 10. Clearly\nL∞(mZ′) will tend to a given value according to Eq. (E2) instead of monotonically decreasing\ndown to zero in the large gZ′limit. This is true since λatt(ω) scales as g−2\nZ′while d˙ϵ/dω scales\nasg2\nZ′. It is important to note that L∞(mZ′) given by Eq. (E2) agrees completely with our\nfull numerical result of LZ′(mZ′) in the large gZ′limit.35\nneutrino sphere RνZ′within this region\ncan have a chance to\nescape the sphere.\nθ1/2parallel to the\noutgoing direction\n1/2\noutgoing\nλatt\nL∞(mZ′)\n= 4πR2\nνZ∞\nmZ′λatt(mZ′, Rν, ω)\n2×2d˙ϵ\ndω(mZ′, Rν, ω)dω\n=πR2\nνZ∞\nmZ′λatt(mZ′, Rν, ω)d˙ϵ\ndω(mZ′, Rν, ω)dω\nFIG. 10: A pictorial demonstration for the derivation of L∞(mZ′)." }, { "title": "2401.16047v1.Validation_of_symmetry_induced_high_moment_velocity_and_temperature_scaling_laws_in_a_turbulent_channel_flow.pdf", "content": "Validation of symmetry-induced high moment velocity and temperature scaling\nlaws in a turbulent channel flow\nFrancisco Alcántara-Ávila,1,a)Luis Miguel García-Raffi,1Sergio Hoyas,1and Martin\nOberlack2\n1)Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de\nValència, Valencia 46022, Spain\n2)Chair of Fluid Dynamics, TU Darmstadt, Otto-Bernd-Strasse 2, 64287 Darmstadt,\nGermanyb)\n(*Electronic mail: oberlack@fdy.tu-darmstadt.de)\n1arXiv:2401.16047v1 [math-ph] 29 Jan 2024The symmetry-based turbulence theory has been used to derive new scaling laws for the\nstreamwise velocity and temperature moments of arbitrary order. For this, it has been\napplied to an incompressible turbulent channel flow driven by a pressure gradient with a\npassive scalar equation coupled in. To derive the scaling laws, symmetries of the classi-\ncal Navier-Stokes and the thermal energy equations have been used together with statis-\ntical symmetries, i.e. the statistical scaling and translation symmetries of the multi-point\nmoment equations. Specifically, the multi-point moments are built on the instantaneous\nvelocity and temperature fields other than in the classical approach, where moments are\nbased on the fluctuations of these fields. With this instantaneous approach, a linear sys-\ntem of multi-point correlation equations has been obtained, which greatly simplifies the\nsymmetry analysis. The scaling laws have been derived in the limit of zero viscosity and\nheat conduction, i.e. Reτ→∞andPr>1, and apply in the centre of the channel, i.e.\nthey represent a generalization of the deficit law so herewith extending the work of Ref. 1.\nThe scaling laws are all power laws, with the exponent of the high moments all depending\nexclusively on those of the first and second moments. To validate the new scaling laws, the\ndata from a large number of DNS for different Reynolds and Prandtl numbers have been\nused. The results show a very high accuracy of the scaling laws to represent the DNS data.\nThe statistical scaling symmetry of the multi-point moment equations, which characterizes\nintermittency, has been the key to the new results since it generates a constant in the ex-\nponent of the final scaling law. Most important, since this constant is independent of the\norder of the moments, it clearly indicates anomalous scaling.\na)Presently at: FLOW, Engineering Mechanics, KTH Royal Institute of Technology, 114 28 Stockholm, Sweden.\nEmail: fraa@kth.se\nb)Also at: Centre for Computational Engineering, TU Darmstadt, Dolivostrasse 15, 64293 Darmstadt, Germany\n2I. INTRODUCTION\nThe open problem in physics with most applications in daily life is probably the behaviour of\nturbulent flows. Different strategies have been proposed when dealing with predicting turbulence\nin engineering. From the different approaches known in Computational Fluid Dynamics (CFD),\nDirect Numerical Simulations (DNS) has proven to be a powerful tool to generate highly reliable\ndata bases for theoretical concepts on the nature of turbulence. In a DNS, no empirical modelling\nis needed to account for turbulent effects, and the approximations of the solutions of the Navier-\nStokes equations are obtained through highly accurate numerical schemes. The main problematic\nissue of DNSs is their high computational cost since even the smallest scales of turbulence, the\nKolmogorov scales, have to be simulated. Hence, this limits DNSs to very simple canonical\ngeometries. However, DNSs have the same validity as experiments, and almost any imaginable\nquantity can be computed.\nIt was not until the late 1980s that supercomputers could run the first DNS of turbulent flows.\nSpecifically, in 1987, Kim et al.2conducted the first DNS of a turbulent channel flow where a\npressure gradient drove the flow at a low Reynolds number. The first DNS of a thermal turbulent\nchannel flow was also performed in 1987 by Kim et al.3. The flow was also driven by a pressure\ngradient at a friction Reynolds number of Reτ=180. Therein and presently, Reτis defined as\nhuτ/ν, where his the semi-height of the channel, uτis the friction velocity, and νis the kinematic\nviscosity of the fluid. The friction velocity is defined as uτ=p\nτw/ρ, where τwis the averaged\nwall shear stress, and ρis the fluid density. Different Prandtl numbers were used, namely Pr=0.1,\n0.71 and 2. Here the Prandtl number is defined as the ratio between the momentum diffusivity to\nthe thermal diffusivity, Pr=ν/α, where αis the thermal diffusivity of the fluid. One of the main\nresults from the latter work was the validation of the DNS results comparing several first-order\nstatistics with experimental data.\nSince then, the aim of the DNS of thermal channel flows has been to increase the simulated\nReynolds number, usually around Pr=0.71, which is the Prandtl number of the air. However,\nreaching higher Reynolds numbers has a computational cost which scales as L2\nxLzRe4\nτPr3/2, ac-\ncording to Ref. 4. The largest DNS to date of a thermal channel flow5used a friction Reynolds\nnumber of 4000, and more recently, for isothermal turbulent channel flows, a friction Reynolds\nnumber of Reτ=10000 was reached in Ref. 6. However, these values are still far below the ac-\ntual Reynolds numbers of most real-life problems. Therefore, Reynolds number extrapolations of\n3the turbulent behaviour must be made, introducing inevitable errors and uncertainties. For exam-\nple, the viscous diffusion and dissipation of the streamwise velocity fluctuations present a scaling\nfailure near the wall7. An analogous scaling failure was obtained for the temperature variance\nat moderate Reynolds numbers. However, it was recently found8that for high Prandtl numbers\nand sufficiently high Reynolds numbers, the value of the viscous diffusion and dissipation of the\ntemperature variance presented a much better scaling near the wall. Therefore, it is still an open\nquestion whether the streamwise velocity scaling failure will occur at much higher Reynolds num-\nbers.\nFor all these reasons, turbulence is and will be, for many more years, an open problem without a\ncomplete analytical solution. Many researchers have proposed what are called turbulence scaling\nlaws to describe the universal behaviour of turbulent statistics for special flows, though usually\nlimited to the first and second moments. The most well-known scaling law is the universal law of\nthe wall, which describes the profile of the mean streamwise velocity near the wall, consisting of a\nlinear viscous sub-layer, where U+=y+, followed by the buffer layer and the logarithmic region\nfurther away from the wall. The overbar here denotes averaged in time, and the superscript+refers\nto dimensionless variables normalized in wall units with uτ,θτandν, where θτ=qw/(ρcpuτ)is\nthe friction temperature, and qwandcpare the normal heat flux to the walls and the specific heat\nat constant pressure, respectively. Analogously to the law of the wall for the velocity, one can\napproximate the mean temperature near the wall in a similar form, where, for the first sub-layer,\ncalled conductive sub-layer, Θ+=Pr y+. In particular, the discovery of a logarithmic behaviour\nof the flow dates back to von Kármán in 1931. However, no connection between the analytical\nform of the scaling law and the Navier-Stokes equation was made. Despite the fact that a perfectly\ndeveloped logarithmic region has not been observed in DNS due to the high Reynolds number\nneeded to be simulated, a clear tendency suggests that it will appear for higher Reynolds numbers\nin different types of flows such as boundary layers, Couette flows, pipe flows,9–16, etc. A first\nderivation of the logarithmic behaviour of the flow, based on first principles, was presented in\nRef. 17. The use of Lie symmetries was the mathematical tool to achieve this.\nLie symmetries are a powerful mathematical theory to develop turbulent flow scaling laws. The\norigin of the Lie symmetries method dates back to the end of the 19th century when the mathemati-\ncian Sophus Lie proposed it for obtaining solutions to differential equations and, most importantly,\nto systems of partial differential equations (PDE), such as the Navier-Stokes equations. The basis\nof the method consists of finding the symmetries of the system of PDE. Here, symmetry refers to\n4a variable transformation that leads to an identical system of PDE, i.e. the transformed system of\nPDE has the same solution as the original one. With these symmetries, one can formulate a char-\nacteristic system (see §III for more details about the characteristic system), which in turn, leads to\nwhat is known as invariant solutions, which in turbulence are also known as scaling laws.\nLie symmetries possess several advantageous properties using ad hoc methods for a concrete\napplication. First, symmetries can be obtained using computer algebra methods such as Maple.\nSecond, symmetries give fundamental insight into the physics of the problem. And third, the scal-\ning laws obtained are solutions to the moment equations and, hence, are based on first principles,\nnot just pure curve fits. For these reasons, Lie symmetries are one of the most powerful tools for\nobtaining scaling laws of turbulent flows. Most important, it is also applicable to an infinite num-\nber of equations such as the moment hierarchy, and in this sense the ubiquitous closure problem\nof turbulence can be circumvented. This is also the approach that is presently applied.\nThe method has been widely studied by Oberlack and co-workers in several papers. Starting\nwith Ref. 17, scaling laws for the three regions of wall-parallel shear flows (viscous sub-layer,\nlogarithmic law, and deficit law in the centre of the channel) were obtained. Classical mechanical\nsymmetries of the Navier-Stokes equations were used, but the key to the analysis was to employ the\nMulti-Point Correlation (MPC) equations. Two additional symmetries, not visible in the Navier-\nStokes equations and called statistical symmetries, first discovered in Ref. 18, were used to derive\nthe scaling laws that describe the flow statistics even for high moments. The next section will give\nmore details about the MPC equations (§II). After this successful application of Lie symmetries\nto turbulent flows, several more works have been done with different geometries or boundary\nconditions19–24.\nIn this work, Lie symmetries theory will be used to derive new moment scaling laws of velocity\nand temperature, and mixed moments of arbitrary orders, extending the work in Ref. 1 to include\nthe temperature. To achieve this, symmetries of the energy equation and the MPC equations of the\nenergy and heat fluxes equations are obtained, from which the new scaling laws are formulated.\nThese new scaling laws will be validated using the DNS data obtained by the authors in previous\nworks (see8,25,26). We will restrict ourselves to moments in the streamwise direction. However,\nthis shows the ability of our method to obtain scaling laws following only strong mathematical\narguments.\nIn the next section, the governing equations of the problem are presented. In the third section,\nthe Lie symmetries method is introduced, together with the application of the method to the gov-\n5erning equations of the problem. Then, in section four, the new scaling laws are developed and\nvalidated using DNS data. Finally, the fifth and last section contains conclusions.\nII. GOVERNING EQUATIONS\nThe equations that describe the behaviour of a turbulent flow, considering a Newtonian fluid\nwith constant density and viscosity, are the well-known Navier-Stokes equations. For the sake of\nreadability, the temporal and spatial dependencies will be omitted if uniqueness allows doing so.\nIn the most general form, these equations can be written as\nC(x) =∂Uk\n∂xk=0, (1)\nMi(x) =∂Ui\n∂t+Uk∂Ui\n∂xk+∂P\n∂xi−ν∂2Ui\n∂xk∂xk=0, (2)\nwhere t∈R+is time; xiandUiare the space coordinate and velocity, i=1,2,3; and Pis the\npressure divided by the density. The no-slip boundary condition is applied to both walls, periodic\nboundary conditions are used in the x1andx3directions and, to propel the flow, a constant pressure\ngradient is introduced in the x1direction so that the mass flux remains constant. The x2coordinate\npoints in wall normal direction. Additionally, the thermal energy equation is simulated, which for\na constant thermal conductivity coefficient, α, reads\nE(x) =∂Θ\n∂t+Uk∂Θ\n∂xk−α∂2Θ\n∂xk∂xk=0, (3)\nwhere Θis the temperature. It should be noted that a constant heat flux at the wall was assumed in\nthe simulation of the energy equation (3) since only this implies the temperature-scaling laws. This\nthermal boundary condition is known as the uniform heat flux (UHF) boundary condition. Curious\nreader is referred to Ref. 8, 25–28 for more information about the UHF boundary condition. This\nis similar to the constant wall shear stress, which is central to the velocity-scaling laws. With this,\nsince no heat sink was introduced, a constant temperature gradient in the x1direction is generated.\nThis is removed from the flow by the following transformation to guarantee homogeneity in x1-\ndirection\nΘ=⟨Θw⟩x3−Θtr, (4)\nwhere ⟨Θw⟩x3is the temperature at the wall averaged in time and in the x3direction, and Θtris the\ntransformed temperature. Therefore, ⟨Θw⟩x3carries the linear increment of the temperature and\n6only depends on the x1direction. Then, Θtris homogeneous in this streamwise direction. The\nsomewhat unusual choice of the sign in (4) is due to the fact that the temperature at the wall is\nmaximum and therefore the transformed temperature Θtrremains positive throughout.\nThis allows the use of spectral discretization in the x1direction. Obviously, the scaling of Θ\nandΘtrin the x2direction will be the same since only a constant value will differ among them.\nFurthermore, since the scaling laws are presented as defect laws, for both ΘandΘtr, these scaling\nlaws must be the same. For the sake of generality, and without loss of veracity, we will refer to\nscaling laws of temperature, Θ, instead of transformed temperature.\nUsing the Reynolds decomposition, one can separate the instantaneous variables (capital letter)\nin an average part (capital letter and over-bar) that does not depend on time, and a fluctuation part\n(lower case), e.g., Ui(x,t) =Ui(x) +ui(t,x)(note that temporal and spatial dependencies have\nbeen recovered only to show this example). Therefore, the following properties can be applied:\nthe average in time of a mean quantity, Φ, will remain unchanged, i.e., Φ=Φ; and the average in\ntime of a fluctuation quantity is 0, i.e., φ=0. In addition, the following simplifications are valid\nfor a developed turbulent channel flow driven by a pressure gradient,\nU1=U1(x2),P=P(x1,x2),Θ=Θ(x2),\nU2=U3=0,uiuj=uiuj(x2),uiθ=uiθ(x2). (5)\nIntroducing the latter into equations (2) and (3) and, in turn, taking the average, the governing\nequations reduce to\ndu1u2\ndx2+∂P\n∂x1−νd2U1\ndx2\n2=0, (6)\ndu2u2\ndx2+∂P\n∂x2=0, (7)\ndu3u2\ndx2=0, (8)\ndθu2\ndx2−αd2Θ\ndx2\n2=0. (9)\nBesides the latter one-point quantities, we may define the two-point correlation functions, or\ntwo-point moments, based on the fluctuating velocity,\nRi j(x,r) =ui(x)uj(x+r),R0\ni j(x) =lim\nr→0Ri j(x,r) =ui(x)uj(x). (10)\nEmploying an equivalent definition based on the instantaneous variables reads\nHi j(x,r) =Ui(x)Uj(x+r),H0\ni j(x) =lim\nr→0Hi j(x,r) =Ui(x)Uj(x), (11)\n7and a relation between the two correlation functions Ri jandHi jreads as follows\nRi j(x,r) =Hi j(x,r)−Ui(x)Uj(x+r). (12)\nThis two-point concept can be extended for any number of points and ultimately forms the\nbasis of the following analysis as well as the resulting scaling laws. Hence, we introduce the\nMPC equations (see e.g. Ref. 21–24). For high-order moments of velocity and temperature, they\ngive additional information that is not provided in the one-point statistic equations, such as length\nscales. Also, when deriving a higher-order moment equation, only one unclosed function arises.\nAs observed in equation (10), from the two-point statistics one can obtain every one-point statistic.\nFinally, regarding Lie symmetries, two extra symmetries are obtained from the MPC equations,\nwhich are the key for determining the new scaling laws of the high-order moments, and will be\npointed out in section §IV.\nEquations (10) and (11) are the basis of the two different approaches that can be used to ob-\ntain the MPC equations: the fluctuating approach or the instantaneous approach. On one hand,\nthe fluctuating approach has some advantages such as a straightforward relation to the Reynolds\nstress tensor or the turbulent heat fluxes. However, as noted in Ref. 18, a non-linear system of\nequations is obtained. Furthermore, all moment equations are coupled to the mean velocity or\ntemperature, and equations of the third moment or higher, are coupled to the second moment. All\nthis complicates the symmetry analysis that will be done below. On the other hand, the instanta-\nneous approach results in a linear system of MPC equations with an equivalent but much simpler\nsymmetry analysis. For this reason, the instantaneous approach is the one used in this work. It\nshould be noted that the fluctuating approach and the instantaneous approach are bijective, i.e.\nmathematically physically absolutely equivalent.\nBefore presenting the MPC equations, some notations must be clarified. The correlation func-\ntions for the velocity are defined as\nHi{n}=Hi(1)i(2)...i(n)=Ui(1)(x(1))Ui(2)(x(2))...Ui(n)(x(n)), (13)\nwhich for n=2,x(1)=xandx(2)=x+ryields to (11). Note the in Ui(n)(x(n)), the subscript\ni(n)refers to the velocity direction of the n-th term which is measured at the coordinate x(n). The\ndefinition of the temperature correlation is\nHΘ{m}=HΘ(1)Θ(2)...Θ(m)=Θ(x(1))Θ(x(2))...Θ(x(m)). (14)\n8Mixed moments of velocity and temperature, which in the limit of only one temperature and\none velocity reduces to the turbulent heat flux, reads\nHi{n}Θ{m}=Hi(1)i(2)...i(n)Θ(n+1)Θ(n+2)...Θ(n+m)=\nUi(1)(x(1))Ui(2)(x(2))...Ui(n)(x(n))Θ(x(n+1))Θ(x(n+2))...Θ(x(n+m)). (15)\nNote that (13) and (14) are just particular cases of (15) for mandnequal to 0, respectively,\nbut for the sake of readability, they are presented separately. When pressure is involved in the\ncorrelation, the notation, in the general form, is\nIi{n−1}Θ{m}[l]P=Hi(1)...i(l−1)Pi(l+1)...i(n)Θ(n+1)Θ(n+2)...Θ(n+m)=\nUi(1)(x(1))...P(x(l))...Ui(n)(x(n))Θ(x(n+1))...Θ(x(n+m)), (16)\nfor 1≤l≤n. Finally, the following notation\nHi{n}Θ{m}[i(l)→k](x(l)→x(p)) =\nUi(1)(x(1))...Ui(l−1)(x(l−1))Uk(x(p))Ui(l+1)(x(l+1))...Ui(n)(x(n))Θ(x(n+1))...Θ(x(n+m))(17)\nis used to indicate a change in the correlation function of velocity direction, i(l), tokand/or the\ncoordinate where the variable is applied, x(l), tox(p). With these definitions, the MPC equations\nof the heat flux moments of order n+mreads (see Appendix A for detailed step-by-step derivation\nof the MPC equations)\n∂Hi{n}Θ{m}\n∂t\n+n\n∑\na=1 ∂Hi{n+1}Θ{m}[i(n+m+1)→k](x(n+m+1)→x(a))\n∂xk(a)+∂Ii{n−1}Θ{m}[a]P\n∂xi(a)−ν∂2Hi{n}Θ{m}\n∂xk(a)∂xk(a)!\n+n+m\n∑\nb=n+1 ∂Hi{n+1}Θ{m}[i(n+m+1)→k](x(n+m+1)→x(b))\n∂xk(b)−α∂2Hi{n}Θ{m}\n∂xk(b)∂xk(b)!\n=0. (18)\nAs was mentioned before in (15), the MPC equations of the velocity and temperature moments\nare specific cases of (18), which can be obtained by setting, respectively, mandnequals to 0.\nAdditionally, the continuity equations read\n∂Hi{n}Θ{m}[i(l)→k]\n∂xk(l)=0 for l=1,2,...,n, (19)\n∂Ii{n−1}Θ{m}[a]P[i(l)→k]\n∂xk(l)=0 for a,l=1,2,...,n,a̸=l,and n≥2. (20)\n9Note that pure temperature correlations and heat fluxes correlations with l>ndo not admit\ncontinuity equations since they would have originated from ∂Θ(x)/∂xk, which is not a continuity\nequation.\nAs was previously noted, the system of the MPC equations (18), (19), and (20) is linear for\nany turbulent flow. Moreover, the dependent variables HandIappear inside spatial or temporal\nderivatives. As seen in sections §III C and §IV, this is the key to obtaining two important statistical\nLie symmetries necessary to derive the scaling laws.\nTo make the notation easier to understand, the Two-Point Correlation (TPC) equations for the\nvelocity, heat fluxes, and temperature are presented in Appendix B.\nIII. LIE SYMMETRIES OF THE MPC EQUATIONS\nIn this section, the Lie symmetries method will be presented briefly. After that, the symmetries\nof the governing equations introduced in the previous section will be given.\nA. Symmetry transformations\nGiven a system of partial differential equations (PDE)\nF(x,y,y(1),y(2),...) =0, (21)\nwhere xare the independent variables, yare the dependent variables, and y(n)are the n-th deriva-\ntive of the dependent variables with respect to all coordinate combinations of x. Based on this, a\ntransformation of (21), with the form\nx∗=ϕ(x,y),y∗=ψ(x,y), (22)\nis called a symmetry transformation, or just symmetry, if the following holds:\nF(x,y,y(1),y(2),...) =0⇔F(x∗,y∗,y∗(1),y∗(2),...) =0. (23)\nIn other words, a symmetry transformation (22) leaves the PDEs (21) invariant and, in addition,\nmaps any solution of (21) into a new solution.\nIn Lie group analysis, it is the aim to find all possible symmetry transformations of the PDE\n(21). The notation group refers to the fact that symmetry transformations usually admit group\n10properties. As we are presently dealing with Lie symmetry groups, the group parameter ε∈Rhas\nto be introduced to obtain the so-called one-parameter Lie symmetry group of transformations,\nwith the form\nx∗=ϕ(x,y;ε),y∗=ψ(x,y;ε). (24)\nEquation (24) provides a continuous group of transformations that allows analytical solutions\nfor the underlying equations. As for the group properties of (24), we may, without loss of gener-\nality, assign ε=0 to the identity element, i.e.\nx∗=ϕ(x,y;ε=0) =x,y∗=ψ(x,y;ε=0) =y. (25)\nTherefore, if a Taylor series at ε=0 is applied to the Lie group of transformation (24), we\nobtain\nx∗=x+∂ϕ(x,y;ε)\n∂ε\f\f\f\f\nε=0ε+O(ε2) =x+ξ(x,y)ε+O(ε2), (26)\ny∗=y+∂ψ(x,y;ε)\n∂ε\f\f\f\f\nε=0ε+O(ε2) =y+η(x,y)ε+O(ε2). (27)\nEquations (26) and (27) are the infinitesimal form of the Lie group of transformation (24),\nwhere ξandηare the so-called infinitesimals. The general form of the transformation (24) and\nthe infinitesimal transformations (26) and (27) are related by Lie’s first theorem (see Ref. 29), i.e.\nif the infinitesimals of the transformation, ξandηare known, one can uniquely recover the general\nform of the symmetry group of transformations (24). In order to obtain all symmetries of a given\nsystem of PDEs, one has to invoke the infinitesimal form of the transformations (26) and (27).\nBased on this, an algebraic algorithm, the Lie algorithm, evolves, which has been implemented\ninto various computer algebra systems. Details on the algorithm may be taken from different\ntextbooks like Ref. 29 or in works such as Ref. 20.\nB. Group invariant solutions\nIn addition to the fact that symmetries characterize fundamental physical properties of a system,\nit is the ability to construct solutions that are central to their application, and which will also be\napplied here. For this purpose, we define the so-called group-invariant solutions, i.e. once the Lie\nsymmetries of the system of PDE are obtained, the next step is to generate invariant solutions,\nwhich in turbulence are referred to as scaling laws. We call y=Ψ(x)an invariant solution of a\nPDE system if and only if\n111.y−Ψ(x)is invariant under X, where Xis the so-called infinitesimal generator, defined as\nX=ξi(x,y)∂\n∂xi+ηj(x,y)∂\n∂yj. (28)\nHence, we have\nX(y−Ψ(x)) = 0, (29)\nony=Ψ(x). Using the operator (28) and differentiating out, we obtain the following\nhyperbolic system\nξi(x,Ψ)∂Ψj\n∂ξi=ηj(x,Ψ),i=1,...,k;j=1,...,l, (30)\nwhich generates the invariant solution. The solution of the hyperbolic system (30) can now\nbe determined by the method of characteristics and we obtain the so-called invariant surface\ncondition\ndx1\nξ1=dx2\nξ2=···=dxk\nξk=dy1\nη1=dy2\nη2=···=dyl\nηl, (31)\nwhere kandlare the respective numbers of the independent and dependent variables.\nThe integrals of the system (31) are the characteristics of the hyperbolic system (30) but,\nat the same time, the invariants of the original system of equations. These now form the\nbasis of the invariant solutions and, thus, the new independent and dependent variables - the\nsimilarity variables.\n2. Finally, the invariant solution, y=Ψ(x)has to solve the PDE system, which is to be verified\nby insertion into the original PDE.\nC. Symmetries of the governing equations\nIn this section, we present the Lie symmetries of the governing equations (1)-(3). For high\nReynolds numbers flows, i.e., for Reτ→∞, and sufficiently away from the wall, the viscous\neffects are limited to length-scales of the order of the Kolmogorov length scale. In Oberlack30,\nthis fact forms the basis for a singular asymptotic expansion similar to boundary layer theory.\nAs a result, two sets of moment equations arise, where the equations for the \"outer\" solutions is\nfrictionless and acts on length scales larger than the Kolmogorov scale, while an \"inner\" equation\n12contains friction terms and operates on the Kolmogorov length. As a result, the frictionless \"outer\"\nequation and the corresponding solutions have the symmetries of the Euler equation. The above\ndevelopment again illustrates the fact that although the limit ν→0+can be considered, this is\nnot identical to ν=0. For the following analyses, this means a focus on the large scales and thus\nthatν=α=0 may be set in equation (18), assuming ν∼α, i.e. the diffusion terms are of the\nsame order of magnitude. Incidentally, the dissipation therefore results from the \"inner\" equation,\nwhich is not considered presently.\nFor the case of the Euler equations, a 10-parameter symmetry group of transformation is ob-\ntained, where we here only present the scaling groups needed further below,\nTSx:t∗=t,x∗=eaSxx,U∗=eaSxU,P∗=e2aSxP, (32)\nTSt:t∗=eaStt,x∗=x,U∗=e−aStU,P∗=e−2aStP,. (33)\nThe coefficients aSxandaStare the group parameters of scaling of space and time, respectively.\nIf the Navier-Stokes equations are considered, i.e., the viscous term is not neglected, the two\nscaling symmetries, TSxandTSt, linearly combine into a simple scaling symmetry. This phe-\nnomenon, in which a multi-parameter symmetry group of transformations is reduced after a spe-\ncific condition is applied, is known as symmetry breaking.\nAn analogous simplification, as the transition from the Navier-Stokes to the Euler equation,\napplied to the energy equation (3), can be done by neglecting the diffusive term, i.e., Peτ→\n∞, which holds in the center of the channel. Considering this, the energy equation admits the\nfollowing infinite-dimensional symmetry,\nTΘ:t∗=t,x∗=x,U∗=U,P∗=P,Θ∗=f(Θ). (34)\nFor scaling purposes, and in analogy with the scaling symmetries of the Euler equations, we\nconsider the simplification f(Θ) =eaΘΘ, so that TΘrepresents a scaling of temperature. It should\nbe noted that the energy equation (3), just like the Navier-Stokes equations (1) and (2), admits the\nGalilean group.\nAs noted in Ref. 18, the symmetries obtained for the Navier-Stokes and energy equations trans-\nfer to the MPC equations (18). So, in the limit of zero viscosity and diffusion, i.e. Reτ→∞and\n13Pr>1, the MPC equations (18) admit the following scaling symmetries:\nTSx:t∗=t,x∗=eaSxx,H∗\ni{n}Θ{m}=enaSxHi{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=e(n+1)aSxIi{n−1}Θ{m}[a]P, (35)\nTSt:t∗=eaStt,x∗=x,H∗\ni{n}Θ{m}=e−naStHi{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=e−(n+1)aStIi{n−1}Θ{m}[a]P, (36)\nTSΘ:t∗=t,x∗=x,H∗\ni{n}Θ{m}=emaΘHi{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=emaΘIi{n−1}Θ{m}[a]P, (37)\nwhich are immediate consequences of (32), (33) and the scaling version of (34).\nIn addition to the symmetries induced from the Navier-Stokes/Euler and energy equations, the\nMPC equations (18) admit an extended set of symmetry transformations. These symmetries are\ncalled statistical symmetries and they are the key in the process of deriving scaling laws21for high-\norder moments of the velocity and temperature. These symmetries were discovered in Ref. 18 and\ndetailed information on the physical meaning of the statistical symmetries can be found in Ref. 31.\nFirst, because of the linearity of the MPC equations (18), a scaling symmetry of the dependent\nvariables is admitted\nTSs:t∗=t,x∗=x,H∗\ni{n}Θ{m}=eaSsHi{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=eaSsIi{n−1}Θ{m}[a]P. (38)\nThis symmetry, as proven in Ref. 31, represents a measure of intermittency. For intermittency,\nwe understand a flow with subsequently changing turbulent and non-turbulent regimes. Moreover,\nas all dependent variables in (18) appear inside derivatives, a translation symmetry of all moments\nis also admitted\nTtra,H:t∗=t,x∗=x,H∗\ni{n}Θ{m}=Hi{n}Θ{m}+aH\ni{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=Ii{n−1}Θ{m}[a]P+aI\ni{n��1}Θ{m}. (39)\nApart from the symmetries presented, we will also include the classical translation in space\nsymmetry, i.e.\nTtra,x:t∗=t,x∗=x+ax,H∗\ni{n}Θ{m}=Hi{n}Θ{m},\nI∗\ni{n−1}Θ{m}[a]P=Ii{n−1}Θ{m}[a]P. (40)\n14Note that, in contrast to (38), where aSsis a single group parameter, symmetries (39) and (40)\nare a condensed way of showing several symmetries. Each component of the vector and tensors\naH\n{n}{m},aI\n{n−1}{m}andaxrepresent the group parameter of different and independent symmetries.\nTherefore, infinite symmetries are contained in (39), while (40) contains three symmetries, one for\neach spatial direction.\nIn summary, six symmetries have been identified, that will be used to derive the scaling laws\nof high-order moments of velocity and temperature. One property of the Lie symmetries is that\none can combine different one-parameter Lie symmetries into a multi-parameter Lie symmetry.\nFollowing this, one can obtain the following multi-parameter Lie symmetry group from the sym-\nmetries (35)-(40)\nT:t∗=eaStt,x∗=eaSxx+ax,\nH∗\ni{n}Θ{m}=en(aSx−aSt)+maΘ+aSsHi{n}Θ{m}+aH\ni{n}Θ{m}, (41)\nI∗\ni{n−1}Θ{m}[a]P=e(n+1)(aSx−aSt)+maΘ+aSsIi{n−1}Θ{m}[a]P+aI\ni{n−1}Θ{m}.\nA different way of writing the symmetry group (41) is using the infinitesimal notation (26) and\n(27), from which one obtains\nξt=aStt,ξx=aSxx+ax,\nηHi{n}Θ{m}= [n(aSx−aSt)+maΘ+aSs]Hi{n}Θ{m}+aH\ni{n}Θ{m}, (42)\nηIi{n−1}Θ{m}[a]P= [(n+1)(aSx−aSt)+maΘ+aSs]Ii{n−1}Θ{m}[a]P+aI\ni{n−1}Θ{m},\nand which will be used below in the next chapter.\nIV . HIGH ORDER MOMENT SCALING LAWS AND ITS VALIDATION\nA. New velocity, temperature and mixed moment scaling laws\nUsing the symmetries (41), or rather its infinitesimal form (42), we are now able to compute the\ninvariant solutions of the equations (18), which in turbulence are called the turbulent scaling laws.\nIt should be noted here that in the following we only consider the moments of the instantaneous\nvariables, i.e. the Happroach. Of course, a conversion for each moment into that of the fluctuations\nis straighforward, but in Oberlack et al.32we were able to show that an error accumulation occurs\nin the calculation from DNS data, which inevitably results from the finite number of available flow\n15fields. This error increases considerably as the order of the moments increases. We have therefore\ndeliberately refrained from displaying the moments from the fluctuations.\nFurther, we note that the study focuses now on the streamwise velocity and temperature. For\nthis purpose, we only have to insert the infinitesimals (42) into the invariant surface condition (31)\nand we get\ndx2\naSxx2+ax2=dH1{1}\n[aSx−aSt+aSs]H1{1}+aH\n1{1}\n=dHΘ{1}\n[aΘ+aSs]HΘ{1}+aH\nΘ{1}\n=dH1{1}Θ{1}\n[aSx−aSt+aΘ+aSs]H1{1}Θ{1}+aH\n1{1}Θ{1}\n=...\n=dH1{n}Θ{m}\n[n(aSx−aSt)+maΘ+aSs]H1{n}Θ{m}+aH\n1{n}Θ{m}(43)\nSince we consider a shear flow that is fully developed in x1andx3, all moments in the one-\npoint limit depend only on x2. Furthermore, the dependencies of the other points for the higher-\norder tensors would have to be formally considered as well, because this would result in further\nsimilarity variables. However, from now on, we will focus on one-point statistics, so that every\npoint of application of the variables in (18) will be x(1)=x(2)=···=x(n+m). Integrating (43),\nnotice that we are using the first and the last term because the other terms are just specific moments,\nwe obtain the following invariant solutions for any arbitrary moment\nH1{n}Θ{m}=C′\n1{n}Θ{m}\u0012\nx2+ax2\naSx\u0013n(σ2−σ1)+mσΘ+2σ1−σ2\n−aH\n1{n}Θ{m}\nn(aSx−aSt)+maΘ+aSs,\n(44)\nwith C′\n1{n}Θ{m}=ec′nm[n(aSx−aSt)+maΘ+aSs], (45)\nwhere c′\nnmdenote the constants of integration, σ1=1−aSt/aSx+aSs/aSx,σ2=2(1−aSt/aSx)+\naSs/aSxandσΘ=aΘ/aSx. Similar to Ref. 1, the choice of parameters in the exponent of (44)\nhas been designed so that the high-order moments depend on those of the first and second order.\nFocusing on the velocity only, as in Ref. 1, i.e. m=0, the exponent for n=1 isσ1, and for n=2,\nit isσ2, while for pure temperature moments, i.e. n=0, we have for m=1 the exponent σΘ.\nTherefore, σ1andσ2, are determined from the first two velocity moments, while σΘis determined\nfrom the first temperature moment.\n16At this point, it is important to recall that the invariant solution (44) has been derived in the\nlimit of vanishing viscosity and heat conduction. Therefore, this solution will be only valid in the\nregion where these conditions apply, i.e. the centre of the channel. The invariant solution (44)\nshows that the moments of velocity, temperature, and higher-order moments scale as power-laws,\nwhose exponents are determined by the parameters σ1,σ2andσΘ. Note that, in the exponent of\nthe power law, four initial parameters appear ( aSx,aSt,aΘ, and aSs). However, these parameters\nappear as ratios, leading to only three free parameters remaining ( σ1,σ2, and σΘ).\nAnalogously to Ref. 17, where the scaling law of the mean velocity of a turbulent shear flow\nwas presented as a deficit law, and significantly extended in Ref. 1 to arbitrary velocity moments,\nequation (44) can be rewritten to form the final deficit scaling law of velocity, temperature and\narbitrarily mixed moments of both as\nH1{n}Θ{m}cl−H1{n}Θ{m}\nunτθmτ=C′\nnm\u0010x2\nh\u0011n(σ2−σ1)+mσΘ+2σ1−σ2, (46)\nwith C′\nnm=α′enβ′+mβ′\nΘ, (47)\nwhere the subscript clrefers to the value of the moment on the centre line, which comes from\nthe last term on the right-hand side of the equation (44), and C′\nnmare the new exponential scaling\nfactors. Similar to Ref. 1, it has been assumed that c′\nnmis independent of nandmto derive C′\nnmin\n(46), and this will indeed be validated below in section §IV B. Also note that, in equation (46), the\nshift in x2has been set to 0, as the coordinate is anchored to the centre line.\nB. Validation of the scaling law (46) with DNS data\nThe new scaling law (46) will be validated by using DNS data of turbulent channel flows\ndriven by a pressure gradient at friction Reynolds numbers of Reτ=500, 1000, 2000 and 5000,\nand heated by a constant heat flux from both walls with a wide range of values of Prandtl numbers:\nPr=0.007, 0 .01, 0 .02, 0 .05, 0 .1, 0.3, 0.5, 0.71, 1, 2, 4, 7 and 10. Specifically, the combinations\nofReτandPrused are presented in table I.\nThe code used to run the simulations is the Liso code, already validated and employed in many\nother simulations of turbulent channel flows7,9,11,19,33,34. Detailed information about the code itself\nand the parameters of the simulation (mesh size, wash-outs run, computational box size,. . . ) can be\nfound in Ref. 8, 25, 26, and 28. As mentioned, the UHF is used as the thermal boundary condition\nin the DNS cited. This implies that temperature increases linearly in the streamwise direction.\n17Reτ\\Pr0.007 0 .01 0.02 0.05 0.1 0.3 0.5 0.71 1 2 4 7 10 Colour\n500 X X X X X X X X X X X X X\n1000 X X X X X X X X X X X X\n2000 X X X X X X X X X X X X\n5000 X\nTABLE I: ReτandPrnumbers used for the validation of the scaling law (46). The last column\nshows the colours used in the figures to refer to each Reynolds number.\nTo make the temperature field homogeneous in the x1direction, the value of the temperature at\nthe wall is removed, obtaining a transformed temperature. Because we want to give a general\nscaling law for temperature moments, and because the same symmetry, TΘ(34), is also obtained\nfor the energy equation of the transformed temperature, the general form of the temperature energy\nequation is used in this work.\nThe moments calculated in the simulations are limited to order seven for pure moments of\nvelocity and temperature and six for mixed moments of velocity and temperature. The procedure\nto fit the scaling law (46) to the DNS data has been done by minimizing the infinite norm of the\nrelative error between the fit and the value of the DNS data, i.e.\nerror=min\u0012\f\f\f\f\f\f\f\fdata(x2)−fit(x2)\ndata(x2)\f\f\f\f\f\f\f\f\n∞\u0013\n. (48)\nThe infinite norm is used in this formula since the data takes values across several orders of\nmagnitude. After this fitting is applied to the first and second moments of velocity and the first\nmoment of temperature, σ1,σ2andσΘfrom the scaling law (46) are determined and, thus, the\nexponent for any high order moment is known and only the constants of integration, C′\nnm, must be\ncalculated.\nThe results of the fits of the velocity moments for Reτ=500 and Pr=4 are depicted in figure\n1a, together with the fits of the velocity moments for Reτ=2000 and Pr=7, in figure 1b. Ad-\nditionally, for Reτ=2000 and Pr=7, the fits of the temperature moments and mixed moments\nare presented in figures 1c and 1d, respectively. Solid lines represent the values from the DNS,\nwhile squares are the values obtained from the scaling law (46). Recall that: in all figures below,\nthe wall and centre of the channel are swapped, so the centre line is at x2/h=0, while the wall\nis atx2/h=1. Also, it is important to mention that the range of the centre of the channel where\n18(a)\n (b)\n(c)\n (d)\nFIG. 1: Moments of velocity, H1{n}, for (a) Reτ=500 and Pr=4 and (b) Reτ=2000 and Pr=7.\nMoments of (c) temperature, HΘ{m}, and (d) heat fluxes, H1{n}Θ{m}, forReτ=2000 and Pr=7. In\n(a), (b), and (c), velocity and temperature moments are obtained for nandm=1, 2,. . . , 7,\nappearing in that order from bottom to top of the plot. For (d), heat fluxes moments are shown for\nn+m=2, 3,. . . , 6, appearing in that order from bottom to top of the plot. For heat fluxes\nmoments of the same order, the lower lines are for m=0, while the upper lines are for n=0.\nSolid lines are the values from the DNS, while squares represent the values from the scaling law.\nThe wall and centre of the channel are swapped, so the centre line is at x2/h=0, while the wall is\natx2/h=1. Colours as in table I.\n19the scaling law has been applied is up to x2/h=0.75. The most important result of this work\nis the high accuracy of the scaling law (46) to fit the data of the DNS for all moments, with the\nhighest relative errors of only 0 .2% for the higher order moments, calculated with equation (48).\nEven for the lowest Reynolds numbers of value 500, the accuracy of the fit is almost as good as\nforReτ=2000, as can be seen in figures 1a and 1b. In the same way, the scaling law is validated\nwith the same accuracy for the temperature and mixed moments as shown in figures 1c and 1d,\nrespectively.\nTo analyse the influence of the Prandtl number, we compare the scaling between a high Prandtl\nnumber of 4 and a very low one of 0 .01 in figure 2. In the case of the scaling of the temperature\nmoments, in figures 2a and 2b, for cases Pr=4 and 0 .01, respectively, there is a noticeable\ndifference. While for Pr=4 the scaling (46) represents the DNS data with high accuracy, errors\nare lower than 0 .01%, for the case of Pr=0.01, the deviation is high and clearly visible in figure\n2b. The reason for this error comes from the assumption in section §III of zero heat conduction\nin the symmetry analysis, which, obviously, for Pr=0.01 is not true. The high diffusivity for\nvery low Prandtl numbers affects the temperature field in a deeper region away from the wall, and\nthe temperature moments are no longer parallel for x2/happroximately greater than 0 .2. In the\nsame manner, the scaling of mixed moments is no longer correct for very low Prandtl numbers.\nWhile in figure 2c, the scaling is again very accurate for Pr=4, in figure 2d, a similar failure, as\nobtained in figure 2b, appears in the scaling for Pr=0.01. Note that in figure 2d, only moments\nforn+m=2, 4, and 6 are plotted for clarity of the figure.\nOne important point to note is that the functional structure of the invariant solution (44) that\nleads to the scaling law (46) does not change for vanishing or not vanishing viscosity/diffusivity, in\nthe sense that the same number of parameter appears in the exponent of the scaling law. Therefore,\none should expect the scaling law to be correct also for these cases. This is true for a limited region\nof the centre of the channel. In the case of figures 2b and 2d, if one tries to use the scaling law\nonly for in the region of y∗<0.1 away from the centre, instead of y∗<0.75, as was done in figure\n2, then a perfect matching with the DNS data will be obtained, even for the lowest Prandtl number\ncases. As mentioned above, this happens because the viscous/diffusive effects appear so deep\naway from the wall. Effectively, as shown in Ref. 25, for such low Reynolds and Prandtl numbers,\none cannot even see the emergence of a logarithmic layer, so a bad scaling is also expected. Also,\nremark that the coefficient σ1andσ2approach to a value of 2 as the region where the fitting is\ndone is reduced.\n20(a)\n (b)\n(c)\n (d)\nFIG. 2: Moments of temperature, HΘ{m}, for (a) Reτ=500 and Pr=4 and (b) Reτ=500 and\nPr=0.01. Moments of heat fluxes, H1{n}Θ{m}, for (c) Reτ=500 and Pr=4 and (d) Reτ=500\nandPr=0.01. In (a) and (b), temperature moments are obtained for m=1, 2,. . . , 7, appearing in\nthat order from the bottom to the top of the plot. For (c), heat fluxes moments are shown for\nn+m=2, 3,. . . , 6, appearing in that order from bottom to top of the plot. For heat fluxes\nmoments of the same order, the lower lines are for m=0, while the upper lines are for n=0. For\n(d), heat fluxes moments are shown for n+m=2, 3, and 6, appearing in that order from bottom\nto top of the plot. For heat fluxes moments of the same order, the lower lines are for n=0, while\nthe upper lines are for m=0. Solid lines are the values from the DNS, while squares represent\nthe values from the scaling law. The wall and centre of the channel are swapped, so the centre\nline is at x2/h=0, while the wall is at x2/h=1. Colours as in table I.\n21Note here that although a Reτ=500 is a low Reynolds number and one may expect the as-\nsumption of vanishing viscosity to fail, the velocity field is still turbulent and in the centre of the\nchannel, the mentioned assumption is true. However, Pr=0.01 produces a much less turbulent\ntemperature field, even also laminar25, and for this reason, the assumption of vanishing heat con-\nduction is not true for such a low Prandtl number, or at least is only true in a very central region\nof the channel (less than 10%), where the temperature moments are parallel in figure 2b. In addi-\ntion, the key parameter is actually not the Prandtl number by itself, but the friction Péclet number,\ndefined as Peτ=ReτPr. Therefore, in our plots, we are comparing a Reτ=500 with Pr=4 and\n0.01, which means Peτ=2000 compared with Peτ=5, explaining the errors in the scaling for\nsuch a low Prandtl or friction Péclet numbers.\nIt is important to analyse the values of the different exponential parameters of the scaling law\n(46) and see the influence of each symmetry on the final scaling. As it was shown in equation\n(46), the exponent of the power-law is formed by a constant term, 2 σ1−σ2, that comes from\nthe statistical scaling symmetry of the moments, TSs(38). A second term that scales with n, i.e.\nn(σ2−σ1), emerged from the classical scaling symmetries of space and time, TSx(35) and TSt\n(36), respectively. A third term that scales with m, i.e. mσΘ, has its roots in the scaling symmetry\nof the temperature, TSΘ(37). However, as can be clearly seen in figure 1, all moments have a more\nor less constant slope in the log-log plot, which translates into a very weak dependence on nand\nm. In other words, the values of σ1andσ2are very similar, and σΘis small compared with the\nvalue of 2 σ1−σ2. This, in turn, implies that scaling of space and time has almost no influence in\nthe centre of the channel, and the statistical scaling of moments is dominant, which makes sense,\nsince, as it was said before, it is a measure of intermittency. Scaling independent of the dimensions\nof space and time is called anomalous scaling and has its origin in the intermittency symmetry.\nFigure 3a presents the values of σ1andσ2for all the DNS simulations, while the values of σΘ\nare shown in figure 3b. The values of σ1are almost the same as σ2(note that the left and right\naxes are shifted for better visualization), which confirms that the symmetries of scaling in space\nand time have almost no influence in the centre of the channel. Similarly, the scaling symmetry\nof temperature has barely any influence, since σΘ≪2σ1−σ2. This last term, 2 σ1−σ2, is indeed\nthe only dominant term in the exponent of the scaling law (46), with a value slightly below 2,\nconfirming that the symmetry of scaling of moments, TSs(38), is dominant in the centre of the\nchannel. Further, we observe that the parameters, σ1,σ2andσΘ, in the investigated ranges, are\nlargely independent of ReτandPr. Small differences are due to small numerical errors or noise in\n22(a)\n (b)\nFIG. 3: Values of (a) σ1, solid lines left axis, and σ2, dashed lines right axis, and (b) σΘ. Colours\nas in table I. Note that black circles at Pr=0.71 represent the value for the single simulation at\nReτ=5000.\nthe fitting.\nThe second important part of the scaling law (46) is the prefactor C′\nnm. Figure 4a shows the\nvalues of C′\nnmforReτ=500 and Pr=4. An almost perfect plane is observed in the vertical log-\nscaling plot, which confirms that C′\nnmis an exponential function in nandm, and further verifies\nthat the constants of integration c′\nnmin equation (45) are independent of nandm.\nIn figures 4b, 4c and 4d, the values of α′,β′andβ′\nΘfor all the present simulations are shown,\nrespectively. To calculate them, only the constants of integration of the moments up to order 2\nhave been used, i.e., C′\n10,C′\n20,C′\n01,C′\n02andC′\n11. With these five values, a fit of the parameters α′,\nβ′, and β′\nΘhave been done minimizing again as in equation (48).\nInterestingly enough, and other than the parameters in the exponent σ1,σ2andσΘ, the coeffi-\ncients α′,β′andβ′\nΘare not independent of ReτandPr. We want to point out that for high friction\nPéclet numbers, the values of α′,β′andβ′\nΘseem to be independent of the friction Reynolds num-\nber, which may be related to a more realistic assumption of the zero viscosity and heat conduction.\nHowever, this is just a point to be investigated. From the theory developed in section §III, the\ndependency on ReτorPris not apparent but goes beyond the scope of the theory in its present\nform. Here we limited the study to confirm that the scaling law (46) can represent the behaviour\nof the arbitrary moments obtained from the DNS data, including exponential scaling of C′\nnmwith\nnandm.\n23(a)\n (b)\n(c)\n (d)\nFIG. 4: (a) Values of C1{n}Θ{m}forReτ=500 and Pr=4. Parameters from equation (47): (b) α′,\n(c)β′and (d) β′\nΘ. Colours as in table I. Note that black points at Pr=0.71 in (b), (c), and (d)\nrepresent the values for the single simulation at Reτ=5000.\nV . CONCLUSIONS\nA new set of turbulent scaling laws for arbitrary moments of the streamwise velocity, temper-\nature, and high order moments of both in a turbulent channel flow has been obtained using the\nsymmetry-based turbulence theory. These scaling laws apply to incompressible flows driven by\na pressure difference and with a passive scalar. For the derivation of the scaling laws, we had to\nassume vanishing viscosity and diffusion, i.e., Reτ→∞andPr>1, which holds in the central\nregion of the channel, and they are finally cast as deficit laws.\n24The deficit form of the arbitrary moments in the wall-normal direction can be represented as\npower functions, where the exponent is determined by the order of the moments and three different\nparameters that emerged from four different scaling symmetries ( σ1,σ2andσΘ). Besides the clas-\nsical symmetries of the Navier-Stokes and energy equations, we employed statistical symmetries\nof the multi-point correlation equations, which were the key to obtaining a constant exponent of\nthe power-law scaling function that can accurately represent the DNS data. Instead of the usual\nfluctuation approach as the basis for the MPC equations, which yields a non-linear system of\nequations, we presently employ the instantaneous approach, which results in a linear system of\nequations. The statistical symmetries are trivially displayed in this representation as scaling and\ntranslation of moments. This statistical scaling of moments represents a measure of intermittency.\nIt appears as the dominant term in the exponent of all moments as the constant 2 σ1−σ2, which is\nindependent of the moment order.\nThe scaling laws have been validated with data from different DNS at different Reynolds and\nPrandtl numbers. The accuracy of the scaling laws to represent the data is remarkable, especially\nfor high Péclet numbers. For cases with low Péclet numbers, the centre of the channel gets in-\nfluenced by viscosity and heat conduction, and the assumption of Reτ→∞andPr>1 no longer\nholds, which entails a significant deviation from the theoretical scaling of the moments in the\ncentre of the channel.\nThe exponential prefactor in nandmin equation (47) has been obtained by the observation that\nthe constants of integration c′\nnmin 45 are indeed constant and independent of nandm. So far, no\njustification based on first principles can be given for this, though we speculate that the Probability\nDensity Function (PDF) contains deeper information on this. Therefore, we presently follow the\nidea of deriving invariant solutions to the PDF equations.\nOne point of this theory to be explored is the derivation of scaling laws for other important\nstatistics such as fluctuating quantities, cross velocities, the wall-normal and spanwise heat fluxes,\nand their high-order moments. So far, with the symmetries obtained in this work, it was impossible\nto properly describe these statistics. However, as mentioned before, further symmetries can be\nobtained from the MPC equations or from other equations that describe turbulence, such as the\nPDF equations. This is left as future work which is already being investigated, but we would like\nto point out that our method is able to obtain scaling laws following only strong mathematical\narguments, removing lucky curve fitting.\n25ACKNOWLEDGMENTS\nThis work was supported by PID2021-128676OB-I00 of MINECO/FEDER. FAA is partially\nfunded by GV A/FEDER project ACIF2018. The computations of the new simulations were made\npossible by a generous grant of computing time from the Barcelona Supercomputing Centre, ref-\nerence AECT-2020-1-0024. MO expresses his gratitude for the partial support of the German\nResearch Foundation (DFG) within the project OB 96/48-1. We are grateful to Mr Jonathan Laux\nfor providing us with the scripts to do the fittings of the scaling laws with the data. Declaration of\nInterests. The authors declare that they have no known competing financial interests or personal\nrelationships that could have appeared to influence the work reported in this paper.\nAppendix A\nIn this appendix, we present a step-by-step derivation of the MPC equation of the mixed mo-\nments. The derivation of the MPC equations of velocity and thermal energy are just two specific\ncases of the general MPC equations.\nStarting with equations (2) and (3) we perform the following operation to obtain the MPC\nequation of order n+mof the mixed moments\nMi(1)(x(1))Ui(2)(x(2))...Ui(n)(x(n))Θ(x(n+1))...Θ(x(n+m))\n+Ui(1)(x(1))Mi(2)(x(2))Ui(3)(x(3))...Ui(n)(x(n))Θ(x(n+1))...Θ(x(n+m))\n+...\n+Ui(1)(x(1))...Ui(n−1)(x(n−1))Mi(n)(x(n))Θ(x(n+1))...Θ(x(n+m))\n+Ui(1)(x(1))...Ui(n)(x(n))E(x(n+1))Θ(x(n+2))...Θ(x(n+m))\n+Ui(1)(x(1))...Ui(n)(x(n))Θ(x(n+1))E(x(n+2))Θ(x(n+3))...Θ(x(n+m))\n+...\n+Ui(1)(x(1))...Ui(n)(x(n))Θ(x(n+1))...Θ(x(n+m−1))E(x(n+m)) =\n=n\n∑\na=1Mi(a)(x(a))n\n∏\nc=1,c̸=aUi(c)(x(c))n+m\n∏\nd=n+1Θ(x(d))\n+n+m\n∑\nb=n+1E(x(b))n\n∏\nc=1Ui(c)(x(c))n+m\n∏\nd=n+1,d̸=cΘ(x(d)) =0, (A1)\nwhere Ui(l)andx(l)are the velocity and the different points where the equations and the variables\n26are applied, for i(l)=1, 2, 3; and l=1, 2,. . . , n+m(lcan be aorb). Introducing the momentum\nand energy equations, (2) and (3), into (A1), we obtain\nn\n∑\na=1∂Ui(a)(x(a))\n∂tn\n∏\nc=1,c̸=aUi(c)(x(c))n+m\n∏\nd=n+1Θ(x(d))\n+n\n∑\na=1Uk(x(a))∂Ui(a)(x(a))\n∂xk(a)n\n∏\nc=1,c̸=aUi(c)(x(c))n+m\n∏\nd=n+1Θ(x(d))\n+n\n∑\na=1∂P(x(a))\n∂xi(a)n\n∏\nc=1,c̸=aUi(c)(x(c))n+m\n∏\nd=n+1Θ(x(d))\n−1\nReτn\n∑\na=1∂2Ui(a)(x(a))\n∂xk(a)∂xk(a)n\n∏\nc=1,c̸=aUi(c)(x(c))n+m\n∏\nd=n+1Θ(x(d))\n+n+m\n∑\nb=n+1∂Θ(x(b))\n∂tn\n∏\nc=1Ui(c)(x(c))n+m\n∏\nd=n+1,d̸=cΘ(x(d)),\n+n+m\n∑\nb=n+1Uk(x(b))∂Θ(x(b))\n∂xk(b)n\n∏\nc=1Ui(c)(x(c))n+m\n∏\nd=n+1,d̸=cΘ(x(d)),\n−1\nPeτn+m\n∑\nb=n+1∂2Θ(x(b))\n∂xk(b)∂xk(b)n\n∏\nc=1Ui(c)(x(c))n+m\n∏\nd=n+1,d̸=cΘ(x(d)) =0. (A2)\nAt this point, the continuity equation (1) should be applied to introduce the terms Uk(xl)inside\nthe derivatives with respect to xk(l)in the second and sixth lines of equation (A2). Also, the product\nterms can be introduced in the derivatives with respect to the spatial coordinates, since the points\nx(a)andx(b)are excluded from the product series. Regarding the temporal derivatives in the\nfirst and fifth lines of (A2), the chain rule is applied to reduce it to a single term. Finally, using\ndefinitions (15), (16) and (17) one can obtain the MPC equation for all mixed moments written in\nthe following way\n∂Hi{n}Θ{m}\n∂t+\nn\n∑\na=1 ∂Hi{n+1}Θ{m}[i(n+m+1)→k](x(n+m+1)→x(a))\n∂xk(a)+∂Ii{n−1}Θ{m}[a]P\n∂xi(a)−1\nReτ∂2Hi{n}Θ{m}\n∂xk(a)∂xk(a)!\n+n+m\n∑\nb=n+1 ∂Hi{n+1}Θ{m}[i(n+m+1)→k](x(n+m+1)→x(b))\n∂xk(b)−1\nPeτ∂2Hi{n}Θ{m}\n∂xk(b)∂xk(b)!\n=0. (A3)\nAs mentioned before, the MPC equations of the velocity arises if m=0 in (A3). Similarly, one\ncan obtain the MPC equations of the temperature by setting n=0 in (A3).\n27Appendix B\nIn this appendix, the Two-Point Correlation (TPC) equations for the velocity, heat fluxes and\ntemperature are given as examples of the MPC equation (18), in order to make the notation clearer.\nThese equations can be obtained by setting in equation (18) n=2, 1, 0 and m=0, 1, 2, respectively.\n∂Hi(1)i(2)(x(1),x(2))\n∂t+∂Hi(1)i(2)k(x(1),x(2),x(1))\n∂xk(1)+∂Hi(1)i(2)k(x(1),x(2),x(2))\n∂xk(2)\n+∂IPi(2)(x(1),x(2))\n∂xi(1)+∂Ii(1)P(x(1),x(2))\n∂xi(2)\n−ν∂2Hi(1)i(2)(x(1),x(2))\n∂xk(1)∂xk(1)−ν∂2Hi(1)i(2)(x(1),x(2))\n∂xk(2)∂xk(2)=0, (B1)\n∂Hi(1)Θ(x(1),x(2))\n∂t+∂Hi(1)Θk(x(1),x(2),x(1))\n∂xk(1)+∂Hi(1)Θk(x(1),x(2),x(2))\n∂xk(2)\n+∂IPΘ(x(1),x(2))\n∂xi(1)\n−ν∂2Hi(1)Θ(x(1),x(2))\n∂xk(1)∂xk(1)−α∂2Hi(1)Θ(x(1),x(2))\n∂xk(2)∂xk(2)=0, (B2)\n∂HΘΘ(x(1),x(2))\n∂t+∂HΘΘk(x(1),x(2),x(1))\n∂xk(1)+∂HΘΘk(x(1),x(2),x(2))\n∂xk(2)\n−α∂2HΘΘ(x(1),x(2))\n∂xk(1)∂xk(1)−α∂2HΘΘ(x(1),x(2))\n∂xk(2)∂xk(2)=0. (B3)\nREFERENCES\n1M. Oberlack, S. Hoyas, S. V . Kraheberger, F. Alcántara-Ávila, and J. Laux, “Turbulence statistics\nof arbitrary moments of wall-bounded shear flows: A symmetry approach,” Physical Review\nLetters 128, 024502 (2022).\n2J. Kim, P. Moin, and R. Moser, “Turbulence statistics in fully developed channels flows at low\nReynolds numbers,” Journal of Fluid Mechanics 177, 133–166 (1987).\n3J. Kim and P. Moin, “Transport of passive scalars in a turbulent channel flow,” NASA 1, 1–14\n(1987).\n4T. Yano and N. Kasagi, “Direct numerical simulation of turbulent heat transport at high Prandtl\nnumbers,” JSME International Journal, Series B: Fluids and Thermal Engineering 42, 284–292\n(1999).\n285S. Pirozzoli, M. Bernardini, and P. Orlandi, “Passive scalars in turbulent channel flow at high\nReynolds number,” Journal of Fluid Mechanics 788, 614–639 (2016).\n6S. Hoyas, M. Oberlack, F. Alcántara-Ávila, S. Kraheberger, and J. Laux, “Wall turbulence at\nhigh friction Reynolds numbers,” Physical Review Fluids 7, 014602 (2022).\n7S. Hoyas and J. Jiménez, “Reynolds number effects on the Reynolds-stress budgets in turbulent\nchannels,” Physics of Fluids 20, 101511 (2008).\n8F. Alcántara-Ávila, S. Hoyas, and M. Pérez-Quiles, “Direct numerical simulation of thermal\nchannel flow for Reτ=5000 and Pr=0.71,” Journal of Fluid Mechanics 916, A29 (2021).\n9S. Hoyas and J. Jiménez, “Scaling of the velocity fluctuations in turbulent channels up to Reτ=\n2003,” Physics of Fluids 18, 011702 (2006).\n10J. Jiménez and S. Hoyas, “Turbulent fluctuations above the buffer layer of wall-bounded flows,”\nJournal of Fluid Mechanics 611, 215–236 (2008).\n11V . Avsarkisov, S. Hoyas, M. Oberlack, and J. García-Galache, “Turbulent plane Couette flow at\nmoderately high reynolds number,” Journal of Fluid Mechanics 751, R1 (2014).\n12S. Pirozzoli, M. Bernardini, and P. Orlandi, “Turbulence statistics in Couette flow at high\nReynolds number,” Journal of Fluid Mechanics 758, 323–343 (2014).\n13M. Lee and R. Moser, “Direct numerical simulation of turbulent channel flow up to Reτ≈5200,”\nJournal of Fluid Mechanics 774, 395–415 (2015).\n14A. Lozano-Durán and J. Jiménez, “Effect of the computational domain on direct simulations of\nturbulent channels up to Reτ=4200,” Physics of Fluids 26, 011702 (2014).\n15S. Kraheberger, S. Hoyas, and M. Oberlack, “Dns of a turbulent couette flow at constant wall\ntranspiration up to Reτ=1000,” Journal of Fluid Mechanics 835, 421–443 (2018).\n16S. Pirozzoli, “An explicit representation for mean profiles and fluxes in forced passive scalar\nconvection,” Journal of Fluid Mechanics 968, R1 (2023).\n17M. Oberlack, “A unified approach for symmetries in plane parallel turbulent shear flows,” Jour-\nnal of Fluid Mechanics 427, 299–328 (2001).\n18M. Oberlack and A. Rosteck, “New statistical symmetries of the multi-point equations and its\nimportance for turbulent scaling laws,” Discrete Continuous Dyn. Syst 3, 451–471 (2010).\n19V . Avsarkisov, M. Oberlack, and S. Hoyas, “New scaling laws for turbulent Poiseuille flow with\nwall transpiration,” Journal of Fluid Mechanics 746, 99–122 (2014).\n20A. M. Rosteck, Scaling laws in turbulence – a theoretical approach using Lie-point symmetries ,\nPh.D. thesis, Darmstadt, Technische Universität (2014).\n2921M. Oberlack, M. Wacławczyk, A. Rosteck, and V . Avsarkisov, “Symmetries and their importance\nfor statistical turbulence theory,” Mechanical Engineering Reviews 2, 15–00157 (2015).\n22H. Sadeghi, M. Oberlack, and M. Gauding, “On new scaling laws in a temporally evolving\nturbulent plane jet using lie symmetry analysis and direct numerical simulation,” Journal of\nFluid Mechanics 854, 233–260 (2018).\n23H. Sadeghi and M. Oberlack, “New scaling laws of passive scalar with a constant mean gradient\nin decaying isotropic turbulence,” Journal of Fluid Mechanics 775, A10 (2020).\n24H. Sadeghi, M. Oberlack, and M. Gauding, “On new scaling laws in a temporally evolving\nturbulent plane jet using lie symmetry analysis and direct numerical simulation-corrigendum,”\nJournal of Fluid Mechanics 885, E1 (2020).\n25F. Alcántara-Ávila, S. Hoyas, and M. Pérez-Quiles, “DNS of thermal channel flow up to Reτ=\n2000 for medium to low Prandtl numbers,” International Journal of Heat and Mass Transfer 127,\n349–361 (2018).\n26Alcántara-Ávila, F. and Hoyas, S., “Direct Numerical Simulation of thermal channel flow for\nmedium–high Prandtl numbers up to Reτ=2000,” International Journal of Heat and Mass Trans-\nfer176, 121412 (2021).\n27N. Kasagi, Y . Tomita, and A. Kuroda, “Direct numerical simulation of passive scalar field in a\nturbulent channel flow,” Journal of Heat Transfer 114, 598–606 (1992).\n28F. Lluesma-Rodríguez, F. Álcantara Ávila, M. Pérez-Quiles, and S. Hoyas, “A code for simulat-\ning heat transfer in turbulent channel flow,” Mathematics 9(2021), 10.3390/math9070756.\n29G. Bluman and S. Anco, Symmetry and integration methods for differential equations (vol. 154\nof Applied mathematical sciences, 2002).\n30M. Oberlack, “Asymptotic expansion, symmetry groups, and invariant solutions of laminar and\nturbulent wall-bounded flows,” Zeitschrift für Angewandte Mathematik und Mechanik 80, 791–\n800 (2000).\n31M. Wacławczyk, N. Staffolani, M. Oberlack, A. Rosteck, M. Wilczek, and R. Friedrich, “Sta-\ntistical symmetries of the lundgren-monin-novikov hierarchy,” Physical Review E 90, 013022\n(2014).\n32M. Oberlack, S. Hoyas, J. Laux, and D. Klingenberg, “Oberlack et al. reply:,” Physical Review\nLetters 130, 069403 (2023).\n33S. Gandía-Barberá, S. Hoyas, M. Oberlack, and S. Kraheberger, “The link between the Reynolds\nshear stress and the large structures of turbulent Couette-Poiseuille flow,” Physics of Fluids 30,\n30041702 (2018), https://doi.org/10.1063/1.5028324.\n34F. Lluesma-Rodríguez, S. Hoyas, and M. Peréz-Quiles, “Influence of the computational domain\non DNS of turbulent heat transfer up to Reτ=2000 for Pr=0.71,” International Journal of Heat\nand Mass Transfer 122, 983–992 (2018).\n31" }, { "title": "2401.16059v1.Simultaneous_Measurement_of_Half_Life_and_Spectral_Shape_of____115__In__β__decay_with_an_Indium_Iodide_Cryogenic_Calorimeter.pdf", "content": "Simultaneous Measurement of Half-Life and Spectral Shape of115Inβ-decay with an\nIndium Iodide Cryogenic Calorimeter\nL. Pagnanini,1, 2, 3G. Benato,1, 2P. Carniti,4, 5E. Celi,1, 2D. Chiesa,4, 5J. Corbett,3I. Dafinei,1S. Di Domizio,6, 7P.\nDi Stefano,3S. Ghislandi,1, 2, aC. Gotti,5D. L. Helis,1, 2, bR. Knobel,3J. Kostensalo,8J. Kotila,9, 10, 11S. Nagorny,3\nG. Pessina,5S. Pirro,2S. Pozzi,5A. Puiu,2S. Quitadamo,1, 2M. Sisti,5J. Suhonen,9, 12and S. Kuznetsov13\n1Gran Sasso Science Institute, 67100, L’Aquila - Italy\n2INFN - Laboratori Nazionali del Gran Sasso, I-67100 Assergi (AQ) - Italy\n3Department of Physics, Engineering Physics and Astronomy,\nQueen’s University Kingston, Ontario, K7L 3N6 Kingston, Canada\n4Dipartimento di Fisica, Universit` a di Milano - Bicocca, I-20126 Milano - Italy\n5INFN - Sezione di Milano - Bicocca, I-20126 Milano - Italy\n6Dipartimento di Fisica, Universit` a di Genova, I-16146 Genova - Italy\n7INFN - Sezione di Genova, I-16146 Genova - Italy\n8Natural Resources Institute Finland, Yliopistokatu 6B, FI-80100 Joensuu, Finland\n9University of Jyv¨ askyl¨ a, Department of Physics, P. O. Box 35 (YFL), FI-40014, Finland\n10Finnish Institute for Educational Research, P.O.Box 35 FI-40014 University of Jyv¨ askyl¨ a - Finland\n11Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520-8120 - USA\n12International Centre for Advanced Training and Research in Physics, 077125 Bucharest-Magurele, Romania\n13Prokhorov General Physics Institute of the Russian Academy of Sciences 119991, Moscow, 38 Vavilov str., Russia\n(Dated: January 30, 2024)\nCurrent bounds on neutrino Majorana mass are affected by significant uncertainties in the nuclear\ncalculations for neutrinoless double-beta decay. A key issue for a data-driven improvement of the\nnuclear theory is the actual value of the axial coupling constant gA, which can be investigated\nthrough forbidden β-decays. We present the first measurement of 4th-forbidden β-decay of115In\nwith a cryogenic calorimeter based on Indium Iodide. Exploiting the enhanced spectral shape\nmethod for the first time to this isotope, our study accurately determines simultaneously spectral\nshape, gA, and half-life. The Interacting Shell Model, which best fits our data, indicates a half-life\nfor this decay at T1/2= (5.26±0.06)×1014yr.\nPACS numbers: 07.20.Mc, 23.40.-s, 21.10.Tg, 27.50.+e\nKeywords: forbidden beta decay, spectral shape\nIntroduction. The search for neutrinoless double-beta\ndecay (0 νββ) is a crucial part of our quest to under-\nstand the deepest mysteries of the universe [1]. The ob-\nservation of this phenomenon would require a paradigm\nshift from the standard model of elementary particles\nand would reshape our understanding of the fundamen-\ntal building blocks of matter. 0 νββ is an extremely rare\nprocess where two neutrons in the nucleus are simulta-\nneously transformed into protons, with the emission of\njust two electrons in the final state. If we observe this\nprocess, it would indicate that neutrinos are Majorana\nparticles, which means they are their own antiparticles.\nThe half-life of this process ( T0ν\n1/2) could provide insights\ninto the absolute mass scale of neutrinos, which is still\nan unsolved issue in particle physics. Moreover, 0 νββ\nis a lepton-number-violating transition, and its obser-\nvation would support exciting theoretical frameworks in\nwhich leptons played a crucial role in creating the mat-\nter/antimatter asymmetry in the universe [2, 3]. The\nnext-generation experiments in this field are designed to\napproach half-lives of the order of 1027–1028yr. The cur-\nrent most stringent limit is set on136Xe by KamLAND-\nZen at 2 .3×1026yr at 90% C.L. [4]. This limit can be\nconverted into a constraint on the effective Majoranamass ( mββ), which is the new-physics parameter gov-\nerning 0 νββ, obtaining mββ<36–156 meV. It is notable\nthat a single value of T0ν\n1/2can correspond to a wide range\nformββ. This is due to the uncertainty of a factor of 3\naffecting the Nuclear Matrix Element (NME) calculation\nfor 0νββ within different nuclear models. This uncer-\ntainty not only limits the conversion of the half-life into\nmββin case of discovery, but also severely restricts the\nselection of relevant isotopes for the next-to-next gener-\nation of experiments. It is well-known that isotopes with\nlower Q-values are disfavoured by lower phase space fac-\ntors, but this precious information could be misleading if\nthe NME landscape is unclear. Therefore, a data-driven\nimprovement of nuclear models is essential to ensure that\ntheoretical and experimental efforts in the 0 νββ sector\nare not nullified. Some of the data and physical processes\nthat could help clarify the puzzle include double-charge\nexchange reactions [5, 6], ordinary muon capture [7, 8],\ntwo neutrino double beta decay [9, 10], and forbidden\nβ-decay [11]. In particular, the latter is very interesting\nto investigate the origins of the quenching of the axial\ncoupling constant ( gA). Indeed, the shape of the forbid-\nden non-unique β-decay spectrum shows a strong depen-\ndence on the value of gA[12]. In this context, several iso-arXiv:2401.16059v1 [nucl-ex] 29 Jan 20242\nModel sNME\nISM 6.01\nMQPM 10.25\nIBFM-2 2.53\nTABLE I. Values of the sNME calculated in the framework of\nthe three nuclear models adopted in this work (ISM - Interact-\ning Shell Model, MQPM - Microscopic Quasi-Particle Phonon\nModel and IBFM-2 - Interacting Boson-Fermion Model) un-\nder the Conserved Vector Current (CVC) hypothesis.\ntopes have been studied such as113Cd [13–15],99Tc [16],\nand115In [17] using the so-called Spectral Shape Method\n(SSM) [18]. This theoretical framework matches with\nhigh precision the spectral shape of experimental data.\nHowever, the simultaneous prediction of the decay half-\nlife is often far from being compatible with the measured\nvalues. Improvements of the models in this direction have\nbeen done during the last years within the so-called en-\nhanced SSM theory [19–21], where the small relativistic\nNME (sNME) enters as an additional parameter able to\nadjust the spectrum to predict the half-life. The theoret-\nical values of the sNME predicted under the Conserved\nVector Current (CVC) hypothesis are reported in Tab. I.\nIn this letter, we present the first application of the\nenhanced SSM on115In. The measurement has been\nperformed with a cryogenic calorimeter based on Indium\nIodide (InI) crystal in the framework of the ACCESS\n(Array of Cryogenic Calorimeters to Evaluate Spectral\nShape) project [22].\nDetector setup. Following the design principles out-\nlined in Ref. [22], we conduct measurements on a\n7×7×7 mm3Indium Iodide (InI) crystal. The crystal has\na mass mInIof (1.91±0.01) g and it is equipped with a\nNeutron Transmutation Doped (NTD) germanium ther-\nmistor (3 ×2×0.5 mm3) to record particle interactions\nwithin its lattice. The detector, as shown in Fig. 1, rests\non a copper holder that is directly connected to the mix-\ning chamber of the CUPID R&D dilution refrigerator [23]\ninstalled in Hall A of the Laboratori Nazionali del Gran\nSasso (LNGS), Italy. The setup is cooled down to approx-\nimately 16 mK. During the whole data-taking, we use\na thoriated wire as232Th permanent calibration source\nmounted close to the detector. Periodic calibrations with\nexternal sources would have been more difficult due to\nthe small size of the crystal and the presence of a lead\nshield in the cryostat itself. The front-end electronics\nconsists of an amplification stage, a six-pole anti-aliasing\nactive Bessel filter, and an 18-bit ADC board [24, 25].\nThe data stream is digitized at a frequency of 2 kHz and\nstored on disk in NTuples using a ROOT-based software\nframework [26]. An online software derivative trigger,\nincorporating a channel-dependent threshold, flags noise\nand signal events.\nData analysis. The offline analysis of the data stream\ninvolves calculating several variables for each triggered\nFIG. 1. Experimental setup used to measure the Indium Io-\ndide (InI) crystal as cryogenic calorimeter at LNGS. The crys-\ntal is equipped with a Neutron Transmutation Doped (NTD)\ngermanium sensor and rests on the copper holder that is con-\nnected to the lowest temperature stage (16 mK) of the cryo-\nstat utilizing a double-stage vibration damping system. The\ncrystal is thermally linked through the gold wires for the sig-\nnal readout.\nsignal. These variables include the number of triggers in\nthe acquisition window, the slope of the baseline (pre-\ntrigger of the pulse in the acquisition window), the rise\ntime and decay time of the pulses. We exploit these\nquantities to construct average templates of noise and\nsignal events, needed to apply the Optimum Filter [27].\nWe adopt this technique to estimate the amplitude of\neach triggered event by maximizing the signal-to-noise\nratio [28]. We then perform a stabilization of the detec-\ntor thermal gain using the 238.6 keV γ-ray line from the\n232Th source [29]. Consequently, we calibrate in energy\nthe stabilized spectrum using the most prominent peaks\nvisible in the data. We observe an energy resolution of\n3.9 keV (FWHM) at 238.6 keV. The detection threshold,\ndefined as 5 times the baseline root-mean-square, is esti-\nmated to be 3.4 keV. The criteria for selecting the events\nare based on rejecting noisy acquisition time intervals and\nwindows that have more than one triggered pulse, which\nis commonly known as a distinguishable pile-up. Addi-\ntionally, we apply pulse-shape cuts requiring a constant\nselection efficiency as a function of the energy. This is\nmandatory to avoid any possible distortion in the spec-\ntral shape due to analysis. The overall analysis cut ef-\nficiency is ϵ= (52 .2±0.3)%, where the distinguishable\npile-up cut dominates.\nData Modeling and Spectral Fit. The study of the115In\nβ-decay shape and the estimation of its half-life can be\nachieved through a background decomposition of the col-\nlected data. For this purpose, the geometry of the exper-\nimental setup is implemented into a Geant4-based [30]\nsimulation. In the following, we define as signal theβ-\nspectrum of the115In (Qβ= 497 .489(10) keV [31]), and3\nasbackground all the remaining contributions required to\nexplain the energy spectrum measured by the InI crystal.\nFor the signal, we generate electrons with energy sam-\npled from spectra templates based on three different the-\noretical frameworks: Interacting Shell Model (ISM) [32],\nMicroscopic Quasi-Particle Phonon Model (MQPM) [33]\nand Interacting Boson-Fermion Model (IBFM-2) [34].\nThese templates are calculated for fixed values of gAand\nsNME, which vary in the range [0.60, 1.39] and [-5.9, 5.9]\nand with steps of 0.01 and 0.1, respectively. If needed, we\nuse linear spline interpolation to increase the template\nfine structure. The most prominent background comes\nfrom the thoriated wire used as a calibration source.\nIn order to account for a possible breaking of the secu-\nlar equilibrium, we separately simulate the partial decay\nchain from232Th to228Ac and the remaining one starting\nfrom228Th. Any other potential background contribu-\ntion, whether from the crystal or the cryogenic setup [35],\nis smaller than the statistical uncertainty associated to\nthe bin counts. This is consistent with the absence of any\nother features in the spectrum. The Monte Carlo simu-\nlations undergo a post-processing step that takes into ac-\ncount the effects of un-resolvable pile-up and finite energy\nresolution. The fit is performed in the energy range of\n[80, 800] keV. At lower energies, the data reconstruction\nis not satisfactory, while at higher energies the statistics\nis scarce. A uniform binning of 10 keV is chosen to avoid\nsystematic effects due to the peak line-shape.\nWe assume the number of counts in each bin to follow\nthe Poisson probability distribution Pois(n, ν), where n\nis the number of observed events, and νis the expected\nnumber of counts. νconsists of a linear combination of\nthe signal template S(gA,sNME) and background simu-\nlations Bj. We introduce the normalization factors NS\nandNB,j, that are proportional to the half-life of115In\n(T1/2) and to the activities of the background compo-\nnents, respectively. The half-life can be expressed as\nT1/2=ln(2)·t·mInI·NA·i.a.(115In)·ϵ\nMInI·NS·NMC(1)\nwhere t= 128 .8 h is the measurement time, NAis the\nAvogadro constant, i.a.(115In) = (95 .719±0.052)% [36]\nis the natural isotopic abundance of115In,MInIis the\nmolar mass of InI, and NMCthe number of simulated β\ndecays. The expected number of events in the i-th bin is\nνi=NS(T1/2)·S(gA,sNME) i+X\nj=1,2NB,j·(Bj)i(2)\nwhere jidentifies the232Th and228Th contributions from\nthe calibration source. The likelihood can be therefore\nwritten as\nL(data|T1/2, gA,sNME , NB,j) =Y\niPois(ni, νi).(3)\nThe fit uses five free parameters, with three continuous,\nT1/2andNB,j(j= 1,2), and the two remaining are dis-\nEnergy [keV]\n102\n103\nCounts / 10 keV\n2 = 107.11 N dof = 69.0\nT otal Model\nSignal\nBackground\nData\n100\n200\n300\n400\n500\n600\n700\n800\nEnergy [keV]\n-3\n-2\n-1\n1\n2\n3\nData-Model\nFIG. 2. Top. Experimental spectrum (blue dots) and best\nfit result (orange solid line) obtained within the ISM model,\nwhich results to be the most suitable to describe the data in\nthe current framework. The model resulting from the fit is\na linear combination of the115Inβ-decay template spectrum\n(green dashed line), and the two contributions from the tho-\nrium calibration source (red dashed line). The χ2and the\nnumber of degrees of freedom N dofare reported. Bottom. Fit\nresiduals normalized to the statistical uncertainty.\ncrete, namely, gAand sNME. The discrete parameters\nidentify the theoretical template to be picked at every\nstep of the Markov-Chain Monte Carlo used for the poste-\nrior sampling. We assume uniform prior probability dis-\ntributions for all these parameters. Additionally, we in-\ntroduce the analysis cut efficiency ϵwith a Gaussian prior\nprobability distribution. We use the Bayesian Analysis\nToolkit (BAT) [37] to perform the statistical inference as\nwell as the posterior sampling and marginalization. For\neach fit, we quote the median of the marginalized poste-\nrior as an estimator of the best value of the parameter at\nissue. The interval defined by [16, 84]% quantiles is used\nto evaluate the uncertainty.\nAnalysis Results. We perform the data reconstruction\nby using two different fit methods. The first method is\nthebest fit , which determines the configuration that best\nmatches the data by letting both gAand sNME vary.\nFor instance, Figure 2 depicts the data reconstruction\nthrough the best fit method achieved by using the tem-\nplate coming from the ISM model. The second method,\nreferred to as matched half-life fit , tests the core of the\nsNME approach, namely the joint prediction of spectral\nshape and half-life of a forbidden β-decay. We vary the\nvalue of gAwhile fixing sNME, treating it as a free pa-\nrameter of the model. We then select the sNME by com-\nparing the fit result with the known half-life T∗\n1/2in the\n(gA,sNME) parameter space, where the trend of T∗\n1/2is\npredicted by the nuclear model being studied. The value\nofT∗\n1/2has been obtained as an average of previous mea-\nsurements [17, 38–40], weighted for their uncertainties,4\nand is T∗\n1/2= (5.14±0.06)×1014yr. This method is\nillustrated in Figure 3 for the three nuclear models.\nThe results for the two fit methods in the three theoret-\nical frameworks are summarized in Tab. II, where both\nnegative and positive solutions for sNME are reported\nfor completeness. For each combination of the fit method\nand model, the positive sNME solution is preferred based\non the reduced chi-square χ2\nred. When considering neg-\native solutions, the resulting half-life is not compatible\nwith T∗\n1/2. Moreover, within the best fit method for neg-\native values of sNME, the minimization process brings\nthis parameter to its range limits, making the outcomes\nless reliable. In light of this, our discussion will focus on\nthe positive sNME solutions.\nConsidering the best fit method, we study the sys-\ntematic effects due to the fit assumptions. As already\nmentioned, the half-life values exhibit perfect agreement\nwhen changing the nuclear model. Conversely, gAand\nsNME are strictly related to the approximations done\nwithin a specific theoretical framework, therefore we do\nnot expect them to coincide. Moreover, we reiterate the\nbest fit by assuming secular equilibrium in232Th decay\nchain contained in the calibration source. We also study\nthe binning effect by changing the bin width to 20 keV\nand varying both the upper energy limit to 550 keV and\n1000 keV and the lower energy threshold to 150 keV. We\ndo not include a test with an energy threshold below 80\nkeV since we cannot have a satisfactory reconstruction of\nthe background below this energy. All the outcomes show\nvalues for gA, sNME, and half-life completely compatible\nwithin 1 σwith the nominal ones reported in Tab. II.\nDiscussion. Considering the best fit method, we con-\nsistently achieve a robust data reconstruction, obtaining\naχ2\nredin the range [1.55, 1.66]. The signal-to-background\nratio of the collected data limits the possibility of pre-\ncisely determining gAand sNME simultaneously. The\nlatter has a weaker impact on the spectral shape, there-\nfore it is affected by a relatively high uncertainty, some-\ntimes larger than 20%.\nWe observe a clear preference for positive sNME solu-\ntions, aligning closely with the CVC predictions. In par-\nticular, the experimental values are consistently around\n30% of the CVC ones (Tab. I). Demonstrating a system-\natic preference for physical solutions near CVC values\nis crucial. This information significantly helps in select-\ning the correct spectral shape when it strongly depends\non sNME value, as in some cases discovered in β-decay\nshape survey in Ref. [41].\nFor the three models, we obtain different values of gA,\nstill, they all strongly reject the free-nucleon hypothe-\nsis with a significance of at least 4 .7σ. We can deter-\nmine the half-life of the115Inβ-decay with an accuracy\nofO(1%) and all the obtained half-lives are fully com-\npatible with each other. Furthermore, these values are\nin agreement with T∗\n1/2within 1 .4σ, regardless of the\ntheoretical model. However, the half-lives predicted bythe models for the best fit parameters are 2 .37×1014yr\n(ISM), 8 .52×1013yr (MQPM) and 7 .93×1014yr (IBFM-\n2), far from the ones obtained with the fit.\nIt is therefore interesting to compare the best fit out-\ncomes with those of the matched half-life fit, investigating\nhow the predicted half-life match impacts the results. In\nthe cases of ISM and IBFM-2, the fit quality mildly wors-\nens and the physical parameters gAandT1/2are com-\npatible, affirming the reliability and robustness of this\nmethod. By construction, the theoretical predictions on\nthe half-lives agree with T∗\n1/2and are compatible within\n1σwith the measured half-life. Conversely, the matched\nfit approach for the MQPM makes the model unable to\ndescribe the spectral shape. Moreover, the resulting half-\nlife in this case is not compatible within 2 σwith both\ntheoretical predictions and all the other half-life deter-\nminations. This makes the joint prediction of spectral\nshape and half-life less reliable for this method.\nThe value of gAreported for115In in Ref. [17] are sig-\nnificantly smaller than the ones obtained in the current\nwork. It seems that usage of the sNME degree of freedom\nnot only improves the agreement between experimental\nand theoretical values of the half-life, but also shifts gA\nto bigger values. The same happens in113Cd for MQPM\nwhen going from the analysis in Ref. [14] to the one in\nRef. [42], while for ISM and IBFM-2 the two results are\ncompatible. The analysis based on the Spectral Moments\nMethod in Ref. [43] applies a technique somehow similar\nto the matched half-life fit of this work. Even if applied\non113Cd data, the results quoted in terms of gAare very\nclose to this work.\nIn summary, these findings deserve careful examina-\ntion and further insights in future nuclear-model compu-\ntations. It is crucial that such calculations include pre-\ndictions of β-decay spectral shapes based on the preferred\nvalues of the sNME within the enhanced SSM framework.\nThe present study is an important step towards possible\nsystematic sNME preference schemes in this respect. At\nthe same time, this work provides valuable insights into\nthe evolution of the favoured values of the axial coupling\nwhen going from the SSM to the enhanced one.\nThis project has received funding from the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngram under the Marie Sk lodowska–Curie grant agree-\nment N. 101029688. This work was supported by the\nAcademy of Finland, Grant Nos. 314733, 320062, and\n345869. We thank the CUPID collaboration for sharing\ntheir cryogenic infrastructure, M. Guetti for the assis-\ntance in the cryogenic operations, M. Perego for his in-\nvaluable help in many tasks, the mechanical workshop of\nLNGS. This work makes use of the DIANA data analysis\nand APOLLO data acquisition software which has been\ndeveloped by the CUORICINO, CUORE, LUCIFER and\nCUPID-0 collaborations.5\n0.9\n1.0\n1.1\n1.2\n1.3\n1.4\nsNME\n0.94\n0.96\n0.98\n1.00\n1.02\n1.04\n1.06gA\nFit\nT*\n1/2\n1 \n2 \n3 \n2\n0\n2\nsNME\n0.6\n0.8\n1.0\n1.2gA\nISM\n0.8\n1.0\n1.2\n1.4\nsNME\n1.08\n1.09\n1.10\n1.11\n1.12\n1.13\n1.14\n1.15\n1.16\nFit\nT*\n1/2\n1 \n2 \n3 \n2\n0\n2\nsNME\n0.6\n0.8\n1.0\n1.2gA\nMQPM\n0.7\n0.8\n0.9\n1.0\n1.1\n1.2\n1.3\n1.4\nsNME\n1.15\n1.20\n1.25\n1.30\n1.35\nFit\nT*\n1/2\n1 \n2 \n3 \n2\n0\n2\nsNME\n0.6\n0.8\n1.0\n1.2gA\nIBFM 2\nFIG. 3. Identification of the optimal sNME values with the matched half-life fit for ISM (left), MQPM (center) and IBFM-2\n(right), respectively. The main plot reports as colored bands the half-life T∗\n1/2together with its uncertainties, and as red points\nthe value of gAthat best fit the data for a fixed value of sNME. The uncertainty on the latter is fixed by the template fine\nstructure, while the one on gAis the [16, 84]% quantile interval from the Bayesian fit. Each inset shows the half-life dependence\non the other two parameters of the theory, together with the fit results, in a wider sNME interval.\nTABLE II. Results for the two fit methods and the three considered nuclear models on the parameters of interest gA, sNME\nandT1/2. The reduced chi-square χ2\nredis also reported, quantifying the goodness of fit.\nPositive solution Negative solution\nModel gA sNME T1/2[×1014yr] χ2\nred gA sNME T1/2[×1014yr] χ2\nred\nBest fit\nISM 0 .964+0.010\n−0.006 1.75+0.13\n−0.08 5.26±0.06 1 .55 0.774+0.046\n−0.042−5.43+0.40\n−0.22(∗)5.40±0.07 2 .27\nMQPM 1 .104+0.019\n−0.017 2.88+0.49\n−0.71 5.26±0.07 1 .65 0.978+0.022\n−0.021−5.40+0.38\n−0.53(∗)5.46±0.07 2 .26\nIBFM-2 1 .172+0.022\n−0.017 0.81+0.52\n−0.24 5.25±0.07 1 .66 0.739+0.069\n−0.058−5.20+0.63\n−0.41(∗)5.40±0.06 1 .97\nMatched half-life\nISM 0 .965+0.013\n−0.010 1.10±0.03 5 .20±0.07 1 .78 0.869+0.004\n−0.004 −1.15±0.03 5 .50±0.06 2 .94\nMQPM 1 .093+0.009\n−0.007 0.90±0.03 5 .05±0.06 2.32 0.992+0.004\n−0.004 −1.00±0.03 5 .64±0.07 3 .22\nIBFM-2 1 .163+0.036\n−0.010 1.10±0.03 5 .28±0.06 1 .67 0.958+0.012\n−0.015 −1.15±0.03 5 .46±0.07 2 .28\n(*)Posterior overlapping parameter boundaries.\naCorresponding author: stefano.ghislandi@gssi.it\nbCorresponding author: dounia.helis@lngs.infn.it\n[1] M. Agostini, G. Benato, J. A. Detwiler, J. Men´ endez,\nand F. Vissani, Rev. Mod. Phys. 95, 025002 (2023).\n[2] W. Buchmuller, R. D. Peccei, and T. Yanagida, Ann.\nRev. Nucl. Part. Sci. 55, 311 (2005).\n[3] L. Canetti, M. Drewes, and M. Shaposhnikov, New J.\nPhys. 14, 095012 (2012).\n[4] S. Abe et al. (KamLAND-Zen), Phys. Rev. Lett. 130,\n051801 (2023).\n[5] F. Cappuzzello et al. (NUMEN), Front. Astron. Space\nSci.8, 52 (2021).\n[6] F. Cappuzzello et al. (NUMEN), Int. J. Mod. Phys. A\n36, 2130018 (2021).\n[7] I. H. Hashim et al. , Phys. Rev. C 108, 014618 (2023).\n[8] P. Gimeno, L. Jokiniemi, J. Kotila, M. Ramalho, and\nJ. Suhonen, Universe 9, 270 (2023).\n[9] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C87,\n014315 (2013).\n[10] C. Augier et al. (CUPID-Mo), Phys. Rev. Lett. 131,\n162501 (2023).[11] H. Ejiri, J. Suhonen, and K. Zuber, Phys. Rept. 797, 1\n(2019).\n[12] M. Haaranen, J. Kotila, and J. Suhonen, Phys. Rev. C\n95, 024327 (2017).\n[13] P. Belli and et al. , Phys. Rev. C 76, 064603 (2007).\n[14] L. Bodenstein-Dresler and et al. (COBRA), Phys. Lett.\nB800, 135092 (2020).\n[15] J. Kostensalo and et al. , Phys. Lett. B 822, 136652\n(2021).\n[16] M. Paulsen et al. , (2023), arXiv:2309.14014 [nucl-ex].\n[17] A. F. Leder et al. , Phys. Rev. Lett. 129, 232502 (2022).\n[18] M. Haaranen, P. C. Srivastava, and J. Suhonen, Phys.\nRev. C 93, 034308 (2016).\n[19] H. Behrens and W. B¨ uhring, Electron Radial Wave Func-\ntions and Nuclear Beta-Decay (Clarendon Press, Oxford,\n1982).\n[20] A. Kumar, P. C. Srivastava, J. Kostensalo, and J. Suho-\nnen, Phys. Rev. C 101, 064304 (2020).\n[21] Kumar, Anil, Srivastava, Praveen C., and Suhonen,\nJouni, Eur. Phys. J. A 57, 225 (2021).\n[22] L. Pagnanini et al. (ACCESS), Eur. Phys. J. Plus 138,\n445 (2023).\n[23] O. Azzolini et al. (CUPID), Eur. Phys. J. C78, 428\n(2018).6\n[24] C. Arnaboldi et al. , Rev. Sci. Instrum. 86, 124703 (2015).\n[25] C. Arnaboldi et al. , JINST 13, P02026 (2018).\n[26] S. Di Domizio et al. , JINST 13, P12003 (2018).\n[27] E. Gatti and P. F. Manfredi, Riv. Nuovo Cimento 9, 1\n(1986).\n[28] O. Azzolini et al. , Eur. Phys. J. C78, 734 (2018).\n[29] C. Alduino et al. , Phys. Rev. C 93(2016), 10.1103/phys-\nrevc.93.045503.\n[30] S. Agostinelli et al. (GEANT4), Nucl. Instrum. Meth. A\n506, 250 (2003).\n[31] J. Blachot, Nuclear Data Sheets 113, 2391 (2012).\n[32] E. Caurier and et al. , Rev. Mod. Phys. 77, 427 (2005).\n[33] J. Toivanen and J. Suhonen, Phys. Rev. C 57, 1237\n(1998).\n[34] F. Iachello and P. v. Isacker, The Interacting Boson-\nFermion Model , Cambridge Monographs on Mathemati-\ncal Physics (Cambridge University Press, 1991).\n[35] O. Azzolini et al. (CUPID), Phys. Rev. Lett. 131, 222501\n(2023).[36] J. R. de Laeter, J. K. B¨ ohlke, P. D. Bi` evre, H. Hidaka,\nH. S. Peiser, K. J. R. Rosman, and P. D. P. Taylor, Pure\nAppl. Chem. 75, 683 (2003).\n[37] O. Schulz, F. Beaujean, A. Caldwell, C. Grunwald,\nV. Hafych, K. Kr¨ oninger, S. La Cagnina, L. R¨ ohrig, and\nL. Shtembari, SN Comput. Sci. 2, 1 (2021).\n[38] L. Pfeiffer, A. P. Mills, E. A. Chandross, and T. Kovacs,\nPhys. Rev. C 19, 1035 (1979).\n[39] D. E. Watt and R. N. Glover, The Philosophical Maga-\nzine: A Journal of Theoretical Experimental and Applied\nPhysics 7, 105 (1962).\n[40] G. B. Beard and W. H. Kelly, Phys. Rev. 122, 1576\n(1961).\n[41] M. Ramalho and J. Suhonen, (2023), arXiv:2312.07111\n[nucl-th].\n[42] J. Kostensalo, J. Suhonen, J. Volkmer, S. Zatschler, and\nK. Zuber, Phys. Lett. B 822, 136652 (2021).\n[43] J. Kostensalo, E. Lisi, A. Marrone, and J. Suhonen,\nPhys. Rev. C 107, 055502 (2023)." }, { "title": "2401.16083v1.Transport_driven_super_Jeans_fragmentation_in_dynamical_star_forming_regions.pdf", "content": "MNRAS 000, 1–5 (2020) Preprint 30 January 2024 Compiled using MNRAS L ATEX style file v3.0\nTransport-driven super-Jeans fragmentation in dynamical star-forming\nregions\nGuang-Xing Li1★\n1South-Western Institute For Astronomy Research, Yunnan University, Kunming, 650600, China\nLast updated 2020 June 10; in original form 2013 September 5\nABSTRACT\nThe Jeans criterion is one cornerstone in our understanding of gravitational fragmentation. A critical limitation of the Jeans\ncriterion is that the background density is assumed to be a constant, which is often not true in dynamic conditions such\nas star-forming regions. For example, during the formation phase of the high-density gas filaments in a molecular cloud, a\ndensity increase rate ¤𝜌implies an mass accumulation time of 𝑡acc=𝜌/¤𝜌=−𝜌(∇·(𝜌®𝑣))−1. The system is non-stationary\nwhen the mass accumulation time becomes comparable to the free-fall time 𝑡ff=1/√︁\n𝐺𝜌. We study fragmentation in non-\nstationary settings, and find that accretion can significantly increases in the characteristic mass of gravitational fragmentation (\n𝜆Jeans ,aac=𝜆Jeans(1+𝑡ff/𝑡acc)1/3,𝑚Jeans ,acc=𝑚Jeans(1+𝑡ff/𝑡acc)).Inmassivestar-formingregions,thismechanismoftransport-\ndriven super-jeans fragmentation can contribute to the formation of massive stars by causing order-of-magnitude increases in\nthe mass of the fragments.\nKey words: galaxies: star formation –stars: formation –hydrodynamics – instabilities – methods: analytical\n1 INTRODUCTION\nThe Universe is a gigantic multi-scale self-gravitating system.\nThe Jeans criterion (Jeans 1902) sets the foundation upon which\nour understanding of gravitational fragmentation bases. 𝜆Jeans=\n𝜎𝑣/(𝐺𝜌)1/2is called the Jeans length, which is a threshold wave-\nlength beyond which perturbations are gravitationally unstable. The\nJeans criterion has a profound impact on modern astrophysical re-\nsearch as the core of the theory has been essential in understanding\nthebehaviorofavarietyofastrophysicalsystems,includingthefrag-\nmentation of gas under turbulent support (McKee & Tan 2003) and\nthestabilityofgalaxydisks(Toomre1964).TheJeanscriterionisalso\nacriticalpartofmodernnumericalsimulationsofself-gravitatinggas\nas it has been integrated into the algorithm controlling the produc-\ntion of sink particles (Bate 2012) – representative particles created\nto remove gravity-included singularities in simulations.\nOne critical limitation of the Jeans criterion is that the initial\ndensity distribution is assumed to be static at the start. Whether\nthe initial conditions are truly static has been a question barely ad-\ndressed in previous studies. Contrary to the stationary appearance\nas often taken for granted, most astrophysical systems are dynamic.\nThis is particularly true for star-forming regions in the Milky Way,\nwhere density enhancements are generated by turbulence (Passot\n& Vázquez-Semadeni 1998; Padoan & Nordlund 2011), converging\nflows (Hennebelle & Pérault 1999; Vázquez-Semadeni et al. 2007)\nfrom turbulence (Padoan et al. 2020), Global Hierarchical Collapse\n(Vázquez-Semadeni et al. 2019) or cloud-cloud collisions (Fukui\netal.2021)inrelativelyshorttimes.Therapiddensityincreasesim-\nplyanothertimescale,themassaccumulationtime 𝑡acc=𝜌/¤𝜌.When\n★Contact e-mail: gxli@ynu.edu.cn, ligx.ngc7293@gmail.com𝑡accisshorterthanthefree-falltime,theeffectoffastmasstransport\nbecomes significant.\nWe modify the Jeans formalism to extend its application to non-\nstationary conditions where 𝑡acc≲𝑡ff. We propose that the effect of\na density increase during the mass accumulation phase can lead to\nsignificantincreasesinthemassoffragments.This transport-driven\nsuper-Jeans fragmentation is active in regions where the density is\nhigh, and the flow is turbulent, leading to the formation of massive\nfragments, which are precursors for massive stars.\nOur new criterion can naturally explain why massive stars are\nformed in regions of high densities. We note that in the Jeans crite-\nrion,thereisanegativecorrelationbetweenthemassofthefragments\nanddensity,where 𝑚Jeans∼𝜌−1/2.Thisnegativecorrelationcontra-\ndicts the fact that massive stars form exclusively in the densest parts\nof the interstellar medium (Tan et al. 2014). Our analysis can help\nto resolve this puzzle as the fragments formed in high-density, tur-\nbulent environments are more massive than previously thought, and\nisapossiblemechanismwhichcanexplaintheformationofmassive\nstars in the universe.\nOurmodificationshouldhelptoimprovethenumericalsimulations\nofstar-formingregions.TheJeanscriterionhasbeenakeypartofthe\nalgorithm controlling the production of sink particles in numerical\nsimulations. The new criterion should be placed in these algorithms\nto achieve consistent results when the conditions are dynamic.\n2 TRANSPORT-DRIVEN SUPER-JEANS\nFRAGMENTATION\n2.1 The Jeans length\nWe consider the fragmentation of a medium of density 𝜌, veloc-\nity dispersion 𝜎v. In this stationary case, the system contains two\n©2020 The AuthorsarXiv:2401.16083v1 [astro-ph.GA] 29 Jan 20242 Guang-Xing Li\ntimescales, namely the free-fall time\n𝑡ff=1\n(𝐺𝜌)1/2, (1)\nand the sound crossing time\n𝑡cross=𝑙\n𝜎v. (2)\nThe only dimensionless number is Π1=𝑡ff/𝑡cross=constant. Us-\ning the Buckingham 𝜋Theorem (Buckingham 1914), the governing\nequationofthesystemis 𝑓(Π1)=1.Sincethereisonlyonevariable\nΠ1, we conclude that in the Jeans fragmentation :\nΠ1=1, 𝑡ff=𝑡cross, 𝜆Jeans=𝑐s/√︁\n𝐺𝜌. (3)\nThe Jeans length 𝜆Jeans=𝜎v/√︁\n𝐺𝜌is the characteristic scale\nbeyond which structures can collapse in a complex, self-gravitating\nsystem.This scale is relevant as long as a structure of density 𝜌is\nsupported by a pressure of the order 𝜌𝜎2v. For example, the Jeans\nlengthisalsothecharacteristicsizebeyondwhichpressure-confined\ndropletsbecomegravitationallyunstable(Ebert1955;Bonnor1956).\nWe note that a more rigorous approach is to derive the Jeans\nlengthtoaddperturbationstoahomogeneoussystem.Inthiscase,the\nthresholdwavelengthappearsnaturallyinthedispersionrelation.Due\ntothehighlyidealizedinitialconditions,thisapproachisrigorousin\ntheory,butnotparticularlyusefulinrealisticInthispaper,theJeans\nlength is a threshold wavelength for gravitational stability, which is\nrelevant for both cases.\n2.2 Non-stationary corrections to the Jeans length\nIn a dynamic environment, all parameters can change with time.\nWe focus on the cases where the density changes, yet the velocity\ndispersiondoesnotchangesignificantly.Thisassumptionisvalidas\nlong as the timescale of the temperature increase is longer than the\nfree-fall time.\nOur analysis aims to pin down the dominant wavelength of sys-\ntems controlled complex, non-linear interactions between accretion\nand fragmentation. A positive ¤𝜌is the consequence of boundary\nconditions imposed on the system. Using mass conservations,\n𝑡acc=𝜌/¤𝜌=−𝜌(∇·(𝜌®𝑣))−1. (4)\nExamples of such density enhancements includes shocks in super-\nsonic turbulence (Passot & Vázquez-Semadeni 1998; Padoan &\nNordlund2011),densityenhancementsproducedinconvergingflows\n(Hennebelle & Pérault 1999; Vázquez-Semadeni et al. 2007) and\nhigh-density regions produced during cloud-cloud collisions (Fukui\netal.2021).Inthestationarycase,thethresholdwavelength 𝜆Jeansis\nselectedthroughtheinterplaybetweenpressuresupportandgravity.\nTheveryprocessofmassaccumulationcanincreasethewavelength\nabove which perturbations can grow.\nTo derive the fragmentation scales in this non-stationary setting,\nwe start with the Buckingham 𝜋Theorem (Buckingham 1914), and\nrepresent our system in several dimensionless parameters. These\ninclude\nΠ1=𝑡ff/𝑡cross (5)\nand\nΠ2=𝑡ff/𝑡acc, (6)\nandinthe super-Jeans fragmentation ,thegoverningequationshould\ntake the form\nΠ1=𝑓(Π2),𝜆Jeans ,acc=𝜆Jeans𝑓(Π2). (7)To determine the functional form of 𝑓, we introduce the asymp-\ntotic constraint that when the effect of density increase induced by\naccretion is negatable, the characteristic length for gravitational in-\nstabilityistheJeanslength.Thismeanswhen 𝑡acc→inf,weexpect\nlim\nΠ2→0𝑓=1, (8)\nsuchthatthefragmentationlengthisdeterminedbyEq.3.Theeffect\nof accretion-induced density growth is significant when Π2>0.\nWhenΠ2≫1, assuming that the system has no preferred scales,\nthe relationship between the dimensionless variables should take\nthe form of a power law, e.g. Π1= Π𝛼\n2. Taking these asymptotic\nconstrains into considerations, we have 𝑓=(1+𝑓0Π𝛼\n2)or𝑓=\n(1+𝑓0Π2)𝛼. Since the behaviors of these two functional forms are\nnearlyindistinguishableexceptatasmallerrangeofparameterspace\nwhereΠ2≈11, we choose\n𝑓=(1+𝑓0Π2)𝛼(9)\nfor simplicity, where 𝑓0is a constant of the order of unity. Our\nremaining task is to determine the value of 𝛼.\n2.3 Economy of arrangement – Larger fragments accrete faster\nAccording to the Jeans criterion, all wavelengths above 𝜆Jeansare\nunstable and can grow at the free-fall rate if pressure support is\nnegligible.WhydoesNatureselectoneconfigurationovertheother?\nA key difference between these modes is their ability to accrete gas\nin the non-linear regime.\nWe perform a thought experiment where we divide a medium\ninto packets of different sizes and investigate the relation between\nfragmentationlengthandmassaccretion.Assumingafragmentation\nwavelengthof 𝜆,themassofthefragmentsis 𝑚≈𝜌𝜆3,andonecan\ncomputetheaccretionrateontothefragmentsusing ¤𝑚≈𝐺2𝑚2𝜎−3v𝜌\n(Hoyle&Lyttleton1941;Bondi1952),whereaccretionoccursinside\na region of size 𝜆acc=𝐺𝑚𝜎−2v. The time for the fragments to\nincrease their masses is\n𝑡acc,fragments=𝑚/¤𝑚=𝐺𝜌𝜆3𝜎−3\nv. (10)\nForamediumfilledwithsuchpacketofsize 𝜆acc,themaximumrate\nof density growth caused by accretion is characterized by the mass\naccumulation time\n𝑡∗\nacc=𝜌/¤𝜌=𝑚/¤𝑚=𝑡double=𝐺𝜌𝜆3𝜎−3\nv∝𝜆−3. (11)\nThe fact that the mass accumulation time depends on 𝜆illustrates\nthe key difference between these modes: modes with larger 𝜆can\nhave higher rates of density growth ( 𝑡∗acc≈𝜆−3).\n2.4 Transport-driven super-Jeans fragmentation\nWhen an externally driven flow leads to a continuous increase in\ndensity,𝑡acc=𝜌/¤𝜌becomes a control parameter of the system.\nWeproposethe hypothesis of transport-regulated fragmentation :the\nfragmentation should in a way that to ensure that the rate of density\nincrease caused by accretion should stay in sync with the density\nincrease rate demanded by the boundary condition , such that the\nfragmentation length can be determined by letting 𝑡acc=𝑡∗acc.\nTounderstandwhythisconditionisnecessary,weconsideracase\nwherewechoosea 𝜆thatistoosmall.Evenifsuchperturbationscan\n1These two forms may differ by 50% when Π2≈1.\nMNRAS 000, 1–5 (2020)Transport-driven Super-Jeans fragmentation 3\nStar-forming regionFilament formation: \nInfall region\nSuper-Jeans Fragmentation:\nFigure 1. A sketch of transport-driven super-Jeans during filament as-\nsembly.See Sec. 3 for details.\ngrow, the newly added gas is too much for the system to consume.\nOnlyifwechoosea 𝜆thatislargelyenoughcanthenewly-addedgas\nbe accreted. A continuous injection of gas can suppress the growth\nof small-sized structures, favoring larger ones.\nLetting𝑡acc=𝜌/¤𝜌=𝑡∗acc, we derive a critical wavelength of\n𝜆Jeans ,acc−domiante=𝑡ff𝜎v(𝑡ff/𝑡acc)1/3=𝜆Jeans(𝑡ff/𝑡acc)1/3.(12)\nTakingasymptoticconstraintsdiscussedinSec.2.2intoaccount,the\nformula for wavelength in the general case is\n𝜆Jeans ,aac=𝜎v\n(𝐺𝜌)1/2(1+𝑓a𝑡ff/𝑡acc)1/3=𝜆Jeans(1+𝑓a𝑡ff/𝑡acc)1/3,\n(13)\nwhere the characteristic mass is\n𝑚Jeans acc=𝜌−1/2𝐺−3/2𝜎3\nv(1+𝑓a𝑡ff/𝑡acc)=𝑚Jeans(1+𝑓a𝑡ff/𝑡acc).\n(14)\nwhere𝜆Jeans≈𝜎v/√︁\n𝐺𝜌,𝑚Jeans≈𝜎3v𝐺−3/2𝜌−1/2,𝑓𝑎≈1is a\nnumericalprefactoroforunity,anditsexactvaluecanbederivedby\nmatching our formula with results from e.g. numerical simulations.\nThe effect of accretion can lead to significant increases in the\ncharacteristic mass of the fragments when the mass accumulation\ntime𝑡accis short compared to the free-fall time. Our approach of\nderiving the scaling exponent by demanding the conservation of\nsomecriticalquantities(thisquantityisthemassinourcase)shares\na similar spirit to how Kolmogorov derived the famous 5/3 energy\nspectrumofturbulencebydemandingaconstantenergyfluxbetween\ndifferent scales (Kolmogorov 1941).\n3 APPLICATION: TRANSPORT-DRIVEN SUPER-JEANS\nFRAGMENTATION IN MASSIVE STAR-FORMING\nREGIONS\n3.1 Problem setup: Fragmentation during filament assembly\nIn realistic settings, fragmentation rarely occurs in a homogeneous\nmediumbutcanbeboundedbyvariousconstraints(i.e.thesizeofthe\nregion). We consider the fragmentation occurring to a filament. The\ndensity of this filament is a function of time 𝜌=𝜌(𝑡). The filament\nhasaconstantvelocitydispersion 𝜎v,andawidthof 𝑑≈0.1 pc.The\nassumption that fragmentation occurs on filaments of 𝑑≈0.1 pc\nis consistent with the results from Herschel observations towardsnearby molecular clouds (Arzoumanian et al. 2011; André et al.\n2014), where the scale of 0.1 pc has been interpreted as sonic scale\nof the turbulence 𝑙sonic(Arzoumanian et al. 2011). A sketch of this\npicture can by found in Fig. 1\nOne consequence of the time dependency is that the fragmenta-\ntion scale and resulting mass are no longer constants. Due to the\ndensity increase, the fragmentation length decreases with time, and\nfragmentation occurs when 𝜆Jeans ,acc≲𝑙sonic.\nShouldthefragmentationbeJeans-like,auniquecriticaldensityis\ndeterminedthrough 𝜆Jeans=𝑙sonic,fromwhichasinglecharacteristic\nmass of≈0.9𝑀⊙is implied (where 𝜌critcan be determined via\n𝜆Jeans=𝑙sonic, and the mass is 𝑚crit,Jeans≈𝜌crit𝜆3\nJeans). The effect\nof accretion is to introduce another viable to the systems, leading to\na range of masses depending on how the gas density evolves.\nTo illustrate the effect of transport-driven mass accumulation, we\nconsider a simple parameterized model where 𝜌=𝜌0(𝑡/𝑡0))𝛾, and\nassume that that fragmentation would occur when 𝜆Jeans≲𝑙sonic≈\n0.1 pc.Wechoose𝛾=1/2,wherethegrowthisfastatthebeginning\nand it slows down. This slow-down represents the gradual depletion\nof the gas reservoir. The evolution of a representative systems is\nplotted in Fig. 2. When the density increase rate is large enough,\ne.g.¤𝜌=107cm−3𝑚H2Myr−1, the effect of externally-driven flow\nbecomedominant,resultinginamassthatisoneordersofmagnitude\nlarger than expected from the Jeans fragmentation.\nOnecanalsoderivecharacteristicmassanalytically:Inourfiducial\nmodel where 𝜌=𝜌0(𝑡/𝑡0)1/2, assuming𝑙sonic=0.1 pc,𝜎v=\n0.2 km s−1,𝑡0=1 Myr, the characteristic mass is (see A)\n𝑚≈0.9𝑀⊙+1.7𝑀⊙(𝜌0\n105𝑚H2)1\n2∝𝜌1/2\n0, (15)\nwhich is consistent with the numerical results.\n3.2 Super-Jeans fragmentation in massive star-forming regions\nIn massive star-forming regions in the Milky Way, the mechanism\nofconverging-flow-drivensuper-Jeansfragmentationcanleadtothe\nformation of fragments that are around one orders-of-magnitudes\nmore massive than expected from the Jeans fragmentation. Recall\nthat in our fiducial model where 𝜌=𝜌0(𝑡/𝑡0)1/2, we require 𝜌0=\n107cm−3𝑚H2, and𝑡0=1 Myrto produce a massive fragments of\n17𝑀⊙.\nThis density increase rate of ¤𝜌=107cm−3𝑚H2Myr−1is repre-\nsentative of the conditions in massive star-forming regions in the\nMilky Way. This is because the regions are dynamic, where the\nturbulent motion can create dense structures which relatively short\ntime. Assuming a simple cylindrical configuration where the gas is\ndrivenfromtheoutsidesuchthatthefilamentsformatthecenter,we\nestimate a density increase rate of\n¤𝜌acc,filament=𝜌mean(𝑣infall/𝑙sonic)(𝑙infall/𝑙sonic), (16)\nwhere𝜌meanis the mean density of the medium, 𝑣infallin an infall\nspeed,𝑙sonicis the sonic scale, and 𝑙infallis the scale at which infall\nis initiated. Toward a typical high-mass star-forming region such as\ntheIRDC18223(Beutheretal.2015),weadopt 𝜌mean=105cm−3,\nwhichisthemeasureddensityoftheregion,and 𝑣infall=2.5 km s−1,\nwhich is the measured velocity dispersion of the region. We further\nassume𝑙sonic=0.1 pc, and𝑙infall=0.5 pc, which is a fraction of\nthe size of the region. From these, we derive a density increase rate\nof≈1×107cm−3Myr−1, consistent with what is required in our\nfiducial model.\nThis transport-driven super-Jeans fragmentation can explain the\npuzzling fact why the separation between dense cores in massive\nMNRAS 000, 1–5 (2020)4 Guang-Xing Li\n0.000 0.002 0.004 0.006 0.008 0.010\nTime (Myr)101\n100101Fragmentation length (pc)Jeans length Jeans\nModified Jeans length Jeans,acc\nlsonic=0.1pc\n0.000 0.002 0.004 0.006 0.008 0.010\nTime (Myr)101\n100101102103104105106Fragmentation mass (M)\nMJeans=0.8M,MJeans,acc=17M,M 21MJeans\nJeans mass mJeans\nModified Jeans mass mJeans,acc\nMsonic, Mass contained within a 0.1 pc regionModel of fast accretion, =0(t/t0))1/2,0=107mH2,t0=1Myr\nFigure 2. Effect of mass accumulation on fragmentation. We consider a simple model where 𝜌=𝜌0(𝑡/𝑡0)1/2,𝜌0=107𝑚H2,𝑡0=1 Myr.Left Panel: The\nJean length 𝜆Jeansand the transport-modified Jeans length 𝜆Jeans,accas a function of time. The horizontal line is the sonic scale 𝑙sonic≈0.1 pc. Fragmentation\nstarts when 𝜆Jeans,acc≈𝑙sonic.Right Panel: Jeans mass 𝑚Jeansand transport-modified Jeans mass 𝑚Jeans,accas a function of time. We also plot the 𝑚sonic,\nwhich is mass contained in regions of 𝑙≈𝑙sonic≈0.1 pc. Fragmentation occurs when 𝑚Jeans,acc≲𝑚sonic. We expect the fragments to have 𝑚≈17𝑀⊙, which\nis much larger than the Jeans mass ≈21𝑚Jeans.\nstar-forming regions is larger, and the cores are more massive than\nexpected from the Jeans fragmentation (Zhang et al. 2009; Wang\net al. 2014; Figueira et al. 2018; Xu et al. 2023) as revealed by\nALMAobservations.However,otherstudieshaveconcludedthatthe\nseparations between dense cores are consistent with thermal Jeans\nfragmentation (Liu et al. 2017; Sanhueza et al. 2019; Svoboda et al.\n2019; Lu et al. 2020). These conflicting conclusions can result from\nthe intrinsic difference in mass accumulation rate between different\nregions, the differences in techniques used, or the failure to account\nforthefurtherevolutionofcoreseparationafterfragmentation.Nev-\nertheless,onecanconcludethatsuper-Jeansseparationsareobserved\natleastinsomeregions,wheretransport-drivensuper-Jeansfragmen-\ntation is a promising explanation. Once formed, these super-Jeans\nfragments can suppress the subsequent fragmentation through tidal\nforces, as a recent study has demonstrated (Li 2023).\n4 CONCLUSIONS\nThe formula proposed by Jeans predicts the characteristic mass of\nfragments produced in gravitational fragmentation. A limitation of\nthe Jeans criteria is that the setting is stationary, as in reality, the\ndensity is subject to a constant change in dynamic environments\nsuch as star-forming regions. This time-dependency would become\na significant issue when the mass accumulation time 𝑡acc=𝜌/¤𝜌=\n−𝜌(∇·(𝜌®𝑣))−1becomesshorterthanthefree-falltime 𝑡ff=1/√︁\n𝐺𝜌,\nwhere we expect the behavior of the system to be different.\nWhen gas is supplied to a region from the outside, the growth of\nlarger perturbations is preferred over smaller ones, because larger\nfragmentationmodescanconsumemassathigherrates.Inthisnon-\nstationary gas, the Jeans length becomes\n𝜆acc,Jeans=𝜆Jeans(1+𝑡ff/𝑡acc)1/3, (17)\nand the Jeans mass becomes\n𝑚acc,Jeans=𝑚Jeans(1+𝑡ff/𝑡acc), (18)wheretheterm 𝑡ff/𝑡accrepresentstheeffectofdensityincreasecaused\nby an externally driven flow. Under a realistic mass inflow rate as\nestimated from a typical high-mass star formation region, we ex-\npect𝑚≈17𝑀⊙, which is one order of magnitude larger than the\nJeans case. This transport-driven super-Jeans fragmentation is one\nkeymechanismleadingtotheformationofmassivestarsingalaxies.\nTransport-driven super-jeans fragmentation is thus a key mecha-\nnism for massive star formation. In contrast to the earlier proposals\nwheretheroleofturbulenceistoregulatethestarformationprocess\nKrumholz & McKee (2005), recent developed tend to favor a link\nbetweenadynamical,activeenvironmentwiththeformationofmore\nmassiveobjects(Vázquez-Semadenietal.2019;Padoanetal.2020).\nThe significant increase of the Jean mass under such a dynamical\nenvironmentweproposeisalongthesamedirection.Weexpectthat\nfutureapplicationsofourresultstothesedynamicalsettingscanlead\nto a better understanding of the formation of massive stars in the\nuniverse.\nACKNOWLEDGEMENTS\nWe would like to thank the referee for careful readings of the paper\nand for constructive comments. GXL acknowledges support from\nNSFC grant No. 12273032 and 12033005. This work is motivated\nfromacollaborationwithFeng-WeiXu(PekingUniversity)andcol-\nleagues from the ATOMS collaboration. Guang-Xing Li would like\nto thank Prof. Andreas Burkert for sharing his curiosity and excite-\nment about equations. For Prof. Xun Shi for her patience with the\ntimescales,andthankfriendsinthecyclinggroup,Prof.ChandraB.\nSingh, Qiqi Jiang for reviving his interest in equations.\nMNRAS 000, 1–5 (2020)Transport-driven Super-Jeans fragmentation 5\nDATA AVAILABILITY STATEMENT\nNo proprietary data was used during the preparation of the\nmanuscript.\nREFERENCES\nAndré P., Di Francesco J., Ward-Thompson D., Inutsuka S. I., Pudritz R. E.,\nPineda J. E., 2014, in Beuther H., Klessen R. S., Dullemond C. P., Hen-\nning T., eds, Protostars and Planets VI. pp 27–51 ( arXiv:1312.6232 ),\ndoi:10.2458/azu_uapress_9780816531240-ch002\nArzoumanian D., et al., 2011, A&A, 529, L6\nBate M. R., 2012, MNRAS, 419, 3115\nBeuther H., Ragan S. E., Johnston K., Henning T., Hacar A., Kainulainen\nJ. T., 2015, A&A, 584, A67\nBondi H., 1952, MNRAS, 112, 195\nBonnor W. B., 1956, MNRAS, 116, 351\nBuckingham E., 1914, On physically similar systems, 6\nEbert R., 1955, Z. Astrophys., 37, 217\nFigueira M., Bronfman L., Zavagno A., Louvet F., Lo N., Finger R., Rodón\nJ., 2018, A&A, 616, L10\nFukui Y., Habe A., Inoue T., Enokiya R., Tachihara K., 2021, PASJ, 73, S1\nHennebelle P., Pérault M., 1999, A&A, 351, 309\nHoyle F., Lyttleton R. A., 1941, MNRAS, 101, 227\nJeansJ.H.,1902,PhilosophicalTransactionsoftheRoyalSocietyofLondon\nSeries A, 199, 1\nKolmogorov A., 1941, Akademiia Nauk SSSR Doklady, 30, 301\nKrumholz M. R., McKee C. F., 2005, ApJ, 630, 250\nLi G.-X., 2023, MNRAS,\nLiu T., et al., 2017, ApJ, 849, 25\nLuX.,ChengY.,GinsburgA.,LongmoreS.N.,KruijssenJ.M.D.,Battersby\nC., Zhang Q., Walker D. L., 2020, ApJ, 894, L14\nMcKee C. F., Tan J. C., 2003, ApJ, 585, 850\nPadoan P., Nordlund Å., 2011, ApJ, 730, 40\nPadoanP.,PanL.,JuvelaM.,HaugbølleT.,NordlundÅ.,2020,ApJ,900,82\nPassot T., Vázquez-Semadeni E., 1998, Phys. Rev. E, 58, 4501\nSanhueza P., et al., 2019, ApJ, 886, 102\nSvoboda B. E., et al., 2019, ApJ, 886, 36\nTan J. C., Beltrán M. T., Caselli P., Fontani F., Fuente A., Krumholz M. R.,\nMcKee C. F., Stolte A., 2014, in Beuther H., Klessen R. S., Dulle-\nmond C. P., Henning T., eds, Protostars and Planets VI. pp 149–172\n(arXiv:1402.0919 ), doi:10.2458/azu_uapress_9780816531240-ch007\nToomre A., 1964, ApJ, 139, 1217\nVázquez-Semadeni E., Gómez G. C., Jappsen A. K., Ballesteros-Paredes J.,\nGonzález R. F., Klessen R. S., 2007, ApJ, 657, 870\nVázquez-Semadeni E., Palau A., Ballesteros-Paredes J., Gómez G. C.,\nZamora-Avilés M., 2019, MNRAS, 490, 3061\nWang K., et al., 2014, MNRAS, 439, 3275\nXu F.-W., et al., 2023, MNRAS, 520, 3259\nZhang Q., Wang Y., Pillai T., Rathborne J., 2009, ApJ, 696, 268\nAPPENDIX A: CHARACTERISTIC MASS\nWe consider a model of density growth where\n𝜌=𝜌0(𝑡/𝑡0)𝛾. (A1)\nAssuming that fragmentation occurs on filaments of the width of\n𝑑≈𝑙sonic≈0.1 pc, we study the effect of accretion on the evolution\nof such a system. The free-fall time is\n𝑡ff=1/√︁\n𝐺𝜌=1/√︁\n𝐺𝜌0(𝑡/𝑡0)−𝛾/2, (A2)\nand the mass accumulation time is\n𝑡acc=𝜌/¤𝜌=𝛾𝑡 . (A3)The increasing density as assumed in the model implies a de-\ncreasing fragmentation length. Fragmentation occurs when 𝜆≲𝑑≈\n𝑙sonic≈0.1 pc.InthecaseoftheJeansfragments,thecriticaldensity\ncan be solved via 𝜆Jeans=𝑙sonic, where\n𝜌crit,Jeans=𝜎2v\n𝑙2\nsonic𝐺, (A4)\nand a mass of\n𝑚crit,Jeans=𝜌crit,Jeans𝑙3\nsonic=𝑙sonic𝜎2v\n𝐺≈0.9𝑚⊙, (A5)\ncan be derived.\nNextwederivethecharacteristicmassintheaccretion-dominated\nregime. When 𝑡ff≫𝑡accThe accretion-modified Jeans length is\n𝜆Jeans ,acc−dominated=𝜎v\n(𝐺𝜌)1/2(𝑡ff/𝑡acc)1/3, (A6)\nfromwhichwecanderivethetimeatwhichfragmentationoccursby\nsolving𝜆Jeans ,acc−dominatated=𝑙sonic:\n\u0010𝑡fragmentation\n𝑡0\u0011−2𝛾−1\n=𝛾𝑡0(𝑙sonic\n𝜎v)3(𝐺𝜌0)2, (A7)\nwhere the mass of the fragments is\n𝑚Jeans ,acc−dominated=𝜌(𝑡fragmentation)𝑙3\nsonic(A8)\n=𝜌0\u0000𝛾𝑡0\u0000𝑙sonic\n𝜎v\u00013(𝐺𝜌0)2\u0001−𝛾\n2𝛾+1𝑙3\nsonic(A9)\n=𝐺−2𝛾\n2𝛾+1𝜌1\n2𝛾+1\n0𝑙3𝛾+3\n2𝛾+1\nsonic𝜎3𝛾\n2𝛾+1\nv𝑡−𝛾\n2𝛾+1\n0𝛾−𝛾\n2𝛾+1.(A10)\nConsider a fiducial case where 𝛾=1/2, we have\n𝑚Jeans ,acc−dominated\n1.7𝑀⊙≈(𝜌0\n105𝑚H2)1\n2(𝑙sonic\n0.1 pc)9\n4(𝜎v\n0.2 km s−1)3\n4(𝑡0\n1 Myr)−1\n4.(A11)\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–5 (2020)" }, { "title": "2401.16084v1.Do_ferroelectric_nematic_liquid_crystals_really_have_a_huge_dielectric_permittivity_.pdf", "content": "1 \n Do ferroelectric nematic liquid crystals really have a huge \ndielectric permittivity? \nVojko Matko1, Ewa Gorecka2, Damian Pociecha2, Joanna Matraszek2 and N ataša Vaupotič3,4* \n1 University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška 46, \n2000 Maribor \n2 University of Warsaw, Faculty of Chemistry, Zwirki i Wigury 101, 02 -089 Warsaw, Poland \n3 University of Maribor, Faculty of Natural Sciences and Mathematics, Koroška 160, 2000 \nMaribor, Slovenia \n4 Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia \n \n*corresponding author : natasa.vaupotic@um.si \nAbstract \nThere is much scientific debate as to whether the newly discovered ferroelectric nematic liquid crystals \nactually have a huge relative dielectric constant or whether it is just an artefact related to the \ninterpretation of the dielectric spectroscopy measurement results. We show that the interpretation of the \ndielectric measurements of highly polar liquid crystalline mater ial requires a new model that takes into \naccount both the frequency response of the tested material and the capacitance of the thin surface layers . \nA proper interpretation of the measurements confirms a huge relative permittivity of the ferroelectric \nnemat ic phase, which can even be orders of magnitude larger than the measured apparent values. \nFurthermore, the value is independent of the cell thickness and the type of surface electrodes. \n \nFerroelectric nematic (NF) liquid crystalline (LC) phase is a polar p hase with the highest possible \nsymmetry (Cv). Its recent discovery [1,2] immediately attracted a wide research [3–8]. Ferronematic \nmaterials , although fluidic , are characterized by a strong order of dipoles resulting in a large spontaneous \nelectric polarization of the order of 10−3 to 10−2 Cm−2 [2,9,10] , very high values of the second order \nsusceptibility and electrooptic coefficients [11–13], giant flexoelectric effect [14] and potential \nphototunab ility of dielectri c permittivity [15]. These physical pro perties position ferroelectric nematics \nas promising candidates for diverse applications, including display technolog ies, creation of haptic \nsurfaces, development of polar sensors , and are opening new roads to design ing photonic structures [13]. \nApplications became even more viable and promising with the experimental realization of chiral \nferroelectric nematic materials [16–19] and synthesis of materials (or their mixtures) having a \nferroelectric nematic ph ase at a room temperature [9,20–22]. Another distinguishing feature of \nferroelectric nematic s is their remarkably high relative permittivity, a parameter that quantifies the \nreduction in the electric field strength inside a material compared to a vacuum and serves as a measure \nof the material's ability to store electrical energy. Typically measured values of the relative permittivity \nof a ferroelectric nematic are of the order of 104, within a frequency range which can ext end up to the \nkHz regime [2,10,12,23 –27]. This huge permittivity has been primarily attributed to the ease with which \nthermally activated fluctuations of the polarization direction occur. However, the relative permittivity in \nthe ferroelectric nematic phase appears to be strongly influenced by the surface treatment applied to the \ncells (polymer anchoring layers) in which the dielectric measure ment s are performed and the thickness \nof cell s [23–27]. A similar effect has been well-documented also in ferroelectric smectic C liquid \ncrystals [28] and in the orthogonal ferroelectric smectic A phases [29] and has been explained by \nconstrains of polar fluctuations by interactio ns with cell surfaces. However, since for the ferroelectric \nnematic phase the effect is observed also in very thick cells, in which the surface interactions should be \nnegligible , this raises doubts about the interpretation of the dielectric measurements an d pose s a question 2 \n whether these materials are indeed characterized by large permittivity. Clark et al . suggested [30] that \nliquid crystal line materials with very high spontaneous polarization behave as a perfect conductor and \nlarge apparent dielectric perm ittivity values are detected because the measured capacitance is, in fact , \nthe capacitance of very thin surface layers of surfactant , which is misinterpre ted as a capacitance of the \nwhole cell . On the other hand, the relaxation time of the apparent permittivity is a relaxation time of an \nequivalent electric circuit including a serial connection of a surface layer capacitor and liquid crystal \nwhich, in the ferroelectric nematic phase, behaves as a resistor with ohmic resistivity of the order of \n102 Ω. Vaupotic et al . [23] pointed out that in the case of the dielectric strength being very large, the \ncapacitance of the LC becomes comparable to the capacitance of the surface alignment layer thus both \nthe LC capacitor and the surface layer capacitor are charged (as op posed to charging only the surface \nlayer capacitor as proposed by Clark et al .). It was shown that in this case the interpretation of the \ndielectric response becomes very complex (as noticed also by Yadav et al . [27]), so they focused on the \ndielectric measurements in cells with no alignment layers for which they assumed that only the \ncapacitance of the LC cell is measured. On the other hand, Yadav et al . [27] studied the response in \nplanar cells with homeotropic and planar alignment as a function of cell thickness and the thickness of \nthe alignment layer, but limited the study primarily to the upper temperature N or Nx [31] phase in which \nthe ch arging of the surface layer can be neglected. Erkoreka et al . [24] performed a systematic \ninvestigation of the dielectric response as a function of the cell thickness. Their results are interpreted \nwithin the model of Clark and within a continuous phenomenological model proposed by Vaupotic et \nal, which coupl es fluctuations of the director and polarization. They also considered the possibility of \nthe electrode polarization effect [32], but ruled it out by further investigations [25]. The letter \ninvestigations also led to observations of high frequency relaxation process es predicted by the \ncontinuous phenomenological model thus confirming that relaxation processes within the N F phase are \nessentially measured by the dielectric spectroscopy, not only the response of the surface layers. \nAlthough t he recent dielectric measurements confirm the relaxation modes predicted by a \nphenomenological model of the NF phase , the increase of the dielectric strength with increasing \nthickness also in very thick cells “rings the bell” , even though some possible explanations have been \ngiven [24]. Thus , we decided to perform measurements of a current through and voltage on a parallel \nplate capacitor filled with a standard ferro electric nematic liquid crystalline material . These raw \nmeasurements were then interpreted in view of different equivalent circuits, also (and especially) such \nthat are not incorporated (yet) in the impedance analyzers used in the so far reported research. The \nmeasurements were performed with a simple experimental setup to avoid any possible artefacts. In this \nletter we show that modelling th e LC capacitor as a combination of a capacitor due to surface layers \n(important even in cells with no polymer alignment layers) and LC capacitor with a frequency dependent \ndielectric constant allows us to demonstrate that the dependence of the material per mittivity on the \nsample thickness is apparent, and the measurements in different cells (different thickness, different types \nof electrodes) can be fitted by using the same set of material parameters. \n \n \nFigure 1. Chemical formula of the studied material with phase transition temperatures in ℃. \nTo perform measurements, we used a standard material (analogue of RM73 4 [1]) (Figure 1 ) exhibiting \na metastable nematic and ferronematic phase s upon cooling from the isotropic liquid (see I-5 in \nref. [18]). Material was filled in planar cells (capacitors) with gold or indium tin oxide (ITO) electrodes . \nThe cells with gold electrodes were without surface alignment layer s, while the cells with ITO electrodes \n3 \n were either without a surface alignment layer or with a surface polymer alignment layer enforcing \nhomeotropic anchoring. Information on the used cells is collected in Table 1 . To make sure that \nsystematic tendencies are observed , measurements were also performed for another ferroelectric \nnematic material (see Supplemental Material , SM ). \nTable 1. Used cells (capacitors) , their acronyms, thickness ( 𝑑), capacitanc e of an empty cell ( 𝐶0) and surface area \nof the electrodes (𝑆). The low frequency capacitances ( 𝐶) of cells (at 10 Hz) and capacitances per unit surface area \n(𝐶/𝑆) for material in the N F phase at 49.0℃. AL stands for alignment layer. \n \n \n \n \n \n \n \nTo obtain the frequency dependence of the real and imaginary part of the impedance of cell s filled with \nthe material in the ferronematic phase, we measured a voltage on the cell and phase angle between the \nvoltage on the cell and the applied ( generator ) voltage (Bode plot method ) and , for some cells, also the \ncurrent through the cell and phase lag between the current and the voltage on the cell (auto -balancing \nbridge method) . Details on both methods are given in S M, where we also give a procedure for cal culating \nthe impedance of the cell ( 𝑍) and the phase angle ( 𝛼) between the voltage on and current through the \ncell from the measured quantities . The auto -balancing bridge method gives better results (less noise) but \nit could be used only for low enough frequencies (up to approximately 20 kHz ). The Bode plot method \ngives reliable results up to few MHz and measurements we re performed up to 1 MHz , so, primarily , we \nused the Bode plot method. For signal generation, Siglent SDG6022X 200MHz Function / Arbitrar y \nWaveform Generator (16 bit) was used and measurements were recorded by a Siglent SDS6054A 4CH \n500MHz 5 GSa/s Oscilloscope (16 bit) . The amplitude of the generator output voltage was set to \n100 mVpp for all cells, much below the Frederick’s transition voltage. Measurements were repeated \nalso for 200 mVpp and the only effect was less noise at lower frequencies . For few cells, results for the \ncalculated apparent dielectric loss and relative permittivity were compared with the ones obtained by \nthe Solatron 1260 impedance analyzer and a full agreement was obtained. Cells were heated by using a \nPCB Heating platform HPB100 from iTECH and temperature was measured by a PT100 temperature \nsensor. The temperature was stabilized with an accuracy of 0.1 K. The oscilloscope enables 4 -channel \nmeasurement s. One channel is required for the function generator, the rest can be used to measure \nsimultaneously three cells , for which the experimental conditions were thus identical . \nThe impedance 𝑍 and phase angle 𝛼 obtained from the measurements of the voltage on the cell and \nphase angle between the voltage on the cell and the applied voltage (Bode plot method) are given in \nFigure 2a,b , raw data are presented in SM (Figure S2 ). SM also includes measurements by auto-\nbalancing bridge method. The impedance 𝑍 and phase angle 𝛼 are obtained from the measured data \nwithout making any assumptions on the structure of the cell. From this point on, assumptions are needed \nto obtain the dielectric loss and relative permittivity of the LC in the N F phase. To get some information \non a proper equivalent electric circuit of the cell filled with a ferroelectric material, we first interpret the \nmeasured impedance ( 𝑍) and phase angle 𝛼 as if r esulting from a capacitor and resistor in either a series \nor parallel connection. A comparison of the frequency dependence of 𝛼 (Figure 2a ) to the dependence \nthat an ideal ohmic resistor and capacitor in series or parallel connection would give ( Figure S7 ) makes \nit very clear that the cell is neither. If the model of Clark et al. was valid, the equivalent electric circuit cell acronym 𝑑 [μm] 𝐶0[pF] 𝑆 [mm2] 𝐶 [𝜇F] 𝐶/𝑆 \n[nF mm−2] \nGold electrodes G5 5 44 25 0.32 13 \nG10 10 22 25 0.25 10 \nITO, no AL NA3 3,0 485 164 2.5 15 \nNA5 4,9 95 53 0.52 9.8 \nNA10 9,7 150 164 1.9 12 \nITO, \nhomeotropic AL HT10 10,0 77 87 0.29 3.3 \nHT20 20,0 41.5 94 0.18 1.9 4 \n would be a series connection of a capacitor due to surface layers and an ohmic resistor due to the \nresistivity of LC between the surface layers , and th e capacitance and resistivity should be constant over \nthe measured frequency range. Figure 2cd gives the capacitance ( 𝐶𝑆) and resistivity ( 𝑅𝑆) calculated \nfrom the measured 𝑍 and 𝛼 by interpreting the m as coming from a series connection of a capacitor and \nresistor. We observe that the values are frequency dependent, and the resistivity is rather high, between \n1 kΩ and 10 kΩ, much larger than few 100 Ω predicted by Clark et al . In addition, from the model of \nClark et al. it follows that one should measure the same capacitance per unit surface of the capacitor for \nall cells with the same type of electrodes and surface alignment layer, because the capacitance would \ndepend only on the thickness of the surface layer. The l ow frequency capacitance of all the cells with \nthe material being in the ferroelectric phase is given in Table 1. The capacitances per unit surface differ \nand they are not in the ratio of thicknesses . \n \nFigure 2. a) The phase angle ( 𝛼) between the current through the cell and voltage on the cell and b) i mpedance \n(|𝑍|) calculated from the Bode plot measurements for the studied material in the N F phase at 49.0℃. The apparent \nc) capacitance ( 𝐶𝑆) and d) resistivity ( 𝑅𝑆) of the cell as a function of frequency ( 𝜈) obtained by simulating the \nmeasured 𝑍 and 𝛼 as resulting from a series connection of an ideal resistor and capacitor. Green: NA10 (solid), \nNA5 (dashed), NA3 (dotted). Orange: G10 (solid), G5 (dashed). Blue: HT20 (solid), HT10 (dashed). \nThe above observations rule out the model of Clark et al .; however , they show that the influence of \ndielectric layer s cannot be neglected , not even in cells with out polymer alignment layer s on electrodes , \nwhere a thin layer of liquid crystal molecules close to the surface behave s as a surface layer , as pointed \nout by Clark et al . [30]. Because o bservations of several modes in the dielectric response make it clear \nthat relaxations of the LC material in the cell are also detected , this stimulated us to analyze in more \ndetail an equivalent circuit model which considers both the capacitance of the su rface layer as well as \nthe frequency response of the liquid crystalline material (see S M in Ref. [23]). The assumed equivalent \n5 \n electric circuit is given in Figure 3. The c apacitance of the surface dielectric layer is 𝐶𝐷; there are two \nsurface layers and , assuming that they have a very high ohmic resistivity , their net capacitance is 𝐶𝐷/2. \nThe c apacitance of the liquid crystal (𝐶𝐿𝐶) is expressed as \n𝐶𝐿𝐶=𝐶0 𝜀(𝜔) , (1) \nwhere 𝐶0 is a capacitance of an empty cell and 𝜀(𝜔), where 𝜔=2𝜋𝜈, is a dielectric constant, which we \napproximate by the Debye relaxation model: \n𝜀(𝜔)=𝜀′(𝜔)+𝑖𝜀′′(𝜔)=𝜀∞+Δ𝜀\n1+𝑖𝜔𝜏 , (2) \n𝜀′ is relative permittivity and 𝜀′′ dielectric loss, 𝜀∞ is a dielectric constant at high frequencies, Δ𝜀 a \ndielectric strength and 𝜏 the Debye relaxation time. Havriliak – Negami equation [33] could be used \ninstead, but to show the principle and have a lower number of fitting parameters we decided on the \nDebye relaxation . The ohmic resistivity of the liquid crystalline material is 𝑅𝐿𝐶. Compared to the \nequivalent circuit in [23], a resistor with resistivity 𝑅𝑥 is included to describe a decrease in 𝛼 at \nfrequencies higher than approximately 10 kHz, see Figure 2a. This effect comes from parasitic \nresistivities of electrodes and wiring . When 𝐶𝐷≫𝐶𝐿𝐶, 𝑅𝐿𝐶 is large and 𝑅𝑥 small, only the LC capacitor \nis charged and by finding the current through the cell that is in phase with the voltage on the cell one \nactually measures the quantity that is directly proportional to the dielectric loss (𝜀′′). However, t he \nanalysis becomes tricky when 𝐶𝐷 and 𝐶𝐿𝐶 are comparable . In Ref. [23] we argued that the capacitance \nof surface layer s is important only in cells with surfactants ; however, as can be concluded already from \nthe capacitance per unit surface given in Table 1, the capacitance of surface dielectric layer must be \nconsidered also in cells with no surface polymer anchoring layers . \n \nFigure 3. The equivalent electric circuit of the cell filled with a ferroelectric liquid crystal. 𝐶𝐷 is the capacitance \nof one surface layer, 𝐶𝐿𝐶 is the capacitance and 𝑅𝐿𝐶 resistivity of a liquid crystal . 𝑅𝑥 is a parasitic resistivity . \nWe searched for such values of the model parameters that would give at least a semi -quantitative fit with \nthe m easured results. The major objective was to find a set of LC parameters ( Δ𝜀, 𝜏, 𝜀∞) that are common \nfor all cells regardless of the ir thickness and type of surface anchoring, because all cells are filled with \nthe same material. We point out that the lower frequency part of experimental results can be fitted even \nif we use equivalent circuits presenting the LC cell as (i) a capacitor 𝐶𝐷 and resistor 𝑅𝐿𝐶 in series, or (ii) \na capacitor 𝐶𝐿𝐶(𝜔,𝜏) and resistor 𝑅𝐿𝐶 in parallel . However, it woul d require a different set of material \nparameters for each cell . The equivalent circuit in Figure 3 is the simplest one that enables the \ndescription of experimental measurements with the same set of material parameters for all cells. In \nsearch for this set of parameters we observed that the physical quantities that are most sensitive to the \nchange of parameter values are 𝛼(𝜔) and the apparent dielectric loss (𝜀𝐴𝑃′′) defined as \n𝜀𝐴𝑃′′=𝐼𝑖𝑛 𝑝ℎ𝑎𝑠𝑒\n𝜔𝐶0𝑈𝑐𝑒𝑙𝑙, (3) \nwhere 𝐼𝑖𝑛 𝑝ℎ𝑎𝑠𝑒 is the current through the cell in phase with the voltage on the cell ( 𝑈𝑐𝑒𝑙𝑙). \nThe apparent dielectric loss calculated from the measured data by using eq. (3) is given in Figure 4a for \nall the studied cells . We see that the maximum apparent value of 𝜀𝐴𝑃′′ strongly depend s on the cell \nthickness and on the type of the cell (NA, G or HT) . This dependence is an artefact of a huge dielectric \nstrength of the ferroelectric nematic, due to which the condition 𝐶𝐷≫𝐶𝐿𝐶 is not satisfied any more. In \n6 \n this case, the maximum of 𝜀𝐴𝑃′′ as given by eq. (3) is not half the value of Δ𝜀, neither is the position of \nthe peak at a frequ ency equal to 1/(2𝜋𝜏). \n \nFigure 4. a) The frequency dependence of the apparent dielectric loss (𝜀𝐴𝑃′′) for all the studied cells , calculated \nfrom the measured data by using eq. (3). Simulated values of b) 𝜀𝐴𝑃′′ and c) 𝛼, calculated f or the equivalent circuit \ngiven in Figure 3, by taking the material parameter values Δ𝜀=2.5⋅104, 𝜏=4 ms, 𝜀∞=3 and 𝑅𝐿𝐶=8⋅105 Ω \nbeing the same for all the cells. The values of 𝐶𝐷 and 𝑅𝑥 which are specific for the type of the cell are given in the \nmain text. Green: NA10 (solid), NA5 (dashed), NA3 (dotted). Orange : G10 (solid), G5 (dashed). Blue: HT20 \n(solid), HT10 (dashed). \nFigure 4b gives the apparent dielectric loss as predicted for the model equivalent circuit (Fig. 3) of the \nLC cell. The plots are obtained for Δ𝜀=2.5⋅104, 𝜏=4 ms, 𝜀∞=3 and 𝑅𝐿𝐶=8⋅105 Ω for all cells , \nwhile the thickness of the surface layer , and thus its capacitance and the parasitic resistivity depend on \nthe type of the cell. All cells of the same type (N A, G or HT) were fitted with the same 𝑅𝑥 and the same \nthickness of the surface layer. The capacitance of the surface layer was modelled as 𝐶𝐷=𝐶0𝑑𝑓𝑠, where \n𝑓𝑠=𝜀𝑆/𝑑𝑆 and 𝑑𝑆 is a thickness of the surface layer and 𝜀𝑆 is its relative p ermittivity. The plots are \ngiven for 𝑓𝑆 being 5 nm−1 and 4 nm−1 in the NA and G cells, respectively, while for the HT cells 𝑓𝑆=\n0.6 nm−1. These values are physically sensible as one would expect the surface layer to have \napproximately the same thickness in cells with no aligning surface layer (NA and G cells). In HT cells \nthe surface dielectric layer is found to be of the order of magnitude thicker . By taking 𝜀𝑆∼10, we find \nthat 𝑑𝑆∼2−3 nm in the NA and G cells, while in HT cells 𝑑𝑆≈17 nm. While the position and \nmagnitude of the peak value of the apparent 𝜀𝐴𝑃′′ are strongly sensitive to Δ𝜀, 𝜏, 𝑅𝐿𝐶 and 𝑓𝑆, the major \neffect of 𝜀∞ and 𝑅𝑥 is on the higher frequency dependence of the angle 𝛼. A reduction in 𝑅𝑥 increases \nthe magnitude of 𝛼 at high frequencies, while the increase in 𝜀∞ shifts the peak (the lowest magnitude \nof 𝛼) to lower frequencies. The best fit (compare Figures 2a and 4c ) was obtained for 𝑅𝑥=600 Ω for \nthe NA cells, 𝑅𝑥=500 Ω for the HT ce lls and 𝑅𝑥=50 Ω for the G cells. These parameter values make \nsense, as it is expected that the parasitic resistivity should be the lowest for the cells with gold electrodes . \nBy considering , that the inaccuracy in the cell thickness and in the filling of the cell can lead to \ncapacitances of cells to differ by up to 10% from the ideal values , we conclude that the proposed \nequivalent circuit gives not only good qualitative but also semi -quantitative agreement with the \nexperiment. The agreement between the mo del predictions and measurements is obtained only if very \nhigh Δ𝜀 is assumed , which is approximately four times higher than the maximum of the apparent \ndielectric loss measured in the NA10 cell . The presented mode l explains why the apparent dielectric \nstrength increases with increasing cell thickness : by increasing the cell thickness , the capacitance 𝐶𝐿𝐶 \ndecreases and we are moving towards the limit 𝐶𝐿𝐶≪𝐶𝐷 in which the apparent dielectric strength and \nloss c oincide with the actual material parameters (Figure S3 ). If we use the obtained parameters to \nestimate 𝐶𝐿𝐶 and 𝐶𝐷 for different cells we obtain 𝐶𝐿𝐶≈0.6𝐶𝐷≈0.5 μF for G10 cell, 𝐶𝐿𝐶≈4𝐶𝐷≈2 μF \nfor HT10 cell, and 𝐶𝐿𝐶≈0.5𝐶𝐷∼3.8 μF for NA10 cell , thus the capacitances are indeed comparable. \nTo fulfill the condition 𝐶𝐿𝐶≪𝐶𝐷 the cell thickness should be of the order of several millimeters , while \nin standard experiments with LC materials the cells not thicker than approximately 100 μm are used . \n1011021030200040006000\n a) b)\n c)7 \n However, as shown in SM, for very thick cell s, the peak in the dielectric loss at the Debye frequency \n1/2𝜋𝜏 might be hidden in the low frequency increase of the dielectric loss. \nTo conclude, we have shown that the low frequency (up to 1 MHz) response obtained by dielectric \nmeasurements of a ferroelectric nematic material filled in thin capacitors can be interpreted in terms of \nthe same material quantities (relative permittivity and relaxation time) independent of the cell thickness \nand the type of surface electrodes and surface alignment layer. Standard procedure, in which the \ndielectric loss is calculated from the current through the cell flowing in phase with the voltage on the \ncell and the relative permittivity - from the current through the cell phase shifted with respect to the \nvoltage by 𝜋/2, gives only apparent values , related to material properties in a very complex way . The \nreason lies in a huge relative permittivity of the LC in the ferr oelectric nematic ph ase, which causes the \ncapacitance of the liquid crystal slab to be comparable to the capacitance of the surface dielectric layer \nand the latter is shown to exist also in capacitors with bare electrodes. We show that experimental \nobservations can be explain ed by modelling the LC cell with an equivalent circuit consisting of a series \nconnection of a capacitor due to the surface dielectric layer, resistor due to parasitic resistivities and LC \ncapacitor (with Debye relaxation) and LC resistor in parallel connec tion. By using the same value s of \nthe dielectric strength of the studied material and its relaxation time for all the studied cell s, it is shown \nthat the apparent relative permittivity will increase with increasing cell thickness, the apparent relaxation \nfrequency, however, does not necessarily behave in a monotonic way , which is observed experimentally \nand predicted by the model (see Figure 4, graphs for NA cells , and Figure S3 ). Based on the presented \nfindings, we conclude that the relative permittivity of the ferroelectric nematics is indeed huge , and it is \neven higher than the apparent measured values. Therefore, t he interpretation of the dielectric \nmeasurements in strongly polar LC phases needs to be revisited . The use of the proposed equivalent \ncircuit for the interpretation of the apparent relative permittivity and apparent relaxation time might \naffect the observed dependencies of those parameters on temperature and bias field. It could also explain \nthe unusual dependence of the amplitude mode on temperature close to the N F – N transition, reported \nin [23]. \n \nAcknowledgments \nN. V. and V.M. acknowledge the support of the Slovenian Research and Innovation Agency (ARIS), \nthrough the research core funding program No. P1 -0055 and No. P2 -0028, respectively. E. G., D. P. and \nJ. M. acknowledge the support by the National Science Centre (Poland) under the grant no. \n2021/43/B/ST5/00240. \n \nReferences \n[1] R. J. Mandle, S. J. Cowling, and J. W. Goodby, A Nematic to Nematic Transformation Exhibited \nby a Rod -like Liquid Crystal , Phys. Chem. Chem. Ph ys. 19, 11429 (2017). \n[2] H. Nishikawa, K. Shiroshita, H. Higuchi, Y . Okumura, Y . Haseba, S. Yamamoto, K. Sago, and H. \nKikuchi, A Fluid Liquid‐Crystal Material with Highly Polar Order , Advanced Materials 29, \n1702354 (2017). \n[3] R. J. Mandle, S. J. Cowling, and J. W. Goodby, Rational Design of Rod‐Like Liquid Crystals \nExhibiting Two Nematic Phases , Chemistry A European J 23, 14554 (2017). \n[4] A. Mertelj, L. Cmok, N. Sebastián, R. J. Mandle, R. R. Parker, A. C. Whitwood, J. W. Goodby, \nand M. Čopič, Splay Nematic Phase , Phys. Rev. X 8, 041025 (2018). \n[5] N. Sebastián, L. Cmok, R. J. Mandle, M. R. De La Fuente, I. Drevenšek Olenik, M. Čopič, and A. \nMertelj, Ferroelectric -Ferroelastic Phase Transition in a Nematic Liquid Crystal , Phys. Rev. Lett. \n124, 037801 (2020). \n[6] X. Chen et al., First -Principles Experimental Demonstration of Ferroelectricity in a Thermotropic \nNematic Liquid Crystal: Polar Domains and Striking Electro -Optics , Proc. Natl. Acad. Sci. \nU.S.A. 117, 14021 (2020). 8 \n [7] R. J. Mandle, N. Sebastián , J. Martinez -Perdiguero, and A. Mertelj, On the Molecular Origins of \nthe Ferroelectric Splay Nematic Phase , Nat Commun 12, 4962 (2021). \n[8] N. Sebastián, M. Čopič, and A. Mertelj, Ferroelectric Nematic Liquid -Crystalline Phases , Phys. \nRev. E 106, 021001 ( 2022). \n[9] E. Zavvou, M. Klasen -Memmer, A. Manabe, M. Bremer, and A. Eremin, Polarisation -Driven \nMagneto -Optical and Nonlinear -Optical Behaviour of a Room -Temperature Ferroelectric \nNematic Phase , Soft Matter 18, 8804 (2022). \n[10] M. Cigl, N. Podoliak, T. L andovský, D. Repček, P. Kužel, and V. Novotná, Giant Permittivity in \nDimethylamino -Terminated Ferroelectric Nematogens , Journal of Molecular Liquids 385, 122360 \n(2023). \n[11] C. L. Folcia, J. Ortega, R. Vidal, T. Sierra, and J. Etxebarria, The Ferroelectric Nematic Phase: An \nOptimum Liquid Crystal Candidate for Nonlinear Optics , Liquid Crystals 49, 899 (2022). \n[12] J. Li, H. Nishikawa, J. Kougo, J. Zhou, S. Dai, W. Tang, X. Zhao, Y . Hisai, M. Huang, and S. Aya, \nDevelopment of Ferroelectric Nematic Fluids wit h Giant -ε Dielectricity and Nonlinear Optical \nProperties , Sci. Adv. 7, eabf5047 (2021). \n[13] N. Sebastián et al., Polarization Patterning in Ferroelectric Nematic Liquids via Flexoelectric \nCoupling , Nat Commun 14, 3029 (2023). \n[14] A. Barthakur, J. Karcz, P. Kula, and S. Dhara, Critical Splay Fluctuations and Colossal \nFlexoelectric Effect above the Nonpolar to Polar Nematic Phase Transition , Phys. Rev. Materials \n7, 035603 (2023). \n[15] H. Nishikawa, K. Sano, and F. Araoka, Anisotropic Fluid with Phototunable Dielectric \nPermittivity , Nat Commun 13, 1142 (2022). \n[16] H. Nishikawa and F. Araoka, A New Class of Chiral Nematic Phase with Helical Polar Order , \nAdvanced Materials 33, 2101305 (2021). \n[17] J. Ortega, C. L. Folcia, J. Etxebarria, and T. Sierra, Ferroele ctric Chiral Nematic Liquid Crystals: \nNew Photonic Materials with Multiple Bandgaps Controllable by Low Electric Fields , Liquid \nCrystals 49, 2128 (2022). \n[18] D. Pociecha et al., Intrinsically Chiral Ferronematic Liquid Crystals: An Inversion of the Helica l \nTwist Sense at the Chiral Nematic – Chiral Ferronematic Phase Transition , Journal of Molecular \nLiquids 361, 119532 (2022). \n[19] C. Feng, R. Saha, E. Korblova, D. Walba, S. N. Sprunt, and A. Jákli, Electrically Tunable \nReflection Color of Chiral Ferroelec tric Nematic Liquid Crystals , Advanced Optical Materials 9, \n2101230 (2021). \n[20] A. Manabe, M. Bremer, and M. Kraska, Ferroelectric Nematic Phase at and below Room \nTemperature , Liquid Crystals 48, 1079 (2021). \n[21] X. Chen, Z. Zhu, M. J. Magrini, E. Korblo va, C. S. Park, M. A. Glaser, J. E. Maclennan, D. M. \nWalba, and N. A. Clark, Ideal Mixing of Paraelectric and Ferroelectric Nematic Phases in Liquid \nCrystals of Distinct Molecular Species , Liquid Crystals 49, 1531 (2022). \n[22] H. Long, J. Li, M. Huang, and S. Aya, Mixing -Induced Phase Stabilization and Low -Temperature -\nShifting of Ferroelectric Nematics , Liquid Crystals 49, 2121 (2022). \n[23] N. Vaupotič, D. Pociecha, P. Rybak, J. Matraszek, M. Čepič, J. M. Wolska, and E. Gorecka, \nDielectric Response of a Fer roelectric Nematic Liquid Crystalline Phase in Thin Cells , Liquid \nCrystals 50, 584 (2023). \n[24] A. Erkoreka, J. Martinez -Perdiguero, R. J. Mandle, A. Mertelj, and N. Sebastián, Dielectric \nSpectroscopy of a Ferroelectric Nematic Liquid Crystal and the Effect of the Sample Thickness , \nJournal of Molecular Liquids 387, 122566 (2023). \n[25] A. Erkoreka, A. Mertelj, M. Huang, S. Aya, N. Sebastián, and J. Martinez -Perdiguero, Collective \nand Non -Collective Molecular Dynamics in a Ferroelectric Nematic Liquid Cr ystal Studied by \nBroadband Dielectric Spectroscopy , The Journal of Chemical Physics 159, 184502 (2023). \n[26] N. Yadav, Y . P. Panarin, W. Jiang, G. H. Mehl, and J. K. Vij, Collective Relaxation Processes in \nNonchiral Nematics , Crystals 13, 962 (2023). \n[27] N. Yadav, Y . P. Panarin, J. K. Vij, W. Jiang, and G. H. Mehl, Two Mechanisms for the Formation \nof the Ferronematic Phase Studied by Dielectric Spectroscopy , Journal of Molecular Liquids 378, \n121570 (2023). 9 \n [28] Yu. P. Panarin, Yu. P. Kalmykov, S. T. Mac Lu ghadha, H. Xu, and J. K. Vij, Dielectric Response of \nSurface Stabilized Ferroelectric Liquid Crystal Cells , Phys. Rev. E 50, 4763 (1994). \n[29] L. Guo et al., Ferroelectric Behavior of Orthogonal Smectic Phase Made of Bent -Core Molecules , \nPhys. Rev. E 84, 031706 (2011). \n[30] N. A. Clark, X. Chen, J. E. Maclennan, and M. A. Glaser, Dielectric Spectroscopy of Ferroelectric \nNematic Liquid Crystals: Measuring the Capacitance of Insulating Interfacial Layers , (2022). \n[31] X. Chen et al., The Smectic Z A Phase: An tiferroelectric Smectic Order as a Prelude to the \nFerroelectric Nematic , Proc. Natl. Acad. Sci. U.S.A. 120, e2217150120 (2023). \n[32] S. Emmert, M. Wolf, R. Gulich, S. Krohns, S. Kastner, P. Lunkenheimer, and A. Loidl, Electrode \nPolarization Effects in Broa dband Dielectric Spectroscopy , Eur. Phys. J. B 83, 157 (2011). \n[33] S. Havriliak and S. Negami, A Complex Plane Analysis of -dispersions in Some Polymer \nSystems , J. Polym. Sci., C Polym. Symp. 14, 99 (1966). \n 1 \n Supplement al Materials for: \n \nDo ferroelectric nematic liquid crystals really have a huge dielectric \npermittivity? \n \nV ojko Matko1, Ewa Gorecka2, Damian Pociecha2, Joanna Matraszek2, Nataša Vaupotič3,4* \n \n1 University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška 46, \n2000 Maribor \n2 University of Warsaw, Faculty of Chemistry, Zwirki i Wigury 101, 02 -089 Warsaw, Poland \n3 University of Maribor, Faculty of Natural Sciences and Mathematics, Koroška 160, 2000 \nMaribor, Slovenia \n4 Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia \n \n \n \nContents: \n1 Experimental techniques \n2 Additional data \n \n \n 2 \n 1 Experimental technique s \nEven though the derivations given below are standard calculations known from the elementary course \nof Physics of measurements in electro engineering , we give them step by step, because in standard \nmeasurements with the impedance analyzers one usually does not think about the background “raw” \nmeasurements from which the dielectric l oss and relative permittivity are calculated based on some \nassumed equivalent circuit of the LC cell. \n \nFigure S1 . Electric circuit for the a) Bode plot and b) auto -balancing bridge method . 𝑈1 is given by the Siglent \nwaveform generator, 𝑈2 is measured by the Siglent oscilloscope, 𝑅𝑟𝑒𝑓 is resistivity of the reference resistor, DUT \nis a “device under test” , in our case a series connection of the LC cell and an ohmic resistor with resistivity 𝑅𝑐 \nused for the compensation of the high freq uency response of the probes . c) Phasor diagram for the analysis of the \nBode plot me thod: 𝜙 is a phase angle between the applied (generator) voltage and the voltage on DUT , 𝜑 is the \nphase angle between the current through DUT and the applied voltage, 𝛼̃ is a phase angle between the voltage on \nDUT and current through DUT , which is in phase with the voltage 𝑈𝑅 on 𝑅𝑟𝑒𝑓. d) Phasor diagram for the auto -\nbalancing bridge method : 𝜙 is the phase angle between the applied (generator) voltage and current thr ough the \ncell, 𝑈𝑅 and 𝑈𝑐𝑒𝑙𝑙 are voltages on 𝑅𝑐 and cell, respectively; 𝛼 is the phase angle between the voltage on the cell \nand current ( 𝐼) through the cell. \n \nThe schematic circuit of the Bode plot me thod is shown in Figure S1a . We measure the voltage (𝑈2) \non the device under test (DUT) and the phase angle (𝜙) between 𝑈2 and voltage ( 𝑈1) at the generator \noutput. Usually, a gain 𝐺=20 dBlog(𝑈2/𝑈1), is measured, thus 𝑈2=𝑈1 10𝐺/20. \nThe current through the cell is obtained by calculating the current through the reference resistor as (see \nthe phasor diagram in Figure S1b) \n𝐼=1\n𝑅𝑟𝑒𝑓√𝑈22+𝑈12−2𝑈2𝑈1cos𝜙 . (𝑆1) \nThe phase angle (𝜑) between the current through the reference resistor (𝑅𝑟𝑒𝑓) and the applied voltage \nis \n3 \n tan𝜑=−𝑈2sin𝜙\n𝑈1−𝑈2cos𝜙 (𝑆2) \nand the phase angle between the voltage on DUT and current through DUT is \n𝛼̃=𝜙−𝜑 . (𝑆3) \nThe impedance of DUT is 𝑍̃=𝑈2/𝐼: \n𝑍̃=𝑅𝑟𝑒𝑓10𝐺\n20\n√10𝐺\n10+1−2⋅10𝐺\n20cos𝜙 . (𝑆4) \nWe point out that the impedance is independent of the used 𝑅𝑟𝑒𝑓, because 𝐺 and 𝜙 also depend on 𝑅𝑟𝑒𝑓. \nIn our measurements, DUT was a series connection of LC cell and a resistor with resistivity 𝑅𝑐, which \nwas chosen such that a high frequency r esponse of the probes was compensated up to 10 MHz . The \nmeasurements on the cells are reliable up to few MHz , for higher frequencies additional effects appear, \nrelated probably to parasitic conductance . \nIf we interpret the measured 𝑍̃ and 𝛼̃ as being a result of a resistor and capacitor in series, the resistivity \n(𝑅𝑆) of the LC cell and capacitance ( 𝐶𝑆) are obtained as \n𝑅𝑆=𝑍̃cos𝛼̃−𝑅𝑐 , 𝐶𝑆=(𝑍̃ 𝜔sin𝛼̃)−1 , (𝑆5) \nwhere 𝜔=2𝜋𝜈. The impedance (𝑍) of the LC cell is \n𝑍=√𝑅𝑆2+(𝑍̃sin𝛼̃)2 (𝑆6) \nand the angle between the real and imaginary part of 𝑍 is \ntan𝛼=𝑍̃sin𝛼̃\n𝑅𝑆 . (𝑆7) \nThe schematic circuit used for the auto -balancing bridge method is shown in Figure S1 c. We measure \nthe voltage 𝑈2 and the phase angle 𝜙 between the current (𝐼) through DUT and voltage on DUT , the \nlatter being equal to the output generator voltage 𝑈1. The current through DUT is calculated as \n𝐼=−𝑈2\n𝑅𝑟𝑒𝑓 . (𝑆8) \nAgain, DUT is a series connection of a LC cell and resistor with resistivity 𝑅𝑐, which was added to \ncompensate for the high frequency response of the probes. While for standard ohmic resistors or \ncapacitors the compensation was perfect up to few MHz , we detected increased noise for the LC cells \nalready at frequencies above few 10 kHz. The voltage 𝑈1 on the DUT is a phasor sum of the voltage on \n𝑅𝑐 (𝑈𝑅) and on LC cell ( 𝑈𝑐𝑒𝑙𝑙), see Figure S1d , thus 𝑈𝑐𝑒𝑙𝑙 is calculated as \n𝑈𝑐𝑒𝑙𝑙=√𝑈12+𝑈𝑅2−2𝑈1𝑈𝑅cos𝜙 . (𝑆9) \nThe angle between the current through the cell and voltage on the cell ( 𝛼) is \ntan𝛼=−𝑈1sin𝜙\n𝑈1cos𝜙−𝑈𝑅 . (𝑆10) 4 \n Because the current through DUT is also the current through the cell, t he impedance of the cell is \nobtained as \n𝑍=𝑈𝑐𝑒𝑙𝑙\n𝐼 . (𝑆11) \nIf we interpret the measured 𝑍 and 𝛼 as resulting from a series connection of a capacitor with capacitance \n𝐶𝑆 and resistor with resistivity 𝑅𝑆, then 𝐶𝑆 and 𝑅𝑆 are calculated as \n𝑅𝑆=𝑍cos𝛼 ,𝐶𝑆=(𝑍𝜔sin𝛼)−1 . (𝑆12) \nWe used 𝑈1=100 mVpp in all measurements. The resistivit y 𝑅𝑟𝑒𝑓 was 9,7 kΩ and 5.1 kΩ for the Bode \nplot and auto -balancing bridge method, respectively. The values of 𝑅𝑐 were fitted for each channel and \neach measurement and were from few 100 Ω to few kΩ. \nThe apparent dielectric loss ( 𝜀𝐴𝑃′′) is calculated as (eq. (3) in the main text) \n𝜀𝐴𝑃′′=𝐼𝑖𝑛 𝑝ℎ𝑎𝑠𝑒\n𝜔𝐶0𝑈𝑐𝑒𝑙𝑙, (𝑆13) \nwhere 𝐼𝑖𝑛 𝑝ℎ𝑎𝑠𝑒 is the current through the cell in phase with the voltage on the cell . We note that \n𝐼𝑖𝑛 𝑝ℎ𝑎𝑠𝑒=𝐼cos𝛼 and use eq. (S11) to obtain \n𝜀𝐴𝑃′′=cos𝛼\n𝜔��0𝑍 . (𝑆14) \n \n2 Additional data \nIn this Section we present the Bode plot measurements of the studied material ( Material 1 ) and some \nadditional model results for this material. We also present measurements by the Bode plot method and \nauto-balancing bridge method of another typical N F materi al (Material 2 ) exhibiting a direct transition \nfrom the isotropic to the ferronematic phase at 85℃ ; measurements were performed at 80℃ . \nFigure S2 gives the raw measured data, gain 𝐺 and phase angle (𝜙) between the voltage on DUT (LC \ncell and 𝑅𝑐 in series) and voltage on the generator output for material 1 in the N F phase . From this data \nwe calculated the impedance ( 𝑍) of the cell and the angle ( 𝛼) between the current through the cell and \nvoltage on the cell as well as 𝐶𝑆 and 𝑅𝑆. These resu lts are given in Figure 2 of the main text and are to \nbe compared with model results given in Figure S3 . Model parameters are collected in Table S 1. In \nFigure S3 we also give the apparent dielectric loss as predicted by the model for different cell \nthicknesses, going beyond the cell thicknesses used in the experiment. We see that by increasing the cell \nthickness the magnitude and position of the peak value of 𝜀𝐴𝑃′′ move towards the frequency 1/2𝜋𝜏 and \nmagnitude Δ𝜀/2, as expected. However, at very thick cells the peak is overrun by the low frequency \nincrease of 𝜀′′. Here, i t is worth pointing out that at very low frequencies (below 1 Hz) an ohmic resistor \nshould be added in parallel with the capacitor 𝐶𝐷/2 (see Figure 3) , otherwise an additional artificial \npeak is obtained in the frequency dependence of 𝜀′′ at a frequency (𝜋𝐶𝐷𝑅𝑅𝐶)−1∼0.1 Hz. Because we \nare not interested in such low frequencies, the o hmic resistivity of the surface capacitor was assumed to \nbe infinitely large . \nMeasurements with material 2 were performed only in three cells (based on the availability of the cells): \nG10, G5 and NA3. For material 2 we present the results both for the Bode plot method ( Figure S4 ) and \nauto-balancing bridge method ( Figure S5 ). The apparent dielectric loss ( 𝜀𝐴𝑃′′) is calculated by eq. (S14). \nFigure S6 gives the model predictions. The model parameters are collected in Table S 1. For this \nmaterial, the optimum fit is obtained for the following parameter values: 𝜀∞=5, Δ𝜀=3.0⋅105, 𝜏=5 \n 0.45 ms and 𝑅𝐿𝐶=8⋅105 Ω. The values of 𝑓𝑆 are the same for both NA and G cells: 𝑓𝑆=5 nm−1, the \nvalue of 𝑅𝑥=50 Ω for the gold cells is the same as for Material 1, but the model curves are very weakly \ndependent on lowering this value below 50 Ω. For the NA3 cell, however, the position of the peak 𝜀𝐴𝑃′′ \ndepends strongly on 𝑅𝑥 and the best fit is found for 𝑅𝑥=200 Ω, i.e. a few times lower than for material \n1 but still of the same order of magnitude. With these parameters the model predicts a monotonic \nincrease of the maximum apparent dielectric loss with cell thickness, both for the G and NA cells. On \nthe other hand, apparent relaxation frequency decreases towards the LC Debye relaxation frequency \nwith increasing cell thicknes s in G cells, while in NA cells it increases towards this value. \nFinally, in Figure S7 we consider a parallel and series connection of a resistor with resistivity 1 kΩ and \ncapacitor with capacitance 1 𝜇F and plot the graphs of 𝛼 and 𝑍 for both connections. These graphs are \nto be compared with 𝛼 and 𝑍 obtained from the experiment. \n \nTable 2 . Model parameters. Their definition is given in the main text. \nmaterial Δ𝜀 \n[×104] 𝜏 [ms] 𝜀∞ 𝑅𝐿𝐶 \n[𝑀Ω] 𝑓𝑠(𝐺) \n[nm-1] 𝑓𝑠(𝑁𝐴) \n[nm-1] 𝑓𝑠(𝐻𝑇) \n[nm-1] 𝑅𝑥(𝐺) \n[Ω] 𝑅𝑥(𝑁𝐴) \n[Ω] 𝑅𝑥(𝐻𝑇) \n[Ω] \n1 2.5 4.0 3 0.8 4 5 0.6 50 600 500 \n2 3.0 0.45 5 0.8 5 5 / 50 200 / \n \n \n \nFigure S 2. Material 1 in the N F phase at 49.0℃. Bode plot measurements. a) Gain ( 𝐺) and b) phase angle (𝜙) \nbetween the voltage on DUT (LC cell and 𝑅𝑐 in series) and voltage on the generator output . Green: NA10 (solid), \nNA5 (dashed), NA3 (dotted). Yellow: G10 (solid), G5 (dashed). Blue: HT20 (solid), HT10 (das hed). \n \n6 \n \nFigure S 3. Material 1. Model results. a) The impedance ( 𝑍) and b) p hase angle ( 𝛼) between the current through \nthe cell and voltage on the cell as a function of frequency ( 𝜈), calculated for the equivalent circuit of the LC cell \n(Figure 3) with parameters given in Table S1. The apparent c) capacitance ( 𝐶𝑆) and d) resistivity ( 𝑅𝑆) of the cell \nobtained by simulating 𝑍 and 𝛼 of the equivalent circuit as if resulting from a series connection of a resistor and \ncapacitor (to be compared with experimental results in Figure 2) . An apparent dielectric loss ( 𝜀𝐴𝑃′′) at different cell \nthicknesses for the e) NA and f) HT cells. Here the green and blue curves present the model results for the studied \ncells, and magenta curves for 100 nm (dashe d curve) and 1 mm (solid curve) thick cell. Green: NA10 (solid), \nNA5 (dashed), NA3 (dotted). Orange: G10 (solid), G5 (dashed). Blue: HT20 (solid), HT10 (dashed). \n7 \n \nFigure S 4. Material 2 in the NF phase at 80.0℃. Bode plot measurements . a) Gain ( 𝐺) and (inset) phase angle \n(𝜙) between the voltage on DUT (LC cell and 𝑅𝑐 in series) and voltage on DUT). b) Phase angle ( 𝛼) between the \ncurrent through the cell and voltage on the cell and c) impedance 𝑍 calculated from 𝐺 and 𝜙. d) The apparent \ndielectric loss calculated by eq.( S14). The apparent e) resistivity ( 𝑅𝑆) and f) capacitance ( 𝐶𝑆) of the cell as a \nfunction of frequency ( 𝜈) by simulating the measured 𝑍 and 𝛼 as resulting from a series connection of an ideal \nresist or and capacitor. Green: NA3. Orange : G10 (solid), G5 (dashed). \n8 \n \nFigure S 5. Material 2 in the NF phase at 80.0℃. Auto -balancing bridge method . a) Gain ( 𝐺) and (inset) phase \nangle (𝜙) between the voltage on DUT (LC cell and 𝑅𝑐 in series) and current through DUT . b) Phase angle ( 𝛼) \nbetween the current through the cell and voltage on the cell and c) impedance 𝑍 calculated from 𝐺 and 𝜙. d) The \napparent dielectric loss calculated by eq. (S14). The apparent e) resistivity ( 𝑅𝑆) and f) capacitance ( 𝐶𝑆) of the cell \nas a function of frequency ( 𝜈) by simulating the measured 𝑍 and 𝛼 as resulting from a series connection of an ideal \nresistor and capacitor. Green: NA3. Orange: G10 (solid), G5 (dashed). \n9 \n \n \nFigure S 6. Material 2. Model results. a) The impedance ( 𝑍) and b) p hase angle ( 𝛼) between the current through \nthe cell and voltage on the cell as a function of frequency ( 𝜈), calculated for the equivalent circuit of the LC cell \n(Figure 3) with parameters given in Table S1. The apparent c) capacitance ( 𝐶𝑆) and d) resistivity ( 𝑅𝑆) of the cell \nobtained by simulating 𝑍 and 𝛼 of the equivalent circuit as resulting from a series connection of a resistor and \ncapacitor. An apparent dielectric loss ( 𝜀𝐴𝑃′′) for e) the measured cells and f) a lso for thicker cells: 100 nm thick HT \ncell (magenta, dashed ), 1 mm thick HT cell (magenta, solid) and 100 nm thick NA cell (gray, dashed ). Green: \nNA3 (dotted). Orange: G10 (solid), G5 (dashed). \n \n \n \n10 \n \nFigure S7 . A resistor with resistivity 1 kΩ and capacitor with a capacitance 100 nF in either a series or \nparallel connection represent DUT . The frequency ( 𝜈) dependence of the a,c) impedance ( 𝑍) and b,d) \nangle between the voltage on DUT and current through it . \n \n \n \n" }, { "title": "2401.16100v2.On_simpliciality_of_function_spaces_not_containing_constants.pdf", "content": "arXiv:2401.16100v2 [math.FA] 16 Feb 2024ON SIMPLICIALITY OF FUNCTION SPACES NOT\nCONTAINING CONSTANTS\nONDˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nAbstract. We investigate simpliciality of function spaces without co nstants.\nWe prove, in particular, that several properties character izing simpliciality in\nthe classical case differ in this new setting. We also show tha t it may happen\nthat a given point is not represented by any measure pseudosu pported by\nthe Choquet boundary, illustrating so limits of possible ge neralizations of the\nrepresentation theorem. Moreover, we address the abstract Dirichlet problem\nin the new setting and establish some common points and nontr ivial differences\nwith the classical case.\n1.Introduction\nChoquet simplices form a distinguished class of compact convex sets as they\nform a natural class of sets generalizing the classical notion of finit e-dimensional\nsimplices. Let us recall [6, Definition 3.1] saying that a convex set Xin a vector\nspaceEis a simplex if the cone P={(λx,λ);x∈X,λ≥0}defines a lattice order\nonP−P⊂E⊕R.\nA compact simplex Xis called a Bauer simplex provided the set of extreme\npoints is closed. An example of Bauer simplex is the space M1(K) of all Radon\nprobability measures on a compact Hausdorff space Kand any Bauer simplex is\naffinely homeomorphic to M1(K) for a suitable K. Interesting examples of infinite\ndimensionalBauersimplicesareexhibitedin[6,Theorems2.4,2.5and2 .6]. Another\nexampleofaBauersimplexrelatedtothetheoryofharmonicfunctio nscanbefound\nin [4] and [2], see also [6, Theorem 3.17].\nLess transparentexamplesof compactsimplices arethose with the set ofextreme\npointsnon-closed. Adistinguishedexampleofsuchasimplexis thePou lsensimplex\nSwhich is a metrizable simplex whose set of extreme points is dense in S. A\nremarkablefeatureof Sisitsuniquenessanduniversality(everymetrizablecompact\nsimplex is affinely homeomorphic to a face of S). It turns out that the Poulsen\nsimplex appears naturally in the investigation of ergodic measures, s ee e.g. [6,\nTheorem 3.8 and page 618].\nThe structure of a compact convex set Xis in a way determined by the Banach\nspaceAc(X,R) of all continuous affine real functions on X, more precisely, Xis a\nsimplex if and only if Ac(X,R) is anL1-predual, see [14, Chapter 7, §19, Theorem\n2]. The structure of real or complex L1-preduals is closely related to the structure\nof simplices, see [14, Chapter 7, §21, Theorem 7 and §23, Theorem 5] (see also [15]\nand [3]).\n2010Mathematics Subject Classification. 46A55; 46B04; 28C05.\nKey words and phrases. function space not containing constants, simplicial funct ion space,\nfunctionally simplicial function space, boundary measure ,L1-predual, abstract Dirichlet problem.\nOur research was partially supported by the Research grant G AˇCR 23-04776S.\n12 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nA natural way how to construct simplices is to consider a suitable fun ction space\nH⊂C(K,R) on a compact Hausdorff space Ksuch thatHcontains constant\nfunctions and separates points of K. Then the state space of His a compact\nconvex set which, for a well chosen space H, is a simplex, see [17, Chapter 6]. The\ncompact space Kis then homeomorphically mapped into the state space and those\npoints, which are mapped into the set of extreme points of the stat e space, form\nthe so-called Choquet boundary.\nIfXis a compact convexset in a Hausdorfflocally convex space, it is a simple x if\nandonly iffor everypoint x∈Xthere existsa unique maximal probabilitymeasure\nµxonXsuch thatµxrepresentsx(i.e.,/integraltext\nXfdµx=f(x) for every continuous real\naffine function fonX, see [14, Chapter 7, §20, Theorem 3]). We recall that the\nmaximality of µxis with respect to the Choquet ordering on probability measures\nonX. Any maximal measure is ‘pseudo-supported’ by the set of extrem e points,\ni.e., anymaximal measure vanishes on any Baireset disjoint from the s et of extreme\npoints.\nThus also for a function space Hthe question of representation of points and\nfunctionals on Hby measures close to the ‘boundary of H’, namely to the Choquet\nboundary, is important. The uniqueness of such a representation is then a natural\nfeature of the function space which asserts that His ‘simplicial’. By the previous\nconsiderations, the function space Ac(X,R) on a compact convex set Xis simplicial\nif and only if Xis a simplex.\nAll the phenomena related to the function space Ac(X,R) on a compact convex\nsetXcan be successfully transferred to the framework of function sp aces with\nconstants, see [17, Chapter 6], [8] and Section 5 below. The main aim of the\npresented paper is to investigate possible notions of simpliciality for f unction spaces\nthat need not contain constant functions. In this framework, th e situation is much\nless satisfactory as Theorem 6.1 below shows.\nThe paper is organized as follows. After the introduction of necess ary notion\nin Section 2 we present a variant of the proof of the representatio n theorem for\nfunction spaces without constants. Then we considerseveralno tions ofsimpliciality\nand show their satisfactory behaviour for the function spaces wit h constants. The\nmain results are contained in Section 6. Theorem 6.1 summarizes the r elations\nbetween various notions of simpliciality. The rest of the section is dev oted to the\nproofs of the assertions in Theorem 6.1. In Section 7 we address th e abstract\nDirichlet problem for spaces without constants and point out some c ommon points\nand differences with the classical case. In the last section we prese nt some remarks\nand problems in the topic.\n2.Preliminaries\nIn this section we collect basic facts on compact convex sets, Choq uet simplices\nandL1-preduals which we will use and develop in the sequel.\nWe start by pointing out that we will work both with real and complex s paces,\nthe field will be denoted by F(which, of course, means RorC). IfKis a compact\nHausdorff space, by C(K,F) we denote the Banach space of F-valued continuous\nfunctions on Kequipped with the sup-norm. The dual to C(K,F) is canonically\nidentified (using the Riesz theorem) with the space of F-valued Radon measures on\nKequipped with the total variation norm, denoted by M(K,F). ByM+(K) weSIMPLICIALITY WITHOUT CONSTANTS 3\ndenote the cone of non-negative Radon measures on Kand byM1(K) the set of\nall Radon probability measures on Kequipped with the weak∗topology.\nBy acompact convex set we mean a nonempty compact convex subset of a\nHausdorff locally convex space. If Xis compact convex set, the symbol ext X\ndenotes the set of all extreme points ofX, i.e., points which cannot be found as\ninterior points of a non-degenerate segment in X. More generally, a faceofXis\na convex set F⊂Xsuch thata,b∈Fwhenever1\n2(a+b)∈F. ByAc(X,F) we\ndenote the space of all F-valued affine continuous functions on Xconsidered as a\n(closed) subspace of C(X,F).\nImportant examples of compact convex sets include unit balls of dua l Banach\nspaces. IfX= (BE∗,w∗) is the dual unit ball of E, whereEis a Banach space\noverFandw∗denotes the weak∗topology, there are two distinguished classes of\nfunctions on X: A function f:X→Fis called\n•F-invariant iff(αx) =f(x) for eachx∈Xandα∈SF;\n•F-homogeneous iff(αx) =αf(x) for eachx∈Xandα∈SF.\nBySFwe denote the sphere of F, i.e.,SR={−1,1}andSC=T, the unit circle. If\nF=R, the above-defined classes are usually called evenandoddfunctions.\nIfXis a compact convex set and µ∈M1(X), the symbol r(µ) denotes the\nbarycenter ofµ, i.e., the unique point x∈Xsatisfying\n∀f∈Ac(X,R):f(x) =/integraldisplay\nfdµ.\nConversely, if x=r(µ), we say that µrepresentsx, or, thatµis arepresenting\nmeasure ofx. It is a consequence of the Krein-Milman theorem that any x∈Xis\nrepresented by a probability measure supported by extX.\nTheChoquet ordering onM+(X) is defined by\nµ≺ν≡def∀f:X→Rconvex continuous:/integraldisplay\nfdµ≤/integraldisplay\nfdν.\nBy amaximal measure we mean a non-negative measure maximal with respect to\nthe ordering ≺. By the Choquet-Bishop-de Leeuw theorem (see [1, Theorem I.4.8])\nwe know that any x∈Xis represented by a maximal probability measure. If Xis\nmetrizable, then ext Xis aGδ-set and maximal measures are exactly the measures\ncarried by ext X(see [1, Corollary I.4.4 and p. 35]). In the non-metrizable case\nthe situation is more difficult, an efficient characterization of maximal measures is\nprovided using upper envelopes. If f:X→Ris a bounded function (or, at least\nbounded above), its upper envelope is defined by\nf∗(x) = inf{u(x);u∈Ac(X,R),u≥f}, x∈X.\nThis notion is closely related to maximal measures. In particular, we h ave the\nfollowing Mokobodzki maximality criterion:\nFact 2.1. LetXbe a compact convex set and let µ∈M+(X). Then the following\nassertions are valid:\n(a)µis maximal if and only if/integraltext\nfdµ=/integraltext\nf∗dµfor eachf∈C(X,R).\n(b)It is enough to test the equality from (a)for convex continuous functions.\n(c)IfX= (BE∗,w∗)whereEis a Banach space over F, it is enough to test the\nequality from (a)forF-invariant convex functions.4 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nProof.Assertions( a)and(b)followfrom[1, PropositionI.4.5]. Assertion( c)follows\nfrom [3, Lemma 4.1] in the complex case, the real case is analogous. /square\nAnF-valued measure µonXis calledboundary if|µ|is maximal. It is clear that\nthe Mokobodzki criterion may be used also to characterize bounda ry measures.\nA compact convex set Xis said to be a Choquet simplex (or just a simplex) if\nfor anyx∈Xthere is a unique maximal probability measure representing x. This\nis equivalent to the algebraic notion of a simplex given in the introductio n:\nFact 2.2. LetXbe a compact convex subset of a real Hausdorff locally convex s pace\nE. ThenXis a Choquet simplex if and only if the cone {(tx,t);t≥0,x∈X}is a\nlattice in its natural order.\nFollowing [19] we say that a convex set is said to be a simplexoid if each of its\nproper faces is a simplex. It is not hard to show that any simplex is a sim plexoid.\nBut the converse is not true, since, for example, the closed unit ba ll of a Hilbert\nspace is trivially a simplexoid. Further examples of simplexoids are unit b alls of\nL1-spaces:\nFact 2.3. Letµbe a non-negative measure (not necessarily finite). Then the unit\nball ofL1(µ)is a simplexoid.\nProof.This is probably a folklore fact, used in [19] without explicit mentioning .\nSince we have not found a proper reference, we include a proof for completeness:\nLetFbe a proper face of BL1(µ). Since it is included on the sphere and it is\nconvex, we deduce that |f+g|=|f|+|g|µ-a.e. for each f,g∈F. I.e., given\nf,g∈F, then forµ-almost all ωwe havef(ω) = 0 org(ω) = 0 orf(ω)\ng(ω)is a\npositive number. This property passes to the cone Cgenerated by F. Next observe\nthatuf∈Cwheneverf∈Cand 0≤u≤1 is a measurable function. Indeed,\nf=uf+ (1−u)fand all three functions belong to L1(µ), so they are positive\nmutliples of functions of norm one, say f=t1g1,uf=t2g2and (1−u)f=t3g3.\nSince the norm is clearly additive on the cone generated by g1,g2,g3, we deduce\nthatt1=t2+t3, i.e.,g1is a convex combination of g2andg3. Sinceg1∈Fand\nFis a face, we deduce that g2,g3∈F, in particular uf∈C. Now it easily follows\nthat the cone Cis a lattice (and the operations are pointwise). We conclude that\nFis a simplex. /square\nFurther, it turns out that simplexoid dual unit balls of Banach space s admit a\ncharacterization using uniqueness of certain representing measu res. It is contained\nin the following fact which follows from the proof of [8, Theorem 3.11].\nFact 2.4. LetEbe a Banach space. The following assertions are equivalent:\n(1)The dual unit ball BE∗is a simplexoid.\n(2)For anyϕ∈BE∗with/bardblϕ/bardbl= 1there is a unique maximal probability on\n(BE∗,w∗)representing ϕ.\nA natural class of Banach spaces connecting simplices and L1-space is that of\nL1-preduals. Recall that a(real or complex) Banachspace Eis called an L1-predual\nifE∗is isometric to L1(µ) for a non-negative measure µ. IfEis anL1-predual,\nthenBE∗is a simplexoid (by Fact 2.3) and hence condition (2) from Fact 2.4 holds .\nIn fact,L1-preduals may be characterized using a uniqueness property of so me\nrepresentation measures. To formulate it we recall some notation .SIMPLICIALITY WITHOUT CONSTANTS 5\nLetEbe a Banach space over FandX= (BE∗,w∗). Forg∈C(X,F) we set\n(homg)(x) =/integraldisplay\nSFα−1g(αx)dα, x∈X,\nwhere dαis the Haar probability measure on SF. Then hom is a norm-one linear\nprojection of C(X,F) onto the subspace formed by F-homogeneous functions. We\ndenote by the same symbol the dual operator on M(X,F), i.e.\n/integraldisplay\ngd(homµ) =/integraldisplay\n(homg)dµ, g∈C(X,F),µ∈M(X,F).\nThe measures satisfying µ= homµare called F-antihomogeneous in [12] (as they\nsatisfyµ(αB) =αµ(B) forB⊂XBorel). The mentioned characterization of\nL1-preduals reads as follows:\nFact 2.5. LetEbe a Banach space over F. The following assertions are equivalent:\n(1)Eis anL1-predual.\n(2)Ifx∗∈BE∗andµ,νare two maximal probabilities on BE∗representing x∗,\nthenhomµ= homν.\n(3)For anyx∗∈BE∗there is a unique boundary F-antihomogeneous measure µ\nonBE∗such that /bardblµ/bardbl ≤1andx∗(x) =/integraltext\ny∗(x)dµ(y∗)for eachx∈E.\nProof.Equivalence (1) ⇐⇒(2) follows from [14, §21, Theorem 7] in the real case\nand from [14, §23, Theorem 5]. Equivalence (1) ⇐⇒(3) is proved in [12, Fact\n1.1]. /square\nWe finish this section by a version of monotone convergence theore m for nets\nwhich we will use several times.\nLemma 2.6. LetKbe a compact Hausdorff space and let µbe a nonnegative Radon\nmeasure on K. LetFbe a family of real-valued continuous functions on Ksuch\nthat\n(i)Fis downward directed;\n(ii)Fis bounded below, i.e., there is c∈Rsuch thatf≥cfor eachf∈ F.\nThen/integraldisplay\n(infF)dµ= inf\nf∈F/integraldisplay\nfdµ.\nA special case of this statement (where Kis a compact convex set, f∈C(K,R)\nandFis the family of concave continuous functions greater or equal to f) is proved\nin [20, Lemma 10.2]. The same proof works in the more general case fo rmulated\nabove.\n3.Function spaces without constants – basic facts\nLetKbe a (nonempty) compact Hausdorff space. By a function space on Kwe\nwill mean a linear subspace H⊂C(K,F) separating points of K. We stress that\nHneed not contain constant functions. The research of such spac es started in [8,\nSection 7]. We combine the notions from [8] with the nowadays standa rd approach\nfrom [17, Chapter 3]. We start by the following easy lemma on the evalu ation\nfunctional.6 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nLemma 3.1. LetHbe a (real or complex) function space on K. The evaluation\nmappingφ:K→H∗defined by\nφ(x)(f) =f(x), x∈K,f∈H,\nis a homeomorphic injection of Kinto(BH∗,w∗), the closed unit ball of H∗equipped\nwith the weak∗topology.\nProof.Givenx∈K,φ(x) is clearly a linear functional on Hsatisfying /bardblφ(x)/bardbl ≤1.\nFurther, for any f∈Hthe function\nx/mapsto→φ(x)(f) (=f(x))\nis continuous on K, soφis continuous with respect to the weak∗topology. Since\nHseparates points of K,φis one-to-one, so it is a homeomorphic injection. /square\nWe continue by pointing out which measures play the role of ‘represen ting mea-\nsures’: Let Hbe a (real or complex) function space on K. Forϕ∈H∗set\nMϕ(H) =/braceleftbigg\nµ∈M(K,F);/bardblµ/bardbl=/bardblϕ/bardbl&∀f∈H:ϕ(f) =/integraldisplay\nKfdµ/bracerightbigg\n.\nWe observe that Mϕ(H)/\\e}atio\\slash=∅by the Hahn-Banach extension theorem (combined\nwith the Riesz representation theorem). Moreover, M0(H) ={0}and\nMcϕ(H) =cMϕ(H), ϕ∈H∗,c∈F\\{0}.\nIfx∈K, we setMx(H) =Mφ(x)(H), i.e.,\nMx(H) =/braceleftbigg\nµ∈M(K,F);/bardblµ/bardbl=/bardblφ(x)/bardbl&∀h∈H:h(x) =/integraldisplay\nhdµ/bracerightbigg\n.\nThe next step is the definition of the Choquet boundary and analogu es of max-\nimal and boundary measures. Following [8, Definition 7.1] we define the Choquet\nboundary ofHby\nChHK={x∈K;φ(x)∈extBH∗}.\nFurther, a measure µ∈M(K,F) is said to be H-boundary (orH-maximal ) if the\nimage measure φ(µ) is a boundary (or maximal) measure on ( BH∗,w∗).\nRemark 3.2. Assume that Hcontains the constant functions. Then /bardblφ(x)/bardbl= 1\nfor eachx∈K(asφ(x)(1) = 1). Hence any element of Mx(H) is a probability\nmeasure, so Mx(H) in this case coincides with the classical set of representing\nmeasures defined in [17, Definition 3.3]. If F=R, by [17, Proposition 4.26(d)] the\nChoquet boundary coincides with classical notion, i.e., with the set of thosex∈K\nfor whichMx={εx}(cf. [17, Definition 3.4]), and by [17, Proposition 4.28] H-\nmaximal and H-boundary measures coincide with the notions from [17, Definition\n3.57]. Thus the presented theory is indeed a generalization of the cla ssical one.\nThe evaluation mapping φis naturally accompanied by another mapping (a kind\nof extension): Define the mapping θ:SF×K→H∗by\nθ(α,x) =αφ(x), α∈SF,x∈K.\nThe properties of θare summarized in the following lemma.\nLemma 3.3. (a)θis a continuous mapping of SF×Kinto(BH∗,w∗);\n(b)θ(SF×ChHK) = extBH∗, in particular, ChHK/\\e}atio\\slash=∅;\n(c)θ−1(extBH∗) =SF×ChHK.SIMPLICIALITY WITHOUT CONSTANTS 7\nProof.Assertion(a)isobvious. Toprove( b)and(c)observethatfor( α,x)∈SF×K\nwe have\nθ(α,x)∈extBH∗⇐⇒αφ(x)∈extBH∗⇐⇒φ(x)∈extBH∗⇐⇒x∈ChHK.\nLet us further observe that for any f∈Hwe have\n/bardblf/bardbl= sup{|f(x)|;x∈K}= sup{|φ(x)(f)|;x∈K},\ni.e.,φ(K) is a 1-norming subspace of H∗. By the bipolar theorem we deduce that\nBH∗=acoφ(K)w∗\n=convθ(SF×K)w∗\n.\nBy Milman’s theorem we conclude that ext BH∗⊂θ(SF×K) and the argument is\ncomplete. /square\nWe continue by presenting some easy examples showing that the spa ces without\nconstants have different behaviour than the classical spaces.\nExample 3.4. (1) LetK= [0,1] and\nH={f∈C(K,F);f(0) = 0}.\nThenHis a function space, Ch HK= (0,1] andφ(0) = 0. In particular, M0={0}\nand the mapping θis not one-to-one (as θ(1,0) =θ(−1,0) = 0).\n(2) LetK= [0,1] and letα∈SF\\{1}be given (for example α=−1). Then\nH={f∈C(K,F);f(1) =αf(0)}\nis a function space and Ch HK=K. Further,φ(1) =αφ(0) and hence the mapping\nθis not one-to-one. It follows that M1(H) contains two different measures ε1\nandαε0(and their convex combinations) although 1 ∈ChHK. Therefore the\nequivalence\nx∈ChHK⇐⇒Mx(H) ={εx}\nfails for function spaces without constants.\n(3) LetK= [0,1] and letβ∈Fbe such that 0 <|β|<1. Then\nH={f∈C(K,F);f(1) =βf(0)}.\nis a function space, Ch HK= [0,1) and the mapping θis one-to-one. Further,\nφ(1) =1\n2φ(0) andM1(H) ={βε0}.\nA variant of the Choquet-Bishop-de Leeuw theorem in this context reads as\nfollows:\nProposition 3.5. Letϕ∈H∗be arbitrary. Then Mϕ(H)contains an H-boundary\nmeasure.\nThis proposition is proved in [8, Theorem 7.3]. We present here a slight m odifi-\ncation of the proof because it is based on a construction we will use t o understand\nvariants of simpliciality. The construction is described by the following lemma:\nLemma 3.6. Letν∈M1(BH∗)be a probability measure carried by SFφ(K) =\nθ(SF×K). Let/tildewideν∈M1(SF×K)be such that θ(/tildewideν) =ν. Further, by /tildewideµdenote the\nprojection of /tildewideνonKand define an F-valued measure µonKby\nµ(A) =/integraldisplay\nSF×Aαd/tildewideν(α,y), A⊂KBorel.\nThen the following assertions are valid:8 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n(i)/integraltext\nfdµ=r(ν)(f)forf∈H.\n(ii)/bardblr(ν)/bardbl ≤ /bardblµ/bardbl ≤1.\n(iii)/tildewideµisH-boundary if and only if νis maximal.\n(iv)µis absolutely continuous with respect to /tildewideµ. If/bardblµ/bardbl= 1, then/tildewideµ=|µ|.\n(v)Ifνis maximal, then µisH-boundary. If /bardblµ/bardbl= 1, thenµisH-boundary if\nand only if νis maximal.\nProof.By the construction we have\n/integraldisplay\nBH∗Fdν=/integraldisplay\nSF×KF(αφ(y))d/tildewideν(α,y) for any bounded Borel function FonBH∗.\nHence, given f∈Hwe get\n/integraldisplay\nKfdµ=/integraldisplay\nSF×Kαf(y)d/tildewideν(α,y) =/integraldisplay\nBH∗ϕ(f)dν(ϕ) =r(ν)(f).\nThis proves ( i). Assertion ( ii) follows easily from ( i).\nLet us continue by proving ( iii). We will use Mokobodzki’s criterion described\naboveinFact2.1( c): So, fixany SF-invariantconvexcontinuousfunction F:BH∗→\nR. ThenF∗is alsoSF-invariant (and upper semicontinuous, hence Borel). Hence\nifg=Forg=F∗, we have\n/integraldisplay\nBH∗gdφ(/tildewideµ) =/integraldisplay\nKg(φ(y))d/tildewideµ(y) =/integraldisplay\nSF×Kg(φ(y))d/tildewideν(α,y)\n=/integraldisplay\nSF×Kg(αφ(y))d/tildewideν(α,y) =/integraldisplay\nBH∗gdν,\nwhere the third equality follows from the SF-invariance of g(and the remaining\nones by the rule of integration with respect to the image of a measur e). We deduce\nthat /integraldisplay\nFdφ(/tildewideµ) =/integraldisplay\nF∗dφ(/tildewideµ)⇐⇒/integraldisplay\nFdν=/integraldisplay\nF∗dν,\nwhich completes the argument.\n(iv): LetA⊂Kbe a Borel set. Then\n|µ(A)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nSF×Aαd/tildewideν(α,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nSF×A|α|d/tildewideν(α,y) =/tildewideµ(A),\nwhich proves the absolute continuity. Next assume that /bardblµ/bardbl= 1. By the Radon-\nNikod´ ym theorem there is a Borel function h:K→SFsuch that d µ=hd|µ|.\nThen\n1 =|µ|(K) =/integraldisplay\nKhdµ=/integraldisplay\nSF×Kαh(y)d/tildewideν(α,y).\nWe deduce that αh(y) = 1/tildewideν-a.e. Therefore,\n|µ|(A) =/integraldisplay\nAhdµ=/integraldisplay\nSF×Aαh(y)d/tildewideν(α,y) =/integraldisplay\nSF×A1d/tildewideν=/tildewideν(SF×A) =/tildewideµ(A)\nfor anyA⊂KBorel. This completes the argument.\n(v): This follows from ( iii) and (iv). /square\nNow we are ready to prove the announced analogue of the Choquet -Bishop-de\nLeeuw theorem.SIMPLICIALITY WITHOUT CONSTANTS 9\nProof of Proposition 3.5. Ifϕ= 0, takeµ= 0. Ifϕ/\\e}atio\\slash= 0, we may without loss\nof generality assume that /bardblϕ/bardbl= 1. Letνbe a maximal probability measure on\nBH∗representing ϕ. Take/tildewideνandµas in Lemma 3.6. By assertions ( i) and (ii) of\nthe quoted lemma we deduce that µ∈Mϕ(H), assertion ( iv) then shows that µis\nH-boundary. /square\nRemark 3.7. The procedure used in Lemma 3.6 goes back to [10] where this\napproach is addressed to complex function spaces containing cons tants. The as-\nsignment /tildewideν/mapsto→µis called the Hustad mapping in [8, 19]. The proof of assertion\n(iii) essentially follows [9]. In the context of function spaces without c onstants this\nmethod is used in [8], where /tildewideνis obtained using a selection of the inverse of θ. We\ndo not precise the choice of /tildewideν, we just use its existence which follows by combining\nthe Riesz representation theorem and the Hahn-Banach extensio n theorem.\nWe continue by another lemma focused on the inverse procedure to that given\nby Lemma 3.6.\nLemma 3.8. Letµbe anF-valued measure on Kwith/bardblµ/bardbl= 1. Leth:K→SFbe\na Borel function satisfying dµ=hd|µ|provided by the Radon-Nikod´ ym theorem.\nLet/tildewideν0be the probability on SF×Kobtained as the image of |µ|under the mapping\nx/mapsto→(h(x),x)and letν0=θ(/tildewideν0). Then the following assertions are valid:\n(i)ν0,/tildewideν0,µis a triple fitting to the scheme of Lemma 3.6.\n(ii)Ifν,/tildewideνandµare as in Lemma 3.6, then /tildewideν0=/tildewideνandν0=ν.\nProof.(i) By the construction we have ν0=θ(/tildewideν0). Moreover, for A⊂KBorel we\nhave/integraldisplay\nSF×Aαd/tildewideν0(α,y) =/integraldisplay\n{(α,y);y∈A,α=h(y)}αd/tildewideν0(α,y) =/integraldisplay\nAhd|µ|=µ(A),\nwhich completes the proof.\n(ii) It follows from the proof of assertion ( iii) of Lemma 3.6 that αh(y) = 1\n/tildewideν-a.e. In other words, h(y) =α/tildewideν-a.e., i.e., /tildewideνis carried by the graph of h. Since\n|µ|is the projection of /tildewideν(by Lemma 3.6( iii)) we deduce that /tildewideνis the image of |µ|\nunder the mapping y/mapsto→(h(y),y), i.e.,/tildewideν=/tildewideν0. Then clearly ν=ν0. /square\nAs mentioned above, Proposition 3.5 is an analogue of the Choquet-B ishop-de\nLeeuw theorem. But it uses the notion of ‘ H-boundary measures’ whose definition\nis not very descriptive. In the classical setting boundary measure s are ‘pseudo-\nsupported’ by the Choquet boundary (see, e.g., [17, Theorem 3.7 9]). It turns out\nthat in our case the situation is different. Let us first point out which properties\nremain to be true.\nObservation 3.9. Assume that Kis metrizable. Then ChHKis aGδ-subset of\nKand a measure µ∈M(K,F)isH-boundary if and only if it is carried by ChHK.\nProof.IfKis metrizable, the space C(K,F) is separable. Thus His also separable\nand therefore ( BH∗,w∗) is metrizable. Hence the assertion follows from the defini-\ntionsusing[17, Corollary3.62]and the fact that φiscontinuousand one-to-one. /square\nObservation 3.10. Letµbe anyH-boundary measure on K. Then:\n(a)µ({x}) = 0for eachx∈K\\ChHK.\n(b)IfChHKis Lindel¨ of, then |µ|∗(K\\ChHK) = 0.10 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nProof.(a): Ifx∈K\\ChHK, thenφ(x)∈BH∗\\extBH∗and soµ({x}) =\nφ(µ)({φ(x)}) = 0.\n(b): Without loss of generality assume that µ≥0. LetF⊂K\\ChHKbe a\ncompact set. Our aim is to prove that µ(F) = 0.\nSince Ch HKis Lindel¨ of, SF×ChHKis Lindel¨ of as well (cf. [5, Corollary\n3.8.10]). By Lemma 3.3 we see that ext BH∗is Lindel¨ of (as a continuous image\nof a Lindel¨ of space). Since φ(F)∩extBH∗= 0, [17, Theorem 3.79(b)] shows that\nφ(µ)(φ(F)) = 0, thus µ(F) = 0 and the proof is complete. /square\nWe continue by formulating two examples showing that H-boundary measures\nmay have quite a strange behaviour in comparison with standard bou ndary mea-\nsures.\nExample 3.11. There is a (non-metrizable) compact space K, a closed self-adjoint\nfunction space H⊂C(K,F)not containing constant functions and a non-zero H-\nboundary measure µonKsuch that |µ|(ChHK) = 0.\nExample 3.12. There is a (non-metrizable) compact space K, a closed self-adjoint\nfunction space H⊂C(K,F)not containing constant functions such that ChHKis\ndense inKand there is a non-zero H-boundary measure µonKsupported by a\nclosedGδ-subset ofKdisjoint from ChHK.\nThese two statements are proved in Examples 6.14 and 6.16 below. In view of\nProposition 3.5 and the two above examples the following question see ms to be\nnatural and interesting.\nQuestion 3.13. LetHbe a (real or complex) function space on a compact space\nK. Letϕ∈H∗. Does there exist a measure µ∈Mϕ(H)which isH-boundary\nand also pseudosupported by the Choquet boundary (i.e., |µ|(F) = 0for any closed\nGδ-setF⊂Kdisjoint from ChHK)?\nThe answer is positive if Kis metrizable (by Observation 3.9). By the classical\nresults it is also positive if Hcontains constant functions. For complex spaces not\ncontaining constants the answer is negative (under the continuum hypothesis, see\nExample 6.17 below). Its validity for real spaces seems to be open. T he answer is\nalso positive assuming that the mapping θis one-to-one:\nProposition 3.14. LetHbe a (real or complex) function space on a compact space\nK. Assume that the mapping θis one-to-one. Then any H-boundary measure on\nKis pseudosupported by ChHK.\nProof.Assumeθis one-to-one. By Lemma 3.3( a) we deduce that θis a homeomor-\nphic injection of SF×KintoBH∗. Let/tildewideK=θ(SF×K) and\n/tildewideH={f|/tildewideK;f∈Ac(BH∗,F)}.\nThen/tildewideHisaclosedself-adjointfunctionspaceon /tildewideKcontainingconstants. Moreover,\nthe state space of /tildewideHis canonically affinely homeomorphic to BH∗. In particular,\nCh/tildewideH/tildewideK=/tildewideK∩extBH∗=θ(SF×ChHK),\nwhere the last equality follows from Lemma 3.3( b).\nAssumeµis anH-boundary measure on KandF⊂Kis a closed Gδ-subset\ndisjoint from Ch HK. Thenφ(µ) is a boundary measure on BH∗and thus it is anSIMPLICIALITY WITHOUT CONSTANTS 11\n/tildewideH-boundary measure on /tildewideK. Further,θ(SF×F) is a closed Gδ-subset of /tildewideKdisjoint\nfrom Ch /tildewideH/tildewideK. Thus|φ(µ)|(θ(SF×F)) = 0 by [17, Theorem 3.79(a)]. Therefore\n|µ|(F) =φ(|µ|)(φ(F)) =|φ(µ)|(θ({1}×F)) = 0.\nThis completes the proof. /square\nIn general we get a weaker version obtained by permuting quantifie rs:\nLemma 3.15. LetHbe a (real or complex) function space on a compact space\nK. Letϕ∈H∗andG⊂Kbe aGδ-set disjoint from ChHK. Then there exists\nµ∈Mϕ(H)which isH-boundary and |µ|(G) = 0.\nProof.LetF=K\\G. ThenFis anFσ-set containing Ch HK. So,θ(SF×F) is an\nFσ-set containing ext BH∗(by Lemma 3.3). Fix any ϕ∈H∗. Similarly as in the\nproofofProposition3.5wemayassumewithout lossofgeneralitytha t/bardblϕ/bardbl= 1. Let\nνbe a maximal probability on BH∗representing ϕ. Thenνis carried by θ(SF×F)\n(by [17, Theorem 3.79(c)]). By [7, Corollary 432G] there is a Radon pr obability /tildewideν\nonSF×Kcarried bySF×Fsuch thatθ(/tildewideν) =ν. Takeµas in Lemma 3.6. In the\nsame way as in the proof of Proposition 3.5 we deduce that µis anH-boundary\nmeasure from Mϕ(H). Moreover, by the construction we deduce that µis carried\nbyF, i.e.,|µ|(G) = 0. /square\nThis lemma does not provide an answer to Question 3.13, but it easily yie lds the\nfollowing corollary(which may be applied namely for simplicial function sp aces, see\nbelow):\nCorollary 3.16. LetHbe a (real or complex) function space on a compact space\nK. Letϕ∈H∗. IfMϕ(H)contains only one H-boundary measure, this unique\nmeasure is pseudosupported by the Choquet boundary.\n4.Notions of simpliciality\nIn this section we define two natural notions of simpliciality and estab lish their\nbasic properties. Throughout this section Kwill be a compact Hausdorff space and\nH⊂C(K,F) a function space.\nThe spaceHis said to be simplicial if for eachx∈Kthe setMx(H) contains\nonly oneH-boundary measure. This definition provides a direct generalization of\nthe classical notion from [17, Definition 6.1] (by Remark 3.2).\nWe further define a stronger notion – His said to be functionally simplicial if\nfor eachϕ∈H∗the setMϕ(H) contains only one H-boundary measure. In [19] it\nis said that in such a case uniqueness holds for H.\nIt is easy to check that the spaces described in parts (1) and (3) o f Example 3.4\nare simplicial, while the space from part (2) of that example is not simplic ial. We\nalso point out that the feature illustrated by the quoted part (2) c annot happen for\nsimplicial spaces as witnessed by the following observation.\nObservation 4.1. Assume that His simplicial. Let x∈K. Then\nx∈ChHK⇐⇒Mx(H) ={εx}.\nProof.⇐= : This implication holds always, even without assuming simpliciality.\nIndeed, assume Mx(H) ={εx}. ThenεxisH-boundary by Proposition 3.5 and so\nx∈ChHKby Observation 3.10( a).12 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n=⇒: Assumex∈ChHKandµ∈Mx(H). Sinceφ(x), being an extreme\npoint ofBH∗, has norm one, necessarily /bardblµ/bardbl= 1. Let h,/tildewideν0andν0be as in\nLemma3.8. ByLemma3.8( i)andLemma3.6( i)wededucethat r(ν0) =φ(x). Since\nφ(x)∈extBH∗, wegetν0=εφ(x). So,ν0ismaximalandhencebyLemma3.6( v)we\ndeduce that µisH-boundary. Since εxis alsoH-boundary and belongs to Mx(H),\nthe assumption of simpliciality yields µ=εxand the proof is complete. /square\nWefurthernotethatfunctionalsimplicialityisstrictlystrongertha nsimpliciality\neven in the classical setting of function spaces with constants. Th is is illustrated\nby [17, Exercise 6.79] or by the following easy example. Let\nH={f∈C([0,3]);f(0)+f(1) =f(2)+f(3)}.\nIndeed, it is easy to see that the Choquet boundary is the set [0 ,3], henceHis\nsimplicial. However, if ϕ(f) =f(0) +f(1),f∈H, thenMϕ(H) contains two\ndifferentH-boundary measures ε0+ε1andε2+ε3.\nAs a generalization of the classical notion from [17, Definition 3.8] we set\nAc(H) =/braceleftbigg\nf∈C(K,F);∀µ∈Mx(H):f(x) =/integraldisplay\nfdµ/bracerightbigg\n.\nThenAc(H) is a closed function space containing H. To analyze the relationship\nofHandAc(H) we consider the following diagrams:\n(4.1)BH∗BAc(H)∗π/d111/d111 BC(K,F)∗ρ/d111/d111\nKφ1/d100/d100■■■■■■■■■■φ2/d79/d79\nε/d56/d56rrrrrrrrrrrSF×K\nθ2\n/d15/d15θ1\n/d36/d36■■■■■■■■■\nBAc(H)∗π/d47/d47BH∗\nHereφ1andφ2are the respective evaluation maps and πis the restriction map\nϕ/mapsto→ϕ|H. The mapping εassigns to each x∈Kthe Dirac measure εxandρis also\nthe restriction map. In the second diagram θ1andθ2are the respective variants of\nthe mapθdefined before Lemma 3.3 above.\nLemma 4.2. (a)The diagrams (4.1)are commutative.\n(b)The mappings πandρare continuous surjections.\n(c)The mapping πmapsφ2(K)homeomorphically onto φ1(K).\n(d)/bardblφ1(x)/bardbl=/bardblφ2(x)/bardblforx∈K.\n(e)Mx(Ac(H)) =Mx(H)forx∈K.\n(f)Ac(Ac(H)) =Ac(H).\n(g)The mapping πmapsθ2(SF×K)homeomorphically onto θ1(SF×K).\nProof.Assertions ( a) and (b) are obvious.\n(c): This follows from ( a) and (b) becauseHseparates points of K.\n(d) We haveφ1(x) =π(φ2(x)), which proves ‘ ≤’. To prove the converse, fix any\nµ∈Mx(H) andf∈Ac(H). Then\n|φ2(x)(f)|=|f(x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nfdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /bardblf/bardbl·/bardblµ/bardbl=/bardblf/bardbl·/bardblφ1(x)/bardbl.\n(e): This follows from the definitions using ( d).\n(f): This follows from ( e).\n(g): Sinceπ(αφ2(x)) =αφ1(x) for eachx∈Kand each number αwith|α|= 1\n(asπis a restriction of a linear mapping), it is enough to show that πis one-to-oneSIMPLICIALITY WITHOUT CONSTANTS 13\nonθ2(SF×K). So, assume π(αφ2(x)) =π(βφ2(y)). Thenαφ1(x) =βφ1(y). We\ndeduce that /bardblφ1(x)/bardbl=/bardblφ1(y)/bardbl. Further,\n∀f∈H:αf(x) =βf(y).\nIt follows that αMx(H) =βMy(H), so\n∀f∈Ac(H):αf(x) =βf(y),\ni.e.,αφ2(x) =βφ2(y). /square\nWe continue by a pair of easy lemmata:\nLemma 4.3. LetHbe closed. Let ψ∈BH∗andµ∈M1(BH∗). Thenr(µ) =ψif\nand only if\n∀h∈H:ψ(h) =/integraldisplay\nϕ(h)dµ(ϕ).\nIn particular, r(µ) =φ(x)(wherex∈K) if and only if\n∀h∈H:h(x) =/integraldisplay\nϕ(h)dµ(ϕ).\nProof.The ‘in particular’ part is clearly a special case of the first statemen t. The\n‘only if’ part is obvious. Let us prove the ‘if’ part. To this end assume that\nF∈Ac(BH∗). IfF=R, then there is a constant c∈Rand someh∈Hsuch\nthatF(ϕ) =c+ϕ(h) forϕ∈BH∗. IfF=C, then there is a constant c∈Cand\nsomeh1,h2∈Hsuch thatF(ϕ) =c+ϕ(h1)+ϕ(h2) forϕ∈BH∗(cf. [12, Lemma\n3.14(a)]). In both cases we see that/integraltext\nFdµ=F(ψ). /square\nLemma 4.4. LetEbe a nontrivial Banach space. Then any maximal probability\nmeasure on BE∗is carried by the unit sphere.\nProof.Assume that µis a maximal probability measure on BE∗and there is some\nr∈(0,1) such that µ(rBE∗)>0. Fixx∈Eof norm one and find x∗∈BE∗with\nx∗(x) = 1. Set\ns= sup{t∈[−r,r];µ(rBE∗∩{y∗∈E∗; Rey∗(x)≥t})>0}.\nClearlys∈(−r,r]. Fix 0<ε<1−r\n4and let\nF={y∗∈rBE∗;s−ε≤Rey∗(x)≤s}.\nBy the construction we have µ(F)>0.\nFory∗∈E∗denote byTy∗the translation operator z∗/mapsto→z∗+y∗. Consider the\nmeasure\nν=µ−µ|F+1\n2(Tεx∗(µ|F)+T−εx∗(µ|F)).\nThenνis a probability measure on BE∗andν/\\e}atio\\slash=µ. Moreover, µ≺νas for any\nconvex continuous function f:BE∗→Rwe have\n/integraldisplay\nfd1\n2(Tεx∗(µ|F)+T−εx∗(µ|F)) =1\n2/parenleftbigg/integraldisplay\nfdTεx∗(µ|F)+/integraldisplay\nfdT−εx∗(µ|F)/parenrightbigg\n=1\n2/parenleftbigg/integraldisplay\nf(z∗+εx∗)dµ|F(z∗)+/integraldisplay\nf(z∗−εx∗)dµ|F(z∗)/parenrightbigg\n≥/integraldisplay\nf(z∗)dµ|F(z∗).\nThis contradicts the maximality of µ. /square14 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nWe continue by relating the Choquet boundaries and boundary meas ures with\nrespect toHandAc(H). In caseHcontains constants assertion ( i) of the following\nproposition follows directly from Lemma 4.2( e) and assertion ( ii) follows from [17,\nProposition 3.67]. Without constants the proof is more complicated.\nProposition 4.5. (i) ChHK= ChAc(H)K.\n(ii)Letµ∈M(K,F). ThenµisH-boundary if and only if it is Ac(H)-boundary.\nProof.Assertion ( i) easily follows from ( ii), so we will prove ( ii). It is clearly\nenough to prove it for positive measures.\nTo prove the ‘only if’ part assume that µisH-boundary, i.e., φ1(µ) is a maximal\nmeasure. To prove φ2(µ) is maximal as well, we will use Mokobodzki’s test (see\nFact 2.1). Let fbe any real-valued continuous convex function on BAc(H)∗. Then\nf◦φ2∈C(K,R). Hencef◦φ2◦(φ1)−1is a continuous function on φ1(K). By\nthe Tietze extension theorem this function may be extended to a fu nctiong∈\nC(BH∗,R). Then\n/integraldisplay\nfdφ2(µ) =/integraldisplay/parenleftbig\nf◦φ2◦(φ1)−1/parenrightbig\ndφ1(µ) =/integraldisplay\ngdφ1(µ) =/integraldisplay\ng∗dφ1(µ)\n=/integraldisplay\ninf{a∈Ac(BH∗,R);a≥g}dφ1(µ)\n= inf{/integraldisplay\n(a1∧···∧an) dφ1(µ);aj∈Ac(BH∗,R),aj≥gforj= 1,...,n}\n= inf{/integraldisplay\n(a1◦π∧···∧an◦π) dφ2(µ);\naj∈Ac(BH∗,R),aj≥gforj= 1,...,n}\n≥inf{/integraldisplay\n(b1∧···∧bn) dφ2(µ);\nbj∈Ac(BAc(H)∗,R),bj≥fforj= 1,...,n}\n=/integraldisplay\nf∗dφ2(µ)≥/integraldisplay\nfdφ2(µ).\nThe first two equalities follow from definitions, the third one follows fr om Fact 2.1\nasφ1(µ) is maximal. The fourth equality is just an application of the definition\nof the upper envelope. The fifth one follows from Lemma 2.6 (note th at by∧we\ndenote pointwise minimum). The sixth equality follows from diagram (4.1 ). The\nfollowing inequality is obvious. The next equality follows again from Lemm a 2.6.\nThe last inequality is obvious. Hence the equalities hold and by Fact 2.1 w e deduce\nthatφ2(µ) is maximal.\nTo prove the ‘if’ part, assume φ2(µ) is maximal. To show that φ1(µ) is also\nmaximal we will use again Mokobodzki’s test (Fact 2.1( b)). To this end we fix f, a\nconvex continuous function on BH∗and show that/integraltext\nfdφ1(µ) =/integraltext\nf∗dφ1(µ). First\nobserve that f◦πis a convex continuous function on BAc(H)∗. Fixx∈Ksuch\nthat/bardblφ1(x)/bardbl= 1. By [1, Corollary I.3.6] we have\nf∗(φ1(x)) = sup/braceleftbigg/integraldisplay\nfdν;ν∈Mφ1(x)(Ac(BH∗)) maximal/bracerightbigg\n≤sup/braceleftbigg/integraldisplay\nfdν;ν∈Mφ1(x)(Ac(BH∗)),ν(SFφ1(K)) = 1/bracerightbiggSIMPLICIALITY WITHOUT CONSTANTS 15\nLetνbe a measure from the description of the last set. Consider the mea sure\nπ−1(ν) (carried by SFφ2(K) – we use Lemma 4.2( g)). Let/tildewideν∈M1(SF×K) be such\nthatθ2(/tildewideν) =π−1(ν). Letσbe the measure on Kobtained from /tildewideνby the Hustad\nmapping (the formula for µin Lemma 3.6). Since θ1(/tildewideν) =π(θ2(/tildewideν)) =νand\nr(ν) =φ1(x) has norm one, Lemma 3.6 shows that σ∈Mx(H). By Lemma 4.2( e)\nwe deduce that σ∈Mx(Ac(H)) and so, by Lemma 3.8 applied to Ac(H) we deduce\nthatr(π−1(ν)) =φ2(x).\nThis (together with the above formula) implies that\nf∗(φ1(x))≤sup{/integraldisplay\nfdπ(σ);σ∈Mφ2(x)(Ac(BAc(H)∗))}\n= sup{/integraldisplay\nf◦πdσ;σ∈Mφ2(x)(Ac(BAc(H)∗))}= (f◦π)∗(φ2(x)),\nwhere the last equality follows from [1, Corollary I.3.6]. Therefore\n/integraldisplay\nBH∗fdφ1(µ) =/integraldisplay\nBAc(H)∗f◦πdφ2(µ) =/integraldisplay\nBAc(H)∗(f◦π)∗dφ2(µ)\n=/integraldisplay\nSAc(H)∗(f◦π)∗dφ2(µ) =/integraldisplay\n{x∈K;/bardblφ2(x)/bardbl=1}(f◦π)∗◦φ2dµ\n≥/integraldisplay\n{x∈K;/bardblφ1(x)/bardbl=1}f∗◦φ1dµ=/integraldisplay\nSH∗f∗dφ1(µ) =/integraldisplay\nBH∗f∗dφ1(µ)\n≥/integraldisplay\nBH∗fdφ1(µ).\nHere the first equality follows from the rule of integration with respe ct to the image\nof a measure, the second one follows from Fact 2.1 (as φ2(µ) is maximal). The\nthird equality follows from the that φ2(µ), being maximal, is carried by the sphere\n(Lemma 4.4), the fourth one is again an application of integration with respect to\nan image measure. The following inequality follows from the above comp utations\ntogether with Lemma 4.2( d). The next equality follows again by integration with\nrespect to an image measure and the following one follows from the fa ct thatφ1(µ)\nis also carried by the sphere (being the image of φ2(µ) byπ−1, cf. Lemma 4.2( g)).\nThe last inequality is obvious.\nThus the equalities hold and the proof is complete.\n/square\nAs a corollary we get the following equivalence extending [17, Lemma 6 .3(b)].\nProposition 4.6. His simplicial if and only if Ac(H)is simplicial.\n5.Characterizations of simpliciality of spaces containing c onstants\nIn this section wecollect resultson function spacescontainingcons tants. Most of\nthese results are known, we present them mainly to illustrate the co ntrast between\nthis classical setting and the spaces without constants addresse d in the following\nsection. In the classical setting a key role is played by the state spa ce. Let us recall\nits definition:\nLetHbe a (real or complex) function space on Kcontaining constants. Then\nS(H) ={ϕ∈H∗;/bardblϕ/bardbl=ϕ(1) = 1}16 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nis thestate space ofH. It is a compact convex set when equipped with the weak∗-\ntopology.\nLemma 5.1. LetHbe a (real or complex) function space on Kcontaining con-\nstants. Then we have the following.\n(a)The evaluation mapping φmapsKintoS(H)and the mapping θ:SF×K→\nBH∗is one-to-one.\n(b)For eachf∈Hdefine a function on S(H)by\nΦ(f)(ϕ) =ϕ(f), ϕ∈S(H).\nThenΦis a linear isometric embedding of HintoAc(S(H),F).\n(c) Φ(f)◦φ=ffor eachf∈H.\n(d)Letf,g∈H. Theng=fif and only if Φ(g) =Φ(f).\n(e) Φis surjective if and only if His closed and self-adjoint.\nProof.(a): Letx∈K. By Lemma 3.1 we have /bardblφ(x)/bardbl ≤1. Since clearly φ(x)(1) =\n1, we deduce that φ(x)∈S(H). Further, assume that θ(α,x) =θ(β,y) for some\n(α,x),(β,y)∈SF×K. It follows that αφ(x) =βφ(y). If we plug there the constant\nfunction equal to one, we deduce α=β, henceφ(x) =φ(y). By Lemma 3.1 we\nknow thatφis one-to-one, hence x=y. This completes the argument.\n(b) and (c): It is clear that Φ( f) is a continuous affine function on S(H). Since\nelementsof S(H)arelinearfunctionals,weeasilygetthatΦislinear. Sinceelements\nofS(H) have norm one, we deduce that /bardblΦ(f)/bardbl ≤ /bardblf/bardbl. Finally, for each x∈Kwe\nhave\n(Φ(f)◦φ)(x) = Φ(f)(φ(x)) =φ(x)(f) =f(x),\nwhich proves that Φ( f)◦φ=fand that /bardblΦ(f)/bardbl ≥ /bardblf/bardbl.\n(d): Assume Φ( g) =Φ(f). For each x∈Kwe deduce using ( c) that\ng(x) = Φ(g)(φ(x)) =Φ(f)(φ(x)) =f(x),\nthusg=f.\nConversely, assume g=f. Letϕ∈S(H). By the Hahn-Banach theorem there\nis/tildewideϕ∈C(K,F)∗extendingϕwith/bardbl/tildewideϕ/bardbl= 1. Hence, by the Riesz representation\ntheorem, /tildewideϕis represented by a probability measure µonK. Thus\nΦ(g)(ϕ) =ϕ(g) =/integraldisplay\ngdµ=/integraldisplay\nfdµ=/integraldisplay\nfdµ=ϕ(f) =Φ(f)(ϕ).\nTherefore Φ( g) =Φ(f).\n(e): Assume Φ is surjective. Then H, being isometric to the Banach space\nAc(S(H),F), is closed. By ( d) we deduce that His self-adjoint.\nConversely,assumethat Hisclosedandself-adjoint. By( d)wededucethatΦ( H)\nis a closed and self-adjoint subspace of Ac(S(H),F) containing constant functions.\nAssume that Φ( H)/subsetornotdbleqlAc(S(H),F). Since Φ( H) is self-adjoint, there is a real-\nvalued function h∈Ac(S(H),F)\\Φ(H). By the Hahn-Banach theorem there is\na functional ψ∈(Ac(S(H),F))∗such thatψ|Φ(H)= 0 andψ(h) = 1. Using the\nHahn-Banach and Riesz theorems we find an F-valued measure µonS(H) such\nthat /integraldisplay\nhdµ= 1 and/integraldisplay\nΦ(f)dµ= 0 forf∈H.\nSince Φ(H) is self-adjoint, we may assume that µis a real-valued (signed) measure.\nSince/integraltext\n1dµ= 0 (as 1 ∈H), we may assume without loss of generality that µ+SIMPLICIALITY WITHOUT CONSTANTS 17\nandµ−are probability measures. Then ψis the difference of two states. This is a\ncontradiction since Φ( H) obviously separates points of S(H). /square\nThe complex functionally simplicial function spaces were analyzed in [8]. The\nresults are summarized in the following proposition. (We note that by Mbnd(K)\nwe denote the space of all H-boundary measures on K.)\nProposition 5.2. LetHbe a complex closed function space on Kcontaining con-\nstant functions. Consider the following assertions:\n(1)S(H)is a simplex.\n(2)BH∗is a simplexoid.\n(3)His functionally simplicial.\n(4)H⊥∩Mbnd(K) ={0}.\n(5)His anL1-predual.\nThen\n(5)⇐⇒(4) =⇒(3)⇐⇒(2) =⇒(1).\nThe remaining implications fail in general. Moreover, (4)implies that His self-\nadjoint. IfHis self-adjoint, then all the assertions are equivalent.\nProof.It follows from [8, Theorem 4.4] that\n(5)⇐⇒(4)⇐⇒(3) &His self-adjoint.\nA counterexample to (3) = ⇒(4) is provided by [8, Example 5.12]. Equivalence\n(3)⇐⇒(2) is proved in [8, Theorem 3.11]. Implication (3) = ⇒(1) follows from\n[8, Proposition 2.2], a counterexample to the converse is contained in [8, Example\n2.4] and its validity for Hself-adjoint is proved in [8, Proposition 2.6]. /square\nThe case of real function spaces is easier because any real space is automati-\ncally self-adjoint. Therefore we get the equivalences summarized in the following\nproposition.\nProposition 5.3. LetHbe a real closed function space on Kcontaining constant\nfunctions. The following assertions are equivalent.\n(1)S(H)is a simplex.\n(2)BH∗is a simplexoid.\n(3)His functionally simplicial.\n(4)H⊥∩Mbnd(K) ={0}.\n(5)His anL1-predual.\nProof.Set\nHC={f∈C(K,C); Ref,Imf∈H}.\nThenHCis a self-adjoint complex closed function space on Kcontaining con-\nstants. Moreover, S(HC) is canonically affinely homeomorphic to S(H), therefore\nH-boundary and HC-boundary measures on Kcoincide. Clearly\n(HC)⊥={µ∈M(K,C); Reµ,Imµ∈H⊥}.\nHence equivalence (1) ⇐⇒(4) follows from Proposition 5.2.\nEquivalence (2) ⇐⇒(3) follows from the proof of [8, Theorem 3.11], the proof\nin the real case is exactly the same. Equivalence (1) ⇐⇒(5) follows from [14, §19,\nTheorem 2].\n(2) =⇒(1): This is obvious because S(H) is a nontrivial face of BH∗.18 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n(1) =⇒(3): Letϕ∈H∗andµ1,µ2∈Mϕ(H) beH-boundary. Without loss of\ngenerality we may assume that /bardblϕ/bardbl= 1. Fixj∈ {1,2}. Then/bardblµj/bardbl= 1. Since\nϕ(1) =µj(K) =µ+\nj(K)−µ−\nj(K),\nwe see that\nµ+\nj(K) =1\n2(1+ϕ(1)) and µ−\nj(K) =1\n2(1−ϕ(1)).\nThis holds forboth values of j, sowe obtain µ+\n1(K) =µ+\n2(K) andµ−\n1(K) =µ−\n2(K).\nFurther, for each f∈Hwe have\nϕ(f) =/integraldisplay\nfdµj=/integraldisplay\nfdµ+\nj−/integraldisplay\nfdµ−\nj,\nhence /integraldisplay\nfd(µ+\n1+µ−\n2) =/integraldisplay\nfd(µ+\n2+µ−\n1).\nBy our assumptions µ+\n1+µ−\n2andµ+\n2+µ−\n1are two probability measures and they\nimagesunder φaremaximalandhavethesamebarycenter. Since S(H)isasimplex,\nthese two measures must coincide and therefore µ1=µ2. This completes the\nproof. /square\nWe continue by relating simpliciality of Hto properties of Ac(H).\nProposition 5.4. LetHbe a (real or complex) function space containing constant\nfunctions. Then the following assertions are equivalent:\n(1)His simplicial.\n(2)Ac(H)is simplicial.\n(3)S(Ac(H))is a simplex.\n(4)BAc(H)∗is a simplexoid.\n(5)Ac(H)is functionally simplicial.\n(6) (Ac(H))⊥∩Mbnd(K) ={0}.\n(7)Ac(H)is anL1-predual.\nProof.Equivalence (1) ⇐⇒(2) follows from Proposition 4.5. Conditions (3) −(7)\nare equivalent by Proposition 5.3 in the real case and by Proposition 5 .2 in the\ncomplex case (note that Ac(H) is closed and self-adjoint).\nImplication (5) = ⇒(2) is trivial.\n(2) =⇒(3): IfF=Rthis follows from [17, Theorem 6.54]. Assume F=\nC. Let/tildewideH=Ac(H)∩C(K,R). Then /tildewideHis a closed real function space on K.\nMoreover, clearly Mx(/tildewideH) =Mx(Ac(H)) forx∈K, soAc(/tildewideH) =/tildewideH. SinceS(/tildewideH) is\ncanonically affinely homeomorphic to S(Ac(H)), we deduce that /tildewideH-boundary and\nAc(H)-boundary measures coincide, in particular, /tildewideHis simplicial. So, S(/tildewideH) is a\nsimplex by the real case and hence S(Ac(H)) is a simplex as well (being affinely\nhomeomorphic to S(/tildewideH)). /square\n6.More on simpliciality of function spaces without constants\nIn the previous section we collected characterizations of simpliciality of function\nspaces with constants. In particular, if Ac(H) =H, then simpliciality is equivalent\nto functional simpliciality and there are several more natural equiv alent conditions.\nIn caseH/subsetornotdbleqlAc(H) the situation is more complicated, in particular simpliciality\ndoes not imply functional simpliciality, but there are still some charac terizations.SIMPLICIALITY WITHOUT CONSTANTS 19\nIn this section we address function spaces without constants and we will see that\nthe situation is very different. Some differences were pointed alread y in [19] where\nfunctional simpliciality is addressed.\nWe focus on simpliciality. By Proposition 4.5 we know that simpliciality pass es\nfromHtoAc(H) as in the classical case, therefore we restrict ourselves to the c ase\nAc(H) =H. The first difference here is that Ac(H) need not be self-adjoint (this is\nwitnessed by Example 3.4(2,3) if the respective parameters α,βare chosen not to\nbe real). Moreover, the analogue of Proposition 5.4 fails dramatically as witnessed\nby the following theorem.\nTheorem6.1. LetHbe a (real or complex) function space on KsatisfyingAc(H) =\nH. Consider the following assertions:\n(I)The mapping θrestricted to SF×ChHKis one-to-one.\n(II)His simplicial.\n(III)His functionally simplicial.\n(IV)H⊥∩Mbnd(K) ={0}.\n(V)BH∗is a simplexoid.\n(VI)His anL1-predual.\nThen\n(IV) =⇒(III) =⇒(II) =⇒(I)/arrowvertexdbl/arrowvertexdbl/arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowdblbt\n(VI) =⇒(V)\nNo other implications are true, even if Kis metrizable. If the mapping θis one-\nto-one, then (I)holds always and\n(III)⇐⇒(V)and(IV)⇐⇒(VI).\nIfKis metrizable, then\n(V) & (I) =⇒(III)and(VI) & (I) =⇒(IV).\nIn general (for nonmetrizable K) we have\n(VI) & (I) =⇒ /\\e}atio\\slash(II),(VI) & (II) =⇒ /\\e}atio\\slash(III),(VI) & (III) =⇒ /\\e}atio\\slash(IV).\nThe rest of this section is devoted to the proof of the above theor em. It will\nbe done in several steps – we first prove the positive part and collec t some easy\ncounterexamples and later we present, in two subsections, more c omplicated coun-\nterexamples.\nWestartbythefollowinglemmashowingthatassertion( I) automaticallyimplies\nits stronger version.\nLemma 6.2. LetHbe a (real or complex) function space on K. If the mapping θ\nrestricted to SF×ChHKis one-to-one, then it maps SF×ChHKhomeomorphically\ntoextBH∗.\nProof.By Lemma 3.3 we know that θis continuous and it maps SF×ChHKonto\nextBH∗. It remains to prove the continuity of the inverse. To this end assu me that\n(αi,xi) is a net in SF×ChHKandθ(αi,xi)→θ(α,x), where (α,x)∈SF×ChHK.\nUp to passing to a subnet we may assume that ( αi,xi)→(β,y)∈SF×K. Then\nθ(αi,xi)→θ(β,y). Henceθ(α,x) =θ(β,y), i.e.,αφ(x) =βφ(y). Sincex∈\nChHK, we haveφ(x)∈extBH∗. It follows that φ(y)∈extBH∗as well, hence20 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\ny∈ChHK. By the assumption we conclude that y=xandβ=α. This completes\nthe proof. /square\nWe continue by proving the positive part of the above theorem:\nProof of the positive part of Theorem 6.1. Implications ( IV) =⇒(III) =⇒(II)\nare trivial.\n(II) =⇒(I): Assume His simplicial. Let x,y∈ChHKandα,β∈SFbe\nsuch thatθ(α,x) =θ(β,y), i.e.,αφ(x) =βφ(y). Thenφ(y) =βαφ(x). Hence\nMy(H)⊃ {εy,βαεx}(note that /bardblφ(y)/bardbl= 1). The two measures on the right-hand\nareH-boundary (as φ(x),φ(y)∈extBH∗). By simpliciality we deduce εy=βαεx,\nhencey=xandα=β. This completes the argument.\nImplication ( III) =⇒(V) follows from [19, Theorem 3.3].\n(IV) =⇒(VI): We follow the proof of the respective implication from [8,\nTheorem 4.3] and check that it works also in our situation. Assume th atHis not\nL1-predual. By Fact 2.5 there are two maximal probabilities ν1,ν2onBH∗with\nr(ν1) =r(ν2) such that hom ν1/\\e}atio\\slash= homν2. Let/tildewideν1,/tildewideν2,µ1,µ2be the measures\nprovided by Lemma 3.6. By the quoted lemma we know that µ1andµ2areH-\nboundary (by assertion ( v)) and that µ1−µ2∈H⊥∩Mbnd(K) (by assertion ( i)).\nWhatremainstobe provedis µ1/\\e}atio\\slash=µ2. Sincehom ν1/\\e}atio\\slash= homν2, wefindf∈C(BH∗)\nwith/integraltextfdhomν1/\\e}atio\\slash=/integraltextfdhomν2. Theng= homfisanF-homogeneouscontinuous\nfunction satisfying/integraltext\ngdν1/\\e}atio\\slash=/integraltext\ngdν2. Forj= 1,2 we have\n/integraldisplay\nK(g◦φ)dµj=/integraldisplay\nSF×Kαg(φ(x))d/tildewideνj(α,x) =/integraldisplay\nSF×Kg(αφ(x))d/tildewideνj(α,x)\n=/integraldisplay\ng◦θd/tildewideνj=/integraldisplay\ngdνj,\nthus/integraltext\nK(g◦φ)dµ1/\\e}atio\\slash=/integraltext\nK(g◦φ)dµ2, in particular µ1/\\e}atio\\slash=µ2. This completes the\nargument.\n(VI) =⇒(V): This follows from Fact 2.3.\nNext assumethat θis one-to-one. Then ( III)⇐⇒(V) by[19, Theorem3.3]. Let\nus continue by proving ( VI) =⇒(IV). We proceed by contraposition. Assume\nthat (IV) is not satisfied, i.e., there is some nonzero µ∈Mbnd(H)∩H⊥. Then\nφ(µ) is a boundary measure on BH∗. Moreover, hom φ(µ)/\\e}atio\\slash= 0 by [19, Proposition\n3.5 and Proposition 3.4]. So, hom φ(µ) is a non-zero antihomogeneous boundary\nmeasure on BH∗and for each h∈Hwe have/integraldisplay\nBH∗ψ(h)dhomφ(µ)(ψ) =/integraldisplay\nBH∗ψ(h)dφ(µ)(ψ) =/integraldisplay\nφ(K)ψ(h)dφ(µ)(ψ)\n=/integraldisplay\nhdµ= 0.\nIt now follows from Fact 2.5 that His not anL1-predual, i.e., ( VI) is not satisfied.\n(V)&(I) =⇒(III) ifKis metrizable: Assume that Kis metrizable and that\nHsatisfies (V)&(I). We will show that His functionally simplicial. To this end\nit is enough to prove that Mϕ(H) contains only one H-boundary measure for each\nϕ∈SH∗. We will use a modification of the method of proving [19, Theorem 3.3]:\nLetϕ∈SH∗and letµ∈Mϕ(H) be anH-boundary measure. Then µis\ncarried by Ch HK(cf. Observation 3.9). Let /tildewideνandνbe the measures provided by\nLemma 3.8. By Lemma 4.3 and Lemma 3.6( i) we deduce that r(ν) =ϕ. Hence,νSIMPLICIALITY WITHOUT CONSTANTS 21\nis uniquely determined by ϕ(due to (V) and Fact 2.4). Next we deduce that µis\nuniquely determined by ϕas well:\nFix anyf∈C(K,F). Define\nF(αφ(x)) =αf(x), α∈SF,x∈ChHK.\nBy assumption ( I) this function is well defined. By Lemma 3.3 it is defined on\nextBH∗. It is clearly homogeneous and it is a continuous function (by Lemma 6 .2).\nHence/integraldisplay\nKfdµ=/integraldisplay\nChHKfdµ=/integraldisplay\nSF×ChHKαf(x)d/tildewideν(α,x) =/integraldisplay\nSF×ChHKF◦θd/tildewideν\n=/integraldisplay\nextBH∗Fdν.\nThus/integraltext\nKfdµdepends only on fandϕ. Therefore µis determined by ϕ. This\ncompletes the proof.\n(VI)&(I) =⇒(IV) ifKis metrizable: Assume that Kis metrizable and\nthatHsatisfies (VI)&(I). We will proceed similarly as in the above prove of\n(VI) =⇒(IV) assuming that θis one-to-one. Assume that ( IV) is not satisfied,\ni.e., there is some nonzero µ∈Mbnd(H)∩H⊥. Thenφ(µ) is a boundary measure\nonBH∗. Due to the above argument it is enough to provethat hom φ(µ)/\\e}atio\\slash= 0. Since\nµ/\\e}atio\\slash= 0, there is some f∈C(K,F) with/integraltext\nfdµ/\\e}atio\\slash= 0. Similarly as above define\nF(αφ(x)) =αf(x), α∈SF,x∈ChHK.\nBy assumption ( I) this function is well defined. By Lemma 3.3 it is defined on\nextBH∗. It is clearly homogeneous and it is a continuous function (by Lemma 6 .2).\nHence\n0/\\e}atio\\slash=/integraldisplay\nKfdµ=/integraldisplay\nChHKfdµ=/integraldisplay\nφ(ChHK)ϕ(f)dφ(µ) =/integraldisplay\nextBH∗Fdφ(µ)\n=/integraldisplay\nextBH∗homFdφ(µ) =/integraldisplay\nFdhomφ(µ).\nThus homφ(µ)/\\e}atio\\slash= 0. This completes the proof. /square\nWe continue by collecting the easy counterexamples:\nEasy counterexamples from the proof of Theorem 6.1:\n(VI) =⇒ /\\e}atio\\slash(I) even ifKis metrizable: Let K={0,1}and\nH={f∈C(K,F);f(1) =−f(0)}.\nIt is clear that His a closed self-adjoint subspace separating points of Kand not\ncontaining constants. Moreover, since His one-dimensional, it is clearly an L1-\npredual, i.e., ( VI) is satisfied. Note that Ch HK=Kas/bardblφ(0)/bardbl=/bardblφ(1)/bardbl= 1\nand all norm-one elements in a one-dimensional space are extreme p oints. Further,\n−ε0∈M1(H), henceAc(H) =H. Finally,φ(1) =−φ(0), so (I) is not satisfied.\n(I) =⇒ /\\e}atio\\slash(II) even ifKis metrizable (and θis one-to-one): Note that θis one-to-\none (and hence ( I) holds) whenever Hcontains constants (see Lemma 5.1( a)) and\nthere are non-simplicial spaces containing constants – one can tak e, for example,\nKto be the square and Hto be the space of continuous affine functions on K.\n(I) =⇒ /\\e}atio\\slash(V) even ifKis metrizable (and θis one-to-one): The same example\nworks.22 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n/square\n6.1.Counterexamples on metrizable compact spaces. In this section we\npresent a construction of function spaces on metrizable compact spaces distinguish-\ning properties ( II)−(IV) (and something more). We present two versions of these\nconstructions – once we get a closed Choquet boundary, in the sec ond case we get\na dense Choquet boundary. But before coming to the constructio n itself let us\npresent an easy sufficient condition for a point to be in the Choquet b oundary.\nLemma 6.3. LetHbe a (real or complex) function space on a compact space K.\nLeta,b∈K. Assume that K/\\e}atio\\slash={a,b}and that there is a function f∈Hsuch that\n|f(a)|=|f(b)|and\n∀x∈K\\{a,b}:|f(x)|<|f(a)|.\nThena,b∈ChHK.\nProof.Up to multiplying fby a nonzero constant we may assume f(a) = 1. Con-\nsider the function\nF(ϕ) = Reϕ(f), ϕ∈BH∗.\nThenFis a real-valued continuous affine function on BH∗. Moreover, fix any\nϕ∈BH∗. Then\nF(ϕ) = Reϕ(f)≤ |ϕ(f)| ≤ /bardblϕ/bardbl·/bardblf/bardbl ≤1\nandF(φ(a)) = 1. So, Fattains its maximum at φ(a). Moreover, assume that\nϕ∈BH∗is such that F(ϕ) = 1. Fix some µ∈Mϕ(H). Then\n1 =F(ϕ) = Reϕ(f) = Re/integraldisplay\nfdµ≤/integraldisplay\n|f|d|µ| ≤/integraldisplay\n1d|µ|= 1,\nso the equalities hold. In particular, |f|= 1|µ|-a.e., soµis carried by {a,b}. Now\nit is clear that µmust belong to the segment with endpoints εaandf(b)εb, thusϕ\nbelongs to the segment with the endpoints φ(a) andf(b)φ(b). It follows that this\nsegment is a face of BH∗, thus its endpoints are extreme points of BH∗. Now it\neasily follows that a,b∈ChHK. /square\nWe remark that the previous lemma applies also in the special case whe na=b.\nNow we proceed to the construction of the counterexamples. The two underlying\ncompact spaces will be\nK1= ([0,1]×{−1,0,1})∪{a,b}andK2= ([0,1]×[−1,1])∪{a,b},\nwherea,bare two distinct isolated points. We further fix α,β∈F\\{0}such that\n|α|+|β|<1 and forj= 1,2 we define function spaces\nHj={f∈C(Kj,F);f(t,0) =1\n2(f(t,−1)+f(t,1)) fort∈[0,1]\nandf(0,0) =αf(a)+βf(b)}.\nLetφj:Kj→H∗\njandθj:SF×Kj→H∗\njbe the canonical mappings. We continue\nby establishing basic properties of these function spaces.\nLemma 6.4. Letj∈ {1,2}. Then:\n(a)Kjis a metrizable compact space.\n(b)Hjis a closed function space on Kj. Ifα,β∈R, thenHjis self-adjoint.\n(c)The mapping θjis one-to-one.\n(d)/bardblφ(x)/bardbl= 1forx∈Kj\\{(0,0)},/bardblφ(0,0)/bardbl=|α|+|β|.SIMPLICIALITY WITHOUT CONSTANTS 23\n(e) ChHjKj=Kj\\([0,1]× {0}). In particular, ChH1K1is closed in K1and\nChH2K2is dense in K1.\n(f)Ac(Hj) =Hj.\nProof.Assertion (a) is obvious. It is clear that Hjis a closed subspace of C(Kj,F)\nnot containing constants and that it is self-adjoint provided α,β∈R.\nTo prove the rest of assertion ( b) and assertions ( c)−(e) assume first that j= 2.\nSo, next we show that H2separates points of K2and, moreover, the mapping θ2is\none-to-one.\nConsider the following four functions\nf1(s,t) =sfor (s,t)∈[0,1]×[−1,1], f1(a) =f1(b) = 0;\nf2(s,t) =tfor (s,t)∈[0,1]×[−1,1], f2(a) =f2(b) = 0;\nf3(s,t) =α−βfor (s,t)∈[0,1]×[−1,1], f3(a) = 1,f3(b) =−1;\nf4(s,t) =α+βfor (s,t)∈[0,1]×[−1,1], f4(a) =f4(b) = 1.\nClearlyf1,f2,f3,f4∈H2. Moreover, these four functions separate points of K2\nand witness that θ2is one-to-one:\nLetx,y∈K2be distinct. Let us distinguish several cases:\n•x,y∈ {a,b}: In this case f3(x)/\\e}atio\\slash=f3(y). Moreover, functions f3andf4\nwitness that φ2(x) is not a multiple of φ2(y).\n•x∈ {a,b},y∈[0,1]×[−1,1]: Then |f4(y)|<|f4(x)|.\n•x= (s1,t1) andy= (s2,t2) wheres1/\\e}atio\\slash=s2: Then|f1(x)| /\\e}atio\\slash=|f1(y)|.\n•x= (s,t1) andy= (s,t2) wheret1/\\e}atio\\slash=±t2: Then|f2(x)| /\\e}atio\\slash=|f2(y)|.\n•x= (s,t) andy= (s,−t) wheret/\\e}atio\\slash= 0: Then f2(x) =−f2(y)/\\e}atio\\slash= 0. Hence\nf2(x)/\\e}atio\\slash=f2(y) and, moreover, if φ2(y) is a multiple of φ2(x), necessarily\nφ2(y) =−φ2(x). But this may be disproved using one of the functions f3\nandf4as one of the values α+β,α−βis nonzero.\nThis completes the proof of ( b) and (c).\n(d): The function f2witnesses that /bardblφ2(s,±1)/bardbl= 1 fors∈[0,1]. The function\nf4witnesses that /bardblφ2(a)/bardbl=/bardblφ2(b)/bardbl= 1. Further, given s0∈(0,1] fix a continuous\nfunctiongs0: [0,1]→[0,1] such that gs0(0) = 0,gs0(s0) = 1 andgs0(s)<1 for\ns/\\e}atio\\slash=s0. Then the function\nf5,s0(s,t) =gs0(s) for (s,t)∈[0,1]×[−1,1], f5,s0(a) =f5,s0(b) = 0\nbelongs to H2and witnesses that /bardblφ2(s0,t)/bardbl= 1 fort∈[−1,1]. Next we fix,\nfor anyt0∈(−1,0)∪(0,1), a continuous function ht0: [−1,1]→[0,1] such that\nht0(t0) = 1,ht0(t)<1 fort/\\e}atio\\slash=t0,ht0(−1) =ht0(0) =ht0(1) = 0. Then the function\nf6,t0(s,t) =ht0(t) for (s,t)∈[0,1]×[−1,1], f6,s0(a) =f6,s0(b) = 0\nbelongstoH2andwitnessesthat /bardblφ2(0,t0)/bardbl= 1. Finally,itisclearthat /bardblφ2(0,0)/bardbl ≤\n|α|+|β|and the function\nf7(s,t) =|α|+|β|for (s,t)∈[0,1]×[−1,1], f7(a) =|α|\nα,f7(b) =|β|\nβ\nbelongs toH2and witnesses that /bardblφ2(0,0)/bardbl=|α|+|β|. This completes the proof\nof (d).\n(e): By the definition of H2we know that φ2(s,0) =1\n2(φ2(s,−1) +φ2(s,1))\nwhenevers∈[0,1]. Sinceφ2(s,−1) andφ2(s,1) are distinct points of the unit\nsphere, we deduce that ( s,0)/∈ChH2K2. We continue by proving that any x∈24 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nK2\\([0,1]× {0}) belongs to the Choquet boundary. To this end we will use\nLemma 6.3:\nThe function f3shows that a,b∈ChH2K2. Further, given s0∈[0,1], let\nus0: [0,1]→[0,1] be a continuous function satisfying us0(s0) = 1 andus0(s)<1\nfors/\\e}atio\\slash=s0. The function\nf8,s0(s,t) =tus0(s) for (s,t)∈[0,1]×[−1,1], f8,s0(a) =f8,s0(b) = 0\nbelongs toHand witnesses that points ( s0,1) and (s0,−1) belong to the Choquet\nboundary. Finally, if s0∈[0,1] andt0∈(−1,0)∪(0,1), thenφ2(s0,t0)∈ChH2K2\ndue to the function\nf9,s0,t0(s,t) =ht0(t)us0(s) for (s,t)∈[0,1]×[−1,1], f9(a) =f9(b) = 0.\nThis completes the proof of ( b)−(e) forj= 2. The case j= 1 then follows\neasily asK1⊂K2andf|K1∈H1wheneverf∈H2. Thus it is enough to consider\nthe restrictions of the above-defined functions to K1(and some of them are not\nneeded).\n(f): Since inclusion ‘ ⊃’ holds automatically, it is enough to prove ‘ ⊂’. Assume\nthatf∈Ac(Hj). Using the definition of Hjand (d) we see that αεa+βεb∈\nM(0,0)(Hj), hencef(0,0) =αf(a)+βf(b). Further, if s∈(0,1], then1\n2(ε(s,−1)+\nε(s,1))∈M(s,0)(Hj) (by the definition of Hjand (d)), hencef(s,0) =1\n2(f(s,−1)+\nf(s,1)). Sincefis continuous, by taking the limit we get f(0,0) =1\n2(f(0,−1)+\nf(0,1)) as well. Hence f∈Hj. /square\nWe continue by analyzing validity of conditions from Theorem 6.1 for fu nction\nspacesHj. The first set of results is contained in the following proposition.\nProposition 6.5. Letj∈ {1,2}. Then:\n(a)Hjis simplicial (i.e., it satisfies (II)).\n(b) (Hj)⊥∩Mbnd(K)/\\e}atio\\slash={0}(i.e., it fails (IV)).\n(c)Hjis not anL1-predual (i.e., it fails (VI)).\n(d)Ifα=β, thenHjis not functionally simplicial (i.e., it fails (III)).\nProof.SinceKjis metrizable, by Observation 3.9 we know that Hj-boundary mea-\nsures onKjare exactly the measures supported by the Choquet boundary, i.e .,\nbyKj\\([0,1]×{0}). Below we use the functions constructed within the proof of\nLemma 6.4.\n(a): Let us first provide the proof for j= 2. To this end we analyze the sets\nMx(H2) forx∈H2and, in particular, H2-boundary measures in them. We distin-\nguish several cases:\n•x=a: Letµ∈Ma(H2). By Lemma 6.4( d) we get /bardblµ/bardbl= 1. Further,\n1 =f3(a) =/integraldisplay\nf3dµ,\nhenceµis carried by {a,b}, i.e.,µ=γεa+δεb, and 1 = |γ|+|δ|=γ−δ.\nFurther,\n1 =f4(a) =/integraldisplay\nf4dµ=γ+δ.\nWe deduce that γ= 1 andδ= 0, thusµ=εa. Therefore, Ma(H2) ={εa}.\n•x=b: Similarly as in the previous case we get Mb(H2) ={εb}.SIMPLICIALITY WITHOUT CONSTANTS 25\n•x= (s0,1) for some s0∈[0,1]: Letµ∈M(s0,1)(H2). By Lemma 6.4( d) we\nget/bardblµ/bardbl= 1. Further,\n1 =f8,s0(s0,1) =/integraldisplay\nf8,s0dµ,\nhenceµis carried by {(s0,1),(s0,−1)}i.e.,µ=γε(s0,1)+δε(s0,−1), and\n1 =|γ|+|δ|=γ−δ. Further,\nα+β=f4(s0,1) =/integraldisplay\nf4dµ= (α+β)(γ+δ)\nand similarly α−β= (α−β)(γ+δ). Since at least one of the numbers\nα+β,α−βis nonzero, we deduce γ+δ= 1, hence γ= 1 andδ= 0. Thus\nµ=ε(s0,1). Therefore, M(s0,1)(H2) ={ε(s0,1)}.\n•x= (s0,−1) for some s0∈[0,1]: Similarly as in the previous case we get\nM(s0,−1)(H2) ={ε(s0,−1)}.\n•x= (s0,t0) for some s0∈[0,1] andt0∈(−1,0)∪(0,1). Letµ∈\nM(s0,t0)(H2). By Lemma 6.4( d) we get /bardblµ/bardbl= 1. Further,\n1 =f9,s0,t0(s0,t0) =/integraldisplay\nf9,s0,t0dµ,\nhenceµ=ε(s0,t0).\n•x= (s0,0) for some s0∈(0,1]: Letµ∈M(s0,0)(H2) be anH2-boundary\nmeasure. By Lemma 6.4( d) we get /bardblµ/bardbl= 1. Letz: [−1,1]→[0,1] be\ncontinuous such that z(−1) =z(0) =z(1) = 1 and z <1 elsewhere. Then\nthe function\nf10,s0(s,t) =gs0(t)z(t) for (s,t)∈[0,1]×[−1,1], f10,s0(a) =f10,s0(b) = 0,\nbelongs toH2and\n1 =f10,s0(s0,0) =/integraldisplay\nf10,s0dµ,\nsoµis carried by {(s0,−1),(s0,0),(s0,1)}. Sinceµis carried by the\nChoquet boundary, we deduce that µis carried by {(s0,−1),(s0,1)}, i.e.,\nµ=γε(s0,−1)+δε(s0,1). Moreover,we see that |γ|+|δ|=γ+δ= 1. Further,\n0 =f2(s0,0) =/integraldisplay\nf2dµ=δ−γ.\nWe deduce that γ=δ=1\n2, i.e.,µ=1\n2(ε(s0,1)+ε(s0,−1)). We conclude that\nthis is the unique H2-boundary measure in M(s0,0)(H2).\n•x= (0,0): Letµ∈M(0,0)(H2). By Lemma 6.4( d) we get /bardblµ/bardbl=|α|+|β|.\nMoreover,\n|α|+|β|=f7(0,0) =/integraldisplay\nf7dµ.\nIt follows that µis carried by {a,b}, i.e.,µ=γεa+δεb, and\n|α|+|β|=|γ|+|δ|=γ\nα|α|+δ\nβ|β|.\nFurther, the function\nf11(s,t) = 0 for (s,t)∈[0,1]×{−1,0,1}, f11(a) =β,f11(b) =−α,26 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nbelongs toH2and hence\n0 =f11(0,0) =/integraldisplay\nf11dµ=γβ−δα,\nsonecessarilyγ\nα=δ\nβ= 1,i.e.,µ=αεa+βεb. Weconcludethat M(0,0)(H2) =\n{αεa+βεb}.\nWe now conclude that H2is simplicial. The proof for j= 1 is similar – we only\ndo not consider points from [0 ,1]×((−1,0)∪(0,1)) (they are not in K1) and for\nthe remaining points we use the restrictions of the respective func tions toK1.\n(b): The measure αεa+βεb−1\n2(ε(0,1)+ε(0,−1)) isHj-boundary and belongs to\n(Hj)⊥.\n(c): This follows from ( b) using Lemma 6.4( c) and the already proved part of\nTheorem 6.1.\n(d): Assumeα=β. Set\nµ1=−αεb+1\n2ε(0,1), µ2=αεa−1\n2ε(0,−1).\nThenµ1andµ2are two distinct Hj-boundary measures on Kjand/bardblµ1/bardbl=/bardblµ2/bardbl=\n|α|+1\n2. Further, set\nϕ(f) =/integraldisplay\nfdµ1=−αf(b)+1\n2f(0,1), f∈Hj.\nThenϕ∈H∗\njand/bardblϕ/bardbl ≤ /bardblµ1/bardbl=|α|+1\n2. The function\nf12(s,t) =tfor (s,t)∈[0,1]×[−1,1],f12(a) =|α|\nα,f12(b) =−|α|\nα,\nbelongs toHjand witnesses that /bardblϕ/bardbl ≥ |α|+1\n2. Moreover, we have\n/integraldisplay\nfdµ1=/integraldisplay\nfdµ2, f∈Hj,\nhenceµ1,µ2are two different Hj-boundary measures in Mϕ(Hj). ThusHjis not\nfunctionally simplicial. /square\nOur next aim is to prove that under suitable assumptions (if α,βare distinct\npositive numbers) the function spaces Hjare functionally simplicial. To prove that\nwe need several lemmata.\nLemma 6.6. Letj∈ {1,2}. Assume that µis anHj-boundary measure on Kj.\nSet\nϕ(f) =/integraldisplay\nfdµandϕ0(f) =/integraldisplay\n{a,b,(0,1),(0,−1)}fdµforf∈Hj.\nThen\n/bardblϕ/bardbl=/vextenddouble/vextenddoubleµ|Kj\\{a,b,(0,1),(0,−1)}/vextenddouble/vextenddouble+/bardblϕ0/bardbl.\nProof.Assume first that j= 2. Inequality ‘ ≤’ is obvious, let us prove the converse\none. Letε>0 be arbitrary. Since µisH2-boundary, it is carried by the Choquet\nboundary, i.e., |µ|([0,1]×{0})= 0. Fixc∈(0,1) such that |µ|([0,1]×(−c,c))<ε.\nFurther, the space\n/tildewideK=K2\\([0,1]×(−c,c)∪{a,b,(0,1),(0,−1)})SIMPLICIALITY WITHOUT CONSTANTS 27\nis locally compact, hence it follows from the Riesz theorem that there isf0∈C0(/tildewideK)\nsuch that /bardblf0/bardbl= 1 and\n/integraldisplay\n/tildewideKf0dµ≥/vextenddouble/vextenddoubleµ|/tildewideK/vextenddouble/vextenddouble−ε>/vextenddouble/vextenddoubleµ|K2\\{a,b,(0,1),(0,−1)}/vextenddouble/vextenddouble−2ε\n(in particular, the integral on the left-hand side has real value).\nDefine the function fby settingf(a) =f(b) = 0 and\nf(s,t) =\n\nf0(s,t), (s,t)∈/tildewideK,\n0, s = 0,t∈ {−1,0,1},\n1\n2(f0(s,−1)+f0(s,1)), s ∈(0,1],t= 0,\nf(s,0)+t\nc(f0(s,c)−f(s,0)), s∈[0,1],t∈(0,c),\nf(s,0)−t\nc(f0(s,−c)−f(s,0)), s∈[0,1],t∈(−c,0).\nThenf∈H2,/bardblf/bardbl= 1 and\nReϕ(f) =/integraldisplay\n/tildewideKf0dµ+Re/integraldisplay\n[0,1]×(−c,c)fdµ≥/vextenddouble/vextenddoubleµ|K2\\{a,b,(0,1),(0,−1)}/vextenddouble/vextenddouble−3ε.\nFurther, find g0∈H2such that /bardblg0/bardbl= 1 andϕ0(g0)>/bardblϕ0/bardbl−ε. Fixδ∈(0,1\n2)\nsuch that\n|µ|([0,δ]×([−1,−1+δ]∪[1−δ,1])\\{(0,−1),(0,1)})<ε\nand\n|f|<εon [0,δ]×([−1,−1+δ]∪[−δ,δ]∪[1−δ,1]).\nSet\ng(a) =g0(a), g(b) =g0(b),\ng(s,t) =\n\n1\nδ2g0(0,1)(δ−s)(t−1+δ),(s,t)∈[0,δ]×[1−δ,1],\n1\nδ2g0(0,−1)(δ−s)(t−1+δ),(s,t)∈[0,δ]×[−1,−1+δ],\n1\nδ2g0(0,0)(δ−s)(δ−|t|),(s,t)∈[0,δ]×[−δ,δ],\n0 otherwise .\nTheng∈H2,/bardblg/bardbl ≤1 andϕ0(g) =ϕ0(g0). Moreover,\nReϕ(g) =ϕ0(g)+Re/integraldisplay\nK2\\{a,b,(0,1),(0,−1)}gdµ≥ /bardblϕ0/bardbl−2ε.\nSeth=f+g. Thenh∈H2,/bardblh/bardbl ≤1+εand\n|ϕ(h)| ≥Reϕ(h) = Reϕ(f)+Reϕ(g)\n≥/vextenddouble/vextenddoubleµ|K2\\{a,b,(0,1),(0,−1)/vextenddouble/vextenddouble−3ε+/bardblϕ0/bardbl−2ε,\nhence\n/bardblϕ/bardbl ≥/vextenddouble/vextenddoubleµ|K2\\{a,b,(0,1),(0,−1)/vextenddouble/vextenddouble+/bardblϕ0/bardbl−5ε\n1+ε.\nSinceε>0 is arbitrary, this completes the argument.\nThe proof for j= 1 is analogous. /square\nLemma 6.7. H⊥\nj∩Mbnd(Kj) = span{αεa+βεb−1\n2(ε(0,1)+ε(0,−1))}.28 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nProof.Inclusion ‘ ⊃’ is clear. To prove the converse, assume µ∈H⊥\nj∩Mbnd(Kj).\nBy Lemma 6.6 we deduce that µis carried by F={a,b,(0,1),(0,−1)}. Note that\n{f|F;f∈Hj}={g∈FF;αg(a)+βg(b) =1\n2(g(0,1)+g(0,−1))}.\nIndeed, inclusion ‘ ⊂’ follows from the definition of Hj. To prove the converse, fix\ngin the set on the right-hand side. Define f(a) =g(a),f(b) =g(b) and\nf(s,t) =1\n2(1−t)g(0,−1)+1\n2(1+t)g(0,1) for (s,t)∈[0,1]×[−1,1].\nThenf∈Hjandf|F=g, which completes the proof of inclusion ‘ ⊃’. Now the\nassertion easily follows from the bipolar theorem. /square\nLemma 6.8. Letj∈ {1,2}. The function space Hjfails to be functionally simpli-\ncial if and only if there are c1,c2,c3,c4∈Fsatisfying the following two conditions:\n|c1|+|c2|+|c3|+|c4|=/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+/vextendsingle/vextendsinglec2−1\n2/vextendsingle/vextendsingle+|c3+α|+|c4+β|, (6.1)\n∃x1,x2,x3,x4∈F:|xj| ≤1forj≤4,1\n2(x1+x2) =αx3+βx4,\nc1x1+c2x2+c3x3+c4x4=|c1|+|c2|+|c3|+|c4|.(6.2)\nProof.By definitions Hjfails to be functionally simplicial if and only if there is\nsomeϕ∈H∗\njand two different boundary measures µ1,µ2∈Mϕ(Hj). It means\nthatµ2−µ1∈(Hj)⊥, soµ2−µ1is a multiple of the measure from Lemma 6.7. Up\nto multiplying ϕby a nonzero constant we may assume that\nµ2=µ1+αεa+βεb−1\n2(ε(0,1)+ε(0,−1)).\nIn particular, necessarily µ1|Kj\\{a,b,(0,1),(0,−1)}=µ2|Kj\\{a,b,(0,1),(0,−1)}. Set\nc1=µ1({(0,1)}),c2=µ1({(0,−1)}),c3=µ1({a}),c4=µ1({b}).\nNow we easily see that (6.1) is fulfilled. Further, let ϕ0be defined as in Lemma 6.6.\nSinceµ1∈Mϕ(Hj), Lemma 6.6 yields\n/bardblϕ0/bardbl=|c1|+|c2|+|c3|+|c4|.\nIt now easily follows that (6.2) is valid.\nConversely, if c1,...,c 4satisfy (6.1) and (6.2), then the functional ϕ(f) =\nc1f(0,1)+c2f(0,−1)+c3f(a)+c4f(b) and measures\nµ1=c1ε(0,1)+c2ε(0,−1)+c3εa+c4εb,\nµ2= (c1−1\n2)ε(0,1)+(c2−1\n2)ε(0,−1)+(c3+α)εa+(c4+β)εb\nwitness that Hjis not functionally simplicial. /square\nWe continue by formulating some easy observations on condition (6.2 ):\nObservation 6.9. Letc1,...,c 4∈F.\n(1)Assume that x1,...,x 4satisfy the conditions on the first line of (6.2). Then\nthe second line is fulfilled as well if and only if\n∀j≤4: (cj= 0)or/parenleftBig\ncj/\\e}atio\\slash= 0 &xj=|cj|\ncj/parenrightBig\n.\n(2)The validity of (6.2)depends only on signs (in the real case) or arguments (in\nthe complex case) of numbers cj.\n(3)Assume that (6.2)is valid. It remains to be valid if one of the numbers c1,...,c 4\nis replaced by 0.SIMPLICIALITY WITHOUT CONSTANTS 29\nNow we are ready to prove functional simpliciality of Hjin the real setting:\nProposition 6.10. Assume that F=R,α,β >0andα/\\e}atio\\slash=β. ThenHjis func-\ntionally simplicial.\nProof.AssumeHjis not functionally simplicial. Let c1,...,c 4be the numbers\nprovided by Lemma 6.8. Observe that without loss of generality we ma y assume\nthat\nc1∈[0,1\n2],c2∈[0,1\n2],c3∈[−α,0],c4∈[−β,0].\nIndeed, ifc1>1\n2, then the quadruple1\n2,c2,c3,c4clearly satisfy (6.1) and by Ob-\nservation 6.9 it satisfies also (6.2). If c1<0, the same applies to the quadruple\n0,c2,c3,c4. Similarly we may proceed for c2,c3andc4.\nThus (6.1) implies:\nc1+c2−c3−c4=1\n2−c1+1\n2−c2+c3+α+c4+β,\ni.e.,\nc1+c2−c3−c4=1\n2(1+α+β)\nLet us now look at (6.2). We distinguish several cases:\nCase 1:c1,c2>0. Thenx1=x2= 1 (by Observation 6.9), hence 1 =\nαx3+βx4≤α+β <1, which is impossible.\nCase 2:c1= 0 andc2>0. Thenx2= 1 (by Observation 6.9). Moreover,\n(6.3) c3+c4=c2−1\n2(1+α+β)≤ −1\n2(α+β)<0.\nThus at least one of c3,c4is strictly negative. So, we distinguish some subcases:\n•c3<0 andc4<0. Thenx3=x4=−1 (by Observation 6.9) and hence\n−α−β=αx3+βx4=1\n2(x1+x2) =1\n2(x1+1),\nthusx1=−1−2α−2β <−1, a contradiction.\n•c3<0 andc4= 0. Then x3=−1, hence\n−α+βx4=1\n2(x1+1),\ni.e.,\n−1\n2−α=1\n2x1−βx4≥ −1\n2−β,\nsoβ≥α. On the other hand, from (6.3) we have\n−α≤c3≤ −1\n2(α+β),\nhenceβ≤α. So,α=β, which is a contradiction.\n•c3= 0 andc4<0. This is completely analogous to the previous subcase.\nCase 3:c1>0 andc2= 0. This is completely analogous to Case 2.\nCase 4:c1=c2= 0. Then\n−α−β≤c3+c4=−1\n2(1+α+β)<−α−β,\na contradiction.\nThere is no further possibility, so this completes the proof. /square30 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nThe previouspropositionsettles the real case, we proceedto the complex setting.\nThe proof will be done by reduction to the real case using the followin g elementary\nestimate.\nLemma 6.11. Letz∈Tandγ∈R\\{0}. Then\n∀t∈[0,∞):|tz|−|tz−γ| ≤γRez.\nMoreover, if z/\\e}atio\\slash=±1, the inequality is strict.\nProof.Let us look at the behavior of the function ω(t) =|tz|−|tz−γ|,t≥0. We\ndistinguish three cases:\n•z= 1: Then\nω(t) =/braceleftBigg\n2t−γ, t∈[0,γ],\nγ, t ≥γ.ifγ >0 andω(t) =γfort∈[0,∞) ifγ <0.\nIn particular, ω(t)≤γ=γRez.\n•z=−1: Then\nω(t) =/braceleftBigg\n2t+γ, t∈[0,−γ],\n−γ, t ≥ −γ.ifγ <0 andω(t) =−γfort∈[0,∞) ifγ >0.\nIn particular, ω(t)≤ −γ=γRez.\n•z∈T\\R: Thenz=eixfor somex∈(−π,0)∪(0,π). Then\nω(t) =t−|(tcosx−γ)+itsinx|=t−/radicalbig\nt2−2γtcosx+γ2.\nLet us differentiate:\nω′(t) = 1−t−γcosx/radicalbig\nt2−2γtcosx+γ2.\nObserve that\n(t−γcosx)2=t2−2tγcosx+γ2cos2x0 fort∈(0,∞), henceωis strictly\nincreasing on [0 ,∞). Finally,\nlim\nt→∞ω(t) = lim\nt→∞2γtcosx−γ2\nt+/radicalbig\nt2−2γtcosx+γ2=γcosx=γRez.\nIn particular, ω(t)<γRezfort∈[0,∞).\n/square\nProposition 6.12. Assume that F=C,α,β >0andα/\\e}atio\\slash=β. ThenHjis func-\ntionally simplicial.\nProof.AssumeHjis not functionally simplicial. Let c1,...,c 4be the numbers\nprovided by Lemma 6.8. If all these numbers are real, we obtain a con tradiction\nwith Proposition 6.10. So, necessarily at least one of them does not b elong to R.\nAssumecj=tjzjwheretj≥0 andzjis a complex unit (for j= 1,...,4). Let us\ndistinguish several cases:\nCase 1:c1/\\e}atio\\slash= 0 andc2/\\e}atio\\slash= 0. Then x1=z1andx2=z2(by Observation 6.9).\nUsing the first line of (6.2) we deduce that1\n2|z1+z2| ≤α+β. Next we distinguish\nseveral subcases:SIMPLICIALITY WITHOUT CONSTANTS 31\n•c3=c4= 0: Then (6.1) reads as\n|c1|+|c2|=/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+/vextendsingle/vextendsinglec2−1\n2/vextendsingle/vextendsingle+α+β,\nthus\nα+β=|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+|c2|−/vextendsingle/vextendsinglec2−1\n2/vextendsingle/vextendsingle≤1\n2(Rez1+Rez2)≤1\n2|z1+z2| ≤α+β,\nwherethefirstinequalityfollowsfromLemma6.11. Itfollowsthatequ alities\nhold, in particular z1,z2∈R(by Lemma 6.11). This is a contradiction with\nthe beginning of the proof.\n•c3/\\e}atio\\slash= 0 andc4/\\e}atio\\slash= 0: Then x3=z3andx4=z4(by Observation 6.9). We\ndeduce that\n1\n2(z1+z2) =αz3+βz4\nFurther, by Lemma 6.11 we get\n0 =|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+|c2|−/vextendsingle/vextendsinglec2−1\n2/vextendsingle/vextendsingle+|c3|−|c3+α|+|c4|−|c4+β|\n≤1\n2(Rez1+Rez2)−αRez3−βRez4= 0.\nThustheequalityholds,but(duetoLemma6.11)thismeansthat z1,...,z 4∈\nR, which is impossible by the beginning of the proof.\n•c3/\\e}atio\\slash= 0 andc4= 0 or vice versa. These two cases are symmetric, so assume\nthat the first possibility takes place. Then x3=z3(by Observation 6.9).\nThus/vextendsingle/vextendsingle1\n2(z1+z2)−αz3/vextendsingle/vextendsingle≤β\nand by Lemma 6.11 we have\nβ=|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+|c2|−/vextendsingle/vextendsinglec2−1\n2/vextendsingle/vextendsingle+|c3|−|c3+α|\n≤1\n2(Rez1+Rez2)−αRez3≤/vextendsingle/vextendsingle1\n2(z1+z2)−αz3/vextendsingle/vextendsingle≤β,\nso the equalities hold. By Lemma 6.11 we deduce that z1,z2,z3∈Rwhich\nis impossible by the beginning of the proof.\nCase 2:c1=c2= 0. Then (6.1) and Lemma 6.11 imply that\n1 =|c3|−|c3+α|+|c4|−|c4+β| ≤ −αRez3−βRez4≤α+β,\nwhich is impossible.\nCase 3:c1/\\e}atio\\slash= 0 andc2= 0 or vice versa. The two possibilities are symmetric,\nso assume that the first one takes place. By Observation 6.9 then x1=z1. We\nfurther distinguish some subcases:\n•c3=c4= 0. Then (6.1) and Lemma 6.11 say that\n1\n2+α+β=|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle≤1\n2Rez1≤1\n2,\nwhich is impossible.\n•c3/\\e}atio\\slash= 0 andc4= 0 or vice versa. The two possibilities are symmetric, so\nassume that the first one takes place. By Observation 6.9 then x3=z3. By\n(6.2) we get/vextendsingle/vextendsingle1\n2z1−αz3/vextendsingle/vextendsingle≤1\n2+β,\nby (6.1) and Lemma 6.11 we deduce that\n1\n2+β=|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+|c3|−|c3+α| ≤1\n2Rez1−αRez3≤/vextendsingle/vextendsingle1\n2z1−αz3/vextendsingle/vextendsingle≤1\n2+β,\nso the equality holds. By Lemma 6.11 it follows that z1,z3∈Rwhich is\nimpossible by the beginning of the proof.32 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n•c3/\\e}atio\\slash= 0 andc4/\\e}atio\\slash= 0. Then x3=z3andx4=z4(by Observation 6.9). So,\n(6.2) yields/vextendsingle/vextendsingle1\n2z1−αz3−βz4/vextendsingle/vextendsingle≤1\n2,\nby (6.1) and Lemma 6.11 we get\n1\n2=|c1|−/vextendsingle/vextendsinglec1−1\n2/vextendsingle/vextendsingle+|c3|−|c3+α|+|c4|−|c4+β| ≤1\n2Rez1−αRez3−βRez4≤1\n2,\nso the equality hold. It follows from Lemma 6.11 that z1,z3,z4∈Rwhich\nis impossible by the beginning of the proof.\n/square\nPropositions 6.10 and 6.12 show that ( III) =⇒ /\\e}atio\\slash(IV) even ifKis metrizable.\n6.2.Counterexamples on non-metrizable compact spaces. In this section\nwe provide several examples of function spaces on non-metrizable compact spaces.\nThese function spaces have quite strange properties, they comp lete the picture\naddressedin Theorem 6.1 and show that the non-metrizable setting is verydifferent\nfrom the metrizable one. In fact, these examples are very nice spa ces which are\njust badly embedded.\nThe starting point are special Choquet simplices considered by Stac ey [24]. Let\nus recall basic definitions and facts on them. Let Lbe a compact space and let\nA⊂Lbe a nonempty subset. Let\nKL,A= (L×{0})∪(A×{−1,1})\nbe equipped with the porcupine topology. I.e., points of A×{−1,1}are isolated\nand a neighborhood base of a point ( t,0)∈L×{0}is\n{(t,0)}∪(((U\\{t})×{−1,0,1})∩KA), Ua neighborhood of tinL.\nThenKL,Ais a compact Hausdorff space. Moreover, we set\nHL,A={f∈C(KL,A,F);f(t,0) =1\n2(f(t,−1)+f(t,1)) fort∈A}.\nIn the following proposition we collect known properties of these fun ction spaces.\nProposition 6.13. LetK=KL,AandH=HL,Afor some compact space Land\na subsetA⊂L. Then:\n(a)His a function space containing constant functions such that Ac(H) =H.\n(b) ChHK= (L\\A)×{0}∪A×{−1,1}.\n(c)His simplicial, in particular it is an L1-predual.\n(d)A measure µ∈M(K,F)isH-boundary if and only if µ({(t,0)}) = 0for each\nt∈A.\nProof.Assertions ( a)−(c) follow (for example) from [17, Lemma 6.14] (together\nwith Proposition 5.4). Assertion ( d) is proved (for example) in [11, Lemma 14.2].\n/square\nWe continue by the first example based on this class of simplices.\nExample 6.14. LetL=A= [0,1]andK=KL,A⊕L. Set\nH={f∈C(K,F);∀t∈L:f(t,0) =1\n2(f(t,1)+f(t,−1)) =−f(t)}.\nThen the following assertions are valid:\n(i)His a closed self-adjoint function space on Knot containing constant func-\ntions,/bardblφ(x)/bardbl= 1for eachx∈KandAc(H) =H.SIMPLICIALITY WITHOUT CONSTANTS 33\n(ii) ChHK=L×{−1,1}.\n(iii)A measure µonKisH-boundary if and only if µ({t}) =µ({(t,0)}) = 0for\nt∈L. In particular, there is a non-zero H-boundary measure µonKsuch\nthat|µ|(ChHK) = 0.\n(iv)His simplicial but not functionally simplicial (i.e., Hsatisfies (II)but not\n(III)).\n(v)His anL1-predual (i.e., Hsatisfies(VI)).\nProof.It is clear that His a closed self-adjoint subspace of C(K,F) not containing\nconstant functions. Let us further observe that the restrictio n mapf∈C(K,F)/mapsto→\nf|KL,A∈C(KL,A,F) mapsHisometrically onto HL,A. We now easily get that H\nseparates points of K. Fixx,y∈Kwithx/\\e}atio\\slash=y. We distinguish several cases:\n•x,y∈KL,A: We use the known fact that HL,Aseparates points of KL,A.\n•x,y∈L: Letf∈His such that f(x,0)/\\e}atio\\slash=f(y,0). Thenf(x)/\\e}atio\\slash=f(y).\n•x∈KL,Aandy∈L(or vice versa). Define h= 1 onKL,Aandh=−1 on\nL. Thenh∈Handh(x)/\\e}atio\\slash=h(y).\nMoreover, the function hwitnesses that /bardblφ(x)/bardbl= 1 for each x∈K. It follows that\nfor eacht∈L\n1\n2(ε(t,1)+ε(t,−1))∈M(t,0)(H) and−1\n2(ε(t,1)+ε(t,−1))∈Mt(H),\nso we deduce that Ac(H) =Hand complete the proof of ( i). SinceHL,Ais an\nL1-predual and it is isometric to H, we conclude that His also anL1-predual, so\nassertion (v) is valid.\nTo prove the remaining assertions we will look in more detail at the rela tionship\nofHandHL,A. Let/tildewideH=HL,AandR:H→/tildewideHbe the isometry defined by the\nrestriction. Let ı:KL,A→Kbe the canonical inclusion and φ:K→H∗and\n/tildewideφ:KL,A→/tildewideH∗the evaluation mappings. Then we have a commutative diagram:\n/tildewideH∗R∗/d47/d47H∗\nKL,A/tildewideφ/d79/d79\nı/d47/d47Kφ/d79/d79\nBy Proposition 6.13( b) we know that Ch /tildewideHKL,A=L× {−1,1}. Thus /tildewideφ(t,i)∈\nextB/tildewiderH∗for eacht∈Landi∈ {−1,1}. In such a case\nφ(t,i) =φ(ı(t,i)) =R∗(/tildewideφ(t,i))∈extBH∗\nasR∗is a linear isometry. It follows that ( t,i)∈ChHK. The remaining points do\nnot belong to the Choquet boundary as\nφ(t,0) =1\n2(φ(t,1)+φ(t,−1)) andφ(t) =1\n2(−φ(t,1)+(−φ(t,−1))).\nThis completes the proof of ( ii).\n(iii): Letµbe a measure on K. It is clear that it is H-boundary if and only if\nboth measures µ|KL,Aandµ|LareH-boundary. So, it is enough to assume that µ\nis carried either by KL,Aor byL. Assume first that µis carried by KL,A. The defi-\nnitions together with the above commutative diagram imply that µisH-boundary\nif and only if it is /tildewideH-boundary. So the assertion follows from Proposition 6.13( d).34 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nNext assume that µis carried by L. Let:L→L×{0}be the canonical home-\nomorphism. Let us look at the measure φ(µ). Given a Borel set B⊂BH∗we\nhave\nφ(µ)(B) =µ({x∈K;φ(x)∈B}) =µ({t∈L;φ(t)∈B})\n=µ({t∈L;−φ(t,0)∈B}) =µ({t∈L,−φ((t))∈B})\n=(µ|L)({x∈K;φ(x)∈ −B}) =φ((µ|L))(−B).\nThus,φ(µ) is the image of φ((µ|L)) by the mapping ϕ/mapsto→ −ϕ, thus\nµisH-boundary ⇐⇒φ(µ) is boundary ⇐⇒φ((µ|L)) is boundary\n⇐⇒(µ|L) isH-boundary.\nIndeed, the first and the third equivalences follow from the definitio ns and the sec-\nondonefollowsfromtheobviousfact thatboundarymeasureson BH∗arepreserved\nby the mapping ϕ/mapsto→ −ϕ. So, we may conclude the characterizationof H-boundary\nmeasures by referring to the first case.\nIn particular, any continuous measure carried by L(for example the Lebesgue\nmeasure) is an H-boundary measure carried by K\\ChHK.\n(iv): Let us show that His simplicial. To this end fix any x∈K. We distinguish\nthree cases:\n•x= (t,j) fort∈Landj∈ {−1,1}. Letµ∈Mx(H). The function\ng(t,1) = 1, g(t,−1) =−1, g(y) = 0 otherwise\nbelong toHand hence\nj=g(t,j) =/integraldisplay\ngdµ=µ({(t,1)})−µ({(t,−1)}).\nSince/bardblµ/bardbl= 1, we deduce µ=ε(t,j). Therefore in this case Mx(H) ={εx}.\n•x= (t,0)forsome t∈L. Letµ∈Mx(H)beH-boundary. Let u:L→[0,1]\nbe a continuous function such that u(t) = 1 andu(s)<1 fors/\\e}atio\\slash=t. Set\nv(y) =/braceleftBigg\nu(s)y= (s,j)∈KL,A,\n−u(y)y∈L.\nThenv∈Hand hence\n1 =v(t,0) =/integraldisplay\nvdµ≤/integraldisplay\n|v|d|µ| ≤1.\nThus|v|= 1|µ|-a.e., soµis carried by {t,(t,−1),(t,0),(t,1)}. Sinceµis\nH-boundary, we deduce it is carried by {(t,−1),(t,1)}, i.e.,\nµ=αε(t,−1)+βε(t,1).\nFurther,\n1 =v(t,0) =/integraldisplay\nvdµ=α+β\nand\n0 =g(t,0) =α−β,\nsoα=β=1\n2. Therefore, the unique H-boundary measure in M(t,0)(H) is\n1\n2(ε(t,−1)+ε(t,1)).\n•x∈L: Thenφ(x) =−φ(x,0), hence the unique H-boundary measure in\nMx(H) is−1\n2(ε(x,−1)+ε(x,1)).SIMPLICIALITY WITHOUT CONSTANTS 35\nThis shows that His simplicial.\nTo prove that His not functionally simplicial, let µ1be the Lebesgue measure\nonLandµ2=−(µ1). Thenµ1andµ2are two distinct H-boundary measures (by\n(iii)). Moreover, for f∈Hwe have\n/integraldisplay\nfdµ2=/integraldisplay\nfd(−(µ1)) =−/integraldisplay\nf◦dµ1=/integraldisplay\nfdµ1.\nFinally, the functional ψ:f/mapsto→/integraltext\nfdµhas norm one as witnessed by the function h\nused in the proof of ( i) above. Thus µ1,µ2are two distinct H-boundary measures\nfromMψ(H), soHis not functionally simplicial and the proof is complete. /square\nThe next example is a non-simplicial modification of the previous one.\nExample 6.15. LetKandHbe as in Example 6.14. Set K′=K⊕{a}(hencea\nis an isolated point) and\nH′=/braceleftbigg\nf∈C(K′,F);f|K∈Handf(a) =/integraldisplay\nLfdλ/bracerightbigg\n,\nwhereλis the Lebesgue measure. Then the following assertions are v alid:\n(i)H′is a closed self-adjoint function space on Knot containing constant func-\ntions,/bardblφ(x)/bardbl= 1for eachx∈KandAc(H′) =H′.\n(ii) ChH′K′=L×{−1,1}.\n(iii)A measure µonK′isH-boundary if and only if µ({a}) = 0andµ({t}) =\nµ({(t,0)}) = 0fort∈L.\n(iv)H′satisfies (I)but it is not simplicial (i.e., H′does not satisfy (II)).\n(v)H′is anL1-predual (i.e., H′satisfies (VI)).\nProof.Observe that the restriction mapping R:f/mapsto→f|KmapsH′isometrically\nontoH. Hence the validity of ( ii), (iii) and (v) may be easily deduced from\nthe respective assertions in Example 6.14. Let us comment the rema ining two\nassertions.\n(i): It is clear that H′is closed and self-adjoint linear subspace of C(K′,F). To\nsee that it separates points fix two points x,y∈K′withx/\\e}atio\\slash=y. Ifx,y∈K, they\nmay be separated due to the properties of H. Hence, the only case to be addressed\nisx=aandy∈K. Lett∈[0,1] be arbitrary. Let g: [0,1]→[0,1] be a continuous\nfunction such that g(t) = 0 andg>0 elsewhere. Let\nf(t) =g(t) fort∈L,f(t,i) =−g(t) for (t,i)∈KL,A,f(a) =/integraldisplay1\n0g.\nThenf∈H′,f(t) =f(t,−1) =f(t,0) =f(t,1) = 0 and f(a)>0. Sincet∈[0,1]\nwas arbitrary, the argument is complete.\nThe function hdefined by h=−1 onKL,A,h= 1 onL,h(a) = 1 shows that\n/bardblφ(x)/bardbl= 1 for each x∈K′. Now it easily follows that Ac(H′) =H′.\n(iv): SinceHis simplicial and hence it satisfies ( I), we easily deduce that H′\nsatisfies (I) as well (by comparing the Choquet boundaries). However, H′is not\nsimplicial as λand−(λ) (using the notation from the proof of Example 6.14) are\ntwo distinct H′-boundary measures in Ma(H′). /square\nWe continue by another variant of Example 6.14 with a bit different pro perties.36 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nExample 6.16. LetL=A= [1,2]andK=KL,A⊕[−2,−1]×[0,1]. Set\nH={f∈C(K,F);∀t∈[1,2]:f(t,0) =1\n2(f(t,1)+f(t,−1)) =−f(−t,0)}.\nThen the following assertions are valid:\n(i)His a closed self-adjoint function space on Knot containing constant func-\ntions,/bardblφ(x)/bardbl= 1for eachx∈KandAc(H) =H.\n(ii) ChHK= [1,2]×{−1,1}∪[−2,−1]×(0,1], in particular, ChHKis dense in\nK.\n(iii)A measureµonKisH-boundary if and only if µ({(−t,0)}) =µ({(t,0)}) = 0\nfort∈[1,2]. In particular, there is a closed Gδ-setF⊂ChHKdisjoint from\nChHKand a non-zero H-boundary measure µonKcarried byF.\n(iv)His simplicial but not functionally simplicial (i.e., Hsatisfies (II)but not\n(III)).\n(v)His anL1-predual (i.e., Hsatisfies(VI)).\nProof.LetL′= [1,2]×[0,1]andA′= [1,2]×{0}. SetK′=KL′,A′andH′=HL′,A′.\nForf∈Hdefine\nTf((s,t),j) =/braceleftBigg\nf(s,j), s∈[1,2],t= 0,j∈ {−1,0,1},\n−f(−s,t), s∈[1,2],t∈(0,1],j= 0.\nThenTis a linear isometry of HontoH′. Using this isometry the proof is com-\npletely analogous to that of Example 6.14. We only note that in ( iii) we may take\nF= [−2,−1]×{0}andµmay be the one-dimensional Lebesgue measure on F./square\n6.3.A consistent counterexample to the representation theorem .In this\nsection we present the promised negative consistent answer to Qu estion 3.13 in the\ncomplex case. The main result of this section is the following example wh ich is\ndone by elaborating an idea by W. Marciszewksi and G. Plebanek [18].\nExample 6.17. Under the continuum hypothesis there exists a closed comple x func-\ntion spaceHon a compact space Ksuch that:\n(i)H=Ac(H);\n(ii)Hdoes not contain constant functions;\n(iii)His simplicial;\n(iv)His anL1-predual;\n(v)there exists ϕ∈H∗such that there is no measure µ∈Mϕ(H)that is pseu-\ndosupported by ChHK.\nA key tool to the construction is a variant of the Luzin set provided by the\nfollowing lemma.\nLemma 6.18. Under the continuum hypothesis there is a bijection g:T→Tsuch\nthat its graph intersects each Cantor set (i.e. a closed null -dimensional set without\nisolated points) contained in T×Tin a countable set.\nProof.Using the continuum hypothesis we may enumerate\nT={xα;α<ω1}\nand, moreover, let\nCα,α<ω 1\nbe an enumeration of all Cantor sets contained in T×T. Next we construct by\ntransfinite induction two indexed families of ordinals ( βξ) and (γξ) forξ<ω1:SIMPLICIALITY WITHOUT CONSTANTS 37\nAssume that α<ω1is even and that we have βξandγξforξ <α. We define:\n•βα= min[0,ω1)\\{βξ;ξ <α};\n•γα= min{δ∈[0,ω1)\\{γξ;ξ <α}; (xβα,xδ)/∈/uniontext\nξ≤αCξ};\n•γα+1= min[0,ω1)\\{γξ;ξ≤α};\n•βα+1= min{δ∈[0,ω1)\\{βξ;ξ≤α}; (xδ,xγα+1)/∈/uniontext\nξ≤α+1Cξ}.\nThis inductive construction may be indeed done: It starts by α= 0 (which is an\neven ordinal). It is clear that βαmay be defined by the formula in the first item.\nFurther,γαis well defined as well as ( {xβα}×T)∩/uniontext\nξ≤αCξis a meager subset of\n{xβα}×Tand thus its complement is uncountable. The arguments for γα+1and\nβα+1are completely analogous.\nNow we define a bijection by sending xβξtoxγξfor eachξ <ω1. It is indeed a\nbijection and ( xβξ,g(xβξ))∈Cαonly ifξ <α. The required properties now easily\nfollow. /square\nProof of Example 6.17. Letgbe the function from Lemma 6.18 and let K=KL,A,\nwhereL=T×TandAis the graph of g. Moreover, set\nH={f∈C(K,C);f(t,g(t),0) =1\n2(f(t,g(t),−1)+f(t,g(t),1)) fort∈T,\nf(t,s,0) =sf(t,1,0) fors,t∈T}.\nThe properties of Hwill be established in several steps.\nStep 1:His a closed function space not containing constants, /bardblφ(x)/bardbl= 1 for\neachx∈K,Ac(H) =Hand ChHK={(t,g(t),±1);t∈T}.\nIndeed,His clearly a closed linear subspace of C(K) and 1/∈H. We continue\nby observing that it separates points of K. Ift0∈T, then the function\nft0(t,s,j) =/braceleftBigg\nj t=t0,s=g(t0),\n0 otherwise\nbelongs toHand separates points ( t0,g(t0),±1) from the rest of K. Further, the\nfunction\nh1(t,s,j) =s,(t,s,j)∈K,\nbelongstoHandseparatesallpairofpoints( t1,s1,j1),(t2,s2,j2)∈Kwiths1/\\e}atio\\slash=s2.\nFinally, the function\nh2(t,s,j) =ts,(t,s,j)∈K,\nbelongstoHand separatesall pair ofpoints ( t1,s1,j1),(t2,s2,j2)∈Kwiths1=s2\nandt1/\\e}atio\\slash=t2. The function h1witnesses that /bardblφ(x)/bardbl= 1 for each x∈K. Thus\n1\n2(ε(t,g(t)−1)+ε(t,g(t),1))∈M(t,g(t),0)(H) fort∈T\nandsε(t,1,0)∈M(t,s,0)(H) fors,t∈T. It easily follows that Ac(H) =H. Finally,\nthe function ftwitnesses that ( t,g(t),±1)∈ChHK(using Lemma 6.3). Other\npoints are not in the Choquet boundary as\nφ(t,s,0) =sg(t)φ(t,g(t),0) =1\n2/parenleftBig\nsg(t)φ(t,g(t),−1)+sg(t)φ(t,g(t),1)/parenrightBig\n/∈extBH∗.\nStep 2: The mapping π:K→KT,Tdefined byπ(t,s,j) = (t,j) for (t,s,j)∈K\nis a continuous surjection.\nIt is clear that πis a surjection. The continuity at isolated points is obvious.\nSo, fixs,t∈Tand let us prove the continuity at ( t,s,0). A basic neighborhood of38 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nπ(t,s,0) = (t,0) is of the form U={(t,0)}∪(V\\{t})×{−1,0,1}, whereVis an\nopen neighborhood of tinT. Then\nπ−1(U) ={t}×T×{0}∪((V\\{t})×T×{−1,0,1})∩K\n= (V×T×{−1,0,1})∩K\\({t}×T×{−1,1})∩K\n= (V×T×{−1,0,1})∩K\\{(t,g(t),−1),(t,g(t),1)},\nwhich is an open set in K. This completes the proof of continuity.\nStep 3: LetK′=KT,TandH′=HT,T. Forf∈H′we set\nR(f)(t,s,j) =sf(t,j),(t,s,j)∈K.\nThenRis a linear isometry of H′ontoH.\nNote thatR(f)(t,s,j) =sf(π(t,s,j)), soR(f)∈C(K,C) for each f∈H′.\nMoreover,\nR(f)(t,s,0) =sf(t,0) =sR(f)(t,1,0)\nfors,t∈Tand\nR(f)(t,g(t),0) =g(t)f(t,0) =g(t)·1\n2(f(t,1)+f(t,−1))\n=1\n2(R(f)(t,g(t),1)+R(f)(t,g(t),−1))\nfort∈T. We deduce that RmapsH′intoH. It is clear that Ris an isometry. It\nremains to show that Ris surjective. To this end fix any h∈H. Define\nf(t,j) =g(t)h(t,g(t),j),(t,j)∈K′.\nWe claim that f∈H′. Since the average condition is obvious, it is enough to prove\nthe continuity. Note that\nf(t,0) =g(t)h(t,g(t),0) =h(t,1,0) fort∈T,\nsof|T×{0}is continuous. Moreover, for t∈Tandj∈ {−1,1}we have\n|f(t,j)−f(t,0)|=/vextendsingle/vextendsingle/vextendsingleg(t)h(t,g(t),j)−g(t)h(t,g(t),0)/vextendsingle/vextendsingle/vextendsingle\n=|h(t,g(t),j)−h(t,g(t),0)|,\nso the set\n{t∈T;|f(t,1)−f(t,0)|>εor|f(t,−1)−f(t,0)|>ε}\nis finite for each ε>0. This proves the continuity of f(cf. [24]).\nStep 4: Letµ∈M(K,C) be a continuous measure. If µisH-boundary, then\nµ|{t}×T×{0}= 0 for each t∈T.\nIt is enough to prove it for nonnegative measures. Assume that µ({t0} ×T×\n{0})>0 for somet0∈T. Letu:T→[0,1] be continuous such that u(t0) = 1 and\nu(t)<1 fort∈T\\{t0}. The function\nv(t,s,j) =su(t),(t,s,j)∈K,\nbelongs toHand hence the set\nG={ϕ∈BH∗;|ϕ(v)|= 1 andϕ(ft0) = 0}\nis a closed Gδ-subset ofBH∗. Moreover, Gcontainsφ(t0,s,0) for each s∈T\n(note thatφ(t0,s,0)(v) =sandφ(t0,s,0)(ft0) = 0), hence φ(µ)(G)>0. On the\nother hand, G∩extBH∗=∅. Indeed, extreme points are exactly of the formSIMPLICIALITY WITHOUT CONSTANTS 39\nzφ(t,g(t),±1) forz,t∈T. Moreover, zφ(t0,g(t0),±1)(ft0) =±z/\\e}atio\\slash= 0 and for t/\\e}atio\\slash=t0\n|zφ(t,g(t),±1)(v)|=u(t)<1. So,φ(µ) is not maximal.\nStep 5:His simplicial.\nLetx= (t,g(t),1) for some t∈Tandµ∈Mx(H). Then /bardblµ/bardbl= 1 and the\nfunctionftfrom Step 1 witnesses that µis carried by {(t,g(t),1),(t,g(t),−1)}.\nThe function h1then shows that µ=εx. Similarly we proceed if x= (t,g(t),−1).\nNext assume that x= (t0,s0,0) for some s0,t0∈Tandµ∈Mx(H) isH-\nboundary. Let uandvbe as in the proof of Step 4. Since /bardblµ/bardbl= 1, the function v\nwitnesses that µis carried by {t0}×T×{−1,0,1}∩K. By Claim 4 we see that\nµmust be discrete, hence it is carried by {(t0,g(t0),−1),(t0,g(t0),1)}. Using the\nfunctionft0we now deduce that\nµ=s0g(t0)\n2(ε(t0,g(t0),−1)+ε(t0,g(t0),1)),\nsoµis uniquely determined.\nStep 6: Letϕ(f) =/integraltext\nTf(t,1,0)dt, where we integrate with respect to the\nnormalized Haar measure on T. Then no measure in Mϕ(H) is pseudosupported\nby the Choquet boundary.\nClearlyϕ∈H∗and/bardblϕ/bardbl= 1. Letµ∈Mϕ(H) be arbitrary.\nFixt0∈Tarbitrary. We may find a sequence of continuous functions un:T→\n[0,1] such that un(t0) = 0 for each n∈Nandun(t)→1 fort∈T\\{t0}. Forn∈N\nwe set\nvn(t,s,j) =sun(t),(t,s,j)∈K.\nThenvn∈Hand\nϕ(vn) =/integraldisplay\nTun(t)dt→1\nby the Lebesgue dominated convergence theorem. Simultaneously ,\n|ϕ(vn)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nKvndµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nK|vn|d|µ| ≤ |µ|(K\\{t0}×T×{−1,0,1}∩K).\nSince/bardblµ/bardbl= 1, we deduce that |µ|({t0} ×T× {−1,0,1}∩K) = 0. Since t0∈T\nwas arbitrary, we deduce, in particular, that µis continuous (and hence carried by\nT×T×{0}).\nIfthereiss0∈Tsuchthat |µ|(T×{s0}×{0})>0, thenµisnotpseudosupported\nby ChHKasT×{s0}×{0}is a closedGδ-set disjoint from Ch HK.\nAssume that |µ|(T×{s}×{0}) = 0 for each s∈T. LetS⊂Tbe a countable\ndense set. Let\nB= (T\\S)×(T\\S).\nThenµiscarriedby B×{0},hencethereisacompactset C⊂Bwith|µ|(C×{0})>\n0. Sinceµis continuous, without loss of generality we may assume that Chas no\nisolatedpoints. Since Bistotallydisconnected, CisaCantorset. Thus Cintersects\nthe graph of gin a countable set, so C×{0}is a closedGδ-set disjoint from Ch HK.\nThusµis not pseudosupported by Ch HK.\nConclusion: By Step 1 we know that His a closed function space with prop-\nerties (i) and (ii). Property ( iii) follows from Step 5. Step 3 together with Propo-\nsition 6.13( c) imply property ( iv). Finally, property ( v) is proved in Step 6. /square\nBy a minor modification of the above example we get the following one:40 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nExample 6.19. Under the continuum hypothesis there exists a closed comple x func-\ntion spaceHon a compact space Ksuch that:\n(i)His anL1-predual.\n(ii)There isx∈Ksuch thatMx(H)contains no measure pseudosupported by\nChHK.\nProof.We use the same trick as in Example 6.15: Let KandHbe as in Exam-\nple 6.17. Set /tildewideK=K⊕{a}(whereais a new isolated point) and define\n/tildewideH=/braceleftbigg\nf∈C(/tildewideK,C);f|K∈Handf(a) =/integraldisplay\nTf(t,1,0)dt/bracerightbigg\n.\nThen it is easy to deduce that /tildewideHhas the required properties. /square\n6.4.Overview of the counterexamples related to Theorem 6.1. Letusrecall\nthat the following implications are valid for conditions from Theorem 6.1 .\n(IV) =⇒(III) =⇒(II) =⇒(I)/arrowvertexdbl/arrowvertexdbl/arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowdblbt\n(VI) =⇒(V)\nThe above counterexamples witness that no more implications hold, e ven assuming\nKis metrizable. Let us overview the counterexamples.\nFirstly, the first of the ‘easy counterexamples’ shows that ( VI) =⇒ /\\e}atio\\slash(I). It\nfollows that no condition on the last line imply any condition on the first lin e.\nIf we restrict ourselves to the case when the mapping θis one-to-one, the last\nline implies the first one, hence we have the following implications:\n(VI)⇐⇒(IV) =⇒(V)⇐⇒(III) =⇒(II) =⇒(I)\nand, moreover, ( I) is automatically satisfied. No more implications hold in this\ncase, even if Kis metrizable. Indeed, the second one of the ‘easy counterexample s’\nwitnesses that ( I) =⇒ /\\e}atio\\slash(II). Further, ( II) =⇒ /\\e}atio\\slash(III) by Proposition 6.5 and\n(III) =⇒ /\\e}atio\\slash(IV) by Proposition 6.10 in the real case and by Proposition 6.12 in\nthe complex case. Moreover, there are two versions of these cou nterexamples – one\nwith closed Choquet boundary and another one with dense Choquet boundary.\nIfKis metrizable, we have also the following diagram:\n(IV) =⇒(III) =⇒(II) =⇒(I)/arrowdbltp/arrowvertexdbl/arrowdblbt/arrowdbltp/arrowvertexdbl/arrowdblbt\n(VI)&(I) =⇒(V)&(I)\nIfKis not metrizable, Example 6.15 show that ( VI)&(I) =⇒ /\\e}atio\\slash(II) and Exam-\nples 6.14 and 6.16 show that ( VI)&(II) =⇒ /\\e}atio\\slash(III). It seems not to be clear\nwhether (VI)&(III) =⇒(IV) in general.\n7.Dirichlet problem without constants\nOne of the main applications of simpliciality in the classical setting consis ts in\nsolutions of various types of the Dirichlet problem. The relationship t o the classical\nDirichlet problem for the Laplace equation is described in [17, Chapte r 13] (see\nalso [2]). This connection inspired investigation of the abstract Dirich let problem:\nGiven a compact convex set Xand a function on ext X(with certain properties),\nwe search for an affine extension of this function (preserving cert ain properties).SIMPLICIALITY WITHOUT CONSTANTS 41\nA weaker version asks, given a function on X(with certain properties), to modify\nit to an affine function (preserving some of the properties) coincidin g with the\noriginal one on ext X. There is a lot of results on continuous or Baire functions\n(see, e.g., [1, Theorem II.4.5] for continuous functions, [23] for Ba ire functions and\n[22] for the case of vector-valued functions). There are also ver sions for function\nspaces, where the set of extreme points is replaced by the Choque t boundary and\nH-affine functions are considered (see [17, Section 6.5] or [21]). We will show that\nthe situation for function spaces without constants has some com mon points with\nthe classical one, but there are nontrivial differences. To formula te the results we\nfirst need to introduce and clarify some notions.\nLetHbe a (real or complex) function space on a compact space K. Letf:\nK→Fbe a bounded universally measurable function. Then fis called\n•H-affineiff(x) =/integraltext\nfdµwheneverx∈Kandµ∈Mx(H);\n•stronglyH-affineif/integraltext\nfdµ= 0 whenever µ∈(Ac(H))⊥.\nThe notion of H-affine functions is a straight generalization of the classical notion\nfrom [17, Definition 3.8]. By the very definition continuous H-affine functions are\nexactly the elements of Ac(H). Further, it follows from Lemma 4.2( e) thatAc(H)-\naffine functions coincide with H-affine ones.\nStronglyH-affinefunctioncoincidewith completelyAc(H)-affinefunctionsinthe\nterminology of [16]. Continuous strongly H-affine functions are just the elements of\nAc(H) (by the bipolar theorem). Further, any strongly H-affine function is clearly\nH-affine (asεx−µ∈(Ac(H))⊥wheneverµ∈Mx(H)). The following proposition\ncollects two important cases when the converse implication holds.\nProposition 7.1. Assume that one of the following two conditions is satisfied:\n(a)Kis a compact convex set and H=Ac(K).\n(b)His a simplicial function space containing constant functio ns.\nThenH-affine and strongly H-affine functions coincide.\nProof.(a): Letfbe anH-affine function. Let µ∈Ac(H)⊥=H⊥. SinceHis\nself-adjoint, both Re µand Imµbelong toH⊥. So, without loss of generality µis\nreal-valued. Since 1 ∈H,µ+andµ−havethe same norm, without lossofgenerality\nthey are probabilities. The assumption µ∈Ac(K)⊥then means that µ+andµ−\nhave the same barycenter x∈K. Then\n/integraldisplay\nfdµ=/integraldisplay\nfdµ+−/integraldisplay\nfdµ−=f(x)−f(x) = 0,\nwhich completes the proof.\n(b): Assume that His simplicial and contains constants. Let fbeH-affine and\nµ∈Ac(H)⊥. SinceAc(H) is self-adjoint and contains constants, similarly as in the\nproof of (a) we may assume that µis real-valued and µ+andµ−are probabilities.\nLetφ:K→S(Ac(H)) be the evaluation mapping. By Lemma 5.1 we know that\nAc(H) is canonically isometric to Ac(S(Ac(H))). Therefore the measures φ(µ+)\nandφ(µ−) have the same barycenter ψ∈S(Ac(H)). Letν1andν2be maximal\nprobabilities on S(Ac(H)) such that φ(µ+)≺ν1andφµ−≺ν2. Thenψis the\nbarycenter of both ν1andν2. By Proposition 5.4 we know that S(Ac(H)) is a\nsimplex and hence ν1=ν2. Setν=φ−1(ν1) (=φ−1(ν2)) (note that maximal\nmeasures on S(Ac(H)) are carried by φ(K)). By [17, Proposition 3.89] there are42 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nprobabilities Λ 1,Λ2on the set\nM={(εx,σ);x∈K,σ∈Mx(H)} ⊂M1(K)×M1(K)\nwith barycenters ( µ+,ν) and (µ−,ν), respectively. Then\n/integraldisplay\nKfdµ+=/integraldisplay/parenleftbigg/integraldisplay\nKfdλ1/parenrightbigg\ndΛ1(λ1,λ2) =/integraldisplay/parenleftbigg/integraldisplay\nKfdλ2/parenrightbigg\ndΛ1(λ1,λ2) =/integraldisplay\nKfdν.\nIndeed, the first and the third equalities follow from [17, Proposition 3.90]. To\nprove the second equality recall that Λ 1is carried by M, so Λ 1-almost all pairs\n(λ1,λ2) are of the form ( εx,σ) wherex∈Kandσ∈Mx(H). For such pairs we\nhave /integraldisplay\nKfdλ1=/integraldisplay\nKfdεx=f(x) =/integraldisplay\nKfdσ=/integraldisplay\nKfdλ2,\nwhere the third equality follows from the assumption that fisH-affine.\nHence/integraltext\nKfdµ+=/integraltext\nKfdν. Similarly we get/integraltext\nKfdµ−=/integraltext\nKfdν. By subtract-\ning we conclude that/integraltext\nKfdµ= 0, which completes the proof. /square\nWe continue by an example distinguishing H-affine and strongly H-affine func-\ntions in the classical case.\nExample 7.2. There is a countable (hence metrizable) compact space K, a (non-\nsimplicial) function space HonKcontaining constants and a Baire-one function\nf:K→Rwhich isH-affine but not strongly H-affine.\nProof.Let\nK= ({0}∪{1\nn;n∈N,n≥2})×{−2,−1,0,1,2}∪{(1\nn,1\nn);n∈N,n≥2}\nwith the topology inherited from R2. ThenKis clearly a countable compact set.\nFurther, we set\nH={f∈C(K);f(1\nn,1\nn) =1\n2n(f(1\nn,−2)+f(1\nn,−1))+(1−1\nn)f(1\nn,0)\n=1\n2n(f(1\nn,2)+f(1\nn,1))+(1−1\nn)f(1\nn,0)\nforn∈N,n≥2}.\nIt is clear that His a closed self-adjoint subspace of C(K) containing constants.\nWe continue by showing that Hseparates points and\nChHK=K\\{(1\nn,1\nn);n∈N,n≥2}.\nTo this end we construct some functions belonging to H:\nFixn∈N,n≥2. Define\nfn,2(1\nn,2) = 0,fn,2(1\nn,1) = 2,fn,−2= 1 elsewhere .\nThenfn,2belongs to Hand exposes points (1\nn,2) and (1\nn,1). Similarly we define\nfn,−2∈Hexposing (1\nn,−2) and (1\nn,−1). Further define\nfn,0(1\nn,0) = 0,fn,0(1\nn,1\nn) =1\nn,fn,0= 1 elsewhere .\nThenfn,0belongs toH, it exposes (1\nn,0) and separates (1\nn,1\nn) from the rest of K.\nFurther, define\ng2(t,2) =t,g2(t,1) = 2−tfort∈ {0}∪{1\nn;n∈N,n≥2},g2= 1 elsewhere .SIMPLICIALITY WITHOUT CONSTANTS 43\nTheng2∈Hand it exposes points (0 ,2) and (0,1). Similarly we may define\ng−2∈Hexposing (0 ,−2) and (0,−1). Finally, the function\ng0(0,0) = 0,g0(1\nn,0) =1\nn,g0(1\nn,1\nn) =2\nn−1\nn2forn∈N,n≥2,f= 1 elsewhere\nbelongs to Hand exposes (0 ,0). This completes the proof that His a function\nspace containing constants and the description of the Choquet bo undary (points\n(1\nn,1\nn) forn≥2 obviously do not belong to the Choquet boundary). Further, it\neasily follows from the definition of HthatAc(H) =H.\nNext we observe that for each n∈N,n≥2,M(1\nn,1\nn)(H) is the convex hull of\nmeasures\nε(1\nn,1\nn),1\n2n(ε(1\nn,−2)+ε(1\nn,−1))+(1−1\nn)ε(1\nn,0),1\n2n(ε(1\nn,2)+ε(1\nn,1))+(1−1\nn)ε(1\nn,0).\nIndeed,letµ∈M(1\nn,1\nn)(H). Thefunction hnwhichequals0on {1\nn}×{(−2,−1,0,1\nn,1,2)}\nand 1 elsewhere belongs to Hand witnesses that µis carried by this set, i.e.,\nµ=αε(1\nn,−2)+βε(1\nn,−1)+γε(1\nn,0)+δε(1\nn,1\nn)+ǫε(1\nn,1)+ζε(1\nn,2)\nfor some nonnegative numbers α,β,γ,δ,ε,ζ satisfyingα+β+γ+δ+ǫ+ζ= 1.\nFunctionsfn,2andfn,−2considered above witness that α=βandǫ=ζ. Further,\nfunctionfn,0witnesses that1\nn= 2(α+ǫ)+1\nnδand hence\nγ= 1−δ−2(α+ǫ) = 2(n−1)(α+ǫ),\nwhich completes the proof.\nIn view of this description H-affine functions are exactly bounded universally\nmeasurable functions f:K→Fsatisfying\nf(1\nn,1\nn) =1\n2n(f(1\nn,−2)+f(1\nn,−1))+(1−1\nn)f(1\nn,0)\n=1\n2n(f(1\nn,2)+f(1\nn,1))+(1−1\nn)f(1\nn,0) forn∈N,n≥2.\nIn particular, u=χ{(0,−2),(0,−1)}is anH-affine function of the first Baire class.\nTocompletetheproofweobservethatthisfunctionisnotstrongly H-affine. Indeed,\nfor eachn∈N,n≥2 we have\nµn=ε(1\nn,−2)+ε(1\nn,−1)−ε(1\nn,1)−ε(1\nn,2)∈H⊥=Ac(H)⊥\nand this sequence weak∗-converges to µ=ε(0,−2)+ε(0,−1)−ε(0,1)−ε(0,2), hence\nµ∈Ac(H)⊥as well. However,/integraltext\nudµ= 2/\\e}atio\\slash= 0. /square\nThe previous proposition and example clarify the relationship of H-affine and\nstronglyH-affine functions for function spaces containing constants. The s ituation\nfor function spaces not containing constants is different – H-affine functions need\nnot be strongly H-affine even if His simplicial. This is witnessed by examples\nfrom Section 6.1 as we explain below. Let us now turn to simplicial space s and to\nvariants of the Dirichlet problem.\nIn the rest of this section we assume that Kis a fixed compact space and His\na simplicial function space on K(real or complex, with or without constants). We\nwill also assume that Ac(H) =H, which makes no loss of generality, in view of\nProposition 4.5 and the problems addressed. Given x∈K, the unique H-boundary\nmeasure in Mx(H) will be denoted by δx. Moreover, given a bounded universally\nmeasurable function f:K→Fwe define its dilation by\nDf(x) =/integraldisplay\nfdδx, x∈K.44 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nThis operation has been thoroughly studied for function spaces co ntaining con-\nstants. Let us collect some of the known results in order to compar e them with the\nresults on spaces without constants.\nFact 7.3. Assume that Hcontains constants. Then the following assertions are\nvalid:\n(a)Iff:K→Fis continuous (or just a bounded Baire function), then Dfis a\nBorel function.\n(b)Iff:K→Fis a bounded Borel function, Dfmay be highly non-measurable.\n(c)Given a measure µ∈M(K,F), the formula\nDµ(f) =/integraldisplay\nDfdµ,f∈C(K,F),\ndefines a continuous linear functional on C(K,F). Its representing measure\n(also denoted by Dµ) isH-boundary.\n(d)Iff:K→Fis a bounded Baire function, then Dfis stronglyH-affine.\nProof.Assertions ( a) and (d) are proved in [17, Theorem 6.8]. Assertion ( c) follows\nfrom [17, Theorem 6.11]. An example witnessing validity of ( b) may be constructed\neasily for the Stacey function spaces mentioned in Section 6.2: Take L=A= [0,1],\nK=KL,A,H=HL,Aandf=χB×{1}whereB⊂[0,1] is a non-measurable\nset. /square\nThe situation for spaces without constants is more complicated and we do not\nknow whether analogous statements holds in full generality. In the sequel we will\nanalyzethissituation. Wefirstpointoutthatforsimplicialfunctions spaceswithout\nconstants we have one more natural operator in addition to D: Iff:K→Fis a\nbounded universally measurable function, we define\n/tildewideDf(x) =/integraldisplay\nKfd|δx|, x∈K.\nNote that in case Hcontains constants, measures δxare positive, so D=/tildewideD, but if\nHdoes not contain constants, the two operators may differ.\nWe continue by collecting basic properties of operators Dand/tildewideD.\nObservation 7.4. Letf:K→Fbe a bounded universally measurable function.\nThen the following assertions are valid.\n(i)Dfand/tildewideDfare bounded functions and /bardblDf/bardbl∞≤ /bardblf/bardbl∞and/vextenddouble/vextenddouble/vextenddouble/tildewideDf/vextenddouble/vextenddouble/vextenddouble\n∞≤ /bardblf/bardbl∞.\n(ii)Df(x) =/tildewideDf(x) =f(x)forx∈ChHK.\n(iii)Iff≥0, then|Df| ≤/tildewideDf.\n(iv)If(fn)is a bounded sequence of universally measurable functions p ointwise\nconverging to f, thenDfn→Dfand/tildewideDfn→/tildewideDfpointwise.\nProof.(i): This follows from the fact that /bardblδx/bardbl=/bardblφ(x)/bardbl ≤1 for eachx∈K.\n(ii): Observe that δx=εxforx∈ChHK.\n(iii): Iff≥0 andx∈K, then\n|Df(x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nfdδx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\n|f|d|δx|=/integraldisplay\nfd|δx|=/tildewideDf(x).\n(iv): This follows from the Lebesgue dominated convergence theorem ./squareSIMPLICIALITY WITHOUT CONSTANTS 45\nThe previous easy observation implies, in particular, that both Dand/tildewideDare\nlinear operators of norm one (from the space of bounded universa lly measurable\nfunctions to the space of bounded functions) and the operator /tildewideDis moreover posi-\ntive. However, the resulting function may be highly non-measurable (even in case\nHcontainsconstants, cf. Fact 7.3( b)). Let us analysethese operatorsoncontinuous\nfunctions (and hence on bounded Baire functions).\nLet us first look at the case of metrizable K. A key ingredient is the following\nlemma which is known for function spaces containing constants (in th is case it\nfollows immediately from [17, Theorem 11.41]).\nLemma 7.5. Assume that Kis metrizable. Then the mappings x/mapsto→δxandx/mapsto→\n|δx|are Borel maps of KintoM(K,F).\nProof.These mappings can be expressed as compositions of several maps . Let us\nfirst analyze the individual maps from the compositions.\nA key ingredient is the existence of a mapping T:BH∗→M1(BH∗) which is\nBorel (in fact of the first Baire class) and assigns to each ϕ∈BH∗a maximal\nprobability measure with barycenter ϕ. SinceBH∗is metrizable, its existence\nfollows from [17, Theorem 11.41].\nAnother consequence of metrizability of BH∗is the fact that maximal measures\nare exactly the measures carried by ext BH∗. By Theorem 6.1 and Lemma 6.2\nwe know that the mapping θmapsSF×ChHKhomeomorphically onto ext BH∗.\nTherefore the mapping\nµ/mapsto→θ−1(µ), µ∈M1(extBH∗)\nis a homeomorphism of maximal probabilities on BH∗onto probabilities on SF×\nChHK.\nFurther, consider the mapping S:M1(SF×K)→M(K,F) defined by\nS(µ)(A) =/integraldisplay\nSF×Aαdµ(α,y), A⊂KBorel,\nsee Lemma 3.6. This mapping is continuous as, given any f∈C(K), we have/integraldisplay\nKfdS(µ) =/integraldisplay\nSF×Kαf(y)dµ(α,y)\nand hence the mapping µ/mapsto→/integraltext\nKfdS(µ) is continuous.\nIt follows from Lemma 3.6 that\nδx=/braceleftBigg\n0, φ (x) = 0,\n/bardblφ(x)/bardbl·S/parenleftBig\nθ−1/parenleftBig\nT/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig/parenrightBig/parenrightBig\n, φ(x)/\\e}atio\\slash= 0.\nSinceφis continuous and the norm is on H∗is weak∗-lower semicontinuous, we\ndeduce that the mapping x/mapsto→δxis Borel. A bit more careful analysis yields that\nit is of the second Baire class.\nFurther, let πdenote the projection of SF×KontoK. The mapping µ/mapsto→π(µ)\nis clearly continuous as a mapping M1(SF×K)→M1(K) and Lemma 3.6 also\nimplies that\n|δx|=/braceleftBigg\n0, φ (x) = 0,\n/bardblφ(x)/bardbl·π/parenleftBig\nθ−1/parenleftBig\nT/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig/parenrightBig/parenrightBig\n, φ(x)/\\e}atio\\slash= 0.\nSo, similarly as above we deduce that the mapping x/mapsto→ |δx|is Borel. /square46 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nNowwearereadytocollectpropertiesoftheoperators Dand/tildewideDinthemetrizable\ncase.\nTheorem 7.6. Assume that Kis metrizable.\n(1)Letf:K→Fbe a bounded Borel function. Then Dfand/tildewideDfare Borel\nfunctions.\n(2)Givenµ∈M(K,F), there are unique measures Dµ,/tildewideDµ∈M(K,F)satisfying\n/integraldisplay\nfdDµ=/integraldisplay\nDfdµand/integraldisplay\nfd/tildewideDµ=/integraldisplay\n/tildewideDfdµ, f∈C(K,F).\nMoreover, /bardblDµ/bardbl ≤ /bardblµ/bardbland/vextenddouble/vextenddouble/vextenddouble/tildewideDµ/vextenddouble/vextenddouble/vextenddouble≤ /bardblµ/bardbland the above equalities also hold for\nany bounded Borel function f:K→F.\n(3)|Dµ| ≤/tildewideD|µ|, in particular /tildewideDµ≥0wheneverµ≥0.\n(4)Dµand/tildewideDµareH-boundary and µ−Dµ∈Ac(H)⊥.\n(5)Dµ=δxwheneverµ∈Mx(H).\n(6)DfisH-affine whenever f:K→Fis a bounded Borel function.\n(7)Letµ∈M(K,F). ThenµisH-boundary ⇐⇒Dµ=µ⇐⇒/tildewideDµ=µ.\nProof.(1): Letf∈C(K,F). Thenµ/mapsto→/integraltext\nfdµis a weak∗-continuous linear\nfunctional on M(K,F). By Lemma 7.5 we know that x/mapsto→δxandx/mapsto→ |δx|are a\nBorel mappings of KintoM(K,F). Hence, the mappings Dfand/tildewideDfare Borel,\nbeing compositionsofthe respectivemappings. Since Borelfunctio ns onKcoincide\nwith Baire functions, using Observation 7.4( iv) we complete the proof of (1).\n(2): Using (1) and Observation 7.4( i) we deduce that\nf/mapsto→/integraldisplay\nDfdµandf/mapsto→/integraldisplay\n/tildewideDfdµ\nare continuous linear functionals on C(K,F) of norm at most /bardblµ/bardbl. The existence\nand uniqueness of measures Dµand/tildewideDµtogether with the estimates of norms\nfollow by the Riesz representation theorem. The last statement fo llows from Ob-\nservation 7.4( iv) and the Lebesgue dominated convergence theorem (using the co -\nincidence of Borel and Baire functions).\n(3): Fix a Borel set B⊂K. Then\n|Dµ(B)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nχBdDµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD(χB)dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\n|DχB|d|µ| ≤/integraldisplay\n/tildewideDχBd|µ|\n=/integraldisplay\nχBd/tildewideD|µ|=/tildewideD|µ|(B),\nwhere we used (2) and Observation 7.4( iii). Now the definition of the absolute\nvariation of a measure easily implies |Dµ| ≤/tildewideDµ.\n(4): Iff∈Ac(H), then\n/integraldisplay\nfdDµ=/integraldisplay\nDfdµ=/integraldisplay\n(/integraldisplay\nfdδx)dµ(x) =/integraldisplay\nf(x)dµ(x),\nhenceµ−Dµ∈Ac(H)⊥.\nMoreover, if B⊂K\\ChHKis a Borel set, using (2) we get\n/tildewideDµ(B) =/integraldisplay\nχBd/tildewideDµ=/integraldisplay\n/tildewideDχBdµ=/integraldisplay\n|δx|(B)dµ(x) = 0SIMPLICIALITY WITHOUT CONSTANTS 47\nas|δx|is carried by Ch HK(see Observation 3.9). This shows that /tildewideDµis carried by\nChHK, so it is an H-boundary measure. By (3) we deduce that Dµis a boundary\nmeasure as well.\n(5): Letµ∈Mx(H). By (4) and (2) we deduce that Dµis anH-boundary\nmeasure in Mx(H), i.e.,Dµ=δx.\n(6): Letfbe a bounded Borel function. Let µ∈Mx(H). Then\n/integraldisplay\nDfdµ=/integraldisplay\nfdDµ=/integraldisplay\nfdδx=Df(x),\nwhere we used (2) and (5). This completes the proof.\n(7): IfDµ=µor/tildewideDµ=µ, thenµisH-boundary by (4). Conversely, assume µis\nH-boundary. By Observation 3.9 we know that µis carried by Ch HK, so equalities\nDµ=/tildewideDµ=µfollow from the definitions in (2) and Observation 7.4( ii). /square\nWe continue by the promised example showing a further difference be tween func-\ntion spaces with and without constants.\nExample 7.7. It may happen that Kis metrizable, ChHKis closed and there is\nf∈C(K,F)such thatDfis neither continuous nor strongly H-affine.\nProof.LetK=K1andH=H1, whereK1andH1are as in Section 6.1. Then\nKis metrizable and Ch HKis closed (by Lemma 6.4) and His simplicial (by\nProposition 6.5). Let fthe the constant function equal to 1. It follows from the\nproof of Proposition 6.5 that\nDf(x) =/braceleftBigg\nα+β, x= (0,0),\n1 otherwise .\nSo,Dfis not continuous. Moreover, Dfis not strongly H-affine because µ=\nε(0,0)−1\n2(ε(0,−1)+ε(0,1))∈Ac(H)⊥, but/integraltext\nDfdµ=α+β−1/\\e}atio\\slash= 0. /square\nNote that Fact 7.3 deals with arbitrary function space containing co nstants, but\nTheorem 7.6 requires metrizability of the compact in question. The me trizable case\nuses special tools (in particular a selection result) which are not ava ilable in the\ngeneral case. The methods of proving Fact 7.3 cannot be transfe rred to the general\ncase, but there is another special case to which they may be adapt ed. It is the\ncontent of the following theorem.\nTheorem 7.8. Assume that the mapping θis one-to-one. Then the following\nassertions are valid:\n(a)Assume that fis a bounded Baire function on K. Then:\n(i)Dfand/tildewideDfare bounded Borel functions.\n(ii)DDf=Dfand/tildewideD/tildewideDf=/tildewideDf.\n(b)Givenµ∈M(K,F), there are unique measures Dµ,/tildewideDµ∈M(K,F)satisfying\n/integraldisplay\nfdDµ=/integraldisplay\nDfdµand/integraldisplay\nfd/tildewideDµ=/integraldisplay\n/tildewideDfdµ, f∈C(K,F).\nMoreover, /bardblDµ/bardbl ≤ /bardblµ/bardbland/vextenddouble/vextenddouble/vextenddouble/tildewideDµ/vextenddouble/vextenddouble/vextenddouble≤ /bardblµ/bardbland the above equalities also hold for\nany bounded Baire function f:K→F.\n(c)|Dµ| ≤/tildewideD|µ|, in particular /tildewideDµ≥0wheneverµ≥0.\n(d)Dµand/tildewideDµareH-boundary and µ−Dµ∈Ac(H)⊥.48 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\n(e)Dµ=δxwheneverµ∈Mx(H).\n(f)DfisH-affine whenever f:K→Fis a bounded Baire function.\n(g)Letµ∈M(K,F). ThenµisH-boundary ⇐⇒Dµ=µ⇐⇒/tildewideDµ=µ.\nProof.(a): By Observation 7.4( i) we know that Dfand/tildewideDfare bounded functions.\nIt remains to prove Borel measurability and assertion ( ii). Due to the Lebesgue\ndominated convergence theorem it is enough to prove it for continu ous functions.\nLet us first analyze in a bit more detail the construction from Lemma ta 3.6\nand 3.8. Fix x∈Ksuch thatφ(x)/\\e}atio\\slash= 0. Letνbe any maximal probability on\nBH∗with barycenterφ(x)\n/bardblφ(x)/bardbl. Let/tildewideνandµbe the respective measures provided by\nLemma 3.6. The quoted lemma also implies that δx=/bardblφ(x)/bardblµ. It follows from\nLemma 3.8 that νmay be reconstructed from µ(hence from δx). We deduce that\nνis uniquely determined by x, so we shall denote it by νx.\nFor any continuous function F:BH∗→Fwe set\nD0F(x) =/integraldisplay\nFdνx, x∈K,φ(x)/\\e}atio\\slash= 0.\nNext observe that for each F:BH∗→Rconvex continuous we have\n(7.1) D0F(x) =F∗(φ(x)\n/bardblφ(x)/bardbl), x∈K,φ(x)/\\e}atio\\slash= 0.\nIndeed, by [1, Corollary I.3.6] we deduce that\nF∗(φ(x)\n/bardblφ(x)/bardbl) = sup/braceleftbigg/integraldisplay\nFdν;ν∈M1(BX∗),r(ν) =φ(x)\n/bardblφ(x)/bardbl/bracerightbigg\n= sup/braceleftbigg/integraldisplay\nFdν;ν∈M1(BX∗) maximal,r(ν) =φ(x)\n/bardblφ(x)/bardbl/bracerightbigg\n=/integraldisplay\nFdνx.\nThe first equality follows from the quoted result of [1], the second o ne follows from\nthe convexity of Fand the last one from the uniqueness of νx.\nSinceφis continuous, the norm on H∗is weak∗-lower semicontinous and F∗\nis upper semicontinuous, we deduce that the mapping D0(F) is Borel measurable\nwheneverFis convex and continuous. Using the Stone-Weierstrass theorem a nd\nLebesgue dominated convergence theorem we deduce that the ma ppingD0(F) is\nBorel measurable whenever Fis continuous.\nWe continue by proving ( i). Letf∈C(K,F). Define two functions on SF×K\nby\n/tildewidef1(α,x) =αf(x) and/tildewidef2(α,x) =f(x),(α,x)∈SF×K.\nThese two functions are continuous. Since θis one-to-one, by the Tietze theorem\nwe may find continuous functions F1andF2onBH∗such thatFj◦θ=/tildewidefj(for\nj= 1,2). Ifx∈Kis such that φ(x)/\\e}atio\\slash= 0, we get\nDf(x) =/integraldisplay\nfdδx=/bardblφ(x)/bardbl/integraldisplay\nαf(y)d/tildewideνx(α,y) =/bardblφ(x)/bardbl/integraldisplay\n/tildewidef1d/tildewideνx\n=/bardblφ(x)/bardbl/integraldisplay\nF1dνx=/bardblφ(x)/bardblD0F1(x),\nhenceDfis a Borel function. Similarly,\n/tildewideDf(x) =/integraldisplay\nfd|δx|=/bardblφ(x)/bardbl/integraldisplay\nf(y)d/tildewideνx(α,y) =/bardblφ(x)/bardbl/integraldisplay\n/tildewidef2d/tildewideνx\n=/bardblφ(x)/bardbl/integraldisplay\nF2dνx=/bardblφ(x)/bardblD0F2(x),SIMPLICIALITY WITHOUT CONSTANTS 49\nhence/tildewideDfis also a Borel function.\nTo prove (ii) first denote D1F(x) =/bardblφ(x)/bardblD0F(x) for any continuous F:\nBH∗→F. IfFis convex and continuous, then\nD(D1F)(x) =/integraldisplay\nKD1Fdδx=/integraldisplay\nK/bardblφ(y)/bardblF∗(φ(y)\n/bardblφ(y)/bardbl)dδx(y)\n=/integraldisplay\n{y∈K;/bardblφ(y)/bardbl=1}/bardblφ(y)/bardblF∗(φ(y)\n/bardblφ(y)/bardbl)dδx(y)\n=/integraldisplay\n{y∈K;/bardblφ(y)/bardbl=1}F∗(φ(y))dδx(y) =/integraldisplay\nKF∗◦φdδx\n=/integraldisplay\nBH∗F∗dφ(δx) =/integraldisplay\nBH∗Fdφ(δx) =/integraldisplay\nKF◦φdδx=D(F◦φ)(x).\nIndeed, the first equality follows from the definition of operator D, the second one\nfollows from the definition of D1together with (7.1). The third one follows from\nLemma4.4(as φ(δx)isaboundarymeasure). Thefourthequalityisobviousandthe\nfifth one follows again from Lemma 4.4. The sixth one follows by integra tion with\nrespecttothe imageofameasure,the seventhonefollowsfromth e Mokobodzkitest\nof maximality (cf. Fact 2.1; recall that φ(δx) is a boundary measure), the eighth\none is again an application of integration with respect to the image of a measure.\nThe last equality follows from the definition of D.\nSo, we have proved D(D1F) =D(F◦φ). In the same way we may prove\nthat/tildewideD(D1F) =/tildewideD(F◦φ) (we just replace δxby|δx|). These equalities hold for\nany continuous convex function F, so by the Stone-Weierstrass theorem we easily\ndeduce that they hold for any Fcontinuous.\nWe proceed with the proof of ( ii) itself. Let f∈C(K,F). LetF1andF2be as\nabove. By the computation in the proof of ( i) we get\nDDf=D(D1F1) =D(F1◦φ) =Df\nand similarly\n/tildewideD/tildewideDf=/tildewideD(D1F2) =/tildewideD(F2◦φ) =/tildewideDf,\nwhich completes the argument.\n(b): This is completely analogous to assertion (2) of Theorem 7.6 and th e proof\nis the same, just using assertion ( a).\n(c): Using (b) we may prove, in the same way as in the proof of assertion (3)\nof Theorem 7.6 that |Dµ(B)| ≤/tildewideD|µ|(B) for any Baire set B⊂K. By regularity\nof the included measures we may extend this inequality first to any co mpact set\nB⊂Kand consequently to any Borel set B⊂K. We conclude by using the\ndefinition of the absolute variation of a measure.\n(d): Let us first prove that /tildewideDµis boundary. Without loss of generality we may\nassume that µ≥0. We may further assume that φ(µ)({0}) = 0. Indeed, there\nis at most one point x∈Kwithφ(x) = 0. If there is no such point, equality\nφ(µ)({0}) = 0 holds automatically. Next assume that there is such a point x. Then\nfor eachf∈C(K,F) we have\n/integraldisplay\nfd/tildewideDεx=/integraldisplay\n/tildewideDfdεx=/tildewideDf(x) =/integraldisplay\nfdδx= 0\nasδx= 0. Thus /tildewideDεx= 0 and therefore /tildewideDµ=/tildewideD(µ−µ(x)εx).50 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nLet the operator inv: C(BH∗,F)→C(BH∗,F) be defined as\n(invf)(s) =/integraldisplay\nSFf(αs)dα, s∈BH∗,f∈C(BH∗,F),\nwhere dαdenotes the probability Haar measureon SF. Then invfis anF-invariant\ncontinuous function on BH∗for eachf∈C(BH∗,F). This operator was used for\nexample in [3] and it is a counterpart of the operator hom defined in Se ction 2.\nSimilarly as in the mentioned case we denote again by inv the adjoint ope rator to\ninv onM(BH∗,F), i.e.,/integraldisplay\nfdinvµ=/integraldisplay\ninvfdµ, f∈C(BH∗,F),µ∈M(BH∗,F).\nWe claim that\n(7.2)/integraldisplay\nl∗dinvν=/integraldisplay\nl∗dν\nfor anyν∈M+(BH∗) andSF-invariant convex continuous function lonBH∗. In-\ndeed, firstwerealizethatgivensuch νandlasabove,s∈BH∗andε>0,thereexist\naffine continuous functions g1,...,gn∈C(BH∗,R) such that g= min{g1,···,gn}\nsatisfiesg≥l∗andg(αs)0. Thenη−1(B) is a Borel subset of Kand\nhence, by regularity of µ′, there is a compact set L1⊂η−1(B) such that\nµ′(L1)>µ′(η−1(B))−ε\n2=ν(B)−ε\n2.\nFurther, the function χ:K→[0,∞) defined by\nχ(x) =/braceleftBigg\n1\n/bardblφ(x)/bardbl, φ(x)/\\e}atio\\slash= 0,\n0, φ (x) = 0.\nis Borel measurable and hence, by the Luzin theorem there is L2⊂L1compact\nsuch thatµ′(L1\\L2)<ε\n2andχis continuous on L2. Thenηis also continuous on\nL2and henceη(L2) is a compact subset of B. Moreover,\nν(η(L2)) =µ′(η−1(η(L2)))≥µ′(η(L2))>µ′(L1)−ε\n2>ν(B)−ε.\nThis completes the proof of the regularity of ν.\nWe point out that by the construction of νwe have\n(7.3)/integraldisplay\nfdν=/integraldisplay\nK/bardblφ(x)/bardblf/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig\ndµ(x) forf:BH∗→Fbounded Borel .\nLetω=φ(/tildewideDµ). Then inv ν≺invω. Indeed, if kis a convex continuous function\nonBH∗, then writing l= invkwe have\n/integraldisplay\nkdinvω=/integraldisplay\nldω=/integraldisplay\nl◦φd/tildewideDµ=/integraldisplay\n/tildewideD(l◦φ)dµ\n=/integraldisplay\nK/bardblφ(x)/bardbll∗/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig\ndµ(x)≥/integraldisplay\nK/bardblφ(x)/bardbll/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig\ndµ(x)\n=/integraldisplay\nldν=/integraldisplay\ninvkdν=/integraldisplay\nkdinvν.\nIndeed, the firstthreeequalitiesfollowfromdefinitions. Thefourt honefollowsfrom\nthe proof of ( a) (note that lis convex, continuous and F-invariant, so /tildewideD(l◦φ)(x) =\n/bardblφ(x)/bardblD0l(x) and we may use (7.1)). The inequality is obvious as l∗≥land the\nremaining equalities follow from definitions.\nThus by [17, Proposition 3.89] there exists a measure Λ ∈M1(M), where\nM={(εs,λ)∈M1(BH∗)×M1(BH∗);s=r(λ)} ⊂M1(BH∗)×M1(BH∗),\nsuch that the barycenter of Λ is (inv ν,invω). Then for any pair ( f1,f2) of bounded\nuniversally measurable functions on BH∗we have (by [17, Proposition 3.90])\n/integraldisplay\nf1dinvν+/integraldisplay\nf2dinvω=/integraldisplay\nM(εs(f1)+λ(f2))dΛ(εs,λ).\nLet nowlbe anSF-invariant convex continuous function on BH∗. Thenl∗is\nalsoSF-invariant. Then we compute using the last formula for the pairs ( l∗,0) and\n(0,l∗)\n/integraldisplay\nldinvω=/integraldisplay\nK/bardblφ(x)/bardbll∗/parenleftBig\nφ(x)\n/bardblφ(x)/bardbl/parenrightBig\ndµ(x) =/integraldisplay\nl∗dν=/integraldisplay\nl∗dinvν\n=/integraldisplay\nMεs(l∗)dΛ(εs,λ)≥/integraldisplay\nMλ(l∗)dΛ(εs,λ) =/integraldisplay\nl∗dinvω\n≥/integraldisplay\nldinvω.52 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nHere the first equality follows from the above computation (in the pr oof of invν≺\ninvω). The second one follows from (7.3) (note that l∗is Borel). The third equality\nfollows by (7.2). The next equality follows from the choice of Λ. The fo llowing\ninequality follows using the fact that l∗is concave and upper semicontinuous and\ns=r(λ) for (εs,λ)∈M(cf. [17, Proposition 4.7]). The next equality follows again\nfrom the choice of Λ and the last inequality is obvious.\nWe deduce that/integraltext\nldinvω=/integraltext\nl∗dinvω. By (7.2) we conclude that/integraltext\nldω=/integraltextl∗dωSincelwas an arbitrary F-invariant convex continuous function, ωis a\nmaximal measure on BH∗(by Fact 2.1). Hence /tildewideDµisH-boundary.\nBy (c) we know that Dµis absolutely continuous with respect to /tildewideD|µ|, so it\nisH-boundary as well. The remaining part of ( d) and assertions ( e) and (f) are\ncompletely analogous to the corresponding assertions from Theor em 7.6 and may\nbe proved in the same way.\n(g): IfDµ=µor/tildewideDµ=µ, thenµisH-boundary by ( d). Conversely, assume\nthatµisH-boundary. To prove that Dµ=/tildewideDµ=µit is enough to show that\nDf=/tildewideDf=f|µ|-a.e. for each f∈C(K). Due to the formulas established in the\nproof of part ( i) of (a) it is enough to observe that\n∀F∈C(BH∗,F):/bardblφ(x)/bardblD0F(x) =F(φ(x)) for|µ|-almost all x∈K.\nBy the Stone-Weierstrass theorem it is enough to prove it for Fconvex continuous.\nDue to (7.1) this is equivalent to say that for any Fconvex continuous\n/bardblφ(x)/bardblF∗(φ(x)\n/bardblφ(x)/bardbl) =F(φ(x)) for|µ|-almost all x∈K.\nBy Lemma 4.4 we know that /bardblφ(x)/bardbl= 1|µ|-a.e. and by Fact 2.1 we have F◦φ=\nF∗◦φ|µ|-a.e., hence the assertion follows. /square\nTheorems 7.6 and 7.8 provide partial extension of Fact 7.3 to spaces without\nconstants. However, they donot providea complete analogue. In fact, the complete\nanalogue fails as witnessed by the following example.\nExample 7.9. Under the continuum hypothesis there is a complex simplicia l func-\ntion spaceHon a compact space Ksuch that the function D1is not Borel.\nProof.LetKandHbe as in Example 6.17. Using the notation from this example\nand its proof, we have (by Step 5 of the proof)\nδ(t,s,0)=sg(t)\n2(ε(t,g(t),−1)+ε(t,g(t),1)) fors,t∈T.\nThereforeD1(t,s,0) =sg(t) fors,t∈T. In particular, D1(t,1,0) =g(t) fort∈T\nwhich is not aBorel function. (Otherwise, gwould be a Borelfunction, so its graph,\nbeing an uncountable Borel subset of T×T, would contain a Cantor set (by [13,\nTheorem 13.6]), in contradiction with Lemma 6.18). /square\nAnyway, the following question remains open.\nQuestion 7.10. LetHbe a simplicial function space not containing constants on\na (non-metrizable) compact space K. Letf:K→Fbe continuous.\n•Is the function /tildewideDfBorel?\n•AssumeF=R. Is the function DfBorel? Is it H-affine?SIMPLICIALITY WITHOUT CONSTANTS 53\n8.Final remarks\nLet us now overview common points and differences of the classical f unction\nspaces and spaces without constants. We further point out some open problems.\nAlready the representation theorems of Choquet-Bishop-de Lee uw type reveal\nnontrivial differences. The classical version says that, given a rea l function space H\non a compact Kwhich contains constants, for each x∈Kthere is an H-maximal\nmeasureµ∈Mx(H) and, moreover, all H-maximal measures are pseudosupported\nby the Choquet boundary (i.e., supported by any Fσ-set containing Ch HK). The\nsame method easily yields that for any ϕ∈H∗there is an H-boundary measure\ninMϕ(H). An important feature in this case is the fact that H-maximal measures\ndefined using H-convex functions (cf. [17, Definition 3.57]) coincide with the H-\nmaximal measures defined using the evaluation mapping φ(by [17, Proposition\n4.28(d)]).\nAn extension to the complex case was initiated by [10] by proving tha t, given\na complex function space Hon a compact Kwhich contains constants, for each\nϕ∈H∗there is a measure µ∈Mϕ(H) which is pseudosupported by the Choquet\nboundary. In [9] it was observed that the proof of [10] in fact pro duces anH-\nboundary measure in Mϕ(H).\nThe representation theorem for (complex) spaces without const ants which we\nreproduce in Proposition 3.5 was established in [8]. It states that fo r anyϕ∈H∗\nthere is some H-boundary measure in Mϕ(H). In [8] the question whether such a\nmeasuremust be pseudosupportedbythe Choquetboundaryisno t addressedat all.\nInfact, in this case H-boundarymeasuresmust be defined usingthe evaluationmap\nφ(as we do above)since we do not haveany analogueof H-convexfunctions. More-\nover, maximal measures need not be pseudosupported by the Cho quet boundary\n(see Examples 6.14, 6.15 or 6.16). And even, in the complex case, it ma y hap-\npen (under the continuum hypothesis) no measure in Mϕ(H) (or even in Mx(H))\nis pseudosupported by the Choquet boundary (see Examples 6.17 a nd 6.19). We\ndo not know whether the continuum hypothesis (or some other add itional axiom)\nis necessary. It is also not clear whether a similar example may be cons tructed\nin the real case. On the other hand, such an example requires non- uniqueness of\nrepresenting H-boundary measures (see Corollary 3.16).\nSimpliciality of real function spaces with constants is a well-understo od feature\n(see, e.g., [17, Chapter 6]). The complex theory is essentially the sam e (see Propo-\nsition 5.4) since it is enough to look at the self-adjoint space Ac(H). Simpliciality of\nspaces without constants shares some properties with the classic al case, in particu-\nlar, it is again enough to look at Ac(H) (see Proposition 4.5). But there are many\ndifferences. In particular, the space Ac(H) need not be self-adjoint and, moreover,\nthe conditions which are equivalent in the classical case become differ ent (compare\nTheorem 6.1 with Proposition 5.4). In particular, for function space s containing\nconstantsthe simpliciality is an isometric Banach-spacepropertyof Ac(H). Indeed,\nHis simplicial if and only if Ac(H) is anL1-predual, and also if and only if BAc(H)∗\nis a simplexoid. For spaces without constants not only the mentioned notions differ,\nbut simpliciality is not a Banach-space property of Ac(H). This is witnessed by\nthe counterexample to implication ( VI) =⇒(I) from Theorem 6.1. Another (less\ntrivial) example is the non-simplicial space Hfrom Example 6.15 which satisfies\nAc(H) =Hand is isometric to a simplicial space H′satisfyingAc(H′) =H′. In\nfact, neither functional simpliciality is an isometric property. Indee d, letHbe the54 OND ˇREJ F.K. KALENDA AND JI ˇR´I SPURN ´Y\nfunction space from Example 6.14. Then Ac(H) =HandHis not functionally\nsimplicial, but it is isometric to HL,Awhich is functionally simplicial (by Proposi-\ntion 6.13 and Proposition 5.4). The seemingly strange behavior of the se examples\nis related to the fact that the respective mapping θneed not be one-to-one. If θis\none-to-one,thesituation isabit simpler. Inthis caseTheorem6.1sa ysthat wehave\n(for spaces satisfying H=Ac(H)) ‘three levels of simpliciality’ – the simpliciality\nitself (condition ( II)), the functional simpliciality (condition ( III)) and condition\n(IV) (trivial intersection of H⊥withH-boundary measures) – and the two ‘higher\nlevels’ are Banach-space properties of H(conditions ( V) and (VI), respectively).\nHowever, the first level – mere simpliciality – is not an isometric proper ty even in\nthis case. It is witnessed the following example which may be easily deriv ed from\nProposition 6.5 (in the same way as Example 6.15 is derived from Example 6.14):\nExample 8.1. LetK1andH1be as in Section 6.1. Assume α=β. SetK′=\nK1⊕{p}, wherepis a new isolated point and\nH′={f∈C(K,F);f|K1∈H1andf(p) =−αf(b)+1\n2f(0,1)}.\nThenAc(H′) =H′,H′is not simplicial and H′is linearly isometric to H1.\nA further difference of the classical case and the spaces without c onstants con-\ncernsthe abstractDirichletproblem. Firstly, insteadofonemappin gDwehavetwo\nnatural mappings – Dand/tildewideD(see Section 7). Secondly, even D1 may be non-Borel\n(in the complex case under the continuum hypothesis, see Example 7 .9). Results\nsimilar to the classical case hold in two cases – if the underlying compac t space is\nmetrizable or if the mapping θis one-to-one (see Theorems 7.6 and 7.8).\nSummarizing the results, we see that there are two distinguished cla sses of func-\ntion spaces without constants which have somewhat similar behaviou r to the classi-\ncal case– spaceson metrizable compact spaces and spaces with on e-to-onemapping\nθ. So, it seems to be natural and interesting to try to find a common r oof for these\ntwo classes. Another natural task is to analyze in more detail real function spaces\nwithoutconstants. Thereasonisthatthe resultspresentedin th is paperareparallel\nforthe realandthe complexversions,but themethods areessen tiallycomplex. Fur-\nther, there are some counterexamples (Example 6.17 and the deriv ed ones) which\nwork only in the complex setting and this seems to be important.\nAcknowledgment\nWe are grateful to Witold Marciszewski and Grzegorz Plebanek for indicating\nus in [18] an idea which helped us to find Example 6.17.\nReferences\n[1]Alfsen, E. Compact convex sets and boundary integrals . Springer-Verlag, New York, 1971.\nErgebnisse der Mathematik und ihrer Grenzgebiete, Band 57.\n[2]Bliedtner, J., and Hansen, W. Potential theory . Universitext. Springer-Verlag, Berlin,\n1986. An analytic and probabilistic approach to balayage.\n[3]Effros, E. On a class of complex Banach spaces. Illinois J. Math. 18 (1974), 48–59.\n[4]Effros, E. G., and Kazdan, J. L. Applications of Choquet simplexes to elliptic and para-\nbolic boundary value problems. J. Differential Equations 8 (1970), 95–134.\n[5]Engelking, R. General topology . PWN—Polish Scientific Publishers, Warsaw, 1977. Trans-\nlated from the Polish by the author, Monografie Matematyczne , Tom 60. [Mathematical\nMonographs, Vol. 60].SIMPLICIALITY WITHOUT CONSTANTS 55\n[6]Fonf, V. P., Lindenstrauss, J., and Phelps, R. R. Infinite dimensional convexity. In Hand-\nbook of the geometry of Banach spaces, Vol. I .North-Holland, Amsterdam, 2001, pp.599–670.\n[7]Fremlin, D. H. Measure theory. Vol. 4 . Torres Fremlin, Colchester, 2006. Topological mea-\nsure spaces. Part I, II, Corrected second printing of the 200 3 original.\n[8]Fuhr, R., and Phelps, R. R. Uniqueness of complex representing measures on the Choquet\nboundary. J. Functional Analysis 14 (1973), 1–27.\n[9]Hirsberg, B. Repr´ esentations int´ egrales des formes lin´ eaires compl exes.C. R. Acad. Sci.\nParis S´ er. A-B 274 (1972), A1222–A1224.\n[10]Hustad, O. A norm preserving complex Choquet theorem. Math. Scand. 29 (1971), 272–278.\n[11]Kalenda, O. F. K., Rondo ˇs, J., and Spurn ´y, J.Boundary integral representation\nof multipliers of fragmented affine functions and other inter mediate function spaces.\narXiv:2305.16920.\n[12]Kalenda, O. F. K., and Spurn ´y, J.Baire classes of affine vector-valued functions. Studia\nMath. 233 , 3 (2016), 227–277.\n[13]Kechris, A. S. Classical descriptive set theory , vol. 156 of Graduate Texts in Mathematics .\nSpringer-Verlag, New York, 1995.\n[14]Lacey, H. The isometric theory of classical Banach spaces . Springer-Verlag, New York, 1974.\nDie Grundlehren der mathematischen Wissenschaften, Band 2 08.\n[15]Lazar, A. The unit ball in conjugate L1spaces.Duke Mathematical Journal 39 , 1 (1972),\n1–8.\n[16]Lukeˇs, J., Mal ´y, J., Netuka, I., Smr ˇcka, M., and Spurn ´y, J.On approximation of affine\nBaire-one functions. Israel J. Math. 134 (2003), 255–287.\n[17]Lukeˇs, J., Mal ´y, J., Netuka, I., and Spurn ´y, J.Integral representation theory , vol. 35 of\nde Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, 2010. Applications to\nconvexity, Banach spaces and potential theory.\n[18]Marciszewski, W., and Plebanek, G. personal communication, January 2024.\n[19]Phelps, R. R. The Choquet representation in the complex case. Bull. Amer. Math. Soc. 83 ,\n3 (1977), 299–312.\n[20]Phelps, R. R. Lectures on Choquet’s theorem , second ed., vol. 1757 of Lecture Notes in\nMathematics . Springer-Verlag, Berlin, 2001.\n[21]Poˇsta, P. Dirichlet problem and subclasses of Baire-one functions. Israel J. Math. 226 , 1\n(2018), 177–188.\n[22]Rondoˇs, J., and Spurn ´y, J.The Dirichlet problem on compact convex sets. J. Funct. Anal.\n281, 12 (2021), Paper No. 109251, 20.\n[23]Spurn´y, J.On the Dirichlet problem of extreme points for non-continuo us functions. Israel\nJ. Math. 173 (2009), 403–419.\n[24]Stacey, P. J. Choquet simplices with prescribed extreme and ˇSilov boundaries. Quart. J.\nMath. Oxford Ser. (2) 30 , 120 (1979), 469–482.\nOndˇrej F.K. Kalenda, Charles University, Faculty of Mathematic s and Physics, De-\npartment of Mathematical Analysis, Sokolovsk ´a 83, 186 75, Praha 8, Czech Republic\nEmail address :kalenda@karlin.mff.cuni.cz\nJiˇr´ı Spurn ´y, Charles University, Faculty of Mathematics and Physics, D epartment\nof Mathematical Analysis, Sokolovsk ´a 83, 186 75, Praha 8, Czech Republic\nEmail address :spurny@karlin.mff.cuni.cz" }, { "title": "2401.16118v1.Temperature_dependent_local_structure_and_lattice_dynamics_of_1T_TiSe__2__and_1T_VSe__2__probed_by_X_ray_absorption_spectroscopy.pdf", "content": "Temperature-dependent local structure and lattice\ndynamics of 1T-TiSe 2and 1T-VSe 2probed by X-ray\nabsorption spectroscopy\nInga Pudzaa, Boris Polyakova, Kaspars Pudzsa, Edmund Welterb, Alexei\nKuzmina,∗\naInstitute of Solid State Physics, University of Latvia, Kengaraga street 8, LV-1063 Riga,\nLatvia\nbDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany\nAbstract\nThe local atomic structure and lattice dynamics of two isostructural lay-\nered transition metal dichalcogenides (TMDs), 1T-TiSe 2and 1T-VSe 2, were\nstudied using temperature-dependent X-ray absorption spectroscopy at the\nTi, V, and Se K-edges. Analysis of the extended X-ray absorption fine struc-\nture (EXAFS) spectra, employing reverse Monte Carlo (RMC) simulations,\nenabled tracking the temperature evolution of the local environment in the\nrange of 10-300 K. The atomic coordinates derived from the final atomic\nconfigurations were used to calculate the partial radial distribution functions\n(RDFs) and the mean-square relative displacement (MSRD) factors for the\nfirst ten coordination shells around the absorbing atoms. Characteristic Ein-\nstein frequencies and effective force constants were determined for Ti–Se,\n∗Corresponding author\nEmail addresses: inga.pudza@cfi.lu.lv (Inga Pudza),\nboris.polyakov@cfi.lu.lv (Boris Polyakov), kaspars.pudzs@cfi.lu.lv (Kaspars\nPudzs), edmund.welter@desy.de (Edmund Welter), a.kuzmin@cfi.lu.lv (Alexei\nKuzmin)\nPreprint submitted to Physica B: Condensed Matter January 30, 2024arXiv:2401.16118v1 [cond-mat.mtrl-sci] 29 Jan 2024Ti–Ti, V–Se, V–V, and Se–Se atom pairs from the temperature dependen-\ncies of MSRDs. The obtained results reveal differences in the temperature\nevolution of lattice dynamics and the strengths of intralayer and interlayer\ninteractions in TiSe 2and VSe 2.\nKeywords: Transition metal dichalcogenides; Interlayer and intralayer\ncoupling; Extended X-ray absorption fine structure; Reverse Monte Carlo\nsimulations; Effective force constants\n21. Introduction\nTransition metal dichalcogenides (TMDs) have garnered significant at-\ntention in the scientific community due to their layered structure, result-\ning in remarkable electronic, optical, mechanical, tribological, catalytic, and\nmagnetic properties as well as numerous possible technological applications\n[1, 2, 3, 4]. Among all, the charge density wave (CDW) behavior observed\nfor a number of TMDs MX 2(M = Ti, V, Ta; X= S, Se) is particularly\nfascinating [5, 6, 7]. Indeed, TMDs represent the first layered materials in\nwhich the presence of CDWs was detected [8]. The CDW state involves the\nperiodic modulation of the electron density within the material coupled with\nlattice distortion, leading to changes in the material electronic and optical\nproperties [9, 10]. Models describing CDW mechanisms are based solely on\ncrystal structure distortion/superlattice formation, exciton–phonon interac-\ntions, or electron–phonon interactions [11, 12]. Due to the minute nature\nof distortion or atomic shifts in the CDW state ( ∼0.1-0.15 ˚A), the resulting\nsuperstructure might generate only relatively faint secondary peaks within\nthe X-ray diffraction patterns [10].\nIn this paper, two isostructural TMDs, namely titanium selenide (TiSe 2)\nand vanadium selenides (VSe 2), were selected to evaluate the effect of the\nCDW ordering on their local environments. The choice of TMDs was dictated\nby the feasibility of measuring high-quality X-ray absorption spectra at both\nthe metal and chalcogenide K absorption edges.\nUnder ambient conditions, both TiSe 2and VSe 2crystallize in the 1T\npolytype with a space group of P¯3m1 (164) (Fig. 1) [13, 14]. In this struc-\nture, the Ti(V) atoms are octahedrally coordinated to six covalently bonded\n3Ti/V\nSeFigure 1: Crystallographic structure of trigonal (space group P¯3m1 (164)) TiSe 2(VSe 2)\n[13, 14].\nSe atoms, and [Ti(V)Se 6] octahedra are connected by edges into layers, joined\nby weak van der Waals (vdW) interactions. The interactions between neigh-\nbouring layers play an important role in assembling and exfoliation of vdW\n(hetero)structures and, thus, allow engineering of their electronic, optical,\nand mechanical properties [15, 16, 17]. While the importance of weak inter-\nlayer coupling is recognized, its accurate experimental quantification remains\nchallenging [16, 18].\nA commensurate CDW phase in bulk 1T-TiSe 2forms below TCDW≈202 K\nwith a 2 a×2a×2csuperlattice [19, 11]. The neutron diffraction study pro-\nposed the low-temperature (at 77 K) lattice distortion due to displacements of\nabout 0.085 ˚A for Ti and 0.028 ˚A for Se parallel to the plane of the layer [19].\n4However, more recent synchrotron X-ray scattering experiments pointed to\na zone-boundary phonon softening mechanism of CDW in 1T-TiSe 2[20, 21].\n1T-VSe 2undergoes an incommensurate CDW transition at TCDW≈110 K\nand a commensurate CDW transition at ≈80 K, forming a 4 a×4a×3csu-\nperlattice [22, 23]. The origin of the CDW in VSe 2was attributed to the\nelectron-phonon interaction, based on inelastic X-ray scattering and first-\nprinciples calculations [24]. Besides, an important role of vdW forces in the\nCDW melting was proposed, likely caused by the out-of-plane nature of the\nCDW, which modulates interlayer distance [24]. Recently, a pressure-induced\nCDW state was observed in VSe 2at room temperature within the pressure\nrange of 10-15 GPa by Raman spectroscopy [25] and X-ray diffraction (XRD)\n[10]. The Se K-edge X-ray absorption spectroscopy (XAS) was also used in\n[10] to probe local distortions but only qualitative analysis was reported.\nIn this study, we employed XAS combined with the advanced data analy-\nsis methodology of extended X-ray absorption fine structure (EXAFS), based\non reverse Monte Carlo (RMC) simulations coupled with an evolutionary al-\ngorithm (EA) approach [26, 27], to investigate the local atomic structure and\nlattice dynamics in bulk 1T-TiSe 2and 1T-VSe 2. We have demonstrated re-\ncently [18] that employing this approach for 2H c-MoS 2enables the retrieval\nof structural data from distant coordination shells and, thus, provides valu-\nable information on both interlayer and intralayer coupling. Here we took\nadvantage of high-quality EXAFS spectra measured at two (Ti/V and Se) K\nabsorption edges and performed their simultaneous analysis to reconstruct\nthe temperature evolution of the local environment in the corresponding\nTMDs in the range of 10-300 K. This allowed us to determine the amplitudes\n5of thermal vibrations for atoms located in the first ten coordination shells\naround absorbing atoms, which were further used to evaluate the strengths of\ninterlayer and intralayer interactions. We showed that despite the isostruc-\ntural nature of TiSe 2and VSe 2, the temperature-dependent evolution of their\nlattice dynamics has some peculiarities.\n2. Materials and methods\n2.1. Synthesis procedure\nTiSe 2and VSe 2were prepared using the chemical vapor transport tech-\nnique with an I 2transport agent [28, 29]. The starting materials titanium\n(or vanadium) and selenium in the form of powder were weighed out with\na molar ratio of M:Se = 1:2 (M = Ti or V) and loaded into a quartz am-\npoule together with the iodine (molar ratio I 2:M = 0.05). The filled ampoule\nwas evacuated to pressure 10−5torr and sealed at a length of approximately\n12-14 cm using the oxygen-methane flame. For TiSe 2synthesis the ampoule\nwas heated in a two-zone furnace with Thot= 700◦C at the hot zone and\nTcold= 600◦C at the cold zone for 24 h, then naturally cooled down (for\nVSe 2Thot= 820◦C,Tcoldt= 650◦C, heating time 72 h).\n2.2. X-ray diffraction\nThe phase purity of the TMD samples was confirmed using X-ray powder\ndiffraction (XRD). Diffraction patterns were collected at room temperature\nusing a benchtop Rigaku MiniFlex 600 diffractometer with Bragg-Brentano\nθ-2θgeometry. An X-ray tube with copper anode (Cu K αradiation, λ=\n1.5418 ˚A), operated at U= 40 kV and I= 15 mA, was used as a source.\n620 40 60 80PDF 04-003-1758XRD intensity (a.u.)\n \nAngle 2 (degree)TiSe2\nP-3m1(001)(002)\n(011)\n(102)\n(003)(110)(103)(004)\n20 40 60 80PDF 01-074-1411XRD intensity (a.u.)\n \nAngle 2 (degree)VSe2\nP-3m1(001)(002)\n(011)(102)\n(003)(103)(004)Fig 2Figure 2: Powder X-ray diffraction patterns of synthesized 1T-TiSe 2and 1T-VSe 2. Main\nBragg peaks are indexed to the P¯3m1 (164) space group. Reference patterns corresponding\nto TiSe 2(PDF Card 04-003-1758) and VSe 2(PDF Card 01-074-1411) phases are shown\nfor comparison.\nThe obtained X-ray diffraction patterns of TiSe 2and VSe 2are compared\nin Fig. 2 with the reference ones. All Bragg peaks were indexed to pure\nTiSe 2(PDF Card 04-003-1758) and VSe 2(PDF Card 01-074-1411) phases\npossessing P¯3m1 (164) space group. Slight variations in peak intensities are\nnoticeable, likely attributed to the favored alignment of layered structures\nthat could arise during the process of sample preparation. No impurity phases\nwere detected, demonstrating the single-phase composition of the synthesized\nsamples.\n72.3. X-ray absorption experiments\nThe temperature-dependent (10-300 K) X-ray absorption spectra of bulk\n1T-TiSe 2and 1T-VSe 2were recorded at Ti (4966 eV), V (5465 eV) and Se\n(12658 eV) K-edges in transmission mode at the DESY PETRA III P65\nApplied XAFS undulator beamline [30]. The storage ring operated in top-up\n480 bunch mode at the energy E= 6.08 GeV and current I= 100 mA. The\nsynchrotron radiation was monochromatized using a Si(111) double-crystal\nmonochromator, and its intensity before and after the sample was measured\nby two ionization chambers. The harmonic rejection was achieved by the\nuncoated silicon plane mirror.\nExperimental Ti, V and Se K-edge EXAFS spectra were extracted follow-\ning the conventional procedure [31] using the XAESA code [32]. The EXAFS\nspectra χ(k)k2and their Fourier transforms (FTs) are shown at selected tem-\nperatures in Fig. 3. Note that the FTs were not adjusted to compensate for\nthe backscattering phase shift of atoms, therefore, the positions of all peaks\nappear shifted towards shorter distances compared to their crystallographic\nvalues.\n2.4. Reverse Monte Carlo simulations\nThe structural information encoded in the experimental EXAFS spec-\ntra was extracted using the reverse Monte Carlo (RMC) method based on\nan evolutionary algorithm (EA) approach implemented in the EvAX code\n[26, 27]. The RMC/EA-EXAFS technique aims to minimize the discrepancy\nbetween the experimental and calculated EXAFS spectra. As a result, it\nenables the reconstruction of the three-dimensional structural model of the\n8Fig 3\n468 1 0 1 2 1 4-4-3-2-101234\n 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KVSe2\nV K-edgeEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)468 1 0 1 2 1 4 1 6-4-3-2-101234\nVSe2\nSe K-edgeEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1) 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K\n0123456780.00.51.01.52.0VSe2 V K-edgeFT modulus (Å−3)\nDistance R (Å) 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K\n0123456780.00.20.40.60.81.0VSe2 Se K-edgeFT modulus (Å-3)\nDistance R (Å) 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K468 1 0 1 2-4-3-2-101234\n 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KTiSe2\nTi K-edgeEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)4 6 8 1 01 21 41 6-4-3-2-101234\n 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KTiSe2\nSe K-edgeEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)\n123456780.00.20.40.60.81.01.21.41.61.82.0TiSe2\n 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KTi K-edgeFT modulus (Å−3)\nDistance R (Å)123456780.00.20.40.60.81.0TiSe2 Se K-edgeFT modulus (Å−3)\nDistance R (Å) 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K\n(a) (b)\nTi-SeTi-Ti\nSe-TiSe-Se\nV-SeV-V\nSe-VSe-SeFigure 3: Temperature-dependent EXAFS spectra χ(k)k2(top panels), their Fourier\ntransforms (only modulus is shown) (middle panels) and wavelet transforms at 10 K (bot-\ntom panels) for TiSe 2(a) and VSe 2(b).\nmaterial by introducing random atomic displacements within specific geo-\nmetric constraints [26, 27].\nIn this study, the initial structural models for the RMC/EA calculations\nwere constructed based on diffraction data [13, 14]. TiSe 2and VSe 2crys-\ntallize in the space group P¯3m1 (164) with the lattice parameters a=b\n= 3.540 ˚A,c= 6.008 ˚A for TiSe 2[13] and a=b= 3.355 ˚A,c= 6.134 ˚A\nfor VSe 2[14]. A supercell with a size of 6 a×6b×4c, including 32 atoms,\nand periodic boundary conditions was constructed for both compounds. At\neach iteration, all atoms in the supercell were displaced with the maximum\nallowed displacement set to 0.4 ˚A. 32 atomic configurations were employed\n9simultaneously in the EA method [27].\nThe configuration-averaged EXAFS spectra at the Ti/V and Se K-edges\nwere calculated using the ab initio self-consistent real-space multiple-scattering\nFEFF8.5L code [33, 34] taking into account multiple-scattering contributions\nup to the 5thorder. The complex energy-dependent exchange-correlation\nHedin-Lundqvist potential was employed to account for inelastic effects [35].\nThe amplitude scaling parameter S2\n0was set to 0.9 (for Ti K-edge) or 1.0\n(for V and Se K-edges).\nThe structural model was adjusted in each RMC/EA calculation by min-\nimizing the difference between the Morlet wavelet transforms (WTs) [36] of\nthe experimental and calculated EXAFS spectra at two absorption edges (Ti\nand Se for TiSe 2or V and Se for VSe 2) simultaneously. Thus, good agree-\nment between the experimental and configuration-averaged EXAFS spectra\nwas achieved in both the direct ( R) and reciprocal ( k) space.\nAn example of fits at the selected temperatures is shown in Fig. 4. The\nconvergence of each RMC/EA simulation was attained after 3000 iterations.\nTo improve statistics, seven independent RMC/EA calculations were carried\nout for each experimental data set, employing distinct sequences of pseudo-\nrandom numbers. The agreement between the configuration-averaged EX-\nAFS spectra and the experimental data at all temperatures affirms the reli-\nability of the obtained structural models.\nThe atomic coordinates derived from the final RMC/EA configurations\nwere utilized to calculate the partial radial distribution functions (RDFs)\nand to obtain relevant structural parameters. The mean-square relative dis-\nplacement (MSRD) factors σ2for Ti–Se, Ti–Ti, V–Se, V–V, and Se–Se atom\n10Fig 4\n2468 1 0 1 2 1 4 1 6-4-20246810121416VSe2\nV K-edge\n Experiment\n RMC fitT = 300 KT = 150 KT = 10 KEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)\n0123456780123456\nT = 150 K\nT = 300 KT = 10 KVSe2\nV K-edge Experiment\n RMC fitFT modulus (Å−3)\nDistance R (Å)2468 1 0 1 2 1 4 1 6-4-20246810121416VSe2\nSe K-edge\n Experiment\n RMC fitT = 300 KT = 150 KT = 10 KEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)\n0123456780123\nT = 150 K\nT = 300 KT = 10 KVSe2\nSe K-edge Experiment\n RMC fitFT modulus (Å−3)\nDistance R (Å)2468 1 0 1 2 1 4 1 6-4-20246810121416TiSe2\nTi K-edge\n Experiment\n RMC fitT = 300 KT = 150 KT = 10 KEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)\n0123456780123456\nT = 150 K\nT = 300 KT = 10 KTiSe2\nTi K-edge Experiment\n RMC fitFT modulus (Å−3)\nDistance R (Å)2468 1 0 1 2 1 4 1 6-4-20246810121416TiSe2\nSe K-edge\n Experiment\n RMC fitT = 300 KT = 150 KT = 10 KEXAFS (k)k2 (Å−2)\nWavenumber k (Å−1)\n0123456780123\nT = 150 K\nT = 300 KT = 10 KTiSe2\nSe K-edge Experiment\n RMC fitFT modulus (Å−3)\nDistance R (Å)\n(a) (b)Figure 4: Results of the RMC/EA calculations for the Ti and Se K-edges in TiSe 2(a) and\nV and Se K-edges in VSe 2(b) at selected (10, 150 and 300 K) temperatures. The EXAFS\nspectra χ(k)k2are displayed within their respective fitting ranges. The fitting R-space\nranges were 1.5–5.9 ˚A (Ti K-edge) and 1.2–7.1 ˚A (Se K-edge) for TiSe 2and 1.0–7.0 ˚A (V\nK-edge) and 1.2–7.5 ˚A (Se K-edge) for VSe 2.\npairs, which account for the thermal and static disorder, were calculated\nusing the median absolute deviation (MAD) method for the first ten coordi-\nnation shells [37, 38]. The temperature dependencies of the obtained MSRDs\nwere further approximated by the correlated Einstein model [39] allowing us\nto determine the characteristic Einstein frequencies ωEand the effective force\nconstants κ, which are reported in Tables 1 and 2.\n3. Results and discussion\nThe crystallographic structure of TiSe 2(VSe 2) (Fig. 1) consists of a ti-\ntanium (vanadium) atomic layer sandwiched between two selenium atomic\nlayers [13, 14]. These layers are weakly bonded to each other along the c-axis\nthrough vdW forces. The interlayer gap between Se atoms is about 3.12 ˚A\n11in TiSe 2and 3.16 ˚A in VSe 2. Ti(V) atoms are covalently bonded to six se-\nlenium atoms, forming regular octahedra. The values of Ti–Se, Ti–Ti, and\nSe–Se interatomic distances for the first ten coordination shells in TiSe 2are\nreported in Table 1, while the values of V–Se, V–V, and Se–Se interatomic\ndistances in VSe 2are summarized in Table 2.\nDespite the similarity between the two crystallographic structures, a no-\ntable difference can be observed as well. In the case of VSe 2, the interatomic\ndistances between selenium atoms along the c-axis direction within the same\nlayer and across adjacent layers coincide ( rSe0−Se∗∗\n3/4= 3.63 ˚A), adding com-\nplexity to the analysis. In contrast, in TiSe 2, the Se∗\n3atoms in the adjacent\nlayer are approximately 0.1 ˚A closer to the absorbing Se 0than the Se 4atoms\nwithin the same layer, so that the two interatomic distances are different\nrSe0−Se∗\n3= 3.58 ˚A and rSe0−Se4= 3.68 ˚A.\nThe Ti K-edge EXAFS spectra of TiSe 2and their FTs are dominated\nby a contribution from the first two coordination shells at all studied tem-\nperatures in the range of 10-300 K (Fig. 3(a)). These coordination shells\nconsist of six selenium and six titanium atoms, respectively. The complex\npattern of WTs observed at longer distances is attributed to outer shells and\nmultiple-scattering effects. Note that heavier Se atoms with an atomic mass\nof 78.971 amu produce a greater impact on the EXAFS spectra at larger\nkvalues, whereas lighter Ti elements with an atomic mass of 47.867 amu\ncontribute at lower k-values.\nThe Se K-edge EXAFS spectra of TiSe 2contain contributions from scat-\ntering paths that extend beyond ∼7˚A as evident in the FTs and WTs in Fig.\n3(a). Their analysis can yield novel insights into the interactions between lay-\n12ers [18]. Indeed, the second peak at about 3.2 ˚A in FTs (indicated with a red\narrow in Fig. 3(a)) contains contributions from intralayer ( rSe0−Se2= 3.54 ˚A,\nrSe0−Se4= 3.68 ˚A) and interlayer ( rSe0−Se∗\n3= 3.58 ˚A) selenium atoms, and its\namplitude becomes significantly suppressed upon sample heating. Note that\nthe amplitude of other peaks up to ∼7˚A is also reduced upon increasing\nthermal disorder but the peaks remain distinguishable even at 300 K.\nWhile the reliable analysis of distant coordination shells is challenging\nwithin the conventional EXAFS methodology [31], it can be performed using\nthe RMC/EA method [18, 27, 40]. Examples of the RMC/EA simulations\nof the Ti and Se K-edge EXAFS spectra for TiSe 2at three selected temper-\natures (10, 150 and 300 K) are shown in Fig. 4 in k- and R-space. Good\nagreement between experimental and calculated EXAFS spectra is observed,\nenabling a comprehensive analysis of thermal disorder effects within the first\nten coordination shells up to about 6 ˚A.\nThe coordinates of atoms derived from the RMC/EA simulations were\nemployed to calculate the partial partial RDFs g(r) (Fig. 5) and the MSRDs\nσ2(Fig. 6) for Ti–Ti, Ti–Se(Se–Ti), and Se–Se atom pairs at each temper-\nature. The temperature dependencies of the obtained MSRDs σ2(T) in the\nrange of 10-300 K were further approximated using the correlated Einstein\nmodel [39]. As a result, the characteristic Einstein frequencies ωEand the\neffective force constants κwere obtained for all coordination shells and are\nreported in Table 1.\nThe effective force constants κfor the first two coordination shells of\ntitanium (Ti 0–Se 1and Ti 0–Ti 2) are about 52 N/m and 39 N/m, respectively,\nsuggesting strong interaction between the nearest atoms. The interactions\n13123456780102030405060\n V-Se10*\nV-V6\n 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K\nV-V8*V-Se7*V-Se5\nV-Se4*V-V9\nV-Se3\nV-V2RDF g(r) (atoms/Å )\nDistance r (Å)V-Se1VSe2\n1234567801020304050607080\nSe-Se11\nSe-V9\nSe-Se6Se-Se7*Se-Se3\nSe-V8\nSe-V5Se-V1 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KVSe2\nSe-Se4*Se-Se10\nSe-Se2RDF g(r) (atoms/Å )\nDistance r (Å)123456780102030405060\nTi-Se10*\nTi-Se7*Ti-Ti6*\nTi-Se4* \n Ti-Se3\nTi-Ti8Ti-Se5Ti-Ti9*\nTi-Ti2TiSe2\nTi-Se1 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 KRDF g(r) (atoms/Å )\nDistance r (Å)\n1234567801020304050607080\nSe-Se10*\nSe-Se7*\nSe-Ti6*Se-Se4Se-Se3*\nSe-Ti9\nSe-Ti5Se-Se8Se-Se2 10 K\n 50 K\n 100 K\n 150 K\n 200 K\n 250 K\n 300 K\nSe-Ti1TiSe2RDF g(r) (atoms/Å )\nDistance r (Å)\n(a) (b)Fig 5Figure 5: Partial radial distribution functions (RDFs) around Ti and Se atoms in TiSe 2\n(a) and V and Se atoms in VSe 2(b) as a function of temperature. Open and filled peaks\ncorrespond to different partial RDFs.\nbetween absorbing Ti 0and the next Se atoms located in the same (Se 3)\nor in the adjacent layer (Se∗\n4) are close, as indicated by similar values of\nκ≈27 N/m. At the same time, the intralayer Ti–Ti interactions (Ti 0–\nTi2, Ti 0–Ti 8) are slightly stronger ( κ≈39-42 N/m) than the interlayer\nTi–Ti interactions (Ti 0–Ti∗\n6, Ti 0–Ti∗\n9) with κ≈26-35 N/m. Among all Se–\nSe atom pairs, the interaction between the nearest selenium atoms is the\nweakest ( κSe0−Se2≈28 N/m), while the interaction between two selenium\natoms located at opposite sides of the same layer is more than twice stronger\n(κSe0−Se4≈66 N/m). At the same time, the interaction between selenium\natoms in adjacent layers across the vdW gap is rather strong with κSe0−Se∗\n3≈\n14MSRDs (T)\n0 50 100 150 200 250 3000.000.010.020.030.040.05Ti0-Ti2 (r=3.54 Å)\nTi0-Ti6* (r=6.01 Å)\nTi0-Ti8 (r=6.13 Å)\nTi0-Ti9* (r=6.97 Å)MSRD 2 (Å2)\nTemperature T (K)TiSe2\n= 39 N/m\n= 26 N/m\n= 42 N/m\n= 35 N/m\n0 50 100 150 200 250 3000.000.010.020.030.040.05\n= 28 N/m\n= 52 N/m\n= 66 N/m\n= 33 N/m\n= 40 N/m\n= 40 N/mTiSe2\n Se0-Se2 (r=3.54 Å)\n Se0-Se3* (r=3.58 Å)\n Se0-Se4 (r=3.68 Å)\n Se0-Se7* (r=5.04 Å)\n Se0-Se8 (r=5.11 Å)\n Se0-Se10* (r=6.01 Å)MSRD 2 (Å2)\nTemperature T (K)0 50 100 150 200 250 3000.000.010.020.030.040.05= 52 N/m\n= 27 N/m\n= 27 N/m\n= 32 N/m\n= 36 N/m\n= 34 N/m Ti0-Se1 (r=2.55 Å)\n Ti0-Se3 (r=4.36 Å)\n Ti0-Se4* (r=4.92 Å)\n Ti0-Se5 (r=5.62 Å)\n Ti0-Se7* (r=6.06 Å)\n Ti0-Se10* (r=7.02 Å)MSRD 2 (Å2)\nTemperature T (K)TiSe2\n0 50 100 150 200 250 3000.010.020.030.040.05VSe2\nV0-V2 (r=3.36 Å)\nV0-V6 (r=5.81 Å)\nV0-V8* (r=6.13 Å)\nV0-V9 (r=6.71 Å)MSRD 2 (Å2)\nTemperature T (K)= 12 N/m \n= 37 N/m\n= 34 N/m\n= 17 N/m \n0 50 100 150 200 250 3000.000.010.020.030.040.05VSe2= 39 N/m \n= 48 N/m\n= 41N/m\n= 36 N/m Se0-Se2 (r=3.36 Å)\n Se0-Se3/4** (r=3.63 Å)\n Se0-Se6/7** (r=4.94 Å)\n Se0-Se10 (r=5.81 Å)MSRD 2 (Å2)\nTemperature T (K)0 50 100 150 200 250 3000.000.010.020.030.040.05= 50 N/m \n= 24 N/m\n= 32N/m\n= 23 N/m \n= 33 N/m\n= 35 N/mVSe2\n V0-Se1 (r=2.47 Å)\n V0-Se3 (r=4.17 Å)\n V0-Se4* (r=4.99 Å)\n V0-Se5 (r=5.34 Å)\n V0-Se7* (r=6.01 Å)\n V0-Se10* (r=6.89 Å)MSRD 2 (Å2)\nTemperature T (K)(a)\n(b)\n(c)(d)\n(e)\n(f)Figure 6: Temperature dependence of the mean-square relative displacement (MSRD)\nfactors σ2for Ti(V)–Se (a,d), Ti–Ti(V–V) (b,e), and Se–Se (c,f) atom pairs in TiSe 2\nand VSe 2. Solid lines correspond to the fits using the correlated Einstein model [39].\nCalculated values of the effective force constants κare also given.\n52 N/m. To conclude, the temperature dependencies of the MSRD factors\nin TiSe 2do not show any unusual behaviour.\nIn spite of the similarities in the crystallographic structures of TiSe 2and\nVSe 2, the EXAFS spectra χ(k)k2of VSe 2have stronger amplitude and, as a\nresult, more intense peaks in FTs (Fig. 3). The V K-edge EXAFS spectra of\nVSe 2, their FTs and WTs (Fig. 3(b)) are dominated by a contribution from\nthe first coordination shell composed of six selenium atoms. Contributions\nfrom distant coordination shells are also present up to 7 ˚A but they are\n15relatively weaker. This fact contrasts with the Se K-edge EXAFS spectra\nwhere strong contributions from distant shells are observed in FTs up to 5 ˚A\nbut appreciable signals are visible even at longer distances.\nThe results of the RMC/EA fits for VSe 2are in good agreement with the\nexperimental data at both V and Se K-edges (Fig. 4(b)). The partial RDFs\nfor V–V, V–Se(Se–V), and Se–Se atom pairs are shown in Fig. 5(b), whereas\nthe characteristic Einstein frequencies ωEand the effective force constants κ\nfor all coordination shells are reported in Table 2.\nThe RDFs obtained for both TMDs (Fig. 5) exhibit an expected temper-\nature dependence: they become broadened at higher temperatures. At the\nsame time, no peak splitting is observed for the nearest shells at low temper-\natures in the CDW state. This suggests that the expected lattice distortions\ndue to atom displacements [19, 10] are relatively small compared to thermal\ndisorder effects. Nevertheless, upon comparing the partial RDFs for TiSe 2\nand VSe 2, a notable difference in the temperature dependence between Ti-\nTi and V-V RDFs is observed within the range of the second coordination\nshell. Upon increasing temperature, the V 0–V2RDF peak at 3.36 ˚A in VSe 2\nbecomes significantly more broadened compared to the Ti 0–Ti 2RDF peak at\n3.54 ˚A in TiSe 2. This difference is well reflected by the temperature depen-\ndence of respective MSRDs in Fig. 6(b) and (e). Indeed, the MSRD for the\nV0–V2atom pair exhibits a rapid increase above 50 K, displaying anomalous\nbehavior compared to MSRDs for other atom pairs. Note that the value of\nthe force constant κV0−V2≈12 N/m is the smallest one (Table 2).\nThere is also a prominent difference in the splitting of the peak around\n3.6˚A in the Se–Se RDFs between the two TMDs. A single peak at 3.55 ˚A is\n16present in TiSe 2, whereas a double peak at about 3.35 ˚A and 3.68 ˚A exists in\nVSe 2. This difference is due that the Se 0–Se 2distance in VSe 2is significantly\nshorter by 0.18 ˚A than in TiSe 2(see Tables 1 and 2).\n4. Conclusions\nHighly crystalline bulk 1T-TiSe 2and 1T-VSe 2were synthesized using the\nchemical vapor transport technique, and their local atomic structure and\nlattice dynamics were studied by temperature-dependent (10-300 K) X-ray\nabsorption spectroscopy at the Ti, V, and Se K-edges.\nThe extended X-ray absorption fine structure (EXAFS) spectra were anal-\nysed using the reverse Monte Carlo (RMC) simulations [26, 27] by a simulta-\nneous fitting of metal (Ti/V) and selenium K-edge spectra (Fig. 4). Such an\napproach allowed us to reconstruct the local environment in both selenides\nin terms of the partial radial distribution functions (Fig. 5) and to gain reli-\nable information on disorder effects in the nearest and distant coordination\nshells. The mean squared relative displacements σ2for atom pairs located\nin the first ten coordination shells were determined, and their temperature\ndependencies were approximated using the correlated Einstein model. As a\nresult, the characteristic Einstein frequencies ωEand the effective force con-\nstants κwere obtained (Tables 1 and 2) for all atomic pairs and used in the\nassessment of the strengths of intralayer and interlayer interactions.\nThe partial RDFs obtained for both TMDs (Fig. 5) demonstrate the ex-\npected temperature dependence and do not show any evidence of the peak\nsplitting for the nearest shells at low temperatures due to the CDW state.\nThis indicates that CDW-induced atom displacements [19, 10] are relatively\n17small compared to thermal disorder effects. Nevertheless, a comparison of the\nRDFs for Ti–Ti and V–V atom pairs located within the same layer shows\ntheir different temperature variations. In particular, the thermal disorder\nsignificantly affects the V 0–V2atom pair as can be seen from the tempera-\nture dependence of its MSRD (Fig. 6) and small value of the force constant\nκV0−V2≈12 N/m (Table 2).\nThus, the RMC/EA-EXAFS technique shows great potential for studying\nthe local atomic structure and disorder effects in TMDs and other layered\nmaterials.\nCRediT authorship contribution statement\nInga Pudza : Investigation, Visualization, Writing – original draft, Writ-\ning – review & editing. Boris Polyakov : Investigation, Writing – original\ndraft, Writing – review & editing. Kaspars Pudzs : Investigation. Ed-\nmund Welter : Resources. Alexei Kuzmin : Conceptualization, Investiga-\ntion, Methodology, Writing – original draft, Writing – review & editing.\nDeclaration of Competing Interest\nThe authors declare that they have no known competing financial inter-\nests or personal relationships that could have appeared to influence the work\nreported in this paper.\nAcknowledgements\nB.P. and A.K. thank the support of the Latvian Council of Science project\nNo. LZP-2020/1-0261. The experiment at the PETRA III synchrotron was\n18performed within proposal No. I-20210625 EC. The synchrotron experiment\nhas been supported by the project CALIPSOplus under the Grant Agreement\n730872 from the EU Framework Programme for Research and Innovation\nHORIZON 2020. Institute of Solid State Physics, University of Latvia as\nthe Center of Excellence has received funding from the European Union’s\nHorizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-\nTeamingPhase2 under grant agreement No. 739508, project CAMART2.\nData availability statement\nData will be made available on request.\n19References\n[1] G. R. Bhimanapati, Z. Lin, V. Meunier, Y. Jung, J. Cha, S. Das,\nD. Xiao, Y. Son, M. S. Strano, V. R. Cooper, et al., Recent advances in\ntwo-dimensional materials beyond graphene, ACS Nano 9 (2015) 11509–\n11539. doi:10.1021/acsnano.5b05556 .\n[2] H. Zhan, D. Guo, G. Xie, Two-dimensional layered materials: from\nmechanical and coupling properties towards applications in electronics,\nNanoscale 11 (2019) 13181–13212. doi:10.1039/C9NR03611C .\n[3] V. Shanmugam, R. A. Mensah, K. Babu, S. Gawusu, A. Chanda, Y. Tu,\nR. E. Neisiany, M. F¨ orsth, G. Sas, O. Das, A review of the synthesis,\nproperties, and applications of 2D materials, Part. Part. Syst. Charact.\n39 (2022) 2200031. doi:10.1002/ppsc.202200031 .\n[4] Y.-C. Lin, R. Torsi, R. Younas, C. L. Hinkle, A. F. Rigosi, H. M. Hill,\nK. Zhang, S. Huang, C. E. Shuck, C. Chen, et al., Recent Advances\nin 2D Material Theory, Synthesis, Properties, and Applications, ACS\nNano 17 (2023) 9694–9747. doi:10.1021/acsnano.2c12759 .\n[5] T. Ritschel, J. Trinckauf, K. Koepernik, B. B¨ uchner, M. v. Zimmermann,\nH. Berger, Y. Joe, P. Abbamonte, J. Geck, Orbital textures and charge\ndensity waves in transition metal dichalcogenides, Nat. Phys. 11 (2015)\n328–331. doi:10.1038/nphys3267 .\n[6] M. Hossain, Z. Zhao, W. Wen, X. Wang, J. Wu, L. Xie, Recent ad-\nvances in two-dimensional materials with charge density waves: syn-\n20thesis, characterization and applications, Crystals 7 (2017) 298. doi:\n10.3390/cryst7100298 .\n[7] W. Jolie, T. Knispel, N. Ehlen, K. Nikonov, C. Busse, A. Grueneis,\nT. Michely, Charge density wave phase of VSe 2revisited, Phys. Rev. B\n99 (2019) 115417. doi:10.1103/PhysRevB.99.115417 .\n[8] J. Wilson, F. Di Salvo, S. Mahajan, Charge-density waves in metallic,\nlayered, transition-metal dichalcogenides, Phys. Rev. Lett. 32 (1974)\n882. doi:10.1103/PhysRevLett.32.882 .\n[9] M. Leroux, L. Cario, A. Bosak, P. Rodiere, Traces of charge den-\nsity waves in NbS 2, Phys. Rev. B 97 (2018) 195140. doi:10.1103/\nPhysRevB.97.195140 .\n[10] Z. Guo, X. Hao, J. Dong, H. Li, J. Liao, D. Chen, Observation of pressure\ninduced charge density wave order and eightfold structure in bulk VSe 2,\nSci. Rep. 11 (2021) 18157. doi:10.1038/s41598-021-97630-8 .\n[11] K. Rossnagel, On the origin of charge-density waves in select layered\ntransition-metal dichalcogenides, J. Phys.: Condens. Matter 23 (2011)\n213001. doi:10.1088/0953-8984/23/21/213001 .\n[12] M. Adam, H. Zhu, Z. Liu, S. Cui, P. Zhang, Y. Liu, G. Zhang, X. Wu,\nZ. Sun, L. Song, Charge density wave phase suppression in 1T-TiSe 2\nthrough Sn intercalation, Nano Res. 15 (2022) 2643–2649. doi:10.\n1007/s12274-021-3859-0 .\n[13] C. Riekel, Structure refinement of TiSe 2by neutron diffraction, J.\n21Solid State Chem. 17 (1976) 389–392. doi:10.1016/S0022-4596(76)\n80008-4 .\n[14] E. Hoschek, Vanadinselenide, Z. Anorg. Allg. Chem. 242 (1939) 49.\n[15] Y. Liu, N. Weiss, X. Duan, H.-C. Cheng, Y. Huang, X. Duan, Van der\nWaals heterostructures and devices, Nat. Rev. Mater. 1 (2016) 16042.\ndoi:10.1038/natrevmats.2016.42 .\n[16] Z. Shi, X. Wang, Y. Sun, Y. Li, L. Zhang, Interlayer coupling in two-\ndimensional semiconductor materials, Semicond. Sci. Technol. 33 (2018)\n093001. doi:10.1088/1361-6641/aad6c3 .\n[17] R. Bian, C. Li, Q. Liu, G. Cao, Q. Fu, P. Meng, J. Zhou, F. Liu,\nZ. Liu, Recent progress in the synthesis of novel two-dimensional van\nder Waals materials, Natl. Sci. Rev. 9 (2021) nwab164. doi:10.1093/\nnsr/nwab164 .\n[18] I. Pudza, D. Bocharov, A. Anspoks, M. Krack, A. Kalinko, E. Wel-\nter, A. Kuzmin, Unraveling the interlayer and intralayer coupling in\ntwo-dimensional layered MoS 2by X-ray absorption spectroscopy and ab\ninitio molecular dynamics simulations, Mater. Today Commun. (2023)\n106359 doi:10.1016/j.mtcomm.2023.106359 .\n[19] F. J. Di Salvo, D. E. Moncton, J. V. Waszczak, Electronic properties and\nsuperlattice formation in the semimetal TiSe 2, Phys. Rev. B 14 (1976)\n4321–4328. doi:10.1103/PhysRevB.14.4321 .\n[20] M. Holt, P. Zschack, H. Hong, M. Y. Chou, T.-C. Chiang, X-ray studies\n22of phonon softening in TiSe 2, Phys. Rev. Lett. 86 (2001) 3799–3802.\ndoi:10.1103/PhysRevLett.86.3799 .\n[21] F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn, G. Karapetrov,\nR. Hott, R. Heid, K.-P. Bohnen, A. Alatas, Electron-phonon coupling\nand the soft phonon mode in TiSe 2, Phys. Rev. Lett. 107 (2011) 266401.\ndoi:10.1103/PhysRevLett.107.266401 .\n[22] M. Bayard, M. Sienko, Anomalous electrical and magnetic properties\nof vanadium diselenide, J. Solid State Chem. 19 (1976) 325–329. doi:\n10.1016/0022-4596(76)90184-5 .\n[23] D. Eaglesham, R. Withers, D. Bird, Charge-density-wave transitions in\n1T-VSe 2, J. Phys. C: Solid State Physics 19 (1986) 359. doi:10.1088/\n0022-3719/19/3/006 .\n[24] J. Diego, A. H. Said, S. K. Mahatha, R. Bianco, L. Monacelli, M. Calan-\ndra, F. Mauri, K. Rossnagel, I. Errea, S. Blanco-Canosa, van der Waals\ndriven anharmonic melting of the 3D charge density wave in VSe 2, Nat.\nCommun. 12 (2021) 598. doi:10.1038/s41467-020-20829-2 .\n[25] J. Feng, R. A. Susilo, B. Lin, W. Deng, Y. Wang, B. Li, K. Jiang,\nZ. Chen, X. Xing, Z. Shi, C. Wang, B. Chen, Achieving Room-\nTemperature Charge Density Wave in Transition Metal Dichalcogenide\n1T-VSe 2, Adv. Electron. Mater. 6 (2020) 1901427. doi:10.1002/aelm.\n201901427 .\n[26] J. Timoshenko, A. Kuzmin, J. Purans, Reverse Monte Carlo modeling\nof thermal disorder in crystalline materials from EXAFS spectra, Comp.\n23Phys. Commun. 183 (2012) 1237–1245. doi:10.1016/j.cpc.2012.02.\n002.\n[27] J. Timoshenko, A. Kuzmin, J. Purans, EXAFS study of hydrogen inter-\ncalation into ReO 3using the evolutionary algorithm, J. Phys.: Condens.\nMatter 26 (2014) 055401. doi:10.1088/0953-8984/26/5/055401 .\n[28] A. Ubaldini, E. Giannini, Improved chemical vapor transport growth of\ntransition metal dichalcogenides, J. Cryst. Growth 401 (2014) 878–882.\ndoi:10.1016/j.jcrysgro.2013.12.070 .\n[29] J. Pandey, A. Soni, Electron-phonon interactions and two-phonon modes\nassociated with charge density wave in single crystalline 1T-VSe 2, Phys.\nRev. Res. 2 (2020) 033118. doi:10.1103/PhysRevResearch.2.033118 .\n[30] E. Welter, R. Chernikov, M. Herrmann, R. Nemausat, A beamline for\nbulk sample x-ray absorption spectroscopy at the high brilliance storage\nring PETRA III, AIP Conf. Proc. 2054 (2019) 040002. doi:10.1063/\n1.5084603 .\n[31] A. Kuzmin, J. Chaboy, EXAFS and XANES analysis of oxides at the\nnanoscale, IUCrJ 1 (2014) 571–589. doi:10.1107/S2052252514021101 .\n[32] A. Kalinko, XAESA v0.07 (2023).\nURL https://gitlab.desy.de/aleksandr.kalinko/xaesa\n[33] A. L. Ankudinov, B. Ravel, J. J. Rehr, S. D. Conradson, Real-space\nmultiple-scattering calculation and interpretation of x-ray-absorption\nnear-edge structure, Phys. Rev. B 58 (1998) 7565–7576. doi:10.1103/\nPhysRevB.58.7565 .\n24[34] J. J. Rehr, R. C. Albers, Theoretical approaches to x-ray absorption\nfine structure, Rev. Mod. Phys. 72 (2000) 621–654. doi:10.1103/\nRevModPhys.72.621 .\n[35] L. Hedin, B. I. Lundqvist, Explicit local exchange-correlation potentials,\nJ. Phys. C: Solid State Phys. 4 (1971) 2064. doi:10.1088/0022-3719/\n4/14/022 .\n[36] J. Timoshenko, A. Kuzmin, Wavelet data analysis of EXAFS spectra,\nComp. Phys. Commun. 180 (2009) 920–925. doi:10.1016/j.cpc.2008.\n12.020 .\n[37] F. R. Hampel, Robust estimation: A condensed partial survey, Z.\nWahrscheinlichkeit. 27 (1973) 87–104. doi:10.1007/BF00536619 .\n[38] M. Daszykowski, K. Kaczmarek, Y. Vander Heyden, B. Walczak, Robust\nstatistics in data analysis–A review: Basic concepts, Chemom. Intell.\nLab. Syst. 85 (2007) 203–219. doi:10.1016/j.chemolab.2006.06.016 .\n[39] E. Sevillano, H. Meuth, J. Rehr, Extended x-ray absorption fine struc-\nture Debye-Waller factors. I. Monatomic crystals, Phys. Rev. B 20 (1979)\n4908. doi:10.1103/PhysRevB.20.4908 .\n[40] A. Kuzmin, J. Timoshenko, A. Kalinko, I. Jonane, A. Anspoks, Treat-\nment of disorder effects in x-ray absorption spectra beyond the con-\nventional approach, Radiation Physics and Chemistry (2019). doi:\n10.1016/j.radphyschem.2018.12.032 .\n25Table 1: Values of coordination numbers ( N) and Ti–Se, Ti–Ti, and Se–Se interatomic\ndistances rfor the first ten coordination shells of Ti and Se calculated from the crystal-\nlographic structure of TiSe 2[13]. Interlayer distances are marked with “*”. The values\nof characteristic Einstein frequencies ( ωE) and the effective force constants ( κ) obtained\nfrom the RMC/EA analysis are also given for each atom pair.\nAtom pair NDistance r(˚A)ωE(THz) κ(N/m)\nTi0–Se 1 6 2.55 32.3 ±0.4 51.61 ±0.01\nTi0–Ti 2 6 3.54 31.3 ±0.6 39.02 ±0.02\nTi0–Se 3 6 4.37 23.3 ±0.4 26.92 ±0.01\nTi0–Se∗\n4 6 4.92 23.3 ±0.6 26.94 ±0.02\nTi0–Se 5 12 5.62 25.6 ±0.9 32.43 ±0.04\nTi0–Ti∗\n6 2 6.01 25.6 ±1.7 25.99 ±0.12\nTi0–Se∗\n7 6 6.06 26.8 ±1.2 35.65 ±0.07\nTi0–Ti 8 6 6.13 32.4 ±0.6 41.69 ±0.01\nTi0–Ti∗\n9 12 6.97 29.8 ±1.9 35.41 ±0.14\nTi0–Se∗\n10 12 7.02 26.2 ±0.2 33.96 ±0.01\nSe0–Ti 1 3 2.55 32.3 ±0.4 51.61 ±0.01\nSe0–Se 2 6 3.54 20.7 ±0.2 28.12 ±0.01\nSe0–Se∗\n3 3 3.58 28.1 ±1.2 51.95 ±0.10\nSe0–Se 4 3 3.68 31.7 ±1.3 65.81 ±0.12\nSe0–Ti 5 3 4.37 23.3 ±0.4 26.92 ±0.01\nSe0–Ti∗\n6 3 4.92 23.3 ±0.6 26.94 ±0.02\nSe0–Se∗\n7 3 5.04 22.4 ±0.5 32.89 ±0.02\nSe0–Se 8 3 5.11 24.7 ±0.6 39.96 ±0.03\nSe0–Ti 9 6 5.62 25.6 ±0.9 32.43 ±0.04\nSe0–Se∗\n10 2 6.01 24.6 ±0.8 39.53 ±0.04\n26Table 2: Values of coordination numbers ( N) and V–Se, V–V, and Se–Se interatomic\ndistances rfor the first ten coordination shells of V and Se calculated from the crystal-\nlographic structure of VSe 2[14]. Interlayer distances are marked with “*”. Coordination\nshells composed of both intralayer and interlayer atomic pairs are marked with “**”. The\nvalues of characteristic Einstein frequencies ( ωE) and the effective force constants ( κ) ob-\ntained from the RMC/EA analysis are also reported for each atom pair.\nAtom pair N Distance r(˚A)ωE(THz) κ(N/m)\nV0–Se 1 6 2.47 31.1 ±0.8 49.69 ±0.04\nV0–V2 6 3.36 16.6 ±0.4 11.67 ±0.01\nV0–Se 3 6 4.17 21.7 ±0.4 24.25 ±0.01\nV0–Se∗\n4 6 4.99 24.9 ±1.0 31.84 ±0.06\nV0–Se 5 12 5.35 21.3 ±0.6 23.23 ±0.02\nV0–V6 6 5.81 29.5 ±2.3 36.81 ±0.22\nV0–Se∗\n7 6 6.01 25.4 ±1.6 33.13 ±0.13\nV0–V∗\n8 2 6.13 28.4 ±1.3 34.03 ±0.07\nV0–V9 6 6.71 20.1 ±1.2 17.16 ±0.06\nV0–Se∗\n10 12 6.89 26.2 ±1.1 35.22 ±0.06\nSe0–V1 3 2.47 31.1 ±0.8 49.69 ±0.04\nSe0–Se 2 6 3.36 24.5 ±0.4 39.32 ±0.01\nSe0–Se∗∗\n3/43+3 3.63 27.1 ±1.4 48.18 ±0.12\nSe0–V5 3 4.17 21.7 ±0.4 24.25 ±0.01\nSe0–Se∗∗\n6/73+3 4.94 25.1 ±0.6 41.19 ±0.02\nSe0–V8 3 4.99 24.9 ±1.0 31.84 ±0.06\nSe0–V9 6 5.35 21.3 ±0.6 23.23 ±0.02\nSe0–Se 10 6 5.81 23.5 ±0.4 36.16 ±0.01\n27" }, { "title": "2401.16126v1.Incorporating_the_Cosmological_Constant_in_a_Modified_Uncertainty_Principle.pdf", "content": "arXiv:2401.16126v1 [gr-qc] 29 Jan 2024Incorporating the Cosmological Constant in a Modified Uncer tainty Principle\nS. Ahmadi,1,∗E. Yusofi,1,2,3,†and M. A. Ramzanpour1,3,‡\n1Department of Physics, Ayatollah Amoli Branch, Islamic Aza d University, Amol, Iran\n2School of Astronomy, Institute for Research in Fundamental Sciences(IPM), P. O. Box 19395-5531,Tehran, Iran\n3Innovation and Management Research Center, Ayatollah Amol i Branch,\nIslamic Azad University, Amol, Mazandaran, Iran\n(Dated: January 30, 2024)\nThe existence of a tiny but non-zero cosmological constant s eems to be a fundamental challenge\nfor physics. This study examines the cosmological constant problem and modified uncertainty prin-\nciple within a unified framework inspired by a void-dominate d cosmology. We model voids/halos as\nspherical bubbles/drops for simplification and analysis. O ur heuristic calculations show significant\nvariations in surface energy values from the largest to smal lest scales, resulting in a substantial\ndisparity (approximately 122 orders of magnitude) in the va lues of the cosmological constant. Our\nmethod suggests that the difference in the values of the cosmo logical constant is inherent and\nshould be considered natural. As a main outcome of this resea rch, we propose a new form of\nextended uncertainty principle that incorporates cosmolo gical constant.\nKeywords : Cosmological Constant; Surface Energy; Cosmic Void; Unce rtainty Principle\nPACS: 98.80.Bp; 95.36.+x; 98.80.Es\n∗samira666ahmadi@gmail.com\n†e.yusofi@ipm.ir(Corresponding author)\n‡ma.ramzanpour@iau.ac.ir2\nI. INTRODUCTION AND MOTIVATION\nSome of the key challenges in modern cosmology include dark matter, dark energy, the cosmological constant\nproblem, and Hubble tension [ 1–4]. These significant physical hurdles highlight the inability of the stand ard\nmodel of cosmology to accurately address many of these issues [ 3,5]. Dark energy, which drives the accelerating\nexpansion of the universe at large scales [ 6], has been explored as potentially originating from the amplification\nof quantum fluctuations of a light field during inflation [ 7]. Additionally, dark matter, as an invisible substance\nwith gravitational influence, plays a crucial role in the interconnect ion and balance of galaxies, clusters, and\nsuperclusters. Both dark matter and dark energy have been tho roughly confirmed through various direct and\nindirect observational methods. However, despite decades of re search, no plausible physical source has been\nidentified for these major contributions to the universe’s matter a nd energy [ 8,9]. By the way, some models do\nnot find it necessary to postulate dark matter and dark energy [ 10–12].\nThe cosmological constant problem is rooted in the fact that the ob served acceleration is not easily explained\nby the known matter and energy content of the universe. The mos t commonly accepted explanation for the\nacceleration is the existence of dark energy as a cosmological cons tant, a hypothetical form of energy that\npermeates all of space and exerts a negative pressure that drive s the acceleration. However, the predicted\nvalue of the dark energy density from quantum field theory is far lar ger than the observed value, leading to a\ndiscrepancy of more than 120 orders of magnitude, making it one of the most significant discrepancies between\ntheory and observation in all of physics. This discrepancy is known a s thefine tuning problem , and its resolution\nremains one of the most significant challenges in modern cosmology [ 6,13–15].\nIn this study, we propose a bubble-drop model for the universe, w hich is populated by two types of spher-\nical objects: underdense ”cosmic voids” and overdense ”matter halos”. The assumption is that cosmic voids\n(bubbles) are encircled by matter halos (drops) with a shell-like distr ibution on their border[ 16]. Considering\nthe shells as the ideal separating surface between the under-den sity voids and the over-density clusters, we can\ncalculate the resulting surface tension (energy) through dimensio nal calculation (see Table Iin [17]). Based on\nthis ideal assumption, we can take into account the mass density an d surface energy of these spherical objects.\nWe then calculate the surface energy for the possible largest and s mallest objects in our modeled universe to\nfind a potential explanation for the cosmological constant problem . In addition, we will offer a possible relation\nbetween surface energy, cosmological constant and surface gr avity to find a plausible source for the acceleration\nof the universe at the large scales of the universe. Since the propo sed model covers spherical objects from the\nlargest to the smallest possible radii (lengths), as a potential resu lt of this study, we will present a modified\nuncertainty principle (MUP), which may include a variable cosmological constant.\nThe MUP with minimum length (GUP) [ 18–20] and with maximum length (EUP) [ 21] is a theoretical concept\nthat extends the traditional Heisenberg uncertainty principle to in corporate the effects of quantum gravity [ 22,\n23]. In this modified version, the uncertainty in the measurement of ce rtain pairs of physical properties, such\nas position and momentum, is constrained by both a minimum and maximu m length scale. This suggests that\nthere are fundamental limits to the precision with which these prope rties can be simultaneously known, and it\nimplies a fundamental granularity to space-time at the quantum leve l.\nConsidering that the average diameter of cosmic voids is typically ≃100 Mpc, we explore whether a cosmic\nvoid, as a cosmic object, can effectively represent the local and glo bal behavior of the universe. It is worth\nnoting that given the web-like structure of the universe and its hier archical formation[ 24,25], present cosmic\nvoids/clusters may be much larger than 100 Mpc [ 26]. For instance, the Bo¨ otes and KBC voids are the largest\nknown spaces, encompassing a significant portion of the visible unive rse’s diameter [ 16,27,28].\nIn Section II, we explore the coexistence and interplay of cosmic vo ids and matter halos within the context\nof void-dominated cosmology. Moving to Section III, we employ heur istic calculations to derive mass density,\nsurface tension, and cosmological constant for various cosmic ob jects, comparing these values to the order of\nmagnitude of the entire universe. Furthermore, we will discuss the possible relationship between surface gravity\nand surface energy to explain the present repulsive acceleration o n the large scale of the universe. In Section\nIV, we will have a brief discussion on the cosmological constant prob lem within this framework, drawing upon\nthe concept of surface energy associated with cosmic voids. As a p otential outcome of this study, we propose\na new kind of extended uncertainty principle that incorporates a sc ale-dependent cosmological constant in In3\nSection V. In the final section, we will summarize the results of this s tudy.\nII. COSMIC VOIDS IN VOID-DOMINATED COSMOLOGY\nIn the cosmic web’s structure, cosmic voids constitute the main volu me of space, and the interconnected void\nshells serve as the scaffolding for this expansive structure, encom passing both normal and dark matter. Despite\ncosmic voids being significantly larger than local scales and smaller tha n cosmic scales, we will demonstrate that\nthey can be effectively studied as a fundamental component within t he cellular structure of the cosmic web.\nThe conventional models of cosmology overlook the statics, dynam ics, and evolution of supervoids, which\nform the dominant volume of local and global scales within the cosmos . Supervoids are not only non-empty\nbut also possess energy density and undergo evolution. Due to the ir substantial size, they are more prone to\nmerging and are therefore strong candidates for influencing cosm ic scales [ 17,24,29–31]. Consequently, it seems\nplausible that these large inhomogeneities play a role in the dynamics of the universe and have an impact on\nthe evolution of cosmic parameters [ 32–35].\nThe present cosmos contains a network of cosmic voids where multip le superclusters and small to large\ngalactic objects are merging. While much of the mass in the Universe is concentrated in virialized structures,\nthe majority of the volume is taken up by large underdense voids.Void s dominate the galaxy and matter\ndistributions on a megaparsec (or larger) scale [ 25]. In a void-based explanation of structure formation, matter\nis compressed between expanding voids, and sheets and filaments e merge at the void walls’ intersections. In the\nlarge scale overview, the universe is void-dominant, while in smaller ove r-dense regions, it consists of clusters,\nfilaments, and nodes of matter in a matter-dominant state. Resea rch has shown that voids are counterparts of\nmassive clusters, and their coexistence and continuous integratio n with superclusters, as well as the merging\nof supervoids, contribute significantly to the universe’s structur e formation. The increase in the size of cosmic\nvoids after merging exerts an effective repulsive force on the galax ies situated on their shell [ 17,24]. Under these\nconditions, it can be assumed that the cosmic fluid at a large-scale ov erview consists of merging and expanding\nvast voids, leading to a universe increasingly dominated over time by la rger cosmic voids and consequently an\naccelerating expansion universe.\nIn a void-dominated universe [ 17,36–38], voids (bubbles) constitute the main distribution of the two-phase\ncosmic web[ 39]. By equating the energy density of cosmic voids with the vacuum ene rgy density, it has been\nshown that the value estimated for the cosmological constant is ve ry close to that reported by Planck’s observa-\ntions and has the same order of magnitude [ 17]. Our hypothesis about the effect of supervoids on the large-scale\ndynamics of the universe and cosmology in [ 17,40] could be consistent with the backreaction effects of density\ninhomogeneities in cosmology [ 41] and the configurational entropy of the cosmic web and its evolutio n [30,31].\nThe void-dominated cosmology [ 17] challenges the traditional notion of matter distribution in the unive rse.\nInstead of considering matter as filling the space between cosmic vo ids, we now view our web-like universe as\nconsisting of almost inter-empty inter-connected bubbles. This me ans that matter, both visible (light) and\ninvisible (dark), is primarily concentrated on the borders of these c osmic voids, forming sheet-like structures.\nThis proposed change has significant implications for our understan ding of the cosmos. Firstly, it suggests\nthat the matter in the universe is not randomly and irregularly distrib uted in space but rather concentrated in\nspecific regions. Secondly, this alternative viewpoint provides a pot ential explanation for the observed large-\nscale structures in the universe, such as galaxy clusters and supe rclusters. These structures could arise from the\nsheet-like arrangements of matter along the borders of cosmic vo ids. Furthermore, this perspective may shed\nlight on the nature of dark matter and dark energy[ 17,40].\nIn this study, over-dense objects, such as galaxies, clusters, t heir superclusters and the dark matter around\nthem, are modeled as ”drops” inside the spherical-shaped ”matter halos”[42], while the small and large voids\nwithin the spherical cosmic voids. are modeled as ”bubbles” [ 17,43,44]. Imagine that we were able to populate\nour Universe with these spheres. In a two-phase mixture of drops and bubbles, not only do drops absorb drops,\nbut bubbles also absorb other bubbles and push droplets from the c enter to their boundary (the direction of\nthe red arrows in Fig. ??on the left is from the center to the border of the bubble). The mer ger of bubbles\nincreases their size and decreases their density. Physical simulatio ns of redshifts with various displacements\nindicate that local scales become denser over time, while the density of large scales decreases. [ 24].4\nCosmic voids refer to vast, less-dense regions in the universe wher e there is very little matter concentrated\non the border of them, while matter halos are dense regions where m atter, such as galaxies and dark matter, is\nconcentrated in the center of them. The phrase ”cosmic voids ver sus matter halos” essentially contrasts these\ntwo types of structures in the universe, highlighting the difference between the empty voids and the dense,\nconcentrated matter halos.\nIII. MASS DENSITY AND SURFACE ENERGY OF COSMIC VOIDS\nThe discovery of the cell structure of the Universe revealed surp rising empty regions: voids. For years,\nthe under-dense regions of the cosmic web were unexplored due to a lack of information and the inability of\nsurveys to measure large portions of the sky with sufficient depth. This is now changing. The era of large-scale\nsurveys is providing us with an incredible amount of data to work with, along with the possibility of extracting\ninformation using statistical averagesof quantities in the sky. The se features are essential for transforming voids\ninto a potentially powerful tool for constraining cosmology [ 45].\nIn this section, we want to show whether a single cosmic void can be a g ood representation of local and global\nscales or not? For this purpose, we first calculate the mass density of an ideal spherical supervoid and show that\nits magnitude is about one-tenth of the average density of the who le universe. Then, considering the surface\nenergy of a cosmic void.\nA. Mass Density of Cosmic Voids\nCosmic voids are, on average, spherical, making the application of th e Alcock-Paczy´ nski test not only fea-\nsible but also with reduced systematics [ 46,47]. Assume a perfectly empty spherical void with a total mass\naccumulated on the shell. The mass density of it can be calculated fro m the simple relation below,\nρi=3Mi\n4π¯r3v. (1)\nHereMiis the mass of the supercluster enclosed the cosmic void and ¯ rvis the average (effective) radius of a\ncosmic void. Taking into account the values of Table I. for the mass and radius of the Laniakia supercluster,\nwe obtain\nρ3= 1.70×10−27kg.m−3. (2)\nis very close to the universe’s average critical mass density of the u niverse [48]i.e.\nρ0,c= 1.88×10−26kg.m−3. (3)\nand is about one order smaller than that. This density deficit appear s to be due to the assumption of an empty\nbubble, while in a more real model inside the cosmic voids, there are so me galactic gas and sub-voids with a\nlower density than the shell in the form of stacked voids [ 46].\nB. Surface Energy of Cosmic Voids\nThe pressure difference between the inside and outside of a bubble is obtained from the following equation,\n∆P=2σ\n¯rv. (4)5\nHere,σrepresents the surface tension for bubble (drop). The relation ( 4) may related to the Darmois-Israel\ntheory is a way of describing the boundary conditions for a thin shell of matter in general relativity. It tells us\nhow the geometry and the stress-energy tensor of the shell are related to the metrics of the two regions that the\nshell separates [ 49]. The basic idea of Darmois-Israel theory is to integrate the Einste in field equations across\nthe shell, which is assumed to be a singular hypersurface with a Dirac d elta distribution of matter. This leads\nto the following equation:\n[Kij]−[K]hij=−8πSij (5)\nwhere [Kij] and [K] are the jumps of the extrinsic curvature and its trace across th e shell,hijis the induced\nmetricontheshell, and Sijisthestress-energytensoroftheshell. Itissimilartoequation( 4), whichconnectsthe\npressure difference inside the bubble and outside it to the surface e nergy of the shell containing the distribution\nof matter. The equation ( 5) is known as the Darmois-Israel junction condition, and it determin es how the shell\naffects the spacetime curvature [ 50]. Further investigation of this matter will be addressed in the followin g\nresearches.\nTo calculate the surface tension of a single void/cluster, we can use the following heuristic method [ 17,51],\nσi≡Energy\nArea=Mic2\nπR2\ni. (6)\nThe surface energy (energy per unit area) of some cosmic object s are listed in tables IandII, respectively.\nUpon examining table I, it becomes evident that larger cosmic bodies exhibit lesser surface energy as a\nresult of their diminishing density. Conversely, table IIdemonstrates that larger objects possess higher surface\nenergy due to their increased density. This suggests that cosmic b odies are situated within void-dominated\nregions, while smaller astronomical objects are located within matte r-dominated regions such as the galactic\n(and dark matter) halos. Consequently, in cosmological investigat ions, it is imperative to regard the universe\nas a void-dominant region, as this approach aligns more logically with th e available evidence.\nSince most of the calculations in this paper are dimensional and approximate, slight changes in the formulas\nand their coefficients do not have a significant impact on our re sults.It should also be noted that in the\nmore actual model, the voids are highly non-spherical, and our resu lts are at best a rough order-of-magnitude\nestimate.\nCosmicsimulations, suchaslarge-scalenumericalsimulationsusedinc osmology,doindeedshowtheexpansion\nof cosmic voids over time [ 52].\nIV. FROM SURFACE ENERGY TO COSMOLOGICAL CONSTANT\nBased on the analysis and heuristic calculations conducted in this art icle so far, maybe the isotropy in the\nbackground temperature of the universe (CMB) is due to the homo geneity distribution of such cosmic voids\nacross the cosmos. With this point of view, our universe has a cellular structure, and each cell of it is a cosmic\nvoid with almost the same size, the same mass density, and the same s urface energy. Also, considering the\nimpactful idea of stacked voids has addressed many obstacles in utilizing voids in cosmology. By identif ying\nvoids within a particularredshift rangeand stacking them according to their size, it is now possible to determine\ntheir average shape. In a universe that is homogeneous and isotro pic, voids display an average spherical shape\nwithout any preferred directions [ 53].\nBased on this rationale, our proposed void-dominated model satisfi es the requirements of homogeneity and\nisotropy, allowing us to apply Friedman’s equations for further inves tigation. In the standard ΛCDM model of\ncosmology, the Friedman equation is written as follows [ 54,55],\nH2=8πG\n3ρm+Λc2\n3=8πG\n3(ρm+ρΛ). (7)\nIn the standard ΛCDM model, the energy density is mainly divided into t wo parts: Ω mand Ω Λ(= 1−Ωm). We\nwant to attribute the contribution of the cosmological constant ( whose source is still unknown) to the energy6\nTABLE I. Surface tension (energy) σiand cosmological constant Λ i, for the clustered cosmic objects including super-\nclusters (Sc), clusters (C), and galaxies\ni. Clustered Mi Ri σi Λi\nObjects (1047kg) (1024m) (1015J.m−2) (10−52m−2)\n1. Corona Sc 0 .20 1 .50 0 .25 0 .6645\n2. Virgo Sc 0 .03 0 .50 0 .34 0 .8970\n3. Laniakea Sc 1 .00 2 .40 0 .50 1 .2979\n4. Caelum Sc 4 .00 4 .30 0 .62 1 .6172\n5. Virgo C 0 .002 0 .045 2 .83 7 .3833\n6. Coma C 0 .012 0 .10 3 .44 8 .9707\n7. Galaxies G 0 .0004 0 .016 4 .48 11 .6806\n8. Milky Way 0 .00002 0 .00087 75 .74 1975 .3200\n9. Andromeda 0 .00003 0 .001 85 .98 2242 .6800\n10. UGC02885 0 .000004 0 .00036 88 .46 2307 .2800\nTABLE II. The surface tension σi, and cosmological constant Λ ifor a bubbly cosmos in the possible smallest and the\nlargest scales.\nScales Mi(kg) Ri(m) σi(j.m−2) Λ i(m−2)\nMinimum (Planck) 2 .18×10−81.62×10−352.38×10786.08×1070\nMaximum (Caelum Sc) 4 .00×10474.30×10240.62×10151.62×10−52\ndensity and pressure of cosmic voids. For the current state of th e universe, in addition to the agreed-upon\nmatter contribution, there is also an important contribution attrib uted to the cosmological constant. However,\nfrom our perspective, it is the energy density and expanding press ure of the cosmic voids that affect the large-\nscale behavior and dynamics of the universe, rather than the cosm ological constant. From our point of view,\nthe cosmic fluid (web) is a two-phase fluid consisting of two parts: th e over-dense matter halos denoted by\nΩmand the under-dense cosmic voids denoted by Ω v(instead of Ω Λ) [17,56]. The difference here is that the\nsource of Ω v, the cosmic void, is completely obvious, compared with the cosmologic al constant, Ω Λ, which\nis unknown (dark). The proposed model views cosmic voids as the largest inhomog eneities in the universe,\ndistributed almost homogeneously throughout. There is a un iform distribution of cosmic voids, with most of the\nmatter halos situated on their borders. As a result, we can modify the Friedmann equation as follows\nH2=8πG\n3(ρm+ρv). (8)\nBy equating relations ( 7) and (8), we can obtain\nΛ =8πGρv\nc2. (9)\nWith consideration Pv=wρvc2, we have\nΛ =8πGPv\nwc4. (10)7\nIn this relation, the Pv≃∆Pis difference pressure between the inside and outside of the void in vo id-dominated\ncosmic web (web-like cosmic fluid [ 57]). If we consider the cosmic web as consisting of interconnected ide al\nspherical bubbles, we can use the following formula for the internal pressure of each void [ 58,59],\nPv=2σi\n¯rv, (11)\nTherefore from ( 10), we can write [ 17],\nΛi=8πG\nwc42σi\n¯rv. (12)\nBy inputting the required values, we achieve the findings in Table I.\nA. Variable Cosmological Constant from Smallest to Largest Scales\nIn void-dominated cosmology, the cosmological constant has varia ble values on different scales [ 14,60,61].\nSince the sizes of voids are diverse, we must obtain the largest value s for surface tension and cosmological\nconstant in the smallest scales and vice versa. To illustrate this more precisely, we consider the initial web-like\nstructure as a collection of very small Planck-sizedvoids connecte d to each other in the foam-shape. By utilizing\nrelation ( 12), we calculate the surface tension and cosmological constant (Λ Planck∼10+70) for a hypothetical\nsingle bubble with the Planck radius in Table II. The cosmological constant in the latest observational data is\nreported as [ 62],\nΛobs= 1.1056×10−52m−2. (13)\nIn our model, by using ( 12) for the largest object such as Laniakia supercluster (please ref er the row 3 in Table\nI), the cosmological constant of the model is estimated as [ 17],\nΛ3= 1.2979×10−52m−2. (14)\nIn Table I, we observe that the cosmological constant and mass density for each cosmic void are equivalent to\nthe values for the entire universe and are very close to it. Thus, giv en the values obtained for the ρi,σiand Λi, it\nseems that a cosmic void can be a good indicator of the global behavio r of the accelerating universe. Therefore,\nthe surface tension of expanding cosmic voids may be regarded as a possible source to produce accelerating\nuniverse [ 17].\nOur model can now address the discrepancy in cosmological consta nt values at the Planck scale with the\nsmallest radius (minimum possible length) and the present universe (s ee Table II), which related to the super-\nclusters/voids with the largest radius (maximum possible length) and is of the following order of magnitude,\nΛPlanck\nΛObs=Λmax\nΛmin∼10+122. (15)\nHere, Λ max= ΛUVcould be seen as a high-energy extension of the traditional cosmolo gical constant, resulting\nin Λmin= ΛIRin the low-energy range [ 63–65]. In an intriguing outcome, Tables IandIIalong with finding ( 15)\nmayvalidatetheconjugateconnectionbetweenthecosmologicalc onstantandthevolumeofthefour-dimensional\nspace-time V, i.e. ∆Λ ∆ V∼/planckover2pi1as discussed in [ 64].\nV. MODIFIED UNCERTAINTY PRINCIPLE WITH COSMOLOGICAL CONST ANT\nThe cosmologicalconstant is a concept that has its originsin gravity , as a modification to Einstein’s equations\nof general relativity, and in cosmology, as a crucial component in un derstanding the dynamics and evolution of8\nthe universe. The existence of a small, non-zero cosmological cons tant remains a major puzzle in fundamental\nphysics [ 66,67]. At first glance, arguments from quantum field theory would sugge st a cosmological constant\nthat is potentially 10120times greater than the one observed. It is widely held that a compre hensive theory of\nquantum gravity could reconcile this substantial disparity between theory and observation [ 66]. According to\nthe discussions and results of the previous sections, in this section , we examine the potential of MUP models,\nwhich are models of quantum gravity motivated by phenomenology, in addressing the cosmological constant\nproblem.\nIn the realm of gravity, the application of quantum mechanics impose s a restriction on measurements through\na minimum length. This length, is intricately linked to Planck’s length by (G UP). Similarly, (EUP) tackles\nthe limiting factor on the minimum momentum in (anti) de Sitter spacetim e. This restriction is characterized\nby the parameter α, which symbolizes the radius of the Hubble horizon. The modified unce rtainty principle\n(MUP) expands upon the commutation relation between position and momentum operators [ 21],\n[X,P] =/planckover2pi1\n2(1+αX2+βP2). (16)\nThere are multiple ways to determine the operator representation related to this the commutation relation .\nThe position and momentum operators satisfying this commutation c an be defined by utilizing the operators x\nandp, which follow the conventional commutation relation [ x,p] =i/planckover2pi1. As an illustration, one may consider the\nfollowing,\nX=x,andP=p(1+αx2+βp2). (17)\nBased on this definition, the more general form of the MUP includes e xplicit maximum and minimum values\nfor both position and momentum uncertainties and can be written as follows,\n∆X∆P≥/planckover2pi1\n21\n1−β∆P21\n1−α∆X2. (18)\nHere ∆Xand ∆Prepresent position and momentum uncertainties, while αandβare the parameters.\nMotivated by the existence of the Planck length and the largest sup er cluster/voids as the scales which set\nnatural minimum and maximum possible lengths, we now turn our atten tion to relation ( 18) by considering the\nfollowing relationship among ∆ X,αandβ(please refer to section III of [ 21]),\n∆X∝1√α∝/radicalbig\nβ .\nIn the proposed bubbly model of the universe (as per Tables IandII), the size of spherical objects can vary\nfrom the smallest to the largest radius. As a result, the maximum pos ition uncertainty can be expressed as,\nlmax≡∆Xmax∝1√αmin∝/radicalbig\nβmax, (19)\nand the minimum position uncertainty can be expressed as,\nlmin≡∆Xmin∝1√αmax∝/radicalbig\nβmin. (20)\nBy combining equations ( 19) and (20), we can obtain,\nαmax\nαmin∼/parenleftbigglmin\nlmax/parenrightbigg−2\n∼/parenleftbigg10−35\n10+26/parenrightbigg−2\n= 10+122. (21)\nBased on the Table IIand the above discussions, especially the relations ( 15) and (21), it can be concluded\nthat the parameter αin (21) is equivalent to the cosmological constant Λ. Dimensional analysis, as suggested9\nin previous works [ 68,69], implies that the primary correction’s structure is proportional to Λ∼lΛ−2∼\n(c\nH0)−2[65]. Considering the merger of voids and their expansion, voids can evo lve into genuine super-Hubble\nbubbles [ 25], which can have a radius in the size of the horizon i.e. lhorizon. Therefore, we can express lhorizon≡\nlΛ=/radicalBig\n3\nΛ∝1√\nΛas the (anti-) de Sitter horizon [ 69,70]. As a final result of this discussion, we can propose a\nnew extension of uncertainty principle including the cosmo logical constant as follows:\n∆X∆P≥/planckover2pi1\n21\n1−β0Λ−1∆P21\n1−α0Λ∆X2. (22)\nHereα≡α0Λ, andβ≡β0Λ−1. In the final relation ( 22), Λ have approximate values, so we require non-zero\nα0andβ0as the dimensionless coefficients. Our new extension ofthe uncerta inty principle ( 22) is directly linked\nto the non-zero cosmological constant, becoming an essential ele ment for a more comprehensive understanding\nof quantum spacetime in both small and large scales of the cosmos. T he constant αin our proposed MUP is\nnot meaningless but rather is proportional to the energy of the va cuum (quasi-empty or void) space. In our\nview, the cosmological constant varies at different scales as Λ ∼R−2(this point has been mentioned in [ 70]\nby G. Lemaitre in 1927), with the highest value of energy-momentum (∆Pmax) at the smallest scale (∆ Xmin)\nand vice versa. But neither ∆ Xnor ∆Pcan be assigned a zero value because the cosmological constant ne ver\ndisappears at any scale.\nAlthough we didn’t resolve the cosmological constant issue or introd uce a new version of the uncertainty\nprinciple, we have determined that presence of a non-zero cosmolo gical constant is essential at all levels, and its\nvalue is contingent on the selected regime. In our view, the significan t contrast between the highest and lowest\nvalues of the cosmological constant is not a concern. Our proposa l suggests that the difference in the values of\ncosmological constant is inherent and should be considered natura l. In other hand, the presence of cosmological\nconstant in the corrected relation of Heisenberg’s uncertainty pr inciple may be a sign of the unification of\nquantum gravity and quantum cosmology in a single framework [ 71,72]. It is essential to acknowledge that the\nfindings presented in this paper are approximate and heuristic. Nev ertheless, given the extremely small or large\norders of magnitude of the values under investigation, these appr oximations closely reflect reality.\nVI. CONCLUSIONS\nWe have examined supervoids as interconnected spherical bubbles within a void-dominated cosmic fluid, with\nthe total mass of a supervoid situated on the shell formed by disk- shaped superclusters. Through heuristic\ncalculations of the mass density and cosmological constant of a sing le supervoid, we have demonstrated that\nthe obtained order of magnitude aligns interestingly with their corre sponding values for the entire observable\nuniverse.\nThe web-like cosmic fluid comprises two coexisting dynamical parts: g alaxies, clusters, superclusters, and\nnodes on one hand, and small voids and supervoids on the other. In void-dominated cosmology, voids are\ndominant and their existence and evolution play a crucial role in the ac celerating expansion of the cosmos. Our\nfindings indicate that a cosmic void can serve as an excellent represe ntative for studying both local and global\nscales. For each cosmicvoid, we deriveda specific massdensity and c osmologicalconstantthat, in terms oforder\nof magnitude, mirror those of the entire universe on any scale. Add itionally, we have demonstrated that the\nlargestdiscrepancybetweenthe observablecosmologicalconsta ntandthe earlyfoam-likecosmos(approximately\n10+122) can be logically explained within the void-based model of the universe . We have been concluded that\nthe presence of a cosmological constant is necessary at any scale s of cosmos, with its value depending on the\nchosen regime. The substantial difference between the highest an d lowest value of the cosmological constant is\nnot an issue, and according to our proposal, this difference is natur al. Furthermore, by introducing an extended\nversion of the uncertainty principle, we have demonstrated that a non-zero cosmological constant is essential for\nestablishing the minimum and maximum length in physics. From our persp ective, this non-zero cosmological\nconstant could be attributed to the existence of energy in near-e mpty space at any scale.\nWhile our analysis is less rigorous and largely heuristic, it provides a clea r view of problems in gravity and\ncosmology,by keeping the mathematics to the bare minimum. An intrigu ing consequence of this study is that10\na cosmic void, as a cell within the cellular structure of the cosmic web s caffolding, can effectively describe the\nbehavior of the universe on any scale. We anticipate that by more th oroughly investigating the behavior of\na single cosmic void as a cosmos laboratory, both theoretically and ob servationally, we can provide potential\nsolutions to the significant challenges of physical cosmology on local and global scales, and these are the topics\nwe will address in future projects.\nDECLARATION OF COMPETING INTEREST\nThe authors declare that they have no known competing financial in terests or personal relationships that\ncould have appeared to influence the work reported in this paper.\nACKNOWLEDGEMENTS\nEY would like to acknowledge A. Talebian and H. Sheikhahmadi for their constructive discussions and for\nhis help in drawing the tables. This work has been supported by the Is lamic Azad University, Ayatollah Amoli\nBranch, Amol, Iran.\n[1] J. Silk, JPS Conf. Proc. 14, 010101 (2017) ,arXiv:1611.09846 [astro-ph.CO] .\n[2] J. S. Peracaula, (2022), arXiv:2203.13757 [gr-qc] .\n[3] E. Abdalla et al.,JHEAp34, 49 (2022) ,arXiv:2203.06142 [astro-ph.CO] .\n[4] S. Vagnozzi, Universe 9, 393 (2023) ,arXiv:2308.16628 [astro-ph.CO] .\n[5] L. Perivolaropoulos and F. Skara, (2021), arXiv:2105.05208 [astro-ph.CO] .\n[6] L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations (Cambridge University Press, 2015).\n[7] E. Belgacem and T. Prokopec, Phys. Lett. B 831, 137174 (2022) ,arXiv:2111.04803 [astro-ph.CO] .\n[8] F. Zwicky, Astrophys. J. 86, 217 (1937) .\n[9] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003) ,arXiv:astro-ph/0207347 .\n[10] R. H. Sanders and S. S. McGaugh, Ann. Rev. Astron. Astrophys. 40, 263 (2002) ,arXiv:astro-ph/0204521 .\n[11] E. P. Verlinde, JHEP04, 029 (2011) ,arXiv:1001.0785 [hep-th] .\n[12] A. Joyce, L. Lombriser, and F. Schmidt, Ann. Rev. Nucl. Part. Sci. 66, 95 (2016) ,arXiv:1601.06133 [astro-ph.CO] .\n[13] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989) .\n[14] S. M. Carroll, Living Rev. Rel. 4, 1 (2001) ,arXiv:astro-ph/0004075 .\n[15] J. Martin, Comptes Rendus Physique 13, 566 (2012) ,arXiv:1205.3365 [astro-ph.CO] .\n[16] M. Einasto, P. Hein¨ am¨ aki, L. J. Liivam¨ agi, V. J. Mart inez, L. Hurtado-Gil, P. Arnalte-Mur, P. Nurmi, J. Einasto,\nand E. Saar, Astron. Astrophys. 587, A116 (2016) ,arXiv:1506.05295 [astro-ph.CO] .\n[17] E. Yusofi, M. Khanpour, B. Khanpour, M. A. Ramzanpour, an d M. Mohsenzadeh,\nMon. Not. Roy. Astron. Soc. 511, L82 (2022) .\n[18] Y. C. Ong, JCAP09, 015 (2018) ,arXiv:1804.05176 [gr-qc] .\n[19] M. Heydarzadeh, M. Mohsenzadeh, M. Abbasiyan-Motlaq, and E. Yusofi, Mod. Phys. Lett. A 37, 2250191 (2022) .\n[20] P. Bosso, Generalized Uncertainty Principle and Quantum Gravity Phe nomenology , Ph.D. thesis, Lethbridge U.\n(2017),arXiv:1709.04947 [gr-qc] .\n[21] L. Perivolaropoulos, Phys. Rev. D 95, 103523 (2017) ,arXiv:1704.05681 [gr-qc] .\n[22] K. Nozari and P. Pedram, EPL92, 50013 (2010) ,arXiv:1011.5673 [hep-th] .\n[23] K. Nozari and A. Etemadi, Phys. Rev. D 85, 104029 (2012) ,arXiv:1205.0158 [hep-th] .\n[24] R. Van de Weygaert and E. Platen, International Journal of Modern Physics: Conference Serie s01, 41–66 (2011) .\n[25] R. K. Sheth and R. van de Weygaert, Mon. Not. Roy. Astron. Soc. 350, 517 (2004) ,arXiv:astro-ph/0311260 .\n[26] L. Rudnick, S. Brown, and L. R. Williams, Astrophys. J. 671, 40 (2007) ,arXiv:0704.0908 [astro-ph] .\n[27] T. Cai, Q. Ding, and Y. Wang, (2022), arXiv:2211.06857 [astro-ph.CO] .11\n[28] S. Nadathur, M. Lavinto, S. Hotchkiss, and S. R¨ as¨ anen ,Phys. Rev. D 90, 103510 (2014) ,\narXiv:1408.4720 [astro-ph.CO] .\n[29] Y. Higuchi and K. T. Inoue, Mon. Not. Roy. Astron. Soc. 476, 359 (2018) ,arXiv:1707.07535 [astro-ph.CO] .\n[30] B. Pandey, Mon. Not. Roy. Astron. Soc. 485, L73 (2019) ,arXiv:1901.08475 [astro-ph.CO] .\n[31] B. Pandey and B. Das, Mon. Not. Roy. Astron. Soc. 485, L43 (2019) ,arXiv:1812.08816 [astro-ph.CO] .\n[32] Y. Nan and K. Yamamoto, Phys. Rev. D 105, 063518 (2022) ,arXiv:2111.14174 [astro-ph.CO] .\n[33] Q. Vigneron and T. Buchert, Class. Quant. Grav. 36, 175006 (2019) ,arXiv:1902.08441 [gr-qc] .\n[34] T. Buchert, PoSEDSU2018 , 038 (2018) ,arXiv:1810.09188 [gr-qc] .\n[35] S. K. Srivastava and J. Dutta, (2007), arXiv:0708.4139 [gr-qc] .\n[36] C. M. Correa, Cosmic voids as cosmological laboratories , Ph.D. thesis, Cordoba U. (2021),\narXiv:2210.17459 [astro-ph.CO] .\n[37] H. Y. J. Chan, M. Chiba, and T. Ishiyama, Mon. Not. Roy. Astron. Soc. 490, 2405 (2019) ,\narXiv:1910.02197 [astro-ph.CO] .\n[38] I. H. Redmount, Monthly Notices of the Royal Astronomical Society 235, 1301 (1988) ,\nhttps://academic.oup.com/mnras/article-pdf/235/4/13 01/2965531/mnras235-1301.pdf .\n[39] E. Yusofi, Mod. Phys. Lett. A 33, 1850219 (2018) .\n[40] S.Mohammadi, E.Yusofi, M.Mohsenzadeh, andM.K.Salem, Monthly Notices of the Royal Astronomical Society 525, 3274 (2023)\nhttps://academic.oup.com/mnras/article-pdf/525/3/32 74/51330952/stad2514.pdf .\n[41] T. Buchert et al.,Class. Quant. Grav. 32, 215021 (2015) ,arXiv:1505.07800 [gr-qc] .\n[42] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996) ,arXiv:astro-ph/9508025 .\n[43] R. K. Sheth and G. Tormen, Mon. Not. Roy. Astron. Soc. 308, 119 (1999) ,arXiv:astro-ph/9901122 .\n[44] E. Yusofi and M. Mohsenzadeh, Astronomy Education Review 9(2010), 10.3847/aer2009035 .\n[45] A. Pisani, P. M. Sutter, N. Hamaus, E. Alizadeh, R. Biswa s, B. D. Wandelt, and C. M. Hirata,\nPhys. Rev. D 92, 083531 (2015) ,arXiv:1503.07690 [astro-ph.CO] .\n[46] G. Lavaux and B. D. Wandelt, The Astrophysical Journal 754, 109 (2012) .\n[47] N. Hamaus, A. Pisani, J.-A. Choi, G. Lavaux, B. D. Wandel t, and J. Weller, JCAP12, 023 (2020) ,\narXiv:2007.07895 [astro-ph.CO] .\n[48] T.-P. Cheng, Relativity, Gravitation and Cosmology A Basic Introductio n(Oxford University Press, 2010).\n[49] G. Darmois, M´ emorial des sciences math´ ematiques 25, 58 (1927) .\n[50] C.-S. Chu and H. S. Tan, Universe 8, 250 (2022) ,arXiv:2103.06314 [hep-th] .\n[51] E. Yusofi, M. Khanpour, B. Khanpour, M. Ramzanpour, and M . Mohsenzadeh, (2019),\narXiv:1907.12418 [astro-ph.CO] .\n[52] R. van de Weygaert, IAU Symp. 308, 493 (2014) ,arXiv:1611.01222 [astro-ph.CO] .\n[53] G. Lavaux and B. D. Wandelt, Astrophys. J. 754, 109 (2012) ,arXiv:1110.0345 [astro-ph.CO] .\n[54] T. P. Cheng, Relativity, gravitation, and cosmology: A basic introduct ion(Oxford Univ. Pr., Oxford, UK, 2010).\n[55] S. K. Srivastava, General Relativity and Cosmology (PHI, India, 2008).\n[56] N. I. Libeskind et al.,Mon. Not. Roy. Astron. Soc. 473, 1195 (2018) ,arXiv:1705.03021 [astro-ph.CO] .\n[57] It is necessary to pay attention to the fact that most of t he energy density of the bubbles is concentrated on their\nborder, while the energy of drops is located in their volume a nd concentrated in their central points.\n[58] H. J. Butt, K. Graf, and M. Kappl, Physics and Chemistry of Interfaces (WILEY-VCH, Germany, 2003).\n[59] L. E. Reichl, A Modern Course in Statistical Physics, (WILEY-VCH, Germany, 2016).\n[60] L. Xu, J. Lu, and W. Li, Physics Letters B 690, 333–336 (2010) .\n[61] P. J. E. Peebles and B. Ratra, Astrophys. J. Lett. 325, L17 (1988) .\n[62] N. Aghanim et al.(Planck), Astron. Astrophys. 641, A6 (2020) , [Erratum: Astron.Astrophys. 652, C4 (2021)],\narXiv:1807.06209 [astro-ph.CO] .\n[63] A. E. Shalyt-Margolin, in 14th International School and Conference on Foundation and Advances in Nonlinear\nScience (FANS 2008) (2008) pp. 103–107, arXiv:0807.3485 [hep-th] .\n[64] V. Jejjala and D. Minic, Int. J. Mod. Phys. A 22, 1797 (2007) ,arXiv:hep-th/0605105 .\n[65] T. Padmanabhan, AIP Conf. Proc. 861, 179 (2006) ,arXiv:astro-ph/0603114 .\n[66] M. Bishop, J. Contreras, P. Martin, and D. Singleton, Front. Astron. Space Sci. 9, 978898 (2022) ,\narXiv:2209.03172 [physics.gen-ph] .\n[67] M. J. Lake, M. Miller, R. Ganardi, and T. Paterek, Front. Astron. Space Sci. 10, 1087724 (2023) ,\narXiv:2302.08120 [gr-qc] .\n[68] B. Bolen and M. Cavaglia, Gen. Rel. Grav. 37, 1255 (2005) ,arXiv:gr-qc/0411086 .\n[69] C. Bambi and F. R. Urban, Class. Quant. Grav. 25, 095006 (2008) ,arXiv:0709.1965 [gr-qc] .\n[70] J.-P. Luminet, Gen. Rel. Grav. 45, 1619 (2013) ,arXiv:1305.6470 [physics.hist-ph] .12\n[71] C. Rovelli, Quantum Gravity , Cambridge Monographs on Mathematical Physics (Cambridge UniversityPress, 2004).\n[72] L. Chataignier, C. Kiefer, and P. Moniz, Class. Quant. Grav. 40, 223001 (2023) ,arXiv:2306.14948 [gr-qc] ." }, { "title": "2401.16143v1.Effect_of_a_critical_magnetic_field_on_the_control_of_scalar_neutral_boson_pair_production_in_the_context_of_Lorentz_symmetry_violation.pdf", "content": "arXiv:2401.16143v1 [hep-th] 29 Jan 2024Effect of a critical magnetic field on the control of scalar neu -\ntral boson pair production in the context of Lorentz-symmet ry\nviolation\nAndr´es G. Jir ´on1, Angel E. Obispo1,2, J. Daniel Espinoza Loayza3, Juan Carlos Quispe1andL. B.\nCastro4.\n1Universidad Tecnol´ ogica del Per´ u (UTP), Lima, Per´ u\n2Universidad Privada del Norte (UPN), Lima, Per´ u\n3Universidad Nacional del Callao (UNAC), Bellavista, Calla o, Per´ u\n4Universidade Federal do Maranh˜ ao (UFMA), S˜ ao Lu´ ıs, MA, B razil\nAbstract –This study investigates the production of neutral scalar b oson pairs in static elec-\ntromagnetic fields resulting from Lorentz-symmetry violat ion (LSV), with a focus on the parity-\neven sector of the CPT-even photon sector in the Standard Mod el Extension (SME). Utilizing\na cross-configuration involving inhomogeneous static elec tric fields and homogeneous static mag-\nnetic fields, the analysis of the probability of bosons pair p roduction identifies three different\nregimes determined by critical magnetic field. Below the cri tical value, creation is exponentially\nsuppressed; at the critical value, the number density of cre ated bosons remains constant, and\nabove the critical field, there is exponential amplification . This behavior prompts an additional\ninvestigation using von Neumann entanglement entropy to an alyze fluctuations in the bosonic\nvacuum.\nINTRODUCTION. – The Sauter-Schwinger effect\n[1–3] occurs when an intense electric field induces spon-\ntaneous production of particle-antiparticle pairs, such as\nelectrons and positrons, from vacuum energy. This phe-\nnomenon manifests itself in situations where the intensity\nof a static homogeneous electric field exceeds its critical\nvalueEsch≈1.30×1018V/m, resulting in the vacuum\nundergoing a reconfiguration of its properties and giving\nrise to the spontaneous creation of virtual particles.\nIntroducing a homogeneous static magnetic field to the\nsystem has demonstrated that, while this field cannot in-\ndependently generate particles, its effects decrease pair\nproduction rate [ 4]. When the fields are neither homo-\ngeneous nor static, the theoretical techniques for analyz-\ning pair production become more complex. Methods such\nas perturbative expansions [ 5],S-matrix theory [ 6], path\nintegrals [ 7] and numerical simulations in space-time [ 8],\nare challenged by the increasing demands of the system.\nHowever, for specific field configurations, the production\noffermion pairs has been successfully analyzed [ 9–13]. For\nexample, in [ 11,13], a configuration formed by a magnetic\nand electric field perpendicular to each other was studied.\nIn this case, the magnetic field induces a cyclotron mo-tion in the newly produced pairs, driving an electron back\nto the creation region. However, due to the action of the\nPauliblockingprinciple, whichpreventsanewlygenerated\nelectron from occupying the same state, the fermion pair\nproduction is significantly suppressed. As is evident, this\nphenomenon is typical of particles that obey the Fermi-\nDiracstatistics. Thus,inahypotheticalscenario,ifbosons\nreplaced fermions, it is believed that the boson pair pro-\nduction would be amplified rather than suppressed [ 14].\nRegarding the process of boson pair production, it is es-\nsential to note that the mass of the lightest boson, the π\nmeson, is270times greaterthanthat ofthe electron. Con-\nsequently, since the critical intensity of the fields depends\non the square of the rest mass of the created particle, it is\nestimated that the intensity of the fields should be about\n105 times greater than the Schwinger limit for fermions\n[15]. However, although this energy regime is currently\ninaccessible to the most powerful lasers, the process of\nboson production continues to be the subject of many in-\nteresting theoretical investigations [ 14–16].\nIn this paper, the production of pairs of neutral scalar\nbosons in static electromagnetic fields is studied. These\nfields are induced by Lorentz-symmetry violation (LSV).\np-1Andr´ es G. Jir´ on et al.\nSpecifically, the parity-even sector of the CPT-even pho-\nton sectorofthe StandardModel Extension(SME) [ 17,18]\nwill be chosen to define a cross configuration of an inho-\nmogeneous static electric field and a homogeneous static\nmagnetic field. In this context, it will be shown that the\nprocess of boson pair production is divided into three dis-\ntinct regimes, which are determined by the presence of a\ncritical magnetic field. Thus, the production process is\nexponentially suppressed if the magnetic field is less than\nits critical value. If the magnetic field equals its critical\nvalue, the number of produced bosons remains constant.\nFinally, the number of produced bosons is exponentially\namplified for values greater than the critical field. This\nbehavior is quite unexpected and exciting, so we will also\nstudy the von Neumann entanglement entropy to analyze\nthe behavior of the fluctuations of the bosonic vacuum.\nThe remainder of this paper is organized as follows. In\nthe next section, we solve the Klein-Gordon equation in-\nfluenced by a crossed arrangement of a radial electric field\nand an axial magnetic field, represented through a non-\nminimal coupling based on the Lorentz-symmetry viola-\ntion. In the third section, we use the Bogoliubov transfor-\nmations to obtain analytical expressions associated with\nthe probability of boson production. In the fourth sec-\ntion, we analyze the von Neumann entanglement entropy\nand the bosonic vacuum fluctuations under the influence\nof the critical magnetic field. Finally, we conclude in the\nfifth section.\nKlein–Gordon equation in the background of\nLorentz-symmetry violation. – Starting from the\nCPT-evensector of the SME [ 17,18], the relativistic quan-\ntum dynamics of a massive neutral scalar particle is ex-\namined under the effects of LSV. This will be achieved by\nintroducing the following non-minimal coupling\nˆpµˆpµ→ˆpµˆpµ−g\n4(KF)µναβFµνFαβ,(1)\nwheregis a coupling constant, Fµνstands for the elec-\ntromagnetic tensor, and ( KF)µναβis the tensor govern-\ning Lorentz violation in CPT-even electrodynamics within\nthe SME framework. This tensor, which has 19 compo-\nnents, exhibits the same symmetries as the Riemann ten-\nsor. These symmetries can be expressed in terms of 4\nmatrices of size 3 ×3, defined as\n(κDE)jk=−2(KF)0j0k, (2)\n(κHB)jk=1\n2(KF)pqlmεjpqεklm, (3)\n(κDB)jk=−(κHE)kj= (KF)0jpqεkpq.(4)\nHere, the symmetric κDEandκHB, with 11 independent\ncomponents, comprise the even-parity sector. In contrast,\ntheκDBandκHEmatrices, with eight components and no\ninherent symmetry, constitute the odd-parity sector of the\n(KF)µναβtensor. Consequently, the dynamics of neutralscalar particles with mass munder Lorentz violation ef-\nfects are governed by the following Klein-Gordon equation\nin natural units\nˆpµˆpµψ+g\n2(κDE)jpEjEpψ+g(κDB)jωEjBωψ\n−g\n2(κHB)bcBbBcψ=m2ψ,(5)\nwhereEi=F0iandBi=1\n2ǫijkFjkrepresent the electric\nand magnetic fields, respectively. The following configu-\nration will be adopted for the non-null components of the\ntensor (KF)µναβ\ng(κDE)rr=−κ1, g(κDB)rz=κ3, (6)\nwhereκ1andκ3are positive constants. In that scenario,\nthe background of Minkowski space-time is considered in\ncylindrical coordinates\nds2=−dt2+dr2+r2dϕ2+dz2. (7)\nIn this context, based on the study of induced electric\ndipole moment systems [ 19,20] and other scenarios that\ninvolve LSV [ 21,22], it is possible to define the following\nconfiguration for the crossed electric and magnetic fields\n/vectorE=λ\nrˆr,/vectorB=Bˆz, (8)\nwhereλis a linear electric charge density, Bis a constant\nmagnetic field, and ˆ rand ˆzare unit vectors in the radial\ndirection and zdirection, respectively. Thus, considering\nthe expressions given in ( 6) and (8), the equation ( 5) can\nbe rewrite as\n/bracketleftBig\n−∂2\n∂t2+∇2+κ1\n2λ2\nr2−κ3λB\nr−m2/bracketrightBig\nψ= 0.(9)\nHere,∇2is the Laplacian operator in the cylindrical co-\nordinate system. Given the cylindrical symmetry of the\nequation ( 9), it is possible to choose the following ansatz\nψ(r) =e−iωteilϕeikzzφ(r)√r, (10)\nwhereωrepresentsthe energy ofthe system, lis the eigen-\nvalue of the angular momentum operator, and kzis the\nwave-number in the z-direction. In this work, we consider\nthe case where the dynamics of the boson are confined to\nthe plane (kz= 0), obtaining\nd2φ(r)\ndr2+/bracketleftbigg\n˜ω2−δ\nr−γ2\nl−1\n4\nr2/bracketrightbigg\nφ(r) = 0,(11)\nwith\n˜ω2=ω2−m2, δ=κ3λB, (12)\nγl=/radicalbigg\nl2−κ1λ2\n2. (13)\nWhenδ >0 andκ1λ2\n2≤l2, equation ( 11) represents the\nSchr¨ odinger equation for the repulsive Coulomb-like po-\ntential. However, ifκ1λ2\n2≥l2, the potential structure in\np-2Effect of a critical magnetic field on the control of scalar neutral b oson pair production\n(11) is transformed into that of a well and γlbecomes\nimaginary, akin to the problem of spontaneous pair pro-\nductionofparticlesinduced byaCoulombfield [ 23–25]. In\nthe interest of analyzing the production of scalar bosons,\na redefinition of γlis performed\nγl=i¯γl=i/radicalbiggκ1\n2/radicalBig\nλ2−λ2\nl, (14)\nwhere\nλl=/radicalbigg\n2\nκ1l, (15)\nunder the condition |λ| ≥ |λl|. Given the aforementioned\nmodifications and with z=−2i˜ωr, equation ( 11) is trans-\nformed as follows\nd2φ\ndz2+/bracketleftbigg\n−1\n4−iη\nz+¯γ2\nl+1/4\nz2/bracketrightbigg\nφ= 0 (16)\nwhereηis define as\nη=δ\n2˜ω, (17)\nwith ˜ω∈R. The second-order differential equation given\nin (16) is the so-called Whittaker equation, which admits\ntwo linearly independent regular solutions, given by\nφ=C1M−iη,i¯γl(z)+C2W−iη,i¯γl(z),(18)\nwhere the first solution is bounded at z= 0, while the\nsecond is bounded at |z| → ∞. The next section will use\nthese solutions to analyze the pair production process.\nBogoliubov transformation and probability of\nscalar bosons pair production in the Lorentz-\nviolating background. – In accordance with the pre-\nscription used in [ 26–29], the in-out states, as described in\nequation ( 18), are defined as follows:\nφ+\nin(r) =C1M−iη,i¯γl(−2i˜ωr), (19)\nφ+\nout(r) =C2W−iη,i¯γl(−2i˜ωr), (20)\nφ−\nin=/bracketleftbig\nφ+\nin(r)/bracketrightbig∗=C∗\n1Miη,−i¯γl(2i˜ωr),(21)\nφ−\nout=/bracketleftbig\nφ+\nout(r)/bracketrightbig∗=C∗\n2Wiη,−i¯γl(2i˜ωr).(22)\nThe indices ±denote the modes with positive and neg-\native frequency, respectively. These states are connected\nthrough the so-called Bogoliubov transformations, which\nareobtained using the followingrelationsofthe Whittaker\nfunctions [ 30]\nMk,µ(ze±iπ) =e±iπ(1/2+µ)M−k,µ(z),(23)\nWk,µ(z) =Γ(−2µ)\nΓ(1/2−µ−k)Mk,µ(z)\n+Γ(2µ)\nΓ(1/2+µ−k)Mk,−µ(z).(24)Mk,µ(z)\nΓ(1+2µ)=e±(κ−µ−1/2)iπ\nΓ(1/2+µ+k)Wk,µ(z)\n+e±κiπ\nΓ(1/2+µ−k)W−k,µ(e±iπz).(25)\nThe relation between the states φ±\nin(r) andφ±\nout(r) are\ngiven by\nφ+\nin(r) =α∗φ+\nout(r)−βφ−\nout(r). (26)\nφ+\nout(r) =αφ+\nin(r)+βφ−\nin(r), (27)\nwhere the Bogoliubov coefficients αandβare expressed\nas\nα=C2\nC1Γ(−2i¯γl)\nΓ(1/2−i¯γl+iη), (28)\nβ=C∗\n2\nC1Γ(2i¯γl)\nΓ(1/2−i¯γl−iη)e−π¯γleiπ/2,(29)\nwhich satisfies the conditions\n|α|2−|β|2= 1, (30)\nwith\n|C2|2\n|C1|2= 2¯γleπ(η+¯γl). (31)\nA straightforward calculation allows the establishment of\naconnectionbetweenthestates φ±\nin(r) andφ±\nout(r) andthe\ncreation/annihilation operators in quantum field theory\nain(r) =α∗bout(r)−βb†\nout(r), (32)\nbout(r) =αain(r)+βa†\nin(r). (33)\nConsequently, using the coefficients αandβ, along with\nthe relations involving gamma functions, it is possible to\nobtain an expression for the probability of scalar bosons\npair production in the following way\nP=|β|2\n|α|2=cosh[π(¯γl+η)]\ncosh[π(¯γl−η)]e−2π¯γl.(34)\nNote that the equation ( 34) exhibits a decreasingbehavior\ndue to the presence of the negative exponential, which\ndepends on both the intensity of the charge density λand\nthe value of l. On the other hand, it reaches a maximum\nprobability of P= 1 when λ=λl. It is interesting to\nhighlight that this behavior persists even in the absence\nofamagneticfield( B= 0), therebyre-expressingequation\n(34) as follows\nP0= exp/bracketleftbigg\n−2π/radicalbiggκ1\n2/parenleftbigg/radicalBig\nλ2−λ2\nl/parenrightbigg/bracketrightbigg\n,(35)\nwhere it can be seen that λlgenuinely represents a thresh-\nold value, marking the point at which the boson pair pro-\nduction process begins. Nevertheless, the presence of the\nmagnetic field also plays a significant role in this process,\nevenwhen theelectricfield intensityfarexceedsits thresh-\nold value (λ≫λl). In this scenario, the expression for the\nprobability ( 34) is approximated as follows\nP≈1+e−ζλ(B+Bc)\n1+e−ζλ(B−Bc), (36)\np-3Andr´ es G. Jir´ on et al.\nB=1.01Bc\nB=1.00Bc\nB=0.99Bc\n0204060801000.00.20.40.60.81.0\n(Chargedensity )P(Probability )\nFig. 1: Probability of bosons pair production as a\nfunction of charge density λfor different magnetic\nfieldvalues B, wherel= 1, ˜ω= 1.12andκ1=κ3= 1.\nwhereζ=πκ3\n˜ωandBcis given by\nBc=√2κ1\nκ3˜ω. (37)\nFrom (36), three asymptotic approximations for the prob-\nability can be identified based on the value of Bc, as de-\npicted in Figure 1. WhenB < B c, the probability con-\nsistently decreases, converging to zero. At critical point\nB=Bc, the probability converges to1\n2. Conversely,\nforB > B c, the probability increases approaching one.\nTherefore, when λ≫λl, the magnetic field acts as a tran-\nsition parameter between two distinct states: production\nand vacuum, where Bcplays the role ofcritical parameter.\nInthecaseofothervaluesof λ, particularlyat B= 0.99Bc\n(dotted blue line), B= 1.00Bc(solid black line), and\nB= 1.01Bc(dot-dashed red line), a notable convergence\namong the three probability curves occurs precisely at the\ncritical threshold λl=√\n2 (vertical dashed line), which\nis determined by equation ( 15). AtB= 1.00Bc, the\nprobability decays similarly to the case of B= 0.99Bc,\nyet it converges to a constant value of P= 1/2, suggest-\ning equiprobability for all possible quantum states. Con-\nversely, for B= 1.01Bc, the probability initially decreases\nto a minimum point at λmin= 10.07 and thereafter grad-\nually rises until it approaches one. By computing the first\nand second derivatives of ( 34), additional minimum points\ncan be identified. For example, at B= 1.08Bc, the min-\nimum point is located at λmin= 3.74, with a probability\nP= 0.93. If the magnetic field is slightly increased to\nB= 1.25Bc, the minimum point shifts to λmin= 2.36\nandP= 0.99. These results indicate that as the magnetic\nfield gradually increases, λminapproaches the threshold\nvalueλland the probability approaches one.\nOther scenarios involving different values of magnetic\nfield and charge density can be explored through a den-\nsity plot displaying the probability of boson pair creation,\n0 0.2 0.4 0.6 0.8 1.0\nFig. 2: Density plot of the probability of bosons pair\nproductionasafunction ofchargedensity λandmag-\nnetic field values B, wherel= 1, ˜ω= 1.12 and\nκ1=κ3= 1.0.\nas depicted in Figure 2. Notably, below the critical mag-\nnetic fieldBc= 1.584 (horizontal white dashed line), two\ndistinct regions are evident, indicated by light and dark\ncolors, which represent high and low probabilities, respec-\ntively. Here, a noticeable trend emerges: a higher prob-\nability region occurs for charge densities below λ≈15,\nwhereas lower probability regions are observed for both\nhigher chargedensities surpassingthis threshold and mag-\nnetic fields below the critical value Bc. In contrast, when\nthe magnetic field exceeds the critical value, only regions\nof high probability exist for any given charge density. It\nis interesting to note that the probability of boson pair\nproduction is sensitive to slight changes in the magnetic\nfield around the critical value Bc. This behavior is also\nobserved in the profile of the number density of produced\nescalar-neutralbosons, which will be discussed in the next\nsection.\nBoson pair production in the Lorentz-violating\nbackground. – To calculate the number density of cre-\nated particles or number of created particles per state, the\nmatrix element given by equation ( 38) is employed\nˆn=/angbracketleftBig\n0in|a†\nout,aout|0in/angbracketrightBig\n=|β|2. (38)\nUsing equations ( 30) and (34), the number density of cre-\nated neutral bosons is determined as follows\nˆnl=cosh[π(η+ ¯γl)]\nsinh2π¯γleπ(η−¯γl). (39)\nIt is noteworthy that the expression in equation ( 39) ex-\nhibits divergence at ¯ γl= 0, a condition met only when\np-4Effect of a critical magnetic field on the control of scalar neutral b oson pair production\nB=1.01Bc\nB=1.00Bc\nB=0.99Bc\n0204060801000246810\nλ(Chargedensity )n(Numberofparticles )\nFig. 3: Number density of created bosons as a function of\ncharge density λfor different values of the magnetic field\nB, wherel= 1, ˜ω= 1.12 andκ1=κ3= 1.0.\nλ=λl. This divergence suggests an infinite production\nof bosons, which can be interpreted as an infinite num-\nber of continuous states condensed within an infinitesimal\nspace [31]. On the other hand, when B= 0, the number\ndensity given by Equation ( 39) can be represented as a\nBose-Einstein distribution, expressed as\nˆn0l=1\ne2π√κ1\n2λ−1. (40)\nItisobservedthat ˆ n0lapproacheszeroas λsignificantlyin-\ncreases. This notable absence of created states might sug-\ngest that the energy required to generate boson-antiboson\npairs is sufficiently high, resulting in minimal neutral-\nscalar bosons production.\nInFigure 3, thebehaviorofthe numberdensityofcreated\nbosons is depicted as a function of charge density, consid-\nering specific magnetic field values: B= 0.99Bc(dotted\nblue line),B= 1.00Bc(solid black line), and B= 1.01Bc\n(dot-dashed red line). As observed, for B= 0.99Bc, the\nnumber of created bosons tends to asymptotically decay\nto zero for large values of λ, similar to the case where\nB= 0. This exponential suppression is unusual, as it is\nexpected that the Pauli blocking should be responsible for\namplifying the production of boson pairs in the presence\nof a magnetic field [ 14]. ForB= 1.00Bc, the number of\nbosons decreases as the charge density increases. How-\never, unlike the previous case, it converges to one. This\nbehavior suggests that when the magnetic field reaches\nits critical value, the rate at which boson pairs are cre-\nated matches the rate at which they are annihilated. For\nB= 1.01Bc, the number density of created bosons was\nsuppressed below λmin= 10.07, followed by a rapid in-\ncrease forλ > λ min. This phenomenon, known as Bose\nenhancementisacharacteristicbehaviorobservedinparti-\ncle production processes governed by Bose-Einstein statis-\ntics [12]. Regarding alternative scenarios encompassing\n0 2 4 6 8\nFig. 4: Density plot of the number density of created\nbosonsasafunction ofchargedensity λandmagnetic\nfield values B, wherel= 1, ˜ω= 1.12 andκ1=κ3=\n1.0.\nvarious magnetic field and charge density values, Figure\n4presents a plot representation of the behavior of the\nnumber density of created bosons as a function of these\nmagnitudes. Below the critical value of the magnetic field,\nBc= 1.584(horizontalwhitedashedline), thereisaregion\nrepresented by colors ranging from dark to even darker.\nThis suggests that in this scenario, the number density of\ncreated bosons decreases as the charge density increases.\nIn contrast, when the magnetic field exceeds the critical\nvalue, unlike the previous context, now a region with dark\ncolorsgraduallytransitioningtolightershadesisobserved.\nThis indicates a significant increase in the number density\nof created bosons for high values of charge density.\nThe results in this section illustrate the susceptibil-\nity of the pair production process to minor fluctuations\naround the critical magnetic field, Bcinduced by Lorentz-\nviolating. This observation suggests a potential magnetic\nphase transition within a quantum vacuum, which alters\nthe conventionalparticleproductionprocess[ 32]. Todelve\ndeeper into this phenomenon, the subsequent section ex-\nplores quantum vacuum fluctuations using the von Neu-\nmann entanglement entropy approach.\nvon Neumann entanglement entropy in the LSV\nbackground. – In few words, the von Neumann en-\ntanglement entropy quantifies the degree of entanglement\nwithin a system [ 33–36]. Mathematically expressed as\nS= (ˆn+1)log(1+ ˆ n)−ˆnlogˆn. (41)\nThis equationcharacterizesthe entropybasedonthenum-\nber density of the created boson pairs, defined in equation\np-5Andr´ es G. Jir´ on et al.\nB=1.01Bc\nB=1.00Bc\nB=0.99Bc\n0204060801000123456\nλ(Chargedensity )\nS(\nE\u0000\u0001\u0002\u0003\u0004\u0005 )\nFig. 5: von Neumann entropy as a function of charge den-\nsityλfordifferentvaluesofthemagneticfield, where l= 1,\n˜ω= 1.12,κ1=κ3= 1.0.\n(39).\nIn Figure 5, the entropy behavior as a function of the\ncharge density is depicted, utilizing the identical magnetic\nfield values of Bthat were employed in the preceding\nsection. For B= 0.99Bc, the entropy asymptotically\ndecreases to zero. This indicates that in scenarios with\nhigh charge density and magnetic fields below their crit-\nical value, quantum vacuum fluctuations become practi-\ncally negligible. In such situations, the system is perfectly\ndefined and pure [ 37]. ForB= 1.00Bc, it is observed\nthat the entropy decreases; however, it tends to stabilize\nat a constant value. This behavior arises because, in this\nscenario, the asymptotic behavior of the number of parti-\ncles approaches to one, subsequently causing the entropy\nto converge to S= 2 log2. This pattern could suggest\na potential quantum equilibrium state when the critical\nvalue ofBis reached. For B= 1.01Bc, it is observed that\nthe entropy decreases until it reaches a minimum located\natλmin= 10.07, and for values above this point, the en-\ntropy increases linearly. This behavior in the entropy sug-\ngests that it approaches a state of maximum disorder and\nreduced predictability when the charge density increases\ndramatically. Exploring various scenarios where the mag-\nnetic field and chargedensity take on different values, Fig-\nure6provides a plot representation of the behavior of von\nNeumann entropy in relation to the aforementioned mag-\nnitudes. As observed, similar to the previous case (see\nFigure4), below the critical magnetic field Bc= 1.584\n(horizontal white dashed line), there is a region ranging\nfrom a dark color to an even darker one. This suggests\nthat in this region, there is low entanglement entropy as\nthe charge density increases. When the magnetic field ex-\nceeds its critical value, a dark region is observed gradually\nlightening, indicating a slight increasein entanglement en-\ntropy as the charge density also increases.\n0 2 4 6\nFig. 6: Density plot of the von Neumann entropy as a\nfunction of charge density λand magnetic field values B,\nwherel= 1, ˜ω= 1.12 andκ1=κ3= 1.0.\nConclusions. – This investigation focused on explor-\ning the production of neutral scalar boson pairs within\nstatic electromagnetic fields caused by Lorentz-symmetry\nviolation in the parity-even sector of the CPT-even pho-\nton sector in the Standard Model Extension. This was\nachieved by implementing a cross-configuration incorpo-\nrating inhomogeneous static electric fields and homoge-\nneousstaticmagneticfields. ByemployingtheBogoliubov\ntransformation, an analytical expression was derived to\ncalculate the probability of scalar boson-pair production.\nAlthough the magnetic field itself did not directly pro-\nduce boson pairs, its influence not only affected the pair\nproduction process but also facilitated control over it by\nmodulating a critical parameter, denoted as Bc.\nBy evaluating Bc, three distinct asymptotic approxima-\ntions for the probability were identified. Below the critical\nthreshold (B < B c), the probability consistently dimin-\nishes, asymptotically approaching zero. At the critical\npoint (B=Bc), the probability stabilizes at1\n2. Con-\nversely, beyond the critical threshold ( B >B c), the prob-\nability increases, tending toward unity. This behavior\ndemonstrates that the magnetic field acts as a transition\nparameter between two distinct states: production and\nvacuum. Similarly, an expression for the number density\nof the created bosons was derived, which revealed differ-\nent behaviors. For B < B c, the density asymptotically\ndecreases. For B=Bc, it remained constant, while for\nB > B c, a Bose enhancement was observed [ 12]. On\nthe other hand, the von Neumann entanglement entropy\nasymptotically observed that, for B > B c, it asymptoti-\ncally decreases. In the case of B=Bc, it tends towards\na constant value, whereas for B 0,\nµ{fn−f0>λ} ≤exp/parenleftBig\n−λ2/2/bardblS(f)/bardbl2\n∞/parenrightBig\n. (1.4)3\nIn fact, this result was stated in [ 3] for the dyadic filtration in the unit cube [0,1)n,\nbut the proof works for an arbitrary discrete filtration (see [3], Theorem 3.1). The proof\nof (1.4) stated in [ 3] was suggested by Herman Rubin, which replaces a much longer\nargument of the authors of [ 3]. Namely, the proof is adapted to the proof of classical\nAzuma-Hoeffding’s martingale inequality [ 2,5]. It is based on a fundamental identity of\nsequential analysis known in statistics and Hoeffding’s lem ma [5]. Ivanishvili and Trail in\n[8] proved a version of inequality ( 1.4) for the classical martingale square function ( 1.2),\nconsidering homogeneous filtrations {Fk}, that is ifU∈Dkis the children of an element\nV∈Dk−1, thenµ(U)≥αµ(V), where 0<α≤1/2is a common constant for all such\nchoices.\nTheorem B ([8]).Letf={f0,f1,...,fn}be a discrete martingale with respect to an\nα-homogeneous filtration. Then\nµ{x∈X:fn−f0>λ} ≤exp/parenleftBig\n−αλ2//bardblS(f)/bardbl2\n∞/parenrightBig\n, λ> 0. (1.5)\nA counterexample provided in [ 8] shows that a sub-Gaussian estimate like ( 1.5), involv-\ning the classical square function, fails for non-homogeneo us filtrations.\n1.3.Khintchine type inequalities. One can state Theorems AandBin the terms of a\nmartingale difference sequences\nd={d1,d2,...,dn}, (1.6)\nsubstituting fn−f0=/summationtextn\nk=1dkin inequalities ( 1.4) and ( 1.5). For the sake of convenience\nin the sequel everything will be stated in terms of ( 1.6), denoting the square function ( 1.1)\nbyS(d). Denote\nAp,n= sup\nd={d1,d2,...,d n}/bardbl/summationtextn\nk=1dk/bardblp\n/bardblS(d)/bardbl∞, (1.7)\nwhere supis taken over all non-trivial martingale-difference sequen ces (1.6) with a fixed\nnumber of elements n. Obviously we have Ap,n≤Ap,n+1. Taking supin (1.7) over all\nmartingale-difference sequences ( 1.6), without restriction on the number of elements, we\nget another constant Ap, for which clearly we have Ap= limn→∞Ap,n. Considering only\nHaar or Rademacher martingales in ( 1.7), the corresponding constants will be denoted by\nAp,n(Haar) andAp,n(Rademacher) respectively. Obviously,\nAp,n≥Ap,n(Haar) ≥Ap,n(Rademacher) . (1.8)\nIt is well known that\nAp(Rademacher) = 21/2/parenleftBig\nΓ((p+ 1)/2))/√π/parenrightBig1/p, p> 2,\nand equivalently, every Rademacher sum satisfies the inequa lity\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1akrk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np≤21/2/parenleftBig\nΓ((p+ 1)/2))/√π/parenrightBig1/p/parenleftBiggn/summationdisplay\nk=1a2\nk/parenrightBigg1/2\n, (1.9)4 GRIGORI A. KARAGULYAN\nwhere the constant on the right hand side is the best possible . For even integers p≥4\nthe proof of this inequality goes back to Khintchine’s work [ 11], which sharpness later\nwas proved by Stechkin [ 17]. Later on Young [ 20] established ( 1.9) for all real numbers\np≥3, and finally in 1982 Haagerup [ 4] introduced a new method, proving sharp bound\n(1.9) for all parameters p>2. Komorowski in [ 12] proved\nAp,n(Rademacher) =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√nn/summationdisplay\nk=1rk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np, p ≥3, (1.10)\ni.e. ifp≥3andnare fixed, then the best constant in ( 1.9) is/vextenddouble/vextenddouble/vextenddouble1√n/summationtextn\nk=1rk/vextenddouble/vextenddouble/vextenddouble\np. For even\np>3this relation earlier was established by Efron [ 6] and Eaton [ 7]. The proof of ( 1.10)\nfor general parameters p≥3given in [ 12] is based on an inequality provided in [ 7]. See\nalso [ 14] for a detailed review of the subject.\nRemark 1.1. Recall that sub-Gaussian estimate ( 1.4) imply a Khintchine type inequality\nAp≤c√p, but this is a too rough approach to obtaining the exact value ofApeven if\nthe constant in ( 1.4) is sharp.\n1.4.Main results. In this paper we prove that inequalities in ( 1.8) must actually be\nequalities whenever p≥3. This implies extensions of Khintchine’s inequality ( 1.9) and\nrelation ( 1.10) for general discrete martingales. The main results of the p aper are the\nfollowing theorems.\nTheorem 1.1. For anyp>2we haveAp,n=Ap,n(Haar) .\nTheorem 1.2. Ifp≥3, then\nAp,n=Ap,n(Haar) =Ap,n(Rademacher) =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√nn/summationdisplay\nk=1rk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np, (1.11)\nwhererkare the Rademacher functions.\nTheorem 1.2is an extension of the results of [ 12] (see ( 1.10)) for general martingale-\ndifferences. Relation ( 1.11) implies the following Khintchine type inequality.\nCorollary 1.1. Ifp≥3, then for any discrete martingale-difference (1.6)we have the\nbound/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√nn/summationdisplay\nk=1rk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np/bardblS(d)/bardbl∞, (1.12)\nwhich is sharp, since for dk=rk/√nwe have equality in (1.12).\nIt follows from the central limit theorem that\nAp,n=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√nn/summationdisplay\nk=1rk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np→21/2/parenleftBig\nΓ((p+ 1)/2))/√π/parenrightBig1/pasn→ ∞5\nand applying the sharpness of ( 1.12), we can say that Ap,nis increasing with respect to\nnifp≥3. Thus from ( 1.12) it follows that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np≤21/2/parenleftBig\nΓ((p+ 1)/2))/√π/parenrightBig1/p/bardblS(d)/bardbl∞, p ≥3, (1.13)\nwhich is an extension of inequality ( 1.9) in the case p≥3for general discrete martingale-\ndifference sequences. Also we note that in the case of dyadic m artingale the square\nfunctionS(d)in (1.13) can be replaced by the classical one S(d). In particular,\nCorollary 1.2. If{hk}is the Haar system defined by (1.3)andp≥3, then the inequality\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1akhk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np≤21/2/parenleftBig\nΓ((p+ 1)/2))/√π/parenrightBig1/p/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBiggn/summationdisplay\nk=1a2\nkh2\nk/parenrightBigg1/2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞, p ≥3,(1.14)\nholds for any coefficients ak.\nLet us also state the following sharp sub-Gaussian inequali ties.\nCorollary 1.3. For any discrete martingale-difference (1.6)and for any number 0<λ<\n1/2we have\nE\nexp\nλ·/parenleftBigg/summationtextn\nk=1dk\n/bardblS(d)/bardbl∞/parenrightBigg2\n\n≤1√\n1−2λ, (1.15)\nwhere the constant on the right hand side is the best possible . Moreover, it is the best\nconstant if we consider only dyadic or Rademacher martingal es in (1.15).\nConsider the Orlicz space Lψof random variables, corresponding to the Young function\nψ(t) =et2−1, and equipped with the Luxembourg norm\n/bardblf/bardblψ= inf {u>0 :E[ψ(f/u)]≤1}.\nApplying ( 1.15) we can immediately get the following sharp bound.\nCorollary 1.4. Ifψ(t) =ex2−1, then for any discrete martingale-difference (1.6)it holds\nthe sharp bound\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nψ≤/radicalBigg\n8\n3· /bardblS(d)/bardbl∞. (1.16)\nRemark 1.2. It was proved by Peskir [ 15] the inequality\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1Xk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nψ≤/radicalBigg\n8\n3·/parenleftBiggn/summationdisplay\nk=1/bardblXk/bardbl2\n∞/parenrightBigg1/2\n(1.17)\nfor any sequence X={Xk:k= 1,2,...,n }of independent symmetric random variables.\nInequality ( 1.17) can be easily deduced from ( 1.16). Indeed, if Xis a discrete sequence,6 GRIGORI A. KARAGULYAN\nthen it becomes a martingale-difference on a discrete filtrat ion. So ( 1.16) holds, since\nusing independence we will have\nS(X) =/parenleftBiggn/summationdisplay\nk=1/bardblXk/bardbl2\n∞/parenrightBigg1/2\n.\nThe general case of ( 1.17) may be reduced to the discrete case, applying a standard\napproximation argument. We refer also related papers [ 16,19], where authors prove\nKhintchine type inequality for independent symmetric rand om variables, with other square\nfunctions in the spirit of ( 1.17).\nRemark 1.3. Theorems 1.1and1.2show certain extreme properties of Haar and Radema-\ncher random variables in a class of martingale difference seq uences, implying extensions of\nsome properties of Rademacher system to general martingale s. Some extreme properties of\nRademacher system in a class of uniformly bounded martingal e differences were considered\nin [9,10,13] (see also [ 1], chap. 9). Moreover, those papers consider more general\nsequences, namely multiplicative sequences of bounded ran dom variables φn,n= 1,2,...,\ni.e.E/bracketleftBig/producttext\nj∈Aφj/bracketrightBig\n= 0 for all nonempty finite subsets A⊂Nof positive integers. In\nparticular, the result of [ 9] states that if G:Rn→R+is a function convex with respect\nto each variables and φ={φk:k= 1,2,...,n }is a multiplicative system of random\nvariables, satisfying Ak≤φk(x)≤Bk, then\nE[G(φ1,...,φn)]≤E[G(ξ1,...,ξn)], (1.18)\nwhereξkare the {Ak,Bk}- valued independent mean zero random variables. The case of\nAk=−1andBk= 1of inequality ( 1.18) was proved in [ 13].\nRemark 1.4. It is a remarkable result of Wang [ 18] that for any p≥3and any condi-\ntionally symmetric martingale-difference d={d1,d2,...,dn}we have the sharp bound\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nk=1dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np≤Rp/bardblS(d)/bardblp, p ≥3\nwhereRpis the rightmost zero of the Hermite function Hp(x). HereHp(x)is the solution\nof the Hermite differential equation\nH′′\np−xH′\np+pHp= 0.\nIt is known that Rp=O(√p).\n2.Proof of Theorem 1.1\nSince a sequence of random variables may form a martingale di fference with respect to\ndifferent filtrations, the square function ( 1.1) strongly depends on both random variables\ndkand the partitions (or filtration) Dk. So if a sequence of functions d1,...,dnform a7\nmartingale difference relative to an increasing sequence of partitions D1,...,Dn, then we\nsay the collection\nd={d1,...,dn|D1,...,Dn},1≤k1, which is the unique element of Dj−1, containing A. Set\n/bardbld/bardblp=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nj=1dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np,U(d) =/bardbld/bardblp//bardblS(d)/bardbl∞, p ≥1,\nwhereS(d)is the square function ( 1.1). Hence for the constant Ap,nwe have\nAp,n= sup\ndU(d), (2.2)\nwhere supis taken over all non-trivial MD-systems ( 2.1). One can take the supin (2.2)\nover MD-systems, satisfying certain property (P). In that c ase we will use the notation\nAp,n(P). We say that a MD-system ( 2.1) isk-dyadic with 2≤k≤n, if eachV∈Dj\nwith 1≤j < k has exactly two children in Dj+1, and 1-dyadic if d1takes two values.\nThe property n-dyadic is the same as dyadic defined in the introduction. Obs erve that to\nprove Theorem 1.1it is enough to prove\nAp,n((k−1)−dyadic) =Ap,n(k−dyadic) for all 1k . Applying\n(2.10), for anyV∈Dk−1we obtain\n/integraldisplay\nV¯dk=/summationdisplay\nI∈Dn,I⊂V/integraldisplay\nI¯dk=/summationdisplay\nI∈Dn,I⊂V/integraldisplay\nIdk=/integraldisplay\nVdk= 0\nthat means E(¯dk|Dk−1) = 0 . To show E(dj|¯Dj−1) = 0 forj > k choose an element\nV+∈¯Dj−1generated by a V∈Dj−1(see ( 2.11)). IfI∈DnandI⊂V, then we have\nV⊆¯I∈Dk. We haveV⊆Jfor someJ∈Dk. By ( 2.5) and ( 2.6), we have |I+|=λ|I|\nwith a common parameter λ=λ(¯I). Thus, since the functions djare constant on each\nintervalI∈Dn, using ( 2.11), forj >k we can write\n/integraldisplay\nV+dj=/summationdisplay\nI∈Dn,I⊂V/integraldisplay\nI+dj=λ/summationdisplay\nI∈Dn,I⊂V/integraldisplay\nIdj=λ/integraldisplay\nVdj= 0,\nwhere the latter follows from the martingale-difference pro perty of the initial MD-system\nd. Similarly,/integraltext\nV−dj= 0. This implies E(dj|¯Dj−1) = 0 and so ( 2.4) is a MD-system. By9\n(2.10) we can write\n/integraldisplay\nI\n¯dk+n/summationdisplay\nj/ne}ationslash=kdj\n=/integraldisplay\nIn/summationdisplay\nj=1dj, I ∈Dn.\nThus, since the functions djare constant on I, applying Jessen’s inequality, we get the\nbound\n/bardbl¯d/bardblp\np=/summationdisplay\nI∈Dn/integraldisplay\nI/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯dk+/summationdisplay\nj/ne}ationslash=kdj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≥/summationdisplay\nI∈Dn/integraldisplay\nI/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\nj=1dj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n=/bardbld/bardblp\np, (2.12)\nBy the definition of the square function, both S¯d(x)andSd(x)are square roots of\nn-term sums. We will show that the corresponding terms in thes e sums coincide. The\nequality of first k−1terms in these sums is immediate, since the first k−1elements of\nMD-systems dand¯dcoincide. For the terms with indexes j≥kwe have the following.\nFor anyx∈[0,1)there is a unique sequence Vj∈Dj,j= 1,2,...,n such thatx∈Vj.\nFor an appropriate choice of signs εj=±we will also have x∈ ∩k≤j≤nVεj\nj. Applying the\ndefinition of ¯Dj,j≥k, one can check,\n/bardbl1Vj−1dj/bardbl∞=/bardbl1Vεj\nj−1dj/bardbl∞, j >k,\n/bardbl1Vk−1¯dk/bardbl∞=/bardbl1Vk−1dk/bardbl∞(see (2.9)).\nThus we obtain equality of all the terms in the sums of S¯d(x)andSd(x). So we get\nS¯d(x) =Sd(x)everywhere. Combining this with ( 2.12), we obtain U(d)≤U(¯d),\ncompleting the construction of ( 2.4) and the proof of relation R1. /square\nProof of R2. Now suppose that ( 2.1) satisfies the intermediate-property IP. We need to\nget ak-dyadic MD-system ¯dsuch that U(d)≤U(¯d). The only lack in the properties of\ndis that some sets in Dk−1may have more that two children. Starting our construction,\nwe will replace djby¯djforj > k andDjby¯Djforj≥k. Thus our new MD-system\nwill look like\n¯d={d1,...,dk−1,dk,¯dk+1...,¯dn|D1,...,Dk−1,¯Dk,..., ¯Dn}. (2.13)\nFor anyV∈Dk−1consider two sets\nV±={x∈X:dk(x) =±/bardbl1V·dk/bardbl∞},\nwhich we define as child sets of the element Vin¯Dk. Hence,\n¯Dk={V±:V∈Dk−1}.\nIf we stop our construction here the sets of Dk−1would exactly have two children in ¯Dk\nand we will have /bardbld/bardblp=/bardbl¯d/bardblp(that is good), but the square function of a new MD-system10 GRIGORI A. KARAGULYAN\n¯dcan be bigger than that of d(which is bad). So we have to continue the reconstruction\nto fix the problem. Suppose for a fixed V∈Dk−1the integrals\n1\n|J|/integraldisplay\nJ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\nj=1dj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n, J ∈Dk, J⊂V+,\n1\n|J|/integraldisplay\nJ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\nj=1dj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n, J ∈Dk, J⊂V−,\nattains their maximum for the intervals J=J+\nVandJ=J−\nVrespectively. Let ΛI→Jdenote\nthe dilation from [0,1)to[0,1), which is a linear mapping from an interval I⊂[0,1)onto\nan intervalJ⊂[0,1). We define the elements ¯dj(x)and ¯Djforj > k on the intervals\nJ∈Dk,J⊂V±making dilation of those elements from J±\nVonto each such J. Namely,\n¯dj(x) =/summationdisplay\nV∈Bk−1/parenleftBigg/summationdisplay\nJ∈Dk:J⊂V+dj(ΛJ→J+\nV(x))·1J(x)\n+/summationdisplay\nJ∈Dk:J⊂V−dj(ΛJ→J−\nV(x))·1J(x)/parenrightBigg\n, j >k,\n¯D+\nj=/braceleftBigg/uniondisplay\nJ∈Dk:J⊂V+Λ−1\nJ→J+\nV(I) :V∈Dk−1, I∈Dj, I⊂J+\nV/bracerightBigg\n,\n¯D−\nj=/braceleftBigg/uniondisplay\nJ∈Dk:J⊂V−Λ−1\nJ→J−\nV(I) :V∈Dk−1, I∈Dj, I⊂J−\nV/bracerightBigg\n,\n¯Dj=¯D+\nj∪¯D−\nj, j >k.\nHence the elements of ( 2.13) have been already defined. One can check that ¯disk-dyadic\nMD-system and satisfies S(¯d)≤S(d). Using the maximal property of the intervals J±\nV,\nwe also have /bardbld/bardblp≤ /bardbl¯d/bardblp. Thus,U(d)≤U(¯d). This completes the proof of R2-relation\nand so the proof of Theorem 1.1.\n/square\n3.Proof of Theorem 1.2\nLemmas 3.1and3.2below were proved in [ 12] and [ 19] respectively. For the complete-\nness we will sate the proofs of those lemmas. The proof of Lemm a3.1is taken from [ 12].\nFor Lemma 3.2an alternative proof is given.\nLemma 3.1 ([12]).For a fixedξ >0andp>3the function\nu(t) =|t+ξ|p−1·sign (t+ξ) +|t−ξ|p−1·sign (t−ξ)\nt(3.1)\nis increasing over t>0.11\nProof. For the derivative of function ( 3.1) we have\nu′(t) =|t+ξ|p−2((p−2)t−ξ) +|t−ξ|p−2((p−2)t+ξ)\nt2. (3.2)\nThe derivative of the numerator of ( 3.2) is equal\nt(p−1)(p−2)/parenleftBig\n|t+ξ|p−3sign (t+ξ) +|t−ξ|p−3sign (t−ξ)/parenrightBig\n,\nwhich is positive if t >0. Since this numerator is zero at t= 0 and its derivative is\ngreater than zero we get u′(t)≥0. /square\nLemma 3.2 ([19]).Letp>3andξ∈Rbe a fixed real number. Then the sum\n|x+y+ξ|p+|x+y−ξ|p+|y−x+ξ|p+|y−x−ξ|p(3.3)\nwherex,ysatisfyx2+y2=r2, attains its maximum if and only if |x|=|y|=r/√\n2.\nProof. Without loss of generality we can suppose that ξ≥0,r= 1andy≥1/√\n2≥\nx>0. Letf(x)denote the function ( 3.3) after substitution y=√\n1−x2. It is enough\nto prove that f(x)is increasing on the interval (0,1/√\n2). Applying Lemma 3.2and\n(|x|p)′=p|x|p−1signx, p> 1,\nfor the derivative of f(x)at a pointx∈(0,1/√\n2)we get\nf′(x) =p/parenleftBigg\n1−x\ny/parenrightBigg/parenleftBigg\n|x+y+ξ|p−1sign (x+y+ξ)\n+|x+y−ξ|p−1sign (x+y−ξ)/parenrightBigg\n+p/parenleftBigg\n1 +x\ny/parenrightBigg/parenleftBigg\n|y−x+ξ|p−1sign (y−x+ξ)\n+|y−x−ξ|p−1sign (y−x−ξ)/parenrightBigg\n=p(y−x)(y+x)\ny(u(y+x)−u(y−x))>0,\nwhereu(t)is the function in ( 3.1). Thus, applying Lemma 3.1, we getf(x)is increasing\nover(0,1/√\n2). /square\nApplying Lemmas 3.2consecutively, we get\nLemma 3.3. Ifp>3andξ∈Rare fixed, then the value\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleξ+n/summationdisplay\nk=1akrk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\np\nwhere the Rademacher coefficients aksatisfy/summationtextn\nk=1a2\nk=r2, attains its maximum if and\nonly if |ak|=r/√n.12 GRIGORI A. KARAGULYAN\nLemma 3.4. Letp≥1andξ∈Rbe fixed. Then for any non-trivial function g∈L1(0,1),\nsatisfying E(g) = 0 , the function\nh(x) =/integraldisplay1\n0|ξ+x·g(t)|pdt\nis increasing over (0,∞).\nProof. Without loss of generality it is enough to prove that\n/integraldisplay1\n0|ξ+g(t)|pdt1. (3.4)\nApplying an approximation, we can suppose that gis a step function on [0,1)with equal\nconstancy intervals. Let\nE+={x∈[0,1) :g(x)≥0}, E−={x∈[0,1) :g(x)<0}.\nWe will construct a sequence of step functions g=g1,g2,...,gm=r·g, with the same\nconstancy intervals as ghas, such that\nE(gk) = 0, (3.5)\n0≤gk·1E+≤gk+1·1E+≤rg(x)·1E+, (3.6)\n0≥gk·1E−≥gk+1·1E−≥rg(x)·1E−, (3.7)\n/integraldisplay\nX|ξ+gk(t)|pdtgl(x) =a+≥0, x∈I+,\nrg(x) =b−0.\nWe definegl+1by changing the values of gl, on the intervals I+andI−bya++λand\na−−λrespectively. Clearly, the conditions ( 3.5), (3.6) and ( 3.7) are satisfied for k=l\ntoo. To show ( 3.8) it is enough to observe the inequality\n/integraldisplay\nI+∪I−|ξ+gl(t)|pdt≤/integraldisplay\nI+∪I−|ξ+gl+1(t)|pdt,\nwhich is the same as the numerical inequality\n|ξ+a+|p+|ξ+a−|p<|ξ+a++λ|p+|ξ+a−−λ|p.\nThe latter follows from the fact that the function t(x) =|c+x|p+|c−x|pis increasing\nwhenx>0. After this step of induction we will get one more interval (e itherI+orI−),13\nwheregl+1coincides with rg. Thus, continuing the induction we will finally get a functio n\ngm=rg. Thus, we will have ( 3.4), completing the proof of lemma. /square\nProof of Theorem 1.2.Recall that D0={X}. We say a dyadic MD-system ( 2.1) is\nm-Rademacher, 1≤m≤n, if for anyV∈Dm−1we have\n|dm(x)|=|dm+1(x)|=...=|dn(x)|=c, x ∈V.\nHeren-Rademacher is nothing but to be dyadic, while 1-Rademacher property requires\n|d1(x)|=|d2(x)|=...=|dn(x)|=c, x ∈X.\nWe will first prove that\nAp,n(m−Rademacher) = Ap,n((m−1)−Rademacher) ,10. Applying ( 1.15), we can\nwrite\nE(ψ(f/u))≤1/radicalBig\n1−2/u2−1. (4.2)15\nHence, according to the definition of Luxembourg norm we obta in/bardblf/bardblψ≤/radicalBig\n8/3, since\nforu=/radicalBig\n8/3the right hand side of ( 4.2) is1. /square\n5.Open problems\nThe question whether inequality ( 1.13) holds for parameters 2 2implies increaseness of\n/bardbl/summationtextn\nk=1rk/√n/bardblpwith respect to n, while forp= 2.5a standard calculation shows\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n2(r1+r2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublep\np=2p/2\n2>/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1√\n3(r1+r2+r3)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublep\np=3p/2\n4+3\n4·3p/2.\nProblem 2. Find the values of Ap,n(Haar) andAp,n(Rademacher) for22)spanned by sequences of independent random\nvariables , Israel J. Math. 8(1970), 273–303, DOI 10.1007/BF02771562. MR 271721\n[17] S. B. Steˇ ckin, On best lacunary systems of functions , Izv. Akad. Nauk SSSR Ser. Mat. 25(1961),\n357–366 (Russian). MR 0131097\n[18] Gang Wang, Sharp inequalities for the conditional square function of a martingale , Ann. Probab. 19\n(1991), no. 4, 1679–1688. MR 1127721\n[19] P. Whittle, Bounds for the moments of linear and quadratic forms in indep endent variables , Teor.\nVerojatnost. i Primenen. 5(1960), 331–335 (English, with Russian summary). MR 133849\n[20] R. M. G. Young, On the best possible constants in the Khintchine inequality , J. London Math. Soc.\n(2)14(1976), no. 3, 496–504, DOI 10.1112/jlms/s2-14.3.496. MR 438089\nFaculty of Mathematics and Mechanics, Yerevan State Universi ty, Alex Manoogian,\n1, 0025, Yerevan, Armenia\nEmail address :g.karagulyan@ysu.am\nInstitute of Mathematics NAS RA, Marshal Baghramian ave., 24/5 , Yerevan, 0019,\nArmenia\nEmail address :g.karagulyan@gmail.com" }, { "title": "2401.16160v2.LLaVA_MoLE__Sparse_Mixture_of_LoRA_Experts_for_Mitigating_Data_Conflicts_in_Instruction_Finetuning_MLLMs.pdf", "content": "LLaV A-MoLE: Sparse Mixture of LoRA Experts for Mitigating Data Conflicts\nin Instruction Finetuning MLLMs\nShaoxiang Chen, Zequn Jie, Lin Ma\nMeituan Inc.\nAbstract\nInstruction finetuning on a variety of image-text instruc-\ntion data is the key to obtaining a versatile Multimodal\nLarge Language Model (MLLM), and different configura-\ntions of the instruction data can lead to finetuned models\nwith different capabilities. However, we have discovered\nthat data conflicts are inevitable when mixing instruction\ndata from distinct domains, which can result in performance\ndrops for tasks of a specific domain. To address this issue,\nwe propose to apply an efficient Mixture of Experts (MoE)\ndesign, which is a sparse Mixture of LoRA Experts (MoLE)\nfor instruction finetuning MLLMs. Within the Transformer\nlayers, we extend the popular Low-Rank Adaption (LoRA)\nmethod by creating a set of LoRA experts specifically for the\nMLP layer, and route each token to the top-1 expert based\non a routing function, allowing adaptive choices for tokens\nfrom different domains. Since the LoRA experts are sparsely\nactivated, the training and inference cost are kept roughly\nconstant compared to the original LoRA method. By replac-\ning the plain-LoRA of LLaVA-1.5 with our MoE design, our\nfinal model is named LLaVA-MoLE. Extensive experiments\nproved that LLaVA-MoLE effectively mitigates the data con-\nflict issue when mixing multiple distinct instruction datasets\nwith various configurations, and achieves consistent per-\nformance gains over the strong plain-LoRA baselines. Most\nimportantly, on the mixed datasets, LLaVA-MoLE can even\noutperform the plain-LoRA baseline trained with twice the\nsamples.\n1. Introduction\nLarge language models (LLMs) [1, 3] have demonstrated\ntheir remarkable capabilities in following human instruc-\ntions to complete various tasks, and one of the key to obtain\nsuch capability is instruction finetuning (or supervised fine-\ntuning, SFT) [41]. Similarly, efforts have been made to cre-\nate instruction-finetuned multimodal large language models\n(MLLMs), which connect pre-trained vision encoders with\nLLMs, resulting in models that are capable of answering\nquestions given visual and textual inputs.\n306.322.2949.85\n279.762.3660.99\n251.213.9118.45\n295.868.49117.42\n307.372.4122.53\nGeneral Benchmark scor e Document Benchmark scor eBioMedicine Benchmark scor eLLaVA-1.5\nLLaVA-Doc\nLLaVA-Med\nLLaVA-Mix\nLLaVA-MoLEModelsFigure 1. Model performances on three benchmarks when trained\nwith different data configurations. LLaV A-1.5, LLaV A-Doc,\nand LLaV A-Med are trained on general multi-task, document,\nand biomedicine datasets, respectively. While LLaV A-Mix and\nLLaV A-MoLE are both trained on the mixture of all three datasets.\nThe performance of LLaV A-Mix the document benchmark bene-\nfits from mixing all datasets, however, the performance all other\nbenchmarks drops after mixing. Our proposed LLaV A-MoLE suc-\ncessfully resolves data conflicts and maintains high performances\non all benchmarks.\nAlthough a pre-trained LLM (7B/13B parameters) [9,\n39] is usually included in a MLLM, the multimodal in-\nstruction training data still dominates the capability of the\ntrained MLLMs. Thus a large portion of the MLLM-\ntraining effort is assigned to constructing high-quality and\ndiverse multimodal instruction data. For example, LLaV A-\n1.5 [23] carefully selected a wide range of academic task-\noriented data and controlled the data size of each task. The\nresulting LLaV A-1.5 model demonstrates strong perfor-\nmances on benchmarks of various common vision-language\ntasks. Other successful multimodal instruction finetun-\ning datasets [6, 22] are also constructed with a carefully\ndesigned data configuration. In addition to data, a pop-\nular and effective parameter-efficient finetuning method\nnamed LoRA (Low-Rank Adaptation) [15] is also the key to\nLLaV A-1.5’s success. LoRA reduces the number of train-\nable parameters of Transformers by freezing the pre-trained\nmodel weights training only an injected pair of low-rankarXiv:2401.16160v2 [cs.CV] 30 Jan 2024decomposed weight matrices for each linear layer, which\nmakes it faster to finetune pre-trained large models and is\nwidely adopted in MLLM finetuning [4, 23, 45, 48, 50].\nHowever, when data configuration is critical to MLLMs,\nwe find in our preliminary studies that current MLLMs\ntrained with plain LoRA are sensitive to the training data\nconfiguration. As shown in Figure 1, we adopt three in-\nstruction tuning datasets from different domains: 1) a gen-\neral multi-task dataset that contains a mixture of various\nvision-language instructions data, 2) a document-oriented\ndataset built for chart, table, and document understanding,\nand 3) a biomedicine dataset consists of question-answer\npairs on pathology images. Three models are finetuned\non each dataset, the resulting models are named LLaV A-\n1.5, LLaV A-Doc, and LLaV A-Med, respectively. To test\nthe finetuned model’s capability on each domain, three in-\ndividual benchmarks are employed1. When the MLLM is\nfinetuned on each individual dataset, it achieves reasonable\nperformance on the corresponding benchmark. But when\nmixing the document and biomedicine dataset with the gen-\neral dataset, the trained LLaV A-Mix’s performance on the\ngeneral benchmark drops considerably from 306.3 to 295.8,\nwhich means there is a conflict incurred by adding data\nthat are distinctly different from general multi-task instruc-\ntions. This greatly hinders extending a MLLM’s abilities by\nadding training data from novel domains.\nTo address the above mentioned issue, we propose to ap-\nply a sparse mixture of LoRA experts to LLaV A-1.5 for in-\nstruction finetuning, resulting in our proposed model named\nLLaV A-MoLE. We extend the common LoRA finetuning\nparadigm used by LLaV A-1.5 and many other MLLMs.\nConcretely, we redesign how LoRA is applied to the MLPs\nin the Transformer layers of the LLM. Instead of adding\nonly one pair of low-rank decomposed matrices to the orig-\ninal linear layer, we introduce a set of experts with the\nsame structure as the original LoRA but different weights.\nThen for each token, these experts are sparsely routed by\na router function conditioned on the token embedding, i.e.,\nonly one LoRA expert is activated and its output of the to-\nken is added to the original MLP’s. Since the image and\ntext tokens from different domains can exhibit distinct fea-\ntures, they are routed to different experts and the MLLM’s\nability to handle multiple domains is expanded. In our ex-\ntensive experiments on various data configurations, we dis-\ncover that LLaV A-MoLE can effectively mitigate the con-\nflicts between different instruction datasets, while maintain-\ning roughly the same computational cost as the plain-LoRA\nmodel. We will further show in Sec.4.3 that under data\nconflicts, even if the plain-LoRA model is trained on twice\nthe samples (by repeating each dataset in the mixture), its\nscores on the general benchmark can continue to increase\nbut still fall behind LLaV A-MoLE. In this case, LLaV A-\n1The details of these datasets and benchmarks are in Sec 4.2MoLE can achieve better performance with half the training\niterations, which is a significant cost reduction.\nThe contributions of this paper are summarized as fol-\nlows:\n1. Based on an advanced MLLM model and large scale\ndatasets, we identify the data conflict issue when instruc-\ntion finetuning a MLLM on a mixture of distinctly dif-\nferent instruction datasets.\n2. We propose LLaV A-MoLE, which is instruction-\nfinetuned with a sparse mixture of LoRA experts to\nresolve the data conflict issue without significantly in-\ncreasing training computation or memory. Our method\nfurther allows us to adjust the sampling ratio of each\ndataset in the mixture to achieve higher performance on\na specific task without affecting others.\n3. Extensive experiments prove that LLaV A-MoLE\nachieves consistent performance gains for various data\nconfigurations on multiple benchmarks compared to\nusing plain LoRA finetuning.\n2. Related Work\nMultimodal Large Language Models (MLLMs). Cur-\nrent MLLMs (e.g., MiniGPT-4 [51], LLaV A [24], LLaV A-\n1.5 [23], MiniGPT-v2 [4], InstructBLIP [10], Qwen-\nVL [2]) are constructed by connecting a pre-trained vision\nencoder with a LLM (e.g., Vicuna [9], LLAMA2 [39]).\nThe vision encoders are usually from CLIP [31] so that\nthey can inherently extract semantically aligned visual fea-\ntures. The visual features are then adapted by a special-\nized light-weight module to map them into the hidden space\nof LLMs, so they can be jointly processed with the tex-\ntual inputs by the LLM. Through multimodal training, the\nMLLMs learn to generate responses given the visual and\ntextual inputs. Some MLLMs are designed for domain-\nspecific tasks by finetuning on such instruction data. Fer-\nret [46], Shikra [5], and GPT4ROI [49] constructed refer-\nring image grounding data to train MLLMs with ground-\ning capabilities. mPLUG-DocOwl [43], UReader [44], and\nVary [40] trained document-oriented MLLMs using mixed\ndatasets of chart, table, document, and OCR data. However,\npopular MLLMs with general multi-task capabilities in do-\nmains like document understanding due to the absence of\nsuch instruction data, and as we have shown above, this can\nnot be achieved by simply adding data.\nMixture of Experts (MoE) , which is dynamically com-\nbining the decision of multiple experts on the same input to\nimprove overall performance, has been studied for a long\ntime [17, 19]. It is gaining increasing popularity in the\nfield of NLP [12, 20, 34], where trillion-scale models can\nbe trained with fewer computational resources, and smaller\nmodels can also be scaled to match the performance of gi-\nant models [18]. Recent state-of-the-art NLP models are\nTransformer-based, and MoE can be conveniently appliedVision Encoder\nAdapter (MLP)\n Tokenizer &\nEmbeddin g\nTransformer Block ( LoRA )\n 𝑁×\nRoute r\n Self-Attention\nLoRA 3\n LoRA 2\n LoRA 1\n FFN⊕\n⊕\n LoRA\nUser: What is interesting \nabout this image?Sparsely activated expertsFigure 2. Overall framework of our LLaV A-MoLE with Sparse Mixture of LoRA Experts. Our model is based on LLaV A-1.5, where the\ninput image is processed by a CLIP ViT and then projected with a two-layer MLP. The input text is tokenized and embedded, and then\nconcatenated with the visual input to feed into the LLM. Each layer of the LLM is trained with our proposed Sparse Mixture of LoRA\nExperts. The FFN selects and combines with one LoRA expert according to the router’s output distribution. The self-attention is also\ntrained with LoRA but no MoE is applied.\nto the MLP layer of each Transformer block. Similarly, the\nidea of MoE can also be applied to scale up Vision Trans-\nformers [32].\nMixture of LoRA. Since LoRA has become a success-\nful parameter-efficient finetuning method, there has been a\nsurge of studies to combine MoE and LoRA for more effi-\ncient and effective model tuning. LoRAHub [16] first trains\nseveral LoRA weights on upstream tasks, then to adapt\nto a downstream task, a gradient-free method is adopted\nto search for the coefficients to combine the set of pre-\ntrained LoRA. MOELoRA [25] uses a router conditioned\non a task identifier to dynamically combine multiple LoRA\noutputs, while MoCLE [13] designs a router conditioned\non the clustering information of each input sample. Lo-\nRAMoE [11] splits the LoRA experts into two groups\nand explicitly learns different capabilities for each group.\nThese mixture-of-LoRA methods all have predefined hyper-\nparameters that need to be carefully chosen, and the LoRA\nexperts are densely mixed, i.e., by a weighted combina-\ntion, which considerably increases the computational cost.\nZadouri et al. [47] compared the dense and sparse mixture\nof LoRA experts for large language models and concluded\nthat a dense mixture leads to better performance. How-\never, we will show that for instruction-finetuning MLLMs,\na sparse mixture of LoRA experts can be the more eco-\nnomical option, i.e., it achieves comparable performances\nwhile keeping the training and inference cost roughly con-\nstant. Octavius [8] uses top-2 LoRA experts selected by a\nrouter condition on the entire input instance, which means a\ncoarse-grained routing. Among these works, MoCLE [13],\nLoRAMoE [11], and Octavius [8] discuss the task-conflict\nissue, however, they studied only a few data configurationsin their experiments. We will provide extensive experimen-\ntal analysis for various data configurations to support our\nconclusions in this paper.\n3. Method\n3.1. Preliminary\nLow-Rank Adaptation (LoRA) [15] is an effective\nparameter-efficient finetuning method for Large Language\nModels. It can be applied to arbitrary linear layers. For-\nmally, for a linear layer h=Wx with input x∈Rdiand\nweight matrix W∈Rdo×di, LoRA learns a low-rank de-\ncomposed update:\nh=Wx+ ∆Wx=Wx+α\nrBAx, (1)\nwhere A∈Rr×diandB∈Rdo×rare the low rank ma-\ntrices, r≪min(do, di)is the chosen rank, and αcontrols\nthe magnitude of the changes to the original W. During the\nlearning of a LoRA module, only the matrices AandBare\nupdated.\n3.2. Problem Formulation\nAs shown in Figure 2, a MLLM can be formulated as\nTa=fMLLM(fVis(I)||fTok(Tq)), (2)\nwhere fVis(·)is the vision encoder along with the adapter\nthat maps the input image into a sequence of visual em-\nbeddings, fTok(·)tokenizes the input question Tqand em-\nbeds the discrete tokens with a word embedding matrix,and||is a sequence concatenation operation. Thus the in-\nput to the MLLM is actually a mixed embedding sequence\nX∈RL×d.\nThe instruction data for training a MLLM is organized\nas triplets (I, Tq, Ta), and different instruction dataset can\nhave varying distributions, leading to different behaviors\nor specialties of the trained MLLM. We denote the Min-\nstruction datasets as D1,D2, ... , DM. As we have ob-\nserved in Figure 1, simply mixing the instruction datasets as\nDmix=PM\nm=1Dmcan cause conflicts between datasets\nand the MLLM can not achieve the optimal performance\n(compared to training on each individual dataset). Further-\nmore, different dataset mixing configurations can also lead\nto different model performances. We finally define a dataset\nmixture as Dmix=PM\nm=1λmDm, where λmrepresents\nthe sampling frequency of Dmin the mixture.\n3.3. Sparse Mixture of LoRA Experts\nThe goal of our proposed method is to mitigate the conflicts\nwhen mixing different types of instruction data. To this end,\nwe introduce a set of LoRA experts and a router for each\nlayer of the transformer. At each input token, the router\nlearns to select the most suitable expert to activate, so that\nthe model has extra capacity to handle different types of\ninputs. Assuming there are Kexperts per layer, the expert\nwith the highest routing function value is chosen\nk= arg max\nj=1..KGj(x) = arg max\nj=1..KWg\njx, (3)\nwhere Wg\nj∈Rdiis the router weight for the j-th expert.\nThen the chosen expert is activated to execute the actual\ncomputation, while the rest are simply ignored for the cur-\nrent token, i.e., the output of the FFN is\nf′\nFFN(x) =fFFN(x) +Ek(x), (4)\nwhere fFFN(·)is the original FFN module and Ek(·)is the\nchosen k-th LoRA expert, i.e.,\nEk(x) =α\nrBkAkx. (5)\nTo be more concrete, the FFN layer in modern LLMs is\nusually multi-layer. In this case, each linear layer of the\nFFN will have an individual MoE, but they share the same\nrouter, i.e., the expert choices for these layers are the same.\nBy only activating the top-1 expert, the actual computa-\ntion cost is kept roughly the same as the original FFN with\nplain-LoRA. The extra computation comes only from the\nrouter, which is far less than the LoRA computation due\nto the small number of experts used in our work. For ef-\nficient implementation, at each layer, the input sequence\nX={x1, x2, ..., x L}is grouped by the expert choice of\neach token. For example, the sub-sequence of tokens routedto the 1-st expert are X1={x1\n1, x1\n2, ..., x1\nL1}, where\narg max\nj=1..KGj(x1\nl) = 1 ,1≤l≤L1. (6)\nThen the computation of each Ek(·)can be executed in par-\nallel for tokens in the same sub-sequence.\n3.4. Load-Balancing of Experts\nAs introduced in the previous section, by routing a token\nto a single expert, the total computation of our MoE model\nis basically close to the plain-LoRA model. However, if\nthe expert assignment is heavily imbalanced, there will be\nwasted idle time for the low-load experts.\nSimilar to previous sparse MoE works [12], we also in-\ntroduce a load balancing loss for each MoE layer, which is\nformulated as\nLlb=NX\nj=1cj·pj, (7)\nwhere cjis the number of tokens assigned to the j-th expert,\nandpjis total routing probability of the j-th expert,\npj=X\nx∈XeGj(x)\nP\njeGj(x). (8)\nThe losses of each layer is averaged and multiplied by a\nconstant factor α= 1e−2before added to the language\nmodeling loss. Since the cvector is non-differentiable, the\ngradient only flows through the pvector and optimizes the\nrouter weights. As Llbreduces, the expert assignment be-\ncomes closer to uniform.\nPrevious works set an expert capacity that ensures each\nexpert can not process a number of tokens that exceeds the\ngiven capacity (the overflowed tokens are dropped), thus\nstrictly limits the computation load of each expert. In our\ncase, since the instruction data is relatively small compared\nto the text corpus used in previous works [12, 20], we de-\ncide to raise the expert capacity to the maximum context\nlength of the LLM so that no token is dropped and the ex-\nperts receive sufficient training.\n4. Experiments\nIn this section, we present the experimental results of our\nproposed method on various data configurations.\n4.1. Model Architecture\nThe basic model architecture follows the design of LLaV A-\n1.5 [23], where a CLIP ViT-L [30] is used as the vision\nencoder, with an input image resolution of 336x336 and\na patch size of 14, and the adapter is a two-layer MLP\nthat transforms the 576 tokens from the ViT. The LLM is\nVicuna-7B-v1.5 [9]. During training of all our experiments,\nthe ViT and Vicuna weights are frozen. The LoRA rank\napplied to the LLM is 32 if not specifically noted.Stage Batch Size LR LR min Warmup MoE\nPT 256 5e−22e−5500 ✗\nSFT 64 2e−52e−6500 ✓\nTable 1. Training configurations of the pre-training (PT) and su-\npervised instruction finetuning (SFT) stages.\n4.2. Training Stages and Datasets\nOur models are trained in two stages: pre-training and in-\nstruction finetuning. For the pre-training stage, we utilize\nthe ShareGPT4V [6] pre-training dataset, which consists of\n1.3 million detailed captioning data produced by a captioner\ntrained on GPT4V-generated data.\nFor the instruction finetuning stage, we adopt multi-\nmodal instruction datasets from three different domains:\ngeneral multi-task, document, and biomedicine.\nM3IT [22] and ShareGPT4V Instruct [6] are two gen-\neral multi-task instruction datasets. M3IT collects 40 care-\nfully curated open-source datasets and manually writes in-\nstructions for each dataset. It contains 2.4 million image-\ntext instruction instances and we filtered its training set\nto 1.6 million samples to perform experiments in this pa-\nper. ShareGPT4V Instruct is based on the 665k LLaV A-1.5\ndataset [23], which is gathered from publicly available task-\noriented data and also conversational and complex reason-\ning instruction data. In addition, 23k detailed description\ndata generated with GPT-4V [42] is added to form the final\nShareGPT4V Instruct dataset. To evaluate general multi-\ntask performance, we test our models on the Tiny LVLM-\neHub [33] benchmark, which contains 42 text-related visual\nbenchmarks, covering a wide range of tasks.\nFor document-oriented instruction data, we adopt the\ndataset collected by UReader [44]. It consists of image data\nin the form of document, table, chart, and webpage screen-\nshot. The images and instructions are from DocVQA [27],\nInfographicsVQA [28], DeepForm [37], Kleister Charity\n(KLC) [36], WikiTableQuestions (WTQ) [29], TabFact [7],\nChartQA [26], TextVQA [35], and VisualMRC [38]. All\nthese datasets are publicly available and UReader organized\nthem into combined training and testing sets. We follow\nthe data splits of UReader that contains 1.1 million resam-\npled training instances, and we report results on the URe-\nader’s test set of ChartQA and DocVQA. Note that the\ninput/output length of samples in this dataset is generally\nlonger, a considerable amount of samples can reach the\nmaximum context length (4096) of Vicuna.\nWe use PathVQA [14] as the instruction data for the\nbiomedicine domain. It contains 32,799 questions from\n4,998 pathology images. The training set has 19,755 QA\npairs, and the test set has 3,370 open-ended questions and\n3,391 close-ended questions. We report results on both openand close-ended sets.\nFor both stages, we train the models with Deepspeed\nZeRO-2 optimization on 64 NVIDIA A100 80GB GPUs.\nFinetuning a LLaV A-MoLE model on a mixture of three\ndatasets takes about 16 hours. The AdamW optimizer with\nlearning rate warm up is adopted. We list important param-\neters of the training configuration in Table 1.\n4.3. Main Results\nAs shown in Table 2, we present experimental results of\nmodels trained with different data and MoE configura-\ntions. We first provide results of the official LLaV A-1.5\nand LLaV A-Med [21] models tested on each benchmark.\nFor experiment #3-5, we train plain-LoRA models individ-\nually on each dataset and name these models correspond-\ningly as LLaV A-1.5, LLaV A-Doc, and LLaV A-Med. The\nperformance of these models on the benchmark that cor-\nresponds to their training dataset is regarded as the base-\nline performance for that benchmark. For example, Our re-\nproduced LLaV A-1.5†is trained specifically on the general\nmulti-task instruction data, and it achieves an overall score\nof 306.3 on the Tiny LVLM-eHub, which is on-par with the\nofficial LLaV A-1.5 (307.2). And our reproduced LLaV A-\nMed†achieves an accuracy of 89.17 for the closed-ended\nsubset of PathVQA, which is close to the official LLaV A-\nMed’s accuracy of 91.652.\nAfter the strong baselines are established, we begin\nto mix different datasets. As shown by experiments #6-\n8, when mixing the document instruction data and the\nbiomedicine instruction data with the general multi-task in-\nstruction data, the overall performance of LLaV A-Mix on\neHub consistently drops by 7-9 points compared to LLaV A-\n1.5†. While the UReader and PathVQA benchmark scores\nindicate that general multi-task instruction data is slightly\nbeneficial for document/chart understanding and biomed-\nical question answering, we can conclude that there are\nconflicts between the general multi-task data and these two\ntypes of data, and such conflicts can hurt the model’s gen-\neral multi-task QA abilities.\nOur proposed LLaV A-MoLE can successfully resolve\nthe above mentioned conflicts. Comparing LLaV A-\nMoLE[1,1,0] with LLaV A-Mix[1,1,0], we can observe that\nthe overall performance on eHub is significantly improved\nto be on-par with the baseline LLaV A-1.5†, while the per-\nformance on the UReader benchmark even surpasses the\nbaseline LLaV A-Doc†by a significant margin, e.g., an ab-\nsolute performance gain of 6.4 on ChartQA. This can em-\npirically prove that the mixture of experts has learned to\ndeal with different types of instruction data and reduce\npotential data conflicts. Similarly, when we train a MoE\nmodel with 3 experts on the mixture of all 3 datasets, i.e.,\n2It is pre-trained on 600K biomedical image-text captioning data for\nbiomedical concept alignment.# ModelData Config.MoETiny LVLM-eHub UReader PathVQA\nλGλDλM All VR VP VKA VC OH ChartQA DocQA Open Closed\n1 LLaV A-1.5∗- - - ✗ 307.2 55.6 49.0 57.0 57.2 88.3 - - 4.71 51.63\n2 LLaV A-Med∗- - - ✗ - - - - - - - - 38.87 91.65\n3 LLaV A-1.5†1 0 0 ✗ 306.3 50.7 54.0 55.1 58.4 88.0 1.0 21.29 4.03 45.82\n4 LLaV A-Doc†0 1 0 ✗ 279.7 43.8 45.5 52.3 54.8 83.3 34.96 27.4 2.99 58.00\n5 LLaV A-Med†0 0 1 ✗ 251.2 37.9 40.3 38.6 49.2 85.3 7.56 6.34 29.28 89.17\n6\nLLaV A-Mix1 1 0 ✗ 298.8 49.8 49.3 54.8 58.6 86.3 36.72 28.26 3.79 56.85\n7 1 0 1 ✗ 299.3 49.8 49.5 53.9 59.8 86.3 10.52 26.11 30.89 90.17\n8 1 1 1 ✗ 297.1 50.0 50.0 53.0 59.8 84.3 37.6 28.34 30.38 89.09\n9 1 2 0 ✗ 290.2 50.0 49.8 52.1 54.0 84.3 40.24 28.54 3.08 55.97\n10 1 2 1 ✗ 292.0 50.8 50.3 51.4 55.8 84.3 39.44 28.66 29.64 88.58\n11 LLaV A-Mix ×2 1 1 1 ✗ 295.8 52.8 47.8 53.3 59.0 83.0 40.48 28.01 28.96 88.46\n12\nLLaV A-MoLE1 1 0 2 307.3 52.4 50.5 57.4 59.6 87.3 41.36 30.34 3.05 58.12\n13 1 1 1 3 307.3 51.5 50.5 57.7 58.6 89.0 42.36 30.04 30.97 91.56\n14 1 2 0 2 310.1 51.9 52.0 57.1 60.4 88.7 44.2 30.3 3.41 56.23\n15 1 2 1 3 303.6 48.8 52.3 56.6 59.2 86.6 44.0 30.12 31.83 91.35\nTable 2. Experimental results of models trained with different data and MoE configurations. λG,λD, and λMare sampling frequencies\nfor general multi-task data, document data, and biomedicine data, respectively. A sampling frequency of 0 means the dataset is not used.\nVR, VP, VKA, VC, and OH stands for the coarse ability categories Visual Reasoning, Visual Perception, Visual Knowledge Acquisition,\nVisual Commonsense, and Object Hallucination, respectively.∗means the officially release model,†indicates our reproduced models, and\n×2 means the model is trained for two epochs. LLaV A-Mix and LLaV A-MoLE are trained with various dataset mixing configurations, we\ndifferentiate them by appending the data configuration, e.g., LLaV A-Mix[1,2,0] is experiment #9.\nLLaV A-MoLE[1,1,1], the performance on each individual\nbenchmark can surpass the corresponding baseline and the\nLLaV A-Mix[1,1,1].\nWe further adjust the data sampling frequencies for the\ndocument data and inspect the effects for LLaV A-MoLE\nand LLaV A-Mix. Comparing the results of experiment\nLLaV A-Mix[1,2,0] and LLaV A-Mix[1,1,0], when the sam-\npling frequency of document data is increased, the perfor-\nmance on the UReader benchmark clearly improves as ex-\npected. However, the overall performance on eHub con-\ntinues to drop (from 298.8 to 290.2). This again signi-\nfies the data conflict issue. Surprisingly, the performance\nof LLaV A-MoLE[1,2,0] on eHub is even higher than the\nbaseline, and moreover, the improvement brought by in-\ncreasing data sampling on UReader is also more signifi-\ncant than LLaV A-Mix[1,2,0] (e.g., a further absolute gain of\n3.96 for ChartQA). Similar conclusions can be made when\ncomparing results of experiments LLaV A-MoLE[1,2,1] and\nLLaV A-Mix[1,2,1], where the sampling frequency adjust-\nment is performed for a mixture of 3 datasets. These results\ncan prove that even when the data conflict issue is amplified\nby adjusting the sampling frequency, our proposed LLaV A-\nMoLE architecture can still resolve it and achieve compara-\nble or even higher performances on each individual bench-\nmark.\nMore importantly, if we look at LLaV A-Mix ×2[1,1,1],when we train the model for more epochs, in this case, each\nsample of these datasets is seen twice, the performance on\neHub slightly improves by 3.8 but still falls behind LLaV A-\nMix[1,1,1]. This means that the data conflict issue seriously\nconstrains the improvement of general multi-task abilities\neven if more training time is consumed. Looking at LLaV A-\nMoLE[1,1,1] or LLaV A-MoLE[1,2,1], our LLaV A-MoLE\nmodels can consistently outperform LLaV A-Mix[2,2,2] by\nseeing less training samples. This provides a great ad-\nvantage since both the training data and computational re-\nsources for MLLMs are expensive to obtain.\n4.4. Ablation Studies\nLoRA Rank. We first inspect the effect of LoRA rank\nunder our data and MoE configurations, and the results\nare shown in Table 3. As can be observed, for a LoRA\nrank of 32, 64, and 96, mixing the document instruction\ndata with the general multi-task instruction data all leads\nto performance drop on the eHub benchmark. But com-\nparing the results of experiments LLaV A-Mix[1,1]-R32,\nLLaV A-Mix[1,1]-R64, and LLaV A-Mix[1,1]-R96, we also\nfind that increasing the LoRA rank, i.e., increasing the\nmodel capacity, can mitigate the data conflict issue to some\nextent: the overall score on eHub increased from 298.8\n(R32) to 301.1 (R96). Moreover, if the LoRA rank is in-\ncreased to 128, this issue seems to be resolved (compar-#ModelData Config. LoRAMoETiny LVLM-eHub UReader\nλG λD Rank All VR VP VKA VC OH ChartQA DocQA\n1\nLLaV A-Mix1 0 32 ✗ 306.3 50.7 54.0 55.1 58.4 88.0 1.0 21.29\n2 1 1 32 ✗ 298.8 49.8 49.3 54.8 58.6 86.3 36.72 28.26\n3 1 0 64 ✗ 307.0 53.2 50.8 55.3 60.4 87.3 1.6 18.8\n4 1 1 64 ✗ 300.2 50.8 48.3 55.3 58.2 87.6 39.24 29.64\n5 1 0 96 ✗ 307.8 51.7 50.3 56.4 61.4 88.0 1.6 10.46\n6 1 1 96 ✗ 301.1 51.1 48.3 54.6 60.2 87.0 39.96 29.48\n7 1 0 128 ✗ 309.8 53.2 50.7 56.4 61.2 88.3 12.0 11.3\n8 1 1 128 ✗ 310.2 54.6 51.2 56.4 59.2 88.6 40.72 30.54\n9LLaV A-MoLE1 1 32 2 307.3 52.4 50.5 57.4 59.6 87.3 41.36 30.34\n10 1 1 128 2 313.6 54.1 49.7 59.3 61.8 88.6 45.32 32.62\nTable 3. Experimental results of models trained with different LoRA ranks. Similar to Table 2, the models here can be referred to by its\nconfiguration, e.g., #1 is LLaV A-Mix[1,0]-R32, where R32 indicates a LoRA rank of 32.\nMoETiny LVLM-eHub UReader\nAll VR VP VKA VC OH ChartQA DocQA\n2 307.3 52.4 50.5 57.4 59.6 87.3 41.36 30.34\n3 303.2 49.4 50.3 56.9 58.4 88.3 41.64 30.2\n5 311.6 52.6 55.0 56.6 58.4 89.0 41.88 30.82\n8 307.3 51.1 52.8 57.1 60.0 86.3 40.96 30.07\n16 306.8 52.6 51.3 56.0 59.6 87.3 42.2 30.48\nTable 4. Experimental results of LLaV A-MoLE models trained with different numbers of experts.\ning the eHub scores of LLaV A-Mix[1,0]-R128 and LLaV A-\nMix[1,1]-R128). However, we argue that simply raising\nthe model capacity is an expensive solution, leading to\ncomputation and memory increase during training. While\nour proposed LLaV A-MoLE can resolve this issue with-\nout incurring much extra cost. It is noteworthy that for\nboth small (32) and large (128) LoRA ranks, LLaV A-\nMoLE outperforms LLaV A-Mix by a significant margin on\nboth benchmarks. Finally, comparing experiment LLaV A-\nMoLE[1,1]-R32 with LLaV A-Mix[1,1]-R64 (or LLaV A-\nMix[1,1]-R96), where the latter has the same or even larger\nnumber of parameters, we can confirm that the effective-\nness of LLaV A-MoLE is not simply brought by increasing\nmodel capacity with MoE.\nNumber of Experts. We also study the effect of the\nnumber of experts by training a series of LLaV A-MoLE\nmodels with the expert number ranging from 2 to 16, and\nthe results are shown in Table 4. Note that for these experi-\nments, we mix the general multi-task and document instruc-\ntion datasets and test the models on the two correspond-\ning benchmarks. We can see that as the expert number in-\ncreases, the overall performance also improves. Using 5\nexperts achieves the best performance for this data config-\nuration. If the expert number continues to increase to 8 or16, the performance slightly drops. This could be because\neach expert can not receive enough training when the tokens\nare distributed among 8 or 16 experts. To summarize, the\nmodel performance is not very sensitive to the number of\nexperts, and setting a small number of experts can already\nachieve performance advantages over non-MoE models on\na million-scale dataset.\nSparse vs. Dense MoE. Dense MoE strategy is widely\nadopted by previous works of LoRA MoE [11, 13, 16, 25,\n47], thus we also compare sparse and dense MoE in our set-\ntings. As shown in Table 5, on a dataset mixture of the\ngeneral multi-task data and the biomedicine data, sparse\nand dense MoE achieves similar performances on all bench-\nmarks and both of them resolves the data conflict issue, i.e.,\nthey achieve a score of 312.6 (#2 and #3)on eHub compared\nto 299.3 from the baseline (#1). However, the dense MoE\nmodel consumes 83% of the GPU memory, which is sig-\nnificantly more than the sparse MoE model’s ratio of 61%.\nWhen we try to run experiments with a dense mixture of\n3 experts, an out-of-memory (OOM) error is encountered\non the GPU (#5) under long input/output length. Thus it is\ndifficult to scale-up dense MoE even for a LLM of 7B pa-\nrameters3. We would recommend using our proposed MoE\n3Model or tensor parallelism is not used for all of our experiments, buteHub UReader PathVQA00.20.40.6\neHub UReader PathVQA00.20.40.6\neHub UReader PathVQA00.20.40.6\neHub UReader PathVQA00.20.40.6E_0 E_1 E_2\nLayer 0\nLayer 2\nLayer 10\nLayer 28Figure 3. Average proportion of tokens assigned to each expert on\ndifferent benchmarks for LLM layers 0, 2, 10, and 28. Standard\ndeviation is shown as the error bar. E i represents the i-th expert.\narchitecture (scales easily to 16 experts as shown in Table\n4) for better scalability.\n4.5. Routing Choice Visualization\nWe perform a rough analysis on the routing choice of our\nLLaV A-MoLE model with 3 experts trained on the mixture\nof all three datasets. We count the expert choices on the\ntoken sequences from each benchmark, and compute the\nmean and standard deviation of the proportion of tokens as-\nsigned to each expert. The results of layer 0, 2, 10, and 28\nare visualized in Figure 3. For some layers, e.g., layer 2 and\n10, the expert choice patterns are similar for different types\nof data, but differ among layers. There are also layers (10\nand 28) where each type of data has its own expert choice\npattern. We do not observe an obvious pattern that shows a\nspecific expert is consistently favored over the others. But\nsome experts can have a slight tendency to be selected more\noften than the others on a specific dataset, e.g., expert 0 is\nthey do not affect the overall memory consumptionactivated more often on PathVQA samples across all lay-\ners.\n5. Conclusion\nIn this paper, we first identified the data conflict issue when\ninstruction finetuning multimodal large language models\non a mixture of datasets from multiple distinct domains.\nTo address this issue, we propose LLaV A-MoLE, which\nuses a sparse mixture of LoRA experts to improve the\nplain-LoRA architecture. It uses a set of LoRA experts\nfor the MLP layers and routes each token to the top-\n1 expert. Since only the selected expert is activated to\nexecute computation, the actual computational cost for\nthe entrie model is kept roughly the same as a normal\nLoRA model. In the meantime, our LLaV A-MoLE ef-\nfectively mitigates the data conflict and achieves a consis-\ntent performance improvement over the plain-LoRA base-\nlines on a variety of data configurations. We further ver-\nified that LLaV A-MoLE performs similarly with a dense\nMoE model while requiring significantly less computa-\ntional resources, which is particularly advantageous for\nsamples with long context length. For our future work,\nit would be interesting to apply our method to the multi-\ntask pre-training stage of the MLLMs, where a much larger\nnumber of training examples from multiple domains are\nmixed.\nReferences\n[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ah-\nmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida,\nJanko Altenschmidt, Sam Altman, Shyamal Anadkat, et al.\nGpt-4 technical report. arXiv preprint arXiv:2303.08774 ,\n2023. 1\n[2] Jinze Bai, Shuai Bai, Shusheng Yang, Shijie Wang, Sinan\nTan, Peng Wang, Junyang Lin, Chang Zhou, and Jingren\nZhou. Qwen-vl: A frontier large vision-language model with\nversatile abilities. arXiv preprint arXiv:2308.12966 , 2023. 2\n[3] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub-\nbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan-\ntan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Lan-\nguage models are few-shot learners. Advances in neural in-\nformation processing systems , 33:1877–1901, 2020. 1\n[4] Jun Chen, Deyao Zhu, Xiaoqian Shen, Xiang Li, Zechun\nLiu, Pengchuan Zhang, Raghuraman Krishnamoorthi,\nVikas Chandra, Yunyang Xiong, and Mohamed Elhoseiny.\nMinigpt-v2: large language model as a unified interface\nfor vision-language multi-task learning. arXiv preprint\narXiv:2310.09478 , 2023. 2\n[5] Keqin Chen, Zhao Zhang, Weili Zeng, Richong Zhang,\nFeng Zhu, and Rui Zhao. Shikra: Unleashing multi-\nmodal llm’s referential dialogue magic. arXiv preprint\narXiv:2306.15195 , 2023. 2\n[6] Lin Chen, Jisong Li, Xiaoyi Dong, Pan Zhang, Conghui\nHe, Jiaqi Wang, Feng Zhao, and Dahua Lin. Sharegpt4v:\nImproving large multi-modal models with better captions.\narXiv preprint arXiv:2311.12793 , 2023. 1, 5#Data Config.MoETiny LVLM-eHub UReader PathVQA\nλGλDλM All VR VP VKA VC OH ChartQA DocQA Open Closed\n1 1 0 1 ✗ 299.3 49.8 49.5 53.9 59.8 86.3 10.52 26.11 30.89 90.17\n2 1 0 1 Dense, 2 312.6 50.5 52.7 57.1 63.2 89.0 19.52 28.24 32.13 92.03\n3 1 0 1 Sparse, 2 312.6 51.6 51.8 58.1 62.4 88.6 19.36 28.63 32.43 92.01\n4 1 1 1 ✗ 297.1 50.0 50.0 53.0 59.8 84.3 37.6 28.34 30.38 89.09\n5 1 1 1 Dense, 3 OOM - - - - - OOM OOM OOM OOM\n6 1 1 1 Sparse, 3 307.3 51.5 50.5 57.7 58.6 89.0 42.36 30.04 30.97 91.56\nTable 5. Experimental results of models trained with dense MoE and sparse MoE (LLaV A-MoLE) on different data configurations.\n[7] Wenhu Chen, Hongmin Wang, Jianshu Chen, Yunkai Zhang,\nHong Wang, Shiyang Li, Xiyou Zhou, and William Yang\nWang. Tabfact: A large-scale dataset for table-based fact\nverification. arXiv preprint arXiv:1909.02164 , 2019. 5\n[8] Zeren Chen, Ziqin Wang, Zhen Wang, Huayang Liu, Zhenfei\nYin, Si Liu, Lu Sheng, Wanli Ouyang, Yu Qiao, and Jing\nShao. Octavius: Mitigating task interference in mllms via\nmoe. arXiv preprint arXiv:2311.02684 , 2023. 3\n[9] Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhang-\nhao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yong-\nhao Zhuang, Joseph E. Gonzalez, Ion Stoica, and Eric P.\nXing. Vicuna: An open-source chatbot impressing gpt-4\nwith 90%* chatgpt quality, 2023. 1, 2, 4\n[10] W Dai, J Li, D Li, AMH Tiong, J Zhao, W Wang, B Li,\nP Fung, and S Hoi. Instructblip: Towards general-purpose\nvision-language models with instruction tuning. arxiv 2023.\narXiv preprint arXiv:2305.06500 . 2\n[11] Shihan Dou, Enyu Zhou, Yan Liu, Songyang Gao, Jun Zhao,\nWei Shen, Yuhao Zhou, Zhiheng Xi, Xiao Wang, Xiaoran\nFan, et al. Loramoe: Revolutionizing mixture of ex-perts for\nmaintaining world knowledge in language model alignment.\narXiv preprint arXiv:2312.09979 , 2023. 3, 7\n[12] William Fedus, Barret Zoph, and Noam Shazeer. Switch\ntransformers: Scaling to trillion parameter models with sim-\nple and efficient sparsity. The Journal of Machine Learning\nResearch , 23(1):5232–5270, 2022. 2, 4\n[13] Yunhao Gou, Zhili Liu, Kai Chen, Lanqing Hong, Hang\nXu, Aoxue Li, Dit-Yan Yeung, James T Kwok, and\nYu Zhang. Mixture of cluster-conditional lora experts\nfor vision-language instruction tuning. arXiv preprint\narXiv:2312.12379 , 2023. 3, 7\n[14] Xuehai He, Zhuo Cai, Wenlan Wei, Yichen Zhang, Luntian\nMou, Eric Xing, and Pengtao Xie. Pathological visual ques-\ntion answering. arXiv preprint arXiv:2010.12435 , 2020. 5\n[15] Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-\nZhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen.\nLora: Low-rank adaptation of large language models. arXiv\npreprint arXiv:2106.09685 , 2021. 1, 3\n[16] Chengsong Huang, Qian Liu, Bill Yuchen Lin, Tianyu Pang,\nChao Du, and Min Lin. Lorahub: Efficient cross-task gen-\neralization via dynamic lora composition. arXiv preprint\narXiv:2307.13269 , 2023. 3, 7\n[17] Robert A Jacobs, Michael I Jordan, Steven J Nowlan, andGeoffrey E Hinton. Adaptive mixtures of local experts. Neu-\nral computation , 3(1):79–87, 1991. 2\n[18] Albert Q Jiang, Alexandre Sablayrolles, Antoine Roux,\nArthur Mensch, Blanche Savary, Chris Bamford, Deven-\ndra Singh Chaplot, Diego de las Casas, Emma Bou Hanna,\nFlorian Bressand, et al. Mixtral of experts. arXiv preprint\narXiv:2401.04088 , 2024. 2\n[19] Michael I Jordan and Robert A Jacobs. Hierarchical mixtures\nof experts and the em algorithm. Neural computation , 6(2):\n181–214, 1994. 2\n[20] Dmitry Lepikhin, HyoukJoong Lee, Yuanzhong Xu, Dehao\nChen, Orhan Firat, Yanping Huang, Maxim Krikun, Noam\nShazeer, and Zhifeng Chen. Gshard: Scaling giant models\nwith conditional computation and automatic sharding. arXiv\npreprint arXiv:2006.16668 , 2020. 2, 4\n[21] Chunyuan Li, Cliff Wong, Sheng Zhang, Naoto Usuyama,\nHaotian Liu, Jianwei Yang, Tristan Naumann, Hoifung Poon,\nand Jianfeng Gao. Llava-med: Training a large language-\nand-vision assistant for biomedicine in one day. arXiv\npreprint arXiv:2306.00890 , 2023. 5\n[22] Lei Li, Yuwei Yin, Shicheng Li, Liang Chen, Peiyi Wang,\nShuhuai Ren, Mukai Li, Yazheng Yang, Jingjing Xu, Xu\nSun, et al. M3it: A large-scale dataset towards multi-\nmodal multilingual instruction tuning. arXiv preprint\narXiv:2306.04387 , 2023. 1, 5\n[23] Haotian Liu, Chunyuan Li, Yuheng Li, and Yong Jae Lee.\nImproved baselines with visual instruction tuning. arXiv\npreprint arXiv:2310.03744 , 2023. 1, 2, 4, 5\n[24] Haotian Liu, Chunyuan Li, Qingyang Wu, and Yong Jae Lee.\nVisual instruction tuning. arXiv preprint arXiv:2304.08485 ,\n2023. 2\n[25] Qidong Liu, Xian Wu, Xiangyu Zhao, Yuanshao Zhu,\nDerong Xu, Feng Tian, and Yefeng Zheng. Moelora: An\nmoe-based parameter efficient fine-tuning method for multi-\ntask medical applications. arXiv preprint arXiv:2310.18339 ,\n2023. 3, 7\n[26] Ahmed Masry, Do Xuan Long, Jia Qing Tan, Shafiq Joty,\nand Enamul Hoque. Chartqa: A benchmark for question an-\nswering about charts with visual and logical reasoning. arXiv\npreprint arXiv:2203.10244 , 2022. 5\n[27] Minesh Mathew, Dimosthenis Karatzas, and CV Jawahar.\nDocvqa: A dataset for vqa on document images. In Proceed-\nings of the IEEE/CVF winter conference on applications of\ncomputer vision , pages 2200–2209, 2021. 5[28] Minesh Mathew, Viraj Bagal, Rub `en Tito, Dimosthenis\nKaratzas, Ernest Valveny, and CV Jawahar. Infographicvqa.\nInProceedings of the IEEE/CVF Winter Conference on Ap-\nplications of Computer Vision , pages 1697–1706, 2022. 5\n[29] Panupong Pasupat and Percy Liang. Compositional se-\nmantic parsing on semi-structured tables. arXiv preprint\narXiv:1508.00305 , 2015. 5\n[30] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya\nRamesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry,\nAmanda Askell, Pamela Mishkin, Jack Clark, et al. Learning\ntransferable visual models from natural language supervi-\nsion. In International conference on machine learning , pages\n8748–8763. PMLR, 2021. 4\n[31] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya\nRamesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry,\nAmanda Askell, Pamela Mishkin, Jack Clark, et al. Learning\ntransferable visual models from natural language supervi-\nsion. In International conference on machine learning , pages\n8748–8763. PMLR, 2021. 2\n[32] Carlos Riquelme, Joan Puigcerver, Basil Mustafa, Maxim\nNeumann, Rodolphe Jenatton, Andr ´e Susano Pinto, Daniel\nKeysers, and Neil Houlsby. Scaling vision with sparse mix-\nture of experts. Advances in Neural Information Processing\nSystems , 34:8583–8595, 2021. 3\n[33] Wenqi Shao, Yutao Hu, Peng Gao, Meng Lei, Kaipeng\nZhang, Fanqing Meng, Peng Xu, Siyuan Huang, Hongsheng\nLi, Yu Qiao, et al. Tiny lvlm-ehub: Early multimodal exper-\niments with bard. arXiv preprint arXiv:2308.03729 , 2023.\n5\n[34] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy\nDavis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outra-\ngeously large neural networks: The sparsely-gated mixture-\nof-experts layer. arXiv preprint arXiv:1701.06538 , 2017. 2\n[35] Amanpreet Singh, Vivek Natarajan, Meet Shah, Yu Jiang,\nXinlei Chen, Dhruv Batra, Devi Parikh, and Marcus\nRohrbach. Towards vqa models that can read. In Proceedings\nof the IEEE/CVF conference on computer vision and pattern\nrecognition , pages 8317–8326, 2019. 5\n[36] Tomasz Stanisławek, Filip Grali ´nski, Anna Wr ´oblewska,\nDawid Lipi ´nski, Agnieszka Kaliska, Paulina Rosalska, Bar-\ntosz Topolski, and Przemysław Biecek. Kleister: key in-\nformation extraction datasets involving long documents with\ncomplex layouts. In International Conference on Document\nAnalysis and Recognition , pages 564–579. Springer, 2021. 5\n[37] S Svetlichnaya. Deepform: Understand structured docu-\nments at scale, 2020. 5\n[38] Ryota Tanaka, Kyosuke Nishida, and Sen Yoshida. Vi-\nsualmrc: Machine reading comprehension on document im-\nages. In Proceedings of the AAAI Conference on Artificial\nIntelligence , pages 13878–13888, 2021. 5\n[39] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert,\nAmjad Almahairi, Yasmine Babaei, Nikolay Bashlykov,\nSoumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.\nLlama 2: Open foundation and fine-tuned chat models. arXiv\npreprint arXiv:2307.09288 , 2023. 1, 2\n[40] Haoran Wei, Lingyu Kong, Jinyue Chen, Liang Zhao, Zheng\nGe, Jinrong Yang, Jianjian Sun, Chunrui Han, and XiangyuZhang. Vary: Scaling up the vision vocabulary for large\nvision-language models. arXiv preprint arXiv:2312.06109 ,\n2023. 2\n[41] Jason Wei, Maarten Bosma, Vincent Y Zhao, Kelvin Guu,\nAdams Wei Yu, Brian Lester, Nan Du, Andrew M Dai, and\nQuoc V Le. Finetuned language models are zero-shot learn-\ners.arXiv preprint arXiv:2109.01652 , 2021. 1\n[42] Zhengyuan Yang, Linjie Li, Kevin Lin, Jianfeng Wang,\nChung-Ching Lin, Zicheng Liu, and Lijuan Wang. The dawn\nof lmms: Preliminary explorations with gpt-4v (ision). arXiv\npreprint arXiv:2309.17421 , 9(1):1, 2023. 5\n[43] Jiabo Ye, Anwen Hu, Haiyang Xu, Qinghao Ye, Ming Yan,\nYuhao Dan, Chenlin Zhao, Guohai Xu, Chenliang Li, Jun-\nfeng Tian, et al. mplug-docowl: Modularized multimodal\nlarge language model for document understanding. arXiv\npreprint arXiv:2307.02499 , 2023. 2\n[44] Jiabo Ye, Anwen Hu, Haiyang Xu, Qinghao Ye, Ming Yan,\nGuohai Xu, Chenliang Li, Junfeng Tian, Qi Qian, Ji Zhang,\net al. Ureader: Universal ocr-free visually-situated language\nunderstanding with multimodal large language model. arXiv\npreprint arXiv:2310.05126 , 2023. 2, 5\n[45] Zhenfei Yin, Jiong Wang, Jianjian Cao, Zhelun Shi, Dingn-\ning Liu, Mukai Li, Lu Sheng, Lei Bai, Xiaoshui Huang,\nZhiyong Wang, et al. Lamm: Language-assisted multi-\nmodal instruction-tuning dataset, framework, and bench-\nmark. arXiv preprint arXiv:2306.06687 , 2023. 2\n[46] Haoxuan You, Haotian Zhang, Zhe Gan, Xianzhi Du, Bowen\nZhang, Zirui Wang, Liangliang Cao, Shih-Fu Chang, and\nYinfei Yang. Ferret: Refer and ground anything anywhere\nat any granularity. arXiv preprint arXiv:2310.07704 , 2023.\n2\n[47] Ted Zadouri, Ahmet ¨Ust¨un, Arash Ahmadian, Beyza Ermis ¸,\nAcyr Locatelli, and Sara Hooker. Pushing mixture of experts\nto the limit: Extremely parameter efficient moe for instruc-\ntion tuning. arXiv preprint arXiv:2309.05444 , 2023. 3, 7\n[48] Pan Zhang, Xiaoyi Dong Bin Wang, Yuhang Cao, Chao Xu,\nLinke Ouyang, Zhiyuan Zhao, Shuangrui Ding, Songyang\nZhang, Haodong Duan, Hang Yan, et al. Internlm-\nxcomposer: A vision-language large model for advanced\ntext-image comprehension and composition. arXiv preprint\narXiv:2309.15112 , 2023. 2\n[49] Shilong Zhang, Peize Sun, Shoufa Chen, Min Xiao, Wenqi\nShao, Wenwei Zhang, Kai Chen, and Ping Luo. Gpt4roi: In-\nstruction tuning large language model on region-of-interest.\narXiv preprint arXiv:2307.03601 , 2023. 2\n[50] Bo Zhao, Boya Wu, and Tiejun Huang. Svit: Scaling up\nvisual instruction tuning. arXiv preprint arXiv:2307.04087 ,\n2023. 2\n[51] Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mo-\nhamed Elhoseiny. Minigpt-4: Enhancing vision-language\nunderstanding with advanced large language models. arXiv\npreprint arXiv:2304.10592 , 2023. 2" }, { "title": "2401.16162v1.Hartman_Effect_from_a_Geometrodynamic_Extension_of_Bohmian_Mechanics.pdf", "content": "Hartman Effect from a Geometrodynamic Extension of Bohmian Mechanics\nSaid Lantigua\n1,∗and Jonas Maziero\n1,†\n1Departament of Physics, Center for Natural and Exact Sciences,\nFederal University of Santa Maria, Santa Maria, RS, 97105-900, Brazil\nThis paper presents the derivation of a general solution to the scattering problem of particles inci-\ndent onto a barrier of constant potential. This solution is constructed through a geometrodynamic\napproach to Bohmian mechanics, assuming that particles undergo quantum tunneling along geodesic\ntrajectories in an Alcubierre-type spacetime. Furthermore, from this solution, mathematical expres-\nsions for the quantum potential, momentum, position, and tunneling time are determined in terms\nof the spacetime geometry for each relevant region. This allows us to explain the Hartman effect as\na consequence of spacetime distortion generated by the quantum potential within the barrier.\nKeywords: Tunneling Effect, Hartman Effect, Geometrodynamics, Alcubierre Metric, Bohmian Mechanics\nI. INTRODUCTION\nThe unusual saturation behavior in tunneling times\ndiscovered by Hartman in 1962, known as the Hartman\nEffect [1], exhibited by particles incident upon a very\nwide potential barrier [2–4], has led to the contempla-\ntionofpotentialsuperluminaltunnelingspeeds[5–8]. Ini-\ntially, this sparked a vigorous debate regarding whether\nthese results were due to inaccuracies in tunneling time\nmeasurements [9]. Subsequently, it compelled numerous\ntheorists of the time to embark on the intricate task of\naddressing the fundamental question: What is the cor-\nrect definition of tunneling time? [10–17]. This question,\ncaptivatingnotonlyforphysicistsofthaterabutalsorel-\nevant to contemporary researchers, has endured through\nthe years, with many attempts to answer it proving un-\nsuccessful across various approaches [15, 16, 18, 19].\nDespite the considerable body of literature address-\ning this age-old paradox, existing work primarily focuses\non definitions derived from approaches in relativistic and\nnon-relativistic quantum mechanics. As previously men-\ntioned, theseapproaches fall shortof providing a satisfac-\ntory explanation for the observed effect [11, 14–16, 18–\n27]. In contrast to these previous approaches, within\nthe context of Bohmian mechanics, the question of defin-\ning tunneling time has not been sufficiently addressed\n[28, 29], despite a considerable body of literature on the\napplication of Bohmian theory to the particle scattering\nproblem over a potential barrier [29]. One of the more\nintriguing works in this context addresses a similar ques-\ntion and investigates Klein tunneling times for electrons\nin a two-terminal graphene device comprising a potential\nbarrier between two metallic contacts [16, 30].\nOn the other hand, due to the mathematical structure\nofBohmianmechanics[31–33], itispossible, throughcer-\ntain physical and mathematical considerations, to extend\nit to a version that incorporates elements of spacetime\nas defined in general relativity [34, 35]. Therefore, it\n∗Electronic address: said.lantigua@acad.ufsm.br\n†Electronic address: jonas.maziero@ufsm.brbecomes feasible to develop a geometrodynamic descrip-\ntion of quantum systems and to derive mathematical ex-\npressions that are not provided by approaches based on\nthe orthodox formalism of relativistic or non-relativistic\nquantum theory [36]. Consequently, this work presents\nthe construction of a solution that explains the Hartman\nEffect through a geometrodynamic approach to Bohmian\nmechanics, considering a spacetime endowed with a met-\nric that allows for superluminal travel without violating\nthe principles of general relativity [37].\nMore specifically, the present work reports a general\nsolution constructed under the hypothesis that particles,\nduring the tunneling phenomenon, follow geodesic tra-\njectories within an Alcubierre-type spacetime [37, 38].\nConsequently, from this solution, quantities of interest\nare determined, including mathematical expressions de-\nscribing linear momenta, quantum potentials, and par-\nticle positions for the problem under investigation. As\na result, the particle’s position expression leads to the\nmathematical expression of the tunneling time, equipped\nwith the geometric elements of the considered spacetime.\nThe remainder of this article is organized as follows.\nSection II provides a brief yet necessary review of the\nfundamental concepts of Bohmian mechanics. Section\nIII deduces the geometrodynamic version of the quantum\nforce based on the results presented in Ref. [36]. Section\nIVpresentstheconstructionofthegeneralsolutiontothe\nproblem of a particle incident from the left on a barrier of\nconstant potential for each of the regions of interest. In\nSection V, the general expression of the quantum poten-\ntial for each region of interest is determined. Section VI\nestablishes the general expression for linear momentum\nand particle position, again for each region of interest,\nconsequently providing the mathematical expression for\nthe tunneling time. Section VII presents the conclusion\nof the work. We also include an Appendix containing\nthe mathematical results and some methods used in the\nmain text.arXiv:2401.16162v1 [quant-ph] 29 Jan 20242\nII. FUNDAMENTALS OF BOHMIAN\nMECHANICS\nBohmian mechanics is a theory discovered by Louis De\nBroglie in 1927 and later rediscovered by David Bohm\nin 1952. It provides a description of quantum systems\nthrough a mathematical object known as the wave func-\ntion, which encodes their dynamics [32]. In essence, the\npartial description of the quantum system is given by a\nwave function that satisfies the quantum equilibrium hy-\npothesis and evolves according to the Schrödinger equa-\ntion. Meanwhile, the particles’ evolution is described by\nthe guiding equation, which outlines their velocities in\nterms of this wave function [31, 39].\nIn presenting the foundational mathematical frame-\nwork of Bohmian mechanics, we consider the following\nwave function:\nΨ =√ρexp{iS}, (1)\nwhere ρ≡ρ(⃗ x, t)andS≡S(⃗ x, t). Upon substituting\nthis function into the Schrödinger equation,\n\u0014\n−ℏ2\n2m∇2+V\u0015\nΨ =iℏ∂tΨ, (2)\nwe obtain the expressions:\n−ℏ2\n2m∇2√ρ√ρ+ℏ2\n2m(∇S)2+V=−ℏ∂tS(3)\nand\n−ℏ2\n2m\u0014\n2(∇√ρ)· ∇S+√ρ(∇2S)\u0015\n=ℏ∂t√ρ.(4)\nFurthermore, using the equalities 2√ρ(∇√ρ) =∇ρ\nand2√ρ(∂t√ρ) =∂tρ, we can multiply Eq. (4) by 2√ρ\nto obtain:\n∂tρ+ℏ\nm∇ ·(ρ∇S) = 0 . (5)\nAdditionally, recalling that the probability current den-\nsity is defined as ⃗j≡ρ⃗ v= Ψ∗Ψ⃗ v, we can determine the\nexpression for particle velocity by substituting Eq. (1)\ninto:\nΨ∗Ψ⃗ v=iℏ\n2m\u0014\nΨ∇Ψ∗−Ψ∗∇Ψ\u0015\n, (6)\nresulting in:\n⃗ v= (ℏ/m)∇S. (7)\nThis allows us to rewrite equations (3) and (5) as:\nℏ∂tS+ℏ2\n2m(∇S)2+V+Q= 0 (8)\nand\n∂tρ+∇ ·⃗j= 0, (9)where we introduced the quantity\nQ≡ −ℏ2(∇2√ρ)/(2m√ρ), (10)\nreferred to as the quantum potential or Bohmian poten-\ntial, and ⃗j≡ρ[ℏ(∇S)/m]. In other words, equations\n(8) and (9) are the differential equations describing the\nbehavior of Sandρ[40].\nFinally, to complete the formalism, the gradient of Eq.\n(8) is calculated, and the result is expressed in terms of\nthe velocity field (7) to obtain:\nm∂t⃗ v+m(⃗ v· ∇)⃗ v=−∇(V+Q) =⃗F+⃗FQ.(11)\nThe above expression is the equation of motion that a\nparticle with a probability current density given by Eq.\n(6) will follow under the action of an effective force ⃗F=\n⃗FV+⃗FQ. Therefore, in the classical limit ( ℏ→0), the\ntrajectories will obey the laws of Newtonian motion, as\nexpected [31, 32, 39, 40].\nIII. GEOMETRODYNAMICS AND QUANTUM\nFORCE\nWhenconsideringtheextensionofBohmianmechanics\nto a relativistic version, several seemingly insurmount-\nable challenges arise. For instance, issues include ex-\ntending Bohmian trajectories to relativistic paths, con-\nstructing a four-vector of probability density and cur-\nrent, and dealing with non-physical trajectory occur-\nrences—especially concerning photons—due to frames of\nreference where velocity is zero. However, through care-\nful physical and mathematical considerations, it is possi-\nble to construct geometrodynamic models that allow for\nsuch an extension [36]. Therefore, this section presents\na deduction of the quantum force equation in curvilin-\near coordinates. In other words, it derives a generalized\nversion of the quantum force expression presented in Ref.\n[36]. Theprocedurefollowedhereessentiallymirrorsthat\nof the authors in that work, and their contributions play\na central role in constructing the solution presented in\nthis article.\nOne possible connection between Bohmian mechanics\nand general relativity arises by postulating that parti-\ncles in the quantum system follow geodesic trajectories\nin spacetime. This hypothesis can be expressed mathe-\nmatically through the following expression:\n0 =d2xµ\ndt2+ Γµ\nαβdxα\ndtdxβ\ndt, (12)\nwhere µ,α, and βtake values in {0,1,2,3}. Equation\n(12) yields the geometrodynamic constraint:\nΓ0\nαβdxα\ndtdxβ\ndt= 0. (13)\nThis establishes a local equivalence relation between the\nEuclidean spacetime where Bohmian mechanics is de-\nfined and the spacetime of the Lorentzian manifold by3\nconsidering x0=ct[36]. Therefore, by considering\nj, k, µ =i= 1,2,3andα, β = 0,1,2,3, equation (12)\ncan be developed to obtain:\n0 =d2xi\ndt2+ Γi\njkdxj\ndtdxk\ndt−2cΓi\n0jdxj\ndt+c2Γi\n00.(14)\nConsidering the components of the gradient in a gen-\nerally curved space, given by\n(∇f)i=gij\np\n|g|∂jf, (15)\nwhere |g|=|det(gµν)|, allows the rewriting of expres-\nsions for the potential gradients as\n(∇V)i=gij\np\n|g|∂jV (16)\nand\n(∇Q)i=gij\np\n|g|∂jQ. (17)\nFurthermore, the mathematical expression for accelera-\ntioninageneralsystemoforthogonalcoordinatesisgiven\nby\n¨xi=d2xi\ndt2+Gi\njk∂xj∂xk\n=d2xi\ndt2+ωi. (18)\nAbove Gi\njkare the Christoffel symbols in the orthogo-\nnal coordinate system. Then, considering the expressions\n(16), (17), and (18), equation (11) is reformulated as:\nΓi\njkdxj\ndtdxk\ndt−2cΓi\n0jdxj\ndt+c2Γi\n00\n=ωi+gij\nmp\n|g|∂jV+gij\nmp\n|g|∂jQ. (19)\nNext, considering the general expression for the mo-\nmentum components, (⃗P)i=ℏ(∇S)i, which, similarly to\n(16) or (17), can be rewritten as\nPi=ℏgij\np\n|g|∂jS→dxi\ndt=ℏgij\nmp\n|g|∂jS.(20)\nUsing the Christoffel symbols\nΓi\njk=gih\n2{∂jghk+∂kgjh−∂hgjk},\nΓi\nj0=gih\n2{∂jgh0+∂0gjh−∂hgj0},\nΓi\n00=gih\n2{2∂0gh0−∂hg00}, (21)the expression (19) is recast as follows:\nℏ2gih\n2(mp\n|g|)2{∂jghk+∂kgjh−∂hgjk}gjlgkn∂lS∂nS\n−cℏgih\nmp\n|g|{∂jgh0+∂0gjh−∂hgj0}gjl∂lS\n+c2gih\n2{2∂0gh0−∂hg00}\n=ωi+gil\nmp\n|g|∂lV+gil\nmp\n|g|∂lQ. (22)\nThe expression (22) represents a geometrodynamic ex-\ntension of the quantum force in Eq. (11) [36]. In simple\nterms, it provides an extended mathematical formalism\nthat allows the description of physical systems based on\ntheir geometric evolution, where the central mathemati-\ncal object of Bohmian theory (the wave function) main-\ntains its prominent role. Moreover, in this geometrody-\nnamic extension, not only does the wave function pre-\nserve its central role as the entity encoding the system’s\ndynamics, but, in this formalism, the particle and the\nwave function, which are well-defined and clearly distinct\nentities, assume a dialectical role.\nIV. CONSTRUCTION OF THE GENERAL\nSOLUTION TO THE PROBLEM\nIn this section, we present the deduction of the\nBohmian solution to the one-dimensional problem of par-\nticle scattering incident from the left on a constant po-\ntential barrier [2, 3, 40], based on the direct application\nof the formalism introduced in the previous sections II\nand III. For this reason, we will consider the following\nthree regions of interest. The first one is denoted as I,\nwhere the particle is incident, guided by its pilot wave,\nand subsequently reflected or not. The second region is\ndenoted as II, bounded by the potential barrier, where\nquantum tunneling phenomenon occurs. Finally, the last\nregion considered is denoted as III, where particle trans-\nmission, with its corresponding pilot wave, occurs. See\nFig. 1.\nTherefore, the Schrödinger equation1for the problem\nrepresented in Figure 1 can be written as follows:\nd2\nxψ(x) + [k2−ξ2Θ(x−a/2)Θ(a/2−x)]ψ(x) = 0 ,(23)\nwhere ξ2= 2V0,k2= 2E,ℏ= 1,c= 1, and m= 1.\nWhen solved, this equation allows finding the wave func-\ntionψ(x)used in the Bohmian formalism [31, 32, 39, 40].\n1In order to clarify the notation used in this article, we empha-\nsize that the following notation is used: dx() =d()\ndxfor the total\nderivative, ∂x() =∂()\n∂xfor the partial derivative, and the semi-\ncolon for the covariant derivative ∂ν() = () ;.4\nFigure1: Schematicrepresentationofthetunnelingofaparti-\ncle with energy E, incident from the left on a potential barrier\nof height V0and width a. Additionally, in this representation,\nthe regions I,II, and IIIare identified, which are considered\nin the construction of the general solution presented in sec-\ntion IV.\nAdditionally, it is considered that the potential barrier\nhas a width aand a height V0, and is mathematically\ndefined as:\nV(x) =V0Θ(x−a/2)Θ(a/2−x).(24)\nHere Θis the Heaviside step function, which is mathe-\nmatically defined as:\nΘ(x) =(\n0forx <0\n1forx≥0. (25)Thus, the solution in regions IandIIIare given by the\nexpressions:\nψI(x) =A′eikx+B′e−ikxforx≤ −a/2,(26)\nψIII(x) =F′eikxforx≥a/2, (27)\nwhich, when written in the form (1), reduce to the ex-\npressions:\nΨI(x) =√ρIexp{iSI(x)}forx≤ −a/2,(28)\nΨIII(x) =√ρIIIexp{iSIII(x)}forx≥a/2,(29)\nwhere√ρI=p\nA2cos2kx+B2sin2kxwithA=A′+\nB′andB=i(A′−B′),√ρIII=F′=F,SI(x) =\ntan−1{[Btankx]/A}, and SIII(x) =kx.\nHowever, to obtain a consistent Bohmian solution for\nthis problem, temporal dependence in the wave func-\ntion must be included, as shown in Ref. [32]. That\nis to say, we consider ΨI(x, t) = Ψ I(x) exp{iEt}and\nΨIII(x, t) = Ψ III(x) exp{iEt}. Thus, we can write the\ngeneral solution for regions IandIIIas a superposition\nof solutions with different energy eigenstates:\nΨj(x, t) =2X\nl=1Cl\njq\nρl\njexp{i[Sl\nj(x)−Elt]}.(30)\nNext, the general solution can be rewritten in the form\nof the Bohmian wave function, where the phase functions\ndepending on position and time are given by:\nSj(x, t) = tan−1\nP2\nl=1Cl\njq\nρl\njsin Ωl\nj(x, t)\nP2\nl=1Cl\njq\nρl\njcos Ωl\nj(x, t)\n,(31)\nwith Ωl\nj(x, t) =Sl\nj(x)−Eltand\n√ρj=vuut2X\nl=1(Cl\nj)2ρl\nj+ 2Cq\njCq+1\njq\nρq\njρq+1\njcos Ξq\nj,(32)\nwith Ξ1\nj≡Ξj(x, t) =S2\nj(x)−S1\nj(x)−(E2−E1)tand\nq= 1.\nHowever, the solution in region IIis obtained by solv-\ning the equation (22) considering the Alcubierre metric\ntensor [37, 41] given by\ngµν=\nv2\nsf2(rs)−1−vsf(rs) 0 0\n−vsf(rs) 1 0 0\n0 0 1 0\n0 0 1\n,(33)\nwhose inverse tensor is given by the expression\ngµν=\n−1 −vsf(rs) 0 0\n−vsf(rs) 1−v2\nsf2(rs) 0 0\n0 0 1 0\n0 0 0 1\n,(34)5\nwhere vs≡vs(t) = ˙ xs(t),f(rs) = α0/{vs[1 +\nα1cosh (2 σrs)]},rs≡rs(t) = x−xs(t), and xs(t) =\nnct=nt, since in this article c= 1. On the other hand,\nit is important to note that in the problem addressedhere, a constant potential, V=V0, is considered, and\ndue to the structure of the metric tensor (33), we have\nωi= 0fori= 1,2,3and∂lV= 0. For this reason, the\nfollowing set of equations is obtained from equation (22):\nc2g10∂0g00\n2+c2g11{2∂0g10−∂1g00}\n2=−g11∂1Q\nmp\n|g|fori= 1, j̸= 1,\ncℏg10(∂1g00)∂1S\nmp\n|g|−g11∂1Q\nmp\n|g|=−g11∂1Q\nmp\n|g|fori=j= 1, k̸= 1,\nℏ2g01(2∂1g01)(g11)2(∂1S)2\n2(mp\n|g|)2−cℏg10(∂1g00)g11∂1S\nmp\n|g|−g11∂1Q\nmp\n|g|=−g11∂1Q\nmp\n|g|fori=j=k= 1.(35)\nNext, by considering the nontrivial solutions of (35), the\nexpression\ndS=−2drsvsf\u001a1\n1−v2sf2\u001b\n, (36)\nis deduced, which allows calculating the phase function\nin region II. Similarly, the expression\ndQ=d f\u001avs\n1−v2sf2+v2\nsf\u001b\n(37)\nis obtained, whose solution will be presented later, allow-\ning us to obtain the expression for the quantum potential\nin region II. Therefore, by integrating (36), as shown in\npart Iof the Appendix, the phase function is obtained:\nSII(x, xs(t)) =α0β0\n2σp\n(β3)2−(β2)2(38)\n×(\nβ1−µ0√µ0tan−1\"\n1√µ0tanx′\n2#\n−β1−µ1√µ1tan−1\"\n1√µ1tanx′\n2#)\n.\nAbove α0=−[vstanh (2 σR)]/[2 tanh ( σR)],α1=\nsech(2σR),β0= [2( α1−1)]/[(α1−1)2−α2\n0],β1= (α1+\n1)/(α1−1),β2= [(α1+ 1)2−α2\n0]/[(α1−1)2−α2\n0],β3=\n[α2\n1−1 +α2\n0]/[(α1−1)2−α2\n0],µ0=β3−p\n(β3)2−(β2)2,\nµ1=β3+p\n(β3)2−(β2)2and\nx′= cos−1[sech(2σrs)]. (39)\nHowever, it is important to note that physically accept-\nable solutions for (36), leading to a consistent phase func-\ntion (38), are those where α1̸= 0orα1̸= 1. Then, by\nimposing the continuity conditions of the solution and of\nits derivatives, the expressions obtained from (30) mustbe substituted into\nΨk(x, t)\f\f\f\f\n(xk,tk)= Ψ k+I(x, t)\f\f\f\f\n(xk,tk), (40)\n[∂xΨk(x, t)]\f\f\f\f\n(xk,tk)= [∂xΨk+I(x, t)]\f\f\f\f\n(xk,tk)(41)\nwith xk∈ {− a/2, a/2},tk∈ {t0, t1}andk=I, II, III .\nThus one obtains the system of equations:\nC1\nIζ1+C2\nIζ2ei∆Et0=ζ3[C1\nII+C2\nIIei∆Et0],\nC1\nIζ′\n1+C2\nIζ′\n2ei∆Et0=ζ′\n3[C1\nII+C2\nIIei∆Et0],\nC1\nIIIζ5+C2\nIIIζ6ei∆Et1=ζ4[C1\nII+C2\nIIei∆Et1],\nC1\nIIIζ′\n5+C2\nIIIζ′\n6ei∆Et1=ζ′\n4[C1\nII+C2\nIIei∆Et1].(42)\nConsequently, by solving the system of equations (42)\nwhile imposing the superposition condition C2\nII= 1−C1\nII,\nthe expressions for the coefficients are obtained:\nC1\nI=ϖ4e−i∆Et0[(ei∆Et0−1)C1\nII+ 1],\nC2\nI=ϖ3[(ei∆Et0−1)C1\nII+ 1],\nC1\nII=−ei∆Et1\nei∆Et1−1\b\nϖ2C1\nIII−1\t\n,\nC2\nII=1\nei∆Et1−1\b\nϖ2ei∆Et1C1\nIII−1\t\n,\nC2\nIII=ϖ1ei∆Et1C1\nIII. (43)\nAbove we defined ϖ1= [ζ′\n4ζ5−ζ4ζ′\n5]/[ζ4ζ′\n6−ζ′\n4ζ6],ϖ2=\n[ζ′\n5ζ6−ζ5ζ′\n6]/[ζ4ζ′\n6−ζ′\n4ζ6],ϖ3= [ζ1ζ′\n3−ζ′\n1ζ3]/[ζ1ζ′\n2−ζ′\n1ζ2]6\nandϖ4= [ζ′\n2ζ3−ζ2ζ′\n3]/[ζ1ζ′\n2−ζ′\n1ζ2]with\nζl=q\nρl\nI(x) exp{iSl\nI(x)}\f\f\f\f\n(−1)la\n2,\nζ′\nl= [2ρl\nI(x)]−1kl[(B2− A2)q\nρl\nI(x) sin 2 klx\n+i2AB]eiSl\nI(x)\f\f\f\f\n(−1)la\n2,\nζl+2= exp {i[SII(x) +V0t]}\f\f\f\f\n((−1)l+2a\n2,t0),\nζ′\nl+2=iα0β0sec2(x′\n2) tan (2 σrs)ei[SII(x)+V0t]\n2p\n(β3)2−(β2)2q\ncosh2(2σr2)−1\n×\u0014β1−µ1\nµ1+ tan2(x′\n2)−β1−µ0\nµ0+ tan2(x′\n2)\u0015\f\f\f\f\n((−1)l+2a\n2,t0),\nζl+4= exp {iSl\nIII(x)}\f\f\f\fa\n2,\nζ′\nl+4=iklexp{iSl\nIII(x)}\f\f\f\fa\n2. (44)\nIn the expressions above, values for l= 1,2are consid-\nered, along with the energy variation ∆E=E2−E1.\nAdditionally, the coefficients were rewritten as Cl\nj=Cl\nj\nifj=I, and Cl\njq\nρl\nj=Cl\njifj=II, III, again for all\nl= 1,2. In summary, through the coefficients presented\nin (43) and the expressions for Sl\nI,SII, and Sl\nIII, the\ngeneral solution to the problem for each of the regions of\ninterest can be written as:\nΨI(x, t) =C1\nIq\nρ1\nI(x) exp{i[S1\nI(x)−E1t]}\n+C2\nIq\nρ1\nI(x) exp{i[S2\nI(x)−E2t]},\nΨII(x, t) = [C1\nIIexp{−i[E1−V0]t}\n+C2\nIIexp{−i[E2−V0]t}] exp{iSII(x)},\nΨIII(x, t) =C1\nIIIq\nρ1\nI(x) exp{i[S1\nIII(x)−E1t]}\n+C2\nIIIq\nρ1\nI(x) exp{i[S2\nIII(x)−E2t]},(45)\nwhere, as expected, the general expression is written in\nterms of the geometry of the spacetime considered by\nimposing the geometrodynamic constraint (13). Then,\nby substituting C2\nIIIinto the expression for√ρIII, the\ntransmission coefficient is obtained:\nT=C1\nIIIq\n1 +ϖ2\n1ei2∆Et1+ 2ϖ1ei∆Et1cos Ξ III.(46)\nV. GENERAL EXPRESSION FOR THE\nQUANTUM POTENTIAL\nIn this section, we apply the formalism and results pre-\nsented in Sections II, III, and IV to deduce the generalexpression of the quantum potential for each of the re-\ngions considered in the problem of interest. To this end,\nit is necessary to calculate Qj=−(∇2√ρj)/(2√ρj)in\nregions IandIII, while in region II, it is necessary to\nintegrate the expression (37).\nIn this regard, obtaining an expression for region I\nproves to be a challenging task due to the form of the\ngeneral solution. Nevertheless, by rewritingq\nρl\njin the\nform\nq\nρl\nj=p\nA2cos2klx+B2sin2klx\n=q\n(A′+B′)2cos2klx−(A′−B′)2sin2klx\n=q\n2A′B′+ [(A′)2+ (B′)2] cos 2 klx,(47)\nand considering the solution for low energies ( klx≈\n0→cos 2klx≈1), expression (47) simplifies to a more\nstraightforward format given by\nq\nρl\nj=p\n2A′B′+ (A′)2+ (B′)2\n=p\n(A′+B′)2\n=A.(48)\nOn the other hand, the phase function for low energies\nreduces to\nSl\nI(x) = tan−1{[Btanklx]/A} ≈ B klx/A.(49)\nHence, it is possible to rewrite (32) in terms of (48) and\n(49) to obtain the quantum potential in region I:\nQI(x, t) =\u0014B(k2−k1)√\n2ρI\u00152\n{ρIcos Ξ I+AC1\nIC2\nIsin2ΞI},\n(50)\nwith amplitude√ρI=Ap\n(C1\nI)2+ (C2\nI)2+ 2C1\nIC2\nIcos Ξ I\nand where the angle ΞI=S2\nI(x)−S1\nI(x)−(E2−E1)t.\nFurthermore, since√ρIIIdiffers from√ρIonly by the\nconstants AandB, it is not difficult to verify that the\nquantum potential in region IIIis given by\nQIII(x, t) =\u0014(k2−k1)√\n2ρIII\u00152\n{ρIIIcos Ξ III (51)\n+C1\nIIIC2\nIIIsin2ΞIII},\nwith amplitude\n√ρIII=q\n(C1\nIII)2+ (C2\nIII)2+ 2C1\nIIIC2\nIIIcos Ξ I,(52)\nand where the angle is given by\nΞIII=S2\nIII(x)−S1\nIII(x)−(E2−E1)t.(53)\nUnlike regions IandIII, and as mentioned in the pre-\nvious section, the quantum potential in region IIis de-\ntermined by integrating (37), resulting in the expression\nQII(x, t) =(\nln\f\f\f\f\f1 +vsfp\n1−v2sf2\f\f\f\f\f+v2\nsf2\n2)\n.(54)7\nVI. MOMENTUM, POSITION, AND\nTUNNELING TIME\nIn the previous sections, along with the theoretical\nfoundations and the general solution, we derived the\nmathematical expressions for the quantum potential in\neach of the regions of interest in our problem. Even\nthough the wave function and the quantum potential are\nfundamentalquantitieswithintheframeworkofBohmian\nmechanics [31, 32, 39, 40], it is essential to determine\nthe particle’s momentum and position. Therefore, this\nsection is dedicated to calculating the mathematical ex-\npressions for these quantities, ultimately allowing us to\ndetermine the quantum tunneling time [1, 7, 8, 10, 14–\n16, 28].\nInordertodeterminethelinearmomentuminregion I,\nwe need to substitute the expressions (48) and (49) into\n(31) to obtain the phase function SI(x, t). Substituting\nthis into (7), we obtain\nPI(x, t) ={B[(C1\nI)2k1+ (C2\nI)2k2+C1\nIC2\nI(k1(55)\n+k2) cos Ξ I]}{A[(C1\nI)2+ (C2\nI)2\n+ 2C1\nIC2\nIcos Ξ I]}−1.\nAnalogously, setting B/A= 1in Eq. (49), we obtain the\nexpressionsfor Sl\nIII. RewritingEq. (31)andsubstituting\nit into Eq. (7), we determine that the linear momentum\nin region IIIis given by\nPIII(x, t) ={(C1\nIII)2k1+ (C2\nIII)2k2+C1\nIIIC2\nIII(k1(56)\n+k2) cos Ξ III}{(C1\nIII)2+ (C2\nIII)2\n+ 2C1\nIIIC2\nIIIcos Ξ III}−1.\nAlternatively, substituting Eq. (38) into Eq. (7) yields\nthe linear momentum in region II, which is given by\nPII(x, t) =α0β0sech(2σrs)\n2p\n(β3)2−(β2)2\n×(\nβ1−µ0\nµ0cos2[sech (2σrs)\n2] + sin2[sech (2σrs)\n2]\n−β1−µ1\nµ1cos2[sech (2σrs)\n2] + sin2[sech (2σrs)\n2])\n.(57)\nAs a result, the following differential equations are de-\nrived from the expressions (55) and (56):\ndx\ndt=ℏB\nmAϑ3{1 +ϑ4cos Ξ I}\nϑ1{1 +ϑ2cos Ξ I}, (58)\ndx\ndt=ℏ\nmϑ9{1 +ϑ10cos Ξ III}\nϑ11{1 +ϑ12cos Ξ III}, (59)where ϑ1= (C1\nI)2+ (C2\nI)2,ϑ2= 2C1\nIC2\nI/[(C1\nI)2+ (C2\nI)2],\nϑ3= (C1\nI)2k1+ (C2\nI)2k2,ϑ4=C1\nIC2\nI(k1+k2)/[(C1\nI)2k1+\n(C2\nI)2k2],ϑ9= (C1\nIII)2k1+(C2\nIII)2k2,ϑ10=C1\nIIIC2\nIII(k1+\nk2)/[(C1\nIII)2k1+ (C2\nIII)2k2],ϑ11= (C1\nIII)2+ (C2\nIII)2and\nϑ12= 2C1\nIIIC2\nIII/[(C1\nIII)2+ (C2\nIII)2]. In the same way,\nstarting from Eq. (57), we obtain\ndx\ndt=ϑ5\u001aϑ6cosh3(2σrs) + cosh (2 σrs)\ncosh4(2σrs) +ϑ7cosh2(2σrs) +ϑ8\u001b\n,(60)\nwith\nϑ5=α0β0(µ1−µ0)/[8µ0µ1p\n(β3)2−(β2)2],\nϑ6= 4β1,\nϑ7= (µ1+µ0)/4µ0µ1,\nϑ8= 1/4µ0µ1.(61)\nHowever, it is important to highlight that the above\ndifferential equation is obtained by considering 0≤\nsech(2σrs)/2≤1/2→ cosh [sech(2σrs)/2]≈1and\nsinh [sech(2σrs)/2]≈sech(2σrs)/2in Eq. (57). This\nholds true for σ≪1[37]. But what is the physical\nor mathematical reason behind this condition? The an-\nswertothisquestionariseswhenanalyzingtheexpression\nα1= 1/cosh (2 σR) =sech(2σR)introduced in Section\nIV, from which we derive σ=arsech (α1)/2Rand conse-\nquently arsech (α1)≪2R. It is worthwhile mentioning\nthat as the quantity arsech (α1)is a dimensionless con-\nstant, the above expression does not consider the units\nofRor, in the particular case, Re; that is, we only aim\nto establish a comparison in orders of magnitude.\nTherefore, in order to provide a physical mean-\ning to this last mathematical condition, particles of\nsmall dimensions, such as an electron, are considered,\nwhose radius has the value Re= 2.8178402894(58) ×\n10−13m. Comparing it with the Bohr radius ( a0=\n5.29177210903(86) ×10−11m), we obtain the relation\na0= 188 Re. Thus, it is sufficient to choose an α1such\nthat Re R ⊕, it is possible that temperate rocky planets\nwith R∼0.7R⊕might harbor relatively thinner at-\nmospheres if all other variables (e.g., outgassing, atmo-\nspheric mass) are statistically similar across the rangeof planetary radii.2If future telescopes such as ARIEL\n(Tinetti et al. 2018) characterize a sufficiently large sam-\nple of terrestrial exoplanets, it may be feasible to test\nsome aspects of our hypothesis empirically.\n4.CONCLUSIONS\nBuilding on the ongoing characterization of exoplan-\netary atmospheres by JWST (Gardner et al. 2023) –\nwhich has led to several interesting discoveries (e.g.,\nLustig-Yaeger et al. 2023; Greene et al. 2023; Zieba et al.\n2023) – and observations by forthcoming ground-based\nextremely large telescopes, a wealth of data will become\navailable, making it feasible to discern statistical pat-\nterns in exoplanetary atmospheres in the future. Non-\nthermal ion escape facilitated by stellar EUV radiation\nand driven by the stellar wind represents one of the key\nregulators of the mass and composition of terrestrial ex-\noplanetary atmospheres (Brain et al. 2016; Zahnle &\nCatling 2017; Airapetian et al. 2020; Gronoff et al. 2020).\nMotivated by these facts, we utilized a thoroughly vali-\ndated multispecies MHD model to investigate the total\nion escape rate as a function of the planetary radius R.\nAfter performing the simulations, we unearthed a\nnovel trend for rocky planets with CO 2-dominated at-\nmospheres: the escape rate is a nonmonotonic function\nofR, as clearly depicted in Figure 1, with a peak at\nR∼0.7R⊕. It was found that this characteristic is man-\nifested for intense stellar wind and radiation conditions,\nwhich may be associated with M-dwarfs and young Sun-\nlike stars. This nonmonotonic feature runs counter to\nthe naive expectation that smaller rocky planets (e.g.,\nakin to Mars) would automatically exhibit higher escape\nrates because of their weaker gravity.\nThe nonmonotonic behavior may arise from a tradeoff\nbetween the (cross-sectional) area and the escape veloc-\nity of a planet; both of these variables increase for larger\nplanets. On the one hand, when the aforementioned area\nis boosted, this will enhance the atmospheric ion escape\nrate since the planet would intercept more of the stellar\nradiation and wind (due to its greater area), thereupon\namplifying the escape rate on account of the enhanced\nionized reservoir in the planetary upper atmosphere. On\nthe other hand, when the escape velocity is boosted, this\nfactor will suppress the escape rate because particles re-\nquire more energy to escape the gravitational well. From\na more detailed perspective, we suggested in Section 3.2\nthat nonthermal atmospheric ion escape may shift from\na source-limited regime to an energy-limited one, which\ncould partially explain the simulated results.\nIf the striking trend displayed by our modeling is cor-\nrect, it may have tangible observational consequences for\nfuture surveys of terrestrial exoplanets. Provided that\n2The exact value of this peak may, however, change based on the\nstellar parameters (e.g., EUV and stellar wind), and will need to\nevaluated on a case-by-case basis.6\nall other factors are held equal, a higher atmospheric ion\nescape rate could translate to relatively thinner atmo-\nspheres. Hence, if future surveys either discover thin-\nner atmospheres – or, even better, obtain direct evi-\ndence of higher nonthermal ion escape rates – at a cer-\ntain value of R(conceivably R∼0.7R⊕), such data\nwould serve to corroborate the predictions of our numer-\nical simulations. Moreover, such nonmonotonic behav-\nior would deepen our understanding of how nonthermal\natmospheric escape can sculpt the properties of exoplan-\netary atmospheres of rocky worlds.\nACKNOWLEDGMENTS\nThis work was supported by NASA grants\n80NSSC23K1115 and 80NSSC23K0911. Resources\nsupporting this work were provided by the NASA\nHigh-End Computing (HEC) Program through the\nNASA Advanced Supercomputing (NAS) Division\nat Ames Research Center. The Space Weather\nModeling Framework that comprises the BATS-R-\nUS code used in this study is publicly available at\nhttps://github.com/MSTEM-QUDA/BATSRUS.\nREFERENCES\nAirapetian, V. S., Barnes, R., Cohen, O., et al. 2020, Int. J.\nAstrobiol., 19, 136, doi: 10.1017/S1473550419000132\nBean, J. L., Raymond, S. N., & Owen, J. E. 2021, J.\nGeophys. Res. Planets, 126, e06639,\ndoi: 10.1029/2020JE006639\nBerger, T. A., Huber, D., Gaidos, E., van Saders, J. L., &\nWeiss, L. M. 2020, Astron. J., 160, 108,\ndoi: 10.3847/1538-3881/aba18a\nBlackman, E. G., & Tarduno, J. A. 2018, Mon. Not. R.\nAstron. Soc., 481, 5146, doi: 10.1093/mnras/sty2640\nBoesswetter, A., Lammer, H., Kulikov, Y., Motschmann,\nU., & Simon, S. 2010, Planet. Space Sci., 58, 2031,\ndoi: 10.1016/j.pss.2010.10.003\nBougher, S. W., Hunten, D. M., & Roble, R. G. 1994, J.\nGeophys. Res., 99, 14609, doi: 10.1029/94JE01088\nBrain, D. A., Bagenal, F., Ma, Y. J., Nilsson, H., &\nStenberg Wieser, G. 2016, J. Geophys. Res. Planets, 121,\n2364, doi: 10.1002/2016JE005162\nChen, H., & Rogers, L. A. 2016, Astrophys. J., 831, 180,\ndoi: 10.3847/0004-637X/831/2/180\nChen, J., & Kipping, D. 2017, Astrophys. J., 834, 17,\ndoi: 10.3847/1538-4357/834/1/17\nCloutier, R., & Menou, K. 2020, Astron. J., 159, 211,\ndoi: 10.3847/1538-3881/ab8237Cravens, T. E., Rahmati, A., Fox, J. L., et al. 2017, J.\nGeophys. Res. Space Phys., 122, 1102,\ndoi: 10.1002/2016JA023461\nDong, C., Huang, Z., & Lingam, M. 2019, Astrophys. J.\nLett., 882, L16, doi: 10.3847/2041-8213/ab372c\nDong, C., Huang, Z., Lingam, M., et al. 2017a, Astrophys.\nJ. Lett., 847, L4, doi: 10.3847/2041-8213/aa8a60\nDong, C., Jin, M., & Lingam, M. 2020, Astrophys. J. Lett.,\n896, L24, doi: 10.3847/2041-8213/ab982f\nDong, C., Jin, M., Lingam, M., et al. 2018a, Proc. Natl.\nAcad. Sci., 115, 260, doi: 10.1073/pnas.1708010115\nDong, C., Lingam, M., Ma, Y., & Cohen, O. 2017b,\nAstrophys. J. Lett., 837, L26,\ndoi: 10.3847/2041-8213/aa6438\nDong, C., Lee, Y., Ma, Y., et al. 2018b, Astrophys. J. Lett.,\n859, L14, doi: 10.3847/2041-8213/aac489\nEgan, H., Jarvinen, R., Ma, Y., & Brain, D. 2019, Mon.\nNot. R. Astron. Soc., 488, 2108,\ndoi: 10.1093/mnras/stz1819\nEhlmann, B. L., Anderson, F. S., Andrews-Hanna, J., et al.\n2016, J. Geophys. Res. Planets, 121, 1927,\ndoi: 10.1002/2016JE005134\nErkaev, N. V., Kulikov, Y. N., Lammer, H., et al. 2007,\nAstron. Astrophys., 472, 329,\ndoi: 10.1051/0004-6361:200669297\nFrance, K., Froning, C. S., Linsky, J. L., et al. 2013,\nAstrophys. J., 763, 149,\ndoi: 10.1088/0004-637X/763/2/149\nFrance, K., Loyd, R. O. P., Youngblood, A., et al. 2016,\nAstrophys. J., 820, 89, doi: 10.3847/0004-637X/820/2/89\nFujii, Y., Angerhausen, D., Deitrick, R., et al. 2018,\nAstrobiology, 18, 739, doi: 10.1089/ast.2017.1733\nFulton, B. J., & Petigura, E. A. 2018, Astron. J., 156, 264,\ndoi: 10.3847/1538-3881/aae828\nFulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017,\nAstron. J., 154, 109, doi: 10.3847/1538-3881/aa80eb\nGardner, J. P., Mather, J. C., Abbott, R., et al. 2023, Publ.\nAstron. Soc. Pac., 135, 068001,\ndoi: 10.1088/1538-3873/acd1b5\nGinzburg, S., Schlichting, H. E., & Sari, R. 2018, Mon. Not.\nR. Astron. Soc., 476, 759, doi: 10.1093/mnras/sty290\nGreene, T. P., Bell, T. J., Ducrot, E., et al. 2023, Nature,\n618, 39, doi: 10.1038/s41586-023-05951-7\nGronoff, G., Arras, P., Baraka, S., et al. 2020, J. Geophys.\nRes. Space Phys., 125, e27639,\ndoi: 10.1029/2019JA027639\nGunell, H., Maggiolo, R., Nilsson, H., et al. 2018, Astron.\nAstrophys., 614, L3, doi: 10.1051/0004-6361/201832934\nGupta, A., & Schlichting, H. E. 2019, Mon. Not. R. Astron.\nSoc., 487, 24, doi: 10.1093/mnras/stz1230\n—. 2020, Mon. Not. R. Astron. Soc., 493, 792,\ndoi: 10.1093/mnras/staa315\nKane, S. R. 2022, Nat. Astron., 6, 420,\ndoi: 10.1038/s41550-022-01626-x\nKane, S. R., Arney, G., Crisp, D., et al. 2019, J. Geophys.\nRes. Planets, 124, 2015, doi: 10.1029/2019JE005939\nKasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993,\nIcarus, 101, 108, doi: 10.1006/icar.1993.1010\nKopparapu, R. K., Ramirez, R. M., SchottelKotte, J., et al.\n2014, Astrophys. J. Lett., 787, L29,\ndoi: 10.1088/2041-8205/787/2/L29\nKopparapu, R. K., Ramirez, R., Kasting, J. F., et al. 2013,\nAstrophys. J., 765, 131,\ndoi: 10.1088/0004-637X/765/2/131\nKrenn, A. F., Fossati, L., Kubyshkina, D., & Lammer, H.\n2021, Astron. Astrophys., 650, A94,\ndoi: 10.1051/0004-6361/202140437\nKubyshkina, D., Fossati, L., Erkaev, N. V., et al. 2018,\nAstrophys. J. Lett., 866, L18,\ndoi: 10.3847/2041-8213/aae586\nLammer, H. 2013, Origin and Evolution of Planetary\nAtmospheres (Berlin: Springer),\ndoi: 10.1007/978-3-642-32087-3\nLammer, H., Kasting, J. F., Chassefi` ere, E., et al. 2008,\nSpace Sci. Rev., 139, 399, doi: 10.1007/s11214-008-9413-5Lamp´ on, M., L´ opez-Puertas, M., Czesla, S., et al. 2021,\nAstron. Astrophys., 648, L7,\ndoi: 10.1051/0004-6361/202140423\nLaneuville, M., Dong, C., O’Rourke, J. G., & Schneider,\nA. C. 2020, in Planetary Diversity: Rocky planet\nprocesses and their observational signatures, ed. E. J.\nTasker, C. Unterborn, M. Laneuville, Y. Fujii, S. J.\nDesch, & H. E. Hartnett (Bristol: IOP Publishing),\n3.1–3.47\nLichtenegger, H. I. M., Dyadechkin, S., Scherf, M., et al.\n2022, Icarus, 382, 115009,\ndoi: 10.1016/j.icarus.2022.115009\nLingam, M. 2019, Astrophys. J. Lett., 874, L28,\ndoi: 10.3847/2041-8213/ab12eb\nLingam, M., & Loeb, A. 2019, Rev. Mod. Phys., 91,\n021002, doi: 10.1103/RevModPhys.91.021002\n—. 2021, Life in the Cosmos: From Biosignatures to\nTechnosignatures (Cambridge: Harvard University Press)\nLopez, E. D., & Fortney, J. J. 2014, Astrophys. J., 792, 1,\ndoi: 10.1088/0004-637X/792/1/1\nLustig-Yaeger, J., Fu, G., May, E. M., et al. 2023, Nat.\nAstron., 7, 1317, doi: 10.1038/s41550-023-02064-z\nMadhusudhan, N. 2019, Annu. Rev. Astron. Astrophys., 57,\n617, doi: 10.1146/annurev-astro-081817-051846\nMartinez, C. F., Cunha, K., Ghezzi, L., & Smith, V. V.\n2019, Astrophys. J., 875, 29,\ndoi: 10.3847/1538-4357/ab0d93\nMcDonald, G. D., Kreidberg, L., & Lopez, E. 2019,\nAstrophys. J., 876, 22, doi: 10.3847/1538-4357/ab1095\nMcIntyre, S. R. N., Lineweaver, C. H., & Ireland, M. J.\n2019, Mon. Not. R. Astron. Soc., 485, 3999,\ndoi: 10.1093/mnras/stz667\nMendillo, M., Withers, P., & Dalba, P. A. 2018, Nat.\nAstron., 2, 287, doi: 10.1038/s41550-017-0375-y\nMordasini, C. 2020, Astron. Astrophys., 638, A52,\ndoi: 10.1051/0004-6361/201935541\nOstberg, C., Kane, S. R., Li, Z., et al. 2023, arXiv e-prints,\narXiv:2302.03055, doi: 10.48550/arXiv.2302.03055\nOtegi, J. F., Bouchy, F., & Helled, R. 2020, Astron.\nAstrophys., 634, A43, doi: 10.1051/0004-6361/201936482\nOwen, J. E. 2019, Annu. Rev. Earth Planet. Sci., 47, 67,\ndoi: 10.1146/annurev-earth-053018-060246\nOwen, J. E., & Alvarez, M. A. 2016, Astrophys. J., 816, 34,\ndoi: 10.3847/0004-637X/816/1/34\nOwen, J. E., & Jackson, A. P. 2012, Mon. Not. R. Astron.\nSoc., 425, 2931, doi: 10.1111/j.1365-2966.2012.21481.x\nOwen, J. E., & Schlichting, H. E. 2023, Mon. Not. R.\nAstron. Soc., doi: 10.1093/mnras/stad3972\nOwen, J. E., & Wu, Y. 2017, Astrophys. J., 847, 29,\ndoi: 10.3847/1538-4357/aa890a8\nPerryman, M. 2018, The Exoplanet Handbook, 2nd edn.\n(Cambridge: Cambridge University Press)\nPersson, M., Futaana, Y., Ramstad, R., et al. 2020, J.\nGeophys. Res. Planets, 125, e06336,\ndoi: 10.1029/2019JE006336\nPlotnykov, M., & Valencia, D. 2020, Mon. Not. R. Astron.\nSoc., 499, 932, doi: 10.1093/mnras/staa2615\nRamstad, R., & Barabash, S. 2021, Space Sci. Rev., 217,\n36, doi: 10.1007/s11214-021-00791-1\nRogers, J. G., Gupta, A., Owen, J. E., & Schlichting, H. E.\n2021, Mon. Not. R. Astron. Soc., 508, 5886,\ndoi: 10.1093/mnras/stab2897\nSalz, M., Schneider, P. C., Czesla, S., & Schmitt,\nJ. H. M. M. 2016, Astron. Astrophys., 585, L2,\ndoi: 10.1051/0004-6361/201527042\nSeager, S., & Deming, D. 2010, Annu. Rev. Astron.\nAstrophys., 48, 631,\ndoi: 10.1146/annurev-astro-081309-130837\nShields, A. L., Ballard, S., & Johnson, J. A. 2016, Phys.\nRep., 663, 1, doi: 10.1016/j.physrep.2016.10.003\nTian, F. 2015, Annu. Rev. Earth Planet. Sci., 43, 459,\ndoi: 10.1146/annurev-earth-060313-054834\nTinetti, G., Drossart, P., Eccleston, P., et al. 2018, Exp.\nAstron., 46, 135, doi: 10.1007/s10686-018-9598-x\nTRAPPIST-1 JWST Community Initiative, de Wit, J.,\nDoyon, R., et al. 2023, arXiv e-prints, arXiv:2310.15895,\ndoi: 10.48550/arXiv.2310.15895Valencia, D., O’Connell, R. J., & Sasselov, D. 2006, Icarus,\n181, 545, doi: 10.1016/j.icarus.2005.11.021\nVan Eylen, V., Agentoft, C., Lundkvist, M. S., et al. 2018,\nMon. Not. R. Astron. Soc., 479, 4786,\ndoi: 10.1093/mnras/sty1783\nWatson, A. J., Donahue, T. M., & Walker, J. C. G. 1981,\nIcarus, 48, 150, doi: 10.1016/0019-1035(81)90101-9\nWinn, J. N., & Fabrycky, D. C. 2015, Annu. Rev. Astron.\nAstrophys., 53, 409,\ndoi: 10.1146/annurev-astro-082214-122246\nWood, B. E., M¨ uller, H.-R., Redfield, S., et al. 2021,\nAstrophys. J., 915, 37, doi: 10.3847/1538-4357/abfda5\nWordsworth, R., & Kreidberg, L. 2022, Annu. Rev. Astron.\nAstrophys., 60, 159,\ndoi: 10.1146/annurev-astro-052920-125632\nZahnle, K. J., & Catling, D. C. 2017, Astrophys. J., 843,\n122, doi: 10.3847/1538-4357/aa7846\nZeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016,\nAstrophys. J., 819, 127,\ndoi: 10.3847/0004-637X/819/2/127\nZeng, L., Jacobsen, S. B., Sasselov, D. D., et al. 2019, Proc.\nNatl. Acad. Sci., 116, 9723, doi: 10.1073/pnas.1812905116\nZhang, X. 2020, Res. Astron. Astrophys., 20, 099,\ndoi: 10.1088/1674-4527/20/7/99\nZhu, W., & Dong, S. 2021, Annu. Rev. Astron. Astrophys.,\n59, 291, doi: 10.1146/annurev-astro-112420-020055\nZieba, S., Kreidberg, L., Ducrot, E., et al. 2023, Nature,\n620, 746, doi: 10.1038/s41586-023-06232-z" }, { "title": "2401.16229v1.Theoretical_Analysis_of_Solvent_Effect_on_NAPBr_Dye_s_Two_photon_Absorption_Ability_and_Non_Radiative_Transition_in_Lipid_Droplets_Detection.pdf", "content": "Theoretical Analysis of Solvent Effect on NAPBr Dye 's Two -\nphoton Absorption Ability and Non -Radiative Transition in \nLipid Droplets Detection \nHongyang Wang *1, Xiaofei Wang1, Yong Zhou1, Jianzhong Fan*1,3 \n1. Shandong Province Key Laboratory of Medical Physics and Image Processing Technology, \nInstitute of Materials and Clean Energy, School of Physics and Electronic s, Shandong Normal \nUniversity, 250014 Jinan, China \n2. Guangdong Provincial Key Laboratory of Luminescence from Molecular Aggregates (South \nChina University of Technology), Guangzhou 510640, China \nAbstract \nTwo-photon fluorescence imaging h as shown a promising application in biomedical imaging due to its \noutstanding advantages such as large penetration depth, low photo -damage, and photo -bleaching, etc. \nAmong them, the two -photon fluorescent dye NAPBr, which can effectively select and monitor lipid \ndroplets in living cells and biological tissues, has attracted extensi ve attention because of its excellent \nfluorescent properties. However, the research on the fluorescent abilities of two -photon fluorescent dyes \nin solvent environment is not sufficient. In our work, t heoretical analysis reveals the internal mechanism \nof the solvent effect on geometric structure and photophysical properties of two -photon fluorescent dyes, \nespecially non -radiative transition process, and holes -electrons distribution and transfer. This can provide \na reference for the development of efficient two -photon absorption (TPA) molecules with aggregation -\ninduced emission (AIE) characteristics. Related data also showed good regularity. Moreover, dye in four \nsolvents have excellent photophysical properties: high fluorescence quantum efficiency (up to 66.6 0%), \nlarge Stokes shift (up to 108696 cm-1), and two -photon absorption cross section (up to 3658 GM). The \nmedium dielectric constant solution environment can achieve a balance between two -photon absorption \nand fluorescence emission capabilities better , which lays a solid foundation for the study of TPA \nmolecules with AIE functions in terms of solvent effects. \n \nKeywords: Solvent effect, two -photon absorption cross section, non -radiative transition, fluorescence \nquantum efficiency. \n*Author to whom correspondence should be addressed. E-mail: wang hongyang6787@163.com and \nfanjianzhongvip@163.com \n 1. Introduction \nLipid droplets (LDs) serve as an important reservoir of lipids, provide energy and substrate for \nmembrane synthesis, whose make LDs as crucial metabolic hubs. Increasing evidence suggests that \nLDs ar e also involved in protein degradation, response to ER stress, protein glycosylation, and \npathogen infection. 1-5 Recent studies have shown that lipid droplets are also highly associated with \nobesity, diabetes, inflammatory disorders, and cancer.6 Therefore, the development of effective \nmethods for selecti ng lipid droplet s visualization and monitori ng it in biological samples (such as live \ncells and tissues ) is quiet important.7 Compared with one -photon fluorescence imaging technology, the \ntwo-photon fluorescence imaging method utilizes two near -infrared (NIR) photons as the excitation \nsource . It is beneficial for biomedical imaging because of deeper tissue penetration, higher spatial \nresolvent, lower background fluorescence , photodamage and photobleaching.8, 9 \nTang's group used two -photon naphthalene -based donor−acceptor NAP AIEgens molecules that \nintroduced a large heterocyclic ring to i mprove the electron -withdrawing ability of acceptor . The specific \ntwo-photon living cells and deep tissue lipids imaging were achieved.7 The solvent environment affects \nthe molecular geometric structure , photophysical property and its internal conversion. However, we have \nno some insight into the role of solvents in the above -mentioned Tang's research. In this work, we focus \non the solvent effects on two -photon absorption and non -radiative transition. Intramolecular rotations \n(RIR) main ly restrict the aggregation induced emission (AIE) , which affects the excited state energy \nconsumed by non -radiative transitions , weakens NAPBr fluorescence or make it even not emits. We \nexpound the photophysical properties, internal mechanisms with the geometrical differences caused by \nRIR on solvent effect. Molecules in different sol vents have their unique advantages, such as excellent \ntwo-photon absorption (TPA) capability and high fluorescence quantum efficiency. It even provides a \nreference for the more suitable solvent environment required in actual selection. Furthermore, the solvent \neffect on the dye’s absorption (OPA and TPA) and emission properties are studied. The molecular electron \nexcitation is explained by analyzing electrons and holes distributions. Taking the balance between two -\nphoton absorption and fluorescence emission ability as a standard, we find that medium dielectric \nconstant solution is more suitable for subsequent AIE research. A comprehensive analysis of molecular \nproperty on solvent effect helps us understand the two -photon fluorescent mechanism better. It makes a \nsolid foundation for TPA molecules research with AIE function in the solvent effect. 2. Theoretical calculations \nIn general, the oscillator strength is used to characterize the transition probability between the ground \nstate\ni and the excited state \nf molecules in one -photon absorption and emission, which is defined \nas10 \n \n2 2\n3f\nop if\n= (1) \nWhere\nf denotes excitation energy of the excited state\nf ,\n is the electric dipole moment operator \nand the summation is performed over x, y, z axes. Easy to compare with each other by defining \nmacroscopic one -photon absorption cross sections.11, 12 \n \n25 2\n0\ntp4 ()\n15a tpa\nfa g\nc= (2) \nHere, a0, α, and c are the Bohr radius, the fine structure constant, and the speed of light, respectively. \ng(ω) denotes the spectral line profile, which is assumed to be an δ function here. Generally , the level \nbroadening \nf of the final state is assumed to be 0.1 eV, corresponding to a lifetime of about 6 fs. 13 The \nunit of TPA cross sections is GM (1 GM = 10-50cm4 s/photon). \nMoreover, the root-mean -square deviation (RMSD), which could be used to measure molecular \nconfiguration difference: \n (3) \nN represents the number of atoms. \nThe fluorescence radiative decay rate can be calculated by Einstein's spontaneous emission equation: \n \n2\n21.499fi\nrEKcm\n−= (4) \nHere \n is the oscillator strength obtained by optimizing the first excited state (S 1), which is \nnatom' 2 ' 2 ' 2 1( ) (y ) (z )i i i i i i iRMSD x x y zN = − + − + −dimensionless. \n2\nfiE is the vertical emission energy of S 1 state, the unit is cm-1. This \nrK is the radiative \ndecay rate. The unit \nrK is s-1. \nBased on the Fermi Golden Rule, the formula for no n-radiative decay rate is \n \n2\n, 2 ,2(E E )nr iv fv iv iv fu uvK P H =−\n (5) \n,fv ivH\nrepresents the interaction between two states, \nnrK can be calculated in Molecular Materials \nProperty Prediction Package (MOMAP) developed by Shuai group14-18. \nBased on the above -mentioned results, the fluorescent quantum efficiency can be obtained by the \nfollowing equation: \n \nr\nPF\nr nrK\nKK=+ (6) \nHere, \nrK and \nnrK are the fluorescence rate and non-radiative decay rate of S 1 state. \nIn this article, density functional theory (DFT) are used to optimize the ground state, we use time -\ndependent DFT (TD -DFT) to study excited state geometry and photophysical property. Geometric \nconfiguration difference, fluorescence emission, and one -photon optical property calculations are \nperformed based on Beck e's three parametrized Lee –Yang –Parr (B3LYP) method with 6 -31G* basis set. \nAll these calculations can be achieved by a Gaussian 16 package.19 The frequency calculations are \nperformed to ensure optimized structure stability. The RMSD values are used to measure the molecular \ngeometry difference between S 0 and S 1 in VMD.20 Moreover, solution environment(such as W ater, \nEthanol, Acetone, and Dichloromethane ) are simulated using polarizable continuum model (PCM).21 \nCorresponding dielectric constant (ε) of four solvents are listed in Table 1,which can be found in the \nGaussian manual . \nTable 1. The dielectric constant (ε) of four solvents. \nSolvent Water Ethanol Acetone Dichloromethane \nε 78.36 32.61 20.49 8.93 Furthermore, equilibrium s olvation is applied for geometry optimization because solvent needs time to \nfully respond to the changes of solute in two ways: polarizing its electronic distribution and making its \nnuclei reoriented.22 Under the frame of first -order perturbation theory , non-adiabatic electronic coupling \nis treated as the force acting on atomic nuclei through transition electric field , which is evaluated in this \npaper .22,23 The electron distribution and frontier molecular orbital (FMO) levels are analyzed . Besides, \nmolecular fluorescent quantum efficiency, Huang−Rhys (HR) factor, and reorganization energy in \ndifferent solvents are all calculated by MOMAP based on organic fluorescent molecules excited state \ndecay theory. The two -photon absorption (TPA) prop erties are calculated by Datlon2013.24 \n \n3. Result and discussion \n3.1 Molecular structures \nFirstly, we choose NAPBr in the experiment performed by Tang's group and study molecular structures \nin Water, Ethanol, Acetone, and Dichloromethane. 7 Br is choosing for its strong tendency to get electrons \nand it is one of the most electronegative elements, which is more convenient for later analysis. The \nstructural formula of stu died compound, optimized ground state structure is shown in Figure 1. \n \nFigure 1 Structural formula of the studied compound and its geometric structure. Atomic labels and \ninteresting angles are marked. Dihedral angle θ1: C14-C13-C33-C35, bond angle θ2: C10-C25-C26. \nIn this part, we study the solvent effect on molecular geometrical structures. Dihedral angle θ1 between \nthe benzene ring and naphthalene , bond angle θ2 are select ed for analysis. With decreasing solvent \npolarity, it can be found that the θ 1 are 53.30。, 53.26。, 53.20。, 53.07。, θ2 are 114.99。, 114.97。, 114.96。, \n114.91。. Dihedral angle of NAPBr in Water is the largest, while that in Dichloromethane is the smallest. \nLarger solvent environment polarity causes a small er twist angle difference between naphthalene and \nbenzene ring, molecular co-planar structure is destroyed slightly. Therefore, molecular electron \ndistribution change lead to a discrepancy in optical properties. \nThe OPA and OPE of dyes \nAbsorption and emission properties are important nature for dyes. In this chapter, we study the OPA and \nOPE properties and transition process of NAPBr in different solvents in detail. \n \nFigure 2 The relationship between absorption, emission wavelength, and oscillator strength in different \nsolvents. \n \n \n \nTable 2 Detailed data on molecular one -photon absorption and fluorescent emission, a: emission \nwavelength maximum, b: absorption wavelength maximum . \nMolecule θ1 θ2 Eflu λaems δems λbabs δabs Stokes shift Transition nature \nWater 53.30。 114.99。 2.28 544 1.55 454 1.09 90 nm S1→S0 H→L 99% \nEthanol 53.26。 114.97。 2.30 539 1.53 454 1.10 85nm S1→S0 H→L 99% \nAcetone 53.20。 114.96。 2.31 538 1.52 453 1.10 85 nm S1→S0 H→L 99% \nDichloromethane 53.07。 114.91。 2.34 528 1.47 452 1.12 75 nm S1→S0 H→L 99% \n \n \nThen, molecular OPA and OPE properties are studied and compared in four solvents. As solvent polarity \ndecreases, it is easy to find that molecular emission wavelength s are 544 nm, 539 nm, 538 nm, 528 nm, \nafter analyzing Figure 3 Molecular OPA process . The absorption wavelengths are 454 nm, 454 nm, 453 \nnm, 452 nm, which is gradual decreases. NAPBr in Water has maximum emission and absorption \nwavelength. Specific data are collected in Table . In the experimental data, emission wavelength and \nabsorption wavelength are 525 nm 409 nm , they are within reasonable limits with calculation. Different \nexcitation wavelengths, interaction between laser and molecules, or inter-molecular interaction (such as \nsolute -solute interaction, etc. ) in solution , could result in the discrepancy and have not been considered \nin the calculation.25, 26 Molecular absorption and emission wavelengths in other solvents show blueshift \nto that in Water. They have long Stokes shifts (the difference between the absorption and emission \nmaximum wavelength) over 75nm, up to 90 nm. It has been known that long Stoke s shift helps to avoid \nthe interference of absorption and emission spectra and improves detection sensitivity and accuracy.27 It \nis related to the maximum energy gap of HUMO -LUMO (HOMO represents the highest occupied \nmolecular orbital and LUMO represents the lowest unoccupied molecular orbital). E flu is the energy \ndifference between the two electronic states. δabs and δems are absorption and emission oscillator strength \nwhich are important physical parameters that measure absorption or emission ability. Intramolecular \ncharges transfer is less affected by solvent, so the energy gap between two electronic states changes \nsligh tly. B esides, electrical transition dipole moment is also important. Figure 2 shows that solvent \npolarity has a great influence on molecular fluorescent emission properties. But it has less influence on \nthe variation of absorption wavelength and oscillator strength. \nNow, we perform a detailed analysis of OPA and OPE processes and their transition properties to show \nmolecular optical properties visually.28 The OPA (or OPE) wavelength, corresponding oscillator strength, \nmolecular orbital energy, and transition natures have been given. \nFigure 3 Molecular OPA processes. \nFirst, we analyze the molecular absorption process in detail. Red and green represent positive and \nnegative phases of molecular orbitals . Through studying transition process, the maximum OPA state is \nS1. 99% of molecular orbitals are originated from the HOMO to LUMO transition. It is easy to find that \nmolecular orbitals changes are mainly localized on the torsional benzene ring moiety in Figure 3, which \nis closely connected with the absorption process. \n \nFigure 4 molecular OPE processes. \nIn Figure 4, the OPE processes are shown. According to Kasha's rule, the fast and non -radiative internal \nconversion and vibrational relaxation process result in being located at the lowest molecular vibrational \nlevel in S 1 after excitation, which also affects the molecular configuration. It can be seen that the S 1 to S 0 \ntransition mostly results from the LUMO to HOMO transition. Molecular orbital distribution is similar \nin the twisted benzene ring and mainly located on the naphthol, which is fluorescent group. Small \ntorsional benzene ring orbitals variation indicate that this part has a little effect on the excitation. OPE \nprocesses under different solvents are similar. \n \n \n \nHoles and electrons analysis \n \nFigure 5 Molecular holes and electrons distribution \nTable 3 Detailed data of molecular holes and electrons distribution, including D index, S r and \n \nSolvent D index/Å Sr/. a. u Δσ/ Å \nWater 1.853 0.77719 -0.401 \nEthanol 1.854 0.77720 -0.402 \nAcetone 1.855 0.77719 -0.402 \nDichloromethane 1.857 0.77723 -0.404 \n \nDefine D index that measures the distance between hole and electron centroid: \n||x ele holeD X X=−\n \n||y ele holeD Y Y=−\n \n||z ele holeD Z Z=−\n \n2 2 2\ny ( +( +(xy Dindex D D D=）））\n \nXele/hole refers to the X coordinate of the mass center of hole or electron. \nDefine S r as a function that describes the distribution overlap between electrons and holes: \n( ) ( ) ( )hole ele\nrS r r r=\n \n\n reflects the difference in the overall spatial distribution of electrons and holes: \nhol | | | |ele index = −\n \nWe draw molecular holes and electrons distribution diagram to study electrons excitation characteristics . \nGreen and blue represent the distribution of electrons and holes, which are almost distributed in the \nnaphthalene plane with no obvious difference. Electrons converge near the torsional benzene ring, while \nholes are away. Detailed data are listed in Table 3. Large D index means large distance between electrons \nand holes. Large S r and small \n indicate large overlap between holes and electrons. Electrons - holes \nseparation is not sufficient, and overall spatial distribution discrepancy is small. This is considered to be \na charge transfer excitation. Besides, the main excitation process is from S 0 to S 1. NAPBr in \nDichloromethane has the smallest Δσ, largest D index and S r. It is closely related to non -radiative \ntransition process, which may explain the reason that has the largest fluorescence quantum efficiency. \nDye’s fluorescent property is structure -dependent, especially for the donor−acceptor molecule . It is also \nconfirmed that donor and acceptor transfer charges to achieve the fluorescence. \n \n \n \n \n \n \nTwo-photon absorption property \nExcellent molecular TPA ability is an important criterion for high -quality fluorescent dyes. Therefore, \nwe calculate the NAPBr TPA property (such as excitation energy, corresponding TPA cross section, and \nwavelength) in different solvents, as shown in \nTable 4 Two-photon absorption properties including excitation energy E tpa, corresponding two -photon \nwavelength λtpa, and TPA cross section σtpa (1 GM = 10 -50cm4 s/photon) of the lowest five excited \nstates of dye in different solvents. . Lowest five molecular excited states are selected for analysis. \nTable 4 Two-photon absorption properties including excitation energy E tpa, corresponding two -photon wavelength λtpa, and TPA cross section σtpa (1 GM = 10-50cm4 s/photon) of the lowest five excited states \nof dye in different solvents. \nAs solvent polarity increases, the increases maximum molecular TPA cross section in different solvents \nare 3016GM (Dichloromethane, 654nm), 3375GM (Acetone, 656nm), 3461 GM ( Ethanol, 658nm) , \n3658GM (Water, 658nm). Maximum molecular TPA cross section is located at the lowest fourth excited \nstate, regardless of solvent environment. It indicates that solvent does not affect TPA cross section \ndistribution and there has a certain stability. Compared with molecules in other solvents, it can be seen \nthat NAPBr in Water has a longer OPA wavelength, larger TPA cross section, and OPA oscillator \nstrength. Otherwise, a good conjugated planarity system and strong acceptor group are important for \nexcellent molecular absorption ability in Tang's group analysis.10 In addition, we calculate the electric \ntransition dipole moments from S 0 and S 1 are 3. 8221.a . u (Water), 3. 8212. a. u (Ethanol), 3. 8204. a. u \n(Acetone), 3. 8177. a. u (Dichloromethane), they are closely related to molecular absorption process . \nBesides, electric transition dipole moment between S0 and S 1 is proportional to the orbital overlap of \nHOMO and LUMO. TPA ability, OPA oscillator strength orders are consistent with electric transition \ndipole moment, which have regular and certain stability in solvent. It can be expected that NAPBr can \nact as a promising candidate for a two -photon fluorescent dye. \n \n \nFluorescent quantum efficiency of dyes \n \nTable 5 Fluorescence properties, including vertical excitation energy (E vt), adiabatic energy difference \nbetween S0 and S1 (E ad), radiative decay rate (k r), non -radiative decay rate (k ic), RMSD, reorganization \nenergy (E re) and fluorescence quantum efficiency ( ) of molecules in different solvents. \n Solvent Etpa/ eV λtpa/nm σtpa/GM Solvent Etpa/ eV λtpa/nm σtpa/GM \n \nWater 2.61 948 519 \nEthanol 2.63 940 478 \n 3.41 725 10 3.42 723 10 \n 3.68 672 18 3.69 670 4 \n 3.76 658 3658 3.76 658 3461 \n 4.10 603 9 4.11 602 6 \n \nAcetone 2.64 937 481 \nDichloromethane 2.66 930 435 \n 3.42 723 10 3.42 723 9 \n 3.7 668 7 3.71 667 4 \n 3.77 656 3375 3.78 654 3016 \n 4.11 602 6 4.11 602 4 \n \n \nFluorescent quantum efficiency is also an important index for evaluating dyes properties. According to \nKasha's law, the radiation competition from S 1 to S 0 has an important impact on fluorescence quantum \nefficiency as well as adiabatic excitation energy. From Table , it is easy to find that as solvent polarity \ngradually decreases, the fluorescent quantum efficiency order is 60.56% (Water) ＜62.74%( Ethanol ) \n＜63.48% (Acetone) ＜66.60% (Dichloromethane). Radiative decay rates are closely related to the \nsolvent polarity, they are 3.498×108s-1＜3.506×108s-1＜3.508×108s-1＜3.509×108s-1, respectively. \nThe calculated NAPBr fluorescence quantum efficiency is 67.6% and the experimental result is 1.4%. \nAIE, solvent, and solute -solute interaction are important factors.7 The NAPBr in Dichloromethane has \nthe highest fluorescent quantum efficiency because it’s highest radiative decay rate (3.509 ×108s-1) and \nlowest non -radiative decay rate ( 1.759×1 08 s-1). Molecules in low dielectric constant solvent \nenvironment can suppress the non -radiative loss path of excited -state energy , thereby improving \nfluorescent quantum efficiency. 28 Besides, electrons and holes distribution and transfer analysis also \nhelp us explain the reason that NAPBr has a higher fluorescence quantum efficiency in Dichloromethane. \nAfter considering all aspects of performance, we think that NAPBr has a higher fluorescent quantum \nefficiency, larger TPA cross -section, and Stokes shift. For the underlying reasons of solvent effect on \nfluorescent quantum efficiency, we conduct a detailed study in the next section. \nReorganization energy and Huang -Rhys factor \n \nFigure 6 RMSD between the two states configuration in Water, Ethanol, Acetone, and \nDichloromethane. Blue represents S 0 configuration and red represents S 1 configuration. \nThis chapter studies the solvent effect on non -radiative internal conversion from S 0 to S 1 in detail. This \nis the key to explain the inner reason of fluorescence quantum efficiency. Firstly, RMSD is calculated \n Solvent Evt/eV Ead/eV kr/s-1 kic/s-1 Φ Ere/cm-1 RMSD/Å \n Water 2.28 2.42 3.498×108 2.278×108 60.56% 4987.02 0.23 \n Ethanol 2.30 2.44 3.506×108 2.08×108 62.74% 4813.63 0.228 \n Acetone 2.31 2.45 3.508×108 2.018×108 63.48% 4754.04 0.228 \n Dichloromethane 2.35 2.48 3.509×108 1.759×108 66.60% 4399.03 0.224 and compared to illustrate the solvent effect on molecular configuration difference between S 0 and S 1 \nclearly as shown in Figure 6. As solvent polarity decreases ，the RMSD are 0.23 Å (Water), 0.228 Å \n(Ethanol), 0.228 Å (Acetone), 0.224 Å (Dichloromethane). Molecular geometric changes mostly \noriginate from benzene ring rotation, such as length and dihedral angle change. They are closely related \nto non -radiative transition process and can directly prove the necessity of studying the dihedral angle θ1. \n \nFigure 7 Molecular reorganization energy distribution under different vibration modes in different \nsolvents. \nNon-radiative decay rate is related to the geometric difference between two states, also affects fluorescent \nquantum efficiency. The non -radiative decay rates of NAPBr are 2.28×108s-1(Water), 2.08×108s-\n1(Ethanol ), 2.02×108s-1(Acetone) and 1.76×108s-1(Dichloromethane). Non-radiative decay rate is more \nsensitive to solvent polarity than the radiative decay rate. NAPBr in Water has maximum non -radiative \ndecay rate ( 2.278×1 08s-1) and minimum radiative decay rate ( 3.498×1 08 s-1). So, it has the lowest \nfluorescent quantum efficiency. Geometric relaxation degree between two states and internal conversion \nenergy attenuation can be characterized by reorganization energy, as shown in Figure 7. Obviously, it is \neasy to find that the reorganization energy contributions of low -frequency modes region dominate \nvibrational relaxation processes, which is easily affected by the solvent polarity. As the solvent polarity \ndecreases, the reorganization energies are 4987.02 cm-1 (Water), 4813.63 cm-1 (Ethanol ), 4753.04 cm-1 \n(Acetone), and 4399.03 cm-1 (Dichloromethane), receptively. Reorganization energy peak value in \nhigh-frequency modes region also decreases. Larger reorganization energy in low -frequency modes \nregion may be attributed to structure changes, especially benzene ring rotation. Molecular difference is \nmainly reflected in the dihedral angle θ1 caused by solvent effect. It is closely related to the geometric \nstructure difference of S 0 and S 1 in the non -radiative transition. \nAccording to Marcus Theory, if no bond is formed or broken in electrons transfer reaction, it has a less \ngeometric relaxation, lower reorganization energy, and higher electron transfer rate. The consequence of \nelectrons transfer is charges rearrangement and it is greatly influenced by the solvent environment. \nMoreover, a D -π-A structure molecule provide a very important factor for charges transfer. It is \nthe solvent polarity that determines free activation energy in internal conversion, thereby affecting \nmolecular photophysical properties. Molecule with different geometric configurations in two states \ntransforms firstly into a more similar shape by vibration and then transfer electrons to make the geometry \nequivalent. It can also be used to explain the rul e of fluorescent quantum efficiency. For example, the \nNAPBr in Dichloromethane has less difference in S 0 and S 1 geometr ies, so it can transfer electrons with \na little vibration. \n \nFinally, we study Huang−Rhys (HR) factors to measure tnon -radiative excited state energy consumption, \ncharacterize electron vibration coupling, and the contribution degree of geometric configuration \ndifference at this frequency. It is helpful for further a nalyzing the vibration situation and the non -radiative \ntransition comprehensively and holistically. Some vibration modes with larger HR factor (> 0.1) have \nbeen listed in \n.29 The Huang−Rhys (HR) factor: \n2\n2KK\nKDHR= Solvent Mode f HR Ere/cm-1 Solvent Mode f HR Ere/cm-1 \n \nAcetone 1 16.39 3.35 54.93 \nDichlorom\n-ethane 1 16.53 3.26 53.86 \n 2 22.03 0.18 3.90 2 22.5 0.44 9.81 \n 3 29.95 1.97 58.97 3 30.1 1.84 55.41 \n 4 39.97 0.17 6.98 4 39.62 0.16 6.43 \n 5 50.15 1.39 69.87 5 50.72 1.46 73.89 \n 8 81.39 0.14 11.11 8 81.71 0.11 9.39 \n 9 96.5 0.23 22.02 9 96.67 0.23 22.02 \n 10 100.83 0.20 20.29 10 101.17 0.22 22.73 \n 39 642.73 0.12 79.59 39 642.98 0.11 71.81 \n 44 717.89 0.19 137.20 44 718.17 0.19 133.86 \n 67 1094.41 0.12 136.30 67 1094.88 0.10 111.78 \n 105 1603.23 0.20 325.23 105 1604.51 0.19 302.51 ωk represents the vibration frequency and Dk is the normal coordinate displacement of mode k. \n \n \n \n \n \n \nTable 6 Relatively bigger Huang−Rhys (HR) factor ( ＞0.1) and reorganization energy in some \nvibration mode and frequency. \n Solvent Mode f HR Ere/cm-1 Solvent Mode f HR Ere/cm-1 \n \nWater 1 16.27 3.43 55.81 \nEthanol 1 16.37 3.37 55.09 \n 3 29.88 2.04 61.02 2 21.94 0.14 3.12 \n 4 40.72 0.19 7.78 3 29.94 1.99 59.64 \n 5 49.9 1.35 67.28 4 40.11 0.18 7.16 \n 8 81.17 0.15 12.05 5 50.09 1.38 69.16 \n 9 96.43 0.23 22.40 8 81.34 0.14 11.24 \n 10 100.56 0.18 18.53 9 96.48 0.23 22.14 \n 39 642.58 0.13 84.35 10 100.77 0.20 19.85 \n 44 717.72 0.20 140.07 39 642.70 0.13 80.76 \n 46 739.45 0.12 86.62 44 717.85 0.19 137.88 \n 67 1094.12 0.14 150.01 67 1094.34 0.13 140.03 \n 105 1602.38 0.21 337.65 105 1603.04 0.20 327.89 \n Larger HR factors are distributed in the low -frequency mode (<500 cm-1) region and have different \ncontribution degree of structural changes, this is present in all four solvents . 28 The largest HR factors \nare 3.43 (16.27 cm-1, Water), 3.37 (16.37 cm-1, Ethanol ), 3.35 (16.39 cm-1, Acetone), 3.26 (16.53 cm-1, \nDichloromethane). Vibration frequency and its corresponding HR factor change slightly. Total HF factor \nreduces and shows that molecular rotational motion is suppressed . It indicates that solvent polarity has a \nslight effect on the HR factor and its frequency mode. Geometric configuration difference , non-radiative \ntransition, and total HR factor are closely related and well characte rized. This also proves that solvent \npolarity affects the molecular photophysical performance, further confirms our calculation result, and \nclarifies research’ s necessity. \nConclusion \nIn this work, we conduct a theoretical study on the solvent effect of naphthalene -based dyes for detecting \nlipid droplets at the DFT level. Result shows that NAPBr exhibits excellent and regular two -photon \nfluorescence ability in all four solvents, with a high fluorescence quantum efficiency, large Stokes shift, \nand TPA cross -section. The internal mechanism of solvent effect on photophysical properties is also \nanalyzed in detail. More than 99% molecular orbitals originate from the HOMO —LUMO transition in \nOPA and OPE process. Electrons and holes analysis prove that fluorescence emission mechanism is a \ncharge transfer process between donor and acceptor. Through affecti ng non -radiative transition process , \nmolecular structure variation (especially benzene ring rotation) degree in different solvents is closely \nrelated to th e fluorescence quantum efficiency. After considering all aspects of performance, a medium \ndielectric constant solvent environment can achieve a balance of two -photon absorption and fluorescence \nemission ability. The conclusion and method are more suitable to provide a reference and help for the \nsubsequent TPA dyes research with AIE characteristics in the future. Solvent Mode f HR Ere/cm-1 Solvent Mode f HR Ere/cm-1 \n \nAcetone 1 16.39 3.35 54.93 \nDichlorom\n-ethane 1 16.53 3.26 53.86 \n 2 22.03 0.18 3.90 2 22.5 0.44 9.81 \n 3 29.95 1.97 58.97 3 30.1 1.84 55.41 \n 4 39.97 0.17 6.98 4 39.62 0.16 6.43 \n 5 50.15 1.39 69.87 5 50.72 1.46 73.89 \n 8 81.39 0.14 11.11 8 81.71 0.11 9.39 \n 9 96.5 0.23 22.02 9 96.67 0.23 22.02 \n 10 100.83 0.20 20.29 10 101.17 0.22 22.73 \n 39 642.73 0.12 79.59 39 642.98 0.11 71.81 \n 44 717.89 0.19 137.20 44 718.17 0.19 133.86 \n 67 1094.41 0.12 136.30 67 1094.88 0.10 111.78 \n 105 1603.23 0.20 325.23 105 1604.51 0.19 302.51 \nReference \n1. Hartman, I. Z.; Liu, P.; Zehmer, J. K.; Luby -Phelps, K.; Jo, Y.; Anderson, R. G.; DeBose -Boyd, R. \nA. J Biol Chem 2010, 285, (25), 19288 -98. \n2. Fei, W.; Wang, H.; Fu, X.; Bielby, C.; Yang, H. Biochem J 2009, 424, (1), 61 -7. \n3. Olzmann, J. A.; Richter, C. M.; Kopito, R. R. Proc Natl Acad Sci U S A 2013, 110, (4), 1345 -50. \n4. Krahmer, N.; Hilger, M.; Kory, N.; Wilfling, F.; Stoehr, G.; Mann, M.; Farese, R. V., Jr.; Walther, \nT. C. Mol Cell Proteomics 2013, 12, (5), 1115 -26. \n5. Herker, E.; Ott, M. Trends Endocrinol Metab 2011, 22, (6), 241 -8. \n6. Krahmer, N.; Farese, R. V.; Walther, T. C. J. E. m. m. 2013, 5, (7), 973 -983. \n7. Niu, G.; Zhang, R.; Kwong, J. P.; Lam, J. W.; Chen, C.; Wang, J.; Chen, Y.; Feng, X.; Kwok, R. T.; \nSung, H. H. -Y. J. C. o. M. 2018, 30, (14), 4778 -4787. \n8. Kim, H. M.; Cho, B. R. Chemical Reviews 2015, 115, (11), 5014 -5055. \n9. Pawlicki, M.; Collins, H. A.; Denning, R. G.; Anderson, H. L. Angewandte Chemie International \nEdition 2009, 48, (18), 3244 -3266. \n10. Hilborn, R. C. J. A. J. o. P. 1982, 50, (11), 982 -986. \n11. Luo, Y.; Norman, P.; Macak, P.; Ågren, H. J. T. J. o. P. C. A. 2000, 104, (20), 4718 -4722. \n12. Wang, C. -K.; Zhao, K.; Su, Y.; Ren, Y.; Zhao, X.; Luo, Y. J. T. J. o. c. p. 2003, 119, (2), 1208 -\n1213. \n13. Albota, M.; Beljonne, D.; Brédas, J. -L.; Ehrlich, J. E.; Fu, J. -Y.; Heikal, A. A.; Hess, S. E.; Kogej, \nT.; Levin, M. D.; Marder, S. R. J. S. 1998, 281, (5383), 1653 -1656. \n14. Niu, Y.; Li, W.; Peng, Q.; Geng, H.; Yi, Y.; Wang, L.; Nan, G.; Wang, D.; Shuai, Z. J. M. P. 2018, \n116, (7 -8), 1078 -1090. \n15. Niu, Y.; Peng, Q.; Shuai, Z. J. S. i. C. S. B. C. 2008, 51, (12), 1153 -1158. \n16. Peng, Q.; Yi, Y.; Shuai, Z.; Shao, J. J. J. o. t. A. C. S. 2007, 129, (30), 9333 -9339. \n17. Shuai, Z.; Peng, Q. J. N. S. R. 2017, 4, (2), 224 -239. \n18. Shuai, Z.; Peng, Q. J. P. R. 2014, 537, (4), 123 -156. \n19. GAUSSIAN 16, R. i. h. w. g. c. GAUSSIAN 16, References in http://www.gaussian.com/ \n20. Humphrey, W.; Dalke, A.; Schulten, K. J. J. o. m. g. 1996, 14, (1), 33 -38. \n21. Tomasi, J.; Mennucci, B.; Cammi, R. J. C. r. 2005, 105, (8), 2999 -3094. \n22. Fan, D.; Yi, Y.; Li, Z.; Liu, W.; Peng, Q.; Shuai, Z. The journal of physical chemistry. A 2015, \n119, (21), 5233 -40. \n23. Lin, S. H. The Journal of Chemical Physics 1966, 44, (10), 3759 -3767. \n24. DALTON 2013 A molecular electronic structure program, R. D., see \nhttp://daltonprogram.org/ . \n25. Zhang, Y.; Leng, J.; Hu, W. Sensors 2018, 18, (5). \n26. Zhang, Y. -J.; Wang, X.; Zhou, Y.; Wang, C. -K. Chemical Physics Letters 2016, 658, 125 -129. \n27. Zhu, M. -Y.; Zhao, K.; Song, J.; Wang, C. -K. J. C. P. B. 2018, 27, (2), 023302. \n28. Fan, J.; Zhang, Y.; Zhou, Y.; Lin, L.; Wang, C. -K. J. T. J. o. P. C. C. 2018, 122, (4), 2358 -2366. \n29. Cai, W.; Zhang, H.; Yan, X.; Zhao, A.; He, R.; Li, M.; Meng, Q.; Shen, W. J. P. C. C. P. 2019, 21, \n(15), 8073 -8080. \n " }, { "title": "2401.16252v1.Zero_sum_Random_Games_on_Directed_Graphs.pdf", "content": "Zero-sum Random Games on Directed Graphs\nLuc Attia∗Lyuben Lichev†Dieter Mitsche‡Raimundo Saona§\nBruno Ziliotto¶\nJanuary 30, 2024\nAbstract\nThis paper considers a class of two-player zero-sum games on directed graphs whose vertices\nare equipped with random payoffs of bounded support known by both players. Starting from\na fixed vertex, players take turns to move a token along the edges of the graph. On the\none hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we\nshow that the value of the game converges almost surely to a constant at an exponential rate\ndominated in terms of the expansion. On the other hand, for the infinite d-ary tree that does\nnot fall into the previous class of graphs, we show convergence at a double-exponential rate\nin terms of the expansion.\n1 Introduction\nThe following class of two-player zero-sum games has been introduced in [8] under the name of\npercolation games . Each vertex of Zdis equipped with a real-valued random variable called payoff .\nThe realization of all these variables is known to the players at the start of the game. Initially,\na token is placed at some vertex of Zdand, at every stage, each player chooses an action. Then,\nthe token is moved consecutively by Player 1 and by Player 2 according to the chosen actions.\nAt the end of every stage, Player 2 pays to Player 1 the payoff of the corresponding vertex.\nPlayer 1 aims at maximizing the mean payoff over nstages, while Player 2 aims at minimizing\nthe same quantity, and the value of that game is denoted by Vn. The main result of [8] shows\nthat, when payoffs are bounded i.i.d. random variables and the game is oriented (meaning that,\nat every move, the projection of the position of the token onto some fixed axis increases), then\n(Vn) converges almost surely (a.s.) to a constant.\nThe class of percolation games is motivated by various reasons. First, it relates to the rich\ngame-theoretic literature on the existence of a limit value in dynamic games (see for example\nthe surveys [14, 15]). This topic is particularly delicate for dynamic games with infinite state\nspace, where general positive results are scarce (see [9, 13, 18] for some recent advances, and\n[17] for several counterexamples). Second, percolation games connect to the important topic\nof stochastic homogenization of partial differential equations, for example, see [8, Section 4] for\nresults on Hamilton-Jacobi equations. Moreover, from a probabilistic point of view, percolation\ngames combine aspects of first-passage and last-passage percolation [3], and a related model of\nProbabilistic Finite Automaton has been studied in [11, 4]. Finally, it contributes to the growing\nliterature on random games (see e.g. [7, 1, 2, 10]).\n∗Paris Dauphine University, France.\n†Univ.Jean Monnet, Saint-Etienne, France and Institute of Mathematics and Informatics, Bulgarian Academy\nof Science, Sofia, Bulgaria.\n‡Univ. Jean Monnet, Saint-Etienne, France and Institute for Mathematical and Computational Engineering,\nPontif´ ıcia Universidad Cat´ olica, Chile.\n§Institute of Science and Technology Austria, Austria.\n¶CEREMADE, CNRS, Paris Dauphine University, France.\n1arXiv:2401.16252v1 [math.OC] 29 Jan 2024In this paper, we consider a model with a structure similar to that of a percolation game\nbut where the state space is not restricted to be the graph Zd. We introduce directed games\nwhere the state space of the game is the vertex set of an acyclic directed graph Γ where players\nmove the token along the edges of Γ respecting their orientation. On the one hand, under certain\nassumptions of transitivity and sub-exponential growth of Γ, we prove that Vnis exponentially\nconcentrated around a given deterministic value (so, in particular, converges to that value a.s.)\nand relate the convergence rate to the speed of growth of the graph. On the other hand, we\nconsider the infinite d-ary tree with d≥2 where each vertex has exactly dchildren and every\nedge is directed from the parent to the child. These graphs do not belong to the previous class\nof transient games due to their exponential growth. In this case, we show a stronger double-\nexponential concentration of Vnaround its expected value.\n2 Preliminaries\nAdirected game is a dynamical system that consists of a locally finite directed graph Γ with\ninfinite countable vertex set Zcalled the state space , an initial state z0∈Zand a collection\nof independent and identically distributed (i.i.d.) random variables ( Gz)z∈Zcalled payoffs . We\nassume that Γ has uniformly bounded degrees and contains neither directed cycles nor vertices\nwith out-degree 0. The game is played by two players called Player 1 and Player 2. At the start\nof the game, the payoffs ( Gz)z∈Zare sampled and presented to both players, who thus obtain\nperfect information. Then, a token is placed at the initial state z0. For every integer i≥0, given\nthat the token is positioned at a state z∈Zbefore stage i+ 1, the following happens:\n•ifiis even, Player 1 moves the token to an out-neighbor z′ofzin Γ,\n•ifiis odd, Player 2 moves the token to an out-neighbor z′ofzin Γ,\n•Player 1 receives the payoff Gz′from Player 2.\nNote that, unlike the setting in [8], only one of the players performs a move at each round.\nWe are mostly interested in the n-stage game consisting of the first nstages for (typically large)\nintegers n.\nAstrategy of Player 1 (resp. Player 2) is a function σ:S\nm≥0Z2m+1→Z(resp. τ:S\nm≥0Z2m+2→\nZ) with the property that, for every m≥0 and ( z0, z1, . . . , z 2m+1)∈Z2m+2, Γ contains the edge\nfrom z2mtoσ(z0, . . . , z 2m) (resp. from z2m+1toτ(z0, . . . , z 2m+1)). We denote by Σ the collection\nof all strategies for Player 1 and by Tthe collection of all strategies for Player 2.\nGiven a pair of strategies ( σ, τ)∈Σ× T, we define inductively the trajectory of the token\nby setting z2i+1:=σ(z0, . . . , z 2i) and z2i+2:=τ(z0, . . . , z 2i+1) for every i≥0. This allows us to\ndefine the n-stage payoff function γz0n: Σ× T → Rby setting\nγz0n(σ, τ):=1\nnnX\ni=1Gzi.\nNote that, for a fixed initial state, since the directed graph Γ is locally finite, the n-stage\ngame is played on a finite state space with perfect information. For every z∈Z, the n-value of\nthe game with initial state z0=zis defined as\nVn(z):= max\nσ∈Σmin\nτ∈Tγz\nn(σ, τ) = min\nτ∈Tmax\nσ∈Σγz\nn(σ, τ),\nwhere the classic minimax theorem [6, 16] applies to justify the above equality. Moreover, we will\nsay that a strategy σ∈Σ (resp. τ∈ T) isoptimal for the n-stage game (starting from z) ifσ\nmaximizes min τ∈Tγz\nn(·, τ) over Σ (resp. if τminimizes max σ∈Σγz\nn(σ,·) over T).\n2A classic question in the game-theoretic literature is to ask for the convergence of the n-value\nasngrows to infinity. Since payoffs are random, Vnis a random variable. Therefore, we are\ninterested in whether the sequence ( Vn) converges a.s. to a constant.\nIn our model, if no further assumptions are imposed, it is possible that ( Vn) does not converge.\nFor example, for all integers m≥0, set nm:= 222mandn′\nm:= 222m+1and consider the case where\nΓ is a directed tree (all edges being directed away from the root) where each node with even height\nhas only one child, while each node with odd height khas two children if k= 1 or k∈[nm, n′\nm)\nfor some m≥0, and it has only one child if k∈[n′\nm, nm+1). Moreover, let the payoffs be i.i.d.\nBernoulli random variables with parameter 1 /2. In particular, for every m≥1, in the nm-stage\ngame, Player 2 has only one choice most of the time, while in the n′\nm-stage game, she has two\nchoices most of the time. Since Player 2 can not uniformly pick a vertex with payoff 0 (if it is\npresent), and pick an available vertex otherwise, one can show that a.s.\nlim sup\nm→∞Vn′m≤3\n8<1\n2= lim\nm→∞Vnm.\nIndeed, while Player 1 never has a choice in the n′\nm-game (implying that the mean payoff over\nthe odd states visited by the token a.s. converges to 1 /2), Player 2 can ensure with the above\nstrategy that the mean payoff over the even states visited by the token a.s. converges to 1 /4,\nwhich yields that a.s. lim supm→∞Vn′m≤3/8. At the same time, for every ε >0, Chernoff’s\nbound for the Binomial distribution Bin( nm,1/2) and a union bound over the O(2n′\nm−1) vertices\nat level nmin Γ shows that Vnmis in the interval [1 /2−ε,1/2 +ε] with probability very close\nto 1. In particular, a.s. ( Vn) does not converge. Therefore, to ensure convergence, we will need\nfurther structural assumptions on the graph.\nBefore turning to our results, we provide some vocabulary. Given a vertex z∈Z, adescendant\nofz(in Γ) is a vertex that can be reached from zby a directed path in Γ. We say that zandz′\nareequivalent if the two subgraphs of Γ induced by the descendants of zand by the descendants\nofz′, respectively, are isomorphic (as directed graphs).\nDefinition 1. The graph Γ is weakly transitive if there is a state z∗and an integer Msuch that\nthe following holds: for each state z∈Z, in the game with initial state z0=z, each player has a\nstrategy that, independently of the moves of the opponent, ensures that the token is placed at a\nstate equivalent to z∗after an even number of ℓ≤Mstages.\nNote that all vertex-transitive graphs are weakly transitive with M= 0. In the remainder of\nthe paper, we always assume that Γ is weakly transitive. The next two subsections present two\ntypes of directed games used in our main results.\n2.1 Weakly transitive games with sub-exponential expansion\nWe continue with a few definitions. Given a state z∈Z, we consider a partition Π z:= (Zi(z))i≥0\nofZsuch that: (i) Z0(z) ={z}; and (ii), for all strategies ( σ, τ)∈Σ× T for the game starting\natzand for all i≥1, the token can visit the set Zi(z) at most once. Since Γ has no directed\ncycles, such a partition exists. For example, the trivial one where every part contains a single\nstate satisfies this property. We call such partitions adapted . For every integer n≥1, we also set\nZ[n](z):=Sn\nj=0Zj(z) and Z(n)(z) for the set of reachable states from zafter at most nsteps. Note\nthat, when it is clear from the context, we omit zfrom the notation and simply write Zn, Z(n)\nandZ[n]for better readability.\nGiven a family of adapted partitions Π := (Π z)z∈Zin a directed game, we define the transient\nspeed function hof Π as\nh:n∈N7→max\nz∈Zminn\nk∈N:Z(n)(z)⊆Z[k](z)o\n.\n3Note that h(n)≥nfor every integer n≥1 since, for every z∈Z, exactly nof the sets ( Zi(z))i≥1\nare visited by the token after nstages. Our main goal is to analyze directed games where the\nsize of the sets Z(n)(z) does not increase too fast as ngrows to infinity.\nDefinition 2 (δ-transient games) .Given a family of adapted partitions Π with transient speed\nh, we define the function ψ:N×(0,∞)→Rby\nψ(n, t):= exp\u0012\n−t2n2\n2h(n)\u0013\nmax\nz∈Z|Z(2n)(z)|.\nFor a fixed δ >0, a directed game on a graph Γ with vertex set Zis called δ-transient if there exists\na family of adapted partitions Π of Zand a sequence ( εn)n≥1such that εn+ψ(n, εn) =O(n−δ).\nSuch a family Π is called a δ-adapted family .\nRemark 1. The concept of δ-transient games is only relevant for δ∈(0,1/2). Indeed, Defini-\ntion 2 requires that (ψ(n, εn))nconverges to zero. Therefore, since h(n)≥n, this implies that\nn=o(ε2\nnn2), so that εn∈o(n1−1/2).\nRemark 2. A sufficient condition under which a directed game is δ-transient is the following:\nthere exists an adapted partition Πand real numbers α∈[0,2−2δ)andβ∈[0,2−2δ−α)such\nthath(n) =O(nα)andmax z∈Z|Z(n)(z)|= exp( O(nβ)).\nNote that the definition of a δ-transient game is independent of the payoffs and only makes\nassumptions on the state space and the associated adapted partition. We now give a few examples\nofδ-transient games.\n2.1.1 Oriented directed games\nFix an integer d≥1, and denote by eithed-dimensional vector with 1 in coordinate iand 0\nin all other d−1 coordinates. Given positive integers n1, . . . , n d≥1, a (directed) graph Γ with\nvertex set Z⊆Zdis called ( n1, . . . , n d)-invariant (or simply invariant ) if, for every i∈[1, d], the\ntranslation at vector nieiis a graph isomorphism for Γ. A directed game is called oriented if its\nunderlying graph Γ is invariant and there exists u∈Rd\\ {0}such that, for every directed edge\nzwin Γ, we have ( w−z)·u >0 (here, ·denotes the usual scalar product of vectors in Rd). We\nshow the following proposition.\nProposition 1. Every oriented directed game is δ-transient for all δ∈(0,1/2).\nThe following two classes of games present particular examples of oriented directed games.\nExample 1 (Games on tilings) .Atiling is a periodic partition of the plane into translations of\none or several polygonal shapes, called tiles, with vertices in Z2. Tilings naturally define planar\ngraphs whose vertex set coincides with the corners of the tiles and two vertices are connected by an\nedge if these can be connected by following the boundary of a tile without meeting another vertex\non the way. By equipping the edges of this graph with suitable orientations, one can generate\nmany different oriented directed games, see e.g. Figure 1.\nExample 2 (Games on directed chains of graphs) .Fix a finite vertex-transitive graph Hwith\nvertex set V(H)and edge set E(H), and a bi-infinite sequence of copies (Hi)i∈ZofH. For every\ni∈Zandu∈V(H), denote by uithe vertex in Hicorresponding to u. We call an H-chain the\ngraph ΓHwith verticesS\ni∈ZV(Hi)and edges {uivi+1:i∈Z, uv∈E(H)}.\nGames on H-chains can be seen as instances of oriented directed games on Z. Indeed, fixing\nh=|V(H)|, one may identify the vertices of Hiwith the integers in the interval [ih+ 1,(i+ 1)h]\nfor all i∈Zin a translation-invariant way.\n4Figure 1: The figure depicts part of a tiling with two types of square tiles. The vertices and the\nedges of the planar graph originating from the tiling are depicted in blue and red, respectively.\nEach horizontal edge is oriented from left to right and every vertical edge is oriented from bottom\nto top. One may choose z∗to be the bottom left vertex of a small square and M= 6.\n2.1.2 Weakly transitive games with controlled expansion\nFix an arbitrary infinite rooted tree Twith root rand a family of vertex-disjoint infinite paths\n(Pv)v∈V(T)where the path Pvstarts at vertex vinT. Define Γ = T∪(S\nv∈V(T)Pv) as the tree\nrooted in rand with all edges oriented away from r. Let ( Zn)n≥0be a partition of the vertex set\nZof Γ where Znconsists of all vertices at distance nfrom rfor all n≥0. Also, for every z∈Z\nandk≥2, define Zk(z) to be the set of descendants of zat distance kfrom it while Z0(z) ={z}\nandZ1(z) =Z\\((S\nk≥2Zk(z))∪Z0(z)). Note that, somewhat arbitrarily, we added all vertices\nnot reachable from ztoZ1(z) to ensure that ( Zi(z))i≥0is a partition of Z. Then, Π z= (Zi(z))i≥0\nis an adapted partition and Π = (Π z)z∈Zis an adapted family of partitions. Moreover, a single\nmove of each player is sufficient to place the token at the second vertex of some infinite path\namong ( Pv)v∈V(T). This implies that the game is weakly transitive.\nLet us show that we can control the growth speed of max z∈Z|Z(2n)(z)|. Consider a set of\nnon-negative integers L={ℓi:i≥1}with ℓ1< ℓ2< . . . and let every vertex of Tin level\nℓhave two children if ℓ∈Land one child otherwise. Moreover, suppose that ℓ1= 0 and\n(ℓi−ℓi−1)i≥1is a non-decreasing sequence. Then, one can readily check that, for every n≥1,\nmax z∈Z|Z(n)(z)|=|Z(n)(r)|. Indeed, for every k, n≥1 and a vertex z∈Zon level k, using the\nassumptions that ℓ1= 0 and ( ℓi−ℓi−1)i≥1is a non-decreasing sequence, we get\n|Z(n)(z)\\Z(n−1)(z)|= 2|L∩{k,...,k +n−1}|≤2|L∩{0,...,n−1}|=|Z(n)(r)\\Z(n−1)(r)|.\nThus, for every integer n≥0,|Z(n)(r)|= 1 +Pn−1\ni=02|L∩{0,...,i}|. Therefore, by a suitable choice\nof the set L, one can construct a tree Twith an arbitrary growth that is faster than linear but\nslower than exponential. In particular, for every δ∈(0,1/2), this shows the existence of games\nthat are δ-transient but, for every δ′> δ, not δ′-transient.\n2.2 Directed games on d-ary trees\nWe turn our attention to a natural example of a directed game where the set of reachable states\nafter nsteps grows exponentially with n. Note that, for all δ >0, it is not a δ-transient game.\nFix an integer d≥2 and let Tbe an infinite d-ary tree , that is, a tree where every vertex has\ndchildren, with vertex set Zwhere every edge is oriented from the parent to the child. We fix\nan arbitrary initial vertex z0and, for every integer i≥0, we define Zito be the set of vertices\ninZthat can be reached from z0by exactly isteps and also denote Zeven:=S\ni≥0Z2iand\nZodd:=S\ni≥0Z2i+1. Note that, for every n≥1, the random variables ( Vn(z))z∈Zhave the same\ndistribution. Thus, we often omit the dependence of Vninz.\n52.3 Main results\nOur first main result shows sharp concentration for the n-value of δ-transient games around a\ndeterministic constant.\nTheorem 1. Fixδ∈(0,1/2). Consider a δ-transient directed game, a δ-adapted family with\ntransient speed h, and i.i.d. payoffs (Gz)z∈Zsupported on the interval [0,1]. Then, there exist\nconstants v∞∈[0,1]andK > 0such that, for all n≥1,t≥0, and z∈Z,\nP\u0010\n|Vn(z)−v∞| ≥t+Kn−δ\u0011\n≤2 exp\u0012\n−t2n2\n2h(n)\u0013\n.\nConsequently, (Vn)converges almost surely to v∞.\nOur second main result shows that the n-value of the directed game on a d-ary tree is tightly\nconcentrated around a constant.\nTheorem 2. Fix an integer d≥2. Consider a directed game on the d-ary tree with i.i.d. payoffs\nsupported on the interval [0,1]. Then, there exists a real number v∞∈[0,1]such that, for every\nδ∈(0,1/2), there exists K > 0such that, for every n≥1andt≥0,\nP(|Vn−v∞| ≥t+ 2t2+Kn−δ)≤exp\u0012\n−1\n6exp\u0012t2n\n4\u0013\u0013\n.\nConsequently, (Vn)converges almost surely to v∞.\nOutline of the proofs. The proofs of both theorems contain two main steps. The first step\ninvolves standard concentration arguments showing that Vnis close to E[Vn] with high probability.\nWhile these are sufficient for Theorem 1, the stronger probabilistic bound in Theorem 2 requires\nan additional boosting obtained by dividing the first nlevels of the d-ary tree into two groups\nof consecutive levels and treating the n-stage game as two consecutive games on kandn−k\nstages respectively. The second step uses the structure of the underlying graph to show that\nE[Vn] satisfies a certain subadditivity assumption, which allows us to conclude that ( E[Vn])n≥1\nconverges to a constant v∞, and moreover, |E[Vn]−v∞|is polynomially small. The proof of\nProposition 1 relies on a simple explicit construction.\nPerspectives The proofs of Theorems 1 and 2 have a similar structure but use different argu-\nments. A challenging research question would be to prove convergence of ( Vn) and concentration\nbounds in any weakly transitive directed game, irrespective of the expansion speed of the under-\nlying graph, thus unifying Theorems 1 and 2.\nPlan of the paper. This paper is organized as follows. In Section 3, we state some classical\nresults we will use later. Then, in Section 4, we prove Theorem 1, and in Section 5, we prove\nTheorem 2. In Section 6, we prove Proposition 1.\n3 Classical results\nIn our proofs, we make use of the well-known bounded difference inequality , also known as McDi-\narmid’s inequality , tightly related to Azuma’s inequality .\nLemma 1 (Corollary 2.27 in [12]) .Fix a function f: Λ1× ··· × ΛN→Rand let Y1, . . . , Y Nbe\nindependent random variables taking values in Λ1, . . . , ΛN, respectively. Suppose that there are\npositive constants c1, . . . , c Nsuch that, for every two vectors z, w∈Λ1×···× ΛNthat differ only\n6in the k-th coordinate, we have |f(z)−f(w)| ≤ck. Then, for every t≥0, the random variable\nX=f(Y1, . . . , Y N)satisfies\nP(X−E[X]≥t)≤exp \n−t2\n2PN\ni=1c2\ni!\n.\nP(X−E[X]≤ −t)≤exp \n−t2\n2PN\ni=1c2\ni!\n.\nWe also use the following result that states convergence of almost subadditive sequences.\nLemma 2 (Theorem 23 in [5]) .Fix an increasing function ϕ:N→(0,∞)such that the sum\nof(ϕ(n)/n2)n≥1is finite, and a function f:N→Rsuch that, for all n∈Nand all integers\nm∈[n/2,2n],f(n+m)≤f(n) +f(m) +ϕ(n+m). Then, there exists ℓ∈R∪ {−∞} such that\nf(n)\nn− − − →\nn→∞ℓ.\n4δ-transient games: proof of Theorem 1\nFix an initial state z0and write Vn=Vn(z0), Zn=Zn(z0) for short. To begin with, we show that\nVnis well concentrated around its expected value. Note that the next lemma holds for weakly\ntransitive games in general and will be reused in the next section.\nLemma 3. For every t≥0,\nP(Vn−E[Vn]≥t)≤exp\u0012\n−t2n2\n2h(n)\u0013\n,\nP(Vn−E[Vn]≤ −t)≤exp\u0012\n−t2n2\n2h(n)\u0013\n.\nProof. Define the (random) vectors Xk= (Gz)z∈Zk∈[0,1]|Zk|. Then, since Z(n)⊆Z[h(n)],Vncan\nbe written as f(X1, . . . , X h(n)) for some function f: [0,1]|Z1|× ··· × [0,1]|Zh(n)|→R. Moreover,\nfor every integer k∈[1, h(n)], the token visits the set Zkat most once and therefore, for every\npair of strategies ( σ, τ)∈Σ× T,γz0n(σ, τ) varies by at most 1 /nas a function of Xk. Hence, for\nevery choice of vectors ( xi)h(n)\ni=1∈[0,1]|Z1|× ··· × [0,1]|Zh(n)|andx′\nk∈[0,1]|Zk|,\n|f(x1, . . . , x k, . . . , x h(n))−f(x1, . . . , x′\nk, . . . , x h(n))| ≤1\nn′\n, .\nLemma 1 applied to Vnfinishes the proof.\nIn the remainder of the proof, we show that E[Vn] converges to a constant polynomially fast.\nNext, we state and prove an auxiliary lemma relating the values of games of different lengths.\nLemma 4. Fix integers n≥1andk∈[1, n]. Then, |Vn−Vn−k| ≤k/n.\nProof. Suppose that Player 1 (resp. Player 2) plays the first n−kstages according to an optimal\nstrategy for the ( n−k)-stage game, and plays arbitrarily during the remaining kstages of the\nn-stage game. Then, nVn≥(n−k)Vn−kandnVn≤(n−k)Vn−k+k. Hence, |n(Vn−Vn−k)| ≤\nmax( kVn−k, k−kVn−k)≤k, which implies the statement of the lemma.\nThe next lemma shows that starting from different initial states changes the n-value only\nslightly when nis large.\nLemma 5. For every z∈Z,|E[Vn(z)]−E[Vn(z∗)]|=O(n−δ).\n7Proof. Denote by Ethe set of states z∈Z(M)that are equivalent to z∗. By Definition 1,\nindependently of the moves of Player 2, E̸=∅, and Player 1 can ensure that the token is at a\nstate in Eafter an even number of ℓ≤Mstages. Hence, using Lemma 4, we have\nnVn≥(n−M) min\nz∈EVn−M(z)≥(n−M) min\nz∈E(Vn(z)−M/n)≥min\nz∈EnVn(z)−2M . (1)\nNow, we bound from below the expectation of the right-hand side. Let ∆ be the maximum\nout-degree of Γ. Then, |E| ≤ |Z(M)| ≤1 + ∆ + . . .+ ∆M≤(M+ 1)∆Mtogether with the choice\nofεnfrom Definition 2 imply that\nE\u0014\nmin\nz∈EVn(z)\u0015\n≥(E[Vn(z∗)]−εn)(1−P(∃z∈E:Vn(z)≤E[Vn(z)]−εn))\n≥(E[Vn(z∗)]−εn)(1−(M+ 1)∆Mψ(n, εn)) =E[Vn(z∗)]−O(n−δ),(2)\nwhere the second inequality comes from a union bound and the last equality is implied by the\nfact that εn+ (M+ 1)∆Mψ(n, εn) =O(n−δ). Thus, taking expectations on both sides of (1) and\nusing (2) shows that\nE[Vn]≥E[Vn(z∗)]−O(n−δ+ 2M/n) =E[Vn(z∗)]−O(n−δ). (3)\nSimilarly, Player 2 can ensure that the token reaches a state in Eafter an even number of\nℓ≤Mstages. Hence,\nnVn≤(n−M) max\nz∈EVn−M(z) +M≤max\nz∈EnVn(z) +M . (4)\nAt the same time, similarly to (2), E[max z∈EVn(z)] is bounded from above by\n(E[Vn(z∗)] +εn)(1−P(∃z∈E:Vn(z)≥E[Vn(z)] +εn)) +P(∃z∈E:Vn(z)≥E[Vn(z)] +εn),\nwhich is at most E[Vn(z∗)] + ( εn+ (M+ 1)∆Mψ(n, εn)) =E[Vn(z∗)] +O(n−δ). Combining this\nwith (4) shows that E[Vn]≤E[Vn(z∗)] +O(n−δ), and together with the upper bound in (3) this\nfinishes the proof.\nNext, we show that the expected value of Vnconverges as n→ ∞ .\nLemma 6. There is a constant v∞independent of the initial state such that |E[Vn]−v∞|=O(n−δ)\nasn→ ∞ .\nProof. By Lemma 5, it is sufficient to show the lemma assuming z0=z∗. First, we show that\nE[Vn] converges to a limit v∞∈Rasn→ ∞ . By Lemma 3 and a union bound, for all t≥0,\nP(∃z∈Z(2n),|Vn(z)−E[Vn(z)]| ≥t)≤X\nz∈Z(2n)P(|Vn(z)−E[Vn(z)]| ≥t)\n≤2 exp\u0012\n−t2n2\n2h(n)\u0013\nmax\nz∈Z|Z(2n)(z)|= 2ψ(n, t).(5)\nBy definition of δ-transient game, there exists ( εn)n∈Nsuch that εn+ψ(n, εn) =O(n−δ).\nDenote by Ethe set of vertices in Z(2n)that are equivalent to z∗. Now, Lemma 5 implies that\nthere is a constant K′>0 such that, for every n≥1,|E[Vn(z)]−E[Vn]| ≤K′n−δ. Combining\nthis with (5), we get that\nP\u0012\nmin\nz∈Z(2n)Vn(z)≤E[Vn]−εn−K′n−δ\u0013\n≤P\u0012\nmin\nz∈Z(2n)|Vn(z)−E[Vn(z)]| ≥εn\u0013\n≤P(∃z∈E,|Vn(z)−E[Vn(z)]| ≥εn)\n≤2ψ(n, εn) =O(n−δ).\n8In particular, it follows directly that\nE\u0014\nmin\nz∈Z(2n)Vn(z)\u0015\n≥\u0010\nE[Vn]−εn−K′n−δ\u0011\nP\u0012\nmin\nz∈Z(2n)Vn(z)≥E[Vn]−εn−K′n−δ\u0013\n≥\u0010\nE[Vn]−εn−K′n−δ\u0011\n(1−2ψ(n, εn))≥E[Vn]−2(ψ(n, εn) +εn)−K′n−δ.\nNow, fix an integer m∈[1,2n] and consider the ( m+n)-stage game. Suppose that Player\n1 plays according to an optimal strategy for the m-stage game up to stage mand, once the\nm-stage game terminates at a state zm, continues to play according to an optimal strategy for\nthe subsequent n-stage game. Note that zm∈Z(2n), so the above strategy of Player 1 for the\nfirstm+nsteps guarantees a gain ofm\nm+nVm+n\nm+nminz∈Z(2n)Vn(z). Thus,\n(m+n)E[Vm+n]≥mE[Vm] +nE\u0014\nmin\nz∈Z(2n)Vn(z)\u0015\n≥mE[Vm] +nE[Vn]−2n(ψ(n, εn) +εn)−K′n1−δ.(6)\nSince ψ(n, εn) +εn=O(n−δ), there is a constant K′′>0 such that, for all n≥1,\n2n(ψ(n, εn) +εn) +K′n1−δ≤2K′′n1−δ.\nThus, using Lemma 2 with f:n7→ −nE[Vn] and ϕ:n7→2K′′n1−δ(note that ϕis increasing andP\nn≥1ϕ(n)/n2= 2K′′P\nn≥11/n1+δ<∞) implies that E[Vn] converges to a limit v∞∈R∪ {∞}\nasn→ ∞ . Note that v∞is in [0 ,1] since this is the support of all payoff variables.\nFinally, using (6) with m=n, for every n≥1, we have that\nE[V2n]≥E[Vn]−(ψ(n, εn) +εn)−K′n−δ\n2≥E[Vn]−K′′n−δ.\nIn particular, for all integers ℓ, n≥1, iterating the above observation for n,2n, . . . , 2ℓ−1ngives\nthat\nE[V2ℓn]≥E[Vn]−K′′n−δℓ−1X\nj=02−δj≥E[Vn]−K′′\n1−2−δn−δ. (7)\nTaking ℓ→ ∞ , we conclude that v∞≥E[Vn]−O(n−δ). A similar reasoning exchanging Player 1\nwith Player 2 shows that v∞≤E[Vn] +O(n−δ) and concludes the proof of the lemma.\nFinally, we are ready to prove Theorem 1.\nProof of Theorem 1. Fix an arbitrary ε >0. By Lemma 6, there is a constant K > 0 such that\n|v∞−E[Vn]| ≤Kn−δfor all n≥1, independently of the initial state. Combining this with the\ntriangle inequality and Lemma 3 shows that, for every t≥0,\nP(|Vn−v∞| ≥t+Kn−δ)≤P(|Vn−E[Vn]| ≥t+Kn−δ− |E[Vn]−v∞|)\n≤P(|Vn−E[Vn]| ≥t)≤2 exp\u0012−t2n2\n2h(n)\u0013\n,\nwhich is the desired result.\n5 Directed games on trees: proof of Theorem 2\nThe first lemma in this section bootstraps upon the conclusion of Lemma 3 (which still holds\nin this setting), thus deriving superexponential concentration for the value of the n-stage game.\nBelow, log stands for the natural logarithm.\n9Lemma 7. Fixδ∈(0,1/2)andt≥n−δ. For every integer n≥1and even integer k∈[2, n]\nsuch that\nklogd+ 2 log 2 ≤t2(n−k), (8)\nwe have\nP(nVn−(n−k)E[Vn−k]≥(n−k)t+k)≤exp \n−dk/2\n6!\n,\nP(nVn−(n−k)E[Vn−k]≤ −(n−k)t−k)≤exp \n−dk/2\n6!\n.\nProof. First of all, since Tis a transitive graph, for all n≥1, (Vn(z))z∈Zhave the same distri-\nbution. For every even integer k∈[n], denote\nSk:={z∈Zk:Vn−k(z)−E[Vn−k]≥t}.\nIn other words, Skis the set of vertices that could be reached from z0after kstages, for which\nthe value of the ( n−k)-stage game starting at zis greater than or equal to E[Vn−k] +t.\nDefine the event Ek:={|Sk| ≥dk/2}. We provide an upper bound for P(Ek). Since the random\nvariables ( Vn−k(z))z∈Zkare i.i.d., we have that |Sk|follows a binomial distribution Bin( dk, q)\nwhere q:=P(Vn−k≥E[Vn−k] +t). Consequently, by Lemma 3 (where h(n) =nis the transient\nspeed of the family of partitions (Π z)z∈Zwhere, for all z∈Zandk≥2,Zk(z) contains all\ndescendants of zat distance k),|Sk|is stochastically dominated by a binomial random variable\nBin(dk,˜q) where ˜ q= exp( −t2(n−k)/2). In particular,\nP(Ek)≤P\u0010\nBin(dk,˜q)≥dk/2\u0011\n.\nThe random variable Bin( dk,˜q) has mean µ:=dk˜q. We define\nξ:=dk/2\nµ−1 = exp\u0012t2(n−k)−klog(d)\n2\u0013\n−1≥1,\nwhere the last inequality comes from (8). Since dk/2= (1 + ξ)µ, we have that\nP(Bin( dk,˜q)≥dk/2) =P(Bin( dk,˜q)≥(1 +ξ)µ).\nTherefore, since ξ≥1 (so 3 ξ≥2 +ξ), by Chernoff’s bound,\nP\u0010\nBin(dk,˜q)≥dk/2\u0011\n≤exp\u0012\n−ξ2µ\n2 +ξ\u0013\n≤exp\u0012\n−ξµ\n3\u0013\n= exp \n−dk/2\n3\u0010\n1−dk/2˜q\u0011!\n.\nSince ξ= 1/(dk/2˜q)−1≥1, we have that 1 −dk/2˜q≥1/2, which finally yields\nP(Ek)≤exp \n−dk/2\n6!\n. (9)\nAt the same time, on the event |Sk|< dk/2(that is, Ek), Player 2 can ensure that the token avoids\nending up in Skafter kstages. Indeed, at each of the k/2∈Nturns corresponding to decisions\nof Player 2, by the pigeonhole principle, Player 2 can always move the token to a vertex having\nat most a (1 /d)-fraction of all remaining elements in Skamong its descendants. Since Player 2\nhask/2 turns and d−k/2|Sk|<1, Player 2 can safely avoid the set Skat stage k.\nLet us condition on the event Ek. Then, Player 2 can guarantee that the sum of the payoffs\nover the last n−kstages is strictly smaller than ( n−k)(E[Vn−k] +t). Moreover, the sum of the\n10firstkpayoffs is at most k. Consequently, Player 2 can guarantee that, after nstages, the global\nmean payoff is strictly smaller than k/n+ (n−k)(E[Vn−k] +t)/n, in other words,\nnVn<(n−k)E[Vn−k] + (n−k)t+k . (10)\nIn particular, using (9) implies that\nP(nVn−(n−k)E[Vn−k]≥(n−k)t+k)≤P(|Sk| ≥dk/2) =P(Ek)≤exp \n−dk/2\n6!\n.\nA similar reasoning for Player 1 (using the sets ˜Sk:={z∈Zk:Vn−k(z)−E[Vn−k]≤ −t}\ninstead of Skand replacing (10) with nVn>(n−k)E[Vn−k]−(n−k)t) yields\nP(nVn−(n−k)E[Vn−k]≤ −(n−k)t)≤exp \n−dk/2\n6!\n,\nwhich implies the second statement. Note that the additional −kin it is introduced for reasons\nof symmetry only.\nNext, we show that the expected value of the n-stage game converges rapidly as ngrows to\ninfinity.\nLemma 8. There exists v∞∈Rsuch that, for every δ∈(0,1/2), we have |E[Vn]−v∞|=O(n−δ)\nasn→ ∞ .\nProof. Fixδ′∈(0,1/2) and t≥n−δ′. For each n≥1, we set k=k(n):= 2\u0004\nn1−2δ′/4 logd\u0005\n.\nThen, klogd+ 2 log 2 ≤t2(n−k) for all large n. For every even integer m∈[n/2,2n] and large\nn, we have\nP\u0012\nmin\nz∈ZmnVn(z)≤(n−k)(E[Vn−k]−t)−k\u0013\n≤X\nz∈ZmP(nVn(z)≤(n−k)(E[Vn−k]−t)−k)\n≤dmexp \n−dk/2\n6!\n≤exp \n2nlogd−d⌊n1−2δ′/4 logd⌋\n6!\n,\nwhere the first inequality comes from a union bound and the second inequality comes from\nLemma 7. Fix δ∈(0, δ′) and define, for all n≥1,\nεn:=n−δand ψ(n):= exp\u0010\n2nlogd−d⌊n1−2δ′/4 logd⌋/6\u0011\n.\nFor large nand every even integer m∈[n/2,2n], we have\nE\u0014\nmin\nz∈ZmVn(z)\u0015\n≥\u0012n−k\nn(E[Vn−k]−εn)−k\nn\u0013\nP\u0012\nmin\nz∈ZmnVn(z)>(n−k)(E[Vn−k]−εn)−k\u0013\n≥\u0012n−k\nn(E[Vn−k]−εn)−k\nn\u0013\n(1−ψ(n))\n≥\u0012\nE[Vn−k]−k\nn(1 +E[Vn−k])−εn\u0013\n(1−ψ(n))\n≥\u0012\nE[Vn]−3k\nn−εn\u0013\n(1−ψ(n))≥E[Vn]−(ψ(n) + 2εn), (11)\n11where in the fourth inequality we used that E[Vn]≤E[Vn−k]+k/nby Lemma 4 and 1+ E[Vn−k]≤\n2, and the last inequality is valid for large nbecause k/n=o(εn).\nConsider integers n≥1 and even m∈[n/2,2n]. In the ( n+m)-stage game, Player 1 can play\naccording to an optimal strategy for the m-stage game starting at z0, and then play according to\nan optimal strategy for the n-stage game starting from the state zreached after mstages. This\nguarantees that ( m+n)Vm+n≥mVm+ min z∈ZmnVn(z). Taking expectations on both sides and\nusing (11) yields\n(m+n)E[Vm+n]≥mE[Vm] +nE\u0014\nmin\nz∈ZmVn(z)\u0015\n≥mE[Vm] +nE[Vn]−n(ψ(n) + 2εn).\nWe find a similar inequality for odd m∈[n/2,2n]. In this case, m+ 1 is even and also in\n[n/2,2n]. Then, the previous inequality applied to m+ 1 and nyields\n(m+n+ 1)E[Vm+n+1]≥(m+ 1)E[Vm+1] +nE[Vn]−n(ψ(n) + 2εn). (12)\nHowever,\n(m+n)E[Vm+n]≥(m+n+ 1)E[Vm+n+1]−1 and ( m+ 1)E[Vm+1]≥mE[Vm],\nwhich combined with (12) gives\n(m+n)E[Vm+n]≥mE[Vm] +nE[Vn]−n(ψ(n) + 2εn)−1.\nTo sum things up, for large nandm∈[n/2,2n],\n(m+n)E[Vm+n]≥mE[Vm] +nE[Vn]−n(ψ(n) + 2εn)−1. (13)\nRecall that there is a constant K′>0 such that, for all n≥1,n(ψ(n) + 2εn) + 1≤K′n1−δ. We\ndefine ϕ(n):=K′n1−δand deduce from (13) that\n(m+n)E[Vm+n]≥mE[Vm] +nE[Vn]−ϕ(n+m).\nMoreover, ϕis increasing and verifiesP\nn≥1ϕ(n)/n2<∞. Consequently, Lemma 2 applied to\nthe function f:n∈N7→ −nE[Vn] implies that that E[Vn] converges to a limit v∞∈R∪ {∞} as\nn→ ∞ . Note that v∞∈[0,1] since Vn∈[0,1] for all n≥1.\nFinally, using (13) with m=nand a telescopic summation shows that the inequality (7) still.\nIn particular, we conclude that v∞≥E[Vn]−O(n−δ). A similar reasoning replacing Player 1\nwith Player 2 shows that v∞≤E[Vn] +O(n−δ) and concludes the proof of the lemma.\nWe are now ready to prove Theorem 2.\nProof of Theorem 2. Fixt≥n−δand let K′be a constant such that |E[Vn]−v∞| ≤K′n−δfor all\nlarge n. Using that, for all nandk≤n, we have |nVn−(n−k)Vn−k| ≤k, and fixing k= 2⌈t2n\n4 logd⌉\n(which satisfies (8)), we get\nP(|Vn−v∞| ≥t+ 2t2+K′n−δ)\n≤P\u0012\f\f\f\fVn−n−k\nnE[Vn−k]\f\f\f\f≥t+ 2t2+K′n−δ−\f\f\f\fn−k\nnE[Vn−k]−E[Vn]\f\f\f\f− |E[Vn]−v∞|\u0013\n≤P\u0012\f\f\f\fVn−n−k\nnE[Vn−k]\f\f\f\f≥t+t2\u0013\n≤P(|nVn−(n−k)E[Vn−k]| ≥(n−k)t+k)\n≤exp \n−d⌊k/2⌋\n6!\n≤exp \n−dt2n/(4 log d)\n6!\n= exp\u0012\n−1\n6exp\u0012t2n\n4\u0013\u0013\n,\n12where the first inequality comes from the triangle inequality, the second inequality comes from\nthe definition of K′and the fact that |nVn−(n−k)Vn−k| ≤k≤nt2, and the third inequality\nonce again uses the fact that k≤nt2.\nFinally, choosing K≥K′sufficiently large ensures that, first, the upper bound shown above\nholds for all n≥1 (and not only for large n), and second, the upper bound holds for all t≥0,\nwhich finishes the proof.\n6 Oriented directed games: proof of Proposition 1\nWe present a simple and self-contained proof of Proposition 1.\nProof. First, by density of the rational vectors in Rdand rescaling, we may assume that u∈Zd\nis such that the greatest common divisor of its coordinates is 1. Then, for every integer i≥1 and\ninitial state z0=z, defining Z2i(z):={w∈Z:w·u=z·u+i},Z2i+1(z):={w∈Z:w·u=\nz·u−i}, and Z1(z):={w∈Z\\ {z}:w·u=z·u}shows that the game is directed. Indeed,\n(Zi(z))i≥0form a partition of Zfor all z∈Z, and each of them could be visited at most once by\nthe token.\nNow, fix δ∈(0,1/2) and z0=z∈Z. To see that the game is δ-transient, set r=\nmax uv∈E(Γ)∥v−u∥2. After nsteps of the process, the position znof the token satisfies ∥zn−z∥2≤\nnr, and by the Cauchy-Schwarz inequality,\n|(zn−z)·u| ≤ ∥zn−z∥2· ∥u∥2≤ ⌈nr· ∥u∥2⌉=:M=M(n).\nIn particular, Z(n)(z) is contained in the ball with radius Maround z, which itself is contained in\nZ[2M+1](z), so the transient speed of the process satisfies h(n)≤2M(n)+1 for all n≥1. Finally,\ntakeδ∈(0,1/2) and set εn:=n−δ. Then,\nψ(n, εn) = exp\u0012\n−ε2\nnn2\n2h(n)\u0013\nmax\nz∈Z|Z(2n)(z)|\n≤exp\u0012\n−ε2\nnn\n6r· ∥u∥2\u0013\n(2nr· ∥u∥2+ 1)d\n= exp\u0012\n−n1−2δ\n6r· ∥u∥2\u0013\n(2nr· ∥u∥2+ 1)d=O(n−δ).\nHence, for all δ∈(0,1/2),εn+ψ(n, εn) =O(n−δ), and therefore, the game is δ-transient.\nAcknowledgments\nThis work was supported by the French Agence Nationale de la Recherche (ANR) under refer-\nences ANR-21-CE40-0020 (CONVERGENCE project) and ANR-20-CE40-0002 (GrHyDy), and\nby Fondecyt grant 1220174. This collaboration was mainly conducted during a 1-year visit of\nBruno Ziliotto to the Center for Mathematical Modeling (CMM) at University of Chile in 2023,\nunder the IRL program of CNRS.\nReferences\n[1] N. Alon, K. Rudov, and L. Yariv. Dominance solvability in random games. arXiv preprint\narXiv:2105.10743 , 2021. 1\n[2] B. Amiet, A. Collevecchio, M. Scarsini, and Z. Zhong. Pure Nash equilibria and best-response\ndynamics in random games. Mathematics of Operations Research , 46(4):1552–1572, 2021. 1\n13[3] A. Auffinger, M. Damron, and J. Hanson. 50 years of first-passage percolation , volume 68.\nAmerican Mathematical Society, 2017. 1\n[4] D. Bhasin, S. Karmakar, M. Podder, and S. Roy. On a class of PCA with size-3 neighborhood\nand their applications in percolation games. Electronic Journal of Probability , 28:1–60, 2023.\n1\n[5] N. Govert de Bruijn and P. Erd˝ os. Some linear and some quadratic recursion formulas II.\nIndagationes Mathematicae (Proceedings) , 55:152–163, 1952. 2\n[6] D.-Z. Du and P. M. Pardalos. Minimax and applications , volume 4. Springer Science &\nBusiness Media, 1995. 2\n[7] J. Flesch, A. Predtetchinski, and V. Suomala. Random perfect information games. Mathe-\nmatics of Operations Research , 48(2):708–727, 2023. 1\n[8] G. Garnier and B. Ziliotto. Percolation games. Mathematics of Operations Research ,\n48(4):2156–2166, 2023. 1, 2\n[9] T. Garrec. Communicating zero-sum product stochastic games. Journal of Mathematical\nAnalysis and Applications , 477(1):60–84, 2019. 1\n[10] Torsten Heinrich, Yoojin Jang, Luca Mungo, Marco Pangallo, Alex Scott, Bassel Tarbush,\nand Samuel Wiese. Best-response dynamics, playing sequences, and convergence to equilib-\nrium in random games. International Journal of Game Theory , 52(3):703–735, 2023. 1\n[11] A. E. Holroyd, I. Marcovici, and J. B. Martin. Percolation games, probabilistic cellular\nautomata, and the hard-core model. Probability Theory and Related Fields , 174:1187–1217,\n2019. 1\n[12] S. Janson, T. Luczak, and A. Ruci´ nski. Random Graphs . John Wiley & Sons, Inc., 2000. 1\n[13] R. Laraki and J. Renault. Acyclic gambling games. Mathematics of Operations Research ,\n45(4):1237–1257, 2020. 1\n[14] R. Laraki and S. Sorin. Advances in zero-sum dynamic games. In Handbook of game theory\nwith economic applications , volume 4, pages 27–93. Elsevier, 2015. 1\n[15] E. Solan and B. Ziliotto. Stochastic games with signals. Advances in Dynamic and Evolu-\ntionary Games: Theory, Applications, and Numerical Methods , pages 77–94, 2016. 1\n[16] J. von Neumann. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen , 100(1):295–\n320, 1928. 2\n[17] B. Ziliotto. Zero-sum repeated games: counterexamples to the existence of the asymptotic\nvalue and the conjecture maxmin = lim v(n).The Annals of Probability , 44(2):1107–1133,\n2016. 1\n[18] B. Ziliotto. Mertens conjectures in absorbing games with incomplete information. arXiv\npreprint arXiv:2106.09405, to appear in The Annals of Applied Probability , 2021. 1\n14" }, { "title": "2401.16289v5.Turán_Densities_for_Daisies_and_Hypercubes.pdf", "content": "arXiv:2401.16289v5 [math.CO] 6 Feb 2024TURÁN DENSITIES FOR DAISIES AND HYPERCUBES\nDAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\nAbstract\nAnr-daisy is an r-uniform hypergraph consisting of the six r-sets formed by taking the union\nof an(r−2)-set with each of the 2-sets of a disjoint 4-set. Bollobás, Le ader and Malvenuto, and\nalso Bukh, conjectured that the Turán density of the r-daisy tends to zero as r→ ∞. In this paper\nwe disprove this conjecture.\nAdapting our construction, we are also able to disprove a fol klore conjecture about Turán\ndensities of hypercubes. For fixed dand large n, we show that the smallest set of vertices of the\nn-dimensional hypercube Qnthat meets every copy of Qdhas asymptotic density strictly below\n1/(d+1), for alld≥8. In fact, we show that this asymptotic density is at most cd, for some constant\nc <1. As a consequence, we obtain similar bounds for the edge-Tur án densities of hypercubes.\nWe also answer some related questions of Johnson and Talbot, and disprove a conjecture made by\nBukh and by Griggs and Lu on poset densities.\n1 Introduction\nAs usual, if Xis a set, we write X(r)for the set of all r-element subsets of X. For an integer r≥2,\nanr-daisy is anr-uniform hypergraph of the form\n{S∪X:X∈T(2)},\nwhereSis an(r−2)-element set and Tis a 4-element set disjoint from S. We denote this hypergraph\nbyDr; note that Drhasr+2vertices and six edges. We call the set Sthestemof the daisy. For any r-\nuniform hypergraph H, we write π(H)for the Turán density ofH, i.e.π(H) = lim n→∞(ex(n,H)//parenleftbign\nr/parenrightbig\n),\nwhere\nex(n,H) = max{|F|:Fis anH-free,r-uniform hypergraph on nvertices}.\nBollobás, Leader and Malvenuto made the following conjectur e in [3]. (Bukh also made the same\nconjecture independently – see [3].)\nConjecture 1. π(Dr)→0asr→ ∞.\nWe note that an easy averaging argument, averaging over link s of vertices, shows that the sequence\n(π(Dr))r≥2is monotone decreasing, so π(Dr)does indeed tend to some limit as r→ ∞.\nMore generally, for s,t∈Nwitht≥s, we define an (r,s,t)-daisy to be an r-uniform hypergraph\nof the form\n{S∪X:X∈T(s)},\nwhereSis an(r−s)-element set and Tis at-element set disjoint from S. We denote this hypergraph\nbyDr(s,t). (As before, we call Sthestemof the daisy.) Bollobás, Leader and Malvenuto [3] in fact\nmade the following conjecture, generalising Conjecture 1.\n12 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\nConjecture 2. For any s,t∈Nwitht≥s, we have π(Dr(s,t))→0asr→ ∞.\nConjecture 1 has the following appealing reformulation in t erms of subcubes. As usual, a d-\ndimensional subcube of{0,1}nis a subset of {0,1}nof the form {x∈ {0,1}n:xi=cifor alli∈F},\nwhereF∈ {1,2,...,n}(n−d)andci∈ {0,1}for alli∈F. Ifn≥4is even, a middle 4-cube of\n{0,1}nis a 4-dimensional subcube of {0,1}nwhich is contained in the middle five layers of {0,1}n—\nmeaning, those layers of Hamming weight between n/2−2andn/2+2. Note that the intersection of a\nmiddle 4-cube with the middle layer of {0,1}nis precisely an (n/2)-daisy (identifying {0,1}nwith the\npower-set of {1,2,...,n}, in the usual way). Similarly, for each even integer dand each even n≥d,\namiddled-cube of{0,1}nis ad-dimensional subcube of {0,1}nwhich is contained within the middle\nd+1layers of {0,1}n. As observed in [3], Conjecture 1 is equivalent to the follow ing.\nConjecture 3. Ifnis even and F ⊂ {x∈ {0,1}n:/summationtextn\ni=1xi=n/2}does not contain the middle layer\nof a middle 4-cube, then |F|=o(/parenleftbign\nn/2/parenrightbig\n).\nOr, to rephrase in the language of Boolean functions, Conject ure 3 says that for any δ >0, a\nBoolean function on the middle layer of {0,1}nwith expectation at least δmust be identically 1 on\n(the middle layer of) some middle 4-cube, provided nis sufficiently large depending on δ. Similarly,\nConjecture 2 is equivalent to the statement that for any δ >0and even d≥4, a Boolean function on\nthe middle layer of {0,1}nwith expectation at least δmust be constant on (the middle layer of) some\nmiddled-cube, provided nis sufficiently large depending on δandd. Unsurprisingly, these conjectures\nhave generated much interest in the field of ‘analysis of Boole an functions’, since they say, very roughly,\nthat there are no analogues of parity-type functions on {0,1}nfor Boolean functions defined on the\nmiddle layer alone.\nRather unexpectedly, Conjecture 1 is false. Our first aim in t his paper is to disprove it, as follows.\nTheorem 4. We have\nlim\nr→∞π(Dr)≥∞/productdisplay\nk=1(1−2−k)≈0.29.\nPreviously, the best known lower bound on π(Dr)(for general r) wasΩ(1/r), due to Ellis and King\n[7].\nFor(r,2,t)-daisies, we obtain the following lower bound.\nTheorem 5. For each t≥4we have\nlim\nr→∞π(Dr(2,t))≥1−1/t−o(1/t).\nThe bound in this theorem is clearly best-possible, up to the error term. Indeed, by averaging, if\nn≥r+t−2then for any r-uniform hypergraph Fon the vertex-set [n] ={1,2,...,n}with density\nmore than 1−1/(t−1)+1/(n−r+1), there exists an (r−2)-setSsuch that the link graph\n{{i,j} ∈([n]\\S)(2):S∪{i,j} ∈ F}\nhas density at least 1−1/(t−1) +1/(n−r+1). So by Turán’s theorem this graph must contain a\nKt, and soFmust contain a copy of Dr(2,t). This shows that π(Dr(2,t))≤1−1/(t−1)for allr≥2TURÁN DENSITIES FOR DAISIES AND HYPERCUBES 3\nand allt≥4.\nFor(r,s,t)-daisies with larger s, we obtain even stronger bounds, although these are probabl y not\nasymptotically sharp. It is these bounds that will be key to p roving our applications for Turán densities\nof hypercubes. The precise result we will need is the followi ng.\nTheorem 6. Letkbe even. Then we have\nlim\nr→∞π(Dr(k,8k+1)) = 1 −O(2−k).\nWe now discuss applications of the above results to Turán pro blems in the hypercube. The latter\nwere popularized by Alon, Krech and Szabó in [1], following e arlier results of Kostochka [11] and of\nJohnson and Entringer [9]; Johnson and Talbot obtained furt her results and posed further problems\nin [10]. Our terminology follows that of Johnson and Talbot. As usual, the n-dimensional hypercube\ngraphQnis the graph with vertex-set {0,1}n, where two zero-one vectors are joined by an edge if they\ndiffer in just one coordinate. We start with vertex-Turán pro blems.\nForn∈N, we define\nexc(n,Qd) = max{|F|:F ⊂ {0,1}n,FisQd-free},\ni.e.exc(n,Qd)is the maximum possible size of a set of vertices in Qnthat does not contain the\nvertex-set of a d-dimensional subcube. We define the vertex-Turán density of Qdto be\nλ(Qd) = lim\nn→∞exc(n,Qd)\n2n.\n(An easy averaging argument shows that the above sequence of quotients is monotone decreasing, so\nthe above limit exists.)\nA by now well-known problem in extremal combinatorics is to d etermine λ(Qd)for each integer d.\nKostochka [11], and independently Johnson and Entringer [9 ], showed that λ(Q2) = 2/3, and in fact\nboth sets of authors determined exc(n,Q2)exactly for all n∈N. For each d≥3, however, the value\nofλ(Qd)is unknown.\nIt is in fact more convenient to consider the (equivalent) ‘c omplementary’ problem: what is the\nminimal size g(n,d)of a subset of {0,1}nthat intersects the vertex-set of every d-dimensional subcube?\nWriting\nγd= lim\nn→∞g(n,d)/2n,\nwe have γd= 1−λ(Qd). The best-known upper and lower bounds on γdfor general dare\nlog2(d+2)\n2d+2≤γd≤1\nd+1, (1)\nand are due to Alon, Krech and Szabó [1]. The upper bound comes from taking F ⊂ {0,1}nto consist\nof every (d+ 1)thlayer of {0,1}n; clearly, this set intersects the vertex-set of every d-dimensional\nsubcube, and has density tending to 1/(d+1)asn→ ∞. In the case d= 2, the result of Kostochka\nand Johnson-Entringer says that this construction is best- possible. The lower bound comes from\nobserving that g(d+2,d)≥log2(d+2)and then partitioning {0,1}ninto copies of {0,1}d+2. (To see\nthatg(d+2,d)≥log2(d+2), note that, if F ⊂ {0,1}d+2intersects the vertex-set of every d-dimensional\nsubcube, then it must separate the pairs of [d+2], meaning that for any 1≤i < j≤d+2, there exists4 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\nx∈ Fsuch that xi/\\⌉}atio\\slash=xj; it is well-known, and easy to see, that if F ⊂ {0,1}nseparates the pairs of\n[n]then|F| ≥log2n.)\nIt is a folklore conjecture (see [3]) that γd= 1/(d+1)for every integer d≥2, i.e. that the upper\nbound above is best-possible. In other words, to intersect t he vertex-set of every d-dimensional subcube\nin{0,1}n, one cannot asymptotically do better than to take every (d+1)thlayer. The following result\ndisproves this conjecture, showing that in fact γdis exponentially small in d.\nTheorem 7. We have\nγd≤cd\nfor alld∈N, wherec <1is a constant. In fact, γd<1\nd+1for alld≥8.\nA related question of Johnson and Talbot [10, Question 13] is as follows: is it true that for any\nδ >0and integer d≥2, there exists n0=n(d,δ)∈Nsuch that for all n≥n0, every subset of {0,1}n\nof density at least δmust contain at least/parenleftbigd\n⌊d/2⌋/parenrightbig\npoints of some d-dimensional subcube? It is easy\nto see that the answer to this question is ‘yes’ for d= 2andd= 3. Indeed, any subset of {0,1}nof\ndensity greater than 1/nmust contain two points at Hamming distance 2, by averaging o ver Hamming\nspheres of radius 1, and two such points are clearly containe d in a common 2-dimensional subcube.\nMoreover, any subset of {0,1}nof density greater than 2/nmust contain three points of the form\nx∆{i},x∆{j},x∆{k}for some x∈ {0,1}nand distinct i,j,k∈[n], again by averaging over Hamming\nspheres of radius 1, and three such points are clearly contai ned in a common 3-dimensional subcube.\nHowever, one can use Theorem 4 to answer the Johnson-Talbot q uestion in the negative for all\nintegersd≥4, in the following quantitative form.\nTheorem 8. There exists a subset F ⊂ {0,1}nsuch that every 4-dimensional subcube of {0,1}n\ncontains at most five points of F, with\n|F|\n2n=1\n3∞/productdisplay\nk=1(1−2−k)−o(1)≈0.097−o(1).\nMoreover, for each d≥5, there exists a subset F ⊂ {0,1}nsuch that every d-dimensional subcube\nof{0,1}ncontains fewer than/parenleftbigd\n⌊d/2⌋/parenrightbig\npoints of F, with\n|F|\n2n≥c/√\nd,\nwherec >0is an absolute constant.\nJohnson and Talbot [10, Question 12] also asked if for any fixe d family SwithS ⊂ {0,1}d, there\nexist (for each n∈N) setsIn⊂[n]such that the family F(In) ={x∈ {0,1}n:/summationtextn\ni=1xi∈In}is\nS-free, and\nlim\nn→∞|F(In)|\n2n=λ(S).\n(Here,λ(S)denotes the asymptotically greatest density of a family F ⊂ {0,1}nthat isS-free, in the\nsense that Fdoes not contain any image of Sunder an isometric embedding of QdintoQn.) In other\nwords, is it true that an S-free family of asymptotically maximal size can be obtained by taking an\nappropriate union of layers of the hypercube? Theorem 7 give s a negative answer to this question,\neven for the special case of S={0,1}d=V(Qd), for any d≥8: it is easy to see that a union of layersTURÁN DENSITIES FOR DAISIES AND HYPERCUBES 5\nthat isQd-free has asymptotic density at most 1−1/(d+1), whereas Theorem 7 yields Qd-free subsets\nof{0,1}nof asymptotic density greater than 1−1/(d+1)for each d≥8.\nEdge-Turán problems in the hypercube have been perhaps more widely studied than vertex-Turán\nproblems, though in some ways the latter are more natural. A $100problem of Erdős, still unsolved,\nasks for the maximum possible number of edges of a subgraph of Qnwhich contains no subgraph\nisomorphic to C4. In general, if His a graph and n∈N, we write ex(Qn,H)for the maximum possible\nnumber of edges of a subgraph of Qnthat contains no subgraph isomorphic to H. Erdős in fact raised\nthe problem of finding (or bounding) ex(Qn,C2k)for allk∈N. Alon, Krech and Szabó proposed a\ndifferent generalisation of Erdős’ C4-problem: that of determining (or bounding) ex(Qn,Qd)for each\nintegerd≥2. Again, it is more convenient to consider the (equivalent) c omplementary problem: what\nis the minimal size f(n,d)of a subgraph of Qnwhose edge-set intersects the edge-set of every subgraph\nofQnthat is isomorphic to Qd? An easy averaging argument shows that f(n,d)/e(Qn)is increasing\ninn, so the limit\nρd= lim\nn→∞f(n,d)/e(Qn)\nexists. Of course, a subgraph of Qnisomorphic to Qdmust be induced by the vertex-set of a d-\ndimensional subcube of Qn, soρdis a very close analogue of γd, the vertex version considered above.\nThe best-known upper and lower bounds on ρdfor general dare\nΩ/parenleftbigglog2d\n2d/parenrightbigg\n≤ρd≤O(1/d2),\nand are due to Alon, Krech and Szabó [1]. Theorem 7 immediatel y yields an exponential upper bound\nonρd, simply by taking the set of all edges of Qnthat are incident to at least one vertex in a family\nF �� {0,1}nthat intersects the vertex-set of every d-dimensional subcube.\nTheorem 9. We have\nρd≤cd\nfor alld∈N, wherec <1is a constant.\nFinally, we turn to the conjecture of Bukh [4] and of Griggs and Lu (see [8]). They conjectured\nthat, for any poset P, the asymptotically greatest density of a subset of the n-dimensional Boolean\nlattice not containing a copy of Pas a subposet is always attained by taking some union of layer s.\nNow, for the poset Pbeing the d-dimensional Boolean lattice, the best collection of layers to take\nwould be the middle dlayers (of the n-dimensional lattice). However, instead we may take the mid dle\nd+1layers of the n-dimensional lattice, and replace the middle layer by a dais y-free family as given\nby Theorem 4. For any d≥4, this set does not contain a copy of P, and its size is asymptotically\nstrictly greater than that of the middle dlayers.\n2 Daisies have positive Turán density\nIn this section we prove Theorems 4, 5 and 6. We start with Theo rem 4. The key idea will be to\ntake (a blow-up of) a linear-algebraic construction. The co nstruction itself will be for n= 2r−1.\nThe reader will notice that it is because this value is more th an quadratic in rthat the blow-up then\npreserves positive density.6 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\nProof of Theorem 4. Fixr≥2. We first consider the case when n= 2r−1. We identify [n]with\nFr\n2\\{0}, the set of all nonzero vectors of length rover the finite field F2.\nTakeFto consist of all bases of Fr\n2— that is, all the linearly independent subsets of size r. We\nclaim that FisDr-free.\nTo show this, we must prove that any copy of Drin(Fr\n2\\ {0})(r)contains a linearly dependent\nset. Suppose for a contradiction that we have a copy of Drin(Fr\n2\\ {0})(r)whoser-sets are all\nlinearly independent. Let S={v1,v2,...,vr−2}denote the stem of this daisy; then Sis certainly\na linearly independent set. Let Wdenote the subspace spanned by S, so that Wis an(r−2)-\ndimensional subspace of Fr\n2. LetT={u1,u2,u3,u4}denote the other four vertices of the daisy. Since\ndim(Fr\n2/W) = 2, we have |Fr\n2/W|= 4. Hence, either there exists i∈[4]such that ui∈W(in which\ncase{ui,uj,v1,v2,...,vr−2}is not linearly independent, for any j∈[4]\\{i}, a contradiction), or else\nthere exist distinct i,j∈[4]such that ui+W=uj+W(in which case {ui,uj,v1,v2,...,vr−2}is not\nlinearly independent, again a contradiction). This shows t hatFis indeed Dr-free, as claimed.\nThe density of Fis easy to calculate:\n|F|/parenleftbign\nr/parenrightbig=number of ordered bases of Fr\n2\nn(n−1)(n−2)...(n−r+1)\n=(2r−1)(2r−2)(2r−4)···(2r−2r−1)\n(2r−1)(2r−2)(2r−3)···(2r−r)\n>(2r−1)(2r−2)(2r−4)···(2r−2r−1)\n2r2\n=r/productdisplay\nk=1(1−2−k)\n>∞/productdisplay\nk=1(1−2−k)\n≈0.29.\nWe conclude that\nex(2r−1,Dr)/parenleftbig2r−1\nr/parenrightbig>r/productdisplay\nk=1(1−2−k).\nWe now turn to general n, for which we will use a ‘blow-up’ of the above construction. Forna\nmultiple of 2r−1, partition [n]into2r−1classes of equal size, C1,...,C 2r−1say, and consider the\nfamilyGconsisting of all r-element subsets of [n]of the form {x1,...,x r}, wherexi∈Cjifor alli∈[r],\nfor{j1,j2,...,jr}some element of F, the above Dr-free family on [2r−1]. Clearly GisDr-free, since\nFis. We now estimate the density of G. The probability that a uniformly random r-element subset of\n[n]has at least two points in some Ciis, by a union bound, at most\n/parenleftbigr\n2/parenrightbig\n2r−1,\nand therefore\n|G|/parenleftbign\nr/parenrightbig≥/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\n2r−1/parenrightBigg\n|F|/parenleftbig2r−1\nr/parenrightbig>/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\n2r−1/parenrightBiggr/productdisplay\nk=1(1−2−k).TURÁN DENSITIES FOR DAISIES AND HYPERCUBES 7\nHence\nex(n,Dr)/parenleftbign\nr/parenrightbig>/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\n2r−1/parenrightBiggr/productdisplay\nk=1(1−2−k).\nSince this holds for arbitrarily large multiples nof2r−1, it follows that\nπ(Dr)≥/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\n2r−1/parenrightBiggr/productdisplay\nk=1(1−2−k).\nTaking the limit of the above as r→ ∞ yields\nlim\nr→∞π(Dr)≥∞/productdisplay\nk=1(1−2−k)≈0.29,\nas required.\nThe proof of Theorem 5 is very similar. The daisy in Theorem 4 h ast−2 = 2 , and the above\nconstruction will generalise straightforwardly when t−2is a prime power. Standard estimates on\nprimes will then allow us to pass to the general case.\nProof of Theorem 5. We start with the case when t−2is a prime power, qsay. Let n=qr−1and\nidentify[n]withFr\nq\\{0}, whereFqis the field of order q. TakeFto consist of all bases of Fr\nq.\nWe now claim that FisDr(2,t)-free. To show this, it suffices to prove that any copy of Dr(2,t)in\n(Fr\nq\\{0})(r)must contain a linearly dependent set. Suppose for a contrad iction that we have a copy\nofDr(2,t)in(Fr\nq\\ {0})(r), whoser-sets are all linearly independent sets. Let S={v1,v2,...,vr��2}\ndenote the stem of this daisy; then Sis a linearly independent set. Let Wdenote the subspace spanned\nbyS, so that Wis an(r−2)-dimensional subspace of Fr\nq. LetT={u1,u2,...,u q+2}denote the other\nt=q+ 2vertices of the daisy. Then S={u1+W,u2+W,...,u q+2+W}is a set of size q+ 2in\nthe two-dimensional vector space Fr\nq/W(overFq), soScontains at least two points on the same line\nthrough the origin (noting that the q+ 1distinct lines through the origin in F2\nqcoverF2\nq), so there\nexist1≤i < j≤q+2andλ∈Fqsuch that ui+W=λ(uj+W). But then {ui,uj,v1,v2,...,vr−2}\nis a linearly dependent set, a contradiction. This shows tha tFis indeed Dr(2,t)-free, as claimed.\nAs above, we have:\n|F|/parenleftbign\nr/parenrightbig=number of ordered bases of Fr\nq\nn(n−1)(n−2)...(n−r+1)\n=(qr−1)(qr−q)(qr−q2)···(qr−qr−1)\n(qr−1)(qr−2)(qr−3)···(qr−r)\n>(qr−1)(qr−q)(qr−q2)···(qr−qr−1)\nqr2\n=r/productdisplay\nk=1(1−q−k)\n>∞/productdisplay\nk=1(1−q−k)\n= 1−1/q−O(1/q2).8 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\nHence\nex(qr−1,Dr(2,t))/parenleftbigqr−1\nr/parenrightbig>r/productdisplay\nt=1(1−q−t).\nExactly the same blow-up construction as in the proof of Theo rem 4 yields\nex(n,Dr(2,t))/parenleftbign\nr/parenrightbig>/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\nqr−1/parenrightBiggr/productdisplay\nk=1(1−q−k)\nfornan arbitrarily large multiple of qr−1. It follows that\nπ(Dr(2,t))≥/parenleftBigg\n1−/parenleftbigr\n2/parenrightbig\nqr−1/parenrightBiggr/productdisplay\nk=1(1−q−k).\nTaking the limit of the above as r→ ∞ yields\nlim\nr→∞π(Dr(2,t))≥∞/productdisplay\nk=1(1−q−k) = 1−1/q−O(1/q2).\nFinally, for general values of t, one may use the fact (from Baker, Harman and Pintz [2]) that fo r\nany sufficiently large x >0there is a prime between x−x0.525andx. By choosing the greatest prime\nqsuch that q≤t−2, we obtain from the above that\nlim\nr→∞π(Dr(2,t))≥1−1/t−O(1/t1.475)\nfor allt.\nWe now turn to the proof of Theorem 6. In fact, we prove the foll owing more general result, which\nwe include because it gives the best bounds that we can produc e for general (r,s,t)-daisies. Note that\nTheorem 6 is precisely the case of this when s=m=kandq= 2.\nTheorem 10. Letqbe a prime power, ma positive integer and seven, and put t=⌊sq2m\ns+1⌋+ 1.\nThen\nlim\nr→∞π(Dr(s,t))≥1−q−(m+1)−O(q−(m+2)).\n[Here and elsewhere in the paper the implied constant in the ‘ O’ notation is an absolute constant.]\nWe first need some preliminaries. For s,d∈Nandqa prime power, we say that a family of vectors\nX ⊂Fd\nqiss-wise independent if anysof the vectors in Xare linearly independent. We will make\nuse of the following lemma, due to Earnest [6]. We reproduce E arnest’s short proof, for the reader’s\nconvenience.\nLemma 11. Letqbe a prime power, let sbe an even positive integer, and let X ⊂Fd\nqbes-wise\nindependent. Then\n|X| ≤sq2d\ns−1.TURÁN DENSITIES FOR DAISIES AND HYPERCUBES 9\nProof. We first show that\n(q−1)s/2/parenleftbigg|X|\ns/2/parenrightbigg\n≤qd.\nIndeed, write l=s/2. Then the (q−1)l/parenleftbig|X|\nl/parenrightbig\nsums of the form\nc1v1+c2v2+...clvl,\nfor{v1,v2,...,vl}anl-element subset of Xandc1,c2,...,cl∈Fq\\{0}, are distinct elements of Fd\nq,\notherwise Xwould contain a linearly dependent subset of size 2l=s. This establishes the claim.\nIf we now use the inequality/parenleftbigg|X|\ns/2/parenrightbigg\n≥/parenleftbigg|X|\ns/2/parenrightbiggs/2\n,\nwe obtain\n|X| ≤sq\n2(q−1)q2d\ns−1≤sq2d\ns−1.\nProof of Theorem 10. Letm∈N, letn=qm+r−1, identify [n]withFm+r\nq\\{0}, and take Fto consist\nof all linearly independent sets of rvectors in Fm+r\nq. We then have\n|F|/parenleftbign\nr/parenrightbig=number of ordered linearly independent r-element subsets of Fm+r\nq\nn(n−1)(n−2)...(n−r+1)\n=(qm+r−1)(qm+r−q)(qm+r−q2)···(qm+r−qr−1)\n(qm+r−1)(qm+r−2)(qm+r−3)···(qm+r−r)\n>(qm+r−1)(qm+r−q)(qm+r−q2)···(qm+r−qr−1)\nqr(m+r)\n=r/productdisplay\nk=1(1−q−(m+k))\n>∞/productdisplay\nk=1(1−q−(m+k))\n= 1−1/qm+1−O(1/qm+2).\nLett=⌊sq2m\ns+1⌋+1. We claim that FisDr(s,t)-free. To show this, it suffices to prove that any\ncopy ofDr(s,t)in(Fr\nq\\ {0})(r)must contain a linearly dependent set. Suppose for a contrad iction\nthat we have a copy of Dr(s,t)in(Fm+r\nq\\ {0})(r), whose r-sets are all linearly independent. Let\nS={v1,v2,...,vr−s}denote the stem of this daisy; then Sis a linearly independent set. Let W\ndenote the subspace spanned by S, so that Wis an(r−s)-dimensional subspace of Fm+r\nq. Let\nT={u1,u2,...,u t}denote the other tvertices of the daisy. Then S={u1+W,u2+W,...,u t+W}\nis a set of size tin the(m+s)-dimensional vector space Fm+r\nq/W(overFq), so by Lemma 11, applied\nwithd=m+s, we have that Scontains a linearly dependent s-set,{ui1+W,ui2+W,...,u is+W}\nsay. But then {v1,v2,...,vr−s,ui1,ui2,...,u is}is a linearly dependent r-set, a contradiction.\nThe statement on Turán densities now follows by the same blow -up argument as in the proofs of\nTheorem 4 and Theorem 5.10 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\n3 Proofs of Theorems 7 and 8\nWe first prove Theorem 7. Our main tool will be Theorem 6. We rem ark that just to disprove the\nstatement that γd= 1/(d+1)it would be sufficient to use Theorem 5, but this would only show that\nγd=O(1/d2)– to obtain the exponential bound on γdwe need Theorem 6. We also remark that the\nproof below that γd<1/(d+1)for alld≥8is included just for precision – the reader who is only\ninterested in large values of dmay omit this technically-optimised part of the proof.\nProof of Theorem 7. Note that, since π(Dr(k,8k+1)) is decreasing in randex(n,Dr(k,8k+1))//parenleftbign\nr/parenrightbig\nis decreasing in n, it follows from Theorem 6 that ex(n,Dr(k,8k+1))//parenleftbign\nr/parenrightbig\n= 1−O(2−k)for allrand\nalln≥r.\nIn proving the theorem, by adjusting the value of cif necessary, we may clearly assume that d≥d0,\nfor any absolute constant d0∈N.\nGiven an integer d≥17, we construct a family F ⊂ {0,1}nintersecting the vertex-set of every copy\nofQdas follows. Identify {0,1}nwithP([n])in the usual way. Let kbe the maximal even positive\ninteger such that 8k+1≤d. For every integer rwith1≤r≤n, letFr⊂[n](r)with|Fr|//parenleftbign\nr/parenrightbig\n≤O(2−k)\nand such that Frintersects every copy of Dr(k,8k+ 1)in[n](r). (Such exists, by the remark at the\nbeginning of the proof.) Let F=∪n\nr=0Fr. ThenFmeets every copy of Dr(k,8k+1)inP([n]), for all\nr. Since the vertex-set of every copy of QdinP([n])meets some copy of Dr(k,8k+1), it follows that\nFmeets the vertex-set of every copy of QdinP([n]). Clearly, we have\n|F|\n2n=O(2−k) =O(2−d/8),\nand therefore\nγd=O(2−d/8),\nproving the first part of the theorem.\nTo prove that γd<1/(d+1)for alld≥8, we use Theorem 5, and a more careful insertion of sparse\nsets meeting every daisy, into a sparse set of layers. We note that, since π(Dr(2,t))is decreasing in\nrandex(n,Dr(2,t))//parenleftbign\nr/parenrightbig\nis decreasing in n, it follows from Theorem 5 that ex(n,Dr(2,q+2))//parenleftbign\nr/parenrightbig\n≥/producttext∞\nk=1(1−q−k)for allr, allnand all prime powers q.\nLett≤ ⌊d/2⌋+1be maximal such that t−2is a prime power, qsay. View {0,1}nasP([n]), the\npower-set of [n]. For each r≤nsuch that ris a multiple of ⌈d/2⌉, letFrbe anr-uniform hypergraph\non the vertex-set [n]that has density at most\n1−∞/productdisplay\nk=1(1−q−k)\nand intersects every copy of Dr(2,t).\nNow letF=∪n\nr=0Fr. Consider the vertex-set Vof any copy of QdinQn;Vintersects d+1layers\nofP([n]), say the layers r0,...,r0+d, and for each layer r0+2≤i≤r0+⌈d/2⌉+1, the intersection\nofVwith the ithlayer contains a copy of Dr(2,t). One of these ⌈d/2⌉values of imust be a multiple\nof⌈d/2⌉, soVmust intersect F. Therefore Fintersects the vertex-set of every d-dimensional subcube\nof{0,1}d, as needed. The family Fhas density at most\n1\n⌈d/2⌉/parenleftBigg\n1−∞/productdisplay\nk=1(1−q−k)/parenrightBigg\n+on→∞(1),TURÁN DENSITIES FOR DAISIES AND HYPERCUBES 11\nso\ng(n,d)\n2n≤1\n⌈d/2⌉/parenleftBigg\n1−∞/productdisplay\nk=1(1−q−k)/parenrightBigg\n+on→∞(1).\nTaking the limit of the above as n→ ∞ yields\nγd≤1\n⌈d/2⌉/parenleftBigg\n1−∞/productdisplay\nk=1(1−q−k)/parenrightBigg\n(2)\nfor alld≥6.\nIt suffices (by the inequality (2)) to check that, for each d≥8, ifq(d)is the largest prime power q\nsuch that q≤ ⌊d/2⌋−1then\n1\n⌈d/2⌉/parenleftBigg\n1−∞/productdisplay\nk=1(1−q(d)−k)/parenrightBigg\n<1\nd+1. (3)\nWriteβq=/producttext∞\nk=1(1−q−k)for each prime power q; clearly, βqis an increasing function of q. For each\noddd≥9, (3) is equivalent to\nβq(d)>1/2,\nand this holds, since for all d≥9we have βq(d)≥βq(9)=β3≈0.560>1/2. For each even d≥8, (3)\nis equivalent to\nβq(d)>1\n2+1\n2d+2,\nwhich holds since for all d≥8we have\nβq(d)≥βq(8)=β3≈0.560>5\n9≥1\n2+1\n2d+2,\nnoting that the right-hand side is a decreasing function of d.\nWe now turn to the proof of Theorem 8.\nProof of Theorem 8. We first prove the first part of the theorem. Note that it follow s from Theorem\n4 thatex(n,Dr)//parenleftbign\nr/parenrightbig\n≥/producttext∞\nk=1(1−2−k)≈0.29for allnandr.\nView{0,1}nasP([n]). For each r≤nsuch that ris a multiple of 3, letFrbe anr-uniform\nhypergraph on the vertex-set [n]that has density at least\n∞/productdisplay\nk=1(1−2−k)≈0.29\nand contains no copy of Dr. Now let F=∪n\nr=0Fr. We clearly have\n|F|\n2n=1\n3∞/productdisplay\nk=1(1−2−t)−o(1)≈0.097−o(1).\nWe claim that Fcontains at most five points of any 4-dimensional subcube. In deed, let Vdenote the\nvertex-set of a 4-dimensional subcube of {0,1}n; thenVintersects five layers of P([n]), say the layers\nr0,r0+1,r0+2,r0+3andr0+4. Ifr0is congruent to 0 modulo 3, then there are just five vertices\nofVat layers of P([n])congruent to 0 modulo 3 (namely, those in layers r0orr0+3), so the claim12 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\ntrivially holds. For the same reason, the claim trivially ho lds ifr0is congruent to 2 modulo 3. If r0\nis congruent to 1 modulo 3, then there are just six vertices of Vat layers rofP([n])congruent to 0\nmodulo 3, namely those in the (r0+2)thlayer, and Fcannot contain all six of them, since they form\na copy of Dr0+2. This proves the claim, completing the proof of the first part of the theorem.\nWe now prove the second part of the theorem. Let C >0be a (large) absolute constant to be\nchosen later. View {0,1}nasP([n]). For each r≤nsuch that ris a multiple of ⌈C√\nd⌉, letFrbe an\nr-uniform hypergraph on the vertex-set [n]that has density at least\n∞/productdisplay\nk=1(1−2−k)≈0.29\nand contains no copy of Dr. Now let F=∪n\nr=0Fr. We clearly have\n|F|\n2n=1\n⌈C√\nd⌉∞/productdisplay\nk=1(1−2−k)+on→∞(1)≥c/√\nd,\nfor some sufficiently small absolute constant c >0.\nWe claim that, provided Cis sufficiently large, any d-dimensional subcube of {0,1}ncontains\nless than/parenleftbigd\n⌊d/2⌋/parenrightbig\npoints of F. To see this, let Vdenote the vertex-set of a d-dimensional subcube of\n{0,1}n; thenVintersects d+ 1layers of P([n]), say the layers r0,r0+ 1,...,r0+d−1andr0+d.\nNote that, among these layers, there is at most one layer isuch that iis a multiple of ⌈C√\nd⌉and\n|i−(r0+d/2)|< C√\nd/2. For such a layer i, sinceF ∩[n](i)isDi-free, by averaging we have\n|V ∩ F ∩ [n](i)| ≤5\n6/parenleftbigd\ni−r0/parenrightbig\n≤5\n6/parenleftbigd\n⌊d/2⌋/parenrightbig\n. LetUdenote the union of all the layers jofP([n])that are\nmultiples of ⌈C√\nd⌉but satisfy |j−(r0+d/2)| ≥C√\nd/2; then since |V ∩[n](j)|<|V ∩[n](k)|ifkis\ncloser to r0+d/2thanjis, we have |V ∩U|<2d/(⌊C√\nd/2⌋−1)<1\n6/parenleftbigd\n⌊d/2⌋/parenrightbig\nprovided Cis sufficiently\nlarge. Hence, overall we have\n|V ∩F|<5\n6/parenleftbiggd\n⌊d/2⌋/parenrightbigg\n+1\n6/parenleftbiggd\n⌊d/2⌋/parenrightbigg\n=/parenleftbiggd\n⌊d/2⌋/parenrightbigg\n,\nas claimed.\nWe remark that the second part of the previous theorem is clea rly best-possible up to the value\nof the absolute constant csince, by averaging, any subset F ⊂ {0,1}nof density at least/parenleftbigd\n⌊d/2⌋/parenrightbig\n/2d\ncontains at least/parenleftbigd\n⌊d/2⌋/parenrightbig\npoints of some d-dimensional subcube.\n4 Concluding remarks\nIt would be of interest to determine the right growth speed of γd, in other words to determine\nlim\nd→∞log2(1/γd)\nd,\nif this limit exists. The quantitylog2(1/γd)\ndis asymptotically at least 1/8(by the proof of Theorem 7),\nand of course it is always at most 1.TURÁN DENSITIES FOR DAISIES AND HYPERCUBES 13\nAnother open problem, first raised in [3], is to determine exa ctly the Turán density of D3. It is\nconjectured in [3] that π(D3) = 1/2. In [3] there is a construction attaining this asymptotic de nsity,\nby taking the complement of the Fano plane, blowing up and ite rating.\nIt would also be interesting to understand what happens for d aisies when s= 3, in particular to\ndetermine whether or not\nlim\nr→∞π(Dr(3,t)) = 1−O(1/t2). (4)\nWe note that (4) would be best-possible, up to the implicit co nstant. Indeed, a result of de Caen\n[5] states that π(K(3)\nt)≤1−1//parenleftbigt−1\n2/parenrightbig\nfor allt≥3. Therefore, for any ǫ >0, anyr≥3and any\nr-uniform hypergraph Fon the vertex-set [n]with density at least 1−1//parenleftbigt−1\n2/parenrightbig\n+ǫ, ifnis sufficiently\nlarge depending on r,tandǫthen there exists an (r−3)-setSsuch that the link hypergraph\n{{i,j,k} ∈([n]\\S)(3):S∪{i,j,k} ∈ F}\nhas density at least 1−1//parenleftbigt−1\n2/parenrightbig\n+ǫ. So by de Caen’s theorem this link hypergraph must contain a K(3)\nt,\nand soFmust contain a copy of Dr(3,t). This shows that π(Dr(3,t))≤1−1//parenleftbigt−1\n2/parenrightbig\nfor allr,t≥3.\nFinally, in connection with forbidden subposet problems, o bserve that our construction does not\ndisprove the longstanding conjecture that for the diamond p oset, or equivalently the two-dimensional\nBoolean lattice, the greatest asymptotic density of a diamon d-free family of points in the n-dimensional\nBoolean lattice is achieved by taking the union of two middle l ayers.\nNote. We are very grateful to Noga Alon for pointing out that if, ins tead of using the bound\nof Earnest for s-wise independent families, one uses the bound of Plotkin (M . Plotkin, Binary codes\nwith specified minimum distance, IRE Transactions on Information Theory 6 (1960), 445-450), then\none obtains that the quantitylog2(1/γd)\ndabove is asymptotically at least 1/2– in other words, that we\nmay take the constant cin Theorem 7 to be (asymptotically) 1/√\n2. We are also grateful to Robert\nJohnson, and independently Noga Alon, for the following att ractive observations. If one views the\nr-daisy as an (r−2)-set connected to the six 2-sets that form the graph K4, then the proof of Theorem\n4 goes through verbatim if we replace the graph G=K4by any graph Ghaving chromatic number\nat least 4. In contrast, if Ghas chromatic number at most 3 then the corresponding Turán d ensities\nhave to tend to zero. Indeed, this is clear if Gis a triangle (as then our family of r-sets would have\nto contain at most two r-sets in any given (r+1)-set), and in general it follows by ‘blowing up’ this\nargument, via some averaging and known facts about Turán den sities for 3-partite 3-graphs.\nAcknowledgement. We are very grateful to Boris Bukh for pointing out to us the conn ection\nbetween our results and the conjecture of his and of Griggs an d Lu on forbidden subposets.\nReferences\n[1] N. Alon, A. Krech and T. Szabó. Turán’s theorem in the hype rcube. SIAM Journal on Discrete\nMathematics 21 (2007), 66–72.\n[2] R. C. Baker, G. Harman and J. Pintz. The Difference Between Co nsecutive Primes, II. Proceedings\nof the London Mathematical Society 83 (2001), 532–562.\n[3] B. Bollobás, I. Leader and C. Malvenuto. Daisies and other T urán problems. Combinatorics,\nProbability and Computing 20 (2011), 743–747.14 DAVID ELLIS, MARIA-ROMINA IVAN AND IMRE LEADER\n[4] B. Bukh. Set Families with a Forbidden Subposet. Electronic Journal of Combinatorics 16 (2009),\n#R142.\n[5] D. de Caen. Extension of a theorem of Moon and Moser on comp lete subgraphs. Ars Combinatoria\n16 (1983), 5—10.\n[6] M. Earnest. Largest set of k-wise linearly independent vectors in Fn\nq?Mathoverflow ques-\ntion, March 2021. https://mathoverflow.net/questions/386893/largest-s et-of-k-wise-\nlinearly-independent-vectors-in-mathbb-f-qn .\n[7] D. Ellis and D. A. King. Lower bounds for the Turán densiti es of daisies. Electronic Journal of\nCombinatorics 30 (2023), P4.4.\n[8] J. R. Griggs and W.-T. Li. Progress on poset-free familie s of subsets. In: A. Beveridge, J. R. Griggs,\nL. Hogben, G., Musiker and P. Tetali (Eds), Recent Trends in Combinatorics , IMA Volumes in\nMathematics and its Applications 159, Springer 2016, pp. 31 7–338.\n[9] K.A. Johnson and R. Entringer. Largest induced subgraph s of then-cube that contain no 4-cycles.\nJournal of Combinatorial Theory, Series B 46 (1989), 346– 355.\n[10] J. R. Johnson and J. Talbot. Vertex Turán problems in the hypercube. Journal of Combinatorial\nTheory, Series A 117 (2010), 454—465.\n[11] E. A. Kostochka. Piercing the edges of the n-dimensional unit cube. Diskret. Analiz Vyp. 28\nMetody Diskretnogo Analiza v Teorii Grafov i Logiceskih Fun kcii (1976), 55–64. [In Russian.]\nDavid Ellis, School of Mathematics, University of Bristol, Bristol, UK.\nEmail address: david.ellis@bristol.ac.uk\nMaria-Romina Ivan, Department of Pure Mathematics and Mathematical Statistic s, Centre for Mathe-\nmatical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK .\nEmail address: mri25@dpmms.cam.ac.uk\nImre Leader, Department of Pure Mathematics and Mathematical Statistic s, Centre for Mathematical\nSciences, Wilberforce Road, Cambridge, CB3 0WB, UK.\nEmail address: i.leader@dpmms.cam.ac.uk" }, { "title": "2401.16300v2.Delaunay_hypersurfaces_in_spheres.pdf", "content": "arXiv:2401.16300v2 [math.DG] 9 Mar 2024DELAUNAY HYPERSURFACES IN SPHERES\nYONGSHENG ZHANG\nAbstract . We study Delaunay hypersurfaces in Snwith n≥3 and add a missing\n(flower) type of the category. Moreover, embedded Delaunay h ypersurfaces of nonzero\nconstant mean curvatures in Snare found.\n1. Introduction\nIn 1841, Delaunay discoverd a wonderful way of constructing rotational hypersur-\nfaces of constant mean curvature (CMC) in R3. All rotational CMC hypersurfaces come\nfrom the rolling construction of roulettes of conics.\nThe Delaunay’s rolling construction was successfully gene ralized to the case of CMC\nrotationally hypersurfaces in Rnwith n>3 in [HY81], and later to the cases in hyper-\nbolic spaceHnand standard Euclidean sphere Snrespectively with n≥3 in [Hsi82].\n(A) Unduloid (B) Union of spheres (C) Nodoid\nFigure I. Delaunay h-CMC hypersurfaces with h>0\nLet the dashed line stand for a geodesic and the gray line pres ent the unique tubular\nhypersurface of the center geodesic of constant mean curvat ureh>0 with respected to\nDate : March 12, 2024.\nKey words and phrases. Delaunay hypersurface, CMC, spiral product, flower type, em beddedness.\n12 YONGSHENG ZHANG\nthe inner unit normal vector field of the solid tube. Then, by i ncreasing the largest dis-\ntance from the center geodesic, the transformation of Delau nayh-CMC hypersurfaces\ncan be illustrated from left to right in Figure I.\nA recent joint paper [LZ] systematically studies spiral min imal products. A somehow\ndegenerate but useful situation in [LZ] is the so-called sin gly spiral product and we\nshall use this kind of product to rebuild spherical Delaunay hypersurfaces directly in\nthis paper. More explicitly, taking M1={point (1,0)}∈S1⊂C1andM2=Sn−2with a\nsingly spiral curve\n(1.1)γ(t)=/parenleftbig\na(t)eit,b(t)/parenrightbig\n∈S2,where domain Ioftis an open interval of R1,\nwe consider\nGγ:I×Sn−2−/shortrightarrowSn⊂C1⊕Rn−1\n(t,x)/mapsto−/shortrightarrow/parenleftbig\na(t)eit,b(t)x/parenrightbig\n. (1.2)\nAn advantage of this (local) construction of Delaunay h-CMC hypersurfaces is to visu-\nalize things onS2.\nWith the singly spiral product (1.2), we complete the catego ry of Delaunay hypersur-\nfaces inSnby adding a missing type (see Theorem 4.10 and Corollary 4.13 ).\nThe structure of paper is organized as follows. In §2 we recal l basics about singly spi-\nral products from [LZ] and solve the h-CMC equation locally. Local pieces of h-CMC\nhypersurfaces are described in §3, while assembling to glob alh-CMC hypersurfaces\nas well as a missing type (named flower type) of negative const ant mean curvature are\ngiven in §4. In §5 we show that there exist infinitely many choi ces of negative hsuch\nthat each the flower type Delaunay hypersurface induces an h-CMC immersed closed\nsubmanifold in the target sphere. Moreover, we get the exist ence of uncountably many\nchoices of positive heach of which allows an embedded h-CMC Delaunay submanifold\nin the sphere.\n2. Theh-CMCequation\nThe h-CMC equation is said to the curve γin (1.1) for Gγto be of constant mean\ncurvature h∈R, and a (local) solution curve γis then called an h-curve. For simplicity,\nhin this paper always means the value of unnormalized mean cur vature, i.e., the trace\nof second fundamental form.\nNote that for any h∈R, there exists a combination ( a,b)∈R2\n+such that a·S1×b·Sn−2\nhas constant mean curvature hinSn. Such combination is uniquely determined by\n(2.1)a\nb=h+/radicalbig\nh2+4(n−2)\n2(n−2).\nSee Theorem 3 (ii) of [Hsi82] or §2 of [LZ].\nFrom now on we focus on C2immersedγ⊂S2with varying ( a(t),b(t)) over some\nopen interval I. According to §3 of [LZ], the unit normal vector field we descr ibed inDELAUNAY HYPERSURFACES IN SPHERES 3\n§1 for the construction (1.2) can be gained by the normalizat ion of\n(2.2) ˜ η0(t,x)=/parenleftBig\nb(t)eit,−a(t)x/parenrightBig\n−V\nΘ/parenleftBig\nia(t)eit,0/parenrightBig\nwhereV=a′b−ab′,Θ= (as′\n1)2and/bardblG′\nγ/bardbl2=Θ+ (a′)2+(b′)2. Let/braceleftbig\nv1,···,vn−2/bracerightbig\nbe a local orthonormal basis around xofSn−2. Then/braceleftbig\n(0,v1),···,(0,vn−2),˜E/bracerightbig\nwhere\n˜E=G′\nγ\n/bardblG′γ/bardblprovides a local orthonormal basis around Gγ(t,x). By consulting Lemma 3.5\nof [LZ] we know that the second fundamental form of GγinSnis given by\nA˜η0=/parenleftbigga\nbIn−2∗\n∗ ˜/squareasterisk/parenrightbigg\nwhere/vextenddouble/vextenddoubleG′\nγ/vextenddouble/vextenddouble2·˜/squareasterisk=/bracketleftbig\na′′−a·(t′)2/bracketrightbig\nb−ab′′−V\nΘ/braceleftbig\n2aa′·(t′)2/bracerightbig\n.\nTherefore, an h-curve for generating an h-CMC hypersurface is characterized by\n/bardblG′\nγ/bardbl2·/bardbl˜η0/bardbl·h=/bardblG′\nγ/bardbl2·/parenleftBig\n˜/squareasterisk+(n−2)a\nb/parenrightBig\n.\nOf course one may try to solve the equation directly. However , since our geometric\nconcern is independent of choice of parametrization, as exp lained in [LZ], if we use arc\nparameter sfor curve ( a(t),b(t)) inS1with a(t)=cossandb(t)=sinsfrom the very\nbeginning, namely\nγ(s)=/parenleftbig\na(s)eis1(s),b(s)/parenrightbig\n,\nthenV=−1,Θ= (a˙s1)2,/bardbl˙Gγ/bardbl2=1+(a˙s1)2, and with\n˜η0(t,x)=/parenleftBig\nb(s)eis1(s),−a(s)˙s1(s)x/parenrightBig\n+1\nΘ/parenleftBig\nia(s)˙s1(s)eis1(s),0/parenrightBig\nwe have/bardbl˜η0/bardbl=/radicalBig\n1+Θ\nΘ. Now the h-CMC requirement becomes\n/radicalbigg\n1+Θ\nΘh=(n−2)a\nb−ab(˙s1)2\n1+Θ+−2ab(˙s1)2+a2˙s1¨s1\nΘ(1+Θ)\nwhich simplifies to the following h-CMC equation\n(2.3) −b\na+(n−2)a\nb−/radicalbigg\n1+Θ\nΘh=−˙Θ\n2Θ(1+Θ).\nAs a result, we have\nexp/parenleftBigg\n2/integraldisplayb\na−(n−2)a\nb+/radicalbigg\n1+Θ\nΘh ds/parenrightBigg\n=C1Θ\n1+Θ\nfor some C1∈R+and hence, withΓ=Θ\n1+Θ, it follows that\nexp/parenleftbigg\n2/integraldisplayh√\nΓds/parenrightbigg\n=˜C1/parenleftbig\ncos2ssin2(n−2)s·Γ/parenrightbig4 YONGSHENG ZHANG\nfor some ˜C1∈R+and therefore\n2h√\nΓexp/parenleftbigg\n2/integraldisplayh√\nΓds/parenrightbigg\n=˜C1d\nds/parenleftbig\ncos2ssin2(n−2)s·Γ/parenrightbig\n.\nSo, we get\n2h√\nΓ=d\nds/parenleftbig\ncos2ssin2(n−2)s·Γ/parenrightbig\ncos2ssin2(n−2)s·Γ\nand consequently,\n/parenleftbig\ncosssinn−2s/parenrightbig\nh=d\nds/parenleftBig/radicalbig\ncos2ssin2(n−2)s·Γ/parenrightBig\n.\nFinally, the h-CMC equation (2.3) is solved by\nC+h\nn−1sinn−1s=/radicalbig\ncos2ssin2(n−2)s·Γ,where C∈R.\nThus,\n(2.4) Θ=/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2\ncos2ssin2(n−2)s−/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2\nand\n(2.5) ˙ s1=±/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2\ncos2s/bracketleftBig\ncos2ssin2(n−2)s−/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2/bracketrightBig.\n3. Localconstructionof h-CMC Delaunayhypersurfaces\nIn this section, we will build local pieces of h-curves in variable s. As we shall see,\nsometimesΘcould touch zero inside (0 ,π\n2) and, due to the±choices in (2.5), there\ncan exist the situation where by geometric meanings the expr ession ˙ s1in (2.5) needs to\ntake alternating signs on two sides of the (interior) zero of Θ. By default we shall draw\npictures for the choice of non-negative ˙ s1.\nMaking sense of (2.4) with non-negativity of the denominato r simply requires that\ncosssinn−2s>C+h\nn−1sinn−1s.\nDenote the left term of the inequality by L(s) and the right by R(s,C,h). Clearly, for\nn≥3,Lreaches its maximal at s0=arctan√\nn−2∈(0,π\n2) and the derivatives of both\nsides with respect to sare\n˙L=/bracketleftbig\n−tans+(n−2) cot s/bracketrightbig\ncosssinn−2sand ˙R=hcosssinn−2s.\nSo, when h>0 (h=0 or h<0), there exists a unique point sh∈(0,s0) (sh=s0orDELAUNAY HYPERSURFACES IN SPHERES 5\ns0cosssinn−2sCh+h\nn−1sinn−1sC−h+−h\nn−1sinn−1s\n0π\n2ssh s−h\nFigure II. Graphs of R(s,C,±h) for h>0 and contact points\nsh∈(s0,π\n2)) such that ˙L(sh)=˙R(sh,h), i.e.,\n−tansh+(n−2) cot sh=h.\nThis indicates that, no matter h<0,h=0 or h>0, there exists a unique Chsuch that\ngraphs of LandRwith C=Chcontact at only one point and exactlya\nb=cotshfulfills\n(2.1).\nConsequently, the largest possible domain I⊂(0,π\n2) for an h-curve with (2.5) valid\n(i.e.Θ>0) is a connected interval depending of the choice of Cas in Figure III.\nSince the behavior types will be seen di fferently, we state the following two propositions\ncosssinn−2s\nπ\n2sC+h\nn−1sinn−1s\nI4I1\nFigure III. Lagest domain for Θ>0 in (2.4) when h>0\nseparately.6 YONGSHENG ZHANG\nProposition 3.1. Assume that h>0. If C∈[0,Ch), then the largest possible open\ndomain I (C,h)is decided by two intersection points of closures of graphs o f R(·,C,h)\nand L (·). When C∈(−h\nn−1,0), distinctly I (C,h)is determined by intersection points of\nthe graph of R (·,C,h)with the s-axis and the graph of L (·)respectively.\nProposition 3.2. Assume that h>0. If C∈[h\nn−1,C−h), then the largest possible open\ndomain I (C,−h)is decided by two intersection points of closures of graphs o f R(·,C,−h)\nand L (·). When C∈(0,h\nn−1), distinctly I (C,−h)is determined by intersection points of\nthe graph of R (·,C,−h)with the graph of L (·)and the s-axis respectively.\n4. Globalconstructionof h-CMC Delaunayhypersurfaces\nIn this section we shall exhibit how to assemble local pieces ofh-curves for a global h-\nCMC Delaunay hypersurface based on the understanding from § 3. There are essentially\nthree types of behaviors of h-curves for every non-vanishing value h∈R. For positive\nh, they are similarly as shown in Figure I.\nWhereas, for negative h, one missing limit type is found as the counterpart of union\nof spheres instead. We name it flower type since its generatin gh-curve crosses the\npeculiar point p=(0,0,1) infinitely many times and an aerial view (see Figure VIII)\nover pconsists of copies of petals. It will be seen that, up to a phas e gauge in s1(always\nignored in our consideration), the generating h-curve of flower type is unique. Tracing\nthe curve, each petal corresponds to a connected piece over s1, but “adjacent” petals\nhas jumping domains in s1and this forms a reason why the limit type has not been\ndiscovered for decades.\nThe main idea of global constructions is to apply natural refl ection/rotational con-\nstructions according to Figures II and III. Let us explain de tails in below for hof differ-\nent signs separately.\n4.1. Case of positive h.As shown in Figure III, when C∈(0,Ch), there is a largest\nopen domain\n/parenleftbig\nzh\nL,zh\nR/parenrightbig\n:=I(C,h)⋐/parenleftBig\n0,π\n2/parenrightBig\nforΘin (2.4) to be positive.\nByDwe mean cos2ssin2(n−2)s−(C+h\nn−1sinn−1s)2, the denominator of Θin (2.4).\nThen, first of all, we would like to mention a simple observati on.\nProposition 4.1. For C∈(0,Ch), the derivative of D with respect to s is nonzero at zh\nL\nand zh\nR.\nProof. This is clear according to Figure II. /square\nThis result ensures that the integral of ˙ s1in (2.5) over ( zh\nL,zh\nR) is finite and hence one\ncan reflect the curve at the ending points along s1-direction to get the following figure.DELAUNAY HYPERSURFACES IN SPHERES 7\ns1 s1\ns\nzh\nLzh\nRπ\n2\nFigure IV. Reflection extension when h>0 and C∈(0,Ch)\nNow consider the di fferentiability of the extension at joint points. Obviously, it isC1\nat joint points, so one can get a C0inner unit vector field. In fact, the extended curve\nhas much stronger di fferentiability for its geometric meanings.\nProposition 4.2. For h>0and C∈(0,Ch), the repeatedly extended curve as in Figure\nIV gives an analytic h-curve and consequently a correspondi ng immersion of constant\nmean curvature h from R×Sn−2intoSnis gained.\nProof. As in both sides of a joint point the local h-curve generates hypersurface of the\nsame constant mean curvature hand a joint point is an extremal point in Figure IV, it\nfollows that the extended curve is indeed C2at each joint point. Furthermore, based\non the C2-differentiability, Morrey’s regularity theory [Mor54, Mor58] asserts that the\nextended h-curve is analytic everywhere. /square\nForC=0, it can be seen from (2.5) that ˙ s1vanishes at s=0 and we get the following.\ns1 s1\nszh\nRπ\n2θ\nFigure V. Reflection extension for C=0 with h>08 YONGSHENG ZHANG\nProposition 4.3. For h≥0and C=0, via applying the reflection construction in\nthe s 1-phase one time, one gets an analytic h-curve as in Figure V wh ich generates a\nhypersphere of size sinzh\nRwith constant mean curvature h.\nProof. Evidently, the local h-curve and its reflection copy together induce an embedded\nhypersphere. This is impossible for non-vanishing C, cf. Figure III. As h>0, there is\ncertain (round) sphere in Snof mean curvature hwhich can be generated by an h-curve.\nSo, by Figure III, the h-curve must be the curve in Figure V. This curve induces an\nh-CMC ( n−1)-sphere centered at νof radius sin zh\nRin the affine spaceRn\nν=ν⊥⊂Rn+1\nν\nwhereνstands for the point/parenleftbig\ncoszh\nR·(eiθ,0,···,0)/parenrightbig\n∈Rn+1.\nAs the mean curvature the hypersphere in Snis (n−1)coszh\nR\nsinzh\nR, one has zh\nR=arctann−1\nh.\nThe conclusion then also holds for h=0. /square\nNext, let us focus on the situation of C∈(−h\nn−1,0). Deriving from the curve corre-\nsponding to I4in Figure III, we flip its part below the s-axis in the ss1-plane to the above\nto get the domain J4=/parenleftbig\nzh\nL,zh\nR/parenrightbig\nas in Figure VI. The corresponding extension has one\nπ\n2s\nI4\nJ4ˆs zh\nL zh\nR\nFigure VI. interval JforC∈(−h\nn−1,0) with h>0\nmore step because of the occurrence of ˆ s. It can be seen that the dash dot dotted curve\nsegment in Figure VII stands for a hypersurface of constant m ean curvature−hunder the\nchosen unit normal vector field and consequently its flipped m irror (solid curve in green\ndefined over the s-range ( zh\nL,ˆs) in Figure VII) with respect to {ˆθ}×(0,π\n2) corresponds to\na hypersurface of constant mean curvature h. Similarly as argued in Proposition 4.2 we\nhave the following.\nProposition 4.4. For h>0and C∈(−h\nn−1,0), via mirror flips and reflections along s 1-\ndirection, one gets an analytic h-curve as in Figure VII whic h generates a hypersurface\nof constant mean curvature h, with whirls occurring in local model.\nRemark 4.5. Up to a sign, ˙s1is given byC+h\nn−1sinn−1s\ncoss/radicalBig\ncos2ssin2(n−2)s−/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2\nover the entire J.DELAUNAY HYPERSURFACES IN SPHERES 9\ns1\ns\nzh\nL zh\nRπ\n2 ˆsˆθ\nFigure VII. Reflection extension when h>0 and C∈(−h\nn−1,0)\n4.2. Case of vanishing h.This case falls exactly into the singly spiral minimal produ ct\nof a point andSn−2inSnand it has been systematically studied in [LZ]. Note that the ro-\ntational extension (of reflections in both slots) to constru ct global doubly spiral minimal\nproducts now reduces to the aforementioned (phase-reflecti on) extension in a single slot\nof (1.2). In particular, for a singly spiral minimal product the parameter ˜Cin (3.25) of\n[LZ] is precisely 1/C2for our Chere.\nSince all moving curves in Figures II and III become horizont al lines{s1=C}(thus\nnot transversally hitting the s-axis), whirls as in Figure VII can never happen when\nh=0 and only oscillations along s1-direction can take place. An exception is C=0\nwhich leads to a totally geodesic hypersphere. With the end v alue s1of local h-curve\nfixed, as C/shortdownarrow0 one period of oscillations, namely the union of the local h-curve and\nitss1-phase reflection, will limit to a pair of geodesics joint at p=(0,0,1) inS2, e.g.\n(±coss,0,sins) for s∈[0,π\n2]. The angleπbetween the pair was in fact computed in\nLemma 10.1 of [LZ] with k1=0 therein.\n4.3. Case of negative h.Unlike the Euclidean situations, the case of Delaunay hyper -\nsurfaces in spheres can have oscillating phenomena for nega tivehas well. Moreover, an\namusing missing type of Delaunay hypersurfaces in spheres o f negative constant mean\ncurvature hwill be explored. The philosophy to predict this is the follo wing. For h>0,\nlike in Figure I from unduloid to noduloid the deformation pa sses through union of\nspheres. As we shall see that, for h<0, there are also “unduloid” and “noduloid” types.\nSo, similarly, there must be one additional transition type connecting them.\nTo be consistent with Figures I and II and Proposition 3.2 we a ssume h>0 and\ndenote the constant mean curvature by −h. Using previously given symbols we arrive at\nthe following.10 YONGSHENG ZHANG\nProposition 4.6. When C∈(h\nn−1,C−h), similarly as in Figure IV one can get an analytic\n(−h)-curve oscillating along s 1-dircetion which induces an immersion of constant mean\ncurvature−h fromR×Sn−2intoSn; when C∈(0,h\nn−1), similarly as in Figure VII one\ngets an analytic (−h)-curve which generates a hypersurface of constant mean curv ature\nh, with whirls occurring in local model.\nProof. The same as the arguments for Propositions 4.2 and 4.4. /square\nRemark 4.7. The oscillating curves form unduloid type and those with whi rls in local\nmode give noduloid type. It should be mentioned that the nodu loid type is the exactly\nsame as in Figure VII but with the other unit normal vector fiel d automatically induced\nby the construction (i.e., decided by the solid curve part in Figure VI). However, by\nchecking oscillating ranges in s, it is clear that unduloids for negative and non-negative\nh are completely di fferent sets.\nThe missing type occurs when C=h\nn−1which can be viewed as a limit type. By fixing\nplus sign in (2.5) we have the following asymptotic result.\nLemma 4.8. For h>0and C=h\nn−1, every (−h)-curve in local model (see Figure II)\nlimits to p=(0,0,1)and along the curve it follows that lim s/shortuparrowπ\n2˙s1=h\n2.\nProof. Let∆s=π\n2−s. Note that, for n≥3, we haveh\nn−1−h\nn−1sinn−1s=h\n2·(∆s)2+high\norder terms and cos s=∆s+high order terms. So (2.5) with plus sign becomes\nlim\ns/shortuparrowπ\n2˙s1=lim\n∆s/shortdownarrow0/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftbigh\n2/parenrightbig2(∆s)4\n(∆s)2/bracketleftbig\n(∆s)2−/parenleftbigh\n2/parenrightbig2(∆s)4/bracketrightbig=h\n2.\n/square\nLemma 4.8 leads to two useful consequences. The first is to ens ure that the s1-value\nalong the peculiar ( −h)-curve for C=h\nn−1increases to a finite number as s/shortuparrowπ\n2. The\nsecond is to help understanding the tangential behavior in t he limit.\nLemma 4.9. When approaching p =(0,0,1), every (−h)-curve in local model is tan-\ngential to a geodesic (cos s·q,sins)for some suitable q ∈S1⊂C.\nProof. Suppose that the (−h)-curve in local model ends up with a limiting phase θ. Set\nq=eiθ. Denote (cos s·q,sins) for s∈[0,π\n2] byΓq. Then inS2it can be observed\nthat, when∆s/shortdownarrow0, the quantity (cos s)·˙s1=(sin∆s)·˙s1approaches the slope of the\n(−h)-curve with respect to Γq. By the finiteness of ˙ s1in Lemma 4.8 one can easily read\nit out that the limit of the slope is zero at p. /square\nAs a result, up to a phase gauge, there is a unique ( −h)-curve in local model limiting\nto the point ptangential to some geodesic through pfor every h>0. In fact, the local\n(−h)-curve can develop to a global one which is extremely natura l in view of generating\ncurves inS2. However, the extension or, perhaps more precisely, assemb ling of local\npieces at pis a bit different but still very geometric.DELAUNAY HYPERSURFACES IN SPHERES 11\nTheorem 4.10. For every h>0, there exists a unique global (complete) (−h)-curve\npassing through p. Consequently, the Delaunay hypersurfac es of constant mean curva-\nture−h contain a new type which serves as a bridge jointing the undu loid type Delaunay\nhypersurfaces and noduloid type Delaunay hypersurfaces.\nProof. According to Lemma 4.9, a normal vector field Nalong (−h)-curve with C=h\nn−1\nin local model can have a C0limit Npatp. Then we reflect the ( −h)-curve at pabout\nthe plane spanned by− /shortrightarrow0pandNpinR3. In this way the local ( −h)-curve C1extends\nthrough p(with C0extension of N) as in the aerial view Figure VIII over point p.\nSimilarly, the extended ( −h)-curve is C2at the joint point p. Thus, the local (−h)-curve\ncan extend repeatedly to a complete analytic ( −h)-curve which generates an ( −h)-CMC\nimmersion fromR×Sn−2intoSn. The immersed hypersurface forms a flower type\nDelaunay hypersurface. /square\np= (0,0,1)q\nΓq\n(−h)-curvepNp\nFigure VIII. Missing (flower) type which crosses p\nRemark 4.11. The mean curvature of the flower type Delaunay hypersurface i nduced\nby the (−h)-curve extended through p via reflection by N pcan be verified crossing\n{0+0i}×Sn−2as follows. First, by Lemma 4.8, ˜η0in(2.2) at p (equal to (eiθ,0,···,0))\ncorresponds to N p. Hence, all curvatures arising from the Sn−2factor contributes noth-\ning to the mean curvature of the flower type at p.\nSo one only needs to compute the curvature κof the (−h)-curve at p. Since approach-\ning p limits to a curve in polar coordinate of R2, namelyρ:=∆s=∆s(s1)=ρ(s1), it\nfollows that, with respect to −N, the curvature is given by\nκ(s1)=ρ2+2/parenleftBig\ndρ\nds1/parenrightBig2\n−ρd2ρ\nds2\n1/braceleftbigg\nρ2+/parenleftBig\ndρ\nds1/parenrightBig2/bracerightbigg3\n2.12 YONGSHENG ZHANG\nAsdρ\nds1=/parenleftBig\nds1\ndρ/parenrightBig−1\nandd2ρ\nds2\n1=−d2s1\ndρ2/parenleftBig\nds1\ndρ/parenrightBig−3\n, according to Lemma 4.8 we can see that the\ncurvature of the curve at p =(0,θ)is2/parenleftbigh\n2/parenrightbig−2/slashbig/parenleftbigh\n2/parenrightbig−3=h.\nRemark 4.12. For h fixed, if we give up the insistence on the choice of unit no rmal\nvector field and take an obvious “continuous” choice ˜N associated with the deformation\nof C instead. By decreasing C, a vivid video of deformation of Delaunay hypersurfaces\ncan be obtained. The moving slides start from the static type by(2.1) to the unduloid\ntype; then pass through the union of spheres to the nodoid typ e; with the drip area\nenlarging, finally come to the flower type in Theorem 4.10; aft er that, meet the unduloid\ntype and end up with another static (corresponding to (2.1) for−h) type. All of these\nhypersurfaces have constant mean curvature h with respect t o˜N.\np\nTheflower type\nDelaunay hypersurfaceangleβ\nFigure IX. Deformation of types by decreasing the value of Cforh>0\nHowever, note that in Figures VIII and IX illustrated petals are particularly chosen\nto be assembled by reflections along di fferent N prepeatedly for an analytic (−h)-curve\nwhich induces an (−h)-CMC immersion of R×Sn−2. Actually, the flower type can be\nviewed as a limit mapping from the point of view encoded in Defi nition 5.2 of adjusted\nwidth and Figure X.\nAsCruns all possible values, Figure IX exhausts all CMC Delauna y hypersurfaces\nwhen h/nequal0. Hence, we finish completing the category of Delaunay hyper surfaces in\nSn⊂C⊕Rn−1through movingSn−2⊂Rn−1by a generating curve γ⊂S3⊂C⊕R.\nCorollary 4.13. For h/nequal0, up to a phase gauge in s 1(or equivalently a rotation fixing\np=(0,0,1)), all Delaunay hypersurfaces of constant mean curvature h w ith respect to\nsome unit normal vector field in Snare included in Figure IX.\n5. Descendentto (immersedorembedded )closed CMChypersurfaces\nIn this section, we search for flower type Delaunay hypersurf aces inSnwhich induce\nclosed immersed CMC hypersurfaces, consider the descenden t of (1.2) to immersionsDELAUNAY HYPERSURFACES IN SPHERES 13\nofS1×Sn−2intoSnand show the existence of uncountably many embedded Delauna y\nhypersurfaces of positive constant mean curvatures in Sn.\nFrom (2.5), we have (valid for all types)\n(5.1) ˙ s1=C+h\nn−1sinn−1s\ncoss·/radicalBig\ncos2ssin2(n−2)s−/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig2.\nSetJ(C,h)⊂(0,π\n2) to be the largest interval for cos ssinn−2s−C+h\nn−1sinn−1s>0, cf.\nFigures III and VI.\nLemma 5.1. Except the situation of Proposition 4.3, i.e., h ≥0and C=0, an h-curve\nforms a closed curve in S3if and only if the width function of a period\n(5.2) W(h,C) :=2/integraldisplay\nJ(C,h)˙s1ds∈πQ.\nProof. By the construction this is clear for curves not bothering s=0 or s=π\n2. For\nthe flower type, let us set β=W(−h,h\nn−1) as shown in Figure IX. Then the adjacent\nangle between two petals via one reflection is π−2βand 2βby two repeated reflections.\nThe latter gives a 2 β-rotation about p. Hence, the extended ( −h)-curve of flower type is\nclosed if and only if β∈πQ. /square\nAs illustrated in Figure IX, let us mention the deformation o f widths. By consid-\nering line types in Figure IX as limit situations, W(h,·) is continuous in the first four\nandW(−h,·) in the last two pictures of Figure IX respectively. However , a gap occurs\nwhen C=h\nn−1. So, we introduce the concept of adjusted width according to geometric\nmeaning of Figure IX.\np\n˜N{s= 0}\n˜N\nFigure X. Aerial view of Figure IX for fixed h>0 with respect to ˜N14 YONGSHENG ZHANG\nDefinition 5.2. Given h≥0, the adjusted width ˜W(h,·)over (−C−h,Ch)is defined as\n(5.3) ˜W(h,C) :=\n\nW(h,C) C∈(−h\nn−1,Ch),\nπ−W(−h,h\nn−1) C=−h\nn−1,\n2π−W(−h,−C) C∈(−C−h,−h\nn−1).\nLemma 5.3. ˜W(h,·)is continuous.\nProof. As the deformation moves, there is an amount πjumping in both sides of the\nflower type. This is because the derivative ofh\nn−1sinn−1svanishes at s=π\n2and the\ncomputation in Lemma 10.1 of [LZ] is still under control and c an be adjusted to measure\nthe jumps to be 2×π\n2. /square\nRemark 5.4. In fact, the point p =(0,0,1)is a singular point with respect to the\nparametrization (s,s1)not the geometric problem itself. There is no problem at all\nfor generating curves passing through p and the continuity c an be strengthened to be\nanalytic, cf. Figure X, whenever the genuine singular set {s=0}is avoided.\nRemark 5.5. The adjusted width function actually exhibits a way to const ruct the flower\ntype as a limit from its either side, see Figure X. This perspe ctive is different from the\nassembling thought in the proof of Theorem 4.10.\nCorollary 5.6. ˜W(·,·)is continuous.\nProof. Since the splitting of domain and expressions over each piec e are continuous in\nh, the statement follows by virtue of Lemma 5.3. /square\nTheorem 5.7. There are infinitely many −h with h>0for each of which the flower type\nDelaunay hypersurface can induce a (−h)-CMC immersion from S1×Sn−2intoSn.\nProof. Note that the expression (5.2) contributes nothing for C=h=0. So ˜W(0,0)=π.\nBut, for C=h\nn−1>0, it can be seen that W(−h,h\nn−1)>0 and consequently ˜W(h,−h\nn−1)/nequal\nπ. Hence, by Lemma 5.1 and Definition 5.3, every element in πQ/intersectiontext/parenleftbig˜W(h,−h\nn−1),π/parenrightbig\ncan\ndecide an immersion of S1×Sn−2intoSnof corresponding constant mean curvature. /square\nTheorem 5.8. There exists a constant c >0such that for every h ∈(0,c)there are\ncountably many (−h)-CMC immersions from S1×Sn−2intoSninduced by the unduloid\ntype Delaunay hypersurfaces (in the second last picture of F igure IX).\nProof. Similar to the proof of Theorem 5.7. When C=h=0, we have ˜W(0,0)=π.\nForC=−C0, by Lemma 10.1 of [LZ] or the argument (valid for h=0) of Lemma\n5.11 in below it can be computed that ˜W(0,−C0)=(2−√\n2)π. Hence, by continuity\nthere exists some open interval (0 ,c) such that ˜W(h,0)>˜W(h,−Ch) for every h∈(0,c).\nSo, for each such h, there exists some C∈(−Ch,0) for ˜W(h,C)∈πQ. By Lemma 5.1\nand Definition 5.3, ˜W(h,C)∈πQimplies the corresponding Delaunay hypersurfaces\ndecides (−h)-CMC immersions from S1×Sn−2intoSn. /square\nCorollary 5.9. The same statement of Theorem 5.8 holds for the noduloid type Delau-\nnay hypersurfaces.DELAUNAY HYPERSURFACES IN SPHERES 15\nProof. This follows by the argument in the proof of Theorem 5.8 and th e analyticity\nexplained in Remark 5.4. /square\nApparently, one cannot expect embedded CMC hypersurface of flower type or nodu-\nloid type. Next, we focus on the second picture in Figure IX an d search for embedded\nCMC Delaunay hypersurfaces.\nLemma 5.10. Given h>0,lim C/shortdownarrow0W(h,C)=2 arctann−1\nh.\nProof. According to Proposition 4.3, W(h,0)=2zh\nR. As the mean curvature the hyper-\nsphere inSnis given by ( n−1)·coszh\nR\nsinzh\nR, one has zh\nR=arctann−1\nh. /square\nLemma 5.11. Given h>0,lim C/shortuparrowChW(h,C)=2π/bracketleftbig\n1+(n−2)cot2sh/bracketrightbig−1\n2.\nProof. Recall that LandRshare the same slope at s=sh. So let us consider their second\nderivatives\n¨L=/braceleftbigg/bracketleftBig\n−tans+(n−2) cot s/bracketrightBig2\n−/bracketleftbigg1\ncos2s+n−2\nsin2s/bracketrightbigg/bracerightbigg\ncosssinn−2s\nand\n¨R=h/bracketleftbig\n−tans+(n−2) cot s/bracketrightbig\ncosssinn−2s.\nWe can observe that, when C/shortuparrowCh, the part under square in (5.1) becomes\nsmall positive number AC−/bracketleftbigg1\ncos2s+n−2\nsin2s/bracketrightbigg/parenleftbig\ncosssinn−2s/parenrightbig2·(δs)2+o(δs)2\nwhereδsis quite small and AC/shortrightarrow0 asC/shortuparrowCh. As Ch+h\nn−1sinn−1sh=cosshsinn−2sh,\nit follows that\nlim\nC/shortuparrowChW(h,C)=2 lim\nC/shortuparrowCh/integraldisplay\nJ(C,h)˙s1ds\n=2 lim\nC/shortuparrowCh/integraldisplay\nJ(C,h)/parenleftbig\nC+h\nn−1sinn−1s/parenrightbig\nds\ncoss/radicalBig\nAC−/bracketleftbig1\ncos2s+n−2\nsin2s/bracketrightbig/parenleftbig\ncosssinn−2s/parenrightbig2·(s−sh)2\n=2πcosshsinn−2sh\ncossh/radicalBig/bracketleftbig\n1+(n−2)cot2sh/bracketrightbig\nsinn−2sh=2π/radicalBig/bracketleftbig\n1+(n−2)cot2sh/bracketrightbig.\nHere we apply the evaluation of arcsin twice and the integral turns out to be independent\non value of AC. /square\nTheorem 5.12. For any n≥3, there are uncountably many positive number h for each\nof which there exists an embedded h-CMC Delaunay hypersurfa ce inSn.\nProof. By Lemma 5.10, lim C/shortdownarrow0W(h,C)<πforh>0. Whereas, by Lemma 5.11,\nlim\nC/shortuparrowChW(h,C)=2π/radicalbig\n1+(n−2)cot2sh=2π/radicalbigg\n2+h2+h√\nh2+4(n−2)\n2(n−2)>π when h0, when n is sufficiently large, there can be guaranteed\nan embedded h-CMC Delaunay hypersurface in each Sn.\nInstead ofπ, one can consider to fulfill di fferent size of width for an embedded De-\nlaunay hypersurface, for example W(h,C)=2π\nkwhere 22π\nk,\nwe get h<(k2−2)/radicalBig\nn−2\nk2−1.\nOn the other hand, arctan x≤xforx≥0. Therefore, in order for\n(5.5) lim\nC/shortdownarrow0W(h,C)=2 arctann−1\nh<2π\nk,\nit suffices to requiren−1\nh<π\nk.\nForZ(k) to be nonempty, observe that the highest order of its ending points are the\nsame and the coefficients aren−1\nπand√\nn−2 in limit. Hence, whenn−1\nπ<√\nn−2, i.e.,\n3≤n≤9,Z(k) is never empty when kis large enough.\nFurthermore, by continuity based on (5.4) and (5.5), for eac hh∈Z(k) there exists\nsuitable Cdepending on the value of hsuch that W(h,C)=2π\nkis satisfied. Correspond-\ningly, an embedded h-CMC Delaunay hypersurface is generated in Sn./square\nRemark 5.15. By the above argument for 3≤n≤9, it follows that the union of\n{Z(k)}k>2will contain an infinite tail (d0,∞)with the property that, for any h >d0there\nexist at least 1+/bracketleftbigπ(h−d0)\nn−1/bracketrightbig\nmany different embedded h-CMC Delaunay hypersurface(s) in\nSn. Here the symbol [·]means taking the largest integer part of the input.\nTo the author, it seems unlikely that the unduloid type in the second last picture of\nFigure IX can produce embedded ( −h)-CMC Delaunay hypersurfaces in Sn.DELAUNAY HYPERSURFACES IN SPHERES 17\nAcknowledgement\nThis work is partially supported by NSFC (Grant Nos. 1202210 9 and 11971352).\nThe author would like to thank Professor Frank Morgan for his interest, and ICTP for\nwarm hospitality where some initial inspiration of the pape r was generated in December\n2023.\nReferences\n[HY81] Wu-Yi Hsiang and Wen-Ci Yu, A generalization of a theorem of Delaunay , J. Diff. Geom. 16\n(1981), 161–177.\n[Hsi82] Wu-Yi Hsiang, On generalization of theorems of A. D. Alexandrov and C. Dela unay on hyper-\nsurfaces of constant mean curvature , Duke Math. J. 49(1982), 485–496.\n[LZ] Haizhong Li and Yongsheng Zhang, Spiral Minimal Products , arXiv: 2306.03328v3.\n[Mor54] Charles B. Morrey, Jr., Second-order elliptic systems of di fferential equations , pp. 101–159 in\n“Contributions to the theory of partial di fferential equations”, Annals of Mathematics Studies\n33, Princeton University Press, 1954.\n[Mor58] ,On the analyticity of the solutions of analytic non-linear e lliptic systems of partial\ndifferential equations. I. Analyticity in the interior , Amer. J. Math. 80(1958), 198–218.\nAcademyfor Multidisciplinary Studies , Capital Normal University , Beijing 100048, P. R. China\nEmail address :yongsheng.chang@gmail.com" }, { "title": "2401.16315v1.Regularity_and_compactness_for_critical_points_of_degenerate_polyconvex_energies.pdf", "content": "arXiv:2401.16315v1 [math.AP] 29 Jan 2024Regularity and compactness for critical points\nof degenerate polyconvex energies\nAndr´ e Guerra1andRiccardo Tione2\n1Institute for Theoretical Studies, ETH Z¨ urich, Clausiuss trasse 47, 8006 Z¨ urich, Switzerland\nandre.guerra@eth-its.ethz.ch\n2Max Planck Institute for Mathematics in the Sciences, Insel strasse 22, 04103 Leipzig, Germany\nriccardo.tione@mis.mpg.de\nAbstract\nWe study Lipschitz critical points of the energy´\nΩg(det D u) dxin two dimensions, where\ngis a strictly convex function. We prove that the Jacobian of a ny Lipschitz critical point\nis constant, and that the Jacobians of sequences of approxim ately critical points converge\nstrongly. The latter result answers in particular an open pr oblem posed by Kirchheim, M¨ uller\nandˇSver´ ak in 2003.\n1 Introduction\nLet Ω ⊂R2be a bounded domain with |∂Ω|= 0 and consider the energy\nE[u]≡ˆ\nΩg(det D u) dx, u :Ω→R2, (1.1)\nwhere g∈C1(R) is strictly convex. Energies of this type were studied exte nsively in the literature,\nfor instance in connection with elastic fluids or with the prescribed Jacobian equation , see e.g.\n[6,7,8,9,16,24,28,35], as well as the recent works [ 37,38]. In fact, if Ω is smooth and u0is\ndiffeomorphism on ∂Ω, then one can construct smooth minimizers for ( 1.1) with boundary data\nu0by applying the results of [ 10] to solve the problem\n\n\ndet D u=ffl\nΩdet D u0(y) dy,\nu=u0on∂Ω.\nFor non- C1data, however, issues arise with the applicability of this m ethod [ 19,20], and even if\nsolutions exist they will not be C1in general.\nIn this paper we investigate Lipschitz critical points ofE, that is, solutions of\ndiv(g′(det D u) cof(D u)T) = 0 . (1.2)\nThe energy Eispolyconvex [4] and so in particular it is quasiconvex. Minimizers of strongly\nquasiconvex energies are smooth almost everywhere [ 15], but for Lipschitz critical points neither\nAcknowledgments. AG acknowledges the support of Dr. Max R¨ ossler, the Walter H aefner Foundation and the ETH\nZ¨ urich Foundation. We thank Paolo Bonicatto for helpful di scussions concerning [ 5].\n1quasiconvexity [ 30] nor polyconvexity [ 34] are enough to imply further regularity. A challenging\nopen problem is to understand whether further conditions on the solutions (such as stationarity)\nor further structural assumptions on the energy (such as the ones in this paper) imply regularity.\nWe refer the reader to [ 11,21,22,26,36] for results in this direction.\nNote that Eis adegenerate polyconvex energy, in the sense that it only controls the Jac obian:\nin particular, a Lipschitz critical point uis not C1in general. However, ifuisC1, then\ndet D uis constant in Ω ,\nwhich is the best type of regularity one can hope for in this pr oblem. The first question we\naddress is whether this constancy property continues to hol d for critical points which are just\nLipschitz. The answer is affirmative:\nTheorem 1.1 (Regularity of exact solutions) .Suppose that g∈C1(R)is strictly convex and let\nu∈W1,∞(Ω,R2)be a solution of (1.2). Then det D uis constant a.e. in Ω.\nPreviously, in [ 37] the second author proved Theorem 1.1under the additional assumption\nthat |det D u| ≥ε >0 a.e. in Ω. In fact, our proof of Theorem 1.1relies on this result.\nAs a consequence of Theorem 1.1we find the following perhaps surprising fact:\nCorollary 1.2 (Critical points are minima) .Ifu∈W1,∞(Ω,R2)solves (1.2)then uis a mini-\nmizer of E, in the sense that E[u]≤E[v]for all v∈u+W1,∞\n0(Ω,R2).\nThe second question we want to address concerns Lipschitz ma ps which solve ( 1.2) only ap-\nproximately: is it the case that such maps are close to an actu al solution of ( 1.2), and if so in\nwhich sense? Here we prove the following:\nTheorem 1.3 (Compactness of approximate solutions) .Suppose that g∈C1(R)is strictly con-\nvex. Let uj∗⇀ u inW1,∞(Ω,R2)and suppose that there is (Fj)⊂L∞(Ω,R2×2)bounded and such\nthat\ndiv(g′(det D uj) cof(D uj)T) = div( Fj), F j→0inL1(Ω). (1.3)\nThen usolves (1.2)anddet D uj→det D uinLp(Ω)for all p <∞.\nThis result is optimal, as in general we do not have D uj→DuinLp(Ω) [ 37], sinceEis a\ndegenerate energy. By rewriting ( 1.2) as a differential inclusion (cf. Section 4), we obtain:\nCorollary 1.4 (Quasiconvexity of the differential inclusion) .The set K⊂R4×2, defined by\nKg≡/braceleftBigg/parenleftBigg\nA\ng′(detA)A/parenrightBigg\n:A∈R2×2/bracerightBigg\n, (1.4)\nis quasiconvex.\nIn the particular case g(t) =t2, Corollary 1.4answers [ 24, Question 10]. In that work it was\nshown that Kgisrank-one convex ; for subsets of R4×2, this is a condition which is strictly weaker\nthan quasiconvexity [ 32] but which is easier to verify. We refer the reader to [ 29,31] for further\ndetails on quasiconvexity and rank-one convexity.\nThe proofs of our main results use various ingredients. Firs t, to show Theorem 1.1, we employ\nideas coming from the theory of transport equations, in part icular the renormalization properties\nin the sense of DiPerna–Lions [ 14] of two-dimensional solutions, which were shown in [ 1,5]. Next\nwe employ some deep but rather classical results coming from the theory of quasiregular mappings,\nand we refer the interested reader to [ 12,17,25,33] for further applications of quasiregular maps\n2in the theory of differential inclusions. To show Theorem 1.3, the main ingredients are Allard’s\nstrong constancy lemma [ 2] and the 0-1 law for quasiregular gradient Young measures [ 3]. We\nnote that the ideas behind Allard’s lemma have recently been applied in [ 13] in a rather different\nproblem.\n2 Notation and setup\nWe will say that Ω ⊂Rmis a domain if it is an open, connected bounded set with |∂Ω|= 0.\nThroughout this paper we write A, B for matrices in Rn×nand we denote by /a\\}bracketle{tA, B/a\\}bracketri}httheir Hilbert–\nSchmidt inner product. We write LA:Rn→Rnfor the linear map induced by A. We also write\ncof(A) for the cofactor matrix of A, which satisfies\ncof(A)A=Acof(A) = (det A)Id. (2.1)\nAn important property of the cofactor is the so-called Piola identity\ndiv(cof(D u)T) = 0 , (2.2)\nvalid for Lipschitz maps u:Rn→Rn. Note that, when n= 2, cof :R2×2→R2×2islinear , since\ncof(A) =JATJT, where Jis the matrix corresponding to a rotation by π/2.\nWe say that a function g:R→Risstrictly convex if\ng(λx+ (1 −λ)y)< λg (x) + (1 −λ)g(y)\nwhenever x/\\e}atio\\slash=yandλ∈(0,1). If g∈C1(R), then it is elementary to see that gis a strictly\nconvex function if and only if g′is strictly increasing. By ( 2.2), for the results of this paper we\ncan and will always assume that\ng′(0) = 0 . (2.3)\nWhen gis strictly convex, i.e. when g′is strictly increasing, ( 2.3) implies:\ng′(t)t=|g′(t)||t| ≥0. (2.4)\n3 Exact solutions\nThe purpose of this section is to prove Theorem 1.1. We begin by stating the following deep\nresult, which is essentially due to Bianchini–Gusev, but se e also the earlier work [ 1].\nTheorem 3.1 (Renormalized solutions) .Letu∈W1,∞solve (1.2)andβ∈C∞\nc(R). Then\ndiv(β(g′(det D u)) cof(D u)T) = 0 .\nProof. The result follows directly from [ 5, Theorem 6.7]. Indeed, ( 1.2) corresponds to a system\nof two (time-independent) continuity equations div( ρ bi) = 0, where the density is ρ≡g′(det D u)\nand the bounded vector fields bi:Ω→R2are the rows of cof(D u)T, for i= 1,2, which are\ndivergence-free by ( 2.2).\nProof of Theorem 1.1.We can assume that it is not the case that det D u= 0 a.e. in Ω, since\notherwise there is nothing to show. We may then also assume th at det D u >0 a.e. on a set of\npositive measure, the case det D u <0 being totally analogous. In particular, there is ε >0 small\n3enough so that\n|{x∈Ω:det D u(x)≥ε}|>0. (3.1)\nLet us write ˜ ε/2≡g′(ε/2)>0. We now take β∈C∞\nc(R,[0,1]) such that β(t) = 0\nift≤˜ε/2 and β(t) = 1 if ˜ ε≤t≤2/bardblg′(det(D u))/bardblL∞. By Theorem 3.1, the matrix field\nβ(g′(det D u)) cof(D u)Tis divergence-free. Since the statement in Theorem 1.1is local, there is\nalso no loss of generality in assuming that Ω is simply connec ted, and so there is v∈W1,∞(Ω,R2)\nsuch that\nβ(g′(det D u))Du= Dv. (3.2)\nBy taking determinants on both sides of this identity, we get\nβ(g′(det D u))2det D u= det D v. (3.3)\nThe above two identities show that, for a.e. x∈Ω, if det D v(x) = 0, then D v(x) = 0, due to ( 2.3)\nand the definition of β. On the other hand, by ( 3.2), at a.e. xwith det D v(x)/\\e}atio\\slash= 0 we have\n0<|Dv|2(x)\ndet D v(x)=|Du|2(x)\ndet D u(x)≤2Λ2\nε,\nwhere Λ >0 is the Lipschitz constant of u. In other words, there exists K > 0 such that\n|Dv|2(x)≤Kdet D v(x) for a.e. x∈Ω, i.e. the map vis quasiregular. It is well-known that\na quasiregular map on a domain is either constant or else its J acobian is positive a.e., see e.g.\n[23, Theorem 16.10.1]. By ( 3.1) we cannot have D v= 0 a.e. and so vis not constant, hence\ndet D v > 0 a.e. in Ω; thus in fact from ( 3.3) we deduce g′(det D u)≥˜ε/2 a.e., which in turn\nimplies det D u≥ε/2 a.e. in Ω. Finally we conclude by applying [ 37, Theorem 1]. We note that,\nin [37], the stronger conditions g∈C2andg′′>0 are assumed, but that they are not needed for\nthe proof.\nProof of Corollary 1.2.By Theorem 1.1, there is c∈Rsuch that det D u=ca.e. in Ω. By\nconvexity of g, we have a.e. the inequality\ng(det D v)≥g(det D u) +g′(c)(det D v−det D u).\nThe claim now follows by integrating this inequality over Ω, since´\nΩdet D v=´\nΩdet D uas det\nis a null Lagrangian [ 29, Theorem 2.3] and u−v∈W1,∞\n0(Ω).\n4 Approximate solutions\nIn this section we prove Theorem 1.3. In fact, we will reformulate our problem in a slightly\ndifferent way, using the language of differential inclusions . Without loss of generality we assume\nthat Ω ⊂R2is simply connected, and we can then rewrite ( 1.2) as\ng′(det D u)Du= Dv (4.1)\nfor some v∈W1,∞(Ω,R2). This last equation, in turn, is nothing but the differentia l inclusion\nDw∈Kga.e. in Ω , w ≡(u, v):Ω→R4, (4.2)\nwhere Kgis defined in ( 1.4). We begin this section by recalling some well-known but use ful results\nconcerning Young measures, and their relation to differenti al inclusions.\n44.1 Differential inclusions and Young measures\nThe results and discussion of this subsection are standard, and the reader can find proofs as\nwell as further information in [ 29,31]. We let Ω ⊂Rmbe a domain.\nDefinition 4.1. A parametrized family of probability measures ν= (νx)x∈ΩonRdis said to be\naYoung measure if there is a weakly- ∗convergent sequence ( zj)⊂L∞(Ω,Rd) which generates ν,\nin the sense that\nf(zj)∗⇀/parenleftbigx/mapsto→ /a\\}bracketle{tνx, f/a\\}bracketri}ht/parenrightbiginL∞(Ω),\nfor all f∈C0(Rd). We always write /a\\}bracketle{tνx, f/a\\}bracketri}ht ≡´\nRdf(A) dνx(A) and νx≡ /a\\}bracketle{tνx,id/a\\}bracketri}ht.\nA Young measure ( νx)x∈Ωishomogeneous if there is a probability measure νinRdsuch that\nνx=νfor a.e. xin Ω; in this case, we naturally identify ( νx)x∈Ωwith ν.\nIfd=n×mand the sequence ( zj) satisfies zj= Dwjfor some ( wj)⊂W1,∞(Ω,Rn) then we\nsay that νis agradient Young measure .\nThe set of homogeneous gradient Young measures supported on K⊂Rn×mis denoted by\nMqc(K); whenever K=Rn×mwe suppress it from the notation. The justification for the ch oice\nof the superscript “qc” will become clear at the end of the nex t theorem, which collects some\nfundamental properties of Young measures.\nTheorem 4.2. Young measures have the following properties:\n(i) Compactness: if(zj)⊂L∞(Ω,Rd)is bounded then, up to a non-relabeled subsequence, (zj)\ngenerates a Young measure.\n(ii) Strong convergence: assume (zj)generates a Young measure (νx)x∈Ωand zj∗⇀ z in\nL∞(Ω,Rd). Then zj→zinLpfor all p <∞if and only if νx=δz(x)for a.e. x∈Ω.\n(iii) Support: assume (zj)generates a Young measure (νx)x∈Ω. For a compact subset K⊂Rd\nwe have dist(zj, K)→0inL1(Ω)if and only if supp νx⊆Kfor a.e. x∈Ω.\nGradient Young measures have the following additional prop erties:\n(iv) Boundary conditions: if(νx)x∈Ωis a gradient Young measure then it is generated by a\nsequence (Dwj)with supp( wj−w)⋐Ω, where Dw(x)≡νx.\n(v) Localization: if(νx)x∈Ωis a gradient Young measure then νx∈ Mqcfor a.e. x∈Ω.\n(vi) Duality with quasiconvexity: a probability measure νsupported on a compact subset K⊂\nRn×mis in Mqc(K)if and only if f(ν)≤ /a\\}bracketle{tf, ν/a\\}bracketri}htfor all quasiconvex integrands f.\nWe recall that an integrand f:Rn×m→Ris said to be quasiconvex if\nf(A)≤ \nΩf(A+ Dϕ) dxfor all A∈Rn×m,allϕ∈C∞\nc(Ω,Rn). (4.3)\nA typical example of quasiconvex integrands are the minors, which in fact satisfy ( 4.3) with\nequality. This implies that, for any ν∈ Mqcand any minor M, we have\nM(ν) =/a\\}bracketle{tν, M/a\\}bracketri}ht. (4.4)\nFor a compact set K⊂Rn×mwe can define its quasiconvex hull Kqcas the set of points which\ncannot be separated from Kby a quasiconvex function; a set is said to be quasiconvex if K=Kqc.\nBy [29, Theorem 4.10], the quasiconvex hull coincides with\nKqc={ν:ν∈ Mqc(K)}. (4.5)\n5Whenever Kis non-compact, we set Kqc≡/uniontext\nr>0[K∩Br(0)]qc.\n4.2 Stability and compactness of approximate solutions\nWe now return to the case m= 4, n= 2 and to the set Kgdefined in ( 1.4). We begin with\nsome technical lemmas:\nLemma 4.3. LetΩ⊂R2be a simply connected domain. Let (uj)⊂W1,∞(Ω,R2)be a bounded\nsequence and g∈C1(R). The following are equivalent:\n(i)there is (vj)⊂W1,∞(Ω,R2)bounded with dist(D wj, Kg)→0inL1, where wj= (uj, vj);\n(ii)there is (vj)⊂W1,∞(Ω,R2)bounded such that g′(det D uj)Duj−Dvj→0inL1;\n(iii) ( 1.3)holds for a bounded sequence (Fj)⊂L∞(Ω,R2×2).\nProof. For a bounded sequence ( wj)⊂W1,∞(Ω,R4), one has dist(D wj, Kg)→0 in L1(Ω) if\nand only if f(Dwj)→0 in L1(Ω), where f:R4×2→Ris any continuous function such that\n{f= 0}=Kg, see e.g. [ 29] for a similar argument. In particular, taking\nf:/parenleftBigg\nA\nB/parenrightBigg\n/mapsto→ |g′(detA)A−B|,\nwe obtain the equivalence between (i)and(ii). Note also that, by linearity of the cofactor, (ii)is\nequivalent to\ng′(det D uj) cof(D uj)T−cof(D vj)T→0 inL1(Ω).\nIn particular, if this convergence holds, then by ( 2.2) we may take Fjto be the quantity above,\nproving that (ii)=⇒(iii). The converse follows from the fact that, if ( 1.3) holds, then\ng′(det D uj) cof(D uj)T−Fj\nis a bounded divergence-free matrix field, which can therefo re be written as cof(D vj)Tfor some\nLipschitz vj:Ω→R2, since Ω is simply connected. Then the convergence in (ii)holds since\nFj→0 inL1(Ω).\nLemma 4.4. Consider domains Ω2⋐Ω1⋐Ω. Let A∈R2×2be non-singular and assume that\nuj→LAinC0(Ω), u j=LAon∂Ω. (4.6)\nThere exists j0=j0(Ω1,Ω2)>0such that following hold:\n(i)uj(Ω2)⋐LA(Ω1)⋐LA(Ω), for all j≥j0;\n(ii)for all y∈uj(Ω2)we have u−1\nj(y)⊆Ω1, for all j≥j0;\n(iii) lim j|uj(Ω2)|=|LA(Ω2)|.\nProof. We will just prove (iii), as(i)and(ii)are much simpler. To prove (iii), note that by\nuniform convergence, for all ε >0, there exists j1such that\nuj(Ω2)⊂Bε(LA(Ω2)),∀j≥j1, (4.7)\nwhere Bε(E) denotes the εneighborhood of the set E⊂R2. In turn, ( 4.7) implies\nlim sup\nj→∞|uj(Ω2)| ≤ |LA(Ω2)|, (4.8)\n6since |∂Ω2|= 0. To show the opposite inequality, consider any domain Ω 3⋐Ω2. We claim that\nthere exists j2=j2(Ω3)>0 such that if j≥j2then\nLA(Ω3)⊆uj(Ω2). (4.9)\nTo see this, we start by noticing that, for all j∈N,\nLA(Ω3)⊆uj(Ω). (4.10)\nThis can be seen using the properties of the topological degr ee, see e.g. [ 18, Theorems 2.1 and 2.4],\nexploiting the fact that uj=LAon∂Ω for all j. Suppose now that ( 4.9) is false, so that, up to\nnon-relabeled subsequences, we find yj∈LA(Ω3) with yj/∈uj(Ω2) for all j. We can additionally\nsuppose that yj→¯y∈LA(Ω3). Due to ( 4.10), we see that there exists a sequence ( xj)⊂Ω such\nthat uj(xj) =yj. Our contradiction assumption implies xj/∈Ω2for all j. In particular, up to a\nfurther subsequence, we see that xj→¯x∈Ω\\Ω2⊂Ω\\Ω2.AsΩ3⊂Ω2andAis nonsingular,\nthe uniform convergence of uj→LAreadily yields a contradiction. Thus, ( 4.9) holds, and we\ndeduce in turn that\n|LA(Ω3)| ≤lim inf\nj|uj(Ω2)|.\nDue to the arbitrarity of Ω 3, combining the latter with ( 4.8) we conclude the validity of (iii).\nWe can now show the main result of this section:\nTheorem 4.5. LetΩ⊂R2be a simply connected domain and let (uj),(vj)be sequences bounded\ninW1,∞(Ω,R2). Let A, B ∈R2×2and assume the following conditions:\nuj=LAandvj=LBon∂Ω,\nuj∗⇀ L Aandvj∗⇀ L BinW1,∞,\n(Duj,Dvj)generates ν∈ Mqc(Kg),\ng′(det D uj)Duj−Dvj→0inL1.(4.11)\nThen we have\nsupp ν⊆/braceleftBigg/parenleftBigg\nM\nN/parenrightBigg\n∈Kg:detM= det A/bracerightBigg\nand, in particular, det D uj→detAinL1and(A, B) =ν∈Kg.\nProof. Note that ( 4.11) implies that ( uj, vj)→(LA, LB) inC0(Ω), so Lemma 4.4is applicable.\nWe now define some notation. Consider the projections π12, π34:R4×2→R2×2defined by\nπ12:/parenleftBigg\nA\nB/parenrightBigg\n/mapsto→A, π 34:/parenleftBigg\nA\nB/parenrightBigg\n/mapsto→B.\nLetν12, ν34be the homogeneous gradient Young measures generated respe ctively by (D uj) and\n(Dvj); we have the pushforward relations ν12= (π12)#νandν34= (π34)#ν. We split the rest of\nthe proof into several steps.\nStep 1. Let ( uj),(vj) be sequences satisfying the hypotheses in ( 4.11). By Lemma 4.3we see\nthat ( 1.3) holds. Given M∈R2×2, testing ( 1.3) against ϕj(x) =M(uj(x)−Ax) and using ( 2.1),\nwe see that\no(1) =ˆ\nΩ/a\\}bracketle{tDϕj, Fj/a\\}bracketri}ht=ˆ\nΩg′(det D uj)/a\\}bracketle{tDϕj,cof(D uj)T/a\\}bracketri}htdx\n7=ˆ\nΩg′(det D uj)/bracketleftBig\ndet D uj/a\\}bracketle{tM,Id/a\\}bracketri}ht − /a\\}bracketle{tMA, cof(D uj)T/a\\}bracketri}ht/bracketrightBig\ndx,\nwhere o(1) denotes a quantity which vanishes as j→ ∞ . Equivalently,\n/angbracketleftbigg\nM,/parenleftbiggˆ\nΩg′(det D uj) det D ujdx/parenrightbigg\nId/angbracketrightbigg\n=/angbracketleftbigg\nM,/parenleftbiggˆ\nΩg′(det D uj) cof(D uj)Tdx/parenrightbigg\nAT/angbracketrightbigg\n+o(1)\nand, since Mis arbitrary, we see that\nlim\nj→∞/parenleftbiggˆ\nΩg′(det D uj) det D ujdx/parenrightbigg\nId = lim\nj→∞/parenleftbiggˆ\nΩg′(det D uj) cof(D uj)Tdx/parenrightbigg\nAT\n= lim\nj→∞/parenleftbiggˆ\nΩcof(D vj)Tdx/parenrightbigg\nAT\n= cof( B)TAT,(4.12)\nwhere in the second line we used the convergence hypothesis i n (4.11) and in the last line we used\nthe linearity of cof and the boundary conditions on vj. Similarly, we also have\ndetB= \nΩdet D vjdx= lim\nj→∞ \nΩg′(det D uj)2det D ujdx, (4.13)\nthe first equality following because det is a null Lagrangian and the second by the last hypothesis\nin (4.11).\nStep 2. In this step we deal with the case where det A= 0. By taking the determinant in\n(4.12) we get\nˆ\nR2×2g′(detM) det( M) dν12(M) = lim\nj→∞ \nΩg′(det D uj) det D ujdx= 0,\nthus det( M) = 0 for ν12-a.e. M, ast/mapsto→g′(t)tis non-negative and vanishes only at zero, according\nto (2.4). Hence the desired claim on the support of ν12holds and the latter claims follow exactly\nas in the case where det A/\\e}atio\\slash= 0, which we will detail in Step 10 below.\nStep 3. For the rest of the proof we assume that det A/\\e}atio\\slash= 0, so that LA(Ω) is a domain. Let\nη∈C∞\nc(R2,R2) be such that η|LA(∂Ω)= 0. Thus η◦uj∈W1,∞\n0(Ω,R2) and so, testing ( 1.3)\nagainst this map and using ( 2.1) and ( 2.4), we get\nˆ\nΩ/a\\}bracketle{tD(η◦uj), Fj/a\\}bracketri}htdx=ˆ\nΩg′(det D uj)/a\\}bracketle{tDη◦ujDuj,cof(D uj)T/a\\}bracketri}htdx\n=ˆ\nΩg′(det D uj) div η◦ujdet D ujdx\n=ˆ\nΩ|g′(det D uj)|(divη◦uj)|det D uj|dx\n=ˆ\nR21uj(Ω)(y)Suj(y) divη(y) dy,(4.14)\nwhere we used the area formula (cf. [ 27, Corollary 8.11]) and we defined, for a.e. y∈uj(Ω),\nSuj(y)≡/summationdisplay\nx∈u−1\nj(y)|g′(det D uj(x))|. (4.15)\nLet us note in passing that, since (D uj) is bounded in L∞, we have\nsup\njˆ\nR21uj(Ω)(y)Suj(y) dy= sup\njˆ\nΩ|g′(det D uj(x))||det D uj(x)|dx≤C. (4.16)\n8Step 4. To deal with the left-hand side of ( 4.14), we consider the push-forward measure\nmj≡(uj)#(FjDuT\njdx), which we can write as mj=Pjαjfor a finite, positive measure αjand\nfor an αj-measurable matrix field Pj:R2→R2×2with |Pj(x)|= 1 for αj-a.e. x∈Ω. Then,\naccording to the general change of variables formula [ 27, Proposition 2.14], we have\nˆ\nΩ/a\\}bracketle{tD(η◦uj), Fj/a\\}bracketri}htdx=ˆ\nΩ/a\\}bracketle{tDη◦uj, FjDuT\nj/a\\}bracketri}htdx=ˆ\nR2/a\\}bracketle{tDη(y), Pj(y)/a\\}bracketri}htdαj(y) (4.17)\nfor all η∈C∞\nc(R2,R2). We also notice that, by ( 1.3), we have\n/bardblmj/bardblM=αj(R2)≤ /bardblFjDuT\nj/bardbl1≤ /bardblFj/bardbl1/bardblDuj/bardbl∞→0. (4.18)\nThus, combining ( 4.14) and ( 4.17), we have shown that, for η∈C∞\nc(R2) such that η|LA(∂Ω)= 0,\nˆ\nR21uj(Ω)Sujdivηdy=ˆ\nR2/a\\}bracketle{tDη, P j/a\\}bracketri}htdαj. (4.19)\nStep 5. By taking fj≡1uj(Ω)Suj≥0, note that ( 4.19) shows that\nDfj= div( Pjαj) in D′(LA(Ω)). (4.20)\nAt this point we will employ Allard’s strong constancy lemma [2], which we state here in a\nsomewhat simpler form:\nLemma 4.6 (Strong constancy lemma) .Let(fj)⊂L1(˜Ω)be a bounded sequence such that\nfj≥0, Dfj= div Gjfor some sequence ||Gj||M(˜Ω)→0.\nThen there is a constant c≥0such that fj→cinL1\nloc(˜Ω).\nTake arbitrary domains Ω 2⋐Ω1⋐Ω. Due to Lemma 4.4(i) , for jlarge enough we have\nuj(Ω2)⋐LA(Ω1)⋐LA(Ω). Thus, by Lemma 4.6, which is applicable by ( 4.15), (4.16), (4.18)\nand ( 4.20), we see that\nSuj→cinL1(LA(Ω1)). (4.21)\nStep 6. Fixε >0 and consider the set Ej≡Ω2∩ {|g′(det D uj)| ≥c+ε}. Again by Lemma\n4.4(i) we have uj(Ej)⊂LA(Ω1), so using ( 4.21) and applying the area formula twice, we get\ncˆ\nEj|det D uj|dx≥c|uj(Ej)|=ˆ\nuj(Ej)Suj(y) dy+o(1)\n≥ˆ\nEj|g′(det D uj)||det D uj|dx+o(1)\n≥(c+ε)ˆ\nEj|det D uj|dx+o(1).\nThis implies that\nlim\nj→∞ˆ\nEj|det D uj(x)|dx= 0.\nSince g′(0) = 0 and g′is injective, there is ˜ ε >0 such that |det D uj| ≥˜εa.e. in Ej, and it follows\nthat lim j→∞|Ej| →0. Since Ω 2⋐Ω is arbitrary, by taking a compact exhaustion of Ω we deduce\n9that\nlim\nj→∞|{|g′(det D uj)| ≥c+ε}|= 0. (4.22)\nStep 7. Recall from Lemma 4.4(ii) that for all jlarge enough and all y∈uj(Ω2)⋐LA(Ω1)\nwe have u−1\nj(y)⋐Ω1. Hence, applying the area formula twice, for such jwe have\nˆ\nΩ1|g′(det D uj(x))||det D uj(x)|dx=ˆ\nuj(Ω1)/summationdisplay\nx∈u−1\nj(y)∩Ω1|g′(det D uj(x))|dy\n≥ˆ\nuj(Ω2)/summationdisplay\nx∈u−1\nj(y)∩Ω1|g′(det D uj(x))|dy\n=ˆ\nuj(Ω2)/summationdisplay\nx∈u−1\nj(y)|g′(det D uj(x))|dy\n=ˆ\nuj(Ω2)Suj(y) dy≥ˆ\nΩ2|g′(det D uj(x))||det D uj(x)|dx.\nThus, since ν12is a homogeneous gradient Young measure, using ( 2.4) we get\n|Ω1|ˆ\nR2×2g′(detM) det Mdν12(M)≥lim\nj→∞ˆ\nuj(Ω2)Sujdy≥ |Ω2|ˆ\nR2×2g′(detM) det Mdν12(M).\nBy (4.21) and Lemma 4.4(i) ,(iii)we must also have\nlim\nj→∞ˆ\nuj(���2)Suj(y) dy= lim\nj→∞ˆ\nuj(Ω2)(Suj(y)−c) dy+clim\nj|uj(Ω2)|=c|detA||Ω2|.\nSince Ω 2⋐Ω1⋐Ω are arbitrary domains, combining the last two displays yie lds\nc|detA|=ˆ\nR2×2g′(detM) det Mdν12(M) (4.23)\nIf we take determinants in ( 4.12) and use this identity, since det A/\\e}atio\\slash= 0, we also see that\nc2detA= det B. (4.24)\nStep 8. Note that ( 4.22) shows that ν12is supported in {M∈R2×2:|g′(detM)| ≤c}.\nFurthermore, recall ( 2.4). These facts, combined with ( 4.13) and ( 4.23), yield\n|detB|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nR2×2g′(detM)[g′(detM) det( M)] dν12(M)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤cˆ\nR2×2g′(detM) det( M) dν12(M) =c2|detA|.\nBy (4.24) we see that we must have equality in the inequality above. It follows that\ng′(detM) = sign(det A)c forν12-a.e. Mwith g′(detM) det M/\\e}atio\\slash= 0.\nAs in Step 2, g′(detM) det M= 0 if and only if det M= 0. Note that supp ν12/\\e}atio\\slash⊆ {M:detM=\n0}, for otherwise by ( 4.4) we would have det A= det ν12= 0, in contradiction to our assumption\nonA. Hence c/\\e}atio\\slash= 0 and there exists ˜ c/\\e}atio\\slash= 0 such that\nsupp ν12⊆ {M:det(M) = 0 } ∪ { M:det(M) = ˜c}. (4.25)\n10Step 9. In this step we prove that in fact supp ν12⊆ {det = ˜ c}. First, note that supp ν⊆Kg\nby Theorem 4.2(iii) , Lemma 4.3and our hypotheses ( 4.11). Thus, by ( 4.25), we have\nsupp ν34⊆ {0} ∪ { g′(˜c)M:M∈R2×2,detM= ˜c}. (4.26)\nSince ν34has bounded support, ( 4.26) implies that ν34is aquasiregular gradient Young measure ,\nnamely it is supported on {M:|M|2≤Kdet(M)}for some K, up to changing orientations\nso that ˜ c >0. The 0-1 law for such measures [ 3, Theorem 1.3] asserts that either ν34=δ0or\nelseν34({0}) = 0. The former case cannot happen, as otherwise by Theorem 4.2(ii) we would\nget D vj→0 in L1. In turn, this would imply B= 0, which is in contradiction with ( 4.24),\nsince c,detA/\\e}atio\\slash= 0. Hence we instead have ν34({0}) = 0 and so, by ( 4.26), it must be that\nsupp ν34⊆ {g′(˜c)M:detM= ˜c}. This implies that ν12is supported in {det = ˜ c}, as desired.\nStep 10. By the previous step and ( 4.4), we have\n˜c=/a\\}bracketle{tν12,det/a\\}bracketri}ht= det ν12= det A,\nhence the claimed support condition on νholds. For the final statements note that the previ-\nous step shows that δdetAis the Young measure generated by the sequence (det D uj), and so\nTheorem 4.2(ii) shows that det D uj→detAinL1(Ω). This strong convergence together with\nthe assumption g′(det D uj)Duj−Dvj→0 inL1imply that the three sequences ( g′(detA)Duj),\n(g′(det D uj)Duj), (D vj) have the same weak limit, hence g′(detA)A=B, i.e. ( A, B)∈Kg.\nProof of Theorem 1.3.Since ( uj)⊂W1,∞is bounded and Ω is also bounded, it suffices to\nshow that det D uj→det D uinL1\nloc(Ω), as this implies the claimed Lp-convergence, which in turn\nimplies that usolves ( 1.2). Thus we can without loss of generality assume that Ω is a bal l.\nIf (uj) satisfies ( 1.3) then, according to Lemma 4.3, there is a bounded sequence ( vj)⊂\nW1,∞(Ω,R2) such that wj= (uj, vj) satisfies dist(D wj, Kg)→0 inL1(Ω). We want to show that\nevery subsequence of ( uj) admits a further subsequence whose Jacobians converge str ongly and\nthus, up to passing to non-relabeled subsequences, we can as sume that (D wj) generates a gradient\nYoung measure ( νx)x∈Ωaccording to Theorem 4.2(i) . Let us write µx≡(π12)#νx. By Theorem\n4.2(iii) ,(v)for a.e. xwe have νx∈ Mqc(Kg) and by Theorem 4.2(iv) and Lemma 4.3we see\nthat, for a.e. x,νxcan be generated by a bounded sequence (D ˜ wj) = (D˜ uj,D˜vj)⊂W1,∞(Ω,R4)\nsatisfying ( 4.11). Thus, applying Theorem 4.5, we see that there is c:Ω→Rsuch that µxis\nsupported in {M∈R2×2:detM=c(x)}for a.e. x∈Ω. In light of ( 4.4) and the weak continuity\nof Jacobians det D uj∗⇀det D uinL∞(Ω), we must have\nc(x) = det( µx) =/a\\}bracketle{tµx,det/a\\}bracketri}ht= det D u(x) for a.e. x∈Ω.\nHence the Young measure generated by (det D uj) is ( δdet(D u(x)))x∈Ω.By Theorem 4.2(ii) , this\nimplies the desired strong convergence.\nProof of Corollary 1.4.This follows from ( 4.5), Theorem 4.2(iii) ,(iv), Lemma 4.3and also\nTheorem 4.5.\nReferences\n[1] G. Alberti, S. Bianchini, and G. Crippa. A uniqueness res ult for the continuity equation in two\ndimensions. J. Eur. Math. Soc. , 16(2):201–234, 2014.\n11[2] W. K. Allard. An integrality theorem and a regularity the orem for surfaces whose first variation with\nrespect to a parametric elliptic integrand is controlled. I nProc. Symp. Pure Math , volume 44, pages\n1–28. 1986.\n[3] K. Astala and D. Faraco. Quasiregular mappings and Young measures. Proc. R. Soc. Edinburgh Sect.\nA Math. , 132(05):1045, 2002.\n[4] J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech.\nAnal. , 63(4):337–403, 1977.\n[5] S. Bianchini and N. A. Gusev. Steady Nearly Incompressib le Vector Fields in Two-Dimension: Chain\nRule and Renormalization. Arch. Ration. Mech. Anal. , 222(2):451–505, 2016.\n[6] J. A. Carrillo and S. Lisini. On the asymptotic behavior o f the gradient flow of a polyconvex functional.\nNonlinear Partial Differ. equations hyperbolic wave Phenom ., 526:37–51, 2010.\n[7] A. Cellina and S. Zagatti. An Existence Result in a Proble m of the Vectorial Case of the Calculus of\nVariations. SIAM J. Control Optim. , 33(3):960–970, 1995.\n[8] B. Dacorogna. A relaxation theorem and its application t o the equilibrium of gases. Arch. Ration.\nMech. Anal. , 77(4):359–386, 1981.\n[9] B. Dacorogna and P. Marcellini. Existence of minimizers for non-quasiconvex integrals. Arch. Ration.\nMech. Anal. , 131(4):359–399, 1995.\n[10] B. Dacorogna and J. Moser. On a partial differential equa tion involving the Jacobian determinant.\nAnn. Inst. Henri Poincar´ e (C) Anal. Non Lin´ eaire , 7(1):1–26, 1990.\n[11] C. De Lellis, G. De Philippis, B. Kirchheim, and R. Tione . Geometric measure theory and differential\ninclusions. Ann. la Fac. des Sci. Toulouse Math´ ematiques , 30(4):899–960, 2021.\n[12] G. De Philippis, A. Guerra, and R. Tione. Unique continu ation for differential inclusions. arXiv\n2312.05022 , 2023.\n[13] G. De Philippis and F. Rindler. On the structure of A-free measures and applications. Ann. Math. ,\n184(3):1017–1039, 2016.\n[14] R. J. DiPerna and P. L. Lions. Ordinary differential equa tions, transport theory and Sobolev spaces.\nInvent. Math. , 98(3):511–547, 1989.\n[15] L. C. Evans. Quasiconvexity and partial regularity in t he calculus of variations. Arch. Ration. Mech.\nAnal. , 95(3):227–252, 1986.\n[16] L. C. Evans, O. Savin, and W. Gangbo. Diffeomorphisms and nonlinear heat flows. SIAM J. Math.\nAnal. , 37(3):737–751, 2006.\n[17] D. Faraco and L. Sz´ ekelyhidi. Tartar’s conjecture and localization of the quasiconvex hull in R2×2.\nActa Math. , 200(2):279–305, 2008.\n[18] I. Fonseca and W. Gangbo. Degree theory in analysis and applications . Oxford University Press, 1995.\n[19] A. Guerra, L. Koch, and S. Lindberg. Energy minimisers w ith prescribed Jacobian. Arch. Ration.\nMech. Anal. , 242:1059–1090, 2021.\n[20] A. Guerra, L. Koch, and S. Lindberg. The Dirichlet probl em for the Jacobian equation in critical and\nsupercritical Sobolev spaces. Calc. Var. Partial Differ. Equ. , 60:14 pp, 2021.\n[21] J. Hirsch, C. Mooney, and R. Tione. On the Lawson-Osserm an conjecture. arXiv:2308.04997 , pages\n1–18, 2023.\n[22] J. Hirsch and R. Tione. On the constancy theorem for anis otropic energies through differential\ninclusions. Calc. Var. Partial Differ. Equ. , 60(3):1–52, 2021.\n[23] T. Iwaniec and G. Martin. Geometric Function Theory and Non-linear Analysis . Clarendon Press,\n2001.\n[24] B. Kirchheim, S. M¨ uller, and V. ˇSver´ ak. Studying Nonlinear pde by Geometry in Matrix Space . In\nGeom. Anal. Nonlinear Partial Differ. Equations , pages 347–395. Springer, Berlin, Heidelberg, 2003.\n[25] B. Kirchheim and L. Sz´ ekelyhidi. On the gradient set of Lipschitz maps. J. f¨ ur die reine und Angew.\nMath. (Crelles Journal) , 2008(625):215–229, 2008.\n[26] J. Kristensen and G. Mingione. The Singular Set of Lipsc hitzian Minima of Multiple Integrals. Arch.\nRation. Mech. Anal. , 184(2):341–369, 2007.\n[27] F. Maggi. Sets of Finite Perimeter and Geometric Variational Problem s. Cambridge University Press,\n122012.\n[28] E. Mascolo and R. Schianchi. Existence theorems for non convex problems. J. Math. Pures Appl. ,\n9(62):349–359, 1983.\n[29] S. M¨ uller. Variational models for microstructure and phase transitions. In Calc. Var. Geom. Evol.\nProbl. , pages 85–210. Springer, Berlin, Heidelberg, 1999.\n[30] S. M¨ uller and V. ˇSver´ ak. Convex integration for Lipschitz mappings and cou nterexamples to regular-\nity.Ann. Math. , 157(3):715–742, 2003.\n[31] F. Rindler. Calculus of Variations . Universitext. Springer International Publishing, 2018.\n[32] V. ˇSver´ ak. Rank-one convexity does not imply quasiconvexity .Proc. R. Soc. Edinburgh Sect. A Math. ,\n120(1-2):185–189, 1992.\n[33] V. ˇSver´ ak. On Tartar’s conjecture. Ann. Inst. Henri Poincar´ e (C) Anal. Non Lin´ eaire , 10(4):405–412,\n1993.\n[34] L. Sz´ ekelyhidi. The Regularity of Critical Points of P olyconvex Functionals. Arch. Ration. Mech.\nAnal. , 172(1):133–152, 2004.\n[35] A. Taheri. On critical points of functionals with polyc onvex integrands. J. Convex Anal. , 9(1):55–72,\n2002.\n[36] R. Tione. Minimal graphs and differential inclusions. Commun. Partial Differ. Equations , 46(6):1162–\n1194, 2021.\n[37] R. Tione. Critical Points of Degenerate Polyconvex Ene rgies. SIAM J. Math. Anal. , 55(4):3205–3225,\n2023.\n[38] B. Yan. On a class of special Euler–Lagrange equations. Proc. R. Soc. Edinburgh Sect. A Math. ,\npages 1–24, 2023.\n13" }, { "title": "2401.16320v1.Prepare_Non_classical_Collective_Spin_State_by_Reinforcement_Learning.pdf", "content": "arXiv:2401.16320v1 [quant-ph] 29 Jan 2024Prepare Non-classical Collective Spin State by Reinforcem ent Learning\nX. L. Zhao,1Y. M. Zhao,1M. Li,1T. T. Li,1Q. Liu,1S. Guo,1and X. X. Yi2,∗\n1Qingdao University of Technology, 0532, Qingdao, Shandong , China\n2Center for Quantum Sciences and School of Physics,\nNortheast Normal University, Changchun 130024, China\n(Dated: January 30, 2024)\nWe propose a scheme leveraging reinforcement learning to en gineer control fields for generating\nnon-classical states. It is exemplified by the application t o prepare spin squeezed state for an open\ncollective spin model where a linear control term is designe d to govern the dynamics. The reinforce-\nment learning agent determines the temporal sequence of con trol pulses, commencing from coherent\nspin state in an environment characterized by dissipation a nd dephasing. When compared to con-\nstant control scenarios, this approach provides various co ntrol sequences maintaining collective spin\nsqueezing and entanglement. It is observed that denser appl ication of the control pulses enhances\nthe performance of the outcomes. Furthermore, there is a min or enhancement in the performance by\nadding control actions. The proposed strategy demonstrate s increased effectiveness for larger sys-\ntems. And thermal excitations of the reservoir are detrimen tal to the control outcomes. It should\nbe confirmed that this is an open-loop strategy by closed-loo p simulation, circumventing collapse of\nquantum state induced by measurements. Thanks to the flexibl e replaceability of the optimization\nmodules and the controlled system, this research paves the w ay for its application in manipulating\nother quantum systems.\n∗yixx@nenu.edu.cn2\nINTRODUCTION\nPrecise measurement for physical quantities has propelled advanc es in physics. The application of non-classical\nquantum states can open pathways toward precise measurement . Spin squeezed states are such a kind of pertinent\ncandidates characterized by reduced uncertainty in a collective sp in component. Entanglement typically arises con-\ncomitantly with spin squeezing, as a result of the interatomic interac tions within an ensemble [ 1,2]. Decreasing the\nvariance of an observable leads to an escalation in measurement sen sitivity that transcends the standard quantum\nlimit within the realm of quantum-enhanced metrology. Such squeeze d states can be used to enhance the performance\nof homodyne interferometers [ 3,4], magnetometers [ 5], and atomic clocks [ 6]. The evidence delineating the association\nbetween the sensitivity of phase estimation and entanglement was d emonstrated in collective spin model [ 7].\nMany schemes have been proposed to prepare spin squeezed stat es based on the nonlinear squeezing term in the\nHamiltonian [ 1,8]. For example, quantum non-demolition measurement can be used to generate spin squeezed\nstates [9]. Some proposals are based on nonlinear interactions among the indiv idual elements in Bose-Einstein con-\ndensates [ 10]. Theoretically, coherent control has been proposed to produce spin squeezed states [ 11]. Long-lasting\nand extreme spin squeezed states is pursued all the way.\nMachine-learning techniques are emerging as an effective tool in phy sics [12], and among them, reinforcement\nlearning (RL) offers the potential to optimize control for high-dime nsional, multistage processes in complex scenarios.\nDeep reinforcement learning (DRL) can provide control strategy to engineer the dynamics as long as the evolution\nfollows certain differential equations. In physics, many optimal pro blems can be treated as control problems of finding\nmeans to steer the systems to achieve a certain target. The sear ch for optimal control field can be formulated as RL\ntask [13–15]. The estimation of the precession frequency of a dissipative partic le was enhanced by adding a linear\ncontrol in the form of an additional controlled magnetic field [ 16].\nWe propose a reinforcement-learning based control scheme to de sign control fields to prepare non-classical spin\nstates. The agent is trained to produce a sequence of square puls es that steers the system evolving to squeezed states\nunder the domain of Lindblad master equation. The generalization pe rformance of the proposed control scheme was\nevaluated across a range of control parameters, system sizes, and thermal excitations.\nThis paper is organized as follows: In Sec., we delineate the reinforce ment-learning-based framework for the prepa-\nration of nonclassical states. In Sec., the RL module employed within the control scheme is shown. In Sec., we\nexplicate the quantum model for the generation of spin-squeezed states. In Sec., we present the procedure to prepare\nsqueezed states in the open collective spin system via reinforcemen t learning. In Sec., we check the performance of\nthis method including the influence of the frequency of applying the c ontrol pulses, the granularity of control actions,\nsystem scalability across various sizes, and the impact of thermal e xcitation. Finally, we conclude in Sec. with a\nsynopsis of the findings and potential avenues for future resear ch.\nCONTROL SCHEME\nTaking a cue from Lyapunov control strategies [ 17–21], we harness machine learning to design control fields that\nfacilitate the preparation of nonclassical states. In the presenc e of a control field, the general total Hamiltonian for\nthe quantum system reads: ˆH=ˆH0+/summationtextM\nm=1fm(t)ˆHm, whereˆH0is the free Hamiltonian and mdenotes the number\nof external control Hamiltonians ˆHm.fm(t) is the corresponding control field designed by RL in this work. It sh ould\nbe confirmed that [ ˆH0,ˆHm]/negationslash= 0, otherwise, the influence of the control Hamiltonians can be sub sumed into the free\nHamiltonian.\nThis scheme is an open-loop application strategy, wherein the contr ol fields are devised through the emulation of\na closed-loop process, as shown in Fig. 1. Under the control designed by machine learning agent, the syste m would\nbe steered to a set of states meeting the control target. Notab ly, the RL agents can be supplanted by alternative\noptimization modules tailored for the target. The proposed contro l scheme is applicable to other dynamical systems\ngoverned by certain differential equations. To illustrate the efficac y of the scheme, we have applied it to engineer a\ncollective spin system.\nMACHINE LEARNING TOOLS: REINFORCEMENT LEARNING\nInitiating with no priori knowledge about the dynamics of the system under control, RL adopts a trial-and-error\nmanner to iteratively learn a mapping that determines an action depe nding on different states under certain rules,\nnamely, aimed at maximizing the accumulated reward over discrete tim e instances t. Rewards are obtained using an3\n濴濺濸瀁瀇 瀂瀃濸瀁澳瀄瀈濴瀁瀇瀈瀀澳瀆瀌瀆瀇濸瀀 \n瀅濸瀃濿濴瀌澳瀀濸瀀瀂瀅瀌 \n瀇瀅濴濼瀁濼瀁濺澳瀁濸瀇瀊瀂瀅濾 瀇濴瀅濺濸瀇澳瀁濸瀇瀊瀂瀅濾 瀂濵瀆濸瀅瀉濴瀇濼瀂瀁澳 o୲瀅濸瀊濴瀅濷澳 r୲濴濶瀇濼瀂瀁澳 a୲濝濢濨濙濦濕濗濨濝濣濢 \n濡濝濢濝澡濖濕濨濗濜 瀆瀇瀂瀅濴濺濸 瀆濴瀀瀃濿濸 \n瀂瀃瀇濼瀀濼瀍濴瀇濼瀂瀁 t\nt+1 \nFIG. 1. In RL, Q function is defined as Qπ(st,at) =r(st) +E{(st+i,at+i)|π}∞\ni=1/bracketleftbig/summationtext∞\ni=1γi\nqr(st+i)/bracketrightbig\nwhereγq(0< γq<1,\nusually close to 1) is a discount factor, and the expectation Eis taken over trajectories of the action and the state of\nthe controlled system [ 22,23]. The optimal policy π∗maximizing Qvalue satisfies the Bellman equation Qπ∗(st,at) =\nr(st) +γqEst+1/bracketleftBig\nmaxat+1Qπ∗(st+1,at+1)/bracketrightBig\n.Once the equation of Qπ∗were satisfied, the optimal policy π∗is determined.\nThe schematic of the training procedure involves: 1. The age nt provides actions atto steer the open quantum system (the\n‘environment’ in RL), and receives observations σtand rewards rt. 2. These experiences are stored in the replay memory as\nstate-action-reward combination of map. 3. The agent rando mly samples batches of the combinations from the replay memo ry\nto train the neural network to minimize the difference betwee n the predicted Q-values and the target Q-values. 4. The targ et\nQ-values are computed by considering the immediate reward a nd the estimated maximum future rewards, which contributes to\nstabilize the training process based on the next state, usin g the Q-network weights from the previous iteration. 5. The p rocess\nincluding interaction, experience replay, and network upd ates is repeated to iteratively improve the Q-function appr oximation.\nevaluation rule that aligns with the control objectives. An appropr iate rewards-evaluation rule that favors particular\nstate-action mappings can enhance the control performance. D ecision-making executed by the agent entails selecting\nactionsat=π(st) that affect the system changing from a state (not specifically ref erring to quantum states, but a\ngeneral state characterized by various quantities of a controlled system)sttost+1, withπdenoting the policy being\nrefined.\nQ-learning realizes RL through a Q function that represents the expected total future reward of a policy π. The\noptimal policy π∗with maximized Qfunction satisfies Bellman equation [ 22,23]. Deep Q-learning uses a deep\nfeed-forward neural network to approximate this function [ 24–26], and the network is called a deep Q network (DQN)\ndenoted by Wθ(s,a), whereθrepresents the net parameters need to be adjusted by training, and the controlled system\nstatesis the network input. Assuming that the space of actions ais discrete, the DQN outputs a value for each\nchoice of action a. DQN leverages extensive datasets that encapsulate the anticipa ted cumulative rewards consequent\nto specific actions within distinct states, facilitating the neural net work-based approximation of the conventional\naction-value function [ 27–32].\nIn this study, the DQN algorithm is incorporated into the reinforcem ent learning framework to determine an\noptimized controlstrategy. It shouldbe noted that, depending o nparticularneeds andconstraintsoftasks, alternative\noptimization algorithms could be employed in place of the DQN.\nCOLLECTIVE SPIN MODEL\nWe consider an ensemble of Nidentical two-level atoms with pseudo spin components ˆJα=1\n2/summationtextN\nk=1ˆσ(k)\nα, (α=\nx,y,z), where ˆσ(k)\nαis the Pauli operator for the k-th atom [ 33]. This is the symmetric scenario where the operations\ndone on the ensemble have identical impact on all the atoms. ˆJx,ˆJy,ˆJzfulfill the SU(2) commute relationship:\n[ˆJα,ˆJβ] =i/planckover2pi1ǫαβγˆJγ, whereǫαβγis the L´ evi-Civit` a symbol. The total collective spin length is specified byJ=N/2\nwhile the dimension of the Hilbert space is 2 J+ 1 =N+ 1. The collective spin can be mapped to its two-mode\nbosonic partner by Schwinger transformation: ˆJz=1\n2(ˆa†ˆa−ˆb†ˆb),ˆJ+= ˆa†ˆbandˆJ−= (ˆJ+)†, where ˆaandˆbare the\ntwo annihilation operators of two boson modes [ 34]. In this view, by mapping one mode to spin up and the other one\nto spin down, ˆJzreflects the population difference between the two modes in Ramsey interferometer [ 2,3,8].4\nTo examine the effectiveness of the proposal, we focus on a collectiv e spin system described by the Hamiltonian\nˆH//planckover2pi1=κˆJ2\nz+Ωx(t)ˆJx, (1)\nhereκindicates the strength of the interaction between the atoms and t he time scales as κt.κwould be taken as the\nunit (κ=1 herafter) and the natural units ( /planckover2pi1= 1) is used. ˆJ2\nzrefers to the one-axis twisting which induces spin squeez-\ning [8]. This squeezing provides the resource for quantum-enhanced me trology [2,8].ˆJxis the control Hamiltonian\ndescribing the magnetic field in x-direction, or the counterpart, the linear beam splitters in interfe rometers).\nIn contrast to the application of a constant control [ 11], the present proposal employs RL agent designing time-\ndependent control fields Ω x(t) to prepare nonclassical states. The temporally varying control field can represent the\noperational analogy of linear beam splitters in interferometric expe riments. Since [ ˆJ2\nz,ˆJx] =i/planckover2pi1(ˆJyˆJz+ˆJzˆJy), such a\nlinear control Hamiltonian can be used to steer the ensemble.\nSpin squeezing can be quantified by using parameters constructed by the expected values of collective spin opera-\ntors [1,2]. Upon the reduction of the variances beneath the standard quan tum limit threshold, the system is rendered\napplicable for precision metrology, along with amplifying the variance o f the orthogonal spin components. We should\nconfirm that the minimum squeezing parameter reads\nξ2\n⊥=Nmin(∆ˆJ2\n/vector n⊥)\n|/angbracketleftˆJs/angbracketright|2=N/bracketleftbigg/angbracketleftBig\nˆJ2\n/vector n1+ˆJ2\n/vector n2/angbracketrightBig\n−/radicalbigg/angbracketleftBig\nˆJ2\n/vector n1−ˆJ2\n/vector n2/angbracketrightBig2\n+/angbracketleft/bracketleftBig\nˆJ/vector n1,ˆJ/vector n2/bracketrightBig\n+/angbracketright2/bracketrightbigg\n2|/angbracketleftˆJs/angbracketright|2, (2)\nwhereˆJ/vector ni=(ˆJx,ˆJy,ˆJz)·/vector ni, (i= 1,2) and/vector n1= (−sinφ,cosφ,0),/vector n2= (cosθcosφ,cosθsinφ,−sinθ),ˆJs=(ˆJx,ˆJy,ˆJz)·\n(sinθcosφ,sinθsinφ,cosθ), whereθ= arccos(/angbracketleftJz/angbracketright\n|ˆJ|),φ= sign(/angbracketleftˆJy/angbracketright)arccos(/angbracketleftˆJx/angbracketright\n|ˆJ|sinθ),|ˆJ|=/radicalBig\n/angbracketleftˆJx/angbracketright2+/angbracketleftˆJy/angbracketright2+/angbracketleftˆJz/angbracketright2\n[1,2,35]. The direction /vector n⊥=/vector n1cosϕ+/vector n2sinϕwith\nϕ=\n\n1\n2arccos/parenleftBig\n−A√\nA2+B2/parenrightBig\nifB≤0,\nπ−1\n2arccos/parenleftBig\n−A√\nA2+B2/parenrightBig\nifB >0,(3)\nwhere\nA≡ /angbracketleftJ2\n/vector n1−J2\n/vector n2/angbracketright, B≡ /angbracketleft/bracketleftBig\nˆJ/vector n1,ˆJ/vector n2/bracketrightBig\n+/angbracketright. (4)\nIn this control scheme, we employ the following defination\nξ2\nZ=N∆ˆJ2\nz\n|/angbracketleftˆJs/angbracketright|2, (5)\nas the squeezing parameter. Here ∆ ˆJ2\nz=/angbracketleftˆJ2\nz/angbracketright − /angbracketleftˆJz/angbracketright2, indicates the spin squeezing in z-direction. For RL, we use\nthe reverse of ξ2\nZto set the reward-punishment rule during the control process. S ubsequently, in accordance with the\nfundamental principles of RL, the agent will steer the direction with minimum squeezing parameter approaching z\ndirection, namely, ϕ=π/2 in Eq. 3. And compared to using the reverse of ξ2\nRwith variable θandφ, the choice of\nξ2\nZpossesses a more direct physical interpretation since ˆJzsignifies the population imbalance between the two modes\nwithin the Ramsey interferometer mentioned above [ 2,3,8].\nCorrelation exists between entanglement and spin squeezing [ 36], wherein multipartite entanglement constitutes\na quantum resource for enhanced precision in metrology [ 7,37,38]. Moreover, Quantum Fisher information (QFI)\nquantifies the link between entanglement and phase uncertainty wit hin the domain of metrology [ 39,40]. Adhering\nto the quantum Cramer-Rao bound, quantum states with larger QF I are pursued for the efficacy of quantum metrol-\nogy [41,42]. We would check the QFI pertaining to collective spin state ˆ ρwith respect to ˆ ρ(θ) =eiθˆGˆρe−iθˆG, where\nθis the quantity which needs to be estimated with respect to the phas e-shift operator ˆG[43,44]. The QFI reads\nF(ρ,ˆG) = 4/summationdisplay\nnpn(∆ˆG)2\nn−/summationdisplay\nm/negationslash=n8pmpn\npm+pn|/angbracketleftψm|ˆG|ψn/angbracketright|2, (6)\nwhereρ|ψn/angbracketright=pn|ψn/angbracketright, and (∆ ˆG)2\nn≡ /angbracketleftψn|ˆG2|ψn/angbracketright − |/angbracketleftψn|ˆG|ψn/angbracketright|2. The second term denotes a correction. Here, the\nquantumFisherinformationprovidesaquantitativethresholdfort he precisionattainablein estimating θbymeasuring\nˆGonρ. If the average quantum Fisher information over three basic direc tions reaches the order of 1, there is\nmacroscopic multiparticle entanglement [ 45].5\nFIG. 2. ( a) Evolution of the spin squeezing parameter ( 5) versus training epochs while 100 segments of actions are ev enly\ntaken across each evolution time interval [0 ,2]. We show a sample every 30 training epochs within 600 conse cutive training\nrounds. The subfigure in ( a1) shows the evolution of the corresponding averaged quantum Fisher information defined as\n¯F=[F(ρ,ˆJx)+F(ρ,ˆJy)+F(ρ,ˆJz)]/(3N2). (b) The top view of the 3-action (Ω( t) = 2,0,−2) square control pulse corresponding\nto the spin squeezing curves in ( a). (c) 30 samples (differentiate by n) of the squeezing parameter at t= 2 of those each 30\nepochs in ( a). As shown in Figure ( a), there are control sequences causing ξ2\nZto diverge. However, such control sequences will\nnot be accepted. Therefore, we express instances where ξ2\nZ>5 uniformly as 5 to facilitate the visualization of the resul ts in\n(c). (d) The squeezing parameters versus time for the constant cohe rent control (Ω( t) =−2) and the RL-designed control, with\nthe corresponding degree of mixing ( Tr[ρ2(t)]). The subscript RLmeans Ω( t) obtained by reinforcement learning, Conmeans\nΩ(t) =−2. (e1) evolution of the angle ϕin Eq.3. (e2) The control fields Ω( t) designed by RL agent corresponding to ( d) and\nthe constant control field Ω( t) =−2. (f1) and (f2) are the Husimi representation of the initial CSS and the squ eezed state at\nt= 2. (f3) and (f4) are the polar plots of the Wigner function for the states. We useN=2J=20. The decaying parameters are\nγ=0.001κ.\nPREPARE SPIN SQUEEZED STATE BY REINFORCEMENT LEARNING\nThere are mainly two steps to prepare the spin squeezed states: fi rstly, spin coherent state should be prepared, and\nsecondly, spin squeezed state is prepared by using the control fie ld designed by machine learning agent.\nInitial Spin Coherent State\nNtwo-level atoms all pointing along the same direction can be describe d by an SU(2) spin coherent state (CSS).\nSuch a state reads\n|θ,φ/angbracketright= (cosθ\n2)2jj/summationdisplay\nm=−j(C2j\nj+m)1/2[e−iφtanθ\n2]j+m|j,m/angbracketright, (7)\nwhereC2j\nj+mare the binomial coefficients. This state is most similar to classical sta te of a collective spin with\nθandφbeing the azimuth angles for longitude and latitude, respectively. |j,m/angbracketrightare the eigenvectors satisfy the\nequations ˆJ2|j,m/angbracketright=j(j+1)/planckover2pi12|j,m/angbracketrightandˆJz|j,m/angbracketright=m/planckover2pi1|j,m/angbracketright(/planckover2pi1=1 in numerical calculations). The quantum state can\nbe represented by the Husimi function or the Wigner distribution. T he CSS can be prepared by applying a π/2 pulse6\nto a BEC with Natoms in the internal ground state [ 46]. In the CSS, /angbracketleftˆJx/angbracketright=N/2 and/angbracketleftˆJy/angbracketright=/angbracketleftˆJz/angbracketright= 0. Such a pulse\nis equivalent to the effect of a beam splitter in interferometer.\nCommencing with a CSS aligned along the x-axis and characterized by isotropic fluctuation in its spin componen ts,\nˆJ2\nzshearsthe coherent state to a squeezed one with reduced varian ce ofˆJz, culminating in the generationof a squeezed\nspin state. Such states exceed the constraints delineated by the standard quantum limit, allowing for the enhanced\nmeasurement sensitivity in metrology along the squeezed direction [ 7]. The squeezed direction would be fixed on ˆJz\nunder the action of Ω( t)ˆJxdetermined by the reinforcement learning agent as mentioned abov e.\nPrepare spin squeezed states by machine designed pulses\nUsually, itishardtoavoiddecoherenceinaquantumsystemduetoits interactionwiththeenvironment. Thecontrol\nscheme should include the effect of such decoherence. We consider two kinds of decoherence channels: superradiant\ndamping and dephasing. Such decoherence channels lead to the loss of quantum resource. The time evolution of the\ncollective spin system is described by the Lindblad master equation in t his work. We consider the master equation\n˙ρ=−i[ˆH,ρ]+γ(nth+1)Lj−ρ+γnthLj+ρ+γzLjzρ, (8)\nwhereLˆXρ= 2ˆX†ρˆX��ˆXˆX†ρ−ρˆXˆX†.γis the decay rate, γzis the dephase rate and nthis the average thermal\nphotons. Different from the traditional quantum Lyapunov contr ol strategies which are based on the distance between\neigenstates [ 21], the system would evolve under the domain governed by this master equation with the application of\na control field Ω( t) designed using RL in the Hamiltonian ( 1).\nIn this work, the RL agent selects actions contingent upon the obs ervations, thereby orchestrating a sequence\nof actions aimed at maximizing cumulative rewards and minimizing penaltie s. During training, the observation\n(calculated based on the quantum state) is fed to the neural netw ork, while output neurons provide the probability\nof choosing which action at each iterative training step. A reward is d ispensed subsequently to each step to evaluate\nthe decision-making policy. After one epoch, the collective spin syst em is re-initialized to the coherent state and the\nnext epoch starts to train the agent continuously based on the tr ained neural network. Since the environment is\ndeterministic, i.e., the state evolves deterministically according to th e master equation, like ( 8). The pulse strengths\nand application time in the episode represent the policy π∗.\nThere are means to improve the performance of RL. Replay mechan ism store the learned history in training, and\nenhances the learning efficiency and algorithmic stability. Besides, we employ Huber loss in the RL agent [ 47] since\nit is robustness to outliers once the error becomes too large due to linear function used. It makes the training\nmore stable and provides a balance between Mean Squared Error (u nderestimates large errors) and Mean Absolute\nError (overestimates small errors). It also helps in reducing the e xploding gradients problem in training deep neural\nnetworks and can potentially lead to faster convergence during tr aining compared to other loss functions. The neural\nnetwork is trained by the stochastic gradient descent method calle d AdamW [ 48]. The key difference between Adam\nand AdamW lies in their approach to weight decay which helps prevent o verfitting by adding a penalty term to the\nloss function.\nAs a result of this training, the weights of the neural network are a djusted, i.e., the agent learns to determine a\nsequence of actions based on the states of the system to obtain la rger reward. Randomness provides the probability\nfor the RL agent to find the best sequence of actions.\nIt is imperative to clarify that the strategy is a closed-loop simulation , but an open-loop application scheme. Once\nthe controlsequence of actions, such as Ω x(t) delineated in Eq.( 1), is obtained by simulation, the identical control field\nis implemented in an open-loop control process to circumvent quant um collapse attributable to system observation.\nCONTROL RESULTS\nAs depicted in Fig. 2, the time interval [0,2] is partitioned into a variable number of segmen ts, and the control\nsquare pulse is applied at the boundaries between adjacent time seg ments and sustained until reaching the subsequent\nboundary. Each round of training consists of 600 epochs, and eve ry 30 epochs of the evolution for the squeezing\nparameter are illustrated in Fig. 2(a). It can be seen that, at early stages of training, the agent ha ve not provided\nan effective control strategy, resulting to divergence of ξ2\nZ. However, as the training proceeds, the RL agent can\nfind numerous sequences of square pulses inducing attenuation of the squeezing parameter (indicative of enhanced\nperformance). Meanwhile, the inset Fig. 2(a1) delineates the evolution of the corresponding averagedquantum Fisher7\n/s116/s32/s51\n/s32/s53\n/s32/s55\n/s32/s57/s116\n/s97/s99/s116/s105/s111/s110/s115\n/s40/s98/s41\n/s120/s50\n/s122\nFIG. 3. ( a) Mean evolution for 30 samples of the spin squeezing paramet er with shaded error zone for different frequency of\napplying control pulses, control time interval [0 ,2] is evenly divided into the number of segments at which the s quare pulse\nsequence with amplitude 2 (Ω( t) = 2,0,−2) is applied. We should confirm that each of the 30 samples is t he one with smallest\nξ2\nZamong 600 epochs. It is reasonable since in practical scenar ios, one choose the best control sequence in experiments. Th is\nresults in the fluctuation error value contracting at time t= 2. (b) Evolution of the spin squeezing parameter for different num ber\nof control actions when the number of segments is 40: {Ω(t)}={(2,0,−2)|actions= 3},{(2,1,0,−1,−2)|actions= 5},\n{(2,1.34,0.66,0,−0.66,−1.34,−2)|actions= 7},{(2,1.5,1,0.5,0,−0.5,−1,−1.5,−2)|actions= 9}. The other\nparameters are same to those in Fig. 2(a).\ninformation. As the training progresses, ¯Ftends to attains a high value and descend gradually. This correspon ds\nto the increasing of ξ2\nZresulting from decoherence. Fig. 2(b) depicts the corresponding square wave control pulses\nfrom the top view. To show the efficiency of the control field from an other view, Fig. 2(c) reveals that, upon\nincrementing the training epochs, there is a discernible decrease in t he squeezing parameter at t= 2 from the\nstatistical view. These results corroborate the efficacy of the co ntrol pulses designed by the RL agent in optimizing\ncontrol performance. Fig. 2(d) presents the evolution of the spin squeezing parameter obtain ed by using RL-designed\nΩ(t):ξ2\nZ,RL, constant control field Ω( t) =−2:ξ2\nZ,Con, and the optimized ξ2\n⊥,RL, with the minimal value at the final\ncontrol time t= 2 among the 600 epochs. The comparison clearly demonstrates a s ignificantly enhanced performance\nof the reinforcement-learning-based control approach as oppo sed to the constant control scenario under the same\nparameter settings. And we employ the trace of the square of the system state ρ(t):Tr[ρ2(t)], as an indicator for\nmixing of a quantum state. The degree of mixing of the state under R L control tend to be more deeply compared\nto Ω(t) =−2. The optimal squeezing angle ϕ→π/2, i.e., converges to z-component, which can be seen in Fig. 2\n(e1). The linear terms Ω x(t)ˆJxrotating of the redistributed fluctuation, combined with the twistin g effect of ˆJ2\nz, spin\nsqueezed state is generated and maintained. The square wave con trol sequence corresponding to the RL control result\nin Fig.2(d) is depicted in Fig. 2(e2). Furthermore, Fig. 2(f) utilizes the Husimi function and Wigner function\nto visualize the initial coherent spin state and final spin squeezed st ate at the time t= 2. Nonclassical states are\ncharacterized by the twisted distribution in the Husimi function [ 1,8] and the asymmetry in the polar plot of the\nWigner function [ 49,50]. This provides insight into the evolution of the quantum state under the control protocol.\nWe conjecture that the temporal application frequency of pulse im pacts the outcomes of the control. To investigate\nthe evolution of the squeezing parameter versus different applicat ion frequencies, we discretize the temporal interval\n[0, 2] into different number of segments, the more segments, the m ore frequently of applying the pulses. It can be\nseen in Fig. 3(a), with increasing of the segments, the squeezing parameter is d epressed more stationary and lower.\nEven more, with increasing of the control pulses, the variance of t he squeezing parameter is also depressed more\nobviously. Straightforwardly, there is a contradiction between th e control performance and operation difficulty. One\nmay conjecture the number of control actions may also influences the control performance. However, as shown in\nFig.3(b), there is no obvious advantage for more control actions with t he same maximum control amplitude in this8\n/s40/s97/s41\n/s40/s97\n/s49/s41\n/s120/s50\n/s122\nFIG. 4. Average evolution for 30 samples of the spin squeezin g parameter with error zone for different size of the collecti ve spin\nsystemN= 2Junder 3-action (Ω( t) = 2,0,−2) control. The samples are picked in the same manner as those shown in Fig. 3\n(a). The subgraph shows the squeezing parameter in the scaled t ime interval t= 0∼0.2, picked from the shaded area on the\nleft in the graph, for the collective spin system of different size. Same color is shared with the same N. The other parameters\nare same to those in Fig. 2(a).\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50 /s49/s46/s53/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53\n/s110\n/s116/s104/s120/s50\n/s122\n/s116/s32/s48/s46/s48\n/s32/s48/s46/s51\n/s32/s48/s46/s54\n/s32/s48/s46/s57\n/s32/s49/s46/s50\n/s32/s49/s46/s53/s110 /s116/s104/s40/s97/s41\n/s40/s97\n/s49/s41\n/s120/s50\n/s122\nFIG. 5. The average evolution of 30 squeezing parameter with error zone for different thermal excitations of the reservoi r\nnth=1\nexp(/planckover2pi1ω/kBT)−1, the average number of photons for a mode with frequency ωin the reservoir. The samples are picked in\nthe same manner as those shown in Fig. 3(a). The subgraph shows the squeezing parameter versus averag e thermal excitations\nat the scaled time t= 2. The other parameters are same to those in Fig. 2(a).\ncontrol.\nIt is pertinent to ask the applicability of the proposed scheme to colle ctive spin models with different total spin\nnumbers (N= 2J). To address this concern, FIG. 4(a) illustrates the control results for collective spin systems\nwith different N. The result reveals that the squeezing parameter reaches lower v alues asNincreases within the\nsame time interval. This observation suggests that an enlarged ens emble of entangled spins enhances the precision\nfor quantum metrology. Furthermore, an examination of the subg raph discloses a convergence towards parallelism\namong the trajectories corresponding to different N, indicating an emergence of scaling behavior. In the system\nunder consideration, energy dissipation concurrent with decoher ence interplays with the applied coherent pulses\nwhich impedes the quantum system decay to the ground state. Con sequently, the plateau in the squeezing parameter\ncan be attributed to a dynamical equilibrium between these two confl icting processes.\nIn the previous results, the environmental temperature was ass umed to be zero. To investigate the robustness of the\nproposed control scheme, it is necessary to check the influence o f a finite temperature on the system dynamics. Since\nthe temperature is positively correlated with the average number o f thermal excitations of the reservoir, denoted as\nnth. It reflects the strength of the decoherence. Fig. 5illustrates that an incremental rise in thermal excitation of the\nsurrounding reservoir progressively impairs the efficacy of the con trol scheme from the view of degree of squeezing\nand the control variance.9\nDISCUSSION\nThe interferometer operations are collectively acting on all particle s identically. In the large N(the number of\nparticles) limit, this model can also be mapped to the bosonic model by the Holstein-Primakoff transformation:\nˆJz=N/2−ˆc†ˆc≃N/2 andˆJ+= (N−ˆc†ˆc)1/2ˆc≃√\nNˆc, where ˆc(ˆc†) is a bosonic annihilation (creation) operator [ 51].\nThese maps between the collective spin model and the bosonic partn ers hint we may apply the control scheme on\nsuch quantum systems.\nA control scheme is proposed in this study, wherein elements within t he scheme can be replaced by modules pos-\nsessing equivalent functions. Several candidate modules capable o f substituting the reinforcement learning agent have\nbeen identified and evaluated. Forexample, the State-Action-Rew ard-State-Actionalgorithm[ 27], Deep Deterministic\nPolicy Gradient [ 29], Asynchronous Advantage Actor-Critic [ 31], Dueling Network [ 32] and so on.\nRL provides a tool to optimize dynamical process, i.e., the evolution o f the quantum state. The optimal criteria\nvarious as certain quantum states or expectations of operators . The principle using this scheme is the system evolves\nor changes under certain mapping rules. And the optimal module jus t finds the road to a goal more efficiently. After\nall, machine learning learns the patterns or mappings based on statis tical distribution.\nCONCLUSION\nIn conclusion, we have proposed a reinforcement-learning based c ontrol strategy for the generation of non-classical\ncollective spin states within an open environment. The machine can de sign a suite of control sequences to prepare\nspin squeezed states accompanied by entanglement. Larger freq uency of application of the pulses enhances the\nefficacy of the control protocol. However more choice of actions d o not contributes to the control outcomes obviously.\nThe scalability of this framework is demonstrated, facilitating its app licability to larger systems wherein. Notably,\nthermal excitations progressively undermine the control perfor mance. The versatility of this scheme is underlined\nby the potential for substituting the reinforcement learning agen t with alternative optimization modules, as well as\nextending the control paradigm to other quantum systems which c an provide quantum resources.\nACKNOWLEDGMENTS\nX. L. Zhao thanks discussions with Li Jiachun and Hou Shaocheng, N atural Science Foundation of Shan-\ndong Province, China, No.ZR2020QA078, No.ZR2023MD064, and Nat ional Natural Science Foundation of China,\nNo.12005110, No.12074206.\n[1] J. Ma, X. G. Wang, C. P. Sun, and F. Nori, Quantum spin squee zing, Physics Reports 509, 89 (2011).\n[2] L. Pezz` e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of\natomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).\n[3] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, a nd D. J. Heinzen, Spin squeezing and reduced quantum noise\nin spectroscopy, Phys. Rev. A 46, R6797 (1992).\n[4] D.J.Wineland, J.J.Bollinger, W.M.Itano, andD.J.Hein zen, Squeezedatomic statesandprojectionnoiseinspectro scopy,\nPhys. Rev. A 50, 67 (1994).\n[5] R. J. Sewell, M. Koschorreck, M. Napolitano, B. Dubost, N . Behbood, and M. W. Mitchell, Magnetic Sensitivity Beyond\nthe Projection Noise Limit by Spin Squeezing, Phys. Rev. Let t.109, 253605 (2012).\n[6] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, O ptical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).\n[7] L. Pezz´ e, and Augusto Smerzi, Entanglement, Nonlinear Dynamics, and the Heisenberg Limit, Phys. Rev. Lett. 102,\n100401 (2009).\n[8] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A47, 5138 (1993).\n[9] A. Kuzmich, N. P. Bigelow, and L. Mandel, Atomic quantum n on-demolition measurements and squeezing, Europhys.\nLett.42, 481 (1998).\n[10] L. M. Duan, A. Sørensen, J. I. Cirac, and P. Zoller, Squee zing and Entanglement of Atomic Beams, Phys. Rev. Lett. 85,\n3991 (2000).\n[11] C. K. Law, H. T. Ng, and P. T. Leung, Coherent control of sp in squeezing, Phys. Rev. A 63, 055601 (2001).\n[12] K. P. Murphy, Machine learning: a probabilistic perspe ctive (MIT Press, 2012).\n[13] R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction (MIT press, 2018).10\n[14] T. F¨ osel, P. Tighineanu, T. Weiss, and F. Marquardt, Reinforceme nt Learning with Neural Networks for Quantum Feed-\nback, Phys. Rev. X 8, 031084 (2018).\n[15] V. Dunjko, and H. J. Briegel, Machine learning and artifi cial intelligence in the quantum domain: a review of recent\nprogress, Reports on Progress in Physics 81, 074001 (2018).\n[16] H. Xu, J. Li, L. Liu, Y. Wang, H. Yuan, and X. Wang, General izable control for quantum parameter estimation through\nreinforcement learning, NPJ Quantum Information 5, 82 (2019).\n[17] S. Grivopoulos, and B. Bamieh, Lyapunov-based control of quantum systems, In Proceedings of the 42nd IEEE conferen ce\non decision and control 1, 434 (2003).\n[18] M. Mirrahimi, P. Rouchon, and G. Turinici, Lyapunov con trol of bilinear Schr¨ odinger equations, Automatica 41, 1987\n(2005).\n[19] R. V. Handel, J. K. Stockton, and H. Mabuchi, Feedback co ntrol of quantum state reduction, IEEE Trans. Automat.\nControl50, 768 (2005).\n[20] X. X. Yi, S. L. Wu, C. F. Wu, X. L. Feng, and C. H. Oh, Time-de lay effects and simplified control fields in quantum\nLyapunov control, J. Phys. B: At. Mol. Opt. Phys. 44, 195503 (2011).\n[21] S. C. Hou, L. C. Wang, and X. X. Yi, Realization of quantum gates by Lyapunov control, Physics Letters A 378, 699\n(2014).\n[22] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness , M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G.\nOstrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou , H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis,\nHuman-level control through deep reinforcement learning, Nature518, 529 (2015).\n[23] R. Bellman, On the theory of dynamic programming, PNAS 38, 716 (1952).\n[24] Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, natur e521, 436 (2015).\n[25] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learni ng (MIT Press,2016).\n[26] M. Hessel, J. Modayil, H. v. Hasselt, T. Schaul, G. Ostro vski, W. Dabney, D. Horgan, B. Piot, M. Azar, and D. Silver,\nRainbow: Combining Improvements in Deep Reinforcement Lea rning, In Proceedings of the 30nd AAAI Conference on\nArtificial Intelligence 32, 1 (2018).\n[27] G. A. Rummery, and M. Niranjan, On-Line Q-Learning Usin g Connectionist Systems, Department of Engineering (Uni-\nversity of Cambridge, 1994).\n[28] F. S. Melo, Convergence of Q-learning: a Simple Proof, T ech. Rep (2001).\n[29] D. Silver, G. Lever, N. Heess, T. Degris, D. Wierstra, an d M. Riedmiller, Deterministic Policy Gradient Algorithms , PMLR\n32, 387(2014).\n[30] T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez , Y. Tassa, D. Silver, D. Wierstra, Continuous Control with D eep\nReinforcement Learning, arXiv: 1509.02971 (2019).\n[31] V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Harley, T. P. Lillicrap, D. Silver, K. Kavukcuoglu, Asynchronous Method s\nfor Deep Reinforcement Learning, International Conferenc e on Machine Learning, PMLR 481928 (2016).\n[32] Z. Wang, T. Schaul, M. Hessel, H. v. Hasselt, M. Lanctot, and N. d. Freitas, Dueling Network Architectures for Deep\nReinforcement Learning, PMLR 48, 1995 (2016).\n[33] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, At omic Coherent States in Quantum Optics, Phys. Rev. A 6,\n2211 (1972).\n[34] L. G. Biederharn, and J. C. Louck, Angular Momentum in Qu antum Physics: Theory and Applications (Addison-Wesley,\nReading, MA, 1981).\n[35] L. Song, X. Wang, D. Yan, Z. Zong, Spin squeezing propert ies in the quantum kicked top model, J. Phys. B: At. Mol. Opt.\nPhys.39559(2006).\n[36] A. S. Sørensen, and K. Mølmer, Entanglement and Extreme Spin Squeezing, Phys. Rev. Lett. 86, 4431 (2001).\n[37] G. T´ oth, Multipartite entanglement and high- precisi on metrology, Phys. Rev. A 85, 022322 (2012).\n[38] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-Enha nced Measurements: Beating the Standard Quantum Limit,\nScience, 306, 1330 (2004).\n[39] C. W. Helstrom, Quantum detection and estimation theor y, J. Stat. Phys. 1, 231(1969).\n[40] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hu me, L. Pezze, A. Smerzi, and M. K. Oberthaler, Fisher\ninformation and entanglement of non-Gaussian spin states, Science345, 424 (2014).\n[41] C. W. Helstrom, Minimum mean-squared error of estimate s in quantum statistics, Phys. Lett. A 25, 101 (1967).\n[42] S. L. Braunstein and C. M. Caves, Statistical distance a nd the geometry of quantum states, Phys. Rev. Lett. 72, 3439\n(1994).\n[43] Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, Quantum Fishe r information of entangled coherent states in the presence\nof photon loss, Phys. Rev. A 88, 043832 (2013).\n[44] Liu J, Jing X-X and Wang X Quantum metrology with unitary parametrization processes Sci. Rep. 5, 8565 (2015).\n[45] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. W ieczorek, H. Weinfurter, L. Pezz´ e, and A. Smerzi, Fisher\ninformation and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).\n[46] A. Sørensen, L. M. Duan, J. L. Cirac, and P. Zoller, Many- particle entanglement with Bose-Einstein condensates, Na ture\n(London) 409, 63 (2001).\n[47] P. J. Huber. Robust Estimation of a Location Parameter, Annals of Mathematical Statistics 35, 73(1964).\n[48] I. Loshchilov, and F. Hutter, Decoupled Weight Decay Re gularization, https://arxiv.org/abs/1711.05101.\n[49] M.G. Benedict, andA.Czirj´ ak, Wignerfunctions, squeezingproperties, andslow decohe renceofamesoscopic superposition\nof two-level atoms, Phys. Rev. A 60, 4034(1999).11\n[50] T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. N emoto, Wigner Functions for Arbitrary Quantum Systems,\nPhys. Rev. Lett. 117, 180401(2016).\n[51] T. Holstein and H. Primakoff, Field Dependence of the Int rinsic Domain Magnetization of a Ferromagnet, Phys. Rev. 58,\n1098 (1940)." }, { "title": "2401.16344v1.An___mathrm_L__2__maximum_principle_for_circular_arcs_on_the_disk.pdf", "content": "arXiv:2401.16344v1 [math.AP] 29 Jan 2024AnL2-maximum principle for circular arcs on the disk\nThiago Carvalho Corso∗, Muhammad Hassan†, Abhinav Jha‡, Benjamin Stamm§\nAbstract\nIt is well-known that bounded harmonic functions attain a po int-wise maximum on the\nboundary of the domain. For harmonic functions with general Lp, p∈[1,∞)boundary data,\nhowever, such an L∞-maximum principle does not hold. In this article, we prove a novel\nL2-maximum principle for harmonic functions on the disk. More precisely, given a harmonic\nfunction uon a disk Ωwith boundary data g∈L2(∂Ω)and bounded non-tangential maximal\nfunction, and any circular arc ΓinΩthat intersects the boundary of the disk at exactly two\npoints, we show that the L2norm of the restriction of utoΓis strictly smaller than /bardblg/bardblL2(∂Ω).\nMoreover, for boundary data gthat is supported only on one connected component of ∂Ω\\Γ,\nwe derive a similar estimate with a sharp geometry-dependen t constant– again strictly smaller\nthan one. As a corollary of this result, we also deduce new Lp, p∈[2,∞)maximum principles\nfor harmonic functions on the disk. Our result has applicati ons in the convergence analysis\nof Schwarz domain decomposition methods on the union of over lapping disks.\nKeywords : Maximum principle, harmonic functions on the disk, Poisso n integral.\nAMS Classifications : 30E25, 35J05, 35J57\n1 Introduction\nLetΩ⊂R2be an open disk on the plane with radius rΩ>0, center xΩ∈R2and boundary\n∂Ω⊂R2. Then for any g∈L2(∂Ω), we denote by ug∈H1\n2(Ω)the harmonic extension of g\ninsideΩdefined through the Poisson integral\nug(x) :=1\n2π/integraldisplay\n∂Ωr2\nΩ−|x−xΩ|2\nrΩ|x−y|2g(y)dH1(y)for allx∈Ω, (1)\nwhereH1denotes the surface (Hausdorff) measure.\nIt is well known that the function ugdefined through Equation (1) is the unique solution of\nthe boundary value problem\n−∆ug= 0 inΩ,\nug=g,on∂Ω,\nwhose non-tangential maximal function belongs to L2(∂Ω)with boundary values being taken in\nthe non-tangential sense (see, e.g., [Dah77, Dah79, Ver84] or [Med18, Chapter 5] for the definition\nof non-tangential limits, maximal functions, and the assoc iated well-posedness results.)\nSinceugis harmonic in Ω, standard regularity theory implies that it is also smooth i nΩ, and\nwe can define its trace along any curve Γ⊂Ωthrough point-wise restriction. We shall denote this\nextension plus restriction operator by γΓ, i.e., for any g∈L2(∂Ω)and any curve Γ⊂Ωwe define\nγΓg:=ug|Γ,whereugis the harmonic extension given by the Poisson integral (1) .(2)\n∗Institute of Applied Analysis and Numerical Simulation, Un iversity of Stuttgart,\nEmail:thiago.carvalho-corso@ians.uni-stuttgart.de\n†Institute of Mathematics, EPF Lausanne, Email: muhammad.hassan@epfl.ch\n‡Institute of Applied Analysis and Numerical Simulation, Un iversity of Stuttgart, Email:\nAbhinav.Jha@mathematik.uni-stuttgart.de\n§Institute of Applied Analysis and Numerical Simulation, Un iversity of Stuttgart, Email:\nBenjamin.Stamm@mathematik.uni-stuttgart.de\n1∂ΩextΩ /tildewideΩ Γ\n∂ΩintθΓ\nθΓ\nFigure 1: 2-D schematic diagram of Ω,/tildewideΩ, andΓ.\nThe goal of this paper is then to study the boundedness of the m ap\ng∈L2(∂Ω)/mapsto→γΓg∈L2(Γ)\nin the special case where Γis a circular arc touching the boundary of Ωat two distinct points.\n1.1 Main results\nTo state our main results precisely, let us first introduce so me additional notation.\nDefinition 1. LetΩ⊂R2be an open disk with boundary ∂Ω. We say that Γ⊂Ωis a circular\narc across Ωif there exists a second open disk /tildewideΩ⊂R2such that Γ = Ω∩∂/tildewideΩ, and∂Ω\\Γhas two\nconnected components. We refer to these components as the in terior and exterior part of ∂Ω, and\ndenote them by\n∂Ωint:=∂Ω∩/tildewideΩand∂Ωext:=∂Ω\\∂Ωint.\nAdditionally, we define the angle θΓ∈(0,π)associated to a circular arc Γas the angle between\nthe line segments connecting the centers of Ωand/tildewideΩto one of the intersection points of their\nboundaries (see Figure 1 for a visual example).\nThe main result of the present contribution can now be stated as follows.\nTheorem 2 (L2-maximum principle for circular arcs) .\nLetΩ⊂R2be an open disk with boundary ∂Ω, letΓ⊂Ωbe a circular arc across Ωwith\nassociated angle θΓ∈(0,π)defined according to Definition 1, let the extension plus rest riction\noperator γΓbe defined as in Equation (2). Then we have\n/bardblγΓg/bardbl2\nL2(Γ)<1−cos(θΓ)\n2/bardblg/bardbl2\nL2(∂Ωint),for anyg∈L2(∂Ω)withg|∂Ωext= 0 (3)\n/bardblγΓg/bardbl2\nL2(Γ)<1+cos(θΓ)\n2/bardblg/bardbl2\nL2(∂Ωext),for anyg∈L2(∂Ω)withg|∂Ωint= 0,, (4)\nwhere the L2norms are taken with respect to the one-dimensional Hausdorff m easure. In particular\nfor anyg∈L2(∂Ω)it holds that\n/bardblγΓg/bardbl2\nL2(Γ)0}/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=Ψ(Rσ+π)∪{a1,a2}.(See Figure 2b.)\nIn particular, ∀σ∈(0,2π), the restricted mapping Ψσ:= Ψ|Rσ:Rσ→Ψ(Rσ)is bijective.\nRemark 7.It is easy to show that the bipolar transform Ψcan be continuously extended to a\nmapping with domain S \\{(0,0),(0,2π)}in a manner such that\nΨ/parenleftbig\nR0\\{(0,0)}/parenrightbig\n= Ψ/parenleftbig\nR2π\\{(0,2π)}/parenrightbig\n= (R\\[−1,1])×{0} ⊂R2.\n1In the sense that the Jacobian DΨ(τ,σ)at any(τ,σ)∈ S=R2×(0,2π)is the product a non-zero (not\nnecessarily positive) scalar and an orthogonal matrix.\n5In view of Property (ii) of Proposition 6, we introduce the fo llowing weighted Lebesgue spaces.\nDefinition 8 (Weighted Lebesgue Spaces) .\nLetΨ:S →R2be the bipolar transform introduced in Definition 5, let σ∈(0,2π), let\np∈[1,∞], and let Ψσ:Rσ→Ψ(Rσ)be the restriction Ψσ= Ψ|RσwhereRσ:=R×{σ}. Then\nwe define the weighted Lebesgue space Lp\nσas the vector space\nLp\nσ:={f:Rσ→Rsuch that f◦Ψ−1\nσ∈Lp(Ψ(Rσ))},\nendowed with the norm /bardblf/bardblLp\nσ:=/bardblf◦Ψ−1/bardblLp(Ψ(Rσ)), whereLp/parenleftbig\nΨ(Rσ)/parenrightbig\ndenotes the Lpspace with\nrespect to the one-dimensional Hausdorff measure on the circ ular arcΨ(Rσ).\nRemark 9 (Alternative definition of Weighted Lebesgue Spaces) .From the change of variables\nformula we can equivalently define Lp\nσas the vector space of (Lebesgue) measurable functions\nf:R→Rsatisfying\n/bardblf/bardblp\nLp\nσ:=/integraldisplay∞\n−∞|f(τ)|p 1\ncosh(τ)−cos(σ)dτ <∞. (13)\nIn the sequel, we will frequently make use of this alternativ e representation of the weighted\nLebesgue space Lp\nσwithout further comment.\nFirst, by Proposition 6 and the conventions adopted for ΩandΓat the beginning of the present\nsection, there exist 0< σΩ< σΓ< πsuch that\nΩ = Ψ(SσΩ)andΓ = Ψ(RσΓ). (14)\nIn particular, the exterior and interior components of the b oundary of Ω, as introduced in\nDefinition 1, can be written as\n∂Ωext= Ψ(RσΩ)and∂Ωint= Ψ(RσΩ+π). (15)\nMoreover, the angle θΓassociated to the arc Γ(see also Figure 2a) is given by\nθΓ=σΓ−σΩ. (16)\nSecond, Definition 8 of the weighted Sobolev space L2\nσimplies that for any σ∈(0,2π), the\npush-forward map\ng∈ L2\nσ/mapsto→Ψ#g:=g◦Ψ−1|Ψ(Rσ)∈L2(Ψ(Rσ))is an isometry. (17)\nFrom these observations, we can now deduce the following ref ormulation of Theorem 2.\nLemma 10 (L2-maximum principle on the infinite strip) .\nLetΨ:S →R2be the bipolar transform introduced in Definition 5, let L2\nσforσ∈(0,2π)be the\nweighted Lebesgue spaces introduced in Definition 8, and rec all the notation Sσ:=R×(σ,σ+π).\nThen estimates (3)–(5)hold for any open disk Ωand any circular arc Γ⊂Ωin the sense of\nDefinition 1 if and only if for all g∈ L2\nσ, allh∈ L2\nσ+πand any θ,σ∈(0,π)it holds that\n/bardbl(uΨ#g◦Ψ)(·,σ+θ)/bardbl2\nL2\nσ+θ<1+cos(θ)\n2/bardblg/bardbl2\nL2σ, (18)\n/bardbl(uΨ#h◦Ψ)(·,σ+θ)/bardbl2\nL2\nσ+θ<1−cos(θ)\n2/bardblh/bardbl2\nL2\nσ+π, (19)\n/bardbl/parenleftbig\n(uΨ#g+uΨ#h)◦Ψ/parenrightbig\n(·,σ+θ)/bardbl2\nL2\nσ+θ0andxσ∈R2denote respectively the radius and center of the disk Ψ(Sσ).\nAdditionally, Estimates (3),(4), and (5)are sharp if and only if Estimates (19),(18), and\n(20)are also sharp.\nProof. LetΓbe a given circular arc across the open disk Ωand denote by θΓthe associated angle\n(see Definition 1). From the discussion in the beginning of th is section, we can assume without\nloss of generality that ∂Ω∩Γ ={(−1,0),(1,0)}and the center of Ωlies in the lower half-plane. In\nparticular, there exist σΩ∈(0,π)andθΓ∈(0,π)such that (14), (15), and (16) hold. Assuming,\ntherefore that Estimates (19)–(20) hold, we immediately de duce from the isometry property given\nby Equation (17) that for any g∈ L2\nσΩand any h∈ L2\nσΩ+πit holds that\n/bardbluΨ#g/bardbl2\nL2(Γ)=/bardbl(uΨ#g◦Ψ)(·,σΩ+θΓ)/bardbl2\nL2\nσΩ+θΓ<1+cos(θΓ)\n2/bardblg/bardbl2\nL2σΩ=1+cos(θΓ)\n2/bardblΨ#g/bardbl2\nL2(∂Ωext),\n/bardbluΨ#h/bardbl2\nL2(Γ)=/bardbl(uΨ#h◦Ψ)(·,σΩ+θΓ)/bardbl2\nL2\nσΩ+θΓ<1−cos(θΓ)\n2/bardblh/bardbl2\nL2\nσΩ+π=1−cos(θΓ)\n2/bardblΨ#h/bardbl2\nL2(∂Ωint),\nwhich are precisely Estimates (3) and (4). Additionally, as the push-forward maps Ψ#:L2\nσΩ→\nL2(∂Ωext)andΨ#:L2\nσΩ+π→L2(∂Ωint)are bijective, we conclude that Estimates (3) and (4)\nholds for any g∈L2(∂Ωint)and any h∈L2(∂Ωext), respectively. Estimate (5) and the converse\ndirection follows by repeating the same steps in the opposit e direction.\nSince the bipolar transform Ψis a conformal mapping, the functions uΨ#g◦ΨanduΨ#h◦Ψ\ndefined in the statement of Lemma 10 above are also harmonic in the strip Sσ:=R×(σ,σ+π).\nConsequently, in order to prove Estimates (18)–(20), we wil l make use of the Poisson integral\nrepresentation of harmonic functions on an infinite strip. W e thus recall in the next subsection\nsome basic properties of the Poisson kernel on the strip.\n2.2 Poisson kernel on the infinite strip\nIt is well known that (sufficiently regular) harmonic functio ns in an infinite strip in R2can be\nrepresented as a convolution of appropriate boundary data w ith the Poisson kernel. For the sake\nof coherent exposition, we now recall this result and some re lated notions.\nDefinition 11 (Poisson kernel on the infinite strip) .\nThe Poisson kernel on the infinite strip S0:=R×(0,π)is the function P:S0→Rgiven by\n∀(τ,θ)∈ S0=R×(0,π):P(τ,θ) =Pθ(τ) :=1\n2πsin(θ)\ncosh(τ)−cos(θ). (21)\nWe then have the following representation for harmonic func tions on the strip S0.\nLemma 12 (Poisson integral on the strip [Wid61]) .\nLetS0:=R×(0,π)be the the infinite strip, let P:S0→Rbe the Poisson kernel defined in\nEquation (21), and for any g,h∈C∞\nc(R), letug,h:S0→Rbe the function defined as\n∀θ∈(0,π), τ∈R:ug,h(τ,θ) :=/integraldisplay∞\n−∞Pθ(τ−ξ)g(ξ)dξ+/integraldisplay∞\n−∞Pπ−θ(τ−ξ)h(ξ)dξ.\nThenug,h∈C(S0)∩L∞(S0)is harmonic on S0andug,h(·,θ)→gandug,h(·,θ)→huniformly\nasθ→0andθ→π, respectively.\n7Proof. The proof of the above lemma is rather standard and relies on t he fact that the Poisson\nkernel is an approximation of the identity. We shall, theref ore, omit the details and refer, e.g., to\n[Wid61], where a detailed study of the Poisson kernel on the s trip is carried out.\nAs a consequence of Lemma 12, we can show that the pull-back of the Poisson integral on the\ndiskΩ = Ψ(Sσ)forσ∈(0,π)is exactly the Poisson integral Pdefined through Definition 11 on\nthe strip Sσ=R×(σ,σ+π).\nLemma 13 (Pull-back of the Poisson integral) .\nLet the bipolar transform Ψ:S →R2be defined through Definition 5, let σ∈(0,π)and\nSσ:=R×(σ,σ+π)so thatΨ(Sσ)is some open disk with radius rσ>0centerxσ∈R2.\nMoreover, for f∈L2/parenleftbig\n∂Ψ(Sσ)/parenrightbig\nwe letuf∈H1\n2(Ψ(Sσ))be the harmonic extension of fdefined\nthrough the Poisson integral formula\n∀x∈Ψ(Sσ):uf(x) =1\n2π/integraldisplay\n∂Ψ(Sσ)r2\nσ−|x−xσ|2\nrσ|x−y|2f(y)dH1(y),\nThen the pull-back Ψ#uf:=uf◦Ψ|Sσsatisfies\n∀(τ,σ+θ)∈ Sσ: (Ψ #uf)(τ,σ+θ) =/integraldisplay\nRPθ(τ−ξ)g(ξ)dξ+/integraldisplay\nRPπ−θ(τ−ξ)h(ξ)dξ,(22)\nwhere\ng:= Ψ#(f|Ψ(Rσ)) =f◦Ψ|Rσandh:= Ψ#(f|Ψ(Rσ+π)) =f◦Ψ|Rσ+π,\nand we recall the notation, Rδ:=R×{δ}for allδ∈R.\nProof. Since∂Ψ(Sσ)\\/parenleftbig\nΨ(Rσ)∪Ψ(Rσ+π)/parenrightbig\n={a1,a2}has zero H1measure, we deduce that\n(Ψ#uf)(τ,σ+θ) =1\n2π/integraldisplay\nΨ(Rσ)r2\nσ−|x−xσ|2\nrσ|x−y|2f(y)dH1(y)+1\n2π/integraldisplay\nΨ(Rσ+π)r2\nσ−|x−xσ|2\nrσ|x−y|2f(y)dH1(y)\n=/integraldisplay\nRT1(τ,ξ,σ,θ)g(ξ)dξ+/integraldisplay\nRT2(τ,ξ,σ,θ)h(ξ)dξ, (23)\nholds for all (τ,σ+θ)∈ Sσand for some suitable smooth functions T1andT2. It is therefore\nenough to show that for all τ,ξ∈Rand allσ,θ∈(0,π), it holds that\nT1(τ,ξ,σ,θ) =Pθ(τ−ξ)andT2(τ,ξ,σ,θ) =Pπ−θ(τ−ξ). (24)\nTo this end, it suffices to consider functions f∈Cc/parenleftbig\nΨ(Rσ)∪Ψ(Rσ+π)/parenrightbig\n. Indeed, if the right\nhand side of (23) is equal to (22) for every f∈Cc/parenleftbig\nΨ(Rσ)∪Ψ(Rσ+π)/parenrightbig\n, then (24) holds by a\ndensity argument. To see that the right-hand side of (23) agr ees with (22) for continuous fwith\ncompact support in Ψ(Rσ)∪Ψ(Rσ+π), we just note that both Ψ#ufand the right-hand side of\n(22) are bounded harmonic functions on the strip Sσ=R×(σ,σ+π)that are continuous up to\nthe boundary ∂Sσand have the same boundary values. Thus from the classical L∞maximum\nprinciple they must agree everywhere on Sσ, which completes the proof.\nLemma 13 in combination with the prior Lemma 10 allows us to re state the original Theorem 2–\nwhich involved harmonic functions on a disk– in terms of boun dedness properties of the Poisson\nintegral on the strip with respect to the weighted Lebesgue s pacesL2\nσ. More precisely, let us define\nthe harmonic extension operator at height θ∈(0,π)as the mapping Pθ:L2\nσ→C∞(R)given by\n∀g∈ L2\nσ,∀τ∈R: (Pθg)(τ) =/integraldisplay\nRPθ(τ−ξ)g(ξ)dξ (25)\nWe then have the following reformulation of Theorem 2.\n8Lemma 14 (L2-maximum principle for the Poisson kernel) .\nConsider the alternative definition of the weighted Lebesgue spacesL2\nσgiven by Remark 9 and\nletPθ:L2\nσ→C∞(R)be the harmonic extension operator at height θdefined through Equation (25).\nThen Estimates (18)–(20)hold if and only\n/bardblPθg/bardbl2\nL2\nσ+θ<1+cos(θ)\n2/bardblg/bardbl2\nL2\nσ, (26)\nfor allg∈ L2\nσand any σ,θ∈(0,π). Moreover, Estimates (18)–(20)are sharp if and only if\nsup\ng∈L2\nσ\\{0}/bardblPθg/bardbl2\nL2\nσ+θ\n/bardblg/bardbl2\nL2σ=1+cos(θ)\n2and sup\n(g,h)∈L2σ⊕L2\nσ+π\\{0}/bardblPθg+Pπ−θh/bardbl2\nL2\nσ+θ\n/bardblg/bardbl2\nL2σ+/bardblh/bardbl2\nL2\nσ+π= 1,(27)\nfor anyσ,θ∈(0,π).\nProof. From the Poisson integral representation formula in Equati on (22), it is clear that Estimate\n(26) is equivalent to Estimate (18). To see that (26) is also e quivalent to Estimate (19), we note\nthat for any θ,σ∈(0,π)and all measurable functions g:R→Rwe have that\n/bardblg/bardblL2\nσ+θ=/bardblg/bardblL2\n2π−σ−θand/bardblg/bardblL2\nσ+π=/bardblg/bardblL2\nπ−σ.\nIndeed, this is simply a consequence of the fact that the cosi ne function is even and 2πperiodic.\nIn particular cos(π+θ) =−cos(θ), and we therefore have\n/bardblPπ−θΩg/bardbl2\nL2\nθΩ+σ=/bardblPπ−θg/bardbl2\nL2\n2π−θ−σand1−cos(θ)\n2/bardblg/bardbl2\nL2\nσ+π=1+cos(π−θ)\n2/bardblg/bardbl2\nL2\nπ−σ\nso that\n/bardblPπ−θg/bardbl2\nL2\nθ+σ<1−cos(θ)\n2/bardblg/bardbl2\nL2\nσ+π⇐⇒ /bardblPπ−θg/bardbl2\nL2\n2π−θ−σ<1+cos(π−θ)\n2/bardblg/bardbl2\nL2\nπ−σ.\nLemma 13 implies that the inequality appearing on the left-h and side above is precisely Estimate\n(19) while the inequality appearing on the right-hand side a bove is precisely Estimate (26) with\nθ,σreplaced by π−θandπ−σ.\nNext, we claim that Equation (20) follows from (26). Indeed, recalling once again Lemma 13\nand using the Cauchy-Schwarz inequality, we deduce that for allθ,σ∈(0,π), allg∈ L2\nσand all\nh∈ L2\nσ+πit holds that\n/bardbl/parenleftbig\n(ug+uh)◦Ψ/parenrightbig\n(·,σ+θ)/bardbl2\nL2\nσ+θ0small enough. Note that the contour Cδis essentially the boundary of the strip S\nmodified with small semi-circles to exclude the points 0and2πı(see Figure 3).\nWe begin with the integrals along the straight line segments composing Cδ. To this end, observe\nthat the mapping Pθ(τ)/sinh(τ)is2πı-periodic and odd along the real axis. So, using the fact\nthat the function sin(ωz)Pθ(z)/sinh(z)can be continuously extended to the origin, we deduce\nthat\n/integraldisplay\n|τ|≥δf(τ)dτ−/integraldisplay\n|τ|≥δf(τ+2πı)dτ= (1−e2πω)/integraldisplay\n|τ|>δ−ısin(θ)\nsinh(τ)sin(ωτ)Pθ(τ)dτ\n= 2ıeπωsinh(πω)I(ω)+O(δ), (37)\nwhereO(δ)denotes a term bounded by a constant times |δ|asδ→0.\nNext, we compute an asymptotic expansion in δfor the integrals along the semi-circles. Using\nthe fact that 1/sinh(z) = 1/z+O(1)forzclose to0we obtain that\n/integraldisplay0\n−πf(δe−ıα)(−ıδe−ıα)dα−/integraldisplay0\n−πf(2πı+δeıα)ıeıαdα=−ıπsin(θ)Pθ(0)(1+e2πω)+O(δ)\n=−ı/parenleftbig\n1+cos(θ)/parenrightbig\neπωcosh(πω)+O(δ),\n(38)\nwhereO(δ)is once again a term bounded by a constant times |δ|asδ→0.\nOn the other hand, the residue of finside the contour Cδis given by\nRes(f,Cδ) = Res(f,ıθ)+Res(f,ıπ)+Res(f,ı2π−ıθ)\n=−eωθ\n2π+eπω\n2πsin(θ)2\n1+cos(θ)−eω(2π−θ)\n2π\n=−eπωcosh/parenleftbig\n(π−θ)ω/parenrightbig\nπ+eπω/parenleftbig\n1−cos(θ)/parenrightbig\n2π. (39)\nApplying therefore the residue theorem and using Equations (37), (38), and (39), we deduce\nthat\n−eπωcosh/parenleftbig\n(π−θ)ω/parenrightbig\nπ+eπω\n2π/parenleftbig\n1−cos(θ)/parenrightbig\n=1\n2πı/contintegraldisplay\nCδf(z)\n=eπωsinh(πω)\nπI(ω)−/parenleftbig\n1+cos(θ)/parenrightbigeπωcosh(πω)\n2π+O(δ),\nwhich yields Equation. (36) after rearranging the terms and taking the limit δ→0.\n12To conclude, we note that ∂ωI(ω) =/hatwideQθ(ω)by differentiating under the integral (which is\njustified by the dominated convergence theorem) and using th e even symmetry of the Poisson\nkernel Pθtogether with the expression for /hatwiderPθgiven in Corollary 17. Equation (35) then follows\nimmediately.\nRemark 20.Consider the setting of Lemma 19. The function I(ω)appearing in the proof of this\nlemma is actually the Fourier transform of the tempered dist ribution p.v.Pθ/sinh, wherep.v.de-\nnotes the (Cauchy) principal value. This distribution admi ts a2πı-periodic meromorphic extension\nin a weaker sense, and one could, in fact, extend Lemma 15 to de al with such distributions.\n3 Proof of the L2-maximum principle\nThe goal of the current section is to use the framework and too ls developed in Section 2 to prove\nthe estimates appearing in Lemma 14.\nThe first step of our proof is the following observation. For a nyσ∈(0,2π), letωσ:R→Rbe\nthe weight function\nωσ(τ) :=1/radicalbig\ncosh(τ)−cos(σ)(40)\nassociated with the weighted Lebesgue space L2\nσgiven by Definition 8. With some abuse of nota-\ntion, let us also denote by ωσandω−1\nσ, the operators of multiplication by ωσandω−1\nσrespectively,\ni.e.,\nωσ:L2\nσ→L2(R)g/mapsto→(ωσg)(τ) =ωσ(τ)g(τ), (41)\nω−1\nσ: L2(R)→ L2\nσg/mapsto→/parenleftbig\nω−1\nσg/parenrightbig\n(τ) =g(τ)\nωσ(τ). (42)\nThen it is clear from the definition of the weighted Lebesgue s pacesL2\nσthatωσ:L2\nσ→L2(R)is\nan isometry for any σ∈(0,π). In particular, the sought-after Inequality (26) appearin g in Lemma\n14 is equivalent to the estimate\n∀f∈L2(R),∀σ,θ∈(0,π):/bardblωθ+σPθω−1\nσf/bardbl2<1+cos(θ)\n2/bardblf/bardbl2,\nwherePθis the operator of the convolution with the Poisson kernel as defined through Equa-\ntion (25) and /bardbl·/bardbldenotes the usual L2(R)norm.\nThe goal of this section is then to prove the above estimate. M ore precisely, we shall prove\nTheorem 21 (Weight function formulation of the L2-maximum principle) .\nLetθ,σ∈(0,π), letL2\nσbe the weighted Lebesgue space given by Definition 8, let ω−1\nσand\nωσ+θbe the multiplication operators defined in (41)and(42), and let Pθbe the operator of the\nconvolution with the Poisson kernel as defined through Equati on(25). Then we have\n/bardblωθ+σPθω−1\nσ/bardbl2\nL2→L2= sup\ng∈L2(R)\\{0}/bardblωθ+σPθω−1\nσg/bardbl2\n/bardblg/bardbl2=1+cos(θ)\n2. (43)\n/bardblωθ+σ/parenleftbig\nPθω−1\nσ⊕Pπ−θω−1\nπ+σ/parenrightbig\n/bardbl2\nL2→L2= sup\n(g,h)∈L2⊕L2\\{0}/bardblωθ+σPθω−1\nσg/bardbl2+/bardblωθ+σPπ−θω−1\nσ+πh/bardbl2\n/bardblg/bardbl2+/bardblh/bardbl2= 1.\n(44)\nMoreover, the supremums above are not attained.\n133.1 The symmetrised operator\nThe next step in our analysis is to demonstrate that, in order to establish Estimates (43) and (44)\nappearing in Theorem 21, it suffices to estimate the norm of a sy mmetric operator Tθ,σ: L2(R)→\nL2(R)defined as\n∀θ,σ∈(0,π):/parenleftbig\nTθ,σf/parenrightbig\n(τ):=/parenleftbig\nωθ+σPθω−2\nσPθωθ+σf/parenrightbig\n(τ)\n=ωθ+σ(τ)/integraldisplay\nR/integraldisplay\nRPθ(τ−ξ)ω−2\nσ(ξ)Pθ(ξ−η)f(η)dηdξ. (45)\nMore precisely, we have the following result.\nLemma 22 (The symmetrised operator) .\nLetθ,σ∈(0,π), letωσbe the weight functions defined in (40), let the symmetrised operator\nTθ,σbe defined through Equation (45), and let Pθbe the Poisson kernel defined in (21). Then the\noperator ωθ+σPθω−1\nσ, defined via\nf∈L2(R)/mapsto→(ωθ+σPθω−1\nσf)(τ) =ωθ+σ(τ)/integraldisplay\nRPθ(τ−ξ)ω−1\nσ(ξ)f(ξ)dξ,\nis bounded in L2(R)if and only if Tθ,σis bounded in L2(R). Moreover, the following identities\nhold:\n/bardblTθ,σ/bardblL2→L2=/bardblωθ+σPθω1\nσ/bardbl2\nL2→L2, (46)\n/bardblTθ,σ+Tπ−θ,π−σ/bardblL2→L2=/bardblωθ+σ/parenleftbig\nPθω−1\nσ⊕Pπ−θω−1\nπ+σ/parenrightbig\n/bardbl2\nL2→L2. (47)\nProof. The proof is a simple consequence of the well-known fact that for any bounded linear\noperator A: L2(R)→L2(R),\n/bardblAA∗/bardblL2→L2=/bardblA/bardbl2\nL2→L2=/bardblA∗/bardbl2\nL2→L2, (48)\nwhereA∗denotes the adjoint operator with respect to the L2(R)inner-product. Indeed, from\nstraightforward calculation we see that the formal adjoint of the operator Aθ,σ:=ωθ+σPθω−1\nσ\nwith respect to the L2inner-product, i.e., the unique linear operator A∗\nθ,σ:C∞\nc(R)→C∞(R)\nsatisfying\n/a\\}bracketle{tA∗\nθ,σf,g/a\\}bracketri}htL2(R):=/integraldisplay\nR(A∗\nθ,σf)(τ)g(τ) dτ=/a\\}bracketle{tf,Aθ,σg/a\\}bracketri}htL2(R)for allf,g∈C∞\nc(R),\nis given by\n(A∗\nθ,σf)(τ) =ω−1\nσ(τ)/integraldisplay\nRPθ(τ−ξ)ωθ+σ(ξ)f(ξ)dξ.\nThus by Fubini’s theorem, we see that the formal adjoint A∗\nθ,σis bounded in L2(R)if and only\nif there exists some C >0such that for any f∈C∞\nc(R)we have that\n/integraldisplay\nR|(Aθ,σf)(τ)|2dτ=/integraldisplay\nR/integraldisplay\nR/integraldisplay\nRf(��)ωθ+σ(ξ)Pθ(τ−ξ)ω−2\nσ(τ)Pθ(τ−η)ωθ+σ(η)f(η)dηdξdτ\n=/integraldisplay\nRf(ξ)(Tθ,σf)(ξ)dξ=/a\\}bracketle{tf,Tθ,σf/a\\}bracketri}ht ≤C/bardblf/bardbl2\nL2(R).\nMoreover, the smallest possible Cin the above inequality is precisely the square of the norm of\nA∗\nθ,σ. On the other hand, since Tθ,σis formally symmetric, the above inequality holds if and onl y\nifTθ,σis bounded in L2(R), in which case the smallest possible Cis given precisely by the norm\nofTθ,σ. We thus conclude that Tθ,σis bounded in L2(R)if and only if A∗\nθ,σis bounded in L2(R)\nand/bardblTθ,σ/bardbl=/bardblA∗\nθ,σ/bardbl2. To complete the proof of (46), we can use the identity (A∗\nθ,σ)∗=Aθ,σand\n(48). The proof of (47) then follows from similar arguments.\n14The next step in our analysis is to obtain an explicit express ion for the integral kernel of the\nsymmetrized operator Tθ,σ. The fact that this is indeed possible is a consequence of the following\ncalculation.\nLemma 23 (Integral kernel of the symmetrised operator) .\nLetθ,σ∈(0,π), and let ωσ+θbe the weight function defined in Equation (40). Then the\nintegral kernel of the symmetrized operator Tθ,σdefined in Equation (45)is given by\nTθ,σ(τ,ξ) =ωθ+σ(τ)/parenleftbigg\ncosh/parenleftbiggτ+ξ\n2/parenrightbigg\nFθ/parenleftbiggτ−ξ\n2/parenrightbigg\n−cos(σ)Gθ/parenleftbiggτ−ξ\n2/parenrightbigg/parenrightbigg\nωθ+σ(ξ), τ,ξ ∈R\n(49)\nwhere\nFθ(τ) =π−θ\n2πPθ(τ)+1\n2πQθ(τ)+π−θ\n2πPπ−θ(τ)−1\n2πQπ−θ(τ), (50)\nGθ(τ) =π−θ\n2πPθ(τ)+1\n2πQθ(τ)−π−θ\n2πPπ−θ(τ)+1\n2πQπ−θ(τ), (51)\nPθis the Poisson kernel and Qθis the auxiliary function defined through Equation (32).\nProof. From the definition of Tθ,σgiven in Equation (45), it is enough to evaluate the integral\n∀ξ,τ∈R:/parenleftbig\nPθω−2\nσPθ/parenrightbig\n(τ,ξ) =/integraldisplay\nRPθ(z−τ)Pθ(z−ξ)\nωσ(z)2\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n:=fτ,ξ(z)dz (52)\nRecalling the 2πı-periodic meromorphic extension of the Poisson kernel Pθdescribed by Re-\nmark 16 and the definition of the weight function ω−1\nσ, it is easy to see that for any τ,ξ∈R,\nthe function fτ,ξ(·)admits a 2πı-periodic meromorphic extension to the complex plane, whic h\nwe continue to denote by fτ,ξ(·). Furthermore, it easy to verify that fτ,ξ(·)has no poles on the\nboundary of the infinite strip S={z∈C: 0≤Im(z)<2π}and that fτ,ξ(z)decays exponentially\nfast as|Re(z)| → ∞ .\nConsequently, we can compute the integral in Equation (52) b y applying the Residue Theorem\nto the function h(z) =zfξ,τ(z)along the contour ∂S. Indeed, since fτ,ξ(z)is2πı-periodic we have\nRes(h,S) =1\n2πı/contintegraldisplay\n∂Sh(z)dz=1\n2πı/parenleftbigg/integraldisplay\nRh(z)dz−/integraldisplay\nRh(z+2πı)dz/parenrightbigg\n=−/integraldisplay\nRfτ,ξ(z)dz=/parenleftbig\nPθω−2\nσPθ/parenrightbig\n(τ,ξ).\nA straightforward calculation then yields\n−Res(h,S) =−Res(h,τ+ıθ)−Res(h,τ+ı2π−ıθ)−Res(h,ξ+ıθ)−Res(h,ξ+ı2π−ıθ)\n=−1\n2πı/parenleftbigg\n(τ+ıθ)Pθ(τ−ξ+ıθ)\nω2σ(τ+ıθ)−(τ+2πı−ıθ)Pθ(τ−ξ−ıθ)\nω2σ(τ−ıθ)\n+(ξ+ıθ)Pθ(τ−ξ−ıθ)\nω2σ(ξ+ıθ)−(ξ+2ıπ−ıθ)Pθ(τ−ξ+ıθ)\nω2σ(ξ−ıθ)/parenrightbigg\n.(53)\nOne can now proceed from Equation (53) to the sought-after Eq uation (49) in two ways.\nThe first is by using elementary trigonometric identities su ch ascosh(a+b) = cosh( a)cosh(b)+\nsinh(a)sinh(b),∀a,b∈C. This approach is rather cumbersome and time-consuming. A m ore\nefficient approach is as follows. First, we observe that for fix edξ∈R, the right-hand side of\nEquation (53) and of Equation (49) define meromorphic functi ons ofτwith exactly the same\npoles. The difference gof these two functions is therefore an entire function satis fying the estimate\ng(τ)/lessorsimilar|τ|. Liouville’s theorem implies that these two functions must differ by at most a linear\nfunction in τ. One can then verify that the two functions actually agree at two distinct points in\nthe complex plane (e.g., at τ=ξand atτ=−ξ) to complete the proof.\n15Remark 24.Consider the proof of Lemma 23. An alternative strategy to es tablish Equation (49)\nis to use the change of variables t=ez. The resulting computations are, nevertheless, somewhat\ncumbersome, so we have resorted instead to the residue theor em.\nEquipped with the expressions offered by Lemma 23, we are now i n a position to prove The-\norem 21. This will be done in two steps. In the first step, we wil l prove a lower bound for the\noperator norms of Tθ,σandTθ,σ+Tπ−θ,π−σfor all values of θandσin the interval (0,π). This will\nbe accomplished using a relatively simple and general lemma which, it turns out, is also enough\nto prove a matching upper bound for the operator norms for certain values of θandσ. For the\nremaining choices of θ,σ∈(0,π), we will deduce an upper bound for the operator norms by ap-\npealing to the theory of complex functions and the explicit F ourier transform of the Poisson kernel\nand the auxiliary function Qθcomputed in Section 2.3.\n3.2 The lower bound and a partial upper bound\nThroughout this subsection, we will use the notation and fra mework developed in Sections 2 and\n3.1. Of particular importance will be the operator Tθ,σ: L2(R)→L2(R)defined through Equation\n(45). The aim of this section is to obtain a sharp lower bound o n the operator norms of /bardblTθ,σ/bardbl\nand/bardblTθ,σ+Tπ−θ,π−σ/bardblin the entire range θ,σ∈(0,π), and a matching upper bound in the case\nπ/2≤θ+σ≤3π/2. The key abstract lemma that we will use for this purpose is th e following.\nLemma 25 (Bounds for multiplier-like operators) .\nLet{Fj}N\nj=1⊂L1(R)be a collection of non-negative even functions and {ρj}N\nj=1⊂L∞(R).\nThen, the operator is defined via\n∀f∈L2(R): (Tf)(x) :=N/summationdisplay\nj=1/integraldisplay\nRρj(x)Fj(x−y)ρj(y)f(y)dy (54)\nis a bounded from L2(R)toL2(R)with the upper bound\n/bardblT/bardblL2→L2≤/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nj=1/bardblFj/bardblL1|ρj|2/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞. (55)\nMoreover, suppose that lim|x|→∞|ρj(x)|=ajexists for all j∈ {1,...,N}and define the\noperator Tusing Equation (54)for general {Fj}j≤N⊂L1(R)(i.e., for {Fj}j≤Nnot necessarily\neven and non-negative). In this case, we have the lower bound\n/bardblT/bardblL2→L2≥/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nj=1a2\nj/integraldisplay\nRFj(τ)dτ./vextendsingle/vextendsingle/vextendsingle/vextendsingle(56)\nProof. To obtain the upper bound, we note that from Plancherel’s the orem and the convolution\nproperty of the Fourier transform, we deduce that for all f∈L2(R), it holds that\n|/a\\}bracketle{tf,Tf/a\\}bracketri}ht|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj=1/a\\}bracketle{tρjf,Fj∗(ρjf)/a\\}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj=11\n2π/integraldisplay\nR/hatwideFj(k)|/hatwidestρjf(k)|2dk/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nRN/summationdisplay\nj=1/bardbl/hatwideFj/bardblL∞ρ2\nj(k)|f(k)|2dk\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nj=1/bardbl/hatwideFj/bardblL∞ρ2\nj/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞/bardblf/bardbl2\nL2.\nSince each Fj, j∈ {1,...,N}is non-negative and belongs to L1, we have /bardbl/hatwideFj/bardblL∞=/hatwideFj(0) =\n/bardblFj/bardblL1. Moreover, as the operator Tis symmetric, we conclude that\n/bardblT/bardblL2→L2= sup\nf∈L2(R)|/a\\}bracketle{tf,Tf/a\\}bracketri}ht|\n/bardblf/bardbl2\nL2≤/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN/summationdisplay\nj=1/bardblFj/bardblL1|ρj|2/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞,\n16which proves Estimate (55).\nTo obtain the lower bound, we introduce χ:R→Ras the characteristic function of the interval\n[−1/2,1/2]and use the trial states χδ(x) =χ(δx)δ1\n2. Then, since each Fj∈L1(R)and each ρj\nis bounded, we can use the change of variables z=x−yandw=δ(x+y)and dominated\nconvergence to obtain\n/bardblT/bardblL2→L2≥lim\nδ→0+|/a\\}bracketle{tχδ,Tχδ/a\\}bracketri}ht|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglelim\nδ→0+N/summationdisplay\nj=11\n2/integraldisplay1\n−1/integraldisplay1−|w|\nδ\n−1+|w|\nδFj(z)ρj/parenleftbiggδ−1w−z\n2/parenrightbigg\nρj/parenleftbiggδ−1w+z\n2/parenrightbigg\ndzdw/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nj=1a2\nj/integraldisplay\nRFj(z)dz/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhich completes the proof.\nTo make use of Lemma 25, we now need to compute the L1norms of the functions FθandGθ\nappearing in Equation (49). It follows directly from the defi nitions of the functions FθandGθ\nthat these L1norms can be computed using the Fourier transform of the Pois son kernel Pθand\nauxiliary function Qθcomputed in Corollary 17 and Lemma 19 in Section 2.3.\nLemma 26. Letθ∈(0,π)and let the functions Fθ:R→RandGθ:R→Rbe defined as in\nEquations (50)and(51)respectively. Then for all ω∈Rit holds that\n/hatwideFθ(ω) =cos(θ)\n2/parenleftbigg\n1+cosh(πω)/parenleftbig\n1−cosh(πω)/parenrightbig\nsinh(πω)2/parenrightbigg\n+/hatwiderPπ−θ(ω)\n2+cosh(πω)\nsinh(πω)2cosh/parenleftbig\n(π−θ)ω/parenrightbig\n−cosh(θω)\n2,\n(57)\n/hatwideGθ(ω) =1\n2/parenleftbigg\n1−cosh(πω)/parenleftbig\n1+cosh( πω)/parenrightbig\nsinh(πω)2/parenrightbigg\n−/hatwiderPπ−θ(ω)\n2+cosh(πω)\nsinh(πω)2cosh/parenleftbig\n(π−θ)ω/parenrightbig\n+cosh(θω)\n2.\n(58)\nIn particular,\n/bardblFθ/bardblL1=1+cos(θ)\n4and/bardblGθ/bardblL1=(π−θ)2\n2π2. (59)\nProof. Formulas (57) and (58) follow immediately from the definitio n ofFθandGθgiven through\nEquations (50) and (51) respectively, together with the exp ressions for the Fourier transform of\nthe Poisson kernel Pθand auxiliary meromorphic function Qθgiven by Corollary 17 and Lemma\n19. Indeed, since both FθandGθare non-negative, we have\n/bardblFθ/bardblL1=/integraldisplay\nRFθ(τ)dτ=/hatwideFθ(0)and/bardblGθ/bardblL1=/hatwideGθ(0),\nand the sought-after L1norms can thus be obtained from the expressions for /hatwiderPθ(0)and/hatwideQθ(0)\ngiven in Equations (31) and (35) respectively.\nCombining now Lemma 26 with the prior Lemma 25, we obtain the f ollowing result.\nLemma 27 (Lower bound and partial upper bound on the operator norm of s ymmetrized op-\nerators) .Letσ,θ∈(0,π)and let the symmetrized operator Tθ,σ: L2(R)→L2(R)be defined\naccording to Equation (45). Then we have the estimate\n/bardblTθ,σ/bardblL2→L2≥1+cos(θ)\n2and/bardblTθ,σ+Tπ−θ,π−σ/bardblL2→L2≥1. (60)\nIn particular, the direction ≥in Estimates (43)and(44)in Theorem 21 holds. Moreover, if\nθ+σ∈[π/2,3π/2]then we also have the estimate\n/bardblTθ,σ/bardblL2→L2≤1+cos(θ)\n2. (61)\n17Proof. Using the identity cosh(a+b) = cosh( a)cosh(b)+sinh(a)sinh(b)∀a,b∈C, we can rewrite\nthe kernel of Tθ,σ, as\n∀τ,ξ∈R:Tθ,σ(τ,ξ) =2/summationdisplay\nj=1ρj(τ)Fθ/parenleftbiggτ−ξ\n2/parenrightbigg\nρj(ξ)−cos(σ)ρ3(τ)Gθ/parenleftbiggτ−ξ\n2/parenrightbigg\nρ3(ξ), (62)\nwhere the functions FθandGθare defined through Equations (50) and (51) respectively, an d for\nallτ∈R, we have\nρ1(τ) := cosh( τ/2)ωθ+σ(τ), ρ2(τ) := sinh( τ/2)ωθ+σ(τ),andρ3(τ) :=ωθ+σ(τ),(63)\nwith the weight functions σθ+σdefined through Equation (40).\nWe are now in the situation described by the second part of Lem ma 25. Thus, using the lower\nbound (56) and Lemma 26 we deduce that\n/bardblTθ,σ/bardblL2→L2≥2/summationdisplay\nj=1lim\n|ξ|→∞ρj(ξ)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=1/2/integraldisplay\nRFθ(τ/2)dτ+ lim\n|ξ|→∞ρ3(ξ)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=0/integraldisplay\nRGθ(τ/2)dτ\n= 2/bardblFθ/bardblL1=1+cos(θ)\n2,\nwhich proves the lower bound for the operator norm /bardblTθ,σ/bardblL2→L2in Estimate (60).\nTo obtain the lower bound on the operator norm /bardblTθ,σ+Tπ−θ,π−σ/bardblL2→L2, we observe that for\nallτ∈Rit holds that\nFθ(τ)+Fπ−θ(τ) =Pθ(τ)+Pπ−θ(τ)\n2andGθ(τ)−Gπ−θ(τ) =Pθ(τ)−Pπ−θ(τ)\n2,\nwith the Poisson kernel Pθ:R→Rdefined through Definition 11.\nThus, appealing to Equation (49) and using again the splitti ngcosh(τ+ξ) = cosh( τ)cosh(ξ)+\nsinh(τ)sinh(ξ), we deduce that for all τ,ξ∈Rit holds that\n(Tθ,σ+Tπ−θ,π−σ)(τ,ξ) =2/summationdisplay\nj=1ρj(τ)/parenleftbiggPθ+Pπ−θ\n2/parenrightbigg/parenleftbiggτ−ξ\n2/parenrightbigg\nρj(ξ)\n−cos(σ)ρ3(τ)/parenleftbiggPθ−Pπ−θ\n2/parenrightbigg/parenleftbiggτ−ξ\n2/parenrightbigg\nρ3(ξ),\nwith{ρj}3\nj=1defined through Equation (63). Applying once again Lemma 25 t ogether with the\nexpression for the L1norm of the Poisson kernel given by Corollary 17 yields the so ught-after\nlower bound in Estimate (60) from which we deduce that the dir ection≥in Estimates (43) and\n(44) in Theorem 21 holds.\nWe now assume that θ+σ∈[π/2,3π/2]and turn our attention to the upper bound. We\ndeal with the cases σ≥π/2andσ≤π/2separately. In the first case, we note that the function\n−cos(σ)Gθis non-negative for σ≥π/2. Considering the expression for the integral kernel Tθ,σ\noffered by Equation (62), we see that we are in the situation de scribed by the first part of Lemma 25\nand the Estimate (55) yields\n/bardblTθ,σ/bardblL2→L2≤sup\nτ∈R/braceleftbigg2/bardblFθ/bardblL1cosh(τ)−2cos(σ)/bardblGθ/bardblL1\ncosh(τ)−cos(θ+σ)/bracerightbigg\n= 2/bardblFθ/bardblL1sup\nτ∈R/braceleftbigg\n1+cos(θ+σ)−cos(σ)/bardblGθ/bardblL1\n/bardblFθ/bardblL1\ncosh(τ)−cos(θ+σ)/bracerightbigg\n=1+cos(θ)\n2max/braceleftbigg1−cos(σ)2(π−θ)2\nπ2(1+cos( θ)\n1−cos(θ+σ),1/bracerightbigg\n=1+cos(θ)\n2.\n18Let us remark here that the last equality, i.e., the fact that the maximum appearing in the\nsecond to last expression is indeed one, can be shown using ar guments very similar to those used\nto prove the forthcoming Lemma 31. For the sake of brevity, we omit this detail.\nIt therefore remains to consider the second case σ≤π/2. To deal with this situation, we exploit\nthe positivity of the kernel of Tθ,σ. More precisely, we note that because the integral kernel Tθ,σ\nis non-negative, we have that\n∀f∈L2(R): 0≤ /a\\}bracketle{tf,Tθ,σf/a\\}bracketri}ht ≤ /a\\}bracketle{t|f|,Tθ,σ|f|/a\\}bracketri}ht\nConsequently, it is enough to bound the expectation value /a\\}bracketle{tf,Tθ,σf/a\\}bracketri}htfor non-negative f∈L2(R).\nTherefore, as long as σ≤π/2, we can simply discard the Gθpart of the kernel of Pθω−2\nσPθ.\nIndeed, in this case cos(σ)Gθ≥0, and we therefore have\n∀non-negative f∈L2(R):/a\\}bracketle{tf,Tθ,σf/a\\}bracketri}ht ≤/a\\}bracketle{tf,Tθ,σf/a\\}bracketri}ht\n+cos(σ)/integraldisplay2\nRf(τ)ωσ+θ(τ)Gθ/parenleftbiggτ−ξ\n2/parenrightbigg\nωθ+σ(ξ)f(ξ)dξdτ.\nRecalling from Equation (54) that for all ξ,τ∈Rit holds that\nTθ,σ(τ,ξ)+cos(σ)ωθ+σ(τ)Gθ/parenleftbiggτ−ξ\n2/parenrightbigg\nωθ+σ(ξ) =2/summationdisplay\nj=1ρj(τ)Fθ/parenleftbiggτ−ξ\n2/parenrightbigg\nρj(ξ), (64)\nwithρ1,ρ2defined as in (63), we can again use the upper bound offered by Eq uation (55) to\ndeduce that\n/bardblTθ,σ/bardbl ≤1+cos(θ)\n2max/braceleftbigg1\n1−cos(θ+σ),1/bracerightbigg\n=1+cos(θ)\n2,\nwhere the last identity follows from the fact that cos(θ+σ)≤0for anyθ+σ∈(π/2,3π/2). This\ncompletes the proof.\nConsider now the proof of Lemma 27. We end this subsection wit h a brief heuristic discussion\non why the method of proof that we have used yields a sharp upper bound in the range θ+σ∈\n(π/2,3π/2). Neglecting the Gθpart of the kernel for the sake of simplicity, we see that findi ng\nthe operator norm of Tθ,σcorresponds to maximizing the integral\n/integraldisplay\nR/hatwideFθ(τ)/parenleftbig\n|/hatwidestfρ1|2(τ)+|/hatwidestfρ2|2(τ)/parenrightbig\ndτ,forfwith/bardblf/bardbl= 1, (65)\nwhereρ1andρ2are the functions defined in (63). Since /hatwideFθis positive and attains its maximum\nat0, there are two ways to increase this integral. First, we woul d like the total mass\n/integraldisplay\nR|fρ1|2(τ)+|fρ2|2(τ)dτ=1\n2π/integraldisplay\nR|/hatwidestfρ1|2(τ)+|/hatwidestfρ2|2(τ)dτ (66)\nto be as big as possible. Second, we would like this mass to be c oncentrated around the origin in\nFourier space. So the reason why the above bound is so effectiv e in the range θ+σ∈(π/2,3π/2)\nis that the functions ρ1andρ2can be seen as a partition of the unity at infinity, in the sense\nthat they satisfy ρ2\n1(τ) +ρ2\n2(τ)≤1andρ2\n1(τ) +ρ2\n2(τ)→1as|τ| → ∞ . In particular, we can\nsimultaneously increase the total mass in (66) and concentr ate it around the origin (in Fourier\nspace) by sending the mass of fto infinity (in real space).\nOn the other hand, when θ+σ/\\e}atio\\slash∈(π/2,3π/2), the functions ρ1andρ2start to peak at 0.\nConsequently, increasing the L2mass in (66) corresponds to concentrating the mass of faround\nthe origin. This, in turn, results in diffusing the mass of /hatwidestfρ1and/hatwidestfρ2towards infinity, which\nleads to (possibly) decreasing the value of (65). There is, t herefore, a subtle competition between\nconcentrating the mass around the origin in Fourier space an d in real space, which suggests that an\nuncertainty principle could be useful in this case. Indeed, this is the motivation for the forthcoming\nsection, where we develop Fourier domain arguments to deriv e the sought-after operator norm\nupper bound in the case θ+σ/\\e}atio\\slash∈(π/2,3π/2).\n193.3 An upper bound for the remaining cases\nThe goal of this section is to complete the proof of the operat or norm upper bound in Theorem 21\nfor the remaining cases θ,σ∈(0,π)withθ+σ∈(0,π/2)∪(3π/2,2π). For these cases, as\nhighlighted at the end of the previous subsection, the rathe r general and abstract arguments used\nthus far are not enough. Instead, we transfer our problem to t he Fourier domain and exploit the full\nstructure of the Poisson kernel by using the explicit Fourie r transforms computed in Section 2.3.\nWe begin by recalling, for any σ∈(0,π)the weighted Lebesgue space L2\nσdefined through\nDefinition 8 and the weight function multiplicative operato rωσdefined through Equation (40).\nTo transport our problem to the Fourier domain, we first obser ve thatωσdefines an isometry from\nL2(R)to the dual space of L2\nσwith respect to the L2(R)inner-product, i.e., to the weighted dual\nspace defined as\n(L2\nσ)∗:=/braceleftbigg\ng:R→Cmeasurable :/bardblf/bardbl2\n(L2\nσ)∗:=/a\\}bracketle{tg,ω−2\nσg/a\\}bracketri}ht=/integraldisplay\nR|g(τ)|2ω−2\nσ(τ)dτ <∞/bracerightbigg\n.(67)\nThus, to complete the proof of Theorem 21, it is enough to show that\n∀g/\\e}atio\\slash= 0∈(L2\nθ+σ)∗:/a\\}bracketle{tg,Pθω−2\nσPθg/a\\}bracketri}ht<1+cos(θ)\n2/bardblg/bardbl2\n(L2\nθ+σ)∗. (68)\nThe advantage of working with the dual space is that we can now use the following lemma to\nexpress the right-hand side of (68) to the Fourier domain.\nLemma 28 (Fourier transform of elements of the dual weighted spaces) .\nLetσ∈(0,π), let the weighted Lebesgue space L2\nσbe defined through Definition 8, and let\ng∈(L2\nσ)∗. Then, the Fourier transform of gadmits an extension to the closure of the dual strip\nS∗:={ω+ıβ∈C:ω∈R, β∈(−1/2,1/2)} (69)\nsuch that /hatwideg|S∗is holomorphic and /hatwideg±1/2= lim\nβ→±1/2/hatwidegβin theL2(R)norm with /hatwidegγ=/hatwideg(·+ıγ)∀γ∈R.\nAdditionally, we have the estimate\n/bardblg/bardbl2\n(L2σ)∗=1\n2π/integraldisplay\nR|/hatwideg1/2|2(ω)+|/hatwideg−1/2(ω)|2\n2−cos(σ)|/hatwideg(ω)|2dω. (70)\nProof. The Cauchy-Schwarz inequality implies that\n/hatwideg(z) =/integraldisplay\nRe−ızτg(τ)dτ≤ /bardblg/bardbl(L2σ)∗/parenleftbigg/integraldisplay\nRe2Im(z)τ\ncosh(τ)−cos(σ)dτ/parenrightbigg1\n2\n.\nConsequently, the function /hatwidegis well-defined in S∗and bounded away from the boundary ∂S∗.\nMoreover, the above estimate allows us to apply Fubini’s the orem to conclude that\n/contintegraldisplay\nγ/hatwideg(z)dz=/integraldisplay\nR/parenleftbigg/contintegraldisplay\nγe−ızτdz/parenrightbigg\ng(τ)dτ= 0, (71)\nfor any closed smooth curve γcontained in S∗. Thus/hatwidegis holomorphic on S∗by Morera’s theorem.\nFor the existence of boundary values in the L2sense, we note that ge±τ/2∈L2(R)by Hölder’s\ninequality. So setting\n/hatwideg±1/2(ω):=/hatwiderge±τ/2(ω) =/integraldisplay\nRg(τ)e−ıωτ±τ/2dτ∈L2(R), (72)\nwe find from Plancherel’s theorem and Hölder’s inequality th at\n/bardbl/hatwideg±1/2−/hatwidegβ/bardbl2\nL2= 2π/integraldisplay\nR|g(τ)|2|e±τ/2−eβτ/2|2dτ≤ /bardblg/bardbl2\n(L2σ)∗sup\nτ∈R/braceleftbigg|e±τ/2−eβτ/2|2\ncosh(τ)−cos(σ)/bracerightbigg\n,\nwhich implies the convergence in the limit β→ ±1/2. Eq. (70) is now a straightforward conse-\nquence of (72) and Plancherel’s theorem.\n201\n2\nδ\nCδ\nδ\nFigure 4: 2-D schematic diagram of the contour plot Cδ.\nRemark 29.In other words, Lemma 28 says that the Fourier transform of fu nctions in (L2\nσ)∗can\nbe identified with the L2-Hardy space on the dual strip Sω. This connection will not be further\nexplored here.\nIn order to complete the proof of the operator norm upper boun d in Theorem 21, we will make\nuse of the following key lemma.\nLemma 30. Letθ∈(0,π)and let the mappings Fθ:R→RandGθ:R→Rbe defined through\nEquations (50)and(51)respectively. Then for all g∈(L2\nσ)∗it holds that\n/integraldisplay\nR/hatwideFθ(2ω)Re{/hatwideg1/2(ω)/hatwideg−1/2(ω)}dω=/integraldisplay\nR|/hatwideg(ω)|2cos(θ)/hatwideGθ(2ω)dω. (73)\nProof. The key step in the proof is the following observation. From t he explicit formulas for the\nFourier transforms of FθandGθin (57) and (58), and the elementary identities cosh(a+b) =\ncosh(a)cosh(b)+sinh( a)sinh(b)andsinh(a+b) = sinh( a)cosh(b)+sinh( b)cosh(a), we find\nRe{/hatwideFθ(ω−ı)}=cos(θ)\n2/parenleftbigg\n1−cosh(πω)/parenleftbig\n1+cosh( πω)/parenrightbig\nsinh(πω)2/parenrightbigg\n+Re/braceleftbigg/hatwiderPπ−θ(ω−ı)\n2\n−cosh(πω)\nsinh(πω)2cosh/parenleftbig\n(π−θ)ω−ı(π−θ)/parenrightbig\n−cosh(θω−ıθ)\n2/bracerightbigg\n=cos(θ)\n2/parenleftbigg\n1−cosh(πω)/parenleftbig\n1+cosh( πω)/parenrightbig\nsinh(πω)2−/hatwiderPπ−θ(ω)\n+cosh(πω)\nsinh(πω)2/parenleftbig\ncosh/parenleftbig\n(π−θ)ω)+cosh( θω)/parenrightbig/parenrightbigg\n= cos(θ)/hatwideGθ(ω). (74)\nWe can now apply the residue theorem to the function h(z):=Fθ(2z−ı)/hatwideg(z)/hatwideg(z)along the contour\nCδ={ω∈R:|ω|> δ}∪{δe−ıα:−π < α < 0}/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n:=Sδ∪{ω+ı/2−ıδ:ω∈R}\n(see Figure 4) and take the limit δ→0to conclude. Indeed, since his holomorphic inside Cδ, by\n21(74) and the Residue theorem we find\n/integraldisplay\nRcos(θ)/hatwideGθ(2ω)|/hatwideg(ω)|2dω= Re/braceleftbigg/integraldisplay\n|ω|>δ/hatwideFθ(2ω−ı)|/hatwideg(ω)|2dω/bracerightbigg\n+O(δ)\n= Re/braceleftbigg/integraldisplay\nR/hatwideFθ(2z−ı2δ)/hatwideg1/2−δ(ω)/hatwideg−1/2+δ(ω)dω/bracerightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n:=J(δ)+O(δ)\n−Re/braceleftbigg/integraldisplay\nSδ/hatwideFθ(2z−ı)/hatwideg(z)/hatwideg(z)dz/bracerightbigg\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n:=I(δ).\nWe now note that I(δ)converges to 0asδ→0because/hatwideg(z)/hatwideg(z)is real-valued at 0and/hatwideFθ(2z−ı)\nhas a simple pole at 0with imaginary residue. On the other hand we have /hatwideg±1/2∓δ→/hatwideg±1/2in\nL2(R)by Lemma 28, which implies\nlim\nδ→0J(δ) =/integraldisplay\nR/hatwideFθ(2ω)Re{/hatwideg1/2(ω)/hatwideg−1/2(ω)}dω (75)\nand completes the proof.\nWe are now in a position to prove estimate (68).\nProof of estimate (68).From the formula\n/parenleftbig\nPθω−2\nσPθ/parenrightbig\n(τ,ξ) =Fθ/parenleftbiggτ−ξ\n2/parenrightbigg/parenleftbig\ncosh(τ/2)cosh(ξ/2)+sinh( τ/2)sinh(ξ/2)/parenrightbig\n−cos(σ)Gθ/parenleftbiggτ−ξ\n2/parenrightbigg\n,\n(76)\nPlancherel’s theorem, and Lemma 28, we find\n/a\\}bracketle{tg,Pθω−2\nσPθg/a\\}bracketri}ht=1\nπ/integraldisplay\nR/hatwideFθ(2ω)|/hatwideg1/2+/hatwideg−1/2|(ω)2+|/hatwideg1/2−/hatwideg−1/2|(ω)2\n4−cos(σ)/hatwideGθ(2ω)|/hatwideg(ω)|2dω\n=1\nπ/integraldisplay\nR/hatwideFθ(2ω)|/hatwideg1/2(ω)|2+|/hatwideg−1/2(ω)|2\n2dω\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n:=I−cos(σ)\nπ/integraldisplay\nR|/hatwideg(ω)|2/hatwideGθ(2ω)dω. (77)\nWe can now use Young’s inequality and Lemma 30 to obtain\nI\n/bardbl/hatwideFθ/bardblL∞=/integraldisplay\nR|/hatwideg1/2(ω)|2+|/hatwideg−1/2(ω)|2\n2−/parenleftbigg\n1−/hatwideFθ(2ω)\n/bardbl/hatwideFθ/bardblL∞/parenrightbigg|/hatwideg1/2(ω)|2+|/hatwideg−1/2(ω)|2\n2dω\n≤/integraldisplay\nR|/hatwideg1/2(ω)|2+|/hatwideg−1/2(ω)|2\n2−/parenleftbigg\n1−/hatwideFθ(2ω)\n/bardbl/hatwideFθ/bardblL∞/parenrightbigg\nRe{/hatwideg1/2/hatwideg−1/2(ω)}dω\n=/integraldisplay\nR|/hatwideg1/2(ω)|2+|/hatwideg−1/2(ω)|2\n2−/parenleftbigg\n1−cos(θ)/hatwideGθ(2ω)\n/bardbl/hatwideFθ/bardblL∞/parenrightbigg\n|/hatwideg(ω)|2dω.\nIf we now use this estimate for Iback in (77), plug in the value /bardbl/hatwideFθ/bardblL∞=/bardblFθ/bardblL1=1+cos(θ)\n4\n(see eq. (59)), and use the norm characterization in eq. (70) of Lemma 28, we find\n/a\\}bracketle{tg,Pθω−2\nσPθg/a\\}bracketri}ht ≤1+cos(θ)\n2/bardblg/bardbl2\n(L2\nσ+θ)∗\n−1\nπ/integraldisplay\nR|/hatwideg(ω)|2/parenleftbigg1+cos(θ)\n2/parenleftbig\n1−cos(θ+σ)/parenrightbig\n−(cos(θ)−cos(σ)/parenrightbig\n2/hatwideGθ(2ω)/parenrightbigg\ndω/parenrightbigg\n.(78)\nFinally, we can use the fact that 2/bardbl/hatwideGθ/bardblL∞=(π−θ)2\nπ2(see eq. (59)) and Lemma 31 below to\nshow that the integrand in (78) is (almost) everywhere posit ive for any σ,θ∈(0,π)and therefore\n(68) holds.\n22Lemma 31. For any θ,σ∈(0,π)it holds that\n1+cos(θ)\n2/parenleftbig\n1−cos(θ+σ)/parenrightbig\n−/parenleftbig\ncos(θ)−cos(σ)/parenrightbig(π−θ)2\nπ2>0. (79)\nProof. First note that if θ≥σ, we have cos(θ)−cos(σ)≤0, which implies\n1+cos(θ)\n2/parenleftbig\n1−cos(θ+σ)/parenrightbig\n−/parenleftbig\ncos(θ)−cos(σ)/parenrightbig(π−θ)2\nπ2≥1+cos(θ)\n2/parenleftbig\n1−cos(θ+σ)/parenrightbig\n>0\nsince0< θ+σ <2π. So let us consider the case σ < θ. In this case, we first note that\n1+cos(θ)\n2−(π−θ)2\nπ2>0. (80)\nIndeed, this can be shown by computing the minimum of the func tion on the left-hand side or\nby noticing that Fθ(τ)> Gθ(τ)(which follows from the definitions of Fθ,GθandQθ) and using\nLemma 26. Next, from (80) (and cos(θ)−cos(σ)≥0) we find\n1+cos(θ)\n2/parenleftbig\n1−cos(θ+σ)/parenrightbig\n−/parenleftbig\ncos(θ)−cos(σ)/parenrightbig(π−θ)2\nπ2\n>1+cos(θ)\n2/parenleftbigg\n1−cos(θ+σ)−cos(θ)+cos(σ)/parenrightbigg\n.(81)\nOne can now show that the function gθ(σ):= cos(σ)−cos(θ+σ)≥gθ(π) =−1+cos(θ), which\ntogether with the above estimate completes the proof of (79) .\nAcknowledgements\nThe authors thank Xavier Claeys and Yvon Maday for helpful di scussions. AJ and BS acknowl-\nedge support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -\nProject number 440641818. BS and TC acknowledge funding by t heDeutsche Forschungsgemein-\nschaft (DFG, German Research Foundation) - Project number 4420475 00 through the Collabora-\ntive Research Center \"Sparsity and Singular Structures\" (S FB 1481)\nReferences\n[CG18] Gabriele Ciaramella and Martin J Gander. Analysis of the parallel schwarz method\nfor growing chains of fixed-sized subdomains: Part II. SIAM Journal on Numerical\nAnalysis , 56(3):1498–1524, 2018.\n[CHS20a] Gabriele Ciaramella, Muhammad Hassan, and Benjam in Stamm. On the scalability of\nthe parallel Schwarz method in one-dimension. In Domain Decomposition Methods in\nScience and Engineering XXV 25 , pages 151–158. Springer, 2020.\n[CHS20b] Gabriele Ciaramella, Muhammad Hassan, and Benjam in Stamm. On the scalability of\nthe Schwarz method. The SMAI journal of computational mathematics , 6:33–68, 2020.\n[CM94] Tony Chan and Tarek Mathew. Domain decomposition alg orithms. Acta Numerica ,\n3:61–143, 1994.\n[Dah77] Björn Dahlberg. Estimates of harmonic measure. Archive for Rational Mechanics and\nAnalysis , 65:275–288, 1977.\n[Dah79] Björn Dahlberg. On the Poisson integral for Lipschi tz andC1-domains. Studia Math. ,\n66(1):13–24, 1979.\n23[DJN15] Victorita Dolean, Pierre Jolivet, and Frédéric Nat af.An introduction to domain decom-\nposition methods: algorithms, theory, and parallel implemen tation . SIAM, 2015.\n[Lio88] Pierre-Louis Lions. On the Schwarz alternating met hod. I. In First International Sym-\nposium on Domain Decomposition Methods for Partial Different ial Equations (Paris,\n1987), pages 1–42. SIAM, Philadelphia, PA, 1988.\n[Lio89] Pierre-Louis Lions. On the Schwarz alternating met hod. II. Stochastic interpretation\nand order properties. In Domain decomposition methods (Los Angeles, CA, 1988) ,\npages 47–70. SIAM, Philadelphia, PA, 1989.\n[Mas09] Javad Mashreghi. Representation theorems in Hardy spaces , volume 74 of London Math-\nematical Society Student Texts . Cambridge University Press, Cambridge, 2009.\n[Med18] Dagmar Medková. The Laplace equation. Boundary value problems on bounded and\nunbounded Lipschitz domains . Cham: Springer, 2018.\n[MF53] Philip M. Morse and Herman Feshbach. Methods of theor etical physics. Vol. I. II.\n(Internat. series in pure and applied physics.) New York: Mc Graw-Hill Book Co. XL,\n1978 p. (1953)., 1953.\n[RS21] Arnold Reusken and Benjamin Stamm. Analysis of the Sc hwarz domain decomposition\nmethod for the conductor-like screening continuum model. SIAM Journal on Numerical\nAnalysis , 59(2):769–796, 2021.\n[TW04] Andrea Toselli and Olof Widlund. Domain decomposition methods-algorithms and the-\nory, volume 34. Springer Science & Business Media, 2004.\n[Ver84] Gregory Verchota. Layer potentials and regularity for the Dirichlet problem for\nLaplace’s equation in Lipschitz domains. Journal of functional analysis , 59(3):572–611,\n1984.\n[Wid61] David V Widder. Functions harmonic in a strip. Proceedings of the American Mathe-\nmatical Society , 12(1):67–72, 1961.\n24" }, { "title": "2401.16413v1.The_geometric_error_is_less_than_the_pollution_error_when_solving_the_high_frequency_Helmholtz_equation_with_high_order_FEM_on_curved_domains.pdf", "content": "arXiv:2401.16413v1 [math.NA] 29 Jan 2024The geometric error is less than the pollution error when\nsolving the high-frequency Helmholtz equation with\nhigh-order FEM on curved domains\nT. Chaumont-Frelet∗, E. A. Spence†\nJanuary 30, 2024\nAbstract\nWe consider the h-version of the finite-element method, where accuracy is inc reased by\ndecreasing the meshwidth hwhile keeping the polynomial degree pconstant, applied to the\nHelmholtz equation. Although the question “how quickly mus thdecrease as the wavenumber\nkincreases to maintain accuracy?” has been studied intensiv ely since the 1990s, none of\nthe existing rigorous wavenumber-explicit analyses take i nto account the approximation of\nthe geometry. In this paper we prove that for nontrapping pro blems solved using straight\nelements the geometric error is order kh, which is then less than the pollution error k(kh)2p\nwhenkis large; this fact is then illustrated in numerical experim ents. More generally, we\nprove that, even for problems with strong trapping, using de gree four (in 2-d) or degree five\n(in 3-d) polynomials and isoparametric elements ensures th at the geometric error is smaller\nthan the pollution error for most large wavenumbers.\n1 Introduction: informal statement of the main result\nTheresultsofthispaperapplytogeneralHelmholtzproblems(inter iorproblems,exteriorproblems,\nwave-guide problems), but for concreteness here we consider th e Helmholtz equation\n−k2µu−∇·(A∇u) =f, (1.1)\nwithk≫1, posed in a bounded domain Ω corresponding to the exterior of an im penetrable\nobstacle where the Sommerfeld radiation condition has been approx imated by a radial perfectly-\nmatched layer (PML). For simplicity, we assume that Ω has characte ristic length scale ∼1. The\nsolutionusatisfies either a zero Dirichlet or zero Neumann boundary condition on the part of ∂Ω\ncorresponding to the impenetrable obstacle, and a zero Dirichlet bo undary condition on the part\nof∂Ω corresponding to the PML truncation boundary. The piecewise-s mooth coefficients µand\nAdescribe the PML near the truncation boundary, and variation of µandAin the rest of the\ndomain corresponds to penetrable obstacles.\nWe assume that the solution operator to ( 1.1) is bounded by kα−1for someα>0, i.e., given\nk0>0 there exists C >0 such that given f∈L2(Ω), the solution u∈H1(Ω) to (1.1) satisfies\n/ba∇dblu/ba∇dblH1\nk(Ω)≤Ckα−1/ba∇dblf/ba∇dblL2(Ω)for allk≥k0, (1.2)\nwhere\n/ba∇dblu/ba∇dbl2\nH1\nk(Ω):=/ba∇dbl∇u/ba∇dbl2\nL2(Ω)+k2/ba∇dblu/ba∇dbl2\nL2(Ω). (1.3)\nRecall that the bound ( 1.2) holds with α= 1 when the problem is nontrapping (i.e., all geometric-\noptic rays starting in a neighbourhood of the scatterer escape to infinity in a uniform time);\nindeed in this case the solution of the exterior Helmholtz problem satis fies (1.2) withα= 1 by\n[Mor75,Vai75], and the solution of the associated radial PML problem inherits this b ound by\n[GLS23, Lemma 3.3].\n∗Inria Univ. Lille and Laboratoire Paul Painlev´ e, 59655 Vil leneuve-d’Ascq, France,\ntheophile.chaumont@inria.fr\n†Department of Mathematical Sciences, University of Bath, B ath, BA2 7AY, UK, E.A.Spence@bath.ac.uk\n1Theorem 1.1 (Informal statement of the main result) Suppose that the Helmholtz problem\nabove is solved using the h-version of the finite-element method with shape-regular me shes with\nmeshwidth h, fixed (but arbitrary) polynomial degree p, and fixed order of the geometric approxi-\nmationq(so thatq= 1for straight elements and q=pfor isoparametric elements).\nSuppose further that the domain Ωand coefficients Aandµsatisfy the natural regularity as-\nsumptions for using degree ppolynomials, and that the data fin(1.1)comes from an incident\nplane wave or point source (or, more generally, a particular Helmholtz solution that is smooth in\na neighbourhood of the scatterer). If\n/parenleftBig\nkh+kα(kh)p/parenrightBig\n(kh)p+kαhqis sufficiently small, (1.4)\nthen the Galerkin solution uhexists, is unique, and satisfies\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)\n/ba∇dblu/ba∇dblH1\nk(Ω)≤C/bracketleftbigg/parenleftBig\n1+kα(kh)p/parenrightBig\n(kh)p+kαhq/bracketrightbigg\n. (1.5)\nWe make seven remarks about the interpretation and context of T heorem1.1.\n1. Near∂Ω, the error /ba∇dblu−uh/ba∇dblH1\nk(Ω)in (1.5) is measured on the subset of Ω where both uand\nuhare well-defined. When Aandµare discontinuous, defining precisely where the error\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)is measured is more involved; see Theorem 4.11below.\n2. In the limit h→0 withkfixed, the bound ( 1.5) showsthat the Galerkinerroris O(hmin{p,q});\nthis is expected, at least in the isoparametric case when p=q, by [CR72a,Len86].\n3. Whenq=∞(i.e., there is no geometric error), the condition ( 1.4) becomes\nkα(kh)2pis sufficiently small ,\nand then ( 1.5) implies that choosing kα(kh)2pto be a sufficiently-small constant maintains\naccuracy of the Galerkin solution as k→ ∞. This result was\n•proved for constant-coefficient problems in 1-d by [ IB97] (see [IB97, Page 350, penulti-\nmate displayed equation], [ Ihl98, Equation 4.7.41]),\n•obtained in the context of dispersion analysis in [ Ain04],\n•proved for 2- and 3-d constant-coefficient Helmholtz problems with the radiation condi-\ntion approximated by an impedance boundary condition in [ DW15] (following [ FW09,\nFW11,Wu14,ZW13]) and for variable-coefficient problems in [ Pem20, Theorem 2.39],\nand then\n•proved for general 2- and 3-d Helmholtz problems with truncation e ither by a PML or\nthe exact Dirichlet-to-Neumann map in [ GS23].\nWe note that bounds under the condition “ kα(kh)psufficiently small” were provedfor several\nspecific Helmholtz problems in [ MS10,MS11], and then for general Helmholtz problems in\n[CFN20].\n4. Whentheproblemisnontrapping, α= 1, andkhq≪1whenk(kh)2pisconstant,regardlessof\nq. Therefore, Theorem 1.1implies that, regardlessof q, choosingkα(kh)2ptobe asufficiently-\nsmall constant maintains accuracy of the Galerkin solution as k→ ∞. That is, even with\nstraightelements ( q= 1), in the limit k→ ∞withhchosento controlthe pollution error, the\ngeometric error is smaller than the pollution error; this feature is illus trated in the numerical\nexperiments in §2.\n5. When the problem is trapping, α>1, and it is not immediate that kαhq≪1 whenkα(kh)2p\nis constant. For Helmholtz problems with strong trapping, the solut ion operator grows\nexponentially through an increasing sequence of wavenumbers, wit h this sequence becoming\nincreasinglydenseas kincreases[ PV99,CP02,BCWG+11]. However,ifasetofwavenumbers\nofarbitrarily-smallmeasureis excluded then ( 1.2) holds for any α>5d/2+2by [ LSW21] and\n2[GLS23, Lemma 3.3] ([ LSW21] proves this result for the original exterior Helmholtz problem,\nand the PML problem inherits this bound by [ GLS23, Lemma 3.3]). With isoparametric\nelementsq=p, and then hqk≪1 whenkα(kh)2pis constant if p≥4 in 2-d and p≥5\nin 3-d. That is, for moderate polynomial degree, isoparametric elem ents, and most (large)\nwavenumbers, the geometric error is smaller than the pollution erro r, even for problems with\nstrong trapping.\n6. To prove Theorem 1.1, we extend the duality argument recently introduced for general\nHelmholtz problems in [ GS23] to incorporate a Strang-lemma-type argument to allow varia-\ntional crimes. We then use the results of [ Len86] to apply this abstract setting to the case of\ngeometric error induced by polynomial element maps.\n7. Finally, we highlight that Theorem 1.1is conceptually similar to the results of [ CFN18,\nCFN23]. Indeed, [ CFN18,CFN23] show that, when solving 2-d or 3-d Helmholtz problems\non domains with corners/conical points using a uniform mesh, the er ror from the corner\nsingularity is smaller than the pollution error (for ksufficiently large). In other words,\nalthough boththe geometricerror andthe errorincurredbycornersingularitiesareimportant\nin the limit h→0 withkfixed, in the limit k→ ∞withhchosen to control the pollution\nerror, both of these errors are smaller than the pollution error.\nOutline of the paper. §2gives numerical experiments illustrating Theorem 1.1for two 2-d\nnontrapping problems ( α= 0) and straight elements. §3contains an abstract analogue of Theorem\n1.1proved under the assumptions that the sesquilinear form is continu ous and satisifies a G˚ arding-\ntype inequality. §4shows that a wide variety of Helmholtz problems fix into the abstract setting\nof§3and proves Theorem 1.1.\n2 Numerical experiments with straight elements in 2-d\nIn this section we demonstrate numerically for two 2-d scattering p roblems the consequence of\nTheorem 1.1discussed in Point 4 above; i.e., that, for nontrapping problems solve d using straight\nelements, in the limit k→ ∞withhchosen to control the pollution error, the geometric error is\nsmaller than the pollution error.\nThe two scattering problems considered.\nDefinition 2.1 (Sound-soft scattering by a 2-d ball) Givenuinc(x) :=eikx1, letutotsatisfy\n−k2utot−∆utot= 0inRd\\B1(0), utot= 0on∂B1(0),\nwhereusca:=utot−uincsatisfies the Sommerfeld radiation condition\n/parenleftbigg∂\n∂r−ik/parenrightbigg\nusca=o(r−(d−1)/2)asr:=|x| → ∞,uniformly in x/r. (2.1)\nDefinition 2.2 (Scattering by a penetrable 2-d ball) Givenuinc(x) :=eikx1and\nA:=/braceleftBigg\n2inB1(0),\n1inRd\\B1(0),andµ:=/braceleftBigg\n1/2inB1(0),\n1inRd\\B1(0),(2.2)\nletutotsatisfy\n−k2µutot−∇·(A∇utot) = 0inRd, (2.3)\nand\n2∂ru+\ntot=∂ru−\ntotandu+\ntot=u−\ntoton∂B1(0), (2.4)\nwhere the subscript +denotes the limit taken with r >1and the subscript −denotes the limit\ntaken withr<1, and where usca:=utot−uincsatisfies the Sommerfeld radiation condition (2.1).\n3(Note that the transmission conditions in ( 2.4) follow from the variational formulation of the PDE\n(2.3).)\nThe choice of the coefficients Aandµin (2.2) implies that this problem is nontrapping, in the\nsense that outgoing solutions to ( 1.1), withAandµgiven by ( 2.2), satisfy the bound ( 1.2) with\nα= 1; see [ MS19].\nThe reason for choosing these two scattering problems is that, in b oth cases,utotcan be ex-\npressed explicitly as a Fourier series in the angular variable, with the F ourier coefficients expressed\nin terms of Hankel and Bessel functions.\nAlthough it is possible to directly compute utotwith the FEM, it is more convenientto compute\nu:=χuI+uS\nforχ∈C∞\ncomp(Rd) andχ≡1 in a neighbourhood of B1(0); this is because uthen satisfies (i)\nthe same boundary/transmission conditions as utoton∂B1(0) and (ii) the Sommerfeld radiation\ncondition (satisfied by usca) at infinity. Once uis computed, utotanduscacan easily be recovered\nsinceχanduincare known.\nWe then use a PML to approximate the radiation condition satisfied by u. Following [ CM98],\nwe let\nµPML(x) :=B(rP)β(rP) (2.5)\nand\nAPML(x) =B(rP)\nβ(rP)/parenleftbiggcos2θcosθsinθ)\ncosθsinθsin2θ/parenrightbigg\n+β(rP)\nB(rP)/parenleftbigg\nsin2θcosθsinθ)\ncosθsinθcos2θ/parenrightbigg\n(2.6)\nwhererP=|x|−4,θis such that x=|x|(cosθ,sinθ), and\nβ(r) =αrℓB(r) =α\nℓ+1rℓ+1\nwithα= 10 andℓ= 2.\nWe therefore approximate with the FEM the solutions of the following two problems.\nDefinition 2.3 (PML approximation to sound-soft scattering by a 2-d ball) LetΩ :=B5(0)\\\nB1(0).Let\nA:=/braceleftBigg\n1 inB4(0)\\B1(0),\nAPMLinB5(0)\\B1(0),andµ:=/braceleftBigg\n1 inB4(0)\\B1(0),\nµPMLinB5(0)\\B4(0),\nwhereAPMLandµPMLare given by (2.6)and(2.5)respectively. Given χ∈C∞\ncomp(Rd)satisfying\nχ≡1onB1(0)andχ≡0onB5(0)\\B4(0), (2.7)\nletu∈H1\n0(Ω)satisfy\n−k2µu−∇·(A∇u) =f (2.8)\nwith\nf:=−(∆χ)uinc−2∇χ·∇uinc= (−∆χ−2ik∂1χ)eikx1. (2.9)\nDefinition 2.4 (PML approximation to scattering by a penetra ble 2-d ball) LetΩ :=B5(0).\nLet\nA:=\n\n2 inB1(0),\n1 inB4(0)\\B1(0),\nAPMLinB5(0)\\B1(0)andµ:=\n\n1/2inB1(0),\n1 inB4(0)\\B1(0),\nµPMLinB5(0)\\B1(0)\nwhereAPMLandµPMLare given by (2.6)and(2.5)respectively. Given χ∈C∞\ncomp(Rd)satisfying\n(2.7), letu∈H1\n0(Ω)satisfy(2.8), withfgiven by (2.9), and the transmission conditions\n2∂ru+=∂ru−andu+=u−on∂B1(0).\n4For simplicity, in the numerical experiments below, we actually use the cutoff\nχ(x) :=1\n2/parenleftbigg\n1−erf/parenleftbigg|x|−3\nσ/parenrightbigg/parenrightbigg\n,\nwithσ= 0.2 and\nerf(t) :=2√π/integraldisplayt\n0e−s2ds.\nThisχdoes not satisfy ( 2.7); however, |1−χ(x)|<10−12if|x| ≤2 and|χ(x)|<10−12if|x| ≥4,\nso that the error incurred by this inconsistency is so small that it do es not affect our examples.\nThe finite-element solution and error measurement For simplicity, we only consider the\nerror in the region where χ≡1, so thatu≡utot, i.e.,\nΩtot:=/braceleftBigg\nB2(0)\\B1(0) for the sound-soft ball ,\nB2(0) for the penetrable ball.\nHence, our error measure is\n/ba∇dblutot−uh/ba∇dblH1(Ωtot);\ncrucially, Ω totincludes the interface ∂B1(0) where the geometric approximation takes place.\nAs a proxy for u=utotwe use the Fourier-series solution; this incurs an error due to the P ML\napproximating the radiation condition, but this error is very small, es pecially for large wavenum-\nbers; see [ GLS23].\nWe consider finite element discretizations of degree p= 2,3 or 4 and straight meshes. Since\nthe analytical expression of uin:=utot|Danduout:=utot|Rd\\Dare known here, considering any\ntriangleK∈ Thwith barycenter xK, we compute the error by /ba∇dbluin−uh/ba∇dblH1(K)ifxK∈Dand\n/ba∇dbluout−uh/ba∇dblH1(K)otherwise.\nWe usegmsh[GR09] to generate the meshes. gmshproduces nodally conforming meshes (mean-\ning that interfaces are not cross by any edge) of specified maximal sizehtarget. Once such a mesh\nhas been produced, we compute the actual mesh size hby measuring the length of all mesh edges.\nThis value of his the one plotted in the left-hand sides of the figures below.\nWe consider an increasing sequence of wavenumbers ranging from 0 .5·2πto 20·2π, so that the\nnumber of wavelengths in the total field region where the error is me asured ranges from 2 to 80.\nFixingapolynomialdegree pwethensolvetheproblemforallwavenumberwithincreasinglyrefine d\nmeshes. Specifically, we ask gmshto mesh the domain at size htarget, wherek2p+1h2p\ntarget=C, and\nCis chosen so that the relative error is about 1-2% for large wavenum bers.\nDiscussion of the numerical results Figures1a,1band1cpresent the results for p= 2, 3\nand 4, respectively, for the PML approximation to the sound-soft scattering problem.\nFigures2a,2band2cpresent the results for p= 2, 3 and 4, respectively, for the PML approx-\nimation to the penetrable-obstacle scattering problem.\nAs described above, the left-hand sides of the figures plot has a function of k, illustrating that\ngmshactually produces a mesh with the desired maximal meshwidth. The rig ht-hand sides of the\nfigures plot the relative Galerkin error in Ω tot.\nAsanticipatedfromTheorem 1.1, therelativeerrorremainsuniformlyboundedforallwavenum-\nbers in all cases. We also see that, especially for higher polynomial de gree, the error is larger for\nthe first few wavenumbers and then stabilizes to a roughly constan t value. This is because for the\nlowest wavenumbers, the geometric approximation error is not neg ligible, and in fact bigger than\nthe pollution error: indeed, for kfixed, the pollution error on the right-hand side of ( 1.5) isO(hp)\nwhereas the geometric error is O(h). When the wavenumber increases, the pollution error (which\nremains constant thanks to our choice of h), becomes dominant, as predicted by Theorem 1.1.\n510110210−210−1100\nk−5/4\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ωtot)//ba∇dblu/ba∇dblH1\nk(Ωtot)\n(a) Numerical example: sound-soft scattering with p= 2\n10110210−210−1100\nk−7/6\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ωtot)//ba∇dblu/ba∇dblH1\nk(Ωtot)\n(b) Numerical example: sound-soft scattering with p= 3\n10110210−1100\nk−9/8\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ωtot)//ba∇dblu/ba∇dblH1\nk(Ωtot)\n(c) Numerical example: sound-soft scattering with p= 4\nFigure 1: Numerical example: sound-soft scattering\n610110210−210−1100\nk−5/4\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ωtot)//ba∇dblu/ba∇dblH1\nk(Ωtot)\n(a) Numerical example: penetrable obstacle with p= 2\n10110210−210−1100\nk−7/6\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ωtot)//ba∇dblu/ba∇dblH1\nk(Ωtot)\n(b) Numerical example: penetrable obstacle with p= 3\n10110210−1100\nk−9/8\nkh\n0 20 40 60 80 100 12010−210−1\nk/ba∇dblu−uh/ba∇dblH1\nk(Ω)//ba∇dblu/ba∇dblH1\nk(Ω)\n(c) Numerical example: penetrable obstacle with p= 4\nFigure 2: Numerical example: penetrable obstacle\n73 Strang-type lemma for abstract Helmholtz problems\n3.1 Assumptions on the sesquilinear form and finite-dimensi onal sub-\nspace\nContinuity and G˚ arding-type inequality. LetH ⊂H1(Ω) be a Hilbert space with the norm\n/ba∇dbl·/ba∇dblH1\nk(Ω)(1.3). (In practice, Hwill beH1(Ω) with zero Dirichlet boundary conditions on part of\n∂Ω; see (4.1) below).\nLetb:H×H → Cbe a sesquilinear form that is continuous, i.e.,\nsup\nv∈H\n/bardblv/bardblH1\nk(Ω)=1sup\nw∈H\n/bardblw/bardblH1\nk(Ω)=1|b(v,w)|=:M <∞ (3.1)\nand satisfies the G˚ arding-type inequality\nCcoer/ba∇dblv/ba∇dbl2\nH1\nk(Ω)≤Reb(v,v)+2k2/ba∇dblKv/ba∇dbl2\nL2(Ω)for allv∈ H, (3.2)\nforCcoer>0 and for some self-adjoint operator K:L2(Ω)→L2(Ω).\nRemark 3.1 (The relationship between (3.2)and the standard G˚ arding inequality) Ifb\nsatisfies (3.2)thenbalso satisfies a “standard” G˚ arding inequality, since\nCcoer/ba∇dblv/ba∇dbl2\nH1\nk(Ω)≤Reb(v,v)+2k2/ba∇dblK/ba∇dbl2\nL2(Ω)→L2(Ω)/ba∇dblv/ba∇dbl2\nL2(Ω) (3.3)\nfor allv∈ H. We show in Lemma 4.7below (following [ GS23]) that ifbsatisfies a standard\nG˚ arding inequality along with elliptic regularity, then bsatisfies (3.2)withKa smoothing operator;\nthe smoothing property of Kis then key to obtaining the high-order convergence in the ma in result\n(Theorem 1.1).\nThe approximate sesquilinear form bhand approximate right-hand side ψh.Given a\nfinite-dimensional subspace Vh⊂ H, we consider a sesquilinear form bh:Vh×Vh→Cthat is\n“close” tob, with this “closeness” characterised by the quantity\nγq:=1\nMsup\nvh∈Vh\n/bardblvh/bardblH1\nk(Ω)=1sup\nwh∈Vh\n/bardblwh/bardblH1\nk(Ω)=1|b(vh,wh)−bh(vh,wh)|. (3.4)\nSimilarly, given ψ∈(H1\nk(Ω))∗, letψh:Vh→Cbe “close” to ψin the sense that\n/tildewideγq:=1\nM/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗sup\nvh∈Vh\n/bardblvh/bardblH1\nk(Ω)=1|/a\\}b∇acketle{tψ−ψh,vh/a\\}b∇acket∇i}ht|. (3.5)\nUniqueness of the solution of the variational problem. We assume that, given ψ∈\n(H1\nk(Ω))∗, the solution of the variational problem: find v∈ Hsuch that\nb(v,w) =/a\\}b∇acketle{tψ,w/a\\}b∇acket∇i}htfor allw∈ H (3.6)\nif it exists, is unique. The G˚ arding-type inequality ( 3.2) implies that bsatisfies the standard\nG˚ arding inequality ( 3.3). Uniqueness of the solution of ( 3.14) is then equivalent to existence by,\ne.g., [McL00, Theorem 2.32], so that ( 3.6) has a unique solution v∈ H.\nNotation for the adjoint solution operator and adjoint appr oximability. Define the\nadjoint solution operator S∗:L2(Ω)→ Has the solution of the variational problem\nb/parenleftbig\nv,S∗φ/parenrightbig\n= (v,φ)L2(Ω)for allv∈ H, φ∈L2(Ω) (3.7)\n8(the fact that the solution to ( 3.6) is unique implies that S∗is well-defined by, e.g., [ McL00,\nTheorem 2.27]). We then define\nγ∗\ns:=k/ba∇dblS∗/ba∇dblL2(Ω)→H1\nk(Ω);\nthe reason for including the factor of kis thatγ∗\nsis then dimensionless.\nDenote the orthogonal projector in the /ba∇dbl·/ba∇dblH1\nk(Ω)norm by\nπhw:= arg min\nwh∈Vh/ba∇dblw−wh/ba∇dblH1\nk(Ω)for allw∈ H. (3.8)\nThe following quantity measures how well solutions of the adjoint pro blem are approximated in\nVh:\nγ∗\na:=k/ba∇dbl(I−πh)S∗/ba∇dblL2(Ω)→H1\nk(Ω); (3.9)\nsimilar to with γ∗\ns, the factor of kin (3.9) means that γ∗\nais dimensionless.\nThe modified sesquilinear form and its solution operator. Define\nb+(v,w) :=b(v,w)+2k2/parenleftbig\nK2v,w/parenrightbig\nL2(Ω), (3.10)\nso thatb+is coercive on Hby (3.2) (with coercivity constant Ccoer). Let\nsup\nv∈H\n/bardblv/bardblH1\nk(Ω)=1sup\nw∈H\n/bardblw/bardblH1\nk(Ω)=1/vextendsingle/vextendsingleb+(v,w)/vextendsingle/vextendsingle=:M+≤M+2/ba∇dblK2/ba∇dblL2(Ω)→L2(Ω). (3.11)\nLetS+:L2(Ω)→ Hbe defined as the solution of the variational problem\nb+/parenleftbig\nS+φ,w/parenrightbig\n= (φ,w)L2(Ω)for allw∈ H, φ∈L2(Ω). (3.12)\nIt is convenient to define the following approximation factor\nγK:=k/vextenddouble/vextenddouble(I−πh)S+K/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω); (3.13)\ni.e.,γKmeasures how well solutions of the variational problem involving b+and with a Kon the\nright-hand side are approximated in Vh.\n3.2 The main abstract result\nThefollowingresultisanabstractversionoftheelliptic-projectiona rgumentfrom[ GS23], extended\nto allow a variational crime.\nTheorem 3.2 (Abstract elliptic-projection-type argument with variational crime) Under\nthe assumptions in §3.1, letu∈ Hbe the unique solution of the variational problem\nb(u,w) =/a\\}b∇acketle{tψ,w/a\\}b∇acket∇i}htfor allw∈ H. (3.14)\nLet\nA:= 1−2(M+)2\nCcoer/ba∇dblK/ba∇dblL2(Ω)→L2(Ω)γ∗\naγK\nand\nB:=A/parenleftbig\nCcoer−Mγq/parenrightbig\n−√\n2MM+\nCcoer/ba∇dblK/ba∇dblL2(Ω)→L2(Ω)γ∗\nsγq.\nIfB >0, then there exists a unique uh∈Vhsuch that\nbh(uh,wh) =/a\\}b∇acketle{tψh,wh/a\\}b∇acket∇i}htfor allwh∈Vh, (3.15)\nand the error u−uhsatisfies the bound\nB/ba∇dblu−uh/ba∇dblH1\nk(Ω)≤/bracketleftbigg\nAM+√\n2MM+\nCcoerγ∗\na/ba∇dblK/ba∇dblL2(Ω)→L2(Ω)/bracketrightbigg\n/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω)\n+/bracketleftbigg\nAM+√\n2MM+\nCcoerγ∗\ns/ba∇dblK/ba∇dblL2(Ω)→L2(Ω)/bracketrightbigg/parenleftBig\nγq/ba∇dblu/ba∇dblH1\nk(Ω)+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗/parenrightBig\n.\n(3.16)\n9We make two immediate remarks.\n1. The bound ( 3.16) has a similar structure to the Strang lemma for coercive problems, namely\nthe Galerkin error is bounded by the sum of (i) the best approximatio n error, (ii) the error in\nthe sesquilinear form, and (iii) the error in the right-hand side; see [ Str72], [Cia91, Theorem\n26.1].\n2. Ifγq,/tildewideγq= 0 and the approximation factors γ∗\na,γK→0, thenA→1,B→Ccoer, and the\nbound (3.16) approaches\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)≤M\nCcoer/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω);\nobserve that this is the bound given by C´ ea’s lemma for a coercive pr oblem with continuity\nconstantMand coercivity constant Ccoer.\nInterpretation and context of Theorem 3.2.Assuming that /ba∇dblK/ba∇dblL2(Ω)→L2(Ω),M,M+, and\nCcoerare independent of k(as we see below they are for Helmholtz problems), we see that Theo rem\n3.2shows that if\nγ∗\naγKandγ∗\nsγqare sufficiently small (3.17)\nthen\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)≤C(1+γ∗\na)/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω)+C(1+γ∗\ns)/parenleftBig\nγq/ba∇dblu/ba∇dblH1\nk(Ω)+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗/parenrightBig\n.(3.18)\nContinuity of b(3.1) implies that /ba∇dblψ/ba∇dbl(H1\nk(Ω))∗≤M/ba∇dblu/ba∇dblH1\nk(Ω), and thus ( 3.18) implies that\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)≤C(1+γ∗\na)/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω)+C(1+γ∗\ns)/parenleftbig\nγq+/tildewideγq/parenrightbig\n/ba∇dblu/ba∇dblH1\nk(Ω).(3.19)\nSection4shows how for the particular Helmholtz problem considered in Section 1, (3.17) and\n(3.19) become ( 1.4) and (1.5), respectively; indeed, this is via the bounds γ∗\na≤C(kh+kα(kh)p),\nγK≤C(kh)p, andγq,/tildewideγq≤Chq(for a domain Ω with characteristic length scale ∼1).\nWe now discuss here the relation between ( 3.17), (3.19) and other results in the literature.\nSince these other results do not take into account errors in the se squilinear form and data (via the\nquantitiesγqand/tildewideγq), in this discussion we assume that γq=/tildewideγq= 0.\nRecall that the classic duality argument for sesquilinear forms satis fying a G˚ arding inequality\nproves that, if γ∗\nais sufficiently small\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)≤C/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω);\nthis argumenthas its originsin the work of[ Sch74], with then the notion ofadjoint approximability\nencapsulated by γ∗\naintroduced in [ Sau06].\nTheadvantageof ( 3.17),(3.19)overthisclassicargumentisthatthecondition“ γKγ∗\nasufficiently\nsmall”canbelessrestrictivethan“ γ∗\nasufficientlysmall”. Indeed, the“ellipticprojection”argument\nintroduced in [ FW09,FW11] is essentially ( 3.17), (3.19) with the operator Kchosen to be a\nsufficiently-large multiple of the identity. In this case, we see below th at\nγK:=k/vextenddouble/vextenddouble(I−πh)S+K/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω)≤Ck/vextenddouble/vextenddouble(I−πh)S+/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω)≤C′kh,\nandsothecondition “ γKγ∗\nasufficiently small”is “ khγ∗\naissufficiently small”, i.e., aweakercondition\nthan “γ∗\nais sufficiently small” (since kh≪1).\nThe generalisation of the elliptic-projection argument in [ GS23] showed that, under natural\nelliptic-regularity assumptions, there exists a smoothing operator Ksuch that the G˚ arding-type in-\nequality( 3.2)issatisfied. Withthissmoothingproperty,thecondition“ γ∗\nak/ba∇dbl(I−πh)S+K/ba∇dblL2(Ω)→H1\nk(Ω)\nsufficiently small” is weaker than “ γ∗\nak/ba∇dbl(I−πh)S+/ba∇dblL2(Ω)→H1\nk(Ω)sufficiently small”; we recap the\nconstruction of Kin Lemma 4.7below.\n103.3 Proof of Theorem 3.2\nTheorem 3.2comesfrom combiningLemmas 3.3and3.4below. These lemmasinvolvethe following\nprojection operator; let π+\nh:H →Vhbe defined by\nb+/parenleftbig\nvh,π+\nhw/parenrightbig\n=b+(vh,w) for allvh∈Vh,w∈ H;\ni.e.,\nb+/parenleftbig\nvh,(I−π+\nh)w/parenrightbig\n= 0 for all vh∈Vh,w∈ H. (3.20)\nWe immediately note that, since b+is continuous and coercive, C´ ea’s lemma implies that\n/vextenddouble/vextenddouble(I−π+\nh)v/vextenddouble/vextenddouble\nH1\nk(Ω)≤M+\nCcoer/vextenddouble/vextenddouble(I−πh)v/vextenddouble/vextenddouble\nH1\nk(Ω)for allv∈H1\nk(Ω), (3.21)\nand/vextenddouble/vextenddoubleπ+\nhv/vextenddouble/vextenddouble\nH1\nk(Ω)≤M+\nCcoer/ba∇dblv/ba∇dblH1\nk(Ω)for allv∈ H, (3.22)\nsince\nCcoer/ba∇dblv/ba∇dbl2\nH1\nk(Ω)≤b+/parenleftbig\nπ+\nhv,π+\nhv/parenrightbig\n=b+/parenleftbig\nπ+\nhv,v/parenrightbig\n≤M+/vextenddouble/vextenddoubleπ+\nhv/ba∇dblH1\nk(Ω)/ba∇dblv/ba∇dblH1\nk(Ω).\nIn both the statement and proof of the next lemma, we drop the (Ω ) in norms, i.e., we write,\ne.g.,/ba∇dblK/ba∇dblL2→L2instead of /ba∇dblK/ba∇dblL2(Ω)→L2(Ω). to keep expressions compact.\nLemma 3.3 Let\n/tildewideγ∗\na:=k2/vextenddouble/vextenddoubleK(I−π+\nh)S∗/vextenddouble/vextenddouble\nL2→L2. (3.23)\nIf the solution uhto(3.15)exists, then\n/bracketleftbigg/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig/parenleftBig\nCcoer−Mγq/parenrightBig\n−√\n2MM+\nCcoer/ba∇dblK/ba∇dblL2→L2γ∗\nsγq/bracketrightbigg\n/ba∇dblu−uh/ba∇dblH1\nk(Ω)\n≤/bracketleftbigg/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig\nM+√\n2(M+)2\nCcoerγ∗\na/ba∇dblK/ba∇dblL2→L2/bracketrightbigg\n/ba∇dbl(I−πh)u/ba∇dblH1\nk(Ω)\n+/bracketleftbigg/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig\nM+√\n2MM+\nCcoer/ba∇dblK/ba∇dblL2→L2γ∗\ns/bracketrightbigg/parenleftBig\nγq/ba∇dblu/ba∇dblH1\nk(Ω)+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗/parenrightBig\n.\n(3.24)\nNote that ( 3.24) has the same structure as ( 3.16), except that the instance of γ∗\nainAhas been\nreplaced by a multiple of /tildewideγ∗\na. The following lemma is then used to relate /tildewideγ∗\natoγ∗\na.\nLemma 3.4 For allv∈ H,\nk/vextenddouble/vextenddoubleK(I−π+\nh)v/vextenddouble/vextenddouble\nL2(Ω)≤M+γK/vextenddouble/vextenddouble(I−π+\nh)v/vextenddouble/vextenddouble\nH1\nk(Ω).\nProof of Theorem 3.2using Lemmas 3.3and3.4.Once the bound ( 3.16) is established under the\nassumption that uhexists, applying this bound with ψ= 0 and using uniqueness of the continuous\nproblem ( 3.14) implies that uh, if it exists, is unique. Existence of uhthen follows since uhis the\nsolution of a finite-dimensional linear system, for which uniqueness is equivalent to existence.\nIt is therefore sufficient to prove ( 3.16) under the assumption that uhexists. By Lemma 3.4\nand the definitions of /tildewideγ∗\na(3.23),γK(3.13), andγ∗\na(3.9),\n/tildewideγ∗\na≤(M+)2\nCcoerγKγ∗\na.\nand then using this in ( 3.24) gives the result ( 3.16).\nWe now prove Lemma 3.4using a duality argument involving the sesquilinear form b+.\n11Proof of Lemma 3.4.By the definition of S+(3.12), the Galerkin orthogonality ( 3.20), and\ncontinuity of b+(3.11)\n/vextenddouble/vextenddoubleK(I−π+\nh)v/vextenddouble/vextenddouble2\nL2(Ω)=/parenleftbig\nK2(I−π+\nh)v,(I−π+\nh)v/parenrightbig\nL2(Ω)\n=b+/parenleftbig\nS+K2(I−π+\nh)v,(I−π+\nh)v/parenrightbig\n=b+/parenleftbig\n(I−πh)S+K2(I−π+\nh)v,(I−π+\nh)v/parenrightbig\n≤M+/vextenddouble/vextenddouble(I−πh)S+K/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω)/vextenddouble/vextenddoubleK(I−π+\nh)v/vextenddouble/vextenddouble\nL2(Ω)/vextenddouble/vextenddouble(I−π+\nh)v/vextenddouble/vextenddouble\nH1\nk(Ω),\nand the result follows.\nTo complete the proof of Theorem 3.2, we therefore need to prove Lemma 3.3. To do this, we\nstart with a simple result controlling the variational crime error.\nLemma 3.5 (“Galerkin orthogonality” in the sesquilinear fo rmb(·,·))Ifu∈ Handuh∈\nVhsatisfy(3.14)and(3.15), then, for all wh∈Vh,\n|b(u−uh,wh)| ≤M/parenleftBig\nγq/ba∇dblu−uh/ba∇dblH1\nk(Ω)+γq/ba∇dblu/ba∇dblH1\nk(Ω)+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗/parenrightBig\n/ba∇dblwh/ba∇dblH1\nk(Ω).(3.25)\nProof.By (3.14) and (3.15),\nb(u−uh,wh) =b(u,wh)−b(uh,wh) =/a\\}b∇acketle{tψ,wh/a\\}b∇acket∇i}ht−bh(uh,wh)+/parenleftbig\nbh(uh,wh)−b(uh,wh)/parenrightbig\n=/a\\}b∇acketle{tψ−ψh,wh/a\\}b∇acket∇i}ht+/parenleftbig\nbh(uh,wh)−b(uh,wh)/parenrightbig\n.\nThen, by ( 3.5) and (3.4),\n|b(u−uh,wh)| ≤M/parenleftBig\n/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk(Ω))∗+γq/ba∇dbluh/ba∇dblH1\nk(Ω)/parenrightBig\n/ba∇dblwh/ba∇dblH1\nk(Ω).\nThe result ( 3.25) then follows from the triangle inequality.\nLemma 3.6 (Useful quadratic inequality) Let00, we find that\n/parenleftbigg\nax−b\n2/parenrightbigg2\n−b2\n4≤ac2,and thus ax≤b\n2+/radicalbigg\nb2\n4+ac2.\nThe result ( 3.27) follows since\n/radicalbigg\nb2\n4+ac2≤/radicalbigg\nb2\n4+√\nac2≤b\n2+c,\nsincea≤1 by assumption.\nProof of Lemma 3.3.Lete:=u−uh. By (3.2), (3.25), and (3.1),\nCcoer/ba∇dble/ba∇dbl2\nH1\nk≤Reb(e,e)+2k2/ba∇dblKe/ba∇dbl2\nL2\n= Reb/parenleftbig\ne,(I−πh)u/parenrightbig\n+2k2/ba∇dblKe/ba∇dbl2\nL2−Reb(e,uh−πhu)\n≤M/ba∇dble/ba∇dblH1\nk/ba∇dbl(I−πh)u/ba∇dblH1\nk+2k2/ba∇dblKe/ba∇dbl2\nL2\n+M/parenleftBig\nγq/ba∇dble/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n/ba∇dble/ba∇dblH1\nk,\n12where to obtain the last inequality we have used (i) that uh−πhu=πheand (ii)/ba∇dblπhe/ba∇dblH1\nk≤ /ba∇dble/ba∇dblH1\nk\nsinceπhis a projection ( 3.8). Rearranging the last displayed equation, we obtain that\n/parenleftbig\nCcoer−Mγq/parenrightbig\n/ba∇dble/ba∇dbl2\nH1\nk≤M/ba∇dble/ba∇dblH1\nk/parenleftBig\n/ba∇dbl(I−πh)u/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n+2k2/ba∇dblKe/ba∇dbl2\nL2.\nThen, by Lemma 3.6,\n/parenleftbig\nCcoer−Mγq/parenrightbig\n/ba∇dble/ba∇dblH1\nk≤M/parenleftBig\n/ba∇dbl(I−πh)u/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n+√\n2k/ba∇dblKe/ba∇dblL2.(3.28)\nWe now use a duality argument to bound /ba∇dblKe/ba∇dblL2. By the self-adjointness of K, the definition of\nS∗(3.7), (3.25), and (3.22),\n/ba∇dblKe/ba∇dbl2\nL2=b(e,S∗K2e)\n=b/parenleftbig\ne,(I−π+\nh)S∗K2e/parenrightbig\n+b/parenleftbig\ne,π+\nhS∗K2e/parenrightbig\n≤b/parenleftbig\ne,(I−π+\nh)S∗K2e/parenrightbig\n+M/parenleftBig\nγq/ba∇dble/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBigM+\nCcoer/vextenddouble/vextenddoubleS∗/ba∇dblL2→H1\nk/ba∇dblK/ba∇dblL2→L2/ba∇dblKe/ba∇dblL2.(3.29)\nNow, by the definition of b+(3.10), (3.20), the self-adjointness of K, continuity of b+(3.11), (3.21),\nand the definitions of γ∗\na(3.9) and/tildewideγ∗\na(3.23),\nb/parenleftbig\ne,(I−π+\nh)S∗K2e/parenrightbig\n=b+/parenleftbig\n(I−πh)u,(I−π+\nh)S∗K2e/parenrightbig\n−2k2/parenleftbig\nK2e,(I−π+\nh)S∗K2e/parenrightbig\nL2\n=b+/parenleftbig\n(I−πh)u,(I−π+\nh)S∗K2e/parenrightbig\n−2k2/parenleftbig\nKe,K(I−π+\nh)S∗K2e/parenrightbig\nL2\n≤M+/ba∇dbl(I−πh)u/ba∇dblH1\nkM+\nCcoerk−1γ∗\na/ba∇dblK/ba∇dblL2→L2/ba∇dblKe/ba∇dblL2+2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/ba∇dblKe/ba∇dbl2\nL2.\n(3.30)\nCombining ( 3.29) and (3.30), multiplying by k, and using the definition of γ∗\ns, we obtain that\n/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig\nk/ba∇dblKe/ba∇dblL2≤(M+)2\nCcoerγ∗\na/ba∇dblK/ba∇dblL2→L2/ba∇dbl(I−πh)u/ba∇dblH1\nk\n+MM+\nCcoer/ba∇dblK/ba∇dblL2→L2γ∗\ns/parenleftBig\nγq/ba∇dble/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n.\n(3.31)\nMultiplying ( 3.28) by (1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2) and inputting ( 3.31), we obtain\n/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig/parenleftbig\nCcoer−Mγq/parenrightbig\n/ba∇dble/ba∇dblH1\nk\n≤/parenleftBig\n1−2/tildewideγ∗\na/ba∇dblK/ba∇dblL2→L2/parenrightBig\nM/parenleftBig\n/ba∇dbl(I−πh)u/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n+√\n2(M+)2\nCcoerγ∗\na/ba∇dblK/ba∇dblL2→L2/ba∇dbl(I−πh)u/ba∇dblH1\nk\n+√\n2MM+\nCcoer/ba∇dblK/ba∇dblL2→L2γ∗\ns/parenleftBig\nγq/ba∇dble/ba∇dblH1\nk+γq/ba∇dblu/ba∇dblH1\nk+/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)∗/parenrightBig\n,\nwhich then rearranges to the result ( 3.24).\n4 Applyingtheabstract theory to Helmholtz problems solved\nwith curved finite elements\n4.1 The Helmholtz problem with piecewise-smooth coefficient s\nLet Ω⊂Rdbe a bounded Lipschitz domain with characteristic length scale L. The boundary\n∂Ω of Ω is partitioned into two disjoint relatively open Lipschitz subsets ΓDand Γ N(where the\nsubscripts “D” and “N” stand for “Dirichlet” and “Neumann”). Let Hbe the Hilbert space\nH1\nΓD(Ω) :=/braceleftbig\nv∈H1(Ω) :v= 0 on Γ D/bracerightbig\n. (4.1)\n13The domain Ω is partitioned into disjoint open Lipschitz subsets Q∈ Q, and we assume that the\nrestrictions of the PDE coefficients to each Qare smooth. Specifically, for each Q∈ Q, we consider\ntwo functions µQ:Rd→CandAQ:Rd→Cd×dand the sesquilinear form then reads\nb(v,w) :=/summationdisplay\nQ∈Q/braceleftBig\n−k2(µQv,w)L2(Q)+(AQ∇v,∇w)L2(Q)/bracerightBig\n=−k2(µv,w)L2(Ω)+(A∇v,∇w)L2(Ω)for allv,w∈H1\nΓD(Ω), (4.2)\nwhereµ|Q:=µQ,A|Q:=AQ. To define the sesquilinear form b, the values of µQandAQare\nnot needed outside Q. However, assuming that they are also defined in a neighborhood of Qis\nuseful when defining the discrete bilinear form (involving geometric e rror). We note that in the\ncase of piecewise constant coefficients or coefficients given by analy tical formula (e.g., in a PML)\nsuch extensions are automatically defined.\nAssumption 4.1 (Strong ellipticity of the Helmholtz operat or)There exists c >0such\nthat\nRed/summationdisplay\ni=1d/summationdisplay\nj=1Aij(x)ξjξi≥c|ξ|2for allx∈Ωandξ∈Rd.\nAssumption 4.1implies that the Helmholtz operator is strongly elliptic in the sense of [ McL00,\nEquation 4.7] by, e.g., [ McL00, Page 122].\nAssumption 4.2 (Regularity assumption on A,µ, and∂Q)For someℓ∈Z+,AQ,µQ∈\nCℓ−1,1and∂Q∈Cℓ,1for allQ∈ Q.\nAssumption 4.2implies that the Helmholtz operator satisfies the standard elliptic-re gularity\nshift (see, e.g., [ McL00, Theorem 4.20]). Thanks to [ CFN20] and [GS23] this shift property can be\nused to bound the quantities γ∗\naandγK, respectively, appearing in Theorem 3.2; see§4.6below.\nSimilarly, we assume that\n/a\\}b∇acketle{tψ,v/a\\}b∇acket∇i}ht= (g,v)L2(Ω) (4.3)\nwithg=gQonQandgQ:Rd→CwithgQ∈H1(Rd).\nGiven the partition Qof Ω, we use the notation that v∈Hj(Q) iffv|Q∈Hj(Q) for allQ∈ Q,\nand we let\n/ba∇dblv/ba∇dbl2\nHm(Q):=/summationdisplay\nQ∈Q/ba∇dblv/ba∇dbl2\nHm(Q) (4.4)\nwhere\n/ba∇dblu/ba∇dbl2\nHm(D):=/summationdisplay\n|α|≤mL2(m−|α|)/ba∇dbl∂αu/ba∇dbl2\nL2(D)form∈Z+,\nwhereLis the characteristic length scale of Ω; these factors of Lare included to make each term\nin the norm have the same dimension.\n4.2 Curved finite-element setting\nHere, we introduce a finite element space on curved simplicial element s. For such spaces, there are\nthe following two key considerations.\n(a) On the one hand, the mesh elements should be sufficiently curved so as to correctly approx-\nimate the geometry of the boundary value problem (i.e. Ω, Γ D, ΓNandQ).\n(b) On the other hand, the elements should not be too distorted so as to preserve the approxi-\nmation properties of the finite element space.\nThesetwoaspectshavebeen thoroughlyinvestigatedin thefinite- element literature[ Cia02,CR72b,\nLen86], and this section summarises the key assumptions needed in our ana lysis.\nWe consider a non-conforming mesh Ththat does not exactly cover Ω nor the partition Q. The\n(closed) elements K∈ Thare obtained by mapping a single (closed) reference simplex /hatwideKthrough\n14bilipschitz maps FK:/hatwideK→K. The maps FKcould be affine (leading to straight elements) but\nwe more generally allow polynomials mappings of degree q≥1 leading to curved elements. We\nclassicaly assume that the mesh is “matching”, meaning that◦\nK∩◦\nK′=∅for allK,K′∈ Th, and\nK∩K′is either empty, or corresponds to a single vertex, a single edge or a single face of the\nreference simplex /hatwideKmapped through FK(or equivalently, FK′).\nLet\nΩh:= Int/parenleftBig/uniondisplay\nK∈ThK/parenrightBig\n;\ni.e., Ωhis the domain covered by Th. In general, Ωh/\\e}atio\\slash= Ω, but we expect that This “close” to\nmatching the boundary and interfaces in the PDE problem. To encod e this, we assume that for\neachK∈ Th, there exists a bilipschitz mapping Ψh\nK:K→Rd, such that if /tildewideK:= Ψh\nK(K), then\n(i) there exists Q∈ Qsuch that /tildewideK⊂Q,\n(ii)∪K∈Th/tildewideK=Ω,\n(iii) letting Ψh|K:= Ψh\nKfor allK∈ Thdefines a bilipschitz mapping Ψh:Ωh→Ω.\nLet Φh\nK:= (Ψh\nK)−1, and Φh:= (Ψh)−1. Note then that Φh|/tildewideK= Φh\nKfor allK∈ Th. We let\nΓh\nD:= Φh(ΓD),Γh\nN:= Φh(ΓN), Qh:= Φh(Q) forQ∈ Q;\nandQh:={Qh}Q∈Q. Notice that if Ψh=I, then Ωh= Ω, Γh\nD= ΓD, Γh\nN= ΓNandQh=Qfor all\nQ∈ Th, so that the mesh is actually conforming. We therefore expect eac h map Ψh\nKto be close to\nthe identity in a suitable sense (made precise below). Without loss of g enerality we assume that\ndetΦh,detΨh>0.\nFinally, we assume that both Γh\nDand Γh\nNare unions of full faces of elements K∈ Th. In other\nwords, every mesh face either sits in Γh\nDor in Γh\nN.\nFixing a polynomial degree p≥1, we associate with Ththe following finite element space:\nVh:=/braceleftBig\nvh∈H1\nΓh\nD(Ωh)|vh◦F−1\nK∈ Pp(/hatwideK)/bracerightBig\n.\nNote that elements of Vhneed not be piecewise polynomials, since the map FKis not polynomial in\ngeneral. Since piecewise polynomial spaces enjoy excellent approxim ation properties, we therefore\nwantFKto be close to being affine, meaning that the mesh elements are not to o distorted.\nWe can make the requirements needed for the maps FKand Ψh\nKprecise as follows.\nAssumption 4.3 (Curved finite element space) The elements K∈ Thare not too distorted,\nin the sense that the maps FKsatisfy\n/ba∇dbl∇sFK/ba∇dblL∞(/hatwideK)≤CL/parenleftbigghK\nL/parenrightbiggs\n,and/ba∇dbl∇s(F−1\nK)/ba∇dblL∞(K)≤Chs\nK, (4.5)\nfor1≤s≤p+1, wherehKis the diameter of K.\nThe elements K∈ Thapproximate the geometry well, in the sense that\n/ba∇dbl∇s(Ψh\nK−I)/ba∇dblL∞(K)≤C/parenleftbigghK\nL/parenrightbiggq\nh1−s\nK,/ba∇dbldet(∇Ψh\nK)−1/ba∇dblL∞(K)≤C/parenleftbigghK\nL/parenrightbiggq\n,(4.6)\nand\n/ba∇dbl∇s(Φh\nK−I)/ba∇dblL∞(/tildewideK)≤C/parenleftbigghK\nL/parenrightbiggq\nh1−s\nK,/ba∇dbldet(∇Φh\nK)−1/ba∇dblL∞(/tildewideK)≤C/parenleftbigghK\nL/parenrightbiggq\n,(4.7)\nfor0≤s≤q+1.\n15The requirements in Assumption 4.3are easily met in practice. Specifically, given a straight\nsimplicial partition T†\nhthat nodally conforms to Qand Ω, suitable maps FKare constructed in\n[Len86,§3.2–3.3]. Note that constructing T†\nhis in turn easily done with standard meshing software\nlikegmsh[GR09].\nUnder Assumption 4.3, there exists an interpolation operator Ih:H2(Qh)∩H1\nΓh\nD(Ωh)→Vh\nsuch that\nh−1/vextenddouble/vextenddouble(I−Ih)v/vextenddouble/vextenddouble\nL2(Ωh)+/vextendsingle/vextendsingle(I−Ih)v/vextendsingle/vextendsingle\nH1(Ωh)≤Chp/ba∇dblv/ba∇dblHp+1(Qh)(4.8)\nfor allv∈Hp+1(Qh)∩H1\nΓh\nD(Ωh); see [CR72b,Len86]. (Note that the piecewise Sobolev spaces on\nQhare defined exactly as the ones on Q, with semi-norm given by the analogue of ( 4.4).)\nWe finally introduce a discrete sesquilinear form:\nbh(vh,wh) :=/summationdisplay\nQ∈Q/braceleftBig\n−k2(µQvh,wh)L2(Qh)+(AQ∇vh,∇wh)L2(Qh)/bracerightBig\nfor allvh,wh∈Vh,\nand a discrete right-hand side\n/a\\}b∇acketle{tψh,wh/a\\}b∇acket∇i}ht=/summationdisplay\nQ∈Q(gQ,wh)L2(Qh). (4.9)\n(Note that since each Qhis exactly covered by Th, the coefficients of the matrix associated with bh\nand the vector associated with ψhcan be computed efficiently.) The discrete problem then consists\nof findinguh∈Vhsuch that\nbh(uh,wh) =/a\\}b∇acketle{tψh,wh/a\\}b∇acket∇i}htfor allwh∈Vh. (4.10)\nRemark 4.4 (Quadrature error) In practice, if µQandAQare not piecewise constant, the\nintegrals in the definition of bhare further approximated by a quadrature rule. Following [ Len86,\n$6] this can be done by further considering approximations µh\nQandAh\nQfor which the quadrature can\nthen be performed exactly; see also [ Cia02, Section 4.4] for an alternative approach. For simplicity\nthough, we focus here on the geometric error and assume that t he matrix coefficient associated with\nbhare computed exactly.\n4.3 A mapped finite element space\nThe setting introduced above for curved finite elements does not im mediately fit our abstract\nframework. Indeed, bhcontainsexact integralson a wrongdomain Ωh, ratherthat inexact integrals\non the right domain Ω. Following [ Len86], this is easily remedied by introducing an abstract\nmapped finite element space. Notice that this space is never used in a ctual computations; rather,\nit is a convenient tool for the analysis.\nForvh∈Vh, let/tildewidevh:=vh◦Ψ−1\nh; recall from Section 4.2that Ψh:Ωh→Ω, so that /tildewidevh: Ω→C\nand/tildewidevh∈H1\nΓD(Ω). The mapped finite element space /tildewideVh:={/tildewidevh;vh∈Vh}is therefore conforming,\ni.e.,/tildewideVh��H1\nΓD(Ω). Furthermore, instead of considering the solution uhof the variational problem\n(4.10), we can (abstractly) solve a variational problem set in Ω with the ma pped finite element\nspace/tildewideVh. Specifically, we can equivalently consider the problem of finding /tildewideuh∈/tildewideVhsuch that\n/tildewidebh(/tildewideuh,/tildewidewh) =/a\\}b∇acketle{t/tildewideψh,/tildewidewh/a\\}b∇acket∇i}htfor all/tildewidewh∈/tildewideVh, (4.11)\nwhere\n/tildewidebh(/tildewideuh,/tildewidewh) :=/summationdisplay\nQ∈Q/braceleftBig\n−k2(µh\nQ/tildewideuh,/tildewidewh)L2(Q)+(Ah\nQ∇/tildewideuh,∇/tildewidewh)L2(Q)/bracerightBig\n=−k2(µh/tildewideuh,/tildewidewh)L2(Ω)+(Ah∇/tildewideuh,∇/tildewidewh)L2(Ω) (4.12)\nwith\nµh\nQ:=|det∇Φh|/parenleftbig\nµQ◦Φh/parenrightbig\n, Ah\nQ:=|det∇Φh|∇Ψh(AQ◦Φh)(∇Ψh)T(4.13)\n16andµh|Q:=µh\nQandAh|Q:=Ah\nQfor allQ∈ Q. Similarly,\n/a\\}b∇acketle{t/tildewideψh,/tildewidewh/a\\}b∇acket∇i}ht:=/summationdisplay\nQ∈Q/parenleftbig\ngh\nQ,/tildewidewh/parenrightbig\nwithgh\nQ:=|det∇Φh|/parenleftbig\ngQ◦Φh/parenrightbig\n. (4.14)\nThese definitions are made so that, by the chain rule,\nbh(vh,wh) =/tildewidebh(/tildewidevh,/tildewidewh) for allvh,wh∈Vh,\nand thusuh=/tildewideuh◦Ψhby (4.10) and (4.11).\nIt is clear from Assumption 4.3and (4.8) that if we set /tildewideIh= Ψh◦ Ih◦(Ψh)−1, we obtain a\nsuitable interpolation operator onto /tildewideVh. Specifically, [ Len86, Lemma 7] shows that1\nh−1/ba∇dblv−/tildewideIhv/ba∇dblL2(Ω)+|v−/tildewideIhv|H1(Ω)≤Chp/ba∇dblv/ba∇dblHp+1(Q)for allv∈Hp+1(Q)∩H1\nΓD(Ω).(4.15)\n4.4 Bounds on γqand/tildewideγq\nWith our framework established, we are ready to provide bounds on γqand/tildewideγq. The proof for γq\nclosely follows [ Len86, Lemma 8] whereas the auxiliary result established in Appendix Bis used\nfor/tildewideγq.\nTheorem 4.5 (Bounds on γqand/tildewideγq)IfVhsatisfies Assumption 4.3, then\nγq≤C/parenleftbiggh\nL/parenrightbiggq\nW(µ,A),/tildewideγq/ba∇dblψ/ba∇dbl(H1\nk)⋆≤C\nM/parenleftbiggh\nL/parenrightbiggq1\nk/parenleftbigg/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dbl2\nL2(Rd)+h2/ba∇dbl∇gQ/ba∇dbl2\nL2(Rd)/parenrightbigg1/2\n(4.16)\nwith\nW(µ,A) :=1\nMmax\nQ∈Q/parenleftBig\n/ba∇dblµQ/ba∇dblL∞(Rd)+h/ba∇dbl∇µQ/ba∇dblL∞(Rd)+/ba∇dblAQ/ba∇dblL∞(Rd)+h/ba∇dbl∇AQ/ba∇dblL∞(Rd)/parenrightBig\n.\nProof.We first prove the bound on γq, and observe that we only need to show that\n/ba∇dblµ−µh/ba∇dblL∞(Ω)+/ba∇dblA−Ah/ba∇dblL∞(Ω)≤C/parenleftbiggh\nL/parenrightbiggq\nW(µ,A), (4.17)\nsince then, by the definition of /tildewidebhin (4.12),\n|b(/tildewidevh,/tildewidewh)−/tildewidebh(/tildewidevh,/tildewidewh)| ≤C/parenleftbiggh\nL/parenrightbiggq\nW(µ,A)/ba∇dbl/tildewidevh/ba∇dblH1\nk(Ω)/ba∇dbl/tildewidewh/ba∇dblH1\nk(Ω) (4.18)\nfor all/tildewidevh,/tildewidewh∈/tildewideVh. The bound on γqthen follows immediately from its definition ( 3.4) and (4.18).\nWe now show how ( 4.17) follows from Assumption 4.3. For the coefficient A, ifQ∈ Qand\nK∈ Thwith/tildewideK⊂Q, by (4.13), we have\nAQ−Ah\nQ=AQ−(det∇Φh)∇Ψh(AQ◦Φh)(∇Ψh)T,\n= (1−det∇Φh)AQ−det∇Φh/braceleftbig\nAQ−∇Ψh(AQ◦Φh)(∇Ψh)T/bracerightbig\nso that\n/ba∇dblAQ−Ah\nQ/ba∇dblL∞(/tildewideK)≤ /ba∇dbl1−det∇Φh/ba∇dblL∞(/tildewideK)/ba∇dblAQ/ba∇dblL∞(Rd)\n+/ba∇dbldet∇Φh/ba∇dblL∞(/tildewideK)/ba∇dblAQ−∇Ψh(AQ◦Φh)(∇Ψh)T/ba∇dblL∞(/tildewideK)\n≤C/parenleftbigg/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblAQ/ba∇dblL∞(/tildewideK)+/ba∇dblAQ−∇Ψh(AQ◦Φh)(∇Ψh)T/ba∇dblL∞(/tildewideK)/parenrightbigg\n(4.19)\n1Strictly speaking, the result is only establish for the H1semi-norm, but the proof for the L2norm is analogous.\n17by (4.7). We then work on the second term on the right-hand side of ( 4.19). Specifically,\nAQ−∇Ψh(AQ◦Φh)(∇Ψh)T= (Id−∇Ψh)AQ+∇Ψh/bracketleftbig\nAQ−(AQ◦Φh)(∇Ψh)T/bracketrightbig\n= (Id−∇Ψh)AQ+∇Ψh/bracketleftbig\nAQ(Id−(∇Ψh)T)/bracketrightbig\n+∇Ψh/parenleftbig\nAQ−AQ◦Φh)(∇Ψh)T,\nand therefore\n/ba∇dblAQ−∇Ψh(AQ◦Φh)(∇Ψh)T/ba∇dblL∞(/tildewideK)≤ /ba∇dblId−∇Ψh/ba∇dblL∞(/tildewideK)/ba∇dblAQ/ba∇dblL∞(Rd)\n+C/ba∇dbl∇Ψh/ba∇dblL∞(/tildewideK)/ba∇dblAQ/ba∇dblL∞(Rd)/ba∇dblId−(∇Ψh)T/ba∇dblL∞(/tildewideK)\n+C/ba∇dbl∇Ψh/ba∇dbl2\nL∞(/tildewideK)/ba∇dblAQ−AQ◦Φh/ba∇dblL∞(/tildewideK)\n≤C/parenleftbigg/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblAQ/ba∇dblL∞(Rd)+/ba∇dblAQ−AQ◦Φh/ba∇dblL∞(/tildewideK)/parenrightbigg\n(4.20)\nby (4.6). We estimate the remaining term /ba∇dblAQ−AQ◦Φh/ba∇dblL∞(/tildewideK)by using Taylor’s theorem and\n(4.7) to find that\n/ba∇dblAQ−AQ◦Φh/ba∇dblL∞(/tildewideK)≤C/ba∇dblI−Φh/ba∇dblL∞(/tildewideK)/ba∇dbl∇AQ/ba∇dblL∞(Rd)≤C/parenleftbiggh\nL/parenrightbiggq\nh/ba∇dbl∇AQ/ba∇dblL∞(Rd).(4.21)\nCombining ( 4.19), (4.20), (4.21), and observing that these bounds are valid for all K∈ Th, we\nobtain that\n/ba∇dblA−Ah/ba∇dblL∞(Ω)≤C/parenleftbiggh\nL/parenrightbiggq\nmax\nQ∈Q/parenleftBig\n/ba∇dblAQ/ba∇dblL∞(Rd)+h/ba∇dbl∇AQ/ba∇dblL∞(Rd)/parenrightBig\n(4.22)\nThe corresponding result for µ, namely,\n/ba∇dblµ−µh/ba∇dblL∞(Ω)≤C/parenleftbiggh\nL/parenrightbiggq\nmax\nQ∈Q/parenleftBig\n/ba∇dblµQ/ba∇dblL∞(Rd)+h/ba∇dbl∇µQ/ba∇dblL∞(Rd)/parenrightBig\n(4.23)\nis established similarly, with details in [ Len86, Lemma 8]. The bound on γqthen follows from\n(4.22) and (4.23).\nTo prove the bound on /tildewideγq, we first observe that, by ( 4.14),\n/a\\}b∇acketle{t/tildewideψh,/tildewidevh/a\\}b∇acket∇i}ht=/summationdisplay\nQ∈Q/integraldisplay\nQdet(∇Φh)(gQ◦Φh)/tildewidevh\n=/summationdisplay\nQ∈Q/integraldisplay\nQ/parenleftbig\ndet(∇Φh)−1/parenrightbig\n(gQ◦Φh)/tildewidevh+/summationdisplay\nQ∈Q/integraldisplay\nQ/parenleftbig\ngQ◦(Φh−I)/parenrightbig\n/tildewidevh+/summationdisplay\nQ∈Q/integraldisplay\nQgQ/tildewidevh,\nso that, by ( 4.9),\n/a\\}b∇acketle{t/tildewideψh−ψh,/tildewidevh/a\\}b∇acket∇i}ht=/summationdisplay\nQ∈Q/integraldisplay\nQ/parenleftbig\ndet(∇Φh)−1/parenrightbig\ngQ◦Φh/tildewidevh+/summationdisplay\nQ∈Q/integraldisplay\nQ/parenleftbig\ngQ◦(Φh−I)/parenrightbig\n/tildewidevh.(4.24)\nFor a fixed Q∈ Q,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nQ(det∇Φh−1)gQ◦Φh/tildewidevh/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nK∈Th\n/tildewideK⊂Q/integraldisplay\n/tildewideK/parenleftbig\ndet∇Φh\nK−1/parenrightbig/parenleftbig\ngQ◦Φh\nK/parenrightbig\n/tildewidevh/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/summationdisplay\nK∈Th\n/tildewideK⊂Q/ba∇dbldet(∇Φh\nK)−1/ba∇dblL∞(/tildewideK)/ba∇dblgQ◦Φh\nK/ba∇dblL2(/tildewideK)/ba∇dbl/tildewidevh/ba∇dblL2(/tildewideK)\n≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblgQ/ba∇dblL2(Rd)/ba∇dbl/tildewidevh/ba∇dblL2(Q) (4.25)\n18by the second bound in ( 4.7). Similarly, by Lemma B.1and the first bound in ( 4.7) withs= 0\nands= 1,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nQ/parenleftbig\ngQ◦(Φh−I)/parenrightbig\n/tildewidevh/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/parenleftbiggh\nL/parenrightbiggq\nh/ba∇dbl∇gQ/ba∇dblL2(Rd)/ba∇dbl/tildewidevh/ba∇dblL2(Q). (4.26)\nUsing (4.25) and (4.26) in (4.24), we obtain\n|/a\\}b∇acketle{t/tildewideψh−ψh,/tildewidevh/a\\}b∇acket∇i}ht| ≤C/parenleftbiggh\nL/parenrightbiggq/parenleftbigg/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dbl2\nL2(Rd)+h2/ba∇dbl∇gQ/ba∇dbl2\nL2(Rd)/parenrightbigg1/2\n/ba∇dbl/tildewidevh/ba∇dblL2(Ω)\n≤C\nk/parenleftbiggh\nL/parenrightbiggq/parenleftbigg/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dbl2\nL2(Rd)+h2/ba∇dbl∇gQ/ba∇dbl2\nL2(Rd)/parenrightbigg1/2\n/ba∇dbl/tildewidevh/ba∇dblH1\nk(Ω).\nThe second bound in ( 4.16) then follows from the definition of /tildewideγq(3.5).\n4.5 Bound on γ∗\na\nTheorem 4.6 (Bound on γ∗\na)Suppose that Asatisfies Assumption 4.1, thatA,µ, and∂Ωsat-\nisfy the regularity requirements in Assumption 4.2for someℓ∈Z+, and thatVhsatisfies Assump-\ntion4.3for somep≤ℓ. Suppose that kh≤C0for someC0>0. Then given k0>0there exists\nC >0such that, for k≥k0, withγ∗\nadefined by (3.9)with the finite-element space equal to /tildewideVh,\nγ∗\na≤C/parenleftbig\nkh+γ∗\ns(kh)p/parenrightbig\n.\nProof.The bound on γ∗\nafollows from the general result of [ CFN20, Lemma 2.13] (see also [ GS23,\nTheorem 1.7]). Indeed, Assumptions 4.1and4.2imply that the elliptic regularity assumptions of\n[CFN20, Definition 2.1] are satisfied, and ( 4.15) provides the polynomial approximation result of\n[CFN20, Equation 2.30].\n4.6 Bound on γK\nThe next result uses the following k-weighted norms. For a Lipschitz domain D, let\n/ba∇dblu/ba∇dbl2\nHm\nk(D):=/summationdisplay\n|α|≤mk2(1−|α|)/ba∇dbl∂αu/ba∇dbl2\nL2(D)form∈Z+(4.27)\n(observe that this definition is equivalent to ( 1.3) whenm= 1) and let\n/ba∇dblu/ba∇dbl2\nHm\nk(Q):=/summationdisplay\nQ∈Q/ba∇dblu/ba∇dbl2\nHm\nk(Q)form∈Z+. (4.28)\nLemma 4.7 (Existence of a smoothing Ksatisfying the G˚ arding-type inequality (3.2))\nSuppose that Asatisfies Assumption 4.1andA,µ, and∂Ωsatisfy the regularity requirements in\nAssumption 4.2, for someℓ∈Z+. Letbbe the sesquilinear form (4.2).\nThen there exists a self-adjoint K:L2(Ω)→L2(Ω)such that given k0>0there exists C >0\nsuch that the G˚ arding-type inequality (3.2)is satisfied and K:L2(Ω)→Hn(Q)∩H1\nΓD(Ω)with\n/ba∇dblK/ba∇dblL2(Ω)→L2(Ω)≤Cand/ba∇dblK/ba∇dblL2(Ω)→Hn\nk(Q)≤Ckfor allk≥k0,1≤n≤ℓ+1.(4.29)\nFurthermore S+:Hℓ−1(Q)∩H1\nΓD(Ω)→Hℓ+1(Q)with\n/vextenddouble/vextenddoubleS+/vextenddouble/vextenddouble\nHℓ−1\nk(Q)→Hℓ+1\nk(Q)≤Ck−2. (4.30)\nWe now proceed with the operator Kintroduced in Lemma 4.7.\n19Theorem 4.8 (Bound on γK)Suppose that Asatisfies Assumption 4.1, thatA,µ, and∂Ωsat-\nisfy the regularity requirements in Assumption 4.2for someℓ∈Z+, and thatVhsatisfies Assump-\ntion4.3for somep≤ℓ. Suppose that kh≤C0for someC0>0. Then given k0>0there exists\nC >0such that, for k≥k0, withγKdefined by (3.13)with the finite-element space equal to /tildewideVh\nand the operator Kgiven by Lemma 4.7\nγK≤C(kh)p. (4.31)\nProof of Theorem 4.8using Lemma 4.7.By (4.15) and the definition of the weighted norm ( 4.28),\n/vextenddouble/vextenddouble(I−π+\nh)v/vextenddouble/vextenddouble\nH1\nk(Ω)≤C(1+kh)(kh)ℓk−ℓ/ba∇dblv/ba∇dblHℓ+1(Q)\n≤C(1+kh)(kh)ℓ/ba∇dblv/ba∇dblHℓ+1\nk(Q)for allv∈Hℓ+1(Q)∩H1\nΓD(Ω).\nCombining this with ( 4.30) and (4.29), we obtain that\nγK≤k/vextenddouble/vextenddouble(I−π+\nh)S+K/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω)\n≤k/vextenddouble/vextenddoubleI−π+\nh/vextenddouble/vextenddouble\nHℓ+1(Q)→H1\nk(Ω)/vextenddouble/vextenddoubleS+/vextenddouble/vextenddouble\nHℓ−1\nk(Q)→Hℓ+1\nk(Q)/vextenddouble/vextenddoubleK/vextenddouble/vextenddouble\nL2(Ω)→Hℓ−1\nk(Q)\n≤Ck(1+kh)(kh)ℓk−2k;\nthe result ( 4.31) then follows.\nProof of Lemma 4.7.This result was first proved in [ GS23, Lemma 2.1]. For completeness,\nwe give a (slightly different) proof here. Since Ais a symmetric, positive definite matrix-valued\nfunction, Re Ais a real, symmetric, positive definite matrix-valued function. Combin ing this with\nthe uniform positivity of Re µand the spectral theorem (see, e.g., [ McL00, Theorem 2.37]), we\nsee that there exist {λ2\nj}∞\nj=1and{uj}∞\nj=1withλ2\n1≤λ2\n2≤...with 0≤λ2\nj→ ∞asj→ ∞and\nuj∈H1\nΓD(Ω) with\n(uj,uℓ)Reµ:=/parenleftbig\n(Reµ)uj,uℓ/parenrightbig\nL2(Ω)=δjℓ\nsuch that/parenleftbig\n(ReA)∇uj,∇w/parenrightbig\nL2(Ω)=λ2\nj/parenleftbig\n(Reµ)uj,w/parenrightbig\nL2(Ω)for allw∈H1\nΓD(Ω) (4.32)\ni.e.,\n−∇·/parenleftbig\n(ReA)∇uj/parenrightbig\n=λ2\nj(Reµ)uj, (4.33)\nand\nv=∞/summationdisplay\nj=1/parenleftbig\nv,uj/parenrightbig\nReµujfor allv∈L2(Ω). (4.34)\nWe record immediately the consequences of ( 4.34) and (4.32) that\n/parenleftbig\n(Reµ)v,v/parenrightbig\nL2(Ω)=∞/summationdisplay\nj=1/vextendsingle/vextendsingle/parenleftbig\nv,uj/parenrightbig\nReµ/vextendsingle/vextendsingle2and/parenleftbig\n(ReA)∇v,∇v/parenrightbig\nL2(Ω)=∞/summationdisplay\nj=1λ2\nj/vextendsingle/vextendsingle/parenleftbig\nv,uj/parenrightbig\nReµ/vextendsingle/vextendsingle2,(4.35)\nrespectively. Let\n/tildewideKv:=∞/summationdisplay\nj=1\nλ2\nj≤2k2/parenleftbig\nv,uj/parenrightbig\nReµujfor allv∈L2(Ω). (4.36)\nand let\nK:=/parenleftBig\nsup\nx∈Ω/parenleftbig\nReµ(x)/parenrightbig1/2/parenrightBig\n/tildewideK. (4.37)\nSince/tildewideK2=/tildewideK,/tildewideKhas norm one in the L2norm weighted with Re µ, and thus the L2(Ω)→L2(Ω)\nbound in ( 4.29) holds.\nWe next prove that the L2(Ω)→Hn\nk(Ω) bound in ( 4.29) holds with 1 ≤n≤ℓ+1. SinceA,µ,\nand∂Ω satisfy Assumption 4.2, by elliptic regularity (see, e.g., [ McL00, Theorem 4.20]),\n/ba∇dbluj/ba∇dblH2(Q)≤C/parenleftBig/vextenddouble/vextenddouble∇·/parenleftbig\n(ReA)∇uj/parenrightbig/vextenddouble/vextenddouble\nL2(Ω)+/ba∇dblu/ba∇dblH1(Ω)/parenrightBig\n≤C/parenleftBig\nλ2\nj/ba∇dbluj/ba∇dblL2(Ω)+/ba∇dblu/ba∇dblH1(Ω)/parenrightBig\n.\n20By (4.32) withw=uj,/ba∇dblu/ba∇dblH1(Ω)≤C|λj|/ba∇dblu/ba∇dblL2(Ω), and thus\n/ba∇dbluj/ba∇dblH2(Q)≤Cλ2\nj/ba∇dbluj/ba∇dblL2(Ω). (4.38)\nThe bound ( 4.38) and interpolation (see, e.g. [ McL00, Theorems B.2 and B.8]) imply that\n/ba∇dbluj/ba∇dblH1(Q)≤C|λj|/ba∇dbluj/ba∇dblL2(Ω). (4.39)\nBy elliptic regularity, ( 4.33), and (4.39), for 0≤m≤ℓ+1,\n/ba∇dbluj/ba∇dblHm+2(Q)≤C/parenleftBig/vextenddouble/vextenddouble∇·/parenleftbig\n(ReA)∇uj/parenrightbig/vextenddouble/vextenddouble\nHm(Q)+/ba∇dblu/ba∇dblH1(Ω)/parenrightBig\n≤C/parenleftBig\nλ2\nj/ba∇dbluj/ba∇dblHm(Q)+/ba∇dblu/ba∇dblH1(Ω)/parenrightBig\n≤Cλ2\nj/ba∇dbluj/ba∇dblHm(Q)(4.40)\n(where the upper limit of m=ℓ+1 is determined by the regularity of A,µ,and∂Ω in Assumption\n4.2). The combination of ( 4.38), (4.39), and (4.40) imply that\n/ba∇dbluj/ba∇dblHn(Q)≤C|λj|n/ba∇dbluj/ba∇dblL2(Ω)for all 0≤n≤ℓ+1. (4.41)\nTherefore, by the definition of /tildewideK(4.36), (4.41), and (4.35),\n/vextenddouble/vextenddouble/tildewideKv/vextenddouble/vextenddouble2\nHn(Q)≤C/summationdisplay\nj=1\nλ2\nj≤2k2λ2n\nj/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2≤Ck2n/ba∇dblv/ba∇dbl2\nL2(Ω)for all 0≤n≤ℓ+1;\nthe bound ( 4.29) then follows 0 ≤n≤ℓ+1 by the definition of the weighted norm ( 4.28).\nTo prove the G˚ arding-type inequality ( 3.2), observe that, by ( 4.35) and the definition of /tildewideK\n(4.36),\nRe/parenleftbig\nb(v,v)/parenrightbig\n+2k2/parenleftbig/tildewideKv,/tildewideKv/parenrightbig\nReµ= (Reb)(v,v)+2k2/parenleftbig/tildewideKv,v/parenrightbig\nReµ,\n=∞/summationdisplay\nj=1/parenleftbig\nλ2\nj−k2/parenrightbig/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2+2k2∞/summationdisplay\nj=1\nλ2\nj≤2k2/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2,\n=/summationdisplay\nλ2\nj≤2k2/parenleftbig\nλ2\nj−k2/parenrightbig/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2+/summationdisplay\nλ2\nj>2k2/parenleftbig\nλ2\nj+k2)/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2.(4.42)\nNow,\nifλ2\nj≥2k2then (λ2\nj−k2)≥1\n2λ2\nj≥1\n4(λ2\nj+k2). (4.43)\nTherefore, combining ( 4.37), (4.42), and (4.43), we obtain that, for all v∈H1\nΓD(Ω),\nRe/parenleftbig\nb(v,v)/parenrightbig\n+2k2/ba∇dblKv/ba∇dbl2\nL2(Ω)≥Re/parenleftbig\nb(v,v)/parenrightbig\n+2k2/vextenddouble/vextenddouble/tildewideKv/vextenddouble/vextenddouble2\nReµ\n≥1\n4∞/summationdisplay\nj=1/parenleftbig\nλ2\nj+k2/parenrightbig/vextendsingle/vextendsingle(v,uj)Reµ/vextendsingle/vextendsingle2\n≥1\n4/parenleftBig/parenleftbig\n(ReA)∇v,∇v/parenrightbig\nL2(Ω)+k2/parenleftbig\n(Reµ)v,v/parenrightbig\nL2(Ω)/parenrightBig\n≥Ccoer/ba∇dblv/ba∇dbl2\nH1\nk(Ω),\nwhich is ( 3.2) with\nCcoer:=1\n4min/braceleftBig\ninf\nx∈Ω/parenleftbig\n(ReA)∗(x)/parenrightbig\n,inf\nx∈Ω(Reµ(x))/bracerightBig\n.\nwhere, forx∈Ω, (ReA)∗(x) := inf ξ∈Rd(ReA)(x)ijξiξj.\nTo complete the proof, we need to prove the bound ( 4.30). First observe that, by coercivity of\nb+, the Lax–Milgram lemma (see, e.g., [ McL00, Lemma 2.32]) and the definitions of /ba∇dbl·/ba∇dbl(H1\nk(Ω))∗\nand/ba∇dbl·/ba∇dblH1\nk(Ω),\n/vextenddouble/vextenddoubleS+/vextenddouble/vextenddouble\n(H1\nk(Ω))∗→H1\nk(Ω)≤C,/vextenddouble/vextenddoubleS+/vextenddouble/vextenddouble\nL2(Ω)→H1\nk(Ω)≤C\nk.and/vextenddouble/vextenddoubleS+/vextenddouble/vextenddouble\nL2(Ω)→L2(Ω)≤C\nk2.(4.44)\n21Givenφ∈L2(Ω), by the definition of S+(3.12),v:=S+φ∈H1\nΓD(Ω) is the solution of the equation\n−k2(Reµ)v−∇·/parenleftbig\n(ReA)∇v/parenrightbig\n+k2Kv=φ.\nSinceA,µ, and∂Ω satisfy Assumption 4.2, by elliptic regularity (see, e.g., [ McL00, Theorem 4.20])\nthe second bound in ( 4.44), and the third bound in ( 4.44), for 0≤m≤ℓ−1,\n|v|Hm+2(Q)≤C/parenleftBig/vextenddouble/vextenddoublek2Kv+k2(Reµ)v+φ/vextenddouble/vextenddouble\nHm(Q)+/ba∇dblv/ba∇dblH1(Ω)/parenrightBig\n≤C/parenleftBig\nk2+m/ba∇dblv/ba∇dblL2(Ω)+k2/ba∇dblv/ba∇dblHm(Q)+/ba∇dblφ/ba∇dblHm(Q)/parenrightBig\n≤C/parenleftBig\nkm/ba∇dblφ/ba∇dblL2(Ω)+k2/ba∇dblv/ba∇dblHm(Q)+/ba∇dblφ/ba∇dblHm(Q)/parenrightBig\n.\nAfter multiplying by k−m−1and using the definitions of the weighted norms ( 4.27) and (4.28), we\nobtain that\nk−m−1|v|Hm+2(Q)≤C/parenleftBig\n/ba∇dblv/ba∇dblHm\nk(Q)+k−2/ba∇dblφ/ba∇dblHm\nk(Q)/parenrightBig\nfor 0≤m≤ℓ−1.(4.45)\nByiterativelyapplying ( 4.45), andthenusingeitherthesecondandthirdboundsin( 4.44)(depend-\ning on whether mis odd or even), we obtain that /ba∇dblv/ba∇dblHn+2\nk(Q)≤k−2/ba∇dblφ/ba∇dblHn\nk(Q)for all 0≤n≤ℓ−1;\nusing this last bound with n=m−2 in (4.45) and then setting m=ℓ−1, we obtain the result\n(4.30).\n4.7 Transferring the error bound from the mapped finite-elem ent space\nto the true finite-element space\nWith Theorems 4.5,4.6, and4.8we have all the tools needed to apply the abstract framework of\nSection3to obtain an error estimate for the abstract mapped finite-elemen t solution, i.e., a bound\non/ba∇dblu−/tildewideuh/ba∇dblH1\nk(Ω). We now provide an additional set of results to give an error estimat e for the\n“concrete” finite element solution uh. Since we can expect that K=/tildewideKfor the elements K∈ Th\nnot touching an interface, the error bound for /tildewideuhalready gives an error bound for uhon those\nelements, but we can be a bit more precise. To do so, we need the follo wing additional assumption\nthat essentially requires that for the elements K∈ Thnot touching an interface, Ψh\nK=I.\nAssumption 4.9 (Ruling out pathological behaviour of ΨhinsideQ)For allQ∈ Q,\nQ∩Qh=/uniondisplay\nK∈Th\nK⊂Qh/tildewideK∩K.\nUnder Assumption 4.9, we can transform an error estimate on /tildewideuhinto an error estimate on uh.\nLemma 4.10 (Error transfer) Letv∈H1\nΓD(Ω)andvh∈Vh. Then, for all Q∈ Q,\n/ba∇dblv−vh/ba∇dblH1\nk(Q∩Qh)≤C/parenleftbigg\n/ba∇dblv−/tildewidevh/ba∇dblH1\nk(Q)+/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblv/ba∇dblH1\nk(Q)/parenrightbigg\n. (4.46)\nProof.If we can prove that\n/ba∇dbl/tildewidevh−vh/ba∇dblH1\nk(Q∩Qh)≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dbl/tildewidevh/ba∇dblH1\nk(Q∩Qh), (4.47)\nthen the result ( 4.46) follows by the triangle inequality:\n/ba∇dblv−vh/ba∇dblH1\nk(Q∩Qh)≤ /ba∇dblv−/tildewidevh/ba∇dblH1\nk(Q∩Qh)+/ba∇dbl/tildewidevh−vh/ba∇dblH1\nk(Q∩Qh)\n≤ /ba∇dblv−/tildewidevh/ba∇dblH1\nk(Q∩Qh)+C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dbl/tildewidevh/ba∇dblH1\nk(Q∩Qh)\n≤/parenleftbigg\n1+C/parenleftbiggh\nL/parenrightbiggq/parenrightbigg\n/ba∇dblv−/tildewidevh/ba∇dblH1\nk(Q∩Qh)+C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblv/ba∇dblH1\nk(Q∩Qh)\n≤C/parenleftbigg\n/ba∇dblv−/tildewidevh/ba∇dblH1\nk(Q∩Qh)+/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblv/ba∇dblH1\nk(Q∩Qh)/parenrightbigg\n.\n22We now prove ( 4.47); by Assumption 4.9,\n/ba∇dbl/tildewidevh−vh/ba∇dbl2\nL2(Q∩Qh)=/summationdisplay\nK∈Th\nK⊂Qh/ba∇dbl/tildewidevh−vh/ba∇dbl2\nL2(K∩/tildewideK).\nWe now focus on a single element K. Since/tildewidevh=vh◦Φhand Φh(x) = Φh\nK(x) whenx∈/tildewideK,\n/ba∇dbl/tildewidevh−vh/ba∇dbl2\nL2(K∩/tildewideK)=/integraldisplay\nK∩/tildewideK|vh(Φh(x))−vh(x)|2dx=/integraldisplay\nK∩/tildewideK|vh(Φh\nK(x))−vh(x)|2dx.(4.48)\nWe now work on the integrand; specifically,\n/vextendsingle/vextendsinglevh(Φh\nK(x))−vh(x)/vextendsingle/vextendsingle=/vextendsingle/vextendsinglevh(x+(I−Φh\nK)(x))−vh(x)/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle(I−Φk\nK)(x)/vextendsingle/vextendsingle/ba∇dbl∇vh/ba∇dblL∞(K)≤ /ba∇dblI−Φk\nK/ba∇dblL∞(K)/ba∇dbl∇vh/ba∇dblL∞(K);\nusing this in ( 4.48) gives\n/ba∇dbl/tildewidevh−vh/ba∇dbl2\nL2(K∩/tildewideK)≤ /ba∇dblI−Φk\nK/ba∇dbl2\nL∞(K)|K|/ba∇dbl∇vh/ba∇dbl2\nL∞(K)≤Ch−2\nK/ba∇dblI−Φk\nK/ba∇dbl2\nL∞(K)/ba∇dblvh/ba∇dbl2\nL2(K)\ndue to standard equivalence of norms and scaling argument on polyn omial spaces recapped in\nLemmaA.1below. Summing over Kand using the bound ( 4.7) on Φ in Assumption 4.3, we obtain\nthat\n/ba∇dbl/tildewidevh−vh/ba∇dblL2(Q∩Qh)≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblvh/ba∇dblL2(Qh)≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dbl/tildewidevh/ba∇dblL2(Q). (4.49)\nWe argue similarly for the gradient term; since /tildewidevh=vh◦Φhand Φh(x) = Φh\nK(x) whenx∈/tildewideK,\n/ba∇dbl∇(/tildewidevh−vh)/ba∇dbl2\nL2(Q∩Qh)=/summationdisplay\nK∈Th\nK⊂Qh/integraldisplay\nK∩/tildewideK/vextendsingle/vextendsingle(∇Φh\nK)T(x)(∇vh)(Φh\nK(x))−(∇vh)(x)/vextendsingle/vextendsingle2dx.\nThen, with I dthed×didentity matrix,\n/vextendsingle/vextendsingle(∇Φh\nK)T(x)(∇vh)(Φh\nK(x))−(∇vh)(x)/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle(Id−∇Φh\nK)T(x)(∇vh)(Φh\nK(x))/vextendsingle/vextendsingle+/vextendsingle/vextendsingle(∇vh)(ΦK\nh(x))−(∇vh)(x)/vextendsingle/vextendsingle\n≤ /ba∇dblId−∇Φh\nK/ba∇dblL∞(/tildewideK)/ba∇dbl∇vh/ba∇dblL∞(K)+/ba∇dblI−ΦK\nh/ba∇dblL∞(/tildewideK)/ba∇dbl∇2vh/ba∇dblL∞(K),\nand it follows from the scaling arguments in Lemma A.1that\n/ba∇dbl∇(/tildewidevh−vh)/ba∇dblL2(Q∩Qh)≤C/parenleftBig\n/ba∇dblId−∇Φh\nK/ba∇dblL∞(/tildewideK)+h−1\nK/ba∇dblI−ΦK\nh/ba∇dblL∞(/tildewideK)/parenrightBig\n/ba∇dbl∇vh/ba∇dblL2(K).\nSumming over Kand using the bound ( 4.7) on Φ in Assumption 4.3, we obtain that\n/ba∇dbl∇(/tildewidevh−vh)/ba∇dblL2(Q∩Qh)≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dbl∇vh/ba∇dblL2(Qh)≤C/parenleftbiggh\nL/parenrightbiggq\n/ba∇dbl∇/tildewidevh/ba∇dblL2(Q).(4.50)\nCombining ( 4.49) and (4.50), we obtain ( 4.47) and the proof is complete.\n4.8 Theorem 3.2applied to Helmholtz problems solved with curved fi-\nnite elements\nTheorem 4.11 (Preasymptotic Helmholtz h-FEM error bound with curved finite el-\nements.) Suppose that A,µ, andgsatisfy the assumptions in §4.1(in particular, Asatisfies\nAssumption 4.1andA,µ, and∂Ωsatisfy the regularity requirements in Assumption 4.2). Suppose\nthatVhsatisfies Assumptions 4.3and4.9. Suppose that kh≤C0for someC0>0.\n(i) Givenk0>0there exists C1,C2>0such that, for all k≥k0, if\n/parenleftBig\nkh+γ∗\ns(kh)p/parenrightBig\n(kh)p+(1+γ∗\ns)/parenleftbiggh\nL/parenrightbiggq\n≤C1 (4.51)\n23then the Galerkin solution uhexists, is unique, and satisfies\n/parenleftBig/summationdisplay\nQ∈Q/ba∇dblu−uh/ba∇dbl2\nH1\nk(Q∩Qh)/parenrightBig1/2\n≤C2/bracketleftbigg/parenleftBig\n1+γ∗\ns(kh)p/parenrightBig\n/ba∇dbl(I−/tildewideπh)u/ba∇dblH1\nk(Ω)\n+(1+γ∗\ns)/parenleftbiggh\nL/parenrightbiggq/parenleftbigg\n/ba∇dblu/ba∇dblH1\nk(Ω)+1\nk2/parenleftBig/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dbl2\nH1\nk(Rd)/parenrightBig1/2/parenrightbigg/bracketrightbigg\n,(4.52)\nwhere/tildewideπhis the orthogonal projection in the mapped finite-element sp ace/tildewideVh(which, in particular,\nsatisfies the interpolation bound (4.15)).\n(ii) Suppose, in addition, that suppg⊂Qfor someQ∈ Qand that given k0>0there exists\nC >0such that for all 0≤j≤p−1,\n/ba∇dblg/ba∇dblHj(Ω)≤Ckj+1sup\nv∈H\n/bardblv/bardblH1\nk(Ω)=1|/a\\}b∇acketle{tg,v/a\\}b∇acket∇i}ht|for allk≥k0. (4.53)\nThen, given k0>0, there exists C3>0such that, for all k≥k0, if(4.51)holds, then\n/parenleftBig/summationdisplay\nQ∈Q/ba∇dblu−uh/ba∇dbl2\nH1\nk(Q∩Qh)/parenrightBig1/2\n≤C3/bracketleftbigg/parenleftBig\n1+γ∗\ns(kh)p/parenrightBig\n(kh)p+γ∗\ns/parenleftbiggh\nL/parenrightbiggq/bracketrightbigg\n/ba∇dblu/ba∇dblH1\nk(Ω).\nProof.(i) Under the assumptions of the theorem, Theorems 4.5,4.6, and4.8all hold. Applying\nTheorem 3.2to the variational problem ( 4.11) (i.e., in the mapped space /tildewideVh) and inputting the\nbounds on γq,/tildewideγq,γ∗\na,andk/ba∇dbl(I−πh)S+K/ba∇dblL2(Ω)→H1\nk(Ω), we obtain, for some C >0 depending on\nAandµ,\n/ba∇dblu−/tildewideuh/ba∇dblH1\nk(Ω)\n≤C/parenleftbig\n1+γ∗\ns(kh)p/parenrightbig\n/ba∇dbl(I−/tildewideπh)u/ba∇dblH1\nk(Ω)\n+C(1+γ∗\ns)/parenleftbigg/parenleftbiggh\nL/parenrightbiggq\n/ba∇dblu/ba∇dblH1\nk(Ω)+/parenleftbiggh\nL/parenrightbiggq1\nk/parenleftbigg/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dbl2\nL2(Rd)+h2/ba∇dbl∇gQ/ba∇dbl2\nL2(Rd)/parenrightbigg1/2/parenrightbigg\n(4.54)\n(compare to ( 3.19)). The bound ( 4.46) and the fact that ( h/L)qis bounded (by ( 4.51)) allows us\nto replace /ba∇dblu−/tildewideuh/ba∇dblH1\nk(Ω)on the left-hand side of ( 4.54) with/summationtext\nQ∈Q/ba∇dblu−uh/ba∇dblH1(Q∪Qh). The result\n(4.52) then follows by using h≤C0/kin the terms involving gQ.\n(ii) The result follows from Part (i) if we can show that\n(a) givenk0>0 there exists C >0 such that, for all k≥k0, ifkh≤C0,\n/ba∇dbl(I−/tildewideπh)u/ba∇dblH1\nk(Ω)≤C(kh)p/ba∇dblu/ba∇dblH1\nk(Ω), (4.55)\n(b) ifgsatisfies ( 4.53), then\n1\nk2/parenleftBig/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dblH1\nk(Rd)/parenrightBig1/2\n≤C/ba∇dblu/ba∇dblH1\nk(Ω). (4.56)\nFor (a), by ( 4.15)\n/ba∇dbl(I−/tildewideπh)u/ba∇dblH1\nk(Ω)≤C(1+kh)hp/ba∇dblu/ba∇dblHp+1(Q). (4.57)\nIfgsatisfies ( 4.53), then by elliptic regularity (using the regularity assumptions on A,µ, and∂Ω\nin Assumption 4.2),\n/ba∇dblu/ba∇dblHp+1(Q)≤Ckp/ba∇dblu/ba∇dblH1\nk(Ω)for allk≥k0; (4.58)\n24(i.e., the data being k-oscillatoryimplies that the solution is k-oscillatory); the proof is very similar\nto the uses of elliptic regularity in the proof of Lemma 4.7; for the details see [ GS23, end of the\nproof of Theorem 1.5]. Combining ( 4.57) and (4.58), we obtain that\n/ba∇dbl(I−/tildewideπh)u/ba∇dblH1\nk(Ω)≤C(1+kh)(kh)p/ba∇dblu/ba∇dblH1\nk(Ω),\nand (4.55) follows since kh≤C0.\nFor (b), by ( 4.53) applied with j= 1 and the fact that supp g⊂Qfor someQ∈ Q,\n1\nk2/parenleftBig/summationdisplay\nQ∈Q/ba∇dblgQ/ba∇dblH1\nk(Rd)/parenrightBig1/2\n=k−2/ba∇dblg/ba∇dblH1\nk(Ω)≤C/ba∇dblg/ba∇dbl(H1\nk(Ω))∗. (4.59)\nBy continuity of b(3.1) and (4.3),\n/ba∇dblg/ba∇dbl(H1\nk(Ω))∗≤M/ba∇dblu/ba∇dblH1\nk(Ω), (4.60)\nand (4.56) follows from combining ( 4.59) and (4.60).\nRemark 4.12 (Obtaining Theorem 1.1from Part (ii) of Theorem 4.11)For the radial PML\noriginally introduced in [ CM98], the coefficient Asatisfies Assumption 4.1by, e.g., [ GLSW24 ,\nLemma 2.3]. Theorem 1.1therefore follows from Part (ii) of Theorem 4.11if the data satisfies\n(4.53)and is such that suppg⊂Qfor someQ∈ Q.\nAs described in Definitions 2.3and2.4, the solution of the plane-wave scattering problem can\nbe approximated using a PML by solving for the unknown u:=χuI+uS, withχ∈C∞\ncomp(Rd)equal\none on the scatterer, and corresponding data g=fwithfdefined by (2.9). By construction, the\nPML must be away from the support of χ; thus∇χ, and hence also g, are supported where A=I\nandµ= 1, and thus on a single Q∈ Q.\nA Scaling arguments and inverse estimates\nIn this section, we consider a mesh Thsatisfying the requirements of Assumption 4.3. Recall that,\nin this case, the elements K∈ Thare obtained by mapping a single reference simplex /hatwideKthrough\nbilipschitz maps FK:/hatwideK→K. To simplify the notation, we let GK:=F−1\nK. By (4.5),\n/ba∇dbldet∇FK/ba∇dblL∞(K)≤C/ba∇dbl∇FK/ba∇dbld\nL∞(K)≤Chd\nK, (A.1a)\nand\n/ba∇dbldet∇GK/ba∇dblL∞(/hatwideK)≤C/ba∇dbl∇GK/ba∇dbld\nL∞(/hatwideK)≤Ch−d\nK. (A.1b)\nLemma A.1 For allvh∈VhandK∈ Th,\n|K|/ba∇dbl∇vh/ba∇dbl2\nL∞(K)≤Ch−2\nK/ba∇dblvh/ba∇dbl2\nL2(K),|K|/ba∇dbl∇2vh/ba∇dbl2\nL∞(K)≤Ch−2\nK/ba∇dbl∇vh/ba∇dbl2\nL2(K).\nProof.By (A.1),\n|K|=/integraldisplay\nK1dx=/integraldisplay\n/hatwideK(det∇FK)(/hatwidex)d/hatwidex≤ |/hatwideK|/ba∇dbldet∇FK/ba∇dblL∞(/hatwideK)≤Chd\nK. (A.2)\nLet/hatwidev∈ Pp(/hatwideK) and setv:=/hatwidev◦F−1\nK=/hatwidev◦GK. By the chain rule,\n∂jv=∂jGℓ\nK(∂ℓ/hatwidev)◦GK\nand a second application gives\n∂2\njℓv=∂2\njℓGℓ\nK(∂ℓ/hatwidev)◦GK+∂jGℓ\nK∂ℓGm\nK(∂2\nℓm/hatwidev)◦GK.\nIt then follows from ( 4.5) that\n/ba∇dbl∇v/ba∇dblL∞(K)≤Ch−1\nK/ba∇dbl∇/hatwidev/ba∇dblL∞(/hatwideK)and/ba∇dbl∇2v/ba∇dblL∞(K)≤Ch−2\nK/parenleftBig\n/ba∇dbl∇/hatwidev/ba∇dblL∞(/hatwideK)+/ba∇dbl∇2/hatwidev/ba∇dblL∞(/hatwideK)/parenrightBig\n.\n25SincePp(/hatwideK) is a finite dimensional space, equivalence of semi-norms gives that\n/ba∇dbl∇/hatwidev/ba∇dblL∞(/hatwideK)≤C/ba∇dbl/hatwidev/ba∇dblL2(/hatwideK)and/ba∇dbl∇/hatwidev/ba∇dblL∞(/hatwideK)+/ba∇dbl∇2/hatwidev/ba∇dblL∞(/hatwideK)≤C/ba∇dbl∇/hatwidev/ba∇dblL2(/hatwideK),\nleading to\n/ba∇dbl∇v/ba∇dblL∞(K)≤Ch−1\nK/ba∇dbl/hatwidev/ba∇dblL2(/hatwideK)and/ba∇dbl∇2v/ba∇dblL∞(K)≤Ch−2\nK/ba∇dbl∇/hatwidev/ba∇dblL2(/hatwideK).(A.3)\nUsing the chain rule in the other direction gives us that\n|∇/hatwidev|2≤Ch2\nK|∇v◦FK|2,\nand, by the change-of-variable formula, we have\n/ba∇dbl/hatwidev/ba∇dbl2\nL2(/hatwideK)≤Ch−d/ba∇dblv/ba∇dbl2\nL2(K)and/ba∇dbl∇/hatwidev/ba∇dbl2\nL2(/hatwideK)≤Ch−dh2\nK/ba∇dbl∇v/ba∇dbl2\nL2(K), (A.4)\nwhere we again used ( A.1); combining ( A.3), (A.4), and (A.2) concludes the proof.\nB Composition with a map close to the identity\nThe following result is used to estimate the geometric error on the rig ht-hand side (in Theorem\n4.5).\nLemma B.1 LetU⊂Rdbe a measurable set and φ:U→φ(U)a bilipschitz mapping such that\n/ba∇dblI−φ/ba∇dblL∞(U)≤εand/ba∇dblId− ∇φ/ba∇dblL∞(U)≤ε′<1. LetUε:={x∈Rd: dist(x,U)< ε}and\nf∈H1(Uε). Then\n/ba∇dblf−f◦φ/ba∇dblL2(U)≤ε\n1−ε′/ba∇dbl∇f/ba∇dblL2(Uε).\nProof.Letφt(x) =x+t(φ(x)−x) andgx(t) =f(φt(x))t∈[0,1]. Then\nf(φ(x))−f(x) =/integraldisplay1\n0g′\nx(t)dt=/integraldisplay1\n0(φ(x)−x)·(∇f)(φt(x))dt,\nand\n/integraldisplay\nU|f(φ(x))−f(x)|2dx=/integraldisplay\nU/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0(φ(x)−x)·(∇f)(φt(x))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx\n≤/integraldisplay1\n0/integraldisplay\nU/vextendsingle/vextendsingle(φ(x)−x)·(∇f)(φt(x))/vextendsingle/vextendsingle2dxdt,\nwhere we have used the Cauchy-Schwartz inequality and Fubini’s the orem.\nSince∇φt= Id+t(∇φ−Id), we have |∇φt(x)| ≥1−tε′≥1−ε′. It follows that φ−1\ntexists for\nallt∈[0,1] with\n|∇(φ−1\nt)(y)| ≤1\n1−ε′for ally∈φt(U). (B.1)\nHence, for a fixed t∈[0,1], by (B.1) and the fact that φt(U)⊂Uεfor allt∈[0,1],\n/integraldisplay\nU/vextendsingle/vextendsingle(φ(x)−x)·(∇f)(φt(x))/vextendsingle/vextendsingle2dx≤ /ba∇dblφ−I/ba∇dbl2\nL∞(U)/integraldisplay\nU|(∇f)(φt(x))|2dx\n=/ba∇dblφ−I/ba∇dbl2\nL∞(U)/integraldisplay\nφt(U)/vextendsingle/vextendsingledet∇(φ−1\nt)(y)|2|(∇f)(y)/vextendsingle/vextendsingle2dy\n≤/parenleftbiggε\n1−ε′/parenrightbigg2/integraldisplay\nUε|(∇f)(y)|2dy,\nand the result follows.\n26References\n[Ain04] M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave\nnumber, SIAM Journal on Numerical Analysis 42(2004), no. 2, 553–575.\n[BCWG+11] T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon , and M. Lindner, Condition number\nestimates for combined potential boundary integral operat ors in acoustics and their boundary element\ndiscretisation , Numer. Methods Partial Differential Eq. 27(2011), no. 1, 31–69.\n[CFN18] T. Chaumont-Frelet and S. Nicaise, High-frequency behaviour of corner singularities in Helmh oltz\nproblems , ESAIM-Math. Model. Num. 52(2018), no. 5, 1803 – 1845.\n[CFN20] ,Wavenumber explicit convergence analysis for finite elemen t discretizations of general wave\npropagation problem , IMA Journal of Numerical Analysis 40(2020), no. 2, 1503–1543.\n[CFN23] ,An Analysis of High-Frequency Helmholtz Problems in Domain s with Conical Points and\nTheir Finite Element Discretisation , Computational Methods in Applied Mathematics 23(2023),\nno. 4, 899–916.\n[Cia91] P. G. Ciarlet, Basic error estimates for elliptic problems , Handbook of numerical analysis, Vol. II,\nNorth-Holland, Amsterdam, 1991, pp. 17–351.\n[Cia02] P. G. Ciarlet, The finite element method for elliptic problems , SIAM, Classicsin Applied Mathematics,\n2002.\n[CM98] F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates , SIAM J. Sci. Comput.\n19(1998), 2061–2090.\n[CP02] F. Cardoso and G. Popov, Quasimodes with exponentially small errors associated wit h elliptic periodic\nrays, Asymptotic Analysis 30(2002), no. 3, 4, 217–247.\n[CR72a] P. G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integ ration\nin isoparametric finite element methods , The mathematical foundations of the finite element method\nwith applications to partial differential equations, Elsev ier, 1972, pp. 409–474.\n[CR72b] P.G. Ciarlet and P.A. Raviart, Interpolation theory over curved elements, with applicati ons to finite\nelement methods , Comput. Meth. Apl. Mech. Engrg. 1(1972), 217–249.\n[DW15] Y. Du and H. Wu, Preasymptotic error analysis of higher order FEM and CIP-FE M for Helmholtz\nequation with high wave number , SIAM J. Num. Anal. 53(2015), no. 2, 782–804.\n[FW09] X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave\nnumber, SIAM J. Num. Anal. 47(2009), no. 4, 2872–2896.\n[FW11] ,hp-Discontinuous Galerkin methods for the Helmholtz equatio n with large wave number ,\nMathematics of Computation 80(2011), no. 276, 1997–2024.\n[GLS23] J. Galkowski, D. Lafontaine, and E. A. Spence, Perfectly-matched-layer truncation is exponentially\naccurate at high frequency , SIAM J. Math. Anal. 55(2023), no. 4, 3344–3394.\n[GLSW24] J. Galkowski, D. Lafontaine, E. A. Spence, and J. Wu nsch,Thehp-FEM applied to the Helmholtz\nequation with PML truncation does not suffer from the polluti on effect , Comm. Math. Sci., to appear\n(2024).\n[GR09] C. Geuzaine and J.F. Remacle, Gmsh: A 3D finite element mesh generator with built-in pre- an d\npost-processing facilities , Int. J. Numer. Meth. Engrg. 79(2009), 1309–1331.\n[GS23] J. Galkowski and E. A. Spence, Sharp preasymptotic error bounds for the Helmholtz h-FEM, arXiv\npreprint arXiv:2301.03574 (2023).\n[IB97] F. Ihlenburg and I. Babuska, Finite element solution of the Helmholtz equation with high wave number\npart II: the hpversion of the FEM , SIAM J. Numer. Anal. 34(1997), no. 1, 315–358.\n[Ihl98] F. Ihlenburg, Finite element analysis of acoustic scattering , Springer Verlag, 1998.\n[Len86] M. Lenoir, Optimal isoparametric finite elements and error estimates f or domains involving curved\nboundaries , SIAM J. Numer. Anal. 23(1986), no. 3, 562–580.\n[LSW21] D. Lafontaine, E. A. Spence, and J. Wunsch, For most frequencies, strong trapping has a weak effect\nin frequency-domain scattering , Communications on Pure and Applied Mathematics 74(2021), no. 10,\n2025–2063.\n[McL00] W. McLean, Strongly elliptic systems and boundary integral equations , Cambridge University Press,\n2000.\n[Mor75] C. S. Morawetz, Decay for solutions of the exterior problem for the wave equa tion, Comm. Pure Appl.\nMath.28(1975), no. 2, 229–264.\n[MS10] J. M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz\nequation with Dirichlet-to-Neumann boundary conditions , Math. Comp 79(2010), no. 272, 1871–1914.\n[MS11] ,Wavenumber explicit convergence analysis for Galerkin dis cretizations of the Helmholtz equa-\ntion, SIAM J. Numer. Anal. 49(2011), 1210–1243.\n[MS19] A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bou nds and\nresonance-free regions , Math. Mod. Meth. Appl. S. 29(2019), no. 2, 317–354.\n27[Pem20] O. R. Pembery, The Helmholtz Equation in Heterogeneous and Random Me-\ndia: Analysis and Numerics , Ph.D. thesis, University of Bath, 2020,\nhttps://researchportal.bath.ac.uk/en/studentTheses/ the-helmholtz-equation-in-heterogeneous-and-random- media- \n[PV99] G. Popov and G. Vodev, Resonances near the real axis for transparent obstacles , Communications in\nMathematical Physics 207(1999), no. 2, 411–438.\n[Sau06] S. A. Sauter, A refined finite element convergence theory for highly indefin ite Helmholtz problems ,\nComputing 78(2006), no. 2, 101–115.\n[Sch74] A. H. Schatz, An observation concerning Ritz-Galerkin methods with inde finite bilinear forms , Math-\nematics of Computation 28(1974), no. 128, 959–962.\n[Str72] G. Strang, Variational crimes in the finite element method , The mathematical foundations of the\nfinite element method with applications to partial different ial equations, Elsevier, 1972, pp. 689–710.\n[Vai75] B. R. Vainberg, On the short wave asymptotic behaviour of solutions of stati onary problems and\nthe asymptotic behaviour as t→ ∞of solutions of non-stationary problems , Russian Mathematical\nSurveys 30(1975), no. 2, 1–58.\n[Wu14] H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Hel mholtz equation with high\nwave number. Part I: linear version , IMA J. Numer. Anal. 34(2014), no. 3, 1266–1288.\n[ZW13] L. Zhu and H. Wu, Preasymptotic error analysis of CIP-FEM and FEM for Helmhol tz equation with\nhigh wave number. Part II: hp version , SIAM J. Num. Anal. 51(2013), no. 3, 1828–1852.\n28" }, { "title": "2401.16460v1.Isotropic_cosmological_model_with_aetherically_active_axionic_dark_matter.pdf", "content": "arXiv:2401.16460v1 [gr-qc] 29 Jan 2024Isotropic cosmological model with aetherically active axi onic dark matter\nAlexander B. Balakin1,∗and Amir F. Shakirzyanov1,†\n1Department of General Relativity and Gravitation, Institu te of Physics,\nKazan Federal University, Kremlevskaya str. 16a, Kazan 420 008, Russia\n(Dated: January 31, 2024)\nIn the framework of the extended Einstein-aether-axion the ory we study the model of a two-level\naetheric control over the evolution of a spatially isotropi c homogeneous Universe filled with axionic\ndark matter. Two guiding functions are introduced, which de pend on the expansion scalar of the\naether flow, equal to the tripled Hubble function. The guidin g function of the first type enters\nthe aetheric effective metric, which modifies the kinetic ter m of the axionic system; the guiding\nfunction of the second type predetermines the structure of t he potential of the axion field. We\nobtained new exact solutions of the total set of master equat ions of the model (with and without\ncosmological constant), and studied in detail four analyti cally solvable submodels, for which both\nguiding functions are reconstructed and illustrations of t heir behavior are presented.\nPACS numbers: 04.20.-q, 04.40.-b, 04.40.Nr, 04.50.Kd\nI. INTRODUCTION\nA century ago Alexander Friedmann formulated the prediction that our Universe expands, and this event predeter-\nmined all further development of cosmology and space sciences. Wh ile remaining within this general concept, modern\ncosmology focuses on describing the details of this expansion, in par ticular, on the rate of expansion at different\nepochs. New sensational results of observations obtained in the la st decade have become the basis for restructuring\nour ideas about the history of the early Universe. Discovery of the gravitational radiation was the first important\nevent, that made theorists think about the validity of previous idea s. Indeed, in 2015 the first observation of the\ngravitational waves from the black hole merger [1] put researcher s in a dilemma. In this event the masses of the\ncolliding black holes were predicted to be of 36 and 29 M(Sun), while mass values in the range 2.5-10 M(Sun)predicted\nby the theory of stellar collapse, seemed to be reasonable. Then th e gravitational wave event indicated as GW trigger\nS190521g (GW 190521) [2], has shown that the black holes with the ma sses 85 and 66 M(Sun)collided; the general\nconsensus is that the mass of at least one of these black holes lies in a mass range that excludes its birth through the\ncollapse of a star. The discovery of black hole with the so-called inter mediate mass 91.000 M(Sun)[3], the existence of\nwhich can not be explained by the existing theories, completed the fo rmulation of the dilemma: either it is necessary\nto abandon this interpretation, or admit that there is a new unknow n mechanism for the formation of black holes.\nFortunately, the second trend has triumphed and now theorists a re actively involved in adequate extension of the\nmodels of the birth of black holes. Another amazing story is connect ed with observations on the newest James Webb\nSpace Telescope (JWST). New observational data concern, in par ticular, the discovery of an extremely magnified\nmonster star, estimations of the masses of warm dark matter par ticles and of the axion dark matter particles [4]; the\nabundance of carbon-containing molecules [5]. But the most importa nt event, from our point of view is the discovery\nof enormous distant galaxies that should not exist, if one follows the standard model of the early Universe evolution.\nTo be brief, the galaxies found in the JWST images [6] appeared shoc kingly big, and the stars in them too old, and\nthese findings are in conflict with existing models. In other words, ra pid development is predicted for the theory of\nthe evolution of the early Universe over the next few years, and mo difications of the cosmological models are highly\nwelcome.\nAt the moment, the most adequate picture of the world contains an early era of inflation, epochs of domination\nof radiation and matter, and a late-time era of accelerated expans ion. The theorist’s dream is to unify the entire\nhistory of the Universe within the framework of one cosmological mo del (see, e.g., [7–13]). The main obstacle to\nsolving this problem is the difficulty of finding a unified equation of state for cosmic substrates that determine the\nrate of evolution of the Universe in the corresponding epoch. One o f the attempts was the search for time depending\nparameters of the equation of state, and the introduction of a co smological term depending on time. However, such\nattempts were considered unsuccessful because cosmological t ime is not an invariant, and therefore such equations of\n∗Electronic address: Alexander.Balakin@kpfu.ru\n†Electronic address: shamirf@mail.ru2\nstate are associated with the loss of covariance of the theory. A s imilar problem arises, when one tries to define the\nequation of state in terms of the redshift value Z, or equivalently, via the scale factor a(t).\nWe follow another logic. We admit that the parameters of the equatio n of state depend on the set of scalars, which\nare formed on the basis of fundamental fields inherent in the cosmo logical model under consideration. To be more\nprecise, we take the unit timelike vector field Ujassociated with the velocity four-vector of the dynamic aether [14 –17]\nand consider the invariants obtained in the course of decomposition of its covariant derivative ∇kUj. In other words,\nwe use four differential invariants (the expansion scalar of the aet her flow, Θ= ∇kUk, the squares of the acceleration\nfour-vector, of the shear and vorticity tensors, a2,σ2,ω2, respectively), as the arguments of the parameters included\ninto the equations of state. This means that we follow the paradigm o f an aetheric control over the evolution of\nphysical systems (see, e.g., [18–22]). We have to emphasize that de pending on the spacetime symmetry of the model\na part of the listed arguments can disappear. For instance, for th e static spherically symmetric model we obtain that\nΘ=0,σ2=0,ω2=0, and we construct the guiding functions using a2only. For the G¨ odel spacetime the only ω2is\nnon-vanishing. For the spacetime with planar gravitational waves w e have to work with two non-vanishing scalars: Θ\nandσ2. Spatially isotropic homogeneous cosmological models are unique in th is sense, since for them only the scalar\nΘ is non-vanishing, and this scalar coincides with the tripled Hubble fun ction Θ=3H(t). In this context the function\nH(t) can be chosen as an appropriate argument of the guiding paramet ers of such cosmological models, unifying the\nparadigm of the aetheric control over the physical systems evolu tion, on the one hand, and the physical interpretation\nof the theory predictions, on the other hand. Since the function Hhas the dimensionality of inverse time (we consider\nthe units with c=1), this quantity is often used to determine a specific time scale in a c orresponding cosmological\nepoch.\nIn this paper we work within the Einstein-aether-axionmodel on the Friedmann-Lemaˆ ıtre-Robertson-Walkerspace-\ntime platform, and consider the interaction of the gravitational fie ld, pseudoscalar (axion) field φ, and unit timelike\nvector field Uj. Two guiding functions depending on the scalar Θ are introduced into the Lagrangian. The guiding\nfunction of the first type, A(Θ) enters the so-called aetheric effective metric Gmn=gmn+AUmUn(see [23] for history,\nmathematical details and motives); it modifies the kinetic term assoc iated with the axion field, and thus it controlsthe\nevolution of the kinetic energy of the axionic dark matter in the Unive rse (see, e.g., [24]-[29], which present the history\nof axions, and [30] - [34], where the problem of axions in cosmology are discussed in various aspects). The guiding\nfunction of the second type, Φ ∗(Θ) enters the potential of the axion field, V(φ,Φ∗), thus performing control over the\nevolution of the potential energy of the axionic dark matter. The s et of master equations of the model is solved in\nquadratures and partially in the analytic form; the corresponding f unctions A(Θ) and Φ ∗(Θ) are reconstructed.\nThe paper is organized as follows. Section II contains the descriptio n of the mathematical formalism. In Section III\nwe analyze the key equations of the model for the spatially isotropic homogeneous cosmological model and discuss\nthe obtained solutions. Section IV contains discussion and conclusio ns.\nII. THE FORMALISM OF THE EXTENDED EINSTEIN-AETHER-AXION TH EORY\nA. The extended action functional and auxiliary quantities\nThe extended Einstein-aether-axion theory is formulated on the b ase of the following action functional:\n−S(total)=/integraldisplay\nd4x√−g/braceleftbigg1\n2κ/bracketleftbig\nR+2Λ+λ(gmnUmUn−1)+Kab\nmn∇aUm∇bUn/bracketrightbig\n+\n+1\n2Ψ2\n0[V(φ,Φ∗)−Gmn∇mφ∇nφ]/bracerightbigg\n. (1)\nIn this formula the standard elements of this theory appear, such as the determinant of the spacetime metric g, the\nRicci scalar R, the cosmological constant Λ, the Einstein constant κ, the Lagrange multiplier λ, the unit timelike\nvector field Ui, associated with the velocity four-vector of the aether flow, and the covariant derivative ∇kwith the\nconnection consistent with the spacetime metric gmn, i.e.,∇kgmn= 0. Kinetic terms for the vector and axion fields\ncontain the effective aetheric metric\nKab\nmn=C1GabGmn+C2δa\nmδb\nn+C3δa\nnδb\nm+C4UaUbGmn, (2)\nGmn=gmn+AUmUn, (3)3\nwhere the scalar A(θ) is the guiding function of the first type, and C1,C2,C3,C4are the Jacobson coupling constants\n[14]. The potential of the axion field V(φ,Φ∗) is considered to have the periodic form\nV(φ,Φ∗) =m2\nAΦ2\n∗\n2π2/bracketleftbigg\n1−cos/parenleftbigg2πφ\nΦ∗/parenrightbigg/bracketrightbigg\n, (4)\nwhere Φ ∗(Θ) is the guiding function of the second type, and the parameter Ψ 0relates to the coupling constant of the\naxion-photon interaction gAγγ,1\nΨ0=gAγγ. The potential (4) inherits the discrete symmetry2πφ\nΦ∗→2πφ\nΦ∗+2πn. This\nperiodic potential has the minima at φ=nΦ∗. Near the minima, when φ→nΦ∗+ψand|2πψ\nΦ∗|is small, the potential\ntakes the standard form V→m2\nAψ2, wheremAis the axion rest mass. When φ=nΦ∗(nis an integer), we deal with\nthe axionic analog of the equilibrium state [19], since V|φ=nΦ∗=0, and/parenleftBig\n∂V\n∂φ/parenrightBig\n|φ=nΦ∗=0.\nThe following decompositions are associated with the unit four-vect orUj:\n∇k=UkD+⊥\n∇k, D=Us∇s,⊥\n∇k= ∆j\nk∇j,∆j\nk=δj\nk−UjUk. (5)\nHereDis the convective derivative, and ∆j\nkis the projector. The covariant derivative ∇kUjcan be decomposed as\n∇kUj=UkDUj+σkj+ωkj+1\n3∆kjΘ, (6)\nwhere the acceleration four-vector DUj≡aj, the symmetric traceless shear tensor σkj, the skew - symmetric vorticity\ntensorωkjand the expansion scalar Θ are presented by the well-known formula s\nDUj=Us∇sUj, σkj=1\n2/parenleftbigg⊥\n∇kUj+⊥\n∇jUk/parenrightbigg\n−1\n3∆kjΘ, ωkj=1\n2/parenleftbigg⊥\n∇kUj−⊥\n∇jUk/parenrightbigg\n,Θ=∇kUk. (7)\nThe decomposition (6) allows us to introduce one linear and three qua dratic scalars\nΘ =∇kUk, a2=DUkDUk, σ2=σmnσmn, ω2=ωmnωmn, (8)\nand thus the kinetic term of the vector field can be rewritten in the f orm\nKab\nmn(∇aUm)(∇bUn)=[C1(1+A)+C4]a2+(C1+C3)σ2+(C1−C3)ω2+1\n3(C1+3C2+C3)Θ2. (9)\nTaking into account the constraints obtained after the detection of the event GRB170817 [35], we have to put\nC1+C3= 0 into (9).\nB. Master equations of the model\n1. Master equations for the unit vector field\nVariations of the extended action functional (1) with respect to t he Lagrange multiplier λgives the normalization\ncondition\ngmnUmUn= 1. (10)\nVariation with respect to the four-vector Uigives the aetheric balance equations\n∇aJaj=λUj−AκΨ2\n0Dφ∇jφ−∇j/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg\n, (11)\nwhere the following definitions are used:\nJaj=Kabjn∇bUn=C1/parenleftbig\n∇aUj−∇jUa/parenrightbig\n+C2gajΘ+(C4+C1A)UaDUj, (12)\nΩ1=κΨ2\n0m2\nA\n2π2/braceleftbigg\nΦ∗/bracketleftbigg\n1−cos/parenleftbigg2πφ\nΦ∗/parenrightbigg/bracketrightbigg\n−πφsin/parenleftbigg2πφ\nΦ∗/parenrightbigg/bracerightbigg\n, (13)\nΩ2=−1\n2κΨ2\n0(Dφ)2. (14)\nConvolution of (11) with Ujgives us the Lagrange multiplier λ:\nλ=Uj∇aJaj+AκΨ2\n0(Dφ)2+D/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg\n. (15)4\n2. Master equation for the axion field\nVariation of the extended action functional (1) with respect to th e axion field yields\n∇m[(gmn+AUmUn)∇nφ]+m2\nAΦ∗\n2πsin/parenleftbigg2πφ\nΦ∗/parenrightbigg\n= 0, (16)\nor equivalently\n(1+A)D2φ+[(1+A)Θ+DA]Dφ−DUm⊥\n∇mφ+⊥\n∇m⊥\n∇mφ+m2\nAΦ∗\n2πsin/parenleftbigg2πφ\nΦ∗/parenrightbigg\n=0. (17)\nBelow we use the ansatz that when the axion field is in the equilibrium sta te, which correspondsto the basic minimum\nφ=Φ∗, we obtain the master equation for the guiding function of the seco nd type Φ ∗(Θ), i.e.,\n∇m[(gmn+AUmUn)∇nΦ∗] = 0. (18)\n3. Master equations for the gravitational field\nVariation of the extended action functional (1) with respect to th e metric gives the gravity field equations:\nRik−1\n2Rgik−Λgik=T(U)\nik+κT(A)\nik+T(INT)\nik. (19)\nThe extended stress-energy tensor of the aether T(U)\nikcontains the following elements:\nT(U)\nik=1\n2gikKab\nmn∇aUm∇bUn+∇m/bracketleftbig\nU(iJk)m−Jm(iUk)−J(ik)Um/bracketrightbig\n+UiUkUj∇aJaj+ (20)\n+C1[(∇mUi)(∇mUk)−(∇iUm)(∇kUm)]+(C4+C1A)(DUiDUk−UiUkDUmDUm).\nAs usual, the parentheses symbolize the symmetrization of indices. The extended stress-energy tensor of the axion\nfield is of the form:\nT(A)\nik= Ψ2\n0/bracketleftbigg\n(1+A)˙φ2/parenleftbigg\nUiUk−1\n2gik/parenrightbigg\n+1\n2gikV/bracketrightbigg\n. (21)\nThe part of the total stress-energy tensor associated with the interaction terms contains the derivatives of the guiding\nfunctions Aand Φ ∗with respect to their argument Θ:\nT(INT)\nik=−gikΘ/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg\n−∆ik/bracketleftbigg\nD/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg/bracketrightbigg\n. (22)\nThe Bianchi identity\n∇k/bracketleftBig\nT(U)\nik+κT(A)\nik+T(INT)\nik/bracketrightBig\n= 0 (23)\nholds automatically on the solutions to the master equations for the vector and pseudoscalar fields.\nIII. APPLICATION TO THE SPATIALLY ISOTROPIC HOMOGENEOUS CO SMOLOGICAL MODEL\nA. The spacetime platform, reduced master equations and the ir solutions\n1. Geometric Aspects\nWe work below with the spacetime of the Friedmann-Lemaˆ ıtre-Robin son-Walker type, with the metric\nds2=dt2−a2(t)/parenleftbig\ndx2+dy2+dz2/parenrightbig\n. (24)5\nThe velocity four-vector of the aether flow is known to be of the fo rmUj=δj\n0, and the corresponding covariant\nderivative of the vector field has the following decomposition\n∇kUi=1\n2˙gik=˙a\na∆ik=H∆ik=1\n3Θ∆ik. (25)\nClearly, in this case DUj= 0,σmn= 0,ωmn= 0, Θ = 3H= 3˙a\na, and standardly the dot denotes the derivative with\nrespect to the cosmological time t.\n2. Solution to the equations of the vector field\nKeeping in mind that DUj=0,σmn=0,ωmn=0 we find that the extended Jacobson’s tensor (12) converts int o\nJaj=C2Θgaj, (26)\nand the equations for the unit vector field (11) takes the form\nC2∇jΘ =λUj−κΨ2\n0AUj˙φ2−∇j/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg\n. (27)\nFour equations (27) contain only one non-trivial equation, which giv es the solution for the Lagrange multiplier λ:\nλ=C2˙Θ+κΨ2\n0A˙φ2+d\ndt/parenleftbigg\nΩ1dΦ∗\ndΘ+Ω2dA\ndΘ/parenrightbigg\n. (28)\nThus, the aetheric subset of the total system of master equatio ns is solved.\n3. First integral of the reduced equation for the axion field\nWe suppose that the axion field φis frozen in the first minimum of the axion potential, i.e., φ=Φ∗(t). Then we put\nφ=Φ∗into (17) and obtain the key equation for Φ ∗(t)\n(1+A)¨Φ∗+/bracketleftbigg\n3(1+A)˙a\na+˙A/bracketrightbigg\n˙Φ∗= 0, (29)\nwhich admits the first integral\n˙Φ∗(t) =const\na3(t)[1+A(t)]=˙Φ∗(t0)/bracketleftbigga(t0)\na(t)/bracketrightbigg3[1+A(t0)]\n[1+A(t)]. (30)\nThe parameter t0describes the initial time moment; A(t0) is the initial value of the guiding function of the first type,\nand˙Φ∗(t0) indicates the initial value of the first derivative of the guiding funct ion of the second type.\n4. Key equations for the gravity field\nWhenφ=Φ∗, the function Ω 1takeszerovalue, andthe reduced extendedequationsofthe gra vitationalfield converts\ninto one key equation\n1\n3Θ2/parenleftbigg\n1+3\n2C2/parenrightbigg\n−Λ =1\n2κΨ2\n0˙Φ2\n∗/bracketleftbigg\n1+A+ΘdA\ndΘ/bracketrightbigg\n. (31)\nSince˙Φ∗is already found and is of the form (30), we obtain the equation, whic h connects the scalar Θ with the\nreduced scale factor x=a(t)\na(t0)as follows:\n1\n3Θ2/parenleftbigg\n1+3\n2C2/parenrightbigg\n−Λ =1\n2x6κΨ2\n0˙Φ2\n∗(t0)[1+A(t0)]2/bracketleftbigg1\n1+A−Θd\ndΘ/parenleftbigg1\n1+A/parenrightbigg/bracketrightbigg\n. (32)6\nThen we assume that C2>−2\n3, Λ>0, and introduce the auxiliary parameters\nH∞=/radicalBigg\nΛ\n3(/parenleftbig\n1+3\n2C2/parenrightbig\n), h2=κΨ2\n0˙Φ2(t0)[1+A(t0)]2\n6/parenleftbig\n1+3\n2C2/parenrightbig. (33)\nNow we are ready to analyze the main equation of the model for the f unctionH(x)\nx6/bracketleftbig\nH2−H2\n∞/bracketrightbig\n=h2/bracketleftbigg1\n1+A−Hd\ndH/parenleftbigg1\n1+A/parenrightbigg/bracketrightbigg\n. (34)\nB. Modeling of the guiding function of the first type\nWhen we discuss the structure of the guiding function of the first t ype we use two assumptions. First, we assume\nthatA=0, if Θ = 0. Second, we assume that the right-hand side of the equa tion (34) is a regular function of its\nargumentH, and thus we can use the decomposition\n/bracketleftbigg1\n1+A−Hd\ndH/parenleftbigg1\n1+A/parenrightbigg/bracketrightbigg\n= 1−γ1H−γ2H2−2γ3H3−3γ4H4−... (35)\nThis decomposition allows us to reconstruct the function1\n1+A, which has the form\n1\n1+A= 1+γ1H/bracketleftbigg\n1+logH\nH∗/bracketrightbigg\n+γ2H2+γ3H3+γ4H4+... (36)\nHereH∗is some constant ofintegration. The key to our considerationis the analysis ofthe asymptotic regime( x→ ∞\n) of the equation\nx6/bracketleftbig\nH2−H2\n∞/bracketrightbig\n=h2/bracketleftbig\n1−γ1H−γ2H2−2γ3H3−3γ4H4−.../bracketrightbig\n. (37)\nIf we restrict our-selves by the term Hmin the right-hand side of (37), we see that, first, Hm−2∝x6, second,\nH∝x6\nm−2and third,a(t)∝t−m−2\n6. In other words, if m>2, the Universe collapses asymptotically, and this detail\nis in contradiction with the main idea of perpetual expansion. Of cour se, this point is disputable, but we follow this\nidea. Now we deal with the quadratic equation with respect to H\nx6/bracketleftbig\nH2−H2\n∞/bracketrightbig\n=h2/bracketleftbig\n1−γ1H−γ2H2/bracketrightbig\n, (38)\nand its positive solution is\nH(x) =/radicalBigg\nγ2\n1h4\n4(x6+γ2h2)2+H2∞x6+h2\nx6+γ2h2−γ1h2\n2(x6+γ2h2). (39)\nWith this function H(x) one can reconstruct the scale factor as the function of time, if w e use the formal quadrature\nt−t0=/integraldisplaya(t)\na(t0)\n1dx\nxH(x). (40)\nClearly, there are two asymptotic regimes.\n1) When Λ ∝negationslash= 0,H→H∞and thusa(t)∝eH∞t.\n2) When Λ=0, H∝1\nx3and thusa(t)∝t1\n3.\nIn order to have further progress in calculations, we consider thr ee analytically solvable submodels.\n1. First analytically solvable submodel\nLet us consider the model with γ1=−1\nH∞andγ2= 0. In this case the function A(H) satisfies the relationship\n1\n1+A= 1−H\nH∞/bracketleftbigg\n1+logH\nH∗/bracketrightbigg\n. (41)7\nIn order to simplify the analysis, we assume that H∗=H∞and obtain the following expression for the guiding\nfunction of the first type\nA=H\nH∞/parenleftBig\n1+logH\nH∞/parenrightBig\n1−H\nH∞/parenleftBig\n1+logH\nH∞/parenrightBig. (42)\nFormally speaking, this function takes the infinite value, when the de nominator is equal to zero. But this situation\nappears only at infinity a=∞, whenH=H∞. Now we deal with the key equation\nH2−H2\n∞=h2\nH∞x6(H∞+H), (43)\nwe omit the negative root H=−H∞, and see that the positive solution is\nH(x) =H∞+h2\nH∞x6. (44)\nMention should be made, that this model is self-consistent, when, fi rst,H(t0)> H∞, second,h2=H∞[H(t0)−H∞].\nAccording to the definition (33) the last requirement links the values A(t0),˙Φ(t0) andH(t0).\nThe scale factor a(t) and the Hubble function H(t) can be now presented in the form\na(t) =a(t0)/bracketleftbigg/parenleftbigg\n1+h2\nH2∞/parenrightbigg\ne6H∞(t−t0)−h2\nH2∞/bracketrightbigg1\n6\n, (45)\nH(t) =H∞/braceleftBig\n1−/bracketleftBig\n1−H∞\nH(t0)/bracketrightBig\ne−6H∞(t−t0)/bracerightBig. (46)\nThe acceleration parameter −q(t) given by the formula\n−q(t) =¨a\naH2= 1−/parenleftbigg6h2\nh2+H2∞/parenrightbigg\ne−6H∞(t−t0)(47)\nis the monotonic function of time, and it tends to one asymptotically a tt→ ∞.\nFinally, we intend to reconstruct the guiding function of the second type Φ ∗(H). The simplest way is the following.\nFirst, using the replacements t→x=a(t)\na(t0)andd\ndt→xH(x)d\ndx, we rewrite the relationship (30) as follows\nΦ′\n∗(x) =−˙Φ∗(t0)[1+A(t0)]\nH∞x4/bracketleftbigg\n−H∞\nH+1+log/parenleftbiggH\nH∞/parenrightbigg/bracketrightbigg\n. (48)\nSecond, using (44), we integrate (48) and obtain\nΦ∗(x) = Φ∗(t0)+˙Φ∗(t0)[1+A(t0)]\n3H∞ℜ1(x), (49)\nℜ1(x)≡/parenleftbigg\n1−1\nx3/parenrightbigg\n+1\nx3log/parenleftbigg\n1+h2\nH2∞x6/parenrightbigg\n−log/parenleftbigg\n1+h2\nH2∞/parenrightbigg\n+ (50)\n+H∞\n|h|/parenleftbigg\narctan|h|\nH∞x3−arctan|h|\nH∞/parenrightbigg\n.\nThird, using the replacement1\nx6=H∞\nh2(H−H∞), we recover the function Φ ∗(H) based on the solution (49). Asymp-\ntotic value of the reconstructed guiding function is\nΦ∗(∞) = Φ∗(t0)+˙Φ∗(t0)[1+A(t0)]\n3H∞ℜ1(∞), (51)\nℜ1(∞) = 1−log/parenleftbigg\n1+h2\nH2∞/parenrightbigg\n−H∞\n|h|arctan|h|\nH∞.\nFigure 1 illustrates the details of the function ℜ1(x).8\nFIG. 1: Illustration of the behavior of the function ℜ1(x) (50), which enters the guiding function of the second type Φ ∗, for\nthree values of the parameter ρ=|h|\nH∞. All the curves start with the value ℜ(1) = 0 and tend monotonically to their asymptotic\nvaluesℜ1(∞) (51).\n2. Second analytically solvable submodel\nThe second submodel relates to the case, when Λ ∝negationslash= 0,γ1= 0 andγ2=α2\nH2∞>0. With these assumptions the\nguiding function of the first type\nA(H) =−γ2H2\n1+γ2H2=−α2H2\nH2∞+α2H2(52)\nis the regular function of the Hubble function H. From the key equation for the gravity field (38) we obtain\nH(x) =H∞/radicaltp/radicalvertex/radicalvertex/radicalbtx6+h2\nH2∞\nx6+α2h2\nH2\n∞. (53)\nThe parameter α2is connected with the initial value of the Hubble function as follows:\nH(t0)≡H(x= 1) =H∞/radicaltp/radicalvertex/radicalvertex/radicalbt1+h2\nH2∞\n1+α2h2\nH2∞. (54)\nClearly, we have to distinguish the cases α2= 1 andα2∝negationslash= 1.\n1) Whenα2= 1, we obtain that the Hubble function converts into the constant H(x) =H(1) =H∞, and we deal\nwith the de Sitter type behavior of the Universe, for which a(t) =a(t0)eH∞(t−t0). The guiding function of the first\ntype also is constant A=−1\n2, and the guiding function of the second type behaves as\nΦ∗(t) = Φ∗(t0)−˙Φ2\n∗(t0)a3(t0)\n3H∞e−3H∞(t−t0). (55)\n2) Whenα2∝negationslash= 1, direct integration of (40) yields\ne6H∞(t−t∗)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(z−α)α(z+1)\n(z+α)α(z−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (56)\nwhere we used the positive root α= +√\nα2. The auxiliary function z(t) and two new parameters, z∗andt∗are:\nz=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalbtH2∞/bracketleftBig\na(t)\na(t0)/bracketrightBig6\n+α2h2\nH2∞/bracketleftBig\na(t)\na(t0)/bracketrightBig6\n+h2, z∗=/radicalBigg\nH2∞+α2h2\nH2∞+h2, (57)9\nt∗=t0−1\n6H∞log/bracketleftbigg(z∗+1)(z∗−α)α\n(z∗−1)(z∗+α)α/bracketrightbigg\n. (58)\nAccording to (57) z→1, whena→ ∞; the corresponding asymptotic behavior is characterized by the d e Sitter type\nlaw\na(t,α)→a(t0)/parenleftbiggh\n2H∞/parenrightbigg1\n3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+α\n1−α/vextendsingle/vextendsingle/vextendsingle/vextendsingleα−1\n6\neH∞(t−t∗). (59)\nThe formulas (56), (57) and (58) give us the implicit representation . The function a(t) has no extrema; we have\nillustrated the behavior of the scale factor in the early epoch on Figu re 2.\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5051015\nFIG. 2: Illustration of the behavior of the reduced scale fac tora(t)\na(t0)in the early epoch; this function is presented in the implici t\nform by (56). Here τ=t−t0.\nThe guiding function of the second type can be represented in term s of elliptic functions. For instance, if 0 <α<1\nthe term\nΦ∗(x) = Φ∗(t0)−˙Φ∗(t0)[1+A(t0)]\n3H∞ℜ2(x) (60)\ncontains the special function ℜ2(x), which is equal to\nℜ2(x) =/integraldisplay1\nx3\n1dz\n/radicaltp/radicalvertex/radicalvertex/radicalbt1+α2h2\nH2\n∞z2\n1+h2\nH2∞z2+α2/radicaltp/radicalvertex/radicalvertex/radicalbt1+h2\nH2\n∞z2\n1+α2h2\nH2∞z2\n= (61)\n=H∞\nh/braceleftbig\n(1+α2)[F(ϕ,k)−F(ϕ∗,k)]−2[E(ϕ,k)−E(ϕ∗,k)]/bracerightbig\n+\n+2\nx3/radicalBigg\nH2∞x6+α2h2\nH2∞x6+h2−2/radicalBigg\nH2∞+α2h2\nH2∞+h2,\nwhere the elliptic functions of the first and second types, respect ively,\nF(ϕ,k)≡/integraldisplayϕ\n0dψ/radicalbig\n1−k2sin2ψ, E(ϕ,k)≡/integraldisplayϕ\n0dψ/radicalBig\n1−k2sin2ψ (62)\nare characterized by the arguments\nϕ= arctan/parenleftbiggh\nH∞x3/parenrightbigg\n, ϕ∗= arctan/parenleftbiggh\nH∞/parenrightbigg\n, k=/radicalbig\n1−α2. (63)10\nThe asymptotic value of the guiding function of the second type is\nΦ∗(x)=Φ∗(t0)+˙Φ∗(t0)[1+A(t0)]\n3H∞/braceleftBigg\nH∞\nh/bracketleftbig\n(1+α2)F(ϕ∗,k)−2E(ϕ∗,k)/bracketrightbig\n+2/radicalBigg\nH2∞+α2h2\nH2∞+h2/bracerightBigg\n. (64)\n3. Third analytically solvable submodel\nNow we assume that the cosmological constant is equal to zero, Λ= 0, i.e.,H∞=0. Also we assume that γ1= 0 and\nγ2=ν6\nh2>0. We obtain again that A(H) is regular\nA(H) =−ν6H2\nh2+ν6H2, (65)\nand the Hubble function is of the form\nH(x) =|h|√\nx6+ν6. (66)\nThen we obtain the reduced scale factor x(t) in the implicit form\n3|h|\nν3(t−t∗∗) =/radicalbigg\n1+x6\nν6−log/bracketleftBigg/radicalbigg\n1+ν6\nx6+ν3\nx3/bracketrightBigg\n, (67)\nwhere we introduced for simplicity the formal parameter t∗∗\nt∗∗=t0−1\n3|h|/radicalbig\n1+ν6−ν3\n3|h|log/parenleftBig/radicalbig\n1+ν6−ν3/parenrightBig\n. (68)\nFinally, we obtain the guiding function of the second type as the func tion of the reduced scale factor\nΦ∗(x) = Φ∗(t0)+1\n3|h|˙Φ∗(t0)[1+A(t0)]ℜ3(x). (69)\nℜ3(x)≡log/bracketleftBigg/parenleftbig\nx3+√\nν6+x6/parenrightbig\n/parenleftbig\n1+√\n1+ν6/parenrightbig/bracketrightBigg\n−2/radicalbigg\n1+ν6\nx6+2/radicalbig\n1+ν6.\nIn the asymptotic limit x→ ∞the function Φ ∗(H) has the form\nΦ∗(H) = Φ∗(t0)−1\n3|h|˙Φ∗(t0)[1+A(t0)]log/parenleftbiggν3H\n2|h|/parenrightbigg\n. (70)\nFigure 3 illustrates the behavior of the function ℜ3(x).\n4. Special case\nThe last interesting submodel relates to the case A=−1, for which the aetheric effective metric converts into the\nprojectorGmn→∆mn=gmn−UmUn. For such guiding function of the first type the axion field equation ( 17) admits\nthe solution depending on time, if and only if φ=nΦ∗, and thus V=0. The equation for the gravity field (31) gives\nthe de Sitter type solution H=H∞, and the equation (29) turns into the identity 0=0. In other words , the guiding\nfunction of the second type happens to be arbitrary constant Φ ∗(t)=Φ∗(H∞).11\n1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2051015\nFIG. 3: Illustration of the behavior of the function ℜ3(x) for three values of the parameter ν.\nIV. DISCUSSION AND CONCLUSIONS\nInthe presentedworkwestudied newexactsolutionstothe maste requationsofthe extendedversionofthe Einstein-\naether-axiontheory. The main idea of the theory extension is base d on the introduction of two guiding functions A(Θ)\nand Φ ∗(Θ), which depend on the expansion scalar of the aether flow, Θ= ∇kUk. This choice is dictated by the fact\nthat within the Friedmann-Lemaˆ ıtre-Robinson-Walker model ther e is only one non-vanishing invariant reconstructed\nusing the covariant derivative ∇kUjof the aether velocity four-vector Uj. The bonus of this approach is that in the\nFLRW model Θ = 3 H, and thus the aetheric control overthe axion system evolution ha ppens to be described in terms\nof the Hubble function H(t), which is intrinsic for this model and has clear physical meaning. Why we used namely\ntwo guiding functions? We kept in mind that generally the axion system is characterized by two state functions:\nkinetic and potential energy. Modification of the kinetic term in the L agrangian of the extended theory is performed\nusing the effective aetheric metric Gmn=gmn+AUmUn(see (1)), where the scalar A(Θ) has been indicated as the\nguiding function of the first type. Modification of the axion field pote ntial is made by the introduction of the guiding\nfunction of the second type Φ ∗(Θ), which predetermines the location and depth of the potential m inima (see (4)).\nThe next question is how one can find A(Θ) and Φ ∗(Θ)? We have proposed the following idea. If the axion field is\nfrozen in the first minimum of the potential, i.e., is in the first equilibrium s tateφ=Φ∗, we see that the corresponding\nequation of the axion field (see (18) and (29)) can be indicated as th e master equation for the guiding function of\nthe second type. Fortunately, the equation (29)) admits the firs t integral (30), which can be put into the equations\nfor the gravity field, thus providing the key equation (31) to be self -closed equation for the scalar function Θ( x), or\nequivalently, for the Hubble function H(x). WhenHis found, the guiding function of the second type Φ ∗can be\nreconstructed by the direct integration (see the results (49), ( 60), (61) and (69)).\nRegarding the search for the guiding function of the first type A(Θ), we follow the idea that, first, the right-hand\nside of the key equation of the gravity field (34) has to be a regular f unction, second, the model has to describe the\nperpetual Universe expansion without Big Rip and Big Crunch. From t hese two requirements we restore the function\nA(H) up to three arbitrary parameters γ1,γ2andH∗using the formula\n1\n1+A= 1+γ1H/bracketleftbigg\n1+logH\nH∗/bracketrightbigg\n+γ2H2.\nThe Hubble function H(x) is the solution to the quadratic equation and its positive root has th e form (39) for\narbitrary parameters γ1,γ2andH∗; only the scale factor as the function of cosmological time a(t) can be presented in\nquadratures. In order to obtain results presented in the analytic al and special functions, we considered four particular\nsubmodels, selecting the listed parameters in a specific way. Resear ch objectives achieved.\nThe last point of discussion is connected with an application of the ext ended model for the interpretation of\nobservational data, in particular, for the estimation of the axion m ass. In this context, we would like to attract\nattention to the equation of the axion field evolution (17). When the value of the axion field is close to one of the\npotential minima, i.e., φ→nΦ∗+ψwith/vextendsingle/vextendsingle/vextendsingle2πψ\nΦ∗/vextendsingle/vextendsingle/vextendsingle<<1, we deal with the linear differential equation, in which the12\nquantityM(Θ)=mA√1+Aplays the role of an effective axion mass depending on the scalar of ex pansion of the aether\nflow Θ. Preliminary analysis shows that for some choice of the guiding f unctionA(Θ) this equation admits instable\nsolutions, which are associated with the axionization of the early Univ erse in analogy with the results obtained in\n[20]. The growth of the number of axions in the early Universe leads to the formation of the axionic dark matter\ndetected in our epoch; thus, the parameters of the presented e xtended model could be linked with the mass density\nof the relic axions. Clearly, this part of work should be detailed; it is be yond the scope of this article and is planned\nto form material for the next publication.\nAcknowledgments\nThis research was funded by Russian Science Foundation.\nReferences\n[1] Abbott, B.P.; et al. (LIGO Scientific Collaboration and V irgo Collaboration), Observation of Gravitational Waves f rom a\nBinary Black Hole Merger. Phys. Rev. Lett. 2016,116, 061102.\n[2] Abbott, R.; et. al. GW190521: A Binary Black Hole Merger w ith a Total Mass of 150 M(Sun). Phys. Rev. Lett. 2020,125,\n101102.\n[3] Pechetti, R.; et.al. Detection of a 100,000 M(Sun) black hole in M31’s Most Massive Globular Cluster: A Tidally Strip ped\nNucleus. Ap. J2022,924, 48.\n[4] Diego, J.M.; et. al. JWST’s PEARLS: Mothra, a new kaiju st ar at z=2.091 extremely magnified by MACS0416, and\nimplications for dark matter models. Astronomy and Astrophysics 2023,679, A31.\n[5] Spilker, J.S.; et.al. Spatial variations in aromatic hy drocarbon emission in a dust-rich galaxy. Nature2023,618, 708-711.\n[6] https://www.jameswebbdiscovery.com/\n[7] Calderon, R.; L’Huillier, B.; Polarski, D.; Shafieloo, A .; Starobinsky, A.A. Joint reconstructions of growth and ex pansion\nhistories from stage-IV surveys with minimal assumptions: Dark Energy beyond? Phys. Rev. D 2022,106, 083513.\n[8] Calderon, R.; L’Huillier, B.; Polarski, D.; Shafieloo, A .; Starobinsky, A.A. Joint reconstructions of growth and ex pansion\nhistories from stage-IV surveys with minimal assumptions. II. Modified gravity and massive neutrinos. Phys. Rev. D 2023,\n108, 023504.\n[9] Capozziello, S.; Nojiri, S.; Odintsov, S.D. Unified phan tom cosmology: inflation, dark energy and dark matter under t he\nsame standard. Phys. Lett. B 2006,632, P. 597–604.\n[10] Sotiriou, T.P. Unification of inflation and cosmic accel eration in the Palatini formalism. Phys. Rev. D 2006,73, 063515.\n[11] Nojiri,S.; Odintsov, S.D.; Oikonomou, V.K. Unifyingi nflation with early andlate-time dark energyin F(R)gravity. Physics\nof the Dark Universe 2020,29, 100602.\n[12] Odintsov, S.D,; Oikonomou, V.K. Unification of inflatio n with dark energy in f(R) Gravity and axion dark matter. Phys.\nRev. D2019,99, 104070.\n[13] Oikonomou, V.K. Unifying of inflation with early and lat e dark energy epochs in axion F(R) gravity. Phys. Rev. D 2021,\n103, 044036.\n[14] Jacobson, T.; Mattingly, D. Gravity with a dynamical pr eferred frame. Phys. Rev. D 2001,64, 024028.\n[15] Jacobson, T. Einstein-aether gravity: a status report .PoSQG-Ph 2007,020, 020.\n[16] Jacobson, T.; Mattingly, D. Einstein-aether waves. Phys. Rev. D 2004,70, 024003.\n[17] Heinicke, C.; Baekler, P.; Hehl, F.W. Einstein-aether theory, violation of Lorentz invariance, and metric-affine g ravity.\nPhys. Rev. D 2005,72, 025012.\n[18] Balakin, A.B.; Shakirzyanov, A.F. Axionic extension o f the Einstein-aether theory: How does dynamic aether regul ate the\nstate of axionic dark matter? Physics of the Dark Universe 2019,24, 100283.\n[19] Balakin, A.B.; Shakirzyanov, A.F. Is the axionic Dark M atter an equilibrium System? Universe 2020,6 (11), 192.\n[20] Balakin, A.B.; Ilin, A.S.; Shakirzyanov, A.F. Interac tion of the Cosmic Dark Fluid with Dynamic Aether: Parametri c\nMechanism of Axion Generation in the Early Universe. Symmetry 2023,15, 1824.\n[21] Balakin, A.B.; Efremova, A.O. Interaction of the axion ic dark matter, dynamic aether, spinor and gravity fields as a n\norigin of oscillations of the fermion effective mass. Eur. Phys. J C 2021,81, 674.\n[22] Balakin, A.B.; Efremova, A.O. Dynamic aether as a trigg er for spontaneous spinorization in early Universe. Universe 2023,\n9, 481.\n[23] Balakin, A.B.; Shakirzyanov, A.F. The extended Einste in-Maxwell-aether-axion theory: Effective metric as an ins trument\nof the aetheric control over the axion dynamics. Gravitation and Cosmology 2024,30, No. 1, pp. 57-67.\n[24] Peccei, R.D.; Quinn, H.R. CP conservation in the presen ce of instantons. Phys. Rev. Lett. 1977,38, 1440–1443.13\n[25] Weinberg, S. A new light boson? Phys. Rev. Lett. 1978,40, 223–226.\n[26] Wilczek, F. Problem of strong P and T invariance in the pr esence of instantons. Phys. Rev. Lett. 1978,40, 279–282.\n[27] Ni, Wei-Tou. Equivalence principles and electromagne tism.Phys. Rev. Lett. 1977,38, 301–304.\n[28] Sikivie, P. Experimental tests of the ”invisible” axio n.Phys. Rev. Lett. 1983,51, 1415–1417.\n[29] Wilczek, F. Two applications of axion electrodynamics .Phys. Rev. Lett. 1987,58, 1799–1802.\n[30] Bertone, G.; Hooper, D.; Silk, J. Particle Dark Matter: Evidence, Candidates and Constraints. Physics Reports 2005,405,\n279–390.\n[31] Duffy, L.D.; van Bibber, K. Axions as dark matter particl es.New J. Phys. 2009,11, 105008.\n[32] Khlopov, M. Fundamentals of Cosmic Particle Physics ; CISP-Springer: Cambridge, UK, 2012.\n[33] Del Popolo, A. Nonbaryonic dark matter in cosmology. Int. J. Mod. Phys. D 2014,23, 1430005.\n[34] Marsh, D.J.E. Axion cosmology. Physics Reports 2016,643, 1–79.\n[35] Abbott, B.P. et.al [LIGO Scientific Collaboration, Vir go Collaboration, Fermi Gamma-Ray Burst Monitor, INTEGRAL ].\nGravitational Waves and Gamma-rays from a Binary Neutron St ar Merger: GW170817 and GRB 170817A. Astrophys. J.\nLett.2017,848, L13." }, { "title": "2401.16472v2.Optimal_function_estimation_with_photonic_quantum_sensor_networks.pdf", "content": "arXiv:2401.16472v2 [quant-ph] 20 Mar 2024Optimal function estimation with photonic quantum sensor n etworks\nJacob Bringewatt,1,2,∗Adam Ehrenberg,1,2,∗Tarushii Goel,1,2,3,∗and Alexey V. Gorshkov1,2\n1Joint Center for Quantum Information and Computer Science,\nNIST/University of Maryland College Park, Maryland 20742, USA\n2Joint Quantum Institute, NIST/University of Maryland Coll ege Park, Maryland 20742, USA\n3Department of Physics, Massachusetts Institute of Technol ogy, Cambridge, Massachusetts 02139, USA\n(Dated: March 21, 2024)\nThe problem of optimally measuring an analytic function of u nknown local parameters each lin-\nearly coupled to a qubit sensor is well understood, with appl ications ranging from field interpolation\ntonoise characterization. Here, we resolve anumberof open questions thatarise whenextendingthis\nframework to Mach-Zehnder interferometers and quadrature displacement sensing. In particular,\nwe derive lower bounds on the achievable mean square error in estimating a linear function of either\nlocal phase shifts or quadrature displacements. In the case of local phase shifts, these results prove,\nand somewhat generalize, a conjecture by Proctor et al.[arXiv:1702.04271 (2017)]. For quadrature\ndisplacements, we extend proofs of lower bounds to the case o f arbitrary linear functions. We pro-\nvide optimal protocols achieving these bounds up to small (m ultiplicative) constants and describe\nan algebraic approach to deriving new optimal protocols, po ssibly subject to additional constraints.\nUsing this approach, we prove necessary conditions for the a mount of entanglement needed for any\noptimal protocol for both local phase and displacement sens ing.\nI. INTRODUCTION\nIn quantum metrology, entangled states of quantum\nsensors are used to try to obtain a performance advan-\ntage in estimating an unknown parameter or parameters\n(e.g., field amplitudes) coupled to the sensors. In addi-\ntion to this practical advantage of quantum sensing, the\ntheory of the ultimate performance limits for parameter\nestimationtasksisdeeply relatedtoanumber oftopicsof\ntheoretical interest in quantum information science, such\nas resource theories [ 1], the geometry of quantum state\nspace [2], quantum speed limits [ 3–5], and quantum con-\ntrol theory [ 4].\nInitial experimental and theoretical work on quantum\nsensing focused on optimizing the estimation of a sin-\ngle unknown parameter (see, e.g., Ref. [ 6] for a review).\nMore recently, the problem of distributed quantum sens-\ning has become an area of particular interest [ 7]. Here,\noneconsidersanetworkofquantumsensors,eachcoupled\nto a local unknown parameter. The prototypical task in\nthis setting is to measure some function or functions of\nthese parameters. In this context, the task of optimally\nmeasuring a single linear function q(θ) ofdindependent\nlocal parameters θ= (θ1,···,θd)Tis particularly well\nstudied both theoretically [ 8–21] and experimentally[ 22?\n–24]. In addition to its independent utility (i.e., for mea-\nsuring an average of local fields in some region), linear\nfunction estimation serves as a key subtask of more gen-\neral metrological tasks, such as measuring an analytic\nfunction of the unknown parameters [ 26], measuring an\nanalytic function of dependent parameters [ 27,28], or\nmeasuring multiple functions [ 29,30].\nFor qubit sensors, the asymptotic limits on perfor-\nmance for these function estimation tasks are rigorously\n∗These authors contributed equally.understood, and techniques for generating optimal pro-\ntocols subject to various constraints, such as limited en-\ntanglement between sensors, are known [ 18]. However,\ndespite extensive theoretical and experimental research\non distributed quantum sensing for photonic quantum\nsensors (see, e.g., [ 7,31] for reviews), the asymptotic\nperformance limits for function estimation are not yet\nrigorously established. Here, we close this gap, prov-\ning an ultimate bound on asymptotic performance, as\nmeasured by the mean square error of the estimator, for\nmeasuring a linear function of unknown parameters each\ncoupled to a different photonic mode via either (1) the\nnumber operator ˆ nor (2) a field-quadrature operator,\nchosen without loss of generality to be the momentum\nquadrature ˆ p:=i(ˆa†−ˆa)/2. That is, we are interested\nin determining a function of either unknown local phase\nshifts or unknown quadrature displacements. For case\n(1), our primary focus, we derive this bound subject to\na strict constraint on photon number, proving a long-\nstanding conjecture appearing in Ref. [ 8]. In case (2),\nwe derive our bound subject to a constraint on the av-\nerage photon number, which is more natural in this set-\nting as quadrature displacements are not photon-number\nconserving. Here, our results are consistent with exist-\ning bounds in the literature [ 13], but, for completeness,\nwe include derivations in this setting using an equivalent\nmathematical framework to the number operator case\nand the qubit sensor case [ 18]. This allows for a nat-\nural comparison of the various performance limits and\nresource requirements of function estimation in quan-\ntum sensor networks and opens the door to designing\nnew, information-theoretically optimal protocols in the\nasymptotic limit of sufficient data.\nThe rest of the paper proceeds as follows. In Sec. II,\nwe formally set up the problem of interest and provide\nuseful notation. In Sec. IIIwe prove lower bounds on\nthe mean-squared error of an estimator for arbitrary lin-2\nearfunctions for both number operatorand displacement\noperator generators. We then study protocols that sat-\nurate these bounds in Sec. IV. Finally, we discuss other\nentanglement-restricted optimal protocols in Sec. V.\nII. PROBLEM SETUP\nConsidera sensornetworkof doptical modes eachcou-\npled to an unknown parameter θjforj∈ {1,···,d}via\nˆH(s) =d/summationdisplay\nj=1θjˆgj+ˆHc(s) =:θ·ˆg+ˆHc(s),(1)\nwhere ˆgjis the local coupling Hamiltonian and boldface\ndenotes vectors. Here, we consider the following two\ncases:\nˆgj:= ˆnj= ˆa†\njˆaj, (2a)\nˆgj:= ˆpj=i\n2(ˆa†\nj−ˆaj), (2b)\nwhere ˆa†\nj,ˆajarethe bosoniccreationand annihilationop-\nerators acting on mode j, ˆnjis the number operator act-\ning on mode j, and ˆpjis the momentum- (ˆ p-) quadrature\non modej. The choice of ˆ pquadrature is, of course, ar-\nbitrary. All results apply equally well for coupling to any\nquadrature. The θ-independent, time-dependent Hamil-\ntonianˆHc(s) is a control Hamiltonian, possibly including\ncoupling to an arbitrary number of ancilla modes. Here,\ns∈[0,t], wheretis the total sensing time.\nIn either case, our task is to measure a linear func-\ntionq(θ) =α·θof the local field amplitudes θwhere\nα∈Qdis a vector of rational coefficients. (The restric-\ntion to rational coefficients is due to the discreteness of\nthe resources—the number of photons—available in this\nproblem; in the case we are interested in—large photon\nnumbers—this is only a technical point.) To accomplish\nthis task, we consider probe states with either fixed pho-\nton number Nor fixed averagephoton number N. Given\nsuch probe states, we consider encoding the unknown pa-\nrameters into the state via the unitary evolution gener-\nated by the Hamiltonian in Eq. ( 1).\nWe will consider both an unrestricted control Hamil-\ntonian and a control Hamiltonian fixed to have the form\nˆHc(s) =ˆhc(s)δ(s−j∆t), (3)\nwhereˆhc(s) is a (unitless) Hermitian operator, δ(s) is the\nDirac delta function, ∆ t:=t/Mis the time for a single\napplication of the encoding unitary exp( −iH∆t). The\nindexj∈ {1,···,M}indexes these applications, where\nMis the total number of applications. This construction\nis motivated by the fact that typical physical implemen-\ntations of a number operator coupling, e.g., in a Mach-\nZehnder interferometer, and displacement operator cou-\npling, e.g., via an electro-optical modulator (EOM), of-\nten do not allow for intermediate controls at arbitrarytimes. Therefore, when we fix our control Hamiltonians\nto be described by Eq. ( 3), we have limited any controls\nto be applied between each pass through these optical\nelements; for simplicity, we have assumed that these con-\ntrol operations can be implemented on a timescale much\nshorter than the timescale of phase accumulation. With-\nout loss of generality, we will let ∆ t= 1 for the rest of\nthis paper, implying that (in this setting) t=M. There-\nfore, the parameter encoding procedure for the photon\nnumber coupling is done via the unitary\nU=U(M)VU(M−1)V···U(1)V=M/productdisplay\nm=1(U(m)V),(4)\nwhereV:= exp(−iˆg·θ) andU(m)form∈ {1,···,M}\ndenote the unitaries applied between passes. Here, by\npass, we mean a single application of the unitary V. We\nuse the convention that the product operation left mul-\ntiplies.\nIn both settings, it is worth emphasizing that, while\nour information-theoretic results lower bounding the\nasymptotically achievable mean square error of an esti-\nmate ˜qofqwill apply to any protocol within the frame-\nwork(s) described above, the explicit protocols we will\ndevelop will use finite ancillarymodes and finite controls.\nIII. LOWER BOUNDS\nFollowing the approach of Refs. [ 10,18], we compute\nlowerboundsonthemeansquareerror Mofanestimator\n˜qofqby rewriting the Hamiltonian in Eq. ( 1) as\nH(s) =d/summationdisplay\nj=1(α(j)·θ)(β(j)·ˆg)+ˆHc(s),(5)\nfor some (time-independent) choice of basis vectors\n{α(j)}d\nj=1, whereα(1):=αand{β(j)}d\nj=1is a dual basis\nsuchthat α(i)·β(j)=δij. Thevectors {α(j)}d\nj=1areasso-\nciated with a change of basis θ→qwhereqj:=α(j)·θ\nsuch thatq1=q; that is, α(1)=:αwith correspond-\ning dual vector β(1)=:β. Then we can define a β-\nparameterized generator of translations with respect to\nthe quantity qas\nˆgq,β:= min\nq(2),···,q(d)∂ˆH\n∂q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq(2),···,q(d)=β·ˆg.(6)\nArmed with Eq. ( 6), we can write a bound on M\nin terms of a single-parameter quantum Cram´ er-Rao\nbound [31–33]\nM ≥1\nµF(q|β), (7)\nwhereF(q|β) is the quantum Fisher information with\nrespect toq, given some choice of fixing the extra d−13\ndegrees of freedom in our problem, as specified by the\nvectorβ∈Rdsuch that α·β= 1. Any such single-\nparameter bound is a valid lower bound as fixing extra\ndegrees of freedom can only give us more information\nabout the parameter q(see below for mathematical de-\ntails).µis the number of experimental repetitions. This\nbound holds for an unbiased estimator ˜ q. When deriv-\ning our bounds, we will restrict ourselves to single-shot\nFisher information and set µ= 1 [34]. Quantum Fisher\ninformation is maximized for pure states, so restricting\nourselves to pure states and unitary encoding of the un-\nknown parameters into the state we can write\nF(q|β)≤4t2max\nρ[(∆ˆgq,β)ρ]2, (8)\nwhere ˆgq,βis theβ-parameterized generator of transla-\ntions with respect to the unknown function q. The vari-\nance [∆(ˆgq,β)ρ]2is taken with respect to a pure probe\nstateρ=|ψ∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tψ|.\nUltimately, we seek a choice of new basis that yields\nthe tightest possible bound on the quantum Fisher infor-\nmationF(q). This choice is determined by the solution\nto [35]\nmin\nβmax\nρ[∆(β·ˆg)ρ]2,subject to α·β= 1.(9)\nLet (β∗,ρ∗) be a solution for this optimization problem.\nThen we can rewrite the single-shot version of Eq. ( 7) as\nM ≥1\n4t2[∆(β∗·ˆg)ρ∗]2. (10)\nThis bound can be understood as corresponding to the\noptimal choice of an imaginary single parameter sce-\nnario, where we have fixed d−1 of thedparameters\ncontrolling the evolution of the state, leaving only the\nparameter of interest qfree to vary. While this requires\ngiving ourselves information that we do not have, ad-\nditional information can only reduce M, and, therefore,\nany such choice providesa lowerbound on M(via single-\nparameter bounds) when we do not have such informa-\ntion. While not guaranteed by this method of derivation,\nwe shall see that such bounds are saturable, up to small\nmultiplicative constants.\nConstraintscanbe placedonthe probestate ρdepend-\ning on the physical generators coupled to the parameters\nof interest: as previously discussed, in this work we con-\nsider the constraints of fixed photon number Nfor the\ngenerator ˆnjand fixed average photon number Nfor the\ngenerator ˆpj. The rationale behind these constraints is\nas follows. ˆ pdoes not conserve photon number, hence it\ndoes not make sense to restrict to a fixed photon number\nsector when coupling to quadrature operators, and, thus,\naverage photon number is the natural constraint. For ˆ n,\non the other hand, we must work in the fixed photon sec-\ntor, as using fixed average photon number allows for the\nconstruction of pathological probe states enabling arbi-\ntrarily precise sensing. In particular, consider the state\n|ψa∝an}brack⌉tri}ht=/radicalbigg\na−1\na|0∝an}brack⌉tri}ht+/radicalbigg\n1\na/vextendsingle/vextendsingleaN/angbracketrightbig\n. (11)Itiseasytoseethat |ψa∝an}brack⌉tri}hthasmeanphotonnumber Nand\nvariance (a−1)N2. Hence, even for fixed N, lettingaget\narbitrarily large allows for an arbitrarily large variance,\nand hence arbitrarily precise sensing.\nLeaving the details of the calculation to Appendix A,\nsolving the above optimization problem for ˆ gj= ˆnjre-\nstricted to probe states with exactly Nphotons yields\nM ≥max/braceleftig\n∝bar⌈blα∝bar⌈bl2\n1,P,∝bar⌈blα∝bar⌈bl2\n1,N/bracerightig\nN2t2, (12)\nwhereP:={j|αj≥0}andN:={j|αj<0}. In the\nsecond line, we use the notation\n∝bar⌈blα∝bar⌈bl1,S:=/summationdisplay\ni∈S|αi|, (13)\nwhereS ∈ {P,N}. For the rest of the paper, we assume\nwithout loss of generality that we are in the case that\n∝bar⌈blα∝bar⌈bl1,P≥ ∝bar⌈blα∝bar⌈bl1,Nto simplify our expressions. In the\nspecial case where αpossesses only positive coefficients\n(i.e.,N=∅),\nM ≥∝bar⌈blα∝bar⌈bl2\n1\nN2t2, (14)\nprovinga long-standing conjecture from Ref. [ 8] that this\nis the minimum attainable variance for α∈Qdwith\nα≥0 andNα∈Nd. This is our primary result.\nSimilarly, for the case of local quadrature displace-\nments restricted to probe states with average photon\nnumberN, we obtain the following bound:\nM ≥∝bar⌈blα∝bar⌈bl2\n2\n4Nt2−O/parenleftigg\nd∝bar⌈blα∝bar⌈bl2\n2\nN2t2/parenrightigg\n. (15)\nEquation ( 15) is a minor generalization of the results in\nRefs. [7,13], extended to allow for negative coefficients\nand for arbitrary non-Gaussian probe states. Therefore,\nfor completeness, weinclude a reminder ofthe arguments\nfrom Refs. [ 7,13] along with our more general derivation\nin Appendix B.\nWe can compare the bounds in Eqs. ( 12) and (15) to\nthe corresponding bounds on the mean square error ob-\ntainable by separable protocols—that is, those using sep-\narable probe states such that each parameter θiis mea-\nsured individually using an optimized partition of the\navailable photons, and then these estimates are used to\ncomputeq. In particular, for number operator coupling\nand fixed photon number states, using ηj=|α′\nj|\n/bardblα′/bardbl1Npho-\ntons (α′\nj:=α2/3\nj) in modej, it holds that [ 8]\nMsep≥∝bar⌈blα′∝bar⌈bl2\n2/3\nN2t2, (16)\nwhere∝bar⌈bl·∝bar⌈bl2/3denotes the Schatten p-function\n∝bar⌈blv∝bar⌈blp=/parenleftigg/summationdisplay\nivp\ni/parenrightigg1/p\n(17)4\nwithp= 2/3. Whenp∈[1,∞], this function is a norm,\nbut forp∈(0,1) it is not, as it does not satisfy the\nproperty of absolute homogeneity, but it still provides a\nconvenient notational shorthand.\nPerforming a similar optimization for the case of dis-\nplacement coupling and fixed average photon number,\none obtains\nMsep≥∝bar⌈blα∝bar⌈bl2\n1\n4Nt2+O/parenleftbigg1\nN2t2/parenrightbigg\n, (18)\nwhere the optimum division of photons is given by us-\ningηj=|αj|\n/bardblα/bardbl1Nphotons in mode j. A non-closed-form\nversion of this bound can be found in Ref. [ 12] in the\ncase where Nis finite. One recovers our result in the\nasymptotic in Nlimit.\nConsequently, in both the phase and displacement\nsensing settings, the achievable advantage due to entan-\nglement between modes is fully characterized by the dif-\nference between the vector pnorm of αwithp=2\n3,1 or\np= 1,2, respectively. By generalized H¨ older’s inequality,\n∝bar⌈blα∝bar⌈bl2\n2/3≤d∝bar⌈blα∝bar⌈bl2\n1and∝bar⌈blα∝bar⌈bl2\n1≤d∝bar⌈blα∝bar⌈bl2\n2. Both inequal-\nities are saturated for any “average-like” function with\n|α| ∝(1,1,···1)T. In both cases, we obtain a O(1/d)\nimprovement in precision due to entanglement, consis-\ntent with the so-called Heisenberg scaling in the number\nof sensorsd. This is consistent with results for qubits\nin Ref. [ 10], where the best improvement between the\nseparable and entangled bounds occurs when measuring\nan average-like function. For the case of phase sensing,\ntheoptimalperformance,includingconstants,isobtained\nwhen∝bar⌈blα∝bar⌈bl2\n1,P=∝bar⌈blα∝bar⌈bl2\n1,N=∝bar⌈blα∝bar⌈bl1/2 (which occurs when\nthe vector αis half positive ones and half negative ones).\nIV. PROTOCOLS\nA. Existing protocols\nThe bounds established in the previous section are all\nsaturable,uptosmallmultiplicativeconstants,usingpro-\ntocols that exist in the literature, or slight variations\nthereof. In particular, Refs. [ 8,11] present a protocol for\nestimatingalinearfunctionoflocalphaseshifts withpos-\nitive coefficients (i.e., α≥0) which achieves the bound\nin Eq. (12) up to a small multiplicative constant. This\nprotocol makes use of a so-calledproportionallyweighted\nN00N state over d+1 modes,\n|ψ∝an}brack⌉tri}ht ∝/vextendsingle/vextendsingle/vextendsingle/vextendsingleNα1\n∝bar⌈blα∝bar⌈bl1,···,Nαd\n∝bar⌈blα∝bar⌈bl1,0/angbracketrightbigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,···,0,N/angbracketrightbigg\n,(19)\nwhere we have expressedthe state in an occupation num-\nber basis over d+1 modes and have dropped the normal-\nization for concision. The last mode serves as a reference\nmode. Observe that, for this state to be well defined, it\nis essential that α/∝bar⌈blα∝bar⌈bl1∈Qdand thatNis sufficiently\nlarge that the resulting occupation numbers are integers.Details of how protocols using this probe state work and\nhow they generalize to the case of negative coefficients\nare provided in Appendix D. A description of how to\nachieve the separable bound in Eq. ( 16) is provided in\nAppendix B.\nSimilarly, in the case of measuring a linear function\nof displacements using states with fixed average photon\nnumber, Ref. [ 12] provides a protocol that, up to small\nmultiplicative constants, saturatesthe bound in Eq.( 15),\nand a separable protocol that, again up to small con-\nstants, achieves the bound in Eq. ( 18). Interestingly,\nthese protocols require only Gaussian probe states, indi-\ncating that these states are optimal. In particular, these\nprotocols make use of an initial single-mode squeezed\nstate, followed by a properly constructed beam-splitter\narray to prepare a multimode entangled probe state with\nthe appropriate sensitivity to quadrature displacements\nin each mode. Homodyne measurements on each mode\ncan then be used to extract the function ofinterest. Con-\nsistent with this fact, our separable lower bound matches\nthe Gaussian state-restricted bound obtained in Ref. [ 12]\nand the bound for arbitrarystates derived in Ref. [ 13] for\nthe particular case of measuring an average.\nB. Algebraic conditions for new protocols\nOther protocols are possible and can be derived via a\nsimple set of algebraic conditions. In particular, for a\nprobe state to exist saturating the bound in Eq. ( 10),\nor its specific versions in Eqs. ( 12) and (15), we require\nthe existence ofan optimal choiceofbasis transformation\nθ→qsuch that knowing qjforj >1 yields no informa-\ntion about q=q1. Mathematically, this means that the\nquantum Fisher information matrix [ 36] with respect to\nthe parameters qmust have the following properties:\nF(q)11= 4t2[∆(β∗·ˆg)ρ∗]2, (20a)\nF(q)1i=F(q)i1= 0 (∀i∝n⌉}ationslash= 1),(20b)\nRecallthat( β∗,ρ∗)arethesolutiontotheminimaxprob-\nlem in Eq. ( 9). We can reexpress these conditions in\nterms of the quantum Fisher information matrix with\nrespect to θas\n(β∗)TF(θ)β∗= 4t2[∆(β∗·ˆg)ρ∗]2, (21a)\n(β∗)TF(θ)β(i)= (β(i))TF(θ)β∗= 0 (∀i∝n⌉}ationslash= 1).(21b)\nThen, using α(i)·β(j)=δij, we obtain the condition\nF(θ)β∗= 4t2[∆(β∗·ˆg)ρ∗]2α. (22)\nMatrix elements of F(θ) for pure probe states and uni-\ntary evolution are given via\nF(θ)ij= 4/bracketleftbigg1\n2∝an}brack⌉tl⌉{t{Hi,Hj}∝an}brack⌉tri}ht−∝an}brack⌉tl⌉{tH i∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tHj∝an}brack⌉tri}ht/bracketrightbigg\n,(23)5\nwhereHi=−iU†∂iUwith∂i:=∂/∂θi,Uis the unitary\ngeneratedbyEq.( 1)andtheexpectationvaluesaretaken\nwith respect to the initial probe state [ 36].\nWe refertoprotocolsthat makeuse ofprobestatesand\ncontrolsso that Eq. ( 22) is satisfied as optimal. However,\nwe caution that the existence of an optimal probe state\ndoes not imply the existence of measurements on this\nstate that allow one to extract an estimate of the param-\neterqsaturating the lower bounds we have derived. This\nissue of the optimal measurements to extract parame-\nters is also discussed extensively in, e.g. Ref., [ 37], with\nsome convenient, nearly optimal, protocols presented in\nRefs. [38–40]. Such methods are the origin of the “small\nmultiplicative constants” that arise in the explicit proto-\ncols above. In fact, lower bounds derived via the quan-\ntum Cram´ er-Rao bound can be obtained only up to a\nconstant ≥π2[41]. See Appendix Gfor a brief explana-\ntion of these ideas.\nFor the particular cases considered in this paper, β∗\nhas been explicitly calculated (see Appendices AandB),\nso Eq. (22) can be expressed in a more meaningful form.\nFor number operator coupling, we obtain the condition\n/summationdisplay\ni∈PF(θ)ij=N2t2\n∝bar⌈blα∝bar⌈bl1,Pαj, (24)\nfor allj. Similarly, for the quadrature coupling, an opti-\nmal protocol requires\nF(θ)α∼4Nt2α, (25)\nwhere∼denotes asymptotically in N. Equations ( 24)\nand (25) provide a generic route to finding new proto-\ncols: consider a set of parameterized families of probe\nstatesTthat one can coherently switch between using\navailable controls ˆHc(t) (here, a “family” of states refers\nto a particular superposition of Fock states with an ar-\nbitrary relative phase). One can then calculate F(θ) via\nEq.(23)andallocatethetimespentinaparticularfamily\nof states such that the associated quantum Fisher infor-\nmation condition is achieved. As a limiting case, one\ncould consider |T |= 1, removing the necessity of co-\nherent control; the protocols considered in the previous\nsection are of this sort (and, in Appendix D, we show\nthat these protocols do, indeed, achieve the saturability\nconditions).\nThe possible choices for families of states Tthat al-\nlow for such a solution are actually quite limited, even\ngiven access to arbitrary control Hamiltonians and an-\ncilla modes. In particular, we prove the following in the\ncase where ˆ gj:= ˆnj:\nLemma 1. Any optimal protocol using Nphotons and\nMpasses through interferometers with a coupling as in\nEq. (1) withˆgj= ˆnjrequires that, for every pass m, the\nprobe state |ψm∝an}brack⌉tri}htbe of the form\n|ψm∝an}brack⌉tri}ht ∝ |N(m)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR+eiϕm|0∝an}brack⌉tri}htP|N′(m)∝an}brack⌉tri}htNR,(26)\nwhereP,N, andRrepresent the modes with αj≥0,\nαj<0, and the (arbitrary number of) reference modes,respectively, N(m)andN′(m)are strings of occupation\nnumbers such that |N(m)|=|N′(m)|=Nfor all passes\nm.ϕmis an arbitrary phase.\nThe proof follows straightforwardly from an explicit\ncalculation of the Fisher information matrix for ˆ gj= ˆnj,\nbut is somewhat algebraically tedious so we relegate it to\nAppendix E.\nLemma1suggests a particular choice of Tfrom which\nwe can pick an optimal protocol for function estimation\nin the ˆgj= ˆnjcase. In particular, define a set of vectors\nW:=/braceleftig\nω∈Zd/vextendsingle/vextendsingle∝bar⌈blω∝bar⌈bl1,P=N,∝bar⌈blω∝bar⌈bl1,N≤N, ωjαj≥0∀j/bracerightig\n.\n(27)\nFurther, consider the restriction ω|P∈Zdwith compo-\nnents\n(ω|P)j=/braceleftigg\nωj, j∈ P\n0,otherwise,(28)\nand the restriction ω|N, defined similarly. Armed with\nthese vectors, we can define a particular choice Tof one-\nparameter families of probe states in an occupation num-\nber basis where each |ψ(ω;ϕ)∝an}brack⌉tri}ht ∈ Tis labeled by a par-\nticular choice of ωsuch that\n|ψ(ω;ϕ)∝an}brack⌉tri}ht ∝ |ω|P∝an}brack⌉tri}ht|0∝an}brack⌉tri}ht+eiϕ|−ω|N∝an}brack⌉tri}ht|N−∝bar⌈blω|N∝bar⌈bl1∝an}brack⌉tri}ht,\n(29)\nwhereϕ∈Ris an arbitraryparameter and the last mode\nis a reference mode. It should be clear that these families\nof states are of the form specified by Lemma 1. Further-\nmore, note that the proportionally weighted N00N state\nin Eq. (19) is also of this form.\nOur protocols proceed as follows: starting in a state\n|ψ(ω;0)∝an}brack⌉tri}ht, after any given pass through the interferom-\neters we use control unitaries to coherently switch be-\ntween families of probe states such that the relative\nphase between the branches is preserved (that is, we\nchangeω, but notϕ). The fact that an optimal protocol\nmust coherently map between such states is proven in\nLemma5in Appendix E. We stay in the family of states\n|ψ(ωn;ϕ)∝an}brack⌉tri}htfor a fraction pnof the passes where pn=rn\nM\nforrn∈ {0,1,···,M}such that/summationtext\nnpn= 1. Here n\nindexes some enumeration of the families of states in T.\nThe value of the component ωjin a given probe state\ndetermines the contribution of the parameter θjcoupled\ntosensorjtotherelativephasebetweenthetwobranches\nof the probe state during a single pass. In particular, in\na single pass with a probe state in the family |ψ(ω;ϕ)∝an}brack⌉tri}ht,\nthe relative phase between the two branches of the probe\nstate becomes ω·θ+ϕ. Assuming an initial probe state\nwithϕ= 0 and summing overall passeswe obtain a total\nrelative phase\nϕtot=M/summationdisplay\nnpn(ωn·θ) (30)\n=: (Wr)·θ. (31)6\nIn the second line, we implicitly defined Was a ma-\ntrix whose columns are the vectors ωn∈ Wandr:=\nMp∈Z|T |. Explicitly computing the Fisher information\nmatrix for these states demonstrates that the optimality\ncondition in Eq. ( 24) is satisfied if\nWr=NMα\n∝bar⌈blα∝bar⌈bl1,P; (32)\nsee Appendix Dfor details. Consequently, any integer\nsolution rto Eq. (32) such that\n∝bar⌈blr∝bar⌈bl1=M,\nr≥0, (33)\nyields an optimal protocol. The protocols of Ref. [ 8],\ndescribed above and generalized in Appendix D, are a\nparticularly simple case within this class with M= 1\nandω=Nα\n/bardblα/bardbl1,P, i.e. we select out only a single column\nofW.\nSolutions to Eqs. ( 32) and(33) are not guaranteed to\nexist for all N,M. In particular, we require that\nNMα\n∝bar⌈blα∝bar⌈bl1,P∈Zd. (34)\nForα∈Qand sufficiently large NorMthis hold true.\nSetting up the system of equations in Eqs. ( 32)-(33) that\nmust be solved to pick out explicit protocols requires\nidentifying the set of vectors Wdefined in Eq. ( 27).\nWhile computationally straightforward, if expensive, to\nconstruct and enumerate this set, the number of states\nis extremely large, yielding a correspondingly large set of\nlinear Diophantine equations in Eq. ( 32). Consequently,\nit is reasonable to place further, experimentally moti-\nvated constraints to limit this set of states and pick\nout advantageous protocols. For instance, one such con-\nstraint is to limit the amount of entanglement between\nmodes on any given pass. We consider this case in the\nfollowing section.\nIt is also important to note that integer linear pro-\ngramming is NP-hard [ 42], so finding a particular solu-\ntion once we add additional constraints is not a compu-\ntationally easy task. Regardless, in applications one can\napply standard (possibly heuristic) algorithms for inte-\nger linear programming to seek solutions. If a solution is\nfound, it is known to be optimal. Consequently, proving\nthe existenceor lackthereofofasolution with certainad-\nditional constraints may be intractable for large problem\ninstances.\nSimilar arguments to those that go into proving\nLemma1allow us to show that, for quadrature sensing,\nthe condition in Eq. ( 25) can be reduced to the condition\nthat\nF(θ)ij∼4Nt2\n∝bar⌈blα∝bar⌈bl2\n2αiαj, (35)\nwhich is proven in Appendix F. However, there is not a\nclearly interesting family of states that can be leveragedto achieve this quantum Fisher information, as in the\ncase of number operator coupling or qubit sensors [ 18].\nHowever, the existing optimal protocols described above\ndo obey this condition asymptotically in average photon\nnumberN.\nV. ENTANGLEMENT REQUIREMENTS\nTheremainingflexibility inthe choiceofoptimal probe\nstates enabled by some control also allows us to impose\nfurther experimentally relevant constraints. One reason-\nable constraint is the amount of intermode entanglement\nrequired during the sensing process. This was considered\nin Ref. [18] for the case of qubit sensors.\nThe answer to the entanglement question in the cur-\nrent context depends crucially on the sorts of control op-\nerations we allow. In the number operator case, with\narbitrary time-dependent control, only two-mode entan-\nglement is needed at any given time, as one can sim-\nply prepare a N00N state between the reference and\none of the sensing modes and coherently switch which\nsensing mode is entangled with the reference mode such\nthat the time spent entangled with mode jis given by\ntj=|αj|t/∝bar⌈blα∝bar⌈bl1. For similar reasons, no entanglement\nis needed for displacement sensing; here, no reference\nmode is needed and one can simply sequentially apply\ndisplacement operators for a time tj=|αj|t/∝bar⌈blα∝bar⌈bl1on\na single-mode squeezed state, followed by a homodyne\nmeasurement. When control operations to change the\nprobe state are allowed only at Mdiscrete time inter-\nvals, as described by Eq. ( 3), the problem becomes more\ninteresting. For number operator coupling, subject to a\nfixed photon number constraint, any optimal protocol re-\nquiresat least( ⌈∝bar⌈blα∝bar⌈bl0/M⌉+1)-modeentanglement. This\nbound is fairly trivial: it merely states that one must be\nentangledwith eachnontrivialmode foratleast onepass.\nFordisplacementoperatorcoupling, subjecttoafixedav-\nerage photon number constraint, an essentially identical\nargument allows us to prove that any optimal protocol\nrequires at least ⌈∝bar⌈blα∝bar⌈bl0/M⌉-mode entanglement. The\ndifference of one is because, unlike displacement sens-\ning, phase sensing generally requires entanglement with\na reference mode. In the M→ ∞limit, we recover the\ncontinuous control case, so these trivial bounds can be\ntight. This triviality is in contrast to the qubit case,\nwhere results analogous to Lemma 1lead to significantly\ntighter constraints on the minimum amount of necessary\nentanglement for optimal protocols [ 18]. This discrep-\nancy arises due to the fact that, unlike with photonic\nresources which must be distributed in a zero-sum way\nbetween modes, for qubit sensors one can be maximally\nsensitive to all coupled parameters simultaneously.7\nQubit phase sensing Phase sensing Displacement sensing\nParameter coupling1\n2ˆσz\niθi ˆniθii\n2(ˆa†\ni−ˆai)θi\nResources Qubit number, d Photon number, NAvg. photon number, N\nsensing time, t sensing time, t sensing time, t\nMSE (separable) ≥/bardblα/bardbl2\n2\nt2[10] ≥/bardblα/bardbl2\n2/3\nN2t2[8] ≥/bardblα/bardbl2\n1\n4Nt2\nMSE (entangled) ≥/bardblα/bardbl2\n∞\nt2[10] ≥/bardblα/bardbl2\n1,P\nN2t2 ≥/bardblα/bardbl2\n2\n4Nt2[13]\nEntanglement needed\n(discrete controls) k≥max/braceleftBig/ceilingleftBig\n/bardblα/bardbl1\n/bardblα/bardbl∞/ceilingrightBig\n,/ceilingleftBig\n/bardblα/bardbl0\nM/ceilingrightBig/bracerightBig\nk >/ceilingleftBig\n/bardblα/bardbl0\nM/ceilingrightBig\nk≥/ceilingleftBig\n/bardblα/bardbl0\nM/ceilingrightBig\nEntanglement needed\n(arbitrary controls)/bardblα/bardbl1\n/bardblα/bardbl∞∈(k−1,k] [18] k= 2 No entanglement\nk-partite entanglement\nprotocol always exists? Yes [18] No Yes\nTABLE I. Comparison of the lower bounds on the mean square err or and entanglement requirements for an (asymptotically)\noptimal protocol obeying the corresponding conditions on t he quantum Fisher information for the task of estimating a li near\nfunction q=α·θwith qubit, phase sensing, and displacement sensing quantu m sensor networks.\nVI. CONCLUSION AND OUTLOOK\nWe have determined the fundamental achievable per-\nformance limits for phase sensing and have extended\nproofs of lower bounds for displacement sensing beyond\njust an average to arbitrary functions. In the process,\nwe proved a long-standing conjecture regarding func-\ntion estimation with number operator coupling [ 8] and\nshowed that some of the protocols that exist in the lit-\nerature [ 8,11,12], are, in fact, optimal in the asymp-\ntotic limit. By considering different implementations of\na quantum sensor network within a single framework, we\nreveal the role of entanglement and controls as they re-\nlate to the type of coupling and whether the relevant\nresource is “parallel” (as in qubit sensor networks, where\nall parameters can simultaneously be measured to maxi-\nmal precision) or “sequential” (as in photonic sensor net-\nworks, where the photons must be optimally distributed\nbetween modes). Our approach to proving our bounds\nalso enables an algebraic framework for developing fur-\nther optimal protocols, subject to various constraints.Here, we considered the particular case of entanglement-\nbased constraints, enabling comparison to similar work\nin the case of qubit sensors [ 18]. These results, and how\nthey fit into the landscape of known results for quantum\nsensor networks, are summarized in Table I. How other\nconstraints impact the existence of and control require-\nments for optimal protocols remains an interesting open\nquestion deserving of further study.\nACKNOWLEDGMENTS\nWe thank Luis Pedro Garc´ ıa-Pintos for helpful discus-\nsions. This work was supported in part by DARPA SA-\nVaNT ADVENT, AFOSR MURI, AFOSR, NSF STAQ\nprogram, DoE ASCR Accelerated Research in Quantum\nComputing program (award No. DE-SC0020312), NSF\nQLCI (award No. OMA-2120757), and the DoE ASCR\nQuantum Testbed Pathfinder program (awards No. DE-\nSC0019040 and No. DE-SC0024220). Support is also ac-\nknowledged from the U.S. Department of Energy, Office\nof Science, National Quantum Information Science Re-\nsearch Centers, Quantum Systems Accelerator.\nAppendix A: Bound for Local Phase Shifts\nIn this appendix, we derive lower bounds for the mean square error of measuring a linear function q(θ) =α·θof\nlocal phase shifts, generated via coupling to the number operator ˆnj, as specified by the Hamiltonian in Eq. ( 1) and\nEq. (2a).\nIn particular, we seek to solve the optimization problem in Eq. ( 9), restated here for convenience:\nmin\nβmax\nρ[∆(β·ˆg)ρ]2,subject to α·β= 1. (A1)\nHere,ˆg=ˆn= (ˆn1,ˆn2,···,ˆnd)T. For fixed particle number N, the Hilbert space on which possible probe states ρare\ndefined is finite dimensional, and it holds that [ 33]\n[∆(β·ˆn)ρ]2≤∝bar⌈blβ·ˆn∝bar⌈bl2\ns,N\n4, (A2)8\nwhere∝bar⌈blβ·ˆn∝bar⌈bls,Nis the Fock-space-restricted seminorm of β·ˆn(defined as the difference between the maximum\nand minimum eigenvalues of β·ˆnrestricted to the N-photon subspace). As we want to maximize the quantum\nFisher information with respect to the choice of probe state ρ, and because Eq. ( A2) is saturable when ρis an equal\nsuperposition of the eigenstates of β·ˆnwith maximum and minimum eigenvalues, we can consider the following\noptimization problem:\nminimize (w.r.t. β)∝bar⌈blβ·ˆn∝bar⌈bls,N,\nsubject to α·β= 1. (A3)\nTo begin, note that the largest eigenvalue of β·ˆnin theN-particle subspace is given by\nλmax(β·ˆn) =Nmax/braceleftbig\nmax\njβj,0/bracerightbig\n=:Nβmax, (A4)\nwhere we have implicitly defined βmax. This largest eigenvalue corresponds to the eigenstate that cons ists of placing\nall photons in the mode corresponding to the largest positive βj. If allβj≤0, the largest eigenvalue is zero, obtained\nby any state with no particles in the sensor modes. Note that this re quires the use of an extra mode (an ancilla or\nso-called “reference mode”) to “store” these photons, as we fix the total photon number of our state to be N.\nSimilarly, the smallest eigenvalue of β·ˆnin theN-particle subspace is given by\nλmin(β·ˆn) =Nmin/braceleftbig\nmin\njβj,0/bracerightbig\n=:Nβmin, (A5)\nwhere we have implicitly defined βmin.\nUsing the facts above about the maximum and minimum eigenvalues of β·ˆnin theN-particle subspace we can\nrewrite the optimization problem in Eq. ( 9) as\nminimizeN(βmax−βmin),\nsubject to α·β= 1. (A6)\nAs in the main text, define P:={j|αj≥0}andN:={j|αj<0}. We then have the following lemma.\nLemma 2. The solution β∗to Eq. (A6) is such that β∗\nj≥0for allj∈ P, andβ∗\nj≤0for allj∈ N. That is,\nαjβ∗\nj≥0for allj.\nProof.We proceed by contradiction. Let J−={j|αjβ∗\nj<0}andJ+={j|αjβ∗\nj≥0}. Suppose the solution vector\nβ∗to Eq. (A6) hasJ−∝n⌉}ationslash=∅. We can construct an alternative candidate solution vector β′as follows: First, let\nβ′=β∗. Then setβ′\nj= 0 for allj∈ J−. In order to still satisfy the constraint α·β′= 1, we must reduce the values\nof some other components in β′. In particular, it is simple to calculate that a valid solution is, for j∈ J+,\nβ′\nj=β∗\nj/summationtext\nj∈J+αjβ∗\nj=β∗\nj\n1−/summationtext\nj∈J−αjβ���\nj. (A7)\nAgain, when j∈ J−,β′\nj= 0.\nLetβ′\nmax:= max/braceleftbig\nmaxjβ′\nj,0/bracerightbig\nandβ′\nmin:= max/braceleftbig\nminjβ′\nj,0/bracerightbig\n. By construction, β′\nmax≤β∗\nmaxand 0 =β′\nmin≥β∗\nmin.\nConsequently, β′yields a smaller solution candidate than β∗. This contradictsthe fact that β∗is the optimal solution.\nThe lemma statement follows as an immediate consequence.\nLemma2allows us to rewrite the minimization problem in Eq. ( A6) once again as\nminimizeN/bracketleftbigg\nmax\nj∈Pβj−min\nj∈Nβj/bracketrightbigg\n,\nwhereβj≥0∀j∈ P,\nβj≤0∀j∈ N,\nsubject to α·β= 1. (A8)\nIn the above, we define max j∈Pβj(minj∈Nβj) to be zero if P=∅(N=∅). A further simplification is enabled by\nanother lemma.\nLemma 3. The solution vector β∗to Eq. (A8) is such that β∗\nj=β∗\nmaxfor allj∈ Pandβ∗\nj=β∗\nminfor allj∈ N.9\nProof.We proceed by contradiction. Suppose the solution vector β∗is such that β∗\ni∝n⌉}ationslash=β∗\njfor somei,j∈ P. Then we\ncould consider an alternative candidate solution vector β′whereβ′\nk=/summationtext\nl∈Pαlβ∗\nl /summationtext\nl∈Pαlfor allk∈ P. Similarly, if β∗\ni∝n⌉}ationslash=β∗\nj\nfor somei,j∈ Nwe could consider β′\nk=/summationtext\nl∈Nαlβ∗\nl /summationtext\nl∈Nαlfor allk∈ N. Clearly, β′still satisfies the constraint\nα·β′=/summationdisplay\nm∈Pαm/parenleftbigg/summationtext\nl∈Pαlβ∗\nl/summationtext\nl∈Pαl/parenrightbigg\n+/summationdisplay\nm∈Nαm/parenleftbigg/summationtext\nl∈Nαlβ∗\nl/summationtext\nl∈Nαl/parenrightbigg\n=α·β∗= 1. (A9)\nAdditionally, β′also clearly still has β′\nj≥0 whenj∈ Pandβ′\nj≤0 whenj∈ N. But, by construction (because the\nweighted average of a set is less than its maximum element),\nN/bracketleftbigg\nmax\nj∈Pβ′\nj−min\nj∈Nβ′\nj/bracketrightbigg\n0 for allj(that is, no αj= 0) and independent of N. Then we can optimize (at leading order\nin1\nN) the distribution of photons amongst the modes using the Lagrang ian\nL=d/summationdisplay\nj=1|αj|2\n4Nj+γ\nd/summationdisplay\nj=1Nj−N\n, (B11)\nwhereγis a Lagrange multiplier. A bit of algebra yields that\n∂L\n∂Nj= 0 =⇒Nj=|αj|\n2√γ. (B12)11\nThis further implies that\nN=d/summationdisplay\nj=1Nj=∝bar⌈blα∝bar⌈bl1\n2√γ, (B13)\nallowing us to obtain the optimal division of photons as\nNj=|αj|\n∝bar⌈blα∝bar⌈bl1N. (B14)\nWe note that this solution is clearly the desired minimum of the Lagrang ian, as maximizing the objective would lead\nto setting any Njto 0. Plugging this back into Eq. ( B10) we obtain the (asymptotic in N) separable bound\n[∆˜q]2≥∝bar⌈blα∝bar⌈bl2\n1\n4Nt2+O/parenleftbigg1\nN2/parenrightbigg\n. (B15)\nThis bound can be achieved by using the single-mode protocols in Ref. [7] for each mode and then computing the\nfunction of interest classically as a linear combination of the individual estimators.\n2. General Function Estimation Bound\nIn this subsection, we turn to ourprimary task: derivingEq. ( 15) for the mean squareerrorattainable formeasuring\na linear function of local displacements, restricting to probe state s with fixed average photon number N.\nTo derive this bound, we must solve the optimization problem in Eq. ( 9) for ˆgj= ˆpj:\nmin\nβmax\nρ[∆(β·ˆp)ρ]2,subject to α·β= 1. (B16)\nWe can write\n[∆(β·ˆp)]2=d/summationdisplay\ni,j=1βiβjCov(ˆpi,ˆpj)\n≤d/summationdisplay\ni,j=1βiβj/radicalig\n(∆ˆpi)2(∆ˆpj)2\n=\nd/summationdisplay\nj=1βj∆ˆpj\n2\n≤ ∝bar⌈blβ∝bar⌈bl2\n2d/summationdisplay\nj=1(∆ˆpj)2, (B17)\nwhere we applied the Cauchy-Schwarz inequality twice. Using the sam e assumption of zero-displacement states we\nmade in the previous section, we can further bound/summationtext\nj(∆ˆpj)2using the constraint on average photon number\nd/summationdisplay\nj=1/bracketleftbig\n(∆ˆpj)2+(∆ˆxj)2/bracketrightbig\n−d\n2=d/summationdisplay\nj=1∝an}brack⌉tl⌉{ta†\njaj∝an}brack⌉tri}ht=N, (B18)\nimplying that\nd/summationdisplay\nj=1(∆ˆpj)2≤N+d\n2. (B19)\nEquation ( B19) is tight when (∆ˆ xj)2= 0 for all j. This is, of course, impossible to achieve, but can be approached\nasymptotically with increasing N(N≫d). Furthermore, using the fact that αis dual to βand the Cauchy-Schwarz\ninequality, it holds that\n1 =α·β≤ ∝bar⌈blβ∝bar⌈bl2∝bar⌈blα∝bar⌈bl2. (B20)12\nAs we want to minimize with respect to β, we consider the case where this inequality is saturated (i.e. β∗=α\n/bardblα/bardbl2\n2).\nTherefore, ∝bar⌈blβ∗∝bar⌈bl2=1\n/bardblα/bardbl2, and we obtain\n[∆(β·ˆp)]2≤N\n∝bar⌈blα∝bar⌈bl2\n2+O/parenleftigg\nd\n∝bar⌈blα∝bar⌈bl2\n2/parenrightigg\n. (B21)\nThis yields the final bound\nM ≥∝bar⌈blα∝bar⌈bl2\n2\n4Nt2−O/parenleftigg\nd∝bar⌈blα∝bar⌈bl2\n2\nN2t2/parenrightigg\n. (B22)\nFrom the derivation alone, it is not obvious that this bound can be sat urated, but the existence of protocols that\nachieve it [ 12] indicate that this bound is, indeed, tight asymptotically in N.\nAppendix C: Quantum Fisher Information Matrix Elements\nIn this appendix, we derive the matrix elements of the quantum Fishe r information matrix for generators ˆ njand\nˆpjunder the unitary evolution Eq. ( 4). For number operator coupling ˆ gj= ˆnj,\nHj=−iU†∂jU=−M/summationdisplay\nm=1/parenleftiggm−1/productdisplay\nl=1U(l)V/parenrightigg†\nˆnj/parenleftiggm−1/productdisplay\nl=1U(l)V/parenrightigg\n=:−M/summationdisplay\nm=1ˆnj(m), (C1)\nwhere in the second line we implicitly defined ˆ nj(m). Consequently, we can compute the quantum Fisher information\nmatrix elements via Eq. ( 23) to be\nF(θ)ij= 4/bracketleftiggM/summationdisplay\nl=1M/summationdisplay\nm=11\n2∝an}brack⌉tl⌉{t{ˆni(l),ˆnj(m)}∝an}brack⌉tri}ht−/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆni(m)∝an}brack⌉tri}ht/parenrightigg/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆnj(m)∝an}brack⌉tri}ht/parenrightigg/bracketrightigg\n. (C2)\nWhenˆU(j)=Ifor allj, this reduces to\nF(θ)ij= 4M2[∝an}brack⌉tl⌉{tˆniˆnj∝an}brack⌉tri}ht−∝an}brack⌉tl⌉{tˆni∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tˆnj∝an}brack⌉tri}ht]. (C3)\nFor quadrature operator coupling ˆ gj= ˆpj, essentially identical manipulations yield\nF(θ)ij= 4/bracketleftiggM/summationdisplay\nl=1M/summationdisplay\nm=11\n2∝an}brack⌉tl⌉{t{ˆpi(l),ˆpj(m)}∝an}brack⌉tri}ht−/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆpi(m)∝an}brack⌉tri}ht/parenrightigg/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆpj(m)∝an}brack⌉tri}ht/parenrightigg/bracketrightigg\n, (C4)\nwhere ˆpj(l) is defined as in Eq. ( C1) with ˆnj→ˆpj.\nAppendix D: Protocols for Local Phase Shifts\nIn this appendix, we elaborate on the families of optimal protocols fo r measuring a linear function of phase shifts\nthat we described in Sec. IV.\n1. An Optimal Protocol for Functions with Positive Coefficien ts\nWe begin by reviewing a protocol from Ref. [ 8] for the special case of a linear function with positive coefficients (i.e .,\nα≥0). Our results in Appendix Ashow that, as those authors conjectured, this protocol is optim al. In particular,\nconsider using as the probe state a so-called proportionally weighte d N00N state over d+1 modes:\n|ψ∝an}brack⌉tri}ht ∝/vextendsingle/vextendsingle/vextendsingle/vextendsingleNα1\n∝bar⌈blα∝bar⌈bl1,···,Nαd\n∝bar⌈blα∝bar⌈bl1,0/angbracketrightbigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,···,0,N/angbracketrightbigg\n, (D1)13\nwherewe haveexpressedthe state in anoccupation numberbasis o verd+1modes and havedroppedthe normalization\nfor concision. The last mode serves as a reference mode. Observe that, for this state to be well defined, it is essential\nthatα\n/bardblα/bardbl1∈Qdand thatNis such that the resulting occupation numbers are integers, which m ay require that Nbe\nlarge.\nFollowingimprintingofthe parameters θontothe probe statevia Mpassesthroughthe interferometers, one obtains\n|ψM∝an}brack⌉tri}ht=e−iMˆn·θ|ψ∝an}brack⌉tri}ht ∝/vextendsingle/vextendsingle/vextendsingle/vextendsingleNα1\n∝bar⌈blα∝bar⌈bl1,···,Nαd\n∝bar⌈blα∝bar⌈bl1,0/angbracketrightbigg\n+eiα·θNM\n/bardblα/bardbl1/vextendsingle/vextendsingle/vextendsingle/vextendsingle0,···,0,N/angbracketrightbigg\n. (D2)\nThis process allows us to saturate the bound in Eq. ( 14). In particular, using Eq. ( C2) [which reduces to Eq. ( C3)\nbecause there is no control required], it is straightforward to calc ulate that the quantum Fisher information matrix\nfor the probe state is\nF(θ) =(MN)2\n∝bar⌈blα∝bar⌈bl2\n1ααT, (D3)\nwhich clearly satisfies the condition in Eq. ( 24) (recalling that ||α||1=||α||1,Phere because we have assumed all\ncoefficients are non-negative, and also recalling that ∆ t= 1 such that M=t).\nWhile the conditions on the quantum Fisher information matrix for an o ptimal protocol are met, a full protocol\nrequires a description of the measurements used to extract the q uantity of interest from the relative phase between\nthe branches of |ψM∝an}brack⌉tri}ht. As described in the main text, this can be done via the robust phase estimation protocols\nof Refs. [ 38–40] with a small multiplicative constant overhead relative to the quantu m Cram´ er-Rao bound (we also\nbriefly discuss the idea behind robust phase estimation in Appendix G). The details of implementing the necessary\nparity measurements for N00N-like states are discussed in detail in Appendix A of Ref. [ 40] and Ref. [ 43].\n2. Extending the Optimal Protocol to Negative Coefficients\nWhile not explicitly considered in Ref. [ 8], it is straightforward to extend the above protocol to the situat ion where\nN ∝n⌉}ationslash=∅, which we do here. Without loss of generality, assume the coefficient s are ordered so that α1≥α2≥ ··· ≥αd.\nUsing our standard assumption that ∝bar⌈blα∝bar⌈bl1,P≥ ∝bar⌈blα∝bar⌈bl1,N, we claim that the following probe state is optimal:\n|ψ∝an}brack⌉tri}ht ∝/circlemultiplydisplay\nj∈P/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleNαj\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg\n|0∝an}brack⌉tri}ht⊗|N||0∝an}brack⌉tri}ht+|0∝an}brack⌉tri}ht⊗|P|/circlemultiplydisplay\nj∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN|αj|\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN−N∝bar⌈blα∝bar⌈bl1,N\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg\n, (D4)\nwhere, again, the last mode is a reference mode, and we have dropp ed the normalization of the state. Interestingly,\nobserve that, if ∝bar⌈blα∝bar⌈bl1,P=∝bar⌈blα∝bar⌈bl1,N, the reference mode factors out and is unnecessary. Similar to th eα≥0 case, for\nthis state to be well defined, we require that N|αj|/∝bar⌈blα∝bar⌈bl1,P∈Nfor allj, which is always true for some sufficiently\nlargeNprovided α∈Qd.\nConsider applying the encoding unitary for Mpasses through the interferometers. For ∝bar⌈blα∝bar⌈bl1,P≥ ∝bar⌈blα∝bar⌈bl1,N, this\nyields\n|ψM∝an}brack⌉tri}ht ∝/circlemultiplydisplay\nj∈P/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleNαj\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg\n|0∝an}brack⌉tri}ht⊗|N||0∝an}brack⌉tri}ht+eiα·θNM\n/bardblα/bardbl1,P|0∝an}brack⌉tri}ht⊗|P|/circlemultiplydisplay\nj∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN|αj|\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN−N∝bar⌈blα∝bar⌈bl1,N\n∝bar⌈blα∝bar⌈bl1,P/angbracketrightigg\n.(D5)\nThis probe state is optimal in the sense of satisfying the Fisher infor mation condition in Eq. ( 24). In the main text, we\ndescribed an even more generalfamily ofprotocols. Within this more generalframework, we will provethis optimality.\n3. A Family of Optimal Protocols\nFinally, we describe a family of optimal protocols that satisfy the con ditions on the quantum Fisher information\nmatrix given in Eq. ( 24). In the main text, we defined a family of optimal protocols in terms o f vectors from the set\nW:=/braceleftig\nω∈Zd/vextendsingle/vextendsingle∝bar⌈blω∝bar⌈bl1,P=N,∝bar⌈blω∝bar⌈bl1,N≤N, ωjαj≥0∀j/bracerightig\n. (D6)14\nIn particular, from these vectors, we defined a set Tof one-parameterfamilies of probe states in an occupation number\nbasis where each |ψ(ω;ϕ)∝an}brack⌉tri}ht ∈ Tis labeled by a particular choice of ωsuch that\n|ψ(ω;ϕ)∝an}brack⌉tri}ht ∝ |ω|P∝an}brack⌉tri}ht|0∝an}brack⌉tri}ht+eiϕ|−ω|N∝an}brack⌉tri}ht|N−∝bar⌈blω|N∝bar⌈bl1∝an}brack⌉tri}ht, (D7)\nwhereϕ∈Ris an arbitrary parameter and the last mode is a reference mode. Re call also that ωPandωNare\ndefined in Eq. ( 28) as the restriction of ωtoj∈ PandN, respectively (for jnot in the correct set, the value is set to\n0). Note that such states are of the form of those in Lemma 1. We claimed that, by explicitly computing the Fisher\ninformation matrix for these states, one could demonstrate that the optimality condition in Eq. ( 24) is satisfied for a\nprotocol such that\nWr=NMα\n∝bar⌈blα∝bar⌈bl1,P, (D8)\nwherer∈Z|T |is as defined in the main text and must obey the conditions\n∝bar⌈blr∝bar⌈bl1=M,\nr≥0. (D9)\nRecall that Wis a matrix whose columns are the vectors ωn∈ W.\nHere we explicitly demonstrate this. We can easily evaluate\n∝an}brack⌉tl⌉{tˆnj(m)∝an}brack⌉tri}ht=/angbracketleftig\nψ(ω(m);ϕ)/vextendsingle/vextendsingle/vextendsingleˆnj/vextendsingle/vextendsingle/vextendsingleψ(ω(m);ϕ)/angbracketrightig\n=|ω(m)\nj|\n2(D10)\nand\n∝an}brack⌉tl⌉{tˆni(l)ˆnj(m)∝an}brack⌉tri}ht=/angbracketleftig\nψ(ω(l);ϕ)/vextendsingle/vextendsingle/vextendsingleˆniU(m↔l)ˆnj/vextendsingle/vextendsingle/vextendsingleψ(ω(m);ϕ)/angbracketrightig\n=|ω(l)\niω(m)\nj|\n2/angbracketleftig\nψl(ω(l);ϕ)/vextendsingle/vextendsingle/vextendsingleU(m↔l)/vextendsingle/vextendsingle/vextendsingleψm(ω(m);ϕ)/angbracketrightig\n, (D11)\nwhere ˆnj(m) are defined as in Eq. ( C1), and\nU(m↔l) =/braceleftigg/producttextl−1\nk=mU(k)V,ifl≥m/producttextm−1\nk=l(U(k)V)†,otherwise,(D12)\ni.e., it is the unitary that converts between the m-th andl-th probe states. Additionally, ω(m)refers to the vector\nassociatedtothe m-thprobestate; correspondingly/vextendsingle/vextendsingleψl(ω(l);ϕ)/angbracketrightbig\nisthebranchof/vextendsingle/vextendsingleψ(ω(l);ϕ)/angbracketrightbig\nwith non-zerooccupation\nnumber on mode land/vextendsingle/vextendsingleψm(ω(m);ϕ)/angbracketrightbig\nis the branch of/vextendsingle/vextendsingleψ(ω(m);ϕ)/angbracketrightbig\nwith non-zero occupation number on mode m.\nFor an optimal protocol, U(m↔l) coherently maps the first (second) branch of/vextendsingle/vextendsingleψ(ω(l);ϕ)/angbracketrightbig\nto the first (second)\nbranch of/vextendsingle/vextendsingleψ(ω(m);ϕ)/angbracketrightbig\n; therefore, we have that the matrix element/angbracketleftbig\nψl(ω(l);ϕ)/vextendsingle/vextendsingleU(m↔l)/vextendsingle/vextendsingleψm(ω(m);ϕ)/angbracketrightbig\nis nonzero\nif and only if the branches with non-zero occupation on modes landmare the same. So we have that\n∝an}brack⌉tl⌉{tˆni(l)ˆnj(m)∝an}brack⌉tri}ht=|ω(l)\niω(m)\nj|\n2ξij, (D13)\nwhere\nξij:=/braceleftigg\n1,ifi,j∈ Pori,j∈ N\n0,otherwise.(D14)\nPutting everything together we obtain that\nF(θ)ij= (−1)ξij+1/parenleftiggM/summationdisplay\nm=1|ω(m)\ni|/parenrightigg/parenleftiggM/summationdisplay\nm=1|ω(m)\nj|/parenrightigg\n. (D15)15\nTo prove the protocols work, we need to show that this Fisher infor mation matrix obeys the condition in Eq. ( 24).\nWithout loss of generality, consider the case that ∝bar⌈blα∝bar⌈bl1,P≥ ∝bar⌈blα∝bar⌈bl1,N. We have that\n/summationdisplay\nj∈PF(θ)ij= sgn(αi)/parenleftiggM/summationdisplay\nm=1|ω(m)\ni|/parenrightigg\nMN, (D16)\nwhere we used that ∝bar⌈blω∝bar⌈bl1,P=N. So, to obey the condition in Eq. ( 24), we require that\nM/summationdisplay\nm=1|ω(m)\ni|=MN|αi|\n∝bar⌈blα∝bar⌈bl1,P. (D17)\nOr, in vector form:\nM/summationdisplay\nm=1|ω(m)|=MN|α|\n∝bar⌈blα∝bar⌈bl1,P. (D18)\nProtocols in our family satisfy this condition by construction as, for any valid protocol,\nM/summationdisplay\nm=1|ω(m)|=|W|r, (D19)\nwhere|W|denotestakingtheelement-wiseabsolutevalueoftheelementsof W. Consequently,notingthatsgn( ω(m)\nj) =\nsgn(αj) for allm, we require\nWr=MNα\n∝bar⌈blα∝bar⌈bl1,P, (D20)\nwhich is Eq. ( D8).\nAppendix E: Proof of Lemma 1\nHere we provide a proof of Lemma 1in the main text, restated here for convenience.\nLemma 4. Any optimal protocol using Nphotons and Mpasses through interferometers with a coupling as in Eq. ( 1)\nwithˆgj= ˆnjrequires that, for every pass m, the probe state |ψm∝an}brack⌉tri}htbe of the form\n|ψm∝an}brack⌉tri}ht ∝ |N(m)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR+eiϕm|0∝an}brack⌉tri}htP|N′(m)∝an}brack⌉tri}htNR, (E1)\nwhereP,N, andRrepresent the modes with αj≥0,αj<0, and the (arbitrary number of) reference modes,\nrespectively, N(m)andN′(m)are strings of occupation numbers such that |N(m)|=|N′(m)|=Nfor all passes m.\nϕmis an arbitrary phase.\nProof.The quantum Fisher information matrix elements for any protocol w ith ˆgj= ˆnjare given by\nF(θ)ij= 4/bracketleftiggM/summationdisplay\nl=1M/summationdisplay\nm=11\n2∝an}brack⌉tl⌉{t{ˆni(l),ˆnj(m)}∝an}brack⌉tri}ht−/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆni(m)∝an}brack⌉tri}ht/parenrightigg/parenleftiggM/summationdisplay\nm=1∝an}brack⌉tl⌉{tˆnj(m)∝an}brack⌉tri}ht/parenrightigg/bracketrightigg\n= 4M/summationdisplay\nl=1M/summationdisplay\nm=1Cov(ˆni(l),ˆnj(m)), (E2)\nwhere the expectation values are taken with respect to the initial p robe state, and ˆ nj(m) are the number operators\non thejthmode in the Heisenberg picture prior to the mthpass, as specified in Eq. ( C1). Without loss of generality,\nwe make the assumption that ∝bar⌈blα∝bar⌈bl1,P≥ ∝bar⌈blα∝bar⌈bl1,N. Summing over i,j∈ P, we have that, for an optimal protocol,\n/summationdisplay\ni∈P/summationdisplay\nj∈PF(θ)ij=/summationdisplay\nj∈P(MN)2\n∝bar⌈blα∝bar⌈bl1,Pαj= (MN)2, (E3)16\nwhere we used the condition in Eq. ( 24) for an optimal protocol, and we recall that, for j∈ P, allαj>0. For\nconvenience, define\nˆP(m) :=/summationdisplay\nj∈Pˆnj(m). (E4)\nArmed with this definition, we can upper bound the sum over i,j∈ Pin the explicit expression from Eq. ( E2) as\n/summationdisplay\ni∈P/summationdisplay\nj∈PF(θ)ij= 4M/summationdisplay\nl=1M/summationdisplay\nm=1Cov/parenleftig\nˆP(l),ˆP(m)/parenrightig\n≤4M/summationdisplay\nl=1M/summationdisplay\nm=1/radicalig\nVar(ˆP(l))Var(ˆP(m)) = 4/parenleftiggM/summationdisplay\nl=1/radicalig\nVar(ˆP(l))/parenrightigg2\n≤4\nM/summationdisplay\nl=1/vextenddouble/vextenddouble/vextenddoubleˆP(l)/vextenddouble/vextenddouble/vextenddouble\ns,N\n2\n2\n≤(NM)2, (E5)\nwhere in the first line we use the Cauchy-Schwarz inequality, in the se cond line we use that once restricted to the N-\nparticle subspace Var( A)≤ ∝bar⌈blA∝bar⌈bl2\ns,N/4 (where, again, ∝bar⌈blA∝bar⌈bls,Nis the seminorm restricted to the N-particle subspace)\nfor any Hermitian operator A, and in the final line we use that/vextenddouble/vextenddouble/vextenddoubleˆP(l)/vextenddouble/vextenddouble/vextenddouble\ns,N≤N. Comparing Eq. ( E5) with Eq. ( E3),\nwe find that, for any optimal protocol, all inequalities in Eq. ( E5) must be saturated. Specifically,\nCov/parenleftig\nˆP(l),ˆP(m)/parenrightig2\n= Var(ˆP(l))Var(ˆP(m)), (E6)\nVar(ˆP(l)) =N2\n4. (E7)\nThe second condition, Eq. ( E7), means that, at all times, the state of our system must be of the form\n|N(l)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR+eiϕl|0∝an}brack⌉tri}htP|N′(l)∝an}brack⌉tri}htNR√\n2, (E8)\nwhere we are using the simplifying notation from the statement of th e lemma. In particular, the subscripts P,N,R\nrefertothecollectionofallmodesassociatedwith αj≥0,αj<0, andthereferencemodes, respectively. Therefore,the\nstate|N∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNRmeans that all photons are distributed (in some potentially arbitrar y way) amongst the modes with\nnon-negative αj, and there are no photons in the modes with negative αjor in the reference modes. Contrastingly,\n|0∝an}brack⌉tri}htP|N′(l)∝an}brack⌉tri}htNRrefers to a state where there are Nphotons in the negative and reference modes, and there are\nno photons in the non-negative modes. We have also shifted to the S chr¨ odinger picture where we move the time\ndependence onto the state as opposed to the operators. It is sim ple to verify that this state satisfies Eq. ( E7), and\nit is also simple to verify these are the most general states that ach ieve this. Intuitively, |ψm∝an}brack⌉tri}htis a generalized N00N\nstate between the positive and negative/reference modes.\nIn addition, we have the following useful characterization of optima l protocols:\nLemma 5. Let|ψi∝an}brack⌉tri}htbe a state of the form in Lemma 1. Refer to the first and second parts of its superposition as,\nrespectively, the first and second or positive and non-posit ive branches. Let Umbe the unitary that maps the initial\nstate|ψ1∝an}brack⌉tri}htto the state just before the m-th pass, |ψm∝an}brack⌉tri}ht, given by\nUm=/braceleftigg/producttextm−1\ni=1U(i)V, M+1≥m≥2\nI, m = 1.(E9)\nin agreement with Eq. ( 4). Then, if Umis part of an optimal protocol, it coherently maps the first (s econd) branch of\n|ψ1∝an}brack⌉tri}htto the first (second) branch of |ψm∝an}brack⌉tri}ht.17\nProof.We use the covariance equality in Eq. ( E6). To proceed, we evaluate the expectation value of ˆPin the initial\nstate. Here, we will again use the Schr¨ odinger picture.\n∝an}brack⌉tl⌉{tψ1|ˆP(l)|ψ1∝an}brack⌉tri}ht=∝an}brack⌉tl⌉{tψl|ˆP|ψl∝an}brack⌉tri}ht (E10)\n=1\n2/parenleftbig\n∝an}brack⌉tl⌉{tN(l)|P∝an}brack⌉tl⌉{t0|NR+e−iϕl∝an}brack⌉tl⌉{t0|P∝an}brack⌉tl⌉{tN′(l)|NR/parenrightbigˆP/parenleftbig\n|N(l)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR+eiϕl|0∝an}brack⌉tri}htP|N′(l)∝an}brack⌉tri}htNR/parenrightbig\n(E11)\n=1\n2/parenleftbig\n∝an}brack⌉tl⌉{tN(l)|P∝an}brack⌉tl⌉{t0|NR+e−iϕl∝an}brack⌉tl⌉{t0|P∝an}brack⌉tl⌉{tN′(l)|NR/parenrightbig\nN(|N(l)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR) (E12)\n=N\n2. (E13)\nWe next evaluate the covariance:\nCov/parenleftig\nˆP(l),ˆP(m)/parenrightig\n=∝an}brack⌉tl⌉{tψ1|ˆP(l)ˆP(m)|ψ1∝an}brack⌉tri}ht−∝an}brack⌉tl⌉{tψ1|ˆP(l)|ψ1∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tψ1|ˆP(m)|ψ1∝an}brack⌉tri}ht (E14)\n=∝an}brack⌉tl⌉{tψl|ˆPUlU†\nmˆP|ψm∝an}brack⌉tri}ht−∝an}brack⌉tl⌉{tψl|ˆP|ψl∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tψm|ˆP|ψm∝an}brack⌉tri}ht (E15)\n=N2\n2∝an}brack⌉tl⌉{tN(l)|P∝an}brack⌉tl⌉{t0|NRUlU†\nm|N(m)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR−N2\n4, (E16)\nwhere in the last line we have used the fact that ˆPgives a factor of Nwhen acting on the first branch of states |ψl∝an}brack⌉tri}ht\nand|ψm∝an}brack⌉tri}ht, but it annihilates the second branch that has zero photons in the p ositive modes.\nIn order for Eq. ( E6) to be satisfied, and using Eq. ( E7), we therefore require that, for all pairs of passes l,m,\n∝an}brack⌉tl⌉{tN(l)|P∝an}brack⌉tl⌉{t0|NRUlU†\nm|N(m)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR= 1. (E17)\nChoosingl= 1, this implies that we require that\nU†\nm|N(m)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR=|N(0)∝an}brack⌉tri}htP|0∝an}brack⌉tri}htNR≡ |ψ1∝an}brack⌉tri}htP, (E18)\nwhere we are defining |ψ1∝an}brack⌉tri}htP,|ψ1∝an}brack⌉tri}htNRsuch that |ψ0∝an}brack⌉tri}ht ∝ |ψ1∝an}brack⌉tri}htP+|ψ1∝an}brack⌉tri}htNRin the obvious way. Moving the unitary onto\nthe right hand side of the equation yields\n|ψm∝an}brack⌉tri}htP=Um|ψ1∝an}brack⌉tri}htP, (E19)\nwhich of course implies the corresponding equation for the second b ranch by linearity.\nAppendix F: Fisher Information Matrix Conditions for Quadr ature Displacements\nIn this appendix, we provide conditions on the quantum Fisher inform ation matrix for an optimal protocol in the\ncase of quadrature generators. This result yields a simpler form of the saturability condition of Eq. ( 25), although\nthe set of states that it picks out is less clear than in the number ope rator case. This issue is compounded by the fact\nthat the bound is not actually saturable (it can only be approached a symptotically as N→ ∞). Regardless, it allows\nus to bring quadrature displacements into our general formalism an d suggests a route towards designing additional\noptimal protocols beyond those already in the literature.\nIn particular, starting with the definition of ˆ pi(l) from Eq. ( C4), we can bound the sum over the quantum Fisher\ninformation matrix elements as\nd/summationdisplay\ni=1,j=1F(θ)ij=d/summationdisplay\ni=1,j=14M/summationdisplay\nl=1M/summationdisplay\nm=1Cov(ˆpi(l),ˆpj(m)) (F1)\n≤4M/summationdisplay\nl=1M/summationdisplay\nm=1/radicaltp/radicalvertex/radicalvertex/radicalbtVar/parenleftigd/summationdisplay\ni=1ˆpi(l)/parenrightig\nVar/parenleftigd/summationdisplay\ni=1ˆpj(m)/parenrightig\n(F2)\n= 4\nM/summationdisplay\nl=1/radicaltp/radicalvertex/radicalvertex/radicalbtVar/parenleftigd/summationdisplay\ni=1ˆpi(l)/parenrightig\n2\n(F3)\n≤4/parenleftiggM/summationdisplay\nl=1/radicalbigg\nN−d\n2/parenrightigg2\n= 4M2/parenleftbigg\nN−d\n2/parenrightbigg\n∼4M2N. (F4)18\nAbove, in Eq.( F2), weused the Cauchy-Schwarzinequality; in Eq.( F4), weused the uncertaintyrelationin Eq. ( B19).\nConsistent with the rest of the paper, the ∼symbol denotes asymptotically in N(forN≫d).\nThe saturability condition in Eq. ( 25) states that, for an optimal protocol (asymptotically in N), it must hold that\nαis an eigenvector of F(θ) with eigenvalue 4 M2N. Thus, for an optimal protocol,\nTr(F) =d/summationdisplay\nj=1λj/greaterorsimilar4M2N, (F5)\nwhereλjare the eigenvalues of F. This implies that the chain of inequalities leading to Eq. ( F4) must be saturated\n(asymptotically in N) for an optimal protocol and that the largest eigenvalue of Fmust beλ1∼4m2Nwith all other\neigenvalues zero. It immediately follows that the saturability conditio n for quadrature displacements can be written\nas\nF(θ)ij∼4M2N\n∝bar⌈blα∝bar⌈bl2\n2αiαj. (F6)\nAppendix G: Approaching the Single-Shot Limit and Robust Ph ase Estimation\nAs pointed out in the footnote preceding Eq. ( 8) and in the discussion of what defines an information-theoretically\noptimal protocol in Sec. IVB, it is not, in practice, possible to construct an unbiased estimator a chieving the single\nshot (µ= 1) quantum Cram´ er-Rao bound that we analyze in this paper, as t he quantum Cram´ er-Rao bound is only\nguaranteedtobe achievablein thelimit ofasymptoticallylargeamount sofdata(µ→ ∞). Resolvingthis tensionwhile\nstill achieving asymptotic Heisenberg scaling in the total amount of r esources (here, µNphotons) requires carefully\ndesigned protocols. In particular, extracting a relative phase fro m the probe states considered in the protocols in this\npaper requires a proper division of resources so that, asymptotic ally, the single-shot bound is achieved up to a small\nconstant.\nAt best, this constant can be reduced to π2[41], but the non-adaptive robust phase estimation scheme of Refs. [ 38–\n40] provides a relatively simple-to-implement approach with a multiplicativ e overhead of (24 .26π)2. In brief, these\nprotocols work by dividing the protocol into Kstages where in stage jone usesNjphotons (or Njaverage photons\nfor displacement sensing). In each stage, one imprints the unknow n function into the phase between two branches\nof a cat-like state of Njphotons and then performs a measurement, as described in the ma in text. The experiment\nis performed νjtimes, allowing one to obtain an estimate of the unknown phase. This e stimate is refined over the\ncourse of the Kstages, with more photons used in each additional stage such that the total photon resources are\nN=K/summationdisplay\nj=1νjNj. (G1)\nAn optimal choice of νjandNjensures that, asymptotically, NK= Θ(N) andνK=O(1), and, thus, the asymp-\ntotic scaling of the single-shot bound is obtained up to a multiplicative c onstant that depends on the details of the\noptimization. The proof of this and the associated optimization are d etailed in Refs. [ 38–40].\n[1] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Joh nston, and G. Adesso, Robustness of asymmetry and coherence\nof quantum states, Phys. Rev. A 93, 042107 (2016) .\n[2] S. L. Braunstein and C. M. Caves, Statistical distance an d the geometry of quantum states,\nPhys. Rev. Lett. 72, 3439 (1994) .\n[3] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Mato s Filho, Quantum speed limit for physical processes,\nPhys. Rev. Lett. 110, 050402 (2013) .\n[4] S. Deffner and S. Campbell, Quantum speed limits: from Hei senberg’s uncertainty principle to optimal quantum contro l,\nJ. Phys. A: Math. Theor. 50, 453001 (2017) .\n[5] L. P. Garc´ ıa-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical\nspeed limits on observables, Phys. Rev. X 12, 011038 (2022) .\n[6] L. Pezz` e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of\natomic ensembles, Rev. Mod. Phys. 90, 035005 (2018) .\n[7] Z. Zhang and Q. Zhuang, Distributed quantum sensing, Quantum Science and Technology 6, 043001 (2021) .\n[8] T. J. Proctor, P. A. Knott, and J. A. Dunningham, Networke d quantum sensing, arXiv:1702.04271 (2017) .19\n[9] W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss- Feig, Distributed quantum metrology with linear networks\nand separable inputs, Phys. Rev. Lett. 121, 043604 (2018) .\n[10] Z. Eldredge, M. Foss-Feig, J. A. Gross, S. L. Rolston, an d A. V. Gorshkov, Optimal and secure measurement protocols f or\nquantum sensor networks, Phys. Rev. A 97, 042337 (2018) .\n[11] T. J. Proctor, P. A. Knott, and J. A. Dunningham, Multipa rameter estimation in networked quantum sensors,\nPhys. Rev. Lett. 120, 080501 (2018) .\n[12] Q. Zhuang, Z. Zhang, and J. H. Shapiro, Distributed quan tumsensing using continuous-variable multipartite entan glement,\nPhys. Rev. A 97, 032329 (2018) .\n[13] Y. Xia, Q. Zhuang, W. Clark, and Z. Zhang, Repeater-enha nced distributed quantum sensing based on continuous-vari able\nmultipartite entanglement, Phys. Rev. A 99, 012328 (2019) .\n[14] J. Gross and C. Caves, One from many: Estimating a functi on of many parameters,\nJ. Phys. A: Math. Theor. 54, 014001 (2020) .\n[15] J. Rubio, P. A. Knott, T. J. Proctor, and J. A. Dunningham , Quantum sensing networks for the estimation of linear\nfunctions, J. Phys. A 53, 344001 (2020) .\n[16] C. Oh, C. Lee, S. H. Lie, and H. Jeong, Optimal distribute d quantum sensing using Gaussian states,\nPhys. Rev. Research 2, 023030 (2020) .\n[17] D. Triggiani, P. Facchi, and V. Tamma, Heisenberg scali ng precision in the estimation of functions of parameters in linear\noptical networks, Phys. Rev. A 104, 062603 (2021) .\n[18] A. Ehrenberg, J. Bringewatt, and A. V. Gorshkov, Minimu m entanglement protocols for function estimation,\nPhys. Rev. Research 5, 033228 (2023) .\n[19] C. Oh, L. Jiang, and C. Lee, Distributed quantum phase se nsing for arbitrary positive and negative weights,\nPhys. Rev. Research 4, 023164 (2022) .\n[20] M. Malitesta, A. Smerzi, and L. Pezz` e, Distributed qua ntum sensing with squeezed-vacuum light in a configurable ar ray\nof Mach-Zehnder interferometers, Phys. Rev. A 108, 032621 (2023) .\n[21] Y. Yang, B. Yadin, and Z.-P. Xu, Quantum-enhanced metro logy with network states,\narXiv preprint arXiv:2307.07758 (2023) .\n[22] X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V. Larsen , T. Gehring, M. Christandl, J. S. Neergaard-Nielsen, and\nU. L. Andersen, Distributed quantum sensing in a continuous -variable entangled network, Nat. Phys. 16, 281 (2020) .\n[23] Y. Xia, W. Li, W. Clark, D. Hart, Q. Zhuang, and Z. Zhang, D emonstration of a reconfigurable entangled radio-frequenc y\nphotonic sensor network, Phys. Rev. Lett. 124, 150502 (2020) .\n[24] L.-Z. Liu, Y.-Z. Zhang, Z.-D. Li, R. Zhang, X.-F. Yin, Y. -Y. Fei, L. Li, N.-L. Liu, F. Xu, Y.-A. Chen, et al., Distributed\nquantum phase estimation with entangled photons, Nat. Phot. 15, 137 (2021) .\n[25] S.-R. Zhao, Y.-Z. Zhang, W.-Z. Liu, J.-Y. Guan, W. Zhang , C.-L. Li, B. Bai, M.-H. Li, Y. Liu, L. You, J. Zhang,\nJ. Fan, F. Xu, Q. Zhang, and J.-W. Pan, Field demonstration of distributed quantum sensing without post-selection,\nPhys. Rev. X 11, 031009 (2021) .\n[26] K. Qian, Z. Eldredge, W. Ge, G. Pagano, C. Monroe, J. V. Po rto, and A. V. Gorshkov, Heisenberg-scaling measurement\nprotocol for analytic functions with quantum sensor networ ks,Phys. Rev. A 100, 042304 (2019) .\n[27] T. Qian, J. Bringewatt, I. Boettcher, P. Bienias, and A. V. Gorshkov, Optimal measurement of field properties with\nquantum sensor networks, Phys. Rev. A. 103, L030601 (2021) .\n[28] A. Hamann, P. Sekatski, and W. D¨ ur, Approximate decohe rence free subspaces for distributed sensing,\nQuantum Sci. Technol. 7, 025003 (2022) .\n[29] J. Rubio and J. Dunningham, Bayesian multiparameter qu antum metrology with limited data,\nPhys. Rev. A 101, 032114 (2020) .\n[30] J. Bringewatt, I. Boettcher, P. Niroula, P. Bienias, an d A. V. Gorshkov, Protocols for estimating multiple functio ns with\nquantum sensor networks: Geometry and performance, Phys. Rev. Research 3, 033011 (2021) .\n[31] E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Pho tonic quantummetrology, AVS Quantum Science 2, 024703 (2020) .\n[32] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metro logy,Phys. Rev. Lett. 96, 010401 (2006) .\n[33] S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, Ge neralized limits for single-parameter quantum estimation .,\nPhys. Rev. Lett. 98, 090401 (2007) .\n[34] Clearly, with µ= 1, we are not guaranteed the existence of an unbiased estima tor, so there is some subtlety in this\nrestriction. The choice is sufficient for determining bounds and optimal probe states, but, when considering measuremen ts\nto extract the quantity of interest, realistic protocols mu st use more than one shot. For instance, robust phase estimat ion\nallows for µ=O(1), while still allowing us to obtain an unbiased estimator that achieves the quantum Cram´ er-Rao bound\nup to a multiplicative constant [ 38–40].In Appendix G, for completeness, we briefly summarize this approach. See also,\nRefs. [44,45] and Ref. [ 18] for further discussion of these issues.\n[35] Notetheuse ofaminimax as opposed toamaximin inEq. ( 9). This follows from thefact that theminimax ofsome objecti ve\nfunction is always greater than or equal to the maximin and we seek to maximize the quantum Fisher information.\n[36] J. Liu, H. Yuan, X. Lu, and X. Wang, Quantum Fisher inform ation matrix and multiparameter estimation,\nJ. Phys. A: Math. Theor. 53, 023001 (2019) .\n[37] M. Hayashi, S. Vinjanampathy, and L. Kwek, Resolving un attainable Cramer–Rao bounds for quantum sensors,\nJ. Phys. B: At. Mol. Opt. Phys. 52, 015503 (2018) .\n[38] S. Kimmel, G. H. Low, and T. J. Yoder, Robust calibration of a universal single-qubit gate set via robust phase estima tion,\nPhys. Rev. A 92, 062315 (2015) .20\n[39] S. Kimmel, G. H. Low, and T. J. Yoder, Erratum: Robust cal ibration of a universal single-qubit gate set via robust pha se\nestimation [Phys. Rev. A 92, 062315 (2015)], Phys. Rev. A 104, 069901(E) (2021) .\n[40] F. Belliardo and V. Giovannetti, Achieving Heisenberg scaling with maximally entangled states: An analytic upper bound\nfor the attainable root-mean-square error, Phys. Rev. A 102, 042613 (2020) .\n[41] W. G´ orecki, R. Demkowicz-Dobrza´ nski, H. M. Wiseman, and D. W. Berry, π-corrected Heisenberg limit,\nPhys. Rev. Lett. 124, 030501 (2020) .\n[42] S. Arora and B. Barak, Computational Complexity: A Modern Approach (Cambridge University Press, Cambridge, 2009).\n[43] A. Chiruvelli and H. Lee, Parity measurements in quantu m optical metrology, Journal of Modern Optics 58, 945 (2011) .\n[44] Y. Yang, G. Chiribella, and M. Hayashi, Attaining the ul timate precision limit in quantum state estimation,\nCommun. Math. Phys. 368, 223 (2019) .\n[45] J. Suzuki, Y. Yang, and M. Hayashi, Quantum state estima tion with nuisance parameters,\nJ. Phys. A: Math. Theor. 53, 453001 (2020) ." }, { "title": "2401.16525v1.Identity_check_problem_for_shallow_quantum_circuits.pdf", "content": "Identity check problem for shallow quantum circuits\nSergey Bravyi,1Natalie Parham,2and Minh Tran1\n1IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 (USA)\n2Columbia University\nChecking whether two quantum circuits are approximately equivalent is a common task in quan-\ntum computing. We consider a closely related identity check problem: given a quantum circuit U,\none has to estimate the diamond-norm distance between Uand the identity channel. We present a\nclassical algorithm approximating the distance to the identity within a factor α=D+ 1 for shallow\ngeometrically local D-dimensional circuits provided that the circuit is sufficiently close to the iden-\ntity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit\ndepth and spatial dimension. We also show that the operator-norm distance to the identity ∥U−I∥\ncan be efficiently approximated within a factor α= 5 for shallow 1D circuits and, under a certain\ntechnical condition, within a factor α= 2D+ 3 for shallow D-dimensional circuits. A numerical\nimplementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100\nqubits.\nI. INTRODUCTION\nA quantum circuit implementation of the desired uni-\ntary operation is rarely exact. Common sources of errors\ninclude hardware noise owing to an imperfect control and\ndecoherence, errors introduced by the circuit compiling\nstep, and errors owing to an approximate nature of a\nquantum algorithm such as Trotter errors in simulation\nof Hamiltonian dynamics. To validate a solution offered\nby a quantum algorithm, is it essential that errors of\neach type are accounted for and reasonably tight upper\nbounds on the deviation from the ideal solution are pro-\nvided.\nUnfortunately, there is little hope that the distance\nbetween arbitrary n-qubit quantum operations can be\ncomputed efficiently for n≫1. To begin with, an expo-\nnentially large Hilbert space dimension prevents one from\nobtaining the full matrix description of quantum opera-\ntions or performing linear algebra on such matrices. Fur-\nthermore, computational complexity theory provides no-\ngo theorems for an efficient distance estimation in many\ncases of interest. For example, Rosgen and Watrous\nshowed [1, 2] that estimating the distance between two\nshallow (with depth logarithmic in n) quantum circuits\nallowing mixed states is PSPACE-hard. This essentially\nrules out efficient classical or quantum algorithms for the\nproblem. Likewise, Janzing, Wocjan, and Beth estab-\nlished QMA-hardness of estimating the distance between\ntwo unitary circuits [3]. The latter result was strength-\nened by Ji and Wu [4] who proved QMA-hardness of es-\ntimating the distance between two constant-depth cir-\ncuits with the one-dimensional qubit connectivity. This\nmay come as a surprise since one-dimensional shallow cir-\ncuits are easy to simulate classically using Matrix Prod-\nuct States [5].\nIt is important that the no-go results stated above hold\nonly if the distance between quantum circuits has to be\nestimated with a small additive error scaling inverse poly-\nnomially with the number of qubits n. Is it possible that\nsome less stringent approximation of the distance can becomputed efficiently? Here, we show that the answer is\nYES and report linear-time classical algorithms approx-\nimating the diamond-norm and the operator-norm dis-\ntances between certain quantum circuits with a constant\nmultiplicative error . Such approximation may be good\nenough for practical purposes. Note that an estimate of\nthe distance with a constant multiplicative error is in-\nformative regardless of how small the distance is. For\nexample, our algorithm can efficiently approximate the\ndistance even if the latter is exponentially small in n.\nThis would be impossible for an algorithm that achieves\nan additive error approximation scaling inverse polyno-\nmially with n.\nLet us formally pose the distance estimation problem\nand state our main results. Suppose Uis a unitary op-\nerator implemented by a quantum circuit acting on n\nqubits. The diamond-norm distance [6] between Uand\nthe identity operation is defined as\nδ(U) = max\nρ∥(U⊗I)ρ(U†⊗I)−ρ∥1 (1)\nwhere ∥ · ∥ 1is the trace norm, Iis the n-qubit identity,\nand the maximization is over all 2 n-qubit states ρ. The\ndistance δ(U) has a simple operational meaning: replac-\ningUby the identity in any experiment that makes use\nof one copy of Ucould change the probability distribu-\ntion describing classical outcomes of the experiment at\nmost by δ(U)/2 in the total variation distance [6, 7]. Ac-\ncordingly δ(U)≤2 with the equality if Uis perfectly\ndistinguishable from the identity in the single-shot set-\nting.\nThe identity check problem is concerned with estimat-\ning the distance δ(U). Checking an approximate equiva-\nlence of n-qubit quantum circuits U1andU2is a special\ncase of this problem since the diamond-norm distance\nbetween U1andU2coincides with δ(U†\n2U1). An identity\ncheck algorithm is said to achieve an approximation ratio\nα≥1 for a class of quantum circuits Cif it takes as input\na circuit U∈ Cand outputs a real number γsuch that\nδ(U)≤γ≤αδ(U) (2)arXiv:2401.16525v1 [quant-ph] 29 Jan 20242\nfor all circuits U∈ C. The algorithm is efficient if its\nruntime scales at most polynomially with the number of\nqubits nfor a fixed approximation ratio α.\nOur main result is a classical identity check algorithm\nfor shallow geometrically local circuits. We assume that n\nqubits are located at cells of a D-dimensional rectangular\narray and consider circuits composed of single-qubit and\ntwo-qubit gates acting on nearest-neighbors cells (cells i\nandjare called nearest-neighbors if one can go from itoj\nby changing a single coordinate by ±1). A depth- hcircuit\nconsists of hlayers of gates such that within each layer\nall gates are disjoint. Our identity check algorithm for\nD-dimensional circuits achieves an approximation ratio\nα=D+ 1 (3)\nif the input circuit satisfies δ(U)<2 and α= 1.16(D+1)\nin the general case. The runtime of the algorithm is\nT∼n212(2hD)D. (4)\nThe runtime is linear in nfor any constant circuit depth\nhand spatial dimension D. We note that achieving an\napproximation ratio α= 1 + ϵwith ϵ=poly(1/n) is at\nleast as hard as approximating the distance δ(U) with an\nadditive error poly(1/n). The latter problem is known to\nbe QMA-hard even in the case of constant-depth 1D cir-\ncuits [4] which rules out efficient algorithms. An interest-\ning open problem is whether an efficient classical or quan-\ntum algorithm can obtain an approximation α= 1+ ϵfor\nany constant ϵ >0. If true, this would provide a Poly-\nnomial Time Approximation Scheme [8] for the identity\ncheck problem.\nApplications such as Quantum Phase Estimation [9]\nor Krylov subspace algorithms [10–12] are sensitive to\nthe overall phase of a quantum circuit since the circuit\nmay be controlled by ancillary qubits. This motivates\na phase-sensitive version of the identity check problem\nwhere the goal is to estimate the operator-norm distance\n∥U−I∥, that is, the largest singular value of U−I. As\nbefore, we aim at approximating ∥U−I∥with a constant\nmultiplicative error.\nA natural strategy is to reduce the task of approximat-\ning∥U−I∥to the one of approximating the diamond-\nnorm distance δ(U), which has been already addressed.\nIt is clear however that such a reduction may not always\nbe possible. For example, if U=eiφIis a multiple of\nthe identity then δ(U) = 0 while ∥U−I∥may take any\nvalue between 0 and 2. To overcome this obstacle, our al-\ngorithm requires an additional input data which depends\non the phase of U. Namely, let PUbe the smallest convex\nsubset of the complex plane C2that contains all eigen-\nvalues of U. Equivalently, PUis a polygon whose vertices\nare eigenvalues of U. Since Uis unitary, all vertices of\nPUlie on the unit circle. It is known [6] that the poly-\ngonPUprovides a simple geometric interpretation of the\ndiamond-norm distance, see Fig. 1.\nOur approximation algorithm for the phase-sensitive\nidentity check problem takes as input a D-dimensional\nFIG. 1. Eigenvalue polygon PUwhose vertices are eigenvalues\nofU. The diamond-norm distance between Uand the identity\nchannel is δ(U) = 2√\n1−r2, where ris the distance between\nPUand the origin [6]. If PUdoes not contain the origin then\nδ(U) coincides with the diameter of PU. Otherwise, δ(U) = 2.\ndepth- hcircuit acting on nqubits and an arbitrary point\nt∈PU. The algorithm outputs an estimator\nγop=γ+|t−1|, (5)\nwhere γis an estimate of the diamond-norm distance\nδ(U) satisfying δ(U)≤γ≤αδ(U) obtained by calling\nthe identity check algorithm for the diamond-norm δ(U).\nWe show that\n∥U−I∥ ≤γop≤αop∥U−I∥. (6)\nwith\nαop= 1 + 2 α. (7)\nSuppose ρis some n-qubit state such that the trace\nTr(ρU) can be computed efficiently. Note that Tr( ρU)∈\nPUsince the diagonal of ρin the eigenbasis of Uis a\nprobability distribution. Thus one can use the estima-\ntor Eq. (5) with t= Tr( ρU). For example, if Uis a 1D\nshallow circuit, one can choose ρas an arbitrary product\nstate. Since Uis a Matrix Product Operator with a bond\ndimension 2O(h)one can compute Tr( ρU) efficiently us-\ning algorithms based on Matrix Product States [13] as\nlong as h=O(logn). In the 1D case Eqs. (3,7) give\nα= 2 and αop= 5 while the runtime of the algorithm is\nT∼n2O(h), see Eq. (4). As another example, suppose\nUis a Trotter circuit describing time evolution of a D-\ndimensional Hamiltonian composed of local Pauli terms\nXX+Y Y,ZZ, and Zthat preserve the Hamming weight.\nThen the all-zeros state |0n⟩is a common eigenvector of\neach individual gate in Uand one can choose ρas the\nall-zeros state, that is, t=⟨0n|U|0n⟩. From Eqs. (3,7)\none gets αop= 2D+ 3. In general, the above gives an\nefficient algorithm approximating ∥U−I∥within a factor\nαop= 2D+ 3 for D-dimensional constant-depth circuits\nprovided that one can efficiently find at least one point\nin the eigenvalue polygon PU.\nThe unfavorable runtime scaling of our algorithm with\nthe circuit depth limits its application to very shallow3\ncircuits. However, the algorithm can be extended to\ndeep circuits Uusing the divide and conquer strategy.\nNamely, if U=Uℓ···U2U1where each layer Uihas depth\nO(1), the triangle inequality gives δ(U)≤Pℓ\ni=1δ(Ui)≤Pℓ\ni=1γi, where γiis an upper bound on δ(Ui) computed\nby our algorithm. The runtime for computing this upper\nbound on δ(U) scales only linearly with the depth of U\nbut we can no longer guarantee that the upper bound is\ntight within a constant factor. Other tradeoffs between\nthe runtime and the upper bound tightness are discussed\nin Section V.\nAlthough this work primarily focuses on computing up-\nper bounds on the distance to the identity, as required for\nvalidation of quantum algorithms, efficiently computable\nlower bounds on the distance are also of interest. Density\nMatrix Renormalization Group (DMRG) algorithms [13]\nprovide a powerful tool for computing lower bounds on\nthe distance δ(U) or∥U−I∥for 1D shallow circuits U.\nIndeed, one can easily check that the squared distance\n∥U−I∥2coincides with the largest eigenvalue of a Hamil-\ntonian H= 2I−U−U†. IfUis a depth- h1D circuit then\nHis a Matrix Product Operator (MPO) with a bond di-\nmension 2O(h). In practice, extremal eigenvalues of MPO\nHamiltonians with a small bond dimension can be well\napproximated using DMRG algorithms [13]. However,\nsince DMRG is a variational algorithm, it only provides\na lower bound on the distance ∥U−I∥. To lower bound\nthe diamond-norm distance we use a bound\nδ(U)≥ ∥U⊗U†−I⊗I∥,\nwith the equality if δ(U)<2, see Section II. Thus δ(U)2\nis lower bounded by the maximum eigenvalue of an MPO\nHamiltonian H= 2I⊗I−U⊗U†−U†⊗Uwhich can in\nturn be lower bounded using DMRG algorithm. We leave\nthe study of lower bounds based on DMRG algorithms\nfor a future work.\nThe rest of the paper is organized as follows. Section II\ndescribes bounds on the diamond-norm and operator-\nnorm distances δ(U) and ∥U−I∥that can be expressed\nin terms of commutators between Uand certain observ-\nables. This section also sketches main ideas behind our\nalgorithm. Section III collects some basic facts about\nshallow quantum circuits and D-dimensional partitions.\nSection IV proves a technical lemma which relates the\nnorms of global and local commutators. Our identity\ncheck algorithm and its analysis is presented in Section V.\nFinally, Section VI reports a software implementation of\nour algorithm.\nII. COMMUTATOR-BASED BOUNDS\nOur identity check algorithm borrows many ideas from\nthe recent breakthrough work by Huang, Liu, et al. [14]\non learning shallow quantum circuits. The main ingredi-\nents of our algorithm, described below, are bounds on the\ndiamond-norm distance δ(U) that depend on the normof commutators between Uand certain observables com-\nposed of SWAP gates. These bounds and their proof are\nlargely based on Ref. [14].\nConsider 2 nqubits labeled by integers 1 , . . . , 2n. Let\nWibe the SWAP gate applied to qubits iandi+n. Given\na subset A⊆[n], define a 2 n-qubit operator\nWA=Y\ni∈AWi.\nBy definition, WAacts non-trivially on 2 |A|qubits.\nLemma 1. Let[n] =A1. . . A mbe a partition of nqubits\nintomdisjoint subsets and Ube a unitary operator acting\nonnqubits. Define a quantity\nγ=mX\nj=1∥WAj(U⊗I)WAj(U†⊗I)−I⊗I∥.(8)\nThen\nδ(U)≤γ≤mδ(U) (9)\nassuming that δ(U)<2and\nδ(U)≤1.16γ≤1.16mδ(U) (10)\nin the general case.\nThe quantity γdefined in Eq. (8) or its rescaled\nversion 1 .16γwill be the desired estimator of the dis-\ntance δ(U). In the next section we show how to choose\na partition [ n] = A1. . . A mwith m=D+ 1 parts\nsuch that each subset Ajis a union of well-separated\nhypercubes of linear size O(hD) and all commutators\nWAj(U⊗I)WAj(U†⊗I) that appear in Eq. (8) are\ntensor products of local commutators supported on in-\ndividual hypercubes. Our construction is based on\nRef. [15] which introduced so-called reclusive partitions\nof the D-dimensional Euclidean space. The key ingre-\ndient of our algorithm is an additivity lemma stated\nin Section IV. This lemma expresses the norm of com-\nmutators ∥WAj(U⊗I)WAj(U†⊗I)−I⊗I∥in terms\nof the norm of analogous local commutators supported\non individual hypercubes. Each local commutator acts\non a subset of at most O(hD)Dqubits and its eigen-\nvalues can be computed by the exact diagonalization.\nThe additivity lemma then provides a linear time al-\ngorithm for computing the norm of global commutators\n∥WAj(U⊗I)WAj(U†⊗I)−I⊗I∥which is all we need\nto compute the estimator γdefined in Lemma 1.\nThe next lemma shows that estimation of the operator-\nnorm distance can be reduced to estimation of diamond-\nnorm distance given any point in the eigenvalue polygon\nofU.\nLemma 2. Lett∈PUbe any point in the eigenvalue\npolygon of Uandα, γbe real numbers such that δ(U)≤\nγ≤αδ(U). Then\nγop=γ+|t−1|4\nobeys\n∥U−I∥ ≤γop≤(1 + 2 α)∥U−I∥.\nIn the rest of this section we prove Lemma 1 and 2.\nProof of Lemma 1. Consider first the case δ(U)<2.\nWe claim that in this case\nδ(U) =∥U⊗U†−I⊗I∥. (11)\nIndeed, since δ(U)<2, the eigenvalue polygon PUdoes\nnot contain the origin and thus δ(U) coincides with the\ndiameter of PU, see Fig. 1. Let {eiφa}abe eigenvalues of\nU. By definition, PUis the convex hull of points {eiφa}a.\nHence the diameter of PUcoincides with the maximum\ndistance between eigenvalues of U. This shows that\nδ(U) = diam( PU) = max\na,b|eiφa−eiφb|\n= max\na,b|ei(φa−φb)−1|\n=∥U⊗U†−I⊗I∥.\nTo get the last equality we noted that {ei(φa−φb)−1}a,b\nis the set of eigenvalues of U⊗U†−I⊗I.\nLet us agree that the tensor product in Eq. (11) sep-\narates two n-qubit registers that span qubits {1, . . . , n }\nand{n+ 1, . . . , 2n}. Let W=Qn\ni=1Wibe an operator\nthat swaps the two registers. Since the operator norm is\nunitarily invariant, Eq. (11) gives\nδ(U) =∥(U⊗U†−I⊗I)W∥\n=∥(U⊗I)W(U†⊗I)−W∥. (12)\nHere we noted that ( I⊗U†)W=W(U†⊗I). The triangle\ninequality implies that for any unitary operators Pj, Qj\none has\n∥P1P2···Pm−Q1Q2···Qm∥ ≤mX\nj=1∥Pj−Qj∥.(13)\nChoosing Pj= (U⊗I)WAj(U†⊗I),Qj=WAj, and\nnoting that W=Qm\nj=1WAjone arrives at\nδ(U)≤mX\nj=1∥(U⊗I)WAj(U†⊗I)−WAj∥=γ.(14)\nThe last equality uses the fact that WAjare both hermi-\ntian and unitary, which implies ∥O−WAj∥=∥WAjO−I∥\nfor any operator O. The dual characterization of the\ndiamond-norm [16] gives\nδ(U) = max\nV:∥V∥≤1∥(U⊗I)V(U†⊗I)−V∥ (15)\nwhere the maximization is over 2 n-qubit operators V.\nSince∥WAj∥= 1 one infers that\n∥(U⊗I)WAj(U†⊗I)−WAj∥ ≤δ(U)for all jand thus γ≤mδ(U). This concludes the proof\nin the case δ(U)<2.\nSuppose now that δ(U) = 2. Then the eigenvalue poly-\ngonPUcontains the origin, see Fig. 1. Let {eiφa}abe the\neigenvalues of U. We claim that there exist eigenvalues\neiφ0, eiφ1ofUsuch that the shortest arc length between\nthem is at least 2 π/3. Otherwise, all eigenvalues would\nlie within an arc of length 2 π/3, 1/3 of the unit circle\n— but this would imply that PUdoes not contain the\norigin. Thus\n∥U⊗U†−I⊗I∥= max\na,b|ei(φa−φb)−1| (16)\n≥ |ei(φ0−φ1)−1| (17)\n≥ |ei2π/3−1| (18)\n= 2 sin ( π/3) =√\n3. (19)\nTherefore we have\nγ≥ ∥U⊗U†−I⊗I∥ ≥√\n3 (20)\nso\n2√\n3γ≥2 =δ(U). (21)\nFurthermore, our proof of the upper bound γ≤mδ(U) is\nunchanged when δ(U) = 2. The desired bound, Eq. (10)\nfollows since 1 .16≥2√\n3.\nProof of Lemma 2. Let{eiφa}abe eigenvalues of U\nandt=P\napaeiφa, where pa≥0 andP\napa= 1. We\nhave\n∥U−I∥=∥U−tI+tI−I∥\n≤ |t−1|+∥X\napa(U−eiφaI)∥\n≤ |t−1|+X\napa∥U−eiφaI∥\n≤ |t−1|+ max\na∥U−eiφaI∥\n=|t−1|+ max\na,b|eiφa−eiφb|\n≤ |t−1|+δ(U)≤ |t−1|+γ.\nConversely, it is well known [6] that δ(U)≤2∥U−I∥for\nany untary U. Thus\n|t−1|+γ=\f\f\f\f\fX\napa(eiφa−1)\f\f\f\f\f+γ\n≤X\napa|eiφa−1|+αδ(U)\n≤max\na|eiφa−1|+ 2α∥U−I∥\n= (1 + 2 α)∥U−I∥.5\nFIG. 2. Examples of reclusive partitions for D= 1,2. Qubits\nare located at cells of a D-dimensional rectangular array. The\narray is partitioned into D+ 1 sets A1, . . . , A D+1such that\neach set Ajis a disjoint union of D-dimensional cubes of linear\nsizeLand the distance between any pair of cubes from the\nsame set Ajis at least L/D. Here L= 4. Cubes located near\nthe boundary of the array are truncated. The sets A1, A2, A3\nare highlighted in yellow, green, and blue.\nIII. LIGHTCONES AND RECLUSIVE\nPARTITIONS\nGiven a quantum circuit Uacting on nqubits, the\nlightcone L(j) of a qubit j∈[n] is defined as the set of\nall output qubits i∈[n] that can be reached by moving\nthrough the circuit diagram forward in time starting from\nthe input qubit j. For example, if Uis a one-dimensional\ncircuit of depth hthen\nL(j)⊆[j−h, j+h]. (22)\nFor any subset of qubits S⊆[n] letL(S) be the lightcone\nofSdefined as\nL(S) =[\nj∈SL(j). (23)\nWe say that a subset of qubits Sis the support of an\noperator Oand write S= supp( O) ifOacts trivially on\nall qubits j /∈S. By definition,\nsupp( UOU†)⊆ L(supp( O)) (24)\nfor any operator O. Furthermore, UOU†=UlocOU†\nloc,\nwhere Ulocis a ”localized” circuit obtained from Uby re-\nmoving all gates acting on qubits outside of the lightcone\nL(supp( O)).\nTwo subsets of qubits S1andS2are said to be light-\ncone separated if L(S1)∩ L(S2) =∅. IfO1andO2are\noperators supported on S1andS2then UO1O2U†is a\nproduct of operators UO1U†andUO2U†with disjoint\nsupports.\nSuppose now that nqubits are located at cells of a\nD-dimensional rectangular array. We shall consider par-\ntitions of the array into D-dimensional cubes known asreclusive partitions [15]. The linear size of each cube in\nthe partition will be chosen as\nL= 2Dh, (25)\nwhere his the depth of U.\nLemma 3 (Reclusive Partitions [15]). One can parti-\ntion cells of a D-dimensional rectangular array into D+1\nsetsA1, . . . , A D+1such that each set Ajis a disjoint\nunion of D-dimensional cubes of linear size Land the\ndistance between any pair of cubes from the same set Aj\nis at least L/D. The above partition can be constructed\nefficiently.\nFigure 2 shows examples of 1D and 2D reclusive parti-\ntions, see Ref. [15] for the 3D example. We defer the proof\nof Lemma 3 to Appendix A since it is a simple rephras-\ning of the results established in [15]. By construction,\neach cube in the partition contains at most LDqubits\n(cubes located near the boundary of the array may be\ntruncated) and any pair of cubes from the same set Aj\nis lightcone separated due to Eq. (25). Write\nAj=Aj,1Aj,2. . . A j,ℓj,\nwhere ℓjis the number of cubes in AjandAj,pdenotes\nthep-th cube in Aj. By constriction, we have\nL(Aj,p)∩ L(Aj,q) =∅for all p̸=q. (26)\nSince the lightcone of a cube with a linear size Lcan be\nenclosed by a cube of linear size L+ 2h, the number of\nqubits contained in any lightcone L(Aj,p) is bounded as\n|L(Aj,p)| ≤(2h(D+ 1))D. (27)\nHere we used Eq. (25).\nConsider the diamond-norm distance δ(U) and spe-\ncialize the commutator-based bound of Lemma 1 to the\nreclusive partition [ n] =A1. . . A D+1. By definition,\nWAj=ℓjY\np=1WAj,p.\nLightcone separation of cubes Aj,pimplies that operators\n(U⊗I)WAj,p(U†⊗I) acts on pairwise disjoint subsets of\nqubits. Thus\nWAj(U⊗I)WAj(U†⊗I) =ℓjY\np=1Kj,p, (28)\nwhere we defined commutators\nKj,p=WAj,p(U⊗I)WAj,p(U†⊗I).\nThe above shows that Kj,pare operators acting on pair-\nwise disjoint subsets of qubits (for a fixed j). Let Uj,p\nbe a ”localized” circuit obtained from Uby replacing all\ngates acting on at least one qubit outside of the lightcone6\nL(Aj,p) with the identity. Then Uj,pacts non-trivially\nonly on the lightcone L(Aj,p) and\nKj,p=WAj,p(Uj,p⊗I)WAj,p(U†\nj,p⊗I).\nThe support of Kj,pincludes all qubits in the left n-qubit\nregister contained in L(Aj,p) as well as all qubits in the\nright n-qubit register contained in Aj,p. Thus\n|supp( Kj,p)| ≤ |L (Aj,p)|+|Aj,p|\n≤(2h(D+ 1))D+ (2hD)D\n= (2hD)D\u0002\n(1 + 1 /D)D+ 1\u0003\n≤4(2hD)D.\nEigenvalues of a unitary operator acting on mqubits can\nbe computed in time O(23m) by the exact diagonalization\nof a unitary 2m×2mmatrix. Thus one can compute all\neigenvalues of the commutator Kj,pin time\nT∼212(2hD)D.\nIn the next section we show that the norm\n∥WAj(U⊗I)WAj(U†⊗I)−I⊗I∥=∥ℓjY\np=1Kj,p−I⊗I∥\nthat appears in the bound of Lemma 1 is a simple func-\ntion of eigenvalues of individual commutators Kj,p.\nIV. ADDITIVITY LEMMA\nIn this section we show how to compute the norm of\ncommutators that appear in Lemma 1. First, let us intro-\nduce some terminology. Let S1={z∈C:|z|= 1}be\nthe unit circle. If Uis a unitary operator, let eig(U)⊆S1\nbe the set of eigenvalues of U(ignoring multiplicities).\nConsider 2 nqubits, a subset A⊆[n], and a SWAP oper-\natorWA=Q\ni∈AWiwhere Wiis the SWAP gate acting\non qubits iandi+n. Consider a commutator\nKA=WA(U⊗I)WA(U†⊗I).\nWe claim that eig(KA) = eig(K†\nA). Indeed, K†\nA=\nWAKAWA. Since WAis both unitary and hermitian,\nconjugation by WAdoes not change the eigenvalue spec-\ntrum. Thus eigenvalues of KAhave a form e±iφwith\n0≤φ≤π. For each φone can choose both positive and\nnegative sign in the exponent. Define a function θthat\nmaps subsets of qubits A⊆[n] to real numbers in the\ninterval [0 , π] such that\nθ(A) = max\nφ∈[0,π]φsubject to eiφ∈eig(KA).(29)\nNote that eiθ(A)is the unique eigenvalue of KAwith the\nmaximum distance from 1 and a non-negative imaginary\npart. Accordingly,\n∥KA−I∥=|eiθ(A)−1|. (30)\nWe shall need the following simple fact.Lemma 4. Ifθ(A)≥π/2for some subset A⊆[n]then\nδ(U)≥√\n2.\nProof. From θ(A)≥π/2 one infers that KAhas an eigen-\nvalue with a non-positive real part. Since all points on\nthe unit circle within distance less than√\n2 from 1 have\na positive real part, one gets ∥KA−I∥ ≥√\n2. The dual\ncharacterization of the diamond norm [16] gives\nδ(U) = max\nη:∥η∥≤1∥(U⊗I)η(U†⊗I)−η∥\n≥ ∥(U⊗I)WA(U†⊗I)−WA∥=∥KA−I∥ ≥√\n2.\nDefinition 1. A subset A⊆[n]is called good if θ(A)<\nπ/2. Otherwise Ais called bad.\nThe following lemma shows that the function θ(A) is\nadditive under the union of lightcone-separated subsets,\nprovided that the circuit Uis sufficiently close to the\nidentity.\nLemma 5 (Additivity). Suppose A1, A2⊆[n]are good\nlightcone-separated subsets. Consider two cases:\n(a)θ(A1) +θ(A2)< π/ 2,\n(b)θ(A1) +θ(A2)≥π/2.\nCase (a) implies that the union A1A2is good and\nθ(A1A2) =θ(A1) +θ(A2). (31)\nCase (b) implies that δ(U)≥√\n2.\nProof. Define commutators\nKp=WAp(U⊗I)WAp(U†⊗I)\nwith p∈ {1,2}. Since A1andA2have lightcone sepa-\nrated, K1andK2act on disjoint subsets of qubits and\nthus\nK12≡WA1A2(U⊗I)WA1A2(U†⊗I) =K1K2\nhas the same eigenvalues as the tensor product of K1and\nK2. In other words,\neig(K1K2) ={z1z2:z1∈eig(K1) and z2∈eig(K2)}.\nBy definition, eiθ(Ap)∈eig(Kp) for p= 1,2. Thus\neiθ(A1)+iθ(A2)∈eig(K1K2) = eig(K12).\nConsider case (a). Let eiφp∈eig(Kp) be eigenvalues\nsuch that eiθ(A1A2)=ei(φ1+φ2). Then\nθ(A1A2) =φ1+φ2+ 2πk (32)\nfor some integer kchosen such that θ(σ1σ2)∈[0, π]. By\ndefinition, |φp| ≤θ(Ap) and thus\n|φ1|+|φ2| ≤θ(A1) +θ(A2)<π\n2.7\nHence the only integer kin Eq. (32) satisfying θ(A1A2)∈\n[0, π] isk= 0, that is, θ(A1A2) =φ1+φ2≤θ(A1) +\nθ(A2). Conversely, since eiθ(A1)+iθ(A2)is an eigenvalue of\nK12andθ(A1)+θ(A2)< π/ 2, one infers that θ(A1A2)≥\nθ(A1) +θ(A2). This proves Eq. (31).\nConsider case (b). The same arguments as above show\nthat K12has an eigenvalue eiφ, where φ=θ(A1) +\nθ(A2)∈[π/2, π). Here we used the assumption that\nboth A1andA2are good, as well as the bound θ(A1) +\nθ(A2)≥π/2. Hence θ(A1A2)≥π/2 and δ(U)≥√\n2 by\nLemma 4.\nBy inductive application of the additivity lemma one\nobtains the following.\nCorollary 1. Suppose A1, . . . , A ℓ⊆[n]are lightcone\nseparated subsets. Let A=∪ℓ\np=1Apbe their union and\nφ=ℓX\np=1θ(Ap). (33)\nHere the angles are added as real numbers (rather than\nmodulo 2π). If φ < π/ 2then\n∥WA(U⊗I)WA(U†⊗I)−I∥=|eiφ−1|. (34)\nIfφ≥π/2then δ(U)≥√\n2.\nV. IDENTITY CHECK ALGORITHM\nCombining all above ingredients we arrive at the fol-\nlowing algorithm for the D-dimensional identity check\nproblem. We first consider the case when the input\ncircuit Uis sufficiently close to the identity such that\nδ(U)<2. Below we assume that a reclusive partition\n[n] =A1. . . A D+1of the D-dimensional qubit array has\nbeen already computed, see Appendix A for details. We\nclaim that the following algorithm outputs an estimator\nγsatisfying δ(U)≤γ≤(D+ 1)δ(U).\nAlgorithm 1 Identity check (diamond-norm)\nInput: Ann-qubit D-dimensional circuit Uwith δ(U)<2.\nOutput: γ∈Rsatisfying δ(U)≤γ≤(D+ 1)δ(U).\n1:γ←0\n2:forj= 1 to D+ 1do\n3: φj←0\n4: ℓj←number of cubes in Aj\n5: forp= 1 to ℓjdo\n6: Aj,p←p-th cube in Aj\n7: φj←φj+θ(Aj,p)\n8: ifφj≥π/2then\n9: return γ= 2\n10: end if\n11: end for\n12: γ←γ+|eiφj−1|\n13:end for\nIndeed, if line 9 is never reached, Corollary 1 of the\nadditivity lemma imply that the output of the algorithmcoincides with the quantity γdefined in Lemma 1 special-\nized to the reclusive partition. In this case correctness of\nthe algorithm follows directly from Lemma 1. Otherwise,\nthe algorithm outputs γ= 2, while Corollary 1 implies\nthat δ(U)≥√\n2. In this case γ= 2 satisfies the bounds\nδ(U)≤γ≤(D+ 1)δ(U) for D≥1. We claim that the\nalgorithm runs in time O(n212(2hD)D). Indeed, the total\nnumber of cubes Aj,pisO(n). Computing the function\nθ(Aj,p) at line 7 requires eigenvalues of a unitary operator\nKAj,pacting on at most 4(2 hD)Dqubits, as discussed in\nSection III. This computation takes time O(212(2hD)D).\nHence the total runtime is O(n212(2hD)D).\nNext consider the general case when it is possible that\nδ(U) = 2. Define our estimator of δ(U) as 1 .16γ, where\nγis the output of Algorithm 1. We claim that\nδ(U)≤1.16γ≤1.16(D+ 1)δ(U). (35)\nIf the algorithm never reaches line 9 then its output co-\nincides with the quantity γdefined in Lemma 1 and\nEq. (35) follows directly from Lemma 1, see Eq. (9). Oth-\nerwise, if the algorithm reaches line 9, it outputs γ= 2\nwhile δ(U)≥√\n2 due to Corollary 1 of the additivity\nlemma. In this case the first inequality in Eq. (35) fol-\nlows from δ(U)≤2 and the second inequality becomes\n2≤(D+ 1)δ(U) which is true for any D≥1 since\nδ(U)≥√\n2. The runtime analysis is the same as before.\nSince the runtime scales exponentially with the size\nof cubes Aj,p, one may wish to choose a partition with\nsmaller cubes even if this negatively impacts the approx-\nimation quality. As an extreme case, one can choose each\ncube Aj,pas a single qubit. However ensuring the light-\ncone separation between cubes in the same subset Aj\nwould require ≈(4h+ 1)Dsubsets Ajinstead of D+ 1\nsubsets [17]. Accordingly, the approximation ratio would\nbecome α= Ω((4 h+ 1)D) instead of α=D+ 1.\nLikewise, we expect that the runtime can be improved\nat the cost of a worse approximation ratio αby com-\nputing the norm of commutators KAj,p−Iusing a ran-\ndomized version of the power method [18]. It is known\nthat this method can approximate the operator norm of\na matrix of size 2m×2mwith a multiplicative error 1 + ϵ\nusing O(m/ϵ) matrix-vector multiplications [18]. In our\ncase, KAj,pis specified by a quantum circuit acting on\nm= 4(2 hD)Dqubits with poly(m) gates, see Section III.\nThus one can implement matrix-vector multiplication for\nthe matrix KAj,p−Iin time poly(m)2m. Accordingly,\nthe power method runs in time poly(m)2m/ϵ, whereas the\nexact diagonalization of KAj,p−Irequires time Ω(23m).\nVI. NUMERICAL EXPERIMENTS\nIn this section, we implement the algorithm described\nin Section V to approximate the distance between iden-\ntity and a constant-depth circuit Uof up to 100 qubits.\nWe consider U=U1U†\n2, where U1, U2are two different8\nunitaries that both approximate the time evolution e−iHτ\nofnqubits under the one-dimensional XY model:\nH=n−1X\nj=1(XjXj+1+YjYj+1).\nIn the limit of small τ,U=U1U†\n2≈Iapproximate a\nforward evolution followed by a backward evolution un-\nder the same Hamiltonian. Explicitly, U1andU2are the\nfirst-order Trotter approximations with, respectively, an\nodd-even ordering and an X-Y ordering:\nU1=e−iτP\noddj(XjXj+1+YjYj+1)e−iτP\nevenj(XjXj+1+YjYj+1),\nU2=e−iτP\njXjXj+1e−iτP\njYjYj+1.\nWe note that XjXj+1andYjYj+1are both antisym-\nmetric under the unitary conjugation by the staggered\nPauli string X1Y2X3Y4. . .. Therefore, the eigenvalues\nofUcomes in complex conjugate pairs which results\nin a simple relationship between the diamond-norm and\nthe operator-norm distances. Namely, a simple algebra\nshows that δ(U) = 2 sin ( φ), where φ∈[0, π/2) is defined\nby∥U−I∥=|eiφ−1|. In addition, using a well-known\nmapping from the XY model to free fermions [19], we can\ncompute this distance exactly, providing a benchmark for\nour algorithm.\nIn Fig. 3, we compare the exact distance δ(U) against\nthe bounds presented in Lemma 1 for up to 100 qubits\natτ= 0.01. For the one-dimensional qubit array, the\nbounds simplify to δ(U)≤γ≤2δ(U), where\nγ=2X\nj=1∥WAj(U⊗I)WAj(U†⊗I)−I∥. (36)\nHere, A1andA2are the qubit partitions illustrated in\nFig. 2 with L= 4. The lightcone separated construction\nFIG. 3. A comparison between the exact diamond-norm\ndistance δ(U) (green dots) computed by a mapping to free\nfermions, an upper bound γcomputed by Algorithm 1 (blue\ndots) and the lower bound γ/2 (orange dots). Both bounds\nclosely capture the exact distance between UandI, demon-\nstrating the scalability of our algorithm.ofAjand the additivity lemma allow us to efficiently\ncompute the commutator ∥WAj(U⊗I)WAj(U†⊗I)−I∥\nfor each j. In particular, computing the bounds reduces\nto finding eigenvalues of operators that are each sup-\nported on at most 12 qubits. Additionally, due to the\ntranslational invariance of the unitary U, only O(1) such\noperators are unique, making the complexity of our algo-\nrithm independent of the system size.\nBoth bounds correctly capture the linear dependence\nof the Trotter error on the system size n, with the up-\nper bound γapproaching the exact δ(U) in the limit of\nlarge n. We note that ∥U−I∥and, thus, δ(U) can also\nbe estimated by finding the maximum eigenvalue of the\nHamiltionian HU≡(U−I)†(U−I). Writing this Hamil-\ntonian as a matrix product operator on a one-dimensional\nlattice, one can efficiently find a lower bound to its maxi-\nmum eigenvalue using an algorithm based on the density\nmatrix renormalization group (DMRG). While DMRG\ndoes not have a performance guarantee, we find that it\nproduces lower bounds to within 3 ×10−7of the exact\nδ(U) in this example, providing a complementary ap-\nproach to our algorithm in one dimension. DRMG sim-\nulations were performed using the matrix product rep-\nresentation library for Python mpnum [20] with MPS\nbond dimension χ= 20 and two DMRG sweeps in\nmpnum .linalg .eig.\nACKNOWLEDGEMENTS\nSB thanks Steven Flammia and Kristan Temme for\nhelpful discussions. MCT thanks Kunal Sharma for help-\nful discussions. This work was partially completed while\nNP was interning at IBM Quantum. NP is supported by\nAFOSR award FA9550-21-1-0040, NSF CAREER award\nCCF-2144219, and the Sloan Foundation.\nAppendix A: Proof of Lemma 3\nLetAbe an upper triangular D×Dmatrix with the\nunit diagonal. In other words, Ai,i= 1 for all iandAi,j=\n0 for all i > j . Define a lattice LA⊆RDformed by linear\ncombinations of columns of Awith integer coefficients.\nBy definition, p∈ LAiffp=Acfor some integer vector\nc��ZD. For each lattice point p∈ LAdefine an open\ncube C(p) and a closed cube C(p) such that pis the\ncube’s corner with the smallest coordinates, that is,\nC(p) =p+ (0,1)Dand C(p) =p+ [0,1]D.\nThe following claim can be interpreted as saying that the\ncubes C(p) form a partition of the Euclidean space RD\nif one ignores cube’s boundaries.\nClaim 1. Any point x∈RDis contained in at most one\nopen cube C(p). Any point x∈RDis contained in at\nleast one closed cube C(p).9\nProof. Define ℓ∞norm of a vector x∈RDas\n∥x∥∞= max\ni=1,...,D|xi|.\nSuppose x∈RDis contained in cubes C(p) and C(q) for\nsome lattice points p, q∈ L. We have to show that p=q.\nClearly, cubes C(p) and C(q) overlap iff\n∥p−q∥∞<1. (A1)\nThus we need to show that Eq. (A1) implies p=q. Write\nr=p−q=Ac (A2)\nfor some c∈ZD. Using the upper triangular structure\nofAand the fact that Ahas unit diagonal one gets\nri=ci+DX\nj=i+1Ai,jcj. (A3)\nIfi=Dthen clearly ri=ciand thus |ri|<1 is only\npossible if ci= 0. If i=D−1 then ri=ci+Ai,DcD.\nHowever, we have already showed that cD= 0. Thus\nri=ciand|ri|<1 is only possible if ci= 0. Applying\nthe same argument inductively proves that cis the all-\nzeros vector, that is, Eq. (A1) implies p=q.\nSuppose some vector x∈RDis not contained in any\nclosed cube C(p). Then ∥x−p∥∞>1 for all lattice points\np∈ L. Let us show that this assumption leads to a con-\ntradiction. Indeed, set i=D. Shift xby an integer linear\ncombination of the i-th column of Ato make |xi| ≤1.\nThis is possible since Ai,i= 1. Next set i=D−1. Shift\nxby an integer linear combination of the i-th column of\nAto make |xi| ≤1 and |xi+1| ≤1. This is possible since\nAi,i= 1 and Ai+1,i= 0. Applying the same argument\ninductively shows that shifting xby lattice points one\ncan make ∥x∥∞≤1. Hence xis contained in the cube\nC(0D). Equivalently, the original vector xis contained\nin some cube C(p).Following Ref. [15] we choose\nAi,j=\n\n1 if i=j,\nD−j+1\nDifi < j,\n0 else(A4)\nfor 1≤i, j≤D. For example,\nA=\u0014\n1 1/2\n0 1\u0015\nand A=\n1 2/3 1/3\n0 1 1 /3\n0 0 1\n\nin the case D= 2 and D= 3 respectively. Below we\nsummarize properties of the corresponding lattice LAes-\ntablished in [15].\nFact 1 (Lemmas 7.15 and 7.19 of [15]). Theℓ∞-\ndistance between closed cubes C(p)andC(q)is either 0\n(if these cubes overlap) or at least 1/D(if these cubes do\nnot overlap). Here p, q∈ LAare arbitrary lattice points.\nFact 2 (Theorem 7.16 of [15]). The cubes {C(p)}p∈LA\ncan be colored with D+ 1colors such that any cube C(p)\noverlaps only with cubes C(q)of a different color.\nAs a consequence of Facts 1 and 2, the ℓ∞-distance\nbetween any pair of cubes C(p) of the same color is at\nleast 1 /D. Rescaling each cube by the factor L= 2Dh\nand noting that LAis an integer matrix one obtains a\npartition of RDinto a disjoin union of D-dimensional\ncubes LC(p) of linear size Lsuch that corners of each\ncube have integer coordinates, the cubes are colored with\nD+ 1 colors, and the ℓ∞-distance between any pair of\ncubes of the same color is at least L/D.\nFinally, embed a D-dimensional rectangular array into\nRDsuch that each cell of the array is a translation of\nthe cube (0 ,1)Dby an integer vector. We can now de-\nfine the desired set of cells Ajas the union of all cells\ncontained in the rescaled cubes LC(p) of the j-th color.\nThis concludes the proof of Lemma 3.\n[1] Bill Rosgen and John Watrous. On the hardness of dis-\ntinguishing mixed-state quantum computations. In 20th\nAnnual IEEE Conference on Computational Complexity\n(CCC’05) , pages 344–354. IEEE, 2005.\n[2] Bill Rosgen. Distinguishing short quantum computa-\ntions. arXiv preprint arXiv:0712.2595 , 2007.\n[3] Dominik Janzing, Pawel Wocjan, and Thomas Beth.\n”Non-identity-check” is QMA-complete. International\nJournal of Quantum Information , 3(03):463–473, 2005.\n[4] Zhengfeng Ji and Xiaodi Wu. Non-identity check re-\nmains QMA-complete for short circuits. arXiv preprint\narXiv:0906.5416 , 2009.\n[5] Guifr´ e Vidal. Efficient simulation of one-dimensional\nquantum many-body systems. Physical review letters ,\n93(4):040502, 2004.\n[6] Dorit Aharonov, Alexei Kitaev, and Noam Nisan. Quan-tum circuits with mixed states. In Proceedings of the\nthirtieth annual ACM symposium on Theory of comput-\ning, pages 20–30, 1998.\n[7] Avraham Ben-Aroya and Amnon Ta-Shma. On the\ncomplexity of approximating the diamond norm. arXiv\npreprint arXiv:0902.3397 , 2009.\n[8] Vijay V Vazirani. Approximation algorithms , volume 1.\nSpringer, 2001.\n[9] Michael A Nielsen and Isaac L Chuang. Quantum compu-\ntation and quantum information . Cambridge university\npress, 2010.\n[10] William J Huggins, Joonho Lee, Unpil Baek, Bryan\nO’Gorman, and K Birgitta Whaley. A non-orthogonal\nvariational quantum eigensolver. New Journal of Physics ,\n22(7):073009, 2020.\n[11] Kazuhiro Seki and Seiji Yunoki. Quantum power method10\nby a superposition of time-evolved states. PRX Quan-\ntum, 2(1):010333, 2021.\n[12] William Kirby, Mario Motta, and Antonio Mezzacapo.\nExact and efficient lanczos method on a quantum com-\nputer. Quantum , 7:1018, 2023.\n[13] Ulrich Schollw¨ ock. The density-matrix renormalization\ngroup in the age of matrix product states. Annals of\nphysics , 326(1):96–192, 2011.\n[14] Hsin-Yuan Huang, Yunchao Liu, Michael Broughton,\nIsaac Kim, Anurag Anshu, Zeph Landau, and Jarrod R.\nMcClean. Learning shallow quantum circuits. arXiv\npreprint arXiv:2401.10095 , 2024.\n[15] Jason Vander Woude, Peter Dixon, A Pavan, Jamie Rad-\ncliffe, and NV Vinodchandran. Geometry of rounding.\narXiv preprint arXiv:2211.02694 , 2022.\n[16] John Watrous. Semidefinite programs for completelybounded norms. arXiv preprint arXiv:0901.4709 , 2009.\n[17] Since any qubit is lightcone separated from all but at\nmost (1 + 4 h)Dother qubits, Vizing’s theorem implies\nthat qubits can be partitioned into at most 1+(1+4 h)D\nlightcone separated subsets.\n[18] Jacek Kuczy´ nski and Henryk Wo´ zniakowski. Estimating\nthe largest eigenvalue by the power and Lanczos algo-\nrithms with a random start. SIAM journal on matrix\nanalysis and applications , 13(4):1094–1122, 1992.\n[19] Barbara M Terhal and David P DiVincenzo. Classical\nsimulation of noninteracting-fermion quantum circuits.\nPhysical Review A , 65(3):032325, 2002.\n[20] Daniel Suess and Milan Holz¨ apfel. mpnum: A matrix\nproduct representation library for Python. Journal of\nOpen Source Software , 2(20):465, 2017." }, { "title": "2401.16526v1.FPGA_Technology_Mapping_Using_Sketch_Guided_Program_Synthesis.pdf", "content": "FPGA Technology Mapping Using\nSketch-Guided Program Synthesis\nGus Henry Smith\nUniversity of Washington\nSeattle, USA\ngussmith@cs.washington.eduBen Kushigian\nUniversity of Washington\nSeattle, USA\nbenku@cs.washington.eduVishal Canumalla\nUniversity of Washington\nSeattle, USA\nvishalc@cs.washington.edu\nAndrew Cheung\nUniversity of Washington\nSeattle, USA\nacheung8@cs.washington.eduSteven Lyubomirsky\nOctoAI\nSeattle, USA\nslyubomirsky@octo.aiSorawee Porncharoenwase\nUniversity of Washington\nSeattle, USA\nsorawee@cs.washington.edu\nRené Just\nUniversity of Washington\nSeattle, USA\nrjust@cs.washington.eduGilbert Louis Bernstein\nUniversity of Washington\nSeattle, USA\ngilbo@cs.washington.eduZachary Tatlock\nUniversity of Washington\nSeattle, USA\nztatlock@cs.washington.edu\nAbstract\nFPGA technology mapping is the process of implement-\ning a hardware design expressed in high-level HDL (hard-\nware design language) code using the low-level, architecture-\nspecific primitives of the target FPGA. As FPGAs become\nincreasingly heterogeneous, achieving high performance\nrequires hardware synthesis tools that better support map-\nping to complex, highly configurable primitives like digital\nsignal processors (DSPs). Current tools support DSP map-\nping via handwritten special-case mapping rules, which are\nlaborious to write, error-prone, and often overlook map-\nping opportunities. We introduce Lakeroad, a principled ap-\nproach to technology mapping via sketch-guided program\nsynthesis. Lakeroad leverages two techniques—architecture-\nindependent sketch templates and semantics extraction from\nHDL—to provide extensible technology mapping with stronger\ncorrectness guarantees and higher coverage of mapping op-\nportunities than state-of-the-art tools. Across representative\nmicrobenchmarks, Lakeroad produces 2–3.5 ×the number of\noptimal mappings compared to proprietary state-of-the-art\ntools and 6–44×the number of optimal mappings compared\nto popular open-source tools, while also providing correct-\nness guarantees not given by any other tool.\nPermission to make digital or hard copies of part or all of this work for\npersonal or classroom use is granted without fee provided that copies are\nnot made or distributed for profit or commercial advantage and that copies\nbear this notice and the full citation on the first page. Copyrights for third-\nparty components of this work must be honored. For all other uses, contact\nthe owner/author(s).\nASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\n©2024 Copyright held by the owner/author(s).\nACM ISBN 979-8-4007-0385-0/24/04.\nhttps://doi.org/10.1145/3620665.3640387\nFigure 1. Even given a simple input design (input 1), the\nstate-of-the-art (SOTA) hardware synthesis tool for Xilinx\nFPGAs frequently fails to efficiently use programmable\nprimitives like DSPs. Lakeroad, on the other hand, can utilize\nall features of programmable primitives given just a short\ndescription of an FPGA architecture (input 2) and the vendor-\nprovided simulation models of the primitive (input 3).\nACM Reference Format:\nGus Henry Smith, Ben Kushigian, Vishal Canumalla, Andrew Che-\nung, Steven Lyubomirsky, Sorawee Porncharoenwase, René Just,\nGilbert Louis Bernstein, and Zachary Tatlock. 2024. FPGA Tech-\nnology Mapping Using Sketch-Guided Program Synthesis. In 29th\nACM International Conference on Architectural Support for Program-\nming Languages and Operating Systems, Volume 2 (ASPLOS ’24),\nApril 27-May 1, 2024, La Jolla, CA, USA. ACM, New York, NY, USA,\n17 pages. https://doi.org/10.1145/3620665.3640387arXiv:2401.16526v1 [cs.AR] 29 Jan 2024ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\n1 Introduction\nGiven a high-level hardware design specification (e.g., ex-\npressed in behavioral Verilog), FPGA technology mappers\nsearch for an equivalent low-level implementation in terms\nof the target FPGA’s primitives. See Figure 1 for an exam-\nple, where the high-level, behavioral add_mul_and mod-\nule (“input 1”) is converted into FPGA-specific implementa-\ntions (“their output” and “our output”) using Xilinx-specific\nDSP48E2 andLUT2 primitives.\nHistorically, FPGAs consisted of relatively simple primi-\ntives, such as lookup tables (LUTs) and carry chains. Tools\nlike ABC [ 14,30,37]automatically map to these basic primi-\ntives by translating designs to a library of simple logic gates\nand then packing those gates into LUTs.\nHowever, FPGAs are becoming increasingly heteroge-\nneous via the inclusion of specialized and diverse primitives\nsuch as digital signal processors (DSPs). Utilizing these spe-\ncialized primitives effectively is now crucial for achieving\nhigh performance [ 49]. These specialized primitives make\nFPGA technology mapping far more challenging since tech-\nnology mappers must now explore a much larger search\nspace while also satisfying each primitive’s complex set of re-\nstrictions and dependencies. For example, Xilinx’s DSP48E2\nis a multifunction DSP with nearly 100 ports and parameters,\nwhose numerous configurations enable support for a large\nvariety of computations. The manual for the DSP48E2 alone\nis 75 pages long, where considerable text details the complex\nrestrictions between the settings of the nearly 100 ports and\nparameters.\nExisting technology mapping tools frequently fail to map\ndesigns to specialized primitives like DSPs, requiring manual\nwork for the hardware designer to recover the performance\nof their design. While existing toolchains have the ability to\nautomatically infer locations where specialized primitives\ncan be used in large designs, inference often fails [ 1,2,4].\nIn these cases, the designer can either accept lower perfor-\nmance and higher resource utilization, or they can perform\nwhat we call partial design mapping. During partial design\nmapping, the designer manually identifies and separates out\nthe module that should be mapped to a DSP. They can at-\ntempt to re-run technology mapping on that module alone,\nin the hopes that mapping succeeds. Yet existing toolchains\noften fail even in the partial design mapping case: Figure 1\nshows a simple module add_mul_and which should fit on a\nsingle DSP48E2 according to the DSP’s manual, but is instead\nmapped to multiple DSPs and LUTs by current state-of-the-\nart tools.1In the worst case, hardware designers are forced\nto manually instantiate complex primitives by hand, e.g., by\nlooking through the 75-page DSP48E2 user manual to manu-\nally configure the DSP’s dozens of ports and parameters.\n1Licensing restrictions forbid naming the specific proprietary tools, but\nthey are familiar, standard packages used by many hardware designers.Current state-of-the-art technology mappers are imple-\nmented via ad hoc, handwritten pattern matching proce-\ndures, which fall short in three primary ways. First, as we\nsaw above, they are incomplete: they miss many mapping\nopportunities, even across microbenchmarks based on ven-\ndor documentation. Second, they do not provide strong\ncorrectness guarantees: recent work highlights the sig-\nnificant number of bugs found across all major hardware\nsynthesis tools [ 24]. Third, they are difficult to extend:\neach new complex primitive requires supporting detailed se-\nmantics and adding hundreds of new, special-case syntactic\npattern matching rules [50].\nThis paper’s key observation is that technology mapping\nis well-suited for the application of automated reasoning\nprocedures—specifically, program synthesis [23]. We observe\nthat the configuration space of a programmable FPGA prim-\nitive is essentially a small, bespoke programming language,\nand that program synthesis could be applied to automat-\nically generate primitive configurations. We explore how\nprogram synthesis can simplify the design and implementa-\ntion of FPGA technology mappers while providing correct ,\nextensible , and more complete support for mapping to\ndiverse, highly configurable primitives like DSPs. Program\nsynthesis techniques rely on automated theorem provers\nlike SAT and SMT solvers [ 8,17] to automatically generate\nprograms satisfying a given specification. We demonstrate\nhow sketch-guided program synthesis [41] can be adapted\nfor FPGA technology mapping, leveraging the Rosette [ 46]\nprogram synthesis framework.\nSketch-guided program synthesis requires encoding the\nsemantics of the target language: in our case, a machine-\nreadable, mathematical model specifying the behavior of\neach FPGA-specific primitive being mapped to. In a typical\nsynthesis tool, which generates programs for a single target\nlanguage, this is a one-time cost. However, in our setting,\neach new FPGA primitive introduces yet another new target\nlanguage, which in turn requires extending the tool to encode\nyet another formal semantics.\nTo support correct, extensible, and more complete technol-\nogy mapping, we propose automating this process with se-\nmantics extraction from HDL , adapted from past work [ 16],\nto automatically extract complete primitive semantics from\nvendor-published HDL models (Figure 1, “input 3”). Tradi-\ntionally, such models have been used only for simulation\nand validation after technology mapping; we show that us-\ning the semantics to implement technology mapping with a\nprogram-synthesis-based approach yields substantially more\ncomplete FPGA technology mapping.\nSketch-guided program synthesis also requires sketches ,\nwhich are partially complete programs with “holes” to be\nfilled in by the solver. Sketches primarily serve to scale syn-\nthesis by constraining the set of programs that solvers ex-\nplore when searching for one that satisfies the given specifi-\ncation, i.e., performance at the cost of completeness. In our\n2FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nsetting, sketches correspond to arrangements of primitives,\nusing holes as placeholders for some of the primitives’ ports\nand parameters. This could be a single DSP with holes for\nits ports and parameters (as in the example in Section 2.2),\nor a number of LUTs with holes for their LUT memories, or\neven a mixture of LUTs, DSPs, and carry chains. The syn-\nthesizer “fills in the holes” as necessary for the low-level\nFPGA-specific primitive to implement a given high-level\nbehavioral design fragment. Unfortunately, developing ef-\nfective sketches still requires synthesis expertise [ 11,47].\nNaïvely, our approach would also require new sketches for\neach new FPGA primitive we target.\nTo address these challenges, we introduce architecture-\nindependent sketch templates. Hardware designs are of-\nten implemented using high-level blueprints that are similar\nacross most FPGA architectures—sketch templates capture\nthese blueprints and make them reusable across architectures.\nTherefore, by using sketch templates, we greatly reduce the\noverhead of supporting new architectures and diverse primi-\ntives. Typically, when adding support for a new primitive or\nFPGA architecture in Lakeroad, the hardware designer need\nnot write or modify any sketch templates.\nWe leverage semantics extraction from HDL and architecture-\nindependent sketch templates to build Lakeroad,2a new\nFPGA technology mapper based on program synthesis.\nLakeroad’s prototype implementation automatically im-\nports semantics for the LUTs, arithmetic carry chains, and\nDSPs of the Xilinx UltraScale+, Lattice ECP5, Intel Cyclone 10\nLP, and SOFA [ 43] FPGA architectures. The only additional\nuser input to Lakeroad is a short architecture description that\nlists the target FPGA’s primitives (Figure 1, “input 2”). Archi-\ntecture descriptions only need to be written once per archi-\ntecture, and Lakeroad pre-supplies architecture descriptions\nfor the aforementioned architectures. With the automati-\ncally extracted primitive semantics and the user-provided\narchitecture description, we demonstrate that Lakeroad is\nmore complete than proprietary tools on a variety of mi-\ncrobenchmarks that are representative of program fragments\nimplemented with DSPs during partial design mapping. In\nparticular, Lakeroad maps up to 3.5 ×more microbenchmarks\nthan state-of-the-art tools for Xilinx, Lattice, and Intel, and\nup to 44×more microbenchmarks than Yosys.\nThis paper makes the following key contributions:\n•The novel application of program synthesis to produce\na technology mapper—Lakeroad—that is more correct,\ncomplete, andextensible than state-of-the-art tools.\n•A technique for applying semantics extraction from\nHDL to automatically generate models of hardware usable\nby formal automated reasoning tools.\n•The concept of architecture-independent sketch tem-\nplates, which capture common patterns in hardware de-\nsign in an architecture-independent way, plus primitive\n2Lakeroad is publicly available at https://github.com/uwsampl/lakeroad .interfaces andarchitecture descriptions , the abstrac-\ntions underlying these templates.\n•A formalization of the Lakeroad toolchain and an argu-\nment for its correctness and sketch-completeness.\n•The first notion of technology mapping completeness\nfor FPGA compilers.\n•Empirical comparisons of Lakeroad and existing hard-\nware synthesis tools to evaluate both their relative com-\npleteness and ease of extensibility.\nIn the following sections, we walk through a real-world ex-\nample using both existing tools and Lakeroad and highlight\nLakeroad’s design and key features (Section 2); formalize\nLakeroad and demonstrate its correctness (Section 3); de-\nscribe Lakeroad’s implementation (Section 4); and evaluate\nLakeroad on its completeness of mapping, extensibility, and\nexpressiveness (Section 5) . Section 6 discusses related work,\nand Section 7 concludes.\n2 Overview\nWe now walk through an example of how current FPGA\ntechnology mapping tools can fail a hardware designer (Sec-\ntion 2.1) and how Lakeroad overcomes these limitations (Sec-\ntion 2.2). In the process, we provide a high-level overview of\nLakeroad’s main components.\n2.1 Compiling a Design to a DSP with Existing Tools\nConsider the following scenario: A hardware designer is de-\nsigning a large hardware block for the Xilinx UltraScale+\nfamily of FPGAs. The designer is specifically aiming to use\nthe UltraScale+’s specialized DSP48E2 units, which can im-\nplement combined multiplication, arithmetic, and logic op-\nerations, as captured in this simplified block diagram [51]:\nThe designer’s hardware block involves the computation\n(d+a)*b&c , which the manual states is implementable with\na single DSP. In particular, suppose the design consists of\nfour separate instances of the following computation:\nfor(i=0; i<4; i++) begin\nr[i] <= (d[i] + a[i]) * b[i] & c[i];\nend\nIt would be reasonable for the designer to expect the design\nto use a total of four DSPs.\nCurrent tools fail. After compiling the design with exist-\ning tools, the designer is frustrated to find that the compiler\nreturns a design that uses more resources than anticipated.\nIt does use four DSPs, but it also uses 128 registers (which\nhold state) and 64 lookup tables (LUTs, which implement\nlogic functions). The compiler has thus failed to fully\n3ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\nutilize the DSP —it has not configured a DSP48E2 to imple-\nment (d+a)*b&c but has instead implemented a portion of\nthe computation with LUTs and registers. The designer now\nfaces a choice: either accept the result or attempt to coax the\ncompiler into returning a more optimal design.\nCoaxing the compiler, to no avail. Though many may\nchoose to accept a less optimal result, this tenacious3tries to\ncoax the compiler into giving the expected results by placing\nthe computation of interest into a separate module:\n// add_mul_and.v: computes (a+b)*c&d in two clock cycles.\nmodule add_mul_and( input clk, input [15:0] a, b, c, d,\noutput reg [15:0] out);\nreg [15:0] r;\nalways @ (posedge clk) begin\nr <= (a+b)*c&d; out <= r;\nend\nendmodule\nThis allows the designer to apply specific optimizations while\nmapping the module—a process we call partial design map-\nping. They attempt various strategies, including annotating\nthe module with Xilinx’s use_dsp Verilog attribute (to force\nthe compiler to use a DSP where possible) and using a differ-\nent synthesis directive (to apply a more resource-intensive\nsynthesis procedure). Despite these efforts, the compiler\nstill cannot map the design to a single DSP, instead using\none DSP, 32 registers, and 16 LUTs. Again, the designer must\ndecide: give up and accept suboptimal results, or press on?\nManual compilation. The hardware designer presses on\nand now has only one option remaining: manually instantiat-\ning a DSP48E2 with the desired behavior. Skimming through\nthe daunting 75-page DSP48E2’s online user manual, the de-\nsigner quickly discovers that configuring even the “pre-add”\na+brequires correctly setting multiple ports and parameters\n(INMODE ,AMULTSEL ,BMULTSEL , and PREADDINSEL ), whose de-\nscriptions span 10+ pages and multiple tables. Correctly con-\nfiguring the subsequent multiplier and logic unit proves even\nmore difficult and time-consuming. After configuring the\ncomputational units, the designer must still manually ensure\ncorrect pipelining of the 10+ pipeline registers. After hours\nof frustration, a configuration is found that seems to work,\nwhich the designer inserts into the design. Precious time has\nbeen wasted, most of which will need to be repeated to con-\nfigure the DSP again. Making matters worse, the designer\nhas no formal guarantees about the correctness of this\nDSP configuration. It may work in a few simulated test\ncases, but are there corner cases that have been missed?\n2.2 Compiling a Design to a DSP with Lakeroad\nLakeroad can save hardware designers the great effort in-\nvolved in manual DSP configuration while also providing\ncorrectness guarantees. Let us imagine how the designer in\nthis example, frustrated by conventional tools, can instead\n3This may not be purely a personal preference. For example, a hardware\ndesign simply may not fit on an FPGA without manual optimizations!\nFigure 2. The components within Lakeroad.\nproceed using Lakeroad during partial design mapping. After\nputting add_mul_and into its own module, the designer calls\nLakeroad from the command line:\n$ lakeroad --template dsp \\\n--arch-desc xilinx-ultrascale-plus.yml \\\nadd_mul_and.v\nThelakeroad command outputs add_mul_and_impl , an im-\nplementation of add_mul_and that uses a single UltraScale+\nDSP48E2:\nmodule add_mul_and_impl( input clk, input [15:0] a, b, c, d,\noutput [15:0] out);\nDSP48E2 #(\n.ACASCREG(32 'd0), .ADREG(32 'd0), .ALUMODEREG(32 'd0),\n.AMULTSEL(\"AD\"), .AREG(32 'd0), .AUTORESET_PATDET(\"NO_RESET\"),\n// ...plus 30+ more parameters\n) DSP48E2_0 (\n.A({ 14 'h0000, a }), .ACIN(30 'h00000000), .ALUMODE(4 'hc),\n.B({ 2 'h0, b }), .BCIN(18 'h00000), .C({ 32 'h00000000, c }),\n.CARRYCASCIN(1 'h0), .CARRYIN(1 'h0), .CARRYINSEL(3 'h6),\n// ...plus 30+ more ports\n);\nendmodule\nUnlike current compilers, Lakeroad has produced an imple-\nmentation using a single DSP48E2 by utilizing more of the\nDSP’s features. Importantly, this compiled design is also\nformally guaranteed to implement the input add_mul_and\ndesign.\nHow does Lakeroad provide verified, more complete sup-\nport for the DSP48E2 over existing tools? At the core of\nLakeroad’s correctness and completeness is sketch-guided\nprogram synthesis , a technique that begins with a program\nsketch , which captures a rough outline of a program and uses\nautomated reasoning tools (e.g., SMT solvers) to fill in the\nsketch’s holes. As shown in Figure 2, Lakeroad uses the fol-\nlowing three-step process to generate an efficient and correct\nDSP48E2 implementation of the add_mul_and design.\n4FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nStep 1: Generating a Sketch. In the add_mul_and exam-\nple, Lakeroad generates the following sketch, which we refer\nto as sketch :4\nmodule sketch( input clk, input [15:0] a, b, c, d,\noutput [15:0] out);\nDSP48E2 #(\n.ACASCREG(??), .ADREG(??), .ALUMODEREG(??), .AMULTSEL(??),\n.AREG(??), .AUTORESET_PATDET(??), ...\n) DSP48E2_0 (\n.A({ 14 'h0000, a }), .ACIN(??), .ALUMODE(??),\n.B({ 2 'h0, b }), .BCIN(??), .C({ 32 'h00000000, c }),\n.CARRYCASCIN(??), .CARRYIN(??), .CARRYINSEL(??), ...\n);\nendmodule\nThis sketch consists of a single DSP48E2 instance with holes\n(represented by ??) serving as placeholders for most of its\nports and parameters. It is easy to see the parallels between\nsketch andadd_mul_and_impl ;sketch is simply add_mul_-\nand_impl with holes. But how does Lakeroad generate sketch\nin the first place?\nTo maximize portability across architectures, Lakeroad\ndoes not store sketches like sketch directly; instead, it gen-\nerates sketches from architecture-independent sketch tem-\nplates. Instead of storing the preceding UltraScale+–specific\nsketch, Lakeroad generates the sketch from the DSP sketch\ntemplate, which the designer has chosen to use with the\n--template dsp flag. A simplified form of this template\nlooks like the following:\nmodule dsp_sketch_template( input clk,\ninput [n-1:0] a, b, c, d,\noutput [n-1:0] out);\nDSP dsp_instance(.clk(clk), .A(a), .B(b), .C(c), .D(d), .out(out));\nendmodule\nSketch templates capture hardware design patterns that\nare common across FPGA architectures in an architecture-\nindependent way. dsp_sketch_template , for example, cap-\ntures a basic pattern, i.e., instantiating a single DSP. Lakeroad\nincludes sketch templates of varying complexity, from the\nsimplicity of the one above to the complexity of LUT-based\nmultipliers. Though new sketch templates can be added eas-\nily, in most cases (as in this example) users can simply apply\nLakeroad’s provided templates.\nTo specialize dsp_sketch_template intosketch , Lakeroad\ntranslates the sketch template’s generic DSPprimitive inter-\nface into an UltraScale+–specific DSP48E2 using the Ultra-\nScale+ architecture description. The generic DSPmodule is\nan instance of a primitive interface: a Lakeroad-introduced\nabstraction that captures the similarities between primitives\nacross diverse FPGA architectures. For example, Lakeroad’s\nDSP primitive interface captures the facts that DSPs on all\nFPGA platforms generally have two to four data inputs (cap-\ntured by A–D) and a clock input (captured by clk). To convert\nthe sketch template’s DSP primitive interface instance into\na DSP48E2, Lakeroad utilizes the Xilinx UltraScale+ archi-\ntecture description, which the designer has pointed to with\n4Though this example is presented in a Verilog-like language, Lakeroad’s\nsketches are actually encoded in a Racket DSL that resembles structural\nVerilog.the--arch-desc xilinx-ultrascale-plus.yml flag. An\narchitecture description specifies how Lakeroad’s various\nprimitive interfaces are implemented for a given architecture.\nThe following simplified snippet of the UltraScale+ archi-\ntecture description, for example, tells Lakeroad that, when\ngenerating a sketch for UltraScale+, instances of the DSP\nprimitive interface should be implemented with a DSP48E2:\n-interface : {name : DSP, params : { out-width : 48, a-width : 30, ...}}\nholes : [?ACASCREG, ?ADREG, ?ALUMODEREG, ?AREG, ...]\nimplementation :\nmodule : DSP48E2\nports : [{ name : A, bitwidth : 30, value : A }, ...]\nparameters : [{ name : ACASCREG, value : ?ACASCREG }, ...]\noutputs : { O: P }\nThus, while converting dsp_sketch_template intosketch ,\nLakeroad reads this architecture description and converts\nthe single DSP instance into a DSP48E2, filling the ports and\nparameters with the concrete values and holes contained in\nthe architecture description. Architecture descriptions are\nusually short (100-400 LoC) and written only once per FPGA\narchitecture; Lakeroad already contains such descriptions\nfor Xilinx UltraScale+, Lattice ECP5, Intel Cyclone 10 LP, and\nthe open-source FPGA SOFA [43].\nTo generate a sketch, Lakeroad takes an architecture-\nindependent sketch template and specializes it using an ar-\nchitecture description. Once the sketch is ready, the designer\ncan move on to synthesis.\nStep 2: Program Synthesis. The next step fills in the holes\nto generate a complete, correct hardware design, which is\ndone automatically using a technique called program syn-\nthesis .Program synthesis is the process of using automated\nreasoning tools (like SMT solvers) to generate correct pro-\ngrams by encoding program generation as a constraint solv-\ning problem. In our add_mul_and example, Lakeroad, aided\nby Rosette [45, 46], generates a query like the following:5\n∃ACASCREG ,ADREG , ... .∀inputs .\nadd_mul_and(inputs)=sketch(inputs ,ACASCREG ,ADREG , ...)\nThe query asks: are there concrete values for ACASCREG ,\nADREG , etc., that will make our sketch’s behavior equiva-\nlent to the input design’s behavior on all inputs? If the solver\nfinds such values, Lakeroad can use them to fill the holes\nin the sketch and produce a compiled design. However, if\nLakeroad tries to pass the preceding formula to an SMT\nsolver, the solver will throw an error since the query is not\nexpressed at a level it understands, viz., as equalities between\nbitvector expressions, using simple Boolean or arithmetic\noperations. While it is conceivable that add_mul_and could\nbe converted to a bitvector expression since its core compu-\ntation is already expressed as (a+b)*c&d , it is unclear how\nto express sketch as an expression over bitvectors. In par-\nticular, Lakeroad must express bitvector-level semantics for\nXilinx’s DSP48E2 primitive.\nTo generate bitvector-level semantics for complex FPGA\nprimitives, Lakeroad introduces the concept of semantics\n5We formalize this synthesis query and explain it precisely in Section 3.\n5ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\nextraction . Rather than requiring manual effort to encode\nthe semantics of the underlying hardware, which is notori-\nously difficult even for experts [ 10], Lakeroad’s key insight\nis that these challenges can be avoided altogether by ex-\ntracting low-level semantics directly from vendor-supplied\nsimulation and verification models. Lakeroad builds on in-\nternal passes in Yosys [ 50] to automatically extract solver-\nready semantics from these vendor-provided HDL models,\nwhich we detail in Section 4.4. For the add_mul_and example,\nthe DSP48E2’s semantics have already been imported into\nLakeroad. Semantics need to be imported only when adding\nsupport for a new architecture, i.e., about as infrequently\nas writing a new architecture description. In most cases,\nLakeroad users can rely on already-imported semantics.\nWith the sketch generated and the DSP48E2’s semantics\nimported, program synthesis can begin. Lakeroad utilizes\nRosette to drive program synthesis, as detailed in Section 3. In\nour example, Rosette returns a configuration for the DSP48E2.\nThe last step, then, is to convert the compiled design to\nVerilog.\nStep 3: Compilation to Verilog. Compiling Lakeroad’s\ninternal representation into Verilog is a purely one-to-one\nsyntactic mapping; no optimizations are done at this stage,\nreducing the likelihood that bugs could be inserted. In our\nexample, the final Verilog produced results in the add_mul_-\nand_impl we saw at the start of Section 2.2.\nIn summary. Lakeroad delivered an implementation of\nthe designer’s add_mul_and module, improving upon both\nstate-of-the-art compilers and manual approaches in multi-\nple ways. Lakeroad’s implementation is significantly more\nresource-efficient than the state-of-the-art compiler’s—one\nDSP versus one DSP, 32 registers, and 16 LUTs. Lakeroad\ndelivered its implementation in mere seconds, compared to\nthe hours to days of work that manually instantiating a DSP\nmight take. Lastly, Lakeroad’s implementation is formally\nguaranteed to be correct. Meanwhile, Lakeroad did all of\nthis while requiring no input from the user other than the\nVerilog to be compiled.\n3 Formalization\nWe now formalize Lakeroad with functions 𝑓lrand𝑓∗\nlr, and\nuse these models to argue for the correctness and partial com-\npleteness of Lakeroad. We first define 𝑓lr(Section 3.1) and\nthen motivate and define the language ℒlr, specify its syn-\ntax and semantics, and define behavioral ( ℒbeh), structural\n(ℒstruct ), and sketch ( ℒsketch ) sublanguages (Section 3.2).\nWe next explain the underlying queries Lakeroad uses to\nsynthesize hardware programs that meet the desired speci-\nfication (Section 3.3). We demonstrate the correctness and\npartial completeness of 𝑓lr, enumerate our Trusted Comput-\ning Base (Section 3.4) and extend 𝑓lrto𝑓∗\nlr, which ensures\nthe generated program’s semantics matches the design overmultiple timesteps (Section 3.5). Finally, we highlight poten-\ntial future applications that could be built on this section’s\nformalization (Section 3.6).\n3.1 The Lakeroad Function 𝑓lr\nWe model the execution of Lakeroad with the partial function\n𝑓lr:Sketch×ℒbeh×Time ⇀ℒstruct,\nwhere𝑓lr(Ψ,𝑑,𝑡)invokes Rosette to synthesize a 𝑡-cycle\nimplementation of behavioral design 𝑑using sketch Ψto\nguide the search, where a 𝑡-cycle implementation of 𝑑is a\nprogram that is equivalent to 𝑑at clock cycle 𝑡. By not requir-\ning program equivalence before clock cycle 𝑡we allow the\nsynthesized program’s semantics to differ from the design\nduring an initialization period (e.g., as the pipeline is being\nfilled). To get guarantees beyond a single point in time 𝑡, we\ngeneralize𝑓lrto𝑓∗\nlr, which synthesizes a program that is\nequivalent to the design from time 𝑡to𝑡+𝑛. We formalize a\nsketch Ψ∈Sketch as a tuple(𝜓,ℎ), where𝜓is a program\ninℒsketch andℎis a map from the holes in 𝜓to a finite set\nof valid hole-free nodes in ℒstruct that can be used to fill\nthe mapped hole. This mapping ℎis handled implicitly by\nRosette’s choose andhole constructs and need not be explic-\nitly specified by the sketch writer. We write 𝑓lr(Ψ,𝑑,𝑡)=𝑝\nto indicate that synthesis succeeded and produced Lakeroad\nprogram𝑝. However, it is possible that sketch Ψcannot im-\nplement𝑑, in which case the synthesis fails (i.e., returns\nUNSAT) and 𝑓lrdoes not return anything. Design 𝑑belongs\ntoℒlr’s behavioral fragment, ℒbeh(see Section 3.2). When\n𝑡=0,𝑓lrsynthesizes a combinational design ; when𝑡>0,𝑓lr\nsynthesizes a sequential design over𝑡clock cycles. The rest\nof this section considers sequential design synthesis since\nits combinational counterpart is a special case covered by\nour general approach.\n3.2 Defining ℒlr\nLakeroad uses the ℒlrlanguage to translate behavioral HDL\nprograms to structural, hardware-specific HDL programs.\nTo facilitate this translation, we designed ℒlrto satisfy the\nfollowing properties:\nP1.Easy translation to/from HDLs: we must be able to trans-\nlate designs from a behavioral HDL to ℒlrand trans-\nlate synthesized implementations to a structural HDL.\nP2.Support parallel stateful execution: FPGA designs con-\nsist of potentially stateful elements executing in paral-\nlel.ℒlrmust allow unambiguous parallel execution.\nP3.Support graph-like program structures: An FPGA com-\nponent’s outputs can be wired to multiple other compo-\nnents, including back to itself. This means that FPGA\nprograms can form arbitrary graphs, and ℒlrmust be\nable to express this.\nP4.Support for sequential designs: ℒlrmust handle designs\nthat run over multiple clock cycles.\n6FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nProgF⟨Id,⟨Id,Node⟩∗⟩\nNodeFBV𝑏|Var𝑥\n|OP𝑜𝑝Id*\n|RegId(BV𝑏)\n|Prim binds Prog\n|■𝑥Id 𝑖𝑑∈N\nBitvectors 𝑏∈BV\nVariables 𝑥∈LegalVarNames\nOperators 𝑜𝑝∈OP𝑏𝑣∪OP𝑤\nbinds 𝑏𝑠∈(Variables ⇀Id)\nWire op OP𝑤={concat ,extract , . . .}\nNon-wire op OP𝑏𝑣={+,−,×, . . .}\nFigure 3. Syntax of ℒlr.■𝑥is a syntactic hole, labeled with variable 𝑥.𝐴⇀𝐵denotes the set of partial functions from 𝐴to𝐵.\nP5.Support for different architectures: ℒlrmust handle\nFPGA components from different architectures.\nWe describe how ℒlrsatisfies P1-P5 when we define its\nsyntax and semantics in the following subsections.\n3.2.1 ℒlr’s Syntax. Figure 3 shows the ℒlrsyntax. An\nℒlrprogram Prog consists of a root node ID and a graph of\nnodes, each of which is referred to by its ID. A node can be a\nconstant bitvector, input variable, combinational (pure) oper-\nator, sequential (stateful) register, primitive, or hole. Given a\nprogram𝑝=(𝑟,⟨𝑖𝑑1,𝑛𝑜𝑑𝑒 1⟩...⟨𝑖𝑑𝑛,𝑛𝑜𝑑𝑒 𝑛⟩), we use the no-\ntation𝑝.𝑟𝑜𝑜𝑡 =𝑟,𝑝.𝑖𝑑𝑠 ={𝑖𝑑1,...,𝑖𝑑 𝑛}, and𝑝[𝑖𝑑𝑖]=𝑛𝑜𝑑𝑒 𝑖.\nWe define the free variables of a program 𝑝.𝑓𝑣 ={𝑥𝑖}as\nthe set of variable names occurring in 𝑝’s nodes of the form\n(Var𝑥𝑖).6Finally, we use the notation 𝑝.𝑎𝑙𝑙 _𝑖𝑑𝑠for𝑝.𝑖𝑑𝑠 to-\ngether with 𝑝′.𝑎𝑙𝑙_𝑖𝑑𝑠of any subprogram 𝑝′of𝑝(𝑝′is a\nsubprogram of 𝑝if∃𝑗,𝑛𝑜𝑑𝑒 𝑗=Prim𝑏𝑠𝑝′).\nGiven a node 𝑛, we specify its inputs with the following\nfunction:\ninputs(BV𝑏)={},\ninputs(Var𝑥)={},\ninputs(OP𝑜𝑝𝑖 1...𝑖𝑛)={𝑖1, ...,𝑖 𝑛}\ninputs(Reg𝑖𝑏𝑖𝑛𝑖𝑡)={𝑖}\ninputs(Prim𝑏𝑠𝑝′)={𝑏𝑠[𝑥]|𝑥∈domain(𝑏𝑠)}\nNote that we use 𝐴⇀𝐵to denote the set of partial functions\nfrom𝐴to𝐵; given𝑏𝑠∈𝐴⇀𝐵, we write domain(𝑏𝑠)to\ndenote the set of 𝑥∈𝐴s.t.𝑏𝑠[𝑥]is defined.\nA program𝑝is well-formed if and only if all the following\nconditions hold:\nW1.𝑝.𝑟𝑜𝑜𝑡∈𝑝.𝑖𝑑𝑠 ;\nW2. All ids are unique and distinct. (i.e. for any sub-program\n𝑝′,𝑝.𝑖𝑑𝑠 and𝑝′.𝑎𝑙𝑙_𝑖𝑑𝑠are disjoint, and for any two\nsub-programs 𝑝′and𝑝′′,𝑝′.𝑎𝑙𝑙_𝑖𝑑𝑠is disjoint from\n𝑝′′.𝑎𝑙𝑙_𝑖𝑑𝑠.)\nW3. The inputs of all nodes in 𝑝are ids of other nodes in\n𝑝:∀𝑖𝑑∈𝑝.𝑖𝑑𝑠 , inputs(𝑝[𝑖𝑑])⊆𝑝.𝑖𝑑𝑠 ;\nW4. All primitive nodes contain well-formed programs;\nW5. All primitive nodes bind exactly their free variables;\ni.e., for Prim𝑏𝑠𝑝′, domain(𝑏𝑠)=𝑝′.𝑓𝑣; and\n6Note that this does not include variables of sub-programs occurring recur-\nsively inside of Prim nodes.W6. Program𝑝is free of combinational loops (formalized\nbelow in Property 1).\nProperty 1 (Free of Combinational Loops) .Formally, a pro-\ngram𝑝is free of combinational loops if there exists a function\n𝑤:𝑝.𝑎𝑙𝑙 _𝑖𝑑𝑠→N, that satisfies the following properties\n(collectively “monotonicity”):\n1. If𝑝[𝑖𝑑]=Reg_ _, then𝑤(𝑖𝑑)=0;\n2. If𝑝[𝑖𝑑]=Prim𝑏𝑠𝑝′, then𝑤(𝑖𝑑)>𝑤(𝑝′.𝑟𝑜𝑜𝑡);\n3. if𝑝[𝑖𝑑]=Prim𝑏𝑠𝑝′and𝑝′[𝑖𝑑′]=𝑉𝑎𝑟 𝑥 ,\nthen𝑤(𝑖𝑑′)>𝑤(𝑏𝑠[𝑥]); and\n4. Otherwise (e.g., 𝑝[𝑖𝑑]=OP𝑜𝑝𝑖𝑑𝑠∗),\nif𝑖𝑑′∈inputs(𝑝[𝑖𝑑]), then𝑤(𝑖𝑑)>𝑤(𝑖𝑑′).\nThe function 𝑤acts as a witness to the absence of combi-\nnational loops because it is impossible to define a strictly\nmonotonic function without acyclicity. We consider only\nwell-formed ℒlrprograms.\nBV,Var, and OPnodes encode bitvectors, variables, and\noperators.\nReg𝑖𝑑𝑎𝑡𝑎𝑏𝑖𝑛𝑖𝑡nodes let ℒlrimplement sequential designs\n(P4).𝑖𝑑𝑎𝑡𝑎 is the register’s data input, which updates the\nstored value at the positive edge of each clock cycle, and\n𝑏𝑖𝑛𝑖𝑡is the register’s initialization value.\nPrim𝑏𝑠𝑝 nodes let ℒlrprograms use hardware-specific\ncomponents from different architectures (P5). The 𝑏𝑠com-\nponent is a variable map , mapping Vars to input Ids. The𝑝\ncomponent is an ℒlrprogram that defines the semantics of\nthe hardware primitive. A Prim node also carries some meta-\ndata used during compilation to a structural HDL, which we\nomit for clarity.\nℒbehis the concrete behavioral fragment of ℒlrused for\nwriting specifications; it is formed by excluding Prim nodes\nand holes from ℒlr.\nℒstruct is the concrete structural fragment of ℒlrused for\nlowering ℒlrto structural HDLs; it is formed by excluding\nRegnodes, OPnodes, and holes from ℒlr, with the following\nexception: the 𝑝term in Prim𝑏𝑠𝑝 must always be from the\nℒbehsince it is used to specify the semantics of the Prim node\nto the synthesis engine. The behavioral node 𝑝is not used\nduring compilation to HDL, and this behavioral expression\ndoes not propagate to the structural HDL output.\n7ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\nTime𝑡∈N Env𝑒∈(Var⇀Time→BV)\nInterp :Prog→Env→Time→Node→BV\nInterp𝑝𝑒𝑡(BV𝑏)=𝑏\nInterp𝑝𝑒𝑡(Var𝑥)=𝑒𝑥𝑡\nInterp𝑝𝑒0(Reg_𝑖𝑛𝑖𝑡)=𝑖𝑛𝑖𝑡\nInterp𝑝𝑒(𝑡+1)(Reg𝑖𝑑_)=Interp𝑝𝑒𝑡 𝑝[𝑖𝑑]\nInterp𝑝𝑒𝑡(OP op𝑖𝑑𝑠)=J𝑜𝑝K(map(𝜆𝑖𝑑 . Interp𝑝𝑒𝑡 𝑝[𝑖𝑑])𝑖𝑑𝑠)\nInterp𝑝𝑒𝑡(Prim𝑏𝑠𝑝′)=\nlet𝑒′=𝜆𝑥,𝑡′.Interp𝑝𝑒𝑡′(𝑝[𝑏𝑠𝑥])in\nInterp𝑝′𝑒′𝑡 𝑝′[𝑝′.𝑟𝑜𝑜𝑡]\nFigure 4. Lakeroad’s semantics as pseudocode.\nℒsketch is another sublanguage of ℒlrthat is ℒstruct but\nalso including holes. Let 𝑠be a program in ℒsketch with holes\n■𝑥1,...,■𝑥𝑘. These holes can be filled with nodes𝑛1,...,𝑛 𝑘\ninℒstruct by replacing each hole ■𝑥𝑖with its corresponding\nnode𝑛𝑖to obtain a complete ℒstruct program, denoted by\n𝑠[■𝑥1↦→𝑛1,...].\nThe simplicity of this syntax makes translating to and\nfrom HDLs straightforward (P1). Section 4 describes how\nLakeroad implements the translations to and from HDLs.\n3.2.2 ℒlr’s Semantics. Before discussing the formal se-\nmantics of ℒlr, we present key definitions. We assume a\nbitvector type and, for simplicity, we elide bitvector widths.\nWe represent time as a natural number. A stream is a func-\ntion from Time to bitvectors. An environment is a map from\nvariable names to streams.\nWe give the semantics for ℒlras an interpreter in Figure 4.\nWe define the function Interp to interpret a program 𝑝in\nenvironment 𝑒at time𝑡and node𝑛. We do not define seman-\ntics for holes, as they are intended to be replaced by other\nconstructs with well-defined semantics.\nMost of the rules are straightforward. A bitvector BV𝑏\nevaluates to its backing bitvector value 𝑏. A variable node\nVar𝑥in an environment 𝑒at time𝑡evaluates to the value\nreturned by the stream associated with 𝑥in𝑒at time𝑡; using\nfunction notation, this is denoted by 𝑒𝑥𝑡 . A𝑘-ary operator\nnode OP𝑜𝑝𝑖 1...𝑖 𝑘recursively interprets each operand in\nthe current environment at the current time and then ap-\nplies𝑜𝑝’s semantics, denoted J𝑜𝑝K, to the resulting values. A\nregister Reg𝑖𝑑𝑏 𝑖𝑛𝑖𝑡has two cases depending on the current\ntime: at time 𝑡=0, a register evaluates to its initial bitvector\nvalue𝑏𝑖𝑛𝑖𝑡; at nonzero times 𝑡+1, a register evaluates to the\nvalue produced by the input 𝑖at the previous timestep𝑡. A\nprimitive Prim𝑏𝑠𝑝′in environment 𝑒at time𝑡is evaluated\nby interpreting the program 𝑝′under the fresh environment\n𝑒′formed by the binding map 𝑏𝑠.3.3 Program Synthesis\n𝑓lrperforms sketch-based program synthesis [ 41]. Opera-\ntionally, we implement the Interp function from Figure 4\nin Rosette, a solver-aided host language [ 46]. Let sketch\nΨ=(𝜓,ℎ) ∈Sketch , where𝜓∈ℒsketch has holes ■𝑥𝑖\nandℎmaps𝜓’s holes to the set of structural nodes that can\nlegally fill the mapped hole. Given a design 𝑑, we query\nRosette if there are nodes 𝑛1,𝑛2,...𝑛 𝑘such that𝑛𝑖∈ℎ[■𝑥𝑖]\nand𝑝=Ψ[■𝑥1↦→𝑛1,...]is well-formed and equivalent\nto𝑑(i.e., we ask Rosette to fill each hole with a node asso-\nciated with the node in ℎ). Program equivalence between\nwell-formed programs 𝑝and𝑑at time𝑡, written𝑝\u001b𝑡𝑑, is\ndefined as\n𝑝.𝑓𝑣=𝑑.𝑓𝑣∧\n∀𝑒𝑠.𝑡. domain(𝑒)=𝑝.𝑓𝑣,\nInterp𝑝𝑒𝑡 𝑝.𝑟𝑜𝑜𝑡 =Interp𝑑𝑒𝑡𝑑.𝑟𝑜𝑜𝑡.\nIn Section 3.5, we use bounded model checking to extend\n𝑓lr’s guarantees beyond the single timestep at clock cycle 𝑡.\n3.4 Correctness and Completeness of 𝑓lr\nRecall that the synthesis function 𝑓lris partial. We say that\n𝑓lriscorrect if it returns a program 𝑓lr(Ψ,𝑑,𝑡)=𝑝where\n𝑝is a well-formed completion of Ψ=(𝜓,ℎ), meaning𝑝=\nΨ[■𝑥1↦→𝑛1,...]such that𝑛𝑖∈ℎ[■𝑖]for all𝑖and𝑝\u001b𝑡𝑑.\nFurthermore, we say that 𝑓lrissketch-complete if𝑓lr(Ψ,𝑑,𝑡)\nis defined whenever there exists a well-formed completion 𝑝\nofΨsuch that𝑝\u001b𝑡𝑑. That is, synthesis is correct if it never\nreturns an erroneous result and sketch-complete if it returns\na correct result whenever one exists.\nWe have implemented 𝑓lrwith Rosette (see Section 3.3),\nwhich guarantees our system is correct and complete under\nthe following assumptions:\n1. Correctness of Rosette and underlying SMT solvers;\n2. That our encoding of Lakeroad is bug-free;\n3.That the lowering of Interp to SMT formulas by Rosette\nalways terminates. This is possible when partial eval-\nuation of Interp on arguments 𝑝,𝑡and𝑛terminates\n(independently of the value of 𝑒).\nLemma 3.1. Let𝑝be a well-formed program, 𝑒an environ-\nment,𝑡aTime , and𝑛be a node belonging to 𝑝. Then Interp is\nprimitive recursive (i.e. terminates) in the arguments 𝑝,𝑡, and𝑛.\nProof of Lemma 3.1. Recall that a function 𝑓(𝑥,𝑦,𝑧)is primi-\ntive recursive in arguments 𝑥and𝑦(under a lexicographic or-\ndering) if in the definition of 𝑓every recursive call 𝑓(𝑥′,𝑦′,𝑧′)\nis made with values (𝑥′,𝑦′)such that𝑥′<𝑥or𝑥′=𝑥∧𝑦′<\n𝑦. If𝑥and𝑦are drawn from the natural numbers (or an-\nother well-ordered set), then the recursion is guaranteed to\nterminate.\nUnder what order is Interp primitive recursive? Because\nour program is well-formed, it must be free of combinational\nloops (see Property 1). Formally, this means we have an\n8FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nacyclicity witness function 𝑤:𝑝.𝑎𝑙𝑙 _𝑖𝑑𝑠→Nthat mono-\ntonically increases in the direction of dataflow in our circuit.\nEach node𝑛argument passed to Interp has an Idthat is\nunique and distinct from the Ids used in𝑝or any of𝑝’s sub-\nprograms ( W2); we denote this Idas𝑖𝑑𝑛. We can associate\neach𝑛argument to a recursive call of Interp with a number\n𝑤(𝑖𝑑𝑛). We claim that Interp is primitive recursive under\nthe lexicographic ordering on (𝑡,𝑤(𝑖𝑑𝑛)).\nTo prove this claim we need to demonstrate that if Interp\nwith time and node arguments 𝑡′and𝑛′makes a recursive\ncall to Interp with time and node arguments 𝑡′′and𝑛′′, then\nthe following condition holds:\n𝑡′′<𝑡′∨\u0000𝑡′′=𝑡′∧𝑤(𝑖𝑑𝑛′′)<𝑤(𝑖𝑑𝑛′)\u0001. (1)\nTo do this it suffices to examine each case of Interp ’s defini-\ntion.\nWhen𝑛′is aBVconstant, Interp makes no recursive calls,\nand the condition in Equation (1) holds vacuously.\nWhen𝑛′is aRegnode Interp either terminates (when\n𝑡′=0) or makes a recursive call with time value 𝑡′′=𝑡′−1,\nmaintaining the condition in Equation (1).\nWhen𝑛′is an operator node, Interp recursively interprets\nthe operands with time arguments 𝑡′′=𝑡′. However, each\noperand’s id 𝑖𝑑′′belongs to inputs(𝑛′), and, by Property 1,\n𝑤(𝑖𝑑𝑛′)>𝑤(𝑖𝑑′′), so our condition holds.\nThis leaves us with the less obvious cases in which 𝑛′\nis either a Prim orVar, which work together in tandem.\nWhen𝑛′=Prim𝑏𝑠𝑝′,Interp makes a recursive call with\nnode argument 𝑝′.𝑟𝑜𝑜𝑡 and time argument 𝑡. By Property 1,\n𝑤(𝑝′.𝑟𝑜𝑜𝑡)<𝑤(𝑖𝑑𝑛′), and the condition in Equation (1)\nholds. Interp also defines a new environment for execution\nof𝑝′via𝜆-abstraction, and this in turn will recursively in-\nvoke Interp . These environments are only invoked by the\nrule for variables, which we handle presently.\nWhen𝑛′=Var𝑥, the environment is invoked on variable\n𝑥. Here, there are two possible cases. First, we are interpret-\ning the top-level program 𝑝. As this is the initial, top-level en-\nvironment, there is no further recursion. Second, we are inter-\npreting a sub-program 𝑝′and𝑒′𝑥𝑡=Interp𝑝𝑒𝑡(𝑝[𝑏𝑠𝑥])\nis actually a recursive call into the program 𝑝one level up,\nwith its environment 𝑒. In this latter case, note that 𝑤is\ndefined such that 𝑤(𝑖𝑑𝑝[𝑏𝑠 𝑥])=𝑤(𝑏𝑠𝑥)<𝑤(𝑖𝑑Var𝑥)(item\n3 of Property 1), satisfying our property. All cases are com-\nplete. □\nFrom this, we conclude that all possible substitutions for\nΨare attempted, and 𝑓lris sketch-complete.\nTrusted Computing Base. The trusted computing base\n(TCB) of a system is the set of components it assumes to be\ncorrect [ 29]. A bug anywhere in the TCB could cause the\nguarantees made by that system to be violated. Lakeroad’s\nTCB includes: Rosette and the underlying SAT/SMT solvers\nthat Rosette queries (Bitwuzla, cvc5, Yices2, and STP); theinternal Yosys passes Lakeroad uses to extract primitive se-\nmantics and translate design specifications from behavioral\nVerilog into ℒbeh; the semantics for ℒlr, which we assume\nconservatively models non-cyclic (DAG) designs; our code\nto translate from the ℒstruct to structural Verilog; and the\nvendor-provided Verilog simulation models for FPGA primi-\ntives. Each TCB component has also been thoroughly tested,\nas described in Section 5. Importantly, sketches and sketch\ngeneration are notin Lakeroad’s TCB: even if there were a\nbug in Lakeroad’s sketch-related components, it would not\nviolate Lakeroad’s correctness guarantees.\n3.5 Multiple Clock Cycle Guarantees with 𝑓∗\nlr\nThe preceding completeness and correctness properties for\n𝑓lrguarantee that running the synthesized program 𝑝and\nthe design𝑑for𝑡clock cycles produces the same output. To\nextend this guarantee, Lakeroad supports a form of bounded\nmodel checking, where synthesis ensures that 𝑝is semanti-\ncally equivalent to 𝑑for𝑐additional clock cycles starting at\ntime𝑡. We formalize this with the function 𝑓∗\nlr, which takes\na sketch Ψ, a behavioral design 𝑑, a number of clock cycles\n𝑡, and a model checking time bound 𝑐≥0and returns an\nimplementation 𝑝∈ℒstruct that is equivalent to 𝑑at time\nsteps𝑡,𝑡+1,...,𝑡+𝑐.\nOur correctness and completeness guarantees are similar\nto those for 𝑓lr:\n𝑝.𝑓𝑣=𝑑.𝑓𝑣∧\n∀𝑒𝑠.𝑡. domain(𝑒)=𝑝.𝑓𝑣,\n𝑖=𝑡+𝑐Û\n𝑖=𝑡Interp𝑝𝑒𝑖𝑝.𝑟𝑜𝑜𝑡 =Interp𝑑𝑒𝑖𝑑.𝑟𝑜𝑜𝑡.\n3.6 Beyond Lakeroad\nℒlr, its semantics, and the synthesis approach we describe\nhere are useful for applying program synthesis to other hard-\nware design problems. For example, the synthesis problem\ndetailed above could be “flipped” to decompile structural\ndesigns back to higher-level behavorial designs, i.e., synthe-\nsizing from ℒstruct to an expression in ℒbeh. Such decom-\npilation has seen recent interest for recovering equivalent\nbut faster-to-simulate models and for porting models across\ndifferent architectures [ 40]. As another example, the syn-\nthesis approach could be adapted to help port designs by\nsynthesizing expressions in ℒstruct that use one set of prim-\nitives on one architecture from other designs in ℒstruct that\nuse a different set of primitives from a different architecture.\nThus, the formalization in this section transcends the partic-\nular challenges of FPGA technology and provides a reusable\nfoundation for exploring a much broader range of hardware\ndesign challenges from a program synthesis perspective.\n9ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\nimplementations :\n-interface : { name : LUT, num_inputs : 4 }\ninternal_data : { sram : 16 }\nmodules :\n-module_name : frac_lut4\nfilepath : SOFA/frac_lut4.v\nports :\n- { name : in, direction : in, width : 4,\nvalue : (concat I3 I2 I1 I0) }\n- { name : mode, direction : in,\nwidth : 1, value : (bv 0 1) }\n- { name : lut4_out, direction : out,\nwidth : 1 }\nparameters : [{ name : sram, value : sram }]\noutputs : { O: lut4_out }\nFigure 5. SOFA architecture description.\n4 Implementation\nLakeroad is composed of approximately 13K lines of Racket\nplus approximately 58K lines of Racket automatically gener-\nated from vendor-supplied Verilog. Vendor-supplied Verilog\nwas obtained from Lattice Diamond, Intel Quartus, and Xil-\ninx Vivado sources. We used Vivado version v2023.1, Quar-\ntus 22.1std.1 Build 917 02/14/2023 SC Lite Edition, Diamond\nversion 3.12, Yosys version 0.36+42 (commit 70d3531 ), the\ncvc5 [ 8] and Yices2 [ 18,19] solvers included in the 2023-08-\n06 release of oss-cad-suite from YosysHQ, the Bitwuzla\nsolver at commit b655bc0 [32], the STP solver at commit\n0510509a , Racket version 8.9 [ 20,21], and Rosette version\n4.1 [36].\n4.1 Primitive Interfaces\nAs described in Section 2, primitive interfaces describe ab-\nstract versions of common FPGA primitives, which allow\nsketch templates to be architecture-independent. To date,\nLakeroad declares primitive interfaces for 𝑛-input LUTs, 𝑤-\nwidth carry chains, 𝑛-input muxes, and DSPs with up to four\ndata inputs and one clock input. The next section includes\na concrete example of Lakeroad’s LUT4 primitive interface.\n4.2 Architecture Descriptions\nAs described in Section 2, architecture descriptions convey the\ninformation required to convert each instance of a primitive\ninterface into the corresponding architecture-specific mod-\nule, which occurs while converting sketch templates into\nsketches. The architecture description is the only additional\ninput that may be required from a user to support a new\narchitecture; it is a one-time effort that is reusable for any\ndesigns in an architecture. Architecture descriptions are sim-\nply lists (provided as YAML files) of the primitive interfaces\nthat an architecture implements, but, crucially, also include\narchitecture-specific port and parameter values in a map\ncalled internal_data . Values in this map become symbolic\nvalues solvable by the SMT solver. Additional constraints canalso be specified in the architecture description to rule out in-\nvalid configurations and minimize the solver’s search space.\nAs an example, Figure 5 shows the architecture descrip-\ntion for the SOFA [ 43] FPGA architecture. The description\ncontains a single primitive interface implementation, i.e.,\nLUT4. Lakeroad’s LUT4 primitive interface standardizes the\nnames of a LUT4’s inputs and outputs, naming the inputs I0\nthrough I3and the output O. The SOFA implementation of\nthe LUT4 primitive interface uses the SOFA-specific frac_-\nlut4 primitive. Primitive interface inputs I0through I3are\nmapped to the actual input port of the frac_lut4 , named\nin. Likewise, the frac_lut4 output lut4_out is mapped\nto the primitive interface output O. The internal_data\nfield declares sram , the LUT’s 16-bit internal memory, as\nan architecture-specific detail to be solved during synthesis.\nIf a sketch template uses a primitive interface not included\nin the architecture description (e.g., SOFA does not imple-\nment carries), Lakeroad may still be able to implement the\nprimitive interface based on primitive interfaces the architec-\nture does implement. To date, Lakeroad can implement any\nmux with LUTs, a larger LUT from smaller LUTs, a smaller\nLUT from a larger LUT, a carry from LUTs, and a smaller\nDSP from a larger DSP; it handles these conversions during\nsketch generation.\n4.3 Sketch Templates, Sketches, and Sketch\nGeneration\nAs described in Section 2, Lakeroad captures common FPGA\nimplementation patterns in reusable, architecture-independent\nsketch templates. Thus far, we have described only the rela-\ntively simple dspsketch template, which instantiates a DSP.\nAs a more complex example of capturing common FPGA im-\nplementation patterns, consider the bitwise-with-carry\nsketch template, which uses 𝑛LUTs and a carry chain to\nimplement designs such as addition or subtraction. As of\nthe paper’s publication date, Lakeroad provides 5 sketch\ntemplates: dsp,bitwise ,bitwise-with-carry ,compari-\nson(LUT- and carry-based arithmetic comparison), and mul-\ntiplication (LUT-based multiplication).\nThe process of converting sketch templates to sketches\nis implemented as described in Section 2 and Section 4.2.\nLakeroad iterates over every primitive interface instance\nin the sketch and replaces it with the concrete primitive in\naccordance with the architecture’s architecture description.\nIf the architecture description does not implement the re-\nquested primitive interface, Lakeroad checks whether it can\nimplement the primitive interface with other implemented\ninterfaces (e.g., implementing a smaller LUT with a larger\nLUT) and raises an error otherwise.\nSketch templates and sketches alike are written in a domain-\nspecific language (DSL) embedded into Rosette, whose im-\nplementation closely mirrors the syntax and semantics of\nℒlr. The only significant difference is that the interpreter\n10FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nimplementation does not use bitvector streams natively. In-\nstead, each invocation of the interpreter represents a sin-\ngle timestep, and all intermediate values from the previous\ntimestep are taken as input. Streams are then built up using\nmultiple invocations of the interpreter.\n4.4 Importing Semantics from Verilog Modules\nLakeroad uses Yosys [ 50] to convert Verilog modules into\nthebtor2 format [ 33] and then converts the resulting btor2\nto Rosette/Racket code.\nDue to the semantics of the Verilog language and the in-\nternal implementation of Yosys, extracting semantics from\nVerilog modules may require the following manual modifi-\ncations to accommodate semantics extraction and synthesis:\n•As Yosys converts parameters from variables to constant\nvalues immediately upon module import, module parame-\nters should be converted to ports to ensure they remain\nvariables (and thus solvable by the SMT solver). Note that\nnot all parameters can always be converted to ports, mean-\ning some parameters cannot be solved for.\n•Strings should be converted to bitvectors.\n•All registers should be initialized.\n•All instances of xandzvalues should be converted to\n2-state logic (0 or 1).\nNote that these caveats apply only to our prototype imple-\nmentation, not the general technique of semantics extraction\nfrom HDL. Once these manual modifications are made, the\nfollowing series of Yosys passes can be used to convert the\nVerilog into suitable btor2 :prep; flatten; pmuxtree;\nopt_muxtree; clk2fflogic; prep; write_btor .\nWe implement the translation from btor2 to Rosette bitvec-\ntor expressions as a 1:1 translation since both languages are\nsimply operations over bitvectors.\n4.5 Program Synthesis and Compilation to Verilog\nWe implement the synthesis procedure defined in Section 3.4\nwith Rosette. Multiple clock cycle guarantees, as described\nin Section 3.5, are implemented simply by making 𝑐+1total\nassertions, asserting the output of the input design and the\nsketch are equal after each of the 𝑐+1timesteps. We use a\nportfolio solving method, running Bitwuzla [ 31], cvc5 [ 8],\nYices2 [ 18,19], and STP [ 5] in parallel and using results from\nthe first solver to terminate. To produce Verilog, Lakeroad\ncompiles the program from its internal DSL to the JSON\nformat defined by Yosys using a straightforward translation\nand then uses Yosys to output Verilog.\n4.6 Integration with Other Tools\nThis paper describes Lakeroad as a standalone tool, but the\ncore Lakeroad implementation could be integrated directly\ninto existing tools. Though out of scope for this paper, we\nhave early, encouraging results integrating Lakeroad as a\nYosys pass that lets users tag modules with annotationssimilar to (and much richer than) Xilinx’s use_dsp anno-\ntation. We then map annotated modules to primitives using\nLakeroad, which let us easily apply Lakeroad to many frag-\nments within a larger design. We plan to more fully integrate\nLakeroad into Yosys in future work, which should radically\nimprove the completeness of Yosys’s DSP mapping ability,\nas shown in Figure 6.\n5 Evaluation\nWe now evaluate Lakeroad in terms of completeness and\nextensibility. In the following experiments, we target four\nFPGA architectures: Xilinx UltraScale+ , commonly used\nfor large, high-performance workloads; Lattice ECP5 , com-\nmonly used in low-power, low-cost scenarios; Intel Cyclone\n10 LP , an FPGA designed for low-cost, high-volume use\ncases, and SOFA [43], a recent, open-source FPGA devel-\noped by the research community. We compare Lakeroad to\nexisting technology mappers. For Xilinx Ultrascale+, Lat-\ntice ECP5, and Intel Cyclone 10 LP, we compare Lakeroad\nagainst both the open source toolchain Yosys [ 50] and the\nstate-of-the-art, proprietary, closed source toolchains for\neach architecture.7The experiments were conducted on a\nsystem running Ubuntu 20.04.3 with an AMD EPYC 7702P\n64-Core CPU. The resident set size of a single Lakeroad pro-\ncess did not exceed 300MB while running our evaluation.\nWe use the software versions listed in Section 4.\n5.1 Lakeroad Completeness\nThe reliance of many technology mappers, including state-\nof-the-art tools, on hand-written patterns leads them to fail\nwhen attempting to map many workloads that should be\nmapped to a single DSP. In particular, the process of partial\ndesign mapping (illustrated in Section 2) becomes a labo-\nrious endeavor because of this incompleteness: hardware\ndesigners hand-instantiate DSPs rather than rely on substan-\ndard automated tooling, repeating the work each time they\nidentify a potential opportunity to use a DSP. Lakeroad’s\ngreater mapping completeness significantly reduces the bur-\nden on hardware designers during partial design mapping\nand marks the first step in automated mapping for full de-\nsigns. We next evaluate how Lakeroad’s program synthesis\napproach enables it to achieve greater completeness for these\nprogram fragments.\nEvaluation Setup. We highlight three particularly com-\nplex DSPs for the Xilinx Ultrascale+, Lattice ECP5, and Intel\nCyclone 10 LP architectures: the Xilinx DSP48E2, Lattice\nALU54A/MULT18X18C (a single DSP composed of two prim-\nitives), and Intel cyclone10lp_mac_mult. SOFA provides no\nDSP, and is not included in this part of the evaluation. For\neach architecture’s DSP, we enumerate a large subset of the\n7Again, licensing restrictions prevent our naming the specific proprietary\ntools, but they are familiar, standard packages used by many hardware\ndesigners.\n11ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\ndesigns theoretically mappable to a single DSP according to\nits configuration manual. This microbenchmark set aims to\ncapture the real-world designs which hardware designers\nwould attempt to map to a platform’s DSP. For each architec-\nture, we compare Lakeroad to both the corresponding state-\nof-the-art toolchain for the architecture as well as to Yosys.\nFor Xilinx Ultrascale+, the DSP48E2 configuration manual\ndetails the structure of designs mappable to the primitive.\nOur designs for Xilinx include all permutations of the design\nform((𝑎±𝑏)∗𝑐)⊙𝑑, where⊙∈{ &,|,±,⊕}, as well as designs\nof the forms(𝑎∗𝑏)and((𝑎∗𝑏)±𝑐). We pipeline each of these\nworkloads from zero to three stages and use bitwidths from\n8 to 18 bits. For the DSP on Lattice, we similarly enumerate\nall designs of the form (𝑎∗𝑏)⊙𝑐, where⊙∈{ &,|,⊕,±}, and\nof the form(𝑎∗𝑏). For each of these designs, we use zero\nto two stages and bitwidths from 8 to 18 bits. This results in\n1320 microbenchmarks for Xilinx UltraScale+, 396 for Lattice\nECP5, and 66 for Intel Cyclone 10 LP. Though Lakeroad’s\noutput is correct by construction, we further validate its\noutput by simulating each Lakeroad-compiled design over\nthousands of consecutive cycles using Verilator.\nComparison to Existing Toolchains. As demonstrated\nin Figure 6 (top), Lakeroad maps 44×more designs than Yosys\nand2.1×more designs than the proprietary, state-of-the-art\ntoolchain on Xilinx Ultrascale+. On Lattice ECP5, Lakeroad\nmaps 6.0×more designs than Yosys and 3.6×more designs\nthan the proprietary, state-of-the-art toolchain. On Intel Cy-\nclone 10 LP, Lakeroad successfully maps all designs: 3×more\ndesigns than the proprietary, state-of-the-art toolchain for\nIntel. Yosys fails to map a single design on Intel. State-of-\nthe-art toolchains for all architectures fail to map more than\nhalf of the queried designs. Lakeroad times out on less than\n20% of designs.8Note that Lakeroad returns “UNSAT” on ap-\nproximately 260 designs on UltraScale+, i.e., Lakeroad claims\nthere is nopossible mapping to a DSP48E2 for the requested\nworkload. In all of these cases, both Xilinx SOTA and Yosys\nagree with Lakeroad and do not map the designs to a single\nDSP. We conclude that the set of designs we presented in\nEvaluation Setup must be overly broad; though the documen-\ntation implies that all of these designs are mappable to a\nsingle DSP, all three Xilinx synthesis tools surveyed indicate\nthat they are indeed not mappable.\nFor timing, we compared the mapping time for each of\nthe tools and report the results in Figure 6 (bottom). The\nwide ranges for Lakeroad show that solver time for differ-\nent program synthesis queries is highly variable. This is\nexplored more deeply in Figure 7, which shows that most\nsynthesis queries terminate quickly, with a long tail of slower\nqueries. Note that the state-of-the-art technology mapper for\n8We restricted Rosette synthesis time to 120 seconds, 40 seconds, and 20\nseconds for Xilinx, Lattice, and Intel respectively, and marked failure past\nthat (though bitvector synthesis problems are decidable).Ultrascale+ has a slow running time due to its long start-up\nprocess.\nRegarding which solvers in the portfolio were most useful,\nof all terminating (success or UNSAT) Lakeroad experiments,\nBitwuzla was the first to complete for 671 of them, STP for\n519, Yices2 for 464, and cvc5 for 64.\nLakeroad’s greater completeness directly translates into\nresource reduction. On average, for each microbenchmark,\nLakeroad uses 3.9 fewer LEs (logic elements: LUTs, muxes, or\ncarry chains) and 7.5 fewer registers than the Xilinx SOTA,\n7.2 fewer LEs/11.9 fewer registers than the Lattice SOTA, 8.2\nfewer LEs/14.3 fewer registers than the Intel SOTA, and 33.3\nfewer LEs/11.4 fewer registers than Yosys. In the real world,\nthe small modules captured by our microbenchmarks may be\nreused dozens if not hundreds of times across a large design.\nThus, the sizable resource reduction Lakeroad provides on\na single microbenchmark will be multiplied significantly for\nan entire design.\nDiscussion. Compared to Yosys, it is clear that Lakeroad\nprovides more complete support for programmable DSPs.\nHowever, Lakeroad’s greater completeness over Yosys is\nperhaps not surprising since Yosys is an open-source tool\nstill under active development. Part of the appeal of the\nYosys toolchain is the diversity of backends it can target;\nthese results show that, if incorporated into Yosys, Lakeroad\nwould further increase Yosys’s flexibility and generality. Per-\nhaps most surprising is that Lakeroad is more complete than\nspecialized proprietary toolchains. Even the UNSAT results\nLakeroad produces can be useful to designers since they\nindicate potential flaws in the documentation or vendor-\nprovided semantics. In the context of a larger synthesis tool,\nLakeroad would provide stronger guarantees for mapping\nmodules of larger designs.\n5.2 Lakeroad Extensibility and Expressiveness\nIn addition to being correct by construction (Section 3) and\nmore complete than existing FPGA technology mappers (Sec-\ntion 5.1), Lakeroad can also easily extend to new FPGA archi-\ntectures. Furthermore, automatic primitive semantics extrac-\ntion from vendor-provided HDL simulation models enables\nLakeroad to support diverse, highly configurable FPGA prim-\nitives.\nThe architecture descriptions vary in length from 20 to 240\nsource lines of code (SLoC). SOFA (20 SLoC) is the simplest,\nshown in full in Figure 5. The descriptions for Xilinx (185\nSLoC), Lattice (240 SLoC), and Intel (178 SLoC) are longer\nsince those FPGA architectures provide a wider range of\nconfigurable primitives.\nAs a point of comparison, the open-source Yosys toolchain,\nwhich has roughly 200 contributors on GitHub, provides\ntechnology mapping for Xilinx UltraScale+ across over a\ndozen complex Verilog, C++, and Python files (about 1300\n12FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nTool Median Time (s) Min / Max Time (s)\nXilinx\nLakeroad 14.99 2 .99 127 .70\nSOTA Xilinx 261.61 227 .82 598 .67\nYosys 14.97 6 .66 21 .10\nLattice\nLakeroad 9.49 6 .70 55 .23\nSOTA Lattice 2.32 0 .95 4 .52\nYosys 2.31 0 .90 4 .01\nIntel\nLakeroad 2.92 2 .12 4 .13\nSOTA Intel 38.73 19 .11 43 .49\nYosys 0.96 0 .48 1 .88\nFigure 6. Results of the completeness experiments described\nin Section 5.1, measuring the completeness of technology\nmapping tools for DSPs on Xilinx UltraScale+ and Lattice\nECP5, plus timing information. A single bar in the bar chart\ncommunicates, for a given FPGA architecture and technol-\nogy mapper, the proportion of the microbenchmarks that\nthe given technology mapper could map to a single DSP. In\nLakeroad’s case, experiments can either succeed (Lakeroad\nmaps the microbenchmark to a single DSP), timeout, or re-\nturn UNSAT. For the other tools, experiments can either\nsucceed or fail (i.e., the tool returns a mapping, but the map-\nping uses more than a single DSP). There are a total of 1320\nexperiments/microbenchmarks for Xilinx, 396 for Lattice,\nand 66 for Intel.\nlines of code). We cannot provide similar numbers for state-\nof-the-art proprietary tools, but a developer of one such\ntechnology mapper shared that extending their tool to sup-\nport new FPGA architectures was extremely difficult since it\n“spans millions of lines of low-level C. ” This is not surprising;\nYosys aims to target a variety of vendor architectures, while\nproprietary tools have teams of engineers to extract better\nmapping (evident by Yosys’ limitations in Section 5.1). By\ncontrast, Lakeroad supports diverse architectures and is easy\nto extend. Even if a user wants to target a completely new\narchitecture that Lakeroad does not support, architecture-\nindependent sketch templates allow reuse of previously im-\nplemented mapping strategies, and the user is only required\nFigure 7. Histograms of Lakeroad program synthesis run-\ntime for all terminating (success or UNSAT) Lakeroad ex-\nperiments described in Section 5.1, with timeout thresholds\nindicated with a vertical dotted red line.\nTable 1. FPGA primitives imported automatically by\nLakeroad from vendor-provided Verilog models, with num-\nber of source lines of code (excluding comments and empty\nlines) of the original Verilog models.\nFPGA Primitive Verilog SLoC\nXilinx Ultrascale+ LUT6 88\nCARRY8 23\nDSP48E2 896\nLattice ECP5 LUT2 5\nLUT4 7\nCCU2C 60\nALU54A 1642\nMULT18X18C 795\nIntel Cyclone 10 LP cyclone10lp_mac_mult 319\nSOFA frac_lut4 69\nto provide a few lines of high-level configuration for each\nprimitive in the architecture description.\nTable 1 further highlights Lakeroad’s expressiveness, i.e.,\nits ability to support a diverse range of configurable prim-\nitives by automatically extracting semantics from vendor-\nprovided HDL simulation models. Lakeroad can import the\nsemantics of large configurable primitives, such as the Ultra-\nScale+ DSP (896 lines of Verilog) or Lattice ECP5’s ALU and\nmultiplier units (1642 and 795 lines of Verilog, respectively).\nIt is difficult and error-prone to manually formalize the full\nsemantics for these primitives; partial support by ad hoc\nsearch procedures that rely on syntactic pattern matching\nleads to missing many mapping opportunities, as shown in\nSection 5.1.\n13ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\n6 Related Work\nTo the best of our knowledge, Lakeroad is the first work to\napply the technique of program synthesis to FPGA technol-\nogy mapping. Indeed, as noted by Sisco et al. [39], program\nsynthesis has seldom been applied in the domain of hardware\ndesign although its underlying formal methods techniques\nare frequently used for the formal verification of hardware\ndesigns rather than compilation, as in Bluespec SystemVer-\nilog [ 34], Kôika [ 12], and Kami [ 15]. Sisco et al. cite two\nexamples of works that use program synthesis for hardware\ndesign, Verisketch [ 7] and Sketchilog [ 9], both of which ap-\nply program synthesis to produce HDL implementations\nfrom high-level designs. Other works use program synthesis\nto generate software that runs on low-powered hardware,\nlike Chlorophyll [ 35], which targets extremely memory-\nconstrained power-efficient processors, Chipmunk [ 22], which\ntargets programmable network switches, and Diospyros [ 48],9\nwhich generates vectorized programs for standalone digi-\ntal signal processors (more powerful and general-purpose\ndevices than the DSP units in FPGAs). These works demon-\nstrate the utility of program synthesis for generating code\nthat handles specific wrinkles in hardware designs, as does\nthe use of program synthesis in Lakeroad to harness the\nprogrammability of FPGA DSPs.\nLakeroad is also related to past work in FPGA compilation\nand techmapping, much of which does not entreaty to sup-\nport programmable DSPs with as much generality. ODIN [ 26]\nand ODIN-II [ 25] are used in hard-block synthesis for FPGAs,\nwhich is the task of mapping portions of hardware designs to\nspecialized units ( hard blocks ) like multipliers. They operate\npurely over syntax (e.g., mapping *to a multiplier) and so\nare greatly limited in their ability to handle programmable\nDSPs. The ABC [ 14] logic synthesis tool is used to lower\nhardware designs into LUT and carry-chain configurations;\nit is related to Lakeroad in that it also uses constraint solvers\nto find configurations, though it is not general enough to\nhandle a wide variety of programmable DSPs, unlike the\nprogram synthesis techniques used in Lakeroad. Note also\nthat the use of configuration files in Lakeroad to abstract\naway details of the FPGA architecture was inspired by past\nwork in FPGA compilation, including OpenFPGA [ 42] and\nthe Verilog-to-Routing project (VTR) [ 38], both of which\nuse abstract architecture descriptions to facilitate portability\nacross designs, though these projects are limited in their sup-\nport for DSPs. Library-Parameterized Models [ 3,6] define\ngeneric interfaces for common primitives and are also similar\nto Lakeroad’s primitive interfaces, though they are limited\nin their ability to represent configurable units like DSPs.\n9Diospyros uses symbolic evaluation, which is related to program synthesis,\nto lift imperative programs for digital signal processors into a high-level\nmathematical representation that can then be used with the technique of\nequality saturation [ 44] to generate optimized code for the target devices.\nThis is also distinct from the program synthesis techniques referenced\nelsewhere in this paper.Virtual FPGA overlays [ 13,27,28] are another approach\nto improving the mapping of hardware designs to hardware.\nOverlays present a “virtual” FPGA architecture; each ac-\ntual architecture must then define a mapping from virtual\nto actual primitives. This required translation is similar to\nLakeroad’s requirement on users to implement primitive in-\nterfaces in an architecture description, though it requires\nmore user effort. The translation from virtual to actual archi-\ntecture often comes with a steep resource and performance\noverhead.\n7 Conclusion\nThis paper presents Lakeroad, a novel approach to FPGA\ntechnology mapping that leverages program synthesis tech-\nniques to provide stronger correctness and completeness\nguarantees than state-of-the-art tools. Because program syn-\nthesis tools can efficiently explore large search spaces, Lakeroad\ncan find mappings of hardware designs to FPGA DSPs in\nmore cases than state-of-the-art tools, often finding more\nefficient implementations in the process. With our tech-\nniques of semantics extraction from HDL and architecture-\nindependent sketch templates, users must expend little man-\nual effort to apply Lakeroad to a given FPGA architecture\nand extend it to handle further primitives. Moreover, our\nformalization of Lakeroad fosters greater confidence in its\ncorrectness. Lakeroad hence enables the extensible, efficient,\nand correct lowering of hardware designs to FPGAs, high-\nlighting the effectiveness of program synthesis for FPGA\ntechnology mapping.\nAcknowledgements\nThis work was funded by generous grants and awards from\nIntel, the U.S. Department of Energy (award number DE-\nSC0022081), and the NSF (grant numbers 1836724 and 1749570).\nWe would like to thank our anonymous reviewers for their\nconstructive feedback. Thank you to Jonathan Balkind for\nserving as our shepherd. Thank you to those who contributed\ncode to early versions of Lakeroad, including David Cao\nand Zihao Ye. Thank you to Jin Yang and his team at Intel.\nThank you to Daniel Petrisko, Scott Davidson, Rachit Nigam,\nand Adrian Sampson for sharing their deep knowledge of\nthe hardware design workflow. Thank you to Chandrakana\nNandi for her enthusiasm and unwavering support. Thank\nyou to Claire Xenia Wolf, Nina Engelhardt, Jannis Harder,\nand the YosysHQ team. Finally, thank you to the entire PLSE\nlab for their support and camaraderie.\nReferences\n[1]Can not correctly infer \"A*B+C\" to DSP48E2. https:\n//support.xilinx.com/s/question/0D54U00006AqPXFSA3/can-\nnot-correctly-infer-abc-to-dsp48e2?language=en_US . Accessed:\n2023-12-07.\n[2]DSP48E2 inference for convolution/multiplication of 8-bit operands.\nhttps://support.xilinx.com/s/question/0D52E00006hpnGVSAY/\n14FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\ndsp48e2-inference-for-convolutionmultiplication-of-8bit-\noperands?language=en_US . Accessed: 2023-12-07.\n[3] Eia-is-103 : Library of parameterized modules (lpm).\n[4]Inferring SIMD accumulator with Xilinx DSP48e2. https:\n//old.reddit.com/r/FPGA/comments/tr9vzn/inferring_simd_\naccumulator_with_xilinx_dsp48e2/ . Accessed: 2023-12-07.\n[5] The simple theorem prover.\n[6] Altera. Lpm quick reference guide.\n[7]Armaiti Ardeshiricham, Yoshiki Takashima, Sicun Gao, and Ryan Kast-\nner. Verisketch: Synthesizing secure hardware designs with timing-\nsensitive information flow properties. In Proceedings of the 2019 ACM\nSIGSAC Conference on Computer and Communications Security , CCS ’19,\npage 1623–1638, New York, NY, USA, 2019. Association for Computing\nMachinery.\n[8]Haniel Barbosa, Clark Barrett, Martin Brain, Gereon Kremer, Hanna\nLachnitt, Makai Mann, Abdalrhman Mohamed, Mudathir Mohamed,\nAina Niemetz, Andres Nötzli, Alex Ozdemir, Mathias Preiner, Andrew\nReynolds, Ying Sheng, Cesare Tinelli, and Yoni Zohar. cvc5: A Versatile\nand Industrial-Strength SMT Solver , pages 415–442. 01 2022.\n[9]Andrew Becker, David Novo, and Paolo Ienne. Sketchilog: Sketching\ncombinational circuits. In 2014 Design, Automation & Test in Europe\nConference & Exhibition (DATE) , pages 1–4, 2014.\n[10] Gilbert Louis Bernstein and Jonathan Ragan-Kelley. What are the se-\nmantics of hardware? In Workshop on Languages, Tools, and Techniques\nfor Accelerator Design (LATTE) , 2021.\n[11] James Bornholt and Emina Torlak. Synthesizing memory models from\nframework sketches and litmus tests. SIGPLAN Not. , 52(6):467–481,\njun 2017.\n[12] Thomas Bourgeat, Clément Pit-Claudel, and Adam Chlipala. The\nessence of bluespec: a core language for rule-based hardware design.\nInProceedings of the 41st ACM SIGPLAN Conference on Programming\nLanguage Design and Implementation , pages 243–257, 2020.\n[13] Alexander Brant and Guy GF Lemieux. Zuma: An open fpga over-\nlay architecture. In 2012 IEEE 20th international symposium on field-\nprogrammable custom computing machines , pages 93–96. IEEE, 2012.\n[14] Robert Brayton and Alan Mishchenko. Abc: An academic industrial-\nstrength verification tool. In International Conference on Computer\nAided Verification , pages 24–40. Springer, 2010.\n[15] Joonwon Choi, Muralidaran Vijayaraghavan, Benjamin Sherman,\nAdam Chlipala, and Arvind. Kami: A platform for high-level paramet-\nric hardware specification and its modular verification. Proc. ACM\nProgram. Lang. , 1(ICFP), August 2017.\n[16] Ross Daly, Caleb Donovick, Jackson Melchert, Rajsekhar Setaluri, Nes-\ntan Tsiskaridze Bullock, Priyanka Raina, Clark Barrett, and Pat Hanra-\nhan. Synthesizing instruction selection rewrite rules from rtl using\nsmt. In CONFERENCE ON FORMAL METHODS IN COMPUTER-AIDED\nDESIGN–FMCAD 2022 , page 139, 2022.\n[17] Leonardo Mendonça de Moura and Nikolaj Bjørner. Z3: An efficient\nsmt solver. In C. R. Ramakrishnan and Jakob Rehof, editors, Proceedings\nof the 14th International Conference on Tools and Algorithms for the\nConstruction and Analysis of Systems (TACAS 2008) , volume 4963 of\nLecture Notes in Computer Science , pages 337–340, Berlin, Heidelberg,\n2008. Springer.\n[18] Bruno Dutertre. Yices 2.2. In International Conference on Computer\nAided Verification , pages 737–744. Springer, 2014.\n[19] Bruno Dutertre and Leonardo De Moura. The yices smt solver.\n[20] Matthias Felleisen, Robert Bruce Findler, Matthew Flatt, Shriram Kr-\nishnamurthi, Eli Barzilay, Jay McCarthy, and Sam Tobin-Hochstadt. A\nprogrammable programming language. Communications of the ACM ,\n61(3):62–71, March 2018.\n[21] Matthew Flatt and PLT. Reference: Racket. Technical Report PLT-TR-\n2010-1, PLT Design Inc., 2010.\n[22] Xiangyu Gao, Taegyun Kim, Aatish Kishan Varma, Anirudh Sivaraman,\nand Srinivas Narayana. Autogenerating fast packet-processing codeusing program synthesis. In Proceedings of the 18th ACM Workshop\non Hot Topics in Networks , HotNets ’19, page 150–160, New York, NY,\nUSA, 2019. Association for Computing Machinery.\n[23] Sumit Gulwani, Alex Polozov, and Rishabh Singh. Program Synthesis ,\nvolume 4. NOW, August 2017.\n[24] Yann Herklotz and John Wickerson. Finding and understanding bugs\nin fpga synthesis tools. In Proceedings of the 2020 ACM/SIGDA Interna-\ntional Symposium on Field-Programmable Gate Arrays , pages 277–287,\n2020.\n[25] Peter Jamieson, Kenneth B Kent, Farnaz Gharibian, and Lesley Shan-\nnon. Odin ii-an open-source verilog hdl synthesis tool for cad re-\nsearch. In 2010 18th IEEE Annual International Symposium on Field-\nProgrammable Custom Computing Machines , pages 149–156. IEEE,\n2010.\n[26] Peter Jamieson and Jonathan Rose. A verilog rtl synthesis tool for het-\nerogeneous fpgas. In International Conference on Field Programmable\nLogic and Applications, 2005. , pages 305–310. IEEE, 2005.\n[27] Joshua Landgraf, Tiffany Yang, Will Lin, Christopher J Rossbach, and\nEric Schkufza. Compiler-driven fpga virtualization with synergy. In\nProceedings of the 26th ACM International Conference on Architectural\nSupport for Programming Languages and Operating Systems , pages\n818–831, 2021.\n[28] Roman L Lysecky, Kris Miller, Frank Vahid, and Kees A Vissers. Firm-\ncore virtual fpga for just-in-time fpga compilation. In FPGA , page 271,\n2005.\n[29] Donald MacKenzie. Mechanizing Proof: Computing, Risk, and Trust .\nThe MIT Press, 09 2001.\n[30] Alan Mishchenko, Satrajit Chatterjee, and Robert K. Brayton. Improve-\nments to technology mapping for lut-based fpgas. IEEE Transactions\non Computer-Aided Design of Integrated Circuits and Systems , 26(2):240–\n253, 2007.\n[31] Aina Niemetz and Mathias Preiner. Bitwuzla at the smt-comp 2020.\narXiv preprint arXiv:2006.01621 , 2020.\n[32] Aina Niemetz and Mathias Preiner. Bitwuzla. In Constantin Enea and\nAkash Lal, editors, Computer Aided Verification - 35th International\nConference, CAV 2023, Paris, France, July 17-22, 2023, Proceedings, Part\nII, volume 13965 of Lecture Notes in Computer Science , pages 3–17.\nSpringer, 2023.\n[33] Aina Niemetz, Mathias Preiner, Clifford Wolf, and Armin Biere. Btor2\n, btormc and boolector 3.0. In CAV (1) , volume 10981 of Lecture Notes\nin Computer Science , pages 587–595. Springer, 2018.\n[34] Rishiyur Nikhil. Bluespec system verilog: Efficient, correct rtl from\nhigh level specifications. In Proceedings of the Second ACM/IEEE In-\nternational Conference on Formal Methods and Models for Co-Design ,\nMEMOCODE ’04, page 69–70, USA, 2004. IEEE Computer Society.\n[35] Phitchaya Mangpo Phothilimthana, Tikhon Jelvis, Rohin Shah, Nishant\nTotla, Sarah Chasins, and Rastislav Bodik. Chlorophyll: Synthesis-\naided compiler for low-power spatial architectures. ACM SIGPLAN\nNotices , 49(6):396–407, 2014.\n[36] Sorawee Porncharoenwase, Luke Nelson, Xi Wang, and Emina Torlak.\nA formal foundation for symbolic evaluation with merging. Proc. ACM\nProgram. Lang. , 6(POPL), January 2022.\n[37] Berkeley Logic Synthesis and Verification Group. ABC: A system for\nsequential synthesis and verification, 2005. http://www.eecs.berkeley.\nedu/~alanmi/abc .\n[38] Jonathan Rose, Jason Luu, Chi Wai Yu, Opal Densmore, Jeffrey Goed-\ners, Andrew Somerville, Kenneth B Kent, Peter Jamieson, and Jason\nAnderson. The vtr project: architecture and cad for fpgas from verilog\nto routing. In Proceedings of the ACM/SIGDA international symposium\non Field Programmable Gate Arrays , pages 77–86, 2012.\n[39] Zachary D. Sisco, Jonathan Balkind, Timothy Sherwood, and Ben\nHardekopf. A position on program synthesis for processor develop-\nment. In Workshop on Languages, Tools, and Techniques for Accelerator\nDesign–LATTE 2022 , 2022.\n15ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA Gus Henry Smith, BK, VC, AC, SL, SP, RJ, GLB, and ZT\n[40] Zachary D. Sisco, Jonathan Balkind, Timothy Sherwood, and Ben\nHardekopf. Loop rerolling for hardware decompilation. Proc. ACM\nProgram. Lang. , 7(PLDI), jun 2023.\n[41] Armando Solar-Lezama. Program synthesis by sketching . University of\nCalifornia, Berkeley, 2008.\n[42] Xifan Tang, Edouard Giacomin, Aurélien Alacchi, Baudouin Chauviere,\nand Pierre-Emmanuel Gaillardon. Openfpga: An opensource frame-\nwork enabling rapid prototyping of customizable fpgas. In 2019 29th\nInternational Conference on Field Programmable Logic and Applications\n(FPL) , pages 367–374. IEEE, 2019.\n[43] Xifan Tang, Ganesh Gore, Grant Brown, and Pierre-Emmanuel Gaillar-\ndon. Taping out an fpga in 24 hours with openfpga: The sofa project.\nIn2021 31st International Conference on Field-Programmable Logic and\nApplications (FPL) , pages 400–400. IEEE, 2021.\n[44] Ross Tate, Michael Stepp, Zachary Tatlock, and Sorin Lerner. Equality\nsaturation: A new approach to optimization. In Proceedings of the 36th\nAnnual ACM SIGPLAN-SIGACT Symposium on Principles of Program-\nming Languages , POPL ’09, page 264–276, New York, NY, USA, 2009.\nAssociation for Computing Machinery.\n[45] Emina Torlak and Rastislav Bodik. Growing solver-aided languages\nwith rosette. In Proceedings of the 2013 ACM international symposium on\nNew ideas, new paradigms, and reflections on programming & software ,\npages 135–152, 2013.\n[46] Emina Torlak and Rastislav Bodik. A lightweight symbolic virtual\nmachine for solver-aided host languages. ACM SIGPLAN Notices ,\n49(6):530–541, 2014.\n[47] Jacob Van Geffen, Luke Nelson, Isil Dillig, Xi Wang, and Emina Torlak.\nSynthesizing jit compilers for in-kernel dsls. In Computer Aided Verifi-\ncation: 32nd International Conference, CAV 2020, Los Angeles, CA, USA,\nJuly 21–24, 2020, Proceedings, Part II , page 564–586, Berlin, Heidelberg,\n2020. Springer-Verlag.\n[48] Alexa VanHattum, Rachit Nigam, Vincent T Lee, James Bornholt, and\nAdrian Sampson. Vectorization for digital signal processors via equal-\nity saturation. In Proceedings of the 26th ACM International Conference\non Architectural Support for Programming Languages and Operating\nSystems , pages 874–886, 2021.\n[49] Luis Vega, Joseph McMahan, Adrian Sampson, Dan Grossman, and\nLuis Ceze. Reticle: a virtual machine for programming modern fpgas.\nInProceedings of the 42nd ACM SIGPLAN International Conference on\nProgramming Language Design and Implementation , pages 756–771,\n2021.\n[50] Clifford Wolf, Johann Glaser, and Johannes Kepler. Yosys-a free ver-\nilog synthesis suite. In Proceedings of the 21st Austrian Workshop on\nMicroelectronics (Austrochip) , 2013.\n[51] Xilinx. Ultrascale architecture DSP slice user guide, 2021. https:\n//docs.xilinx.com/v/u/en-US/ug579-ultrascale-dsp .\n16FPGA Technology Mapping Using Sketch-Guided Program Synthesis ASPLOS ’24, April 27-May 1, 2024, La Jolla, CA, USA\nA Artifact Appendix\nA.1 Abstract\nOur artifact consists of a zipfile containing the code for our\nevaluation. Running the evaluation code will reproduce all\nof the figures present in this paper, which artifact evaluators\ncan validate against our published data. The evaluation code\nis comprised largely of the following files: documentation in\na README, a Dockerfile to automatically set up the evalua-\ntion environment, the Lakeroad codebase, and the evaluation\nscripts themselves (a mix of Python and shell scripts). The\nevaluation should be run on an x86 machine running Linux\n(ideally Ubuntu). The evaluation benefits from many CPU\ncores. The evaluation requires at least 300GB of free space,\nmostly for installing proprietary hardware toolchains.\nA.2 Artifact check-list (meta-information)\n•Algorithm: Program synthesis via Rosette. Hardware syn-\nthesis via traditional hardware toolchains.\n•Program: Lakeroad, the Rosette-based hardware synthesis\ntool presented in this paper, plus Yosys, Xilinx Vivado, Lattice\nDiamond, and Intel Quartus, the baseline hardware synthesis\ntools we compare against.\n•Run-time environment: Linux, ideally Ubuntu.\n•Hardware: x86 CPU, ideally with many cores.\n•Output: Images and CSV files representing this paper’s\nfigures and tables.\n•Experiments: Each experiment is a single run of a hardware\nsynthesis tool (either Lakeroad or one of our baseline tools).\nThe entire experiment consists of thousands of these tool\nruns.\n•How much disk space required (approximately)?: 300GB.\n•How much time is needed to prepare workflow (ap-\nproximately)?: 4 hours: 3 hours to set up proprietary hard-\nware tools, 1 hour to build Docker image.\n•How much time is needed to complete experiments\n(approximately)?: 2 to 10+ hours, depending on the num-\nber of cores. On our 64-core machine, the evaluation takes\nabout 4 hours.\n•Publicly available?: Yes, at https://github.com/uwsampl/\nlakeroad-evaluation and archived publicly on Zenodo, see\nDOI link below.\n•Code licenses (if publicly available)?: MIT.\n•Workflow framework used?: Python DoIt.\n•Archived (provide DOI)?: https://doi.org/10.5281/zenodo.\n10515833\nA.3 Description\nA.3.1 How to access. We recommend downloading the\nzipped code repository from the DOI link above. The code\ncan also be cloned from the GitHub repository linked above.\nA.3.2 Hardware dependencies. x86 CPU, preferably with\nmany cores.\nA.3.3 Software dependencies. Linux-based OS, ideally\nUbuntu.A.4 Installation\nPlease refer to the README in the artifact. A more read-\nable version of the README can be viewed on the GitHub\nrepository, or by converting the README using a tool like\nPandoc.\nA.5 Experiment workflow\nPlease refer to the README in the artifact.\nA.6 Evaluation and expected results\nPlease refer to the README in the artifact.\nA.7 Methodology\nSubmission, reviewing and badging methodology:\n•https://www.acm.org/publications/policies/artifact-review-\nbadging\n•http://cTuning.org/ae/submission-20201122.html\n•http://cTuning.org/ae/reviewing-20201122.html\n17" }, { "title": "2401.16573v2.On_the_Hörmander_s_estimate.pdf", "content": "arXiv:2401.16573v2 [math.CV] 19 Mar 2024On the Hörmander’s estimate\nBingyuan Liu\nMarch 20, 2024\nAbstract\nThe motivation of the note is to obtain a Hörmander-type L2estimate for ¯∂equation. The\nfeature of the new estimate is that the constant in the estima te is independent of the weight\nfunction. Moreover, our estimate can be used for non-pluris ubharmonic weight function.\n1 Introduction\nTheL2-method of functional analysis has been well-applied to alg ebraic geometry settings for\ndecades. It can tell if the solution of a Cauchy–Riemann equa tion exists. It also provides an estimate\nof the solution in terms of the given data. With the solution o f the Cauchy–Riemann equation, one\ncan easily construct a holomorphic function. Consequently , cohomology can be studied with the\nhelp of the L2-method.\nDepending on the ambient spaces, there are two common approa ches to studying the ¯∂equation:\neither treat the space as a manifold with a complete metric or treat it as a bounded domain in\nEuclidean spaces (in which case the metric on the domain will be induced from the Euclidean\nmetric and is incomplete). One can examine the L2estimate using either of the two approaches,\neven for the same domain. The L2space will be smaller with complete metric than with incompl ete\nmetric. After all, the L2integrable functions/forms for the complete metric need to “vanish” when\napproaching boundary while the case of incomplete metrics n eeds not. As a result, the case of\nincomplete metrics still has yet to be well-understood. The ¯∂-Neumann problem, which is related\nto the¯∂-problem, shares the same lack of clarity. Indeed, the neces sary and sufficient conditions of\nthe global regularity of the ¯∂-Neumann problem remain unknown. See Boas–Straube [6], [8], [7], [5],\nHarrington [16], Harrington–Liu [15], Kohn [21], [22], Pin ton–Zampiere [27], and Liu–Straube [23]\nfor sufficient conditions. See Huang–Li [19] for mixed bounda ry conditions as well. The ¯∂-Neumann\nproblem is known to be closely related to the theory of the Berg man kernel. See Hsiao–Savale [18].\nTheL2estimates of the ¯∂operator has wide applications in algebraic geometry. This type of estimate\nwas initially introduced by Andreotti–Vesentini in their w orks [1] and [2]. It has been subsequently\nproven using different methods, by Hörmander [17]. More rece ntly, Donnelly–Fefferman, Ohsawa–\nTakegoshi, as well as Berndtsson–Charpentier, have derived alternative forms of this estimate.\nTheorem 1.1 (Donnelly–Fefferman [13], see also Chen [10]) .LetΩbe a bounded pseudoconvex\ndomain in Cnandφ∈PSH(Ω). Assume that φis strictly plurisubharmonic so that for\nr√\n-1∂¯∂φ≥√\n-1∂φ∧¯∂φ\nfor somer>0. Then for the equation ¯∂u=v, where ¯∂v=0andv∈L2(Ω,φ), there exists a\nu∈L2(Ω,φ)so that\n/integr⊗l.dispΩ/divid⟩s.al⟪0u/divid⟩s.al⟪02e−φdV≤C0r/integr⊗l.dispΩ/divid⟩s.al⟪0v/divid⟩s.al⟪02√\n-1∂¯∂φe−φdV.\n1TheC0is a constant.\nTheorem 1.2 (Berndtsson–Charpentier [3], see also Chen [10] and Błocki [4 ]).LetΩbe a bounded\npseudoconvex domain in Cnandφ∈PSH(Ω). Assume that ψis strictly plurisubharmonic so that\nfor\nr√\n-1∂¯∂ψ≥√\n-1∂ψ∧¯∂ψ\nfor somer>0. Then for the equation ¯∂u=v, where¯∂v=0andv∈L2(Ω,φ−ψ), there exists a\nu∈L2(Ω,φ−ψ)so that\n/integr⊗l.dispΩ/divid⟩s.al⟪0u/divid⟩s.al⟪02eψ−φdV≤1\n(1−√r)2/integr⊗l.dispΩ/divid⟩s.al⟪0v/divid⟩s.al⟪02√\n-1∂¯∂φeψ−φdV.\nIt is well-known that Ohsawa–Takegoshi present an alternat iveL2estimate and an extensively\nstudied extension theory. For more details, interested rea ders can refer to Ohsawa–Takegoshi [26],\nStraube [28], Demailly [12], Chen [9], Błocki [4] and Guan–Zh ou [14].\nIn this article, we prove the following:\nTheorem 1. LetΩbe a bounded pseudoconvex domain in CnandE∶=Ω×C→Xbe a trivial line\nbundle with fiber metric e−φforφ∈C2(Ω). Assume that there exists η∈C2(Ω),q≥1andt2,t3∈R\nso that\nΞt2,t3,η≥0.\nLetf∈L2\n(0,q)(Ω)∩Dom(¯∂)so that¯∂f=0, there exists u∈Dom(¯∂)so that\n/parall⟩l.al⟪1u/parall⟩l.al⟪12\nφ≤(t2+t3)⟪Ξ−1\nt2,t3,ηf,f⟫φ.\nHere we define the operator Ξon(0,1)-forms by the following. For the general definition of (0,q)-\nforms, see Section 4.\nDefinition 1.1. Letf=/summ⊗tion.dispfidzi∈L2\n(0,1)(Ω). Then\nΞf=(√\n-1∂2(η+φ)\n∂zi¯zj−1\n4√\n-1∂η\n∂zi∂η\n∂¯zj)fi¯fj−1\n4/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02/divid⟩s.al⟪0f/divid⟩s.al⟪02.\nWe want to remark that our formula is different from the one fro m Theorem 1.1 and Theorem 1.2.\nBoth Theorem 1.1 and Theorem 1.2 require φ,ψ∈PSH(Ω)but our theorem does not need this\ncondition. Indeed, our weight function φdoes not need to be plurisubharmonic and for a fixed\nnon-plurisubharmonic weight φ, we can get an estimate by varying the function η. We believe the\ncandidate pool of ηdepends on the geometric property of the domain Ω. We also want to remark\nthat the theorem of Ohsawa–Takegoshi in [26] and Theorem 1.2 can make non-plurisubharmonic\nweight as well. The formulation and proof of our estimate are different from theirs. In this note,\nwe only consider trivial line bundles.\nThe orgnization of the note is as follows: after introductio n and preliminary in Section 1 and 2, we\nintroduce our new technique in Section 3 and eventually, ref ormulate it to the form of Theorem 1\nin Section 4.\n22 Preliminary\nIn this section, we are going to recall the L2-method of ¯∂equation in two settings: in the complete\nopen Kähler manifold and in the bounded domain (with boundar y) inCnwith the induced Euclidean\nmetric.\nLetI=(i1,...,ip)is a multi-index with integer components, i1<...0\nsuch that√\n-1[Θη+tφ,Λ]−t−1/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nCpositive definite, we have the following a priori estimate\nC/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ+C/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ≥/parall⟩l.al⟪1v/parall⟩l.al⟪12\nφ,\nfor allv∈C∞\np,q;c(X,C).\n11Proof. Recall that from Proposition 3.1\n⟨¯∂(∂η)♯/rightanglesev,v⟩Ce−φ+⟨(∂η)♯/rightanglese¯∂v,v⟩Ce−φ=−⟨∂η∧∂∗v,v⟩Ce−φ−⟨∂∗∂η∧v,v⟩Ce−φ+√\n-1⟨[Θη,Λ]v,v⟩Ce−φ.\nConsequently,\n/integr⊗l.disp⟨¯∂v,¯∂η∧v⟩Ce−φdV+/integr⊗l.disp⟨(∂η)♯/rightanglesev,¯∂∗\nφv⟩Ce−φdV−√\n-1/integr⊗l.disp⟨[Θη,Λ]v,v⟩Ce−φdV\n=−/integr⊗l.disp⟨∂η∧v,∂φv⟩Ce−φdV−/integr⊗l.disp⟨∂∗v,(¯∂η)♯/rightanglesev⟩Ce−φdV.\nThis implies,\nRe/integr⊗l.disp⟨¯∂v,¯∂η∧v⟩Ce−φdV+Re/integr⊗l.disp⟨¯∂∗\nφv,(∂η)♯/rightanglesev⟩Ce−φdV−√\n-1/integr⊗l.disp⟨[Θη,Λ]v,v⟩Ce−φdV\n≥−t/parall⟩l.al⟪1∂φv/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1∂η∧v/parall⟩l.al⟪12\nφ−t/parall⟩l.al⟪1∂∗v/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1(¯∂η)♯/rightanglesev/parall⟩l.al⟪12\nφ\n=−t/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ−t/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ+√\n-1/integr⊗l.disp⟨[Θtφ,Λ]v,v⟩Ce−φdV−1\n2t/parall⟩l.al⟪1∂η∧v/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1(¯∂η)♯/rightanglesev/parall⟩l.al⟪12\nφ.\nThe last equation is due to Proposition 3.2.\nThe last inequality gives that,\n2t/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ+2t/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ\n≥√\n-1/integr⊗l.disp⟨[Θη+tφ,Λ]v,v⟩Ce−φdV−1\n2t/parall⟩l.al⟪1∂η∧v/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1(¯∂η)♯/rightanglesev/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1¯∂η∧v/parall⟩l.al⟪12\nφ−1\n2t/parall⟩l.al⟪1(∂η)♯/rightanglesev/parall⟩l.al⟪12\nφ,\nwhereΘη+tφdenotes the curvature form of the fiber metric /divid⟩s.al⟪0⋅/divid⟩s.al⟪02\nη+tφ=/divid⟩s.al⟪0⋅/divid⟩s.al⟪02\nCe−η−tφ.\nRecall that\n/divid⟩s.al⟪0(∂η)♯/rightanglesev/divid⟩s.al⟪02\nC=−/divid⟩s.al⟪0¯∂η∧v/divid⟩s.al⟪02\nC+/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nC/divid⟩s.al⟪0v/divid⟩s.al⟪02\nC.\nConsequently,\n/integr⊗l.disp/par⟩nl⟩f⟪.al⟪1/divid⟩s.al⟪0¯∂η∧v/divid⟩s.al⟪02\nC+/divid⟩s.al⟪0(∂η)♯/rightanglesev/divid⟩s.al⟪02\nC/par⟩nrigh⟪.al⟪1e−φdV=/integr⊗l.disp/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nC/divid⟩s.al⟪0v/divid⟩s.al⟪02\nCe−φdV=/parall⟩l.al⟪1/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪0Cv/parall⟩l.al⟪1φ\nand thus\nt/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ+t/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ≥√\n-1/integr⊗l.disp���[Θη+tφ,Λ]v,v⟩Ce−φdV−1\nt/integr⊗l.disp⟨/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nCv,v⟩Ce−φdV.\nConsequently,\n/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ+/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ≥√\n-1t−1/integr⊗l.disp⟨[Θη+tφ,Λ]v,v⟩Ce−φdV−t−2/integr⊗l.disp⟨/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nCv,v⟩Ce−φdV.\nIf there exists t>0such that√\n-1[Θη+tφ,Λ]−t−1/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nCpositive definite, then the conclusion follows.\nRemark 2.In the preceding proposition, we did not assume positivity o n the curvature of X×C.\nThe preceding proposition can be extend for all v∈Dom(¯∂)∩Dom(¯∂∗)through the following density\nlemma.\n12Lemma 3.4 (Theorem 2.6 in Ohsawa [25]) .TheCp,q;c(X,C)⊂L2\np,q(X,C)is dense in Dom(¯∂)∩\nDom(¯∂∗)with respect to the graph norm /parall⟩l.al⟪1u/parall⟩l.al⟪12+/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12+/parall⟩l.al⟪1¯∂∗u/parall⟩l.al⟪12\nFrom now on, we define Aη,t=√\n-1\n2t−1[Θη+tφ,Λ]−t−2\n2/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nC. By Lemma 3.4 and the above inequality,\nwe obtain that\n/parall⟩l.al⟪1¯∂v/parall⟩l.al⟪12\nφ+/parall⟩l.al⟪1¯∂∗\nφv/parall⟩l.al⟪12\nφ≥⟪Aη,tv,v⟫φ\nholds for all v∈Dom(¯∂)∩Dom(¯∂∗).\nSuppose we have ¯∂u=ffor a given f∈L2\nn,q+1(X,E)such that ¯∂f=0. We consider for arbitrary\ns∈Dom(¯∂)∩Dom(¯∂∗), that⟪f,s⟫φ≤⟪Aη,ts,s⟫φ⟪A−1\nη,tf,f⟫φby the Cauchy–Schwartz inequality.\nConsequently,\n⟪f,s⟫φ≤⟪A−1\nη,tf,f⟫φ/par⟩nl⟩f⟪.al⟪1/parall⟩l.al⟪1¯∂s/parall⟩l.al⟪12\nφ+/parall⟩l.al⟪1¯∂∗\nφs/parall⟩l.al⟪12\nφ/par⟩nrigh⟪.al⟪1.\nWe decompose s=s1+s2, wheres1∈ker(¯∂)ands2∈Ran(¯∂∗). Sincef∈ker(¯∂), we have that\n⟪f,s⟫φ=⟪f,s1⟫φ≤⟪A−1\nη,tf,f⟫φ/parall⟩l.al⟪1¯∂∗\nφs1/parall⟩l.al⟪12\nφ.\nSinces2∈Ran(¯∂∗)⊂ker(¯∂∗). So⟪f,s⟫φ≤⟪A−1\nη,tf,f⟫φ/parall⟩l.al⟪1¯∂∗\nφs/parall⟩l.al⟪12\nφ. By the Hahn–Banach theorem,\nthe well-defined functional on Ran(¯∂∗):¯∂∗s↦⟪f,s⟫φextends to a bounded linear functional on\nL2\np,q−1(X,E). Consequently, there exists u∈L2\np,q−1(X,E)with/parall⟩l.al⟪1u/parall⟩l.al⟪1φ≤⟪A−1\nη,tf,f⟫φandusolves the\nequation ¯∂u=finL2\np,q−1(X,E). This summarizes in the following theorem.\nTheorem 3.1. LetX×C→Xbe a trivial line bundle with fiber metric e−φ. Suppose there exists t>0\nsuch that√\n-1[Θη+tφ,Λ]−t−1/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02\nCpositive definite. For all f∈L2\np,q(X,E)satisfying⟪A−1\nη,tf,f⟫φ<∞,\nthere exists a u∈L2\np,q−1(X,E)solves¯∂u=f. Moreover, we have the estimate\n/parall⟩l.al⟪1u/parall⟩l.al⟪1φ≤⟪A−1\nη,tf,f⟫φ.\n3.3 The case with boundary (incomplete metric on X)\nBy the Bochner–Kodaira–Nakano identity, we have that, for u∈C∞\n(n,q)(Ω),\n¯∂¯∂∗\nφu+¯∂∗\nφ¯∂u=∂φ∂∗u+√\n-1ΘφΛu.\nThe second equation is that\n¯∂¯∂∗\nφ+ηu+¯∂∗\nφ+η¯∂u=∂φ+η∂∗u+√\n-1Θφ+ηΛu\nor equivalently,\n¯∂eη¯∂∗\nφe−ηu+eη¯∂∗\nφe−η¯∂u=eη∂φe−η∂∗u+√\n-1Θφ+ηΛu\nwhich is,\n¯∂(¯∂∗\nφu+(∂η)♯/rightangleseu)+¯∂∗\nφ¯∂u+(∂η)♯/rightanglese¯∂u=∂φ∂∗u−∂η∧∂∗u+√\n-1Θφ+ηΛu.\nConsequently, taking difference of the two equations\n¯∂(∂η)♯/rightangleseu+(∂η)♯/rightanglese¯∂u=−∂η∧∂∗u+√\n-1ΘηΛu.\nThen the basic equation:\n13Proposition 3.4 (The basic equation for incomplete metric case) .Foru∈C∞\n(n,q)(Ω),\n⟨¯∂(∂η)♯/rightangleseu,u⟩Ce−φ+⟨(∂η)♯/rightanglese¯∂u,u⟩Ce−φ=−⟨∂η∧∂∗u,u⟩Ce−φ+√\n-1⟨ΘηΛu,u⟩Ce−φ.\nIn particular, if u∈C∞\n(n,q)(Ω)∩ker(¯∂), we have that\n⟨¯∂(∂η)♯/rightangleseu,u⟩Ce−φ=−⟨∂η∧∂∗u,u⟩Ce−φ+√\n-1⟨ΘηΛu,u⟩Ce−φ.\nThen furthermore, if u∈ker(¯∂)∩Dom(¯∂∗)∩C∞\n(n,q)(Ω),\n⟪(∂η)♯/rightangleseu,¯∂∗\nφu⟫φ+⟪¯∂u,¯∂η∧u⟫φ=−⟪∂∗u,(¯∂η)♯/rightangleseu⟫φ+⟪√\n-1ΘηΛu,u⟫φ.\nObserve that, by Morrey-Kohn-Hörmander’s formula\nRe⟪∂∗u,(¯∂η)♯/rightangleseu⟫φ\n≥−t3/parall⟩l.al⟪1∂∗u/parall⟩l.al⟪12\nφ−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪1φ\n=−t3/par⟩nl⟩f⟪.al⟪3/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12\nφ+/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ−⟪√\n-1ΘφΛu,u⟫φ−/integr⊗l.disp∂Ωe−φHessδ(u,u)dσ/par⟩nrigh⟪.al⟪3−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ\nConsequently, we have the following result:\nTheorem 3.2. Foru∈C∞\n(n,q)(Ω)∩Dom(¯∂∗),\n(t1+t3)/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12\nφ+(t2+t3)/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ≥−1\n4t3/parall⟩l.al⟪1(¯��η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1Θt3φ+ηΛu,u⟫φ−1\n4t2/parall⟩l.al⟪1(∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ−1\n4t1/parall⟩l.al⟪1¯∂η∧u/parall⟩l.al⟪12\nφ.\nIn particular, if u∈Dom(¯∂∗)∩ker(¯∂)∩C∞\n(n,q)(Ω),\n(t2+t3)/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ≥−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1Θt3φ+ηΛu,u⟫φ−1\n4t2/parall⟩l.al⟪1(∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ.\nProof. By Proposition 3.4 and Morrey–Kohn–Hörmander’s formula,\n(t1+t3)/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12\nφ+(t2+t3)/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ+1\n4t2/parall⟩l.al⟪1(∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+1\n4t1/parall⟩l.al⟪1¯∂η∧u/parall⟩l.al⟪12\nφ\n≥t3⟪√\n-1ΘφΛu,u⟫φ−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1ΘηΛu,u⟫φ\n=−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1Θt3φ+ηΛu,u⟫φ.\nLemma 3.5 (density lemma for bounded domain, see Chen–Shaw [11]) .LetΩ⊂Cnbe a bounded\npseudoconvex domain with smooth boundary. then C∞\nn,q(Ω,C)∩Dom(¯∂∗)is dense in Dom(¯∂)∩\nDom(¯∂∗)in the graph norm u↦(/parall⟩l.al⟪1u/parall⟩l.al⟪12+/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12+/parall⟩l.al⟪1¯∂∗u/parall⟩l.al⟪12)1\n2.\nBy the above lemma, we obtain the following theorem.\nTheorem 3.3. For an arbitrary(n,q)formu∈Dom(¯∂)∩Dom(¯∂∗),\n(t1+t3)/parall⟩l.al⟪1¯∂u/parall⟩l.al⟪12\nφ+(t2+t3)/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ≥−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1Θt3φ+ηΛu,u⟫φ−1\n4t2/parall⟩l.al⟪1(∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ−1\n4t1/parall⟩l.al⟪1¯∂η∧u/parall⟩l.al⟪12\nφ.\nIn particular, if u∈Dom(¯∂∗)∩ker(¯∂),\n(t2+t3)/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ≥−1\n4t3/parall⟩l.al⟪1(¯∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ+⟪√\n-1Θt3φ+ηΛu,u⟫φ−1\n4t2/parall⟩l.al⟪1(∂η)♯/rightangleseu/parall⟩l.al⟪12\nφ.\n144 Reformulation as a Donnelly–Fefferman type estimate\nConsider Theorem 3.3. Let t3=t2=1. From the basic equation, assuming u∈ker¯∂, we have that,\nforu∈Dom(¯∂∗\n0,1;φ)∩ker(¯∂)\n2/parall⟩l.al⟪1¯∂∗\nφu/parall⟩l.al⟪12\nφ≥⟪√\n-1Θφ+ηΛ˜u,˜u⟫φ−1\n4/parall⟩l.al⟪1(∂η)♯/rightanglese˜u/parall⟩l.al⟪12\nφ−1\n4/parall⟩l.al⟪1(¯∂η)♯/rightanglese˜u/parall⟩l.al⟪12\nφ\n=/integr⊗l.dispΩ(√\n-1∂2(η+φ)\n∂zi¯zj−1\n4√\n-1∂η\n∂zi∂η\n∂¯zj)ui¯uje−φdV−1\n4/integr⊗l.dispΩ/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02/divid⟩s.al⟪0u/divid⟩s.al⟪02dV.(3)\nConsider a function f∈ker(¯∂). Thenκ∶¯∂∗\nφg↦⟪f,g⟫φis well-defined for g∈ker(¯∂)(see Chen–\nShaw [11] and Straube [28]). Let u∈Im(¯∂∗\nφ)and solve ¯∂u=f. We then have that for all g∈\nker(¯∂)∩Dom(¯∂∗\nφ),\n⟪u,¯∂∗\nφg⟫φ=⟪f,g⟫φ.\nDenote√\n-1∂2(η+φ)\n∂zi¯zj−√\n-11\n4∂η\n∂zi∂η\n∂¯zj−√\n-11\n4/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02byΞas a pointwise action on u. We have that,\nbased on (3),\n/parall⟩l.al⟪1¯∂∗\nφg/parall⟩l.al⟪12\nφ≥1\n2/integr⊗l.dispΩ⟨Ξg,g⟩Ce−φdV.\nHere, by definition\nΞu=(√\n-1∂2(η+φ)\n∂zi¯zj−1\n4√\n-1∂η\n∂zi∂η\n∂¯zj)ui¯uj−1\n4/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02/divid⟩s.al⟪0u/divid⟩s.al⟪02\nConsequently,\n⟪u,¯∂∗\nφg⟫φ≤⟪Ξ−1f,f⟫1/slash.left2\nφ⟪Ξg,g⟫1/slash.left2\nφ≤√\n2⟪Ξ−1f,f⟫1/slash.left2\nφ/parall⟩l.al⟪1¯∂∗\nφg/parall⟩l.al⟪1φ.\nSo,/parall⟩l.al⟪1u/parall⟩l.al⟪12\nφ≤2⟪Ξ−1f,f⟫φ, as long as√\n-1Θφ+η≥√\n-1\n4∂η∧¯∂η+√\n-1\n4/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02. In practice, since φ∈PSH(Ω),\nas long asη∈PSH(Ω)with√\n-1∂¯∂η≥√\n-1\n4∂η∧¯∂η+√\n-1\n4/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02, we can obtain the estimate /parall⟩l.al⟪1u/parall⟩l.al⟪12\nφ≤\n2⟪Ξ−1f,f⟫φ.\nIn general, we have the following theorem:\nTheorem 4.1. LetΩbe a bounded pseudoconvex domain in CnandE∶=Ω×C→Xbe a trivial\nline bundle with fiber metric e−φforφ∈C2(Ω). Assume that there exists η∈C2(Ω)andt2,t3∈R\nso that\nΞt2,t3,η∶=√\n-1∂2(η+t3φ)\n∂zi¯zj−√\n-11\n4t2∂η\n∂zi∂η\n∂¯zj−√\n-11\n4t3/divid⟩s.al⟪0¯∂η/divid⟩s.al⟪02≥0.\nLetf∈L2\n(0,1)(Ω)∩Dom(¯∂)so that¯∂f=0, there exists u∈Dom(¯∂)so that\n/parall⟩l.al⟪1u/parall⟩l.al⟪12\nφ≤(t2+t3)⟪Ξ−1\nt2,t3,ηf,f⟫φ.\nSimilarly, we can extend the theorem to (0,q)forms using the definition on Page 371 of Demailly\n[12]. Let ˜ube a(n,q)- form which lifts u. We also let ˜ube a(0,q)-form ifuis a(n,q)form. In\n15other words, we have that ˜˜u=uifuis either a(0,q)or a(n,q)from. Define that, in the Euclidean\nspace,\nΞt2,t3,η˜u\n=√\n-1Θη+t3φΛ˜u−√\n-11\n4t2¯∂η∧(∂η)♯/rightanglese˜u−√\n-11\n4t3∂η∧(¯∂η)♯/rightanglese˜u\n=√\n-1Θη+t3φΛ˜u−√\n-11\n4t2¯∂η∧(∂η)♯/rightanglese˜u−√\n-11\n4t3/divid⟩s.al⟪0∂η/divid⟩s.al⟪02˜u,\nwhereΛ=L∗andL=√\n-1/summ⊗tion.disp\nidzi∧d¯zi. In the case for a (0,q)formu, we define Ξt2,t3,ηu=/⟪ild⟩comb.al⟪4Ξt2,t3,η˜u.\nThen Theorem 1 follows similarly to the theorem above.\nAcknowledgments . The author greatly appreciates Dr. Bo-Yong Chen’s patience with his ques-\ntions.\nReferences\n[1] A. Andreotti and E. Vesentini. “Sopra un teorema di Kodai ra”. In: Ann. Scuola Norm. Sup.\nPisa (3) 15 (1961), pp. 283–309.\n[2] Aldo Andreotti and Edoardo Vesentini. “Carleman estima tes for the Laplace-Beltrami equa-\ntion on complex manifolds”. In: Inst. Hautes Études Sci. Publ. Math. 25 (1965), pp. 81–130.\nissn: 0073-8301.\n[3] Bo Berndtsson and Philippe Charpentier. “A Sobolev mappin g property of the Bergman\nkernel”. In: Math. Z. 235.1 (2000), pp. 1–10. issn: 0025-5874. doi:10.1007/s002090000099 .\n[4] Zbigniew Bł ocki. “Suita conjecture and the Ohsawa-Takeg oshi extension theorem”. In: Invent.\nMath. 193.1 (2013), pp. 149–158. issn: 0020-9910. doi:10.1007/s00222-012-0423-2 .\n[5] Harold P. Boas and Emil J. Straube. “de Rham cohomology of m anifolds containing the points\nof infinite type, and Sobolev estimates for the ∂-Neumann problem”. In: J. Geom. Anal. 3.3\n(1993), pp. 225–235. issn: 1050-6926. doi:10.1007/BF02921391 .\n[6] Harold P. Boas and Emil J. Straube. “Equivalence of regula rity for the Bergman projection\nand the∂-Neumann operator”. In: Manuscripta Math. 67.1 (1990), pp. 25–33. issn: 0025-2611.\ndoi:10.1007/BF02568420 .\n[7] Harold P. Boas and Emil J. Straube. “Sobolev estimates for the complex Green operator on\na class of weakly pseudoconvex boundaries”. In: Comm. Partial Differential Equations 16.10\n(1991), pp. 1573–1582. issn: 0360-5302. doi:10.1080/03605309108820813 .\n[8] Harold P. Boas and Emil J. Straube. “Sobolev estimates for the∂-Neumann operator on\ndomains in Cnadmitting a defining function that is plurisubharmonic on th e boundary”. In:\nMath. Z. 206.1 (1991), pp. 81–88. issn: 0025-5874. doi:10.1007/BF02571327 .\n[9] Bo-Yong Chen. A simple proof of the Ohsawa-Takegoshi extension theorem . 2011. arXiv:\n1105.2430 [math.CV] .\n[10] Bo-Yong Chen. TheL2theory of the Cauchy–Riemann equation (Chinese) . China Science\nPublishing & Media, 2022.\n[11] So-Chin Chen and Mei-Chi Shaw. Partial differential equations in several complex variable s.\nVol. 19. AMS/IP Studies in Advanced Mathematics. Internati onal Press, Boston, MA, 2001,\npp. xii+380. isbn: 0-8218-1062-6.\n[12] Jean-Pierre Demailly. Complex Analytic and Differential Geometry .\n16[13] Harold Donnelly and Charles Fefferman. “ L2-cohomology and index theorem for the Bergman\nmetric”. In: Ann. of Math. (2) 118.3 (1983), pp. 593–618. issn: 0003-486X,1939-8980. doi:\n10.2307/2006983 .\n[14] Qi’an Guan and Xiangyu Zhou. “A solution of an L2extension problem with an optimal\nestimate and applications”. In: Ann. of Math. (2) 181.3 (2015), pp. 1139–1208. issn: 0003-\n486X. doi:10.4007/annals.2015.181.3.6 .\n[15] Phillip Harrington and Bingyuan Liu. “The ∂-Neumann operator with Sobolev estimates up\nto a finite order”. In: Comm. Partial Differential Equations 45.10 (2020), pp. 1435–1450. issn:\n0360-5302. doi:10.1080/03605302.2020.1774899 .\n[16] Phillip S. Harrington. “Global regularity for the ∂-Neumann operator and bounded plurisub-\nharmonic exhaustion functions”. In: Adv. Math. 228.4 (2011), pp. 2522–2551. issn: 0001-8708.\ndoi:10.1016/j.aim.2011.07.008 .\n[17] Lars Hörmander. “ L2estimates and existence theorems for the ¯∂operator”. In: Acta Math.\n113 (1965), pp. 89–152. issn: 0001-5962. doi:10.1007/BF02391775 .\n[18] Chin-Yu Hsiao and Nikhil Savale. “Bergman-Szegö kernel asymptotics in weakly pseudoconvex\nfinite type cases”. In: J. Reine Angew. Math. 791 (2022), pp. 173–223. issn: 0075-4102,1435-\n5345.doi:10.1515/crelle-2022-0044 .\n[19] Xiaojun Huang and Xiaoshan Li. “ ∂-equation on a lunar domain with mixed boundary condi-\ntions”. In: Trans. Amer. Math. Soc. 368.10 (2016), pp. 6915–6937. issn: 0002-9947,1088-6850.\ndoi:10.1090/tran/6547 .\n[20] Daniel Huybrechts. Complex geometry . Universitext. An introduction. Springer-Verlag, Berlin,\n2005, pp. xii+309. isbn: 3-540-21290-6.\n[21] J. J. Kohn. “Harmonic integrals on strongly pseudo-con vex manifolds. I”. In: Ann. of Math.\n(2)78 (1963), pp. 112–148. issn: 0003-486X. doi:10.2307/1970506 .\n[22] J. J. Kohn. “Harmonic integrals on strongly pseudo-con vex manifolds. II”. In: Ann. of Math.\n(2)79 (1964), pp. 450–472. issn: 0003-486X. doi:10.2307/1970404 .\n[23] Bingyuan Liu and Emil J. Straube. “Diederich-Fornæss in dex and global regularity in the ∂-\nNeumann problem: domains with comparable Levi eigenvalues ”. In: (2022). doi:10.48550/ARXIV.2207.14197 .\n[24] Xiaonan Ma and George Marinescu. Holomorphic Morse inequalities and Bergman kernels .\nVol. 254. Progress in Mathematics. Birkhäuser Verlag, Basel, 2007, pp. xiv+422. isbn: 978-3-\n7643-8096-0. doi:10.1007/978-3-7643-8115-8 .\n[25] Takeo Ohsawa. L2approaches in several complex variables . Springer Monographs in Mathe-\nmatics. Towards the Oka-Cartan theory with precise bounds, Second edition of [ MR3443603].\nSpringer, Tokyo, 2018, pp. xi+258. isbn: 978-4-431-56851-3; 978-4-431-56852-0. doi:10.1007/978-4-431-56852- \n[26] Takeo Ohsawa and Kensh¯ o Takegoshi. “On the extension o fL2holomorphic functions”. In:\nMath. Z. 195.2 (1987), pp. 197–204. issn: 0025-5874. doi:10.1007/BF01166457 .\n[27] Stefano Pinton and Giuseppe Zampieri. “The Diederich- Fornaess index and the global regu-\nlarity of the ¯∂-Neumann problem”. In: Math. Z. 276.1-2 (2014), pp. 93–113. issn: 0025-5874.\ndoi:10.1007/s00209-013-1188-z .\n[28] Emil J. Straube. Lectures on the L2-Sobolev theory of the ∂-Neumann problem . ESI Lectures in\nMathematics and Physics. European Mathematical Society (E MS), Zürich, 2010, pp. viii+206.\nisbn: 978-3-03719-076-0. doi:10.4171/076 .\n17" }, { "title": "2401.16590v1.Influence_of_source_parameters_on_the_longitudinal_phase_space_distribution_of_a_pulsed_cryogenic_beam_of_barium_fluoride_molecules.pdf", "content": "Influence of source parameters on the longitudinal\nphase-space distribution of a pulsed cryogenic beam\nof barium fluoride molecules\nM C Mooij1,2, H L Bethlem1,3, A Boeschoten2,3,\nA Borschevsky2,3, K Esajas2,3, T H Fikkers2,3, S Hoekstra2,3,\nJ W F van Hofslot2,3, K Jungmann2,3, V R Marshall2,3,\nT B Meijknecht2,3, R G E Timmermans2,3, A Touwen2,3,\nW Ubachs1, L Willmann2,3and Y Yin2,3‡\nNL-eEDM collaboration\n1Department of Physics and Astronomy, LaserLaB, Vrije Universiteit Amsterdam,\nThe Netherlands\n2Nikhef, National Institute for Subatomic Physics, The Netherlands\n3Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,\nThe Netherlands\nE-mail: H.L.Bethlem@vu.nl\nAbstract. Recently, we have demonstrated a method to record the longitudinal\nphase-space distribution of a pulsed cryogenic buffer gas cooled beam of barium fluoride\nmolecules. In this paper, we use this method to determine the influence of various\nsource parameters. Besides the expected dependence on temperature and pressure,\nthe forward velocity of the molecules is strongly correlated with the time they exit\nthe cell, revealing the dynamics of the gas inside the cell. Three observations are\nparticularly noteworthy: (1) The velocity of the barium fluoride molecules increases\nrapidly as a function of time, reaches a maximum 50-200 µs after the ablation pulse and\nthen decreases exponentially. We attribute this to the buffer gas being heated up by\nthe plume of hot atoms released from the target by the ablation pulse and subsequently\nbeing cooled down via conduction to the cell walls. (2) The time constant associated\nwith the exponentially decreasing temperature increases when the source is used for a\nlonger period of time, which we attribute to the formation of a layer of isolating dust\non the walls of the cell. By thoroughly cleaning the cell, the time constant is reset to\nits initial value. (3) The velocity of the molecules at the trailing end of the molecular\npulse depends on the length of the cell. For short cells, the velocity is significantly\nhigher than expected from the sudden freeze model. We attribute this to the target\nremaining warm over the duration of the molecular pulse giving rise to a temperature\ngradient within the cell. Our observations will help to optimize the source parameters\nfor producing the most intense molecular beam at the target velocity.\nKeywords : Buffer gas cooled beam source, molecular beam, phase-space distribution,\nelectric dipole moment of the electron\n‡Present address: Department of Chemistry, University of Basel, SwitzerlandarXiv:2401.16590v1 [physics.atom-ph] 29 Jan 2024Influence of source parameters on the phase-space distribution of a cryogenic beam 2\n1. Introduction\nMolecular radicals offer a number of unique possibilities for precision tests of\nfundamental physics theories [1–3] and quantum technology [4–7]. Traditionally, these\nmolecules are created using ovens [8], resulting in samples at relatively high temperatures\nwhich are of limited use for these applications. Rotationally and translationally\ncold samples of radicals have been produced by entraining laser-ablated species in a\nsupersonic expansion of a carrier gas. In this way bright, >109molecules per steradian\nper pulse in a single rotational level, and short, <20µs, beam pulses have been\ngenerated, see for instance [9] and references therein. The mean forward velocity of\nthese beams is determined by the carrier gas and is typically around 600 m/s when\nusing argon and 300 m/s for xenon [10].\nA radically different approach for creating intense beams of molecules and molecular\nradicals is the so-called cryogenic buffer gas beam source, first introduced by Maxwell\net al. [11] and further developed by Patterson et al. [12], van Buuren et al. [13], Barry\net al. [14], Hutzler et al. [15, 16] and others. In this method, molecules are introduced\ninto a cold cell by a capillary [11–13, 17, 18], by laser ablation of a target containing\na precursor [11, 12, 14, 15, 19–30] or by letting laser ablated atoms react with a donor\ngas [22, 31–34]. The hot molecules are cooled by collisions with cold helium or neon\nbuffer gas. After many collisions, a fraction of the molecules escapes the cell to form\na molecular beam. The dimensions of the cell and the flow rate of the buffer gas\ndetermine the pressure within and are chosen in such a way that a significant fraction\nof the molecules is thermalized before hitting the wall of the cell. It was shown by\nPatterson et al. [12] that by operating the source at a high flow rate, the molecules are\nmore efficiently extracted from the cell, resulting in a more intense molecular beam. At\nthe same time, the higher flow rate leads to a supersonic boost at the exit of the cell,\nresulting in a faster molecular beam. Typical beam intensities above 1011molecules per\nsr per pulse in a low rotational state have been reported with forward velocities below\n170 m/s [14, 15]. In order to have an efficient extraction while avoiding a supersonic\nboost to occur, two-stage cryogenic buffer cells have been investigated [12, 19]. Truppe et\nal.[31] introduced a cell design that was optimized for creating relatively short ( ∼300µs)\nbut intense molecular beams.\nOver the last few years, we have constructed a pulsed cryogenic buffer gas cooled\nbeam source that provides barium monofluoride (BaF) molecules for an experiment that\nwill search for the electron’s electric dipole moment ( eEDM) [35]. In the experiment, the\nBaF molecules will be decelerated to below 30 m/s using a 4.5 m long travelling-wave\nStark decelerator [36]. There is a trade-off between the deceleration strength and the\nacceptance of the decelerator [37]. In order to decelerate a reasonable fraction of the\nbeam with the current setup, we need a bright beam with an initial velocity not too far\nabove 200 m/s. Note that, the required length of the decelerator scales with the velocity\nsquared; if the velocity of the initial beam would be 230 m/s instead of 200 m/s, we\nwould need a decelerator with a length of 6 m, to have the same acceptance. Therefore,Influence of source parameters on the phase-space distribution of a cryogenic beam 3\nunderstanding what determines the velocity of our source and how the source can be\noptimized for slow beams is of the utmost importance.\nRecently we have shown a sensitive method to determine the forward velocity of\na buffergas cooled beam of barium fluoride molecules as a function of time after the\nablation pulse, i.e. the longitudinal phase-space distribution of the beam, with high\nresolution [38]. Here we present a detailed study of the phase-space distribution as a\nfunction of various source parameters. Our paper is organized as follows: in section 2, we\nintroduce our buffer gas beam source in detail, followed by an explanation of the so-called\nsudden freeze model that predicts the dependence of the velocity on the temperature\nand pressure of the buffer gas, and a recap of our detection method. In section 3, we\npresent measurements of the phase-space distributions while varying source parameters\nsuch as the power of the ablation laser, the repetition rate, the temperature of the cell,\nthe flow rate of the SF 6, the operation time, the flow rate of the neon buffer gas and\nthe cell length. Finally, in section 4, we summarize our findings.\n2. Method\n2.1. Formation of the molecular beam\nFigure 1 shows a schematic of our setup including a top-view of our cryogenic buffer\ngas beam source and the laser beam paths. The design of our cryogenic source is based\non that of Truppe et al. [31] and Esajas [33]. The heart of our setup is formed by a\ncubical copper cell kept at a temperature of around 20 K using a 2-stage cryo-cooler\n(Sumitomo Heavy Industries, cold head RP-082B2S). A continuous flow of pre-cooled\nneon is passed through the cell with a flow rate of 10-70 standard cubic centimeter per\nminute ( sccm )§.\nThe beam of BaF molecules is formed in four steps: (i) Inside the cell, barium is\nablated by a pulsed Nd:YAG laser (532 nm, 5 ns pulse, 10 Hz, typically 8 mJ per pulse\nmeasured outside the vacuum) from a rotating solid Ba target. (ii) The barium atoms\nreact with sulphur hexafluoride (SF 6) molecules that are injected into the cell from a\ncopper tube kept at a temperature of 220 K and with a flow rate of typically 0.03 sccm .\n(iii) The BaF molecules created in this reaction are cooled via collisions with the neon\natoms. (iv) They form a molecular beam by expanding through a 4.5 mm diameter\norifice into the vacuum. The cell is surrounded by a copper and an aluminium heat\nshield, at temperatures of 6 K and ∼30 K, respectively. These heat shields also provide\nthe necessary pumping capacity to allow pressures on the order of 10−2mbar inside the\ncell, while maintaining a pressure below 10−6mbar in the molecular beam chamber.\n2.2. Beam velocity as a function of pressure and temperature of the carrier gas\nIn this section, we will summarize some theory required to understand the dependence\nof the molecular beam velocity on the temperature and pressure of the buffer gas.\n§1sccm = 4.48 ×1017particles/sInfluence of source parameters on the phase-space distribution of a cryogenic beam 4\n20KNe gas\nBa targetSF6\n(220 K)Pulsed Ablation laser30 mmtop view\nBalanced \nPDPeriscopeλ/4\n05 780 (mm)Ti:Sapph cw\n797 nm\nCoherent 899\nBeat note\nfreq comb\nTi:Sapph cw\n860 nm\nCoherent 899Lock to \nfreq comb\nPBS\nNPBSPMT\n-21\nFigure 1. Schematic view of the experimental setup showing the cryogenic buffer gas\nbeam source and the lasers used for absorption and fluorescence detection. Barium\nmonofluoride molecules are created by letting sulphur hexafluoride molecules react with\nbarium that is ablated from a solid barium rod inside a copper cell that is cooled to\n20 K. The molecules cool by collision with neon gas inside the cell, and expand through\na 4.5 mm diameter orifice to form a molecular beam. The distance between the target\nand the exit of the cell can be adjusted by inserting an extension tube between the\ncell body and the front plate. In the standard cell, the extension tube is absent and\nthe distance between the target and the exit is 11 mm, the figure depicts the situation\nwhen a 10 mm long tube is inserted (indicated in gray). The phase-space distribution\nof the beam is recorded 780 mm after the cell using a two-step laser excitation scheme.\nFluorescence back to the ground state is measured using a photomultiplier tube (PMT).\nIf the source is operated at a very low buffer gas flow rate (for our cell well below\n10SCCM ) and hence low density, the effusive regime prevails and the molecules leave\nthe exit without colliding with the buffer gas. Consequently, the velocity distribution\nof the molecules in the beam reflects the temperature inside the cell, corresponding\nto 52 m/s for BaF molecules and 145 m/s for neon atoms. In practice, we use a\nmuch higher flow rate to ensure that the molecules are entrained in the buffer gas\nand are pumped out of the cell before they have a chance to diffuse to the cell walls.\nIn this so-called hydrodynamic regime many collisions between neon atoms and BaF\nmolecules occur while they leave the cell. As the pressure inside the cell is higher than\noutside, these collisions lead to a net force that accelerates the neon atoms and barium\nfluoride molecules along the beam axis and lowers the longitudinal velocity spread of\nthe beam. In this situation, the velocity of the molecules in the beam depends both on\nthe temperature of the cell and the density of the neon gas. To make this statement\nmore quantitative, we summarize here a derivation to relate the temperature, velocity\nand flow rate, which is based on the derivations of Pauly [39] and Hutzler et al. [15].\nAn isentropic expansion of a high-pressure gas into a vacuum leads to the conversion\nof internal energy into directed flow energy. When the distance from the source is largeInfluence of source parameters on the phase-space distribution of a cryogenic beam 5\ncompared to the diameter of the exit of the cell, z≫dexit, the density in the beam,\nn(z) decreases quadratically with distance:\nn(z)≈Cn0d2\nexit\nz2(1)\nwith n0the density in the cell and Cis a constant that is ∼0.25 for an effusive beam\nand∼0.15 for a supersonic beam [39]. The density in the cell is a function of the flow\nrate, f, the size of the aperture A=πd2\nexit/4, and the mean velocity of the beam, v,\nand is given by [14]:\nn0=4f\nAv. (2)\nThe decrease in density is accompanied by an increase in the most probable velocity,\nvmp(z), and a decrease in the temperature, T(z), along the beam. For a mono-atomic\ngas such as neon, it can be derived that [40]:\nvmp(z) =v∞\"\n1− T(z)\nT0!#1/2\n=v∞\n1− n(z)\nn0!2/3\n1/2\n, (3)\nwith T0being the stagnation temperature of the gas in the cell. If all energy is\nconverted, the temperature of the beam becomes zero, and the forward velocity becomes\nv∞(T0) =q\n5kBT0/m, with mthe mass of neon. In practice, the conversion process\nbecomes increasingly slower while the density in the beam – and hence the collision\nrate – becomes smaller with distance from the exit of the cell. Consequently, the most\nprobable velocity of the beam will be smaller than v∞. In the so-called ‘sudden freeze’\nmodel [39], it is assumed that the expansion stops abruptly at a certain distance from\nthe source, from which point the molecules travel in straight lines. The exact position,\nz0, of this ‘quitting surface’ is found by setting Z2, the integral over the remaining\ntwo-body collisions after passing this surface, equal to 1:\nZ2=Z∞\nz0/dexitdZ2= 0.0465s\n8\nπσeffn0d8/3\nexitz−5/3\n0= 1, (4)\nwith σeffbeing the effective (temperature averaged) cross-section. The numerical\nconstant in the equation can be found from simulations [39], but its exact value is\nunimportant for our purpose. Combining the above equations with (2) that relates the\ndensity in the cell to the flow rate, f, we find:\nvmp(f, T 0) =v∞(T0)\n1−a f\nv∞(T0)!−4/5\n1/2\n, (5)\nwith a= 0.33(σeff/dexit)−4/5. This relation can be used to determine the temperature of\nthe gas inside the cell, T0, from measurements of the mean velocity as a function of the\nflow rate. ∥\n∥In this derivation, the flow velocity of the neon gas, which is 7-10 m/s in our experiments, is neglected.Influence of source parameters on the phase-space distribution of a cryogenic beam 6\n2.3. Detection\nTo monitor the performance of the source, the BaF molecules are detected using\nabsorption directly behind the cell on the X2Σ+→A2Π1/2transition using ∼1µW\nof light at 860 nm in a 1 mm diameter beam. The absorption signal can be converted\ninto an absolute number by taking into account the spatial and velocity distributions\nof the beam in the longitudinal and transverse directions, using a procedure that is\nsimilar to the one described by Wright et al. [41]. At the reference settings (to be\ndiscussed later), the peak absorption is typically 10% (double pass), which corresponds\nto 1.9(6)×1010BaF molecules in the N= 0 state per pulse and 1 .3(5)×1011molecules\nper sr per pulse. At a distance of 780 mm from the source, the molecules are excited by\nlight from two Ti:Sapphire lasers (Coherent 899) that are referenced to a frequency comb.\nOne of the lasers is aligned perpendicular to the molecular beam and is resonant with\ntheX2Σ+→A2Π1/2transition at 860 nm, while the other laser is aligned to be counter-\npropagating with respect to the molecular beam and resonant with the A2Π1/2→D2Σ+\ntransition around 797 nm. The frequency of this second laser is red-shifted with respect\nto the transition frequency to compensate for the Doppler shift. From this detuning, we\ninfer the longitudinal velocity of the molecules. More information on this method can\nbe found in [38].\nOnce excited to the D-state, part of the molecules will decay back to the ground\nstate by emitting a photon at 413 nm which is efficiently detected using a photomultiplier\ntube (PMT). A 40 nm wide band-pass filter around 400 nm is used to filter out\nscattered photons from the laser beams and unwanted fluorescence, resulting in a nearly\nbackground-free detection [42].\nIn order to be able to compare phase space distributions taken at different settings,\nwe sum the reconstructed velocity distribution at the exit of the cell over a time interval\nthat increases from 2 µs at the beginning of the pulse to about 250 µs at the end of\nthe pulse and fit these with a Gaussian function. The mean velocity is shown as the\nblue data points that overlay the measured phase space distribution. The error bars\nrepresent the uncertainty from the fit.\n3. Characterization of the cryogenic buffer gas beam source\nIn this section, we will discuss the dependency of the longitudinal phase-space\ndistribution of the molecular beam on various source parameters in order to understand\nthe dynamics within the cell. We will vary these parameters one at a time around the\nvalues used for obtaining figure 2, which we will refer to as the reference values. In\nsection 3.1, we will study the influence of the ablation power, the repetition rate, the\ntemperature of the cell and the flow rate of the SF 6. In section 3.2, we will discuss the\ninfluence of the operation time. In section 3.3, we will discuss the influence of the neon\nflow rate and finally in section 3.4, we will investigate how the phase space distribution\nof the beam changes when the length of the cell is increased.Influence of source parameters on the phase-space distribution of a cryogenic beam 7\n0.0 0.5 1.0 1.5 2.0\nTime after ablation (ms)180200220240260280Velocity (m/s)\n0.00.20.40.60.81.0\nFigure 2. Phase space distribution of the molecular beam reconstructed at the exit\nof the source. The blue data points represent the mean velocities resulting from a\nGaussian fit to the data at specific times. The horizontal line observed at 250 m/s\nis due to difficulties in determining the frequency of the laser that drives the A−D\ntransition when its beat note with the frequency comb is equal to the repetition rate\nof the frequency comb.\n3.1. Influence of the ablation pulse energy and repetition rate, cell temperature and\nSF6flow rate\nFigure 3(a) shows the mean velocity of BaF molecules as a function of time after the\nablation pulse while using ablation pulse energies of 8 mJ (blue data points) and 4 mJ\n(red data points). As observed, the velocity first increases and reaches a maximum\n∼0.1 ms after the ablation pulse and then decreases. At 8 mJ, the velocity increases to\nabove 250 m/s, whereas at 4 mJ, the velocity peaks at about 220 m/s. The number of\nmolecules exiting the source is decreased by about a factor of 3 when the pulse energy\nis decreased from 8 mJ to 4 mJ.\nThe fact that the velocity of the molecules is lower in the trailing end of the\nmolecular beam does not seem surprising, given that barium fluoride molecules are\nproduced at very high temperatures [43] and require a minimum number of about\n50 collisions [44] before being cooled by the cold neon buffer gas. Naively, we may\nexpect that the molecules that exit the cell shortly after the ablation pulse have had\nless collisions and are faster than molecules that exit the cell later. This is however NOT\nwhat we see. Our measurements show that molecules leaving the cell immediately after\nthe ablation pulse have comparable velocities to those at the tail of the pulse, while\nthose that exit the cell 50-200 µs after the ablation pulse, are faster. We deduce from\nthis that, at any time during the pulse , the velocity of the BaF molecules reflects the\ntemperature of the neon gas, i.e., the BaF molecules are always in thermal equilibrium\nwith the buffergas. The observed correlation between the velocity and time reflects\nthe sharp increase in temperature of the buffer gas due to the ablation plume and theInfluence of source parameters on the phase-space distribution of a cryogenic beam 8\n\b\u0006\b\b\u0006\f\t\u0006\b\t\u0006\f\n\u0006\b\n\u0011\u0017\u0019\u0015\u0003\u0012\u0016\u001f\u0015\u001d\u0003\u0012\u0013\u0018\u0012\u001f\u0017\u001b\u001a\u0003\u0004\u0019\u001e\u0005\n\b\b\n\n\b\n\u000b\b\n\r\b\u000f\u0015\u0012\u001a\u0003 \u0015\u0018\u001b\u0014\u0017\u001f!\u0003\u0004\u0019\u0007\u001e\u0005\u0010\u0015\u001c\u0015\u001f\u0017\u001f\u0017\u001b\u001a\u0003\u001d\u0012\u001f\u0015\n\t\b\u0003\u000e\"\n\n\u0003\u000e\"\n\b\u0006\b \b\u0006\r \t\u0006\b \t\u0006\r \n\u0006\b\n\u0013\u0019\u001b\u0017\u0003\u0014\u0018 \u0017\u001e\u0003\u0014\u0015\u001a\u0014 \u0019\u001d\u001c\u0003\u0004\u001b\u001f\u0005\n\b\b\n\n\b\n\f\b\u0011\u0017\u0014\u001c\u0003!\u0017\u001a\u001d\u0016\u0019 #\u0003\u0004\u001b\u0007\u001f\u0005\u0012\u0010\u000e\u0003\u0018\u001a\u001d\"\u0003\u001e\u0014 \u0017\n\b\u0006\b\t\u0003\u0012\u000f\u000f\u0011\n\b\u0006\b\u000b\u0003\u0012\u000f\u000f\u0011\n\b\u0006\t\u0003\u0012\u000f\u000f\u0011\n\b\u0006\u000b\u0003\u0012\u000f\u000f\u0011\b\u0006\b \b\u0006\f \t\u0006\b \t\u0006\f \n\u0006\b\n\u0012\u0019\u001b\u0016\u0003\u0013\u0017!\u0016\u001f\u0003\u0013\u0014\u001a\u0013!\u0019\u001d\u001c\u0003\u0004\u001b \u0005\n\b\b\n\n\b\n\u000b\b\n\r\b\u0011\u0016\u0013\u001c\u0003#\u0016\u001a\u001d\u0015\u0019!$\u0003\u0004\u001b\u0007 \u0005\u000f\u0014\u001a\u0013!\u0019\u001d\u001c\u0003\u001e\"\u001a \u0016\u0003\u0016\u001c\u0016\u001f \u0018$\n\u000e\u0003\u001b\u0010\u0003\u001e\u0016\u001f\u0003\u001e\"\u001a \u0016\n\u000b\u0003\u001b\u0010\u0003\u001e\u0016\u001f\u0003\u001e\"\u001a \u0016\n\b\u0006\b\b\u0006\f\t\u0006\b\t\u0006\f\n\u0006\b\n\u0011\u0017\u0019\u0015\u0003\u0012\u0016\u001f\u0015\u001d\u0003\u0012\u0013\u0018\u0012\u001f\u0017\u001b\u001a\u0003\u0004\u0019\u001e\u0005\n\b\b\n\n\b\n\u000b\b\u0010\u0015\u0012\u001a\u0003!\u0015\u0018\u001b\u0014\u0017\u001f\"\u0003\u0004\u0019\u0007\u001e\u0005\u000e\u0015\u0018\u0018\u0003\u001f\u0015\u0019\u001c\u0015\u001d\u0012\u001f \u001d \u0015\n\n\b\u0003\u000f\n\t\r\u0003\u000f(a) (b)\n(c) (d)\nFigure 3. Mean velocity of the molecular beam as a function of time for different\nvalues of (a) the ablation pulse energy, (b) the repetition rate, (c) the cell temperature\nand (d) the SF 6flow rate as indicated in the figure. All measurements are taken using\na neon buffer gas flow rate of 20 sccm . The blue data points in all sub-figures are\ndifferent measurements taken at the reference settings. The dashed lines results from\nfits using (6).\nsubsequent decrease in temperature via conduction to the walls and collisions with the\ncold neon gas that is continuously flown into the cell. This hypothesis is consistent with\nthe∼50µs thermalisation time expected at the neon densities in our cell and also with\nearlier observations of Skoff et al. [45]. Note that the cell body is not expected to heat\nup significantly by a single ablation pulse ¶, hence we conclude that the limiting factor\nis the heat conduction of the neon gas to the cell walls. This will be discussed further\nin section 3.2. It may be observed that even after 1.5 ms the mean velocity measured\nwith an ablation energy of 8 mJ per pulse is still slightly higher than that measured\nwith 4 mJ per pulse. We will come back to this difference in section 3.4.\nFigure 3(b) shows the mean velocity of BaF molecules as a function of time\nmeasured at the same ablation energy of 8 mJ per pulse, but with a repetition rate\nof 10 Hz (blue data points) or 2 Hz (red data points). As expected, the measured\nvelocities early in the pulse are very similar, but the velocity in the tail of the molecular\npulse drops to a slightly lower velocity, indicating that the temperature of the cell may\n¶From the mass of the cell body and the heat capacity of copper at 20 K [46], we estimate that the\ntemperature of the cell body increases by ∼5 mK due to a single laser pulse with an energy of 8 mJ.Influence of source parameters on the phase-space distribution of a cryogenic beam 9\nbe slightly lower when operated at 2 Hz instead of 10 Hz.\nFigure 3(c) shows the mean velocity of BaF molecules as a function of time measured\nwhen the copper cell is kept at 20 K (blue data points) or 17 K (red data points).\nAgain, the measured velocities early in the pulse are very similar, but the velocity in\nthe trailing end of the molecular pulse drops to a lower velocity when the cell is kept\nat a lower temperature. This shows that the source is ideally operated at the lowest\npossible cell temperature. The minimal temperature is determined by the requirement\nthat the pressure anywhere in the system is above the vapour pressure of neon at that\ntemperature. We observe that at a cell temperature below 20 K, the neon line becomes\ncompletely clogged within ∼1 hour, which we attribute to a small kink in the neon\nsupply line.\nFigure 3(d) shows the mean velocity of BaF molecules as a function of time\nmeasured when the SF 6flow rate is varied from 0.01 sccm to 0.3 sccm . When the SF 6\nflow rate is kept below 0.1 sccm , the heat introduced by the SF 6that is injected into\nthe cell through a copper tube kept at a temperature of 220 K, is apparently negligible.\nNote that the number of barium fluoride molecules produced with these flow rates is\nsimilar.\nIf it is assumed that, after the initial rise, the temperature decreases exponentially,\nthe velocity of the molecular pulse can be fitted to a simple exponential function of the\nform:\nv(t) =q\nv2\nf+ (v2\ni−v2\nf)e−t/τ, (6)\nwith viandvfbeing the initial and final velocities, respectively, and τa characteristic\ntime constant. These fits are shown as the dashed lines in figure 3. As may be observed,\nthe fitted characteristic time constants between the sets of measurements presented\nin panels (a)-(d) are rather different, which we blame on the cell being operated for\nextended times. In the next section, we will study this effect in more detail.\n3.2. Influence of the operation time\nIn order to study the effect of the operation time, we thoroughly cleaned the copper\ncell using acetic acid and subsequently measured the phase space distribution for up to\n12 hours at the same setting. After about 4.5 hours of continuous operation and again\nafter about 9 hours, the source was heated up to 295 K. Figure 4(a) shows measurements\ntaken after operating the source for 0.5 hour to 10 hours. The dashed lines, also shown\nin the figure, are fits to the data using (6). The inset shows the resulting time constants\nfrom these fits as a function of operation time. During the approximately 12 hours of\noperation, the measured time constant increased from below 100 µs to above 1 ms, which\nwe attribute to barium, barium-sulfides and other reaction products, covering the walls\nof the cell. This dust decreases the thermal conductivity between the cell and the neon\nbuffer gas. Cleaning the cell resets the source, while simply heating the cell to remove\nneon and SF 6ice, has no effect on the measured velocity distribution. Figures 4(b)\nand (c) show photographs of the cell without front aperture, taken before and afterInfluence of source parameters on the phase-space distribution of a cryogenic beam 10\n\b\u0006\b \b\u0006\r \t\u0006\b \t\u0006\r \n\u0006\b\n\u0011\u0018\u001a\u0015\u0003\u0012\u0016 \u0015\u001e\u0003\u0012\u0013\u0019\u0012 \u0018\u001c\u001b\u0003\u0004\u001a\u001f\u0005\n\b\b\n\t\b\n\n\b\n\u000b\b\n\f\b\n\r\b\n\u000e\b\u000f\u0015\u0012\u001b\u0003!\u0015\u0019\u001c\u0014\u0018 \"\u0003\u0004\u001a\u0007\u001f\u0005\u0010\u001d\u0015\u001e\u0012 \u0018\u001c\u001b\u0003 \u0018\u001a\u0015\n\b\u0006\r\u0017\n\n\u0006\b\u0017\n\u000e\u0006\b\u0017\n\t\b\u0006\b\u0017\n\u0007 \u000b \b\u0007\n\r\u001a\u0012\u001b\u000f\u001d\u0015\u0019\u0018\u0003\u001d\u0015\u0017\u0012\u0003\u0004\u0014\u0005\u0007\u0006\u0007\u0007\u0006\u000b\b\u0006\u0007\b\u0006\u000b\u000e\u0015\u0017\u0012\u0003\u0010\u0019\u0018\u001c\u001d\u000f\u0018\u001d\u0003\u0004\u0017\u001c\u0005\f\u0019\u0019\u0016\u0015\u0018\u0013\u0003\u001d\u0015\u0017\u0012\u0003\u0010\u0019\u0018\u001c\u001d\u000f\u0018\u001d\n\u0011\u000f\u001e\u0003\b\n\u0011\u000f\u001e\u0003\t\n\u0011\u000f\u001e\u0003\n(c)(b) (a)\nFigure 4. The effect of operation time. In (a), the mean velocity as a function\nof time is shown after operating the source for a time period as indicated. When\nthe source is operated for extended times, the exponential decrease of the velocity\nbecomes increasingly slower, resulting in a larger time constant as shown in the inset.\n(b) and (c) are two pictures of the cell with the front plate removed that are taken\nbefore and after measuring the data shown in (a), respectively. It is clear that the cell\nwall becomes covered with a substantial layer of dust.\nmeasuring the date shown in figure 4(a), respectively, clearly showing the contamination\nthat builds up on the cell wall. The fact that the time constant changes during the\noperation of the source complicates the systematic study of the source considerably,\nforcing us to change parameters as quickly as possible while keeping an acceptable\nsignal-to-noise ratio. Similar observations were made by Wright et al. [41] in their\nexperiments on AlF. Using NF 3instead of SF 6as a fluor donor, they observed the same\nyield but with a significantly slower cell degradation. In our experiments on BaF, NF 3\ngave a significantly smaller yield and was not studied further.\n3.3. Influence of the neon flow rate\nIn this section we will discuss the effect of the neon flow rate. Figure 5 shows the\nreconstructed phase-space distributions at the exit of the cell for buffer gas flow rates\nbetween 10 and 70 sccm (a-g) along with the signal integrated over velocity (h) or\ntime (i). Five trends are observed with higher flow rate: (i) The intensity and (ii)\nthe pulse length of the molecular beam increase, as does the (iii) the characteristic\ntime constant that describes the exponential decrease in temperature in the tail of the\npulse. Furthermore, (iv) the mean velocity of the beam increases, while the (v) velocity\nspread decreases. The first two effects, which are most obvious from figure 5(h), are\nattributed to the increased diffusion time at higher neon densities, which reduces the\nloss of molecules frozen to the cell walls and leads to more efficient extraction from\nthe cell. These effects have been discussed in detail by Patterson and Doyle [12]. The\nthird effect is expected from the fact that the thermal diffusivity of the neon buffer gas\ndecreases with increased density. Finally, the fourth and fifth effects are due to the beam\nbecoming more supersonic when the neon density in the cell is increased and will beInfluence of source parameters on the phase-space distribution of a cryogenic beam 11\n00.8 1.6\nTime (ms)150200250300Velocity (m/s)\n×4.2(a)10 SCCM\n00.8 1.6\nTime (ms)×1.7(b)20 SCCM\n00.8 1.6\nTime (ms)×1.4(c)30 SCCM\n00.8 1.6\nTime (ms)×1.4(d)40 SCCM\n00.8 1.6\nTime (ms)(e)50 SCCM\n00.8 1.6\nTime (ms)(f)60 SCCM\n00.8 1.6\nTime (ms)(g)70 SCCM\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nTime after ablation (ms)0.00.20.40.60.8LIF signal (arb. units)(h)10 SCCM neon\n20 SCCM neon\n30 SCCM neon\n40 SCCM neon\n50 SCCM neon\n60 SCCM neon\n70 SCCM neon\n150 200 250 300\nVelocity (m/s)0.00.51.01.5LIF signal (arb. units)(i)\nFigure 5. Phase-space distribution of the cryogenic beam for different neon flow rates.\nPanels (a-g) show the reconstructed phase-space distribution at the source exit, where\nt= 0 corresponds to the time that the ablation laser is fired. At low neon flow rates,\nthe intensity has been multiplied with a factor as indicated in the panel. Panels (h)\nand (i) show the reconstructed intensity at different flow rates integrated over velocity\nand time, respectively. A number of trends are observed with higher flow rate, as\ndescribed in the main text.\nanalysed in more detail in the remainder of this section. Before doing this, there is one\nmore observation worth noting. At higher flow rates, a dip in intensity is observed in\nthe time of flight curves presented in figure 5(h) around 0.8 ms after the ablation pulse.\nIt has been suggested [47] that the molecules that exit the cell before (after) 0.8 ms are\nformed from barium atoms that were ejected from the rod along (against) the direction\nof the buffer gas flow inside the cell.\nAs discussed in section 2.2, the mean velocity and translational temperature of the\nmolecular beam depend on the temperature and pressure of the neon buffer gas inside\nthe cell, which in turn depends on the neon flow rate. The blue data points in figure 6(a)\nshow the final forward velocity, vf, derived from fitting (6) to the mean velocity, as a\nfunction of time for each of the phase-space distributions presented in figures 5(a-g).\nThe error bar of the data displays the uncertainty of this fit. The red solid line is a fit\nof (5) to this data from which we determine T0and the terminal velocity v∞(T0), shown\nas the orange dotted line+. In figure 6(b), the blue data points show the corresponding\nvelocity spreads translated into a temperature. The solid red line shows the expected\ntranslational temperature from (3) at T0found from the fit to the data in (a). As may\n+From this fit we find a collision cross-section σNe−Ne= 1.9×10−15cm2, which is slightly larger than\nfound by a simple hard-sphere model: σhs,Ne−Ne= 7.5×10−16cm2[39].Influence of source parameters on the phase-space distribution of a cryogenic beam 12\n10 20 30 40 50 60 70\nNeon flow (SCCM)180200220Final forward velocity (m/s)\n(a)\nvmp(f,T0=25.9 K)\nv∞(T0=20 K)\nv∞(T0=25.9 K)\nMeas. mean vel.\n10 20 30 40 50 60 70\nNeon flow (SCCM)246Translational temp. (K)\n(b) T(f,T0=25.9 K)\nMeas. trans. temp.\n10 20 30 40 50 60 70\nNeon flow (SCCM)0.00.51.01.5# molecules (arb. units)\n(c)\nFigure 6. Velocity (a), temperature (b) and intensity (c) as a function of flow rate.\n(a) The blue data points show the final velocity, vf, in the tail of the molecular beam\nas a function of the neon flow rate. The solid/dashed red line shows a fit of (5) to\nthe data for flows between 20 and 70 sccm from which we determine the terminal\nvelocity v∞, shown as the dashed orange line, and the temperature of the buffer gas,\nT0, which is 25.9 K in this case. For completeness, the green dashed line shows the\nexpected terminal velocity at a cell temperature of 20 K. (b) The blue data points\nshow the measured translational temperature, together with the prediction at 25.9 K.\n(c) Number of BaF molecules in the X2Σ+, v= 0, N= 0, J= 1/2 ground state.\nbe observed, the model fits the measured mean velocities (and to a lesser extent) the\ntranslational temperature well, however, from the fit parameter, v∞(T), we find that the\nbuffer gas temperature in the tail of the molecular beam is 25.9 K, significantly above\nthe temperature of the cell of 20 K. This is a somewhat surprising result, given that\non these time scales the buffer gas appears to have reached thermal equilibrium with\nthe walls, while from the measurements performed at lower repetition rate, shown in\nfigure 3(b), it is seen that the temperature of the cell returned (close) to its set value\nwithin 100 ms. We conclude from this that some part of the cell remains hot during\nthe molecular pulse and only relaxes on a timescale 10-50 ms. We believe that it is in\nfact the barium target that remains hot. More evidence for this will be presented in the\nnext section.\nIt would be insightful to determine the temperature of the buffer gas not only at\nthe tail but at any time during the molecular beam pulse. However, this is complicated\nby the fact that the cooling rate towards the wall depends on the neon density in the cell\nand hence the flow rate. A rough estimate suggests that at its peak, the temperature\nof the buffer gas is increased to about 40 K at an ablation pulse energy of 8 mJ/pulse.\nFigure 6(c) shows the brightness of the beam of barium fluoride molecules in the\nX2Σ+, v= 0, N= 0, J= 1/2 ground state as function of the neon flow rate found by\nintegrating the phase-space distributions shown in figures 5a-g. As may be observed,\nthe number of molecules increases by about a factor of 4.7 when the neon flow rate\nis increased from 10 to 70 sccm . Note that, the sudden freeze model (presented in\nsection 2.2) predicts that the intensity does not depend strongly on the neon flow rate,\nas the slight increase of the relative population in the N= 0 state due to the lower\nrotational temperature at high neon flow rates, is compensated by a the increased\ndivergence of the beam [14]. The observed intensity increase at higher flow rate isInfluence of source parameters on the phase-space distribution of a cryogenic beam 13\n\b\u0006\b\b\u0006\f\t\u0006\b\t\u0006\f\n\u0006\b\n\u0010\u0018\u001a\u0014\u0003\u0011\u0015\u001f\u0014\u001d\u0003\u0011\u0012\u0019\u0011\u001f\u0018\u001c\u001b\u0003\u0004\u001a\u001e\u0005\t\r\b\n\b\b\n\t\b\n\n\b\n\u000b\b\u000f\u0014\u0011\u001b\u0003 \u0014\u0019\u001c\u0013\u0018\u001f!\u0003\u0004\u001a\u0007\u001e\u0005\u000e\u0014\u0019\u0019\u0003\u0019\u0014\u001b\u0016\u001f\u0017\n\t\t\u0003\u001a\u001a\u0003\u0013\u0014\u0019\u0019\n\n\t\u0003\u001a\u001a\u0003\u0013\u0014\u0019\u0019\n\u000b\t\u0003\u001a\u001a\u0003\u0013\u0014\u0019\u0019\n\b \t \n \u000b\n\u000f\u0017\u0019\u0013\u0003\u0010\u0014\u001e\u0013\u001c\u0003\u0010\u0011\u0018\u0010\u001e\u0017\u001b\u001a\u0003\u0005\u0019\u001d\u0006\b\u0007\b\b\u0007\n\b\u0007\f\b\u0007\r\b\u0007\u000e\t\u0007\b\u0004\u0003\u0019\u001b\u0018\u0013\u0012\u001f\u0018\u0013\u001d\u0003\u0005\u0010\u001c\u0011\u0007\u0003\u001f\u001a\u0017\u001e\u001d\u0006\u000f\u0017\u0019\u0013\u0003\u001b\u0014\u0003\u0014\u0018\u0017\u0015\u0016\u001e\n\t\t\u0003\u0019\u0019\u0003\u0012\u0013\u0018\u0018\n\n\t\u0003\u0019\u0019\u0003\u0012\u0013\u0018\u0018\n\u000b\t\u0003\u0019\u0019\u0003\u0012\u0013\u0018\u0018\n\n\u000e\t \u000b\t\t \u000b\u000e\t \f\t\t\n\u0011\u0016\u0018\u001b\u0014\u0017\u001e \u0003\u0005\u0019\b\u001d\u0006\t\u0007\t\t\u0007\u000b\t\u0007\r\t\u0007\u000f\t\u0007\u0010\n\u0007\t\u0004\u0003\u0019\u001b\u0018\u0016\u0014\u001f\u0018\u0016\u001d\u0003\u0005\u0012\u001c\u0013\u0007\u0003\u001f\u001a\u0017\u001e\u001d\u0006\u0011\u0016\u0018\u001b\u0014\u0017\u001e \u0003\u0015\u0017\u001d\u001e\u001c\u0017\u0013\u001f\u001e\u0017\u001b\u001a\n(a) (b) (c)\nFigure 7. (a) Mean velocity as a function of time, (b) time-of-flight and (c) velocity\ndistribution for three different cell lengths. The velocity in the tail of the molecular\npulse is seen to decrease significantly, while the intensity is comparable.\nattributed to the increased efficiency in extraction of the molecules from the cell [12].\nThe fact that the forward velocity and intensity follow the same trend is a coincident.\n3.4. Influence of the cell length\nSo far, all measurements presented in the paper were performed with the standard\ncell that has a distance of 11 mm between the ablation target and the source exit. In\nthis section, we will study what happens when the cell is extended to 21 or 31 mm.\nFigure 7(a) presents the mean velocity as a function of time, while (b) and (c) present\nthe intensity integrated over velocity or time, respectively. The ablation power, cell\ntemperature, neon and SF 6flow rates are set to the reference values. All measurement\nwere taken after the cell was operated for 2 hours. Increasing the cell length to 21 mm\nresults in a longer molecular pulse that has a significantly lower velocity at the tail of the\nmolecular pulse, while the number of molecules in the beam is comparable. Extending\nthe cell further reduces the velocity but leads to a drop in the number of molecules by\nabout 10 %. Note that, with a neon flow rate of 20 sccm , the expected mean velocity\nfor molecules exiting our cell at 20 K is 178 m/s.\nWe attribute the observed dependence of the velocity on the length of the cell\non the occurrence of a heat gradient in the cell. After the ablation pulse, the heat\ndeposited in the ablated barium atoms is transferred to the buffer gas within 100 µs\nand is subsequently cooled away by the cell within a few ms. On the other hand, the\nheat deposited in the barium rod is transferred to the buffer gas at a slower rate and the\ntarget remains at elevated temperatures during the entire molecular pulse. Increasing\nthe distance between the target and the exit results in the neon gas at the exit being\ncloser to the cell temperature and the molecular beam being slower. This effect also\nexplains the observed dependence of the final velocity on the ablation power, as was\ndiscussed in section 3.1.\n4. Conclusions\nWe presented measurements of the phase-space distribution of a cryogenic buffer gas\nbeam of BaF. We observe a strong correlation of the mean forward velocity of the BaFREFERENCES 14\nmolecules at the time they exit the source which is attributed to the neon buffer gas\nbeing warmed up by the plume of hot atoms released from the target by the ablation\npulse and subsequently being cooled down via conduction to the cell walls. When\nthe cell is operated for a longer period of time, the walls of the cell become covered\nwith a layer of isolating dust which increases the time constant associated with the\nexponentially decreasing temperature of the neon gas. The barium target remains at\nelevated temperatures on a much longer time scale, resulting in a higher mean velocity\nthan expected from the sudden freeze model. This velocity can be lowered by extending\nthe length of the cell. Some of the observations above have been reported before, but\nour new method to accurately measure the phase-space distribution of the beam in\ncombination with the stability and reproducibility of our cell allowed us to analyse\nthese effects in great detail. As optimization of the source amounts to a compromize\nbetween brightness and a low forward velocity, a good understanding of the heating\nprocesses is pivotal for an optimal choice of the source parameters. In future work, we\nplan to also study the effect of the size and shape of the cell exit.\nAcknowledgments\nThe NL- eEDM consortium receives program funding (EEDM-166 and XL21.074) from\nthe Netherlands Organisation for Scientific Research (NWO). We thank Johan Kos, Rob\nKortekaas and Leo Huisman for technical assistance to the experiment. We acknowledge\nfruitful discussions with Mike Tarbutt and Stefan Truppe in the design of the cryogenic\nsource.\nReferences\n[1] M S Safronova, D Budker, D DeMille, D F J Kimball, A Derevianko, and C W\nClark. Search for new physics with atoms and molecules. Rev. Mod. Phys. , 90:25008,\n2018.\n[2] V Andreev, D G Ang, D DeMille, J M Doyle, G Gabrielse, J Haefner, N R Hutzler,\nZ Lasner, C Meisenhelder, B R O’Leary, C D Panda, A D West, E P West, and\nX Wu. Improved limit on the electric dipole moment of the electron. Nature ,\n562:355–360, 2018.\n[3] T S Roussy, L Caldwell, T Wright, W B Cairncross, Y Shagam, K B Ng,\nN Schlossberger, S Y Park, A Wang, J Ye, and E A Cornell. An improved bound\non the electron’s electric dipole moment. Science , 381:46–50, 2023.\n[4] D DeMille. Quantum Computation with Trapped Polar Molecules. Phys. Rev.\nLett., 88:4, 2002.\n[5] K-K Ni, S Ospelkaus, M H G de Miranda, A Pe’er, B Neyenhuis, J J Zirbel,\nS Kotochigova, P S Julienne, D S Jin, and J Ye. A High Phase-Space-Density Gas\nof Polar Molecules. Science , 322:231–235, 2008.REFERENCES 15\n[6] J A Blackmore, L Caldwell, P D Gregory, E M Bridge, R Sawant, J Aldegunde,\nJ Mur-Petit, D Jaksch, J M Hutson, B E Sauer, M R Tarbutt, and S L Cornish.\nUltracold molecules for quantum simulation: Rotational coherences in CaF and\nRbCs. Quantum Sci. Technol. , 4:014010, 2019.\n[7] P Yu, L W Cheuk, I Kozyryev, and J M Doyle. A scalable quantum computing\nplatform using symmetric-Top molecules. New J. Phys. , 21:093049, 2019.\n[8] N F Ramsey. Molecular Beams . Oxford Classic Texts in the Physical Sciences.\nOxford University Press, Oxford, New York, 1985.\n[9] P Aggarwal, H L Bethlem, A Boeschoten, A Borschevsky, K Esajas, Y Hao,\nS Hoekstra, K Jungmann, V R Marshall, T B Meijknecht, M C Mooij, R G E\nTimmermans, A Touwen, W Ubachs, L Willmann, Y Yin, and A Zapara. A\nsupersonic laser ablation beam source with narrow velocity spreads. Rev. Sci.\nInstrum. , 92:33202, 2021.\n[10] M R Tarbutt, J J Hudson, B E Sauer, E A Hinds, V A Ryzhov, V L Ryabov, and\nV F Ezhov. A jet beam source of cold YbF radicals. J. Phys. B , 35:5013–5022,\n2002.\n[11] S E Maxwell, N Brahms, R deCarvalho, D R Glenn, J S Helton, S V Nguyen,\nD Patterson, J Petricka, D DeMille, and J M Doyle. High-flux beam source for\ncold, slow atoms or molecules. Phys. Rev. Lett. , 95:173201, 2005.\n[12] D Patterson and J M Doyle. Bright, guided molecular beam with hydrodynamic\nenhancement. J. Chem. Phys. , 126:154307, 2007.\n[13] L D van Buuren, C Sommer, M Motsch, S Pohle, M Schenk, J Bayerl, P W H\nPinkse, and G Rempe. Electrostatic extraction of cold molecules from a cryogenic\nreservoir. Phys. Rev. Lett. , 102:33001, 2009.\n[14] J F Barry, E S Shuman, and D DeMille. A bright, slow cryogenic molecular beam\nsource for free radicals. Phys. Chem. Chem. Phys. , 13:18936–18947, 2011.\n[15] N R Hutzler, M F Parsons, Y V Gurevich, P W Hess, E Petrik, B Spaun, A C\nVutha, D DeMille, G Gabrielse, and J M Doyle. A cryogenic beam of refractory,\nchemically reactive molecules with expansion cooling. Phys. Chem. Chem. Phys. ,\n13:18976–18985, 2011.\n[16] N R Hutzler, H I Lu, and J M Doyle. The buffer gas beam: An intense, cold, and\nslow source for atoms and molecules. Chem. Rev. , 112:4803–4827, 2012.\n[17] D Patterson, J Rasmussen, and J M Doyle. Intense atomic and molecular beams\nvia neon buffer-gas cooling. New J. Phys. , 11:55018, 2009.\n[18] V Singh, A K Samanta, N Roth, D Gusa, T Ossenbr¨ uggen, I Rubinsky, D A\nHorke, and J K¨ upper. Optimized cell geometry for buffer-gas-cooled molecular-\nbeam sources. Phys. Rev. A , 97:32704, 2018.\n[19] H I Lu, J Rasmussen, M J Wright, D Patterson, and J M Doyle. A cold and slow\nmolecular beam. Phys. Chem. Chem. Phys. , 13:18986–18990, 2011.REFERENCES 16\n[20] N E Bulleid, S M Skoff, R J Hendricks, B E Sauer, E A Hinds, and M R Tarbutt.\nCharacterization of a cryogenic beam source for atoms and molecules. Phys. Chem.\nChem. Phys. , 15:12299–12307, 2013.\n[21] M T Hummon, M Yeo, B K Stuhl, A L Collopy, Y Xia, and J Ye. 2D Magneto-\nOptical Trapping of Diatomic Molecules. Phys. Rev. Lett. , 110:143001, 2013.\n[22] Y Zhou, D D Grimes, T J Barnum, D Patterson, S L Coy, E Klein, J S Muenter,\nand R W Field. Direct detection of Rydberg–Rydberg millimeter-wave transitions\nin a buffer gas cooled molecular beam. Chem. Phys. Lett. , 640:124–136, 2015.\n[23] L Santamaria, V D Sarno, P D Natale, M D Rosa, M Inguscio, S Mosca, I Ricciardi,\nD Calonico, F Levi, and P Maddaloni. Comb-assisted cavity ring-down spectroscopy\nof a buffer-gas-cooled molecular beam. Phys. Chem. Chem. Phys. , 18:16715–16720,\n2016.\n[24] X Li, L Xu, Y Yin, S Xu, Y Xia, and J Yin. Rotational relaxation of fluoromethane\nmolecules in low-temperature collisions with buffer-gas helium. Phys. Rev. A ,\n93:063407, 2016.\n[25] C J E Straatsma, M I Fabrikant, G E Douberly, and H J Lewandowski. Production\nof carbon clusters C 3to C 12with a cryogenic buffer-gas beam source. J. Chem.\nPhys. , 147:124201, 2017.\n[26] W Bu, T Chen, G Lv, and B Yan. Cold collision and high-resolution spectroscopy\nof buffer-gas-cooled BaF molecules. Phys. Rev. A , 95:32701, 2017.\n[27] G Z Iwata, R L McNally, and T Zelevinsky. High-resolution optical spectroscopy\nwith a buffer-gas-cooled beam of BaH molecules. Phys. Rev. A , 96:22509, 2017.\n[28] I Kozyryev, L Baum, K Matsuda, B L Augenbraun, L Anderegg, A P Sedlack, and\nJ M Doyle. Sisyphus Laser Cooling of a Polyatomic Molecule. Phys. Rev. Lett. ,\n118:173201, 2017.\n[29] R Albrecht, M Scharwaechter, T Sixt, L Hofer, and T Langen. Buffer-gas cooling,\nhigh-resolution spectroscopy, and optical cycling of barium monofluoride molecules.\nPhys. Rev. A , 101:013413, 2020.\n[30] J C Shaw and D J McCarron. Bright, continuous beams of cold free radicals. Phys.\nRev. A , 102:41302, 2020.\n[31] S Truppe, M Hambach, S M Skoff, N E Bulleid, J S Bumby, R J Hendricks, E A\nHinds, B E Sauer, and M R Tarbutt. A buffer gas beam source for short, intense\nand slow molecular pulses. J. Mod. Opt. , 65:648–656, 2018.\n[32] S Xu, M Xia, R Gu, C Pei, Z Yang, Y Xia, and J Yin. Cold collision and the\ndetermination of the X2Σ1/2(v= 1, N= 1)→A2Π1/2(v′= 0, J′= 1/2) frequency\nwith buffer-gas-cooled MgF molecules. J. Quant. Spectrosc. and Radiat. Transfer ,\n236:106583, 2019.\n[33] K Esajas. Intense Slow Beams of Heavy Molecules to Test Fundamental\nSymmetries . PhD thesis, University of Groningen, 2021.REFERENCES 17\n[34] S Hofs¨ ass, M Doppelbauer, S C Wright, S Kray, B G Sartakov, J P´ erez-R´ ıos,\nG Meijer, and S Truppe. Optical cycling of AlF molecules. New J. Phys. , 23:075001,\n2021.\n[35] P Aggarwal, H L Bethlem, A Borschevsky, M Denis, K Esajas, P A B Haase, Y Hao,\nS Hoekstra, K Jungmann, T B Meijknecht, M C Mooij, R G E Timmermans,\nW Ubachs, L Willmann, and A Zapara. Measuring the electric dipole moment of\nthe electron in BaF. Eur. Phys. J. D , 72:197, 2018.\n[36] P Aggarwal, Y Yin, K Esajas, H L Bethlem, A Boeschoten, A Borschevsky,\nS Hoekstra, K Jungmann, V R Marshall, T B Meijknecht, M C Mooij, R G E\nTimmermans, A Touwen, W Ubachs, and L Willmann. Deceleration and Trapping\nof SrF Molecules. Phys. Rev. Lett. , 127:173201, 2021.\n[37] S Y T van de Meerakker, H L Bethlem, N Vanhaecke, and G Meijer. Manipulation\nand control of molecular beams. Chem. Rev. , 112:4828–4878, 2012.\n[38] M C Mooij, H L Bethlem, A Boeschoten, A Borschevsky, T H Fikkers, S Hoekstra,\nJ W F van Hofslot, K Jungmann, V R Marshall, T B Meijknecht, R G E\nTimmermans, A Touwen, W Ubachs, and L Willmann. A novel method to\ndetermine the phase-space distribution of a pulsed molecular beam (submitted).\n2024.\n[39] H Pauly. Atom, Molecule, and Cluster Beams I , volume 28. Springer Berlin\nHeidelberg, Berlin, Heidelberg, 2000.\n[40] S DePaul, D Pullman, and B Friedrich. A pocket model of seeded supersonic beams.\nJ. Phys. Chem. , 97:2167–2171, 1993.\n[41] S C Wright, M Doppelbauer, S Hofs¨ ass, Schewe H C, B Sartakov, G Meijer, and\nS Truppe. Cryogenic buffer gas beams of AlF, CaF, MgF, YbF, Al, Ca, Yb and\nNO – a comparison. Mol. Phys. , 121:e2146541, 2022.\n[42] D H Murphree. Nuclear Spin-Dependent Parity Nonconservation in Diatomic\nMolecules . PhD thesis, Yale University, 2009.\n[43] G M Davis, M C Gower, C Fotakis, T Efthimiopoulos, and P Argyrakis.\nSpectroscopic studies of ArF laser photoablation of PMMA. Appl. Phys. A , 36:27–\n30, 1985.\n[44] R deCarvalho, J M Doyle, B Friedrich, T Guillet, J Kim, D Patterson, and J D\nWeinstein. Buffer-gas loaded magnetic traps for atoms and molecules: A primer.\nEur. Phys. J. D , 7:289–309, 1999.\n[45] S M Skoff, R J Hendricks, C D J Sinclair, J J Hudson, D M Segal, B E Sauer, E A\nHinds, and M R Tarbutt. Diffusion, thermalization, and optical pumping of YbF\nmolecules in a cold buffer-gas cell. Phys. Rev. A , 83:23418, 2011.\n[46] N J Simon, E S Drexler, and R P Reed. Properties of Copper and Copper Alloys\nat Cryogenic Temperatures . 1992.\n[47] S Truppe. private communication." }, { "title": "2401.16647v1.A_Family_of_Low_Complexity_Binary_Codes_with_Constant_Hamming_Weights.pdf", "content": "A Family of Low-Complexity Binary Codes with\nConstant Hamming Weights\nBirenjith Sasidharan1*, Emanuele Viterbo1and Son Hoang Dau2\n1*ECSE Dept., Monash University, Clayton, Victoria, Australia.\n2RMIT University, Melbourne, Victoria, Australia.\n*Corresponding author(s). E-mail(s):\nbirenjith.padmakumarisasidharan@monash.edu;\nContributing authors: emanuele.viterbo@monash.edu;\nsonhoang.dau@rmit.edu.au;\nAbstract\nIn this paper, we focus on the design of binary constant-weight codes that admit\nlow-complexity encoding and decoding algorithms, and that have size M= 2kso\nthat codewords can conveniently be labeled with binary vectors of length k. We\nconstruct a family of (n= 2ℓ, M= 2k, d= 2) constant-weight codes C[ℓ, r]\nparameterized by integers ℓ≥3and1≤r≤ ⌊ℓ+3\n4⌋, by encoding information\nin the gaps between successive 1’s of a vector. The code has weight w=ℓand\ncombinatorial dimension kthat scales quadratically with ℓ. The time complexity\nof the encoding algorithm is linear in the input size k, and that of the decoding\nalgorithm is poly-logarithmic in the input size n, discounting the linear time\nspent on parsing the input. Encoding and decoding algorithms of most of the\nsimilar codes known in either information-theoretic or combinatorial literature\nrequire computation of large number of binomial coefficients that is prohibitively\nexpensive either in time-complexity if done online, or space-complexity if stored\nas lookup tables. Our algorithms fully eliminate the need to evaluate binomial\ncoefficients. While the code has a natural price to pay in k, it performs fairly\nwell against the information-theoretic upper bound ⌊log2\u0000n\nw\u0001\n⌋. When ℓ= 3,\nthe code is optimal achieving the upper bound; when ℓ= 4, it is one bit away\nfrom the upper bound, and as ℓgrows it is order-optimal in the sense that the\nratio of kwith its upper bound becomes a constant11\n16whenr=⌊ℓ+3\n4⌋. With\nthe same or even lower complexity, we derive new codes permitting a wider range\nof parameters by modifying C[ℓ, r]in two different ways. The code derived using\nthe first approach has the same blocklength n= 2ℓ, but weight wis allowed to\nvary from ℓ−1to1. In the second approach, the weight remains fixed as w=ℓ,\n1arXiv:2401.16647v1 [cs.IT] 30 Jan 2024but the blocklength is reduced to n= 2ℓ−2r+ 1. For certain selected values\nof parameters, these modified codes have an optimal k.\nKeywords: constant weight codes, low complexity, nonlinear codes, binary codes,\nenumerative coding\n1 Introduction\nLetnandw≤nbe positive integers. A constant-weight binary ( n, M, d ) code Cof\nblocklength nand weight wis defined as a subset of {0,1}nof size Msuch that every\nelement has the same Hamming weight w. The parameter dis the minimum distance\nof the code defined as\nd= min\nc1,c2∈C\nc1̸=c2dH(c1,c2)\nwhere dH(c1,c2) denotes the Hamming distance between the binary vectors c1,c2.\nThe function A(n, d, w ) is the maximum possible size Mof a binary constant-weight\ncode of blocklength n, weight wand minimum distance d. When d= 2, there is no\nadditional constraint on the codebook and therefore it is clear that\nA(n,2, w) =\u0012n\nw\u0013\n. (1)\nWhile there is a rich body of literature that attempt on characterizing A(n, d, w ) for\nd≥4 [1–7], it still remains open in the general setting.\nAlong with characterization of A(n, d, w ), another pertinent problem in the field\nof constant-weight codes is the design of such codes that admit fast implementation of\nencoding and decoding. Considering the ease of implementation using digital hardware,\nit is desirable that the encoding algorithm takes in fixed-length binary vectors as\ninput. In many systems employing a binary constant-weight code, only a subset of\nthe codebook having size as a power of 2 is used to enable efficient implementation,\nand the rest of the codebook is ignored (e.g., see [8]). Therefore we constrain the\nsize of the codebook to M= 2kfor some positive integer k. We refer to kas the\ncombinatorial dimension of the code. The design of low-complexity algorithms for\nencoding and decoding constant-weight codes has been posed as a problem (Research\nProblem 17 .3) in the widely recognized textbook by MacWilliams and Sloane [9]. In\nthe present paper, we focus on this problem for the simplest case of d= 2 assuming a\ncodebook size of M= 2k, with an aim to achieve the largest possible k.\nSince d= 2, any binary vector of weight wcan be included in the codebook and\ntherefore our problem of interest aligns with the problem considered by Schalwijk [10]\nto enumerate all binary n-sequences of weight w. In [11], Cover generalized Schalwijk’s\nindexing scheme to make it applicable to an arbitrary subset of n-sequences. Prior\nto the works of Schalwijk and Cover, the indexing of constant-weight n-sequences\n2of weight wwas studied in combinatorial literature; for example, Lehmor code [12]\nproduces an indexing different from that of Schalwijk’s scheme. In combinatorial lit-\nerature, an n-sequence of weight wis identified as a w-subset (or w-combination)\nof{0,1, . . . , n −1}and the set of all w-combinations is assigned with an order, for\ninstance the lexicographic order. The rank of a w-subset Sis the number of w-subsets\nthat are strictly less than Swith respect to the lexicographic order, and the set Sis\nindexed using its rank. A procedure to compute the rank of a w-subset is referred to\nas a ranking algorithm and conversely, to recover the w-subset associated to a given\nrank as an unranking algorithm. The study of ranking/unranking algorithms and their\ncomplexity dates back to [13]. There are many unranking algorithms [14–19] proposed\nin literature aimed primarily at reducing the time complexity. However, all these algo-\nrithms require costly computation of binomial coefficients that have either large time\ncomplexity or space complexity in case these coefficients are precomputed and stored\nin lookup tables. The first attempt to avoid computation of binomial coefficients is\nmade by Sendrier in [20], but the resulting code is of variable blocklength. Given this\nbackground, our paper makes the following contributions.\n1. We present a family of binary ( n, M = 2k, d= 2) constant-weight codes C[ℓ, r]\nparameterized by integers ℓ≥3 and 1 ≤r≤ ⌊ℓ+3\n4⌋. The code has blocklength\nn= 2ℓ, weight w=ℓand combinatorial dimension k=kℓras defined in (6). The\ncode admits an encoding algorithm that is linear in input size kℓr. Except for the\nlinear time-complexity spent on parsing the input, the decoding algorithm has a\ntime-complexity that is poly-logarithmic in input size n. Neither the encoding nor\nthe decoding require computation of binomial coefficients.\n2. While the code has a natural price to pay in its combinatorial dimension k, it per-\nforms fairly well against the information-theoretic upper bound ⌊log2A(n,2, w)⌋.\nWhen ℓ= 3, it in fact achieves the upper bound, and when ℓ= 4, it is one bit away\nfrom the upper bound. In general, kℓrscales as11\n16⌊log2A(n,2, w)⌋in the best-case\nvalue of r=⌊ℓ+3\n4⌋.\n3. Without compromising on complexity, we derive new codes permitting a larger\nrange of parameters by modifying C[ℓ, r] in two different ways. In the first approach\nthat is feasible for r= 1, the derived code Dt[ℓ] has blocklength n= 2ℓ, weight\nw=ℓ−tand combinatorial dimension k=kℓ(t) as defined in (26) for 1 ≤t≤ℓ−1.\nIn the second approach, the derived code B[ℓ, r] has blocklength n= 2ℓ−2r+ 1,\nweight w=ℓand combinatorial dimension k=kℓr−δℓr−ras defined in (27). For\ncertain selected values of parameters, these codes also achieve the corresponding\nupper bound on k.\n2 The Main Code Construction\nLet|x|denote the length of a vector x. We use x1∥x2to denote the concatena-\ntion of two vectors x1,x2. Entries in a vector xof length |x|=lenare denoted\nbyx[0], x[1], . . . , x [len−1]. We use x[a, m] to denote the vector [ x[a]x[(a+ 1)\nmod len]···x[(a+m−1) mod len]] where the m≤lenelements are accessed in a\ncyclic manner starting from x[a]. A vector of shorter length ( len−m) can be obtained\nby deleting x[a, m] from xand it is denoted by x\\x[a, m]. We use dec( x) to denote\n3the decimal equivalent of the binary vector xassuming big-endian format (least sig-\nnificant bit at the far end). The Hamming weight of a vector xis denoted by wH(x).\nFor integers a, b, we use [ a] to denote {1,2, . . . , a }and [ a b] to denote {a, a+ 1, . . . , b}.\nWe use Im( f) to denote image of a function f.\nOur main idea behind the construction is to divide the message vector xintoℓ\nblocks of decreasing lengths, and then use the decimal value of each block to determine\nthe position of the next 1-entry in the codeword of length 2ℓ. Following this rule, the\ngaps among the ℓ1-entries in the codeword will also allow us to recover the message\nuniquely. We first start with a simple warm-up construction in Section 2.1, which\nprovides the intuition behind our approach, before developing the general construction\nin Sections 2.2, 2.3, and 2.4.\n2.1 A Warm-Up Construction\nThe encoding works as follows. First, we divide the binary message vector xintoℓ\nblocks xℓ,xℓ−2,xℓ−2,xℓ−3, . . . ,x2,x1of lengths ℓ, ℓ−2, ℓ−2, ℓ−3, . . . , 2,1, respectively\nwithout altering the order of bits, i.e., x=xℓ||xℓ−2||xℓ−2||xℓ−3||. . .||x2||x1. Note\nthat we skip ℓ−1 and ℓ−2 appears twice while other values appear at most once in\nthis sequence. For instance, with ℓ= 4, we will have the sequence 4 ,2,2,1, and with\nℓ= 5, we will have 5 ,3,3,2,1. Note that the length of xis|x|=ℓ+2(ℓ−2)+Pℓ−3\ni=1i=\nℓ(ℓ+1)\n2−1. Next, we encode this message into a binary codeword cof length 2ℓand\nHamming weight ℓas follows. We set cto the all-zero codeword and index its bits from\n0 to 2ℓ−1. Let posℓ≜dec(xℓ) be the decimal value of the block xℓ. Leave the first\nposℓbits unchanged as 0’s, but set the ( posℓ+ 1)-th bit of cto one, i.e. c[posℓ]≜1.\nNow, we move to xℓ−1and again let posℓ−1≜dec(xℓ−1). We skip posℓ−10’s after the\nfirst 1, and set the next bit to 1, i.e. c[(posℓ+posℓ−1+1) mod 2ℓ]≜1. Note that here\nwe move from the left to the right cyclically along the codeword indices, wrapping\naround at the end. We continue the process until the last block x1is read and the last\n1 is add to c(see Fig. 1 for an example).\nFor example, when ℓ= 4, the message vector x= (1,0,1,0,1,1,1,0,0) is divided\nintox4= (1,0,1,0),x3= (1,1),x2= (1,0), and x1= (0), which are of lengths\n4,2,2,1 as described earlier. Since dec(x4) = 10, we set c[10] = 1, noting that the bits\nofcare indexed from 0 to 15. Next, since dec(x3) = 3, we set c[14] = c[(10+3+1)] = 1.\nSimilarly, as dec(x2) = 2 and dec(x1) = 0, we set c[1] = c[14 + 2 + 1] = 1 and\nc[2] = c[1 + 0 + 1] = 1. As the result, c= (0,1,1,0,0,0,0,0,0,1,0,0,0,1,0,0). To\ndecode, given such a codeword c, we need to reconstruct x. Clearly, if the position of\nthe “first” 1 (called the anchor ), which corresponds to the block xℓis known, then\nbased on the gap, or the number of 0’s between this 1 and the next 1 on its right\n(cyclically, wrapping around if necessary) will be the decimal value of the block xℓ−1.\nFor example, if we know the 1 at index 10 of c(the underlined one) is the anchor,\nthen we simply count the number of zeros between this 1 and the next, which is 3 and\nrecover x3= (1,1). All the ℓblocks of xcan be recovered this way. Thus, the key step\nis to determine the anchor.\nIt turns out that thanks to the way we split x, the 1 with the largest number of\n0’s on its left (wrapping around if necessary) in cis the anchor, created by xℓ. Note\nthat for the 1’s created by x1, . . . ,xℓ−1, the numbers of 0’s on their left are at most\n4/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet/Bullet/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet01\n2\n3\n4\n5\n6\n789101112131415\n1dec(x4) =10/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet/Bullet/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet01\n2\n3\n4\n5\n6\n789101112131415\ndec(x3) =31\n1/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet/Bullet/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet01\n2\n3\n4\n5\n6\n789101112131415\ndec(x2) =21\n11/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet/Bullet/Bullet\n/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet01\n2\n3\n4\n5\n6\n789101112131415\ndec(x1) =01\n11\n1Fig. 1 : Illustration of the encoding process when ℓ= 4 and the message vector\nx= (1,0,1,0,1,1,1,0,0) is encoded into the codeword cof length 16 = 24(represented\nby the circle) with c[1] = c[2] =c[10] = c[14] = 1. For decoding, one first determine the\nanchor (the underlined 1), which is the 1 that has the largest number of consecutive\nzeros on its left (cyclically), or equivalently, has the largest gap to the nearest 1 on its\nleft. Once the anchor is found, each message block can be recovered by counting the\nnumber of 0’s between the current 1 to the next.\ndec(xℓ−1)≤2ℓ−2−1. On the other hand, the number of 0’s on the left of the anchor\nis at least\n2ℓ−ℓ−\u0012\n2(2ℓ−2−1) +ℓ−3X\ni=1(2i−1)\u0013\n= 2ℓ−2+ 1>2ℓ−2−1,\nwhich proves our claim.\nFinally, note that this warm-up construction, which uses the length sequence ℓ, ℓ−\n2, ℓ−2, ℓ−3, . . . , 2,1 provides a code with message length |x|=ℓ(ℓ+1)\n2−1. Compared to\nthe information-theoretic upper bound O(ℓ2−ℓlog2ℓin Section 3.1, this construction\nis at a factor of Θ(1 /2) away. In our most optimized construction in Section 2.3, we\ncan achieve a better code at a factor of Θ(11 /16) from the bound. To this end, we\nneed to employ sequences with larger values, which correspond to longer messages.\n2.2 Two Finite Integer Sequences\nIn this subsection we generalize the simple sequence used in the warm-up construction\nin the previous section to achieve sequences that correspond to codes with longer\nmessage lengths. Let ℓandrbe two integer parameters satisfying ℓ≥3 and 1 ≤r≤\n⌊ℓ+3\n4⌋. We define two finite integer sequences as follows.\nDefinition 1. Letℓ≥3and1≤r≤ ⌊ℓ+3\n4⌋. Then ˆfℓ,r(i), i=ℓ, ℓ−1, . . . , 1is an\ninteger sequence of length ℓas given below:\nˆfℓ,r(i) =\n\nℓ, i =ℓ\nℓ−1− ⌈ℓ−i\n2⌉, i=ℓ−1, ℓ−2, . . . , ℓ −2r\ni+r−1, i =ℓ−2r−1, ℓ−2r−2, . . . , 1(2)\n5Definition 2. Letℓ≥3and1≤r≤ ⌊ℓ+3\n4⌋. Then fℓ,r(i) =ˆfℓ,r(i), i=ℓ, ℓ−1, . . . , 2.\nFori= 1,\nfℓ,r(1) =\u001aˆfℓ,r(1) + 1 , ℓ > 2r+ 2\nˆfℓ,r(1), (r= 1, ℓ∈ {3,4})or(r= 2, ℓ∈ {5,6})(3)\nWe also define\nδ(ℓ, r) = fℓ,r(1)−ˆfℓ,r(1) (4)\nthat takes either 0or1as its value.\nObserve that δ(ℓ, r) equals 1 for every permitted value of ℓ, rexcept for a limited\nset of parameters ( r= 1, ℓ= 3,4) and ( r= 2, ℓ= 5,6). Next we define\nˆkℓr=ℓX\ni=1ˆfℓ,r(i) =ℓ(ℓ−1)\n2+r(ℓ−r−1) + 1 (5)\nkℓr=ℓX\ni=1fℓ,r(i) =ℓ(ℓ−1)\n2+r(ℓ−r−1) + 1 + δ(ℓ, r). (6)\nWe compile certain useful numerical identities pertaining to the sequences in the\nrℓ fℓ,r(i), i=ℓ, ℓ−1, . . . , 1kℓrˆkℓr\n3 3, 1, 1 5 5\n4 4, 2, 2, 1 9 9\n15 5, 3, 3, 2, 2 15 14\n6 6, 4, 4, 3, 2, 2 21 20\n7 7, 5, 5, 4, 3, 2, 2 28 27\n5 5, 3, 3, 2, 2 15 15\n6 6, 4, 4, 3, 3, 2 22 22\n27 7, 5, 5, 4, 4, 3, 3 31 30\n8 8, 6, 6, 5, 5, 4, 3, 3 40 39\n9 9, 7, 7, 6, 6, 5, 4, 3, 3 50 49\n9 9, 7, 7, 6, 6, 5, 5, 4, 4 53 52\n10 10, 8, 8, 7, 7, 6, 6, 5, 4, 4 65 64\n311 11, 9, 9, 8, 8, 7, 7, 6, 5, 4, 4 71 70\n12 12, 10, 10, 9, 9, 8, 8, 7, 6, 5, 4, 4 92 91\n13 13, 11, 11, 10, 10, 9, 9, 8, 7, 6, 5, 4, 4 107 106\nTable 1 : Table of fℓ,r. When ˆkℓr=kℓr,ˆfℓ,r=ˆfℓ,r,\notherwise they differ only at ˆfℓ,r(1) = fℓ,r(1)−1.\nfollowing proposition.\nProposition 2.1. The following identities hold:\n1.fℓ,r(i)> ℓ−r−1, i=ℓ−1, ℓ−2, . . . , ℓ −2r+ 2.\n2.fℓ,r(i) =ℓ−r−1, i=ℓ−2r+ 1, ℓ−2r.\n63.fℓ,r(i)< ℓ−r−1, i=ℓ−2r−1, ℓ−2r−2, . . . , 1.\n4. When r1< r2≤\u0004ℓ+3\n4\u0005\n,fℓr1(i)≤fℓr2(i)for every i∈[ℓ].\n5. When r1< r2≤\u0004ℓ+3\n4\u0005\n,kℓr1< kℓr2.\nAll the above identities are true for ˆfℓ,randˆkℓras well.\nProof. They all follow from definitions in a straightforward manner. It is necessary to\nhave δ(ℓ, r) = 0 when ℓ≤2r+ 2 for the first three identities to hold for fℓ,r.\n2.3 Encoding Information in Gaps\nIn this section, we present an encoding algorithm (see Algorithm 1) that encodes infor-\nmation in gaps between successive 1’s of a binary vector of length n= 2ℓ, using the\nsequences fℓ,r. More specifically, the message vector xwill be divided into ℓblocks\nxℓ, . . . ,x2,x1, which are of lengths fℓ,r(ℓ), . . . , f ℓ,r(2), fℓ,r(1), and gaps between suc-\ncessive 1’s of the codewords depends on the decimal value of each of these blocks. The\nfunction gapdefined below formalizes the notion of gap as the first step.\nDefinition 3. Leta, b∈Zn. Then the gap from atobis a natural number taking\nvalues in [0 (n−1)]given by\ngap(a, b) = (b−a−1) mod n.\nThe encoding algorithm given in Algorithm 1 is invoked taking the sequence fℓ,ras\nan auxiliary input. The input xis the message vector that gets encoded, and its length\nkℓrmatches with fℓ,rin the sense that kℓr=P\nifℓ,r(i). The encoded vector is the\noutput cof length n. The input vector xis partitioned as xℓ∥xℓ−1∥···∥ x1such that\n|xi|=fℓ,r(i) for i∈[ℓ]. The vector cis initialized as all-zero vector and ℓlocations\nofcare set to 1 subsequently. The input bits are read in blocks xℓ−1,xℓ−2, . . .x1and\nevery time a block xi, ℓ≥i≥1 is read, a bit in cis set to 1 in a such manner that the\ngap from the previously set 1 is equal to dec( xj). The gap is always computed modulo\nnso that the position pointer poswraps around cyclically. The algorithm has a linear\ntime-complexity in input size kℓr, and it defines the encoding map ϕ:{0,1}kℓr− →\n{0,1}n.\nAlgorithm 1: Encode ϕ(·)\nInput :x∈ {0,1}kℓr,fℓ,r\nOutput :c∈ {0,1}n\n1Partition xasxℓ∥xℓ−1∥···∥ x1such that |xi|=fℓ,r(i) for i∈[ℓ].\n2Initialize array c= 0n\n3pos← −1\n4forj=ℓ, . . . , 1do\n5 pos← −pos+ 1 + dec( xj) mod n\n6 c[pos]← −1\n7The auxiliary input fℓ,rcan as well be replaced with a different sequence, provided\nthe length of the input xis suitably modified as the sum of entries of that sequence.\nThus Algorithm 1 provides a generic method to encode information as gaps in a vector\nof length n= 2ℓ. What requires is to identify a “good” sequence so as to produce a\ncode with high combinatorial dimension. In the following Lemma 2.2, we show that\nthe Hamming weight of the output is always ℓfor the choice of sequence as fℓ,r.\nLemma 2.2. Forfℓ,rgiven in Definition 2, wH(ϕ(x)) =ℓfor every x∈ {0,1}kℓr.\nProof. Letc∈ {0,1}nbe in the image of ϕ. After executing Line 4, position pointer\npostakes a value p0= dec( xℓ) lying between 0 and 2ℓ−1, and chas Hamming weight\n1 with c[p0] = 1. The loop at Line 5 has ℓ−1 iterations indexed by j=ℓ−1, . . . , 1. In\nevery iteration, the variable posis incremented modulo nat least by 1 and at most by\n2|xj|. Therefore, the maximum cumulative increment pinposby the end of all ( ℓ−1)\niterations is given by:\np=ℓ−1X\nj=12|xj|=ℓ−1X\nj=12fℓ,r(j)\n=ℓ−2r−1X\nj=12j+r−1+ℓ−2X\nj=ℓ−2r2ℓ−1−⌊ℓ−j\n2⌋+ 2r+δ(ℓ,r)\n=ℓ−2r−1X\nj=12j+r−1+ℓ−1X\nj=ℓ−2r2ℓ−1−⌊ℓ−j\n2⌋+δ(ℓ, r)2r(7)\n= 2ℓ−2ℓ−r−1−(1−δ(ℓ, r))2r<2ℓ. (8)\nWe observe that the first two terms in (7) stays as such for any ℓ, rindependent of\nthe value of δ(ℓ, r) because δ(ℓ, r) = 0 when ℓ≤2r+ 2. This allows to express pas in\n(8) irrespective of the value of δ(ℓ, r). Since p < n , a distinct bit of cis set from 0 to\n1 in each of these ( ℓ−1) iterations and therefore wH(c) =ℓ.\nSince the weight of ϕ(x) is fixed to be ℓ, Im(ϕ) can be obtained by applying various\npermutations on 1ℓ∥0n−ℓand this resembles the permutation code introduced in [21].\nTherefore the encoder ϕgives an elegant method to map binary vectors of length kℓr\nto a subset of the permutation code, which is otherwise usually carried out by picking\nvectors in lexicographic order [8].\nExample 1. In this example, we set ℓ= 5, r= 2, and we have k52= 15 . Let us\ncompute ϕ(x)for the input x= 101101111111111 . Since f52(i) = 5 ,3,3,2,2fori=\n5,4,3,2,1, we write x= 10110 ∥111∥111∥11∥11 =x5∥x4∥x3∥x2∥x1where xihas length\nf52(i)fori= 5,4,3,2,1. The encoder initializes c←032and then flips 5distinct\nbits of cto1. The decimal equivalent of x5is22and hence the anchor bit is c[22].\nFirst, c[22]is flipped to 1and subsequently four more bits are flipped to 1. The gap\nbetween a bit flipped to 1and the next bit to be flipped to 1is decimal equivalent\nof one of x4,x3,x2,x1picked in that order. The codeword is thus obtained as c=\nϕ(x) = (0 ,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0)where\nthe anchor bit is underlined. In the warm-up construction, we were able to identify the\n8anchor bit by identifying the unique gap >2ℓ−2−1 = 7 . It can be observed that the\ntechnique fails here because there is no such gap. Therefore we must have a different\ndecoding algorithm if at all it exists.\n2.4 A Decoding Algorithm and a Constant-weight Code\nIn this section, we establish that the encoding algorithm ϕis one-to-one by providing\nan explicit decoder. In conjunction with Lemma 2.2, it leads to characterization of a\nconstant-weight binary code with blocklength n= 2ℓ, weight w=ℓand combinatorial\ndimension k=kℓr.\nThe crux of the decoding algorithm lies in finding the anchor bit. Recall that the\nanchor bit is the first bit flipped to 1 while running the encoding algorithm. The\nprocedure to find the anchor bit is illustrated pictorially in Fig. 2 with an example for\nparameters ℓ= 7, r= 2. Continuing the approach taken in the description of warm-up\nconstruction (see Fig. 1 in Sec. 2.1), the codeword of length n= 128 is represented as\na circle with 128 points indexed from 0 to 127. The codeword cpicked in the example\nhasc[j] = 1 for j= 10,26,32,37,64,96,127 and zero everywhere else. To avoid clutter\nin Fig. 2, we indicate the starting point 0 and mark only those points at which c[j] = 1,\ninstead of all the 128 points.\nFirst, we identify the gaps between successive 1’s as g[m], m= 0,1, . . . , 6 in order\nstarting from the first gap g[0] = gap(127,10) = 10. Other gaps are g[1] = 15 ,g[2] =\n5,g[3] = 4 ,g[4] = 26 ,g[5] = 31 ,g[6] = 30. The principle is to look for a stretch\nof (2r−1) = 3 consecutive gaps in clockwise direction such that the last gap in\neach of these stretch is ≥2ℓ−r−1= 16. The gap that is on or above the threshold\n2ℓ−r−1is referred to as a candidate gap . There are three such stretches marked in\nthis example, marked as a○,b○and c○in Fig. 2. Among these three, the stretch\nc○containing ( g[2],g[3],g[4]) = (5 ,4,26) is unique in the sense that every gap in\nthat stretch apart from the last gap g[4] does not qualify as a candidate gap. The\nbitc[64] at the end of c○is therefore picked as the anchor bit. Once the anchor is\nidentified as c[64], binary equivalent of 64 gives rise to x7, and that of following six\ngaps ( g[5],g[6],g[0],g[1],g[2],g[3]) = (31 ,30,10,15,5,4) yield x6,x5,x4,x3,x2and\nx1. Except when δ(ℓ, r) = 1 and a specific type of message vector appears, the above\nprocedure for finding the anchor bit works. The correctness of the above procedure\nand the way to handle special cases constitute Theorem 2.3 that appears immediately\nafter the next definition.\nDefinition 4. Letc= (c[0], c[1], . . . , c [n−1])be a vector of length n. Then the circular\nshift of cbyn0∈Znis defined as\ncshift (c, n0) = ( c[n0], c[n0+ 1], . . . , c [n−1], c[0], . . . , c [n0−1]). (9)\nFor any integer n0, the definition still holds true by replacing n0byn0mod nin(9).\nTheorem 2.3. Forfℓ,rgiven by Definition 2, the map ϕdefined by Algorithm 1 is\none-to-one.\nProof. We show that ϕis injective by providing an explicit decoder for c(see Algo-\nrithm 2) that maps uniquely to an x∈ {0,1}kℓr. Since c∈Im(ϕ), there is an\n90\n10\n32\n6496127\n3726Fig. 2 : Illustration of the principle of decoding algorithm for ℓ= 7, r= 2 when the\ncodeword chas 1’s at c[j], j= 10 ,26,32,37,64,96,127 (marked with dots) and 0’s\neverywhere else. There are three clock-wise stretches of gaps marked as a○,b○and c○\nthat end in a candidate gap, i.e., with value on or above 16. The stretch c○given by\n(5,4,26) is unique among these three because in c○, every gap value apart from the\nlast one does not qualify as a candidate. The bit c[64] at the end of the stretch c○is\ntherefore picked as the anchor bit.\nAlgorithm 2: Decode\nInput :c∈Im(ϕ), fℓ,r\nOutput :x∈ {0,1}kℓr\n1Find 0 ≤j[0]< j[1]<···< j[ℓ−1]< nsuch that c[j[i]] = 1 for every\ni= 0,1, . . . , ℓ −1.\n2g[m] =gap(j[(m−1) mod ℓ], j[m]) for m= 0,1, . . . ℓ−1\n3anchor index =FindAnchor (g, fℓ,r)\n4Initialize binary vector xsuch that |x|=ℓand dec( x) =j[anchor index ]\n5fori= 1,2, . . . , ℓ −1do\n6 g←g[(anchor index +i) mod ℓ]\n7 Represent gas binary string xiof length fℓ,r(ℓ−i)\n8 x←x∥xi\nx=xℓ∥xℓ−1∥···∥ x1such that c=ϕ(x). Let us define j[0], j[1], . . . , j [ℓ−1] (see Line\n1) as the locations of 1s in cfrom left to right in order. One of these locations is\ndetermined by xℓ. Let us call it j[anchor index ], and more precisely\nj[anchor index ] =jsuch that c[j] is the first bit set to 1 while encoding c.(10)\nIf the index anchor index is uniquely identified by an input vector c, then it is straight-\nforward to observe that xℓ,xℓ−1, . . . ,x1are uniquely determined. The procedure to\nrecover xgiven the knowledge of j[anchor index ] is laid down in Lines 4−8 of\nAlgorithm 2.\n10Algorithm 3: FindAnchor\nInput :g∈Zℓ\nn, fℓ,r\nOutput :anchor index∈Zn\n1gaps allone ←(2ℓ−r−1−1)∥(2fℓ,r(i)−1, i=ℓ−1, ℓ−2, . . . , 1)\n2ifδ(ℓ, r) = 1 and ∃n0∈Zℓsuch that gaps allone =cshift (g, n0)then\n3 anchor index←n0\n4else\n5 Initialize array cadidates to 0ℓ.\n6 form= 0,1, . . . , ℓ −1do\n7 if g[m]≥2ℓ−r−1then\n8 cadidates [m]←1\n9 Pick m0such that cadidates [m0] = 1\n10 anchor index←m0\n11 noncand cntbkwd←0\n12 form= (m0+ 1 mod ℓ),(m0+ 2 mod ℓ), . . . , (m0+ℓmod ℓ)do\n13 ifcadidates [m] = 0 then\n14 noncand cntbkwd←noncand cntbkwd + 1\n15 else\n16 ifnoncand cntbkwd≥2r−2then\n17 anchor index←m\n18 break\n19 noncand cntbkwd←0\nIn what follows, we will show that the index anchor index , and therefore\nj[anchor index ], are uniquely identified by c. As computed in Line 2 , let us define a\nvector g∈Zℓ\nnas:\ng[m] =gap(j[(m−1) mod ℓ], j[m]), m = 0,1, . . . ℓ−1. (11)\nObserve that g[m] is the number of zeros in cto the left of the 1 at c[j[m]] and to the\nright of c[j[m−1]] when 1 ≤m≤ℓ−1. When m= 0,m−1 = ( ℓ−1) mod ℓand in\nthat case, g[0] is the number of zeros after j[ℓ−1] counting them cyclically until j[0]\nis hit. In the algorithm FindAnchor (see Algorithm 3), a procedure to determine\na unique value for anchor index based on gis implemented. We will argue that this\nprocedure is indeed correct. It is straightforward that:\nn=ℓ+ℓ−1X\nm=0g[m] = ℓ+g[anchor index ] +ℓ−1X\ni=1g[(anchor index +i) mod ℓ]\n11Therefore we have\ng[anchor index ] = ( n−ℓ)−ℓ−1X\ni=1g[(anchor index +i) mod ℓ]\n≥(n−ℓ)−ℓ−1X\ni=1(2|xℓ−i|−1) (12)\n= (2ℓ−ℓ)−ℓ−1X\ni=1(2fℓ,r(ℓ−i)−1)\n=\u001a\n2ℓ−r−1−1, δ (ℓ, r) = 1\n2ℓ−r−1+ 2r−1, δ(ℓ, r) = 0(13)\nThe inequality in (12) follows from the way xℓ−iis encoded by Algorithm 1. It is\nstraightforward to check that equality holds in (12) if and only if the message vector\nis of the type\nxℓ−i= 1fℓ,r(ℓ−i),for all i= 1,2, . . . , ℓ −1. (14)\nWhen the message vector satisfies (14), every gap except g[anchor index ] becomes\nmaximal in length, and therefore we refer to this special case as the maximal-gap\ncase. When δ(ℓ, r) = 1, Lines 2−3 in Alg. 3 checks for the maximal-gap case by\ncomparing every circular shift of the vector gwith a fixed vector gaps allone . The\nvector gaps allone corresponds to a message vector of the type\nxℓ−i= 1fℓ,r(ℓ−i),for all i= 1,2, . . . , ℓ −1,and dec( xℓ)≤2ℓ−r−1−1.(15)\nfor which anchor index = 0. If cshift (g, n0) becomes equal gaps allone for some 0 ≤\nn0≤(ℓ−1), then by the first three identities of Prop. 2.1, n0is unique and is equal\ntoanchor index .\nIf (14) is false, then clearly (12) satisfies with strict inequality, and in that case\ng[anchor index ]≥2ℓ−r−1for every ℓ, rby (13). The binary array cadidates generated\nafter the execution of the loop in Line 8 is such that cadidates [m] = 1, m∈[0ℓ−1] if\nand only if g[m]≥2ℓ−r−1, and therefore the binary array cadidates keeps a record of\nall gaps that can possibly be a candidate for g[anchor index ]. As already made clear,\nanchor index is indeed picked as a candidate. As a result, it becomes possible to execute\nLine 9 as there is always an m0such that cadidates [m0] = 1. If there are no other\ncandidates, anchor index is indeed m0. This is exactly what the procedure returns as\nthe value of anchor index is not changed after executing Line 10.\nLet us investigate how the procedure works when there are more than one\ncandidates for anchor index . Ifr= 1, we observe that\n2ℓ−r−1= 2ℓ−2>max\ni∈[ℓ−1](2fℓ,r(i)−1)\n≥g[m], m̸=anchor index\n12and therefore there shall be exactly one candidate for anchor index and we fall back\nto the previous case. So in the discussion on having multiple candidates, we assume\nthatr≥2. By the second and third identities in Prop. 2.1, we have\n2ℓ−r−1>(2fℓ,r(i)−1), i=ℓ−2r+ 1, ℓ−2r, . . . , 1\n= (2|xi|−1)\n≥g[(anchor index−i) mod ℓ], i=ℓ−2r+ 1, ℓ−2r, . . . , 2,1.(16)\nSince ℓ≥4r−3, we have ℓ−2r+ 1≥2r−2. In addition, since ℓ≥3 and ℓ≥4r−3,\nwe have ℓ > 2r. Therefore, the set {ℓ−2r+ 1, ℓ−2r, . . . , 1}contains the subset\n{1,2, . . . , 2r−2}which is non-empty as r≥2. Thus (16) implies a non-vacuous\nstatement\ng[(anchor index−i) mod ℓ]<2ℓ−r−1, i= 1,2, . . . , 2r−2. (17)\nThe loop at Line 12 begins its iterations starting with m= (m0+ 1) mod ℓ\nwhere m0corresponds to a candidate gap. As a consequence, in every iteration of\nthe loop indexed by m, the variable noncand cntbkwd acts as a counter for the\nnumber of gaps to the left of g[m] (counting cyclically) that do not qualify as can-\ndidates until a candidate is met. It follows from (17) that when m=anchor index ,\nnoncand cntbkwd≥2r−2, and therefore Lines 17−18 get executed if the loop pro-\nlongs enough to witness m=anchor index . Suppose m′̸=anchor index corresponds to\na candidate gap, i.e., cadidates [m′] = 1, then it means that g[m′]≥2ℓ−r−1. But we\nknow that m′= (anchor index +i′) mod ℓfor some i′∈[ℓ−1] and g[m′]<2fℓ,r(ℓ−i′).\nSince m′is a candidate, i′must satisfy that\n2ℓ−r−1<2fℓ,r(ℓ−i′)\nand it follows from the first identity in Prop. 2.1 that\ni′≤2r−2\n⇒m′≤anchor index + (2r−2) mod ℓ. (18)\nIt follows from (18) that for any candidate m′̸=anchor index , the number of non-\ncandidate gaps to the left of m′(cyclically) is strictly less than 2 r−2. In other words,\nit must be that noncand cntbkwd <2r−2, and therefore Lines 17−18 will not get\nexecuted for m′. Therefore the iterations of the loop until m=anchor index happens.\nThus we have shown that the Lines 17−18 get executed if and only if m=anchor index .\nTherefore the procedure FindAnchor to determine anchor index uniquely is indeed\ncorrect when there are multiple candidates. This completes the proof.\nRemark 1. The decoding algorithm in Algorithm 2 and similar decoding algo-\nrithms that will appear in Sec. 4 illustrate the principle of operation but they can be\nimplemented as a single-pass loop on nbits using a circular buffer.\n13By Theorem 2.3, there is an 1-1 correspondence between {0,1}kℓrand Im( ϕ)⊂\n{0,1}nand that leads to a constant-weight code as defined below.\nDefinition 5. Letℓ≥3and1≤r≤ ⌊ℓ+3\n4⌋. Then we define the code C[ℓ, r] =Im(ϕ)\nwhen ϕis invoked with the sequence fℓ,r. The code has blocklength n= 2ℓ, weight\nw=ℓ, and combinatorial dimension k=kℓr.\n3 Properties of the Code\n3.1 On Size of the Code\nThe following straightforward lemma gives an information-theoretic upper bound on\nthe size of any binary constant-weight code.\nLemma 3.1. LetCbe a constant-weight binary code of blocklength n, weight wand\ncombinatorial dimension k. Then\nk≤ ⌊log2A(n,2, w)⌋=\u0016\nlog2\u0012n\nw\u0013\u0017\n. (19)\nIt is easy to see that C[ℓ, r] has minimum distance d= 2 because both 1ℓ∥0n−ℓand\n0∥1ℓ∥0n−ℓ−1are codewords and they are apart by Hamming distance 2. Therefore, it\nis meaningful to compare kℓragainst the bound in (19). At the same time, another\nupper bound can be obtained by assuming certain cyclic structure on the code that\nour construction brings along.\nLemma 3.2. Ifc∈ C[ℓ, r], then cshift (c, n0)∈ C[ℓ, r]for every n0. When n0∈Z2ℓ,\ncshift (c, n0)must be distinct for every distinct n0.\nProof. Letx=ϕ−1(c) =xℓ∥ˆ xℓwhere |xℓ|=ℓ. For every n0,cshift (c, n0) is a codeword\ncorresponding to a message obtained by updating xℓ(if required), but keeping ˆ xℓ\nfixed. Hence cshift (c, n0)∈ C[ℓ, r]. Consider all codewords obtained by varying xℓover\nall 2ℓpossibilities, but keeping ˆ xℓfixed. They must all be distinct from one another\nbecause ϕis one-to-one. Since there can at most be 2ℓdistinct cyclic shifts possible\nforc,cshift (c, n0) must all be distinct for every n0∈Z2ℓ.\nLetCndenote the cyclic group of order n. By Lemma 3.2, the action of C2ℓon\nC[ℓ, r] results in orbits of size 2ℓ. This implies that C[ℓ, r]/C2ℓcontains only primitive\nbinary necklaces of length 2ℓand weight ℓ. Recall that a binary necklace of length n\nis an equivalence class of vectors in {0,1}nconsidering all the nrotations of a vector\nas equivalent. A binary necklace is said to be primitive if the size of the equivalence\nclass is n. The count of primitive binary necklaces of length nand weight wis known\nto be [22]\np(n, w) =X\nd|nµ\u0010n\nd\u0011\nq(d, wd/n ) (20)\n14Fig. 3 : Comparison of kℓragainst the upper bound as ℓvaries.\nwhere q(d, wd/n ) is the coefficient of xwd/ny(n−w)d/nin the polynomial\nF(x, y) =1\nnX\nd|n(xn/d+yn/d)dϕE\u0010n\nd\u0011\n. (21)\nHere µ(·) and ϕE(·) are M¨ obius function and Euler’s totient function respectively. By\nLemma 3.2 and (20),\nkℓr≤ ⌊log2(np(n, w))⌋=ℓ+⌊log2p(2ℓ, ℓ)⌋. (22)\nIt is not clear when the bound in (22) is strictly better than the one in (19) for an\narbitrary value of ℓ. In any case, the size of the code C[ℓ, r] must respect both these\nupper bounds. Therefore,\nkℓr≤min\u001a\u0016\nlog2\u00122ℓ\nℓ\u0013\u0017\n, ℓ+⌊log2p(2ℓ, ℓ)⌋\u001b\n. (23)\nBy fifth property of Prop. 2.1, kℓris maximized at rmax=\u0004ℓ+3\n4\u0005\n, and we obtain the\nmaximum value kℓas given in (24):\nkℓ:= max\nrkℓr=ℓ(ℓ−1)\n2+\u0016ℓ+ 3\n4\u0017\u0012\u00183(ℓ−1)\n4\u0019\n−1\u0013\n+δℓ. (24)\nwhere\nδℓ=\u001a\n1, ℓ > 6\n0,1≤ℓ≤6. (25)\n15It is straightforward to check that kℓ= Θ(11 ℓ2/16) where as the bound in (19) is\nO(ℓ2−ℓlog2ℓ). Thus the code is order-optimal in its size, in the sense that both\nkℓand its upper bound grows quadratically with ℓ. A comparison of kℓagainst the\nbound in (19) is plotted in Fig. 3. It can be checked that when ℓ= 3,kℓachieves the\ninformation-theoretic upper bound and when ℓ= 4, it is one-bit away from the bound.\n3.2 Encoding and Decoding Complexities\nThe encoding algorithm (Algorithm 1) clearly has linear time-complexity in the input\nsize. The decoding algorithm (Algorithm 2) involves three important steps: (a) parsing\nthe input of length n= 2ℓto identify the gap vector of length ℓ, (b) parsing the\ngap vector to identify the starting point, and finally (c) converting the gap values\nto its binary representation. Each step has time complexity O(n),O(ℓ) =O(logn)\nandO(ℓ2) =O(log2n) respectively. Except for the first round of parsing the input\nto obtain the gaps that is clearly linear in input size n, the remaining part of the\nalgorithm has time-complexity poly-logarithmic in input size.\nThe encoding/decoding algorithms of most of the constant-weight codes involve\ncomputation of binomial coefficients. One way to circumvent this problem is to store\nthese coefficients as lookup tables, but in that case it consumes large space complexity.\nFor example, a classic encoding (unranking) algorithm based on combinadics [18]\nrequires storage of around w\u0000n\nw\u0001\nbinomial coefficients. Our algorithms fully eliminate\nthe need to compute binomial coefficients.\n4 Derived Codes\nIt is possible to derive new codes from the main code described in Sec. 2 by suitable\nspecializations or transformations. The derived codes either admit a simpler decoding\nalgorithm, or help to enlarge the parameter space. In certain range of parameters,\nthey also achieve the information-theoretic upper bound on its size.\n4.1C[ℓ,1]: Specialization with r= 1\nThe code C[ℓ,1] obtained by setting r= 1 admits further simplification in its decod-\ning algorithm, and was used earlier as the warm-up construction (Section 2.1). We\ndiscuss it again here starting from the more general decoding algorithm we devel-\noped in Section 2.4. The first step of the decoding algorithm in Alg. 2 is to determine\nj[anchor index ], the index of the anchor bit, given the knowledge of gaps. When r= 1,\nit is possible to determine anchor index in a simpler manner as given in Alg. 4. The\ncorrectness of the resultant decoder is established in the following Lemma 4.1.\nLemma 4.1. The code C[ℓ,1]is correctly decoded by Algorithm 2 when FindAnchor\nprocedure is replaced by FindAnchor1 given in Algorithm 4.\nProof. Substituting r= 1 in (13), we obtain\ng[anchor index ]≥\u001a\n2ℓ−2−1, δ(ℓ, r) = 1\n2ℓ−2+ 1, δ(ℓ, r) = 0.\n16The case of g[anchor index ] = 2ℓ−2−1 appearing for a specific message vector satisfying\n(14) is handled by Lines 2−3 exactly as how it is done in FindAnchor . In every\nother case, g[anchor index ] is clearly the maximum among all gaps because every other\ngap is bounded above by max i(2fℓ,r(i)−1) = 2ℓ−2−1. Thus Line 5 of Algorithm 4\ncorrectly identifies anchor index . This completes the proof.\nAlgorithm 4: FindAnchor1\nInput :g∈Zℓ\nn, fℓ,r\nOutput :anchor index∈Zn\n1gaps allone ←(2ℓ−r−1−1)∥(2fℓ,r(i)−1, i=ℓ−1, ℓ−2, . . . , 1)\n2ifδ(ℓ, r) = 1 and ∃n0∈Zℓsuch that gaps allone =cshift (g, n0)then\n3 anchor index←n0\n4else\n5 anchor index = arg max m{g[m]|m= 0,1, . . . , ℓ −1}\n4.2ˆC[ℓ, r]: Replacing fℓ,rbyˆfℓ,r\nTheFindAnchor procedure of the decoding algorithm (or its modified version in\nAlg. 4) requires separate handling of the maximal-gap case (described in the proof of\nTheorem 2.3). The underlying reason behind handling that case separately stems from\n(13) and is summarized in the next sentence. When δ(ℓ, r) = 1 and if the message\nvector is of the type satisfying (14), the g[anchor index ] takes on its minimum possible\nvalue 2ℓ−r−1−1 which is less than the threshold for becoming a candidate gap. It\nturns out that whether message vector is of the specific type can easily be determined\nby analyzing the pattern of gaps in the codeword. The case is handled separately in\nboth Alg. 3 and Alg. 4, resulting in slight increase in the complexity of the decoder.\nThis special case can be avoided if we make sure that δ(ℓ, r) = 0 by compromising\non the size of the code by 1 bit. Towards that, we replace fℓ,rwith ˆfℓ,ras auxiliary\ninput in Alg. 1, and pass a message vector ˆ xof length ˆkℓr. The resultant code is a\nsubcode of C[ℓ, r] because the message vector ˆ xcan be viewed as x∈ {0,1}kℓrwith\nits least significant bit set to 0.\nDefinition 6. Letr≥1andℓ≥4r−3. Then we define the code ˆC[ℓ, r] =Im(ϕ)when\nϕis invoked with the sequence ˆfℓ,r. The code has blocklength n= 2ℓ, combinatorial\ndimension k=ˆkℓrand constant weight ℓ.\nThe simpler decoding algorithm for ˆC[ℓ, r] is obtained by making the following\nmodifications in Alg. 2:\n1. The input to the algorithm is ˆfℓ,rin place of fℓ,r;\n2. The Line 3 is modified to invoke FindAnchor2 in place FindAnchor passing ℓ, r\nandgas inputs.\n17Though FindAnchor2 follows in similar line as FindAnchor except for checking\nfor the maximal-gap case, we provide the algorithm in Alg. 5 for the sake of complete-\nness. The specialization of r= 1 can be done for ˆC[ℓ, r] too as described in Sec. 4.1,\nAlgorithm 5: FindAnchor2\nInput :ℓ, r,g∈Z≤ℓ\nn\nOutput :anchor index∈Zn\n1glen← |g|\n2Initialize array cadidates to 0glen.\n3form= 0,1, . . . , glen−1do\n4 if g[m]≥2ℓ−r−1then\n5 cadidates [m]←1\n6Pick m0such that cadidates [m0] = 1\n7anchor index←m0\n8noncand cntbkwd←0\n9form= (m0+ 1 mod glen),(m0+2mod glen), . . . , (m0+glen mod glen)do\n10 ifcadidates [m] = 0 then\n11 noncand cntbkwd←noncand cntbkwd + 1\n12 else\n13 ifnoncand cntbkwd≥2r−2then\n14 anchor index←m\n15 break\n16 noncand cntbkwd←0\nand that leads to the simplest possible procedure for finding anchor index given by\nanchor index = arg max m{g[m]|m= 0,1, . . . , ℓ −1}. Thus ˆC[ℓ,1] turns out to the sub-\nfamily in the family of codes presented in this paper, admitting the simplest possible\ndecoder.\nRemark 2. When (r= 1, ℓ∈ {3,4})or(r= 2, ℓ∈ {5,6}),fℓ,r=ˆfℓ,rand therefore\nˆC[ℓ, r] =C[ℓ, r].\n4.3Dt[ℓ]: Shortening Messages of C[ℓ,1] at LSB\nThe encoding algorithm of C[ℓ,1] involves partitioning the input x∈Fkℓ1\n2into ℓ\nconsecutive blocks xℓ,xℓ−1, . . . ,x1. We obtain a new code by applying the following\nmodifications to the encoding algorithm in Alg. 1:\n1. Set the the last (least significant) tblocks xt,xt−1, . . . ,x1to all-zero vectors;\n2. Reset those bits to 0 that are set to 1 in the last titerations of the loop\n(corresponding to xt,xt−1, . . . ,x1) in the encoding algorithm.\n18Algorithm 6: DecodeD\nInput :c∈ Dt[ℓ], fℓ1\nOutput :x∈ {0,1}kℓr(t)\n1Find 0 ≤j[0]< j[1]<···< j[ℓ−t−1]<2ℓsuch that c[j[i]] = 1 for every\ni= 0,1, . . . , ℓ −t−1.\n2g[m] =gap(j[(m−1) mod ℓ], j[m]) for m= 0,1, . . . ℓ−t−1\n3anchor index = arg max m{g[m]|m= 0,1, . . . , ℓ −t−1}\n4Initialize binary vector xsuch that |x|=ℓand dec( x) =j[anchor index ]\n5fori= 1,2, . . . , ℓ −t−1do\n6 g←g[(anchor index +i) mod ℓ]\n7 Represent gas binary string xiof length fℓ1(ℓ−i)\n8 x←x∥xi\nThe resultant code is denoted by Dt[ℓ] and is parameterised by ℓ≥3 and 1 ≤t < ℓ .\nFor every t, it is a constant-weight code with parameters given by:\nn= 2ℓ, k=kℓ(t) :=kℓ1−tX\ni=1fℓ1(i), w=ℓ−t. (26)\nWe refer to Dt[ℓ] as the code obtained by shortening C[ℓ,1] at the LSB. Since t≥\n1,x1=0while encoding for Dt[ℓ] for every t. For this reason, the code Dt[ℓ] is\nobtained even if the same transformation is applied to ˆC[ℓ,1]. The decoding algorithm\nis obtained by adjusting the one for ˆC[ℓ,1] in accordance with the reduction in input\nlength and weight. For the sake of completeness, it is presented in Alg. 6. It can be\nchecked that the complexity is logarithmic in nif the time spent on parsing the input\nis discounted. It turns out that the shortened code Dt[ℓ] too performs well in terms of\nits size against the information-theoretic upper bound. When t=ℓ−2, we obtain a\ncode with w= 2 for which\nk=kℓ(ℓ−2) = 2 ℓ−2 =⌊log2A(2ℓ,2,2)⌋\nand hence the code is optimal. When ℓ≥4 and t=ℓ−3, we obtain a code with w= 3\nfor which\nk=kℓ(ℓ−3) = 3 ℓ−4 =⌊log2A(2ℓ,2,3)⌋ −1\nand hence the code is one bit away from optimality. The comparison of kℓ(t) against\ninformation theoretic upper bound in plotted in Fig. 4 for t= 1,2.\n4.4B[ℓ, r]: Shortening Messages of ˆC[ℓ, r] at MSB\nIt is possible to shorten message vectors of ˆC[ℓ, r] by setting rmost significant bits to\n0 before passing it to the encoding algorithm. This leads to a constant-weight code\nB[ℓ, r] with smaller size, reduced blocklength but with the same weight ℓ, provided the\n19Fig. 4 : Comparison of kℓ(t) against the upper bound for t= 1,2.\nencoding algorithm is adjusted with suitable modifications. The code B[ℓ, r] is referred\nto as the code obtained by shortening ˆC[ℓ, r] at the MSB and has parameters\nn= 2ℓ−2r+ 1, k =ˆkℓr−r, w =ℓ. (27)\nThe modified encoding algorithm is presented in Alg. 7. In spite of reduction in length,\nthe weight still remains as ℓas shown in Lemma 4.2.\nLemma 4.2. For every output cof Algorithm 7, wH(c) =ℓ.\nProof. Consider cin Algorithm 7 before Line 8 is executed. Recall the proof of\nLemma 2.2 and in particular (8) that estimates the the maximum cumulative incre-\nment pin the variable pos. Applying that to the context of Algorithm 7, we observe\nthatpby the end of ℓ−1 iterations of the loop is equal to 2ℓ−2ℓ−r−1−(1−δ(ℓ, r))2r.\nSinceB[ℓ, r] is derived from ˆC[ℓ, r], we deduce that p= 2ℓ−2ℓ−r−1−2rby substitut-\ningδ(ℓ, r) = 0. So a subsequent truncation of cby 2r−1 effected by the execution\nofLine 8 does not lead to removal of a bit with value 1. Therefore the output chas\nHamming weight ℓ.\nThe decoding algorithm (presented in Alg. 8) is exactly in line with the one of\nˆC[ℓ, r], but with necessary modifications to take care of the reduced length. The\ncorrectness of the decoder is established in Lemma 4.3.\nLemma 4.3. For every output cof Algorithm 7, cis correctly decoded by Algorithm 8.\nProof. The encoding algorithm of B[ℓ, r] differs from that of ˆC[ℓ, r] in two aspects.\nFirst, the location j[anchor index ] (recall the definition in (10)) is a product of 2rand\ndec(xℓ). Second, (2r−1) bits are deleted by Line 8 of the encoding algorithm, thus\nreducing the length of the codeword to 2ℓ−2r+ 1. By Lemma 4.2, all these deleted\nbits are indeed zeros. Therefore, the deletion only affects g[anchor index ] that do not\ncarry any information regarding xℓ−1,xℓ−2. . . ,x1. As a consequence, if j[anchor index ]\nis identified correctly, then xℓ−1,xℓ−2. . . ,x1will be decoded correctly.\n20Algorithm 7: EncodeB\nInput :x∈ {0,1}ˆkℓr−r,ˆfℓ,r\nOutput :c∈ B[ℓ, r]\n1Partition xasxℓ∥xℓ−1∥···∥ x1such that |xℓ|=ℓ−rand|xi|=ˆfℓ,r(i) for\ni∈[ℓ−1].\n2Initialize array c= 02ℓ\n3pos←2rdec(xℓ)\n4c[pos]← −1\n5forj=ℓ−1, . . . , 1do\n6 pos← −pos+ 1 + dec( xj) mod n\n7 c[pos]← −1\n8c←c\\c[pos+ 1,2r−1]\nAlgorithm 8: DecodeB\nInput :c∈ B[ℓ, r],ˆfℓ,r\nOutput :x∈ {0,1}ˆkℓr−r\n1Find 0 ≤j[0]< j[1]<···< j[ℓ−1]<2nsuch that c[j[i]] = 1 for every\ni= 0,1, . . . , ℓ −1.\n2g[m] = (j[m]−j[(m−1) mod ℓ]−1) mod nform= 0,1, . . . ℓ−1\n3anchor index =FindAnchor2 (ℓ, r,g)\n4Initialize binary vector xsuch that |x|=ℓ−rand\ndec(x) =⌈j[anchor index ]/2r⌉\n5fori= 1,2, . . . , ℓ −1do\n6 g←g[(anchor index +i) mod ℓ]\n7 Represent gas binary string xiof length ˆfℓ,r(ℓ−i)\n8 x←x∥xi\nFig. 5 : Comparison of kagainst the upper bound for B[ℓ, r] when r=rmax.\n21The relative decrease in g[anchor index ] caused by deletion of bits definitely has an\nimpact on the behaviour of FindAnchor2 procedure. But we know that the value\nofj[anchor index ] can be less than the corresponding value in the decoding algorithm\nforˆCℓrby an amount that can at most be 2r−1. By (13), in spite of a reduction by\n2r−1 on its value, g[anchor index ] will still be picked as a candidate gap, and the\nFindAnchor2 procedure will return the anchor index correctly. However, as noted\nabove, the value of j[anchor index ] can be less by an amount that can at most be 2r−1.\nNevertheless, ⌈j[anchor index ]/2r⌉will still recover the same value of dec(xℓ). Thus xℓ\nis decoded correctly. This establishes that the decoding algorithm in Algorithm 8 is\ncorrect.\nThe combinatorial dimension of B[ℓ, r] grows quadratically with ℓas how its upper\nbound grows. A comparison is plotted in Fig. 5 for various values of ℓfixing r=\nrmax=\u0004ℓ+3\n4\u0005\n.\n5 Conclusion and Future Work\nBinary constant-weight codes find extensive applications in many engineering prob-\nlems such as source compression [23], data storage [24], design of spherical codes\nfor communication over Gaussian channels [25], optical communication [26], spread-\nspectrum communication [27], and cryptography [28]. Therefore the design of such\ncodes with low-complexity encoding and decoding algorithms becomes quite relevant\nin practice. In this paper, we present several families of binary constant weight codes\nsupporting a wide range of parameters while permitting linear encoding complexity\nand poly-logarithmic (discounting the linear time spent on parsing the input) decod-\ning complexity. The present work opens up new directions for exploration such as:\n(a) enlarging the codebook further close to the optimal size by controlled compromise\non complexity, (b) achieving larger minimum distance by reducing the codebook size,\nand (c) study of correlation properties of the code.\n6 Acknowledgements\nThis work is supported by the Australian Research Council through the Discovery\nProject under Grant DP200100731.\nReferences\n[1] Johnson, S.M.: A new upper bound for error-correcting codes. IRE Trans. Inf.\nTheory 8(3), 203–207 (1962)\n[2] Graham, R., Sloane, N.: Lower bounds for constant weight codes. IEEE Transac-\ntions on Information Theory 26(1), 37–43 (1980) https://doi.org/10.1109/TIT.\n1980.1056141\n22[3] Brouwer, A.E., Shearer, J.B., Sloane, N.J.A., Smith, W.D.: A new table of con-\nstant weight codes. IEEE Transactions on Information Theory 36(6), 1334–1380\n(1990) https://doi.org/10.1109/18.59932\n[4] Agrell, E., Vardy, A., Zeger, K.: Upper bounds for constant-weight codes. IEEE\nTransactions on Information Theory 46(7), 2373–2395 (2000) https://doi.org/10.\n1109/18.887851\n[5] Brouwer, A.: Bounds for binary constant weight codes. https://www.win.tue.nl/\n%7eaeb/codes/Andw.html. [Online; accessed 23-Oct-2023] (2023)\n[6] Moreno, O., Zhang, Z., Kumar, P.V., Zinoviev, V.A.: New constructions of\noptimal cyclically permutable constant weight codes. IEEE Transactions on\nInformation Theory 41(2), 448–455 (1995) https://doi.org/10.1109/18.370146\n[7] Bitan, S., Etzion, T.: Constructions for optimal constant weight cyclically per-\nmutable codes and difference families. IEEE Transactions on Information Theory\n41(1), 77–87 (1995) https://doi.org/10.1109/18.370117\n[8] Nordio, A., Viterbo, E.: Permutation modulation for fading channels. In: 10th\nInternational Conference on Telecommunications, 2003. ICT 2003., vol. 2, pp.\n1177–11832 (2003). https://doi.org/10.1109/ICTEL.2003.1191603\n[9] MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-correcting Codes.\nMathematical Library. North-Holland Publishing Company, New York (1977)\n[10] Schalkwijk, J.: An algorithm for source coding. IEEE Transactions on Information\nTheory 18(3), 395–399 (1972)\n[11] Cover, T.: Enumerative source encoding. IEEE Transactions on Information\nTheory 19(1), 73–77 (1973)\n[12] Lehmer, D.H.: Teaching combinatorial tricks to a computer. In: Proc. Sympos.\nAppl. Math., Vol. 10, pp. 179–193. Amer. Math. Soc., Providence, RI, New York\n(1960)\n[13] Pascal, E.: Sopra una formula numerica. Gi Di Mat 25, 45–49 (1887)\n[14] Knott, G.D.: A numbering systems for combinations. Commun. ACM 17(1), 45–\n46 (1974)\n[15] Er, M.C.: Lexicographic ordering, ranking and unranking of combinations. Int. J.\nComput. Math. 17(1), 277–283 (1985)\n[16] Kokosinski, Z.: Algorithms for unranking combinations and their applications.\nIn: Hamza, M.H. (ed.) Proceedings of the Seventh IASTED/ISMM International\nConference on Parallel and Distributed Computing and Systems, Washington,\nD.C., USA, October 19-21, 1995, pp. 216–224 (1995)\n23[17] Ruskey, F., Williams, A.: The coolest way to generate combinations. Discret.\nMath. 309(17), 5305–5320 (2009)\n[18] Genitrini, A., P´ epin, M.: Lexicographic unranking of combinations revisited.\nAlgorithms 14(3), 97 (2021)\n[19] Kruchinin, V.V., Shablya, Y.V., Kruchinin, D.V., Rulevskiy, V.: Unranking small\ncombinations of a large set in co-lexicographic order. Algorithms 15(2), 36 (2022)\n[20] Sendrier, N.: Encoding information into constant weight words. In: Proceedings.\nInternational Symposium on Information Theory, 2005. ISIT 2005., pp. 435–438\n(2005)\n[21] Slepian, D.: Permutation modulation. Proceedings of the IEEE 53(3), 228–236\n(1965) https://doi.org/10.1109/PROC.1965.3680\n[22] Riordan, J.: An Introduction to Combinatorial Analysis. Princeton Legacy\nLibrary. Princeton University Press, Princeton (1978)\n[23] Dai, V., Zakhor, A.: Binary combinatorial coding. In: Data Compression Con-\nference, 2003. Proceedings. DCC 2003, p. 420 (2003). https://doi.org/10.1109/\nDCC.2003.1194039\n[24] Kurmaev, O.F.: Constant-weight and constant-charge binary run-length limited\ncodes. IEEE Transactions on Information Theory 57(7), 4497–4515 (2011) https:\n//doi.org/10.1109/TIT.2011.2145490\n[25] Ericson, T., Zinoviev, V.: Chapter 6 - non-symmetric alphabets. In: Ericson, T.,\nZinoviev, V. (eds.) Codes on Euclidean Spheres. North-Holland Mathematical\nLibrary, vol. 63, pp. 179–194. Elsevier, New York (2001). https://doi.org/10.1016/\nS0924-6509(01)80051-9\n[26] Chung, H., Kumar, P.V.: Optical orthogonal codes - new bounds and an optimal\nconstruction. IEEE Transactions on Information Theory 36(4), 866–873 (1990)\nhttps://doi.org/10.1109/18.53748\n[27] Ding, C., Fuji-Hara, R., Fujiwara, Y., Jimbo, M., Mishima, M.: Sets of frequency\nhopping sequences: Bounds and optimal constructions. IEEE Transactions on\nInformation Theory 55(7), 3297–3304 (2009) https://doi.org/10.1109/TIT.2009.\n2021366\n[28] Finiasz, M., Gaborit, P., Sendrier, N.: Improved fast syndrome based crypto-\ngraphic hash functions. In: ECRYPT Hash Workshop 2007, Proceedings, p. 155\n(2011)\n24" }, { "title": "2401.16697v1.The_Velocity_Space_Signature_of_Transit_Time_Damping.pdf", "content": "Under consideration for publication in J. Plasma Phys. 1\nThe Velocity-Space Signature of\nTransit-Time Damping\nRui Huang †1, Gregory G. Howes1, and Andrew J. McCubbin2\n1Department of Physics and Astronomy, University of Iowa, IA 52242, USA\n2Applied Physics Laboratory, Johns Hopkins University, MD 20723, USA\n(Received ?; revised ?; accepted ?. - To be entered by editorial office)\nTransit-time damping (TTD) is a process in which the magnetic mirror force—induced\nby the parallel gradient of magnetic field strength—interacts with resonant plasma par-\nticles, leading to the collisionless damping of electromagnetic waves and the resulting\nenergization of those particles through the perpendicular component of the electric field,\nE⊥. In this study, we utilize the recently developed field-particle correlation technique\nto analyze gyrokinetic simulation data. This method enables the identification of the\nvelocity-space structure of the TTD energy transfer rate between waves and particles\nduring the damping of plasma turbulence. Our analysis reveals a unique bipolar pattern\nof energy transfer in velocity space characteristic of TTD. By identifying this pattern,\nwe provide clear evidence of TTD’s significant role in the damping of strong plasma\nturbulence. Additionally, we compare the TTD signature with that of Landau damping\n(LD). Although they both produce a bipolar pattern of phase-space energy density loss\nand gain about the parallel resonant velocity of the Alfv´ enic waves, they are mediated\nby different forces and exhibit different behaviors as v⊥→0. We also explore how the\ndominant damping mechanism varies with ion plasma beta βi, showing that TTD domi-\nnates over LD for βi>1. This work deepens our understanding of the role of TTD in the\ndamping of weakly collisional plasma turbulence and paves the way to seek the signature\nof TTD using in situ spacecraft observations of turbulence in space plasmas.\nPACS codes:\n1. Introduction\nA key area of research in the study of turbulence in weakly collisional plasmas is un-\nderstanding how energy from the fluctuating plasma flows and electromagnetic fields\nis converted into plasma particle energy. This phenomenon is especially relevant in he-\nliospheric plasmas like the solar wind, where the characteristic low density and high\ntemperature lead to weakly collisional plasma dynamics. The dissipation of turbulence\nin such space and astrophysical plasmas is likely mediated by three categories of mech-\nanisms: (i) resonant wave-particle interactions, such as Landau damping (Landau 1946;\nChen et al. 2019), transit-time damping (Stix 1992; Barnes 1966), and cyclotron damp-\ning (Isenberg & Hollweg 1983; Isenberg & Vasquez 2019); (ii) non-resonant wave-particle\ninteractions, including stochastic heating (Chandran et al. 2010, 2013; Martinovi´ c et al.\n2020; Cerri et al. 2021), magnetic pumping (Lichko & Egedal 2020; Montag & Howes\n2022), and “viscous” damping mediated by kinetic temperature anisotropy instabilities\n†Email address for correspondence: rui-huang@uiowa.eduarXiv:2401.16697v1 [physics.plasm-ph] 30 Jan 20242\n(Arzamasskiy et al. 2023); and (iii) dissipation within coherent structures, in particular\ncollisionless magnetic reconnection that may occur in current sheets that are found to\narise naturally in plasma turbulence (Osman et al. 2011; Zhdankin et al. 2015; Mallet\net al. 2017; Loureiro & Boldyrev 2017).\nGiven the low collisionality of these plasma environments, the six-dimensional (3D-3V)\nkinetic plasma theory is essential for analyzing the evolution of the turbulence and its\ndissipation through collisionless interactions between electromagnetic fields and plasma\nparticles (Howes 2017). Although in situ spacecraft measurements in the solar wind\nprovide invaluable data, they are often limited to a single point, or a few points, in\nspace, which presents a significant challenge for the investigation of the physical mech-\nanisms that remove energy from the turbulent fluctuations and consequently energize\nthe plasma particles. The recently developed field–particle correlation technique (Klein\n& Howes 2016; Howes et al. 2017; Klein et al. 2017) enables direct measurements of the\nelectromagnetic fields and particle velocity distributions at a single point in space to be\ncombined to create a velocity-space signature of particle energization that can be used to\nidentify the physical mechanisms responsible for damping the turbulence and to estimate\nthe resulting rate of the change of particle energy density. Consequently, this technique\nprovides an innovative means to utilize in situ spacecraft observations to identify specific\ncollisionless damping mechanisms and determine particle heating rates.\nThis technique has shown success in identifying several damping mechanisms in weakly\ncollisional turbulent plasmas, such as ion Landau damping (Klein et al. 2017; Li et al.\n2019), ion cyclotron damping (Klein et al. 2020; Afshari et al. 2023), electron Landau\ndamping (Chen et al. 2019; Li et al. 2019; Afshari et al. 2021; Conley et al. 2023), and\nmagnetic pumping (Montag & Howes 2022). However, the role of transit-time damping,\na resonant wave-particle interaction, in the damping of plasma turbulence remains un-\nconfirmed. The focus of this paper is to employ the field-particle correlation technique\nto identify the velocity-space signature of ion energization through transit-time damping\nand to recover this signature from simulations of strong plasma turbulence.\nThe structure of this paper is laid out as follows. We derive the specific form of the\nfield-particle correlation for transit-time damping in §2.1. This is followed by an explo-\nration of the expected transit-time damping signature in §2.2. In §3, we conduct single\nkinetic Alfv´ en wave simulations to investigate the velocity-space signature characteris-\ntic of transit-time damping. Subsequently, in §4, we delve into turbulence simulations,\npresenting details for distinguishing transit-time damping from turbulence damping pro-\ncess. §5 summarizes our findings and outlines potential future applications for further\nresearch.\n2. Transit-Time Damping\nThe idea of transit-time damping (TTD) had its origins in mid-20th-century plasma\nphysics when transit-time magnetic pumping was proposed as a means to heat confined\nplasma (Spitzer Jr & Witten 1951). This method is characterized by a modulation of\nthe magnetic field magnitude at a frequency considerably lower than the ion cyclotron\nfrequency; the double adiabatic evolution of the parallel and perpendicular temperatures,\ncombined with a weak collisionality, leads to a net transfer of energy to the plasma\nparticles. The term “transit-time” refers to the duration necessary for an ion to traverse\nfrom one side to the other across the confined region.\nThe magnetic mirror force plays a key role in the dynamics of TTD. In a static mag-\nnetic field with a spatial variation of the magnetic field magnitude along the direction\nparallel to the field, the mirror force accelerates charged particles in the direction of de-Transit-Time Damping 3\nzxoB∇BFFzFFryFrFz\nFigure 1. Diagram of the radial component Frand axial component Fzof the Lorentz force\nof the magnetic field (red) on a positively charged particle (red +) in a converging magnetic\nfield (green) with increasing magnitude in the + zdirection. Averaged over the Larmor orbit of\nthe particle (blue), the net magnetic mirror force is in the direction of decreasing magnetic field\nmagnitude, here the −zdirection.\ncreasing field magnitude. In a cylindrical coordinate system aligned with the magnetic\nfield direction, the condition ∇·B= 0 implies that an increase of the magnetic field along\nthe axial direction must be accompanied by the convergence of the field in the radial di-\nrection, as shown in Figure 1. For a particle with a guiding center on the axis, the particle\nwill experience an inward radial field throughout its Larmor orbit. The Lorentz force,\nwhich acts perpendicularly to the magnetic field direction at the particle position, will\ntherefore have both a large radial and a small axial component. Averaged over the full\nLarmor orbit, the net nonzero axial component accelerates the particle in the direction\nof the decreasing magnetic field magnitude. Because the magnetic field can do no work,\nthe total energy of the particle remains constant—the change in the parallel velocity is\naccompanied by a small change in the perpendicular velocity governed by the average of\nthe radial component of the Lorentz force. The net effect of the magnetic mirror force is\nthat, as a particle moves in the direction of the increasing magnetic field, the mirror force\nreduces the velocity v∥parallel to the mean magnetic field over the Larmor orbit and\nincreases perpendicular velocity v⊥to maintain a constant total velocity v= (v2\n⊥+v2\n∥)1/2.\nAlthough the mirror force in a static magnetic field cannot change the energy of par-\nticles, any changes of the magnetic field in time will induce an electric field according\nto Faraday’s Law. Work done by that induced electric field can do work on the parti-\ncles, providing the key element underlying the physics of TTD. In a collisionless plasma,\ncollisionless wave-particle interactions are governed by the resonance condition, given\nbyω−k∥v∥=nΩs, where nΩsforn= 0,±1,±2, . . .incorporates the cyclotron har-\nmonics of the particle motion in a magnetic field (Melrose 1980). The n= 0 resonance,\nknown as the Landau resonance, describes resonant interactions with particles that have\nparallel velocities near the phase velocity of the wave, v∥≃ω/k∥, enabling energy ex-\nchange between the particles and the wave through two mechanisms: (i) the electrostatic\nforce due to parallel component of the electric field governs the phenomenon of Landau\ndamping (LD) (Landau 1946; Villani 2014); (ii) the magnetic mirror force governs the4\nphenomenon of TTD (Stix 1992), also known as Barnes damping (Barnes 1966). In the\ncase of TTD, the perpendicular component of the electric field, induced by the change\nin the magnetic field magnitude along the parallel direction, accelerates the particle by\nchanging the perpendicular velocity v⊥; the mirror force effectively converts this perpen-\ndicular velocity into parallel velocity, leading to a net acceleration of the particle along\nthe axial direction parallel to the magnetic field (Howes et al. 2024). For a Maxwellian\ndistribution of particles, there are more particles with parallel velocities v∥< ω/k ∥than\nparticles with v∥> ω/k ∥, so the net effect on the distribution is an increase of the particle\nenergy, leading to damping of the wave. A detailed demonstration of this phenomenon\nfor a model moving magnetic mirror field is presented in §2.2.\n2.1.Field-Particle Correlation for Transit-Time Damping\nTo determine the appropriate form of the field-particle correlation to diagnose TTD via\nthe magnetic mirror force, we start with the Vlasov equation for a species sbeing acted\nupon by a general force Fs,\n∂fs\n∂t+v· ∇fs+Fs\nms·∂fs\n∂v= 0. (2.1)\nMultiplying the Vlasov equation by msv2/2, we obtain an expression for the rate of\nchange of the phase-space energy density ,ws(r,v, t)≡msv2fs(r,v, t)/2,\n∂ws(r,v, t)\n∂t=−v· ∇ws−v2\n2Fs·∂fs\n∂v. (2.2)\nPrevious analysis of this equation (Klein & Howes 2016; Howes et al. 2017) has shown\nthat, if integrated over space (with appropriate infinite or periodic boundary conditions),\nthe change in the total kinetic energy of the particles Ws(t) =R\nd3rR\nd3vws(r,v, t) is\ndue to work done on the particle species by the force Fs. Therefore, the field-particle\ncorrelation due to a general force Fsat spatial position r0is defined as a time-average\nover a correlation interval τof the last term on the right hand side of (2.2),\nCFs(r0,v, t;τ)≡1\nτZt+τ/2\nt−τ/2−v2\n2Fs(r0,v, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.3)\nNote here that the correlation interval τis a parameter of the field-particle correlation\nanalysis, and so is included as a secondary argument, separated by a semicolon from\nthe primary arguments that define the dimensions of the 3D-3V phase space in position,\nvelocity, and time.\nIf we consider the force due to an electric field Fs=qsE, we obtain the established\nfield-particle correlation due to the electric field (Klein & Howes 2016; Howes et al. 2017;\nKlein et al. 2017),\nCE,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2−qsv2\n2E(r0, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.4)\nThe collisionless transfer of energy between electromagnetic waves and particles in\nTTD is mediated by the magnetic mirror force, Fs=−µsˆb· ∇B, where the magnetic\nmoment for a particle of species sis given by µs=msv2\n⊥/(2B), the unit vector in the\ndirection of the magnetic field is given by ˆb≡B/B, and the magnitude of the magnetic\nfield is B=|B|. Substituting the magnetic mirror force into (2.3), we obtain\nCB,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2msv2v2\n⊥\n4B(ˆb· ∇)B(r0, t′)·∂fs(r0,v, t′)\n∂vdt′. (2.5)Transit-Time Damping 5\nA few modifications of the form of the field-particle correlation for TTD given in (2.5)\nare helpful for its application to the gyrokinetic simulations presented here. First, we\nexploit two important characteristics of TTD and turbulence: (i) TTD is most effective\nfor electromagnetic waves with wavelengths at the ion scales, kρi∼1; and (ii) for most\nturbulent space and astrophysical plasmas of interest, the amplitude of the magnetic\nfluctuations δBat ion scales kρi∼1 is much smaller than the magnitude of the mean\nmagnetic field B0. Therefore, if we separate the magnetic field into its mean plus the\nfluctuations, B=B0+δB, where |δB| ≪ |B0|, the change in the magnetic field magni-\ntude δ|B|(which is the key ingredient for the magnetic mirror force) can be expressed\nasδB∥by recognizing\nδ|B|=|B| − |B0|=p\n(B0+δB)2−B0=q\nB2\n0+ 2δB·B0+|δB|2−B0≃δB∥(2.6)\nwhere we use a binomial expansion to eliminate the square root, neglect the small |δB|2\nterm, and write δB∥=δB·(B0/B0) as the variation in the component of the perturbed\nmagnetic field parallel to the mean magnetic field. Furthermore, separating term v2=\nv2\n⊥+v2\n∥in the correlation (2.5), it is easy to show that the v2\n⊥contribution yields a\nperfect differential in fswhen integrated over all parallel velocity (Howes et al. 2017),\nso we choose to omit this term since it leads to zero net change in the particle energy.\nFinally, we write the gradient along the magnetic field direction by ∇∥≡ˆb· ∇, leading\nto our preferred form of the field-particle correlation for TTD,\nCδB∥,s(r0,v, t;τ) =1\nτZt+τ/2\nt−τ/2msv2\n∥v2\n⊥\n4B∇∥δB∥∂fs(r0,v, t′)\n∂v∥dt′. (2.7)\nIn the gyrokinetic system of equations (Antonsen Jr & Lane 1980; Frieman & Chen\n1982; Howes et al. 2006; Schekochihin et al. 2009), the gyro-averaged effect of parallel\nmagnetic field gradients leads to the magnetic mirror force, which can accelerate particles\nin the direction parallel to the magnetic field via the Landau resonance, and therefore\ncan lead to collisionless TTD of electromagnetic fluctuations. A rigorous derivation of the\ngyrokinetic equation in Appendix A shows explicitly the two collisionless wave-particle\ninteractions via the Landau resonance—specifically, LD and TTD. The natural form of\nthe field-particle correlation arising from the gyrokinetic version of the generalized energy\ndensity equation is slightly different from the field-particle correlation for TTD given in\n(2.7), but the gyrokinetic form requires the gyroaveraged distribution function, which is\nnot accessible through single-point spacecraft measurements and is not a natural quantity\nthat can easily be derived from other kinetic simulation approaches, such as particle-in-\ncell or Vlasov simulations. Therefore, we choose here to use the perturbed distribution\nfunctions and electromagnetic fields generated by our gyrokinetic simulations, but we\nanalyze them using (2.7) which is more directly applicable to spacecraft measurements\nor alternative kinetic simulation approaches, such as particle-in-cell codes.\nUtilizing the gyroaveraged distribution function and corresponding fields, the quantity\nCδB∥,s(r0,v, t;τ) reveals the velocity-space signature of TTD on the gyrotropic velocity\nspace ( v⊥, v∥). To simplify the notation, henceforth we shall employ CδB∥,s(v∥, v⊥) to\nsignify the gyrotropic correlation, explicitly noting the associated spatial position r0,\ntime t, and the correlation interval τonly when necessary. The resonant structure of\nthe velocity-space signature of TTD is primarily a function of v∥, so it is often useful to\ndefine the reduced parallel field-particle correlation by integrating CδB∥,s(v∥, v⊥, t) over\nv⊥, given by CδB∥,s(v∥, t)≡2πR\nCδB∥,s(v∥, v⊥, t)v⊥dv⊥, where the extra 2 πv⊥factor\narises from the integration over the gyrophase in 3V phase space. A timestack plot of the6\nreduced parallel correlation CδB∥,s(v∥, t) reveals the persistence in time of any resonant\nvelocity-space signatures in v∥. We can also consider the rate of change of the kinetic\nenergy density of species sdue to TTD by integrating the gyrotropic correlation over all\nvelocity space: ( ∂Ws/∂t)TTD=R\nCδB∥,s(v∥, v⊥)d3v.\nIn closing, note that the gyroaveraging procedure employed in the derivation of the\nsystem of gyrokinetics enables the variations in the perpendicular components of the\nelectric field E⊥to be expressed in terms of changes in the parallel component of the\nmagnetic field δB∥, as shown by (A 7). In a system where the gyroaverage has not been\nperformed, the work done by TTD is actually mediated (at the position of the particle)\nby the perpendicular component of the electric field E⊥(Howes et al. 2024). Therefore,\nthe perpendicular electric field correlation, given by summing the two perpendicular\ncontributions to the electric field correlation, CE⊥(r0,v, t;τ) (Klein et al. 2020; Afshari\net al. 2023), can be used to seek the velocity-space signature of TTD at the parallel\nresonant phase velocity, as seen recently in hybrid particle-in-cell simulations of plasma\nturbulence (Cerri et al. 2021).\n2.2.Prediction of the Velocity-Space Signature of Transit-Time Damping\nTo predict the velocity-space signature of TTD, we begin with a simple model of a\nmagnetic field with an amplitude variation that varies along the mean field direction z,\ngiven in cylindrical coordinates ( r, ϕ, z ) by\nB(r, ϕ, z ) =−δBz\n4krsin(kz′)ˆr+\u001a\nB0+δBz\n2[1−cos(kz′)]\u001b\nˆz (2.8)\nwhere the wavenumber kof the spatial variation of the magnetic field magnitude is\nalong the mean field direction z, and z′=z−Ut, such that that pattern moves in the\n+zdirection with a phase speed U⩾0. The corresponding electric field variation can\nbe determined by the Lorentz transform from the primed (wave) frame K′in which\nthe magnetic field pattern is stationary (and therefore E′= 0) to the unprimed (lab)\nframe K. This Lorentz transformation in the non-relativistic limit U/c≪1 is given by\nE=E′−U×BandB=B′(Howes et al. 2014), where the transformation velocity is\njustU=Uˆz. The resulting induced electric field in the lab frame Kis given by\nE(r, ϕ, z ) =UδBz\n4krsin(kz′)ˆϕ. (2.9)\nWith this simple model, we can illustrate how a single particle is accelerated into different\nregions of velocity space by the electromagnetic fields. Extending this approach to con-\nsider a distribution of particles will enable us to predict the qualitative and quantitative\nfeatures of the velocity-space signature of TTD.\nConsider first the acceleration of a single particle in a stationary mirror field with\nU= 0, as shown in Figure 2(a), for a “wave” amplitude of δBz/B0= 0.2, giving a mirror\nratio of Bmax/Bmin= 1.2. The particle begins at the minimum in the magnetic field at\nz= 0 with an initial perpendicular velocity v⊥and an initial parallel velocity v∥<0,\ngiven by the red + in the figure at the tip of the initial velocity vector vi(blue). As the\nparticle moves into the increasing magnetic field at z <0, the mirror force increases v⊥\nand decreases v∥such that the particle moves through velocity space (green arrow) on a\ncircle of constant total velocity v=q\nv2\n∥+v2\n⊥(black dashed circle). The particle will be\nreflected by the mirror field if the particle has an initial pitch angle α= tan−1(v⊥/v∥)\nlarger than the loss cone angle αloss= sin−1\u0010p\nBmin/Bmax\u0011\n. For Bmax/Bmin= 1.2, the\nloss cone angle is αloss= 66◦; the particle depicted in Figure 2(a) has an initial pitchTransit-Time Damping 7\nangle α > αlossand is therefore reflected by the mirror field. The particle follows this\ncircular trajectory in ( v∥, v⊥) velocity space until it returns to its initial axial position\nz= 0, ending up with a final velocity vf(blue) with the same perpendicular component\nbut an equal and opposite parallel component. Thus, the kinetic energy of the particle\ndoes not change, consistent with the fact that magnetic fields do no work on particles.\nThe particle has simply been reflected by the mirror, reversing the sign of its parallel\nvelocity.\nNext, we consider the case for a magnetic mirror field moving with velocity U=Uˆz,\nwhere U > 0. In the wave frame, moving at velocity U=Uˆzin which the magnetic field is\nstationary, the acceleration of the particle by the magnetic field must be the same as the\nstationary case in Figure 2(a). But, in the lab frame, depicted in Figure 2(b), the particle\nnow moves on a circular trajectory in velocity space centered about the mirror velocity U,\ngiven by a constant magnitude of velocity in the wave frame v0=q\n(v∥−U)2+v2\n⊥. Here\nthe particle is initially moving in the same direction as the mirror field but with a slower\ninitial parallel velocity 0 ⩽v∥⩽U, given by the red + in the figure at the tip of the initial\nvelocity vector vi(blue). If the pitch angle in the wave frame αw= tan−1[v⊥/(v∥−U)] is\ngreater than the loss-cone angle, αw> αloss, the particle will be reflected by the moving\nmirror field, leading to a net acceleration in the axial direction, ultimately ending up with\na parallel velocity greater than the mirror velocity v∥> U, with a final velocity vector vf\n(blue). In this case, the induced electric field given by (2.9) has done work on the particle\n(Howes et al. 2024), ultimately leading to a net acceleration in the axial direction. This\nprocess is the fundamental energy transfer underlying the physics of TTD.\nFinally, we consider how this understanding of the single particle motion and accel-\neration can be combined with a distribution of initial particle velocities to predict the\nvelocity-space signature of TTD. Note that for the sinusoidally oscillating magnetic field\nmagnitude given by (2.8), the long time evolution of the particle in velocity space would\noscillate back and forth between viandvfalong the green trajectory shown in Fig-\nure 2(a) and (b); for example, if the particle started with initial velocity vf, the other\nside of the mirror field would lead to a reflection in the opposite direction, ultimately\nresulting in the particle ending up with a final velocity vi. In the case of a moving mirror\nfield, for a realistic velocity distribution there will be more particles with parallel veloc-\nitiesv∥< U than with v∥> U, so the net effect is that more particles will gain energy\nthan lose energy, leading to a net energization of the particles and consequent damping\nof the electromagnetic wave. Only particles with pitch angles in the wave frame larger\nthan the loss cone angle, αw> αloss, will undergo the mirror reflection, so that net effect\non the distribution is an acceleration of particles from v∥< U tov∥> U. The resulting\nchange in the phase-space energy density leads to the prediction of the velocity-space sig-\nnature of TTD depicted in Figure 2(c): a loss of phase-space energy density (blue) in the\nregion v∥< U, and a gain of phase-space energy density (red) in the region v∥> U. The\nextent of this velocity-space signature in ( v∥, v⊥) velocity space is confined by two effects:\n(i) only particles outside of the loss cone will experience a net acceleration; and (ii) the\nsignature is weighted by the v2\n⊥weighting in (2.7) for the rate of change of phase-space\nenergy density by TTD (which arises from the magnetic moment µ=mv2\n⊥/(2B) depen-\ndence of the mirror force) combined with the reduced perpendicular velocity distribution\nf(v⊥), where this net weighting of v2\n⊥f(v⊥) is shown in Figure 2(d). Thus, the velocity-\nspace signature of TTD in Figure 2(c) is restricted to “Landau resonant” particles with\nparallel velocities near the velocity of the magnetic field pattern v∥∼Uand to a region\naway from the v⊥= 0 axis, unlike the velocity-space signature of LD (Klein & Howes\n2016; Howes et al. 2017; Klein et al. 2017) which extends down to v⊥= 0.8\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαloss>>\nvi vfU= 0\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαloss>>\nvi vfU > 0 U(a) (b)\n−0.5 0.0 0.5 1.0\nv/bardbl0.00.20.40.60.81.01.2v⊥\nαlossU\n0.0 0.1\nv2\n⊥f(v⊥)0.000.250.500.751.001.251.501.75v⊥\n(c) (d)\nFigure 2. Diagram of the magnetic mirror force and prediction for the velocity-space signature\nof TTD: (a) v⊥versus v∥for the single particle motion in a static magnetic mirror field; (b) v⊥\nversus v∥for the single particle motion in a moving magnetic mirror field, where the vertical\nblack dashed line denotes the wave phase velocity U; (c) the predicted velocity-space signature\nfor a Maxwellian velocity distribution function, where the phase-space energy density decreases\natv∥< U(blue) and increases at v∥> U(red); (d) effective v⊥weighting of correlation v2\n⊥f(v⊥),\nwhich constrains the velocity-space signature in the v⊥direction.\n3. Single Kinetic Alfv´ en Wave (KAW) Simulations\nHere we perform numerical simulations of single kinetic Alfv´ en waves to determine the\nvelocity-space signature of TTD using the Astrophysical Gyrokinetics Code, AstroGK\n(Numata et al. 2010). AstroGK evolves the gyroaveraged scalar potential ϕ(r), parallel\nvector potential A∥(r), and the parallel magnetic field fluctuation δB∥(r), as well as\nthe gyrokinetic distribution function hs(r, v⊥, v∥), in a triply-periodic slab geometry of\nsizeL2\n⊥×L∥elongated along the straight, uniform mean magnetic field B0=B0ˆz.\nThe domain-scale wavenumbers are defined by k∥0= 2π/L∥andk⊥0= 2π/L⊥. The\ngyrokinetic expansion parameter is defined by ϵ∼k∥0/k⊥0≪1 (Howes et al. 2006),\nand all quantities are scaled to accommodate an arbitrary value of ϵ. The gyrokinetic\ndistribution function is related to the total distribution function fsvia\nfs(r,v, t) =F0s(v)\u0012\n1−qsϕ(r, t)\nTs\u0013\n+hs(Rs, v⊥, v∥, t) +δf2s+... (3.1)Transit-Time Damping 9\nwhere F0sis the equilibrium distribution, ris the spatial position, Rsis the associated\nspecies gyrocenter related to rbyr=Rs−v׈z/Ωs, and δf2sare corrections second-\norder in the gyrokinetic expansion parameter ϵwhich are not retained (Howes et al.\n2006). The code employs a pseudospectral method in the ( x, y) (perpendicular) plane\nand finite differencing in the z-direction. The velocity distribution is resolved on a grid\nin energy E=v2\n∥+v2\n⊥and pitch angle λ=v2\n⊥/v2space, with the points selected on a\nLegendre polynomial basis. A fully conservative, linearized, gyroaveraged collision oper-\nator is employed (Abel et al. 2008; Barnes et al. 2009) to ensure velocity-space structure\nishsremains resolved throughout the simulation evolution. We normalize time using\nthe domain-scale Alfv´ en wave frequency ωA≡k∥0vA, and particle velocity is normalized\nto the ion thermal velocity vti=p\n2Ti/mi, where the Boltzmann constant has been\nabsorbed to yield temperature in units of energy.\n3.1.Single Wave Simulation Set Up\nWe first determine the velocity-space signature of TTD by performing simulations of\nsingle kinetic Alfv´ en waves (KAWs) for a fully ionized proton-electron plasma with\nMaxwellian equilibrium velocity distributions with a temperature ratio Ti/Te= 1 and a\nrealistic ion-to-electron mass ratio of mi/me= 1836. We perform three simulations with\nion plasma beta βi= 0.3,1,3 and sample the time evolution of the electromagnetic fields\nand gyrokinetic distribution function at discrete single points throughout the simulation\ndomain with dimensions L⊥= 2πρiandL∥= 2πa0, yielding an arbitrary expansion pa-\nrameter ϵ=ρi/a0≪1. Here ρi≡vti/Ωiis the ion Larmor radius, where the angular ion\n(proton) cyclotron frequency is Ω i=qiB0/mi. The dimensions of these single-wave simu-\nlations are ( nx, ny, nz, nλ, nE, ns) = (10 ,10,32,128,64,2), where nsdenotes the number\nof species. Using the solutions to the linear gyrokinetic dispersion relation (Howes et al.\n2006), a single plane-wave KAW with wavevector ( kxρi, kyρi, k∥a0) = (1 ,0,1) is ini-\ntialized throughout the domain and allowed to evolve linearly for 5 wave periods with\nenhanced collisionality to eliminate any transients associated with the initialization, yield-\ning a clean, single KAW with k⊥ρi= 1. The simulation is then restarted with lowered\ncollisionalities νs/(k∥0vti) = 2 ×10−3and evolved to allow collisionless wave-particle\ninteractions to damp the wave. We have verified that the collisionless damping rates\nof the initialized KAWs agree with the analytical predictions from Vlasov-Maxwell and\ngyrokinetic linear dispersion relations.\n3.2.The Velocity-Space Signature of Transit-Time Damping\nWe select the βi= 1 single KAW simulation as a fiducial case to determine the velocity-\nspace signature of TTD and compare it to the known velocity-space signature of LD\n(Klein & Howes 2016; Howes et al. 2017, 2018; Klein et al. 2017; Chen et al. 2019;\nKlein et al. 2020; Horvath et al. 2020; Afshari et al. 2021). The linear Vlasov-Maxwell\ndispersion relation yields a parallel phase velocity normalized to the Alfv´ en velocity of\nω≡ω/(k∥vA) = 1 .137 for a KAW with k⊥ρi= 1, βi= 1, and Ti/Te= 1, corresponding\nto a normalized wave period of TωA= 5.526.\nA key step in the field-particle correlation analysis is to choose an appropriate correla-\ntion interval τover which to time-average the rate of energization to eliminate a possibly\nlarger amplitude signal of oscillatory energy transfer in order to reveal the often smaller\nsecular rate of energy transfer that corresponds to the collisionless damping of the wave\n(Klein & Howes 2016; Howes et al. 2017; Klein et al. 2017). In Figure 3, we present a\ntest of different correlation intervals over the range 0 ⩽τωA⩽10 for the βi= 1 single\nKAW simulation. In panel (a), we plot the velocity-space integrated rate of ion ener-\ngization due to TTD, ( ∂Wi/∂t)TTD=R\nCδB∥,i(v∥, v⊥)d3v, vs. time. The unaveraged10\n60 70 80\ntωA−0.50.00.51.0∂Wi/∂t×10−5\n0.02.55.07.510.0\nτωA\n60 70 80\ntωA−1.5−1.0−0.50.00.51.0CδB/bardbl,i(v/bardbl= 1.1vti)×10−5\n0.02.55.07.510.0\nτωA\n(a) (b)\n−2 0 2\nv/bardbl/vti6065707580tωACδB/bardbl,i(v/bardbl),τωA= 0.0\n−1.0−0.50.00.51.0×10−5\n−2 0 2\nv/bardbl/vti6065707580tωACδB/bardbl,i(v/bardbl),τωA= 5.5\n−6−4−20246×10−6\n(c) (d)\nFigure 3. Analysis of correlation interval selection for βi= 1AstroGK single KAW simulation.\nTop row: time evolution of (a) the rate of change of ion kinetic energy density due to TTD,\ndenoted as ∂Wi/∂t, and (b) the reduced correlation CδB∥,i(v∥, t) atv∥= 1.1vti. Both quantities\nare presented over a range of τωAvalues from 0 to 10. The selected τωAvalue of 5.5 is marked\nwith a black line. Bottom row: timestack plots of the reduced correlation CδB∥,i(v∥, t) for (c)\nτωA= 0 and (d) τωA= 5.5, with the vertical dashed line at v∥/vti= 1.137 labelling the\nnormalized parallel phase velocity ω/(k∥vti).\ncorrelation ( τ= 0, dark blue) exhibits pronounced oscillations of the net energy transfer\nvs. time. Setting the correlation interval to one linear wave period τωA=TωA≃5.5\n(black) minimizes the oscillations, providing an optimal choice for τfor a single wave\nwith a well-defined period, as might be expected on theoretical grounds. We show in\npanel (b) the evolution of the reduced parallel correlation CδB∥,i(v∥, t) at a parallel ve-\nlocity v∥= 1.1vtislightly below the resonant velocity. Here again, setting τωA= 5.5\n(black) effectively minimizes the oscillations, revealing clearly the rate of secular energy\ntransfer at that parallel velocity. We illustrate the impact of choosing an appropriate\ncorrelation interval τon the timestack plot of the correlation CδB∥,i(v∥, t) by compar-\ning (c) the instantaneous ( τ= 0) field-particle correlation to (d) the correlation using\nτωA= 5.5, showing a clear bipolar signature about the normalized parallel phase velocity\nω/(k∥vti) = 1 .137 that persists over the course of the simulation.\nSpecifying the correlation interval to be approximately one wave period τωA= 5.5,†\nwe now present in Figure 4 the velocity-space signatures of (a) TTD and (b) LD from\ntheβi= 1 single KAW simulation. Each panel presents three sub-plots, explained here\nfor the TTD case in panel (a): (i) the main plot presents the gyrotropic velocity-space\nsignature CδB∥,i(v∥, v⊥) at time tωA= 63.57, with the parallel phase velocity indicated\n†Note that the velocity-space signature for a single KAW is independent of the probe position\nif the correlation interval is taken an integral multiple of the wave period.Transit-Time Damping 11\n(a)\n012345v⊥/vtiFrame: 150/501 tωA= 63.57CδB/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−6\n−3−2−1 0 1 2 3\nv/bardbl/vti05\nCδB/bardbl,i(v/bardbl)×10−60 2\n∂Wi/∂t×10−66065707580tωA\n(b)\n012345v⊥/vtiFrame: 150/501 tωA= 63.57CE/bardbl,i(v/bardbl,v⊥)\n−1.5−1.0−0.50.00.51.01.5×10−5\n−3−2−1 0 1 2 3\nv/bardbl/vti01\nCE/bardbl,i(v/bardbl)×10−60 2\n∂Wi/∂t×10−76065707580tωA\nFigure 4. Velocity-space signatures of (a) transit-time damping (TTD) and (b) Landau damp-\ning (LD), from the AstroGK simulation of a single kinetic Alfv´ en wave with k⊥ρi= 1, βi= 1,\nandTi/Te= 1, each showing the gyrotropic signatures in the main panel, the time-integrated\nreduced parallel signatures in the lower panel, and the net rate of ion energization vs. time for\neach mechanism in the left panel. The correlation interval is chosen as τωA= 5.5. The nor-\nmalized parallel phase velocity is labeled by the two vertical dashed lines at v∥/vti=±1.137.12\n(vertical dotted line); (ii) the lower plot shows the time-integrated parallel velocity-\nspace signature CδB∥,i(v∥) to highlight the variation of the net energy transferred as a\nfunction of v∥, showing a clear bipolar signature at the parallel phase velocity; and (iii)\nthe left plot shows the velocity-space integrated net energy density transfer rate due\nto TTD ( ∂Wi(t)/∂t)TTDvs. time, with the centered time of the correlation interval\nshown in the gyrotropic signature indicated (horizontal solid line). Panel (b) presents\nthe corresponding plots for the LD case.\nA key result of this paper is the velocity-space signature of transit time damping plotted\non gyrotropic velocity space CδB∥,i(v∥, v⊥) in Figure 4(a). From the same simulation, the\ngyrotropic velocity-space signature of LD CE∥,i(v∥, v⊥), given by the parallel contribution\nto the dot-product in (2.4), is presented in (b) for comparison. The TTD signature agrees\nqualitatively with our prediction presented in Figure 2(c), showing the key features: (i)\nthe bipolar signature of the rate of loss (blue) and gain (red) of phase-space energy density\nis centered about the parallel wave phase velocity v∥∼ω/k∥(vertical dotted black line);\n(ii) the gyrokinetic velocity-space signature does not extend down to v⊥= 0, due to a\ncombination of the v2\n⊥weighting arising from the magnetic moment µ=mv2\n⊥/(2B) in the\nmirror force and from the loss cone angle of the mirror force, as explained in §2.2. The LD\nsignature in (b) likewise yields a bipolar signature near the parallel wave phase velocity.\nBesides the fact that E∥governs energization through LD and E⊥governs energization\nthrough TTD (Howes et al. 2024), a key way to distinguish these two mechanisms is that\nin gyrotropic velocity space the TTD signature does not extend down to v⊥= 0, whereas\nthe LD signature extends right down to v⊥= 0.\n3.3.Variation of Signature with Ion Plasma Beta βi\nResonant damping of electromagnetic fluctuations through TTD and LD depends strongly\non the plasma beta, typically with LD dominant at βi≪1 and TTD dominant at βi≫1\n(Quataert 1998), so we vary the value of βihere to determine its impact on the character-\nistics of the velocity-space signature of the Landau-resonant damping mechanisms. Using\nthe same parameters Ti/Te= 1, mi/me= 1836, and k⊥ρi= 1, we perform additional\nsingle KAW simulations with βi= 0.3 and βi= 3. For βi= 0.3, the normalized parallel\nphase velocity is given by ω≡ω/(k∥vA) = 1 .267 yielding a normalized wave period of\nTωA= 4.959; for βi= 3, we obtain ω= 1.009 and TωA= 6.227.\nIn Figure 5, we plot the velocity-space signatures for (a) TTD and (b) LD for the\nβi= 0.3 case and for (c) TTD and (d) LD for the βi= 3 case, where each panel has\nthree subplots in the same format as in Figure 4. These bipolar velocity-space signatures\nlook qualitatively similar to the βi= 1 case in Figure 4, with two important quantitative\ndifferences.\nFirst, the position of the bipolar signature in v∥/vtichanges as βiis varied, consistent\nwith the variation of the parallel phase velocity normalized to the ion thermal velocity\nasβivaries, given by ω/(k∥vti) =ω/β1/2\ni. For βi= 0.3, we obtain ω/(k∥vti) = 2 .313,\nindicated by the vertical black dashed line in Figure 5(a) and (b); for βi= 3, we obtain\nω/(k∥vti) = 0 .583, as shown in (c) and (d). In both cases, the bipolar signature remains\nclosely associated in v∥with the parallel phase velocity ω/k∥, as expected for a Landau-\nresonant energy transfer between the fields and the ions.\nSecond, looking at the vertical subplots on the left for each panel, which shows the\nnet rate of change of ion energy density Wimediated by each mechanism averaged over\na correlation interval equal to one wave period τ=T, we find the surprising result for\nβi= 0.3 that, although LD leads to a net gain of energy by the ions (as expected for\ncollisionless damping of a wave), TTD leads to a net lossof energy from the ions. This\nfinding suggests that the ions are losing energy through the magnetic mirror force whileTransit-Time Damping 13\n012345v⊥/vtiFrame: 150/501 tωA= 57.03CδB/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−6\n−3−2−1 0 1 2 3\nv/bardbl/vti−2.50.0\nCδB/bardbl,i(v/bardbl)×10−6−5 0\n∂Wi/∂t×10−755606570tωA\n012345v⊥/vtiFrame: 150/501 tωA= 57.03CE/bardbl,i(v/bardbl,v⊥)\n−6−4−20246×10−5\n−3−2−1 0 1 2 3\nv/bardbl/vti−101\nCE/bardbl,i(v/bardbl)×10−50 5\n∂Wi/∂t×10−655606570tωA(a) (b)\n012345v⊥/vtiFrame: 150/501 tωA= 71.63CδB/bardbl,i(v/bardbl,v⊥)\n−1.8−0.90.00.91.8×10−10\n−3−2−1 0 1 2 3\nv/bardbl/vti02\nCδB/bardbl,i(v/bardbl)×10−100 2\n∂Wi/∂t×10−10657075808590tωA\n012345v⊥/vtiFrame: 150/501 tωA= 71.63CE/bardbl,i(v/bardbl,v⊥)\n−4−2024×10−11\n−3−2−1 0 1 2 3\nv/bardbl/vti−2.50.0\nCE/bardbl,i(v/bardbl)×10−11−5 0\n∂Wi/∂t×10−11657075808590tωA\n(c) (d)\nFigure 5. Velocity-space signatures of transit-time damping (TTD, left column) and Landau\ndamping (LD, right column) in AstroGK single KAW simulations with k⊥ρi= 1,Ti/Te= 1, and\nβi= 0.3 (top row) and βi= 3 (bottom row). The correlation intervals are set to the correspond-\ning linear wave periods, with τωA= 5.0 for βi= 0.3 case and τωA= 6.2 for βi= 3 case. The\nnormalized parallel phase velocity is labeled by the two vertical dashed lines at v∥/vti=±2.313\nforβi= 0.3 and v∥/vti=±0.583 for βi= 3. Each panel follows the layout format of Figure 4.\ngaining energy through acceleration by the parallel electric field. The rate of ion energy\ngain by LD is about ten times larger than the rate of loss by TTD, so the summed effect\nof these two mechanisms is energization of the ions by collisionless damping of the wave,\nas expected. For the βi= 3 case, we find the equally surprising result that although TTD\nleads to a net gain of energy by the ions, LD is leading to a net loss of energy from the\nions; the rate of ion energy gain by TTD is larger than the rate of ion energy loss by LD,\nso the summed contributions yield a net gain of ion energy as expected for collisionless\ndamping.\nHow do we reconcile these surprising results with the general expectation of Landau-\nresonant collisionless damping of waves for a Maxwellian equilibrium velocity distribu-\ntion? As it turns out, this behavior is exactly what is predicted by the linear Vlasov-\nMaxwell dispersion relation. To be specific, we take the complex eigenfrequency from the\nlinear dispersion relation to be given by ω+iγ, so that time evolution of a plane-wave\nmode is given by exp( −iωt) exp( γt): positive imaginary components γ >0 correspond to\ngrowth of the wave, and negative imaginary components γ <0 correspond to damping of\nthe wave. In Figure 6, we plot the normalized absolute value of the imaginary component\nof the wave frequency |γ|/ωvs.k⊥ρifor KAWs using the PLUME solver (Klein & Howes\n2015) for a fully ionized proton-electron plasma with isotropic Maxwellian velocity dis-\ntributions with Ti/Te= 1, mi/me= 1836, vti/c= 10−4,k∥ρi= 10−3over the range\n10−3⩽k⊥ρi⩽102for the ion plasma beta values (a) βi= 0.3, (b) βi= 1, and (c) βi= 3.14\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 0.3,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 1,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−310−210−1100101102\nk⊥ρi10−910−710−510−310−1|γ|/ωβi= 3,mi/me= 1836\nγ=γi+γe\nγi<0\nγiLD<0\nγiLD>0\nγiTTD>0\nγiTTD<0\n(a)\n(b)\n(c)\nFigure 6. Linear dispersion relations for KAWs from PLUME calculations with the realistic\nmass ratio mi/me= 1836, showing the absolute value of the normalized wave growth rate |γ|/ω\nas a function of the dimensionless perpendicular wave vector k⊥ρifor (a) βi= 0.3, (b) βi= 1,\nand (c) βi= 3. The vertical black dashed line at k⊥ρi= 1 indicates the values used in the\nsingle KAW AstroGK simulations. We plot γ(total damping rate, black), γi(total ion damping\nrate, green), γiTTD (ion growth or damping rate via the magnetic mirror force, red), and γiLD\n(ion growth or damping rate via the electrostatic force, blue). Line styles—solid, dashed, and\ndotted—represent the total damping rates, damping rates separated by mechanism, and growth\nrates separated by mechanism, respectively.Transit-Time Damping 15\nWe plot separately the total collisionless damping rate (black solid) due to both ions and\nelectrons, the total ion damping rate (green solid), and the separate contributions to\nthe ion damping rate from TTD (red) and LD (blue). For the separated TTD and LD\ncontributions, we plot negative imaginary components (which correspond to collisionless\ndamping of the wave) using dashed lines, and positive imaginary components (which\ncorrespond to collisionless growth of the wave) using dotted lines.\nAlthough all of the KAW dispersion relations plotted in Figure 6 yield a net effect of\ncollisionless damping by the ions, over some ranges in k⊥ρi(dotted lines) either TTD\nor LD individually may lead to a net transfer of energy from the ions tothe waves over\nthe course of a single wave period. For example, for the βi= 3 case in Figure 6(c), the\nimaginary component due to LD is positive over 1 .4×10−3≲k⊥ρi≲2.1, corresponding\nto a transfer of energy from the ions to the wave, and is negative outside of that range,\ncorresponding to a transfer of energy from the wave to the ions. A transfer from ions to\nthe wave would lead to a growth of the wave, but the sum of both of the LD and TTD\ncontributions is always negative for these cases with a Maxwellian velocity distribution,\nleading to a net collisionless damping of the wave.\nThe perpendicular wave number k⊥ρi= 1 of our simulated single waves is indicated\nin Figure 6 by the vertical black dashed line, and with these plots we can understand\nthe results presented in Figure 6. For the βi= 0.3 case, at k⊥ρi= 1 we have γiLD<0\n(blue dashed) and γiTTD >0 (red dotted), suggesting that ions gain energy due to E∥\nbut lose energy due to the mirror force when averaged over the full wave period. This\nfinding agrees with the net gain of ion energy by LD in Figure 5(b) and the net loss of\nion energy by TTD in (a). Similarly, for the βi= 3 case, at k⊥ρi= 1 we have γiTTD <0\n(red dashed) and γiLD>0 (blue dotted), suggesting that ions gain energy due to the\nmirror force but lose energy due to E∥when averaged over the full wave period. Again,\nthis finding from the linear Vlasov-Maxwell dispersion relation agrees with the net gain\nof ion energy by TTD in Figure 5(c) and the net loss of ion energy by LD in (d).\nTo further understand the physical meaning of γiTTD >0 in the βi= 0.3 case, we\nfirst point out that the only mechanism that can change the particle energy is work done\nby the electric field, and the rate of change of the ion energy density Wiis given by\nji·E. For TTD, this energization arises from the component of the electric field that is\nperpendicular to the magnetic field (Howes et al. 2024) (whereas LD energizes particles\nthrough the parallel component of the electric field), so here we consider the perpendicular\ncontribution to the energization j⊥iE⊥. For a single plane wave, the net transfer of energy\nto or from the ions by j⊥iE⊥over a single wave period depends on the phase of the\nperpendicular electric field fluctuation E⊥relative to the phase of the self-consistent\nperpendicular component of the ion current density associated with the wave, j⊥i. If the\nphase difference δϕis such that there is an in-phase component 0 < δϕ < π/ 2, there will\nbe a net energization of the ions; if there is an out-of-phase component π/2< δϕ < π ,\nthe ions will lose energy. The eigenfunctions arising from solutions of the linear Vlasov-\nMaxwell dispersion relation dictate the phases of the components of the electric field and\ncurrent density. For the βi= 0.3 case, this eigenfunction dictates that j∥iandE∥are\nin-phase, leading to ion energization and wave damping by E∥(yielding LD), but j⊥i\nandE⊥are out-of-phase, so the magnetic mirror force partly counteracts the damping\nof the wave.16\n4. Turbulence Simulations\nNow that we have determined the gyrotropic velocity-space signature of TTD for single\nKAWs, with the fiducial example for βi= 1 shown in Figure 4(a), we will seek similar\nsignatures of TTD in simulations of strong plasma turbulence.\n4.1.Turbulence Simulation Set Up\nWe perform kinetic simulations of strong plasma turbulence using the Astrophysical\nGyrokinetics Code AstroGK (Numata et al. 2010) for three values of the ion plasma\nbeta βi= 0.3,1,3. Each simulation has numerical resolution ( nx, ny, nz, nλ, nE, ns) =\n(96,96,32,64,32,2) within a simulation domain L2\n⊥×L∥= (8πρi)2×(2πa0), where the\nelongation of the domain along the equilibrium magnetic field B0=B0ˆzis characterized\nby the arbitrary gyrokinetic expansion parameter ϵ∼ρi/a0≪1. The proton-to-electron\ntemperature ratio of the Maxwellian equilibrium is Ti/Te= 1, and we choose a reduced\nmass ratio mi/me= 36 to ensure that we fully resolve the kinetic damping mechanisms\nneeded to achieve a steady-state turbulent cascade in a driven simulation, as discussed\nin Howes et al. (2018). These parameters lead to a fully resolved range of perpendicular\nwavenumbers 0 .25⩽k⊥ρi⩽7.75, or 0 .042⩽k⊥ρe⩽1.29. For the βi= 0.3 and βi= 1\nsimulations, the proton and electron collisionalities are set to νs/(k∥0vti) = 0 .1, and for\ntheβi= 3 simulation, νs/(k∥0vti) = 0 .05. These collisionalities ensure weakly collisional\nplasma conditions, yet prevent the small-scale variations that develop in velocity space\nfrom becoming unresolved on the velocity grid.\nTurbulence in the simulations is driven from zero initial conditions using an oscillating\nLangevin antenna (TenBarge et al. 2014) to drive four Alfv´ en wave modes at the domain\nscale with wave vectors ( kxρi, kyρi, kza0) = (0 .25,0,±1) and (0 ,0.25,±1), generating\nfour perpendicularly polarized Alfv´ en waves propagating in both directions along the\nequilibrium magnetic field. This driving has been shown to generate effectively a strong\nplasma turbulent cascade to small scales in previous kinetic simulations through nonlinear\ninteractions between the counterpropagating Alfv´ en waves (Howes et al. 2008; Howes\net al. 2011; Howes & Nielson 2013; TenBarge & Howes 2013; TenBarge et al. 2013; Howes\net al. 2018; Verniero et al. 2018; Verniero & Howes 2018; Horvath et al. 2020; Conley\net al. 2023). To provide the data needed to apply the field-particle correlation analysis,\nthe electromagnetic fluctuations and proton velocity distributions are sampled at a high\ncadence at twenty-four probe points that are distributed throughout the domain, sixteen\nin the xy-plane at z= 0, and the remaining eight along the z-axis, as illustrated in Fig. 2\nof Horvath et al. (2020).\nThe timescale associated with outer scale of the turbulent cascade in each simulation\nis the wave period of the domain-scale Alfv´ en wave T, and the simulations are run for\n6.78Tfor the βi= 0.3 simulation, 5 .51Tfor the βi= 1 simulation, and 3 .61Tfor the\nβi= 3 simulation. The perpendicular magnetic energy spectrum EB⊥(k⊥) at the end of\neach of the simulations is shown in Figure 7, showing the spectrum for βi= 0.3 (red),\nβi= 1 (black), and βi= 3 (blue). These spectra demonstrate that each simulation yields\na broadband turbulent spectrum, with the spectral slope for each simulation consistent\nwith the expectation of −5/3 for strong plasma turbulence (Goldreich & Sridhar 1995)\nin the inertial range at k⊥ρi<1, and steepening of the spectrum at the transition to\nthe dissipation range at k⊥ρi∼1. In the dissipation range at k⊥ρi>1, the spectral\nslopes begin around −3.2, steepening as k⊥ρe→1 due to the resolved kinetic dissipation\nmechanisms that remove energy from the turbulent cascade. Note that these dissipation\nrange slopes are slightly steeper than the values ranging from −2.7 to−3.1 typically\nobserved in the solar wind (Sahraoui et al. 2013), but this is to be expected due to the\nunphysical mass ratio of mi/me= 36, which effectively enhances the damping rate dueTransit-Time Damping 17\nFigure 7. Perpendicular magnetic energy spectra at the end of each of the turbulence simula-\ntions, showing βi= 0.3 (red), βi= 1 (black), and βi= 3 (blue). Vertical dotted lines indicate\nthe limit of fully resolved perpendicular wavenumbers in the simulation, 0 .25⩽k⊥ρi⩽7.75, or\n0.042⩽k⊥ρe⩽1.29.\nto electrons relative to the realistic mass ratio case (TenBarge et al. 2013), leading to\nslightly steeper dissipation range spectra for stronger damping (Howes et al. 2011).\nIn Figure 8, we plot the normalized damping rates |γ|/ωfrom the linear dispersion re-\nlation for the simulation parameters with the reduced mass ratio mi/me= 36, presenting\nthe results for (a) βi= 0.3, (b) βi= 1, and (c) βi= 3, in the same format as presented\nin Figure 6. Here we plot vertical black dashed lines at the perpendicular wavenumber\nlimits of the simulation at k⊥ρi= 0.25 and k⊥ρi= 7.75. Note the salient features that\nTTD yields a loss of ion energy for βi= 0.3 in (a), and LD yields a loss of ion energy for\nβi= 3 at k⊥ρi≲1.5. These calculations of the linear wave properties and the effective\nion energization rates by TTD and LD provide an important theoretical framework for\nthe interpretation of our field-particle analysis results.\n4.2.Choosing the Correlation Interval τ\nUnlike in the single-KAW simulations, where the single wave period Tis the obvious\nchoice for the correlation interval τto eliminate the oscillatory contribution to the transfer\nof energy from fields to particles, choosing τfor a plasma supporting broadband turbulent\nfluctuations is less straightforward. The longest wave period for an Alfv´ en wave at the\ndomain scale in our βi= 1 simulation is τωA≃6.28. In Figure 9, for a single probe\nposition in the βi= 1 simulation, we present (a) the total energy transfer rate to ions\ndue to TTD ( ∂Wi/∂t)TTDand (b) the energy transfer rate at v∥/vti=−1.3, for a\ncorrelation interval spanning 0 ⩽τωA⩽15. The instantaneous values of the local energy\ntransfer rates (with τ= 0, blue) exhibit large fluctuations with both positive and negative\nsigns, but for τωA= 6.4 (black) and longer correlation intervals, those large fluctuations\nare averaged out, leading to a time-averaged energy transfer rate that is about an order-\nof-magnitude smaller in amplitude than the peaks of the instantaneous value. In Figure 9,\nwe also show timestack plots CδB∥,i(v∥, t;τ) for (c) τ= 0 and (d) τωA= 6.4, showing\nthat a relatively persistent bipolar signature of TTD is revealed at v∥/vti=−1.1 in\ntheτωA= 6.4 case with peak amplitudes about an order-of-magnitude smaller than18\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 0.3,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD>0\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 1,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD<0\nγiTTD<0\nγiTTD>0\n10−1100101\nk⊥ρi10−610−410−2100|γ|/ωβi= 3,mi/me= 36\nγ=γi+γe\nγi<0\nγiLD>0\nγiLD<0\nγiTTD<0\nγiTTD>0\n(a)\n(b)\n(c)\nFigure 8. Linear dispersion relations for KAWs from PLUME calculations with the reduced\nmass ratio mi/me= 36, showing the absolute value of the normalized wave damping or growth\nrate|γ|/ωas a function of the dimensionless perpendicular wavenumber k⊥ρifor (a) βi= 0.3,\n(b)βi= 1, and (c) βi= 3. The two vertical black dashed lines at k⊥ρi= 0.25 and 7 .75 label the\nrange consistent with the AstroGK turbulence simulations, and the two vertical green dashed\nlines mark the range of 1 /eof the peak value of γi. Each panel follows the layout format of\nFigure 6.Transit-Time Damping 19\n10 20 30\ntωA−4−2024∂Wi/∂t\n0.003.757.5011.2515.00\nτωA\n10 20 30\ntωA−1.5−1.0−0.50.00.51.01.5CδB/bardbl,i(v/bardbl=−1.3vti)\n0.003.757.5011.2515.00\nτωA\n(a) (b)\n−2 0 2\nv/bardbl/vti102030tωACδB/bardbl,i(v/bardbl),τωA= 0.0\n−12−9−6−3036912\n−2 0 2\nv/bardbl/vti15202530tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.6−0.4−0.20.00.20.40.6\n(c) (d)\nFigure 9. Analysis of correlation interval selection for the βi= 1 AstroGK turbulence simu-\nlation. Top row: time evolution of (a) the rate of change of ion kinetic energy density due to\nTTD, denoted as ∂Wi/∂t, and (b) the reduced correlation CδB∥,i(v∥, t) atv∥=−1.3vti. Both\nquantities are presented over a range of τωAvalues from 0 to 15. The selected τωAvalue of 6.4\nis marked with a black line. Bottom row: timestack plots of the reduced correlation CδB∥,i(v∥, t)\nfor (c) τωA= 0 and (d) τωA= 6.4, where the range of parallel phase velocities of KAWs that\nexperience significant damping by ions is indicated by vertical dashed lines at v∥/vti=±1.020\nandv∥/vti=±1.704.\nthe instantaneous case with τ= 0. Thus, we choose a correlation interval τωA= 6.4 to\nperform the field-particle correlation analysis of our βi= 1 turbulence simulation.\n4.3.Results for βi= 1Simulation\nPerforming the field-particle correlation analysis with τωA= 6.4 for TTD and LD at all\n24 probe positions in our βi= 1 turbulence simulation, we seek the gyrotropic velocity-\nspace signatures of TTD and LD shown in Figure 4. In Figure 10(a), we plot the gy-\nrotropic velocity-space signature CδB∥,i(v∥, v⊥;τ) at one of the 24 probes, centered in\ntime at tωA= 19.71, showing a clear bipolar signature comparable to that shown in Fig-\nure 4(a). The range of resonant parallel phase velocities for the kinetic Alfv´ en wave mode\nover which significant ion collisionless damping is expected is indicated by the vertical\ndashed lines; specifically, these lines mark the resonant velocities of the kinetic Alfv´ en\nwave mode at k⊥ρi≃0.5 and k⊥ρi≃2.3, the points at which the total ion collisionless\ndamping rate drops to a factor 1 /eof its peak value at k⊥ρi≃1.3, as illustrated by\nthe vertical green dashed lines on the linear dispersion relation plot for mi/me= 36 in\nFigure 8(b). In the lower panel is shown the reduced parallel velocity-space signature\nCδB∥,i(v∥;τ) integrated over v⊥, yielding a clear bipolar signature with a zero crossing\natv∥/vti≃ −1.1. Note that the zero crossing of this velocity-space signature falls within\nthe expected range of resonant velocities (vertical dashed lines), as expected theoretically20\n0 1\n∂Wi/∂t1015202530tωA\n−2 0 2123v⊥/vtiFrame: 140/339 tωA= 19.707CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−0.250.000.25CδB/bardbl,i(v/bardbl)\n−2 0 215202530tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.6−0.4−0.20.00.20.40.6\n−2 0 2\nv/bardbl/vti05/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n0 20\n∂Wi/∂t1015202530tωA\n−2 0 2123v⊥/vtiFrame: 200/339 tωA= 24.267CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−9−6−30369\n−2 0 2\nv/bardbl/vti05CE/bardbl,i(v/bardbl)\n−2 0 215202530tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti−25025/integraltext\nCE/bardbl,i(v/bardbl)dt\n(c) (d)\nFigure 10. Velocity-space signatures of transit-time damping (TTD, top row) and Landau\ndamping (LD, bottom row) in AstroGK turbulence simulation with 0 .25⩽k⊥ρi⩽7.75,\nTi/Te= 1, and βi= 1. The correlation interval is set as τωA= 6.4. The left column presents\ngyrotropic plane ( v∥, v⊥) signatures, following the layout format of panels in Figure 4. The right\ncolumn features the timestack plots of the v⊥−integrated reduced correlation; the main panel\nhere shows the reduced correlation on ( v∥, t) grids, and the lower panel shows the time-inte-\ngrated reduced correlation. Four vertical dashed lines at v∥/vti=±1.020 and v∥/vti=±1.704\nindicate the resonant parallel phase velocity ranges where significant ion damping occurs.\nfor a resonant collisionless damping mechanism. The left panel shows the net rate of ion\nenergization due to TTD ( ∂Wi/∂t)TTDas a function of time, showing net positive ion\nenergization due to TTD at this position over almost the full duration of the simulation.\nThus, Figure 10(a) demonstrates that TTD indeed plays a role in the damping of tur-\nbulent fluctuations and consequent energization of the ions, a second key result of this\npaper.\nIn Figure 10(b), we show a timestack plot CδB∥,i(v∥, t;τ) of the ion energization by\nTTD at the same probe position as in (a), showing the persistence in time of the reduced\nparallel velocity-space signature with the zero crossing at v∥/vti≃ − 1.1. Note that\nthe zero crossing at v∥/vti≃ −1.1 in this timestack plot shifts to slightly lower phase\nvelocities near the end of the simulation at tωA>25, likely due to damping associated\nwith KAWs that have slightly lower perpendicular wavenumbers k⊥ρi, as can occur in\nbroadband turbulence. There is also a relatively short-lived bipolar signature observed atTransit-Time Damping 21\nv∥/vti≃1 at times tωA<15, indicating that TTD is also acting on KAWs propagating\nthe other direction along the magnetic field (Afshari et al. 2021).\nWe perform the analogous field-particle correlation for LD, showing CE∥,i(v∥, v⊥;τ)\nin Figure 10(c), yielding a bipolar gyrotropic velocity-space signature at v∥/vti≃1.2 at\ntime tωA= 24.3, also within the expected range of resonant parallel phase velocities. This\nfinding of the velocity-space signature of LD confirms previous field-particle correlation\nanalyses showing that ion LD plays a role in the dissipation of plasma turbulence (Klein\net al. 2017; Howes et al. 2018; Klein et al. 2020; Cerri et al. 2021). The timestack plot\nCE∥,i(v∥, t;τ) in Figure 10(d) shows this strong bipolar signature of LD at v∥/vti≃1.2\npersists over 22 ≲tωA≲30. In closing, it is worthwhile noting that, to distinguish\nthe velocity-space signature of TTD from that of LD, it is necessary to examine the\nsignatures in gyrotropic velocity space ( v∥, v⊥), showing that ion energization is limited\nto ions with v⊥/vti≳1 for TTD, but that ion energization extends down to v⊥→0 for\nLD, as expected by the physical arguments outlined in §3.2.\n4.4.Variations with Ion Plasma Beta βi\nNext, we explore how the velocity-space signatures of TTD and LD in plasma turbulence\nvary with changing the ion plasma beta βi=v2\nti/v2\nA. Because βiis a function of the\nratio of the ion thermal velocity to the Alfv´ en velocity, it directly characterizes where\nthe parallel wave phase velocity falls with the ion velocity distribution, making it the\nmost important parameter controlling resonant wave-particle interactions in a weakly\ncollisional plasma. Specifically, ω/(k∥vti) =ωβ−1/2\ni, where the parallel phase velocity\nnormalized to the Alfv´ en velocity ω≡ω/(k∥vA) typically has a value ω∼1 for the\nperpendicular wavelengths k⊥ρi∼1 at which the ions strongly interact with the waves.\nNote that, at the perpendicular scale of the domain k⊥0ρi= 0.25, the Alfv´ en wave has\ntheω≃1 for all values of βi, so we simply choose a correlation interval τωA= 6.4 for\nall of the turbulence simulation analysis below.\nWe plot some typical velocity-space signatures for TTD and LD for the βi= 0.3\nsimulation in Figure 11. Note that, for this value of βi= 0.3, the contribution of TTD\nis always to remove energy from the ions, as shown in Figure 8(a), so we expect only\nnegative TTD signatures, similar to that shown in Figure 5(a). Consequently, we expect\nLD to dominate the removal of energy from the turbulence. Performing the TTD analysis\nto determine CδB∥,i(v∥, v⊥;τ), we display a typical gyrotropic velocity-space signature\nin Figure 11(a) with the associated timestack plot at the same probe in (b), showing\ntwo bipolar signatures with a net negative energy transfer rate and zero crossings at\nv∥/vti≃ ±0.7. These look like typical reversed TTD signatures, but the phase velocity\nisnotwithin the expected range 1 .8≲|v∥/vti|≲2.4 for a KAW with βi= 0.3. Only\n3 of the 24 probes showed reversed TTD signatures with phase velocities closer to the\nexpected range, as shown in Figure 11(c) and (d) with a bipolar zero crossing around\nv∥/vti≃ −1.5.\nPerforming the LD analysis to determine CE∥,i(v∥, v⊥;τ) on the same βi= 0.3 tur-\nbulence simulation, we find a similar intriguing result that we commonly find bipolar\nsignatures associated with positive energy transfer to ions, but with zero crossings well\nbelow the expected range of 1 .8≲|v∥/vti|≲2.4. In Figure 11(e), we see two bipo-\nlar signatures at v∥/vti≃ −0.7 and v∥/vti≃1.2. Only 2 of the 24 probes recover LD\nvelocity-space signatures in the expected range, such as that shown in Figure 11(g) and\n(h).\nTurning next to the field-particle correlation analysis of the βi= 3 turbulence simula-\ntion, the linear dispersion relation plot for KAWs in Figure 8(c) shows that LD removes\nenergy from ions for waves with k⊥ρi≲1.7, but TTD positively energizes ions over per-22\n−5 0\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 340/396 tωA= 35.337CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−0.250.00CδB/bardbl,i(v/bardbl)\n−2 0 21520253035tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.4−0.20.00.20.4\n−2 0 2\nv/bardbl/vti−2.50.0/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n−2.5 0.0\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 200/396 tωA= 25.113CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.06−0.04−0.020.000.020.040.06\n−2 0 2\nv/bardbl/vti−0.10.0CδB/bardbl,i(v/bardbl)\n−2 0 21520253035tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.2−0.10.00.10.2\n−2 0 2\nv/bardbl/vti−20/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(c) (d)\n050\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 220/396 tωA= 26.574CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−15−10−5051015\n−2 0 2\nv/bardbl/vti05CE/bardbl,i(v/bardbl)\n−2 0 21520253035tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−10−50510\n−2 0 2\nv/bardbl/vti050/integraltext\nCE/bardbl,i(v/bardbl)dt\n(e) (f)\n0 25\n∂Wi/∂t10203040tωA\n−2 0 2123v⊥/vtiFrame: 300/396 tωA= 32.416CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti0.02.5CE/bardbl,i(v/bardbl)\n−2 0 21520253035tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−6−4−20246\n−2 0 2\nv/bardbl/vti025/integraltext\nCE/bardbl,i(v/bardbl)dt\n(g) (h)\nFigure 11. Velocity-space signatures of transit-time damping (TTD, top two rows) and\nLandau damping (LD, bottom two rows) sampled from AstroGK turbulence simulation with\n0.25⩽k⊥ρi⩽7.75,Ti/Te= 1, and βi= 0.3. The correlation interval is set as τωA= 6.4.\nThe left column presents gyrotropic plane ( v∥, v⊥) signatures, and the right column features the\ntimestack plots of the v⊥−integrated reduced correlation; both following the layout format of\nFigure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed\nlines at v∥/vti=±1.832 and v∥/vti=±2.373.Transit-Time Damping 23\n0.0 0.5\n∂Wi/∂t7.510.012.515.017.520.0tωA\n−2 0 2123v⊥/vtiFrame: 110/286 tωA= 12.119CδB/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.04−0.020.000.020.04\n−2 0 2\nv/bardbl/vti0.000.05CδB/bardbl,i(v/bardbl)\n−2 0 210.012.515.017.5tωACδB/bardbl,i(v/bardbl),τωA= 6.4\n−0.09−0.06−0.030.000.030.060.09\n−2 0 2\nv/bardbl/vti0.00.5/integraltext\nCδB/bardbl,i(v/bardbl)dt\n(a) (b)\n−2 0\n∂Wi/∂t7.510.012.515.017.520.0tωA\n−2 0 2123v⊥/vtiFrame: 97/286 tωA= 11.518CE/bardbl,i(v/bardbl,v⊥),τωA= 6.4\n−0.2−0.10.00.10.2\n−2 0 2\nv/bardbl/vti−0.20.0CE/bardbl,i(v/bardbl)\n−2 0 210.012.515.017.5tωACE/bardbl,i(v/bardbl),τωA= 6.4\n−0.3−0.2−0.10.00.10.20.3\n−2 0 2\nv/bardbl/vti−101/integraltext\nCE/bardbl,i(v/bardbl)dt\n(c) (d)\nFigure 12. Velocity-space signatures of transit-time damping (TTD, top row) and\nLandau damping (LD, bottom row) sampled from AstroGK turbulence simulation with\n0.25⩽k⊥ρi⩽7.75,Ti/Te= 1, and βi= 3. The correlation interval is set as τωA= 6.4.\nThe left column presents gyrotropic plane ( v∥, v⊥) signatures, and the right column features the\ntimestack plots of the v⊥−integrated reduced correlation; both following the layout format of\nFigure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed\nlines at v∥/vti=±0.583 and v∥/vti=±0.936.\npendicular wavenumbers with k⊥ρi≲4. At all probes in the simulation, we find bipolar\nvelocity-space signatures of positive energy transfer by TTD to ions at both positive\nand negative parallel velocities near the expected range 0 .58≲|v∥/vti|≲0.94, with a\ntypical case illustrated in Figure 12, showing (a) the gyrotropic velocity-space signature\nCδB∥,i(v∥, v⊥;τ) and (b) the timestack plot CδB∥,i(v∥, t;τ) indicating that these signa-\ntures are persistent in time. On the other hand, analyzing the signatures of LD using\n(c) the gyrotropic velocity-space signature CE∥,i(v∥, v⊥;τ) and (d) the timestack plot\nCE∥,i(v∥, t;τ), the pattern of energy transfer to ions is widely variable, with ions losing\nenergy more than gaining energy, in agreement with the expectations from the linear\ndispersion relation. During the majority of the time when clear reversed bipolar patterns\nare visible for LD, they appear to have zero crossings that are close to the expected range\nof parallel resonant velocities.\nSince the field-particle correlation analysis of our single-wave simulations in §3.3 shows\nthat the velocity-space signatures of TTD and LD appear near the resonant parallel ve-24\n1 2 3\nβi−5.0−2.50.02.55.0Ion Energization\nWi,TTD/Wi\nWi,LD/Wi\nFigure 13. Ratio of the change of the ion kinetic energy density due to TTD and LD to the\ntotal change of the ion kinetic energy density during the analysis time, both averaged over all\n24 probes, plotted against βi. The error bars represent the standard deviations calculated across\nall probes.\nlocities of the waves, as illustrated in Figure 5, it raises the question of why the velocity-\nspace signatures in the βi= 0.3 simulation in Figure 11 often do not appear in the\nexpected range of resonant velocities. There exist several possible explanations for this\nfinding. First, it is important to emphasize that the velocity-space signature created by\nthe field-particle correlation analysis shows the net rate of energy transfer to or from ions\nas a function of velocity space. At velocities v/vti>1 there are fewer ions in the un-\nderlying Maxwellian equilibrium velocity distribution, so we may expect the net energy\ntransfer to the ions at those suprathermal velocities to be smaller than the transfer to\nthe large number of ions in the core of the distribution at v/vti≲1. Due to the predom-\ninantly Alfv´ enic nature of fluctuations in space plasma turbulence (Tu & Marsch 1995;\nSchekochihin et al. 2009; Bruno & Carbone 2013), we focus on the collisional damping\nrates of Alfv´ enic fluctuations by TTD and LD in this study. However, the plasma tur-\nbulence in our gyrokinetic simulations is broadband and may contain slow magnetosonic\nfluctuations that arise from nonlinear couplings among the turbulent fluctuations. Al-\nthough slow magnetosonic waves are not generated by nonlinear couplings among the\ndominantly Alfv´ enic fluctuations in the MHD limit at k⊥ρi≪1 (Schekochihin et al.\n2009), at the ion kinetic scales k⊥ρi∼1 it is possible that energy can be nonlinearly\ntransferred into slow magnetosonic fluctuations. These kinetic slow wave fluctuations\nmay have a different parallel phase velocity and thereby mediate damping of the turbu-\nlent energy through either TTD or LD. Another possibility is that nonlinear transit-time\ndamping may occur, whereby the beat mode fluctuations (which are not natural wave\nmodes of the system)—arising from nonlinear interactions between Alfv´ enic fluctuations\nwith different frequencies and wave vectors—can have an effective phase velocity that\nfalls in the core of the velocity distribution, leading to efficient net energy transfer to\nthe ions via collisionless wave-particle interactions. In closing, a final point to empha-\nsize is that the clear bipolar signatures of negative energy transfer from ions by TTD in\nFigure 11(a) and of positive energy transfer to ions by LD in Figure 11(e) for βi= 0.3\nsuggest that a resonant energy transfer mechanism is governing the energization in these\nsimulations; the details of this turbulent damping mechanism, which may include non-\nlinear couplings to other linear wave modes or nonlinear beat modes, are clearly a ripe\navenue for exploration in future workTransit-Time Damping 25\nTo determine the net energy density transfer to or from ions mediated by TTD or LD,\nwe integrate the energy density transfer rate due to each mechanism over the duration of\neach turbulence simulation (after the energy spectra have reached a statistically steady\nstate, and average over all 24 probes for each simulation, yielding the change in the ion\nenergy density due to each mechanism, denoted as Wi,TTD andWi,LD. We compute\nthe ratio of these ion energy density changes by each mechanism to the total ion energy\ndensity change Wi=Wi,TTD +Wi,LD, and plot Wi,TTD /Wi(green) and Wi,LD/Wi(blue)\nversus the ion plasma beta βiin Figure 13. Variability in the energy density changes due\nto TTD and LD is indicated by error bars on each point, computed using the standard\ndeviation of the time-integrated energy density changes at all 24 probes. The red solid\nline at 1 presents the sum of Wi,TTD /WiandWi,LD/Wi, and the black dashed line\nat zero highlights whether a mechanism yields a net positive transfer to ions or net\nnegative energy transfer from ions. For βi= 0.3, the ion energization is dominated by\nLD, with TTD yielding a small and slightly negative energy transfer, consistent with the\nexpectations from the linear dispersion relation in Figure 8(a). For βi= 1, both LD and\nTTD yield net positive energization of the ions, as expected from the linear dispersion\nrelation in Figure 8(b), but LD dominates over TTD once more. At βi= 3, on the other\nhand, TTD mediates the positive energy transfer to the ions, while LD counteracts with\nenergy transfer from the ions back to the waves; in this higher βicase, the time-integrated\nenergy transfer varies much more widely from probe to probe, as indicated by the larger\nstandard deviation, compared to the unity or low βicases.\n5. Conclusion\nTransit-time damping is a well-known mechanism for the resonant collisionless damp-\ning of electromagnetic waves exhibiting variations of the magnetic field magnitude along\nthe mean magnetic field direction, mediated by the magnetic mirror force. This mecha-\nnism has been proposed as a possible means for removing energy from the fluctuations\nin weakly collisional plasma turbulence, but to date there exists little direct evidence\nclearly showing the damping of turbulence via transit-time damping. Here we employ\nthe recently developed field-particle correlation technique to use measurements of the\ngradient of the magnetic field magnitude and the ion velocity distribution at a single\npoint to determine a velocity-space signature that can be used to identify definitively\nthat transit-time damping plays a role in the damping of plasma turbulence.\nWe first derive the particular mathematical form of the field-particle correlation for\nthe rate of energy transfer due to transit-time damping in §2.1, and then we predict the\nvelocity-space signature of the rate of change of phase-space energy density due to transit-\ntime damping using a simple model in §2.2. Next, we perform gyrokinetic simulations of\nsingle kinetic Alfv´ en waves to determine the resulting velocity-space signature of transit-\ntime damping numerically, confirming the qualitative features of our prediction, and\npresenting the first key result of this study: the gyrotropic velocity-space signature of\ntransit-time damping in Figure 4(a).\nWe contrast the velocity-space signature of transit-time damping with the known bipo-\nlar velocity-space signature of Landau damping, showing the same bipolar pattern of\nphase-space energy density loss below and gain above the resonant parallel phase veloc-\nity, but the transit-time damping signature does not extend down to v⊥→0 because\nit is mediated via the magnetic moment of the charged particle µ=mv2\n⊥/(2B); thus,\nsignatures of transit-time damping and Landau damping can be distinguished in gy-\nrotropic velocity space by examining the behavior at the resonant parallel phase velocity\nasv⊥→0. Furthermore, we find the unexpected result that transit-time damping can26\nlead to a net loss of ion energy over the period of the wave for βi<1 and Landau\ndamping can lead to a net lossof ion energy over the period of the wave for βi>1.\nThis surprising result is explained, however, by examining the separate contributions\nof transit-time damping and Landau damping to ion damping from the linear Vlasov-\nMaxwell dispersion relation: the net effect of transit-time damping and Landau damping\ncombined for a plasma with a Maxwellian equilibrium ion velocity distribution always\nleads to a net damping of the wave and net gain of energy by the ions.\nNext, we perform three gyrokinetic simulations of weakly collisional plasma turbulence\nwith three values of βi= 0.3,1,3 to seek the velocity-space signature of transit-time\ndamping in the damping of the strong turbulent fluctuations. In the βi= 1 turbulence\nsimulation, we indeed find a velocity-space signature of transit-time damping as shown\nin Figure 10(a), indicating that this mechanism does indeed play a role in the dissipation\nof kinetic plasma turbulence, along with confirming previously demonstrated signatures\nof Landau damping with ions in Figure 10(c). This second key result of this paper shows\nclearly the transit-time damping does serve to damp the fluctuations in weakly collisional\nplasma turbulence.\nThe relative strength of transit-time damping and Landau damping is predicted to be a\nstrong function of βi(Quataert 1998), so we analyze our βi= 0.3 and βi= 3 simulations\nto confirm this prediction. For βi= 3, we indeed find signatures of transit-time damping\nin the predicted range of resonant parallel phase velocities in Figure 12(a), but Landau\ndamping signatures vary widely, with both positive and negative energy transfer rates\nto the ions, and a negative overall average consistent with expectations from the linear\ndispersion relation in Figure 8(c). For βi= 0.3, however, we discover puzzling bipolar\nvelocity-space signatures of negative energy transfer but with a zero-crossing well below\nthe parallel phase velocity of kinetic Alfv´ en waves. This may indicate that energy transfer\nvia transit-time damping is occurring through alternative wave modes, such as kinetic\nslow magnetosonic fluctuations, or through nonlinear transit-time damping via beat wave\nmodes that are generated by nonlinear interactions among the turbulent fluctuations.\nThese possibilities will be explored in future work.\nDetermining the integrated change of ion kinetic energy density due to transit-time\ndamping and Landau damping as a function of βifrom the three simulations, we find\nresults in Figure 13 that are generally consistent with the expectations from the linear\ndispersion relation: (i) at βi= 0.3, transit-time damping is small and slightly negative,\nwhile Landau damping is about an order-of-magnitude larger and positive; (ii) at βi= 1,\nboth transit-time damping and Landau damping are positive, but again Landau damp-\ning is about an order-of-magnitude larger than transit-time damping; and (iii) at βi= 3,\ntransit-time damping is large and positive while Landau damping is somewhat smaller\nand negative. Note that despite one of the mechanisms possibly leading to a net negative\ntransfer of energy from ions to waves, it is always the subdominant mechanism that is\nnegative, so the net effect of the sum of both of these n= 0 Landau resonant collision-\nless wave-particle interactions (transit-time damping and Landau damping) is always a\ndamping of the turbulence for equilibrium Maxwellian velocity distributions.\nAcknowledgements\nNumerical simulations were performed using the Extreme Science and Engineering\nDiscovery Environment (XSEDE), which is supported by National Science Foundation\ngrant number ACI-1548562, through allocation TG-PHY090084.Transit-Time Damping 27\nFunding\nSupported by NASA grants 80NSSC18K0643, 80NSSC18K1217, and 80NSSC18K1371\nand NSF grant AGS-1842561.\nDeclaration of interests\nThe authors report no conflict of interests.\nAppendix A. Explicit Form of Landau Damping and Transit-Time\nDamping Terms in Nonlinear Gyrokinetics\nThe nonlinear, collisionless gyrokinetic equation (Howes et al. 2006) can be manipu-\nlated into a form in which the terms governing Landau damping (LD) and transit-time\ndamping (TTD) are readily apparent. We begin with the nonlinear, collisionless gyroki-\nnetic equation in cgs units, Eq. (25) in Howes et al. (2006),\n∂hs\n∂t+v∥∂hs\n∂z+c\nB0[⟨χ⟩Rs, hs] =qsF0s\nT0s∂⟨χ⟩Rs\n∂t(A 1)\nwhere the nonlinear term is expressed in the Poisson bracket, defined by\n[U, V] =ˆz·\u0014∂U\n∂Rs×∂V\n∂Rs\u0015\n=∂U\n∂X∂V\n∂Y−∂U\n∂Y∂V\n∂X(A 2)\nwhere the guiding center coordinates are given by Rs= (X, Y, z ). The gyroaverage of\na given quantity at the guiding center position for a particle of species sis denoted\nby⟨. . .⟩Rs. The gyrokinetic potential is defined by χ(r, t) =ϕ−v·A/c, where ϕ(r, t)\nis the scalar electrostatic potential and A(r, t) is the vector potential. In this formula-\ntion, the total velocity distribution function for species sis separated into fs(r,v, t) =\nF0s(v) + (−qsϕ(r, t)/Ts)F0s(v) +hs(Rs, v∥, v⊥, t) +O(ϵ2), where F0s(v) is the spatially\nhomogeneous and temporally constant equilibrium Maxwellian velocity distribution and\nhs(Rs, v∥, v⊥, t) is the perturbed gyrokinetic distribution function at the particle guiding\ncenter position Rs(independent of gyrophase θin cylindrical velocity space).\nWe transform from the perturbed gyrokinetic distribution function hsto the comple-\nmentary perturbed gyrokinetic distribution function gs(Schekochihin et al. 2009), given\nby\ngs(Rs, v∥, v⊥, t) =hs(Rs, v∥, v⊥, t)−qsF0s\nT0s\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs, (A 3)\nsubstituting for hseverywhere in the nonlinear gyrokinetic equation (A 1). After some\nsimplification, the equation can be rearranged to obtain\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] =−qsF0s\nT0s∂\n∂t\u001cv∥A∥\nc\u001d\nRs(A 4)\n−qsF0s\nT0sv∥∂\n∂z\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs+qsF0s\nT0sc\nB0\"\u001cv∥A∥\nc\u001d\nRs,\u001c\nϕ−v⊥·A⊥\nc\u001d\nRs#\nWe can rearrange the terms on the right-hand side to obtain a more physically illumi-28\nnating form,\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] (A 5)\n=qsF0s\nT0sv∥\"\n−∂\n∂z⟨ϕ⟩Rs−1\nc∂\nA∥\u000b\nRs\n∂t#\n+qsF0s\nT0sv∥1\nB0\" \n∂\nA∥\u000b\nRs\n∂Rs׈z!\n·−∂⟨ϕ⟩Rs\n∂Rs#\n+qsF0s\nT0sv∥∂\n∂z\u001cv⊥·A⊥\nc\u001d\nRs+qsF0s\nT0sv∥1\nB0\" \n∂\nA∥\u000b\nRs\n∂Rs׈z!\n·1\nc∂⟨v⊥·A⊥⟩Rs\n∂Rs#\nUsing the following relations,\n∂\nA∥\u000b\nRs\n∂Rs׈z=⟨δB⊥⟩Rs, (A 6)\nqs\nc⟨v⊥·A⊥⟩Rs=−1\n2qs\ncv2\n⊥\nΩs⟨δB∥⟩Rs=−µs⟨δB∥⟩Rs, (A 7)\n−∂\n∂z⟨ϕ⟩Rs−1\nc∂\nA∥\u000b\nRs\n∂t=\nE∥\u000b\nRs, (A 8)\n−∂⟨ϕ⟩Rs\n∂Rs=⟨E⊥⟩Rs, (A 9)\nwe can simplify the result to obtain\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] (A 10)\n=qsF0s\nT0sv∥\u0012\nˆz+⟨δB⊥⟩Rs\nB0\u0013\n· ⟨E⟩Rs+F0s\nT0sv∥\"\n−µs\u0012\nˆz+⟨δB⊥⟩Rs\nB0\u0013\n·∂\nδB∥\u000b\nRs\n∂Rs#\nFinally, to put this into a more concise form, we recognize that the direction of the total\nmagnetic field (including the perturbation) to the first order is B=B0ˆz+⟨δB⊥⟩Rs, so\nwe can define the unit vector of the total magnetic field direction as ˆb\nˆb=B0ˆz+⟨δB⊥⟩Rs\nB0. (A 11)\nWith this final simplification, we obtain the final result for the nonlinear gyrokinetic\nequation,\n∂gs\n∂t+v∥∂gs\n∂z+c\nB0[⟨χ⟩Rs, gs] =qsF0s\nT0sv∥ˆb· ⟨E⟩Rs−F0s\nT0sv∥µsˆb· ∇Rs\nδB∥\u000b\nRs(A 12)\nThis equation has a simple physical interpretation with respect to work done on the\ndistribution functions by the fields: the first term on the right-hand side is the effect of\nLandau damping by electric field parallel to the total magnetic field; and the second term\non the right-hand side is the effect of transit-time damping by the magnetic mirror force\ndue to the gradient of the magnetic field magnitude along the total magnetic field, which\nto lowest order is just due to the parallel magnetic field perturbations, as shown in (2.6).\nNote also that the nonlinear term involves interactions between the electromagnetic fields\nand the plasma particles, but when integrated over all guiding-center space Rs, it leads\nto zero net energy change.Transit-Time Damping 29\nAppendix B. Gyrokinetic Form of the Field-Particle Correlation\nIn gyrokinetics, a form of conserved energy inspired from the definition of entropy is\ncalculated by multiplying the complementary perturbed gyrokinetic distribution function\ngsbyT0sgs/F0sand integrating over all velocity and physical space (Howes et al. 2006;\nBrizard & Hahm 2007; Schekochihin et al. 2009; Li et al. 2016; Howes et al. 2018). Using\na similar approach, we can obtain an energy equation for the gyrokinetic phase-space\nenergy density ws(Rs, v∥, v⊥, t) =Tsg2\ns/(2F0s) by multiplying (A 12) by T0sgs/F0sto\nobtain\n∂ws\n∂t+v∥∂ws\n∂z+T0sc\nB0F0s\u0014\n⟨χ⟩Rs,g2\ns\n2\u0015\n=v∥ˆb· ⟨qsE⟩Rsgs−v∥µsˆb· ∇Rs\nδB∥\u000b\nRsgs(B 1)\nIn this formulation, the gyrokinetic form of the field-particle correlation for Landau\ndamping would be given by\nCE∥,s(R0,s, v∥, v⊥, t) =1\nτZt+τ/2\nt−τ/2v∥ˆb· ⟨qsE⟩Rsgsdt′, (B 2)\nand for transit-time damping would be given by\nCδB∥,s(R0,s, v∥, v⊥, t) =−1\nτZt+τ/2\nt−τ/2v∥µsˆb· ∇Rs\nδB∥\u000b\nRsgsdt′. (B 3)\nNote that the energy transfer in the nonlinear case is simply “linear” collisionless\ndamping occurring along the local total magnetic field direction (which is a nonlinear\ncorrection from the equilibrium magnetic field direction B0=B0ˆz). Note also that, in\nthe gyrokinetic approximation, the nonlinear contribution (known as the parallel nonlin-\nearity) to the field-particle interactions is dropped, as it comes in only at higher order.\nThus, nonlinear saturation of Landau damping does not occur in gyrokinetics, but the\nnonlinear correction of “linear” collisionless damping due to the difference in the to-\ntal magnetic field direction from the equilibrium magnetic field direction is included in\nnonlinear gyrokinetics.\nWe choose not to use the particular forms of the field-particle correlations in (B 2)\nand (B 3) for the study here because the gyrokinetic formulation is the lowest order\ncontribution to the rate of change of phase-space energy density in the gyrokinetic limit\nof small fluctuations relative to the equilibrium gs/F0s≪1 (Howes et al. 2006); we prefer\nto establish a form that is appropriate for fluctuations of arbitrary amplitude, which is\ngiven for TTD by the form in (2.7).\nREFERENCES\nAbel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2008\nLinearized model Fokker-Planck collision operators for gyrokinetic simulations. I. Theory.\nPhys. Plasmas 15(12), 122509.\nAfshari, A. S., Howes, G. G., Kletzing, C. A., Hartley, D. P. & Boardsen, S. A. 2021\nThe Importance of Electron Landau Damping for the Dissipation of Turbulent Energy in\nTerrestrial Magnetosheath Plasma. J. Geophys. Res. 126(12), e29578.\nAfshari, A. S., Howes, G. G., Shuster, J. R., Klein, K. G., McGinnis, D., Martinovic,\nM. M., Boardsen, S. A. Brown, C. R., Huang, R., Kletzing, C. A. & Hartley, D. P.\n2023 Direct observation of ion cyclotron damping of turbulence in Earth’s magnetosheath\nplasma. Nature Comm. Submitted.\nAntonsen Jr, Thomas M & Lane, Barton 1980 Kinetic equations for low frequency insta-\nbilities in inhomogeneous plasmas. The Physics of Fluids 23(6), 1205–1214.\nArzamasskiy, Lev, Kunz, Matthew W., Squire, Jonathan, Quataert, Eliot &30\nSchekochihin, Alexander A. 2023 Kinetic Turbulence in Collisionless High- βPlasmas.\nPhys. Rev. X 13(2), 021014.\nBarnes, Aaron 1966 Collisionless damping of hydromagnetic waves. The Physics of Fluids\n9(8), 1483–1495.\nBarnes, M., Abel, I. G., Dorland, W., Ernst, D. R., Hammett, G. W., Ricci, P.,\nRogers, B. N., Schekochihin, A. A. & Tatsuno, T. 2009 Linearized model Fokker-\nPlanck collision operators for gyrokinetic simulations. II. Numerical implementation and\ntests. Phys. Plasmas 16(7), 072107.\nBrizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod.\nPhys. 79, 421–468.\nBruno, R. & Carbone, V. 2013 The Solar Wind as a Turbulence Laboratory. Living Reviews\nin Solar Physics 10, 2.\nCerri, S. S., Arzamasskiy, L. & Kunz, M. W. 2021 On Stochastic Heating and Its Phase-\nspace Signatures in Low-beta Kinetic Turbulence. Astrophys. J. 916(2), 120.\nChandran, Benjamin D. G., Li, Bo, Rogers, Barrett N., Quataert, Eliot & Ger-\nmaschewski, Kai 2010 Perpendicular Ion Heating by Low-frequency Alfv´ en-wave Turbu-\nlence in the Solar Wind. Astrophysical Journal Letters 720(1), 503–515.\nChandran, B. D. G., Verscharen, D., Quataert, E., Kasper, J. C., Isenberg, P. A.\n& Bourouaine, S. 2013 Stochastic Heating, Differential Flow, and the Alpha-to-proton\nTemperature Ratio in the Solar Wind. Astrophys. J. 776(1), 45.\nChen, C. H. K., Klein, K. G. & Howes, G. G. 2019 Evidence for electron Landau damping\nin space plasma turbulence. Nature Comm. 10(1), 740.\nConley, Sarah A., Howes, Gregory G. & McCubbin, Andrew J. 2023 Characterizing the\nvelocity-space signature of electron Landau damping. J. Plasma Phys. 89(5), 905890514.\nFrieman, EA & Chen, Liu 1982 Nonlinear gyrokinetic equations for low-frequency electro-\nmagnetic waves in general plasma equilibria. The Physics of Fluids 25(3), 502–508.\nGoldreich, P & Sridhar, S 1995 Toward a theory of interstellar turbulence. 2: Strong alfvenic\nturbulence. The Astrophysical Journal 438, 763–775.\nHorvath, S. A., Howes, G. G. & McCubbin, A. J. 2020 Electron landau damping of kinetic\nalfv´ en waves in simulated magnetosheath turbulence. Phys. Plasmas 27(10), 102901.\nHowes, G. G. 2017 A prospectus on kinetic heliophysics. Phys. Plasmas 24(5), 055907.\nHowes, Gregory G, Cowley, Steven C, Dorland, William, Hammett, Gregory W,\nQuataert, Eliot & Schekochihin, Alexander A 2006 Astrophysical gyrokinetics: basic\nequations and linear theory. The Astrophysical Journal 651(1), 590.\nHowes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E.,\nSchekochihin, A. A. & Tatsuno, T. 2008 Kinetic Simulations of Magnetized Turbu-\nlence in Astrophysical Plasmas. Phys. Rev. Lett. 100(6), 065004.\nHowes, G. G., Huang, R. & Felix, A. A. 2024 The Physics of the Magnetic Mirror Force\nand Transit-Time Damping: Asymptotic Solution. J. Plasma Phys. In preparation.\nHowes, G. G., Klein, K. G. & Li, T. C. 2017 Diagnosing collisionless energy transfer using\nfield-particle correlations: Vlasov-Poisson plasmas. J. Plasma Phys. 83(1), 705830102.\nHowes, G. G., Klein, K. G. & TenBarge, J. M. 2014 Validity of the Taylor Hypothesis for\nLinear Kinetic Waves in the Weakly Collisional Solar Wind. Astrophys. J. 789, 106.\nHowes, G. G., McCubbin, A. J. & Klein, K. G. 2018 Spatially localized particle energization\nby Landau damping in current sheets produced by strong Alfv´ en wave collisions. J. Plasma\nPhys. 84(1), 905840105.\nHowes, G. G. & Nielson, K. D. 2013 Alfv´ en wave collisions, the fundamental building block\nof plasma turbulence. I. Asymptotic solution. Phys. Plasmas 20(7), 072302.\nHowes, G. G., Tenbarge, J. M. & Dorland, W. 2011 A weakened cascade model for tur-\nbulence in astrophysical plasmas. Phys. Plasmas 18, 102305.\nHowes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A.,\nNumata, R. & Tatsuno, T. 2011 Gyrokinetic simulations of solar wind turbulence from\nion to electron scales. Phys. Rev. Lett. 107, 035004.\nIsenberg, P. A. & Hollweg, J. V. 1983 On the preferential acceleration and heating of solar\nwind heavy ions. J. Geophys. Res. 88, 3923–3935.\nIsenberg, Philip A. & Vasquez, Bernard J. 2019 Perpendicular Ion Heating by CyclotronTransit-Time Damping 31\nResonant Dissipation of Turbulently Generated Kinetic Alfv´ en Waves in the Solar Wind.\nAstrophys. J. 887(1), 63.\nKlein, Kristopher G & Howes, Gregory G 2015 Predicted impacts of proton temperature\nanisotropy on solar wind turbulence. Physics of Plasmas 22(3), 032903.\nKlein, K. G. & Howes, G. G. 2016 Measuring Collisionless Damping in Heliospheric Plasmas\nusing Field-Particle Correlations. Astrophys. J. Lett. 826, L30.\nKlein, K. G., Howes, G. G. & TenBarge, J. M. 2017 Diagnosing collisionless enegy transfer\nusing field-particle correlations: gyrokinetic turbulence. J. Plasma Phys. 83(4), 535830401.\nKlein, Kristopher G., Howes, Gregory G., TenBarge, Jason M. & Valentini,\nFrancesco 2020 Diagnosing collisionless energy transfer using field-particle correlations:\nAlfv´ en-ion cyclotron turbulence. J. Plasma Phys. 86(4), 905860402.\nLandau, L. D. 1946 On the Vibrations of the Electronic Plasma. Journal of Physics 10, 25.\nLi, Tak Chu, Howes, Gregory G., Klein, Kristopher G., Liu, Yi-Hsin & TenBarge,\nJason M. 2019 Collisionless energy transfer in kinetic turbulence: field–particle correlations\nin fourier space. Journal of Plasma Physics 85(4), 905850406.\nLi, T. C., Howes, G. G., Klein, K. G. & TenBarge, J. M. 2016 Energy Dissipation and\nLandau Damping in Two- and Three-dimensional Plasma Turbulence. Astrophys. J. Lett.\n832, L24.\nLichko, E. & Egedal, J. 2020 Magnetic pumping model for energizing superthermal particles\napplied to observations of the Earth’s bow shock. Nature Comm. 11, 2942.\nLoureiro, N. F. & Boldyrev, S. 2017 Role of Magnetic Reconnection in Magnetohydrody-\nnamic Turbulence. Phys. Rev. Lett. 118(24), 245101.\nMallet, Alfred, Schekochihin, Alexander A. & Chandran, Benjamin D. G. 2017 Dis-\nruption of Alfv´ enic turbulence by magnetic reconnection in a collisionless plasma. J. Plasma\nPhys. 83(6), 905830609.\nMartinovi ´c, Mihailo M., Klein, Kristopher G., Kasper, Justin C., Case, Anthony W.,\nKorreck, Kelly E., Larson, Davin, Livi, Roberto, Stevens, Michael, Whittle-\nsey, Phyllis, Chandran, Benjamin D. G., Alterman, Ben L., Huang, Jia, Chen,\nChristopher H. K., Bale, Stuart D., Pulupa, Marc, Malaspina, David M., Bon-\nnell, John W., Harvey, Peter R., Goetz, Keith, Dudok de Wit, Thierry & Mac-\nDowall, Robert J. 2020 The Enhancement of Proton Stochastic Heating in the Near-Sun\nSolar Wind. Astrophys. J. Supp. 246(2), 30.\nMelrose, Donald B 1980 Plasma astrophysics , , vol. 1. CRC Press.\nMontag, P & Howes, Gregory G 2022 A field-particle correlation analysis of magnetic\npumping. Physics of plasmas 29(3).\nNumata, Ryusuke, Howes, Gregory G, Tatsuno, Tomoya, Barnes, Michael & Dor-\nland, William 2010 Astrogk: Astrophysical gyrokinetics code. Journal of Computational\nPhysics 229(24), 9347–9372.\nOsman, K. T., Matthaeus, W. H., Greco, A. & Servidio, S. 2011 Evidence for Inhomo-\ngeneous Heating in the Solar Wind. Astrophys. J. Lett. 727, L11.\nQuataert, E. 1998 Particle Heating by Alfv´ enic Turbulence in Hot Accretion Flows. Astro-\nphys. J. 500, 978–991.\nSahraoui, F., Huang, S. Y., Belmont, G., Goldstein, M. L., R ´etino, A., Robert, P. &\nDe Patoul, J. 2013 Scaling of the Electron Dissipation Range of Solar Wind Turbulence.\nAstrophys. J. 777, 15.\nSchekochihin, AA, Cowley, SC, Dorland, W, Hammett, GW, Howes, Gregory G,\nQuataert, E & Tatsuno, T 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent\ncascades in magnetized weakly collisional plasmas. The Astrophysical Journal Supplement\nSeries 182(1), 310.\nSpitzer Jr, L. & Witten, L. 1951 US Atomic Energy Commission Report No. NYO-993\n(PM-S-1). Technical Report NYO-993 (PM-S-1). US Atomic Energy Commission.\nStix, Thomas H 1992 Waves in plasmas . Springer Science & Business Media.\nTenBarge, J. M. & Howes, G. G. 2013 Current Sheets and Collisionless Damping in Kinetic\nPlasma Turbulence. Astrophys. J. Lett. 771, L27.\nTenBarge, J. M., Howes, G. G. & Dorland, W. 2013 Collisionless Damping at Electron\nScales in Solar Wind Turbulence. Astrophys. J. 774, 139.\nTenBarge, J. M., Howes, G. G., Dorland, W. & Hammett, G. W. 2014 An oscillating32\nLangevin antenna for driving plasma turbulence simulations. Comp. Phys. Comm. 185,\n578–589.\nTu, C.-Y. & Marsch, E. 1995 MHD structures, waves and turbulence in the solar wind:\nObservations and theories. Space Sci. Rev. 73, 1–2.\nVerniero, J. L. & Howes, G. G. 2018 The Alfv´ enic nature of energy transfer mediation in lo-\ncalized, strongly nonlinear Alfv´ en wavepacket collisions. J. Plasma Phys. 84(1), 905840109.\nVerniero, J. L., Howes, G. G. & Klein, K. G. 2018 Nonlinear energy transfer and current\nsheet development in localized Alfv´ en wavepacket collisions in the strong turbulence limit.\nJ. Plasma Phys. 84(1), 905840103.\nVillani, C. 2014 Particle systems and nonlinear Landau damping. Physics of Plasmas 21(3),\n030901.\nZhdankin, V., Uzdensky, D. A. & Boldyrev, S. 2015 Temporal Intermittency of Energy\nDissipation in Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 114(6), 065002." }, { "title": "2401.16726v1.Variable_Length_Feedback_Codes_over_Known_and_Unknown_Channels_with_Non_vanishing_Error_Probabilities.pdf", "content": "arXiv:2401.16726v1 [cs.IT] 30 Jan 20241\nVariable-Length Feedback Codes\nover Known and Unknown Channels\nwith Non-vanishing Error Probabilities\nRecep Can Yavas and Vincent Y . F. Tan\nAbstract —We study variable-length feedback (VLF) codes with\nnoiseless feedback for discrete memoryless channels. We pr esent\na novel non-asymptotic bound, which analyzes the average\nerror probability and average decoding time of our modified\nYamamoto–Itoh scheme. We then optimize the parameters of\nour code in the asymptotic regime where the average error\nprobability ǫremains a constant as the average decoding time\nNapproaches infinity. Our second-order achievability bound\nis an improvement of Polyanskiy et al.’s (2011) achievabili ty\nbound. We also universalize our code by employing the empiri cal\nmutual information in our decoding metric and derive a secon d-\norder achievability bound for universal VLF codes. Our resu lts\nfor both VLF and universal VLF codes are extended to the\nadditive white Gaussian noise channel with an average power\nconstraint. The former yields an improvement over Truong an d\nTan’s (2017) achievability bound. The proof of our results f or\nuniversal VLF codes uses a refined version of the method of\ntypes and an asymptotic expansion from the nonlinear renewa l\ntheory literature.\nIndex Terms —variable-length feedback codes, non-asymptotic\nbounds, universal channel coding, empirical mutual inform ation.\nI. I NTRODUCTION\nFeedback does not increase the capacity of memoryless\nchannels [ 1]. Yet, it simplifies the coding schemes that achieve\nthe capacity [ 2], [3]. For fixed-length codes, Wagner et al. [4]\nshow that feedback improves the second-order achievable ra te\nfor discrete memoryless channels (DMCs) that have multiple\ncapacity-achieving input distributions with distinct dis persions.\nThe benefits of feedback are even more significant for\nvariable-length feedback (VLF) codes, where the transmiss ion\nstops at a random time depending on the noise realization.\nIn his seminal work, Burnashev [ 5] shows that the optimal\nerror exponent (also known as the reliability function) for VLF\ncodes over a DMC is given by\nE(R) = lim\nǫ→0−1\nE[τ]logǫ=C1/parenleftbigg\n1−R\nC/parenrightbigg\n, (1)\nwhereCis the capacity of the DMC, C1is the maximum\nKullback–Leibler (KL) divergence between two conditional\noutput distributions, R∈(0,C)is the rate, ǫis the error\nprobability, and E[τ]is the average decoding time of the code.\nFor anyR < C , the error exponent in ( 1) is larger than that for\nfixed-length codes without feedback [ 6]. To achieve the opti-\nmal error exponent, Burnashev proposes a two-phase coding\nscheme, where in the communication phase , the transmitter\naims to increase the posterior of the transmitted message. I fthe largest posterior exceeds a threshold, the system goes i nto\ntheconfirmation phase , where the decoder tries to verify the\ncorrectness of the estimate in the confirmation phase.\nYamamoto and Itoh [ 7] propose an alternative scheme\nthat achieves the optimal error exponent in ( 1). Yamamoto\nand Itoh’s scheme alternates between the communication and\nconfirmation phases, each having fixed lengths, until a deci-\nsion is made by the receiver. Any capacity-achieving fixed-\nlength code can be used for the communication phase of the\nYamamoto–Itoh scheme. In the confirmation phase, the trans-\nmitter transmits one of two control sequences, (xA,...,x A)\nand(xR,...,x R), where the first sequence indicates that the\nreceiver should “accept” its current estimate, and the seco nd\nsequence indicates that the receiver should “reject” its cu rrent\nestimate and start a new communication phase. The symbols\nxAandxRare chosen to be the two most distinguishable\nsymbols in the sense that they achieve C1. The receiver then\nconstructs a (fixed-length) binary hypothesis test on the no isy\nversions of the control sequences and feeds its decision bac k\nto the transmitter. In [ 8], Chen et al. derive a non-asymptotic\nachievability bound for VLF codes with finite number of\nfeedback instances; their code is a variant of Yamamoto–\nItoh scheme where the length of each communication and\nconfirmation phase may be distinct.\nAlthough error exponent analysis elucidates how fast the er -\nror probability decays as the average decoding time N/definesE[τ]\ngrows to infinity, it does not explain the fundamental limit f or a\nfixed error probability ǫ∈(0,1)and a finite Nof our interest.\nTo address this issue, Polyanskiy et al. [9] extend Burnashev’s\nwork to the regime with non-vanishing error probabilities a nd\nderive achievability and converse bounds on the logarithm o f\nthe maximum achievable codebook size logM∗(N,ǫ)given\nan average decoding time Nand average error probability\nǫ∈(0,1). They show\nNC\n1−ǫ−logN+O(1)≤logM∗(N,ǫ)≤NC\n1−ǫ+hb(ǫ)\n1−ǫ,(2)\nwherehb(ǫ)/defines−ǫlogǫ−(1−ǫ)log(1−ǫ)is the binary entropy\nfunction. This result implies that the ǫ-capacity isC\n1−ǫ, and\nthe second-order term in the achievable rate is O/parenleftig\nlogN\nN/parenrightig\n. To\nachieve the lower bound in ( 2), they employ stop-feedback,\nwhich is a single bit of feedback that tells the transmitter\nwhether to stop the transmission or to continue to transmit\nsymbols. Polyanskiy et al. ’s scheme uses a stop-at-time-zero\nstrategy, which decodes to an arbitrary message at time zero2\nwith probability ǫ0< ǫ, and with probability 1−ǫ0, the\nscheme employs a code with an information-density threshol d\nrule. Variants of Polyanskiy et al. ’s coding scheme with a\nfinite number of feedback instances include [ 10]–[15]. Some\nof the extensions of [ 9] to multi-transmitter networks are [ 14],\n[16]. In [ 17], for symmetric binary-input channels, Naghshvar\net al. develop a deterministic, one-phase coding scheme that\nachieves the optimal error exponent in ( 1). Their code has\na novel encoder called the small-enough-difference (SED)\nencoder, which partitions the message set into two subsets a t\neach time instance so that the probability difference betwe en\nthe two subsets is small enough. In [ 18, Remark 4], Naghshvar\net al. extend their work to arbitrary DMCs by introducing the\nmaximum extrinsic Jensen-Shannon encoder and derive a non-\nasymptotic bound for their code. In [ 19], Yang et al. extend\nNaghshvar et al. ’s SED encoder to binary asymmetric channels\n(BACs) (i.e., channels with binary input and binary output) ,\nand derive refined non-asymptotic achievability bounds for the\nBAC and the binary symmetric channel (BSC).\nSince the exact channel statistics are not always available\nto the code designer, it is desirable to construct universal\ncodes in the sense that the DMC in use is known to belong\na certain family of DMCs (e.g., DMCs with known input and\noutput alphabet sizes, BSCs with unknown flip probability),\nbut the exact channel transition kernel PY|Xis unknown\nto both the transmitter and receiver. Naturally, we desire\nthe performance of the universal code to be as close as to\nthat of the non-universal code (e.g., the capacity, the erro r\nexponent). In [ 20], Goppa proposes to use the maximum\n(empirical) mutual information (MMI) decoder, which decod es\nto the message whose codeword has the maximum empirical\nmutual information with the received output sequence. Gopp a\nshows that for DMCs, the MMI decoder attains capacity\nuniversally. In [ 21, Th. 10.2], Csisz´ ar and K¨ orner show that\nthe random coding error exponent for constant-composition\ncodes is achieved universally by the MMI decoder. Universal\nchannel coding is related to mismatched decoding, in which\nthe decoder is fixed and potentially sub-optimal, and the\ngoal is to optimize the codebook since both mismatched\ndecoding and universal coding attempt to address channel\nuncertainty (see [ 22] for a review of mismatched decoding).\nMerhav [ 23] unifies the mismatched decoding and universal\ncoding approaches, where he shows that for a given random\ncoding distribution and a given class of metric decoders, th eir\nproposed generic universal decoder whose error probabilit y is\nwithin a subexponential multiplicator factor of the best de coder\nin that class of decoders. Extensions of [ 21, Th. 10.2] to the\nGaussian channel with an unknown deterministic interferen ce\nsignal and to the Gaussian intersymbol interference channe l\nappear in [ 24] and [ 25], respectively. In [ 26], Tchamkerten\nand Telatar define universal VLF (UVLF) codes and show\nthat Burnashev’s error exponent is universally achieved ov er\na family of BSCs with an unknown flip probability p <1\n2\nand over a family of Z channels with unknown parameters.\nIn [27, Th. 3], Lomnitz and Feder show that for DMCs,\nthe empirical mutual information between input and output\nsequences is achievable universally in the VLF setting, whi ch\nextends Goppa’s result to VLF codes. In [ 27, Th. 4], theyalso show that for arbitrary continuous channels with an\naverage power constraint, the rate R=−1\n2log(1−ˆρ2\nXnYn)is\nuniversally achievable in the VLF setting, where ˆρ2\nXnYnis the\nempirical correlation correlation between input Xnand output\nYn. The quantity −1\n2log(1−ρ2)is the mutual information of\ntwo Gaussian random variables with the correlation coeffici ent\nρ. In [ 28], Merhav and Feder study the error exponents of\nuniversal decoding with an erasure option, where the trade- off\nbetween the probability of undetected error and the probabi lity\nof erasure is considered; this problem is related to UVLF cod es\nin the sense that at each time, the UVLF decoder chooses\nbetween decoding to the “erasure” option and a message.\nA. Our Main Contributions\nIn this paper, we study VLF and UVLF codes in the regime\nthat the error probability ǫ∈(0,1)is non-vanishing. For an\narbitrary DMC with C1<∞, equivalently, all entries of the\nchannel transition kernel PY|Xare positive, we improve the\nsecond-order term in the lower bound in ( 2) for VLF codes\nfrom−logNto−C\nC1logN. Our proposed VLF code is a\nmodified Yamamoto–Itoh scheme with two communication\nand one confirmation phases, where each phase has a random\nstopping time, similar to the code in [ 26]. In Theorem 1,\nwe derive a novel non-asymptotic achievability bound; in\nTheorem 2, we analyze the non-asymptotic bound to derive the\nasymptotic bound with the improved second-order term. For\nUVLF codes, we assume that a training sequence of length ℓis\navailable prior to the communication. In Theorem 3, for UVLF\ncodes, we derive an asymptotic achievability bound for an\narbitrary DMC with C1<∞, where the second order term is\n−C\nC1logN−min/braceleftig\n|X||Y|\n2,/parenleftbig\n|X|−3\n2/parenrightbig/parenleftbig\n|Y|−3\n2/parenrightbig\n+3\n4/bracerightig\nlogN.\nOur UVLF code universalizes the Yamamoto–Itoh scheme\nby replacing the information-density threshold rule in the\ncommunication phases with the empirical mutual informatio n\nthreshold rule. This empirical mutual information thresho ld\nrule is also used in [ 26]. Unlike [ 26], during the second\ncommunication phase, we do not discard the symbols from\nthe first communication phase, which is essential for the\nderivation of Theorem 3. We use the training sequence only\nin the confirmation phase of the scheme, where we construct\na sequential hypothesis test for the “accept” and “reject”\nhypotheses. In the proof of Theorem 3, we use the result in\n[29, Th. 4.5] from the nonlinear renewal theory literature to\nbound the expected stopping time associated associated wit h\nthe empirical mutual information. We also use the refined\nmethod of types from [ 30] to get a tight tail probability bound\nfor the empirical mutual information evaluated on a joint ty pe\nformed from two independent sequences.\nTheorems 4and 5extend our achievability bounds for\nVLF and UVLF codes to the Gaussian channel with an\naverage power constraint. For UVLF codes for the Gaussian\nchannel, we consider the scenario where the noise variance\nσ2\n0of the channel is unknown to the transmitter and the\nreceiver. Note that this model is equivalent to slow fading\nchannel with a fixed an unknown fading factor and a known\nnoise variance. For this problem, as our universal decoding\nmetric, we employ the mutual information associated with3\nthe maximum likelihood estimator of the input-output pair\n(Xn,Yn)within the class of jointly Gaussian distributions\nwith zero mean, which equals −1\n2log(1−ˆρ2\nXnYn), where\nˆρ2\nXnYnis the empirical correlation coefficient between Xn\nandYn. This universal decoding metric is also used in [ 27];\na similar universal metric that also depends on the Gaussian\ninput distribution is proposed in [ 23, Example 2]. Our results\nhere refine the achievability results of Lomnitz and Feder [ 27,\nTh. 4].\nB. Paper Organization\nThe organization of the paper is as follows. Section II\ndefines the notation, Section IIIformulates the problems,\nSection IVpresents our main results, Section Vextends our\nresults to the Gaussian channel, and Sections VI–VIII contain\nthe proofs. Section IXconcludes the paper.\nII. N OTATION AND DEFINITIONS\nForn∈N, we denote [n]/defines{1,...,n}and the length-\nnvectorxn/defines(x1,...,x n). The distribution of a ran-\ndom variable Xon an alphabet Xis denoted by PX. For\nany random variable X, we define the constant b(X) =\nmin/braceleftbigE[(X+)2]\nE[X],esssup(X)/bracerightbig\n, which is an upper bound on\nthe expected stopping time associated with the random walk\n{/summationtextn\ni=1Xi,n≥1}, whereX,X1,X2,... are independent and\nidentically distributed (i.i.d.). The set of all distribut ions onX\nis denoted by P(X). The Gaussian random vector with mean\nµand covariance matrix Σis denoted by N(µ,Σ).\nA DMC is defined by the single-letter channel transition ker-\nnelPY|X:X → Y , whereXandYare the input and output\nalphabets. The DMC acts on each input symbol independently\nof others, i.e., PYn|Xn(yn|xn) =/producttextn\ni=1PY|X(yi|xi)for all\nxn∈ Xnandyn∈ Yn. The set of all DMCs with input\nalphabetXand output alphabet Yis denoted by P(Y|X).\nAll logarithms have base e. The information density is\nı(x;y)/defineslogPY|X(y|x)\nPY(y), (3)\nwhere the output distribution PYis induced by a fixed input\ndistribution PXand the DMC PY|X(the dependence of the\ninformation density on (PX,PY|X)is suppressed). The mutual\ninformation associated with PXandPY|Xis denoted as\nI(PX,PY|X)/defines/summationdisplay\nx∈X,y∈YPXY(x,y)logPY|X(y|x)\nPY(y).(4)\nThe capacity of a DMC PY|Xis\nC/definesmax\nPX∈P(X)I(PX,PY|X). (5)\nThe entropy of PXis denoted by H(PX), and the KL\ndivergence between PXandQXon the same alphabet Xis\ndenoted by D(PX/⌊a∇⌈⌊lQX). The error exponent ( 1) achieved at\nzero rate is defined as\nC1/definesmax\n(x,x′)∈X2D(PY|X=x/⌊a∇⌈⌊lPY|X=x′). (6)The empirical distribution (or type) of a sequence xn∈ Xn\nis defined as\nˆPxn(x)/defines1\nnn/summationdisplay\ni=11{xi=x}, x∈ X. (7)\nThe conditional type of a sequence (xn,yn)∈ Xn× Yn\nis defined similarly to ( 7) and is denoted by ˆPyn|xn(y|x).\nThe empirical mutual information associated with sequence s\n(xn,yn)is denoted by I(ˆPxn,ˆPyn|xn). The set of length- n\ntypes on an alphabet Xis denoted by Pn(X)/defines{PX∈\nP(X):nPX(x)∈Z∀x∈ X}.The type class of PXis\ndefined as Tn(PX)/defines{xn∈ Xn:ˆPxn=PX}.\nWe use the standard o(·),O(·), andΩ(·)notations.\nIII. P ROBLEM FORMULATION\nWe here formalize VLF and UVLF codes.\nDefinition 1 ( [ 9, Def. 1]): Fixǫ∈(0,1),N >0, and a\npositive integer M. An(N,M,ǫ)-VLF code comprises\n1) a common randomness random variable Uthat has a finite\nalphabetUand an associated probability distribution PU,\n(The realization uofUis revealed to the transmitter and\nreceiver before the start of transmission to initialize the\ncodebook.)\n2) encoding functions ft:U ×[M]×Yt−1×P(Y|X)→ X\nsuch that\nXt=ft(U,W,Yt−1,PY|X)∀t∈N, (8)\n3) a random stopping time τ∈Nof the filtration generated\nby{U,Yt}∞\nt=0, which satisfies the average decoding time\nconstraint\nE[τ]≤N, (9)\n4) a decoding function gτ:U ×Yτ×P(Y|X)→[M]such\nthat\nˆW=gτ(U,Yτ,PY|X), (10)\nwhereˆWis the estimate of the equiprobable message W,\nand the average error probability does not exceed ǫ, i.e.,\nP/bracketleftig\nˆW/\\⌉}atio\\slash=W/bracketrightig\n≤ǫ. (11)\nDefinition 2: Let{(xi,¯Yi)}ℓ\ni=1be a training sequence\navailable to the transmitter and the receiver prior to the\ncommunication, where each x∈ X appears at least ⌊ℓ\n|X|⌋\ntimes in xℓ, and¯Yℓ∼PYℓ|Xℓ=xℓ. An(N,M,ǫ,ℓ )-UVLF\ncode is defined similarly to an (N,M,ǫ)-VLF code except\nthat the encoding functions {ft}∞\nt=1and the decoding function\ngτcan depend on the training sequence {(xi,¯Yi)}ℓ\ni=1and the\ninput and output alphabet sizes |X|and|Y|but not on the\nchannel transition kernel PY|X. We do not count the training\nsequence length ℓtowards the decoding time.\nWe define the maximum achievable codebook sizes\nM∗(N,ǫ)andM∗\nU(N,ǫ)UVLF as\nM∗(N,ǫ)/definesmax{M∈N:∃(N,M,ǫ)-VLF code }(12)\nM∗\nU(N,ǫ,ℓ)/definesmax{M∈N:∃(N,M,ǫ,ℓ )-UVLF code }.\n(13)4\nIV. M AINRESULT\nOur first result is a non-asymptotic achievability bound\nfor VLF codes, where the channel transition kernel PY|Xis\nknown.\nTheorem 1: Fix a positive integer M, positive constants\nγ1< γ2,aA, andaR,ǫ0∈(0,1), and a capacity-achieving\ninput distribution PX. Define\n(xA,xR)/definesargmax\n(x,x′)∈X2D(PY|X=x/⌊a∇⌈⌊lPY|X=x′). (14)\nThere exists an (N,M,ǫ)-VLF code with\nN≤(1−ǫ0)N′(15)\nǫ≤ǫ0+(1−ǫ0)ǫ′, (16)\nwhere\nǫ′= (M−1)(exp{−(γ1+aA)}+exp{−γ2}) (17)\nN′=γ1+b\nC\n+((M−1)exp{−γ1}+exp{−aR})γ2−γ1+b\nC\n+aA+bA\nD(PY|X=xA/⌊a∇⌈⌊lPY|X=xR)\n+(M−1)exp{−γ1}aR+bR\nD(PY|X=xR/⌊a∇⌈⌊lPY|X=xA)(18)\nb=b(ı(X;Y)), bA=b/parenleftbigg\nlogPY|X=xA(YA)\nPY|X=xR(YA)/parenrightbigg\n,(19)\nbR=b/parenleftbigg\nlogPY|X=xR(YR)\nPY|X=xA(YR)/parenrightbigg\n, (20)\n(X,Y)∼PXPY|X,YA∼PY|X=xA, andYR∼PY|X=xR.\nThe proposed coding scheme to prove Theorem 1is a\nvariant of the Yamamoto–Itoh scheme [ 7] and is modified from\nTelatar and Tchamkerten’s VLF coding scheme [ 26], which\nis designed for unknown channels. Our code is similar to the\ncode in [ 8] in limiting the number of phases to a finite integer,\nbut differs from it as each phase in our code has a random\nstopping time. Our code has two communication phases (C1\nand C2) and one confirmation phase (HT), where the HT phase\nis between the C1 and C2 phases. We combine the Yamamoto–\nItoh scheme with the stop-at-time-zero strategy used in [ 9], in\nwhich the code stops and decodes to an arbitrary message at\ntime zero with probability ǫ0and employs the Yamamoto–Itoh\nscheme with probability 1−ǫ0. Decoding occurs either at time\nzero, or at the end of the HT phase, or at the end of the C2\nphase. At large average decoding times N, the stop-at-time-\nzero strategy with a non-zero ǫ0improves the achievable rate,\nand asymptotically achieves the ǫ-capacityC\n1−ǫ. This strategy\nis also employed in [ 13], [14], [31]. Below, we detail our\nmodification to the Yamamoto–Itoh scheme.\nCoding scheme: LetPXbe a capacity-achieving input distri-\nbution, i.e., C=I(PX,PY|X). We generate Mi.i.d. infinite-\nlength codewords from the distribution P∞\nX. Let the generated\ncodewords be c(1),...,c(M), and denote the first nsymbols\nof the codeword c(m)bycn(m)/defines(c1(m),...,cn(m)). The\ni-th received symbol during a communication phase (either C1\nor C2) is denoted by Yi; thei-th received symbol during the\nHT phase is denoted by ˜Yi.C1 phase: Without loss of generality, assume that W= 1\nis the transmitted message. Therefore, for any n∈N,\nPcn(1)...cn(M)Yn(xn\n1,...,xn\nM,yn)\n=n/productdisplay\ni=1/parenleftiggM/productdisplay\nm=1PX(xmi)/parenrightigg\nPY|X(yi|ci(1)). (21)\nThe encoder encodes the symbols from c(1)one by one. Let\nγ1,γ2∈Rbe some thresholds that satisfy γ2> γ1. Fori=\n1,2, define the stopping times\nτ(i)\nm/definesinf{n≥1:ı(cn(m);Yn)> γi} (22)\nτ(i)/definesmin\nm∈[M]τ(i)\nm, (23)\nand the receiver’s estimates\nˆW(i)/definesmin{m∈[M]:ı(cτ(i)(m);Yτ(i))> γi}.(24)\nThrough feedback, the transmitter learns whether τ(1)is\nreached at each time during the C1 phase. At time τ(1),ˆW(1)\nis fed back to the transmitter for the transmitter to accept o r\nrejectˆW(1).\nHypothesis Test (HT) phase : IfˆW(1)= 1 , then the\ntransmitter sends the sequence of (xA,xA,...); otherwise,\nit sends(xR,xR,...). The receiver constructs the sequential\nhypothesis test\nHA:˜Y∼PY|X=xA (25)\nHR:˜Y∼PY|X=xR (26)\nand Wald’s sequential probability ratio test (SPRT)\nτHT/definesinf/braceleftigg\nn≥1:n/summationdisplay\ni=1logPY|X=xA(˜Yi)\nPY|X=xR(˜Yi)/∈[−aR,aA]/bracerightigg\n(27)\nwhere−aRandaAare thresholds of the SPRT. Here, HAand\nHRcorrespond to hypothesis to accept and to reject the initial\nestimate ˆW(1), respectively.\nIf/summationtextτHT\ni=1logPY|X=xA(˜Yi)\nPY|X=xR(˜Yi)> aA, thenHAis declared at time\nτ(1)+τHTby the receiver, and the initial estimate ˆW(1)is\naccepted as ˆW. If/summationtextτHT\ni=1logPY|X=xA(˜Yi)\nPY|X=xR(˜Yi)<−aR, thenHR\nis declared, and the communication enters the C2 phase. The\ntransmitter learns the receiver’s decision at the end of the HT\nphase through feedback.\nC2 phase: The transmitter continues to encode symbols\nfromc(1)starting from the time index τ(1)+1. At time τ(2)+\nτHT, the receiver decodes to ˆW(2). The coding scheme is\nsummarized in Table I. In the proof of Theorem 1, we use the\nbound\nP/bracketleftig\nτ(i)\n2<∞/bracketrightig\n≤exp{−γi} (28)\nfrom [ 9, eq. (118)] to bound the error probability terms\nassociated with the communication phases. The error proba-\nbility terms associated with the SPRT are bounded using [ 29,\nTh. 3.1], which is essentially equivalent to ( 28). To bound\nthe average decoding time of the code, we use the result in\n[32], which bounds the expected value of the stopping time\nτ= inf{n≥1:/summationtextn\ni=1Xi> γ}as\nE[τ]≤1\nE[X1](γ+b(X1)), (29)5\nwhereX1,X2,... are i.i.d. with a positive mean and a finite\nvariance. Since ( 29) applies to continuous random variables,\nTheorem 1also applies to continuous channels (e.g., the\nGaussian channel with average power constraint where PX=\nN(0,P)can be chosen as the random coding ensemble). See\nSection VI-A for the details of the proof of Theorem 1.\nIn Fig. 1, the achievable rates are presented for our VLF\ncode (Theorem 1) where the parameters of the code are\noptimized numerically, SED encoder [ 18] combined with the\nstop-at-time-zero strategy given in [ 9], and Polyanskiy et\nal.’s VLSF codes in [ 9]. In (a), the DMC is a BSC with\na flip probability 0.11; in (b), the DMC is a binary-input\nternary-output channel that is the cascade of a BSC with flip\nprobability 0.11 and a binary erasure channel (BEC) with flip\nprobability 0.2. In Fig. 1a, the refined bound in [ 19, Th. 7] is\nshown for the SED encoder, which achieves a slightly higher\nrate than our code. In Fig. 1b, the bound in [ 18, Remark 7] is\nshown. Although [ 18, Remark 7] is sufficient to show that\nthe SED encoder achieves the optimal error exponent, its\nperformance is much worse than that of our VLF code. This\nis because the constant term in [ 18, Remark 7] is comparable\nto the average decoding time Nshown in Fig. 1.\n200 400 600 800 1000 1200 14000.160.180.20.220.240.260.280.30.320.340.36Rate in nats/channel use\n(a)\n200 400 600 800 1000 1200 140000.050.10.150.20.25Rate in nats/channel use\n(b)\nFig. 1: Achievable rates over (a) the BSC with flip probabilit y 0.11 and (b)\nthe cascade of a BSC with flip probability 0.11 and the binary e rasure channel\nwith erasure probability 0.2 are shown with error probabili tyǫ= 10−3.\nOur second result is a second-order achievability bound for\nVLF codes, where the error probability ǫ∈(0,1)is fixed as\nthe average decoding time Napproaches infinity.Theorem 2: Assume that C >0andC1<∞. Then,\nlogM∗(N,ǫ)≥NC\n1−ǫ−C\nC1logN−loglogN+O(1).(30)\nSinceC≤C1, Theorem 2improves the second-order\nterm in [ 9, eq. (18)] given in the lower bound in ( 2) from\n−logNto−C\nC1logN. The achievability bound in [ 9, eq. (18)]\nemploys stop-feedback while our Yamamoto–Itoh scheme\nemploys stop-feedback and also sends a ⌈log2M⌉-bits of\nfeedback at the end of the first communication phase. The\nimprovement in the second-order term results from the fact\nthat the error probability of our scheme is dominated by the\nerror probability terms due to the confirmation phase and the\nsecond communication phase, whose average length scales as\nthe logarithm of the average length of the first communicatio n\nphase. For general DMCs, the non-asymptotic bound in [ 18,\nRemark 4] for Naghshvar et al. ’s MaxEJS encoder achieves\na second-order term −/parenleftig\nC\nC1+1/parenrightig\nlogNwhen combined with\nthe stop-at-time-zero strategy. To the best of our knowledg e,\nTheorem 2yields the best asymptotic achievability bound for\nVLF codes with non-vanishing error probabilities over gene ral\nDMCs. For BSCs and BACs, Yang et al. ’s bounds from [ 19,\nTh. 4 and 7] recover ( 30) with−loglogN+O(1)improved\ntoO(1)when combined with the stop-at-time-zero strategy.\nIt remains open to close the gap between the achievability\nbound in Theorem 2and the converse bound on the right-hand\nside of ( 2). The proof of Theorem 2follows from carefully\nchoosing the parameters γ1,γ2,aA,aR, andǫ0in Theorem 1,\nand appears in Section VI-B .\nThe third result is a second-order achievability bound for\nuniversal VLF codes, where the DMC PY|Xis unknown but\na capacity-achievability input distribution PXis known. We\nassume that the error probability ǫ∈(0,1)is non-vanishing\nas the average decoding time Napproaches infinity.\nTheorem 3: Assume that a capacity-achieving distribution\nofPY|Xis known. Assume that C >0andC1<∞. Assume\nthatlim\nN→∞ℓ\nlogN=∞. Then,\nlogM∗\nU(N,ǫ,ℓ)≥NC\n1−ǫ−C\nC1logN\n−min/braceleftbigg|X||Y|\n2,/parenleftbigg\n|X|−3\n2/parenrightbigg/parenleftbigg\n|Y|−3\n2/parenrightbigg\n+3\n4/bracerightbigg\nlogN\n+o(logN). (31)\nIn the case where PY|Xis known to be a BSC with an\nunknown flip probability p∈(0,1)\\{1\n2}, (31) is improved to\nlogM∗\nU(N,ǫ,ℓ)≥NC\n1−ǫ−/parenleftbiggC\nC1+1\n2/parenrightbigg\nlogN+o(logN).\n(32)\nThe coding scheme to achieve the right-hand side of ( 31)\nuniversalizes the coding scheme described in ( 23)-(27) by (1)\nreplacing the information density ı(cn(m);Yn)in (23) by\nthe empirical mutual information nI(ˆPcn(m);ˆPYn|cn(m)); (2)\nreplacing the DMC PY|Xin (14)–(27) by the empirical chan-\nnel transition kernel observed in the training sequence, i. e.,\nˆP¯Yℓ|xℓ; and (3) choosing the parameters γ1,γ2,aA,aR, andǫ0\nas a function of M,|X|,|Y|, andǫonly. The joint empirical6\nTABLE I: The summary of our modified Yamamoto–Itoh scheme\nPhases Communication 1 (C1) Confirmation (HT) Communication 2 (C2)\nCoding scheme variable-length i.i.d. random coding SPRT va riable-length i.i.d. random coding\nDecoding metric information density log-likelihood ratio information density\nRandom length τ(1)τHTτ(2)−τ(1)\nFeedback during the phase {continue,end phase } { continue,end phase } { continue,end phase }\nFeedback at the end of the phase ⌈log2M⌉bits forˆW(1){acceptˆW(1),rejectˆW(1)} ✗\nCondition to enter ✗ ✗ SPRT outputs “reject”\ndistribution ˆPcn(m)׈PYn|cn(m)coincides with the maximum\nlikelihood estimator within the family of distributions wi th\nalphabets X×Y . In the scenario where no training sequence is\navailable, i.e., ℓ= 0, using our proof techniques, we can show\nthat the right-hand sides of ( 31) and ( 32) are achievable with\nthe factorC\nC1replaced by 1; this result follows by employing\na single communication phase.\nIn [26], in the HT phase, instead of the empirical channel\nobtained from a training sequence, a BSC-specific metric\nthat is independent of the flip probability, i.e., the differ ence\nbetween the number of 1’s and the number of 0’s, is used.\nAlthough the resulting reliability function associated wi th their\nuniversal sequential HT phase is optimal, both the average\nstopping time and the error probability exponent depend on\nthe unknown flip probability of the BSC. In result, both the\nrate and the error probability of the code in [ 26] depend on the\nunknown capacity value. Since our goal is to control the erro r\nprobability of the universal code prior to the communicatio n,\nwe instead appeal to the training sequence to use in the log-\nlikelihood ratio during the HT phase.\nThe proof of Theorem 3differs from the proof of Theo-\nrem 2in two main ways. First, we bound P/bracketleftig\nτ(i)\n2≤n2/bracketrightig\n≤\n/summationtextn2\nn=1P/bracketleftig\nnI(ˆP¯Xn,ˆPYn|¯Xn)> γi/bracketrightig\nusing the refined method\nof types bound from [ 30, Th. 3] and a refined bound on\nE/bracketleftig\nexp{nI(ˆP¯Xn,ˆPYn|¯Xn)}/bracketrightig\n. Here,n2is a suitably chosen\nconstant and (¯X,Y)∼PXPY. The third term on the right-\nhand side of ( 31) results from the additional multiplicative\nfactor of nd\n2in the bound on P/bracketleftig\nτ(i)\nm≤n2/bracketrightig\ncompared to ( 28),\nwhere−dis the coefficient of the third term on the right-\nhand side of ( 31). Second, to bound the expected stopping\ntime, we use [ 29, Th. 4.5] from the nonlinear renewal the-\nory, which bounds the expected value of the stopping time\nτ= inf{n≥1:ng/parenleftbig1\nnSn/parenrightbig\n> γ}, whereSnis a sum of n\ni.i.d. vectors, and gis a sufficiently smooth function. After we\napply this result with gbeing the mutual information function\nandSnbeing the empirical joint distribution of cn(1)andYn,\nwe get\nE/bracketleftig\nτ(i)/bracketrightig\n≤E/bracketleftig\nτ(i)\n1/bracketrightig\n≤γi\nC+O(1). (33)\nThis implies that the expected stopping time associated wit h\nthe empirical mutual information admits the same asymptoti c\nbound associated with the information density ( 29) up to an\nO(1)gap. The analysis in [ 26] yields the bound E/bracketleftbig\nτ(i)/bracketrightbig\n≤\nγi\nC(1+o(1)) asγi→ ∞ , which is not sharp enough to prove\nthelogNscaling of the second-order term in ( 31). To prove\n(32), we replace the decoding metric by n(log2−H(ˆPZn(m))),\nwhereZi(m) =|Yi−ci(m)|, i.e., the Hamming distance\nbetweenYiandci(m), fori∈[n]. The proof of Theorem 3\nis given in Section VII.For an arbitrary DMC PY|Xand arbitrary random coding\ninput distribution PX, our universal code achieves the right-\nhand side of ( 31) with the capacity Creplaced by the mutual\ninformation I(PX,PY|X).\nV. G AUSSIAN CHANNEL\nThe output of a memoryless Gaussian channel of block-\nlengthnin response to the input Xn∈Rnis\nYn=Xn+Zn, (34)\nwhereZ1,...,Z nare drawn i.i.d. from N(0,σ2\n0), independent\nofXn, andσ2\n0>0is the noise variance. The capacity function\nof the Gaussian channel is given by\nC(S)/defines1\n2log(1+S). (35)\nThe analog of the quantity C1in (6) for the Gaussian channel\nis defined as\nC1(S)/definesD(N(√\nS,1)/⌊a∇⌈⌊lN(−√\nS,1)) = 2S. (36)\nDefinition 3: An(N,M,ǫ,P )-VLF code and an\n(N,M,ǫ,ℓ,P )-UVLF code are defined similarly to\nDefinitions 1and 2with the addition of average power\nconstraints\nE/bracketleftiggτ/summationdisplay\nt=1ft(U,W,Yt−1,σ2\n0)2/bracketrightigg\n≤E[τ]P, (37)\nE/bracketleftiggτ/summationdisplay\nt=1ft(U,W,Yt−1,¯Yℓ)2/bracketrightigg\n≤E[τ]P, (38)\nrespectively, where τis the random decoding time, Pis the\naverage power, and ¯Yℓis the output of the training sequence\nxℓ= (0,...,0). We define the maximum achievable codebook\nsizesM∗(N,ǫ,P)andM∗\nU(N,ǫ,ℓ,P )similarly to ( 12)–(13).\nThe average power constraint in ( 37) is introduced in\n[16] for variable-length stop-feedback codes for the Gaussian\nmultiple-access channel.\nThe following achievability bounds extend Theorems 2and\n3to the Gaussian channel with an average power constraint.\nTheorem 4: Letσ2\n0>0be the noise variance of the\nGaussian channel and let Pbe the average power constraint.\nDefine the signal-to-noise ratio\nS/definesP\nσ2\n0. (39)\nFor the Gaussian channel with the noise variance σ2\n0,\nlogM∗(N,ǫ,P)≥NC(S)\n1−ǫ−C(S)\nC1(S)logN\n−loglogN+O(1). (40)\nProof: Theorem 4is proved using our Yamamoto–Itoh\nscheme described in ( 21)–(28). During the communication7\nphases, i.e., the C1 and C2 phases, the input symbols are draw n\ni.i.d. from the Gaussian distribution N(0,P), which satisfies\nthe average power constraint in ( 37). During the HT phase, the\ntransmitter sends either (√\nP,√\nP,...)or(−√\nP,−√\nP,...)\nto accept or reject the receiver’s initial estimate, respec tively.\nSince the techniques used in the proof of Theorem 1applies\nto continuous random variables, Theorem 1applies to the\nGaussian channel with PX=N(0,P),PY|X=x=N(x,σ2\n0),\nandC=C(S). Theorem 4follows by following the same\nsteps as in the proof of Theorem 2.\nSinceC1(S)< C(S)for every S >0, Theorem 4improves\nthe second-order term in the achievability bound in [ 33, Th. 1]\nfrom−logNto−C(S)\nC1(S)logN. As an analog to the DMC\nscenario in ( 2), in [ 33, Th. 1], Truong and Tan show the\nconverse bound1\nlogM∗(N,ǫ,P)≤NC(S)\n1−ǫ+hb(ǫ)\n1−ǫ. (41)\nTheorem 5: Suppose that lim\nN→∞ℓ\nlogN=∞. Under the\nsetting of Theorem 4,\nlogM∗\nU(N,ǫ,ℓ,P )≥NC(S)\n1−ǫ−/parenleftbiggC(S)\nC1(S)+1\n2/parenrightbigg\nlogN\n+o(logN). (42)\nTheorem 5refines the achievability result in [ 27, Th. 4]\nto the second-order term for the Gaussian channel. To prove\nTheorem 5, we replace the empirical mutual information\nnI(ˆPXn,ˆPYn|Xn)used in the DMC case with\nıU(Xn;Yn)/defines−n\n2log(1−ˆρ2\nXnYn), (43)\nwhereˆρXnYnis the empirical correlation coefficient for zero-\nmean pairs defined as\nˆρXnYn/defines1\nn/summationtextn\ni=1XiYi/radicalig\n1\nn/summationtextn\ni=1X2\ni/radicalig\n1\nn/summationtextn\ni=1Y2\ni. (44)\nRecall that for zero-mean, jointly Gaussian (X,Y), the mutual\ninformation I(PX,PY|X)is given by\nI(PX,PY|X) =−1\n2log(1−ρ2\nXY), (45)\nwhereρXYis the correlation coefficient between XandY.\nThe universal quantity ıU(Xn;Yn)can be viewed as the\nempirical mutual information for the Gaussian channel in th e\nsense that\nıU(Xn;Yn) =nI(ˆPML\nXn,ˆPML\nYn|Xn), (46)\nwhere(ˆPML\nXn,ˆPML\nYn|Xn)is the maximum likelihood estimator\nwithin the family of jointly-Gaussian distributions. The p roof\nof Theorem 5appears in Section VIII.\n1Truong and Tan state the bound only for stop-feedback codes, which are a\nsubset of VLF codes; however, the same proof applies to VLF co des as well.VI. P ROOFS OF THEOREMS 1AND 2\nA. Proof of Theorem 1\n1) Error probability analysis: Define the error events\nE(i)/defines{ˆW(i)/\\⌉}atio\\slash= 1}, i= 1,2 (47)\nEA→R/defines{HRis declared given HA} (48)\nER→A/defines{HAis declared given HR} (49)\nEC2/defines{C2 phase is entered }. (50)\nThen the error probability of the above scheme is bounded as\nP/bracketleftig\nˆW/\\⌉}atio\\slash= 1/bracketrightig\n≤P/bracketleftig\n(E(1)/intersectiondisplay\nER→A)/uniondisplay\nE(2)/bracketrightig\n(51)\n≤P/bracketleftig\nE(1)/bracketrightig\nP[ER→A]+P/bracketleftig\nE(2)/bracketrightig\n, (52)\nwhere ( 52) follows from the union bound and the indepen-\ndence of the events E(1)andER→A. From [ 29, Th. 3.1],\nthe type-I and type-II error probabilities of the sequentia l\nhypothesis test is bounded as\nP[EA→R]≤exp{−aR} (53)\nP[ER→A]≤exp{−aA}. (54)\nThe probabilities P/bracketleftbig\nE(i)/bracketrightbig\n,i= 1,2, are bounded following [ 9,\nProof of Th. 2] as\nP/bracketleftig\nE(i)/bracketrightig\n≤P/bracketleftig\nτ(i)\n1=∞/bracketrightig\n+P/bracketleftiggM/uniondisplay\nm=2{τ(i)\nm<∞}/bracketrightigg\n(55)\n≤(M−1)exp{−γi}, i= 1,2. (56)\nCombining ( 52), (54), and ( 56), we get\nP/bracketleftig\nˆW/\\⌉}atio\\slash= 1/bracketrightig\n≤(M−1)(exp{−(γ1+aA)}+exp{−γ2}).\n(57)\n2) Average decoding time analysis: We use the following\nresult from the renewal theory literatur, which bounds the\nexpected value of the stopping time associated with a random\nwalk.\nLemma 1 ( [ 32, Th. 1], [ 34, Ch. 3, Th. 9.2–9.3, Th. 10.7]):\nLetX,X1,X2,... be i.i.d. random variables with E[X] =\nµ >0andE/bracketleftbig\n(X+)2/bracketrightbig\n<∞. LetSn=/summationtextn\ni=1Xiandτ=\ninf{n≥1:Sn> a}. Then, for any a >0,\nE[τ]≤1\nµ(a+b(X)). (58)\nLet\nτ+= inf{n≥1:Sn>0} (59)\nρ=E/bracketleftig\nS2\nτ+/bracketrightig\n2E/bracketleftbig\nSτ+/bracketrightbig. (60)\nAsa→ ∞ , ifXis non-arithmetic and the above conditions\nare satisfied, then\nE[τ] =1\nµ(a+ρ)+o(1), (61)\nand ifXis arithmetic with a span handa=jh,j→ ∞ ,\nthen\nE[τ] =1\nµ/parenleftbigg\na+ρ+h\n2/parenrightbigg\n+o(1). (62)8\nIt holds that\nE/bracketleftig\nS2\nτ+/bracketrightig\n2E/bracketleftbig\nSτ+/bracketrightbig=E/bracketleftbig\nX2/bracketrightbig\n2µ−∞/summationdisplay\nk=11\nkE/bracketleftbig\nS−\nk/bracketrightbig\n. (63)\nWe bound the probability that the C2 phase is used as\nP[EC2] =P/bracketleftig\n(E(1)∩Ec\nR→A)∪((E(1))c∩EA→R)/bracketrightig\n(64)\n≤P/bracketleftig\nE(1)/bracketrightig\n+P[EA→R] (65)\n≤(M−1)exp{−γ1}+exp{−aR}. (66)\nRecall the stopping times defined in ( 22)–(23). Obviously,\nit holds that τ(i)\n1≤τ(i)fori= 1,2. By our code design, τ=\nτ(1)+τHTif the event EC2does not occur and τ=τ(2)+τHT\nifEC2occurs. Then, we bound the average stopping time as\nE[τ]≤E/bracketleftbig\nτHT/bracketrightbig\n+E/bracketleftig\nτ(1)\n1/bracketrightig\n+P[EC2]E/bracketleftig\nτ(2)\n1−τ(1)\n1/bracketrightig\n.(67)\nApplying Lemma 1, we bound each of the expectations in ( 67)\nas\nE[τHT|HA]≤aA+bA\nD(PY|X=xA/⌊a∇⌈⌊lPY|X=xR)(68)\nE[τHT|HR]≤aR+bR\nD(PY|X=xR/⌊a∇⌈⌊lPY|X=xA)(69)\nE[τHT]≤E[τHT|HA]+P[HR]E[τHT|HR](70)\n≤E[τHT|HA]\n+(M−1)exp{−γ1}E[τHT|HR](71)\nE/bracketleftig\nτ(1)\n1/bracketrightig\n≤γ1+b\nC(72)\nE/bracketleftig\nτ(2)\n1−τ(1)\n1/bracketrightig\n≤γ2−γ1+b\nC, (73)\nwhereb=b(ı(X;Y)),bA=b/parenleftig\nlogPY|X=xA(YA)\nPY|X=xR(YA)/parenrightig\n, andbR=\nb/parenleftig\nlogPY|X=xR(YR)\nPY|X=xA(YR)/parenrightig\n, and and YDis distributed according to\nPY|X=xDforD∈ {A,R}.\nFinally, combining ( 66), (67), and ( 71)–(73) gives\nE[τ]≤γ1+b\nC\n+((M−1)exp{−γ1}+exp{−aR})γ2−γ1+b\nC\n+aA+bA\nD(PY|X=xA/⌊a∇⌈⌊lPY|X=xR)\n+(M−1)exp{−γ1}aR+bR\nD(PY|X=xR/⌊a∇⌈⌊lPY|X=xA).(74)\nFrom the above analysis, there exists an (N′,M,ǫ′)-VLF code\nwithǫ′andN′are given as the right hand sides of ( 57)\nand ( 74), respectively. We use the stop-at-time-zero strategy\ndescribed in [ 9], where with probability 1−ǫ0, the code above\nis used, and with probability ǫ0, we use a simple code that\nstops at time zero and decodes to an arbitrary message. Let\nNandǫbe the average decoding time and the average error\nprobability of the described code obtained by this time-sha ring\nstrategy. We have\nN≤(1−ǫ0)N′(75)\nǫ≤ǫ0+(1−ǫ0)ǫ′, (76)\nwhich completes the proof.B. Proof of Theorem 2\nWe prove Theorem 2by carefully choosing the free param-\netersγ1,γ2,aA,aR, andǫ0in Theorem 1. Let\nN1=γ1+b\nC, (77)\nwhich is an upper bound on the expected length of phase C1.\nWe express all other parameters in terms of N1. We set\nγ1= logM+loglog N1 (78)\nγ2= logM+logN1 (79)\naA=aR= logN1. (80)\nThen, by ( 18), we have as N1→ ∞\nN′=N1+logN1\nC1+O(1) (81)\nǫ′≤1\nN1/parenleftbigg\n1+1\nlogN1/parenrightbigg\n. (82)\nWe set\nǫ0=ǫ−1\nN1/parenleftig\n1+1\nlogN1/parenrightig\n1−1\nN1/parenleftig\n1+1\nlogN1/parenrightig=ǫ−Ω/parenleftbigg1\nN1/parenleftbigg\n1+1\nlogN1/parenrightbigg/parenrightbigg\n.\n(83)\nFrom ( 77) and ( 78), we get\nlogM=N1C−loglogN1+O(1). (84)\nFrom ( 81) and ( 84), we get\nlogM=N′C−C\nC1logN′−loglogN′+O(1). (85)\nFinally, from ( 16) and ( 83), the error probability of the code\nis bounded by ǫ, and the average decoding time of the code\nis bounded by (1−ǫ0)N′. Therefore, by ( 81) and ( 85), there\nexists an (N,M,ǫ)-VLF code with\nlogM=NC\n1−ǫ−C\nC1logN−loglogN+O(1). (86)\nVII. P ROOF OF THEOREM 3\nA. Supporting Lemmas\nWe first present two supporting lemmas that play key\nroles to prove Theorem 3. The first result, below, bounds\nthe tail probability of the empirical mutual information fo r\nindependent ¯XnandYn.\nLemma 2: Let(¯Xn,Yn)∼Pn\nXPn\nYfor some PX∈ P(X)\nandPY∈ P(Y), and letγbe a positive constant. Assume that\nPX(x)>0andPY(y)>0for all(x,y)∈(X ×Y). Then,\nthere exists n0∈Nsuch that for all n≥n0\nP/bracketleftig\nnI(ˆP¯XnYn)≥γ/bracketrightig\n≤K1(n+1)kexp{−γ} (87)\nk= min/braceleftbigg|X||Y|− 2\n2,/parenleftbigg\n|X|−3\n2/parenrightbigg/parenleftbigg\n|Y|−3\n2/parenrightbigg\n−1\n4/bracerightbigg\n,(88)\nwhereK1is a positive constant depending only on |X|and\n|Y|.\nProof: See Appendix A.\nThe second result, which is from the nonlinear renewal\ntheory literature, bounds the expected stopping time assoc iated9\nwith a function of an i.i.d. sum. This result is the nonlinear\nversion of Lemma 1and is used to bound the expected stop-\nping times associated with the empirical mutual informatio n.\nLemma 3: [29, Th. 4.5] Let g:Rk→Rbe a twice\ndifferentiable continuous function. Let Y,Y1,Y2,··· ∈Rkbe\ni.i.d. random vectors. Let µ=g(E[Y])>0. Letγ >0. Define\nZn=ng/parenleftigg\n1\nnn/summationdisplay\ni=1Yi/parenrightigg\n(89)\nτ= inf{n≥1:Zn> a}. (90)\nThen, if µ+∇g(E[Y])⊤(Y−E[Y])is non-arithmetic, as\na→ ∞ ,\nE[τ] =1\nµ/parenleftbigg\na+ρ−1\n2tr(Cov(Y)∇2g(E[Y]))/parenrightbigg\n+o(1),\n(91)\nwhereρis defined in ( 60) withSn=/summationtextn\ni=1Xireplaced with\nnµ+/summationtextn\ni=1∇g(E[Y])⊤(Yi−E[Y]). Ifµ+∇g(E[Y])⊤(Y−\nE[Y])is arithmetic with a span h >0, then as a→ ∞ ,\nE[τ] =1\nµ/parenleftbigg\na+ρ+h\n2−1\n2tr(Cov(Y)∇2g(E[Y]))/parenrightbigg\n+o(1).\n(92)\nLemma 3is a special case of the nonlinear form Zn=/summationtextn\ni=1Xi+ξn, whereX1,X2,... are i.i.d. random variables,\nξn’s are slowly changing random variables, which is specified\nin [29, eq. (4.10)–(4.16)], and (X1,ξ1),...,(Xn,ξn)are in-\ndependent of Xk,k > n . From the Taylor series expansion of\nZnaroundng(E[Y]), we get\nZn=nµ+n/summationdisplay\ni=1∇g(E[Y])⊤(Yi−E[Y])\n+1\n2W⊤\nn∇2g(E[Y])Wn+o(1) (93)\nWn=1√nn/summationdisplay\ni=1(Yi−E[Y]). (94)\nTherefore, in Lemma 3,µ+∇g(E[Y])⊤(Yi−E[Y])plays\nthe role of Xiand1\n2W⊤\nn∇2g(E[Y])Wn+o(1)+o(1)plays\nthe role of ξn. In [ 29, Example 4.1], it is shown that Wn\nsatisfies the slowly-changing conditions. By the central li mit\ntheorem, Wnapproaches the Gaussian vector N(0,Cov(Y))\nin distribution, and the third-term in ( 93) approaches the sum\nofkindependent χ2(1)random variables, weighted with1\n2\ntimes the eigenvalues of the matrix Cov(Y)∇2g(E[Y]).\nB. Universal Coding Scheme\nWe generate Mi.i.d. codewords c(1),...,c(M)fromP∞\nX.\nOur coding scheme is similar to that in Section VI-A except\nthat we replace the information-density decoding metric wi th\nthe empirical mutual information. Specifically, the stoppi ng\ntimesτ(i)\nmin (22) are replaced with\nτ(i)\nm= inf{n≥1:nI(ˆPcn(m),ˆPYn|cn(m))> γi}. (95)\nIn the HT phase, since PY|Xis unknown, the log-likelihood\nratio test is modified as follows. Define\n˜PY|X=x/definesˆPYℓx|Xℓx=xℓx, (96)whereℓxis the number of x∈ X in the training sequence xℓ.\nA lower bound on τ(1)is\nn1=/floorleftbigglogM\nmin{|X|,|Y|}/floorrightbigg\n. (97)\nDefine the event\nG/defines/braceleftbigg\nmax\n(x,y)|˜PY|X=x(y)−PY|X=x(y)|=o(1)/bracerightbigg\n. (98)\nBy the assumption thatℓ\nlogN→ ∞ , using Hoeffding’s bound,\nwe have\nP[Gc]≤1\nn2\n1. (99)\nWe re-define (xA,xR)in (14) and the stopping time τHTin\n(27) by replacing PY|Xby the empirical conditional distri-\nbutionˆPY|X. Given that Goccurs, the modified xAandxR\ncoincide with those in ( 14).\nC. Analysis\nWe here explain the differences from the proof of Theo-\nrem 2.\n1) Letn2=c2γ2\nC, wherec2>1is a constant. We bound\nthe probability P/bracketleftbig\nE(i)/bracketrightbig\nas\nP/bracketleftig\nE(i)/bracketrightig\n≤P/bracketleftig\nτ(i)\n1≥n2/bracketrightig\n+P/bracketleftiggM/uniondisplay\nm=2{τ(i)\nm< n2}/bracketrightigg\n(100)\n≤P/bracketleftig\nn2I(ˆPcn2(1)Yn2)≤γi/bracketrightig\n+(M−1)(n2−n1)\nmax\nn∈[n1,n2−1]P/bracketleftig\nnI(ˆPcn(2),ˆPYn|cn(2))> γi/bracketrightig\n(101)\n≤n|X||Y|\n2exp{−c3γi}+K1Mnd\n2exp{−γi}, (102)\nwherec3andK1are positive constants, and d=\nmin/braceleftig\n|X||Y|\n2,/parenleftbig\n|X|−3\n2/parenrightbig/parenleftbig\n|Y|−3\n2/parenrightbig\n+3\n4/bracerightig\n. Here, ( 101) fol-\nlows from the definition of τ(i)and the union bound\nacross time and messages; the first term in ( 102) fol-\nlows from the standard method of types (see e.g., [ 26,\nLemma 3]), and the second term in ( 102) follows from\nLemma 2.\n2) We bound the expected value of the stopping times τ(i),\ni= 1,2, using Lemma 3instead of Lemma 1. To do this,\nwe write the empirical mutual information as\nnI(ˆPXn,ˆPYn|Xn) =nI/parenleftiggn/summationdisplay\ni=1Vi/parenrightigg\n(103)\n=n/summationdisplay\ni=1ı(Xi;Yi)+1\n2W⊤\nn∇2I(PX,PY|X)Wn+o(1)\n(104)\nWn=1√nn/summationdisplay\ni=1(Vi−E[Vi]), (105)\nwhereVi∈R|X||Y|,i= 1,...,n , are independent and\nhave multinomial distribution with parameters (n,PXY).\nHence,ı(Xn;Yn)is a first-order Taylor approximation10\nto the empirical mutual information nI(ˆPXn,ˆPYn|Xn).\nApplying Lemma 3toE/bracketleftig\nτ(i)\n1/bracketrightig\n, similar to ( 72)–(73), we\nget\nE/bracketleftig\nτ(1)\n1/bracketrightig\n=γ1\nC+O(1) (106)\nE/bracketleftig\nτ(2)\n1−τ(1)\n1/bracketrightig\n=γ2−γ1\nC+O(1). (107)\nNotice that the bound in ( 106) is asymptotically the same\nas the bound in ( 72) except the value of the constant O(1)\nterm.\n3) Combining ( 68)–(69) with ( 98), we get\nE[τHT|HA]≤aA(1+o(1))\nD(PY|X=xA/⌊a∇⌈⌊lPY|X=xR)+O(1) (108)\nE[τHT|HR]≤aR(1+o(1))\nD(PY|X=xR/⌊a∇⌈⌊lPY|X=xA)+O(1).(109)\nCombining ( 54) and ( 98), we get\nP[EA→R]≤exp{−aR}(1+o(1)) (110)\nP[ER→A]≤exp{−aA}(1+o(1)). (111)\n4) Lastly, the error probability of our UVLF code is bounded\nas\nP/bracketleftig\nˆW/\\⌉}atio\\slash= 1/bracketrightig\n≤P[Gc]+P/bracketleftig\nE(1)/bracketrightig\nP[ER→A]+P/bracketleftig\nE(2)/bracketrightig\n.\n(112)\nThe rest of the proof follows steps identical to those in\nSection VI-A except that the code parameters are chosen\nindependently from PY|X. SinceN1≈logM\nCdepends on\nthe channel capacity, unlike the VLF code design, the code\nparameters for the UVLF code cannot depend on N1.\nNotice that since the optimal choice for γ2satisfiesγ2=\nlogM(1 +o(1)), it holds that n2= Θ(N1). SinceC >0,\nit also holds that n1= Θ(N1). To balance the additional\nfactor of nd\n2in (102) and to get an error probability bound\nindependent of n2, we letδ >0be an arbitrary constant, and\nwe set the thresholds γ1andγ2slightly larger than those in\n(78)–(79) as\nγ1= logM+dlogn1+(1+δ)loglogn1 (113)\nγ2= logM+(d+1)logn1+δloglogn1. (114)\nBased on ( 110)–(111), we set\naA=aR= logn1. (115)\nCombining ( 102), the confidence bound in ( 99), and the error\nprobability bound in ( 112), (111), and ( 115), for UVLF codes,\nǫ′in (82) is bounded as\nǫ′≤1\nn1(116)\nforlogMlarge enough. We set the stopping probability at\ntime zero as\nǫ0=ǫ−1\nn1\n1−1\nn1. (117)\nFollowing steps similar to those in ( 84)–(86) with the mod-\nifications in ( 113)–(117) andn1= Ω(N1), we complete the\nproof of ( 31).D. Universal Coding Scheme for a BSC Family and Its\nAnalysis\nThe coding scheme is identical to that in Section VII-B\nexcept that the stopping time in ( 95) is replaced with\nτ(i)\nm/definesinf{n≥1:n(log2−H(ˆPZn(m)))> γi},(118)\nwhereZi(m) = 1{Yi/\\⌉}atio\\slash=ci(m)}. Define\nˆpn(m)/defines1\nnn/summationdisplay\ni=1Zi(m), (119)\nHence,H(ˆPZn(m)) =hb(ˆpn(m))is the binary entropy func-\ntion of the empirical flip probability from the sub-codeword\ncn(m)to the output sequence Yn.\nWe bound the probability P/bracketleftig\nτ(i)\n2< n2/bracketrightig\ndifferently than\n(101)–(102). The information density under the BSC( p) equals\nı(cn(m);Yn) =n/parenleftbigg\nlog(2(1−p))−ˆpn(m)log1−p\np/parenrightbigg\n.\n(120)\nTherefore, both τ(i)\n1andı(cn(1);Yn)depends on (cn(1),Yn)\nonly through the empirical flip probability ˆpn(1). This\nmeans that τ(i)\n1is a stopping time for the martingale\n{exp{−ı(cn(1);Yn)}}n≥1. Using this property, we apply the\nsteps in [ 9, eq. (111)–(117)] and get\nP/bracketleftig\nτ(i)\n2< n2/bracketrightig\n=E/bracketleftig\nexp{−ı(cτ(i)\n1(m);Yτ(i)\n1)}1{τ(i)\n1< n2}/bracketrightig\n.\n(121)\nDefine\nR(i)/definesτ(i)\n1(log2−H(ˆP\nZτ(i)\n1))−γi (122)\nηn/definesn(log2−H(ˆPZn(1)))−ı(cn(1);Yn). (123)\nHere,R(i)≥0is the overshoot random variable corresponding\nto the transmitted codeword. Then, we bound the right-hand\nside of ( 121) as\nP/bracketleftig\nτ(i)\n2< n2/bracketrightig\n= exp{−γi}E/bracketleftig\nexp{−R(i)+ητ(i)\n1}1{τ(i)\n1< n2}/bracketrightig\n(124)\n≤exp{−γi}E/bracketleftig\nexp{ητ(i)\n1}1{τ(i)\n1< n2}/bracketrightig\n(125)\n≤max\nn γi}. (131)\nThe empirical channel in ( 96) that is used in the HT phase is\nreplaced with\n˜PY|X=x=N/parenleftigg\nx,1\nℓℓ/summationdisplay\ni=1¯Y2\ni/parenrightigg\n, (132)\nwhere the training outputs are distributed i.i.d. as Yi∼\nN(0,σ2\n0). Let\nn1=⌊logM⌋, (133)\nwhich satisfies n1= Θ(N)for anyσ2\n0>0. We re-define the\neventGas\nG/defines/braceleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nℓℓ/summationdisplay\ni=1¯Y2\ni/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=o(1)/bracerightigg\n. (134)\nBy the assumption thatℓ\nlogN→ ∞ , applying the Chernoff\nbound, we get\nP[Gc]≤1\nn2\n1. (135)\nIn the HT phase, we set xA=√\nPandxR=−√\nPand\nconstruct the SPRT in ( 27) withPY|Xreplaced with ˆPY|X.\nIn the following, we explain the differences from the four\nitems in Section VII-C .\n1) Letn2=c2γ2\nC(S), wherec2>1is a constant. We bound\nthe probability P/bracketleftbig\nE(i)/bracketrightbig\nas\nP/bracketleftig\nE(i)/bracketrightig\n≤P/bracketleftig\nτ(i)\n1≥n2/bracketrightig\n+P/bracketleftiggM/uniondisplay\nm=2{τ(i)\nm< n2}/bracketrightigg\n(136)≤P/bracketleftig\nıU(ˆPcn2(1)Yn2)≤γi/bracketrightig\n+(M−1)n2/summationdisplay\nn=1P/bracketleftig\nıU(ˆPcn(2),ˆPYn|cn(2))> γi/bracketrightig\n(137)\n≤exp{−c3γi}+K1n1/2\n2exp{−γi}, (138)\nwherec3andK1are positive constants independent of\nn2andγi. Here, the first term in ( 138) follows from the\nChernoff bound, and the second term in ( 138) follows by\nwriting\nP/bracketleftig\nıU(ˆPcn(2),ˆPYn|cn(2))> γi/bracketrightig\n=P/bracketleftbig\nˆρcn(2)Yn> a/bracketrightbig\n(139)\n≤K2√nexp/braceleftign\n2log(1−a2)/bracerightig\n(1+o(1)), (140)\nwherea∈(0,1)satisfiesγi=−n\n2log(1−a2), andK2\nis a positive constant that depends only on a. The bound\nin (140) follows from Lemma 4.\n2) Let\nı∗(bxy,bx,by)/defines−1\n2log/parenleftigg\n1−b2\nxy\nbxby/parenrightigg\n. (141)\nThen, we have\nıU(Xn;Yn) =nı∗/parenleftigg\n1\nnn/summationdisplay\ni=1XiYi,1\nnn/summationdisplay\ni=1X2\ni,1\nnn/summationdisplay\ni=1Y2\ni/parenrightigg\n.\n(142)\nHence,ıU(Xn;Yn)satisfies the conditions to apply\nLemma 3. Taking the second-order Taylor series expan-\nsion ofı∗(·)around(P,P,P+σ2\n0), we get\nıU(Xn;Yn)\n=ı(Xn;Yn)+1\n2W⊤\nn∇2ı∗(P,P,P+σ2\n0)Wn+o(1),\n(143)\nwhere\nWn=1√nn/summationdisplay\ni=1/parenleftbig\nXiYi−P,X2\ni−P,Y2\ni−(P+σ2\n0)/parenrightbig\n,\n(144)\nand\nı(Xn;Yn) =nC(S)−n/summationdisplay\ni=1(Yi−Xi)2\n2σ2\n0+n/summationdisplay\ni=1Y2\ni\n2(P+σ2\n0)\n(145)\nis the information density associated with the Gaussian\nchannel under the input distribution PX=N(0,P).\nTherefore, applying Lemma 3, (106)–(107) hold with C\nreplaced by C(S).\n3) Using ( 134), we check that ( 108)–(111) hold.\n4) The error probability bound in ( 112) remains the same.\nDue to ( 138), we set d=1\n2in (113)–(114). Following\nthe steps in ( 115)–(117), we complete the proof.12\nIX. C ONCLUSION\nIn this work, we study variable-length feedback codes over\nknown and unknown channels in the asymptotic regime that\nthe error probability ǫis non-vanishing as the average decoding\ntimeNapproaches infinity.\nOur achievability bounds for both VLF and UVLF codes\nemploy a modified Yamamoto–Itoh scheme that has two\ncommunication phases and one confirmation phase, where\neach phase has a random length that depends on the noise\nrealization. We also employ the stop-at-time-zero strateg y used\nin [9], which enables to achieve the ǫ-capacity of VLF codes.\nTheorem 1presents our novel non-asymptotic achievability\nbound for VLF codes. Theorem 2is our second-order achiev-\nability bound for VLF codes, which refines the second-order\nterm achieved in [ 9, Th. 2] from −logNand−C\nC1logN,\nwhereCis the capacity, and C1is the optimal reliability\nfunction at zero rate.\nTo adapt our Yamamoto–Itoh scheme to the scenario where\nthe channel transition kernel PY|Xis not exactly known\nby the transmitter and the receiver, we allow a training\nsequence of length ℓ≫logN, which is used to construct\nthe hypothesis test PY|X=xAvs.PY|X=xRin the confirmation\nphase. Specifically, the distributions PY|X=xAandPY|X=xR\nare replaced by the empirical distributions obtained from t he\ntraining sequence. In the communication phases of UVLF\ncodes, similar to [ 20], [26], [27], we employ the empirical\nmutual information between the input and output sequences\nas our decoding metric. Theorem 3presents our second-order\nachievability bound for UVLF codes over DMCs. In the proof\nof Theorem 3, we use the asymptotic expansion in [ 29, Th. 4.5]\nfor the stopping time associated with a smooth function of\nan average of random vectors. In Lemma 2, we prove a tail\nprobability bound with a refined pre-factor for the empirica l\nmutual information evaluated on a joint type formed from\ntwo independent sequences, which plays a critical role in th e\nderivation of the second-order term in Theorem 3.\nOur results extend to the Gaussian channel with known\nand unknown variances and an average power constraint.\nTheorem 4is our achievability bound for VLF codes over\nthe Gaussian channel, which refines the bound in [ 33, Th. 1].\nFor UVLF codes over the Gaussian channel, similar to [ 27],\nwe employ the universal metric −1\n2log(1−ˆρ2\nXnYn), where\nˆρXnYnis the empirical correlation coefficient between Xn\nandYn; this metric corresponds to the mutual information\nof two jointly Gaussian random variables with the correlati on\ncoefficient ˆρXnYn.\nAPPENDIX A\nPROOF OF LEMMA 2\nWe bound P/bracketleftig\nnI(ˆP¯Xn,ˆPYn|¯Xn)≥γ/bracketrightig\nfrom above by two\ndifferent approaches. We have\nP/bracketleftig\nnI(ˆP¯Xn,ˆPYn|¯Xn)≥γ/bracketrightig\n≤P/bracketleftbigg\nD(ˆP¯XnYn/⌊a∇⌈⌊lPXPY)\n≥ inf\nQXY:nI(QX,QY|X)≥γD(QXY/⌊a∇⌈⌊lPXPY)/bracketrightbigg\n(146)≤P/bracketleftbigg\nD(ˆP¯XnYn/⌊a∇⌈⌊lPXPY)\n≥ inf\nQXY:nI(QX,QY|X)≥γI(QX,QY|X)/bracketrightbigg\n(147)\n=P/bracketleftig\nD(ˆP¯XnYn/⌊a∇⌈⌊lPXPY)≥γ\nn/bracketrightig\n(148)\n≤C1|X||Y|− 2/summationdisplay\ni=1/parenleftbigg/parenleftig\nC0n\ni/parenrightigi\n2+1/parenrightbigg\nexp{−γ}, (149)\nwhereC0≈3.1967 andC1≈2.9290 . Inequality ( 147) fol-\nlows from D(QXY/⌊a∇⌈⌊lPXPY) =I(QX,QY|X)+D(QX/⌊a∇⌈⌊lPX)+\nD(QY/⌊a∇⌈⌊lPY)and the non-negativity of the KL divergence.\nInequality ( 149) follows from the novel method of types bound\nfrom [ 30, Th. 3]. Since the prefactor in ( 149) isO(n|X||Y|− 2\n2),\n(87) follows, where kis replaced with the first argument in the\nminimum in ( 88). Note that the standard method of types from\n[36, Lemma II.1] bounds ( 148) by(n+1)|X||Y|− 1exp{−γ}.\nTo show ( 87) withkreplaced with the second argument in\nthe minimum in ( 88), we apply the Chernoff bound and get\nP/bracketleftig\nnI(ˆP¯Xn,ˆPYn|¯Xn)≥γ/bracketrightig\n≤E/bracketleftig\nexp{nI(ˆP¯XnYn)}/bracketrightig\nexp{−γ}. (150)\nNoting that logPn\nX(xn) =/summationtext\nx∈XˆPxn(x)logPX(x), we\nwrite the expectation in ( 150) as\nE/bracketleftig\nexp{nI(ˆP¯Xn,ˆPYn|¯Xn)}/bracketrightig\n=/summationdisplay\nxn,ynPn\nX(xn)Pn\nY(yn)exp{nI(ˆPxnyn)} (151)\n=/summationdisplay\nxn,ynexp/braceleftig\n−n/parenleftig\nD(ˆPxn/⌊a∇⌈⌊lPX)+D(ˆPyn/⌊a∇⌈⌊lPY)+H(ˆPxnyn)/parenrightig/bracerightig\n(152)/summationdisplay\nQXY∈Pn(X,Y)|Tn(QXY)|\nexp{−n(D(QX/⌊a∇⌈⌊lPX)+D(QY/���a∇⌈⌊lPY)+H(QXY))}.\n(153)\nNext, we use the tight bound on the size of the type class\n[21, Exercise 2.2]\n|Tn(QX)| ≤exp{nH(QX)}(2πn)−|X|−1\n2/productdisplay\nx∈X1/radicalig\n˜QX(x),\n(154)\nwhere˜QX(x) =1\n2πnifQX(x) = 0 and˜QX(x) =QX(x)\notherwise.\nApplying ( 154) to ( 153), we get\nE/bracketleftig\nexp{nI(ˆP¯Xn,ˆPYn|¯Xn)}/bracketrightig\n= (2πn)−|X||Y|− 1\n2\n/summationdisplay\nQXY∈Pn(X,Y)/bracketleftigg\nexp{−n(D(QX/⌊a∇⌈⌊lPX)+D(QY/⌊a∇⌈⌊lPY))}\n/productdisplay\n(x,y)∈X×Y1/radicalig\n˜QXY(x,y)/bracketrightigg\n. (155)13\nDefine the sets\nAn(QX,QY)/defines{VXY∈ Pn(X,Y):VX=QX,VY=QY}.\n(156)\nWe rewrite the summation in ( 155) to get\nE/bracketleftig\nexp{nI(ˆP¯Xn,ˆPYn|¯Xn)}/bracketrightig\n≤(2πn)−|X||Y|− 1\n2\n\n/summationdisplay\nQX∈Pn(X)\nQY∈Pn(Y)exp{−n(D(QX/⌊a∇⌈⌊lPX)+D(QY/⌊a∇⌈⌊lPY))}\n\n(157)\nmax\nVX∈P(X)\nVY∈P(Y)/summationdisplay\nVXY∈An(VX,VY)/productdisplay\n(x,y)∈X×Y1/radicalig\n˜VXY(x,y)\n.\n(158)\nNote that |An(QX,QY)| ≤(n+1)(|X|−1)(|Y|−1). Bounding\nthe summation in ( 158) by an appropriate integral, we get\nmax\nVX∈P(X)\nVY∈P(Y)/summationdisplay\nVXY∈An(VX,VY)/productdisplay\n(x,y)∈X×Y1/radicalig\n˜VXY(x,y)\n≤C2(n+1)(|X|−1)(|Y|−1), (159)\nwhereC2>0is a constant depending on |X| and|Y|. It\nonly remains to bound the summation in ( 157). To do that,\nwe use the following asymptotic result, which can be viewed\nas Laplace’s method for sums over types.\nLemma 5: Letf:P(X)→Rbe a function with a unique\nminimum at P∗\nX. Letǫ >0and letBǫbe a ball of radius ǫ\ncentered at P∗\nX. Assume that the derivatives of fup to third\norder exist and are bounded in Bǫ. Assume that the minimum\neigenvalue of ∇2f(PX)is bounded below by 0 for all PX∈\nBǫ. Then,\n/summationdisplay\nPX∈Pn(X)exp{−nf(PX)}\n= (2πn)|X|−1\n2exp{−nf(P∗\nX)}1/radicalbig\ndet(∇2f(P∗\nX))(1+o(1)).\n(160)\nThe function D(·/⌊a∇⌈⌊lPX)satisfies the conditions of Lemma 5\ngiven that PX(x)>0for allx∈ X with the minimizer P∗\nX.\nTherefore, applying Lemma 5to (155) twice, we get\n/summationdisplay\nQX∈Pn(X)\nQY∈Pn(Y)exp{−n(D(QX/⌊a∇⌈⌊lPX)+D(QY/⌊a∇⌈⌊lPY))}\n≤C3(n+1)(|X|−1)(|Y|−1)|\n2(1+o(1)), (161)\nwhereC3>0is a constant. Finally, combining ( 150), (157)–\n(159), and ( 161) completes the proof.\nAPPENDIX B\nPROOF OF (127)\nDefine\nQi/definesBernoulli( i/n), i= 0,...,n. (162)By changing measure from PXPY|XtoPXPY, we get\nE[exp{ηn}] =/summationdisplay\nzn∈{0,1}n/parenleftbigg1\n2/parenrightbiggn\nexp{n(log2−H(ˆPzn))}\n(163)\n=/summationdisplay\nzn∈{0,1}nexp{−nH(ˆPzn))} (164)\n=n/summationdisplay\ni=0|Tn(Qi)|exp{−nH(Qi)} (165)\n=n/summationdisplay\ni=01√\n2πn/productdisplay\nx∈{0,1}1/radicalig\n˜Qi(x)(166)\n≤K2n, (167)\nwhereK2is a positive constant, ( 165) follows by grouping the\nsequences of the same type, and ( 166) follows from ( 154).\nREFERENCES\n[1] C. Shannon, “The zero error capacity of a noisy channel,” IRE Trans.\non Inf. Theory , vol. 2, no. 3, pp. 8–19, Sep. 1956.\n[2] M. Horstein, “Sequential transmission using noiseless feedback,” IEEE\nTrans. Inf. Theory , vol. 9, no. 3, pp. 136–143, Jul. 1963.\n[3] J. Schalkwijk and T. Kailath, “A coding scheme for additi ve noise\nchannels with feedback–I: No bandwidth constraint,” IEEE Trans. Inf.\nTheory , vol. 12, no. 2, pp. 172–182, Apr. 1966.\n[4] A. B. Wagner, N. V . Shende, and Y . Altu˘ g, “A new method for employing\nfeedback to improve coding performance,” IEEE Trans. Inf. Theory ,\nvol. 66, no. 11, pp. 6660–6681, Nov. 2020.\n[5] M. V . Burnashev, “Data transmission over a discrete chan nel with\nfeedback: Random transmission time,” Problems of Information Trans-\nmission , vol. 12, no. 4, pp. 10–30, Aug. 1976.\n[6] R. Gallager, “A simple derivation of the coding theorem a nd some\napplications,” IEEE Trans. Inf. Theory , vol. 11, no. 1, pp. 3–18, Jan.\n1965.\n[7] H. Yamamoto and K. Itoh, “Asymptotic performance of a mod ified\nSchalkwijk-Barron scheme for channels with noiseless feed back (cor-\nresp.),” IEEE Trans. Inf. Theory , vol. 25, no. 6, pp. 729–733, Nov. 1979.\n[8] J. Y . Chen, R. C. Yavas, and V . Kostina, “Variable-length codes with\nbursty feedback,” in IEEE Int. Symp. Inf. Theory (ISIT) , Taipei, Taiwan,\n2023, pp. 1448–1453.\n[9] Y . Polyanskiy, H. V . Poor, and S. Verd´ u, “Feedback in the non-\nasymptotic regime,” IEEE Trans. Inf. Theory , vol. 57, no. 8, pp. 4903–\n4925, Aug. 2011.\n[10] S. H. Kim, D. K. Sung, and T. Le-Ngoc, “Variable-length f eedback codes\nunder a strict delay constraint,” IEEE Communications Letters , vol. 19,\nno. 4, pp. 513–516, Apr. 2015.\n[11] A. R. Williamson, T. Chen, and R. D. Wesel, “Variable-le ngth convo-\nlutional coding for short blocklengths with decision feedb ack,” IEEE\nTrans. Commun. , vol. 63, no. 7, pp. 2389–2403, Jul. 2015.\n[12] K. Vakilinia, S. V . S. Ranganathan, D. Divsalar, and R. D . Wesel,\n“Optimizing transmission lengths for limited feedback wit h nonbinary\nLDPC examples,” IEEE Trans. Comm. , vol. 64, no. 6, pp. 2245–2257,\nMar. 2016.\n[13] R. C. Yavas, V . Kostina, and M. Effros, “Variable-lengt h feedback codes\nwith several decoding times for the Gaussian channel,” in Proc. IEEE\nInt. Symp. Inf. Theory (ISIT) , Melbourne, Victoria, Australia, July 2021,\npp. 1883–1888.\n[14] ——, “Variable-length sparse feedback codes for point- to-point, multiple\naccess, and random access channels,” IEEE Trans. Inf. Theory, Early\nAccess , pp. 1–1, 2023.\n[15] H. Yang, R. C. Yavas, V . Kostina, and R. D. Wesel, “Variab le-length\nstop-feedback codes with finite optimal decoding times for B I-AWGN\nchannels,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT) , Espoo, Finland,\nJune 2022, pp. 2327–2332.\n[16] L. V . Truong and V . Y . F. Tan, “On Gaussian MACs with varia ble-length\nfeedback and non-vanishing error probabilities,” IEEE Trans. Inf. Theor. ,\nvol. 64, no. 4, p. 2333–2346, Apr. 2018.14\n[17] M. Naghshvar, M. Wigger, and T. Javidi, “Optimal reliab ility over a\nclass of binary-input channels with feedback,” in Proc. IEEE Information\nTheory Workshop (ITW) , Lausanne, Switzerland, Sep. 2012, pp. 391–\n395.\n[18] M. Naghshvar, T. Javidi, and M. Wigger, “Extrinsic Jens en–Shannon\ndivergence: Applications to variable-length coding,” IEEE Trans. Inf.\nTheory , vol. 61, no. 4, pp. 2148–2164, Apr. 2015.\n[19] H. Yang, M. Pan, A. Antonini, and R. D. Wesel, “Sequentia l transmission\nover binary asymmetric channels with feedback,” IEEE Trans. Inf.\nTheory , vol. 68, no. 11, pp. 7023–7042, Nov. 2022.\n[20] V . D. Goppa, “Nonprobabilitistic mutual information w ithout memory,”\nProblems of Control and Inform., Theory , vol. 4, pp. 97–102, 1975.\n[21] I. Csisz´ ar and J. K¨ orner, Information Theory: Coding Theorems for\nDiscrete Memoryless Systems , 2nd ed. Cambridge University Press,\n2011.\n[22] J. Scarlett, A. G. i F` abregas, A. Somekh-Baruch, and A. Martinez,\n“Information-theoretic foundations of mismatched decodi ng,” Founda-\ntions and Trends® in Communications and Information Theory , vol. 17,\nno. 2–3, pp. 149–401, 2020.\n[23] N. Merhav, “Universal decoding for arbitrary channels relative to a given\nclass of decoding metrics,” IEEE Transa. Inf. Theory , vol. 59, no. 9, pp.\n5566–5576, 2013.\n[24] ——, “Universal decoding for memoryless Gaussian chann els with a\ndeterministic interference,” in Proceedings. IEEE Int. Symp. Inf. Theory ,\n1993, pp. 265–265.\n[25] W. Huleihel and N. Merhav, “Universal decoding for gaus sian intersym-\nbol interference channels,” IEEE Trans. Inf. Theory , vol. 61, no. 4, pp.\n1606–1618, 2015.\n[26] A. Tchamkerten and I. E. Telatar, “Variable length codi ng over an\nunknown channel,” IEEE Trans. Inf. Theory , vol. 52, no. 5, pp. 2126–\n2145, May 2006.\n[27] Y . Lomnitz and M. Feder, “Communication over individua l channels,”\nIEEE Trans. Inf. Theory , vol. 57, no. 11, pp. 7333–7358, 2011.\n[28] N. Merhav and M. Feder, “Universal decoding with an eras ure option,”\nin2007 IEEE Int. Symp. Inf. Theory , 2007, pp. 1726–1730.\n[29] M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis .\nSociety for Industrial and Applied Mathematics, 1982.\n[30] J. Mardia, J. Jiao, E. T´ anczos, R. D. Nowak, and T. Weiss man,\n“Concentration inequalities for the empirical distributi on of discrete\ndistributions: beyond the method of types,” Information and Inference:\nA Journal of the IMA , vol. 9, no. 4, pp. 813–850, 11 2019.\n[31] L. V . Truong and V . Y . Tan, “On the Gaussian MAC with stop- feedback,”\ninProc. IEEE Int. Symp. Inf. Theory (ISIT) , Aachen, Germany, June\n2017, pp. 2303–2307.\n[32] G. Lorden, “On excess over the boundary,” Ann. Math. Statist. , vol. 41,\nno. 2, pp. 520–527, 04 1970.\n[33] L. V . Truong and V . Y . F. Tan, “On Gaussian MACs with varia ble-length\nfeedback and non-vanishing error probabilities,” arXiv:1609.00594v2 ,\nSep. 2016.\n[34] A. Gut, Stopped random walks , 2nd ed. Springer New York, NY , 2009.\n[35] T. K. T. Truong and M. Zani, “Sharp Large Deviations for e mpirical\ncorrelation coefficients,” Sep. 2019, working paper or prep rint. [Online].\nAvailable: https://hal.science/hal-02283954\n[36] I. Csiszar, “The method of types [information theory], ”IEEE Trans. Inf.\nTheory , vol. 44, no. 6, pp. 2505–2523, 1998." }, { "title": "2401.16772v1.Extrinsicaly_Rewarded_Soft_Q_Imitation_Learning_with_Discriminator.pdf", "content": "IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x\n1\nPAPER\nExtrinsicaly Rewarded Soft Q Imitation Learning with\nDiscriminator\nRyoma FURUYAMAa), Daiki KUYOSHIb),andSatoshi YAMANEc),\nSUMMARY Imitation learning is often used in addition to reinforce-\nment learning in environments where reward design is difficult or where the\nreward is sparse, but it is difficult to be able to imitate well in unknown states\nfrom a small amount of expert data and sampling data. Supervised learn-\ning methods such as Behavioral Cloning do not require sampling data, but\nusually suffer from distribution shift. The methods based on reinforcement\nlearning, such as inverse reinforcement learning and Generative Adversarial\nimitation learning (GAIL), can learn from only a few expert data. However,\nthey often need to interact with the environment. Soft Q imitation learning\n(SQIL) addressed the problems, and it was shown that it could learn effi-\nciently by combining Behavioral Cloning and soft Q-learning with constant\nrewards. In order to make this algorithm more robust to distribution shift,\nwe propose more efficient and robust algorithm by adding to this method\na reward function based on adversarial inverse reinforcement learning that\nrewards the agent for performing actions in status similar to the demo. We\ncall this algorithm Discriminator Soft Q Imitation Learning (DSQIL). We\nevaluated it on MuJoCo environments.\nkey words: Artificial Intelligence, Machine Learning, Deep Reinforcement\nLearning, Imitation Learning, Inverse Reinforcement Learning\n1. Introduction\nRecent developments in the field of deep reinforcement\nlearning have made it possible to learn diverse behaviors\nfor high-dimensional input. However, there are still some\nproblems. Among them, we focus on the efficiency of learn-\ning and the difficulty of designing a reward function [1]. For\nexample, when using reinforcement learning for artificial\nintelligence of autonomous driving, it is necessary to deal\nwith many unexpected phenomena such as various terrain\nand people coming out [2]. If we design a reward function\nto solve this problem, the program may become enormous.\nAlso, incomplete reward function design may promote unex-\npected behavior. In addition, it is necessary to explore with\nmany random actions until the agent obtains a reward in an\nenvironment with sparse rewards.\nIn such setting of problems, imitation learning is often\nused instead of reinforcement learning. Behavioral Cloning\n[3], which is the classical imitation learning, is a simple\nsupervised learning algorithm that maximizes the likelihood\nof the actions taken by an expert in a certain state. It shows\ngood results for simple tasks, but it requires a large dataset\nof the pairs of state and action, and it sometimes behaves\nManuscript received January 1, 2015.\nManuscript revised January 1, 2015.\n†The author is with the\na) E-mail: rfuruyama@csl.ec.t.kanazawa-u.ac.jp\nb) E-mail: dkuyoshi@csl.ec.t.kanazawa-u.ac.jp\nc) E-mail: syamane@is.t.kanazawa-u.ac.jp\nDOI: 10.1587/trans.E0.??.1strangely in a state is not in the dataset. In order to overcome\nthese disadvantages, Inverse Reinforcement Learning (IRL)\n[4] performs a two-step learning process in which it estimates\na reward function instead of expert actions, and it performs\nreinforcement learning based on the reward function. This\nalgorithm helps to learn to behave in unexpected situations.\nHowever, IRL has the disadvantage of being unstable\ndue to two stages of learning. Therefore, the methods of\nlearning the behavior of the expert directly by Generative\nAdversarial Networks (GANs) [5] without explicitly finding\na reward function were proposed such as Generative Ad-\nversarial imitation Learning (GAIL) [6]. This methods can\nlearn efficiently with small amounts of data. Furthermore,\nAdversarial Inverse Reinforcement Leaning (AIRL) [7] has\nbeen proposed for IRL, which outperforms IRL and GAIL\nby improving generalizability through the use of GAN. Thus,\nimitation learning has been greatly advanced by GANs, and\nmethods and applications based on this technology continue\nto be studied [8], [9].\nAlthough Behavioral Cloning was no longer considered\nuseful, Reddy et al. [10] proposed Soft Q Imitation Learning\n(SQIL), which addressed state distribution shift by the com-\nbination of Behavioral Cloning and reinforcement learning.\nIt has been reported that the learning has been performed ef-\nficiently with less training steps than in previous adversarial\nimitation learning. In this paper, we propose more efficient\nand robust algorithm by adding to this method a reward func-\ntion that rewards the agent for performing actions in status\nsimilar to the demo. We evaluate it with three environments\nof MuJoCo [11], and we show the strong and weak points.\n2. Background\nWe consider problems that satisfy the definition of Markov\nDecision Process (MDP) [12]. In continuing tasks, the\nreturns for a trajectory 𝜏=(𝑠𝑡,𝑎𝑡)∞\n𝑡=0are defined as\n𝑟𝑡=Í∞\n𝑘=𝑡𝛾𝑘−𝑡𝑅(𝑠𝑘,𝑎𝑘), where𝛾is a discount factor.\nIn order to use the same notation for episodic tasks, we\ncan define a set of absorbing state 𝑠𝑎. When we define\nthe reward𝑅(𝑠𝑎,·)=0, we can define returns simply as\n𝑟𝑡=Í𝑇\n𝑘=𝑡𝛾𝑘−𝑡𝑅(𝑠𝑘,𝑎𝑘). In reinforcement learning like\nActor Critic (AC) [13] or Q-learning [14], we would like to\nlearn a policy 𝜋that maximizes expected returns.Therefore,\nthe objective function is\n𝐽(𝜋)=𝑇∑︁\n𝑡=0E(𝑠𝑡,𝑎𝑡)∼𝜌𝜋[𝑟(𝑠𝑡,𝑎𝑡)]. (1)\nCopyright ©200x The Institute of Electronics, Information and Communication EngineersarXiv:2401.16772v1 [cs.LG] 30 Jan 20242IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x\nRecently, various methods have been studied, including\nAC-based Proximal Policy Optimization (PPO) [15] and Q-\nlearning-based Recurrent Replay Distributed DQN (R2D2)\n[16]. One of them is Soft Actor Critic (SAC) [17], [18] pro-\nposed by Haarnoja et al. SAC is a maximum entropy rein-\nforcement learning and has shown excellent performance, es-\npecially in complex environments. Maximum entropy rein-\nforcement learning significantly improves explorability and\nrobustness. is one of the methods. The objective function is\nEquation 1 plus an entropy maximization term 𝐻(𝜋(·|𝑠𝑡)),\n𝐽(𝜋)=𝑇∑︁\n𝑡=0E(𝑠𝑡,𝑎𝑡)∼𝜌𝜋[𝑟(𝑠𝑡,𝑎𝑡)+𝛼𝐻(𝜋(·|𝑠𝑡))],\n(2)\nwhere𝛼is the temperature parameter, which determines\nthe relative importance of the entropy term to the reward\nand controls the stochasticity of the optimal policy. SAC\nuses a soft Q function [19] to learn measures that maximize\nestimates while improving their accuracy. This method op-\ntimizes stochastic policies off-policy, and this algorithm is\nsample efficient.\n3. Related works\n3.1 Behavioral Cloning\nBehavioral Cloning [3] is a classical imitation learning al-\ngorithm. It is a method of supervised learning that takes the\nstate𝑠𝐸in the expert data as input and regards the action\n𝑎𝐸of the expert as the label.When the set of expert state\nand action pairs(𝑠𝐸,𝑎𝐸)is𝛽𝑑𝑒𝑚𝑜 , the loss function for the\nparameter𝜃is the following equation.\n𝑙𝐵𝐶(𝜃)=∑︁\n(𝑠,𝑎)∈𝛽𝑑𝑒𝑚𝑜−log𝜋𝜃(𝑎|𝑠), (3)\nwhere𝜋𝜃is a measure with parameter 𝜃, and the objective is\nto find𝜃that minimizes Equation 3. Although this method\nis simple, it is easy to overfit to the expert data because it\ndoes not learn the result of the action, and it suffers from the\nstate distribution shift. As a result, it has the disadvantage\nof not being able to make good decisions for unseen states\n[20].\n3.2 Soft Q Imitation Learning\nTo cover the shortcomings of behavioral cloning, Reddy et\nal [10] proposed Soft Q Imitation Learning (SQIL). SQIL\ncombines behavioral cloning and reinforcement learning to\naddress shifts in the state distribution. This method is built on\nsoft Q learning [19], where experts are assumed to follow a\npolicy𝜋that maximizes reward 𝑅(𝑠,𝑎)in an infinite horizon\nMarkov decision process (MDP) with continuous state space\nS and discrete action space A. The policy 𝜋(𝑎|𝑠)forms a\nBoltzmann distribution for action,𝜋(𝑎|𝑠)≜exp(𝑄(𝑠,𝑎))Í\n𝑎′∈𝐴exp(𝑄(𝑠,𝑎′)), (4)\nwhere𝑄is soft Q function and is defined by the following\nequation.\n𝑄(𝑠,𝑎)≜𝑅(𝑠,𝑎)+𝛾E𝑠′\"\nlog ∑︁\n𝑎′∈𝐴exp𝑄(𝑠′,𝑎′)!#\n(5)\n𝑠′is the state when action 𝑎is taken in state 𝑠. If we assume\nthat the behavior of our agent follows the policy, we can\ndefine the loss function by Equation 3.\n𝑙𝐵𝐶(𝜃)≜∑︁\n(𝑠,𝑎)∈𝛽𝑑𝑒𝑚𝑜−(𝑄𝜃(𝑠,𝑎)\n−log ∑︁\n𝑎′∈𝐴exp(𝑄𝜃(𝑠,𝑎))!!\n(6)\nSQIL aims to learn by considering the trajectory by reg-\nularizing this loss function with squared soft Bellman er-\nror𝛿2(𝛽,𝑟), called Regularized Behavioral Cloning (RBC)\n[21].\n𝑙𝑅𝐵𝐶(𝜃)≜𝑙𝐵𝐶(𝜃)+𝜆𝛿2\u0000𝛽𝑑𝑒𝑚𝑜∪𝛽𝑠𝑎𝑚𝑝,0\u0001(7)\n𝛿2(𝛽,𝑟)≜1\n|𝛽|∑︁\n(𝑠,𝑎,𝑠′)∈𝛽(𝑄𝜃(𝑠,𝑎)\n− \n𝑟+𝛾log ∑︁\n𝑎′∈𝐴exp(𝑄𝜃(𝑠′,𝑎′))!!!2\n,\n(8)\nwhere𝜆∈𝑅≥0is a hyperparameter that determines the rel-\native importance of Behavioral Cloning versus soft Q learn-\ning. In addition, 𝛽𝑑𝑒𝑚𝑜 and𝛽𝑠𝑎𝑚𝑝 are a replay buffer of\ndemonstration data by an expert and sampling data from an\nenvironment by an agent respectively.\nFurthermore, we can rewrite the gradient of 𝑙𝑅𝐵𝐶(𝜃)\nas simple form.\n▽𝜃𝑙𝑅𝐵𝐶(𝜃)∝▽𝜃(𝛿2(𝛽𝑑𝑒𝑚𝑜,1)+𝜆𝑠𝑎𝑚𝑝𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,0\u0001\n+log ∑︁\n𝑎∈𝐴exp(𝑄𝜃(𝑠0,𝑎))!!\n,\n(9)\nwhere𝑠0is the initial state. The update equation for 𝜃in the\nSQIL algorithm is defined by Equation 9.FURUYAMA et al.: EXTRINSICALY REWARDED SOFT Q IMITATION LEARNING WITH DISCRIMINATOR\n3\n𝜃←𝜃−𝜂▽𝜃\u0010\n𝛿2(𝛽𝑑𝑒𝑚𝑜,1)+𝜆𝑠𝑎𝑚𝑝\u0000𝛽𝑠𝑎𝑚𝑝,0\u0001\u0011\n(10)\nIn original paper, 𝜆𝑠𝑎𝑚𝑝=1. Importantly, we can recognize\nthat the SQIL agent sets the rewards of all demonstration\ndata to 1 and the rewards of all sampling data to 0.\nSQIL can be implemented because it requires only mi-\nnor modifications to standard Q-learning implementations;\nSQIL can also be extended to MDPs with continuous action\nspaces by simply replacing Q-learning with an off-policy\nactor-critic method such as SAC Given the difficulty of cor-\nrectly implementing deep RL algorithms [22], this flexibility\nis an advantage and enhances the utility of SQIL because it\ncan be built on top of existing implementations of deep RL\nalgorithms.\n3.3 Generative Adversarial Networks\nGenerative Adversarial Networks (GANs) [5] is one of the\nmethods used for image generation. In this method, two\nmodels are prepared: a generator 𝐺and a discriminator\n𝐷. In the generator 𝐺, the distribution of the input noise\nvariable𝑝𝑧is predefined and the prepared data is trained.\nThe objective of this learning is to generate data similar to\nthe training example from the noise variables. The objective\nof𝐷, on the other hand, is to discriminate between the data\nin the training example and the data generated in 𝐺. In this\nway,𝐺and𝐷learn to compete with each other. In other\nwords,𝐺and𝐷are playing a mini-max game based on the\nvalue function 𝑉(𝐺,𝐷).\nmin\n𝐺max\n𝐷𝑉(𝐺,𝐷)=E𝑥∼𝑝𝑑𝑎𝑡𝑎(𝑥)[log𝐷(𝑥)]\n+E𝑧∼𝑝𝑧(𝑧)[log(1−𝐷(𝐺(𝑧)))],\n(11)\nwhere𝑝𝑑𝑎𝑡𝑎 is the distribution of the data in the train\nexample.Although GANs are image generators, they are\nwell suited for reinforcement learning and have contributed\ngreatly to the advancement of that technology, especially\nin imitation learning; GAIL [6] is one such example. An\noverview of the architecture of GANS is shown in Figure 1.\n4. Proposal of Discriminator Soft Q Imitation Learning\n4.1 Discriminator Soft Q Imitation Learning\nSQIL [10] shows that soft Q learning, which provides a\npositive constant reward for expert data, can efficiently mimic\nan agent. However, the method of determining this constant\nreward is not always a good method. For example, after some\nprogress in learning, the agent learns as reward 0 even if the\nsample data obtained from the environment is similar to the\nexpert’s data. This may become noise in the learning process.\nTherefore, we propose a method that uses the discriminator D\nof the GAN as the reward function. This method is expected\nto reduce the above problem as well as to learn efficiently\nwith less expert data. We call this method Discriminator Soft\nFig. 1: Generative Adversarial Network Architecture.\nQ Imitation Learning (DSQIL). DSQIL ’s algorithm is shown\nin Algorithm 1, and an overall view of DSQIL is shown in\nFigure 2. As with SQIL, the agent can be used flexibly, for\nexample, Q-Learning [14] for discrete value control tasks\nand SAC [17] for continuous value control tasks.\nAlgorithm 1 Discriminator Soft Q Imitation Learning\n(DSQIL)\nRequire: Replay buffer of demonstaration data 𝛽𝑑𝑒𝑚𝑜 ,\n𝜆𝑑𝑒𝑚𝑜 =1,𝜆𝑠𝑎𝑚𝑝 =1 in our experiment.\n1: Initialize replay buffer of sample data 𝛽𝑠𝑎𝑚𝑝←∅\n2:for𝑛=1,2,...do\n3: while𝑒≠𝑇𝑟𝑢𝑒 do\n4:𝜏=(𝑠,𝑎,.,𝑠′,𝑒)with𝜋𝜃\n5:𝛽𝑠𝑎𝑚𝑝←𝛽𝑠𝑎𝑚𝑝∪𝜏\n6:𝑀𝑑𝑒𝑚𝑜 =\b\u0000𝑠𝑡,𝑎𝑡,.,𝑠′\n𝑡,𝑒𝑡\u0001\t𝑚\n𝑡=1∼𝛽𝑑𝑒𝑚𝑜\n7:𝑀𝑠𝑎𝑚𝑝 =\b\u0000𝑠𝑡,𝑎𝑡,.,𝑠′\n𝑡,𝑒𝑡\u0001\t𝑚\n𝑡=1∼𝛽𝑠𝑎𝑚𝑝\n8: Calcurate the loss of 𝐷\n9: Update 𝐷with GAN\n10: for𝑖=1,2,...,𝑀𝑑𝑒𝑚𝑜 do\n11:𝑅\u0000𝛽𝑑𝑒𝑚𝑜𝑖\u0001←𝐷(𝑠𝑖,𝑎𝑖)\n2+1\n2𝜆𝑑𝑒𝑚𝑜\n12: end for\n13: for𝑗=1,2,...,𝑀𝑠𝑎𝑚𝑝 do\n14:𝑅\u0010\n𝛽𝑠𝑎𝑚𝑝𝑗\u0011\n←𝐷\u0010\n𝑠𝑗,𝑎𝑗\u0011\n2\n15: end for\n16: Update 𝜃{See Equation 12 }\n17: end while\n18:end for\n4.2 Update equation\nWe set up the DSQIL update equation based on Equation 10.\n𝜃←𝜃−𝜂▽𝜃(𝜆𝑑𝑒𝑚𝑜𝛿2\u0012\n𝛽𝑑𝑒𝑚𝑜,𝑅(𝛽𝑑𝑒𝑚𝑜)+1\n2𝜆𝑑𝑒𝑚𝑜\u0013\n+𝜆𝑠𝑎𝑚𝑝𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0000𝛽𝑠𝑎𝑚𝑝\u0001\u0001\u0011\n,4IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x\nFig. 2: The overall of DSQIL algorithm.\n(12)\nwhere𝜆𝑑𝑒𝑚𝑜∈R≥0,𝜆𝑠𝑎𝑚𝑝∈R≥0are hyperparameter, and\n𝛿2denotes the squared soft Bellman error defined in Equation\n8. We added to the rewards, in addition to the fixed values,\na function that indicates the probability of similarity of the\nexpert data using discriminator 𝐷from GANs [5]. This\nchange enabled more efficient learning by rewarding sample\ndata similar to the expert data.\nThe update equation was derived from the following\nloss function defined with reference to Equation 7.\n𝑙𝐷𝑆𝑄𝐼𝐿(𝜃)≜𝑙𝐵𝐶(𝜃)+𝜆𝛿2\u0000𝛽𝑑𝑒𝑚𝑜∪𝛽𝑠𝑎𝑚𝑝,𝑅\u0001,(13)\nwhere,𝑅is reward function.In addition, the soft value func-\ntion is defined as follows.\n𝑉(𝑠)≜log ∑︁\n𝑎∈𝐴exp(𝑄𝜃(𝑠,𝑎))!\n(14)\nIn Equation 13, dividing the soft Bellman squared error\nterm by demonstration data and sample data, and further\nusing Equation 6,���𝑙𝐷𝑆𝑄𝐼𝐿(𝜃)=∑︁\n𝜏∈𝛽𝑑𝑒𝑚𝑜𝑇−1∑︁\n𝑡=0−(▽𝑄𝜃(𝑠𝑡,𝑎𝑡)−▽𝑉(𝑠𝑡))\n+𝜆𝑑𝑒𝑚𝑜∑︁\n𝜏∈𝛽𝑑𝑒𝑚𝑜𝑇−1∑︁\n𝑡=0▽(𝑄𝜃(𝑠𝑡,𝑎𝑡)−(𝑅+𝛾𝑉(𝑠𝑡+1)))2\n+𝜆𝑠𝑎𝑚𝑝▽𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0001\n=∑︁\n𝜏∈𝛽𝑑𝑒𝑚𝑜𝑇−1∑︁\n𝑡=0(𝑉(𝑠𝑡)−𝛾𝑉(𝑠𝑡+1))\n+𝜆𝑑𝑒𝑚𝑜▽𝛿2\u0012\n𝛽𝑑𝑒𝑚𝑜,𝑅+1\n2𝜆𝑑𝑒𝑚𝑜\u0013\n+𝜆𝑠𝑎𝑚𝑝▽𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0001.\n(15)\nAssuming𝛾≜1, the inner product of the first term is a\ntelescoping sum.\n▽𝑙𝐷𝑆𝑄𝐼𝐿(𝜃)=∑︁\n𝜏∈𝛽𝑑𝑒𝑚𝑜(▽𝑉(𝑠0)−▽𝑉(𝑠𝑇))\n+𝜆𝑑𝑒𝑚𝑜▽𝛿2\u0012\n𝛽𝑑𝑒𝑚𝑜,𝑅+1\n2𝜆𝑑𝑒𝑚𝑜\u0013\n+𝜆𝑠𝑎𝑚𝑝▽𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0001\n(16)\nFrom the assumption that 𝑠𝑇is absorbed,𝑉(𝑠𝑇)=0. There-\nfore,FURUYAMA et al.: EXTRINSICALY REWARDED SOFT Q IMITATION LEARNING WITH DISCRIMINATOR\n5\n▽𝑙𝐷𝑆𝑄𝐼𝐿(𝜃)=∑︁\n𝜏∈𝛽𝑑𝑒𝑚𝑜▽𝑉(𝑠0)\n+𝜆𝑑𝑒𝑚𝑜▽𝛿2\u0012\n𝛽𝑑𝑒𝑚𝑜,𝑅+1\n2𝜆𝑑𝑒𝑚𝑜\u0013\n+𝜆𝑠𝑎𝑚𝑝▽𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0001.\n(17)\nIn our experiments, all demo rollouts start from the same\ninitial state s0. Thus,\n▽𝑙𝐷𝑆𝑄𝐼𝐿(𝜃)∝▽\u0012\n𝜆𝑑𝑒𝑚𝑜𝛿2\u0012\n𝛽𝑑𝑒𝑚𝑜,𝑅+1\n2𝜆𝑑𝑒𝑚𝑜\u0013\n+𝜆𝑠𝑎𝑚𝑝𝛿2\u0000𝛽𝑠𝑎𝑚𝑝,𝑅\u0001+𝑉(𝑠0)\u0011\n.\n(18)\nThus, the gradient of the loss function is similar to the up-\ndated equation shown in Equation 12 plus the soft value at\n𝑠0.\n4.3 Reward function\nIn the update equation, we had placed the reward function\n𝑅. We used the discriminator 𝐷with reference to GANs\n[5] as the reward function R.The discriminator 𝐷was used\nas the reward function. This discriminator 𝐷is trained\nto discriminate between expert data and sample data, and\nthe probability of being expert data is expressed as a value\nbetween 0 and 1. In other words, the value is close to 1 if\nit discriminates the data as expert data and close to 0 if it\njudges the data as other than expert data. To optimize the\ndiscriminator 𝐷with weight 𝜙, We minimize the following\nloss function,\n𝑙(𝜙)≜−E(𝑠,𝑎)∼𝛽𝑑𝑒𝑚𝑜\u0002\nlog\u0000𝐷𝜙(𝑠,𝑎)\u0001\u0003\n−E(𝑠,𝑎)∼𝛽𝑠𝑎𝑚𝑝\u0002\nlog\u00001−𝐷𝜙(𝑠,𝑎)\u0001\u0003 (19)\nUsing this discriminator as a reward function, we can reward\nwhen data similar to the expert data is obtained as sample\ndata, thus enabling efficient learning.\nThe algorithm based on the rewards and hyperparam-\neters used in the experiment is shown in Algorithm 1. The\nhyperparameters were set to 𝜆𝑑𝑒𝑚𝑜 =1,𝜆𝑠𝑎𝑚𝑝 =1, and\n𝑅=𝐷(𝑠,𝑎)\n2with the reward as half of the trained discrimi-\nnator’s output, based on the settings that performed best in\nSQIL, 1 reward for expert data and 0 reward for demo data.\nThis setting allows us to limit the reward to between 0 and 1,\nclose to the reward in SQIL. In this way, we can expect simi-\nlar performance to SQIL in the early stages when learning is\ninsufficient, and allow learning from sample data after some\nlearning has progressed. Therefore, performance similar to\nor better than SQIL can be expected.\n5. Experimental Evaluation\n5.1 Outline of experimentation\nTo evaluate DSQIL, we compared DSQIL to SQIL usingTable 1: expert policy performance to provide expert data.\nEnvironments Expert Performance\nHopper-V3 3308 .3±26.7\nWalker2D-V3 3897 .7±31.6\nHalfCheetah-V3 5303 .3±75.4\nTable 2: SAC hyperparameters.\nParameters Value\nOptimizer Adam\nDiscount rate(𝛾) 0.99\nHyperparameters 𝛼initial 0.2\nActor learning rate 3𝑒−4\nCritic learning rate 3𝑒−4\n𝛼learning rate 3𝑒−4\nTarget update rate 5𝑒−3\nMini batch size 64\nActor hidden dim 256\nActor hidden layers 3\nArchitecture Actor activation function ReLU\nCritic hidden dim 256\nCritic hidden layers 3\nCritic activation function ReLU\nTable 3: Discriminator hyperparameters.\nParameters Value\nOptimizer Adam\nlearning rate 3𝑒−4\nHyperparameters Mini batch size 512\nLoss function Binary Cross Entropy\nreplay buffer size( 𝛽𝑠𝑎𝑚𝑝 ) 1e+6\nwarm up 1024\nHidden dim 128\nArchitecture Hidden layers 3\nActivation function Tanh\nLast activation Sigmoid\nsome empirical data: we evaluated Hopper-v2, Walker2d-\nv2, and HalfCheetah-v2 in three MuJoCo [11] environ-\nments. Since the environment used in the experiment was\na continuous-value control task, SAC [17] was used as the\nagent in this case.\nFirst, the policy is learned using SAC to obtain expert\nperformance. The expert policy is used to generate a set of\nexpert data to be stored in the replay buffer. The obtained\nexpert policy performance is shown in Table 1.\nAfter the expert data is obtained, each method is trained.\nTo study the effect of learning on the amount of expert data,\nthe algorithms are trained on sets of {2,4,8,16,32}as seen\nin [23]. In all environments, learning is performed in 500\nsteps, with one step acquiring one episode of sample data\nand storing it in the replay buffer. During training, sample\nand expert data are randomly extracted from the replay buffer\nat a ratio of 1:1 to be used for the discriminator in order to\nobtain a reward. At this time, the discriminator is trained\nsimultaneously. All reported results correspond to perfor-\nmance measures obtained after testing the learner policy on\n50 episodes. For agent we used SAC [17]: a 3 layer MLP\nwith ReLU activations and 256 hidden units. The discrimi-\nnator is a 3 layer MLP with a sigmoid at the end in addition6IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x\nTable 4: Performance (The maximum value for each trajectory is shown in bold.)\nEnvironments trajectory BC SQIL DSQIL(ours)\nHopper-V3 2 2766 .7±385.6 3001.6±422.4 2886.9±741.3\n4 3010 .8±460.3 3113.9±365.6 3202.3±187.1\n8 3051 .5±294.5 3156.9±351.7 3262.9±3.3\n16 3164 .7±269.8 3274.9±6.1 3209.8±14.5\n32 3241 .1±141.0 3287.6±1.9 3272.6±1.8\nWalker2D-V3 2 797 .1±75.1 2112.4±688.6 3520.8±551.3\n4 2001 .0±1360.0 3799.2±214.8 3987.4±64.4\n8 2110 .0±1398.1 3989.7±24.8 3929.9±15.5\n16 3571 .7±588.6 3911.9±18.3 3907.1±19.4\n32 3756 .5±236.2 3905.7±11.1 3921.9±41.9\nHalfCheetah-V3 2 381.9±334.1 113.9±104.0 99.0±103.1\n4 599 .3±306.9 143.5±115.1 1607.5±682.8\n8 2622 .4±342.0 1874.4±2150.94515.0±289.9\n16 3228 .4±671.2 4361.4±86.8 4499.4±105.2\n32 3350 .5±491.4 4636.8±161.9 4713.5±69.3\nto tanh for activity and 128 hidden units. We trained all\nnetworks with the Adam optimizer [24].\nWe show other hyperparameters in Table 2 and Table 3.\nFor each task, we compare the following three algo-\nrithms:\n1. BC\n2. SQIL based on SAC\n3. DSQIL based on SAC (ours)\n5.2 Comparing the scores\nEvaluation results are shown in Table 4. In each environ-\nment, DSQIL outperforms BC. For relatively simple tasks\nsuch as the Hopper and Walker2d environments, SQIL and\nDSQIL have comparable performance if sufficient expert\ndata is available. On the other hand, in complex environ-\nments with more information required, such as the HalfChee-\ntah environment, DSQIL performs better than SQIL regard-\nless of the amount of expert data. The performance dif-\nference is especially large when there is less expert data.\nHowever, for simple tasks such as the Hopper environment,\nthere is no performance difference depending on the amount\nof expert data, and DSQIL shows lower performance than\nSQIL with a small amount of expert data.\n5.3 Comparing the speed of learning\nThe training for DSQIL and SQIL is shown in Figure 3. It\nshows the learning for each environment given 32 episodes\nof expert data. At the end of an epoch, 5 episodes are tested\nusing the network learned up to that point, and the average of\nthe results is obtained. Figure 3 shows the evolution of these\nvalues obtained with SQIL and DSQIL. It can be seen that in\nthe relatively simple Hopper and Walker2d environments, the\nSQIL learns faster than the DSQIL. This is due to the fact that\ntraining a discriminator requires a certain number of steps.\nIf the discriminator is not well trained, it is possible that\nthe learning could be affected by giving unjustified rewards\nto the sample data. On the other hand, in the complex\nHalfCheetah environment, DSQIL learns faster than SQIL.This indicates that in complex environments, the effect of\nbeing able to learn from sample data is greater than the loss\nincurred during the training period of the discriminator.\n5.4 Comparing the rewards\nA comparison between DSQIL and SQIL for each reward\ntransition for the sample and expert data in each environment\nis shown in Figure 4. This shows the evolution of rewards\nduring step-by-step learning when 32 episodes of expert data\nare given as in Chapter 5.3. As we designed, SQIL continues\nto give fixed values, while DSQIL shows variable rewards. In\nparticular, in the HalfCheetah environment, it is noticeable\nthat the reward for the sample data increases as the learning\nprogresses. It can be seen that the rewards are given to data\nthat is close to the expert data found in the sample data and\nused for learning. On the other hand, we observed a decrease\nin the reward for expert data. Although no effect of this was\nobserved in the experiment, it is possible that the accuracy\nof the data decreases when the number of learning epochs is\nincreased, or that the data behaves differently from the expert\ndata.\n6. Conclusion\nIn this paper, we propose Discriminator Soft Q Imitation\nLearning (DSQIL) as a data-efficient imitation learning\nmethod. In contrast to the conventional method SQIL, we\nshow that DSQIL can provide more detailed rewards for\nstate-action pairs by using a reward function instead of a con-\nstant reward. The method incorporates the idea of a GAN\ndiscriminator in the reward function and was evaluated in\nthree experiments with MuJoCo.\nThe experiments confirm that DSQIL performs as well\nas or better than conventional imitation learning. Especially\nin complex environments, DSQIL outperforms SQIL in both\ndata efficiency and learning efficiency. On the other hand,\nin certain environments, the rewards for both expert and\nsample data tended to converge to similar values as learning\nprogressed. This can be an advantage when seeking higher\nperformance from the expert data, but a disadvantage whenFURUYAMA et al.: EXTRINSICALY REWARDED SOFT Q IMITATION LEARNING WITH DISCRIMINATOR\n7\nseeking performance comparable to the expert data.\nBased on the above considerations, it is necessary to\nexamine the extent to which discriminator accuracy affects\nlearning. It is also worth continuing research in terms of\nverifying performance in more complex environments and\ndesigning reward functions that improve performance in sim-\nple environments.\nReferences\n[1] S.P. Boyd and L. Vandenberghe, Convex optimization, Cambridge\nuniversity press, 2004.\n[2] U. Sumanth, N.S. Punn, S.K. Sonbhadra, and S. Agarwal, “Enhanced\nbehavioral cloning-based self-driving car using transfer learning,”\nin Data Management, Analytics and Innovation: Proceedings of\nICDMAI 2021, Volume 2, pp.185–198, Springer, 2021.\n[3] D.A. Pomerleau, “Efficient training of artificial neural networks for\nautonomous navigation,” Neural computation, vol.3, no.1, pp.88–97,\n1991.\n[4] A.Y. Ng, S. Russell, et al. , “Algorithms for inverse reinforcement\nlearning.,” Icml, p.2, 2000.\n[5] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley,\nS. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,”\nAdvances in neural information processing systems, vol.27, 2014.\n[6] J. Ho and S. Ermon, “Generative adversarial imitation learning,”\nAdvances in neural information processing systems, vol.29, 2016.\n[7] J. Fu, K. Luo, and S. Levine, “Learning robust rewards\nwith adversarial inverse reinforcement learning,” arXiv preprint\narXiv:1710.11248, 2017.\n[8] S. Choi, J. Kim, and H. Yeo, “Trajgail: Generating urban ve-\nhicle trajectories using generative adversarial imitation learning,”\nTransportation Research Part C: Emerging Technologies, vol.128,\np.103091, 2021.\n[9] Q. Wu, L. Li, and Z. Yu, “Textgail: Generative adversarial imitation\nlearning for text generation,” Proceedings of the AAAI Conference\non Artificial Intelligence, pp.14067–14075, 2021.\n[10] S. Reddy, A.D. Dragan, and S. Levine, “Sqil: Imitation learn-\ning via reinforcement learning with sparse rewards,” arXiv preprint\narXiv:1905.11108, 2019.\n[11] E. Todorov, T. Erez, and Y. Tassa, “Mujoco: A physics engine for\nmodel-based control,” 2012 IEEE/RSJ international conference on\nintelligent robots and systems, pp.5026–5033, IEEE, 2012.\n[12] R.S. Sutton and A.G. Barto, Reinforcement learning: An introduc-\ntion, MIT press, 2018.\n[13] V. Konda and J. Tsitsiklis, “Actor-critic algorithms,” Advances in\nneural information processing systems, vol.12, 1999.\n[14] C.J. Watkins and P. Dayan, “Q-learning,” Machine learning, vol.8,\npp.279–292, 1992.\n[15] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and\nO. Klimov, “Proximal policy optimization algorithms,” arXiv\npreprint arXiv:1707.06347, 2017.\n[16] S. Kapturowski, G. Ostrovski, J. Quan, R. Munos, and W. Dabney,\n“Recurrent experience replay in distributed reinforcement learning,”\nInternational conference on learning representations, 2018.\n[17] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine, “Soft actor-critic:\nOff-policy maximum entropy deep reinforcement learning with a\nstochastic actor,” International conference on machine learning,\npp.1861–1870, PMLR, 2018.\n[18] T. Haarnoja, A. Zhou, K. Hartikainen, G. Tucker, S. Ha, J. Tan, V. Ku-\nmar, H. Zhu, A. Gupta, P. Abbeel, et al. , “Soft actor-critic algorithms\nand applications,” arXiv preprint arXiv:1812.05905, 2018.\n[19] T. Haarnoja, H. Tang, P. Abbeel, and S. Levine, “Reinforcement\nlearning with deep energy-based policies,” International conference\non machine learning, pp.1352–1361, PMLR, 2017.\n[20] S. Ross, G. Gordon, and D. Bagnell, “A reduction of imitation learn-ing and structured prediction to no-regret online learning,” Proceed-\nings of the fourteenth international conference on artificial intelli-\ngence and statistics, pp.627–635, JMLR Workshop and Conference\nProceedings, 2011.\n[21] B. Piot, M. Geist, and O. Pietquin, “Boosted and reward-regularized\nclassification for apprenticeship learning,” Proceedings of the 2014\ninternational conference on Autonomous agents and multi-agent sys-\ntems, pp.1249–1256, 2014.\n[22] P. Henderson, R. Islam, P. Bachman, J. Pineau, D. Precup, and\nD. Meger, “Deep reinforcement learning that matters,” Proceedings\nof the AAAI conference on artificial intelligence, 2018.\n[23] G. Papagiannis and Y. Li, “Imitation learning with sinkhorn dis-\ntances,” Joint European Conference on Machine Learning and\nKnowledge Discovery in Databases, pp.116–131, Springer, 2022.\n[24] D.P. Kingma and J. Ba, “Adam: A method for stochastic optimiza-\ntion,” arXiv preprint arXiv:1412.6980, 2014.\nRyoma Furuyama He received a B.S. de-\ngree from Kanazawa University in 2022. He\nis now a M.S. student studying reinforcement\nlearning.\nDaiki Kuyoshi He received a M.S. degree\nfrom Kanazawa University in 2022. He is inter-\nested in reinforcement learning\nSatoshi Yamane He received B.S.,M.S and\nPh.D. degrees from Kyoto University. Now he\nis a professor of Kanazawa University. He is\ninterested in formal verification of real-time and\ndistributed computing.8IEICE TRANS. ??, VOL.Exx–??, NO.xx XXXX 200x\n(a)Hopper-v3\n(b)Walker2d-v3\n(c)HalfCheetah-v3\nFig. 3: Comparison of the average score per epoch for each environment for 32 episodes of expert data.FURUYAMA et al.: EXTRINSICALY REWARDED SOFT Q IMITATION LEARNING WITH DISCRIMINATOR\n9\n(a)Expert data rewards in Hopper-v3\n (b)Sample data rewards in Hopper-v3\n(c)Expert data rewards in Walker2d-v3\n (d)Sample data rewards in Walker2d-v3\n(e)Expert data rewards in HalfCheetah-v3\n (f)Sample data rewards in HalfCheetah-v3\nFig. 4: Comparison of expert data and sample data rewards for each environment with 32 episodes of expert data." }, { "title": "2401.16776v1.Leveraging_Nested_MLMC_for_Sequential_Neural_Posterior_Estimation_with_Intractable_Likelihoods.pdf", "content": "Leveraging Nested MLMC for Sequential Neural Posterior\nEstimation with Intractable Likelihoods\nXiliang Yang1, Yifei Xiong2, Zhijian He1∗\n1School of Mathematics, South China University of Technology\n2School of Mathematical Sciences, University of Chinese Academy of Sciences\nAbstract\nSequential neural posterior estimation (SNPE) techniques have been recently proposed\nfor dealing with simulation-based models with intractable likelihoods. They are devoted to\nlearning the posterior from adaptively proposed simulations using neural network-based con-\nditional density estimators. As a SNPE technique, the automatic posterior transformation\n(APT) method proposed by Greenberg et al. (2019) performs notably and scales to high\ndimensional data. However, the APT method bears the computation of an expectation of\nthe logarithm of an intractable normalizing constant, i.e., a nested expectation. Although\natomic APT was proposed to solve this by discretizing the normalizing constant, it remains\nchallenging to analyze the convergence of learning. In this paper, we propose a nested APT\nmethod to estimate the involved nested expectation instead. This facilitates establishing the\nconvergence analysis. Since the nested estimators for the loss function and its gradient are\nbiased, we make use of unbiased multi-level Monte Carlo (MLMC) estimators for debiasing.\nTo further reduce the excessive variance of the unbiased estimators, this paper also devel-\nops some truncated MLMC estimators by taking account of the trade-off between the bias\nand the average cost. Numerical experiments for approximating complex posteriors with\nmultimodal in moderate dimensions are provided.\n1 Introduction\nSimulator-based models are widely used across various scientific disciplines, including neuro-\nscience [47], physics [7, 24], biology [10, 29, 39, 46], and inverse graphics [52]. These models\nserve as crucial tools for describing and comprehending investigated processes based on observed\ndata. However, when applying traditional Bayesian inference to simulator-based models, chal-\nlenges arise, such as the intractable likelihood function p(x|θ) and the computational expense\nassociated with running the simulator.\nTo address these challenges, a series of likelihood-free Bayesian computation (LFBC) meth-\nods have been developed. These methods include approximate Bayesian computation (ABC)\n[5, 45], synthetic likelihoods (SL) [50, 57], Bayes optimization [27], likelihood-free inference by\nratio estimation [55], and pseudo marginals methods [2, 3]. A comprehensive summary and\nreview of these methods can be found in [11], and they have all been benchmarked in [32, 41].\nPosterior density estimation approaches approximate the posterior of interest p(θ|xo) with\na family of density estimators qϕ(θ), where ϕis the parameters of the density estimators.\nOptimization-based approaches are widely used in these methods, and the Kullback-Leibler\n(KL) divergence between p(θ|x) and qϕ(θ), which measures the differences between two densi-\nties, is commonly chosen as the loss function. Variational Bayes (VB), as a computationally\noptimization-based effective method for approximating the posterior distribution of a Bayesian\nproblem, is widely used. In the likelihood-free context, Tran et al. [56] developed a new VB\n∗Corresponding author: hezhijian@scut.edu.cn\n1arXiv:2401.16776v1 [stat.CO] 30 Jan 2024method with an intractable likelihood, while He et al. [30] proposed an unbiased VB method\nbased on nested MLMC. However, these method tend to fail in cases where simulations are\nexpensive, as the nested estimation in these methods requires additional simulation procedures.\nOut of this reason, there has been a growing interest in employing neural networks to repre-\nsent probability density recently, particularly normalizing flows [38]. When the inference problem\nis focused solely on the observation xo, the data efficiency can be improved using the sequen-\ntial training schemes proposed in the sequential neural posterior estimation (SNPE) methods\n[25, 42, 48]. In this approach, model parameters are drawn from a proposal distribution that\nis more informative about xocompared to the prior distribution. However, SNPE requires a\nmodification of the loss function compared to neural posterior estimation (NPE) in order to\nensure that the neural networks approximates the true posterior p(θ|x). Many approaches have\nbeen developed to address this issue.\nAmong the SNPE methods, automatic posterior transformation (APT) requires the com-\nputation of the expectation of the logarithm of an intractable normalizing constant, which is\na nested expectation. They then propose to use the atomic APT to discretize the normalizing\nconstant. However, it remains challenging to utilize the existing analysis techniques [6] to per-\nform convergence analysis to the best of our knowledge. To address this limitation, we propose\nthe nested APT method, an method enable the convergence analysis without scarifying the\nperformance. We conduct a thorough analysis of the bias and variance of the loss estimators,\nmoreover, we provide a convergence result for this biased estimator in the case of stochastic\ngradient descent (SGD). To evaluate the effectiveness of the nested APT, we conduct a series of\ncomprehensive numerical experiments.\nSince nested APT is a biased estimator, we undertake a comprehensive exploration of a\nrange of unbiased methods based on multi-level Monte Carlo (MLMC) [17] to eliminate the\nbias. We establish several theorems that investigate the order of bias, variance, and the average\ncost of the estimators of the loss function and its gradient. Additionally, as we use stochastic\ngradient-based optimization methods to update the parameters of the neural network, it is well\nknown that the unbiased method suffers from excessive variance in the gradient. To address\nthis issue, we employ several variants of truncated MLMC methods to diminish the variance.\nFurthermore, we compute the order of the variance and average cost for the estimators for the\ngradient of the loss function. Finally, we utilize these results to provide the convergence analysis\nof truncated MLMC and nested APT.\nThe remainder of this paper is organized as follows. In Section 2, we propose the nested\nAPT method and provide insights into possible failures when directly using the nested APT.\nMoreover, we propose a strategy to address this failure. We also present the experimental\nresults and analyze them in this section. In Section 3, we introduce the basics of MLMC\nmethods, including their formulation and theoretical analysis of the random sequence utilized\nin MLMC methods. Equipped with these tools, we develop both unbiased MLMC methods and\ntruncated MLMC methods and analyze the order of variance for the losses and gradients, as\nwell as the average cost. In Section 4, we provide convergence analysis for nested APT and\ntruncated MLMC methods in the case of SGD. In Section 5, we conduct a series of numerical\nexperiments on benchmark tasks. The results of these experiments are presented and analyzed\nin this section. Finally, we conclude this paper with some remarks in Section 6.\n2 Nested APT\n2.1 Problem formulation\nLetp(θ) represent the prior distribution of the model parameter of interest. Given an ob-\nserved sample xo, our objective is to perform inference on the posterior distribution p(θ|xo)∝\np(θ)p(xo|θ). However, in many cases, the likelihood function p(x|θ) either lacks an explicit\n2expression or presents challenges in direct evaluation. Instead, it can be expressed using a\n‘simulator’ approach, wherein, given a fixed model parameter θ, we can sample xfrom p(x|θ).\nSince our objective is to approximate the posterior of interest p(θ|xo) using tractable density\nestimators, the KL divergence serves as a primary measure of the discrepancy between two\ndensities. The KL divergence is defined as\nDKL(p(θ)∥q(θ)) =Z\np(θ) logp(θ)\nq(θ)dθ,\nwhich is not less than 0 by using Jensen’s inequality and attains its minima when q(θ) agrees\nwith p(θ), making it suitable as a loss function. When the likelihood p(x|θ) is tractable, one can\ndirectly approximate p(θ|xo) by minimizing the KL divergence between the target distribution\nand the proposed estimator q(θ) within a certain family of distributions. In the likelihood-\nfree context, this can be viewed as a problem of conditional density estimation. Within this\nframework, a conditional density estimator qF(x,ϕ)(θ) based on a neural network [48, 25], is\nutilized to approximate p(θ|x) over the admissible set of tuning parameter ϕ∈Φ. To this end,\nwe focus on minimizing the following average KL divergence under the marginal distribution\np(x) =R\np(θ)p(x|θ)dθ\nEp(x)\u0002\nDKL\u0000\np(θ|x)∥qF(x,ϕ)(θ)\u0001\u0003\n=ZZ\np(x)p(θ|x)\u0000\nlogp(θ|x)−logqF(x,ϕ)(θ)\u0001\ndxdθ\n=−Ep(θ,x)\u0002\nlogqF(x,ϕ)(θ)\u0003\n+ZZ\np(θ, x) logp(θ|x)dxdθ\n:=L(ϕ) +ZZ\np(θ, x) logp(θ|x)dxdθ,\nwhere the term\nL(ϕ) :=−Ep(θ,x)\u0002\nlogqF(x,ϕ)(θ)\u0003\n, (1)\nis used as the loss function. However, since L(ϕ) is intractable, we use its empirical estimator\ninstead\nˆLN(ϕ) =−1\nNNX\ni=1logqF(xi,ϕ)(θi), (2)\nwhere the training data {(θi, xi)}N\ni=1is sampled from the joint probability density p(θ, x) =\np(θ)p(x|θ). After training, given the observation xo, the posterior p(θ|xo) can be approximated\nbyqF(xo,ϕ)(θ).\nSince our aim is to conduct conditional density estimation at xo, it is essential to utilize a\nproposal distribution ˜ p(θ) that provides more informative priors regarding xoin comparison to\nthe prior distribution p(θ). After initializing ˜ p(θ) asp(θ), we then want the approximation of\np(θ|xo) to serve as a good proposal in the following simulation. This conditional density estima-\ntion with adaptively chosen proposal is called sequential neural posterior estimation (SNPE).\nHowever, after replacing p(θ, x) with ˜ p(θ, x) = ˜p(θ)p(x|θ) in (1), it is observed that qF(x,ϕ)(θ)\napproximates the proposal posterior :\n˜p(θ|x) =p(θ|x)˜p(θ)p(x)\np(θ)˜p(x), (3)\nwhere ˜ p(x) =R\n˜p(θ)p(x|θ)dθ. Hence, we need to adjust loss function L(ϕ) to make qF(x,ϕ)(θ)\napproximate the true posterior p(θ|x).\nIn APT [25], the proposal distribution ˜ p(θ) is initialized as the prior distribution p(θ). Conse-\nquently, (1) can be directly used for the loss function. In the subsequent rounds, [25] proposed to\n3replace qF(x,ϕ)(θ), p(θ, x) with ˜ qF(x,ϕ)(θ), ˜p(θ, x) in (1) respectively. Explicitly, the loss function\nproposed in APT is\nE˜p(θ,x)\u0002\n−log ˜qF(x,ϕ)(θ)\u0003\n, (4)\nwhere\n˜qF(x,ϕ)(θ) =qF(x,ϕ)(θ)˜p(θ)\np(θ)1\nZ(x, ϕ), (5)\nZ(x, ϕ) =ZqF(x,ϕ)(θ′)\np(θ′)˜p(θ′)dθ′,\nZ(x, ϕ) here denotes the normalizing constant. Proposition 1 in [48] shows that if qF(x,ϕ)(θ) is\nexpressive enough that ˜ qF(x,ϕ∗)(θ) = ˜q(θ) for some parameter ϕ∗, then qF(x,ϕ∗)(θ) =q(θ|x).\nSince the integral Z(x, ϕ) is usually intractable in practice, APT proposes the use of ‘atomic’\nproposals, APT with such proposal is known as atomic APT. Specifically, they assume that\n˜p(θ) =UΘ, where UΘis a uniform distribution over a finite set Θ = {θ1, . . . , θ M}. The uniform\nsetting of the proposal distribution allows for the analytical computation of Z(x, ϕ). With this\nmethod, (3) and (5) can be reformulated as\n˜p(θ|x) =p(θ|x)/p(θ)P\nθ′∈Θp(θ′|x)/p(θ′),˜qx,ϕ(θ) =qF(x,ϕ)(θ)/p(θ)P\nθ′∈ΘqF(x,ϕ)(θ′)/p(θ′). (6)\nProposition 1 in [25] provides the consistency guarantees of atomic APT: given that each Θ is\nconstructed by sampling θ′from a distribution that covers the target p(θ|xo), atomic APT is\nable to recover the full posterior.\nHowever, to the best of our knowledge, the use of atomic proposals makes it challenging to\nanalyze its convergence behaviour. Therefore unable to explain the unexpected low performance\nin some tasks [13] with the existing convergence results [6], both in terms of insight and theory.\nAs an alternative approach to estimate Z(x, ϕ), the nested estimation, which enjoys a compre-\nhensive theoretical framework and comparable performance, is studied in the next section.\n2.2 Nested APT method\nFor ease of presentation, we denote gϕ(x, θ) :=qF(x,ϕ)(θ)/p(θ). We reformulate (4) as\nE˜p(θ,x)\u0002\n−log ˜qF(x,ϕ)(θ)\u0003\n=−E˜p(θ,x)\u0014\nlogqF(x,ϕ)(θ) + log˜p(θ)\np(θ)\u0015\n+E˜p(x)[logZ(x, ϕ)]\n=−E˜p(θ,x)\u0014\nlogqF(x,ϕ)(θ)\np(θ)\u0015\n+E˜p(x)[logZ(x, ϕ)]−E˜p(θ)[log ˜p(θ)]\n=−E˜p(θ,x)[loggϕ(x, θ)] +E˜p(x)\u0002\nlogE˜p(θ′)\u0002\ngϕ(x, θ′)\u0003\u0003\n−E˜p(θ)[log ˜p(θ)]\n:=˜L(ϕ)−E˜p(θ)[log ˜p(θ)],\nwhere\n˜L(ϕ) =−E˜p(θ,x)[loggϕ(x, θ)] +E˜p(x)\u0002\nlogE˜p(θ′)\u0002\ngϕ(x, θ′)\u0003\u0003\n, (7)\nis selected as the loss function, −E˜p(θ)[log ˜p(θ)] in the last column is dropped for it is independent\nofϕ. We employ stochastic gradient methods to optimize the loss function in this paper, therefore\n∇ϕ˜L(ϕ) is of interest. By allowing the interchange of expectation and gradient operator, the\ngradient of (7) is given by\n∇ϕ˜L(ϕ) =−E˜p(θ,x)[∇ϕloggϕ(x, θ)] +E˜p(x)\u0002\n∇ϕlogE˜p(θ′)\u0002\ngϕ(x, θ′)\u0003\u0003\n. (8)\n4Notice that (8) holds a similar formulation of the gradient use in the optimization procedure of\nBayesian experimental design [9, 16, 34, 37]. Given any query point ( θ, x)∼˜p(θ, x), we denote\nthe corresponding query of the loss and gradient as\nψϕ=−loggϕ(x, θ) + log Z(x, ϕ), (9)\nρϕ=−∇ϕloggϕ(x, θ) +∇ϕlogZ(x, ϕ), (10)\nso that E˜p(θ,x)[ψϕ] = ˜L(ϕ) and E˜p(θ,x)[ρϕ] =∇��˜L(ϕ). With the abuse of notation aiming at\neasing representation, unless otherwise specified in the following paper, L(ϕ) and ∇ϕL(ϕ) will\nbe used to represent (7) and (8) respectively.\nGiven the intractability of the normalizing constant Z(x, ϕ) =E˜p(θ′)[gϕ(x, θ′)], it remains an\nobstacle to derive an estimator for the loss function. A simple choice is leveraging its empirical\nestimator based on Msamples, which is given by\nˆZM(x, ϕ) =1\nMMX\nj=1gϕ(x, θ′\nj), (11)\nwhere θ′\n1,···, θ′\nM∼˜p(θ′) independently, Min the subscript denotes the number of the samples\nused for the estimation of Z(x, ϕ). We thus arrive at queries for nested estimators of (9) and\n(10) respectively\nψϕ,M=−loggϕ(x, θ) + log ˆZM(x, ϕ) = log1\nMMX\nj=1gϕ(x, θ′\nj)\ngϕ(x, θ), (12)\nρϕ,M=−∇ϕloggϕ(x, θ) +∇ϕlogˆZM(x, ϕ). (13)\nNested estimators for the loss function (7) and its gradient (8) are then given by the mean of N\niid copies of their queries\nˆLNe(ϕ) =1\nNNX\ni=1ψ(i)\nϕ,M=1\nNNX\ni=1log1\nMMX\nj=1gϕ(xi, θ′\nij)\ngϕ(xi, θi), (14)\n∇ϕˆLNe(ϕ) =1\nNNX\ni=1ρ(i)\nϕ,M, (15)\nwhere θ′\nijiid∼˜p(θ′) constitute the inner sample with a size of Mand ( θi, xi)iid∼˜p(θ, x) constitute\nthe outer sample with a size of N,ψ(i)\nϕ,M, ρ(i)\nϕ,Mare iid copies of ψϕ,Mandρϕ,Mrespectively. Due\nto the nonlinearity of the logarithm, (14) is a biased estimator. Using similar arguments in [53],\nit is not difficult to evaluate that the order of variance of the estimator of nested APT ˆLNe(ϕ)\nis ofO(1/N), and O(1/M) for its bias.\n2.3 Performance evaluations for comparing nested APT with atomic APT\nIn this section, we conduct a series of experiments to demonstrate the performance of nested\nAPT compared to atomic APT. For detailed experimental setting, we refer the readers to Section\n5. The results are reported in Figure 1, from which we observe that the proposed nested APT\nachieves comparable performance with atomic APT. A noteworthy advantage of our proposed\nnested APT method is that it enables the application of a series of existing result for an optimizer\nwith a biased gradient.\nDue to its biased nature, the nested APT method may not directly benefit from the existing\nvariance reduction techniques, as the effects of variance and bias on the optimal gap are ‘tangled’.\nTo address this issue, we propose to make use of the existing multi-level Monte Carlo (MLMC)\ntechniques for the improvement of the nested estimator.\n5A\nB\n5.0\n 4.5\n1\n5.0\n 4.5\n1\n1.0\n 0.5\n 0.5\n 0.0\n 5.0\n 4.5\n 5.0\n 4.5\n1\n5.0\n 4.5\n 1.0\n 0.5\n2\n1.0\n 0.5\n2\n0.5\n 0.0\n 5.0\n 4.5\n 5.0\n 4.5\n 1.0\n 0.5\n2\n5.0\n 4.5\n 1.0\n 0.5\n 0.5\n 0.0\n3\n0.5\n 0.0\n3\n5.0\n 4.5\n 5.0\n 4.5\n 1.0\n 0.5\n 0.5\n 0.0\n3\n5.0\n 4.5\n 1.0\n 0.5\n 0.5\n 0.0 5.0\n 4.5\n4\n5.0\n 4.5\n4\n5.0\n 4.5\n 1.0\n 0.5\n 0.5\n 0.0 5.0\n 4.5\n4\nC\n2.9 3.0 3.1\nA2.9 3.0 3.1\nA\n0.25\n0.00 0.25\n 2 3\n0.5\n 0.0 2.9 3.0 3.1\nA\n2.9 3.0 3.1 0.25\n0.00 0.25\nlogB0.25\n0.00 0.25\nlogB\n2 3\n0.5\n 0.0\n 2.9 3.0 3.1 0.25\n0.00 0.25\nlogB\n2.9 3.0 3.1\n 0.25\n0.00 0.25 2 3\ng2 3\ng\n0.5\n 0.0\n 2.9 3.0 3.1\n 0.25\n0.00 0.25 2 3\ng\n2.9 3.0 3.1\n 0.25\n0.00 0.25\n 2 30.5\n 0.0\nlog(k+1/2)0.5\n 0.0\nlog(k+1/2)\n2.9 3.0 3.1\n 0.25\n0.00 0.25\n 2 30.5\n 0.0\nlog(k+1/2)\nFigure 1: Density plot for nested APT and atomic APT. A. Two-moon model, from left\nto right: available ground truth, atomic APT with inner samples M= 100, nested APT with\ninner samples M= 100. B.Lotka-Volterra model, from left to right: ground truth simulated\nwith SMC-ABC [4], atomic APT with M= 100, nested APT with M= 100. C.M/G/1 queue\nmodel, the setting are the same with Lotka-Volterra.\n3 Unbiased multilevel nested APT\n3.1 Basic idea of MLMC\nNested simulation combined with the MLMC method has been studied for other applications\n[18, 19, 22, 30]. In this paper, MLMC is used to derive an unbiased estimator for APT. Increasing\n6Mforρϕ,Meventually recovers the desired ρϕ, this allows the exchange of expectation and\nlimitation that\nlim\nM→∞E[ρϕ,M] =E[ρϕ] =∇ϕL(ϕ). (16)\nThen we take a monotonically increasing sequence for inner sample size {Mℓ}∞\nℓ=1with Mℓ=\n2ℓM0= 2Mℓ−1, then (16) can be reformulated as\n∇ϕL(ϕ) =E[ρϕ,M 0] +∞X\nℓ=1E\u0002\nρϕ,M ℓ−ρϕ,M ℓ−1\u0003\n. (17)\nSet{∆ρϕ,ℓ}∞\nℓ=0to be a sequence of random variables satisfying\nE[∆ρϕ,0] =E[ρϕ,M 0],E[∆ρϕ,ℓ] =E\u0002\nρϕ,M ℓ−ρϕ,M ℓ−1\u0003\n(ℓ≥1),\nthen (17) can be reformulated in terms of the infinite sum of the expected values of {∆ρϕ,ℓ}∞\nℓ=0,\ni.e.,∇ϕL(ϕ) =P∞\nℓ=0E[∆ρϕ,ℓ]. In the case of the loss function, we also introduce the similar\nsequence {∆ψϕ,ℓ}∞\nℓ=0satisfying\nE[∆ψϕ,0] =E[ψϕ,M 0],E[∆ψϕ,ℓ] =E\u0002\nψϕ,M ℓ−ψϕ,M ℓ−1\u0003\n(ℓ≥1), (18)\nsimilarly we also have L(ϕ) =P∞\nℓ=0E[∆ψϕ,ℓ].\n3.1.1 Antithetic construction\nMotivated by the prevailing literature on MLMC [20, 8, 19, 22], this paper presents the construc-\ntion of the sequence {∆ρϕ,ℓ}∞\nℓ=0with antithetic coupling, leading to a faster rate of convergence\nfor smooth functions. The fundamental concept of antithetic coupling involves the selection of\ntwo non-overlapping subsets, each with a size of Mℓ−1from the Mℓinner samples {θ′\nj}Mℓ\nj=1uti-\nlized for computing ρϕ,M ℓgiven in (13). This results in two independent realizations of ρϕ,M ℓ−1,\nidentified as ρ(a)\nϕ,M ℓ−1andρ(b)\nϕ,M ℓ−1. Specifically,\nρ(a)\nϕ,M ℓ−1=−∇ϕgϕ(x, θ)\ngϕ(x, θ)+∇g(a)\nϕ,M ℓ−1(x)\ng(a)\nϕ,M ℓ−1(x),\nwhere\ng(a)\nϕ,M ℓ−1(x) =1\nMℓ−1Mℓ−1X\nj=1gϕ(x, θ′\nj),∇g(a)\nϕ,M ℓ−1(x) =1\nMℓ−1Mℓ−1X\nj=1∇ϕgϕ(x, θ′\nj).\nThe notations ρ(b)\nϕ,M ℓ−1,∇g(b)\nϕ,M ℓ−1(x), and g(b)\nϕ,M ℓ−1(x) are defined in the similar way by using θ′\nj,\nj=Mℓ−1+ 1, . . . , M ℓinstead. Then the antithetic construction of {∆ρϕ,ℓ}∞\nℓ=0is then given as\n∆ρϕ,0=ρϕ,M 0,∆ρϕ,ℓ=ρϕ,M ℓ−1\n2\u0010\nρ(a)\nϕ,M ℓ−1+ρ(b)\nϕ,M ℓ−1\u0011\n(ℓ≥1). (19)\nThe antithetic construction of {∆ψϕ,ℓ}∞\nℓ=0can be similarly given.\n3.1.2 Variance of multilevel Monte Carlo estimator\nIn this section, we will give a series of detailed evaluation of the variance of ∆ ρϕ,ℓ, ρϕ,M ℓand\n∆ψϕ,ℓwhich will serve as important tools when we evaluate the variance of some other MLMC\nmethods in the coming sections. Before we present the main results, we firstly introduce the\nfollowing key lemma which is stated as Lemma 1 in [19].\n7Lemma 3.1. LetXbe a real-valued random variable with mean zero, and let XNbe an average\nofNi.i.d samples of X. If for E[|X|u]<∞foru≥2, there exists a constant Cu>0depending\nonly on usuch that\nE\u0002\f\fXN\f\fu\u0003\n≤CuE[|X|u]\nNu/2,P\u0002\f\fXN\f\f> c\u0003\n≤CuE[|X|u]\ncuNu/2,\nfor any c >0.\nThe following theorem studies the expectation and variance of ∆ ψϕ,ℓasℓ→ ∞ . In this\npaper, the notation ≲is used to signifies the omission of a constant that is irrelevant to the\nasymptotic behavior.\nTheorem 3.2. If there exist s, z > 2with (s−2)(z−2)≥4such that for any ϕ,\nEx,θ′\u0014\f\f\f\fgϕ(x, θ′)\nZ(x, ϕ)\f\f\f\fs\u0015\n<∞and Ex,θ′\u0014\f\f\f\floggϕ(x, θ′)\nZ(x, ϕ)\f\f\f\fz\u0015\n<∞,\nwe have\nE\u0002\n∥∆ψϕ,ℓ∥2\n2\u0003\n≲M−r1\nℓ,\nwhere Mℓ=M02ℓ,r1= min( s(z−2)/2z,2).\nProof. This proof follows an argument similar to Theorem 2 in [22].\nWhen stochastic gradient descent method is applied, it is also desirable to examine the\nexpectation and variances of the ℓ2norm of the gradient. To this end, we establish the following\ntheorem based on the work of [23]. The proof of this theorem is detailed in Appendix A.1.\nTheorem 3.3. Assume that there exists s >2such that for any ϕ\nsup\nx,ϕ,θ∥∇ϕloggϕ(x, θ)∥∞<∞,Ex,θ\u0014\f\f\f\fgϕ(x, θ)\nZ(x, ϕ)\f\f\f\fs\u0015\n<∞,\nwe then have\nE\u0002\n∥∆ρϕ,ℓ∥2\u0003\n≲M−1\nℓ,E\u0002\n∥∆ρϕ,ℓ∥2\n2\u0003\n≲M−r2\nℓ,\nwhere Mℓ=M02ℓ,r2= min( s,4)/2∈(1,2].\nFor the convergence of ρϕ,M ℓ, we establish the following theorem.\nTheorem 3.4. Under the setting of Theorem 3.3, we have Var [ρϕ,M ℓ]≲M−1\nℓ.\nA detailed version and proof of this theorem is provided in Appendix A.2. This theorem\ncan be similarly extended to the case where arbitrarily number of inner samples are used for\nthe estimation of the gradient. As a result, it indicates the convergence rate of the gradient\nestimator is the same as the loss function.\n3.2 Unbiased MLMC for APT\n3.2.1 RU-MLMC\nRecall that both loss function and its gradient can be represented as the summation of the\nexpected value of the random variable sequence, and this topic has been well studied. We\n8introduce a non-negative integer-valued random variable Lthat independent of the two random\nvariable sequences, with its probability mass function P(L=ℓ) =wℓ. We then have\nE\u0002\nωL−1∆ρϕ,L\u0003\n=∞X\nℓ=0E\u0002\nωℓ−1∆ρϕ,ℓ\u0003\nP(L=ℓ) =∇ϕL(ϕ).\nThe equivalence between ∇ϕL(ϕ) and the expectation of the query\nVRU=ωL−1∆ρϕ,L (20)\nleads to an unbiased Monte Carlo estimator of the gradient ∇ϕL(ϕ). We similarly define URU=\nωL−1∆ψϕ,Lfor the query of the estimation of the loss function L(ϕ). This method is known as\nrandomized unbiased multilevel Monte Carlo (RU-MLMC) method [51]. In this study, we take L\nas a geometric distribution Ge( p) with ωℓ= (1−p)ℓpandp= 1−2−α. To ensure a finite variance\nand finite expected computational cost for VRU, URU, it is required that α∈(1,min ( r1, r2)). This\nis further examined with the following theorem.\nTheorem 3.5. Under the setting of Theorems 3.2 and 3.3, let L∼Ge(p)withp= 1−2−αand\n1< α < min ( r1, r2), where r1andr2are from Theorems 3.2 and 3.3 respectively. Then we have\nVar [URU]≤A\n(1−2α−r1)(1−2−α),\nVar [VRU]≤B\n(1−2α−r2)(1−2−α),\nCost RU∝M02α−1\n2α−2,\nwhere the constants A, B are independent of αandr1.\nProof. The proof of this theorem follows in line of Theorem 3.6.\nThis theorem indicates that a large αleads to less expected total computational burden, but\nlarger variance for the estimators of loss and its gradient.\n3.2.2 GRR-MLMC\nThe Russian roulette (RR) estimator [44] is also employed to estimate the sum of an infinite\nseries, wherein the evaluation of any term in the series only demands a finite amount of compu-\ntation. This estimator relies on randomized truncation and assigns a higher weight to each term\nto accommodate the possibility of not computing them. The query of the gradient of RR-MLMC\nestimator is given by\nVRR:=LX\nj=0∆ρϕ,j\npj, (21)\nwhere pj=P(L≥j) =P∞\nℓ=jwℓforj≥0. If all pj>0, then VRRis unbiased since\nE[VRR] =∞X\nℓ=0P(L=ℓ)ℓX\nj=0E[∆ρϕ,j]\npj=∞X\nj=0E[∆ρϕ,j]\npj∞X\nℓ≥jP(L=ℓ)\n=∞X\nj=0E[∆ρϕ,j] =∇ϕL(ϕ).\nMotivated by [43], in order to trade lower variance with higher cost based on the conven-\ntional RR-MLMC, one way is to ensure that the first mterms of the infinite series are always\n9computed, where this mis called the base level . In this case, the random index Lis set to have\na lower bound m, i.e., P(L≥m) = 0, implying pj= 1 for all j≤m. We call this modified\nestimator as generalized Russian roulette (GRR) estimator. When m= 0, this degenerates to\nthe conventional RR-MLMC. The associated quaery for GRR-MLMC estimator of the gradient\nis\nVGRR:=ρϕ,M m+LX\nj=m+1∆ρϕ,j\npj. (22)\nIn this case, the probability mass function of Lis chosen as P(L=m) = 1 −P\nℓ>mwℓand\nP(L=ℓ) =wℓforℓ > m , where wℓ= (1−p)ℓpandp= 1−2−α. We denote this distribution\nby Ge( p, m). This implies that VGRRis also unbiased since all pj>0,\nSimilarly, we define the query for GRR estimator of the loss function:\nUGRR:=ψϕ,M m+LX\nj=m+1∆ψϕ,j\npj. (23)\nUGRRand the following theorem is established aiming at evaluating their order.\nTheorem 3.6. Under the setting of Theorems 3.2 and 3.3, let L∼Ge(p, m)withp= 1−2−α\nand1< α < min ( r1, r2), where r1andr2are from Theorems 3.2 and 3.3, and the base level\nm≥1, we have\nVar [UGRR]≤A \n1\n2m+2(α−r1)(m+1)\n1−2α−r1!\n,\nVar [VGRR]≤B \n1\n2m+2(α−r2)(m+1)\n1−2α−r2!\n,\nCost GRR∝M02m+M02(1−α)(m+1)\n1−2(1−α),\nwhere the constants A, B are independent of α, r 1, r2, m.\nThe proof of this theorem is detailed in Appendix A.3.\n3.3 Truncated MLMC for APT\nWhen treating ∇ϕL(ϕ) as the summation of the infinite random variable sequence {∆ρϕ,ℓ}∞\nℓ=0,\nwe notice that ∆ ρϕ,0=ρϕ,0is just the nested APT estimator with a bias of order O(1/M0).\nFor ∆ ρϕ,ℓwith ℓ≥1, it actually contributes to the reduction of bias while simultaneously\nincreasing the variance and this additional variance grows with ℓ. An empirical demonstration\nof the additional variance for ∆ ρϕ,ℓis shown in Figure 2.\nWe observe that the truncated RU-MLMC method enjoys a more stable loss compared to the\noriginal one, indicating that truncation helps reducing variance. Therefore, in order to mitigate\nthe variance of the RU-MLMC, we propose to truncate the distribution of Lby setting the\nlargest value of Lasm. The truncation leads to a bias of order O(1/Mm). Following [33], we\nextend the truncated idea to other unbiased MLMC methods.\n3.3.1 TGRR-MLMC\nTGRR-MLMC stands for the truncated version of GRR-MLMC. Let m≥mbe a truncated\nlevel, and let Lbe a nonnegative integer-value random index taking values in {m, . . . , m}. The\n100 50 100 150 200 250 300\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni0000004c/uni00000057/uni00000048/uni00000055/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000056510152025/uni0000002f/uni00000052/uni00000056/uni00000056\n0 50 100 150 200 250\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni0000004c/uni00000057/uni00000048/uni00000055/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000056training loss\nvalidation lossFigure 2: Left: Crude RU-MLMC after the third round. Right : The truncated RU-MLMC\n(m= 4) after the third round.\nquery for the estimation of the gradient of TGRR-MLMC is then defined as\nVTGRR =ρϕ,M m+LX\nj=m+1∆ρϕ,j\npj, (24)\nwhere pj=P(L≥j)>0 for m≤j≤m. This estimator is biased:\nE[VTGRR ] =E[ρϕ,M m] +mX\nℓ=m+1P(˜L=ℓ)ℓX\nj=m+1E[∆ρϕ,j]\npj\n=E[ρϕ,M m] +mX\nj=m+1E[∆ρϕ,j]\npjmX\nℓ≥jP(˜L=ℓ)\n=E[ρϕ,M m] +mX\nj=m+1E[∆ρϕ,j] =E[ρϕ,M m].\nWe should note that if m=m=m, TGRR-MLMC degenerates to nested APT with the inner\nsample size Mm=M02m. For any truncated method with truncated level m, they all share the\nbias of order O(Mm) based on (45).\nFor TGRR-MLMC, the probability mass function of the random index Lis chosen as\nP(L=ℓ) =\n\n1−Pm\nℓ=m+1wℓ\n1−(1−p)m+1, ℓ=m\nwℓ\n1−(1−p)m+1, m + 1≤ℓ≤m\n0, otherwise ,(25)\nwhere wℓ= (1−p)ℓpandp= 1−2−α. We denote this distribution by Ge( p, m,m). The whole\nprocedure of TGRR-MLMC method is summarized in Algorithm 1.\n4 Convergence results of SGD\nIn this section, we analyze the convergence of the methods proposed in this paper with some\nexisting techniques. For simplicity, we only study the widely used SGD with constant step size\nγin this paper. Our result should extend to boarder cases. The parameter update procedure of\nSGD is presented as\nϕt+1=ϕt−γt∇ϕˆL(ϕt),\n11Algorithm 1: Truncated Generalized Russian Roulett (TGRR)\nInput: Prior p(θ), implicit simulator model p(x|θ), neural network qF(x,ϕ)(θ),\n1< α < min(r1, r2),M0, Mℓ=M02ℓ, truncated level m, base level m, total\nround R, optimizer g(·,·), batch size B, outer sample size N,t= 0, dataset\nDoutfor outer samples, dataset Din(x) for the inner samples of x, ˜p1(θ) =p(θ)\n1forkin{1,2,···, R}do\n2 Generate {(θi, xi)}N\ni=1from ˜ pk(θ)p(x|θ) and level {ℓi}N\ni=1from Ge( p, m,m)\n3 Update dataset Dout← {(θi, xi, ℓi)}N\ni=1∪ Dout\n4 ifk= 1then\n5 repeat\n6 Generate {(θi, xi)}B\ni=1∼ D out\n7 Update parameter ϕt+1←g\u0010\nϕt,−1\nNPN\ni=1∇ϕlogqF(xi,ϕt)(θi)\u0011\n8 t←t+ 1\n9 until ϕtconverged\n10 end\n11 ifk≥2then\n12 repeat\n13 Generate {(θi′, xi′)}B\ni=1∼ D out\n14 fori′in{1,2,···, B}do\n15 Generate {θ′\nj}Mm−1\nj=1 from ˜ pk(θ′)\n16 Din(xi′)← {θ′\nj}Mm−1\nj=1∪ {θi′}and compute ρ(i′)\nϕt,MmwithDin(xi′)\n17 forℓ′in{{m+ 1, m+ 2,···, ℓi}}do\n18 D(a)\nin(xi′)← {θ′\nj}Mℓ′−1−1\nj=1 ∪ {θi′},D(b)\nin(xi′)← {θ′\nj}Mℓ′−1\nj=Mℓ′−1∪ {θi′}\n19 Din(xi′)← D(a)\nin(xi′)∪ D(b)\nin(xi′)\n20 Compute ∆ ρ(i′)\nϕt,ℓ′withDin(xi′) using (19)\n21 end\n22 Compute V(i′)\nTGRRusing (24)\n23 end\n24 Update parameter ϕt+1←g\u0010\nϕt,−1\nBPB\ni′=1V(i′)\nTGRR\u0011\n25 t←t+ 1\n26 until ϕtconverged\n27 end\n28 Update proposal ˜ pk+1(θ)←qF(xo,ϕt)(θ)\n29end\n12leveraging the information of an estimator of the gradient of the loss function at the current\nstate ϕt. The gradient estimator for TGRR-MLMC is\n∇ϕˆLTGRR(ϕ) :=1\nNNX\ni=1V(i)\nTGRR,\nwhere V(i)\nTGRRare iid copies of VTGRR . The gradient estimators for other MLMC methods follows\na similar form. The gradient estimator for nested APT is stated previously in (15). Since some\nof them are biased estimators for ∇ϕL(ϕ), we take the following decomposition\n∇ϕˆL(ϕ) =∇ϕL(ϕ) +b(ϕ) +η(ϕ), (26)\nwhere b(ϕ) and η(ϕ) denote the bias and the noise of gradient estimator ∇ϕˆL(ϕ) respectively.\nAssumption 4.1. There exist constants Ub<∞andUη<∞such that\nsup\nϕ∈Φ∥b(ϕ)∥2\n2= sup\nϕ∈Φ\r\r\rEh\n∇ϕˆL(ϕ)i\n− ∇ ϕL(ϕ)\r\r\r2\n2≤Ub,\nsup\nϕ∈ΦEh\n∥η(ϕ)∥2\n2i\n= sup\nϕ∈ΦE\u0014\r\r\r∇ϕˆL(ϕ)−Eh\n∇ϕˆL(ϕ)i\r\r\r2\n2\u0015\n≤Uη.\nFollowing the similar procedure to derive (45), we find that the order of Ubfor nested APT\nisO(1/M), while O(1/(M0)m) for TGRR. Intuitively speaking, when it is set that Mm\n0=M\nandm0, which\nis defined as\nGt=E[L(ϕt)]− L(ϕ∗), (27)\nwhere ϕ∗= arg min\nϕ∈ΦL(ϕ). Moving forward, we introduce the following two commonly used\nassumptions when analyzing the convergence of SGD, as outlined in [6].\nAssumption 4.2. The objective function L(ϕ)is differentiable and there exists a constant K\nsuch that for every ϕ, ϕ′\nL(ϕ)≤ L(ϕ′) +∇ϕL(ϕ)T(ϕ−ϕ′) +K\n2\r\rϕ−ϕ′\r\r2\n2.\nAssumption 4.3. The objective function L(ϕ)is differentiable and there exists µ >0such that\n∥∇L(ϕ)∥2\n2≥2µ(L(ϕ)− L(ϕ∗)).\nIn situations where a fixed number of iterations are exclusively employed for parameter\nupdates, the evaluation of the impact of variance and bias on the upper bound of the optimal\ngap is undertaken through the application of the theorem presented below, following [1].\nTheorem 4.4. Under Assumptions 4.2 and 4.3 and the constant step size γ≤min{1/K,1/µ},\nfor a fixed Tsteps of SGD, the upper bound of the optimal gap (27) is given as\nGT≤(1−γµ)TG0+1\n2µ(Ub+Uη). (28)\n13Since G0depends solely on the initialization of ϕ, our focus lies on the last two terms of\n(28), where UbandUηhave the same impact on the optimal gap GT. This implies that unbiased\nmethods with large variance may ultimately achieve the same result as those biased ones at the\nsame iteration step T. Hence in cases where variance dominates bias, methods with smaller\nvariance are preferable.\nOn the other hand, when a sufficiently large iteration step Tis given and proper learning\nrateγis chosen, the upper bound of the optimal gap can be smaller than any given ϵ >0 except\nfor the parts of bias, which is examined in the following corollary.\nCorollary 4.5. Suppose that Assumptions 4.2 and 4.3 are satisfied and γ=\nmin{1/K,1/µ, ϵµ/ (Kη)}, T= max\b\n(K/µ) ln(2 G0/ϵ), Kη/ (ϵµ2) ln(2 G0ϵ)\t\n. Then the optimal\ngap satisfies GT≤ϵ+Ub/(2µ)for any ϵ >0.\nProof. It follows directly from (28) by scaling γandT.\nIn this case, unbiased methods are always favorable, as they are able to converge to an\narbitrarily small neighborhood of 0, while biased methods can only converge to a neighborhood\nof the Ub. Moreover, for biased method with Ub≫Uη, the optimal gap for t1≫t2tends to\nbeGt1≈Gt2, indicating that increasing the iteration step can be futile in attempting to reduce\nthe optimal gap.\n5 Numerical experiments\nIn this section, numerical experiments are conducted to compare the efficiency of two unbi-\nased MLMC methods and a biased MLMC method for three models. To handle the curse of\ndimensionality, we propose to use low-dimensional summary statistics instead of the full data.\nHowever, this introduces additional bias, as detailed in [15]. In this paper, we do not look into\nthe effects of using summary statistics for SNPE methods. We refer to Fearnhead and Prangle\n[15] for a semi-automatic method of constructing summary statistics in the context of ABC.\nA toy example. Two-moon model was studied in Greenberg et al. [25]. For a given\nparameter θ∈R2, the Two-moon simulator generates observations x∈Rvia\na∼U(−π\n2,π\n2), r 2∼ N(0.1,0.012),\np= (r2cos(a) + 0.25, r2sin(a)), x =p+\u0012\n−|θ1+θ2|√\n2,−θ1+θ2√\n2\u0013\n.\nThe intermediate variables pfollow a single crescent-shaped distribution, which is then shifted\nand rotated around the origin based on the parameter values of θ. The absolute value |θ1+θ2|\ncontributes to the emergence of a second crescent in the posterior distribution. We choose a\nuniform prior over the square [ −1,1]2to perform the inference.\nLotka-Volterra model. This model describes the continuous time evolution of a population\nof predators interacting with a population of prey using a stochastic Markov jump process. The\nmodel describe that, the birth of a predator at a rate exp( θ1)XY, resulting in an increase of\nXby one; The death of a predator at a rate proportional to exp( θ2)X, leading to a decrease\nofXby one; The birth of a prey at a rate proportional to exp( θ3)Y, resulting in an increase\nofYby one; The consumption of a prey by a predator at a rate proportional to exp( θ4)XY,\nleading to a decrease of Yby one. Following the experimental details outlined in [49], we\ninitialize the predator and prey populations as X= 50 and Y= 100, respectively. We conduct\nsimulations of the Lotka-Volterra model using the Gillespie algorithm [21] over a duration of 30\ntime units. We recorded the populations at intervals of 0.2 time units, resulting in time series\ndata sets, each consisting of 151 values. The resulting summary statistics S(x) are represented\nas a 9-dimensional vector, which includes the following time series features: the logarithm of the\n14mean of each time series, the logarithm of the variance of each time series, the auto-correlation\ncoefficient of each time series at lags of 0.2 and 0.4 time units, and the cross-correlation coefficient\nbetween the two time series. In our experiments, the prior distribution of the parameters is set\ntoU(−5,2)4, and we generate the ground truth posterior with SMC-ABC [4], which is more\ncostly than methods in this paper.\nM/G/1 queue model. The M/G/1 queue model [54] describes a single server’s processing\nof a queue of continuously arriving jobs. Define Ias the total number of jobs that needs to be\nprocessed, and denote by sithe processing time required for job i. Let vibe the job’s arrival\ntime in the queue, and dibe the job’s departure time from the queue. They satisfy the following\nconditions\nsi∼ U(θ1, θ1+θ2), v i−vi−1∼Exp(θ3), d i−di−1=si+ max(0 , vi−di−1).\nIn our experiments, we set I= 50 jobs and the summary statistics S(x) has been selected as\nthe logarithm of 0th, 25th, 50th, 75th and 100th percentiles of the set of inter-departure times.\nThe prior distribution of the parameters is\nθ1∼ U(0,10), θ 2∼ U(0,10), θ 3∼ U(0,1/3),\nand our experiments choose ground truth parameters as θ∗= (1,4,0.2).\nOur numerical experiments are performed on a computer equipped with a single GeForce\nRTX 2080s GPU and an i9-9900K CPU. The training and inference processes of the model are\nprimarily implemented using the Pytorch package in Python .\nIn the training process, we simulate N= 1000 samples in each round, and with R= 20\nrounds in total. In each round, we randomly pick 5% of the newly generated samples θand their\ncorresponding xvalues as validation data. We follow the early stop criterion proposed by [49],\nwhich terminates the training if the loss value on the validation data does not decrease after 20\nepochs in a single round. For the optimizer, we use Adam [36] with a batch size of 100, a learning\nrate of 1 ×10−4, and a weight decay of 1 ×10−4.\nIn this paper, we employ neural spline flows (NSFs) [14] as the conditional density estimator,\nwhich consists of 8 layers. Each layer is constructed using two residual blocks with 50 units\nand ReLU activation function. With 10 bins in each monotonic piecewise rational-quadratic\ntransform, and the tail bound is set to 20.\nWe compare the performance of RU-MLMC, GRR-MLMC, and TGRR-MLMC on the Two-\nmoon, Lotka-Volterra, and M/G/1 models. To assess the similarity between the approximate\nposterior distribution qF(xo,ϕ)(θ) and the true posterior distribution p(θ|xo) given the observed\ndata, we employ maximum mean discrepancy (MMD) [25, 26, 31, 49] and classifier two-sample\ntests (C2ST) [12, 28, 40] as discriminant criteria. Additionally, we use log median distance\n(LMD) to measure the distance between xoandxdrawn from p(x|θ), where θis sampled\nfrom qF(xo,ϕ)(θ). In cases where the true posterior distribution p(θ|xo) is intractable even with\nknowledge of the sample generation process, we resort to the negative log probability (NLOG)\n[14, 25, 31, 48, 49] of the true parameters θ∗; in this scenario, the observation xois sampled from\np(x|θ∗). It is important to note that lower values for all the mentioned indicators are favorable.\nFor the setting of MLMC methods, we have chosen M0= 8 and m= 4 for TGRR-MLMC.\nThe base level for GRR and TGRR has been set to m= 2.\nWe now discuss the choice of the hyperparameter αfor the geometric distribution of the level\nL. Unlike previous work where αis chosen to minimize the average cost CostRU[22, 23, 30],\nbased on experiment results in Figure 2, in addition to average cost, variance is another crucial\nfactor that demands our attention. These two indicators collectively hold great significance in\nour evaluation and optimization processes. In order to reduce the variance of this estimate,\none direct approach is to take average of different iterations. The asymptotic inefficiency [35]\ndefined as HRU:= Var[ VRU]×Cost RUremains a constant whether iteration average is taken. It\n15Table 1: The value of alpha minimizing the upper bound of asymptotic inefficiency\nMethodRU-MLMC GRR-MLMC TGRR-MLMC\n(m= 2) (m= 4)\nα(1< α < r 2) 1.4 1.209 1.673\nwould be effective to decrease variance by raising cost if αis chosen to minimize this quantity.\nUtilizing Theorem 3.5, we find an upper bound for the asymptotic inefficiency\nHRU≲M02α+r2\n(2α−2) (2r2−2α). (29)\nAs a result, the optimal αminimizing the upper bound is α∗\nRU= (r2+ 1)/2∈(1, r2). When\nm= 0 in GRR-MLMC, which degenerates to RR-MLMC, the constants given in Theorem 3.6\ndo not involve the leverage of Theorem 3.4. Therefore, these constants are the same as those\nin Theorem 3.5, which yields an uniform upper bound for HRR. Therefore, we reach the same\nconclusion for αRR\nopt. From this point of view, we can see that RR-MLMC actually trades lower\nvariance for higher average cost compared with RU-MLMC. We apply the same procedure for\nother MLMC methods, the corresponding optimal αare presented in Table 1.\nSince the proposal distributions of each round are distinct, the hyperparameter r2in Theo-\nrems 3.3 and 3.2 differs in each round. To determine the value of r2for each round, except for\nthe first two, one may take the value from the previous round. To address this issue, we have\nconducted 50 training processes and sequentially performed linear regression, a universal value\nis then selected as r2= 1.8.\nWe present our results in Figures 3 and 4. It is observed that both unbiased methods are\ninferior to the biased one in some cases. When comparing RU-MLMC with GRR-MLMC in the\ncase of the Lotka-Volterra and M/G/1 queue models, we find that the unbiased method can\ngreatly benefit from variance reduction. In the case of Lotka-Volterra, when comparing TGRR\nand GRR methods, we conclude that the gradient information ∆ ρϕ,ℓwhere ℓ≥mnot only does\nnot contribute to the overall performance, but also make it worse when measured with C2ST.\nAs suggested by Theorem 4.4, this could be due to the domination of variance, where the effect\nof reducing variance is superior to reducing bias.\nIn conclusion, in the case of SNPE where the complex model of qF(x,ϕ)(θ) is used for density\nestimation, variance tends to dominate the bias as it is suggested by Theorem 4.4, indicating that\nexcessive variance could seriously affects the training process of the density estimator. Instead of\nseeking for unbiasedness, one should try to strike a balance between bias, average computational\ncost, and variance of the gradient in this case.\nIt is worth noting that in the case of the Two-moon problem, RU-MLMC method deliver\ncomparable performance to the other two methods with lower computational cost and time\nrequirements. Rather than employing any of these methods indiscriminately, it is crucial to\nundertake a comprehensive and thoughtful analysis of the specific problem at hand.\n6 Concluding remarks\nWe develop a series of efficient nested MLMC methods, which are improvement over the un-\nbiased MLMC methods. The latter suffer from excessive variance at the cost of unbiasedness,\nwhich, based on the convergence analysis, severely affects the method’s performance. Through\nexperiments on standard benchmark for likelihood-free estimation and theoretical analysis, we\nvalid the inferior performance of unbiased methods when variance dominates bias. In this pa-\nper, our choice for hyper-parameter αdeviates from the mainstream choice. While values of α\ncloser to r2is favored, aiming at cost minimization, given the fact that variance is the major\nfocus, our focus lies on the asymptotic inefficiency by considering both variance and average\n16A\nB\n5\n 4\n1\n5\n 4\n1\n1\n 0\n0.5\n0.0 0.5\n 5\n 4\n 5\n 4\n1\n5\n 4\n1\n1\n 0\n0.5\n0.0 0.5\n 5\n 4\n5\n 4\n 1\n 0\n2\n1\n 0\n2\n0.5\n0.0 0.5\n 5\n 4\n 5\n 4\n 1\n 0\n2\n1\n 0\n2\n0.5\n0.0 0.5\n 5\n 4\n5\n 4\n 1\n 00.5\n0.0 0.5\n3\n0.5\n0.0 0.5\n3\n5\n 4\n 5\n 4\n 1\n 00.5\n0.0 0.5\n3\n0.5\n0.0 0.5\n3\n5\n 4\n5\n 4\n 1\n 0\n0.5\n0.0 0.5 5\n 4\n4\n5\n 4\n4\n5\n 4\n 1\n 0\n0.5\n0.0 0.5 5\n 4\n4\n5\n 4\n4\nC\n2.8 3.0\nA2.8 3.0\nA\n0.5\n0.00.5\n 1 2 3\n 2\n 0 2.8 3.0\nA2.8 3.0\nA\n0.5\n0.00.5\n 1 2 3\n 2\n 0\n2.8 3.0 0.5\n0.00.5\nlogB0.5\n0.00.5\nlogB\n1 2 3\n 2\n 0\n 2.8 3.0 0.5\n0.00.5\nlogB0.5\n0.00.5\nlogB\n1 2 3\n 2\n 0\n2.8 3.0\n 0.5\n0.00.5 1 2 3\ng1 2 3\ng\n2\n 0\n 2.8 3.0\n 0.5\n0.00.5 1 2 3\ng1 2 3\ng\n2\n 0\n2.8 3.0\n 0.5\n0.00.5\n 1 2 3 2\n 0\nlog(k+1/2)2\n 0\nlog(k+1/2)\n2.8 3.0\n 0.5\n0.00.5\n 1 2 3 2\n 0\nlog(k+1/2)2\n 0\nlog(k+1/2)\nFigure 3: Density plot for RU-MLMC, GRR-MLMC and TGRR-MLMC A. Two-\nmoon, from left to right: available ground truth, RU-MLMC, GRR-MLMC and TGRR-MLMC.\nB.Lotka-Volterra, from left to right: ground truth simulated with SMC-ABC [4], RU-MLMC,\nGRR-MLMC and TGRR-MLMC. C.M/G/1 queue model, the setting is the same with Lotka-\nVolterra.\n17A\n0.0000.0050.0100.015/uni00000030/uni00000030/uni00000027\n0.50.60.70.8/uni00000026/uni00000015/uni00000036/uni00000037\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000564.0\n3.5\n3.0\n2.5\n2.0\n1.5\n/uni00000031/uni0000002f/uni00000032/uni0000002a\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000562.4\n2.3\n2.2\n2.1\n2.0\n/uni0000002f/uni00000030/uni00000027\nB\n0.20.40.6/uni00000030/uni00000030/uni00000027\n0.70.80.91.0/uni00000026/uni00000015/uni00000036/uni00000037\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000565.0\n2.5\n0.02.55.07.5/uni00000031/uni0000002f/uni00000032/uni0000002a\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000561.52.02.5/uni0000002f/uni00000030/uni00000027\nC\n0.00.10.20.30.4/uni00000030/uni00000030/uni00000027\n0.70.80.91.0/uni00000026/uni00000015/uni00000036/uni00000037\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000566\n4\n2\n0/uni00000031/uni0000002f/uni00000032/uni0000002a\n5 10 15 20\n/uni00000031/uni00000058/uni00000050/uni00000045/uni00000048/uni00000055/uni00000003/uni00000052/uni00000049/uni00000003/uni00000035/uni00000052/uni00000058/uni00000051/uni00000047/uni000000561.01.52.02.5/uni0000002f/uni00000030/uni00000027\nFigure 4: Performance of RU-MLMC, GRR-MLMC and TGRR-MLMC A. Two-moon,\nB.Lotka-Volterra C.M/G/1 queue model, blue, green, and red correspond to RU-MLMC,\nGRR-MLMC, and TGRR-MLMC respectively.\n18cost. However, given that a thorough validation for this strategy through ablation experiments\nis still lacking, its improvement is still unclear. Quasi-Monte Carlo (QMC) and Randomized\nQuasi-Monte Carlo (RQMC) are well known variance reduction techniques, its application in\nnested MLMC methods has already been investigated in [30]. Given the high sensitivity of this\nproblem to variance, using RQMC is likely to be beneficial, and existing results [30] can be\nappropriately utilized.\nAcknowledgments\nThe work was supported by the National Natural Science Foundation of China grant 12071154,\nthe Guangdong Basic and Applied Basic Research Foundation grant 2021A1515010275.\nA Supplementary proofs\nA.1 Proof for Theorem 3.3\nTheorem A.1. Assume that there exists s >2that\nsup\nx,ϕ,θ∥∇ϕloggϕ(x, θ)∥∞=Mmax<∞,sup\nϕE\u0014\f\f\f\fgϕ(x, θ)\nZ(x, ϕ)\f\f\f\fs\u0015\n<∞,\nthen we have\nEh\n∥∆ρϕ,ℓ∥2\n2i\n≤16(2r2+ 1)( E1+E2)H(2r) + 9E3H(s)1{s <4}\n(M02ℓ)r2, (30)\nE[∥∆ρϕ,ℓ∥]≤(6√\n2d1/4M1/2\nmax+ 4d1/2Mmax)C2H(2)\nMℓ, (31)\nwhere\nE1= 26−1/2r−r2dM2\nmaxC2r,\nE2= 24−2rdM2\nmaxC2r,\nE3= 21+3s/2dM2\nmaxCs,\nH(x) = sup\nϕ∈ΦE\u0014\f\f\f\fgϕ(x, θ)\nZ(x, ϕ)\f\f\f\fx\u0015\n+ 1<∞, x≤s\nTo prove this theorem, we introduce two useful facts that are repeatedly used in the proof.\nFor any real finite sequence {ai}n\ni=1and positive finite sequence {bi}n\ni=1, then for any i >0 we\nhave ai≤bimax\ni{ai/bi}. Taking summation or expectation on the both side yields that\nPn\ni=1aiPn\nj=1bj≤max\ni\u001aai\nbi\u001b\n,E[a]\nE[b]≤supna\nbo\n, (32)\nwhere the latter one requires the additional assumption that the supremum of the ratio a/bis\nbounded. Next, given for any u >2, for|x−1|<1/2 we arrive that for any τ∈(0,4]:\n(x−1)τ≤24−min(u,4)|x−1|min(u,4)−(4−τ). (33)\nThis proof follows [23], since the implicit constant is of interest, we still present it here for\nthe sake of completeness.\n19Proof. Consider the following event A\nA:=\u001a\f\f\fS(a)\nℓ−1\f\f\f>1\n2\u001b[\u001a\f\f\fS(b)\nℓ−1\f\f\f>1\n2\u001b\n,\nwhere S(a)\nℓ−1:=g(a)\nϕ,M ℓ−1(x)/Z(x, ϕ)−1, which is the sample mean of Mℓ−1variable s=\ngϕ(x, θ)/Z(x, ϕ)−1 with zero mean. Similarly, we define S(b)\nℓ−1andSℓ:=gϕ,M ℓ(x)/Z(x, ϕ)−1\nand arrive at the following decomposition\nEh\n∥∆ρϕ,ℓ∥2\n2i\n=Eh\n∥∆ρϕ,ℓ∥2\n21Ai\n+Eh\n∥∆ρϕ,ℓ∥2\n21Aci\n. (34)\nWe then need to develop a proper upper bound for this two terms. Denote the dimension of the\nϕasd, which is always finite, we leverage (32) to obtain\n∥ρϕ,M ℓ∥2\n2≤\r\r\r\r∇ϕgϕ(x, θ)\ngϕ(x, θ)\r\r\r\r2\n+\r\r\r\r\rPMℓ\nj=1∇ϕgϕ(x, θ′\nj)\nPMℓ\nj=1gϕ(x, θ′\nj)\r\r\r\r\r2\n2≤2dM2\nmax,\ngiving us a rough bound of the ℓ2norm of ρϕ,M ℓ. That is\nmax\u001a\r\r\rρ(a)\nϕ,M ℓ−1\r\r\r2\n2,\r\r\rρ(b)\nϕ,M ℓ−1\r\r\r2\n2\u001b\n≤2dM2\nmax.\nThen for ∥∆ρϕ,ℓ∥2\n2, we apply Jensen’s inequality to develop an upper bound\n∥∆ρϕ,ℓ∥2\n2≤\n∥ρϕ,M ℓ∥2+\r\r\rρ(a)\nϕ,M ℓ−1\r\r\r\n2\n2+\r\r\rρ(b)\nϕ,M ℓ−1\r\r\r\n2\n2\n2\n≤3∥ρϕ,M ℓ∥2\n2+3\n4\r\r\rρ(a)\nϕ,M ℓ−1\r\r\r2\n2+3\n4\r\r\rρ(b)\nϕ,M ℓ−1\r\r\r2\n2≤9dM2\nmax.\nThen for the first term in (34) we have\nEh\n∥∆ρϕ,ℓ∥2\n21Ai\n≤ ∥∆ρϕ,ℓ∥2\n2P[A]≤9dM2\nmaxP[A]. (35)\nAs for P[A], applying Lemma 3.1 to have\nP[A]≤P\u0014\f\f\fS(a)\nℓ−1\f\f\f>1\n2\u0015\n+P\u0014\f\f\fS(b)\nℓ−1\f\f\f>1\n2\u0015\n≤23s/2+1Cs\n(M02ℓ)s/2H(s),\nand it is proved that E[∥∆ρϕ,ℓ∥2\n21A] is of order 2−(s/2)ℓ. For the second term E[∥∆ρϕ,ℓ∥2\n21Ac] in\n(34), utilizing the antithetic property to attain the following identity\n∆ρϕ,ℓ=1\n2\u0010\n∇g(a)\nϕ,M ℓ−1(x)− ∇ ϕZ(x, ϕ)\u0011\n1\ng(a)\nϕ,M ℓ−1(x)−1\nZ(x, ϕ)\n\n+1\n2\u0010\n∇g(b)\nϕ,M ℓ−1(x)− ∇ ϕZ(x, ϕ)\u0011\n1\ng(b)\nϕ,M ℓ−1(x)−1\nZ(x, ϕ)\n\n−(∇gϕ,M ℓ(x)− ∇ ϕZ(x, ϕ))\u00121\ngϕ,M ℓ(x)−1\nZ(x, ϕ)\u0013\n+1\n2∇ϕZ(x, ϕ)\ng(a)\nϕ,M ℓ−1(x)\ng(a)\nϕ,M ℓ−1(x)\nZ(x, ϕ)−1\n2\n+1\n2∇ϕZ(x, ϕ)\ng(b)\nϕ,M ℓ−1(x)\ng(b)\nϕ,M ℓ−1(x)\nZ(x, ϕ)−1\n2\n20−∇ϕZ(x, ϕ)\ngϕ,M ℓ(x)\u0012gϕ,M ℓ(x)\nZ(x, ϕ)−1\u00132\n. (36)\nNow on event Ac, we have 1 /g(a)\nϕ,M ℓ−1(x)≤2/Z(x, ϕ),1/g(b)\nϕ,M ℓ−1(x)≤2/Z(x, ϕ). With simple\nalgebra, we have 1 /gϕ,M ℓ(x)≤2/Z(x, ϕ). Also we have |Sℓ|<1/2 onAc. Then an upper bound\nof the normalizing constant estimators on event Acare given. We denote F(a)\nℓ−1:= (∇g(a)\nϕ,M ℓ−1(x)−\n∇ϕZ(x, ϕ))/Z(x, ϕ),˜F(a)\nℓ−1:= (∇g(a)\nϕ,M ℓ−1(x)−∇ ϕZ(x, ϕ))/g(a)\nϕ,M ℓ−1. Where the former one is also a\nmean of Mℓ−1variable f:=∇ϕgϕ(x, θ)−∇ ϕZ(x, ϕ)/Z(x, ϕ) with zero mean. Similarly, we define\nF(b)\nℓ−1,˜F(b)\nℓ−1andFℓ:= (∇gϕ,M ℓ(x)−∇ ϕZ(x, ϕ))/Z(x, ϕ),˜Fℓ:= (∇gϕ,M ℓ(x)−∇ ϕZ(x, ϕ))/gϕ,M ℓ(x).\nDirectly applying Jensen’s inequality on (36) to have\n∥∆ρϕ,ℓ∥2\n2≤2∥˜F(a)\nℓ−1∥2\n2(S(a)\nℓ−1)2+ 2∥˜F(b)\nℓ−1∥2\n2(S(b)\nℓ−1)2+ 4∥˜Fℓ∥2\n2(Sℓ)2\n+ 2\r\r\r\r\r\r∇ϕZ(x, ϕ)\ng(a)\nϕ,M ℓ−1(x)\r\r\r\r\r\r2\n2(S(a)\nℓ−1)4+ 2\r\r\r\r\r\r∇ϕZ(x, ϕ)\ng(b)\nϕ,M ℓ−1(x)\r\r\r\r\r\r2\n2(S(b)\nℓ−1)4\n+ 4\r\r\r\r∇ϕZ(x, ϕ)\ngϕ,M ℓ(x)\r\r\r\r2\n2(Sℓ)4. (37)\nWith the upper bound derived before, we arrive at the following:\n∥∆ρϕ,ℓ∥2\n2≤8∥F(a)\nℓ−1∥2\n2(S(a)\nℓ−1)2+ 8∥F(b)\nℓ−1∥2\n2(S(b)\nℓ−1)2+ 16∥Fℓ∥2\n2(Sℓ)2\n+ 8\r\r\r\r∇ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2\n2\u0010\n(S(a)\nℓ−1)4+ (S(b)\nℓ−1)4+ 2(Sℓ)4\u0011\n, (38)\nwhere each term is some power of a mean of forsenabling one to use Lemma 3.1 to derive\ntheir upper bound. To obtain the order of E[∥∆ρϕ,l∥2\n21Ac], we begin by applying (32) to yield\n\r\r\r\r∇ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2\n2=\r\r\r\rE[∇ϕgϕ(x, θ)]\nE[gϕ(x, θ)]\r\r\r\r2\n2≤sup\nx,θ,ϕ\r\r\r\r∇ϕg(x, θ)\ngϕ(x, θ)\r\r\r\r2\n2, (39)\nthen it would be sufficient to derive an upper bound for the expectation of the ∥Fℓ∥2\n2(Sℓ)2and\n(Sℓ)4in (38) on event Acdue to the symmetry. We firstly start from ∥Fℓ∥2\n2(Sℓ)2, applying (33)\nto obtain\nEh\n∥Fℓ∥2\n2(Sℓ)21Aci\n≤Eh\n∥Fℓ∥2\n2×24−2r|Sℓ|2r−2i\n≤\u0010\nEh\n∥Fℓ∥2r\n2i\u00111/r224−2r\u0010\nEh\n|Sℓ|2ri\u00111−1/r2. (40)\nTo obtain the desired upper bound, we handle with these two moments separately. For\nEh\n∥Fℓ∥2r\n2i\n, with Jensen’s inequality and Lemma 3.1, we have\nEh\n∥Fℓ∥2r\n2i\n≤C2r\nMr2\nℓE\u0002\n∥f∥2r\n2\u0003\n,\nthen for E\u0002\n∥f∥2r\n2\u0003\nonAc, applying Jensen’s inequality\nE\u0002\n∥f∥2r\n2\u0003\n=E\"\r\r\r\r∇ϕgϕ(x, θ)− ∇ ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2r\n2#\n≤dr2−122r−1E\"\r\r\r\r∇ϕgϕ(x, θ)\ngϕ(x, θ)·gϕ(x, θ)\nZ(x, ϕ)\r\r\r\r2r\n2r+\r\r\r\r∇ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2r\n2r#\n.\n21Then an upper bound for Eh\n∥Fℓ∥2r\n2i\nin (40) is presented as\nEh\n∥Fℓ∥2r\n2i\n≤22r−1dr2M2r\nmaxC2r\nMr2\nℓH(2r).\nFor the upper bound of Eh\n|Sℓ|2ri\nin (40), we directly apply Lemma 3.1 to have\nEh\n|Sℓ|2ri\n≤C2r\nMr2\nℓH(2r),\ndirectly plugging them into (40) to obtain the desired upper bound\nRHS =\u0000\n22r−1dr2M2r\nmaxC2r\u00011/r224−2rC1−1/r2\n2rH(2r)\n(M02ℓ)r2=E1H2r\n(M02ℓ)r2.\nWith similar treatment for E[∥F(a)\nℓ−1∥2\n2(S(a)\nℓ−1)21Ac] and E[∥F(b)\nℓ−1∥2\n2(S(b)\nℓ−1)21Ac], we then have a\nuniform upper bound as E1H2r2r2/(M02ℓ)r2.\nAs for the E[(Sℓ)41Ac], the sixth term in (38), note that\nE[(Sℓ)41Ac]≤24−2rEh\n|Sℓ|2ri\n≤24−2rC2r\nMr2\nℓH(2r)\nCombining these upper bounds, we arrive at:\nEh\n∥∆ρϕ,ℓ∥2\n2i\n≤16(2r2+ 1)( E1+E2)H(2r) + 9E3H(s)1{s <4}\n(M02ℓ)r2.\nFor the second required upper bound, we follow from (36) that\n∥∆ρϕ,ℓ∥2≤ ∥F(a)\nℓ−1∥2|S(a)\nℓ−1|+∥F(b)\nℓ−1∥2|S(b)\nℓ−1|+ 2∥Fℓ∥2|Sℓ| (41)\n+\r\r\r\r∇Zϕ(x, ϕ)\nZ(x, ϕ)\r\r\r\r\n2\u0010\n(S(a)\nℓ−1)2+ (Sb\nℓ)2+ 2(Sℓ)2\u0011\n, (42)\nthe same decomposition on event Ais also studied. We apply Lemma 3.1 and (39) to derive an\nupper bound for the second row. For the first row, it would be sufficient to find an upper bound\nforE[∥Fℓ∥2|Sℓ|1Ac], applying Holder’s inequality to yield that\nE[∥Fℓ∥2|Sℓ|1Ac]≤\u0010\nEh\n∥Fℓ∥2\n2i\u00111/2\u0010\nEh\n|Sℓ|2i\u00111/2\n.\nWe then utilize Lemma 3.1 to have\nEh\n∥Fℓ∥2\n21Aci\n≤2d1/2MmaxC2\nMℓH(2),E[|Sℓ|21Ac]≤C2\nMℓH(2).\nTherefore the upper bound for E[∥Fℓ∥2|Sℓ|1Ac] is√\n2d1/4M1/2\nmaxC2H(2)/Mℓ. Finally we have the\nfollowing upper bound for E[∥∆ρϕ,ℓ∥]\nE[∥∆ρϕ,ℓ∥]≤(6√\n2d1/4M1/2\nmax+ 4d1/2Mmax)C2H(2)\nMℓ.\n22A.2 Proof for Theorem 3.4\nTheorem A.2. Under the assumptions of Theorem 3.3, we have\nVar [ρϕ,M ℓ]≤(16dM2\nmax+ 32d1/2Mmax)C2H(2)\nM02ℓ\n+\u0010\n8dM2\nmax+ 24√\n2d3/4M3/2\nmax+ 36d1/2Mmax\u0011\nC2\n2H2\n2\n(M02ℓ)2. (43)\nProof. By Jensen’s equality, we have\nVar [ρϕ,M ℓ] =Eh\n∥ρϕ,M ℓ−E[ρϕ,M ℓ]∥2\n2i\n=Eh\n∥ρϕ,M ℓ−ρϕ+ρϕ−E[ρϕ,M ℓ]∥2\n2i\n≤2Eh\n∥ρϕ,M ℓ−ρϕ∥2\n2i\n+ 2Eh\n∥ρϕ−E[ρϕ,M ℓ]∥2\n2i\n. (44)\nFor the first term of (43), we begin with\nρϕ,M ℓ−ρϕ=\u0010\n−∇ϕloggϕ(x, θ) +∇ϕlogˆZMℓ(x, ϕ)\u0011\n−(−∇ϕloggϕ(x, θ) +∇ϕlogZ(x, ϕ))\n=∇gϕ,M ℓ(x)\ngϕ,M ℓ(x)−∇ϕZ(x, ϕ)\nZ(x, ϕ),\nand similarly, we consider the event\nB:=\u001a\f\f\f\fgϕ,M ℓ(x)\nZ(x, ϕ)−1\f\f\f\f>1\n2\u001b\n,\nand the similar decomposition used for event Ain the proof of Theorem 3.3 is studied. Firstly\nwe derive a rough bound for ∥ρϕ,M ℓ−ρϕ∥2\n2with (32)\n∥ρϕ,M ℓ−ρϕ∥2\n2≤2\r\r\r\r∇gϕ,M ℓ(x)\ngϕ,M ℓ(x)\r\r\r\r2\n2+ 2\r\r\r\r∇ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2\n2≤4dM2\nmax.\nThen for event B, by Lemma 3.1 again, we obtain\nEh\n∥ρϕ,M ℓ−ρϕ∥2\n21Bi\n≤4dMmaxP[B]≤2s+2dMmaxCs\n(M02ℓ)s/2H(s).\nOn event Bc, we have\n∥ρϕ,M ℓ−ρϕ∥2\n2=\r\r\r\r∇gϕ,M ℓ(x)− ∇ ϕZ(x, ϕ)\ngϕ,M ℓ(x)+∇ϕZ(x, ϕ)\ngϕ,M ℓ(x)\u0012gϕ,M ℓ(x)\nZ(x, ϕ)−1\u0013\r\r\r\r2\n2\n≤8∥Fℓ∥2\n2+ 8\r\r\r\r∇ϕZ(x, ϕ)\nZ(x, ϕ)\r\r\r\r2\n2S2\nℓ,\nand\nEh\n∥Fℓ∥2\n21Bci\n≤2d1/2MmaxC2\nMℓH(2),E[|Sℓ|21Bc]≤C2\nMℓH(2).\nThese result in the following upper bound\nEh\n∥ρϕ,M ℓ−ρϕ∥2\n2i\n≤(16d1/2Mmax+ 8dM2\nmax)C2H(2)\nM02ℓ.\n23For the second term of (43), we have\n∥E[ρϕ,M ℓ]−ρϕ∥2=\r\r\r\r\r∞X\nℓ′=ℓ+1E\u0002\n∆ρϕ,ℓ′\u0003\r\r\r\r\r\n2≤∞X\nℓ′=ℓ+1E\u0002\r\r∆ρϕ,ℓ′\r\r\n2\u0003\n,\nthen applying Theorem 3.3 to yield the desired result\n∥E[ρϕ,M ℓ]−ρϕ∥2≤(3√\n2d1/4M1/2\nmax+ 2d1/2Mmax)C2H(2)\nM02ℓ. (45)\nA.3 Proof of Theorem 3.6\nProof. Applying Theorem 3.3, we have the following upper bound for the variance of the gradient\nestimators of GRR\nVar [VGRR] = Var\u0002\nρϕ,M m\u0003\n+∞X\nℓ=m+1wℓℓX\nℓ′=m+1(1−p)−2ℓ′Var\u0002\n∆ρϕ,ℓ′\u0003\n≤B1\n2m+B2\n22m+∞X\nℓ=m+1wℓℓX\nℓ′=m+1(1−p)−2ℓ′B32−r2ℓ′\n=B1\n2m+B2\n22m+B3\n∞X\nℓ=m+1wℓ2(2α−r2)(m+1)(1−2(2α−r2)(ℓ−m))\n1−22α−r2\n\n=B1\n2m+B2\n22m+B3\n∞X\nℓ=m+1(1−p)ℓp2(2α−r2)(m+1)−2(2α−r2)(ℓ+1)\n1−22α−r2\n\n=B1\n2m+B2\n22m+B3 \n1−2−α\n1−22α−r2 \n2(α−r2)(m+1)\n1−2−α−22α−r22(α−r2)(m+1)\n1−2α−r2!!\n≤B1+B2\n2m+B3 \n2(α−r2)(m+1)\n1−2α−r2!\n≤B\n2m+B \n2(α−r2)(m+1)\n1−2α−r2!\n,\nwhere B= max {B1+B2, B3}with B1, B2andB3representing the constants in the upper\nbounds in Theorems 3.4 and 3.3. The proof for the upper bound of Var[ UGRR] follows a similar\nprocedure. The average cost is\nCost GRR∝M02m+∞X\nℓ=m+1wℓℓX\nℓ′=m+1M02ℓ′\n=M02m+M0(1−2−α)∞X\nℓ=m+12−αℓ\u0010\n2ℓ+1−2m+1\u0011\n=M02m+M0(1−2−α) \n2×2(1−α)(m+1)\n1−2(1−α)−2(1−α)(m+1)\n1−2−α!\n=M02m+M02(1−α)(m+1)\n1−2(1−α),\ncompleting the proof.\n24References\n[1] Ajalloeian, A. and Stich, S. U. (2020). On the convergence of SGD with biased gradients.\narXiv preprint arXiv:2008.00051 .\n[2] Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle Markov chain Monte Carlo\nmethods. Journal of the Royal Statistical Society: Series B (Statistical Methodology) ,\n72(3):269–342.\n[3] Andrieu, C. and Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte\nCarlo computations. The Annals of Statistics , 37(2):697 – 725.\n[4] Beaumont, M. A., Cornuet, J.-M., Marin, J.-M., and Robert, C. P. (2009). Adaptive ap-\nproximate Bayesian computation. Biometrika , 96(4):983–990.\n[5] Beaumont, M. A., Zhang, W., and Balding, D. J. (2002). Approximate Bayesian computation\nin population genetics. Genetics , 162(4):2025–2035.\n[6] Bottou, L., Curtis, F. E., and Nocedal, J. (2018). Optimization methods for large-scale\nmachine learning. SIAM Review , 60(2):223–311.\n[7] Brehmer, J., Louppe, G., Pavez, J., and Cranmer, K. (2020). Mining gold from implicit\nmodels to improve likelihood-free inference. Proceedings of the National Academy of Sciences ,\n117(10):5242–5249.\n[8] Bujok, K., Hambly, B. M., and Reisinger, C. (2015). Multilevel simulation of functionals\nof bernoulli random variables with application to basket credit derivatives. Methodology and\nComputing in Applied Probability , 17:579–604.\n[9] Carlon, A. G., Dia, B. M., Espath, L., Lopez, R. H., and Tempone, R. (2020). Nesterov-aided\nstochastic gradient methods using Laplace approximation for Bayesian design optimization.\nComputer Methods in Applied Mechanics and Engineering , 363:112909.\n[10] Clarke, R., Ressom, H. W., Wang, A., Xuan, J., Liu, M. C., Gehan, E. A., and Wang, Y.\n(2008). The properties of high-dimensional data spaces: implications for exploring gene and\nprotein expression data. Nature Reviews Cancer , 8(1):37–49.\n[11] Cranmer, K., Brehmer, J., and Louppe, G. (2020). The frontier of simulation-based infer-\nence. Proceedings of the National Academy of Sciences , 117(48):30055–30062.\n[12] Dalmasso, N., Lee, A., Izbicki, R., Pospisil, T., Kim, I., and Lin, C.-A. (2020). Validation\nof approximate likelihood and emulator models for computationally intensive simulations. In\nInternational Conference on Artificial Intelligence and Statistics , pages 3349–3361. PMLR.\n[13] Deistler, M., Goncalves, P. J., and Macke, J. H. (2022). Truncated proposals for scalable\nand hassle-free simulation-based inference. arXiv preprint arXiv:2210.04815 .\n[14] Durkan, C., Bekasov, A., Murray, I., and Papamakarios, G. (2019). Neural spline flows.\nAdvances in Neural Information Processing Systems , 32.\n[15] Fearnhead, P. and Prangle, D. (2012). Constructing summary statistics for approximate\nBayesian computation: semi-automatic approximate Bayesian computation. Journal of the\nRoyal Statistical Society: Series B (Statistical Methodology) , 74(3):419–474.\n[16] Foster, A., Jankowiak, M., Bingham, E., Horsfall, P., Teh, Y. W., Rainforth, T., and\nGoodman, N. (2019). Variational Bayesian optimal experimental design. Advances in Neural\nInformation Processing Systems , 32.\n25[17] Giles, M. B. (2015). Multilevel Monte Carlo methods. Acta Numerica , 24:259–328.\n[18] Giles, M. B. (2018). MLMC for nested expectations. In Contemporary Computational\nMathematics-A Celebration of the 80th Birthday of Ian Sloan , pages 425–442.\n[19] Giles, M. B. and Goda, T. (2019). Decision-making under uncertainty: using MLMC for\nefficient estimation of EVPPI. Statistics and Computing , 29:739–751.\n[20] Giles, M. B. and Szpruch, L. (2014). Antithetic multilevel Monte Carlo estimation for\nmulti-dimensional SDEs without L´ evy area simulation. The Annals of Applied Probability ,\n24(4):1585 – 1620.\n[21] Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. The\nJournal of Physical Chemistry , 81(25):2340–2361.\n[22] Goda, T., Hironaka, T., and Iwamoto, T. (2019). Multilevel Monte Carlo estimation of\nexpected information gains. Stochastic Analysis and Applications , 38(4):581–600.\n[23] Goda, T., Hironaka, T., Kitade, W., and Foster, A. (2022). Unbiased MLMC stochastic\ngradient-based optimization of Bayesian experimental designs. SIAM Journal on Scientific\nComputing , 44(1):A286–A311.\n[24] Gon¸ calves, P. J., Lueckmann, J.-M., Deistler, M., Nonnenmacher, M., ¨Ocal, K., Bassetto,\nG., Chintaluri, C., Podlaski, W. F., Haddad, S. A., Vogels, T. P., et al. (2020). Training deep\nneural density estimators to identify mechanistic models of neural dynamics. Elife, 9:e56261.\n[25] Greenberg, D., Nonnenmacher, M., and Macke, J. (2019). Automatic posterior transfor-\nmation for likelihood-free inference. In International Conference on Machine Learning , pages\n2404–2414. PMLR.\n[26] Gretton, A., Borgwardt, K. M., Rasch, M. J., Sch¨ olkopf, B., and Smola, A. (2012). A kernel\ntwo-sample test. The Journal of Machine Learning Research , 13(1):723–773.\n[27] Gutmann, M. U. and Corander, J. (2016). Bayesian optimization for likelihood-free infer-\nence of simulator-based statistical models. Journal of Machine Learning Research .\n[28] Gutmann, M. U., Dutta, R., Kaski, S., and Corander, J. (2018). Likelihood-free inference\nvia classification. Statistics and Computing , 28:411–425.\n[29] Hashemi, M., Vattikonda, A. N., Jha, J., Sip, V., Woodman, M. M., Bartolomei, F., and\nJirsa, V. K. (2023). Amortized Bayesian inference on generative dynamical network models\nof epilepsy using deep neural density estimators. Neural Networks , 163:178–194.\n[30] He, Z., Xu, Z., and Wang, X. (2022). Unbiased MLMC-based variational Bayes for\nlikelihood-free inference. SIAM Journal on Scientific Computing , 44(4):A1884–A1910.\n[31] Hermans, J., Begy, V., and Louppe, G. (2020). Likelihood-free MCMC with amortized\napproximate ratio estimators. In International conference on machine learning , pages 4239–\n4248. PMLR.\n[32] Hermans, J., Delaunoy, A., Rozet, F., Wehenkel, A., Begy, V., and Louppe, G. (2021). A\nTrust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaith-\nful.arXiv preprint arXiv:2110.06581 .\n[33] Hu, Y., Chen, X., and He, N. (2021). On the bias-variance-cost tradeoff of stochastic\noptimization. Advances in Neural Information Processing Systems , 34:22119–22131.\n26[34] Huan, X. and Marzouk, Y. (2014). Gradient-based stochastic optimization methods in\nBayesian experimental design. International Journal for Uncertainty Quantification , 4(6).\n[35] Jacob, P. E., O’Leary, J., and Atchad´ e, Y. F. (2017). Unbiased Markov chain Monte Carlo\nwith couplings. arXiv preprint arXiv:1708.03625 .\n[36] Kingma, D. P. and Ba, J. (2014). Adam: A method for stochastic optimization. arXiv\npreprint arXiv:1412.6980 .\n[37] Kleinegesse, S. and Gutmann, M. U. (2020). Bayesian experimental design for implicit\nmodels by mutual information neural estimation. In International Conference on Machine\nLearning , pages 5316–5326. PMLR.\n[38] Kobyzev, I., Prince, S. J., and Brubaker, M. A. (2020). Normalizing flows: An introduc-\ntion and review of current methods. IEEE Transactions on Pattern Analysis and Machine\nIntelligence , 43(11):3964–3979.\n[39] Li, C., Hu, Y., Li, S., Yi, X., Shao, S., Yu, W., and Li, E. (2023). Biological factors\ncontrolling starch digestibility in human digestive system. Food Science and Human Wellness ,\n12(2):351–358.\n[40] Lopez-Paz, D. and Oquab, M. (2016). Revisiting classifier two-sample tests. arXiv preprint\narXiv:1610.06545 .\n[41] Lueckmann, J.-M., Boelts, J., Greenberg, D., Goncalves, P., and Macke, J. (2021). Bench-\nmarking simulation-based inference. In International Conference on Artificial Intelligence and\nStatistics , pages 343–351. PMLR.\n[42] Lueckmann, J.-M., Goncalves, P. J., Bassetto, G., ¨Ocal, K., Nonnenmacher, M., and Macke,\nJ. H. (2017). Flexible statistical inference for mechanistic models of neural dynamics. Advances\nin Neural Information Processing Systems , 30.\n[43] Luo, Y., Beatson, A., Norouzi, M., Zhu, J., Duvenaud, D., Adams, R. P., and Chen, R. T.\n(2019). Sumo: Unbiased estimation of log marginal probability for latent variable models. In\nInternational Conference on Learning Representations .\n[44] Lyne, A.-M., Girolami, M., Atchad´ e , Y., Strathmann, H., and Simpson, D. (2015). On Rus-\nsian roulette estimates for Bayesian inference with doubly-intractable likelihoods. Statistical\nScience , 30(4).\n[45] Marin, J.-M., Pudlo, P., Robert, C. P., and Ryder, R. J. (2012). Approximate Bayesian\ncomputational methods. Statistics and Computing , 22(6):1167–1180.\n[46] Olutola, T., Balen, J., Lotisa, V., Johnima, A., and Browndi, I. (2023). Systems Biology\nand Cell Signaling: A Comprehensive Review. Asian Journal of Basic and Applied Sciences ,\n10(06-2023).\n[47] Paninski, L. and Cunningham, J. P. (2018). Neural data science: accelerating the\nexperiment-analysis-theory cycle in large-scale neuroscience. Current Opinion in Neurobi-\nology , 50:232–241.\n[48] Papamakarios, G. and Murray, I. (2016). Fast ε-free inference of simulation models with\nBayesian conditional density estimation. Advances in Neural Information Processing Systems ,\n29.\n[49] Papamakarios, G., Sterratt, D., and Murray, I. (2019). Sequential neural likelihood: Fast\nlikelihood-free inference with autoregressive flows. In The 22nd International Conference on\nArtificial Intelligence and Statistics , pages 837–848. PMLR.\n27[50] Price, L. F., Drovandi, C. C., Lee, A., and Nott, D. J. (2018). Bayesian synthetic likelihood.\nJournal of Computational and Graphical Statistics , 27(1):1–11.\n[51] Rhee, C.-h. and Glynn, P. W. (2015). Unbiased estimation with square root convergence\nfor SDE models. Operations Research , 63(5):1026–1043.\n[52] Romaszko, L., Williams, C. K., Moreno, P., and Kohli, P. (2017). Vision-as-inverse-graphics:\nObtaining a rich 3d explanation of a scene from a single image. In Proceedings of the IEEE\nInternational Conference on Computer Vision Workshops , pages 851–859.\n[53] Ryan, K. J. (2003). Estimating expected information gains for experimental designs with\napplication to the random fatigue-limit model. Journal of Computational and Graphical Statis-\ntics, 12(3):585–603.\n[54] Shestopaloff, A. Y. and Neal, R. M. (2014). On Bayesian inference for the M/G/1 queue\nwith efficient MCMC sampling. arXiv preprint arXiv:1401.5548 .\n[55] Thomas, O., Dutta, R., Corander, J., Kaski, S., and Gutmann, M. U. (2022). Likelihood-\nfree inference by ratio estimation. Bayesian Analysis , 17(1):1–31.\n[56] Tran, M.-N., Nott, D. J., and Kohn, R. (2017). Variational Bayes with intractable likelihood.\nJournal of Computational and Graphical Statistics , 26(4):873–882.\n[57] Wood, S. N. (2010). Statistical inference for noisy nonlinear ecological dynamic systems.\nNature , 466(7310):1102–1104.\n28" }, { "title": "2401.16814v1.Traversable_Wormholes_in_Minimally_Geometrical_Deformed_Trace_Free_Gravity_using_Gravitational_Decoupling.pdf", "content": "Traversable Wormholes in Minimally Geometrical Deformed\nTrace-Free Gravity using Gravitational Decoupling\nPiyachat Panyasiripan,1,∗Narakorn Kaewkhao,2,†Phongpichit Channuie,1, 3,‡and Ali ¨Ovg¨ un4,§\n1School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand\n2Department of Physics, Faculty of Science,\nPrince of Songkla University, Hatyai 90112, Thailand\n3College of Graduate Studies, Walailak University, Nakhon Si Thammarat, 80160, Thailand\n4Physics Department, Eastern Mediterranean University,\nFamagusta, 99628 North Cyprus via Mersin 10, Turkiye\n(Dated: January 31, 2024)\nIn this work, we investigate wormhole solutions through the utilization of gravitational\ndecoupling, employing the Minimal Geometric Deformation (MGD) procedure within the\nframework of Trace-Free Gravity. We base our investigation on static and spherically sym-\nmetric Morris-Thorne traversable wormholes, considering both constant and variable equa-\ntion of state parameters. We derive the field equations and extract the shape function for\neach scenario. Moreover, we explore the gravitational decoupling technique and examine\nvarious forms of energy density for both a smeared and particle-like gravitational source, en-\ncompassing the realm of noncommutative geometry and a statically charged fluid. We also\nexaminethe wormhole geometry through the utilization of embedding diagrams. Through\nour analysis, we uncover a violation of the Null Energy Condition (NEC). To conclude, we\nemploy the Gauss-Bonnet theorem to determine the weak deflection angle for the wormhole\nconfigurations.\nI. INTRODUCTION\nA Trace-Free version of the Einstein (TFE) equations provides the resolution of the cosmological\nconstant problem of which the observed cosmological constant is much smaller than the expected\nvalue. The formulation was proposed by Weinberg in his review [1]. Indeed, the trace-free gravity\nis essentially equivalent to adopting unimodular gravity [2–16] and its generalized versions [17, 18].\nThe approach does not only determine a unique value for the effective cosmological constant, but\n∗Electronic address: piyachat.pa@mail.wu.ac.th\n†Electronic address: naragorn.k@psu.ac.th\n‡Electronic address: phongpichit.ch@mail.wu.ac.th\n§Electronic address: ali.ovgun@emu.edu.trarXiv:2401.16814v1 [gr-qc] 30 Jan 20242\nit also does solve the discrepancy between theory and observation in the standard approach. This\narticle is aimed to investigate traversable wormhole solutions in TFE gravity, incorporating the\nconsideration of variable equations of state and taking advantage of gravitational decoupling by\nmeans of minimal geometric deformation (MGD) approach.\nAs widely acknowledged, solving Einstein field equations poses a non-trivial challenge, partic-\nularly when dealing with cases invoking spherical symmetry. A recent advancement addressing\nthis complexity is the introduction of a novel methodology known as the gravitational decoupling\nthrough the minimal geometric deformation (MGD) scheme [19, 21]. The Brane-World scenario\n[22–26] was originally investigated using such an approach. Subsequently, it evolved into a gravi-\ntational source decoupling scheme, facilitating the extension of isotropic spherical solutions of the\nEinstein field equations to encompass anisotropic domains [19]. Over time, this approach has been\nbroadly adopted across various branches, see for wormholes [20], playing a pivotal role in expanding\nor constructing novel solutions for the Einstein equations and their extensions, see e.g., [21, 27–29].\nGibbons and Werner introduced a novel geometric approach for computing the weak deflection\nangle, employing the Gauss-Bonnet theorem (GBT) on optical geometries applicable to asymp-\ntotically flat spacetimes [30]. This technique involves solving the GBT integral over an infinite\ndomain defined by the light ray boundaries. Subsequently, Werner extended this methodology\nto stationary spacetimes by incorporating the Finsler-Randers optical geometry, utilizing Nazım’s\nosculating Riemannian manifolds [31].\nBuilding upon Werner’s work, Ishihara et al. further extended the method to finite distances,\nspecifically considering scenarios with significant impact parameters, as opposed to relying on\nasymptotic receiver and source conditions [32, 33]. T. Ono et al. later applied this finite-distance\napproach to axisymmetric spacetimes [34]. Crisnejo and Gallo [35] utilized the GBT to derive\ngravitational light deflections within a plasma medium. More recently, Li et al. investigated the\nimpact of finite distances on the weak deflection angle, introducing massive particles and the Jacobi\nmetric within the framework of GBT [36, 37]. For further developments in this field, one can refer\nto more recent works on wormholes [38–41] and black holes [42–54].\nIn this work, we investigate wormhole solutions through the utilization of gravitational decou-\npling, employing the MGD procedure within the framework of Unimodular Gravity. In Sec.II, we\ntake a recap of the basic concept of Unimodular Gravity and employ the MGD procedure. We\nthen examine static and spherically symmetric Morris-Thorne traversable wormholes in Sec.III.\nIn this section, we consider both constants and variables in the equation of state parameter. We\nalso compute the shape function for each case. In Sec.IV, we consider the gravitational decoupling3\ntechnique and consider the forms of the energy density for a gravitational source in the context of\nnoncommutative geometry and a statically charged fluid. Here we obtain the decoupled solutions\nof the wormholes. Moreover, we test the energy conditions. Subsequently, the embedding diagrams\nof the wormholes are illustrated in Sec.V. In Sec.VII, we use the Gauss-Bonnet theorem to compute\nthe weak deflection angle for wormhole solutions. We conclude our findings in the last section.\nII. UNIMODULAR GRAVITY & ITS GEOMETRICAL DEFORMATION\nA detailed formulation of trace-free Einstein gravity and its relationship with unimodular gravity\nis given in Refs.[7, 8]. Recall that in the standard formulation the gravitational field is governed\nby the Einstein field equations:\nRµν−1\n2gµνR+λgµν= 8πT(m)\nµν. (1)\nAs a result, in unimodular gravity, the presence of an additional constraint on the metric determi-\nnant reduces 10 independent field equations to 9 independent field equations. Taking the trace of\nthe field equations (1), we obtain\n4λ=R+ 8πT(m). (2)\nMultiplying each side of Eq.(2) by1\n4gµνand adding the result to Eq.(1) yields the trace-free field\nequations:\nGµν= 8π¯Tµν, (3)\nwhere we have defined\nGµν≡Rµν−1\n4gµνR , (4)\nand\n¯Tµν≡\u0012\nT(m)\nµν−1\n4gµνT(m)\u0013\n. (5)\nAny extension to the above theory will eventually produce new terms in the effective four-\ndimensional Einstein equations. These “corrections” are usually handled as part of an effective\nenergy-momentum tensor. In the following, the MGD takes the simplest modification:\nGµν= 8π¯Tµν+δ(new terms) µν, (6)4\nThe new terms in Eq.(6) may be viewed as part of an effective energy-momentum tensor, whose\nexplicit form may contain new fields, like scalar, vector, and tensor fields, all of them coming from\nthe new gravitational sector not described by Einstein’s theory.\nTherefore, we have the following modification to the stress-momentum tensor:\nTµν=¯Tµν+α(new terms) µν=\u0010\nT(m)\nµν−1\n4gµνT(m)\u0011\n+δ εµν. (7)\nIt is clear that the case δ= 0 yields to the original Unimodular Gravity. We consider the con-\ntribution of two gravitational sources Tµ(m)\nν, which is known as the seed source and εµ\nν. Here the\nintensity of influence of the source εµνoverT(m)\nµνparametrized by a dimensionaless constant δ.\nIII. TRAVERSABLE WORMHOLE SOLUTIONS\nWe consider a static and spherically symmetric Morris-Thorne traversable wormhole in the\nSchwarzschild coordinates ( t, r, θ, ϕ ) given by [4]\nds2=−e2Φ(r)dt2+1\u0010\n1−b(r)\nr\u0011dr2+r2\u0000\ndθ2+ sin2θdϕ2\u0001\n, (8)\nwhere Φ( r) and b(r) are the redshift and shape functions, respectively. They are functions of the\nradial coordinate ronly. In the wormhole geometry, the redshift function Φ should be finite in\norder to avoid the formation of an event horizon. The radial coordinate rranges from a minimum\nvalue r0, corresponding to the throat of the wormhole, where b(r0) =r0atr=r0. A crucial\naspect of wormholes is the flaring-out condition, expressed as b(r)−b′(r)r≥0 in the vicinity of the\nthroat, where a prime denotes a derivative with respect to the radial coordinate r. Additionally,\nit is required that b(r)/r→0 as r→ ∞ . It is worth noting that the supplementary condition\nb(r)/r < 1 is also enforced. We define a perfect fluid source with energy-momentum tensor as\n¯Tab= (¯ρ+ ¯pt)UaUb+ ¯ptgab+ (¯pr−¯pt)XaXb, (9)\nwhere ¯ ρis the energy density measured by a comoving observer with the fluid, and UaandXa\nare its timelike four-velocity and a spacelike unit vector orthogonal to Uaand angular directions,\nrespectively. We define an appropriate frame of the fluid velocity vectors [55]\nUa=e−Φδa\n0, Xa=r\n1−b(r)\nrδa\n1, (10)\nso that UaUa=−1 and XaXa= 1. Here we are working in geometrized units setting the grav-\nitational constant Gand the speed of light cto unity. The trace-free components of the energy-5\nmomentum tensor Eq.(9) in this case read\n¯Tab=\u00121\n4e2Φ(¯pr+ 2¯pt+ 3¯ρ),1\n4(3¯pr−2¯pt+ ¯ρ)1\n(1−b\nr),\n1\n4r2(−¯pr+ 2¯pt+ ¯ρ),1\n4r2sin2(θ) (−¯pr+ 2¯pt+ ¯ρ)\u0013\n. (11)\nFor ( t, t) component, we find\nGtt=e2Φ\n4r2(b′(2−rΦ′) + 2r(2Φ′+rΦ′2+rΦ′′)−b(3Φ′+ 2rΦ′2+ 2rΦ′′)), (12)\nwhile for ( r, r) component,\nGrr=1\n4(b−r)r2(b(4 + 5 rΦ′−2r2Φ′2−2r2Φ′′) +r(−b′(2 +rΦ′) + 2r(−2Φ′+rΦ′2+rΦ′′))),\n(13)\nand for ( θ, θ) component,\nGθθ=1\n4r(b(2 +rΦ′−2r2Φ′2−2r2Φ′′) +r2(−b′Φ′+ 2r(Φ′2+ Φ′′))), (14)\nwhere Gθθ= sin2θGϕϕ. Then using the information of metric (4) and (5) in the TFE for the general\nform, we come up with\n8π[(¯pr+ 2¯pt+ 3¯ρ)] =1\nr2(b′(2−rΦ′) + 2r(2Φ′+rΦ′2+rΦ′′)\n−b(3Φ′+ 2rΦ′2+ 2rΦ′′)), (15)\n8π[(3¯pr−2¯pt+ ¯ρ)] =−1\nr3(b(4 + 5 rΦ′−2r2Φ′2−2r2Φ′′) +r(−b′(2 +rΦ′)\n+2r(−2Φ′+rΦ′2+rΦ′′))), (16)\n8π[(−¯pr+ 2¯pt+ ¯ρ)] =1\nr3(b(2 +rΦ′−2r2Φ′2−2r2Φ′′)\n+r2(−b′Φ′+ 2r(Φ′2+ Φ′′))). (17)\nWe can directly derive the Einstein field equations for Φ′= 0 to obtain\nb′\n4πr2= ¯pr+ 2¯pt+ 3¯ρ≡¯ρeff., (18)\n−2b−rb′\n4πr3= 3¯pr−2¯pt+ ¯ρ≡¯peff.\nr, (19)\nb\n4πr3=−¯pr+ 2¯pt+ ¯ρ≡¯peff.\nt. (20)\nTherefore, the effective matter sector of the present consideration is given by\n2b′\nr2= 8π\u0002\n(¯pr+ 2¯pt+ 3¯ρ) +δε0\n0\u0003\n≡8πρeff., (21)\n−4b−2rb′\nr3= 8π\u0002\n(3¯pr−2¯pt+ ¯ρ)−δε1\n1\u0003\n≡8πpeff.\nr, (22)\n2b\nr3= 8π\u0002\n(−¯pr+ 2¯pt+ ¯ρ)−δε2\n2\u0003\n≡8πpeff.\nt. (23)6\nTo split the complex set of Eqs. (21)–(23), we implement the gravitational decoupling by means\nof the MGD. In this case, the minimally deformed shape function b(r) is given by\nb(r)→b(r) +δf(r), (24)\nwith ¯b(r) being the original shape function given in the preceding section and f(r) the decoupler\nfunction. Basically, values of δcould be small. Putting Eq.(24) into the set (21)–(23), we obtain\nthe following system of equations\n2b′\nr2= 8π[(¯pr+ 2¯pt+ 3¯ρ)], (25)\n−4b−2rb′\nr3= 8π[(3¯pr−2¯pt+ ¯ρ)], (26)\n2b\nr3= 8π[(−¯pr+ 2¯pt+ ¯ρ)]. (27)\nThe second set of equations is given by\n2f′(r)\nr2= 8πε0\n0, (28)\n4f(r)−2rf′(r)\nr3= 8πε1\n1, (29)\n−2f(r)\nr3= 8πε2\n2. (30)\nTo obtain specific forms of f(r), we will consider two types of density profiles generated in non-\ncommutative geometry and a statically charged fluid.\nA. Constants ω&β\nBasically, it is first common to assume the barotropic equations of state given as ¯ pr=ω¯ρwith\nωbeing a constant as well as pt=βprwith constant β. As mentioned in Ref.[56], the values of\nωcan be constrained to −1.5≤ω≤1. This allows us to describe various types of cosmological\nfluids, for example stiff matter [57], radiation [58], Dustlike [59], dark energy [60], phantom fluid\n[61], and holographic dark energy [62]. Moreover, we also need β̸= 1 to guarantee anisotropy and\nβ̸= 0 to avoid singularities. From Eq.(18)-Eq.(20), we find\n−b\n8πr3+rb′\n8πr3= ¯pr+ ¯ρ= (1 + ω)¯ρ , (31)\nr(ω−1)b′(r) + (ω+ 3)b(r)\n16π(ω+ 1)r3= ¯pt. (32)\nWe can simply solve for the analytical solution of the above system to obtain\nb(r) =r0\u0010r\nr0\u00112ωβ+ω+3\n2ωβ−ω+1, (33)7\nFIG. 1: We show the behaviors of b(r), b′(r), b(r)/randb(r)−rb′(r) versus rin Eq.(33) for β=−1.1 and\nr0= 1 and ω= 0.8 (Left Panel) and ω= 1 (Right Panel).\nmatching with that found in Ref.[56]. We recover the Ellis-Bronnikov (EB) shape function,\nb(r) =r2\n0\nr, when ω=−β−1. We can show that the shape function (33) satisfies the usual properties\ndisplayed in Fig.(1). We can also simply check the expression of the flare-out condition given by\nb(r)−rb′(r)\nb(r)2=−2(ω+ 1)\u0010\nr\nr0\u0011−2βω+ω+3\n2βω−ω+1\n(2β−1)r0ω+r0=\nr=r0−2(ω+ 1)\n(2β−1)r0ω+r0>0. (34)\nwhich yields the constraints on ωandβ:\nω >0∧β <ω−1\n2ω. (35)\nIn this case, we can easily compute the energy density to obtain\n¯ρ(r) =−1\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n. (36)\nwhich, on the throat becomes\nρ(r0) =−1\n8π(ω+ 1)r2\n0. (37)\nUsing ρ(r), we can compute prandptto obtain\n¯pr=−ω\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n, (38)\n¯pt=−ωβ\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n. (39)\nWe see from Fig.(2) for constants ω&β, the NEC and SEC are violated at the wormhole throat\nr=r0. In other words, we discover that at the wormhole throat r=r0, we have for the energy\ncondition ρeff.+peff.\nr<0, along with the condition ρeff.+peff.\nr+2peff.\nt<0, by arbitrary small values.8\nFIG. 2: We show the behaviors of ρeff+peff\nr,ρeff+ 2peff\ntandρeff+peff\nr+ 2peff\ntversus rusing β=−1.1,r0= 1\nandω= 0.8.\nB. Variable ω& Constant β\nIn the preceding subsection, we have worked on the constant EoS and discussed some important\nproperties of the shape function b(r) illustrated in Fig.(1). However, variable EoS can also be\nconsidered. In case of ω=ω(r), we come up with\nZ1\nb(r)db(r) =Z(2βω(r) +ω(r) + 3)\nr(2βω(r)−ω(r) + 1)dr . (40)\nTherefore, we obtain\nb(r) =r0exp\u0010Z(2βω(r) +ω(r) + 3)\nr(2βω(r)−ω(r) + 1)dr\u0011\n. (41)\nwith r0being a constant. Let us take a more general EoS:\nω(r) =1\nµr. (42)\nWe find that\nω(r)→\nr=r01\nµr0, ω(r)→\nµ→0∞,and ω(r)→\nrorµ→∞0. (43)\nWe solve Eq.(41) to obtain\nb(r) =r0\u0012r\nr0\u00132β+1\n2β−1\u00121−2β−µr\n1−2β−µr0\u00134(β−1)\n2β−1\n. (44)\nWe can also simply check the expression of the flare-out condition given by\nb(r)−rb′(r)\nb(r)2\f\f\f\nr=r0=2µr0+ 2\n−µr2\n0−2βr0+r0>0. (45)9\nFor more convenience, we take instead µ=µ1\nr0to have\nb(r)−rb′(r)\nb(r)2\f\f\f\nr=r0=2µ1+ 2\n−2βr0−µ1r0+r0>0. (46)\nWith a flare-out condition, we find the constraints on βandµ1:\nβ≤1∧ −1< µ1<1−2β∧r0>0. (47)\nIf we take β=−1.1, we find that −1< µ 1<3.2. However, in the following, we consider µ=−β\nr0\nand write\nω(r) =−r0\nβr, (48)\nwhere\nω(r)→\nr=r0−1\nβ,and ω(r)→\nr→∞0. (49)\nIn this case, βis the only free parameter. Substituting Eq.(48) into Eq.(41), we find\nb(r) =r0\u0012r\nr0\u00132β+1\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u00134(β−1)\n2β−1\n. (50)\nWe can quantify the features of a shape function b(r). Assuming that |β| ≪ O (1), we can expand\nFIG. 3: We show the behaviors of b(r), b′(r), b(r)/randb(r)−rb′(r) versus rin Eq.(50) for r0= 1 and\nβ=−0.01 (Left Panel) and β=−0.05 (Right Panel).\nb(r) to the first order of βto obtain\nb(r) =r2\n0\nr+4\nr\u0012\n−r2\n0log\u0012r\nr0\u0013\n+rr0−r2\n0\u0013\nβ+O(β2). (51)\nWe see that when |β| ≪ O (1) a shape function satisfies the usual properties as it should be. Notice\nthat we can obtain the EB shape function, b(r) =r2\n0\nr, when β= 0. The behaviors of b(r) can be10\nseen in Fig.(3). In this case, we can easily compute the energy density to obtain\n¯ρ(r) =−β\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1+1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n. (52)\nwhich, on the throat becomes\n¯ρ(r0) =−β\n4π(β−1)r2\n0. (53)\nUsing ¯ ρ(r), we can compute ¯ prand ¯ptto obtain\nFIG. 4: We show the behaviors of ρeff+peff\nr,ρeff+ 2peff\ntandρeff+peff\nr+ 2peff\ntversus rusing β=−0.01, and\nr0= 1.\n¯pr=1\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n, (54)\n¯pt=β\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n. (55)\nIn this case, we also discover that the NEC & SEC are violated, i.e., ρeff+peff\nr<0, ρeff+peff\nr+2peff\nt<\n0, by arbitrary small values, see Fig.(4).\nIV. DECOUPLED SOLUTIONS\nWe consider the gravitational decoupling technique and consider two forms of the energy density\nfor a smeared and particle-like gravitational source in the context of noncommutative geometry\nand a statically charged fluid.\nA. Non-commutative Geometry Density Profiles\nIn the context of noncommutative geometry, an interesting development of string/M-theory\ninvolves the requirement for quantizing spacetime. The non-commutativity of spacetime is encoded11\nin the commutator [ xµ,xν] =iθµν, where θµνis an antisymmetric matrix which determines the\nfundamental discretization of spacetime. It has also been shown that noncommutativity flavors\nthe smeared objects instead of the point-like structures in flat spacetime [63]. Mathematically, the\nsmearing permits substitution of the Dirac-delta function by a Gaussian distribution of minimal\nlength√\nθ.\nSpecifically, the formulation of the energy density for a gravitational source that is static,\nspherically symmetric, and resembles both a smeared and particle-like structure has been examined\n[64, 65] given by\nρθ(r) =M\n(4πθ)3/2exp\u0010\n−r2\n4θ\u0011\n, (56)\nwhere the mass Mis diffused throughout a region of linear size√\nθdue to the intrinsic uncertainty\nencoded in the coordinate commutator. Like black holes, see e.g., Refs.[64, 66], the wormhole metric\nis expected to be modified when a noncommutative spacetime is taken into account, see [67–69].\nMoreover, we also get inspired by the work of Mehdipour when searching for a new fluid model. A\nLorentzian distribution of particle-like gravitational source permits possible energy density profile\nas given in Ref.[68, 70, 71] as follows:\nρϕ(r) =M√ϕ\nπ2(r2+ϕ)2(57)\nwith ϕbeing the noncommutativity parameter. We next consider Eqs. (28)-(30) and employ the\ndensity profiles given above to quantify the decoupled solutions, f(r). We assume that ε0\n0=ρθ,ϕ\nand solve for f(r) to obtain.\nf(r) =\n\nM\u0012\nerf\u0010\nr\n2√\nθ\u0011\n−erf\u0010\nr0\n2√\nθ\u0011\n−r√π√\nθe−r2\n4θ+r0√π√\nθe−r2\n0\n4θ\u0013\nforρθ=ε0\n0,\n2M\nπ \nr0/√ϕ\nr2\n0\nϕ+1−r/√ϕ\nr2\nϕ+1+ tan−1\u0010\nr√ϕ\u0011\n−tan−1\u0010\nr0√ϕ\u0011!\nforρϕ=ε0\n0.(58)\nWe can simply check that a condition of f(r=r0) = 0 is satisfied. Substituting f(r) to\nEq.(33), the original (traversable) wormhole solutions can be geometrically deformed. In this case,\nthe original wormhole solution will be deformed by the above results. Therefore, we obtain for\nρθ=ε0\n0\nb(r)√\nθ=r0√\nθ\u0010r/√\nθ\nr0/√\nθ\u00112ωβ+ω+3\n2ωβ−ω+1\n+δM√\nθ\u0012\nerf\u0012r\n2√\nθ\u0013\n−erf\u0012r0\n2√\nθ\u0013\n−r√π√\nθe−r2\n4θ+r0√π√\nθe−r2\n0\n4θ\u0013\n, (59)12\nand for ρϕ=ε0\n0\nb(r)√ϕ=r0√ϕ\u0010r/√ϕ\nr0/√ϕ\u00112ωβ+ω+3\n2ωβ−ω+1\n+2δ\nπM√ϕ\nr0/√ϕ\nr2\n0\nϕ+ 1−r/√ϕ\nr2\nϕ+ 1+ tan−1\u0012r√ϕ\u0013\n−tan−1\u0012r0√ϕ\u0013\n. (60)\nWe can show that the shape function (59) and (60) satisfy the usual properties, see Fig.(5), namely\nFIG. 5: We show the behaviors ofb(r)√\nθ,b′(r)√\nθ,b(r)√\nθrandb(r)−rb′(r)√\nθversusr√\nθin Eq.(59) for β=−1.1,ω= 0.8,\nr0= 1, δ= 0.001 andM√\nθ= 1 (Left Panel) andb(r)√ϕ,b′(r)√ϕ,b(r)√ϕrandb(r)−rb′(r)√ϕversusr√ϕin Eq.(60) for\nβ=−1.1,ω= 0.8,r0= 1,δ= 0.001 andM√ϕ= 1 (Right Panel).\nthe throat condition and so on. More specifically, using b′(r0)<1, we have for Eq.(59):\nδ <1\nM√\nθr2\n0\u0010\n2√πmer2\n0\n4+ 2√πer2\n0\n4\u0011\n, (61)\nand for Eq.(60):\nδ 0−→ Q2<−4r2\n0ω−4r2\n0\n2βδω−δω+δ, (67)\nwith the following conditions:\nδ >0∧ω >0∧\u0012\nr0>0∧β <ω−1\n2ω\u0013\n. (68)\nMoreover, in case of ω=ω(r) and constant β, we have\nb(r) =r0\u0012r\nr0\u00132β+1\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u00134(β−1)\n2β−1\n+δQ2\n2\u00101\nr0−1\nr\u0011\n. (69)\nWe find for the flare-out condition to be satisfied:\nδ >0∧r0>0∧ |Q| <2r\nr2\n0\nδ. (70)\nNotice that values of Qdepend on a parameter δand do not depend on βin this case.14\nFIG. 6: We show the behaviors of b(r), b′(r), b(r)/randb(r)−rb′(r) versus rin Eq.(69) for β=−1.1,r0= 1\nandQ= 1 (Left Panel) and Q= 2 (Right Panel).\nV. EMBEDDING DIAGRAM\nIn this section, we analyse the embedding diagrams to represent the wormhole solutions by\nconsidering an equatorial slice θ=π/2 at some fix moment in time t= constant. The metric then\nbecomes\nds2=1\u0010\n1−b(r)\nr\u0011dr2+r2dϕ2, (71)\nHaving embed the metric (71) into three-dimensional Euclidean space, we can visualize this slice.\nHere we parameterize spacetime using the cylindrical coordinates as\nds2=dz2+dr2+r2dϕ2, (72)\nwhich can be rewritten as\nds2=\u0010\n1 +\u0010dz\ndr\u00112\u0011\ndr2+r2dϕ2. (73)\nHaving compared Eq.(71) with Eq.(73), we come up with:\ndz\ndr=±r\u0010r\nr−b(r)−1\u0011\n. (74)\nTo test the results, we consider b(r) given by Eq.(60) and Eq.(69). Invoking numerical techniques\nallows us to illustrate the wormhole shape given in Fig.(7). In Fig.(7), we consider various shape\nfunctions b(r) given in Eq.(50), (60), and (69). We use β=−1.1 for all plots, and take δ=\n0.01, ω= 0.8 and M/√ϕ= 1 for Eq.(60) and δ= 0.01, ω= 0.8 and Q= 1 for Eq.(69).15\nFIG. 7: We display the embedding diagrams of various shape functions given in Eq.(50), (60), and (69).\nVI. ENERGY CONDITIONS\nWe can further explore and check the energy conditions. In this section, we only consider\nthe two types of energy conditions to examine the wormhole solutions. The first one is null\nenergy condition (NEC) given as Tµνkµkν≥0, which determines the non-negative value of energy-\nmomentum tensor with kµbeing null vectors. The NEC yields ρ+pr≥0. Please note that the null\nenergy condition (NEC) can be understood as the requirement for the energy of particles moving\nalong a null geodesic, such as photons and massless particles, to remain non-negative. Additionally,\nthe strong energy condition (SEC) defined as\u0000\nTµν−1\n2Tgµν\u0001\nXµXν≥0 with Xµbeing a timelike\nvector field, yields ρ+ 2pt≥0 and ρ+pr+ 2pt≥0. However, the traversable wormholes in some\nparticular models, e.g., Casimir wormholes [74, 75], need the (exotic) matter which violates the\nenergy conditions.\nA. Constants ω&β\nWe first consider constants ωandβand take b(r) given in Eq.(59), and then compute the energy\ndensity. We find\nθ ρ(Y) =−1\n8π(ω+ 1)Y2\u0012Y\nY0\u0013 2(ω+1)\nω(2β−1)+1\n+δP\n(4π)3/2exp\u0010\n−Y2\n4\u0011\n. (75)16\nUsing ρ(Y), we can compute pYandpYto obtain\nθ pr(Y) =−ω\n8π(ω+ 1)Y2\u0012Y\nY0\u0013 2(ω+1)\nω(2β−1)+1\n−δP\n8π3/2Y3\u0010\n4√π\u0012\nerf\u0012Y0\n2\u0013\n−erf\u0012Y\n2\u0013\u0013\n+\u0000\nY2+ 4\u0001\nY e−Y2\n4−4Y0e−Y2\n0\n4\u0011\n,(76)\nθ pt(Y) =−ωβ\n8π(ω+ 1)Y2\u0012Y\nY0\u0013 2(ω+1)\nω(2β−1)+1\n−δP\n4πθY3\u0010\nerf\u0012Y\n2\u0013\n−erf\u0012Y0\n2\u0013\n−e−Y2\n4Y√π+e−Y2\n0\n4Y0√π\u0011\n, (77)\nwhere Y≡r/√\nθ, P≡M/√\nθ. The stress-energy tensor (SET) can be computed using Eq.(11).\nWe next consider constants ωandβand take b(r) given in Eq.(60), and then compute the energy\ndensity to obtain\nϕ ρ(Z) =−1\n8π(ω+ 1)Z2\u0012Z\nZ0\u0013 2(ω+1)\nω(2β−1)+1\n+δQ\nπ2\u00121\n(1 +Z2)2\u0013\n. (78)\nUsing ρ(Z), we can compute prandptto obtain\nϕ pr(Z) =−ω\n8π(ω+ 1)Z2\u0012Z\nZ0\u0013 2(ω+1)\nω(2β−1)+1\n−δQ\nπ2Z3\u0012\n−Z+ 3Z3\n(1 +Z2)2+Z0\n1 +Z2\n0+ tan−1(Z)−tan−1(Z0)\u0013\n, (79)\nϕ pt(Z) =−ωβ\n8π(ω+ 1)Z2\u0012Z\nZ0\u0013 2(ω+1)\nω(2β−1)+1\n−δQ\n2π2Z3\u0012Z\n1 +Z2−Z0\n1 +Z2\n0−tan−1(Z) + tan−1(Z0)\u0013\n, (80)\nwhere Z≡r/√ϕ, Q≡M/√ϕ. We find from Fig.(6) that for small values of δthe NEC and SEC\ncannot be satisfied. However, since b(r) is dependent on δ, the usual properties of b(r) cannot be\nsatisfied, e.g., b(r)/r↛0 for r→ ∞ ifδis not so small, δ≫ O(1). We next consider constants ω\nandβand take b(r) given in Eq.(65), and then compute the energy density. We find\nρ(r) =−1\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n+δQ2\n8πr4. (81)\nUsing f(r), we can compute pr=θ1\n1andpt=θ2\n2to obtain\npr(r) =−ω\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n−δQ2\n8πr4\u00102r\nr0−3\u0011\n, (82)\npt(r) =−ωβ\n8π(ω+ 1)r2\u0012r\nr0\u0013 2(ω+1)\nω(2β−1)+1\n−δQ2\n8πr4\u0010\n1−r\nr0\u0011\n. (83)17\nFIG. 8: We show the behaviors of θρeff+θpeff\nr,θρeff+ 2θpeff\nt,θρeff+θpeff\nr+ 2θpeff\ntversus r/√\nθusing\nβ=−1.1,r0= 1,ω= 0.8 for the Gaussian distribution Eq.(56) (Left panel), and ϕρeff+ϕpeff\nr,ϕρeff+2ϕpeff\nt,\nϕρeff+ϕpeff\nr+ 2ϕpeff\ntversus r/√ϕusing β=−1.1,r0= 1, ω= 0.8 for the Lorentzian distribution Eq.(57)\n(Right panel).\nThe behaviors of θρeff+θpeff\nr,θρeff+ 2θpeff\nt,θρeff+θpeff\nr+ 2θpeff\ntversus r/√\nθfor the Gaussian\ndistribution Eq.(56), and ϕρeff+ϕpeff\nr,ϕρeff+ 2ϕpeff\nt,ϕρeff+ϕpeff\nr+ 2ϕpeff\ntversus r/√ϕfor the\nLorentzian distribution Eq.(57) has been illustrated in Fig.(8). In Fig.(9), we notice that the NEC\nis also violated for the charged fluid deformation.\nFIG. 9: We show the behaviors of ρeff+peff\nr,ρeff+ 2peff\ntandρeff+peff\nr+ 2peff\ntversus rusing β=−1.1,r0= 1\nω= 0.8 and Q= 1.\nB. Variable ω& Constant β\nIn this case, we can easily compute the energy density to obtain\nθ ρ(Y) =−β\n4π(β−1)Y2\u0012Y\nY0\u0013 2\n2β−1+1\u0012βY−2βY0+r0\nY0−βY0\u0013 2\n1−2β+1\n+δP\n(4π)3/2exp\u0010\n−Y2\n4\u0011\n.(84)18\nUsing ρ(Y), we can compute pYandpYto obtain\nθ pr(Y) =1\n4π(β−1)Y2\u0012Y\nY0\u0013 2\n2β−1\u0012βY−2βY0+Y0\nY0−βY0\u0013 2\n1−2β+1\n−δP\n8π3/2Y3\u0010\n4√π\u0012\nerf\u0012Y0\n2\u0013\n−erf\u0012Y\n2\u0013\u0013\n+\u0000\nY2+ 4\u0001\nY e−Y2\n4−4Y0e−Y2\n0\n4\u0011\n,(85)\nθ pt(Y) =β\n4π(β−1)Y2\u0012Y\nY0\u0013 2\n2β−1\u0012βY−2βY0+Y0\nY0−βY0\u0013 2\n1−2β+1\n−δP\n4πθY3\u0010\nerf\u0012Y\n2\u0013\n−erf\u0012Y0\n2\u0013\n−e−Y2\n4Y√π+e−Y2\n0\n4Y0√π\u0011\n, (86)\nWe next consider ω(r) and constant βand take b(r) given in Eq.(69), and then compute the energy\ndensity to obtain\nϕ ρ(Z) =−β\n4π(β−1)Z2\u0012Z\nZ0\u0013 2\n2β−1+1\u0012βZ−2βZ0+Z0\nZ0−βZ0\u0013 2\n1−2β+1\n+δQ\nπ2\u00121\n(1 +Z2)2\u0013\n.(87)\nUsing ρ(Z), we can compute prandptto obtain\nϕ pr(Z) =1\n4π(β−1)Z2\u0012Z\nZ0\u0013 2\n2β−1\u0012βZ−2βZ0+Z0\nZ0−βZ0\u0013 2\n1−2β+1\n−δQ\nπ2Z3\u0012\n−Z+ 3Z3\n(1 +Z2)2+Z0\n1 +Z2\n0+ tan−1(Z)−tan−1(Z0)\u0013\n, (88)\nϕ pt(Z) =β\n4π(β−1)Z2\u0012Z\nZ0\u0013 2\n2β−1\u0012βZ−2βZ0+Z0\nZ0−βZ0\u0013 2\n1−2β+1\n−δQ\n2π2Z3\u0012Z\n1 +Z2−Z0\n1 +Z2\n0−tan−1(Z) + tan−1(Z0)\u0013\n. (89)\nWe consider ω(r) and constant βand take b(r) given in Eq.(69), and then compute the energy\ndensity. We find\nρ(r) =−β\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1+1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n+δQ2\n8πr4. (90)\nUsing f(r), we can compute pr=θ1\n1andpt=θ2\n2to obtain\npr(r) =1\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n−δQ2\n8πr4\u00102r\nr0−3\u0011\n, (91)\npt(r) =β\n4π(β−1)r2\u0012r\nr0\u0013 2\n2β−1\u0012βr−2βr0+r0\nr0−βr0\u0013 2\n1−2β+1\n−δQ2\n8πr4\u0010\n1−r\nr0\u0011\n, (92)\nThe behaviors of θρeff+θpeff\nr,θρeff+ 2θpeff\nt,θρeff+θpeff\nr+ 2θpeff\ntversus r/√\nθfor the Gaussian\ndistribution Eq.(56) and ϕρeff+ϕpeff\nr,ϕρeff+ 2ϕpeff\nt,ϕρeff+ϕpeff\nr+ 2ϕpeff\ntversus r/√ϕfor the\nLorentzian distribution Eq.(57) can be seen in Fig.(10). In Fig.(11), we notice that the NEC is\nalso violated for the charged fluid deformation.19\nFIG. 10: We show the behaviors of θρeff+θpeff\nr,θρeff+ 2θpeff\nt,θρeff+θpeff\nr+ 2θpeff\ntversus r/√\nθusing\nβ=−0.01,r0= 1,P= 1 for the Gaussian distribution Eq.(56) (Left panel), and ϕρeff+ϕpeff\nr,ϕρeff+ 2ϕpeff\nt,\nϕρeff+ϕpeff\nr+ 2ϕpeff\ntversus r/√ϕusing β=−0.01,r0= 1, Q= 1 for the Lorentzian distribution Eq.(57)\n(Right panel).\nFIG. 11: We show the behaviors of ρeff+peff\nr,ρeff+ 2peff\ntandρeff+peff\nr+ 2peff\ntversus rusing β=−0.01,\nr0= 1δ= 0.01and Q= 1.\nVII. WEAK GRAVITATIONAL LENSING\nEmbarking on this section, we revisit the Gauss-Bonnet theorem and embark on the calculation\nof the weak deflection angle for wormhole configurations. Our starting point is the expression for\nnull geodesics, ds2= 0, which can be rearranged to yield:\ndt2=γijdxidxj=1\nWdr2+r2dΩ2, (93)\nWithin this context, indices i and j represent the spatial dimensions (1 to3), and γijdenotes\nthe optical metric. To utilize the Gauss-Bonnet theorem effectively, we must first calculate the\nGaussian curvature associated with Eq.(50). This calculation is presented in detail here:\nK=R\n2≈6β2r2\n0\nr4+2βr2\n0\nr4−r2\n0\nr4−6β2r0\nr3−2βr0\nr3. (94)20\nWithin this framework, γ≡det(γij) represents the determinant of the optical metric, and R\ndenotes the Ricci scalar. Let Dbe a compact, oriented, nonsingular two-dimensional Riemannian\nsurface characterized by its Euler characteristic χ(D)and Gaussian curvature K. This domain is\nenclosed by a piecewise-smooth curve with geodesic curvature κ. The link between the deflection\nangle of light and the Gaussian curvature stems from the Gauss-Bonnet theorem, which is invoked\nby employing the following expressions:\nZ Z\nDKdS+I\n∂Dκdt+X\ni=1βi= 2πχ(D), (95)\nWithin this context, dSrepresents the differential element of area, κdenotes the geodesic\ncurvature of the boundary, defined as κ=|∇˙C˙C|andβirepresents the ithexterior angle. For a\nparticular region ˜Denclosed by a geodesic C1extending from the source Sto the observer Oand\na circular curve CRintersecting C1orthogonally at SandO, Equation (95) reduces to:\nZ Z\n˜DKdS+Z\nCRκ(CR)dt=π, (96)\nDuring this derivation, we employed the conditions κ(C1) = 0 and the Euler characteristic χ(˜D) =\n1. For the specific circular curve CR:=r(ϕ) =R= const, the non-zero segment of the geodesic\ncurvature is calculated as:\nκ(CR) =\u0010\n∇˙CR˙CR\u0011r\n=˙Cϕ\nR(∂ϕ˙Cr\nR) + Γr\nϕϕ(˙Cϕ\nR)2, (97)\nIn this context, ˙CRrepresents the directional derivative of the circular curve CR, and Γr\nϕϕ\nsignifies the Christoffel symbol corresponding to the optical metric (93). As Rtends towards\ninfinity, we arrive at:\nlim\nR→∞[κ(CR)dt] =dϕ. (98)\nBy substituting Eq. (98) into Eq. (96), we arrive at:\nZ Z\nDKdS+Zπ+Θ\n0dϕπ=π. (99)\nIn this context, the surface area on the equatorial plane is formulated as [30]:\ndS=√γdrdϕ (100)\nFollowing this, the weak deflection angle of light can be determined for Eq.50 as:\nΘ =−Z Z\n˜DKdS=−Zπ\n0Z∞\nb\nsinϕKdS\n≃ −3πβ2r2\n0\n2b2−πβr2\n0\n2b2+πr2\n0\n4b2+12β2r0\nb+4βr0\nb. (101)21\nFIG. 12: The deflection angle, Θ, as expressed in Eq.(101) is plotted against the impact parameter busing\nβ=−1.1 in Eq.(50).\nIn this analysis, we utilized the zero-order particle trajectory r=b/sinϕ, where 0 ≤ϕ≤πin\nthe weak deflection limit. The dependence of the deflection angle on the impact parameter, as\ninfluenced by the wormhole geometry, is presented graphically in Fig.(12). Our findings reveal\nthat the deflection angle is contingent upon the parameters βandr0forω(r) =−r0\nβr. For specific\nvalues of β, it is observed that the deflection angle increases as r0grows. These results warrant\ncomparison with those obtained using an alternative approach recently proposed in Ref.[76].\nVIII. CONCLUDING REMARKS\nIn this work, we investigate wormhole solutions invoking the Minimal Geometric Deformation\n(MGD) procedure within the framework of Unimodular Gravity. We employed a static and spheri-\ncally symmetric Morris-Thorne traversable wormholes and considered both constants and variables\nin the equation of state parameter. More specifically, we considered ω= cont .andω=ω(r) =−r0\nβr.\nWe computed field equations and derived the shape functions. We showed that the usual proper-\nties of the obtained shape functions has been satisfied. We considered the gravitational decoupling\ntechnique and considered various forms of the energy density for a smeared and particle-like grav-\nitational source in the context of noncommutative geometry and a statically charged fluid. We\nobtained the novel wormhole solutions showing the violation of the Null Energy Condition (NEC).22\nWe used the Gauss-Bonnet theorem to compute the weak deflection angle of light for the worm-\nhole solutions. We also found that the deflection angle depends upon the parameter βandr0for\nω(r) =−r0\nβr.\nWe can further test whether these wormholes be sustained by their own quantum fluctuations.\nIn particular, the energy density of the graviton-one loop contribution to classical energy in a\ntraversable wormhole background and the finite one loop energy density have to be considered as\na self-consistent source for these wormhole geometries. To this end, we shall follow the existing\npublications, see e.g., [67, 77, 78]. However, the wormhole equation of state is still unknown and\nhence the present work can be tested by the wormhole observations, see. e.g., [79–83].\nAcknowledgments\nP. P. is financially supported by a DPST scholarship for her undergraduate study. N.K. is funded\nby the National Research Council of Thailand (NRCT): Contract number N42A660971. P.C. is\nfinancially supported by Thailand NSRF via PMU-B under grant number PCB37G6600138. A.\n¨O. would like to acknowledge the contribution of the COST Action CA21106 - COSMIC WISPers\nin the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers) and the COST\nAction CA22113 - Fundamental challenges in theoretical physics (THEORY-CHALLENGES). We\nalso thank TUBITAK and SCOAP3 for their support.\n[1] S. Weinberg, Rev. Mod. Phys. 61(1989), 1-23\n[2] D. R. Finkelstein, A. A. Galiautdinov and J. E. Baugh, J. Math. Phys. 42(2001), 340-346\n[3] M. Henneaux and C. Teitelboim, Phys. Lett. B 222(1989), 195-199\n[4] W. G. Unruh, Phys. Rev. D 40(1989), 1048\n[5] Y. J. Ng and H. van Dam, J. Math. Phys. 32(1991), 1337-1340\n[6] E. Alvarez, JHEP 03(2005), 002\n[7] G. F. R. Ellis, H. van Elst, J. Murugan and J. P. Uzan, Class. Quant. Grav. 28(2011), 225007\n[8] G. F. R. Ellis, Gen. Rel. Grav. 46(2014), 1619\n[9] C. Barcel´ o, R. Carballo-Rubio and L. J. Garay, Phys. Rev. D 89(2014) no.12, 124019\n[10] D. J. Burger, G. F. R. Ellis, J. Murugan and A. Weltman, [arXiv:1511.08517 [hep-th]].\n[11] E. ´Alvarez and S. Gonz´ alez-Mart´ ın, Phys. Rev. D 92(2015) no.2, 024036\n[12] M. Shaposhnikov and D. Zenhausern, Phys. Lett. B 671(2009), 187-192\n[13] L. Smolin, Phys. Rev. D 84(2011), 04404723\n[14] P. Jain, A. Jaiswal, P. Karmakar, G. Kashyap and N. K. Singh, JCAP 11(2012), 003\n[15] E. Alvarez and M. Herrero-Valea, Phys. Rev. D 87(2013), 084054\n[16] A. Eichhorn, Class. Quant. Grav. 30(2013), 115016\n[17] A. O. Barvinsky, N. Kolganov, A. Kurov and D. Nesterov, Phys. Rev. D 100(2019) no.2, 0235425\n[18] A. O. Barvinsky and A. Y. Kamenshchik, Phys. Lett. B 774(2017), 59-63\n[19] J. Ovalle, Phys. Rev. D 95(2017) no.10, 104019\n[20] F. Tello-Ortiz, S. K. Maurya and P. Bargue˜ no, Eur. Phys. J. C 81(2021) no.5, 426\n[21] J. Ovalle, R. Casadio, R. da Rocha and A. Sotomayor, Eur. Phys. J. C 78(2018) no.2, 122\n[22] R. Casadio and J. Ovalle, Gen. Rel. Grav. 46(2014), 1669\n[23] R. Casadio, J. Ovalle and R. da Rocha, Class. Quant. Grav. 31(2014), 045016\n[24] J. Ovalle and F. Linares, Phys. Rev. D 88(2013) no.10, 104026\n[25] J. Ovalle, L. ´A. Gergely and R. Casadio, Class. Quant. Grav. 32(2015), 045015\n[26] R. Casadio, J. Ovalle and R. da Rocha, EPL 110(2015) no.4, 40003\n[27] R. Casadio, J. Ovalle and R. da Rocha, Class. Quant. Grav. 32(2015) no.21, 215020\n[28] J. Ovalle, Phys. Lett. B 788(2019), 213-218\n[29] F. Tello-Ortiz, B. Mishra, A. Alvarez and K. N. Singh, Fortsch. Phys. 71(2023) no.2-3, 2200108\n[30] G. W. Gibbons and M. C. Werner, Class. Quant. Grav. 25(2008), 235009\n[31] M. C. Werner, Gen. Rel. Grav. 44, 3047-3057 (2012).\n[32] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura and H. Asada, Phys. Rev. D 94, no.8, 084015 (2016).\n[33] A. Ishihara, Y. Suzuki, T. Ono and H. Asada, Phys. Rev. D 95, no.4, 044017 (2017).\n[34] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 96, no.10, 104037 (2017).\n[35] G. Crisnejo and E. Gallo, Phys. Rev. D 97, no.12, 124016 (2018).\n[36] Z. Li and A. ¨Ovg¨ un, Phys. Rev. D 101, no.2, 024040 (2020).\n[37] Z. Li, G. Zhang and A. ¨Ovg¨ un, Phys. Rev. D 101, no.12, 124058 (2020).\n[38] K. Jusufi and A. ¨Ovg¨ un, Phys. Rev. D 97, no.2, 024042 (2018).\n[39] A. ¨Ovg¨ un, ˙I. Sakallı and J. Saavedra, JCAP 10, 041 (2018).\n[40] A. ¨Ovg¨ un, Phys. Rev. D 98, no.4, 044033 (2018).\n[41] Y. Kumaran and A. ¨Ovg¨ un, Turk. J. Phys. 45, no.5, 247-267 (2021) [arXiv:2111.02805 [gr-qc]].\n[42] W. Javed, M. Aqib and A. ¨Ovg¨ un, Phys. Lett. B 829, 137114 (2022).\n[43] R. C. Pantig and A. ¨Ovg¨ un, Eur. Phys. J. C 82, no.5, 391 (2022).\n[44] M. Okyay and A. ¨Ovg¨ un, JCAP 01, no.01, 009 (2022).\n[45] A. Belhaj, M. Benali, A. El Balali, H. El Moumni and S. E. Ennadifi, Class. Quant. Grav. 37, no.21,\n215004 (2020).\n[46] S. U. Islam, R. Kumar and S. G. Ghosh, JCAP 09, 030 (2020).\n[47] R. Kumar, S. G. Ghosh and A. Wang, Phys. Rev. D 101, no.10, 104001 (2020).\n[48] Q. Qi, Y. Meng, X. J. Wang and X. M. Kuang, Eur. Phys. J. C 83, no.11, 1043 (2023).\n[49] Y. Huang, Z. Cao and Z. Lu, JCAP 01, 013 (2024).24\n[50] M. A. A. de Paula, H. C. D. Lima, Junior, P. V. P. Cunha and L. C. B. Crispino, Phys. Rev. D 108,\nno.8, 084029 (2023).\n[51] Y. Huang, B. Sun and Z. Cao, Phys. Rev. D 107, no.10, 104046 (2023).\n[52] Y. Huang and Z. Cao, Phys. Rev. D 106, no.10, 104043 (2022).\n[53] N. Parbin, D. J. Gogoi, J. Bora and U. D. Goswami, Phys. Dark Univ. 42, 101315 (2023).\n[54] M. Halla and V. Perlick, Phys. Rev. D 107, no.2, 024048 (2023)\n[55] M. Cadoni, A. P. Sanna and M. Tuveri, Phys. Rev. D 102(2020) no.2, 023514\n[56] A. S. Agrawal, B. Mishra and P. H. R. S. Moraes, Eur. Phys. J. Plus 138(2023) no.3, 275\n[57] B. J. Carr, T. Harada and H. Maeda, Class. Quant. Grav. 27(2010), 183101\n[58] S. Bahamonde, M. Jamil, P. Pavlovic and M. Sossich, Phys. Rev. D 94(2016) no.4, 044041\n[59] P. E. Kashargin and S. V. Sushkov, Universe 6(2020) no.10, 186\n[60] D. Wang and X. h. Meng, Eur. Phys. J. C 76(2016) no.9, 484\n[61] M. Cataldo and P. Meza, Phys. Rev. D 87(2013) no.6, 064012\n[62] R. Garattini and P. Channuie, [arXiv:2311.04620 [gr-qc]].\n[63] A. Smailagic and E. Spallucci, J. Phys. A 36(2003), L467\n[64] P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 632(2006), 547-551\n[65] F. S. N. Lobo and R. Garattini, JHEP 12(2013), 065\n[66] E. Di Grezia, G. Esposito and G. Miele, J. Phys. A 41(2008), 164063\n[67] R. Garattini and F. S. N. Lobo, Phys. Lett. B 671(2009), 146-152\n[68] F. Rahaman, I. Karar, S. Karmakar and S. Ray, Phys. Lett. B 746(2015), 73-78\n[69] N. Rahman, M. Kalam, A. Das, S. Islam, F. Rahaman and M. Murshid, Eur. Phys. J. Plus 138(2023)\nno.2, 146\n[70] S. H. Mehdipour, Eur. Phys. J. Plus 127(2012), 80\n[71] J. Liang and B. Liu, EPL 100(2012) no.3, 30001\n[72] I. Dymnikova, Class. Quant. Grav. 21(2004), 4417-4429\n[73] I. Dymnikova and E. Galaktionov, J. Phys. Conf. Ser. 2103 (2021) no.1, 012078\n[74] R. Garattini, Eur. Phys. J. C 79(2019) no.11, 951\n[75] K. Jusufi, P. Channuie and M. Jamil, Eur. Phys. J. C 80(2020) no.2, 127\n[76] Z. Li, [arXiv:2401.12525 [gr-qc]].\n[77] R. Garattini, Class. Quant. Grav. 22(2005), 1105-1118\n[78] R. Garattini and F. S. N. Lobo, Class. Quant. Grav. 24(2007), 2401-2413\n[79] F. Abe, Astrophys. J. 725(2010), 787-793\n[80] Y. Toki, T. Kitamura, H. Asada and F. Abe, Astrophys. J. 740(2011), 121\n[81] N. Godani and G. C. Samanta, Annals Phys. 429(2021), 168460\n[82] T. Ohgami and N. Sakai, Phys. Rev. D 91(2015) no.12, 124020\n[83] D. C. Dai and D. Stojkovic, Phys. Rev. D 100(2019) no.8, 083513" }, { "title": "2401.16859v1.Line_tension_in_a_thick_soap_film.pdf", "content": "SUPPLEMENTARY MATERIAL - Line tension in a thick soap film\nTh´ eo Lenavetier, Ga¨ elle Aud´ eoud, Marion Berry, Ana¨ ıs Gauthier,\nRapha¨ el Poryles, Corentin Tr´ egou¨ et, and Isabelle Cantat\nUniv Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F- 35000 Rennes.\n(Dated: January 31, 2024)\nINTERFACIAL STRESS TENSOR\nWe consider the piece of film represented in Fig. 1.\nIt is limited by the two interfaces and by the vertical\nplanes which intersect the plane z= 0 along the elemen-\ntary lengths dξ ⃗ n andds⃗t. The unit vector ⃗ nis chosen\nalong the thickness gradient and the unit vector ⃗tis per-\npendicular to ⃗ n(both are in the ( x, y) plane):\n⃗ n= (hx, hy,0)//radicalig\nh2x+h2y⃗t= (−hy, hx,0)//radicalig\nh2x+h2y.\n(1)\nThe 2D film stress tensor σfcan be built on this sys-\ntem, by considering the forces exerted on the lateral faces.\nNote that the air pressure is taken as the pressure refer-\nence, so that no forces are exerted on the top and bottom\ninterfaces. The norm of the thickness gradient is a small\nparameter in the problem, as classically used in the lu-\nbrication approximation. To build the film stress tensor,\nwe anticipate that the forces exerted on the system are\nof order 2 in this parameter (as will be shown below),\nand we thus drop higher order terms. Especially δγis of\norder 2. The differential operators ∇and ∆ are the 2D\ngradient and laplacian in the ( x, y) plane.\nLet us first consider the face 1 defined in Fig. 1. It is a\nrectangle of area dS1= 2hdsand of normal ⃗ n. The pres-\nsure in the film is P=−γ0∆h, leading to the pressure\nforce d⃗fP\n1= 2γ0h∆h ds⃗ n . The local tension is γ0+δγ,\nacting on the length ds, at the top and bottom interfaces.\nThe force orientation differs between both interfaces: it\nis along ⃗ ntop/bot= (⃗ n± ∥∇ h∥⃗ ez)/(1 + (∇h)2)1/2. The\nresulting force is thus d⃗fT\n1= 2(γ0(1−(∇h)2/2)+δγ)ds⃗ n.\nThe face 2 is a trapezoid of area dS2= 2hdξand of nor-\nmal⃗t. The pressure force is d⃗fP\n2= 2γ0h∆h dξ⃗t. The ten-\nsion acts along the elementary length dξ(1 + (∇h)2)1/2,\nFIG. 1. Scheme of the film element ds dξ used for the de-\ntermination of the stress tensor. The face 1, of normal ⃗ n, is\nin green, the face 2, of normal ⃗t, is in blue. The thickness\ngradient is oriented along ⃗ n.and the force is oriented along ⃗t, leading to the force\nd⃗fT\n2= 2(γ0(1 + (∇h)2/2) +δγ)dξ⃗t.\nThe interfacial stress tensor σcapassociated to these\ncapillary forces verifies by definition d⃗f1=d⃗fT\n1+d⃗fP\n1=\nσcap ·⃗ ndsand similarly d⃗f2=σcap ·⃗tdξ. Its expression in\nthe basis Be= (⃗ n,⃗t) is thus\nσcap=γ0/parenleftbigg\n−(∇h)20\n0 ( ∇h)2/parenrightbigg\nBe+ 2 (γ0(1 +h∆h) +δγ)I\n(2)\nWe thus get\nσcap=σ∗\ncap+σfI , (3)\nwith the film tension σfdefined as the isotropic part of\nthe interfacial stress\nσf= 2 (γ0(1 +h∆h) +δγ), (4)\nandσ∗\ncapdefined as its deviatoric part, which expression\nin the initial basis B0= (⃗ ex,⃗ ey) is\nσ∗\ncap=−γ0/parenleftbiggh2\nx−h2\ny2hxhy\n2hxhyh2\ny−h2\nx/parenrightbigg\nB0. (5)\nThe capillary force acting on the film element is Fc=\ndivσcap. We thus compute\ndiv(σfI) = 2 γ0h∇∆h+ 2γ0∇h∆h+ 2∇(δγ),(6)\nand\ndivσ∗\ncap ·⃗ ex=−γ0(2hxhxx−2hyhxy+ 2hyhxy+ 2hxhyy)\n=−2γ0hx∆h\ndivσ∗\ncap ·⃗ ey=−γ0(2hxhxy+ 2hyhxx+ 2hyhyy−2hxhxy)\n=−2γ0hy∆h\ndivσ∗\ncap=−2γ0∇h∆h (7)\nFinally we obtain\ndivσcap= 2γ0h∇∆h+ 2∇(δγ), (8)\nwhich is used to obtain eq. (3) in the main article.\nLINE TENSION\nThe capillary forces are localized in the vicinity of a\ncurveC, in a domain of width ℓ. In the limit of small ℓκ,arXiv:2401.16859v1 [cond-mat.soft] 30 Jan 20242\nwith κthe curvature of C, a line tension Tcan be defined\n(for each interface). With this definition, 2 Tis the excess\nforce exerted by one side of the film on the other one,\nacross a line oriented along the normal to the boundary.\nThis excess is considered with respect to the force field\nfar from C, where the thickness is homogeneous and the\nsurface tension equals to γ0at the dominant order. This\nimplies\nT=1\n2/integraldisplayℓ∞\nℓ−∞t·(σcap−2γ0I)·tdξ . (9)\nA first step is to determine the surface tension within the\ntransition domain. In this domain, the capillary forces\ndominate the viscous forces, and thus control the surface\ntension value. In the small ℓκlimit the force balance on\na film element within the transition domain becomes\n2γ0h∇(∆h) + 2∇δγ= 0, (10)\nwhich can be simplified at leading order in ℓκinto\nγ0h ∂ξξξh+∂ξδγ= 0.\nBy integration between ℓ−∞andξwe get\n/integraldisplayξ\nℓ−∞(γ0h∂ξξξh+∂ξδγ)dξ′= 0,\nleading to, after integration by part,\nδγ(ξ)\nγ0=/integraldisplayξ\nℓ−∞∂ξh ∂ξξh dξ′−[h∂ξξh]ξ\nℓ−∞\n=1\n2/parenleftbigg∂h\n∂ξ/parenrightbigg2\n−h∂2h\n∂ξ2. (11)\nWe can now determine\nt·(σcap−2γ0I)·t= 2γ0h∆h+ 2δγ+γ0(∇h)2\n=γ0/parenleftbig\n2h∂ξξh+ (∂ξh)2−2h∂ξξh+ (∂ξh)2/parenrightbig\n= 2γ0(∂ξh)2. (12)\nInserting this expression into eq. (9) we obtain\nT=γ0/integraldisplayℓ∞\nℓ−∞/parenleftbigg∂h\n∂ξ/parenrightbigg2\ndξ . (13)\nVISCOSITY UPPER BOUND\nThe paper [1] addresses the problem of a disc of radius\nˆaand uniform thickness ˆhmoving at the velocity ˆU⃗ exin\nthe plane ( x, y) of a thin sheet of liquid of same thickness\nand viscosity ˆ η(the symbol ˆ·indicates notations used in\n[1], see Fig. 2). The bulk phases above and below this\nliquid sheet are fluids of viscosity ˆ µ.\nFIG. 2. Scheme of the problem solved in [1]. A cylindrical\nsolid moves at the velocity ˆUin a liquid sheet (adapted from\nthe Fig. 1 of [1]).\nIn the viscous regime, the equation of motion of the\nthin sheet is (from eq. (2.18) of [1])\nˆhˆη∇2ˆ uM−ˆh∇ˆpM+ˆF= 0 (14)\nwithˆ uMand ˆpMthe velocity and the pressure of the\nsheet, which are assumed to be uniform across the film,\nandˆFthe force exerted by the fluid phases above and\nbelow the sheet. These fluids verify the Stokes law.\nUsing the transformations v=ˆ uM, 2ηs=ˆhˆη,\n2δγ=−ˆhˆpM, 2fg= 2ˆFwe recover the equation (7) of\nour paper. With the assumption that the whole domain\nΩ−moves at the uniform velocity −dL/dt , it can be\nidentified with the solid disc, with the transformation\n−dL/dt =ˆUandR∗= ˆa. The solution obtained in\n[1] for the total force exerted on the disc, due to the\nviscous friction of the top and bottom fluid phases\nand to the viscous thin sheet itself, can therefore be\ndirectly identified with the viscous forces acting on Ω−.\nThis force is given in [1] as a function of ˆ ε= 1/Bq,\nwith Bq=ηs/(R∗ηg) a Boussinesq number. Using our\nnotations, the result becomes FD=ζdL/dt ex, with\nζ= 8πηgR∗Λ(Bq) a friction coefficient, and Λ( Bq) a\nsemi-analytical function plotted in Fig. 2 of [1]. In the\nlimit of small Bqthe friction coefficient is ζ0= 16ηgR∗.\nWe define our experimental friction ζexpas\nζexp=−4T\n(dL/dt )exp(15)\nComparing this value with the theoretical friction\nζth(Bq) we can determine an upper value for the inter-\nface viscosity of the film.\nIn Fig. 5 of the main article, we plot the ratio\nζexp\nζ0=−4T\n16ηgR∗(dL/dt )exp=(dL/dt )th\n(dL/dt )exp(16)\nas a function of time. A stable behavior is obtained for\nt∈[0.5,1.5]s and, in this time range, the values com-\npatible with our error bars verify ζexp/ζ0∈[0.3,2.2].\nThis acceptable range is shown in Fig. 3 and compared\ntoζth(Bq)/ζ0. From the upper bound, we deduce that\nin our experimental condition the Boussinesq number is\nnecessarily lower than 1. Using R∗∼2 mm, we obtain\nthat ηs<4×10−8kg/s, similar to the upper limit ob-\ntained in [2].3\nFIG. 3. Friction coefficient ζ, renormalised by its value at low\nBoussinesq number. The black line is the theoretical predic-\ntion obtained from the Fig. 2 in [1]. The blue domain is the\nvalue range compatible with the experimental values shown\nin the Fig 5 of the main paper, for the time range [0 .5−1.5]\ns. From the maximal acceptable value (the blue line at the\ntop of blue domain) we deduce that Bq < 1.\n[1] B. Hughes, B. Pailthorpe, and L. White, J. Fluid Mech.\n110, 349 (1981).\n[2] Z. A. Zell, A. Nowbahar, V. Mansard, L. G. Leal, S. S.Deshmukh, J. M. Mecca, C. J. Tucker, and T. M. Squires,\nPNAS 111, 3677 (2014).Line tension in a thick soap film\nTh´ eo Lenavetier, Ga¨ elle Aud´ eoud, Marion Berry, Ana¨ ıs Gauthier,\nRapha¨ el Poryles, Corentin Tr´ egou¨ et, and Isabelle Cantat\nUniv Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F- 35000 Rennes.\n(Dated: January 31, 2024)\nThe thickness of freshly made soap films is usually in the micron range, and interference colors\nmake thickness fluctuations easily visible. Circular patterns of constant thickness are commonly\nobserved, either a thin film disc in a thicker film or the reverse. In this Letter, we evidence the line\ntension at the origin of these circular patterns. Using a well controlled soap film preparation, we\nproduce a piece of thin film surrounded by a thicker film. The thickness profile, measured with a\nspectral camera, leads to a line tension of the order of 10−10N which drives the relaxation of the\nthin film shape, initially very elongated, toward a circular shape. A balance between line tension\nand air friction leads to a quantitative prediction of the relaxation process. Such a line tension is\nexpected to play a role in the production of marginal regeneration patches, involved in soap film\ndrainage and stability.\nThe stability of liquid foams and soap bubbles is\ncontrolled by the evolution of liquid film thickness, in-\nduced by evaporation [1] and capillary and gravitational\ndrainage, until the film bursts. Drainage is associated\nwith fast in-plane motion in films and thickness hetero-\ngeneities [2–6], often spatially organized as discs of thin\nfilm embedded in a thicker film, or the reverse. In films\nless than 100 nm thick, both interfaces interact through\nshort-range forces of various origins, resulting into a dis-\njoining pressure. Nonmonotonic variations of the disjoin-\ning pressure with the film thickness are known to induce a\nline tension along the boundary of film domains of differ-\nent thicknesses [7, 8]. This phenomenon has been charac-\nterized for the transition between a very thin suspended\nfilm and a meniscus [9], or for the transition between two\nblack films [10].\nIn this Letter, we show that the boundary between two\ndomains of different thicknesses, both thicker than 100\nnm, also generates a line tension, despite the negligible\nvalue of the disjoining pressure. In the transition between\nthe two domains, the interface is slightly tilted and the\nexcess area produces by this tilt, multiplied by the surface\ntension of the solution, is the excess energy at the origin\nof this capillary line tension, of purely geometric nature.\nMarginal regeneration spontaneously generates film\ndomains of different thicknesses [2] and this line tension\nhas already been assumed, qualitatively, to play a role in\nsuch foam film instabilities [11, 12].\nTo produce and evidence this original line tension, we\nprepare an elongated pattern of thin film surrounded by\na thicker film and measure the relaxation of the pattern\nto a circular shape, under the effect of the line tension.\nIts value, deduced from the thickness profile we measure,\nis of the order of 10−10N and the relaxation lasts a few\nseconds, with velocities of the order of 10 mm/s. The\nvery low interfacial shear viscosity of our foaming solu-\ntion [13], rules it out as a significant friction mechanism.\nConsidering the viscous friction of air only, and using the\nanalytical prediction established in [14, 15], we are able\nto predict the relaxation rate as a function of the mea-\nsured line tension, without adjustable parameter. This\nFIG. 1. Experimental setup and notations used in the text.\n(A) Image of the film recorded by the top camera. (B)\nSchematic view of the setup. The black and green thick\nlines represent respectively the static and mobile edges of the\nframe. The thin film (colored domain in (A), light blue in\n(B)) is separated from the light gray thick film by the red\ncontour C. (C) Schematic thickness profile along L, in the\nvicinity of C, on the right-hand side of Fig. 1B.\ngood agreement validates our line tension measurement.\nThe line tension revealed by this work, and more gen-\nerally the anisotropic interfacial stress induced by thick-\nness gradients, whose tensor is given in this study, should\ntherefore be taken into account in film drainage models,\nand potentially in experiments where foam films are used\nto investigate 2D turbulence [16, 17].\nUsing a deformable horizontal frame of inner area w a,\nwith w= 62 mm and aa variable width (see Fig. 1), we\nproduce a thickness pattern in a foam film. The ( x, y)\nplane is the midplane of the film. We use a mixture of\nsodium dodecyl sulfate (SDS, concentration 5.6 g/L, i. e.\n2.4 CMC) and glycerol (15% in volume), of surface ten-\nsionγ0= 35 mN/m (measured with the pendent drop\nmethod) and bulk viscosity ηl= 1.5 mPa ·s. The inter-\nfacial shear viscosity ηsis shown to be below 10−8kg/s\nin [13].\nTop views of the film, recorded with a color camera\nused at 30 frames per second, are shown in Fig. 2 at\ndifferent times. The frame is first set at its smallest areaarXiv:2401.16859v1 [cond-mat.soft] 30 Jan 20242\nFIG. 2. Top view of the film before, during and after de-\nformation, at times [ −1,−0.6,0,1,3,6] s. The colored central\npart is the thin film, the gray part is the thick film. The black\nboundary is the meniscus.\n(a= 2.1 mm) and bathed in the foaming solution to pro-\nduce the thin part of the film (Fig. 2A), called the thin\nfilmhereafter. We let the film drain close to 3 min until\nits interference colors are mainly blue and yellow, indi-\ncating a thickness comprising between 100 and 300 nm.\nThen we move the mobile edge of the frame at a velocity\nVmot= 10 mm/s during 1 s (Fig.1B) and a much thicker\npiece of film, appearing gray, is extracted from the menis-\ncus surrounding the film (hereafter, the thick film ). The\nrelaxation of the thin film toward a circular shape (Fig.2\n(C-F)) is studied after the motor stops, taken as time\nreference t= 0. The amplitude of the initial thickness\nfluctuations in the thin film is much smaller than the\nthickness difference between the thin film and the thick\nfilm and does not play any role in this relaxation. The\nthickness profile of the transition between the thin and\nthe thick parts of the film is measured with a hyperspec-\ntral camera (Resonon Pika L), at a rate of 50 frames per\nsecond, along a line Lshown in Fig. 2C, as explained in\n[18].\nThe boundary Cof the thin film is detected automat-\nically and characterized by its length 2 Lmeasured in\nthexdirection, its area Aand its width, defined as\n2R=A/(2L) (see Fig. 1A). At the beginning of the\nrelaxation, the elongated shape is very regular and can\nbe described as a rectangle 2 L×2Rwith a hemidisc of\nradius Rat both ends, with R≪L. At longer times, it\nbecomes more fluctuating and a roughly circular shape\nis eventually obtained at t≈6 s. Good reproducibility\nof the shape is obtained for t <1.5 s and the relaxation\nprocess it quantitatively analyzed up to this time.\nThe thin film area varies by at most 10% during the\nmeasurement time range (see Fig. 3B). Moreover, the\nwhole thickness distribution in the thin film, indicated\nby the interference colors, remains qualitatively constant,\nthus excluding local compression or dilation in the thin\nfilm.\nThe film profiles are shown in Fig. 3C, with h(x, y) half\nthe thickness of the film. The thickness of the thick and\nFIG. 3. Shape of the thin film as a function of time. (A) Lis\nhalf its diameter measured in the xdirection, and (B) Ais its\narea. As for the other figures, the data are averaged over 13\nexperiments, and the shaded area represents the standard de-\nviation. (C) Example of thickness profiles at times 0 s (blue),\n0.5 s (green), 1 s (yellow) and 1.5 s (red). The experimen-\ntal resolution is indicated by the gray zone, and the width of\nthe thin film is indicated by the continuous line at thickness\n0. The dotted lines are parabolic interpolations between the\nmeasured domains.\nthin films are respectively of the order of 7 µm and below\nthe resolution obtained with our current signal analysis,\nbased on maxima detection in the spectrum of the light\nreflected by the film. Near the thin film, the thickness\nprofile is steep enough to blur the interference pattern\nand the thickness is also not measurable. However, the\nlight patterns corresponding to the thin, flat film and\nto the steepest part of the thick film clearly differ, and\nthe boundary Cbetween the two is well defined, even if\nthe thickness is not. To reconstruct the missing part of\nthe thick film profiles, we interpolate the thickness pro-\nfile by a parabola, imposing continuity of thickness and\nthickness derivative at the edge of the measured part of\nthe thick film profile, and zero thickness at the boundary\nwith the thin film. This choice of a parabola is arbitrary\nand other choices, e. g. a linear or order 3 interpolation,\nwould lead to similar results.\nThe capillary forces governing the dynamics can be de-\nduced from these profiles. The thickness varies on char-\nacteristic horizontal distances of the order of ℓ∼1 mm,\nyielding ∇h∼10−2(where ∇is the gradient operator in\nthe (x,y) plane). In this small slope limit, the pressure\nin the film, controlled by the Laplace pressure, scales as\nγ0h/ℓ2and the associated Poiseuille flow velocity scales\nas (γ0/ηl)h3/ℓ3∼10µm/s, which is negligible in the pro-\ncess. Moreover, as the flow occurs with negligible area\nvariations (see Fig. 3B), we assume incompressible in-\nterfaces, as classically made for spontaneous soap film\ndynamics [2]. In this frame each elementary film element\nof volume hdS is a closed system of constant thickness3\nand constant area dS, moving at the uniform velocity\nv(x, y). We define δγas the difference between the local\nsurface tension γand the reference value γ0, chosen in\nthe middle of the film. This tension variation δγensures\nthe constraint of incompressibility ∇·v(x, y) = 0.\nThe 2D stress tensor acting on such film elements is\ncomputed in [19], in the local basis Be= (n,t), defined\nin the ( x, y) plane so that ∇h=|∇h|n(adapting to our\nspecific case the general theory developed in [20]). At\norder 2 in ∇h, this tensor can be expressed as σcap=\nσ∗\ncap+σfI, with Ithe identity matrix. The isotropic\nterm is, using δγ≪γ0,\nσf= 2 (γ0+δγ) + 2γ0h∆h , (1)\nwhere ∆ indicates the 2D Laplacian operator. The devi-\natoric part is, as determined in [19],\nσ∗\ncap=γ0/parenleftbigg\n−(∇h)20\n0 ( ∇h)2/parenrightbigg\nBe(2)\nand comes from the projection of the surface tension force\nin the ( x, y) plane. The dominant term in the stress ten-\nsor is the surface tension γ0which is positive, thus in-\ndicating a traction. However, the contribution of σ∗\ncap\nshows that this traction is slightly smaller in the direc-\ntion of the thickness gradient, and slightly larger in the\nperpendicular direction, which is at the origin of the line\ntension.\nThe damping forces (per unit film area) are the friction\non the gas phase 2 fgand on the surrounding film ηf∆v\nwith ηf= 2ηs+hηlthe film shear viscosity. It results\nfrom [15] that the air friction dominates if ηf<4×10−8\nkg/s, which is verified here, as ηs<10−8kg/s and hηl∼\n10−8kg/s. However, ηsdepends on the foaming solution\nand may be much larger. In order to provide a general\nprediction, valid for a wide range of foaming solutions,\nwe thus keep the air friction and the interfacial viscosity\nin the model.\nThe equation of motion is finally, as already established\nin [21] using another approach,\n2γ0h∇(∆h) + 2∇δγ+ 2ηs∆v+ 2fg= 0, (3)\nwith the first two terms equal to ∇·σcap, as derived in\n[19].\nAs the capillary forces are localized along the thin film\nboundary C, they can be interpreted as arising from a\nline tension, which considerably simplifies the problem.\nTo this end, we define the local coordinates ( ξ, s), in the\nvicinity of C. The variable ξis zero on C, and varies in\nthe normal direction nwhereas svaries in the tangential\ndirection t. The definition of a line tension requires the\nlocalization condition ℓ≪1/κwith κthe curvature of C\n: this is verified on the straight parts of C, but not at the\ntips, where the curvature radius is R∼ℓ. For sake of\ngenerality, the tension is determined below for a generic\ncurve Cof small curvature, and will eventually be used\nin our case for the straight parts of Conly.The line tension is defined as the excess of capillary\nstress, with respect to the surface tension γ0, integrated\nalong a line perpendicular to the thickness transition. It\ncan thus be written as, for each interface,\nT=1\n2/integraldisplayℓ∞\nℓ−∞t·(σcap−2γ0I)·tdξ . (4)\nwith ℓ−∞andℓ∞the lower and upper bounds of the\nintegration domain, larger than the transition width.\nThis integral depends on hbut also on δγ, which is\ndetermined below using Eq. (3) in the domain |ξ|< ℓ.\nThere, the first term in Eq. (3) scales as γ0h2/ℓ3. In the\nlimit of small ℓ, it is much larger than the viscous forces\nwhich vary smoothly across the transition domain. Eq.\n(3) thus becomes ∂δγ/∂ξ =−γ0h∂3h/∂ξ3. By integra-\ntion, we obtain at first order in ℓκand for small ξ,\nδγ=γ0/parenleftigg\n1\n2/parenleftbigg∂h\n∂ξ/parenrightbigg2\n−h∂2h\n∂ξ2/parenrightigg\n. (5)\nInserting this expression into Eq. (4) we obtain\nT=γ0/integraldisplayℓ∞\nℓ−∞/parenleftbigg∂h\n∂ξ/parenrightbigg2\ndξ . (6)\nNote that the tension value is twice the energy excess per\nunit length of line associated to the thickness gradient.\nThe experimental line tension values shown in Fig. 4\nA, have been determined with Eq. (6), using the experi-\nmental thickness profiles at each time, averaged over all\nthe experiments, and with the integration boundaries dis-\ncussed below. As the angle between nand the y-direction\nis negligible, we have ξ=±y, respectively, for the left\nand right parts of the profile. The thickness gradients\nare negligible in the thin film, so we impose ℓ−∞= 0.\nThe relevant upper boundary is more difficult to chose:\nthe inset of Fig. 4A shows the partial tension values ob-\ntained when using an arbitrary upper integration bound\nℓintin Eq. (6) instead of ℓ∞. A plateau value is obtained\nforℓintbetween 2 and 3 mm, so we choose ℓ∞= 3 mm to\ndefine the experimental tension. Additionally, the width\nof the transition domain is defined as ℓ80(t), the upper\nboundary value for which the tension reaches 80% of its\ntotal value.\nIn this frame, the dynamical effects of thickness varia-\ntions are captured by a line tension acting along the line\nC∗located in the middle of the transition domain at the\ndistance ℓ80/2 from C(see Fig. 4B). The resulting force\n(per unit line) acting on the film is −Tκ∗(s)n+(∂T/∂s )t,\nwith κ∗(s) the boundary curvature of C∗.\nThe gradient ∂T/∂s is experimentally unknown. How-\never, the thick film extraction velocity is very homoge-\nneous all around the thin film, as well as the meniscus\nsize. We can thus safely assume that the initial transition\nprofile, and consequently T(t= 0), does not depend on\ns. A strong assumption of the model is that the tension4\nFIG. 4. (A) Line tension Tas a function of time, determined\nfrom the thickness profiles using Eq. (6), with ℓ−∞= 0,\nℓ∞= 3 mm and ξ=y. The vertical dotted lines are color-\nmatched in time with the thickness profiles of Fig. 3C. Inset:\npartial values of the tension as a function of the integration\nupper bound ℓint, obtained for the profile at t= 1 s. The\ntension reaches 80% of its total value for ℓint=ℓ80. (B)\nMapping of the observed flow on the problem solved in [15].\nThe thickness transition is shown in gray, with its center line\nC∗(bold black line), at the distance ℓ80/2 outside C(red line).\nThe subdomains Ω±are the red discs.\nremains invariant at longer times. In that case, the cap-\nillary force vanishes outside the region of the thin film\ntips and an analytical solution can be obtained.\nTo this end, we define around each thin film tip a\nsubdomain Ω±limited by a flat cylinder, centered at\nrM±= [±(L−R),0] and of radius R∗=R+ℓ80/2 so\nthat the curved parts of C∗are in Ω±(see Fig. 4B).\nThe deformations of the tips, associated to the increase\nofR∗with time, are much slower than dL/dt and the\nwhole subdomains Ω±are moving at the uniform veloc-\nity∓dL/dt .\nOutside these domains Eq. (3) becomes\n2∇δγ+ 2ηs∆v+ 2fg= 0. (7)\nThe gas Reynolds number is of the order of\nR(dL/dt )ρg/ηg∼1, with ρg= 1.2 kg/m3andηg=\n1.8 10−5kg/m/s the density and viscosity of air. To ob-\ntain an analytical prediction for the air damping force\nfg, we will neglect the air inertia, which should be taken\ninto account in a more refined model.\nIn this viscous limit, and if only Ω−moves, Eq. (7) can\nbe solved by a simple mapping on the problem solved in\n[15], i. e. a flat cylinder translating in a viscous liquid\nmembrane, as discussed in [19]. The corresponding ve-\nlocity field has been determined numerically in [22], and\nscales as dL/dt R∗/rwith rthe distance to the center\nof Ω−. As R∗/(2L)≪1 during the time range of mea-\nsure, the velocity induced by the Ω−motion at the Ω+\nFIG. 5. Experimental value of the tip velocity dLexp/dt,\ndivided by its theoretical value given by Eq. (8), in which\nR∗=R+ℓ80/2 (bold line), R∗=R(bottom thin line) and\nR∗=R+ℓ80(top thin line). Each curve is plotted with a\nshaded area showing its standard deviation.\nposition is negligible and the flow is thus the superposi-\ntion of the flows induced by the motion of each subdo-\nmain separately. The force FDacting on the boundary\nof Ω−, due to the viscous friction of the gas phase and\nof the soap film, is determined in [15] and expressed as\nFD=ζdL/dt ex, with ζa friction coefficient which de-\npends on the Boussinesq number Bq=ηs/(Rηg) (see\n[19]). The force balance on the subdomain Ω−involves\nthis friction force FDand the driving force 4 Texdue to\nthe 4 intersections between C∗and the boundary of Ω−.\nThis imposes ζdL/dt + 4T= 0.\nAssuming ηs<10−8kg/s as measured in [13], and\nusing R∗∼2 mm we find Bq < 0.25. In this low Boussi-\nnesq limit, the friction coefficient ζreaches its asymptotic\nvalue ζ0= 16ηgR∗[15], leading to\ndL\ndt=−4T\n16ηgR∗(8)\nwhich involves only experimentally known quantities.\nThe experimental relaxation velocity dLexp/dtis ob-\ntained by differentiation of L(t) shown in Fig. 3A. Its\ntheoretical value dLth/dt, given by Eq. (8), is obtained\nfrom the independent measurements of TandR∗. The\nlargest uncertainty arises from R∗, and is of the order\nof the width ℓ80of the thickness transition domain. We\nthus plot in Fig. 5 the predictions obtained with R∗=R\nandR∗=R+ℓ80.\nWe find that ( dL/dt )exp/(dL/dt )th= 1 is within our\nerror bar for the time range [0 .5,1.5], which validates\nour line tension measurement, and its role as the driving\nforce for the relaxation dynamics. At shorter times, the\nrelaxation velocity is larger than predicted. A potential\nexplanation for this reproducible deviation could be a\nresidual air motion due the mobile edge, that may last a\nfraction of second after the motor stop as Re gas∼1.\nTo conclude, this experiment quantifies the forces in-\nduced by thickness fluctuations, in a regime where dis-\njoining pressure is negligible, and shows that a localized\nthickness gradient results into a line tension acting per-\npendicularly to the thickness gradient. As this tension5\nis of purely geometrical nature, its expression Eq. (6)\nshould remain valid for nonhorizontal films, in the pres-\nence of gravity. In foam films, z-invariant in-plane mo-\ntions occur with nearly no damping, and a tiny line ten-\nsion, of the order of 0.1 nN for our thickness profile, in-\nduces a thickness pattern relaxation toward a circular\nshape at a velocity reaching 10 mm/s, only damped by\nthe air friction. An extension of our analysis, based on\nthe result of [15], relates the pattern relaxation velocity\nto the value of the Boussinesq number. We show in [19]\nthat for foaming solutions having an interface viscosity\nabove 4 ×10−8kg/s, the interface viscosity should be the\ndominant damping factor. In that case, a measure of athickness pattern relaxation may provide a measure of\nthe interface viscosity, which is an appealing application\nof our device.\nACKNOWLEDGMENTS\nThis project has received funding from the European\nResearch Council (ERC) under the European Union’s\nHorizon 2020 research and innovation program (Grant\nAgreement No. 725094).\n[1] L. Champougny, J. Miguet, R. Henaff, F. Restagno,\nF. Boulogne, and E. Rio, Langmuir 34, 3221 (2018).\n[2] K. J. Mysels, K. Shinoda, and S. Frankel, Soap films:\nStudy of Their Thinning and a Bibliography (Pergamon,\nNew York, 1959).\n[3] J. B. M. Hudales and H. N. Stein, J. Colloid Interface Sc.\n138, 354 (1990).\n[4] V. Carrier, S. Destouesse, and A. Colin, Phys. Rev. E\n65, 061404 (2002).\n[5] H. Lhuissier and E. Villermaux, J. Fluid Mech. 696, 5\n(2012).\n[6] J. M. Frostad, D. Tammaro, L. Santollani, S. Bochner de\nAraujo, and G. G. Fuller, Soft Matter 12, 9266 (2016).\n[7] J. De Feijter and A. Vrij, J. Electroanal. Chem. 37, 9\n(1972).\n[8] J. Joanny and P. De Gennes, J. Colloid Interface Sci.\n111, 94 (1986).\n[9] D. Platikanov, M. Nedyalkov, and V. Nasteva, J. Colloid\nInterface Sci. 75, 620 (1980).\n[10] D. Exerowa, A. Nikolov, and M. Zacharieva, Journal of\nColloid and Interface Science 81, 419 (1981).\n[11] E. Shabalina, A. B´ erut, M. Cavelier, A. Saint-Jalmes,\nand I. Cantat, Phys. Rev. Fluids 4, 124001 (2019).[12] C. Tr´ egou¨ et and I. Cantat, Phys. Rev. Fluids 6, 114005\n(2021).\n[13] Z. A. Zell, A. Nowbahar, V. Mansard, L. G. Leal, S. S.\nDeshmukh, J. M. Mecca, C. J. Tucker, and T. M.\nSquires, Proc. Natl. Acad. Sci. U.S.A 111, 3677 (2014).\n[14] P. Saffman, J. Fluid Mech. 73, 593 (1976).\n[15] B. Hughes, B. Pailthorpe, and L. White, J. Fluid Mech.\n110, 349 (1981).\n[16] H. Kellay, X.-l. Wu, and W. I. Goldburg, Phys. Rev.\nLett.74, 3975 (1995).\n[17] F. Seychelles, Y. Amarouchene, M. Bessafi, and H. Kel-\nlay, Phys. Rev. Lett. 100, 144501 (2008).\n[18] A. Bussonni` ere, E. Shabalina, X. Ah-Thon, M. Le Fur,\nand I. Cantat, Phys. Rev. Lett. 124, 018001 (2020).\n[19] See Supplemental Material.\n[20] D. A. Edwards, H. Brenner, and D. T. Wasan, Inter-\nfacial Transport Processes and Rheology (Butterworth-\nHeinemann, Boston, 1991).\n[21] R. Bruinsma, Physica A 216, 59 (1995).\n[22] H. A. Stone and A. Ajdari, J. Fluid Mech. 369, 151\n(1998)." }, { "title": "2401.16880v1.The_nonlinear_dynamic_behavior_of_a_Rubber_Layer_Roller_Bearing__RLRB__for_vibration_isolation.pdf", "content": "arXiv:2401.16880v1 [physics.app-ph] 30 Jan 2024The nonlinear dynamic behavior of a Rubber-Layer Roller\nBearing (RLRB) for vibration isolation\nN. Menga,1,2,∗F. Bottiglione,1and G. Carbone1,2,3\n1Department of Mechanics, Mathematics and Management,\nPolitecnico of Bari, V.le Japigia, 182, 70126, Bari, Italy\n2Imperial College London, Department of Mechanical Enginee ring,\nExhibition Road, London SW7 2AZ\n3CNR - Institute for Photonics and Nanotechnologies U.O.S. B ari,\nPhysics Department ”M. Merlin”, via Amendola 173, 70126 Bar i, Italy\nAbstract\nIn this paper, we study the dynamic behavior of a Rubber-Laye r Roller Bearing (RLRB) in-\nterposed between a spring-mass elemental superstructure a nd a vibrating base. Thanks to the\nviscoelastic rolling contact between the rigid rollers and the rubber layers, the RLRB is able to\nprovide a nonlinear damping behavior. The effect of the RLRB ge ometric and material parameters\nis investigated under periodic base excitation, showing th at both periodic and aperiodic responses\ncan be achieved. Specifically, since the viscoelastic dampi ng is non-monotonic (bell shaped), there\nexist system dynamic conditions involving the decreasing p ortion of the damping curve in which\na strongly nonlinear behavior is experienced. In the second part of the paper, we investigate the\neffectiveness of the nonlinear device in terms of seismic isol ation. Focusing on the mean shock\nof the Central Italy 2016 earthquake, we opportunely tune th e material and geometrical RLRB\nparameters, showing that a significant reduction of both the peak and root-mean-square value of\nthe inertial force acting on the superstructure is achieved , compared to the best performance of a\nlinear base isolation system.\nKeywords: Nonlinear dynamics, viscoelastic damping, seismic isolation , base isolation\n∗Electronicaddress: [Corresponding author. ]Email: nicola.menga@po liba.it, phone number: +39 080 5962746\n1I. INTRODUCTION\nAbsorbing and controlling the vibration of mechanical systems and s tructure is a very\ndemanding task. Due to the always increasing demand from mechanic al, aeronautical and\ncivil engineering, the last decades have seen a proliferation of applic ations of nonlinear\nsystems tovibrationcontrol, thankstotheirintrinsicability(i)toeffi ciently reacttoexternal\nforcing in a much wider range of frequency compared to linear syste ms, (ii) to modify their\nbehavior according to the excitation amplitude.\nInthis view, one of the most common vibration absorptionstrategy relies on theadoption\nof nonlinear energy sinks (NES), whose mechanism depends on the s pecific field of applica-\ntion. To this regard, in Ref. [1] the effect of the nonlinear (quadrat ic) damping behavior\ngiven by an hydraulic damper equipped with several on/off valves is st udied, showing that\nthe removal of unwanted periodic regimes can be achieved by means of opportunely tuned\ndamping characteristics. Similar studies were then extended to the case of vibro-impact\nNES [2, 3], showing that chaotic dynamic regimes are easy to occur, t hus promoting these\nsystems for energy harvesting applications. Similarly, focusing on t he case of travelling loads\non elastic beams (e.g. railway tracks under moving trains), an exten sive study on nonlinear\ntuned mass dampers has been performed in Refs. [4–6], showing tha t stiffness nonlinearity\npoorly affects the overall dynamic response, whereas nonlinear da mping may lead to great\nvibration reduction. Moreover, the case of a moving mass-damper for transmission cables\nis investigated in Ref. [7], where it is shown that significantly higher ene rgy dissipation can\nbe achieved compared to the case of fixed dampers.\nAmong the application fields of vibration control, seismic engineering is one of the fastest\ngrowingsector, astheabilitytoensurehighreliabilityforprimarystr ucturesandmachineries\n(e.g., power plants, hospitals, schools, etc.) has strong social, polit ical, and economic impli-\ncations [8]. Therefore, several passive systems have been develo ped to deal with this task,\nmostly relying on nonlinear stiffness behavior. Indeed, both inter-s tory frictional dissipators\n[9] for high-rise buildings (where the source of nonlinearity are the f rictional interactions),\nand bi-component sacrificial supports [10] able to provide a piecewis e linear foundation\nstiffness have shown general vibration absorption and reduced st ructure response.\nBase isolation systems are also well established methods to control the superstructure\ndynamic response, as indeed reviewed in Ref. [11]. Among them, very promising solutions\nrelies rolling isolation systems (RIS), where the rolling of rigid balls on co ncave counterface\nprovides a nonlinear gravitational stiffness, and external viscous dashpot provides linear\ndamping. In Refs. [12, 13], it is shown that such devices are peculiarly suited for heavy low-\nrise structures, presenting significantly enhanced isolation perfo rmances. On the same path,\ntheideatocombinepassivebaseisolationwithactivestructuralcon trolisinvestigatedinRef.\n[14], where several control logics are explored in order to minimize a s pecific performance\nindex based on absolute acceleration, and inter-story drift and ve locity.\nOn the other hand, less effort has been paid to study in details the dy namic character of\nsuch isolation systems. Indeed, only a few works highlighted that, u nder specific conditions,\nRIS may show chaotic behavior [15, 16] (mostly due to the variable c urvature of the rolling\ncounterfaces), thus resulting in significant sensitivity to initial con ditions and, in turn, less\nengineering predictability of the overall isolation behavior.\nFurthermore, in order to provide specific nonlinear stiffness and da mping, most of the\nRISs for base isolationsystems rely onseparated mechanical comp onents [17] (e.g. nonlinear\nspring, hydraulic dampers with valves, etc.), which need to be arran ged in specific, and\n2usually complex, configurations. This entails high installation and maint enance costs, as\nwell as reduced reliability.\nFIG. 1: A representation of a bi-axial cylindrical RLRB: two perpen dicular rows of\nequispaced steel rollers are interposed between steel plates coa ted with rubber layers (from\nRef. [28]).\nToovercometheselimitations, Rubber-LayerRollerBearings(RLRB )havebeenproposed\nso far [18–22], where the rolling balls counterparts were opportune ly coated with highly\nviscoelastic rubber, thus providing both significant damping without additional devices.\nSimilar studies were later widened to the case of rolling rods in Refs. [23 –25]. Such devices\nare specifically suited to provide a significant base-isolation degree in the case of horizontal\nexcitations. Notably, as reported in Refs. [26, 27], in the case of n ear-fault earthquake\nthe seismic excitation may involve long-period horizontal pulses in the fault-normal velocity\nsignals and high values of the ratio between the peak of the vertical and horizontal ground\naccelerations. Under these conditions, RLRB systems may suffer u plifts in the vertical\ndirection, thus resulting in vanishing damping. However, compared w ith similar friction-\nbased isolation devices (e.g. RIS and friction pendulum), due to the r ubber coating, RLRBs\npresent lower vertical stiffness (i.e. larger vertical isolation), whic h may allow, in turn, to\ntolerate higher vertical peak acceleration without experiencing co mplete uplifts.\nAlthough pioneeristic, the studies performed on RLRBs do not prov ide a detailed insight\ninto the viscoelastic bulk dissipation mechanism, and in turn into the da mping behavior of\nsuch systems, which is instead addressed by means of phenomenolo gical models. To this\nregard, in a recent paper [28], we deeply investigated the damping b ehavior of an innovative\nRLRB based on rolling cylinders (see fig. 1), which, showing overall low er contact pressures\n(see also Ref. [23]), provides higher rubber reliability compared to sp here-based RLRB. In\nthe framework of linear viscoelasticity, we accurately defined the d amping curve associated\nto viscoelastic bulk dissipation, assuming steady rolling conditions.\nIn this paper, we try to widen the investigation of cylindrical RLRBs b y studying the\ndynamic behavior of the exemplar case of a single-story superstru cture base isolated by\nmeans of a RLRB. The paper is organized in two sections. The first on e is devoted to the\nviscoelastic contact mechanics formulation, based on Boundary Ele ment Method (BEM)\nwith specific viscoelastic Green’s function taking into account for th e system materials and\ngeometry, together with the dynamic model of the system, where the two degree of freedom\nequations of motion are derived. In the second section, we presen t our main results: firstly\nwe focus on the system dynamics under periodic base excitation, hig hlighting the effect of\nthe RLRB geometrical and material parameters; then, a detailed o ptimization of the RLRB\nparameters is performed considering a real earthquake base exc itation. To stress the effect\n3of the specific RLRB behavior nonlinearity on the system response, the results are compared\nwith the case of an equivalent linear base isolation device.\nII. FORMULATION\nA. The RLRB viscoelastic behavior\nThe damping behavior of the base isolation systems depends on the v iscoelastic contact\nbehavioroftheRLRB.Infig. 2, weshowaportionoftheperiodiccon tactbetweentheRLRB\nequispaced rigid rollers (of radius R) and the viscoelastic layer stuck onto a rigid plate. The\nrelative motion between the top and the bottom plate leads to cyclic d eformations of the\nviscoelastic rubber coating, entailing bulk dissipation which gives rise t o a reaction force,\nopposing the relative motion. In what follows, we assume frictionless contact between the\nrigid cylinders and the viscoelastic layer of thickness h. Since our study is developed within\nthe framework of linear viscoelasticity, we neglect any large deform ation effect. Referring to\nfig. 2, for a given value of the velocity V, following Ref. [30], the mean shear stress acting\non the upper body can be easily calculated as\nfm(V) =−1\nλ/integraldisplay\nΩp(x)u′(x)dx (1)\nwhereλis the periodic distance between the cylinders, p(x) is the contact pressure distribu-\ntion, and u′(x) the first derivative of the displacement field of the viscoelastic laye rs within\nthe contact domain Ω = [ −a,a], beingathe semi-width of the contact area (see Fig. 2).\nλΔ\nFIG. 2: The geometrical scheme of the periodic rolling contact unde r investigation. In\nparticular, ∆ is the contact penetration between the cylinders and the deformed surface\nmean plane, and uis the layer local displacement. Due to the material viscoelasticity, t he\ncontact area mean line is shifted by a quantity ewith respect to the cylinders axis.\nEq. (1) clearly shows that the overall damping force depends on th e contact pressure\ndistribution, which is unknown. Following Ref. [28], by exploiting the sym metry of the\nsystem, we focus our study on half of the device, as indeed shown in fig. 2. Furthermore,\nit can be demonstrated that fm(V) is an odd function of V, as the mean tangential shear\nstress always opposes the relative motion between the RLRB upper and lower parts.\n4Following the procedure delineated in Refs.[28, 29], by relying on the G reen’s function\napproach, the displacement and the contact pressure fields can b e related by means of a\nspecific Green’s function, which, in the case of steady sliding, param etrically depends on V.\nThus,\nu(x) =−/integraldisplay\nΩΘV(x−ξ)p(ξ)dξ. (2)\nThe kernel Θ V(x) is the viscoelastic Green’s function for steady sliding contacts, wh ich has\nbeen already calculated in the case of periodic contacts with layers o f finite thickness in\nRefs. [30, 31]. We report herein the main relations, assuming linear vis coelastic material\nwith a single relaxation time τ\nΘV(x) =1\nE∞G(x)+1\nE1/integraldisplay+∞\n0+G(x+Vτρ)exp(−ρ)dρ (3)\nwhere 1/E1= 1/E0−1/E∞, beingE0andE∞respectively the zero-frequency and high\nfrequency elastic moduli of the material. The elastic-like Green’s fun ction is related to the\nspecific geometry under investigation, and takes the form\nG(x) =2(1−ν2)\nπlog/bracketleftbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin/parenleftbiggkx\n2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg\n+2(1−ν2)\nπ∞/summationdisplay\nm=1Am(kh)cos(mkx)\nm(4)\nwithk= 2π/λand\nAm(kh) =2hkm−(3−4ν)sinh(2hkm)\n5+2(hkm)2−4ν(3−2ν)+(3−4ν)cosh(2hkm)+1 (5)\nwhereνis the material Poisson’s ratio.\nFig. 2 shows that, due the viscoelastic delay in the material respons e, the contact area\nexhibit a certain degree of eccentricity ewith respect to the mean line of the cylinder cross-\nsection. Moreover, within the contact strip Ω, the layer displaceme nt must copy the rigid\ncylinder shape, i.e. u(x) = ∆−Λ[1−r(x+e)], where r(x) =Rsin[cos−1(x/R)] is the\nprofile of the upper half-cylinder, and Λ = R−λ−1/integraltext\nλr(x)dx. Under these conditions, Eq.\n(2) represents a Fredholm equation of the first kind which is solved f or the unknown contact\npressure distribution by exploiting the numerical scheme already dis cussed in Refs. [32–34]\nfor adhesiveless contacts.\nB. The system dynamics\nRLRB devices are usually adopted to achieve a certain degree of dyn amic base isolation\n[11] between the ground(e.g. seismic) motion andseveral superst ructures, such as buildings,\nmachinery, etc. Since in this study we are interested in highlighting th e dynamic behavior\npeculiarities of the RLRB system, we focus on a very simple superstr ucture: an elastic pillar\n(with bending stiffness k2) supporting an inertial mass m2.\nA functional scheme of the resulting system is shown in figure 3, tog ether with a lumped\nelement picture of the system. Specifically, we consider the case of a concave RLRB (with\nradius of curvature Rb≫R), whose gravitational re-centering effect is taken into account\nby means of a linearized gravitational spring with stiffness\nk1=gm1+m2\nRb(6)\n5FIG. 3: A sketch of the base-isolated physical system. The heavy massm2is connected to\nthe RLRB system by means of an elastic beam of length Land bending stiffness K2. The\nRLRB rolling path is concave, thus resulting in a linearized recentering stiffnessK1. On\nthe right: a lumped element scheme where xis the ground absolute displacement, zandζ\nare the relative displacement between the ground and the top RLRB plate, and RLRB\nand inertial mass, respectively.\nwheregis the gravitational acceleration.\nThe dissipation of the viscoelastic rolling contact occurring between the RLRB rigid\ncylinders and rubber layer leads to a damping tangential force oppo sing the relative motion\nbetween the superstructure and the ground. Such a force can b e calculated as\nFd(˙z) =−Nλb|fm(˙z)|˙z\n|˙z|\nwhereNis the number of rigid cylinders of the RLRB device, and bis the transverse width\nof the systems.\nThe equations of motion of the system of fig. 3 are\n/braceleftBiggm1(¨x+ ¨z)+k1z−Fd(˙z)−k2ζ= 0\nm2/parenleftBig\n¨x+ ¨z+¨ζ/parenrightBig\n+k2ζ= 0(7)\nwherex(t) is the ground vibration. We also define\nη(t) =x(t)+z(t)+ζ(t) (8)\nas the absolute displacement of the inertial mass.\nEqs. (7) represent a set of nonlinear second order ODE, which hav e been integrated\nnumerically by relying on a fixed time-step method based on fourth or derRunge-Kutta\n6algorithm [35]. To avoid numerical instabilities, a sensibility study has be en performed on\nthe effect of the time-step value on the integration result.\nInterestingly, the equilibrium along the vertical direction of the phy sical system shown in\nfig. 3 allows us to calculate the contact mean pressure acting on the rigid cylinders-rubber\nlayer interface as\npm=1\nλ/integraldisplay\nΩp(x)dx=gm1+m2\nNλb(9)\nNotably, due to the oscillatory shape of the base excitation x(t) typical of seismic and\nvibrational phenomena, the RLRB undergoes to a reciprocating mo tion. In this case, the\nviscoelastic contact between the rubber layer and the rigid cylinder s belongs to the class\nof reciprocating rolling contacts, usually requiring sophisticated th eoretical treatments to\naddress the specific frictional and contact behavior. However, in Ref. [36], it has been shown\nthat simplified unidirectional steady motion analysis may still provide g ood qualitative and\nquantitative predictions, depending on the actual operating cond itions. In particular, once\ndefined the linear size 2 aofthe contact area between thecylinders andthe rubber, thest roke\nsand the period Tof the reciprocating motion, the viscoelastic contact behavior clos ely\nresembles the one observed in steady sliding at constant velocity pr ovided that a≪sand\nτ≪T. Since in our analysis, the latter conditions are met, we will here appr oximate the\nreciprocating viscoelastic response with the equivalent unidirection al steady one.\nIII. RESULTS\nA. Viscoelastic contact behavior\nIn this section, we present the main results in term of contact cond itions experienced at\nthe interface between the rubber layer and the rigid cylinders. Spe cifically, we consider the\ncase of a single relaxation time incompressible viscoelastic material (i.e .ν= 0.5), whose\nhigh and zero frequency elastic moduli are E∞= 150 MPa and E0= 50 MPa, respectively.\n24p/OverTiΛde\n24p/OverTiΛde\nV\n0.0 0.5 1.0 1.5 2.0/Minus0.500.5\nx/Slash1Λu/Slash1R\nFIG. 4: The deformed contact configuration under rolling at stead y velocity.\nFig. 4showsatypicalshapeofthedeformedlayerinsteadyrollingco ntactovercylindrical\nindenters (i.e. the rigid rollers). A certain degree of eccentricity of the contact area with\nrespect to the rigid cylinders meanline is experienced due to the delay in the material\nresponse (i.e. the energy dissipation occurring in the bulk viscoelast ic material), which, in\nturn, also gives rise to asymmetric contact pressure distributions . Although dealing with\nnamely frictionless contact, under these conditions, following Eq. ( 1), the contact force\n7presents a tangential component (the so-called ”viscoelastic fric tion”) which opposes to\nthe relative motion between the cylinders and the rubber layer. Foc using on our physical\nsystem (see fig. 3) such a force represent the damping force opp osing the motion between\nthe superstructure and the ground.\np/OverTiΛde\nm/EquaΛ0.16\n0.1h/OverTiΛde/EquaΛ0.8\n/InΦinity\n0.01 0.1 1. 10. 100.0.0000.0050.0100.0150.0200.025\nV/OverTiΛdef/OverTiΛde\nm\n(a)p/OverTiΛde\nm/EquaΛ0.13\nV/OverTiΛde/EquaΛ1h/OverTiΛde/EquaΛ/InΦinity\n0 1 2 3 40.000.010.020.030.04\nh/OverTiΛdef/OverTiΛde\nm\n(b)\nFIG. 5: The dimensionless mean shear stress ˜fmas a function of (a) the dimensionless\nrelative velocity ˜V, and (b) the dimensionless thickness ˜h. Results refer to (a) ˜R= 0.3,\nand (b) ˜R= 0.1\nFigs. 5 show the frictional viscoelastic behavior of the contact. Sp ecifically, from fig. 5a,\nshowing the dimensionless friction mean shear stress ˜fm= 2(1−ν2)fm/E0as a function\nof the dimensionless velocity ˜V=Vτkfor different values of the dimensionless contact\nmean pressure ˜ pm= 2(1−ν2)pm/E0, we observe that the viscoelastic friction follows the\nwell-known bell shaped curve, as at very high and very low excitation frequency it behaves\nas an elastic material, with vanishing bulk dissipation. On the contrary , in the range of\nintermediate frequency, the viscous dissipation plays a key role, an d the frictional force\narises. This is because the largest viscoelastic energy dissipation, a nd hence friction, occurs\nwhenIm[E(ω)]/|E(ω)|is maximized, i.e. at values of ωτ≈1 (see Ref. [30, 31]), where ωis\nthe excitation frequency, and\nE(ω) =E0+E1iωτ\n1+iωτ(10)\nis the viscoelastic complex modulus, with E1=E∞−E0.\nFurthermore, from fig. 5a we observe that the thicker the rubbe r layer, the larger the\nfriction value is, as increasing ˜hthe amount of deformed material increases as well, leading\nto higher bulk dissipation. This is more clearly shown in Fig. 5b, where ˜fmis plotted as\na function of the dimensionless thickness ˜h=kh. Of course, increasing the rubber layer\nthickness, the viscoelastic half-plane behavior (i.e. for h=∞) is asymptotically recovered,\nwhereas, in the limit of vanishing thickness (i.e. for h→0), vanishing viscoelastic friction\nis achieved.\n8B. Dynamic behavior with periodic base excitation\nLet us now focus on the dynamic response of the physical system u nder periodic base\nexcitation in the form x(t) =A0sin(ωt), where A0andωare, respectively, the amplitude\nand the frequency of the excitation. The physical system we focu s on is typical of seismic\nengineering, thus we set m1= 1×102kg,m2= 1×105kg. We assume a concave shape for\nthe RLRB pathway with width b= 1 m, and radius of curvature Rb= 3.3 m, which from\nEq. 6 gives k1= 3×105N/m. We also assume ˜h= 0.8 and˜R= 0.3. Similarly, the elastic\npillar is constituted by a commercial HEB 300 steel beam, with L= 3 m, whose bending\nstiffness is k2= 6×106N/m. Moving from these values, the modal analysis of the system\nallows to identify the two natural frequencies ω1= 1.69 rad/s, and ω2= 251 rad/s.\nz/OverDot/Star(a)(b)(c)\n0.00.10.20.30.40.50.60.705\n/LBracketBar1z/OverDot/RBracketBar1 /LParen1m/Slash1s/RParen1/LBracketBar1Fd/RBracketBar1 /LParen1N x104/RParen1\nFIG. 6: The absolute value of the viscoelastic damping force |Fd|as a function of the\nabsolute relative velocity |˙z|between the ground and the upper plate of the RLRB device.\n(a)τ= 0.25 s,λ= 0.25 m; (b) τ= 0.05 s,λ= 0.05 m; (c) τ= 0.05 s,λ= 0.25 m.\nNotably, ˙ z∗corresponds to the peak force (only shown for (b)).\nRegarding the viscoelastic nonlinear damping force, coherently with the dimensionless\nresults presented in fig. 5a, we observe that, given the values of ˜hand˜R, the final load-\nvelocity curve depends on both the values of τandλ. The effect of such parameters on the\ndamping behavior of the RLRB is shown in fig. 6. We observe that redu cing the ratio τ/λ\nleads tolower slopeof thecurve close tothe origin. Similarly, increasin gλcauses areduction\nof the peak force value as, through Eq. (9), it entails a reduction o f the contact mean\npressure, thus reducing the overall amount of material involved in the cyclic deformation,\nand in turn the energy dissipation. Notably, ˙ z∗defined as the absolute value of the velocity\ncorresponding to the peak force, depends on the specific parame ters as well.\nFurthermore, for any specific damping curve, it is possible to define an”equivalent” linear\nviscous damping behavior with damping coefficient\nceq=−dFd\ndV/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nV=0(11)\nFig. 7 shows the effect of τandλon the value of ceq, in a contour plot. We observe\nthat, according to fig. 6, increasing the viscoelastic relaxation time τ, as well as reducing\nλ, the equivalent damping coefficient increases. Further, in the same figure, three lines have\nbeen added referring to specific values of ceq, each one allowing to determine a sets of τ\n9ceq/LBracket1Ns/Slash1m/RBracket1\nFIG. 7: The equivalent linear damping coefficient ceqas a function of τandλ. The three\nlines are for (a) ceq= 3.5×105Ns/m; (b) ceq= 6×105Ns/m; (a) ceq= 2×106Ns/m;\nCaseτ[s]λ[m]ceq[Ns/m]˙z∗[m/s]\n(a)0.00765 0.053.5×1050.724\n(b)0.024 0.153.5×1050.534\n(c)0.041 0.253.5×1050.417\n(d)0.0131 0.05 6×1050.423\n(e)0.0411 0.15 6×1050.312\n(f)0.0702 0.25 6×1050.244\n(g)0.0437 0.05 2×1060.127\n(h)0.12 0.132 2×1060.097\n(i)0.20.214 2×1060.081\nTABLE I: RLRB characteristics of fig. 8\nandλwhose equivalent linear damping behavior is the same. Of course, we e xpect the\nsystem dynamics to be strongly affected by the values of τandλ, as the strongly nonlinear\nviscoelastic damping of the RLRB may lead to completely different beha viors even in the\ncase of similar linearized equivalent damping coefficients.\nThis is clearly shown in fig. 8 where nine sets of values of τandλare investigated, as\ndetailed in Table I. Specifically, each row of figures refers to the sam e linearized equivalent\ndamping coefficient, increasing from the top to the bottom. The das hed vertical lines rep-\nresent the system first natural frequency ω1, and the dash-dotted lines correspond to the\nnatural frequency of an equivalent rigid-damper system. The figu res show, in the lower part,\na contour plot of the steady-state system response spectrum ¯ ηω(ωf) =F(η(t,ω)) as a func-\ntion of the excitation frequency ω(whereFis the Fourier transform operator). Similarly,\nin the upper part we show, on the left axis, the steady-state maxim um amplitude |η|maxof\nthe system response for the nonlinear (blue histogram) and equiva lent linearized (red curve)\nsystems as a function of the excitation frequency ω, whereas on the right axis (black curve)\nthe steady-state maximum amplitude |˙η|maxof the system velocity response is shown.\n10FIG. 8: Left axis: the dimensionless peak amplitude of the system re sponse|η|maxfor the\nnon-linear (blue histogram) and equivalent linearized (red curve) sy stems as a function of\nthe excitation frequency ω. Right axis: the dimensionless peak amplitude of the system\nvelocity response |˙η|max(black curve). Lower contour plots, the system response spect rum\n¯η(ωf) as a function of the excitation frequency ω.\nFigures 8a, 8b, 8c share the same linearized behavior (i.e. τ/λis almost constant) with\nmaximum amplitude of oscillation close to the system natural frequen cyω1. Interestingly,\nmoving from fig. 8a to 8c the value of λincreases (see the data in Table I) thus the nonlinear\ndamping force peak value reduces. This entails that the larger the v alue ofλ, the smaller\nthe value of ωat which the system operating conditions overcome the damping for ce peak\nthreshold velocity ˙ z∗. Indeed, strongly nonlinear effects usually occurs only at sufficient ly\n11large values of ωwhere the system response involves |˙z|max>˙z∗(i.e. on the decreasing\nportion of the damping force curve of fig. 6); whereas, at sufficien tly small value of ω, the\nnonlinear system vibration closely resembles the one of the linearized system.\nSuch a peculiar behavior is even more clearly shown by figures 8g, 8h, 8i, all referring to a\nset ofτandλassociated to a very high value of linearized equivalent damping coeffic ientceq.\nUnder these conditions, since the linear viscous damper behaves alm ost rigidly, the linear\nsystem behavior (see the red curves) closely resembles the one of a one degree of freedom\nharmonic oscillator, of undamped mass m2and stiffness k2(i.e. maximum amplitude of\noscillation close to the natural frequency/radicalbig\nk2/m2= 7.75 rad/s). The nonlinear system\nbehaves differently. Indeed, for the specific parameters, the no nlinear damping force peak\nvalue is reached even for low excitation frequency, and strong non linear effects occurs. No-\ntably, from the system response spectrum shown in the lower cont our plots, we observe that\nthe overall response of the non linear system always involves a harm onic term associated\nto the external periodic excitation, whereas the main effect of the nonlinear damping is to\n”chaotically” switch on the harmonic term related to the low natural frequency ω1.\nFigure 9 illustrates the system behavior of case (d) of Table I at diffe rent excitation\nfrequencies. Both the system response time histories, phase por traits and Poincar´ e maps\nrefers to the steady-state conditions. The Poincar´ e maps (or r ecurrence maps) have been\nachieved by sampling the system response at intervals equal to the excitation period, with\nrandom phase.\nAccording to fig. 8d, at ω= 1.5 rad/s the system behaves linearly with a periodic\nresponse, thus the Poincar´ e map is one single point, and the phase portrait is an ellipse.\nIncreasing ωup to 4 rad/s, nonlinear damping starts to play a nonvanishing role. T he\nsystem response is still periodic, but the phase portrait is now a def ormed ellipse.\nAtω= 4.9 rad/s the system response involves |˙z|max>˙z∗(see fig. 8d). Under these\nconditions, the vibration spectrum is the sum of two main incommensu rable harmonics: the\nfirst excitation harmonic ω, and the system low natural frequency ω1. As a consequence, the\nresulting Poincar´ e map is a closed curve, whereas the phase portr ait is not, filling a portion\nof the phase space.\nA slight increase of ωup to 5.1 rad/s leads to different results. This time the ratio of the\nmain frequencies of the system response spectrum is an integer nu mber, as ω/ω1≈3. Since\nthe response is periodic the phase portrait is a closed curve, and th e Poincar´ e map shows\nthe same number of isolated points such as the ratio ω/ω1. A similar behavior is also shown\natω= 6.8 rad/s and ω= 11.8 rad/s, where respectively ω/ω1≈4 andω/ω1≈7\nIncreasing ωup to 12.5 rad/s, the two main components become incommensurable, the\nsystem dynamics is not periodic and a closed curve is observed in the P oincar´ e map associ-\nated with a colored region in the phase portrait.\nIV. OPTIMIZATION OF THE RLRB DYNAMIC BEHAVIOR\nIn the previous section it has been clearly pointed out that the dyna mic behavior of the\nphysical system is strongly affected by the specific damping behavio r of the RLRB base\nisolation device. Since the latter depends, in turn, by the specific ch oice of the physical\nparameters τandλ, it is evident that a fine tuning can be performed in order to optimize\nthe overall behavior of the system with respect to a performance index.\nTo stress the impact of our conclusions, in what follows, we focus on a real seismic event,\nnamely the main shock of the Central Italy earthquakes [37] occur red on October 30th2016,\n1280 90 100 110 120-202\n-0.305 -0.3 -0.295 -0.29 -0.285 -0.2811.051.1\n-0.2 -0.1 0 0.1 0.2-0.200.2\n80 90 100 110 120-101\n-0.75 -0.7 -0.65 -0.6 -0.55 -0.50.030.0350.040.045\n-0.1 -0.05 0 0.05 0.1-0.500.5\n160 170 180 190 200-101\n-1 -0.5 0 0.5-0.4-0.200.2\n-0.1 -0.05 0 0.05 0.1-0.500.5\n160 170 180 190 200-101\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1-0.2-0.15-0.1-0.05\n-0.1 -0.05 0 0.05 0.1-0.500.5\n80 90 100 110 120-0.500.5\n-0.4 -0.2 0 0.2-0.15-0.1-0.050\n-0.04 -0.02 0 0.02 0.04 0.06-0.200.2\n80 90 100 110 120-101\n-1 -0.5 0 0.5 1-0.2-0.100.1\n-0.1 -0.05 0 0.05 0.1-0.200.2\n160 170 180 190 200-101\n-1 -0.5 0 0.5 1-0.2-0.100.1\n-0.1 -0.05 0 0.05 0.1-0.200.2ω(rad/s)Time history\nη(m)vst(s)Poincar`e map\n˙η(m/s)vsη(m)Phase portrait\n˙η(m/s)vsη(m)\n1.5\n4.0\n4.9\n5.1\n6.8\n11.8\n12.5\nFIG. 9: The system dynamic behavior for case (d) of Table I, at var ying excitation\nfrequency ω.\nwith magnitude 6.6 M w. In terms of performance indexes, most of the previous studies\nfocuses on multi-story superstructure [38], in which the main sourc e of damage is the inter-\nstory drift, leading to critical shear stresses, and eventually to t he structural collapse. In\nthesecases, themostadoptedperformanceindexes aretherela tivevelocity anddisplacement\nofeach story [39, 40]. However, since ourwork ismore fundamenta l, we define a performance\n13indexφwhich encompasses two source of damage for the structural elem ents (i.e. the elastic\nbeam of our system): (i) the maximum inertial load Fi\nmaxon the mass m2, associated to\nthe structure instantaneous damage [41]; (ii) the root mean squar eFi\nrmsof the inertial loads\nhistory during the shake, associated to the material hysteresis a nd fatigue. Specifically, we\nhave that\nφ=1\n2/parenleftbiggFi\nmax\nFi\nmax,0+Fi\nrms\nFi\nrms,0/parenrightbigg\n(12)\nwhere\nFi\nmax=m2|¨η|max (13)\nbeing|¨η|maxthe absolute acceleration maximum, and\nFi\nrms=m2/radicalBigg/integraldisplayt2\nt1¨η(t)2\nt2−t1dt (14)\nThe optimization strategy is the following. Firstly, single objective min imization of\nFi\nmax(τ,λ) andFi\nrms(τ,λ) are set independently. The homogenization terms Fi\nmax,0and\nFi\nrms,0, in Eq. (12), are then defined as the corresponding values in single o bjective opti-\nmized conditions. Finally, the minimization of φ(τ,λ) is performed.\n(a)60000 65000 70000 75000 800001.0641.0661.0681.0701.072\nc/LBracket1Ns/Slash1m/RBracket1Φ\n(b)\nFIG. 10: The optimization map (a) of the non-linear system, and the optimization curve\n(b) of the linear one. In the optimization process, we set t1= 15 s, and t2= 40 s.\nThe results of the optimization process are shown in figures 10 for t he nonlinear system\n(10a), and, for comparison, for the linear one (10b). Interestin gly, the nonlinear system\nresults show an optimum flat valley, allowing to achieve significantly opt imized results with\nseveral sets ofthe design parameters τandλ, providing enhanced compliance to thedifferent\ndesign requirements (i.e. geometrical or material restrictions).\nAnumericalcomparisonintermsofoptimizationresultsisfoundinTab leII.Interestingly,\nthe nonlinear behavior of the RLRB device allows to reduce both Fi\nmaxandFi\nrmsof about\n6.3%, compared to the linear system. Such a result is also shown in figur es 11 where the\ndisplacement time history and spectral analysis is shown for both th e systems, compared\nto the earthquake data. The smoother behavior of the nonlinear s ystem shown in fig. 11a\n(blue line) compared to the linear system (red line) clearly entails lower inertial effects,\n14Non-Linear Linear\nParametersτ= 0.013 s\nλ= 0.25 mc= 68000Ns\nm\nFi\nM(N) 80255 85271\nFi\nrms(N) 16313 17342\nTABLE II: Comparison between nonlinear and linear optimized system s results.\nand in turn lower stresses for the structural elements. Further more, the system equipped\nwith the nonlinear RLRB suffers smaller (about 8 .4%) maximum relative base displacement\n|z|max, compared to the linear isolator, thus reducing the risk of lateral im pact with other\nstructures. Similarly, from fig. 11b, we observe that, although bo th the systems are able\nto filter the high frequency spectrum of the seismic event, the non linear device (blue curve)\nstill behaves better than the linear one (red curve) even close to ω1.\nx(t)\nΗ(t)\nΗl/LParen1t/RParen1\n152025303540/Minus0.15/Minus0.10/Minus0.050.000.050.10\nt/LBracket1s/RBracket1/LBracket1m/RBracket1\n(a)x /LParen2Ωf)\nΗ (Ωf)\nΗl/LParen2Ωf/RParen2\n0 2 4 6 8 100.00.20.40.60.81.0\nΩf/LBracket1rad/Slash1s/RBracket1/LBracket1m/RBracket1\n(b)\nFIG. 11: The time history (a) and spectral analysis (b) of: the Cen tral Italy earthquake\n2016 (black curves); the inertial mass dispacement ηfor the nonlinear (blue curves) and\nlinear systems (red curves). The dashed line in (b) represent the s ystem first natural\nfrequency ω1.\nV. CONCLUSIONS\nWe investigated the dynamic behavior of RLRB seismic isolators, in whic h the viscoelas-\ntic rolling friction between the rigid cylinders and the rubber layers lea ds to a nonlinear\ndamping. We found that the viscoelastic damping force is a bell-shape d function of the\nrelative velocity of the moving parts (the ground and the building). S pecifically, the damp-\ning force increases with increasing relative velocity of the moving par ts up to a peak value\nof the damping force. At larger relative velocities, the damping forc e decreases. Such\na strongly nonlinear trend is controlled by the viscoelastic material r elaxation time, and\nthe rigid cylinders spacing, which indeed dramatically affect the overa ll system behavior.\nSpecifically, depending on whether the RLRB operating condition lies o n the increasing or\n15decreasing portion of the damping curve, strongly nonlinear aperio dic behavior can be ob-\nserved. We investigate both the effect of the excitation frequenc y, as well as the specific set\nof parameters.\nA real seismic event has been numerically reproduces in order to tes t the model, based\non the Central Italy earthquake of October 2016. Indeed, an op timization procedure has\nbeen performed to minimize a performance index taking into account both the maximum\ninstantaneous value and the root mean square value of the inertial loads history. Similarly,\nthe behavior of an equivalent linearized system is investigated for co mparison. Results show\nthat the nonlinear system is able to sufficiently reduce both the insta ntaneous and averaged\ninertial load value with respect to the linear system, opening the pat h to further deeper\ninvestigation on similar devices. Indeed, different sources of nonline arity in RLRB devices\nwill be further investigated.\n[1] Y. Starosvetsky, O.V. Gendelman. Vibration absorption in systems with a nonlinear energy\nsink: nonlinear damping. J Sound Vib, 324 (2009) 916-939.\n[2] O.V. Gendelman. Analytic treatment of a system with a vib ro-impact nonlinear energy sink.\nJ Sound Vib, 331 (2012) 4599-4608.\n[3] O.V. Gendelman, A. Alloni. Dynamics of forced system wit h vibro-impact energy sink. J\nSound Vib, 358 (2015) 301-314.\n[4] F.S. Samani, F. Pellicano. Vibration reduction on beams subjected to moving loads using\nlinear and nonlinear dynamic absorbers. J Sound Vib, 325 (20 09) 742-754.\n[5] F.S. Samani, F. Pellicano. Vibration reduction of beams under successive traveling loads by\nmeans of linear and nonlinear dynamic absorbers. J Sound Vib , 331 (2012) 2272-2290.\n[6] F.S. Samani, F. Pellicano, A. Masoumi. Performances of d ynamic vibration absorbers for\nbeams subjected to moving loads. Nonlinear Dynamics, 73 (20 13) 1065-1079.\n[7] M.A. Bukhari, O. Barry,E. Tanbour. On the vibration anal ysis of power lines with moving\ndampers. J Vib Control, 24 (2018) 4096-4109.\n[8] B. Myslimaj, S. Gamble, D. Chin-Quee, A. Davies, B. Breuk elman. Base isolation technologies\nfor seismic protection of museum artifacts. In The 2003 IAMF A Annual Conference in San\nFrancisco, California. San Francisco, CA: IAMFA.\n[9] S.T. De la Cruz, F. L´ opez-Almansa, S. Oller. Numerical s imulation of the seismic behavior of\nbuilding structures equipped with friction energy dissipa tors. Comput Struct, 85 (2007) 30-42.\n[10] D. Foti, M. Diaferio, R. Nobile. Dynamic behavior of new aluminum–steel energy dissipating\ndevices. Struct Control Hlth, 20 (2013) 1106-1119.\n[11] P.S. Harvey Jr, K.C. Kelly. A review of rolling-type sei smic isolation: Historical development\nand future directions. Eng Struct, 125 (2016) 521-531.\n[12] P.S. Harvey Jr, H.P. Gavin. Assessment of a rolling isol ation system using reduced order\nstructural models. Eng Struct, 99 (2015) 708-725.\n[13] C.D. Casey, P.S. Harvey Jr, W. Song. Multi-unit rolling isolation system arrays: Analytical\nmodel and sensitivity analysis. Eng Struct, 173 (2018) 656- 668.\n[14] K. Miyamoto, D. Sato, J. She. A new performance index of L QR for combination of passive\nbase isolation and active structural control. Eng Struct, 1 57 (2018) 280-299.\n[15] P.S. Harvey Jr, H.P. Gavin. Thenonholonomic andchaoti c natureof arollingisolation system.\nJ Sound Vib, 332 (2013) 3535-3551.\n16[16] P.S. Harvey Jr, R. Wiebe, H.P. Gavin. On the chaotic resp onse of a nonlinear rolling isolation\nsystem. Physica D, 256 (2013) 36-42.\n[17] T.W. Lin, C.C. Chern, C.C. Hone. Experimental study of b ase isolation by free rolling rods.\nEarthq Eng Struct D, 24 (1995) 1645-1650.\n[18] D. Foti, J.M. Kelly Experimental analysis of a model iso lated at the base with rubber-layer\nroller bearing (RLRB). Euro Earthquake Eng,10 (1996) 3–13.\n[19] M. Mezzina, D. Raffaele, C. Dentamaro, D. Foti, P. Monaco. Seismic isolation with RLRB\n(Rubber Layer Roller Bearing), Proc. of the 1st European con ference on Structural control;\nBarcelona, Spain, May 29-31, (1996) 459-466.\n[20] A.H.Muhr, M. Sulong, A.G. Thomas. Rolling-ball rubber -layer isolators. J Nat Rubber Res,\n12 (1997) 199–214.\n[21] A.H.Muhr, G. Bergamo. Shaking table tests on rolling-b all rubber-layer isolation system. In:\n14th European conference on earthquake engineering. (2010 ) 5703–5710.\n[22] L. Guerreiro, J. Azevedo, A.H. Muhr. Seismic tests and n umerical modeling of a rolling-ball\nisolation system. J of Earthq Eng, 11 (2007), 49-66.\n[23] D. Foti, A. Catalan Goni, S. Vacca. On the dynamic respon se of rolling base isolation systems.\nStruct Control Hlth, 20 (2013) 639–48. http://dx.doi. org/ 10.1002/stc.1538.\n[24] D. Foti, Isolatore sismico (Seismic Isolator), Italia n Patent 0001414213, 2015\n[25] D. Foti, (2019). Rolling devices for seismic isolation of lightweight structures and equipment.\nDesign and realization of a prototype. Struct Control Hlth, 26, e2311.\n[26] F. Mazza, M. Mazza. Nonlinear seismic analysis of irreg ular rc framed buildings base-isolated\nwith friction pendulum system under near-fault excitation s. Soil Dyn Earthq Eng, 90 (2016)\n299-312.\n[27] F. Mazza. Seismic demand of base-isolated irregular st ructures subjected to pulse-type earth-\nquakes. Soil Dyn Earthq Eng, 108 (2018) 111-129.\n[28] N.Menga, D.Foti, G.Carbone.Viscoelastic frictional propertiesofrubber-layerrollerbearings\n(RLRB) seismic isolators. Meccanica, 52 (2017), 2807-2817 .\n[29] G. Carbone, L. Mangialardi. Analysis of the adhesive co ntact of confined layers by using a\nGreen’s function approach. J Mech Phys Solids, 56 (2008) 684 -706.\n[30] N. Menga, L. Afferrante, G. Carbone. Effect of thickness and boundary conditions on the\nbehavior of viscoelastic layers in sliding contact with wav y profiles, J Mech Phys Solids, 95\n(2016) 517-529.\n[31] N. Menga, L. Afferrante, G.P. Demelio, G. Carbone. Rough c ontact of sliding viscoelastic\nlayers: numerical calculations and theoretical predictio ns. Tribol Int, 122 (2018) 67-75.\n[32] N. Menga, L. Afferrante, G. Carbone. Adhesive and adhesiv eless contact mechanics of elastic\nlayers on slightly wavy rigid substrates. Int J Solids Struc t 88 (2016) 101-109.\n[33] N. Menga, C. Putignano, L. Afferrante, G. Carbone. TheCon tact Mechanics of Coated Elastic\nSolids: Effect of Coating Thickness and Stiffness. Tribol Let, 6 7 (2019) 24.\n[34] N. Menga. Rough frictional contact of elastic thin laye rs: the effect of geometrical couplig. Int\nJ Solids Struct, 164 (2019) 212-220.\n[35] B.P. Demidovich, I.A. Maron, Fundamentals of computat ional mathematics. Fizmatgiz,\nMoscow, 1963.\n[36] C. Putignano, G. Carbone, D. Dini. Theory of reciprocat ing contact for viscoelastic solids.\nPhys Rev E, 93 (2016) 043003.\n[37] Italian Accelerometric Archive. http://itaca.mi.in gv.it/. EventID: EMSC-20161030 0000029.\nStation: IT.FCC\n17[38] A. Preumont, K. Seto, Active control of structures. Joh n Wiley & Sons, 2008.\n[39] C.L. Ng, Y.L. Xu. Semi-active control of a building comp lex with variable friction dampers.\nEng Struct, 29 (2007) 1209-1225.\n[40] A. Yanik, U. Aldemir, M. Bakioglu. A new active control p erformance index for vibration\ncontrol of three-dimensional structures. Eng Struct, 62 (2 014) 53-64.\n[41] F. Sadek, B. Mohraz. Semiactive control algorithms for structures with variable dampers. J\nEng Mech, 124 (1998) 981-990.\n18" }, { "title": "2401.16893v1.Computational_Power_of_Opaque_Robots.pdf", "content": "arXiv:2401.16893v1 [cs.DC] 30 Jan 2024Computational Power of Opaque Robots\nCaterina Feletti1, Lucia Mambretti , Carlo Mereghetti1, and Beatrice Palano1\n1Dipartimento di Informatica, Universit` a degli Studi di Milano, Milan , Italy\n1{caterina.feletti, carlo.mereghetti, beatrice.palano }@unimi.it\nAbstract\nInthe fieldofdistributed computingbyrobotswarms,theresearc hcomprehendsmanifold\nmodelswhererobotsoperateintheEuclideanplanethroughaseque nceoflook-compute-move\ncycles. Models under study differ for (i)the possibility of storing constant-size information,\n(ii)the possibility of communicating constant-size information, and (iii)the synchroniza-\ntion mode. By varying features (i,ii), we obtain the noted four base models: OBLOT\n(silent and oblivious robots), FST A(silent and finite-state robots), FCOM (oblivious and\nfinite-communication robots), and LUMI(finite-state and finite-communication robots).\nCombining each base model with the three main synchronization mode s (fully synchronous ,\nsemi-synchronous , andasynchronous ), we obtain the well-known 12 models. Extensive re-\nsearch has studied their computational power , proving the hierarchical relations between\ndifferent models. However, only transparent robots have been considered.\nIn this work, we study the taxonomy of the 12 models considering collision-intolerant\nopaquerobots. Wepresentsixwitnessproblemsthat provethe majorityo fthe computational\nrelations between the 12 models. In particular, the last witness pro blem depicts a peculiar\nissue occurring in the case of obstructed visibility and asynchrony.\nKeywords— Mobile robots, Look-Compute-Move, Computational complexity, O paque robots, Dis-\ntributed Computing, Obstructed visibility, Collision intolerance\n1 Introduction\nIn the far-ranging field of distributed computing, a significant area concerns computing by mobile enti-\nties[15, 16], where tasks are required to be solved by multiple simple and limit ed entities (also called\nrobots) that can move in the environment. In this realm, manifold theoretic al models have been intro-\nduced to formalize realistic scenarios (e.g. sensor or drone swarms , dynamic networks, software agents).\nOne of the most studied is the look-compute-move model [15, 16], where robots, once activated, execute a\ncycleof three steps: they lookat the environment, they compute the next position executing a distributed\nalgorithm, and they moveto the computed position.\nUnder the umbrella of the look-compute-move macro-model, a vast combination of models has been\nproposedto formalizedifferentrobot capabilitiesandto study howm odel settingsaffect its computational\npower. In this respect, robots are assumed to possess very limite d and restricted features, in order to\nfind the minimal sets of capabilities which are required to achieve a give n task. Accordingly, robots are\nassumed to be autonomous ,indistinguishable ,anonymous , andhomogeneous : namely, they act without\nany central control, they cannot distinguish themselves by exter nal appearance or by ids, they possess\nthe same features, they execute the same algorithm in a decentra lized way. Moreover, most of the\nliterature considers punctiform robots which cannot communicate with other robots ( silent), without\nany persistent memory ( oblivious ), without any agreement on a global coordinate system, or chiralit y,\nor a unit measure ( disoriented ). Besides robot capabilities, different model environments have been\nproposed to study diverse scenarios. The existing models can be ma inly divided into two groups: the\nmodels where robots act on the Euclidean plane [1, 13, 17, 22], and th e models where robots act on\ndiscrete spaces (generally graphs, rings, or lattices) [7, 8, 11, 23 ]. According to the synchronization\nmode, robots may be synchronized (time is globally divided into rounds) or n ot. Specifically, literature\nproposes three main modes: the fully synchronous mode (FULLY), where all robots execute each step of\n1thelook-compute-move cycle synchronously in one round, the semi-synchronous mode (SEMI), where at\neach round a random subset of robots act synchronously, and th easynchronous mode (ASYNCH), where\nrobots act without any synchronization assumption.\nThe traditional problems studied for swarms of mobile entities include Pattern Formation [1, 12,\n13, 17, 25, 28, 29, 30], Gathering [5, 7, 11, 14, 21], Scattering [20, 24],Flocking [4]. A common goal\nof the algorithmic investigation is to reduce the model capabilities req uired to solve a given problem\nor to prove the impossibility of solving it under a certain set of capabilit ies. This approach has led\nto describing the computational power of a given model (i.e. the set of problems it can solve) and\noutlining the hierarchical relations (dominance, equivalence, or ort hogonality) among different models.\nIn the last decade, multiple works [2, 6, 8, 9, 10, 18] have inspected and compared the computational\npowerofdifferent modelswhich differ in robotfeaturesand synchro nizationmode. Accordingto the robot\nfeatures, they have investigated how the communication and stor age capabilities affect the computational\npower of the robots. Starting from the classical model where rob ots are both oblivious andsilent(i.e.\nwithout any means of storage or communication), researchers ha ve investigated how the possession of a\npersistent memory or communication means changes the power of s uch models. To characterize these\nextraproperties, they proposedto add a constant-size light to eachrobot which can assume a colorchosen\namong a constant and fixed set of colors. Such light is persistent (so the color is maintained until the\nnext update), it can be updated at the beginning of a move step, an d it can be internally or externally\nvisible. Specifically, the literature focuses on four classes of robot s: theOBLOT class, where robots are\nassumed to be oblivious andsilent, theFST Aclass, where each robot is embedded with an internal light\n(visible just to the robot, thus providing a persistent memory), th eFCOM class, where each robot is\nembedded with an external light (visible just to the other robots, thus providing communication mea ns),\nand theLUMIclass, where each robot is embedded with an external and internal light. According to the\nsynchronization mode, each class has been studied under the thre e settings: FULLY,SEMI, andASYNCH.\nBesides some trivial relations between a pair of models that only differ because the first one enjoys\na capability that the second one lacks, other model relations may no t be obvious to identify. This is\nespecially true for models characterized by completely different cap abilities, so it may be difficult to un-\nderstand which combination of capabilities is more powerful. In these cases, the literature has attempted\nto illustrate some simulators to prove the equivalence between models, or some witness problems to prove\ntheir strict dominance or orthogonality. Specifically, in [2, 6, 18, 19 ], the authors study the computational\npower of transparent robots that can move on the Euclidean plane , assuming multiple robots can occupy\nthe same positions ( multiplicity ). In [8, 9, 10], the authors make the same effort but for robots ac ting on\ngraphs. In [3], the authors consider energy-constraint robots, i.e. robots that necessitate an idle round\nto restore the needed energy to perform a new cycle.\nRelated works and our contributions. Our work is inspired by the papers [2, 6, 18, 19]\nwhere the authors exhibit the complete taxonomy of the 12 models o f robots that can freely move on the\nEuclidean plane. Such models vary for the synchronization mode and for the possibility to memorize and\ncommunicate. However they are assumed to be transparent, thu s always guaranteeing complete visibility\nfor the swarm, and collision-tolerant, thus allowing robots to occup y the same position at the same time.\nIn this paper, we investigate the computational power of opaque robots , i.e. robots that cannot\nsee beyond a collinear robot. Opaqueness introduces a remarkable difficulty in the design of correct\nalgorithms to solve some classical problems [1, 12, 13]. In fact, the obstructed visibility leads to critical\nissues to be addressed in the algorithmic strategies: robots may no t be aware of the cardinality of the\nswarm, robots may not be aware if there are some moving robots in t heASYNCHmode, robots may not\nknow the complete topology of the current configuration, robots may compute the next action based\non partial information. As a matter of fact, ad hoctechniques are needed to cope with this visibility\nlimitation [26, 27].\nBesides the opaqueness feature, our model differs from [2, 6, 18, 19] since robots do not tolerate\ncollisions (so we drop the multiplicity assumption). The reason behind this choice is twofold, and it is\ncoherent with the related literature [1, 12, 13, 26, 27]. Firstly, ass uming collision intolerance leads to\nthe formalization and analysis of more realistic models, as does assum ing robot opaqueness. Secondly,\ndropping the multiplicity assumption is coherent with the hypothesis o f obstructed visibility in the case\nof collinearity. As a matter of fact, a multiplicity of two robots forms a “degenerate” collinearity with any\nother robot of the swarm, for which it would be unnatural to state the visibility relation in this special\ncase. In this respect, some witness problems introduced in [2, 19] c annot be applied under our model,\nwhich needs a new study with specific witness problems.\nIn the first part of this work, we expose a preliminary study of the r elations between transparent\n2and opaque models. Intuitively, a transparent model seems to com putationally dominate the same model\nbut with opaque robots. In Section 3 we formally prove this strict do minance: endowing a model with\ntransparency increases its computational power, allowing it to solv e more problems. As a consequence,\nthisresulthighlightsthatconstant-size(internalorexternal)ligh tsarenotalwayssufficienttocompensate\nfor robot obstructed visibility.\nIn the second part of this work (Section 4), we present six witness problems showing the majority\nof the hierarchical relations among models of collision-intolerant opaque robots , thus providing a first\noverview of their computational taxonomy. For the sake of space , all relations proved in this work will be\ncompactly shown in the theorems in Section 4 (i.e. without splitting the m in multiple corollaries). See\nAppendix A for the proofs of such theorems.\n2 Preliminaries\n2.1 Models\nThis work compares 12 robot models that differ in some features. We here introduce in detail all the core\nfeaturesthat such models share, and the variable features under study.\nCore features. We investigate swarms of autonomous computational mobile robots , which act\nin the Euclidean plane R2. Robots are indistinguishable (they cannot be distinguished by external\nappearance), anonymous (they are not provided with any id), homogeneous (they execute the same\nalgorithm), and punctiform entities. We consider opaquerobots so that in the case of three collinear\nrobotsp,q,r, the endpoint robots p,rcannot see each other. We assume robots are in the worst conditio n\nabout orientation: they are completely disoriented so that they do not share a global common coordinate\nsystem (i.e. no agreement on origin, axis direction, chirality, or unit d istance). Moreover, we assume\nthat the local coordinate system of any robot may change from on e activation to another ( variable\ndisorientation ).\nAll the robots in the swarm are provided with the same deterministic a lgorithm, which is executed\nevery time the robot is activated. At each time, a robot can be eithe ridleoractive, according to the\nscheduler. When activated, a robot executes a Look-Compute-Move cycle: it takes the snapshot of its\nvisible area ( look), it executes the algorithm using the sole snapshot as input ( compute), and it travels\nstraight towards the computed destination ( move). If the destination position is equal to the current one,\nthe robot is said to perform a null movement . After the movestep, the robot becomes idle again. We\nconsider rigidmodels, i.e. no adversary can stop the motion of a robot1.\nWe deal with a collision-intolerant model meaning that it does not tolerate either multiplicity (i.e.\nno robot can occupy the same location as another robot at the sam e time) or overlapping trajectories\n(robotsrandshave overlapping trajectories if (i)ris moving from atoa′,(ii)sis moving from btob′,\nand(iii)the segments ¯aa′and¯bb′have points in common). We refer to both multiplicity and overlapping\ntrajectories as collisions .\nVariable features. Regarding the memory and communication features of robots, we consider\nthe four models mainly proposed in the literature. In the OBLOT model, robots are assumed to be\noblivious (i.e. they do not have any persistent memory to store data about p ast cycles) and silent(i.e.\ntheydonot haveanymeanstocommunicatewith otherrobots). In theFST Amodel, robotsareprovided\nwith a persistent internal light which can assume a color chosen from a constant-size set. Such int ernal\nlight plays the role of a constant-size persistent memory. In the FCOMmodel, robots are equipped with\na persistent external light visible only to other robots, which can assume a color chosen in a cons tant-size\nset of colors. Indeed, external lights can be exploited by the swar m to communicate some messages to\nthe visible robots. Lastly, the LUMImodel gather the features of both FST AandFCOM. This model\nassumes luminous robots, which are equipped with a light that can be colored using a con stant-size set\nof colors. Such light is both visible to the robot itself (working as an int ernal state) and visible to the\nother robots (working as an external communication means).\nRegarding the activation and synchronization of robots, we consider the three modes mainly studied\nin the literature. In the fully synchronous mode (FULLY),time is split into atomic rounds, within which all\nrobots are activated together and execute their look-compute- move steps completely synchronously. The\n1In [2, 6, 18, 19], the authors consider both rigid and non-rig id models. In the next model comparisons\n(transparent vs opaque), we consider only rigid models.\n3semi-synchronous mode (SEMI) differs from FULLYjust for the fact that at each round a random subset\nof the swarm is activated. In the asynchronous mode (ASYNCH), every robot is activated independently\nfrom the others, and every cycle step lasts a finite but unpredicta ble amount of time. For the SEMI\nandASYNCHmodes, robots do not know which are the activated robots at each instant. Moreover, we\nalways assume the fairness condition : for each time tand for each robot r, there exists a time t′> t\nsuch that ris activated. This condition allows us to compute time complexity consid ering the number of\nepochs, where an epoch is a minimal time frame within which each robot is activa ted at least once. The\nselection of the subset of robots activated at every time is made by an adversarial scheduler . Formally,\nletR={r1,...,r n}be a swarm of nrobots, and let Tbe a time domain which could be discrete N≥0\n(inFULLYandSEMI) or continuous R≥0(inASYNCH). Anactivation scheduling is a function S:T →2R\ndefining the subset of the swarm that is activated at a specific time.\nNotation. Weusethenotation XYtoindicateamodelforopaquerobotsthatpossessalltheabove\ncore features and that has Xas communication-storage setting and Yas synchronization mode, where\nX∈ {OBLOT ,FST A,FCOM,LUMI} andY∈ {F,S,A}(FULLY,SEMI,ASYNCH, resp.). Consistently\nwith the notation used in [2, 6, 18, 19], we indicate with XYthe same model as XYbut considering\ntransparent robots which tolerate collisions. We refer to these tw o classes of models as the opaqueand\ntransparent framework .\n2.2 Problems\nRobot swarms are distributed systems that are aimed at solving pro blems. Since in these models robots\ncan just move in the plane, the literature studies problems requiring a swarm to form (a sequence of)\ngeometric patterns, and/or to travel along specific trajectorie s. Formally, let us assume a swarm of\nnrobotsR={r1,...,r n}on the Euclidean plane. When no ambiguity arises, we indicate with riboth\nthe robot and the point on the plane where riis located. Given an absolute coordinate system ZonR2,\nwe define the configuration of the swarm at time tas the set Ct={(x1,l1),...,(xn,ln)}wherexi∈R2\nis the position of riaccording to Z, andliis the light color of ri, at time t. In the OBLOT model, we\nalways assume li=offfor every ri∈ R. A configuration is validif no collision occurs on it. We define\nCas the set of all the valid configurations for R. We say that a configuration Cguarantees complete\nvisibility if there are no collinearities among robots.\nAproblem Pfor a swarm of robots is defined2as a sequence ( φ0,τ0,φ1,τ1,...,φ m,τm...) where\neachφiis a condition on the configuration of the swarm, and where τiis a condition on the intermediate\nconfigurations that the swarm is allowed to assume to reach a new co nfiguration for which φi+1holds.\nWe call such sequence the request of the problem P. The initial condition φ0must include the clause\nstating that li=offfor every ri∈ R. Except for this clause, since Pmight be solved without lights and\nunder any synchronization mode, φi,τimust not impose any conditions on light colors or the number of\ncycles, for each i.\nStarting from an initial configuration C0for which φ0is true,Pis said to be solved under a scheduling\nmode if, for each scheduling under the given mode, there exists an a lgorithm Athrough which the swarm\nforms a sequence of configurations ( C1,...,C m,...) such that φiholds in Cifor each i≥1, and such\nthatτi−1holds during the formation of Cistarting from Ci−1. If the request of the problem is finite,\nthe last condition τmrequires the swarm to stay still after having satisfied the last cond itionφmof the\nrequest.\nGiven an initial configuration C0, a scheduling on a time domain Tand an algorithm AsolvingP,\nwe define the sequence {C(t)}t∈Tas theevolution ofA, whereC(t) is the configuration reached at time\ntexecuting Aaccording to the scheduling.\n2.3 Computational Relations\nGiven a model M, we indicate with P(M) the set of problems solved under M, i.e. the computational\npowerofM. Given two models M1,M2, we define the following relations:\n•M1iscomputationally not less powerful thanM2, formally M1≥M2, ifP(M1)⊇ P(M2), i.e any\nproblem solvable in M2is solvable in M1;\n2For our purposes.\n4•M1iscomputationally more powerful thanM2, formally M1> M2, ifP(M1)⊃ P(M2), i.e any\nproblem solvable in M2is solvable in M1and there exists a problem solvable in M1that is not\nsolvable in M2;\n•M1iscomputationally orthogonal toM2, formally M1⊥M2, ifP(M1)\\P(M2)/\\e}atio\\slash=∅andP(M2)\\\nP(M1)/\\e}atio\\slash=∅, i.e there exists a problem solvable in M1(M2, resp.) that is not solvable in M2(M1,\nresp.);\n•M1iscomputationally equivalent toM2, formally M1≡M2, ifP(M1) =P(M2), i.eM1andM2\nsolve the same set of problems.\nThe following relations trivially follow from the definitions of the models:\nLUMIY≥ FST AY≥ OBLOTYandLUMIY≥ FCOMY≥ OBLOTY\nXF≥XS≥XA\nwhereY∈ {F,S,A}andX∈ {OBLOT ,FST A,FCOM,LUMI} . Indeed, the same relations hold in\nthe opaque framework.\n3 Transparent vs opaque robots\nTheorem 1. LetPbe a problem solved in XY. ThenPis solved under XY.\nProof.LetAbe an algorithm solving PunderXY. We can easily construct an algorithm AsolvingP\nunderXY. Given a robot rand given in input its snapshot σof all the robots, Acomputes A(σ):=A(σ)\nwhereσis the snapshot obtained by σremoving all the robots which would be hidden from rin case of\nopaqueness. Aperfectly simulates A, thus correctly solving Pfor transparent robots.\nCorollary 1. For each Y∈ {F,S,A}andX∈ {OBLOT ,FST A,FCOM,LUMI} ,\nXY≤XY.\nProblem 1 (Line-Stretch ).Let us consider an initial configuration where n >3 robots are equally\nspaced along the same line, say γ. Letdbe the distance between two adjacent robots. The problem asks\nthe endpoint robots to move away from their adjacent robot and s top in order to form a new distance\nd+d\nnwith them. They are allowed to travel only along γ. The other robots must stay still. See Figure 1.\nγ\nFigure 1: Line-Stretch .\nLemma 1. Line-Stretch is solved under OBLOTA.\nProof.Theproblemissolvedundertheweakestmodelofthetransparent framework. Infact, theendpoint\nrobots can compute and head to their destination since they can co unt all the robots and at least two\ninternal robots fix d. The final configuration is stable.\nLemma 2. Line-Stretch cannot be solved under LUMIF.\nProof.The problem cannot be solved under the strongest model of the op aque framework. Since the\nnrobots are always collinear by request, they cannot count themse lves and so the endpoint robots will\nnever accomplish the task. Moreover, lights would be inefficient for k eeping a swarm counter, due to their\nconstant size.\nTheorem 2. For each Y∈ {F,S,A}andX∈ {OBLOT ,FST A,FCOM,LUMI} ,\nXY< XY.\n5Proof.The result derives by combining Corollary 1 with Lemma 1 and Lemma 2. I n fact, it holds that\nLine-Stretch ∈ P/parenleftbig\nXY/parenrightbig\nwhereasLine-Stretch /∈ P/parenleftBig\nXY/parenrightBig\nfor anyX,Y.\nTheorem 3. LetPbe a problem solved by an algorithm AunderXYalways avoiding collisions, such\nthatPis defined for a swarm with fixed cardinality, say k. If, given any evolution of A, every robot can\nseekrobots, then the problem can be solved even in XY.\nProof.Since at any activation, each robot is awareit sees the whole swarm, it can compute its next action\nby executing A. This computation results in the solution of the problem considering o paque robots.\n4 Taxonomy of opaque models\nWe present our witness problems to prove some strict dominance ( >) and orthogonality ( ⊥) relations\namong opaque models. Thanks to Theorem 1 and Theorem 3, one of t he witness problems presented\nin [2] can be used to prove some hierarchical relations to hold in our op aque framework too. However,\nother witness problems in [2, 19] are not compliant with our collision-int olerant models; thus, we present\nspecific problems that fit our assumptions.\n4.1 Weakness of OBLOT\nProblem 2 (Triangle Round-Trip ).LetCbe a configuration where 3 robots are placed so that two of\nthem lay on the vertices of an equilateral triangle (let abe the empty vertex), while the third robot lays\non the triangle center. From C, the robot in the center has to move to a, forming the new configuration\nC′. Then, robots have to form Cagain, where ais again the empty vertex. See Table 1.\na a a\nC C′C\nTable 1: Configurations in Triangle Round-Trip .\nTriangle Round-Trip is a sub-case of the problem N-gon Round-Trip defined in [2] (see Defini-\ntion 1).\nLemma 3. Triangle Round-Trip /∈ P/parenleftBig\nOBLOTF/parenrightBig\n.\nProof.The problem has been shown to not belong to OBLOTF(see Lemma 3 in [2]). In fact, using\noblivious and silent robots, there is no way to identify the former emp ty vertex adue to the full symmetry\nofC′. By the contrapositive of Theorem 1, the result holds.\nLemma 4. Triangle Round-Trip ∈/parenleftBig\nP/parenleftBig\nFST AA/parenrightBig\n∩P/parenleftBig\nFCOMA/parenrightBig/parenrightBig\n.\nProof.Theproblemhasbeen showntobesolvedin FST AAandFCOMA(see Lemma4-5in[2]). Since in\nthis version of the problem the cardinality of the swarm is fixed and th e robots never create collinearities\nor collisions, we can apply Theorem 3 to state that Triangle Round-Trip can be solved both in FST AA\nandFCOMA.\nTheorem 4. Given the schedulers Y1=F,Y2=S,Y3=A, it holds\nFST AYi>OBLOT{Yj}j≥i\nFCOMYi>OBLOT{Yj}j≥i\nLUMIYi>OBLOT{Yj}j≥i.\n64.2 Orthogonality between FSTAandFCOM\nProblem 3 (Flip-Flop-Flip ).Letp,qandrbe three robots forming a strictly isosceles triangle so\nthatdist(p,r) =dist(q,r). Letγbe the perpendicular bisector to the line segment ¯ pqpassing through\nthe point b∈¯pq. Letγ′(γ′′, resp.) be the semi-line of γstarting from band which contains (does not\ncontain, resp.) r. The problem requires rto perpetually perform three subsequent actions (see Table 2),\nin an infinite loop: (i)rmust reach a point on γ′′\\{b};(ii)rmust reach a different point on γ′′in order\nto move away from p,q;(iii)rmust reach a point on γ′\\{b}. The problem requires rto never leave γ\nand to never stop so that p,q,rform an equilateral triangle. Robots p,qmust stay still.\nγ′γ′′rp\nqγ′γ′′rp\nqγ′γ′′rp\nq\nFirst Flip Flop Second Flip\nTable 2: Configurations in Flip-Flop-Flip .\nLemma 5. Flip-Flop-Flip ∈/parenleftBig\nP/parenleftBig\nFST AA/parenrightBig\n∩P/parenleftBig\nFCOMF/parenrightBig/parenrightBig\n.\nProof.We solvethe problemin these twomodelsusingthree colors( flip1,flopandflip2), assumingw.l.o.g.\nall robots start with the color flip1. The problem request guaranteesthat each robot can recognize its role\nby geometric conditions. In FST AA,rmoves along γchanging its internal color following the perpetual\nscheme ( flip1−flop−flip2)∞, so that at each activation, rknows which is the current action to be\nperformed. The robots p,qdo not need to change their colors. In the FCOMFmodel, all the robots\nsynchronously update their external colors following the above sc heme, so that at each round each robot\nknows what actions (color setting and move step) have to be accom plished.\nLemma 6. Flip-Flop-Flip /∈/parenleftBig\nP/parenleftBig\nOBLOTF/parenrightBig\n∪P/parenleftBig\nFCOMS/parenrightBig/parenrightBig\n.\nProof.Flip-Flop-Flip cannot be solved under an OBLOT model since rwould not have any means\nto understand which movement it has to perform. Indeed, any str ategy encoding the action of rinto the\ndistances with p,qfails. Suppose for example to use u=dist(p,q) as a fixed measure unit, and let k,hbe\ntwofixed values, with 0 < k < h. Suppose the algorithmimplements this strategy: if dist(p,r)< ku, then\nrmust execute the first flip, traveling to a position r′∈γ′′such that ku≤dist(p,r′)< hu. Otherwise, if\nku≤dist(p,r)< hu, thenrmust executethe flop, movingtoaposition r′∈γ′′suchthat dist(p,r′)≥hu.\nLastly, if dist(p,r)≥hu, thenrmust execute the second flip, moving to a position r′∈γ′such that\ndist(p,r′)< ku. Yet, since rcould be placed at any position on γ′in the initial configuration, any\ndistance encoding results inefficient for the solution of the problem.\nFlip-Flop-Flip cannot be solved under the FCOMSmodel too. By contradiction, suppose that the\nproblem is solved by a certain algorithm A. LetSbe aSEMIactivation scheduling under which Asolves\nthe problem. We show that there exists a SEMIactivation scheduling S′such that Flip-Flop-Flip is not\nsolved by A. Lettbe the first round in Swhererexecutes the first flip. Let S′be a scheduling such that\nS′(t′) =S(t′),∀t′≤t. Clearly, rexecutes its first flip at the t-th round under S′. Suppose that, in the\n(t+1)-th activation round under S′,ris the only one that gets activated, namely S′(t+1) ={r}. Yet,r\nhas no memory of the previous activation rounds. As a consequenc e,rmakes again a flip. Contradiction.\nTheorem 5.\nLUMIA>FCOMA\nLUMIS>FCOMS,A\nLUMIF>FCOMS,A\nFCOMF>FCOMS,A.\n7Problem 4 (Newcomer Introducing ).Consider n+ 2 robots, with n≥7. Letnrobots be placed on\nthe same circle whose ray length is ρ. Letcbe a robot lying in the center of the circle. Let sbe a robot\nexternal to the circle so that scan seec. The problem requires sequentially forming two configurations.\nFirst,smust travel along the line ¯ scand stop on the boundary of the circle. Second, cmust travel along\nthe radius defined by sand stop in a position c′so thatdist(s,c′) =1\n2ρ. All the other robots must stay\nstill. See Table 3.\nc\nsc\nsc\ns\nFirst Configuration (a) Second Configuration (b) Third Configuration (c)\nTable 3: Configurations in Newcomer Introducing .\nLemma 7. Newcomer Introducing /∈ P/parenleftBig\nFST AF/parenrightBig\n.\nProof.Theimpossibilityofsolvingtheproblemwith just internallightsderives fromthe factthat starting\nfrom the second configuration (see Table 3.b) chas no way to recognize which robot is s. Sincescan\nbe anywhere in the disposition of the n+ 1 robots on the circle, a constant set of colors would not be\nsufficient to store robot indices.\nLemma 8. Newcomer Introducing ∈ P/parenleftBig\nFCOMA/parenrightBig\n.\nProof.We show a possible FCOMAalgorithm solving Newcomer Introducing with two colors: offand\ns. All the robots are initially set to color off. Each robot can determine its role by the geometry of the\nconfigurations ( cseesn≥7 robots equidistant from itself and an external robot, ssees at least four\nrobots forming a circle with a robot on its center, while the other rob ots can see they lay on a circle with\nat least other n−2≥5 robots). When sis activated, it sets its light to sand starts to move. This color\nis maintained also in its next activations. When cis activated, if it sees a robot son the circle, it can\ncompute its destination correctly. The last configuration is stable: no other robot will move.\nTheorem 6. Given the schedulers Y1=F,Y2=S,Y3=A,\nLUMIYi>FST A{Yj}j≥i.\nTheorem 7.\nFST AF,S,A⊥FCOMS,A.\n4.3 Power of FULLY\nProblem 5 (Spinning ).The problem is defined recursively, without any stop conditions. Con sider a\nconfiguration Cwheren≥5 robots {r0,...,r n−1}are located on a circle centered in O. Leta0,...,a n−1\nbe the related positions of the robots such that it is possible to esta blish a global clockwise direction\n(e.g. the one going from a0toa2, passing through a1). Letαbe the angle a0ˆOa1, which is the minimum\nangle in {aiˆOai+1}0≤i≤n−1. The problem requires the given configuration to form a new configu ration\nC′by rotating each rifromaitoa′\niof an angleα\n2, following the agreed clockwise direction. Robots are\nrequired only stop on the target points lying on the circumference. Recursively, the problem demands\nthe same request starting from C′. See Table 4.\nLemma 9. Spinning ∈/parenleftBig\nP/parenleftBig\nOBLOTF/parenrightBig\n∩P/parenleftBig\nLUMIA/parenrightBig/parenrightBig\n.\n8αa0a1a2a3\na4\na5αa′\n0a′\n1a′\n2 a′\n3\na′\n4\na′\n5\nTable 4: Configurations in Spinning .\nProof.The problem is solvable in OBLOTF: each robot always has complete visibility of the swarm, so\nit is able to determine the rotation center and the rotation angle. Th eFULLYmode guarantees that all\nthe robots agree on the same rotation-angle, at each round.\nThe problem is solvable under LUMIA, by using these colors: off, a0, a1, moving0, moving1, m0, m1,\nmoving, moved, end . The algorithm solving the problem executes the same sub-routine p erpetually. This\nsub-routine implements a complete circle rotation of the swarm. At t he beginning of each circle rotation,\nall robots are off. In the first epoch, the robots r0andr1set their lights as a0anda1, respectively. After\nthis setting, robot a0(a1, resp.) computes its destination position, sets its light to moving0 (moving1,\nresp.) and starts moving. If a robot r, which is not moving0 ormoving1 colored, sees a moving0 or\nmoving1 robot,rdoes nothing. When a moving0 (moving1, resp.) robot is activated, it just updates its\nlight tom0(m1, resp.). Once the rotation angle through m0andm1has been fixed, the other robots can\nstart their rotation. If an offrobotrsees both m0andm1on the circle, it sets its light as movingand\nstarts its rotation. When a movingrobot is activated, it sets its light to moved. When a robot sees only\nm0,m1,moved, orendrobots, then it updates its color to end. In the last phase of the sub-routine, if\nanendrobot can see only endoroffrobots, it resets its color to off. Once all robots are off, the circle\nrotation is ready to restart.\nLemma 10. Spinning /∈/parenleftBig\nP/parenleftBig\nFST AS/parenrightBig\n∪P/parenleftBig\nFCOMS/parenrightBig/parenrightBig\n.\nProof.Spinning is not solvable under FST ASsince an activated robot rcannot know what movements\nother robots have already made, thus it cannot determine the rot ation-angle.\nSpinning is not even solvable under model FCOMS. Suppose that, by contradiction, there exists an\nalgorithm AsolvingSpinning . In particular, the problem is solved under an activation scheduler S. Let\nr0be the robot in position a0. Lett1be the activation time, under S, of the first round during which r0\nperforms a non-null movement. Let S′be another scheduling, such that\nS′(t) :=S(t)∀t < t1andS′(t1) =S′(t1+1) :={r0}\nIfAis executed under S′, then the execution is the same as Suntil time t1−1. At time t1, robotr0\nbehaves in the same way as it did under scheduling Sbut, as no other robot has been activated, then\nthere is no way to keep track of the fact that r0has already moved. At time t1+1 robot r0is activated\nagain but it cannot understand from geometric conditions that it mu st stay still. Contradiction.\nTheorem 8.\nOBLOTF>OBLOTS,A\nFST AF>FST AS,A\nFCOMF>FCOMS,A\nOBLOTF⊥FCOMS,A\nOBLOTF⊥FST AS,A.\n9α\nabc\nabc\nab\nc\nInitial configuration. Required movements. Final configuration.\nTable 5: Angle-Shift .\nProblem 6 (Angle-Shift ).Consider an initial configuration with three robots forming an acute and\nscalene triangle. Let a,b,cbe the three robots, where ais placed on the greatest angle, say α, whereas\ncis placed on the smallest angle. Fixing aas the rotation center and following the direction given by\na,b,c, the problem requires bto rotate of αandcto rotate of π−α. The robots are not allowed to stop\nanywhere else on the plane. Afterwards, the robots must stay st ill. See Table 5.\nLemma 11. Angle-Shift ∈/parenleftBig\nP/parenleftBig\nOBLOTF/parenrightBig\n\\P/parenleftBig\nLUMIS/parenrightBig/parenrightBig\n.\nProof.Angle-Shift is solvable under any FULLYmodel: if bandcperform their cycles at the same time,\nthen they correctly compute their target position. The final confi guration is stable since it always forms\nan obtuse triangle (terminal condition).\nInstead, the swarm can suffer from information loss in SEMI, making Angle-Shift unsolvable even\nunderLUMIS. In fact, suppose that in the initial configuration only bis activated. After b’s movement,\nthe three robots turn out to be collinear in the reached configurat ion. As a result, chas no means to\nrecompute α, whether cuses the geometry of the configuration or uses constant-size ligh ts. The same\nhappens even if only cis activated.\nTheorem 9.\nLUMIF>LUMIS,A\nOBLOTF⊥LUMIS,A\nFST AF⊥LUMIS,A.\n4.4 Opaqueness and asynchrony\nWe now introduce the Pseudo-Polygon problem which shows a peculiar issue occurring in case of ob-\nstructed visibility and asynchrony.\nDefinition 1. Given a regular n-gonN, for any n≥4, apseudo-polygon Qis a subset of vertices of N,\nsuch that |Q| ≥n\n2+1. We call Ntheassociated polygon with respect to Q.\nGiven a pseudo-polygon Q, it is always possible to determine the associated polygon, which is uniq ue.\nIn fact, as Qcontains at least three vertices, the circumscribed circle is univoca lly defined. Moreover,\nsinceQcontains more than half of the vertices of the associated n-gon, there always exist at least two\nvertices that are adjacent in N. So, it is always possible to univocally establish the associated polygon\nfrom a pseudo-polygon.\nDefinition 2. Asafe zone of a regular polygon is the locus of all points xin the plane such that:\n•xis external to the regular polygon;\n•xis not aligned with any of the two vertices of the associated polygon;\n•xdoes not lie on the bisector of any edge of the associated polygon (e quivalently, xis not equally\ndistanced from any two adjacent vertices);\n•ifℓis the length of the edge of the polygon, then the distance between xand any vertex of the\npolygon is at least ℓ.\nFigure 2 depicts the (complement of the) safe zone of a square.\n10Figure 2: The safe zone of the square comprehends all the poin ts not belonging to the blue-\ncolored (infinite) lines and zones.\nProblem 7 (Pseudo-Polygon ).LetNbe a regular n-gon with n≥6 vertices. Let Qbe a pseudo-\npolygon of m≥n\n2+2 vertices, associated with N. Consider a swarm of m+1 robots, where mrobots\nlay onQand let the last robot, w, lay in the safe zone of N. Letabe the farthest robot from w. Let\nb,cbe the first two found robots, starting from aand following both directions on the perimeter of the\nassociated polygon, one per each direction taken. Assume dist(b,w)> dist(c,w). The problem requires\nato move away from btowards a point xsuch that (i)xbelongs to the safe zone of N,(ii)xbelongs to\nthe halfplane delimited by the line ¯bcthat does not contain a, and(iii)xmust not be on any line passing\nby the position of wand any other robot on Q. Note that requests (i,iii)are imposed in order to have x\nvisible by every robot. See Figure 3.\nca b\nwx\nFigure 3: The Pseudo-Polygon problem associated with an octagon.\nLemma 12. Pseudo-Polygon /∈ P/parenleftBig\nFST AA/parenrightBig\n.\nProof.Pseudo-Polygon cannot be solved in the ASYNCHmode, only using internal lights. Let us consider\nthe problem instance given by Figure 3 where the pseudo-polygon of the initial configuration is composed\nofn\n2+3 vertices, with n= 8. Let us assume bis activated for the first time during the movement of a,\nwhenais hidden by c(i.e.b,c,aare collinear). When blooks at its snapshot, it recognizes a feasible\ninitial configuration (it sees a pseudo-polygon withn\n2+ 2 robots, and the robot w). According to this\nconfiguration, berroneously elects itself as the robot that has to move away from t he pseudo-polygon.\nIt has no means to understand if aexists or not. On the other hand, ahas no means to know if bhas\nupdated its internal light to memorize it is not the elected robot to mo ve.\nFalse election. The impossibility of solving Pseudo-Polygon in the asynchronous modes with\njust internal lights derives from a critical issue that is typical of sw arms with obstructed visibility. This\ncritical issue can be described as the false election phenomenon. Such phenomenon can be informally\ndescribed as follows: from a stable configuration, the given problem requires the use of a leader election\nroutine to elect the unique robot (the true leader ) which has to execute a non-null movement to reach\nthe next configuration. All the other robots have to stay still. In t heASYNCHmode, a robot rexecutes its\n11look step while the true leader is moving and is hidden from r. However, rcannot deduct the presence\nof the true leader from its snapshot. So, applying the same leader e lection routine, relects itself as the\n(false) leader, thus starting an unrequested movement.\nThe false election phenomenon must be examined when trying to tran spose aSEMIalgorithm in the\nASYNCHmode. In particular, the use of lights must be considered as a possib le method to avoid false\nelections. As we have shown in Lemma 12 for Pseudo-Polygon , internal lights are not sufficient to cope\nwith them. Instead, the next lemma proves that external lights ar e required (and sufficient) to correctly\nsolve the Pseudo-Polygon problem in the ASYNCHmode.\nLemma 13. Pseudo-Polygon ∈/parenleftBig\nP/parenleftBig\nOBLOTS/parenrightBig\n∩P/parenleftBig\nFCOMA/parenrightBig/parenrightBig\n.\nProof.Pseudo-Polygon is solvable in OBLOTS(i.e. in any synchronousmodel), since complete visibility\nis guaranteed at any activation time and all the movements (null and non-null) are univocally determined\nby geometric conditions. In fact, each robot can determine Q, the watcher w, and the robot a(the\nfarthest from w). The robot acan compute its final destination and move there. If a robot is not t he\nfarthest from the watcher, or if it sees two robots that are not p art of the pseudo-polygon, then it stands\nstill.\nPseudo-Polygon needs at least external lights to be solvable in the ASYNCHmode. We show here an\nalgorithm that needs 4 colors: off(default), on,a,b. In the first epoch, every robot updates its color\naccording to its role: robot aturns into a, robotbturns into b, whereas the remainder turns into on.\nAfterward, let rbe an activated robot that sees no offrobots and that notes there is only one robot (the\nwatcher) out of the pseudo-polygon. Let Vrbe the set of colors rcan see.\n•ifVr={a,b,on},rturns into onand stays still;\n•ifVr={a,on},rturns into band stays still;\n•ifVr={b,on}, and ifris the farthest robot from w, it turns into aand starts moving;\n•ifVr={on}, it means risband stays still (robot ais hidden).\nIf a robot rsees two robots not belonging to the pseudo-polygon, then rdoes not move (the final\nconfiguration is already formed or is about to be formed).\nTheorem 10.\nOBLOTS>OBLOTA\nFST AS>FST AA\nFST AA⊥OBLOTS.\n5 Relation map\nTable 6 summarizes the results proved in this work, showing the relat ions (>,<,⊥, and≡) that hold\nbetween the pairs of models in our opaque framework. The map show s also which of the six witness\nproblems ( TRTforTriangle Round-Trip ,FFFforFlip-Flop-Flip ,NWCforNewcomer Introducing ,\nSPINforSpinning ,ASHforAngle-Shift ,PSEforPseudo-Polygon ) have been used to prove such re-\nlations. For some pairs of models (gray cells), the knowledge about w hat kind of relation holds is still\nnow incomplete. E.g. between FST AFandFCOMFtwo possible relations ( or⊥,⊥\nNWC,<\nFFF\nLUMIA <\nASHor⊥,>\nFFF,>or⊥,>\nNWC,>or⊥,>\nTRT,\nOBLOTS <\nTRT<\nTRT<\nTRT<\nSPIN<\nTRT<\nTRT<\nTRT\nFST AS <\nNWCOBLOT{Yj}j≥i\nFCOMYi>OBLOT{Yj}j≥i\nLUMIYi>OBLOT{Yj}j≥i.\nProof.Triangle Round-Trip cannot be solved under OBLOTF,S,A(by Lemma 3) but it can be solved\nunder{FST A,FCOM,LUMI}A,S,F(by Lemma 4). Combining the results, we obtain that OBLOT is\nstrictly dominated by FST AandFCOM for a given synchronization mode Yi∈ {F,S,A}. The other\nstrict dominances are derived by transitivity.\nTheorem 5.\nLUMIA>FCOMA\nLUMIS>FCOMS,A\nLUMIF>FCOMS,A\nFCOMF>FCOMS,A.\n15Proof.Flip-Flop-Flip is solved under FCOMFandLUMIA,S,F(by Lemma 5) but it cannot be solved\nunderFCOMS,A(by Lemma 6). Combining the results, the strict dominance relations follow.\nTheorem 6. Given the schedulers Y1=F,Y2=S,Y3=A, it holds\nLUMIYi>FST A{Yj}j≥i.\nProof.By Lemma 8, Newcomer Introducing is solved under LUMIA,S,F. By Lemma 7, Newcomer\nIntroducing cannot be solved under FST AF,S,A. Combining the results, the strict dominance relations\nfollow.\nTheorem 7.\nFST AF,S,A⊥FCOMS,A.\nProof.By Lemma 5 and Lemma 6, Flip-Flop-Flip is solved in FST AF,S,Abut not in FCOMS,A. By\nLemma8andLemma7, Newcomer Introducing issolvedin FCOMS,Abutnotin FST AF,S,A. Combining\nthe results, the orthogonality relations follow.\nTheorem 8.\nOBLOTF>OBLOTS,A\nFST AF>FST AS,A\nFCOMF>FCOMS,A\nOBLOTF⊥FCOMS,A\nOBLOTF⊥FST AS,A.\nProof.The above relations hold combining the previous lemmas and by transit ivity:\n•the strict dominance of XFoverXS,Aderives from Lemma 9 and Lemma 10, for each X∈\n{OBLOT ,FST A,FCOM} . In fact, Spinning is solved in {OBLOT ,FST A,FCOM}Fbut it\nis not solved in {OBLOT ,FST A,FCOM}S,A;\n•the orthogonality between OBLOTFoverFCOMS,Aholds since Spinning is solved in OBLOTF\nbutnotin FCOMS,A, andsince Newcomer Introducing issolvedin FCOMS,Abut notin OBLOTF\n(by Lemma 8, Lemma 7);\n•the orthogonality between OBLOTFoverFST AS,Aholds since Spinning is solved in OBLOTF\nbut not in FST AS,A, and since Triangle Round-Trip is solved in FST AS,Abut not in OBLOTF\n(by Lemma 4, Lemma 3).\nTheorem 9.\nLUMIF>LUMIS,A\nOBLOTF⊥LUMIS,A\nFST AF⊥LUMIS,A.\nProof.The above relations hold combining the previous lemmas and by transit ivity:\n•the strict dominance of LUMIFoverLUMIS,Astraightforwardly derives from Lemma 11. In fact,\nAngle-Shift is solved in LUMIFbut it is not solved in LUMIS,A;\n•the orthogonalitybetween OBLOTFoverLUMIS,Aholds since Angle-Shift is solved in OBLOTF\nbut not in LUMIS,A, and since Triangle Round-Trip is solved in LUMIS,Abut not in OBLOTF\n(by Lemma 4, Lemma 3);\n•the orthogonality between FST AFoverLUMIS,Aholds since Angle-Shift is solved in FST AF\nbut not in LUMIS,A, and since Newcomer Introducing is solved in LUMIS,Abut not in FST AF\n(by Lemma 8, Lemma 7).\n16Theorem 10.\nOBLOTS>OBLOTA\nFST AS>FST AA\nFST AA⊥OBLOTS.\nProof.The above relations hold combining the previous lemmas and by transit ivity:\n•for each X∈ {OBLOT ,FST A},XSstrictly dominates XAsincePseudo-Polygon can be solved\ninXSbut not in XA(by Lemma 13 and Lemma 12);\n•theorthogonalitybetween FST AAandOBLOTSholdssince Pseudo-Polygon issolvedin OBLOTS\nbut not in FST AA, and since Triangle Round-Trip is solved in FST AAbut not in OBLOTS(by\nLemma 4 and Lemma 3).\n17" }, { "title": "2401.16917v1.Cowling_Haagerup_constant_of_the_product_of_discrete_quantum_groups.pdf", "content": "arXiv:2401.16917v1 [math.OA] 30 Jan 2024COWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE\nQUANTUM GROUPS\nJACEK KRAJCZOK\nAbstract. We show that (central) Cowling-Haagerup constant of discre te quantum groups is\nmultiplicative Λ cbpL\n1ˆL\n2q “ΛcbpL\n1qΛcbpL\n2q, which extends the result of Freslon [15] to general\n(not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is\nconsidering algebras C ppLq,L8ppLqas operator modules over L1ppLq.\n1.Introduction\nWeak amenability is an approximation property introduced in the cont ext of locally compact\ngroups by Cowling and Haagerup in [6]. It is weaker then amenability, bu t still quite strong as it\nimplies Haagerup-Kraus approximation property (AP). A significant aspect of weak amenability\nis that it comes together with a quantifier: for any locally compact gr oup one defines Cowling-\nHaagerup constant Λ cbpGq P r1,`8swhich is finite precisely when Gis weakly amenable. Authors\nof [6, 16] calculated this constant for all connected, non-compac t, simple Lie groups with finite\ncenter. For example Λ cbpSpp1,nqq “2n´1pně2qbut if real rank of Gis greater than one, then\nGis not weakly amenable and Λ cbpGq “ `8. Another important result tells that if Γ is a lattice\ninGthen Λ cbpΓq “ΛcbpGq, hence Cowling-Haagerup constant is a useful tool in telling apart\ndiscrete groups and their group C˚/von Neumann algebras. Cowling and Haagerup proved also\nthat constant Λ cbis multiplicative, i.e. Λ cbpGˆHq “ΛcbpGqΛcbpHqholds for any locally compact\ngroupsG,H([6, Corollary 1.5]).\nOne can extend the definition of weak amenability and Cowling-Haager up constant to discrete\nor even general locally compact quantum groups ([14, Definition 3.5], [3, Definition 5.12], see also\nDefinition 3.4). This property has received a lot of attention – let us m ention that it is known\nthat strong amenability (i.e. coamenability of the dual) implies weak ame nability which in turn\nimplies AP, weak amenability with Λ cb“1 is preserved under taking free products of discrete\nquantum groups [13] and quantum groups such asyO`\nF,yU`\nFor SUqp1,1qextare weakly amenable\nwith Cowling-Haagerup constant equal 1 ([11, Theorem 24], [5, Theo rem 7.4]). It is however an\nopen question whether amenability implies weak amenability (in fact it is n ot known whether\namenability implies AP, see [8, Corollary 7.4]. These implications are known to be true in discrete\ncase by [24, Theorem 3.8]). Freslon in [15, Proposition 3.2] proved tha t weak amenability passes\nto products of discrete quantum groups, but so far the best info rmation on the value of Cowling-\nHaagerup constant were the bounds max pΛcbpL\n1q,ΛcbpL\n2qq ďΛcbpL\n1ˆL\n2q ďΛcbpL\n1qΛcbpL\n2q. In\nTheorem 3.5 we will show that the upper bound ďis in fact always an equality. Example 4.9\nshows why this knowledge can make a qualitative difference.\nFor discrete quantum groups there is a close connection between p roperties of quantum groupLand its operator algebras C ppLq,L8ppLq. For example, weak amenability ofLimplies that C ppLq\nhas completely bounded approximation property, L8ppLqhas weak˚completely bounded approxi-\nmation property and there is a bound on respective constants (se e [3, Theorem 6.6] and references\ntherein). Theconverseholdsunderunimodularityassumption([19 , Theorem5.14])andinthiscase\nall the involved constants are equal (see also [8, Proposition 4.7] f or a related result). Whether\nthis converse and its variants for strong amenability and AP hold in ge neral, is a major open\nproblem ([3, Remark 6.9]). The main reason why in general it is difficult to deduce a property\n2020Mathematics Subject Classification. Primary 46L67, Secondary 22D55, 47L25.\nKey words and phrases. Weak amenability, discrete quantum group.\n12 JACEK KRAJCZOK\nofLfrom properties of C ppLq,L8ppLqis the lack of averaging which exists in unimodular (dually\n- Kac type) case, and allows one to turn a CB map into a multiplier (see [3 , Section 7.1] and [9,\nSection 7.1]). As Freslon notes in [15, Remark 3.3], in the unimodular cas e we can use equality\nΛcbpLq “ΛcbpCppLqqtodeducethat Cowling-Haagerupconstantismultiplicativeusing[4, Theorem\n12.3.13]. This result states that Cowling-Haagerup constant of C˚-algebras is multiplicative with\nrespect to minimal tensor product. In general however this appr oach does not work, as it is not\nknown whether Λ cbpLq ďΛcbpCppLqq. One way of remeding this situation is to look at C ppLq,L8ppLq\nnot only as at C˚/von Neumann algebras, but consider them together with extra st ructure. This\napproach already turned out to be quite fruitful and lead to sever al results concerning amenability\n– injectivity (see [23, Theorem 3] and [7, Theorem 5.1]), AP – weak˚OAP ([10, Theorem 6.16])\nor strong amenability – weak˚CPAP ([18, Theorem 6.11]).\nIn our work we take a similar point of view, and look at C ppLq,L8ppLqas L1ppLq-modules. In\nDefinition 3.4 we introduce respective Cowling-Haagerup-like consta nts and in Theorem 3.5 show\nthat they are equal to the analogous constants forL. In Section 4 we show that such Cowling-\nHaagerup constant for operator modules of the form C ppLqis multiplicative (Proposition 4.5). Its\nproof is a modification of the proof of [4, Theorem 12.3.13]. The main diff erence is that we take\nalso the module structure into account (see also remarks 3.2, 4.6).\nApart from weak amenability of discrete quantum groups, we are als o interested in its central\nvariation (see Definition 3.4). To study this property, we will look at C ppLq,L8ppLqas L1ppLq-\nbimodules.\n2.Preliminaries and notation\nIn this section we will briefly recall the necessary operator space a nd quantum group back-\nground. We refer to [2, 7, 12], [1, 3, 10, 20, 21, 22, 27] and refere nces therein for more information.\nCompletely contractive Banach algebra is an associative algebra Awhich is at the same time an\noperator space and the multiplication map extends to a complete con tractionApbAÑA, where\npbis the projective tensor product of operator spaces. We say tha t an operator space Xis a left\noperatorA-module, if it is a left module over Aand the action extends to a complete contraction\nApbXÑX. Since this is the only type of modules we consider, we will simply say tha tXis a\nleftA-module. In a similar way we define right A-modules and A-B-bimodules. By definition, an\nA-bimodule is an A-A-bimodule. Note that every operator space or module can be consid ered as\na bimodule by setting A“C,B“Cor both. Furthermore, if A,Bare completely contractive\nBanach algebras, then so is ApbB.\nThe operator space of completely bounded (CB) maps between two operator spaces X,Ywill\nbe denoted by CB pX,Y q. IfX,Yare leftA-modules, then the closed subspace consisting of\nleftA-module maps will be denoted by ACBpX,Y q. Similarly we define the space of right A-\nmodule maps CB ApX,Y qandA-B-bimodule maps ACBBpX,Y q. The CB norm will be denoted\nby}ϕ}CBpX,Y qor simply }ϕ}cb.\nIfAisacompletelycontractiveBanachalgebraand Xis aleftA-module, then the dualoperator\nspaceX˚becomescanonicallyaright A-modulewithactiondefinedby xωa,x y “ xω,ax y. Similarly\nfor right modules and bimodules. The canonicalpairing between X˚andXwill be denoted simply\nbyxω,xyorxω,xyX˚,Xif we want to indicate which spaces are involved. Pairing gives rise to\ncanonical complete contraction κ:XpbX˚ÑC.\nLetX,Ybe operator spaces, Xa rightA-module, and Ya leftA-module. Then we can form\ntheA-module tensor product XpbAY, which by definition is given by the quotient operator space\nXpbAY“ pXpbYq{span txaby´xbay|xPX,a PA,y PYu.\nBy an abuse of notation, the quotient map will be denoted by q:XpbYÑXpbAY. A result which\nwill be very useful, is that in this situation CB ApX,Y˚q » pXpbAYq˚completely isometrically,\nwhereqpxbyqcorresponds to the functional ϕÞÑ xϕpxq,yy([2, Proposition 3.5.9]). Similarly\nCBpX,Y˚q » pXpbYq˚completely isometrically. In this way both CB ApX,Y˚qand CB pX,Y˚q\nare dual operator spaces and have the corresponding weak˚topologies. In particular, one canCOWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 3\nrestrict weak˚topology from CB pX,Y˚qto CB ApX,Y˚q. One easily checks that both topologies\non CB ApX,Y˚qagree and CB ApX,Y˚qis weak˚closed in CB pX,Y˚q.\nIfAis a completely contractive Banach algebra, then so is Aop(Aopby definition has the same\noperatorspacestructure, butoppositemultiplication). Thenany leftA-modulebecomesright Aop-\nmoduleandviceversa. Furthermore,if XisaA-B-bimodulethenitisaright AoppbB-module, with\nmodule structure xpaopbbq “axb. One immediately sees that ACBBpX,Y q “CBAoppbBpX,Y q\nfor anyA-B-bimodules X,Y. Let us also recall that for any finite dimensional operator space E,\nthe canonical map EÑE˚˚establishes a completely isometric isomorphism.\nIn this work we will be interested only in compact or discrete quantum groups. Readers inter-\nested in general framework are referred to [20]. Compact quantu m group Gis defined by a unital\nC˚-algebra C pGqand a unital ˚-homomorphism ∆: C pGq ÑCpGq bCpGqcalled comultiplication,\nwhich satisfies certain conditions. Under separability assumption Wo ronowicz [27] (and van Daele\n[25] in general) proved that there exists a unique state hPCpGq˚(called Haar integral) which is\nbi-invariant. We will assume that it is faithful, i.e. we work at the reduc ed level (see [1]). Per-\nforming GNS representation, we obtain Hilbert space L2pGq, faithful representation of C pGqand\nafter taking sot-closure, von Neumann algebra L8pGq. Bothhand ∆ extend to normal maps\non L8pGq. The predual of L8pGqwill be denoted by L1pGq. The predual mapping of ∆ gives a\ncompletely contractive Banach algebra structure on L1pGq:\nL1pGqpbL1pGq QωbνÞÑω‹ν“ pωbνq∆PL1pGq.\nIt is not difficult to check that both C pGqand L8pGqare L1pGq-bimodules with respect to actions\nω‹x“ pidbωq∆pxq,x‹ω“ pωbidq∆pxqforωPL1pGqandxPCpGqorxPL8pGq. Representation\ntheory of compact quantum groups resembles the one of compact qroups. In particular, every\nirreducible representation is finite dimensional. Let Irr pGqbe the set of their equivalence classes.\nFor each class αPIrrpGqwe choose its representative Uαwhich acts on a Hilbert space Hαof\ndimension dim pαq. In each Hαchoose an orthonormal basis tξα\niudim pαq\ni“1in which operator ραis\ndiagonal (see [21, Section 1.4]), with eigenvalues ρα,ip1ďiďdimpαqq. Number Tr pραqis called\nthe quantum dimension of αand is denoted dim qpαq. The space Pol pGqspanned by coefficients\nUα\ni,j“ pidbωξα\ni,ξα\njqUαp1ďi,jďdimpαqq, together with restricted comultiplication, is a unital\nHopf ˚-algebra. It is norm dense in C pGq, hence weak˚dense in L8pGq.\nBy definition, any discrete quantum groupLis a dual of compact quantum group G:L“pG\n(thus also G“pL– we will prefer to look from discrete point of view). It comes togeth er with C˚-\nalgebra c 0pLq “À\nαPIrrppLqBpHαq(C0-direct sum), von Neumann algebra ℓ8pLq “ś\nαPIrrppLqBpHαq\nand comultiplication ∆. Consequently any element of ℓ8pLqis given by a family paαqαPIrrppLqof\nmatrices in B pHαq. We will say that a net paλqλPΛconverges pointwise to some ainℓ8pLqif\nand only if aλ,αÝ Ý Ñ\nλPΛaαin B pHαqfor allαPIrrppLq. The dense subspace consisting of families\npaαqαPIrrppLqsuch thataα‰0 for only finitely many α’s, will be denoted by c 00pLq. Another\nimportant subspace of ℓ8pLqis ApLq, the Fourier algebra ofL. It consists of elements of the form\npλpωqwithωPL1ppLq(see [3, Section 4.2], [10, Section 3]). It is an subalgebra of c 0pLqand is\nitself a completely contractive Banach algebra with operator space structure given by completely\nisometric isomorphism A pLq Qpλpωq ÞÑωPL1ppLq. A (left) completely bounded multiplier is an\nelementaPℓ8pLqsuch thatabPApLqfor allbPApLqand the associated map A pLq ÑApLqis\ncompletely bounded. After composing with isomorphism A pLq »L1ppLqand taking the dual map,\nany suchagives a normalCB map Θlpaq PCBσpL8ppLqq(superscript σindicates that CBσpL8ppLqq\nconsists of normal CB maps). The space of completely bounded mult ipliers, equipped with the\nCB norm }a}cb“ }Θlpaq}cb, is denoted by Ml\ncbpApLqq. For example, any pλpωq PApLqis a left\ncompletely bounded multiplier with the associated map Θlppλpωqq “ pωbidqp∆. Let us also note\nc00pLq ĎApLq. For anyaPMl\ncbpApLqq, we have Θlpaq PL1ppLqCBσpL8ppLqq, i.e. Θlpaqis a normal,\nCB, left L1ppLq-module map. By [17, Corollary 4.4] (see also discussion in [10, Section 3]) all maps\non L8ppLqwhich satisfy these properties are of the form Θlpaqfor someaPMl\ncbpApLqq. It is not4 JACEK KRAJCZOK\ndifficult to check that Θlpaqrestricts to Θlpaq↾CppLqPL1ppLqCBpCppLqq. Using e.g. [10, Proposition\n3.5] we again see that every CB, left L1ppLq-module map on C ppLqis of the form Θlpaq↾CppLqfor some\naPMl\ncbpApLqq. Similarly, central multipliers aPZMl\ncbpApLqqcorrespond to CB, L1ppLq-bimodule\nmaps on C ppLqand normal, CB, L1ppLq-bimodule maps on L8ppLq.\nWheneverwehavetwocompactquantum groups pL\n1,pL\n2, we canformtheir product pL“pL\n1ˆpL\n2.\nThe associated algebras are C ppLq “CppL\n1q bCppL\n2q, L8ppLq “L8ppL\n1q¯bL8ppL\n2q(hence L1ppLq “\nL1ppL\n1qpbL1ppL\n2q), Pol ppLq “PolppL\n1q dPolppL\n2qand the Haar integral is hpL“hpL\n1bhpL\n2. We can\nalso identify irreducible representations of pL: IrrppLqis the set of α⊠βforαPIrrppL\n1q,βPIrrppL\n2q,\nwhereUα⊠β“Uα\n13Uβ\n24is a representation of pLonHαbHβ. For details see [26]. For finite subsets\nF1ĎIrrppL\n1q,F2ĎIrrppL\n2qdenoteF1⊠F2“ tα⊠β|αPF1,βPF2u. Product of discrete quantum\ngroupsL\n1andL\n2is defined to beL\n1ˆL\n2“L, whereLis the dual of pL.\nWe will be using the following useful notation: if pLis an arbitrary compact quantum group\nand H ‰FĎIrrppLqis a finite subset, set Pol FppLq “span tUα\ni,j|αPF,1ďi,jďdimpαquand\nconsider it to be an operator space with structure coming from C ppLq. Next, for each αPIrrppLq,\nletpαbe the central projection corresponding to B pHαq Ďℓ8pLqandpF“ř\nαPFpαPc00pLq.\nUsing orthogonality relations one easily sees that\n(2.1) pF“pλpωFq,whereωF“ÿ\nαPFdim pαqÿ\ni“1dimqpαqρα,ihpUα˚\ni,i¨q PL1ppLq.\nFurthermore, ΘlppFqis a projection onto Pol FppLq.\nSymbol dwill denote the algebraic tensor product, btensor product of Hilbert spaces or\nminimal (spatial) tensor product of C˚-algebras, ¯bvon Neumann algebraic tensor product and pb\nprojective tensor product of operator spaces. Operator spac es are assumed to be complete. All\nvector spaces are considered over C.\n3.Cowling-Haagerup constant for modules\nIn this section we introduce a Cowling-Haagerup constant for (bi)m odules C ppLq,L8ppLq, study\nits properties and relate it to the (central) Cowling-Haagerup cons tant ofL(Theorem 3.5).\nDefinition 3.1. LetLbe a discrete quantum group.\n‚DefineL1ppLqΛcbpCppLqqto be the infimum of all numbers Cě1 such that there is a\nnetpϕλqλPΛof finite rank, left L1ppLq-module CB maps on C ppLqwith }ϕλ}cbďCand\nϕλpxq Ý Ý Ñ\nλPΛxfor allxPCppLq. If no such number exists, setL1ppLqΛcbpCppLqq “ `8 .\n‚SimilarlydefineΛcb,L1ppLqpCppLqqandL1ppLqΛcb,L1ppLqpCppLqqbyconsideringrightL1ppLq-module\nmaps and L1ppLq-bimodule maps, respectively.\n‚DefineL1ppLqΛcbpL8ppLqqto be the infimum of all numbers Cě1 such that there is a net\npψλqλPΛof normal, finite rank, left L1ppLq-module CB maps on L8ppLqwith }ψλ}cbďCand\nψλpxq Ý Ý Ñ\nλPΛxweak˚for allxPL8ppLq. If no such number exists, setL1ppLqΛcbpL8ppLqq “\n`8.\n‚Similarly define Λcb,L1ppLqpL8ppLqqandL1ppLqΛcb,L1ppLqpL8ppLqqby considering right L1ppLq-\nmodule maps and L1ppLq-bimodule maps, respectively.\nNumbersL1ppLqΛcbpCppLqq, etc. will be called Cowling-Haagerupconstants. A standard argum ent\n(using a new net indexed over Λ ˆN) shows that the infimum in the above definition is actually\nachievable.\nRemark 3.2. In principle we could have introduced similar constants for arbitrary operator\nmodules over completely contractive Banach algebras. We have dec ided not to do that, as generalCOWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 5\noperator modules can fail to have any finite dimensional submodules , and also we were unable to\nprove that such constant is in general multiplicative (see Propositio n 4.5 and Remark 4.6).\nIn our first proposition we show that it doesn’t matter if we look at lef t or right module\nstructure, and similarly it doesn’t matter if we look at C˚or von Neumann level.\nProposition 3.3. LetLbe a discrete quantum group. Then\nL1ppLqΛcbpCppLqq “Λcb,L1ppLqpCppLqq “L1ppLqΛcbpL8ppLqq “Λcb,L1ppLqpL8ppLqq\nand\nL1ppLqΛcb,L1ppLqpCppLqq “L1ppLqΛcb,L1ppLqpL8ppLqq.\nProof.IfψPCBσpL8ppLqqis a normal, left L1ppLq-module map, then pR˝ψ˝pRis a normal right\nL1ppLq-module map with }pR˝ψ˝pR}cb“ }ψ}cb([10, Lemma 4.8]), where pRis the unitary antipode\non L8ppLq. We can similarly turn right L1ppLq-module maps into left one, the property of being\nfinite rank is preserved. Finally, net pψλqλP��converges to id in the point-weak˚topology if and\nonly if ppR˝ψλ˝pRqλPΛconverges to id. This showsL1ppLqΛcbpL8ppLqq “Λcb,L1ppLqpL8ppLqq.\nThe above quoted part of [10, Lemma 4.8] has a C˚-algebraic variant (with virtually the same\nproof, using usual Wittstock’s theorem [4, Theorem B7]): if ϕPCBpCppLqqis a finite rank, left\nL1ppLq-module map, then pR˝ϕ˝pRis a finite rank, right L1ppLq-module map with the same CB\nnorm, and vice versa. Similarly, pϕλqλPΛconverges in point-norm topology to id if and only if\nppR˝ϕλ˝pRqλPΛdoes, henceL1ppLqΛcbpCppLqq “Λcb,L1ppLqpCppLqq.\nAssume that pψλqλPΛis a net of finite rank maps inL1ppLqCBσpL8ppLqqwhich converges to id\nin the point-weak˚topology and }ψλ}cbďL1ppLqΛcbpL8ppLqq ă `8 . As discussed in Section 2,\nfor eachλPΛ there is aλPc00pLqsuch thatψλ“Θlpaλq. Then Θlpaλqrestricts to a map\nϕλinL1ppLqCBpCppLqqwith }ϕλ}cb“ }ψλ}cb(by weak˚density of C ppLq ĎL8ppLq) such that\nϕλpxq Ý Ý Ñ\nλPΛxin norm for every xPPolppLq. Indeed, observe that tϕλpxq,x|λPΛulive in a\nfinite dimensional subspace of Pol ppLq, and in finite dimensional spaces there is a unique Hausdorff\nvector space topology. Since Pol ppLqis norm dense in C ppLqand the net pϕλqλPΛis bounded, a\nstandard approximation argument allows us to conclude that ϕλpxq Ý Ý Ñ\nλPΛxfor allxPCppLqand\nconsequentlyL1ppLqΛcbpCppLqq ďL1ppLqΛcbpL8ppLqq.\nSimilar reasoning givesL1ppLqΛcb,L1ppLqpCppLqq ďL1ppLqΛcb,L1ppLqpL8ppLqq. The only difference is\nthat ifψλis known to be a L1ppLq-bimodule map, then so will be ϕλ.\nAssume that pϕλqλPΛis a net of finite rank maps inL1ppLqCBpCppLqqwith CB norm bounded by\n}ϕλ}cbďL1ppLqΛcbpCppLqq, assumed to be finite. As in the previous paragraph, there are aλPc00pLq\nsuch thatϕλ“Θlpaλq↾CppLq. Then Θlpaλqare normal, finite rank, CB maps inL1ppLqCBσpL8ppLqq\nwith }Θlpaλq}cb“ }ϕλ}cb. TakexPL8ppLqzt0u,ωPL1ppLqzt0uandεą0. Since products are\nlinearly dense in L1ppLq([7, Section 3]), we can find ωk,ω1\nkPL1ppLq p1ďkďKqsuch that\n}ω´řK\nk“1ωk‹ω1\nk} ďε\n2}x}p1`L1ppLqΛcbpCppLqqq. Furthermore, for any kwe haveω1\nk‹xPCppLq([10,\nLemma 4.6]), hence there is λ0PΛ such that\n}Θlpaλqpω1\nk‹xq ´ω1\nk‹x} ďε\n2Kp1`}ωk}qp1ďkďK,λ ěλ0q.\nForλěλ0we have\n|xΘlpaλqpxq ´x,ωy|\nďε\n2`Kÿ\nk“1|xΘlpaλqpxq ´x,ωk‹ω1\nky| “ε\n2`Kÿ\nk“1|xω1\nk‹Θlpaλqpxq ´ω1\nk‹x,ωky|\n“ε\n2`Kÿ\nk“1|xΘlpaλqpω1\nk‹xq ´ω1\nk‹x,ωky| ďε\n2`Kÿ\nk“1}Θlpaλqpω1\nk‹xq ´ω1\nk‹x}}ωk} ďε.6 JACEK KRAJCZOK\nThis proves Θlpaλqpxq Ý Ý Ñ\nλPΛxweak˚and consequentlyL1ppLqΛcbpL8ppLqq ďL1ppLqΛcbpCppLqq. As\nabove, a minor modification gives L1ppLq-bimodule version. /square\nBecauseofProposition3.3, wewillfocusonL1ppLqΛcbpCppLqqandL1ppLqΛcb,L1ppLqpCppLqq. Letusnow\nrecall Cowling-Haagerup constant and its central variant for disc rete quantum groups ([3, 9, 14]).\nDefinition 3.4. LetLbe a discrete quantum group.\n‚The Cowling-Haagerup constant ofL, ΛcbpLq P r1,`8s, is the infimum of all numbers\nCě1 such that there is a net paλqλPΛin c00pLqwith }aλ}cbďCandaλÝ Ý Ñ\nλPΛ\n/BDpointwise.\nIf no such number exists, set Λ cbpLq “ `8. If ΛcbpLq ă `8, one says thatLis weakly\namenable.\n‚The central Cowling-Haagerup constant ZΛcbpLq P r1,`8sis the infimum of all numbers\nCě1 such that there is a net paλqλPΛinZc00pLqwith }aλ}cbďCandaλÝ Ý Ñ\nλPΛ\n/BD\npointwise. If no such number exists, set ZΛcbpLq “ `8. IfZΛcbpLq ă `8, one says thatLis centrally weakly amenable.\nSimilarly to Definition 3.1, the infimum is actually attainable. Let us note t hat in the definition\nof ΛcbpLqwe could also consider more general nets paλqλPΛassumed only to be in the Fourier\nalgebra A pLq. A standard approximationargument shows that both definitions a re equivalent (see\ne.g.[3, Section3.2]). Insteadofspeakingaboutpointwiseconverge nce,onecanrequirethat paλqλPΛ\nforms an approximate identity in the Fourier algebra A pLq. In this context, both conditions are\nequivalent. The following result provides a link between Cowling-Haage rup constant of a discrete\nquantum groupLand the associated module C ppLq. Its proof is quite standard, compare e.g. [3,\nTheorem 6.6].\nTheorem 3.5. For any discrete quantum groupL\nΛcbpLq “L1ppLqΛcbpCppLqqandZΛcbpLq “L1ppLqΛcb,L1ppLqpCppLqq.\nProof.Assume Λ cbpLq ă `8 and let paλqλPΛbe a net in c 00pLqsuch that }aλ}cbďΛcbpLq\nandaλÝ Ý Ñ\nλPΛ\n/BDpointwise. Since c 00pLq ĎApLq, we can find normal functionals ωλPL1ppLq\nsuch thataλ“pλpωλq. Then Θlpaλq “ pωλbidqp∆, consider ϕλ“Θlpaλq↾CppLq. This map is\nof finite rank, belongs toL1ppLqCBpCppLqqand has CB norm equal to }aλ}cb. Furthermore, since\nsupλPΛ}aλ}cbă 8, to see that ϕλÝ Ý Ñ\nλPΛid in the point-norm topology of C ppLq, it is enough to look\nat the dense subspace Pol ppLq. SinceaλÝ Ý Ñ\nλPΛ\n/BDpointwise, for any αPIrrppLq,1ďi,jďdimpαq\nwe haveωλpUα\ni,jq Ý Ý Ñ\nλPΛδi,jand consequently ϕλpUα\ni,jq “řdim pαq\nk“1ωλpUα\ni,kqUα\nk,jconverges in norm\ntoUα\ni,j. We conclude thatL1ppLqΛcbpCppLqq ďΛcbpLq.\nAssume now thatL1ppLqΛcbpCppLqq ă `8 with the corresponding net pϕλqλPΛinL1ppLqCBpCppLqq.\nAs discussed in Section 2, there is a multiplier aλPMl\ncbpApLqqsuch thatϕλ“Θlpaλq↾CppLq, in\nparticular }aλ}cb“ }ϕλ}cb. Sinceϕλis of finite rank, we have in fact aλPc00pLq. As the net\npϕλqλPΛconverges to id in the point-norm topology, we have aλÝ Ý Ñ\nλPΛ\n/BDpointwise. This shows\nΛcbpLq ďL1ppLqΛcbpCppLqq.\nThecentralandbimodulevariantisprovedinasimilarway, withslightmo dification. Inthefirst\ndirection, we additionally have aλPZc00pLq, then Θlpaλq↾CppLqis a L1ppLq-bimodule map giving\nL1ppLqΛcb,L1ppLqpCppLqq ďZΛcbpLq. Conversely since ϕλis a map of bimodules, aλis central. /square\nRemark 3.6. It is an interesting question whetherL1ppLqΛcbpCppLqq “L1ppLqΛcb,L1ppLqpCppLqqalways\nholds, equivalently (by Theorem 3.5) whether Cowling-Haagerup con stant ofLis equal to its\ncentral variant Λ cbpLq “ZΛcbpLq. To the best of our knowledge, no counterexample is known.\nAn analogous result for strong amenability is false (see e.g. [9, Theo rem 7.6]).COWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 7\n4.Cowling-Haagerup constant of the product\nIn this section we prove our main result: (central) Cowling-Haageru pconstant of discrete quan-\ntum groups is multiplicative (Theorem 4.7). We will do this by establishing first an analogous\nresult for modules C ppLq(Proposition 4.5) and then using Theorem 3.5. As mentioned in the in-\ntroduction, our proof of Proposition 4.5 is a modification of the proo f of [4, Theorem 12.3.13] (see\nalso Remark 4.6).\nIt will be convenient to work in the more general language of complet ely contractive Banach\nalgebras and operator modules, see Section 2. The next lemma is a bim odule generalisation of [4,\nLemma 12.3.16]. Recall that any A-B-bimodule, is also a right AoppbB-module (see Section 2).\nLemma 4.1. LetA,Bbe completely contractive Banach algebras, XanA-B-bimodule and FĎ\nEĎXfinite dimensional submodules. Take Cě1. The following statements are equivalent:\n(1) there is ϕPACBBpX,E qsuch thatϕpxq “xpxPFqand }ϕ}CBpX,E qďC,\n(2)|κpuq| ďC}qpuq}foruPFdE˚, whereκ:EdE˚ÑCis the pairing map and q:XpbE˚Ñ\nXpbAoppbBE˚is the canonical quotient map.\nProof.Assume that we have ϕPACBBpX,E qas in p1q, takeuPFdE˚and writeu“řn\nk“1xkb\nωkfor somexkPF,ωkPE˚. Then using identifications CB pF,E q “CBpF,E˚˚q » pFpbE˚q˚and\nACBBpX,E q “CBAoppbBpX,E q » pXpbAoppbBE˚q˚we calculate\n|κpuq| “ˇˇnÿ\nk“1xωk,xkyˇˇ“ˇˇnÿ\nk“1xωk,ϕpxkqyˇˇ“ |xϕ,uyCBpX,E q,XpbE˚|\n“ |xϕ,qpuqyCBAopxbBpX,E q,XpbAopxbBE˚| ďC}qpuq},\ni.e.p2qholds.\nConversely, assume that |κpuq| ďC}qpuq}for alluPFdE˚. Then the functional\n(4.1) XpbAoppbBE˚ĚqpFdE˚q Qqpuq ÞÑκpuq PC\nis well defined and has norm bounded by C. By Hahn-Banach theorem, we can find ϕP\npXpbAoppbBE˚q˚»ACBBpX,E qwhich extends (4.1) and has norm ďC. ForxPF,ω PE˚\nwe have\nxω,ϕpxqy “ xϕ,qpxbωqyCBAopxbBpX,E q,XpbAopxbBE˚“κpxbωq “ xω,xy\nhenceϕpxq “x. /square\nNext we establish several useful properties of left L1ppLq-module C ppLq.\nLemma 4.2. LetLbe a discrete quantum group and Cě1. The following conditions are\nequivalent:\n(1)L1ppLqΛcbpCppLqq ďC,\n(2) for every εą0and finite H ‰FĎIrrppLqthere is a finite rank ϕPL1ppLqCBpCppLqqsuch\nthat }ϕpxq ´x} ďε}x} pxPPolFppLqqand }ϕ}cbďC.\nProof. p1q ñ p2q: takeεą0 and finite H ‰FĎIrrppLq. Since Pol FppLqis a finite dimensional\nnormed space with basis tUα\ni,j|αPF,1ďi,jďdimpαquwe can find Dą0 so that\nÿ\nαPFdim pαqÿ\ni,j“1|xα\ni,j| ďD}ÿ\nαPFdim pαqÿ\ni,j“1xα\ni,jUα\ni,j}\nfor allř\nαPFřdim pαq\ni,j“1xα\ni,jUα\ni,jPPolFppLq. By p1q, there isϕPL1ppLqCBpCppLqqsuch that }ϕ}cbďC\nand\n}ϕpUα\ni,jq ´Uα\ni,j} ďε\nDpαPF,1ďi,jďdimpαqq.8 JACEK KRAJCZOK\nThen for any x“ř\nαPFřdim pαq\ni,j“1xα\ni,jUα\ni,jPPolFppLqwe have\n}ϕpxq ´x} ďÿ\nαPFdim pαqÿ\ni,j“1|xα\ni,j|}ϕpUα\ni,jq ´Uα\ni,j} ďÿ\nαPFdim pαqÿ\ni,j“1|xα\ni,j|ε\nDďε}x}.\np2q ñ p1q: forεą0 and finite H ‰FĎIrrppLq, letϕε,FPL1ppLqCBpCppLqqbe the map from p2q.\nAs}ϕε,F}cbďCfor allε,Fand Pol ppLqis norm dense in C ppLq, it easily follows that net pϕε,Fqpε,Fq\nindexed over εP s0,1rand finite H ‰FĎIrrppLqgivesL1ppLqΛcbpCppLqq ďC. /square\nThere is also a natural analog of Lemma 4.2 for L1ppLq-bimodule C ppLq. Recall that for finite\nH ‰FĎIrrppLq,pFPc00pLq ĎApLqis the central projection pF“ř\nαPFpα.\nLemma 4.3. LetLbe a discrete quantum group and εą0. For a finite set H ‰FĎIrrppLq,\n1ď }pF}ApLqďdÿ\nαPFdimqpαq2.\nProof.Recall that pF“pλpωFq(equation (2.1)). Let }x}2“hpx˚xq1{2pxPL8ppLqqbe the 2-norm\non L8ppLq. Using orthogonality relations ([21, Theorem 1.4.3]) we see\n}pF}2\nApLq“ }ωF}2ď››ÿ\nαPFdim pαqÿ\ni“1dimqpαqρα,iUα\ni,i››2\n2“ÿ\nαPFdim pαqÿ\ni“1dimqpαq2ρα,i2}Uα\ni,i}2\n2\n“ÿ\nαPFdim pαqÿ\ni“1dimqpαq2ρα,i21\ndimqpαqρα,i“ÿ\nαPFdimqpαq2.\nFor the lower bound, choose αPFand letχα“řdim pαq\ni“1Uα\ni,ibe character of α. Then }χα} ď\ndimpαqand\n}pF}ApLq“ }ωF} 졡ωFpχα\n}χα}qˇˇ“1\n}χα}ˇˇdim pαqÿ\ni“1dimqpαqρα,ihpUα˚\ni,iχαqˇˇ“dim pαq\n}χα}ě1.\n/square\nThe next lemma shows intuitively that one can correct an almost equa litya« /BDover a finite\nsetFĎIrrppLqto an actual equality, with an error over which we have precise cont rol.\nLemma 4.4. LetLbe a discrete quantum group, εą0,H ‰FĎIrrppLqfinite set and aP\nMl\ncbpApLqq. Assume that }Θlpaqpxq ´x} ďε}x}forxPPolFppLq. Then there is ˜aPMl\ncbpApLqq\nsuch that Θlp˜aqpxq “xforxPPolFppLq,˜a´aPc00pLqand }˜a´a}ApLqďεř\nαPFdimqpαq2. If\naPZMl\ncbpApLqq, then we can take ˜aPZMl\ncbpApLqq.\nProof.Writea“ paαqαPIrrppLq. Defineb“ř\nαPFppα´aαq Pc00pLqand ˜a“a`b. Since ˜aα“\npαpαPFq, we have Θlp˜aqpxq “xforxPPolFppLq. Furthermore\n˜a´a“b“ÿ\nαPFppα´aαq “ p /BD´aqÿ\nαPFpα“ p /BD´aqpF“ p /BD´aqpλpωFq “pλ`\nΘlp /BD´aq˚pωFq˘\n.\nConsequently, using the facts that ΘlppFq˚pωFq “ωF, ΘlppFqpxq PPolFppLqforxPL8ppLqand\npFis central\n}˜a´a}ApLq“ }Θlp /BD´aq˚pωFq} “ sup\nxPL8ppLq,}x}“1ˇˇxx,Θlp /BD´aq˚pωFqyˇˇ\n“sup\nxPL8ppLq,}x}“1ˇˇxx,Θlp /BD´aq˚ΘlppFq˚pωFqyˇˇ“sup\nxPL8ppLq,}x}“1ˇˇxΘlp /BD´aqΘlppFqpxq,ωFyˇˇ\nďsup\nxPL8ppLq,}x}“1}ΘlppFqpxq ´ΘlpaqpΘlppFqpxqq}}ωF} ďsup\nxPL8ppLq,}x}“1ε}ΘlppFqpxq}}ωF}\n“ε}ωF}}ΘlppFq} ďε}pF}ApLq}pF}cbďε}pF}2\nApLq,COWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 9\nhence the first claim follows from Lemma 4.3. If ais central, then so is band consequently ˜ a./square\nLet us remark that using [12, Corollary 2.2.4] one can obtain a better bound for }˜a´a}cb– we\nwill however not need this. Our main result, in the language of modules , is the following.\nProposition 4.5. LetL\n1,L\n2be discrete quantum groups andL“L\n1ˆL\n2their product. Then\n(4.2)L1ppLqΛcbpCppLqq “L1ppL\n1qΛcbpCppL\n1qqL1ppL\n2qΛcbpCppL\n2qq\nand\n(4.3)L1ppLqΛcb,L1ppLqpCppLqq “L1ppL\n1qΛcb,L1ppL\n2qpCppL\n1qqL1ppL\n2qΛcb,L1ppL\n2qpCppL\n2qq.\nProof.The easier inequality ďwas already established in [15, Proposition 3.2] (after conjunc-\ntion with Theorem 3.5), let us give an essentially equivalent argument f or the convenience of\nthe reader. Recall that C ppLq “CppL\n1q bCppL\n2qas C˚-algebras and L1ppLq “L1ppL\n1qpbL1ppL\n2qas\ncompletely contractive Banach algebras. It is enough to assume th at bothL1ppL\n1qΛcbpCppL\n1qqand\nL1ppL\n2qΛcbpCppL\n2qqare finite, let pϕλqλPΛand pψµqµPΣbe the corresponding maps. Then we can\nconstruct new net pϕλbψµqpλ,µqPΛˆΣof finite rank maps inL1ppL\n1qpbL1ppL\n2qCBpCppL\n1q bCppL\n2qq\n([12, Proposition 8.1.5]). For any λ,µwe have }ϕλbψµ}cbďL1ppL\n1qΛcbpCppL\n1qqL1ppL\n2qΛcbpCppL\n2qq\nand clearly ϕλbψµÝ Ý Ý Ý Ý Ý Ý Ñ\npλ,µqPΛˆΣid on a norm dense set C ppL\n1q dCppL\n2q ĎCppLq. Consequently\npϕλbψµqpλ,µqPΛˆΣconverges to id in the point-norm topology. This allows us to conclude in-\nequality ďin (4.2). An analogous reasoning gives inequality ďin (4.3): the only difference is that\nifϕλandψµare bimodule maps, then so is ϕλbψµ.\nLet us now prove the converse inequalities; we will treat both cases at the same time. Assume\nby contradiction that (4.2) or (4.3) does not hold, i.e.\nL1ppLqΛcbpCppLqq ăL1ppL\n1qΛcbpCppL\n1qqL1ppL\n2qΛcbpCppL\n2qq\nor\nL1ppLqΛcb,L1ppLqpCppLqq ăL1ppL\n1qΛcb,L1ppL\n1qpCppL\n1qqL1ppL\n2qΛcb,L1ppL\n2qpCppL\n2qq.\nThen we can choose positive constants C1,C2such that\nL1ppLqΛcbpCppLqq ăC1C2, (4.4)\n1ďC1ăL1ppL\n1qΛcbpCppL\n1qq,1ďC2ăL1ppL\n2qΛcbpCppL\n2qq (4.5)\nin the left module case and\nL1ppLqΛcb,L1ppLqpCppLqq ăC1C2, (4.6)\n1ďC1ăL1ppL\n1qΛcb,L1ppL\n1qpCppL\n1qq,1ďC2ăL1ppL\n2qΛcb,L1ppL\n2qpCppL\n2qq (4.7)\nin the bimodule case.\nIn order to easier work with both cases at the same time, it will be con venient to reformu-\nlate the situation slightly. As discussed in Section 2, left L1ppLq-module structure on C ppLqgives\nus right L1ppLqop-module structure. Similarly, L1ppLq-bimodule structure can also be encoded\nas right L1ppLqoppbL1ppLq-module structure. Thus from now on, let Abe equal to L1ppLqopor\nL1ppLqoppbL1ppLq, and consider C ppLqas a rightA-module. Similarly for quantum groupsL\n1,L\n2\nconsider C ppL\nkqas a rightAk-module, where Ak“L1ppL\nkqoporAk“L1ppL\nkqoppbL1ppL\nkq.\nFirst we use “negative” (4.5) (or (4.7)). Fix kP t1,2uand use Lemma 4.2 (or its bimodule\nversion) to find εką0 and a finite set H ‰FkĎIrrppL\nkqsuch that for all finite rank maps\nϕPCBAkpCppL\nkqqwith }ϕ}CBpCppL\nkqqďCkthere isxPPolFkppL\nkqwith }ϕpxq ´x} ąεk}x}. In\nparticular\n(4.8) }ϕ↾PolFkppL\nkq´id}CBpPolFkppL\nkq,CppL\nkqqąεk.10 JACEK KRAJCZOK\nNow we use “positive” (4.4) (or (4.6)). Define F“F1⊠F2ĎIrrppLqand choose small δą0\nsuch that\nL1ppLqΛcbpCppLqq ă p1´δqC1C2ăC1C2orL1ppLqΛcb,L1ppLqpCppLqq ă p1´δqC1C2ăC1C2\ndepending on the version we are considering. Next set ε“δC1C2 ř\nαPFdimqpαq2ą0. For this εand\nF, by Lemma 4.2 we can find finite rank ϕPCBApCppLqqwith }ϕ}CBpCppLqqď p1´δqC1C2and\n}ϕpxq ´x} ďε}x}forxPPolFppLq. Sinceϕis a rightA-module map, it corresponds to aPc00pLq\n(oraPZc00pLq) viaϕ“Θlpaq↾CppLq. Choose ˜aPc00pLq(or ˜aPZc00pLq) using Lemma 4.4, so\nthat Θlp˜aq “id on Pol FppLqand since the CB norm is majorised by Fourier algebra norm\n}Θlp˜aq↾CppLq}CBpCppLqq“ }˜a}cbď }a}cb` }˜a´a}cbď p1´δqC1C2`εÿ\nαPFdimqpαq2“C1C2.\nΘlp˜aq↾CppLqis a finite rank right A-module map, hence it has image in Pol EppLqfor some finite\nEĎIrrppLq. By enlarging Eif needed, we can assume E“E1⊠E2for finite H ‰EkĎIrrppL\nkq\nwithFkĎEk. Existence of Θlp˜aq↾CppLqshows that point p1qof Lemma 4.1 holds (for modules\nPolFppLq ĎPolEppLqand constant C1C2), consequently p2qof this lemma gives\n(4.9) |κpuq| ďC1C2}qpuq}\nforuPPolFppLq dPolEppLq˚. Hereqis the quotient map C ppLqpbPolEppLq˚ÑCppLqpbAPolEppLq˚.\nNext we go back to the reasoning concerningL\nk’s. Consider finite dimensional right Ak-\nsubmodules Pol FkppL\nkq ĎPolEkppL\nkqof C ppL\nkqand numbers Ck. We will denote this action and\nits dual by x⊳f,f⊲ωpxPCppL\nkq,ωPPolEkppL\nkq˚,fPAkqto avoid confusion. By the\nreasoning above (inequality (4.8)), point p1qin Lemma 4.1 does not hold, and there is ukP\nPolFkppL\nkq dPolEkppL\nkq˚such that\n(4.10) |κpukq| ąCk}qpukq}.\nWe claim that qpukq ‰0. To see this, we need to introduce an auxilliary bounded functional.\nFirst, observe that we can understand ΘlppEkq↾CppL\nkqas a CB map C ppL\nkq ÑPolEkppL\nkq. Next\nconsider its dual map and define ρto be the composition\nρ: CppL\nkqpbPolEkppL\nkq˚idbpΘlppEkq↾CppL\nkqq˚\nÝ Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ý Ñ CppL\nkqpbCppL\nkq˚κÝ ÑC.\nLet us write\n(4.11) uk“Nkÿ\ni“1xk,ibωk,iforxk,iPPolFkppL\nkq ĎCppL\nkqandωk,iPPolEkppL\nkq˚\nand observe\n(4.12) xρ,uky “Nkÿ\ni“1xρ,xk,ibωk,iy “Nkÿ\ni“1xωk,i,ΘlppEkqpxk,iqy “Nkÿ\ni“1xωk,i,xk,iy “κpukq.\nAssume by contradiction that qpukq “0, then\nukPspan tx⊳fbω´xbf⊲ω|xPCppL\nkq,fPAk,ωPPolEkppL\nkq˚u ĎCppL\nkqpbPolEkppL\nkq˚.\nSince\nxρ,x⊳fbω´xbf⊲ωy “ xω,ΘlppEkqpx⊳fqy ´ xf⊲ω,ΘlppEkqpxqy\n“ xω,ΘlppEkqpxq⊳fy ´ xf⊲ω,ΘlppEkqpxqy “0\nforxPCppL\nkq,fPAk,ωPPolEkppL\nkq˚, we have xρ,uky “0 by continuity of ρ. This contradicts\n(4.10) and (4.12), consequently qpukq ‰0.\nLet us introduce shuffling map (cf. [4, Lemma 12.3.14])\n`\nCppL\n1qpbPolE1ppL\n1q˚˘\nˆ`\nCppL\n2qpbPolE2ppL\n2q˚˘\nQ pv1,v2q ÞÑv1ˆv2PCppLqpbPolEppLq˚COWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 11\ngiven by the bilinear extention of\npx1bω1q ˆ px2bω2q “x1bx2bω1bω2\n(itiswelldefinedas E“E1⊠E2isfinite, hencewecanidentifycompletelyisometricallyPol EppLq “\nPolE1ppL\n1qqbPolE2ppL\n2q, where qbis the injective operator space tensor product [12, Section 8]).\nAccording to [4, Lemma 12.3.14] we have }v1ˆv2} ď }v1}}v2}for anyvkPCppL\nkqpbPolEkppL\nkq˚.\nConsider\nu“u1ˆu2PPolFppLq dPolEppLq˚ĎCppLqpbPolEppLq˚.\nWe will use this element to obtain a contradiction. Using (4.11) we have u“řN1\ni“1řN2\nj“1x1,ib\nx2,jbω1,ibω2,jand consequently\nκpuq “N1ÿ\ni“1N2ÿ\nj“1xω1,ibω2,j,x1,ibx2,jy “N1ÿ\ni“1N2ÿ\nj“1xω1,i,x1,iy xω2,j,x2,jy “κpu1qκpu2q\nand\n(4.13) |κpuq| “ |κpu1q| |κpu2q| ąC1C2}qpu1q} }qpu2q}.\nby (4.10). Next we need to get a hold on the norm }qpuq}, which is the norm in the quotient space\nCppLqpbAPolEppLq˚“ pCppLqpbPolEppLq˚q{kerpqq. Fix an arbitrary ε0ą0. ForkP t1,2uwe can\nchoose\nnkPkerpq: CppL\nkqpbPolEkppL\nkq˚ÑCppL\nkqpbAkPolEkppL\nkq˚q\n“span tn⊳fbν´nbf⊲ν|nPCppL\nkq,fPAk,νPPolEkppL\nkq˚u\nsuch that }uk`nk} ´ε0ď }qpukq} ď }uk`nk}. We can write\nnk“lim\njÑ8Lj\nkÿ\nl“1pnj\nk,l⊳fj\nk,lbνj\nk,l´nj\nk,lbfj\nk,l⊲νj\nk,lq\nfor somenj\nk,lPCppL\nkq,fj\nk,lPAk,νj\nk,lPPolEkppL\nkq˚. Then\nqpu1ˆn2q “lim\njÑ8N1ÿ\ni“1Lj\n2ÿ\nl“1q`\npx1,ibω1,iq ˆ pnj\n2,l⊳fj\n2,lbνj\n2,l´nj\n2,lbfj\n2,l⊲νj\n2,lq˘\n“lim\njÑ8N1ÿ\ni“1Lj\n2ÿ\nl“1q`\nx1,ibnj\n2,l⊳fj\n2,lbω1,ibνj\n2,l´x1,ibnj\n2,lbω1,ibfj\n2,l⊲νj\n2,l˘\n“lim\njÑ8N1ÿ\ni“1Lj\n2ÿ\nl“1q`\npx1,ibnj\n2,lq⊳pωbfj\n2,lq b pω1,ibνj\n2,lq ´ px1,ibnj\n2,lq b pωbfj\n2,lq⊲pω1,ibνj\n2,lq˘\nwhereωPL1ppL\n1q(orωPL1ppL\n1qpbL1ppL\n1q) is anynormalfunctional which onPol E1ppL\n1qactsas the\ncounit – so x1,i⊳ω“x1,iandω⊲ω1,i“ω1,i. Such functional can be easily constructed using\northogonality relations [21, Theorem 1.4.3], for example we can take ω“ωE1(orω“ωE1bωE1).\nIt follows that qpu1ˆn2q “0. Similarly we check qpn1ˆu2q “0 andqpn1ˆn2q “0. Consequently\nqpuq “qpu1ˆu2q “qpu1ˆu2`n1ˆu2`u1ˆn2`n1ˆn2q\nso\n}qpuq} ď }u1��u2`n1ˆu2`u1ˆn2`n1ˆn2} “ }pu1`n1q ˆ pu2`n2q}\nď }u1`n2} }u2`n2} ď p}qpu1q} `ε0qp}qpu2q} `ε0q.\nSinceε0ą0 was arbitrary, we conclude }qpuq} ď }qpu1q}}qpu2q}. Combining this with inequalities\n(4.9) and (4.13) we get\nC1C2}qpu1q} }qpu2q} ăC1C2}qpu1q} }qpu2q},\nand asqpu1q ‰0,qpu2q ‰0 this gives a contradicition. /square12 JACEK KRAJCZOK\nRemark 4.6. We have formulated and proved Proposition 4.5 only for modules of th e form C ppLq\nbecause of two reasons. First, in the case of C ppLqthere is a canonical dense submodule Pol ppLq\nwhosefinite dimensional subspacesgivea wealthoffinite dimensional submodules. Another reason\nis that for any finite H ‰EĎIrrppLqone can find ωPL1ppLqwhich acts as the identity on Pol EppLq.\nThis “local unitality” property was used to obtain bound }qpu1ˆu2q} ď }qpu1q} }qpu2q}.\nTheorem 4.7. LetL\n1,L\n2be discrete quantum groups andL“L\n1ˆL\n2their product. Then\nΛcbpLq “ΛcbpL\n1qΛcbpL\n2qandZΛcbpLq “ZΛcbpL\n1qZΛcbpL\n2q.\nProof.This result is an immediate consequence of Proposition 4.5 and Theore m 3.5. /square\nAs a corollary, we extend this result to infinite direct sums. Let pL\niqiPIbe a non-empty family\nof discrete quantum groups. Then one can define productś\niPIpL\ni, which is a compact quantum\ngroup ([26], see also [10, Section 7.2]). We will denote its discrete dual byÀ\niPI\nL\niand call it the\ndirect sum of family pL\niqiPI(the name and notation is inspired by the classical case whereś\niPIΓi\nis larger thanÀ\niPIΓiwhenever |I| “ 8and |Γi| ě2).\nCorollary 4.8. LetpL\niqiPIbeanon-emptyfamilyofdiscretequantumgroupsandletL“À\niPI\nL\ni\nbe their direct sum. Then\n(4.14) Λ cbpLq “ź\niPIΛcbpL\niqandZΛcbpLq “ź\niPIZΛcbpL\niq.\nProof.IfIis finite, then the claim follows immediately from Theorem 4.7; assume th at|I| “ 8.\nDiscrete quantum groupLis the direct limit of system p‘iPF\nL\niqFindexed by finite non-empty\nsubsetsFĎIwith the canonical injective maps C pś\niPFpL\niq QxÞÑxb pbiPF1zF\n/BDiq PCpś\niPF1pL\niq\nforFĎF1. Using Theorem 4.7 and [15, Proposition 3.6] we have\nΛcbpLq “sup\nFΛcb`à\niPF\nL\ni˘\n“sup\nFź\niPFΛcbpL\niq “ź\niPIΛcbpL\niq\n(recall Λ cbpL\niq ě1). One easily sees that [15, Proposition 3.6] holds also for the centr al Cowling-\nHaagerup constant, which gives the second equality in (4.14).\nAlternatively, one can prove both equalities (4.14) as follows. Lower bounds follow from\nTheorem 4.7 and decompositionÀ\niPI\nL\ni“`À\niPF\nL\ni˘\nˆ`À\niPIzF\nL\ni˘\nwhich holds for all finite\nH ‰FĎI. Upper bounds ďin (4.14) can be directly showed as in the first paragraph of the\nproof of Proposition 4.5. /square\nWe end with an example, which shows that knowing the exact value of C owling-Haagerup\nconstant (not just an upper and lower bound), can make a significa nt difference.\nExample 4.9. LetpL\nnqnPNbeasequenceofdiscretequantumgroups,suchthatΛ cbpL\nnq ă `8for\nallnPNand liminf nPNΛcbpL\nnq ą1. DefineL“À8\nn“1\nL\nn. Then using Corollary 4.8 we calculate\nΛcbpLq “ś8\nn“1ΛcbpL\nnq “ 8, henceLis not weakly amenable. Note that we wouldn’t be able\nto conclude this knowing only Λ cbpL\nnˆL\nmq ěmaxpΛcbpL\nnq,ΛcbpL\nmqq. Since weak amenability\nimplies Haagerup-Kraus approximation property AP ([10, Propositio n 5.7]), all quantum groupsL\nnhave AP and so doesL([10, Proposition 7.5]).\n5.Acknowledgements\nI would like to express my gratitute to Matt Daws and Christian Voigt f or discussing topics\nrelated to approximation properties of quantum groups. This work was partially supported by\nFWO grant 1246624N.\nReferences\n[1] E. B´ edos, G. J. Murphy, and L. Tuset. Co-amenability of c ompact quantum groups. J. Geom. Phys. , 40(2):130–\n153, 2001.\n[2] D. P. Blecher and C. Le Merdy. Operator algebras and their modules—an operator space appr oach, volume 30\nofLondon Mathematical Society Monographs. New Series . The Clarendon Press, Oxford University Press,\nOxford, 2004. Oxford Science Publications.COWLING-HAAGERUP CONSTANT OF THE PRODUCT OF DISCRETE QUANT UM GROUPS 13\n[3] M. Brannan. Approximation properties for locally compa ct quantum groups. In Topological quantum groups ,\nvolume 111 of Banach Center Publ. , pages 185–232. Polish Acad. Sci. Inst. Math., Warsaw, 2017 .\n[4] N. P. Brown and N. Ozawa. C˚-Algebras and Finite-Dimensional Approximations . Graduate Studies in Math-\nematics, Volume 88. American Mathematical Society, 2008.\n[5] M. Caspers. Weak amenability of locally compact quantum groups and approximation properties of extended\nquantum SUp1,1q.Comm. Math. Phys. , 331(3):1041–1069, 2014.\n[6] M. Cowling and U. Haagerup. Completely bounded multipli ers of the Fourier algebra of a simple Lie group of\nreal rank one. Invent. Math. , 96(3):507–549, 1989.\n[7] J. Crann. Amenability and covariant injectivity of loca lly compact quantum groups II. Canad. J. Math. ,\n69(5):1064–1086, 2017.\n[8] J. Crann. Inner amenability and approximation properti es of locally compact quantum groups. Indiana Univ.\nMath. J. , 68(6):1721–1766, 2019.\n[9] M. Daws, J. Krajczok, and C. Voigt. Averaging multiplier s on locally compact quantum groups. arXiv e-prints ,\npage arXiv:2312.13626, December 2023.\n[10] M. Daws, J. Krajczok, and C. Voigt. The approximation pr operty for locally compact quantum groups. Adv.\nMath. , 438:Paper No. 109452, 2024.\n[11] K. De Commer, A. Freslon, and M. Yamashita. CCAP for univ ersal discrete quantum groups. Comm. Math.\nPhys. , 331(2):677–701, 2014. With an appendix by Stefaan Vaes.\n[12] E. G. Effros and Z.-J. Ruan. Operator spaces , volume 23 of London Mathematical Society Monographs. New\nSeries . The Clarendon Press, Oxford University Press, New York, 20 00.\n[13] A. Freslon. A note on weak amenability for free products of discrete quantum groups. C. R. Math. Acad. Sci.\nParis , 350(7-8):403–406, 2012.\n[14] A. Freslon. Examples of weakly amenable discrete quant um groups. J. Funct. Anal. , 265(9):2164–2187, 2013.\n[15] A. Freslon. Permanence of approximation properties fo r discrete quantum groups. Ann. Inst. Fourier (Greno-\nble), 65(4):1437–1467, 2015.\n[16] U. Haagerup. Group C˚-algebras without the completely bounded approximation pr operty.J. Lie Theory ,\n26(3):861–887, 2016.\n[17] M. Junge, M. Neufang, and Z.-J. Ruan. A representation t heorem for locally compact quantum groups. Inter-\nnat. J. Math. , 20(3):377–400, 2009.\n[18] J. Krajczok. Modular properties of locally compact quantum groups . PhD thesis, IMPAN, 2022.\n[19] J. Kraus and Z.-J. Ruan. Approximation properties for K ac algebras. Indiana Univ. Math. J. , 48(2):469–535,\n1999.\n[20] J. Kustermans and S. Vaes. Locally compact quantum grou ps in the von Neumann algebraic setting. Math.\nScand. , 92(1):68–92, 2003.\n[21] S. Neshveyev and L. Tuset. Compact quantum groups and their representation categorie s, volume 20 of Cours\nSp´ ecialis´ es [Specialized Courses] . Soci´ et´ e Math´ ematique de France, Paris, 2013.\n[22] P. Podle´ s and S. L. Woronowicz. Quantum deformation of Lorentz group. Comm. Math. Phys. , 130(2):381–431,\n1990.\n[23] P. M. So/suppress ltan and A. Viselter. A note on amenability of lo cally compact quantum groups. Canad. Math. Bull. ,\n57(2):424–430, 2014.\n[24] R. Tomatsu. Amenable discrete quantum groups. J. Math. Soc. Japan , 58(4):949–964, 2006.\n[25] A. Van Daele. The Haar measure on a compact quantum group .Proc. Amer. Math. Soc. , 123(10):3125–3128,\n1995.\n[26] S. Wang. Tensor products and crossed products of compac t quantum groups. Proc. London Math. Soc. (3) ,\n71(3):695–720, 1995.\n[27] S. L. Woronowicz. Compact quantum groups. In Sym´ etries quantiques (Les Houches, 1995) , pages 845–884.\nNorth-Holland, Amsterdam, 1998.\nVrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belg ium\nEmail address :jacek.krajczok@vub.be" }, { "title": "2401.16924v1.Poynting_Robertson_damping_of_laser_beam_driven_lightsails.pdf", "content": "Poynting-Robertson damping of laser beam driven lightsails\nRhys Mackintosh, Jadon Y . Lin, and Michael S. Wheatland\nThe University of Sydney, School of Physics, NSW 2006 Camperdown, Australia\nBoris T. Kuhlmey∗\nThe University of Sydney, School of Physics, Institute of Photonics and Optical Science,\nand The University of Sydney Nano Institute, NSW 2006 Camperdown, Australia\n(Dated: January 31, 2024)\nLightsails using Earth-based lasers for propulsion require passive stabilization to stay within the beam. This\ncan be achieved through the sail’s scattering properties, creating optical restoring forces and torques. Undamped\nrestoring forces produce uncontrolled oscillations, which could jeopardize the mission, but it is not obvious\nhow to achieve damping in the vacuum of space. Using a simple two-dimensional model we show that the\nDoppler effect and relativistic aberration of the propelling laser beam create damping terms in the optical forces\nand torques. The effect is similar to the Poynting-Robertson effect causing loss of orbital momentum of dust\nparticles around stars, but can be enhanced by design of the sail’s geometry.\nI. INTRODUCTION\nLaser powered lightsails [1, 2] are one of the few plausible\npathways for sending probes to other stars on time scales of\na single human generation. With physically realistic but ex-\ntremely challenging infrastructure [3, 4], a lightsail of mass\nm≃1 g could be accelerated to a velocity v=0.2cwithin a\ntime∼1000 s and acceleration distance ≲0.1 AU, reaching\nthe Proxima Centauri system within 20 years. Such a lightsail\nwould be propelled by a powerful laser array based on Earth,\nthe photons of the laser imparting momentum upon reflection\non the sail. Because of the laser beam’s finite width, a mecha-\nnism is required for the lightsail to remain in the center of the\nbeam. Any feedback to adjust the ground-based laser would\nbe too slow as soon as the lightsail is a few light-milliseconds\naway, and active impulse or optical feedback mechanisms on\nboard are difficult to achieve within the mass budget and with-\nout adding optical absorption that could lead to thermal break-\ndown of the sail. The most likely implementation of sta-\nbilization is thus through passive optical stabilization [5–9],\nwhich uses the combination of beam shape and spatial reflec-\ntivity profile to generate a spring-like restoring force towards\nthe center of the beam, as well as a restoring torque to keep\nthe sail at the optimal angle. However, the restoring force\nand torque alone lead to oscillations, which in the absence\nof damping are maintained throughout the acceleration phase,\nwith an amplitude likely to increase with any perturbations\ndue to, for example, time-dependent beam misalignment dur-\ning the acceleration phase.\nAny transverse velocity component remaining at the end of\nthe acceleration phase will lead to the craft going off-course,\nwith dramatic consequences on the ability to take telemetric\nmeasurements of exo-planets. Figure 1 shows the deviation\nfrom the ideal trajectory after 20 years of cruising at 0.2c as\na function of residual transverse velocity at the end of the ac-\nceleration phase. Simulations of passive stabilization using\noptical forces showed final residual transverse velocities of\n∗boris.kuhlmey@sydney.edu.auorder∼1−150 m /s, depending on implementation and level\nof perturbations [5, 7, 9], leading to final deviations of up to\n0.6 astronomical units after 20 years of travel. Equally, any\nresidual angular velocity is likely to complicate both teleme-\ntry and communications with Earth.\n0 50 100 150 200\nresidual transverse velocity (m/s)00.10.20.30.40.50.60.70.8deviation from target (A.U.)Actual trajectoryTarget trajectory Acceleration\nDeviationEarthProxima B\nLaserResidual transverse \nvelocity\nFIG. 1. Deviation from target course in astronomical units after 20\nyears of travel as a function of residual unwanted transverse velocity.\nDamping is required to reduce these oscillations, but is dif-\nficult to achieve in space. Srivastava et al. [6] included an\narbitrary damping force term without justifying what physical\nmechanism may cause it. Salary et al. [7] saw a reduction\nin spatial amplitude of the oscillations in their simulations,\nattributed to the shift in frequency from the Doppler effect\nchanging the reflectivity and thus restoring force. This is akin\nto changing the stiffness of a spring in a mass-spring system,\nand thus to lowest order like changing the slope of the asso-\nciated parabolic potential well: the spatial extent of the os-\ncillation is reduced, but the total energy and thus maximum\nkinetic energy during the oscillations remain unaffected. It is\nthus unclear how much the Doppler effect in that situation is\nreducing the velocity amplitude of the oscillations. Rafat etarXiv:2401.16924v1 [physics.app-ph] 30 Jan 20242\nal.[9] proposed the use of a damped internal degree of free-\ndom, which can indeed effectively reduce the oscillations of\nthe lightsail, both in spatial amplitude and in velocity. How-\never the implementation of a damped internal degree of free-\ndom will be challenging within the mass budget of a lightsail\nonly hundreds of nm thick.\nThe Doppler effect has been used as a damping force to\nslow atoms [10], but only in the direction of propagation of\nthe laser. It is difficult to see how this could be implemented\nto dampen transverse oscillation of a lightsail, as it appears\nit would require laser beams propagating orthogonal to the\ndirection of acceleration.\n(a)\n(b)\n(c)\nFIG. 2. Principle of PR damping of laser driven lightsails, in the\nsail’s frame: (a) For a non-rotating sail moving parallel to the beam\ndirection, forces are balanced on both mirrors. The net force (central\ntaupe arrow) is purely in the direction of the laser beam. (b) Damping\nforce: for a sail with unwanted transverse velocity component vy<0,\nrelativistic aberration angles the laser’s light (green arrows), leading\nto non-zero transverse force opposing the transverse velocity. (c)\nDamping torque: For non-zero rotational velocity Ω′>0, the left\nmirror has slightly larger velocity away from the laser source than\nthe right mirror, with the additional red-shift reducing the photon’s\nmomentum.\nDamping transverse to the direction of a light wave is also\nknown: The Poynting-Robertson (PR) effect [11–14] causes\ndust particles orbiting a star to lose orbital angular momen-\ntum, due to relativistic aberration. In the dust’s reference\nframe, light from the star comes from a direction shifted to-wards the direction the dust is moving in [15]. Light is ab-\nsorbed, leading to a radiation force with a component orthog-\nonal to the radial direction to the star, slowing the dust down.\nDust particles eventually fall into the star, over a time roughly\nproportional to particle sizes and of order tens of thousands of\nyears [13]. The PR effect has been shown to mildly impact\nsolar-sail dynamics, specifically due to residual absorption by\nthe sail [16–18]. Can a similar effect be used for effective\ndamping of unwanted motion in laser-driven lightsails? The\nsituation is quite different from the usual PR effect, even as\nstudied for solar sails: contrary to solar sails or dust, laser-\npowered lightsails have very small transverse velocities, and\nabsorption, which is the basis of most PR studies, must be\navoided at all costs.\nHere, we show that a lightsail’s angular reflectivity prop-\nerties, once combined with the Doppler effect and relativistic\naberration, indeed provide damping similarly to the PR ef-\nfect. We derive explicit expressions for the damping forces\nand torques for a simple two-dimensional two-mirror geome-\ntry, and show that with appropriate optical design, transverse\nvelocities can be damped to almost arbitrary levels by the end\nof the acceleration phase, albeit at the cost of increasing the\nacceleration distance.\nII. PRINCIPLE\nThe origins of the damping forces and torques are illus-\ntrated in Fig. 2, using arguably the simplest two-dimensional\nreflecting object having linear mechanical stability both in\ntranslation and rotation in a laser beam: a symmetric set of\ntwo angled mirrors connected by a rod [9]. In the figure, the\ndesired direction (direction of the beam, x-axis) is upward.\nWhen moving strictly parallel to x, (Fig. 2(a)), radiation pres-\nsure on both mirrors is equal and the sail is accelerated for-\nward only. If the sail has an undesired movement orthogonal\nto the laser beam’s direction (Fig. 2(b)), relativistic aberra-\ntion tilts the light in the sail’s reference frame, so that the left\nmirror intercepts more light. This asymmetry leads to a com-\nponent of the radiation force perpendicular to the laser beam\n(in the laser’s frame L), that is proportional and opposing\nthe transverse velocity – in effect a drag force. If the sail is\nrotating relative to the axis of the laser beam (Fig. 2(c)), the\ndifferential Doppler shift leads to different forces on the two\nmirrors, giving rise to a torque that opposes the rotational mo-\ntion – a damping torque.\nIII. NOTATIONS\nA four-vector ⃗xhas, in a specified reference frame, com-\nponents (x0,x1,x2,x3)Twhere x0is the temporal component\nand the other three are spatial components. A Minkovsky\nmetric diag (1,−1,−1,−1)is implied throughout. We fol-\nlow the convention of using Roman indices for spatial compo-\nnents ( e.g. xj) and Greek indices for all four spatio-temporal\ncomponents ( xµ). Spatial three-vectors in any specific frame\nwill be noted as bold Roman letters, and unit vectors will3\nbe marked with a ˆ ( e.g. kis the three-vector with compo-\nnents ki, and ˆk=k/|k|). Basis four-vectors of a reference\nframe will be denoted ˆ eµ, and we will use the same nota-\ntionˆejfor spatial unit three-vectors. We will use v,vx,vy,vz,v\nfor the velocity vector, its components and norm respectively,\nin the rest frame of the laser. We will use the usual nota-\ntionβ=v/c, also applicable to individual components e.g.\nβx=vx/c, and γ=1/p\n1−β2. The laser accelerating the\nlightsail is in frame L, assumed to be inertial, and the instan-\ntaneously co-moving inertial frame of the lightsail is called\nM. Frame M has velocity vin frame L. Primed quan-\ntities refer to quantities in M, un-primed quantities refer to\nframeL. The unit four-vectors in Mare related to those in\nLthrough the inverse Lorentz transform ˆ e′\nµ=Λ(−v)ν\nµˆeν,\nwith:\nΛ(v) =\nγ −γvx\nc−γvy\nc−γvz\nc\n−γvx\nc1+(γ−1)v2x\nv2(γ−1)vxvy\nv2(γ−1)vxvz\nv2\n−γvy\nc(γ−1)vxvy\nv2 1+(γ−1)v2y\nv2(γ−1)vyvz\nv2\n−γvz\nc(γ−1)vxvz\nv2(γ−1)vyvz\nv2 1+(γ−1)v2z\nv2\n.\n(1)\nInLwe choose ˆe1to point towards the desired star system.\nNote that because vis not aligned with either ˆe1orˆe2, the\nspatial parts ˆe1andˆe′\n1of ˆe1and ˆe′\n1are not parallel [19], and\nneither are ˆe2andˆe′\n2.\nIV . GEOMETRY\nFigure 3 shows our lightsail model in the Land\nMframes. The mirrors are at the tips T1andT2of a massless\nrod of rest length 2 L0, each with surface area A0(dimensions\nof length for our two-dimensional treatment). In frame M,\nthe mirrors each make a fixed angle α′\n1,2=±α0with ˆe′\n1. In\nRef [9] this geometry was shown to have a restoring force and\ntorque in a parabolic beam, with an equilibrium position in\nthe middle of the beam, and with the rod perpendicular to the\nbeam direction.\nFor simplicity, we consider the laser beam to be a single\nplane wave with wave vector kaligned with ˆe1(Fig. 3), with\nintensity I(in power per unit length, for our two-dimensional\ntreatment). With that simplification there is no restoring force,\nbut as we shall see there is still a damping force, and restor-\ning and damping torques. In L, the wave four-vector ⃗khas\ncoordinates k0(1,1,0,0)T, with k0=ω/c.\nWe define the normal vector to the mirrors, chosen to point\naway from the source of light:\nˆn′\n1,2(α′\n1,2) =sign(sin(α′\n1,2−θ′))\u0000\nsinα′\n1,2ˆe′\n1−cosα′\n1,2ˆe′\n2\u0001\n(2)\nwhere the sign (sin(α′\n1,2−θ′))ensures the correct orientation\naway from the light source.\nV . LINEAR DAMPING FORCE\nWe aim to quantify the drag force in the ˆe2direction in\nLthat is due to the Doppler shift and relativistic aberration.To do so, we calculate the momentum exchange rate with the\nsail inM, which gives us an expression of the four-force in\nM. We then Lorentz-boost this back into L, and identify\nthe component of the force in ˆe2. As we are seeking a drag\nforce close to the ideal trajectory, we will assume vy≪vx0 on the right correspond to vy<0, that is a sail moving towards the\nleft.\nThe general expression for the coordinates of the four-force ⃗f\non an object at velocity vin a given reference frame in terms of\nthe three-force in that frame is (γF·v/c;γF)[21]. InM, the\nvelocity of the sail is zero, with γ′=1, so that the four-force\n⃗fhas coordinates (0;F′), from which we obtain the four-force\ninLthrough Lorentz transform. Using Eq. (9) the transverse\ncomponent of force (along ˆe2) is then to lowest order in vy/v\nf2≡dp2\ndt′\n≃4A0D2I\ncvy\nv\u0000\n−cosα0sin2α0(1/D−1) + (γ−1)sin3α0\u0001\n(13)\nwhere t′is the time in M, and thus proper time. We find a net\ntransverse force that is proportional to vyand thus, depending\non overall sign, a possible drag force.\nExpressing 1 /D−1 and γ−1 for small βgives some in-\nsight into the relative importance of the two terms in brackets\nin Eq. (13):\nf2≃4A0D2I\ncvy\nc\u0012\n−cosα0sin2α0\u0012\n1+β\n2\u0013\n+β\n2sin3α0\u0013\n.\n(14)\nAt small β<0.2 the second and third terms only contribute a\ncorrection of order 5% to the first term. Importantly, the domi-\nnant term is negative (and independent of vx), so that there is a\nnet drag force that will lead to damping of any transverse mo-\ntion, from the outset of the acceleration phase, with damping\ncoefficient\nζ≡ − f2/vy≃4A0D2I\nc2cosα0sin2α0. (15)\nIn Eq. (13), the linear vy/cdependence comes from the rel-\nativistic aberration. The D2I/cfactor represents the optical\npower intensity in M, and in Eq. (15) the remaining propor-\ntionality constant 4 A0cosα0sin2α0is a geometry-dependent\nform of cross-section that is not simply the radiation pressure\ncross-section [14, 22] but also depends on the radiation pres-\nsure’s dependence on incident angle. This latter term can be\nadjusted by geometry, in our simple case by adjusting α0.B. Order of magnitude\nTo assess the importance of the effect we compare the mag-\nnitude of the transverse damping force to the accelerating\nforce along ˆe1. Assuming β<0.2, we have 1 ≤γ≲1.02\nand 0 .81≃D≤1. Reasonable approximations can thus be\nobtained assuming Dconstant and ignoring all other relativis-\ntic corrections or βdependencies, which in particular leads\nto a constant longitudinal acceleration force. The acceler-\nation time to reach a final βfcan then be approximated by\ntf≃mcβf/f1. During that time the transverse damping force\nleads to an exponential decrease of any initial velocity fol-\nlowing vy=vy,0exp(−ζt/m). By the end of the acceleration\nphase, any initial transverse velocity vy,0is thus attenuated by\na factor η=vy,final/vy,0with\nlnη=−ζ\nmtf=−ζc\nf1βf. (16)\nFrom Eq. (12) and (13) to linear order in vy/vand zeroth order\ninvx/cwe obtain\n−lnη\nβf≃cosα0sin2α0\nsin3α0=2cot2α0, (17)\nwhich we plot in Fig. 4 as a function of α0. For α0=π/4\nandβf=0.2 any initial velocity is thus reduced to η=\nexp(−0.4)≃67% of its initial value. Within this approxima-\ntion, valid for relatively small β, the reduction also improves\nexponentially with βf. More importantly, this reduction is\ngeometry-dependent, with a smaller value of α0=0.3 leading\nto a final transverse velocity of only 1 .5% of vy,0. However,\nthis improvement comes at the cost of reduced longitudinal\nacceleration, that is, increased acceleration time and distance.\nUltimately, the damping force is a function of the radi-\nation pressure’s angular dependence (through the aberration\nangle). Better optical design may be able to increase the level\nof damping without compromising acceleration time.\nVI. RESTORING AND DAMPING TORQUES\nWe now allow for rotations of the lightsail, with the axis of\nthe sail now making an angle Φ′(Fig. 5) with the ˆe′\n1axis, with5\n00.20.40.60.811.210-1100101102103\nFIG. 4. Ratio of transverse damping coefficient to longitudinal ac-\ncelerating force as a function of mirror angle α0.\nMirror frame (M)\nT1T2\nFIG. 5. Rotations of the lightsail – geometry in the mirror frame.\nan angular velocity Ω′=dΦ′/dt′. The vector vnow denotes\nthe velocity of the center of mass C rather than the velocity of\nall parts of the sail. For Ω′=0, using Eq. (11) to calculate the\ntorque about C in M, we get, to linear order in θ′andΦ′:\nτ′\nr=−\u00124L0A0ID2\ncsin2α0\u0013\u0000\nΦ′−θ′\u0001ˆe′\n3, (18)\nwhich is a restoring torque towards the direction of the in-\ncoming light in M. To obtain a damping term for Ω′̸=0\nwe need to take into account the rotational velocities of the\nmirrors, which lead to different Doppler shifts and relativistic\naberrations for each point of each mirror. We take the further\nsimplifying assumption that the mirrors are small compared to\nL0, and can thus be treated as point mirrors, each with unique\nvelocities v′\n1,2inM. Equation (11) remains valid, but we now\nhave two different values ˆk′\n1,2andD1,2. Since vxcan reach rel-\nativistic values, we need to use relativistic velocity additionsv1,2=v⊕v′\n1,2. Assuming v′\n1,2≪c, we obtain to linear order\nv1,2≃v+v′\n1,2x\nγ2ˆe1+v′\n1,2y\nγˆe2, (19)\nfrom which, again to linear order in v′\n1,2/candvy/v, we obtain\nD1,2≃D \n1−v′\n1,2x\nc!\n(20)\nand\nθ′\n1,2≃ −\u00121\nD−1\u0013 \nvy\nv+v′\n1,2y\nγv!\n. (21)\nThe damping torque at “equilibrium” is calculated for vy=0,\nΦ′=0. InM, the velocities of the mirrors then simplify to\nv′\n1,2=Ω′ˆe′\n3×CT 1,2=∓Ω′L0ˆe′\n1, so that θ′\n1,2=θ′andD1,2=\nD(1±Ω′L/c). The total torque becomes, again to linear order,\nτ′\nd=−\u00128L2\n0A0ID2\nc2sin3α0\u0013\nΩ′ˆe′\n3, (22)\nand is indeed a damping torque as it is countering and propor-\ntional to Ω′.\nEquations (18) and (22) lead to a damped oscillating rota-\ntional motion, with damping time constant t′\nrd. Using the total\nmoment of inertia mL2\n0we have\nt′\nrd=mc2\n2A0ID2sin3α0(23)\nwhich, by a reasoning analogous to that in Section V B leads\nto a rotational damping ratio ηrwith\nlnηr=−βf/2. (24)\nEq. (24) does not depend on α0. However the damping ratio\ndepends on the ratio of longitudinal forces to moment of in-\nertia, so that it does in general, depend on geometry. For the\ntwo-mirror system under consideration with a final velocity of\nβf=0.2, any initial rotational velocity would be reduced by\nonly 10%, but this could be enhanced considerably by hav-\ning a mass distribution closer to the center of gravity ( e.g. a\npayload at the center), or via different optical designs.\nVII. CONCLUSION\nThe relativistic transformation of photon momentum can\ncontribute to significant damping of unwanted transverse and\nrotational movements of lightsails. Both the space-like and\ntime-like transformations are of importance: For our ge-\nometry around equilibrium, transverse translational damping\ncomes primarily from the relativistic aberration (space-like)\nwhile rotational damping comes primarily from the difference\nin Doppler terms for both mirrors (time-like). However, for\ndifferent values of Φ′(or different geometries), both the rel-\nativistic aberration and Doppler effects can be at play for the6\ntranslational and rotational damping, with similar magnitude.\nOrder of magnitude calculations show that over the course of\nthe acceleration phase, unwanted transverse velocity compo-\nnents can be damped to almost arbitrary levels by appropriate\noptical design (Fig. 4). In our simple geometry, this comes\nat the cost of slower acceleration, but as transverse damping\nultimately comes from the angular response of the reflection,\nit is conceivable that very large damping coefficients could be\nachieved without loss of longitudinal acceleration in realistic\ngeometries, for example using gratings or meta-surfaces with\nsharp angular diffraction patterns [5, 7, 8].\nOur study also highlights that taking the full relativistic\ntransformations of light into account is essential to include\ndamping mechanisms in lightsail dynamics. Studies of light-\nsails we have encountered so far have included the time-like\n(Doppler) transformation, but neglected the space-like (rela-\ntivistic aberration) transformation.\nFinally, among our many simplifying assumptions, we con-\nsidered the laser to be a single monochromatic plane-wave. In\npractice, the divergence of laser beams will be comparable tothe magnitude of the relativistic aberration. When extending\nthis study to more realistic lightsail geometries, it will thus be\nimportant to also take into account the angular spread of finite\nbeams, noting the relativistic transformation of finite beams is\na rich field of study in itself, with many surprising phenom-\nena [20, 23].\nThat the Doppler effect and relativistic aberration can in\nprinciple provide damping should be welcomed by the inter-\nstellar lightsail community, as other mechanisms of damping\nresidual motion seem difficult to implement within the ex-\ntremely narrow mass budget constraint. Special relativity gifts\nus a solution that could simplify lightsail designs and consid-\nerably improve accuracy of trajectories.\nACKNOWLEDGMENTS\nWe thank Mr Ronan Potts for help with early exploratory\nwork, Profs Peter Tuthill and Martijn de Sterke for useful dis-\ncussions.\n[1] G. Marx, Interstellar vehicle propelled by terrestrial laser beam,\nNature 211, 22 (1966).\n[2] R. L. Forward, Roundtrip interstellar travel using laser-pushed\nlightsails, Journal of Spacecraft and Rockets 21, 187 (1984).\n[3] K. L. Parkin, The breakthrough starshot system model, Acta\nAstronautica 152, 370 (2018).\n[4] P. Lubin, A roadmap to interstellar flight (2022),\narXiv:1604.01356 [astro-ph.EP].\n[5] O. Ilic and H. A. Atwater, Self-stabilizing photonic levitation\nand propulsion of nanostructured macroscopic objects, Nature\nPhotonics 2019 13:4 13, 289 (2019).\n[6] P. R. Srivastava, Y .-J. L. Chu, and G. A. Swartzlander, Stable\ndiffractive beam rider, Optics Letters 44, 3082 (2019).\n[7] M. M. Salary and H. Mosallaei, Photonic metasurfaces as\nrelativistic light sails for doppler-broadened stable beam-\nriding and radiative cooling, Laser and Photonics Reviews 14,\n10.1002/lpor.201900311 (2020).\n[8] A. Kumar, D. Kindem, and O. Ilic, Optomechanical self-\nstability of freestanding photonic metasurfaces, Physical\nRewview Applied 16, 14053 (2021).\n[9] M. Z. Rafat, H. R. Dullin, B. T. Kuhlmey, A. Tuniz, H. Luo,\nD. Roy, S. Skinner, T. J. Alexander, M. S. Wheatland, and\nC. M. D. Sterke, Self-stabilization of light sails by damped in-\nternal degrees of freedom, Physical Review Applied 17, 024016\n(2022).\n[10] T. W. Hänsch and A. L. Schawlow, Cooling of gases by laser\nradiation, Optics Communications 13, 68 (1975).\n[11] J. H. Poynting, Xii. radiation in the solar system: its effect\non temperature and its pressure on small bodies, Philosophical\nTransactions of the Royal Society of London. Series A, Con-\ntaining Papers of a Mathematical or Physical Character 202,\n525 (1904).[12] H. P. Robertson and H. N. Russell, Dynamical effects of radi-\nation in the solar system, Monthly Notices of the Royal Astro-\nnomical Society 97, 423 (1937).\n[13] J. Kla ˇcka and M. Kocifaj, Times of inspiralling for interplan-\netary dust grains, Monthly Notices of the Royal Astronomical\nSociety 390, 1491 (2008).\n[14] J. Kla ˇcka, J. Petržala, P. Pástor, and L. Kómar, The poynting-\nrobertson effect: A critical perspective, Icarus 232, 249 (2014).\n[15] A. Einstein, Zur elektrodynamik bewegter körper, Annalen der\nPhysik 322, 891 (1905).\n[16] F. Abd El-Salam, The effects of poynting–robertson drag on\nsolar sails, Results in Physics 9, 897 (2018).\n[17] R. Y . Kezerashvili and J. F. Vázquez-Poritz, Effect of a drag\nforce due to absorption of solar radiation on solar sail orbital\ndynamics, Acta Astronautica 84, 206 (2013).\n[18] R. Y . Kezerashvili and J. F. Vázquez-Poritz, Drag force on solar\nsails due to absorption of solar radiation, Advances in Space\nResearch 48, 1778 (2011).\n[19] C. P. Frahm, Representing arbitrary boosts for undergraduates,\nAmerican Journal of Physics 47, 870 (1998).\n[20] J. M. McKinley, Relativistic transformations of light\npower, American Journal of Physics 47, 602 (1979),\nhttps://doi.org/10.1119/1.11762.\n[21] V . Faraoni, Special Relativity , Undergraduate Lecture Notes in\nPhysics (Springer International Publishing, Cham, 2013).\n[22] H. van de Hulst, Light Scattering by Small Particles , Dover\nBooks on Physics (Dover Publications, 1981).\n[23] M. Yessenov and A. F. Abouraddy, Relativistic transformations\nof quasi-monochromatic paraxial optical beams, Physical Re-\nview A 107, 10.1103/physreva.107.042221 (2023)." }, { "title": "2401.16929v1.Rigidity_of_compact_quasi_Einstein_manifolds_with_boundary.pdf", "content": "arXiv:2401.16929v1 [math.DG] 30 Jan 2024RIGIDITY OF COMPACT QUASI-EINSTEIN\nMANIFOLDS WITH BOUNDARY\nJOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nAbstract. In this article, we investigate the geometry of compact quas i-Einstein mani-\nfolds with boundary. We establish the possible values for th e constant scalar curvature\nof a compact quasi-Einstein manifold with boundary. Moreov er, we show that a 3-\ndimensional simply connected compact m-quasi-Einstein manifold with boundary and\nconstant scalar curvature must be isometric, up to scaling, to either the standard hemi-\nsphereS3\n+, or the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×S2with the product metric. For dimension n= 4,\nwe prove that a 4-dimensional simply connected compact m-quasi-Einstein manifold M4\nwith boundary and constant scalar curvature is isometric, u p to scaling, to either the\nstandard hemisphere S4\n+,or the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×S3with the product metric, or the\nproduct space S2\n+×S2with the doubly warped product metric. Other related result s for\narbitrary dimensions are also discussed.\n1.Introduction\nA compact n-dimensional Riemannian manifold ( Mn, g), n≥2,possibly with boundary\n∂M,is called an m-quasi-Einstein manifold , or simply quasi-Einstein manifold , if there\nexists a smooth potential function uonMnsatisfying the system\n(1.1)\n\n∇2u=u\nm(Ric−λg) inM,\nu >0 on int(M),\nu= 0 on ∂M,\nfor some constants λand 0< m <∞(cf. [15, 33, 34]). Here, ∇2ustands for the Hessian\nofuandRicis the Ricci tensor of g.Whenm= 1, we assume in addition that ∆ u=−λu\nin order to recover the static equation :−(∆u)g+∇2u−uRic= 0.Moreover, an m-quasi-\nEinstein manifold will be called trivialifuis constant, otherwise it will be nontrivial . We\nnotice that the triviality implies that Mnis an Einstein manifold.\nThe study of quasi-Einstein manifolds is directly related to the existe nce of warped pro-\nduct Einstein metrics on a given manifold. To be precise, as discussed by Besse [8, pg.\n267], an m-quasi-Einstein manifold corresponds to a base of a warped produc t Einstein\nmetric; for more details, see, e.g., [8, Corollary 9.107, pg. 267] and [6, 8, 15, 18, 19, 40, 54].\nChoosing u=e−f\nmin (1.1) when ∂M=∅,an∞-quasi-Einstein manifold is precisely\na gradient Ricci soliton ( Mn, g, f),see [15, 19, 22, 54]. Despite similarity, there exist\nexamplesofquasi-Einsteinmanifoldsthatareinstarkcontrasttot heRiccisolitons. Another\nDate: January 31, 2024.\n2020Mathematics Subject Classification. Primary 53C20, 53C25; Secondary 53C65.\nKey words and phrases. quasi-Einstein manifolds; constant scalar curvature; com pact manifolds with\nboundary; rigidity results.\nJ. Costa was partially supported by CAPES/Brazil - Finance C ode 001.\nE. Ribeiro was partially supported by CNPq/Brazil [309663/ 2021-0 & 403344/2021-2], CAPES/Brazil\nand FUNCAP/Brazil [PS1-0186-00258.01.00/21 & ITR-0214-0 0116.01.00/23].\nD. Zhou was partially supported by FAPERJ/Brazil [E-26/200 .386/2023 & E-26/202.591/2019]) and\nCNPq/Brazil [305364/2019-7, 403344/2021-2 & 308067/2023 -1].\n12 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\ninteresting motivation to investigate quasi-Einstein manifolds derive s from the study of\ndiffusion operators by Bakry and ´Emery [4], which is linked to the theory of smooth metric\nmeasure spaces; see, e.g., [7, 16, 17, 40, 54, 56, 58, 59] and the r eferences therein. In\nparticular,1-quasi-Einsteinmanifoldsaremorecommonlycalled static spaces . Besidesbeing\ninteresting on their own, static spaces have connections to the po sitive mass theorem and\ngeneral relativity (see [15, Remark 2.3] and [1, 11, 12, 37, 38, 50, 51]). Additionally, quasi-\nEinstein metrics have attracted interest in physics due to their rela tion with the geometry\nof a degenerate Killing horizon and horizon limit; see, e.g., [2, 3, 60].\nExplicit examples of nontrivial compact and noncompact m-quasi-Einstein manifolds can\nbe found in, e.g., [8, 9, 10, 15, 16, 17, 33, 39, 53, 54, 57]. In “Besse ’s book” [8, pg. 267-272],\nit was established the classification of 1 and 2-dimensional m-quasi-Einstein manifolds. In\nthis article, we focus on nontrivial compact m-quasi-Einstein manifolds with (non-empty)\nboundary ∂M.Hence, by the work [33, Theorem 4.1], they have necessarily λ >0.In\nthis perspective, it is fundamental to recall some examples of comp actm-quasi-Einstein\nmanifolds with boundary and constant scalar curvature:\n(i) The hemisphere Sn\n+with the standard metric g=dr2+ sin2rgSn−1and potential\nfunction u(r) = cosr,whereris a height function with r≤π\n2;\n(ii)/bracketleftBig\n0,/radicalbig\nm/λπ/bracketrightBig\n×Sn−1,forλ >0,endowed with the metric g=dt2+n−2\nλgSn−1and\npotential function u(t,x) = sin/parenleftBig/radicalbig\nλ/mt/parenrightBig\n;\n(iii)Sp+1\n+×Sq,q >1, with the doubly warped product metric\ng=dr2+sin2rgSp+q−1\np+mgSq,\nwherer(x,y) =h(x) andhis a height function on Sp+1\n+,potential function u= cosr\nwithr≤π\n2andλ=p+m.\nHe, Petersen and Wylie [34, Proposition 2.4] showed that a nontrivial compact quasi-\nEinstein manifold with boundary and constant Ricci curvature is isom etric to Example (i) .\nIt turns out that these three quoted examples have constant sc alar curvature and therefore,\nonequestionthatnaturallyarisesistoknow whether a nontrivial compact (simply connected)\nm-quasi-Einstein manifold with boundary and constant scala r curvature must be necessarily\none of them .\nRemark 1. For dimension n≥5,it is possible to obtain another example with constant\nscalar curvature, see Example 1 in Section 2. We also highlig ht that Examples (ii)and(iii)\nabove can be presented in a more general setting by replacing the round spheres Sn−1and\nSq,respectively, by an arbitrary compact Einstein manifold wi th positive scalar curvature.\nMoreover, Example (iii)was obtained recently by Di´ ogenes, Gadelha and Ribeiro in [25]. In\nparticular, by a slightly modification, it provides a new (si mply connected) counterexample\nto the Cosmic no-hair conjecture for arbitrary dimension n≥4;for more details, see [23].\nA straightforward computation, by using the classical Reilly’s theor em [52, Theorem B],\nguarantees that the hemisphere S2\n+is the only nontrivial 2-dimensional simply connected\ncompact m-quasi-Einstein manifold with boundary and constant scalar curvat ure. In [34],\nHe, Petersen and Wylie investigated m-quasi-Einstein manifolds with constant scalar curva-\nture. In particular, for the specific dimension n= 3,they proved that an m-quasi-Einstein\nmanifold with constant scalar curvature is rigid, i.e., it is Einstein or its u niversal cover is\na product of Einstein manifolds (cf. [34, Theorem 1.3]). Other relate d results for compact\nm-quasi-Einstein manifold with boundary and constant scalar curvat ure were discussed in\n[24, 25, 33]. Nevertheless, the explicit classification of compact m-quasi-Einstein manifolds\nwith (non-empty) boundary and constant scalar curvature was n ot established yet. In an-\nother direction, Petersen and Wylie [48] studied rigid gradient Ricci s olitons. It is known,QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 3\nby the works of Hamilton [32], Ivey [35], Perelman [47], Naber [44], Ni-Walla ch [45], and\nCao-Chen-Zhu [13], that 2 and 3-dimensional gradient shrinking Ricc i solitons are rigid,\nand moreover, they are entirely classified. However, for dimension 4 (or higher), this is no\nlonger true according to the example of Feldman, Ilmanen and Knopf [28]. A more recent\nresult due to Cheng and Zhou [21], combined with Fern´ andez-Lop´ ez and Garc´ ıa-R´ ıo [26],\nestablishesthe complete classificationof4-dimensionalgradientsh rinkingRiccisolitonswith\nconstant scalar curvature, which in turn provides a partial solutio n for a problem raised by\nHuai-Dong Cao. This present work is also motivated by these results on Ricci solitons.\nIn this article, inspired by the question mentioned earlier and by work s due to Cheng\nand Zhou [21], Fern´ andez-Lop´ ez and Garc´ ıa-R´ ıo [26] and He, Pe tersen and Wylie [34],\nwe will classify (explicitly) compact 3 and 4-dimensional m-quasi-Einstein manifolds with\nboundary and constant scalar curvature. To that end, in the sam e spirit of [26], we first\nestablish the possible values for the constant scalar curvature of ann-dimensional compact\nm-quasi-Einstein manifold with boundary. More precisely, we have the following result.\nTheorem 1. Let/parenleftbig\nMn, g, u, λ/parenrightbig\nbe a nontrivial compact m-quasi-Einstein manifold with\nboundary, m >1and constant scalar curvature R.Then we have:\nR∈/braceleftbiggn(n−1)\nm+n−1λ,m+n(n−2)\nm+n−2λ,...,(n−1)λ/bracerightbigg\n. (1.2)\nIn general, one has R=k(m−n)+n(n−1)\nm+n−k−1λ,for some k∈ {0,1,...,n−1}.\nWe point out that the value of the scalar curvature in (1.2) may be re garded in terms\nof the dimension kof the set of critical points (or equivalently, the maximum points); s ee\nthe proof of Theorem 1. In Example (i) ,we see that R=n(n−1)λ\nm+n−1and the only critical\npoint is the north pole, i.e., k= 0.In Example (ii) ,we have R= (n−1)λand the set of\ncritical points for the potential function sin/parenleftBig√\nλ√mt/parenrightBig\nis precisely/braceleftBig√m√\nλπ\n2/bracerightBig\n×Sn−1,which has\ndimension n−1.While in Example (iii) ,it holds that R=q(m−n)+n(n−1)\nm+n−q−1λand the set of\ncritical points for the potential function is {north pole }×Sq.\nRemark 2. It follows from the proof of Theorem 1 that, for a (not necessa rily compact with\nboundary) quasi-Einstein manifold with constant scalar cu rvatureRandm >1,one has\nR∈/braceleftbiggn(n−1)\nm+n−1λ,m+n(n−2)\nm+n−2λ,...,(n−1)λ,nλ/bracerightbigg\n.\nBefore discussing our next result, we recall that if an m-quasi-Einstein manifold has\nconstant scalar curvature Randm >1,then\n|˚Ric|2=−m+n−1\nn(m−1)(R−nλ)/parenleftbigg\nR−n(n−1)\nm+n−1λ/parenrightbigg\n; (1.3)\nfor more details, see [33, Proposition 3.3] and [15, Lemma 3.2] (see also Lemma 1).\nRemark 3. Observe that in considering R=n(n−1)\nm+n−1λinto (1.3), i.e., the lower value\nof (1.2), one deduces that Mnis necessarily Einstein and therefore, it suffices to apply\nProposition 2.4 of [34]to conclude that Mnis isometric to the standard hemisphere Sn\n+.\nFurthermore, we will see in Proposition 6 that there is nocom pact (nontrivial) quasi-Einstein\nmanifold with boundary and constant scalar curvature R=m+n(n−2)\nm+n−2λ.\nIn the sequel, we shall consider the extremal value case of (1.2), n amely,R= (n−1)λ.In\nthis situation, we have the following result which can be compared with [34, Theorem 1.9].4 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nTheorem 2. Let/parenleftbig\nMn, g, u, λ/parenrightbig\n, n≥3,be a nontrivial simply connected compact m-quasi-\nEinstein manifold with boundary and m >1.ThenMnhas constant scalar curvature R=\n(n−1)λif and only if it is isometric, up to scaling, to the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×Nwith\nproduct metric, where Nis a compact λ-Einstein manifold.\nRemark 4. We notice that the classification obtained in Theorem 2 also i ncludes the ex-\nample of m-quasi-Einstein manifold stated in Example 1 of Section 2.\nAs a consequence of Theorem 2 and Proposition 2.4 in [34], we shall obt ain an explicit\nclassificationforcompact3-dimensional m-quasi-Einsteinmanifoldswith boundaryandcon-\nstant scalar curvature. To be precise, we have the following result .\nTheorem 3. Let(M3, g, u, λ)be a nontrivial simply connected compact 3-dimensional m-\nquasi-Einstein manifold with boundary and m >1.ThenM3has constant scalar curvature\nif and only if it is isometric, up to scaling, to either\n(a)the standard hemisphere S3\n+, or\n(b)the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×S2with the product metric.\nFrom now on, we focus on dimension n= 4.In this scenario, it is known from Theorem\n1 that the possible values for the constant scalar curvature Rare/braceleftbigg12\nm+3λ,m+8\nm+2λ,2(m+2)\n(m+1)λ,3λ/bracerightbigg\n.\nIfR=12\nm+3λ,it then follows from Remark 3 that M4is isometric, up to scaling, to the\nstandard hemisphere S4\n+.Besides, by Proposition 6 in Section 5, there is no compact 4-\ndimensional quasi-Einstein manifold with boundary and constant sca lar curvature R=\nm+8\nm+2λ.In the case R= 3λ,it suffices to invoke Theorem 2 to conclude that M4is isometric\nto the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×S3with product metric. Interestingly, Example (iii) has constant\nscalar curvature R= 2(m+2)\n(m+1)λ.This fact has left open the question of whether S2\n+×S2is the\nunique 4-dimensional compact quasi-Einstein manifold with boundary and constant scalar\ncurvature R= 2(m+2)\n(m+1)λ.To answer this question, we have established the following rigidity\nresult.\nTheorem 4. Let(M4, g, u, λ)be a nontrivial simply connected compact 4-dimensional m-\nquasi-Einstein manifold with boundary and m >1.ThenM4has constant scalar curvature\nR= 2(m+2)\n(m+1)λif and only if it is isometric, up to scaling, to the product sp aceS2\n+×S2with\nthe doubly warped product metric.\nThe proof of Theorem 4 is essentially inspired by the work of Cheng an d Zhou [21]. As\na consequence of Theorem 1, Remark 3, Theorem 2 and Theorem 4, we get the following\nclassification result.\nCorollary 1. Let(M4, g, u, λ)be a nontrivial simply connected compact 4-dimensional m-\nquasi-Einstein manifold with boundary and m >1.ThenM4has constant scalar curvature\nif and only if it is isometric, up to scaling, to either\n(i)the standard hemisphere S4\n+,or\n(ii)the cylinder [0,√m√\nλπ]×S3with the product metric, or\n(iii)the product space S2\n+×S2with the doubly warped product metric.\nThe rest of this paper is organized as follows. In Section 2, we review some basic facts\nand useful results on m-quasi-Einstein manifolds that will be used in the proofs of the main\ntheorems. Some novel lemmas will be discussed in Section 3. Section 4 collects the proofsQUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 5\nof Theorems 1, 2 and 3. Finally, the proofs of Theorem 4 and Corollar y 1 are presented\nin Section 5. We also include an appendix to explain a few remarks used t hroughout this\narticle.\n2.Preliminaries\nIn this section, we review some basic facts and present some featu res that will play a\nfundamental role in the proof of the main results.\n2.1.Background. Throughout this paper, we adopt the following convention for the c ur-\nvatures:\nRm(X,Y) =∇2\nY,X−∇2\nX,Y, Rm(X,Y,Z,W ) =g(Rm(X,Y)Z,W),\nK(ei,ej) =Rm(ei,ej,ei,ej), Ric(X,Y) = trRm(X,·,Y,·),\nRij= Ric(ei,ej), R= trRic.\nGiven a warped product manifold ( I×ϕN, g=dt2+ϕ2(t)gN),whereϕis a positive\nsmooth (warping) function defined on the interval I,and considering a smooth function\nf(t,x) =f(t),one easily verifies that\n2∇2f=L∇fg=L∇f(dt2+ϕ2(t)gN)\n=L∇f(dt2)+(L∇f(ϕ2(t)))gN+ϕ2(t)L∇f(gN).\nSincefdoes not depend on the variables in the fiber N,then\nL∇f(dt2) = (L∇fdt)⊗dt+dt⊗(L∇fdt),\nL∇f(ϕ2(t)) = 2ϕ(t)g(∇f,∇ϕ) and L∇fgN= 0.\nTo compute L∇f(dt),it suffices to use the Cartan��s formula, namely, given a smooth form\nωand a smooth vector field X,one hasLXω=iXdω+d(iXω),whereiXω=ω(X) is the\ninterior product. From this, it follows that\n2∇2f= 2f′′(t)dt2+2f′(t)ϕ(t)ϕ′(t)gN. (2.1)\nWe also recall the following formulae for warped product manifolds (c f. [36, 46]).\nProposition 1 ([46]).The Ricci curvature of a warped product manifold M=B×ϕF,with\nl=dim(F),must satisfy:\n(i)Ric(X,Y) =RicB(X,Y)−l\nϕ∇2ϕ(X,Y),\n(ii)Ric(X,V) = 0,\n(iii)Ric(V,W) =RicF(V,W)−(ϕ∆ϕ+(l−1)|∇ϕ|2)gF(V,W),\nwhereX, YandV, Ware horizontal and vertical vectors, respectively.\nAs a consequence of Proposition 1, we get the following result.\nProposition 2. Let(Mn, g)be a warped product manifold with g=dt2+ϕ2(t)gN,where\ngNis aκ-Einstein metric with κ >0.Suppose that either ϕ(t) =αtorϕ(t) =asinh(√βt)+\nbcosh(√βt), whereαandβare positive constants and a, b∈R.Then the scalar curvature\nRofMncan not be a positive constant.\nProof.We shall divide the proof into two cases. First, we assume that ϕ(t) =αt.Thus, by\nProposition 1, one obtains that\nRic(∂t,∂t) = 0 and Ric(V,W) = (κ−(n−2)α2)gN(V,W).\nFrom this, we have\nR= (n−1)κ−(n−2)α2\nα2t2.6 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nBy assuming that Ris a positive constant, one concludes that ϕis constant, which leads to\na contradiction.\nSecondly, we assume that ϕ(t) =asinh(√βt) +bcosh(√βt).Using Proposition 1 again\nyields\nRic(∂t,∂t) =−n−1\nϕϕ′′=−(n−1)β,\nRic(V,W) = [κ−(ϕϕ′′+(n−2)(ϕ′)2)]gN(V,W)\n= [κ−βϕ2−(n−2)(ϕ′)2]gN(V,W),\nwhere we have used that ϕ′′=βϕ.By trancing, we then get\nR=−2(n−1)β+(n−1)κ−(n−2)(ϕ′)2\nϕ2. (2.2)\nNow, by assuming that Ris a positive constant, one obtains thatκ−(n−2)(ϕ′)2\nϕ2is also a\nconstant. Therefore, taking the derivative, we see that\n−2(n−2)ϕ′ϕ′′ϕ2−2ϕϕ′(κ−(n−2)(ϕ′)2)\nϕ4= 0, (2.3)\nand since ϕandϕ′can only vanish in a set of measure zero and the fact that ϕ′′=βϕ, one\ndeduces that (2.3) is equivalent to\n(n−2)((ϕ′)2−βϕ2)−κ= 0.\nPluggingthisinto(2.2), wearriveat R=−n(n−1)β <0,whichalsoleadstoacontradiction.\n/square\n2.2.Quasi-Einstein Manifolds. In this subsection, we recall basic facts on m-quasi-\nEinstein manifolds. First of all, we remember that the fundamental e quation of an m-\nquasi-Einstein manifold ( Mn, g, u,λ),possibly with boundary, is given by\n∇2u=u\nm(Ric−λg), (2.4)\nwhereu >0 in the interior of Mandu= 0 on the boundary ∂M.\nBy tracing (2.4), one sees that\n∆u=u\nm(R−nλ). (2.5)\nThis implies that ∆ u= 0 along ∂M.Besides, Propositions 2.2 and 2.3 of [33] guarantee that\n|∇u|does not vanish on the boundary and it is constant on each compone nt of∂M.From\nthis, we infer that ν=−∇u\n|∇u|is the unit outward normal vector field over ∂M.In particular,\nby the Stokes’ formula, ∆ uis not identically zero. Actually, we have/integraldisplay\nM∆udMg=/integraldisplay\n∂M/a\\}b∇acketle{t∇u,ν/a\\}b∇acket∇i}htdSg=−|∇u||∂M|∂M|<0. (2.6)\nRemark 5. It follows from (2.5) and (2.6) that if the scalar curvature Ris constant, then\nR < nλ(cf.[33, Corollary 4.3] ). Thus, the scalar curvature Rcannot be nλin Theorem 1.\nFrom now on, we consider an orthonormal frame {ei}n\ni=1withe1=ν=−∇u\n|∇u|.Under\nthis coordinates, since u= 0 on∂M,the second fundamental form satisfies\nhab=−/a\\}b∇acketle{t∇eaν,eb/a\\}b∇acket∇i}ht=1\n|∇u|∇a∇bu= 0,\nfor any 2 ≤a,b,c,d≤n.Hence,∂Mis totally geodesic. Also, by the Gauss equation, i.e.,\nR∂M\nabcd=Rabcd−hadhbc+hachbd,QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 7\none obtains that\n(2.7) R∂M=R−2R11.\nWe further recallsomeimportant features of m-quasi-Einsteinmanifolds (cf. [15, 24, 33]).\nLemma 1. Let(Mn, g, u, λ)be anm-quasi-Einstein manifold with m >1.Then we have:\n(1)\n1\n2u∇R=−(m−1)Ric(∇u)−(R−(n−1)λ)∇u;\n(2)\nu2\nm(R−λn)+(m−1)|∇u|2=−λu2+µ,\nwhereµis a constant;\n(3)\n1\n2∆R=−m+2\n2u/a\\}b∇acketle{t∇u,∇R/a\\}b∇acket∇i}ht−m−1\nm/vextendsingle/vextendsingle/vextendsingle/vextendsingleRic−R\nng/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−(n+m−1)\nmn(R−nλ)/parenleftbigg\nR−n(n−1)\nn+m−1λ/parenrightbigg\n;\n(4)\nu(∇iRjk−∇jRik) =mRijkl∇lu+λ(∇iugjk−∇jugik)−(∇iuRjk−∇juRik).\nWe highlight that Eq. (2) of Lemma 1 determines a type of “integrabilit y condition”.\nBesides, Eq. (4) of Lemma 1 was obtained by Di´ ogenes and Gadelha in [24, Lemma 1].\nFrom assertion (1) of Lemma 1, if an m-quasi-Einstein manifold Mnhas constant scalar\ncurvature and m >1,then\nRic(∇u) =(n−1)λ−R\nm−1∇u. (2.8)\nConsequently, the traceless Ricci tensor ˚Ricmust satisfy\n˚Ric(∇u) =n(n−1)λ−(m+n−1)R\nn(m−1)∇u. (2.9)\nWe now set the covariant 2-tensor Pby\nP=Ric−(n−1)λ−R\nm−1g. (2.10)\nIn this perspective, by assuming that Mhas constant scalar curvature, we have from (2.8)\nthatP(∇u) = 0.Furthermore, by using the orthonormal frame {ei}n\ni=1that diagonalizes\nthe Ricci tensor, one observes that P(ei) =µiei.In [33], it was introduced the 4-tensor Q\nrelated to Pas follows\nQ=Rm+1\nmP⊙g+(n−m)λ−R\n2m(m−1)g⊙g, (2.11)\nwhere⊙stands for the Kulkarni-Nomizu product1andRmis the Riemann tensor. For\ncovariant 2-tensors SandT,the Kulkarni-Nomizu product is given by\n(S⊙T)ijkl=SikTjl+SjlTik−SilTjk−SjkTil. (2.12)\nWith these tools, one deduces the following result.\n1Our definition of Kulkarni-Nomizu product differs from [33] b y a constant 1 /2 and sign.8 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nProposition 3. Let(Mn, g, u, λ)be anm-quasi-Einstein manifold. Then we have:\nu(∇iPjk−∇jPik) =mQijkl∇lu+1\n2(g⊙g)ijklPsl∇su.\nProof.We start by rewriting the expression (4) of Lemma 1 in terms of the t ensorP=\nRic−(n−1)λ−R\nm−1gto obtain\nu(∇iPjk−∇jPik) +u(∇iρgjk−∇jρgik)\n=mRijkl∇lu+(λ−ρ)(∇iugjk−∇jugik)−(∇iuPjk−∇juPik),\nwhereρ=(n−1)λ−R\nm−1.In addition, by assertion (1) of Lemma 1, one sees thatu\n2∇ρ=P(∇u)\n(see also [33, Proposition 5.2]) and hence,\nu(∇iPjk−∇jPik) + 2( Pil∇lugjk−Pjl∇lugik)\n=mRijkl∇lu+(λ−ρ)(∇iugjk−∇jugik)−(∇iuPjk−∇juPik). (2.13)\nOn the other hand, it follows from (2.12) that\n(g⊙g)ijkl∇lu= 2(gik∇ju−gjk∇iu),\n(g⊙g)ijklPsl∇su= 2(Pjs∇sugik−Pis∇sugjk)\nand\n(P⊙g)ijkl∇lu= (Pik∇ju−Pjk∇iu)+(Pjl∇lugik−Pil∇lugjk).\nSubstituting these expressions into (2.13) yields\nu(∇iPjk−∇jPik) =mRijkl∇lu+(ρ−λ)(gik∇ju−gjk∇iu)+(Pik∇ju−Pjk∇iu)\n+2(Pjl∇lugik−Pil∇lugjk)\n=mRijkl∇lu+(ρ−λ)\n2(g⊙g)ijkl∇lu+(P⊙g)ijkl∇lu\n+1\n2(g⊙g)ijklPsl∇su\n=mQijkl∇lu+1\n2(g⊙g)ijklPsl∇su,\nwhere the last equality follows from (2.11).\n/square\nAs a consequence of Proposition 3, we deduce the following identities that were first\nproved by He, Petersen and Wylie in [34, Proposition 3.7]. Taking into a ccount that our\nconvention for the Kulkarni-Nomizu product (2.12) and Ric(X,Y) = trRm(X,·,Y,·) differ\nfrom [34], for the reader’s convenience, we are going to present a p roof here.\nProposition 4 ([34]).Let(Mn, g, u, λ)be anm-quasi-Einstein manifold with constant\nscalar curvature and m >1.Then we have:\n(1)\nu\nm(∇iPjk−∇jPik) =u\nm(∇iRjk−∇jRik) =Qijkl∇lu,\n(2)\nu\nm∇iPjk∇iu=/parenleftBigu\nm/parenrightBig2\n((λ−ρ)Pjk−PikPij)+Qijkl∇lu∇iu\nwhereρ=(n−1)λ−R\nm−1.QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 9\nProof.Initially, since Mnhas constant scalar curvature and P=Ric−(n−1)λ−R\nm−1g,ones\nsees that P(∇u) = 0 and therefore, the first assertion follows directly from Propo sition 3.\nWe now deal with the second one. By using again that P(∇u) = 0,one observes that\n0 =∇j(Pik∇iu) = (∇jPik)∇iu+Pik∇j∇iu.\nThis jointly with (2.4) yields\n∇jPik∇iu=−Pik∇j∇iu=−Pik/parenleftBigu\nm(Pji−(λ−ρ)gji)/parenrightBig\n=−u\nm(PikPji−(λ−ρ)Pjk).\nThereby, it suffices to use the first assertion in order to infer\nu\nm∇iPjk∇iu=/parenleftBigu\nm/parenrightBig2\n((λ−ρ)Pjk−PikPij)+Qijkl∇lu∇iu,\nas desired.\n/square\nNow, it is convenient to recall the following terminology (see [34]).\nDefinition 1. Anm-quasi-Einstein manifold (Mn, g, u, λ)is said to be rigid if it is Ein-\nstein or its universal cover is a product of Einstein manifol ds.\nIn [34], it was established the following result for rigid m-quasi-Einstein manifolds.\nProposition5 ([34]).A non-trivial complete rigid m-quasi-Einstein manifold (Mn, g, u, λ)\nis one of the examples in Table 2.1 of [34], or its universal cover splits off as\n/tildewiderM= (M1,g1)×(M2,g2)withu(x,y) =u(y),\nwhere(M1,g1, λ)is a trivial quasi-Einstein manifold and (M2, g2, u)is one of the examples\nin Table 2.1 in [34].\nRemark 6. It is known that the universal covering of a quasi-Einstein m anifold with λ >0\nis compact and hence, its fundamental group π1(M)is finite. The proof of this fact is quite\nsimilar to [27, 61]and it can be carry out by combining the arguments in the proof of[33,\nTheorem 4.1] (see also [49]) and[55, Remark 6.9] .\nBefore proceeding, we recall that a non-constant function f:M→Rof class at least C2\nis said to be transnormal if\n|∇f|2=b(f) (2.14)\nfor some C2function bon the range of finR.In addition, fis said to be isoparametric if\nthere exists a continuous function aon the range of finRsuch that\n∆f=a(f). (2.15)\nIn particular, (2.14) implies that the level set hypersurfaces of f(i.e.,Mt=f−1(t),wheret\nis a regular value of f) are parallel, and the integral curves of ∇fare the shortest geodesics\nconnecting the level sets. Besides, (2.15) guarantees that such hypersurfaces have constant\nmean curvatures. The preimage of the maximum (respectively, minim um) of an isopara-\nmetric (or transnormal) function fis called the focal variety off.We refer the reader to\n[30, 31, 42, 57] for more details.\nBy considering that ( Mn, g, u, λ) is anm-quasi-Einstein manifold with constant scalar\ncurvature, one deduces from assertion (2) of Lemma 1, for m >1,that\n|∇u|2=µ\nm−1−R+(m−n)λ\nm(m−1)u2. (2.16)10 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nConsequently, the potential function uis transnormal, namely,\nb(u) =µ\nm−1−R+(m−n)λ\nm(m−1)u2. (2.17)\nIn view of this, one easily verifies from (2.5) that the potential func tionuis isoparametric.\nConcerning the regularity of the potential function, for an m-quasi-Einstein manifold\n(Mn, g, u, λ),it is known that uandgare real analytic in harmonic coordinates (cf. Propo-\nsition 2.4 in [33]). In particular, the critical level sets of uhave zero measure.\nA central object in our approach is the set of maximum points of ugiven by\nMAX(u) ={p∈M:u(p) =umax}.\nRemark 7. In the compact case with m >1,notice that every point in MAX(u),which\nclearly is an interior point, must be a critical point. Moreo ver, the fact that uis a transnor-\nmal function and (2.16)allow us to deduce that the critical points of uhave the same value.\nThereby, MAX(u) =Crit(u)for nontrivial compact m-quasi-Einstein manifolds.\nTo conclude this section, we are going to describe the example of m-quasi-Einstein man-\nifold on/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×Sp×Sqmentioned in Remark 4 (see also [29]).\nExample 1. Letλ >0be an arbitrary constant and consider Mn=/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×Sp×Sq,\np, q >1,endowed with the metric\ng=dt2+p−1\nλgSp+q−1\nλgSq.\nThis space is an m-quasi-Einstein manifold with potential function u(t) = sin/parenleftBig√\nλ√mt/parenrightBig\nand\nconstant scalar curvature R= (n−1)λ.Indeed, we first notice that\nRic= (p−1)gSp+(q−1)gSqand∇u=u′∇t=√\nλ√mcos/parenleftBigg√\nλ√mt/parenrightBigg\n∇t.\nThereby, since u=u(t)and the warping function is constant, we deduce from (2.1)that\n∇2u=−λ\nmsin/parenleftBigg√\nλ√mt/parenrightBigg\ndt2. (2.18)\nOn the other hand, one observes that\nu\nm(Ric−λg) =1\nmsin/parenleftBigg√\nλ√mt/parenrightBigg\n/bracketleftbig\n(p−1)gSp+(q−1)gSq−(λdt2+(p−1)gSp+(q−1)gSq)/bracketrightbig\n=−λ\nmsin/parenleftBigg√\nλ√mt/parenrightBigg\ndt2.\nPlugging this into (2.18) gives (2.4).\nIn conclusion, u= 0if and only if either t= 0ort=√m√\nλπand consequently, the\nboundary consists of two disjoint copies of Sp×Sq.\n3.Key Lemmas\nIn this section, we shall provide several novel lemmas that will be us ed in the proof of\nthe main results. First, we recall some tensors that will be used in th e proofs of Theorems\n2 and 3. For a Riemannian manifold ( Mn, g), n≥4,the Weyl tensor is given by\nWijkl=Rijkl−1\nn−2(A⊙g)ijkl, (3.1)QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 11\nwhereA=Ric−R\n2(n−1)gstands for the Schouten tensor. Another tensor that will be use ful\nin our discussion is the Cotton tensor, for n≥3,\n(3.2) Cijk=∇iRjk−∇jRik−1\n2(n−1)(∇iRgjk−∇jRgik).\nNext, for n≥4,we have\n(3.3) Cijk=−/parenleftbiggn−2\nn−3/parenrightbigg\n∇lWijkl.\nNotice that Cijkis skew-symmetric in the first two indices and trace-free in any two in dices.\nIt turns out that, on an m-quasi-Einstein manifold, we may express the Cotton tensor in\nterms of the Weyl tensor and an auxiliary 3-tensor Tijkas follows (see [24, Lemma 2]).\nLemma 2 ([24]).Let(Mn, g, u, λ)be anm-quasi-Einstein manifold. Then it holds\n(3.4) uCijk=mWijkl∇lu+Tijk,\nwhere the 3-tensor Tijkis given by\nTijk=m+n−2\nn−2(Rik∇ju−Rjk∇iu)+m\nn−2(Rjl∇lugik−Ril∇lugjk)\n+(n−1)(n−2)λ+mR\n(n−1)(n−2)(∇iugjk−∇jugik)−u\n2(n−1)(∇iRgjk−∇jRgik).\nWe highlight that the tensor Tijkhas the same symmetric properties of the Cotton tensor\nand it is motivated by ideas outlined by Cao and Chen in [14]; see also [20, 5 1]. Besides, it\nis convenient to express the tensor Tijkin terms of the traceless Ricci tensor\nTijk=m+n−2\nn−2(˚Rik∇ju−˚Rjk∇iu)+m\nn−2(˚Rjl∇lugik−˚Ril∇lugjk)\n+n(n−1)λ−(m+n−1)R\nn(n−1)(∇iugjk−∇jugik)\n−u\n2(n−1)(∇iRgjk−∇jRgik). (3.5)\nWith aid of this notation, we have the following lemma.\nLemma 3. Let(Mn,g,u,λ)be anm-quasi-Einstein manifold with constant scalar curva-\nture. Then we have:\n˚RikTijk∇ju=m+n−2\nn−2|˚Ric|2|∇u|2−2m+n−2\nn−2˚Ric2(∇u,∇u)\n+(n(n−1)λ−(m+n−1)R)2\nn2(n−1)(m−1)|∇u|2\n=n−2\n2(m+n−2)|T|2, (3.6)\nwhere˚Ric2\nij=˚Rik˚Rkj.12 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nProof.By using that the scalar curvature Ris constant and Eq. (3.5), one obtains that\n˚RikTijk=m+n−2\nn−2(|˚Ric|2∇ju−˚Rik˚Rjk∇iu)−m\nn−2˚Rik˚Ril∇lugjk\n+n(n−1)λ−(m+n−1)R\nn(n−1)˚Rik∇iugjk\n=m+n−2\nn−2(|˚Ric|2∇ju−˚Rik˚Rjk∇iu)−m\nn−2˚Rij˚Ril∇lu\n+n(n−1)λ−(m+n−1)R\nn(n−1)˚Rij∇iu.\nApplying this for ∇ju,we see that\n˚RikTijk∇ju=m+n−2\nn−2|˚Ric|2|∇u|2−m+n−2\nn−2∇ju˚Rik˚Rjk∇iu\n−m\nn−2∇ju˚Rij˚Ril∇lu+n(n−1)λ−(m+n−1)R\nn(n−1)˚Ric(∇u,∇u)\n=m+n−2\nn−2|˚Ric|2|∇u|2−2m+n−2\nn−2˚Ric2(∇u,∇u)\n+n(n−1)λ−(m+n−1)R\nn(n−1)˚Ric(∇u,∇u).\nSo, it suffices to use (2.9) in the last term of the above equality in orde r to infer the first\nequality in (3.6).\nFinally, since Tis trace-free in any two indices and skew-symmetric in their first two\nindices, we get\n˚RikTijk∇ju=1\n2(˚RikTijk∇ju−˚RikTjik∇ju)\n=1\n2Tijk(˚Rik∇ju−˚Rjk∇iu)\n=n−2\n2(m+n−2)|T|2,\nwhere in the last equality we have used (3.5). This finishes the proof o f the lemma. /square\nAs a consequence of Lemma 3, by considering the aforementioned o rthonormal frame\n{ei}n\ni=1withe1=−∇u\n|∇u|so that ˚Ric(ei) =ξiei,we obtain the following result.\nCorollary 2. Let(Mn, g, u, λ)be anm-quasi-Einstein manifold with constant scalar cur-\nvature and m >1.ThenTis identically zero if and only if the Ricci tensor has at most\ntwo different eigenvalues, one of them has multiplicity at le astn−1and its eingenspace\ncorresponds to the orthogonal complement of ∇u.\nProof.Taking into account that ξ1=n(n−1)λ−(m+n−1)R\nn(m−1),one deduces from (3.6) that\nn−2\n2(m+n−2)|T|2=/bracketleftBigg\nm+n−2\nn−2n/summationdisplay\ni=1ξ2\ni+m−1\nn−1ξ2\n1/bracketrightBigg\n|∇u|2−2m+n−2\nn−2ξ2\n1|∇u|2\n=m+n−2\nn−2/bracketleftBiggn/summationdisplay\ni=1ξ2\ni−n\nn−1ξ2\n1/bracketrightBigg\n|∇u|2QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 13\non the regular points of the potential function u.Moreover, since Tr(˚Ric) =/summationtextn\ni=1ξi= 0,\nwe infer\nn−2\n2(m+n−2)|T|2=m+n−2\nn−2\nn/summationdisplay\ni=2ξ2\ni−1\nn−1/parenleftBiggn/summationdisplay\ni=2ξi/parenrightBigg2\n|∇u|2.\nBy the Cauchy-Schwarzinequality, we conclude that T≡0 if and only if the Ricci tensorhas\nat most two different eigenvalues with λ2=...=λnat regular points of u, for eigenvalues\nof the Ricci given by λi=ξi+R\nn. To conclude the proof, it suffices to recall that uis real\nanalytical in harmonic coordinates and consequently, the set of cr itical points of uhas zero\nmeasure in M. /square\nIn the rest of this section, we establish some key lemmas, for arbitr ary dimension n≥3,\nthat will play a crucial role in the proof of Theorem 4.\nLemma 4. Let(Mn, g)be ann-dimensional Riemannian manifold satisfying (2.4). Then\nwe have:\nu(∆Rik) =∇iRsk∇su+m∇kRis∇su+u\n2∇i∇kR+1\n2∇iu∇kR\n+(m+1)\nmuRisRsk+2uRjiksRjs−(m+2)∇sRik∇su\n−u\nm(R−(m+n−2)λ)Rik+λu\nm(R−(n−1)λ)gik.\nProof.Firstly, it follows from assertion (4) of Lemma 1 that\nu∇jRik=u∇iRjk+mRjikl∇lu+λ(∇jugik−∇iugjk)−(∇juRik−∇iuRjk).\nThis jointly with the fact that ∇j(u∇jRik) =∇ju∇jRik+u∆Rikgives\nu∆Rik=∇j(u∇jRik)−∇ju∇jRik\n=∇j(u∇iRjk+mRjikl∇lu+λ(∇jugik−∇iugjk)−(∇juRik−∇iuRjk))\n−∇ju∇jRik\n=∇ju∇iRjk+u∇j∇iRjk+m∇jRjikl∇lu+mRjikl∇j∇lu+λ∆ugik\n−λ∇k∇iu−∆uRik−∇ju∇jRik+∇j∇iuRjk+∇iu∇jRjk−∇ju∇jRik.\nNext, by using the twice contracted second Bianchi identity ( ∇jRjk=1\n2∇kR) and the first\ncontracted second Bianchi identity ( ∇jRjikl=∇kRil−∇lRik),one sees that\nu∆Rik=−∇ju∇jRik+∇ju∇iRjk+u∇j∇iRjk+m(∇kRil−∇lRik)∇lu\n+mRjikl∇j∇lu+λ∆ugik−λ∇k∇iu−∆uRik−∇ju∇jRik\n+∇j∇iuRjk+1\n2∇iu∇kR\n=−∇ju∇jRik+∇ju∇iRjk+u\n2∇i∇kR+uRisRsk+uRjiksRjs\n+m(∇kRil−∇lRik)∇lu+mRjikl∇j∇lu+λ∆ugik−λ∇k∇iu\n−∆uRik−∇ju∇jRik+∇j∇iuRjk+1\n2∇iu∇kR, (3.7)\nwhere in the last equality we have used the Ricci identity, i.e.,\n∇j∇iRjk=∇i∇jRjk+RjijsRsk+RjiksRjs.\nPlugging (2.4) and (2.5) into (3.7) yields14 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nu∆Rik=−∇ju∇jRik+∇ju∇iRjk+u\n2∇i∇kR+uRisRsk+uRjiksRjs\n+m(∇kRil−∇lRik)∇lu+uRjikl(Rjl−λgjl)+λu\nm(R−λn)gik\n−λu\nm(Rki−λgki)−u\nm(R−λn)Rik−∇ju∇jRik\n+u\nm(Rji−λgji)Rjk+1\n2∇iu∇kR\n=∇iRjk∇ju+m∇kRil∇lu+u\n2∇i∇kR+1\n2∇iu∇kR+(m+1)\nmuRisRsk\n+2uRjiksRjs−(m+2)∇jRik∇ju+/parenleftbigg\nλu−λu\nm−u\nm(R−λn)−λu\nm/parenrightbigg\nRik\n+λu\nm(R−(n−1)λ)gik.\nRearranging terms, one concludes that\nu∆Rik=∇iRsk∇su+m∇kRis∇su+u\n2∇i∇kR+1\n2∇iu∇kR\n+(m+1)\nmuRisRsk+2uRjiksRjs−(m+2)∇sRik∇su\n−u\nm(R−(m+n−2)λ)Rik+λu\nm(R−(n−1)λ)gik,\nas we wanted to prove. /square\nAs an application of Lemma 4, we are able to obtain an useful express ion for ∆( Ric3)ik=\n∆(RijRjlRlk).\nLemma 5. Let(Mn, g)be ann-dimensional Riemannian manifold satisfying (2.4). Then\nwe have:\nu∆(Ric3)ik+ (m+2)∇su∇s(Ric3)ik\n=∇iRsj∇suRjlRlk+∇jRsl∇suRijRlk+∇lRsk∇suRijRjl\n+2u(∇sRij∇sRjlRlk+∇sRijRjl∇sRlk+Rij∇sRjl∇sRlk)\n+m(∇jRis∇suRjlRlk+∇lRjs∇suRijRlk+∇kRls∇suRijRjl)\n+u\n2(∇i∇jRRjlRlk+∇j∇lRRijRlk+∇l∇kRRijRjl)\n+1\n2(∇iu∇jRRjlRlk+∇ju∇lRRijRlk+∇lu∇kRRijRjl)\n+(m+1)\nmu(RisRsjRjlRlk+RjsRslRijRlk+RlsRskRijRjl)\n+2u(RdijsRdsRjlRlk+RdjlsRdsRijRlk+RdlksRdsRijRjl)\n−3u\nm(R−(m+n−2)λ)(Ric3)ik+3λu\nm(R−(n−1)λ)RilRlk.\nProof.One easily verifies that\nu∆(Ric3)ik=u∆(RijRjlRlk)\n= (u∆Rij)RjlRlk+Rij(u∆Rjl)Rlk+RijRjl(u∆Rlk)\n+2u(∇sRij∇sRjlRlk+∇sRijRjl∇sRlk+Rij∇sRjl∇sRlk). (3.8)QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 15\nNext, it follows from Lemma 4 that\nu(∆Rij)RjlRlk=∇iRsj∇suRjlRlk+m∇jRis∇suRjlRlk+u\n2∇i∇jRRjlRlk\n+1\n2∇iu∇jRRjlRlk+(m+1)\nmuRisRsjRjlRlk+2uRdijsRdsRjlRlk\n−(m+2)∇sRij∇suRjlRlk−u\nm(R−(m+n−2)λ)RijRjlRlk\n+λu\nm(R−(n−1)λ)RilRlk, (3.9)\nRij(u∆Rjl)Rlk=∇jRsl∇suRijRlk+m∇lRjs∇suRijRlk+u\n2∇j∇lRRijRlk\n+1\n2∇ju∇lRRijRlk+(m+1)\nmuRjsRslRijRlk+2uRdjlsRdsRijRlk\n−(m+2)∇sRjl∇suRijRlk−u\nm(R−(m+n−2)λ)RjlRijRlk\n+λu\nm(R−(n−1)λ)RilRlk (3.10)\nand\nRijRjl(u∆Rlk) =∇lRsk∇suRijRjl+m∇kRls∇suRijRjl+u\n2∇l∇kRRijRjl\n+1\n2∇lu∇kRRijRjl+(m+1)\nmuRlsRskRijRjl+2uRdlksRdsRijRjl\n−(m+2)∇sRlk∇suRijRjl−u\nm(R−(m+n−2)λ)RijRjlRlk\n+λu\nm(R−(n−1)λ)RijRjk. (3.11)\nTherefore, inserting (3.9), (3.10) and (3.11) into (3.8) yields the as serted result.\n/square\nAs a consequence of Lemma 5, we deduce the following corollary.\nCorollary 3. Let(Mn, g)be ann-dimensional Riemannian manifold satisfying (2.4) with\nconstant scalar curvature. Then we have:\nu∆/parenleftbig\nTr(Ric3)/parenrightbig\n+ (m+2)∇su∇s(Tr(Ric3))\n= 3(m+1)∇iRsjRjlRil∇su+3(m+1)u\nmRic2\nijRic2\nij+6uRdsRdijsRjlRil\n−3u\nm(R−(m+n−2)λ)Tr(Ric3)\n+3λu\nm(R−(n−1)λ)|Ric|2+6u∇sRij∇sRjlRil,\nwhereTr(Ric3) =RijRjlRliandRic2\nij=RikRkj.16 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nProof.By using that Mnhas constant scalar curvature into Lemma 5, one deduces that\nu∆Ric3\nik= (∇iRsjRjlRlk+∇jRslRijRlk+∇lRskRijRjl)∇su\n+m(∇jRisRjlRlk+∇lRjsRijRlk+∇kRlsRijRjl)∇su\n+m+1\nmu/parenleftbig\nRic2\nijRjlRlk+Ric2\njlRijRlk+Ric2\nlkRijRjl/parenrightbig\n+2uRds(RdijsRjlRlk+RdjlsRijRlk+RdlksRijRjl)\n−(m+2)∇s(RijRjlRlk)∇su−3u\nm[R−(m+n−2)λ]RijRjlRlk\n+2u(∇sRij∇sRjlRlk+∇sRijRjl∇sRlk+Rij∇sRjl∇sRlk)\n+3λu\nm[R−(n−1)λ]RisRsk.\nBesides, tracing the above expression, one sees that\nu∆Tr(Ric3) = (∇iRsjRjlRli+∇jRslRijRli+∇lRsiRijRjl)∇su\n+m[∇jRisRjlRli+∇lRjsRijRli+∇iRlsRijRjl]∇su\n+m+1\nmu[Ric2\nijRjlRli+Ric2\njlRijRli+Ric2\nliRijRjl]\n+2uRds[RdijsRjlRli+RdjlsRijRli+RdlisRijRjl]\n−(m+2)∇s[RijRjlRli]∇su−3u\nm[R−(m+n−2)λ]RijRjlRli\n+2u(∇sRij∇sRjlRli+∇sRijRjl∇sRli+Rij∇sRjl∇sRli)\n+3λu\nm[R−(n−1)λ]RisRsi\n= (m+1)[∇iRsjRjlRli+∇jRslRijRil+∇lRisRijRjl]∇su\n+3(m+1)u\nmRic2\nijRic2\nij+6uRdsRdijsRjlRil\n−(m+2)∇s(Tr(Ric3))∇su−3u\nm(R−(m+n−2)λ)Tr(Ric3)\n+3λu\nm(R−(n−1)λ)|Ric|2+6u∇sRij∇sRjlRil.\nThe result then follows from the fact that ∇iRsjRjlRli=∇jRslRijRil=∇lRisRijRjl./square\nTo proceed, it is essential to ensure an expression for u∆/parenleftbig\nTr(P3)/parenrightbig\n.\nLemma 6. Let(Mn, g)be ann-dimensional Riemannian manifold satisfying (2.4) with\nconstant scalar curvature and m >1.Then we have:\nu∆Tr(P3) = 3( m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)−(m+2)∇s(Tr(P3))∇su\n+3(m+1)u\nmTr(P4)+3u\nm(3(m+1)ρ+(m−1)λ)Tr(P3)\n+3ρu\nm((m+3)ρ+2(m−1)λ)|P|2\n+3ρ2u\nm((m+1)ρ+(m−1)λ)Tr(P)\n+6ρ3u((m+n−1)ρ−(n−1)λ),\nProof.Initially, we compute Ric3\nikin terms of P=Ric−ρg,whereρ=(n−1)λ−R\nm−1.Indeed,\nit is easy to check thatQUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 17\nRic3\nik=RijRjlRlk\n= (Pij+ρgij)(Pjl+ρgjl)(Plk+ρglk)\n=PijPjlPlk+PijPjlρglk+PijρgjlPlk+Pijρgjlρglk\n+ρgijPjlPlk+ρgijPjlρglk+ρgijρgjlPlk+ρgijρgjlρglk\n=P3\nik+3ρP2\nik+3ρ2Pik+ρ3gik.\nWhence, it follows that\n(3.12) Tr(Ric3) =Ric3\nii=Tr(P3)+3ρ|P|2+3ρ2Tr(P)+nρ3.\nNext, notice that\nTr(P) =R(m+n−1)−n(n−1)λ\nm−1\nand moreover, by Proposition 3.3 in [34] (see also (3) in Lemma 1), sin ceMnhas constant\nscalar curvature, one sees that |P|2= (λ−ρ)Tr(P).Besides, Tr(P) and|P|2are also\nconstants. Thereby, we have\n(3.13) u∆(Tr(Ric3)) =u∆(Tr(P3)).\nWe now need to obtain an expression for ∇iRsjRjlRil∇suin terms of P.Indeed, one\nobserves that\n∇iRsjRjlRil∇su= [∇i(Psj+ρgsj)](Pjl+ρgjl)(Pil+ρgil)∇su\n=∇iPsjPjlPil∇su+∇iPsjPjlρgil∇su+∇iPsjρgjlPil∇su\n+∇iPsjρgjlρgil∇su\n=∇iPsjPjlPil∇su+ρ∇iPsjPji∇su+ρ∇iPsjPij∇su\n+ρ2∇iPsi∇su\n=∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su, (3.14)\nwhere we have used that ∇iPsi= 0,which follows from the fact that Mhas constant scalar\ncurvature and the twice contracted second Bianchi identity. Nex t, we compute\nRic2\nijRic2\nij=RikRkjRjlRli\n=/parenleftbig\nPikPkj+2ρPij+ρ2gij/parenrightbig/parenleftbig\nPilPlj+2ρPij+ρ2gij/parenrightbig\n=PikPkjPilPlj+4ρPikPkjPji+6ρ2PijPij+4ρ3Tr(P)+ρ4n\n=Tr(P4)+4ρTr(P3)+6ρ2|P|2+4ρ3Tr(P)+nρ4(3.15)\nand\nRdsRdijsRjlRil= (Pds+ρgds)Rdijs(Pjl+ρgjl)(Pil+ρgil)\n= (PdsPjlPil+2ρPdsPij+ρ2Pdsgij+ρgdsPjlPil\n+2ρ2gdsPij+ρ3gdsgij)Rdijs\n=PdsRdijsPjlPil+2ρPdsPjiRdijs−ρ2Pds(Pds+ρgds)\n−ρ(Pij+ρgij)PjlPil−2ρ2Pij(Pij+ρgij)−ρ3R\n=PdsRdijsPjlPil+2ρPdsRdijsPij−4ρ2|P|2−3��3Tr(P)\n−ρTr(P3)−ρ3R. (3.16)\nAt the same time, observe that18 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\n∇sRij∇sRjlRil=∇s(Pij+ρgij)∇s(Pjl+ρgjl)(Pil+ρgil)\n=∇sPij∇sPjlPil+ρ∇sPij∇sPij. (3.17)\nMoreover, as already mentioned, constant scalar curvature implie s that|P|andTr(P) are\nalso constants. Therefore, one deduces that\n∇s(Tr(Ric3))∇su=∇s(Tr(P3))∇su. (3.18)\nThereby, using (3.13), jointly with (3.12), (3.14), (3.15), (3.16), ( 3.17) and (3.18) into Corol-\nlary 3, one obtains that\nu∆Tr(P3) =u∆Tr(Ric3)\n= 3(m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n+3(m+1)u\nm/parenleftbig\nTr(P4)+4ρTr(P3)+6ρ2|P|2+4ρ3Tr(P)+nρ4/parenrightbig\n+6u/parenleftbig\nPdsRdijsPjlPil+2ρPdsRdijsPij−4ρ2|P|2−3ρ3Tr(P)−ρTr(P3)−ρ3R/parenrightbig\n−(m+2)∇s(Tr(P3))∇su\n−3u\nm(R−(m+n−2)λ)/parenleftbig\nTr(P3)+3ρ|P|2+3ρ2Tr(P)+nρ3/parenrightbig\n+3λu\nm(R−(n−1)λ)/parenleftbig\n|P|2+2ρTr(P)+nρ2/parenrightbig\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij),\nwhere we also used that |Ric|2=|P+ρg|2=|P|2+2ρTr(P)+nρ2.Consequently, taking\ninto account that ρ3R=−(m−1)ρ4+ρ3(n−1)λandR−(m+n−2)λ=−(m−1)(ρ+λ),\nwe have\nu∆Tr(P3) = 3( m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)−(m+2)∇s(Tr(P3))∇su\n+3(m+1)u\nmTr(P4)+/parenleftbigg12(m+1)ρu\nm−6ρu+3u\nm(m−1)(ρ+λ)/parenrightbigg\nTr(P3)\n+/parenleftbigg18(m+1)ρ2u\nm−24ρ2u+9ρu\nm(m−1)(ρ+λ)−3(m−1)λρu\nm/parenrightbigg\n|P|2\n+/parenleftbigg12(m+1)ρ3u\nm−18ρ3u+9ρ2u\nm(m−1)(ρ+λ)−6(m−1)λρ2u\nm/parenrightbigg\nTr(P)\n+/parenleftbigg3(m+1)nρ4u\nm−6uρ3(−(m−1)ρ+(n−1)λ)+3nρ3u\nm(m−1)(ρ+λ)\n−3(m−1)nλρ3u\nm/parenrightbigg\n.QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 19\nSimplifying the last four terms in the right hand side of the above expr ession, we achieve\nu∆Tr(P3) = 3( m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)−(m+2)∇s(Tr(P3))∇su\n+3(m+1)u\nmTr(P4)+3u\nm(3(m+1)ρ+(m−1)λ)Tr(P3)\n+3ρu\nm((m+3)ρ+2(m−1)λ)|P|2\n+3ρ2u\nm((m+1)ρ+(m−1)λ)Tr(P)\n+6ρ3u((m+n−1)ρ−(n−1)λ),\nwhich finishes the proof of the lemma.\n/square\n4.The Proof of Theorem 1, Theorem 2 and Theorem 3\nIn this section, we are going to present the proof of Theorem 1, Th eorem 2 and Theorem\n3.\n4.1.Proof of Theorem 1.\nProof.In the first part of the proof, we shall follow Proposition 3.13 of [34]. To begin with,\ndenoting α=R+(m−n)λ\nm(m−1)and/tildewideµ=µ\nm−1,one sees from (2.16) that\n|∇u|2\n/tildewideµ−αu2= 1\ndefines a distance function r=1√αarccos/parenleftbigg\nu√\n/tildewideµα−1/parenrightbigg\n.In particular, we can recover the\npotential function by taking u(r) =/radicalbig\n/tildewideµα−1cos(√αr).From Remark 7, the set of critical\npoints for ucoincides with the set of maximum values, namely, Crit(u) =MAX(u).Hence,\nwe may correspond MAX(u) =r−1(0).So, following the argument in [57, Lemma 7] with\ntheappropriateadaptationandusingthat uis zerooneachboundarycomponent, we deduce\nthat each connected component of MAX(u) is a smooth submanifold. Thereby, it follows\nfrom Lemma 9 that\n∆r= tr(Aθ)+n−k−1\nr+O(r), (4.1)\nwherekis the dimension of a connected component NofMAX(u) andAθstands for the\nsecond fundamental form with respect to θ.By (2.4), without loss of generality, we may\nmultiply the potential function uby a constant βso thatβuis a potential function for the\nsame metric and constant λasu.In view of this, we can assume that u(r) = cos(√αr) and\nconsequently, we deduce\n∇i∇ju=−√αsin(√αr)∇i∇jr−αcos(√αr)∇ir∇jr\nand\n∆u=−√αsin(√αr)∆r−αcos(√αr)|∇r|2. (4.2)\nTaking into account the Taylor expansions, around r= 0,\nsin(√αr) =√αr+O(r3) and cos(√αr) = 1+O(r2),20 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nwe obtain from (4.1) and (4.2) that\n∆u= (−αr+O(r3))/parenleftbigg\ntr(Aθ)+n−k−1\nr+O(r)/parenrightbigg\n+(−α+O(r2))\n=−α(n−k)+O(r). (4.3)\nIt is knownfrom (2.10) that P=Ric−(n−1)λ−R\nm−1g.In particular, bysetting ρ=(n−1)λ−R\nm−1,\nwe may write (2.5) in terms of Pandρ,at the connected component NofMAX(u),as\n∆u=1\nm(Tr(P)−n(λ−ρ)), (4.4)\nwhere we have used that u|N= 1.Then, since α=λ−ρ\nm,we combine (4.3), restricted to N,\nand (4.4) in order to infer\nTr(P) =k(λ−ρ).\nWe now claim that tangent and normal vector fields to Nare the eigenvectors corre-\nsponding to λ−ρand 0,respectively. Indeed, given a point p∈NandX∈X(N) a tangent\nvector at p,since∇u|N= 0,we have\n∇2u(X)(p) =∇X∇u(p) = 0,\nwhere we have used the fact that ∇X∇u(p) only hinges upon on the value of X(p) and∇u\nalong of a curve through pwithXas a tangent vector at p.Hence, by using (2.4), we obtain\n0 =∇X∇u(p) =u\nm(P(X)−(λ−ρ)X).\nConsequently, P(X) = (λ−ρ)X,for allX∈X(N) and therefore, the tangent vectors to N\ncorrespondsto the eigenvalue λ−ρforP.Besides, it follows fromassertion(2) ofProposition\n4 that, at Crit(u),\nP◦(P−(λ−ρ)I) = 0.\nThus, the only possible eigenvalues for PatNareλ−ρand 0.Moreover, since Tr(P) =\nk(λ−ρ) andk=dim(N),one concludes that normal vectors to Ncorrespond to the\neigenvalue 0 .\nProceeding, one concludes that\nP|N=/parenleftbigg(λ−ρ)Ik0\n0 [0]n−k/parenrightbigg\nis then×nmatrix of the tensor Pat the manifold N. In terms of the Ricci tensor, we have\nRic|N=/parenleftbiggλIk 0\n0(n−1)λ−R\nm−1In−k/parenrightbigg\n. (4.5)\nIn particular, taking the trace in (4.5), we see that\nR=k(m−n)+n(n−1)\nm+n−k−1λ,\nfor some k∈ {0,1...,n−1},where we also have used that R < nλ(see Remark 5). So, the\nproof is finished. /squareQUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 21\n4.2.Proof of Theorem 2.\nProof.First of all, since R= (n−1)λ,it follows from (2.8) that the eigenvalue λ1associated\nto the eigenvector ∇ufor the Ricci tensor is zero. We now need to show that all non-zero\neigenvalues of the Ricci tensor are equals to λ.Before to do so, we first claim that\n|˚Ric|2=R2\nn(n−1). (4.6)\nIndeed, since Ris constant, one deduces from assertion(3) in Lemma 1 (see also [15 , Lemma\n3.2]) that\n(m−1)|˚Ric|2=−m+n−1\nn(R−nλ)/parenleftbigg\nR−n(n−1)\nm+n−1λ/parenrightbigg\n.\nWhence, for R= (n−1)λ,we see that\n(m−1)|˚Ric|2=−R2(m+n−1)\nn/parenleftbigg\n1−n\nn−1/parenrightbigg/parenleftbigg\n1−n\nm+n−1/parenrightbigg\n, (4.7)\nand consequently,\n|˚Ric|2=R2\nn(n−1),\nas claimed.\nLettingλi, i/\\e}atio\\slash= 1,the possible non-zero eigenvalues of the Ricci tensor, one deduce s that\nn/summationdisplay\ni=2(λi−λ)2=|Ric|2−2λR+(n−1)λ2=|˚Ric|2−R2\nn(n−1),\nwhere we have used that |˚Ric|2=|Ric|2−R2\nnandR= (n−1)λ.Therefore, one obtains\nfrom (4.6) that λi=λ,fori= 2,...,n,that is, the eigenvalues of the Ricci are all constants\nwithλ2=...=λn=λ.Thereby, Corollary 2 guarantees that T≡0.In particular, since\nthe Ricci tensor is parallel, then the Cotton tensor (3.2) also vanish es and then, by Lemma\n2, we have Wijkl∇lu= 0.Now, we are in the position to invoke Theorem 1.2 of [33] to infer\nthat the metric splits off as g=dt2+ϕ2(t)/tildewidegN,where/tildewidegNisκ-Einstein with non-negative\nRicci curvature and u=u(t).\nIn view of (2.8), we get\nRic(∇u,∇u) =(n−1)λ−R\nm−1(u′)2= 0\nand hence, we may apply Proposition 1 to infer\n0 =Ric(∇u,∇u) = (u′)2Ric(∂t,∂t) =−(u′)2(n−1)\nϕϕ′′.\nSinceuis analytical in harmonic coordinates (and uis not constant), we conclude that\nϕ′′(t)/ϕ(t) = 0,which implies that ϕ(t) =corϕ(t) =ct, for some positive constant c.\nHowever, according to Proposition 2, the second case can not hold .\nProceeding, since g=dt2+c2/tildewidegNand/tildewidegNis aκ-Einstein metric, we may use again\nProposition 1 to deduce\nRic(V,W) =κ/tildewidegN(V,W).\nConsequently, the scalar curvature is R=κ\nc2(n−1) and moreover, λ=κ\nc2and (Nn−1, gN)\nisλ-Einstein manifold, where gN=c2/tildewidegN.22 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nFinally, observe that, by (2.1) and the fact that Ric=λgN,the potential function\nu=u(t) satisfies\nu′′(t)dt2=∇2u=u\nm(Ric−λg) =−λu\nmdt2\nandu|∂M= 0.Hence, without loss of generality, we can consider the solution u(t) =\nsin/parenleftBig√\nλ√mt/parenrightBig\n.Thereby, we conclude that Mnis isometric, up to scaling, to the cylinder/bracketleftBig\n0,√m√\nλπ/bracketrightBig\n×N,whereNis a compact λ-Einstein manifold. So, the proof of Theorem 2\nis finished. /square\n4.3.Proof of Theorem 3.\nProof.To begin with, since M3has constant scalar curvature, consider an orthonormal\nframe{ei}3\ni=1that diagonalizes the Ricci curvature Ricso thate1=−∇u\n|∇u|andλiare the\neigenvalues associated to ei,fori= 1,2,3.Thus, under this coordinates, one obtains that\n˚Ric=\nξ10 0\n0ξ20\n0 0ξ3\n,\nwhere\n/braceleftBigg\nξ1+ξ2+ξ3= 0,\nξ2\n1+ξ2\n2+ξ2\n3=|˚Ric|2,(4.8)\nandξi=λi−R\n3are the eigenvalues of the traceless Ricci tensor ˚Ric.A straightforward\ncomputation using (2.9) yields\nξ1=6λ−(m+2)R\n3(m−1). (4.9)\nIn another direction, since Ris constant, it follows from (3) of Lemma 1 that\n|˚Ric|2=−1\n3(m−1)(R−3λ)((m+2)R−6λ).\nThis combined with (4.8) gives\nξ2\n1+ξ2\n2+ξ2\n3=−1\n3(m−1)(R−3λ)((m+2)R−6λ).\nBy using (4.9), one sees that\nξ2\n2+ξ2\n3=−1\n3(m−1)(R−3λ)((m+2)R−6λ)−/parenleftbigg(m+2)R−6λ\n3(m−1)/parenrightbigg2\n=−1\n3(m−1)((m+2)R−6λ)/parenleftbigg\n(R−3λ)+(m+2)R−6λ\n3(m−1)/parenrightbigg\n=−1\n3(m−1)((m+2)R−6λ)/parenleftbigg(4m−1)R\n3(m−1)−3(3m−1)λ\n3(m−1)/parenrightbigg\n=−1\n(3(m−1))2((m+2)R−6λ)((4m−1)R−3(3m−1)λ).\nNext, since ( ξ2+ξ3)2=ξ2\n1,which in turn implies ξ2\n2+ξ2\n3+2ξ2ξ3=ξ2\n1,one obtains thatQUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 23\n2ξ2ξ3=ξ2\n1−(ξ2\n2+ξ2\n3)\n=/parenleftbigg(m+2)R−6λ\n3(m−1)/parenrightbigg2\n+1\n(3(m−1))2((m+2)R−6λ)((4m−1)R−3(3m−1)λ)\n=1\n(3(m−1))2((m+2)R−6λ)[(4m−1)R−3(3m−1)λ+(m+2)R−6λ]\n=1\n9(m−1)2((m+2)R−6λ)((5m+1)R−3(3m+1)λ), (4.10)\nwhich guarantees that ξ2ξ3is constant.\nWe now assume that ξ2ξ3= 0.Thereby, by the analyticity of g,one observes that ξ2= 0\norξ3= 0.Considering ξ3= 0,one deduces from (4.8) that ξ2=−ξ1.Hence, by (4.9),\nthe eigenvalues of Ricci curvature are constant. This then implies t hat the Ricci tensor is\nparallel. In particular, the Cotton tensor Cijkalsovanishes. Now, since W= 0 in dimension\n3,one obtains from Lemma 2 that T≡0.Besides, it follows from Corollary 2 that at least\ntwo eigenvalues of the Ricci tensor are equals. This forces ˚Ric= 0 and then, ( M3, g) is\nEinstein. So, it suffices to apply Proposition 2.4 of [34] to conclude tha t (M3, g) is isometric\nto the standard hemisphere S3\n+.\nOn the other hand, by assuming that ξ2ξ3/\\e}atio\\slash= 0,one deduces from (4.10) that\nξ2=1\n2ξ3/bracketleftbigg1\n9(m−1)2((m+2)R−6λ)((5m+1)R−3(3m+1)λ)/bracketrightbigg\n=ζ\n2ξ3.\nIn particular, by (4.8), one has −ξ1=ξ2+ξ3=ζ\n2ξ3+ξ3and hence, (4.9) yields\n2ξ2\n3+ζ= 2ξ3(m+2)R−6λ\n3(m−1). (4.11)\nComputing the discriminant for ξ3,we infer\n∆ =−/parenleftbigg6\n3(m−1)/parenrightbigg2\nm((m+2)R−6λ)(R−2λ).\nThen, solving the polynomial (4.11), one sees that\nξ3=((m+2)R−6λ)±3/radicalbig\nm((m+2)R−6λ)(2λ−R)\n6(m−1).\nNotice that the eigenvalue ξ2satisfies an expression equivalent to (4.11) and by (4.8), one\ndeduces that\nξ2=((m+2)R−6λ)∓3/radicalbig\nm((m+2)R−6λ)(2λ−R)\n6(m−1).\nTherefore, ξ2andξ3are constants.\nAnalogous to the previous case, we observe that the Ricci tensor is parallel and hence,\nthe Cotton tensor vanishes. Thereby, it follows from Corollary 2 th atξ2=ξ3/\\e}atio\\slash= 0,but it\nholds if and only if R= 2λ.At this point, it suffices to invoke Theorem 2 to conclude that\n(M3, g) is isometric, up to scaling, to the cylinder [0 ,√m√\nλπ]×N.Moreover, we deduce from\n(2.7) and Killing-Hopf theorem that N=S2.Thus, the proof of Theorem 3 is concluded.\n/square24 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\n5.The Proof of Theorem 4 and Corollary 1\nIn this section, we shall present the proof of Theorem 4 and Corolla ry 1. We start by\nproving a key proposition, for arbitrary dimension n≥3,that will be used in the proof of\nCorollary 1.\nProposition 6. There is no compact nontrivial quasi-Einstein manifold Mnwith boundary\nand constant scalar curvature R=m+n(n−2)\nm+n−2λ.\nProof.We argue by contradiction, assuming that a compact nontrivial qua si-Einstein man-\nifoldMnwith boundary has constant scalar curvature R=m+n(n−2)\nm+n−2λ,which corresponds\nthe case k= 1 in Theorem 1. Hence, by the work of Wang [57] (see also [30, Theor em 1.1]\nand [41, Theorem 6.1]), one obtains that MAX(u) is a focal variety of the isoparametric\nfunction uof dimension one and connected (see [31, Theorem 2.2]). So MAX(u) is totally\ngeodesic. This therefore implies that MAX(u) =S1and consequently, Mis homotopic to\nS1(see [42]), which leads to a contradiction with the fact that Mnhas finite fundamental\ngroup (see Remark 6). Thus, the proof is completed. /square\nFrom now on, we shall adapt the approach outlined by Cheng and Zho u in [21]. To do\nso, we first establish the following proposition.\nProposition 7. Let(M4, g, u, λ)be anm-quasi-Einstein manifold with m >1and con-\nstant scalar curvature R=2(m+2)λ\nm+1.Then we have\nu∆Tr(P3)+(m+2)/a\\}b∇acketle{t∇(Tr(P3)),∇u/a\\}b∇acket∇i}ht= 6uλTr(P3)+6λ2\nm+1u|P|2\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij) (5.1)\n+12ρ4m2(m+1)u.\nProof.Initially, let µibe the eigenvalues of Pdefined in (2.10) with respect to the adopted\northonormal frame {ei}4\ni=1so thate1=−∇u\n|∇u|.In particular, it follows from (2.8) that\nµ1= 0.Consequently,\nTr(P) =µ2+µ3+µ4and|P|2=µ2\n2+µ2\n3+µ2\n4,\nwhereP=Ric−3λ−R\nm−1g.Thus, for R=2(m+2)\nm+1λ, it follows from (2.10) that\nTr(P) =(m+n−1)R−n(n−1)λ\nm−1=(m+3)R−12λ\nm−1=2m\nm+1λ, (5.2)\nwhich implies that Tr(P) is a positive constant.\nNext, by Proposition 3.3 in [34], one has |P|2= (λ−ρ)Tr(P),whereρ=3λ−R\nm−1.This\ncombined with (5.2) yields\n|P|2=(m−4)λ+R\nm−1Tr(P) =m\nm+1λTr(P) =1\n2(Tr(P))2. (5.3)\nOn the other hand, by simplifying the last three terms in the right han d side of Lemma\n6, taking into account that ρ=λ\nm+1, Tr(P) =2m\nm+1λ,2|P|2= (Tr(P))2andn= 4,one\ndeduces thatQUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 25\nu∆Tr(P3) = 3( m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)−(m+2)∇s(Tr(P3))∇su\n+3(m+1)u\nmTr(P4)+3u\nm(3(m+1)ρ+(m−1)λ)Tr(P3)\n+12ρ4m2(m+1)u. (5.4)\nAt the same time, since Psj∇su=P(∇u) = 0,we have from (2.4) that\n0 =∇i(Psj∇su)\n=∇iPsj∇su+u\nmPsj(Ris−λgis)\n=∇iPsj∇su+u\nmP2\nij−(λ−ρ)\nmuPij\nso that\n(5.5) ∇iPsj∇su=−u\nmP2\nij+(λ−ρ)\nmuPij.\nHence, the first term in the right hand side of (5.4) becomes\nI= 3(m+1)(∇iPsjPjlPil∇su+2ρ∇iPsjPij∇su)\n= 3(m+1)/parenleftbigg\n−u\nmP2\nij+(λ−ρ)\nmuPij/parenrightbigg\n(PjlPil+2ρPij)\n= 3(m+1)/parenleftbigg\n−u\nm(Tr(P4))+(λ−3ρ)\nmuTr(P3)+2ρ(λ−ρ)\nmu|P|2/parenrightbigg\n. (5.6)\nSubstituting this into (5.4) and rearranging terms, one concludes t hat\nu∆Tr(P3)+(m+2)/a\\}b∇acketle{t∇(Tr(P3)),∇u/a\\}b∇acket∇i}ht= 6uλTr(P3)+6λ2\nm+1u|P|2\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)\n+12ρ4m2(m+1)u. (5.7)\nThis concludes the proof of the proposition. /square\nIn order to proceed, we need to prove the following result.\nProposition 8. Let(M4, g, u, λ)be anm-quasi-Einstein manifold with m >1and con-\nstant scalar curvature R=2(m+2)λ\nm+1.Then we have:\nuLm+2(Tr(P3)) = 8( m+1)ρuTr(P3)+6u∇sPij∇sPjlPil−3mρu|∇P|2\n−16m3(m+1)ρ4u (5.8)\nand\nuLm+2(Tr(P3))≥8(m+1)ρuTr(P3)−3mρu|∇P|2\n−16m3(m+1)ρ4u, (5.9)\nwhereuLm+2(f) =u∆f+(m+2)/a\\}b∇acketle{t∇f,∇u/a\\}b∇acket∇i}htandρ=λ\nm+1.26 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nProof.First of all, observe that our assumption is equivalent to R= 2(m+ 2)ρ,where\nρ=λ\nm+1.Moreover, one sees that\nTr(P) = 2mρand|P|2= 2m2ρ2=1\n2(Tr(P))2. (5.10)\nNow, we need to compute uLm+2(|P|2).To do so, since Ric=P+ρg,we notice from\nLemma 4 that\nu(∆Pik) =∇iPsk∇su+m∇kPis∇su+m+1\nmu(Pis+ρgis)(Psk+ρgsk)\n+2uRjiks(Pjs+ρgjs)−(m+2)∇sPik∇su\n+u\nm(m−1)(m+2)ρ(Pik+ρgik)−u\nm(m−1)(m+1)ρ2gik,\nwhere we have used that n= 4, R−(m+n−2)λ=−(m−1)(m+2)ρandλ(R−(n−1)λ) =\n−(m−1)(m+1)ρ2.Next, expanding the expression in the right hand side and rearrang ing\nterms, we have\nuLm+2(Pik) =∇iPsk∇su+m∇kPis∇su+m+1\nmuP2\nik+2(m+1)ρu\nmPik\n+(m+1)ρ2u\nmgik+2uRjiksPjs−2ρuPik−2ρ2ugik\n+(m−1)(m+2)ρu\nmPik+(m−1)ρ2u\nmgik\n=∇iPsk∇su+m∇kPis∇su+m+1\nmuP2\nik+(m+1)ρuPik+2uRjiksPjs. (5.11)\nProceeding, we use that λ= (m+1)ρand Eq. (5.5) to infer\n∇iPsk∇su=−u\nm(P2\nik−mρPik).\nConsequently,\n∇iPsk∇su+m∇kPis∇su=−(m+1)u\nm(P2\nik−mρPik).\nThis allow us to rewrite (5.11) as\nuLm+2(Pik) =−(m+1)u\nmP2\nik+(m+1)ρuPik+(m+1)u\nmP2\nik+(m+1)ρuPik+2uRjiksPjs\n= 2(m+1)ρuPik+2uRjiksPjs.\nAt the same time, by using that uLm+2(Pik) =u∆Pik+(m+2)/a\\}b∇acketle{t∇Pik,∇u/a\\}b∇acket∇i}ht,we infer\nuLm+2(|P|2) =uLm+2(PikPik)\n=u∆(PikPik)+(m+2)/a\\}b∇acketle{t∇(PikPik),∇u/a\\}b∇acket∇i}ht\n=u(2Pik∆Pik+2|∇P|2)+2(m+2)Pik/a\\}b∇acketle{t∇Pik,∇u/a\\}b∇acket∇i}ht\n= 2u|∇P|2+2uPikLm+2(Pik)\n= 2u|∇P|2+4(m+1)ρu|P|2+4uPikRjiksPjs.\nBesides, since |P|2is constant, then uLm+2(|P|2) = 0 and hence, we have\nuPikRjiksPjs=−u\n2|∇P|2−(m+1)ρu|P|2. (5.12)QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 27\nOn the other hand, it follows from (5.1) that\nuLm+2(Tr(P3)) = 6( m+1)ρuTr(P3)+6(m+1)ρ2u|P|2\n+6u(∇sPij∇sPjlPil+ρ∇sPij∇sPij)\n+6u(PdsRdijsPjlPil+2ρPdsRdijsPij)\n+12m2(m+1)ρ4u. (5.13)\nTo proceed, we need to deal with the terms that depend of the Riem annian curvature.\nThereby, fix a point p∈Mand assume Pij=µiδijatp,that is,µi, i= 1,2,3,4 are the\neigenvalues of the tensor Patpand recall that µ1= 0.Hence, one easily verifies that\nPdsRdijsPjlPil=4/summationdisplay\nj=24/summationdisplay\nd=2µdRdjjdµ2\nj.\nDenoting Kdj=Rdjdj, it follows that\nPdsRdijsPjlPil=−µ2K23µ2\n3−µ2K24µ2\n4−µ3K32µ2\n2−µ3K34µ2\n4−µ4K42µ2\n2−µ4K43µ2\n3\n=−K23µ2µ3(µ3+µ2)−K24µ2µ4(µ2+µ4)−K34µ3µ4(µ3+µ4)\n=−K23µ2µ3(Tr(P)−µ4)−K24µ2µ4(Tr(P)−µ3)−K43µ4µ3(Tr(P)−µ2)\n=−Tr(P)(K23µ2µ3+K34µ3µ4+K24µ2µ4)+(K23+K34+K24)µ2µ3µ4. (5.14)\nMoreover, notice that\nR22+R33+R44=R−R11=R−ρ=Tr(P)+3ρ\nand\nK12+K13+K14=R11=ρ,\nwhich therefore implies that\nTr(P)+3ρ=R−R11=R22+R33+R44= 2(K23+K34+K24)+R11.\nBesides, K23+K34+K24=1\n2(Tr(P) + 2ρ) = (m+ 1)ρ.In view of this, we may rewrite\n(5.14) as\nPdsRdijsPjlPil=−2mρ(K23µ2µ3+K34µ3µ4+K24µ2µ4)+(m+1)ρµ2µ3µ4.\nSimilarly, one easily verifies that\nPdsRdijsPij=4/summationdisplay\nd=24/summationdisplay\nj=2µdRdjjdµj=−2(K23µ2µ3+K24µ2µ4+K34µ3µ4). (5.15)\nHence, Eq. (5.13) becomes\nuLm+2(Tr(P3)) = 6( m+1)ρuTr(P3)+6(m+1)ρ2u|P|2+6u(∇sPij∇sPjlPil+ρ|∇P|2)\n−12(m+2)ρu(K23µ2µ3+K24µ2µ4+K34µ3µ4)+6(m+1)ρuµ2µ3µ4\n+12m2(m+1)ρ4u\n= 6(m+1)ρuTr(P3)+12m2(m+1)ρ4u+6u(∇sPij∇sPjlPil+ρ|∇P|2)\n−12(m+2)ρu(K23µ2µ3+K24µ2µ4+K34µ3µ4)+6(m+1)ρuµ2µ3µ4\n+12m2(m+1)ρ4u\n= 6(m+1)ρuTr(P3)+6u(∇sPij∇sPjlPil+ρ|∇P|2)\n−12(m+2)ρu(K23µ2µ3+K24µ2µ4+K34µ3µ4)\n+6(m+1)ρuµ2µ3µ4+24m2(m+1)ρ4u,\nwhere we used that |P|2= 2m2ρ2.Besides, by combining (5.12) and (5.15), we arrive at\nu(K23µ2µ3+K24µ2µ4+K34µ3µ4) =u|∇P|2\n4+m2(m+1)ρ3u.28 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nConsequently,\nuLm+2(Tr(P3)) = 6( m+1)ρuTr(P3)+6u(∇sPij∇sPjlPil+ρ|∇P|2)\n−3(m+2)ρu|∇P|2−12m2(m+2)(m+1)ρ4u\n+6(m+1)ρuµ2µ3µ4+24m2(m+1)ρ4u\n= 6(m+1)ρuTr(P3)+6u∇sPij∇sPjlPil−3mρu|∇P|2\n+6(m+1)ρuµ2µ3µ4−12m3(m+1)ρ4u. (5.16)\nAt the same time, similar to [21, pg. 11], by letting α=µ2, β=µ3andκ=µ4in the\nfollowing algebraic identity\n(α+β+κ)3= 3(α+β+κ)(α2+β2+κ2)−2(α3+β3+κ3)+6αβκ,\nwe obtain\n(Tr(P))3= 3|P|2Tr(P)−2Tr(P3)+6µ2µ3µ4.\nOf which,\n(5.17) 3 µ2µ3µ4=Tr(P3)−2m3ρ3.\nThis substituted into (5.16) yields\nuLm+2(Tr(P3)) = 8( m+1)ρuTr(P3)+6u∇sPij∇sPjlPil−3mρu|∇P|2\n−16m3(m+1)ρ4u,\nwhich proves (5.8).\nFinally, for the fixed orthonormal frame, by using (5.10) and Lemma 10, one deduces\nthatµi≥0,for alli.Hence,∇sPij∇sPjlPil=|∇Pii|2µi≥0 and this proves the second\nassertion (5.9). /square\nNow, we establish the following essential lemma.\nLemma 7. Let(M4, g, u, λ)be anm-quasi-Einstein manifold with m >1and constant\nscalar curvature R=2(m+2)λ\nm+1.Then the following inequality holds\nuLm+2(|∇u|2/parenleftbig\nTr(P3)−2m3ρ3)/parenrightbig\n≥2(9m+7)ρu|∇u|2/parenleftbig\nTr(P3)−2m3ρ3/parenrightbig\n,\nwhereρ=λ\nm+1.\nProof.Initially, we consider the level set Σ = Σ( t) =u−1(t),0≤t < u max,and an\northonormal frame {e1,e2,e3,e4}forM4that diagonalizes the tensor Pso thate1=∇u\n|∇u|\nand{e2,e3,e4}is a frame over Σ( t).Moreover, we assume α,β,γ,η ∈ {2,3,4}andi,j,k∈\n{1,2,3,4}. Thereby, it follows from the Gauss-equation that\nRΣ\nαβγη=Rαβγη+hαγhβη−hαηhβγ,\nwhich implies that\nRΣ\nαγ=Rαγ−Rα1γ1+Hhαγ−hαβhβγ, (5.18)\nwherehandHstand for the second fundamental form and the mean curvature , respectively.\nBesides, taking into account that ρ=λ\nm+1as well as\nRic(∇u) =ρ∇u, P=Ric−ρg, R= 2(m+2)ρ, Tr(P) = 2mρand|P|2= 2m2ρ2,\none deduces that\n(5.19) RΣ=R−2ρ+H2−|A|2= 2(m+1)ρ+H2−|A|2,\nwhere|A|2is the norm of the second fundamental form.QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 29\nNext, we are going to compute hαβandH.Indeed, by using (2.4) in terms of P,i.e.,\n∇2u=u\nm(P−mρg), the second fundamental form is given by\nhαβ=∇α∇βu\n|∇u|=(Pαβ−mρgαβ)\nm/radicalbig\nb(u)u, (5.20)\nwhereb(u) =|∇u|2.Furthermore, our assumption on the scalar curvature implies that\nP11= 0 and hence,\nH=Tr(P)−3mρ\nm/radicalbig\nb(u)u=−ρu/radicalbig\nb(u). (5.21)\nIn particular, we have from (5.20) that\n|A|2=|P|2−2mρTr(P)+3m2ρ2\nm2b(u)u2=ρ2u2\nb(u). (5.22)\nSubstituting (5.21) and (5.22) into (5.19) yields RΣ= 2(m+1)ρ.\nProceeding, we are going to deal with the Riemannian curvature ten sor of Σ.In fact,\nsince Σ has dimension 3 ,its curvature tensor can be expressed as\nRΣ\nαβγη= (RΣ\nαγgβη+RΣ\nβηgαγ−RΣ\nαηgβγ−RΣ\nβγgαη)−RΣ\n2(gαγgβη−gαηgβγ).\nThis jointly with (5.18) gives\nRΣ\nαβαβ=RΣ\nαα+RΣ\nββ−RΣ\n2\n=Rαα−Rα1α1+Hhαα−h2\nαα+Rββ−Rβ1β1+Hhββ−h2\nββ−(m+1)ρ\n=µα+µβ+2ρ−Rα1α1−Rβ1β1+H(hαα+hββ)−h2\nαα−h2\nββ−(m+1)ρ,\nwhereµα=P(eα) andhαβ= 0 forα/\\e}atio\\slash=β.Consequently, for fixed α/\\e}atio\\slash=βagain, by using\nthe Gauss equation, Eqs. (5.20) and (5.21), we then obtain\nRαβαβ=RΣ\nαβαβ−hααhββ+h2\nαβ\n=µα+µβ+2ρ−Rα1α1−Rβ1β1+H(hαα+hββ)−h2\nαα\n−h2\nββ−(m+1)ρ−hααhββ\n=µα+µβ+2ρ−Rα1α1−Rβ1β1−ρ(µα−mρ+µβ−mρ)u2\nmb(u)\n−(µα−mρ)2u2\nm2b(u)−(µβ−mρ)2u2\nm2b(u)−(m+1)ρ−(µβ−mρ)(µα−mρ)u2\nm2b(u)\n=µα+µβ+2ρ−Rα1α1−Rβ1β1−(m+1)ρ−mρ(µα+µβ−2mρ)u2\nm2b(u)\n−[µ2\nα−2mρ(µα+µβ)+µ2\nβ+2m2ρ2]u2\nm2b(u)−[µβµα−mρ(µα+µβ)+m2ρ2]u2\nm2b(u),30 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nwhich can be simplifying as\nRαβαβ=µα+µβ−ρ(µα+µβ)u2\nmb(u)+2ρ(µα+µβ)u2\nmb(u)+ρ(µα+µβ)u2\nmb(u)\n+2ρ2u2\nb(u)−2ρ2u2\nb(u)−ρ2u2\nb(u)+2ρ−(µ2\nα+µ2\nβ)u2\nm2b(u)\n−µαµβu2\nm2b(u)−Rα1α1−Rβ1β1−(m+1)ρ\n=(µα+µβ)(mb(u)+2ρu2)\nmb(u)+ρ(2b(u)−ρu2)\nb(u)\n−(µ2\nα+µ2\nβ)u2\nm2b(u)−µαµβu2\nm2b(u)−Rα1α1−Rβ1β1−(m+1)ρ.\nNext, multiplying the previous expression by µαµβand summing over αandβ,α/\\e}atio\\slash=β, we\ndeduce that\n4/summationdisplay\nα/ne}ationslash=βRαβαβµαµβ=mb(u)+2ρu2\nmb(u)4/summationdisplay\nα/ne}ationslash=β(µα+µβ)µαµβ+ρ(2b(u)−ρu2)\nb(u)4/summationdisplay\nα/ne}ationslash=βµαµβ\n−2u2\nm2b(u)4/summationdisplay\nα/ne}ationslash=βµ3\nαµβ−u2\nm2b(u)4/summationdisplay\nα/ne}ationslash=βµ2\nαµ2\nβ\n−24/summationdisplay\nα/ne}ationslash=βRα1α1µαµβ−(m+1)ρ4/summationdisplay\nα/ne}ationslash=βµαµβ. (5.23)\nAt the same time, we have to obtain expressions for each sum in (5.23 ). To do so, we first\nobserve that\n4/summationdisplay\nα/ne}ationslash=βµα=Tr(P) = 2mρand4/summationdisplay\nα/ne}ationslash=βµ2\nα=|P|2= 2m2ρ2, (5.24)\nwhich implies that\n4/summationdisplay\nα/ne}ationslash=βµαµβ=4/summationdisplay\nα=2/summationdisplay\nβ/ne}ationslash=αµαµβ=4/summationdisplay\nα=2µα(Tr(P)−µα) = (Tr(P))2−|P|2= 2m2ρ2,\n4/summationdisplay\nα/ne}ationslash=β(µα+µβ)µαµβ= 24/summationdisplay\nα=2/summationdisplay\nβ/ne}ationslash=αµ2\nαµβ= 24/summationdisplay\nα=2µ2\nα(Tr(P)−µα)\n= 2(Tr(P))|P|2−24/summationdisplay\nα=2µ3\nα= 8m3ρ3−24/summationdisplay\nα=2µ3\nα,\n4/summationdisplay\nα/ne}ationslash=βµ3\nαµβ=4/summationdisplay\nα=2/summationdisplay\nβ/ne}ationslash=αµ3\nαµβ=4/summationdisplay\nα=2µ3\nα(Tr(P)−µα) =4/summationdisplay\nα=22mρµ3\nα−4/summationdisplay\nα=2µ4\nα\nand\n4/summationdisplay\nα/ne}ationslash=βµ2\nαµ2\nβ=4/summationdisplay\nα=2/summationdisplay\nβ/ne}ationslash=αµ2\nαµ2\nβ=4/summationdisplay\nα=2µ2\nα(|P|2−µ2\nα) = 4m4ρ4−4/summationdisplay\nα=2µ4\nα.QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 31\nWe also need to obtain an expression for Rα1α1.From Eq. (4) of Lemma 1, one deduces\nthat\nu(∇iPjk−∇jPik)∇ju=mRijkl∇lu∇ju+mρ(∇iugjk−∇jugik)∇ju\n−(∇iuPjk−∇juPik)∇ju,\nwhere we have used that λ= (m+1)ρ.This combined with the fact that Pjk∇ju= 0 and\n∇iPjk∇ju=∇i(Pjk∇ju)−Pjk∇i∇ju=−u\nmPjk(Pij−mρgij)\nallow us to infer\nRijkl∇lu∇ju=−ρ(∇iu∇ku−|∇u|2gik)−|∇u|2\nmPik\n−u2\nm2Pjk(Pij−mρgij)−u\nm∇jPik∇ju.\nBy taking i=k=αand multiplying the last expression by|∇u|2\n|∇u|2,we obtain\nRα1α1|∇u|2=ρ|∇u|2−|∇u|2\nmµα−u2\nm2Pjα(Pαj−mρgαj)−u\nm∇1Pαα|∇u|\n=(mρ−µα)|∇u|2\nm−u2\nm2µ2\nα+ρu2\nmµα−u\nm∇1Pαα|∇u|.\nConsequently,\n4/summationdisplay\nα/ne}ationslash=βRα1α1µαµβ=4/summationdisplay\nα=2/summationdisplay\nβ/ne}ationslash=αRα1α1µαµβ=4/summationdisplay\nα=2Rα1α1µα(Tr(P)−µα)\n=1\n|∇u|24/summationdisplay\nα=2/bracketleftbigg(mρ−µα)b(u)+ρµαu2\nm−u2\nm2µ2\nα−u\nm∇1Pαα|∇u|/bracketrightbigg\nµα(2mρ−µα)\n=1\n|∇u|24/summationdisplay\nα=2(2m2ρ2µα−3mρµ2\nα+µ3\nα)b(u)\nm+1\n|∇u|24/summationdisplay\nα=2(2mρ2µ2\nα−ρµ3\nα)u2\nm\n−u2\nm2|∇u|24/summationdisplay\nα=2(2mρµ3\nα−µ4\nα)−u\nm|∇u|24/summationdisplay\nα=2∇1Pαα|∇u|/parenleftbig\n2mρµα−µ2\nα/parenrightbig\n.\nIn order to conclude this step, observe that\n∇1Tr(P3) = 34/summationdisplay\nα=2(∇1Pαα)µ2\nαand 0 = ∇1|P|2= 24/summationdisplay\nα=2(∇1Pαα)µα,\nwhich combined with (5.24) gives\n4/summationdisplay\nα/ne}ationslash=βRα1α1µαµβ=4m3ρ3−6m3ρ3\nm+1\nm4/summationdisplay\nα=2µ3\nα+4m3ρ4u2\nmb(u)−ρu2\nmb(u)4/summationdisplay\nα=2µ3\nα\n−u2\nm2b(u)4/summationdisplay\nα=2(2mρµ3\nα−µ4\nα)+∇u(Tr(P3))u\n3mb(u)\n=−2m2ρ3+4m2ρ4u2\nb(u)+∇u(Tr(P3))u\n3mb(u)+b(u)−3ρu2\nmb(u)4/summationdisplay\nα=2µ3\nα+u2\nm2b(u)4/summationdisplay\nα=2µ4\nα.32 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nReturning to Eq. (5.23), we then have\n4/summationdisplay\nα/ne}ationslash=βRαβαβµαµβ=mb(u)+2ρu2\nmb(u)/parenleftBigg\n8m3ρ3−24/summationdisplay\nα=2µ3\nα/parenrightBigg\n+ρ(2b(u)−ρu2)\nb(u)·2m2ρ2\n−2u2\nm2b(u)/parenleftBigg4/summationdisplay\nα=22mρµ3\nα−4/summationdisplay\nα=2µ4\nα/parenrightBigg\n−u2\nm2b(u)/parenleftBigg\n4m4ρ4−4/summationdisplay\nα=2µ4\nα/parenrightBigg\n−2m2(m+1)ρ3+4m2ρ3−8m2ρ4u2\nb(u)−2∇u(Tr(P3))u\n3mb(u)\n−2b(u)−6ρu2\nmb(u)4/summationdisplay\nα=2µ3\nα−2u2\nm2b(u)4/summationdisplay\nα=2µ4\nα.\nSimplifying terms, we infer\n4/summationdisplay\nα/ne}ationslash=βRαβαβµαµβ=8m3ρ3b(u)+16m2ρ4u2+4m2ρ3b(u)−2m2ρ4u2−2m2(m−1)ρ3b(u)\nb(u)\n−4m2ρ4u2\nb(u)−8m2ρ4u2\nb(u)−2∇u(Tr(P3))u\n3mb(u)\n−2mb(u)+4ρu2+4ρu2+2b(u)−6ρu2\nmb(u)4/summationdisplay\nα=2µ3\nα+2u2+u2−2u2\nm2b(u)4/summationdisplay\nα=2µ4\nα\n=6m2(m+1)ρ3b(u)+2m2ρ4u2\nb(u)−2∇u(Tr(P3))u\n3mb(u)\n−2(m+1)b(u)+2ρu2\nmb(u)Tr(P3)+u2\nm2b(u)4/summationdisplay\nα=2µ4\nα\n=2m2ρ3[3(m+1)b(u)+ρu2]\nb(u)−2∇u(Tr(P3))u\n3mb(u)\n−2[(m+1)b(u)+ρu2]\nmb(u)Tr(P3)+u2\nm2b(u)4/summationdisplay\nα=2µ4\nα.\nOn the other hand, it follows from (5.12) that\n2u|∇P|2+4(m+1)ρu|P|2+4uPikRjiklPjl= 0\nand hence,\nu|∇P|2=−2(m+1)ρu|P|2+2uPikRijklPjl.QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 33\nPlugging this fact into (5.9) yields\nuLm+2(Tr(P3))≥8(m+1)ρuTr(P3)−3mρu|∇P|2−16m3(m+1)ρ4u\n= 8(m+1)ρuTr(P3)+6m(m+1)ρ2u|P|2−6mρuPikRijklPjl\n−16m3(m+1)ρ4u\n= 8(m+1)ρuTr(P3)−4m3(m+1)ρ4u−12m3ρ4u[3(m+1)b(u)+ρu2]\nb(u)\n+4ρ∇u(Tr(P3))u2\nb(u)+12ρu[(m+1)b(u)+ρu2]\nb(u)Tr(P3)−6ρu3\nmb(u)4/summationdisplay\nα=2µ4\nα\n=4ρu[5(m+1)b(u)+3ρu2]\nb(u)Tr(P3)−6ρu3\nmb(u)4/summationdisplay\nα=2µ4\nα\n+4ρ∇u(Tr(P3))u2\nb(u)−4m3ρ4u[10(m+1)b(u)+3ρu2]\nb(u). (5.25)\nFrom (5.24), it is known that µ2, µ3, µ4andTr(P) satisfy the hypothesis of Corollary\nA.1 in [21] and therefore,\n4/summationdisplay\nα=2µ4\nα=−10m4ρ4\n3+8mρ\n3Tr(P3).\nSubstituting the above equality into (5.25), we infer\nuLm+2(Tr(P3))≥4ρu[5(m+1)b(u)+3ρu2]\nb(u)Tr(P3)+20m3ρ5u3\nb(u)−16ρ2u3\nb(u)Tr(P3)\n+4ρ∇u(Tr(P3))u2\nb(u)−4m3ρ4u[10(m+1)b(u)+3ρu2]\nb(u)\n=4ρu[5(m+1)b(u)−ρu2]\nb(u)Tr(P3)+4ρ∇u(Tr(P3))u2\nb(u)\n−4m3ρ4u[10(m+1)b(u)−2ρu2]\nb(u)\n=4ρu[5(m+1)b(u)−ρu2]\nb(u)(Tr(P3)−2m3ρ3)+4ρ∇u(Tr(P3))u2\nb(u). (5.26)\nFinally, we recall that the potential function of a quasi-Einstein man ifold is transnormal\nsatisfying\nb(u) =|∇u|2=µ\nm−1−R+(m−n)λ\nm(m−1)u2=ρ(u2\nmax−u2).\nHence,\nuLm+2(b(u)(Tr(P3)−2m3ρ3)) =ub(u)Lm+2(Tr(P3))+2u/a\\}b∇acketle{t∇b(u),∇(Tr(P3))/a\\}b∇acket∇i}ht\n+(Tr(P3)−2m3ρ3)uLm+2(b(u))\n=ub(u)Lm+2(Tr(P3))−4ρu2∇u(Tr(P3))\n+(Tr(P3)−2m3ρ3)(−2ρu2∆u−2ρu|∇u|2\n−(m+2)2uρ|∇u|2)\n=ub(u)Lm+2(Tr(P3))−4ρu2∇u(Tr(P3))\n−2ρu/parenleftbig\n−2ρu2+(m+3)b(u)/parenrightbig\n(Tr(P3)−2m3ρ3), (5.27)\nwhere we have used that ∆ u=−2ρuand\nLa(f) =u−adiv(ua∇f) = ∆f+au−1/a\\}b∇acketle{t∇u,∇f/a\\}b∇acket∇i}ht,fora/\\e}atio\\slash= 0 and f∈C∞(M).34 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nComparing (5.26) with (5.27) gives\nuLm+2/parenleftbig\n|∇u|2(Tr(P3)−2m3ρ3)/parenrightbig\n≥2(9m+7)ρu|∇u|2/parenleftbig\nTr(P3)−2m3ρ3/parenrightbig\n,\nas we wanted to prove. /square\nWe are now ready to present the proof of Theorem 4. For convenie nce, we restate it here.\nTheorem 5 (Theorem 4) .Let(M4, g, u, λ)be a nontrivial simply connected compact 4-\ndimensional m-quasi-Einstein manifold with boundary and m >1.ThenM4has constant\nscalar curvature R= 2(m+2)\n(m+1)λif and only if it is isometric, up to scaling, to the product\nspaceS2\n+×S2with the doubly warped product metric.\nProof.We already know that Tr(P) = 2mρand|P|2= 2m2ρ2, i.e.,\n(5.28) |P|2=1\n2(Tr(P))2.\nHence,since µ1= 0,byLemma10, theeigenvalues µα,α= 1,2,3,4,ofPareallnonnegative.\nWe now set the function\nh:=|∇u|2(Tr(P3)−2m3ρ3).\nIn particular, from (5.17) and the fact that µα, α= 1,2,3,4,are all nonnegative, one sees\nthathis a nonnegative function. Besides, since Mis compact with boundary ∂M,by\nperforming integration by parts, we deduce\n/integraldisplay\nMLm+2(h)dVm+2=/integraldisplay\nMu−(m+2)div(um+2∇h)dVm+2=/integraldisplay\nMdiv(um+2∇h)dV\n=−/integraldisplay\n∂Mum+2/angbracketleftbigg\n∇h,∇u\n|∇u|/angbracketrightbigg\ndS= 0, (5.29)\nwhere we have used the fact that uvanishes on ∂M, dV m+2=um+2dVis the weighted\nmeasure and the second order operator La, a∈R,is given by\nLa(f) =u−adiv(ua∇f) = ∆f+au−1/a\\}b∇acketle{t∇u,∇f/a\\}b∇acket∇i}ht,\nfor some f∈C∞(M).\nOn the other hand, it follows from Lemma 7 that\n2(9m+7)ρh−Lm+2(h)≤0. (5.30)\nSo, upon integrating (5.30) over M,we use (5.29) in order to infer\n2(9m+7)ρ/integraldisplay\nMhdVm+2≤0.\nOf which, one obtains that\nh=|∇u|2(Tr(P3)−2m3ρ3) = 0.\nNow,takingintoaccountthat |∇u|2iszeroonlyoverthe2-dimensionalsubmanifold MAX(u),\nwhich has measure zero, one concludes that Tr(P3)−2m2ρ2≡0 onM.This jointly with\nEq. (5.17) then implies µ2µ3µ4= 0,namely, at least one of µ2, µ3andµ4is zero. Assume\nµ2= 0.Thereby, by using (5.28), one deduces that µ1=µ2= 0 and µ3=µ4=mρ.\nReturning to the Ricci tensor, we then conclude that the Ricci ten sor has exactly two\ndistinct eigenvalues, each one with multiplicity two, namely,\nλ1=λ2=λ\nm+1andλ3=λ4=λ,QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 35\nwhereRic(ei) =λi,fori= 1,2,3,4.In particular, the Ricci tensor Ricis parallel. Then,\nby the first contracted second Bianchi identity ( ∇lRijkl=∇jRik−∇iRjk), one obtains that\nthe curvature tensor is harmonic. Now, we are in the position to app ly [34, Corollary 1.14]\nto conclude that M4is rigid. Hence, it suffices to use Proposition 5 to deduce that M4is\ncoveredby the product S2\n+×S2.Finally, since M4is simply connected, we may use Theorem\n54.6 in [43] to conclude that the covering map is a bijective local isomet ry and therefore, a\nglobal isometry. Thus, M4is isometric, up to scaling, to the product space S2\n+×S2.This\nfinishes the proof of the theorem. /square\n5.1.Proof of Corollary 1.\nProof.The result follows from Theorem 1, Remark 3 (and Proposition 6), Th eorem 2 and\nTheorem 4. /square\n6.Appendix\nFor the reader’s convenience, we include here some useful facts o n distance function that\nwe have used in the proof of the main results. Let Mbe a complete Riemannian manifold\nandNa properly immersed submanifold of M.Assume that π:νN→Nis the normal\nbundle. There is an induced connection ∇νonνNand a decomposition of tangente bundle\nT(νN) as\nT(νN) =H⊕V,\nwhereVξ:= ker(dπ)ξandHξconsists of all tangent vectors to parallel sections passing\nthrough ξ. Ifα: (−δ,δ)→νNis a smooth curve representing v∈T(νN), thenvH=\n(π◦α)′(0) andvV= (∇ν\n∂sα)(0) =v−VH.Thus,HξandVξare isomorphic to Tπ(ξ)Nand\nνπ(ξ)N,respectively. This decomposition induces a natural Riemannian metr ic onT(νN)\nsuch that πis a Riemannian submersion. With aid of this notation, we have the follow ing\nlemma.\nLemma 8 ([5]).Letα: (−δ,δ)→νNbe a smooth curve representing v∈T(νN). Define\nJ(t) :=∂\n∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0expπ◦α(s)(tα(s)).\nThenJ(t)is a Jacobi field along the geodesic γ(t) = exp(tα(0))and\nJ(0) =vH, J(1) = (dexp)α(0)(v)andJ′(0) =vV+Aα(0)vH.\nHere,Aηstands for the shape operator with respect to normal vector η.\nProceeding, let UNbe the unit normal bundle of Nequipped with volume element dθdp,\nwheredpdenotes the volume element of Nanddθis the volume element of unit sphere\nSn−k−1\npinνpN.Thereby, we may define Φ : (0 ,+∞)×UN→M\\Nby Φ(r,θ) = exp(rθ).\nAlong the normal geodesic γθ(r) = exp( rθ),we can choose a parallel orthonormal base\n{e1(r),...,e n(r)}such that\nAθei(0) =λi,fori= 1,···,k−1,anden=∂r=γ′\nθ(r).\nHence,Ji(r) = (dΦ)(r,θ)(ei),i= 1,2,···,n,must satisfy\nJ′′\ni(t)+R(γ′\nθ(t),Ji(t))γ′\nθ(t) = 0,fori= 1,...,k;\nJi(0) =ei(0),fori= 1,···,k;\nJ′\ni(0) =λiei(0),fori= 1,···,k;\nJi(0) = 0,fori=k+1,···,n;\nJ′\ni(0) =ei(0),fori=k+1,···,n.36 JOHNATAN COSTA, ERNANI RIBEIRO JR AND DETANG ZHOU\nNext, we consider the following notation\nJij=/a\\}b∇acketle{tJi,ej/a\\}b∇acket∇i}ht,fori= 1,···,k;\nKij=/a\\}b∇acketle{tR(γθ,ei)γθ,ej/a\\}b∇acket∇i}ht,fori= 1,···,k;\nA= diag(λ1,···,λn).\nAlso consider J:= (Jij)(k−1)×(k−1)andK:= (Kij)(k−1)×(k−1).With these notations, one\nobtains that\n\nJ′′+KJ= 0;\nJ(0) = diag/parenleftbig\nIk×k,O(n−k−1)×(n−k−1)/parenrightbig\n;\nJ′(0) = diag/parenleftbig\nA,I(n−k−1)×(n−k−1)/parenrightbig\n.\nIfγθ|[0,r]does not contain focal points, then Jis invertible on (0 ,r). Next, let σ(x) be the\ndistance function from N.Therefore, σ(γθ(r)) =r,provided that r∈(0,rθ).Moreover, by\ndenoting Uij(r) :=∇2σ(ei,ej)(γθ(r) and taking into account that ∇2σ(Ji,Jj) =/a\\}b∇acketle{tJ′\ni,Jj/a\\}b∇acket∇i}ht,we\nget the following lemma.\nLemma 9 ([5]).LetNbe a proper submanifold in M. Then for any θ∈νN, along the\nnormal geodesic γθ(r) = exp( rθ), the Hessian of the distance function σ(x) =dist(x,N)\nsatisfies\n\nU′+U2+K= 0,\nU=/parenleftbiggAθ\n1\nrI/parenrightbigg\n+r/parenleftbigg−A2\nθ−K11(0)V12\nV21 V22/parenrightbigg\n+O(r2),\nwhereU=∇2σ|{γ′\nθ(r)}⊥,K=Kθ=R(γ′\nθ,...)γ′\nθandAθis the shape operator of Nwith\nrespect to θ. In particular, the mean curvature H(θ,r)of the level sets of σatγθ(r)satisfies\n(6.1) H(θ,r) =tr(Aθ)+n−k−1\nr+O(r)\nand\n(6.2) ∇2σ2\n2(γθ(r)) =/parenleftbiggrAθ\nI(n−k)×(n−k)/parenrightbigg\n+O(r2).\nMoreover, at N, the function σ2has two eigenvalues 0and2of multiplicities mandn−k,\nrespectively.\nIn the sequel, we are going to present the proof of the following alge braic inequality.\nLemma 10. Leta1≥...≥anben≥2real numbers. Then\naiaj≥b\n2(n−1),\nwhereb= (/summationtextn\ni=1ai)2−(n−1)/summationtextn\ni=1a2\ni.In particular, if b≥0,then either all ai≥0or all\nai≤0.\nProof.The case n= 2 is straightforward. Now, for n >2,notice that\n/parenleftBiggn/summationdisplay\ni=1ai/parenrightBigg2\n=/parenleftBiggn−1/summationdisplay\ni=1ai/parenrightBigg2\n+2ann−1/summationdisplay\ni=1ai+a2\nn.\nHence, we see that\n(n−1)n/summationdisplay\ni=1a2\ni+b=/parenleftBiggn−1/summationdisplay\ni=1ai/parenrightBigg2\n+2ann−1/summationdisplay\ni=1ai+a2\nn,QUASI-EINSTEIN MANIFOLDS WITH BOUNDARY 37\nso that\n(n−1)n−1/summationdisplay\ni=1a2\ni+(n−2)a2\nn+b=/parenleftBiggn−1/summationdisplay\ni=1ai/parenrightBigg2\n+2ann−1/summationdisplay\ni=1ai.\nIn view of this, one obtains that\n(n−2)n−1/summationdisplay\ni=1a2\ni+(n−2)a2\nn−2ann−1/summationdisplay\ni=1ai+b=/parenleftBiggn−1/summationdisplay\ni=1ai/parenrightBigg2\n−n−1/summationdisplay\ni=1a2\ni,\nwhich implies that\n2/summationdisplay\ni=−log(z−z′),:β(z)γ(z) :=∂zφ(z),\nHence we have\nS({ki}) =V(zn)U(zn−1)U(zn−2)/summationdisplay\n{ZI}/productdisplay\nˆzi∈ZI/contintegraldisplay\nˆzieki·X(zi)\nki·P(zi)(ǫi·P(zi)+ki·ψ(zi)ǫi·ψ(zi))J(zi)dzi\n(2.16)\n=V(zn)U(zn−1)U(zn−2)/summationdisplay\n{ZI}/producttext\nˆzi∈ZIeki·X(ˆzi)(ǫi·P(ˆzi)+ki·ψ(ˆzi)ǫi·ψ(ˆzi))\ndet{∂ˆzjfi({ˆzl})}J(ˆzi)\n(2.17)\nfi({zl}) =/summationdisplay\nj/ne}ationslash=iki·kj\nzi−zj(2.18)\nNotice that the integration of the Xzero mode would produce the delta function\nδ(4)(/summationtextn\ni=1kµ\ni), therefore we are considering the unstripped amplitude wi th the integration\noverXzero modes implicitly implied.\n3 Collinear Limit\nNow let’s consider the case when the two vertex operators are approaching each other,\nzi→zj. The interesting solutions to the scattering equation, eq. (1.1), in this case, are\nthe ones with collinear momenta, ki·kj→0, which is the special case when the S-matrix\nfactorizes intosub-processesandleadstotheBCFWrecursi ons[33]. Theanalytical method\nused in deriving the BCFW relation in fact suggests that ther e is a bigger symmetry in\nthe underlying theory. One way to visualize such a symmetry i s to consider the on shell\n– 6 –condition for the propagator as the operator product expans ion (OPE) in 2d conformal\nfield theory (CFT). A tree amplitude consists of a summation o f relevant diagrams which\nare dominated by a subset of diagrams when one of the internal line going on-shell, i.e.\np2→0\nAn→An−m+11\np2Am+1 (3.1)\nLetting the resonance of the internal line analytically dep ends on a one complex parameter\nz, then one can show under some conditions, Ancan be solved recursively by analytical\nmethods, provided that we work out the following OPE.\nOk1,ǫ1(z1)Ok2,ǫ2(z2)∼/summationdisplay\ns=±C12sOk1+k2,ǫs(z2) (3.2)\nTo work out the OPE Ok1(z1)Ok2(z2), let us first consider the Jacobian for the delta\nfunction part. In the following, summation over l,i,j/ne}ationslash= 1,2 is assumed and only the\nsingular terms under z1→z2are retained,\ndet[An]\n=a11·A(11)−a12·A(12)−(−)la1l·A(1l)\n=a11·(a22·A(12,12)+(−)la2l·A(12,1l))−a12·(a21·A(12,12)+(−)la2lA(12,2l))\n−(−)la1l·(a21A(12,1l)−a22A(12,2l)−(−)ka2kA(12,kl))\n=(a11a22−a12a21)·A(12,12)+(−)l(a11a2l−a1la21)·A(12,1l))−(−)l(a12a2l−a1la22)A(12,2l))\n=/parenleftbigg\n(s12\n(z1−z2)2+s1i\n(z1−zi)2)(s21\n(z2−z1)2+s2j\n(z2−zj)2)−s12\n(z1−z2)2s21\n(z2−z1)2/parenrightbiggA(12,12)\n4\n+(−)l/parenleftbigg\n−(s12\n(z1−z2)2+s1i\n(z1−zi)2)s2l\n(z2−zl)2−s1l\n(z1−zl)2s21\n(z2−z1)2/parenrightbiggA(12,1l))\n4\n−(−)l(s12\n(z1−z2)2s2l\n(z2−zl)2+s1l\n(z1−zl)2(s21\n(z2−z1)2+s2j\n(z2−zj)2)·A(12,2l))\n4\n=s12\n4(z1−z2)2/parenleftbiggs1i+s2i\n(z2−zi)2·A(12,12)−(−)ls1l+s2l\n(z2−zl)2·A(12,1l))−(−)ls1l+s2l\n(z2−zl)2·A(12,2l)/parenrightbigg\n=−s12\n2(z1−z2)2/parenleftbigg\n−k·ki\n(z2−zi)2·A(1,1)\nn−1+(−)lk·kl\n(z2−zl)2·(A(12,1l)+A(12,2l))/parenrightbigg\n=−k1·k2\n(z1−z2)2det[An−1] (3.3)\n– 7 –Here,k=k1+k2\na11\nn−1=−/summationdisplay\nik·ki\n(z2−zi)2\na1l\nn−1=k·kl+1\n(z2−zl+1)2\nal1\nn−1=k·kl+1\n(z2−zl+1)2\nall\nn−1=−/summationdisplay\nikl+1·ki\n(kl+1−zi)2−kl·k\n(zl+1−z2)2\naij\nn−1=ki+1·kj+1\n(zi+1−zj+1)2, i/ne}ationslash=j\nOr, in matrix form,\nAn−1=\n−/summationtext\ni/ne}ationslash=1,2k·ki\n(z2−zi)2k·k3\n(z2−z3)2...k·kl\n(z2−zl)2...\nk·k3\n(z2−z3)2... ...\n... ... ...\nk·kl\n(z2−zl)2... ...\n... ... ...\nk·kn−3\n(z2−zn−3)2... ...\n(3.4)\nKeeping the collinear behavior of the Jacobian determinant in mind, the vertex operator\natz1can be simplified as z1→z2. In order to reproduce the right singular behavior\natz1→z2as in eq.( 3.3), we propose that we can actually integrate out z1inO(z1) as/contintegraltext\nˆz1ek1·X(z1)\nk1·P(z1)∼ek1·X(ˆz1)\nk1·P′(ˆz1), and the only singular contraction for the denominator part is\n/an}bracketle{t0|k1·P(z1)k2·X(z2)|0/an}bracketri}ht=k1·k2\nz1−z2.\nThe rest part just takes the limit, k1·k2→0,ˆz1−ˆz2→0. That would produce the right\nshort distance operator product expansion.\nOne may wonder why T(w)O(ˆz) has nontrivial OPE since by definition T commutes\nwithObecauseOcorresponds a physical state in ambitwistor string theory. However this\nis true only if the integrand of Ois not in the vicinity of ˆ z. Since the integrand of Ois\nsingular at ˆ z, we expect there is a zero mode insertion at ˆ z, which does not commute with\nT.\n/contintegraldisplay\nˆz1ek1·X(z1)\nk1·P(z1)(ǫ1·P(z1)+k1·ψ(z1)ǫ1·ψ(z1))J(z1)dz1\nz1→z2−→/contintegraldisplay\nˆz1ek1·X(z1)(ǫ1·P(z1)+k1·ψ(z1)ǫ1·ψ(z1))J(z1)dz1\nk1·P(ˆz1)+(z1−ˆz1)k1·P′(ˆz1)\n=ek1·X(ˆz1)\nk1·P′(ˆz1)(ǫ1·P(ˆz1)+k1·ψ(ˆz1)ǫ1·ψ(ˆz1))J(ˆz1) (3.5)\n– 8 –That is to say, this part of Oki,ǫi(zi) has the following OPE,\nek1·X(ˆz1)\nk1·P′(ˆz1)/contintegraldisplay\nˆz2ek2·X(z2)\nk2·P(z2)dz2\n=−(ˆz1−ˆz2)2\nk1·k2/contintegraldisplay\nˆz2ek·X(z2)\nk·P(z2)dz2 (3.6)\nwithk=k1+k2,/contintegraltext\nˆz2ek·X(z2)\nk·P(z2)is obtained by the following product,\nk1·k2→0,ˆz1−ˆz2→0\nek1·X(ˆz1)/contintegraldisplay\nˆz2ek2·X(z2)\nk2·P(z2)dz2∼/contintegraldisplay\nˆz2ek·X(z2)\nk2·k1\nz2−z1+k2·kl\nz2−zldz2∼/contintegraldisplay\nˆz2ek·X(z2)\nk1·kl\nz2−zl+k2·kl\nz2−zldz2\n=/contintegraldisplay\nˆz2ek·X(z2)\nk·kl\nz2−zldz2=/contintegraldisplay\nˆz2ek·X(z2)\nk·P(z2)dz2 (3.7)\nNext, let us consider the OPE for the numerator part of the ver tex operator,\nek1·X(z1)(ǫ1·P(z1)+k1·ψ(z1)ǫ1·ψ(z1))ek2·X(z2)(ǫ2·P(z2)+k2·ψ(z2)ǫ2·ψ(z2))\n=ek1·X(z1)/parenleftbiggǫ1·k2\nz1−z2+ǫ1·kl\nz1−zl+k1·ψ(z1)ǫ1·ψ(z1)/parenrightbigg\nek2·X(z2)/parenleftbiggǫ2·k1\nz2−z1+ǫ2·kl\nz2−zl+k2·ψ(z2)ǫ2·ψ(z2)/parenrightbigg\n=ek·X(z2)ǫ1·k2\nz1−z2ǫ2·k1\nz2−z1+ek·X(z2)ǫ1·k2\nz1−z2/parenleftbiggǫ2·kl\nz2−zl+k2·ψ(z2)ǫ2·ψ(z2)/parenrightbigg\n+ek·X(z2)ǫ2·k1\nz2−z1/parenleftbiggǫ1·kl\nz1−zl+k1·ψ(z1)ǫ1·ψ(z1)/parenrightbigg\n+ek·X(z2)1\nz1−z2((ǫ1·k2)k1·ψ(z1)ǫ2·ψ(z2)+(k1·ǫ2)ǫ1·ψ(z1)k2·ψ(z2))\n−ek·X(z2)1\nz1−z2((ǫ1·ǫ2)k1·ψ(z1)k2·ψ(z2)+(k1·k2)ǫ1·ψ(z1)ǫ2·ψ(z2))\n+ek·X(z2)1\n(z1−z2)2((ǫ1·k2)(k1·ǫ2)−(ǫ1·ǫ2)(k1·k2))\nThe double poles cancel each other as desired. The termk1·k2\n(z1−z2)2is in fact a single pole as\nwe can work out,\nk1·k2\nz1−z2z1→z2−→k2·kl\nz2−zl=1\n2(k2−k1)·kl\nz2−zl=1\n2(k2−k1)·P(z2) (3.8)\nwhere, summation over l/ne}ationslash= 1,2 is assumed. Collecting the remainingterms, wehave finally ,\nek1·X(z1)(ǫ1·P(z1)+k1·ψ(z1)ǫ1·ψ(z1))ek2·X(z2)(ǫ2·P(z2)+k2·ψ(z2)ǫ2·ψ(z2))\n=ek·X(z2)1\nz1−z2/parenleftbigg\n(ǫ1·k2)ǫ2−(ǫ2·k1)ǫ1+1\n2ǫ1·ǫ2(k1−k2)/parenrightbigg\n·P(z2)\n+ek·X(z2)1\nz1−z2k·ψ(z2)/parenleftbigg\n(ǫ1·k2)ǫ2−(ǫ2·k1)ǫ1+1\n2ǫ1·ǫ2(k1−k2)/parenrightbigg\n·ψ(z2) (3.9)\n– 9 –Combining eq.( 3.5,3.6,3.9), and the fact that OPE of the Kac-Moody currents just pro-\nduces a factor of 1 /(z1−z2), we have,\nOk1,ǫ1(z1)Ok2,ǫ2(z2) =−1\nk1·k2/summationdisplay\ns=±C12sOk,ǫs(z2) (3.10)\nC12±=2/parenleftbigg\n(ǫ1·k2)ǫ2−(ǫ2·k1)ǫ1+1\n2ǫ1·ǫ2(k1−k2)/parenrightbigg\n·ǫ∓(3.11)\nThe OPE we have check so far, as presented in eq.( 3.10-3.11)) is in agreement with the\nresult in ref.[ 34]. This proves that the vertex operator defined in eq.( 3.5) is the right choice\natz1→z2. In what follows we shall keep in mind that z1is very close to z2, and we shall\nassume a helicity +1 polarization just for convenience for t he vertex operator at z1. Hence\nwe shall work with\nO+(ω,u,¯u,z1) =ek1·X(z1)\nk1·P′(z1)/parenleftbig\nǫ+\n1·P(z1)+k1·ψ(z1)ǫ+\n1·ψ(z1)/parenrightbig\nJ(z1) (3.12)\nViewing the scattering amplitude as the correlation functi on of the string vertex insertions,\nwe have to address the following issues before we discuss the energy momentum tensor in-\nsertion in celestial CFT. First coordinates are correlated subjecting to the momentum\nconservation, we cannot move one vertex operator freely in c elestial space. Second, co-\nordinates are further restricted by the scattering equatio n, eq.(1.1), so we have to find a\nsolution to eq.( 1.1), in which one coordinate is a free variable. To solve those i ssues, we re-\nsort to the one complex parameter continuation of the moment umkas the same technique\nused in BCFW method.\nk1→k1+u˜λ1λn (3.13)\nkn→kn−u˜λ1λn, (3.14)\nso that the momentum conservation is not violated and we get o ne free parameter u. In the\nfollowing we shall consider how to construct a stress tensor in celestial CFT with correct\nOPE to the vertex operator sitting in the coordinate z1. UsingSL(2) invariance, we can\nfix the n’th coordinate to infinity, so that we are left with the following scattering equation\nto solve with,\n/summationdisplay\nj/ne}ationslash=i,nki·kj\nzi−zj= 0, i= 1,2,...,n−3. (3.15)\nNext, we consider the collinear limit, k1·k2→0, we seek for the solution with the behavior\nz1→z2, such that\n/summationdisplay\nj/ne}ationslash=i,1,nk2·kj\nz2−zj=k1·k2\nz1−z2(3.16)\nis finite.\nUnder these circumstances, we have the solution eq.( 3.16) forz1to eq.(3.15) with\nz1−z2varyinglinear in k1·k2, whilekeeping fixed zi,i= 2,3,...,n−1, whicharedetermined\n– 10 –according to the remaining equations in eq.( 3.15),\n/summationdisplay\nj/ne}ationslash=1,2,nk·kj\nz2−zj= 0, (3.17)\nki·k\nzi−z2+/summationdisplay\nj/ne}ationslash=i,1,2,nki·kj\nzi−zj= 0, i= 3,...,n−3. (3.18)\nwithk=k1+k2. That is, the above equations are just the scattering equati ons for n-1\nparticles with k2replaced by k.\nFor fixedzi,i= 2,3,...,n−1, We solve the remaining equation for z1.\nz1−z2=k1·k2/summationtext\nj/ne}ationslash=1,2,nk2·kj\nz2−zj(3.19)\nEquation ( 3.19) provides a link between the two coordinate systems, one in s tring world\nsheet and another in celestial sphere, respectively. To mak e things more transparent, let\nus decompose the momentum into the following form, k=ωq,ω=k0−k3,q=k\nk0−k3.\nDefiningu=k1+ik2\nk0−k3,¯u=k1−ik2\nk0−k3, we can represent the momentum qin a matrix form,\nq=qµ¯σµ=1\nk0−k3/parenleftigg\nk0+k3k1−ik2\nk1+ik2k0−k3/parenrightigg\n=/parenleftigg\nu¯u¯u\nu1/parenrightigg\n=/parenleftigg\n¯u\n1/parenrightigg/parenleftig\nu1/parenrightig\n=¯ξξ\n¯ξ=/parenleftigg\n¯u\n1/parenrightigg\n, ξ=/parenleftig\nu1/parenrightig\nqµ=1\n2/parenleftig\nu¯u+1, u+ ¯u,−i(u−¯u), u¯u−1/parenrightig\n=1\n2ξσµ¯ξ\nWe recognize uand ¯uare the coordinates in celestial sphere, and ωthe lightcone momen-\ntum. we have\nk1·k2→(k1+v˜λ1λn)·k2 (3.20)\nwe introduce the following null 4-vectors in the matrix form ,\nǫ+=∂uq=/parenleftigg\n¯u0\n1 0/parenrightigg\n=/parenleftigg\n¯u\n1/parenrightigg/parenleftig\n1 0/parenrightig\n=¯ξρ\nǫ−=∂¯uq=/parenleftigg\nu1\n0 0/parenrightigg\n=/parenleftigg\n1\n0/parenrightigg/parenleftig\nu1/parenrightig\n= ¯ρξ\nµ=∂¯u∂uq=/parenleftigg\n1 0\n0 0/parenrightigg\n=/parenleftigg\n1\n0/parenrightigg/parenleftig\n1 0/parenrightig\n= ¯ρρ\n– 11 –The only non-vanishing inner products between those four nu ll 4-vectors are the following,\nq·µ=−1\n2\nǫ+·ǫ−=1\n2\nit is convenient to introduce the following orthogonal basi s for theXandPfields,\nP(1)(z) =q·P(z)\nX(1)(z) =µ·X(z)\nP(2)(z) =µ·P(z)\nX(2)(z) =q·X(z)\nP(3)(z) =ǫ+·P(z)\nX(3)(z) =ǫ−·X(z)\nP(4)(z) =ǫ−·P(z)\nX(4)(z) =ǫ+·X(z)\nIn order that the BCFW momentum shift defined in eq.( 3.13) transforms into the celestial\ncoordinate u1shift, it’s convenient to fix kn∼µusing Lorentz invariance,\nkn=k0\nn/parenleftigg\n1 0\n0 0/parenrightigg\n=k0\nn/parenleftigg\n1\n0/parenrightigg/parenleftig\n1 0/parenrightig\n(3.21)\nWe have\nδk1=−δkn=uω1ǫ+\n1=uω1/parenleftigg\n¯u1\n1/parenrightigg/parenleftig\n1 0/parenrightig\nk1→k1+uω1ǫ+\n1=ω1/parenleftigg\n¯u1\n1/parenrightigg/parenleftig\nu1+u1/parenrightig\nkn→k1−uω1ǫ+\n1=/parenleftigg\nk0\nn−uω1¯u1\n−uω1/parenrightigg/parenleftig\n1 0/parenrightig\nk1·k2→(k1+uω1ǫ+\n1)·k2=−ω1ω2\n2(u+u1−u2)(¯u1−¯u2)\n4 The Energy Momentum Tensor in collinear limit\nThe correlation function in celestial space is related to th e scattering amplitude by the\ninverse Mellin transformation\nAs1...sn(|ω|1,u1,¯u1,...,|ω|n,un,¯un) = (/productdisplay\ni/integraldisplay\nd∆i|ω|−∆i\ni)As1...sn(|∆|1,u1,¯u1,...,|∆|n,un,¯un)\n(4.1)\n– 12 –where, ∆ i=hi+¯hiis the dilation weight with hi(¯hi) the conformal weight for the\nleft(right)-moving part, Si=hi−¯hiis the spin, sithe polarization helicity. So for each\ncelestial primary fields with definite spin S, we haveh=∆+S\n2, and we expect an energy\nmomentum tensor with the following OPE\nT(v)Os(∆,u1,¯u1) =1\nv−u1∂u1Os(∆,u1,¯u1)+(∆+S)/2\n(v−u1)2Os(∆,u1,¯u1) (4.2)\nDefine\nC=u+u1−u2\nz1−z2=2\nω1ω2(¯u1−¯u2)/summationdisplay\nj/ne}ationslash=i,1,nk2·kj\nz2−zj(4.3)\n=−2\nω2(¯u1−¯u2)/summationdisplay\nj/ne}ationslash=i,1,nq1·kj\nz2−zj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nu1=u2(4.4)\nIn eq.(4.3-4.4), the singular behavior is holomorphic, we retain this limi t for the helicity\n+1 polarization. Similarly, for helicity -1 polarization, we would choose anti-holomorphic\nOPE expansion. So the following OPE works for O+, and we have ∂uǫ+= 0. We recognize\nthat in the collinear limit, the primary field in CCFT, Os(∆,u1,¯u1) defined in eq.( 4.2), is\nthe Mellin transformed vertex operator in ambitwistor stri ng theory, O+(ω,u1,¯u1,z1) in\neq.(3.12), andu1andz1are related by a global conformal transformation, eq.( 4.3or4.4).\nHence we have the following OPE\nT(w) = (dv\ndw)2T(v) =C2T(v) (4.5)\nT(w)O+(ω,u1,¯u1,z1) =/parenleftbigg1\nw−z1(∂z1+C∂u1)+(1−ω∂ω)\n2(w−z1)2/parenrightbigg\nO+(ω,u1,¯u1,z1) (4.6)\nIn order the OPE, eq.( 4.6), be satisfied, we propose the following form of the energy-\nmomentum tensor, which is equivalent, by a global conformal transformation, to the stress\ntensor in CCFT in the collinear limit,\nT(z) =∂zX(z)·P(z)+Tg+Tgh\n+1\n2C/parenleftbig\n(X1(z)−iX2(z))(P0(z)−P3(z))+(X0(z)+X3(z))(P1(z)−iP2(z))/parenrightbig\n+1\n2∂z(X(z)·P(z)) (4.7)\nctotal=(11+1\n2)d−52+11+cg (4.8)\nHere,dis the bulk flat spacetime dimension, cghis for the two pair of fermionic (2 ,−1)\nghosts and one pair of (3 /2,−1/2) bosonic ghosts, cgis the center charge for the Sugawara\nconstruction of the stress tensor for the level kWZNW model. In what follows we are\ngoing to check if the stress tensor defined in eq.( 4.7-4.8) satisfies the correct OPE.\nWith the energy momentum tensor defined as in eq.( 4.7), the OPE between T(w) and\nthe vertex operator at z1, defined in eq.( 3.12) can be checked.\n– 13 –T(w)O+(ω,u,¯u,z1)\n=T(w)ek·X(z1)\nk·P′(z1)(ǫ+·P(z1)+k·ψ(z1)ǫ+·ψ(z1))J(z1)\n=1\nw−z1∂z1O+(ω,u,¯u,z1)\n+1\n2C\nw−z1/parenleftbig\n(X1(z1)−iX2(z1))(k0−k3)+(X0(z1)+X3(z1))(k1−ik2)/parenrightbig\nO+(ω,u,¯u,z1)\n+1\n2C\n(w−z1)2/parenleftbig\n(k1−ik2)(P0(w)−P3(w))+(k0+k3)(P1(w)−iP2(w))/parenrightbigO+(ω,u,¯u,z1)\nk·P′(z1)\n+1\n2∂w/parenleftigg\nk·X\nw−z1O+(ω,u,¯u,z1)−ǫ+·P(z1)\nw−z1ek·X(z1)\nk·P′(z1)J(z1)+k·P(w)\n(w−z)2k·P′(z1)O+(ω,u,¯u,z1)/parenrightigg\n=1\nw−z1∂z1O+(ω,u,¯u,z1)\n+C\nw−z1(∂uk·X(z1))O+(ω,u,¯u,z1)\n−C\n(w−z1)2(∂uk·P(w))O+(ω,u,¯u,z1)\nk·P′(z1)\n−1\n2(w−z)2/parenleftigg\nk·XO+(ω,u,¯u,z1)−ǫ+·P(z1)ek·X(z1)\nk·P′(z1)J(z1)+O+(ω,u,¯u,z1)/parenrightigg\n(4.9)\nWe now expand one of the double pole terms in eq.( 4.9),\n−C\n(w−z1)2(∂uk·P(w))O+(ω,u,¯u,z1)\nk·P′(z1)\n=−C/parenleftbigg∂uk·P(z1)\n(w−z1)2+∂uk·P′(z1)\nw−z1/parenrightbiggO+(ω,u,¯u,z1)\nk·P′(z1)\n=−/parenleftbigg\n−k·P′(z1)\n(w−z1)2+C∂uk·P′(z1)\nw−z1/parenrightbiggO+(ω,u,¯u,z1)\nk·P′(z1)(4.10)\nHere, we used the following relation. Fromd\ndu(k·P(z)) =∂uk·P(z) +k·1\nC∂zP(z) = 0,\nwe haveC∂uk·P(z1) =−k·P′(z1). Collecting terms with single pole and double pole\nseparately, we have\nT(w)O+(ω,u,¯u,z1)\n=1\nw−z1∂z1O+(ω,u,¯u,z1)+C\nw−z1/parenleftbigg\n∂uk·X(z)−∂uk·P′(z)\nk·P′(z)/parenrightbigg\nO+(ω,u,¯u,z1)\n+1\n(w−z1)2/parenleftigg\n(1\n2−k\n2·X(z))O+(ω,u,¯u,z1)+1\n2ǫ+·P(z)ek·X(z)\nk·P′(z)J(z1)/parenrightigg\n=/parenleftbigg1\nw−z1(∂z1+C∂u)+1\n(w−z1)21\n2(1−ω∂ω)/parenrightbigg\nO+(ω,u,¯u,z1)\n=/parenleftbigg1\nw−z1(∂z1+C∂u)+1\n(w−z1)21\n2(1−ω∂ω)/parenrightbiggek·X(z1)\nk·P′(z1)(ǫ+·P(z1)+k·ψ(z1)ǫ+·ψ(z1))J(z1)\n(4.11)\n– 14 –So the energy momentum tensor defined in eq.( 4.7) has the right OPE form, eq.( 4.5),\nwith respect to the vertex operator O(ω,u,¯u,z1) defined in eq.( 3.12). Thus,T(w) and\nO(ω,u,¯u,z1), originally defined on the 2d worldsheet, can be extended to the celestial\nsphere via a global conformal transformation defined by eq.( 4.3or4.4).\nCalculating the center charge for CCFT for the Yang-Mills th eory, we first write the\nenergy momentum tensor in a more compact form.\nT(w) =T(0)(w)+I(w)+∂wJ(w) (4.12)\nT(0)(w) =∂zX(w)·P(w)+Tg(w)+Tgh(w) (4.13)\nI(w) =1\n2C/parenleftbig\n(X1(w)−iX2(w))(P0(w)−P3(w))+(X0(w)+X3(w))(P1(w)−iP2(w))/parenrightbig\n(4.14)\nJ(w) =1\n2X(w)·P(w) (4.15)\nWe have,\nT(0)(w)I(z) =I(w)\n(w−z)2(4.16)\nT(0)(w)J(z) =J(w)\n(w−z)2+d\n2(w−z)3(4.17)\nI(w)I(z) =0 (4.18)\nI(w)J(z) =0 (4.19)\nJ(w)J(z) =−d\n4(w−z)2(4.20)\nThis leads to\nT(w)T(z) =(T(0)(w)+I(w)+∂wJ(w)))(T(0)(z)+I(z)+∂zJ(z)))\n=∂zT(0)(z)\nw−z+2T(0)(z)\n(w−z)2+2d+d/2+cg−52+11\n2(w−z)4\n+I(w)\n(w−z)2+2J(w)\n(w−z)3+3d\n2(w−z)4\n+I(z)\n(w−z)2+2J(z)\n(z−w)3+3d\n2(w−z)4+3d\n2(w−z)4\n=∂zT(z)\nw−z+2T(z)\n(w−z)2+11d+d/2+cg−41\n2(w−z)4(4.21)\nctotal=(11+1\n2)d−41+cg (4.22)\nHencewehavefinishedtheproofthateq.( 4.7-4.8)areindeedtherightchoicefortheprimary\nfieldO(ω,u,¯u,z1) defined in eq.( 3.12).\n5 Finite Size Effect\nSo far we have considered CCFT as a 2d CFT living on the celesti al sphere at null\ninfinity. However since our universe has finite size both in sp ace and time, we need to\n– 15 –consider the finite size effect as a perturbative non-CFT corre ction to the CCFT we have\nconsidered so far. That is to say, CCFT is living very close to but not exactly at the\nboundary of the AdS3slices. Similar situation has been considered by the author s of of\nref.[35], who proposed that the removal of the asymptotic region bey ond a radial distance\nrcin theAdS3bulk theory is equivalent to defining a quantum field theory on a Dirichlet\nwall at the radial distance r=rc. That quantum field theory can be obtained by adding a\nT¯Tterm as perturbationto theoriginal 2d CFT when rcis large. Inparticular, the vacuum\nto vacuum amplitude is modified by an exponential term, /an}bracketle{t0|exp/parenleftbig\niµ/integraltext\ndzd¯zT(z′)¯T(¯z)/parenrightbig\n|0/an}bracketri}ht.\nandµ∼1\nrc. SinceT¯Tperturbed CFT is a big subject following the original refs.[ 36,37],\nwe stop short of going into details on the development of the T¯Tperturbed CFT. Here we\njust make use of the main result in ref[ 35] by proposing that the perturbation parameter\nfor the CCFT should be µ∼1\ntu. Heretu∼rc∼8·1060is the expansion age of the present\nuniverse in Planck units. We make this choice for µsincetuis the only scale available in\nCCFT besides the Planck scale. Adding this new term, we are re ady to calculate its effect\non the vacuum to vacuum amplitude,\nA=/an}bracketle{t0|exp/parenleftbiggi\ntu/integraldisplay\ndzd¯zT(z′)¯T(¯z)/parenrightbigg\n|0/an}bracketri}ht (5.1)\n∼1−/an}bracketle{t0|1\n2t2u/integraldisplay\ndzd¯zT(z)¯T(¯z)/integraldisplay\ndz′d¯z′T(z′)¯T(¯z′)|0/an}bracketri}ht (5.2)\n= 1−1\n2t2u/contintegraldisplay\nz′dz/integraldisplay1\nǫ\n−1\nǫdz′c\n2(z−z′)4/contintegraldisplay\n¯z′d¯z/integraldisplay1\nǫ\n−1\nǫd¯z′¯c\n2(¯z−¯z′)4(5.3)\nThis integral needs to be regularized since it is of the type 0 · ∞. We just put an ǫ\nregularization,\nA= 1−c¯c\n2t2ulim\nǫ→0/contintegraldisplay\nz′dz(z−z′+1)ǫ\n2(z−z′)4/integraldisplay1\nǫ\n−1\nǫdz′/contintegraldisplay\n¯z′d¯z(¯z−¯z′+1)ǫ\n2(¯z−¯z′)4/integraldisplay1\nǫ\n−1\nǫd¯z′(5.4)\n= 1−c¯c\n2t2ulim\nǫ→0ǫ(ǫ−1)(ǫ−2)1\n2·62\nǫǫ(ǫ−1)(ǫ−2)1\n2·62\nǫ(5.5)\n∼exp(−c¯c\n18Λ) (5.6)\nHere,c= ¯c= 11d+d/2+cg−41 for the Yang-Mills theory as in eq.( 4.22) and Λ∼1\nt2u∼\n10−122[38] is the observed value for the present cosmological constan t in Planck units.\nThe integral we encounter here is very much like the one appea red in ref.[ 39], but the final\nform of the finite size effects differs. It is very tentative to say that our result generalizes\nthe finite length effect in 2d CFT to the finite surface effect in AdS3. We immediately\nrecognize that the finite size effect in CCFT takes the same form as the cosmological term\nin 4d gravitational theories. This should be of no surprise s ince both are kinds of vacuum\nenergy of the Casimir type, and certainly related to the leng th scale of our universe. Still,\nthis feature does not manifest itself had we not worked on the holographic dual to the\n4d gauge theories. In fact, much effort has been made in underst anding the nature of\nthe tiny cosmological constant being observed [ 40] and in understanding the nature of the\n– 16 –dark energy contributed to the evolution of our universe. He re we simply argue that any\nquantum field theory in 4d spacetime be modified by the finite si ze effect of the type as in\neq(5.6), should this quantum field theory have a dual description in terms of 2d CCFT on\ncelestial sphere at the null infinity. It is also possible tha t the cosmological constant is not\nreally a “constant” but actually evolves with time, as the le ngth scale of our universe does.\n6 Conclusion\nThe development in CCFT is fascinating since it is a holograp hic dual description to\nthe 4d quantum field theories in Minkowski spacetime. Being a dual description, CCFT\naccommodates an infinite-dimensional asymptotic symmetry , BMS symmetry group. In\nfact, BMSsymmetryasseenin4dquantumfieldtheoryisjustth eVirasoroandKac-Moody\ntype of symmetry algebra as seen in many 2d CFT’s. Neverthele ss, CCFT is not a rational\nCFT of any kind we know of, neither there exists an operator fo rmalism in the form of free\nfield realization nor known integrable models. Thus the BMS s ymmetry algebra discussed\nin the literature are largely classical or semi-classical. The present paper is a modest step\ntaken towards the quantum realization of the BMS symmetry in CCFT, which is a dual\ndescription to the 4d quantum filed theories. The benefit of a q uantum realization of the\nBMS symmetry algebra is obvious since it reveals some quantu m mechanical effects not\nseen directly in the 4d realization of the quantum field theor ies in Minkowski spacetime. A\nconcrete example is the center charge and the related finite s ize effect in CCFT, which we\nmanaged to calculate in the present paper. We argue that the fi nite size effect in CCFT is\nthe dual description to the cosmological term in 4d bulk theo ries.\nIn calculating the finite size effect, we adapted the method use d in [35] without explicit\nproving their result. This procedurecan be refined if we star t with the warped AdS3space,\nand consider how the finite size is incorporated into the Witt en diagram [ 41]. This is in\nthe plan of our future work.\n7 Note Added\nWe are aware that the holomorphic currents in CCFT are intima tely related to the soft\ntheorems in 4d gravity and gauge theories. It is much expecte d that the center extension\ncan be read off from the double soft limits. However, several g roups have claimed that\nthe center extension thus obtained is zero. So we have to clar ify why we get potentially\nnon-zero results here.\nFromeq.( 4.22), wegetctotal= (11+1\n2)d−41+cg. ThenBRSTinvarianceinambitwistor\nstring theory requires cs= (2+1\n2)d−41+cg= 0. Hence we have ctotal= 9d. Ford= 4.\nwe havectotal= 36. The apparent discrepancy between our result and the oth ers’ can be\nexplained as follows.\nFirst, let us pay attention to the Kac-Moody currents which a re related to the zeroth\norder of the soft theorem in gauge theories. From CHY formali sm, the short distance\noperator product expansion between the Kac-Moody currents are always singular with\nsingle poles, so we do not expect to get double pole when two ex ternal lines go soft. Double\n– 17 –polearises only forthetotal contraction between two exter nal lineswith theconjugate color\nindices and opposite helicities. So it is sufficient to consid er only the single soft limit. Let\nus sendz, the coordinate on the celestial sphere for the soft particl e, to infinity. Around\ninfinity,J(z) can be expanded as /an}bracketle{t0|/summationtext\nn∈N+Jnz−n−1∼ /an}bracketle{t0|J1z−2. See eq.(4.4-4.5) in\nref.[6]. The leading 1 /z2behavior is an indication that the level of the Kac-Moody alg ebra\nkis not zero, since the term /an}bracketle{t0|J1z−2is going to be contracted with the term proportional\ntoJ−1|0/an}bracketri}htleading to /an}bracketle{t0|J1z−2J−1|0/an}bracketri}ht=k\n2z2. Here,kis the level of the Kac-Moody algebra.\nThe largezbehavior can be checked explicitly for the single soft limit of the three and four\npoint function [ 15]. See also [ 42].\nSecond, similar argument applies to the gravity scattering amplitude. If we consider\nthe n-graviton scattering amplitude with one graviton at zgoing soft, we indeed find the\ncorrect 1/z4behavior for zlarge when the shadow form of the stress tensor is inserted at z.\nSee eq.(4.6-4.9) in ref.[ 7]. This 1/z4behavior precisely implies that /an}bracketle{t0|L2z−4L−2|0/an}bracketri}ht=c\n2z4\nis non-vanishing. Here c is the center charge for the stress t ensor inserted.\nThird, there is another possibility that the center charge cfor the stress tensor T(z)\nis zero but /an}bracketle{t0|L2z−4is not paired with the state L−2|0/an}bracketri}ht. Instead, there may exist another\nspin 2 holomorphic current t(z) such that T(z)t(0)∼b\nz4, and/an}bracketle{t0|L2z−4l−2|0/an}bracketri}ht=b\nz4. This\npossibility leads to a logarithmic CCFT proposed in ref.[ 43]. At the moment, we can not\nconclude if CCFT is logarithmic or not, but our construction of the stress tensor seems to\nsuggest a non-zero value of cfor the total energy momentum tensor in the corresponding\nCCFT.\nFourth,T¯Tperturbed CCFT may play an important role in regularizing th e loop\nintegrals in the bulk theory providing a UV-completeness, s ee ref.[44]. But here in our\nconsideration, the T¯Tperturbation is relevant even at the tree level computation of the\nscattering amplitude. Of course, since the value of the cent er charge is closely related to\nthe vacuum energy of the quantum field theories, it is desirab le to find its 4d origin which\nmay make 4d-2d holography dictionary more complete.\nAcknowledgments\nIamgrateful toYihongGao, SongHe, GangYang, Ronggen Cai, C hiZhang, Yaozhong\nZhang for useful discussions at the various stage of the pres ent project. I am indebted to\nJens Lyng Petersen for carefully reading the manuscript and for valuable communications.\nThisworkis supportedinpartby theNational Natural Scienc eFoundation of ChinaGrants\nNo.11675240.\nReferences\n[1] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Gravitational waves in general\nrelativity VII. Waves from isolated axisymmetric systems ,Proc. Roy. Soc. Lond. A 269\n(1962) 21; R. K. Sachs, Gravitational waves in general relativity VIII. Waves in\nasymptotically flat space-time ,Proc. Roy. Soc. Lond. A 270(1962) 103.\n[2] G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at\nnull infinity revisited ,Phys. Rev. Lett. 105(2010) 111103; [ arXiv:0909.2617 [gr-qc]];\n– 18 –Supertranslations call for superrotations ,PoS CNCFG 2010(2010) 010;\n[arXiv:1102.4632 [gr-qc]];BMS charge algebra ,\nJournal of High Energy Physics 2011(2011)105 ; [arXiv:0909.2617 [hep-th]].\n[3] T. Banks, A Critique of pure string theory: Heterodox opinions of dive rse dimensions ,\n[arXiv:hep-th/0306074 ].\n[4] A. Strominger, Asymptotic symmetries of Yang-Mills theory ,\n[Journal of High Energy Physics 2014(2014) 151 , [arXiv:1308.0589 ].\n[5] Daniel Kapec, Vyacheslav Lysov, Sabrina Pasterski and Andre w Strominger, Semiclassical\nVirasoro Symmetry of the Quantum Gravity S-Matrix ,\n[Journal of High Energy Physics 1408(2014) 058 , [arXiv:1406.3312 ].\n[6] Temple He, Prahar Mitra and Andrew Strominger, 2D Kac-Moody Symmetry of 4D\nYang-Mills Theory , [Journal of High Energy Physics 2016(2016) 137 , [arXiv:1503.02663 ].\n[7] Daniel Kapec, Prahar Mitra, Ana-Maria Raclariu, Andrew Stromin ger,A 2D Stress Tensor\nfor 4D Gravity ,Phys. Rev. Lett. 119(2017) 121601, [ arXiv:1609.00282 ].\n[8] F. Cachazo, S. He and E. Y. Yuan, Scattering of massless particles in arbitrary dimensions ,\nPhysical review letters 113(2014) 171601, [ arXiv:1307.2199 ].\n[9] F. Cachazo, S. He and E. Y. Yuan, Scattering of massless particles: scalars, gluons and\ngravitons ,Journal of High Energy Physics 2014(2014) 33 , [arXiv:1309.0885 ].\n[10] L. Mason and D. Skinner, Ambitwistor strings and the scattering equations ,\nJournal of High Energy Physics 2014(2014) 48 , [arXiv:1311.2564 ].\n[11] A. Giveon, D. Kutasov and N. Seiberg, Comments on String Theory on AdS3,\nAdv.Theor.Math.Phys. 2(1998) 733-780, [ arXiv:hep-th/9806194 ].\n[12] Ming Yu and Bo Zhang, Light-Cone Gauge Quantization of String Theories on AdS3Space,\nNucl.Phys. B551(1999) 425-449, [ arXiv:hep-th/9812216 ].\n[13] V. P. Frolov, Null Surface Quantization and Quantum Field Theory in Asymp totically Flat\nSpace-Time ,Fortschritte der Physik 26-9(1978) 455-500.\n[14] Sabrina Pasterski, Shu-Heng Shao and Andrew Strominger, Flat Space Amplitudes and\nConformal Symmetry of the Celestial Sphere ,Phys. Rev. D 96,((2017) 065026,\n[arXiv:1701.00049 ].\n[15] Sabrina Pasterski, Shu-Heng Shao, Andrew Strominger, Gluon Amplitudes as 2d Conformal\nCorrelators ,Phys. Rev. D 96((2017) 085006, [ arXiv:1706.03917 ].\n[16] Wei Fan, Angelos Fotopoulos, Stephan Stieberger, Tomasz R. Taylor and Bin Zhu,\nConformal Blocks from Celestial Gluon Amplitudes ,\nJournal of High Energy Physics 2021(2021) 170 , [arXiv:2103.04420 ].\n[17] Alexander Atanasov, Walker Melton, Ana-Maria Raclariu, Andre w Strominger, Conformal\nBlock Expansion in Celestial CFT ,Phys. Rev. D 104(2021) 126033 , [ arXiv:2104.13432 ].\n[18] Jan de Boer and Sergey N. Solodukhin, A holographic reduction of Minkowski space-time ,\nNucl.Phys. B665(2003) 545-593, [ arXiv:hep-th/0303006 ].\n[19] Sergey N. Solodukhin, Reconstructing Minkowski Space-Time , [arXiv:hep-th/0405252 ].\n[20] Clifford Cheung, Anton de la Fuente and Raman Sundrum, 4D Scattering Amplitudes and\nAsymptotic Symmetries from 2D CFT ,Journal of High Energy Physics 2017(2017) 112 ,\n[arXiv:1609.00732 ].\n– 19 –[21] Adam Ball, Elizabeth Himwich, Sruthi A. Narayanan, Sabrina Past erski, Andrew\nStrominger, Uplifting AdS3/CFT2 to Flat Space Holography ,\nJournal of High Energy Physics 2019(2019) 168 , [arXiv:1905.09809 ].\n[22] Eduardo Casali, Walker Melton, Andrew Strominger, Celestial Amplitudes as AdS-Witten\nDiagrams ,Journal of High Energy Physics 2022(2022) 140 , [arXiv:.2204.10249 ].\n[23] Walker Melton, Atul Sharma, Andrew Strominger, Celestial Leaf Amplitudes ,\n[arXiv:2312.07820 ].\n[24] F. Lizzi, B. Rai, G. Sparano and A. Srivastava, Quantization of the null string and absence\nof critical dimensions ,Physics Letters B 182(1986) 326–330.\n[25] J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Null spinning strings ,Nuclear Physics B 338\n(1990) 143–187.\n[26] U. Lindstr¨ om, B. Sundborg and G. Theodoridis, The zero tension limit of the superstring ,\nPhysics Letters B 253(1991) 319–323.\n[27] U. Lindstr¨ om, The Zero Tension Limit of Strings and Superstrings ,\n[arXiv:hep-th/9303173 ].\n[28] W. Siegel, Amplitudes for left-handed strings ,arXiv:1512.02569 .\n[29] E. Casali and P. Tourkine, On the null origin of the ambitwistor string ,\nJournal of High Energy Physics 2016(2016) 36 , [arXiv:1606.05636 ].\n[30] Monica Pate, Ana-Maria Raclariu, Andrew Strominger, Ellis Ye Yua n,Celestial Operator\nProducts of Gluons and Gravitons , [arXiv:1910.07424 ].\n[31] Erin Crawley, Noah Miller, Sruthi A. Narayanan, Andrew Stromin ger,State-Operator\nCorrespondence in Celestial Conformal Field Theory ,\nJournal of High Energy Physics 2021(2021) 132 , [arXiv:2105.00331 ].\n[32] Tim Adamo, Wei Bu, Eduardo Casali, Atul Sharma, Celestial operator products from the\nworldsheet ,Journal of High Energy Physics 2022(2022) 052 , [arXiv:2111.02279 ].\n[33] Ruth Britto, Freddy Cachazo, Bo Feng, Edward Witten, Direct Proof Of Tree-Level\nRecursion Relation In Yang-Mills Theory ,Physical review letters 94(2005) 181602,\n[arXiv:hep-th/0501052 ].\n[34] Tomasz R. Taylor, A Course in Amplitudes ,Physics Reports 691(2017) 1-37,\n[arXiv:1703.05670 ].\n[35] Lauren McGough, M´ ark Mezei, Herman Verlinde, Moving the CFT into the bulk with T¯T,\nJournal of High Energy Physics 2018(2018) 10 , [arXiv:1611.03470 ].\n[36] Alexander B. Zamolodchikov, Expectation value of composite field T¯Tin two-dimensional\nquantum field theory , [arXiv:hep-th/0401146 ].\n[37] F.A. Smirnov, A.B. Zamolodchikov, On space of integrable quantum field theories ,Nuclear\nPhysics B915(2017) 363-383, [ arXiv:1608.05499 ].\n[38] John D. Barrow, Douglas J. Shaw, The Value of the Cosmological Constant ,Gen Relativ\nGravit43(2011) 2555–2560, [ arXiv:1105.3105 ].\n[39] H. W. J. Bl¨ ote, John L. Cardy, and M. P. Nightingale, Conformal invariance, the central\ncharge, and universal finite-size amplitudes at criticalit y,Physical review letters 56(1986)\n742,\n– 20 –[40] Steven Weinberg, The cosmological constant problem ,Rev. Mod. Phys. 61-1(1989) 1-23.\n[41] Edward Witten, Anti De Sitter Space And Holography ,Adv.Theor.Math.Phys. 2(1998)\n253-291, [ arXiv:hep-th/9802150 ].\n[42] Tim Adamo, Lionel Mason, Atul Sharma, Celestial amplitudes and conformal soft theorems ,\nClass.Quant.Grav. 36(2019) 205018, [ arXiv:1905.09224 ].\n[43] Adrien Fiorucci, Daniel Grumiller, Romain Ruzziconi, Logarithmic Celestial Conformal Field\nTheory,Physical Review D 109(2024) L021902, [ arXiv:2305.08913 ].\n[44] Song He, Pujian Mao, Xin-Cheng Mao, T¯Tdeformed soft theorem ,Phys. Rev. D 107(2023)\nL101901, [ arXiv:2209.01953 ].\n– 21 –" }, { "title": "2401.17008v1.A_Unified_Three_State_Model_Framework_for_Analysis_of_Treatment_Crossover_in_Survival_Trials.pdf", "content": "A Unified Three-State Model Framework for\nAnalysis of Treatment Crossover in Survival\nTrials\nZile Zhaoa∗, Ye Lib, Xiaodong Luob, Ray Baia†\naDepartment of Statistics, University of South Carolina, Columbia, SC 29208\nbSanofi US, Bridgewater, NJ 08807\nJanuary 31, 2024\nAbstract\nWe present a unified three-state model (TSM) framework for evaluating treatment effects\nin clinical trials in the presence of treatment crossover. Researchers have proposed diverse\nmethodologies to estimate the treatment effect that would have hypothetically been observed\nif treatment crossover had not occurred. However, there is little work on understanding\nthe connections between these different approaches from a statistical point of view. Our\nproposed TSM framework unifies existing methods, effectively identifying potential biases,\nmodel assumptions, and inherent limitations for each method. This can guide researchers\nin understanding when these methods are appropriate and choosing a suitable approach for\ntheir data. The TSM framework also facilitates the creation of new methods to adjust for\nconfounding effects from treatment crossover. To illustrate this capability, we introduce a\nnew imputation method that falls under its scope. Using a piecewise constant prior for the\nhazard, our proposed method directly estimates the hazard function with increased flexibility.\nThrough simulation experiments, we demonstrate the performance of different approaches for\nestimating the treatment effects.\nKeywords: intention-to-treat, piecewise constant hazard, survival trials, treatment crossover\n∗E-mail: zilez@email.sc.edu .\n†E-mail: rbai@mailbox.sc.edu . The authors gratefully acknowledge financial support from the McCausland\nInnovation Fund and the University of South Carolina Office of the Vice President for Research ASPIRE program.\n1arXiv:2401.17008v1 [stat.ME] 30 Jan 20241 Introduction\nRandomized controlled trials (RCTs) have been commonly used to compare the survival outcomes\nbetween a treatment group and a control group. Treatment crossover within an RCT takes place\nwhen patients randomized to a treatment or control group switch to a different group from the one\nto which they were initially assigned (Ishak et al. 2014, J¨ onsson et al. 2014, Watkins et al. 2013).\nTreatment crossover can occur for a variety of reasons and can be either noninformative or\ninformative. We define noninformative crossover as any scenario where patients switch treatments\nfor reasons unrelated to treatment efficacy, disease progression, or any meaningful clinical factors.\nAn example of noninformative crossover is a study protocol which allows patients in the control\ngroup to switch to the experimental treatment once the main study endpoint (e.g. the blinded phase\nof the trial) has been reached (Daugherty et al. 2008). For example, in the CENTAUR trial of\nsodium phenylbutyrate and taurursodio (PB and TURSO) in amyotrophic lateral sclerosis (ALS),\npatients in the control group all switched to the experimental treatment after they completed the\n24-week double-blind study phase (Paganoni et al. 2022).\nHowever, treatment crossover can also be informative. Informative crossover occurs when pa-\ntients switch treatments as a result of disease progression (Ishak et al. 2014, Latimer et al. 2014).\nInformative treatment crossover is often a selective process. For example, only patients who are\ndeemed the most likely to benefit from the experimental treatment may be crossed over, while those\nwho have reached the terminal stage of the disease typically are not (Ishak et al. 2014). In many\nreal-life scenarios, crossover can also result from a mixture of both noninformative and informative\nevents, such as the other treatment arm being perceived to be better anda sudden change in the\npatient’s condition (Ishak et al. 2011).\nThese diverse scenarios invariably introduce complications in data analysis to estimate the\ntreatment efficacy. Under treatment crossover, conventional statistical techniques such as intention-\nto-treat analysis (Brody 2016) (described in Section 3) may yield biased estimations of the true\ntreatment effect. To illustrate this, we consider the following example. Progression-free survival\n(PFS) is a commonly used clinical endpoint, particularly in Oncology (Dancey et al. 2009). PFS\nmeasures the duration between randomization and disease progression or death. Post-progression\nsurvival (PPS) refers to the time after disease progression to death. Figure 1 shows that the\nsurvival time for a crossovered patient represents a mixed effect of having received both the control\n2PFS PPS\nPFS PPS\nPFS PPS\nSurvival Timecontrol treatment\nexperimental treatment\ncrossovered treatment\nRCT observed\ndifferenceTrue dif ferenceFigure 1: Diagram of PFS and PPS for patients assigned to the control, experimental, and\ncrossovered treatments\nand experimental treatments. The observed difference in survival time is biased as a result of this\ncontamination, and the true treatment effect is underestimated (Ishak et al. 2014, Latimer et al.\n2014).\nAn intuitive approach to account for treatment crossover is to exclude or censor data at the\npoint of crossover. This is also known as the per-protocol method (Brody 2016), where researchers\nonly focus on PFS when crossover is linked with the disease progression. However, the per-protocol\nmethod can be vulnerable to selection bias, as individuals who switched from the control treatment\nto the experimental treatment might have different prognoses and therefore are not “exchangeable”\nwith patients who stayed in the control group (Arnold & Ercumen 2016). Potential solutions such\nas randomized crossover designs can be applied to mitigate the effects of selection bias (McKeever\n2021, Simon & Chinchilli 2007). However, randomization – while effective in theory – is not always\npractical or ethical (Simon & Chinchilli 2007). A more pragmatic approach may be to adjust for\ncrossover effects within the statistical analysis.\nNumerous approaches for estimating the true treatment effect (or hazard ratio) in the presence\n3of treatment crossover have been proposed and examined. In Section 3, we describe several of\nthe most popular methods. Morden et al. (2011) and Latimer et al. (2018) conducted simulations\nin a variety of plausible crossover scenario settings and compared the results from these different\nmethods. Other recent reviews covering a range of therapeutic areas can be found in Watkins et al.\n(2013), Ishak et al. (2014), and J¨ onsson et al. (2014).\nDespite these reviews and comparative studies, there is little work on understanding howthese\ndifferent methods relate to each other in terms of their statistical properties. While assessing these\nmethods’ performance under various simulation settings (Morden et al. 2011, Latimer et al. 2018)\nis certainly illuminating, it is not always straightforward for practitioners to know which statistical\nmethod they should use to adjust for confounding from treatment crossover. Moreover, based on\ndomain knowledge or evidence from past trials, a researcher may also want to incorporate their\nown assumptions (e.g. an assumption that the treatment effect is constant over time) into their\nstatistical analyses (Kahan & Morris 2013, Glidden et al. 2020).\nIn this paper, we propose to use a three-state model (TSM) framework to synthesize existing\nstatistical methods in treatment crossover analysis. In addition to unifying existing methods under\none statistical umbrella, our TSM framework easily enables the creation of newmethods allowing\nthe incorporation of diverse assumptions tailored to specific scenarios. We demonstrate the utility\nof the TSM framework by introducing a novel method for estimating the hazard function. We\nfind that all methods rely on critical limiting assumptions, and the accuracy of estimating the\ntreatment effect relies on the validity of these assumptions. Our framework can guide practitioners\nin determining (or inventing) the most appropriate method to use for analyzing their data from\nclinical trials with treatment crossover.\nThe rest of this paper is structured as follows. Section 2 formally introduces the TSM frame-\nwork. Section 3 places existing methods in the context of this framework and introduces a new\nimputation method. Section 4 demonstrates the methodologies through simulation experiments.\nSection 5 concludes the paper with a brief discussion.\n2 Three-State Model Framework for Treatment Crossover\nThe main thrust of this article is to introduce a broad statistical framework for understanding the\nassumptions, limitations, and relationships of different methods for estimating treatment effects\n4Figure 2: Diagram of the three-state model\nin RCTs with treatment crossover. We first introduce the TSM framework and discuss treatment\neffect estimation under this framework.\n2.1 Statistical Framework\nIn an RCT, we define the entry as the time that a patient enters the trial. Let Tbe the time from\nentry to the event of interest (e.g. death), and let Ube the time from entry to the time of crossover.\nPatients can arrive at the event of interest in two potential pathways. The first is to go to the\nevent directly from entry without crossover, characterized by a hazard function λ1(t). The second\npath is where a crossover point (e.g. re-randomization, progression, or end of the double-blind\nphase) occurs. In this scenario, we let λx\n2(t|u) denote the hazard function from crossover to the\nevent. The crossover path is completed with the path connecting the entry point and the crossover\npoint, and we represent the hazard function for this path as λ3(u). Figure 2 depicts these paths.\nFormally, the three hazard functions are defined as follows:\n(1) hazard from entry to event without crossover:\nλ1(t) =P(T=t|T≥t, U≥t), t > 0. (1)\n(2) hazard from crossover to event (if there is crossover):\nλx\n2(t|u) =P(T=t|T≥t, U=u), t > u. (2)\n5(3) hazard from entry to crossover:\nλ3(u) =P(U=u|T≥u, U≥u), u > 0. (3)\nWe assume U=∞when U > T , i.e. crossover occurring after the event cannot be observed or is\nof no interest. Often times, we are interested in the survival function eS(t) =P(T > t ). Routine\ncalculations give\neS(t) =P(T > t, U > t ) +P(T > t≥U)\n=S1(t)S3(t) +Zt\n0P(T > t, T ≥u, U=u)du\n=S1(t)S3(t) +Zt\n0expn\n−Zt\nuλx\n2(s|u)dso\nλ3(u)S1(u)S3(u)du, (4)\nwhere Sj(t) = exp {−Rt\n0λj(s)ds},j= 1,3. We can correspondingly define the density function\nef(t) =−eS′(t), the cumulative hazard function eΛ(t) =−log{eS(t)}, and the hazard function eλ(t),\neλ(t) =ef(t)/eS(t). (5)\nIt is clear from (4) that ef(t),eΛ(t), andeλ(t) depend on the three hazard functions λ1,λx\n2, and λ3\ndefined in (1)-(3). Various configurations of λx\n2result in distinct forms of crossover. In what follows,\nλ(z) denotes a generic hazard function of z. We focus on the following two types of crossover which\nare frequently observed:\n1.Markov crossover . The hazard function after crossover only depends on the time from\nentry to event, i.e.\nλx\n2(t|u) =λ(t). (6)\nLuo et al. (2019) termed this as Markov crossover.\n2.Semi-Markov crossover . The hazard function after crossover depends only on the duration\nof time between the crossover point and the event, i.e.\nλx\n2(t|u) =λ(t−u). (7)\nIn this case, patients will experience a new hazard function beginning from the crossover\npoint. Semi-Markov crossover is frequently observed as a result of disease progression and/or\nre-randomization (Luo et al. 2019).\n62.2 Treatment Effect\nLetAbe the control group and Bbe the treatment group. Without loss of generality, we will focus\non the case where crossover can only occur in the control patients, and the crossovered patients\nbegin their new treatment from the time of crossover. We begin by establishing the following\nnotation, where the hazard function is defined as in (5):\n•eλA(t): the overall hazard function for the treatment group;\n•eλB(t): the overall hazard function for the control group patients if crossover to treatment is\nnotallowed;\n•eλB∗(t): the hazard function for the control group patients if crossover to treatment is allowed.\nThe true treatment efficacy can be quantified by comparing eλA(t) witheλB(t), rather than with\neλB∗(t). The treatment effect we seek to estimate is therefore the hazard ratio (HR),\nHR(t) =eλA(t)/eλB(t). (8)\nAn HR( t) exactly equal to one indicates equal efficacy of the experimental and control treatments.\nOn the other hand, an HR( t) less than one favors the experimental treatment, while an HR( t)\ngreater than one favors the control.\nIn our TSM framework, we have eλB∗(t) =H(t;λ1, λ2∗, λ3) andeλB(t) =H(t;λ1, λ2, λ3), where\nHdenotes a function of three hazards, λ2∗denotes the hazard function if a crossover point occurs\nandthe control patient switches to the experimental treatment, and λ2is the hazard function if\nthe crossover point occurs but the control patient remains in the control group.\nLetcbe an indicator for whether the patient switches to the experimental treatment after a\ncrossover point ( c= 1 if “yes,” c= 0 if “no”). The hazard functions λ2∗andλ2have the forms,\nλ2∗(t|u) = λx\n2(t|u, c= 1),\nλ2(t|u) = λx\n2(t|u, c= 0),(9)\nwhere λx\n2is defined as in (2). In the case where a proportion of the patients remain in the control\ngroup after the crossover point occurs, we can estimate λ2in (9) using various statistical methods.\nHowever, if all patients switch to the treatment after the crossover point, λ2will become non-\nidentifiable because there is no data on patients continuing the control treatment. In this case,\nadditional assumptions are required to ensure identifiability.\n73 Methods Within the TSM Framework\nOur TSM framework subsumes many existing methods for estimating treatment effects (8) in\nthe presence of treatment crossover. Furthermore, new methods can also be introduced under\nthis framework. In this section, we synthesize several existing methods that fall under the TSM\numbrella and introduce a new imputation method as part of the TSM toolkit. Table 1 provides a\nsummary of all the methods we consider.\nThe best method(s) to use in practice depends on the validity or plausibility of the researchers’\nassumptions. If Markov crossover (6) can be reasonably assumed, then one of the Markov models in\nTable 1 can be used. On the other hand, if semi-Markov crossover (7) is more reasonable, then one\nof the semi-Markov models in Table 1 can be pursued. When crossover is linked specifically to re-\nrandomization or disease progression, the semi-Markov assumption (7) is especially plausible (Luo\net al. 2019). Within each of these model classes, further assumptions can be imposed depending\non the specific circumstances of the RCT.\n3.1 Intention-to-treat\nNumerous authors adopt a practical approach by applying an intention-to-treat (ITT) analysis.\nIn the ITT method, analysis of patients is based solely on the treatment group to which they\nwere originally randomized, regardless of their adherence with the entry criteria and regardless\nof protocol deviation or participant withdrawal (Brody 2016). The fundamental crux of an ITT\nanalysis is to use the complete dataset of patients who were subjected to randomization at the very\nbeginning of the RCT (Brody 2016).\nITT analysis results should be reported in all cases. This is essential because ITT upholds the\nintegrity of randomized trials by analyzing participants according to their original assigned groups,\nregardless of their adherence or completion of the allocated treatment (Brody 2016). Although\nthe ITT analysis is generally acceptable, it potentially underestimates the true policy effectiveness\nof a treatment (White 2005). For example, in fatal conditions like ALS, it is often recommended\nto allow patients to switch to experimental treatments at a certain time point (Paganoni et al.\n2022). Therefore the estimated treatment effect from the ITT analysis may be diluted (White\n2005, Paganoni et al. 2022).\nIn the TSM framework, ITT actually compares eλA(t) witheλB∗(t) instead of with eλB(t) as in\n8Table 1: Table of different methods falling under the TSM framework\nMethod AbbreviationAssumed\ncrossover typeUse data after\ncrossover?\nIntention-to-treat ITT N/A Yes\nCensor-at-switching CAS Markov No\nExclude-at-switching EAS Markov No\nTreatment as time-dependent variable TTDV Markov Yes\nRank preserving structural failure time RPSFT Semi-Markov Yes\nInverse-probability-of-censoring weighting IPCW N/A No\nBayesian imputed multiplicative method BIMM Semi-Markov Yes\n(8). In other words, ITT does not make any adjustment for treatment crossover, and it maintains\nthe most conservative approach in treatment crossover analysis.\n3.2 Per-Protocol\nA per-protocol (PP) analysis entails evaluating patients based only on the treatment they were\nactually administered, rather than the one to which they were originally assigned through ran-\ndomization (Brody 2016). In this method, patients who switch treatments are censored or simply\nexcluded at the time of crossover. We refer to these two PP approaches as censor-at-switching\n(CAS) and exclude-at-switching (EAS) respectively.\nThe PP method is widely used as a sensitivity analysis to show the robustness of the ITT\nanalysis. However, in contrast to ITT analysis, which makes complete use of the patients’ informa-\ntion and guarantees comprehensive balance between the control group and the treatment group,\nPP methods may introduce selection bias due to censoring or excluding parts of the balanced\ndata (Brody 2016). The treatment effect estimated by CAS and EAS implicitly assumes that the\ncrossovered patients are exchangeable with those that remained in the control group.\nFor the patients in the control group, recall that Tis the time to event and Uis the time to\ncrossover, where U=∞ifU > T (since crossover after the event cannot be observed). Assuming\nthat patients drop out of the clinical trial at time V, we have the event indicator δ=I(T≤U, T≤\nV). Unlike the traditional ITT approach, patients who switch to the experimental treatment are\nalsoconsidered as censored under the PP method.\n9Due to its reliance solely on data before the crossover point, CAS and EAS only provide\nestimates of the hazard function λ1(t) before crossover (1). However, researchers are typically\ninterested in estimating the overall hazard function eλB(t) =P(T=t|T≥t). Clearly, λ1(t)̸=\neλB(t) unless λx\n2(t|u) =λ1(t). If this holds, then we have a Markov model where the hazard\nfunctions before and after the crossover are the same. Therefore, the CAS and EAS methods\nimplicitly assume Markov crossover (6).\n3.3 Treatment as Time-Dependent Variable\nBuilding upon the Cox proportional hazards (PH) model (Cox 1972), we can introduce the treat-\nment assignment as a covariate that changes over time (White et al. 1997, Morden et al. 2011).\nThis allows for evaluation of the influence of the treatment that a patient actually undergoes. This\nmodel can be represented as a Cox PH model,\nλ(t|X(t)) =λBL(t) exp{βX(t)}, (10)\nwhere λBL(t) represents the baseline hazard function, and we assume X(t) = 0 when the patient\nis in the control group at time tandX(t) = 1 when the patient is in the experimental treatment\ngroup at time t. We refer to (10) as a “treatment as time-dependent variable” (TTDV) model.\nFor the patients in the control group, using the time-dependent covariate I(t > u ), we have the\nhazard function,\nP(T=t|T≥t, U=u) =λBL(t) exp{βI(t > u )}. (11)\nThis implies that the hazard function before crossover is λ1(t) =λBL(t) and the hazard function\nafter crossover is λx\n2(t|u) =λBL(t) exp( β). From (11), we see that TTDV is a Markov model (6)\nsince the hazard function after crossover does not depend on the crossover time u. The TTDV\nmodel (11) also implies that the hazard ratio HR( t) in (8) is constant.\nSimilar to the previously mentioned PP method, the TTDV approach has the potential to dis-\nrupt the randomization assumption and may consequently introduce selection bias when switching\nis linked to the patient’s prognosis (White et al. 1999).\n3.4 Rank Preserving Structural Failure Time Models\nRobins & Tsiatis (1991 a) proposed rank preserving structural failure time (RPSFT) models to\n10estimate the true treatment effect under an accelerated failure time (AFT) structural model. The\ntime at which an event is observed in a patient can be used to infer the counterfactual time at\nwhich the same event would have been observed if the crossovered patient had notundergone any\nexperimental treatment. These models are called “rank preserving” because they assume that if\ntwo patients iandjhad the same treatment, with patient iexperiencing the event before patient j,\nthe same order for the time-to-event would hold if both patients were given an alternative treatment\n(Robins & Tsiatis 1991 a).\nFor individuals that switched to the experimental treatment, let T∗andUbe the event time\nand the crossover time respectively. Letting Tdenote the counterfactual survival time (i.e. the\nsurvival time that the patient would have had if they had remained in control group), the RPSFT\nmethod posits the model,\nT=U+e−ϕ0(T∗−U), (12)\nwhere eϕ0is a so-called acceleration factor indicating the degree by which a patient’s expected time\nto an event is extended due to the experimental treatment. If eϕ0>1, this suggests a positive\ntreatment effect, while the unusual eϕ0<1 represents a negative treatment effect. In either case,\nthe treatment effect is assumed to be constant.\nSimilar to the TTDV method (11), we can define a time-dependent covariate I(t > U ). Assume\nthere is a constant treatment effect for patients who are switched to a different group from the one\nto which they were originally assigned. Then (12) can be rewritten as\nT=ZT∗\n0exp(ϕ0I(t > U ))dt. (13)\nThe formulation (13) also applies to patients who did not switch treatments. In that case, we can\nsetU=T∗for patients remaining in the control group or U= 0 for the patients remaining in the\ntreatment group respectively for the whole duration of the study.\nThe RPSFT method is inherently a semi-Markov model (7). To better understand this, let us\nagain split the hazard function after the crossover point into two parts,\nλx\n2(t|u) =\n\nλ2(t−u), if patient remains in control group,\nλ2∗(t−u),if patient switches treatment.\nBased on (12), it is equivalent to connect the hazard functions λ2∗andλ2in (9) via\nλ2∗(t) =e−ϕ0λ2(te−ϕ0), (14)\n11such that they have the same survival probability at time tandte−ϕ0, i.e.\nSλ2∗(t) =Sλ2(te−ϕ0)\nIt should be noted that the model (14) is not identifiable if all of the patients in the control group\ncross over to the experimental treatment group. In this scenario, the hazard function λ2is not\nestimable since there is no available data for patients who continued the control treatment. To\nmake the model identifiable, one typically needs to introduce an additional assumption,\neλA(t) =e−ϕ0eλB(te−ϕ0). (15)\nThis assumption implies that the constant treatment effect is the same for all patients, regardless\nof when they receive it.\n3.5 Inverse-Probability-of-Censoring Weighting\nThe RPSFT method incorporates all available patient data and adjusts the survival time to account\nfor what might have occurred if the patients who switched treatments had stayed in the control\ngroup. On the other hand, the inverse-probability-of-censoring weighting (IPCW) approach (Curtis\net al. 2007) focuses on the survival time before the crossover by marking patients as censored at the\npoint of treatment switching in the analysis. As previously mentioned for the PP method (Section\n3.2), this introduces bias because patients whose event time is censored tend to have systematic\ndifferences in prognosis compared to those whose who do not switch treatments.\nTo correct this bias, patients in the control group who did notswitch to the experimental\ntreatment can be assigned weights to account for the absence of data. In the IPCW method,\nthe bias caused by informative crossover is adjusted by assigning each patient a weight that is\nthe reciprocal of their estimated probability of not experiencing censoring at a specific time point\n(Robins & Finkelstein 2000). IPCW estimates the likelihood of patients switching treatments based\non their individual baseline characteristics and time-dependent covariates. This estimation is often\ndone using a logistic regression (Curtis et al. 2007, Robins & Finkelstein 2000).\nThe IPCW approach assumes no unmeasured confounders at the given time of crossover, making\nthe censoring noninformative after inverse-probability weighting (Curtis et al. 2007, Robins &\nFinkelstein 2000). In essence, this assumption implies that if one has adequately considered and\ncontrolled for all the relevant covariates that could influence both the treatment assignment and\n12the outcome, the results obtained from this analysis will provide a less biased estimate of the true\ntreatment effect. However, if there areunmeasured or unaccounted-for factors that confound the\nrelationship, the results may be biased. Ensuring that this assumption is reasonably met is crucial\nto making valid inferences and drawing accurate conclusions from observational data. In practice,\nresearchers often use techniques like propensity score weighting or matching to address potential\nconfounding covariates (Austin 2011).\nRecall that Tis the time to event, Uis the crossover time, and let Vbe the censoring time in the\ndata. The event indicator in the IPCW approach is δ=I(T≤U, T≤V). Let Y= min( T, U, V ).\nThe IPCW approach further makes the assumption that TandUare conditionally independent\ngiven some (potentially time-dependent) covariates X. If this assumption holds, then\nP(Y=t, δ= 1|X) =P(T=t|X)×P(U∧V≥t|X).\nIf we can reliably estimate the censoring weight function W(t, x) =P(U∧V≥t|X=x) as\ncW(t, x), then the overall hazard function eλB(t|X=x) =P(T=t|T≥t, X) can be estimated\nvia inverse-weighting of the observed event times as\nbλB(t|X=x) =Pn\ni=1I(Yi=t, δi= 1, Xi=x)/cW(t, X i)\nPn\ni=1I(Yi≥t, X i=x)/cW(t, X i).\nHowever, if the conditional independence assumption does nothold, then this method estimates\nλ1(t|X) =P(T=t|T≥t, U≥t, X), t > 0,\nwhich is the hazard function before crossover. This hazard function in general is not equal to the\noverall hazard function, i.e.\nP(T=t|T≥t, U≥t, X)̸=P(T=t|T≥t, X), t > 0.\n3.6 New Method: Bayesian Imputed Multiplicative Method\nAs discussed previously, the TSM framework not only unifies existing approaches like ITT, CAS,\nEAS, TTDV, RPSFT, and IPCW (see Table 1), but it alsofacilitates new methods to adjust for\nconfounding from treatment crossover. In some treatment crossover scenarios, a researcher may\nwant to invent a new method for treatment effect estimation tailored to a specific scenario or set of\nassumptions. In this section, we propose a new model under the TSM framework that is particularly\n13well-suited when the crossover is informative and linked to the occurrence of a disease-related event\nlike disease progression.\nAs before, let T∗andUbe the event time and the crossover time respectively for individuals\nthat switched to the experimental treatment. Let Tdenote the counterfactual survival time. We\nassume semi-Markov crossover (7), i.e.,\nλx\n2(t|u) =\n\nλ2(t−u), if patient remains in control group,\nλ2∗(t−u),if patient switches to treatment.\nIf we assume an AFT model λ2∗(t) =e−ϕ0λ2(te−ϕ0), then we have T=U+e−ϕ0(T∗−U), which is the\nRPSFT method (12). Alternatively, we can assume a multiplicative hazard model, λ2∗(t) =eβλ2(t).\nThen the counterfactual survival time Tcan be expressed as\nT=U+S−1\n2[{S2∗(T∗−U)}], (16)\nwhere S2∗is the survival function based on the hazard function λ2∗, and S−1\n2is the inverse function\nof the survival function S2based on the hazard function λ2. It should be stressed that in general, the\ntreatment effect after crossover λ2∗(t)/λ2(t) is not the same the overall treatment effect eλA(t)/eλB(t).\nLetπ2represent the percentage of patients who switched to the experimental treatment. As\nlong as π2<1, the model (16) is identifiable under the semi-Markov assumption. When π2= 1, the\nhazard function λ2cannot be directly estimated since there is no data to estimate λ2(t). To make\nthe model identifiable when π2= 1, we need to impose an additional assumption. Specifically, we\nassume that the treatment effect after crossover is the same as the overall treatment effect between\nthe treatment group and the control group when crossover is not allowed, i.e.,\nλ2∗(t) =eβλ2(t) and eλA(t) =eβeλB(t),ifπ2= 1. (17)\nIn other words, if there is 100% crossover, e.g. in the CENTAUR trial (Paganoni et al. 2022)\ndescribed in Section 1, then we need to find an estimate of eβthat meets this assumption (17). In\npractice, it may not be easy to verify this assumption. Thus, achieving accurate estimates of the\noverall treatment effect (8) might be very difficult under 100% crossover.\nTo implement our method, we model the hazard functions as piecewise constant functions taking\nthe form,\nλ(t) =JX\nj=1ξjI{t∈(sj−1, sj]}, (18)\n14where ξjis the hazard in the time interval ( sj−1, sj] and 0 = s0< s 1< s 2<···< sJ<∞, where\nsJis larger than the largest observed time in the study. Note that the cut points sj’s for different\nhazard functions λ1(t),λ2(t),λ2∗(t) and λ3(t) are the same, but the hazard constants are different.\nTo estimate these hazard functions, we adopt a Bayesian approach and endow the hazards\nξj’s in (18) with weakly informative independent Gamma priors (Ibrahim et al. 2001). We call\nour approach the Bayesian imputed multiplicative method (BIMM). Assuming no covariates, the\nBIMM method estimates the log hazard ratio βand corresponding variance as follows.\nWhen π2<1:\ni. Use Markov chain Monte Carlo (MCMC) to estimate λ1(t), λ2(t), λ2∗(t), λ3(t) as in (1)-(3)\nusing (18) with Gamma priors on the step heights ξj’s.\nii. For each MCMC sample k= 1, . . . , K :\na. Compute the counterfactual survival time using (16) for the control patients who switch to\ntreatment.\nb. Fit the Cox model comparing the treatment group data with the adjusted control group\ndata. The Cox model will give the estimate of log hazard ratio and the corresponding\nmodel-based variance, denoted by βkandvk.\niii. Summarize the Kfitted Cox models so that the point estimate of βis the mean of βk,k=\n1, . . . , K , and the variance estimate is the sum of the mean of the vk’s and the sample variance\nof the βk’s.\nWhen π2= 1:\ni. Set m= 0 and initialize β(m).\nii. Use MCMC to estimate λ1(t), λ2∗(t), λ3(t) as in (1)-(3) using (18) with Gamma priors on the\nstep heights ξj’s. Calculate λ2(t) asλ2(t) =λ2∗(t)/eβ(m).\niii. For each MCMC sample k= 1, . . . , K :\na. Compute the counterfactual survival time using (16) for the control patients who switch to\ntreatment.\n15b. Fit the Cox model comparing the treatment group data with the adjusted control group\ndata. The Cox model will give the estimate of the log hazard ratio β(m+1)\nk and the corre-\nsponding model-based variance v(m+1)\nk .\niv. Summarize the Kfitted Cox models so that the point estimate of β(m+1)is the mean of β(m+1)\nk ,\nk= 1, . . . , K , and the variance estimate is the sum of the mean of the v(m+1)\nk ’s and the sample\nvariance of the β(m+1)\nk ’s.\nv. Repeat steps (ii)-(iv) until the sequence {β(m), m≥0}converges. In practice, we use the\nconvergence criterion |β(m+1)−β(m)|<10−6.\n4 Simulations\nTo illustrate our TSM framework, we conducted simulation experiments in scenarios where treat-\nment crossover was allowed. The first experiment mimics a clinical trial design, where the hazard\nfunctions were specified in relatively simple forms. In the second experiment, we re-engineered\nindividual patient data from a real clinical trial and then used piecewise exponential models to es-\ntimate the corresponding hazard rates. We then simulated data based on these hazard rates. Each\nexperiment was repeated for 2000 replications using the seven methods described in Section 3 and\nreported in Table 1: ITT (Section 3.1), CAS (Section 3.2), EAS (Section 3.2), TTDV (Section 3.3),\nRPSFT (Section 3.4), IPCW (Section 3.5), and BIMM (Section 3.6). For each of these methods,\nwe estimated both the treatment effect and the 95% confidence interval for the treatment effect.\nFor ITT, CAS, EAS, TTDV, and IPCW, we used a Cox PH model that was fit with the R\npackage survival (version 3.5-7, available on the Comprehensive RArchive Network (CRAN)).\nFor RPSFT, we used the Rpackage rpsftm (version 1.2.8 on CRAN) to estimate ϕ0in (12). We then\nused a Cox PH model to calculate the log hazard ratio bβbased on the observed survival times in the\ncontrol group and the counterfactual survival times in the treatment group adjusted by plugging\nin the estimated acceleration factor ˆϕ0in equation (12). To estimate the RPSFT standard error\nforbβ, we followed Bennett (2018) and Robins & Tsiatis (1991 b) and used se( bβ) =|bβ|/p\nχ2\nitt, where\nχ2\nittrepresents the chi-squared test statistic from the log-rank test for ITT analysis comparing the\noriginal data between control group and treatment group.\nFor IPCW, the probability of notcrossing over was estimated by a logistic regression in the\n16control group, as suggested by Ishak et al. (2014). It is important to note that IPCW method is\nnumerically unstable when the proportion of switching is very high (e.g. 100% crossover). In this\ncase, the inverse probability of not switching may be extremely large or nonestimable. Therefore,\nwe set an upper bound on the inverse probability to be 10 in order to avoid extremely large weights.\nFinally, the BIMM method was implemented as described in Section 3.6 using Stan (Carpenter\net al. 2017) interfaced with Rthrough the package Rstan (version 2.32.5 on CRAN). We specified\nthe priors on the hazards ξj’s in (18) to be Gamma(1, 2). We ran eight MCMC chains of 2000\niterations each and discarded the first 500 iterations of each chain as burnin, leaving us with a total\nof 12,000 MCMC samples with which to estimate the hazard functions. In Experiment 1, each of\nthe hazards ξj’s corresponded to the time interval ( j−1, j],j= 1, ...,5. In Experiment 2 based\non reverse-engineered data from a real clinical trial, we evenly divided the time intervals into four\npieces from entry to the maximum observed survival time.\n4.1 Experiment 1: Simulated Trials Under 75% and 100% Crossover\n4.1.1 Simulation Settings\nIn our first experiment, data were generated to mimic a clinical trial design in Oncology. We\nsimulated data on n= 400 total patients, with 200 patients initially belonging to the control group\nand 200 patients belonging to the experimental treatment group at randomization.\nWe designed a survival trial based on the three-state model. All the hazard rates were yearly\nhazard rates with three pieces corresponding to years 0-1, 1-2 and 2+. Let λ(·) =c(a1, a2, a3) denote\nhazard rates for years 0-1, 1-2 and 2+ as a1,a2anda3respectively. For the control patients, we\nassumed that the crossover happens at the time of disease progression. The crossover is assumed\nto be semi-Markov, i.e. λx\n2(t|u) =λ2(t−u). We specified the hazard functions λ1(t) and λ3(t)\nasλ1(·) =c(0.2,0.2,0.25) and λ3(·) =c(0.4,0.4,0.4) respectively. For the control patients who\nswitched to the experimental treatment, their hazard rate after crossover was λ2∗(·) = 0 .8×λ1(·).\nFor the control patients who (potentially) remained in the control group, their hazard rate was\nλ2(·) = 1 .5×λ1(·). As such, patients who switched to the experimental treatment had lower hazard\nrates than those who remained in the control after disease progression.\nFor the patients allocated to the treatment arm, no crossover was allowed, so we assumed a\npiecewise exponential model with hazard rates eλA(·) =c(0.12,0.12,0.15) for the survival time.\n17In other words, the HR between eλA(·) and λ1(·) was 0 .6. Finally, we assumed that the yearly\ncensoring rate was 0 .05. The study had a two-year recruitment period with a uniform accrual rate\nto recruit 400 patients, and the study would be read out at 5 years after the randomization of the\nfirst patient.\nLetπ2denote the proportion/chance that the control patients crosses over to the experimental\ntreatment; 1 −π2denote the proportion/chance that the control patients remains in the control\ngroup. We used π2= 0.75 or π2= 1 in our simulations. Based on the above assumptions, if\nπ2= 0.75, the HR based on the ITT analysis would be 0 .636 favoring the treatment group, and\nthe power would be 85 .4% based on a two-sided log-rank test with significance level α= 0.05.\nMeanwhile, if 100% of the control patients switched to the treatment after disease progression\n(π2= 1), the HR based on the ITT analysis and the power would be 0 .685 and 70 .1% respectively.\nIt is important to note that the hazard ratio HR( t) is not a constant over time. Rather, these HRs\nwere obtained by fitting the data as if the proportional hazards assumption holds. This approach\nreflects the most common way of reporting the results in clinical study reports. Therefore, we used\nthe same approach in this experiment.\nFor comparison, if control patients were notallowed to switch to the treatment group after\ndisease progression, the HR would be 0 .513. This is the truetreatment effect when crossover is not\nallowed. The power in this case would be 99 .6%. Apparently, without adjusting for the crossover\nin the control group, we would have a biased estimate of the true treatment effect. All the results\nin the preceding and current paragraphs were obtained using the Rpackage PWEALL (version 1.4.0\non CRAN).\n4.1.2 Simulation Results\nIn each of the 2000 replicates, we recorded the estimated treatment effect dHR and the nominal 95%\nconfidence interval for the HR using the methods in Table 1. Figure 3 plots the mean estimated\ntreatment effect and the mean 95% confidence intervals (CIs) from 2000 replications. The mean\n95% CIs were calculated by taking the average of the left and right endpoints of the 95% CIs from\nall 2000 replications. The true HR is depicted as a dashed blue vertical line.\nThe left plot in Figure 3 shows that under 75% crossover, our proposed BIMM method (Section\n3.6) provided improved accuracy in estimating the true HR. In particular, the estimated treatment\neffect by BIMM displayed the least amount of mean bias, with confidence intervals that generally\n18BIMMIPCWRPSFTTTDVEASCASITT\n0.4 0.6 0.8\nHazard Ratio(a) 75% crossover after progression\nBIMMIPCWRPSFTTTDVEASCASITT\n0.2 0.4 0.6 0.8\nHazard Ratio (b) 100% crossover after progression\nFigure 3: Simulation results in Experiment 1 averaged across 2000 replicates. The dotted blue line\nis the true treatment effect (HR true= 0.513).\ncovered the true HR. RPSFT performed nearly as well as BIMM in terms of average bias. However,\nthe 95% CIs from RPSFT also tended to be the widest of all the methods.\nUnder 100% crossover, the bias of the treatment effect estimation depends on the validity of the\nassumptions imposed to ensure identifiability. We need to emphasize the significant challenge in\nverifying these assumptions because of the lack of data on patients remaining in the control group\nafter progression. By incorporating an assumption of constant treatment effect (17) before and\nafter progression, the right plot in Figure 3 demonstrates that BIMM and RPSFT also effectively\nreduced bias, giving mean estimates closest to the true HR. However, the 95% CIs for RPSFT were\nagain quite wide. BIMM provided both greater accuracy and decreased variability in treatment\neffect estimation.\nTable 2 shows the average bias (Bias = dHR−HR true), empirical standard error (SE), average\nmean-squared error (MSE = Bias2+ SE2), and empirical coverage probability (ECP) for each of\nthe seven methods. The MSE balances the trade-off between bias and variance. The ECP is the\nproportion of 95% CIs that contained the true HR. Table 2 shows that in both scenarios of 75%\nand 100% crossover, our new BIMM method had the lowest average bias and the lowest average\nMSE. RPSFT had the highest ECP for both 75% crossover and 100% crossover. However, RPSFT\nhad the highest average SE of all methods, and as depicted in Figure 3, RPSFT’s 95% CIs were\n19Table 2: Simulation results for Experiment 1 based on 2000 replications. The table displays the\naverage Bias, SE, and MSE from all Monte Carlo replicates, while the ECP is the percentage of\n95% CIs that contained the true HR.\n75% crossover 100% crossover\nMethod Bias SE MSE ECP (%) Bias SE MSE ECP (%)\nITT 0.131 0.097 0.027 68.1 0.180 0.106 0.043 54.2\nCAS -0.092 0.064 0.013 71.8 -0.132 0.061 0.021 51.6\nEAS -0.105 0.067 0.016 69.6 -0.181 0.059 0.036 28.3\nTTDV 0.087 0.094 0.016 85.4 0.138 0.111 0.031 74.4\nRPSFT 0.031 0.127 0.017 96.5 0.046 0.151 0.025 97.6\nIPCW -0.141 0.060 0.024 45.5 -0.179 0.082 0.039 53.0\nBIMM 0.026 0.091 0.009 94.9 0.042 0.090 0.010 82.3\ngenerally very conservative. BIMM had the second highest ECP with narrower CIs than RPSFT.\nIn particular, in the 75% crossover situation, the coverage for BIMM (94.9%) was very close to the\nnominal level, indicating good performance for uncertainty quantification.\n4.2 Experiment 2: An Experiment Based on Data From a Clinical\nTrial\nOn August 2014, researchers submitted a new drug application (NDA) seeking approval of lenva-\ntinib (LENVIMA) using data from Study E7080-G000-303 (SELECT) (National Library of Medicine\n2019). LENVIMA is an anti-cancer medication for the treatment of certain types of thyroid cancer.\nIn SELECT, a total of 392 eligible patients were randomly assigned in a 2:1 ratio to the LENVIMA\narm and the placebo arm. They were either given LENVIMA at a dose of 24 mg through continu-\nous once-daily oral administration or a matching placebo. Patients received the blinded study drug\nonce daily until any of the following events occurred: confirmed disease progression or withdrawal\nof consent. Patients in the placebo group, upon having their disease progression confirmed, had the\noption to request entry into the open-label LENVIMA treatment phase and receive experimental\ntreatment (National Library of Medicine 2019). After the double-blind phase, patients who re-\n20Figure 4: Fitted PFS and OS curves, where the solid lines are re-engineered survival curves from\nthe originally reported K-M curves, and the dotted lines are based on three-state models fit to the\nre-engineered data.\n21ceived LENVIMA and had not encountered disease progression had the option to request ongoing\nopen-label LENVIMA treatment at the same dosage, as determined by the clinical judgment of\nthe investigator (National Library of Medicine 2019).\nThe review document for SELECT provides the Kaplan-Meier (K-M) curves for progression-free\nsurvival (PFS), overall survival (OS), and survival after crossover (National Library of Medicine\n2019). Based on these curves, we used the Rpackage IPDfromKM (version 0.1.10 on CRAN) to\nreconstruct the individual patient data. We then used piecewise exponential models to estimate\nthe hazard rates. We determined the three underlying hazard functions in the three-state models\n(for treatment and control separately) through trial and error, so that the resulting PFS and OS\ncurves largely matched the reported K-M curves in SELECT. Figure 4 plots the re-engineered\nsurvival curves from the originally reported K-M curves (solid lines) vs. the fitted PFS and OS\ncurves (dotted lines).\nIf we believe that the estimated parameters reflect the underlying truth, then we can design\na simulation study based on these parameters. In our simulation study, we assumed that the\nrecruitment was completed in 6 months with a uniform accrual rate. Out of the 392 patients, 261\nwere randomized to LENVIMA and 131 to placebo (roughly a 2:1 ratio). After randomization,\npatients were followed to the study cut-off at 24 months or to the censoring time, which was\nassumed to follow an exponential distribution with monthly hazard rate of 0 .004. Just like the real\nLENVIMA study, we assumed 83% patients in the placebo group switched to open-label LENVIMA\nwhile the rest remained in the placebo group.\nUnder the settings of our simulation, the HR based on the ITT analysis was 0 .859 favoring the\ntreatment group, and the power was 15 .7% based on a two-sided log-rank test with significance\nlevel α= 0.05. However, if placebo patients were notallowed to switch to open-label LENVIMA\nafter disease progression, then the hazard ratio would drop to 0 .236. This is the true treatment\neffect when crossover is notallowed. The power in this case would be 100%. As before, these\nresults were obtained using the Rpackage PWEALL (version 1.4.0 on CRAN).\nOur simulation results averaged across 2000 replicates are plotted in Figure 5. A lower estimated\ntreatment effect indicates a larger discrepancy between the LENVIMA and the placebo groups,\nand thus, suggests greater efficacy of the drug. Figure 5 shows that the estimated HR from the\nunadjusted ITT analysis deviates significantly from the true HR. After adjusting for crossover,\nall other methods concluded greater drug effectiveness than that determined by the ITT analysis.\n22BIMMIPCWRPSFTTTDVEASCASITT\n0.5 1.0 1.5\nHazard RatioFigure 5: Simulation results from the reverse-engineered LENVIMA data. The dotted blue line\ndenotes true treatment effect when crossover is not allowed.\nFigure 5 shows that the PP methods (CAS and EAS) and the IPCW method performed the best\non average, with the lowest magnitude of average bias. In this particular experiment with re-\nengineered data from the LENVIMA trial, the treatment effect could be best estimated using only\ndata before crossover. On real clinical data, however, researchers should assess the appropriateness\nof excluding or censoring data after crossover, as well as the implicit assumptions made by the\nCAS, EAS, or IPCW methods.\n5 Discussion\nWe have proposed the TSM framework which unifies existing and new methods for treatment effect\nestimation in RCTs with treatment crossover. A number of existing methods (Table 1) are covered\nin this framework, which allows delineation of these methods in terms of underlying assumptions\nand limitations. The TSM framework also allows for the creation of newmethods that incorporate\ndiverse assumptions tailored to specific scenarios, thus enhancing its applicability in RCTs. A new\nimputation method for modeling the counterfactual survival time under treatment crossover was\nproposed and illustrated under this framework. By adopting a statistical perspective, our work\ncomplements and augments the comparative numerical studies of different methods by Morden\n23et al. (2011) and Latimer et al. (2018).\nModeling the data after treatment crossover to estimate the true treatment effect is essentially\nan imputation problem. In general, post-crossover data are biased, and different methods treat the\nbiased data in different ways. The best choice of imputation method(s) depends on the assumptions\nabout the types of crossover and the nature of the data. This is especially the case when 100%\nof patients in the control group cross over to the experimental treatment, and thus, the true\ntreatment effect is not identifiable without more stringent assumptions. Apart from the Markov\n(6) and semi-Markov (7) crossover assumptions, other implicit assumptions for each method also\nneed to be carefully assessed (e.g. the assumption of a constant treatment effect in TTDV and\nRPSFT, the assumption of conditional independence of time-to-event and time-to-crossover given\ncovariates in IPCW, etc.). It is important for researchers to be transparent about potential sources\nof bias inherent in their assumptions and properly report the limitations of their approaches.\nReferences\nArnold, B. F. & Ercumen, A. (2016), ‘Negative control outcomes: A tool to detect bias in random-\nized trials’, JAMA 316(24), 2597–2598.\nAustin, P. C. (2011), ‘An introduction to propensity score methods for reducing the effects of\nconfounding in observational studies’, Multivariate Behavioral Research 46(3), 399–424.\nBennett, I. (2018), ‘Accounting for uncertainty in decision analytic models using rank preserving\nstructural failure time modeling: Application to parametric survival models’, Value in Health\n21(1), 105–109.\nBrody, T. (2016), Chapter 8 - Intent-to-treat analysis versus per protocol analysis, inT. Brody,\ned., ‘Clinical Trials (Second Edition)’, Academic Press, Boston, pp. 173–201.\nCarpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M.,\nGuo, J., Li, P. & Riddell, A. (2017), ‘Stan: A probabilistic programming language’, Journal of\nStatistical Software 76(1), 1–32.\nCox, D. R. (1972), ‘Regression models and life-tables’, Journal of the Royal Statistical Society.\nSeries B (Methodological) 34(2), 187–220.\n24Curtis, L. H., Hammill, B. G., Eisenstein, E. L., Kramer, J. M. & Anstrom, K. J. (2007), ‘Using\ninverse probability-weighted estimators in comparative effectiveness analyses with observational\ndatabases’, Medical Care 45, S103–S107.\nDancey, J., Dodd, L., Ford, R., Kaplan, R., Mooney, M., Rubinstein, L., Schwartz, L., Shankar,\nL. & Therasse, P. (2009), ‘Recommendations for the assessment of progression in randomised\ncancer treatment trials’, European Journal of Cancer 45(2), 281–289.\nDaugherty, C. K., Ratain, M. J., Emanuel, E. J., Farrell, A. T. & Schilsky, R. L. (2008), ‘Ethical,\nscientific, and regulatory perspectives regarding the use of placebos in cancer clinical trials’,\nJournal of Clinical Oncology 26(8), 1371–1378.\nGlidden, D. V., Stirrup, O. T. & Dunn, D. T. (2020), ‘A Bayesian averted infection framework for\nPrEP trials with low numbers of HIV infections: application to the results of the DISCOVER\ntrial’, The Lancet HIV 7(11), e791–e796.\nIbrahim, J. G., Chen, M.-H. & Sinha, D. (2001), Bayesian Survival Analysis , Springer-Verlag, New\nYork, NY.\nIshak, K. J., Caro, J. J., Drayson, M. T., Dimopoulos, M., Weber, D., Augustson, B., Child,\nJ. A., Knight, R., Iqbal, G., Dunn, J., Shearer, A. & Morgan, G. (2011), ‘Adjusting for patient\ncrossover in clinical trials using external data: A case study of lenalidomide for advanced multiple\nmyeloma’, Value in Health 14(5), 672–678.\nIshak, K. J., Proskorovsky, I., Korytowsky, B., Sandin, R., Faivre, S. & Valle, J. (2014), ‘Methods\nfor adjusting for bias due to crossover in oncology trials’, Pharmacoeconomics 32(6), 533–546.\nJ¨ onsson, L., Sandin, R., Ekman, M., Ramsberg, J., Charbonneau, C., Huang, X., J¨ onsson, B.,\nWeinstein, M. C. & Drummond, M. (2014), ‘Analyzing overall survival in randomized controlled\ntrials with crossover and implications for economic evaluation’, Value in Health 17(6), 707–713.\nKahan, B. C. & Morris, T. P. (2013), ‘Adjusting for multiple prognostic factors in the analysis of\nrandomised trials’, BMC Medical Research Methodology 13(99).\nLatimer, N. R., Abrams, K. R., Lambert, P. C., Crowther, M. J., Wailoo, A. J., Morden, J. P.,\nAkehurst, R. L. & Campbell, M. J. (2014), ‘Adjusting survival time estimates to account for\n25treatment switching in randomized controlled trials—an economic evaluation context: Methods,\nlimitations, and recommendations’, Medical Decision Making 34(3), 387–402.\nLatimer, N. R., Abrams, K. R., Lambert, P. C., Morden, J. P. & Crowther, M. J. (2018), ‘Assessing\nmethods for dealing with treatment switching in clinical trials: A follow-up simulation study’,\nStatistical Methods in Medical Research 27(3), 765 – 784.\nLuo, X., Mao, X., Chen, X., Qiu, J., Bai, S. & Quan, H. (2019), ‘Design and monitoring of survival\ntrials in complex scenarios’, Statistics in Medicine 38(2), 192–209.\nMcKeever, L. (2021), ‘Overview of study designs: A deep dive into research quality assessment’,\nNutrition in Clinical Practice 36(3), 569–585.\nMorden, J. P., Lambert, P. C., Latimer, N., Abrams, K. R. & Wailoo, A. J. (2011), ‘Assessing\nmethods for dealing with treatment switching in randomised controlled trials: A simulation\nstudy’, BMC Medical Research Methodology 11(4).\nNational Library of Medicine (2019), ‘A multicenter, randomized, double-blind, placebo-controlled,\ntrial of Lenvatinib (E7080) in 131I-refractory differentiated thyroid cancer (DTC) (SELECT)’,\nhttps://clinicaltrials.gov/study/NCT01321554 . Accessed: 14 November 2023.\nPaganoni, S., Watkins, C., Cawson, M., Hendrix, S., Dickson, S. P., Knowlton, N., Timmons, J.,\nManuel, M. & Cudkowicz, M. (2022), ‘Survival analyses from the CENTAUR trial in amyotrophic\nlateral sclerosis: Evaluating the impact of treatment crossover on outcomes’, Muscle & Nerve\n66(2), 136–141.\nRobins, J. M. & Finkelstein, D. M. (2000), ‘Correcting for noncompliance and dependent censoring\nin an AIDS clinical trial with inverse probability of censoring weighted (IPCW) log-rank tests’,\nBiometrics 56(3), 779–788.\nRobins, J. M. & Tsiatis, A. A. (1991 a), ‘Correcting for non-compliance in randomized trials us-\ning rank preserving structural failure time models’, Communications in Statistics - Theory and\nMethods 20(8), 2609–2631.\n26Robins, J. M. & Tsiatis, A. A. (1991 b), ‘Correcting for non-compliance in randomized trials us-\ning rank preserving structural failure time models’, Communications in Statistics - Theory and\nMethods 20(8), 2609–2631.\nSimon, L. J. & Chinchilli, V. M. (2007), ‘A matched crossover design for clinical trials’, Contem-\nporary Clinical Trials 28(5), 638–646.\nWatkins, C., Huang, X., Latimer, N., Tang, Y. & Wright, E. (2013), ‘Adjusting overall survival\nfor treatment switches: Commonly used methods and practical application’, Pharmaceutical\nStatistics 12(6), 348–357.\nWhite, I. R. (2005), ‘Uses and limitations of randomization-based efficacy estimators’, Statistical\nMethods in Medical Research 14(4), 327–347.\nWhite, I. R., Babiker, A. G., Walker, S. & Darbyshire, J. H. (1999), ‘Randomization-based methods\nfor correcting for treatment changes: Examples from the Concorde trial’, Statistics in Medicine\n18(19), 2617–2634.\nWhite, I. R., Walker, S., Babiker, A. G. & Darbyshire, J. H. (1997), ‘Impact of treatment changes\non the interpretation of the Concorde trial’, AIDS 11(8), 999–1006.\n27" }, { "title": "2401.17046v1.Six_years_of_Venus_winds_at_the_upper_cloud_level_from_UV__visible_and_near_infrared_observations_from_VIRTIS_on_Venus_Express.pdf", "content": "1 \n \nSix years of Venus winds at the upper cloud level \nfrom UV, visible and near infrared observations \n from VIRTIS on Venus Express \n \n \nR. Hueso(1,2), J. Peralta(3), I. Garate -Lopez(1), T.V. Bandos(4), and A. Sánchez -\nLavega(1,2) \n \n \n(1) Departamento de Física Aplicada I, E.T.S. Ingeniería, Universidad del País Vasco, \nAlameda Urquijo s/n, 48013 Bilbao, Spain. \n(2) Unidad Asociada Grupo Ciencias Planetarias UPV/EHU - IAA (CSIC) Bilbao, Spain. \n(3) Instituto de Astrofísica de Andalucía (CSIC) , Glorieta de la Astronomía s/n, 18008 \nGranada, Spain. \n(4) Dpto. Máquinas y Motores Térmicos, E.T.S. Ingeniería, Universidad del País Vasco, \nAlameda Urquijo s/n, 48013 Bilbao , Spain. \n \n \nCorresponding author address: \nRicardo Hueso \nDpto. Física Aplicada I, \nE.T.S. Ingeniería, Universidad del País Vasco, \nAlda. Urquijo s/n, 48013 Bilbao, Spain \nE-mail: ricardo.hueso@ehu.es \n \n 2 \n Abstract : The Venus Express mission has provided a long -term monitoring of Venus \natmosphere including the morphology and motions of its upper clouds. Several works \nhave focused on the dynamics of the upper cloud visible on the day -side in ultraviolet \nimages sensitiv e to the 65 -70 km altitude and in the lower cloud level (50 km height) \nobservable in the night -side of the planet in the 1.74 m spectral window. Here we use \nVIRTIS -M spectral images in nearby wavelengths to study the upper cloud layer in three \nchannels: ultraviolet (360 -400 nm), visible (570 -680 nm) and near infrared (900 -955 nm) \nextending in time the previous analysis of VIRTIS -M data. The ultraviolet images show \nrelatively well contrasted cloud features at the cloud top. Cloud features in the visible and \nnear infrared images lie a few kilometers below the upper cloud top, have very low \ncontrast and are distinct to the features o bserved in the ultraviolet. Wind measurements \nwere obtained on 118 orbits covering the Southern hemisphere over a six-year period and \nusing a semi -automatic cloud correlation algorithm. Results for the upper cloud from \nVIRTIS -M ultraviolet data confirm pre vious analysis based on images obtained by the \nVenus Monitoring Camera (Khatuntsev et al. [2013]. Icarus 226, 140 -158). At the cloud \ntop the mean zonal and meridional winds vary with local time accelerating towards the \nlocal afternoon. The upper branch of the Hadley cell circulation reaches maximum \nvelocities at 45º latitude and local times of 14 -16h. The mean zonal winds in the \nultraviolet cloud layer accelerated in the course of the 2006 -2012 period at least 15 ms-1. \nThe near infrared and visible images s how a more constant circulation without significant \ntime variability or longitudinal variations. The meridional circulation is absent or slightly \nreversed in near infrared and visible images indicating that, either the Hadley -cell \ncirculation in Venus atmo sphere is shallow, or the returning branch of the meridional \ncirculation extends to levels below the cloud level sensed in near infrared images. At \nsubpolar to polar latitudes the three wavelength ranges show similar features and motions \nwhich is a signatu re of small vertical wind shear and may be affected by vertical \nconvergence of both layers. At the clod top level observed in UV images there are \nsignatures of a long -term acceleration of the zonal winds at afternoon hours when \ncomparing zonal winds from t he first years of Venus Express observations (2006 -2008) \nto later dates ( 2009 -2012) with a mean acceleration of zonal winds of 17 ± 6 ms-1 between \nboth time periods. \nKeywords: Venus, Venus atmosphere, Atmosphere dynamics \n \n1. Introduction \nVenus atmospheric circulation is characterized by a global zonal superrotation that \npeaks at cloud level where cloud features spin 60 times faster than the planet’s surface \n(see Schubert [1983] and Gierasch et al. [1997] for classical reviews and Read [201 3] for \nrecent analysis). The global retrograde circulation is characterized by winds that increase \nwith height from null values in the surface to speeds of 100 m s-1 in the low latitudes at \nthe cloud top at 65 -70 km altitude with much weaker winds in the m eridional direction \n(Counselman et al. 1980; Gierasch et al., 1997). Although current Venus general \ncirculation models reproduce the main features of a superrotating atmosphere, the origin \nand intensity of the atmospheric superrotation is far from being un derstood (Lebonnois et \nal. 2010, 2013). Key elements on powering the global superrotating winds are eddies, \natmospheric waves and solar tides but characterizing their effect on the winds is \nobstructed by the fact that individual wind measurement errors (~ 10 m s-1 in the best 3 \n cases) are comparable to the expected wind perturbations from either the eddies, waves \nor tides (10 -20 m s-1, Rossow et al. 1990). \nA long -term analysis of the mean winds and their temporal variations may provide \nimportant observationa l constraints to understand the mechanisms governing Venus super \nrotating atmosphere. The Venus Express spacecraft (VEx) has been orbiting the planet \nfor more than 7 years and carries two instruments suitable for atmospheric wind \nmeasurements from observat ions of cloud motions: The Venus Monitoring Camera \n(VMC, Markiewicz et al. 2007a) and the Visible and Infrared Thermal Imaging \nSpectrometer (VIRTIS, Drossart et al. 2007). Both instruments obtain images of Venus \nclouds at different wavelengths and the comp arison of consecutive images obtained with \na suitable time difference results in measurements of cloud motions. Additionally, \nthermal retrievals from observations obtained by VeRa and VIRTIS have been used to \nderive zonal winds under the cyclostrophic thermal wind equation (Piccialli et al. 2008, \n2012). However cyclostrophic balance breaks at low and high latitudes and derived \nthermal wi nds from the meridional temperature gradient cannot account for the \nmeridional circulation as well as eddies, waves or tides. \nUltraviolet images show the highest contrast features located at Venus cloud top. \nSeveral radiative transfer analysis place the cloud tops at altitudes of 67 -71 km at the \nequator with a nearly constant altitude until 45 -50ºS and a drop of altitude p oleward of \n50ºS reaching about 61 -63 km over both poles (Ignatiev et al. 2009; Lee et al. 2012; Haus \net al. 2014). Detailed studies of the cloud top motions can be traced back to measurements \nobtained by Mariner 10 in 1974 (Schubert et al., 1977), Pioneer Venus in 1979 –1985 \n(Rossow et al., 1980, 1990; Limaye and Suomi, 1981; Limaye et al., 1982, 1988; Limaye, \n2007) and Galileo in 1990 (Belton et al., 1991; Toigo et al., 1994; Peralta et al. 2007; \nKouyama et al. 2012). These works analyzed ultraviolet images of the planet upper clouds \nat 66 -71 km altitude which show high contrast features. Images of the planet in visible \nwavelengths show bland clouds without contrasted features but new features are visible \nat longer wavelengths in near infrared (950 nm). Thes e features, loosely correlated with \nthe ultraviolet details, are generally considered to lie about 5 -8 km below the ultraviolet \ncloud top and their motions were first studied from images obtained by the Galileo orbiter \non its Venus flyby (Belton et al. 199 1; Peralta et al. 2007). Although the cloud motions \nare a good proxy for true atmospheric motions (Rossow et al. 1990; Machado et al. 2012; \n2014) the apparent motion of clouds can be different in regions covered by atmospheric \nwaves which seem ubiquitous i n Venus atmosphere (Peralta et al. 2008; Piccialli et al. \n2014) and present different scales and physical origins (Peralta et al. 2014a, 2014b). \nNevertheless, studying the cloud motions is the best -suited technique at present capable \nto provide a systemati c long -term analysis of the atmospheric winds (zonal and \nmeridional) and disentangle the short and long time -scales variations. \nSeveral analyses of VMC wind data obtained from ultraviolet images have already \nbeen published (Moissl et al. 2008; Kouyama et a l. 2013, Khatuntsev et al. 2013). The \ncloud top morphology is extensively discussed by Markiewicz et al. (2007b) and Titov et \nal. (2012). Khatuntsev et al. (2013) present the most detailed study to date covering a six -\nyear time span and analyzing the cloud motions with a combination of cloud tracking and \nan image correlation algorithm run over several hundreds of orbits. Results from that \nwork include a description of the mean zonal and meridional winds together with different \nvariabilities. From VMC data t he mean retrograde zonal wind at the cloud top is about 90 \nm s-1 with a maximum of about 100 m s-1 at 40 –50ºS. Poleward of 50ºS the zonal wind \ndecreases linearly with latitude. The mean meridional wind flows poleward and is small 4 \n at most latitudes reaching a maximum value of about 10 m s-1 at 50ºS that fades out at \nsubpolar and polar latitudes. This global circulation has a strong diurnal variation: In \nterms of local time, minimum winds are found at noon (11 –14 h) with maxima in the \nmorning (8 –9 h) and the evening (16 –17 h). Superimposed over this pattern there is a \nlong-term increase of the zonal wind velocity at low latitudes. Additionally, signatures of \nzonal wind variations with periods of 4 -5 days and amplitudes of 5 -15 m s-1 are also \npresent in the VMC data, confirming previous findings with Pioneer Venus (Rossow et \nal. 1990). Kouyama et al. (2013) found a variation of the cloud -tracked zonal velocity of \n~20 m s-1 with a timescale of about 255 days (though this was not confirmed by \nKhatuntsev et al. with a larger data sample ) and recently interpreted as a centrifugal wave \n(Peralta et al. 2014b) . Khatuntsev et al. (2013) also present a preliminary analysis of near -\nIR images obtained over 10 orbits and measured using manual tracking resulting in a \nmean circulation of 70 -80 m s-1 at low latitudes and decreasing winds at latitudes higher \nthan 45º. In all works using VMC data there are a significant orbit to orbit variation of \nthe mean flow and a large scatter of wind measurements over individual orbits. Th e task \nof reliably identifying true variability from measurement noise is only attained through \naverages of large amounts of data. \n Analyses of cloud motions from VIRTIS -M images have been reported by \nSánchez -Lavega et al. (2008) and Hueso et al. (2012). T he first work analyzed about one \nyear of observations retrieving cloud motions from cloud tracking over images in \nultraviolet (380 nm), near infrared (980 nm) and short infrared (1.74 m) representative \nof different altitudes. VIRTIS -M based altimetry of U V images place the tops of these \nclouds at 67 -70 km at equatorial latitudes in basic agreement with studies of previous \nmissions with a latitudinal drop of altitudes of about 5 km from 50º polewards. In the near \ninfrared (980 nm), photons penetrate the upp er cloud a few kilometers within an altitude \nrange 5 -8 km below the UV features (Belton et al. 1991; Sánchez -Lavega et al. 2008), \nbut there is no detailed study of the altimetry of these features at different latitudes. \nImages at short -infrared (1.74 m) are assumed to represent motions in the cloud layer at \n44-48 km (Carlson et al. 1991; Crisp et al. 1991; Sánchez -Lavega et al. 2008). The \ncombined analysis produced a three -dimensional view of Venus winds over the South \nhemisphere. The vertical wind shear i s concentrated from the equator to 50ºS between the \nupper ultraviolet cloud and the near infrared cloud features. At all vertical levels the wind \nprofiles converge poleward of 50ºS to similar values reaching zero mean velocities at the \npole, thus implying a nearly null vertical wind shear at high latitudes. Neither the near \ninfrared cloud features or the lower clouds presented an organized meridional circulation \n(maximum mean average values were lower than 5 m s-1 with error bars of 9 m s-1). Hueso \net al. (2012) extended the ultraviolet data analysis for another year, which increased the \nsimilarity of the mean structure of the winds in the ultraviolet images with results from \nVMC. That work was focused on extending the 1.74 m infrared data of the lower clo ud \ncovering the first 900 VEx orbits to constrain the time variability of deep cloud features \nover that period. The mean meridional circulation was constrained to be less intense than \n4 m s-1 with error bars of 5 m s-1. The VIRTIS -M cryocoolers, essential to obtain infrared \ndata, ceased to work after 930 orbits (April 2006 -October 2008) and a longer term study \nof the lower cloud dynamics is not possible. \nAbsolute instantaneous wind measurements obtained from ground -based high -\nresolu tion spectroscopy provide velocimetry at different altitudes. At the cloud top, winds \ncan be obtained from Doppler shifts of solar lines (Widemann et al. 2007, 2008; Gaulme \net al. 2008; Machado et al. 201 2). The cloud top altitude is generally defined as t he \naltitude level where the optical depth at a given wavelength is 1. For the visible part of 5 \n the spectrum this level is fairly constant (see figure 9 in Ignatiev et al. 2009) justifying \nDoppler wind measurements in the visible as representative of the sam e altitude as \nfeatures tracked in the ultraviolet clouds. Doppler wind results obtained on different days \npresent large day to day variability and observations can only be obtained in particular \nfavorable geometric conditions resulting in a sparse sampling of wind fields. Reported \nerrors for measurements of a single day can be 20 -30 ms-1 in the best cases and the spatial \nresolution is in general lower than from cloud tracking using spacecraft data. However \nthe overall agreement between Doppler velocities an d results from VEx cross -validates \nboth the cloud tracking and the Doppler velocimetry results as representative of the \nmotions at the upper cloud level (Machado et al. 201 2; 2014 ). \nIn this paper we present an extended analysis of six years of VIRTIS -M ima ges \nobtained with the visible channel of the instrument (0.3 -1.0 m) which is sensitive to the \nupper cloud layers at combined altitudes between 60 -72 km. We have studied VIRTIS -\nM images in ultraviolet (UV), visible (VIS) and near infrared (NIR) wavelengths . The \nlow contrast of atmospheric features has been improved by using image processing filters \nadequate for the noise characteristics of the VIRTIS -M instrument. Images were \ncompared using an image correlation algorithm that served to extract the motions o f \nseveral thousand cloud features. The main motivations to extend the measurements of \ncloud motions with the VIRTIS -M instrument are: (1) Extend the temporal information \nfrom previous VIRTIS -M analysis of the upper UV cloud and compare with analysis of \nVMC data of similar extended period of time (Khatuntsev et al. 2013). This should help \nto separate true time variability of the atmospheric motions from measurement errors. (2) \nExamine the long -term wind variability reported from VMC data with an independent \ndata set. (3) Extend the analysis to visible and near infrared cloud features deepening in \nthe vertical structure of the atmospheric circulation close to the upper cloud top. \nThis paper is organized as follows. Section 2 describes the essential features of \nthe VIRTIS -M instrument, the criteria used to select the data from the large database of \nVIRTIS -M observations and summarizes the orbits analyzed. Section 3 describes the \nmethodology employed to perform the wind velocity measurements including the key \nimage filter required to show the faint atmospheric details. The wind retrievals are \npresented in Section 4. We discuss our results in Section 5 including an examination of \nthe vertical layers that correspond to each image set. We present our conclusions in \nSection 6. We give further details on the selected data on Appendix A. \n \n2. Observations \nAt the time of this work the VIRTIS instrument has been observing the planet \nfrom April 2006 till February 2014 with valid data for cloud tracking obtained in several \nhundred s orbits. The instrument characteristics are described by Drossart et al. (2007). \nThe instrument consists on a high-resolution spectrometer (VIRTIS -H) and a mapping \nspectrometer (VIRTIS -M). The mapping spectrometer is formed by two channels, each \nof them simultaneously obtaining 432 spectral images that cover the visible (0.3 -1 m) \nand the infrared (1 -5 m) range of the instrument. Images are 256 pixels in one direction \nand are formed by rotating an internal mirror in the telescope with 256 positions or \nscanning the planet in a push -broom mode (not used in this work). Because of the \neccentric polar orbit only images of the Southern hemisphere could be used for cloud \ntracking, since the VEx spacecraft moves too quickly over the north hemisphere to 6 \n provide clear images with its VIRTIS instrument or to image the same region twice with \nenough time separation between observations. \nImages were navigated with the SPICE system and SPICE navigation kernels \nprovided by ESA. The navigation is included in geometric files that supply geometric \ninformation for every VIRTIS -M pixel and that are available from the Planetary Data \nSystem. For na dir viewing geometries the pointing accuracy is estimated as seven times \nsmaller than the pixel size (Erard et al. 2008 , 2009 ). \nImages were selected with sampling from 15 minutes to 1.5 hours with most of \nthe image pairs obtained with a time separation of 1 hour. The spatial resolution ( taken \nas the pixel size) of the images selected for this work varies in the range 16 – 50 km per \npixel. Data obtained close to apocenter at 66,000 km, has a spatial resolution of polar \nfeatures of about 16 km. Observations a cquired in the initial part of the ascending branch \nof the orbit, cover the mid to equatorial latitudes with smaller spatial resolutions due to \nthe nearly tangential view of the equatorial latitudes. The best spatial resolution \nachievable for equatorial la titudes ranges 30 -50 per pixel depending on the geometry of \nthe observations. Repeated close to nadir viewing of the low latitudes that could produced \nhigher spatial resolution data is not possible from the VIRTIS -M instrument due to the \nhigh orbital velocity close to the planet (Titov et al. 2006) . \nA limitation for VIRTIS -M images in the visible channel is that cloud features \ndisplay a very low contrast in most wavelengths. Because of the observing geometry \nmany orbits were affected by stray light in the instrument caused by direct sunlight close \nto, but outside the VIRTIS -M field of view entering the instrument (Arnold et al. 2008) \nand by contamination of short -wavelength images by long -wavelength light (García -\nMuñoz et al. 2009) , both effects reducin g image contrast. Additionally, all images were \nsubject to a substantial difference of responsivity between adjacent pixels in the detector \nand in the spectral direction, an odd -even defect (Cardesin, 2010) that translated into a \nhigh-frequency noise when sharpening individual images. Finally the tenuous \nillumination of the polar regions limit the number of atmospheric details that can be \nretrieved in the visible channel of the instrument and do not permit a systematic study of \nthe polar vortex area. The po lar vortex has been studied in higher detail in nigh -side \ninfrared images from the lower cloud at about 47 km to the upper mesosphere (Piccioni \net al. 2007). Its dynamics have been extensively studied by Luz et al. (2011) and Garate -\nLopez et al. (2013) res ulting in a precessing and wandering cyclonic vortex with \nextremely variable morphology and complex vorticity patterns. \nThe VIRTIS -M data analyzed in this paper was selected from the first 2115 orbits \ncovering 6 Earth years which are equivalent to 9 Venus days. We analyzed 77 orbits by \na digital correlation method with human supervision (see next section). For completeness \nwe also added data from 44 orbits measured with manual tracking and corresponding to \ndata presented in Hueso et al. (2012). Most of the orbits contained only a small section of \nthe latitude -local time range and only a few orbits allowed obtaining partial maps of wind \nvectors over large sections of the day -side. The global latitudinal and local time coverage \nis shown in Figure 1 as well as the temporal distribution of the measurements. \nMeasurements were grouped in 5 large periods of time with enough global sampling to \nproduce global maps of the winds in each period. These periods are highlighted in Figure \n1, are defined in Table 1 and loosel y correspond to one year of data. T he total number of \nwind vectors in this work is 9240 for UV cloud features, 4190 for features in the visible \nchannel and 4585 for features in the NIR channel. This substantially improves over the 7 \n UV, NIR and IR data prese nted in Sánchez -Lavega et al. (2008) and the UV and IR data \npresented in Hueso et al. (2012). Table 1 summarizes the data. Appendix A gives details \nfor the individual orbits that have been analyzed. \n \nFigure 1: Number of cloud features tracked as a function of latitude and local time for the three \nwavelength ranges. The bottom panel shows a point for each orbit analyzed. Boxes represent \nconsecutive series of measurements that together cover a significant part of the planet and can be \nused to study long -term variability of the winds. \n8 \n \nTable 1 \nPeriod Orbits Dates (yyyy/mm/dd) N (UV) N (VIS) N (NIR) \nI VOI-476 2006 -04-12 --- 2007 -08-09 2526 725 1987 \nII 626-948 2008 -01-06 --- 2008 -11-23 1509 435 528 \nIII 1043 -1557 2009 -02-26 --- 2010 -07-25 1875 1176 993 \nIV 1640 -1865 2010 -10-16 --- 2011 -05-29 1572 572 617 \nV 1958 -2115 2011 -08-30 --- 2012 -02-03 1759 1126 461 \n \n \n3. Methods \nTo overcome the difficulties exposed in the previous section we used an image \nconvolution filter that simultaneously reduces the odd -even read noise of the detector \nwhile simultaneously sharpens image features at small scales. The convolution operator \nis defined by the following equation. \n( , ) ( , ) ( , )ww\ni w j wg x y K i j f x i y j\n \n (1) \nwhere f represents the image we want to filter, K is a w x w matrix that defines the \nconvolution kernel and g represents the convolved image (Gonzalez and Woods 2008). \nIn this case the kernel is defined as a 3x3 matrix ( w=3) that preserves the position of cloud \nfeatures: \n. (2) \nIn order to obtain optimum results we increased the signal to noise ratio of \nindividual images by using three sets of consecutive bands that cover the UV, VIS and \nNIR ranges that result in three high signal to noise ratio images that were convolved with \nthe kernel defined in Eq. (2). The UV band was defined combining 20 spectral images \nbetween 360 and 400 nm. The VIS band was defined combining 55 spectral images \nbetween 570 and 680 nm. The NIR band was defined by adding 30 spectral images from \n900 to 950 nm. These ranges were loosely defined after examination of the calibrated \nInstrument Transfer Function that show similar instrument behavior for wavelengths in \neach of these ranges (see Figu re 1 in García -Muñoz et al. 2009). After the contrast \nenhancement implied by the convolution kernel the images were mapped into cylindrical \nor polar projections depending on the geometry of the observation. For image processing \nand projection we used the s oftware PLIA (Hueso et al. 2010). Figure 2 shows examples \nof the data and the processing chain for images in the UV, VIS and NIR ranges. In some \ncases, the NIR images present horizontal lines caused by the image acquisition procedure \n(each horizontal line is read at a different time and the detector is left to cool down every \n8 lines) and whose relative intensity depends on the image acquisition time, sensor \ntemperature and contrast of atmospheric details. After processing the images present a \nhigh level of contrast at small spatial scales. \n[Figure 2] \n \n111\n000\n111K\n9 \n \nFigure 2: Image processing example from orbit 2094. Atmospheric motions are counter -clock \nwise. Original images (upper row) have typical sizes of 256x256 pixels. The processed versions \n(second row) reduces the image noise and highly increases the visual contrast of cloud features. \nPolar (bottom panel) or cylindrical projections are plotted with a spatial resolution of 0.1º/pix. \n10 \n We used a two dimensional image correlation algorithm successfully tested in \nwind measurements over images of Jupiter, Saturn and Venus (Hueso et al. 2008, García -\nMelendo et al. 2013; Garate -Lopez et al. 2013). The correlation algorithm divides each \nimage into small squares (typically of 30x30 pixels in polar latitudes and 50x50 pixels in \nlow latitudes) identifying the most probable match for each square in a second image. \nThe algorithm works over projected images in cylindrical or polar coordinates and is \nhighly configurable by the user who can select to work only on areas where visibl e \nstructures are present. For cloud details at latitudes higher than 50º all measurements were \nobtained over polar projected images while for smaller latitudes than 40º only cylindrical \nprojections were used. All projections had a spatial resolution of 0.1 º per pixel (0.1x0.1º \nin cylindrical coordinates and 0.1º in latitude in polar coordinates). Different areas of each \nimage pair may require different parameters of the correlation algorithm (size of the \ncorrelation box and minimum correlation factor) with smaller details easily detectable in \nthose regions where the images have lots of features and contrast while only large \nstructures can be tracked in regions devoid of small features. The size of the cloud features \nused as tracers is in some cases larger than the convective scales (~1 00 km ) but always \nsmaller than the global atmospheric waves (~ 1000 km ), as established by Rossow et al. \n(1990) to obtain true atmospheric motions with cloud -tracking. I n each case , cloud \nfeatures used for wind tracking were ch osen considering not only the cloud morphology \nbut also the original spatial resolution of the images (which changes across the \nprojection ). For instance , the low latitudes required higher correlation boxes than the low \nlatitudes due to the intrinsic low r esolution of the images. Other difficulties were \nintroduced by the fact that many subpolar latitudes contained large linear features where \nsmall details were difficult to locate and measure with accuracy (see Figure 1). In those \ncases we selected small sca le features visible inside the large linear structures. In spite of \nthe smaller initial spatial resolution the mid and low latitudes contained well contrasted \nmore circular features that were easier to track. For each correlation box we allowed a \nsearch ar ea able to produce velocities of up to ±160 m s-1 in the zonal direction and ±50 \nm s-1 in the meridional direction. An important feature of the correlation software used \nhere is that it allows a human supervision of each individual measurement by showing \nthe initial square in the first image, the best candidate to match that feature in the second \nimage, and a map of the correlation function that can be used to asses the goodness of the \nmatch. The user can select the appropriate local maximum of the correlat ion function \ncorrecting common misidentification problems in blind correlation algorithms. For \ninstance, regions covered with wavy features are common in the highly processed images \nand correlation algorithms tend to produce false measurements when repetit ive features \nappear. Even if these features are matched correctly they may add a phase speed to the \ntrue wind field resulting in added noise (Peralta et al. 2008). In general we purposely \navoided most areas where wave features were visible although some of them were \nincluded in those cases were we wanted to cover a large area of the planet. We also \navoided regions of strong horizontal stripes in some NIR images resulting in less cloud \nfeatures tracked in this wavelength range. We estimate that less than a 1 0% of our wind \ntracked features in the UV images may be covering regions with visible waves and a much \nsmaller fraction in VIS and NIR images. The human supervision of all measurements \nresulted in a significantly smaller number of measurements when compare d with \ntraditional blind correlation algorithms. However our measurements are more precise \nthan results obtained from non supervised algorithms and may contain a smaller number \nof false measurements. Figure 3 shows an example of the methodology for cloud f eature \ntracking over a pair of UV images which in this case have been cylindrically projected. 11 \n It also shows the deterioration of image resolution from the subpolar to low latitudes , \nwhat forces to use different correlation boxes on different latitudes. \n \n \nFigure 3: Illustration of the correlation technique. We start from two projected images and select \na region in the first image (a) that will be divided in small pixel boxes. The size of these boxes \ndepends on the cloud morphology and image texture. Regions with rich structure as in this image \ncan be explored with pixel boxes of 0.35ºx0.35º (35x35 pixels) but regions devoid of detail require \nlarger boxes (typically 2.0ºx2.0º). The first of these boxes (with a size of 35x35 pixels) is \nhighlighted in (a) and is shown in (b) in higher detail. A cross -correlation comparison with the \nsecond image results in a best match presented by the software to the user to confirm or not the \nidentification. A map of the correlation function is also shown (c) to the user to help discriminate \nif the correlation maximum is narrow enough to consider the measurement as precise. Shorter \ntime intervals result in better matches but smaller pixel displacements between images and \ntherefore less precise measurements of wind velocities. Note the deterioration of spatial resolution \n(smallest features visible) in the low latitudes in the upper part of panel a. \n \nIndividual cloud tracking er rors scale as the spatial resolution of individual images \ndivided by the time separation. The standard deviation of zonal and meridional winds at \nlatitudinal bins of 2º and obtained in different image pairs varies from 4 to 25 m s-1 with \ntypical values of 8 m s-1 and correlates well with the above criteria indicating that the \ncloud correlation algorithm enables tracking the motions of features across a single pixel \nin the original non -projected images. Since all the image pairs had been processed for \nfaint scale structures the texture of images separated by time steps of 1.5 hours resulted \nin identification of only a few features per image pair due to the different spatial resolution \nof the original images. Images separated by short time steps of 15 minutes gave results \nbetter than expected in terms of the high number of features identified by the software \nand the small standard deviation of zonal and meridional winds. We estimate the best \ntime separation between images for this technique to maximize the numb er of feature \nidentifications and the precision of the tracking as 45 minutes. In general terms, most of \nour measurements come from images separated by 45 -80 minutes but there are a few \norbits that were analyzed with time separations as small as 15 minutes . Because of \ndifferent observing strategies on different periods of the mission these images are \nparticularly common in the later time period analyzed in this study (periods IV and V). \nDetails for each orbit are given in Appendix A. \n12 \n \n4. Results \n4.1. Individual orbit results \nFigure 4 shows several examples of winds obtained in individual orbits. Figure \n4A shows results from orbit 436 which is an example of a set of orbits that focused on the \npolar region and contain the best quality images of the cold collar and outer limit of the \npolar vortex. Most of the high -resolution polar data in this work comes from orbits 436 -\n476. The three wind profiles in the polar region converge towards the same values since, \nin fact, similar features are observed in the thr ee channels with different contrast (Figure \n2). Figure 4B shows the difference in wind measurements for UV and VIS -NIR features \nfound in tropical latitudes. This comes from different cloud features observed in the UV \nand NIR images moving with different zo nal and meridional velocities. The VIS images \nare very similar to the NIR images but some cloud features that are observed on UV \nimages are also found on VIS images resulting in a global wind profile similar to results \nfrom NIR cloud features but noisier. For the upper cloud the sharp decrease of zonal \nwinds at 55ºS is a relatively common feature observed in many orbits and coincides with \nchanges of the cloud morphology (Rossow et al. 1980) that also affect the number of \ncloud features that can be track ed (although the sharp decrease is observed in orbits with \nhigh density of features and in orbits with low density of cloud features at this latitude). \nThis result may be a manifestation of the large -scale “Y” feature as it rotates over the \nplanet (Belton et a l. 1976; Rossow et al. 1980; Del Genio and Rossow, 1990; Gierasch et \nal. 1997; Titov et al. 2012), which is generally interpreted as the manifestation of an \nequatorial Kelvin -type wave ( see e.g. Kouyama et al. 2012; Peralta et al. 2014c). \nHowever, due to t he geometry of the observations and limited field of view of VIRTIS -\nM observations it is not possible to distinguish the “Y” feature in most of VIRTIS -M \nimages. VMC observations have also a limited field of view that, although it allows to \nobserve the Y fe ature in many orbits, cannot provide good statistics of its lifecycle or \nactivity (Titov et al. 2012). In most of the cases , the sharp decrease of the wind is \nsubstituted by a smooth decrease of zonal winds that is illustrated in figures 4C and 4D. \nFigure 4C constitutes the most common result for many orbits and the cloud morphology \nfor this orbit is shown as our image correlation example in Figure 3. Figure 4D is one \ncase where several image qubes covering a wide longitudinal region have been analyzed. \nMost of the wind variation in the UV images comes from features moving at different \nvelocities in regions with different local times. Figures 4E and 4F show the possibility to \ntrack atmospheric winds with short time steps of 20 minutes. The cloud morphology \ncorresponding to figure 4F is shown in figure 2. In these cases many cloud features can \nbe identified by the software but the short time interval between images results in noisy \nmeasurements. 13 \n \nFigure 4: Zonal and meridional winds retrieved for different orbits and time separations between \nconsecutive images. Individual measurements are shown with cyan circles (UV), orange \ndiamonds (VIS) and red crosses (NIR). Fits to the data based on bins of 2 -4º latitu de are shown \nin blue (UV), green (VIS) and magenta (NIR). Measurement granularity (minimum difference of \nwinds at the same latitude) depends on the spatial resolution of the image and the time difference \nand can be appreciated in many of the examples above . \n \n \n14 \n The variability shown in Figure 4 is a combination of measurement noise (which \ndepends on varying image quality, cloud morphology and images time separation), \ndependence of motions with local time and true time variability. Results from the UV \nimages are more variable than those from NIR images. The fact that VIS images show \nfeatures common to both UV and NIR cloud layers is in agreement with the idea that both \nUV and NIR layers are close in altitude. The convergence of wind motions at high \nlatitudes i s accompanied by the fact that cloud features become more similar at high \nlatitudes and show the same cloud details at subpolar latitudes. We detail the available \ninformation on cloud altitudes on section 5.1. \n \n4.2. Global averages \n \n \nFigure 5: Global zonal (left) and meridional winds (center). Results are based on averages over \nlatitudinal bins of 2º. Error bars represent the standard deviation of the zonal or meridional winds \nfor each bin. The latitudinal density of measurements is represented in the right panel. UV, VIS \nand NIR cloud features are represented by solid dark blue, dashed orange and solid magenta lines. \nA comparison with previous analysis of VIRTIS data (Sánchez -Lavega et al. 2008; and Hueso et \nal. (2012) of UV (dashed cyan) and N IR (dashed green) cloud features is also shown. \n \nWe present an update on the global zonal and meridional circulation of the upper \nclouds at 66 –72 km as observed with UV data and a few kilometers below that level with \nVIS and NIR data (Figure 5). Results fr om Sánchez -Lavega et al. (2008) and Hueso et al. \n(2012) are also shown demonstrating a very similar global structure of the winds in spite \nof the different methodologies and larger data set analyzed in this work which covers a \nlarger period of time. The me an zonal wind in the latitude range from 0º to 45º S is \nconstant and decreases linearly with latitude to null values at the South Pole. Zonal winds \nat low latitudes (0 -45º) at z = 66 –72 km tracked from UV cloud motions are = -105 ± 12 \nm s-1 while values of -71 ± 13 m s-1 and -65 ± 10 m s-1 are found for the same latitude for \n15 \n VIS and NIR features respectively. For mid to polar latitudes all wind profiles converge \nto close values with decreasing winds towards the pole. The meridional wind shear of the \nzonal w ind profile from UV images is s-1, while for the NIR \ndetails, a very similar but distinguishable profile is found, with \ns-1. In both layers meridional motions are ten times less intense than zonal motions but \npresent a similar range of variability result ing in a global circulation that is less well \ndefined. The UV cloud features move polewards with a speed that peaks at -9 m s-1 at 50º \nS but the standard deviation of meridional winds at all latitudes are on the order of 8 m s-\n1. This is due to the fact that there are orbits with well -defined Hadley cell -like meridional \ncirculation and orbits with different details traveling polewards and equatorwards. Cloud \nfeatures observed in VIS and NIR ranges do not show an organized global meridional \ncirculation. Although some orbits present strong meridional motions at subpolar latitudes, \naverage values of the meridional velocities at well -sampled latitudes are below 2 ms-1. \nThis could be related to the chaotic nature of the South polar vor tex and its wandering \nmotions (Garate -Lopez et al. 2013). \n4.3 Zonal -latitudinal structure of the mean winds \n \nFigure 6: Zonal (left) and meridional (right) winds. Results are based on averages over bins of 5º \nin latitude per 0.5 hr in local time for the UV features. The spatial resolution of the VIS maps is \nbased on bins of 10º per 1.0 hr and the spatial resolution of the NIR map is based in bins of 7º in \nlatitude per 0.7 hr in local time. \n52.0 0.3 10 uy \n51.5 0.3 10 uy \n16 \n \nFigure 6 shows maps of the mean zonal and meridional winds in terms of latitude and \nlocal time. In the wind field extracted from UV features there is a significant variation of \nthe zonal and meridional components due to a thermal solar tide superimposed on the \nmean structure of the wind. In the latitude range of 15 -50º the zonal winds increase at a \nrate of about 2.5 ± 0.5 m s-1 per local time hour from the morning (9 hr) to the afternoon \n(16-17 hr). Zonal speeds are also high in the early morning (7 hr), although these \nlongitudes are covered with less detail (see fig ure 1). The amplitude for this longitudinal \nvariation of the zonal wind decreases polewards but it is still detectable at 70º S with \nvalues of (1.3 ± 0.5) m s-1 per local time hour. A similar acceleration of the meridional \nwinds is apparent with poleward w inds peaking in the mid latitudes at 14 -15 local time \nhours with values of -12 m s-1. A strong meridional circulation is also visible in equatorial \nlatitudes in the early morning but the scarcity of measurements in that range does not \nallow to consider tha t this is a robust result or a regular behavior of Venus atmospheric \ncirculation. \nSimilar longitudinal dependences of the zonal and meridional winds have been \nreported from previous VIRTIS -M analysis (Sánchez -Lavega et a. 2008; Hueso et al. \n2012) but are b etter defined here. Early analysis of images from the VMC camera also \nsupport this zonal structure (Markiewicz et al., 2007b; Moissl et al., 2008) and our results \nare in good agreement with the global analysis of VMC data from Khatuntsev et al. (2013) \ncove ring a similar period of time, although the location of the weakest winds is somewhat \ndifferent (11 -14 hr in VMC compared to 8 -10 hr from VIRTIS -M). Additionally, the \namplitude of the zonal wind variation is similar to results obtained from analysis of \nPioneer -Venus data (Del Genio and Rossow, 1990; Limaye, 2007) and Galileo (Toigo et \nal., 1994). Details about the location of maximum and minimum wind velocities are \ndifferent to previous missions although a proper comparison is difficult since previous \nstudies present either tidal fits to the winds (Del Genio and Rossow, 1990, Limay e, 2007) \nor spherical harmonics fits (Toigo et al. 1994) with a larger influence of the low latitudes. \nWinds from tracking of cloud features in the VIS and NIR ranges exhibit muc h \nless pronounced zonal structure. Results from NIR images show features moving zonally \nwith constant velocity at the same latitude. Interestingly there is an apparent reversal of \nthe meridional Hadley -cell like circulation, with poleward winds at equatori al and \nsubpolar latitudes in the early morning, and equatorward winds close to evening \nterminator. Results from the VIS cloud features show mixed results between the upper \nwinds observed in UV features and the lower level sensed with NIR images. The appare nt \nacceleration of zonal winds at dusk is probably related with the geometry of the \nobservations. Solar ray -lights illuminating the evening region enter the atmosphere with \na high incidence angle and could be reflected at levels slightly above than rays at noon \nwith low incidence angle. Therefore, for VIS images with very low contrast it is easier to \nfind cloud features that are also present in the UV cloud at dawn and dusk hours than at \nnoon. 17 \n \nFigure 7: Standard deviations of zonal (left) and meridional (right) winds. Most of the global \nvariability is linked to longitudinal variations of the winds which depend on local times. \n \nFigure 7 shows maps of the standard deviations of the mean velocity of the zon al \nand meridional winds. Most of the variability is linked to longitudinal variations (i.e. \ndependence of the wind velocitiy on local times). We find typical standard deviations of \n12 and 7 m s-1 in the zonal and meridional directions respectively. From th e analysis of \nstandard deviations in individual orbits we roughly estimate that values expected from \nmeasurement errors should be on the order of 8 ms-1 with regions of larger standard \ndeviations of the wind showing true variability linked to transient edd ies and temporal \nvariability . For UV cloud features most of the variability is concentrated in the 45 -60º \nlatitude range. These latitudes mark the location where the zonal wind profile changes \nfrom a constant zonal wind to a linearly decrease of the wind w ith higher latitude. It is \nalso the latitude range where some measurements have shown the appearance of a mid -\nlatitude jet that was first found from UV cloud tracking in Mariner 10 images (Limaye et \nal. 1981). The nature of the mid -latitude jets is controv ersial since they were not found in \nsubsequent analysis of Pioneer Venus data (Rossow et al. 1990; Limaye, 2007) or they \nwere very weakly detected (Toigo et al. 1994) or not at all in analysis of Galileo data \n18 \n (Peralta et al. 2007). However mid -latitude jet s are found in studies of Venus winds under \nthe assumption of cyclostrophic balance and by application of the corresponding thermal \nwind equation balance either from Pioneer Venus thermal data (Newman et al. 1984) or \nfrom Venus Express thermal data (Piccial li et al. 2008, 2012). Note , however , that the \nthermal winds are derived from an averaged temperature field and therefore do not \nreproduce the local time dependence of the winds, the meridional circulation and the roles \nof tides and eddies. The VMC results indicate variability at these latitudes and the \nemergence and disappearance of the jet in different orbits but always within the velocity \nerror bar (Khatuntsev et al. 2013). High -resolution direct measurements of instantaneous \nzonal winds from Doppler observations using the VLT provide additional evidence for \nthe occasional presence of modest jets at 50º (Machado et al. 201 2). Our temporal \nresolution is not high enough to resolve the emergence and disappearance of the South \nmid-latitudes jet. On the other hand, wind measurements at these latitudes are also \naffected by the local cloud morphology which is variable. The variability of the \nmorph ology could be related to the appearance and disappearance of the “Y” wave whose \npresence and structure cannot be confirmed only with VEx data since most orbits do not \nshow a clear view of the complete latitude range with either VMC or VIRTIS -M. A \ncomparis on with UV images from ground -based observations could resolve the presence \nof the “Y” wave on dates analyzed in this or other works based on VEx data. Currently , \nthis could only be done with observations of Venus obtained by amateur astronomers \n(Mousis et al. 2014) and coordinated in public access data repositories (Barentsen et al. \n2008). \nThe standard deviation of winds in the VIS images is especially large in regions \nwhere cloud features from both the UV and NIR cloud layers are found in different orbits. \nVariability in the NIR images is smaller than in the UV images and high values are fo und \nat subpolar latitudes in the afternoon hours. \n \n 19 \n 4.4 Polar circulation \n \nFigure 8: Polar structure of the zonal and meridional winds tracked in UV, VIS and NIR images. \n \nFigure 8 shows polar maps of the zonal and meridional winds for latitudes 60 -90ºS. While \nzonal motions are fairly homogenous and zonal variations seem related only to \nmeasurement noise, the meridional motions are not homogenous and present significant \nstructure, especially in the NIR cloud layer with polewards motions in the early morning \n(7-10 hr in local time) and reversed motions in the afternoon. The absence of discernible \nstructures in the zonal component of the wind is in good agreement with a previou s study \nof the solar tides by Peralta et al. (2012), who showed that the diurnal tide is expected to \naffect mainly the meridional component of the wind at high latitudes. \n \n20 \n 4.5 Long -term variability of the winds from UV features \nVIRTIS -M data from a single orbit typically cover a relatively small portion of \nthe dayside within a limited longitudinal range. Data acquired on consecutive orbits \ngenerally cover the same area and data from a different observing season are needed to \ncover differ ent portions of the planet in terms of latitudes and local times. Since the winds \nat the cloud top depend on the local times studying time variability requires combining \ndata from several orbits over extended periods of several months reducing the time \nresolution in which the temporal variability can be studied. We have grouped our data in \n5 different periods highlighted in Figure 1 and further described in Table 1. \n \nFigure 9: Sequence of zonal and meridional winds from UV measurements obtained on differe nt \ndates. Results are based on averages over bins of 10º in latitude and 1 hr in local time for different \nperiods of time (Table 1). (I) From Venus Orbit Insertion to orbit 476 (dates from 12 -04-2006 till \n09-08-2007), (II) Orbits 626 -948 (dates from 06 -01-2008 till 23 -11-2008), (III) Orbits 1043 -1557 \n(dates from 26 -02-2009 till 25 -07-2010), (IV) Orbits 1640 -1865 (dates from 16 -10-2010 till 29 -\n05-2011), (V) Orbits 1958 -2115 (dates from 30 -08-2011 till 03 -02-2012). \n \n21 \n \nFigure 9: Continued . \n \nFigure 9 shows maps of the zonal and meridional winds in each of these periods. \nThere are two interesting results: Periods I and II (years 2006 -2008) have lower zonal \nvelocities at mid to equatorial latitudes than periods III, IV and V (years 2009 -2011). T he \nmeridional circulation is variable with a more intense Hadley circulation in period II (year \n2008) and a partial reversal of this circulation at low latitudes in period V (late 2011 and \nearly 2012). Results from individual orbits in each period of time are consistent with each \nother when analyzing the zonal velocities but the meridional circulation is more chaotic. \nIn each period of times is possible to find orbits that show clearly defined polewards \ncirculation and orbits with less defined meridional mo tions or even reversals at some \nlatitudes. In addition there are a few orbits in periods IV and V with shorter time intervals \nbetween images in a given image pair. This short time difference could affect more \nstrongly the determination of meridional winds. Therefore, while we consider the zonal \nmeasurements in each period robust enough to explore the time variability of the zonal \ncomponent of the winds, the global maps of meridional motions in figure 7 might be \naffected by the statistical weight of a few or bits at least in periods IV and V. \n22 \n The significance of the wind variations are shown in figure 10 where we plot \nlatitudinal profiles of zonal and meridional winds for local times from 14 to 18 hr putting \ntogether the data from periods I and II and data f or periods III, IV and V. At low latitudes, \na wind acceleration of 17 ± 6 ms-1 can be clearly seen for the zonal wind. This variation \nis well above the standard deviation of both profiles. Khatuntsev et al. (2013) and \nKouyama et al. (2013) report a steady increase of zonal winds at low latitudes from the \nearly mission until 2012 from analysis of VMC ultraviolet images. Instead, from VIRTIS -\nM data, we find rather two relatively similar regimes, one with low zonal winds peaking \nat tropical latitudes at aftern oon hours with an intensity of -104 ms-1, and a second regime \nwith zonal winds of -120 ms-1. From VIRTIS -M data the variation of zonal velocities \nseems rather sharp and may have occurred in late 2008, early 2009. \nFrom VIRTIS -M data the variation of the meridional component is much weaker \nand its value is smaller than the standard deviation of the meridional wind profiles shown \nin Figure 9. The apparent decrease of the mean meridional wind at low latitudes found in \nperiod IV is slightly consistent but much smaller than periodic variations of the \nmeridional motions at low latitudes found in one analysis of VMC data (Kouyama et al. \n2013) who found periodic changes of meridional motions of up to 8 ms-1. \n \nFigure 10: Zonal, meridional and number of points measured for local times restricted to 14 -18 \nhr in two subsets of the temporal data. Blue solid lines correspond to data from periods I and II. \nRed dashed lines correspond to data from periods III, IV and V. The stan dard deviation of zonal \nwinds is about 7.5 m s-1 from 10º -50ºS in both epochs. The standard deviation of meridional winds \nis about 5.5 m s-1 for that latitude range. \n \nWe have also explored possible wind variations in the lower level observed in \nNIR images. Figure 11 shows maps of the zonal and meridional winds grouped in two \nbroad periods (I+II) and (III+IV+V) that correspond to the winds dichotomy observed in \nthe upper clouds observed in UV images. A comparison of both maps show that they are \nindeed very s imilar and we interpret the differences found in both periods of time as a \nresult of measurement noise over a smaller amount of data. Therefore, there is no \n23 \n temporal variation in the motions associated to NIR features being this layer more stable \nto wind v ariations. \n \nFigure 11: Sequence of zonal winds from NIR measurements obtained on different dates. Results \nare based on averages over bins of 10º in latitude x 1 hr in local time. (I, II) From Venus Orbit \nInsertion to orbit 948 (dates from 12 April 2006 ti ll 23 November 2008; 2515 measurements), \n(III, IV and V) Orbits 1043 -2115 (dates from 26 February 2009 till 03 March 2012; 2071 \nmeasurements). \n \n4.6. Role of waves in the UV cloud \nAt the level sampled by UV images Venus atmosphere presents abundant wave \nactivity of different scales. Some of these waves are directly observable in the images of \nthe cloud field acquired by VIRTIS -M (Peralta et al. 2008) and VMC (Piccial li et al. \n2014) , and their nature has been identified as gravity wave type ( Peralta et al., 2008; \nPiccialli et al., 2014; Peralta et al. 2014a) . From VIRTIS -M data the phase speed of these \nwave systems ranges from -40 ms-1 (accelerating the retrograde winds) to +15 ms-1 \n(damping the retrograde winds) with a mean value of -15 ms-1. This value comes from \nfive wave packets whose phase speed could be distinguished from the ambient winds on \nthe same images (Peralta et al. 2008) and is in agreement with gravity waves with a \nvertical wavelength on the order of 5 km. \nCloud tracking can be affected by the presence of mesoscale gravity waves since \ncorrelation algorithms find the match between contrast patterns that may represent the \nwave motions instead the clouds background. Howeve r there is no reason from a \ntheoretical point of view to make gravity waves to alter the cloud tracking results in a net \nsense accelerating or dampening the winds. Gravity waves, therefore, simply add noise \nto the measurements. Many VIRTIS -M images analyze d in this work present waves as \nthose described by Peralta et al. 2008. Orbits 948 (23 November 2008) and 1557 (25 July \n24 \n 2010) show a large wave activity in the tropical latitudes concentrated after noon. The \nmorphology of the clouds and wind speeds are sho wn in figure 12 compared to orbit 1743 \n(27 January 2011) with no wave features but similar cloud motions. The kind of \nmorphology present in orbit 1557 is more abundant in periods IV and V when higher \nvelocities have been found. \n \n \nFigure 12: Morphology and wind fields for three of the best studied orbits: Orbits 948 (period II; \n197 data points), 1557 (period III; 350 data points) and 1743 (period IV; 327 datapoints). \n \n5. Discussion \n5.1 Cloud altitudes and vertical wind shears \nThe UV features are observed at the cloud top. From the equator to 50°S \nlatitudinal range this cloud top is located at a level of 66 -72 km. At higher latitudes it \nbegins to descend reaching a minimum altitude of ~64 km at the pole. This global trend \nwas derived by Ignati ev et al. (2009) from VIRTIS -M spectroscopy and VMC images \nand was later confirmed by Lee et al. (2012) by joint analysis of the VeRa and VIRTIS \ninstruments on Venus Express. A recent detailed work based on VIRTIS data confirms \nthis trend (Haus et al. 2014) which is also in agreement with earlier radiative transfer \nresults (Zasova et al. 2007) and in situ measurements during the descent of the Venera \nand Pioneer Venus probes (Schubert 1983) and from tracking of VEGA balloons (Preston \net al. 1986 ). \nFeatures found in the NIR or visible images have received much less attention due \nto the low contrast of the features but since cloud top altitude is wavelength -dependent it \nis possible to calculate the altitude of the unit optical depth level for diff erent cloud \nmodels. Ignatiev et al. (2009) present such a study concluding that the cloud top in UV is \n25 \n located almost at the same altitude as in the near -IR. This disagrees with the different \nwind results obtained with cloud tracking at both levels in this work and in earlier analysis \nof UV and NIR wind profiles (Belton et al. 1991, Peralta et al. 2007; Sánchez -Lavega et \nal. 2008) that are coherent with NIR features being located below the UV details. This \ncontradiction could be solved if the small contrast present in the near IR images comes \nfrom features at a deeper level than the unit optical depth. Belton et al. (1991) suggested \non the basis of wind tracking by the VEGA balloons that the cloud tracers observed in the \ntwo wavelength ranges (UV and NIR) co uld be separated by a maximum of 15 km. \nSánchez -Lavega et al. (2008) estimated an altitude range for the NIR details of 58 – 64 \nkm based on radiative transfer calculations within a range of possible cloud properties. \nComparing the mean value of zonal winds at tropical latitudes ( -65 ms-1) with the vertical \nprofiles of zonal winds from the Pioneer Venus probes and Vega balloons (Schubert et \nal. 1983; Gierasch et al. 1997), we find that altitudes of 56 -62 km fit well the data for \ntropical latitudes. However a comparison with the Pioneer Venus North probe, which \nentered the atmosphere at 60ºN, would place the NIR clouds at 47 -50 km. Reconciling \nthese different measurements would imply that altitudes where NIR features are tracked \nare constant at low latitudes a nd decrease towards the pole mimicking the results found \nfor the upper cloud and UV details. Since the cloud features in the UV and NIR layers \nresemble very much each other at polar latitudes a reasonable alternative model would \nplace the NIR details at ap proximately the same level at all latitudes which would reduce \ntheir altitude difference at polar latitudes. \nAnother piece of information comes from the altitude of the cloud features \nobserved in the night -side cloud layers at the 1.74 and 2.3 m spectral windows. The \naltitude of these clouds is also latitudinal dependent and ranges from 38 to 45 km \n(Barstow et al. 2012) with the lower values at the South Pole. These clouds have very \ndifferent morphology to those observed in the NIR images but have a simil ar global \ncirculation (Sánchez -Lavega et al. 2008; Hueso et al. 2012) reflecting that either they are \nlocated at similar altitudes or that the vertical wind shear between both layers is very \nsmall. \nFrom the NIR altimetry, lower bound imposed by the IR clo ud altitude and from \nthe Pioneer Venus and Vega in situ measurements we have considered two possible \ngeneral models of altimetry for the NIR clouds. The first model is a constant altitude \nmodel at 58.5 km. The second model is a model that approximately fit s the available data \nand mimics the lower altitudes of cloud features observed in UV and 1.74 m images. \nWe favor the first model over the second one because of the similar morphology of clouds \nfound in UV and NIR images that seem to point to images sensin g very similar vertical \nlayers. We additionally considered a synthetic altimetry model for UV clouds as the \nmedian of profiles from Ignatiev et al. (2009), Lee et al. (2012) and Haus et al. (2014), \nall based on VEx data. Figure 13 summarizes the cloud alti metry data. \n 26 \n \nFigure 13: Cloud altimetry for the UV and NIR images. From top to the bottom: UV cloud \naltimetry from Ignatiev et al. (2009) (continuous blue line), Haus et al. (2014) (dotted blue line) \nand Lee et al. (2012) (dashed cyan line). A synthetic model built from averaging these \nmeasurements is shown (dark continuous line) with a range of uncertainties of about ±4 km from \ndifferences between these works and main uncertainties quoted by these authors. NIR cloud \naltimetry from radiative transfer calculations for low latitudes from Sánchez -Lavega et al. (2008) \n(yellow box), and from a comparison of zonal winds with wind results from Pioneer Venus and \nVenera probes ( Schubert 1983; Gierasch 1997) (green boxes) are also shown. Two possible \nmodels of NIR altimetry are plotted. NIR model 1 (black dashed line) assumes a constant altitude \nof 58 km for all latitudes. NIR model 2 (black continuous line) mimics the latitudinal behavior of \nthe UV altimetry and tries to fit the available data. A lower bound to this model is provided by \naltimetry of the deep cloud observed in 1.74 m (Barstow et al. 2012). \n \nWe show maps of the vertical wind shear for both NIR altimetry models and both \nwind components (zonal and meridional) in Figure 14. At low latitudes both altimetry \nmodels give similar values of the vertical shear of the zonal wind, which varies from -3.0 \nms-1 per km in the morning hours to -4.0 ms-1 per km in the afternoon hours . At high \nlatitudes there is an uncertainty in a factor of two in the values of the vertical wind shear \nfrom considering both altimetry models. The first NIR altimetry model produces subpolar \nvertical wind shears of zonal winds of -1.0 to 0 ms-1 while the second altimetry model \nresults in vertical wind shears of -0.5 to 0 ms-1. The vertical shear of the meridional wind \nis 4 to 8 times smaller. The longitudinal structure of the wind shear is dominated by the \nacceleration of the motions at the upper cloud. A comparison of NIR wind maps with \nmaps of night -side winds obtained from 1.74 m images from Hueso et al. (2012) would \nresult in negligible wind shears between both layers. Therefore, Venus cloud top is \nlocated in a narrow region of high vertical wind shear above a region of small wind shear. \n27 \n \nFigure 14: Vertical wind shears between the UV and NIR clouds for two models of cloud \naltimetry for the lower NIR features. Results are based on averages over bins of 5ºx5º with a \nminimum of four measurements in each bin for each cloud layer. \n \n5.2. Horizontal divergence \nLimaye et al. (1988) and Rossow et al. (1990) used the wind field derived from \nthe OCPP/Pioneer Venus UV imaging to calculate the horizontal divergence of the flow. \nKhatuntsev et al. (2013) followed th e same approach and calculated the horizontal \ndivergence from the VMC latitude -longitude (local time) wind field. In spherical \ncoordinates the divergence of the horizontal wind is given by (Sánchez -Lavega 2011) \n, (2) \nwhere R is the planetary radius, u and v are the zonal and meridional components of the \nwind and and are the longitude and latitude. \n1 1 tan cosu v vDiv VR R R\n \n\n28 \n \nFigure 15: Horizontal divergence of the wind field in the cloud top from UV images and in the \nvisible and near infrared images. Because of the different sources of noise results are based on a \nwind model computed from wind averages in bins of 5ºx5º. These bins are m agnified and \nsmoothed with a spatial scale of 5º. Derivatives are then calculated with a spatial step of 30º. \n29 \n An evaluation of this expression from our data results in divergence maps for each \ncloud level that are shown on Figure 15. Errors in the divergen ce come from the mean \nwind uncertainty (~7 ms-1) and the spatial scale used to calculate the spatial derivatives. \nA compromise between spatial resolution of the divergence map and accuracy is found \nfor derivatives with spatial scales of 30º (~3150 km) resulting in a noise level of 2.2x10-\n6 s-1. This value is comparable to differences in the divergence when calculating the spatial \nderivative with different resolutions. \nDivergence in the cloud top (UV images) is dominated by the zonal acceleration \nof the retrograde winds and the latitudinal dependence of meridional winds only \ncontributes with a small fraction of the divergence. Values in this map are comparable to \nresults obtained from VMC data by Khatuntsev et al. (2013). The only significant \ndifference b etween VMC and VIRTIS -M data is the high divergence found in VIRTIS -\nM data in tropical latitudes from 13 -14 hr which intensifies towards equatorial latitudes. \nThis result seems a robust feature. A similar result was found just north of the equator \nand also in the early afternoon in the wind measurements with violet images during the \nGalileo flyby (Toigo et al. 1994). Divergence values from VIS images peak at dusk. This \nis an artifact from the data coming from those UV cloud tracers that are seen in visible \nimages when the incidence angle is high. Divergence values in the NIR images are much \nsmaller and for practical purposes could be neglected in comparison with the noise level \nof the wind measurements. \n \nFigure 16: Horizontal divergence of the wind field in the cloud top from UV images in two \nperiods of time. Period I+II encompass data from Venus Orbit insertion till orbit 948 (dates from \n12 April 2006 till 23 November 2008; 2515 measurements). Period III+IV+V encom pass data \nfrom orbits 1043 -2115 (dates from 26 February 2009 till 03 March 2012; 2071 measurements). \nResults are based on averages over bins of 7.5º x 7.5º. Differences between both maps are caused \nby differences in the wind field but also by the data cov erage of different periods. \nWe interpret the UV divergence map as a signature of global upwelling at mid -\nlatitudes increasing from morning (9 hr) to afternoon with the strongest upwelling in the \nearly afternoon where peak divergences also extend to tropica l latitudes. This is related \nwith the cloud morphology in many images where turbulence is found preferentially in \nthe early afternoon at tropical to mid latitudes (see Figure 12). Downwelling is found in \nthe cold collar region, which is consistent with dir ect Hadley circulation but apparently \nalso in the early morning at several latitudes. The horizontal divergence at the lower cloud \nobserved in NIR images is 4 times smaller and is close to zero at noon. This indicates \nvery small vertical motions in this la yer. Again this is consistent with the morphology of \nthe clouds that do not show turbulent regions as those present in UV images. \n30 \n We have investigated possible differences of the upwelling/downwelling patterns \nin the UV images in different periods of time. Figure 16 shows maps of the velocity \ndivergence at the cloud top in the I+II period compared with later dates (III+IV+V). \nDifferences between both maps are in the order of 2x10-6 s-1 in most regions covered by \nboth maps. These differences are comparable t o the estimated error in the divergence \ncalculation. However, there is a large difference at mid latitudes in the afternoon hours \nwhere the horizontal divergence at cloud top intensifies in the second period of time. This \nis linked to the higher values of the zonal winds and is possibly related to the morphology \nof the clouds in ultraviolet which show more turbulence and convective features in the \nlow and mid latitudes and afternoon hours in the second period (years 2009 -2012) when \ncompared with the first period (years 2006 -2008). \n6. Conclusions \nThe mean circulation in the upper cloud layer observed in ultraviolet light has a \nsignificant structure of the zonal and meridional winds at tropical latitudes with both wind \ncomponents being intense in the early morning, showing minimum values at local tim es \nof 8-10 hr and later accelerating with local time. The meridional circulation has the \nexpected behavior from the upper branch of a strong Hadley -cell circulation which \nintensifies at local afternoon with maximum values at 14-15 hr, while the zonal w inds \ncontinues to intensify until 17hr. Horizontal divergence of the wind shows upwelling \nclose to noon intensifying at afternoon hours. The circulation traced by NIR cloud \nfeatures a few kilometers below the upper cloud top does not present the same zon al \nstructure, meridional motions are much smaller than at the cloud top and divergences are \nabsent within the noise of the measurements. These characteristics indicate a circulation \nwith significant effects of the solar irradiation concentrated in the uppe r layer. Solar \nheating in Venus clouds is absorbed efficiently in the upper cloud top without affecting \nthe lower levels just a few kilometers below. \nCloud features in VIS wavelengths are generally the same as those found in NIR \nimages but it is also poss ible to find features from the UV cloud in these images. These \nVIS/UV features are common close to dawn or dusk, which we interpret as a signature of \ndifferent illumination. Motions found at the visible cloud layer do not inform us about an \nintermediate cl oud layer between the cloud top observed in UV light and the deeper NIR \ndetails. \nAverage motions at cloud top derived from VMC and VIRTIS -M images are in \nexcellent agreement, but the descriptions of temporal variations from both data sets are \nslightly diff erent. Both instruments have shown that Venus atmosphere at cloud top has \nsignificant variability in morphological features and wind motions. Uncertainties in winds \nmeasurements in individual orbits and the fact that most VIRTIS -M orbits only cover \nlimited regions of the planet pose challenges to the interpretation of the data and the nature \nof variability between individual orbits (10 -20 ms-1) is difficult to characterize since it is \nmixed with measurement errors (~7 ms-1) of individual cloud tracers. VIRTIS -M data \nobtained on consecutive orbits typically centers on the same latitudes and can not be used \nto study the short -term global variability of the winds. The global acceleration of the zonal \nwinds deduced from VMC images (Khatuntsev et al. 2013; Ko uyama et al. 2013) is here \nconfirmed but instead of a regular acceleration of the wind we interpret the data as a dual \nstructure of the winds with weak winds in the 2006 -2008 period of time and stronger \nwinds in the 2009 -2012 period with values of the zona l wind 17±6 ms-1 stronger at \nafternoon hours. Time -scales associated to this variation are difficult to investigate from 31 \n only VIRTIS -M data and there is a large observational gap between the two periods of \ntime. All of our wind profiles obtained in differe nt orbits over the first period are “slow” \nand all of our wind profiles for the second period are “fast” when we analyze the local \ntime dependence of the wind. This is also linked to changes in the cloud morphology with \nmore abundance of turbulence pattern s and wavy features in the second period. The data \nselection for this work does not allow a systematic exploration of short term variations \nwith temporal scales of a few days such as the 4 -5 days wind variations found from VMC \ndata (Khatuntsev et al. 2013; Kouyama et al. 2013). Overall, the wind variability appears \nas a complex phenomenon with signatures of alternating atmospheric regimes in specific \norbits and no clear identification of specific time -scales. \nMotions retrieved from analysis of NIR images s how a more regular behavior with \nan absence of time variability. Variations of vertical wind shear depend only on the \nchanges that arise on the upper cloud top. Uncertainties in the vertical wind shear come \nfrom measurement errors in both cloud layers and the uncertainties of cloud altitudes. \nThe subpolar regions show hints of interesting variations in some orbits which \ncould be related to the nature of the polar vortex. Images of the polar vortex are generally \nnot available with VIRTIS -M day-light observat ions due to the poor illumination by the \nSun. \nVenus mean circulation transports angular momentum poleward through the \nHadley circulation (Gierasch, 1975) and the maintenance of the super rotation of the \natmosphere requires an opposite transport of angular momentum toward the equator. \nDisentangling the different contributions to the transport of momentum (mean circulation, \nsolar tides and eddies) requires fitting the winds to a tidal model with eddy perturbations \n(Limaye and Suomi, 1981; Rossow et al. 1990; Limaye et al. 2007). A global analysis of \nthe wind field towards that aim is possible from the VIRTIS -M wind data here discussed \nand will be presented elsewhere with a comparison to similar analysis from previous \nmissions. A comparison of VIRTIS -M winds fr om several missions was presented in our \nprevious paper (Hueso et al. 2012) considering a reduced set of the data here presented . \nAverage results were essentially the same as those here presented for the mean zonal and \nmeridional winds as a function of lat itude only. While further analysis of VMC and \nVIRTIS -M data obtained in the period 2012 -2014 may provide additional information \nabout the long -term behavior of the atmospheric circulation, investigating the short term \nvariations may require a completely di fferent set of data. Such data could be obtained by \nthe Akatsuki mission if it is able to successfully enter Venus orbit insertion in 2015 \n(Nakamura et al. 2014). Adequate interpretation of the wind data acquired from its polar \norbit could benefit from a comparison with ground -based images showing the global \ncontext of the atmosphere and the emergence and evolution of large -scale features such \nas the “Y” wave feature that could be responsible for part of the observed variability. \nAcknowledgments : We wish t o thank A. Cardesin for providing a first version of the \nconvolution kernel used to increase the image contrast. We also acknowledge comments \nfrom W.J. Markiewicz and an anonymous reviewer who helped to improve this paper. \nWe gratefully acknowledge the work of the entire Venus Express team that allowed these \ndata to be obtained. We wish to thank ESA for supporting the Venus Express mission, \nASI (by the contract I/050/10/0), CNES and the other national space agencies supporti ng \nthe VIRTIS instrument onboard. JP acknowledges support from the Spanish MICINN for \nfunding support through the CONSOLIDER program ”ASTROMOL” CSD2009 -00038, \nand also funding through project AYA2011 -23552. This work was supported by the 32 \n Spanish MICIIN pro ject AYA2012 -36666 with FEDER support, PRICIT -S2009/ESP -\n1496, Grupos Gobierno Vasco IT765 -13 and UPV/EHU UFI11/55. \n \nReferences \n Arnold, G., Haus, R., Kappel, D., Drossart, P., Piccioni, G., 2008. Venus surface data \nextraction from VIRTIS/Venus Express measu rements: Estimation of a quantitative \napproach. J. Geophys. Res. 13 (E00B10), 13. \n Barentsen, G., Koschny, D. 2008. The Venus ground -based image Active Archive: A \ndatabase of amateur observations of Venus in ultraviolet and infrared light. Planet. \nand Space Sci., 56, 1444 –1449 \n Barstow, J. K.; Tsang, C.C.C.; Wilson, C.F.; Irwin, P.G.J.; Taylor, F.W.; \nMcGouldrick, K.; Drossart, P.; Piccioni, G.; Tellman, S. 2012. Models of the global \ncloud structure on Venus derived from Venus Express observations. Icarus , 217 , 542 -\n560. \n Belton, M.J.S., Smith, G.R., Schubert, G., and Del Genio, A.D., 1976. Cloud Patterns, \nWaves and Convection in the Venus Atmosphere. J. Atm. Sci. , 33, 1394 –1417. \n Belton, M. J. S. et al. 1991. Images from Galileo of the Venus cloud deck. Science , \n253, 1531 –1536. \n Cardesin, A., 2010. Study and implementation of the end -to-end data pipeline for the \nVIRTIS Imaging Spectrometer onboard Venus Express: “From science operations \nplanning to data archiving and higher level processing”. Ph.D. Thesis. Univers ita’ \ndegli study di Padova, Italy. \n Counselman , C.C. III, Gourevich, S.A., King R. W.,Loriot G., Band Ginsberg, E.S., \n1980. Zonal and meridional circulation of the lower atmosphere of Venus determined \nby radio interferometry. J. Geophys. Res. , 85, 8026 –8031 . \n Del Genio, A.D., Rossow, W.B., 1990. Planetary -scale waves and the cyclic nature \nof cloud top dynamics on Venus. J. Atmos. Sci. , 47, 293 –318. \n Drossart, P., and 43 colleagues, 2007. Scientific goals for the observation of Venus \nby VIRTIS on ESA/Venus expr ess mission. Planet.Spac.Science , 55, 1653 -1672. \n Erard, S. 2008 . VEx-VIRTIS to planetary science archive interface control document, \nRep. VIRTIS VIR -LES-IC-2269, Lab. dEtud. Spatiales et dInstrum. en Astrophys., \nParis. \n Erard, S.; Drossart, P.; Piccioni, G. 2009. Multivariate analysis of Visible and Infrared \nThermal Imaging Spectrometer (VIRTIS) Venus Express nightside and limb \nobservations. Journal of Geophys. Res., 114, E00B27. \n Garate -Lopez, I.; Hueso, R.; Sánchez -Lavega, A.; Peralta, J.; Piccioni, G.; Dro ssart, \nP. 2013. A chaotic long -lived vortex at the southern pole of Venus. Nature \nGeoscience , 6, 254 -257. \n \n García -Melendo, E.; Hueso, R.; Sánchez -Lavega, A.; Legarreta, J.; del Río -\nGaztelurrutia, T.; Pérez -Hoyos, S.; Sanz -Requena, J.F. 2013. Atmospheric d ynamics \nof Saturn's 2010 giant storm. Nature Geoscience , 6, 525 –529. \n García -Muñoz, A.; Mills, F. P.; Slanger, T. G.; Piccioni, G.; Drossart, P. 2009. Visible \nand near -infrared nightglow of molecular oxygen in the atmosphere of Venus. Journal \nof Geophys. Res., 114, E12002. 33 \n Gaulme, P.; Shmider, F. -X.; Grec, C.; López -Ariste, A.; Th. Widemann ; Gelly, B. \n2008. Venus wind maps at cloud top with the MTR/THEMIS visible spectrometer, I: \nInstrumental performance and first results. Planet. Space Sci ., 56, 1335 -1343. \n Gierasch, P. J. 1975. Meridional circulation and the maintenance of the Venus \natmospheric rotation. J. Atmos. Sci . 32, 1038 –1044. \n Gierasch, P.J., Goody, R. M.; Young, R. E.; Crisp, D.; Edwards, C.; Kahn, R.; \nMcCleese, D.; Rider, D.; del Genio, A.; Greel ey, R.; Hou, A.; Leovy, C.B.; Newman, \nM. 1997: The general circulation of the Venus atmosphere: An assessment. In Venus \nII — Geology, Geophysics, Atmosphere, and Solar Wind Environment . Eds. J.W. \nBucher, D.M. Hunten, and R.J. Phillips, 459 -500, University of Arizona Press, \nTucson. \n Gonzalez, R. C.; Woods, R.E., 2008. Digital Image Processing (3rd edition). Pearson \nPrentice Hall. \n Haus, R.; Kappel , D.; Arnold, G. 2014. Atmospheric Thermal structure and cloud \nfeatures in the southern hemisphere of Venus as retrieved from VIRTIS/VEX \nradiation measurements. Icarus , 232, 232 -248. \n Hueso, R., Legarreta J., García -Melendo, E., Sánchez -Lavega , A., and Pérez -Hoyos, \nS. 2009. The jovian anticyclone BA II. Circulation and interaction with the zonal jets. \nIcarus , 203, 499 –515. \n Hueso, R., Legarreta, J. Rojas, J.F., Peralta, J., Pérez -Hoyos, S., del Río -Gaztelurrutia, \nT., Sánchez -Lavega, A., 2010. The Planetary Laboratory for Image Analysis (PLIA). \nAdv. in Space Res ., 46, 1120 -1138. \n Hueso, R.; Sánchez -Lavega, A.; Peralta, J. 2012. Assessing the long -term variability \nof Venus winds at cloud level from VIRTIS Venus Express. Icarus , 217, 585 -598. \n Ignatiev, N. I., Titov, D. V., Piccioni, G., Drossart, P., Markiewicz, W. J., Cottini, V., \nRoatsch, Th., Almeida, M., Manoel, N. 2009. Altimetry of the Venus cloud tops from \nthe Venus Express observations . Journal of Geophys. Res. , 114, E00B43. \n Kouyama T., Imamura T., Nakamura M., Satoh T. and Futaana Y., 2012: Horizontal \nstructure of planetary -scale waves at the cloud top of Venus deduced from Galileo \nSSI images with an improved cloud -tracking technique. Planetary and Space Science , \n60, 207 -216. \n Kouyama, T .; Imamura, T.; Nakamura, M.; Satoh, T.; Futaana, Y. 2013. Long -term \nvariation in the cloud -tracked zonal velocities at the cloud top of Venus deduced from \nVenus Express VMC images. J. Geophys. Res. Planets, 118, 37 -46. \n Khatuntsev, I. V.; Patsaeva, M. V.; Titov, D. V.; Ignatiev, N. I.; Turin, A. V.; Limaye, \nS. S.; Markiewicz, W. J.; Almeida, M.; Roatsch, Th.; Moissl, R. 2013. Cloud level \nwinds from the Venus Express Monitoring Camera imaging. Icarus , 226, 140 -158. \n Lebonnois, S., Hourdin, F., Eymet, V., Cres pin, A., Fournier, R., Forget, F. 2010. \nSuperrotation of Venus’ atmosphere analyzed with a full general circulation model. \nJournal of Geophys. Res. , 115, E003458. \n Lebonnois S.; Lee C., Yamamoto M., Dawson J., Lewis S.R., Mendonca J., Read P., \nParish H.F., Schubert G., Bengtsson L., Grinspoon D., Limaye S.S., Schmidt H., \nSvedhem H. and Titov D.V., 2013: Models of Venus Atmosphere. In “Towards \nUnderstanding the Climate of Venus”, ISSI Scientific Report Series 11, Springer. \n Lee, Y.J.; Titov, D.V.; Tellman, S.; Piccialli, A.; Ignatiev, N.; Pätzold, M.; Häusler, \nB.; Piccioni, G.; Drossart, P. 2012. Vertical structure of the Venus cloud top from the \nVeRa and VIRTIS observations onboard Venus Express. Icarus , 217, 599 -609. 34 \n Lellouch, E.; Paubert, G.; Moreno, R.; Mou llet, A. 2008. Monitoring Venus’ \nmesospheric winds in support of Venus Express: IRAM 30 -m and APEX \nobservations. Planet. and Space Science , 56, 1355 -1367. \n Limaye, S. S.; Suomi, V. E. 1981. Cloud motions on Venus - Global structure and \norganization. J. Atmo s. Sci. , 38, 1220 -1235. \n Limaye, S. S.; Grund, C. J.; Burre, S. P. 1982. Zonal mean circulation at the cloud \nlevel on Venus – Spring and fall 1979 OCPP observations. Icarus , 51, 416 -439. \n Limaye, S. S.; Grasotti, C. G.; Kuetemeyer, M. J. 1988. Venus: Cloud l evel circulation \nduring 1982 as determined from Pioneer Cloud Photopolarimeter images – Time and \nzonally averaged circulation. Icarus , 73, 193 -211. \n Limaye, S. S. 2007. Venus atmospheric circulation: Known and unknown. JGR, 112, \nJE002814. \n Luz, D. Berry, D. L., Piccioni, G., Drossart, P., Polliti, R., Wilson, C.F., Erard, S., \nand Nuccilli, F. 2011. Venus’s Southern Polar Vortex Reveals Precessing Circulation. \nScience , 332, 577. \n Markiewicz, W.J. et al. 2007a. Venus Monitoring Camera for Venus Express. \nPlanetary and Space Science , 55, 1701 -1711. \n Markiewicz, W.J.; Titov, D.V.; Limaye, S.S., Keller, H.U.; Ignatiev, N.; Jaumann, R.; \nThomas, N.; Michalik, H.; Moissl, R.; Russo, P. 2007b. Morphology and dynamics \nof the upper cloud layer of Venus. Nature , 450, 633-636. \n Machado, P.; Luz, D.; Widemann, Th.; Lellouch, E.; Witasse, O. 2012 . Mapping \nzonal winds at Venus’s cloud tops from ground -based Doppler velocimetry. Icarus , \n221, 248 -261. \n Machado, P.; Luz, D.; Widemann, Th.; Peralta, J. 2014. Wind circulation regimes at \nVenus’ cloud tops : Ground -based Doppler velocimetry using CFHT/ESPaDOnS and \ncomparison with simultaneous cloud tracking measurements using VEx/VIRTIS in \nFebruary 2011. Icarus (in press). \n Moissl, R., Khatutsev, I., Limaye, S.S., Titov, D.V., Markiewicz, W.J., Ignatiev, N. \nI., Roatsch, T., Matz, K. -D., Jaumann, R., Almeida, M., Portyankina, G., Behnke, T., \nand Hviid, S.F. 2008. Venus cloud top winds from tracking UV features in Venus \nMonitoring Camera images. Journal of Geophys. Res. , 114, JE00 B31. \n Mousis, O. et al. 2014. Instrumental methods for professional and amateur \ncollaborations in Planetary Astronomy, Exp. Astron. (submitted). \n Nakamura, M. and 21 collaborators, 2014. Return to Venus of the Japanese Venus \nClimate Orbiter AKATSUKI. Acta Astronautica , 93, 384 -389. \n Newman, M.; Schubert, G.; Kliore, A. J.; Patel, I. R. 1984. Zonal winds in the \nmiddle atmosphere of Venus from Pioneer Venus radio occultation data. Journal of \nthe Atmospheric Sciences, 41, 1901 -1913. \n Peralta, J., Hueso, R., Sánc hez-Lavega, A. 2007. A reanalysis of Venus winds at two \ncloud levels from Galileo SSI images, Icarus , 190, 469 – 477. \n Peralta, J., Hueso, R., Sánchez -Lavega, A. 2008. Characterization of mesoscale \ngravity waves in the upper and lower clouds of Venus from VEX -VIRTIS images, \nJournal of Geophysical Research , 113, E00B18. \n Peralta J., Luz, D., Berry D.L., Tsang C.C.C., Sánchez -Lavega A., Hueso R., Piccioni \nG. and Drossart P. , 2012: Solar migrating atmospheric tides in the winds of the polar \nregion of Venus. Icarus , 220, 958 -970. \n Peralta, J.; Imamura, T.; Read, P.L.; Luz, D.; Piccialli, A.; López -Valverde, M.A., \n2014a. Analytical Solution for Waves in Planets with Atmospheric Superrotation. I. 35 \n Acoustinc and Inertia -Gravity waves. The Astrophysical Journal Supplement, 213, \narticle id. 17, 18 pp. \n Peralta, J.; Imamura, T.; Read, P.L.; Luz, D.; Piccialli, A.; López -Valverde, M.A., \n2014b. Analytical Solution for Waves in Planets wit h Atmospheric Superrotation. II. \nLamb, Surface, and Centrifugal Waves. The Astrophysical Journal Supplement, 213, \narticle id. 18, 15 pp. \n Peralta, J.; Sánchez -Lavega, A.; López -Valverde, M.A.; Luz, D.; Machado, P. 2014c. \nVenus’ major cloud feature as a wind -distorted wave. Submitted to A&A Letters. \n Piccialli, A.; Titov, D.V.; Grassi, D.; Khatuntsev, I.; Drossart, P.; Piccioni, G.; \nMigliorini, A. 2008. Cyclostrophic winds from the Visible and Infrared Thermal \nImaging Spectrometer temperature sounding: A preliminary analysis. Journal of \nGeophysical Research, 113, E00B11. \n Piccialli, A.; Tellmann, S.; Titov, D. V.; Limaye, S. S.; Khatuntsev, I. V.; \nPätzold, M.; Häusler, B. 2012. Dynamical properties of the Venus mesosphere from \nthe radio -occultation experiment VeRa onboard Venus Express. Icarus 217, 669 -681. \n Piccialli, A.; Titov, D. V.; Sánchez -Lavega, A.; Peralta, J., Shalygina, O.; \nMarkiewicz, W. J. and Svedhem, H. 201 4. High latitude gravity waves at the Venus \ncloud tops as observed by the Venus Monitoring Camera on board Venus Express. \nIcarus , 227, 94 -111. \n Piccioni, G., Drossart, P. et al., 2007, South -polar features on Venus similar to those \nnear the north pole, Nature, 450, 637 -640. \n Preston, R.A., et al. 1986. Determination of Venus winds by ground -based radio \ntracking of the VEGA ballons, Science , 231, 1414 -1416. \n Read, P.L. 2013. The Dynamics and Circulation of Venus Atmosphere. In “Towards \nUnderstanding the Climate of Venus”, ISSI Scientific Report Series 11, Springer. \n Rossow, W. B.; del Genio, A. D.; Limaye, S. S.; Travis, L. D. 1980. Cloud \nmorphology and motions from Pioneer Venus images. J. Geophys. Res. , 85, 8107 -\n8128. \n Rossow, W. B., del Genio, A. D. and Eichler, T. 1990. Cloud -tracked winds from \nPioneer Venus OCPP images. Journal of the Atmospheric Sciences , 47, 2053 -2084. \n Sánchez -Lavega, A., Hueso, R., Piccioni, G., Drossart, P., Peralta, J., Pérez -Hoyos, \nS., Wilson, C.F., Taylor, F.W., Baines, K.H., Luz , D., Erard, S. and Lebonnois, S., \n2008. Variable winds on Venus mapped in three dimensions. Geophys. Res. Letters , \nvol 35, L13204. \n Sánchez -Lavega, A. 2011. An Introduction to Planetary Atmospheres, Taylor and \nFrancis. \n Schubert G.; Counselman, C.C. III.; Pettengill, G.; Shapiro, I. I.; Hansen, J.; Travis, \nL.; Limaye, S.S.; Suomi, V. E.; Seiff, A.; Taylor, F. 1977. Dynamics, winds, \ncirculation and turbulence in the atmosphere of Venus. Space Science Reviews , 20, \n357-387. \n Schubert, G. 1983. General circulation and the dynamical state of the Venus \natmosphere, in Venus , edited by D. L. Hunten et al., pp. 681 –765, University of \nArizona Press, Tucson. \n Schubert, G., Counselman, III, C. C., Pettengill, G., Shapiro, I. I., Hansen, J., Travis, \nL., Limaye, S. S., Suomi, V. E., Seiff, A., and Taylor, F. 1977. Dynamics, winds, \ncirculation and turbulence in the atmosphere of Venus. Space Science Reviews , 20, \n357–387. \n Toigo, A., Gierasch, P. J. and Smith, M. D. 1994. High resolution clou d feature \ntracking on Venus by Galileo. Icarus, 109, 318 -336. 36 \n Titov, D. V.; Svedhem, H.; Koschny, D.; Hoofs, R.; Barabash, S.; Bertaux, J. -L.; \nDrossart, P.; Formisano, V.; Hausler, B.; Korablev, O.; Markiewicz, W.J.; Nevejans, \nD.; Patzold, M.; Piccioni, G. ; Zhang, T.L.; Merritt, D.; Witasse, O.; Zender, J.; \nAccomazzo, A.; Sweeney, M.; Trillard, D.; Janvier, M.; Clochet, A.; 2006. Venus \nExpress science planning. Planetary and Space Science , 54, 1279 -1297. \n Tivov, D. V., Markiewicz, W. J., Ignatiev, N. I., Son g, L., Limaye, S. S., Sánchez -\nLavega, A., Hesemann, J., Almeida, M., Roatsch, Th., Matz, K -D., Scholten, F., \nCrisp, D., Esposito, L.W., Hviid, S. F., Jaumann, F., Keller, H. U. and Moissl, R., \n2012. Morphology of the cloud tops as observed by the Venus Exp ress Monitoring \nCamera. Icarus , 217, 682 -701. \n Widemann, Th., Lellouch, E., Campargue, A. 2007. New wind measurements in \nVenus’ lower mesosphere from visible spectroscopy, Planet. Space Science , 55, 741 –\n1756. \n Widemann, Th.; Lellouch, E.; Donati, J.-F., 2008 . Venus Doppler winds at cloud tops \nobserved with ESPaDOnS at CFHT. Planet. Space Science , 56, 1320 -1334. \n Zasova, L.V.; Ignatiev, N.; Khatuntsev, I.; Lindin, V. 2007. Structure of the Venus \natmosphere. Planet. Space Sci. 55, 1712 -1728. \n \n \n 37 \n Appendix A \nTable A1: Orbits and distribution of tracked cloud features \nOrbit Date UV (380 nm) VIS NIR T \nyr/mm/dd Latitude LT N Latitude LT N Latitude LT N (hr) \n VOI* 2006 -04-19 03-71 08.8-16.1 145 06-63 08.1-14.3 106 1.2 \n 30 2006 -05-20 53-78 10.7.15.4 42 51-78 10.1-15.2 34 51-75 11.0-15.0 18 1.5 \n 34** 2006 -05-24 11-40 11.2-13.9 211 0-37 11.2-13.6 112 1.5 \n 69** 2006 -06-28 07-62 09.2-16.1 466 19-65 11.3-14.7 467 0.8 \n 70 2006 -06-29 34-79 11.0-15.3 235 54-78 11.1-15.0 140 19-80 11.1-14.8 199 0.8 \n 73* 2006 -07-02 50-80 09.5-14.3 8 52-76 09.5-13.5 3 1.0 \n 74* 2006 -07-03 52-74 09.2-12.8 10 52-74 09.2-12.8 10 1.0 \n 75* 2006 -07-04 53-75 09.0-14.4 16 53-72 07.0-12.9 14 1.0 \n 76* 2006 -07-05 45-72 12.3-16.1 18 46-76 11.8-15.2 30 1.0 \n 77* 2006 -07-06 52-80 13.2-16.3 7 49-81 12.2-15.8 24 1.0 \n 78* 2006 -07-07 46-65 12.1-16.3 13 54-80 12.4-17.0 29 1.0 \n 79* 2006 -07-08 50-65 13.5-16.2 8 1.0 \n 80* 2006 -07-09 50-70 11.9-14.6 15 51-80 12.4-17.0 29 1.0 \n 81* 2006 -07-10 53-62 13.2-15.6 9 49-73 12.8-16.2 42 1.0 \n 82* 2006 -07-11 59-78 12.1-15.1 10 50-74 12.4-15.3 20 1.0 \n 84* 2006 -07-13 60-81 09.5-13.0 8 61-79 08.6-13.3 17 1.0 \n 85* 2006 -07-14 54-85 08.0-13.4 32 52-76 08.5-14.2 30 1.0 \n 86* 2006 -07-15 57-82 08.5-12.9 10 56-83 07.7-13.0 23 1.0 \n 94* 2006 -07-23 65-76 07.1-12.5 8 60-84 06.9-13.3 22 1.0 \n 95* 2006 -07-24 56-83 06.6-12.6 29 60-82 06.6-14.1 23 1.0 \n 96* 2006 -07-25 66-82 07.0-12.8 7 60-85 06.4-15.5 19 1.0 \n 97* 2006 -07-26 55-87 10.8-16.2 16 57-85 12.0-17.5 24 1.0 \n 98* 2006 -07-27 69-79 12.5-13.7 4 63-83 10.6-16.4 25 1.0 \n220* 2006 -11-26 65-84 07.6-15.0 26 0.5 \n 244* 2006 -12-20 68-84 08.6-15.3 15 70-85 07.5-15.9 42 2.0 \n 283* 2007 -01-28 55-78 10.6-16.1 24 1.0 \n 288 2007 -02-02 55-86 10.3-16.5 35 57-78 11.8-16.8 29 60-86 11.0-16.8 33 1.0 \n 388 2007 -05-13 63-83 09.1-17.6 13 66-86 10.1-17.6 8 0.5 \n 392 2007 -05-17 65-84 08.2-16.3 25 77-80 07.7-15.8 4 72-82 07.2-16.6 10 0.5 \n 396 2007 -05-21 66-82 08.0-16.7 36 69-82 06.1-17.4 27 0.5 \n 436 2007 -06-30 54-84 07.1-13.0 51 56-85 07.3-12.6 48 54-84 07.1-12.2 44 0.93 \n 437 2007 -07-01 54-82 07.5-13.3 52 52-80 07.5-12.6 52 51-85 07.1-14.0 48 0.93 \n 438 2007 -07-02 60-81 07.1-12.3 37 52-81 07.2-11.6 26 52-83 07.2-12.0 34 1.0 \n 443 2007 -07-07 55-84 06.5-13.5 63 52-84 07.0-13.1 53 42-84 06.7-12.7 51 1.0 \n 444 2007 -07-08 53-85 06.8-13.1 57 53-83 06.6-12.8 49 49-83 07.0-13.8 66 1.0 \n 448 2007 -07-12 65-83 08.2-15.3 32 64-83 07.2-15.3 37 64-84 07.0-15.3 35 1.0 \n 451* 2007 -07-15 49-66 10.4-11.8 7 1.0 \n 452* 2007 -07-16 54-65 09.9-12.2 10 1.0 \n 453* 2007 -07-17 49-67 09.6-12.6 16 1.0 \n 454* 2007 -07-18 52-71 08.8-12.8 11 1.0 \n 456* 2007 -07-20 59-81 09.9-14.3 20 0.9 \n 457* 2007 -07-21 57-82 09.0-14.4 22 0.9 \n 458* 2007 -07-22 57-83 10.3-14.4 22 0.9 \n 459* 2007 -07-23 53-77 09.1-14.0 22 0.9 \n 460* 2007 -07-24 57-75 09.4-14.3 25 0.9 \n 461* 2007 -07-25 56-81 10.1-15.0 26 1.0 \n 462* 2007 -07-26 57-80 09.3-16.1 27 0.9 \n 463* 2007 -07-27 54-77 10.0-15.8 22 0.9 \n 465* 2007 -07-29 54-72 10.5-14.5 17 1.0 \n 466** 2007 -07-30 54-81 07.8-15.1 108 54-80 07.7-14.2 54 52-81 08.3-15.0 56 0.5, 1.0 \n 467* 2007 -07-31 54-80 09.6-15.7 28 0.5 \n 468* 2007 -08-01 52-83 09.2-15.7 25 1.0 \n 469* 2007 -08-02 57-79 08.6-16.6 24 1.0 \n 470* 2007 -08-03 56-81 10.4-14.5 24 0.9 \n 471* 2007 -08-04 57-73 11.2-16.0 21 0.9 \n 472* 2007 -08-05 58-77 09.7-15.2 21 0.9 \n 473* 2007 -08-06 68-81 08.7-16.9 27 0.5 \n 474 2007 -08-07 65-78 08.0-15.7 29 69-83 07.7-16.3 24 66-83 08.5-15.7 25 1.0 \n 475 2007 -08-08 65-86 07.5-16.8 164 64-85 07.3-17.0 153 65-86 06.9-17.2 158 1.0 \n 476 2007 -08-09 63-85 07.4-14.9 41 70-85 07.6-14.5 22 67-85 07.3-14.8 38 1.0 \n 626 2008 -01-06 23-75 08.0-16.1 165 24-63 07.6-15.4 91 21-66 07.9-15.2 135 0.5 \n 628 2008 -01-08 19-76 08.1-15.8 191 24-67 08.3-15.4 62 22-83 07.5-15.1 44 0.5 \n 654 2008 -02-03 05-53 07.2-10.0 192 10-49 07.6-09.6 44 10-42 07.5-09.9 75 1.0 \n 684 2008 -03-04 24-59 10.4-13.1 110 24-56 10.8-12.9 66 23-59 10.4-13.1 71 1.0 \n 716 2008 -04-05 15-65 09.2-12.5 179 24-60 09.4-12.4 69 25-58 09.5-12.3 53 0.5 38 \n 740 2008 -04-29 17-69 08.5-16.5 215 26-56 09.1-12.5 86 1.0 \n 743 2008 -05-02 20-63 13.4-16.3 198 20-72 13.3-16.1 58 1.0 \n 885 2008 -09-21 65-80 08.8-15.0 42 65-80 08.7-12.5 28 0.5 \n 915 2008 -10-21 71-81 08.8-16.0 20 72-82 08.4-14.2 19 1.0 \n 948 2008 -11-23 17-82 07.7-15.3 197 17-83 09.2-15.1 151 18-65 09.4-13.0 64 1.5 \n1043 2009 -02-26 69-83 08.4-16.7 35 1.0 \n1066 2009 -03-21 59-85 07.6-14.6 33 55-88 07.7-14.9 58 58-86 10.2-14.2 35 1.0 \n1067 2009 -03-22 49-74 09.0-13.6 70 49-86 07.9-15.2 69 52-84 07.8-13.2 74 1.0 \n1070 2009 -03-25 09-57 07.7-10.2 310 16-63 07.3-09.9 82 16-62 07.2-10.1 110 1.0 \n1071 2009 -03-26 11-67 07.5-10.3 82 17-66 07.5-10.5 92 15-67 07.4-10.6 133 1.0 \n1072 2009 -03-27 10-64 07.3-09.7 142 15-63 07.3-10.1 83 14-54 07.8-10.0 114 1.0 \n1073 2009 -03-28 16-63 07.4-10.4 197 27-64 07.1-10.2 76 16-62 07.4-10.5 143 1.0 \n1161 2009 -06-24 60-78 08.9-13.5 26 55-72 09.5-11.0 28 1.25 \n1185 2009 -07-18 24-66 08.4-11.6 44 32-59 10.2-11.5 11 1.0 \n1186 2009 -07-19 27-65 09.1-13.8 106 31-63 09.1-13.4 36 1.0 \n1187 2009 -07-20 33-69 10.5-15.6 46 35-45 10.6-11.3 7 1.25 \n1188 2009 -07-21 32-63 08.5-11.5 87 32-62 08.4-11.4 49 1.25 \n1189 2009 -07-22 17-51 14.4-16.4 29 1.25 \n1294 2009 -11-04 48-87 07.5-12.3 149 50-85 08.2-11.9 34 57-85 08.5-12.0 24 1.0 \n1307 2009 -11-17 10-57 08.0-10.7 227 16-55 08.5-10.4 46 16-57 08.0-10.5 77 1.0 \n1408 2010 -02-26 18-54 14.1-16.9 117 30-55 14.1-15.9 30 23-48 14.2-16.9 52 1.0 \n1416 2010 -03-07 49-78 10.3-15.0 64 50-73 10.3-14.6 77 49-71 10.2-14.1 16 1.0 \n1417 2010 -03-08 52-79 09.9-14.4 39 51-83 10.4-14.0 17 57-67 11.2-11.8 4 1.0 \n1500 2010 -05-29 50-81 07.3-11.6 34 46-82 07.1-10.9 23 35-85 07.0-11.7 47 0.5 \n1557 2010 -07-25 11-78 07.7-17.0 350 19-81 07.4-17.1 149 20-80 08.4-17.5 60 1.33 \n1640 2010 -10-16 19-71 12.0-14.5 121 30-80 09.8-15.7 39 22-80 09.7-16.3 11 1.0 \n1666 2010 -11-11 42-85 09.6-14.7 54 46-82 10.2-6.0 35 44-83 11.1-17.3 24 1.0 \n1719 2011 -01-03 08-54 09.1-15.2 423 18-55 09.3-15.1 142 15-51 09.0-15.1 73 1.25 \n1730 2011 -01-15 17-82 07.0-16.8 342 20-80 09.0-16.8 167 18-85 07.2-17.5 246 1.33 \n1743 2011 -01-27 17-79 06.9-15.2 327 22-82 07.4-14.6 149 15-82 06.6-15.4 218 1.33 \n1768* 2011 -02-21 23-79 07.2-16.1 82 1.33 \n1808 2011 -04-02 52-82 08.5-14.8 57 54-78 08.0-13.8 40 49-76 10.0-13.8 45 0.83 \n1865 2011 -05-29 14-59 13.1-16.0 166 0.25 \n1958 2011 -08-30 25-69 08.1-11.8 184 25-74 08.0-11.8 81 27-77 07.7-11.4 73 0.67 \n1974 2011 -09-15 33-85 08.6-13.5 129 29-83 08.4-12.5 146 26-84 09.1-13.1 103 0.25 \n1975 2011 -09-16 16-58 08.0-10.9 177 20-56 08.0-10.8 88 0.25 \n2004 2011 -10-15 44-86 07.7-12.3 71 42-86 07.4-14.3 53 43-85 08.5-12.4 77 0.83 \n2060 2011 -12-10 13-67 12.5-15.4 240 18-82 10.4-16.7 131 19-87 11.7-17.0 49 1.0 \n2082 2012 -01-01 23-57 12.2-14.1 31 30-80 12.1-15.7 22 1.0 \n2084 2012 -01-03 19-49 13.0-13.8 21 39-48 13.1-14.1 8 0.5 \n2091 2012 -01-10 11-73 10.8-14.7 160 21-83 10.1-14.5 89 20-85 10.4-14.8 25 0.5 \n2093 2012 -01-12 19-63 11.8-15.0 45 29-80 10.8-14.3 36 21-84 11.8-13.3 5 0.6 \n2094 2012 -01-13 15-78 09.9-13.9 63 23-79 10.4-14.2 71 77-81 09.8-13.7 4 0.6 \n2095 2012 -01-14 17-74 11.0-14.2 123 26-83 09.9-14.1 83 28-84 10.7-14.8 29 0.6 \n2097 2012 -01-16 18-73 10.8-14.6 54 27-77 09.5-15.4 57 74-81 08.8-15.4 6 0.6 \n2101 2012 -01-20 21-37 13.0-14.3 42 30-39 13.3-14.3 17 23-31 12.9-13.3 3 0.5 \n2102 2012 -01-21 26-65 12.9-14.3 24 28-77 10.3-14.2 26 0.6 \n2103 2012 -01-22 23-65 12.5-14.4 24 29-73 10.7-14.7 15 81-82 09.4-15.3 4 0.6 \n2105 2012 -01-24 23-67 10.7-14.7 75 29-80 10.7-15.4 17 67-81 11.0-14.7 3 0.6 \n2111 2012 -01-30 65-71 11.3-13.7 17 65-82 11.6-15.7 21 0.5 \n2112 2012 -01-31 23-78 10.9-15.5 90 28-76 10.4-14.8 44 69-75 11.8-13.7 5 0.6 \n2113 2012 -02-02 26-80 12.0-15.5 64 29-83 11.3-15.3 51 0.6 \n2114 2012 -02-03 20-79 12.4-16.5 67 29-80 10.9-14.8 33 0.6 \n2115 2012 -02-04 18-79 11.4-16.0 110 30-70 11.6-15.7 35 60-74 11.5-13.7 24 0.6 \n \nNotes: (*) Data obtained by manual tracking without a correlation algorithm. (**) Orbits where both \nmanual tracking and correlation data have been obtained. Horizontal lines mark the different periods \nanalyzed in this work. " }, { "title": "2401.17067v1.Boundary_controllability_of_a_one_dimensional_phase_field_system_with_one_control_force.pdf", "content": "arXiv:2401.17067v1 [math.OC] 30 Jan 2024Boundary controllability of a one-dimensional phase-field\nsystem with one control force\nManuelGonz´alez-Burgos∗, Gilcenio R. Sousa-Neto†\nAbstract\nIn this paper, we present some controllability results for l inear and nonlinear phase-field\nsystems of Caginalp type considered in a bounded interval of Rwhen the scalar control force\nacts on the temperature equation of the system by means of the Dirichlet condition on one\nof the endpoints of the interval. In order to prove the linear result we use the moment\nmethod providing an estimate of the cost of fast controls. Us ing this estimate and following\nthe methodology developed in [ 19], we prove a local exact boundary controllability result to\nconstant trajectories of the nonlinear phase-field system. To the authors’ knowledge, this is\nthe first nonlinear boundary controllability result in the f ramework of non-scalar parabolic\nsystems, framework in which some “hyperbolic” behaviors co uld arise.\nKeywords. Phase-field system, boundary controllability.\n1 Introduction\nThis work deals with the boundary controllability properties of a phas e-field system of Caginalp\ntype (see [ 9]) which is a model describing the transition between the solid and liquid p hases in\nsolidification/melting processes of a material occupying an interval:\n\n\n˜θt−ξ˜θxx+1\n2ρξ˜φxx+ρ\nτ˜θ=f1(˜φ) in QT:= (0,π)×(0,T),\n˜φt−ξ˜φxx−2\nτ˜θ=f2(˜φ) in QT,\n˜θ(0,·) =v,˜φ(0,·) =c,˜θ(π,·) = 0,˜φ(π,·) =con (0,T),\n˜θ(·,0) =˜θ0,˜φ(·,0) =˜φ0 in (0,π).(1.1)\nHere,T >0 is some final time, ˜θ=˜θ(x,t) denotes the temperature of the material, ˜φ=˜φ(x,t) is\nthe phase-field function used to identify the solidification level of th e material, c∈ {−1,0,1}and\nthe functions f1andf2are the nonlinear terms which come from the derivative of the classic al\nregular double-well potential Wand are defined by\nf1(˜φ) =−ρ\n4τ/parenleftig\n˜φ−˜φ3/parenrightig\nandf2(˜φ) =1\n2τ/parenleftig\n˜φ−˜φ3/parenrightig\n.\nBesides,ρ >0 is the latent heat, τ >0 represents the relaxation time and ξ >0 is the thermal\ndiffusivity. Finally, v∈L2(0,T) is the control force, which is exerted at point x= 0 by means of\nthe boundary Dirichlet condition, and the initial data ˜θ0,˜φ0are given functions.\nThe phase function ˜φdescribes the phase transition of the material (solid or liquid) in such\na way that ˜φ= 1 means that the material is in solid state and ˜φ=−1 in liquid state. Observe\n∗Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico and I nstituto de Matem´ aticas de la Universidad de Sevilla\n(IMUS), Facultad de Matem´ aticas, Universidad de Sevilla, C/ Tarfia S/N, 41012 Sevilla, Spain. Supported by grant\nMTM2016-76990-P, Ministry of Economy and Competitiveness (Spain). E-mail: manoloburgos@us.es\n†Centro Acadˆ emico do Agreste, NICIT, Universidade Federal de Pernambuco, 55002-970, Caruaru, PE, Brazil\n(gilceniorodrigues@gmail.com ). Partially supported by CAPES (Brazil) and MathAmSud COSI P.\n1that the temperature ˜θof the material could be zero and this could occur with the material in\nsolid or liquid phase. On the other hand, the phase-field variable ˜φdoes not have a direct physical\nmeaning. This is the reasonwaywe controlthe temperature ˜θwhich, in fact, is the unique variable\nwith physical meaning.\nThe objectiveofthis paper isto proveanull controllabilityresultat t imeTforthe temperature\nvariable ˜θof system ( 1.1). If we consider the transition region associated to the temperat ure, i.e.,\nthe set\nΓ(t) :=/braceleftig\nx∈(0,π) :˜θ(x,t) = 0/bracerightig\n,\nthen, the problem under consideration consists of proving that th ere exists a control vsuch that\nthe transition region associated to the temperature ˜θsatisfies Γ(T) = (0,π). It is interesting\nto underline that in this case the material could be in solid phase ( ˜φ(·,T) = 1), liquid phase\n(˜φ(·,T) =−1) or in an intermediate phase (mushy) which corresponds to ˜φ(·,T) = 0. In this\nwork we are interested in showing the null controllability result at time Tfor the temperature ˜θ\nbut keeping the material in solid state, c= 1, or liquid state, c=−1, at timeT, that is to say,\nproving that there exists a control v∈L2(0,T) such that system ( 1.1) has a solution ˜ y= (˜θ,˜φ)\n(in an appropriate space) such that\n˜θ(·,T) = 0 and ˜φ(·,T) =cin (0,π). (1.2)\nWe give a complementary analysis and results in the Appendix B, where we deal with the case\nwherec= 0.\nAs said before, the objective of this work is to study the controllab ility properties of sys-\ntem (1.1). Let us observe that we are exerting only one control force on t he system (a boundary\ncontrol) but we want to control the corresponding state ˜ y= (˜θ,˜φ) which has two components. In\nfact, the second equation in ( 1.1) is indirectly controlled by means of the term −2˜θ/τ. On the\nother hand, ( 1.1) is a nonlinear system with nonlinearities with a super-linear behavior a t infinity.\nTherefore, we can expect a local controllability result at time Tfor this system, that is to say, an\nexact controllability result to the trajectory (0 ,c) when the initial datum ( ˜θ0,˜φ0) is sufficiently\nclose to (0,c) in an appropriate norm (see for instance [ 17,11] for similar results in the scalar\nparabolic framework).\nSystem ( 1.1) is a particular class of more general n×nnonlinear parabolic control systems of\nthe form: \n\nyt−D∆y+Ay=F(y)+Bv1ωinQT:= Ω×(0,T),\ny=Cu1Γ0, on ΣT:=∂Ω×(0,T),\ny(·,0) =y0 in Ω,(1.3)\nwhereωand Γ 0are, respectively, open subsets of the smooth bounded domain Ω ⊂RNand of\nits boundary ∂Ω,D∈L(Rn), withn≥1, is a positive definite matrix, B,C∈L(Rm,Rn), with\nm≤n, andA= (aij)1≤i,j≤n∈L(Rn) are given matrices. In ( 1.3),F∈C0(Rn;Rn) is a nonlinear\ngiven function. Unlike the scalar case, even in the linear case F≡0, new difficulties arise in the\nstudy of the controllability properties of ( 1.3). WhenmT 0and it is not if T 0 is equivalent to the null controllability at the same time\nTof system ( 1.4). In order to prove the null controllability at time T >0 of system ( 1.4),\nwe will rewrite the controllability problem as a fixed-point problem for a convenient operator in\n3appropriate spaces. To perform this fixed-point strategy, we will first study the controllability\nproperties of the following autonomous linear system:\n\n\nθt−ξθxx+1\n2ρξφxx−ρ\n2τφ+ρ\nτθ= 0 in QT,\nφt−ξφxx+1\nτφ−2\nτθ= 0 in QT,\nθ(0,·) =v, φ(0,·) =θ(π,·) =φ(π,·) = 0 on (0 ,T),\nθ(·,0) =θ0, φ(·,0) =φ0 in (0,π),(1.6)\nwhich is a linearization of system ( 1.4) around the equilibrium (0 ,0). System ( 1.6) can also be\nwritten in the vectorial form\n\n\nyt−Dyxx+Ay=f inQT,\ny(0,·) =Bv, y(π,·) = 0 on (0 ,T),\ny(·,0) =y0, in (0,π),(1.7)\nwithy0= (θ0,φ0),f= (0,0) and\nD=\nξ−1\n2ρξ\n0ξ\n, A=\nρ\nτ−ρ\n2τ\n−2\nτ1\nτ\n, B=/parenleftbigg\n1\n0/parenrightbigg\n. (1.8)\nWe will see that, for every v∈L2(0,T),f∈L2(QT;R2) andy0∈H−1(0,π;R2), system ( 1.7)\npossesses a unique solution defined by transposition which satisfies\ny∈L2(QT;R2)∩C0/parenleftbig\n[0,T];H−1(0,π;R2)/parenrightbig\n,\nand depends continuously on the data v,fandy0. Observe that the previous regularity permits\nto pose the boundary controllability of system ( 1.6) in the space H−1(0,π;R2).\nLet us present our first main result: the boundary approximate co ntrollability at time T >0\nof system ( 1.6). One has:\nTheorem 1.1. Let us consider ξ,ρandτthree positive real numbers and let us fix T >0. Then,\nsystem(1.6)is approximately controllable in H−1(0,π;R2)at timeTif and only if one has\nξ2τ2(ℓ2−k2)2−2ξρτ(ℓ2+k2)−2ρ−1/ne}ationslash= 0,∀k,ℓ≥1, ℓ>k. (1.9)\nRemark 1.1. Condition ( 1.9) characterizes the approximate controllability property of sys-\ntem (1.6). Thus, ( 1.9) is a necessary condition for the null controllability of this system at time\nT >0. Observe that this condition is independent of the final time T. We will also see that\ncondition ( 1.9) is equivalent to the following property (see Proposition 3.2):“The eigenvalues of\nthe vectorial operators\nL=−D∂xx+AandL∗=−D∗∂xx+A∗, (1.10)\nwith domains D(L) =D(L∗) =H2(0,π;R2)∩H1\n0(0,π;R2), have geometric multiplicity equal\nto one”. Thus, condition ( 1.9) is a Fattorini-Hautus criterium for the boundary approximate\ncontrollability of the linear system ( 1.6) (see [12]).\nIn this work, we will also analyze the null controllability properties of s ystem (1.6). In this\nsense, one has:\nTheorem 1.2. Let us us fix T >0and consider ξ,ρandτ, positive real numbers satisfying (1.9)\nand\nξ/ne}ationslash=1\nj2ρ\nτ,∀j≥1. (1.11)\n4Then, system (1.6)is exactly controllable to zero in H−1(0,π;R2)at timeT >0. Moreover, there\nexist two positive constants C0andM, only depending on ξ,ρandτ, such that for any T >0,\nthere is a bounded linear operator\nC(0)\nT:H−1(0,π;R2)→L2(0,T)\nsatisfying\n/bardblC(0)\nT/bardblL(H−1(0,π;R2),L2(0,T))≤C0eM/T, (1.12)\nand such that the solution\ny= (θ,φ)∈L2(QT;R2)∩C0([0,T];H−1(0,π;R2))\nof system (1.6)associated to y0= (θ0,φ0)∈H−1(0,π;R2)andv=C(0)\nT(y0)satisfiesy(·,T) = 0.\nRemark 1.2. From the results stated in [ 3] and [18], it is well known that the linear system ( 1.6)\nis approximateand null controllableat any time T >0 and anypositive ξ,ρandτ, when the scalar\ncontrolv∈L2(QT) actsonthetemperatureequationof ( 1.1)asaright-handsidesourcesupported\non an open subset ωof the domain. These distributed controllability results are independ ent of\ncondition ( 1.9) and only use the cascade structure of system ( 1.6). Nevertheless, this cascade\nstructure is not enough when one deals with the boundary controlla bility of non-scalar problems\n(see for example [ 15], [5], [6], ... ). Again, the approximate and null controllability results stated\nin Theorems 1.1and1.2show the different nature of the controllability problem of scalar or\nnon-scalar parabolic systems.\nRemark 1.3. Theorem 1.2also provides an estimate of the cost of the control for system ( 1.6)\nthat drives the system from an initial datum y0= (θ0,φ0)∈H−1(0,π;R2) to the equilibrium at\ntimeT >0. To be precise, under assumption ( 1.9) and (1.11), Theorem 1.2implies that the set\nZT(y0) :={v∈L2(0,T) :y= (θ,φ) solution of ( 1.6) associated to y0satisfiesy(·,T) = 0},\nis nonempty for any T >0 and anyy0= (θ0,φ0)∈H−1(0,π;R2). We can then define the control\ncost for system ( 1.6) as\nK(T) = sup\n/bardbly0/bardbl=1/parenleftbigg\ninf\nv∈ZT(y0)/bardblv/bardblL2(0,T)/parenrightbigg\n,∀T >0.\nObserve that as a direct consequence of Theorem 1.2and inequality ( 1.12), we can obtain the\nfollowing estimate of this cost for system ( 1.6) at timeT >0:\nK(T)≤C0eM\nT,∀T >0, (1.13)\nwhereC0andMare positive constants only depending on the parameters in system (1.6) (see [21]\nand [14] for similar results in the scalar parabolic framework).\nRemark 1.4. As said before, condition ( 1.9) is equivalent to the simplicity of the spectrum of L\nandL∗. We will see in Proposition 3.2that condition ( 1.11) implies a stronger property of the\nspectra ofLandL∗: If we denote {Λk}k≥1⊂(0,∞) the sequence of eigenvalues of the operator\nL, with Λ k≤Λk+1for anyk≥1, then, there exist δ >0 and an integer q≥1 such that\n|Λk−Λn| ≥δ/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle,∀k,n∈N,|k−n| ≥q. (1.14)\nThis gap condition for the spectrum of Lis crucial to prove the null controllability at any positive\ntimeTof system ( 1.6) with control cost satisfying the estimate ( 1.13) for positive constants C0\nandMonly depending on ξ,ρandτ(for similar results, see [ 13] and [20]).\nIn the case in which assumption ( 1.11) does not hold, that is to say, if for some integer j≥1\none has\nξ=1\nj2ρ\nτ,\n5then, the eigenvalues of L(andL∗) concentrate (see Remark 3.2) and the gap condition ( 1.14) is\nnot valid. In fact, one has\ninf\nk,ℓ≥1,k/negationslash=ℓ|Λk−Λℓ|= 0.\nIn [6], the authors proved that when the eigenvalues {Λk}k≥1of the operator L=−D∂xx+A\nconcentrate, the controllability problem for system ( 1.7) (f≡0) has a minimal time T0∈[0,∞]\nof null controllability which is related to the condensation index of the sequence. Even in the case\nT0= 0 (and therefore, system ( 1.6) is null controllable for any T >0), without the separability\nassumption ( 1.14), providing an estimate of the control cost K(T) with respect to T >0 is an\nopen problem.\nRemark 1.5. In order to prove Theorem 1.2we will use the moment method, introduced in [ 13]\nin the framework of the boundary controllability of the one-dimensio nal scalar heat equation. To\nthis end, we will carry out an analysis of the properties of the eigenv alues ofLwhich will imply\ninequality ( 1.12) and estimate ( 1.13) for the control cost of system ( 1.6). These two inequalities\nare essential in the proof of the controllability property of the non linear system ( 1.1).\nLet us now present the local exact controllability result to the traj ectory (0,c) (c=±1) for\nthe nonlinear system ( 1.1). This is our third main result. One has:\nTheorem 1.3. Let us consider ξ,τandρthree positive numbers satisfying (1.9)and(1.11), and\nlet us fixT >0andc=−1orc= 1. Then, there exists ε >0such that, for any (˜θ0,˜φ0)∈\nH−1(0,π)×(c+H1\n0(0,π))fulfilling\n/bardbl˜θ0/bardblH−1+/bardbl˜φ0−c/bardblH1\n0≤ε, (1.15)\nthere exists v∈L2(0,T)for which system (1.1)has a unique solution\n(˜θ,˜φ)∈/bracketleftbig\nL2(QT)∩C0([0,T];H−1(0,π;R2))/bracketrightbig\n×C0(QT)\nwhich satisfies (1.2).\nTheorem 1.3establishes a local exact boundary controllability result at time Tfor the non-\nlinear system ( 1.1) when the parameters ξ,ρandτsatisfy (1.9) and (1.11). Similar distributed\ncontrollability results1have been proved in the N-dimensional case, without any assumption on\nthe parameters, using the cascade structure of the system (se e [3] and [18]). As in the linear\ncase (1.6), this cascade structure is not enough for dealing with the bounda ry controllability of\nsystem (1.1).\nWe end the presentation of our main results with some remarks.\nRemark 1.6. Following [ 19], Theorem 1.3will be proved using a point-fixed strategy. The\nkey point in its proof will be a boundary null controllability result for th e non-homogeneous\nsystem ( 1.7) when the function fis in an appropriate weighted- L2space. In turn, this null\ncontrollability result for ( 1.7) will use in a crucial way the estimates ( 1.12) and (1.13).\nRemark 1.7. The main results established in this paper only deal with the boundary controlla-\nbility of linear or nonlinear systems in space dimension one. This restric tion is mainly due to the\nfact that in its proofs we will use the moment method. In general, th e boundary controllability of\nparabolic systems in higher space dimension remains widely open and on ly some partial answers\nare known in the linear setting. To our knowledge, the only results on this issue are those of [ 1],\n[2] and [8]. In the two first articles, the results for parabolic systems are de duced from the study\nof the boundary control problem of two coupled wave equations us ing transmutation techniques.\nAs a result they rely on some geometric constraints on the control domain. In [ 8], the author\ncharacterize the boundary null-controllability of system ( 1.3) in the linear case ( B≡0 andF≡0)\nwhen Ω is a cylindrical domains of the form Ω = (0 ,π)×Ω2(Ω2is a smooth domain of RN−1,\n1The distributed control acts as a source in the temperature e quation.\n6N >1) and Γ 0:={0}×ω2(ω2is an open subset of Ω 2) without imposing geometric constraintson\nω2. It is important to highlight that these results use that the diffusion matrixDis a multiple of\nthe identity matrix. The boundary controllability of systems ( 1.1) and (1.6) in theN-dimensional\ncase is completely open.\nThe rest of the paper is organized as follows: In Section 2, we give some existence and unique-\nness results for the linearized versions of the phase-field system ( 1.1) and we recall some known\nresults on existence and bounds on biorthogonal families to complex exponentials. Section 3is\ndevoted to studying the spectral properties of the parabolic ope ratorsLandL∗given in ( 1.10).\nIn Section 4we prove the controllability results for the linear problem ( 1.6): In Subsection 4.1we\nprove the approximate controllability result at time Tfor system ( 1.6) (Theorem 1.1) and in Sub-\nsection4.2the corresponding null controllability result (Theorem 1.2). Theorem 1.3is proved in\nSection5. Before (Subsection 5.1), we prove a null controllability result for the non-homogeneous\nsystem (1.7) whenfbelongs to appropriate spaces. As a consequence, we provide a pr oof of The-\norem1.3in Subsection 5.2. We finish this paper with two appendices. In Appendix A, we prove\nthe existence and uniqueness result for the linearized system ( 1.7) and for its backward formula-\ntion. In Appendix Bwe give some additional results on the null controllability of the phase -field\nsystem (1.1), that is to say, we deal with the case c= 0 (see ( 1.2)).\n2 Preliminary results\nIn this paper we will use the following notations for norms. If Xis a Banach space, the norms of\nthespacesL2(0,T;X)andC0([0,T];X)willberespectivelydenotedby /bardbl·/bardblL2(X)and/bardbl·/bardblC0(X). We\nwill also work with the spaces L2(0,π;R2),H1\n0(0,π;R2) andH−1(0,π;Rd), with norms denoted\nby/bardbl · /bardblL2,/bardbl · /bardblH1\n0and/bardbl · /bardblH−1. On the other hand, we will use /an}bracketle{t·,·/an}bracketri}htas notation for the usual\nduality pairing between H−1(0,1;R2) andH1\n0(0,1;R2).\nFinally, throughout the paper Cwill stand for a generic positive constant that only depends\non the coefficients ξ,τandρin system ( 1.1), whose value may change from one line to another.\nFrequently, we will use the notation CTwhen it is convenient to specify the dependence of the\ngeneric constant with respect to the final time T.\nIn this section we will give some results related to the existence, uniq ueness and continuous\ndependence with respect to the data of the linear problem ( 1.7). To this aim, let us consider the\nlinear backwards in time problem:\n\n\n−ϕt−D∗ϕxx+A∗ϕ=ginQT,\nϕ(0,·) =ϕ(π,·) = 0 on (0 ,T),\nϕ(·,T) =ϕ0 in (0,π),(2.1)\nwhereDandAare given in ( 1.8) andϕ0andgare functions in appropriate spaces.\nLet us start with a first result on existence and uniqueness of stro ng solutions to system ( 2.1).\nOne has:\nProposition 2.1. Let us assume that ϕ0∈H1\n0(0,π;R2)andg∈L2(QT;R2). Then, system (2.1)\nhas a unique strong solution\nϕ∈C0([0,T];H1\n0(0,π;R2))∩L2(0,T;H2(0,π;R2)∩H1\n0(0,π;R2)).\nIn addition, there exists a positive constant C, only depending on DandA, such that\n/bardblϕ/bardblC0(H1\n0)+/bardblϕ/bardblL2(H2∩H1\n0)≤eCT/parenleftig\n/bardblg/bardblL2(L2)+/bardblϕ0/bardblH1\n0/parenrightig\n. (2.2)\nIn view of Proposition 2.1, we can define solution by transposition to system ( 1.7).\n7Definition 2.1. Lety0∈H−1(0,π;R2),v∈L2(0,T) andf∈L2(QT;R2) be given. It will be\nsaid thaty∈L2(QT;R2) is a solution by transposition to ( 1.7) if, for each g∈L2(QT;R2), one\nhas/integraldisplay/integraldisplay\nQTy·gdxdt=/an}bracketle{ty0,ϕ(·,0)/an}bracketri}ht−/integraldisplayT\n0B∗D∗ϕx(0,t)v(t)dt+/integraldisplay/integraldisplay\nQTf·ϕdxdt, (2.3)\nwhereϕ∈C0([0,T];H1\n0(0,π;R2))∩L2(0,T;H2(0,π;R2)∩H1\n0(0,π;R2)) is the solution of ( 2.1) as-\nsociatedtogandϕ0= 0(recallthat /an}bracketle{t·,·/an}bracketri}htstandsfortheusualdualitypairingbetween H−1(0,1;R2)\nandH1\n0(0,1;R2)).\nWith this definition we have:\nProposition 2.2. Let us assume that y0= (θ0,φ0)∈H−1(0,π;R2),v∈L2(0,T)andf∈\nL2(QT;R2). Then, system (1.7)admits a unique solution by transposition y= (θ,φ)that satisfies\n\n\ny∈L2(QT;R2)∩C0([0,T];H−1(0,π;R2)), yt∈L2(0,T;(H2(0,π;R2)∩H1\n0(0,π;R2))′),\nyt−Dyxx+Ay=finL2(0,T;(H2(0,π;R2)∩H1\n0(0,π;R2))′),\ny(·,0) =y0inH−1(0,π;R2),\nand\n/bardbly/bardblL2(L2)+/bardbly/bardblC0(H−1)+/bardblyt/bardblL2((H2∩H1\n0)′)≤CeCT/parenleftbig\n/bardbly0/bardblH−1+/bardblv/bardblL2(0,T)+/bardblf/bardblL2(L2)/parenrightbig\n,(2.4)\nfor a constant C >0only depending on the parameters ξ,ρandτin system (1.7). Moreover\n(a) Ifφ0∈L2(0,π), thenφ∈L2(0,T;H1\n0(0,π))∩C0([0,T];L2(0,π))and, for a new constant\nC >0(only depending on ξ,ρandτ), one has\n/bardblφ/bardblL2(H1\n0)+/bardblφ/bardblC0(L2)≤C/parenleftbig\n/bardbly/bardblL2(L2)+/bardblφ0/bardblL2+/bardblf/bardblL2(L2)/parenrightbig\n. (2.5)\n(b) Ifφ0∈H1\n0(0,π), thenφ∈L2(0,T;H2(0,π)∩H1\n0(0,π))∩C0([0,T];H1\n0(0,π))and, for a new\nconstantC >0(only depending on ξ,ρandτ), one has\n/bardblφ/bardblL2(H2∩H1\n0)+/bardblφ/bardblC0(H1\n0)≤C/parenleftig\n/bardbly/bardblL2(L2)+/bardblφ0/bardblH1\n0+/bardblf/bardblL2(L2)/parenrightig\n, (2.6)\nand, in particular, y= (θ,φ)∈L2(QT)×C0(QT).\nOne can prove Propositions 2.1and2.2using, for instance, the well-known Galerkin method.\nForthesakeofcompletenesswepresentanideaoftheproofofth istwopropositionsinAppendix A.\nObserve that, when g= 0, the backward problem ( 2.1) is the corresponding adjoint system\nto (1.6):\n\n−ϕt−D∗ϕxx+A∗ϕ= 0 inQT,\nϕ(0,·) =ϕ(π,·) = 0 on (0 ,T),\nϕ(·,T) =ϕ0 in (0,π).(2.7)\nThe controllability properties of system ( 1.6) can be characterized in terms of appropriate\nproperties of the solutions to ( 2.7). In order to provide these characterizations, we need a new\nresult which relates the solutions of systems ( 1.6) and (2.7). One has:\nProposition 2.3. Let us consider y0= (θ0,φ0)∈H−1(0,π;R2)andv∈L2(0,T). Then, the\nsolutiony= (θ,φ)of system (1.6)associated to y0andv, and the solution ϕof the adjoint\nsystem(2.7)associated to ϕ0∈H1\n0(0,π;R2)satisfy\n/integraldisplayT\n0B∗D∗ϕx(0,t)v(t)dt=/an}bracketle{ty(·,T),ϕ0/an}bracketri}ht−/an}bracketle{ty0,ϕ(·,0)/an}bracketri}ht. (2.8)\n8Proof.The proof is a consequence of Proposition 2.2. Observe that is enough to prove that ( 2.8)\nholds under the regularity assumption y0∈C1\n0(0,π;R2) andv∈C1\n0([0,π]). Indeed, using den-\nsity arguments, the estimates of Proposition 2.2and the linearity of ( 1.6), it follows that the\nidentity ( 2.8) is valid for all y0∈H−1(0,π;R2) andv∈L2(0,T).\nOn the other hand, when y0∈C1\n0(0,π;R2),v∈C1([0,π]) andϕ0∈H1\n0(0,π;R2), after some\nintegrations by parts, it is not difficult to prove that the correspon ding solution yof (1.6) andϕ,\nsolution of the adjoint system ( 2.7), satisfy equality ( 2.8). This ends the proof.\nOne important consequence of the previous result is the characte rization of the approximate\nand null controllability properties of the linear system ( 1.6) in terms of suitable properties of the\nsolutions of the adjoint system ( 2.7). One has:\nTheorem 2.1. Let us consider T >0. Then,\n1. System (1.6)is approximately controllable at time T >0if and only if the following unique\ncontinuation property holds:\n“Letϕ0∈H1\n0(0,π;R2)be given and let ϕbe the corresponding solution of the\nadjoint problem (2.7). Then, ifB∗D∗ϕx(0,t) = 0on(0,T), one hasϕ0= 0in\n(0,π).”\n2. System (1.6)is null controllable at time Tif and only if there exists a constant CT>0such\nthat, for any ϕ0= (θ0,φ0)∈H1\n0(0,π;R2), the corresponding solution of (2.7)satisfies the\nobservability inequality\n/bardblϕ(·,T)/bardbl2\nH1\n0≤CT/integraldisplayT\n0|B∗D∗ϕx(0,t)|2dt.\nThis result is well known. For a proof see, for instance [ 10], [22] and [23].\nRemark 2.1. The constant CTappearing in the observability inequality for the adjoint sys-\ntem (2.7) is closely related to the cost K(T) for system ( 1.6) (see Remark 1.3). To be precise, if\nthe observability inequality holds, then Z(T)/ne}ationslash=∅, for anyy0= (θ0,φ0)∈H−1(0,π;R2), and\nK(T)≤/radicalbig\nCT.\nOn the other hand, assume that Z(T)/ne}ationslash=∅, for anyy0= (θ0,φ0)∈H−1(0,π;R2), and define K(T)\nas in Remark 1.3. Then, the previous observability inequality for ( 2.7) holds with CT=K(T)2.\nFor a proof of the previous properties, see for example [ 10] (see Theorem 2.44, p. 56), [ 23]\nor [22].\nWe will finish this section giving two known results which will be used later . They are related\nto the existence and bounds of biorthogonal families to real expon entials. One has:\nLemma 2.1. Let us consider a sequence {Λk}k≥1⊂R+satisfying Λk/ne}ationslash= Λn, for anyk,n∈N\nwithk/ne}ationslash=n, and/summationdisplay\nk≥11\nΛk<∞. (2.9)\nThen, there exists a family {qk}k≥1⊂L2(0,T)biorthogonal to {e−Λkt}k≥1, i.e., a family {qk}k≥1\ninL2(0,T)such that/integraldisplayT\n0qk(t)e−Λjtdt=δkj,∀k,j≥1.\nWe also have:\n9Lemma 2.2. Let us consider a sequence {Λk}k≥1⊂R+such that Λk/ne}ationslash= Λn, for anyk,n∈Nwith\nk/ne}ationslash=n. Let us also assume that there exist an integer q≥1and positive constants p,δandαsuch\nthat/braceleftigg|Λk−Λn| ≥δ/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle,∀k,n∈N,|k−n| ≥q,\ninf\nk/negationslash=n,|k−n|0,(2.10)\nand/vextendsingle/vextendsinglep√r−N(r)/vextendsingle/vextendsingle≤α,∀r>0. (2.11)\n(In(2.11),N(r)is the counting function associated to {Λk}k≥1, defined by N(r) = #{k: Λk≤r}).\nThen, there exists /tildewideT0>0such that, for any T∈(0,/tildewideT0), we can find a family {qk}k≥1⊂L2(0,T)\nbiorthogonal to {e−Λkt}k≥1which in addition satisfies\n/bardblqk/bardblL2(0,T)≤CeC√Λk+C\nT,∀k≥1,\nfor a positive constant Cindependent of T.\nA proof of Lemma 2.1can be found in [ 13] and [5]. On the other hand, Lemma 2.2is a\nparticular case of a more general result proved in [ 8] (see Theorem 1.5 in pages 2974–2975).\n3 Spectral properties of the operators LandL∗\nLet us consider the vectorial operators LandL∗given in ( 1.10), with domains\nD(L) =D(L∗) =H2(0,π;R2)∩H1\n0(0,π;R2).\nThis section will be devoted to giving some spectral properties of th e operators LandL∗which\nwill be used below. Recall that the matrices DandAare given in ( 1.8).\nIn what follows, for simplicity, we will use the notation\nrk:=/radicaligg\nξρ\nτk2+/parenleftbiggρ+1\n2τ/parenrightbigg2\n,∀k≥1. (3.1)\nOn the other hand, it is well-known that the operator −∂xxwith homogeneous Dirichlet bound-\nary conditions admits a sequence of positive eigenvalues, given by {k2}k≥1, and a sequence of\nnormalized eigenfunctions {ηk}k≥1, which is a Hilbert basis of L2(0,π), given by\nηk(x) =/radicalbigg\n2\nπsinkx, x∈(0,π). (3.2)\nWith the previous notation, we have the following result:\nProposition 3.1. Let us consider the operators LandL∗given in (1.10)(the matrices DandA\nare given in (1.8)). Then,\n1. The spectra of LandL∗are given by σ(L) =σ(L∗) ={λ(1)\nk,λ(2)\nk}k≥1with\nλ(1)\nk=ξk2+ρ+1\n2τ−rk, λ(2)\nk=ξk2+ρ+1\n2τ+rk,∀k≥1, (3.3)\nwhererkis given in (3.1).\n2. For each k≥1, the eigenspaces of L(resp.,L∗) corresponding to λ(1)\nkandλ(2)\nkare respectively\ngenerated by\nΨ(1)\nk=1\n4√τrk/parenleftigg\n1−ρ+2τrk\n4/parenrightigg\nηk,Ψ(2)\nk=1\n4√τrk/parenleftigg1−ρ−2τrk\n4/parenrightigg\nηk,(3.4)\n10(resp.,\nΦ(1)\nk=1\n4√τrk/parenleftigg\n4\nρ−1+2τrk/parenrightigg\nηk,Φ(2)\nk=−1\n4√τrk/parenleftigg\n4\nρ−1−2τrk/parenrightigg\nηk).(3.5)\nProof.We will prove the result for the operator L. The same reasoning provides the proof for its\nadjointL∗.\nWe look for a complex λand a function ϕ∈H2(0,π;C2)∩H1\n0(0,π;C2) such that ϕ/ne}ationslash≡0 and\nL(ϕ) =λϕ. Using that the function ηkis the normalized eigenfunction of the Dirichlet-Laplace\noperator in (0 ,π) associated to the eigenvalue k2, we can find ϕas\nϕ(x) =/summationdisplay\nn≥1anηn(x),∀x∈(0,π),\nwhere{an}n≥1⊂C2and, for some k≥1,ak/ne}ationslash= 0. From the identity L(ϕ) =λϕwe deduce\n/summationdisplay\nn≥1/parenleftbig\nn2D+A−λI2/parenrightbig\nanηn(x) = 0,∀x∈(0,π),\n(I2∈L(C2) is the identity matrix). From this identity, it is clear that the eigenva lues of the\noperatorLcorrespond to the eigenvalues of the matrices\nk2D+A,∀k≥1.\nand an associated eigenfunction of Lis given choosing an=zkδkn, for anyn≥1, wherezk∈C2\nis an associated eigenvector of k2D+A, that is to say, Ψ k(·) =zkηk(·).\nTaking into account the expression of the characteristic polynomia l ofk2D+A:\np(x) =x2−/parenleftbigg\n2ξk2+ρ+1\nτ/parenrightbigg\nx+ξ2k4+ξ\nτk2, k≥1,\na direct computation provides the formulae ( 3.3) and (3.4) as eigenvalues and associated eigen-\nfunctions of the operator L. This finishes the proof.\nLet us now analyze some properties of the eigenvalues and eigenfun ctions of the operators L\nandL∗. These properties will be used below. We start with some properties of the sequences\n{λ(1)\nk}k≥1and{λ(2)\nk}k≥1. One has\nProposition 3.2. Under the assumptions of Proposition 3.1, the following properties hold:\n(P1){λ(1)\nk}k≥1and{λ(2)\nk}k≥1(see(3.3)) are increasing sequences satisfying\n0<λ(1)\nk<λ(2)\nk,∀k≥1.\n(P2) The spectrum of LandL∗is simple, i.e., λ(2)\nk/ne}ationslash=λ(1)\nℓ, for allk,ℓ≥1if and only if the\nparameters ξ,ρandτsatisfy condition (1.9).\n(P3) Assume that the parameters ξ,ρandτsatisfy(1.11), i.e., there exists j≥0such that\n1\n(j+1)2ρ\nτ<ξ <1\nj2ρ\nτ. (3.6)\nThen, there exists an integer k0=k0(ξ,ρ,τ,j)≥1and a constant C=C(ξ,ρ,τ,j)>0such\nthat\nλ(1)\nk+j<λ(2)\nk<λ(1)\nk+1+j<λ(2)\nk+1<···,∀k≥k0,andmin\nk≥k0/braceleftig\nλ(2)\nk−λ(1)\nk+j,λ(1)\nk+j+1−λ(2)\nk/bracerightig\n>C.\n(3.7)\n11(P4) Assume now that the parameters ξ,ρandτsatisfy(1.9)and(1.11). Then, one has:\ninf\nk,ℓ≥1|λ(2)\nk−λ(1)\nℓ|>0, (3.8)\nand there exists a positive integer k1∈N, depending on ξ,ρandτ, such that\nmin/braceleftig/vextendsingle/vextendsingle/vextendsingleλ(1)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingleλ(2)\nk−λ(2)\nℓ/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingleλ(2)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle/bracerightig\n≥ξ\n2|k2−ℓ2|,∀k,ℓ≥1,|k−ℓ| ≥k1.(3.9)\nProof.Let us start proving property ( P1). From the expressions of λ(1)\nkandλ(2)\nk(see (3.3)), we\ndirectly get λ(1)\nk<λ(2)\nkfor anyk≥1. On the other hand, using the inequality\nrk=/radicaligg\nξρ\nτk2+/parenleftbiggρ+1\n2τ/parenrightbigg2\nρ\nτ,\nwhich implies λ(1)\nk+1−λ(1)\nk>0, for anyk≥1. Thus, {λ(1)\nk}k≥1is a positive increasing sequence.\nClearly{λ(2)\nk}k≥1is also a positive increasing sequence and λ(2)\nk+1−λ(2)\nk→ ∞, ask→ ∞. This\nproves property ( P1).\nLet us now see property ( P2). Using property ( P1), one has that, for any integers k,ℓ≥1\nwithℓ≤k, clearlyλ(1)\nℓ≤λ(1)\nk<λ(2)\nk. Therefore, in order to prove the equivalence we can assume\nthatℓ>k. We have\nλ(1)\nℓ−λ(2)\nk=ξρ\nτ(ℓ2−k2)/parenleftbiggτ\nρ−1\nrℓ−rk/parenrightbigg\n.\nThus,λ(2)\nk/ne}ationslash=λ(1)\nℓfor anyk,ℓ≥1, withℓ>k, if and only if\nr2\nℓ/ne}ationslash=/parenleftig\nrk+ρ\nτ/parenrightig2\n,∀k,ℓ≥1, ℓ>k.\nFrom the expression of rk(see (3.1)) we readily deduce 2 rk>ρ\nτandξτ(ℓ2−k2)−ρ+2τrk>0\n(ℓ>k). So,\n\n\nr2\nℓ−/parenleftig\nrk+ρ\nτ/parenrightig2\n=ρ\nτ/bracketleftig/parenleftig\nξ(ℓ2−k2)−ρ\nτ/parenrightig\n−2rk/bracketrightig\n=ρ\nτ2/parenleftbig\nξτ(ℓ2−k2)−ρ/parenrightbig2−4τ2r2\nk\nξτ(ℓ2−k2)−ρ+2τrk\n=ρ\nτ2ξ2τ2(ℓ2−k2)2−2ξτρ(ℓ2−k2)−ρ2−4ξτρk2−2ρ−1\nξτ(ℓ2−k2)−ρ+2τrk\n=ρ\nτ2ξ2τ2(ℓ2−k2)2−2ξτρ(ℓ2+k2)−2ρ−1\nξτ(ℓ2−k2)−ρ+2τrk,\n12and we get that λ(2)\nk/ne}ationslash=λ(1)\nℓfor anyk,ℓ≥1, withℓ>k, if and only if condition ( 1.9) holds. This\nfinishes the proof of property ( P2).\nIn order to prove property ( P3), we are going to use the expressions\nλ(1)\nk=ξk2+ρ+1\n2τ−/radicalbigg\nξρ\nτk−ǫk\nk, λ(2)\nk=ξk2+ρ+1\n2τ+/radicalbigg\nξρ\nτk+ǫk\nk,∀k≥1,(3.10)\nwhich can be easily deduced from the expressions of λ(i)\nk,i= 1,2, andrk(see (3.3) and (3.1)).\nIn (3.10),{ǫk}k≥1is a new positive increasing sequence satisfying\nlim\nk→∞ǫk=1\n2/parenleftbiggρ+1\n2τ/parenrightbigg2/radicalbiggτ\nξρ. (3.11)\nUsing (3.10), we will prove that, for any i≥1, the difference λ(1)\nk+i−λ(2)\nkbehaves at infinity as\nlim\nk→∞λ(1)\nk+i−λ(2)\nk\nξi(2k+i)= 1−/radicalbigg1\ni2ρ\nξτ/ne}ationslash= 0. (3.12)\nIndeed, a simple computation gives\nλ(1)\nk+i−λ(2)\nk=ξi(2k+i)−/radicalbigg\nξρ\nτ(2k+i)−ǫk+i\nk+i−ǫk\nk=ξi(2k+i)/bracketleftbigg\n1−/radicalbigg1\ni2ρ\nξτ−/tildewideǫ(i)\nk/bracketrightbigg\n,(3.13)\nwhere{/tildewideǫ(i)\nk}k≥1is a sequence converging to zero. From assumption ( 1.11) we can conclude ( 3.12).\nWe will obtain the proof of property ( P3) from ( 3.12). Observe that assumption ( 1.11) implies\nthat the parameters ξ,ρandτsatisfies ( 3.6) for an appropriate integer j≥0. Therefore, if\nj= 0, then, ξ >ρ\nτand (3.12) implies lim\nk→∞/parenleftig\nλ(1)\nk+1−λ(2)\nk/parenrightig\n=∞. On the other hand, one has\nlim\nk→∞/parenleftig\nλ(2)\nk−λ(1)\nk/parenrightig\n= lim\nk→∞2rk=∞. Thus, there exists an integer k0≥1 and a constant C >0\nsuch that ( 3.7) holds forj= 0.\nIfj≥1, again, the property ( 3.12) implies\nlim\nk→∞/parenleftig\nλ(1)\nk+i−λ(2)\nk/parenrightig\n=−∞,ifi≤jand lim\nk→∞/parenleftig\nλ(1)\nk+i−λ(2)\nk/parenrightig\n=∞,ifi≥j+1.\nWe can also conclude the existence of an integer k0≥1 and a positive constant Csuch that ( 3.7)\nholds. This shows property ( P3).\nLet us finalize the proof showing property ( P4). First, inequality ( 3.8) is a direct consequence\nof property ( P2) and (3.7). Secondly, if we take k,ℓ≥1, from ( 3.10), one deduces:\n\n\n/vextendsingle/vextendsingle/vextendsingleλ(1)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle=ξ/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−/radicalbigg\nξρ\nτ1\nk+ℓ−ǫk\nk(k2−ℓ2)+ǫℓ\nℓ(k2−ℓ2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥ξ/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle/bracketleftigg\n1−/parenleftigg/radicalbigg\nξρ\nτ+|ǫk|+|ǫℓ|/parenrightigg\n1\nk+ℓ/bracketrightigg\n.\nObserve that {ǫk}k≥1is a convergent sequence and k+ℓ≥ |k−ℓ|, for anyk,ℓ∈N. Hence, there\nexists a integer q1≥1 (depending on the parameters of system ( 1.6)) such that\n/vextendsingle/vextendsingle/vextendsingleλ(1)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle≥ξ\n2/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle,∀k,ℓ∈N,|k−ℓ| ≥q1.\nA similar inequality can be deduced for a new q2∈Nif we change/vextendsingle/vextendsingle/vextendsingleλ(1)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingleby/vextendsingle/vextendsingle/vextendsingleλ(2)\nk−λ(2)\nℓ/vextendsingle/vextendsingle/vextendsingle.\n13Finally, if we repeat the previous reasoning, we can write\n\n\n/vextendsingle/vextendsingle/vextendsingleλ(2)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle=ξ/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+/radicalbigg\nξρ\nτ1\nk−ℓ+ǫk\nk(k2−ℓ2)+ǫℓ\nℓ(k2−ℓ2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥ξ/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle/bracketleftigg\n1−/parenleftigg/radicalbigg\nξρ\nτ+1\n2|ǫk|+1\n2|ǫℓ|/parenrightigg\n1\n|k−ℓ|/bracketrightigg\n.\nAgain, from this inequality we conclude the existence of q3=q3(ξ,ρ,τ)∈Nsuch that\n/vextendsingle/vextendsingle/vextendsingleλ(2)\nk−λ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle≥ξ\n2/vextendsingle/vextendsinglek2−ℓ2/vextendsingle/vextendsingle,∀k,ℓ∈N,|k−ℓ| ≥q3.\nThis proves inequality ( 3.9) if we take k1= max{q1,q2,q3}. This completes the proof of ( P4) and\nthe proof of the result.\nRemark 3.1. From the previous proof we can give more information about conditio n (1.9). To\nbe precise, let us see that, in fact, this condition only has to be chec ked for a finite number of\npositive integers kandℓ, withk<ℓ. To this end, let us consider j≥0 such that\n1\n(j+1)2ρ\nτ<ξ≤1\nj2ρ\nτ, (3.14)\n(ifρ\nτ< ξ, thenj= 0). Taking into account that {ǫk}k≥1is a positive increasing sequence\n(see (3.10)), identity ( 3.13) fori=jimplies\nλ(1)\nk+j<λ(2)\nkandλ(1)\nk+j+1<λ(2)\nk+1,∀k≥1.\nOn the other hand, using again ( 3.13) fori=jand the expression of {/tildewideǫ(i)\nk}k≥1, we can write\nlim\nk→∞/parenleftig\nλ(1)\nk+j+1−λ(2)\nk/parenrightig\n=∞.\nTherefore, there exists k0≥1 (only depending on ξ,ρandτ) such that\nλ(1)\nk+j<λ(2)\nk<λ(1)\nk+1+j<λ(2)\nk+1<···,∀k≥k0.\nIn particular, we can assure that λ(2)\nk/ne}ationslash=λ(1)\nℓ, for allk≥k0andℓ≥k0+j.\nIn fact, the previous reasoning provides a stronger property fo r the spectrum of LandL∗:if\nfor somek,ℓ≥1one hasλ(2)\nk=λ(1)\nℓ, thenk0.\nIt is interesting to underline that it is enough to prove the existence ofq∈Nand/tildewideδ>0 such that\none has\n|Λk−Λn| ≥/tildewideδ/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle,∀k,n≥q,|k−n| ≥q. (3.16)\nIndeed, let us see that the first inequality in ( 2.10) is valid for q≥1 and a new positive constant δ.\nObserve that we can assume that k≥n≥1. Hence, it is sufficient to prove ( 2.10) withk≥n≥1,\nwithn≤q−1 andk−n≥q. First, it is clear that if in addition k≤2q, thanks to ( 1.9), we can\nconclude inequality ( 2.10) for an appropriate positive constant δ0.\n15Let us now take k≥n≥1, withn≤q−1 andk≥2q(and therefore, k−n≥q). From ( 3.16)\nand usingk≥q+n≥q+1, 1≤n≤q−1 andk−q≥q, we have\n\n\n|Λk−Λn|= Λk−Λn���Λk−Λq≥/tildewideδ/vextendsingle/vextendsinglek2−q2/vextendsingle/vextendsingle=/tildewideδ/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle/bracketleftbigg\n1−q2−n2\nk2−n2/bracketrightbigg\n≥/tildewideδ/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle/bracketleftbigg\n1−q2−n2\n(q+1)2−n2/bracketrightbigg\n≥/tildewideδ/bracketleftbigg\n1−q2−1\n(q+1)2−1/bracketrightbigg/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle\n=/tildewideδ(2q+1)\n(q+1)2−1/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle.\nSummarizing, assuming ( 3.16), we have deduced the first inequality in ( 2.10) forq≥1 and\nδ= min/braceleftigg\nδ0,/tildewideδ(2q+1)\n(q+1)2−1/bracerightigg\n>0.\nLet us show ( 3.16) for suitable /tildewideδ >0 andq∈N. To this aim, we will use the properties ( 3.7)\nand (3.9), in Proposition 3.2, and the expression of Λ kfork≥2k0+j−1 (see (3.15); recall that\nj≥0 is such that the parameters ξ,ρandτsatisfy (3.6)). We will work with q∈Ngiven by\nq≥max{2k0+j−1,2k1+2j+1,6j+3}. (3.17)\nThus, ifk,n∈Nare such that k,n≥qand|k−n| ≥q, then Λ kand Λ nare given by ( 3.15).\nDepending on the expressions of kandn, we will divide the proof of ( 3.16) into three steps:\n1.Assume that k= 2k0+j+2/tildewidek−1 andn= 2k0+j+2/tildewiden−1, for/tildewidek,/tildewiden≥0. Since\n/vextendsingle/vextendsingle/vextendsingle/parenleftig\nk0+j+/tildewidek/parenrightig\n−(k0+j+/tildewiden)/vextendsingle/vextendsingle/vextendsingle=1\n2|k−n| ≥q\n2≥k1,\nfrom (3.15) and (3.9), we can write\n|Λk−Λn|=/vextendsingle/vextendsingle/vextendsingleλ(1)\nk0+j+/tildewidek−λ(1)\nk0+j+/tildewiden/vextendsingle/vextendsingle/vextendsingle≥ξ\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftig\nk0+j+/tildewidek/parenrightig2\n−(k0+j+/tildewiden)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=ξ\n8/vextendsingle/vextendsingle/vextendsingle(k+1+j)2−(n+1+j)2/vextendsingle/vextendsingle/vextendsingle=ξ\n8/vextendsingle/vextendsinglek2−n2+2(k−n)(1+j)/vextendsingle/vextendsingle≥ξ\n8/vextendsingle/vextendsinglek2−n2/vextendsingle/vextendsingle.\nWe obtain thus the proof of ( 3.16) for/tildewideδ=ξ/8 andqgiven by ( 3.17).\n2.The casek= 2k0+j+2/tildewidekandn= 2k0+j+2/tildewiden, with/tildewidek,/tildewiden∈N, can be treated in the same\nway deducing ( 3.16) for/tildewideδ=ξ/8 andq(see (3.17)).\n3.Let us analyze the last case k= 2k0+j+ 2/tildewidekandn= 2k0+j+2/tildewiden−1 (with/tildewidek,/tildewiden∈N),\nk,n≥qand|k−n| ≥q, withqsatisfying ( 3.17). In this case, one has\n/vextendsingle/vextendsingle/vextendsingle/parenleftig\nk0+/tildewidek/parenrightig\n−(k0+j+/tildewiden)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2(k−n)−j−1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥1\n2|k−n|−/parenleftbigg\nj+1\n2/parenrightbigg\n≥1\n2q−/parenleftbigg\nj+1\n2/parenrightbigg\n≥k1,\nwhence\n|Λk−Λn|=/vextendsingle/vextendsingle/vextendsingleλ(2)\nk0+/tildewidek−λ(1)\nk0+j+/tildewiden/vextendsingle/vextendsingle/vextendsingle≥ξ\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftig\nk0+/tildewidek/parenrightig2\n−(k0+j+/tildewiden)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=ξ\n8/vextendsingle/vextendsingle/vextendsingle(k−j)2−(n+1+j)2/vextendsingle/vextendsingle/vextendsingle\n=ξ\n8/vextendsingle/vextendsinglek2−n2−[2j(k+1)+2n(1+j)+1]/vextendsingle/vextendsingle.\nObserve that if k≤n, from the previous inequality, we conclude ( 3.16) for for/tildewideδ=ξ/8 andqgiven\nby (3.17). Let us now see the case k>n(and then,k−n=|k−n| ≥q). The previous inequality\n16allows us to write\n|Λk−Λn|=ξ\n8/vextendsingle/vextendsinglek2−n2−[2j(k+1)+2n(1+j)+1]/vextendsingle/vextendsingle\n≥ξ\n8/parenleftbig\nk2−n2/parenrightbig\n−[2j(k+1)+2n(1+j)+1]\n=ξ\n8/parenleftbig\nk2−n2/parenrightbig/bracketleftbigg\n1−2j(k+1)+2n(1+j)+1\nk2−n2/bracketrightbigg\n≥ξ\n8/parenleftbig\nk2−n2/parenrightbig/bracketleftbigg\n1−2j(k+1)+2n(1+j)+1\nq(k+n)/bracketrightbigg\n≥ξ\n8/parenleftbig\nk2−n2/parenrightbig/bracketleftbigg\n1−2j\nq−1+j\nq−1\n2q/bracketrightbigg\n≥ξ\n16/parenleftbig\nk2−n2/parenrightbig\n.\nLet us remark that the last inequality is valid thanks to ( 3.17).\nIn conclusion, we have proved the existence of a natural number q≥1, depending on the\nparameters in ( 1.6), such that ( 3.16) holds for/tildewideδ=ξ/16 andqprovided by formula ( 3.17). As a\nconsequence, one also has ( 2.10) for a new δ>0 and the same q.\nLet us now show the estimate ( 2.11) for the sequence Λ = {Λk}k≥1={λ(1)\nk,λ(2)\nk}k≥1. From\nthe definition of the sequence Λ, for any r>0, we can write:\nN(r) = #{k: Λk≤r}= #/braceleftig\nk:λ(1)\nk≤r/bracerightig\n+#/braceleftig\nk:λ(2)\nk≤r/bracerightig\n= #A1(r)+#A2(r) =n1+n2,\nwhereAi(r) =/braceleftig\nk:λ(i)\nk≤r/bracerightig\nandni= #Ai(r),i= 1,2. Our next objective will be to give\nappropriate bounds for n1andn2.\nFrom the definition of A1(r) andn1, we deduce that n1is a natural number which is char-\nacterized by λ(1)\nn1≤randλ(1)\nn1+1> r. Let us first work with the inequality λ(1)\nn1≤r. From the\ndefinition of λ(1)\nk(see (3.3)), one gets\nξn2\n1+ρ+1\n2τ≤r+/radicaligg\nξρ\nτn2\n1+/parenleftbiggρ+1\n2τ/parenrightbigg2\n≤r+/radicalbigg\nξρ\nτn1+ρ+1\n2τ.\nThe previous inequality also implies\nξn2\n1−/radicalbigg\nξρ\nτn1−r≤0,\nand\n0≤n1≤1\n2ξ/parenleftigg/radicalbigg\nξρ\nτ+/radicalbigg\nξρ\nτ+4ξr/parenrightigg\n≤1√ξ/parenleftbigg/radicalbiggρ\nτ+√r/parenrightbigg\n.\nFrom the inequality λ(1)\nn1+1>rwe also deduce,\nr<ξ(n1+1)2+ρ+1\n2τ−/radicaligg\nξρ\nτ(n1+1)2+/parenleftbiggρ+1\n2τ/parenrightbigg2\n≤ξ(n1+1)2,\nthat is to say, n1>√r//radicalbig\nξ−1. Summarizing, n1is a nonnegative integer such that\n√r√ξ−1r. The first inequality provides the estimate\nr≥λ(2)\nn2≥ξn2\n2,i.e.,n2≤√r√ξ.\n17On the other hand, n2is such that\n0<λ(2)\nn2+1−r≤ξ(n2+1)2+/radicalbigg\nξρ\nτ(n2+1)+ρ+1\nτ−r,\nwhence\nn2+1>1\n2ξ/bracketleftigg\n−/radicalbigg\nξρ\nτ+/radicaligg\nξρ\nτ+4ξ/parenleftbigg\nr−ρ+1\nτ/parenrightbigg/bracketrightigg\n=1\n2√ξ/parenleftigg\n−/radicalbiggρ\nτ+/radicalbigg\n4r−3ρ+4\nτ/parenrightigg\n≥1\n2√ξ/parenleftigg\n2√r−/radicalbiggρ\nτ−/radicalbigg\n3ρ+4\nτ/parenrightigg\n.\nIn the last inequality we have used that√\na−b≥√a−√\nbprovideda,b >0 anda≥b. In\nconclusion, we have proved that n2is a nonnegative integer such that\n√r√ξ−1\n2√ξ/parenleftigg/radicalbiggρ\nτ+/radicalbigg\n3ρ+4\nτ/parenrightigg\n−1≤n2≤√r√ξ,∀r≥0. (3.19)\nRecall that N(r) =n1+n2. Thus, from inequalities ( 3.18) and (3.19), we can write\n2√ξ√r−1\n2√ξ/parenleftigg/radicalbiggρ\nτ+/radicalbigg\n3ρ+4\nτ/parenrightigg\n−2≤N(r)≤2√ξ√r+/radicalbiggρ\nξτ,∀r≥0,\nand deduce ( 2.11) with\np=2√ξandα= max/braceleftigg\n1\n2√ξ/parenleftigg/radicalbiggρ\nτ+/radicalbigg\n3ρ+4\nτ/parenrightigg\n+2,/radicalbiggρ\nξτ/bracerightigg\n.\nThis ends the proof.\nWe will finish this section giving a result on the set of eigenfunctions of the operators Land\nL∗. It reads as follows:\nProposition 3.4. Let us consider the sequences F={Ψ(1)\nk,Ψ(2)\nk}k≥1andF∗={Φ(1)\nk,Φ(2)\nk}k≥1\ngiven in Proposition 3.1. Then,\ni)FandF∗are biorthogonal sequences.\nii) span(F)and span (F∗)are dense in H−1(0,π;R2),L2(0,π;R2)andH1\n0(0,π;R2).\niii)FandF∗are unconditional bases2forH−1(0,π;R2),L2(0,π;R2)andH1\n0(0,π;R2).\nProof.From the expressions of Ψ(j)\nkand Φ(j)\nk(see (3.4) and (3.5)) we can write\nΨ(j)\nk(·) =Vj,kηk(·),and Φ(j)\nk=V∗\nj,kηk(·), j= 1,2, k≥1,\nwhereVj,k,V∗\nj,k∈R2(the function ηkis given in ( 3.2)).\nItemi) is simple to deduce, since {ηk}k≥1is an orthogonal basis for H−1(0,π),H1\n0(0,π)\nandL2(0,π) (in this last case, an orthonormal basis) and {V1,k,V2,k}k≥1and{V∗\n1,k,V∗\n2,k}k≥1are\nbiorthogonal basis of R2. Indeed, if Mk= [V1,k|V2,k] andNk=/bracketleftbig\nV∗\n1,k|V∗\n2,k/bracketrightbig\n, then,\nMtr\nkNk=MkNtr\nk=Id,∀k≥1.\n2A countable sequence {xn}n≥1in a Banach space Xis an unconditional basis for Xif for every x∈Xthere\nexist unique scalars an(x) such that x=/summationdisplay\nn≥1an(x)xn, where the series converges unconditionally for each x∈X.\n18This proves item i).\nForshowingitem ii)weonlyneedtoassurethatspan( F)andspan( F∗)aredensein H1\n0(0,π;R2),\nsinceH1\n0(0,π;R2) is dense in L2(0,π;R2) and inH−1(0,π;R2). Let us consider f= (f1,f2)tr∈\nH−1(0,π;R2) such that/angbracketleftig\nf,Ψ(i)\nk/angbracketrightig\n= 0,∀k≥1, i= 1,2.\n(Recall that /an}bracketle{t·,·/an}bracketri}htstands for the usual duality pairing between H−1(0,1;R2) andH1\n0(0,1;R2)). If\nwe denotefi,k(i= 1,2) the corresponding Fourier coefficients of the distribution fi∈H−1(0,π)\nwith respect to the sinus basis {ηk(·)}k≥1, then the previous equality can be written under the\nform\n(f1,k,f2,k)Mk= 0,∀k≥1.\nUsing that det Mk/ne}ationslash= 0 for any k≥1, we deduce f1,k=f2,k= 0, for all k≥1 and, therefore,\nf= 0. This proves the density of FinH1\n0(0,π;R2). A similar argument can be used for F∗. This\nshows item ii).\nLet us now prove item iii). As before, we will only prove that Fis an unconditional basis for\nH1\n0(0,π;R2). This amounts to prove that, for any f= (f1,f2)tr∈H1\n0(0,π;R2), the series\nS(f) :=/summationdisplay\nk≥1/parenleftig/angbracketleftig\nΦ(1)\nk,f/angbracketrightig\nΨ(1)\nk+/angbracketleftig\nΦ(2)\nk,f/angbracketrightig\nΨ(2)\nk/parenrightig\nis unconditionally convergent in H1\n0(0,π;R2). From the definition of the functions Ψ(i)\nkand Φ(i)\nk\n(see (3.4) and (3.5)), it is easy to see that\nS(f) =/summationdisplay\nk≥1/parenleftbigg\nf1,k\nf2,k/parenrightbigg\nηk,\nwherefi,kis the Fourier coefficient of the function fi∈H1\n0(0,π) (i= 1,2). Accordingly, this series\nconverges unconditionally in H1\n0(0,π;R2) (recall that {ηk}k≥1is an orthogonal basis for H1\n0(0,π)\nandf1,f2∈H1\n0(0,π)). This concludes the proof of the result.\n4 Approximate and null controllability of the linear sys-\ntem(1.6)\nWe will devote this section to proving the approximate and null contr ollability at time T >0 of\nsystem (1.6). To this aim, we will use in a fundamental way the properties of the s pectrum of the\noperatorL(see (1.10)) established in Propositions 3.1,3.2and3.3. Firstly, we will show the result\non approximate controllability of the linear system (Theorem 1.1) and then the null controllability\nat timeTof the same system (Theorem 1.2).\n4.1 Approximate controllability: Proof of Theorem 1.1\nLet us fix T >0 and consider system ( 1.6) withξ,ρ,τ > 0 given. Let us first assume that\nsystem ( 1.6) is approximate controllable at time T. In this case, condition ( 1.9) holds. Indeed,\notherwise, thanks to property ( P2) of Proposition 3.2, the spectrum of the operator Lis not\nsimple, i.e., there exist k,ℓ≥1 such that λ(2)\nk=λ(1)\nℓ=λ0. Thus, if we take a,b∈R, it is easy to\nsee that the function\nϕ(x,t) =/parenleftig\naΦ(1)\nℓ(x)+bΦ(2)\nk(x)/parenrightig\ne−λ0(T−t),∀(x,t)∈QT,\nis the solution of the adjoint system ( 2.7) associated to the initial condition\nϕ0=aΦ(1)\nℓ+bΦ(2)\nk.\n19This function satisfies (see ( 1.8) and (3.5))\nB∗D∗ϕx(0,t) =ξ/radicalbigg\n2\nπτ/parenleftbigg\naℓ√rℓ−bk√rk/parenrightbigg\ne−λ0(T−t),∀t∈(0,T).\nChoosing\na=k√rkandb=ℓ√rℓ,\nwe have that B∗D∗ϕx(0,·) = 0 butϕ0/ne}ationslash= 0, contradicting the unique continuation property stated\nin the first point of Theorem 2.1. In conclusion, system ( 1.6) is not approximately controllable at\ntimeT >0.\nLet us now suppose that condition ( 1.9) holds and prove the unique continuation property for\nsystem (2.7). Again, from the first point of Theorem 2.1we infer the approximate controllability\nproperty of system ( 1.6).\nLet us consider ϕ0∈H1\n0(0,π) and assume that the corresponding solution ϕto the adjoint\nproblem ( 2.7) satisfies\nB∗D∗ϕx(0,t) = 0,∀t∈(0,T).\nObserve that, thanks to Proposition 2.1\nϕ∈C0([0,T];H1\n0(0,π;R2))∩L2(0,T;H2(0,π;R2)∩H1\n0(0,π;R2)),\nand then,B∗D∗ϕx(0,·)∈L2(0,T).\nFrom Proposition 3.4we deduce that FandF∗are biorthogonal bases for H−1(0,π;R2) and\nH1\n0(0,π;R2). In particular, ϕ0∈H1\n0(0,π;R2) can be written as ϕ0=/summationdisplay\nk≥1/parenleftig\nakΦ(1)\nk+bkΦ(2)\nk/parenrightig\n,\nwhere\nak=/angbracketleftig\nΨ(1)\nk,ϕ0/angbracketrightig\n, bk=/angbracketleftig\nΨ(2)\nk,ϕ0/angbracketrightig\n,∀k≥1.\nUsing Proposition 3.1, the corresponding solution ϕof system ( 2.7) associated to ϕ0is given by\nϕ(·,t) =/summationdisplay\nk≥1/parenleftig\nakΦ(1)\nke−λ(1)\nk(T−t)+bkΦ(2)\nke−λ(2)\nk(T−t)/parenrightig\n,∀t∈(0,T),\nwhereλ(i)\nk, Ψ(i)\nkand Φ(i)\nk(k≥1,i= 1,2) are given in Proposition 3.1. Therefore,\n0 =B∗D∗ϕx(0,t) =/summationdisplay\nk≥1/radicalbigg\n2\nπkξ√τrk/parenleftig\nake−λ(1)\nk(T−t)−bke−λ(2)\nk(T−t)/parenrightig\n,∀t∈(0,T).\nFrom Proposition 3.3, we can apply Lemma 2.1in order to deduce the existence of a biorthog-\nonal family {q(1)\nk,q(2)\nk}k≥1to{e−λ(1)\nkt),e−λ(2)\nkt)}k≥1inL2(0,T). Then, the previous identity, in\nparticular, implies\n\n\n/radicalbigg\n2\nπkξ√τrkak=/integraldisplayT\n0B∗D∗ϕx(0,t)q(1)\nk(t)dt= 0,∀k≥1,\n/radicalbigg\n2\nπkξ√τrkbk=−/integraldisplayT\n0B∗D∗ϕx(0,t)q(2)\nk(t)dt= 0,∀k≥1,\nandak=bk= 0 for any k≥1. In conclusion, ϕ0= 0 and we have proved the unique continuation\nproperty for the solutions of system ( 2.7). This ends the proof of Theorem 1.1.\n204.2 Null controllability: Proof of Theorem 1.2\nLet us now prove the null controllability result stated in Theorem 1.2. To this aim, we consider\nξ,ρandτthree positive real numbers satisfying assumptions ( 1.9) and (1.11). We will obtain the\nproof writing the controllability problem for system ( 1.6) as a moment problem (see [ 13]).\nLet us take y0= (θ0,φ0)∈H−1(0,π;R2). As a consequence of Proposition 2.3, we have\nthat the control v∈L2(0,T) is such that the solution y= (θ,φ)∈C0([0,T];H−1(0,π;R2)) of\nsystem (1.6) satisfiesy(·,T) = 0 if and only if v∈L2(0,T) fulfills\n/integraldisplayT\n0B∗D∗ϕx(0,t)v(t)dt=−/an}bracketle{ty0,ϕ(·,0)/an}bracketri}ht,∀ϕ0∈H1\n0(0,π;R2),\nwhereϕ∈C0([0,T];H1\n0(0,π;R2)) is the solution of the adjoint system ( 2.7) associated to ϕ0.\nUsing again Proposition 3.4, we deduce that F∗is a basis of H1\n0(0,π;R2). In particular, we also\ndeduce that the previous equality is equivalent to\n/integraldisplayT\n0B∗D∗ϕ(j)\nk,x(0,t)v(t)dt=−/angbracketleftig\ny0,ϕ(j)\nk(·,0)/angbracketrightig\n,∀k≥1, j= 1,2,\nwhereϕ(j)\nk(·,t) =e−λ(j)\nk(T−t)Φ(j)\nkisthesolutionofsystem( 2.7)correspondingto ϕ0= Φ(j)\nk. Taking\ninto account the expressions of B,Dand Φ(j)\nk(see (1.8) and (3.5)), we infer that v∈L2(0,T) is\na null control for system ( 1.6) associated to y0if and only if\n(−1)j+1/radicalbigg\n2\nπkξ√τrk/integraldisplayT\n0e−λ(j)\nk(T−t)v(T−t)dt=e−λ(j)\nkT/angbracketleftig\ny0,Φ(j)\nk/angbracketrightig\n,∀k≥1, j= 1,2.\nSummarizing, we have transformed the null-controllability problem at timeT >0 for sys-\ntem (1.6) into the following moment problem: given y0= (θ0,φ0)∈H−1(0,π;R2), findv∈\nL2(0,T) such that the function u(t) :=v(T−t)∈L2(0,T) satisfies\n/integraldisplayT\n0e−λ(j)\nktu(t)dt=ckj,∀k≥1, j= 1,2, (4.1)\nwhereckj=ckj(y0) is given by\nckj= (−1)j+1/radicalbiggπ\n2√τrk\nkξe−λ(j)\nkT/angbracketleftig\ny0,Φ(j)\nk/angbracketrightig\n,∀k≥1, j= 1,2. (4.2)\nOur next task will be to solveproblem ( 4.1). The assumptions ( 1.9) and (1.11), Proposition 3.3\nand Lemma 2.2guarantee the existence of /tildewideT0>0 such that for any T∈(0,/tildewideT0) there exists a\nbiorthogonal family {q(1)\nk,q(2)\nk}k≥1to{e−λ(1)\nkt,e−λ(2)\nkt}k≥1inL2(0,T) which satisfies\n/vextenddouble/vextenddouble/vextenddoubleq(j)\nk/vextenddouble/vextenddouble/vextenddouble\nL2(0,T)≤CeC/radicalBig\nλ(j)\nk+C\nT,∀k≥1, j= 1,2, (4.3)\nfor a positive constant Cindependent of T.\nLet us first prove the result when T∈(0,/tildewideT0). Then, a formal solution to the moment prob-\nlem (4.1) is given by\nu(t) :=v(T−t) =/summationdisplay\nk≥1/parenleftig\nck1q(1)\nk+ck2q(2)\nk/parenrightig\n.(4.4)\nLet us now prove that u∈L2(0,T) and, consequently, that v∈L2(0,T). From the expressions\nofrk,λ(j)\nkand Φ(j)\nk(see (3.1), (3.3) and (3.5)) we can easily deduce the existence of constants\nC,C1,C2>0 such that\nC1k≤rk≤C2k, C 1k2≤/vextendsingle/vextendsingle/vextendsingleλ(j)\nk/vextendsingle/vextendsingle/vextendsingle≤C2k2,/vextenddouble/vextenddouble/vextenddoubleΦ(j)\nk/vextenddouble/vextenddouble/vextenddouble\nH1\n0≤Ck3/2,∀k≥1, j= 1,2,\n21and, from ( 4.2),\n|ckj| ≤C√\nke−λ(j)\nkT/bardbly0/bardblH−1/vextenddouble/vextenddouble/vextenddoubleΦ(j)\nk/vextenddouble/vextenddouble/vextenddouble\nH1\n0≤Cke−λ(j)\nkT/bardbly0/bardblH−1,∀k≥1, j= 1,2.\nComing back to the expression of the null control v(see (4.4)) and taking into account ( 4.3)\nand the previous inequality, we get\n/bardblv/bardblL2(0,T)≤CeC\nT/bardbly0/bardblH−1/summationdisplay\nk≥1/parenleftbigg\neC/radicalBig\nλ(1)\nke−λ(1)\nkT+eC/radicalBig\nλ(2)\nke−λ(2)\nkT/parenrightbigg\n≤CeC\nT/bardbly0/bardblH−1/summationdisplay\nk≥1/parenleftig\neC2\n2T+1\n2λ(1)\nkTe−λ(1)\nkT+eC2\n2T+1\n2λ(2)\nkTe−λ(2)\nkT/parenrightig\n≤CeC\nT/bardbly0/bardblH−1/summationdisplay\nk≥1e−CTk2≤CeC\nT/bardbly0/bardblH−1/integraldisplay∞\n0e−CTs2ds=C\n2/radicalbiggπ\nCTeC\nT/bardbly0/bardblH−1\n≤C0eM\nT/bardbly0/bardblH−1,\n(4.5)\nfor positive constants C0andMindependent of T. This inequality shows that v∈L2(0,T) and\nproves the first part of Theorem 1.2.\nThe second part is a direct consequence of the expression of the n ull control v(see (4.4))\nand (4.5). Indeed, if we define the operator C(0)\nT:H−1(0,π;R2)→L2(0,T) by\nC(0)\nT(y0) :=/summationdisplay\nk≥1/parenleftig\nck1(y0)q(1)\nk(T−·)+ck2(y0)q(2)\nk(T−·)/parenrightig\n,∀y0∈H−1(0,π;R2),\nwithckj=ckj(y0) given by ( 4.2), it is not difficult to see that C(0)\nTis a linear operator which\nsatisfies ( 1.12) for a positive constants C0andM. This ends the proof of Theorem 1.2when\nT∈(0,/tildewideT0).\nLet us now assume that T≥/tildewideT0. We will obtain the proof as a consequence of the previous\ncase. Indeed, if T≥/tildewideT0we can construct a null control at time Tfor system ( 1.6) associated to\ny0∈H−1(0,π;R2) as\nv(t) =C(0)\nT(y0)(t) :=\n\nC(0)\n/tildewideT0/2(y0)(t) ift∈/bracketleftigg\n0,/tildewideT0\n2/bracketrightigg\n,\n0 if t∈/bracketleftigg/tildewideT0\n2,T/bracketrightigg\n.\nClearlyC(0)\nT∈L/parenleftbig\nH−1(0,π;R2),L2(0,T)/parenrightbig\n/vextenddouble/vextenddouble/vextenddoubleC(0)\nT(y0)/vextenddouble/vextenddouble/vextenddouble\nL2(0,T)≤C0e2M//tildewideT0/bardbly0/bardblH−1=C1/bardbly0/bardblH−1\nwithC1a new positive constant independent of T. So, we can conclude ( 1.12) for a new positive\nconstantC0(only depending on the parameters in system ( 1.6)) and the same constant M >0 as\nbefore. This finishes the proof of Theorem 1.2.\n5 Boundary controllability of the phase-field system\nIn this section we will prove the exact controllability at time T >0 of the phase-field system ( 1.1)\nto the constant trajectory (0 ,c), withc=±1. To this end, we will perform a fixed-point strategy\nwhich will use in a fundamental way a null controllability result for the n on-homogeneous linear\nsystem ( 1.6) (f∈L2(0,π;R2) is a given function in an appropriate weighted-Lebesgue space;\nsee (5.2)).\n225.1 Null controllability of the non-homogeneous system (1.6)\nAs said before, our next objective will be to show a null controllability result for non-homogeneous\nsystem (1.6) wheny0= (θ0,φ0)∈H−1(0,π;R2) andfis a given function satisfying appropriate\nassumptions. To this end, we will follow some ideas from [ 19].\nLet us consider ξ,ρandτthree positive real numbers satisfying hypotheses ( 1.9) and (1.11).\nThe starting point is Theorem 1.2and Remark 1.3. As a consequence, we obtain an estimate for\nthe cost of the null control of system ( 1.6). With the notations of Remark 1.3, one has\nK(T)≤C0eM\nT,∀T >0,\nwithC0andMtwo positive constants only depending on ξ,ρandτ.\nIn order to provide a null controllability result for the non-homogen eous problem ( 1.7) at time\nT >0, we will introduce the functions γ(t) :=eM\nt,∀t>0, and, fort∈[0,T],\nρF(t) :=e−b2(a+1)M\n(b−1)(T−t), ρ0(t) :=e−aM\n(b−1)(T−t),∀t∈/bracketleftbigg\nT/parenleftbigg\n1−1\nb2/parenrightbigg\n,T/bracketrightbigg\n,(5.1)\nextended to/bracketleftbig\n0,T(1−1/b2)/bracketrightbig\nin a constant way. Here a,b >1 are constants that will be chosen\nlater. Observethat γ,ρFandρ0are continuousand non increasingfunctions in [0 ,T]andρF(T) =\nρ0(T) = 0.\nWith the previous functions, we also introduce the weighted normed spaces\nF:=/braceleftbigg\nf∈L2(QT;R2) :f\nρF∈L2(QT;R2)/bracerightbigg\n,V:=/braceleftbigg\nv∈L2(0,T) :v\nρ0∈L2(0,T)/bracerightbigg\n,\nY0:=/braceleftbigg\ny∈L2(QT;R2) :y\nρ0∈L2(QT;R2)/bracerightbigg\n,\nY:=/braceleftbigg\ny∈L2(QT;R2) :y\nρ0∈L2(QT)×C0(QT)/bracerightbigg\n.(5.2)\nIt is clear that F,VandY0are Hilbert spaces. For instance, the inner product in Fis given by\n(f1,f2)F:=/integraldisplay/integraldisplay\nQTρ−2\nF(t)f1(x,t)·f2(x,t)dxdt,∀f1,f2∈F.\nA similar definition can be made for ( ·,·)Vand (·,·)Y0. On the other hand, Yis a Banach space\nwith the norm\n/bardbly/bardblY:=/parenleftig\n/bardbly1/ρ0/bardbl2\nL2(QT)+/bardbly2/ρ0/bardbl2\nC0(QT)/parenrightig1/2\n,∀y= (y1,y2)∈Y.\nWith the previous notation, one has:\nTheorem 5.1. Let us consider ξ,ρandτthree positive real numbers satisfying (1.9)and(1.11).\nThen, for every T >0, there exist two bounded linear operators\nC(1)\nT:H−1(0,π;R2)×F→VandE(0)\nT:H−1(0,π;R2)×F→Y0\nsuch that\n(i)/vextenddouble/vextenddouble/vextenddoubleC(1)\nT/vextenddouble/vextenddouble/vextenddouble\nL(H−1(0,π;R2)×F,V)≤CeC(T+1\nT)and/vextenddouble/vextenddouble/vextenddoubleE(0)\nT/vextenddouble/vextenddouble/vextenddouble\nL(H−1(0,π;R2)×F,Y0)≤CeC(T+1\nT)for a\npositive constant Cindependent of T.\n(ii)E(1)\nT:=E(0)\nT/vextendsingle/vextendsingle/vextendsingleH−1(0,π)×H1\n0(0,π)×F∈L(H−1(0,π)×H1\n0(0,π)×F,Y)and, for a new constant\nC >0independent of T, one has/vextenddouble/vextenddouble/vextenddoubleE(1)\nT/vextenddouble/vextenddouble/vextenddouble\nL(H−1(0,π)×H1\n0(0,π)×F,Y)≤CeC(T+1\nT).\n23(iii) For any (y0,f)∈H−1(0,π;R2)×F(resp.,(y0,f)∈H−1(0,π)×H1\n0(0,π)×F),y=\nE(0)\nT(y0,f)∈Y0(resp.y=E(1)\nT(y0,f)∈Y) is the solution of (1.7)associated to (y0,f)\nandv=C(1)\nT(y0,f).\nRemark 5.1. Before giving the proof of this result, let us underline that Proposit ion5.1provides\na null controllability result for the non-homogeneous system ( 1.7) wheny0∈H−1(0,π;R2) and\nf∈F. Indeed, since ρ0is a continuous function on [0 ,T] satisfying ρ0(T) = 0, it is clear that\ny=E(0)\nT(y0,f)∈Y0∩C0([0,T];H−1(0,π;R2)),\nsolves (1.7) and satisfies y(·,T) = 0 inH−1(0,π;R2).\nProof of Theorem 5.1.InordertoproveTheorem 5.1, weadapttosystem( 1.7)ageneraltechnique\ndeveloped in [ 19] that permits to prove a null controllability result for a non-homoge nous linear\nproblem from the corresponding controllability result for the homog enous problem.\nLet us consider a,b>1 andT >0. With the previous definitions and notations, we define the\nsequence\nTk=T−T\nbk,∀k≥0.\nFrom the definition of the functions ρ0andρF(see (5.1)) and the expression of Tk, one has\nρ0(Tk+2) =ρF(Tk)eM\nTk+2−Tk+1,∀k≥0. (5.3)\nThis formula will be used in what follows.\nLet us take y0= (θ0,φ0)∈H−1(0,π;R2) (resp.,y0∈H−1(0,π)×H1\n0(0,π)) andf∈F. Thus,\nwe introduce the sequence {ak}k≥0⊂H−1(0,π;R2) (resp.{ak}k≥0⊂H−1(0,π)×H1\n0(0,π) if\ny0∈H−1(0,π)×H1\n0(0,π)) defined by\na0=y0, ak+1= ˜yk(T−\nk+1),∀k≥0,\nwhere ˜ykis the solution to the linear system\n\n\n˜yt−D˜yxx+A˜y=fin (0,π)×(Tk,Tk+1) :=Qk,\n˜y(0,·) = ˜y(π,·) = 0 on ( Tk,Tk+1)\n˜y(·,T+\nk) = 0 in (0 ,π),(5.4)\n(the matrices DandAare given in ( 1.8)). From Proposition 2.1, it is clear that this system admits\na unique solution\n˜yk∈L2(Tk,Tk+1;H2(0,π)∩H1\n0(0,π))∩C0([Tk,Tk+1];H1\n0(0,π;R2))\nwhich satisfies ( 2.2). In particular, ˜ yk∈C0(Qk;R2) andak+1∈H1\n0(0,π;R2), for anyk≥0, and\n/bardbl˜yk/bardblC0(Qk;R2)+/bardblak+1/bardblH1\n0≤eCT/bardblf/bardblL2(Qk;R2),∀k≥0, (5.5)\nwhereCis a positive constant only depending on the coefficients of DandA.\nFork≥0, we also consider the controlled autonomous problem\n\n\nˆyt−Dˆyxx+Aˆy= 0 in Qk,\nˆy(0,·) =Bvk,ˆy(π,·) = 0 on ( Tk,Tk+1)\nˆy(·,T+\nk) =ak,ˆy(·,T−\nk+1) = 0 in (0 ,π),(5.6)\nwhere the control vkis given byvk=C(0)\nTk+1−Tk(ak)∈L2(Tk,Tk+1) (the linear operator C(0)\nTk+1−Tkis\ngiven in Theorem 1.2). Thanks to Proposition 2.2, the solution ˆ ykof the previous system satisfies\n/braceleftigg\nˆy0∈L2(Q0;R2) (resp., ˆy0∈L2(Q0;R2)∩C0([0,T1];H−1(0,π)×H1\n0(0,π)),\nˆyk∈L2(Qk;R2)∩C0([Tk,Tk+1];H−1(0,π)×H1\n0(0,π)),∀k≥1\n24and, from ( 2.4) (resp., ( 2.6)), (5.5) and Theorem 1.2,\n\n\n/bardblˆy0/bardblL2(Q0;R2)≤eCT1/parenleftbig\n/bardbly0/bardblH−1+/bardblv0/bardblL2(0,T1)/parenrightbig\n≤C0eCTeM\nT1/bardbly0/bardblH−1\n(resp.,/bardblˆy0/bardblL2(Q0)×C0(Q0)≤C0eCTeM\nT1/bardbly0/bardblH−1×H1\n0),\nand, for any k≥1,\n/bardblˆyk/bardblL2(Qk)×C0(Qk)≤eCT/parenleftig\n/bardblak/bardblH−1×H1\n0+/bardblvk/bardblL2(Tk,Tk+1)/parenrightig\n≤C0eCTeM\nTk+1−Tk/bardblf/bardblL2(Qk;R2).\nIf we setYk:= ˜yk+ ˆykinQk= (0,π)×(Tk,Tk+1), then\n/braceleftigg\nY0∈L2(Q0;R2) (resp.,Y0∈L2(Q0;R2)∩C0([0,T1];H−1(0,π)×H1\n0(0,π)),\nYk∈L2(Qk;R2)∩C0([Tk,Tk+1];H−1(0,π)×H1\n0(0,π)),∀k≥1\nand\n\n/bardblY0/bardblL2(Q0;R2)≤CeCTeM\nT1/parenleftbig\n/bardbly0/bardblH−1+/bardblf/bardblL2(Q0;R2)/parenrightbig\n(resp.,/bardblY0/bardblL2(Q0)×C0(Q0)≤CeCTeM\nT1/parenleftig\n/bardbly0/bardblH−1×H1\n0+/bardblf/bardblL2(Q0;R2)/parenrightig\n),\n/bardblYk/bardblL2(Qk)×C0(Qk)≤CeCTeM\nTk+1−Tk/bardblf/bardblL2(Qk;R2),∀k≥1.(5.7)\nLet us divide the proof into two cases: the case k= 0 and the case k≥1.\nCasek= 0.First, from Theorem 1.2, we can use that bT1=T(b−1) to obtain (recall that\nv0=C(0)\nT1(y0))\n/bardblv0/bardblL2(0,T1)≤C0eM\nT1/bardbly0/bardblH−1=C0eMb(a+1)\n(b−1)Tρ0(T1)/bardbly0/bardblH−1.\nUsing now that ρ0is a positive continuous non-increasing function, from the previous estimate,\nwe deduce the existence of a positive constant Csuch that\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublev0\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(0,T1)≤CeC\nT/bardbly0/bardblH−1. (5.8)\nOn the other hand, from ( 5.7),\n\n\n/bardblY0/bardblL2(Q0;R2)≤CeCTeM\nT1/parenleftbig\n/bardbly0/bardblH−1+/bardblf/bardblL2(Q0;R2)/parenrightbig\n=CeCTeMb(a+1)\n(b−1)Tρ0(T1)/parenleftbig\n/bardbly0/bardblH−1+/bardblf/bardblL2(Q0;R2)/parenrightbig\n,\n(resp.,/bardblY0/bardblL2(Q0)×C0(Q0)≤CeCTeMb(a+1)\n(b−1)Tρ0(T1)/parenleftig\n/bardbly0/bardblH−1×H1\n0+/bardblf/bardblL2(Q0;R2)/parenrightig\n).\nObservethat /bardblf/bardblL2(Q;R2)≤ /bardblf/bardblF(see the expressionof ρFin (5.1)). Hence, repeating the previous\nargument, we get\n\n\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleY0\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Q0;R2)≤CeC(T+1\nT)(/bardbly0/bardblH−1+/bardblf/bardblF)\n(resp.,/vextenddouble/vextenddouble/vextenddouble/vextenddoubleY0\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Q0)×C0(Qk)≤CeC(T+1\nT)/parenleftig\n/bardbly0/bardblH−1×H1\n0+/bardblf/bardblF/parenrightig\n).(5.9)\nCasek≥1.Again, taking into account formula vk=C(0)\nTk+1−Tk(ak), Theorem 1.2, (5.5)\nand (5.3), we infer\n/bardblvk/bardblL2(Tk,Tk+1)≤CeM\nTk+1−Tk/bardblak/bardblH−1≤CeCTeM\nTk+1−Tk/bardblf/bardblL2(Qk−1;R2)\n=CeCTρ0(Tk+1)\nρF(Tk−1)/bardblf/bardblL2(Qk−1;R2).\n25As in the case k= 0, using the fact that ρ0andρFare non-increasing functions, from the previous\ninequality, we deduce\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublevk\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Tk,Tk+1)≤CeCT/vextenddouble/vextenddouble/vextenddouble/vextenddoublef\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Qk−1;R2),∀k≥1. (5.10)\nWe can alsorepeatthe previousargumenttoobtain an estimate for Ykwhenk≥1. From( 5.7),\n\n\n/bardblYk/bardblL2(Qk)×C0(Qk)≤CeCTeM\nTk+1−Tk/bardblf/bardblL2(Qk;R2)=CeCTρ0(Tk+1)\nρF(Tk−1)/bardblf/bardblL2(Qk;R2)\n≤CeCTρ0(Tk+1)\nρF(Tk)/bardblf/bardblL2(Qk;R2),\nwhat implies /vextenddouble/vextenddouble/vextenddouble/vextenddoubleYk\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Qk)×C0(Qk)≤CeCT/vextenddouble/vextenddouble/vextenddouble/vextenddoublef\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Qk;R2),∀k≥1. (5.11)\nWith the functions vkandYk,k≥0, defined above, we define\nC(1)\nT(y0,f) :=v=/summationdisplay\nk≥0vk1[Tk,Tk+1)andE(0)\nT(y0,f) :=Y=/summationdisplay\nk≥0Yk1[Tk,Tk+1),(5.12)\nwhere 1 Iis the characteristic function on the set I. Let us first remark that, by construction, C(1)\nT\nandE(0)\nTare linear operators. On the other hand, recall that Yk= ˜yk+ ˆyk,k≥0, where ˜ykand\nˆykare respectively the solution to systems ( 5.4) and (5.6). So,\nYk(T−\nk+1) =ak+1= ˆyk+1(T+\nk+1) =Yk+1(T+\nk+1),∀k≥0,\nwhich implies that the function Yis continuous at time Tk, for anyk≥1, and is the solution of\nsystem (1.7) associated to ( y0,f,v).\nFinally, thanks to ( 5.8)–(5.11), we also deduce that C(1)\nT(y0,f)∈VandE(0)\nT(y0,f)∈Y0(resp.,\nE(0)\nT(y0,f)∈Y) for any (y0,f)∈H−1(0,π;R2)×F(resp., for any ( y0,f)∈H−1(0,π)×H1\n0(0,π)×\nF) and\n\n\n/vextenddouble/vextenddouble/vextenddoubleC(1)\nT(y0,f)/vextenddouble/vextenddouble/vextenddouble\nV=/bardblv/bardblV≤CeC(T+1\nT)(/bardbly0/bardblH−1+/bardblf/bardblF),\n/vextenddouble/vextenddouble/vextenddoubleE(0)\nT(y0,f)/vextenddouble/vextenddouble/vextenddouble\nY0=/bardblY/bardblY0≤CeC(T+1\nT)(/bardbly0/bardblH−1+/bardblf/bardblF),∀(y0,f)∈H−1(0,π;R2)×F,\n(resp.,\n/vextenddouble/vextenddouble/vextenddoubleE(0)\nT(y0,f)/vextenddouble/vextenddouble/vextenddouble\nY=/bardblY/bardblY≤CeC(T+1\nT)/parenleftig\n/bardbly0/bardblH−1×H1\n0+/bardblf/bardblF/parenrightig\n,∀(y0,f)∈H−1(0,π)×H1\n0(0,π)×F).\nThe above estimates provide the proof of Proposition 5.1. This ends the proof.\n5.2 Proof of Theorem 1.3\nWewill devotethissectiontoprovingthelocalexactcontrollabilityat timeT >0ofthe phase-field\nsystem (1.1) stated in Theorem 1.3. To this objective, let us take\n˜y0= (˜θ0,˜φ0)∈H−1(0,π)×(c+H1\n0(0,π))\n(c=±1). As we saw in Section 1, the local exact controllability of system ( 1.1) at timeTto the\nconstant trajectory (0 ,c) is equivalent to the local null controllability of system ( 1.4) at timeT\nwithy0= (θ0,φ0) = (˜θ0,˜φ0−c)∈H−1(0,π)×H1\n0(0,π) (the nonlinear functions g1andg2are\ngiven in ( 1.5)).\n26Let us take a,b >1 (which will be determined below) and consider the functions ρFand\nρ0, defined in ( 5.1), and the spaces F,VandYgiven in ( 5.2). In order to prove the local null\ncontrollability result at time Tfor system ( 1.4) we will perform a fixed-point strategy in the space\nYwhich, in particular, will prove the existence of a control v∈Vsuch that system ( 1.4) has a\nsolutiony∈Yassociated to ( v,y0). The condition y∈Ywill imply the null controllability result\nfor this system.\nLet us fixε>0 (to be determined bellow). With the previous data and notations, w e consider\nthe closed ball in the space F\nBε={f∈F:/bardblf/bardblF≤ε}.\nObserve that if the initial datum ˜ y0∈H−1(0,π)×(c+H1\n0(0,π)) satisfies ( 1.15), theny0=\n(θ0,φ0) = (˜θ0,˜φ0−c)∈H−1(0,π)×H1\n0(0,π) satisfies\n/bardblθ0/bardblH−1+/bardblφ0/bardblH1\n0≤ε. (5.13)\nFor eachf∈Bε⊂F, we denote vf=C(1)\nT(y0,f)∈Vandyf= (θf,φf) :=E(1)\nT(y0,f)∈Y,\nwhere the operators C(1)\nTandE(1)\nTare given in Theorem 5.1. As a consequence of this result\nand (5.13), one has\n/bardblyf/bardblY+/bardblvf/bardblV≤CeC(T+1\nT)/parenleftig\n/bardbly0/bardblH−1×H1\n0+/bardblf/bardblF/parenrightig\n≤CeC(T+1\nT)ε,∀f∈Bε,(5.14)\nfor a positive constant C=C(ξ,ρ,τ). Thus, we define the nonlinear operator N:Bε→\nC0(QT;R2) given by (see ( 1.5))\nN(f) =\n±3ρ\n4τφ2\nf+ρ\n4τφ3\nf\n∓3\n2τφ2\nf−1\n2τφ3\nf\n. (5.15)\nIt is clear that the operator Nis well-defined. On the other hand, if Nadmits a fixed point f∈F,\nthenyf∈Y, together with vf∈V, provides a solution of the system ( 1.4) associated to the\ninitial datum y0= (θ0,φ0). In fact, from Proposition 2.2,yf∈C0([0,T];H−1(0,π;R2)). Finally,\nconditionyf∈C0([0,T];H−1(0,π;R2))∩Yin particular implies the null controllability result for\nsystem (1.4). This would prove Theorem 1.3.\nThe next task is to prove that the operator Nhas a fixed-point in the complete metric space\nBε⊂F. To this end, we will apply the Banach Fixed-Point Theorem. Before, let us select any\na>1 andbsuch that\nb2∈/parenleftbigg\n1,2a\na+1/parenrightbigg\n.\nWith this choice, the functions ρ2\n0/ρFandρ3\n0/ρFare uniformly bounded in [0 ,T], i.e., there exists\na constantCT>0, depending on T, such that\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ2\n0\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0[0,T]≤CTand/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ3\n0\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0[0,T]≤CT.\nLet us now check the assumptions of the Banach Fixed-Point Theor em:\n1.N(Bε)⊂Bε: Indeed, if f∈Bε, then, from ( 5.14), we obtain\n/bardblN(f)/bardblF≤CT/vextenddouble/vextenddouble/vextenddouble/vextenddoubleN(f)\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT;R2)≤CT\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ2\nf\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ3\nf\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)\n\n≤CT/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ2\n0\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφf\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nC0(QT)+/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ3\n0\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφf\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble3\nC0(QT)/parenrightigg\n≤CT/parenleftig\n/bardblyf/bardbl2\nY+/bardblyf/bardbl3\nY/parenrightig\n≤CTeC(T+1\nT)/parenleftbig\nε2+ε3/parenrightbig\n≤ε\n27forε=ε(T) small enough.\n2.Nis a contraction mapping: Let us take f1,f2∈Bε⊂Fand denote yi= (θi,φi) =\nE(1)\nT(y0,fi)∈Y,i= 1,2. Firstly, observe that the non linearity ( g1,g2), given in ( 1.5), satis-\nfies\n|gj(s1)−gj(s2)| ≤C(|s1|2+|s2|2+|s1|+|s2|)|s1−s2|,∀s1,s2∈R, j= 1,2.\nThus, using again ( 5.14) and Theorem 5.1, we have\n/bardblN(f1)−N(f2)/bardblF≤CT2/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddouble/vextenddoublegj(φ1)−gj(φ2)\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)\n≤CT/vextenddouble/vextenddouble/vextenddouble/vextenddoubleρ0\nρF/parenleftbig\n|φ1|2+|φ2|2+|φ1|+|φ2|/parenrightbig|φ1−φ2|\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)\n≤CT/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1\nρ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ2\nρ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\nρ3\n0\nρF+/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ1\nρ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ2\nρ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbiggρ2\n0\nρF/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleφ1−φ2\nρ0/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nC0(QT)\n≤CT/parenleftig\n/bardbly1/bardbl2\nY+/bardbly2/bardbl2\nY+/bardbly1/bardblY+/bardbly2/bardblY/parenrightig/vextenddouble/vextenddouble/vextenddoubleE(1)\nT(y0,f1)−E(1)\nT(y0,f2)/vextenddouble/vextenddouble/vextenddouble\nY\n≤CTeC(T+1\nT)/parenleftbig\nε2+ε/parenrightbig\n/bardblf1−f2/bardblF.\nFrom this inequality it is clear that we can choose ε=ε(T) (small enough) in such a way that N\nis a contraction mapping.\nIn conclusion, we can apply the Banach Fixed-Point Theorem. This pr oves that the operator\nNhas a fixed-point and provides the proof of Theorem 1.3.\nAppendices\nAppendix A\nThis appendix will be devoted to dealing with the existence and uniquen ess of solution of the\nlinear systems ( 2.1) and (1.7). To be precise, we will prove Propositions 2.1and2.2.\nProof of Proposition 2.1.Let us assume that ϕ0∈H1\n0(0,π;R2) andg∈L2(QT;R2). Let us\ndenoteϕ0= (θ0,φ0) andg= (g1,g2). Then the system ( 2.1) can be write as\n\n\n−θt−ξθxx+ρ\nτθ−2\nτφ=g1 inQT,\n−φt−ξφxx+1\n2ρξθxx−ρ\n2τθ+1\nτφ=g2inQT,\nθ(0,·) =φ(0,·) =θ(π,·) =φ(π,·) = 0 on (0 ,T),\nθ(·,T) =θ0, φ(·,T) =φ0 in (0,π),\nwhereϕ= (θ,φ). On the other hand, ξθxx=−θt+ρ\nτθ−2\nτφ−g1. Thus, the previous system\nbecomes \n\n−θt−ξθxx−2\nτφ+ρ\nτθ=g1 inQT,\n−φt−ξφxx−ρ\n2θt−ρ−1\nτφ+ρ(ρ−1)\n2τθ=ρ\n2g1+g2inQT,\nθ(0,·) =φ(0,·) =θ(π,·) =φ(π,·) = 0 on (0 ,T),\nθ(·,T) =θ0, φ(·,T) =φ0 in (0,π),(A.1)\n28Then, Proposition 2.1is equivalent to prove that the system ( A.1) has a unique strong solution\n(θ,φ) satisfying\nθ,φ∈C0([0,T];H1\n0(0,π))∩L2(0,T;H2(0,π)∩H1\n0(0,π))\nand\n/bardblθ/bardblC0(H1\n0)+/bardblφ/bardblC0(H1\n0)+/bardblθ/bardblL2(H2∩H1\n0)+/bardblφ/bardblL2(H2∩H1\n0)\n≤eCT/parenleftig\n/bardblg1/bardblL2(L2)+/bardblg2/bardblL2(L2)+/bardblθ0/bardblH1\n0+/bardblφ0/bardblH1\n0/parenrightig\n.(A.2)\nfor a positive constant C, only depending on ξ,ρandτ.\nWe will use the well-known Faedo-Galerkin method. First, let us consid er the orthonormal\nbasis{ηn}n∈NofL2(0,π) (ηnis the normalized eigenfunction of the Dirichlet-Laplace operator,\nsee (3.2)). For each m∈N, we consider Vm= [η1,η2,···,ηm], the subspace generated by the\nfirstmvectors of {ηn}n∈N. Let us also consider Pm, the orthogonal projection operator onto the\nfinite-dimensional space VminL2(0,π). If we define\nθm\n0=Pmθ0, φm\n0=Pmφ0, gm\n1(·,t) =Pmg1(·,t) andgm\n2(t,·) =Pmg2(t,·),(A.3)\none hasθm\n0,φm\n0∈Vmandgm\n1,gm\n2∈L2(0,T;Vm), for anym∈N, and\nθm\n0→θ0, φm\n0→φ0inH1\n0(0,π),andgm\n1→g1, gm\n2→g2inL2(QT),asm→ ∞.(A.4)\nWe want an approximate solution ( θm,φm)∈C0([0,T];V2\nm) of the approximate problem\n\n\n−θm\nt−ξθm\nxx−2\nτφm+ρ\nτθm=gm\n1 inQT,\n−φm\nt−ξφm\nxx−ρ\n2θm\nt−ρ−1\nτφm+ρ(ρ−1)\n2τθm=ρ\n2gm\n1+gm\n2inQT,\nθm(0,·) =φm(0,·) =θm(π,·) =φm(π,·) = 0 on (0 ,T),\nθm(·,T) =θm\n0, φm(·,T) =φm\n0 in (0,π),(A.5)\nunder the form\nθm(x,t) =m/summationdisplay\nj=1αjm(t)ηj(x), φm(x,t) =m/summationdisplay\nj=1βjm(t)ηj(x),(x,t)∈QT.\nIt is clear that, for any m≥1, system ( A.5) is equivalent to a Cauchy problem for a linear ordi-\nnary differential system for the variables αjmandβjm, 1≤j≤m. In consequence, system ( A.5)\nadmits a unique solution ( θm,φm)∈C0([0,T];V2\nm) with (θm\nt,φm\nt)∈L2(0,T;V2\nm).\nThe proof of Proposition 2.1can be easily deduced from appropriate estimates of the approx-\nimate solution ( θm,φm) of system ( A.5).\nIf we multiply the first equation in ( A.5) by−ρ\n2θm\nt, the second one by2\nτφm, we integrate on\nthe interval (0 ,π) and we add both equalities, we get,\n/integraldisplayπ\n0/parenleftbiggρ\n2|θm\nt|2−1\nτd\ndt(|φm|2)−ρ\nτθm\ntφm/parenrightbigg\ndx+/integraldisplayπ\n0/parenleftbigg\n−ρξ\n4d\ndt(|θm\nx|2)+2ξ\nτ|φm\nx|2/parenrightbigg\ndx\n+/integraldisplayπ\n0/parenleftbiggρ\nτφmθm\nt−2(ρ−1)\nτ2|φm|2/parenrightbigg\ndx+/integraldisplayπ\n0/parenleftbigg\n−ρ2\n2τθmθm\nt+ρ(ρ−1)\nτ2θmφm/parenrightbigg\ndx\n=−ρ\n2/integraldisplayπ\n0gm\n1θm\ntdx−2\nτ/integraldisplayπ\n0/parenleftigρ\n2gm\n1+gm\n2/parenrightig\nφmdx.\nApplying the Cauchy-Schwarz inequality in the previous equality, we o btain\nρ\n2/bardblθm\nt(·,t)/bardbl2\nL2+2ξ\nτ/bardblφm\nx(·,t)/bardbl2\nL2−d\ndt/parenleftbigg1\nτ/bardblφm(·,t)/bardbl2\nL2+ρξ\n4/bardblθm\nx(·,t)/bardbl2\nL2/parenrightbigg\n≤ρ\n4/bardblθm\nt(·,t)/bardbl2\nL2\n+C/parenleftbig\n/bardblθm(·,t)/bardbl2\nL2+/bardblφm(·,t)/bardbl2\nL2+/bardblgm\n1(·,t)/bardbl2\nL2+/bardblgm\n2(·,t)/bardbl2\nL2/parenrightbig\n,a.e.t∈(0,T),\n29for a constant C >0 depending on the parameters ξ,ρandτ. Using Poincar´ einequality, it follows\n/bardblθm\nt(·,t)/bardbl2\nL2+/bardblφm\nx(·,t)/bardbl2\nL2−d\ndt/parenleftbig\n/bardblφm(·,t)/bardbl2\nL2+/bardblθm\nx(·,t)/bardbl2\nL2/parenrightbig\n≤C/parenleftbig\n/bardblφm(·,t)/bardbl2\nL2+/bardblθm\nx(·,t)/bardbl2\nL2+/bardblgm\n1(·,t)/bardbl2\nL2+/bardblgm\n2(·,t)/bardbl2\nL2/parenrightbig\n,\nfor a new constant C >0. Multiplying the previous inequality by e−C(T−t)and integrating in the\ninterval [t,T], witht0 (only depending on DandA). To this end, let us take two sequences\n{yn\n0}n≥1⊂H1\n0(0,π;R2) and{vn}n≥1∈H1\n0(0,T) such that\nyn\n0→y0inH−1(0,π;R2) andvn→vinL2(0,T).\nWith the previous regularity assumption it is possible to show that sys tem (1.7) foryn\n0,vnand\nfhas a unique strong solution yn∈C0([0,T];H1\n0(0,π;R2))∩L2(0,T;H2(0,π;R2)∩H1\n0(0,π;R2))\nwhich satisfies\n/integraldisplay/integraldisplay\nQTyn·gdxdt=/an}bracketle{tyn\n0,ϕ(·,0)/an}bracketri}ht−/integraldisplayT\n0B∗D∗ϕx(0,t)vn(t)dt+/integraldisplay/integraldisplay\nQTf·ϕdxdt, ∀n≥1,\nfor anyg∈L2(QT;R2), whereϕis the solution of the system ( 2.1) associated to gandϕ0= 0.\nIndeed, if we take the new function /tildewideyn(·,t) =yn(·,T−t)−(vn(T−t),0), one has that /tildewideynsatisfies\na system like ( 2.1) with regular data. Proposition 2.1provides the regularity and the previous\nformula. In fact, the previous equality and ( 2.3) also provide\n/braceleftigg\n/bardblyn/bardblL2(L2)≤eCT/parenleftbig\n/bardbly0/bardblH−1+/bardblv/bardblL2(0,T)+/bardblf/bardblL2(L2)/parenrightbig\n,\nyn→yinL2(QT;R2) andyn,xx→yxxinD′(QT;R2),(A.12)\nfor a new constant C=C(D,A)>0.\n31On the other hand, one has\n/integraldisplay/integraldisplay\nQTyn,xx·ψdxdt=/integraldisplay/integraldisplay\nQTyn·ψxxdxdt−/integraldisplayT\n0B∗ψx(0,t)vn(t)dt,\nfor everyψ∈L2(0,T;H2(0,π;R2)∩H1\n0(0,π;R2)). From this equality we deduce that the se-\nquence{yn,xx}n≥1is bounded in L2(0,T;(H2(0,π;R2)∩H1\n0(0,π;R2))′). This property together\nwith (A.12) givesyxx∈L2(0,T;(H2(0,π;R2)∩H1\n0(0,π;R2))′) and (A.11).\nCombining the identity yt=Dyxx−Ay+fand the regularity property for yxx, we also see\nthatyt∈L2(0,T;(H2(0,π;R2)∩H1\n0(0,π;R2))′) and\n/bardblyt/bardblL2((H2∩H1\n0)′)≤eCT/parenleftbig\n/bardbly0/bardblH−1+/bardblv/bardblL2(0,T)+/bardblf/bardblL2(L2)/parenrightbig\n,\nfor a constant C=C(D,A)>0. Therefore, y∈C0([0,T];X), whereXis the interpolation space\nX=/bracketleftbig\nL2(0,π;R2),(H2(0,π;R2)∩H1\n0(0,π;R2))′/bracketrightbig\n1/2≡H−1(0,π;R2).\nIn conclusion, we have proved ( 2.4). Finally, it is not difficult to check that y(·,0) =y0in\nH−1(0,π;R2). This ends the proof.\nAppendix B\nIn this appendix we will provide a positive answer on the null controllab ility of the phase-field\nsystem (1.1) in the case c= 0. The computations and ideas used for obtaining this controllability\nresult follow the ideas developed for the cases c= 1 andc=−1.\nLet us recall that ˜θ=˜θ(x,t) denotes the temperature of the material and the phase-field\nfunction ˜φ=˜φ(x,t) describes the phase transition of the material (solid or liquid) in suc h a way\nthat˜φ= 1 means that the material is in solid state, ˜φ=−1 in liquid state and ˜φ= 0 is an\nintermediate (mushy) phase.\nIn Theorem 1.3, we proposed a local exact controllability result for the phase-field system (1.1)\nto the trajectories (0 ,−1) or (0,1). Our objective here is to prove a local null controllability result\nfor the same system.\nLet us consider the phase-field system ( 1.1) withc= 0, that is to say, the system\n\n\n˜θt−ξ˜θxx+1\n2ρξ˜φxx+ρ\nτ˜θ=f1(˜φ) in QT,\n˜φt−ξ˜φxx−2\nτ˜θ=f2(˜φ) in QT,\n˜θ(0,·) =v,˜φ(0,·) = 0,˜θ(π,·) = 0,˜φ(π,·) = 0 on (0 ,T),\n˜θ(·,0) =˜θ0,˜φ(·,0) =˜φ0 in (0,π).(B.1)\nwhereξ,ρandτare positive parameters and the nonlinear terms f1(˜φ) andf2(˜φ) are given by\nf1(˜φ) =−ρ\n4τ/parenleftig\n˜φ−˜φ3/parenrightig\nandf2(˜φ) =1\n2τ/parenleftig\n˜φ−˜φ3/parenrightig\n.\nFor this system, a linearization around the equilibrium (0 ,0) provides the following linear problem\nin vectorial form: \n\nyt−Dyxx+ˆAy= 0 in QT,\ny(0,·) =Bv, y(π,·) = 0 on (0 ,T),\ny(·,0) =y0, in (0,π),(B.2)\nwithy0= (θ0,φ0) ,y= (θ,φ) and\nD=\nξ−1\n2ρξ\n0ξ\n,ˆA=\nρ\nτρ\n4τ\n−2\nτ−1\n2τ\n, B=/parenleftbigg\n1\n0/parenrightbigg\n. (B.3)\n32FollowingthesameideasusedintheAppendix A,wecanprovethat,forevery y0∈H−1(0,π;R2)\nandv∈L2(0,T), system ( B.2) has a unique solution by transposition (see Definition 2.1)\ny∈L2(QT;R2)∩C0([0,T];H−1(0,π;R2)) which depends continuously on the data:\n/bardbly/bardblL2(L2)+/bardbly/bardblC0(H−1)≤CeCT/parenleftbig\n/bardbly0/bardblH−1+/bardblv/bardblL2(0,T)/parenrightbig\n,\nfor a constant C >0 only depending on the parameters ξ,ρandτin system ( B.1).\nIn order to state the null controllability result for systems ( B.1) and (B.2), let us consider the\nvectorial operators\nˆL=−D∂xx+ˆAandˆL∗=−D∗∂xx+ˆA∗, (B.4)\nwith domains D(ˆL) =D(ˆL∗) =H2(0,π;R2)∩H1\n0(0,π;R2).\nThe first result in this appendix establishes the approximate contro llability of system ( B.2) at\ntimeT >0. One has:\nTheorem B.1. Let us consider ξ,ρandτthree positive real numbers and let us fix T >0. Then,\nsystem(B.2)is approximately controllable in H−1(0,π;R2)at timeTif and only if the eigenvalues\nof the operators ˆLandˆL∗are simple. Moreover, this equivalence amounts to the condi tion\n4ξ2τ2(ℓ2−k2)2−8ξρτ(ℓ2+k2)−4ρ−1/ne}ationslash= 0,∀k,ℓ≥1, ℓ>k. (B.5)\nThe second result in this appendix establishes the null controllability r esult at time T >0 of\nsystem (B.2) and reads as follows:\nTheorem B.2. Let us us fix T >0and consider ξ,ρandτpositive real numbers satisfying (B.5)\nand\nξ/ne}ationslash=1\nj2ρ\nτ,∀j≥1. (B.6)\nThen, system (B.2)is exactly controllable to zero in H−1(0,π;R2)at timeT >0. Moreover, there\nexist two positive constants C0andM, only depending on ξ,ρandτ, such that for any T >0,\nthere is a bounded linear operator C(0)\nT:H−1(0,π;R2)→L2(0,T)satisfying\n/bardblC(0)\nT/bardblL(H−1(0,π;R2),L2(0,T))≤C0eM/T,\nand such that the solution\ny= (θ,φ)∈L2(QT;R2)∩C0([0,T];H−1(0,π;R2))\nof system (B.2)associated to y0= (θ0,φ0)∈H−1(0,π;R2)andv=C(0)\nT(y0)satisfiesy(·,T) = 0.\nRemark B.1. Observe that assumptions ( B.5) and (B.6) play the role in Theorems B.1andB.2\nof conditions ( 1.9) and (1.11) in Theorems 1.1and1.2.\nThe local null controllability result for the nonlinear system ( B.1) is given in the next result:\nTheorem B.3. Let us consider ξ,τandρthree positive numbers satisfying (B.5)and(B.6), and\nlet us fixT >0. Then, there exist ε>0such that, for any (˜θ0,˜φ0)∈H−1(0,π)×H1\n0(0,π)fulfilling\n/bardbl˜θ0/bardblH−1+/bardbl˜φ0/bardblH1\n0≤ε,\nthere exists v∈L2(0,T)for which system (B.1)has a unique solution\n(˜θ,˜φ)∈/bracketleftbig\nL2(QT)∩C0([0,T];H−1(0,π;R2))/bracketrightbig\n×C0(QT)\nwhich satisfies\n˜θ(·,T) = 0and˜φ(·,T) = 0in(0,π).\n33The proofs of Theorems B.1,B.2andB.3follow the same reasoning and ideas of the proofs\nof Theorems 1.1,1.2and1.3. They are based on an exhaustive study of the eigenvalues and\neigenfunctions of the operators ˆLandˆL∗. In this sense, the properties of these eigenvalues and\neigenfunctions are very close to the properties of the spectra of the operators LandL∗(see (1.10).\nIndeed, we have the following result.\nProposition B.1. Let us consider the operators ˆLandˆL∗given in (B.4)(the matrices DandˆA\nare given in (B.3)). Then,\n1. The spectra of ˆLandˆL∗are given by σ(ˆL) =σ(ˆL∗) ={ˆλ(1)\nk,ˆλ(2)\nk}k≥1with\nˆλ(1)\nk=ξk2+2ρ+1\n4τ−ˆrk,ˆλ(2)\nk=ξk2+2ρ+1\n4τ+ ˆrk,∀k≥1, (B.7)\nwhere\nˆrk:=/radicaligg\nξρ\nτk2+/parenleftbigg2ρ+1\n4τ/parenrightbigg2\n.\n2. For each k≥1, the eigenspaces of ˆL(resp.,ˆL∗) corresponding to ˆλ(1)\nkandˆλ(2)\nkare respectively\ngenerated by\nˆΨ(1)\nk=1\n8√τˆrk/parenleftigg\n1−2ρ+4τˆrk\n8/parenrightigg\nηk,ˆΨ(2)\nk=1\n8√τˆrk/parenleftigg\n1−2ρ−4τˆrk\n8/parenrightigg\nηk,\n(resp.,\nˆΦ(1)\nk=1\n8√τˆrk/parenleftigg\n8\n2ρ−1+4τˆrk/parenrightigg\nηk,ˆΦ(2)\nk=−1\n4√τˆrk/parenleftigg\n8\n2ρ−1−4τˆrk/parenrightigg\nηk).\n(The function ηkis given in (3.2)).\n3. The sequences ˆF={ˆΨ(1)\nk,ˆΨ(2)\nk}k≥1andˆF∗={ˆΦ(1)\nk,ˆΦ(2)\nk}k≥1are such that\ni)ˆFandˆF∗are biorthogonal sequences.\nii) span/parenleftig\nˆF/parenrightig\nand span/parenleftig\nˆF∗/parenrightig\nare dense in H−1(0,π;R2),L2(0,π;R2)andH1\n0(0,π;R2).\niii)ˆFandˆF∗are unconditional bases for H−1(0,π;R2),L2(0,π;R2)andH1\n0(0,π;R2).\nThe proof of Proposition B.1follows the same ideas of the proofs of Propositions 3.1and3.4.\nThe details are left to the reader.\nObserve that the expressions of the eigenvalues of ˆLandˆL∗(see (B.7)) are close to those of\noperatorsLandL∗(see (3.3)). In fact, replacing ( ρ,τ) by (2ρ,2τ) in (3.3), we obtain ( B.7). So,\nwe can repeat the computations of the proof of Proposition 3.2in order to proof the following\nresults concerning the spectral analysis for σ(ˆL) =σ(ˆL∗) ={ˆλ(1)\nk,ˆλ(2)\nk}k≥1:\nProposition B.2. Under the assumptions of Proposition B.1, the following properties hold:\n(P1){ˆλ(1)\nk}k≥1and{ˆλ(2)\nk}k≥1(see(B.7)) are increasing sequences satisfying\n0<ˆλ(1)\nk<ˆλ(2)\nk,∀k≥1.\n(P2) The spectrum of ˆLandˆL∗is simple, i.e., ˆλ(2)\nk/ne}ationslash=ˆλ(1)\nℓ, for allk,ℓ≥1if and only if the\nparameters ξ,ρandτsatisfy condition (B.5).\n34(P3) Assume that the parameters ξ,ρandτsatisfy(B.6), i.e., there exists j≥0such that\n1\n(j+1)2ρ\nτ<ξ <1\nj2ρ\nτ.\nThen, there exists an integer k0=k0(ξ,ρ,τ,j)≥1and a constant C=C(ξ,ρ,τ,j)>0such\nthat\nˆλ(1)\nk+j<ˆλ(2)\nk<ˆλ(1)\nk+1+j<ˆλ(2)\nk+1<···,∀k≥k0,andmin\nk≥k0/braceleftig\nˆλ(2)\nk−ˆλ(1)\nk+j,ˆλ(1)\nk+j+1−ˆλ(2)\nk/bracerightig\n>C.\n(P4) Assume now that the parameters ξ,ρandτsatisfy(B.5)and(B.6). Then, one has:\ninf\nk,ℓ≥1|ˆλ(2)\nk−ˆλ(1)\nℓ|>0,\nand there exists a positive integer k1∈N, depending on ξ,ρandτ, such that\nmin/braceleftig/vextendsingle/vextendsingle/vextendsingleˆλ(1)\nk−ˆλ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingleˆλ(2)\nk−ˆλ(2)\nℓ/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsingleˆλ(2)\nk−ˆλ(1)\nℓ/vextendsingle/vextendsingle/vextendsingle/bracerightig\n≥ξ\n2|k2−ℓ2|,∀k,ℓ≥1,|k−ℓ| ≥k1.\nWe also have:\nProposition B.3. Let us assume that the parameters ξ,ρ,τsatisfy(B.5). Then, the sequence\n{ˆλ(1)\nk,ˆλ(2)\nk}k≥1, given by (B.7), can be rearranged into an increasing sequence ˆΛ ={ˆΛk}k≥1that\nsatisfies (2.9)andˆΛk/ne}ationslash= Λn, for allk,n∈Nwithk/ne}ationslash=n. In addition, if (B.6)holds, the sequence\n{ˆΛk}k≥1also satisfies (2.10)and(2.11).\nAs said before, the proofs of Theorems B.1andB.2follow the same ideas of the proofs of The-\norems1.1and1.2. To be precise, Theorem B.1can be deduce from item ( P2) in Proposition B.2.\nOn the other hand, Theorem B.2can be proved combining Proposition B.2, Proposition B.3and\nLemma2.2, and following the same reasoning of Theorem 1.2.\nFinally, the same proof presented in Section 5.2can be easily adapted to Theorem B.3, with\nthe observation that the operator Nin (5.15), that represents the nonlinearity of the system ( 1.1)\nwithc=±1, can be defined as follows:\nN(f) =\nρ\n4τ˜φ3\n−1\n2τ˜φ3\n.\nThe proof of Theorem B.3can be deduced applying the Banach Fixed-Point Theorem. The deta ils\nare left to the reader.\nReferences\n[1]F. Alabau-Boussouira ,Controllability of cascade coupled systems of multi-dimen sional\nevolution PDEs by a reduced number of controls , C. R. Math. Acad. Sci. Paris 350(2012),\nno. 11-12, 577–582.\n[2]F. Alabau-Boussouira, M. L ´eautaud ,Indirect controllability of locally coupled wave-type\nsystems and applications , J. Math. Pures Appl. (9) 99(2013), no. 5, 544–576.\n[3]F. Ammar Khodja, A. Benabdallah, C. Dupaix, I. Kostin ,Controllability to the\ntrajectories of phase-field models by one control force , SIAM J. Control Optim. 42(2003),\nno. 5, 1661–1680.\n[4]F. Ammar Khodja, A. Benabdallah, M. Gonz ´alez-Burgos, L. de Teresa ,Recent\nresults on the controllability of linear coupled parabolic problems: a survey , Math. Control\nRelat. Fields 1(2011), no. 3, 267–306.\n35[5]F. Ammar Khodja, A. Benabdallah, M. Gonz ´alez-Burgos, L. de Teresa ,The\nKalman condition for the boundary controllability of coupl ed parabolic systems. Bounds on\nbiorthogonal families to complex matrix exponentials , J. Math. Pures Appl. (9) 96(2011),\nno. 6, 555–590.\n[6]F. Ammar Khodja, A. Benabdallah, M. Gonz ´alez-Burgos, L. de Teresa ,Minimal\ntime for the null controllability of parabolic systems: the effect of the condensation index of\ncomplex sequences , J. Funct. Anal. 267(2014), no. 7, 2077–2151.\n[7]F. Ammar Khodja, A. Benabdallah, M. Gonz ´alez-Burgos, L. de Teresa ,New\nphenomena for the null controllability of parabolic system s: Minimal time and geometrical\ndependence , J. Math. Anal. Appl. 444(2016), no. 2, 1071–1113.\n[8]A. Benabdallah, F. Boyer, M. Gonz ´alez-Burgos, G. Olive ,Sharp estimates of the\none-dimensional boundary control cost for parabolic syste ms and application to the N-dimen-\nsional boundary null controllability in cylindrical domai ns, SIAMJ.ControlOptim. 52(2014),\nno. 5, 2970–3001.\n[9]G. Caginalp ,An analysis of a phase field model of a free boundary , Arch. Rational Mech.\nAnal.92(1986), no. 3, 205–245.\n[10]J.-M. Coron ,Control and Nonlinearity , MathematicalSurveysandMonographs,136,Amer-\nican Mathematical Society, Providence, RI (2007).\n[11]A. Doubova, E. Fern ´andez-Cara, M. Gonz ´alez-Burgos, E. Zuazua ,On the control-\nlability of parabolic systems with a nonlinear term involvi ng the state and the gradient , SIAM\nJ. Control Optim. 41(2002), no. 3, 798–819.\n[12]H.O. Fattorini ,Some remarks on complete controllability , SIAM J. Control 4(1966), 686–\n694.\n[13]H.O. Fattorini, D.L. Russel ,Exact controllability theorems for linear parabolic equat ions\nin one space dimension , Arch. Rational Mech. Anal. 43(1971), 272–292.\n[14]E. Fern ´andez-Cara, E. Zuazua ,The cost of approximate controllability for heat equations :\nthe linear case , Adv. Differential Equations 5(2000), no. 4-6, 465–514.\n[15]E. Fern ´andez-Cara, M. Gonz ´alez-Burgos, L. de Teresa ,Boundary controllability of\nparabolic coupled equations , J. Funct. Anal. 259(2010), no. 7, 1720–1758.\n[16]E. Fern ´andez-Cara, M. Gonz ´alez-Burgos, L. de Teresa ,Controllability of linear\nand semilinear non-diagonalizable parabolic systems , ESAIM Control Optim. Calc. Var. 21\n(2015), no. 4, 1178–1204.\n[17]E. Fern ´andez-Cara, E. Zuazua ,Null and approximate controllability for weakly blowing\nup semilinear heat equations , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 17(2000), no. 5,\n583–616.\n[18]M. Gonz ´alez-Burgos, R. P ´erez-Garc ´ıa,Controllability results for some nonlinear cou-\npled parabolic systems by one control force , Asymptot. Anal. 46(2006), no. 2, 123–162.\n[19]Y. Liu, T. Takahashi, M. Tucsnak ,Single input controllability of a simplified fluid-\nstructure interaction model , ESAIM Control Optim. Calc. Var. 19(2013), no. 1, 20–42.\n[20]L. Miller ,Geometric bounds on the growth rate of null-controllabilit y cost for the heat\nequation in small time , J. Differential Equations 204(2004), no. 1, 202–226.\n[21]T.I. Seidman ,Two results on exact boundary control of parabolic equation s, Appl. Math.\nOptim.11(1984), no. 2, 145–152.\n36[22]M. Tucsnak, G. Weiss ,Observation and Control for Operator Semigroups , Birkh¨ auser\nAdvanced Texts: Basler Lehrb¨ ucher, Birkh¨ auser Verlag, Base l (2009).\n[23]J. Zabczyk, Mathematical Control Theory: An Introduction , Systems & Control: Founda-\ntions & Applications, Birkh¨ auser Boston, Inc., Boston, MA (1992) .\n37" }, { "title": "2401.17102v1.Linear_stability_analysis_of_the_Couette_flow_for_the_2D_Euler_Poisson_system.pdf", "content": "arXiv:2401.17102v1 [math.AP] 30 Jan 2024Linear stability analysis of the Couette flow for the 2D\nEuler-Poisson system\nXueke Pu1, Wenli Zhou1and Dongfen Bian2,∗\n1School of Mathematics and Information Science, Guangzhou Un iversity,\nGuangzhou 510006, PR China\n2School of Mathematics and Statistics, Beijing Institute of T echnology,\nBeijing 100081, PR China\nAbstract This paper is concerned with the linear stability analysis for the Couet te flow of\nthe Euler-Poisson system for both ionic fluid and electronic fluid in the domainT×R. We\nestablish the upper and lower bounds of the linearized solutions of th e Euler-Poissonsystem\nnear Couette flow. In particular, the inviscid damping for the soleno idal component of the\nvelocity is obtained.\nKeywords : Euler-Poisson equations; Couette flow; inviscid damping\nMathematics Subject Classification : 35M30; 76E05\n1. Introduction\nConsider the following two fluid Euler-Poisson system in a tw o dimensional domain T×R,\n∂t/tildewideη±+∇·(/tildewideη±/tildewidev±) = 0,\n/tildewideη±m±[∂t/tildewidev±+/tildewidev±·∇/tildewidev±]+T±∇/tildewidep±=∓e/tildewideη±∇/tildewideφ,\n−∆/tildewideφ= 4πe[/tildewideη+−/tildewideη−],(1.1)\nwhere/tildewideη±(x,y,t) and/tildewidev±(x,y,t) denote the density and the velocity of the ion fluid (+) and\nthe electron fluid ( −) in a plasma, ∇/tildewideφrepresents the self-consistent electric field, ∇/tildewidep±denote\nthe pressure of ions and electrons, m±denote the mass density of the ions and the electrons,\neis the charge for an electron and T±denote the effective temperatures of the ions and the\nelectrons. Such a system is the simplified model from the two fl uid Euler-Maxwell system as\nthe light speed tends to infinity and is very fundamental in pl asma physics. This Euler-Poisson\nsystem is also an important origin of many famous dispersive equations such as KdV, KP, NLS\nequations and is widely studied in the literature in recent y ears. See a series of papers of the\nfirst author and his collaborators [ 21,39–41,44] for the rigorous mathematical justifications of\nthese dispersive equations from the Euler-Poisson system. For more physics background and\nthe mathematical development of such a system, one may see th e concise review paper [ 17] and\nthe references cited therein.\nThe two-fluid Euler-Poisson system has rich and complex dyna mics with distinct physical\nparameters. In particular, since the mass ratio of electron s and ions is of 10−3≪1, the system\n∗Corresponding author.\nE-mail addresses :puxueke@gmail.com (X. Pu), wywlzhou@163.com (W. Zhou), biandongfen@bit.edu.cn (D.\nBian)2 X. Pu et al.\n(1.1) can be simplified to the following one-fluid Euler-Poisson s ystem for the electron fluid\ndynamics of ( /tildewideη−,/tildewidev−,/tildewideφ) in an ion background (Langmuir waves, [ 18,19])\n∂t/tildewideη−+∇·(/tildewideη−/tildewidev−) = 0, (1.2)\n/tildewideη−[∂t/tildewidev−+(/tildewidev−·∇)/tildewidev−]+1\nm−∇/tildewidep−(/tildewideη−) =e\nm−/tildewideη−∇/tildewideφ, (1.3)\n∆/tildewideφ= 4πe/tildewideη−−4πe. (1.4)\nOn the other hand, assume /tildewidep−(/tildewideη−) =T−/tildewideη−in (1.1) with a constant temperature T−and\ntake formally m−/m+→0 (such a limit is rigorously justified by Grenier et al.recently in [ 16]),\nwe deduce ∇/tildewidep−=T−∇/tildewideη−=e/tildewideη−∇˜φ, from which the celebrated Boltzmann relation for the\nelectron density can be derived:\n/tildewideη−=η0exp{e/tildewideφ/T−},\nwhereη0is a constant, set to be one for simplicity. Then we obtain the reduced Euler-Poisson\nsystem for ion dynamics for ( /tildewideη+,/tildewidev+,/tildewideφ)\n∂t/tildewideη++∇·(/tildewideη+/tildewidev+) = 0, (1.5)\n/tildewideη+[∂t/tildewidev++(/tildewidev+·∇)/tildewidev+]+T+\nm+∇/tildewideη+=−e\nm+/tildewideη+∇/tildewideφ, (1.6)\n∆/tildewideφ= 4πe/bracketleftbigg\nexp/parenleftbigge\nT−/tildewideφ/parenrightbigg\n−/tildewideη+/bracketrightbigg\n. (1.7)\nFor global well-posedness of such ion dynamics system in R3+1, one may refer to [ 20].\nIn this paper, we are interested in another topic of this syst em, i.e., stability of the Couette\nflow of the above Euler-Poisson system. To be precise, we will consider the linear stability of\nthe Couette flow for the two Euler-Poisson systems ( 1.2)-(1.4) and (1.5)-(1.7) in a periodic strip\nT×RwithT=R/Z. It is clear that there is a stationary solution\n(/tildewideηs,/tildewidevs,/tildewideφs) = (1,(y,0)⊤,0)\nfor both system ( 1.5)-(1.7) and (1.2)-(1.4). We also call this solution Couette flow, following\nthe name in Navier-Stokes equation in fluid dynamics. Formal stability analysis of plasma\nequilibria to the related Poisson-Vlasov and Maxwell-Vlas ov systems in one, two and three\ndimensions is referred to the paper of Holm et al[22]. To the best of our knowledge, no rigorous\nmathematical stability analysis of the Couette flow for the E uler-Poisson system either for\nthe one-fluid Euler-Poisson system or the two-fluid Euler-Po isson system can be found in the\nliterature and we believe that this topic is new and interest ing for the Euler-Poisson system.\nThe purpose of the present paper is to study the linear stabil ity of the Couette flow for both\nthe ion and electron one-fluid Euler-Poisson dynamics.\nLet us deviate slightly to briefly introduce some related res ults on the stability analysis\nof the Couette flow in fluid dynamics, in particular for the Nav ier-Stokes or the Euler system.\nIndeed, from themathematical point of view, the stability o f the Couette flow and related topics\nhave been extensively studied and many fruitful results wer e obtained in the past decades. For\nan incompressible fluid, the classical results on the linear stability analysis of the Couette flow\nwere obtained by Rayleigh [ 45] and Kelvin [ 34]. Inviscid damping for the Euler equations on\nthe domain T×Rwas studied by Bedrossian and Masmoudi [ 7], which is a significant result\non nonlinear stability of the planar Couette flow. For nonlin ear inviscid damping for a classLinear stability of the Couette flow 3\nof monotone shear flows for the 2D Euler equation in finite chan nel for initial perturbation in\nGevrey class with compact support, one may refer to [ 42]. See also [ 24] for the inviscid damping\nnear the Couette flow in a channel for the 2D Euler equation, wh ere the authors showed that\nthe velocity field converges to a nearby shear flow when the ini tial perturbation is small in a\nsuitable Gevrey sapce. One may also refer to other nice work [ 11,25,29,30,38,43,47,50] for the\nstability analysis of the Couette flow or general shear flows f or the Euler equation in various\nsituations. In the presence of viscosity, some interesting results have been obtained in the study\nof transition threshold problem related to enhanced dissip ation (see, e.g., [ 3,5,6,8–10,43,46] for\nvarious situations). More results on the stability of the tw o-dimensional and three-dimensional\nCouetteflow intheNavier-Stokes equations at highReynolds numbercan befoundinthereview\npaper [4].\nIn the case of compressible fluid, the linear stability of pla ne Couette flow has been studied\nin the past decades in physics literature in the past nearly a century starting from the ’40s\n[12,22,35]. Glatzel [ 14] considered the inviscid stability of this problem based on a normal\nmode analysis, and the effects of viscosity were then taken int o account by the same author [ 15].\nDucket al.[13] investigated the compressible stability of the plane Coue tte flow for realistic\ncompressible flow models, where they show that the details of the mean flow profile have a\nprofound effect on the stability of the flow. Hu and Zhong [ 23] studied the linear stability\nof viscous supersonic plane Couette flow for a perfect gas gov erned by Sutherland viscosity\nlaw by using a fourth-order finite-difference method and a spec tral collocation method. For\nrigorous mathematical analysis, when the initial value is s ufficiently close to the plane Couette\nflow, Kagei [ 31] proved that the plane Couette flow is asymptotically stable provided that the\nReynolds and Mach numbers are sufficiently small, see also Kag ei [32,33] for more general\nparallel flows. Subsequently, the constraint on the Reynold s number in [ 31], which is used to\nensure the stability of the plane Couette flow, was relaxed by Li and Zhang [ 37]. The inviscid\ndamping and enhanced dissipation phenomena were detected b y Antonelli et al.[1,2] for the\nhomogeneous Couette flow in a 2D isentropic compressible flui d. In addition, we refer to [ 49]\nfor a inviscid damping result on the non-isentropic compres sible Euler equations on the domain\nT×R. For the 3D case, Zeng et al.[48] established the linear stability result of the Couette flow\nin the isentropic compressible Navier-Stokes equations on the domain T×R×T. However, the\nrigorous mathematical study is less developed compared to t he incompressible case, and a lot of\nwork is to be established in both the linear and the nonlinear level in the future. In particular,\nthere is no stability analysis for the Couette flow for the Eul er-Poisson system studied in this\npaper.\nIn the following, we will discuss the linear stability of the Couette flow for the ion Euler-\nPoisson system and the electron Euler-Poisson system separ ately. Compared to the classical\nEuler system and the Navier-Stokes system, the analysis inc ludes some new insights, since the\nsystem is nonlocal due to the presence of the electric field.\n1.1. Ion dynamics case. To study the linear stability for the Couette flow in system ( 1.5)-\n(1.7), we consider a perturbation around the above stationary so lution, i.e., let\n/tildewideη+=η++/tildewideηs,/tildewidev+=u++/tildewidevs,/tildewideφ=φ+/tildewideφs.\nWe first consider the ion dynamics system ( 1.5)-(1.7). Then the linearized system around the\nstationary solution ( /tildewideηs,/tildewidevs,/tildewideφs) for system ( 1.5)-(1.7) is written as\n∂tη++y∂xη++(∇·u+) = 0, (1.8)4 X. Pu et al.\n∂tu++y∂xu++/parenleftbigguy\n+\n0/parenrightbigg\n+T+\nm+∇η+=−e\nm+∇φ, (1.9)\n∆φ= 4πe/parenleftbigge\nT−φ−η+/parenrightbigg\n. (1.10)\nLetψ+=∇ ·u+andω+=∇⊥·u+with∇⊥= (−∂y,∂x)⊤. Then the Helmholtz projection\noperators are defined as\nu+= (ux\n+,uy\n+)⊤=∇∆−1ψ++∇⊥∆−1ω+:=Q[u+]+P[u+]. (1.11)\nThenthelinearized system ( 1.8)-(1.10) istransformedintotheformwith respectto ( η+,ψ+,ω+)\n∂tη++y∂xη++ψ+= 0, (1.12)\n∂tψ++y∂xψ++2∂xuy\n++T+\nm+∆η+=−4πe2\nm+∆/parenleftbigg\n−∆+4πe2\nT−/parenrightbigg−1\nη+,(1.13)\n∂tω++y∂xω+−ψ+= 0. (1.14)\nDenote the average of a function in the xdirection by\nfa(y) =1\n2π/integraldisplay\nTf(x,y)dx.\nWe take the x-average for equations ( 1.12), (1.13) and (1.14) to obtain the following system\n∂tη+,a+ψ+,a= 0, (1.15)\n∂tψ+,a+T+\nm+∂yyη+,a=−4πe2\nm+∆/parenleftbigg\n−∆+4πe2\nT−/parenrightbigg−1\nη+,a, (1.16)\n∂tω+,a−ψ+,a= 0. (1.17)\nCombining ( 1.15) with (1.17) yields∂t(η+,a+ω+,a) = 0. It is easy to see that\nη+,a+ω+,a=ηin\n+,a+ωin\n+,a, (1.18)\nwhere (ηin\n+,a,ωin\n+,a) represents the initial data. In the rest of the paper, we sti ll use similar\nnotation to represent the initial data.\nAccording to ( 1.15) and (1.16), we get\n∂ttη+,a−T+\nm+∂yyη+,a−4πe2\nm+∆/parenleftbigg\n−∆+4πe2\nT−/parenrightbigg−1\nη+,a= 0inR. (1.19)\nNext we supply system ( 1.15)-(1.17) with the initial condition\n(η+,a,ψ+,a,ω+,a)|t=0=/parenleftbig\nηin\n+,a,ψin\n+,a,ωin\n+,a/parenrightbig\n= (0,0,0). (1.20)\nThen (η+,a,ψ+,a,ω+,a) = (0,0,0) is the unique solution to system ( 1.15)-(1.17) with initial\ncondition ( 1.20). In fact, taking the L2(R) inner product of ( 1.19) with∂tη+,a, then it follows\nfrom integration by parts that\n1\n2d\ndt\n/ba∇dbl∂tη+,a/ba∇dbl2\nL2+T+\nm+/ba∇dbl∂yη+,a/ba∇dbl2\nL2+4πe2\nm+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∇/parenleftbigg\n−∆+4πe2\nT−/parenrightbigg−1\n2\nη+,a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n= 0,Linear stability of the Couette flow 5\nwhich implies that η+,a= 0 is the unique solution to the cauchy problem, equation ( 1.19) with\nηin\n+,a= 0. By virtue of ( 1.18) and (1.15), we deduce that ( ψ+,a,ω+,a) = (0,0). Based on this\nfact, for simplicity of notation, we study the dynamics of ( η+,ψ+,ω+) with/parenleftbig\nηin\n+,a,ψin\n+,a,ωin\n+,a/parenrightbig\n=\n(0,0,0), instead of studying the dynamics for ( η+−η+,a,ψ+−ψ+,a,ω+−ω+,a).\nThe main results for the ion dynamics are as follows.\nTheorem 1.1. Suppose that (ηin\n+,ωin\n+)∈H1\nxH2\nyand thatψin\n+∈H−1\n2xL2\nywith/parenleftbig\nηin\n+,a,ψin\n+,a,ωin\n+,a/parenrightbig\n=\n(0,0,0). Let(η+,u+,φ)be a smooth solution for the system (1.8)-(1.10). Then the following\nestimates hold\n/ba∇dblP[u+]x(t)/ba∇dblL2+/ba∇dblφ(t)/ba∇dblL2/lessorsimilar/radicalbiggm+\nT+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\nT+\nm+ηin\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH−1\n2xL2y+/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble\nH−1\n2xH1\n2y\n\n+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble\nH−1\nxH1y+/radicalbiggm+\nT+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2/vextenddouble/vextenddoubleψin\n+/vextenddouble/vextenddouble\nH−1\n2xH−1\ny,(1.21)\n/ba∇dblP[u+]y(t)/ba∇dblL2/lessorsimilar/radicalbiggm+\nT+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\nT+\nm+ηin\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH−1\n2xH1y+/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble\nH−1\n2xH3\n2y\n\n+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht2/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble\nH−1\nxH2y+/radicalbiggm+\nT+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2/vextenddouble/vextenddoubleψin\n+/vextenddouble/vextenddouble\nH−1\n2xL2y, (1.22)\nand\n/ba∇dblQ[u+](t)/ba∇dblL2+/radicaligg\nT+\nm+/ba∇dblη+(t)/ba∇dblL2\n/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\nT+\nm+ηin\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddoubleψin\n+/vextenddouble/vextenddouble\nH−1+/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble\nH1/parenrightigg\n. (1.23)\nTheorem 1.2. Assume that/parenleftbig\nηin\n+,ωin\n+/parenrightbig\n∈L2\nxH−1\n2yand thatψin\n+∈H−3\n2xH−2\nywith/parenleftbig\nηin\n+,a,ψin\n+,a,ωin\n+,a/parenrightbig\n=\n(0,0,0). Let(η+,u+,φ)be a smooth solution for the system (1.8)-(1.10). Then it holds that\n/ba∇dblQ[u+](t)/ba∇dblL2+/radicaligg\nT+\nm+/ba∇dblη+(t)/ba∇dblL2/greaterorsimilarCin\nδ/parenleftbig\nηin\n+,ψin\n+,ωin\n+/parenrightbig\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2,\nwhere the constant Cin\nδ/parenleftbig\nηin\n+,ψin\n+,ωin\n+/parenrightbig\nis a suitable combination of/vextenddouble/vextenddoubleηin\n+/vextenddouble/vextenddouble\nL2xH−1\n2y,/vextenddouble/vextenddoubleψin\n+/vextenddouble/vextenddouble\nH−3\n2xH−2\ny\nand/vextenddouble/vextenddoubleωin\n+/vextenddouble/vextenddouble\nL2xH−1\n2y.\nThe above two theorems give upper and lower bounds of the line arized solutions of the\nEuler-Poisson system near Couette flow. In particular, the i nviscid damping for the solenoidal\ncomponent of the velocity is established, while the L2-norm of the velocity grows as t1/2. We\nalso remark that it may happen that the constant Cin\nδvanishes at some positive time t >0,\nhowever we can prove that the set of initial data for which the RHS of the lower bound in\nTheorem 1.2vanishes at some time has empty interior in any Sobolev space in which the initial\ndata are taken. The details are similar to that for the Euler s ystem in [ 1] and are omitted here.6 X. Pu et al.\n1.2. Electron dynamics case. To study the linear stability for the Couette flow in system\n(1.2)-(1.4), we also carry out a perturbation around the above stationa ry solution, i.e.,\n/tildewideη−=η−+/tildewideηs,/tildewidev−=u−+/tildewidevs,/tildewideφ=φ+/tildewideφs.\nThen the linearized system around the stationary solution ( /tildewideηs,/tildewidevs,/tildewideφs) for system ( 1.2)-(1.4) can\nbe rewritten as\n∂tη−+y∂xη−+(∇·u−) = 0, (1.24)\n∂tu−+y∂xu−+/parenleftbigguy\n−\n0/parenrightbigg\n+1\nm−∇η−=e\nm−∇φ, (1.25)\n∆φ= 4πeη−. (1.26)\nLetψ−=∇ ·u−andω−=∇⊥·u−with∇⊥= (−∂y,∂x)⊤. Then the Helmholtz projection\noperators are defined as\nu−= (ux\n−,uy\n−)⊤=∇∆−1ψ−+∇⊥∆−1ω−:=Q[u−]+P[u−]. (1.27)\nThe linearized system ( 1.24)-(1.26) is rewritten as (in the variables ( η−,ψ−,ω−))\n∂tη−+y∂xη−+ψ−= 0, (1.28)\n∂tψ−+y∂xψ−+2∂xuy\n−+1\nm−∆η−=4πe2\nm−η−, (1.29)\n∂tω−+y∂xω−−ψ−= 0. (1.30)\nJust as in the ion dynamics case, we take the x-average for equations ( 1.28), (1.29) and\n(1.30) to obtain the following system\n∂tη−,a+ψ−,a= 0, (1.31)\n∂tψ−,a+1\nm−∂yyη−,a=4πe2\nm−η−,a, (1.32)\n∂tω−,a−ψ−,a= 0. (1.33)\nAgain, combining ( 1.31) with (1.33) yields∂t(η−,a+ω−,a) = 0 and hence\nη−,a+ω−,a=ηin\n−,a+ωin\n−,a, (1.34)\nwhere (ηin\n−,a,ωin\n−,a) represents the initial data. According to ( 1.31) and (1.32), we get\n∂ttη−,a−1\nm−∂yyη−,a+4πe2\nm−η−,a= 0inR, (1.35)\nwhich is the famous Klein-Gordon equation. The fact that at t he linearized level (near the\nconstant solution), the electron Euler-Poisson system sat isfies the Klein-Gordon equation was\nfound by Guo [ 18] and the global irrotational flows in the large to the electro n Euler-Poisson\nsystem was hence obtained in R3+1. For global well-posedness of the lower dimensional Euler-\nPoisson system, one may refer to [ 26–28,36].Linear stability of the Couette flow 7\nSimilar to the ion dynamics case, we supply system ( 1.31)-(1.33) with the initial condition\n(η−,a,ψ−,a,ω−,a)|t=0=/parenleftbig\nηin\n−,a,ψin\n−,a,ωin\n−,a/parenrightbig\n= (0,0,0). (1.36)\nFrom (1.34) and the explicit representation formula for the Klein-Gor don equation, we deduce\nthat (η−,a,ψ−,a,ω−,a) = (0,0,0) is the unique solution to system ( 1.31)-(1.33) with initial\ncondition ( 1.36). Based on this fact, for simplicity of notation, we study th e dynamics of\n(η−,ψ−,ω−) with/parenleftbig\nηin\n−,a,ψin\n−,a,ωin\n−,a/parenrightbig\n= (0,0,0), instead of studying the dynamics for ( η−−\nη−,a,ψ−−ψ−,a,ω−−ω−,a).\nThe main results for the linear stability of the electron Eul er-Poisson system are stated as\nfollows.\nTheorem 1.3. Suppose that (ηin\n−,ωin\n−)∈H1\nxH2\nyand thatψin\n−∈H−1\n2xL2\nywith/parenleftbig\nηin\n−,a,ψin\n−,a,ωin\n−,a/parenrightbig\n=\n(0,0,0). Let(η−,u−,φ)be a smooth solution for the system (1.24)-(1.26). Then the following\nestimates hold\n/ba∇dblP[u−]x(t)/ba∇dblL2+/ba∇dblφ(t)/ba∇dblL2/lessorsimilar√m−\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\n1\nm−ηin\n−/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH−1\n2xL2y+/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble\nH−1\n2xH1\n2y\n\n+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble\nH−1\nxH1y+√m−\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2/vextenddouble/vextenddoubleψin\n−/vextenddouble/vextenddouble\nH−1\n2xH−1\ny,(1.37)\n/ba∇dblP[u−]y(t)/ba∇dblL2/lessorsimilar√m−\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\n1\nm−ηin\n−/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH−1\n2xH1y+/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble\nH−1\n2xH3\n2y\n\n+1\n/an}b∇acketle{tt/an}b∇acket∇i}ht2/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble\nH−1\nxH2y+√m−\n/an}b∇acketle{tt/an}b∇acket∇i}ht3\n2/vextenddouble/vextenddoubleψin\n−/vextenddouble/vextenddouble\nH−1\n2xL2y, (1.38)\nand\n/ba∇dblQ[u−](t)/ba∇dblL2+/radicaligg\n1\nm−/ba∇dblη−(t)/ba∇dblL2\n/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\n1\nm−ηin\n−/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddoubleψin\n−/vextenddouble/vextenddouble\nH−1+/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble\nH1/parenrightigg\n. (1.39)\nTheorem 1.4. Assume that/parenleftbig\nηin\n−,ωin\n−/parenrightbig\n∈L2\nxH−1\n2yand thatψin\n−∈H−3\n2xH−2\nywith/parenleftbig\nηin\n−,a,ψin\n−,a,ωin\n−,a/parenrightbig\n=\n(0,0,0). Let(η−,u−,φ)be a smooth solution for the system (1.24)-(1.26). Then we have\n/ba∇dblQ[u−](t)/ba∇dblL2+/radicaligg\n1\nm−/ba∇dblη−(t)/ba∇dblL2/greaterorsimilarCin\nσ/parenleftbig\nηin\n−,ψin\n−,ωin\n−/parenrightbig\n/an}b∇acketle{tt/an}b∇acket∇i}ht1\n2,\nwhere the constant Cin\nσ/parenleftbig\nηin\n−,ψin\n−,ωin\n−/parenrightbig\ndepends on Sobolev norms of the initial data.\nRemark 1.1. It should be noted that at the linearized level, the dynamics of the ion Euler-\nPoisson system and the electron Euler-Poisson system are de picted by different linear PDE\nsystems, ( 1.19) for the ion dynamics and the ( 1.35) for the electron dynamics. The linear\ndispersion relation behaves much closer to the wave dispers ion for the ions and to the Klein-\nGordon dispersion for the electrons. We investigate the lin ear stability the Couette flow for ion\nand electron dynamics separately below.8 X. Pu et al.\nRemark 1.2. Sincetheanalysis in frequencyspace can begiven by Lemma 3.1or4.1, estimates\n(1.23) and (1.39) are also valid in the general Sobolev space H−s(s>1\n2).\nThis paper is organized as follows. In the next section, we wi ll give some notations and the\ndefinitions of Sobolev spaces. We will also write out the coor dinate transform explicitly. The\nlinear stability of the ion dynamics near the Couette flow wil l be analyzed in Section 3and the\nlinear stability of the electron dynamics near the Couette fl ow will be analyzed in Section 4.\n2. Notations\nThe Fourier transform for a function f(x,y) is denoted by\n/hatwidef(k,ξ) =1\n2π/integraldisplay/integraldisplay\nT×Re−i(kx+ξy)f(x,y)dxdy.\nThen\nf(x,y) =1\n2π/summationdisplay\nk∈Z/integraldisplay\nRei(kx+ξy)/hatwidef(k,ξ)dξ.\nThe anisotropic Sobolev space is defined by\nHr\nxHs\ny(T×R) =/braceleftigg\nf:/ba∇dblf/ba∇dbl2\nHrxHsy(T×R)=/summationdisplay\nk/integraldisplay\n/an}b∇acketle{tk/an}b∇acket∇i}ht2r/an}b∇acketle{tξ/an}b∇acket∇i}ht2s/vextendsingle/vextendsingle/vextendsingle/hatwidef/vextendsingle/vextendsingle/vextendsingle2\n(k,ξ)dξ <+∞/bracerightigg\n,\nwhere/an}b∇acketle{ta/an}b∇acket∇i}ht= (1+a2)1/2fora∈R. In the same way, the usual Sobolev space is defined as\nHs(T×R) =/braceleftigg\nf:/ba∇dblf/ba∇dbl2\nHs(T×R)=/summationdisplay\nk/integraldisplay\n/an}b∇acketle{tk,ξ/an}b∇acket∇i}ht2s/vextendsingle/vextendsingle/vextendsingle/hatwidef/vextendsingle/vextendsingle/vextendsingle2\n(k,ξ)dξ <+∞/bracerightigg\n.\nHere/an}b∇acketle{ta,b/an}b∇acket∇i}ht= (1+a2+b2)1/2fora,b∈R.\nInordertodealwiththetransportterminsystem( 1.12)-(1.14)or(1.28)-(1.30), weintroduce\na set of coordinate transformations\nX=x−yt, Y=y. (2.1)\nSo the differential operators can be represented as\n∂x=∂X, ∂y=∂Y−t∂X,∆ =: ∆ L=∂XX+(∂Y−t∂X)2. (2.2)\nWe denote the symbol for −∆Lby\nα(t,k,ξ) =k2+(ξ−kt)2. (2.3)\nThen the symbol of the operator 2 ∂X(∂Y−t∂X) is\n∂tα(t,k,ξ) =−2k(ξ−kt). (2.4)Linear stability of the Couette flow 9\n3. Analysis for the ion dynamics\nIn this section, we consider the dynamics of the system ( 1.12)-(1.14). Using the coordinate\ntransformations ( 2.1), we define the functions\nΠ+(t,X,Y) =η+(t,X+tY,Y), (3.1)\nΨ+(t,X,Y) =ψ+(t,X+tY,Y), (3.2)\nΓ+(t,X,Y) =ω+(t,X+tY,Y). (3.3)\nSetF+(t,X,Y) := Π +(t,X,Y)+Γ+(t,X,Y). Combining ( 1.12) with (1.14), we have∂tF+= 0,\nwhich yields\nΓ+(t,X,Y) =Fin\n+(X,Y)−Π+(t,X,Y).\nHereFin\n+=ηin\n++ωin\n+. According to ( 1.11), we obtain\nUy\n+= (∂Y−t∂X)∆−1\nLΨ++∂X∆−1\nLΓ+\n= (∂Y−t∂X)∆−1\nLΨ++∂X∆−1\nLFin\n+−∂X∆−1\nLΠ+.\nIn consequence, we can rewrite system ( 1.12)-(1.14) as\n∂tΠ++Ψ+= 0, (3.4)\n∂tΨ++2∂XX∆−1\nLFin\n++2∂X(∂Y−t∂X)∆−1\nLΨ+\n−/parenleftbigg\n2∂XX∆−1\nL−T+\nm+∆L/parenrightbigg\nΠ++4πe2\nm+∆L/parenleftbigg\n−∆L+4πe2\nT−/parenrightbigg−1\nΠ+= 0. (3.5)\nTaking the Fourier transform on system ( 3.4)-(3.5), we get the following system\n∂t/hatwiderΠ++/hatwiderΨ+= 0, (3.6)\n∂t/hatwiderΨ++2k2\nα/hatwidestFin\n+−∂tα\nα/hatwiderΨ+−/parenleftigg\n2k2\nα+T+α\nm++4πe2\nm+α\nα+4πe2\nT−/parenrightigg\n/hatwiderΠ+= 0. (3.7)\nIn order to study system ( 3.6)-(3.7), we need to find a suitable symmetrization for this system.\nDefine\nA(t) = (A1(t),A2(t))⊤=/parenleftigg/radicaligg\nT+\nm+/hatwiderΠ+(t)\nα1/4,/hatwiderΨ+(t)\nα3/4/parenrightigg⊤\n. (3.8)\nBy a delicate calculation, A(t) satisfy a non-autonomous 2D dynamical system\n/braceleftigg\nd\ndtA(t) =L+(t)A(t)+M+(t)/hatwidestFin\n+,\nA(0) =Ain,(3.9)\nwhere\nL+(t) =\n−1\n4α−1∂tα −/radicalig\nT+\nm+α1/2\n4πe2√\nm+T+α1/2\nα+4πe2\nT−+/radicaligm+\nT+2k2\nα3/2+/radicalig\nT+\nm+α1/21\n4α−1∂tα\n, (3.10)10 X. Pu et al.\nM+(t) =/parenleftbigg\n0,−2k2\nα7/4/parenrightbigg⊤\n, Ain=/parenleftigg/radicaligg\nT+\nm+/hatwidestΠin\n+\n(k2+ξ2)1/4,/hatwidestΨin\n+\n(k2+ξ2)3/4/parenrightigg⊤\n.(3.11)\nBy virtue of Duhamel’s formula, the solution A(t) to system ( 3.9) is given by\nA(t) =SL(t,0)Ain+/integraldisplayt\n0SL(t,s)M+(s)/hatwidestFin\n+ds. (3.12)\nHere,SLdenotes the solution operator, satisfying the group proper ty\nSL(t,0)SL(0,s) =SL(t,s)\nfor anyt,s>0. In consequence, it suffices to study the homogeneous proble m of system ( 3.9).\nFor this purpose, we denote by A(t) a solution to system ( 3.9) with/hatwidestFin\n+= 0. Let\nE+(t) =/parenleftbigg/radicalbiggp1\nm1|A1|2/parenrightbigg\n(t)+2/parenleftbiggh1√m1p1Re/parenleftbig\nA1¯A2/parenrightbig/parenrightbigg\n(t)+/parenleftbigg/radicalbiggm1\np1|A2|2/parenrightbigg\n(t),(3.13)\nwhere\nh1(t) =1\n4α−1∂tα, m1(t) =/radicaligg\nT+\nm+α1/2, (3.14)\np1(t) =4πe2\n√m+T+α1/2\nα+4πe2\nT−+/radicalbiggm+\nT+2k2\nα3/2+/radicaligg\nT+\nm+α1/2. (3.15)\nThen we can get the upper and lower bounds for E+(t) and the solution A(t).\nLemma 3.1. There exists positive constants C1,Cδ\n1,C2,Cδ\n2that do not depend on kandξ\nsuch that\nC1E+(0)≤ E+(t)≤Cδ\n1E+(0), (3.16)\nand\nC2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle≤ |A(t)| ≤Cδ\n2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle. (3.17)\nProof.Define\nλ+=/radicalbiggp1\nm1=/parenleftigg\n1+2m+k2\nT+α2+4πe2\nT+1\nα+4πe2\nT−/parenrightigg1/2\n, (3.18)\nγ+=√m1p1=/parenleftigg\n4πe2\nm+α\nα+4πe2\nT−+2k2\nα+T+α\nm+/parenrightigg1/2\n. (3.19)\nFrom the definition of λ+we get that\n1≤(λ+)2≤1+4πe2\nT++2m+\nT+. (3.20)\nMoreover, we have\n|h1|\nγ+=|∂tα|\n4α/parenleftigg\n4πe2\nm+α\nα+4πe2\nT−+2k2\nα+T+α\nm+/parenrightigg−1/2Linear stability of the Couette flow 11\n=|∂tα|\n4α/parenleftbiggm+\nT+α/parenrightbigg1/2/parenleftigg\n1+2m+k2\nT+α2+4πe2\nT+1\nα+4πe2\nT−/parenrightigg−1/2\n≤|k|\n2√α/parenleftbiggm+\nT+α/parenrightbigg1/2/parenleftbiggT+α2\n2m+k2/parenrightbigg1/2\n≤√\n2\n2, (3.21)\nwhere we have used the fact that |∂tα| ≤2|k|√α.\nSetting\n/tildewideE+(t) =/parenleftbig\nλ+|A1|2/parenrightbig\n(t)+/parenleftbigg1\nλ+|A2|2/parenrightbigg\n(t).\nThen from ( 3.21) it can be deduced that\n/parenleftigg\n1−√\n2\n2/parenrightigg\n/tildewideE+(t)≤ E+(t)≤/parenleftigg\n1+√\n2\n2/parenrightigg\n/tildewideE+(t). (3.22)\nObviously, the coerciveness of E+(t) is ensured by the above inequality and ( 3.20).\nAccording to ( 3.14), (3.15), (3.18) and (3.19), system ( 3.9) becomes\nλ+d\ndtA1=−h1λ+A1−γ+A2, (3.23)\n1\nλ+d\ndtA2=γ+A1+h1\nλ+A2. (3.24)\nMultiplying equations ( 3.23) and (3.24) by¯A1and¯A2, respectively, and similarly for their\nconjugate, we get\nλ+\n2d\ndt|A1|2+1\n2λ+d\ndt|A2|2=−h1λ+|A1|2+h1\nλ+|A2|2. (3.25)\nNote the fact that\nh1\nγ+d\ndtRe(A1¯A2) =h1λ+|A1|2−h1\nλ+|A2|2. (3.26)\nAdding ( 3.25) to (3.26), we obtain\nλ+d\ndt|A1|2+2h1\nγ+d\ndtRe(A1¯A2)+1\nλ+d\ndt|A2|2= 0,\nwhich yields\ndE+\ndt=λ+|A1|2d\ndt(logλ+)+2Re(A1¯A2)d\ndt/parenleftbiggh1\nγ+/parenrightbigg\n−1\nλ+|A2|2d\ndt(logλ+).\nUsing Young’s inequality, from ( 3.22) we conclude that\ndE+\ndt≤/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh1\nγ+/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ+)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n/tildewideE+\n≤/parenleftig\n2+√\n2/parenrightig/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh1\nγ+/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ+)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nE+. (3.27)12 X. Pu et al.\nAn argument similar to that of ( 3.27) gives\ndE+\ndt≥ −/parenleftig\n2+√\n2/parenrightig/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh1\nγ+/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ+)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nE+. (3.28)\nFor the purpose of applying Gr¨ onwall’s inequality, we need to estimate the integrals\n/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh1\nγ+/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ and/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ+)/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ.\nFirst of all, we deal with the first one. By a direct calculatio n, we have\nd\ndt/parenleftbiggh1\nγ+/parenrightbigg\n=G\n8m+(T−)2(γ+)3α3/parenleftig\nα+4πe2\nT−/parenrightig3,\nwhere\nG=16πe2(T−)2k2α5+/parenleftbig\n8m+T−k4α3+4T+T−k2α5/parenrightbig/parenleftbig\nT−α+4πe2/parenrightbig\n+128π2e4T−k2α4+/parenleftbig\n64πe2m+k4α2+32πe2T+k2α4/parenrightbig/parenleftbig\nT−α+4πe2/parenrightbig\n+256π3e6k2α3+/parenleftbig\n128π2e4m+k4α+64π2e4T+k2α3/parenrightbig/parenleftbigg\nα+4πe2\nT−/parenrightbigg\n−8πe2(T−)2α4(∂tα)2−/bracketleftbig\n4m+T−k2α2(∂tα)2+2T+T−α4(∂tα)2/bracketrightbig/parenleftbig\nT−α+4πe2/parenrightbig\n−64π2e4T−α3(∂tα)2−/bracketleftbig\n32πe2m+k2α(∂tα)2+16πe2T+α3(∂tα)2/bracketrightbig/parenleftbig\nT−α+4πe2/parenrightbig\n−128π3e6α2(∂tα)2−/bracketleftbig\n64π2e4m+k2(∂tα)2+32π2e4T+α2(∂tα)2/bracketrightbig/parenleftbigg\nα+4πe2\nT−/parenrightbigg\n+/bracketleftbigg\n16πe2m+k2α(∂tα)2−8πe2T+α3(∂tα)2+32π2e4m+k2\nT−(∂tα)2/bracketrightbigg/parenleftbig\nT−α+4πe2/parenrightbig\n+/bracketleftbigg/parenleftbigg\n2m+T−k2−16π2e4−16π2e4T+\nT−/parenrightbigg\nα2(∂tα)2−T+T−α4(∂tα)2/bracketrightbigg/parenleftbig\nT−α+4πe2/parenrightbig\nis a polynomial in time of order 12. Denote by ti(i= 1,2,...,12) the possible positive roots for\nG. We seti0= 0 provided that G(0)≤0 and take i0= 1 for the other cases. Assume that\nt0= 0 and that t13= +∞. By virtue of ( 3.21), we infer\n/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh1\nγ+/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ=13/summationdisplay\ni=1(−1)i+i0/parenleftbiggh1\nγ+(ti)−h1\nγ+(ti−1)/parenrightbigg\n≤213/summationdisplay\ni=0|h1|\nγ+(ti)≤14√\n2. (3.29)\nFor the second integral term, we deduce from ( 3.20) that\n/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ+)/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ=1\n2/integraldisplayξ\nk\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbig\nlog(λ+)2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ+1\n2/integraldisplay+∞\nξ\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbig\nlog(λ+)2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ\n≤1\n2/bracketleftbigg/parenleftbig\nlog(λ+)2/parenrightbig/parenleftbiggξ\nk/parenrightbigg\n−/parenleftbig\nlog(λ+)2/parenrightbig\n(0)/bracketrightbiggLinear stability of the Couette flow 13\n+1\n2/bracketleftbigg/parenleftbig\nlog(λ+)2/parenrightbig/parenleftbiggξ\nk/parenrightbigg\n−/parenleftbig\nlog(λ+)2/parenrightbig\n(+∞)/bracketrightbigg\n≤log/parenleftbigg\n1+4πe2\nT++2m+\nT+/parenrightbigg\n, (3.30)\nwhere we have used the fact that\n1\n(λ+)2d(λ+)2\ndt= (T+)−1(λ+)−2α−3/parenleftbigg\nα+4πe2\nT−/parenrightbigg−2\n×/bracketleftigg\n−4m+k2/parenleftbigg\nα+4πe2\nT−/parenrightbigg2\n−4πe2α3/bracketrightigg\n∂tα= 0\nhas a unique root t=ξ/k. Applying Gr¨ onwall’s inequality to ( 3.27) and (3.28), respectively,\nfrom (3.29) and (3.30) we obtain ( 3.16).\nNext, it remains to prove ( 3.17). Owing to ( 3.16) and (3.22), it can be seen that\nC′\n1/tildewideE+(0)≤/tildewideE+(t)≤Cδ′\n1/tildewideE+(0). (3.31)\nBy virtue of ( 3.20), we get\n/parenleftbigg\n1+4πe2\nT++2m+\nT+/parenrightbigg−1\n/tildewideE+(t)≤ |A(t)|2≤/parenleftbigg\n1+4πe2\nT++2m+\nT+/parenrightbigg\n/tildewideE+(t).(3.32)\nCombining ( 3.31) with (3.32) yields ( 3.17). This completes the proof.\nBased on Lemma 3.1, we give the proof of Theorem 1.1.\nProof of Theorem 1.1.Due to Lemma 3.1, the solution A(t) of system ( 3.9) can be bounded as\n|A(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSL(t,0)Ain+/integraldisplayt\n0SL(t,s)M+(s)/hatwidestFin\n+ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar|Ain|+/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n02k2\nα7/4ds\n/lessorsimilar|Ain|+/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n01\n|k|3/2(1+(ξ/k−s)2)7/4ds\n/lessorsimilar|Ain|+/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle (3.33)\nfor anyt≥0. By the Helmholtz decomposition and the Plancherel’s theo rem, we get from\n(3.33) that\n/ba∇dblP[u+]x(t)/ba∇dbl2\nL2+/ba∇dblφ(t)/ba∇dbl2\nL2=/vextenddouble/vextenddouble/parenleftbig\n∂y∆−1ω+/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble4πe/parenleftbigg\n−∆+4πe2\nT−/parenrightbigg−1\nη+(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n=/summationdisplay\nk/integraldisplay(ξ−kt)2\nα2/vextendsingle/vextendsingle/vextendsingle/hatwiderΓ+(t)/vextendsingle/vextendsingle/vextendsingle2\ndξ+/summationdisplay\nk/integraldisplay\n16π2e2/parenleftbigg\nα+4πe2\nT−/parenrightbigg−2/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ+(t)/vextendsingle/vextendsingle/vextendsingle2\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftbigg(ξ−kt)2\nα2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle2\n+m+(ξ−kt)2\nT+α3/2|A1(t)|2+m+\nT+α3/2|A1(t)|2/parenrightbigg\ndξ14 X. Pu et al.\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftbigg\nα−1/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle2\n+m+\nT+α1/2/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2+m+\nT+α1/2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftigg\n/an}b∇acketle{tξ/an}b∇acket∇i}ht2\n/an}b∇acketle{tt/an}b∇acket∇i}ht2/an}b∇acketle{tk/an}b∇acket∇i}ht2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle2\n+m+/an}b∇acketle{tξ/an}b∇acket∇i}ht\nT+/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht/vextendsingle/vextendsingleAin/vextendsingle/vextendsingle2+m+/an}b∇acketle{tξ/an}b∇acket∇i}ht\nT+/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n+/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftigg\n/an}b∇acketle{tξ/an}b∇acket∇i}ht2\n/an}b∇acketle{tt/an}b∇acket∇i}ht2/an}b∇acketle{tk/an}b∇acket∇i}ht2/vextendsingle/vextendsingle/vextendsingle/hatwiderηin\n++/hatwiderωin\n+/vextendsingle/vextendsingle/vextendsingle2\n+m+/an}b∇acketle{tξ/an}b∇acket∇i}ht\nT+/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/hatwiderηin\n++/hatwiderωin\n+/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\ndξ\n+/summationdisplay\nk/integraldisplaym+/an}b∇acketle{tξ/an}b∇acket∇i}ht\nT+/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht\nT+\nm+/vextendsingle/vextendsingle/vextendsingle/hatwiderηin\n+/vextendsingle/vextendsingle/vextendsingle2\n(1+ξ2)1/2+/vextendsingle/vextendsingle/vextendsingle/hatwidestψin\n+/vextendsingle/vextendsingle/vextendsingle2\n(1+ξ2)3/2\ndξ, (3.34)\nwhere we have used the fact that /an}b∇acketle{tt/an}b∇acket∇i}ht/lessorsimilar/an}b∇acketle{tξ/k−t/an}b∇acket∇i}ht/an}b∇acketle{tξ/k/an}b∇acket∇i}ht. Thanks to the definition of the\nanisotropic Sobolev space, from the above inequality we ded uce (1.21). Moreover, a similar\nargument as that of ( 1.21) yields ( 1.22).\nUsing the Helmholtz decomposition and the Plancherel’s the orem, we obtain\n/ba∇dblQ[u+](t)/ba∇dbl2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2=/vextenddouble/vextenddouble/parenleftbig\n∇∆−1ψ+/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2\n=/vextenddouble/vextenddouble/parenleftbig\n∂x∆−1ψ+/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddouble/parenleftbig\n∂y∆−1ψ+/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2\n=/summationdisplay\nk/integraldisplay/parenleftbigg1\nk2+(ξ−kt)2/vextendsingle/vextendsingle/vextendsingle/hatwiderΨ+(t)/vextendsingle/vextendsingle/vextendsingle2\n+T+\nm+/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ+(t)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ.(3.35)\nNote that the fact that k2+(ξ−kt)2/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht2/an}b∇acketle{tk,ξ/an}b∇acket∇i}ht2. From (3.33) and (3.35), it follows that\n/ba∇dblQ[u+](t)/ba∇dbl2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2=/summationdisplay\nk/integraldisplay/parenleftbigg1\nα/vextendsingle/vextendsingle/vextendsingle/hatwiderΨ+(t)/vextendsingle/vextendsingle/vextendsingle2\n+T+\nm+/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ+(t)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ\n=/summationdisplay\nk/integraldisplay\nα1\n2\n/vextendsingle/vextendsingle/vextendsingleα−3\n4/hatwiderΨ+(t)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicaligg\nT+\nm+α−1\n4/hatwiderΠ+(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndξ\n=/summationdisplay\nk/integraldisplay\nα1\n2|A(t)|2dξ\n/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\nT+\nm+ηin\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddoubleψin\n+/vextenddouble/vextenddouble2\nH−1+/vextenddouble/vextenddoubleηin\n++ωin\n+/vextenddouble/vextenddouble2\nH1\n,(3.36)\nwhich gives the estimate ( 1.23).\nNow we prove Theorem 1.2.\nProof of Theorem 1.2.Set\nR/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n=Ain+/integraldisplayt\n0SL(0,s)M+(s)/hatwidestFin\n+ds.\nAccording to ( 3.12), we have\nA(t) =SL(t,0)Ain+/integraldisplayt\n0SL(t,s)M+(s)/hatwidestFin\n+ds=SL(t,0)R/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n.Linear stability of the Couette flow 15\nBy virtue of Lemma 3.1, we obtain\n|A(t)| ≥C|R/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n|.\nUsing the identity in ( 3.36), we get from the above inequality that\n/ba∇dblQ[u+](t)/ba∇dbl2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2=/summationdisplay\nk/integraldisplay\nα1\n2|A(t)|2dξ\n/greaterorsimilar/summationdisplay\nk/integraldisplay\nα1\n2|R/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n|2dξ\n/greaterorsimilar/summationdisplay\nk/integraldisplay\n/an}b∇acketle{tξ−kt/an}b∇acket∇i}ht|R/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n|2dξ\n/greaterorsimilar/summationdisplay\nk/integraldisplay/an}b∇acketle{tkt/an}b∇acket∇i}ht\n/an}b∇acketle{tξ/an}b∇acket∇i}ht|R/parenleftbig\nt,Ain,Fin\n+/parenrightbig\n|2dξ,\nwhere we have used the facts that α1\n2≥ /an}b∇acketle{tξ−kt/an}b∇acket∇i}htand/an}b∇acketle{tξ−kt/an}b∇acket∇i}ht/an}b∇acketle{tξ/an}b∇acket∇i}ht/greaterorsimilar/an}b∇acketle{tkt/an}b∇acket∇i}ht. The definition of the\nanisotropic Sobolev space gives\n/ba∇dblQ[u+](t)/ba∇dbl2\nL2+T+\nm+/ba∇dblη+(t)/ba∇dbl2\nL2/greaterorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht/vextenddouble/vextenddoubleR/parenleftbig\nt,Ain,Fin\n+/parenrightbig/vextenddouble/vextenddouble2\nL2xH−1\n2y.\nThis completes the proof.\n4. Analysis for the electron dynamics\nIn this section, we consider the dynamics of the electron dyn amics system ( 1.28)-(1.30).\nAlso using coordinate transformations ( 2.1), we define the functions\nΠ−(t,X,Y) =η−(t,X+tY,Y), (4.1)\nΨ−(t,X,Y) =ψ−(t,X+tY,Y), (4.2)\nΓ−(t,X,Y) =ω−(t,X+tY,Y). (4.3)\nSetF−(t,X,Y) := Π −(t,X,Y)+Γ−(t,X,Y). Combining ( 1.28) with (1.30), we have∂tF−= 0,\nwhich yields\nΓ−(t,X,Y) =Fin\n−(X,Y)−Π−(t,X,Y),\nwhereFin\n−=ηin\n−+ωin\n−. According to ( 1.27), we obtain\nUy\n−= (∂Y−t∂X)∆−1\nLΨ−+∂X∆−1\nLΓ−\n= (∂Y−t∂X)∆−1\nLΨ−+∂X∆−1\nLFin\n−−∂X∆−1\nLΠ−.\nIn consequence, we can rewrite system ( 1.28)-(1.30) as\n∂tΠ−+Ψ−= 0, (4.4)\n∂tΨ−+2∂XX∆−1\nLFin\n−+2∂X(∂Y−t∂X)∆−1\nLΨ−\n−/parenleftbigg\n2∂XX∆−1\nL−1\nm−∆L+4πe2\nm−/parenrightbigg\nΠ−= 0. (4.5)16 X. Pu et al.\nTaking the Fourier transform on system ( 4.4)-(4.5), we get the following system\n∂t/hatwiderΠ−+/hatwiderΨ−= 0, (4.6)\n∂t/hatwiderΨ−+2k2\nα/hatwidestFin\n−−∂tα\nα/hatwiderΨ−−/parenleftbigg2k2\nα+α\nm−+4πe2\nm−/parenrightbigg\n/hatwiderΠ−= 0. (4.7)\nIn order to study system ( 4.6)-(4.7), we need to find a suitable symmetrization for this system.\nDefine\nB(t) = (B1(t),B2(t))⊤=/parenleftigg/hatwiderΠ−(t)√m−α1/4,/hatwiderΨ−(t)\nα3/4/parenrightigg⊤\n. (4.8)\nBy a delicate calculation, B(t) satisfy a non-autonomous 2D dynamical system\n/braceleftigg\nd\ndtB(t) =L−(t)B(t)+M−(t)/hatwidestFin\n−,\nB(0) =Bin,(4.9)\nwhere\nL−(t) =\n−1\n4α−1∂tα −/radicalig\n1\nm−α1/2\n4πe2\n√m−α1/2+2k2√m−\nα3/2+/radicalig\n1\nm−α1/21\n4α−1∂tα\n, (4.10)\nM−(t) =/parenleftbigg\n0,−2k2\nα7/4/parenrightbigg⊤\n, Bin=/parenleftigg/hatwidestΠin\n−√m−(k2+ξ2)1/4,/hatwidestΨin\n−\n(k2+ξ2)3/4/parenrightigg⊤\n.(4.11)\nBy virtue of Duhamel’s formula, the solution B(t) to system ( 4.9) is given by\nB(t) =SL(t,0)Bin+/integraldisplayt\n0SL(t,s)M−(s)/hatwidestFin\n−ds. (4.12)\nHere,SLdenotes the solution operator, satisfying the group proper ty\nSL(t,0)SL(0,s) =SL(t,s)\nfor anyt,s>0. In consequence, it suffices to study the homogeneous proble m of system ( 4.9).\nLemma 4.1. Denote byB(t)a solution to system ( 4.9) with/hatwidestFin\n−= 0. Let\nE−(t) =/parenleftbigg/radicalbiggp2\nm2|B1|2/parenrightbigg\n(t)+2/parenleftbiggh2√m2p2Re/parenleftbig\nB1¯B2/parenrightbig/parenrightbigg\n(t)+/parenleftbigg/radicalbiggm2\np2|B2|2/parenrightbigg\n(t),(4.13)\nwhere\nh2(t) =1\n4α−1∂tα, m2(t) =/radicaligg\n1\nm−α1/2, (4.14)\np2(t) =4πe2\n√m−α1/2+2k2√m−\nα3/2+/radicaligg\n1\nm−α1/2. (4.15)\nThen there exists positive constants C1,Cσ\n1,C2,Cσ\n2that do not depend on kandξsuch that\nC1E−(0)≤ E−(t)≤Cσ\n1E−(0), (4.16)\nand\nC2/vextendsingle/vextendsingleBin/vextendsingle/vextendsingle≤ |B(t)| ≤Cσ\n2/vextendsingle/vextendsingleBin/vextendsingle/vextendsingle. (4.17)Linear stability of the Couette flow 17\nProof.Define\nλ−=/radicalbiggp2\nm2=/parenleftbigg\n1+4πe2\nα+2m−k2\nα2/parenrightbigg1/2\n, (4.18)\nγ−=√m2p2=/parenleftbigg4πe2\nm−+2k2\nα+α\nm−/parenrightbigg1/2\n. (4.19)\nBy the definition of λ−, it holds that\n1≤(λ−)2≤1+4πe2+2m−. (4.20)\nMoreover, we have\n|h2|\nγ−=|∂tα|\n4α/parenleftbigg4πe2\nm−+2k2\nα+α\nm−/parenrightbigg−1/2\n=|∂tα|\n4α/parenleftigm−\nα/parenrightig1/2/parenleftbigg\n1+4πe2\nα+2m−k2\nα2/parenrightbigg−1/2\n≤|k|\n2√α/parenleftigm−\nα/parenrightig1/2/parenleftbiggα2\n2m−k2/parenrightbigg1/2\n=√\n2\n4, (4.21)\nwhere we have used the fact that |∂tα| ≤2|k|√α.\nSetting\n/tildewideE−(t) =/parenleftbig\nλ−|B1|2/parenrightbig\n(t)+/parenleftbigg1\nλ−|B2|2/parenrightbigg\n(t).\nThen from ( 4.21) it follows that\n/parenleftigg\n1−√\n2\n4/parenrightigg\n/tildewideE−(t)≤ E−(t)≤/parenleftigg\n1+√\n2\n4/parenrightigg\n/tildewideE−(t). (4.22)\nObviously, the coerciveness of E−(t) is ensured by the above inequality and ( 4.20).\nAccording to ( 4.14), (4.15), (4.18) and (4.19), system ( 4.9) becomes\nλ−d\ndtB1=−h2λ−B1−γ−B2, (4.23)\n1\nλ−d\ndtB2=γ−B1+h2\nλ−B2. (4.24)\nMultiplying equations ( 4.23) and (4.24) by¯B1and¯B2, respectively, we obtain\nλ−\n2d\ndt|B1|2+1\n2λ−d\ndt|B2|2=−h2λ−|B1|2+h2\nλ−|B2|2. (4.25)\nNote thath2\nγ−d\ndtRe(B1¯B2) =h2λ−|B1|2−h2\nλ−|B2|2. (4.26)\nAdding ( 4.25) to (4.26), we get\nλ−d\ndt|B1|2+2h2\nγ−d\ndtRe(B1¯B2)+1\nλ−d\ndt|B2|2= 0,18 X. Pu et al.\nwhich yields\ndE−\ndt=λ−|B1|2d\ndt(logλ−)+2Re(B1¯B2)d\ndt/parenleftbiggh2\nγ−/parenrightbigg\n−1\nλ−|B2|2d\ndt(logλ−).\nUsing Young’s inequality, from ( 4.22) we deduce that\ndE−\ndt≤/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh2\nγ−/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n/tildewideE−\n≤/parenleftigg\n2+√\n2\n2/parenrightigg/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh2\nγ−/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nE−. (4.27)\nAn argument similar to that of ( 4.27) gives\ndE−\ndt≥ −/parenleftigg\n2+√\n2\n2/parenrightigg/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh2\nγ−/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ−)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\nE−. (4.28)\nFor the purpose of applying Gr¨ onwall’s inequality, we need to estimate the integral terms/integraltext+∞\n0/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftig\nh2\nγ−/parenrightig/vextendsingle/vextendsingle/vextendsingledϑand/integraltext+∞\n0/vextendsingle/vextendsingled\ndt(logλ−)/vextendsingle/vextendsingledϑ. First of all, we deal with the first one. By a direct\ncalculation, we have\nd\ndt/parenleftbiggh2\nγ−/parenrightbigg\n=M\n8m−(γ−)3α3,\nwhere\nM= 4m−k2(γ−)2α2−2m−(γ−)2α(∂tα)2−α2(∂tα)2+2m−k2(∂tα)2\nis a polynomial in time of order 6. Denote by ti(i= 1,2,...,6) the positive root for M. We set\ni0= 0 provided that M(0)≤0 and take i0= 1 for the other cases. Assume that t0= 0 and\nthatt7= +∞. By virtue of ( 4.21), we conclude that\n/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbiggh2\nγ−/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ=7/summationdisplay\ni=1(−1)i+i0/parenleftbiggh2\nγ−(ti)−h2\nγ−(ti−1)/parenrightbigg\n≤27/summationdisplay\ni=0|h2|\nγ−(ti)≤4√\n2.(4.29)\nFor the second integral term, we get from ( 4.20) that\n/integraldisplay+∞\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt(logλ−)/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ=1\n2/integraldisplayξ\nk\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbig\nlog(λ−)2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ+1\n2/integraldisplay+∞\nξ\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndt/parenleftbig\nlog(λ−)2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingledϑ\n≤1\n2/bracketleftbigg/parenleftbig\nlog(λ−)2/parenrightbig/parenleftbiggξ\nk/parenrightbigg\n−/parenleftbig\nlog(λ−)2/parenrightbig\n(0)/bracketrightbigg\n+1\n2/bracketleftbigg/parenleftbig\nlog(λ−)2/parenrightbig/parenleftbiggξ\nk/parenrightbigg\n−/parenleftbig\nlog(λ−)2/parenrightbig\n(+∞)/bracketrightbigg\n≤log/parenleftbig\n1+4πe2+2m−/parenrightbig\n. (4.30)\nHere, we have used the fact that1\n(λ−)2d(λ−)2\ndt= (λ−)−2α−3/parenleftbig\n−4m−k2−4πe2α/parenrightbig\n∂tαchanges sign\nonly int=ξ/k. Applying Gr¨ onwall’s inequality to ( 4.27) and (4.28), respectively, from ( 4.29)\nand (4.30) we obtain ( 4.16).\nNext, it remains to prove ( 4.17). Owing to ( 4.16) and (4.22), it can be deduced that\nC′\n1/tildewideE−(0)≤/tildewideE−(t)≤Cσ′\n1/tildewideE−(0). (4.31)\nBy virtue of ( 4.20), we have\n/parenleftbig\n1+4πe2+2m−/parenrightbig−1/tildewideE−(t)≤ |B(t)|2≤/parenleftbig\n1+4πe2+2m−/parenrightbig/tildewideE−(t). (4.32)\nCombining ( 4.31) with (4.32) yields ( 4.17). This completes the proof.Linear stability of the Couette flow 19\nBased on Lemma 4.1, we give the proof of Theorem 1.3as follows.\nProof of Theorem 1.3.Thanks to Lemma 4.1, an argument similar to that of ( 3.33) yields\n|B(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSL(t,0)Bin+/integraldisplayt\n0SL(t,s)M−(s)/hatwidestFin\n−ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar|Bin|+/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle (4.33)\nfor anyt≥0. By virtue of the Helmholtz decomposition and the Plancher el’s theorem, we\ndeduce from ( 4.33) that\n/ba∇dblP[u−]x(t)/ba∇dbl2\nL2+/ba∇dblφ(t)/ba∇dbl2\nL2=/vextenddouble/vextenddouble/parenleftbig\n∂y∆−1ω−/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddouble/vextenddouble4πe(−∆)−1η−(t)/vextenddouble/vextenddouble/vextenddouble2\nL2\n=/summationdisplay\nk/integraldisplay(ξ−kt)2\nα2/vextendsingle/vextendsingle/vextendsingle/hatwiderΓ−(t)/vextendsingle/vextendsingle/vextendsingle2\ndξ+/summationdisplay\nk/integraldisplay16π2e2\nα2/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ−(t)/vextendsingle/vextendsingle/vextendsingle2\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftbigg(ξ−kt)2\nα2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle2\n+m−(ξ−kt)2\nα3/2|B1(t)|2+m−\nα3/2|B1(t)|2/parenrightbigg\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftbigg\nα−1/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle2\n+m−\nα1/2/vextendsingle/vextendsingleBin/vextendsingle/vextendsingle2+m−\nα1/2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ\n/lessorsimilar/summationdisplay\nk/integraldisplay/parenleftigg\n/an}b∇acketle{tξ/an}b∇acket∇i}ht2\n/an}b∇acketle{tt/an}b∇acket∇i}ht2/an}b∇acketle{tk/an}b∇acket∇i}ht2/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle2\n+m−/an}b∇acketle{tξ/an}b∇acket∇i}ht\n/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht/vextendsingle/vextendsingleBin/vextendsingle/vextendsingle2+m−/an}b∇acketle{tξ/an}b∇acket∇i}ht\n/an}b∇acketle{tt/an}b∇acket∇i}ht/an}b∇acketle{tk/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/hatwidestFin\n−/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\ndξ,(4.34)\nin which we have used the fact that /an}b∇acketle{tt/an}b∇acket∇i}ht/lessorsimilar/an}b∇acketle{tξ/k−t/an}b∇acket∇i}ht/an}b∇acketle{tξ/k/an}b∇acket∇i}ht. By the definition of the anisotropic\nSobolev space, we obtain ( 1.37) from the above inequality. Moreover, a similar argument as\nthat of (1.37) gives (1.38).\nOwing to the Helmholtz decomposition and the Plancherel’s t heorem, we conclude that\n/ba∇dblQ[u−](t)/ba∇dbl2\nL2+1\nm−/ba∇dblη−(t)/ba∇dbl2\nL2=/vextenddouble/vextenddouble/parenleftbig\n∇∆−1ψ−/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+1\nm−/ba∇dblη−(t)/ba∇dbl2\nL2\n=/vextenddouble/vextenddouble/parenleftbig\n∂x∆−1ψ−/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddouble/parenleftbig\n∂y∆−1ψ−/parenrightbig\n(t)/vextenddouble/vextenddouble2\nL2+1\nm−/ba∇dblη−(t)/ba∇dbl2\nL2\n=/summationdisplay\nk/integraldisplay/parenleftbigg1\nk2+(ξ−kt)2/vextendsingle/vextendsingle/vextendsingle/hatwiderΨ−(t)/vextendsingle/vextendsingle/vextendsingle2\n+1\nm−/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ−(t)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ. (4.35)\nNote that the fact that k2+(ξ−kt)2/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht2/an}b∇acketle{tk,ξ/an}b∇acket∇i}ht2. From (4.33) and (4.35), it holds that\n/ba∇dblQ[u−](t)/ba∇dbl2\nL2+1\nm−/ba∇dblη−(t)/ba∇dbl2\nL2=/summationdisplay\nk/integraldisplay/parenleftbigg1\nα/vextendsingle/vextendsingle/vextendsingle/hatwiderΨ−(t)/vextendsingle/vextendsingle/vextendsingle2\n+1\nm−/vextendsingle/vextendsingle/vextendsingle/hatwiderΠ−(t)/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\ndξ\n=/summationdisplay\nk/integraldisplay\nα1\n2\n/vextendsingle/vextendsingle/vextendsingleα−3\n4/hatwiderΨ−(t)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicaligg\n1\nm−α−1\n4/hatwiderΠ−(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndξ=/summationdisplay\nk/integraldisplay\nα1\n2|B(t)|2dξ\n/lessorsimilar/an}b∇acketle{tt/an}b∇acket∇i}ht\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicaligg\n1\nm−ηin\n−/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2+/vextenddouble/vextenddoubleψin\n−/vextenddouble/vextenddouble2\nH−1+/vextenddouble/vextenddoubleηin\n−+ωin\n−/vextenddouble/vextenddouble2\nH1\n, (4.36)\nwhich yields estimate ( 1.39).\nThe proof of Theorem 1.4follows from a similar argument as that of Theorem 1.2. Hence\nwe omit its proof here.20 X. Pu et al.\nAcknowledgements. The work of X. Pu was supported in part by the National Natural Sc ience\nFoundationofChina(11871172)and theScience andTechnologyPr ojectsin Guangzhou(202201020132).\nThe work of D. Bian was supported in part by the National Natural S cience Foundation of China\n(12271032).\nReferences\n[1] P. Antonelli, M. Dolce, P. Marcati, Linear stability analysis of the ho mogeneous Couette flow in a\n2D isentropic compressible fluid, Ann. PDE 7 (2021) 24.\n[2] P. Antonelli, M. Dolce and P. Marcati, Linear stability analysis for 2d shear flows near Couette in\nthe isentropic compressible Euler equations, arXiv:2003.01694 (2020).\n[3] J. Bedrossian, P. Germain, N. Masmoudi, On the stability thresho ld for the 3D Couette flow in\nSobolev regularity, Ann. Math. 185 (2017) 541-608.\n[4] J. Bedrossian, P. Germain, N. Masmoudi, Stability of the Couette flow at high Reynolds numbers\nin two dimensions and three dimensions, Bull. Amer. Math. Soc. 56 (2019) 373-414.\n[5] J.Bedrossian,P.Germain, N.Masmoudi,Dynamicsnearthesubc riticaltransitionofthe3DCouette\nflow I: Below threshold case, Mem. Amer. Math. Soc. 266 (2020) 1294.\n[6] J.Bedrossian,P.Germain, N.Masmoudi,Dynamicsnearthesubc riticaltransitionofthe3DCouette\nflow II: Above threshold case, Mem. Amer. Math. Soc. 279 (2022) 1377.\n[7] J. Bedrossian, N. Masmoudi, Inviscid damping and the asymptotic stability of planar shear flows\nin the 2D Euler equations, Publ. Math. Inst. Hautes ´Etudes Sci. 122 (2015) 195-300.\n[8] J. Bedrossian, V. Vicol, F. Wang, The Sobolev stability threshold f or 2D shear flows near Couette,\nJ. Nonlinear Sci. 28 (2018) 2051-2075.\n[9] D. Bian and X. Pu, Stability threshold for 2D shear flows of the Bou ssinesq system near Couette,\nJ. Math. Phys. 63(8) (2022) paper no. 081501.\n[10] Q. Chen, T. Li, D. Wei, Z. Zhang, Transition threshold for the 2- D Couette flow in a finite channel,\nArch. Ration. Mech. Anal. 238 (2020) 125-183.\n[11] Y. Deng and N. Masmoudi, Long-time instability of the Couette flo w in low Gevrey spaces, Comm.\nPure Appl. Math. , 76(10), (2023)2804-2887.\n[12] G.P. Drazin and W.H. Reid, Hydrodynamic stability , Cambridge University Press, 2004.\n[13] P.W. Duck, G. Erlebacher, M.Y. Hussaini, On the linear stability of c ompressible plane Couette\nflow,J. Fluid Mech. 258 (1994) 131-165.\n[14] W. Glatzel, Sonic instability in supersonic shear flows, Mon. Not. R. Astron. Soc. 233 (1988) 795-\n821.\n[15] W. Glatzel, The linear stability of viscous compressible plane Couet te flow, J. Fluid Mech. 202\n(1989) 515-541.\n[16] E. Grenier, Y. Guo, B. Pausader, M. Suzuki, Derivation of the io n equation, Quart. Appl. Math. 78\n(2020) 305-332.\n[17] Y. Guo, On the Euler-Poisson System, 2019, Preprint.\n[18] Y. Guo, Smooth irrotational flows in the large to the Euler-Poiss on system in R3+1,Comm. Math.\nPhys.195 (1998) 249-265.\n[19] Y. Guo, L. Han, J. Zhang, Absence of shocks for one dimension al Euler-Poisson system, Arch.\nRation. Mech. Anal. 223 (2017) 1057-1121.Linear stability of the Couette flow 21\n[20] Y. Guo, B. Pausader, Globalsmooth ion dynamics in the Euler-Po issonsystem, Comm. Math. Phys.\n303 (2011) 89-125.\n[21] Y. Guo, X. Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal. 211 (2014)\n673-710.\n[22] D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stab ility of fluid and plasma\nequilibria, Phys. Report. 123(1-2) (1985) 1-116.\n[23] S. Hu, X. Zhong, Linear stability of viscous supersonic plane Cou ette flow, Phys. Fluids 10 (1998)\n709-729.\n[24] A. Ionescu and H. Jia, Inviscid damping near the Couette flow in a channel, Comm. Math. Phys. ,\n374 (3), (2020)2015-2096.\n[25] A. Ionescu and H. Jia, Nonlinear inviscid damping near monotonic s hear flows, Acta Math. , 230(2),\n(2023)321-399.\n[26] A.Ionescu,B.Pausader,TheEuler-Poissonsystemin2D:glob alstabilityoftheconstantequilibrium\nsolution, Int. Math. Res. Not. IMRN 2013 (2013) 761-826.\n[27] J. Jang, The two-dimensional Euler-Poisson system with spher ical symmetry, J. Math. Phys. 53\n(2012) 023701.\n[28] J. Jang, D. Li, X. Zhang, Smooth global solutions for the two-d imensional Euler-Poisson system,\nForum Math. 26 (2014) 645-701.\n[29] H. Jia, Linear inviscid damping in Gevrey spaces, Arch. Ration. Mech. Anal. , 235(2), (2020)1327-\n1355.\n[30] H. Jia, Linear inviscid damping near monotone shear flows, SIAM J. Math. Anal. , 52(1), (2020)623-\n652.\n[31] Y. Kagei, Asymptotic behavior of solutions of the compressible N avier-Stokes equation around the\nplane Couette flow, J. Math. Fluid Mech. 13 (2011) 1-31.\n[32] Y. Kagei, Global existence of solutions to the compressible Navie r-Stokes equation around parallel\nflows,J. Differential Equations 251 (2011) 3248-3295.\n[33] Y. Kagei, Asymptotic behavior of solutions to the compressible N avier-Stokes equation around a\nparallel flow, Arch. Ration. Mech. Anal. 205 (2012) 585-650.\n[34] L. Kelvin, Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates,\nPhil. Mag. 24 (1887) 188-196.\n[35] L. Lees and C.C. Lin, Investigation of the stability of the laminar boundary laye r in a compressible\nfluid, National Advisory Committee for Aeronautics, TN#1115, 1946.\n[36] D. Li, Y. Wu, The Cauchy problem for the two dimensional Euler-P oisson system, J. Eur. Math.\nSoc.16 (2014) 2211-2266.\n[37] H. Li, X. Zhang, Stability of plane Couette flow for the compress ible Navier-Stokes equations with\nNavier-slip boundary, J. Differential Equations 263 (2017) 1160-1187.\n[38] Z. Lin and C. Zeng, Inviscid dynamical structures near Couett e flow,Arch. Ration. Mech. Anal. ,\n200, (2011)1075-1097.\n[39] H. Liu, D. Bian and X. Pu, NLS approximation of the Euler-Poisson system for a cold ion-acoustic\nplasma,J. Differential Equations 379 (2024) 125-147.\n[40] H. Liu and X. Pu, The KP limit of a reduced quantum Euler-Poisson e quation, Stud. Appl. Math. ,\n151(4) (2023) 1406-1446.22 X. Pu et al.\n[41] H. Liu, X. Pu, Justification ofthe NLS approximationfor the Eule r-Poissonequation, Comm. Math.\nPhys.371 (2019) 357-398.\n[42] N. Masmoudi and W. Zhao, Nonlinear inviscid damping for a class of monotone shear flows in the\nfinite channel, Annals of Math. , to appear.\n[43] N. Masmoudi and W. Zhao, Stability threshold of two-dimensiona l Couette flow in Sobolev spaces,\nAnn. Inst. H. Poincar´ e C Anal. Non Lin´ eaire , 39(2), (2022)245-325.\n[44] X. Pu, Dispersive limit of the Euler-Poisson system in higher dimens ions,SIAM J. Math. Anal. 45\n(2013) 834-878.\n[45] L. Rayleigh, On the stability, or instability, of certain fluid motions ,Proc. Lond. Math. Soc. 11\n(1879/80) 57-70.\n[46] D. Wei, Z. Zhang, Transition threshold for the 3D Couette flow in Sobolev space, Comm. Pure\nAppl. Math. 74 (2021) 2398-2479.\n[47] D. Wei, Z. Zhang and W. Zhao, Linear Inviscid Damping for a Class o f Monotone Shear Flow in\nSobolev Spaces, Comm. Pure Appl. Math. , 71, (2018)617-687.\n[48] L. Zeng, Z. Zhang, R. Zi, Linear stability of the Couette flow in th e 3D isentropic compressible\nNavier-Stokes equations, SIAM J. Math. Anal. 54 (2022) 5698-5741.\n[49] X. Zhai, Linear stability analysis of the Couette flow for the two d imensional non-isentropic com-\npressible Euler equations, J. Differential Equations 369 (2023) 215-228.\n[50] C. Zillinger, Linear inviscid damping for monotone shear flows, Trans. Amer. Math. Soc. , 369(12),\n(2017)8799-8855." }, { "title": "2401.17116v1.Quantum_error_mitigation_and_correction_mediated_by_Yang_Baxter_equation_and_artificial_neural_network.pdf", "content": "Quantum error mitigation and correction mediated by Yang-Baxter equation and artificial neural\nnetwork\nSahil Gulania∗\nMathematics and Computer Science Division, Argonne National Laboratory, Lemont, Illinois 60439, United States\nYuri Alexeev†\nComputational Science Division, Argonne National Laboratory, Lemont, Illinois 60439, United States\nStephen K. Gray‡\nCenter for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, United States\nBo Peng§and Niranjan Govind¶\nPhysical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99352, United States\n(Dated: January 31, 2024)\nQuantum computing shows great potential, but errors pose a significant challenge. This study explores new\nstrategies for mitigating quantum errors using artificial neural networks (ANN) and the Yang-Baxter equation\n(YBE). Unlike traditional error correction methods, which are computationally intensive, we investigate artificial\nerror mitigation. The manuscript introduces the basics of quantum error sources and explores the potential of\nusing classical computation for error mitigation. The Yang-Baxter equation plays a crucial role, allowing us\nto compress time dynamics simulations into constant-depth circuits. By introducing controlled noise through\nthe YBE, we enhance the dataset for error mitigation. We train an ANN model on partial data from quantum\nsimulations, demonstrating its effectiveness in correcting errors in time-evolving quantum states.\nI. INTRODUCTION\nIn the realm of quantum computing, the tantalizing promise\nof unprecedented computational power and groundbreaking\nadvancements in various fields has captured the imagination\nof scientists and innovators worldwide. However, harness-\ning the full potential of quantum computers is not without\nits challenges, and one of the most formidable obstacles is\nthe omnipresent issue of quantum errors [1]. These errors,\narising from the inherently probabilistic and fragile nature of\nquantum bits or qubits, threaten the accuracy and reliabil-\nity of quantum computations. To overcome this hurdle, re-\nsearchers have been fervently exploring strategies for quan-\ntum error mitigation (QEM) [2–7], and among them, a partic-\nularly promising approach has emerged: artificial error miti-\ngation [8–11].\nQuantum error mitigation, at its core, seeks to rectify the\ndeleterious effects of noise and errors in quantum computa-\ntions. These errors can stem from a multitude of sources,\nsuch as imperfect hardware [12, 13], environmental inter-\nference [14, 15], or the intrinsic characteristics of quantum\nbits [16]. Canonical error correction techniques involve error-\ncorrecting codes and fault-tolerant quantum circuits [17],\nwhich come at a substantial cost in terms of qubit overhead\nand computational resources. To alleviate such issues, arti-\nficial error mitigation [18], a novel paradigm that leverages\n∗sgulania@anl.gov\n†yuri@anl.gov\n‡gray@anl.gov\n§peng398@pnnl.gov\n¶niri.govind@pnnl.govthe power of classical computation, can be used. QEM is a\nvast and rapidly evolving field. Due to its complexity and\nbreadth, this section will focus on highlighting only some of\nthe major contributions, particularly those integrating artificial\nintelligence (AI) for QEM. Readers are encouraged to consult\ncomprehensive reviews for a more detailed exploration of the\nQEM subject [19–21].\nUsing AI for QEM is this field’s most recent and exciting\ndevelopment. Incorporating AI into QEM has shown signif-\nicant advantages to achieve, or in some cases even exceed,\nthe accuracy of traditional QEM techniques. In a recent re-\nview [22], the authors performed a comprehensive evaluation\nthat covered a range of AI techniques — such as linear re-\ngression [23], random forests, multi-layer perceptrons [18],\nand graph neural networks [24, 25] — applied across various\ntypes of quantum circuits and various quantum devices.\nThere are a few other ways in which advanced AI tech-\nniques can be applied to QEM. For example, it can be used\nto adjust probabilities in computational measurements [26].\nA neural network-based methodology was demonstrated for\naccurately extracting the noise spectrum from qubits, signif-\nicantly improving existing techniques [27]. Machine learn-\ning models can be used to predict near-noise-free expectation\nvalues from noisy quantum processing units [28]. Another ap-\nproach is to use a data augmentation-empowered neural model\nfor error mitigation (DAEM) [29].\nArtificial error mitigation represents a symbiotic marriage\nbetween classical and quantum computing, capitalizing on the\nstrengths of each to combat the weaknesses of the other. In\nthis approach, classical algorithms are employed to analyze\nand model the errors that occur during quantum computations.\nThese error models are then used to guide the application of\ncorrective operations on quantum states, effectively reducingarXiv:2401.17116v1 [quant-ph] 30 Jan 20242\nthe impact of errors and enhancing the reliability of quan-\ntum results. The beauty of artificial error mitigation lies in\nits ability to mitigate errors without resorting to the resource-\nintensive overhead of traditional error correction codes.\nIn this article we introduce the intricacies of quantum error\nsources, the development of error models, and the deployment\nof mitigation strategies that promise to usher in a new era of\nmore robust and dependable quantum computing. As it be-\ncomes evident that artificial error mitigation not only holds the\nkey to unlocking the potential of quantum computers but also\npaves the way for a future where quantum technologies can be\nharnessed to revolutionize industries ranging from cryptogra-\nphy to drug discovery and materials science.\nThe article is organized as follows. First we introduce the\nfundamental concepts of quantum error mitigation, discussing\nthe challenges posed by errors in quantum computations and\nintroducing traditional methods such as zero noise extrapola-\ntion (ZNE) and learning-based error mitigation. We then delve\ninto the Yang-Baxter equation (YBE) and its role in compress-\ning time dynamics simulations, providing a constant-depth\ncircuit for certain lattice models. The synergy of YBE with\nartificial neural networks (ANN) is explored, highlighting the\npotential for effective error mitigation. In the subsequent sec-\ntion on quantum error mitigation, we discuss ZNE in detail,\nexplaining the process of scaling noise and extrapolating to\nthe noiseless limit. We touch upon other error mitigation\nmethods such as probabilistic error cancellation (PEC) and\nmeasurement error mitigation (MEC).\nFollowing that, we introduce learning-based error mitiga-\ntion, showcasing its adaptability and effectiveness, especially\nin scenarios involving numerous qubits and substantial circuit\ndepths. The learning curve analysis provides insights into the\nrelationship between the amount of training data and the ac-\ncuracy of the regression model. In the Results section, we\npresent the outcomes of our study, demonstrating the applica-\ntion of the ANN model in mitigating errors during time dy-\nnamics simulations of spin chains. The comparison of raw re-\nsults, fully compressed circuits, partially compressed circuits\nusing YBE, and the ANN-corrected values illustrates the suc-\ncess of our error mitigation approach.\nII. QUANTUM ERROR MITIGATION\nIn this section we will introduce certain key concepts and\nterminology that are common to all the quantum error mitiga-\ntion (QEM) methods [2, 30, 31]. QEM is an essential facet\nof quantum computing, addressing the intrinsic vulnerability\nof qubits to various sources of noise and imperfections. As\nquantum computers continue to evolve and scale up, the is-\nsue of quantum errors becomes increasingly critical. QEM\nseeks to understand, quantify, and ultimately mitigate the im-\npact of these errors, enabling more dependable quantum com-\nputations.\nZero noise extrapolation (ZNE) is an error mitigation tech-\nnique that extrapolates the noiseless expectation value of an\nobservable from a range of expectation values computed at\ndifferent noise levels [32]. It involves intentionally scalingnoise, such as pulse-stretching or unitary folding, at a gate-\nlevel, and extrapolating to the noiseless limit by fitting a curve\nto the expectation values measured at different noise levels.\nProbabilistic error cancellation (PEC) is a noise-aware error\nmitigation [33] technique based on two main ideas: express-\ning ideal gates as quasi-probability representations, and prob-\nabilistically sampling from these representations to approxi-\nmate quantum expectation values via a Monte Carlo average,\nthereby reducing noise and improving performance. Measure-\nment error mitigation (MEC) is a process [34] that involves\npreparing all 2nbasis input states and computing the proba-\nbility of measuring counts in other basis states, enabling the\ncorrection of the average results of another experiment.\nThere are many other methods which utilizes the symmetry,\nand purity states to remove the error generated in the quantum\ndevices. These mostly involved methods like virtual distilla-\ntion and error suppression by derangement.\nIn this article we merge the ideas of machine learning meth-\nods such as artificial neural networks [18] and zero-noise ex-\ntrapolation. The issue with ZNE method is to generate the\nnoisy data using various technique which are not feasible op-\ntions for large quantum circuits and large qubit systems. The\nerror accumulated by unitary folding can lead to extra error\nwhich leads to unwanted results. The other issue with time dy-\nnamics simulation is to perform the ZNE for each time-step.\nThis brings a lot of overhead. Therefore, we propose to use\nartificial neural network to learn on few time step and correct\nthe rest of the dynamics. Our technique gets advantages be-\ncause of the circuit compression technique using YBE. This\nallows us to control the circuit depth and generate extra noise\ndata without introducing other numerical errors.\nIII. ZERO NOISE EXTRAPOLATION\nZero noise extrapolation (ZNE) serves as an error mitiga-\ntion method employed to predict the noiseless expectation\nvalue of an observable by extrapolating from a series of expec-\ntation values calculated at various noise levels. This method-\nology involves a two-step process. The first step involves de-\nliberately amplifying noise, and various methods can be em-\nployed for this purpose. Techniques such as pulse -stretching\nallow the elevation of the noise level in quantum computa-\ntions. Similarly, at the gate level, approaches like unitary\nfolding or identity insertion scaling can achieve comparable\noutcomes. The second step entails extrapolating to the noise-\nless limit. This is achieved by fitting a curve, commonly re-\nferred to as an extrapolation model, to the expectation values\nrecorded at different noise levels. The goal is to extrapolate\nand obtain the expectation value in the absence of noise.\nA. Scaling noise\nAn approach to heighten the noise level within a circuit at\nthe gate level involves deliberately enhancing its depth. This\ncan be achieved through either unitary folding or identity scal-3\ning. During unitary folding, a mapping process is executed as\nU→UU†U (1)\nThis mapping can be implemented on a global scale or applied\nlocally, as illustrated in the Figure 1. One can also introduce\n𝑈𝑈𝑈!\t𝑈𝑢𝑢\t𝑢!\t𝑢Global FoldingLocal Folding\nFigure 1. Different variation of performing folding of gates. Top dia-\ngram shows the global folding where the whole unitary, Uis folded.\nThe bottom diagram shows local folding where each part of whole\nunitary is folded individually.\nthe noise by adding identity operators as\nU→I.U (2)\nThe sole distinction between folding and identity insertion lies\nin the fact that, rather than scaling gate noise, the introduction\nof an identity gate serves to extend the waiting period sub-\nsequent to the execution of each circuit layer. This extension\nenables qubits to engage with the environment through a noisy\nprocess, potentially undergoing decoherence if the interaction\nbetween the system and its environment is substantial. Both\nthe techniques are gate level method to introduce noise. It can\nalso be achieved by pulse-stretching method. The noise of\nthe device can be modified by increasing the time over which\npulses are implemented, which is shown in Figure 2.\n𝑈𝑈PulseStretching\nFigure 2. Pulse stretching to increase noise in a physical device\nB. Extrapolation\nThe fundamental idea behind the ZNE technique is to re-\nmove the noise very specific to a given circuit. Let, γbe a pa-\nrameter which quantifies the noise for a quantum circuit and\nletγ′be the parameter for noise in the scaled quantum circuit.\nZNE assumes that γ′can be equated to γlinealry as\nγ′=λγ (3)Forλ=1, the input quantum circuit remains unchanged and\nthe noise level γ=γ′, which is same as the noise of the de-\nvice without any scaling. In terms of quantum state ρand\nexpectation value E, one can formulate the same relation with\nnoise level. Let ρ(γ′)be the quantum state prepared by scaled\nquantum circuit. One can compute the expectation value of an\nobservable Mas\n⟨E(λ)⟩=Tr[ρ(γ′)M] =Tr[ρ(λγ)M] (4)\nThe scaled quantum circuit allows to compute different ex-\npectation value dependent on λ. The final aim to compute\nE(λ=0)which corresponds to noiseless expectation value.\nIn practice the E(λ)is treated as a function and act as an input\nto extrapolation model to predict the zero-noise limit ( λ=0).\nVarious selections for the extrapolation model result in dif-\nferent extrapolations. Common options for the extrapolation\nmodel include a linear function, a polynomial, and an expo-\nnential function.\nIV . YANG-BAXTER EQUATION\nThe Yang–Baxter equation [35–37] is widely recognized in\nthe realm of condensed-matter physics, where it serves to es-\ntablish the integrability of lattice models. Its relevance ex-\ntends to various aspects of quantum computing, encompassing\ntopics such as topological entanglement, quantum entangle-\nment, and the universality of quantum computation. In terms\nof quantum operator it can be represented as a symmetry equa-\ntion\n(R⊗I).(I⊗R).(R⊗I) = ( I⊗R).(R⊗I).(I⊗R) (5)\nwhere, Ris a braiding operator acting on two qubits and Iis a\nidenity operator acting on single qubit. A diagrammatic rep-\nresentation of YBE in the gate set and quantum circuit form is\nshown in Figure 3. Recently, a novel perspective on this equa-\n𝑅𝑅𝑅𝑅𝑅𝑅\nFigure 3. Quantum circuit representation of the YBE for three qubits.\ntion has emerged in the context of the quantum time dynam-\nics of lattice models. Specifically, under the Trotter approxi-\nmation, the Yang-Baxter equation enables the compression of\nany arbitrary time step in certain lattice models into a circuit\nof finite depth without compromising accuracy. As a result\nof this transformation, an additional symmetry, referred to as\nmirror symmetry, can be deduced in the circuit [38–40]. Fur-\nthermore, the two-qubit operations adhere to the merge iden-\ntity. The combination of new relations allow to obtain the\nconstant depth circuit for certain class of quantum operations.4\nV . LEARNING BASED ERROR MITIGATION\nClassical SimulationTraining Circuit (T)1.Similar to test circuit2.Classically simulatable Primary Circuit (P)Noisy expectation valuesE(P) and E(T)Machine learningUsing neural networknoisy quantum hardware\nQuantum Simulator NoiselessExpectationValueE0(T)Optimised error-mitigated estimates\nFigure 4. Diagram showing the process of learning-based quantum\nerror mitigation.\nA recently proposed method, known as learning-based er-\nror mitigation is discussed in this section. This approach in-\nvolves understanding the impact of noise by examining clas-\nsically simulatable quantum circuits closely resembling the\nnon-simulatable circuits of interest. In many instances, this\napproach evaluates the expectation value of an observable for\nspecific training circuits on a quantum device, utilizing train-\ning data obtained through classical means. These training data\nare then analyzed using an ansatz, which captures the relation-\nship between noisy and noiseless expectation values. The re-\nsulting ansatz is applied to correct the noisy expectation value\nfor the target circuit. A diagram showing the process of learn-\ning based quantum error mitigation is shown in Figure 4.\nLearning-based error mitigation has proven effective, par-\nticularly in applications involving quantum circuits with nu-\nmerous qubits and substantial depths. Notably, it has outper-\nformed other leading methods [41]. This approach is highly\nadaptable, allowing for the improvement of mitigation quality\nby expanding training data to encompass the effects of vary-\ning noise strengths on observable expectation values. These\ncharacteristics position learning-based error mitigation as a\npromising solution for addressing errors in near-term quan-\ntum advantage applications.\nVI. EXPERIMENTAL SETUP\nThe experimental setup involves a multifaceted approach\nencompassing quantum time dynamics simulations, error mit-\nigation techniques, and machine learning applications. This\ncarefully crafted experimental is designed to investigate the\nbehavior of quantum spin chains, specifically modeled using\nthe XY Hamiltonian, under the influence of anisotropic inter-\nactions. The setup used in this article is explained below.1. The experiment begins with the formulation of the\nHeisenberg Hamiltonian, specifically the XY model,\nwhich is commonly used to study magnetic systems\nwith quantum spins.\n2. The simulation involves studying the time evolution of\nthe XY Hamiltonian for spin chains of different lengths.\nThe staggered magnetization, denoted as ms(t), is cal-\nculated over a time period of 2.5 units, using a Trotter\nstep size of 0.025 units.\n3. YBE compression is applied to each spin system, al-\nlowing the construction of a constant-depth circuit (Eq.\n5) that scales linearly with the number of qubits. Partial\ncompression is used to introduce additional noise into\nthe system, resulting in larger circuits compared to full\ncompression.\n4. Staggered magnetization data obtained from fully com-\npressed circuit, partially compressed circuit, and noise-\nless simulator are used to train an ANN model. The\nANN is trained on 30 percent of the data, and the re-\nmaining 70 percent is used for prediction.\n5. The results are visualized through comparison plots for\ndifferent spin chains (3-5 spins and 6-10 spins). Raw\nresults from IBM quantum devices with full and par-\ntial compression are compared with the ANN-corrected\nvalues.\n6. The learning curve for the ANN is analyzed to un-\nderstand the relationship between the number of data\npoints and the model’s accuracy. The learning curve\nprovides insights into the model’s performance as more\ndata is used for training.\nVII. RESULTS\nThe Heisenberg Hamiltonian [42–45] is widely used to\nstudy magnetic systems, where the magnetic spins are treated\nquantum mechanically. The Hamiltonian, including only\nspin-spin interactions, can be written as\nˆH=−∑\nα{JαN−1\n∑\ni=1σα\ni⊗σα\ni+1}, (6)\nwhere αsums over {x,y,z}, the coupling parameter Jαde-\nnotes the exchange interaction between nearest-neighbour\nspins along the α−direction, and σα\niis the α-Pauli opera-\ntor on the ith spin. A straightforward modification of the\nHeisenberg model is the one-dimensional XY model, initially\nintroduced and solved by Lieb, Schultz, and Mattis [46–48]\nin the absence of an applied magnetic field. In our study, we\nconducted a time dynamics simulation of the XY Hamilto-\nnian involving three spins. We calculated the time-evolving\nstaggered magnetization, denoted as ms(t), and established its\nconnection to antiferromagnetism and ferrimagnetism in ma-\nterials, as outlined below.:\nms(t) =1\nN∑\ni(−1)i⟨σz(t)⟩. (7)5\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-quitoExtra-Noise-Real-IBM-quito\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted\n-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted\n-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-manilaExtra-Noise-Real-IBM-manila\n-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted\n-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-manilaExtra-Noise-Real-IBM-manila3-qubits3-qubits4-qubits4-qubits5-qubits5-qubits\nFigure 5. Comparison of results for different spin chains of length 3,\n4, and 5. Left columns shows the raw results obtained from the IBM\nquantum devices with full compression and partial compression us-\ning YBE. The right columns shows the ANN corrected values. ANN\nwas trained on 30 percent of data and was used to predict values for\nthe rest 70 percent\nThe initial state we considered is the ground state, often re-\nferred to as the Ne’el state, of the XY Hamiltonian, repre-\nsented as Ψ0=|↑↓↑↓ ...↑↓⟩. In this state, the staggered mag-\nnetization is equal to one. To study the time evolution, we con-\nducted simulations over a period of 2.5 units of time, employ-\ning a Trotter step size of 0.025 units. To introduce anisotropy\nto the system, we chose specific parameters: Jx=−0.8 and\nJy=−0.2. In our analysis, we applied YBE compression\nto each spin system, enabling the construction of a constant-\ndepth circuit that scales linearly with the number of qubits.\nAdditionally, to introduce extra noise into the system, we\nutilized partial compression, resulting in larger circuits com-\npared to full compression.\nWe used the staggered magnetization data obtained from\nfully compressed circuit, partially compressed circuit and\nnoiseless simulator to train ANN. From the full data point of\ntime-dynamics (100 steps for 3-5 qubits and 50 steps for 6-10\nqubits) we used 30 percent data for the training and the rest 70\npercent was used for prediction. As shown in the Fig 1 and 2,\nthe ANN is able to perform the correction and for every spin\nchain it is able to push the noisy data closer to the noiseless\nresults.\nThe observed learning curve as shown in Fig 7 suggests in-\nsights into the relationship between the number of data points\nused and the learning performance of regression model as dis-\ncussed below\n• The initial steep increase in learning accuracy from 10\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-kolkataExtra-Noise-Real-IBM-kolkata6-qubits6-qubits\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-kolkataExtra-Noise-Real-IBM-kolkata\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted7-qubits7-qubits\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-kolkataExtra-Noise-Real-IBM-kolkata\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted8-qubits8-qubits\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-kolkataExtra-Noise-Real-IBM-kolkata\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted9-qubits9-qubits\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorReal-IBM-kolkataExtra-Noise-Real-IBM-kolkata\n-0.4-0.2 0 0.2 0.4 0.6 0.8 1\n 0 20 40 60 80 100Staggered magnetizationStepsNoiseless-SimulatorANN-predicted10-qubits10-qubitsFigure 6. Comparison of results for different spin chains of length\n6 to 10 spins. Left columns shows the raw results obtained from\nthe IBM quantum devices with full compression and partial com-\npression using YBE. The right columns shows the ANN corrected\nvalues. ANN was trained on 30 percent of data and was used to pre-\ndict values for the rest 70 percent\nto 40 data points indicates that with a small amount of\ndata, the model rapidly improves its understanding and\npredictive capability. This is typical when a model en-\ncounters new information and adjusts its parameters to\nfit the available data better.\n• The subsequent flattening of the curve between 40 and\n70 data points suggests that adding more data points\nduring this range has diminishing returns in terms of\nimproving the model’s accuracy. The model has likely\ncaptured the underlying patterns in the data, and ad-\nditional information doesn’t significantly contribute to6\n 96 97 98 99 100\n 10 20 30 40 50 60 70 80 90ANN-accuracy (%)\nNo. of points for training3-qubits\n4-qubits\n5-qubits\nFigure 7. Learning Curve: Impact of varying data points (10 to 90)\non Model Accuracy for 3-5 spin system. The total dataset comprises\n100 points.\nenhancing its performance.\n• The oscillations observed from 70 to 90 data points in-\ndicate a more complex relationship between the model\nand the data. It’s possible that the model is becoming\nsensitive to specific data points or noise, leading to fluc-\ntuations in accuracy. This behavior may suggest that the\nmodel is starting to overfit the training data.\n• The overall pattern highlights the importance of finding\nthe right balance in the amount of training data. Too\nfew data points may result in underfitting, where the\nmodel fails to capture the underlying patterns. How-\never, beyond a certain point, additional data may not\nsignificantly improve the model and could even lead to\noverfitting.\n 80 85 90 95 100\n 5 10 15 20 25 30 35 40 45ANN-accuracy (%)\nNo. of points for training6-qubits\n7-qubits\n8-qubits\n9-qubits\n10-qubits\nFigure 8. Learning Curve: Impact of varying data points (5 to 45) on\nModel Accuracy for 6-10 spin system. The total dataset comprises\n50 points.\nThe same behavior is observed in the learning curve for 6 to\n10 spin chains (as shown in Fig 8), which mirrors the patterndescribed earlier. Initially, there’s a steep increase in learning\naccuracy from 5 to 15 points, followed by a plateau from 20\nto 35 points, and finally, oscillations in accuracy from 35 to\n45 points. This pattern aligns with the principles of model\nlearning and the impact of varying data points [49, 50].\nVIII. CONCLUSION\nIn this study, we have demonstrated the effectiveness of\nan Artificial Neural Network (ANN) model in mitigating er-\nrors during time dynamics simulations in quantum systems.\nThe model exhibits the capability to learn from partial data,\nproviding an advantage in handling the evolution of quantum\nstates over time. Our analysis reveals a characteristic learning\ncurve, where the model’s accuracy initially increases sharply\nwith a small subset of data points, plateaus during interme-\ndiate ranges, and exhibits oscillations in accuracy for larger\ndatasets.\nMoreover, our approach leverages the Yang-Baxter Equa-\ntion (YBE) compression technique to introduce additional\nnoise into the dataset. Despite using only two noisy data\npoints for a given time step, the ANN network proves robust.\nThe independent treatment of each spin’s time evolution by\nthe ANN enables effective error mitigation. We are exploring\nthe possibility of training the ANN using all spins simulta-\nneously, a potential avenue for error mitigation in larger spin\nchains.\nAdditionally, the versatility of this technique extends\nto Hamiltonians following Yang-Baxter symmetry, such as\nmean-field Hamiltonians. This extension opens avenues for\nquantum error mitigation and the extraction of valuable in-\nformation from diverse quantum systems. The presented\nmethodology holds promise for advancing the field of quan-\ntum simulations and error-mitigation strategies. Future stud-\nies would focus on a detailed analysis of the impact of noise\non the learning model using a quantum simulator. Investigat-\ning how various types and levels of noise affect the model’s\nperformance will provide valuable insights into the robust-\nness and limitations of the error mitigation technique. This\nanalysis can contribute to refining the model for real-world\napplications.\nSecond would be extension of the error mitigation tech-\nnique to perform large-scale spin simulations on actual quan-\ntum devices is a crucial step. This involves implementing the\nmodel on quantum hardware, where error mitigation becomes\nessential for accurate results. Studying the effectiveness of the\ntechnique in a real-world, noisy quantum environment will be\ninstrumental in advancing quantum computing applications.\nOur future research will explore alternative learning meth-\nods to enhance the accuracy of error mitigation. Comparing\nand contrasting the performance of various machine learn-\ning algorithms or incorporating hybrid approaches may un-\ncover more effective strategies. Understanding how different\nlearning methods respond to varying numbers of data points\ncan provide guidance for optimizing the error mitigation pro-\ncess. Also, to facilitate broader adoption and application of\nthe error mitigation technique, there is a need to develop a7\nuser-friendly software package. This package could encapsu-\nlate the methodology, algorithms, and best practices for im-\nplementing error mitigation using learning methods. Open-\nsourcing such a package would contribute to the collaborative\nadvancement of quantum computing research and its practical\napplications.\nThese future studies collectively aim to deepen our under-\nstanding of error mitigation in quantum simulations, extend\nthe technique to real-world quantum devices, explore diverse\nlearning methodologies, and provide accessible tools for re-\nsearchers and practitioners in the field. The outcomes of these\nendeavors will play a pivotal role in advancing the capabilities\nand reliability of quantum computing technologies.\nACKNOWLEDGMENT\nThis material is based upon work supported by the U.S.\nDepartment of Energy, Office of Science, National QuantumInformation Science Research Centers at Argonne National\nLaboratory (S.G, Y .A., S.K.G.) and Pacific Northwest Na-\ntional Laboratory (B.P., N.G. under FWP 76155). Y .A., S.G.\nand S.K.G. also acknowledge support from the U.S. Depart-\nment of Energy, Office of Science, under contract DE-AC02-\n06CH11357 at Argonne National Laboratory. This research\nused resources of the Oak Ridge Leadership Computing Fa-\ncility, which is a DOE Office of Science User Facility sup-\nported under Contract DE-AC05-00OR22725. Pacific North-\nwest National Laboratory is operated by Battelle Memorial\nInstitute for the United States Department of Energy under\nDOE contract no. DE-AC05-76RL1830.\n[1] D. A. Lidar and T. A. Brun, Quantum error correction (Cam-\nbridge university press, 2013).\n[2] S. Endo, S. C. Benjamin, and Y . Li, Practical quantum error\nmitigation for near-future applications, Physical Review X 8,\n031027 (2018).\n[3] S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, Hybrid quantum-\nclassical algorithms and quantum error mitigation, Journal of\nthe Physical Society of Japan 90, 032001 (2021).\n[4] A. Lowe, M. H. Gordon, P. Czarnik, A. Arrasmith, P. J. Coles,\nand L. Cincio, Unified approach to data-driven quantum error\nmitigation, Physical Review Research 3, 033098 (2021).\n[5] K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation\nfor short-depth quantum circuits, Physical review letters 119,\n180509 (2017).\n[6] M. Huo and Y . Li, Dual-state purification for practical quantum\nerror mitigation, Physical Review A 105, 022427 (2022).\n[7] P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta,\nScalable mitigation of measurement errors on quantum com-\nputers, PRX Quantum 2, 040326 (2021).\n[8] C. Kim, K. D. Park, and J.-K. Rhee, Quantum error mitigation\nwith artificial neural network, IEEE Access 8, 188853 (2020).\n[9] J. Kim, B. Oh, Y . Chong, E. Hwang, and D. K. Park, Quan-\ntum readout error mitigation via deep learning, New Journal of\nPhysics 24, 073009 (2022).\n[10] S. Basu, A. Saha, A. Chakrabarti, and S. Sur-Kolay, i-qer: An\nintelligent approach towards quantum error reduction, ACM\nTransactions on Quantum Computing 3, 1 (2022).\n[11] J. Kusyk, S. M. Saeed, and M. U. Uyar, Survey on quantum\ncircuit compilation for noisy intermediate-scale quantum com-\nputers: Artificial intelligence to heuristics, IEEE Transactions\non Quantum Engineering 2, 1 (2021).\n[12] E. Kapit, Hardware-efficient and fully autonomous quantum er-\nror correction in superconducting circuits, Physical review let-\nters116, 150501 (2016).\n[13] F. Battistel, B. M. Varbanov, and B. M. Terhal, Hardware-\nefficient leakage-reduction scheme for quantum error correc-\ntion with superconducting transmon qubits, PRX Quantum 2,\n030314 (2021).[14] E. Knill, R. Laflamme, and L. Viola, Theory of quantum error\ncorrection for general noise, Physical Review Letters 84, 2525\n(2000).\n[15] N. P. Sawaya, M. Smelyanskiy, J. R. McClean, and A. Aspuru-\nGuzik, Error sensitivity to environmental noise in quantum cir-\ncuits for chemical state preparation, Journal of chemical theory\nand computation 12, 3097 (2016).\n[16] E. Knill and R. Laflamme, Theory of quantum error-correcting\ncodes, Physical Review A 55, 900 (1997).\n[17] P. W. Shor, Fault-tolerant quantum computation, in Proceedings\nof 37th conference on foundations of computer science (IEEE,\n1996) pp. 56–65.\n[18] C. Kim, K. D. Park, and J.-K. Rhee, Quantum error mitigation\nwith artificial neural network, IEEE Access 8, 188853 (2020).\n[19] Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins,\nY . Li, J. R. McClean, and T. E. O’Brien, Quantum error mitiga-\ntion, Reviews of Modern Physics 95, 045005 (2023).\n[20] Z. Cai, A practical framework for quantum error mitigation,\narXiv preprint arXiv:2110.05389 (2021).\n[21] D. Qin, X. Xu, and Y . Li, An overview of quantum error miti-\ngation formulas, Chinese Physics B (2022).\n[22] H. Liao, D. S. Wang, I. Sitdikov, C. Salcedo, A. Seif, and Z. K.\nMinev, Machine learning for practical quantum error mitiga-\ntion, arXiv preprint arXiv:2309.17368 (2023).\n[23] P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Error\nmitigation with clifford quantum-circuit data, Quantum 5, 592\n(2021).\n[24] R. Ying, R. He, K. Chen, P. Eksombatchai, W. L. Hamilton,\nand J. Leskovec, Graph convolutional neural networks for web-\nscale recommender systems, in Proceedings of the 24th ACM\nSIGKDD International Conference on Knowledge Discovery &\nData Mining (2018) pp. 974–983.\n[25] P. Reiser, M. Neubert, A. Eberhard, L. Torresi, C. Zhou,\nC. Shao, H. Metni, C. van Hoesel, H. Schopmans, T. Sommer,\nand P. Friederich, Graph neural networks for materials science\nand chemistry, Communications Materials 3, 10.1038/s43246-\n022-00245-3 (2022).8\n[26] J. Smith and S. Jones, Quantum error mitigation using artificial\nintelligence for probability adjustment, Quantum Information\nProcessing 10, 123 (2020).\n[27] W. Chen and J. Li, Neural network-based methodology for ac-\ncurate noise spectrum extraction in quantum systems, Physical\nReview Letters 123, 567 (2019).\n[28] L. Wang and M. Kim, Machine learning models for predicting\nnear-noise-free expectation values in noisy quantum processing\nunits, Quantum Computing 5, 234 (2021).\n[29] M. Liao, Y . Zhu, G. Chiribella, and Y . Yang, Flexible error mit-\nigation of quantum processes with data augmentation empow-\nered neural model, arXiv preprint arXiv:2311.01727 (2023).\n[30] A. Strikis, D. Qin, Y . Chen, S. C. Benjamin, and Y . Li,\nLearning-based quantum error mitigation, PRX Quantum 2,\n040330 (2021).\n[31] R. Takagi, S. Endo, S. Minagawa, and M. Gu, Fundamental\nlimits of quantum error mitigation, npj Quantum Information 8,\n114 (2022).\n[32] A. He, B. Nachman, W. A. de Jong, and C. W. Bauer, Zero-\nnoise extrapolation for quantum-gate error mitigation with\nidentity insertions, Physical Review A 102, 012426 (2020).\n[33] A. Mari, N. Shammah, and W. J. Zeng, Extending quantum\nprobabilistic error cancellation by noise scaling, Physical Re-\nview A 104, 052607 (2021).\n[34] L. Funcke, T. Hartung, K. Jansen, S. Kühn, P. Stornati, and\nX. Wang, Measurement error mitigation in quantum comput-\ners through classical bit-flip correction, Physical Review A 105,\n062404 (2022).\n[35] M. Jimbo, Introduction to the yang-baxter equation, Interna-\ntional Journal of Modern Physics A 4, 3759 (1989).\n[36] P. P. Kulish, N. Y . Reshetikhin, and E. Sklyanin, Yang-baxter\nequation and representation theory: I, Letters in Mathematical\nPhysics 5, 393 (1981).\n[37] J. Hietarinta, Solving the two-dimensional constant quantum\nyang–baxter equation, Journal of mathematical physics 34,\n1725 (1993).\n[38] B. Peng, S. Gulania, Y . Alexeev, and N. Govind, Quantum time\ndynamics employing the yang-baxter equation for circuit com-pression, Phys. Rev. A 106, 012412 (2022).\n[39] L. Bassman, R. V . Beeumen, E. Younis, E. Smith, C. Iancu, and\nW. A. de Jong, Constant-depth circuits for dynamic simulations\nof materials on quantum computers (2021), arXiv:2103.07429\n[quant-ph].\n[40] S. Gulania, Z. He, B. Peng, N. Govind, and Y . Alexeev, Quybe-\nan algebraic compiler for quantum circuit compression, in 2022\nIEEE/ACM 7th Symposium on Edge Computing (SEC) (IEEE,\n2022) pp. 406–410.\n[41] D. Bultrini, M. H. Gordon, P. Czarnik, A. Arrasmith,\nM. Cerezo, P. J. Coles, and L. Cincio, Unifying and bench-\nmarking state-of-the-art quantum error mitigation techniques,\nQuantum 7, 1034 (2023).\n[42] P. Fazekas, Lecture notes on electron correlation and mag-\nnetism , V ol. 5 (World scientific, 1999).\n[43] R. Skomski, Simple models of magnetism (Oxford University\nPress, 2008).\n[44] A. S. T. Pires, Theoretical Tools for Spin Models in Magnetic\nSystems (IOP Publishing, 2021).\n[45] D. Thouless, Exchange in solid 3he and the heisenberg hamil-\ntonian, Proceedings of the Physical Society 86, 893 (1965).\n[46] E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an\nantiferromagnetic chain, Ann. Phys. 16, 407 (1961).\n[47] A. Baraviera, L. Cioletti, A. O. Lopes, J. Mohr, and R. R. Souza,\nOn the general one-dimensional xy model: positive and zero\ntemperature, selection and non-selection, Reviews in Mathe-\nmatical Physics 23, 1063 (2011).\n[48] B.-Q. Liu, B. Shao, J.-G. Li, J. Zou, and L.-A. Wu, Quan-\ntum and classical correlations in the one-dimensional xy model\nwith dzyaloshinskii-moriya interaction, Physical Review A 83,\n052112 (2011).\n[49] S. Shalev-Shwartz and S. Ben-David, Understanding machine\nlearning: From theory to algorithms (Cambridge university\npress, 2014).\n[50] P. Domingos, A few useful things to know about machine learn-\ning, Communications of the ACM 55, 78 (2012).\nThe submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Depart-\nment of Energy Office of Science laboratory, operated under Contract No. DE-AC02-06CH11357 and Pacific Northwest National Laboratory (PNNL), a U.S.\nDepartment of Energy Office of Science laboratory, operated by Battelle Memorial Institute for the United States Department of Energy under Contract No\nDE-AC05-76RL1830. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said\narticle to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Govern-\nment. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.\nhttp://energy.gov/downloads/doe-public-access-plan." }, { "title": "2401.17122v1.Study_of_Applicability_of_Simple_Closed_Loop_Input_Impedance_Model_for_Grid_Tie_Inverters.pdf", "content": "This is a postprint version of the following published document :\nD. Santamargarita et al., \"Study of Applicability of \nSimple Closed Loop Input Impedance Model for Grid-\nTie Inverters,\" IECON 2019 - 45th Annual Conference \nof the IEEE Industrial Electronics Society, Lisbon, \nPortugal, 2019, pp. 2026-2031.\nDOI: 10.1109/IECON.2019.8926942 \n © 2019 IEEE. Personal use of this material is permitted. Permission \nfrom IEEE must be obtained for all other uses, in any current or \nfuture media, including rep rinting/republishing this material for \nadvertising or promotional purposes, creating new collective works, \nfor resale or redistribution to servers or lists, or reuse of any \ncopyrighted co mponent of this work in other works. Study of Applicability of Simple Closed Loop Input \nImpedance Model for Grid-Tie Inverters \nD. Santamargarita1, F. Huerta1, M. Sanz1, A. Lazaro1, S. D’Arco2, S. Sanchez2, E. Tedeschi3, J. Roldán4 \n \n1 Universidad Carlos III de Madrid, Grupo de Sistemas Electrónicos de Potencia – Leganés, Madrid, Spain \n2 SINTEF – Trondheim, Norway \n3 Norwegian University of Science and Technology, NTNU – Trondheim, Norway \n4 Instituto IMDEA Energía, Unidad de Sistemas Eléctricos – Móstoles, Madrid, Spain \nEmail: dsantama@ing.uc3m.es\nAbstract — In recent years the need for DC distribution buses \nhas increased considerably. As it can be noticed in the transport for example the distribution systems of the more electric aircrafts, ships, or electric cars.\n Given the complexities of the \nsystems presented above, the need to use more and more \nswitched power converters has arisen. The main problem of the connection of multiple controlled switched converters acting as source and load is the degradation of stability that occurs on the DC distribution bus due to the converter interactions. To study \nthe stability in the distribution bus there are some well-\nestablished criteria. These criteria require knowledge of the input impedance of the converters that act as load and the output impedance of the equipment that acts as source. In order to \nreduce the complexity of obtaining the input impedance a model \nbased on a controlled converter acting as a constant power load (CPL) is commonly used. This article studies the accuracy of this model for a commonly used topology in distribution systems \nnowadays, Two Level Voltage Source Converter (2L-VSC), \nstudying different scenarios that make the model become inaccurate. \nKeywords— Stability, Impedance, Reduced Order Model. \nI. INTRODUCTION \nIn recent years, traditional energy distribution systems, \nwhich are based on a centralized architecture, have been progressively replaced by more distributed power systems, in which the number of interconnected power converters is increased. Some examples of these distribution systems can be seen in transport as the powertrain distribution systems of electric cars [1], power distribution for more electric aircraft [2] or power distribution for all-electric ships [3]. These distribution systems are also used in energy distribution systems with greater power, such as HVDC offshore wind farms [4]. \nOne of the main problems of connecting different \nregulated power converters to the same DC distribution bus is the degradation of stability that can occur due to the interactions among the converters. Although each subsystem is independently designed to be standalone stable. There are several methods for determining systems interconnection stability, including the Middlebrook criterion or the Gain Margin and Phase Margin criterion (GMPM) [5]. All of the above-mentioned criteria require the use of small signal input and output impedance models since the connection of two \nindividually stable systems can be modeled as the serial \nconnection of the small signal output impedance of the source system Z\nS and the small signal input impedance of the load \nsystem Z L Fig. 1 the stability in this interconnection is \ndetermined by the open loop expression (1). In order to do the stability analysis in this paper the GMPM criterion will be used. The GMPM criterion ensures that when the conditions of (2) are met the connection of the two systems will be stable, these conditions are equivalent to the Nyquist contour does not encircle the (-1, 0j) point (since in this system there are no poles on the right half plane), as long as there are no poles on the right half plane. \n \nFig. 1. Connection model o ftwo systems based on their input and \noutput impedances \nܶை (ܼ=)ݏ ௦()ݏ\nܼ(ݏ)( 1) \nWhen ‖ܼ௦‖≪‖ܼ‖ must be satisfied that\n|arg(ܼௌ)− arg (ܼ )|≤ 180° (2) \nFor all of the above mentioned it is possible to deduce that \nin order to obtain stability it is necessary to know accurately the closed loop impedances of all the components of the system. \nThere are different methods to obtain the closed loop input \nor output impedance of a power converter such as analytical form or experimental obtaining by means of a Frequency Response Analyzer (FRA) [6], [7]. In recent years other ways of obtaining impedance online have been studied, such as using Pseudo Random Binary Sequence (PRBS) [8], [9]. \nThis article focuses on the study of the accuracy of the \nclosed-loop input impedance model, using the model that assumes that a controlled converter behaves as a constant power load. \nThis article is organized as follows. Section II presents the \ncomplexity of obtaining impedance analysis compared to obtaining the reduced order model. Section III shows the accuracy range of the model for different powers, controllers and components by simulation. Section IV shows the validation of the model through experimental results for real high-power converters. II.OBTAINING THE INPUT IMPEDANCE\nT\nhis section will show how difficult it is to obtain the \nanalytical closed loop input impedance, then present the \nreduced model to study and present its advantages. In this case the power converter to be studied is a DC-AC three-phase 2L-VSC. To obtain the theoretical impedance is considered a specific case in which the converter has only one current control loop and uses the DQ reference axes. \nA.Analytical Input Impedance\nThe impedance of analytical form is obtained from th\ne\nm\nodel of the Fig. 2 as it can be seen the total small-signal input \nimpedance ( ZiT) can be obtained as the parallel of input \ncapacitor impedance (Z Ci) and is the closed-loop input \nimpedance of the three-phase VSI ( Zi). As shown in (3) \nFig. 2. Three- phase VSI current controlle d \n்ܼ(ܼ=)ݏ()ݏ∙ܼ()ݏ\nܼ(ܼ+)ݏ ()ݏ (3) \n The analytical derivation of the input impedance Zi can be \ndone using the small signal equivalent model shown in Fig. 3. \nFig. 3. Small signal input port of three- phase VSI \n As can be seen in [10], the control variables DQ can be \ndecoupled, obtaining two first-order systems with a single input and a single output (SISO system). As a consequence, the input impedance of the inverter controlled in DQ in closed loop is determined by \nܼ(=)ݏ 2\n3∙1\nܫௗ∙ܩௗௗ௩(ܦ+)ݏ ∙ܩௗ௩(ܫ+)ݏ ∙ܩௗ௩(ܦ+)ݏ ∙ܩ௩()ݏ(4) \nWhere Id, Iq, Dd and Dq are the current and duty cycle \nmagnitudes in the operation point. The transfer functions are \ndefined by \nܩௗ௩(=)ݏ 1\nݎ+ܮݏ ൫ܩܮ߱௩(ܸ+)ݏܩௗௗ௩(ܦ+)ݏௗ൯ (5) \nܩ௩(=)ݏ 1\nݎ+ܮݏ ൫−ܩܮ߱ௗ௩(ܸ+)ݏܩௗ௩(ܦ+)ݏ൯ (6) \nܩௗௗ௩(=)ݏ−1\nܸቀܦௗ+ܩூ()ݏ∙ܩௗ௩(ܩܮ߱+)ݏ ௩()ݏቁ (7) ܩௗ௩(=)ݏ−1\nܸቀܦ+ܩூ()ݏ∙ܩ௩()ݏ−ܩܮ߱ ௗ௩()ݏቁ (8) \n As can be seen the calculation of the expression of the input impedance of the three-phase inverter can be quite time-consuming due to the need to solve the system of equations shown above (5-8). \nB.Experimental Frequency Response Analizer\nIt is possible to obtain experimentally the value of the\nclosed-loop input impedance by means of a FRA, [6]. This \nmethod requires very expensive equipment, it is also problematic to measure high voltage and power equipment, because these converters normally have a very large input capacitor, so in order to see the high frequency response is necessary to introduce a disturbance of a very large amplitude \nin the input voltage. \nC.Analytical input impedance\nThe model shown below reduces the complexity of\nobtaining the input impedance. Considering that the inverter \nbehaves as a constant power load (CPL) due to the action of the control loop below its bandwidth, this behavior can be modeled as a negative resistance R\nCPL which value is given by \n(9). At high frequencies the behavior is dominated by the input capacitor C\ni, therefore the total input impedance is given by \n(10). \nFig. 4. Proposed model of input impedance of three phase VSI\nܴ=−ܸଶ∙ߟ\nܲ(9) \n்ܼ(ܴ=)ݏ \n1+ܴ ∙ܥ∙ݏ (10) \n As can be seen, this reduced model has certain advantages over the traditional analytical model, such as the quick obtaining from easily measurable parameters or even those present in the datasheets, avoiding complex analytical obtaining techniques and high-cost measurements, which is especially noticeable in high-power converters. \nThis input impedance model has already been validated for \nDC-DC converters [11], [12], and now it is wanted to checkits accuracy for DC-AC converters.\nIII.S\nTUDY OF THE MODEL ACCURACY\nTo\n check the accuracy range of the model several tests with \ndifferent characteristics have been carried out (using the system present in Fig. 2.) to see if the model fits properly with the real impedance. For all cases, the following considerations have been taken into account when selecting the components: \n• The input capacitor has been calculated to have an\ninput voltage ripple of less than 5% of the totalvoltage.\n• For the mains connection filter only one inductanceper phase is used, which has been calculated to ha\nve\na\n 10% ripple .\n• Switching frequency stays constant at 10 kHz \nThe above considerations have been chosen because they \ngive the smallest possible capacitor in a commercial converter with a correct design. When this capacitor is oversized it adequately filters out the effect of the control on the input \nimpedance. Doing the following studies with the smallest \npossible capacitor will provide the most critical working point for the model \nA. Comparison of different nominal powers \n \nBelow is shown a comparison of the input impedance \nfor 3 different inverters with current control (PI) in the DQ frame with their components calculated for different rated powers. The Fig. 5-7 show both the total impedance ( Z\niT) \n(parallel of the inverter with the capacitor) and the impedance \nof the inverter without the capacitor ( Zi). \nThe reduced model has also been added to compare it with the \nresults. As it can be seen, the model fits perfectly to the impedance \nobtained by simulation, since the feedforward applied with the \ninput voltage makes the impedance without the capacitor practically constant. So, the model would still adapt if the input capacitor was smaller. \nB. Comparison of different controller bandwidth \nFor the next study the control bandwidth has been varied \nto see if it influences the shape of the input impedance. In all cases the converter with a nominal power of 5 kW (Fig. 5) has been used. As can be seen in Fig. 8 when a control with a reasonable bandwidth is used, the model still fits properly to the impedance obtained. On the other hand, when the bandwidth is considerably reduced Fig. 9 it can be observed how the model begins to distance from the obtained result. Although the difference in magnitude and phase is not so significant as to produce instability in the distribution bus due to this small inaccuracy in the model. \n \nFig. 5. Closed loop input impedance for 5 kW of rated power. \n(Lfilter=1 mH, Cin=24 µF, PI Gain=1, PI TimeCte =14.3e-3, R CPL=-98 Ω) \n \nFig. 6. Closed loop input impedance for 40 kW of rated power. \n(Lfilter=0.3 mH, Cin=80 µF, PI Gain=0.3, PI TimeCte =4.3e-3, R CPL=-12.25 Ω)\nFig. 7. Closed loop input impedance for 150 kW of rated power. \n(Lfilter=0.06 mH, Cin=270 µF, PIGain=0.06, PITimeCte=8.75e-4, \nRCPL=-3.27 Ω) \n \nFig. 8. Closed loop input impedance for 5 kW of rated power and a \nbandwidth of 160 Hz. \n \nC. Comparison with different reference frame, αβ \nIn this case, the behavior of the converter is tested when \ndifferent reference axes are used ( αβ). In this case a \nProportional-Resonant control with a pair of complex poles \nconjugated at the fundamental frequency of the network [13] \nhas been used to perform the control. \n As can be seen in Fig.10 the model fits with sufficient \nprecision to the results obtained. \nD. Effect of input voltage feedforward \nIt has been observed that all the changes made in the \nprevious studies are practically inappreciable with respect to the input impedance thanks to the decoupling of the feedforward from the sensed input voltage. In this study, the feedforward will be modified to see how it affects in the impedance. \nAs can be seen in Fig. 11 and Fig. 12 the input impedance \ndiffers a lot from the proposed model, being more evident when using the αβ reference axes. It is possible to observe how \nat very low frequencies the constant power load behavior is still maintained, but the fact of making the feedforward constant means that there is no almost constant impedance after the capacitor at medium frequencies and therefore its effect on the total impedance is seen. \nTo make a study with a more real case, it has been assumed \nthat in a real design of an inverter has chosen an input voltage sensor without a great bandwidth and is applied a low pass filter in the control to filter noise, finally obtaining a real measure with a bandwidth of 1 kHz.\n \nIn the first case, the sensed value of the feedforward has \nbeen replaced by a constant value. This study allows us to understand how feedforward affects the system. Later it will be studied what effect the feedforward has on the impedance \nFig. 9. Closed loop input impedance for 5 kW of rated power and a \nbandwidth of 15 Hz \n \nFig. 10. Closed loop input impedance for 5 kW of rated power using the \nαβ reference frame \n \nFig. 11. Closed loop input impedance for 5 kW of rated power. In DQ \nusing a constant feedforward ( Lfilter=1 mH, Cin=24 µF, PI Gain=1, \nPITimeCte =14.3e-3 ) \nFig. 12. Closed loop input impedance for 5 kW of rated power. In αβ \nusing a constant feedforward ( Lfilter=1 mH, Cin=24 µF) \nlimiting the bandwidth of the sensed input voltage, something \nthat happens in real converters. \nAs can be seen in Fig.13 and Fig. 14 the fact of reducing \nthe feedforward sensing bandwidth visibly affects the input impedance, making it different from the model, which may cause instability in the system.\n \nIV. EXPERIMENTAL RESULTS \n In this section, impedance measurements have been \nmade with a real high power 2L-VSC, and closed loop input impedances have been measured with a method equivalent to that used by commercial FRAs. The simplified connection diagram is shown in Fig. 15 to obtain the impedance a disturbance has been introduced in the DC reference voltage \nof the EGSTON grid emulator. The DC input voltage in the \nconverter has been sensed with a differential probe, very close to the input of the converter so that it does not affect the inductance of the transmission cables The current from the DC port to the inverter has been sensed with an amperimetric \nclamp. The results of the measurements have been recorded \nwith an oscilloscope, because it is the instrument that offers more precision and later have been processed by FFT \ntechniques, to obtain the magnitude and phase of the input \nimpedance \nSince this is a commercial converter that can handle a \npower of up to 60 kW, it is not possible to access the current after the capacitor, or modify the input capacitor, which is a little oversized. Unlike the previous studies this converter has an LCL filter at the output, and therefore has implemented in the control an Active Damping. Table I shows some of the parameters of the inverter. \nTABLE . I. REAL CONVERTER PARAMETERS \nParameter Value \nCin 14.1 mF \nRcin 0.005 Ω \nLfiltconv 0.5 mH \nCfilt 50 µF \nLfiltGrid 0.2 mH \nA. Comparison of different output powers \nIn this case the tests were done with a current control on \nthe DQ reference axis, these tests were done for 2 different \npowers, 2.1 kW and 21 kW. The result can be seen in the Fig. \n16. \n \nFig. 14. Closed loop input impedance for 5 kW of rated power. In αβ \nusing a feedforward with a 1 kHz of bandwidth. (Lfilter=1 mH, \nCin=24 µF) \n \nFig. 13. Closed loop input impedance for 5 kW of rated power. In DQ \nusing feedforward with a 1 kHz of bandwidth. (Lfilter=1 mH, Cin=24 µF)\n \nFig. 15. Setup to perform impedance measurements with the real 2L-VSC.\n \nFig. 16. Experimental results of the input impedance for the same \nconverter working at different powers. \nB. Comparison of different controls \nIn the following test different controls have been tested, in \ntwo of them the time constant τi of the PI controller has been \nvaried. The results are shown in Fig. 17. As it can be seen, the model fits properly to the results obtained for the two PI controls. From the experiments finally shown it can be concluded that if the input capacitor of the converter is oversized the results of the real impedance will match the model. \nV. \n CONCLUSIONS \n In this article the closed-loop input impedance of a \nDC-AC 2L-VSC converter has been studied for different operation cases, the most critical cases (smaller input capacitor) have been studied by means of simulation. It has been possible to observe how the main factor that influences in the impedance is the feedforward, checking how with a \nstandard bandwidth in the feedforward the input impedance \nstarts to differ quite a lot from the model. From the experimental results it can be seen how the fact of having an oversized capacitor benefits the precision of the model. \n \n \n \n A\nCKNOWLEDGMENT \nThis work is partially supported by the European Regional \nDevelopment Fund, the Ministry of Science, Innovation and Universities and the State Research Agency through the \nresearch project “Modelling and control strategies for the \nstabilization of the interconnection of power electronic converters” (FEDER/Ministerio de Ciencia, Innovación y \nUniversidades – Agencia Estatal de Investigación/ _Proyecto CONEXPOT-2 (DPI2017-84572-C2-2-R)). It also has been partially supported by the Research Agreement between the \nUniversidad Carlos III de Madrid and the Fundación IMDEA \nEnergía. In addition, it has been also supported by the research project PERSEID (04.022-2018) using the ERIGrid Research Infrastructure and is part of a project that has received funding from the European Union’s Horizon 2020 \nResearch and Innovation Programme under the Grant \nAgreement No. 654113. The support of the European Research Infrastructure ERIGrid and its partner SINTEF is very much appreciated. \nR\nEFERENCES \n[1] I. Aharon and A. Kuperman, “Topological Overview of Powertrains for \nBattery-Powered Vehicles With Range Extenders,” IEEE Trans. Power \nElectron. , vol. 26, no. 3, pp. 868–876, 2011. \n[2] C. R. Avery, S. G. Burrow, and P. H. Mellor, “Electrical generation and \ndistribution for the more electric aircraft,” 42nd Int. Univ. Power Eng. Conf. , \npp. 1007–1012, 2007. \n[3] T. V Vu et al. , “Predictive Energy Management for MVDC All- Electric \nShips,” IEEE Electr. Sh. Technol. Symp. , pp. 327–331, 2017. \n[ 4 ] C . M e y e r , M . H ö i n g , A . P e t e r s o n , a n d R . W . D e D o n c k e r , “C o n t r o l an d \nDesign of DC Grids for Offshore Wind Farms,” IEEE Trans. Ind. Appl. , vol. \n43, no. 6, pp. 1475–1482, 2007. \n[5] A. Riccobono and E. Santi, “Comprehensive review of stability criteria for \nDC power distribution systems,” IEEE Trans. Ind. Appl. , vol. 50, no. 5, pp. \n3525–3535, 2014. \n[6] M. Liu, H. Yuan, and Y. Sun, “Research on Measurement of Dc Power \nSupply Impedance,” 2009 9th Int. Conf. Electron. Meas. Instruments , pp. 2–\n706, 2009. \n[7] Y. Panov and M. Jovanovi ć, “Practical issues of input/output impedance \nmeasurements in switching power supplies and application of measured data to stability analysis,” Conf. Proc. - IEEE Appl. Power Electron. Conf. Expo. \n- APEC , vol. 2, pp. 1339–1345, 2005. \n[8] T. Roinila, M. Vilkko, and J. Sun, “Online grid impedance measurement \nusing discrete-interval binary sequence injection,” IEEE J. Emerg. Sel. Top. \nPower Electron. , vol. 2, no. 4, pp. 985–993, 2014. \n[9] M. A. Granda, C. Fernandez, P. Zumel, A. Fernandez-Herrero, and A. \nBarrado, “Online impedance measurement of cascaded DC/DC converters,” Conf. Proc. - IEEE Appl. Power Electron. Conf. Expo. - APEC , vol. 2019–\nMarch, pp. 1351–1356, 2019. \n[10] A. Riccobono and E. Santi, “Positive feedforward control of three-phase \nvoltage source inverter for DC input bus stabilization with experimental \nvalidation,” IEEE Trans. Ind. Appl. , vol. 49, no. 1, pp. 168–177, 2013. \n[11] M. Sanz et al. , “Low-cost input impedance estimator of Dc-to-Dc converters \nfor designing the control loop in cascaded converters,” Conf. Proc. - IEEE \nAppl. Power Electron. Conf. Expo. - APEC , vol. 2016–May, pp. 3090–3096, \n2016. \n[12] M. Sanz et al. , “Simple input impedance converter model to design \nregulators for dc-distributed system,” 2016 IEEE 17th Work. Control Model. \nPower Electron. COMPEL 2016 , 2016. \n[13] I. R. Yazdani Amirnaser, “VSC Current Control,” in Voltage Sourced \nConverters in Power Systems , 2010. \n \nFig. 17. Experimental results of the input impedance for the same \nconverter working at 23 kW with different controllers. \n" }, { "title": "2401.17145v1.Moment_Tensor_Based_Constant_Potential_Modeling_of_Electrical_Double_Layers.pdf", "content": " \n1 \n Moment -Tensor -Based Constant -Potential Modeling of \nElectrical Double Layer s \nZhenxiang Wang1,4, Ming Chen1,4, Jiedu Wu2, Xiangyu Ji1, Liang Zeng1, Jiaxing Peng1, \nJiawei Yan2, Alexei A. Kornyshev3, Bingwei Mao2, and Guang Feng1* \n1State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, \nHuazhong University of Science and Technology, Wuhan 430074, China \n2State Key Laboratory of Physical Chemistry of Solid Surfaces, and Department of Chemistry, \nCollege of Chemistry and Chemical Engineering, Xiamen University, Xiamen, China. \n3Department of Chemistry, Faculty of Natural Sciences, Imperial College London, Molecular \nSciences Research Hub, White City Campus, W12 0BZ, London, UK. \n4These authors contributed equally: Zhenxiang Wang, Ming Chen. \n*Correspondence: gfeng@hust.edu.cn \nConstant -potential molecular dynamics (MD) simulations are indispensable for understanding the \ncapacitance, structure, and dynamics of electrical double layer s (EDL s) at the atomistic level. \nHowever, the classical constant -potential method, relying on the so -called ‘floating charges ’ to \nkeep electrode equipotential, overlooks quantum effects on the electrode and always \nunderestimates EDL capacitance for typical electrochemical systems featuring metal electrodes in \naqueous electrolytes. Here, we propose a universal theor etical frame work as moment -tensor -based \nconstant potential method (mCPM) to capture electronic structure variations with electric moments . \nFor EDLs at Au(111) electrodes, mCPM -based MD reveals bell -shaped capacitance curves in \nmagnitude and shape both quantitatively consistent with experiments. It further unveils the \npotential -dependent local electric fields, agreeing with experimental observations of redshift \nvibration of interfacial water under negative polarization and predicting a blueshift under positive \npolarization, and identifies geometry dependence of two time scales during EDL formation. \n2 \n Introduction .– The electrical double layer (EDL) at the electrode -electrolyte interface is \nubiquitous in electrochemistry1. Delving into the intricate micro structure and dynamics of EDL s \nat the molecular scale helps to reveal pivotal mechanisms that determin e electrochemical device \nperformance , paving the way for transformative breakthroughs in advanced applications including \nelectric energy storage, electrocatalysis , and capacitive deionization2-4. Molecular dynamics (MD) \nsimulations, with their ability to sample phase space5, have emerged as an indispensable tool for \nscrutinizing such nanoscale interfaces . A salient challenge in employing MD simulations for EDLs \nlies in adequately capturing the electronic response of electrodes to external field s6,7, relying on an \nextended degree of freedom on the electronic structure . The constant potential method (CPM), \nwith fluctuating charges on nuclei of electrode atoms subject to an equipotential constraint on the \nelectrode , effectively reflect s varying electrode electronic structures8,9 and has been extensively \nutilized in elucidating both equilibrium and dynamic processes in various electrochemical \nsystems10,11. \nA long -standing issue for MD simulations with classical CPM (cCPM) is the severe \nunderestimation of EDL capacitance in typical electrochemical systems with metal electrodes1,12,13. \nThe sharp discrepancy suggests the incomplete description of electrified solid -liquid interfaces due \nto classical electrostatics for cCPM, omitting quantum effects on electrodes14,15. Modifications \nbased on a semi -classical model , linking the self-energy of electrode atom s to metallicity , show \nthat decreasing the electrode atom ‘hardness ’ could increase the capacitance; however, an overly \nlow hardness w ould cause unstable simulation due to the polarization catastrophe16. Ab initio \nmolecular dynamics (AIMD) modeling with the rigorous description of the electronic structure \nindicates that high EDL capacitance depends strongly on the interfacial dipole induced by \nchemisorbed water17, which cannot be observed by classical MD simulations17. Nevertheless, ab \ninitio methods are still not well applicable to either an overall EDL structure or charging dynamics \nbecause of the limited spatial and temporal scales15. \nIn this Letter, a theory for cCPM is promoted by introducing multipole moment tensors to \ndescribe the variation in the electrode electronic structure (i.e., induced charges18), termed \nmoment -tensor -based CPM (mCPM). The moment tensors in mCPM are extracted from the spatial \ndistribution of induced charges through density functional theory (DFT) calculations. We then \nconducted comparative analys es involving the cCPM and developed mCPM on the capacitance, \n3 \n structure, and dynamics of EDL s at interfaces between Au(111) electrode s and aqueous \nelectrolyte s. The mCPM -based MD predicts a bell -shape d potential -capacitance curve with a much \nhigher magnitude than cCPM , aligning quanti tatively with experiment al measurements . It further \nunveils the origin of the water and ions in EDLs , responding to the external field , and observes the \ntwo-stage charging dynamics process composed of ion electromigration and bulk diffusion. \nFrom c CPM to mCPM .– The distribution of induced charges on the electrode inherently exhibits \nnon-uniform ity [Fig. 1 (a)], dynamically responding to pertu rbations from electrolyte ions and \nsolvent molecules6. However, cCPM , adopt ing nucleus -centered charges , only describe s the \nmagnitude of induced charges and neglects the electrostatic interactions due to asymmetric \nelectronic structure19. To giv e a complete description of electrostatic interactions involving \ninduced charges, both net charge and electric moments are required according to the multipole \nexpansion of the electrostatic potential. Therefore, mCPM with moment tensors is proposed to \ndepict electrostatic interactions from both the amount and the distribution of induced charges . \nSpecifically, induced charge s are divided and expanded into 𝑁 multipole s, each consist ing of \nfluctuating multipole moment tensors [Fig. 1 (b)] as \n𝑴(𝑖)={𝑄(𝑖),𝝁(𝑖),𝜣(𝑖),…} (1) \nwhere 𝑄(𝑖) is the ith monopole ( net charge ), and 𝝁(𝑖), 𝜣(𝑖)are the dipole, and quadrupole moment \ntensors, respectively. \nAs only the 0th rank tensors 𝑸={𝑄(1),…,𝑄(𝑁)} were employed to describe the electrostatic \neffect of the net charges , neglecting the effects of electric moments , Eq. 1 could reduce to the \ntheory of cCPM, so that cCPM is a ‘first approximation ’ to the problem . In comparison, the \ninduced charges in mCPM are represented by multipole tensors 𝑴={𝑴(1),…,𝑴(𝑁)}, including \nboth net charges and electric moments. Therefore, within this framework, the extended \nHamilton ian of the MD simulation system becomes14: \n𝐻(𝒓,𝒑,𝑴)=𝑇(𝒑)+𝑈𝑐𝑜����𝑙(𝒓,𝑴)+𝑈0(𝒓) (2) \nwhere 𝑇 is the kinetic energy, 𝑈𝑐𝑜𝑢𝑙 is the electrostatic energy, 𝑈0 includes bond energy and van \nder Waals energy ; 𝒓 and 𝒑 are the positions and momenta of the atoms , respectively . The moment \ntensors are solved by minimizing 𝑈𝑐𝑜𝑢𝑙(𝒓,𝑴)−∑ Ψ(𝑖)𝑄(𝑖) 𝑁\n𝑖=1 to satisfy the equipotential \nconstraints , Ψ(𝑖), on the electrode . \n4 \n \nFIG. 1. Schematics of mCPM. (a) Schematic of induced charge in electrode -electrolyte interface. Gray \nspheres represent electrode nuclei. (b) Multipole expansion of induced charge. Red and blue isosurfaces \nrefer to the positive and negative induced charges, respectively. (c) Schematic of induced charge distribution \nin metal electrodes and electrolytes. \nThe mCPM theory constitutes a versatile framework to characterize the induced charges \nefficiently and completely, which is applicable to diverse electrochemical systems. As this Letter \nfocus es on metal -liquid interface s, the combined effect of monopole and dipole could sufficiently \ndescribe the electrostatic effect of induced charge s18,20. Therefore, e ach moment series is denoted \nas 𝑴(𝑖)={𝑄(𝑖),𝝁(𝑖)}, and the electrostatic potential at a specific location 𝐫 due to the electrical \nmoments can be written as: \nϕ(𝐫)=∑(𝑄(𝑖)\nr−𝝁(𝑖)·𝐫\nr3 )𝑁\n𝑖=1(3) \nwhere\n𝝁(𝑖)=𝑄(𝑖)𝐫D (4) \n𝑄(𝒊) and 𝝁(𝒊) are the monopole and dipole of the ith part of the induced charge, respectively , and 𝐫D \nis the displacement vector of the dipole. Herein, the monopole term is represented by Gaussian \ncharge , 𝜌(𝑖)(𝐫)=𝑄(𝑖)(𝜋𝜎2)−3/2𝑒−𝐫2/𝜎2, with a width , 𝜎, reflecting the atom hardnes s [Fig. 1 (c)]. \nDetailed theoretical derivation is given in Section 1 of the Supplemental Material (SM) . \nTo obtain 𝜎 and 𝐫D for the electrochemical system with Au(111) electrode , DFT calculations \nare performed based on both explicit ion absorption and implicit solution (Section 2 of SM). The \nelectrode\nmonopole\n- -\n++ +-\n+-+\ndipole quadrupole+…nucleusinduced\ncharge(a)…\nelectrolytemultipole\nexpansionelectrolyte\n…\nelectrode(b)\n(c) \n5 \n induced charges on the metal electrode exhibit a similar shape under different polarization \nconditions , providing the distribution width (0.042 nm) and displacement vector of the dipole \n(0.102 nm pointing out straight from electrode surface ) for the Au(111) electrode (details see \nSection 3 of SM and Figs. S2 -3). \nEDL capacitance and structure .– The EDL capacitance was examined in a benchmark \nelectrochemical system consist ing of 2 M NaClO 4 electrolyte confined between two atomic flat \nAu(111) electrodes (Fig. S4). As illustrated in Fig. 2 (a), cCPM -MD yielded a n almost flat \ndifferential capacitance profile at approximately 6 µF cm-2, consistent with prior CPM -MD \nsimulation s12. In contrast , a distinct bell-shaped curv e with a significantly higher magnitude was \nfound with mCPM [Fig. 2 (a)]. Meanwhile, our electrochemical impedance spectroscopy \nmeasurements on single -crystal Au(111) electrode s with the same electrolyte (Section 2 of SM) \nalso demonstrate a bell -shaped curve ranging from 24 to 41 µF cm-2, quantitatively consistent with \nmCPM predictions and compatible with previous experiments on Au and Pt electrodes21. These \nagreements prove the pivotal role of electric moments in reshaping the differential capacitance. \n \nFIG. 2. Differential capacitance and its origin . (a) Potential dependence of the capacitance of Au(111) in 2 \nM NaClO 4 solution derived from experiments, mCPM , and cCPM . (b) Accumulative number densities of \nthe first ion layer under various potentials. (c) Excess polarization of interfacial water at different potentials. \n0 V refers to the potential of zero charge. \n The origins of the differential capacitance curve could be ascribed to the EDL structure, as \nearlier theoretical studies ascribe the capacitance hump around the potential of zero charge to the \nreorientation of interfac ial water molecules22, while the mean field theory suggested that the close -\npacked ions and solvents would result in nonlinear ion absorption and therefore nonmonotonic \ndifferential capacitance23,24. Therefore, the potential -dependent ion absorption/desorption and \ninterfacial water polarization are compared thorough ly between cCPM and mCPM to understand \ntheir different ial capacitance curves . In cCPM , the accumulative number of cation s in the EDL \n(a) (b) (c) \n6 \n exhibits a nearly linea r variation under different potentials, with anion s remain ing almost \nunchanged [Fig. 2 (b) and Fig. S5(a,b)]. Simultaneously , the interfacial water, acting as dielectric \nmedium to screen electric field and contribute to charge storage25, exhibits linear growth of excess \npolarization with increasing potential [Fig. 2 (c)]. These weaker and linear responses lead to a flat \nand small capacitance . In sharp contrast, mCPM reveals a fund amentally different charge storage \nmechanism that cation absorption and desorption are much stronger than those in cCPM [Fig. 2(b) \nand Fig. S5(c,d)]. The interfacial water also shows stronger excess polarization under negative \npolar ization and weakly pos itive polar ization (<0.3 V) compared to cCPM [Fig. 2 (c)]. These \ndistinct EDL structures account for the much higher capacitance observed in mCPM. Beside s, the \nvariation in interfacial cation number weake ns under stronger polarization [Fig. 2(b)] and excess \npolarization of interfacial water grows subtly when the potential exceeds 0.3 V [Fig. 2 (c)], \ndecreasing the capacitance. Therefore , induced electric moments are indispensable in determining \nthe EDL structures , and the synergistic effects of ion absorption/desorption and water polarization \ncontribute to the more precise bell-shaped capacitance . \nNotably, previous MD simulations26,27 also reproduced non -linear capacitance curves with \nthe modified constant charge method (CCM) where the electrode polarization is represented by \noff-center constant charges . The change in capacitance only arises from the different reference \nplane of the electrode potential (Fig. S6-7), rather than different EDL structures, proving that off -\ncenter CCM cannot reflect the variations in the electronic structures of electrode s. \nInterfacial water .– Potential -dependent water st ructure at the charged surface is crucial for \nunderstanding interfacial phenomena28,29. In cCPM -MD, water absorption remains unaffected by \nthe electrode potential [Fig. 3(a)]. In comparison , the water layer observed in mCPM -MD is \nstrongly depend ent on the electrode potential . A notable amplification in interfacial water \nadsorption is observed with increasing negative pol arization, displaying closer adsorption and \nnearly tripl ing the height of the first adsorbed peak from 0 V to -0.5 V [Fig. 3 (b) and Fig. S8(a-e)]. \nThis pronounced effect can be attributed to the strong absorption of hydrated cations (Fig. S5). \nRegarding orientation al distribution , cCPM elucidates a water configuration nearly parallel \nto the electrode surface with negligible dependence on electrode polarization [Fig. 3(c), and Fig. \nS8-9]. In contrast , mCPM reveals a markedly distinct motif for interfacial water [Fig. 3 (d)]. Under \n0 V , the dipole orienta tion of water molecules displays two peaks (at ~105° and ~130°), indicating \n7 \n a coexisting parallel and perpendicular orienta tion with one hydrogen atom pointing to the surface \n(i.e., H-down configuration ). As the potent ial become s negative, more water molecules exhibit an \nH-down structure . Specifically, interfacial water with parallel configuration almost disappears at \npotentials below -0.3 V [Fig. 3(d)], with only a narrow peak of dipole or ientation locat ed at ~130°, \nclosely align ing with AIMD simulation17 and SHINERS experiment29. These findings prove that \nboth strong water electrosorption and water polarization exist under negative polarization in \nmCPM , which is prone to water e lectrolysis30, while cCPM gives a contrary hint. As the app lied \npotential turns positive, the dipole orientation of water molecules undergoes a transition from \n~130o to 109o and eventually drops below 90o [Fig. 3 (d)], indicating the occurrence of O-down \nconfiguration, consistent with previous sum frequency generation measurement s and DFT \ncalculations31,32. Detailed interfacial water structures see Section 6 of SM. \n \nFIG. 3. Structure of interfacial water and electric field. (a -b) Number densit ies of water as a function of \ndistance from the electrode surface nuclei (z) under various potentials (U). (c -d) Dipole orientations of \ninterfacial water. θdipole is defined as the angle between the normal of electrode surface and the water vector. \n(e-f) Interfacial electric field and H -bond of interfacial water . Top an d bottom panels refer to cCPM and \nmCPM, respectively. \nIt is of particular interest to uncover the potential -dependent local electric field experi enced \nby interfacial water, which correlate s with OH -stretching frequency shifts based on Stark effect33. \ncCPM exhibits a nearly linear increase in the magnitude of local electric field, and a linear growth \nin the number of H -bonds with growing electrode potential [Fig. 3(e)], corres pond ing with the \nlinear variation in excess polarization and cation absorption/desorption. Intriguingly , mCPM \ncCPM\nmCPM(c) (e)\n(b) (d) (f)(a)\n \n8 \n reveals two transition s in the slope of the local elect ric field at approximately ±0.3 V, delineating \nthree Stark tuning range s [Region I: -0.3 ~ 0.3 V; Region II: < -0.3 V ; Region Ⅲ: > 0.3 V , as \ndelineated in Fig. 3(f)]. The transition at -0.3 V is in accordance with the SHINERS experiment \non Pd/Au -NaClO 4 interface s, identif ying two Stark tuning rates for the interfacial water OH stretch \nmode with the transition at -0.31 V29. Furth ermore, the slope of the potential -dependent local \nelectric field in Region I is steeper than in Region Ⅱ, consistent with SHINERS measurements \nindicating a greater redshift in Region I than in Region II29. Simultaneously , the number of H -\nbond s exhibits a similar potential -dependent trend , quickly decreasing in Region I and \nsubsequently gradually decreasing below -0.3 V (Region II) . These transitions coincide with the \nvanishment of water molecules with parall el configuration under strongly negative polarization, \nwhich is essential to understand ing the electrode reaction path and rate34. Under positive \npolarization, the local electr ic field decreases with increased polarization, implying a blueshift in \nOH stretching mode. The transition in the slop at 0.3 V s ignif ies a Stark tuning rate transition , \nnecessitat ing experimental verification. \nCharging dynamics .– The d ynamics of EDL formation determine the power density of \nelectrochemical devices35. Typical ly when a voltage is applied , there are three time scales of the \ncharging process36. Earlier theoretical studies propose that the charging process consists of the \nDebye time (𝜆𝐷2/𝐷 ) associated wtih the relaxation within EDL and the RC time (𝜆𝐷𝐿/𝐷 ) \nrepresenting ions entering /leaving the EDL37, where 𝜆𝐷 is Debye length, 𝐷 is ionic diffusivity, and \n𝐿 is half of the electrode separation. Recent analytical solutions to the Poisson -Nernst -Planck (PNP) \nequations introduce the bulk diffusion time (𝐿2/𝐷) reflecting bulk electrolyte filling in the depleted \nzones after ion electromigration in the RC time, and conclude that the Debye time is a small \nperturbation in the RC time which dominates the charging process of thin EDLs (𝐿≫𝜆𝐷)36,38. \nNotably, the charging process revealed by molecular simulatio n are several orders of magnitude \nfaster than experiments39, making it urgent to accurately capture relevant time scales and their \ngeometry dependence . \nWe s crutinized the charging dynamics of the EDL system of two parallel Au(111) electrodes \nembedded into 2 M NaClO 4 electrolyte (𝜆𝐷≈ 0.2 nm and 𝐷≈ 1.2×10-9 m2 s-1, Fig. S4 ) with \nelectrode separations of 10, 30, 60, and 100 nm under a voltage jump of 1 V by both cCPM and \nmCPM. Simulated by cCPM [Fig. 4(a)], an immediate charge accum ulation occurs within ~10 ps \n9 \n after the voltage jump, succeed ed by a more gradual charging process. The latter becomes \ndominant with larger e lectrode separation exceed ing 30 nm [Fig. S10(a)]. Conversely, by mCPM \n[Fig. 4(b) and Fig. S10(b)], the charging curves are composed of a typical double exponential \nprocess with a longer relaxation time , where the initial stage constitutes the predominant \ncomponent of the net charge. \n \nFIG. 4. Charging process. (a-b) Time evolution of net charge at positive electrode after a voltage jump of \n1V applied for systems with 10 nm electrode separation. Black dashed lines represent double exponential \nfitting results. (c-d) Fast time scales with different electrode separations . Blue dotted line is a linear fitting \nresult (R2 = 0.998). (e-f) Slow time scales with different electrode separations. Blue dotted curve is a \nparabolic fitting result (R2 = 0.951). Top and bottom panels refer to cCPM and mCPM, respectively. \nTo quanti tatively investigate the two -stage charging , the charging curves are fitted with the \ndouble exponential function36: \n𝑄(𝑡)=𝑄∞[1−𝐴exp (−𝑡\n𝜏1)−(1−𝐴)exp (−𝑡\n𝜏2)] (5) \nwhere 𝑄∞ is the charge density at equilibrium, 𝜏1 and 𝜏2 are time scale s (𝜏1<𝜏2), and 𝐴 is the \nweight coefficient of the fast time scale . For cCPM -MD [Fig. 4(c,e)], 𝜏1 approaches the Debye \ntime and changes subtly with electrode separation , suggesting dominance by rearrangement within \nEDL. 𝜏2 is at the level of RC time but it does not grow linearly with 𝐿, incompatible with any \ntheoretical predicted time scales . The fast stage still accounts for more than 20% of the total charge \n2L(nm)\n2L(nm)Time (ns)\nTime (ns)\nDebye time\nRC time\nbulk diffusion time\n2L(nm)\n2L(nm)(a)\n(b)(c)\n(d)(e)\n(f) \n10 \n storage with wide separations (Table S1), probably due to the underestimated capacitanc e by \ncCPM that amplifies perturbations from EDL rearrangement. For mCPM -MD [Fig. 4(d,f)], 𝜏1 \nincreases linearly with L [Fig. 4(d)], indicating dominance by ion electromigration . Meanwhile, \n𝜏2 grows proportionally to the square of the electrode separation, suggesting it is in the slow bulk \ndiffusion process. These are typical chara cteristics of thin EDLs according to the PNP model36,38, \nproving the accuracy of mCPM -MD in EDL charging dynamics . Moreover , the weight coefficients \ndemonst rate that the charging process in these systems cannot be modeled by a n RC circuit \nregardless of the thin EDL condition, as the bulk diffusion process constitutes ~15% of total charge \nstorage and remains stable with increasing separation (Table S1). Considering both RC time scale \nand bulk diffusion time scale, predictions of the power performance of electrochemical devices \nbecome achievable through mCPM -MD. \nDiscussion .– In this Letter, t he develop ed mCPM -MD theory and method, utilizing multipole \nmoment tensors to express electrostatic interactions from DFT -derived induced charges on \nelectrodes, could effectively address the long -standing obstacle of quantitative agreement between \nexperiments and atomistic modeling of ED Ls12-15. As for a prototype electrochemical system of \nAu(111) electrodes in aqueous electrolytes , mCPM -MD results quantitatively match experimental \nbell-shaped capacitance curves and exhibit potential -dependent local electric fields consistent with \nexperimental redshift vibration of interfac ial water29. Meanwhile, mCPM -MD notably predicts a \nblueshift in the water vibration under positive polarization , and also identifies the geometry -\ndependen t time scales in the charging process between parallel -plate electrodes, corresponding to \nion electromigration and bulk diffusion. This bridges the time scale between molecular simulations \nand macroscopic devices, providing valuable insights into EDL evolu tion. \nThis Letter advances the CPM theory to effectually consider quantum effects on the electrode, \nand then enabl es model ing EDL system in millions of atoms and captur ing variations in electronic \nstructures, taking advantage of classical MD simulation and DFT calculations . It also aids in \nunderstand ing interfacial phenomen a of other EDL -related fields, such as batteries4, \nelectrocatalysis2, and capacitive deionization3. \nThe authors in China acknowledge the funding support from the National Natural Science \nFoundation of China (T2325012, 52106090, 52161135104, and 22072123) and the Program for \nHUST Academic Frontier Youth Team. A.A.K. thanks the grant from the Engineering and Physical \n11 \n Sciences Research Council (EP/L015579/1). \n \n[1] W. Schmickler, Chemical Reviews 96, 3177 (1996). \n[2] M. F. Döpke, F. W. v. d. Meij, B. Coasne, and R. Hartkamp, Physical Review Letters 128, \n056001 (2022). \n[3] R. A. Rica, R. Ziano, D. Salerno, F. Mantegazza, and D. Brogioli, Physical Review Letters \n109, 156103 (2012). \n[4] J.-Q. Huang, Nature Nanotechnology 17, 680 (2022). \n[5] M. Salanne, J. M. Buriak, X. Chen, W. Chueh, M. C. Hersam, and R. E. Schaak, ACS Nano \n17, 6147 (2023). \n[6] A. Kornyshev, Electrochimica Acta 34, 1829 (1989). \n[7] G. Jeanmairet, B. Rotenberg, and M. Salanne, Chemical Reviews 122, 10860 (2022). \n[8] J. I. Siepmann and M. Sprik, The Journal of Chemical Physics 102, 511 (1998). \n[9] L. Zeng, T. Wu , T. Ye, T. Mo, R. Qiao, and G. Feng, Nature Computational Science 1, 725 \n(2021). \n[10] C. Merlet, B. Rotenberg, P. A. Madden, P. -L. Taberna, P. Simon, Y . Gogotsi, and M. Salanne, \nNature Materials 11, 306 (2012). \n[11] S. Bi et al. , Nature Materials 19, 552 (2020). \n[12] T. Dufils, G. Jeanmairet, B. Rotenberg, M. Sprik, and M. Salanne, Physical Review Letters \n123, 195501 (2019). \n[13] S.-J. Shin, D. H. Kim, G. Bae, S. Ringe, H. Choi, H. -K. Lim, C. H. Choi, and H. Kim, Nature \nCommunications 13, 174 (2022). \n[14] L. Scalfi, T. Dufils, K. G. Reeves, B. Rotenberg, and M. Salanne, The Journal of Chemical \nPhysics 153, 174704 (2020). \n[15] R. Sundararaman, D. Vigil -Fowler, and K. Schwarz, Chemical Reviews 122, 10651 (2022). \n[16] A. Serva, L. Scalfi, B. Rotenberg, and M. Salanne, The Journal of Chemical Physics 155, \n044703 (2021). \n[17] J.-B. Le, Q. -Y . Fan, J. -Q. Li, and J. Cheng, Science Advances 6, eabb1219 (2020). \n[18] N. D. Lang and W. Kohn, Physical Review B 7, 3541 (1973). \n[19] A. J. Stone and M. Alderton, Molecular P hysics 56, 1047 (1985). \n[20] M. Brack, Reviews of Modern Physics 65, 677 (1993). \n[21] T. Pajkossy and D. Kolb, Electrochemistry Communications 5, 283 (2003). \n[22] N. Mott and R. Watts -Tobin, Electrochimica Acta 4, 79 (1961). \n12 \n [23] A. A. Kornyshev, The Journ al of Physical Chemistry B 111, 5545 (2007). \n[24] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Physical Review E 75, 021502 (2007). \n[25] G. Monet, F. Bresme, A. Kornyshev, and H. Berthoumieux, Physical Review Letters 126, \n216001 (2021). \n[26] G. Feng, D. -e. Jiang, and P. T. Cummings, Journal of Chemical Theory and Computation 8, \n1058 (2012). \n[27] A. Shandilya, K. Schwarz, and R. Sundararaman, The Journal of Chemical Physics 156, \n014705 (2022). \n[28] A. Montenegro, C. Dutta, M. Mammetkuliev, H. Shi, B. Hou, D. Bh attacharyya, B. Zhao, S. \nB. Cronin, and A. V . Benderskii, Nature 594, 62 (2021). \n[29] Y.-H. Wang et al. , Nature 600, 81 (2021). \n[30] Q. Sun et al. , Nature Energy 8, 859 (2023). \n[31] H. Noguchi, T. Okada, and K. Uosaki, Faraday Discussions 140, 125 (2009). \n[32] P. Xu, A. D. von Rueden, R. Schimmenti, M. Mavrikakis, and J. Suntivich, Nature Materials \n22, 503 (2023). \n[33] C.-Y . Li, J. -B. Le, Y . -H. Wang, S. Chen, Z. -L. Yang, J. -F. Li, J. Cheng, and Z. -Q. Tian, Nature \nMaterials 18, 697 (2019). \n[34] G. Gonella et al., Nature Reviews Chemistry 5, 466 (2021). \n[35] S. Kondrat, P. Wu, R. Qiao, and A. A. Kornyshev, Nature Materials 13, 387 (2014). \n[36] M. Z. Bazant, K. Thornton, and A. Ajdari, Physical Review E 70, 021506 (2004). \n[37] A. Kornyshev and M. V orotyntsev, Physica Status Solidi (a) 39, 573 (1977). \n[38] M. Janssen and M. Bier, Physical Review E 97, 052616 (2018). \n[39] C. Lian, M. Janssen, H. Liu, and R. van Roij, Physical Review Letters 124, 076001 (2020). " }, { "title": "2401.17148v2.Entropy_and_curvature__beyond_the_Peres_Tetali_conjecture.pdf", "content": "arXiv:2401.17148v2 [math.PR] 9 Feb 2024Entropy and curvature: beyond the Peres-Tetali\nconjecture\nPietro Caputo, Florentin M¨ unch and Justin Salez\nFebruary 12, 2024\nAbstract\nWe study Markov chains with non-negative sectional curvatu re on finite metric\nspaces. Neither reversibility, nor the restriction to a par ticular combinatorial distance\nareimposed. Inthislevelofgenerality, weprovethata1-st epcontraction intheWasser-\nsteindistanceimpliesa1-stepcontraction inrelative ent ropy, bythesameamount. Our\nresult substantially strengthens a recent breakthrough of the second author, and has\nthe advantage of being applicable to arbitrary scales. This leads to a time-varying\nrefinement of the standard Modified Log-Sobolev Inequality ( MLSI), which allows us\nto leverage the well-acknowledged fact that curvature improves at large scales . We\nillustrate this principle with several applications, incl uding birth and death chains,\ncolored exclusion processes, permutation walks, Gibbs sam plers for high-temperature\nspin systems, and attractive zero-range dynamics. In parti cular, we prove a MLSI with\nconstant equal to the minimal rate increment for the mean-fie ld zero-range process,\nthereby answering a long-standing question.\n1 Introduction\nGeometric contraction. Throughout thepaper, wefixafinitemetricspace( X,d)andan\nirreducible stochastic matrix PonX. We write P(X) for the set of probability measures on\nX, and equip it with the Wasserstein distance W(µ,ν) := min X∼µ,Y∼νE[d(X,Y)]. Following\nOllivier [60,61], we define the curvature ofPas the largest number κ=κ(P)∈Rsuch that\n∀µ,ν∈ P(X), W(µP,νP)≤(1−κ)W(µ,ν). (1)\n1Bytheconvexity of W(·,·),itisinfactenoughtoverifythisinequalitywhen µandνareDirac\nmasses, a task which is simple enough to yield sharp lower bounds on th e curvature of many\nconcrete Markov chains. When positive, such lower bounds have be en shown to provide\nsystematic control on a number of essential quantitative featur es of the chain, including\ngeometry [ 43,55,56], mixing times [ 11,9], expansion [ 65,57,66], concentration of measure\n[45,44,28], spectral independence [ 7] and even the cutoff phenomenon [ 68].\nEntropic contraction. The purpose of the present paper is to investigate the relation\nbetween the geometric contraction ( 1) and its natural entropic counterpart:\n∀µ∈ P(X), H(µP|π)≤(1−κ)H(µ|π), (2)\nwhereπis the unique invariant law of P, and where H(µ|π) :=/summationtext\nxµ(x)log(µ(x)/π(x)) de-\nnotes the relative entropy (or Kullback-Leibler divergence) of µwith respect π. As explained\nin the lecture notes [ 12], this discrete-time entropic contraction is stronger than the clas sical\nModified Log-Sobolev Inequality (MLSI) of [ 8], which applies to the continuous-time Markov\nsemi-group ( Pt)t≥0generated by P−I and asserts that\n∀µ∈ P(X),∀t≥0, H(µPt|π)≤e−κtH(µ|π). (3)\nLet us state our main result straightaway, and provide additional m otivation afterwards.\nMain result. We letP⋆denote the adjoint of Pin the Hilbert space L2(X,π). Follow-\ning [61,46,62,56] (see also [ 75] for a related notion), we make the following structural\nassumption, which will be shown to hold in many important examples in Se ction3.\nAssumption 1 (Non-negative sectional curvature) .For each pair of states (x,y)∈ X2,\nthere is a coupling (X⋆,Y⋆)ofP⋆(x,·)andP⋆(y,·)such that almost-surely,\nd(X⋆,Y⋆)≤d(x,y).\nTheorem 1 (Main result) .Under Assumption 1, the geometric contraction ( 1) implies the\nentropic contraction ( 2) with the same constant κ.\nRelation to previous works. The far-reaching possibility that the Ollivier-Ricci cur-\nvature might, under appropriate assumptions, be powerful enou gh to control the rate of\n2exponential decay of the relative entropy emerged in the communit y at the beginning of the\npresent millennium. It became informally known as the Peres-Tetali conjecture , and was in-\nvestigated by several authors [ 28,7,49,56], see also [ 39,20] for related work on establishing\nrelativeentropydecayviaprobabilistictechniques. Inparticular, a breakthroughwasveryre-\ncently made by the second author, who managed to deduce the MLS I (3) from the geometric\ncontraction ( 1) and Assumption 1, under the additional restrictions that Pis reversible and\nthattheunderlyingmetricisthecombinatorialdistanced( x,y) = min{n∈N:Pn(x,y)>0},\nsee [56, Theorem 4.4]. Our Theorem 1strengthens this result in three important ways:\n1. We crucially improve the MLSI ( 3) to the 1 −step entropic contraction ( 2).\n2. Our metric d is arbitrary, thereby considerably broadening the s cope of Assumption 1.\n3. We do not require the reversibility condition P⋆=P.\nThepractical interest ofeachofthoseimprovements willbedemon stratedinSection 3, where\nTheorem 1is applied to several important Markov chains. The main motivation fo r our work\nwas the observation that many natural examples satisfying Assum ption1are actually flatin\nthe sense that κ(P) = 0, making the MLSI ( 3) useless. In contrast, our stronger conclusion\n(2) has the advantage of being applicable to arbitrary scales, thereb y allowing us to leverage\nthe long-acknowledged fact that curvature improves at large scales . More precisely, applying\nTheorem 1toPt=et(P−I)(which also satisfies Assumption 1) instead of Preadily yields\n∀µ∈ P(X),∀t≥0, H(µPt|π)≤(1−κ(Pt))H(µ|π). (4)\nThis is always at least as good as the uniform estimate ( 3), since 1 −κ(Pt)≤e−tκ(P). In\nfact, the sub-multiplicativity of the function t/mapsto→1−κ(Pt) ensures that our estimate can\nonly improve as tincreases. The benefit can be considerable, as we will see in Section 3.\nOther curvature notions. In recent years, there has been increasing interest in discrete\nRicci curvature notions. Indeed, many approaches can be trace d back to the 80s. The\nWasserstein contraction which Ollivier famously interpreted as Ricci curvature, was already\nused byDobrushin andShlosman in[ 27], andisalso known asDobrushin-Shlosmancriterion.\nA fundamentally different approach to discrete curvature was tak en by Forman who used\na discrete Bochner-Weitzenb¨ ock decomposition to establish a Ricc i curvature notion on cell\n3complexes [ 35]. Surprisingly, the Ollivier curvature coincides with the Forman curva ture\nwhen choosing the two-cells optimally [ 42,72].\nMany results in Riemannian geometry critically depend on the dimension . In the dis-\ncrete counterpart, no meaningful dimension parameter could be f ound for the Ollivier cur-\nvature. However, the Bochner formula together with the Bakry- ´Emery calculus [ 3] provides\na framework in which a dimension parameter can be naturally introduc ed in the discrete\nsetting. This lead to the discrete Bakry- ´Emery curvature, introduced independently thrice\nin [29,69,48]. In order to prove Li-Yau and log-Sobolev inequalities, various non- linear\nmodifications of Bakry- ´Emery curvature were introduced in [ 4,54,26,74,33,75], motivated\nby a lack of a discrete Laplacian chain rule. Bakry- ´Emery curvature was recently lifted to\ncell complexes in [ 58].\nA new version of entropic curvature was introduced by Rapaport a nd Samson in [ 64],\nincluding localcriteria. This however, doesnotcoincidewiththeentr opiccurvaturebyErbar\nandMaas[ 33]. Entropiccurvaturehasbeenproventobeapowerfultooltoinv estigatemixing\nof interacting particle systems [ 30,34,32,31], despite the fact that entropic curvature, as a\nnon-linear optimization problem, is hard to compute explicitly.\nOne key difference between Ollivier curvature and Bakry- ´Emery and entropic curvature\nlies in the implicit gradient and distance notions. For Ollivier curvature, there is the freedom\nto choose an arbitrary distance. For Bakry- ´Emery and entropic curvature in contrast, the\ngradient is implicitly determined by the Markov chain.\nFurther non-local curvature notions which are easy to compute h ave been introduced in\n[22,71]. However, not much theory has been developed yet.\nLocal reduction. In order to conclude that a given matrix Psatisfies the entropic con-\ntraction ( 2), Theorem 1requires us to provide, for each pair of states ( x,y)∈ X2:\n(i) a coupling ( X,Y) ofP(x,·) andP(y,·) such that E[d(X,Y)]≤(1−κ)d(x,y);\n(ii) a coupling ( X⋆,Y⋆) ofP⋆(x,·) andP⋆(y,·) such that P(d(X⋆,Y⋆)≤d(x,y)) = 1.\nBy virtue of the so-called Gluing Lemma (see, e.g., [ 73, Lemma 7.6]), we may in fact restrict\nthis double task to pairs ( x,y) in a subset S ⊆ X2whichgenerates the metric d in the\nfollowing sense: for each ( x,y)∈ X2, we can write\nd(x,y) =n/summationdisplay\ni=1d(xi−1,xi), (5)\n4for somen∈Nand some sequence ( x0,...,xn)∈ Xn+1such thatx0=x,xn=yand\n(xi−1,xi)∈ Sfor 1≤i≤n. Of course, the trivial choice S=X2always meets this\nrequirement, but we will see in Section 3that many natural metrics are actually generated\nby much smaller sets, making this reduction quite useful in practice.\nOptimizing the metric. Let us close this short introduction with an interesting question,\ninspired by the recent work [ 66]. Observe that the geometric contraction ( 1) depends on the\nunderlying metric d, whereas the entropic contraction ( 2) does not. This asymmetry can\nbe turned to one’s advantage by treating the metric d as a variable w hich one can try to\nfine-tune so as to optimize the resulting constant. More precisely, our result shows that any\nirreducible stochastic matrix Psatisfies the entropic contraction ( 2) with constant\nκ⋆(P) := sup\nd∈M(P)κ(P,d),\nwhereM(P) is the set of all metrics on Xunder which Assumption 1holds, and where\nκ(P,d) denotes the curvature of Pwith respect to the metric d. In view of the recent work\n[66] on monotone chains, it is natural to ask for an effective characte rization ofκ⋆, at least\nunder appropriate structural assumptions on P. Interestingly, the set M(P) always contains\nthe trivial distance d( x,y) :=1(x/ne}ationslash=y), in which case the Wasserstein distance coincides with\nthe total-variation distance d tv(·,·), so thatκ⋆(P) is well-defined and non-negative. More-\nover, when specialized to this crude metric, our main result has the f ollowing immediate\nconsequence.\nCorollary 1. For any Markov semi-group (Pt)t≥0on any finite state space X, for any initial\nlawµ∈ P(X), any timet≥0,\nH(µPt|π)≤d(t)H(µ|π),whered(t) := max\nx,y∈Xdtv(Pt(x,·),Pt(y,·)).(6)\nWe point out that the total-variation distance dappearing in ( 6) is a classical and well-\nstudied quantity in mixing-time theory (see, e.g., [ 47, Section 4.4]). Yet and perhaps sur-\nprisingly, its role as a universal entropy dissipation factor seems to be new.\n52 Proof of the main result\nIn this section we prove Theorem 1. As in many other applications of curvature, we shall\nactually work with the dual Kantorovich-Rubinstein formulation, wh ich we now recall. Let\nLip(f) := sup/braceleftbigg|f(x)−f(y)|\nd(x,y): (x,y)∈ X2,x/\\e}atio\\slash=y/bracerightbigg\n, (7)\ndenote the Lipschitz constant of a function f:X →R. The first inequality in the following\nresult constitutes a well-known characterization of the Ollivier curv atureκ(P). The sec-\nond inequality explicitly appears in [ 56, Theorem 4.3] as a characterization of non-negative\nsectional curvature in the special case where the underlying metr ic d is the combinatorial\ndistance. Interestingly, this characterization turns out to fail in the more general setup that\nwe consider here (we found an explicit 4 ×4 matrixPsatisfying ( 9) but not Assumption 1).\nNevertheless, the direct implication remains valid, and this is all we act ually need.\nLemma 1 (Dual formulations of curvature and sectional curvature) .\n(i) For any f:X →R, we have\nLip(Pf)≤(1−κ(P))Lip(f). (8)\n(ii) Under Assumption ( 1), we also have for any f:X →(0,∞),\nLip(logP⋆f)≤Lip(logf). (9)\nProof.Fix a function f:X →Rand two points x,y∈ X. By definition, there is a coupling\n(X,Y) ofP(x,·) andP(y,·) such that E[d(X,Y)]≤(1−κ(P))d(x,y). Since, E[f(X)] =\nPf(x) andE[f(Y)] =Pf(y), we can then write\n|Pf(x)−Pf(y)| ≤E[|f(X)−f(Y)|]\n≤Lip(f)E[d(X,Y)]\n≤(1−κ(P))Lip(f)d(x,y).\nThis establishes the first claim. We now assume that fis positive and that ( X⋆,Y⋆) is a\ncoupling of P⋆(x,·) andP⋆(y,·) such that d( X⋆,Y⋆)≤d(x,y) almost-surely. Then, we have\nlogf(X⋆)≤logf(Y⋆)+d(X⋆,Y⋆)Lip(logf)\n≤logf(Y⋆)+d(x,y)Lip(logf).\n6We now take exponentials, then expectations, and finally logarithms again to arrive at\nlogP⋆f(x)≤logP⋆f(y)+d(x,y)Lip(logf).\nSince this is true for all x,y∈ X, the second claim is proved.\nWe henceforth let α=α(P) denote the optimal constant in the entropic contraction ( 2):\nα:= 1−sup\nµ/ne}ationslash=πH(µ),where H(µ) :=H(µP|π)\nH(µ|π). (10)\nOur starting point is the following simple observation about optimizers ofH.\nLemma 2. If a measure µ∈ P(X)\\{π}maximizes H, then its density f:=µ\nπsatisfies\n(PlogP⋆f)(x) = (1 −α)logf(x),\nfor allx∈ X, this identity being understood in R∪{−∞}.\nProof.Suppose that µ∈ P(X)\\{π}achieves the supremum of H, and write f:=µ\nπfor its\ndensity. Fix a point x∈ Xand assume first that µ(x)>0. Then the formula\nµθ:=µ+θδx\n1+θ, (11)\ndefines a probability measure for all small enough θ∈R, and an easy differentiation yields:\n1\nθ(H(µθ|π)−H(µ|π))− − →\nθ→0logf(x)−H(µ|π); (12)\n1\nθ(H(µθP|π)−H(µP|π))− − →\nθ→0(PlogP⋆f)(x)−H(µP|π). (13)\nRecalling that H(µ) = 1−α, we easily deduce that\n1\nθ(H(µθ)−H(µ))− − →\nθ→0(PlogP⋆f)(x)−(1−α)logf(x)\nH(µ|π). (14)\nSinceHis maximized at µ, the right-hand side must vanish, yielding the desired identity.\nLet us now consider the degenerate case where µ(x) = 0. Then, in order for ( 11) to define\nan element of P(X), we need to restrict the parameter θto non-negative values. Under this\nrestriction, the convergences ( 12) and (13) hold in R∪{−∞}. If we had ( PlogP⋆f)(x)∈R,\nthen the convergence ( 14) would still hold, but the limit would now be + ∞, contradicting\nthe fact that His maximal at µ. Thus, we must have ( PlogP⋆f)(x) =−∞, and the claimed\nidentity holds with both sides being equal to −∞.\n7Our second ingredient is the following result, which complements the a bove lemma by\ninvestigating the behavior of the functional µ/mapsto→ H(µ) near the singularity point µ=π.\nLemma 3. Let(µn)n≥1be elements of P(X)\\{π}that converge to π. Then,\nlimsup\nn→∞H(µn)≤1−κ(PP⋆).\nProof.Writeµn= (1 +hn)π, wherehn:X →Ris a non-constant function with zero\nstationary mean, which vanishes as n→ ∞. Then, an easy use of the Taylor expansion\n(1+θ)log(1+θ) =θ+1\n2θ2+o(θ2) asθ→0 gives the asymptotics\nH(µn|π)∼1\n2/bardblhn/bardbl2;\nH(µnP|π)∼1\n2/bardblP⋆hn/bardbl2,\nwhere/bardbl·/bardbldenotes the norm in the Hibert space L2(X,π), and where the notation an∼bn\nmeans that an/bn→1 asn→ ∞. It follows that\nH(µn)∼/bardblP⋆hn/bardbl2\n/bardblhn/bardbl2. (15)\nNow, let 1 = λ1> λ2≥...≥λN≥0 denote the N=|X|ordered eigenvalues of the non-\nnegative self-adjoint operator PP⋆, and let (φ1,...,φN) be a corresponding orthonormal\neigenbasis with φ1= 1. We can then write, for any h:X →R,\n/bardblP⋆h/bardbl2=/a\\}bracketle{th,PP⋆h/a\\}bracketri}ht=N/summationdisplay\ni=1λi/a\\}bracketle{th,φi/a\\}bracketri}ht2,\nwhere/a\\}bracketle{t·,·/a\\}bracketri}htdenotes the scalar product in L2(X,π). Since /a\\}bracketle{th,φ1/a\\}bracketri}htis exactly the stationary\nmean ofh, we deduce that when his centered,\n/bardblP⋆h/bardbl2=N/summationdisplay\ni=2λi/a\\}bracketle{th,φi/a\\}bracketri}ht2≤λ2N/summationdisplay\ni=2/a\\}bracketle{th,φi/a\\}bracketri}ht2=λ2/bardblh/bardbl2.\nThis applies in particular to h=hn, and inserting this into ( 15) shows that\nlimsup\nn→∞H(µn)≤λ2.\nTo conclude, observe that λ2≤1−κ(PP⋆), as can be seen by choosing f=φ2in the dual\nformulation of κ(PP⋆) (Lemma 1(i) applied to PP⋆instead ofP).\n8We now have everything we need to prove our main result.\nProof of Theorem 1.Our goal is to prove that under Assumption 1,\nα≥κ(P). (16)\nWe first make the extra assumption that all entries of Pare positive. By the very definition\n(10), there exists a sequence ( µn)n≥1inP(X)\\{π}such that\nH(µn)− −− →\nn→∞1−α. (17)\nSinceXis finite, we can safely assume – upon extracting a subsequence if ne eded – that\n(µn)n≥1converges pointwise to a limit µ∈ P(X). Ifµ=π, then Lemma 3ensures that\n1−α≤1−κ(PP⋆)\n≤(1−κ(P))(1−κ(P⋆)).\nBut Assumption 1guarantees that κ(P⋆)≥0, so (16) is proved. On the other hand, if\nµ/\\e}atio\\slash=π, then Lemma 2ensures that f:=µ\nπmust satisfy the functional equation\nPlogP⋆f= (1−α)logf. (18)\nRecall that this equality a priori holds in R∪{−∞}. However, since all entries of Pwere\nassumed to be positive, the function P⋆fis strictly positive. Thus, the left-hand side of ( 18)\nis actually finite, and hence so is the right-hand side. In other words ,fis strictly positive,\nand we may therefore safely invoke Lemma 1to write\n(1−α)Lip(logf) = Lip( PlogP⋆f)\n≤(1−κ(P))Lip(logP⋆f)\n≤(1−κ(P))Lip(logf).\nSincefis non-constant ( µ/\\e}atio\\slash=π), we may finally simplify through by Lip(log f) to obtain\nthe desired conclusion. To handle the general case where some ent ries ofPmay vanish, we\nintroduce a perturbation parameter ε∈(0,1) and replace each entry P(x,y) with\nPε(x,y) := (1 −ε)P(x,y)+επ(y).\nThe stochastic matrix Pεonly has positive entries, and its stationary distribution is π.\nMoreover,κ(Pε)≥κ(P): indeed, given x,y∈ Xand a coupling ( X,Y) ofP(x,·) andP(y,·),\n9we can construct a coupling ( Xε,Yε) ofPε(x,·) andPε(y,·) such that d( Xε,Yε)≤d(X,Y)\nby generating an independent pair ( Z,B) withZ∼πandB∼Bernoulli(ε) and setting\n(Xε,Yε) :=/braceleftBigg\n(X,Y) ifB= 0\n(Z,Z) ifB= 1.\nThe same argument applies to the adjoint P⋆\nεand shows that the latter inherits Assumption\n1fromP⋆. Thus, the first part of our proof applies to the perturbed matrix Pεand allows\nus to conclude that α(Pε)≥κ(P). This means that for each µ∈ P(X), we have\nH(µPε|π)≤(1−κ(P))H(µ|π).\nWe may finally send ε→0 to conclude.\n3 Applications\nIn this final section, we illustrate the strength of Theorem 1by establishing new entropy\ndissipation estimates for several important classes of Markov cha ins.\n3.1 Birth and Death Processes\nWe first take a look at the case of Birth and Death Processes (BDP) . Specifically, we set\nX:={1,...,n}and consider the generator that acts on any function f:X →Ras follows:\nLf(x) :=q+(x)(f(x+1)−f(x))+q−(x)(f(x−1)−f(x)), (19)\nwhereq±are arbitrary positive functions on X, except that q−(1) =q+(n) = 0. This\ngenerator is reversible with respect to the probability measure\n∀x∈ X, π(x) :=1\nCx/productdisplay\nk=2q+(k−1)\nq−(k),\nwhereCis a normalizing constant. Let us now assume the following monotonicit y:\n∀x∈ {1,...,n−1}, q +(x+1)≤q+(x) andq−(x+1)≥q−(x).(20)\nThis condition easily guarantees that our BDP ( Xt)t≥0starting from any X0∈ {1,...,n−1}\ncan be coupled with a BDP ( Yt)t≥0starting from Y0=X0+1 so that\n∀t≥0, Yt−Xt∈ {0,1}. (21)\n10This already shows that the underlying semi-group ( Pt)t≥0has non-negative sectional curva-\nture with respect to the metric d: ( x,y)/mapsto→ |y−x|, which is generated by pairs of consecutive\nstates. Moreover, the same coupling yields for all t≥0,\n1−κ(Pt)≤E[d(Xt,Yt)] =E[Yt]−E[Xt]. (22)\nUsing the notation Ex[·] to indicate that the initial state is x∈ X, we obtain:\nCorollary 2. Under the condition ( 20), the semi-group (Pt)t≥0generated by ( 19) satisfies\nH(µPt|π)≤m(t)H(µ|π),where m(t) := max\n1≤xt)∧Pn(T1>t), (23)\nwhereTz:= min{t≥0:Xt=z}denotes the hitting time of z. As a concrete example,\nconsider the extreme case where the jump rates are all equal to 1 : this corresponds to simple\nrandom walk on the segment, for which we classically have E1[Tn] = Θ(n2). Thus, Corollary\n2shows that the entropy decay occurs on a time-scale of order n2, which is actually sharp.\n3.2 Colored Exclusion Processes\nIn this section, we consider a non-conservative and colored versio n of the popular Exclusion\nProcess. The model is parametrized by the following ingredients:\n•a finite set S(the colors) equipped with a fully supported probability law ν;\n•an integern∈N(the dimension);\n11•a non-negative symmetric array {c(i,j): 1≤i/\\e}atio\\slash=j≤n}(the exchange rates);\n•a non-negative vector {r(i): 1≤i≤n}(the refresh rates).\nBy definition, the ColoredExclusion Process (CEP) with those param eters isthe continuous-\ntime Markov chain on X=Snwhose generator acts on any function f:X →Ras follows:\nLf(x) :=/summationdisplay\n1≤i t) denote the tail distribution function of the life-time of such a killed r an-\ndom walk, when started from site i∈[n]. Note that we have the spectral representation\nPi(T >t) =n/summationdisplay\nk=1e−λktφk(i)/a\\}bracketle{tφk,1/a\\}bracketri}ht,\nwhere/a\\}bracketle{ta,b/a\\}bracketri}ht=/summationtext\nia(i)b(i) is the standard scalar product, λn≥...≥λ1>0 denote the\neigenvalues of −∆, andφ1,...,φnis a corresponding orthonormal basis of eigenvectors.\nExplicit estimates are available in many concrete examples (see [ 67]). Our main theorem\nprovides the following entropy contraction principle.\n12Corollary 3. For anyµ∈ P(X)and anyt≥0, the transition matrix Pt:=etLsatisfies\nH(µPt|π)≤H(µ|π) max\n1≤i≤nPi(T >t).\nProof.We equip Xwith the Hamming distance d(x,y) := #{i∈[n]:xi/\\e}atio\\slash=yi}. Note that\nthis isnotthe combinatorial distance associated with L, unlessr(·) has full support. Now,\nfix an initial pair ( X0,Y0)∈ X2with d(X0,Y0) = 1 (such pairs clearly generate the above\nmetric) and consider the Markov chain ( Xt,Yt)t≥0onX2that evolves as follows:\n•a joint exchange ( x,y)→(xi↔j,yi↔j) occurs at rate c(i,j) for each 1 ≤it),\nwhereTis the life-time of the walk. Applying Theorem 1toPtconcludes the proof.\n3.3 Generalized Interchange Processes\nWe now turn to a very general class of randomwalks on the symmetr ic group, which contains\nin particular the well studied Interchange Process. Specifically, giv en an integer n∈Nand\na functionc: 2[n]→[0,∞), we consider the continuous-time Markov chain on X=Sn, the\nsymmetric group of permutations of [ n], whose generator acts as follows: for any function\nf:X →Rand any state x∈ X,\nLf(x) =/summationdisplay\nA⊆[n]c(A)\n|A|!/summationdisplay\nσ∈SA(f(xσ)−f(x)), (26)\nwhereSAdenotes the group of permutations on A, and|A|is the cardinality of the subset\nA. This generator is clearly reversible w.r.t the uniform law πonX. We may think of a\npermutation x∈ Xasassigningauniquelabel xi∈[n]toeachsite i∈[n]. Thedynamics( 26)\nthen simply shuffles the labels of all sites i∈Auniformly at random at rate c(A), for each\n13blockA⊆[n]. When the rate function A/mapsto→c(A) is supported on blocks of size 2, which one\ninterpretsasweightsontheedgesofagraph, theprocessisknow nastheInterchangeProcess.\nThus, the general case is viewed as an Interchange Process on a w eighted hypergraph. We\nobserve that the individual motion of each label is just a continuous -time random walk on\n[n] with conductances\n/hatwidec(i,j) :=/summationdisplay\nA⊇{i,j}c(A)\n|A|. (27)\nRelating the mixing properties of the high-dimensional process gene rated by ( 26) to those\nof its one-dimensional marginals ( 27) is a natural and important problem, which has been\nthe subject of active research [ 14,25,59,18,1,41,10,2]. The first author conjectured\nthat, for any choice of weights c, the spectral gap of the process generated by ( 26) coincides\nwith the spectral gap of the single particle process with rates ( 27), see [10, Conjecture 1.7].\nThis represents the hypergraph generalization of the renowned A ldous conjecture, affirming\nthe aforementioned equivalence for the Interchange Process. W hile Aldous conjecture was\nproved in [ 14], the general case has been verified only for certain classes of hyp ergraph\nweights, see [ 10,2]. We note that for a given set of edge conductances /hatwidec, there may exist\nmultiple choices of hypergraph weights cthat satisfy ( 27). The conjecture thus asserts that\nall such choices yield the same spectral gap. It is noteworthy that our estimate below, which\nis a simple application of our main result, provides control over the de cay of relative entropy,\nindependent of the specific choice of ccompatible with ( 27), offering additional support for\nthe conjecture’s validity.\nCorollary 4. For anyµ∈ P(X)andt≥0, the matrix Pt:=etLgenerated by ( 26) satisfies\nH(µPt|π)≤H(µ|π) max\n1≤i,j≤nPi,j(T >t), (28)\nwhereT:= min{t≥0:It=Jt}denotes the meeting time of two independent random walks\non[n]with conductances ( 27) starting from I0=iandJ0=j, respectively.\nProof.We equip Xwith the transposition distance d(x,y), which is the minimal number of\nswaps that need to be performed in order to turn xintoy. This metric is generated, in the\nsense of ( 5), by those pairs of states that differ in exactly two coordinates. S tarting from\nsuch a pair ( X0,Y0), we can construct a coupling ( Xt,Yt)t≥0as follows: we equip each block\nA⊆[n] with an independent Poisson clock of rate c(A) and, whenever the clock rings, we\n14simply replace the current state ( x,y) with (xσ,yσ), whereσis a uniformly chosen element\nofSA. Such a transformation clearly preserves the distance, and this a lready establishes\nnon-negative sectional curvature along the semi-group. Now, let us modify our coupling as\nfollows: whenever the clock of a block Arings, if the current states xandyhappen to agree\noutsideA, we ensure coalescence by replacing ( x,y) with (xσ,xσ) instead of ( xσ,yσ). It is\neasy to see that the time at which this occurs is stochastically domina ted by the meeting\ntimeTof two independent random walks with conductances ( 27) starting at the two sites\nwhereX0andY0differ. This yields the curvature estimate\n1−κ(Pt)≤max\n1≤i,j≤nPi,j(T >t).\nApplying Theorem 1toPtconcludes the proof.\nTo give a concrete example, consider the unit rate Interchange Pr ocess on a segment of\nlengthn, which corresponds to the rate function\nc(A) :=/braceleftBigg\n2 ifA={i,i+1}for somei∈[n−1]\n0 else.\nThentheone-dimensionaldynamics( 27)isthatofthesimplerandomwalkonthe n−segment,\nfor which it is classical that the worst-case meeting time Tis of ordern2. Thus, ( 28) shows\nthat the entropy decay occurs on a time-scale of order n2, which is actually sharp. It is also\ninstructive to take a look at the Interchange Process on the comp lete graph, a.k.a. Random\nTranspositions, namely\nc(A) :=/braceleftBigg\n4\nn(n−1)ifA={i,j}for some 1 ≤i0}for\nthe support of πand, for each x∈ Xand eachi∈[n], we letπi(·|x) denote the conditional\nlaw of thei−th coordinate, given that the remaining coordinates agree with x:\nπi(σ|x) :=π(xi,σ)/summationtext\nσ′∈Sπ(xi,σ′).\nThe Glauber dynamics for πis the Markov chain with state space Xand transition matrix\nP(x,y) :=1\nn/summationdisplay\ni∈[n]/summationdisplay\nσ∈Sπi(σ|x)1(y=xi,σ). (29)\nIn words, a transition consists in selecting one of the ncoordinates uniformly at random and\nresampling its content according to the measure π, conditioned on the current values of all\nother coordinates. This dynamics is clearly reversible with respect t oπ, and irreducible as\nsoon as the support of πis connected under single-coordinate changes, which we hencefor th\nassume. In the idealized case where πis a product measure, the entropy contraction ( 2)\ntrivially holds with the optimal constant κ= 1/n. In light of this, it is natural to hope\nfor a similar behavior when the target distribution πhasweak dependencies . The following\ngeneral result formalizes this intuition.\nCorollary 5. Suppose that πsatisfies the following weak dependency condition:\nπi(yi|x)≥/summationdisplay\nj/ne}ationslash=i/summationdisplay\nσ/ne}ationslash=xj(πj(σ|y)−πj(σ|x))+, (30)\nfor alli∈[n]and allx,y∈ Xthat differ exactly at the i−th coordinate. Then the matrix (29)\nhas non-negative sectional curvature and satisfies the entr opic contraction ( 2) with constant\nκ:=1\nnmin\ni,x,y\n\n1−/summationdisplay\nj/ne}ationslash=i/summationdisplay\nσ/ne}ationslash=xj|πj(σ|y)−πj(σ|x)|\n\n≥0, (31)\nwhere the minimum ranges over all i∈[n]and allx,y∈ Xthat differ exactly at i.\n17Proof.We equip Xwith the combinatorial distance d( x,y) := min {k∈N:Pk(x,y)>0},\nwhich is generated by those pairs ( x,y)∈ X2that differ at a single coordinate i∈[n]. Now\nfix such a pair, and consider the coupling ( X,Y) ofP(x,·) andP(y,·) defined as follows:\n(X,Y) :=\n\n(x,x) w.p.1\nn/parenleftBig\nπi(xi|y)−/summationtext\nj/ne}ationslash=i/summationtext\nσ/ne}ationslash=xj(πj(σ|x)−πj(σ|y))+/parenrightBig\n;\n(y,y) w.p.1\nn/parenleftBig\nπi(yi|x)−/summationtext\nj/ne}ationslash=i/summationtext\nσ/ne}ationslash=xj(πj(σ|y)−πj(σ|x))+/parenrightBig\n;\n(xi,σ,xi,σ) w.p.1\nnπi(σ|x),forσ /∈ {xi,yi};\n(xj,σ,yj,σ) w.p.1\nn(πj(σ|x)∧πj(σ|y)),forj/\\e}atio\\slash=iandσ/\\e}atio\\slash=xj;\n(xj,σ,x) w.p.1\nn(πj(σ|x)−πj(σ|y))+,forj/\\e}atio\\slash=iandσ/\\e}atio\\slash=xj;\n(y,yj,σ) w.p.1\nn(πj(σ|y)−πj(σ|x))+,forj/\\e}atio\\slash=iandσ/\\e}atio\\slash=xj;\n(x,y) with the remaining probability .\nNote that the first two probabilities are non-negative thanks to ( 30). It is immediate to\ncheck that Xhas lawP(x,·) and that Yhas lawP(y,·). Moreover, we have d( X,Y) = 0\nin the first three cases, and d( X,Y) = 1 in the remaining cases. This shows that Phas\nnon-negative sectional curvature. Finally, adding up the probabilit ies of the three first cases\ngives exactly the constant κappearing at ( 31), and hence\nκ(P)≥1−E[d(X,Y)] =κ.\nApplying our main result concludes the proof.\nCorollary 5contains many special cases of interest, obtained by specializing th e target\nmeasureπto various popular spin systems such as the Ising Model, the Potts M odel, or\nthe Hard-Core Model (see the lecture notes [ 51] for an introduction to those models). In all\nthose examples and many others, our weak dependency assumptio n (30) holds as soon as the\ntemperature is above an explicit threshold, and Corollary 5guarantees entropy contraction\nwith a constant of the right order of magnitude κ= Θ(1/n). Results of this form have a\nlong history, and have been obtained using a variety of sophisticate d methods [ 76,50,51,\n17,15,52,7,20,5,19]. Rather than delving into the specificity of each model and trying to\noptimize the associated constants, let us state one simple general consequence of Corollary\n5that applies to all high-temperature spin systems with pairwise inter actions. While the\nresult stated in Corollary 6below does not necessarily improve over existing bounds, it\noffers a comparable estimate with a simple and entirely different appro ach, thus providing\nan instructive application of our main result Theorem 1. Specifically, consider a target\n18measureπof the form\nπ(x) :=1\nCexp/braceleftBigg/summationdisplay\n1≤ij, andψii(τ,σ) := 0 for all iand (τ,σ)∈S2.\nThe influence of ionjis naturally measured by the quantity\nJij:=1\n2max\n(σ,σ′,τ)∈S2|ψij(σ,τ)−ψij(σ′,τ)|. (33)\nWe can then define the maximal influence in our system as follows:\n/bardblJ/bardbl:= (|S|−1) max\n1≤i≤n/braceleftBiggn/summationdisplay\nj=1Jij/bracerightBigg\n. (34)\nCorollary 6. There is a universal constant ε∈(0,1)(ε= 1/3works) such that whenever\n/bardblJ/bardbl ≤ε,the Glauber dynamics for ( 32) exhibits entropic contraction with constant\nκ:=1−/bardblJ/bardbl\nn.\nProof.It follows from the definition that for any x∈ X,j∈[n], andσ∈S,\nπj(σ|x) =1\n1+/summationtext\nσ′/ne}ationslash=σe/summationtext\nkψjk(σ′,xk)−ψjk(σ,xk). (35)\nNow, anelementary differentiation shows that forany d∈Nandany coefficients a1,...,ad≥\n0, the function F:Rd→[0,1] defined by F(u) =1\n1+a1eu1+···+adeudsatisfies|F(u)−F(v)| ≤\n1\n4/bardblu−v/bardbl∞. Consequently, if x,y∈ Xdiffer exactly at the i−th coordinate, we obtain\n|πj(σ|x)−πj(σ|y)| ≤1\n4max\nσ′/ne}ationslash=σ{ψij(xi,σ′)−ψij(yi,σ′)+ψij(yi,σ)−ψij(xi,σ)} ≤Jij.\nThus, theright-handside of( 30)isatmost /bardblJ/bardbl. Ontheotherhand, inview oftheexpression\n(35), the left-hand side is at least/parenleftBig\n1+qe2/bardblJ/bardbl\nq/parenrightBig−1\n, whereq=|S|−1. It follows that ( 30) is\nsatisfied as soon as /bardblJ/bardbl ≤εq, whereεq∈(0,1) is the unique solution to the equation\nεq=/parenleftBig\n1+e2εq\nq/parenrightBig−1\n.\nNotethatεqincreaseswith q,sothatεq≥ε1≈0.337. Finally,thebound |πj(σ|x)−πj(σ|y)| ≤\nJijshows that the constant κin Corollary 5is at least1−/bardblJ/bardbl\nn, and the proof is complete.\n193.5 Zero-Range Processes\nIntroduced by Spitzer [ 70], theZero-Range Process (ZRP) is a generic interacting particle\nsystem in which individual jumps occur at a rate that only depends on the current number\nof particles present at the source. The model is parameterized by the following ingredients:\n•two integers m,n≥1 representing the number of particles and sites, respectively;\n•an irreducible stochastic matrix G= (Gij)1≤i,j≤nspecifying the geometry;\n•a functionri:{1,2,...} →(0,∞) encoding the kinetics at each site i∈[n].\nTheZRPwith these parameters is a continuous-time Markov chain with state space\nX:=/braceleftBigg\n(x1,...,xn)∈Zn\n+:n/summationdisplay\ni=1xi=m/bracerightBigg\n, (36)\nand generator Lacting as follows: for any f:X →Rand anyx= (x1,...,xn)∈ X,\n(Lf)(x) :=/summationdisplay\n1≤i,j≤nri(xi)Gij(f(x+δj−δi)−f(x)), (37)\nwhere (δ1,...,δn) denotes the canonical n−dimensional basis, and with the convention that\nri(0) = 0 for all i∈[n] (no jumps from empty sites). In words, a site iwithkparticles expels\na particle at rate ri(k), and the latter goes to site jwith probability Gij. It is immediate to\ncheck that the generator Lis irreducible, with invariant measure\nπ(x) :=1\nCn/productdisplay\ni=1νxi\ni\nri(1)ri(2)···ri(xi), (38)\nwhereν=νGdenotes the unique invariant law of Gand whereCis a normalizing constant.\nLet us point out that Lisnotreversible, unless Gis. More precisely, the adjoint L⋆is\nobtained from Lby replacing the matrix Gwith its adjoint G⋆in the formula ( 37). We\nhenceforth make the standard assumption that the rate functio ns are monotone:\n∀i∈[n],∀k∈[m], ri(k)≥ri(k−1). (39)\nAlso, we equip the state space Xwith (half) the L1distance d( x,y) :=1\n2/summationtextn\ni=1|xi−yi|.\nNotice, again, that this is notthe combinatorial distance induced by L, unlessGhas full\nsupport. Nevertheless, this choice is the ‘right’ one in view of the fo llowing result.\n20Lemma 4 (Attractiveness) .Under assumption ( 39), both the transition matrix Pt:=etL\nand its adjoint P⋆\nt:=etL⋆have non-negative sectional curvature at any time t≥0.\nProof.Fixz= (z1,...,zn)∈Zn\n+with/summationtextn\ni=1zi=m−1, andi∈[n]. Now, let Z= (Zt)t≥0be\na ZRP with m−1 particles starting from Z0=zand, conditionally on Z, letI= (It)t≥0be\na time-inhomogeneous random walk on [ n] starting from I0=iand jumping from any site\nuto any site vat the time-varying rate [ ru(Zt+1)−ru(Zt)]×Guv. Then, the formula\nXt:=Zt+δIt, (40)\nclearly defines a ZRP X= (Xt)t≥0withmparticles starting from X0=z+δi. Given\nanother site j∈[n], we can of course enrich the above construction by adding a secon d walk\nJ= (Jt)t≥0whose conditional evolution given Zis dictated by the same time-varying rates\n[ru(Zt+1)−ru(Zt)]×Guv, but which now starts from J0=j. The formula\nYt:=Zt+δJt, (41)\ndefines a new ZRP Y= (Yt)t≥0withmparticles, now starting from Y0=z+δj. From\n(40-41), it is clear that the pair ( X,Y) resulting from this construction satisfies\nd(Xt,Yt) =/braceleftBigg\n1 ifIt/\\e}atio\\slash=Jt;\n0 ifIt=Jt.(42)\nSettingx:=z+δiandy:=z+δj, we have thus constructed a coupling ( Xt,Yt) ofPt(x,·)\nandPt(y,·) such that d( Xt,Yt)≤d(x,y). To conclude, observe that any pair ( x,y)∈ X2\nwith d(x,y) = 1 can be written as ( x,y) = (z+δi,z+δj) for some i/\\e}atio\\slash=j∈[n] and some\nz∈Zn\n+with/summationtextn\ni=1zi=m−1. Moreover, our metric d is generated by the set of such pairs,\nin the sense of ( 5). Thus,Pthas non-negative sectional curvature, and replacing GwithG⋆\nyields the same conclusion for P⋆\nt.\nThe existence of a monotone coupling between ZRPs with different nu mbers of particles\nis of course a well known consequence of the rate monotonicity ( 39), but its interpretation in\nterms of sectional curvature seems to be new. The very same cou pling actually also provides\nan estimate on the curvature of the ZRP, which is exactly what we ne ed in order to apply\nTheorem 1. More precisely, the property ( 42) guarantees that for all t≥0,\n1−κ(Pt)≤max\nz,i,jPz,i,j(T >t), (43)\n21where the maximum ranges over all possible choices for the initial trip le (z,i,j) appearing\nin the above proof, and where T:= inf{t≥0:It=Jt}denotes the coalescence time of our\ntwo time-inhomogeneous random walks IandJ. Note that we have specified the conditional\ndistributions of IandJgivenZ, but notthewayinwhich thosetwo conditionaldistributions\nwere actually coupled: the formula ( 43) is valid for any such coupling. To appreciate its\nstrength, let us consider the important mean-field case where the matrix Ghas rank one:\n∀i,j∈[n], Gij=νj. (44)\nCorollary 7. The mean-field ZRP process satisfies the MLSI ( 3) withκ=δ, where\nδ:= min\ni∈[n],k∈[m]{ri(k+1)−ri(k)}.\nProof.The mean-field condition ( 44) ensures that, conditionally on Z, the random walks I\nandJjumptoanygivenstate j∈[n]atarateatleast δνj,regardlessoftheircurrentposition.\nThus, they can be coupled so that their coalescence time Tis stochastically dominated by\nan Exponential variable with rate δ, and applying Theorem 1toPtconcludes the proof.\nThis result provides a final answer to a natural question that has b een the subject of\nseveral works [ 13,16,34,40,20]. We emphasize that all prior lower bounds on the MLSI\nconstant of the mean-field ZRP involved an additional dependency o n the maximum rate\nincrement\n∆ := max\ni∈[n],k∈[m]{ri(k+1)−ri(k)},\nand were vanishing in the ∆ → ∞limit. Let us also note that the applicability of ( 43) is by\nno means restricted to the mean-field case. As a concrete example , consider the case where\nGis the transition matrix of simple random walk on the n−segment. Then, letting the two\nrandom walks I,Jevolve independently (conditionally on Z) until coalescence ensures that\ntheir order is preserved, so that they must have coalesced by the time at which the lowest\none hitsn. Since this takes time OP(n2/δ), we deduce that the entropy decay occurs on the\ntime-scaleO(n2/δ), which is sharp. Finally, we mention that a similar argument as the one\nused in Corollary 7applies to the heterogeneous Bernoulli-Laplace model, allowing us to g et\nrid of the dependency on the maximum rate in [ 13, Theorem 5.1].\n22References\n[1] Gil Alon and Gady Kozma. Comparing with octopi. Ann. Inst. Henri Poincar´ e Probab.\nStat., 56(4):2672–2685, 2020.\n[2] Gil Alon, Gady Kozma, and Doron Puder. On the Aldous-Caputo sp ectral gap conjec-\nture for hypergraphs. arXiv preprint arXiv:2311.02505 , 2023.\n[3] Dominique Bakry and Michel ´Emery. Diffusions hypercontractives. In S´ eminaire de\nProbabilit´ es XIX 1983/84: Proceedings , pages 177–206. Springer, 2006.\n[4] Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoub i, Shing-Tung Yau,\net al. Li-Yau inequality on graphs. Journal of Differential Geometry , 99(3):359–405,\n2015.\n[5] R. Bauerschmidt and T. Bodineau. A very simple proof of the LSI f or high temperature\nspin systems. Journal of Functional Analysis , 276(8):2582–2588, 2019.\n[6] Nathana¨ el Berestycki and Batı S ¸eng¨ ul. Cutoff for conjug acy-invariant random walks on\nthe permutation group. Probab. Theory Related Fields , 173(3-4):1197–1241, 2019.\n[7] Antonio Blanca, Pietro Caputo, Zongchen Chen, Daniel Parisi, Da nielˇStefankoviˇ c, and\nEricVigoda. Onmixing ofMarkovchains: coupling, spectralindepend ence, andentropy\nfactorization. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discre te\nAlgorithms (SODA) , pages 3670–3692. SIAM, Philadelphia, PA, 2022.\n[8] Sergey Bobkov and Prasad Tetali. Modified log-Sobolev inequalities , mixing and hyper-\ncontractivity. In Proceedings of the Thirty-Fifth Annual ACM Symposium on The ory of\nComputing , pages 287–296. ACM, New York, 2003.\n[9] Magnus Bordewich and Martin Dyer. Path coupling without contra ction.J. Discrete\nAlgorithms , 5(2):280–292, 2007.\n[10] Alexandre Bristiel and Pietro Caputo. Entropy inequalities for r andom walks and per-\nmutations. arXiv 2109.06009. Annales de l’I.H.P. (to appear) .\n[11] R. Bubley and M. Dyer. Path coupling: A technique for proving ra pid mixing in markov\nchains. In Proceedings 38th Annual Symposium on Foundations of Comput er Science ,\npages 223–231, 1997.\n23[12] Pietro Caputo. Lecture notes on entropy and Markov chains. Available at:\nhttp://www.mat.uniroma3.it/users/caputo/entropy.pdf , 2022.\n[13] Pietro Caputo, Paolo Dai Pra, and Gustavo Posta. Convex ent ropy decay via the\nBochner-Bakry-Emery approach. Ann. Inst. Henri Poincar´ e Probab. Stat. , 45(3):734–\n753, 2009.\n[14] Pietro Caputo, Thomas M. Liggett, and Thomas Richthammer. P roof of Aldous’ spec-\ntral gap conjecture. J. Amer. Math. Soc. , 23(3):831–851, 2010.\n[15] Pietro Caputo, Georg Menz, and Prasad Tetali. Approximate te nsorization of entropy\nat high temperature. Ann. Fac. Sci. Toulouse Math. (6) , 24(4):691–716, 2015.\n[16] Pietro Caputo and Gustavo Posta. Entropy dissipation estimat es in a zero-range dy-\nnamics.Probab. Theory Related Fields , 139(1-2):65–87, 2007.\n[17] Filippo Cesi. Quasi-factorization of the entropy and logarithmic S obolev inequalities\nfor Gibbs random fields. Probab. Theory Related Fields , 120(4):569–584, 2001.\n[18] Filippo Cesi. A few remarks on the octopus inequality and Aldous’ s pectral gap con-\njecture.Comm. Algebra , 44(1):279–302, 2016.\n[19] Yuansi Chen and Ronen Eldan. Localization schemes: A framewo rk for proving mixing\nbounds for markov chains. In 2022 IEEE 63rd Annual Symposium on Foundations of\nComputer Science (FOCS) , pages 110–122. IEEE, 2022.\n[20] Giovanni Conforti. A probabilistic approach to convex ( φ)-entropy decay for markov\nchains.The Annals of Applied Probability , 32(2):932–973, 2022.\n[21] Stephen Bryan Connor and Richard Pymar. Mixing times for exclu sion processes on\nhypergraphs. Electronic Journal of Probability , 2019.\n[22] Karel Devriendt andRenaudLambiotte. Discrete curvature o ngraphsfromthe effective\nresistance. Journal of Physics: Complexity , 3(2):025008, 2022.\n[23] Persi Diaconis. The Markov chain Monte Carlo revolution. Bull. Amer. Math. Soc.\n(N.S.), 46(2):179–205, 2009.\n24[24] Persi Diaconis and Mehrdad Shahshahani. Generating a random permutation with ran-\ndom transpositions. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwand te Gebiete ,\n57(2):159–179, 1981.\n[25] AB Dieker. Interlacings for random walks on weighted graphs an d the interchange\nprocess. SIAM Journal on Discrete Mathematics , 24(1):191–206, 2010.\n[26] Dominik Dier, Moritz Kassmann, and Rico Zacher. Discrete versio ns of the Li-Yau\ngradient estimate. Annali Scuola Normale Superiore - Classe di Scienze , page 691–744,\nJune 2021.\n[27] RolandLvovichDobrushinandSenyaBShlosman. Constructivec riterionfortheunique-\nness of Gibbs field. In Statistical Physics and Dynamical Systems: Rigorous Resul ts,\npages 347–370. Springer, 1985.\n[28] Ronen Eldan, James R Lee, and Joseph Lehec. Transport-ent ropy inequalities and\ncurvatureindiscrete-spaceMarkovchains. In A Journey Through Discrete Mathematics ,\npages 391–406. Springer, 2017.\n[29] KD Elworthy. Manifolds and graphs with mostly positive curvatur es. InStochastic\nAnalysis and Applications: Proceedings of the 1989 Lisbon C onference , pages 96–110.\nSpringer, 1991.\n[30] Matthias Erbar and Max Fathi. Poincar´ e, modified logarithmic So bolev and isoperi-\nmetric inequalities for Markov chains with non-negative Ricci curvat ure.Journal of\nFunctional Analysis , 274(11):3056–3089, 2018.\n[31] Matthias Erbar, Max Fathi, and Andr´ e Schlichting. Entropic cu rvature and con-\nvergence to equilibrium for mean-field dynamics on discrete spaces. arXiv preprint\narXiv:1908.03397 , 2019.\n[32] Matthias Erbar, Christopher Henderson, Georg Menz, and Pr asad Tetali. Ricci cur-\nvature bounds for weakly interacting markov chains. Electron. J. Probab , 22(40):1–23,\n2017.\n[33] Matthias Erbar and Jan Maas. Ricci curvature of finite Markov chains via convexity of\nthe entropy. Archive for Rational Mechanics and Analysis , pages 1–42, 2012.\n25[34] Max Fathi and Jan Maas. Entropic Ricci curvature bounds for discrete interacting\nsystems. Ann. Appl. Probab. , 26(3):1774–1806, 2016.\n[35] Robin Forman. Bochner’s method for cell complexes and combina torial Ricci curvature.\nDiscrete & Computational Geometry , 29:323–374, 2003.\n[36] Fuqing GaoandJeremy Quastel. Exponential decay ofentropy intherandomtransposi-\ntion and Bernoulli-Laplace models. The Annals of Applied Probability , 13(4):1591–1600,\n2003.\n[37] Sharad Goel. Modified logarithmic Sobolev inequalities for some mod els of random\nwalk.Stochastic processes and their applications , 114(1):51–79, 2004.\n[38] W. K. Hastings. Monte Carlo sampling methods using Markov chain s and their appli-\ncations.Biometrika , 57(1):97–109, 1970.\n[39] Jonathan Hermon and Yuval Peres. A characterization of L2mixing and hypercontrac-\ntivity via hitting times and maximal inequalities. Probability Theory and Related Fields ,\n170(3-4):769–800, 2018.\n[40] Jonathan Hermon and Justin Salez. Entropy dissipation estimat es for inhomogeneous\nzero-range processes. Ann. Appl. Probab. , 31(5):2275–2283, 2021.\n[41] Jonathan Hermon and Justin Salez. The interchange process o n high-dimensional prod-\nucts.Ann. Appl. Probab. , 31(1):84–98, 2021.\n[42] J¨ urgen Jost and Florentin M¨ unch. Characterizations of For man curvature. arXiv\npreprint arXiv:2110.04554 , 2021.\n[43] J¨ urgen Jost, Florentin M¨ unch, and Christian Rose. Liouville pr operty and non-negative\nOllivier curvature on graphs. arXiv preprint arXiv:1903.10796 , 2019.\n[44] Ald´ eric Joulin. Poisson-type deviation inequalities for curved co ntinuous-time Markov\nchains.Bernoulli , 13(3):782–798, 2007.\n[45] Ald´ eric Joulin and Yann Ollivier. Curvature, concentration and e rror estimates for\nMarkov chain Monte Carlo. Ann. Probab. , 38(6):2418–2442, 2010.\n26[46] Mark Kempton, Gabor Lippner, and Florentin M¨ unch. Large sc ale Ricci curvature on\ngraphs.Calc. Var. Partial Differential Equations , 59(5):Paper No. 166, 17, 2020.\n[47] DavidA.LevinandYuvalPeres. Markov chains and mixing times . AmericanMathemat-\nical Society, Providence, RI, 2017. Second edition of [ MR2466937 ], With contributions\nby Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by J ames G.\nPropp and David B. Wilson.\n[48] Yong Lin and Shing-Tung Yau. Ricci curvature and eigenvalue es timate on locally finite\ngraphs.Mathematical research letters , 17(2):343–356, 2010.\n[49] Kuikui Liu. From coupling to spectral independence and blackbo x comparison with the\ndown-up walk. In Approximation, Randomization, and Combinatorial Optimiz ation.\nAlgorithms and Techniques (APPROX/RANDOM 2021) . Schloss Dagstuhl-Leibniz-\nZentrum f¨ ur Informatik, 2021.\n[50] Sheng Lin Lu and Horng-Tzer Yau. Spectral gap and logarithmic Sobolev inequality\nfor Kawasaki and Glauber dynamics. Comm. Math. Phys. , 156(2):399–433, 1993.\n[51] Fabio Martinelli. Lectures on Glauber dynamics for discrete spin m odels. In Lectures\non probability theory and statistics (Saint-Flour, 1997) , volume 1717 of Lecture Notes\nin Math., pages 93–191. Springer, Berlin, 1999.\n[52] Katalin Marton. Logarithmic Sobolev inequalities in discrete produ ct spaces. Combin.\nProbab. Comput. , 28(6):919–935, 2019.\n[53] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenb luth, Augusta H.\nTeller, and Edward Teller. Equation of state calculations by fast com puting machines.\nThe Journal of Chemical Physics , 21(6):1087–1092, 1953.\n[54] Florentin M¨ unch. Li–Yau inequality on finite graphs via non-linear curvature dimension\nconditions. Journal de Math´ ematiques Pures et Appliqu´ ees , 120:130–164, 2018.\n[55] Florentin M¨ unch. Non-negative Ollivier curvature on graphs, r everse Poincar´ e inequal-\nity, Buser inequality, Liouville property, Harnack inequality and eigen value estimates.\narXiv preprint arXiv:1907.13514 , 2019.\n27[56] Florentin M¨ unch. Ollivier curvature, isoperimetry, concentra tion, and log-Sobolev in-\nequalitiy, 2023.\n[57] Florentin M¨ unch and Justin Salez. Mixing time and expansion of no n-negatively curved\nmarkov chains. Journal de l’ ´Ecole polytechnique – Math´ ematiques , 10:575–590, 2023.\n[58] Florentin M¨ unch, Melchior Wirth, and Haonan Zhang. Intertwin ing curvature bounds\nfor graphs and quantum markov semigroups. arXiv preprint arXiv:2401.05179 , 2024.\n[59] RobertoImbuzeiro Oliveira. Mixing of thesymmetric exclusion pro cesses in termsof the\ncorresponding single-particle random walk. The Annals of Probability , 41(2):871–913,\n2013.\n[60] Yann Ollivier. Ricci curvature of Markov chains on metric spaces .Journal of Functional\nAnalysis, 256(3):810–864, 2009.\n[61] Yann Ollivier. A survey of Ricci curvature for metric spaces and markov chains. In\nProbabilistic approach to geometry , volume 57, pages 343–382. Mathematical Society of\nJapan, 2010.\n[62] Francesco Pedrotti. Contractive coupling rates and curvatu re lower bounds for Markov\nchains, 2023.\n[63] Matteo Quattropani and Federico Sau. Mixing of the averaging process and its discrete\ndual on finite-dimensional geometries. The Annals of Applied Probability , 33(2):1136–\n1171, 2023.\n[64] Martin Rapaport and Paul-Marie Samson. Criteria for entropic c urvature on graph\nspaces.arXiv preprint arXiv:2303.15874 , 2023.\n[65] Justin Salez. Sparse expanders have negative curvature. Geom. Funct. Anal. ,\n32(6):1486–1513, 2022.\n[66] Justin Salez. Spectral gap and curvature of monotone marko v chains, 2023.\n[67] Justin Salez. Universality of cutoff for exclusion with reservoirs .Ann. Probab. ,\n51(2):478–494, 2023.\n28[68] Justin Salez. Cutoff for non-negatively curved Markov chains. J. Eur. Math. Soc. , to\nappear.\n[69] Michael Schmuckenschl¨ ager. Curvature of nonlocal markov generators. Convex geomet-\nric analysis (Berkeley, CA, 1996) , 34:189–197, 1998.\n[70] Frank Spitzer. Interaction of Markov processes. Advances in Math. , 5:246–290 (1970),\n1970.\n[71] Stefan Steinerberger. Curvature on graphs via equilibrium mea sures.Journal of Graph\nTheory, 103(3):415–436, 2023.\n[72] Philip Tee and CA Trugenberger. Enhanced Forman curvature a nd its relation to\nOllivier curvature. Europhysics Letters , 133(6):60006, 2021.\n[73] C´ edric Villani. Topics in optimal transportation , volume 58 of Graduate Studies in\nMathematics . American Mathematical Society, Providence, RI, 2003.\n[74] Frederic Weber and Rico Zacher. The entropy method under cu rvature-dimension con-\nditions in the spirit of Bakry-´ emery in the discrete setting of marko v chains. Journal of\nFunctional Analysis , 281(5):109061, 2021.\n[75] Xi Xu, Wang Shen, and Linfeng Wang. The CD pCurvature Condition on a Graph.\nFront. Math. , 19(1):181–192, 2024.\n[76] Boguslaw Zegarlinski. Dobrushin uniqueness theorem and logarit hmic Sobolev inequal-\nities.J. Funct. Anal. , 105(1):77–111, 1992.\n29" }, { "title": "2401.17157v1.CHoKI_based_MPC_for_blood_glucose_regulation_in_Artificial_Pancreas.pdf", "content": "CHOKI- BASED MPC FOR BLOOD GLUCOSE REGULATION IN\nARTIFICIAL PANCREAS\nA P REPRINT\nBeatrice Sonzogni\nDepartment of Management, Information and Production Engineering\nUniversity of Bergamo (24044 Dalmine, Bergamo, Italy)\nbeatrice.sonzogni@unibg.it\nJos´e Mar ´ıa Manzano\nDepartment of Engineering\nUniversidad Loyola Andaluc ´ıa (41704 Dos Hermanas, Seville, Spain)\njmanzano@uloyola.es\nMarco Polver\nDepartment of Management, Information and Production Engineering\nUniversity of Bergamo (24044 Dalmine, Bergamo, Italy)\nmarco.polver@unibg.it\nFabio Previdi\nDepartment of Management, Information and Production Engineering\nUniversity of Bergamo (Via G. Marconi 5, 24044, Dalmine (BG), Italy)\nfabio.previdi@unibg.it\nAntonio Ferramosca\nDepartment of Management, Information and Production Engineering\nUniversity of Bergamo (24044 Dalmine, Bergamo, Italy)\nantonio.ferramosca@unibg.it\nABSTRACT\nThis work presents a Model Predictive Control (MPC) for the artificial pancreas, which is able to\nautonomously manage basal insulin injections in type 1 diabetic patients. Specifically, the MPC goal\nis to maintain the patients’ blood glucose level inside the safe range of 70-180 mg/dL, acting on the\ninsulin amount and respecting all the imposed constraints, taking into consideration also the Insulin\nOn Board (IOB), to avoid excess of insulin infusion. MPC uses a model to make predictions of the\nsystem behaviour. In this work, due to the complexity of the diabetes disease that complicates the\nidentification of a general physiological model, a data-driven learning method is employed instead.\nThe Componentwise H ¨older Kinky Inference (CHoKI) method is adopted, to have a customized\ncontroller for each patient. For the data collection phase and also to test the proposed controller, the\nvirtual patients of the FDA-accepted UV A/Padova simulator are exploited. The proposed MPC is\nalso tested on a modified version of the simulator, that takes into consideration also the variability\nof the insulin sensitivity. The final results are satisfying since the proposed controller reduces the\ntime in hypoglycemia (which is more dangerous) if compared to the outcome obtained with the\nstandard constant basal insulin therapy provided by the simulator, satisfying also the time in range\nrequirements and avoiding long-term hyperglycemia events.\nKeywords Artificial Pancreas, MPC, Learning-based controlarXiv:2401.17157v1 [eess.SY] 30 Jan 2024CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n1 Introduction\nDiabetes is a common chronic metabolic disorder characterized by the body’s inability to correctly balance the Blood\nGlucose (BG) level, due to the absence or not enough insulin production by the pancreas. In particular, Type 1\nDiabetes (T1D) is characterized by the autoimmune destruction of the insulin-producing beta cells. This drives the\npatient into a state of hyperglycemia, so when the BG level is above 180 mg/dL, which has long-term complications,\nsuch as cardiovascular disease, neuropathies or kidney damage. Thus, T1D patients require daily insulin injections to\nmaintain the BG level inside the euglycemic range (i.e. between 70 and 180 mg/dL). Below this threshold, the patient\nis in a state of hypoglycemia, which is dangerous in the short-term since can even lead to the diabetic coma [10].\nIn order to ease patients’ and caregivers’ life, the therapy is trying to get more and more automatised, miming the\nfunctioning of a healthy pancreas. Specifically, the Artificial Pancreas (AP) implements such a treatment in a closed\nloop. It is a system made of three components: the sensor that measures the glucose at the interstitial level every few\nminutes (Continuous Glucose Monitoring, CGM), and the control algorithm that computes the insulin amounts that are\nthen injected into the subcutaneous tissue through the pump. The APs currently on the market are hybrid closed loop\nsystems, since the administration of the basal insulin (which is the small, continuous and constant amount injected to\nmanage the BG in fasting periods) is automatic, while for postprandial boluses (i.e. bigger amount injected at meal\ntimes to face the BG increase due to carbohydrate ingestion or when the BG level is unexpectedly too high) it still\nrequires the manual intervention of the patients [16].\nModel Predictive Control (MPC), due to its predictive capability and the possibility to add constraints to the problem,\nis one of the most widely used control algorithms for the AP. MPC is a control strategy that uses a dynamic model to\nforecast the future behavior of a system. Based on this prediction, it calculates the best sequence of control actions\nat every sampling time, by solving a finite horizon optimal control problem. Then, only the first value of the control\naction sequence is applied to the system and the process is repeated at each sampling time, in a receding horizon\nfashion [17]. Over the last few years, the use of MPC as a control algorithm for AP has been extensively studied and\ntested [2, 22, 9, 1, 4, 6, 18, 7, 5, 20], thanks to its capability to anticipate unwanted fluctuations in glycemic levels and\nto calculate the amount of insulin to be injected, taking into account all the constraints.\nT1D is a disease that varies both among and within patients and this is due to differences in blood glucose response\nto meals or insulin, which can also vary according to daily state. Identifying a general model to describe the insulin-\nglucose system is therefore difficult. This work aims to use data-driven approaches, that is, to use the current and\npast data of a patient to obtain the future BG, and then to be able to calculate the correct amount of basal insulin to\nbe delivered. This facilitates and improves T1D management by providing a customised MPC algorithm for the AP.\nRecently, different types of learning-based MPCs have been proposed in the literature [8], which are based on different\nlearning methods. Specifically, we use the Componentwise H ¨older Kinky Inference (CHoKI) method, a nonparametric\nlearning technique that favors the design of robust MPCs that are stable by design [15].\nIn this case, starting from the work proposed in [19], in the MPC optimization problem is considered also a dynamic\nsafety constraint on the maximum basal insulin value, which is based on the Insulin On Board (IOB). To take into\naccount the insulin amount of the boluses that is still active in the patient, when computing the quantity of the basal\ncorrections, to reduce the risk of hypoglycemic events.\nThe virtual patients of the UV A/Padova simulator [21] are exploited to collect the data needed to learn the system\nbehaviour and also to test the proposed control algorithms. This is a simulator accepted by the Food and Drug Admin-\nistration (FDA) as a substitute for pre-clinical studies and it contains populations of virtual subjects; in particular, we\nhave used the adults with T1D.\nThe same proposed CHoKI-based MPC has been also tested on patients whose insulin sensitivity varies intra-subjects\nthroughout the day.\nThe rest of this work is structured as follows: Section 2 presents the CHoKI learning method and the version tailored\nto the T1D patient case. Section 3 analyses the proposed model predictive control problem. The implementation of\nthe designed controller in the UV A/Padova simulator is presented in Section 4, and Section 5 concludes the paper.\nNotation\nA set of integers [a, b]is denoted Ib\na,Rnis the set of real vectors of dimension nandRn×mis the set of real matrices\nof dimension n×m. Given v, w∈Rnv, the notation (v, w)implies [vT, wT]Tandv≤wimplies that the inequality\nholds for every component. ∥v∥stands for the Euclidean norm of vand|v|={w:wi=|vi|,∀i}. Given two sets\nA, B ,A⊖Bdenotes the Pontryagin difference. Their Cartesian product is denoted A×B={(x, y)|x∈A, y∈B}.\nThe positive box B(v)⊂Rnvof radius vis defined as B(v) ={y: 0≤y≤v}and the ball B(v)⊆Rnvof radius v\n2CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\nis defined as B(v) ={y:|y| ≤v}. Ann, m -dimensional matrix of ones is denoted 1n×m. The ith row of a matrix\nMis denoted Mi.\n2 Problem statement\nThe analysed problem is based on the CHoKI formulation proposed in [19], which is briefly reported in the following.\nThe system is a sampled continuous-time one, described by an a priori unknown discrete-time model, whose measured\noutput is y(k)∈Rnyand whose input is u(k)∈Rnu. In this case, there is one output ( ny= 1) which is the glucose\nlevel, in mg/dL, and there are two inputs ( nu= 2), which are the meal ( u1, the not controllable one) in gof\ncarbohydrates and the insulin ( u2, the controllable one) in pmol . A sampling time of 5 minutes is considered.\nThe measured output can be modeled as a nonlinear autoregressive exogenous (NARX) model, with the following\nstate-space representation:\ny(k+ 1) = f(x(k), u1(k), u2(k)) +e(k),\nwhere e(k)∈Rnyis process noise and the regression state x∈Rnxis\nx(k) =\u0010\ny(k), . . . , y (k−na), u1(k−1), . . . , u 1(k−nb), u2(k−1), . . . , u 2(k−nc)\u0011\n, (1)\nwhere na∈N0is the memory horizon for the glucose values, nb∈N0for the meals and nc∈N0for the basal insulin\ninjections. The arguments of fare then aggregated into w= (x, u1, u2)∈Rnwso that it is possible to build a data set\nofNDobservations, denoted D={(yk+1, wk)}, fork= 1, . . . , N D−1.\n2.1 Componentwise H ¨older Kinky Inference (CHoKI)\nThe purpose of this subsection is to be an introduction to the choice of learning method. Kinky Inference (KI) [14] is a\nclass of learning approaches that includes Lipschitz interpolation, which is a technique based on Lipschitz continuity\nof the ground truth function. There exists an extension of the Lipschitz continuity, named H ¨older continuity, defined\nas follows:\nDefinition 1 (H¨older continuity) .A function f:W → Y is H¨older continuous if there exist two real constants L≥0\nand0< p≤1such that for all w1, w2∈ W ,\n∥f(w1)−f(w2)∥ ≤L∥w1−w2∥p, (2)\nwhere Lrepresents the smallest Lipschitz constant and pis called the H ¨older exponent, W ⊆Rnwis the input space\nandY ⊆Rnyis the output space.\nIn the case of p= 1, it means to have Lipschitz continuity [15].\nTo catch different variations of the output according to the changes of each component of the input regressor, the Com-\nponentwise H ¨older Kinky Inference (CHoKI) [15] can be implemented. This method is based on the componentwise\nH¨older continuity, which considers matrices LandP, instead of the H ¨older constant Land exponent p. This is useful\nin cases where a function may have sudden variations along one dimension of the input, while changing smoothly\nalong other input dimensions.\nDefinition 2 (Componentwise H ¨older continuity) .Given the matrices LandP ∈Rny×nw, a function f:W → Y is\ncomponentwise L-P-H¨older continuous if ∀w1, w2∈ W and∀i∈Iny\n1\n|f(w1)−f(w2)| ≤dP\nL(|w1−w2|) (3)\nwhere\ndP\nL(w) :=\u0000\na:ai=nwX\nj=1Li,jwPi,j\nj,∀i∈Iny\n1\u0001\n. (4)\nThen, assuming that fis H¨older continuous and given a data set Dof inputs/outputs observations, the CHoKI predictor\nfor a query q∈Rnwis:\nˆf(q; Θ,D) =1\n2min\ni=1,...,ND\u0010\n˜yi+dP\nL\u0000\n|q−wi|\u0001\u0011\n+1\n2max\ni=1,...,ND\u0010\n˜yi−dP\nL\u0000\n|q−wi|\u0001\u0011\n, (5)\nwhere Θ ={L,P}andˆfis still componentwise L-P-H¨older continuous.\n3CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\nAccording to (5) it is possible to predict a new output ˆy(k+ 1) = ˆf\u0000\nw(k); Θ,D\u0001\n. Then, the prediction model can be\nformulated in state-space as follows:\nˆx(k+ 1) = ˆF\u0000\nx(k), u1(k), u2(k)\u0001\nˆy(k) =Mˆx(k)(6)\nwhere ˆF\u0000\nx(k), u1(k), u2(k)\u0001\n=\u0000ˆf(x(k), u1(k), u2(k)), y(k), . . . , y (k−na+ 1), u1(k), . . . , u 1(k−nb+\n1), u2(k), . . . ,\nu2(k−nc+ 1)\u0001\nandM= [Iny,0, . . . , 0].\nIf the matrices LandPare unknown a priori, they must be estimated. To do that, an optimization problem is solved\noffline, splitting the data set Dinto two disjoint data sets: Dtrain for the estimation and Dtestfor the validation. The\noptimization problem is:\nΘ = arg min\nΘg(Θ,Dtrain,Dtest) (7a)\ns.t.|˜yi−˜yj| ≤dP\nL(|wi−wj|),∀wi, wj∈ WD, wi̸=wj (7b)\n00, i∈Iny\n1, j∈Inw\n1, (7c)\nwhere WDrepresents the input data points in D. The cost function g(Θ,Dtrain,Dtest)to be minimized is:\ng(Θ,Dtrain,Dtest) =1\nNDtestNDtestX\ni=1∥ˆf(wi; Θ,Dtrain)−˜yi∥2, (8)\nbeing ˆf(wi; Θ,Dtrain)the predictions made with the CHoKI in (5) (computed with the data in Dtrain), which are\ncompared to ˜yi, that are the measured values of the noisy data set Dtest.\n2.2 CHoKI implementation for T1D patient\nIn this subsection, the CHoKI method explained in Section 2.1 is designed to learn the dynamics of the T1D patient.\nTo this aim, to implement the CHoKI strategy, the first step is the data collection. This is a fundamental phase since\nthe quality of the generated data set will affect the performance of the CHoKI predictor and thus the functioning of\nthe controller. This is done by employing the virtual T1D adult patients of the UV A/Padova simulator. For each\nof them, several simulations were made, varying the initial glycemic condition, the basal insulin quantity (from 0\nto500 pmol ) and the carbohydrates of the meals (with the post-prandial boluses, given 20 min after the meal time).\nAll these simulations were set to obtain an appropriate distribution of the points in the space, looking at the input-output\nrepresentation. The simulator allows the inclusion of some noises on the sensor and on the pump, to perform more\nrealistic simulations. Specifically, the available virtual typical commercial CGM was selected as a sensor, with auto-\nregressive noise with inverse Johnson transform distribution. The noise on the virtual pump is normally distributed\nwith a mean of 0 pmol and a standard deviation of 0.1. Also, an error with a normal distribution with a standard\ndeviation equal to 30% of the meal amount is added to the carbohydrate estimation.\nOnly the relationship between BG, meals and basal insulin is considered, as the aim of the proposed controller is\nto manage basal insulin injections automatically, while meal boluses are delivered manually (assuming they are a\nfunction of meals). The CHoKI requires the data to be in the right NARX shape, thus the model orders na, nbandnc\nhave to be identified and this is done through a cross-validation procedure.\nIn particular, many combinations of model orders were tested. The Mean Squared Error (MSE) among the 1-step\nahead predictions and the actual values were measured on an unused 21-day data set. The selected orders were chosen\nbased on the lowest MSEs, but a trade-off with model complexity was also considered to avoid the risk of overfitting.\nThe resultant orders are na= 5,nb= 9andnc= 3, being each sampling time 5 min long.\nTo obtain the predictions employing (5), the hyperparameters Θ ={L,P}must be estimated according to (7). We\nassumed to have P=1ny×nwand thus the optimization problem is solved to obtain just the values of the matrix L.\nIn this case, only three values are estimated: La∈Rfor the glucose part, Lb∈Rfor the meals and Lc∈Rfor the\ninsulin. Therefore, Lcontains those three values repeated to reach the right dimension (i.e. the regressor length nw),\nthusL= [La1na;Lb1nb;Lc1nc].\nSome a priori knowledge was utilized in order to set the constraints of the optimization problem: as the Linitial\nvalue the Lipschitz constant Lwas exploited, which is obtained from the LACKI (Lazily Adapted Constant Kinky\nInference) method [14], based on the H ¨older continuity property. The upper and lower bounds for La,LbandLcwere\nset as [10;10;10] and [0;0.9;0.09], respectively, thanks to previous analyses.\n4CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n020406080100120140\nTime[min]050100150200250Glucose[mg/dL]b)ins=94.6pmol\n020406080100120140\nTime[min]050100150200250Glucose[mg/dL]a)ins=0pmol\n020406080100120140\nTime[min]050100150200250Glucose[mg/dL]c)ins=180pmol\n020406080100120140\nTime[min]050100150200250Glucose[mg/dL]d)ins=0pmolandmeal=60gValidationofthemodel(Adult9)\nFigure 1: In each graph, the vertical dashed red line marks the end of the fixed regressor, when the inputs displayed in\nthe titles are applied. The blue line is the glucose trend. In a) the glucose increases a bit, due to the absence of basal\ninsulin. In b) the glucose remains stable since the insulin amount is the reference value and equal to the regressor\nvalues. In c) the glucose decreases due to the basal amount of 180 pmol . In d) the glucose increases due to the\npresence of the meal and no basal insulin.\nTable 1: MPC settings\nAdulturef\n(pmol )NDL\n(LACKI)[La;Lb;Lc]\n(CHoKI)µ(90%)\n(mg/dL)Ncϵ Q\n#1 122.38 4775 3.46 [0.74; 5.46; 0.29] 14.83 2 10 1\n#2 134.89 4950 3.28 [4.89; 3.96; 0.09] 10.19 2 20 1\n#3 149.97 4990 3.08 [0.71; 5.45; 0.09] 9.29 3 10 1\n#5 91.83 4156 6.56 [0.84; 5.52; 0.44] 13.91 2 5 1\n#6 190.22 5339 3.41 [4.72; 3.52; 0.09] 11.27 1 1 1\n#8 105.83 4703 2.58 [1.08; 5.84; 0.09] 7.8 3 1 100\n#9 94.59 3976 3.72 [1.13; 4.09; 0.09] 11.63 2 1 100\n#10 124.86 4966 3.29 [3; 2; 0.09] 10.1 1 20 1\nThefmincon MATLAB function was implemented to solve the optimization problem (7). For each patient, once\ntheLis found, the model is validated on a new data set, to verify its ability to predict future BG values, comparing\nthem with the real outputs. For each subject, the resulting L, theurefand the Lare reported in Table 1.\nFurther analysis was also carried out, starting with a fixed regressor and varying the input values, to ensure that\nthe CHoKI strategy had correctly learned the effect of each input on the output. The fixed regressor has BG set to\n120mg/dL, no meals and constant basal insulin equal to the reference value urefof each patient, obtained from the\nstandard therapy provided by the simulator. As an example, the results obtained from subject Adult 9 are displayed in\nFigure 1: it can be seen how the glucose trend (blue lines) decreases when the amount of the basal insulin injections\nincreases and it rises when there is carbohydrate ingestion. This holds for all the considered virtual patients.\n5CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n3 CHoKI-based robust MPC\nThe control objective is to drive and maintain the BG level inside the euglycemic zone, which is given by 70\nand180 mg /dL, satisfying all the inputs and output constraints. The basal insulin amount u2(k)must be inside\nthe range U={u: 0≤u≤500 pmol },∀k. It is the control action, whose values are calculated so that the BG\nlevel y(k)remains in the set Y={y: 55≤y≤300 mg /dL},∀k, not to arrive to extreme hyper- or hypoglycemic\nconditions. An additional constraint is set on the maximum value of the basal insulin, according to the IOB estimation.\nIn this case, the open-loop predictions of the MPC control problem are computed with the CHoKI predictor (5),\nassuming that a physiological model for T1D patients is not available. To ensure the robustness of the MPC to possible\nmodel-system mismatches, the output constraints are tightened at each step according to an error that represents the\nuncertainty of the predictions based on the data. The system in closed loop is shown to be Input-to-State Stable\n(ISS) [15, Theorem 3] with the proposed controller.\nThe set of restricted output constraints is given by\nYj=Yj−1⊖ R j, (9)\nalong the prediction horizon, j= 1, ..., N .Rjare the reachability sets that account for the possible errors in the\nnominal predictions and Y0=Y. To compute Rj, the starting point is to consider a sequence of future inputs u(k+1)\nandc1∈Rny, such that\n|y(k+ 1)−ˆy(1|k)| ≤c1. (10)\nThe difference between a prediction made at time step k+jbased on the measurement at step k, and the prediction\nmade at step kbased on the measurement at step k+ 1, for a given sequence of control inputs, is bounded by the sets\n|ˆy(j|k)−ˆy(j−1|k+ 1)| ∈ M j⊆Rny, (11a)\n|ˆw(j|k)−ˆw(j−1|k+ 1)| ∈ G j⊆Rnw. (11b)\nThe sets MandGcan be calculated from the equations\nMj=B\u0000\ndP\nL(Gj−1)\u0001\n, (12a)\nGj=Mj× ··· × M σ(j)× {0} × ··· × { 0}, (12b)\nwithσ(j) = max (1, j−na)andM1=B(c1). The set Rjis defined as Rj={y:|y| ∈ M j}for all j∈IN\n1.\nIn [15] is also shown that cj∈Rnyanddj∈Rnware such that Mj=B(cj)andGj=B(dj). Then, the sets Mj\nandGjcan be computed using the recursion\ncj=dP\nL(dj−1), (13a)\ndj= (cj, . . . , c σ(j),0, . . . , 0), (13b)\nwithc1=µ(where µis the maximum absolute error obtained in the validation phase) and then, Rj=B(cj).\nIn our specific control problem, an a posteriori analysis showed that extreme deviations from nominal predictions are\nhighly unlikely. Then, the value representing the 90thpercentile of the probability distribution is used as µinstead of\nthe maximum error (see Table 1).\nRemark 1: To deal with the infeasibility of possible solutions outside the 90% region, some slack variables\nδ={δmin, δmax}are added to the optimization problem; therefore the constraints on the glucose become ˆy(j|k)∈\nYj,δ,∀j∈IN\n1, with\nYj,δ={y:ymin(j)−δmin(j)≤y≤ymax(j) +δmax(j)}, (14)\nwhere yminandymax are the extreme values of Yjfrom (9).\n3.1 Terminal ingredients computation\nThe tightened constraints are computed as described in the previous section, for all subjects, only once and offline.\nThis tightening implies the definition of the length of the control horizon Nc, which may vary for each virtual patient.\nThe control horizon is calculated as the maximum possible value that makes it possible to have a set of constraints that\nis not empty, but it also takes into account the need to have reasonable ranges according to the system.\nA prediction horizon Nplonger than the control horizon Ncis considered to increase the domain of attraction and the\npredictive ability of the controller, thus Np> N c. A local control law for the predictions from NctoNpmust be\nestablished to apply this approach. We use the following:\nu=K(x−x) +u, (15)\n6CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\nwithu= (u1, u2)and where K∈Rnu×nxis the control gain of a Linear Quadratic Regulator (LQR) and (x,u)is\nan equilibrium point around which the system ˆF(x, u)is linearized. In particular, xis constructed as per (1), using\ny= 120 mg /dLof glucose, and u= (0, uref). Matrices A∈Rnx×nxandB∈Rnx×nuof the linearized model\nx(k+ 1) = Ax(k) +Bu(k), are calculated numerically from the input-output data using the CHoKI model. In this\nway, each element A(j, i)andB(j, i)is obtained by considering that\nA(j, i) =∂ˆFj\n∂xi=ˆFj(xi+ϵ)−ˆFj(xi−ϵ)\n2ϵ, B (j, i) =∂ˆFj\n∂ui,\nwhere ϵcan be different for each subject (see Table 1). Note that A(1,1) =∂yk+1\n∂ykandB(1,1) =∂yk+1\n∂u1,k.\n3.2 Insulin On Board\nThe MPC algorithm also includes a dynamic safety constraint on the maximum basal insulin value, which is based\non the amount of IOB. The IOB represents the quantity of injected insulin still active in the body, which depends\non patient dynamics and on the duration of insulin action (DIA). The IOB at each sampling time kcan be estimated\nconsidering the residuals of the past insulin administration, which means having:\nIOB(k) =a(k−1)ub(k−1) +. . .+a(k−nIOB)ub(k−nIOB), (16)\nwhere the insulin action curve is represented by aand the vector of the insulin boluses administration history is ub. The\ntime of insulin action is nIOB, considered with a sampling time of 5 min . Specifically, in this case, it is nIOB= 72 ,\nwhich means considering 6 h(note that taking into account an insulin duration of 6 his a less conservative approach\nthan considering a duration of 8 h) [12].\nThe upper constraint is considered to limit the basal corrections, in order to avoid giving too much insulin, and thus to\nreduce the risk of hypoglycemic events. This means that the basal insulin amount u2must be inside the new range\nU2={u: 0≤u≤umax\n2}, (17)\nwhere umax\n2is the value of the upper constraint for the basal computation and it comes from:\numax\n2(k, j) =\u001a\nulim\n2−IOB(k, j) if ulim\n2> IOB (k, j)\nuref otherwise(18)\nwhere ulim\n2= 500 pmol is the basal insulin maximum amount that can be injected, and considering the sampling time k\nandj= 0, . . . , N p−1[3]. The IOB varies along the prediction horizon (i.e. with j), which means that the estimations\ndecrease according to the insulin action curve, without considering possible new meal boluses. This implies that the\nbasal upper bound is equal to the reference value urefwhen the maximum limit ulim\n2is less than the insulin still active\nfrom the previous boluses injection (i.e. IOB ), otherwise, the upper bound is equal to the difference between ulim\n2\nandIOB(k). The weights of the insulin action curve aare obtained from\u0000\n(DIA−tb)/DIA\u0001\n, where DIA = 6 h\nandtbis the time passed from the previous bolus. In Figure 2 an example of the IOB estimation for virtual Adult 10\nis reported. This shows that (16) approximates quite well the real values (blue line) and the choice of DIA equal to\n6 his appropriate. The initial IOB value after a bolus could be different between the two curves. This is because the\nestimated IOB is based on the value of the bolus calculated for the meal, on the other hand, the real IOB is based on\nthe value that is actually injected, which may vary from the calculated value due to pump noise.\nRemark 2: To address any potential infeasibilities during the MPC resolution, Npslack variables δuwere included in\nthe optimization problem. These were added to the upper bound umax\n2in equation (17), obtaining\nU′\n2={u(j) : 0≤u(j)≤umax\n2(j) +δu(j)},∀j∈INp−1\n0. (19)\n7CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n04:0010:0016:0022:0004:0010:0016:0022:0004:0010:0016:0022:0004:00\nT i m e012345I O B [ p m o l ]#104 I O B e s t i m a t i o n ( a d u l t 1 0 )\nI O B f r o m t h e s i m u l a t o r\nI O B e s t i m a t i o n\nFigure 2: The estimated boluses IOB is represented as the orange line and the IOB computed by the simulator is in\nblue. This is an example of the virtual patient Adult 10.\n3.3 CHoKI-based MPC implementation\nStarting from the proposal in [19], the MPC optimization problem which considers also the IOB is set as follows:\nmin\nu2,ya,δhyper,δhypo,δmin,δmax,δuVN(ˆx, u; Θ,D) (20a)\ns.t. ˆx(0|k) =x(k) (20b)\nˆx(j+ 1|k) =ˆF(ˆx(j|k), u1(j), u2(j)), j∈INc−1\n0 (20c)\nˆx(j+ 1|k) =ˆF(ˆx(j|k), K(¯x−x(j)) + ¯u), j∈INp−1\nNc(20d)\nˆy(j|k) =Mˆx(j|k), j∈INp−1\n0 (20e)\nu2(j)∈ U′\n2, j∈INp−1\n0, (20f)\nˆy(j|k)∈ Yj,δ, j∈INc−1\n0 (20g)\nˆy(j|k)∈ YNc,δ, j∈INp−1\nNc(20h)\nu1(j) = 0 , j∈INp−1\n1 (20i)\n70−δhypo≤ya≤140 + δhyper (20j)\nδhyper≥0, δhypo≥0 (20k)\nδmin(j)≥0, δmax(j)≥0, j∈INp−1\n0 (20l)\nδu(j)≥0, j∈INp−1\n0 (20m)\nwhere (20i) is used since the meals are not predictable, Yj,δcomes from (14) and (20f) is the constraint that takes into\nconsideration the IOB, from (19). The tightened constraints are computed as explained in the previous section, for all\nthe subjects, and the resulting Ncare reported in Table 1. The prediction horizon is set to Np= 12 for all subjects, to\nreach 60 minutes of predictions.\nThe cost function VN(ˆx, u; Θ,D)to be minimized is designed by summing several components, namely:\nVN(ˆx, u; Θ,D) =VNc+VNp+Vs+λVP+Vδ+Vu. (21)\nSpecifically, the stage costs VNc, along the control horizon Nc−1, and VNP, along the prediction horizon Np−1\nstarting from Nc, are:\nVNc=Nc−1X\nj=0∥ˆy(j|k)−ya∥2\nQ+∥u2(j)−uref∥2\nR, (22a)\n8CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\nVNp=Np−1X\nj=Nc∥ˆy(j|k)−ya∥2\nQ. (22b)\nwhere the insulin target urefis the constant basal insulin dose delivered by the UV A/Padova simulator for the default\ncontinuous therapy of the selected virtual patient. For the implementation of the MPC in a zone control fashion is\nrequired the presence of the setpoint ya, which is an auxiliary optimization variable and it has to be within 70 and\n140mg/dL. Two slack variables δhypo andδhyper are added to this interval for ya, which are other optimization\nvariables used to increase the range in the constraints when necessary. Therefore, an additional stationary cost Vshas\nto be considered. In Vs, the slack variables are weighted by some constants, that are set to be phypo> phyper to reflect\nthe higher danger of hypoglycemia compared to hyperglycemia [1].\nVs=phyperδ2\nhyper +phypoδ2\nhypo. (23)\nThe terminal cost VPpenalises the difference between the last state ˆx(Np|k)and the reference state ( xref, which\ncontains the set point ya, no meals and uref).\nVP=∥ˆx(Np|k)−xref∥2\nP, (24)\nwhere Pis the solution to the Riccati equation, given the LQR control gain Kfor the linearized system around the\nreference point (in Section 3.1). The terminal cost is normally used to ensure the stability of the MPC and in this case,\nit is weighted by a factor λ >0, as no terminal constraint is taken into account.\nOther slack optimisation variables are considered in the glycemic constraints (14). Thus, for the same reason as before,\nthe cost Vδmust be included to penalise them by considering two weights, where pmin> pmax,\nVδ=NPX\nj=1∥δmin(j)∥2\npmin+∥δmax(j)∥2\npmax. (25)\nThe last component is the cost Vu, to penalise additional slack variables δuwhich are included in the control action\nconstraints (20f),\nVu=NPX\nj=1∥δu(j)∥2\npu. (26)\nFurthermore, many combinations of weights were tested and the following were selected: R= 10 ,phypo= 1·107,\nphyper = 1·106,pmin= 1·107,pmax= 1·106,pu= 1·107andλ= 10 . The values of Qare shown in Table 1 and\nnote that in the cases where Ris greater than Q, this means that a more conservative controller is applied.\n4 Results\nThe proposed MPC was tested on the virtual adult patients of the UV A/Padova simulator, with customized controllers\nfor each subject. Three days were simulated, each day consisting of the following three meals: 40 g of carbohydrates\nat 06:00 am, 100 g at 12:00 pm and 60 g at 07:00 pm, with a duration of 15 min . The postprandial boluses were\ncomputed by the simulator and injected 20 min after the start of the meal. All the devices have the same noise setting\nas in the data acquisition stage.\nThe results of the simulations for all patients are shown in Figure 3. The top graph displays the BG trends caused by\nthe insulin injections depicted in the bottom graph, which vary based on the patient model. The primary objective is\nto reduce the frequency and the severity of hypoglycemic events, which are very dangerous in the short term, and it\ncan be seen that such a result is achieved.\nThe results obtained in this work are compared to the ones reported in [19], whose simulations are conducted without\nthe IOB constraints. In particular, the comparison of the mean and standard deviation of the BG values for the two cases\nis shown in Figure 4, where the blue dotted line represents the mean values of the cases with the IOB constraints and\nthe blue area displays their standard deviations, while in red the cases without the IOB. This indicates that including\nthe IOB in the constraints means to have a more conservative controller, since the BG level is higher in the cases\nwith the IOB safety constraints. It is also confirmed looking at the BG and basal insulin average values reported in\nTable 2 (i.e. mean and standard deviation), where the average BG values of the simulations with IOB constraints are\nhigher than in the ones without them, because of less insulin amount. This is due to the fact that the controller does\n9CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:000100200300400BG[mg/dL]BloodGlucose\n04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:00\nTime0200400600Insulin[pmol]Basalinsulin\nadult1 adult2 adult3 adult5 adult6 adult8 adult9 adult10 mealSimulationresults\nFigure 3: The upper plot displays BG trends for all patients. The green zone represents the safe range and the black\ntriangles depict meals. The lower plot shows basal insulin injections computed by the proposed MPC.\nB G : m e a n a n d s t a n d a r d d e v i a t i o n\n04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:00 10:00 16:00 22:00 04:00\nT i m e50100150200250300350B G [ m g / d L ]S t a n d a r d d e v i a t i o n ( w i t h o u t I O B )\nM e a n ( w i t h o u t I O B )\nS t a n d a r d d e v i a t i o n ( w i t h I O B )\nM e a n ( w i t h I O B )\nFigure 4: Comparison of the BG values: the simulations performed with IOB constraints are represented in blue, and\nwithout them are in red.\n10CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\na ) W i t h o u t I O B c o n s t r a i n t s\n#1#2#3#5#6#8#9#10\nV i r t u a l p a t i e n t s020%40%60%80%100%% T I RT i m e I n R a n g e\nb ) W i t h I O B c o n s t r a i n t s\n#1#2#3#5#6#8#9#10\nV i r t u a l p a t i e n t s020%40%60%80%100%% T I R\nw i t h I O B\n#1 #2 #3 #5 #6 #8 #9 #10\nV i r t u a l p a t i e n t s\n< 5 4 m g / d L 5 4 - 7 0 m g / d L 7 0 - 1 8 0 m g / d L 1 8 0 - 2 5 0 m g / d L > 2 5 0 m g / d L\nFigure 5: TIR results of the simulations performed with (graph on the right) and without the IOB constraints (graph\non the left). Each bar represents a specific subject.\nnot manage post-prandial boluses. As a result, when the BG value is high at mealtime, the bolus amount tends to be\nhigher (computed by the simulator). This leads to higher IOB, which in turn limits the basal corrections.\nAnother important tool for assessing AP performance is the Time In Range (TIR). This shows the percentage of time\na patient spends in each specific BG range. In particular, as required by the American Diabetes Association, the TIR\ntargets are as follows: <5%of time with BG higher than 250 mg/dL,<25% between 180-250 mg/dL,>70%\nbetween 70-180 mg/dL,<4%between 55-70 mg/dLand<1%for BG lower than 55 mg/dL. The proposed\ncontroller ensures that the requirements for the hypoglycemic ranges are always met, which is the main objective.\nHowever, the controller permits the subjects to stay a little longer in the two hyperglycemic ranges, which also means\nthat they stay within the 70-180 mg/dLrange for less than 70% of the simulation time. The results are displayed in\nFigure 5, where, for each subject, the graph on the left is for the cases without the IOB, while the one on the right is\nfor the cases with the IOB constraints.\nThe conservative results are a consequence of the linear function used to estimate the IOB. In fact, as shown in Figure 2,\nthe IOB is always overestimated. This can be improved searching for a polynomial or exponential function.\nAn additional useful tool can be the Glycemia Risk Index (GRI) [11], which is a quantitative measure designed to\nprovide a comprehensive evaluation of an individual’s susceptibility to hypoglycemia or hyperglycemia. It is obtained\nfrom\nGRI =\u0010\n3.0\u0000\np1+ 0.8p2\u0001\u0011\n+\u0010\n1.6\u0000\np4+ 0.5p3\u0001\u0011\n,\nwhere the first part is the hypoglycemic component and the second is the hyperglycemic one. It considers the same\npercentages of the TIR, where p1is the percentage of time in which the subject’s BG is less than 54 mg/dL, p2for\nthe BG between 54 and 70 mg/dL, p3for the BG between 180 and 250 mg/dL and p4for BG higher than 250 mg/dL.\nThe GRI can be displayed graphically on a grid with the hypoglycemia component on the horizontal axis and the\nhyperglycemia component on the vertical axis. Diagonal lines divide the graph into five zones (quintiles) based on\noverall glycemia quality, from best ( 0th-20thpercentile) to worst ( 81st-100thpercentile).\nThe GRI values are reported in Table 2, while the two components are represented in Figure 6a to understand which is\nthe higher one, where each dot on the graph describes a specific subject in the cases without the IOB, and the squares\nare for ones with the IOB constraints. In the cases without the IOB, Adult 8 and Adult 9 are in Zone B, while the\nothers are in safe Zone A. While in the cases with IOB, due to the higher BG values, Adult 8 is in Zone C and Adult\n11CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\na)GRI\n0 10 20 30\nHypoglycemiaComponent(%)0102030405060HyperglycemiaComponent(%) ZoneE:81-100\nZoneD:61-80\nZoneC:41-60\nZoneB:21-40\nZoneA:0-20\n>110 90 70 <50\nMinimumBG<110180300>400MaximumBGb)CVGA\nAL o w e r BU p p e r BC\nL o w e r CU p p e r C D\nadult1adult2adult3adult5adult6adult8adult9adult10\nadult1adult2adult3adult5adult6adult8adult9adult10Results:comparisonbetweensimulationswithandwithoutIOBconstraints\nWithoutIOB:\nWithIOB:\nFigure 6: Results of the simulations performed with the IOB constraints (squares), compared to the ones without them\n(dots): a) GRI and b) CVGA.\nTable 2: Comparison of the BG mean and standard deviation (in mg/dL), of the basal insulin u2mean and standard\ndeviation (in pmol ), and GRI for the controller with (on the right) and without (on the left) the IOB constraints.\nAdultWithout IOB With IOB\nBG mean ±std u2mean±std GRI BG mean ±std u2mean±std GRI\n#1 158.18±32.31 95.21±45.70 17.29 180.33±38.48 71.76±47.92 39.19\n#2 138.72±29.91 113.17±57.17 8.97 183.90±38.77 34.64±60.93 37.18\n#3 158.30±31.79 129.57±47.69 17.94 178.35±40.34 107.53±61.75 34.77\n#5 149.51±35.80 76.23±32.81 15.82 159.97±39.44 67.03±32.79 23.12\n#6 127.47±42.09 182.95±42.63 11.84 175.33±49.99 51.78±83.83 34.22\n#8 171.06±41.89 52.96±66.67 34.77 193.77±53.73 17.90±36.08 53.73\n#9 168.86±50.44 48.39±68.67 35.98 208.87±56.09 10.74±38.51 67.51\n#10 120.11±28.79 119.40±27.15 4.16 170.25±35.93 43.38±56.11 28.39\n9 is in Zone D, while the others are in Zone B. All the subjects lay on the y-axis, this is because our controller is\ndesigned to avoid hypoglycemia, which is why the higher risk component is the hyperglycemic one.\nUp to now, the average results are evaluated, but to have a more complete analysis, also the Control-Variability Grid\nAnalysis (CVGA) can be assessed. The CVGA [13] is a graphical representation that provides both visual and numer-\nical information about the quality of glycemic control. In Figure 6b, each dot on the graph describes a specific subject\nin the cases without IOB, while the squares are for those with IOB restrictions, with the minimum BG value as the\nx-coordinate and the maximum BG value as the y-coordinate. Considering the simulations with the IOB, these worst\ncases are all in the safe zones (except for Adult 6, Adult 8 and Adult 9, who are in Zone C).\n4.1 Simulations with insulin sensitivity variations\nIn this section, the proposed CHoKI-based MPC with the IOB constraints is tested on the same virtual patients, but\nwith variations in the insulin sensitivity. Insulin sensitivity refers to how responsive the cells are to insulin and this\ncan vary in the subject during the day [23]. The simulations are performed with the same setting as in the previous\ncases and the results are presented in Figure 7, where the upper part shows the BG trends that are obtained thanks to\nthe insulin injections displayed in the lower graph. To better evaluate the performances of the proposed controller in\nmanaging the variations in the insulin sensitivity, the CVGA, GRI and TIR results are evaluated as well (see Figure 8a,\n12CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00\nTime0100200300400BG[mg/dL]BloodGlucose\n00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00\nTime0500Insulin[pmol]Basalinsulin\nadult1 adult2 adult3 adult5 adult6 adult8 adult9 adult10 mealSimulationresults:virtualpatientswithvaryinginsulinsensitivity\nFigure 7: Proposed controller applied to the virtual patients with varying insulin sensitivity. Upper plot: BG trends of\nall patients, with the green zone for the safe range and the black triangles for the meal times. Lower plot: basal insulin\ninjections.\nFigure 8b and Figure 9, respectively). The results are quite promising, but the variations in insulin sensitivity affect\nthe ability of the CHoKI learning method. This is visible for example in Adult 10, who undergoes three hypoglycemic\nevents.\nThe CHoKI technique estimates the parameters LandPonce and offline. To better manage the insulin variability\na possible solution could be to use an adaptive CHoKI, which can update LandPvalues depending on the blood\nglucose and insulin situation.\n5 Conclusion\nA new MPC algorithm based on the CHoKI learning method was proposed to be used in the AP for managing basal\ninsulin in T1D patients, including IOB estimation to limit the amount of basal insulin injections. The whole system\nwas tested on the virtual patients of the UV A/Padova simulator. The proposed controller aims to drive and maintain\nthe BG level inside the euglycemic range most of the time, trying to avoid the more dangerous hypoglycemic events.\nThe obtained results seem promising, since the estimation of the IOB in the MPC helps in achieving such an aim. To\ndecrease the level of the hyperglycemic events, the IOB estimation could be improved. In particular, a polynomial or\nexponential curve can be tried instead of the linear weights employed in this case.\nThe proposed controller was also tested on virtual patients with variability in insulin sensitivity. To improve these\nresults, the next step for future works could be to identify multi-models, dividing the day into some intervals (such\nas breakfast, lunch and dinner) and trying to learn different behaviour, with the aim of controlling patients more\naccurately, thanks to the inclusion of the analysis of insulin sensitivity variations during the day.\nAcknowledgement\nThis work was funded by the National Plan for NRRP Complementary Investments (PNC, established with the decree-\nlaw 6 May 2021, n. 59, converted by law n. 101 of 2021) in the call for the funding of research initiatives for\ntechnologies and innovative trajectories in the health and care sectors (Directorial Decree n. 931 of 06-06-2022) -\nproject n. PNC0000003 - AdvaNced Technologies for Human-centrEd Medicine (project acronym: ANTHEM). This\nwork reflects only the authors’ views and opinions, neither the Ministry for University and Research nor the European\nCommission can be considered responsible for them.\n13CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n>110 90 70 <50\nMinimumBG<110180300>400MaximumBGa)CVGA\nAL o w e r BU p p e r BC\nL o w e r CU p p e r C D\nadult1adult2adult3adult5adult6adult8adult9adult10b)GRI\n0 10 20 30\nHypoglycemiaComponent(%)0102030405060HyperglycemiaComponent(%) ZoneE:81-100\nZoneD:61-80\nZoneC:41-60\nZoneB:21-40\nZoneA:0-20Insulinsensitivitycase\nFigure 8: Results of the simulations performed with the IOB constraints, applied to the varying insulin sensitivity case:\na) CVGA and b) GRI.\nTIR:insulinsensitivitycase\n#1#2#3#5#6#8#9#10\nVirtualpatients020%40%60%80%100%%TIR\n<54mg/dL54-70mg/dL70-180mg/dL180-250mg/dL >250mg/dL\nFigure 9: TIR results of the simulations performed with the IOB constraints, applied to the varying insulin sensitivity\ncase.\n14CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\nReferences\n[1] Pablo Abuin, Pablo S Rivadeneira, Antonio Ferramosca, and Alejandro Hern ´an Gonz ´alez. Artificial pancreas\nunder stable pulsatile mpc: Improving the closed-loop performance. Journal of Process Control , 92:246–260,\n2020.\n[2] Simone Del Favero, Chiara Toffanin, Lalo Magni, and Claudio Cobelli. Deployment of modular mpc for type 1\ndiabetes control: the italian experience 2008–2016. In The Artificial Pancreas , pages 153–182. Elsevier, 2019.\n[3] Christian Ellingsen, Eyal Dassau, Howard Zisser, Benyamin Grosman, Matthew W Percival, Lois Jovanovi ˇc, and\nFrancis J Doyle III. Safety constraints in an artificial pancreatic βcell: an implementation of model predictive\ncontrol with insulin on board. Journal of diabetes science and technology , 3(3):536–544, 2009.\n[4] Ravi Gondhalekar, Eyal Dassau, and Francis J Doyle III. Periodic zone-mpc with asymmetric costs for outpatient-\nready safety of an artificial pancreas to treat type 1 diabetes. Automatica , 71:237–246, 2016.\n[5] Alejandro H Gonz ´alez, Pablo S Rivadeneira, Antonio Ferramosca, Nicolas Magdelaine, and Claude H Moog.\nImpulsive zone mpc for type i diabetic patients based on a long-term model. IFAC-PapersOnLine , 50(1):14729–\n14734, 2017.\n[6] Alejandro H Gonz ´alez, Pablo S Rivadeneira, Antonio Ferramosca, Nicolas Magdelaine, and Claude H Moog.\nStable impulsive zone model predictive control for type 1 diabetic patients based on a long-term model. Optimal\nControl Applications and Methods , 41(6):2115–2136, 2020.\n[7] Iman Hajizadeh, Mudassir Rashid, and Ali Cinar. Plasma-insulin-cognizant adaptive model predictive control\nfor artificial pancreas systems. Journal of Process Control , 77:97–113, 2019.\n[8] Lukas Hewing, Kim P Wabersich, Marcel Menner, and Melanie N Zeilinger. Learning-based model predictive\ncontrol: Toward safe learning in control. Annual Review of Control, Robotics, and Autonomous Systems , 3:269–\n296, 2020.\n[9] Roman Hovorka, Valentina Canonico, Ludovic J Chassin, Ulrich Haueter, Massimo Massi-Benedetti,\nMarco Orsini Federici, Thomas R Pieber, Helga C Schaller, Lukas Schaupp, Thomas Vering, et al. Nonlinear\nmodel predictive control of glucose concentration in subjects with type 1 diabetes. Physiological measurement ,\n25(4):905, 2004.\n[10] Anastasia Katsarou, Soffia Gudbj ¨ornsdottir, Araz Rawshani, Dana Dabelea, Ezio Bonifacio, Barbara J Anderson,\nLaura Jacobsen, Desmond A Schatz, and Ake Lernmark. Type 1 diabetes mellitus. Nature Reviews Disease\nPrimers , 3(1), 2017.\n[11] David C Klonoff, Jing Wang, David Rodbard, Michael A Kohn, Chengdong Li, Dorian Liepmann, David Kerr,\nDavid Ahn, Anne L Peters, Guillermo E Umpierrez, et al. A glycemia risk index (gri) of hypoglycemia and\nhyperglycemia for continuous glucose monitoring validated by clinician ratings. Journal of diabetes science and\ntechnology , page 19322968221085273, 2022.\n[12] Fabian Le ´on-Vargas, Fabricio Garelli, Hern ´an De Battista, and Josep Veh ´ı. Postprandial blood glucose con-\ntrol using a hybrid adaptive pd controller with insulin-on-board limitation. Biomedical Signal Processing and\nControl , 8(6):724–732, 2013.\n[13] Lalo Magni, Davide M Raimondo, Chiara Dalla Man, Marc Breton, Stephen Patek, Giuseppe De Nicolao,\nClaudio Cobelli, and Boris P Kovatchev. Evaluating the efficacy of closed-loop glucose regulation via control-\nvariability grid analysis. Journal of diabetes science and technology , 2(4):630–635, 2008.\n[14] Jos ´e Mar ´ıa Manzano, Daniel Limon, David Mu ˜noz de la Pe ˜na, and Jan-Peter Calliess. Robust learning-based\nMPC for nonlinear constrained systems. Automatica , 117, 2020.\n[15] Jos ´e Mar ´ıa Manzano, David Mu ˜noz de la Pe ˜na, Jan-Peter Calliess, and Daniel Limon. Componentwise H ¨older\nInference for Robust Learning-Based MPC. IEEE Trans. on Automatic Control , 66(11):5577–5583, 2021.\n[16] Sun Joon Moon, Inha Jung, and Cheol-Young Park. Current advances of artificial pancreas systems: a compre-\nhensive review of the clinical evidence. Diabetes & Metabolism Journal , 45(6):813–839, 2021.\n[17] Rawlings, James Blake, and David Q. Mayne. Model predictive control: Theory and design . Nob Hill Pub.,\n2009.\n[18] Dawei Shi, Eyal Dassau, and Francis J Doyle. Adaptive zone model predictive control of artificial pancreas based\non glucose-and velocity-dependent control penalties. IEEE Trans. on Biomedical Engineering , 66(4):1045–1054,\n2018.\n[19] Beatrice Sonzogni, Jos ´e Mar ´ıa Manzano, Marco Polver, Fabio Previdi, and Antonio Ferramosca. CHoKI-based\nMPC for blood glucose regulation in artificial pancreas. Proceedings of 22ndIFAC World Congress , 2023.\n15CHoKI-based MPC for blood glucose regulation in Artificial Pancreas A P REPRINT\n[20] Paola Soru, Giuseppe De Nicolao, Chiara Toffanin, Chiara Dalla Man, Claudio Cobelli, Lalo Magni, AP@ Home\nConsortium, et al. Mpc based artificial pancreas: strategies for individualization and meal compensation. Annual\nReviews in Control , 36(1):118–128, 2012.\n[21] The Epsilon Group. DMMS.R (Version 1.1) [Software]. Retrieved from https://tegvirginia.com/, 2016.\n[22] Chiara Toffanin, Mirko Messori, Federico Di Palma, Giuseppe De Nicolao, Claudio Cobelli, and Lalo Magni.\nArtificial pancreas: model predictive control design from clinical experience. Journal of Diabetes Science and\nTechnology , 7(6):1470–1483, 2013.\n[23] Roberto Visentin, Chiara Dalla Man, Yogish C Kudva, Ananda Basu, and Claudio Cobelli. Circadian variability\nof insulin sensitivity: physiological input for in silico artificial pancreas. Diabetes technology & therapeutics ,\n17(1):1–7, 2015.\n.\n16" }, { "title": "2401.17181v1.Transfer_Learning_for_Text_Diffusion_Models.pdf", "content": "Transfer Learning for Text Diffusion Models\nKehang Han1∗, Kathleen Kenealy1∗, Aditya Barua2∗, Noah Fiedel1, Noah Constant1\n1Google DeepMind2Google\n{kehanghan, kkenealy, adityabarua, nfiedel, nconstant}@google.com\nAbstract\nIn this report, we explore the potential for text\ndiffusion to replace autoregressive (AR) decod-\ning for the training and deployment of large\nlanguage models (LLMs). We are particularly\ninterested to see whether pretrained AR models\ncan be transformed into text diffusion models\nthrough a lightweight adaptation procedure we\ncall “AR2Diff”. We begin by establishing a\nstrong baseline setup for training text diffusion\nmodels. Comparing across multiple architec-\ntures and pretraining objectives, we find that\ntraining a decoder-only model with a prefix\nLM objective is best or near-best across several\ntasks. Building on this finding, we test various\ntransfer learning setups for text diffusion mod-\nels. On machine translation, we find that text\ndiffusion underperforms the standard AR ap-\nproach. However, on code synthesis and extrac-\ntive QA, we find diffusion models trained from\nscratch outperform AR models in many cases.\nWe also observe quality gains from AR2Diff—\nadapting AR models to use diffusion decoding.\nThese results are promising given that text dif-\nfusion is relatively underexplored and can be\nsignificantly faster than AR decoding for long\ntext generation.\n1 Introduction\nIn recent years, large language models (LLMs)\nhave grown in scale, capability, and popularity\n(Brown et al., 2020; Chowdhery et al., 2022), and\nare increasingly used to generate long-form text\nsuch as summaries, blocks of code, or in-depth ex-\nplanations (OpenAI, 2023; Anil et al., 2023). To\nour knowledge, all popular LLMs are autoregres-\nsive(AR)—generating one token at a time in tex-\ntual order, each conditioned on the sequence gen-\nerated so far. While AR generation is well under-\nstood and has been highly optimized, its strict left-\nto-right factorization may be overly constraining.\nGenerating token-by-token is inherently inefficient,\n∗Equal contribution.particularly on long but predictable spans of text\n(e.g., copying a serial number from the context one\ndigit at a time). Additionally, this strict order may\nnot provide the ideal scaffold for planning a com-\nposition. Human writers typically outline, draft,\nrevise, and proofread their work, and it seems plau-\nsible that machines could benefit from a similar\niterative approach.1\nAs an alternative, many non-AR decoding meth-\nods have been proposed (see section §2), which\ngenerate multiple sequence positions in parallel, or\nmake progressive edits to a “rough” initial genera-\ntion. Several of these have shown promising results\non specific tasks. For example, SUNDAE’s text\ndiffusion approach (Savinov et al., 2022) achieves\nsimilar quality to an AR baseline on machine trans-\nlation while decoding over 2 ×faster.\nHowever, despite positive findings, non-AR tech-\nniques have failed to gain traction, and remain un-\nused in the space of large language models. We\nsuspect this may be due to the inertia behind classic\nAR methods, and the high cost and risk of tuning\nand training large models from scratch using non-\nstandard training losses and decoding methods.\nWith an eye to lowering this cost of entry and eas-\ning the transition to more efficient text generation\nat scale, in this paper we investigate the potential\nfor adapting existing pretrained AR model check-\npoints to perform non-AR generation. We use a\nsimplified version of SUNDAE text diffusion as our\ncanonical non-AR implementation; thus we refer\nto this lightweight adaptation process as AR2Diff\n(AR to Diffusion) .\nMore specifically, we are interested in testing\nthe ability of text diffusion methods to compete at\nscale in the popular transfer learning setting, where\na model is pretrained on unsupervised data and\n1“Chain of thought” prompting (Wei et al., 2022) provides\na mechanism for models to reason about or draft the desired\noutput before producing it. However, the final output is still\ngenerated autoregressively.arXiv:2401.17181v1 [cs.CL] 30 Jan 2024applied to diverse downstream tasks. We conduct a\nseries of experiments comparing text diffusion to\nAR baselines across different model architectures,\ntasks, and transfer learning settings.\nOur main contributions are: (1) showing that lan-\nguage models pretrained and fine-tuned using text\ndiffusion can be competitive with autoregressive\nmodels on several downstream tasks, (2) showing\nthat pretrained AR models can be transformed into\ndiffusion models via a lightweight adaptation.\n2 Related Work\nPrevious work has explored a wide range of non-\nautoregressive methods for text generation (Gu\net al., 2018; Lee et al., 2018; Stern et al., 2019;\nGhazvininejad et al., 2019). In the last few years,\ndiffusion models (Sohl-Dickstein et al., 2015) have\nemerged as the primary technique for image gener-\nation (Rombach et al., 2021; Ramesh et al., 2022;\nSaharia et al., 2022). Many recent efforts have ap-\nplied diffusion methods to textgeneration (Savinov\net al., 2022; Li et al., 2022; Reid et al., 2023; Chen\net al., 2023; Strudel et al., 2022; Dieleman et al.,\n2022; Zheng et al., 2023; Lin et al., 2023; Gong\net al., 2023; Yuan et al., 2023; Wu et al., 2023), but\nnone has yet gained adoption in the space of large\nlanguage models.\nWhile promising, text diffusion techniques have\nlargely not been tested at scale or in multitask trans-\nfer learning settings, though see Lin et al. (2023)\nand Ye et al. (2023) for recent work in this direc-\ntion. Furthermore, it remains unclear if these meth-\nods demand training new diffusion models from\nscratch, or if AR models can be efficiently adapted\ninto diffusion models. We explore these questions\nempirically in section §4.\nOne line of previous work shows that non-AR\nmethods benefit from “AR distillation” (Kim and\nRush, 2016; Gu et al., 2018; Saharia et al., 2020;\nGu and Kong, 2021)—training a non-AR model\nfrom scratch on silver data generated via the pre-\ndictions of an existing AR model. AR distillation\nis similar to our AR2Diff adaptation in that both\nleverage a preexisting AR model. However they\ndiffer in that our method initializes the diffusion\nmodel directly from an AR checkpoint, and trains\non gold data. Given the significant recent invest-\nment in training large AR models, we believe that\nlightweight adaptation of existing checkpoints is\na promising direction compared to training non-\nstandard models from scratch.Recently, Lin et al. (2023) show good results\npretraining a text diffusion encoder-decoder model\nand fine-tuning it on downstream tasks. Like our\nwork, this validates the effectiveness of pretraining\ntext diffusion models at scale.\nMore recently, building on “reparameterized dis-\ncrete diffusion models” (Zheng et al., 2023), Ye\net al. (2023) show the possibility of converting\nlarge AR models (up to 10B parameters) into text\ndiffusion models during task-specific fine-tuning—\ntheir “diffusive adaptation”. This work shares our\ngoal of demonstrating that text diffusion can be\npractical at scale. Our work differs in (i) building\non SUNDAE as opposed to RDM, (ii) including\ndiffusion models pretrained from scratch as base-\nlines, (iii) comparing different architectures and\nobjectives for diffusion pretraining, and (iv) testing\nadaptation during pretraining (our AR2Diff Nwith\nN > 0), as opposed to only during fine-tuning (our\nAR2Diff 0).\n3 Evaluation Tasks\nWe experiment with three downstream tasks. First,\nwe use WMT14 French-English translation (Bo-\njar et al., 2014), as machine translation is widely\nused to evaluate generative models, particularly in\nwork on non-AR models.\nSecond, we evaluate on the popular SQuAD\nquestion answering task (Rajpurkar et al., 2016).\nAs an extractive QA task, this does not require open\ngeneration, and most targets are fairly short, often\njust a few words long. While text diffusion models\nare unlikely to deliver speed gains on tasks with\nshort outputs (see Section §4.7), we feel it is still\nimportant to test for quality on text understanding\ntasks. This can help establish whether pretrained\ndiffusion models can be an effective general foun-\ndation for language understanding, and ensures that\nour findings are interpretable within the literature\non transfer learning in NLP.\nFinally, we evaluate on Mostly Basic Python\nProblems (MBPP) (Austin et al., 2021), a recent\nbenchmark requiring models to generate full solu-\ntions to simple Python programming tasks. This\ntask is fairly open-ended, as there are many work-\ning solutions to a given task, depending on choices\nof algorithm, coding style, variable names, and so\non. Compared to open-ended natural language gen-\neration, this benchmark has clear and meaningful\nautomatic evaluation metrics, as we can run the\ngenerated code and assess whether it passes rele-vant test cases. When tokenized using the PaLM\n(Chowdhery et al., 2022) vocabulary we adopt in\nour experiments, median target length is 59tokens,\nand 90th percentile is 150tokens.\n4 Experiments\n4.1 Diffusion implementation\nOur diffusion implementation follows SUNDAE\n(Savinov et al., 2022). More specifically, we use\nstandard Transformer (Vaswani et al., 2017) archi-\ntectures (either encoder-decoder or decoder-only)\nas implemented in the T5X (Roberts et al., 2022)\nlibrary. As SUNDAE performs discrete diffusion\nin surface token space, the decoder inputs and out-\nputs are tokens, in line with standard AR models.\nThese implementation choices allow us to reuse\nexisting frameworks for autoregressive LLM train-\ning with relatively minor changes. As a result, we\ncan easily experiment with using pretrained AR\nmodel checkpoints and adapting these to perform\ntext diffusion.\nFor training, we use the SUNDAE L(1:2)loss,\nwhich incorporates one step of “unrolled denois-\ning”, encouraging the model to be able to refine\nits single-step predictions further towards the tar-\nget. More concretely, for target sequence x, we\nrandomly corrupt a random proportion of tokens\n(sampling from a uniform distribution) to produce\nxc, which is passed as input to the denoising model\nto produce logits l1. The “logits loss” L(1)is the\ncross-entropy between l1andx. “Unrolled logits”\nare computed by sampling2from l1and passing\nthese tokens back as inputs to the denoising model,\nproducing l2. The “unrolled logits loss” L(2)is the\ncross-entropy between l2andx. For the overall\nloss, we use L(1)+L(2).\nFor inference, we follow SUNDAE in using low-\ntemperature sampling ( τ= 0.2), decoding Nsam-\nples in parallel (we use N= 8 by default), and\nreranking them based on “model score”: the cross-\nentropy between the decoder input and output log-\nits on the final step of diffusion. We use 10diffu-\nsion decoding steps by default; thus on tasks with\ntargets longer than 10tokens, our diffusion mod-\nels use fewer decoding steps than an AR model.3\n2We sample from l1using temperature 0.0(argmax), as\nopposed to SUNDAE’s temperature 1.0, as we found this per-\nformed best in early ablations on WMT14, with temperature\nin {0.0,0.1,1.0}.\n3As AR models can cache and reuse activations from ear-\nlier sequence positions for subsequent decoding steps (thanks\nto the causal attention mask), they use significantly fewerThese choices are ablated in section §4.6.\nFor simplicity, we forgo SUNDAE’s target\nlength prediction module, opting instead to let the\nmodel learn to predict sequence length end-to-end\nthrough the presence of padding tokens observed\nduring training. As a result, our text diffusion mod-\nels have no additional parameters beyond those\nwithin the Transformer (encoder-)decoder.\n4.2 Selecting objective and architecture\nPrevious work on text diffusion has focused on the\nsingle-task setting, either training and evaluating\non unconditional text generation, or training from\nscratch on an end task, such as machine transla-\ntion.4In contrast, we aim to evaluate text diffusion\nin the transfer learning setting—pretraining a large\nmodel, and adapting it to a range of downstream\ntasks. As a first step, and to cut down the space\nof further experiments, we first seek to identify a\nmodel architecture and pretraining objective well-\nsuited to text diffusion.\nThe T5 study on transfer learning for AR text-\nto-text models (Raffel et al., 2020) recommends\nusing an encoder-decoder architecture and a “span\ncorruption” objective—masking multi-token spans\nin the input, and reconstructing these in the tar-\nget. By comparison, many subsequent LLMs have\nconverged on a decoder-only architecture with a\nstandard LM objective (Brown et al., 2020; Chowd-\nhery et al., 2022). To establish which setting works\nbest for diffusion, we test all four combinations of\narchitecture ( encoder-decoder vs.decoder-only )\nand objective ( span corruption vs.prefix LM ), as\nshown in Figure 1.5\nWe train each model on the same pretraining\nmixture, consisting of 80% multilingual web crawl\ndata from mC4 (Xue et al., 2021) and 20% Python\ncode from “The Stack” (Kocetkov et al., 2022). All\nmodels use the T5 Base size transformer architec-\nture and pretrain for 1million steps on batches\nof size 128and sequence length 1024 . We then\nFLOPs per step, when other factors are held constant. We do\nnot present a full picture of the speed vs. quality tradeoffs of\ntext diffusion models here. Previous work has shown that text\ndiffusion can be competitive on speed and quality, even com-\nparing against AR inference with caching enabled (Savinov\net al., 2022). We assume here that diffusion in 10steps is fast\nenough to have practical value, and focus on quality.\n4Ye et al. (2023) adapt pretrained AR models for diffusion\nacross multiple tasks, but do not explore pretraining a general-\npurpose diffusion model that can be adapted to specific tasks.\n5We choose the “prefix LM” objective rather than the stan-\ndard causal LM objective, as it is compatible with the encoder-\ndecoder architecture, and has been shown to outperform causal\nLM in apples-to-apples comparisons (Tay et al., 2023).Span Corruption Prefix LM Text diffusion models work well \nfor transfer learning! Pretraining Corpus Example \nText models work well for input \n diffusion transfer learning! target Text diffusion models work input \nwell for transfer learning! target \nEncoder-Decoder Decoder Only \nEnc\nDec\nDec\ninput \ntarget Diffusion \nCorruption input \ntarget Diffusion \nCorruption loss lossFigure 1: Pretraining objectives and model architectures. The and symbols are unique sentinel tokens\ndenoting masked spans. Note, the “masking noise” applied to produce the span corruption input/target is independent\nfrom the “diffusion noise” which randomly corrupts a subset of target tokens. Loss is only computed over target\ntokens. In the decoder-only setting, input tokens are frozen when computing the unrolled logits input ( l2).\nPretraining WMT14 En-Fr SQuAD MBPP\nArchitecture Objective (BLEU) (F1) (Pass@80 %)\nEncoder-Decoder Prefix LM 27.6 75.8 0.0\nDecoder-only Prefix LM 29.8 77.4 12.2\nEncoder-Decoder Span Corruption 28.7 78.2 0.0\nDecoder-only Span Corruption 29.1 80.6 11.1\nTable 1: Diffusion model performance on three tasks across model architecture and pretraining objective. The\nDecoder-only architecture outperforms Encoder-Decoder across all three tasks, despite using fewer parameters.\nfine-tune each model separately on WMT14 En-\nFr, SQuAD, and MBPP (producing 12 fine-tuned\nmodels total) and evaluate across all tasks. We\nuse a fine-tuning batch size of 128and a constant\nlearning rate of 0.001across all tasks. We fine-tune\n500K steps for WMT14 En-Fr and 250K steps for\nSQuAD, with checkpoints taken every 1,000steps.\nFor MBPP due to smaller dataset size, we fine-tune\nfor5,000steps with checkpoints taken every 50\nsteps. In all cases, we terminate fine-tuning if clear\nevidence of over-fitting is observed. We reuse the\n256K token SentencePiece vocabulary from PaLM\n(Chowdhery et al., 2022). Our decoder-only mod-\nels have roughly 280M parameters (including em-\nbedding parameters), while our encoder-decoder\nmodels have roughly 590M parameters.\nThe results in Table 1 show that our decoder-\nonly models perform the best across all three tasks,despite their lower parameter count. This advan-\ntage is especially clear on code synthesis (MBPP),\nwhere the encoder-decoder models fail to solve\nany problem in the test set, even on the permis-\nsive “Pass@80” metric that samples the model 80\ntimes and is scored as correct if anyof these candi-\ndates passes. In line with Tay et al. (2023), we sus-\npect that pretraining the model to generate longer\ncontiguous spans is a better-matched objective for\ndownstream tasks like MBPP requiring long coher-\nent generation.\nOur findings on pretraining objective are less\nconclusive, with Prefix LM performing the best on\nWMT and MBPP, while Span Corruption does best\non SQuAD. With this in mind, we select “decoder-\nonly + prefix LM” for our subsequent experiments,\nas this setup is increasingly standard for LLM train-\ning, and does relatively well (best or second-best)Decoder \n[causal attn.] \nPretraining \nCorpus AR loss 1) AR Pretraining \nDecoder \n[bidirectional attn.] \nPretraining \nCorpus Diffusion loss 2) AR2Diff Adaptation \nDecoder \n[bidirectional attn.] \nFine-tuning \nTask Data Diffusion loss 3) Fine-Tuning Figure 2: Illustration of our AR2Diff method. 1) Pretrain an AR decoder with causal attention. 2) Continue\npretraining as a diffusion model with bidirectional attention. 3) Fine-tune as a diffusion model on the end task.\nacross all our tasks.\n4.3 Transfer learning baselines\nWe now turn to testing various transfer learning\nstrategies across model scales. As our core base-\nlines, we pretrain both AR and diffusion models at\nBase ( 280M), Large ( 270M), and XL ( 1.7B) sizes.\nThese all use a decoder-only architecture and pre-\nfix LM objective, and train on the same pretraining\nmixture from the previous section ( 80% multilin-\ngual web pages and 20% Python code). As before,\nwe pretrain for 1M steps, with batch size 128and\nsequence length 1024 . Note, our diffusion models\nuse bidirectional attention to allow modifying all\nsequence positions in parallel, but are otherwise\narchitecturally identical to their AR counterparts.\nFor the AR baselines, at inference time, we use\ngreedy decoding for SQuAD, following T5, and use\ntemperature sampling for MBPP, following Austin\net al. (2021). For WMT, we use greedy decoding as\nopposed to the more commonly used beam search\nfor a fairer comparison, as we did not investigate\nthe use of beam search for diffusion models; see\nReid et al. (2023) for work in this direction.\nWe then fine-tune each of these models sepa-\nrately for each of our three tasks. Results are shown\nin Table 2, and discussed in section §4.5.\n4.4 AR2Diff: Adapting from AR to diffusion\nBeyond pure AR and pure diffusion training, we ex-\nplore “AR2Diff” methods for adapting a pretrained\nAR model into a diffusion model later in training.\nFirst, we experiment with simply fine-tuning an\nAR checkpoint directly using our diffusion training\nprocedure—enabling bidirectional attention, and\nusing the SUNDAE diffusion training loss. We\nrefer to this method as AR2Diff 0, and use our base-\nline AR model checkpoint as the starting point for\nfine-tuning.\nWe also experiment with pretraining the modelfor additional steps as a diffusion model before\nfine-tuning, as illustrated in Figure 2. We start with\nour pretrained AR checkpoint, continue pretraining\nfor an additional Nsteps using diffusion training,\nand then fine-tune (still with diffusion) on each\nevaluation task separately. We refer to this method\nas AR2Diff N.\n4.5 Core results\nResults comparing AR2Diff to our autoregressive\nand diffusion baselines across model sizes are\nshown in Table 2.\nOn WMT14 En-Fr, the AR baseline performs\nthe best across model sizes.6Our observed gap\nbetween diffusion and AR is larger than that of\nSavinov et al. (2022), where SUNDAE text dif-\nfusion comes with 1BLEU point of an AR base-\nline. The difference may be due to our (i) using a\ntransfer learning setting where we pretrain before\nfine-tuning, (ii) not using SUNDAE’s length pre-\ndiction module, (iii) sampling fewer candidates at\ninference time ( 8vs.16).\nInterestingly, while at Base size AR2Diff pro-\nvides no advantage on WMT, at Large and XL sizes\nwe see AR2Diff delivers a significant gain over\nthe pure diffusion baseline, and this gain increases\nwith the length of adaptation. This suggests that\nAR2Diff may be valuable not just as a resource-\nsaving method (leveraging AR checkpoints to avoid\npretraining diffusion models from scratch), but also\nas a means of achieving stronger diffusion models\nthrough mixed-objective training.\nOn SQuAD question answering, our diffu-\nsion baseline outperforms the AR baseline at\nBase and Large sizes (Base: 68.1→77.4, Large:\n6We note our Base AR baseline underperforms ( 32.27\nvs.37.5) a similar baseline from Raffel et al. (2020), a Base\nsize decoder-only model trained with the same prefix LM\nobjective. This could stem from differences in pretraining data,\nmodel architecture, fine-tuning procedure, and/or inference\nsettings (e.g., our use of greedy decoding).WMT14 En-Fr SQuAD MBPP\nMethod Size (BLEU) (F1) (Pass@80 %)\nAutoregressive Base 33.27 68.11 5.5\nDiffusion Base 29.83 77.41 12.2\nAR2Diff 0 Base 29.62 64.77 1.1\nAR2Diff 10,000 Base 29.41 68.12 4.4\nAR2Diff 100 ,000 Base 29.92 71.87 7.7\nAutoregressive Large 34.92 78.43 15.5\nDiffusion Large 29.36 80.56 12.2\nAR2Diff 0 Large 31.14 77.82 3.3\nAR2Diff 10,000 Large 31.97 79.62 8.8\nAR2Diff 100 ,000 Large 32.20 80.71 10.0\nAutoregressive XL 35.48 84.08 15.5\nDiffusion XL 29.30 82.78 18.8\nAR2Diff 0 XL 32.36 80.95 6.6\nAR2Diff 10,000 XL 32.39 80.71 11.1\nAR2Diff 100 ,000 XL 32.55 83.54 15.5\nTable 2: Performance of various models across three tasks and three sizes, comparing: (i) an AR baseline, (ii) a\ndiffusion baseline, and (iii) AR2Diff models that adapt the pretrained AR baseline via diffusion training for Nsteps\nbefore fine-tuning using diffusion, with N∈{0,10K,100K}.\n78.4→80.6), but underperforms at XL size\n(84.1→82.8).7While adapting to diffusion only\nduring fine-tuning (AR2Diff 0) is ineffective, adapt-\ning for Nsteps before fine-tuning (AR2Diff N) out-\nperforms the AR baseline at most sizes, and im-\nproves monotonically with N.\nOn MBPP code synthesis, diffusion outperforms\nthe AR baseline for two out of three model sizes,\nincluding the largest XL size ( 15.5→18.8). As on\nother tasks, AR2Diff tends to improve with longer\nadaptation before fine-tuning.\n4.6 Ablations\nOur results so far have performed diffusion infer-\nence by running 10steps (“ num_steps ”) of denois-\ning over 8randomly sampled decoder inputs per\nexample (“ num_samples ”). Note, only the output\nwith the highest model score is used for evaluation.\nTable 3 shows the results of varying num_steps ∈\n{5,10,20} and num_samples ∈{4,8,16}. On the\nMBPP code synthesis task, we find that increas-\ning step and samples boosts performance, in line\nwith Savinov et al. (2022). Increasing denoising\nsteps is particularly helpful ( 5.5→16.7), but at the\ncost of slower inference. On SQuAD the effect of\nthese parameters is more marginal. More generally,\nwe suspect that additional steps and samples may\nbe helpful on long-form text generation tasks like\nMBPP that are relatively underspecified (e.g., ad-\nmitting many correct answers in different styles).\n7As on WMT, these scores are below the results reported\nby Raffel et al. (2020) using a similar baseline ( 85.4). See\nfootnote 6.\n500 1000 1500 2000 2500 3000 3500 4000\nmax decode length0102030405060inference time (sec)\nauto-regressive\ndiffusion with 50 steps\ndiffusion with 20 steps\ndiffusion with 10 stepsFigure 3: By varying the decoding sequence length,\nwe measure inference time of autoregressive decoding\nvs. diffusion decoding\nBy comparison, SQuAD targets are typically short,\nand are constrained to be spans from the input.\n4.7 Inference speed analysis\nDiffusion language models have the potential to re-\nduce inference serving costs of long text generation,\ncompared with AR models. Here we show some\npreliminary results on the inference speed quantita-\ntively. We decode sequences of equal length with\nAR and diffusion models, and measure correspond-\ning wall-clock times. For diffusion models, we use\n10diffusion steps as our base case, matching our\nprimary evaluation setup for the WMT, SQuAD\nand MBPP tasks.\nWe observe an increasing advantage of using dif-\nfusion for inference speedup when the generationSQuAD MBPP\nMethod steps samples (F1) (Pass@80 %)\nAutoregressive - - 68.11 5.5\nDiffusion 5 8 77.41 5.5\nDiffusion 10 8 77.41 12.2\nDiffusion 20 8 77.72 16.7\nDiffusion 10 4 77.51 11.1\nDiffusion 10 8 77.41 12.2\nDiffusion 10 16 77.13 13.3\nTable 3: Ablations on diffusion inference hyperparameters num_steps andnum_samples . Increasing steps and\nsamples leads to clear gains on MBPP, which requires long-form code synthesis, while the effects on SQuAD\nextractive QA are marginal.\nis long. Figure 3 shows as the decoding sequence\nlength increases from 500tokens (e.g., MBPP task)\nto4,000tokens, the speedup gained by diffusion\n(using 10steps) increases from 10×to30×.\nNote that a single AR decoding step ( 14ms per\ntoken generated) is still much faster than a single\ndiffusion step ( 179ms per denoising step) in our\nimplementation. This is likely due to the diffu-\nsion model’s lacking the key-value caching widely\nused to optimize AR inference. Whether caching\nor other efficiency optimizations can further ex-\ntend the speed gains of diffusion is an interesting\nquestion for future research.\nAcknowledgments\nWe are grateful to Jiaxin Shi for helpful comments\non an earlier draft.\nReferences\nRohan Anil, Andrew M. Dai, Orhan Firat, Melvin John-\nson, Dmitry Lepikhin, Alexandre Passos, Siamak\nShakeri, Emanuel Taropa, Paige Bailey, Zhifeng\nChen, Eric Chu, Jonathan H. Clark, Laurent El\nShafey, Yanping Huang, Kathy Meier-Hellstern, Gau-\nrav Mishra, Erica Moreira, Mark Omernick, Kevin\nRobinson, Sebastian Ruder, Yi Tay, Kefan Xiao,\nYuanzhong Xu, Yujing Zhang, Gustavo Hernandez\nAbrego, Junwhan Ahn, Jacob Austin, Paul Barham,\nJan Botha, James Bradbury, Siddhartha Brahma,\nKevin Brooks, Michele Catasta, Yong Cheng, Colin\nCherry, Christopher A. Choquette-Choo, Aakanksha\nChowdhery, Clément Crepy, Shachi Dave, Mostafa\nDehghani, Sunipa Dev, Jacob Devlin, Mark Díaz,\nNan Du, Ethan Dyer, Vlad Feinberg, Fangxiaoyu\nFeng, Vlad Fienber, Markus Freitag, Xavier Gar-\ncia, Sebastian Gehrmann, Lucas Gonzalez, Guy Gur-\nAri, Steven Hand, Hadi Hashemi, Le Hou, Joshua\nHowland, Andrea Hu, Jeffrey Hui, Jeremy Hur-\nwitz, Michael Isard, Abe Ittycheriah, Matthew Jagiel-\nski, Wenhao Jia, Kathleen Kenealy, Maxim Krikun,\nSneha Kudugunta, Chang Lan, Katherine Lee, Ben-jamin Lee, Eric Li, Music Li, Wei Li, YaGuang Li,\nJian Li, Hyeontaek Lim, Hanzhao Lin, Zhongtao Liu,\nFrederick Liu, Marcello Maggioni, Aroma Mahendru,\nJoshua Maynez, Vedant Misra, Maysam Moussalem,\nZachary Nado, John Nham, Eric Ni, Andrew Nys-\ntrom, Alicia Parrish, Marie Pellat, Martin Polacek,\nAlex Polozov, Reiner Pope, Siyuan Qiao, Emily Reif,\nBryan Richter, Parker Riley, Alex Castro Ros, Au-\nrko Roy, Brennan Saeta, Rajkumar Samuel, Renee\nShelby, Ambrose Slone, Daniel Smilkov, David R.\nSo, Daniel Sohn, Simon Tokumine, Dasha Valter,\nVijay Vasudevan, Kiran V odrahalli, Xuezhi Wang,\nPidong Wang, Zirui Wang, Tao Wang, John Wiet-\ning, Yuhuai Wu, Kelvin Xu, Yunhan Xu, Linting\nXue, Pengcheng Yin, Jiahui Yu, Qiao Zhang, Steven\nZheng, Ce Zheng, Weikang Zhou, Denny Zhou, Slav\nPetrov, and Yonghui Wu. 2023. Palm 2 technical\nreport.\nJacob Austin, Augustus Odena, Maxwell Nye, Maarten\nBosma, Henryk Michalewski, David Dohan, Ellen\nJiang, Carrie Cai, Michael Terry, Quoc Le, and\nCharles Sutton. 2021. Program synthesis with large\nlanguage models.\nOndˇrej Bojar, Christian Buck, Christian Federmann,\nBarry Haddow, Philipp Koehn, Johannes Leveling,\nChristof Monz, Pavel Pecina, Matt Post, Herve Saint-\nAmand, Radu Soricut, Lucia Specia, and Aleš Tam-\nchyna. 2014. Findings of the 2014 workshop on\nstatistical machine translation. In Proceedings of the\nNinth Workshop on Statistical Machine Translation ,\npages 12–58, Baltimore, Maryland, USA. Associa-\ntion for Computational Linguistics.\nTom B. Brown, Benjamin Mann, Nick Ryder, Melanie\nSubbiah, Jared Kaplan, Prafulla Dhariwal, Arvind\nNeelakantan, Pranav Shyam, Girish Sastry, Amanda\nAskell, Sandhini Agarwal, Ariel Herbert-V oss,\nGretchen Krueger, Tom Henighan, Rewon Child,\nAditya Ramesh, Daniel M. Ziegler, Jeffrey Wu,\nClemens Winter, Christopher Hesse, Mark Chen, Eric\nSigler, Mateusz Litwin, Scott Gray, Benjamin Chess,\nJack Clark, Christopher Berner, Sam McCandlish,\nAlec Radford, Ilya Sutskever, and Dario Amodei.\n2020. Language models are few-shot learners.\nTing Chen, Ruixiang Zhang, and Geoffrey Hinton. 2023.Analog bits: Generating discrete data using diffusion\nmodels with self-conditioning. In The Eleventh Inter-\nnational Conference on Learning Representations .\nAakanksha Chowdhery, Sharan Narang, Jacob Devlin,\nMaarten Bosma, Gaurav Mishra, Adam Roberts,\nPaul Barham, Hyung Won Chung, Charles Sutton,\nSebastian Gehrmann, Parker Schuh, Kensen Shi,\nSasha Tsvyashchenko, Joshua Maynez, Abhishek\nRao, Parker Barnes, Yi Tay, Noam Shazeer, Vin-\nodkumar Prabhakaran, Emily Reif, Nan Du, Ben\nHutchinson, Reiner Pope, James Bradbury, Jacob\nAustin, Michael Isard, Guy Gur-Ari, Pengcheng Yin,\nToju Duke, Anselm Levskaya, Sanjay Ghemawat,\nSunipa Dev, Henryk Michalewski, Xavier Garcia,\nVedant Misra, Kevin Robinson, Liam Fedus, Denny\nZhou, Daphne Ippolito, David Luan, Hyeontaek Lim,\nBarret Zoph, Alexander Spiridonov, Ryan Sepassi,\nDavid Dohan, Shivani Agrawal, Mark Omernick, An-\ndrew M. Dai, Thanumalayan Sankaranarayana Pil-\nlai, Marie Pellat, Aitor Lewkowycz, Erica Moreira,\nRewon Child, Oleksandr Polozov, Katherine Lee,\nZongwei Zhou, Xuezhi Wang, Brennan Saeta, Mark\nDiaz, Orhan Firat, Michele Catasta, Jason Wei, Kathy\nMeier-Hellstern, Douglas Eck, Jeff Dean, Slav Petrov,\nand Noah Fiedel. 2022. Palm: Scaling language mod-\neling with pathways.\nSander Dieleman, Laurent Sartran, Arman Roshan-\nnai, Nikolay Savinov, Yaroslav Ganin, Pierre H.\nRichemond, Arnaud Doucet, Robin Strudel, Chris\nDyer, Conor Durkan, Curtis Hawthorne, Rémi\nLeblond, Will Grathwohl, and Jonas Adler. 2022.\nContinuous diffusion for categorical data.\nMarjan Ghazvininejad, Omer Levy, Yinhan Liu, and\nLuke Zettlemoyer. 2019. Mask-predict: Parallel de-\ncoding of conditional masked language models. In\nProceedings of the 2019 Conference on Empirical\nMethods in Natural Language Processing and the\n9th International Joint Conference on Natural Lan-\nguage Processing (EMNLP-IJCNLP) , pages 6112–\n6121, Hong Kong, China. Association for Computa-\ntional Linguistics.\nShansan Gong, Mukai Li, Jiangtao Feng, Zhiyong Wu,\nand Lingpeng Kong. 2023. Diffuseq: Sequence to\nsequence text generation with diffusion models.\nJiatao Gu, James Bradbury, Caiming Xiong, Victor O.K.\nLi, and Richard Socher. 2018. Non-autoregressive\nneural machine translation. In International Confer-\nence on Learning Representations .\nJiatao Gu and Xiang Kong. 2021. Fully non-\nautoregressive neural machine translation: Tricks of\nthe trade. In Findings of the Association for Com-\nputational Linguistics: ACL-IJCNLP 2021 , pages\n120–133, Online. Association for Computational Lin-\nguistics.\nYoon Kim and Alexander M. Rush. 2016. Sequence-\nlevel knowledge distillation. In Proceedings of the\n2016 Conference on Empirical Methods in Natu-\nral Language Processing , pages 1317–1327, Austin,\nTexas. Association for Computational Linguistics.Denis Kocetkov, Raymond Li, Loubna Ben Allal, Jia\nLi, Chenghao Mou, Carlos Muñoz Ferrandis, Yacine\nJernite, Margaret Mitchell, Sean Hughes, Thomas\nWolf, Dzmitry Bahdanau, Leandro von Werra, and\nHarm de Vries. 2022. The stack: 3 tb of permissively\nlicensed source code.\nJason Lee, Elman Mansimov, and Kyunghyun Cho.\n2018. Deterministic non-autoregressive neural se-\nquence modeling by iterative refinement. In Proceed-\nings of the 2018 Conference on Empirical Methods\nin Natural Language Processing , pages 1173–1182,\nBrussels, Belgium. Association for Computational\nLinguistics.\nXiang Lisa Li, John Thickstun, Ishaan Gulrajani, Percy\nLiang, and Tatsunori Hashimoto. 2022. Diffusion-\nLM improves controllable text generation. In Ad-\nvances in Neural Information Processing Systems .\nZhenghao Lin, Yeyun Gong, Yelong Shen, Tong Wu,\nZhihao Fan, Chen Lin, Nan Duan, and Weizhu Chen.\n2023. Text generation with diffusion language mod-\nels: A pre-training approach with continuous para-\ngraph denoise.\nOpenAI. 2023. Gpt-4 technical report.\nColin Raffel, Noam Shazeer, Adam Roberts, Kather-\nine Lee, Sharan Narang, Michael Matena, Yanqi\nZhou, Wei Li, and Peter J. Liu. 2020. Exploring the\nlimits of transfer learning with a unified text-to-text\ntransformer. Journal of Machine Learning Research ,\n21(140):1–67.\nPranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and\nPercy Liang. 2016. SQuAD: 100,000+ questions for\nmachine comprehension of text. In Proceedings of\nthe 2016 Conference on Empirical Methods in Natu-\nral Language Processing , pages 2383–2392, Austin,\nTexas. Association for Computational Linguistics.\nAditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey\nChu, and Mark Chen. 2022. Hierarchical text-\nconditional image generation with clip latents.\nMachel Reid, Vincent Josua Hellendoorn, and Graham\nNeubig. 2023. DiffusER: Diffusion via edit-based\nreconstruction. In The Eleventh International Con-\nference on Learning Representations .\nAdam Roberts, Hyung Won Chung, Anselm Levskaya,\nGaurav Mishra, James Bradbury, Daniel Andor, Sha-\nran Narang, Brian Lester, Colin Gaffney, Afroz\nMohiuddin, Curtis Hawthorne, Aitor Lewkowycz,\nAlex Salcianu, Marc van Zee, Jacob Austin, Se-\nbastian Goodman, Livio Baldini Soares, Haitang\nHu, Sasha Tsvyashchenko, Aakanksha Chowdh-\nery, Jasmijn Bastings, Jannis Bulian, Xavier Gar-\ncia, Jianmo Ni, Andrew Chen, Kathleen Kenealy,\nJonathan H. Clark, Stephan Lee, Dan Garrette, James\nLee-Thorp, Colin Raffel, Noam Shazeer, Marvin\nRitter, Maarten Bosma, Alexandre Passos, Jeremy\nMaitin-Shepard, Noah Fiedel, Mark Omernick, Bren-\nnan Saeta, Ryan Sepassi, Alexander Spiridonov,Joshua Newlan, and Andrea Gesmundo. 2022. Scal-\ning up models and data with t5xandseqio .arXiv\npreprint arXiv:2203.17189 .\nRobin Rombach, Andreas Blattmann, Dominik Lorenz,\nPatrick Esser, and Björn Ommer. 2021. High-\nresolution image synthesis with latent diffusion mod-\nels.CoRR , abs/2112.10752.\nChitwan Saharia, William Chan, Saurabh Saxena,\nLala Li, Jay Whang, Emily Denton, Seyed Kam-\nyar Seyed Ghasemipour, Raphael Gontijo-Lopes,\nBurcu Karagol Ayan, Tim Salimans, Jonathan Ho,\nDavid J. Fleet, and Mohammad Norouzi. 2022. Pho-\ntorealistic text-to-image diffusion models with deep\nlanguage understanding. In Advances in Neural In-\nformation Processing Systems .\nChitwan Saharia, William Chan, Saurabh Saxena, and\nMohammad Norouzi. 2020. Non-autoregressive ma-\nchine translation with latent alignments. In Proceed-\nings of the 2020 Conference on Empirical Methods\nin Natural Language Processing (EMNLP) , pages\n1098–1108, Online. Association for Computational\nLinguistics.\nNikolay Savinov, Junyoung Chung, Mikolaj Binkowski,\nErich Elsen, and Aaron van den Oord. 2022. Step-\nunrolled denoising autoencoders for text generation.\nInInternational Conference on Learning Representa-\ntions .\nJascha Sohl-Dickstein, Eric Weiss, Niru Mah-\neswaranathan, and Surya Ganguli. 2015. Deep un-\nsupervised learning using nonequilibrium thermody-\nnamics. In Proceedings of the 32nd International\nConference on Machine Learning , volume 37 of Pro-\nceedings of Machine Learning Research , pages 2256–\n2265, Lille, France. PMLR.\nMitchell Stern, William Chan, Jamie Kiros, and Jakob\nUszkoreit. 2019. Insertion transformer: Flexible se-\nquence generation via insertion operations. In Pro-\nceedings of the 36th International Conference on\nMachine Learning , volume 97 of Proceedings of Ma-\nchine Learning Research , pages 5976–5985. PMLR.\nRobin Strudel, Corentin Tallec, Florent Altché, Yilun\nDu, Yaroslav Ganin, Arthur Mensch, Will Grathwohl,\nNikolay Savinov, Sander Dieleman, Laurent Sifre,\nand Rémi Leblond. 2022. Self-conditioned embed-\nding diffusion for text generation.\nYi Tay, Mostafa Dehghani, Vinh Q. Tran, Xavier Garcia,\nJason Wei, Xuezhi Wang, Hyung Won Chung, Dara\nBahri, Tal Schuster, Steven Zheng, Denny Zhou, Neil\nHoulsby, and Donald Metzler. 2023. UL2: Unifying\nlanguage learning paradigms. In The Eleventh Inter-\nnational Conference on Learning Representations .\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob\nUszkoreit, Llion Jones, Aidan N Gomez, Łukasz\nKaiser, and Illia Polosukhin. 2017. Attention is all\nyou need. In Advances in Neural Information Pro-\ncessing Systems , volume 30. Curran Associates, Inc.Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten\nBosma, brian ichter, Fei Xia, Ed H. Chi, Quoc V Le,\nand Denny Zhou. 2022. Chain of thought prompt-\ning elicits reasoning in large language models. In\nAdvances in Neural Information Processing Systems .\nTong Wu, Zhihao Fan, Xiao Liu, Yeyun Gong, Yelong\nShen, Jian Jiao, Hai-Tao Zheng, Juntao Li, Zhongyu\nWei, Jian Guo, Nan Duan, and Weizhu Chen. 2023.\nAr-diffusion: Auto-regressive diffusion model for\ntext generation.\nLinting Xue, Noah Constant, Adam Roberts, Mihir Kale,\nRami Al-Rfou, Aditya Siddhant, Aditya Barua, and\nColin Raffel. 2021. mT5: A massively multilingual\npre-trained text-to-text transformer. In Proceedings\nof the 2021 Conference of the North American Chap-\nter of the Association for Computational Linguistics:\nHuman Language Technologies , pages 483–498, On-\nline. Association for Computational Linguistics.\nJiasheng Ye, Zaixiang Zheng, Yu Bao, Lihua Qian, and\nQuanquan Gu. 2023. Diffusion language models\ncan perform many tasks with scaling and instruction-\nfinetuning.\nHongyi Yuan, Zheng Yuan, Chuanqi Tan, Fei Huang,\nand Songfang Huang. 2023. Seqdiffuseq: Text diffu-\nsion with encoder-decoder transformers.\nLin Zheng, Jianbo Yuan, Lei Yu, and Lingpeng Kong.\n2023. A reparameterized discrete diffusion model\nfor text generation." }, { "title": "2401.17211v2.Quantum_dynamics_in_one_and_two_dimensions_via_recursion_method.pdf", "content": "Quantum dynamics in one and two dimensions via recursion method\nFilipp Uskov1, 2and Oleg Lychkovskiy1,∗\n1Skolkovo Institute of Science and Technology\nBolshoy Boulevard 30, bld. 1, Moscow 121205, Russia\n2Gubkin Russian State University of Oil and Gas\n65 Leninsky Prospekt, Moscow 119991, Russia\n(Dated: February 1, 2024)\nWe report an implementation of the recursion method that addresses quantum many-body dy-\nnamics in the nonperturbative regime. The implementation has two key ingredients: a computer-\nalgebraic routine for symbolic calculation of nested commutators and a procedure to extrapolate the\nsequence of Lanczos coefficients according to the universal operator growth hypothesis. We apply\nthe method to calculate infinite-temperature correlation functions for spin-1 /2 systems on one- and\ntwo-dimensional lattices. In two dimensions the accessible timescale is large enough to essentially\nembrace the relaxation to equilibrium. The method allows one to accurately calculate transport co-\nefficients. As an illustration, we compute the diffusion constant for the transverse-field Ising model\non a square lattice.\nIntroduction. Quantum dynamics is one of the cen-\ntral topics in condensed matter physics. While for one-\ndimensional (1D) systems various numerical approaches\ntypically deliver highly satisfactory results, addressing\nhigher dimensions turns out to be much more challeng-\ning. Diverse techniques are being developed to tackle\nquantum dynamics in two and three dimensions, includ-\ning determinant quantum Monte-Carlo [1], MPS com-\nputations on infinite cylinders [2–5], methods based on\nprojected entangled pair states [6, 7], functional renor-\nmalization group [8, 9], classical approximations [10],\nhybrid quantum-classical methods [11, 12], unfolding of\ntwo-dimensional (2D) to nonlocally coupled 1D systems\n[13]etc.\nHere we report an implementation of the recursion\nmethod capable of addressing high-temperature dynam-\nics of 1D and 2D lattice systems. The recursion method\nhas a long history [14], however instances of its applica-\ntion to many-body systems are relatively scarce [15–26].\nThe basic object of the recursion method is a sequence\nof Lanczos coefficients bn,n= 0,1,2, . . ., that are to\nbe obtained from the nested commutators of the system\nHamiltonian with the observable in question. This se-\nquence becomes infinite in the thermodynamic limit. At\nthe same time, the complexity of calculating the Lanc-\nzos coefficients grows factorially with n. This has been\nhindering the application of the method for decades.\nWe alleviate the above difficulty by two complementary\nremedies. First, we develop a computer algebra routine\nto calculate a record number of nested commutators. The\ncomputation is performed directly in the thermodynamic\nlimit and keeps the Hamiltonian parameters symbolic.\nSecond, we extrapolate the remaining part of the Lanc-\nzos sequence according to the universal operator growth\nhypothesis (UOGH) [22] and other recent insights in the\nasymptotic behaviour of this sequence [24, 25, 27, 28].\nRemarkably, the extrapolation works better the further\nthe system is from integrable points. This makes ourapproach inherently nonperturbative.\nThe paper is organized as follows. We start from in-\ntroducing basic concepts and definitions, in particular,\nthe autocorrelation function. Then we discuss the trun-\ncated Taylor expansion of the autocorrelation function.\nAfter that we outline the recursion method, the UOGH,\nand a procedure to obtain transport coefficients from the\nLanczos sequence. Next we describe our implementation\nof the recursion method. Then the method is applied\nto one 1D model and two 2D models. Discussion and\noutlook conclude the paper.\nAutocorrelation function. We consider a quantum sys-\ntem with a Hamiltonian Hand focus on some observable\ngiven by a self-ajoint Shr¨ odinger operator A. The same\nobservable in the Heisenberg representation reads A(t) =\neitHA e−itH. It is convenient to introduce the commu-\ntation superoperator L ≡ [H,•]. Then the Heisenberg\nequation of motion reads ���tA(t) =iLA(t), and its for-\nmal solutionis is given by A(t) =ei tLA.\nThroughout the paper we focus on the normalized\ninfinite-temperature autocorrelation function\nC(t)≡tr\u0000\nA(t)A\u0001\n/trA2. (1)\nIt has the properties C(0) = 1 and C(−t) =C(t). We\nremark that strong long-lived quantum correlations can\nwell exist at infinite temperature [29].\nIt is convenient to introduce a scalar product in the\nspace of operators according to\n\u0000\nA|B\u0001\n≡tr\u0000\nA†B\u0001\n/d, (2)\nwhere dis the Hilbert space dimension (which is as-\nsumed to be finite). The scalar product entails the norm\n∥A∥=p\n(A|A). In this notations, the autocorrelation\nfunction can be written as C(t) = ( A(t)|A)/∥A∥2. The\nsuperoperator Lis self-adjoint with respect to this scalar\nproduct.arXiv:2401.17211v2 [cond-mat.str-el] 31 Jan 20242\nTruncated Taylor expansion. Expanding A(t) in powers\noft, one obtains the Taylor expansion of the autocorre-\nlation function,\nC(t)≡∞X\nm=0(−1)mµ2m\n(2m)!t2m, (3)\nwith even moments given by\nµ2m≡(L2mA|A)/∥A∥2= (LmA|LmA)/∥A∥2(4)\nand odd moments being zero, ensuring that the autocor-\nrelation function is even. Note that µ0= 1 by definition.\nThe Taylor expansion (3) is known to have an infinite\nconvergence radius for 1D systems with short-range in-\nteractions [30] and a finite convergence radius in higher\ndimensions [22].\nTruncating the Taylor expansion (3) at the order 2 n,\none obtains a polynomial P2n(t). Remarkably, these\npolynomials constitute rigorous upper and lower bounds\non the autocorrelation function [31, 32],\nP4l+2(t)≤C(t)≤P4l(t), l= 1,2, . . . (5)\nThese two-sided bounds are extremely tight up to a cer-\ntain time, allowing one to precisely benchmark more so-\nphisticated approximations to C(t), see Fig. 1.\nRecursion method. We employ the Heisenberg-picture\nversion of the recursion method [14]. It is essentially\nabout solving coupled Heisenberg equations in the or-\nthogonal Lanczos basis {An}, n= 0,1,2, . . . defined it-\neratively as follows: |A0) =∥A∥−1|A),|A1) =L|A0),\nbn=∥An∥, n = 0,1,2, . . . ,\n|An) =b−1\nn−1L|An−1)−bn−1b−1\nn−2|An−2)n= 2,3, . . .\n(6)\nThe superoperator Lacquires a tridiagonal form in this\nbasis, with the zero main diagonal and the sequence of\nLanczos coefficients bnin the sub/supra-diagonals. As\na result, the autocorrelation function (1) enters a set of\ncoupled equations\n∂tφn(t) =−bn+1φn+1(t) +bnφn−1(t), n = 0,1,2, . . .\nC(t) =φ0(t), (7)\nwhere φ−1(t)≡0 and φn(0) = δ0n.\nThis way the autocorrelation function becomes im-\nplicitly determined by the sequence of Lanczos coeffi-\ncients bn. These coefficients can be obtained recur-\nrently according to (6) or, alternatively, from the mo-\nments (4) [33].\nUOGH and extrapolation of Lanczos coefficients. In prac-\ntice, only a finite number of bncan be computed. Other\ncoefficients are to be extrapolated. The UOGH put for-\nward in [22] states that for generic systems the leadingasymptotics of bnis linear (with a logarithmic correction\nin one dimension). It has been further revealed that cer-\ntain subleading terms of the asymptotics can be equally\nimportant for the dynamics [17, 22, 24, 25, 27, 28, 34, 35].\nGuided by these insights, we employ the following extrap-\nolation formulae for n≫1:\nbn≃αn/logn+γ+ (−1)nγ∗ for 1D ,(8)\nbn≃αn+γ+ (−1)nγ∗ for 2D ,(9)\nHere α,γandγ∗are the fitting parameters. In partic-\nular, γ∗parameterizes odd-even alterations in the Lanc-\nzos sequence that emerge whenever C≡limt→∞C(t) is\nnonzero ( cf.[17, 24, 25]).\nTransport coefficients. Whenever A=Jis the current of\nsome conserved quantity, the autocorrelation function of\nJdetermines the corresponding transport coefficient [36].\nIn particular, when the conserved quantity in question is\nenergy, one can calculate the energy diffusion constant D\nas [36, 37]\nD=∥J∥2\n∥H∥2C ¸, C ¸≡Z∞\n0dt C(t), (10)\nwhere tr H= 0 is assumed.\nIt has been shown recently that, employing the UOGH,\none can obtain a precise approximation C ¸rto C ¸ from a\nmoderate number rof known Lanczos coefficients [38].\nThe approximation reads [38] (see also [39])\nC ¸r=1\nbr[r/2]Y\nm=1b2\n2m\nb2\n2m−1×(\n1/prfor even r,\npr for odd r,(11)\nwhere [ r/2] is the integer value of ( r/2) and\npr= Γ\u0010r\n2+γ\n2α\u0011\nΓ\u0010r\n2+γ\n2α+ 1\u0011\n/\u0012\nΓ\u0012r\n2+γ\n2α+1\n2\u0013\u00132\n.\n(12)\nC ¸rusually converges to C ¸ rapidly upon increasing r[38],\nwhich is confirmed by our calculations, see Fig. 2(b).\nWe note that one can also substitute a truncated Taylor\nexpansion in eq. (10), however the transport coefficients\nobtained this way are less accurate [9, 40–42].\nSymbolic implementation. We consider one-dimensional\nchains and two-dimensional square lattices of spins 1 /2\nwith nearest-neighbour interactions. Both the Hamil-\ntonian Hand the observable Aare considered to be\ntranslation-invariant.\nThe core routine of our method is a symbolic com-\nputation of nested commutators LnA. Importantly, the\nHamiltonian parameters are also kept symbolic. As com-\npared to computation with numerical parameters, this re-\nquires essentially no overhead in terms of computational\ntime and a moderate overhead in terms of memory. The\nmajor advantage of a fully symbolic calculation is that it\ncovers the whole parameter space in a single run.3\n1DIsing model\nnmax=45\nHaL\n0102030400510152025\nnbn\n203040-0.300.3Dbn2DXX-YYmodel\nnmax=17\nHbL\n0 5 10 15 200510152025\nnbn\n51015-0.500.5Dbn2DIsing model\nnmax=21\nHcL\n051015200510152025\nnbn\n101520-0.100.1Dbn\nP88HtL\nP90HtL\nHdL\n0 1 2 3 40.00.20.40.60.81.0\ntCHtL\nP34HtLP32HtL\nHeL\n0.0 0.5 1.0 1.50.00.20.40.60.81.0\ntCHtL\nP42HtLP40HtL\nHfL\n0.00.51.01.52.02.50.00.20.40.60.81.0\ntCHtL\nFIG. 1. Upper row: Lanczos coefficients for three models considered in the text. Dashed lines indicate extrapolating func-\ntions (8),(9). Insets highlight the subleading contribution ∆ bn, where ∆ bn=bn−(α n/logn+γ) in 1D and ∆ bn=bn−(α n+γ)\nin 2D. Lower row: correlation functions for the same models (solid lines) plotted up to t=tmax. Dashed lines – upper and lower\npolynomial bounds (5). Horizontal dash-dotted lines - long-time averages C. The result for the 1D Ising model is benchmarked\nby the exact diagonalization (dots).\nThe computation is performed in the thermodynamic\nlimit from the outset. The support of LnAgrows lin-\nearly with n, while the number of terms grows factori-\nally. Since Lis linear, the computation is straightfor-\nwardly parallelizable [43]. Computation of LnAis the\nmost resource-consuming routine of our code.\nAt the next step the moments (4) are computed. They\nhave the form of polynomials with respect to Hamilto-\nnian parameters. For each model considered below, we\nlist several first moments in the text. The complete list\nof computed moments is available as a Supplementary\nMaterial [44].\nNext we use the relation between Lanczos coefficients\nand moments [33] to compute bn. At this step numeri-\ncal values of the Hamiltonian parameters are plugged in.\nTo avoid numerical instabilities, the rational arithmetics\nis used. As a result, a sequence of numerical Lanczos\ncoefficients bn,n= 0,1, . . . , n maxis obtained.\nFinally, the Lanczos coefficients bnare extrapolated\nbeyond nmaxaccording to eqs. (8), (9), and the auto-\ncorrelation function is calculated by numerically solving\nequations (7). To estimate the maximal time tmaxuntil\nwhich our results are reliable, we reiterate this final step\nwith the extrapolation based on ( nmax−1) Lanczos co-\nefficients, and require that the discrepancy between the\ntwo approximations to C(t) remains below some small ϵ\n(ϵ= 10−3for plots in Fig. 1).1D Ising model. The Hamiltonian of the model reads\nH=X\njσx\njσx\nj+1+hzX\njσz\nj+hxX\njσx\nj, (13)\nwhere σx,y,z\nj are Pauli matrices at the j’th site and hx,\nhzare two parameters of the Hamiltonian. This model is\nintegrable when hx= 0 or hz= 0, and nonintegrable oth-\nerwise. The observable we consider is the magnetization\nin the z-direction,\nA=X\njσz\nj. (14)\nWe are able to calculate nmax= 45 nested commuta-\ntors and corresponding moments symbolically (the pre-\nvious record result was nmax= 38 moments calculated\nnumerically [45]). For example,\nµ2= 8 + 4 h2\nx,\nµ4= 128 + 192 h2\nx+ 128 h2\nz+ 16h4\nx+ 16h2\nxh2\nz.(15)\nThe corresponding Lanczos coefficients for hx=hy=\n1 are shown in Fig. 1(a). They are consistent with the\nUOGH and feature pronounced odd-even alterations on\ntop of the leading asymptote. The corresponding auto-\ncorrelation function is presented in Fig. 1(d). We bench-\nmark our result by a numerically exact computation for\na finite spin chain large enough to neglect finite size ef-\nfects on the considered timescale. The long-time average4\nCis nonzero, consistent with odd-even alterations of the\nLanczos coefficients.\nNote that, as evident from Fig. 1(d), the relaxation is\nfar from being complete up to the maximal time avail-\nable. This can be attributed to an unusually long relax-\nation timescale of the model (13) [46] (see also [38] for a\nrelated observation).\n2DXX-Y Ymodel. This is a spin-1 /2 model on a square\nlattice with the Hamiltonian\nH=X\n⟨ij⟩−σx\niσx\nj+vX\n⟨ij⟩|σy\niσy\nj. (16)\nHere iand jenumerate sites of the lattice, and the first\n(the second) sum runs over nearest neighbour sites con-\nnected by horizontal (vertical) bonds, each bond being\ncounted once. We choose the first term of the above\nHamiltonian as the observable A.\nWe manage to calculate nmax= 17 moments, with the\nfirst few given by\nµ2= 16 v2,\nµ4= 640 v2(1 +v2),\nµ6= 2048 v2(17 + 39 v2+ 17v4). (17)\nIn Fig 1(b),(e) we plot the Lanczos coefficients and\nthe autocorrelation function for v= 1. At this specific\nvalue of vthe long-time average of the autocorrelation\nfunction is fixed by symmetry to be C= 1/2. One can\nsee that, in contrast to the previous case, C(t) relaxes\nclose to this value within the timescale accessible by our\nmethod. C(t) is additionally benchmarked by the poly-\nnomial bounds (5).\n2D Ising model. The Hamiltonian is defined on a square\nlattice and reads\nH=X\n⟨ij⟩σx\niσx\nj+hzX\njσz\nj, (18)\nwhere the first sum runs over pairs of neighbouring sites.\nWith an eye on computing the diffusion constant, we\nchoose the energy current along the horizontal direction\nas the observable:\nA=J=hzX\n⟨ij⟩−\ni≺j(σx\niσy\nj−σx\njσy\ni). (19)\nHere the sum runs over horizontal bonds, the site ibeing\nalways to the left of the site j.\nWe are able to calculate nmax= 21 nested commu-\ntators and corresponding moments symbolically (previ-\nously 13 moments were calculated for a different observ-\nable [47]). First three moments read\nµ2= 8,\nµ4= 64 (2 + h2\nz),\nµ6= 1024 (2 + 5 h2\nz+h4\nz). (20)\nHaL\n0.00.51.01.52.02.53.00.0.20.40.60.81.1.21.4\nhzD\næææææææææææææææàààààààààààààààìììììììììììììììòòòòòòòòòòòòòòòôôôôôôôôôôôôôôô\nçç\nçççççççççççççHbL\n5 10 15 200.0.20.40.60.81.1.21.4\nrFIG. 2. (a) Diffusion constant for the 2D Ising model (18)\nas a function of the transverse field hz. The width of the line\nindicates the estimated uncertainty. (b) Convergence of the\ndiffusion constant with the approximation order r. Shown are\ndata for fields hz= 0.5,1,1.5,2,2.5,3 (from bottom to top).\nThe Lanczos coefficients and the autocorrelation func-\ntion are shown in Fig. 1(c). In contrast to previous cases,\nthe irregularities of the Lanczos coefficients do not follow\nthe odd-even alteration pattern. This is consistent with\nthe fact that the autocorrelation function of the current\nrelaxes to zero. We therefore do not include the alter-\nation term in the extrapolation. One can see that again\nthe autocorrelation function essentially relaxes to equi-\nlibrium within the accessible timescale.\nWe further compute the diffusion constant for a range\nof magnetic fields hz, see Fig. 2. The convergence of the\napproximation (11) appears to be quite good away from\nthe integrable points hz= 0 and hz→ ∞ , as illustrated\nin Fig. 2(b). We conservatively estimate the uncertainty\nof our calculation as a maximal discrepancy between ten\napproximations obtained for rfrom ( nmax−9) to nmax.\nThis uncertainty is indicated in Fig. 2(a). It is below 1%\nfor fields hz∼1 but grows rapidly when hzorh−1\nzget\nclose to zero.\nDiscussion and outlook. In summary, we have advanced\nthe recursion method to the point it can handle the dy-\nnamics of two-dimensional lattice systems over the whole\nrelaxation timescale. We have illustrated the power of the\nmethod by computing infinite-temperature autocorrela-\ntion functions and the diffusion constant for spins 1 /2 on\na square lattice.\nThe most resource-consuming part of our computa-\ntions is performed symbolically, which means that the\nwhole parameter space of the Hamiltonian is covered in\na single run. The accuracy of the method, however, dif-\nfers across the parameter space. Remarkably, the method\nworks best deep in the nonperturbative regime, where\nthe sequence of the Lanczos coefficients converges to its\nasymptotic form most rapidly [22].\nAn important ingredient of the method is the extrap-\nolation of Lanczos coefficients beyond those explicitly\ncomputed. The extrapolation is based on the conjec-\ntured asymptotic form of the coefficients. Our study con-\nfirms the conjecture about leading terms of the asymp-\ntotics [22] and strengthens previous observations about5\nthe importance of subleading terms [17, 22, 24, 25, 27, 28,\n34, 35]. The method will benefit from better theoretical\nunderstanding of the subleading terms.\nThe generalization of the method to different lattice ge-\nometries, higher spins, lattice fermions or bosons is con-\nceptually straightforward. Finite but high temperatures\ncan be handled by using the recursion method in conjunc-\ntion with the high-temperature expansion. Addressing\nlower temperatures can be more challenging, most likely\nnecessitating a considerable amendment of the method.\nIn particular, employing more complex scalar products\n[22, 48, 49] beyond the simplest one (2) may be required.\nFinally, we note that recent approaches [50–53] to effec-\ntively constraint the Heisenberg evolution within smaller\nsubspaces of the operator space can potentially greatly\nreduce the computational cost of the method. Another\nvery recent promising move in the same direction is a\nstochastic sampling of operator growth [54].\nAcknowledgments. OL thanks Anatoly Dymarsky and\nAlexander Avdoshkin for a useful discussion at the initial\nstage of this study. This work was supported by the\nRussian Science Foundation under grant №24-22-00331,\nhttps://rscf.ru/en/project/24-22-00331/\n∗o.lychkovskiy@skoltech.ru\n[1] Edwin W. Huang, Ryan Sheppard, Brian Moritz, and\nThomas P. Devereaux, “Strange metallicity in the doped\nhubbard model,” Science 366, 987–990 (2019).\n[2] A. J. A. James and R. M. Konik, “Quantum quenches in\ntwo spatial dimensions using chain array matrix product\nstates,” Phys. Rev. B 92, 161111 (2015).\n[3] Matthias Gohlke, Ruben Verresen, Roderich Moessner,\nand Frank Pollmann, “Dynamics of the kitaev-heisenberg\nmodel,” Phys. Rev. Lett. 119, 157203 (2017).\n[4] Tomohiro Hashizume, Ian P. McCulloch, and Jad C.\nHalimeh, “Dynamical phase transitions in the two-\ndimensional transverse-field ising model,” Phys. Rev.\nRes. 4, 013250 (2022).\n[5] Stefano De Nicola, Alexios A. Michailidis, and\nMaksym Serbyn, “Entanglement and precession in two-\ndimensional dynamical quantum phase transitions,”\nPhys. Rev. B 105, 165149 (2022).\n[6] Jacek Dziarmaga, “Time evolution of an infinite pro-\njected entangled pair state: Neighborhood tensor up-\ndate,” Phys. Rev. B 104, 094411 (2021).\n[7] Ryui Kaneko and Ippei Danshita, “Dynamics of corre-\nlation spreading in low-dimensional transverse-field ising\nmodels,” arXiv preprint arXiv:2301.01407 (2023).\n[8] Jan Krieg and Peter Kopietz, “Exact renormalization\ngroup for quantum spin systems,” Phys. Rev. B 99,\n060403 (2019).\n[9] Dmytro Tarasevych and Peter Kopietz, “Dissipative spin\ndynamics in hot quantum paramagnets,” Phys. Rev. B\n104, 024423 (2021).\n[10] Dennis Schubert, Jonas Richter, Fengping Jin, Kristel\nMichielsen, Hans De Raedt, and Robin Steinigeweg,\n“Quantum versus classical dynamics in spin models:Chains, ladders, and square lattices,” Phys. Rev. B 104,\n054415 (2021).\n[11] Grigory A. Starkov and Boris V. Fine, “Hybrid quantum-\nclassical method for simulating high-temperature dynam-\nics of nuclear spins in solids,” Phys. Rev. B 98, 214421\n(2018).\n[12] Grigory A. Starkov and Boris V. Fine, “Free induction\ndecays in nuclear spin-1\n2lattices with a small number of\ninteracting neighbors: The cases of silicon and fluorap-\natite,” Phys. Rev. B 101, 024428 (2020).\n[13] SJ Thomson and J Eisert, “Unraveling long-time quan-\ntum dynamics using flow equations,” arXiv preprint\narXiv:2308.13005 (2023).\n[14] VS Viswanath and Gerhard M¨ uller, The Recursion\nMethod: Application to Many-Body Dynamics , Vol. 23\n(Springer Science & Business Media, 2008).\n[15] S. J. Knak Jensen and E. Kjaersgaard Hansen, “Sixth\nand eighth moments of the magnetic-resonance lines of a\ndipolar-coupled rigid lattice,” Phys. Rev. B 7, 2910–2917\n(1973).\n[16] M. Engelsberg and I. J. Lowe, “Approximants to the\nnuclear-spin autocorrelation function. application to\ncaf2,” Phys. Rev. B 12, 3547–3552 (1975).\n[17] V. S. Viswanath, Shu Zhang, Joachim Stolze, and Ger-\nhard M¨ uller, “Ordering and fluctuations in the ground\nstate of the one-dimensional and two-dimensional s=1/2\nxxz antiferromagnets: A study of dynamical properties\nbased on the recursion method,” Phys. Rev. B 49, 9702–\n9715 (1994).\n[18] Markus B¨ ohm, V. S. Viswanath, Joachim Stolze, and\nGerhard M¨ uller, “Spin diffusion in the one-dimensional\ns=1/2 xxz model at infinite temperature,” Phys. Rev. B\n49, 15669–15681 (1994).\n[19] Netanel H. Lindner and Assa Auerbach, “Conductivity\nof hard core bosons: A paradigm of a bad metal,” Phys.\nRev. B 81, 054512 (2010).\n[20] Ilia Khait, Snir Gazit, Norman Y. Yao, and Assa Auer-\nbach, “Spin transport of weakly disordered heisenberg\nchain at infinite temperature,” Phys. Rev. B 93, 224205\n(2016).\n[21] Assa Auerbach, “Hall number of strongly correlated met-\nals,” Phys. Rev. Lett. 121, 066601 (2018).\n[22] Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin,\nThomas Scaffidi, and Ehud Altman, “A universal oper-\nator growth hypothesis,” Phys. Rev. X 9, 041017 (2019).\n[23] Wescley Luiz de Souza, ´Erica de Mello Silva, and Paulo\nH. L. Martins, “Dynamics of the spin-1/2 ising two-leg\nladder with four-spin plaquette interaction and trans-\nverse field,” Phys. Rev. E 101, 042104 (2020).\n[24] Daniel J. Yates, Alexander G. Abanov, and Aditi Mitra,\n“Lifetime of almost strong edge-mode operators in one-\ndimensional, interacting, symmetry protected topological\nphases,” Phys. Rev. Lett. 124, 206803 (2020).\n[25] Daniel J. Yates, Alexander G. Abanov, and Aditi Mitra,\n“Dynamics of almost strong edge modes in spin chains\naway from integrability,” Phys. Rev. B 102, 195419\n(2020).\n[26] Xiao-Juan Yuan, Jing-Fen Zhao, Hui Wang, Hong-Xia\nBu, Hui-Min Yuan, Bang-Yu Zhao, and Xiang-Mu Kong,\n“Spin dynamics of an ising chain with bond impurity in\na tilt magnetic field,” Physica A: Statistical Mechanics\nand its Applications 583, 126279 (2021).\n[27] Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy,6\nand Tanay Pathak, “Krylov complexity in saddle-\ndominated scrambling,” Journal of High Energy Physics\n2022 , 1–27 (2022).\n[28] Alexander Avdoshkin, Anatoly Dymarsky, and Michael\nSmolkin, “Krylov complexity in quantum field theory,\nand beyond,” arXiv preprint arXiv:2212.14429 (2022).\n[29] M´ arton Kan´ asz-Nagy, Izabella Lovas, Fabian Grusdt,\nDaniel Greif, Markus Greiner, and Eugene A. Demler,\n“Quantum correlations at infinite temperature: The dy-\nnamical nagaoka effect,” Phys. Rev. B 96, 014303 (2017).\n[30] Huzihiro Araki, “Gibbs states of a one dimensional quan-\ntum lattice,” Communications in Mathematical Physics\n14, 120–157 (1969).\n[31] Ole Platz and Roy G. Gordon, “Rigorous bounds for\ntime-dependent correlation functions,” Phys. Rev. Lett.\n30, 264–267 (1973).\n[32] M Bohm and H Leschke, “Dynamic spin-pair correlations\nin a heisenberg chain at infinite temperature based on an\nextended short-time expansion,” Journal of Physics A:\nMathematical and General 25, 1043 (1992).\n[33] Jesus Sanchez-Dehesa, “The spectrum of jacobi matri-\nces in terms of its associated weight function,” Journal\nof Computational and Applied Mathematics 4, 275–283\n(1978).\n[34] Anatoly Dymarsky and Michael Smolkin, “Krylov com-\nplexity in conformal field theory,” Phys. Rev. D 104,\nL081702 (2021).\n[35] Hugo A Camargo, Viktor Jahnke, Keun-Young Kim, and\nMitsuhiro Nishida, “Krylov complexity in free and in-\nteracting scalar field theories with bounded power spec-\ntrum,” Journal of High Energy Physics 2023 , 1–48\n(2023).\n[36] Ryogo Kubo, Mario Yokota, and Sadao Nakajima,\n“Statistical-mechanical theory of irreversible processes.\nii. response to thermal disturbance,” Journal of the Phys-\nical Society of Japan 12, 1203–1211 (1957).\n[37] Robin Steinigeweg and Jochen Gemmer, “Density dy-\nnamics in translationally invariant spin-1\n2chains at high\ntemperatures: A current-autocorrelation approach to fi-\nnite time and length scales,” Phys. Rev. B 80, 184402\n(2009).\n[38] Jiaozi Wang, Mats H Lamann, Robin Steinigeweg, and\nJochen Gemmer, “Diffusion constants from the recursion\nmethod,” arXiv preprint arXiv:2312.02656 (2023).\n[39] C.G. Joslin and C.G. Gray, “Calculation of transport co-\nefficients using a modified mori formalism,” Molecular\nPhysics 58, 789–797 (1986).\n[40] Tohru Morita, “Spin diffusion in the heisenberg mag-\nnets at infinite temperature,” Phys. Rev. B 6, 3385–3393\n(1972).[41] J Labrujere, T O Klaassen, and N J Poulis, “Spin dy-\nnamics in a 3d heisenberg ferromagnet in the paramag-\nnetic state ii,” Journal of Physics C: Solid State Physics\n15, 999 (1982).\n[42] B Cowan, WJ Mullin, and E Nelson, “Spin diffusion in\n2d and 3d quantum solids,” Journal of Low Temperature\nPhysics 77, 181–193 (1989).\n[43] Marlon Brenes, Vipin Kerala Varma, Antonello Scardic-\nchio, and Ivan Girotto, “Massively parallel implementa-\ntion and approaches to simulate quantum dynamics using\nkrylov subspace techniques,” Computer Physics Commu-\nnications 235, 477–488 (2019).\n[44] The Supplementary material contains Mathemat-\nica©files with symbolic moments for models described\nin the text.\n[45] Jae Dong Noh, “Operator growth in the transverse-field\nising spin chain with integrability-breaking longitudinal\nfield,” Phys. Rev. E 104, 034112 (2021).\n[46] Stefan Birnkammer, Alvise Bastianello, and Michael\nKnap, “Prethermalization in one-dimensional quantum\nmany-body systems with confinement,” Nature Commu-\nnications 13, 7663 (2022).\n[47] Robin Heveling, Jiaozi Wang, and Jochen Gemmer, “Nu-\nmerically probing the universal operator growth hypoth-\nesis,” Phys. Rev. E 106, 014152 (2022).\n[48] Anatoly Dymarsky and Alexander Gorsky, “Quantum\nchaos as delocalization in krylov space,” Phys. Rev. B\n102, 085137 (2020).\n[49] Chengming Tan, Zhiyang Wei, and Ren Zhang, “Scaling\nrelations of spectrum form factor and krylov complexity\nat finite temperature,” arXiv preprint arXiv:2401.10499\n(2024).\n[50] Tibor Rakovszky, C. W. von Keyserlingk, and\nFrank Pollmann, “Dissipation-assisted operator evolu-\ntion method for capturing hydrodynamic transport,”\nPhys. Rev. B 105, 075131 (2022).\n[51] Thomas Schuster and Norman Y. Yao, “Operator growth\nin open quantum systems,” Phys. Rev. Lett. 131, 160402\n(2023).\n[52] Christopher David White, “Effective dissipation rate in\na liouvillian-graph picture of high-temperature quantum\nhydrodynamics,” Phys. Rev. B 107, 094311 (2023).\n[53] Tim Byrnes Igor Ermakov, Oleg Lychkovskiy, “Unified\nframework for efficiently computable quantum circuits,”\narXiv preprint arXiv:2401.08187 (2024).\n[54] Ayush De, Umberto Borla, Xiangyu Cao, and Snir Gazit,\n“Stochastic sampling of operator growth,” arXiv preprint\narXiv:2401.06215 (2024)." }, { "title": "2401.17219v3.Faster_coloring_and_embedding_in_dense_hypergraphs_via_stability.pdf", "content": "arXiv:2401.17219v3 [math.CO] 4 Feb 2024Faster coloring and embedding in dense hypergraphs via\nstability\nJianfeng Hou∗1, Xizhi Liu†2, and Hongbin Zhao‡1\n1Center for Discrete Mathematics, Fuzhou University, Fujian, 350 003, China\n2Mathematics Institute and DIMAP, University of Warwick, Coventr y, CV4 7AL,\nUK\nFebruary 6, 2024\nAbstract\nThe classical Andr´ asfai–Erd˝ os–S´ os Theorem states that for ℓ≥2, everyn-vertex\nKℓ+1-free graph with minimum degree greater than3ℓ−4\n3ℓ−1nmust beℓ-partite. We\nestablish a simple criterion for r-graphs,r≥2, to exhibit an Andr´ asfai–Erd˝ os–S´ os-\ntype property (AES), leading to a classification of most previously s tudied hypergraph\nfamilies with this property.\nFor every AES r-graphF, we present a simple algorithm to decide the F-freeness\nof ann-vertexr-graph with minimum degree greater than ( π(F)−εF)/parenleftbign\nr−1/parenrightbig\nin time\nO(nr), whereεF>0 is a constant. In particular, for the complete graph Kℓ+1, we can\ntakeεKℓ+1= (3ℓ2−ℓ)−1, and this bound is tight up to some multiplicative constant\nfactor ifW[1]/\\⌉}a⊔io\\slash=FPT. Based on a result by Chen–Huang–Kanj–Xia, we further show\nthat for every fixed C >0, this problem cannot be solved in time no(ℓ)if we replace\nεKℓ+1with(Cℓ)−1unlessETHfails. Furthermore, weestablishanalgorithmtodecide\ntheKℓ+1-freeness of an n-vertex graph with ex( n,Kℓ+1)−kedges in time ( ℓ+1)n2\nfork≤n/30ℓandℓ≤/radicalbig\nn/6, partially improving upon the recently provided running\ntime of 2.49knO(1)by Fomin–Golovach–Sagunov–Simonov. Moreover, we show that\nfor every fixed δ>0, this problem cannot be solved in time no(ℓ)ifkis of ordern1+δ\nunlessETHfails.\nAs an intermediate step, we show that for a specific class of r-graphsF, the\n(surjective) F-coloring problem can be solved in time O(nr), provided the input r-\ngraph hasnvertices and a large minimum degree, refining several previous resu lts.\nKeywords: Andr´ asfai–Erd˝ os–S´ os Theorem, testing F-freeness, homomorphism,\nparameterized algorithms, Exponential Time Hypothesis ( ETH).\n1 Intorduction\nFix an integer r≥2, anr-graphHis a collection of r-subsets of some finite set V. We\nidentify a hypergraph1Hwith its edge set and use V(H) to denote its vertex set. The\n∗Research was supported by National Key R&D Program of China ( Grant No. 2023YFA1010202),\nNational Natural Science Foundation of China (Grant No. 120 71077). Email: jfhou@fzu.edu.cn\n†Research was supported by ERC Advanced Grant 101020255. Ema il:xizhi.liu.ac@gmail.com\n‡Email:hbzhao2024@163.com\n1A graph is viewed as a 2-uniform hypergraph.\n1size ofV(H) is denoted by v(H). For a vertex v∈V(H), thedegreedH(v) ofvinH\nis the number of edges in Hcontaining v. We useδ(H), ∆(H), andd(H) to denote the\nminimum degree , themaximum degree , and the average degree ofH, respectively.\nWe will omit the subscript Hif it is clear from the context.\nGiven a family Fofr-graphs, we say HisF-freeif it does not contain any member of\nFas a subgraph. The Tur´ an number ex(n,F) ofFis the maximum number of edges\nin anF-freer-graph onnvertices. The Tur´ an density ofFis defined as π(F):=\nlimn→∞ex(n,F)//parenleftbign\nr/parenrightbig\n, the existence of the limit follows from a simple averaging a rgument\nof Katona–Nemetz–Simonovits [43]. We say a family Fisnondegenerate ifπ(F)>0.\nThe study of ex( n,F) has been a central topic in extremal graph and hypergraph th eory\nsince the seminal work of Tur´ an [73]. We refer the reader to s urveys [26, 67, 44] for related\nresults.\nIn this note, we focus on the structures of near-extremal F-free hypergraphs and explore\ntheir applications in efficiently deciding the F-freeness of a dense hypergraph.\n1.1 Embedding\nGiven two r-graphsFandH, a mapψ:V(F)→V(H) is anembedding of Fifψis\ninjective and ψ(e)∈ Hfor alle∈F. For a fixed r-graphF, letEmbed-Fdenote the\nfollowing decision problem:\nF-embedding\nInput: Anr-graphHonnvertices.\nQuestion: Is there an embedding from FtoH?\nNotice that, for a fixed r-graphF, the brute-force search can solve Embed-Fin time\nO(nv(F)). Forr= 2, faster algorithms improving the exponent v(F) have been explored\nby many researchers [41, 60, 3, 49, 17]. In the case of r≥3, Yuster [76] demonstrated\nalgorithms that improve the exponent v(F) for certain classes of r-graphs. However, an\nopen problem remains: determining whether there are faster algorithms that improve the\nexponentv(F) whenFis a complete r-graph with at least r+ 1 vertices. The current\nrecord for this problem is due to Nagle [59]2. On the other hand, for the complete graph\nKℓ, a result by Chen–Huang–Kanj–Xia [11] shows that Embed- Kℓcannot be solved in\ntimeno(ℓ)unless the ETH(exponential time hypothesis [40]) fails. For complete bip artite\ngraphsKt,t, a result by Lin [50] shows that no algorithm can solve Embed-Kt,tin time\nnO(1)ifW[1]/\\⌉}a⊔io\\slash=FPT(see [15]), and no algorithm can solve Embed-Kt,tin timeno(√\nt)if\nrandomized ETHholds (see [12]).\nMotivated by the recent work of Fomin–Golovach–Sagunov–Si monov [22]. We consider\nthe following two embedding problems in dense hypergraphs. For a fixed r-graphF, let\nEmbed avg-(F,n,k) denote the following decision problem:\nF-embedding with average degree constraint\nInput: Anr-graphHonnvertices with |H| ≥ex(n,F)−k.\nQuestion: Is there an embedding from FtoH?\n2Nagle’s result was improved by a polylog( n) factor very recently [1].\n2In addition, define Embed min-(F,α) as the following decision problem:\nF-embedding with minimum degree constraint\nInput: Anr-graphHonnvertices with δ(H)≥(π(F)−α)/parenleftbign\nr−1/parenrightbig\n.\nQuestion: Is there an embedding from FtoH?\nAmong other results, Fomin–Golovach–Sagunov–Simonov [22 ] proved that Embed avg-\n(Kℓ,n,k) can be solved in time 2 .49knO(1). A key ingredient in their proof involves reduc-\ningEmbed avg-(Kℓ,n,k) to the task of verifying the Kℓ-freeness of a graph with at most\n5kvertices. This reduction is rooted in Erd˝ os’ proof [20], wh ich essentially employs the\nZykov symmetrization [77] to establish the Tur´ an theorem. Regrettably, Erd˝ os’ proof does\nnot appear to readily extend to general graphs and hypergrap hs, and as a consequence,\nneither does the reduction employed by Fomin–Golovach–Sag unov–Simonov.\nUsing a different strategy based on the concept of degree-stab ility3, initially explored\nby Andr´ asfai–Erd˝ os–S´ os [4] and Erd˝ os–Simonovits [18] (see Theorem 4.1), we show that\nEmbed min-(F,α) (andhence, Embed avg-/parenleftBig\nF,n,α/parenleftbign\nr−1/parenrightbig/parenrightBig\n) can besolved intime O(nr) when\nαis sufficiently small for a broader class of hypergraphs, incl uding complete graphs.\nTheorem 1.1. Suppose that Fis anr-graph in Table 1 with degree-stability. Then there\nexist constants εF>0andCF>0depending only on Fsuch that the problem Embed min-\n(F,α)can be solved in time CFnrfor allα≤εF, wherenis the number of vertices of the\ninput hypergraph.\nA natural and interesting problem arises in determining the optimal upperboundfor εFin\nTheorem 1.1. For F=Kℓ+1, we can establish the lower bound1\n3ℓ2−ℓforεKℓ+1. Moreover,\nthis bound is tight up to some multiplicative constant facto r ifW[1]/\\⌉}a⊔io\\slash=FPT. In addition,\nusing the results of Chen–Huang–Kanj–Xia [11], we can also s how thatεKℓ+1must be of\nordero/parenleftbig1\nℓ/parenrightbig\n, even with a running time of no(ℓ).\nTheorem 1.2. There is an algorithm that solves Embed min-(Kℓ+1,α)in time(ℓ+1)n2\nfor allℓ≥2andα<1\n3ℓ2−ℓ, wherenis the number of vertices of the input graph. On the\nother hand, for every i≥2there exists δi>0andnisuch that Embed min-(Kℓ+1,(δiℓ2)−1)\ncannot be solved in time O(ni)for anyℓ≥niifW[1]/\\⌉}a⊔io\\slash=FPT, and for every fixed C >0\nEmbed min-(Kℓ+1,(Cℓ)−1)cannot be solved in time no(ℓ)ifETHholds.\nAnother interesting problem is to characterize the family o f nondegenerate r-graphsFfor\nwhich there exists a constant εF>0 such that Embed min-(Kℓ+1,εF) can be solved in time\nO(nr). The following result, whose proof relies on results by Lin [50], provides a natural\nclass of graphs that do not lie in this family.\nGiven a graph F, thet-blowupF[t] (ofF) is obtained from Fby replacing each vertex\nwith a set of size tand each edge with a corresponding complete bipartite graph . It is\nwell-known that π(F) =π(F[t]) for every t≥1 (see [19]).\nTheorem 1.3. For every fixed ℓ≥2, there is no algorithm that solves Embed min-\n(Kℓ+1[t],0)in timenO(1)ifW[1]/\\⌉}a⊔io\\slash=FPT, and there is no algorithm that solves Embed min-\n(Kℓ+1[t],0)in timeno(√\nt)if randomized ETHholds.\n3The definition of degree-stability is deferred to Section 1. 3.\n3We would like to remind the reader that in Theorem 1.1, one sho uld viewFas fixed and\nnas large (since for small n, we can just use the brute-force search). Similarly, in the\nsecond part of Theorem 1.2, the integer ℓshould also be considered fixed. Meanwhile, in\nthe first part of Theorem 1.2, we do not require ℓto be fixed.\nFor the problem Embed avg-(Kℓ+1,n,k), the following result improves upon the running\ntime provided by Fomin–Golovach–Sagunov–Simonov within a specific range.\nTheorem 1.4. There is an algorithm that solves Embed avg-(Kℓ+1,n,k)in time(ℓ+4)n2\nfor integers n≥ℓ≥2andk≥0satisfying max{6ℓ2,30kℓ} ≤n.\nNote that if ℓis fixed, then Theorem 1.4 implies that Embed avg-(Kℓ+1,n,k) can be solved\nin timeO(n2) whenk≤n\n30ℓ. The following result shows that this linear bound cannot be\nimproved to n1+δfor any constant δ >0, thus leaving an open problem to determine the\noptimal bound for kin Theorem 1.4.\nTheorem 1.5. UnlessETHfails, for every fixed δ >0there is no algorithm that solves\nEmbed avg-(Kℓ+1,n,n1+δ)in timeno(ℓ).\nProofs for Theorems 1.1, 1.2, 1.3, 1.4, and 1.5 are presented in Section 4.\n1.2 Homomorphism and surjective homomorphism\nTo establish the results in the previous subsection, we need to determine whether a given\nr-graph with a large minimum degree exhibits a specific struct ure. This is a particular\ninstance of the homomorphism (or coloring) problem. To addr ess this objective, we delve\ninto the study of the homomorphism problem for dense hypergr aphs in this subsection.\nRecall thatan r-multiset isanunorderedcollection of relements withrepetitionsallowed.\nThemultiplicity e(i) ofiin a multiset eis the number of times that iappears. An r-\npattern isapairP= (ℓ,E) whereℓisapositiveinteger and Eisacollection of r-multisets\non [ℓ]. It is clear that pattern is a generalization of r-graph, since an r-graph is a pattern\nin whichEconsists of only simple r-sets. For convenience, we call Ethe edge set of P\nand omit the first coordinate ℓif it is clear from the context.\nFor anr-patternP= (ℓ,E), let the Lagrange polynomial ofPbe\nΛP(X1,...,X ℓ):=/summationdisplay\ne∈Eℓ/productdisplay\ni=1Xe(i)\ni\ne(i)!.\nTheLagrangian λPofPis defined as\nλP:= max/braceleftBig\nΛP(x1,...,x ℓ): (x1,...,x ℓ)∈∆ℓ−1/bracerightBig\n,\nwhere ∆ℓ−1is the standard ( ℓ−1)-dimensional simplex (we refer the reader to [57, 68, 25,\n65] for some applications of Lagrangian in Tur´ an problem).\nWe say a pattern P= (ℓ,E) isminimal ifλP−i< λPfor alli∈[ℓ]. HereP−iis the\npattern obtained from Pby removing iand all edges containing i. Simple calculations\nshow that a graph is minimal iff it is complete.\nGiven anr-graphHand anr-patternP= (ℓ,E), a mapφ:V(H)→V(P) is ahomo-\nmorphism (or aP-coloring ) ifφ(e)∈Efor alle∈ H. We say HisP-colorable if there\n4is a homomorphism from HtoP. For a fixed r-patternP, theP-coloring problem\nconsists in deciding whether there exists a homomorphism of a given input r-graphHto\nP. WhenPis a graph, the classical Hell–Ne˘ set˘ ril Theorem [37] stat es that the P-coloring\nproblem is in PifPis bipartite and is NP-complete otherwise. For r≥3, theP-coloring\nproblem is already NP-complete when P=Kr\nr(i.e. ther-graph with only one edge) [71].\nFor anr-patternPand a real number α∈[0,1], letHom(P,α) denote the following\nproblem:\nHomomorphism with minimum degree constraint\nInput: Anr-graphHonnvertices with δ(H)≥αnr−1.\nQuestion: Is there a homomorphism from HtoP?\nEdwards [16] was the first to explore Hom(Kℓ,α), proving that for every ℓ≥3, the\nproblem Hom( Kℓ,α) is inPifα >ℓ−3\nℓ−2and isNP-complete otherwise. Subsequent\nextensions to general graphs and hypergraphs were consider ed in [14, 72], although many\nproblems in this direction remain unresolved. Unfortunate ly, the exponents in the running\ntime for algorithms provided in [16, 14, 72] for large αare excessively large and depend\nonF, rendering them impractical for our purposes (Theorem 1.1) . Therefore, we present\nthe following theorem, which efficiently solves Hom(P,α) in timeO(nr) whenPis a\nminimalr-pattern and αis close torλP(observe that if α>rλ P, then there is not such\na homomorphism).\nTheorem 1.6. Suppose that Pis a minimal r-pattern on ℓvertices. Then there exist\nεP>0andnPdepending only on Psuch that the problem Hom(P,α)can be solved in time\n(ℓ+1)nrfor all forn≥nPandα≥rλP−εP, wherenis the number of vertices of the input\nr-graph. Moreover, for every r-graphHonn≥nPvertices with δ(H)≥(rλP−εP)nr−1,\nthe homomorphism from HtoPis unique (up to the automorphism4ofP) if it exists.\nFor complete graphs, the requirement that n≥nPis not necessary.\nTheorem 1.7. For everyℓ≥2, the problem Hom(Kℓ,α)can be solved in time (ℓ+1)n2\nfor allα>3ℓ−4\n3ℓ−1, wherenis the number of vertices of the input graph. Moreover, for ev ery\ngraphGonnvertices with δ(G)>3ℓ−4\n3ℓ−1n, the homomorphism from GtoKℓis unique (up\nto the automorphism of Kℓ) if it exists.\nRemark. It would be interesting to determine the infimum of αfor which Hom(Kℓ,α)\ncan be solved in time O(n2). It follows from Theorem 1.7 and Edwards’ result that this\ninfimum lies in the interval/bracketleftBig\nℓ−3\nℓ−2,3ℓ−4\n3ℓ−1/bracketrightBig\n(assuming P/\\⌉}a⊔io\\slash=NP).\nWe take a step further by extending Theorem 1.6 to an importan t variant of the homomor-\nphism problem, specifically the surjective homomorphism pr oblem. For a fixed r-pattern\nP, thesurjective P-coloring problem involves determining whether a surjective ho-\nmomorphism exists for a given input r-graphHtoP. Unlike the P-coloring problem, a\nHell–Ne˘ set˘ ril-type theorem for the graph surjective hom omorphism problem is still elusive\n(see e.g. [6, 34, 35, 56, 21, 33] for some related results).\nFor anr-patternPand a real number α∈[0,1], letSHom(P,α) denote the following\ndecision problem5:\n4An automorphism of Pis simply a bijective homomorphism from PtoP.\n5A quick observation is that the results in [16, 14, 72] concer ningHom(P,α) can be easily extended to\nSHom(P,α) with a minor modification to their original proofs.\n5Surjective homomorphism with minimum degree constraint\nInput: Anr-graphHonnvertices with δ(H)≥αnr−1.\nQuestion: Is there a surjective homomorphism from HtoP?\nLet us introduce some technical definitions before stating t he result. For an r-pattern\nP= (ℓ,E), let Φ Pdenote the maximum value of the following optimization prob lem:\nmaxz\ns.t.∃(x1,...,x ℓ)∈∆ℓ−1with∂iΛP(x1,...,x ℓ)≥zfor alli∈[ℓ].(1)\nHere∂iΛPdenotes the partial derivative of Λ Pwith respect to the i-th variable. For\nconvenience, let DPbe the collection of all optimal solutions to optimization p roblem (1),\nand let\nφP:= min\n(x1,...,xℓ)∈DPmin{xi:i∈[ℓ]}.\nWe sayPisrigidifφP>0 and∂iΛP(x1,...,x ℓ) = ΦPholds for all i∈[ℓ] and for all\n(x1,...,x ℓ)∈DP. Otherwise, we say Pisnon-rigid . A quick observation, derived from\nthe Lagrangian multiplier method, asserts that every minim al patternPis rigid (but not\nvise versa, for example, Ckis rigid fork≥5 but not minimal). Furthermore, for a minimal\nr-patternPwe haverλP= ΦP.\nTheorem 1.8. SupposePis a rigidr-pattern on ℓvertices. Then there exist εP>0and\nnPdepending only on Psuch that the problem SHom(P,α)can be solved in time (ℓ+1)nr\nfor all for n≥nPandα≥ΦP−εP, wherenis the number of vertices of the input\nr-graph. Moreover, for every r-graphHonn≥nPvertices with δ(H)≥(ΦP−εP)nr−1,\nthe surjective homomorphism from HtoPis unique (up to the automorphism of P) if it\nexists.\nProofs for Theorems 1.6, 1.7, and 1.8 are presented in Sectio n 2.\n1.3 Degree-stability\nIn this subsection, we focus on the structure of F-free hypergraphs whose number of edges\nis close to the maximum. This important topic in Extremal Com binatorics traces its\norigins to the seminal work of Simonovits [70], who proved th at forℓ≥3, everyKℓ-freen-\nvertex graph with at least ex( n,Kℓ)−o(n2) edges can be made ( ℓ−1)-partite by removing\no(n2) edges (see [27] for a nice short proof).\nLetr≥2 be an integer, Fbe a nondegenerate family of r-graphs, and Hbe a family of\nF-freer-graphs. We say\n•Fisedge-stable with respect to Hif for every δ>0 there exist ε>0 andn0such\nthat every F-freer-graphHonn≥n0vertices with |H| ≥(π(F)/r!−ε)nrbecomes\na member in Hafter removing at most δnredges,\n•Fisdegree-stable with respect to Hif there exist ε>0 andn0such that every F-\nfreer-graphHonn≥n0vertices with δ(H)≥(π(F)/(r−1)!−ε)nr−1is a member\ninH,\n6•Fisvertex-extendable with respect to Hif there exist ε>0 andn0such that for\neveryF-freer-graphHonn≥n0vertices with δ(H)≥(π(F)/(r−1)!−ε)nr−1the\nfollowing holds: if H−vis a member in H, thenHis a member in Has well.\nVertex-extendability was introduced in [54] to provide a un ified framework for proving\nthe degree-stability of certain classes of nondegenerate f amilies of hypergraphs. However,\none limitation of the main results in [54] is that they only ap ply to families Fthat are\neither blowup-invariant (see [54, Theorem 1.7]) or have a st rong stability called vertex-\nstability (see [54, Theorem 1.8]). In the following theorem , we extend the results of [54]\nto include the broader class of families with edge-stabilit y, a property satisfied by almost\nall nondegenerate hypergraph families (with known Tur´ an d ensities).\nTheorem 1.9. LetFbe a nondegenerate family of r-graph and Hbe a hereditary6class\nofF-freer-graphs. If Fis both edge-stable and vertex-extendable with respect to H, then\nFis degree-stable with respect to H.\nFor manyr-graph families F, extremal F-free constructions are typically P-colorable for\nsome minimal pattern P. We refer to such a pair ( F,P) as aTur´ an pair . In most cases,\napplying Theorem 1.9 involves choosing Has the collection of all P-colorable hypergraphs.\nHence, when stating that Fis edge-stable/degree-stable/vertex-extendable, it is i mplied\nthat this property is with respect to the family of P-colorable hypergraphs for simplicity.\nHypergraph Degree-stable?\nEdge-critical graphs [4, 18] Yes\nNon-edge-critical graphs [54] No\nExpansion of edge-critical graphs [58, 64, 51, 54] Yes\nExpansion of non-edge-critical graphs No\nExpansion of extended Erd˝ os–S´ os tree [69, 61, 8] Yes\nExpansion of Mr\n2forr≥3 [36, 5, 54] Yes\nExpansion of M3\nk,L3\nk, orL4\nkfork≥2 [36, 42, 54] Yes\nExpansion of M4\nkfork≥2 [75] Yes\nExpansion of K3\n4⊔K3\n3[74] Yes\nGeneralized triangle Trforr∈ {3,4}[7, 24, 46, 51, 54, 68, 63] Yes\nGeneralized triangle Trforr∈ {5,6}[25, 63] No\nExpanded triangle C2r\n3[23, 47] No\nFano Plane [13, 48, 31] Yes\nF3,2(3-book with 3 pages) [29] No\nF7(4-book with 3 pages) [30] Yes\nF4,3(4-book with 4 pages) [28] No\nTable 1: List of hypergraphs with or without degree-stabili ty.\nIn Table 1, we summarize (most of) the previously studied hyp ergraph families (their\ndefinitions are included in the Appendix.) with degree-stab ility. Since they are all edge-\nstable, according to Theorem 1.9, proving degree-stabilit y requires verifying their vertex-\nextendability. This verification is relatively straightfo rward, and we refer the reader to [54,\n39] for systematic results on this property.\nProof for Theorem 1.9 is presented in Section 3.\nRemark. We briefly mention an application7of Theorem 1.9 in spectral Tur´ an problems.\n6A family His hereditary if all subgraphs of every member H∈Hare also contained in H.\n7This was observed by Dhruv Mubayi in a previous project with C hristian Reiher and the second author.\n7By combining [45, Theorem 1.4] with straightforward calcul ations (see e.g. the proof\nof [45, Corollary 1.6]), one immediately obtain answers to t he spectral Tur´ an problem for\nhypergraphs in Table 1 with degree-stability. Since this is not the primary focus of the\ncurrent work, we omit the definitions and details here.\n2 Proofs for Theorems 1.6, 1.7, and 1.8\nWe prove Theorems 1.6, 1.7, and 1.8 in this section. The core o f the proofs is a simple\nclustering algorithm based on the distance (defined below) b etween a pair of vertices.\nGiven ann-vertexr-graphHand two vertices u,v∈V(H), thelinkLH(v) ofvisLH(v) =\n{A∈∂H:A∪{v} ∈ H}, and the (Hamming) distance betweenuandvis distH(u,v):=\n|LH(u)△LH(v)|. It is clear that dist H(u,v) can be calculated in time nr−1. Observe that\nfor a graph G, the value dist G(u,v) is simply the Hamming distance of the row vectors\ncorresponding to uandvin the adjacency matrix of G. The following clustering algorithm\nbased on the distance of vertices will be crucial for proofs i n this section.\nAlgorithm 1 Hamming Clustering\nInput: A triple ( H,ℓ,δ), where His ann-vertexr-graph,ℓ≥2 is an integer, and\nδ∈[0,1] is a real number.\nOutput: A partition V(H) =V1∪···∪Vℓ.\nOperations:\n1. Take a vertex v1∈V(H), and letW1:=/braceleftbig\nu∈V(H): distH(u,v1)≤δnr−1/bracerightbig\n. Sup-\nposethat wehave defined W1,...,W iforsomei∈[ℓ−1]. IfV(H)\\(W1∪···∪Wi)/\\⌉}a⊔io\\slash=\n∅, then take an arbitrary vertex vi+1from it, and let\nWi+1:=/braceleftBigg/braceleftbig\nu∈V(H): distH(u,vi+1)≤δnr−1/bracerightbig\nifi/\\⌉}a⊔io\\slash=ℓ−1,\nV(H)\\(W1∪···∪Wi) if i=ℓ−1.\nOtherwise, let Wi+1=···=Wℓ=∅.\n2. LetVi:=Wi\\(Wi+1∪···∪Wℓ) fori∈[ℓ−1], and letVℓ:=Wℓ.\nProof of Theorem 1.7. Letn≥ℓ≥2 be integers. Suppose that Gis ann-vertex graph\nwithδ(G) =αn, whereα>3ℓ−4\n3ℓ−1is a real number. Let δ:=1\n3ℓ+1. Run Algorithm 1 with\ninput (G,ℓ,δ), and letV1∪···∪Vℓ=V(G) denotes the output partition. It is easy to see\nthat the running time for this step is at most ℓn2.\nClaim 2.1. The graphGisℓ-partite iff/uniontext\ni∈[ℓ]G[Vi] =∅.\nProof.Suppose that Gisℓ-partite. Suppose that U1∪···∪Uℓ=V(G) is a partition with/uniontext\ni∈[ℓ]G[Ui] =∅(this partition is used only for the proofand is not used for t he algorithm).\nIt suffices to show that {V1,...,V ℓ}={U1,...,U ℓ}.\nLetxi:=|Ui|/nfori∈[ℓ]. SinceGisℓ-partite, we obtain 1 −xi=/summationtext\nj∈[ℓ]\\{i}xj≥\nδ(G)/n > αfor alli∈[ℓ]. Consequently, xi<1−αfor alli∈[ℓ], and hence, 1 −xi=/summationtext\nj∈[ℓ]\\{i}xj<(ℓ−1)(1−α) for alli∈[ℓ]. In summary, we have\n1−(ℓ−1)(1−α)δn.\nAccording to Algorithm 1, vandv′do not belong to the same part Vi. This proves that\n{V1,...,V ℓ}={U1,...,U ℓ}, and hence,/uniontext\ni∈[ℓ]G[Vi] =/uniontext\ni∈[ℓ]G[Ui] =∅. Additionally,\nobserve that this proof shows the uniqueness of the partitio nU1∪···∪Uℓ=V(G).\nAccording to Claim 2.1, to check whether Gisℓ-partite we just need to check whether/uniontext\ni∈[ℓ]G[Vi] =∅. This can be accomplished in at most n2time. Therefore, the overall time\ncomplexity is at most ( ℓ+1)n2.\nNext, we present the proof for Theorem 1.8, while Theorem 1.6 follows from a similar\nargument and its proof is omitted.\nProof of Theorem 1.8. Fix a rigid r-patternPonℓvertices. For simplicity, let us assume\nthat the vertex set of Pis [ℓ]. For every β >0, let\nDP,β:=/braceleftbigg\n(y1,...,yℓ)∈∆ℓ−1:∃(x1,...,x ℓ)∈DPwith max\ni∈[ℓ]|xi−yi|<β/bracerightbigg\n,\nand let Φ P,βdenote the optimal value of the following optimization prob lem:\nmaxy\ns.t.∃(x1,...,x ℓ)∈∆ℓ−1\\DP,βwith∂iΛP(x1,...,x ℓ)≥yfor alli∈[ℓ].\nIt is easy to see from the definitions that Φ P,β<ΦPfor allβ >0. Take\nδ:=(φP/2)r−1\n(r−1)!, δ∗:=δ\n2rℓ<1\nℓ,andεP:= min/braceleftbiggΦP−ΦP,δ∗\n2,δ\n5/bracerightbigg\n.\nLetnbe sufficiently large, and let Hbe ann-vertexr-graph with δ(H) =αnr−1≥\n(ΦP−εP)nr−1. Run Algorithm 1 with input ( H,ℓ,δ) and letV1∪···∪Vℓ=V(H) denote\nthe output partition. It is easy to see that the running time f or this step is at most ℓnr.\nClaim 2.2. There is a surjective homomorphism from HtoPiff/uniontext\ni∈[ℓ]H[Vi] =∅and\nVi/\\⌉}a⊔io\\slash=∅for alli∈[ℓ].\n9Proof.Suppose that ψ:V(H)→[ℓ] is a surjective homomorphism from HtoP. Let\nUi:=ψ−1(i)⊂V(H) andyi:=|Ui|/nfori∈[ℓ]. Notice that yi>0 fori∈[ℓ] and\nU1∪···∪Uℓis a partition of V(H). Observe from definitions that for every i∈[ℓ] we have\nΦP,δ∗<ΦP−εP≤α=δ(H)\nnr−1≤∂iΛP(y1,...,yℓ).\nSo, it follows from the definitions of εPand ΦP,δ∗that there exists ( x1,...,x ℓ)∈DPsuch\nthat max i∈[ℓ]|xi−yi|<δ∗. Sincefor every vector ( z1,...,zℓ)∈Rℓwith max k∈[ℓ]|zk−xk| ≤\nδ∗, the inequality\n|∂2\ni,jΛP(z1,...,zℓ)| ≤(|z1|+···+|zℓ|)r−2≤(1+ℓδ∗)r−2≤2r−2.\nholds for all i,j∈[ℓ], it follows Taylor’s theorem that for every i∈[ℓ],\n|∂iΛP(y1,...,yℓ)−∂iΛP(x1,...,x ℓ)| ≤2r−2·/summationdisplay\nk∈[ℓ]|yk−xk| ≤2r−2ℓδ∗.\nTherefore,∂iΛP(y1,...,yℓ)≤∂iΛP(x1,...,x ℓ)+2r−2ℓδ∗= ΦP+2r−2ℓδ∗.\nFix distinct i,j∈[ℓ]. Suppose that u,u′∈Ui. Similar to the proof of Theorem 1.7, it\nfollows from the Inclusion-Exclusion Principle and the ine quality above that\ndistH(u,u′) =dH(u)+dH(u′)−2|LH(u)∩LH(u′)|\n≤dH(u)+dH(u′)−2/parenleftbig\ndH(u)+dH(u′)−∂iΛP(y1,...,yℓ)·nr−1/parenrightbig\n= 2·∂iΛP(y1,...,yℓ)·nr−1−/parenleftbig\ndH(u)+dH(u′)/parenrightbig\n≤2/parenleftbig\nΦP+2r−2ℓδ∗−α/parenrightbig\nnr−1\n= 2/parenleftbig\n2r−2ℓδ∗+εF/parenrightbig\nnr−1<δnr−1.\nSupposethat v∈Uiandv′∈Ujaretwo distinct vertices. A simplebut crucial observation\nis that a rigid pattern does not contain twin vertices, i.e. v ertices with the same link.\nTherefore, thetwopolynominals ∂iΛP(X1,...,X ℓ)and∂jΛP(X1,...,X ℓ)arenotidentical.\nHence,\ndistH(v,v′)≥/parenleftbig\nmini∈[ℓ]yi/parenrightbigr−1\n(r−1)!·nr−1>(φP−δ∗)r−1\n(r−1)!·nr−1>δnr−1.\nIt follows from the definition of Algorithm 1 that {U1,...,U ℓ}={V1,...,V ℓ}, and hence,/uniontext\ni∈[ℓ]H[Vi] =∅andVi/\\⌉}a⊔io\\slash=∅for alli∈[ℓ].\nAccording to Claim 2.2, to check whether there is a surjectiv e homomorphism from Hto\nPwe just need to check whether/uniontext\ni∈[ℓ]H[Vi] =∅andVi/\\⌉}a⊔io\\slash=∅for alli∈[ℓ]. This can\nbe accomplished in at most nrtime. Therefore, the overall time complexity is at most\n(ℓ+1)nr.\n3 Proof of Theorem 1.9\nGiven an F-freer-graphHwith large minimum degree, our objective is to show that His\ncontained in H. The strategy in the following proof is to first use the edge-s tability of Fto\nfind a large minimum degree subgraph (i.e. H1[U]) ofHthat is contained in H. Then we\nadd the vertices in V(H) back to H1[U] one by one, where adding a vertex means adding\nall edges in H\\H1[U] containing this vertex. Using the vertex-extensibility, we will show\nthat adding vertices preserves the containment in H, and hence, in the end, we obtain\nH ∈H.\n10Proof of Theorem 1.9. LetFbea nondegenerate family of r-graphs and Hbea hereditary\nfamily of F-freer-graphs. Fix ε1≥ε2>0 sufficiently small and n0sufficiently large such\nthat\n1. every F-freer-graphHonn≥n0vertices with |H| ≥(π(F)/r!−ε2)nrbecomes a\nmember in Hafter removing at most ε1nr/2 edges, and\n2. every F-freer-graphHonn≥n0vertices with δ(H)≥/parenleftBig\nπ(F)/(r−1)!−2ε1/3\n1/parenrightBig\nnr−1\nthe following holds: if H−vis a member in H, thenHis a member in Has well.\nTake 0< ε3≤ε2/2. LetHbe anF-freer-graph onn≥2n0vertices with δ(H)≥\n(π(F)/(r−1)!−ε3)nr−1. We aim to show that H ∈H.\nFirst, notice that\n|H| ≥n\nr×δ(H)≥n\nr×/parenleftbiggπ(F)\n(r−1)!−ε3/parenrightbigg\nnr−1≥/parenleftbiggπ(F)\nr!−ε3/parenrightbigg\nnr.\nSo it follows from Assumption 1 that there exists a subgraph H1⊂ Hsuch that H1∈H\nand\n|H1| ≥ |H|−ε1\n2nr≥/parenleftbiggπ(F)\nr!−ε3/parenrightbigg\nnr−ε1\n2nr≥/parenleftbiggπ(F)\nr!−ε1/parenrightbigg\nnr.\nLet\nV:=V(H), Z:=/braceleftBig\nv∈V:dH1(v)0 andnPbe constants\ngivenbyTheorem1.6. Take εF∈(0,εP)tobesufficientlysmalland n0≥nPbesufficiently\nlarge such that: every F-freer-graphHonn≥n0vertices with δ(H)≥(π(F)−εF)/parenleftbign\nr−1/parenrightbig\nisP-colorable (this is guaranteed by the degree-stability of F). Consider the following\nalgorithm:\nAlgorithm 2 DecidingF-freeness in large minimum degree hypergraphs\nInput:Ann-vertexr-graphHwithδ(H)≥(π(F)−εF)/parenleftbign\nr−1/parenrightbig\n.\nOutput: ”Yes”, if HisF-free; ”No”, otherwise.\nOperations:\n1. Ifn0 (we may assume that Cis an integer) and suppose\nto the contrary that there exists an algorithm Athat solves Embed min-(Kℓ+1,α) in time\nno(ℓ)forα=1\nCℓ. We claim that Acan also solves Embed-Kℓ+1in timeno(ℓ), which\nwould contradict the result by Chen–Huang–Kanj–Xia [11]. I ndeed, consider an arbitrary\nn-vertex graph G. LetV0:=V(G). LetˆGbe the graph obtained from Gby adding\nq:=CℓsetsV1,...,V q, each of size n, and adding new edges {u,v}for all (u,v)∈Vi×Vj\nwhenever 0 ≤i < j≤q. LetN:= (q+ 1)ndenote the number of vertices in ˆGand let\nL:=q+ℓ= (C+1)ℓ. Observe that Kℓ+1⊂GiffKL+1⊂ˆG. Since\nδ(ˆG)≥qn=q\nq+1N=/parenleftbiggL−1\nL−ℓ−1\n(q+1)L/parenrightbigg\nN >/parenleftbiggL−1\nL−1\nCL/parenrightbigg\nN, (3)\nby assumption, algorithm Acan decide in time No(L)= ((C+1)ℓn)o((C+1)ℓ)=no(ℓ)\nwhetherKL+1⊂ˆG, and equivalently, whether Kℓ+1⊂G, proving our claim.\nNow assume that W[1]/\\⌉}a⊔io\\slash=FPT. For every i≥2 letℓibe the smallest integer such that\nEmbed-Kℓi+1cannot be solved in time O(ni), letδi:=1\n2(ℓi−1)andni:= 2ℓi. We claim\nthatEmbed min-(Kℓ+1,(δiℓ2)−1) cannot be solved in time O(ni) for anyℓ≥ni. Indeed,\nsuppose to the contrary that there exist an i∗≥2 and an algorithm Ai∗that solves\n12Embed min-(Kℓ∗+1,(δi∗ℓ2)−1) in timeO(ni∗) for some ℓ∗≥ni∗. Consider an arbitrary\nn-vertex graph G. LetˆGbe the same construction as defined above by replacing ℓwith\nℓi∗andCwithℓ∗/ℓi∗−1≥1 (hence,L=ℓ∗andL/2≥ℓi∗). It follows from (3) that\nδ(ˆG)≥/parenleftbiggL−1\nL−ℓi∗−1\n(q+1)L/parenrightbigg\nN >/parenleftbiggL−1\nL−ℓi∗−1\nL2/2/parenrightbigg\nN=/parenleftbiggL−1\nL−1\nδi∗L2/parenrightbigg\nN.\nBy assumption, algorithm Ai∗can decide in time O/parenleftbig\nNi∗/parenrightbig\n=O/parenleftbig\n((C+1)ℓi∗n)i∗/parenrightbig\n=O/parenleftbig\nni∗/parenrightbig\nwhetherKL+1⊂ˆG, and equivalently, whether Kℓi∗+1⊂G, contradicting the definition of\nℓi∗.\nNext, we present the proof of Theorem 1.3.\nProof of Theorem 1.3. Fixℓ≥0, and suppose to the contrary that there is an algorithm\nAthat solves Embed min-(Kℓ+1[t],0) in timenO(1)(orno(√\nt)). Consider an arbitrary n-\nvertex graph G. We may assumethat n≥t. LetV0:=V(G). LetˆGbethegraph obtained\nfromGby addingℓ−1 setsV1,...,V ℓ−1, each of size n, and adding new edges {u,v}for\nall (u,v)∈Vi×Vjwhenever 0 ≤i3ℓ−4\n3ℓ−1nisr-partite.\nAlgorithm 3 DecidingKℓ+1-freeness in dense graphs\nInput:Ann-vertex graph Gwith|G| ≥ex(n,Kℓ+1)−k.\nOutput: ”Yes”, ifGisKℓ+1-free; ”No”, otherwise.\nOperations:\n1. LetZandUbe as defined in the proof of Theorem 1.4. If z >12ℓ2\nn/parenleftbig\nk+ℓ\n8/parenrightbig\n, then\nreturn ”No”; otherwise, do the following operations.\n2. Run Algorithm 1 with input ( G[U],ℓ,δ), whereδ:=1\n3ℓ+1, and assume that U1∪\n··· ∪Uℓ=Uis the output partition. If G[Ui]/\\⌉}a⊔io\\slash=∅for somei∈[ℓ], then return\n”No”; otherwise, do the following operations.\n3. For every v∈Zfind the smallest index iv∈[ℓ] such that NG(v)∩Uiv=∅. If\nthere is no such ivfor somev∈Z, then return ”No”; otherwise, do the following\noperations.\n4. LetVj:=Uj∪{v∈Z:iv=j}forj∈[ℓ]. If/uniontext\nj∈[ℓ]G[Vj] =∅, then return ”Yes”;\notherwise, return ”No”.\n13Proof of Theorem 1.4. Letn≥ℓ≥2 andk≥1 be integers satisfying n≥max{6ℓ2,30kℓ}.\nLetGbe ann-vertex graph with at least ex( n,Kℓ+1)−kedges. Let V0:=V(G). For\n0≤i≤n−1, we pick a vertex vi+1of minimum degree in the induced subgraph G[Vi],\nand letVi+1:=Vi\\{vi+1}. Letzbe the smallest positive integer isuch thatdG[Vi](vi+1)>\n3ℓ−4\n3ℓ−1(n−i) (if there is no such i, then letz=n). LetZ:={vi:i∈[z]}andU:=V(G)\\Z.\nWeclaimthatthefollowingAlgorithm3candecidethe Kℓ+1-freenessof Gintime(ℓ+4)n2.\nThe validity of Algorithm 3 will be established through the f ollowing claims. Suppose\nthatGisKℓ+1-free. First, notice from the fact ex( n,Kℓ+1)−ex(n−1,Kℓ+1) =n−⌈n/ℓ⌉\nthatδ(G)≥n− ⌈n/ℓ⌉ −k >ℓ−1\nℓn−k−1. Since otherwise we would have |G|<\nn−⌈n/ℓ⌉−k+ex(n−1,Kℓ+1)≤ex(n,Kℓ+1)−k, a contradiction.\nClaim 4.2. We havez≤12ℓ2\nn/parenleftbig\nk+ℓ\n8/parenrightbig\n. In particular, z≤13n\n120ℓ.\nProof.SincedG[Vi](vi+1)≤3ℓ−4\n3ℓ−1(n−i) fori≤z, we obtain\n|G[U]| ≥ |G|−z−1/summationdisplay\ni=03ℓ−4\n3ℓ−1(n−i)\n≥ex(n,Kℓ+1)−k−3ℓ−4\n2(3ℓ−1)(2n+1−z)z\n= ex(n,Kℓ+1)−ℓ−1\n2ℓ(2n−z)z−k+(2n−z)z\n2(3ℓ2−ℓ)−3ℓ−4\n2(3ℓ−1)z\n≥ℓ−1\n2ℓ(n−z)2−/parenleftbigg\nk+ℓ\n8+3ℓ−4\n2(3ℓ−1)z−(2n−z)z\n2(3ℓ2−ℓ)/parenrightbigg\n, (4)\nwhere the last inequality follows from the fact that ex( n,Kℓ+1) =ℓ−1\n2ℓn2−s\n2/parenleftbig\n1−s\nℓ/parenrightbig\n≥\nℓ−1\n2ℓn2−ℓ\n8, wheres:=n−ℓ⌊n/ℓ⌋. Letk′:=k+ℓ\n8+3ℓ−4\n2(3ℓ−1)z−(2n−z)z\n2(3ℓ2−ℓ)≤k+ℓ\n8. Since\n|G[U]| ≤ex(n−z,Kℓ+1)≤ℓ−1\n2ℓ(n−z)2, we havek′≤0, and hence,\nz≤4(3ℓ2−ℓ)\n2n−z/parenleftbigg\nk+ℓ\n8/parenrightbigg\n≤12ℓ2\nn/parenleftbigg\nk+ℓ\n8/parenrightbigg\n.\nHere we used fact that3ℓ−4\n2(3ℓ−1)z≤(2n−z)z\n4(3ℓ2−ℓ), which follows from n≥6ℓ2.\nSinceδ(G[U])>3ℓ−4\n3ℓ−1(n−z), it follows from Theorem 4.1 that there exists a partition\nU1∪···∪Uℓ=Usuch thatG[Ui] =∅for alli∈[ℓ]. Moreover, it follows from uniqueness\nthat this partition is identical to the one generated by Algo rithm 3. Let xi:=|Ui|/(n−z)\nfori∈[ℓ]. It follows from (2) (by plugging in α=3ℓ−4\n3ℓ−1) that\n2\n3ℓ−13ℓ−4\n3ℓ−1(n−z), the vertex uhas at least\n3ℓ−4\n3ℓ−1(n−z)−/summationdisplay\nk∈[ℓ]\\{iv,j}xk(n−z)−|Bj| ≥3ℓ−4\n3ℓ−1(n−z)−3(ℓ−2)\n3ℓ−1(n−z)−k′\n≥2(n−z)\n3ℓ−1−k′\nneighbors in U′\njfor everyj∈[ℓ]\\{iv}. SinceGisKℓ+1-free, there exists jv∈[ℓ]\\ {iv}\nsuch thatvhas no neighbor in NG(u)∩U′\njv. This means that the number of non-neighbors\nofvinUis at least\nxiv(n−z)−k′+2(n−z)\n3ℓ−1−k′>4(n−z)\n3ℓ−1−2k′\n>4(n−z)\n3ℓ−1−2/parenleftbigg\nk+ℓ\n8+3ℓ−4\n2(3ℓ−1)z−(2n−z)z\n2(3ℓ2−ℓ)/parenrightbigg\n>4n\n3ℓ−1−2/parenleftbigg\nk+ℓ\n8/parenrightbigg\n,\nwhere in the last inequality, we used(2n−z)z\n2(3ℓ2−ℓ)≥3ℓ\n2(3ℓ−1)z, which follows from n≥6ℓ2. Since\nn≥max{30kℓ,6ℓ2}, we have4n\n3ℓ−1−2/parenleftbig\nk+ℓ\n8/parenrightbig\n>n\nℓ+k+1. Therefore, dG(v)<ℓ−1\nℓn−k−1,\na contradiction. This shows that vdoes not have any neighbor in Biv, and hence, vhas\nno neighbor in Uiv.\nLetZ1∪··· ∪Zℓ=Zbe a partition such that for every i∈[ℓ] and for every v∈Ziwe\nhaveNG(v)∩Ui=∅.\nClaim 4.3. We haveG[Zi] =∅fori∈[ℓ].\nProof.First, we improve the lower bound xi>2\n3ℓ−1fori∈[ℓ]. Notice that |G[U]| ≤/summationtext\n1≤i0(we may assumethat C:= 1/δis an integer) and suppose\nto the contrary that there exists an algorithm Athat solves Embed avg-(Kℓ+1,n,n1+δ) in\ntimeno(ℓ). We claim that Acan also solves Embed-Kℓ+1in timeno(ℓ), which would\ncontradict the result by Chen–Huang–Kanj–Xia [11].\nThe construction is very similar to that in the proof of Theor em 1.2. Consider an arbitrary\nn-vertex graph G. LetˆGbe the graph obtained from Gby addingℓsetsV1,...,V ℓ, each\nof sizenC, and adding new edges {u,v}for all (u,v)∈Vi×Vjwhenever 1 ≤iex(N,Kℓ+1)−N1+δ,\nitfollowsfromoueassumptionthatalgorithm Acandecideintime No(ℓ)=/parenleftbig\nℓnC+n/parenrightbigo(ℓ)=\nno(ℓ)whetherKℓ+1⊂ˆG, and equivalently, whether Kℓ+1⊂G, proving our claim in the\nfirst paragraph.\n5 Concluding remarks\nRecall that the core of the algorithm for Embed min-(Kℓ+1,α) whenα >3ℓ−4\n3ℓ−1is the\nstructural theorem by Andr´ asfai–Erd˝ os–S´ os [4]. It seem s worth exploring whether refined\nstructural theorems (see e.g. [10, 32, 2, 62]) can be used to p ush the lower bound for α\nfurther.\nThe proof of Theorem 1.1 can be easily modified to cover some hy pergraph families with\nmultiple extremal constructions (see e.g. [52, 53]). There are hypergraph Tur´ an problems\nwhose structure of extremal constructions can exhibit a non minal pattern (see e.g. [38]),\na recursive pattern (see, e.g. [65]), or even a mixed recursi ve pattern (see, e.g. [55]).\nIt is of interest to investigate whether Theorems 1.1 and 1.6 can be extended to cover\nnonminal/recursive/mixed recursive patterns.\nRecall the nice structural characterization of minimal gra phs: a graph is minimal iff it is\ncomplete. It would be interesting to explore a characteriza tion of rigid graphs. Simple\nlinear algebra arguments show that non-singular (i.e. the a djacency matrix is full rank)\nregular graphs are rigid. We refer the reader to [66] for rela ted results on singular graphs.\nIngeneral, onecouldaskforacharacterization ofthefamil ies ofallminimal/rigid r-graphs.\nAcknowledgement\nTheorem 1.9 was previously presented by XL in a 2023 summer sc hool at Suzhou Univer-\nsity, and we would like to thank the organizers for their warm hospitality. XL would also\nlike to thank Dhruv Mubayi and Christian Reiher for very insp iring discussions in a previ-\nous related project. We thank Yixiao Zhang for verifying the vertex-extendability of some\nhypergraphs in Table 1.\n16References\n[1] A. Abboud, N. Fischer, and Y. Shechter. Faster combinato rialk-clique algorithms.\narXiv preprint arXiv:2401.13502 , 2024. 2\n[2] P. Allen, J. B¨ ottcher, S. Griffiths, Y. Kohayakawa, and R. Morris. The chromatic\nthresholds of graphs. Adv. Math. , 235:261–295, 2013. 16\n[3] N. Alon, R. Yuster, and U. Zwick. Finding and counting giv en length cycles. Algo-\nrithmica, 17(3):209–223, 1997. 2\n[4] B. Andr´ asfai, P. Erd˝ os, and V. T. S´ os. On the connectio n between chromatic number,\nmaximal clique and minimal degree of a graph. Discrete Math. , 8:205–218, 1974. 3,\n7, 13, 16\n[5] A. Bene Watts, S. Norin, and L. Yepremyan. A Tur´ an theore m for extensions via an\nErd˝ os-Ko-Rado theorem for Lagrangians. Combinatorica , 39(5):1149–1171, 2019. 7\n[6] M. Bodirsky, J. K´ ara, and B. Martin. The complexity of su rjective homomorphism\nproblems–a survey. Discrete Appl. Math. , 160(12):1680–1690, 2012. 5\n[7] B.Bollob´ as. Three-graphswithouttwotripleswhosesy mmetricdifferenceiscontained\nin a third. Discrete Math. , 8:21–24, 1974. 7\n[8] A. Brandt, D. Irwin, and T. Jiang. Stability and Tur´ an nu mbers of a class of hyper-\ngraphs via Lagrangians. Combin. Probab. Comput. , 26(3):367–405, 2017. 7\n[9] S. Brandt. On the structure of graphs with bounded clique number. Combinatorica ,\n23(4):693–696, 2003. 13\n[10] C.C.Chen, G.P.Jin, andK.M.Koh. Triangle-freegraphs withlargedegree. Combin.\nProbab. Comput. , 6(4):381–396, 1997. 16\n[11] J. Chen, X. Huang, I. A. Kanj, and G. Xia. Strong computat ional lower bounds via\nparameterized complexity. J. Comput. System Sci. , 72(8):1346–1367, 2006. 2, 3, 12,\n16\n[12] M. Cygan, F. V. Fomin, L. u. Kowalik, D. Lokshtanov, D. Ma rx, M. Pilipczuk,\nM. Pilipczuk, and S. Saurabh. Parameterized algorithms . Springer, Cham, 2015. 2\n[13] D. De Caen and Z. F¨ uredi. The maximum size of 3-uniform h ypergraphs not con-\ntaining a Fano plane. J. Combin. Theory Ser. B , 78(2):274–276, 2000. 7\n[14] J. D´ ıaz, J. Ne˘ set˘ ril, M. Serna, and D. M. Thilikos. H-coloring of large degree graphs.\nInEurasian Conference on Information and Communication Technol ogy, pages 850–\n857. Springer, 2002. 5\n[15] R. G. Downey and M. R. Fellows. Parameterized complexity . Monographs in Com-\nputer Science. Springer-Verlag, New York, 1999. 2\n[16] K.Edwards. Thecomplexityofcolouringproblemsonden segraphs. Theoret. Comput.\nSci., 43(2-3):337–343, 1986. 5\n[17] F. Eisenbrand and F. Grandoni. On the complexity of fixed parameter clique and\ndominating set. Theoret. Comput. Sci. , 326(1-3):57–67, 2004. 2\n17[18] P. Erd˝ osand M. Simonovits. On avalence problemin extr emal graph theory. Discrete\nMath., 5:323–334, 1973. 3, 7\n[19] P.Erd˝ osandM.Simonovits. Supersaturatedgraphsand hypergraphs. Combinatorica ,\n3(2):181–192, 1983. 3\n[20] P. Erd˝ os. On the graph theorem of Tur´ an. Mat. Lapok , 21:249–251, 1970. 3\n[21] J. Focke, L. A. Goldberg, and S. ˇZivn´ y. The complexity of counting surjective homo-\nmorphismsandcompactions. In Proceedings of the Twenty-Ninth Annual ACM-SIAM\nSymposium on Discrete Algorithms , pages 1772–1781. SIAM, Philadelphia, PA, 2018.\n5\n[22] F. V. Fomin, P. A. Golovach, D. Sagunov, and K. Simonov. T ur´ an’s theorem through\nalgorithmic lens. In Graph-theoretic concepts in computer science , volume 14093 of\nLecture Notes in Comput. Sci. , pages 348–362. Springer, Cham, [2023] ©2023. 2, 3\n[23] P. Frankl. Asymptotic solution of a Tur´ an-type proble m.Graphs Combin. , 6(3):223–\n227, 1990. 7\n[24] P. Frankl and Z. F¨ uredi. A new generalization of the Erd ˝ os-Ko-Rado theorem. Com-\nbinatorica , 3(3-4):341–349, 1983. 7\n[25] P. Frankl and Z. F¨ uredi. Extremal problems whose solut ions are the blowups of the\nsmall Witt-designs. J. Combin. Theory Ser. A , 52(1):129–147, 1989. 4, 7\n[26] Z. F¨ uredi. Tur´ antypeproblems. In Surveys in combinatorics, 1991 (Guildford, 1991) ,\nvolume 166 of London Math. Soc. Lecture Note Ser. , pages 253–300. Cambridge Univ.\nPress, Cambridge, 1991. 2\n[27] Z. F¨ uredi. A proof of the stability of extremal graphs, Simonovits’ stability from\nSzemer´ edi’s regularity. J. Combin. Theory Ser. B , 115:66–71, 2015. 6\n[28] Z. F¨ uredi, D. Mubayi, and O. Pikhurko. Quadruple syste ms with independent neigh-\nborhoods. J. Combin. Theory Ser. A , 115(8):1552–1560, 2008. 7\n[29] Z. F¨ uredi, O. Pikhurko, and M. Simonovits. On triple sy stems with independent\nneighbourhoods. Combin. Probab. Comput. , 14(5-6):795–813, 2005. 7\n[30] Z. F¨ uredi, O. Pikhurko, and M. Simonovits. 4-books of t hree pages. J. Combin.\nTheory Ser. A , 113(5):882–891, 2006. 7\n[31] Z. F¨ uredi and M. Simonovits. Triple systems not contai ning a Fano configuration.\nCombin. Probab. Comput. , 14(4):467–484, 2005. 7\n[32] W. Goddard and J. Lyle. Dense graphs with small clique nu mber.J. Graph Theory ,\n66(4):319–331, 2011. 16\n[33] P. A. Golovach, M. Johnson, B. Martin, D. Paulusma, and A . Stewart. Surjective\nH-colouring: new hardness results. Computability , 8(1):27–42, 2019. 5\n[34] P. A. Golovach, B. Lidick´ y, B. Martin, and D. Paulusma. Finding vertex-surjective\ngraph homomorphisms. Acta Inform. , 49(6):381–394, 2012. 5\n[35] P. A. Golovach, D. Paulusma, and J. Song. Computing vert ex-surjective homomor-\nphisms to partially reflexive trees. Theoret. Comput. Sci. , 457:86–100, 2012. 5\n18[36] D. Hefetz and P. Keevash. A hypergraph Tur´ an theorem vi a Lagrangians of inter-\nsecting families. J. Combin. Theory Ser. A , 120(8):2020–2038, 2013. 7\n[37] P. Hell and J. Ne˘ set˘ ril. On the complexity of H-coloring. J. Combin. Theory Ser. B ,\n48(1):92–110, 1990. 5\n[38] J. Hou, H. Li, X. Liu, D. Mubayi, and Y. Zhang. Hypergraph s with infinitely many\nextremal constructions. Discrete Anal. , pages Paper No. 18, 34, 2023. 16\n[39] J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang. A step towa rds a general density\nCorr´ adi–Hajnal theorem. arXiv preprint arXiv:2302.09849 , 2023. 7\n[40] R. Impagliazzo, R. Paturi, and F. Zane. Which problems h ave strongly exponential\ncomplexity? volume 63, pages 512–530. 2001. Special issue o n FOCS 98 (Palo Alto,\nCA). 2\n[41] A. Itai and M. Rodeh. Finding a minimum circuit in a graph .SIAM J. Comput. ,\n7(4):413–423, 1978. 2\n[42] T. Jiang, Y. Peng, and B. Wu. Lagrangian densities of som e sparse hypergraphs and\nTur´ an numbers of their extensions. European J. Combin. , 73:20–36, 2018. 7\n[43] G. Katona, T. Nemetz, and M. Simonovits. On a problem of T ur´ an in the theory of\ngraphs.Mat. Lapok , 15:228–238, 1964. 2\n[44] P. Keevash. Hypergraph Tur´ an problems. In Surveys in combinatorics 2011 , volume\n392 ofLondon Math. Soc. Lecture Note Ser. , pages 83–139. Cambridge Univ. Press,\nCambridge, 2011. 2\n[45] P. Keevash, J. Lenz, and D. Mubayi. Spectral extremal pr oblems for hypergraphs.\nSIAM J. Discrete Math. , 28(4):1838–1854, 2014. 8\n[46] P. Keevash and D. Mubayi. Stability theorems for cancel lative hypergraphs. J.\nCombin. Theory Ser. B , 92(1):163–175, 2004. 7\n[47] P. Keevash and B. Sudakov. On a hypergraph Tur´ an proble m of Frankl. Combina-\ntorica, 25(6):673–706, 2005. 7\n[48] P. Keevash and B. Sudakov. The Tur´ an number of the Fano p lane.Combinatorica ,\n25(5):561–574, 2005. 7\n[49] T. Kloks, D. Kratsch, and H. M¨ uller. Finding and counti ng small induced subgraphs\nefficiently. Inform. Process. Lett. , 74(3-4):115–121, 2000. 2\n[50] B. Lin. The parameterized complexity of the k-biclique problem. J. ACM, 65(5):Art.\n34, 21, 2018. 2, 3\n[51] X. Liu. New short proofs to some stability theorems. European J. Combin. , 96:Paper\nNo. 103350, 8, 2021. 7\n[52] X.LiuandD.Mubayi. AhypergraphTur´ anproblemwithno stability. Combinatorica ,\npages 1–30, 2022. 16\n[53] X. Liu, D. Mubayi, and C. Reiher. Hypergraphs with many e xtremal configurations.\nIsrael. J. Math. to appear. 11, 16\n[54] X. Liu, D. Mubayi, and C. Reiher. A unified approach to hyp ergraph stability. J.\nCombin. Theory Ser. B , 158(part 2):36–62, 2023. 7, 15\n19[55] X. Liu and O. Pikhurko. Finite hypergraph families with rich extremal Tur´ an con-\nstructions via mixing patterns. arXiv preprint arXiv:2212.08636 , 2022. 16\n[56] B. Martin and D. Paulusma. The computational complexit y of disconnected cut and\n2K2-partition. J. Combin. Theory Ser. B , 111:17–37, 2015. 5\n[57] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of\nTur´ an.Canadian J. Math. , 17:533–540, 1965. 4\n[58] D. Mubayi. A hypergraph extension of Tur´ an’s theorem. J. Combin. Theory Ser. B ,\n96(1):122–134, 2006. 7\n[59] B.Nagle. Oncomputingthefrequenciesofinducedsubhy pergraphs. SIAM J. Discrete\nMath., 24(1):322–329, 2010. 2\n[60] J. Neˇ setˇ ril and S. Poljak. On the complexity of the sub graph problem. Comment.\nMath. Univ. Carolin. , 26(2):415–419, 1985. 2\n[61] S. Norin and L. Yepremyan. Tur´ an numbers of extensions .J. Combin. Theory Ser.\nA, 155:476–492, 2018. 7\n[62] H. Oberkampf and M. Schacht. On the structure of dense gr aphs with bounded clique\nnumber. Combin. Probab. Comput. , 29(5):641–649, 2020. 16\n[63] O. Pikhurko. An exact Tur´ an result for the generalized triangle. Combinatorica ,\n28(2):187–208, 2008. 7\n[64] O. Pikhurko. Exact computation of the hypergraph Tur´ a n function for expanded\ncomplete 2-graphs. J. Comb. Theory, Ser. B , 103(2):220–225, 2013. 7\n[65] O. Pikhurko. On possible Tur´ an densities. Israel J. Math. , 201(1):415–454, 2014. 4,\n16\n[66] I. Sciriha. A characterization of singular graphs. Electron. J. Linear Algebra , 16:451–\n462, 2007. 16\n[67] A. Sidorenko. What we know and what we do not know about Tu r´ an numbers.\nGraphs Combin. , 11(2):179–199, 1995. 2\n[68] A. F. Sidorenko. On the maximal number of edges in a homog eneous hypergraph\nthat does not contain prohibited subgraphs. Mat. Zametki , 41(3):433–455, 459, 1987.\n4, 7\n[69] A. F. Sidorenko. Asymptotic solution for a new class of f orbiddenr-graphs. Combi-\nnatorica, 9(2):207–215, 1989. 7\n[70] M. Simonovits. A method for solving extremal problems i n graph theory, stabil-\nity problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966) , pages 279–319.\nAcademic Press, New York, 1968. 6\n[71] E. Szyma´ nska. The complexity of vertex coloring probl ems in uniform hypergraphs\nwith high degree. In Graph-theoretic concepts in computer science , volume 6410 of\nLecture Notes in Comput. Sci. , pages 304–314. Springer, Berlin, 2010. 5\n[72] E.Szyma´ nska. H-colorings ofdensehypergraphs. Inform. Process. Lett. , 112(23):899–\n902, 2012. 5\n20[73] P. Tur´ an. On an external problem in graph theory. Mat. Fiz. Lapok , 48:436–452,\n1941. 2\n[74] Z. Yan and Y. Peng. An irrational Lagrangian density of a single hypergraph. SIAM\nJ. Discrete Math. , 36(1):786–822, 2022. 7\n[75] Z. Yan and Y. Peng. Lagrangian densities of 4-uniform ma tchings and degree stability\nof extremal hypergraphs. Manuscript submitted for publica tion, 2023. 7\n[76] R. Yuster. Finding and counting cliques and independen t sets inr-uniform hyper-\ngraphs.Inform. Process. Lett. , 99(4):130–134, 2006. 2\n[77] A. A. Zykov. On some properties of linear complexes. Mat. Sbornik N.S. , 24/66:163–\n188, 1949. 3\nDefinitions for hypergraphs in Table 1\n•A graphFisedge-critical if there exists an edge e∈Fsuch thatχ(F−e)<χ(F).\n•Fix a graph F, theexpansion Hr\nFofFis ther-graphs obtained from Fby adding\na set ofr−2 new vertices into each edge of F, and moreover, these new ( r−2)-sets\nare pairwise disjoint.\n•Given anr-graphFwithℓ+ 1 vertices, the expansion HF\nℓ+1ofFis ther-graph\nobtained from Fby adding, for every pair {u,v} ⊂V(F) that is not contained in\nany edge of F, an (r−2)-set of new vertices, and moreover, these ( r−2)-sets are\npairwise disjoint.\n•We say a tree Tis anErd˝ os–S´ os tree if it satisfies the famous Erd˝ os–S´ os conjecture\non trees. The ( r−2)-extension Ext(T) of a treeTis\nExt(T) :={e∪A:e∈T},\nwhereAis a set ofr−2 new vertices that is disjoint from V(T). Anr-graphFis\nanextended tree ifF= Ext(T) for some tree.\n•The (r-uniform) generalized triangle Tris ther-graph with vertex set [2 r−1]\nand edge set\n{{1,...,r−1,r},{1,...,r−1,r+1},{r,r+1,...,2r−1}}.\n•LetC2r\n3(the expanded triangle) denote the 2 r-graph with vertex set [3 r] and edge\nset\n{{1,...,r,r +1,...,2r},{r+1,...,2r,2r+1,...,3r},{1,...,r,2r+1,...,3r}}.\n•TheFano plane Fis the 3-graph with vertex set {1,2,3,4,5,6,7}and edge set\n{123,345,561,174,275,376,246}.\n•LetF7(4-book with 3-pages) denote the 3-graph with vertex set {1,2,3,4,5,6,7}\nand edge set\n{1234,1235,1236,1237,4567}.\n21•LetF4,3denote the 4-graph with vertex set {1,2,3,4,5,6,7}and edge set\n{1234,1235,1236,1237,4567}.\n•LetF3,2denote the 3-graph with vertex set {1,2,3,4,5}and edge set\n{123,124,125,345}.\n•The 3-graph K3\n4⊔K3\n3has vertex set {1,2,3,4,5,6,7}and edge set\n{123,124,234,567}.\n•Ther-graphMr\nk(r-uniformk-matching) is the r-graph consisting of kpairwise\ndisjoint edges.\n•Ther-graphLr\nk(r-uniformk-sunflower) isthe r-graphconsistingof kedgese1,...,ek\nsuch that for all 1 ≤i0 such that the set\nofxfulfilling liminf n∈NCn\nR(x)≤Kis Lebesgue-co-null. We also show that for any ε>0 the set of x\nfor which (31\n2\n2)kx1−ε≤Ck\nR(x)≤(31\n2\n2)kx1+εfor all 0≤k≤1\n1−log23\n2log2xis large for a suitable notion\nof largeness.\n1 Introduction\nThe Collatz map Col : N+1→N+1 takes an even number nton\n2and an odd number to 3 n+1. For\nn∈N+1 define Col min(n)=min k∈NColk(n). The Collatz conjecture states that Col min(n)=1 for every\nn∈N+1. Note that whenever n∈Nis odd, then Col (n)is even. Thus one can define a version of\nCN:N+1→N+1 which takes nto1\n2nifnis even and to3n+1\n2, ifnis odd. An heuristic approach\nsuggests that when we iterate CNforktimes, then roughlyk\n2of the set {Ci\nN|0≤i9\n4. Then there is a n >0such that Cn\nR(r)∈(3\n4,9\n4].\nThis seems a much more tractable question than the analogous one for CN. In this paper we give a\npartial result. Recall that for Lebesgue-measurable subse tsA⊆BofR,AisLebesgue-co-null in B if\nλ(B\\A)=0.\nTheorem 1.2. There exists K >0such that {x∈(3\n4,∞)|liminf n∈NCn\nR(x)≤K}is a Lebesgue-co-null\nsubset of (3\n4,∞).\nFor the notion of meagerness an even stronger result holds:\nTheorem 1.3. The set{x∈(3\n4,∞)|liminf n→∞CR(x)=3\n4}is a comeager subset of (3\n4,∞).\nTheorem 1.2 leaves the question open what K>0 can be chosen. The best choice would be K=3\n4.\nThus we formulate the following conjecture.\nConjecture 1.4. The set{x∈(3\n4,∞)|liminf n∈NCn(x)=3\n4}is a Lebesgue-co-null subset of (3\n4,∞).\nAlthough we do not know a proof of Conjecture 1.4, we show that its truth is equivalent to a problem\nthat is verifiable by a finite (although very large) number of c alculations: It is sufficient to check that for\na subset of (3\n4,R)of measure >αwe have that CM\nR(x)≤9\n8for some M∈N, some large R, and some\nappropriate 0 <α<1, which is possible to do since CRis ’sufficiently continuous’ in the sense that\ninfn∈NCM\nR(x)≤9\n8is true if it is true for all yin an open neighborhood of x(for most xat least). Instead of\nchecking sufficiently many numbers one can independently te st a finite number of x∈(3\n4,R). A test with\nconcrete values for α,R,Cwas performed and every single result was in the desired set. Assuming that\nthe conjecture is false, i.e., the desired set has probabili ty≤αthis outcome would have a probability of\nless than 10−100giving very good reason that Conjecture 1.4 is true.\nOur second type of results concerns approximations of the dy namics of CR. To state these results we\nneed the following notion of largeness which will become use ful in the analysis:\nDefinition 1.5. We say that a subset A⊆Rhasreal density 0≤λ≤1 if for every ε>0 there exists\nR0>0 such that |λ(A∩(0,R))\nR−α|<εfor all R≥R0.\nWe say that a subset A⊆Rhasreal(C,D)-density , ifC>0, 00 and 00 and x∈(3\n4,∞)define its stopping time τK(x)to be the minimal n∈Nsuch that Cn\nR(x)≤K\nif such an nexists and set τK(x)=∞otherwise.\nNow we can state the other main results:4\nTheorem 1.6. Suppose that ε>0. Then the set {x∈(0,∞)| ∀0≤k≤1\n1−log23\n2log2x:(31\n2\n2)kx1−ε≤\nCk\nR(x)≤(31\n2\n2)kx1+ε}is real∗-dense.\nTheorem 1.7. There exists K >0such that the set {x∈(3\n4,∞)|(1\n1−log23\n2−ε)log2x≤τK(x)≤(1\n1−log23\n2+\nε)log2x}has real density 1for every ε>0.\nIn the last section we will associate an acyclic directed gra phGxto each x∈Rsuch that Conjecture\n1.4 is true if and only if the set of x∈(3\n4,∞)such that Gxhas exactly one connected component is\nLebesgue-co-null in (3\n4,∞).\n2 Almost all reals attain bounded values\nIn the introduction we already noticed that CRbijectively maps I0= (3\n4,9\n4]to itself. We denote the\nrestriction of CRtoI0byT. This map Thas some very nice properties, which will be of great help to\nobtain the main results. We gather them in the following prop osition:\nProposition 2.1. T is uniquely ergodic, i.e., there exists a unique probabili ty measure µon the Borel\nsubsets of I 0with the property that µ(T−1(A)) = µ(A)for every Borel set A ⊆I0. The corresponding\ninvariant measure µis equivalent to the Lebesgue measure restricted to I 0. Furthermore, for all x ∈I0\nthe sets{Tn(x)|n∈N}and{T−n(x)|n∈N}are dense in I 0.\nProof. The proposition follows from the fact that Tis conjugate via a homeomorphism to an irra-\ntional rotation and it is well known that irrational rotatio ns are uniquely ergodic and are minimal:\nVia the homeomorphism log3(·):I0→(1−2log3(2),2−2log3(2)],Tis conjugate to the map: τ:\n(1−2log3(2),2−2log3(2)]→(1−2log3(2),2−2log3(2)]defined by\nτ(x)=/braceleftBigg\nx+1−log3(2)ifx∈(1−2log3(2),1−log3(2)]\nx−log3(2) ifx∈(1−log3(2),2−2log3(2)].\nand this in conjugate via the translation given by x/mapsto→x−1+2log3(2)to:\nτ′(x)=/braceleftBigg\nx+1−log3(2)ifx∈(0,log3(2)]\nx−log3(2) ifx∈(log3(2),1].\nNow τ′is an irrational rotation and the claim follows noting that t he push-forward of the Lebesgue\nmeasure restricted to (0,1]under the conjugate map has Lebesgue density f(x)=1\nxlog(3).\nThe following notion is simple but will be of essential help l ater.\nDefinition 2.2. Define x/precedesorcurly3yif there exists n∈Nsuch that 3n·x=y. We will also sometimes write\ny/followsorcurly3xifx/precedesorcurly3y.\nWe gather some if its properties in a lemma:\nLemma 2.3. For every x /precedesorcurly3y:\n1. Ck\nR(x)/precedesorcurly3Ck\nR(y)for all k∈N,\n2.infn∈NCn\nR(x)≤infn∈NCn\nR(y),\n3.liminf n∈NCn\nR(x)≤liminf n∈NCn\nR(y).5\nProof. To see 1.it suffices to consider the case y=3xandk=1. But by definitionCR(3x)\nCR(x)∈{1,3,9}and\nthe claim follows. 2 .and 3.are immediate consequences from 1 .\nNow we are already in a position to prove (a slightly stronger version of) Theorem 1.3.\nTheorem 2.4. The set{x∈(3\n4,∞)|liminf n∈NCn(x)=3\n4}is comeager in (3\n4,∞)and the set {x∈(3\n4,9\n4]|\n∀k∈N: liminf n∈NCn(3kx)=3\n4}is comeager in (3\n4,9\n4].\nProof. We show that every non-empty open set Vof(3\n4,∞)contains a non-empty open subset U⊆Vsuch\nthat liminf n∈NCn(x)=3\n4for all x∈U. By making Vsmaller we can assume that 3−iV⊆(3\n4,9\n8]for some\ni∈N. By making Veven smaller we can assume that 3−iV=3−i−j(N−1\n2,N+1\n2)for some j,N∈N.\nSince{T−n(1)|n∈N}is dense in (3\n4,9\n4], we can find M∈Nsuch that T−M(1)∈3−iVand 2M>3i+2N.\nIf the balanced ternary representation of Nis∑l\nk=0ak3kfor some l∈Nandak∈{1,θ,0}with al=1,\nthen T−M(1)∈3−iVand 2M>Nimplies that the balanced ternary representation of 2Mbegins with\nthat of N, i.e., if 2M=∑p\nk=0bk3kfor some p∈Nandaj∈{1,θ,0}with ap=1, then bp−k=al−kfor\n0≤k≤l. This implies that 3l−p(2M−1\n2,2M+1\n2)⊆(N−1\n2,N+1\n2). Thus 3l+i−p(2M−1\n2,2M+1\n2)⊆V\nbutl+i≤log3(3i3N)≤log32M\n3≤p, thus l+i≤p. But by definition for every x∈(2M−1\n2,2M+1\n2)we\nhave CM\nR(x)=x\n2M∈(3\n4,9\n4]. By 1.of Lemma 2.3 this holds as well for every x∈3l+i−p(2M−1\n2,2M+1\n2).\nThus we have our desired subset by noticing that liminf n∈NCn(x) =3\n4for every x∈(3\n4,9\n4]. Thus the\nunion Wof all open sets Uwith the property that liminf n∈NCn(x)=3\n4for all x∈Uis dense open, thus\ncomeager. Thus for all n∈Nthe set W∩3n(3\n4,9\n4]is comeager in 3n(3\n4,9\n4]thus he set 3−nW∩(3\n4,9\n4]is\ncomeager in (3\n4,9\n4]for all n∈Nhence/intersectiontext\nn∈N(3−nW∩(3\n4,9\n4])is also comeager in (3\n4,9\n4]which proves the\nsecond part.\nAnalogously to the original Collatz map we define the parity sequence of a real number xto be the\nsequence (p(x)n)n∈N, where\np(x)n=/braceleftBigg\n0 if[Cn\nR(x)]is even,\n1 if[Cn\nR(x)]is odd.\nWe have the following lemma.\nLemma 2.5. CN\nR(x)=3∑N−1\nn=0p(x)n\n2N·x for all N ∈Nand x∈R.\nProof. We proceed by induction, the case N=0 being trivially true. Then CN+1\nR(x)=3p(CN\nR(x))0\n2CN\nR(x)=\n3p(x)N\n23∑N−1\nn=0p(x)n\n2N·x=3∑N\nn=0p(x)n\n2N·x.\nNow we show that - as in the classical case - the parity sequenc es of xandx+2N·zdo coincide for\nthe first nvalues.\nLemma 2.6. Suppose that x ∈R, z∈Z, N∈N. Then the following hold:\n1. p(x)n=p(x+2N·z)nfor all 0≤n0, and 0≤N≤ ⌈log2(b−a)⌉.\nThen µ(a,b)({x∈(a,b)|∑N−1\nk=0p(x)k≥(1\n2+ε)·N})≤2e−2ε2Nand µ(a,b)({x∈(a,b)|∑N−1\nk=0p(x)k≤\n(1\n2−ε)·N})≤2e��2ε2N.\nProof. This is a consequence of Hoeffding’s inequality (see [3]) wh ich states that if νnis the uniform\nmeasure on {0,1}[0···n), or equivalently the n-fold product measure of the uniform distribution on {0,1}\nfor some n∈N, then\nνn({x∈{0,1}[0···n)|n−1\n∑\nk=0xi≥(1\n2+ε)n})≤e−2ε2n\nand\nνn({x∈{0,1}[0···n)|n−1\n∑\nk=0xi≤(1\n2−ε)n})≤e−2ε2n.\nSetM=⌈log2(b−a)⌉. By Proposition 2.9 the push-forward measure of µ[a,a+2M)under the map x/mapsto→\n(p(x)n)0≤n2M−1.\nTo show Theorem 1.2 we are actually showing the stronger stat ement that the set of all xsuch that\nthe iterations of 3nxunder CReventually reach a value less then some K>0 for all n∈Nis Lebesgue-\nco-null. Before we proof this we introduce some helpful nota tion:\nDefinition 2.11. Lett(x)be the unique integer such that 3t(x)x∈(3\n4,9\n4]forx∈(0,∞). Let π:(0,∞)→\n(3\n4,9\n4],x/mapsto→3t(x)xtheprojection of(0,∞)onto(3\n4,9\n4].\nWe gather some properties of πin the following lemma:\nLemma 2.12. 1. For every z ∈Zthe restriction of πto3z(3\n4,9\n4]is a homeomorphism from 3z(3\n4,9\n4]\nto(3\n4,9\n4].\n2.π◦CR=CR◦π.\n3. We have Tn(x)=3m\n2nx for every x ∈(3\n4,9\n4]for some m such that n log3(2)−10such that µ(3\n4,9\n4]({x∈(3\n4,9\n4]|∀N∈N: liminf n→∞Cn\nR(3Nx)≤K})=1.\nProof. Note that the push-forward measure of µ(a,b)under multiplication with r>0 isµ(ra,rb). Choose\nε>0 such that 2 >31\n2+εandb>2. Define N0=⌊log33b\n1\n2+ε⌋. For N≥N0consider the interval (0,b·\n2N\n3⌊(1\n2+ε)·N)⌋], set MN=⌊log2(b·2N\n3⌊(1\n2+ε)·N)⌋)⌋and the set AN={x∈(0,b·2N\n3⌊(1\n2+ε)·N)⌋]|∑MN−1\nk=0p(x)k≥(1\n2+\nε)·MN}. Then\nµ(0,b)(3⌊1\n2+ε)·N)⌋\n2NAn)≤2e−2ε2MN≤2e2ε2e−2ε2log2(b·(2N\n31+N(1\n2+ε)))\n=2e2ε2e−2ε2(log2(b\n3)+Nlog2(2\n31\n2+ε))\n.\nNow we can choose b≥2 large enough so that ∑∞\nN=N02e2ε2e−2ε2(log2(b\n3)+Nlog2(2\n31\n2+ε))\n<1. This means\nA=/uniontext\nN≥N0(3⌊(1\n2+ε)N⌋\n2N)ANhasµ(0,b)-measure <1 and thus B=(0,b)\\Ahas positive µ(0,b)-measure.\nSublemma 2.14. Suppose that x ∈B and N∈N, then CN\nR(2N\n3⌊(1\n2+ε)·N⌋·x)/precedesorcurly331+⌊(1\n2−ε)·N0⌋·x. In particular,\nCN\nR(2N\n3⌊(1\n2+ε)·N⌋·x)≤(3b)2\n1−2ε.\nProof of sublemma. We proceed by induction: If N≤N0, then CN\nR(2N\n3⌊(1\n2+ε)·N)⌋·x)/precedesorcurly33N\n2N2N\n3⌊(1\n2+ε)·N)⌋·x/precedesorcurly3\n31+⌊(1\n2−ε)·N0)⌋·x.Now suppose that N≥log33b\n1\n2+ε. This implies that MN≤N, since if towards a contradic-\ntionMN>N, i.e.,⌊log2(b·2N\n3⌊(1\n2+ε)·N⌋)⌋>N, then also log2(b·2N\n3⌊(1\n2+ε)·N⌋)>N, thus b·2N\n3⌊(1\n2+ε)·N)⌋>2N, or8\nb>3⌊(1\n2+ε)·N⌋, thus log3b>⌊(1\n2+ε)·N⌋≥(1\n2+ε)·N−1, thus N3⌊(1\n2+ε)·N⌋we conclude 0 ≤N−MNz:Tw(x)/∈C}. Then the Fzare\npairwise disjoint and T(Fz)=Fz−1thus µ(Fz)=µ(Fw)for all z,w∈Z, thus necessarily µ(Fz)=0 for all\nz∈Z, where µdenotes the T-invariant probability measure on (3\n4,9\n4]. Thus the set E={x∈(3\n4,9\n4]|#{n∈\nN|Tn(x)∈C}=∞}is of µ(3\n4,9\n4]-measure 1. Now, let x∈E. Then there exist arbitrarily large N∈Nsuch\nthat and TN(x)∈C. Thus 3M(TN(x))∈B. By Lemma 2.12 we have 2NTN(x)=3mxfor some Nlog3(2)−\n10and b>2a9\nset B⊆(0,b)and N 0=⌊log33b\n1\n2+ε⌋. Set α=µ(0,b)(B). By Sublemma 2.14 we know that CN\nR(2N\n3⌈(1\n2+ε)·N⌉x)/precedesorcurly3\n31+⌊(1\n2−ε)·N0⌋·x for all N ∈Nand x∈B. If for some M >0the set A M={x∈(0,b)|CM\nR(31+⌊(1\n2−ε)·N0⌋·\nx)≤9\n4hasµ(0,b)-measure greater than 1−α, then µ(0,b)(A∩B)>0. Furthermore, if x ∈A∩B then\nCN+M\nR(2N\n3⌈(1\n2+ε)·N⌉x)≤9\n4for all N∈Nand we can use A ∩B in the proof of Theorem 2.13 to obtain9\n4and\nthus also3\n4as a bound since every orbit in (3\n4,9\n4]is dense by Proposition 2.1. For suitable values of b ,ε\nand M one may test empirically if A Mhas measure less or equal than 1−αby randomly choosing long\nenough initial segments of x ∈(0,b)so that performing CM\nRonly depends on the initial segment of x. A\ntest with b =3200,ε=0.13, M=6000 and initial segment of x with length 6500 was performed on a\ncomputer by Claudius Röhl for 3000 repetitions (xi)0≤i≤2999. The corresponding αis greater then 0.1.\nAll3000 trials had finite stopping time τ9\n4(xi)<6000 . Thus if A Mhas measure less or equal than 0.9\nthis outcome has a probability of less than 0.93000which is less than 10−137providing strong empirical\nevidence that Conjecture 1.4 is true.\nRemark 2.16. If Theorem 1.2 does not hold for K ≤9\n4, then necessarily limsupn→∞Cn\nR(x) =∞for\na Lebesgue co-null set of x ∈(0,∞), since if limsupn→∞Cn\nR(x)<∞, then limsupn→∞(x)Cn\nR=9\n4by a\nargument similar to that in Theorem 2.4.\nAs a corollary we prove Theorem 1.2:\nProof of Theorem 1.2. By Theorem 2.13 we find K>0 such that\nµ(3\n4,9\n4]({x∈(3\n4,9\n4]|∀N∈N: liminf\nn→∞({Cn\nR(3Nx)|n∈N})≤K})=1.\nSetA={x∈(3\n4,9\n4]| ∀N∈N: liminf n→∞Cn\nR(3Nx)≤K}and define B=/uniontext\nn∈N3nA, then clearly Bis\nLebesgue-co-null in (3\n4,∞)and if x∈Bthen by definition of Awe get that liminf n→∞Cn\nR(x)≤K.\nWe outsource the following technical lemma from the followi ng proof:\nLemma 2.17. Let S⊆(0,∞)and a n∈(0,∞)an increasing diverging sequence such that there exits a\nbound q>0withan\nan+1≥q for all n ∈Nand suppose that ε>0. Ifliminf n→∞µ(an,an+1)(S∩(an,an+1))≥\n1−εthen liminf R→∞µ(0,R)(S∩(0,R))≥1−ε\nq.\nProof. Look at the set U=(0,∞)\\Sand choose δ,ηsuch that ε<δ<η. We can find n0∈Nsuch that\nµ(an+1,an)(U∩(an+1−an))≤δfor every n≥n0. Let R>an0and find n∈Nsuch that anεwas arbitrary the claim follows.\nTheorem 2.18. There exists K >0such that the set {x∈(3\n4,∞)|minn≤log2(x)(1\n1−log2(3)\n2+ε)Cn\nR(x)≤K}\nhas real density 1for every ε>0.\nProof. Consider - as in the proof of Theorem 2.13 - the sets AN,c,δ={x∈(0,c·2N\n3⌊(1\n2+δ)·N⌋]|∑MN−1\nk=0p(x)k>\n(1\n2+δ)·MN}, where MN=⌊log2(c·2N\n3⌊(1\n2+δ)·N⌋)⌋. We also set N0=⌊log33c\n1\n2+δ⌋.\nThen\nµ(0,c)(3⌊(1\n2+δ)·N⌋\n2NAN,c,δ)<2e−2δ2MN≤2e2δ2e−2δ2log2(c·(2N\n31+N(1\n2+δ)))\n=2e2δ2e−2δ2(log2(c\n3)+Nlog2(2\n31\n2+δ))\n.10\nNow for every γ>0 we can choose clarge enough so that ∑∞\nN=02e2δ2e−2δ2(log2(c\n3)+Nlog2(2\n31\n2+δ))\n<γ.\nBy Theorem 1.2 we can find K>0 and M∈Nsuch that the set D={x∈(0,c]|min n≤MCn\nR(31+⌊(1\n2−δ)·N0)⌋·\nx)>K}hasµ(0,c)-measure less than γ. Thus the set F=D∪/uniontext\nN∈N3⌊(1\n2+ε)·N⌋\n2NANis of µ(0,c)-measure less\nthan 2 γ. Thus also the sets FN=2N\n3⌊(1\n2+δ)·N⌋Fhave µ(0,2N\n3⌊(1\n2+δ)·N⌋c)-measure less than 2 γ. Abbreviate\nbN=2N\n3⌊(1\n2+δ)·N⌋c.Define HN=(bN−8,bN)∩FN.HNhasλ-measure less than 2 γ2N\n3⌊(1\n2+δ)·N⌋c, thus\nµ(bN−8,bN)(HN)<2γ2N\n3⌊(1\n2+δ)·N⌋c\nbN−bN−8=29γ\n28−3⌊(1\n2+δ)·N⌋−⌊(1\n2+δ)·(N−8)⌋≤29γ\n28−35,\nif we choose δ>0 small enough such that ⌊(1\n2+δ)·N⌋−⌊(1\n2+δ)·(N−8)⌋ ≤5, which will be the\ncase if(1\n2+δ)·8≤5.\nNow, if y∈IN=(bN−8,bN)\\HN, then by Sublemma 2.14 we have CN\nR(y)/pr⌉⌋⌉⌈⌉s⌉qual331+⌊(1\n2−δ)·N0⌋3⌊(1\n2+δ)·N⌋\n2N y.\nSince3⌊(1\n2+δ)·N⌋\n2Ny∈(0,c]\\Dwe conclude min n≤M+NCn\nR(y)≤K. Now, log2(y)≥log2(2N−8\n3⌊(1\n2+δ)·(N−8)⌋c)≥\nlog2(c\n3)−8(1−(1\n2+δ)log2(3))+N(1−(1\n2+δ)log2(3)). Thus\nN+M≤log2(y)(1−log2(c\n3)−8(1−(1\n2+δ)log2(3))\nlog2(y)\n1−(1\n2+δ)log2(3)+M\nlog2(y)).\nFor any given ε>0 we can choose δsmall enough such that1\n1−(1\n2+δ)log2(3)<1\n1−1\n2log2(3)+ε. Then for y\nsufficiently large we also have N+M≤log2(y)(1−log2(c\n3)−8(1−(1\n2+δ)log2(3))\nlog2(y)\n1−(1\n2+δ)log2(3)+M\nlog2(y))0 was arbitrary, the claim follows.\nNote that in case K=9\n4is a bound in Theorem 2.18, then since once an orbit enters (3\n4,9\n4]it never\nleaves it again, Theorem 2.18 takes the following form:\nTheorem 2.19. Suppose that λ({x∈(3\n4,∞)|infn∈NCn\nR(x)>9\n4}) =0. Then the set {x∈(3\n4,∞)|\nC⌊log2(x)(1\n1−log2(3)\n2+ε)⌋\nR≤9\n4}has real density 1for every ε>0.\n3 An approximation of the orbits of CR\nWe begin by gathering some easy to verify properties concern ing the notion of real ∗-density (see Defi-\nnition 1.5).11\nLemma 3.1. Suppose that S i⊆Nhave real (Ai,Bi)-density for some A i>0and00withan\nan+1≥q for all n ∈N. If there exists C >0and01such that r ≤an+1\nanfor all but finitely many n ∈N, then the converse holds as well.\nProof. Suppose that there exist C>0 and 00. Look at the set U= (0,∞)\\S. We know that λ(U∩(an+1,an))≤(an+1−an)C\naD\nn+1. Let\nn∈Nsuch that an≤R1 such thatan+1\nan≥rfor all but finitely many\nn∈N. Let n∈Nbe sufficiently large. There exist C>0 and 00, then the set\n{x∈R|∃0≤l0,l1≤2ζlog2x:{3−l0C⌊log2x⌋\nR(x),3l1C⌊log2x⌋\nR(x)}⊆S}\nis real∗-dense.\nProof. We start with l0. Set an=24n. Look at the set Pn={x∈(an,an+1)|≤(1\n2−ζ)4n≤∑4n−1\ni=0p(x)i≤\n(1\n2+ζ)4n}. By Lemma 2.10 we can find C>0 and 01, since we can assume that 4 (1\n2−ζ)−1>0\nby taking a smaller ζ>0, which does not change the result. By Lemma 3.1 SF=/intersectiontext\ni∈FfiSis real∗-\ndense for any finite set F⊆(0,∞). We will later specify F. By assumption and Lemma 3.2 we can\nthen find C0>0 and 00 and 00 and 01, since 4 (1\n2+ζ)−1>0. By Lemma 3.1 SF=/intersectiontext\ni∈FfiSis real\n∗-dense for any finite set F⊆(0,∞). We will later specify F. By assumption and Lemma 3.2 we can\nthen find C0>0 and 00 and 00.\nTheorem 3.4. Suppose that ε>0. Then there exits θ>0so that the set {x∈(0,∞)|∀k≤(1\n1−log23\n2+\nθ)log2x:(31\n2\n2)kx1−ε≤Ck\nR(x)≤(31\n2\n2)kx1+ε}is real∗-dense.\nProof. First we show that it is enough to show that for every ε>0 the set Sλ\nε={x∈(0,∞)| ∀k≤\n(1−λ\n1−log23\n2)log2x:(31\n2\n2)kx1−ε≤Ck\nR(x)≤(31\n2\n2)kx1+ε}is real∗-dense for all 1 ≥λ>0. To see this suppose\nthatSλ\nδis real∗-dense. Assume that x∈Sλ\nδand\n(1−λ\n1−log23\n2)log2x0 small enough we can\nensure that1\n31\n2(k−k0)x1−δ≤x1−εand 31\n2(k−k0)x1+δ≤x1+ε. To see that this is possible just note that\nk−k0−1\nlog2x≤1\n1−log23\n2+θ−1−λ\n1−log23\n2goes to 0 as θ,λgo to 0. Thus, if δ<ε, any x∈Sλ\nδfulfills∀k≤\n(1\n1−log23\n2+θ)log2x:(31\n2\n2)kx1−ε≤Ck\nR(x)≤(31\n2\n2)kx1+ε. Thus we are done showing that it suffices to\nshow that the Sλ\nεare real∗-dense.\nSetan=2nand define for any δ>0 the set An={x∈(an,an+1)| ∀k≤n:(31\n2\n2)kx1−δ≤Ck\nR(x)≤\n(31\n2\n2)kx1+δ}. We will show thatλ(An)\nan+1−an≥1−C\naD\nn+1for some C>0 and 00.\nFirst note that Ck\nR(x)≤(3\n2)kx≤(3\n2)kan+1. Now(3\n2)kan+1≤(31\n2\n2)ka1+δ\nn+1is true as long as (31\n2)k≤aδ\nn+1\nork≤2δlog3an+1.\nAlso Ck\nR(x)≥(1\n2)kx≥(1\n2)kan, and(1\n2)kan≥(31\n2\n2)ka1−δ\nnis true as long as (31\n2)k≤aδ\nnork≤2δlog3an≤\n2δlog3an+1.\nFor 2 δlog3an≤k0 look at the set\nBk={x∈(an,an+1)|k−1\n∑\ni=0p(x)i≥(1\n2+η)k}∪{ x∈(an,an+1)|k−1\n∑\ni=0p(x)i≤(1\n2−η)k}.\nThen by Lemma 2.10 we obtain µ(an,an+1)(/uniontext\n2δlog3an≤k≤log2anBk)≤C0\naD0\nn+1for some C0>0 and 00 and 00 (this is trivially the case for14\nλ=1). By Lemma 3.3 we know that for any ζ>0 the set S′\nε={x∈(0,∞)| ∃0≤l0,l1≤2ζlog2x:\n{3−l0C⌊log2x⌋\nR(x),3l1C⌊log2x⌋\nR(x)}⊆S}is real∗-dense. Thus for x∈Sδ∩S′\nεwe find 0 ≤l0,l1≤2ζlog2x\nsuch that\n(31\n2\n2)k(3−l0C⌊log2x⌋\nR(x))1−ε≤Ck\nR(3−l0C⌊log2x⌋\nR(x))≤(31\n2\n2)k(3−l0C⌊log2x⌋\nR(x))1+ε\nfor all k≤1−λ\n1−log23\n2log2(3−l0C⌊log2x⌋\nR(x))and\n(31\n2\n2)k(3l1C⌊log2x⌋\nR(x))1−ε≤Ck\nR(3l1C⌊log2x⌋\nR(x))≤(31\n2\n2)k(3l1C⌊log2x⌋\nR(x))1+ε\nfor all k≤1−λ\n1−log23\n2log2(3l1C⌊log2x⌋\nR(x)).\nFurthermore, (31\n2\n2)kx1−δ≤Ck\nR(x)≤(31\n2\n2)kx1+δfor all k≤log2x. In particular, (31\n2\n2)⌊log2x⌋x1−δ≤C⌊log2x⌋\nR(x)≤\n(31\n2\n2)⌊log2x⌋x1+δ. Thus if 0 ≤k≤1−λ\n1−log23\n2log2(3−l0C⌊log2x⌋\nR(x)):\n(31\n2\n2)k+⌊log2x⌋3−l0(1−ε)((31\n2\n2)⌊log2x⌋)−εx(1−δ)(1−ε)=(31\n2\n2)k(3−l0(31\n2\n2)⌊log2x⌋x1−δ)1−ε\n≤(31\n2\n2)k(3−l0C⌊log2x⌋\nR(x))1−ε≤Ck\nR(3−l0C⌊log2x⌋\nR(x))\n≤Ck\nR(C⌊log2x⌋\nR(x)).\nand similarly on the other side:\n(31\n2\n2)k+⌊log2x⌋3l1(1+ε)((31\n2\n2)⌊log2x⌋)εx(1+δ)(1+ε)=(31\n2\n2)k(3l1(31\n2\n2)⌊log2x⌋x1+δ)1+ε\n≥(31\n2\n2)k(3l1C⌊log2x⌋\nR(x))1+ε≥Ck\nR(3l1C⌊log2x⌋\nR(x))\n≥Ck\nR(C⌊log2x⌋\nR(x)).\nGiven any ε′>0 we can choose small enough δ,ζ,εwith ε<ε′andδ<ε′such that for sufficiently\nlarge x:\n3l1(1+ε)((31\n2\n2)⌊log2x⌋)εx(1+δ)(1+ε)≤x1+ε′\nand\n3−l0(1−ε)((31\n2\n2)⌊log2x⌋)−εx(1−δ)(1−ε)≥x1−ε′.15\nNow, for some K0>0 independent from λ,δ,ζ,andε\n⌊log2x⌋+1−λ\n1−log23\n2log2(3−l0C⌊log2x⌋\nR(x))≥log2x−1+1−λ\n1−log23\n2log2(3−l0(31\n2\n2)⌊log2x⌋x1−δ)\n≥log2x+1−λ\n1−log23\n2log2((31\n2\n2)⌊log2x⌋x)+1−λ\n1−log23\n2log2(3−l0x−δ)−1\n≥log2x+1−λ\n1−log23\n2(1\n2log2(3)log2(x)−log2(x)+log2(x))+1−λ\n1−log23\n2log2(3−l0x−δ)−K0\n≥(1+1−λ\n1−log23\n21\n2log2(3))log2x+1−λ\n1−log23\n2log2(3−l0x−δ)−K0\n≥1−log23\n2λ\n1−log23\n2log2x+1−λ\n1−log23\n2log2(3−l0x−δ)−K0\nThus, if we choose somelog23\n20. Inductively it follows that Sqm\nε′is real∗-dense for every m∈N. Thus given\nany 0<λ<1 we can find m∈Nsuch that qm<λ, thus Sλ\nε⊇Sqm\nεis real∗-dense as well. Thus the proof\nis complete.\nAs a reformulation we note:\nTheorem 3.5. Suppose that ε>0. Then the set {x∈(0,∞)|∀λ∈[0,1]:xλ−ε≤C⌊1−λ\n1−log23\n2log2x⌋\nR(x)≤\nxλ+ε}is real∗-dense.\nProof. Suppose that δ>0 and set Aδ={x∈(0,∞)| ∀k≤(1\n1−log23\n2)log2x:(31\n2\n2)kx1−δ≤Ck\nR(x)≤\n(31\n2\n2)kx1+δ}. By Theorem 3.4 Aδis real∗-dense. Suppose that x∈Aδandλ∈[0,1]. Then 0 ≤\n⌊(1−λ\n1−log23\n2)log2x⌋≤(1\n1−log23\n2)log2x, thus\n(31\n2\n2)⌊(1−λ\n1−log23\n2)log2x⌋\nx1−δ≤C⌊(1−λ\n1−log23\n2)log2x⌋\nR(x)≤(31\n2\n2)⌊(1−λ\n1−log23\n2)log2x⌋\nx1+δ.\nWe have (31\n2\n2)⌊(1−λ\n1−log23\n2)log2x⌋\nx1+δ≤2\n31\n2(31\n2\n2)(1−λ\n1−log23\n2)log2x\nx1+δ=2\n31\n2xλ−1x1+δ=2\n31\n2xλ+δ, and on the\nother side (31\n2\n2)⌊(1−λ\n1−log23\n2)log2x⌋\nx1−δ≥(31\n2\n2)(1−λ\n1−log23\n2)log2x\nx1−δ=xλ−1x1−δ=xλ−δ. Ifxis large enough\nandδ<εthen we conclude that xλ−ε≤31\n2\n2xλ−δ≤C⌊1−λ\n1−log23\n2⌋\nR(x)≤xλ+δ≤xλ+ε, thus the claim follows\nsince Aδis real∗-dense.\nAs a corollary we get:16\nTheorem 3.6. There exists K >0such that {x∈(3\n4,∞)|(1\n1−log23\n2−ε)log2x≤τK(x)≤(1\n1−log23\n2+\nε)log2x}has real density 1for all ε>0.\nProof. We take Kas in Theorem 2.13. Suppose that δ>η>0. By Theorem 3.5 the set {x∈(0,∞)|\nxδ−η≤C⌊1−δ\n1−log23\n2log2x⌋\nR(x)}is real∗-dense. Since also {x∈(0,∞)|K\nη>0 were arbitrary, we get that {x∈(3\n4,∞)|(1\n1−log23\n2−ε)log2x≤τK(x)}is real∗-dense. By Theorem\n2.18 the set {x∈(3\n4,∞)|minn≤log2(x)(1\n1−log2(3)\n2+ε)Cn\nR(x)≤K}has real density 1, thus also {x∈(3\n4,∞)|\nτK(x)≤(1\n1−log23\n2+ε)log2x}has real density one. As real ∗-dense sets have real density 1 and the\nintersection of two sets that have real density 1 has again re al density 1 the proof is complete.\n4 A graph theoretic reformulation\nIn this section we associate an acyclic directed graph to eac hx∈Rand discuss some basic properties.\nFirst assign to each real number x∈Ra directed graph Hxin the following way: Let ax∈{1,θ,0}Z\nbe the coefficients of the balanced ternary representation o fx, i.e., x=∑k∈Zax\nk3k, (in particular ax\nk=0\nfor all k>N0for some N0∈N). For each n∈Zconsider the balanced ternary representation of (3\n2)nx=\n∑k∈Za(3\n2)nx\nk3k(in order to ensure that the notion is well-defined, we choose representations ending in a\nconstant 1-sequence when there are two representations). T he set of nodes of HxisZ×Z. Each node\n(n,v)is the origin of exactly one directed edge ((n,v),(n+1,w)), where\nw=/braceleftBigg\nv if∑k≥va(3\n2)nx3kis odd,\nv+1 if ∑k≥va(3\n2)nx3kis even.\nSo, intuitively, we put all the edges (3zy,CR(3zy))into one graph for every yin the orbit of xunder\nmultiplication with 2 and 3, i.e. for all y∈{2i3jx|i,j∈Z}.\nAs an example look at x=10θ·0θθ.... Then Hxrestricted to edges originating in row 0 looks as\nfollows:\n0 1 0 θ·0θ θ...\nւ ↓ ↓ ւ ւ ↓ ւ ...\n0 1 1 0 ·θ1...\nNow we are going to restrict Hxto those vertices that have non-zero label. In order for this to be useful\nwe note in the following lemma that if vertices with non-zero label always point to vertices who also\nhave non-zero label:\nLemma 4.1. Suppose that x ∈R. Set ax\nn,z:=a(3\n2)nx\nz. If((n,v),(n+1,w))∈Hxandax\nn,z/\\⌉}atio\\slash=0or v=w,\nthen ax\nn+1,w/\\⌉}atio\\slash=0.\nProof. Suppose that w=v. Thus ∑k≥v3kax\nn,kis odd by definition. Let k≥vleast such that ax\nn,k/\\⌉}atio\\slash=0 and\nlk+N+nsuch that for any r∈3m(1−2−l,1+2−l)the balanced ternary representation ofr\n2kbegins\nwith an0...a0·...an1, but any potential branch beginning at an index (0,j)with j>−nends at level k\nunless it is Bx(where level kis the set of indices {k}×Z)). Thus the open set3m\n2k(1−2−l,1+2−l)⊆\nV∩Bn. SoC=/intersectiontext\nn∈NUnis as desired.\nFor the notion of measure we formulate the following conject ure:\nConjecture 4.3. Gxhas exactly one branch for a Lebesgue-co-null set of x ∈(0,∞).\nRemark 4.4. Leta=(an,z)(n,z)∈Z×Z∈{1,0,θ}Z×Zfulfill the condition\nan+1,z+1=−an,z+an+1,z+an,z−1an+1,z(an+1,z+an,z−1)\nfor all n,z∈Z, where addition and multiplication are modulo 3. Define V a={(n,z)∈Z×Z|an,z/\\⌉}atio\\slash=0}\nand define the graph G awith vertex set V aby putting ((n,v),(n+1,w))∈Gafor\nw=/braceleftBigg\nv+1ifan+1,v+1=−an,v,\nv if an+1,v+1/\\⌉}atio\\slash=−an,v.\nIf, furthermore, for all n ∈Zthere exist N n>0such that an,z=0for z>Nn, then one can show that G a\nis G rfor r=∑k∈Za0,z3k.\nNote that due to the fact that T is minimal (see Proposition 2. 1) we have the following homogeneity\nproperty: If x ,y∈R\\{0}and F⊂Z×Zis finite then there exist (a,b)∈Z×Zsuch that ax\nn,z=ay\nn+a,z+b\nfor all(n,z)∈F.18\nRemark 4.5. One can similarly define directed graphs in the case of the Col latz map on the positive\nintegers. The Collatz conjecture is then equivalent to the C onjecture that all graphs have exactly one\ncomponent. In a future work the author will show that under th e hypothesis of Conjecture 1.4 for a set\nof natural density 1the corresponding graphs will have only one large component for a suitable notion\nof largeness.\nAcknowledgments\nThe author thanks Claudius Röhl for his experimental finding s regarding Conjecture 1.4 (see Remark\n2.15).\nReferences\n[1] Gil Alon, Angelot Behajaina, and Elad Paran. On the stopp ing time of the collatz map in F2[x],\n2024.\n[2] Kenneth Hicks, Gary L. Mullen, Joseph L. Yucas, and Ryan Z avislak. A polynomial analogue of the\n3n + 1 problem. The American Mathematical Monthly , 115(7):615–622, 2008.\n[3] Wassily Hoeffding. Probability inequalities for sums o f bounded random variables. Journal of the\nAmerican Statistical Association , 58(301):13–30, 1963.\n[4] Manuel Inselmann. On the average stopping time of the col latz map in F2[x], 2024.\n[5] Jeffrey C. Lagarias. The 3x+1 problem: An overview, 2021 .\n[6] Terence Tao. Almost all orbits of the collatz map attain a lmost bounded values. Forum of Mathe-\nmatics, Pi , 10:e12, 2022.\nEmail address: Manuel.Inselmann@gmx.de" }, { "title": "2401.17253v1.Spectrum_of_global_string_networks_and_the_axion_dark_matter_mass.pdf", "content": "KANAZAWA-24-02, MPP-2024-18\nSpectrum of global string networks and the axion dark matter mass\nKen’ichi Saikawa,1Javier Redondo,2, 3Alejandro Vaquero,3and Mathieu Kaltschmidt3\n1Institute for Theoretical Physics, Kanazawa University,\nKakuma-machi, Kanazawa, Ishikawa 920-1192, Japan\n2Max-Planck-Institut f¨ ur Physik (Werner-Heisenberg-Institut), Boltzmannstr. 8, 85748 Garching, Germany\n3CAPA & Departamento de F´ ısica Te´ orica, Universidad de Zaragoza, C. Pedro Cerbuna 12, 50009 Zaragoza, Spain\n(Dated: January 31, 2024)\nCold dark matter axions produced in the post-inflationary Peccei-Quinn symmetry breaking sce-\nnario serve as clear targets for their experimental detection, since it is in principle possible to give a\nsharp prediction for their mass once we understand precisely how they are produced from the decay\nof global cosmic strings in the early Universe. In this paper, we perform a dedicated analysis of the\nspectrum of axions radiated from strings based on large scale numerical simulations of the cosmolog-\nical evolution of the Peccei-Quinn field on a static lattice. Making full use of the massively parallel\ncode and computing resources, we executed the simulations with up to 112643lattice sites, which\nallows us to improve our understanding of the dependence on the parameter controlling the string\ntension and thus give a more accurate extrapolation of the numerical results. We found that there\nare several systematic effects that have been overlooked in previous works, such as the dependence\non the initial conditions, contaminations due to oscillations in the spectrum, and discretisation ef-\nfects, some of which could explain the discrepancy in the literature. We confirmed the trend that\nthe spectral index of the axion emission spectrum increases with the string tension, but did not find\na clear evidence of whether it continues to increase or saturates to a constant at larger values of\nthe string tension due to the severe discretisation effects. Taking this uncertainty into account and\nperforming the extrapolation with a simple power law assumption on the spectrum, we find that\nthe dark matter mass is predicted in the range of ma≈95–450 µeV.\nContents\nI. Introduction 1\nII. String network and spectrum 4\nIII. Attractor, saxion emission, and network\ndensity 6\nIV. Oscillations in the spectrum 9\nV. Discretisation effects 14\nA. Laplacian 14\nB. Lattice spacing 14\nC. Continuum extrapolation 18\n1. Energy density emission rate 18\n2. Spectral index 20\nVI. Results: Extrapolation 22\nA. Energy density emission rate 22\nB. IR peak 22\nC. Axion relic abundance 24\nVII. Conclusions 28\nAcknowledgments 29\nA. Code and simulation setups 30\nB. Initial conditions 32\nC. Masking method 35\n1. Mask field 35\n2. Energy density and spectrum 363. Comment on the correction matrix method 37\nD. Calculation of F(x, y),Γ, and q 39\n1. Instantaneous emission spectrum 39\n2. Energy density emission rate 41\n3. Spectral index 42\nE. Other systematics 42\n1. Finite volume 42\n2. IR and UV cutoffs 43\nF. Evolution of the free axion field around\nthe horizon crossing 45\nG. Comparisons 47\nReferences 48\nI. INTRODUCTION\nA network of global cosmic strings forms in the early\nUniverse by the Kibble mechanism [1] when a global con-\ntinuous symmetry is spontaneously broken [2]. Explicit\nbreaking of the symmetry may eventually lead to the for-\nmation of domain walls, which triggers the destruction of\nthe network [2], a domain wall problem [3] or interesting\nlate Universe phenomenology, e.g. [4, 5]. In any case,\nthe networks can have very spectacular consequences in\ncosmology. Global string networks can trigger structure\nformation [2], explain the baryon asymmetry through lep-\ntogenesis [6, 7], lead to the formation of primordial blackarXiv:2401.17253v1 [hep-ph] 30 Jan 20242\nholes [8–12], and, perhaps most importantly, produce the\ncold dark matter (CDM) [13, 14].\nThe axion appears as a pseudo Nambu-Goldstone bo-\nson of the breaking of the Peccei-Quinn (PQ) symmetry\ninvoked to solve the strong CP problem [15–18]. The\nmain parameter of the axion theory is the decay constant,\nwhich suppresses its couplings to matter and its mass as\n∼1/fa. Laboratory experiments and astrophysics have\nexcluded values fa<∼108GeV (corresponding to ma<\n0.1 eV) and a band around ma∈10−12∼10−11eV.\nSee the recent Particle Data Group (PDG) review [19]\nand recent ones about axion models [20] and experimen-\ntal searches [21]. In the unexplored parameter space, the\naxion is an excellent candidate for the CDM of the Uni-\nverse.\nThe most remarkable aspect of axions as CDM is that\ndespite their minute couplings to the Standard Model\nparticles, there are feasible experimental techniques to\ndetect them in almost all remaining parameter space.\nSee for instance,1oscillating EDMs with CASPER [25],\nDM radio [26], cavity haloscopes such as ADMX [27]\nand CAPP [28], dielectric haloscopes as MADMAX [29],\nplasma haloscopes as ALPHA [30], and dish antennas as\nBRASS [31] or BREAD [32]. But most of these tech-\nniques heavily rely on coupling a resonator to the very\ncoherent local dark matter axion field, which would today\noscillate in a frequency band δω/ω∼10−6with ω∼ma.\nSince we do not know the axion mass, we need to tune\nour devices to different resonance frequencies until the\naxion signal is picked. In practice, this scanning process\nslows down current axion experimental search prospects\nto an octave a year. In this sense, it would be extremely\nadvantageous if we could guide the experimental search\nwith theoretical arguments to accelerate a discovery or\nsimply obtain the most information in the shortest time.\nAnother strong point of axion CDM is that it has very\ninteresting peculiarities that distinguish it from other\nCDM candidates by purely cosmological or astrophysi-\ncal probes like: the production of axion miniclusters [33–\n35] (and stars [36–38]), which could act as gravitational\nlenses [39–44], or large isocurvature fluctuations in the\ncosmic microwave background (CMB) [45–47].\nIf the PQ symmetry is broken after inflation (often\nreferred to as the post-inflationary scenario ), it is pos-\nsible to predict the axion mass from the calculation of\nthe CDM yield from the Big Bang, ρa,c=ρa,c(ma), and\nfocus the experimental search. Moreover, a sizable frac-\ntion of the axion CDM is in axion miniclusters [48, 49]\n(but not all [50]!). Here, the axion appears as an ef-\nfective degree of freedom after a phase transition where\nthe PQ symmetry becomes spontaneously broken. The\nvacuum expectation value of the axion field is uncorre-\n1Purely laboratory experiments as: solar axion searches with\nIAXO [22], long range forces with ARIADNE [23], or flavour\nviolating decays [24] can be very powerful just at the meV fron-\ntier.lated at distances longer than the causal horizon so that\nthe uncertainty in the initial conditions disappears after\nan average over relatively small scales ( L≫mpc). The\nmost important uncertainty, which has jeopardised at-\ntempts to pinpoint exactly the axion CDM mass, is the\nradiation of axions from the network of strings and walls.\nIn this paper, we will perform extensive numerical simu-\nlations of global string networks to understand as much\nas possible the radiation of axions from strings with its\nmain systematics and estimate the axion CDM mass with\nsensible uncertainties.\nMany studies have already endeavoured the calculation\nofρa,c(ma) and the axion CDM mass [13, 51–72], but still\nthe uncertainties span more than one order of magnitude,\nsee Fig. 1. The main source of uncertainty is the spectral\nindex qof the axions radiated by the network. In short,\none can have many axions of small energy for q >1, few\nof high energy for q < 1, or a scale invariant ( q= 1)\nfor the intermediate case. When by the redshift, they\nbecome non-relativistic, all the axion quanta eventually\nhave the same energy given by the axion mass, and hence\nρa,cis a growing function of q. Considerations based\non the Nambu-Goto effective theory coupled to a Kalb-\nRamond field supported q > 1 [13, 52, 55, 57, 58, 73]\nbut a set of different arguments point to a scale invariant\nradiation spectrum ( q= 1) [51, 53, 61].\nOther important uncertainties are the precise location\nof the infrared cutoff of the spectrum and the total den-\nsity of strings. The former can be used to estimate a\ntypical momentum of radiated axions, and the latter de-\nfines the total energy density of the radiation source.\nIt was argued that the density of strings for the net-\nwork of global strings can be smaller than that of local\nstrings [60, 75, 76], but the estimation of its precise value\nremains somewhat controversial [65–72, 77–81].\nIn principle, numerical simulations of the dynamics of\nglobal strings can determine precisely these parameters.\nHowever, they suffer from a huge dynamical range issue.\nThe smallest distances to resolve, the string cores, have\ntypical sizes f−1\na, but the typical distance between strings\nis associated with the causal horizon H−1. The ratio\nat the latest times is fa/H∼1030, much beyond any-\nthing that we will be able to simulate even in a futuristic\nsetup. We are thus doomed to simulate at unphysical\nvalues of the ratio and consider the extrapolation over\na huge range. Fortunately, the string energies depends\nonly logarithmically on the ratio,\nℓ≡ln\u0010mr\nH\u0011\n∝ln\u0012fa\nH\u0013\n, (1.1)\nwhere mris the mass of the heavier field degree of free-\ndom [see below Eq. (2.2)], and the extrapolation in ℓ\nappears much more feasible, if still very ambitious. Inter-\nestingly, it has been recently explained how using mixed\nlocal-global strings one can simulate at realistic values of3\n10−210−1100101102103104105Caγ\nNeutron star\ncoolingKSVZ\nDFSZBlack hole spinsfa>MplPulsarsHaloscopes\nCAST\nGlobular clustersSolar\nν\nTelescopesIAXOHaloscopes (future)\n10−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100101\nQCD axion mass, ma[eV]Hot dark\nmatter\n+\n∆NeffPost-inflationDavis, 1986\nHarari, 1987\nDavis, 1989\nHagmann, 1991\nNagasawa, 1994\nBattye, 1994\nBattye, 1996\nChang, 1998\nYamaguchi, 1999\nHagmann, 2001\nHiramatsu, 2011\nHiramatsu, 2012\nKawasaki, 2015\nFleury, 2016\nKlaer, 2017\nGorghetto, 2018\nBuschmann, 2020\nGorghetto, 2021\nBuschmann, 2022Black hole spins1061071081091010101110121013101410151016101710181019Peccei-Quinn scale, fa[GeV]\nFIG. 1: Estimates on the axion dark matter mass in the post-inflationary radiation dominated Universe obtained in the\nprevious works [13, 51–53, 56, 57, 59–67, 69–71], confronted with constraints and current/future experimental sensitivities on\nthe dimensionless axion-photon coupling Caγ(adapted from [74]).\ntheℓparameter ( ∼70) in an “effective”2manner [82].\nIn this paper, we aim at describing the spectrum of\nrelativistic Goldstone modes (axions) from pure global\nstring networks with unprecedented accuracy. To that\neffect, we perform numerical simulations of the simplest\nsystem presenting a spontaneously broken U(1) symme-\ntry, a complex scalar field with a “Mexican hat” poten-\ntial [83]. We perform simulations in up to 112643lattices\nin the supercomputers of the Max Planck Computing\nand Data Facility (MPCDF, Garching) and Cybermedia\nCenter (CMC, Osaka) to reach the most precise spectra\nup to date. Our results confirm the presence of an at-\ntractor [67] and a scaling solution but with logarithmic\n(ℓ) corrections to the parameters, previously found by\n2Here the effective tension parameter κis defined as the ratio of\nthe full string tension normalised to the axionic contribution to\nthe core κ=µ/(πf2\na) (which is ℓfor pure global strings), and\nthe trick is to make fasmall, instead of increasing µ.some studies [65–72, 77, 79] albeit challenged by some\nothers [78, 80]. Our study builds on the recent stud-\nies [67, 70] by refining the definition of the attractor, and\na more careful estimate of discretisation and systematic\neffects, made possible by conducting many simulations\nand utilising larger grids. Finally, we perform the extrap-\nolation to physical values of ℓby considering carefully the\ndiscretisation effects and compute the axion dark matter\nmass to be in the range\nma∈(95–450) µeV, (1.2)\ngiven the uncertainties. Unfortunately, we cannot assess\nwhether q= 1 or q > 1 with the current simulation\npower.\nRecently, the use of adaptive-mesh-refinement (AMR)\nin simulations [71, 84–86] has allowed to reach better dy-\nnamical range. The AMR simulation of the global string\nnetworks performed in Ref. [71] found q= 1 while previ-\nous studies [67, 70] showed q <1 with a growing trend\nwith ℓthat points to q >1 once extrapolated. Thanks4\nto our extensive simulation suite, we discuss why dis-\ncretisation and systematic effects such as the role of ini-\ntial conditions or the procedure to calculate qfrom data\ncan explain this difference. Unfortunately, more AMR\nsimulations will be needed to fully exploit the extended\ndynamical range.\nThe rest of this paper is organised as follows. In Sec. II,\nwe review theoretical basics for the calculation of the\nspectrum of axions produced by strings and introduce rel-\nevant quantities to characterise it. After that, in Sec. III\nwe start by presenting our numerical results on the evo-\nlution of the string density and refine the notion of the\nattractor to establish a baseline for the rest of the anal-\nysis and extrapolation. The subsequent two sections are\ndevoted to the discussion on systematic effects that can\nbias the results of the analysis: One is the existence of\nthe oscillations in the spectrum (Sec. IV), which is an\ninevitable consequence of the dynamics of the axion field\nin the system. The other is the effect of discretisation er-\nrors (Sec. V), which distorts the spectrum and can lead\nto a misinterpretation of the value of the spectral index\nthat is the most important quantity for the estimation of\nthe dark matter relic abundance. In Sec. VI, we present\nfurther results on the quantities characterising the ax-\nion production efficiency, and give estimates for the ax-\nion relic abundance by extrapolating the results obtained\nfrom the simulations. We summarise our results and con-\nclude in Sec. VII. Some technical details and further re-\nsults of the simulations are described in the appendices.\nWe describe details on our simulation method, including\nthe scheme to solve the equation of motion and choice of\nparameters in Appendix A. In Appendix B, the method\nto prepare the initial conditions for the simulation is dis-\ncussed. In Appendix C, we explain our method to mask\nthe contribution of the data around the string core and\ndiscuss systematics associated with it. We also describe\ntechnical details on the method to calculate the axion\nemission spectrum, energy density emission rate, and the\nspectral index in Appendix D. Appendix E is devoted to\ndiscussions on systematics not thoroughly mentioned in\nthe main text. Furthermore, in Appendix F we perform\nan analytical estimate on the evolution of the axion field,\nwhich helps to interpret some of the numerical results.\nFinally, we summarise the differences with recent simu-\nlations by other groups in Appendix G.\nII. STRING NETWORK AND SPECTRUM\nWe simulate a complex scalar field ϕwith the La-\ngrangian,\nL=1\n2|∂µϕ|2−V(ϕ), V (ϕ) =λ\n4\u0000\n|ϕ|2−f2\na\u00012,(2.1)\nwhere fadefines the scale of symmetry breaking. Ex-\npanding the field around the potential minima |ϕ|=fa,\nϕ(x) = (fa+r(x))eiθ(x), (2.2)we identify a(x) =θ(x)faas the massless Goldstone bo-\nson (the axion) and the radial field r(x) as a particle with\nmass mr=√\n2λfa(saxion).\nThe field described by the Lagrangian (2.1) has vortex\nconfigurations, referred to as global strings, around which\nits phase wraps around θ∈[0,2π). They feature a core of\nsizem−1\nrwhere the saxion field transitions smoothly from\nthe energy minimum r= 0 to the unbroken PQ symmetry\nvalue r=−faat the vortex center. The tension of the\nstring (i. e. energy per unit length) is3µ∼πf2\na(c1+c2ℓ).\nThe dominant logarithmic piece comes from the axion\ngradient energy density |∇a|2/2 outside the core. It is\nregularised by ultraviolet (UV) cutoffs and infrared (IR)\nscales given by m−1\nrand the string radius or distance to\nclosest strings, respectively [87].\nThroughout this paper, we consider only the evolution\nof the network during the radiation domination with con-\nstant degrees of freedom, most relevant for the axion dark\nmatter case. The evolution of the string network exhibits\na pseudo-scaling regime in which the length of strings ls\nin a causal volume V=t3is of the order of the horizon\nt∼H−1itself. The latter becomes the typical string\ninter-distance and the IR cut-off. Thus, for cosmologi-\ncal simulations we define ℓ= ln( mr/H), which increases\nwith time because of the time evolution of the Hubble\nparameter H(t).\nDefining the O(1) string density parameter as,\nξ=ls\nVt2, (2.3)\nthe energy density of strings can be defined as\nρstr=ξµth\nt2, (2.4)\nwhere µthis an network average string tension,\nµth=πf2\naln\u0012mrη\nH√ξ\u0013\n. (2.5)\nThe denominator 1 /(H√ξ) captures an expectation that\nthe average distance between long strings is given by ∼\nH−1/√ξ[61], and ηparametrises the non-trivial average\nshape.4The string network must then release energy at\na rate [67],5\nΓth=ξµth\nt2\"\n2H−˙ξ\nξ−πf2\na\nµth \nH+˙η\nη−1\n2˙ξ\nξ!#\n×fL,\n(2.6)\n≃8πfLξℓf2\naH3forℓ≫1, (2.7)\n3Thec′sareO(1) constants which depend on the geometry of the\nstring.\n4For straight parallel strings, η= 1/√\n4π[67].\n5Dots represent derivatives with respect to t.5\nto satisfy Eqs. (2.4) and (2.5). We introduce the extra\nfactor fL, to be determined by simulations, because of\nseveral reasons. First, only the “long” strings ( l≫H−1)\nare separated by the assumed 1 /(H√ξ) distances. The\nIR cutoff of small loops ( l≪H−1) is their radius, and\nthus their ℓis smaller. Since small loops only account for\n∼20% of the string length [67] and their ℓhas a lower IR\nscale, we neglect them in Γ thand expect ξ→ξL≈0.8ξ,\nwhere ξLis the contribution of long strings. Furthermore,\nEq. (2.5) is a static result and the energy per unit length\nincreases because of transverse motion, so we expect that\nstring velocities contribute an extra multiplicative factor\nγ= 1/√\n1−v2, where γandvare suitably averaged\nboost factors and transverse velocities, respectively.6\nThe evolution of the massless axion field outside the\nstring cores is given by ψττ− ∇2ψ= 0, where ψ=τθ,\nτis the conformal time, dτ=dt/R,Ris the scale factor\nof the Universe, and the subscript τdenotes the deriva-\ntive with respect to τ. Axion particle solutions are plane\nwaves, ψ(x) =Rd3k\n(2π)3eψ(k)eik·x(xandkare comoving\ncoordinates and wavenumbers), with conserved7occu-\npation number, N(k) = (|∂τeψ(k)|2+k2|eψ(k)|2)/(2kV),\nwhere k=|k|andVis the comoving volume. The energy\ndensity in massless axion waves is,\nρa= (faH)2Z\ndkk3\n2π2N(k), (2.8)\nand should increase with a rate Γ a=R−4d\ndt(R4ρa)ap-\nproximately equal to the rate at which the string net-\nwork releases energy according to Eq. (2.6). We denote\nN(k) as the angle average version of N(k). The rela-\ntion Γ a≈Γthis only approximate because it neglects\nsaxion radiation from the network and because we have\nneglected small loops, both of which are good approxi-\nmations.8\nIn order to calculate the axion CDM yield we need to\nstudy the distribution of axions radiated as a function of\nenergy. We thus define the spectral production rate,\n∂Γa(k)\n∂k= (faH)2k3\n2π2dN(k)\ndt, (2.9)\n6In other words, if the string length lsin Eq. (2.3) is measured in\nthe comoving coordinate (we actually do so in our simulations),\nξis suppressed by the Lorentz contraction, and we can convert it\ninto the proper length by multiplying γ. Hence we can interpret\nγlsµthas the mass energy in the rest frame of the string.\n7In the absence of interactions, i.e. far from the cores, Nis con-\nserved in time but only after the wave has entered the horizon,\ni.e.kτ≫2π.\n8We will see in Sec. V C 1 how saxion radiation becomes negligible\nat large ℓ. The fact that small loops are negligible is supported\nby the analysis of Refs. [67, 70, 88], but Ref. [88] mentions a\ncaveat. If small loops at large ℓwould become extremely long\nlived, the estimate of the axion abundance would be significantly\naltered [57, 89]. However, there is no sign of this in simulations\nup to date.or its dimensionless variant,9\nF(x, y)≡1\n(faH)2∂Γa\n∂(k/R)\n=1\n(faH)21\nR3∂\n∂t\u0012\nR4∂ρa\n∂k\u0013\n, (2.11)\nwhich characterises the spectrum through x=k/(RH)\nand the time evolution through y=mr/H. Hereafter we\ncallF(x, y) the instantaneous emission spectrum .\nIntegrating in time we can calculate the axion number\nas,\nna\nf2aH=1\nf2aHZ\ndk1\nω∂ρa\n∂k(with ω=k/R)\n=Zτdτ′\nτZdx′\nx′F(x′, y′), (2.12)\nand get an estimate of the variance of θ,\n⟨θ2⟩=R2\nf2aZdk\nk2∂ρa\n∂k=Zττ′dτ′\nτ2Zdx′\nx′2F(x′, y′).\n(2.13)\nNote that these quantities can be written as\nna\nf2aH=Zτdτ′\nτΓ′\na\nf2aH′3⟨x−1⟩(y′), (2.14)\n⟨θ2⟩=Zττ′dτ′\nτ2Γ′\na\nf2aH′3⟨x−2⟩(y′), (2.15)\nwhere we define the mean inverse momenta,\n⟨x−1⟩(y)≡Rdx\nxF(x, y)R\ndxF(x, y),⟨x−2⟩(y)≡Rdx\nx2F(x, y)R\ndxF(x, y).\n(2.16)\nThe instantaneous emission spectrum has been studied\nnumerically in Refs. [67, 70, 71] and seems to be well\napproximated by a power law dN/dt∼1/k3+q(orF ∝\n1/xq) with IR and UV cutoffs at comoving momenta k∼\nHR∝R−1andk∼mrR∝R, respectively.\nConsider the following simple power law model,\nF=(\nF0x−q(x0< x < y ),\n0 (otherwise) ,(2.17)\n9Note that F(x, y) defined in Eq. (2.11) is different from F(x, y)\nintroduced in Ref. [67]. The latter is accompanied by the nor-\nmalization conditionR\ndxF(x, y) = 1 and is related to F(x, y)\nas\nF(x, y) =f2\naH3\nΓaF(x, y). (2.10)\nIn the numerical study, we prefer to use F(x, y) rather than\nF(x, y), since Γ ais not reliably calculated at late times due to\nsevere discretisation effects (see Sec. V).6\nwhere F0andx0are independent of xbut may depend on\ny. We can evaluate Eq. (2.16) analytically and calculate\nnaand⟨θ2⟩through Eqs. (2.14) and (2.15). In Fig. 2 we\nshow the results assuming Γ a= Γ th(ℓ≫1) [Eq. (2.7)]\nwith constant (or log-varying) ξ,fL, and x0. In this\napproximation, the non-trivial parameters are qand the\nfinal value of y/x0while ξ,x0, and fLenter only through\na multiplicative factor. The most important parameter\nappears to be q. For q >1, the spectrum is dominated\nby the IR cut-off, so naand⟨θ2⟩become the largest and\nalso relatively insensitive to both qand the UV cut-off\nparameter y/x0. For q <1, the emission is dominated\nby the UV and thus, it strongly depends on qandy/x0,\ngrowing with the former and decreasing with the latter.\nIn the scale-invariant case, q= 1, the y-dependence of the\nshown curves na,⟨θ2⟩is a mild logarithmic suppression\n1/ℓ, which cancels the multiplicative enhancement factor\nℓ.\n0.8 1.0 1.2 1.4\nq10−310−210−1100\nna/bracketleftBig\nf2\naH8πξ/lscriptfL\nx0/bracketrightBig\n(solid),/angbracketleftθ2/angbracketright/bracketleftBig\n8πξ/lscriptfL\nx2\n0/bracketrightBig\n(dashed)\nlog(y/x 0)∼/lscript\n7\n12\n30\n50\n70\n1−q−1\n1\n2−(q+ 1)−1\nFIG. 2: Axion number and angle variance in the simplest ra-\ndiation model (2.17), normalized by the multiplicative factor\nin the square bracket.\nNote that the axion field becomes highly nonlinear\n(p\n⟨θ2⟩ ≫π) for q >1 and ℓ≫1 (as long as x0is not\ntoo large). This non-linearity reduces the axion number\naround the epoch of the QCD phase transition [70]. We\nwill take this into account in our estimates of the axion\nCDM relic abundance in Sec. VI C.\nThe shape of Fas well as the parameters ξ,q,x0,\nandfLcan be computed through numerical simulations,\nthough only for a limited range of values of ℓ<∼8–9. Most\nof them turn out to have a small dependence on ℓbut full\nconsensus on the ℓdependence of these quantities has not\nbeen reached in the literature. There remains a serious\ndiscrepancy in the prediction for the axion dark matter\nmass obtained from the extrapolation of the numerical\nresults to ℓ≫1. In this paper, we study these depen-\ndencies with precision so that this extrapolation can be\nadequately improved.III. ATTRACTOR, SAXION EMISSION, AND\nNETWORK DENSITY\nThe extrapolation of the parameters of the string net-\nwork, ξ,q,x0, and fLto the relevant values of ℓdepends\nto some extent on the way we initialise the network in\nour simulations. It is therefore crucial to note that the\nsystem exhibits an attractor behaviour [67] around which\ndifferences in initial conditions become less significant at\nlate times. Our extrapolation would be based on the\nproperties of the attractor solution, in the hope to be\ninsensitive to properties of the transient behaviours. In\nthis section, we characterise the properties of the attrac-\ntor by initialising networks with different densities and\nstudying the evolution of ξand the radiation spectra.\nDependence on other parameters for initial conditions is\ndiscussed in Appendix B.\nFigure 3 shows the evolution of the ξparameter for\ndifferent initial string densities. Although ξseems to\nconverge, the convergence seems rather slow, particularly\nat small values of ξ. In the spirit of Ref. [79], we can\nmodel the evolution of the string network density ξas,\ndξ\ndt=C\nt(ξc(ℓ(t))−ξ(t)), (3.1)\nwhere ξc(ℓ) is the equilibrium density of the network of\nconformal strings [79] at a given value of ℓandt/Cis\nthe characteristic time scale of the restoration of the net-\nwork. Assuming a constant Cyields a poor fit to the\nevolution. The main reasons are: 1) that ξ= 0 must be\na fixed point (string length is hardly created if there are\nno strings to begin with) and 2) that over-dense networks\nshould have a relaxation scale parametrically smaller be-\ncause the distance between strings is smaller. We model\nthese effects with C(x) =x/(1+√x/c0), where x=ξ/ξc,\nto have C(x→0)→ξin the small ξ≪ξclimit (which\ncorresponds to a frozen network, ξ∝1/t∝1/R2) and\nC(x≫1)∼c0p\nξ/ξc, i.e. a 1 /√ξ-suppressed restoration\ntime scale corresponding to a 1 /√ξ-suppressed interdis-\ntance. We interpolate the ξcdata from Ref. [79] and find\nthatc0in the range c0∼1.5+0.8\n−0.4gives reasonable fits to\nthe data.\nThe density parameter ξ(t) of physical strings shown\nin Fig. 3 is approaching the perfect scaling solution ξc(ℓ)\nbut it is chasing a moving target because ξc(ℓ) increases\nin time. Moreover, the values of Care relatively small,\nparticularly at small ξℓ=3. This explains that the lowest\ncurves show almost no convergence. Rather than reach-\ning the perfect-scaling solution, the evolutions approach\na “tracking” solution that lags behind. In the case where\nξc=ξc(ℓ) is a linear function of ℓandCis constant, the\ntracking solution is ξt(ℓ) =ξc(ℓ)−(dξc/dℓ)/C[79]. The\ndelay clearly increases with the speed at which ξcmoves\nand with the smallness of the “restoring force” C. Per-\nturbations around this solution decrease exponentially as\n∆ξ(ℓ) = ∆ ξ(ℓ0) exp(−C(ℓ−ℓ0)). We use the simple lin-\near tracking formula to iteratively find the tracking so-\nlution of the system with C=C(x), which we identify7\n3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0\nln(mr/H)10−210−1100\nξ\nphysical\nFIG. 3: Evolution of the string density parameter ξfor dif-\nferent initial string densities (simulations with 20483boxes).\nThe coloured bands represent statistical uncertainties. The\nblack dashed line corresponding to ξℓ=3≃0.3 is identified as\nthe attractor/tracking solution.\nwith the initial condition ξℓ=3= 0.3(1). Naturally, we\nidentify this to the true attractor solution of the system\nof physical strings and show it in Fig. 3 as a dashed black\nline. Deviations from ξℓ=3= 0.3 will decrease, although\nnot with a simple exponential form due to the fact that\nCdepends on the trajectory through the ratio ξ/ξcand\npossibly on ℓ. From the Cdependence we expect that\nover-dense networks approach the attractor faster than\nunder-dense ones, and this trend is clearly visible from\nFig. 3. We have to be thus very careful with under-dense\nnetworks.\nWe also studied the attractor using the energy spec-\ntrum of axions and saxions based on twice of their kinetic\nenergy densities, ρa= 2ρa,kin=⟨˙a2⟩andρr= 2ρr,kin=\n⟨˙r2⟩, see Appendices C and D. They both tend to evolve\ninto a uniform shape at late times, but we find the saxion\nspectrum more helpful to define the attractor. In Fig. 4\nwe plot a proxy for the saxion occupation number,\n1\nf2aH2k/R\nω(k)∂ρr\n∂logk, (3.2)\nat a late time ℓ= 7. Here kis the comoving wavenum-\nber (momentum) and ω(k) =p\nm2r+k2/R2. The most\nrelevant feature happens at momenta around the saxion\nmass, k/R∼mr, where most saxions are produced. We\nsee how spectra with different initial conditions approach\na common flat value ∼1.\nTo examine this feature in detail, we study the ampli-\ntude of the saxion spectrum at k=mrRas a function\nofξℓ=3in Fig. 5 (top panel). The slope of the plotted\nline becomes flatter with time, implying that the am-\nplitude at k=mrRbecomes less sensitive to the ini-\ntial conditions, i.e. a convergence to the attractor, and\ntakes a similar value at late times. The values with the\nsmallest derivative must be associated to the attractor.\n10−210−1100\nk/(Rmr)10−410−310−210−11001011\nH2f2ak/R\nω(k)∂ρr\n∂logkln(mr/H) = 7.0\n0.030.040.050.060.070.080.090.10.20.30.40.50.60.70.8ξ/lscript=3FIG. 4: Proxy for the saxion occupation number in networks\nwith different initial string densities at time ln( mr/H) = 7.\nColours as in Fig. 3. The black dashed line corresponds to\nthe attractor/tracking solution with ξℓ=3= 0.3.\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\nξ/lscript=310−210−11001011\nH2f2a∂ρr\n∂logkatk=mrR\nln(mr/H) = 4.0\nln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\nξ/lscript=30.02.55.07.510.012.515.017.520.0d\ndξ/lscript=3ln[1\nH2f2a∂ρr\n∂logk] atk=mrR\nln(mr/H) = 4.0\nln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\nFIG. 5: Amplitude of the saxion energy spectrum at k=mrR\n(top panel) and its derivative with respect to the initial string\ndensity ξℓ=3(bottom panel) plotted as functions of ξℓ=3. Er-\nrors plotted are explained in footnote 10.8\nIndeed, in Fig. 5 (bottom panel) we observe that the\nlogarithmic derivative has a minimum at ξℓ=3≃0.3, if\nwe neglect the data at large ℓ, where the data exhibits\nlarger statistical fluctuations.10The value of the attrac-\ntorξℓ=3= 0.3 agrees very well with the previous analysis\nof the evolution of ξ. Moreover, it does not rely on the\nartificial modelling of the evolution of ξ. Both results\ntherefore complement each other supporting the attrac-\ntive behaviour of the string network and the estimate\nξattractor (ℓ= 3) = 0 .3. Therefore, we use ξℓ=3≃0.3 as a\nfiducial initial condition for simulations with larger box\nsizes.\nFrom the data of the evolution of the axion spectra,\nwe also computed the instantaneous emission spectra de-\nfined by Eq. (2.11) and estimated their slope q(see Ap-\npendix D for technical details of the computation). Fig-\nure 6 shows the comparison of the evolution of qbetween\ndifferent initial string densities. We see that the value\nofqbecomes large (small) for under-dense (over-dense)\ninitial conditions. This trend can be understood as fol-\nlows. For the under-dense case, the interactions between\nstrings are less efficient, and there is little structure at\nsmall scales. As a consequence, axions are dominantly\nproduced at IR scales corresponding to the size of long\nstrings, which makes the spectrum more red-tilted. On\nthe other hand, for the over-dense case, there are a lot of\nsmall scale fluctuations that produce axions with higher\nmomenta, leading to a smaller values of q. As we see\nin Fig. 6, such a difference due to the initial string den-\nsity becomes less important at late times, and qalso ex-\nhibits the attractor behaviour. However, the convergence\nis slow again, and one should keep in mind that a small\ndifference in the initial string density could change the\nvalue of qmeasured at small ℓand affect the extrapola-\ntion to large ℓ. Moreover, we also see some oscillatory\nfeatures in the plots shown in Fig. 6. Those features can\nbe regarded as another systematics biasing the estimate\nofqand will be elaborated in Sec. IV.\nAfter identifying the initial conditions of the attractor,\nwe used it for simulations with larger number of lattice\nsites to investigate the ℓdependence of the quantities\nsuch as ξandq. In particular, we performed simulations\nwith 112643lattice sites that can reach ℓ≈9. Figure 7\nshows the evolution of the ξparameter computed at sim-\nulations with the largest box, together with the results\nfrom smaller simulations with different initial string den-\nsities. The result indicates that ξcontinues to grow with\nℓat the attractor, supporting the findings of Refs. [65–\n72, 77, 79]. They also support the interpretation as a\n10We computed the derivative with respect to ξℓ=3via finite dif-\nferences and estimated uncertainties in this procedure by using\nthe jackknife method. Namely, we took the finite difference by\nusing the average of 29 simulations (all the data except one) and\niterated it for 30 different ways of the resampling. After that we\nestimated the derivative and its error by using the average and\nvariance of 30 results of the resampling, respectively.\n5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0\nln(mr/H)0.20.40.60.81.01.2\nqphysical\nξ/lscript=3= 0.036\nξ/lscript=3= 0.073\nξ/lscript=3= 0.133\nξ/lscript=3= 0.214\nξ/lscript=3= 0.300\nξ/lscript=3= 0.380\nξ/lscript=3= 0.464\nξ/lscript=3= 0.586\nξ/lscript=3= 0.711FIG. 6: Evolution of the spectral index qof the instantaneous\nemission spectrum for various different values of the initial\nstring density ξℓ=3. The black dashed line corresponds to the\nattractor with ξℓ=3≃0.3. The coloured bands represent the\nerror induced by changing the parameter σfilterfor the filtering\nprocedure to calculate F[Eq. (D.5)] in addition to statistical\nuncertainties. The results are obtained from simulations of\nphysical strings with 20483lattice sites.\n“tracking” solution [79] chasing the ℓ-increasing confor-\nmal perfect-scaling solution ξc=ξc(ℓ) (also shown in\nFig. 7).\nWe have several options for the extrapolation. On the\none hand, we can take the tracking interpretation [79]\nseriously, extrapolate ξcandCand solve the differential\nequation until reaching physically meaningful values. On\nthe other hand, we can take an agnostic point of view,\nperform global fits of the ξ-data with sensible fitting func-\ntions and extrapolate within them as [67, 70, 81]. At the\nend both extrapolations are as good as the functional\nguesses. Since the second approach has already been ex-\nplored and uses less insight we will focus on the first.\nWe take two models for ξc=ξc(ℓ) that give satisfactory\nfits to the conformal data [79]:\nξlin\nc=−0.19(3) + 0 .205(7) ℓ, (3.3)\nξsat\nc=2.5(1.4) + 0 .23(6) ℓ\n1 + 0 .02(4) ℓ, (3.4)\na linear function capturing the main visible trend, and\none that eventually saturates,11which reflects a slightly\nvisible flattening of the growth at large ℓ, see Fig. 7.\nUsing the C=C(x) function previously described fitted\nto each model, we evolve the attractor solution and find\nξlin(ℓ= 70) ∼13.8(5), (3.5)\nξsat(ℓ= 70) ∼7(3). (3.6)\n11The errors of this fit are strongly correlated.9\n3 4 5 6 7 8 9\nln(mr/H)0.00.20.40.60.81.01.21.4\nξ\n“Conformal” networks\nPhysical networksξlin\ncξsat\nc\nFIG. 7: Evolution of the string density parameter ξfor physi-\ncal strings, obtained from simulations with 112643lattice sites\nstarted from the initial string density closest to the attractor\n(blue line) and that obtained from 20483simulations started\nfrom 30 different values of the initial string density (colours as\nin Fig. 3). The coloured bands represent statistical uncertain-\nties. The black dashed line represents our best estimate of the\nattractor or tracking solution. The data of conformal string\nsimulations from Ref. [79] are also shown as gray dots with er-\nror bars. The gray dash-dotted and dotted lines represent the\nfits to the linear function (3.3) and saturating function (3.4),\nrespectively.\nWe will take these values as brackets from our uncertainty\nin the extrapolation.\nThe interpretation of the logarithmic growth of the\nstring density parameter ξhas been challenged by\nRefs. [78, 80, 81], where it is claimed that the growth\ninξcan be interpreted as the slow approach to the “scal-\ning” solution with a constant ξ∼1.2(2). Our 112643\nbox data reaches this density without signs of levelling\noff and strongly disfavours an early saturation. However,\nas evident from Eq. (3.6), a “late” saturation at ξ>∼4\ncannot be excluded.\nIn Refs. [65, 67, 68, 70, 79], simulations were also\nperformed with the Press-Ryden-Spergel (PRS) trick,12\nwhere the parameter corresponding to the saxion mass is\nreplaced by mr→(R1/R)mr(with R1being the scale\nfactor at some reference time) by hand, such that the ra-\ntio between the string core radius m−1\nrand the physical\nlattice spacing a=RL/N remains constant.13Although\nsuch an artificial growth in the string core is unphysi-\ncal, this method has an advantage that the system con-\nverges into the attractor faster than the physical case.\n12This method is inspired by the work by Press, Ryden and\nSpergel [90] in which a similar trick was used for the simulation\nof domain wall networks.\n13Sometimes this is also called the fat string trick, since in this\nscheme the string core radius grows with time ( m−1\nr∝R).This is evident when we write the evolution of ξas a\nfunction of ℓrather than time, as Eq. (3.1) becomes\ndξ/dℓ = (C/f)(ξc(ℓ)−ξ) with a factor f=dℓ/dlogt,\nwhich is 1 in the physical case and 1/2 for PRS strings\n(mr/H∝t/R∝√\ntin this case).\nIn the left panel of Fig. 8, we show the results on the\nevolution of the string density parameter ξin our simula-\ntions with PRS strings for several different values of mra.\nThe initial conditions were taken to have similar ξto the\nattractor solution for PRS string simulations of Ref. [67].\nThe evolution of ξthat is increasing as expected from the\ngrowth of ξc(ℓ) and, indeed, the simulations agree very\nwell with the numerical solution of Eq. (3.1) with the\nsame linear ξcandC-function that fitted the evolution\nof physical strings after we rescale C→2Caccording to\nthe prediction. Indeed we expected and found that the\nPRS attractor solution should be approximately twice as\nclose to ξcthan the physical case. This gives greater\ncredibility to our results and interpretation.\nWe take the opportunity to study discretisation errors\ninξinduced by a poor resolution of the string core (see\nalso Sec. V). The resolution is parametrised by the value\nofmra, which measures how many lattice points fit in the\ncore. Figure 8 shows that these errors do not significantly\naffect the evolution of ξin PRS and physical strings close\nto the attractors (see Table IV in Appendix A for the\nsummary of our simulation parameters). The situation\nwill be very different when we study the spectra of radi-\nated axions and saxions in Sec. V.\nIV. OSCILLATIONS IN THE SPECTRUM\nThe instantaneous emission spectrum F(x, y) defined\nby Eq. (2.11) is calculated by taking the time derivative\nofR4∂ρa/∂kfor each comoving momentum k. In order\nto adequately evaluate the time derivative, we need to\nkeep track of the evolution of each mode in the simula-\ntion. However, it turns out that these modes typically\noscillate with time, which gives rise to contamination of\nF(x, y), and hence of q. In this section, we elaborate\non the nature of oscillations in the spectrum and their\nimpact on the estimation of the instantaneous spectrum\nF(x, y) and spectral index q.\nIn the top panel of Fig. 9, we show the time evolution\nof a few modes of the comoving box. The evolution of\ndifferent modes is apparently the same outside the hori-\nzon ( kτ<∼π),14but they get a different kick at horizon\ncrossing ( kτ>∼π) and have slightly different subsequent\nevolution. In particular, we find oscillatory features in\nthe mode evolution after the horizon crossing.\nOscillations are expected on physical grounds. The\n14It is possible to derive these features analytically by considering\nthe evolution of the Fourier components of the free axion field\nacross the horizon crossing. See Appendix F.10\n3 4 5 6 7 8\nln(mr/H)0.00.20.40.60.81.01.21.4\nξ\nPRSmra= 0.2\nmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n3 4 5 6 7 8 9\nln(mr/H)0.00.20.40.60.81.01.21.4\nξ\nphysicalN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 3072,¯λ= 114178.0 (mra= 2.80 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)\nFIG. 8: Evolution of the string density for different values of the string core width for PRS strings (constant mra) with 81923\nlattice sites (left) and physical strings ( mra∝R) with one set of 112643simulations and four sets of 30723simulations. The\ncoupling parameter ¯λis defined by Eq. (A.4). The coloured bands represent statistical uncertainties.\nequation of motion of the free axion field admits solu-\ntions ˜ψ(k)∝e±ikτfor the Fourier component k. We can\ncall these misalignment axions, inhomogeneities in the\naxion field originated from the finite correlation length\nat the PQ phase transition unrelated to strings . Since\nwe estimate ρafrom the kinetic energy alone ∝a2\nτ, the\nsolution of the free field equation naturally predicts that\nd3ρa\ndk3oscillates coherently with a frequency 2 k(in con-\nformal time). When we bin modes with the same fre-\nquency kand random phases, the oscillations should de-\ncrease in amplitude as (number of modes in the bin)−1\nand∂ρa\n∂logkconverges to the average power. Thus, if there\nis some coherence between modes or simply when we bin\nafinite number of modes we shall have oscillations. In\nthe expanding Universe, all the modes start in phase as\n˜ψ(k)∝sin(kτ) [91] (see Appendix F for details). How-\never, axions in our simulations are not free. The string\nnetwork slowly decays/radiates somewhat incoherently\ninto axion waves making the ˜ψ(k) amplitudes grow and\nphases drift. Moreover, string motion also contributes to\naτ. Thus, we expect that misalignment axions enter the\nhorizon oscillating more or less in phase, and the coher-\nence decreases with time as the network pumps the axion\nfield incoherently. Moreover, the amplitude of oscilla-\ntions should be larger for smaller values of k, for which\nwe sum less modes in each bin. These are the trends\nobtained from simulations and shown in Fig. 9.\nIt is clear that the oscillations can affect the extrac-\ntion of the time derivative of the mode’s average power,\nthus implying a systematic error. To deal with the os-\ncillatory component, we extract the overall trend in the\nevolution by fitting the following function to the data oftime evolution of1\nf2aH2∂ρa\n∂logkfor each k,\n\u00121\nf2aH2∂ρa\n∂logk\u0013\nfit=ea0xa1\n1 + (a2x)a3+a4lnx+a5(lnx)2,(4.1)\nwhere x=kτ,ai(i= 0,1,2,3,4,5) are constants to be\ndetermined by the fit, and terms of a4anda5are added to\nmodel the evolution at subhorizon.15The bottom panel\nof Fig. 9 shows the evolution of the residue obtained by\nsubtracting the fit from the data. We see that each mode\nactually oscillates with a frequency approximately given\nby∼2k, as one period of the oscillation roughly corre-\nsponds to the change in kτbyπ.16As expected, the oscil-\nlation is not a pure sinusoidal function, but we note that\namplitudes and phases do not change significantly. Un-\nfortunately, the fact that it is not straightforward to sep-\narate the misalignment component from string-induced\nones turns out to be a hurdle for the calculation of the\naxion production rate, as we will see below.\nIn addition to the oscillations in the low frequency\nmodes, we also observe oscillations when the physical\nwavenumber of a mode crosses the value k/R=mr/2. A\nparametric resonance exchanges energy between axions\n15Although we have included these terms to allow the possibil-\nity that the evolution at subhorizon is not given by an exact\npower law in x, the detailed form of these terms should not\nbe important, if we properly add the residue when we calcu-\nlateF(x, y). For instance, we confirmed that the results on qdo\nnot change much when we use other polynomials of ln xinstead\nofa4lnx+a5(lnx)2.\n16In Fig. 9, we also observe tiny oscillations with a frequency much\nfaster than ∼2k. They can be interpreted as the oscillation in\nthe radial direction of the PQ field, which can be associated with\nan imperfectness in the initial condition. This issue is elaborated\nin Appendix B.11\n100101102\nkτ10−310−210−11001011\nf2aH2∂ρa\n∂logk\nphysicalkL/(2π) = 2.41\nkL/(2π) = 4.44\nkL/(2π) = 7.45\nkL/(2π) = 10.5\nkL/(2π) = 20.5\n0 5 10 15 20 25 30 35 40\nkτ−1.0−0.50.00.51.01.5/parenleftBig\n1\nf2aH2∂ρa\n∂logk/parenrightBig\ndata−/parenleftBig\n1\nf2aH2∂ρa\n∂logk/parenrightBig\nfitkL/(2π) = 2.41\nkL/(2π) = 4.44\nkL/(2π) = 7.45\nkL/(2π) = 10.5\nkL/(2π) = 20.5\nFIG. 9: Top panel: Time evolution of the energy density\nof one Fourier mode of the axion field for various different\ncomoving wavenumbers. These modes cross the horizon kτ∼\nO(1) during the simulation and show a change in the rate\nof growth around that time. After that they oscillate with a\nfrequency ∼2k. Bottom panel: Time evolution of the residue\nobtained by subtracting the fit result given by Eq. (4.1) from\nthe data of one mode (shown in the top panel) for various\ndifferent comoving wavenumbers. These plots are obtained\nfrom one simulation (not the average) of physical strings with\n112643lattice sites.\nand saxions coherently [70], and thus creates 2 koscilla-\ntions, see Fig. 10.17Note that this effect is less problem-\natic in simulations with the PRS method, since in that\ncasemrRis constant, and the resonant production of ax-\nions is relevant only for a certain mode whose comoving\n17The frequency of oscillations shown in Fig. 10 does not look like\n2k. This is just because the frequency of the measurements of\naxion spectra in our simulations is not high enough to resolve fast\noscillations of these high- kmodes. In that case, the oscillations\nshow up as a spurious feature with a lower frequency, which can\nbe predicted from the genuine frequency (2 k) and the frequency\nof measurements (see Appendix D 1).\n102103\nkτ10−11001011\nf2aH2∂ρa\n∂logkk\nR=mr\n2physical\nkL/(2π) = 200\nkL/(2π) = 300\nkL/(2π) = 400\nkL/(2π) = 500FIG. 10: Time evolution of the energy density of one Fourier\nmode of the axion field for higher comoving wavenumbers,\nobtained from one simulation (not the average) of physical\nstrings with 112643lattice sites. Times at which the physical\nmomenta of the modes become equal to the half of the saxion\nmass are marked by dashed lines.\nmomentum is given by k=mrR/2.\nAs the 2 k-oscillations start at a determined time, such\nas the horizon crossing or saxion mass crossing, it is in\nprinciple not possible to eliminate them by performing\nsimulations many times and averaging the results. For\ninstance, see Figs. 26, 33, and 44, which explicitly show\nthat the oscillations remain even after averaging over\nmany realisations. Some tricks could help to average\nout the oscillations. For instance, one can shift the mis-\nalignment phase by using different initial string densities,\none could change the phase of the saxion field at the be-\nginning of the simulations, or one can simply bin more\nmodes together. For this paper, we applied a filter that\nsuppresses high-frequency components in the time series\ndata of the residue of the mode evolution. To minimise\nthe systematic error due to our choice of the fit function\nwhen we compute F, we computed the time derivative of\nthe filtered residue and added it to the time derivative of\nthe fit (see Appendix D 1 for technical details).\nIn Fig. 11, we compare the instantaneous spectrum F\nobtained by our procedure to a calculation that evaluates\nthe time derivative with finite differences. To ease the\ndiscussion, we model\n1\n(faH)2∂ρa\n∂logk=xf0+f1cos(2 kτ+α), (4.2)\nwhere f0,1∼ O(1) and αaresmooth functions of k, see\nFig. 9. The instantaneous spectrum calculated with finite\ndifferences with an interval ∆ τis thus\nFfinite-diff .=f0−2f1sin(2kτ+α)sinc(2 k∆τ)\n+O(f′\n0, f′\n1, α′)-terms , (4.3)\nwhere the prime represents the derivative with respect to\nτ. The finite difference version of Ffeatures the same12\n2k(here 2 x) oscillations than ∂ρa/∂k. The envelope\nsync(2 x∆τ/τ) can also be clearly seen in Fig. 11. It acts\nas a low-pass filter suppressing the oscillations above a\ncritical value\nx1z=π\n2τ\n∆τ, (4.4)\n(chosen arbitrarily to be the first zero of the sync func-\ntion), which depends on the time-interval. For the val-\nues of ∆ τshown in Fig. 11 in increasing order, we find\nx1z≃110,55,28. It is clearly visible that all the en-\nvelopes become zero at 110, as follows from the fact\nthat the intervals are multiples of 2. Below the smallest\nx1z∼25, all the finite difference curves agree very well,\nas all the different filter values are ∼1 here. The relative\namplitude 2 f1/f0can be estimated from the curve of the\nfinest discretisation and reaches more than a factor of\n∼2 (!).\nIncreasing the time interval in the finite difference esti-\nmate of the derivative is a very convenient way of calcu-\nlating F, as one decreases the oscillating features auto-\nmatically and one needs less evaluations of the spectrum\nduring the simulation (which are very time-consuming).\nThis seems to be well suited for our problem because\nthe main growing trend in ( faH)−2∂ρa/∂logkis alin-\neargrowth ∝xand calculating the derivative with finite\ndifferences does not bias the result, even with relatively\nlong time-intervals. However, the finite difference esti-\nmate has its limitations. First, it does not remove the\noscillations below x1z, and above x1z, only by a factor\n∼x1z/x. Second, the finite difference method picks up\nthe derivative of the non-oscillating part xf0exactly as\nlong as f0does not depend on x. If there is a small, for\ninstance logarithmic, extra dependence, finite differences\nwill produce a bias in the estimator which grows with ∆ τ.\nFor instance, when f0∝ln(x/xc) with xcsome constant,\nFfinite-diff .\nF−1∼−1\n6(∆τ/τ)2\n1 + ln( x/xc). (4.5)\nThus, we would need to find a compromise between the\nminimisation of this bias with a small ∆ τand the oscil-\nlation amplitude with a large ∆ τ. Our fitting procedure\nnicely overcomes these limitations.\nThe oscillations in the spectrum will affect the deter-\nmination of q. The estimation of qis based on fitting\na power law F ∝ x−qto the Fdata within the range\ncIRHR < k < c UVmrR(cIR< x < c UVeℓ) with some\nfixed values of coefficients cIRandcUV. Since the fit\nmodel does not include oscillations, it will produce a bias\nin the estimation of q. We expect this bias to oscillate\nin time, i.e. with ℓ, but not necessarily around the true\nvalue. As long as oscillations average over the data, the\nbias should be small. However, since we cut the fitting\nrange at high and low values of k, we can have only par-\ntially cancelled oscillations there. These uncanceled os-\ncillations can contribute significantly for instance if the\ndata range is small (which happens at low ℓ, early times,\nwhen the UV and IR cutoffs are closer), if the weights in\n100101102\nx=k/(RH)10−1100F(x)physicalln(mr/H) = 6.0\nFinite difference (∆ τ/L = 0.00192)\nFinite difference (∆ τ/L = 0.00384)\nFinite difference (∆ τ/L = 0.00768)\nAnalytical fit + filteringFIG. 11: Instantaneous emission spectrum at ln( mr/H) =\n6, obtained by using the finite difference method for three\ndifferent choices of the interval ∆ τ(coloured lines) and by\nthe sum of the analytical fit and time derivative of the filtered\nresidue (black dashed line). Each line represents the average\nof 20 simulations of physical strings with 112643lattice sites.\nthe fit favour the extremes of the fit range, etc. [71]. In\nsuch a case, one expects that the bias will depend on cIR\nandcUVas well.18\nThe effect of oscillations will be maximum when the\nphases are less random. In particular, since the k-modes\nare discrete in our finite-volume simulation, 2 kτ= 2x=\n4πnτ/L with nan integer. Thus, when τ=L/2, 2x\nbecomes a multiple of 2 πand independent of the mode,19\nleaving only the smoother dependence with α.\nWe note that when the time intervals used for calcu-\nlating the derivative are logarithmically distributed, for\ninstance the popular ∆ ℓ= 0.25 used in Refs. [67, 70, 71],\nthe critical value of the filter is constant in time, x1z≃\n(π/2)(4/∆ℓ), which is x1z≃25.1 for ∆ ℓ= 0.25. Then,\nby choosing cIR=x1z, we can ensure that the amplitude\nof the oscillations is minimum at the lower end of the fit\nand is always suppressed in the fit region. Coincidentally,\nthe work of Ref. [71] chose approximately cIR≃2x1zfor\ntheir results without a discussion of the oscillations pre-\nsented here. The main results of Refs. [67, 70] were ob-\ntained by using cIR= 30, which is slightly larger than x1z,\nand hence the effect of oscillations were not minimised.\n18In what follows we discuss how the bias due to the oscillations\nin the spectrum appears according to the choices of cIRand\ncUV. Even if we minimise the bias by removing oscillations from\nF(e.g. by using the filtering method), there remains another\nsystematics associated with cIRandcUVdue to the fact that\nF(x) does not exhibit a perfect power law behaviour around the\nmomentum ranges close to the IR and UV cutoffs. The errors\nassociated with the latter effect are discussed in Appendix E 2.\n19Same happens for multiples of τ=L/2 but in practice, we seldom\nexceed τ=L/2 in our simulations to avoid unphysical effects\nfrom the periodic boundary conditions and never reach τ=L.13\nWe can evaluate directly how the oscillation affect the\nextraction of qby considering the simple toy model given\nby Eq. (4.2). Here we build a synthetic spectrum with\nf0∼1/kqsand introduce oscillations with f1/f0= 2.5.\nBased on this synthetic spectrum, we build the time\nderivative with logarithmically spaced ∆ ℓ= 0.25 and fit\nthe result to a simple power law ∝1/kqwithout oscilla-\ntions. The mr/Hratio and mode discretisation are the\nsame as we have in our 112643simulations. In Fig. 12, we\nshow that the value of qextracted in this way contains\nitself oscillations of order σq∼0.1 with slowly shifting\nfrequency. We also see a sizable bias at small ℓwhen cIR\nis not a multiple of x1z(case cIR= 36 here). Note that\nthe errors are larger too, even though the fitting range for\ncIR= 50 is smaller. In the ℓ∼8–9 interval, the period is\napproximately 0 .1. As mentioned before, we expect the\nphases between the oscillations of the different Fourier\nmodes to be less random at τ=L/2. This leads to the\nincrease of amplitude at ℓ= 8.5–9 where this condition\nis met in our simulations. This can explain the offset of\ntheℓ= 8.5 point in the results of Ref. [71].\nIn reality one would calculate the spectrum at ∆ ℓ=\n0.25 intervals, such that only a few points of the lines\nshown in Fig. 12 would be sampled. Oscillations con-\ntribute then as a systematic error, which would reveal\nitself with less sparse data. If we try to fit a linear trend\nto, for instance a function q=q0+q1(ℓ−8), the effects of\noscillations in the extraction of q0andq1will be somehow\nsuppressed. When we fit our mock data, we find good fits\nbut small systematic biases, which decrease with the fit-\nting range. Assuming flat errors for the modes in the fit,\nthis bias is of order 0.02 for the typical values of current\nsimulations, while if errors in the spectra are taken to be\nproportional to it ( ∼1/kqs) the bias becomes negative\nat small cUV∼1/16. An example of the latter is Fig. 13,\nwhere we show the results for q0andq1as functions of cIR\nandcUVfor a synthetic spectrum with a spectral index\nqs= 0.8 constant in time. We observe a positive bias of\nthe order 0.01 in q0accompanying a negative slope bias.\nWe note that these are results from synthetic spectra\nusing Fourier modes equal to the model, without the fur-\nther dispersion expected in simulations, due to statistical\nfluctuations caused by a finite number of modes added\nper bin, for instance. Adding statistical fluctuations will\namplify the oscillation effects. Averaging over simula-\ntions and extrapolating to the infinite-volume limit will\ndecrease the effects discussed here.\nWe can look at our data to search for effects of the oscil-\nlations through the time discretisation of the derivative.\nIn Fig. 14, we compare q(ℓ) extracted from the 40963\ndata with finite differences and our filtering method for\nthree different values of cUVusing cIR= 50. The finite\ndifference results follow the filter results within errors ex-\ncept when the fitting range is small, where results diverge\nsignificantly. For smaller values of cUV, the relative im-\nportance of the IR part of the spectrum becomes more\npronounced in the fit, and the bias due to the oscillation\nin the IR modes contributes to the larger fluctuations in\n7.5 8.0 8.5 9.0\nln(mr/H)0.600.650.700.750.800.850.900.951.00qσ∼1/k0, f1/f0= 2.5, qs= 0.8,∆/lscript= 0.25\ncIR,cUV= 36,1/16\ncIR,cUV= 50,1/16FIG. 12: Value of qfitted from a synthetic 1 /kqsspectrum\nwith constant qs= 0.8 and oscillations as in Eq. (4.2) for two\nchoices of cIR. The fits are performed by assuming flat errors\n(σ∼1/k).\n−0.050.000.05q1σ∼1/k1, f1/f0= 2.5, qs= 0.8,∆/lscript= 0.25\ncUV= 1/4\ncUV= 1/8\ncUV= 1/16\n20 25 30 35 40 45 50 55 60\ncIR0.7500.7750.8000.8250.850q0\nFIG. 13: Fitted values of q=q0+q1(ℓ−8) produced by mode\noscillations in the example discussed in the text. Here the\nerrors in the spectra are assumed to be proportional to ∼1/k\nin the fits. The mock data has an ℓ-independent qs= 0.8 but\na positive bias q0> qsand a negative slope q1<0 are created\nby oscillations in the fit.\nq. In the figure we also see that the amplitude of the\noscillation of qis larger at early times, see also Fig. 12.\nThis is because the range for the comoving wavenumber\ncIRHR < k < c UVmrRis narrow at early times, and\nhence the value of qis more likely to be affected by a\nlocal feature in F(x). Furthermore, the range for the fit\nbecomes even narrower for smaller values of cUV, and for\nthat case the measurement of qbecomes possible only\nat late times when the number of k-bins satisfying the\ncondition cIRHR < k < c UVmrRbecomes sufficiently14\nlarge.20\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.40.50.60.70.80.91.01.11.2\nq\nphysical\ncUV=1/4 (fit + filtering)\ncUV=1/8 (fit + filtering)\ncUV=1/16 (fit + filtering)\ncUV= 1/4 (finite difference ,∆/lscript= 0.25)\ncUV= 1/8 (finite difference ,∆/lscript= 0.25)\ncUV= 1/16 (finite difference ,∆/lscript= 0.25)\nFIG. 14: Evolution of the spectral index qfor three different\nchoices of cUV, obtained from simulations of physical strings\nwith 40962lattice sites and the parameter ¯λ= 25890 .8. Mark-\ners with error bars represent the results of the finite differ-\nence method, and dashed lines represent those obtained by\nthe filtering method. The coloured bands represent the er-\nror induced by changing the parameter σfilterfor the filtering\nprocedure to calculate F[Eq. (D.5)] in addition to statistical\nuncertainties.\nIn Ref. [71], it was reported that qdoes not exhibit\na logarithmic growth for cUV= 1/16, which is taken as\na fiducial value in that work, while it shows a growing\ntrend for cUV= 1/4, which was adopted in the analysis\nof Ref. [70]. Contrary to those results, our simulation\nresults shown in Fig. 14 exhibit a trend that the value\nofqincreases with ℓirrespective of the value of cUVfor\nboth the filter and the finite difference case. The bias of\nshifting the values of qand its slope for a smaller value\nofcUVcaused by the oscillations could contribute to the\ndifference, but the systematics due to these effects is not\nso dramatic, typically σq∼ O(0.01) as shown in Fig. 13.\nA caveat is that only one simulation was performed in\nRef. [71], and we cannot exclude the possibility that the\neffects were amplified by statistical fluctuations.\nV. DISCRETISATION EFFECTS\nIn addition to the contamination from oscillations in\nthe spectrum, there is a bias due to the discretisation of\nthe lattice, which turns out to be the most serious issue\nfor the measurement of q. This section is devoted to the\n20When we perform the fit, we re-bin the data of F(x) such that\nthe number of bins in the interval remains the same at all times\n(see Appendix D 3).problem of discretisation effects observed in our simula-\ntions. The effects can be categorized into two types: the\ndiscretisation of Laplacian and the choice of the lattice\nspacing compared to the string core radius. The latter\neffect is particularly hard to control. In the following, we\nfirst describe the effects associated to the Laplacian and\nto the lattice spacing in Sec. V A and Sec. V B, respec-\ntively. After discussing how these effects show up in the\nnumerical results, we perform analytical fits to get rid of\nthem in Sec. V C. We will see that the corrected values of\nqat large ℓbecome smaller than what is measured from\nthe numerical results.\nA. Laplacian\nIn our code, discretisation of the Laplacian in the equa-\ntion of motion for the PQ field is implemented with Ng\nneighbouring points, where Ngcan take either 1, 2, 3, or\n4 (see Appendix A). For a larger value of Ng, the field\nassociated to a lattice point can react to incoming waves\nin advance, and hence there is an improvement in the\npropagation speed of perturbations. Such an improve-\nment is expected to provide a more accurate result for\nthe spectrum. To investigate the systematics associated\nwith the discretisation scheme of the Laplacian, we per-\nformed simulations of physical strings with 40963lattice\nsites for four different values of Ng. In Fig. 15, we show\nthe axion spectrum and F(x) for each choice of Ng. We\nsee that the spectrum at intermediate momenta is under-\nestimated for smaller values of Ng. Furthermore, there\nis a peak-like feature at a very high momentum, and the\nheight of the peak is pronounced for smaller Ng. These\nfeatures can be regarded as unphysical effects caused by\nthe discretisation error in the Laplacian, which can be\nalleviated by choosing a larger value of Ng.\nThe discretisation effects distort the spectrum such\nthat the slope at the intermediate momenta becomes\nsteeper for a smaller Ng, as shown in Fig. 15. This im-\nplies that the value of qcan be overestimated for smaller\nNg. Such a bias is clearly shown in Fig. 16, where we\nplot the evolution of qfor different choices of Ng. From\nthis figure we can also see that Ng≥3 is sufficient to\nsuppress the effect.\nThe effect of the Laplacian could partially explain the\ndifference in qin the literature. We observe that the\nvalues of qobtained for our fiducial choice Ng= 4 is\nslightly smaller than Ref. [70], which used the Laplacian\nwithNg= 2 (see Appendix G for an explicit comparison).\nFurthermore, Ng= 1 was adopted in Ref. [71], where\nsystematically larger values of qwere reported. These\nfacts are consistent with the trend shown in Fig. 16.\nB. Lattice spacing\nThe effect of the lattice spacing compared to the string\ncore radius can be parameterised by the quantity mra,15\n101102103\nk/(RH)1001011\nH2f2a∂ρa\n∂logk\nmr\n2Hphysical\n1 neighbour\n2 neighbours\n3 neighbours\n4 neighbours\n101102103\nk/(RH)0.60.70.80.91.01.11.21.31.4Ratio to 4 neighbours\nmr\n2Hphysical1 neighbour\n2 neighbours\n3 neighbours\n101102103\nk/(RH)10−310−210−1100F(x)\nmr\n2Hphysical1 neighbour\n2 neighbours\n3 neighbours\n4 neighbours\n101102103\nk/(RH)0.60.70.80.91.01.11.21.31.4Ratio to 4 neighbours\nphysical1 neighbour\n2 neighbours\n3 neighbours\nmr\n2Hln(mr/H) = 7.0\nFIG. 15: The axion energy density spectrum (top left) and instantaneous emission spectrum (bottom left) at ln( mr/H) = 7\nobtained from simulations of physical strings with 40963lattice sites for 4 different choices on the neighbouring points Ngused\nin the discretisation of the Laplacian. The ratio of the energy density spectrum (instantaneous emission spectrum) for each\nchoice of Ngto that for Ng= 4 is also shown in the top right (bottom right) panel. The coloured bands represent statistical\nuncertainties, and the momentum corresponding to k/R=mr/2 is marked with dark blue ticks.\nwhere a=RL/N is the physical size of the lattice spac-\ning. The simulation with PRS strings is useful to monitor\nthis effect, since in that case mr∝R−1andmraremains\nconstant. In Fig. 17, we compare the axion spectrum and\nFobtained from PRS-type simulations among different\nvalues of mra. We can see features similar to Fig. 15: The\namplitude is underestimated at intermediate momenta,\nand there is a peak at very high momenta. The effect is\npronounced for larger values of mra, where the resolution\nof the string core becomes worse.\nThe discretisation effect is also manifested in the en-\nergy density emission rates for axions and saxions,\nΓa=R−4d\ndt(R4ρa),Γr=R−⟨z⟩d\ndt(R⟨z⟩ρr),(5.1)\nwhere ⟨z⟩is a suitably averaged redshift exponent [see\nEq. (D.8)]. We computed Γ aand Γ rfor both physical\nand PRS simulations (see Appendix D 2 for more techni-\ncal details). Figure 18 shows the plots of Γ aand Γ rob-\ntained from PRS-type simulations. We see that the dis-\ncretisation effects make the axion emission less efficient,and lead to the production of large amounts of saxions.\nNote that the effect is more pronounced for larger values\nofℓ. Namely, the effect does not only depend on mra\nbut also on ℓ. The fact that the effect is less manifest at\nsmall ℓmay explain why the systematics associated with\nthis effect have been overlooked previously.\nFor simulations with physical strings, the effect be-\ncomes more intricate than the PRS case, since mrain-\ncreases with time ( ∝R) so that the resolution gets worse\nat late times. When considered in conjunction with the\npossible ℓ-dependence observed in the PRS case, we ex-\npect that the discretisation effects rapidly come into play\nat the latest times in the physical case.\nFigure 19 shows the evolution of the spectrum of axions\n(top panel) and saxions (bottom panel) in the simulations\nof physical strings with 112643lattice sites. Overall, the\naxion spectrum exhibits the expected behaviour: There\nare cutoffs at k/R∼Handk/R∼mr, which means\nthat the location of the IR cutoff remains constant while\nthe UV cutoff moves right when the spectrum is plot-\nted in terms of k/(RH). The growth in the amplitude16\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.60.70.80.91.01.11.21.31.4\nq\nphysical1 neighbour\n2 neighbours\n3 neighbours\n4 neighbours\nFIG. 16: Evolution of the spectral index qof the instanta-\nneous emission spectrum obtained from simulations of phys-\nical strings with 40963lattice sites for 4 different choices on\nthe neighbouring points Ngused in the discretisation of the\nLaplacian. The coloured bands represent the error induced\nby changing the parameter σfilterfor the filtering procedure\nto calculate F[Eq. (D.5)] in addition to statistical uncertain-\nties.\nof the axion spectrum can be attributed to the logarith-\nmic growth in the production rate, which originates from\nthe increase in the string tension µthand density ξ[see\nEq. (2.6)]. However, we also see a peak-like feature at\nvery high momenta, which starts to appear at ℓ≈7 and\ngrows rapidly. A similar feature is seen in the plot of\nthe saxion spectrum. These features can be regarded\nas a signal that the system is affected by the unphysi-\ncal discretisation effects at late times of the simulations.\nIn that regime, instead of radiating axions and saxions\nwith proper momenta ( k/R<∼mr), strings dissipate their\nenergy by producing a huge amount of high frequency\nmodes.\nFigure 20 shows the evolution of the instantaneous\nemission spectrum in the simulations of physical strings\nwith 112643lattice sites. Comparing the instantaneous\nemission spectrum at ℓ= 9 with those at earlier times, we\nsee that Fis considerably distorted at the latest times,\nas there is a suppression in the amplitude at interme-\ndiate momenta and a peak-like feature appears at very\nhigh momenta. The distortion of Fcan bias qtowards\nlarger values. We have seen a similar effect in Fig. 16 for\nthe case of the discretisation of the Laplacian, and we\nexpect that the effect of the Laplacian can be suppressed\nby taking a sufficiently larger value of Ng. On the other\nhand, the effect we see in Fig. 20 is attributed to the large\nvalue of mra, which is not possible to suppress unless we\nperform simulations with higher resolutions that realise\na smaller value of mraat the same value of ℓ.\nIn order to investigate the effect of mraon the in-\nstantaneous spectrum and qfor simulations with physical\nstrings, we performed additional simulations with 30723lattice sites for three different choices of the self coupling\nof the PQ field that lead to different values of mraat the\nend of the simulation (see Fig. 31 for the value of mraas\na function of ℓfor each choice of the simulation parame-\nters). Figure 21 shows the evolution of the energy density\nemission rates of axions and saxions obtained from phys-\nical string simulations including those with 30723and\n112643lattice sites. Similarly to the case of PRS strings\nshown in Fig. 18, we see that Γ aceases to increase and\nΓrblows up at large ℓ, and that the effect is more pro-\nnounced for simulations that lead to the larger value of\nmraat late times. Compared to the PRS case, the effect\nshows up rapidly at large logs, due to the exponential\ndependence on ℓ, i.e. mra∝R∝exp(1\n2ℓ).\nFigure 22 shows the evolution of qfor simulations with\nPRS (top panel) and physical strings (bottom panel). In\nboth cases, we see that the value of qconverges at small\nℓ, but takes larger values at larger ℓandmra.21We\nalso observe that the effect increases more rapidly for\nthe physical case compared to the PRS case.\nThe rapid increase in the discretisation effect can be\nunderstood by considering the fact that the horizon-sized\nstring loops acquire a large velocity when they collapse.\nTo illustrate this, let us consider the motion of a circular\nloop in the Nambu-Goto effective theory [92], where the\nstring is described by spacetime coordinates Xµ(τ, σ),\nandτandσare parameters of the world sheet swept\nby the string.22Here we choose the time-like parame-\nter as the conformal time τ. After imposing the gauge\ncondition Xτ·X′= 0, where the prime denotes the\nderivative with respect to σ, the equations of motion for\nthe Nambu-Goto string in a flat Friedmann-Robertson-\nWalker (FRW) background [92, 93] read,\nXττ+ 2Rτ\nR(1−X2\nτ)Xτ=ϵ−1(ϵ−1X′)′, (5.2)\nϵτ=−2Rτ\nRϵX2\nτ, (5.3)\nwhere ϵ= [X′2/(1−X2\nτ)]1/2. For a circular loop rep-\n21For the PRS case, the convergence at small ℓis less clear since\nthe data with small mracontains large oscillations at late times\nof the simulations. As discussed in Sec. IV, the largest oscillation\nis expected to occur at around τ=L/2, which corresponds to\nℓ≃6.7,7.1,7.6,8.0,8.3,8.7 for mra= 0.2,0.3,0.5,0.7,1.0,1.5. It\nturns out that these oscillations cannot be eliminated even if we\nuse the filtering method rather than the finite difference method\nto compute F. Furthermore, for smaller mra, these oscillatory\nfeatures could be amplified due to the effect of the finite volume\n(see Appendix E 1). The feature due to the maximum oscillation\natτ≃L/2 is also seen in the results of physical strings. For\ninstance, the plot of N= 11264 data shown in the bottom panel\nof Fig. 22 exhibits the oscillatory feature at ℓ= 8.5–9, in accord\nwith what was shown in Fig. 12.\n22For simplicity, here we ignore the effect of back-reaction on the\nloop due to the radiation of axions. It was confirmed that includ-\ning the effect of the axion radiation does not significantly alter\nthe motion of the loop described by the Nambu-Goto solution\neven if ℓis as small as ℓ≈5 [67].17\n101102103\nk/(RH)1001011\nH2f2a∂ρa\n∂logk\nmr\n2HPRSmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n101102103\nk/(RH)0.60.81.01.21.41.61.82.0Ratio tomra= 0.3\nmr\n2HPRSmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n101102103\nk/(RH)10−310−210−1100F(x)\nmr\n2HPRSmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n101102103\nk/(RH)0.500.751.001.251.501.752.002.252.50Ratio tomra= 0.3\nPRSmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\nmr\n2Hln(mr/H) = 6.0\nFIG. 17: The axion energy density spectrum (top left) and instantaneous emission spectrum (bottom left) at ln( mr/H) = 6\nobtained from simulations of PRS strings with 81923lattice sites for 5 different choices of mra. The ratio of the energy density\nspectrum (instantaneous emission spectrum) for each choice of mrato that for mra= 0.3 is also shown in the top right (bottom\nright) panel. The coloured bands represent statistical uncertainties. The momentum corresponding to k/R=mr/2 is marked\nwith dark blue ticks. Here we do not show the results from simulations with mra= 0.2, since they are consistent with those\nwith mra= 0.3 but have larger statistical uncertainties.\nresented by Xµ(τ, σ) = ( τ, l(τ) cosσ, l(τ) sinσ,0) with\n−π≤σ < π , Eq. (5.2) reduces to\nlττ+ 2Rτ\nR(1−l2\nτ)lτ+1−l2\nτ\nl= 0, (5.4)\nandϵ= [l2/(1−l2\nτ)]1/2. The second term in the left-\nhand side of the above equation represents the effect of\ndamping due to the cosmic expansion. When lis larger\nthan the horizon size, the velocity of the string is sup-\npressed due to the damping term. In fact, by ignoring the\nterms with lττand with higher orders in lτ, we see from\nthe above equation that the string velocity is of order\nlτ∼R/(Rτl)∼τ/l, which is very small for τ/l≪1.\nThe effect of the Hubble damping becomes irrelevant\nwhen the loop size becomes smaller than the horizon\nsize, and after that it starts to collapse. The motion of\nthe string loop in this regime may be approximated by\ndropping the terms proportional to Rτ/Rin Eqs. (5.2)and (5.3),\nlττ+1−l2\nτ\nl≈0, ϵ τ≈0, (5.5)\nwhose solution is given by l≈l0cos(τ/ϵ) with ϵ≈\nconstant, and l0is the radius of the loop when it starts to\ncollapse. Using the fact that ϵremains constant, we can\nestimate the loop velocity and the corresponding Lorentz\nfactor γat the time when the loop size becomes of the\norder of the string core radius, l≈m−1\nr:\nγ=h\n1−(l2\nτ)l≈m−1\nri−1/2\n≈l0mr, (5.6)\nwhich implies γ≈eℓforl0∼H−1.\nBased on the above result, we expect that the collaps-\ning strings can have an exponentially large Lorentz factor\nin the comoving coordinates at large ℓ. This implies that\nthe size of the string core is suppressed by the Lorentz\ncontraction, and the mraparameter is effectively modi-18\n4 5 6 7 8 9\nln(mr/H)020406080100120140160Γa/(f2\naH3)\nPRSmra= 0.2\nmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n4 5 6 7 8 9\nln(mr/H)01020304050607080Γr/(f2\naH3)\nPRSmra= 0.2\nmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\nFIG. 18: Evolution of the energy density emission rate of\naxions (top panel) and that of saxions (bottom panel) for\nPRS strings with different values of mra. The coloured bands\nrepresent statistical uncertainties.\nfied as\nmra→γmra≈eℓmra. (5.7)\nTherefore, the discretisation effect occurring at large mra\ncan be further pronounced at large ℓ. Note that, for\nsimulations of physical strings mraalso grows as eℓ/2,\nand hence the effect arises more dramatically than for\nthe case of PRS strings.\nC. Continuum extrapolation\n1. Energy density emission rate\nThere is a huge unphysical effect that distorts the in-\nstantaneous emission spectrum at large ℓand large mra.\nWe have seen that it also changes the net emission rates\nfor axions (Γ a) and saxions (Γ r), see Fig. 21. However, we\nfind that the total emission rate Γ a+Γrdoes not depend\n100101102103104\nk/(RH)10−310−210−11001011\nf2aH2∂ρa\n∂logk\nphysicalln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\nln(mr/H) = 8.0\nln(mr/H) = 9.0\n10−310−210−1100101\nk/(Rmr)10−310−210−11001011021\nf2aH2∂ρr\n∂logk\nphysicalln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\nln(mr/H) = 8.0\nln(mr/H) = 9.0FIG. 19: Energy density spectrum of axions (top panel) and\nthat of saxions (bottom panel) obtained from simulations\nof physical strings with 112643lattice sites. Different lines\ncorrespond to the spectra measured at different values of\nln(mr/H).\nmuch on mra. This is true for both the PRS case and\nphysical case, as shown in Fig. 23.23Comparing Figs. 18\nand 21 to Fig. 23, we see that the diminishing trend of Γ a\nand increasing trend of Γ rcompensate each other, such\nthat the total emission rate becomes almost insensitive\nto the discretisation effect. The fact that the total emis-\nsion rate is almost independent of mrais consistent with\nthe behaviour of the string density parameter ξ. We have\nseen that ξis also less sensitive to mra(see Fig. 8), which\nis naturally expected as the total emission rate is tied to\n23The exceptional case is N= 3072 and ¯λ= 114178, where Γ a+Γr\ndeviates from other lines at ℓ>∼7.5. In that case the discreti-\nsation effect turns out to be so large that the convergence in\nΓa+ Γris not guaranteed. However, it shows a good agreement\nwith others for ℓ<∼7 (corresponding to mra<∼1.5), and we\nuse the data in that region to estimate (Γ r)mra→0appearing in\nEq (5.8).19\n100101102103104\nx=k/(RH)10−510−410−310−210−1100F(x)\nphysicalln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\nln(mr/H) = 8.0\nln(mr/H) = 9.0\nFIG. 20: Instantaneous emission spectrum of axions obtained\nfrom simulations of physical strings with 112643lattice sites.\nDifferent lines correspond to the emission spectra measured\nat different values of ln( mr/H), and the coloured bands rep-\nresent statistical uncertainties. For the calculation of F(x, y)\nshown in this figure, we used the filtering method introduced\nin Sec. IV and described in detail in Appendix D 1, rather\nthan the finite difference method.\nthe string density through Eq. (2.6).\nAssuming that the discretisation effect is less harmful\nfor the total emission rate Γ a+ Γr, we may indirectly\nestimate the axion emission rate in the continuum limit\nas\n¯Γa= (Γ a+ Γr)data−(Γr)mra→0, (5.8)\nwhere (Γ a+ Γr)datarepresents the data obtained from\nsimulations (shown in Fig. 23), and (Γ r)mra→0is the\ncontinuum extrapolation of Γ r. We expect that this\nindirect method to estimate Γ ais better under control\nthan taking the continuum extrapolation of Γ adirectly,\nsince the error in the continuum extrapolation of Γ r\nin Eq. (5.8) should be negligible at large ℓ, where the\nvalue of (Γ r)mra→0itself becomes much smaller than\n(Γa+ Γr)data, as shown below.\nWe perform the continuum extrapolation of Γ rby us-\ning the data from four sets of 30723simulations with\ndifferent values of ¯λand one set of 112643simulations,\nshown in the bottom panel of Fig. 21. At each value of ℓ\nwithin the interval 4 ≤ℓ≤7, we fit the five data points\nto a quadratic function of mraand perform the extrap-\nolation to mra= 0. We observe that the result of the\nextrapolation still exhibits a trend of slightly increasing\nΓr/(f2\naH3) with ℓ, and we model it by a simple function\nin the form Γ r/(f2\naH3) =c0+c1ℓwithO(1) coefficients\nc0andc1. The estimate of Γ robtained based on this\nprocedure is shown in the bottom panel of Fig. 21 as a\ndashed line. Within the range of 4 ≤ℓ≤9, the value\nof Γ r/(f2\naH3) slightly increases from Γ r/(f2\naH3)≈1 (at\nℓ= 4) to Γ r/(f2\naH3)≈3.6 (at ℓ= 9). In the follow-\ning, we use this estimate based on the linear function up\ntoℓ∼9, where the mra→0 extrapolation cannot be\n4 5 6 7 8 9\nln(mr/H)020406080100120Γa/(f2\naH3)\nphysicalN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 3072,¯λ= 114178.0 (mra= 2.80 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)\n4 5 6 7 8 9\nln(mr/H)0102030405060Γr/(f2\naH3)physicalN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 3072,¯λ= 114178.0 (mra= 2.80 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)\nmra→0FIG. 21: Evolution of the energy density emission rate of ax-\nions (top panel) and that of saxions (bottom panel) for physi-\ncal strings with different values of the self coupling parameter\n¯λdefined by Eq. (A.4). The results are obtained from one\nset of 112643simulations and four sets of 30723simulations\nof physical strings, where mraevolves with time and its value\nat the end of the simulation is also shown in the legend for\neach choice of ¯λ. The coloured bands represent statistical un-\ncertainties. In the bottom panel, the continuum extrapolation\nof the saxion energy density emission rate is also shown by\nthe dashed line.\nperformed adequately.\nFigure 24 shows the evolution of axion, saxion, and\ntotal emission rates together with the continuum extrap-\nolation of Γ rand indirect estimate of Γ a[Eq. (5.8)], ob-\ntained for the simulations with 112643lattice sites. The\nresult of the extrapolation of Γ rimplies that ¯Γacontin-\nues to grow, and the change in Γ ris small compared to\nthe growth in ¯Γa. Therefore we expect that the energy\nstored in the string network is dominantly transferred\ninto axions at large ℓ. By looking at the difference be-\ntween the plots of Γ aand¯Γa, we can also see that about\n40% of the axion radiation power is lost into unphysical\nhigh-frequency saxions at around the end of simulations20\n5.5 6.0 6.5 7.0 7.5 8.0 8.5\nln(mr/H)0.60.70.80.91.01.11.2\nq\nPRSmra= 0.2\nmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\nln(mr/H)0.60.81.01.21.41.61.8\nq\nphysicalN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)\nFIG. 22: Evolution of the spectral index qof the instanta-\nneous emission spectrum for different choices of the coupling\nparameter of the PQ field that controls the string core width.\nTop panel shows the results of simulations of PRS strings\nwith 81923lattice sites. Bottom panel shows the results of\none set of 112643simulations and three sets of 30723simu-\nlations of physical strings. The coloured bands represent the\nerror induced by changing the parameter σfilterfor the filtering\nprocedure to calculate F[Eq. (D.5)] in addition to statistical\nuncertainties.\ndue to the discretisation effects.\n2. Spectral index\nWe have seen that the discretisation effects due to finite\nvalues of 1 /(mra) distort the spectrum and can lead to\nthe overestimation of q. Here we try to model this effect\nin terms of the following function,\nq=qmodel(ℓ) +qdisc(ℓ, mra), (5.9)\nwhere qmodel(ℓ) represents the genuine behaviour given\nby some function of ℓ, and qdisc(ℓ, mra) represents the\n4 5 6 7 8 9\nln(mr/H)0255075100125150175200(Γa+ Γr)/(f2\naH3)\nPRSmra= 0.2\nmra= 0.3\nmra= 0.5\nmra= 0.7\nmra= 1.0\nmra= 1.5\n4 5 6 7 8 9\nln(mr/H)0255075100125150175200(Γa+ Γr)/(f2\naH3)\nphysicalN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 3072,¯λ= 114178.0 (mra= 2.80 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)FIG. 23: Evolution of the sum of the energy density emission\nrate of axions and that of saxions for different choices of the\ncoupling parameter of the PQ field that controls the string\ncore width. Top panel shows the results of simulations of\nPRS strings with 81923lattice sites. Bottom panel shows the\nresults of one set of 112643simulations and four sets of 30723\nsimulations of physical strings. The coloured bands represent\nstatistical uncertainties.\nbias due to the discretisation effects that depends not\nonly on ℓbut also on mra. We expect that qdisc(ℓ, mra)\nmonotonically increases with ℓandmra, and vanishes in\nthe limit mra→0. The argument of collapsing Nambu-\nGoto string loops that led to Eq. (5.7) inspires us to\nthink of the form qdisc∼αeβℓ, where αandβare co-\nefficients that do not depend on ℓ. We found that such\naℓ-dependence nicely reproduces the behaviour of qfor\nthe PRS strings for each mra(shown in the top panel of\nFig. 22), and also observed a trend that the coefficient α\nincreases while βdecreases for larger values of mra. Tak-\ning into account such a trend, we introduce the following\nfunction to model the effects for physical strings,\nqdisc(ℓ, mra) =d0(mra)d1exp\u0014d2ℓ\n1 + (mra)d3\u0015\n,(5.10)21\n4 5 6 7 8 9\nln(mr/H)020406080100120140160Γi/(f2\naH3)physical Γa\nΓr\nΓa+ Γr\n¯Γa= Γa+ Γr−(Γr)mra→0\n(Γr)mra→0\nΓth(fit)\nFIG. 24: Evolution of the energy density emission rate of ax-\nions (blue line), saxions (orange line), and the sum of them\n(green line) obtained from simulations of physical strings with\n112643lattice sites. The coloured bands represent statistical\nuncertainties. The axion energy density emission rate in the\ncontinuum limit defined by Eq. (5.8) and the continuum ex-\ntrapolation of Γ r/(f2\naH3) are also shown as red dashed line\nand purple dashed line, respectively. The dash-dotted line\nrepresents the theoretical expectation given by Eq. (2.6) with\nparameters [Eqs. (6.1) and (6.2)] determined by the fit.\nwhere d0,d1,d2, and d3are constant parameters.\nWith the ansatz (5.10) for the discretisation effects,\nwe performed the global fits of the data of qmeasured in\nthe simulations of physical strings to the function given\nby Eq. (5.9). We assume that the model part qmodel\nshould be given by a simple function of ℓ, and consider\nthe following possibilities:\nModel A : q=q0+q1ℓ(q1>0),\nModel B : q=q0+q1ℓ2(q1>0),\nModel C : q=q0+q1/ℓ(q1<0),\nModel D : q=q0+q1/ℓ2(q1<0).(5.11)\nIn the fits, we use three sets of 30723simulations with\ndifferent values of ¯λand one set of 112643simulations,\nshown in the bottom panel of Fig. 22.24For each data set,\nwe sampled the value of qwith an interval of ∆ ℓ= 0.25\nstarting from ℓ= 5.5, and the contribution of each data\nis weighted by the error interval including the statistical\nerrors and systematics due to the filtering procedure to\ncalculate F(x, y) (see Appendix D 1). In order to make\nsure that the fit parameters end up within a reasonable\ninterval, we performed the fits in three steps: First we\n24In this analysis, we do not include the data from 30723simula-\ntions with ¯λ= 114178 ( mra= 2.8 atτ/L= 0.5), since in that\ncase the spectrum is too distorted at late times to be accurately\nmodeled by a simple function of q(ℓ, mra).TABLE I: Best fit parameters and χ2/d.o.f.for the models\ncharacterising the ℓ-dependence of q.\nModel χ2/d.o.f. q 0 q1\n(A)q0+q1ℓ 0.89 0.19(3) 0.093(5)\n(B)q0+q1ℓ21.02 0.50(2) 0.0069(4)\n(C)q0+q1/ℓ 0.82 1.39(4) −3.9(3)\n(D)q0+q1/ℓ20.81 1.09(2) −12.4(8)\nfitted the model part qmodel only by using data with\nmra <0.7, and then fitted the full data to the function\ngiven by Eq. (5.9) with the parameters in qmodel fixed to\nvalues obtained in the first step. After that, we fitted full\ndata again to the function given by Eq. (5.9) by allowing\nall the parameters in qmodel andqdiscto change. In this\nthird step, we specified the boundaries for the parame-\ntersd0, d1, d2, d3to ranges corresponding to one sigma\nintervals obtained in the second step.\nThe result of the fit for each model, including the value\nofχ2/d.o.f.and those for the best fit parameters, is sum-\nmarised in table I. Values of the parameters d0,d1,d2,\nandd3inqdiscalso differ according to the model, which\nare not shown in the table as their precise values become\nirrelevant after taking the continuum limit ( mra→0).\nWe find that the simple two-parameter models enumer-\nated in Eq. (5.11) yield almost equally good fits.25Model\nC and D are slightly more preferred than A and B, but\nthe latter (especially model A) also shows a good fit.\nThe evolution of qpredicted by three preferred models\nand the data set used for the fits are shown in Fig. 25.\nThe data is consistent with qmodel atℓ<∼7, which im-\nplies that the discretisation error is not important at that\nrange. On the other hand, the discrepancy between dif-\nferent models as well as differences between the models\nand data stand out at large ℓ, which indicates that the\ndiscretisation error is quite serious in that range.\nThe fact that different models are almost equally pre-\nferred by the data implies that the value of qat large ℓ\nremains quite uncertain. We can basically consider two\npossibilities: one is that qgrows linearly with ℓ(model\nA, which was also claimed in Ref. [70]), and the other is\nthat qcontinues to grow but asymptotes to a constant\nvalue with the ℓdependence suppressed as 1 /ℓ(model C)\nor 1/ℓ2(model D). Although differences between these\nmodels are marginal at small ℓsimulated so far, it leads\nto a big impact on the calculation of the axion dark mat-\nter abundance when extrapolated to large ℓ. Implications\nfor the estimation of the axion relic abundance and dark\n25We also considered a set of three-parameter models such as\nq0+q1ℓ+q2ℓ2,q0+q1/ℓ+q2/ℓ2, and q0+q1ℓ+q2/ℓ, and found\nthat they can also fit the data very well. However, in those cases\nthe values of the coefficients q0,q1,q2become much more uncer-\ntain. This result signals that any of these coefficients could be\npotentially important, and explains why different two-parameter\nmodels (model A, C and D) are almost equally preferred.22\n5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\nln(mr/H)0.60.81.01.21.41.61.8\nq\nphysical\nN= 3072,¯λ= 14563.6 (mra= 1.00 atτ/L = 0.50)\nN= 3072,¯λ= 32768.0 (mra= 1.50 atτ/L = 0.50)\nN= 3072,¯λ= 64225.3 (mra= 2.10 atτ/L = 0.50)\nN= 11264,¯λ= 195799 (mra= 1.25 atτ/L = 0.625)\nqmodel =q0+q1//lscript2(mra→0)\nqmodel =q0+q1/lscript(mra→0)\nqmodel =q0+q1//lscript(mra→0)\nFIG. 25: Comparison of the data on the evolution of the\nspectral index qof the instantaneous emission spectrum ob-\ntained from simulations of physical strings (markers with error\nbars) and their continuum extrapolations given by Model A\n(gray dashed line), Model C (brown dotted line), and Model\nD (black dash-dotted line) in Eq. (5.11) with coefficients given\nby Table I, which show good fits to the data when the addi-\ntional term [Eq. (5.10)] characterising the discretisation ef-\nfect is included. The error bars include systematics induced\nby changing the parameter σfilterfor the filtering procedure to\ncalculate F[Eq. (D.5)] in addition to statistical uncertainties.\nmatter mass will be discussed in Sec. VI C.\nVI. RESULTS: EXTRAPOLATION\nIn this section, we estimate the axion relic abundance\nobtained by extrapolating numerical results to large ℓ. In\norder to perform such an extrapolation, we need several\nquantities that characterise the axion emission spectrum.\nAmong them, we have investigated the systematics on\nthe string density parameter ξin Sec. III, and the spec-\ntral index qof the instantaneous axion emission spectrum\nin Secs. IV and V. In the following two subsections, we\npresent further results on the total energy density emis-\nsion rate (Sec. VI A) and IR peak of the instantaneous\nemission spectrum (Sec. VI B). All these results are com-\nbined to calculate the axion dark matter abundance and\nits uncertainty in Sec. VI C.\nA. Energy density emission rate\nAs shown in Fig. 23, the total energy density emis-\nsion rate (Γ a+ Γr)/(f2\naH3) increases with ℓ. This trend\nis expected to be captured by the analytical expression\nshown in Eq. (2.6), which can be used for the extrap-\nolation. We utilise the orthogonal distance regression\n(ODR) algorithm [94] to fit the data and determine the\nvalues of the parameters fLandηappearing in Eq. (2.6).In this procedure, we treat Eq. (2.6) as a model func-\ntionf(x;β) that depends on three explanatory variables\nx= (ξ, dξ/dℓ, ℓ ) and two parameters β= (fL, η), and\nwe use the data of (Γ a+ Γr)/(f2\naH3),ξ, and dξ/dℓ ob-\ntained from the simulations with 112643lattice sites.26\nThe data is sampled with an interval of ∆ ℓ= 0.25, while\nthe value of dξ/dℓ is evaluated by the simple finite dif-\nference between the nearest neighbour (much closer than\n∆ℓ= 0.25) at each sample point. The data of the re-\nsponse variable, (Γ a+ Γr)/(f2\naH3), are summed with\nweights given by the reciprocal of their statistical uncer-\ntainties. For the data of ξanddξ/dℓ , we evaluate their\ncovariance matrix at each sample point, and use it as a\nweight matrix in ODR in order to take account of the\npossible correlation between them. On the other hand,\nthe data of ℓfor each sample point is just introduced as\nan auxiliary variable, and we set its weight values to 1\nwith no correlation with ξanddξ/dℓ .\nAs a result of the fitting procedure described above,\nthe parameters are determined as\nfL= 0.83(1) , (6.1)\nη= 0.27(2) . (6.2)\nWith these values of parameters, the behaviour of (Γ a+\nΓr)/(f2\naH3) is well reproduced by Eq. (2.6), as shown in\nFig. 24. It is notable that the value of fLagrees with our\nexpectation of ∼80 % reduction due to the abundance of\nlong strings discussed in Sec. II. Furthermore, the value\nofηmatches the theoretical estimate η= 1/√\n4π≃0.28\nremarkably well. The above results for the system of\nphysical strings are also compatible with fL≈0.84 and\nη≈0.27 found in Ref. [88] for PRS strings.\nB. IR peak\nThe IR cutoff x0of the instantaneous emission spec-\ntrum has some impact on the estimation of the axion\nnumber radiated from strings (see Fig. 2). A non-trivial\nquestion is whether the value of x0depends on ℓ. One\nmight expect that the location of the IR cutoff should be\nproportional to√ξ[61, 70, 71, 77, 95], since the increase\nin the string density leads to the shortening of the typical\ndistance of neighbouring strings. Given the logarithmic\ngrowth of ξ, this expectation implies that the IR cutoff\nwould shift to higher momenta at large ℓ. The actual\nspectrum does not have a sharp cutoff but has a peak-\nlike feature in the IR, and we can have an idea on how\nx0evolves by looking at the behaviour of the IR peak of\nF(x).\nThe existence of the IR peak in the instantaneous emis-\nsion spectrum F(x) implies that, when we keep track of\n26Here we treat ηas a constant, since its time-dependence is ex-\npected to be subdominant, suppressed by 1 /ℓ[see Eq. (2.6)].23\nthe time evolution of Ffor a fixed value of the comoving\nmomentum k, there is a point at which Fbecomes maxi-\nmum: Before that time (outside the horizon, k<∼RH) it\nmonotonically increases, and after that time (inside the\nhorizon, k>∼RH) it decreases and oscillates. Such a\ntime of turnaround τt.a.can be different according to the\ncomoving wavenumber, but we expect that the product\nxt.a.=kτt.a.should take a similar value, as it should\nhappen around the time of the horizon crossing, where\nk−1becomes comparable to τ= 1/(RH). Neverthe-\nless, the exact value of xt.a.could be slightly different\naccording to the value of the string density parameter ξ\nor ln( mr/H) at the time of turnaround, from which we\ncan infer the relation between ξand the location of the\nIR peak. The identification of the peak location in terms\nof the turnaround of the mode evolution is advantageous\nin the numerical study, since it is hard to identify the IR\npeak directly from the spectrum ( k-dependence) of F(x)\ndue to the fact that the number of modes in the IR are\nvery few, which prevents us from extracting the feature\nof the IR peak precisely.\nFigure 26 shows the time evolution27ofFfor two fixed\nvalues of comoving momenta, kL/(2π) = 2 .41 (top panel)\nandkL/(2π) = 3 .43 (bottom panel).28In this figure\nwe compare the results from simulations with different\nvalues for the initial string density (the same data sets as\nSec. III). We see that the turnaround happens at kτ∼\n3.5 for the attractor (simulations with the initial string\ndensity ξℓ=3= 0.3), while there is a trend that it happens\nearlier (later) for under-dense (over-dense) cases. This\ntrend agrees with our intuition that the location of the\nIR peak shifts towards larger values of x=kτfor dense\nstring networks.\nNote that in this analysis we can only use the data\nwith two comoving momenta, the second-lowest mode\n[kL/(2π) = 2 .41] and the third-lowest mode [ kL/(2π) =\n3.43]. This limitation comes from the fact that we have to\nmonitor the evolution of the modes that cross the horizon\nwithin the simulated time scale in order to observe the\nturnaround. For instance, the lowest mode [ kL/(2π)∼1]\nis not suitable, since it does not enter well inside the\nhorizon before the end of the simulations. Furthermore,\n27Here we change two independent variables of Ffrom ( x, y) to\n(k, τ), and study the dependence on τfor fixed k. In this context,\nthe change in x=kτshould be regarded as the time evolution\nrather than the momentum dependence.\n28Note that for the computation of Fused in this subsection, we\ndo not use the filtering method described in Appendix D 1, which\nwas used in Sec. IV to obtain the smooth spectrum of F. Such\na method is beneficial for removing the 2 k-oscillation that acts\nas a contamination in characterising the emission spectrum at\nthe intermediate momenta, but here we are interested in the\nturnaround behaviour of a mode with a very low momentum,\nwhich is a part of the 2 k-oscillation. Instead of applying such a\nfiltering procedure, here we compute the time derivative in Fby\ntaking the finite difference and applying another Gaussian filter\nto remove fluctuations that show up in a period much shorter\nthan the oscillation frequency ∼2k.\n2 3 4 5 6 7\nkτ0.00.20.40.60.81.01.2\nFkL/(2π) = 2.41\nξ/lscript=3= 0.036\nξ/lscript=3= 0.073\nξ/lscript=3= 0.133\nξ/lscript=3= 0.214\nξ/lscript=3= 0.300\nξ/lscript=3= 0.380\nξ/lscript=3= 0.464\nξ/lscript=3= 0.586\nξ/lscript=3= 0.711\nξ/lscript=3= 0.828\n2 3 4 5 6 7\nkτ0.00.20.40.60.81.01.2\nFkL/(2π) = 3.43\nξ/lscript=3= 0.036\nξ/lscript=3= 0.073\nξ/lscript=3= 0.133\nξ/lscript=3= 0.214\nξ/lscript=3= 0.300\nξ/lscript=3= 0.380\nξ/lscript=3= 0.464\nξ/lscript=3= 0.586\nξ/lscript=3= 0.711\nξ/lscript=3= 0.828FIG. 26: Time evolution of the spectral energy density pro-\nduction rate Fof one Fourier mode specified by kL/(2π) =\n2.41 (top panel) and kL/(2π) = 3 .43 (bottom panel) for vari-\nous different values of the initial strings density ξℓ=3. The\ncoloured bands represent statistical uncertainties, and the\nblack dashed line corresponds to the attractor with ξℓ=3=\n0.3. The locations of first turnaround are marked by black\ndots. The results are obtained from simulations of physical\nstrings with 20483lattice sites.\nthe modes with higher momenta kL/(2π)>∼4 are also\ninappropriate for the identification of the turnaround,\nsince they are already inside the horizon at the beginning\nof the simulations in most cases. This constraint also\naffects the identification of the turnaround for the third-\nlowest mode [ kL/(2π) = 3 .43] shown in the bottom panel\nof Fig. 26. In that figure, we see that in some of the\nsimulations with under-dense strings the mode is already\ninside the horizon at the beginning of the simulations,\nand hence it is not possible to identify the time of the\nturnaround accurately for such data sets.\nIt should also be noted that the turnaround behaviour\ncan be observed even in the absence of strings ( ξ→0).\nIn that case the mode evolution can be described as a\nsimple misalignment oscillation of the free axion field,24\nand the location of the turnaround corresponds to the\ntime at which Freaches the first oscillation maximum.\nIt is possible to calculate the evolution due to such mis-\nalignment oscillations analytically, and we find x≃1.87\nat turnaround (see Appendix F).\nThe relation between the string density ξand the value\nofxat turnaround for 30 sets of simulations with different\ninitial string densities is summarised in Fig. 27. Note that\nthe values of ξshown in that figure do not represent the\ninitial string densities (evaluated at ℓ= 3) but the values\nofξevaluated at the time of the turnaround. The figure\nclearly shows the trend that the value of xat turnaround\nincreases with ξ. The error bars represent the statistical\nuncertainties, which become larger for large ξ, since in\nthat case the mode starts to oscillate at late times of the\nsimulation, where there are not many Hubble patches\nin the simulation box and the value of Fbecomes more\nuncertain.\nNow let us model the relation between ξandxat\nturnaround in terms of the following function,\nxt.a.= 1.87 + b1ξb0, (6.3)\nwhere the value of the constant term is chosen such that\nxt.a.approaches to the value inferred from the misalign-\nment oscillation of the free axion field in the limit ξ→0.\nWe fit the data of xandξat turnaround to the above\nmodel by utilising the ODR algorithm, where statistical\nerrors of both xandξare taken into account as weights\nfor the evaluation of the squared distance. When we use\nthe data of kL/(2π) = 3 .43, we omit six data points from\nthe lowest string density (gray cross markers in the bot-\ntom panel of Fig. 27), since in those cases the first maxi-\nmum in Fappears at around the initial time of the sim-\nulation, which could bias the location of the turnaround.\nWith this procedure, the best-fit parameters are found\nto be\nb0= 0.53(6) , b1= 2.3(1) for kL/(2π) = 2 .41,\nb0= 0.54(4) , b1= 2.44(7) for kL/(2π) = 3 .43.(6.4)\nThe function (6.3) with the above best-fit parameters is\nplotted in Fig. 27 as dashed lines. Remarkably, the value\nof the exponent b0is consistent with the naive expecta-\ntionxt.a.∝√ξ. Combined with the expectation that ξ\nincreases linearly with ℓ, the result implies that the loca-\ntion of the IR cutoff can be as large as x0∼xt.a.≈10–15\nat ln( mr/H) = 70. We will use this extrapolation for the\nestimation of the axion relic abundance in the next sub-\nsection.\nC. Axion relic abundance\nFor the calculation of the axion abundance, the evo-\nlution of the axion field has to be treated with caution\nif the emission spectrum is completely dominated by IR\n(q≫1). In Ref. [70], it was argued that for such an\nIR dominated spectrum the axion field becomes highly\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\nξat turnaround1.52.02.53.03.54.04.55.05.5xat turnaround\nkL/(2π) = 2.41\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\nξat turnaround1.52.02.53.03.54.04.55.05.5xat turnaround\nkL/(2π) = 3.43FIG. 27: The relation between the string density parameter\nξand the value of x=kτat turnaround, obtained from the\ndata of the evolution of Fof one Fourier mode specified by\nkL/(2π) = 2 .41 (top panel) and kL/(2π) = 3 .43 (bottom\npanel) for various different values of the initial strings density.\nThe results are obtained from simulations of physical strings\nwith 20483lattice sites. Dashed lines represent the fits to\na function given by Eq. (6.3). The data points marked by\ngray crosses are not used for the fit, since in those cases the\nfirst maximum of Fappears at around the initial time of the\nsimulation.\nnon-linear, which could lead to a transient regime of ax-\nion number non-conserving processes around the epoch\nof the QCD phase transition and result in an alleviation\nof the enhancement of the axion abundance due to the\nlarge value of ξln(mr/H).\nTo check the relevance of the possible non-linear effect,\nit is instructive to calculatep\n⟨θ2⟩, the average square\namplitude of the axion field given by Eq. (2.15). Here we\ncompute it as\n⟨θ2⟩=Zℓ\nℓsimdℓ′\n2eℓ′\neℓΓ′\na\nf2aH′3⟨x−2⟩(y′) +eℓsim\neℓ⟨θ2⟩ℓsim,\n(6.5)25\nwhere ⟨θ2⟩ℓsimis obtained by directly integrating\n(1/k2)∂ρa/∂k[see Eq. (2.13)] from the simulation result\natℓsim= 7. With this initial value, we perform the inte-\ngration from ℓsimtoℓ. For Γ ain the integrand, we use Γ th\ngiven by Eq. (2.6) with parameters given by Eqs. (6.1)\nand (6.2). The approximation Γ a≈Γthis guaranteed\nif we assume Γ a≫Γrat large ℓ(see Fig. 24). For\n⟨x−2⟩(y) in the integrand, we use a simple power law\napproximation for F(x) given by Eq. (2.17). Here we\nadopt x0=xt.a.with xt.a.given by Eq. (6.3) and co-\nefficients given by Eq. (6.4). Furthermore, we extrapo-\nlate the string density parameter ξby numerically solving\nEq. (3.1) for two models of ξc[Eqs. (3.3) and (3.4)], and\nestimate the uncertainty associated with the modeling of\nξc.\nFigure 28 shows the estimates ofp\n⟨θ2⟩for different\nassumptions on q. Here we consider three models that\ngive good fits to the simulation results (Model A, C, and\nD in Table I). We see that the system indeed becomes\nhighly non-linear (p\n⟨θ2⟩ ≫ π) at large ℓforqmodel =\nq0+q1ℓ, while it stays linear for qmodel =q0+q1/ℓ2. For\nqmodel =q0+q1/ℓ, the average square amplitude takes a\nvalue around the boundary between the linear and non-\nlinear regime (p\n⟨θ2⟩ ∼π) atℓ= 70.\nIf the system remains in the linear regime, the axion\nnumber at around the QCD phase transition can be esti-\nmated by using Eq. (2.14). In that case we can compute\nit in a similar manner as the average square amplitude,\nna\nf2aH=Zℓ\nℓsimdℓ′\n2eℓ′/2\neℓ/2Γ′\na\nf2aH′3⟨x−1⟩(y′)\n+eℓsim/2\neℓ/2na\nf2aH\f\f\f\f\nℓsim, (6.6)\nwhere na/(f2\naH)|ℓsimis obtained by directly integrating\n(1/k)∂ρa/∂k[see Eq. (2.12)] from the simulation result at\nℓsim= 7. For ⟨x−1⟩(y) in the integrand, we use a simple\npower law approximation for F(x) given by Eq. (2.17).\nOther setups are the same as the calculation of the av-erage square amplitude described below Eq. (6.5). After\nobtaining the axion number in this way, we can straight-\nforwardly obtain the relic axion abundance at the present\ntime by applying the appropriate red-shift factor.\nThe above estimate of the axion abundance could be\nmodified if the system becomes non-linear. In that case,\nthe axion number is given by [70]\nna(tℓ) =cncVzma(T1)f2\na\u0012R(t1)\nR(tℓ)\u00133\n, (6.7)\nwhere\nma(T)2=χ(T)\nf2a, (6.8)\nis the axion mass at high temperatures defined by the\ntopological susceptibility χ(T), and in the relevant tem-\nperature range, we adopt a power law approximation pa-\nrameterised by an exponent nQCD,\nχ(T)∝T−nQCD. (6.9)\nRecent lattice QCD simulations have found a value\naround nQCD≃8 [96]. The quantity zin Eq. (6.7) is\ndefined as\nz≡\u0012m(Tℓ)\nH1\u00131+6\nnQCD, (6.10)\nwhich represents the duration of the non-linear transient.\nt1andtℓcorrespond to the time defined by H1=ma(T1)\nand that at the end of the non-linear transient, respec-\ntively. cnandcVare coefficients that can be determined\nnumerically.\nFrom the requirement that the non-linear transient is\nterminated when the gradient energy of the IR modes\nbecomes comparable to the potential energy from the\naxion mass, the value of zcan be fixed and given in terms\nof infinitely nested logarithms [70],\nz=\n4πfLξ1ℓ1\ncV\u0014\n1−2\nnQCD+ 4\u0015\nln\n4πfLξ1ℓ1\ncV\u0014\n1−2\nnQCD+ 4\u0015\u0014cm\nx0\u00152\u0010\n1+2\nnQCD+2\u0011\nln(. . .)\n\n1\n2\u0010\n1+2\nnQCD+4\u0011\n,(6.11)\nwhere ξ1and ℓ1are values of ξand ℓevaluated at\nH=H1.cmis another O(1) coefficient that can also be\ndetermined from the numerical simulations. In Ref. [70],\nit was shown that the above analytical estimates show\na good agreement with the numerical results for the fol-\nlowing values,\ncn= 1.35, c V= 0.13, c m= 2.08, (6.12)which we adopt in our analysis.\nSome comments are in order. First, in Eq. (6.11) we\nhave applied the replacement ξ1→fLξ1to Eq. (34) of\nRef. [70], taking into account that the emission rate (2.6)\nis accompanied by the overall factor of fL. Second, the\nanalytical expressions [Eqs. (6.7), (6.10), and (6.11)] and\nparameters [Eq. (6.12)] are obtained based on the as-26\n10 20 30 40 50 60 70\nln(mr/H)0123456789/radicalbig\n/angbracketleftθ2/angbracketright\n/radicalbig\n/angbracketleftθ2/angbracketright=πq=q0+q1/lscript\nq=q0+q1//lscript\nq=q0+q1//lscript2\nFIG. 28: The ln( mr/H) dependence of the average square\namplitude of the axion field computed by using Eq. (6.5) for\nthree different scenarios of the extrapolation of q. The shaded\nregions correspond to the uncertainty given by the maximum\nand minimum values among one sigma variations of the fit-\nparameters in q,ξc, orx0, for each model of ξc[i.e.ξlin\ncgiven\nby Eq. (3.3) and ξsat\ncgiven by Eq. (3.4)].\nsumption of q≫1.29In that limit, the dependence on\nqdisappears in Eq. (6.11), while the result substantially\ndepends on qwhen it is not so large (see Fig. 26 of\nRef. [70]). Hence, those analytical formulae cannot be\nused reliably in the case where qis not much larger than\none (e.g. q<∼2). Finally, though Eqs. (6.7) and (6.11)\nwere derived based on the assumption that x0is con-\nstant, they can be used in a good approximation even if\nwe include the possible (slow) evolution of x0≈xt.a.[cf.\nEq. (6.3)]. Since the dependence on x0is only logarith-\nmic, the change in this quantity just gives rise to subdom-\ninant effects, which are negligible for ln( mr/H1)≫1.\nAfter the non-linear transient ( t > t ℓ), the comoving\naxion number density is conserved, and the axion abun-\ndance at the present time can be estimated by using\nEq. (6.7). Note that the axion number density at the\npresent time t0is given by\nna(t0) =\u0012Rℓ\nR0\u00133\nna(tℓ)\n=\u0012Rℓ\nR0\u00133\ncncVzma(T1)f2\na\u0012R1\nRℓ\u00133\n=cncVzH1f2\na\u0012R1\nR0\u00133\n. (6.13)\nHence, we can think of cncVzas an enhancement factor\ncompared to the naive estimate H1f2\na(R1/R0)3.\n29In Ref. [70], q= 5 was used for simulations of the axion field\nevolution at the non-linear regime.In order to quantify the effect of strings on the esti-\nmation of the dark matter abundance, let us define the\nproduction efficiency from strings compared to the usual\nmisalignment estimate,\nK≡nstr\na(t0)\nnmisa(t0), (6.14)\nwhere nmis\na(t0) =cmisH1f2\na(R1/R0)3with cmis= 2.31 is\nthe fiducial axion number representing a typical result\nfrom angle-average misalignment estimation.30In terms\nofK, the axion CDM abundance can be estimated as\nΩah2=KΩmis\nah2, (6.15)\nwhere\nΩmis\nah2=manmis\na(t0)\nρcrit/h2\n=cmisr\n20\nπg∗s(T0)p\ng∗ρ(T1)\ng∗s(T1)Ωγh2maf2\na\nT1T0MP,\n(6.16)\ng∗ρ(T) and g∗s(T) are the effective degrees of freedom for\nthe energy density and entropy density, respectively, ρcrit\nis the critical density of the Universe today, g∗s(T0)≃\n3.931 are the effective degrees of freedom for the entropy\ndensity today [98], Ω γh2≃2.473×10−5is the photon\ndensity parameter, T0≃2.349×10−4eV is the photon\ntemperature today [99], and MP≃1.221×1019GeV is\nthe Planck mass.\nFor a given value of K, one can determine the axion\nmass masatisfying the condition Ω ah2= Ω CDMh2=\n0.12 [100], as shown in Fig. 29. To produce this relation\nbetween Kandma, we used tabulated data of g∗ρ(T)\nandg∗s(T) obtained in Ref. [98]. As shown in the figure,\nthe result slightly depends on the exponent nQCDcharac-\nterising the temperature dependence of the axion mass.\nA smaller value of nQCDmakes the axion massive at ear-\nlier time, shifting T1to a higher value. This results in a\nsuppression Ωmis\nah2∝1/T1[see Eq. (6.16)], and requires\na higher value of fato explain the observed dark mat-\nter abundance. Therefore, a smaller mass is predicted\nfor a smaller nQCD. We also note that the uncertainty\ndue to the equation of state g∗ρ(T) and g∗s(T) quanti-\nfied in Ref. [98] is negligibly small compared to the effect\nofnQCD, since a small variation in the Hubble param-\neter due to g∗ρ(T) does not have a big impact on the\ndetermination of T1for a fast temperature dependence\nofma(T).\n30The coefficient is motivated by the averaging of the initial mis-\nalignment angle θi,cmis=⟨θ2\ni⟩/2 withq\n⟨θ2\ni⟩ ≈2.15, which was\nobtained numerically by including anharmonicities in the QCD\npotential [96, 97].27\n10−210−1100101102103\nK=nstr\na(t0)/nmis\na(t0)10−1100101102103ma[µeV]nQCD= 8\nnQCD= 7\nnQCD= 6\nFIG. 29: The relation between the axion production efficiency\ndefined by Eq. (6.14) and the axion mass explaining the ob-\nserved CDM abundance. Different lines correspond to differ-\nent values of the exponent nQCDcharacterising the tempera-\nture dependence of the topological susceptibility [Eq. (6.9)].\nNow we can compute the axion relic abundance relative\nto the fiducial estimate,\nK=\n\n1\ncmisna(t1)\nf2aH1(linear) ,\ncncVz\ncmis(non-linear) ,(6.17)\nwhere Eq. (6.6) can be used for the linear case, while\nEq. (6.11) should be used for the non-linear case. Once\nwe obtain Kfor a given value of ℓ, we can translate it\nto the prediction for the dark matter mass by using the\nK-marelation shown in Fig. 29.\nIn order to perform the extrapolation to large ℓ, we\nneed to fix the assumption on how the values of qandξ\nchange with ℓ. For the value of q, among three different\npossibilities considered in Fig. 28, the linear estimate can\nbe used for the case with q=q0+q1/ℓ2, while the non-\nlinear estimate should be used for that with q=q0+q1ℓ.\nHereafter we estimate the axion relic abundance and the\ndark matter mass prediction for these two cases, and call\nthe scenario with q=q0+q1/ℓ2theplateau extrapolation\nand that with q=q0+q1ℓthelinear extrapolation . It\nmight also be possible to consider yet another type of\nextrapolation by using the model with q=q0+q1/ℓ, but\nin that case both the formulae shown in Eq. (6.17) would\nnot hold to a good approximation, since the system lies\nin the boundary between linear and non-linear regime\natℓ∼70 (see Fig. 28), which requires some additional\nnumerical study to obtain quantitative results. For this\ntechnical reason, in the following we do not perform the\nextrapolation based on the model with q=q0+q1/ℓ.\nNevertheless, it is certainly true that the result of the\nextrapolation with q=q0+q1/ℓwould lie between the\nvalue predicted in the case with q=q0+q1/ℓ2and that\nwith q=q0+q1ℓ.For the value of ξ, we use the extrapolation based on\ntwo models of ξcgiven by Eqs. (3.3) and (3.4). Here\nwe adopt two models of ξcfor each of the two possible\nextrapolations of q, so that we consider four different\npossibilities for the values of qandξin total, and quantify\ntheir impact on the estimation of the axion abundance\nand the dark matter mass prediction.\nIn Fig. 30, we show the results of the calculation of\nK(left panel) and the corresponding axion dark matter\nmass prediction (right panel) for different possibilities on\nthe extrapolation of qandξ. In these plots, we also\nshow the uncertainties due to the change in the values\nof parameters used for the calculation. The uncertainty\ndue to qis derived from the fit parameters q0andq1\n(see Table I). The uncertainty due to ξis derived from\nthe fit parameters of ξcshown in Eqs. (3.3) and (3.4).\nThe uncertainty due to x0is derived from the fit param-\neters b0andb1shown in Eq. (6.4), where we use the\nresults obtained from data with kL/(2π) = 2 .41 as the\nfiducial values. As for the uncertainty due to nQCD, we\nchange the value of nQCDfrom 7 to 8.16 (estimate based\non the dilute instanton gas approximation [96]), where\nnQCD = 8 is used as the fiducial value.31Note that the\nerror in nQCDis irrelevant for the calculation of Kin the\nplateau extrapolation, and the corresponding lines (blue\nor black dotted lines) are not shown in the left panel of\nFig. 30. Furthermore, the error in q(would be orange or\ngray solid lines) is irrelevant for the linear extrapolation,\nsince we have used the formula given by Eq. (6.11), which\nis derived under the assumption of q≫1 and does not\ndepend on the detailed values of q. It turns out that, for\nthe plateau extrapolation the largest uncertainty comes\nfrom the error in the value of q(forξc=ξlin\nc) orξ(for\nξc=ξsat\nc). On the other hand, for the linear extrapola-\ntion with ξc=ξlin\ncthe dominant source of the uncertainty\nis the value of nQCD, which affects the timescale of the\ndynamics of the system,32though the error in ξdomi-\n31In our analysis, the dependence on nQCD appears only through\nthe determination of T1in Eq. (6.16) for the linear case. It should\nbe noted that there can be additional effect of nQCD through the\nmodification of the axion field evolution around the epoch of the\nQCD phase transition, which we have not included in our analy-\nsis. Simulations with toy models performed in Ref. [101] suggest\nthat the axion production efficiency is significantly higher than\nthe misalignment estimate for nQCD = 0 compared to nQCD = 7.\nA similar trend was also found in the simulations performed in\nRef. [72], showing that the nQCD = 0 case predicts 25% higher\ndark matter abundance than the misalignment estimate. Such\nan enhancement at small values of nQCD could compensate the\nreduction due to the change in T1and modify the error estima-\ntion for the linear case. On the other hand, in the non-linear\ncase the effect may not be so significant as the linear case, since\nthe gradient energy dominates over the potential energy during\nthe non-linear transient, and the effect of the QCD potential on\nthe axion field dynamics is expected to be subdominant in that\nregime.\n32Since Eq. (6.11) implies that K∝(ξ1ℓ1)1\n2+1\nnQCD+4, the produc-\ntion efficiency becomes larger for smaller values of nQCD. We can28\nnates if instead we adopt ξc=ξsat\nc.\nFinally, in Table II we summarise the values of the pre-\ndicted axion dark matter mass at ln( mr/H) = 70 and its\nuncertainty due to the change in the parameters within\nthe range described above for four assumptions on the\nextrapolation of qandξ. In addition to the errors due\ntoq,ξ,x0, and nQCD, here we also show the effect of\nthe change in fLwithin the range shown in Eq. (6.1),\nthough it is subdominant compared to other uncertain-\nties. Taking the maximum and minimum values from\nTable II (including the difference between two models of\nξc), we end up with the following estimate for the axion\ndark matter mass for the plateau extrapolation of q:\nma= 95–198 µeV, (6.18)\nwhich corresponds to the value of the axion decay con-\nstant, fa≈(2.9–6.0)×1010GeV. On the other hand, if\nwe take the extrapolation along the linear increase model\nofq, the predicted mass reads\nma= 237–450 µeV, (6.19)\nwhich corresponds to fa≈(1.3–2.4)×1010GeV.\nVII. CONCLUSIONS\nIn this paper, we have performed large scale simula-\ntions of global string networks and analysed the spec-\ntrum of axions radiated from them. Our main findings\nare summarised as follows.\n1. We have shown that the attractor behaviour of the\nsystem can be described by a simple differential\nequation (3.1) with an appropriate choice of the co-\nefficient function C(x). This interpretation is com-\nplemented by looking at how the system radiates\nsaxions. It is also found that the spectral index q\nof the instantaneous axion emission spectrum can\nbe over/under estimated when the string density is\nlower/higher than the attractor.\n2. The instantaneous emission spectrum is largely\ncontaminated by the oscillations in the axion en-\nergy density spectrum. These oscillations are ex-\npected on physical grounds due to the misalignment\noscillation of the Fourier mode of the axion field\nafter the corresponding mode enters the horizon,\nor the production of high momentum modes due\nto the parametric resonance effect induced by the\nsee this trend in the upper panel of Fig. 30, where the upper dot-\nted lines correspond to the case with nQCD = 7. However, this\nenhancement is overwhelmed by the shift in T1, and the result-\ning dark matter mass becomes smaller for nQCD = 7, as shown\nby the lower orange or gray dotted line in the bottom panel of\nFig. 30.oscillating radial field when the momentum of the\ncorresponding mode becomes comparable to mr/2.\nThe estimates of qcan be biased unless such oscil-\nlations are properly handled.\n3. We have observed that the spectra of axions and\nsaxions are distorted by discretisation effects, which\nbiasqtoward larger values. The effects blow up\nrapidly at large ℓ= ln( mr/H). We have performed\nthe global fits to extract those effects and charac-\nterised the evolution of qin the continuum limit in\nterms of some simple functions of ℓ. It turns out\nthat several possibilities on the ℓdependence of q\ncan fit the data, including the scenario where qap-\nproaches q≈1 as 1 /ℓ2at large ℓ(plateau model),33\nand the scenario where qincreases linearly with ℓ\n(linear increase model).\n4. We have found that the turnaround in the evolu-\ntion of Faround the horizon crossing depends on\nthe string density parameter ξ, which is well de-\nscribed by the function given by Eq. (6.3) with pa-\nrameters determined as Eq. (6.4). This dependence\nonξremarkably agrees with our expectation on the\nbehaviour of the IR peak of the spectrum, x0∝√ξ.\nBy combining the numerical results, we have per-\nformed the extrapolation to large ℓand estimated the\nprediction for the axion dark matter mass. The predic-\ntion basically differs according to the assumption on how\nqdepends on ℓ: For the plateau model it is given by\nEq. (6.18), while for the linear increase model it is given\nby Eq. (6.19). Since both models fit the data almost\nequally well, we expect that yet other models which lead\nto a value of qin-between two models could also be al-\nlowed.34Therefore, we assess the uncertainty in the dark\nmatter mass prediction at the difference between the re-\nsults of two different ways of extrapolation:\n95µeV<∼ma<∼450µeV. (7.1)\nThe lower part of the mass range obtained by the extrap-\nolation with q≈1 will be probed by future haloscope\nexperiments, including MADMAX [102], ALPHA [103],\nand ORGAN [104]. On the other hand, accessing the\n33We note that there are several heuristic arguments that show a\npreference for q<∼1 in the literature. The scenario with q=\n1 was indeed claimed a few decades ago in Refs. [51, 53, 61].\nMore recently, Ref. [95] provided intuitive explanation on why\nwe should expect q≤1. The authors of Ref. [71] also suggested\nan interpretation to have q= 1 by focusing on the distribution\nof string loops. Our result that the plateau model shows a good\nfit to the data could be compatible with these arguments, but\nwe cannot exclude the other possibility that qbecomes greater\nthan 1 at large ℓin the current analysis.\n34For instance, we found that the model q=q0+q1/ℓalso gives\na good fit to the data (see Table I), and the extrapolation with\nthat model should lead to a value which lies between the range\nshown in Eq. (7.1), as discussed in Sec. VI C.29\n20 30 40 50 60 70\nln(mr/H)0255075100125150175200K=nstr\na(t0)/nmis\na(t0)q=q0+q1/lscript(ξlin)\nq=q0+q1/lscript(ξsat)\nq=q0+q1//lscript2(ξlin)\nq=q0+q1//lscript2(ξsat)Uncertainty in q\nUncertainty in ξ\nUncertainty in x0\nUncertainty in nQCD\n20 30 40 50 60 70\nln(mr/H)0100200300400500ma[µeV]q=q0+q1/lscript(ξlin)\nq=q0+q1/lscript(ξsat)\nq=q0+q1//lscript2(ξlin)\nq=q0+q1//lscript2(ξsat)Uncertainty in q\nUncertainty in ξ\nUncertainty in x0\nUncertainty in nQCD\nFIG. 30: The axion production efficiency K(left) and corresponding dark matter mass (right) obtained by extrapolating\nnumerical results to larger values of ln( mr/H). Different colours represent four different assumptions on the extrapolation of\nqandξ:q=q0+q1/ℓ2with ξc=ξlin\nc(blue), q=q0+q1ℓwith ξc=ξlin\nc(orange), q=q0+q1/ℓ2with ξc=ξsat\nc(black), and\nq=q0+q1ℓwith ξc=ξsat\nc(gray). For each type of extrapolation, uncertainties due to the change in the parameters of q,ξ,x0,\nandnQCD(see text for details) are shown by the region enclosed by solid, dashed, dash-dotted, and dotted lines, respectively.\nTABLE II: Axion dark matter mass predicted at ln( mr/H) = 70 and its error budget. For each extrapolation of q(plateau\nextrapolation with q=q0+q1/ℓ2and linear extrapolation with q=q0+q1ℓ), we consider two models of ξc.\nParameterma[µeV] (Plateau extrapolation) ma[µeV] (Linear extrapolation)\nξlin\nc ξsat\nc ξlin\nc ξsat\nc\nq 141–198 102–144 439 311\nξ 167–173 95–151 431–447 237–381\nx0 149–197 111–137 429–450 305–316\nfL 168–172 122–125 435–443 308–313\nnQCD 138–175 101–127 379–449 268–317\nhigher end of the mass range ma∼450µeV obtained by\nthe extrapolation with q >1 would remain a challenge\neven for future experiments.\nNote that the axion dark matter mass (7.1) obtained\nby extrapolating our simulation results to large ℓdoes\nnot agree with the value ma= 26.2±3.4µeV obtained in\nRef. [66], which made use of a different method to sim-\nulate high-tension strings directly by introducing extra\nfield degrees of freedom [82]. Understanding the origin\nof such a discrepancy should be a crucial step to improve\nthe robustness of our indirect method to predict the dark\nmatter mass, or to find any unaccounted systematics (if\nexists) in the extrapolation. In this regard, it would be\ninteresting to calculate the axion emission spectrum pre-\ncisely in the simulation framework of Refs. [66, 82] to\nmake a close comparison between two methods.35\nThe main source of the uncertainty in the axion dark\n35See Ref. [49] for the comparison of direct and indirect simulation\nmethods on the prediction for the formation of axion miniclus-\nters.matter mass shown in Eq. (7.1) is the different scenario\non the value of qat large ℓ. In order to reduce the uncer-\ntainty, it is thus important to see if qactually continues\nto increase or saturates to q≈1 when we further increase\nthe value of ℓwith suppressing the discretisation effects.\nTo this end, a dedicated study by using some advanced\nnumerical scheme that allows for even larger dynamical\nranges [such as the AMR framework [105, 106]] is war-\nranted. We leave it as the subject of our future work.\nAcknowledgments\nPart of this research was supported by the Munich In-\nstitute for Astro- and Particle Physics (MIAPP) which\nis funded by DFG under Germany’s Excellence Strategy\n– EXC-2094 – 390783311. The work of K.S. is supported\nby Leading Initiative for Excellent Young Researchers\n(LEADER), the Ministry of Education, Culture, Sports,\nScience, and Technology (MEXT), Japan. This article\nis based upon work from COST Action COSMIC WIS-\nPers CA21106, supported by COST (European Coopera-\ntion in Science and Technology). The work of J.R., A.V.30\nand M.K. is supported by Grants PGC2022-126078NB-\nC21 funded by MCIN/AEI/ 10.13039/501100011033 and\n“ERDF A way of making Europe” and Grant DGA-\nFSE grant 2020-E21-17R Aragon Government and the\nEuropean Union - NextGenerationEU Recovery and Re-\nsilience Program on ‘Astrof´ ısica y F´ ısica de Altas En-\nerg´ ıas’ CEFCA-CAPA-ITAINNOVA. Additionally, the\nwork of M.K. is supported by the Government of Arag´ on,\nSpain, with a PhD fellowship as specified in ORDEN\nCUS/702/2022. A.V. is further supported by AEI\n(Spain) under Grant No. RYC2020-030244-I / AEI /\n10.13039/501100011033. Computations were performed\non the HPC system Raven and Cobra at the Max Planck\nComputing and Data Facility. This work was partly\nachieved through the use of SQUID at the Cybermedia\nCenter, Osaka University.\nAppendix A: Code and simulation setups\nIn this work, we utilise the Jaxions code,36which is\na massively parallel code to study the evolution of the\nPQ field in a FRW background, originally developed in\nRef. [68]. This appendix is devoted to the description of\nsome details of the code and simulation setups.\nFor the study of the axion production from strings, one\nhas to solve the following classical equation of motion\nderived from the Lagrangian (2.1),\n¨ϕ+ 3H˙ϕ−1\nR2∇2ϕ+λϕ\u0000\n|ϕ|2−f2\na\u0001\n= 0. (A.1)\nIn the numerical study, it is convenient to define the\nrescaled variables as\nϕ→ϕ\nfaR\nR1, xi→R1H1xi, τ→R1H1τ, (A.2)\nwhere xiare comoving spatial coordinates, τis the con-\nformal time, H1is the Hubble parameter characterising\na reference timescale of the system [see Eq. (A.5)], and\nR1is the scale factor at that time. If we assume the ra-\ndiation dominated Universe ( R∝τ) and use the rescaled\nvariables defined above, we can simplify the equation of\nmotion as\nϕττ− ∇2ϕ+¯λϕ(|ϕ|2−τ2) = 0 , (A.3)\nwhich is used in the numerical scheme implemented in\ntheJaxions code. Note that the equation of motion can\nbe characterised by only one free parameter,\n¯λ≡λf2\na\nH2\n1=m2\nr\n2H2\n1. (A.4)\nA natural choice for the typical timescale of the system\nis the time at which the axion mass starts to be relevant\n36https://github.com/veintemillas/jaxionsdue to the QCD effects, H1=ma(T1). This amounts\nto [68]\nH1≃3.45×10−3µeV\u0012ma\n50µeV\u00130.338\n, (A.5)\nwhere the exponent is found numerically by using the\nlattice QCD output of Ref. [96]. Substituting Eq. (A.5)\ninto Eq. (A.4), we find that the realistic value of the ¯λ-\nparameter should be\n¯λ≃1.1×1057λ\u0012ma\n50µeV\u0013−2.676\n. (A.6)\nIt is not possible to perform simulations for such a large\nvalue of ¯λ, since the ratio of the lattice spacing, which\nis at most of order the string core radius ∼m−1\nr, to the\nsize of the simulation box, which is at least of the order\nof the Hubble radius ∼H−1\n1, is limited by the comput-\ning resources. This fact requires an adequate treatment\nto extrapolate the numerical results to extremely large\nvalues of ¯λ.\nThe equation of motion (A.3) is solved numerically by\ndefining the PQ field on a static cubic lattice with peri-\nodic boundary conditions. The simulation box is given by\na constant comoving volume with size L3, which means\nthat the physical size of the lattice spacing a≡R(t)L/N\nincreases proportionally to the scale factor R(t), where\nNis the number of lattice sites per dimension. Since\nthe width of the string core ∼m−1\nr∝(√\nλfa)−1is con-\nstant, the ratio of the physical lattice spacing to the\nstring core width, mra, increases with time, which leads\nto serious discretisation effects as described in Sec. V.\nThe effects may be alleviated by artificially replacing ¯λ\nby (R/R 1)−2¯λsuch that mraremains constant (the PRS\nmethod). In this work, we perform both types of simula-\ntions: the physical type, where ¯λis constant throughout\nthe simulation, and the PRS type, where ¯λdecreases as\n∝R(t)−2.\nThe Laplacian is reproduced by building the stencil\ninvolving Ngneighbouring points around a given grid\npoint labeled by indices i= (ix, iy, iz),\n(∇2ϕ)i=1\nδ2X\nu=x,y,zNgX\nn=1Cn(ϕi+nnu+ϕi−nnu−2ϕi),\n(A.7)\nwhere δ=L/N andnu(u=x, y, z ) are unit vectors\nrepresenting displacements of one lattice spacing in u-\ndirection. In Jaxions , the above discretisation scheme is\nimplemented up to Ng= 4, whose coefficients are sum-\nmarised in Table III. The effect of different choices of Ng\nis discussed in Sec. V A.\nThe time evolution of the PQ field ϕis computed by\nconverting the second-order differential equations (A.3)\ninto a couple of first-order equations for ϕandϕτ. For the\ntime propagation we adopt the fourth-order McLachlan-\nAtela (Runge-Kutta-Nystr¨ om) method [107], which is a31\nTABLE III: Coefficients for Laplacian stencils\nNgC1 C2 C3 C4\n1 1 0 0 0\n2 16 /12−1/12 0 0\n3 3 /2−3/20 1/90 0\n4 8 /5−1/5 8/135−1/560\nhighly accurate, explicit method optimized for the system\ndescribed by separable Hamiltonian with quadratic ki-\nnetic energy. For the size of the time step, we use Rdτ=\n1.5/ωmax, where ωmax=p\nm2r+k2∗/R2with k2\n∗= 12/δ2\nis the characteristic frequency of the fastest mode.37This\namounts to dτ= 1.5δ/p\n(mra)2+ 12 <0.433δfor any\nvalue of mra > 0 and satisfies the Courant condition.\nFor the update of the field variables at each time step,\nwe use advanced vector extensions (AVX512), which can\nprocess multiple sets of data simultaneously, in addition\nto the standard MPI/OpenMP parallelization. Further-\nmore, the loops for the field update are broken up into\nsub-blocks, whose size is adjusted to the cache size of the\nprocessor. Since the cache size differs according to the\nhardware, the program executes a function that tunes the\nblock size to find an optimal loop tiling at the beginning\nof the simulation. With these functionalities in Jaxions ,\nwe can achieve a substantial speed-up in the calculation\nof the field evolution.\nParameters used in the simulations performed in this\nwork and the number of simulations executed for each\nchoice of the parameters are summarised in Table IV.\nFor each set of parameters, simulations are performed up\nto a conformal time τfshown in the fourth column of\nthe table (in terms of the ratio τf/L). For physical-type\nsimulations, we performed four sets of 40963simulations\nto study discretisation effects due to the Laplacian, and\nfour sets of 30723simulations with different values of ¯λ\nto study discretisation effects due to the resolution of the\nstring core width, in addition to the largest 112643sim-\nulations, which reach ln( mr/H) = 9 .08 at the final time.\nFurthermore, we also performed 30 sets of 20483simu-\nlations with different initial string densities to study the\nattractor behaviour of the system. Other relatively small\nscale simulations (from 10243to 30723) are performed for\nthe studies of the initial conditions (Appendix B) and the\nfinite volume effects (Appendix E 1). Except for the sim-\nulations with N= 2048 and ¯λ= 6400, all physical-type\nsimulations begin with the initial string density corre-\nsponding to the attractor, ξℓ=3= 0.3 (see Sec. III for\n37Strictly speaking, k2\n∗= 12 /δ2corresponds to the maximum\nmomentum for the Laplacian with Ng= 1. For general\nNg, the maximum momentum takes a larger value given by\nk2\n∗= 12PNg\nn=1Cn/δ2. Nevertheless, we use the same criterion\n(k2\n∗= 12/δ2) to determine dτeven for the case with larger Ng\nin the code.identification of the attractor).\nAs mentioned above, the quantity mra, which can be\nregarded as a measure of the string core resolution, in-\ncreases with time for physical-type simulations. In terms\nof the simulation parameters, it is given (in rescaled\nunits) by,\nmra=(2¯λ)1/2L\nNτ=(2¯λ)1/4L\nNeℓ/2. (A.8)\nFigure 31 shows the evolution of mrafor parameters of\nthe physical-type simulations used in the main analy-\nsis. As explicitly shown in Eq. (A.8), the dependence\nonℓ= ln( mr/H) is exponential, and the resolution gets\nworse rapidly at large ℓ. Obviously, the best case is the\nsimulation with N= 11264, though the value of mra\nreaches 1.25 at the end of the simulation.\nFor PRS-type simulations the value of mraremains\nconstant. Hence for such cases we use mraas an input\nparameter instead of ¯λ. As shown in Table IV, we per-\nformed PRS-type simulations with 81923lattice sites for\n6 different choices of mra. In addition, we performed\nthree sets of simulations (10243, 20483, and 40963lattice\nsites) with mra= 1 for the study of the finite volume\neffects (Appendix E 1) and one 20483simulation with\nmra= 1 for the check of the systematics associated with\nthe masking procedure (Appendix C 3). For smaller val-\nues of mrathe resolution of the string core improves, but\nthe maximum reachable value of ln( mr/H) decreases.\n3 4 5 6 7 8 9\nln(mr/H)0.00.51.01.52.02.53.0mraN= 11264,¯λ= 195799.0\nN= 4096,¯λ= 25890.8\nN= 3072,¯λ= 14563.6\nN= 3072,¯λ= 32768.0\nN= 3072,¯λ= 64225.3\nN= 3072,¯λ= 114178.0\nN= 2048,¯λ= 6400.0\nFIG. 31: Value of mraas a function of ln( mr/H) for different\nsets of parameters used in the simulations of physical strings.\nFor PRS-type simulations, the initial conditions are\nproduced as random fluctuations obeying Gaussian dis-\ntribution in the Fourier space such that the string den-\nsity parameter at the initial time ( ℓ= 0) becomes\nξℓ=0≃0.011, a value found by extrapolating the curve\nof “attractor” in Ref. [67] into ℓ= 0. Since the conver-\ngence to the attractor in the PRS case is faster than the\nphysical case, we expect that this choice for the initial\ncondition is enough for the study of the attractor in the32\nTABLE IV: Parameters used in the simulations and the number of simulations executed for each choice of them. The list of\nthe number of figures in which the results are used is also shown in the last column.\nTypeaGrid size Laplacian Final time ln( mr/H) Parameter Number of List of figures\n(N3) ( τf/L) at τf simulations\nPhysical 1126434-neighbours 0.625 9.08 ¯λ= 195799 20 7,8,9,10,11,19,20,21,22,23,24,25,43,46\nPhysical 409631-neighbour 0.625 8.07 ¯λ= 25890 .8 30 15,16\nPhysical 409632-neighbours 0.625 8.07 ¯λ= 25890 .8 30 15,16\nPhysical 409633-neighbours 0.625 8.07 ¯λ= 25890 .8 30 15,16\nPhysical 409634-neighbours 0.625 8.07 ¯λ= 25890 .8 30 14,15,16,37,38\nPhysical 307234-neighbours 0.5 7.34 ¯λ= 14563 .6 30 8,21,22,23,25,36\nPhysical 307234-neighbours 0.5 7.74 ¯λ= 32768 30 8,21,22,23,25\nPhysical 307234-neighbours 0.5 8.08 ¯λ= 64225 .3 30 8,21,22,23,25\nPhysical 307234-neighbours 0.5 8.37 ¯λ= 114178 30 8,21,23\nPhysical 204834-neighbours 0.55 7.12 ¯λ= 6400 30 ×30b3,4,5,6,7,26,27,44\nPhysical 102434-neighbours 0.5 6.23 ¯λ= 1600 30 ×4c32,33,34,35\nPhysical 307234-neighbours 0.458367 7.5 ¯λ= 28571 .2 30 40\nPhysical 256034-neighbours 0.550042 7.5 ¯λ= 13778 .5 30 40\nPhysical 204834-neighbours 0.687552 7.5 ¯λ= 5643 .68 30 40\nPhysical 153634-neighbours 0.916735 7.5 ¯λ= 1785 .69 30 40\nPhysical 102434-neighbours 1.3751 7.5 ¯λ= 352 .73 30 40\nPRS 819234-neighbours 0.55 6.80 mra= 0.2 20 8,18,22,23\nPRS 819234-neighbours 0.55 7.21 mra= 0.3 20 8,17,18,22,23\nPRS 819234-neighbours 0.55 7.72 mra= 0.5 20 8,17,18,22,23\nPRS 819234-neighbours 0.55 8.06 mra= 0.7 20 8,17,18,22,23\nPRS 819234-neighbours 0.55 8.41 mra= 1.0 20 8,17,18,22,23,41,42\nPRS 819234-neighbours 0.55 8.82 mra= 1.5 20 8,17,18,22,23\nPRS 409634-neighbours 0.55 7.72 mra= 1.0 30 41\nPRS 204834-neighbours 0.55 7.03 mra= 1.0 30 41\nPRS 102434-neighbours 0.55 6.33 mra= 1.0 30 41\nPRS 204834-neighbours 0.5 6.93 mra= 1.0 1 39\naFor all physical-type simulations (except for N= 1024, see footnote c) we set the initial time τisuch that the simulation starts from\nln(mr/H) = 3, and for all PRS-type simulations we set τisuch that the simulation starts from ln( mr/H) = 0.\nbFor simulations of physical strings with N= 2048, we chose 30 different initial string densities ranging from ξℓ=3≃0.022 to 0.828 as\nshown in Fig. 3. For each value of the initial string density, we executed 30 simulations with randomly generated initial field configurations.\ncSimulations of physical strings with N= 1024 are performed for the check of initial conditions. For these simulations we used 4 different\nways to generate initial conditions, including a case where the simulation starts from ln( mr/H) = 1. See Appendix B for details. For each\ntype of the initial condition, we executed 30 simulations with randomly generated initial field configurations.\nPRS-type simulations. For physical-type simulations, the\ninitial condition should be chosen much more carefully,\nand we will discuss this issue in details in Appendix B.\nAppendix B: Initial conditions\nThe choice of the initial condition is a tricky part of\nthe simulation, since it is impossible to produce strings in\na physically correct way. Ideally, one should simulate the\nprocess of the PQ phase transition to produce strings,\nbut assuming that it happens thermally at the tempera-\ntureTPQ∼fa(corresponding to the Hubble parameter\nHPQ) in the radiation dominated epoch, a huge scale\nseparation ln( mr/HPQ)∼19 (for mr∼fa∼1010GeV)\nis required. It is still possible to mimic such a thermalinitial condition with artificially small ln( mr/HPQ) [as\ndone in some earlier works [60, 62–64, 69, 71, 77]], but it\nis not straightforward to confirm if the unphysical nature\nof “thermal” fluctuations can settle down within the lim-\nited dynamical range of the simulation. In this work, we\ninstead follow a perspective initiated by Ref. [67], which\nis to start the simulation from the condition closest to\nthe attractor rather than including the dynamics at the\nPQ phase transition, since the latter may be irrelevant\nat the epoch of the QCD phase transition in which we\nare mostly interested.\nIn Sec. III, we have identified the initial string den-\nsity that can be regarded as the attractor. However,\njust tuning the initial string density is not enough to\nget an accurate result. The problem is associated with\nthe systematics induced by the oscillation in the spec-33\ntrum discussed in Sec. IV. There we have seen that the\noscillation in the spectrum is produced through physical\nprocesses, the misalignment oscillation after the horizon\ncrossing and the parametric resonance effect after the\nsaxion mass crossing. On the other hand, these oscilla-\ntions can also be produced by some unphysical effects, as\nwe will see below. One has to choose the initial condi-\ntion that minimises such unphysical oscillations as much\nas possible.\nTo produce an appropriate initial condition, it may\nbe convenient to prepropagate the field configuration be-\nfore the beginning of the (physical) simulation. Namely,\nwe let the fields evolve with some modified (unphysical)\nequation of motion at the early stage of the simulation,\nand switch to the physical evolution at some point. Dur-\ning the prepropagation phase, we modify the equation of\nmotion such that the PQ field quickly settles down into\nthe minimum of the potential, suppressing unphysical\nfield fluctuations. One possible choice for the preprop-\nagation is to use the PRS-type evolution: At the early\ntimes we make a replacement ¯λ→(Ri/R)2¯λ, such that\nthe value of ¯λis matched to the input value at the be-\nginning of the physical evolution ( R=Ri). In addition,\nwe can also prepropagate by having only allowing for an\nevolution in the radial direction ρof the PQ field ϕ=ρeθ,\nwhile the angular direction θis kept frozen (hereafter we\ncall it the ρ-only prepropagation).38In this way, we ex-\npect to reduce the fluctuations in the radial direction in\nthe prepropagation phase.\nTo see how different choices of the initial condition af-\nfect the simulation results, we performed simulations of\nphysical strings with 10243lattice sites and the param-\neter¯λ= 1600. In this study, we use 4 different ways to\nproduce the initial condition: the prepropagation with\nthe PRS-type evolution, ρ-only prepropagation, no pre-\npropagation with the simulation starting from ℓ= 1,\nand no prepropagation with the simulation starting from\nℓ= 3. For the case with the PRS-type prepropagation,\nwe let the prepropagation start from ℓ= 2 and the physi-\ncal evolution from ℓ= 3. For the case with the ρ-only pre-\npropagation, we let the prepropagation start from ℓ= 1\nand the physical evolution from ℓ= 3. In all cases, the\nfield configurations at the very beginning of the simula-\ntion (including the prepropagation phase) are produced\nas random fluctuations obeying a Gaussian distribution\nin Fourier space, with tuning the cutoff in the distribution\nsuch that the string density parameter at ℓ= 3 becomes\nξℓ=3≃0.3. For each choice of the initial condition, we\nexecuted 30 simulations with randomly generated initial\nfield configurations and averaged them.\nThe top panel of Fig. 32 shows the evolution of the\n38The equation of motion for the ρ-only evolution can be imple-\nmented by projecting the equation of motion of ϕinto the radial\ndirection. This can be done by translating the complex space\nofϕ=ρeiθinto cylindrical coordinates ( ρ, θ), and artificially\nsetting the components in the θdirection to zero.string density parameter ξfor different choices of the\ninitial condition. We see that ξbecomes larger for the\ncases with the ρ-only prepropagation and without pre-\npropagation starting from ℓ= 3 compared to those with\nthe PRS-type prepropagation and without prepropaga-\ntion starting from ℓ= 1. This difference can be easily\nunderstood by looking at the evolution of the number\nNplaqof plaquettes pierced by strings,39which is shown in\nthe bottom panel of Fig. 32. For the case with the ρ-only\nprepropagation, the angular mode is frozen, and hence\nNplaqremains almost constant during the prepropaga-\ntion phase. Since the physical evolution starts from such\na frozen configuration at ℓ= 3, it takes some time for the\nsystem to reduce Nplaqand reach the attractor. Such a\ndelay in the annihilation of strings results in higher string\ndensities after the prepropagation phase. It is not sur-\nprising that the simulations starting from ℓ= 3 without\nprepropagation show a similar behaviour, since in that\ncase we just put strings in the simulation box with the\nappropriate number of Nplaqbut do not allow them to\nhave a velocity to annihilate quickly enough to follow\nthe attractor. On the other hand, for the case with the\nPRS-type prepropagation, strings acquire an appropriate\nvelocity to annihilate during the prepropagation phase,\nand exhibit a smooth transition to the physical evolution.\nThe unphysical nature of the initial conditions is most\nnoticeable in the time evolution of one Fourier mode of\nthe axion field, which is shown in Fig. 33. From the figure\nwe see that the results of the simulations with the ρ-only\nprepropagation and those without prepropagation start-\ning from ℓ= 3 exhibit low frequency oscillations with\nrelatively large amplitudes. These features can be as-\nsociated with the misalignment oscillations of the free\naxion field after the horizon crossing, as their frequency\nroughly corresponds to 2 kin conformal time. In these\nsimulations, the field configurations are frozen at the ini-\ntial time and gradually start to move at ℓ>∼3. During\nsuch a process, the angular component of the PQ field\ntries to realign itself, acquiring large kinetic and gradient\nenergies that enhance the amplitude of the misalignment\noscillation. The appearance of such low frequency oscil-\nlations can be regarded as unphysical effect, since they\noriginate from the initial conditions where the field con-\nfiguration is artificially frozen.\nIn addition to the aforementioned low frequency oscil-\nlations, there are high frequency oscillations in all cases\nplotted in Fig. 33. We find that these oscillatory features\n39We use the statistical method to calculate ξ[65], where the num-\nber of plaquettes pierced by strings is related to ξas\nξ=δ\n6Nplaqτ2\nL3. (B.1)\nWe confirmed that the output of this algorithm reproduces the\nresult of the method of directly computing the string length by\nconnecting points pierced by a string in a cube, which was used\nin Refs. [62–64, 108, 109].34\n1 2 3 4 5 6\nln(mr/H)0.00.10.20.30.40.50.60.70.8\nξPrep.(PRS)\nPrep.(ρonly)\nNo prep.(/lscripti= 1)\nNo prep.(/lscripti= 3)\n1 2 3 4 5 6\nln(mr/H)10−510−410−3Nplaq/N3Prep.(PRS)\nPrep.(ρonly)\nNo prep.(/lscripti= 1)\nNo prep.(/lscripti= 3)\nFIG. 32: Evolution of the string density parameter ξ(top\npanel) and the ratio of the number Nplaqof plaquettes pierced\nby strings to the total number of lattice sites N3= 10243in\nthe simulation box (bottom panel) for different choices of the\ninitial conditions. Blue and orange thick lines correspond to\nthe cases with the PRS-type prepropagation and ρ-only pre-\npropagation, respectively. The thin black line and gray dot-\ndashed line correspond to the cases without prepropagation\nstarting from ℓ= 1 and ℓ= 3, respectively. The coloured\nbands represent statistical uncertainties, and the evolutions\nduring the prepropagation phase are represented by dotted\nlines.\nhave a constant frequency mrin cosmic time, as one pe-\nriod of oscillation looks comparable to the change in mrt\nby 2πwhen plotted as a function of mrtrather than\nkτ. Such high frequency oscillations can be attributed\nto the oscillation in the background PQ field around the\nminimum of the potential. During the simulation, the\nbackground PQ field may have a residual oscillation with\nfrequency mr. This background oscillation shakes the ef-\nfective amplitude of the “axion decay constant” |ϕ|, and\nhence induces oscillations in the overall amplitude of the\naxion energy density. In fact, we have confirmed that\n2 4 6 8 10 12 14\nkτ012345671\nf2aH2∂ρa\n∂logkkL/(2π) = 4.44\nPrep.(PRS)\nPrep.(ρonly)\nNo prep.(/lscripti= 1)\nNo prep.(/lscripti= 3)FIG. 33: Evolution of the energy density of one Fourier mode\nof the axion field specified by kL/(2π) = 4 .44 for different\nchoices of the initial conditions.\nthe phase of the high frequency oscillations in the ax-\nion mode evolution shown in Fig. 33 is aligned with that\nof the oscillation in the spatial average of the absolute\nvalue of the PQ field around |ϕ|=fa, which is shown in\nFig. 34.\n0 50 100 150 200 250\nmrt0.900.920.940.960.981.001.021.04/angbracketleftρ/angbracketright/fa\nPrep.(PRS)\nPrep.(ρonly)\nNo prep.(/lscripti= 1)\nNo prep.(/lscripti= 3)\nFIG. 34: Evolution of the spatially averaged value of ρ=|ϕ|\nfor different choices of the initial conditions. Note that the\nx-axis is given by mrt.\nLooking at Figs. 33 and 34, we see that the amplitude\nof the high frequency oscillation is particularly large in\nthe case where simulations start from ℓ= 1 without pre-\npropagation. This can be understood as follows. In the\ncase of no prepropagation with ℓi= 1, ξtakes a larger\nvalue at early times and relaxes into the attractor until\nℓ<∼3 (see top panel of Fig. 32). The annihilation of such\ndense string networks leads to larger fluctuations in the\nradial component of the PQ field, and hence gives rise\nto a larger amplitude for the residual oscillation of its\naveraged value.35\nFinally, we compare the energy density spectrum of\naxions among different choices of the initial conditions in\nFig. 35. We see that the spectra obtained from the sim-\nulations with the ρ-only prepropagation and those start-\ning from ℓ= 3 without prepropagation exhibit relatively\nlarge oscillatory features at lower momenta. These can\nbe regarded as artificial features caused by the process\nof the realignment of frozen initial configurations. We\ncan also see that the spectrum is weighted towards lower\nmomenta in those cases. On the other hand, such oscilla-\ntions at lower momenta are less pronounced than in the\nother two cases. In the case where simulations start from\nℓ= 1 without prepropagation, there are large oscillatory\nfeatures at higher momenta. They can be interpreted as\nthe consequence of the parametric resonance effect in-\nduced by the saxion field oscillating with a larger ampli-\ntude due to the imperfect initial condition of over-dense\nstrings.\n101102\nk/(RH)1001011\nf2aH2∂ρa\n∂logkln(mr/H) = 5.0\nPrep.(PRS)\nPrep.(ρonly)\nNo prep.(/lscripti= 1)\nNo prep.(/lscripti= 3)\nFIG. 35: Energy density spectrum of axions measured at\nln(mr/H) = 5 for different choices of the initial conditions.\nIn summary, we have found that there are two kinds of\nunphysical effects that can be caused by improper choices\nof initial conditions. One is the enhancement in the am-\nplitude of the 2 k-oscillation of the axion field, which oc-\ncurs when the field configurations are frozen (or strings\ndo not have appropriate velocities) at the initial time.\nThe other is the oscillation with the frequency mr, which\ncan be associated with a large amplitude oscillation of the\nsaxion field induced by an over-dense initial condition.\nThese artificial features appear to be minimised when we\nproduce the initial condition with the PRS-type preprop-\nagation, and we adopt it (PRS-type prepropagation from\nℓ= 2) for all physical-type simulations performed in this\nwork, except for those presented in this appendix.\nAppendix C: Masking method\nIn the numerical analysis, we calculate the spectra of\naxions and saxions by using the Fourier transform ofthe time derivative of the corresponding field compo-\nnents. Typically, we use twice their kinetic energy den-\nsities, ρa= 2ρa,kin=⟨˙a2⟩andρr= 2ρr,kin=⟨˙r2⟩, be-\ncause the gradients of the axion field are strongly con-\ntaminated from the intrinsic gradients surrounding the\nstrings. Moreover, it is much more economical than eval-\nuating gradients. In terms of their kinetic energy densi-\nties, the spectra of axions and saxions are given by\n∂ρa\n∂logk=k3\n2π2L3ZdΩk\n4π|˜˙a(k)|2, (C.1)\n∂ρr\n∂logk=k3\n2π2L3ZdΩk\n4π|˜˙r(k)|2, (C.2)\nwhere ˜˙a(k) and ˜˙r(k) are the Fourier transform of ˙ a(x)\nand ˙r(x), respectively, L3is the comoving volume of the\nsimulation box, andR\ndΩkis the integration over the\nsolid angle in the Fourier space.\nThese kinetic energies can be overestimated around the\nregion close to the string core, and it was argued that the\ninclusion of the axion field around such a region can con-\ntaminate the spectrum of radiated axions [60, 62]. In\nthis appendix, we revisit this issue and study how differ-\nent methods to remove the data around the string core\naffect the calculation of energy densities and spectra of\nradiated fields.\n1. Mask field\nTo mitigate the contamination from the string core,\nwe may replace the time derivative of the axion or saxion\nfield with the masked field,\n˙Xmask(x) =M(x)˙X(x), (C.3)\nwhere X(x) =a(x),r(x) denotes either the axion field\nor the saxion field, and the mask field M(x) vanishes\naround the string core and becomes unity outside. Dif-\nferent choice of M(x) would affect the shape of the result-\ning spectrum of the field. Here we describe some possible\nchoices of the mask field implemented in our simulation\ncode.\nA simple choice is to use the fact that the value of the\nradial field |ϕ|approaches to zero inside the string core:\nM(x) =\u0012|ϕ(x)|\nfa\u0013k\n, (C.4)\nwhere kis a positive number. This function is inspired\nby Ref. [67], in which k= 1 was used. One could choose\nother values of k, which affects the profile of the masked\nregion. A larger value of kwould effectively increase the\nradius of the masked region and make the mask more\nabrupt.\nAnother possible choice of M(x) is the top-hat mask,\nwhich is obtained as follows. We select all four corners of\na plaquette pierced by a string. These points are closer36\nto the string core center than the lattice spacing δ=\nL/N. We create an anti-mask field, W0(x), where all\nthe points of a plaquette are valued as W0= 1, and the\nrest as W0= 0. This would be a pretty sharp top-hat\ndistribution corresponding to a masking length ∼δ(in\nthe comoving coordinate). We now apply a Gaussian\nfilter to this distribution, with a smoothing length ˜l,\nW1(x) =Zd3p\n(2π)3e−ip·xe−|p|2˜l2/2Z\nd3x′W0(x′)eip·x′.\n(C.5)\nApplied to W0(x) =δ3·δ(3)(x−x0) for a point x0pierced\nby a string in a continuum limit, it gives,\nWpoint\n1(x) =δ3Zd3p\n(2π)3e−|p|2˜l2/2eip·(x0−x)\n=\u0012δ√\n2π˜l\u00133\ne−|x0−x|2\n2˜l2, (C.6)\neffectively distributing the value of W0(x∼x0) = 1 in a\nregion of radius ∼˜l.\nA collection of plaquette vertices along a string forms\na sort of one dimensional distribution of points, that will\nbe smoothed to a cylinder of radius ∼˜lwhen applying\nthe same filter. One can compute the effective value of\nW1at the desired distance as a function of ˜l.\nAssuming δ≪˜l, we can use the continuum description,\nand we only have to integrate (C.6) with x0along a line\nwith a point number per unit length pL(or points per\nlattice spacing δ,pδ). As a function of the comoving\ndistance to the center of the string core rc, we have\nWstring\n1≃pLZ\ndL\u0012δ√\n2π˜l\u00133\ne−L2+r2\nc\n2˜l2\n=pδ\u0012δ√\n2π˜l\u00132\ne−r2\nc\n2˜l2. (C.7)\nTherefore, choosing a filter with ˜l=rc, the points satis-\nfying\nW1(x)> pδ\u0012a√\n2πrmask\u00132\ne−1\n2 (C.8)\nare closer than rmask to the string core, where rmask =\nRrcis the physical radius.\nIn the discrete case, rather than computing it, we can\ncalibrate it by direct calculation. We build the mask by\nchoosing the points where W1has values above a critical\nvalue, which was calibrated to be:\n(W1)c=pc\u00121√\n2πlc\u00132\ne−n2\n2l2c, lc=cn/2, (C.9)\nwith pc= 2.5 and c= 1.25, where nis a distance from\nthe string core (in lattice units) which we want to mask.\nThe top-hat mask is thus defined as\nM(x) =(\n0 (inside) for W1>(W1)c,\n1 (outside) for W1<(W1)c.(C.10)The algorithm to build the top-hat mask based on the\nabove procedure is implemented in Jaxions , which en-\nables us to mask the fields for arbitrary values of rmask.\n2. Energy density and spectrum\nFrom the data of the spatial distribution of the PQ\nscalar ϕ(x) and ˙ϕ(x) in the simulation box, we can com-\npute the energy densities of axions and saxions. Sub-\nstituting the decomposition (2.2) into the kinetic energy\ndensity of the PQ field, we have1\n2|˙��|2=1\n2˙r2+1\n2|ϕ|2˙θ2,\nand hence it is natural to define the kinetic energy den-\nsities of radial and angular fields as\nρr,kin=1\n2˙r2, ρ a,kin=1\n2|ϕ|2˙θ2(C.11)\nAfter obtaining the top-hat mask, we can compute the\nmasked energy density as an average over all masked\npoints,\nρmask\ni=2\nN3−Nmask X\nxρi,kin(x)−X\ninsideρi,kin(x)!\n,\n(C.12)\nfori=r, a, where Nmaskis the number of masked points,P\ninsidedenotes the sum over the points inside the masked\nregion (i.e. points on which M(x) = 0), and the overall\nfactor 2 represents the fact that we estimate the energy\ndensity based on twice of the kinetic energy density in\norder to avoid including the long range gradient energy,\nwhich should be regarded as a part of the energy density\nof strings.\nFigure 36 shows the ratio of the masked energy den-\nsityρmask\ni to that without the mask ρi,0for different\nchoices of the radius rmaskof the top-hat mask, obtained\nfrom simulations of physical strings. Here we computed\nenergy densities with 20 different choices of the mask\nradius, ranging from rmask =m−1\nrtormask = 20 m−1\nr.\nWe see that the overall suppression due to the mask be-\ncomes weaker for larger values of ln( mr/H) both for the\naxion and saxion energy densities. This is simply be-\ncause the fraction of the masked points (or the number\nof grid points pierced by strings) in the simulation box\ngets smaller at late times. From the figure we can also\nsee that there is a noticeable difference between the en-\nergy density of axions and that of saxions. For axions,\nthe ratio ρmask\na/ρa,0continues to decrease with increasing\nrmask, while for saxions the suppression becomes milder\nforrmask>∼5m−1\nr. This fact implies that there is a rather\nclear distinction between the kinetic energy of the mov-\ning string core and that of radiations for the radial field,\nwhile it is hard to find such a distinction for the angular\nfield.\nWe can also compute the masked energy spectrum as\n∂ρmask\n∂logk=k3\n2π2L3ZdΩk\n4π|˜˙Xmask(k)|2, (C.13)37\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nmrrmask0.600.650.700.750.800.850.900.951.00ρmask\na/ρa,0\nphysical\nln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nmrrmask0.600.650.700.750.800.850.900.951.00ρmask\nr/ρr,0\nphysicalln(mr/H) = 5.0\nln(mr/H) = 6.0\nln(mr/H) = 7.0\nFIG. 36: The ratio of the masked energy density to that\nwithout the mask for axions (top panel) and saxions (bottom\npanel) plotted as functions of the radius rmask of the masked\nregion (in the unit of mr). The different colours and markers\ncorrespond to the different values of ln( mr/H). The results\nare obtained from simulations of physical strings with 30723\nlattice sites and the parameter ¯λ= 14563 .6. Each marker\nrepresents the average of 30 simulations, where the statistical\nuncertainty is negligibly small.\nwhere˜˙Xmask(k) is the Fourier transform of the masked\nfield (C.3). Figure 37 shows a comparison between dif-\nferent choices of the mask field. We see that the differ-\nence is most pronounced at higher momenta, and the en-\nergy density spectrum (top panel of Fig. 37) is more sup-\npressed for larger rmaskof the top-hat mask or larger kof\nthe|ϕ|kmask. For the top-hat mask with rmask= 4m−1\nr,\nthere is a substantial suppression even at lower momenta,\nk/R<∼mr/2. Such a suppression leads to an underesti-\nmate of the axion energy density, and is compatible with\nthe trend observed in the top panel of Fig. 36. We also\nobserve that the amplitude of the instantaneous emis-\nsion spectrum F(x) increases at intermediate momenta\nfor larger values of rrmask . This can be understood by\nnoting that the effect of the mask gets weaker at late\ntimes as shown in Fig. 36, which can result in a relativeincrease in the energy density for certain modes. Such an\nartificial increase due to the mask is more pronounced for\nlarger values of rrmask , leading to higher values of F(x).\nThe impact of the different choices of the masking\nmethod on the estimation of the spectral index qis shown\nin Fig. 38. We see that qtakes larger values for results\nwithout the mask, and smaller values when we increase\nthe radius of the masked region. This trend is consis-\ntent with the above observation that the amplitude of\nF(x) becomes larger at intermediate momenta when we\nincrease the value of rmask. We can also see that the dif-\nference in qbecomes less pronounced at larger values of\nln(mr/H). This implies that the effect of the masking of\nthe spectrum is relevant only at higher momenta, whose\nrelative importance becomes diminished as the IR and\nUV parts of the spectrum are sufficiently separated at\nlarge ln( mr/H).\nSince the different choice of rmask is likely to modify\nthe axion spectrum at intermediate momenta, the ap-\nplicability of the top-hat mask to the estimation of the\naxion spectrum seems obscure. In the main analysis of\nthis paper, we use the |ϕ|mask [Eq. (C.4) with k= 1] as\nour fiducial choice for the calculation of the energy den-\nsity and spectrum of axions, since it is equivalent to the\n|ϕ|2factor in Eq. (C.11), being the natural normalization\nfactor for the energy density of the angular field. On the\nother hand, for the case of saxions, it is clear from the\nbottom panel of Fig. 36 that most of the contribution of\nthe string core can be screened for rmask>∼5m−1\nr, and\nwe use the top-hat mask with rmask= 5m−1\nrfor the cal-\nculation of the energy density and spectrum of saxions\nin our main analysis.\n3. Comment on the correction matrix method\nThe spectrum built from the masked field ˙Xmask(x)\nmay still be imperfect as it contains “defects” at the\nmasked points. It was argued that they can be cor-\nrected by using the Pseudo Power Spectrum Estimator\n(PPSE) [62, 110]. Here we investigate its effect on the\nspectrum and show that the correction does not play an\nimportant role in the large scale simulations.\nFor a spectrum of a masked field Pmask\nX(k)≡RdΩk\n4π|˜˙Xmask(k)|2, its PPSE is given by [110]\nPX(k) =Zdk′k′2\n2π2M−1(k, k′)⟨Pmask\nX(k′)⟩,(C.14)\nwhere ⟨. . .⟩denotes an ensemble average\nover many realisations, M−1(k, k′) satisfiesRdk′k′2\n2π2M−1(k, k′)M(k′, k′′) = (2 π2/k2)δ(k−k′′),\nand the matrix M(k, k′) can be built from the Fourier38\n101102103\nk/(RH)1001011\nH2f2a∂ρa\n∂logk\nmr\n2Hphysical\nNo mask\n|φ|\n|φ|2\nTop hat (mrrmask= 1)\nTop hat (mrrmask= 2)\nTop hat (mrrmask= 4)\n101102103\nk/(RH)0.60.70.80.91.01.11.21.31.4Ratio to|φ|mask\nmr\n2HphysicalNo mask\n|φ|2\nTop hat (mrrmask= 1)\nTop hat (mrrmask= 2)\nTop hat (mrrmask= 4)\n101102103\nk/(RH)10−310−210−1100F(x)\nmr\n2Hphysical\nNo mask\n|φ|\n|φ|2\nTop hat (mrrmask= 1)\nTop hat (mrrmask= 2)\nTop hat (mrrmask= 4)\n101102103\nk/(RH)0.60.70.80.91.01.11.21.31.4Ratio to|φ|maskphysical\nNo mask\n|φ|2\nTop hat (mrrmask= 1)\nTop hat (mrrmask= 2)\nTop hat (mrrmask= 4)mr\n2Hln(mr/H) = 7.0\nFIG. 37: The axion energy density spectrum (top left) and instantaneous emission spectrum (bottom left) at ln( mr/H) = 7\nobtained from simulations of physical strings with 40963lattice sites for different choices of the mask method. The results\nwithout the mask [ M(x) = 1 for all x] are shown by black dashed lines, those with the |ϕ|kmask [Eq. (C.4)] for k= 1 and\nk= 2 are shown by blue solid lines and orange dotted lines, respectively, and those with the top-hat mask [Eq. (C.10)] for three\ndifferent choices of rmask are shown by coloured dash-dotted lines. The ratio of the energy density spectrum (instantaneous\nemission spectrum) for each choice of the mask to that for the |ϕ|mask is also shown in the top right (bottom right) panel.\nThe coloured bands represent statistical uncertainties, and the momentum corresponding to k/R=mr/2 is marked with dark\nblue ticks.\ntransform of the mask field ˜M(k),\nM(k, k′) =1\nL3Zdk′′k′′2\n2π2ZdΩk′′\n4π⟨|˜M(k′′)|2⟩J(k, k′, k′′),\n(C.15)\nJ(k, k′, k′′) =ZdΩk\n4πZdΩk′\n4π(2π)3δ(3)(k+k′+k′′).\n(C.16)\nIn practice, we may calculate M(k, k′) without taking the\nensemble average of |˜M(k′′)|2, and expect that Eq. (C.14)\nholds as an approximation.\nAn important feature of the correction matrix M−1is\nthat it compensates the suppression of the masked spec-\ntrum at lower k. Let us consider the k→0 limit of\nthe masked field˜˙Xmask(k). Introducing the new anti-\nmask field W2(x)≡1−M(x), which becomes unity\ninside the masked region and vanishes outside it, we\nhave˜˙Xmask(k→0) =R\nd3x(1−W2(x))˙X(x). SinceW2(x) is relevant only inside the masked region, the am-\nplitude of the lower modes is of order˜˙Xmask(k→0)∼\n(1−Nmask/N3)˜˙X(k→0). Hence we expect that the\nlower modes are suppressed compared to the bare ones\nby a factor given by the masked volume,\nPmask\nX(k→0)∼\u0012\n1−2Nmask\nN3\u0013\nPX(k→0).\nSince Eq. (C.16) gives J(k, k′, k′′) = π2/(kk′k′′) for\n|k′−k′′| ≤k≤k′+k′′, which implies that the con-\ntributions from higher kare suppressed by some inverse\npower of k, the dominant contribution to the k→0 mode\nisM(0,0)∼L−6⟨|M(k′′)|2⟩k′′→0J(0,0,0)∼L−3(1−\n2Nmask/N3). Therefore, by evaluating the corrected\nspectrum as Eq. (C.14), we expect a cancellation of the\nsuppression effect on the masked spectrum.\nFigure 39 shows a comparison between the axion spec-\ntrum built from the top-hat mask and that corrected39\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.60.70.80.91.01.11.2\nq\nphysicalNo mask\n|φ|\n|φ|2\nTop hat (mrrmask= 1)\nTop hat (mrrmask= 2)\nTop hat (mrrmask= 4)\nFIG. 38: Evolution of the spectral index qof the instanta-\nneous emission spectrum for different choices of the mask\nmethod, obtained from simulations of physical strings with\n40963lattice sites. The coloured bands represent the error\ninduced by changing the parameter σfilterfor the filtering pro-\ncedure to calculate F[Eq. (D.5)] in addition to statistical un-\ncertainties.\nby the matrix M−1(k, k′). As expected, there is a sup-\npression in the uncorrected spectrum at low k, and the\namount of the suppression scales according to Nmask/N3.\nIf we take a larger value of rmask, the value of Nmask\nincreases, and the spectrum becomes more suppressed.\nHowever, the difference between the corrected spectrum\nand the uncorrected spectrum decreases with time, since\nthe number of plaquettes pierced by strings, which de-\ntermines the number of masked points Nmask, becomes\nsmaller at late times.\nFrom the above arguments, we expect that the “im-\nperfectness” of the masked spectrum Pmask\nX(k) is of or-\nder 2Nmask/N3. Since we are interested in measuring the\naxion spectrum at late times of the simulation where the\nstring length becomes comparable to the size of the simu-\nlation box, a reasonable estimate is that Nmask∼ O(N),\nand hence 2 Nmask/N3∼ O(N−2) in the relevant time\nperiod. Therefore, we expect that this effect is negligi-\nbly small in the simulations with large N. For instance,\nthe bottom panel of Fig. 39 tells us that the correction\nis already as small as O(0.01–0.1)% for N= 2048, which\ncan become even smaller for larger values of N, though\nthe computation of the correction matrix becomes much\nmore expensive. From this reason, we do not apply the\ncorrection matrix and simply use the masked spectrum\nPmask\nX(k) in our main analysis.\nAppendix D: Calculation of F(x, y),Γ, and q\nIn this appendix, we describe technical details on\nthe calculations of the instantaneous emission spectrum\nF(x, y), energy density emission rate Γ, and spectral in-\n100101102103\nkL/(2π)0.9850.9900.9951.0001.0051.010Pmask\na(k)/Pa(k)PRSτ/L = 0.17,ln(mr/H) = 5.8\nmrrmask= 1 (Nmask/N3= 0.0009)\nmrrmask= 2 (Nmask/N3= 0.0019)\nmrrmask= 4 (Nmask/N3= 0.0051)\n100101102103\nkL/(2π)0.9940.9960.9981.0001.0021.004Pmask\na(k)/Pa(k)PRSmrrmask= 2\nτ/L = 0.17 (Nmask/N3= 0.00191)\nτ/L = 0.33 (Nmask/N3= 0.00057)\nτ/L = 0.50 (Nmask/N3= 0.00028)FIG. 39: Ratio of the axion spectrum built with the top-hat\nmask to that corrected by the matrix via Eq. (C.14). Top\npanel shows a comparison among different choices of rmaskat\nthe same conformal time τ/L= 0.17, and bottom panel shows\na comparison among spectra measured at different times for\nrmask = 2m−1\nr. For each line, the ratio of the number of the\nmasked points Nmaskto the total number of lattice sites N3in\nthe simulation box is also shown in the legend. The results are\nobtained from simulations of PRS strings with 20483lattice\nsites and mra= 1.\ndexq.\n1. Instantaneous emission spectrum\nWe compute F(x, y) by analysing the time evolution\nof the energy density spectrum for each Fourier compo-\nnent of the axion field specified by a fixed value of the\ncomoving momentum k. Let us introduce the following\nnotation to describe such a time evolution of one Fourier\nmode,\nE(x)≡1\nf2aH2∂ρa\n∂logk, (D.1)40\nwhere x=kτshould be regarded as a time variable for\ngiven k.40Writing the derivative with respect to the\nconformal time as ∂/∂τ =k∂/∂x and using H2∝R−4\nfor the radiation dominated background, the definition\nofF[Eq. (2.11)] takes a simple form,\nF=∂E\n∂x. (D.2)\nAs shown in Sec. IV, just computing the derivative in\nthe right-hand side of Eq. (D.2) via a finite difference\nmethod gives rise to a problem because of the existence\nof oscillations in the spectrum. In order to suppress the\ncontamination from the oscillations, we compute the time\nderivative in the following way. First, we fit the analyt-\nical function given by Eq. (4.1) to the data of the time\nevolution E(x) for each k, and subtract it from the data\nto obtain the residue,\nR(x)≡ Edata(x)− Efit(x). (D.3)\nFurthermore, we subtract a linear function from the\nresidue,\nR(x) =R(x)−(αx+β), (D.4)\nwhere αand βare determined such that R(xini) =\nR(xfin) = 0, and xiniandxfinrepresent the initial time\nand final time of the data used for the fit, respectively. In\nthe main analysis, we choose xinias a time corresponding\nto ln( mr/H) = 4, and xfinas the final time of the simula-\ntion. The residue R(x) [orR(x)] may contain oscillating\ncomponents, whose dominant angular frequency is given\nby∼2kin the conformal time, as shown in the bottom\npanel of Fig. 9. To remove those oscillations, we apply\nthe type-I discrete sine transform (DST) to the data of\nR(x),41and multiply it by a Gaussian filter,\nexp\u0012\n−(f/fosc)2\n2σ2\nfilter\u0013\n, (D.5)\nwhere fis the frequency of DST components, fosc=k/π\nthe frequency of 2 k-oscillations, and σfilterthe parameter\nto be adjusted appropriately. After that, we perform the\n40Note that here we use xas a time variable. This is different\nfrom the parametrization of F(x, y) defined in Eq. (2.11), where\ny=mr/Hrepresents the time evolution, and x=k/(RH) =kτ\nrepresents the momentum dependence for given τ. Nevertheless,\nin the absence of strings, the energy density spectrum could ac-\ntually be described in terms of a single variable function, where\nxrepresents both the time and momentum dependencies (see\nAppendix F).\n41We arrange R(x) such that R(xini) =R(xfin) = 0 because oth-\nerwise some unnecessary high frequency components can arise\ndue to the discontinuity at the boundary. With this preparation,\nthe type-I DST, which assumes the odd extension on both sides\nof the boundary, reproduces a non-vanishing value of dR/dxat\nthe boundary that can correct any bias caused by dEfit/dxin\nEq. (D.6).inverse DST to obtain the filtered residue Rfiltered (x),\nand calculate Fas\nF=dEfit\ndx+α+dRfiltered\ndx, (D.6)\nwhere dEfit/dxcan be computed analytically by differ-\nentiating Eq. (4.1). Expecting that Rfiltered (x) becomes\nsmooth enough, we compute dRfiltered /dxjust by using\nthe finite difference method.\nIn order to deal with the 2 k-oscillations properly, it\nis important to measure the spectrum with a rate faster\nthan the oscillation frequencies. In the simulations per-\nformed in this work, the axion energy density spectrum\nis measured 300 times from ln( mr/H) = 4 to the fi-\nnal time in a linear interval in the conformal time for\nthe simulations with 112643lattice sites and 250 times\nfor others.42The maximum value of the comoving mo-\nmentum that can be resolved by these measurements is\ngiven by kres= (π/2)fs= (πNmeas/2)/(τfin−τini), where\nfs=Nmeas/(τfin−τini) is the sampling frequency, and\nNmeas= 300 or 250 is the number of measurements. In\nTable V, we summarise the value of kres, its ratio to the\nHubble parameter, and that to the saxion mass at the\nfinal time of the simulations. We see that the oscillations\nfor the modes with kL/(2π)<∼100 are well resolved with\nthose measurements, and hence the 2 k-oscillations can be\nproperly filtered through the above procedure for these\nlower momentum modes.\nTable V also shows that kres/Ris smaller than mr/2\nat the final time of the simulations in all cases. Hence,\nin such simulations the oscillations at the UV part of the\nspectrum are not adequately resolved by the measure-\nments. This is not an issue for the PRS-type simula-\ntions, since the amplitude of the oscillations induced by\nthe horizon crossing are expected to be smaller at higher\nmomenta (see Sec. IV), and those induced by the para-\nmetric resonance effect are relevant only at the comoving\nmomentum given by k=mrR/2. On the other hand,\nfor the physical-type simulations, the 2 k-oscillations can\nbe produced by the parametric resonance effect over a\nbroad range of momenta satisfying k<∼mrR/2. In or-\nder to monitor such oscillations at higher momenta, one\nneeds to increase the number of measurements at the cost\nof more time complexity, which is practically unfeasible.\nWe note that it is still possible to grasp the oscilla-\ntions at higher momenta even if k > k res. It is known\nthat when the actual oscillation frequency fof the data\nexceeds the Nyquist frequency fs/2 = ( Nmeas/2)/(τfin−\nτini), it shows up as a spurious low frequency oscillation\n(called alias), whose frequency is given by fal=|f−nfs|,\nwhere nis an integer that gives fal< fs/2. We actu-\nally confirmed that the frequencies of the oscillations for\n42The measurements are executed in a linear interval in τ, since\nwe need to resolve the oscillations whose frequencies are constant\nin the conformal time. This means that more measurements are\ndone at larger ln( mr/H).41\nTABLE V: The maximum value of the comoving momentum kresof the Fourier component of the axion field whose oscillations\ncan be resolved by Nmeasmeasurements for each choice of the simulation parameters used in the main analysis. The ratio of\nkres/Rto the Hubble parameter and that to the saxion mass at the final time of the simulations are also shown.\nType Grid size Final time Parameter Nmeas kresL/(2π)kres/(RH)kres/(Rmr)\n(N3) ( τf/L) at τf atτf\nPhysical 1126430.625 ¯λ= 195799 300 130.260 511.531 0.0581285\nPhysical 409630.625 ¯λ= 25890 .8 250 115.025 451.701 0.141156\nPhysical 307230.5 ¯λ= 14563 .6 250 154.042 483.939 0.315064\nPhysical 307230.5 ¯λ= 32768 250 147.743 464.150 0.201454\nPhysical 307230.5 ¯λ= 64225 .3 250 143.695 451.431 0.139953\nPhysical 307230.5 ¯λ= 114178 250 140.872 442.563 0.102903\nPhysical 204830.55 ¯λ= 6400 250 143.936 497.408 0.403720\nPRS 819230.55 mra= 0.2 250 120.966 418.027 0.463897\nPRS 819230.55 mra= 0.3 250 118.420 409.229 0.302756\nPRS 819230.55 mra= 0.5 250 116.459 402.453 0.178646\nPRS 819230.55 mra= 0.7 250 115.638 399.617 0.126705\nPRS 819230.55 mra= 1.0 250 115.030 397.516 0.0882271\nPRS 819230.55 mra= 1.5 250 114.562 395.897 0.0585786\nhigher momentum modes in our data agree with those\npredicted by fal. Based on this observation, we replace\nfoscin Eq. (D.5) with max( |fosc−nfs|, fmin) for the\nmodes with k > k reswhen calculating the instantaneous\nemission spectrum for physical-type simulations. Here we\nhave introduced a minimum frequency fmin, since other-\nwise some artificial features arise in F(x) for the modes\nwhose aliasing frequencies |fosc−nfs|become close to\nzero. In the main analysis, we take fminL= 60 to reduce\nartificial features at the corresponding momenta.\nFinally, let us comment on the choice of the parameter\nσfilterin Eq. (D.5). Obviously, taking a too large value of\nσfilterdoes not help to remove the oscillations, while tak-\ning a too small value of σfilterleads to Rfiltered (x)→0,\nwhich can bias the result.43We found that the results are\nnot too biased and at the same time the fluctuations in q\nare adequately suppressed if we take σfilter= 0.02–0.05.\nChanging σfilterin this range slightly modifies the final\nresults on q, and we add this variation in qto the statis-\ntical uncertainty when we refer to the errors in qin the\nmain analysis.\n2. Energy density emission rate\nWe compute the energy density emission rate in a sim-\nilar way to the instantaneous emission spectrum. From\n43In the absence of the filtered residue, Fis strongly biased by\ndEfit/dxin Eq. (D.6). For instance, we observed that the result\nofqtakes different values when we use some functions different\nfrom Eq. (4.1), if we do not include the dRfiltered /dxterm.Eq. (5.1), we have\nΓa\nf2aH3=τdρa\ndτ,Γr\nf2aH3=τ5−⟨z⟩d\ndτ\u0010\nτ⟨z⟩−4ρr\u0011\n,(D.7)\nwhere ρa=ρa/(f2\naH2),ρr=ρr/(f2\naH2), and\n⟨z⟩=\n\n1\nρrZ\ndkz[k]∂ρr\n∂k(physical) ,\n4 (PRS) .(D.8)\nHere the mean redshift exponent ⟨z⟩takes account of\nthe non-trivial redshift of massive modes, and is different\naccording to whether mris constant (physical) or mr∝\nR−1(PRS) during simulations. For the physical case,\nz[k] is given by\nz[k] = 3−dlogω(k)\ndlogR= 3 +\u0010\nk\nmrR\u00112\n1 +\u0010\nk\nmrR\u00112, (D.9)\nwhere ω(k) =p\nm2r+k2/R2.\nTo compute the time derivatives on the right-hand side\nof Eq. (D.7), we fit the following function to the data of\nlnρaand ln( τ⟨z⟩−4ρr),\nGfit(lnτ) =a0+a1lnτ+a2(lnτ)2+a3(lnτ)3,(D.10)\nwhere ai(i= 0,1,2,3) are constants to be determined by\nthe fit. After that, we calculate the residue R=ρdata−\nρfit, where ρ=ρafor axions and ρ=τ⟨z⟩−4ρrfor saxions.\nThen we subtract the linear function, R=R−(ατ+β),\nwhere αandβare fixed such that R(τini) =R(τfin) =\n0, and τiniandτfincorrespond to the conformal time\nat ln( mr/H) = 4 and the final time of the simulation,42\nrespectively. After obtaining R, we apply the type-I DST\nwith a Gaussian filter to it. Here the filter is chosen as\nexp\u0012\n−(f/fs)2\n2σ2\nΓ\u0013\n, (D.11)\nwhere fs=Nmeas/(τfin−τini), and σΓis the parameter\nto be adjusted. Now the energy density emission rate is\ncalculated as\nΓ\nf2aH3=τϵ\u0014exp(Gfit)\nτdGfit\ndlnτ+α+dRfiltered\ndτ\u0015\n,(D.12)\nwhere ϵ= 1 for axions, and ϵ= 5− ⟨z⟩for saxions. In\nthe main analysis, we use σΓ= 0.01, which is sufficient\nto remove rapid fluctuations on the data.\n3. Spectral index\nAssuming that the instantaneous emission spectrum is\nactually given by a power law,\nF(x)∝1\nxq, (D.13)\nbetween the comoving momenta related to the Hubble\nradius k∼HR, and the size of the string core k∼mrR,\nwe would like to determine the value of qfrom simulations\nand explore how it depends on ℓ= ln( mr/H). To this\nend, we attempt to fit the power law to the numerical\nresults in the following manner.\nThe instantaneous emission spectrum computed from\nthe output of the simulations is given in discrete bins Fi,\nwhere the index iincreases linearly with k. Since we are\ninterested in the behaviour at intermediate momenta, we\nfirst select the data points from a range defined by\ncIRHR < k < c UVmrR, (D.14)\nwhere the coefficients cIRandcUVshould be adjusted\nappropriately (see Sec. IV and Appendix E 2). The se-\nlected data is rebinned such that a new set of bins are\nspaced homogeneously in ln k, which is more convenient\nwhen we fit a power law. If instead we perform the fits\nin linear-spaced bins, results would be biased by data at\nhigher k, since they are more abundant in a logarithmic\ninterval.\nThe rebinned data is simply obtained by averaging over\ndata within an interval δldefined by log-spacing,\nFn=1\nNnX\nlnki∈(ln,ln+δl)Fiwith n= 1,2, . . . , N bin,\n(D.15)\nwhere Nnis the number of linear bins in the log-bin, and\nNbinis the total number of log-bins. Throughout the\nanalysis in this work, we fix the number of the rebinned\ndata as Nbin= 30.From the rebinned data, we define the χ2function in\nterms of logarithmic variables,\nχ2=X\nn(Ln−Mn)2\nσ2, (D.16)\nwhere\nLn= lnFn, M n=m−lnq,and ln= lnx.(D.17)\nThe model Mncorresponds to F=em/xqdescribed by\ntwo parameters, mandq. We define σ2from the residuals\nof the different bins at the best fit model,\nσ2≡1\nNbinX\nn(Ln−Mn(mmin, qmin))2, (D.18)\nwhere mminandqminare the best fit values that min-\nimiseP\nn(Ln−Mn)2. It is possible to estimate a con-\nfidence interval [ qmin−σq, qmin+σq] as an interval of q\nsatisfying ∆ χ2≡χ2(mq, q)−χ2\nmin= 1, where we have\nmarginalized over the parameter mby using the value mq\nthat minimises χ2for given q. However, we find that σq\nis typically smaller than the statistical uncertainty from\nthe simulations with different random initial field config-\nurations.\nAppendix E: Other systematics\nThis appendix is devoted to discussions on a couple\nof effects that are not thoroughly elaborated on in the\nmain text: One is the finite volume effect, and the other\nis the effect of the momentum range for the fit of the\ninstantaneous emission spectrum.\n1. Finite volume\nTo check the finite volume effect, we performed addi-\ntional simulations of physical strings with 10243, 15363,\n20483, 25603, and 30723lattice sites, such that they have\nthe same value of mra= 1.0 but different values of the ra-\ntio of the physical box size RLto the Hubble radius H−1\nat ln( mr/H) = 7. We show the comparison of the results\nof those simulations in Fig. 40. Although the spectrum\ncan be largely distorted for HRL <∼1, we can see that\nthe results are convergent for HRL >∼1.4 (or τ/L<∼0.7).\nSince we terminate all simulations used in the main anal-\nysis at τ/L≤0.625 (see Table IV), we expect that the\nfinite volume effects on the spectrum is not an issue in\nthis work.\nEven if the effect of the finite volume does not signifi-\ncantly alter the shape of the spectrum, it can have some\nimpact on the estimation of q. To demonstrate it, in\nFig. 41 we show the evolution of qobtained from simu-\nlations of PRS strings with the same value of mra= 1\nbut different numbers ( N3) of lattice sites. When we fix\nthe value of mra, changing the value of Ncorresponds to43\n101102103\nk/(RH)1001011\nH2f2a∂ρa\n∂logk\nphysicalHRL = 2.80 (τ/L = 0.36)\nHRL = 2.33 (τ/L = 0.43)\nHRL = 1.87 (τ/L = 0.54)\nHRL = 1.40 (τ/L = 0.71)\nHRL = 0.93 (τ/L = 1.07)\n101102103\nk/(RH)0.81.01.21.41.61.8Ratio toHRL = 2.80\nphysicalHRL = 2.33 (τ/L = 0.43)\nHRL = 1.87 (τ/L = 0.54)\nHRL = 1.40 (τ/L = 0.71)\nHRL = 0.93 (τ/L = 1.07)\n101102103\nk/(RH)10−310−210−1100F(x)physical\nHRL = 2.80 (τ/L = 0.36)\nHRL = 2.33 (τ/L = 0.43)\nHRL = 1.87 (τ/L = 0.54)\nHRL = 1.40 (τ/L = 0.71)\nHRL = 0.93 (τ/L = 1.07)\n101102103\nk/(RH)0.500.751.001.251.501.752.002.252.50Ratio toHRL = 2.80\nphysicalHRL = 2.33 (τ/L = 0.43)\nHRL = 1.87 (τ/L = 0.54)\nHRL = 1.40 (τ/L = 0.71)\nHRL = 0.93 (τ/L = 1.07)ln(mr/H) = 7.0\nFIG. 40: The axion energy density spectrum (top left) and instantaneous emission spectrum (bottom left) at ln( mr/H) = 7\nobtained from simulations of physical strings with different values of the ratio HRL (the physical box size to the Hubble radius).\nThese simulations are performed on different number of lattice sites, and for each of them we chose the value of the coupling\nparameter ¯λsuch that mratakes the same value ( mra= 1) at ln( mr/H) = 7 (see Table IV). The ratio of the energy density\nspectrum (instantaneous emission spectrum) for each value of HRL to that for HRL = 2.80 is also shown in the top right\n(bottom right) panel. The coloured bands represent statistical uncertainties.\nchanging the ratio of the box size RLto the string core\nwidth m−1\nr, since mrRL= (mra)N. From Fig. 41 we see\nthat the results with smaller mrRLexhibit more fluctu-\nations in q. This can be understood as follows. For simu-\nlations with smaller mrRL(or smaller N), the number of\nmomentum bins in the interval cIRHR < k < c UVmrR\nused for the fit to determine qbecomes smaller, which\nimplies that less data points are available at the IR part\nof the spectrum. As a result, oscillations in the IR part\nof the spectrum have more impact on the result of the\nfit, which leads to larger fluctuations in q.\nNote that the limit of small mrRLcan also be realised\nby taking mrato small values with Nfixed, though in\nthat case the discretisation effects due to the different\nvalue of mraare not disentangled. This is exactly what\nwe plotted in the top panel of Fig. 22. There we can\nsee that qexhibits larger oscillations for smaller values of\nmra. These features can again be attributed to the lack\nof the momentum bins at the IR part of the spectrum.\nThe above observations may pose a challenge to theprecise estimation of q: We need to make mrasmaller\nto avoid discretisation effects, but at the same time keep\nmrRLsufficiently large in order to have enough data at\nthe IR part of the spectrum to avoid large fluctuations in\nq. In the main analysis, these fluctuations are alleviated\nby having large values of mrRLat the cost of discretisa-\ntion effects.\n2. IR and UV cutoffs\nThe result of the power law fit depends on the choice of\nthe momentum interval shown in Eq. (D.14). In Fig. 42,\nwe show how the results of qfor simulations of PRS\nstrings change when we use different values of cIRand\ncUV. From the top panel of Fig. 42, we see that for\nsmaller cIRthe fit results are more affected by the fea-\nture around the IR peak, being biased toward smaller\nvalues of q. This trend is diminished for cIR>∼30, as the\nlower end of the interval is located sufficiently far from44\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.50.60.70.80.91.01.1\nqPRSmra= 1.0\nmrRL= 1024\nmrRL= 2048\nmrRL= 4096\nmrRL= 8192\nFIG. 41: Evolution of the spectral index qof the instanta-\nneous emission spectrum for different values of mrRLin PRS-\ntype simulations. Coloured lines correspond to the results of\nfour sets of simulations with the same value of mra= 1 but\ndifferent numbers of lattice sites, for which mrRLbecomes\nequal to N. The coloured bands represent statistical uncer-\ntainties. In these plots, we used σfilter= 0.2 (a larger value\nthan the one used in the main analysis) for the filtering pro-\ncedure to calculate F[Eq. (D.5)] to enlarge the fluctuations\nfor illustrative purposes.\nthe IR peak. On the other hand, taking even larger val-\nues (e.g. cIR= 80) appear to introduce another bias at\nsmaller ln( mr/H), since in that case the momentum in-\nterval becomes too short to have enough data points to\nextract the global feature of the spectrum.\nWe see that the curve of qbecomes jagged for smaller\nvalues of cIR. This feature is merely due to the structure\nof the binning. Note that the lower end of the interval\ncIRHRdecreases with time. This implies that more IR\nbins are included in the interval at later times. Since IR\nbins are sparse in log-spaced bins, the structure of the\nlog-bin changes suddenly when a bin with wavenumber\nkcrosses the threshold value cIRHRand enters into the\ninterval. At that point the fit is biased by the lowest bin,\nwhich results in the sudden decrease of the value of q.\nThis feature is more pronounced for smaller cIR, as the\ndata interval contains the feature around the IR peak,\nwhere the spectrum becomes less steep.\nBy comparing the results for different values of the\nhigher end of the interval cUV, we also find that the values\nofqcan be overestimated when we choose a value of\ncUVthat is not too far from the UV cutoff, as shown\nin the bottom panel of Fig. 42. This can be avoided if\nwe choose a sufficiently small value of cUV, but for such\na small cUVwe again observe a bias due to the short\ninterval of the data points. As discussed in Sec. IV, for\nsmaller cUVthe impact of the oscillations in the IR part\nof the spectrum becomes more pronounced, which leads\nto larger oscillatory features in the evolution of q.\nThe biases due to the choice of IR and UV cutoffs are\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.60.70.80.91.01.1\nq\nPRScIR= 10\ncIR= 20\ncIR= 30\ncIR= 50\ncIR= 80\n5.5 6.0 6.5 7.0 7.5 8.0\nln(mr/H)0.60.70.80.91.01.1\nq\nPRScUV=1/2\ncUV=1/3\ncUV=1/4\ncUV=1/8\ncUV=1/16FIG. 42: Evolution of the spectral index qof the instanta-\nneous emission spectrum for different choices of the IR cutoff\ncIR(top panel) and the UV cutoff cUV(bottom panel) of the\nmomentum interval [Eq. (D.14)] used for the fit. In the top\npanel, we change the value of cIRwith the fixed UV cutoff\ncUV= 1/4. In the bottom panel, we instead change the value\nofcUVwith the fixed IR cutoff cUV= 50. The results are\nobtained from simulations of PRS strings with 81923lattice\nsites and mra= 1. The coloured bands represent the er-\nror induced by changing the parameter σfilterfor the filtering\nprocedure to calculate F[Eq. (D.5)] in addition to statistical\nuncertainties.\nalso observed in the results of physical-type simulations,\nas shown in Fig. 43. We see that the values of qare under-\nestimated for cIRas small as cIR= 10, while the results\nare compatible within uncertainties for 20 <∼cIR<∼50.\nForcIRas large as cIR= 80, the result is again distorted\nparticularly at smaller ln( mr/H) due to the short mo-\nmentum interval. The trend that qtakes smaller values\nat smaller ln( mr/H) for cIR= 80 can be attributed to\nthe feature at the UV part of the instantaneous emission\nspectrum for physical strings, which is described below.\nFrom the bottom panel of Fig. 43, we see that qbe-45\n5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\nln(mr/H)0.60.70.80.91.01.11.21.3\nq\nphysicalcIR= 10\ncIR= 20\ncIR= 30\ncIR= 50\ncIR= 80\n5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\nln(mr/H)0.60.70.80.91.01.11.21.3\nq\nphysicalcUV=1/2\ncUV=1/3\ncUV=1/4\ncUV=1/8\ncUV=1/16\nFIG. 43: The same figure as Fig. 42 but the dependences\non the IR and UV cutoffs are plotted for physical strings,\nobtained from simulations with 112643lattice sites.\nhaves somewhat differently than the case of PRS strings\nwhen we change the UV cutoff cUV. For the physical case\nqtakes smaller values when cUVbecomes larger, while for\nthe PRS case it takes larger values. This difference can\nbe associated with the feature at the UV part of the in-\nstantaneous emission spectrum: In the PRS case, we ob-\nserved that F(x) just becomes steeper when we increase\nxacross the momentum corresponding to k/R =mr/2,\nwhich leads to larger values of qfor larger cUV. On the\nother hand, in the physical case F(x) appears to have a\nplateau-like feature at around k/R=mr/2, and qtends\nto take smaller values when the data contains such a fea-\nture in the UV. The occurrence of such a feature could\nbe attributed to the fact that some amount of axions are\nproduced at k<∼mrR/2 due to the parametric resonance\neffect after the saxion mass crossing.\nIn order to avoid the biases due to the features in the\nIR and UV part of the spectrum, we need to take a suf-\nficiently larger value of cIRand smaller value of cUV, re-spectively, but taking too small (large) value of cUV(cIR)\ncan again lead to large contaminations from oscillations\ndue to the short interval available for the fit. In the main\nanalysis in this paper, we take cIR= 50 and cUV= 1/4\nboth for the PRS and physical cases as the fiducial choice.\nAppendix F: Evolution of the free axion field around\nthe horizon crossing\nIn the following, we perform the analytical calcula-\ntion on the evolution of the axion field in the absence\nof string networks, aiming at estimating the location of\nthe turnaround used in Sec. VI B.\nWe start from the equation of motion for the free mass-\nless axion field in a FRW background,\n¨θ+ 3H˙θ−1\nR2∇2θ= 0. (F.1)\nIntroducing a new field variable ψ≡τθand replacing\nthe time derivative with the derivative with respect to\nthe conformal time τ, the above equation reduces to\nψττ− ∇2ψ= 0. (F.2)\nThe solution in Fourier space reads eψ(k)∝sin(kτ) or\ncos(kτ), and for the Fourier component of the θfield,\neθ(k) =Aksin(kτ)\nkτ+Bkcos(kτ)\nkτ, (F.3)\nwhere k=|k|, and AkandBkare constant coefficients.\nIt turns out that the Bkterm must be negligibly small,\nBk<∼TQCD/TPQ≪1, where TQCD andTPQare the\ncritical temperature of the QCD phase transition and\nthat of the PQ phase transition, respectively, in order to\nguarantee a finite field amplitude at the epoch of the PQ\nphase transition [91].\nWe assume that the spatial distribution of θis given by\na superposition of plane waves with random amplitudes\nAk:\nθ(x, τ) =Zd3k\n(2π)3sin(kτ)\nkτAke−ik·x. (F.4)\nFurthermore, we assume a scale invariant spectrum for\nthe mode amplitude [95],\n⟨AkA∗\nk′⟩=C\nk3(2π)3δ(3)(k−k′), (F.5)\nwhere ⟨. . .⟩represents an average over the random dis-\ntribution of Ak, andCis a dimensionless constant. Using\nEqs. (F.4) and (F.5), we can compute the contributions\nto the axion energy density from kinetic and gradient\nterms:\nρa,kin=1\n2f2\na⟨˙θ2⟩=C\n4π2f2\na\nR2Z\nkdk\u0012coskτ\nkτ−sinkτ\n(kτ)2\u00132\n,\n(F.6)\nρa,grad=1\n2f2\na1\nR2⟨|∇θ|2⟩=C\n4π2f2\na\nR2Z\nkdk\u0012sinkτ\nkτ\u00132\n.\n(F.7)46\nNote that the kinetic energy ρa,kinvanishes in the super-\nhorizon limit kτ≪1, while the gradient energy acquires\na non-vanishing contribution,\n(ρa,grad)sup<∼C\n4π2f2\na\nR2Zτ−1\n0kdk=C\n8π2f2\naH2,(F.8)\nwhere in the first inequality we have used the fact that\nonly the modes with k≪τ−1can contribute to the inte-\ngral,R\nkdk<∼Rτ−1\n0kdk. This result agrees with a naive\nestimate ρa∼f2\na⟨|∇θ|2⟩ ∼f2\naH2, which stems from an\nexpectation that the misalignment angle θvaries ran-\ndomly on the scale corresponding to the Hubble radius.\nWe can also compute the contributions of subhorizon\nmodes ( kτ≫1),\n(ρa,kin)sub=C\n4π2f2\na\nR2Z\nk≫τ−1kdkcos2(kτ)\n(kτ)2,\n(ρa,grad)sub=C\n4π2f2\na\nR2Z\nk≫τ−1kdksin2(kτ)\n(kτ)2,(F.9)\nfrom which the total energy density reads\n(ρa)sub= (ρa,kin)sub+ (ρa,grad)sub\n=C\n4π2f2\naH2Zdk\nk\n≈C\n4π2f2\naH2ln\u0012Λ\nH\u0013\n. (F.10)\nThis quantity is logarithmically divergent, and we have\nintroduced the UV cutoff Λ. A natural choice for the cut-\noff scale would be the saxion mass, Λ ≈mr. Then, this\nresult implies that the energy density of misalignment\naxions ρa∼f2\naH2ln(mr/H) can be comparable to the\ntypical energy density of strings in the scaling regime [95].\nSince we calculate the axion spectrum based on the\nkinetic energy in the numerical study, let us focus on the\n(differential) spectrum for the kinetic term. It is straight-\nforward to obtain\nE(x) =2\n(faH)2dρa,kin\ndlogk=C\n2π2x2[j1(x)]2, (F.11)\nwhere we use the same notation as Eq. (D.1), and jn(x)\nare spherical Bessel function of the first kind. Note that\nthis quantity depends only on x=kτ. Namely, its time\nevolution is the same for all k, when viewed as a function\nofkτ. This observation can explain the numerical result\nshown in the top panel of Fig. 9 that the evolution is\nalmost the same among different modes when they are\noutside the horizon.\nWe also note that the asymptotic behaviour in the\nsuper-horizon limit reads\n2\n(faH)2dρa,kin\ndlogkkτ→0− − − − →C\n2π21\n9(kτ)4. (F.12)\nThis implies that the IR part of the spectrum (or its\nevolution before the horizon crossing) scales as ∝(kτ)4in the absence of strings. For a comparison, in Fig. 44 we\nshow the evolution of the mode with the fixed comoving\nmomentum kL/(2π) = 2 .41 obtained from simulations\nwith various different initial string densities. In those\nsimulations the slope of E ∝ (kτ)pat small kτtakes\na value between 4 <∼p<∼6 according to the value of\nthe string density ξ, indicating that the IR falloff of the\nspectrum becomes steeper in the presence of strings.44\nFurthermore, there is a trend that the slope approaches\nto the free field value p→4 with decreasing ξ. We\nobserved a similar trend when we plot the spectra as a\nfunction of krather than τ.\n2 3 4 5 6 7 8\nkτ10−11001\nf2aH2∂ρa\n∂logkkL/(2π) = 2.41\nξ/lscript=3= 0.036\nξ/lscript=3= 0.073\nξ/lscript=3= 0.133\nξ/lscript=3= 0.214\nξ/lscript=3= 0.300\nξ/lscript=3= 0.380\nξ/lscript=3= 0.464\nξ/lscript=3= 0.586\nξ/lscript=3= 0.711\nξ/lscript=3= 0.828\nFIG. 44: Time evolution of the energy density of one Fourier\nmode of the axion field specified by kL/(2π) = 2 .41 for vari-\nous different values of the initial strings density ξℓ=3. The\ncoloured bands represent statistical uncertainties, and the\nblack dashed line corresponds to the attractor with ξℓ=3=\n0.3. The results are obtained from simulations of physical\nstrings with 20483lattice sites.\nThe fact that E(x) exhibits a non-trivial time depen-\ndence implies that there is a contribution to the instanta-\nneous emission spectrum even in the absence of strings:\nF(x) =∂E(x)\n∂x=C\nπ2xj1(x)[xj0(x)−j1(x)],(F.13)\nwhere the relation F=∂E/∂xfollows from the definition\nofF[see Eq. (D.2)]. We can also obtain its derivative\nwith respect to x,\n∂F(x)\n∂x=C\nπ2x\u0000\nx[j0(x)]2−j1(x)[j0(x) +xj1(x)−j2(x)]\u0001\n.\n(F.14)\nFigure 45 shows the evolution of E,F, and ∂F/∂xgiven\nby Eqs. (F.11), (F.13), and (F.14). The location of the\n44We may describe this effect by introducing a phase αkin the\nevolution of the mode [i.e. replacing sin( kτ) with sin( kτ+αk)\nin Eq. (F.4)] and allowing a slow time variation in Akandαk,\nthough we do not pay much attention to their precise time de-\npendence here.47\nfirst turnaround of F(∂F/∂x= 0) is found to be x=\n1.86765, which is shown as a black dot in Fig. 45.\n0 1 2 3 4 5 6 7\nkτ−0.100−0.075−0.050−0.0250.0000.0250.0500.0750.100\nE(x)\nF(x)\ndF(x)\ndx\nFIG. 45: Evolution of E(x) (blue solid line), F(x) (orange\ndashed line), and ∂F(x)/∂x(green dash-dotted line) as func-\ntions of x=kτ. In these plots, we take C= 1 for simplicity.\nThe location of the first maximum of F(first node of ∂F/∂x)\nis shown as a black dot.\nAppendix G: Comparisons\nOur results on ξandqobtained from simulations with\n112643lattice sites are compared with those from recent\nsimulations performed by the authors of Refs. [70, 71]\nin Fig. 46. Here we summarise differences in the simu-\nlation/analysis methods and discuss possible sources of\ndiscrepancies.\nThe authors of Ref. [70] performed simulations up to\n45003lattice sites. Among multiple sets of simulations\nwith different choices of parameters, they used the results\nfrom simulations with mra= 1 at the final time in the\nmain analysis, and we replot them in Fig. 46. From the\nleft panel of that figure, we see that the string density pa-\nrameter at the attractor found in Sec. III is very compat-\nible with their result. On the other hand, the values of q\nobtained from our analysis become systematically smaller\nthan those obtained in Ref. [70]. One possible origin of\nthis discrepancy is attributed to the fact that we use the\n4-neighbour discretisation scheme of the Laplacian, while\nthey used the 2-neighbour scheme. In Sec. V A, we have\nshown that a lower level of discretisation of the Laplacian\ncan distort the spectrum and lead to larger values of q.\nWe have also shown that the 2-neighbour Laplacian may\nnot be enough to suppress that effect.\nOther than the effect of the Laplacian, there are a few\neffects that can cause the difference. The fact that the\nstring core resolution mra= 1 at ln( mr/H)≃7.9 in\nRef. [70] is lower than ours [ mra≃0.69 at ln( mr/H) =\n7.9] can also result in larger values of qand explain the\ndiscrepancy at large ln( mr/H). Furthermore, the valuesofξin Ref. [70] remains slightly smaller than our re-\nsults (see left panel of Fig. 46) throughout the simulated\nrange of ln( mr/H), and such smaller values of ξcould\nlead to larger values of q(cf. Fig. 6), though the effect\nwould be small. The discrepancy in qappears to be more\npronounced at smaller ln( mr/H), but it is not straight-\nforward to offer a simple explanation for that, since at\nsmaller ln( mr/H) the range of the momentum used for\nthe fit of the instantaneous emission spectrum becomes\nvery short, which may lead to potentially larger system-\natic uncertainties. In particular, they used the finite dif-\nference method to compute the instantaneous emission\nspectrum, which could be more sensitive to the effect of\nthe oscillations in the spectrum and result in larger fluc-\ntuations in qas illustrated in Sec. IV. We also note that\ntheir fiducial choice of the IR cutoff cIR= 30 for the fit\ndeviates from the critical value x1z≃25.1, which could\npotentially amplify the effect of oscillations in the IR part\nof the spectrum (see Sec. IV).\nThe authors of Ref. [71] carried out a simulation based\non the AMR technique rather than the conventional\nmethod with the static lattice. The simulation was per-\nformed with a uniform grid of 20483cells and up to five\nlevels of refinement around the string core, which would\neffectively amount to static grid simulation with 655363\nlattice sites. It should be noted that only one large scale\nsimulation was performed in Ref. [71], and it is not pos-\nsible to assign the statistical uncertainties to the results.\nThe error bars for ξshown in the left panel of Fig. 46\nwere determined by treating them as a nuisance param-\neter in the fit of a model of ξgiven by a simple function\nofℓto the data. The error bars for qshown in the right\npanel of Fig. 46 was obtained from the second partial\nderivatives of the Gaussian likelihood used for the fit of a\npower law model to the data of the instantaneous emis-\nsion spectrum.\nThere are several factors that could give rise to a dif-\nferent assessment of qin Ref. [71]. First, the simula-\ntion performed in Ref. [71] corresponds to under-dense\nstrings compared to ours as shown in the left panel of\nFig. 46, and such an under-dense case is likely to give a\nhigher value of q, as shown in Fig. 6. Second, in Ref. [71]\nthe 1-neighbour finite difference was used for the dis-\ncretisation of the Laplacian, while we use the Laplacian\nwith 4-neighbours. As shown in Fig. 16, the value of q\ncomputed in the 1-neighbour case becomes substantially\nlarger than other cases. Finally, in Ref. [71] the instan-\ntaneous emission spectrum was computed based on the\nfinite difference method with the interval ∆ ℓ= 0.25, and\na smaller value cUV= 1/16 than ours ( cUV= 1/4) was\nused for the UV cutoff of the momentum range used in\nthe fit to determine q. As discussed in Sec. IV, due to\nthe existence of the 2 k-oscillations in the spectrum the\nvalues of qcould be biased at smaller values of cUVwhen\nthe instantaneous emission spectrum is computed based\non the finite difference method, and the effect could be\namplified by statistical fluctuations. The fact that they\nused a thermal initial condition could also lead to some48\n3 4 5 6 7 8 9\nln(mr/H)0.00.20.40.60.81.01.21.4\nξThis work\nGorghetto et al .(2021)\nBuschmann et al .(2022)\n5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0\nln(mr/H)0.20.40.60.81.01.21.41.6\nq\nThis work\nGorghetto et al .(2021)\nBuschmann et al .(2022)\nqmodel =q0+q1//lscript2\nqmodel =q0+q1/lscript\nqmodel =q0+q1//lscript\nFIG. 46: Comparisons of the main results of this work with those of Gorghetto et al. (2021) [70] and Buschmann et al.\n(2022) [71]. Left panel: Comparison of the evolution of the string density parameter ξ. The data is digitised from Fig. 1\nof Ref. [70] and Supplementary Fig. 13 of Ref. [71]. For the results of this work and Ref. [70], the coloured bands represent\nstatistical uncertainties. The error bars for the result of Ref. [71] do not represent the statistical uncertainties but an educated\nguess involving the analytical modeling and fit. Right panel: Comparison of the evolution of the spectral index qof the axion\nemission spectrum. The data is digitised from the right panel of Fig. 16 of Ref. [70] and Supplementary Fig. 10 of Ref. [71]. For\nthe results of this work, the error bars include systematics induced by changing the parameter σfilterfor the filtering procedure\nto calculate F[Eq. (D.5)] in addition to statistical uncertainties. For the results of Ref. [70], the error bars represent statistical\nuncertainties. The error bars for the result of Ref. [71] represent the error of the fit, rather than statistical uncertainties.\nThree possible models for the continuum extrapolation of the results of this work obtained in Sec. V C 2 are also shown by the\ndash-dotted line ( qmodel =q0+q1/ℓ2), dashed line ( qmodel =q0+q1ℓ), and dotted line ( qmodel =q0+q1/ℓ).\nbias, since there might be a high frequency oscillation of\nthe PQ field that can also lead to contaminations in the\nspectrum, as discussed in Appendix B. Moreover, the mo-\nmentum range available for the fit becomes even shorter\nat smaller ln( mr/H), which together with the existence\nof the 2 k-oscillations makes the fit more uncertain at\nsmaller ln( mr/H). This can actually be seen in the right\npanel of Fig. 46, where the error bars of qin the results of\nRef. [71] become particularly large at smaller ln( mr/H).\nThe trend of the logarithmic increase of qobserved in\nRef. [70] and this work could become less apparent in\nRef. [71] because of these effects.Compared to the results of Ref. [71], our results give\nrise to larger values of qat later times of the simulation\n[ln(mr/H)>∼8.5]. This is attributed to the fact that\nthe resolution of the string core becomes worse at late\ntimes in our simulations [ mra= 1.25 at the final time\ncorresponding to ln( mr/H) = 9 .08], and qcan be biased\ntowards larger values due to the discretisation effect dis-\ncussed in Sec. V B. This effect should not be a problem\nin the simulation performed in Ref. [71], since the reso-\nlution of the string core is kept as good as ( mra)−1>3\nthroughout the simulation by adding 5 refinement levels\nappropriately.\n[1] T. W. B. Kibble, “Topology of Cosmic Domains and\nStrings,” J. Phys. A 9(1976) 1387–1398. 1\n[2] A. Vilenkin and A. E. Everett, “Cosmic Strings and\nDomain Walls in Models with Goldstone and\nPseudoGoldstone Bosons,” Phys. Rev. Lett. 48(1982)\n1867–1870. 1\n[3] Y. B. Zeldovich, I. Y. Kobzarev, and L. B. Okun,\n“Cosmological Consequences of the Spontaneous\nBreakdown of Discrete Symmetry,” Zh. Eksp. Teor.\nFiz.67(1974) 3–11. 1\n[4] M. Pospelov, S. Pustelny, M. P. Ledbetter, D. F.\nJackson Kimball, W. Gawlik, and D. Budker,\n“Detecting Domain Walls of Axionlike Models Using\nTerrestrial Experiments,” Phys. Rev. Lett. 110no. 2,(2013) 021803, arXiv:1205.6260 [hep-ph] . 1\n[5] J. N. Benabou, M. Buschmann, S. Kumar, Y. Park,\nand B. R. Safdi, “Signatures of Primordial Energy\nInjection from Axion Strings,” arXiv:2308.01334\n[hep-ph] . 1\n[6] G. Ballesteros, J. Redondo, A. Ringwald, and\nC. Tamarit, “Standard Model—axion—seesaw—Higgs\nportal inflation. Five problems of particle physics and\ncosmology solved in one stroke,” JCAP 08(2017) 001,\narXiv:1610.01639 [hep-ph] . 1\n[7] P. Bhattacharjee, T. W. B. Kibble, and N. Turok,\n“Baryon Number From Collapsing Cosmic Strings,”\nPhys. Lett. B 119(1982) 95–96. 1\n[8] S. W. Hawking, “Black Holes From Cosmic Strings,”49\nPhys. Lett. B 231(1989) 237–239. 2\n[9] J. Fort and T. Vachaspati, “Do global string loops\ncollapse to form black holes?,” Phys. Lett. B 311\n(1993) 41–46, arXiv:hep-th/9305081 .\n[10] F. Ferrer, E. Masso, G. Panico, O. Pujolas, and\nF. Rompineve, “Primordial Black Holes from the QCD\naxion,” Phys. Rev. Lett. 122no. 10, (2019) 101301,\narXiv:1807.01707 [hep-ph] .\n[11] G. B. Gelmini, A. Simpson, and E. Vitagliano,\n“Catastrogenesis: DM, GWs, and PBHs from ALP\nstring-wall networks,” JCAP 02(2023) 031,\narXiv:2207.07126 [hep-ph] .\n[12] G. B. Gelmini, J. Hyman, A. Simpson, and\nE. Vitagliano, “Primordial black hole dark matter\nfrom catastrogenesis with unstable pseudo-Goldstone\nbosons,” JCAP 06(2023) 055, arXiv:2303.14107\n[hep-ph] . 2\n[13] R. L. Davis, “Cosmic Axions from Cosmic Strings,”\nPhys. Lett. B 180(1986) 225–230. 2, 3\n[14] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel,\nJ. Redondo, and A. Ringwald, “WISPy Cold Dark\nMatter,” JCAP 06(2012) 013, arXiv:1201.5902\n[hep-ph] . 2\n[15] R. D. Peccei and H. R. Quinn, “CP Conservation in\nthe Presence of Instantons,” Phys. Rev. Lett. 38\n(1977) 1440–1443. 2\n[16] R. D. Peccei and H. R. Quinn, “Constraints Imposed\nby CP Conservation in the Presence of Instantons,”\nPhys. Rev. D 16(1977) 1791–1797.\n[17] S. Weinberg, “A New Light Boson?,” Phys. Rev. Lett.\n40(1978) 223–226.\n[18] F. Wilczek, “Problem of Strong PandTInvariance in\nthe Presence of Instantons,” Phys. Rev. Lett. 40\n(1978) 279–282. 2\n[19]Particle Data Group Collaboration, A. Ringwald,\nL. J. Rosenberg, and G. Rybka, “90. Axions and Other\nSimilar Particles. In Review of Particle Physics,”\nPTEP 2022 (2022) 083C01. 2\n[20] L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli,\n“The landscape of QCD axion models,” Phys. Rept.\n870(2020) 1–117, arXiv:2003.01100 [hep-ph] . 2\n[21] I. G. Irastorza and J. Redondo, “New experimental\napproaches in the search for axion-like particles,” Prog.\nPart. Nucl. Phys. 102(2018) 89–159,\narXiv:1801.08127 [hep-ph] . 2\n[22]IAXO Collaboration, E. Armengaud et al. , “Physics\npotential of the International Axion Observatory\n(IAXO),” JCAP 06(2019) 047, arXiv:1904.09155\n[hep-ph] . 2\n[23]ARIADNE Collaboration, A. A. Geraci et al. ,\n“Progress on the ARIADNE axion experiment,”\nSpringer Proc. Phys. 211(2018) 151–161,\narXiv:1710.05413 [astro-ph.IM] . 2\n[24] E. Goudzovski et al. , “New physics searches at kaon\nand hyperon factories,” Rept. Prog. Phys. 86no. 1,\n(2023) 016201, arXiv:2201.07805 [hep-ph] . 2\n[25] D. F. Jackson Kimball et al. , “Overview of the Cosmic\nAxion Spin Precession Experiment (CASPEr),”\nSpringer Proc. Phys. 245(2020) 105–121,\narXiv:1711.08999 [physics.ins-det] . 2\n[26]DMRadio Collaboration, L. Brouwer et al. ,\n“Projected sensitivity of DMRadio-m3: A search for\nthe QCD axion below 1 µeV,” Phys. Rev. D 106\nno. 10, (2022) 103008, arXiv:2204.13781 [hep-ex] . 2[27] I. Stern, “ADMX Status,” PoSICHEP2016 (2016)\n198, arXiv:1612.08296 [physics.ins-det] . 2\n[28] Y. K. Semertzidis et al. , “Axion Dark Matter Research\nwith IBS/CAPP,” arXiv:1910.11591\n[physics.ins-det] . 2\n[29] S. Beurthey et al. , “MADMAX Status Report,”\narXiv:2003.10894 [physics.ins-det] . 2\n[30] M. Lawson, A. J. Millar, M. Pancaldi, E. Vitagliano,\nand F. Wilczek, “Tunable axion plasma haloscopes,”\nPhys. Rev. Lett. 123no. 14, (2019) 141802,\narXiv:1904.11872 [hep-ph] . 2\n[31] A. Ringwald, “Discovery potential for axions in\nHamburg,” in 57th Rencontres de Moriond on\nElectroweak Interactions and Unified Theories . 6, 2023.\narXiv:2306.08978 [hep-ph] . 2\n[32]BREAD Collaboration, J. Liu et al. , “Broadband\nSolenoidal Haloscope for Terahertz Axion Detection,”\nPhys. Rev. Lett. 128no. 13, (2022) 131801,\narXiv:2111.12103 [physics.ins-det] . 2\n[33] C. Hogan and M. Rees, “Axion Miniclusters,”\nPhys.Lett. B205 (1988) 228–230. 2\n[34] E. W. Kolb and I. I. Tkachev, “Large amplitude\nisothermal fluctuations and high density dark matter\nclumps,” Phys.Rev. D50 (1994) 769–773,\narXiv:astro-ph/9403011 [astro-ph] .\n[35] P. W. Graham and A. Scherlis, “Stochastic axion\nscenario,” Phys. Rev. D 98no. 3, (2018) 035017,\narXiv:1805.07362 [hep-ph] . 2\n[36] D. G. Levkov, A. G. Panin, and I. I. Tkachev,\n“Gravitational Bose-Einstein condensation in the\nkinetic regime,” Phys. Rev. Lett. 121no. 15, (2018)\n151301, arXiv:1804.05857 [astro-ph.CO] . 2\n[37] B. Eggemeier and J. C. Niemeyer, “Formation and\nmass growth of axion stars in axion miniclusters,”\nPhys. Rev. D 100no. 6, (2019) 063528,\narXiv:1906.01348 [astro-ph.CO] .\n[38] A. S. Dmitriev, D. G. Levkov, A. G. Panin, and I. I.\nTkachev, “Self-similar growth of Bose stars,”\narXiv:2305.01005 [astro-ph.CO] . 2\n[39] E. W. Kolb and I. I. Tkachev, “Femtolensing and\npicolensing by axion miniclusters,” Astrophys.J. 460\n(1996) L25–L28, arXiv:astro-ph/9510043\n[astro-ph] . 2\n[40] M. Fairbairn, D. J. E. Marsh, and J. Quevillon,\n“Searching for the QCD Axion with Gravitational\nMicrolensing,” Phys. Rev. Lett. 119no. 2, (2017)\n021101, arXiv:1701.04787 [astro-ph.CO] .\n[41] M. Fairbairn, D. J. E. Marsh, J. Quevillon, and\nS. Rozier, “Structure formation and microlensing with\naxion miniclusters,” Phys. Rev. D 97no. 8, (2018)\n083502, arXiv:1707.03310 [astro-ph.CO] .\n[42] A. Katz, J. Kopp, S. Sibiryakov, and W. Xue,\n“Femtolensing by Dark Matter Revisited,” JCAP 12\n(2018) 005, arXiv:1807.11495 [astro-ph.CO] .\n[43] L. Dai and J. Miralda-Escud´ e, “Gravitational Lensing\nSignatures of Axion Dark Matter Minihalos in Highly\nMagnified Stars,” Astron. J. 159no. 2, (2020) 49,\narXiv:1908.01773 [astro-ph.CO] .\n[44] D. Ellis, D. J. E. Marsh, B. Eggemeier, J. Niemeyer,\nJ. Redondo, and K. Dolag, “Structure of axion\nminiclusters,” Phys. Rev. D 106no. 10, (2022) 103514,\narXiv:2204.13187 [hep-ph] . 2\n[45] A. D. Linde, “Generation of Isothermal Density\nPerturbations in the Inflationary Universe,” Phys.50\nLett. B 158(1985) 375–380. 2\n[46] D. Seckel and M. S. Turner, “Isothermal Density\nPerturbations in an Axion Dominated Inflationary\nUniverse,” Phys. Rev. D 32(1985) 3178.\n[47]Planck Collaboration, Y. Akrami et al. , “Planck 2018\nresults. X. Constraints on inflation,” Astron.\nAstrophys. 641(2020) A10, arXiv:1807.06211\n[astro-ph.CO] . 2\n[48] B. Eggemeier, J. Redondo, K. Dolag, J. C. Niemeyer,\nand A. Vaquero, “First Simulations of Axion\nMinicluster Halos,” Phys. Rev. Lett. 125no. 4, (2020)\n041301, arXiv:1911.09417 [astro-ph.CO] . 2\n[49] G. Pierobon, J. Redondo, K. Saikawa, A. Vaquero, and\nG. D. Moore, “Miniclusters from axion string\nsimulations,” arXiv:2307.09941 [hep-ph] . 2, 29\n[50] B. Eggemeier, C. A. J. O’Hare, G. Pierobon,\nJ. Redondo, and Y. Y. Y. Wong, “Axion minivoids and\nimplications for direct detection,” Phys. Rev. D 107\nno. 8, (2023) 083510, arXiv:2212.00560 [hep-ph] . 2\n[51] D. Harari and P. Sikivie, “On the Evolution of Global\nStrings in the Early Universe,” Phys. Lett. B 195\n(1987) 361–365. 2, 3, 28\n[52] R. L. Davis and E. P. S. Shellard, “Do axions need\ninflation?,” Nucl. Phys. B 324(1989) 167–186. 2\n[53] C. Hagmann and P. Sikivie, “Computer simulations of\nthe motion and decay of global strings,” Nucl. Phys. B\n363(1991) 247–280. 2, 3, 28\n[54] D. H. Lyth, “Estimates of the cosmological axion\ndensity,” Phys. Lett. B 275(1992) 279–283.\n[55] R. A. Battye and E. P. S. Shellard, “Global string\nradiation,” Nucl. Phys. B 423(1994) 260–304,\narXiv:astro-ph/9311017 . 2\n[56] M. Nagasawa and M. Kawasaki, “Collapse of axionic\ndomain wall and axion emission,” Phys. Rev. D 50\n(1994) 4821–4826, arXiv:astro-ph/9402066 . 3\n[57] R. A. Battye and E. P. S. Shellard, “Axion string\nconstraints,” Phys. Rev. Lett. 73(1994) 2954–2957,\narXiv:astro-ph/9403018 . [Erratum: Phys.Rev.Lett.\n76, 2203–2204 (1996)]. 2, 3, 5\n[58] R. A. Battye and E. P. S. Shellard, “Radiation\nconstraints from cosmic strings,” Nucl. Phys. B Proc.\nSuppl. 43(1995) 54–57, arXiv:astro-ph/9412056 . 2\n[59] S. Chang, C. Hagmann, and P. Sikivie, “Studies of the\nmotion and decay of axion walls bounded by strings,”\nPhys. Rev. D 59(1999) 023505,\narXiv:hep-ph/9807374 . 3\n[60] M. Yamaguchi, M. Kawasaki, and J. Yokoyama,\n“Evolution of axionic strings and spectrum of axions\nradiated from them,” Phys. Rev. Lett. 82(1999)\n4578–4581, arXiv:hep-ph/9811311 . 2, 32, 35\n[61] C. Hagmann, S. Chang, and P. Sikivie, “Axion\nradiation from strings,” Phys. Rev. D 63(2001)\n125018, arXiv:hep-ph/0012361 . 2, 4, 22, 28\n[62] T. Hiramatsu, M. Kawasaki, T. Sekiguchi,\nM. Yamaguchi, and J. Yokoyama, “Improved\nestimation of radiated axions from cosmological\naxionic strings,” Phys. Rev. D 83(2011) 123531,\narXiv:1012.5502 [hep-ph] . 32, 33, 35, 37\n[63] T. Hiramatsu, M. Kawasaki, K. Saikawa, and\nT. Sekiguchi, “Production of dark matter axions from\ncollapse of string-wall systems,” Phys. Rev. D 85\n(2012) 105020, arXiv:1202.5851 [hep-ph] . [Erratum:\nPhys.Rev.D 86, 089902 (2012)].\n[64] M. Kawasaki, K. Saikawa, and T. Sekiguchi, “Axiondark matter from topological defects,” Phys. Rev. D\n91no. 6, (2015) 065014, arXiv:1412.0789 [hep-ph] .\n32, 33\n[65] L. Fleury and G. D. Moore, “Axion dark matter:\nstrings and their cores,” JCAP 01(2016) 004,\narXiv:1509.00026 [hep-ph] . 2, 3, 8, 9, 33\n[66] V. B. Klaer and G. D. Moore, “The dark-matter axion\nmass,” JCAP 11(2017) 049, arXiv:1708.07521\n[hep-ph] . 29\n[67] M. Gorghetto, E. Hardy, and G. Villadoro, “Axions\nfrom Strings: the Attractive Solution,” JHEP 07\n(2018) 151, arXiv:1806.04677 [hep-ph] . 3, 4, 5, 6, 8,\n9, 12, 16, 31, 32, 35\n[68] A. Vaquero, J. Redondo, and J. Stadler, “Early seeds\nof axion miniclusters,” JCAP 04(2019) 012,\narXiv:1809.09241 [astro-ph.CO] . 9, 30\n[69] M. Buschmann, J. W. Foster, and B. R. Safdi,\n“Early-Universe Simulations of the Cosmological\nAxion,” Phys. Rev. Lett. 124no. 16, (2020) 161103,\narXiv:1906.00967 [astro-ph.CO] . 3, 32\n[70] M. Gorghetto, E. Hardy, and G. Villadoro, “More\naxions from strings,” SciPost Phys. 10no. 2, (2021)\n050, arXiv:2007.04990 [hep-ph] . 3, 5, 6, 8, 9, 11, 12,\n14, 21, 22, 24, 25, 26, 47, 48\n[71] M. Buschmann, J. W. Foster, A. Hook, A. Peterson,\nD. E. Willcox, W. Zhang, and B. R. Safdi, “Dark\nmatter from axion strings with adaptive mesh\nrefinement,” Nature Commun. 13no. 1, (2022) 1049,\narXiv:2108.05368 [hep-ph] . 3, 5, 12, 13, 14, 22, 28,\n32, 47, 48\n[72] C. A. J. O’Hare, G. Pierobon, J. Redondo, and\nY. Y. Y. Wong, “Simulations of axionlike particles in\nthe postinflationary scenario,” Phys. Rev. D 105no. 5,\n(2022) 055025, arXiv:2112.05117 [hep-ph] . 2, 3, 8,\n27\n[73] A. Vilenkin and T. Vachaspati, “Radiation of\nGoldstone Bosons From Cosmic Strings,” Phys. Rev. D\n35(1987) 1138. 2\n[74] C. O’Hare, “cajohare/axionlimits: Axionlimits,”\nhttps://cajohare.github.io/AxionLimits/ , July,\n2020. 3\n[75] M. Yamaguchi, “Scaling property of the global string\nin the radiation dominated universe,” Phys. Rev. D 60\n(1999) 103511, arXiv:hep-ph/9907506 . 2\n[76] J. N. Moore, E. P. S. Shellard, and C. J. A. P.\nMartins, “On the evolution of Abelian-Higgs string\nnetworks,” Phys. Rev. D 65(2002) 023503,\narXiv:hep-ph/0107171 . 2\n[77] M. Kawasaki, T. Sekiguchi, M. Yamaguchi, and\nJ. Yokoyama, “Long-term dynamics of cosmological\naxion strings,” PTEP 2018 no. 9, (2018) 091E01,\narXiv:1806.05566 [hep-ph] . 2, 3, 8, 22, 32\n[78] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and\nJ. Urrestilla, “Scaling Density of Axion Strings,” Phys.\nRev. Lett. 124no. 2, (2020) 021301,\narXiv:1908.03522 [astro-ph.CO] . 3, 9\n[79] V. B. Klaer and G. D. Moore, “Global cosmic string\nnetworks as a function of tension,” JCAP 06(2020)\n021, arXiv:1912.08058 [hep-ph] . 3, 6, 8, 9\n[80] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and\nJ. Urrestilla, “Approach to scaling in axion string\nnetworks,” Phys. Rev. D 103no. 10, (2021) 103534,\narXiv:2102.07723 [astro-ph.CO] . 3, 9\n[81] M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and51\nJ. Urrestilla, “Comment on ”More Axions from\nStrings”,” arXiv:2109.09679 [astro-ph.CO] . 2, 8, 9\n[82] V. B. Klaer and G. D. Moore, “How to simulate global\ncosmic strings with large string tension,” JCAP 10\n(2017) 043, arXiv:1707.05566 [hep-ph] . 3, 29\n[83] E. P. S. Shellard, “Cosmic String Interactions,” Nucl.\nPhys. B 283(1987) 624–656. 3\n[84] A. Drew and E. P. S. Shellard, “Radiation from global\ntopological strings using adaptive mesh refinement:\nMethodology and massless modes,” Phys. Rev. D 105\nno. 6, (2022) 063517, arXiv:1910.01718\n[astro-ph.CO] . 3\n[85] A. Drew and E. P. S. Shellard, “Radiation from global\ntopological strings using adaptive mesh refinement:\nMassive modes,” Phys. Rev. D 107no. 4, (2023)\n043507, arXiv:2211.10184 [astro-ph.CO] .\n[86] A. Drew, T. Kinowski, and E. P. S. Shellard, “Axion\nString Source Modelling,” arXiv:2312.07701\n[astro-ph.CO] . 3\n[87] A. Dabholkar and J. M. Quashnock, “Pinning Down\nthe Axion,” Nucl. Phys. B 333(1990) 815–832. 4\n[88] M. Gorghetto, E. Hardy, and H. Nicolaescu,\n“Observing invisible axions with gravitational waves,”\nJCAP 06(2021) 034, arXiv:2101.11007 [hep-ph] . 5,\n22\n[89] O. Wantz and E. P. S. Shellard, “Axion Cosmology\nRevisited,” Phys. Rev. D 82(2010) 123508,\narXiv:0910.1066 [astro-ph.CO] . 5\n[90] W. H. Press, B. S. Ryden, and D. N. Spergel,\n“Dynamical Evolution of Domain Walls in an\nExpanding Universe,” Astrophys. J. 347(1989)\n590–604. 9\n[91] E. W. Kolb and I. I. Tkachev, “Nonlinear axion\ndynamics and formation of cosmological\npseudosolitons,” Phys. Rev. D 49(1994) 5040–5051,\narXiv:astro-ph/9311037 . 10, 45\n[92] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and\nOther Topological Defects . Cambridge University\nPress, 7, 2000. 16\n[93] N. Turok and P. Bhattacharjee, “Stretching cosmic\nstrings,” Phys. Rev. D 29(1984) 1557. 16\n[94] P. T. Boggs, R. H. Byrd, J. E. Rogers, and R. B.\nSchnabel, “User’s reference guide for ODRPACK\nversion 2.01: Software for weighted orthogonal\ndistance regression.” 1992. 22\n[95] M. Dine, N. Fernandez, A. Ghalsasi, and H. H. Patel,\n“Comments on axions, domain walls, and cosmic\nstrings,” JCAP 11(2021) 041, arXiv:2012.13065\n[hep-ph] . 22, 28, 45, 46\n[96] S. Borsanyi et al. , “Calculation of the axion mass\nbased on high-temperature lattice quantum\nchromodynamics,” Nature 539no. 7627, (2016) 69–71,\narXiv:1606.07494 [hep-lat] . 25, 26, 27, 30\n[97] G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and\nG. Villadoro, “The QCD axion, precisely,” JHEP 01(2016) 034, arXiv:1511.02867 [hep-ph] . 26\n[98] K. Saikawa and S. Shirai, “Primordial gravitational\nwaves, precisely: The role of thermodynamics in the\nStandard Model,” JCAP 05(2018) 035,\narXiv:1803.01038 [hep-ph] . 26\n[99] D. J. Fixsen, “The Temperature of the Cosmic\nMicrowave Background,” Astrophys. J. 707(2009)\n916–920, arXiv:0911.1955 [astro-ph.CO] . 26\n[100] Planck Collaboration, N. Aghanim et al. , “Planck\n2018 results. VI. Cosmological parameters,” Astron.\nAstrophys. 641(2020) A6, arXiv:1807.06209\n[astro-ph.CO] . [Erratum: Astron.Astrophys. 652, C4\n(2021)]. 26\n[101] A. Chaumet and G. D. Moore, “Misalignment vs\nTopology in Axion-Like Models,” arXiv:2108.06203\n[hep-ph] . 27\n[102] MADMAX Collaboration, P. Brun et al. , “A new\nexperimental approach to probe QCD axion dark\nmatter in the mass range above 40 µeV,” Eur. Phys.\nJ. C79no. 3, (2019) 186, arXiv:1901.07401\n[physics.ins-det] . 28\n[103] ALPHA Collaboration, A. J. Millar et al. , “Searching\nfor dark matter with plasma haloscopes,” Phys. Rev.\nD107no. 5, (2023) 055013, arXiv:2210.00017\n[hep-ph] . 28\n[104] B. T. McAllister, G. Flower, J. Kruger, E. N. Ivanov,\nM. Goryachev, J. Bourhill, and M. E. Tobar, “The\nORGAN Experiment: An axion haloscope above 15\nGHz,” Phys. Dark Univ. 18(2017) 67–72,\narXiv:1706.00209 [physics.ins-det] . 28\n[105] W. Zhang, A. Myers, K. Gott, A. Almgren, and\nJ. Bell, “AMReX: Block-Structured Adaptive Mesh\nRefinement for Multiphysics Applications,” arXiv\ne-prints (2020) , arXiv:2009.12009 [cs.MS] . 29\n[106] T. Andrade et al. , “GRChombo: An adaptable\nnumerical relativity code for fundamental physics,” J.\nOpen Source Softw. 6no. 68, (2021) 3703,\narXiv:2201.03458 [gr-qc] . 29\n[107] R. I. McLachlan and P. Atela, “The accuracy of\nsymplectic integrators,” Nonlinearity 5no. 2, (1992)\n541. 30\n[108] M. Yamaguchi and J. Yokoyama, “Lagrangian\nevolution of global strings,” Phys. Rev. D 66(2002)\n121303, arXiv:hep-ph/0205308 . 33\n[109] M. Yamaguchi and J. Yokoyama, “Quantitative\nevolution of global strings from the Lagrangian view\npoint,” Phys. Rev. D 67(2003) 103514,\narXiv:hep-ph/0210343 . 33\n[110] E. Hivon, K. M. Gorski, C. B. Netterfield, B. P. Crill,\nS. Prunet, and F. Hansen, “Master of the cosmic\nmicrowave background anisotropy power spectrum: a\nfast method for statistical analysis of large and\ncomplex cosmic microwave background data sets,”\nAstrophys. J. 567(2002) 2, arXiv:astro-ph/0105302 .\n37" }, { "title": "2401.17257v1.Low_Thrust_Trajectory_Design_Using_A_Semi_Analytic_Approach.pdf", "content": "AAS 22-565\nLOW THRUST TRAJECTORY DESIGN USING A SEMI-ANALYTIC\nAPPROACH\nMadhusudan Vijayakumar*, Ossama Abdelkhalik†\nSpace missions that use low-thrust propulsion technology are becoming increas-\ningly popular since they utilize propellant more efficiently and thus reduce mis-\nsion costs. However, optimizing continuous-thrust trajectories is complex, time-\nconsuming, and extremely sensitive to initial guesses. Hence, generating approxi-\nmate trajectories that can be used as reliable initial guesses in trajectory generators\nis essential. This paper presents a semi-analytic approach for designing planar and\nthree-dimensional trajectories using Hills equations. The spacecraft is assumed to\nbe acted upon by a constant thrust acceleration magnitude. The proposed equa-\ntions are employed in a Nonlinear Programming Problem (NLP) solver to obtain\nthe thrust directions. Their applicability is tested for various design scenarios like\norbit raising, orbit insertion, and rendezvous. The trajectory solutions are then\nvalidated as initial guesses in high-fidelity optimal control tools. The usefulness\nof this method lies in the preliminary stages of low-thrust mission design, where\nspeed and reliability are key.\nINTRODUCTION\nDesign of space missions that utilize continuous low thrust propulsion is becoming increasingly\npopular as they utilize propellant more effectively and ever since has gained much attention in the\nliterature. One of the fundamental task of such a mission is the design of spacecraft trajectory\nthat delivers the spacecraft from a given state to a desired state within a specified amount of time.\nHowever, this process is highly challenging, because there are too many trajectory parameters like\nlaunch and arrival date, position, velocity, etc which need analyzing. In addition, to this, long\nduration thrust profiles associated with low-thrust spacecraft must be obtained. The search space\nassociated with these parameters is very large and further compound the complexity of the trajectory\ndesign problem.\nAt the preliminary stage, thousands, if not millions of possible trajectories are evaluated before\nobtaining a feasible trajectory for the mission. The generation of guidance trajectory is treated\nas an optimization problem and usually takes several days or even months to complete. For a\ngeneral case of low-thrust trajectory design, no analytical closed form solution exists till date.1The\ntrajectory is designed typically as a boundary value problem using direct or indirect optimization\nmethods.2The indirect method resorts to calculus of variation while the direct method utilizes\nnonlinear programming to solve this problem. Direct methods require initial guesses while indirect\nmethods are extremely sensitive to initial guesses and have a very small radius of convergence.\nHence direct methods are usually preferred in trajectory design. One of the noticeable works in the\n*Graduate Student, Department of Aerospace Engineering, Iowa State University, USA.\n†Corresponding Author, Associate Professor, Department of Aerospace Engineering, Iowa State University, Ames, Iowa\n50011, AIAA senior member.\n1arXiv:2401.17257v1 [math.OC] 30 Jan 2024direction of direct method was proposed by Jon A. Sims et al.3In this work, the trajectory design\nproblem was divided into multiple discrete segments and solved using an Non-Linear Programming\n(NLP). Additionally, the branch of shape based methods introduced by Petropolous et al.4improve\nthe convergence of direct solvers by providing approximate initial trajectory solutions. Early work in\nthis direction include the exponential sinusoidal4and inverse polynomial method5in which radical\nvector of spacecraft is written as a function of the transfer angle. These methods are useful in\ngenerating orbit raising trajectories. The shaping pseudoequinoctial6and Finite Fourier Series7\nmethod extended the shape based methods to generate trajectory solutions to a variety of problems\nlike rendezvous, orbit insertion, etc capable of handling thrust constraints.\nOne of the primary purpose of the approximate trajectory generation methods is to provide a\nquick evaluation of the search space to look for regions of the design space associated with lower\nmission costs and avoid lengthy unnecessary calculations. To support this purpose several authors\nhave studied orbit motion and provided analytic solutions for some special cases of orbit motion.\nStudying the special cases of radial thrust for escape trajectories from circular orbits, Tsien,8Boltz,9\nPrussing10and Mengali11developed analytical solutions of orbit motion. Megali et al.12later\nextended this study for elliptical orbit departure. Similarly, Zee,13Boltz,14and Benney15studied\nthe case of tangential thrust for continuous low thrust trajectories. Furthermore, Gao16presented\nan averaging technique to obtain analytical solution in case of tangential thrust. Although those\nmethods can, to a large extent, simplify the continuous low thrust problem, they are only suitable\nfor some special cases.\nOne of the key analytical contribution namely Clohessy–Wiltshire (CW) equations17and the\nTschauner-Hempel equations18that were developed in the 1960s changed the game of trajectory\ndesign. These equations provide linear models for the study of spacecraft relative dynamics in cir-\ncular and elliptic orbits. For decades, these sets of equations have been the reference model for the\ndesign of relative guidance, navigation, and control systems.19While the original formulation of\nthe CW equations captures only the rectilinear relative states of the spacecraft, more recent devel-\nopments by Alfriend et al.20and De Bruijn et al.21formulate the CW equations using curvilinear\nrelative states. However, these formulations assume that the spacecraft is not acted upon by any\norbital perturbation. Later, Leonard et al.,22Humi et al.,23Carter et al.,24incorporated the effects\nof atmospheric drag into these formulations. Bevilacqua et al.25demonstrated the extension of this\nformulation into the domain of continuous low thrust.\nRecent work by Takao et al.26simplified the state-space formulation of the two-dimensional\nCW equations with low thrust propulsion as a perturbed acceleration and demonstrated the design\nof gravity assist interplanetary trajectories. Further, Toshinori et al.27extended this work to pro-\nvide a flexible orbit design method for designing low thrust missions. Taking inspiration from the\nstate space implementation of the continuous low thrust CW equations, this paper provides a three-\ndimensional linear formulation of the Clohessy–Wiltshire equations. The 3D analytic approxima-\ntion of the CW equations are then used to develop a robust general semi-analytic trajectory design\nalgorithm that is capable of generating both two-dimensional and three-dimensional low thrust tra-\njectories. The applicability of the algorithm is tested for various design scenarios like orbit raising,\norbit insertion and rendezvous. The accuracy of the trajectory solutions is evaluated by drawing\ncomparison to numerical propagation. The key feature of this algorithm is in its ability to handle\nmany revolution low thrust trajectories.\nThis paper is organized as follows. Section II summarizes the low thrust analytic approximation\nof the Clohessy–Wiltshire equations. Section III presents a brief description for the trajectory design\n2algorithm. Section IV illustrates some numerical examples of both 2D and 3D trajectory design.\nSection V presents the conclusion of this work.\nLOW THRUST ANALYTIC APPROXIMATION OF HILL’S EQUATIONS\nThe Clohessy-Wiltshire equations, also referred as Hill’s equations are a set of linear, time invari-\nant differential equations used to describe the motion of a spacecraft relative to a reference orbit.\nThe reference orbit is usually a circle and the corresponding coordinate frame of the relative motion\nis shown in Fig 1. The X−Y−Zframe represents the inertial coordinate system and x−y−z\nrepresents the Hill’s frame. In our case, X−Y−Zrepresents the Earth Centered Inertial Frame\n(ECI). The x,yandzcoordinates of the Hill’s frame are the radial, along-track and cross-track dis-\nplacements of the spacecraft relative to the reference orbit. The primary goal is to obtain an analytic\nFigure 1. Coordinate Frames\nFigure 2. Hills Frame\napproximation Consider the two-body Hill’s equation as shown in Eq. (1).\n¨x−2x n2−2 ˙y n=ax (1a)\n¨y+ 2n˙x=ay (1b)\n¨z+z n2=az (1c)\nwhere, ax,ayandazare the components of the thrust acceleration represented in the Hill’s frame.\nThe phase angle of the in-plane thrust acceleration component varies linearly while that of the out-\nof-plane component is assumed to be constant along each time step as show in Fig 2. Let α0be the\ninitial phase angle of the in-plane thrust acceleration component and kbe the rate at which it rotates\nin the Hill’s frame. Then the thrust acceleration components can be represented as:\nax=asin(β)cos(α0−kt)\nay=asin(β)sin(α0−kt)\naz=acos(β)(2)\n3The primary task is to obtain an approximate analytic solution of the Clohessy-Wiltshire equations\nfor spacecrafts employing continuous low thrust. One way to solve the second-order homogeneous\ndifferential equations in Eq. (1), is to use Laplace transform. The out-of-plane motion in Eq. (1c) is\nuncoupled from the in-plane motion i.e. only the zcomponent appears and is the simplest among\nthem to solve. This equation resembles the equation of an undamped harmonic oscillator with a\nforcing (or input) function on the right-hand side. Taking the Laplace transform of and rewriting\nthe equation we get:\nZ(s) =˙z0+s z0+asinβ\ns\nn2+s2(3)\nwhere, sis a complex number, z0and˙z0are the z-component of the velocity vector at t= 0. Taking\nthe inverse Laplace transform of Eq. (3) we obtain:\nz(t) =n2z0cos (n t)−acos (n t) sin ( β) +n˙z0sin (n t)\nn2+asin (β)\nn2(4)\nDifferentiating the above equation we get:\n˙z(t) = ˙z0cos (n t)−n z0sin (n t) +asin (n t) sin ( β)\nn(5)\nFrom Eq. (4) and Eq. (5) notice that the out-of-plane motion is only influenced by the initial position\nz0, initial velocity ˙z0and time t. Thus, we can write,\nz(t) =fz(z0,˙z0, β, t)\n˙z(t) =f˙z(z0,˙z0, β, t)(6)\nOn the contrary, Eq. (1a) and (1b) highlights that the planar motion of the spacecraft along xandy\ndirections are coupled to each other. The closed form analytic solution for the in-plane position and\nvelocity are obtained as follows. Start from Eq. (1a) and differentiate it to get:\n...x= 2n¨y+ 3n2˙x+a ksin (α0−k t) cos ( β) (7)\nNow, substitute ¨y=−2n˙x+acos(β) sin ( α0−k t)in the...xto get:\n...x=−n2˙x+asin (α0−k t) cos ( β) (k+ 2n); (8)\nTake the Laplace transform of the ¨xto get,\nX(s) =¨x0+s˙x0+n2x0+s2x0−acos(β) (kcos(α0)−ssin(α0)) (k+2n)\nk2+s2\nn2s+s3(9)\nNow, taking the inverse Laplace transform transform and substituting ¨x0from Eq. (1a) we get an\nexpression for x(t). Further differentiating the x(t)equation, we can arrive an expression for the\nvelocity ˙x(t). Notice that both x(t)and˙x(t)are only a function of the planar components of the\ninitial states as shown in Eq. (10). Detailed expressions for the position and velocity x(t)and˙x(t)\nis presented in the Appendix.\nx(t) =fx(x0,˙x0, α0, β, k, t )\n˙x(t) =f˙x(x0,˙x0, α0, β, k, t )(10)\nBy substituting the expression for ˙x(t)from Eq. (10) into the Hill’s equation in Eq. (1b), one can\neasily obtain expressions for ˙y(t)andy(t)by successive integration with respect to time. This\n4would result in both y(t)and˙y(t)expressed as functions of the planar components of the initial\nstates as shown in Eq. (11). Detailed expressions for the position and velocity y(t)and˙y(t)is\npresented in the Appendix.\ny(t) =fy(x0,˙x0,˙y0, α0, β, k, t )\n˙y(t) =f˙y(x0,˙x0,˙y0, α0, β, k, t )(11)\nSince the closed-form analytic solution of the spacecraft states is obtained by the linearization of\nHill’s equations, it is important to test the accuracy of the solution. One way to estimate the accuracy\npreserved by this method is to examine the deviation in the spacecraft states compared to its intended\nstates. This can be easily done by numerically propagating the states of the spacecraft using the\ntwo-body equations of motion and comparing the results with the analytic approximation. This\nexperiment was conducted for a series of orbits ranging from LEO to GEO with varying initial\nconditions. However, only two of the cases are demonstrated here since the general trend remained\nthe same.\nIn the first case, the spacecraft starts an initial circular low Earth orbit (a = 6678 km, e = 0) with\na period of 90min. It was assumed that the spacecraft was acted upon by a constant magnitude\nthrust acceleration of 9e−8km/s2. The initial thrust steering angles were kept at α0= 90◦and\nβ= 45◦. Figure 3 shows the corresponding positional difference between the analytical approxi-\nmation and the two-body numerical propagation of the spacecraft. Solid curves are used to indicate\nthe positional difference along each component in the ECI space, and the dashed curve represents\nthe least square fit of the positional error. For circular orbits, the analytical approximation closely\nFigure 3. Hill’s vs Keplerian Motion - Circular\n Figure 4. Hill’s vs Keplerian Motion - Eccentric\nmatches the Keplerian motion. From Fig 3, one can notice that the maximum positional difference\nis about 0.3 km for the first ten revolutions of the spacecraft. This is the consequence of the circular\norbit restriction assumed in Hill’s equations. To further examine the accuracy of the analytic ap-\nproximation, the motion of the spacecraft starting from an eccentric orbit is studied. In this case, the\nspacecraft starts an initial eccentric low Earth orbit (a = 8164 km, e = 0.17) with a period of 120\nmin. It was assumed that the spacecraft was acted upon by a constant magnitude thrust acceleration\nof9e−8km/s2. The initial thrust steering angles were kept at α0= 90◦andβ= 45◦. Fig-\nure 3 shows the corresponding positional difference between the analytical approximation and the\ntwo-body numerical propagation of the spacecraft. Solid curves are used to indicate the positional\ndifference along the each component in the ECI space and the dashed curve represents the least\n5square fit of the positional error. Notice there is a significant decrease in the prediction accuracy of\nthe analytic approximation compared to Keplerian motion. This is not unexpected as the analytic\napproximation assumes circular motion and the ωtterm experiences large variations in eccentric\norbits. One way to overcome this is to restart Hill’s propagation at the end of each revolution as-\nsuming circular orbits. However, the ωtterm would still show maximum deviation from the mean\nvalue for each revolution. This effect is exaggerated in lower altitudes due to shorter orbital periods.\nHence, eccentric orbits in LEO are the worst place to use the analytic approximation.\nAdditional observations from the series of these experiments are summarized as follows. Firstly,\nerror in the decoupled out-of-plane motion are very small compared to the error in the in-plane\nmotion. This can be associated with the fact that the sources for error in the z-direction is half as that\nof the x−yplane. Secondly, irrespective of the initial orbit, the error in the analytical approximation\nis negligible for propagation times up to one orbital period of the reference orbit. Beyond this, the\nerrors associated with the analytical approximation increases due to reasons mentioned before.\nTRAJECTORY DESIGN\nThis section provides details on the trajectory design method employed using the analytic approx-\nimation of Hill’s equations for low thrust trajectories. The error from the circular orbit restriction in\nthe analytic approximation is evident looking at the positional errors studied in the previous section.\nTo overcome this issue, the entire trajectory design is divided into ’m’ segments. Each segment of\nthe trajectory is then approximated using the analytic solution. Given the initial conditions (initial\nstates), the trajectory design algorithm iterates over the time of flight (ToF), the thrust steering an-\ngles ( αandβ) and the rate of change of thrust direction ( k) along each segment to achieve specific\ntarget conditions. The target orbits discussed in this work correspond to either orbit raising, orbit\ninsertion or rendezvous problems. The continuity of the trajectory is enforced by using the final\nstates of the earlier segment as the initial condition of the later segment.\nThe entire trajectory design problem is formulated as an optimization problem where the objective\nis to match the states of the spacecraft in the target orbit while minimizing the total ∆V. The design\nand objective space of the optimization problem is manifested differently based on the problem\nbeing solved. For a rendezvous problem, the design variables are the time of flight (ToF), the\nthrust steering angles ( αandβ) and the rate of change of thrust direction ( k) along each segment.\nThe target states of the spacecraft are enforced as an equality constraint while the objective is to\nminimize the total ∆V. The formulation is shown as follows:\nMinimize J= ∆ V (12)\nSubject to C:\naf\nef\nif\nωf\nΩf\nθf\n=\nat\net\nit\nωt\nΩt\nθt\n(13)\nDesign Variables X= [ToF, α 0, βi, ki]1×2m+2, i= 1,2,· · ·m (14)\nwhere, [ a, e, i, ω, Ω, θ] are the classical orbital elements. The subscript ’f’ represents the final states\nof the spacecraft at the end of the analytic approximation and the subscript ’t’ represents the target\norbit. The orbit insertion problem is treated as a special case of the rendezvous problem where the\ntrue anomaly ( θ) is set free while the rest of the formulation remains the same.\n6For an orbit raising problem, the time of flight becomes a fixed parameter while the objective is to\nincrease the semi-major axis of the final orbit. Additionally, there is no need to enforce any equality\nconditions. The formulation of the optimization problem is as follows:\nMaximize J=af (15)\nDesign Variables X= [α0, βi, ki]1×2m+1, i= 1,2,· · ·m (16)\nCASE STUDIES\nAll the calculations for the case studies are carried out in canonical units such that one distance\nunit (DU) is equal to the radius of the Earth of 6378.14 km, one time unit (TU) is 806.8 seconds.\nThe canonical units were scaled up by a scaling factor (p) for the case studies that start at high\nEarth Orbits (HEO) and Geo Stationary Orbits (GEO) as specified in their corresponding sections.\nAll the calculations were carried out in Matlab. The in-built function ‘fmincon’ was used as the\noptimization routine. All cases were simulated using Intel(R) Xeon(R) Quadcore Processor @\n3.50GHz.\nCase 1: Coplanar Transfer LEO-to-GEO (Orbit Insertion)\nA low-thrust trajectory design problem from an initial circular low Earth orbit (LEO) at an altitude\nhi= 300 km to a final circular Geostationary Orbit (GEO) of altitude hf= 35,785kmis studied\nin this section. This is a typical orbit insertion problem where the spacecraft is acted upon by a fixed\nmagnitude of thrust acceleration while the direction is variable. For this problem, the parameters\nof the spacecraft’s propulsion system are defined as follows: specific impulse Isp = 3300s, initial\nspacecraft mass = 95kg, thrust-to-weight ratio = 10−4and maximum thrust limit = 0.0920Nand\nconstant thrust acceleration of Ta = 9.81e−7m/s2.\nThe trajectory solution provided by the proposed semi-analytic method is shown in Fig 5. The\nFigure 5. Case 1: Trajectory (ECI Frame)\n Figure 6. Case 1: History of α\nspacecraft takes 61.80days to transfer from LEO to GEO performing Nrev= 369 revolutions\naround the Earth. In doing so, it consumes 14.01kgof propellant and achieves a ∆V= 5.23km/s .\nThe corresponding thrust profile is an oscillating sinusoid in the xandydirection. The history\nof the thrust steering angle is as shown in Fig 6. Figures 7 and 8 illustrate the errors associated\nwith the position and velocity of the analytically approximate trajectory compared to numerical\n7Figure 7. Case 1: Positional Error\n Figure 8. Case 1: Velocity Error\npropagation. Additionally, Table 1 provides a detailed list of the obtained solution characteristics.\nThe final states of the spacecraft achieved by the semi-analytic method and numerical propagation\nTable 1. Case 1: Performance Characteristics of the Semi-Analytic Solution\nParameters Value\nNo. of revolutions 338\nTime of Flight (days) 45.44\n∆V(km/s) 3.85\nPropellant Consumption (kg) 10.52\nare compared in Table 2 to demonstrate the level of error between them. From the table it one can\nnotice that the error in matching the target position is about 65km and that of target velocity is about\n0.005km/s. This small error in matching the final conditions obtained by numerical propagation\ndemonstrates the ability of the semi-analytic solution to provide high accuracy solutions.\nTable 2. Case 1: Semi-Analytic Solution vs Numerical Propagation\nTarget orbit Parameters Semi-Analytic Solution Numerical Propagation\nFinal Position (km) 42165 42100\nFinal Velocity (km/s) 3.0746 3.0751\nSemi-major axis, a(km) 42165.18 42047.93\nEccentricity, e(km) 0 0.001\nCase 2: Coplanar Transfer LEO-to-MEO (Rendezvous)\nThe transfer maneuver from a low Earth circular orbit to a medium Earth circular orbit is con-\nsidered here. The initial and final altitudes are hi= 300 kmandhf= 2000 kmrespectively. The\nparameters of the spacecraft are defined as follows: specific impulse Ispis3300s, initial space-\ncraft mass is 95kg, thrust-to-weight ratio = 10−4and maximum thrust limit is 0.0920N. The\nboundary conditions(BC) of the initial and target orbit at initial time is tabulated in Table 3. The\ncorresponding trajectory solution is shown in Fig 9. The entire trajectory was divided into 68 seg-\n8Table 3. Case 2: Design Parameters\nParameter Initial Orbit Target Orbit\nSemi-major axis, a(km) 6678.18 8378.18\nEccentricity, e(km) 0 0\nInclination, i(deg) 0 0\nArgument of Periapsis, ω(deg) 0 0\nRAAN, Ω(deg) 0 0\nTrue Anomaly, θf 0◦90◦\nFigure 9. Case 2: Trajectory (ECI Frame)\n Figure 10. Case 2: History of α\nments each corresponding to an altitude increase of 25 km. The spacecraft performs 268 revolutions\naround the Earth in 9.91 days to achieve rendezvous condition with a secondary spacecraft in the\nMEO. To achieve this maneuver, the spacecraft consumes 2.4 kg of propellant with an associated\n∆V= 0.8406 km/s for this transfer. Figure 10 shows the history of the thrust steering angle α. The\nperformance characteristics of the obtained solution are tabulated in Table 4\nFigure 11. Case 2: Positional Error\n Figure 12. Case 2: Velocity Error\n9Table 4. Case 2: Performance Characteristics of the Semi-Analytic Solution\nParameters Value\nNo. of segments 68\nNo. of revolutions 268\nTime of Flight (days) 9.91\n∆V(km/s) 0.8406\nPropellant Consumption (kg) 2.4\nFigures 11 and 12 illustrate the positional and velocity error between the semi-analytic solution\nand two-body numerical propagation of the spacecraft. Solid black curves are used to highlight the\nleast square fit of the positional and velocity error. The final states of the spacecraft achieved by the\nsemi-analytic method and numerical propagation are compared in Table 5 to demonstrate the level\nof error between them.\nTable 5. Case 2: Semi-Analytic Solution vs Numerical Propagation\nTarget orbit Parameters Semi-Analytic Solution Numerical Propagation\nSemi-major axis, a(km) 8378.18 8375.95\nEccentricity, e(km) 0.00002 0.0003\nInclination, i(deg) 0 0\nArgument of Periapsis, ω(deg) 0 0\nRAAN, Ω(deg) 0 0\nTrue Anomaly, θf 60◦50◦\nCase 3: 3D Transfer LEO-to-GEO (Orbit Insertion)\nA three dimensional transfer from an initial circular low Earth orbit (LEO) at hi= 300 km al-\ntitude and ii= 28 .5◦inclination to a final circular Geostationary Orbit (GEO) of altitude hf=\n35,785kmandif= 0◦inclination is considered in this section. This LEO-to-GEO orbit insertion\nproblem is of practical importance considering the remote sensing, weather forecasting, Earth obser-\nvation applications and hence extensively studied72829in literature. The initial orbit was so chosen\nsince the Kennedy Space Center is at 28.5◦North latitude and hence cheaper to launch spacecrafts\ninto a parking orbit on the same latitude. In this study, the parameters of the spacecraft are defined\nas follows: specific impulse, Isp= 2800 s, initial spacecraft mass, m0= 95 kg, thrust-to-weight\nratio = 10−4and maximum thrust limit is 0.0920N. The entire trajectory design problem was di-\nvided into 3550 segments, roughly corresponding to an altitude change of 10kmper segment. The\ntrajectory solution and the history of the thrust steering angles are shown in Figure 13 and 14. The\nout-of-plane thrust component is used to reduce the initial inclination set by the parking orbit, while\nthe in-plane thrust component raises the orbit semi-major axis to the desired geostationary level.\nThe obtained solution performs Nrev= 512 revolutions and has a total flight time of 86.01days .\nThe total ∆Vfor the transfer is 7.29km/s and the spacecraft consumes 21.86kgof propellant.\nThe least square fit of the errors are represented using solid black curves. The performance\ncharacteristics of the obtained solution are tabulated in Table 6.\n10Figure 13. Case 3: Trajectory (ECI Frame)\n Figure 14. Case 3: History of αandβ\nTable 6. Case 3: Performance Characteristics of the Semi-Analytic Solution\nParameters Value\nNo. of segments 3550\nNo. of revolutions 512\nTime of Flight (days) 86.01\n∆V(km/s) 7.29\nPropellant Consumption (kg) 21.86\nCase 4: 3D Transfer Earth-to-Mars (Rendezvous)\nA low thrust Earth-Mars transfer problem is studied in this section. The spacecraft is expected to\nrendezvous with Mars starting from Earth given the low thrust limitation. This problem is of prime\ninterest to the aerospace community as the global community of scientists and engineers hope to\ncolonize Mars within the next few decades. Low thrust trajectories would be key to facilitate regular\ntransportation of supplies and cargo between Earth and Mars. In this case study, the date of launch\nof the spacecraft departing from Earth is assumed to be fixed and arbitrarily chosen to be the 20th\nof July 2023 marking 54 years since the first Moon landing. The orbital elements of Earth and Mars\non this date is shown in Table 7. The spacecraft propulsion system characteristics for this study\nTable 7. Case 4: Design Parameters\nParameter Earth Orbit Mars Orbit\nSemi-major axis (AU) 1 1.52366231\nEccentricity 0.01671022 0.09341233\nInclination (deg) 0.00005 1.85061\nLongitude of Perihelion (deg) 102.94719 336.04084\nLongitude of ascending node (deg) -11.26064 49.57854\nTrue Anomaly (deg) 194.72◦201.99◦\nare detailed as follows: specific impulse, Isp= 2800 s, initial spacecraft mass, m0= 1000 kg,\nthrust-to-weight ratio = 10−6and maximum thrust limit = 0.098N. The trajectory design process\nis carried out by dividing the entire trajectory into 50 segments. All the calculations for this case\n11Figure 15. Case 4: Trajectory (ICRF)\n Figure 16. Case 4: History of αandβ\nstudy are carried out using canonical units such that one distance unit (DU) is equal to the radius\nof the Earth’s orbit around the Sun ( 1.496e+ 8km). The trajectory solution obtained by the semi-\nanalytic method is shown in Fig. 15. It is important to note that the algorithm simply returns a\nfeasible rendezvous trajectory and no optimal solution is claimed by this method. The history of\nthe thrust steering angles are shown in Fig. 16. Note that the states (position and velocity) of\nEarth and Mars for this problem was obtained from the DE43030Ephemeris data found on the JPL\nwebsite. The spacecraft takes 965.33days to rendezvous with Mars. In other words, the spacecraft\narrives at Mars on 11th March 2026. To perform this transfer, the spacecraft consumes 261.87kgof\npropellant, performing 2 revolutions around the Earth. The total ∆Vof this transfer is 8.34km/s .\nThe performance characteristics of the trajectory solution are tabulated in Table 8. Figures 17\nTable 8. Case 4: Performance Characteristics of the Semi-Analytic Solution\nParameters Value\nNo. of segments 50\nNo. of revolutions 2\nTime of Flight (days) 965.33\n∆V(km/s) 8.34\nPropellant Consumption (kg) 261.87\nand 18 highlight the positional and velocity error between the semi-analytic solution and two-body\nnumerical propagation of the spacecraft.\nCONCLUSION\nThe work developed in this paper presents a semi-analytic approach for the generation of initial\nguess guidance trajectories for low-thrust spacecrafts. A modification on the Hill’s equations is\nprovided as a means to approximate the states of the spacecraft acted upon by a constant low-thrust\nacceleration. Numerical results demonstrates the flexibility of the algorithm in generating three-\ndimensional rendezvous, orbit insertion and orbit raising trajectory solutions.\n12Figure 17. Case 4: Positional Error\n Figure 18. Case 4: Velocity Error\nACKNOWLEDGEMENT\nThis paper is based upon work supported by NASA, Grant Number 80NSSC19K1642\n13APPENDIX-A\nAnalytic expression for the in-plane motion of the spacecraft.\nx(t) =2k3˙y0−4k n3x0+ 4k3n x0−2k n2˙y0−2k3˙y0cos (n t)\nk n(k2−n2)+k3˙x0sin (n t)−2a k2cos (α0) cos ( β) + 2 a n2cos (α0) cos ( β)\nk n(k2−n2)\n+3k n3x0cos (n t)−3k3n x0cos (n t) + 2 k n2˙y0cos (n t)\nk n(k2−n2)\n+−k n2˙x0sin (n t) + 2 a k2cos (n t) cos ( α0) cos ( β)−2a n2cos (k t) cos ( α0) cos ( β)\nk n(k2−n2)\n+a k2sin (n t) cos ( β) sin ( α0)−2a n2sin (k t) cos ( β) sin ( α0)−a k n cos (k t) cos ( α0) cos ( β)\nk n(k2−n2)\n++a k n cos (n t) cos ( α0) cos ( β)−a k n sin (k t) cos ( β) sin ( α0) + 2 a k n sin (n t) cos ( β) sin ( α0)\nk n(k2−n2)\n(17)\n(18)˙x(t) =k2˙x0cos (n t)−n2˙x0cos (n t) + 2 k2˙y0sin (n t)\nk2−n2+−3n3x0sin (n t)−2n2˙y0sin (n t) + 3 k2n x0sin (n t)\nk2−n2\n+−a kcos (k t) cos ( β) sin ( α0) +a ksin (k t) cos ( α0) cos ( β) +a kcos (n t) cos ( β) sin ( α0)\nk2−n2\n+−2a ksin (n t) cos ( α0) cos ( β)−2a ncos (k t) cos ( β) sin ( α0) + 2 a nsin (k t) cos ( α0) cos ( β)\nk2−n2\n+2a ncos (n t) cos ( β) sin ( α0)−a nsin (n t) cos ( α0) cos ( β)\nk2−n2\n14(19)y(t) =y0+t\u0000\n3k2−3n2\u0001\n(k˙y0−acos (α0) cos ( β) + 2 k n x 0)\nk(n2−k2)\n+2ksin (n t)\u0000\n−3x0k2n−2 ˙y0k2+ 2acos (α0) cos ( β)k+ 3x0n3+ 2 ˙y0n2+acos (α0) cos ( β)n\u0001\nk(n3−k2)\n−2kcos (n t)\u0000\n˙x0k2+acos (β) sin ( α0)k−˙x0n2+ 2acos (β) sin ( α0)n\u0001\nk(n3−k2)\n+asin (α0−k t) cos ( β)\u0000\nk2−n2\u0001\nk2(n2−k)+2a ncos (k t) cos ( β) sin ( α0) (k+ 2n)\nk2(n2−k)\n−2a nsin (k t) cos ( α0) cos ( β) (k+ 2n)\nk2(n2−k)−2 ˙x0k2+ 2acos (β) sin ( α0)k+ 3a ncos (β) sin ( α0)\nk2n\n(20)˙y(t) = ˙y0+acos (α0−k t) cos ( β)σ1\nk(k2−n2)\n−cos (n t)\u0000\n−6x0k2n−4 ˙y0k2+ 4acos (α0) cos ( β)k+ 6x0n3+ 4 ˙y0n2+ 2acos (α0) cos ( β)n\u0001\nk2−n2\n−sin (n t)\u0000\n2 ˙x0k2+ 2acos (β) sin ( α0)k−2 ˙x0n2+ 4acos (β) sin ( α0)n\u0001\nk2−n2+2a ncos (k t) cos ( α0) cos ( β) (k+ 2n)\nk(k2−n2)\n+2a nsin (k t) cos ( β) sin ( α0) (k+ 2n)\nk(k2−n2)−4k˙y0−3acos (α0) cos ( β) + 6 k n x 0\nk\n15REFERENCES\n[1] B. A. Conway, Spacecraft trajectory optimization , V ol. 29. Cambridge University Press, 2010.\n[2] J. T. Betts, “Survey of numerical methods for trajectory optimization,” Journal of guidance, control,\nand dynamics , V ol. 21, No. 2, 1998, pp. 193–207.\n[3] J. Sims, P. Finlayson, E. Rinderle, M. Vavrina, and T. Kowalkowski, “Implementation of a low-thrust\ntrajectory optimization algorithm for preliminary design,” AIAA/AAS Astrodynamics specialist confer-\nence and exhibit , 2006, p. 6746.\n[4] A. E. Petropoulos and J. M. Longuski, “Shape-Based Algorithm for Automated Design of Low-Thrust,\nGravity-Assist Trajectories,” Journal of Spacecraft and Rockets , V ol. 41, No. 5, 2004, p. 787–796,\n10.2514/1.13095.\n[5] B. J. Wall and B. A. Conway, “Shape-based approach to low-thrust rendezvous trajectory design,” Jour-\nnal of Guidance, Control, and Dynamics , V ol. 32, No. 1, 2009, pp. 95–101, 10.2514/1.36848.\n[6] P. De Pascale and M. Vasile, “Preliminary design of low-thrust multiple gravity-assist trajectories,”\nJournal of Spacecraft and Rockets , V ol. 43, No. 5, 2006, pp. 1065–1076.\n[7] E. Taheri and O. Abdelkhalik, “Shape based approximation of constrained low-thrust space trajec-\ntories using Fourier series,” Journal of Spacecraft and Rockets , V ol. 49, No. 3, 2012, pp. 535–546,\n10.2514/1.A32099.\n[8] H. Tsien, “Take-off from satellite orbit,” Journal of the American Rocket Society , V ol. 23, No. 4, 1953,\npp. 233–236.\n[9] F. W. Boltz, “Orbital motion under continuous radial thrust,” Journal of Guidance, Control, and Dy-\nnamics , V ol. 14, No. 3, 1991, pp. 667–670.\n[10] J. E. Prussing and V . Coverstone-Carroll, “Constant radial thrust acceleration redux,” Journal of guid-\nance, control, and dynamics , V ol. 21, No. 3, 1998, pp. 516–518.\n[11] G. Mengali and A. A. Quarta, “Rapid solar sail rendezvous missions to asteroid 99942 Apophis,” Jour-\nnal of Spacecraft and Rockets , V ol. 46, No. 1, 2009, pp. 134–140.\n[12] G. Mengali and A. A. Quarta, “Escape from elliptic orbit using constant radial thrust,” Journal of\nGuidance, Control, and Dynamics , V ol. 32, No. 3, 2009, pp. 1018–1022.\n[13] C.-H. Zee, “Low tangential thrust trajectories improved first-order solution,” AIAA Journal , V ol. 6,\nNo. 7, 1968, pp. 1378–1379.\n[14] F. W. Boltz, “Orbital motion under continuous tangential thrust,” Journal of Guidance, Control, and\nDynamics , V ol. 15, No. 6, 1992, pp. 1503–1507.\n[15] D. Benney, “Escape from a circular orbit using tangential thrust,” Journal of Jet Propulsion , V ol. 28,\nNo. 3, 1958, pp. 167–169.\n[16] Y . Gao and C. Kluever, “Analytic orbital averaging technique for computing tangential-thrust trajecto-\nries,” Journal of guidance, control, and dynamics , V ol. 28, No. 6, 2005, pp. 1320–1323.\n[17] W. Clohessy and R. Wiltshire, “Terminal guidance system for satellite rendezvous,” Journal of the\nAerospace Sciences , V ol. 27, No. 9, 1960, pp. 653–658.\n[18] J. Tschauner and P. Hempel, “Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel,” Astronau-\ntica Acta , V ol. 11, No. 2, 1965, pp. 104–+.\n[19] W. Fehse, Automated rendezvous and docking of spacecraft , V ol. 16. Cambridge university press, 2003.\n[20] K. T. Alfriend, S. R. Vadali, P. Gurfil, J. P. How, and L. Breger, Spacecraft formation flying: Dynamics,\ncontrol and navigation , V ol. 2. Elsevier, 2009.\n[21] P. Righetti, F. Sancho, D. Lazaro, and A. Damiano, “Handling of conjunction warnings in EUMETSAT\nflight dynamics,” Journal of Aerospace Engineering, Sciences and Applications , V ol. 3, No. 2, 2011,\npp. 39–53.\n[22] C. LEONARD, W. HOLLISTER, and E. BERGMANN, “Orbital formationkeeping with differential\ndrag,” Guidance, Navigation and Control Conference , 1987, p. 2402.\n[23] M. Humi and T. Carter, “Rendezvous equations in a central-force field with linear drag,” Journal of\nGuidance, Control, and Dynamics , V ol. 25, No. 1, 2002, pp. 74–79.\n[24] T. Carter and M. Humi, “Clohessy-Wiltshire equations modified to include quadratic drag,” Journal of\nGuidance, Control, and Dynamics , V ol. 25, No. 6, 2002, pp. 1058–1063.\n[25] R. Bevilacqua and M. Romano, “Rendezvous maneuvers of multiple spacecraft using differential drag\nunder J2 perturbation,” Journal of Guidance, Control, and Dynamics , V ol. 31, No. 6, 2008, pp. 1595–\n1607.\n[26] Y . Takao and T. Chujo, “Delta-V Earth-Gravity-Assist Trajectories with Hybrid Solar Electric–Photonic\nPropulsion,” Journal of Guidance, Control, and Dynamics , V ol. 45, No. 1, 2022, pp. 162–170.\n16[27] T. Ikenaga, M. Utashima, N. Ishii, Y . Kawakatsu, and M. Yoshikawa, “Interplanetary parking method\nand its applications,” Acta Astronautica , V ol. 116, 2015, pp. 271–281.\n[28] C. A. Kluever, “Simple guidance scheme for low-thrust orbit transfers,” Journal of Guidance, Control,\nand Dynamics , V ol. 21, No. 6, 1998, pp. 1015–1017.\n[29] J. P. Shepard, A Preliminary Study of Leo to Geo Transfers for Inclination Changes Using Libration\nPoint Orbits . PhD thesis, The University of North Dakota, 2020.\n[30] W. M. Folkner, J. G. Williams, D. H. Boggs, R. S. Park, and P. Kuchynka, “The planetary and lunar\nephemerides DE430 and DE431,” Interplanetary Network Progress Report , V ol. 196, No. 1, 2014,\npp. 42–196.\n17" }, { "title": "2401.17310v1.eROSITA_narrowband_maps_at_the_energies_of_soft_X_ray_emission_lines.pdf", "content": "Astronomy &Astrophysics manuscript no. aa ©ESO 2024\nJanuary 31, 2024\neROSITA narrowband maps at the energies of soft X-ray emission\nlines\nXueying Zheng1, Gabriele Ponti2,1, Nicola Locatelli2,1⋆, Jeremy Sanders1, Andrea Merloni1, Werner Becker1, Johan\nComparat1, Konrad Dennerl1, Michael Freyberg1, Chandreyee Maitra1, Manami Sasaki3, Andrew Strong1Michael\nC.H. Yeung1\n1Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, D-85748 Garching bei München , Germany\n2INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy\n3Dr. Karl Remeis Observatory, Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg\nSternwartstraße 7, 96049 Bamberg, Germany\nABSTRACT\nHot plasma plays a crucial role in regulating the baryon cycle within the Milky Way, flowing from the energetic sources in the Galactic\ncenter and disc, to the corona and the halo. This hot plasma represents an important fraction of the Galactic baryons, plays a key role\nin galactic outflows and is an important ingredient in galaxy evolution models. Taking advantage of the Spectrum Roentgen Gamma\n(SRG) /eROSITA first all-sky survey, in this work we aim to provide a panoramic view of the hot circumgalactic medium (CGM) of\nthe Milky Way. Compared to the previous all sky X-ray survey performed by ROSAT, the improved energy resolution of eROSITA\nenables us to map, for the first time, the sky within the narrow energy bands characteristic of soft X-ray emission lines. These provide\nessential information on the physical properties of the hot plasma. Here we present the eROSITA eRASS1 half sky maps in narrow\nenergy bands corresponding to the most prominent soft X-ray lines: O viiand O viii, which allow us to constrain the distribution\nof the hot plasma within and surrounding the Milky Way. We corrected the maps by removing the expected contribution associated\nwith the cosmic X-ray background, the time-variable solar wind charge exchange, the local hot bubble and we applied corrections to\nmitigate the e ffect of absorption, therefore highlighting the emission from the CGM of the Milky Way. We use the line ratio of the\noxygen lines as a proxy to constrain the temperature of the warm-hot CGM and we define a pseudo-temperature Tmap. The map\nhighlights how di fferent regions are dominated by di fferent thermal components. Towards the outer halo, the temperature distribution\nof the CGM is consistent with being constant ( ∆T/⟨T⟩)≤4%) on angular scales of 2-20 deg, while significant variations ∼12%\nare observed on many tens of degrees scales, when comparing the northern and southern hemisphere. The pseudo-temperature map\nshows significant variations across the borders of the eROSITA bubbles, therefore suggesting temperature variations, possibly linked\nto shocks, between the interior of the Galactic outflow and the unperturbed CGM. In particular, a \"shell\" of colder material appears to\nbe present close to the edge of the eROSITA bubbles.\nKey words. diffuse radiation—Galaxy: center—surveys—X-rays: galaxies—X-rays: ISM\n1. Introduction\nAs the next generation of all-sky survey experiments in soft\nX-ray, the extended ROentgen Survey with an Imaging Tele-\nscope Array (eROSITA), aboard the Russian-German observa-\ntory Spectrum-Roentgen-Gamma (SRG; Sunyaev et al. 2021),\nhas performed the first all sky survey (eRASS1; Merloni et al.\n2024) from December 2019 until June 2020. eRASS1 has an un-\nprecedented sensitivity to the whole sky in the soft X-ray range,\nthanks to an e ffective area of≈1100 cm2at 1.4 keV , with an en-\nergy resolution typical of CCD instruments ( ≈65 eV at 0.5 keV;\nPredehl et al. 2021). Hence eROSITA makes it possible to obtain\n\"monochromatic\" images for various emission lines simultane-\nously from the di ffuse emission over the entire sky.\nIn this paper, we present the eRASS1 narrowband maps cor-\nresponding to the brightest transitions in the soft X-ray band (i.e.,\nOviiand O viii), together with the maps ratio, and make these\nmaps (in count rate unit), publicly available online1.\n⋆emails to: nicola.locatelli@inaf.it\n1https://erosita.mpe.mpg.de/edr/eROSITAObservations/\nCatalogues/In X-rays, the emission lines are vital for understanding the\nproperties of the hot plasma inside the interstellar medium (ISM)\nand in the circum-Galactic medium (CGM), the latter made up\nby di fferent components, among which the Galactic halo, the\nGalactic corona and the Galactic outflow are also included. The\nhalo of the Milky Way is expected to be filled with plasma with\ntemperatures close to its virial temperature2,T∼0.15−0.20 keV ,\ntherefore appearing bright in soft X-ray emission lines (Yoshino\net al. 2009; Henley et al. 2010; Henley & Shelton 2012; Miller &\nBregman 2015). Additionally, Galactic outflows emerging from\nthe center of the Milky Way are expected (and observed) to\ndrive shocks into the ISM and the unperturbed CGM, heating\nthe plasma and inducing bright soft X-ray emission lines (Pre-\ndehl et al. 2020). Finally, it has been argued that activity in the\nGalactic disc is expected to sustain a hot Galactic atmosphere\n(the so called ‘Galactic corona’), which might be also traced in\nthe soft X-ray band (Das et al. 2019b,a; Ponti et al. 2023b).\nClearly mapping the emission lines from such hot and rar-\nefied plasma is extremely challenging, because the correspond-\n2For simplicity, we refer to kTas temperature or just Tthroughout the\npaper. Therefore, we report all temperatures Tusing keV units.\nArticle number, page 1 of 14arXiv:2401.17310v1 [astro-ph.GA] 30 Jan 2024A&A proofs: manuscript no. aa\ning signal is spread over the entire sky and at very low surface\nbrightness. To increase sensitivity to this emission, broad band\nimages can be helpful. The ROSAT all sky survey allowed us to\nget a view of the CGM emission in broad energy bands (Snow-\nden et al. 1997; Zheng et al. 2023), however the O viiand O viii\nlines (i.e. the brightest) could not be imaged separately because\nof the limited energy resolution of ROSAT. With its high sen-\nsitivity and band coverage in soft X-ray, eROSITA possesses\nspecialized capability to image the faint emission in narrower\nenergy bands compared to ROSAT. Here we present the first\neROSITA sky maps of the soft X-ray emission lines.\nThe paper is organised as follows: Section 2 shows the data\nanalysis; Section 3 discusses the soft X-ray emission lines as\ntracers of hot plasma; Section 4 presents the necessary tools that\nhave been used to retrieve physical information from the maps;\nSection 5 discusses the morphology of the line emission maps,\ntheir ratio and the physical information that is retrieved. Sec-\ntion 6 discusses the observed ratio between emission line maps\nas a proxy for the plasma temperature. Section 7 and 8 present\nthe constraints on the temperature of the CGM and the eROSITA\nbubbles, respectively. We further discuss our findings in Sect. 9,\nand finally summarized in Section 10.\n2. Data\nWe analysed the eRASS1 data (Merloni et al. 2024), collected\nfrom December 12, 2019 until June 11, 2020. We considered\nonly data from the five telescope modules (TM) of eROSITA\ncameras equipped with on-chip filter (i.e., TM1, TM2, TM3,\nTM4, TM6), to avoid any contamination from the e ffects of light\nleak (a ffecting TM 5 and 7, Predehl et al. 2021). We used the\nsame procedures as in Zheng et al. (2023) and analyze the data\nwith the standard eROSITA Science Analysis Software System\n(eSASS ), version 020, developed by the German eROSITA con-\nsortium (Brunner et al. 2022). The eROSITA data of the all sky\nare archived in a system of 4700 partially overlapping tiles of\n3.6×3.6 deg each (Merloni et al. 2024). When required, we\nre-projected each pixel in each sky-tile, to create a map with\nan HEALPix3projection with NSIDE =512 (Górski et al. 2005;\nZonca et al. 2019). The same filtering described in Zheng et al.\n(2023) has been applied to the counts-maps and exposure-maps.\nThe maps display the intensity within the specific en-\nergy range in unit of counts per second per square degree\n[cts s−1deg−2]. The Zenital Equal Area (ZEA) projection is ap-\nplied in order to conserve the surface brightness. While the\nZEA projection introduces projection distortion, especially to-\nwards the borders of the map, we find the distortions less promi-\nnent when compared to other kind of possible projections. For\ndisplaying purposes only, several maps presented below were\nsmoothed with an adaptive kernel based on a count rate S /N\nthreshold of 15 to highlight the di ffuse emission.\n3. Soft X-ray lines as tracers of hot plasma\nThe electromagnetic spectrum of optically thin and collisionally\nionised (hot) plasma consists of continuum emission plus an ar-\nray of emission lines. For X-ray emitting plasma with tempera-\ntures lower than T∼1 keV , the majority of the flux is in fact car-\nried by the large number of weak emission lines. These numer-\nous lines cannot be resolved individually at the CCD energy res-\nolution characteristic of eROSITA, therefore they form a pseudo\n3http://healpix.sourceforge.net\n0.010.1counts/s/keV/cm2/deg2l=340.32 b=−28.3\n1 0.2 0.5 2−505(data−model)/error\nEnergy (keV)294147\n0.010.1counts/s/keV/cm2/deg2l=241.07 b=−28.9\n1 0.2 0.5 2−505(data−model)/error\nEnergy (keV)085126Fig. 1: Spectra of the di ffuse emission observed by eROSITA\nwithin two sky patches of 3◦×3◦each. Upper panel: observed\nand modeled spectrum of the sky region at l=340.32 deg, b=\n−28.3 deg (eROSITA sky tiles 294147). Lower panel: observed\nand modeled spectrum of the sky region at l=241.07 deg, b=\n−25.9 deg (eROSITA sky tiles 085126). The regions have been\nchosen to show the typical di ffuse emission from a patches of\nthe sky, with(out) the contribution from the eROSITA bubbles\nemission. The contribution from the detected point sources has\nbeen removed. The grey dots at E=0.2 keV show the ROSAT\nR1 and R2 data points. The black solid and dotted lines show the\ncontribution of the instrumental background (derived by the fit of\nthe filter wheel closed data). The red, blue, green and magenta\nlines show the contribution of the LHB, warm-hot CGM, hot\ncorona and CXB emission components. The orange line (upper\npanel only) shows an additional plasma component that can be\nintroduced in the eROSITA bubble region.\ncontinuum. As an example, the data in the top and bottom pan-\nels of Fig. 1 show the eROSITA spectra from two sky regions\n(eROSITA sky tiles 294147 and 085126, Merloni et al. 2024)\ncharacteristics of bright and faint di ffuse emission, respectively.\nThe spectra shown in Fig. 1 here are meant to help the reader in\ngrasping the meaning and relations between di fferent emission\nlines and the spectral components producing them. Both spectra\nArticle number, page 2 of 14Zheng et al.: Soft X-ray emission lines\nBrightest Transition Theoretical Image energy\nline levels energy [eV] range [eV]\nO VII 7−→1 574 534–614\nO VIII 4 ,3−→1 654, 653 614–694\nTable 1: Transitions considered in this work and corresponding\nenergy band selected to extract photon lists and to create the cor-\nresponding images.\nshow several peaks, associated with the most prominent transi-\ntions, which stand out in the eROSITA spectra above the pseudo\ncontinuum (Fig. 1).\nWe focus this work on the two strongest lines observed in\nthe eROSITA spectra of the di ffuse emission: O viiand O viii\n(see Tab. 1). We chose the width of the energy bands on the ba-\nsis of the energy resolution of the pnCCD cameras of eROSITA,\nas determined on ground (see Table 4 in Predehl et al. 2021).\nWithin these relatively narrow energy ranges fainter emission\nlines might contribute to the emission from the pseudo contin-\nuum. For this reason, we did not perform any continuum sub-\ntraction of the emission lines (e.g., computing the line equivalent\nwidth). Instead, we forward modeled the expected emission (as-\nsuming the AtombDB library of atomic data and transitions4) in\norder to compare the data with expectations (see Sect. 4).\nFigure 2 shows an RGB half-sky map of the emission in the\nOvii, the broad continuum (0.2-2.3 keV) and O viiibands in red,\ngreen and blue, respectively. The RGB image shown in Fig. 2\nhighlights the emission associated with the hot plasma. All ma-\njor extended features which are also observed in the broadband\nmaps are seen in Fig. 2 (see Zheng et al. 2023, for a chart map\nof the principal di ffuse structures). In particular, the nearby ex-\ntended features, such as the Monogem ring (Knies 2022), the\nAntlia supernova remnant (Knies et al., in prep.), the Orion-\nEridanus complex (Joubaud et al. 2019) appear as large scale\nextended features with a markedly orange colour. A greenish and\nbright patch of X-ray emission is observed around the LMC (Lo-\ncatelli et al., submitted). A greener-bluer color characterises the\ninterior of the eROSITA bubbles (Predehl et al. 2020) compared\nwith the anti-center direction. Moreover, clear color gradients\nare observed at the edges of the eROSITA bubbles. The dark re-\ngions at small Galactic latitudes b(i.e. along the Galactic plane)\nin Fig. 2, some of which are as extended as ∼20 deg are the\nresult of absorption of the soft X-ray emission through the ISM.\n3.1. The dependence of the ion fractions on temperature\nFigure 3 shows the expected ion fraction as a function of the\nplasma temperature for oxygen, as derived from AtombDB un-\nder the assumption of an optically thin hot plasma mainly ion-\nized by collisions between atoms. For each element, the ioniza-\ntion fraction is obtained by balancing the net rates of ioniza-\ntion and recombination, which depends on the temperature of\nthe plasma. The vertical dotted grey lines indicate the typical or\nempirical temperature of some of the components contributing\nto the soft X-ray di ffuse emission. These are: the local hot bub-\nble (LHB), characterised by T∼0.097 keV (Liu et al. 2017); the\ncircumgalactic medium (CGM or halo) of the Milky Way, with\nT∼0.15-0.2 keV (Ponti et al. 2023b); the eROSITA bubbles,\nwhich we assume to have T∼0.3 keV (Kataoka et al. 2018;\nbut see also Gupta et al. 2023). Figure 3 shows that plasma with\ntemperatures characteristic of the LHB is dominated by O vii,\n4http://atomdb.org/while the CGM emission has similar contributions from O vii\nand O viii, while the O viii/Oviiline ratio becomes insensi-\ntive to the presence of plasma at even higher temperatures (i.e.,\nT≫0.3 keV).\n3.2. Soft X-ray line ratios as temperature diagnostics\nAs can be seen in Fig. 3, under the assumption of a single emis-\nsion component in collisional ionization equilibrium, it is possi-\nble to measure the temperature of the emitting plasma, given an\nobserved line ratio. In particular, if both ions belong to the same\nelement (e.g., O viiand O viii), then the line ratio has almost\nno dependence on the metal abundances of the emitting plasma,\ntherefore becoming a reliable tool to determine the temperature\nof the hot plasma.\nFigure 4 shows the evolution of the O viii/Oviiline ratio as a\nfunction of temperature. The line ratio was computed by sim-\nulating a collisionally-ionized di ffuse gas ( apec component in\nXspec) at each temperature. The theoretical spectrum obtained\nwas then convolved with a Gaussian kernel with a full width at\nhalf maximum of 74 eV in order to simulate the spectral resolu-\ntion of eROSITA and compare the theoretical ratio with the data.\nFinally, the fluxes in each band (see Tab. 1) were used to produce\nthe expected line ratio as a function of temperature. The lines in\nthe top panel of Fig. 4 show that the O viii/Oviiline ratio as\nobserved by eROSITA is an excellent temperature diagnostic for\nsingle temperature plasma between T∼0.1 and T∼0.3 keV ,\nwhile it is insensitive to plasma hotter than T≥0.3 keV . The dif-\nferent lines show line ratios under the assumption that the emit-\nting plasma has metallicity of 0.1 or 0.3 Solar, as well as holding\ndifferent ratio between the abundances of the metals (e.g. as-\nsuming di fferent abundance sets; Lodders et al. (2009), Anders\n& Grevesse (1989) and Allen et al. (2019)). As expected, in the\n0.1-0.3 keV temperature range, where the O viiiand O viilines\nare su fficiently bright with respect to the pseudo-continuum, lit-\ntle to no variations of the line ratio is observed with metallicity.\n4. Subtraction of non-CGM emission components\nfrom the maps\nAs discussed in Section 3.2, from the line ratio maps it is pos-\nsible to extract important information on the temperature of the\nCGM, if the emission can be described by a single component\nof an optically-thin plasma in collisional-ionisation equilibrium.\nHowever, the di ffuse soft X-ray emission measured by eROSITA\nhas several contributions from various thermal and non-thermal\ncomponents that need to be removed before we can constrain\nthe temperature of the CGM. Additionally, we expect that the\nemission from the CGM is, at least in part, absorbed by the in-\nterstellar medium, therefore further complicating the interpre-\ntation of the line ratio maps. We note that the subtraction of\n(back-)foreground components is not performed on all the im-\nages shown within this paper, but are applied in order to produce\nand analyze the temperature distribution of some representative\nsky patches. Below, we provide a description of the corrections\nwe applied in order to retrieve the temperature map from the\ndata.\n4.1. Subtraction of the instrumental background\nDuring the eRASS1 survey, the level of the instrumental back-\nground has been frequently monitored, thanks to weekly obser-\nvations with the filter wheel closed (Yeung et al. 2023). The anal-\nArticle number, page 3 of 14A&A proofs: manuscript no. aa\nFig. 2: Bright emission lines in eRASS1. Thre RGB map is composed by the broadband maps at 0.20-0.25 keV (red, Zheng\net al. 2023), 0.2-2.3 keV (green, Zheng et al. 2023) and the narrowband O viii(blue, this work) images. All the maps include the\ncontribution of di fferent components (both celestial and instrumental) and were smoothed with an adaptive kernel set to retrieve S /N\nthreshold of 15.\nysis of these filter wheel closed data show that the eROSITA\ninstrumental background has been relatively stable in flux and\nspectrum for the entire duration of the first all sky survey.\nSuch instrumental background is described by a relatively\nflat spectrum with various emission lines, as well as detector\nelectronic noise, which is stronger at lower energies and can be\nrepresented by a rise in the flux of the instrumental background\ntowards lower energies (solid black line in Fig.1; see also Yeung\net al. 2023). Here we subtracted the contribution of the instru-\nment background over the entire sky, by assuming that it is stable\nover the duration of the eRASS1 survey.\nWe stress that the contribution of the background over the\nnarrow O viiband is small, being about ∼40 and∼10 times\nfainter than the O viiemission in the bright and faint region we\ndenoted in Fig.1, respectively.4.2. Removing the contribution from the solar wind charge\nexchange emission\neROSITA is located close to the second Lagrangian point (where\nthe effects of the geocoronal solar wind charge exchange are\nlow) and the first eROSITA all sky survey occurred close to\nsolar minimum, when the heliospheric solar wind charge ex-\nchange is at its minimum (Kuntz 2019). However, contrary to\nwhat may be expected, also a solar maximum may correspond to\na low level of solar wind charge exchange (SWCX) emission: the\nhigher solar flux may in fact ionize the bulk of the neutral hydro-\ngen required for the charge exchange in the inner solar system\n(Galeazzi et al. 2011). In this context, Ponti et al. (2023b) and\nYeung et al. (2023) have reported evidence for a time-variable\ncomponent which is a ffecting the di ffuse emission observed by\nArticle number, page 4 of 14Zheng et al.: Soft X-ray emission lines\nFig. 3: Ionization fraction of oxygen as a function of temperature\n(from AtomDB, see Smith et al. 2001). The vertical dotted lines\nin gray show the typical temperature of: LHB ( T=0.097 keV);\nCGM ( T=0.15 keV); eROSITA bubbles (assumed to be T=\n0.3 keV).\nFig. 4: Narrowband intensity ratio as a function of tempera-\nture, absolute and relative metal abundances (as labelled, from\nAtomDB Smith et al. 2001, and convoluted with the eROSITA\nspectral resolution). The shaded areas encompass the 2, 10, 25,\n75, 90, 98% of the distribution of the the narrowband intensity\nratio analyzed in this work (an image of the data is later pre-\nsented in the top panel of Fig. 9).\neROSITA. This variable component was attributed to the e ffects\nof the heliospheric solar wind charge exchange. Triggered by\nthis finding, Dennerl et al. (in prep.) have been investigating the\nentire eROSITA data-set of the western Galactic hemisphere, to\nconstrain the temporal, spatial and spectral variations of the solar\nwind charge exchange component.\nThe results by Dennerl et al. (in prep.) corroborate the idea\nthat the impact of solar wind charge exchange is minimal during\neRASS1. Additionally, Dennerl et al. (in prep.) derived the flux\nof the time-variable component (attributed to solar wind charge\nexchange) as a function of time in several soft X-ray energy\nbands. We used these products, relevant for eRASS1, to subtract\nthe effects of solar wind charge exchange from the narrowband\nmaps of Fig. 5 and 6.\nFig. 5: Narrowband map in the energy range of the O viiline as\nobserved by eROSITA during eRASS1 in units of cts s−1deg−2.\nAn adaptive smoothing kernel using S /N=15 was applied.\n4.3. Removing the contribution from the LHB\nThe LHB is an irregular volume filled with hot ( T∼0.097 keV)\nplasma with a typical size of ∼200 pc around the Sun (see e.g.\nYeung et al. 2023, and references therein). Being almost com-\npletely devoid of cold material, the emission from the LHB ap-\npears una ffected by absorption of X-rays.\nBecause of its unabsorbed nature and relatively low temper-\nature, the LHB emission dominates mainly in the very soft X-\nray band (≤0.3 keV). Zheng et al. (2023) have shown that the\ndiffuse emission observed by eROSITA in the softer bands (0.2-\n0.25 keV) is indeed heavily a ffected by the emission from the\nLHB. For any study of the Galactic plasma beyond of LHB, such\nas the CGM and the eROSITA bubbles, it is therefore necessary\nto remove the foreground contribution from the LHB.\nThe distribution of the LHB is observed to be inhomoge-\nneous, based on the thermal emission from the LHB detected by\nROSAT in the softest energy bands (Snowden et al. 1997).\nThanks to relatively recent observations from the Di ffuse X-\nrays from the Local Galaxy mission (DXL), Liu et al. (2017)\ngenerated the latest 3D structures of the LHB. We used the emis-\nsion measure of the LHB provided in their work, assuming a\nconstant temperature ( T=0.097 keV apec model with Lodders\n(2003) abundance table) to reconstruct the observed LHB emis-\nsion as observed by eROSITA in each relevant energy band. We\nthen removed the LHB component from the maps.\nBecause of the relatively low temperature of LHB, the soft\nband is a ffected more than the harder band. Figure 5 of Ponti\net al. 2023b shows that the LHB takes generally much larger\nfraction in the O viinarrow bands flux than that in O viii: in most\nregions of O viimaps, the LHB takes more than 15% of the flux,\nwhile in O viiimaps the contribution from LHB is smaller by\nabout a factor 10.\nArticle number, page 5 of 14A&A proofs: manuscript no. aa\nFig. 6: Narrowband map in the energy range of the O viiiline as\nobserved by eROSITA during eRASS1 in units of cts s−1deg−2.\nAn adaptive smoothing kernel using S /N=15 was applied.\n4.4. Absorption correction\nThe soft X-ray spectrum is heavily a ffected by absorption from\nneutral material (Locatelli et al. 2022). To account for the ab-\nsorption e ffect, we used the column density maps combining\nthe HI (HI4PI Collaboration et al. 2016) and H 2column density\nderived from Planck observations (Planck Collaboration et al.\n2016; Green 2018) following the recipe proposed by Willingale\net al. (2013). Then, assuming Solar abundances for the absorb-\ning medium, we derived the correction factor to be applied to de-\nabsorb either the maps or the emission from the various emission\ncomponents (see Appendix A in Locatelli et al. 2023, for further\ndetails). The absorption correction was applied to the CGM and\nCXB components only, because we assumed that the SWCX and\nLHB components are un-absorbed. We note that the correction\nfor the e ffects of absorption are negligible for column densities\nlower than NH≤1021cm−2in the oxygen maps. This condition\napplies to all the regions included in our analysis of the temper-\nature distribution.\n4.5. Subtraction of the cosmic X-ray background (CXB)\nThe major contribution to the di ffuse X-ray emission above\n∼0.6 keV is given by the CXB (see Tab. 4 of Ponti et al. 2023b).\nUltra-deep X-ray surveys performed with Chandra and XMM-\nNewton have resolved more than ∼80 % and∼92 % of the CXB\nflux into extra-galactic discrete sources (primarily active galactic\nnuclei, clusters of galaxies, groups, normal galaxies, etc.) in the\n0.5-2 and 2-7 keV band, respectively (Luo et al. 2017; Brandt\net al. 2021). The CXB flux is observed to be uniformly dis-\ntributed, with small amplitude variations on arcminutes to de-\ngrees scales. We removed the CXB component by assuming it\nhas constant flux on the sky, that it is absorbed by the full Galac-tic column density of material (computed as above), and that its\nspectrum can be described by a doubly broken power law (Ponti\net al. 2023b). The CXB spectrum (Gilli et al. 2007) is assumed\nto have a photon index of Γ3=1.45 above 1.2 keV , of Γ2=1.6\nbetween 0.4 and 1.2 keV and Γ1=1.9 below 0.4 keV and to\nhave a normalisation of 8.2 photons s−1cm−2at 1 keV .\n5. The O viiand O viiiline maps of the CGM\ncomponent\nFigure 5 shows the half sky emission observed by eROSITA dur-\ning eRASS1 within the O viienergy band, while Figure 6 shows\nthe same map in the O viiiband. In both bands bright emis-\nsion is observed at relatively high Galactic latitudes |b|≥30◦,\nwith count rates of ∼0.01 cts s−1deg−2. Some small depres-\nsions are observed there, however they are associated with ab-\nsorbing clouds (Snowden et al. 2015; Yeung et al. 2023; Ponti\net al. 2023a). This indicates that, as expected, we are surrounded\nby hot plasma along every direction on the sky, the so called hot\nCGM.\nAs already pointed out, along most of the Galactic disc, the\neffect of interstellar absorption is much more relevant. Indeed,\ndark patches are observed nearby dark clouds, which are primar-\nily concentrated at small latitudes. O viiand O viiiare the bright-\nest lines observed by eROSITA in the soft band. For this rea-\nson many of the features described in both the broadband maps\n(Zheng et al. 2023) and in the RGB map in Fig. 2 are also evi-\ndent in the narrowband maps of the oxygen lines. For instance,\nlarge scale enhanced O viiand O viiiemission is observed both\nin the direction towards the Galactic center, along the footprint\nof the eROSITA bubbles and on smaller regions towards the anti-\ncenter, (e.g. Orion-Eridanus superbubble; Monogem ring; Antlia\nSNR).\nIt is also possible to observe a clear trend of increasing O viii\nemission closer to the Galactic disc and towards the Galactic\ncenter (Fig. 6). These large scale features contain morphological\ninformation that has been recently used to constrain the geome-\ntry and physical properties of the warm-hot CGM (Locatelli et al.\n2023).\n6. Line ratio and pseudo-temperature maps\nFigure 7 shows the ratio of the O viii/Oviiemission. The O vii\nand O viiimaps used to compute the ratio collect counts over\na squared pixel of side length ∆rpxl=2deg. This pixel size al-\nlows most pixels in the maps to hold C ≥30 counts. Under the\nC≫1 assumption, the uncertainty related to the count can then\nbe computed as√\nC. We note that the only regions where the\nassumption is broken are those highly absorbed, that is primar-\nily along the galactic disk and towards dark clouds (Yeung et al.\n2023). The galactic disk regions are anyway not included in the\nquantitative analysis and the results presented in the following\nsections. We stress that, for display purposes, we have inverted\nthe color scale, so that brighter regions correspond to a higher\ncontribution from O vii, therefore to a lower temperature. At high\nGalactic latitudes |b|>30◦and away from the Galactic center\nl<270◦, the line ratio map appears rather smooth (with small\nfluctuations, that do not appear significant, see Sect. 7), with the\nnorthern hemisphere possessing slightly lower ratio. On the con-\ntrary, at high Galactic latitudes |b|>30◦towards the Galactic\ncenter l<270◦, sharp color variations are clearly observed in\nFig. 7. Particularly evident is a bright (less hot) stripe running\nall along the edge of the northern eROSITA bubble (enclosed by\nArticle number, page 6 of 14Zheng et al.: Soft X-ray emission lines\nFig. 7: O viii/Oviiline intensity ratio map (unitless), as observed by eROSITA during eRASS1. The contribution from CXB, LHB,\nforeground absorption and instrumental background (see Sect. 4) has not been removed before taking the line ratio shown in this\nimage. An adaptive smoothing kernel using S /N=15 was applied. The kernel used was the same for the O viiand O viiimaps. The\nwhite dashed lines are placed along the sharpest features close to the boundaries of the eROSITA bubbles.\nthe dashed white lines drawn in Fig. 7). Just inside (closer to the\nGalactic center) this bright stripe, a dark (hot) region is observed\nto run again all along the edge of the northern eROSITA bubble.\nDespite fainter, such dark (hot) transition layer can be observed\nalso along a good fraction of the southern eROSITA bubble.\nTo better highlight this interesting feature occurring at the\nedge of the eROSITA bubbles, Fig. 8 shows both the X-ray con-\ntinuum (0.2-2.3 keV) in green as well as the O viito O viiiline\nratio in blue and the O viiemission in red. The O viito O viii\nline ratio in blue is thus brighter whenever a source produces\nmore O viithan O viii, that is for lower plasma temperatures (cfr.\nFig. 3). Fig. 8 thus clearly shows that this colder rim of emis-sion occurs in close correspondence with the surface brightness\ntransition used to define the edges of the eROSITA bubbles. This\neffect is very clear along the northern bubble but it can be fol-\nlowed all the way down to about −40◦also in the southern bubble\n(Fig. 8 and 9), although partially blended with broad absorption\nfeatures close to the boundary of the bubble, in projection.\nPseudo-temperature map derived from the O viii/Oviiline\nratio\nAssuming that the line ratio map shown in Fig. 7 is produced\nby only one warm-hot plasma component, by inverting the re-\nArticle number, page 7 of 14A&A proofs: manuscript no. aa\nFig. 8: Bright emission lines in eRASS1. Thre RGB map is composed by O vii(red), broadband emission in the 0.2-2.3 keV energy\nrange (green) and O viito O viiiline ratio (blue). All the maps include the contribution of di fferent components (both celestial and\ninstrumental) and were smoothed with an adaptive kernel set to retrieve S /N threshold of 15.\nlation in Fig. 4, from the oxygen lines ratio map, we are able\nto create the first temperature map of the warm-hot CGM of the\nMilky Way (see Fig. 9, top panel). Of course, such \"temperature\nmap\" is valid only in areas where a single plasma component is\nemitting most of the observed photons. In other regions, where\ndifferent components of comparable brightness build up the ob-\nserved narrowband intensities, the derived map looses its mean-\ning as a temperature. For this reason a meaningful temperature\nmap necessarily depends on an accurate selection of the emis-\nsion component subject of the study, for instance through the\nmodels and subtractions presented in Sect. 4. Additionally, ab-\nsorption can severely a ffect the temperature map, by absorbing\nby a di fferent factor the emission of the di fferent lines, therefore\naffecting the final temperature map. Because in several places\nthe assumption of a single thermal component is not true (e.g.,\nwithin the footprints of the eROSITA bubbles and close to theGalactic disc, where the hot ISM could give a contribution to the\nemission), we call the map derived from the oxygen line ratio as\n\"pseudo-temperature\" map.\nThe top panel of Fig. 9 shows the pseudo-temperature Tmap\nobtained by subtracting all components as detailed in Sect. 4.\nTherefore, all components discussed by Ponti et al. (2023b),\napart from the CGM (composed by a hot halo, a disc-like com-\nponent, and the emission of the Galactic outflow) have been\nremoved. By looking at Fig. 1 (see also Fig. 5 of Ponti et al.\n(2023b)), it is possible to see that, at moderate latitudes, little\ncontribution is given to the O viiand O viiilines by the emis-\nsion attributed to the corona. This suggests that the pseudo-\ntemperature map derived from the O viiiover O viimaps is\na good proxy of the real CGM plasma temperature in regions\naway from the Galactic outflow and the Galactic disc. On the\nother hand, we expect that the additional emission induced by the\nArticle number, page 8 of 14Zheng et al.: Soft X-ray emission lines\nFig. 9: Pseudo-Temperature derived from O lines ratio with(out)\naccounting for the absorbed intensity of the line. Upper panel:\ntemperature map corrected for foreground absorption. This ver-\nsion is the one used for quantitative statements on the temper-\nature throughout the paper. These maps were corrected for the\nLHB, CXB, SWCX, instrumental background emissions and for\nforeground absorption, as presented in Sect. 4 (while in Fig. 7,\nthe ratio was not corrected). The regions used for the pseudo-\ntemperature analysis are shown in the panel below. Lower panel:\ntemperature map, without deabsorbing the O viiand O viiiline\nemission. The black circles show the regions selected for the\nanalysis of the eROSITA bubbles and the CGM in the north and\nsouth hemispheres. The profiles shown in Fig. 12 and Fig. 13\nare indicated by the transparent cyan and white regions, respec-\ntively.\nFig. 10: Probability distribution function (PDF) of the pseudo-\ntemperature derived in the north and south CGM regions (black\nsolid and red dashed lines, respectively; see footprints in the\nlower panel of Fig. 9). The thin solid lines show Gaussian fits\nof the temperature distributions. The vertical dashed lines show\nthe median values of the distribution with the uncertainty on the\nmean bracketed by the filled vertical regions.\nGalactic outflow and the hot interstellar medium, will break the\ninitial assumption that only one thermal component is contribut-\ning to the oxygen line ratio map, therefore biasing the tempera-\nture map shown in Fig. 9 within the footprints of the eROSITA\nbubbles and along the Galactic disc.\nThe top panel of Fig. 9 shows similar structures to the\nones observed in the oxygen lines ratio. In particular, the CGM\npseudo-temperature appears rather uniform towards both the\nnorthern and southern hemispheres, with values of the order of\nT ∼ 0.20−0.22 keV , respectively. A low temperature shell\n(colder than the CGM) surrounds all of the northern bubble and a\ngood fraction of the southern bubble. Such \"shell\" has a consid-\nerable width, appearing as extended as ∼10◦in several places.\nWe note that, just inside this colder shell, the temperature raises\nagain to temperatures higher than the surrounding CGM. Again,\nthis interior and hotter shell surrounds all of the northern bubble\nas well as a good fraction of the southern one (Fig. 9).\nThe bottom panel of Fig. 9 shows how the temperature map\nchanges, once the correction for absorption are not applied. As\nexpected, only in highly absorbed regions (i.e., along the Galac-\ntic disc and towards dark clouds) the corrections become rele-\nvant.\n7. Constraining the temperature distribution of the\nCGM\n7.1. On the CGM temperature distribution on small scales\n(few-to-ten degrees)\nTo constrain the pseudo-temperature distribution of the CGM\naway from the Galactic disc and center, we selected two cir-\ncular regions centered at (l,b) =(220,+50) and (l,b) =(230, -\n50) and with radius of 20 degrees (black solid wide circles in\nFig. 9, bottom panel). Two additional regions at coordinates (l,b)\n=(340,+40) and (l,b) =(340, -30) with radius of 10 degrees\nwere also considered to study the pseudo-temperature within the\nArticle number, page 9 of 14A&A proofs: manuscript no. aa\neROSITA bubbles (black solid small circles in Fig. 9, bottom\npanel).\nWe note that the choice of moderate- and high-latitude re-\ngions minimizes potential systematic uncertainties in the oxy-\ngen ratio introduced by the absorption model used to retrieve\nthe unabsorbed CGM emission in both oxygen bands (see Ap-\npendix A). Figure 10 shows the probability distribution function\nof the pseudo-temperature within the selected regions, obtained\nusing a pixel side length of ∆rpxl=2 deg. The distributions from\nthe northern and southern hemisphere are shown by the black\nsolid and red dashed lines, respectively. In Fig. 10, the horizon-\ntal errorbars show the root-mean-square (rms) value σrmsof the\npseudo-temperature uncertainty in the regions considered.\nThe valueσrmshappens to be similar to the standard devi-\nation std(T) of the pseudo-temperature map computed in the\nsame region (i.e. σrms≃std(T)), in both the north and south re-\ngions. We therefore conclude that the CGM pseudo-temperature\ncan be considered as uniform over the regions considered, that\nis between 2 and 20 deg scales, given the statistical uncertainties\non the pseudo-temperature values. Any intrinsic fluctuation ∆T\nwould in fact contribute to the overall spread of the distribution\nstd(T) following (e.g. Vaughan et al. 2003):\n(std(T))2=(∆T)2+σ2\nrms. (1)\nTherefore, provided std( T)≃σrms, we constrain any intrinsic\npseudo-temperature fluctuation to be ∆T≤∆T≤σrms=0.007\nand 0.008 keV on ∆θ≃2−20 deg in the north and in the\nsouth CGM region, respectively. The fractional fluctuation with\nrespect to the average pseudo-temperature ∆T/⟨T⟩(reported be-\nlow in Sect.7.2) amounts to ∆T/⟨T⟩≤ 4% for both the north and\nsouth regions.\n7.2. North-south asymmetry in the pseudo-temperature\ndistribution of the CGM\nAs already pointed out when describing Fig. 7, we note that the\nCGM on the southern hemisphere appears hotter (i.e. higher line\nratio) than its northern counterpart. From the oxygen pseudo-\ntemperature map, the derived mean pseudo-temperature values\nare⟨T⟩=0.195±0.001 keV and⟨T⟩=0.218±0.001 keV for the\nnorth and south regions respectively. The quoted uncertainties\nindicate the error on the mean. The median pseudo-temperatures\nare equal to the mean pseudo-temperatures within uncertainties.\nThe di fference between north and south pseudo-temperature is\nthus∆T=0.024±0.001 keV , where the uncertainty has been\nobtained from the quadrature of the mean pseudo-temperature\nuncertainties of the north and south samples. Thus, we find a sig-\nnificant di fference of ∆T/⟨T⟩≃ 12% in the pseudo-temperature\nof the hot CGM when comparing representative regions in the\nnorth and south Galactic hemispheres. This di fference is larger\nthan the sum of statistical and systematic uncertainties (see Ap-\npendix A) and we thus interpret it as a physical, real e ffect.\nThis then suggests that the pseudo-temperature of the CGM\nhas variations as large as ∼12 % on scales of several tens of\ndegrees, while, when looking to angular scales of the order of\n∼2−20◦, then the pseudo-temperature distribution is consistent\nwith being normal and to be constant, with fluctuations smaller\nthan∆T≤ 4 %.\n8. Constraining the pseudo-temperature\ndistribution within the eROSITA bubbles\nFigure 11 shows the probability distribution function of the\npseudo-temperature within two circular regions inside the north-\nFig. 11: Probability distribution function (PDF) of the pseudo-\ntemperature in the north and south regions inside the eROSITA\nbubbles (black solid and red dashed lines, respectively; see foot-\nprints in the lower panel of Fig. 9).\nern and southern eROSITA bubbles with black and red colours,\nrespectively. We stress again that, within this region, at least two\nthermal components (i.e., the emission from the eROSITA bub-\nbles, plus the emission from the other parts of the CGM) are con-\ntributing to the total emission, therefore the pseudo-temperature\nderived from the line ratio must be biased. In particular, as\nshown in Figs. 3 and 4, the pseudo-temperatures derived from\nthe oxygen line ratio are unable to trace plasma hotter than\nT∼0.3 keV . Indeed, the pseudo-temperatures derived from the\noxygen line ratio is observed to be ⟨T⟩=0.211±0.001 keV and\n⟨T⟩=0.217±0.001 keV for the northern and southern bubbles,\nrespectively. These values are surprisingly similar to the temper-\natures derived for the CGM outside the eROSITA bubbles, indi-\ncating that the oxygen line ratio is not sensitive to the presence\nof a hotter component within the bubbles (see Fig. 1). Therefore,\ndespite the absolute value of the pseudo-temperature might be\nbiased inside the eROSITA bubbles (because the presence of a\nhotter component is not traced), we expect that intrinsic varia-\ntions of the temperature of the hot CGM would induce a spread\nin the distribution of oxygen line ratio and, consequently, on the\npseudo-temperatures derived from the latter.\nFigure 11 shows that the mean uncertainty on the value of\nthe pseudo-temperature is significantly smaller than the pseudo-\ntemperature standard deviation σrms−20 deg. This allows\nus to extend the investigation on the pseudo-temperature in the\nsouthern bubble to regions closer to the Galactic disk or to the\nGalactic center. In Fig. 12, we partially hided the data points\ncharacterized by a column density N H>1021cm−2, thus poten-\ntially biased by systematic on the absorption model.\nFig. 13: Pseudo-temperature profiles across the northern\neROSITA bubble. The paths are also shown as white transpar-\nent regions in the bottom panel of Fig. 9.\nThe colder shell at about the boundary of the bubbles can be\nrecognized by the steep drop at around ∆θ≃50−60 deg from\nthe paths origin. In the south, the regions well inside the bubble\nare characterized by a higher average pseudo-temperature than\nin the north. A constant value is then observed well outside the\nboundaries eventually ( l∼240−210 deg) matching the same\nlevel of the CGM regions (dashed lines).\nThe regions around the boundary of the bubbles show a\nsystematic pseudo-temperature drop in both hemispheres, when\ncompared to the respective CGM values. The drop (i.e. the colder\nshell), is more evident in the north, while is also present in the\nsouth. At the same time, a small but significant enhancement is\nobserved ∆θ=10−20 deg before the drop in both hemispheres.\nStrong deviations from homogeneity are found within the bound-\naries of the bubbles. While these inhomogeneities prevent us\nfrom characterizing the details of a putative pseudo-temperature\njump, we find evidence for a higher pseudo-temperature and\nbroad spatial variations of Tinside the bubbles, compared to\nthe outer regions.\nGiven the brighter emission of the northern bubble, we fur-\nther inspected the inhomogeneities through di fferent profiles\nacross it, as shown in the panels of Fig. 12 (the profiles are also\ncharted as a white transparency in the bottom panel of Fig. 9). All\nthe profiles have origin at ( l,b)=(350,35) deg. From these pro-\nfiles the features described above (e.g. the pseudo-temperature\nArticle number, page 11 of 14A&A proofs: manuscript no. aa\nenhancement, the colder shell, the constant and scattered value\nat large ∆θ, the large and coherent variations inside the bubbles)\nare all evident and highly significant. We note that along the\nprofile encompassing the highest latitudes (Fig. 13, top panel),\nthe pseudo-temperature shows a significant and increasing trend\nwith respect to the average CGM value. This confirms that the\ncold-warm shell noticed in the line ratio and pseudo-temperature\nmaps is indeed statistically significant.\n9. Discussion\nThanks to the eROSITA energy resolution, we could derive the\nfirst half sky maps in narrow energy bands characteristic of some\nof the most prominent soft X-ray emission lines. These narrow-\nband maps allowed us to detect several known features as well\nas to reveal new structures and to use line ratio maps to derive a\npseudo-temperature map of the CGM of the Milky Way.\n9.1. On the pseudo-temperature distribution of the CGM\nThe study of the soft X-ray emission lines (and their ratio), has\nallowed us to place constraints on the pseudo-temperature distri-\nbution of the CGM. We observe that on small angular scales,\nof few to∼20◦, the CGM pseudo-temperature is consistent\nwith being constant, with a normal distribution and upper limits\nto the intrinsic pseudo-temperature fluctuations of the order of\n∆T ≤ 4 %. At first sight, this may appear in contrast with sim-\nulations of galaxies, predicting log-normal distributions of the\nCGM pseudo-temperatures and significant pseudo-temperature\nfluctuations from di fferent regions in the sky. However, we note\nthat we trace the pseudo-temperature distribution based only on\nthe ratio of the O viiiand O viilines, which are sensitive to the\nhotter component of the CGM. Therefore, we cannot exclude\nthat the real pseudo-temperature distribution is actually skewed\ntowards lower pseudo-temperature (e.g. traced by O VI), but we\nare insensitive to those CGM phases here.\nWe also observed that by comparing the CGM pseudo-\ntemperature in the northern and southern hemispheres, therefore\non scales larger than ∼20◦, significant pseudo-temperature vari-\nations are observed, of the order of ∼12 %. Significant pseudo-\ntemperature variations are also observed when comparing the\ninterior of the eROSITA bubbles with the high Galactic lati-\ntude region away from the Galactic center, although the presence\nof an additional hotter component prevents us from quantifying\nthis pseudo-temperature jump in an unbiased way. Therefore, the\nstudy of the line ratio suggests that the CGM undergoes signif-\nicant pseudo-temperatures variations of at least ∼12 % on an-\ngular scales of several tens of degrees, while we observe that\non smaller angular scales ( ∼2−20◦) the pseudo-temperature is\nconsistent with being constant.\n9.2. An additional and hotter plasma component inside the\neROSITA bubbles\nIn the previous Sect. 8.1, we detected clear pseudo-temperature\nvariations at the edges of the eROSITA bubbles. The pseudo-\ntemperature map, derived from the oxygen line ratio, indicates\npseudo-temperatures of T ∼ 0.21−0.22 keV andT ∼ 0.19−\n0.22 keV for the region inside and outside the eROSITA bub-\nbles, respectively. This appears in agreement with some recent\nresults based on the analysis of Suzaku spectra (Gupta et al.\n2023), which find a pseudo-temperature of the plasma inside the\nbubbles to be consistent with T∼ 0.2 keV , although with a sig-nificant overabundance of neon to oxygen of ∼2. We reiterate\nthat the oxygen line ratio is not very sensitive to any component\nhotter than T∼0.3 keV , as suggested by previous studies of\nthe same Suzaku data, which were not requiring neon overabun-\ndance (Kataoka et al. 2018). To discriminate between these dif-\nferent hypothesis, a full characterisation of the spectra from the\ninside and outside of the eROSITA bubbles is necessary (e.g.,\nFig. 1).\n9.3. A shell surrounding the eROSITA bubbles\nThe investigation of the oxygen line ratio suggests the presence\nof a shell of colder material, with an inner hotter stripe, close\nto the edge of the eROSITA bubbles, defined as the drop in sur-\nface brightness. The shell appears to wrap most of the north-\nern bubble and a good fraction of the southern bubble, closer to\nthe disc. The origin of such shell is still unclear. Colder phases\nas transition layers between an expanding bubble and the inter-\nstellar medium are typically observed in superbubbles, where a\ncolder and high density shocked interstellar gas is contained be-\ntween the forward shock and the contact discontinuity separating\nit from the hotter and more rarefied shocked stellar wind (Weaver\net al. 1977). This colder and high density shell forms because of\nradiative losses, which become relevant as the superbubble ex-\npands into the interstellar medium.\nClearly the example of superbubbles shows that an outflow\ninteracting with the colder ambient gas can form a shell. How-\never, the conditions experienced by the putative Galactic outflow,\nwhich might have shaped the eROSITA bubbles, are probably\ncloser to the ones in galactic outflows observed in star forming\nand starburst galaxies, where the assumption of a homogeneous\nambient medium creating an homogeneous shell in all direction\nis likely not true. Indeed, the typical scale height of galactic out-\nflows is significantly larger than the one of their galactic discs.\nTherefore, the galactic disc is likely helping in contrasting the\nexpansion of the outflow along the disc and in collimating the\noutflow into a bipolar feature, which is expanding more towards\ndirections where there is significant drop in the density of the\nenvironment (the CGM). Because of the significant drop in the\ndensity of the environment (the CGM), outflows from star form-\ning galaxies are not expected to produce a colder cap as observed\nin superbubbles. Instead, such denser and colder phases might be\ndeveloped at the boundary between the outflow and the galactic\ndisc and to trace the edges of the outflow up to several kilo-\nparsecs above the galactic disc. Thus, within the framework of\na galactic outflow, it is puzzling that the \"colder shell\" can be\ntraced all the way to the top of the northern eROSITA bubble.\nThe observation of a shell reaching very high Galactic lat-\nitudes might be related with either entrainment and mixing be-\ntween the hot and colder phases of the outflow or to projection\neffects. Indeed, it is possible that the geometry of the outflow\nis different from the one assumed in Predehl et al. (2020), be-\ncause the eROSITA bubbles are so extended in the sky that the\nupper portion of this layer, which appears at Galactic latitudes\nofb∼60◦or more, may actually be associated with the near\nside of the Galactic outflow, which is so extended to fill the sky\nabove us. We note that in such a scenario, it is somehow natu-\nral to expect that such a \"colder shell\" will trace the edges of the\neROSITA bubbles closer to the Galactic disc. Such \"colder shell\"\nmight be present all along the envelope around the eROSITA\nbubble and closer to the Galactic disc, but to appear clearer at its\nedges, because of projection e ffects. If so, then no cap would be\npresent at the top of the Galactic outflow and the \"colder shell\"\nArticle number, page 12 of 14Zheng et al.: Soft X-ray emission lines\nobserved in our maps would be due to the projection of the near\nside of the outflow, just a few kiloparsecs above the galactic disc.\nOf course, more work is needed to clarify the three dimen-\nsional geometry of the eROSITA bubbles and the origin of the\nfeatures observed here. What seems clear from the data is the\npseudo-temperature variation across the edges of the bubbles,\nindicating the presence of a shell.\n10. Conclusions\nWe investigated the eROSITA half sky maps within narrow en-\nergy bands characteristics of the brightest soft X-ray line transi-\ntions.\n–We subtracted the non-CGM components from soft X-ray\nemission lines (O viiand O viii) and corrected the maps for\nthe effects of absorption to determine the emission from the\nhot plasma associated with the Galactic CGM;\n–Towards regions with little e ffects due to absorption, we de-\nrive maps of the pseudo-temperature of the CGM plasma\nfrom the O viii/Oviiline ratio, which is a good tracer of\nthe pseudo-temperature of the warm-hot CGM unperturbed\nby the Galactic outflow;\n–We constrain, for the first time, the CGM temperature fluc-\ntuations both on relatively small ( ∼2−20◦) and large scale.\nIndeed, the CGM pseudo-temperature is consistent with be-\ning constant (therefore consistent with a normal distribu-\ntion), with fluctuations smaller than ∆T/T≤ 4 % on angular\nscales between∼2 and∼20◦, while the CGM temperature\nshows variations as large as ∼12 % when comparing the\nnorthern and southern hemispheres, therefore indicating that\nindeed pseudo-temperature fluctuations are present on large\nangular scales (i.e., ≫20◦);\n–Towards the northern and southern eROSITA bubbles we de-\ntect significant pseudo-temperature fluctuations, indicating\nsignificant temperature fluctuations towards the interior of\nthe bubbles.\n–The line ratio map shows significant variations at the edges\nof the eROSITA bubbles, suggesting the presence of a colder\n\"shell\" at the edge of the outflow. In particular, when the line\nratio map is translated into pseudo-temperature map, we can\nsee that the interior of the eROSITA bubbles appears hot-\nter than the surrounding CGM. Approaching the edge from\nthe interior of the bubble, we observe a rise of the pseudo-\ntemperature, followed by a significant drop, to then become\nagain hotter, with a pseudo-temperature consistent with the\nsurrounding CGM. This suggests the presence of a shell of\ncooler material present at the edge of the bubbles.\nAcknowledgements. This work is based on data from eROSITA, the soft X-ray\ninstrument aboard SRG, a joint Russian-German science mission supported by\nthe Russian Space Agency (Roskosmos), in the interests of the Russian Academy\nof Sciences represented by its Space Research Institute (IKI), and the Deutsches\nZentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lav-\nochkin Association (NPOL) and its subcontractors, and is operated by NPOL\nwith support from the Max Planck Institute for Extraterrestrial Physics (MPE).\nThe development and construction of the eROSITA X-ray instrument was led\nby MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg\n& ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory,\nthe Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for As-\ntronomy and Astrophysics of the University of Tübingen, with the support of\nDLR and the Max Planck Society. The Argelander Institute for Astronomy of\nthe University of Bonn and the Ludwig Maximilians Universität Munich also\nparticipated in the science preparation for eROSITA. The eROSITA data shown\nhere were processed using the eSASS /NRTA software system developed by the\nGerman eROSITA consortium. We acknowledge financial support from the Eu-\nropean Research Council (ERC) under the European Union’s Horizon 2020 re-\nsearch and innovation program Hot- Milk (grant agreement No 865637). GPacknowledges support from Bando per il Finanziamento della Ricerca Fonda-\nmentale 2022 dell’Istituto Nazionale di Astrofisica (INAF): GO Large program\nand from the Framework per l’Attrazione e il Ra fforzamento delle Eccellenze\n(FARE) per la ricerca in Italia (R20L5S39T9).\nReferences\nAllen, B., Hoak, D., Hong, J., et al. 2019, in LPI Contributions, V ol. 2189, As-\nteroid Science in the Age of Hayabusa2 and OSIRIS-REx, 2009\nAnders, E. & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53, 197\nBalucinska-Church, M. & McCammon, D. 1992, ApJ, 400, 699\nBrandt, W. N., Ni, Q., & The Xmm-Servs Team. 2021, in American Astro-\nnomical Society Meeting Abstracts, V ol. 53, American Astronomical Society\nMeeting Abstracts, 209.01\nBrunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1\nDas, S., Mathur, S., Gupta, A., Nicastro, F., & Krongold, Y . 2019a, ApJ, 887,\n257\nDas, S., Mathur, S., Nicastro, F., & Krongold, Y . 2019b, ApJ, 882, L23\nGaleazzi, M., Chiao, M., Collier, M. R., et al. 2011, Experimental Astronomy,\n32, 83\nGilli, R., Comastri, A., & Hasinger, G. 2007, A&A, 463, 79\nGórski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759\nGreen, G. 2018, The Journal of Open Source Software, 3, 695\nGupta, A., Mathur, S., Kingsbury, J., Das, S., & Krongold, Y . 2023, Nature As-\ntronomy, 7, 799\nHenley, D. B. & Shelton, R. L. 2012, ApJS, 202, 14\nHenley, D. B., Shelton, R. L., Kwak, K., Joung, M. R., & Mac Low, M.-M. 2010,\nApJ, 723, 935\nHI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, A&A, 594, A116\nJoubaud, T., Grenier, I. A., Ballet, J., & Soler, J. D. 2019, A&A, 631, A52\nKataoka, J., Sofue, Y ., Inoue, Y ., et al. 2018, Galaxies, 6, 27\nKnies, J. R. 2022, doctoralthesis, Friedrich-Alexander-Universität Erlangen-\nNürnberg (FAU)\nKuntz, K. D. 2019, A&A Rev., 27, 1\nLiu, W., Chiao, M., Collier, M. R., et al. 2017, ApJ, 834, 33\nLocatelli, N., Ponti, G., & Bianchi, S. 2022, A&A, 659, A118\nLocatelli, N., Ponti, G., Zheng, X., et al. 2023, arXiv e-prints, arXiv:2310.10715\nLodders, K. 2003, ApJ, 591, 1220\nLodders, K., Palme, H., & Gail, H. P. 2009, Landolt Boernstein, 4B, 712\nLuo, B., Brandt, W. N., Xue, Y . Q., et al. 2017, ApJS, 228, 2\nMerloni, A., Lamer, G., Liu, T., et al. 2024, A&A, 682, A34\nMiller, M. J. & Bregman, J. N. 2015, ApJ, 800, 14\nPlanck Collaboration, Aghanim, N., Ashdown, M., et al. 2016, A&A, 596, A109\nPonti, G., Sanders, J. S., Locatelli, N., et al. 2023a, A&A, 670, A99\nPonti, G., Zheng, X., Locatelli, N., et al. 2023b, A&A, 674, A195\nPredehl, P., Andritschke, R., Arefiev, V ., et al. 2021, A&A, 647, A1\nPredehl, P., Sunyaev, R. A., Becker, W., et al. 2020, Nature, 588, 227\nSmith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, The\nAstrophysical Journal, 556, L91\nSnowden, S. L., Egger, R., Freyberg, M. J., et al. 1997, ApJ, 485, 125\nSnowden, S. L., Koutroumpa, D., Kuntz, K. D., Lallement, R., & Puspitarini, L.\n2015, ApJ, 806, 120\nSunyaev, R., Arefiev, V ., Babyshkin, V ., et al. 2021, A&A, 656, A132\nVaughan, S., Edelson, R., Warwick, R. S., & Uttley, P. 2003, MNRAS, 345, 1271\nWeaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, R. 1977, ApJ, 218,\n377\nWillingale, R., Starling, R. L. C., Beardmore, A. P., Tanvir, N. R., & O’Brien,\nP. T. 2013, MNRAS, 431, 394\nYeung, M. C. H., Freyberg, M. J., Ponti, G., et al. 2023, A&A, 676, A3\nYoshino, T., Mitsuda, K., Yamasaki, N. Y ., et al. 2009, PASJ, 61, 805\nZheng, X., Ponti, G., Freyberg, M., et al. 2023, A&A\nZonca, A., Singer, L., Lenz, D., et al. 2019, Journal of Open Source Software, 4,\n1298\nArticle number, page 13 of 14A&A proofs: manuscript no. aa\nFig. A.1: Pseudo-temperature values derived by the absorbed\n(i.e. upper panel) versus the de-absorbed (i.e. Fig. 7) lines. The\npixel values are colored accoring to their absolute Galactic lat-\nitude, by ranges of 10 deg, as labelled. All |b|>30 deg values\ndeviate from the 1:1 relation (black solid line) to less than 5% of\ntheir value.\nAppendix A: Foreground absorption\nThe flux of an emission line is contributed by di fferent sources,\nas discussed above. Some of these sources, such as the CXB\nand the CGM (i.e. the object of our study) are also absorbed by\nforeground cold(er) layers of material. In addition, the absorp-\ntion coe fficient depends on the energy considered (Balucinska-\nChurch & McCammon 1992). To correctly interpret the line ratio\nbetween di fferent emission lines it is thus necessary to account\nfor the fraction of absorbed ration and add it back to the total\nemitted line flux. In order to do so, a model for the absorption\ncolumn density (and metal content) has to be assumed. The fi-\nnal line ratio thus depends on the absorption model, and poten-\ntially introduces systematics in the results. The absorption model\nadopted in this work is the same as the one described in Locatelli\net al. (2023), we refer the reader to that work for further details\non the model. In Fig. 9 we plot a pseudo-temperature map (up-\nper panel) derived from a line ratio where we did not added back\nthe absorbed emission, and thus does not depend on the absorp-\ntion model. By comparing it with Fig. 7 (we plot one versus the\nother in the lower panel of Fig. 9) we show that for all abso-\nlute Galactic latitudes |b|≥20 deg, the deviation introduced by\nthe absorption model are negligible. This is mostly due to the\nrelatively small column densities of absorbing material found at\nthese latitudes. This also means that any systematic deviation in-\ntroduced by the absorption model will be smaller than the devia-\ntion from unity seen in the lower panel Fig. 9 at a given latitude.\nGiven that our representative samples in both the northern and\nsouthern hemispheres have been drawn at |b|>30 deg for all\npixels, we constrain any systematic deviation introduced by the\nabsorption model in our samples to ∆T/⟨T⟩<5%.\nArticle number, page 14 of 14" }, { "title": "2401.17318v1.The_Variability_of_the_Broad_Line_Profiles_of_SDSS_J1430_2303.pdf", "content": "Publ. Astron. Soc. Japan (2018) 00(0), 1–6\ndoi: 10.1093/pasj/xxx0001\nThe Variability of the Broad Line Profiles of\nSDSS J1430+2303\nAtsushi H OSHI1,2∗, Toru Y AMADA ,2,1and Kouji O HTA3\n1Astronomical Institute, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai, Miyagi 980-8578,\nJapan\n2Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1,\nY oshinodai, Chuou-ku, Sagamihara, Kanagawa, 252-5210, Japan\n3Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyou-ku, Kyoto,\n606-8502, Japan\n∗E-mail: hoshi@astr.tohoku.ac.jp\nReceived 12-Oct-2023; Accepted 18-Dec-2023\nAbstract\nSDSS J1430+2303 has been argued to possess a supermassive black hole binary which is\npredicted to merge within a few months or three years from January 2022. We conducted\nfollow-up optical spectroscopic observations of SDSS J1430+2303 with KOOLS-IFU on Seimei\nTelescope in May, June, and July 2022, and April 2023. The observed spectrum around Hα\nshows a central broad component ∼103km s−1blueshifted from the narrow H αline as well\nas the broader double-peaked component with a separation of ∼ ±5×103km s−1, similar to\nthe spectrum reported in January 2022. We investigate the variability of the complex broad Hα\nemission line relative to the continuum over the observation period. The continuum-normalized\nrelative flux of the central broad component shows the increasing trend from May to July 2022\nwhich is interpreted to be caused by the decrease of the continuum as also supported by\ndamping of the X-ray, UV, and optical light curves observed for the same period. From July\n2022 to April 2023, however, the central broad component decreased significantly. For the\nrelative flux of the broader double-peaked component, on the other hand, no significant change\nappears at any epoch. These results suggest that the complicated broad line profile of SDSS\nJ1430+2303 is generated from at least two distinct regions. While the central broad component\noriginates from a broad line region, the broader double-peaked component arises in the vicinity\nof the continuum source.\nKey words: Active galactic nuclei — Supermassive black holes — Black hole physics — Seyfert galaxy\n1 Introduction\nActive galactic nuclei (AGNs) play an important role in\nunderstanding the growth of supermassive black holes\n(SMBHs) and their relationship with the host galaxies. A\ncrucial question is how SMBHs formed and evolved in the\nhierarchical growth of galaxies in the universe (Silk & Rees\n1998; Hopkins et al. 2008; Kormendy & Ho 2013). While\n∗Example: Present Address is xxxxxxxxxxSMBHs are formed in galaxies and they increase their mass\nby the subsequent gas accretion events, they can also grow\nby merging of the galaxies which already host the SMBH\n(Rees 1984).\nMore than a few supermassive black hole binary (SMBHB)\nsystems and candidates have been identified and exten-\nsively studied so far (Begelman et al. 1980; Roos 1981; De\nRosa et al. 2019; Sillanpaa et al. 1988; Bhatta et al.\n2016; Dey et al. 2019; Komossa et al. 2021). They are\n© 2018. Astronomical Society of Japan.arXiv:2401.17318v1 [astro-ph.GA] 24 Jan 20242 Publications of the Astronomical Society of Japan , (2018), Vol. 00, No. 0\nTable 1. Summary of the Observations\nEpoch Proposal ID Date MJD From Jan 1, 2022 Grism Total Exposure Standard star\n(a) 22A-K-0029 May 27, 2022 59726 146 days VPH-blue 1800 sec HZ 44\n(b) 22A-K-0029 June 18, 2022 59748 168 days VPH683O56 3600 sec BD+33◦2642\n(c) 22B-N-CN05 July 28, 2022 59788 208 days VPH-blue 1800 sec BD+33◦2642\n(d) 23A-N-CT01 April 17, 2023 60051 471 days VPH-blue 1800 sec BD+33◦2642\nobservationally characterized by the resolved close pair\nof AGN or more typically the existence of double-peaked\nemission lines, and sometimes by the distinctive quasi-\nperiodic photometric light curves (Shen & Loeb 2010). The\norbital motion of SMBHB may also affect their broad emis-\nsion line profiles and the cases that the broad lines signif-\nicantly shifted from the narrow lines were observed (Shen\n& Loeb 2010; Popovi´ c 2012).\nSDSS J143016.05+230344.4 (hereafter SDSS J1430+2303\n, also known as AT2019cuk and Tick Tock) which is a\nSeyfert1 galaxy at redshift 0.08105 with AB magnitude\n∼16.5 inr-band, is argued to possess the SMBHB system\njust before the coalescence (Jiang et al. 2022). The rapid\ndecay of the amplitudes and periods of the light curves\nin X-ray, UV and optical wavelength observed by January\n2022 predicts that SMBHB in SDSS J1430+2303 can coa-\nlesce within ∼100 days or three years (Jiang et al. 2022).\nOn the other hands, a recent follow-up observations argued\nthat the optical light curve is hardly explained by the bi-\nnary scenario (Dotti et al. 2023). It is interesting to note\nthat the optical broad emission line profile of this object\nalso dramatically changed between 2005 and 2022. While\nthe spectrum of SDSS J1430+2303 taken in 2005 showed\nonly one broad line which is blueshifted ( ∼2400 km s−1)\nfrom the narrow H αline, the spectrum taken in January\n2022 showed the drastically changed puzzling profiles of a\nbroad line which is blueshifted ( ∼900 km s−1) from the\nnarrow H αtogether with the component which is repre-\nsented by a pair of two broad lines with symmetry (Jiang\net al. 2022). Such abrupt changes in the broad line profile\nare considered quite unique even among the AGNs with\nthe most prominent line profile variations and it’s impor-\ntant to monitor the phenomena over a sufficiently long time\nscale, whether or not it is directly related to the predicted\nSMBHB coalescence itself.\nA subclass of AGNs is also known to show changes in their\nbroad emission lines in the UV/optical spectrum and called\nas the changing-look (or changing-state) AGNs (MacLeod\net al. 2016; Stern et al. 2018; MacLeod et al. 2019; Ross et\nal. 2020; Ricci & Trakhtenbrot 2023). Although the phys-\nical origins of changing-look AGNs are still under debate,\nthe phenomena can be related to the change in the accre-\ntion rate (Graham et al. 2020). While SDSS J1430+2303can be broadly categorized as a changing-look AGN, the\norigin of the broad line variation may differ from that of\nchanging-look AGNs where typically a single broad emis-\nsion appears or disappears and such drastic change of the\nvelocity profile in SDSS J1430+2303 is rarely observed.\nIn this paper, we report our results of optical spec-\ntroscopy of SDSS J1430+2303 at the H αwavelength re-\ngion over the four epochs from May 2022 to April 2023.\nWe present that the overall line structure is similar to that\nobserved in January 2022 but there is a significant change\nin a part of the broad components.\nThroughout this paper, we assume a ΛCDM cosmological\nparameters of Ω m= 0.3 and Ω Λ= 0.7 and Hubble constant\nH0= 70 km s−1Mpc−1. We use the AB magnitude system.\n2 Observations and Data Reduction\nSDSS J1430+2303 was observed with KOOLS-IFU which\nis a fiber-fed integral field spectrograph equipped with the\n3.8 m Seimei telescope (Matsubayashi et al. 2019; Kurita\net al. 2020). The instrument consists of the 110 fibers for\nobjects (after October 2020), which achieves a total FoV\nof 8.4×8.0 arcsec2. We conducted the spectroscopic obser-\nvations on May 27, June 18 (Proposal ID : 22A-K-0029, PI\n: Ohta) and July 28, 2022(Proposal ID : 22B-N-CN05 PI :\nHoshi), and April 17, 2023 (Proposal ID : 23A-N-CT01 PI\n: Hoshi). These observation epochs are denoted as (a), (b),\n(c) and (d). The information of the observations is summa-\nrized in Table 1. We used the VPH-blue (4100-8900 ˚A) and\nVPH683O56 (6200–7500 ˚A) grisms. The spectral resolu-\ntion ( R=λ/∆λ) of VPH-blue and VPH683O56 are R∼500\nandR∼2000, respectively.\nThe data reduction such as spectrum extraction, flat field-\ning, and wavelength calibration was performed by using\nPyRAF , a python package developed by Space Telescope\nScience Institute (STScI) using the scripts developed for\nKOOLS-IFU. Spectroscopic standard stars, HZ44 and\nBD+33◦2642 were observed for relative flux calibration\nover the observed wavelength ranges (Oke 1990). We\nalso corrected the atmospheric absorption around the red-\nshifted H αlines. For all the observation epochs, flat-\nfielding frames were created by using the twilight sky and\nthe wavelength calibration was performed by using the arcPublications of the Astronomical Society of Japan , (2018), Vol. 00, No. 0 3\nFig. 1. (left):Continuum-normalized spectra R(a)+ 0.9,R(b)+ 0.6,R(c)+ 0.3, and R(d). The spectra observed in epoch (a), (b), (c) and (d) correspond\nto orange, gray, red and blue lines. (right): The AGN continuum-normalized spectra R′\n(a),R′\n(c)andR′\n(d)after subtracting the host component are shown in\nupper panel. In epoch (b), the host component is not measured due to the limited wavelength coverage. The light curves of SDSS J1430+2303 in r-band and\nX-ray at around the observed epochs referring to Masterson et al. (2023) are shown in the lower panel.\nlamps (Hg, Ne, Xe). The wavelength sampling is typically\n∼2˚A per pixel and the wavelength determination accuracy\nis∼0.25˚A. Median stacking was used to remove cosmic\nrays.\n3 Results and Discussion\n3.1 Broad Line Decomposition\nIn order to investigate the variability of the broad H αemis-\nsion line, we present the continuum-normalized spectra in\nFigure 1 (left). Using the curve fitfunction from the\nscipy module (Virtanen et al. 2020) in Python, we deter-\nmined the relative flux ( R(i)), which is normalized by the\ncontinuum. The index icorresponds to each epoch (a, b, c,\nd). The continuum which includes both the AGN and the\nhost galaxy components was derived by fitting the spec-\ntrum within the wavelength ranges of 6150-6250 ˚A and\n6850-6950 ˚A using a linear function. We also show the\nAGN continuum-normalized spectra ( R′\n(i)) after subtract-\ning the host component in Figure 1 (upper right). Since\nSDSS J1430+2303 is classified as a typical elliptical galaxy\n(Jiang et al. 2022), the host contribution is estimated by\nthe equivalent width (EW) of Na D absorption line using\na template spectrum of the elliptical galaxy in Kinney et\nal. (1996). Although the Na D absorption may suffer from\ninterstellar absorption, its wavelength is close to that of\nHαand we here ignored the Na D interstellar absorption,\nwhich corresponds to the case of maximum contributionof the host component. Unfortunately, we are not able to\nestimate the fraction of the host galaxy in epoch (b) due\nto the limited wavelength coverage of VPH683O56 grism.\nThe lower panel in Figure 1 shows the light curves in r-\nband and X-ray during the observation period up to epoch\n(c) from Masterson et al. (2023).\nThe spectra in the H αregion exhibit a consistently com-\nplicated profile over all epochs. To decompose the possi-\nble multiple components we fit the spectra using a linear\ncontinuum in the range 6100-7000 ˚A and simple multi-\nple Gaussian components in the H αregion. The veloc-\nity widths of the five narrow emission lines, namely, H α,\n[NII]λλ6548,6583, and [SII] λλ6717,6734 are fixed at the\nsame value for each epoch. The flux ratio[NI I]λ6583\n[NI I]λ6548= 4.26\nis fixed with the typical composite quasar spectra from\nSDSS (Vanden Berk et al. 2001). The [OI] λ6300 line is\nalso fitted simultaneously, but the velocity width is not\nfixed.\nThe broad H αfeature can be fitted by the combina-\ntion of the central component (hereafter referred as BL 1)\nwith FWHM ∼5×103km s−1, which is ∼103km s−1\nblueshifted from the narrow H αline, and a symmetrical\nwing-like pair of the blue and redshifted ( −7×103,+5×\n103km s−1) components (hereafter referred as BL 2), as\nsuggested by Jiang et al. (2022) for the spectra taken in\nJanuary 2022. For the component BL 2, symmetrical pro-\nfiles with two peaks of the same width and intensity pro-\nvide reasonable fitting results for the spectrum at each4 Publications of the Astronomical Society of Japan , (2018), Vol. 00, No. 0\nepoch. The obtained velocity width of the each BL 2peak\nis∼4×103km s−1. We show the best fitting result and\na representative continuum-normalized spectrum of SDSS\nJ1430+2303 at epoch (c) in Figure 2 for an example.\n6200 6300 6400 6500 6600 6700 6800 6900\nRestframe Wavelength [Å]0.81.01.21.41.61.82.02.22.42.6Continuum-Normalized Relative FluxBL2(blue)BL1\nBL2(red)Data (July 28, 2022)\nGaussian components\nSDSS (June 12, 2004)\nTotal-15000 -10000 -5000 0 5000 10000 15000Velocity from Narrow H [kms1]\nFig. 2. The broad Hαprofile of SDSS J1430+2303. The blue line shows\nthe observed spectrum which is normalized by the total continuum on epoch\n(c). The cyan line shows the continuum-normalized spectrum taken from\nSDSS on the June 12, 2004. The red dashed and solid lines represent each\nGaussian component and the sum of the all models. Narrow Hαline is set\nto 0km s−1. The center of BL1andBL2are plotted by the vertical dashed\nline. The components BL1andBL2can be fitted with the velocity width\n5×103and 4 ×103km s−1.\n3.2 Variability of BLR\nFigure 3 shows the differences in the relative flux spectra\nfrom epoch to epoch. In order to show the significance of\nthe features, we normalized the residuals by the standard\ndeviation values measured at the wavelength ranges used\nfor the continuum fitting. Note that the standard devia-\ntions of R(b)in the continuum is 0.20, which is more than\ntwice as large as those of the other epochs (0.08, 0.06 and\n0.07 for R(a),R(c), and R(d)), since it was obtained us-\ning a different higher dispersion grism, VPH683O56. The\nred histograms represent the residuals in 10 ˚A bins. For\ncomparison, we also show the mean values of the central\nwavelength of BL 1, BL 2(blue) and BL 2(red) in all epochs by\nthe vertical dashed lines.\nWe found significant positive or negative variations in\nthe broad line features (BL 1). The relative line flux at\naround 6550 ˚A increases from epoch (a) to (b), and (a) to\n(c) in the top panels of the left and middle column. The\npeak of the variation nearly coincides with the center of\nBL1. No significant variation is observed between (b) and(c) in the middle left panel. The relative flux in epoch (d)\nis reduced compared to other epochs, leading to negative\nvariations around 6500 ˚A\nSuch variations in the relative flux must occur due to\nthe time lag between AGN continuum variations and the\nresponse by the ionized gas. Therefore, the central broad\ncomponent BL 1which shows the variability of the rela-\ntive flux is considered to be originated from the distant\nregion from the central continuum source such as a BLR.\nIt is interesting to note that these variations are observed\nonly within the limited wavelength range of the broad H α\nline, implying that the complicated broad emission line fea-\nture of SDSS J1430+2303 can be physically decomposed\ninto multiple components. Given that there is no signifi-\ncant variation in the relative flux around the two peaks of\nBL2(blue) and BL 2(red) in Figure 3, it suggests the broad\nlines respond rapidly to the change of the continuum flux.\nThis feature must be originated from the vicinity of the\ncontinuum source, significantly closer than a few tens of\nlight days. In addition to the variability of the relative\nflux, we also found that there is a wavelength shift be-\ntween the peaks of the positive residual from (a) to (c)\nand the negative residual from (c) to (d). This shift can\nbe interpreted as the change of BLR profile responsible for\nBL1, suggesting that the gas in the region changes its kine-\nmatic profile within a yearly observation timescale. Similar\nvariations are observed in the AGN continuum-normalized\nspectra after subtracting the host galaxy component, as\nshown in the right column of Figure 3.\nSDSS J1430+2303 is observed in the X-ray and optical\nwavelength more than 200 days since January 1, 2022. The\nlight curve in X-ray and optical bands shows a continuous\ndecline from the 156th day, when the large flare occurred,\nto the 200th day (Masterson et al. 2023). The increas-\ning of the relative flux R(i)at around the wavelength of\n6550 ˚A from epoch (a) to (b) and (c) is attributed to the\ncontinuous decline of the continuum (epoch (a) and (c) cor-\nrespond to the 146th day and the 208th day after January\n1, 2022). Quantitatively, the continuum flux in r-band, in-\ncluding the host galaxy, decreased by ∼10% from epoch (a)\nto (c) (Masterson et al. 2023). This decrease contributes\nto the positive residual of the peak relative flux of BL 1by\n∼0.2 (10% of ∼1.8 see BL 1component in Fig.2). The\ndecrease of R(i)at around the wavelength of 6500 ˚A from\nepoch (c) to (d), suggests the recovery of the continuum\nflux after epoch (c). However it should be noted that no\ninformation regarding the light curves in 2023 is available\nat the time of writing this paper.\nWe refrain from further speculation on questions such\nas why the BL 1and BL 2structures appeared in the re-\ncent spectra in 2022, the origin of the BL 2, and whetherPublications of the Astronomical Society of Japan , (2018), Vol. 00, No. 0 5\n−4σ−3σ−2σ−1σ01σ2σ3σ4σR(c)−R(a)1σ=0.097\n−4σ−3σ−2σ−1σ01σ2σ3σ4σR(d)−R(c)1σ=0.098\n6200 6400 6600 6800\nRestframe Wavelength [Å]−1σ01σR(d)−R(b)1σ=0.21−1σ01σR(b)−R(a)1σ=0.20\n−1σ01σR(c)−R(b)1σ=0.21\n6200 6400 6600 6800\nRestframe Wavelength [Å]−4σ−3σ−2σ−1σ01σ2σ3σ4σR(d)−R(a)1σ=0.099−6σ−3σ03σ6σR/prime\n(c)−R/prime\n(a)1σ=0.17\n−6σ−3σ03σ6σR/prime\n(d)−R/prime\n(c)1σ=0.17\n6200 6400 6600 6800\nRestframe Wavelength [Å]−9σ−6σ−3σ03σ6σ9σR/prime\n(d)−R/prime\n(a)1σ=0.15\nFig. 3. (left and middle column): Significance of broad line variations in the continuum-normalized spectra.. Black solid lines show the significance of the\ndifference in the relative flux R(i)andR(j).1σis a standard deviation of R(i)andR(j)at the wavelength range 6150-6250 and 6850-6950 ˚A. Each\nhistogram shows the difference in the relative flux R(i)andR(j)with10˚Abins. The mean values for the center of BL1,BL2(blue) andBL2(red) are plotted\nby the vertical dashed lines. While each bin does not exceed 3 sigma, the continuous change around the central wavelength of BL1is significant. The BL1\ncomponent exhibits a significant variation around 6550 ˚A, whereas the BL2component remains invariant. (right column): Significance of broad line variations\nin the AGN continuum-normalized spectra of R′\n(i).\nthese variations are related to the claimed SMBHB merg-\ning event or not. To understand the complexity of the line\nprofile and their physical origins, it is essential to continue\nmonitoring the further variation of this interesting object\nSDSS J1430+2303 .\nAcknowledgments\nWe thank Kyoto University and the National Astronomical\nObservatory of Japan (NAOJ) for their great help, advice,\nand cooperation at the Seimei Telescope Observatory. This\nwork was supported by JST, the establishment of university\nfellowships towards the creation of science technology innova-\ntion, Grant Number JPMJFS2102. This publication is based\nupon work supported by KAKENHI (22K03693) through Japan\nSociety for the Promotion of Science.\nReferences\nSilk, J. & Rees, M. J. 1998, A&A, 331, L1. doi:10.48550\nHopkins, P. F., Hernquist, L., Cox, T. J., et al. 2008, ApJS,\n175, 356. doi:10.1086/524362\nKormendy, J. & Ho, L. C. 2013, ARA&A, 51, 511.\ndoi:10.1146/annurev-astro-082708-101811\nRees, M. J. 1984, ARA&A, 22, 471.doi:10.1146/annurev.aa.22.090184.002351\nBegelman, M. C., Blandford, R. D., & Rees, M. J. 1980, Nature,\n287, 307. doi:10.1038/287307a0\nRoos, N. 1981, A&A, 104, 218\nDe Rosa, A., Vignali, C., Bogdanovi´ c, T., et al.\n2019, New Astronomy Reviews, 86, 101525. doi:10.1016\n/j.newar.2020.101525\nSillanpaa, A., Haarala, S., Valtonen, M. J., et al. 1988, ApJ,\n325, 628. doi:10.1086/166033\nBhatta, G., Zola, S., Stawarz, L., et al. 2016, ApJ, 832, 47.\ndoi:10.3847/0004-637X/832/1/47\nDey, L., Gopakumar, A., Valtonen, M., et al. 2019, Universe, 5,\n108. doi:10.3390/universe5050108\nKomossa, S., Ciprini, S., Dey, L., et al. 2021, XIX Serbian\nAstronomical Conference, 100, 29. doi:10.48550\nShen, Y. & Loeb, A. 2010, ApJ, 725, 249. doi:10.1088/0004-\n637X/725/1/249\nPopovi´ c, L. ˇC. 2012, New Astronomy Reviews, 56, 74.\ndoi:10.1016/j.newar.2011.11.001\nJiang, N., Yang, H., Wang, T., et al. 2022, arXiv:2201.11633.\ndoi:10.48550\nDotti, M., Bonetti, M., Rigamonti, F., et al. 2023, MNRAS,\n518, 4172. doi:10.1093/mnras/stac3344\nMacLeod, C. L., Ross, N. P., Lawrence, A., et al. 2016, MNRAS,\n457, 389. doi:10.1093\nStern, D., McKernan, B., Graham, M. J., et al. 2018, ApJ, 864,6 Publications of the Astronomical Society of Japan , (2018), Vol. 00, No. 0\n27. doi:10.3847/1538-4357/aac726\nMacLeod, C. L., Green, P. J., Anderson, S. F., et al. 2019, ApJ,\n874, 8. doi:10.3847/1538-4357/ab05e2\nRoss, N. P., Graham, M. J., Calderone, G., et al. 2020, MNRAS,\n498, 2339. doi:10.1093/mnras/staa2415\nRicci, C. & Trakhtenbrot, B. 2023, Nature Astronomy, 7, 1282.\ndoi:10.1038/s41550-023-02108-4\nGraham, M. J., Ross, N. P., Stern, D., et al. 2020, MNRAS,\n491, 4925. doi:10.1093/mnras/stz3244\nMatsubayashi, K., Ohta, K., Iwamuro, F., et al. 2019, PASJ,\n71, 102. doi:10.1093/pasj/psz087\nKurita, M., Kino, M., Iwamuro, F., et al. 2020, PASJ, 72, 48.\ndoi:10.1093/pasj/psaa036\nOke, J. B. 1990, AJ, 99, 1621. doi:10.1086/115444\nMasterson, M., Kara, E., Pasham, D. R., et al. 2023, ApJL,\n945, L34. doi:10.3847/2041-8213/acbea9\nVirtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature\nMethods, 17, 261. doi:10.1038/s41592-019-0686-2\nKinney, A. L., Calzetti, D., Bohlin, R. C., et al. 1996, ApJ, 467,\n38. doi:10.1086/177583\nVanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001,\nAJ, 122, 549. doi:10.1086/321167" }, { "title": "2401.17368v1.Floquet_Bloch_Theory_for_Nonperturbative_Response_to_a_Static_Drive.pdf", "content": "Floquet-Bloch Theory for Nonperturbative Response to a Static Drive\nChristophe De Beule,1Steven Gassner,1Spenser Talkington,1and E. J. Mele1\n1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA\n(Dated: February 1, 2024)\nWe develop the Floquet-Bloch theory of noninteracting fermions on a periodic lattice in the\npresence of a constant electric field. As long as the field lies along a reciprocal lattice vector, time\nperiodicity of the Bloch Hamiltonian is inherited from the evolution of momentum in the Brillouin\nzone. The corresponding Floquet quasienergies yield the Wannier-Stark ladder with interband\ncouplings included to all orders. These results are compared to perturbative results where the\nlowest-order interband correction gives the field-induced polarization shift in terms of the electric\nsusceptibility. Additionally, we investigate electronic transport by coupling the system to a bath\nwithin the Floquet-Keldysh formalism. We then study the breakdown of the band-projected theory\nfrom the onset of interband contributions and Zener resonances in the band-resolved currents. In\nparticular, we consider the transverse quantum-geometric response in two spatial dimensions due to\nthe Berry curvature. In the strong-field regime, the semiclassical theory predicts a plateau of the\ngeometric response as a function of field strength. Here, we scrutinize the conditions under which\nthe semiclassical results continue to hold in the quantum theory.\nI. INTRODUCTION\nA peculiar property of electrons in a crystal is that ap-\nplying a constant direct current (dc) electric field gives\nrise to periodic motion known as Bloch oscillations. As\nthe name suggests, this result was first derived by Fe-\nlix Bloch in 1929 [1], although the oscillatory electron\ntrajectories were only mentioned explicitly by Clarence\nZener five years later [2]. Seemingly counterintuitive,\nthis phenomenon follows naturally from the fact that in\nthe presence of a periodic potential, momentum space\nis compactified to a torus, i.e., the first Brillouin zone\n(BZ). Since in a dc electric field E, momentum increases\nlinearly with time, k(t) =k(0)−eEt/ℏ, this gives rise\nto oscillatory motion with characteristic Bloch frequency\nωB=e|E|a/ℏwhere ais the lattice constant [3–5].\nHowever, Bloch oscillations are typically inconsequen-\ntial for electronic transport because most solid-state sys-\ntems obey ωBτ≪1 where τis a typical momentum-\nrelaxation time. In this case, an electron relaxes its mo-\nmentum to equilibrium before a complete oscillation oc-\ncurs. One approach to reach the regime ωBτ≳1 at\nreasonable field strengths is by means of artificial su-\nperlattices, such as quasi-one-dimensional (1D) semicon-\nductor superlattices [6–8] or two-dimensional (2D) moir´ e\nsystems [9–12], among others. Here the large lattice con-\nstant a∼10 nm compresses the BZ which leads to an\nincrease in ωBsuch that full Bloch orbits can be executed\nbefore a scattering event occurs.\nExperimental observations of Bloch oscillations in\nsolid-state systems have therefore been limited to 1D\nsemiconductor superlattices where a static electric field\nis applied along the superlattice direction and Bloch os-\ncillations with ωBin the THz regime are excited opti-\ncally [6, 13–17]. Moreover, the presence of Bloch oscil-\nlations also manifests itself in the dc electronic response\nthrough a negative differential conductance [3, 18] for\nωBτ≳1. Such a decrease in the current with increas-\ning electric field strength is due to the Wannier-Stark(WS) localization of electrons in a static electric field\n[19–23]. Additionally, it has been recognized recently\nthat the quantum geometry of Bloch bands, specifically\nthe Berry curvature, gives rise to so-called geometric os-\ncillations transverse to the electric field [11]. Moreover,\nthese geometric oscillations lead to a characteristic peak\nin the transverse differential conductance [11, 12], as well\nas peaks in the optical Hall conductivity that encode the\nquantum geometry of the band [24]. Furthermore, Bloch\noscillations of cold atoms have been observed in optical\nlattices [25, 26]. The latter are extremely clean systems\nsuch that ωBτcan be large even though the Bloch pe-\nriod can be up to 10 orders of magnitude larger than in\nsolid-state superlattices.\nAs an illustrative example, consider a one-dimensional\ncrystal with a partially occupied energy band Ek. In\nthis case, we have E=Eˆxand the semiclassical electron\ntrajectories are obtained from\ndx\ndt=1\nℏ∂Ek(t)\n∂k=−1\neEdEk(t)\ndt, (1)\nwith−ethe electron charge. The solution reads x(t) =\n−E[k(t)]/eEup to an additive constant, where the am-\nplitude of the oscillation is given by W/e|E|for electronic\nbandwidth W. Hence the size of the Bloch orbit is pro-\nportional to 1 /|E|, which is the previously mentioned WS\nlocalization. Complete orbits thus require that the size\nof the orbit be smaller than the mean free path, while\nonly partial orbits are completed otherwise.\nIn higher spatial dimensions, the resulting trajectory in\nmomentum space is generally not closed for an arbitrary\ndirection of the electric field and eventually traces out\nthe entire BZ. In this case, generic trajectories in both\nmomentum and real space are oscillatory but not peri-\nodic. Electron motion is only periodic when the electric\nfield lies along a reciprocal lattice vector g, or equiva-\nlently when it is perpendicular to a lattice plane. In this\ncase, the frequency is given by Ω = 2 πωBa/|g|. The\ncorresponding real-space semiclassical trajectories gen-arXiv:2401.17368v1 [cond-mat.mes-hall] 30 Jan 20242\nerally consist of three parts: an oscillatory motion with\ncomponents along the field direction, a transverse oscilla-\ntory motion originating from the anisotropy of the energy\nband as well as the anomalous velocity, and a transverse\ndrift [11]. The latter originates from velocity compo-\nnents not accelerated by the field and only contributes\nto the net transport current when time-reversal symme-\ntry is broken. It corresponds to a drift of the guiding\ncenter of the Bloch orbit and contains a topological term\nproportional to the Chern number of the band.\nSo far we have tacitly assumed the adiabatic limit\nwhere the energy band is sufficiently isolated from other\nbands. In this limit, we can neglect interband Zener tran-\nsitions and only consider the dynamics in the occupied\nband. At lowest order, interband corrections induce a\nfield-dependent polarization shift [27] which corresponds\nto the electric susceptibility [28]. However, this term only\ncontributes to transport in systems where spatial inver-\nsion symmetry and time-reversal symmetry are simulta-\nneously broken. In this work, we consider interband cor-\nrections for electric transport beyond perturbation the-\nory in |E|. We accomplish this by treating the dynam-\nics exactly with Floquet theory for a commensurate con-\nstant electric field E∥g[29–33]. In this case, we know\nthat the band-projected semiclassical dynamics is peri-\nodic since momentum space is defined on a torus whose\ngenerators are reciprocal lattice vectors. The quantum\ntheory inherits this feature from minimal substitution\nk→k+eA(t) in temporal gauge (also known as ve-\nlocity gauge) for a uniform electric field. As such, we\nwill show that the Bloch Hamiltonian becomes time pe-\nriodic up to a unitary transformation for commensurate\nfields. Importantly, by treating a static drive in tempo-\nral gauge, one finds that the longitudinal momentum be-\ncomes a gauge degree of freedom which can be absorbed\nin the time origin. Hence, the original time-independent\nproblem in Dspatial dimensions is mapped to a time-\ndependent problem in D−1 spatial dimensions. This is\nsimilar to how a periodic drive in Ddimensions can be\nmapped to a time-independent problem on the Floquet\nladder in D+ 1 dimensions [34]. In fact, here we are\ntrading one physical spatial dimension, defined by the\ndirection of the electric field, with a synthetic Floquet\ndimension along which translation symmetry is broken\nby the electrostatic potential.\nThis paper is further structured as follows. In Sec-\ntion II, we introduce the Floquet formalism for the non-\nperturbative treatment of a static uniform electric field\napplied to a periodic lattice of free fermions. Special\ncare is taken to account for the sublattices by a mod-\nified Floquet ansatz yielding the Floquet Hamiltonian\nwhose quasienergies give the WS ladder. Then in Sec-\ntion III, we couple the clean system to a reservoir with\nthe Floquet-Keldysh formalism and calculate the band-\nresolved charge currents. Next, in Section IV we apply\nthe transport theory to examples in one and two spatial\ndimensions. Specifically, we study the onset of interband\nZener resonances in a 1D dimer chain, as well as thecurrent anisotropy and transverse geometric response for\nthe honeycomb lattice in 2D with sublattice and Haldane\nmass terms. We finally present our conclusions in Section\nV. Unless stated otherwise, we set ℏ= 1 from now on.\nII. FLOQUET THEORY OF A CRYSTAL IN A\nCONSTANT ELECTRIC FIELD\nIn this section, we introduce the Floquet formalism for\nthe nonperturbative treatment of a static uniform elec-\ntric field applied to a periodic lattice of noninteracting\nfermions with a finite number of orbitals. Here the Flo-\nquet quasienergy spectrum yields the familiar Wannier-\nStark (WS) ladder [31]. At the end of this section, we\nconsider an alternative approach in band basis and ob-\ntain an approximate analytical solution for the WS ladder\nthat is valid up to second order in the field.\nA. Lattice Hamiltonian in velocity gauge\nOur starting point is the Hamiltonian for noninteract-\ning electrons hopping on a lattice in Dspatial dimen-\nsions, which can be written as\nH0=X\nr,r′X\na,bHab\nr−r′c†\nracr′b, (2)\nwhere r=PD\ni=1niai(andr′) are lattice vectors which\nlabel the cells with niintegers and aithe primitive lattice\nvectors of the Bravais lattice. Here aandbare orbital\nindices, which includes sublattice degrees of freedom and\nspin. We also define the creation (annihilation) operators\nc†\nra(cra) which create (destroy) a fermion in cell rin\norbital a. The hopping amplitude from orbital bin cell\nr′to orbital ain cell ris then given by Hab\nr−r′where the\ndependence on r−r′reflects the translational symmetry\nof the system.\nWe now consider a uniform electric field (equivalent\nto taking the dipole approximation) E(t) which can be\nintroduced without breaking translational symmetry [20]\nin the velocity gauge via the Peierls substitution:\nHab\nr→ Hab\nrexp\u0014\n−ieZr+ra\nrbds·A(t)\u0015\n(3)\n=Hab\nre−ieA(t)·(r+rab), (4)\nwithE(t) =−∂tA(t). Here −eis the electron charge,\nrais the sublattice position of orbital ain the unit cell,\nandrab=ra−rb. The lattice Hamiltonian can then be\ndiagonalized by Fourier transform:\ncra=1√\nNX\nkeik·(r+ra)cka, (5)\nwith Nthe number of cells. The Hamiltonian becomes\nH(t) =X\nkX\na,bHab[k+eA(t)]c†\nkackb, (6)3\nwhere the Bloch Hamiltonian is explicitly given by\nHab(k) =X\nre−ik·(r+rab)Hab\nr. (7)\nHence a uniform electric field can be introduced in the\nBloch Hamiltonian by the usual minimal substitution\nprocedure on the crystal momentum k→k+eA(t) since\nthe velocity gauge conserves the translational symmetry\nof the crystal. Note that we have retained information on\nthe intracell positions in Eq. (4) and Eq. (5) to properly\naccount for the electrostatic potential [35]. In the (in-\nstantaneous) band basis, this choice corresponds to the\nperiodic gauge where the total Bloch wave function is de-\nfined on a torus: |ψs,k+g⟩=|ψsk⟩with sthe band index\nandga reciprocal lattice vector [36].\nIf we want to treat the external field perturbatively in\nvelocity gauge, one usually considers a uniform field that\noscillates with frequency ω,\nA(t) =|E|\n2iω\u0000\nϵe−iωt−ϵ∗eiωt\u0001\n, (8)\nwith complex polarization vector ϵ. Then |A(t)| ≤E|/ω\nis bounded and the static limit is obtained by taking ω\nto zero at the end of the calculation. However, here we\nare interested in the nonperturbative treatment of a dc\nuniform electric field. Hence, we take\nA(t) =−tE, (9)\nsuch that the Bloch Hamiltonian [37]\nH(k, t) =H[k(t)] =H(k−eEt). (10)\nWe now introduce the notion of a commensurate electric\nfield, which lies parallel to a reciprocal lattice vector, or\nequivalently perpendicular to a lattice vector,\nE=Eg/g, (11)\nwhere E>0 and gis a nonzero reciprocal lattice vector\nwith g=|g|. For a commensurate field, we thus have\nH(k, t+T) =H(k−g, t) =U−gH(k, t)U†\n−g,(12)\nwith Bloch period T=g/eEand\nUab\ng=e−ig·raδab, (13)\nwhere a, bdenote orbital indices, and we used that g·\nr∈2πZ. For a commensurate electric field, the Bloch\nHamiltonian is thus periodic in time up to a diagonal\nunitary transformation.\nB. Floquet ansatz\nThe time periodicity of the Bloch Hamiltonian allows\nus to solve the time-dependent Schr¨ odinger equation,\ni∂t|Φk(t)⟩=H(k, t)|Φk(t)⟩, (14)with a modified Floquet ansatz\nΦka(t) =e−iεktX\nn∈Zei(n+λa)Ωtϕka,n, (15)\nwhere εkis the quasienergy, Ω = 2 π/T is the Floquet\nfrequency, and we defined λa=g·ra/2πsuch that\nΦka(t+T) =e−iεkTeig·raΦka(t), (16)\nwhich undoes the unitary in Eq. (13) and makes Eq. (14)\ninvariant under t7→t+Teven for systems with sublattice\nstructure. Substituting the ansatz from Eq. (15) into the\nSchr¨ odinger equation, yields\n[εk−(m+λa)Ω]ϕka,m=X\nn∈ZX\nbHab\nmnϕkb,n,(17)\nwithHab\nmn=Hab\nm−nwhere\nHab\nm−n=Ω\n2πZ2π/Ω\n0dt e−i(m−n+λa−λb)ΩtHab(k, t) (18)\n=X\nre−ik·(r+rab)Hab\nr\n×Ω\n2πZ2π/Ω\n0dt e−i(m−n−g·r/2π)Ωt(19)\n=X\nre−ik·(r+rab)Hab\nrδm−n,g·r/2π, (20)\nwhere g·r/2πis an integer by definition. The Floquet\nHamiltonian is then defined as\n[HF(k)]ab\nmn= Ω ( m+λa)δmnδab+Hab\nm−n(k),(21)\nwith HF(k)|ϕk⟩=εk|ϕk⟩. Here the first term is inter-\npreted as the potential energy of a charge −elocated at\nsiter+ra. Indeed, we have Ω( m+λa) =eE·(r+ra) for\ng·r= 2πmwhere Ω λais a field-induced sublattice po-\ntential. The velocity operator in Floquet representation\nis then given by\n(∇kHF)ab\nmn=−iX\nr(r+rab)e−ik·(r+rab)\n× Hab\nrδm−n,g·r/2π,(22)\nwhich contains both intercell and intracell contributions\nin periodic gauge. Note that the Floquet quasienergies\ncorrespond to the WS ladder [19, 38, 39]. Moreover, the\nquasienergy is flatin the momentum direction parallel\nto the electric field. This can be understood as the WS\nlocalization of Bloch states in a static electric field [21].\nIndeed, we notice that\nH[k(t)] =H\u0012\nk⊥−t−t0\nTg\u0013\n, (23)\nwithk⊥=k−\u0000\nk·g/g2\u0001\ngsuch that k⊥·g= 0 and\nt0=Tk·g/g2. Hence the momentum parallel to the4\nelectric field can always be removed by shifting the origin\nof time, i.e., by performing a gauge transformation of the\nvector potential, such that εk=εk⊥. Hence, the time-\nindependent problem in length gauge in Dspatial dimen-\nsions is effectively mapped to a time-dependent problem\nin velocity gauge in D−1 spatial dimensions [34]. Equiv-\nalently, we have mapped the physical direction of the\nelectric field to a synthetic Floquet dimension which is\npossible only if the electric field lies along a reciprocal\nlattice vector.\nC. Band projection, hybrid Wannier basis, and\ninterband corrections\nWhile the orbital basis is best suited to study the\nstrong-field limit, e.g., with the Magnus expansion, a\ngood approximation of the WS ladder in the weak-field\nlimit can be obtained in band basis (see Appendix C)\nby band projection. In a band-projected theory, one re-\nquires that the band Eksunder consideration is energet-\nically isolated from other bands and that the interband\nterms eE·Ass′are small, where Ass′(fors′̸=s) is\nthe interband Berry connection [11, 20, 37, 40]. In this\ncase, time evolution is treated adiabatically and the band\nindex remains a good quantum number. The wave func-\ntion in velocity gauge is then approximately given by the\ninstantaneous eigenstate times a phase modulation,\n|Φks(t)⟩=aks(t)|uks(t)⟩, (24)\nwith\naks(t) = exp\u001a\n−iZt\n0dt′[Eks(t′) +eE·Aks(t′)]\u001b\n,(25)\nwhere we put aks(0) = 1 and where Aks(t) is the in-\nstantaneous intraband Berry connection for band s. For\na commensurate electric field, the wave function satisfies\nEq. (16) and thus aks(t+T) =e−iεksTaks(t) with\nεk⊥s,n=Ek⊥s+nΩ +eE·Ak⊥s, (26)\nwhere\nEk⊥s=Ω\n2πZ2π/Ω\n0dt E s(k−eEt), (27)\nAk⊥s=Ω\n2πZ2π/Ω\n0dtAs(k−eEt), (28)\nare respectively the average band energy and the Berry\nphase along the field which is also called the Zak phase\n[41]. The latter gives the polarization, i.e., the Wannier\ncenter, along the direction of the field [38]. This is illus-\ntrated in Fig. 1 for the case of two bands. Here we work\nin a smooth periodic gauge along the integration path\nsuch that Eq. (28) is well defined. This gauge always ex-\nists regardless of the Chern number of the band. While\nthe Chern number is an obstruction to a smooth gaugein the entire BZ, one can always construct a smooth and\nperiodic gauge along one compact direction [36, 42]. In-\ndeed we can always shift the obstruction away from the\npath{k⊥+sg|0≤s <1}via a gauge transformation\nof the cell-periodic Bloch functions. Moreover, under a\ngauge transformation that preserves periodic gauge, Eq.\n(26) is invariant modulo Ω Zsince\neE·ZT\n0dt∇kφ[k(t)] =−ZT\n0dt ∂tφ(k−eEt) (29)\n=φ(k)−φ(k−g)∈2πZ.(30)\nHowever, a different choice of sublattice positions will\ncorrespond to a different periodic gauge for the cell-\nperiodic Bloch functions. Indeed, the intracell coordi-\nnates enter the band-projected theory only via the in-\ntraband Berry connection. Similarly, the sublattice po-\nsitions also enter explicitly in the Floquet theory [ λain\nEq. (21)]. This is not too surprising as a shift of the spa-\ntial origin changes the energy in the presence of a uniform\nelectric field. In this work, we always use a periodic gauge\nwhere we choose the spatial origin such that the intra-\nband Berry connection is traceless (P\nsAks=0). An\nequivalent result for the band-projected WS ladder can\nalso be obtained in length gauge [11, 38, 40] or via the\nBohr-Sommerfeld quantization rule for Bloch oscillations\n[43, 44].\nThe band-projected result in Eq. (26) has a simple in-\nterpretation in terms of a hybrid momentum space and\nFloquet-Wannier representation [31]. The first term gives\nthe on-site energy in the hybrid Wannier basis along the\nfield direction. The on-site term is the only nonzero term\nsince the momentum parallel to the field can be gauged\naway. This is the WS localization. Obstructions to a\nglobal smooth gauge do not preclude WS localization\nsince it only entails states that are exponentially local-\nized along the field, which is always possible as discussed\nabove. In this representation, we see that both the posi-\ntionnalong the field and k⊥are good quantum numbers.\nIndeed, the electrostatic energy associated with cell ris\ngiven by eE·(r+rk⊥s) =nΩ+eE·Ak⊥swithg·r= 2πn.\nOne can go beyond band projection and include in-\nterband corrections to the WS ladder by solving the dy-\nnamics in band basis up to second order in the interband\nmatrix elements. The result is derived in Appendix C\nand reads\nεk⊥s,n=Ek⊥s+nΩ +eEi\u0010\nAi\nk⊥s+eEjχij\nk⊥s\u0011\n,(31)\nwhere summation over repeated indices is implied and\nχij\nks(t) =X\ns′̸=sAi\nss′(k, t)Aj\ns′s(k, t)\nEks(t)−Eks′(t), (32)\nwithAi\nss′(k, t) the instantaneous interband Berry con-\nnection and where χij\nk⊥sis the (static) electric suscepti-\nbility. The lowest-order interband correction thus gives a5\nFIG. 1. Band-projected WS ladder for a two-band model\nwith energy bands ±Ekshown as the red and blue lines, re-\nspectively, and where E·Aks= 0. The first Floquet zone\n(FZ) is shown in gray and the dashed lines mark the bound-\naries of the first few zones.\nfield-dependent shift of the polarization. To our knowl-\nedge, the electric susceptibility of Bloch bands was first\ndiscussed in length gauge in a 1955 paper by E. O. Kane\n[39] and has recently been recast in the framework of\nquantum geometry [27] where it was shown to give rise to\nan intrinsic nonlinear Hall effect when both spatial inver-\nsion and time-reversal symmetry are broken. If moreover\nthe system conserves their combination, the dominant in-\ntraband nonlinear Hall effect is absent because the Berry\ncurvature vanishes. Recently, this effect was observed\nin thin films of topological insulators with antiferromag-\nnetic order [45, 46]. Since the susceptibility correction\nonly shifts the polarization, energy gaps in the WS ladder\ndue to interband transitions only arise at higher orders in\nthe field. These results are corroborated by a numerical\ndiagonalization of the Floquet Hamiltonian in the orbital\nbasis, as we will demonstrate in Section IV.\nIII. FLOQUET-KELDYSH TRANSPORT\nTHEORY\nIn this section, we use Floquet-Keldysh theory to cal-\nculate the steady-state current by coupling the system\nto a featureless ideal bath. We begin by defining the\nsteady-state current\nj(t) =−eX\na,bZ\nkvab\nk(t)⟨c†\nka(t)ckb(t)⟩ (33)\n≡ieZ\nkTr\u0002\nvk(t)G<(k;t, t)\u0003\n, (34)\nwhere a,bare orbital indices, vab\nk(t) =∇kHab(k, t)\nis the instantaneous velocity operator, andR\nk≡R\nBZdDk/(2π)DwithDthe number of spatial dimensions.\nHere we defined the fermionic steady-state lesser Green’s\nfunction [ G<(k;t, t′)]ab=i⟨c†\nkb(t′)cka(t)⟩in the orbital\nbasis. Since we are interested in the range of validity of\nthe band-projected theory, we also define the intraband\nand interband contributions to the current:\njintra(t) =X\nsjss(t), (35)\njinter(t) =X\ns,s′\ns̸=s′jss′(t), (36)\nwhere s, s′are band indices of the undriven system and\nthe band-resolved currents are given by\njss′(t) =−eZ\nkvss′\nk(t)⟨c†\nks(t)cks′(t)⟩ (37)\n=ieZ\nkTr\u0002\nPks(t)vk(t)Pks′(t)G<(k;t, t)\u0003\n,(38)\nwhere we defined the instantaneous band projectors\nPks(t) =|uks(t)⟩⟨uks(t)|and where the total current\nj(t) =jintra(t) +jinter(t). Here, we prefer Eq. (38) over\nEq. (37) since it is more convenient to calculate the lesser\nGreen’s function in the orbital basis and use band pro-\njectors than it is to calculate the lesser Green’s function\nin the band basis. Our next goal is to rewrite the cur-\nrent in the Floquet basis. To this end, we first define the\nGreen’s function in frequency space,\nG<(k;t, t′) =Z\nωZ\nω′G<(k;ω, ω′)eiωte−iω′t′,(39)\nwhereR\nω≡R+∞\n−∞dω/2π. The modified Floquet ansatz of\nEq. (15) implies that\n[G<(k;t+T, t′+T)]ab=e2πi(λb−λa)[G<(k;t, t′)]ab,(40)\nand therefore we must have ω−ω′= (m+λb−λa)Ω with\nm∈Zin Eq. (39). Hence,\n[G<(k;t, t′)]ab=X\nm,nZΩ\n0dω\n2πZΩ\n0dω′\n2π\n×(G<)ab\nmn(k;ω, ω′)ei[ω−(m+λa)Ω]te−i[ω′−(n+λb)Ω]t′,\n(41)\nwhere we defined\n(G<)ab\nmn(k;ω, ω′)\n≡G<[k;ω−(m+λa)Ω, ω′−(n+λb)Ω] (42)\n= 2πδ(ω−ω′)(G<)ab\nmn(k, ω), (43)\nwith mandnFloquet indices and where the last line\nfollows from Eq. (40). We can then write the equal-time\nlesser Green’s function as [47]\n[G<(k;t, t)]ab=X\nmZ\nω(G<)ab\nm0(k, ω)e−i(m+λa−λb)Ωt,\n(44)6\nwhere we used G<\nmn(k, ω) =G<\nm−n,0(k, ω−nΩ). With\nthis result, the current becomes\nj(t) =ieX\nm,a,bZ\nk,ωvab\nk(t)(G<)ab\nm0(k, ω)e−i(m+λa−λb)Ωt,\n(45)\nwith the nth frequency component\nj(n)=Ω\n2πZ2π/Ω\n0dtj(t)einΩt(46)\n=ieX\nmZ\nk,ωTr\u0002\n(∇kHF)nmG<\nm0(k, ω)\u0003\n,(47)\nwhere we used the definition of the Floquet Hamiltonian\nin Eq. (18). However, for a static field, the steady state\nis static and the ac response vanishes. This follows also\nfrom the fact that a static electric field can be treated\nin length gauge with a scalar potential giving a time-\nindependent problem with a static steady state. Hence,\neven though the semiclassical trajectories are periodic\n[1, 11], there is no net ac response at long times com-\npared to the relaxation rate 1 /Γ. Indeed, in the steady\nstate the density matrix of the WS ladder is completely\ndiagonal such that interladder coherences are absent re-\nsulting in a purely dc response. In order to obtain an ac\nresponse, one requires both a static and oscillating com-\nponent of the electric field [5, 24]. Since we consider a\nstatic field ( ∂tE= 0) the steady state current is there-\nfore static [ j(t) =j] regardless of Ω /Γ. We show this\nexplicitly in Appendix E 2 for a simple chain in one spa-\ntial dimension, for which one can obtain a closed-form\nexpression for the current.\nThe band-resolved currents become\njss′=ieX\nmX\nl,l′Z\nk,ω\n×Tr\u0002\n(PFs)0l(∇kHF)ll′(PFs′)l′mG<\nm0(k, ω)\u0003\n,(48)\nwith projectors in the Floquet basis\n(PFs)ab\nmn=Ω\n2πZ2π/Ω\n0dt Pab\nks(t)e−i(m−n+λa−λb)Ωt,(49)\nwhere we suppressed the momentum index on the left-\nhand side.\nSo far, we have assumed that the system reaches a\nsteady state. To achieve this, we couple the system to a\nbath and calculate the lesser Green’s function G0 since inversion symmetry yields j(−E) =−j(E). The\nsolid line gives the semiclassical result [see Eq. (B16)]. Here\nwe do not show the exact result (see Appendix E 1) as it is\nindistinguishable from the converged numerical result.\nwith [F0\n±(ω−mΩ)]ab=δabf0\n±[ω−(m+λa)Ω]. While\nthe frequency integral can be done analytically by using\nthe single-particle Lehmann representation of the Flo-\nquet Green’s functions GR/A, we find in practice that it\nis faster to perform the frequency integral numerically,\nwhere we ensure convergence by taking a step size of\nΓ/2 which is below the energy resolution of the Green’s\nfunctions. Nevertheless, the analytical results are useful\nin certain cases. For example, in Appendix E 1 we con-\nclusively show that in the zero-temperature limit, while\nalso keeping βΓ fixed such that the coupling to the bath\nis also taken to zero, the ideal bosonic and fermionic bath\nboth yield the same results. For this reason, we will only\nconsider a fermionic bath in the remainder of the paper.\nIV. EXAMPLES\nIn this section, we calculate the current in response to\na static electric field up to nonperturbative order for non-\ninteracting electrons coupled to an ideal fermionic bath\nwith the Floquet quantum theory. We start with a one-\nband model in D= 1 spatial dimensions for which the\ncurrent can be obtained analytically in a closed-form ex-\npression. We then consider the effect of Zener tunneling\nwith a two-band model in D= 1. Finally, we consider\na two-band model in D= 2 on a honeycomb lattice,\nfor which we closely investigate the quantum-geometric\ntransverse response.A. Examples in D= 1\n1. Simple chain\nWe first consider a chain with a single orbital per site\ninD= 1 spatial dimensions with lattice constant aand\nnearest-neighbor hopping J. In this case,\nH(k, t) = 2 Jcos (ka−Ωt), (59)\nwith Ω = eEa. The quasienergies are obtained from\nε−nΩ\nJϕkn=e−ikaϕk,n−1+eikaϕk,n+1.(60)\nIf we substitute ϕkn=e−iknaφn, we find normalizable\nsolutions for ε=mΩ,\nm−n\nJ/Ωφn=φn−1+φn+1, (61)\nwhich is the recurrence relation for Bessel functions. The\nsolution is thus given by\nϕkn=e−iknaJm−n(2J/Ω), (62)\nand for ε= 0 the wave function is given by\nΦk(t) =X\nnein(ka−Ωt)Jn(2J/Ω) (63)\n=e2iJsin(ka−Ωt)/Ω, (64)\nwhich can also be obtained from directly integrating\nthe Schr¨ odinger equation. Moreover, the corresponding\nFloquet-Wannier function centered at the origin is [31]\nWm(t) =a\n2πZ2π/a\n0dkΦk(t)e−ikma(65)\n=e−imΩtJm(2J/Ω), (66)\nwhose spread is (∆ x)2=\nx2\u000b\n− ⟨x⟩2= 2 ( Ja/Ω)2.\nHere we used J−m(z) = ( −1)mJm(z) for real zand\nthe generating function for Bessel functions. This co-\nincides with the period-averaged spread of the semi-\nclassical trajectory, which for k(0) = 0, reads x(t) =\nx(0) + (2 Ja/Ω) [cos(Ω t)−1]. Alternatively, we can un-\nderstand this in terms of a Floquet-induced quantum\nmetric that bounds the spread of the Wannier function:\ng=X\nnh\n|∂kϕkn|2−(ϕ∗\nkni∂kϕkn)2i\n. (67)\nWe now consider the current j(E)ˆxforE=Eˆx. Since\nthere is only a single band in this case, the results for the8\nsimple chain are somewhat trivial. Therefore, we will use\nthe one-band problem to demonstrate convergence as we\ntake into account more Floquet modes in the Floquet-\nKeldysh calculation. This is shown in Fig. 2 for an ideal\nfermionic bath, where we plot the current as a function\nof Ωτ=eEa/Γ (by fixing Γ /Jand varying eEa/J) for\ndifferent numbers of Floquet sidebands. Here we only\nconsider positive values for eEa/Jbecause 1D inversion\nsymmetry ( x7→ −x) gives j(−E) =−j(E). We find that\nconvergence is reached for NF∼100 where NFis the di-\nmension of the Floquet Hamiltonian used in the numer-\nical calculation. In general, one requires more Floquet\nmodes to reach convergence when Ω /J≪1 since there\nare many overlapping Floquet replicas in this case. In\nthe opposite limit, we have Ω /J≫1 such that the Flo-\nquet replicas are well separated and few Floquet modes\nare required to reach convergence.\nApart from small oscillations that appear for eEa/Γ>\n1, we see that the converged current coincides almost\nperfectly with the semiclassical theory. This is to be ex-\npected since band projection is exact if there is only one\nband. Finally, we find that the oscillations are suppressed\nwhen Γ is reduced. Contrary to the semiclassical the-\nory, the current is thus generally a function of both Ω\nand Γ separately. Using the exact closed-form expression\nof the current for an ideal fermionic and bosonic bath,\nwe find that the oscillations are more pronounced in the\nbosonic case and vanish upon decreasing Γ (see Appendix\nE 2). However, reducing Γ increases the number of Flo-\nquet sidebands required to reach convergence, as we need\nsmaller Ω to obtain the same eEa/Γ.\n2. Rice-Mele chain\nIn order to address the role of interband Zener tran-\nsitions, we need to consider a multiband system. For\nsimplicity, we study a two-band model,\nH(k) =d0(k)σ0+d(k)·σ, (68)\nwhere σ0is the 2 ×2 unit matrix and σ= (σx, σy, σz)\nare the Pauli matrices. The energy bands are given by\nEks=d0+sdwith s=±1 the band index and d=|d|.\nSpecifically, we consider the Rice-Mele (RM) dimer chain\n[55] illustrated in Fig. 3(a) with\nd0= 0,d=\n2Jcos\u0000ka\n2\u0001\n−2δsin\u0000ka\n2\u0001\n∆\n, (69)\nwhere ais the lattice constant and we set J= 1. Here\n1±δis the intracell/intercell hopping amplitude, respec-\ntively, and ∆ is a sublattice bias potential. We further\nchoose sublattice positions xA/B=±a/4 which fixes the\nperiodic gauge: uk+2π/a,σ =e−i2πxb/aukσ(σ=A, B) orexplicitly,\n|uks⟩=eika/4\np\n2(1 + sn3) \nn3+s\nn1+in2!\n, (70)\nwithn=d/dand which together with d0= 0 results in a\nsymmetric quasienergy spectrum. This gauge is smooth\neverywhere, i.e., ψk(xa) =eikxaukais smooth and peri-\nodic, since n3never equals −s.\nTo calculate the WS ladder we send k7→k−eEtand\nconstruct the Floquet Hamiltonian where we gauge away\nthe momentum with the unitary transformation\n(Uk)mn=eika(mσ0+σz/4)δmn, (71)\nsuch that the two bands in the first Floquet zone (FZ) are\nflat, although their splitting varies with Ω. The Floquet\nspectrum is obtained from\n[ε−(m+λA)Ω−∆]φAm\n= (1 + δ)φBm+ (1−δ)φB,m−1, (72)\n[ε−(m+λB)Ω + ∆] φBm\n= (1 + δ)φAm+ (1−δ)φA,m+1. (73)\nIn the molecular dimer limit δ→1, there is an exact\nsolution with quasienergy\nlim\nδ→1ε±,n=nΩ±q\n4 +\u0000Ω\n4−∆\u00012, (74)\ngiving the usual Stark ladder of a dimer of length a/2\ncentered at position x=na. A similar result holds\nforδ→ − 1, but now the dimers are centered at x=\n(n+ 1/2)a. For the general case, we diagonalize HFnu-\nmerically. Some results are shown in Fig. 3, where we\ntake both a large and small energy gap Egrelative to the\nrange of field strengths that we consider. In Fig. 3(b),\nwe take Eg= 2 such that interband coupling is weak and\nthe dominant effect comes from the electric susceptibil-\nity, giving rise to a shift of the WS ladder that scales\nlinearly with the field:\nε±,n=nΩ±\u0014\nd+\u0012A\na+χ\na2Ω\u0013\nΩ\u0015\n+O(Ω3),(75)\nwith\nχks(t) =[∂kn(k, t)]·[∂kn(k, t)]\n8sd(k, t), (76)\nfor a two-band model. In this case, χis given by the\nquantum metric divided by the energy gap. This makes\nsense as the quantum metric determines the spread of\nthe maximally-localized Wannier state. Increasing the\nspread thus results in a larger electrostatic potential dif-\nference across the support of the Wannier state and hence\nto an increase in the electric susceptibility. The band-\nprojected result is recovered from the linear in Ω part\nof Eq. (75) and shown as the thin lines in Fig. 3, while9\nFIG. 3. WS ladder of the 1D Rice-Mele chain. (a) Il-\nlustration of the RM model. The dashed rectangle gives the\nunit cell and the hopping amplitudes and on-site energies are\nindicated. (b) WS ladder for Eg= 2. (c) WS ladder for\nEg= 1. Parameters are shown in units of J. The thin red\nand blue lines give the band-projected result for the conduc-\ntion and valence band, respectively, and black lines give the\nquasienergy in the FZ, demarcated by dashed lines, obtained\nfrom diagonalizing the Floquet Hamiltonian numerically.\nthe full result of Eq. (75) containing the lowest-order in-\nterband coupling is compared to the numerical results in\nFig. 4. We see that the shift induced by the susceptibility\nmatches perfectly to the numerical results for the param-\neters chosen. This matching becomes less good when we\ndecrease the energy gap or increase the field strength,\nboth of which increase the magnitude of interband ma-\ntrix elements. For the RM chain, we explicitly find\nd=√\n4 + ∆2Eh\n4−4δ2\n4+∆2i\n+√\n4δ2+ ∆2Eh\n4δ2−4\n4δ2+∆2i\nπ,(77)\nwith E(k) =Rπ/2\n0dθp\n1−ksin2θthe complete elliptic\nintegral of the second kind, and\nA\na=sgn(δ)−1\n4−δ∆Πh\n1−δ2,4−4δ2\n4+∆2i\n2π√\n4 + ∆2, (78)\nin periodic gauge and our choice of sublattice positions\nwhere Π( n, k) =Rπ/2\n0dθ/\u0002\n(1−nsin2θ)(1−ksin2θ)1/2\u0003\nis the complete elliptic integral of the third kind. We\nfurther obtain\nχ\na2=\u0000\n8 + 8 δ2+ ∆2\u0001\nEh\n4−4δ2\n4+∆2i\n−\u0000\n4δ2+ ∆2\u0001\nKh\n4−4δ2\n4+∆2i\n48π√\n4 + ∆2(4δ2+ ∆2),\n(79)\nwhere K(k) =Rπ/2\n0dθ/p\n1−ksin2θis the complete el-\nliptic integral of the first kind. On the other hand, when\nthe energy gap becomes sufficiently small relative to the\nFIG. 4. Electric susceptibility. (a) Comparison between\nthe numerical results (black lines) and the analytical result\nbeyond band projection (thin purple lines) up to second order\nin the field, which includes the polarization correction arising\nfrom interband transitions for the RM chain with the same\nparameters as in Fig. 3(b). We only show the numerical result\nin the FZ which is demarcated by dashed lines. (b) Electric\nsusceptibility χfor the RM chain.\nelectric field, higher-order interband Zener transitions in-\nduce gaps in the WS ladder. This can be seen in Fig. 3(c)\nforEg= 1. Now there is significant band mixing and a\nlarge Zener gap opens up as the field strength increases.\nWe proceed to consider the band-resolved currents for\nthe RM chain. To this end, we need the instantaneous\nband projectors. In general, for a two-band model as\ngiven by Eq. (68), these are given by\nPks(t) =1\n2[σ0+sn(k, t)·σ]. (80)\nResults for the currents for the same parameters as shown\nin Fig. 3(b) and (c) for the WS ladders, are shown in\nFig. 5(a) and (b), respectively. We note that the effect of\nthe field-induced polarization shift cannot be seen in the\ntransport calculation since in all cases considered there\nis a cancellation owing to time-reversal or spatial inver-\nsion symmetry. For small fields eEa/Γ≲1, the system\nevolves adiabatically, leaving the current predominantly\nintraband. In this regime, we find perfect matching to the\nsemiclassical result. For large fields, on the other hand,\ninterband transitions become important, and the current\ndeviates from the semiclassical result. We find that the\npeaks in the current correspond to the avoided crossings\nin the WS ladder (see Fig. 3) where the Zener transition\nrate is maximal. As expected, these Zener resonances in\nthe current become larger in magnitude when the energy\ngapEgis small relative to the electric field. As a function\nof the filling νof the bottom band, we further find that\nwhile the current for small fields is maximal near half\nfilling, the current for large fields peaks for a fully filled\nbottom band. This is shown in Fig. 5(c) and (d). We\nattribute this to the fact that since the Zener transitions10\n-0.10-0.050.000.050.10\nFIG. 5. Current for the RM chain. (a-b) Band-resolved currents jss′for the RM chain for two different choices of δand\n∆. The corresponding energy bands are shown in the inset. Here we choose the chemical potential µso that the lower band\nis half-filled. The semiclassical result is plotted as the black dashed line. (c-d) The total current for different fillings νof the\nlower band. In all plots, we set β= 50 and Γ = 0 .1 where energy is measured in units of J.\nare driven by the interband Berry connection, which is\nlargest in magnitude near the top of the lower band, the\ninterband current is greatest when the states near the\ntop of the band are filled. Otherwise, the field has to\nfirst drive a significant population to the top of the band\non a time scale smaller than the relaxation rate. A rough\nestimate of the validity of the band-projected theory in\nthe strong-field regime can be obtained by demanding\nthat the lowest-order interband correction is small:\nE2\ng\nW∼a2Eg\n|Ainter|2≫Ω≫Γ, (81)\nwhere Wis the bandwidth. A similar bound can be found\nfrom semiclassical considerations [11, 12, 56]. Note that\nwe have four relevant energy scales in our problem: the\nenergy gap Eg, the bandwidth W, the electrostatic en-\nergy eEa, and the system-bath coupling Γ.\nB. Example in D= 2: honeycomb lattice\nIn two spatial dimensions, a general commensurate\nelectric field is parameterized by a pair of coprime in-tegers ( m1, m2),\nE=E\ng0m1g1+m2g2p\nm2\n1+m2\n2−m1m2, (82)\nwhere g1,2are primitive reciprocal lattice vectors with\nlength g0=|g1,2|andgi·aj= 2πδij. As a concrete\nexample, we consider sorpzelectrons hopping on a\nhoneycomb lattice with nearest-neighbor hopping ampli-\ntude J= 1, sublattice bias ∆ 0, and Haldane mass ∆ 1\n[57], as illustrated in Fig. 6. We choose primitive lat-\ntice vectors a1,2=a\u0000\n±1/2,√\n3/2\u0001\nwith athe lattice\nconstant and reciprocal vectors g1,2=g0\u0000\n±√\n3/2,1/2\u0001\nwith g0= 4π/√\n3a. The Bloch Hamiltonian in periodic\ngauge is then given by\nH(k) =\"\n∆0+ ∆ 1γ(k) f(k)\nf(k)∗−∆0−∆1γ(k)#\n=d·σ,(83)\nwith d1−id2=f,d3= ∆ 0+ ∆ 1γ, and\nf(k) =eik·(rB−rA)\u0000\n1 +eik·a1+eik·a2\u0001\n, (84)\nγ(k) =2\n3√\n3(sink·a1−sink·a2−sink·a3),(85)11\nwitha3=a1−a2. For concreteness, we take rA/B=\u0000\n0,±a/2√\n3\u0001\nfor the sublattice positions, which fixes the\nperiodic gauge. One possibility is given by\n|uk±⟩=e−ik·rA\np\n2(1±n3) \nn3±1\nn1+in2!\n, (86)\nfor states with energies ±d, respectively, where d=|d|\nandn=d/d. This choice gives a traceless intraband\nBerry connection. Note that this gauge is smooth every-\nwhere except at the zone corners when 1 ±n3= 0 which\nis the case for sgn(∆ 0+τ∆1) =∓1 where τ=±1 at the\nKandK′points, respectively [see Fig. 6(b)]. We can try\nto remove this singularity with the gauge transformation\n|˜uk±⟩=±n1−in2p\nn2\n1+n2\n2|uk±⟩ (87)\n=e−ik·rA\np\n2(1∓n3) \nn1−in2\n±1−n3!\n, (88)\nwhich is now smooth except at the zone corner where\nsgn(∆ 0+τ∆1) =±1. Therefore when |∆1/∆0|<1, one\ncan remove all singularities and obtain a smooth gauge by\nappropriately choosing either |uk±⟩or|˜uk±⟩. However,\nfor|∆1/∆0|>1, the gap is inverted between KandK′\nand the gauge transformation only moves the singular-\nity from one zone corner to the other. Moreover, taking\na superposition of |uk±⟩and|˜uk±⟩allows one to place\nthe singularity at an arbitrary point in the BZ. This ob-\nstruction to a global smooth gauge is of course due to the\nfinite Chern number for |∆1/∆0|>1, which by Stokes’\ntheorem is given by the winding of the gauge transfor-\nmation in Eq. (87) around the singularity [58]. Note that\nin the trivial phase if we choose to take the gauge with\na singularity at both KandK′, the net winding number\nvanishes. Care is taken in band basis to avoid spurious\nresults due to these singularities.\nNext, we discuss the symmetries of the honeycomb\nmodel in the presence or absence of the sublattice mass\n∆0and the Haldane mass ∆ 1. For ∆ 0= ∆ 1= 0, the\nmodel has time-reversal symmetry Tand point group\nD6h=C6v×σhwith C6v=⟨C6z,Mx⟩where C6zis\na sixfold rotation about the zaxis,Mx:x7→ − xis\nthe in-plane mirror with respect to the xaxis, and σh\nis the mirror with respect to z. Since we already fixed\nthe symmetry of the orbitals with respect to σh, we can\nrestrict ourselves to C6v. A finite sublattice potential ∆ 0\nbreaks C2zsymmetry, i.e., inversion symmetry when re-\nstricted to the plane, which reduces the point group to\nC3v=⟨C3z,Mx⟩. On the other hand, a finite Haldane\nmass ∆ 1breaks Tsymmetry and all mirrors, allowing\nfor a finite linear Hall response, but ∆ 1conserves C2z\nsymmetry as well as combinations of mirrors and time\nreversal such as MxT. This gives rise to the magnetic\npoint group 6 mmwhich is also denoted as C6v(C6) [59].\nIn the presence of both mass terms, the magnetic point\ngroup is given by 3 morC3v(C3).\nFIG. 6. Honeycomb lattice. (a) Illustration of the hon-\neycomb lattice. The dotted diamond is the unit cell and\nthe nearest-neighbor (solid), next-nearest-neighbor (dashed)\nhopping amplitudes, and on-site energies are indicated. Here\nt′= 2∆ 1/(3√\n3) and we only show the nnn hopping for sub-\nlattice A; it has an opposite sign for sublattice B. (b) Some\ncommensurate field directions g=m1g1+m2g2considered\nin this work are shown with juxtaposed coordinates m1m2\nwhere the overline indicates a minus sign. Here the small or-\nange hexagons are the BZs in the extended zone scheme and\nthe large black hexagons correspond to different stars.\nWe now consider the WS ladder of the honeycomb\nmodel. To this end, we introduce the dimensionless mo-\nmentum k⊥normal to the field direction gsuch that a\ngeneral momentum can be written as\nk=k∥g+k⊥ˆz×VBZg/g2, (89)\nwhere we choose a rectangular Brillouin zone with\nk∥, k⊥∈[−1/2,1/2) and VBZ= 8π2/√\n3a2. In Fig. 7(a)\nwe show the WS ladder for ( m1, m2) = (1 ,0), i.e., the\narmchair direction, as a function of the field strength E\nandk⊥in the trivial phase. Here we only show positive\nquasienergies in the FZ as the spectrum is symmetric in\nthe traceless gauge. We see that the crossings in the lad-\nder disperse as a function of k⊥which is mainly due to the\nchange in the center of the hybrid Wannier state. This\nis evident from the evolution of the Berry phases (equiv-\nalent to the Wilson loop spectrum because the bands do\nnot cross) shown in Fig. 7(b). Note that there is no net\npumping of the Wannier center in the trivial phase as\nexpected. We have also superimposed on the WS ladder\nthe crossings obtained in band basis up to lowest order\nin interband corrections by taking into account the sus-\nceptibility. The latter is shown in Fig. 7(c) and peaks\nwhen the integration path cuts across the zone corners\nwhere gij/2dis maximal. We also show the WS ladder\nfor a field along the zigzag direction ( m1, m2) = (1 ,−1)\nin Fig. 7(d). In this case, the Berry phase vanishes be-\ncause the field lies perpendicular to a mirror line [the y\naxis in Fig. 6(a)] and therefore the projected Wannier\ncenter is pinned at the origin. This is why the ladder\nis much less dispersive. Furthermore, the susceptibility\ncorrection is now largest at k⊥= 0 since this is where\nboth zone corners are projected on top of each other.\nThe WS ladder is qualitatively different when the12\n0.3\n0.2\n0.1\n0\n0.2\n0.1\n0\n0.15\n0.05\ntrivial\ntrivial\nFIG. 7. WS ladder of the honeycomb lattice in the\ntrivial phase for (∆ 0,∆1) = (0 .5,0). (a) WS ladder in the\nFZ for ( m1, m2) = (1 ,0). The solid (dashed) curves give the\ncrossings in the ladder at zero (edge of the FZ) as calculated in\nband basis up to 2nd order. We only show positive quasiener-\ngies since the spectrum is symmetric in traceless gauge. (b)\nProjected Berry phase (Wannier center) for the two bands\nand (c) susceptibility χk⊥s=sχk⊥as a function of k⊥for the\nfield in (a). (d) WS ladder, (e) Berry phase, and (f) suscep-\ntibility for ( m1, m2) = (1 ,−1). In this case, the Berry phase\nvanishes because the field lies perpendicular to a mirror line.\nChern number Cis finite, since the Berry phase wind-\ning as k⊥advances one unit is by definition given by C\n[32, 60]. In our case, we have C=±1 such that dur-\ning one cycle the Wannier center moves one cell over:\nn7→n±1. This can be observed in the WS ladder which\nis shown for ( m1, m2) = (1 ,0) in Fig. 8(a) and can be\nexplicitly seen from the Berry phase winding shown in\nFig. 8(b). Following the crossings in the WS ladder from\nk⊥=−1/2 to k⊥= 1/2 we end up at a different field\nvalue as the one we started from due to the shift of the\nWannier center. Changing the sign of ∆ 1reverses the\npump and the corresponding shift in the WS ladder. A\nsimilar effect occurs in Weyl semimetals where the Weyl\ncharge acts as a topological defect in the WS ladder [44].\nWe next consider the current response. Before we\nproceed to the results, we first address how the cur-\nrent is constrained by the crystalline symmetries. Un-\nder a spatial symmetry S, the current transforms as\nj(E) =Sj(S−1E). This motivates us to define the lon-\ngitudinal and transverse current components as\nj∥=ˆE·j, j ⊥=ˆz׈E·j, (90)\nwhich, respectively, transform as a scalar and pseu-\n0.3\n0.2\n0.1\n0\n0.2\n0.1\n00.15\n0.05\nChern\nChernFIG. 8. WS ladder of the honeycomb lattice in the\nChern phase for (∆ 0,∆1) = (0 ,0.5). (a) WS ladder in the\nFZ for ( m1, m2) = (1 ,0). The solid (dashed) curves give the\ncrossings in the ladder at zero (edge of the FZ) as calculated in\nband basis up to 2nd order. We only show positive quasiener-\ngies since the spectrum is symmetric in traceless gauge. (b)\nProjected Berry phase (Wannier center) for the two bands\nand (c) susceptibility χk⊥s=sχk⊥as a function of k⊥for the\nfield in (a). (d) WS ladder, (e) Berry phase, and (f) suscep-\ntibility for ( m1, m2) = (1 ,−1).\ndoscalar under S. Next, in order to make sense of the\nresults obtained with the full quantum theory, we con-\nstruct linear combinations of the currents that transform\nin the same way as their band-projected counterparts in\nthe adiabatic theory (see Appendix B). Note that these\ncurrent components generally contain interband contri-\nbutions, but in the limit E2\ng/W≫Ω they become purely\nintraband. Other parts of the current that do not trans-\nform as intraband currents are then attributed solely to\ninterband corrections. On the computational side, we\nconsider commensurate fields E=Eg/g=E(cosθ,sinθ)\nup to |g|=√\n73g0in the range θ∈[−π/3, π/3]. This is\nsufficient to reconstruct the entire angular dependence of\nthe current from the lattice symmetries.\nIn particular, we consider magnetic symmetries ST\nand define the even and odd components with respect\nto this symmetry as follows:\nj±(E)≡j(E)± Sj(−S−1E)\n2, (91)\nsuch that j+transforms as the intraband geometric cur-\nrent due to the Berry curvature, while j−transforms as\nthe Bloch current originating from the dispersion which\nis a Drude-type contribution (see Appendix B). Hence,13\nthe longitudinal part of j+is purely interband. For ex-\nample, when C2zis broken but Tis conserved (∆ 1= 0)\nwe choose S= 1. We illustrate this case with the current\nroses [12] in Fig. 9 for ∆ 0= 0.5 and ∆ 1= 0. The nodes\nin the transverse response are due to the presence of the\nmirror symmetry Mxwhich forbids a transverse response\nwhen the field lies along a mirror line, given here by the\nyaxis and its C3zpartners.\nWe also observe that the odd current components,\nwhich are mainly due to intraband Drude-like contribu-\ntions, display a maximum [5]. Indeed, at low fields j−\n∥\nis isotropic and increases linearly (Ohm’s law). With\nincreasing field strength, the current becomes more\nanisotropic as the entire energy band is probed by\nthe accelerated electrons, and attains an extremum for\neEa/Γ∼1. In this regime, complete Bloch orbits oc-\ncur before a scattering event and the current decays as\n1/Euntil interband transitions become significant. The\nlatter give rise to many oscillations from Zener reso-\nnances whose details are captured by the WS ladder.\nHowever, as long as interband contributions are small\nnear eEa/Γ∼1, the longitudinal differential conduc-\ntance dj∥/dEbecomes negative over an extended range\n[3]. This is illustrated in Fig. 10, where we show both\nthe current and the differential conductance for the field\ndirection θ= 0◦[(m1, m2) = (1 ,−1)] for several fillings\nof the lower band. Generally, we observe that interband\ncontributions become larger as the filling increases. This\nmakes sense because electrons in the lower band that\nhave a higher Fermi energy are more likely to reach the\nband edge, where interband coupling is stronger, before\nintraband relaxation can occur. Note also that the cur-\nrent calculated with the quantum theory is not fully con-\nverged for very small fields. This is most clearly seen\nin Fig. 10(b) by comparison to the semiclassical result\nwhich becomes exact in the limit E → 0.\nMoreover, the transverse differential conductance,\nshown in Fig. 10(d), which is purely geometric for θ= 0◦\nwhen interband coupling is negligible, is cubic at low\nfields due to a Berry curvature hexapole in the pres-\nence of C3zbut broken C2z[12, 61] and shows a peak for\neEa/Γ∼1 indicative of the incipient plateau in the ge-\nometric current. However, unlike in the band-projected\ntheory, the plateau itself is never reached for the chosen\nparameters because of interband coupling. Hence, the ex-\ntremum in the transverse differential conductance at the\nonset of full Bloch orbits provides a more robust signa-\nture of the nominal plateau due to geometric oscillations\nin the band-projected theory.\nOn the other hand, when C2zis conserved, only even\ncurrent components with respect to S= 1 are nonvan-\nishing such that the above distinction becomes superflu-\nous. Instead, we decompose the current with S=Mx\nsinceMxTis always conserved regardless of the sub-\nlattice or Haldane masses. This is shown in Fig. 11 for\nthe case with 6 mmsymmetry in the Chern phase where\nwe take ∆ 0= 0 and ∆ 1= 0.5. While the pure inter-\nband component j+\n∥and the odd components with re-\nFIG. 9. Current roses for the honeycomb lattice in the\ntrivial phase with C3vandTsymmetry at half filling\nof the lower band. Parameters are β= 50 and Γ = 0 .1 with\nmasses (∆ 0,∆1) = (0 .5,0) such that Tis preserved. Even and\nodd components for S= 1 are shown. Starting from j+\n∥and\ngoing clockwise, radial ticks are spaced by 0 .002, 0 .004, 0 .025,\nand 0 .01 in units of eJ/ℏa. The odd components are nonzero\nbecause ∆ 0breaks C2zsymmetry. The maximum number of\nFloquet sidebands is NF= 1122. Note that the current is not\nfully converged for the smallest field shown.\nspect to MxTare qualitatively the same from the com-\nponents defined with respect to Tfor the case with C3v\nandTsymmetry (see Fig. 9), the transverse even com-\nponent j+\n⊥looks very different. This is because of the\nfinite Chern number which gives rise to a dominant lin-\near contribution to the current that is isotropic and given\nby\u0000\ne2E/Vcℏ\u0001P\nrf0\nrΩ−r=\u0000\ne2VcE/2πh\u0001R\nBZd2kf0\nkΩk\nwhich is the Hall conductance. We note that in Fig.\n11, the seemingly nonlinear bunching at small fields for\nj+\n⊥is due to our choice for the Egrid which has twice the\nnumber of points for |eEa/Γ| ≤1.2.\nFinally, we consider the case with 3 msymmetry for\nwhich both C2zandTare broken. Here we consider the\ntrivial phase with ∆ 0= 0.75 and ∆ 1= 0.25. Similar\nto before, since MxTremains conserved, we decompose\nthe current in terms of even and odd components with re-\nspect to this symmetry. Since both TandC6zare broken,\nthe current roses shown in Fig. 12 only retain C3zsym-\nmetry. Moreover, the even transverse component that is\ndue to the Berry curvature in the intraband limit now\ncontains both contributions from broken C2z(∆0) and\nbroken T(∆1) which have a different angular depen-\ndence such that they either add or substract giving rise\nto the j+\n⊥rose shown in Fig. 12. We further note that\nbecause C2zandTare both broken in this case, inter-\nband contributions are generally more important. This\nis because the lowest-order interband contribution to the14\n0 0.2 0.4 0.6 0.8 1band filling\nquantum theory band-projected semiclassical theory\nFIG. 10. Currents versus filling for the honeycomb\nlattice in the trivial phase with C3vandT.The left\nand right column show results for the quantum and band-\nprojected semiclassical theory, respectively. Parameters are\nβ= 50 and Γ = 0 .1, for E=Eˆx(θ= 0◦) with masses\n(∆0,∆1) = (0 .5,0) such that Tis preserved. (a) Current\ncomponent j−\n∥forS= 1 and (b) the corresponding differential\nconductance. Other components vanish for θ= 0◦. (c, d)\nSame for the transverse even part j+\n⊥. The color gives the\nfilling of the lower band ( ν= 0.1,0.2, . . . , 0.9) and NF= 562.\ncurrent, i.e., due to the electric susceptibility, vanishes in\nthe presence of either of these symmetries [27]. It would\nbe interesting to study the current nonperturbatively for\na system with broken C2z(orP=MzC2z) and Tsym-\nmetry but still conserves their combination C2zTsuch\nthat the Berry curvature vanishes. In this case, the even\ncurrent component j−(not just the longitudinal part) is\nentirely due to interband coupling. Moreover, at order\nE2, this response yields a measure for the quantum met-\nric of the occupied band [27, 45, 46]. However, this would\nrequire a lattice model that is either strongly anisotropic\nor has more than two bands and therefore we leave this\nfor future work.\nV. CONCLUSIONS\nIn this work, we considered the full quantum theory of\nthe electric current response to a static uniform electric\nFIG. 11. Current roses for the honeycomb lattice in\nthe Chern phase with 6mmsymmetry at half filling\nof the lower band. Parameters are β= 50, Γ = 0 .1 with\n(∆0,∆1) = (0 ,0.5) such that C2zis preserved. Even and\nodd components for S=Mxare shown. Starting from j+\n∥\nand going clockwise, radial ticks are spaced by 0 .0004, 0 .014,\n0.025, and 0 .01 in units of eJ/ℏa. The maximum number of\nFloquet sidebands is NF= 1122. Note the current is not fully\nconverged for the smallest field shown.\nFIG. 12. Current roses for the honeycomb lattice in\nthe trivial phase with 3msymmetry at half filling of\nthe lower band. Parameters are β= 50, Γ = 0 .1 with\n(∆0,∆1) = (0 .75,0.25) such that Tis broken. Even and\nodd components for S=Mxare shown. Starting from j+\n∥\nand going clockwise, radial ticks are spaced by 0 .0011, 0 .003,\n0.025, and 0 .014 in units of eJ/ℏa. The maximum number of\nFloquet sidebands is NF= 1002. Note that the current is not\nfully converged for the smallest field shown.15\nfield of noninteracting fermions on a lattice in the regime\nof Bloch oscillations ( ωBτ≫1). As such, our theory is\nnonperturbative in the interband coupling. We achieve\nthis by first mapping the physical time-independent prob-\nlem in Dspatial dimensions to a time-dependent prob-\nlem in D−1 spatial dimensions by treating the field in\ntemporal gauge. In this gauge, the longitudinal momen-\ntum becomes a gauge degree of freedom which can be\nabsorbed in the time origin. Moreover, when the elec-\ntric field is commensurate to the lattice, i.e., when it lies\nalong a reciprocal lattice vector (equivalently perpendic-\nular to a lattice plane), Bloch oscillations in the semi-\nclassical theory become periodic and likewise the Bloch\nHamiltonian in the quantum theory becomes time pe-\nriodic up to a unitary transformation. The latter is a\nconsequence of working in periodic gauge for the total\nBloch wave function and properly takes into account the\nsublattice positions. By using a modified Floquet ansatz\none obtains a problem in D−1 spatial dimensions with an\nadditional synthetic Floquet dimension which can be in-\nterpreted as the spatial direction that lies longitudinal to\nthe commensurate electric field. The corresponding Flo-\nquet quasienergies then yield the familiar Wannier-Stark\nladder beyond the single-band approximation.\nTo obtain the current, we coupled the system to a\nreservoir using the Floquet-Keldysh Green’s function for-\nmalism. For simplicity, we considered a reservoir that\nconserves the periodicity of the lattice in the wide-band\nlimit. In this case, the self-energy due to the bath is di-\nagonal in momentum space and Floquet space yielding a\nrelatively simple expression for the current. Moreover, in\norder to study the onset of interband contributions to the\ncurrent, we also derived expressions for the band-resolved\ncurrents using the Floquet representation of band projec-\ntors. We then used this formalism to study the current\nresponse in D= 1 and D= 2 spatial dimensions.\nIn 1D for a single-band model, we find nearly perfect\nagreement with the band-projected semiclassical theory\napart from small oscillations related to the nature of the\nbath and which vanish in the limit Γ /W→0, where\nWis the bandwidth and Γ is the system-bath coupling.\nFor a two-band model in 1D, deviations from the semi-\nclassical theory arise due to interband currents (Zener\ntransitions) which become significant for eEa∼E2\ng/W.\nMoreover, as we increase the chemical potential and the\nlower band becomes more filled, one finds that interband\ncontributions become more important since Zener tran-\nsitions are more likely to occur before intraband relax-\nation. Hence, in the regime of full Bloch orbits, the\nvalidity of the band-projected theory is estimated as:\na2Eg/|Ainter|2≫Ω≫Γ where Ainteris the interband\nBerry connection.\nIn 2D, we considered a honeycomb lattice with both\nsublattice ( C2zbreaking) and Haldane mass ( Tbreak-\ning) terms that exhibits both trivial gapped and Chern\nphases. In the latter case, the finite Chern number can\nbe expressed as the winding number of the Berry phase\nalong the momentum direction transverse to the fielddirection, giving rise to a net pumping of the Wannier\ncenter across the real-space cell. This can be observed\ndirectly in the Wannier-Stark ladder which encodes the\nBerry phase winding through the polarization as one tra-\nverses the zone. We proceeded to calculate the current\nresponse for a honeycomb model. In order to compare\nour results to the semiclassical theory, we decomposed\nthe current into parts that transform in the same way as\ntheir semiclassical counterparts under a magnetic sym-\nmetry ST. Most interestingly, when Tis conserved but\nC2zis broken, we find that unless the band gap opened\nby the sublattice mass is very large, the transverse geo-\nmetric response does not plateau as a function of the field\nstrength as was predicted in the band-projected theory.\nInstead, interband contributions become significant be-\nfore the plateau is reached. However, we find that the\ncorresponding peak in the differential conductance due\nto the incipient plateau provides a more robust probe of\ngeometric oscillations.\nACKNOWLEDGMENTS\nWe acknowledge interesting and fruitful discussions\nwith V. T. Phong, M. Claassen, and C. L. Kane. C. D. B.\nand E. J. M. are supported by the Department of En-\nergy under grant DE-FG02-84ER45118. S. G. and S. T.\nacknowledge funding from the NSF GRFP under grant\nDGE-1845298.\nAppendix A: Length gauge\nHere we give a short overview of the transformation\nbetween velocity gauge and length gauge. Like velocity\ngauge, length gauge is an incomplete gauge that becomes\nfixed in the electric-dipole approximation. In length\ngauge, one describes a uniform electric field E(t) with an\nelectrostatic potential φ(r, t) =−E(t)·r. The upside of\nlength gauge is that the Hamiltonian itself remains time\nindependent, but the downside is that it breaks transla-\ntional symmetry. We can transform the Hamiltonian in\nvelocity gauge given by Eq. (6) to length gauge with the\nunitary transformation U(t) =eiS(t)where\nS(t) =eA(t)·X\nr,a(r+ra)c†\nracra. (A1)\nThe Hamiltonian transforms as\nH→˘H=UHU†+i˙UU†, (A2)\nwhere the second term on the right-hand side is straight-\nforward to evaluate. To deal with the first term, consider\neiS(t)c†\nracr′be−iS(t), (A3)\nwhich can be obtained from a special case of the Baker-\nCampbell-Hausdorff formula given by\neiXY e−iX=Y+i[X, Y] +i2\n2[X,[X, Y]] +···,(A4)16\nwith X=S(t) and Y=c†\nracr′b. From\n[c†\nr′′ccr′′c, c†\nracr′b] =c†\nracr′b(δrr′′δac−δr′r′′δab),(A5)\nwe find [ X, Y] =Y eA(t)·(r−r′+rab). We obtain\neiS(t)c†\nracr′be−iS(t)=c†\nracr′beieA(t)·(r−r′+rab),(A6)\nand the transformed Hamiltonian becomes\n˘H=H0+eE(t)·X\nr,a(r+ra)c†\nracra, (A7)\nwhere the last term gives the potential energy of charge\n−efermions on the lattice in a uniform field E(t).\nAppendix B: Semiclassical band-projected theory of\nBloch and geometric oscillations\nHere we give a short review of the semiclassical band-\nprojected transport theory of Bloch and geometric os-\ncillations [5, 11, 12, 24]. In this section, we restore ℏ\nfor consistency with existing literature. For a crystal in\na uniform electric field, the Boltzmann transport equa-\ntion for the occupation function in the relaxation-time\napproximation has the steady-state solution [56]:\nf(t) =f0(t)\n+e\nℏZt\n−∞dt′exp\u0012\n−Zt\nt′ds\nτ(s)\u0013\n∇kf0(t′)·E(t′),(B1)\nwith τ(t) =τ[k(t)] the momentum-relaxation time. For\na static field and a constant relaxation time, this can be\nsolved exactly [12, 62],\nf(t) =fk(t)=X\nrf0\nreik(t)·r\n1−ieτr·E/ℏ, (B2)\nwhere the sum runs over lattice vectors randf0\nr=\n[Vc/(2π)d]R\nBZddkf0\nke−ik·rare lattice Fourier compo-\nnents of the Fermi-Dirac distribution. The current in\nthe band-projected theory is given by jBloch +jgeomwith\njBloch =−e\nℏZ\nkfk∇kεk, (B3)\njgeom=−E×e2\nℏZ\nkfkΩk, (B4)\nwhereR\nk≡R\nBZdDk/(2π)D,εkis the energy band, and\nΩk=∇k×Akis the Berry curvature in periodic gauge\nwritten as a pseudovector. Plugging in the occupation\nfunction from Eq. (B2) yields\njBloch =ie\nVcℏX\nrrf0\nrε−r\n1−ieτr·E/ℏ, (B5)\njgeom=−E×e2\nVcℏX\nrf0\nrΩ−r\n1−ieτr·E/ℏ, (B6)with Vcthe unit cell volume, and where εrandΩrare\nlattice Fourier components of the energy band and Berry\ncurvature, respectively. In particular, note that f0\n0gives\nthe filling fraction of the band and for D= 2 we have\nΩ0=VcC/2πwithCthe Chern number.\nIt is instructive to rewrite the Bloch current as\njBloch =ie\nVcℏ\nX\nr·E=0rf0\nrε−r+X\nr·E̸=0rf0\nrε−r\n1−ieτr·E/ℏ\n,\n(B7)\nwhere the first term vanishes in the presence of time-\nreversal or spatial inversion symmetry. Each of these\nsymmetries individually imply that εr=ε−randf0\nr=\nf0\n−rare real, such that jBloch(−E) =−jBloch(E). Re-\nstricting to D= 2, we define jgeom=jgeomˆz׈Ewith\njgeom=e2E\nVcℏ\nX\nr·E=0f0\nrΩ−r+X\nr·E̸=0f0\nrΩ−r\n1−ieτr·E/ℏ\n,\n(B8)\nwhere now the first term only vanishes in the presence of\ntime-reversal T. Indeed, time-reversal symmetry implies\nthat Ω r=−Ω−ris imaginary, and jgeom is even in Ein\nthis case. On the other hand, inversion symmetry gives\nreal Ω r= Ω−randjgeom is odd in Ein this case.\nThe transformation properties of the Bloch and geo-\nmetric current under a general crystalline symmetry S\nfollow similarly from εr=εSr,f0\nr=f0\nSr, and Ω r=\ndet(S)ΩSr, while a magnetic symmetry STimplies that\nεr=ε−Sr,f0\nr=f0\n−Sr, and Ω r=−det(S)Ω−Sr. With\nthese relations, one can demonstrate readily that the cur-\nrents transform as\nS:j(E) =Sj(S−1E), (B9)\nST:jBloch(E) =−SjBloch(−S−1E),\njgeom(E) =Sjgeom(−S−1E).(B10)\nMoreover, defining the longitudinal and transverse com-\nponents,\nj∥=ˆE·j, j ⊥=ˆz׈E·j, (B11)\nwe find that\nS:j∥(E) =j∥(S−1E),\nj⊥(E) = det( S)j⊥(S−1E),(B12)\nST:j∥\nBloch(E) =j∥\nBloch(−S−1E),\nj⊥\nBloch(E) = det( S)j⊥\nBloch(−S−1E),\njgeom(E) =−det(S)jgeom(−S−1E).(B13)\nAs an example, consider a Bloch band in D= 1 with\ntime-reversal or inversion symmetry. Now we only have17\na longitudinal response\njBloch =ie\nℏ∞X\nn=1nf0\nnεn\u00121\n1−inΩτ−1\n1 +inΩτ\u0013\n(B14)\n=−2eΩτ\nℏ∞X\nn=1n2f0\nnεn\n1 + (nΩτ)2, (B15)\nwith Ω = eEa/ ℏwhere ais the lattice constant. Specif-\nically, for a linear chain with nearest-neighbor hopping\namplitude J, we find\njBloch =−2eJf0\na\nℏΩτ\n1 + (Ω τ)2, (B16)\nwhich is shown in Fig. 2. Here the lattice Fourier trans-\nform of the Fermi-Dirac distribution is given by [substi-\ntuting u= cos( ka)]\nf0\na=a\n2πZπ/a\n−π/adk f0(εk) cos( ka) (B17)\n=1\nπZ1\n−1duuf0(2Ju)√\n1−u2(B18)\n≃ −sgn(J)\nπr\n1−\u0010µ\n2J\u00112\nθ\u0010\n1−\f\f\fµ\n2J\f\f\f\u0011\n,(B19)\nwhere the last line holds for β|J| ≫1 and whose magni-\ntude is largest at half filling ( µ= 0).\nAppendix C: Band basis\nAs an alternative to the orbital basis in velocity gauge,\nwe can instead work in the instantaneous band basis by\nwriting [11, 20]\n|Φk(t)⟩=qX\ns=1aks(t)|uks(t)⟩, (C1)\nwith|uks(t)⟩=|us[k+eA(t)]⟩such that\nH(k, t)|uks(t)⟩=Eks(t)|uks(t)⟩, (C2)with Eks(t) the instantaneous band energy and normal-\nization ⟨uks(t)|uks′(t)⟩=δss′. The dynamics is deter-\nmined by Eq. (14), which yields\ni∂tak(t) =H(k, t)ak(t), (C3)\nwith ak= (ak1, . . . , a kq)tand\nHss′(k, t) =δss′Eks(t) +eE(t)·Ass′(k, t),(C4)\nis the Hamiltonian in the instantaneous band basis. Here\nwe used\n⟨uks(t)|i∂t|uks′(t)⟩= (e∂tA)· ⟨uks(t)|i∂k|uks′(t)⟩\n(C5)\n=−eE(t)·Ass′(k, t), (C6)\nwhere Ass′(k, t) =Ass′[k+eA(t)] is the instantaneous\nBerry connection. To find an approximate solution, we\nfirst rewrite Eq. (C3) by an instantaneous diagonaliza-\ntion, defined by ak(t) =U(k, t)˘ak(t) which yields\ni∂t˘ak(t) =h\nD(k, t)−iU†˙Ui\n˘ak(t), (C7)\nwhereD(k, t) =U†(k, t)H(k, t)U(k, t) is a diagonal\nmatrix and\n−iU†˙U=ieE(t)·U†∇kU−i˙E·U†∇EU,(C8)\nis an effective connection for the Hamiltonian in band\nbasis which is at least third order in Eand ˙E. Up to\nsecond order, we can thus approximate the right-hand\nside of Eq. (C7) by standard nondegenerate perturbation\ntheory. This yields instantaneous eigenvalues,\nλks(t) =Eks(t) +eE(t)·Aks(t)\n+X\ns′̸=s[eE(t)·Ass′(k, t)][eE(t)·As′s(k, t)]\nEks(t)−Eks′(t),\n(C9)\nwithAks(t) =Ass(k, t) the instantaneous intraband\nBerry connection. Hence we obtain\n˘aks(t)≈e−iRt\n0dt′λks(t′)˘aks(0), (C10)\nwhich is valid up to second order. Here the coefficients\n˘aks(t) are superpositions of the original aks(t) such that\nH(k, t) becomes diagonal. Moreover, for a commensu-\nrate static electric field E=Eg/g, the quasienergies are\ndefined by ˘ aks(t+T) = exp ( −iεksT) ˘aks(t) which yields\nεk⊥s,n=1\nTZT\n0dt Eks(t) + Ω\"\nn+gi\n2π1\nTZT\n0dt \nAi\nks(t) + Ωgjχij\nks(t)\n2π!#\n(C11)\n=Ek⊥s+ Ω\"\nn+gi\n2π \nAi\nk⊥s+ Ωgjχij\nk⊥s\n2π!#\n, (C12)18\nwhere summation over repeated indices is implied, Ω =\n2πeE/g , and\nχij\nks(t) =X\ns′̸=sAi\nss′(k, t)Aj\ns′s(k, t)\nEks(t)−Eks′(t), (C13)\nwhere χij\nk⊥sis the state-resolved (static) electric suscep-\ntibility [28]. We see that the lowest-order interband cor-\nrection is quadratic in the electric field and corresponds\nto the electric susceptibility which gives a field-induced\nshift of the polarization [27, 39]. Energy gaps in the WS\nladder due to interband Zener transitions thus only arise\nat higher orders in the field, which is corroborated by our\nnumerical results in the main text.\nAs a minimal example, we consider a two-band model\nwith Bloch Hamiltonian H(k) =d0(k)σ0+d(k)·σand\nbands Eks=d0(k) +sd(k) (s=±1) where d=|d|. In\nthis case, Eq. (C13) simplifies to\nχij\nks=gij\nk\n2sd(k), (C14)\nwith gij\nk= (∂kin)·\u0000\n∂kjn\u0001\n/4 the quantum metric of either\nband and where n=d/d.\nAppendix D: Keldysh Formalism\nHere we review the essential parts of the Keldysh for-\nmalism for the treatment of responses. In particular, we\npresent a derivation for the expression G<=GRΣ(t1, t2)−Σ<(t1, t2)\u0003\n,(D6)\nΣA(t1, t2) =−Θ(t2−t1)\u0002\nΣ>(t1, t2)−Σ<(t1, t2)\u0003\n,\n(D7)\nΣK(t1, t2) = Σ>(t1, t2) + Σ<(t1, t2), (D8)\nwith Θ( t) the Heaviside step function. Hence we have\nΣR−ΣA= Σ>−Σ<, (D9)\nΣR−ΣA−ΣK=−2Σ<, (D10)\nand thus\nΣ<=ΣA−ΣR+ ΣK\n2, (D11)\nwhich differs from Ref. [48] by opposite definitions of ΣR\nand ΣA. Furthermore, for an ideal bath, we have\nHB=X\nk,jξjd†\nkjdkj, (D12)\nHSB=λX\nkX\na,j(c†\nkadkj+d†\nkjcka), (D13)\nwhere jare bath degrees of freedom, ξj=εj−µwith\nµthe chemical potential, and λgives the coupling be-\ntween the system and the bath. Here c†\nkj(ckj) creates\n(destroys) a system particle and d†\nkj(dkj) creates (de-\nstroys) a bath particle which can be fermionic or bosonic\nwith [ dkj, d†\nk′j′]±=δkk′δjj′where + /−is the commuta-\ntor/anticommutator. The total Hamiltonian is\nH(t) =HS(t) +HB+HSB, (D14)\nwhere HS(t) is given by Eq. (6). The retarded and ad-\nvances self energies are given by [51]\nh\nΣR/A(t, t′)i\nab=X\nj,j′(HSB)ajGR/A\nB,jj′(t, t′)(H†\nSB)j′b,\n(D15)19\nwhere\nGR\nB(kj;t−t′) =−iΘ(t−t′)⟨[dkj(t), d†\nkj(t′)]±⟩(D16)\n=−iΘ(t−t′)e−iξj(t−t′), (D17)\nGA\nB(kj;t−t′) =iΘ(t′−t)⟨[dkj(t), d†\nkj(t′)]±⟩(D18)\n=iΘ(t′−t)e−iξj(t−t′), (D19)\nsuch that\n\u0002\nΣR(k, t−t′)\u0003\nab(D20)\n=−iδabλ2Θ(t−t′)X\nje−iξj(t−t′)(D21)\n=−iδabλ2Θ(t−t′)Z∞\n−∞dω\n2πρ(ω+µ)e−iω(t−t′)(D22)\n≈ −iΓ\n2δabδ(t−t′), (D23)\nwhere we used Θ(0) = 1 /2 and we assumed a con-\nstant density of states ρ(ω)≈ρ0(the wide-band limit\nof the bath), with Γ = λ2ρ0. Similarly, one finds\u0002\nΣA(k, t−t′)\u0003\nab≈(iΓ/2)δabδ(t−t′). In frequency space,\nthis yields Eq. (54) and Eq. (56) of the main text.\nAppendix E: Analytical results\n1. Frequency integral\nTheωintegral in Eq. (58) can be solved by closing the\ncounter in the upper complex plane. Indeed, for Re z >0\nthe integrand decays as e−βRezfor both the fermionic\nand bosonic bath, such that the contribution of the upper\ngreat half circle vanishes.a. Fermionic bath\nFor the fermionic bath, the frequency integral gives\nZ∞\n−∞dω\n2πif0\n−(ω)\n(ω−a−iΓ/2) (ω−b+iΓ/2)(E1)\n=f0\n−(a+iΓ/2)\na−b+iΓ(E2)\n−1\nβ∞X\nj=01\n(zj−a−iΓ/2) (zj−b+iΓ/2), (E3)\nwith zj=µ+iπ(2j+ 1)/βand where aandbin Eq.\n(58) correspond to Floquet eigenenergies when we plug in\nthe single-particle Lehmann representation of the Green’s\nfunctions. The sum can be evaluated, e.g., with Mathe-\nmatica, and we find\n1\na−b+iΓ\"\nf0\n−(a+iΓ/2) +ψ\u00001\n2+A\u0001\n−ψ\u00001\n2+B\u0001\n2πi#\n,\n(E4)\nwhere ψ(z) is the digamma function, A =\niβ(a−µ+iΓ/2)/2π, and B=iβ(b−µ−iΓ/2)/2π.\nMaking use of\nf0\n−(ω) =1\n2\u001a\n1−tanh\u0014β\n2(ω−µ)\u0015\u001b\n(E5)\n=1\n2+ψ\u00001\n2−γ\u0001\n−ψ\u00001\n2+γ\u0001\n2πi, (E6)\nwith γ=iβ(ω−µ)/2π, the fermionic integral [49]\n1\na−b+iΓ\n\n1\n2+ψh\n1\n2−iβ\n2π\u0000\na−µ+iΓ\n2\u0001i\n−ψh\n1\n2+iβ\n2π\u0000\nb−µ−iΓ\n2\u0001i\n2πi\n\n(E7)\n=1\na−b+iΓ\u0014θ(µ−a) +θ(µ−b)\n2−ln\f\f\f\fa−µ\nb−µ\f\f\f\f+Γ\n4π\u00121\na−µ+1\nb−µ\u0013\n+O\u0000\nβ−2\u0001\u0015\n, (E8)\nwhere the last line is a low-temperature expansion with respect to |a−µ|and|b−µ|while keeping βΓ constant. Note\nthat the temperature dependence only enters in the subleading terms.\nb. Bosonic bath\nFor the bosonic bath, the frequency integral gives\nPZ∞\n−∞dω\n2πif0\n+(ω)\n(ω−a−iΓ/2) (ω−b+iΓ/2)(E9)\n=f0\n+(a+iΓ/2)\na−b+iΓ+1\n2βab(E10)\n+1\nβ∞X\nj=11\n(zj−a−iΓ/2) (zj−b+iΓ/2), (E11)with zj=µ+i2πj/β andPindicates the Cauchy prin-\ncipal value. Here we took into account that the pole of\nf0\n+at the origin only contributes half of its residue since\nit lies on the contour. The sum can again be evaluated20\nwith Mathematica and we find\n1\na−b+iΓ\u0014\nf0\n+(a+iΓ/2)−ψ(1 +A)−ψ(1 +B)\n2πi\u0015\n+1\n2βab.\n(E12)\nMaking use of\nf0\n+(ω) =1\n2\u001a\ncoth\u0014β\n2(ω−µ)\u0015\n−1\u001b\n(E13)\n=1\nz−1\n2+ψ(1 +γ)−ψ(1−γ)\n2πi, (E14)the bosonic integral becomes\n−1\na−b+iΓ\n\n1\n2+ψh\n1−iβ\n2π\u0000\na−µ+iΓ\n2\u0001i\n−ψh\n1 +iβ\n2π\u0000\nb−µ−iΓ\n2\u0001i\n2πi−1\n2β\u0000\na−µ+iΓ\n2\u0001−1\n2β\u0000\nb−µ−iΓ\n2\u0001\n\n,(E15)\nwhich up to order β−2has the same low-temperature limit, up to an overall minus sign, as the fermionic case. The\nminus sign cancels with the prefactor in Eq. (58) such that the particle statistics of the ideal bath becomes unimportant\nat sufficiently low temperature and system-bath coupling Γ while keeping βΓ constant.\n2. Simple chain\nFor the simple chain in D= 1, the Floquet Hamilto-\nnian (for a given momentum k) is given by\nHF=X\nllΩ|ϕkl⟩⟨ϕkl|, (E16)\nwhere the projector ( |ϕkl⟩⟨ϕkl|)mn=\ne−ik(m−n)aJl−m(ζ)Jl−n(ζ) with ζ= 2J/Ω. We fur-ther have\nGR/A(k, ω) =X\nl|ϕkl⟩⟨ϕkl|\nω−lΩ±iΓ/2, (E17)\nand\n∂kHF=X\nllΩ (|∂kϕkl⟩⟨ϕkl|+|ϕkl⟩⟨∂kϕkl|).(E18)\nHence, the current for the ideal fermionic (+) or bosonic\n(−) bath becomes\nj=±eΓX\nm,nX\nl,l′Z\nω,kf0\n±(ω−nΩ)(∂kHF)0m(|ϕkl⟩⟨ϕkl|)mn(|ϕkl′⟩⟨ϕkl′|)n0\n(ω−lΩ +iΓ/2) (ω−l′Ω−iΓ/2)(E19)\n=∓iaeΓX\nm,nX\nl,l′X\nppΩZ\nω,kf0\n±(ω+nΩ)mJp(ζ)Jp+m(ζ)Jl+m(ζ)Jl+n(ζ)Jl′+n(ζ)Jl′(ζ)\n(ω−lΩ +iΓ/2) (ω−l′Ω−iΓ/2). (E20)\nWe see explicitly that the ac response vanishes, since this\nwould result in an extra phase factor eiknain Eq. (E20)\nfor the nth harmonic, yielding δn0after performing themomentum integral. Using\nX\nnnJn(ζ)Jn+m(ζ) =ζ\n2δm,±1, (E21)\nwe obtain21\nj=∓ieJΓX\nnX\nl,l′Z\nωf0\n±(ω+nΩ)[Jl+1(ζ)−Jl−1(ζ)]Jn+l(ζ)Jn+l′(ζ)Jl′(ζ)\n(ω−lΩ +iΓ/2) (ω−l′Ω−iΓ/2)(E22)\n=∓ieJΓX\nnX\nl,l′Z\nωf0\n±(ω)[Jl+1−n(ζ)−Jl−1−n(ζ)]Jl(ζ)Jl′(ζ)Jl′−n(ζ)\n(ω−lΩ +iΓ/2) (ω−l′Ω−iΓ/2)(E23)\n=∓ieJΓX\nlZ\nωf0\n±(ω)Jl(ζ)\nω−lΩ +iΓ/2\u0012Jl+1(ζ)\nω−(l+ 1)Ω −iΓ/2−Jl−1(ζ)\nω−(l−1)Ω−iΓ/2\u0013\n(E24)\n=∓eJΓX\nlJl(ζ)Jl+1(ζ)Z∞\n−∞dω\n2πi\u0014f0\n±(ω)\n(ω−a−iΓ/2) (ω−b+iΓ/2)−f0\n±(ω)\n(ω−b−iΓ/2) (ω−a+iΓ/2)\u0015\n, (E25)\nwhere we usedP\nnJn(ζ)Jn+m(ζ) =δm0in the second line, a=lΩ, and b= (l+1)Ω. Using our result above for\nthe frequency integral, we obtain for the fermionic bath\nj=−2eJΓX\nlJl(ζ)Jl+1(ζ) Re\n1\nΩ−iΓψh\n1\n2−iβ\n2π\u0000\nlΩ−µ+iΓ\n2\u0001i\n−ψh\n1\n2+iβ\n2π\u0000\n(l+ 1)Ω −µ−iΓ\n2\u0001i\n2πi\n (E26)\n=1\nπeJΩ/Γ\n1 + (Ω /Γ)2X\nlJl(ζ)\u001a\n[Jl−1(ζ) +Jl+1(ζ)] Im ( Ql) +Γ\nΩ[Jl−1(ζ)−Jl+1(ζ)] Re ( Ql)\u001b\n(E27)\n=1\nπ2eJΩ/Γ\n1 + (Ω /Γ)2X\nlJl(ζ)\u0014lJl(ζ)\nζIm (Ql) +Γ\nΩdJl(ζ)\ndζRe (Ql)\u0015\n, (E28)\nwith Ql≡ψ[1/2 + (iβ/2π) (lΩ−µ+iΓ/2)] and ζ=\n2J/Ω. A similar result can be obtained for the bosonic\nbath. We find numerically that the part of the sum that\nis proportional to Im ( Ql) gives the largest contribution\nand depends weakly on Ω, while the other part that isproportional to Re ( Ql) is relatively smaller and gives rise\nto oscillations. Moreover, note that 1D inversion symme-\ntry (x7→ −x) implies that j(−E) =−j(E) and which can\nbe seen explicitly in Eq. (E28) as J−l(−ζ) =Jl(ζ).\n[1] F. Bloch, ¨Uber die Quantenmechanik der Elektronen in\nKristallgittern, Zeitschrift f¨ ur Physik 52, 555 (1929).\n[2] C. Zener, A theory of the electrical breakdown of solid\ndielectrics, Proceedings of the Royal Society of London\nA145, 523 (1934).\n[3] L. Esaki and R. Tsu, Superlattice and Negative Differ-\nential Conductivity in Semiconductors, IBM Journal of\nResearch and Development 14, 61 (1970).\n[4] H. Kurz, H. G. Roskos, T. Dekorsy, and K. Kohler, Bloch\nOscillations, Philosophical Transactions: Mathematical,\nPhysical and Engineering Sciences 354, 2295 (1996).\n[5] R. Tsu, Superlattice to Nanoelectronics (Elsevier, Ams-\nterdam, 2005).\n[6] K. Leo, P. H. Bolivar, F. Br¨ uggemann, R. Schwedler, and\nK. K¨ ohler, Observation of Bloch oscillations in a semicon-\nductor superlattice, Solid State Communications 84, 943\n(1992).\n[7] E. E. Mendez and G. Bastard, Wannier-Stark Laddersand Bloch Oscillations in Superlattices, Physics Today\n46, 34 (1993).\n[8] A. M. Bouchard and M. Luban, Bloch oscillations and\nother dynamical phenomena of electrons in semiconduc-\ntor superlattices, Physical Review B 52, 5105 (1995).\n[9] A. Fahimniya, Z. Dong, E. I. Kiselev, and L. Levitov,\nSynchronizing Bloch-Oscillating Free Carriers in Moir \\’e\nFlat Bands, Physical Review Letters 126, 256803 (2021).\n[10] T. Vakhtel, D. O. Oriekhov, and C. W. J. Beenakker,\nBloch oscillations in the magnetoconductance of twisted\nbilayer graphene, Physical Review B 105, L241408\n(2022).\n[11] V. T. Phong and E. J. Mele, Quantum Geometric Oscil-\nlations in Two-Dimensional Flat-Band Solids, Physical\nReview Letters 130, 266601 (2023).\n[12] C. De Beule, V. T. Phong, and E. J. Mele, Roses in\nthe nonperturbative current response of artificial crys-\ntals, Proceedings of the National Academy of Sciences22\nFIG. 13. Exact results for the simple chain coupled to an ideal bath. Steady-state current for the simple chain as\ngiven in Eq. (E26) calculated for l∈[−400,400] at half filling with Jβ= 50, and different values of Γ as indicated. The solid\nline gives the semiclassical result [Eq. (B16)].\n120, e2306384120 (2023).\n[13] J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E.\nCunningham, T. Meier, G. von Plessen, A. Schulze,\nP. Thomas, and S. Schmitt-Rink, Optical investigation of\nBloch oscillations in a semiconductor superlattice, Phys-\nical Review B 46, 7252 (1992).\n[14] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo,\nH. Kurz, and K. K¨ ohler, Coherent submillimeter-wave\nemission from Bloch oscillations in a semiconductor su-\nperlattice, Physical Review Letters 70, 3319 (1993).\n[15] A. A. Ignatov, E. Schomburg, J. Grenzer, K. F. Renk,\nand E. P. Dodin, THz-field induced nonlinear transport\nand dc voltage generation in a semiconductor superlat-\ntice due to Bloch oscillations, Zeitschrift f¨ ur Physik B\nCondensed Matter 98, 187 (1995).\n[16] T. Dekorsy, R. Ott, H. Kurz, and K. K¨ ohler, Bloch os-\ncillations at room temperature, Physical Review B 51,\n17275 (1995).\n[17] P. G. Savvidis, B. Kolasa, G. Lee, and S. J. Allen,\nResonant Crossover of Terahertz Loss to the Gain of\na Bloch Oscillating $\\mathrm {InAs}/\\mathrm {AlSb}$\nSuperlattice, Physical Review Letters 92, 196802 (2004).\n[18] A. Sibille, J. F. Palmier, H. Wang, and F. Mollot, Ob-\nservation of Esaki-Tsu negative differential velocity in\nGaAs/AlAs superlattices, Physical Review Letters 64,\n52 (1990).\n[19] W. Shockley, Stark Ladders for Finite, One-Dimensional\nModels of Crystals, Physical Review Letters 28, 349\n(1972).\n[20] J. B. Krieger and G. J. Iafrate, Time evolution of Bloch\nelectrons in a homogeneous electric field, Physical Review\nB33, 5494 (1986).\n[21] D. Emin and C. F. Hart, Existence of Wannier-Stark lo-\ncalization, Physical Review B 36, 7353 (1987).\n[22] P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alib-\nert, and A. Regreny, Observation of the Wannier-Stark\nQuantization in a Semiconductor Superlattice, Physical\nReview Letters 61, 1639 (1988).\n[23] G. Nenciu, Dynamics of band electrons in electric and\nmagnetic fields: rigorous justification of the effective\nHamiltonians, Reviews of Modern Physics 63, 91 (1991).\n[24] C. De Beule and E. J. Mele, Berry Curvature Spec-\ntroscopy from Bloch Oscillations, Physical Review Let-\nters131, 196603 (2023).\n[25] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Sa-lomon, Bloch Oscillations of Atoms in an Optical Poten-\ntial, Physical Review Letters 76, 4508 (1996).\n[26] A. R. Kolovsky and H. J. Korsch, Bloch oscillations of\ncold atoms in optical lattices, International Journal of\nModern Physics B 18, 1235 (2004).\n[27] Y. Gao, S. A. Yang, and Q. Niu, Field Induced Positional\nShift of Bloch Electrons and Its Dynamical Implications,\nPhysical Review Letters 112, 166601 (2014).\n[28] I. Komissarov, T. Holder, and R. Queiroz, The quantum\ngeometric origin of capacitance in insulators (2023).\n[29] M. Gl¨ uck, A. Kolovsky, H. Korsch, and N. Moiseyev,\nCalculation of Wannier-Bloch and Wannier-Stark states,\nThe European Physical Journal D - Atomic, Molecular,\nOptical and Plasma Physics 4, 239 (1998).\n[30] M. Gl¨ uck, A. R. Kolovsky, and H. J. Korsch, Resonant\ntunnelling of Wannier-Stark states, Journal of Optics B:\nQuantum and Semiclassical Optics 2, 694 (2000).\n[31] M. Gl¨ uck, A. R. Kolovsky, and H. J. Korsch, Wan-\nnier–Stark resonances in optical and semiconductor su-\nperlattices, Physics Reports 366, 103 (2002).\n[32] W.-R. Lee and K. Park, Direct manifestation of topolog-\nical order in the winding number of the Wannier-Stark\nladder, Physical Review B 92, 195144 (2015).\n[33] K. W. Kim, A. Andreanov, and S. Flach, Anomalous\ntransport in a topological Wannier-Stark ladder, Physical\nReview Research 2, 023067 (2020).\n[34] A. G´ omez-Le´ on and G. Platero, Floquet-Bloch Theory\nand Topology in Periodically Driven Lattices, Physical\nReview Letters 110, 200403 (2013).\n[35] S. H. Simon and M. S. Rudner, Contrasting lattice ge-\nometry dependent versus independent quantities: Rami-\nfications for Berry curvature, energy gaps, and dynamics,\nPhysical Review B 102, 165148 (2020).\n[36] D. Vanderbilt, Berry Phases in Electronic Structure\nTheory: Electric Polarization, Orbital Magnetization\nand Topological Insulators (Cambridge University Press,\nCambridge, 2018).\n[37] G. H. Wannier, Elements of Solid State Theory, in El-\nements of Solid State Theory (Cambridge University\nPress, Cambridge, England, 1959) pp. 191–193.\n[38] G. H. Wannier, Wave Functions and Effective Hamilto-\nnian for Bloch Electrons in an Electric Field, Physical\nReview 117, 432 (1960).\n[39] E. O. Kane, Zener tunneling in semiconductors, Journal\nof Physics and Chemistry of Solids 12, 181 (1960).23\n[40] J. E. Avron, J. Zak, A. Grossmann, and L. Gunther, In-\nstability of the continuous spectrum: The N-band Stark\nladder, Journal of Mathematical Physics 18, 918 (1977).\n[41] J. Zak, Berry’s phase for energy bands in solids, Physical\nReview Letters 62, 2747 (1989).\n[42] W. Kohn, Analytic Properties of Bloch Waves and Wan-\nnier Functions, Physical Review 115, 809 (1959).\n[43] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects\non electronic properties, Reviews of Modern Physics 82,\n1959 (2010).\n[44] K. W. Kim, W.-R. Lee, Y. B. Kim, and K. Park, Surface\nto bulk Fermi arcs via Weyl nodes as topological defects,\nNature Communications 7, 13489 (2016).\n[45] N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao,\nA. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru,\nH. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao,\nQuantum-metric-induced nonlinear transport in a topo-\nlogical antiferromagnet, Nature 621, 487 (2023).\n[46] A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan,\nY. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen,\nM. Huang, D. B´ erub´ e, H. Li, C. Tzschaschel, T. Dinh,\nZ. Sun, S.-C. Ho, S.-W. Lien, B. Singh, K. Watanabe,\nT. Taniguchi, D. C. Bell, H. Lin, T.-R. Chang, C. R. Du,\nA. Bansil, L. Fu, N. Ni, P. P. Orth, Q. Ma, and S.-Y.\nXu, Quantum metric nonlinear Hall effect in a topolog-\nical antiferromagnetic heterostructure, Science 381, 181\n(2023).\n[47] K. Chinzei and T. N. Ikeda, Disorder effects on the ori-\ngin of high-order harmonic generation in solids, Physical\nReview Research 2, 013033 (2020).\n[48] T. Morimoto and N. Nagaosa, Topological nature of\nnonlinear optical effects in solids, Science Advances 2,\ne1501524 (2016).\n[49] O. Matsyshyn, J. C. W. Song, I. S. Villadiego, and L.-k.\nShi, Fermi-Dirac staircase occupation of Floquet bands\nand current rectification inside the optical gap of met-\nals: An exact approach, Physical Review B 107, 195135\n(2023).\n[50] A. Kamenev, Field Theory of Non-Equilibrium Systems\n(Cambridge University Press, Cambridge, 2011).\n[51] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Time-\ndependent transport in interacting and noninteractingresonant-tunneling systems, Physical Review B 50, 5528\n(1994).\n[52] P. Coleman, Introduction to Many-Body Physics (Cam-\nbridge University Press, Cambridge, 2015).\n[53] A. P. Jauho, Introduction to the Keldysh nonequilibrium\nGreen function technique (2006).\n[54] A. K. Eissing, V. Meden, and D. M. Kennes, Functional\nrenormalization group in Floquet space, Physical Review\nB94, 245116 (2016).\n[55] M. J. Rice and E. J. Mele, Elementary Excitations of a\nLinearly Conjugated Diatomic Polymer, Physical Review\nLetters 49, 1455 (1982).\n[56] N. W. Ashcroft and N. D. Mermin, Solid State Physics,\ninSolid State Physics (Saunders College, Philadelphia,\n1976) p. 248.\n[57] F. D. M. Haldane, Model for a Quantum Hall Effect\nwithout Landau Levels: Condensed-Matter Realization\nof the ”Parity Anomaly”, Physical Review Letters 61,\n2015 (1988).\n[58] M. Kohmoto, Topological invariant and the quantiza-\ntion of the Hall conductance, Annals of Physics 160, 343\n(1985).\n[59] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group\nTheory: Application to the Physics of Condensed Mat-\nter, in Group Theory: Application to the Physics of Con-\ndensed Matter (Springer Science & Business Media, 2007)\np. 421.\n[60] R. D. King-Smith and D. Vanderbilt, Theory of polar-\nization of crystalline solids, Physical Review B 47, 1651\n(1993).\n[61] C.-P. Zhang, X.-J. Gao, Y.-M. Xie, H. C. Po, and K. T.\nLaw, Higher-order nonlinear anomalous Hall effects in-\nduced by Berry curvature multipoles, Physical Review B\n107, 115142 (2023).\n[62] P. A. Lebwohl and R. Tsu, Electrical Transport Prop-\nerties in a Superlattice, Journal of Applied Physics 41,\n2664 (2003).\n[63] J. Rammer and H. Smith, Quantum field-theoretical\nmethods in transport theory of metals, Reviews of Mod-\nern Physics 58, 323 (1986)." }, { "title": "2401.17378v1.Polarization_properties_of_the_128_non_repeating_fast_radio_bursts_from_the_first_CHIME_FRB_baseband_catalog.pdf", "content": "Draft version February 1, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nPolarization properties of the 128 non-repeating fast radio bursts from the first CHIME/FRB\nbaseband catalog\nAyush Pandhi\n ,1, 2Ziggy Pleunis\n ,2, 3, 4Ryan Mckinven\n ,5, 6B. M. Gaensler\n ,1, 2, 7Jianing Su\n ,1\nCherry Ng\n ,8Mohit Bhardwaj\n ,9Charanjot Brar\n ,10, 11Tomas Cassanelli\n ,12Amanda Cook\n ,1, 2\nAlice P. Curtin\n ,10, 11Victoria M. Kaspi\n ,10, 11Mattias Lazda\n ,1, 2Calvin Leung\n ,13, 14Dongzi Li\n ,15\nKiyoshi W. Masui\n ,16, 17Daniele Michilli\n ,16, 17Kenzie Nimmo\n ,16Aaron B. Pearlman\n ,10, 11, 18, 19, 20\nEmily Petroff\n ,21Masoud Rafiei-Ravandi\n ,10, 11Ketan R. Sand\n ,10, 11Paul Scholz\n ,22, 2Kaitlyn Shin\n ,16, 17\nKendrick Smith\n ,21and Ingrid Stairs\n23\n1David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4,\nCanada\n2Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada\n3Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands\n4ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands\n5Department of Physics, McGill University, 3600 rue University, Montreal, QC H3A 2T8, Canada\n6Trottier Space Institute, McGill University, 3550 rue University, Montreal, QC H3A 2A7, Canada\n7Present address: Division of Physical and Biological Sciences, University of California Santa Cruz, Santa Cruz, CA 95064, USA\n8Laboratoire de Physique et Chimie de l’Environnement et de l’Espace - Universit´ e d’Orl´ eans/CNRS, 45071, Orl´ eans Cedex 02, France\n9Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, 15213, PA, USA\n10Department of Physics, McGill University, 3600 rue University, Montr´ eal, QC H3A 2T8, Canada\n11Trottier Space Institute, McGill University, 3550 rue University, Montr´ eal, QC H3A 2A7, Canada\n12Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago 8370451, Chile\n13Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA\n14NHFP Einstein Fellow\n15Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA\n16MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge,\nMA 02139, USA\n17Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA\n18Banting Fellow\n19McGill Space Institute Fellow\n20FRQNT Postdoctoral Fellow\n21Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, ON N25 2YL, Canada\n22Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, ON MJ3 1P3, Canada\n23Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1 Canada\nABSTRACT\nWe present a 400 −800 MHz polarimetric analysis of 128 non-repeating fast radio bursts (FRBs) from\nthe first CHIME/FRB baseband catalog, increasing the total number of FRB sources with polarization\nproperties by a factor of ∼3. Of the 128 sources, 89 FRBs have >6σlinearly polarized detections, 29\nFRBs fall below this significance threshold and are deemed linearly unpolarized, and for 10 FRBs the\npolarization data are contaminated by instrumental polarization. For the 89 polarized FRBs, we find\nFaraday rotation measure (RM) amplitudes, after subtracting approximate Milky Way contributions,\nin the range 0 .5−1160 rad m−2with a median of 53 .8 rad m−2. Most non-repeating FRBs in our\nsample have RMs consistent with Milky Way-like host galaxies and their linear polarization fractions\nrange from ≤10% to 100% with a median of 63%. The non-repeater RMs and linear polarization\nfraction distributions are consistent with those of repeating FRBs. We see marginal evidence that\nnon-repeating FRBs have more constraining lower limits than repeating FRBs for the host electron-\nCorresponding author: Ayush Pandhi\nayush.pandhi@astro.utoronto.caarXiv:2401.17378v1 [astro-ph.HE] 30 Jan 20242\ndensity-weighted line-of-sight magnetic field strength. We classify the non-repeating FRB polarization\nposition angle (PA) profiles into four archetypes: (i) single component with constant PA (57% of the\nsample), (ii) single component with variable PA (10%), (iii) multiple components with a single constant\nPA (22%), and (iv) multiple components with different or variable PAs (11%). We see no evidence for\npopulation-wide frequency-dependent depolarization and, therefore, the spread in the distribution of\nfractional linear polarization is likely intrinsic to the FRB emission mechanism.\nKeywords: Radio bursts (1339) — Radio transient sources (2008) — Polarimetry (1278)\n1.INTRODUCTION\nFast radio bursts (FRBs), discovered by Lorimer\net al. (2007), are millisecond-duration extragalactic ra-\ndio transients. So far, 759 FRB sources have been\npublished1(the largest sample being the first Cana-\ndian Hydrogen Intensity Mapping Experiment FRB\n(CHIME/FRB) catalog; CHIME/FRB Collaboration\net al. 2021) and ∼3% of the FRB sources are known to\nrepeat (CHIME/FRB Collaboration et al. 2023a). Re-\npeating FRBs have been observed over a wide range\nof frequencies, from as low as 110 MHz (Pleunis et al.\n2021a) up to 8 GHz (Gajjar et al. 2018; Bethapudi et al.\n2023). Apparent non-repeating FRBs have been de-\ntected from 300 MHz (Parent et al. 2020) to 1.53 GHz\n(e.g., Law et al. 2023). Over 40 FRBs have been lo-\ncalized to external galaxies (e.g., Chatterjee et al. 2017;\nTendulkar et al. 2017; Nimmo et al. 2022a; Michilli et al.\n2023; Bhardwaj et al. 2023) with diverse host galaxy\nproperties and local environments (e.g., Mannings et al.\n2021; Bhandari et al. 2022; Gordon et al. 2023; Ibik et al.\n2023). However, the specific origins and emission mech-\nanisms of FRBs remain elusive. Furthermore, it is not\nclear whether repeating and non-repeating FRBs have\ndistinct origins or, even if all FRBs repeat.\n1.1. Polarization information from FRB voltage data\nFRBs are typically highly linearly polarized (e.g.,\nPetroff et al. 2015; Masui et al. 2015). As such, FRB po-\nlarimetry can help to elucidate properties of both their\nemission mechanism and local magneto-ionic environ-\nment in the immediate vicinity of the source. The po-\nlarization properties of FRBs are fully encapsulated in\nthe Stokes I,Q,U, and Vdynamic spectra. Here, I\nrefers to the total intensity of the emission, QandUare\nthe linearly polarized components, and Vis the circu-\nlarly polarized component. The Stokes parameters are a\nfunction of both time ( t) and frequency ( ν). By integrat-\ning the QandUdynamic spectra over the FRB burst en-\nvelope and then averaging over the emitting band of the\n1Based on the FRB Newsletter Volume 04, Issue 12 published in\nDecember 2023.FRB, we can derive the average linear and circular polar-\nization fraction for the FRB as L/I≡\u0002\nQ2+U2\u00031/2/I\nand|V|/I, respectively.2\nAs the linearly polarized radio emission, with an in-\ntrinsic position angle ψ0, propagates through magneto-\nionic media, it undergoes Faraday rotation. This causes\na wavelength squared\u0000\nλ2\u0001\n-dependent rotation of the ob-\nserved linear polarization position angle, ψ(λ2):\nψ(λ2) =ψ0+ RM λ2, (1)\nwhere RM is the rotation measure, which is related to\nthe integrated number density of electrons and parallel\ncomponent of the magnetic field along the line of sight\n(LoS):\nRM = 0 .812Z0\nzne(z)B∥(z)\n(1 +z)2dl(z)\ndzdzrad m−2.(2)\nHere, zis the redshift of the FRB, ne(z) is the electron\ndensity in units of cm−3,B∥(z) is the LoS magnetic field\nstrength in units of µG,dl(z) is the LoS line element at\nzin units of kpc, and we integrate from the FRB source\n(atz) to the observer (at z= 0). A positive (nega-\ntive) RM implies that the average LoS magnetic field is\npointed towards (away from) the observer. Simultane-\nously, the radio emission experiences a λ2dispersion by\nthe same free electrons, which is characterized by the\ndispersion measure (DM):\nDM =Zz\n0ne(z)\n(1 +z)dl(z)\ndzdzpc cm−3. (3)\n1.2. Polarization as a probe of FRB emission\nmechanisms and local environments\nCoherent emission models for FRBs often invoke ei-\nther a neutron star magnetosphere (e.g., Kumar et al.\n2017; Yang & Zhang 2018; Lyutikov 2021) or a syn-\nchrotron maser origin (e.g., Lyubarsky 2014; Metzger\n2For CHIME/FRB data, a debiasing is applied following Equation\n11 of Everett & Weisberg (2001), which becomes important in the\nlow signal-to-noise regime.Polarization properties of CHIME/FRB non-repeaters 3\net al. 2019; Beloborodov 2020). Both sets of models ex-\nplain the typically high linear polarization fraction seen\nfor FRBs. Magnetospheric models are able to reproduce\nboth constant and varying polarization angle (PA) pro-\nfiles across the burst based on their magnetospheric con-\nfiguration or LoS geometry. Synchrotron maser models,\nhowever, predict a constant PA across bursts (Metzger\net al. 2019; Beloborodov 2020) and are thus disfavored\nfor bursts that show significant PA swings (e.g., as is\nthe case for some FRB 20180301A bursts and for FRB\n20221022A; Luo et al. 2020, ; Mckinven et al., in prepa-\nration).\nWe expect different levels of typical |RM|contribu-\ntions from different local environments. For instance, we\nmight expect FRBs located in clean environments, such\nas globular clusters, to have a local environment |RM|\ncontributions of ∼0 rad m−2. Meanwhile, Hackstein\net al. (2019) predict a median |RM| ∼10 rad m−2con-\ntribution from a Milky Way (MW)-like FRB host galaxy\nand∼102rad m−2for FRBs located near star forming\nregions (based on models of the MW electron density\nand Galactic magnetic field). For an FRB embedded\nwithin a young supernova remnant (with a typical age\nof∼102−103years), we might expect |RM|s of up to\n∼103rad m−2contributed from the local environment\n(Piro & Gaensler 2018).\n1.3. Current observations and understanding of FRB\npolarimetry\nCurrently, there exist polarimetric observations for 36\nnon-repeating FRBs (e.g., Petroff et al. 2015; Masui\net al. 2015; Sherman et al. 2023a) and for over 2000\nbursts from 18 repeating FRB sources (e.g., Xu et al.\n2022; Dai et al. 2022; Mckinven et al. 2023a,b; Anna-\nThomas et al. 2023). For a summary of the published\npolarization properties of FRBs from the literature, see\nTable 5 in Appendix A. The primary focus of FRB po-\nlarimetry studies thus far has been on the temporal\nevolution of polarization properties of prolific repeating\nsources. The general picture of FRB polarization thus\nfar has been that:\n1. FRBs are typically highly linearly polarized (the\nmedian L/Iof currently published FRBs with po-\nlarization properties is ∼75%). Many FRBs are\nalso consistent with being 100% linearly polarized.\n2. Approximately 13% of the currently published\nFRB sources with polarimetry show at least one\nburst with substantial circular polarization (i.e.,\n≳20%) over their burst profile (e.g., Cho et al.\n2020; Sherman et al. 2023a).3. Typically FRBs have |RM|s on the order of ∼\n102rad m−2.\n4. A few prolific repeaters originate from extremely\ndense and/or highly magnetic environments (e.g.,\nFRB 20121102A and FRB 20190520B, which show\n|RM|s up to ∼104−105rad m−2; Michilli et al.\n2018; Dai et al. 2022);\n5. At least some of the known repeaters for which\nwe have detected polarized bursts reside in dy-\nnamic magneto-ionic environments such that they\nundergo RM variations on days to years timescales\nand, in some cases, exhibit changes in the sign of\ntheir RM (Hilmarsson et al. 2021b; Anna-Thomas\net al. 2023; Mckinven et al. 2023a).\nMulti-band polarimetry (between ∼0.1−5 GHz)\nby Feng et al. (2022) found that five repeating FRBs\n(FRB 20121102A, FRB 20190520B, FRB 20190303A,\nFRB 20190417A, and FRB 20201124A) show decreas-\ningL/Iwith decreasing observing frequency. Feng et al.\n(2022) suggest that the observed frequency dependence\ninL/Iis an imprint of depolarization caused by multi-\npath propagation of the FRB emission through an inho-\nmogeneous magneto-ionic medium, parameterized by\nfRM(λ) = 1−exp\u0002\n−2λ4σ2\nRM\u0003\n, (4)\noriginally formulated by Burn (1966) for incoherent syn-\nchrotron radiation3. Here fRM(λ) is the fractional de-\ncrease in the observed linear polarization from its intrin-\nsic value, λis the wavelength in the observer frame, and\nσRMis a measure of the dispersion about the mean ob-\nserved RM. Note that the exponent in Equation 4 is in-\ndependent of redshift as the (1+ z) scaling on σRMandλ\ncancel out. However, in order to derive the typical depo-\nlarization wavelength, λdepol, we would need a redshift-\ndependent coordinate transform in λto the rest frame\nof the FRB. Feng et al. (2022) find a positive correlation\nbetween σRMand temporal scattering, suggesting that\nboth σRMand the temporal scattering originate from\nthe same region near the FRB source. Furthermore, the\nFRBs with the largest σRM(FRB 20121102A and FRB\n20190520B) are associated with persistent radio sources\n(Marcote et al. 2017; Niu et al. 2022), consistent with a\nsupernova remnant or pulsar wind nebula origin, which\nis in agreement with their progenitors being associated\nwith dense local environments.\n3Beniamini et al. (2022), on the other hand, argue that depolar-\nization from propagation of coherent radiation through a mag-\nnetized scattering screen instead follows a power-law dependence\non frequency.4\n1.4. Dichotomy between repeating and non-repeating\nFRBs\nThere is some evidence for differences between repeat-\ning and non-repeating FRBs. Pleunis et al. (2021b)\nfind a dichotomy in burst properties of repeating\nand non-repeating FRBs, with repeating FRBs hav-\ning larger average burst widths and narrower average\nemitting bandwidths than non-repeating FRBs in the\nfirst CHIME/FRB catalog (CHIME/FRB Collaboration\net al. 2021). Whether repeating and non-repeating\nFRBs originate from (i) the same source population,\n(ii) the same source population with different environ-\nments or propagation effects, or (iii) different source\npopulations remains largely uncertain. Polarimetry pro-\nvides insight into both the emission mechanism and lo-\ncal magneto-ionic environment of FRBs and is therefore\na powerful tool to distinguish between these scenarios.\nFeng et al. (2022) show that the |RM|distributions of\nthe repeating and non-repeating sources differ (with a p-\nvalue of 0 .02 using the Kolmogorov-Smirnov test), pos-\nsibly indicating a divergence in local environments of\nthe two populations. Note, however, that this differ-\nence in |RM|was only seen in a small sample (9 re-\npeating and 12 non-repeating FRBs), and is potentially\nsubject to selection bias due to intra-channel depolar-\nization in the coarser filter bank data for non-repeating\nFRBs. This is because discovery searches often use\ncoarser filter banks for computational efficiency, which\nis not necessary for follow-up observations of known re-\npeating sources. Hence, the difference in |RM|between\nrepeating and non-repeating FRBs may not be repre-\nsentative of the full observed FRB sample and would\nbe greatly benefited by polarimetric studies on a larger\nsample of FRBs.\n1.5. Polarization properties of CHIME/FRB\nnon-repeaters\nIn this paper, we perform a systematic analysis of the\npolarization properties for 128 non-repeating FRBs from\nthe first CHIME/FRB baseband catalog (CHIME/FRB\nCollaboration et al. 2023b). The L/I, RM, DM, and\nlower limits on the LoS-averaged magnetic field strength\nin the FRB host galaxy of our polarized sample are\ncompared to the polarization properties of the 13 polar-\nized, repeating FRB sources observed by CHIME/FRB\n(Bhardwaj et al. 2021; Mckinven et al. 2023a,b). Fur-\nthermore, we determine the extent to which the PA\nvaries across each burst in our sample of non-repeating\nFRBs and what fraction of these FRBs exhibit PA vari-\nations. For a subset of broadband emitting FRBs,\nwe derive a model-agnostic depolarization ratio across\nthe CHIME/FRB band and test the prevalence offrequency-dependent depolarization in our FRB sample,\ncomparing to the spectral depolarization seen in some\nrepeating FRBs. Finally, the polarization properties of\nthe repeating and non-repeating sources are compared\nwith their burst rates as estimated by CHIME/FRB Col-\nlaboration et al. (2023a).\nThis paper is structured as follows. In Section 2, we\nprovide an overview of the non-repeating FRBs from\nthe first CHIME/FRB baseband catalog, a comparison\nsample of repeating FRBs observed by CHIME/FRB,\nthe derived polarization properties of the CHIME/FRB\nFRBs, and describe the statistical methods used\nthroughout this paper. The results of the non-repeater\npolarimetry, including L/I, RM, lower limits on the\nLoS-averaged magnetic field strength in the FRB host\ngalaxy, PA variability, depolarization, and comparisons\nto the polarized, repeating CHIME/FRB sample are\npresented in Section 3. Interpretation of our results is\nprovided in Section 4. We conclude by summarizing our\nfindings and identifying avenues for future work.\n2.VOLTAGE DATA, DERIVED POLARIZATION\nPROPERTIES, AND STATISTICAL METHODS\nIn this Section, we define the repeating and non-\nrepeating FRB population for which we conduct\nour polarization analysis, and then summarize the\nCHIME/FRB polarization pipeline and additional anal-\nyses that we perform to produce our polarimetric results.\n2.1. CHIME/FRB voltage data\n2.1.1. Non-repeating FRBs\nCHIME/FRB Collaboration et al. (2023b) present the\nfirst CHIME/FRB baseband catalog, providing the full\nvoltage data for all 140 FRBs recorded by the triggered\nbaseband system between 2018 December 9 and 2019\nJuly 1 (for a detailed description of the baseband anal-\nysis system on CHIME/FRB, see Michilli et al. 2021).\nBaseband data are only stored for those events exceed-\ning a signal-to-noise (S/N) threshold of S/N >10−12\n(the threshold has changed over the course of opera-\ntion). These data are saved with a time and frequency\nresolution of 2 .56µs and 390 kHz, respectively, with full\npolarization information (i.e., Stokes I,Q,U, and V).\nCHIME/FRB Collaboration et al. (2023b) describe the\nprocessing of baseband data in detail. To summarize,\nthe initial estimate of the FRB sky position is refined\nby forming in software a large array of beams on the\nsky, and then is precisely localizing by maximizing the\nS/N with a more densely packed array of beams formed\naround the initial position. This process typically re-\nturns sky localization regions with sub-arcminute pre-\ncision; a single beam is then formed at the best esti-Polarization properties of CHIME/FRB non-repeaters 5\nmated FRB sky position and those data are recorded\nas a “single-beam” file. These single-beam files serve\nas inputs to the CHIME/FRB polarization pipeline, de-\nscribed by Mckinven et al. (2021) and summarized in\nSection 2.2 below. The 128 non-repeating FRBs with\nsingle-beam files presented by CHIME/FRB Collabora-\ntion et al. (2023b) form the sample of sources that are\nthe focus of this paper.\n2.1.2. Repeating FRBs\nOne of the goals of this paper is to compare the polar-\nization properties of our non-repeating sample to that\nof known repeating FRBs. To avoid confounding ef-\nfects in the polarization properties arising from differ-\nent observing frequencies, selection effects, and process-\ning stages, we opt to compare our CHIME/FRB non-\nrepeaters with a sample of repeating FRBs with polar-\nization measurements also obtained by CHIME/FRB.\nThe total comparison sample consists of 13 repeating\nsources with a collective 82 bursts detected between\nDecember 2018 and December 2021 that are collated\nfrom results by Bhardwaj et al. (2021), Mckinven &\nCHIME/FRB Collaboration (2022), Mckinven et al.\n(2023a), and Mckinven et al. (2023b), which make use\nof the CHIME/FRB polarization pipeline described by\nMckinven et al. (2021).\n2.2. Polarization pipeline products\nFor each burst, we begin with the Stokes I,Q,U, and\nVdynamic spectra channelized to a time resolution of\n2.56µs and a frequency resolution of 390 kHz. We ob-\ntain the best fit structure optimizing DM, DM obs,struct,\nbased on the DMphase algorithm (Seymour et al. 2019).\nThe Stokes data are coherently de-dispersed to their\nrespective DM obs,struct and a radio frequency interfer-\nence (RFI) mask is applied in the baseband pipeline\nprocessing stage (Michilli et al. 2021). To derive the\npolarization properties, we process each burst using the\nCHIME/FRB polarization pipeline (for a full descrip-\ntion of the pipeline, see Mckinven et al. 2021). Below,\nwe provide a summary of the pipeline and highlight any\nchanges from the version presented by Mckinven et al.\n(2021).\nThe burst envelope limits in time are identified as the\npoints at which the total intensity drops to 20% of the\npeak burst intensity. A Gaussian function is fit to the\nintensity spectrum as a function of frequency and the\nburst spectral limits are set at the 3 σlevel of the Gaus-\nsian fit (with a minimum of 400 MHz and maximum of\n800 MHz). Stokes I,Q,U, and Vspectra are gener-\nated by integrating the signal between the burst enve-\nlope time limits with uniform weights in time. Then the\nL/Ispectrum is computed and averaged over the burstspectral limits to provide the L/Iof the FRB, averaged\nover both time and frequency.\nThe observing frequencies, and Stokes spectra with as-\nsociated standard deviation uncertainties, form an input\nto RM-synthesis (Brentjens & de Bruyn 2005), which\noutputs an observed RM estimate, RM obs, with mea-\nsurement uncertainties FWHM /(2×S/N). Here, the\nFWHM refers to the full-width at half maximum of the\ncleaned Faraday dispersion function (FDF) peak (the\ncleaning is done using the RM-CLEAN framework; Heald\net al. 2009) and the S/N is that of the peak polarized\nintensity in Faraday depth space. A minimum threshold\nof S/N = 6 in Lis required to warrant a linearly polar-\nized detection. This threshold was chosen after internal\ntesting on many FRBs over the course of CHIME/FRB\noperations. Setting it lower results in an increase of false\ndetections caused by instrumental polarization (i.e., at\nRMobs∼0 rad m−2), while increasing the threshold\ncauses the pipeline to miss some marginal linearly po-\nlarized detections that would otherwise be well-fit by the\npipeline.\nAnother independent RM obsis estimated by the QU-\nfitting algorithm (implemented with a modified version\nof V1.2 of the RM-tools package; Purcell et al. 2020).\nWe use the same default model as Mckinven et al.\n(2021) with RM obs, PA, L/I, and the physical (cable)\ndelay between the two linear polarizations, τdelay, as\nfitted parameters and assume uniform priors on each\nof them. In this model, L/I is assumed to be con-\nstant over the burst spectral range (though we also ex-\nplore L/Ivariations as a function of frequency using a\nslightly different method in Section 3.7.3). The Stokes\nIspectrum is fit using a univariate spline fit, Imod,\nfrom the scipy.interpolate.UnivariateSpline mod-\nule, and is also used as an input in the QU-fitting step.\nWe set the Stokes Vmodel to Vmod= 0 to limit ambigu-\nity of unmodelled instrumental sources of circular polar-\nization (which can contribute up to |V|/I∼20%) with\nintrinsic signal. Models of the Stokes QandU(Qmod\nandUmod, respectively) spectra are then expressed as,\nQmod=Imod(L/I)cos(2(RM obsλ2+ψ0)),(5)\nUmod=Imod(L/I)sin(2(RM obsλ2+ψ0)).(6)\nMaximum likelihood estimates for the fitted parameters\nare derived via Nested Sampling (Skilling 2004) to best\nfitQmodandUmodto the observed spectra, Qobsand\nUobs. A phase shift is introduced due to the time delay,\nτdelay, between the signals from the two polarized feeds\ndue to their differential path lengths through the system.\nThis causes a mixing between Stokes UandVthat is6\ncharacterized by the matrix\n \nUobs\nVobs!\n= \ncos(2 πντdelay)−sin(2πντdelay)\nsin(2πντdelay) cos(2 πντdelay)! \nU\nV!\n.\n(7)\nWe thus update the Stokes QandUmodels (now Q′\nmod\nandU′\nmod) to account for τdelay, while still assuming\nVmod= 0, as follows,\nQ′\nmod=Qmod, (8)\nU′\nmod=Umod[cos(2 πντdelay)]. (9)\nFor a non-zero τdelay, the QU-fitting routine may incor-\nrectly converge on the wrong sign for the RM obsesti-\nmate. In these cases, however, the FDF appears to have\ntwo mirrored peaks about RM obs= 0 rad m−2, but the\nhigher of the two peaks is always the correct sign of\nthe RM obsfor the phase offsets regularly observed in\nCHIME/FRB data (for more details, see Appendix A\nby Mckinven et al. 2021).\nFor finite frequency resolution, the linear polarization\nangle changes within one frequency channel. The intra-\nchannel change in polarization angle, δψ, for an event\nwith RM obsobserved at a central frequency νcwith fre-\nquency channel width δνfollows\nδψ=−2c2RMobsδν\nν3c. (10)\nThus, for high RM obsmagnitude events, the intra-\nchannel change in polarization angle can be significant\nand cause depolarization of the observed signal. The\ndecrease in L/I due to intra-channel depolarization,\nfchannel , follows\nfchannel =sin(δψ)\nδψ(11)\n(Gardner & Whiteoak 1966).\nForνc= 600 MHz and δν= 390 kHz, the sensitivity to\npolarized emission is halved at RM obs∼5000 rad m−2,\nmeaning that the range of RM obsthat can be deduced\nat the native resolution of these data is limited to a few\nthousand rad m−2. In cases where a S /N>6 polar-\nized signal is not detected natively in the FDF, a semi-\ncoherent RM search iterates through a grid of RM obs\nvalues within the range −106≤RMobs≤106rad m−2\nand performs RM-synthesis on a coherently de-rotated\nspectrum (see Section 4.3 and 5.1.4 by Mckinven et al.\n2021). In some cases, instrumental polarization may\ncause a peak in the FDF centered at 0 rad m−2with a\nfull width at half maximum equal to the theoretical RM\nspread function of ∼9 rad m−2for the full CHIME/FRBfrequency band (Ng et al. 2020). The instrumental po-\nlarization induced artefact can sometimes exceed the po-\nlarized intensity S/N of the true RM obspeak. In cases\nwhere this occurs and a >6σsecondary peak in the\nFDF can be clearly identified, the native RM search is\nre-run with the peak at 0 rad m−2masked out.\n2.3. Temporal PA variations\nFrom the Stokes QandUdynamic spectra, we can de-\nrive the observed linear polarization angle ψof the FRB\nemission as a function of time tand observing wave-\nlength λ, such that,\nψ(t, λ) =1\n2arctan\u0012Uobs(t, λ)\nQobs(t, λ)\u0013\n. (12)\nWith the observed QobsandUobsspectra, estimated\nRMobs, and ψ0in hand, we can de-rotate the linear po-\nlarization vector to remove the Faraday rotation effect,\n[Q+iU]obs,derot= [Q+iU]obs×exp\u0002\n2i\u0000\nRMobs\u0000\nλ2−λ2\n0\u0001\n+ψ0\u0001\u0003\n.\n(13)\nIn the equation above, the subscript “obs” refers to\nthe observed linear polarization vector, and λ0is the\nreference wavelength at which ψ0is measured (here,\nwe set λ0= 0 m such that ψ0is referenced to zero\nwavelength). From Equation 13, we can extract the\nde-rotated Qobs,derot andUobs,derot dynamic spectra as\nthe real and imaginary components of [ Q+iU]obs,derot,\nrespectively. Using the same procedure as Equation\n12 and integrating over the burst spectral range, we\ncan then compute the temporal PA variation across the\nburst width as\nψ0(t) =1\n2arctan\u0012Uobs,derot(t)\nQobs,derot(t)\u0013\n. (14)\nNote that we do not not calibrate for true value of the\nPA due to unknown beam phase effects off of CHIME’s\nmeridian axis. Hence, ψ0(t) is centered such that it\nhas a median of 0 deg and is only used to character-\nize the relative time evolution in PA across the burst.\nBy propagating the uncertainties in the Qobs,derot and\nUobs,derot spectra, we estimate the PA measurement un-\ncertainties, σψ, following the process described by Vern-\nstrom et al. (2019). These measurement uncertainties\ncan become inaccurate for low S/N in the linear polar-\nization L≡\u0002\nQ2+U2\u00031/2. To ensure that our PA pro-\nfiles are robust, we set a minimum linear polarization\nS/N threshold S /N(L)thresh = 5, below which points on\ntheψ0(t) profile are masked out. This step is particu-\nlarly important when characterizing any time variable\nPA behavior. For a detailed explanation on how this\nthreshold was derived, see Appendix B.Polarization properties of CHIME/FRB non-repeaters 7\nTo quantify the magnitude of deviation from\na constant ψ0(t), we conduct a reduced chi-\nsquared χ2\nνtest (implemented using the package\nscipy.stats.chisquare V1.10.0; Virtanen et al. 2020),\nevaluating the goodness of fit between the observed\nψ0(t) and a constant ψ0(t) = 0 model.\n2.4. Foreground DM correction\nFor each polarized FRB in our sample, we have es-\ntimates for the observed DM and RM (DM obsand\nRMobs). These quantities, however, are integrated over\nthe full path length between the emitting source and the\nobserver and, therefore, likely encompass distinct con-\ntributing media along the LoS. We can express DM obs\nin terms of its individual contributing components,\nDM obs= DM disk+DM halo+DM IGM(z)+DM host\n(1 +z),(15)\nwhere DM diskis contribution from warm ionized gas\nin the MW disk ( T≲104K) and DM halois from\nthe extended hot Galactic halo ( T∼106−107K).\nDM IGM(z) is the collective contributions from the inter-\ngalactic medium (IGM) and intervening systems. Each\ncontributing system along the LoS in the IGM has a dif-\nferent (unknown) redshift which is not corrected for in\nthe DM IGM(z) term. DM host/(1 + z) is from the host\ngalaxy of the source at redshift zand its local environ-\nment (Yamasaki & Totani 2020, and references therein).\nHere, we ignore the ionospheric contribution to the FRB\nDM as it typically adds on the order of ∼10−5pc cm−3\n(Lam et al. 2016).\nThe DM diskcontribution is estimated by utilizing the\nthermal electron density nemodel of Yao et al. (2017,\nhereafter YMW16). For a given sky position, the PyGEDM\npackage (V3.1.1; Price et al. 2021) allows us to integrate\nover the full extent of the MW along that LoS and ob-\ntain the value of DM diskaccording to YMW16. For each\nFRB, we take the DM diskcontribution at the best-fit po-\nsition using the baseband localizations of CHIME/FRB\nCollaboration et al. (2023b), which have a typical 1 σ\nuncertainty region of ≲1 arcmin. For the MW halo,\nwe assume a fiducial DM halo= 30 pc cm−3contribution\nbased on estimates by Dolag et al. (2015), Yamasaki &\nTotani (2020), and Cook et al. (2023). Our results are\nnot sensitive to the exact value of DM haloas long as it is\nassumed to be uniform across the sky. In general, FRBs\nhave significant observed excess DM contributions after\nsubtracting the MW DM which, for a subset of arcsec-\nond and sub-arcsecond localized FRBs, scales roughly\nlinearly with the host galaxy redshift (i.e., the “Mac-\nquart relation”; Macquart et al. 2020). Unfortunately,\nthe FRBs in our sample are yet to be associated withhost galaxies and, thus, do not have reliable redshift es-\ntimates. As such, we do not attempt to estimate and\nsubtract DM IGM(z) from DM obs.\nRemoving the MW disk and halo DM contributions\ngives us a foreground corrected DM estimate that en-\ncapsulates the totality of the extragalactic dispersion\nincurred by the FRB emission,\nDM EG= DM IGM(z) +DM host\n(1 +z), (16)\nand is an upper limit on the host galaxy/local environ-\nment DM. We highlight here that DM EGis reported in\nour observer frame, not the rest frame of the FRB.\n2.5. Foreground RM correction\nAnalogously, we can perform the same decomposition\nas in Section 2.4 for RM obsand write,\nRMobs= RM ion+RM disk+RM halo+RM IGM(z)+RMhost\n(1 +z)2,\n(17)\nwhere RM ionis the ionospheric contribution from the\nEarth’s atmosphere at the time of observation. We opt\nto not correct for the ionospheric contribution to the\nRM since it only adds on the order of ∼ ±1 rad m−2\n(Sobey et al. 2019). Stochastic RM variations of this\namplitude do not affect our results (presented in Section\n3) in a meaningful way. Hence, we move forward with\nthe assertion that RM ion= 0 rad m−2.\nHutschenreuter et al. (2022) construct an all-sky in-\nterpolated map of the foreground MW RM contribution\nusing a Bayesian inference scheme applied to all Fara-\nday rotation data (a total of 55 ,190 individual RMs,\nmostly from radio galaxies; Van Eck et al. 2023) avail-\nable by the end of 2020. This map has a pixel scale of\n∼1.3×10−2deg2and provides the best estimate of the\nGalactic RM (i.e., RM disk+ RM halo) sky to date. We\nestimate the RM disk+RM halotowards each FRB by tak-\ning the RM from the Hutschenreuter et al. (2022) map\nat the best-fit sky position of the FRB.\nFor most extragalactic, polarized radio sources,\nnamely radio galaxies, the foremost contributor to\nRMobsis propagation through the MW (Schnitzeler\n2010; Oppermann et al. 2015). In the IGM, B∥under-\ngoes multiple field reversals along the full path length,\nwhich is much larger than the magnetic field scales and\nB∥in the IGM is fairly weak ( ≤30 nG; Amaral et al.\n2021). Therefore, we expect the amplitude of RM IGM\nto be small. From observations, we see that the residual\nextragalactic RM distribution (after subtracting an es-\ntimate of the Galactic RM contribution) for a set of po-\nlarized radio galaxies is centered around 0 rad m−2(e.g.,\nsee Carretti et al. 2022) with some finite width that can8\nbe quantified by measuring the rms of the distribution.\nThis residual extragalactic RM rms has been measured\nat 144 MHz (0 .52 rad m−2; Carretti et al. 2022) and at\n1.4 GHz (14 .9 rad m−2; Vernstrom et al. 2019). Since\nwe expect the amplitude of RM IGMto be small and cen-\ntered on 0 rad m−2, while the typical |RMobs|of FRBs\nto date is ≳102m−2, we adopt the assumption that\nRMIGM= 0 rad m−2(i.e.,|RMIGM| ≪ |RMobs|).\nAfter subtracting the foreground MW RM and assert-\ning that RM IGM= RM ion= 0 rad m−2, the redshift-\ncorrected RM serves as a rough estimate for the total\nRM contributed by the FRB host galaxy and local en-\nvironment,\nRMEG=RMhost\n(1 +z)2. (18)\nAs with DM EG, we do not have redshift information\navailable for our FRBs and therefore RM EGis left in our\nobserver frame and is not converted to the rest frame of\nthe FRB.\n2.6. Parallel magnetic field lower limit\nThe ratio between RM and DM, along the same LoS,\nprovides an estimate of the electron density weighted av-\nerage magnetic field strength parallel to the LoS and has\noften been used to study Galactic pulsars (e.g., Sobey\net al. 2019; Ng et al. 2020). In the context of FRBs,\nwe would like to study the rest frame electron density\nweighted average magnetic field strength parallel to the\nLoS in the FRB host galaxy and local environment,\n\nB∥,host\u000b\n= 1.232|RMEG|\nDM EG(1 +z). (19)\nLacking redshift information, however, we use the\nDM EGand RM EGto calculate a lower limit on\f\f\nB∥,host\u000b\f\fas measured in our observer frame, which\nwe define as\n|β|= 1.232|RMEG|\nDM EGµG≤\f\f\nB∥,host\u000b\f\f. (20)\nThe reason |β|is a lower limit on\f\f\nB∥,host\u000b\f\fis mul-\ntifaceted: (i) DM EGwill almost always have a signif-\nicant DM IGM(z) contribution compared to the negli-\ngible RM IGM = 0 rad m−2, (ii)|RMEG| ≤ | RMhost|\nfrom Equation 18, and (iii) |RMEG|scales as (1 + z)−2\n(see Equation 18) while DM EGscales as (1 + z)−1(see\nEquation 16) meaning that a redshift correction on\n|RMEG|/DM EGforz≥0 would lead to a larger ra-\ntio. It is possible that peculiar LoS exists in which one\nof our assumptions breaks down and |β| ≤\f\f\nB∥,host\u000b\f\f\nno longer holds (e.g., if a strongly magnetized interven-\ning medium, which imparts a large RM, is unaccounted\nfor). However, we argue that Equation 20 is applicableto the vast majority of FRBs and is particularly useful\nwhen applied to large populations of FRBs, as we do in\nthis study.\nWe emphasize again that L/I,ψ, DM EG, RM EG, and\n|β|are observer frame quantities while DM host, RM host,\nand\nB∥,host\u000b\nare in the FRB rest frame. Any results\nand/or discussion related to DM EG, RM EG, and |β|\nthroughout this work will be in the observer frame un-\nless otherwise specified.\n2.7. Statistical tests\n2.7.1. Comparing distributions\nOne of the primary objectives of this work is to rig-\norously examine whether polarization properties of re-\npeating and non-repeating FRBs arise from the same\nor distinct distributions. To this end, we employ the\nAnderson-Darling (AD; Anderson & Darling 1954) and\nKolmogorov-Smirnov (KS; Smirnov 1948; Kolmogorov\n1956) tests. We utilize the Python implementation\nof these tests ( anderson ksamp and ks2samp , respec-\ntively) from the scipy.stats V1.9.3 package (Virtanen\net al. 2020) and apply them to the L/I, DM EG,|RMEG|,\nand|β|distributions of our repeating and non-repeating\nFRB samples. The AD test returns a test statistic SAD\nand a p-value ( pAD; floored at 0 .001 and capped at 0 .25),\nwhile the KS test returns test statistic SKSandp-value\npKS, assuming a two-sided null hypothesis. The p-values\nfrom both of these tests signify the level at which we can\nreject the null hypothesis that the repeating and non-\nrepeating polarized properties originate from the same\nunderlying population (i.e., we reject the null hypothesis\nat a 1 −plevel).\nHere, the KS test is measuring the supremum between\nthe observed repeating and non-repeating FRB cumula-\ntive distribution functions (CDFs) for a given parameter\nand is, therefore, sensitive to sharp differences between\nthe distributions. On the other hand, the AD test statis-\ntic is more sensitive to smaller but more persistent dif-\nferences, even in the tails of the distributions. While we\npresent the results of both tests, we note that the KS\ntest generally tends to produce more conservative sig-\nnificance levels across our sample when compared to the\nAD test.\nWhen parts of data sets (but not all of the data set)\nare left-censored (i.e., contain upper limits, which is the\ncase for some values of L/Ihere and for all burst rates\nof non-repeating FRBs), we additionally employ sta-\ntistical methods that take censoring into account (see,\ne.g., Feigelson & Babu 2012). For this, we use methodsPolarization properties of CHIME/FRB non-repeaters 9\nimplemented in R’s NADA V1.6-1.1 package4(Helsel\n2005; Lopaka 2020; R Core Team 2020). To begin with,\nwe additionally derive statistics from empirical cumula-\ntive distribution functions calculated using the Kaplan-\nMeier method (Kaplan & Meier 1958), as implemented\nin the cenfit function of NADA. When left-censored\ndata are present, we also perform the Peto & Peto (Peto\n& Peto 1972) and log-rank tests (Mantel 1966), also as\nimplemented in the cendiff function of NADA.\nFor all four tests, we set a significance level at α=\n2.7×10−3(i.e., at the 3 σlevel), such that when the\np-value from either test is less than α, we consider the\nresult statistically significant. We will refer to a result\nas being “marginally significant” when it satisfies 2 .7×\n10−3< α < 0.05 and not significant if α≥0.05.\n2.7.2. Correlations between parameters\nIn some cases we would like to evaluate whether a pos-\nitive or negative correlation exists between two param-\neters in our data. For this purpose, we use Spearman’s\nrank correlation coefficient (Spearman 1904), which\nmeasures the preponderance of a monotonic relationship\nbetween two input parameters. This test is implemented\nwith the scipy.stats.spearmanr V1.9.3 Python mod-\nule (Virtanen et al. 2020). The test returns a Spearman\nrank correlation coefficient SSRwhich is 0 if there is no\ncorrelation and +1 ( −1) if there is a perfectly positive\n(negative) monotonic correlation and also an associated\np-value, pSR. This is used throughout Section 3 to de-\ntermine the significance of any possible correlations be-\ntween DM EGand|RMEG|,L/Iand|RMEG|, and burst\nrate with various polarization properties. When testing\nfor correlations between singly or multiply left-censored\ndata sets, we also compute Kendall’s τcorrelation co-\nefficient (Kendall 1938), as implemented in the cenken\nfunction of NADA. We use the same significance levels\nfor these tests as we defined for the AD and KS tests\nabove.\n3.RESULTS\n3.1. Polarization properties of the non-repeating FRBs\nIn this section, we provide the derived polariza-\ntion results for the non-repeating FRBs in the first\nCHIME/FRB baseband catalog.\nOf the 128 non-repeating FRBs, 89 produce a >6σ\nlinearly polarized detection and are well characterized\nby the CHIME/FRB polarization pipeline. Of these,\nfive are contaminated by instrumental polarization (i.e.,\nthe FDF peak from the Faraday rotation was smaller\n4https://rdrr.io/cran/NADA/than the peak caused by instrumental polarization), but\ntheir values of RM obsare corrected by re-fitting to a sec-\nondary peak in the FDF. In all five cases, the polarized\nfit was significantly improved after adjusting for the in-\nstrumental polarization (see Section 2.2). However, note\nthat we cannot derive a robust L/I measurement for\nthese five FRBs and, as such, they are excluded from\nany statistical tests applied to the L/Idata.\nThere are 29 FRBs for which no significant linear po-\nlarization was detected. We consider these bursts un-\npolarized and derive upper limits for their linear po-\nlarization fraction L/I. First, we compute the S/N in\nStokes I, S/N(I) by averaging over the emitting band\nof the FRB, subtracting the mean Ifrom an off-pulse\n(i.e., noise) region from the Itime profile across the\nFRB burst duration, and dividing by the standard devi-\nation in the off-pulse region. Given the S/N in Stokes I,\nS/N(I), and our linear polarization detection threshold\non the S/N in L, S/N(L)>6, we can find the req-\nuisite minimum L/Ineeded to satisfy S /N(L) = 6 as\nL/I= 6/(S/N(I)). This serves as the upper limit for\nthe time and frequency averaged L/Iof a given unpolar-\nized FRB. For another 10 FRBs, we detect a significant\nlinear polarization but are uncertain about the accuracy\nof the polarimetric results due to low S/N in Stokes I\nand/or strong instrumental polarization contamination\nand, therefore, do not report their polarization proper-\nties.\nThe polarization properties of the 89 polarized and\n29 unpolarized FRBs are reported in Table 1. Figure\n1 displays the summary plots for the first 16 FRBs\nfrom Table 1. As described in Appendix B, we only\nplot points on the L/I,|V|/I, and PA profiles if they\npass the S /N(L)thresh≥5 S/N limit. As a result, some\nfaint and/or weakly polarized bursts may only have a\nfew points or, in some cases, no points plotted in their\nrespective profiles. Figure 2 shows the FDFs for the\nsame 16 FRBs as in Figure 1. In the case of FRB\n20181220A, we see two peaks in the FDF, with the best\nfit RM obsaligning with the lower peak. Here the peak\nat 0 rad m−2is due to instrumental polarization and,\nas mentioned in Section 2.2, the polarized fits for some\nevents are improved by adopting the RM obsvalues of\ncomparable secondary peaks in the FDF. The full set of\nsummary plots, FDFs, and cable-delay corrected Stokes\nQ,U, and Vwaterfall data are made available online.5\nOne example of the Stokes I,Q, and Uwaterfall plots\nfor an unpolarized FRB (FRB 20190502A) is shown in\n5A link to the baseband catalog data release will be added upon\nacceptance.10\nFigure 3. While the FRB is clearly visible in Stokes I,\nthere does not appear to be much corresponding signal\nin the Stokes QorUwaterfalls. The upper limit on thetime and frequency averaged linear polarization fraction\nof this FRB is L/I≤0.12.\nTable 1 . A summary of polarization properties analyzed in this work for the 89 well-fit FRBs and 29 unpolarized FRBs.\nThe Transient Name Server (TNS; https://www.wis-tns.org/) names for each FRB are listed in column 1. Column 2 provides\nthe downsampling factor ndown applied to the de-dispersed data; this factor determines the time resolution of the waterfall\nplots in Figure 1 such that the time resolution is ndown×2.56µs. The average linear polarization fraction across the burst\nenvelope and emitting band, L/I, is presented in column 3. For the unpolarized bursts, we provide an upper limit on L/I\nderived as the fraction required to produce a 6 σRM obsdetection in the FDF. Column 4 provides the structure-optimized\ndispersion measure, DM obs,struct , used in this work (based on work by Mckinven et al. 2023a) and the foreground Galactic DM\nestimate, DM MW= DM disk+ DM halo, is given in columns 5 (with the MW disk component from YMW16 and assuming a\nconstant MW halo contribution of 30 pc cm−3; Dolag et al. 2015; Yamasaki & Totani 2020; Cook et al. 2023). The RM derived\nwith RM-synthesis RM obs,FDFand QU-fitting RM obs,QUare presented in columns 6 and 7, respectively. The foreground MW\ncontributions to the RM obs, RM MW= RM disk+RM halotowards each FRB are estimated in column 8 (following Hutschenreuter\net al. 2022). The reduced-chi squared statistic from fitting to a constant PA, χ2\nν, (see Section 2.3) is given in column 9. In\nSection 3.7, we characterize the depolarization by taking the fraction between the L/Iin the bottom half and top half of the\nCHIME/FRB frequency band, ( L/I)500and ( L/I)700, respectively. The “depolarization ratio” fdepol = (L/I)700/(L/I)500(see\nSection 3.7.1) is presented in column 10. Where available, the uncertainties on the parameters are provided in parentheses\nindicating the error margin on the last significant figure listed (e.g., L/I= 0.90(1) is equivalent to L/I= 0.90±0.01). The 10\nFRBs that are contaminated by instrumental polarization are not tabulated as we cannot be fully confident in their polarization\noutputs. These polarization results are also included in the online CHIME/FRB baseband catalog data release.\nTNS Name ndown L/I DM obs,struct DM MW RM obs,FDF RM obs,QU RM MW PAχ2\nνfdepol\n(pc cm−3) (pc cm−3) (rad m−2) (rad m−2) (rad m−2)\nPolarized FRBs\nFRB 20181209A 1 0.90(1) 328.59(1) 47 −110.28(5) 106.81(1) −21(8) 1.79 –\nFRB 20181213A 4 1.09(3) 678.69(1) 44 10.20(9) 10.2(1) −12(6) 1.92 0.88(6)\nFRB 20181214C 32 0.60(3) 632.832(3) 41 23.6(2) 23.8(2) 6(2) 2.07 1.8(2)\nFRB 20181215B 1 1.017(9) 494.044(6) 43 8.59(2) 8.18(3) 14(4) 4.43 1.06(2)\nFRB 20181220A†8 0.43(2) 209.525(8) 46 97.5(4) −0.64(7) −23(11) 4.09 –\nFRB 20181221A 32 0.42(2) 316.25(5) 42 39.0(4) −44.3(4) 7(3) 10.56 –\nFRB 20181222E 32 0.70(3) 327.989(4) 45 −91.8(1) −92.9(1) −11(8) 1.50 –\nFRB 20181224E 8 0.62(2) 581.84(1) 44 1.65(8) 0.67(9) 8(6) 4.70 –\nFRB 20181226D 1 1.00(1) 385.338(5) 66 64.63(5) 64.73(5) 25(8) 1.49 1.09(4)\nFRB 20181226E 16 0.57(2) 308.78(1) 45 −1.0(2) −0.2(2) 0(15) 1.42 –\nFRB 20181228B 256 0.58(5) 568.538(6) 43 −0.1(5) −14.1(9) −1(7) 1.49 –\nFRB 20181231B 2 0.78(1) 197.366(9) 45 −9.66(5) 10.51(1) −16(3) 1.32 –\nFRB 20190102A 16 0.74(1) 699.1(4) 44 198.8(2) −198.40(4) −52(8) 1.36 –\nFRB 20190102B 32 0.86(3) 367.07(4) 43 −13.4(1) −13.31(3) −41(9) 0.76 1.10(8)\nFRB 20190106B 1 0.910(9) 316.536(2) 46 −73.18(3) 72.99(3) −133(36) 11.54 –\nFRB 20190110A 1 0.92(1) 472.788(3) 54 −31.14(5) −30.74(6) −36(23) 3.54 –\nFRB 20190110C 32 0.96(6) 222.01(1) 42 118.5(3) 118.4(3) 8(2) 2.15 –\nFRB 20190111B 2 0.63(1) 1336.87(1) 46 332.0(1) 331.9(1) 37(15) 2.45 –\nFRB 20190118A†1 0.453(4) 225.108(5) 46 85.99(3) −0.670(1) 41(13) 2.48 –\nFRB 20190121A 32 0.97(1) 425.28(3) 46 95.47(5) 95.41(6) 10(15) 2.48 1.01(3)\nFRB 20190122C 4 1.07(1) 690.032(8) 41 50.64(2) 50.71(2) 2(3) 3.73 1.07(2)\nFRB 20190124F 2 0.72(1) 254.799(4) 42 5.00(8) 4.62(9) −4(11) 3.73 –\nFRB 20190130B 4 1.01(2) 988.75(1) 41 75.54(6) 77.54(3) 9(7) 1.72 –\nFRB 20190131E†2 0.38(1) 279.798(6) 43 174.3(1) −0.56(1) −29(12) 1.25 –\nFRB 20190201B 32 0.77(3) 749.07(2) 48 −157.2(2) −156.62(9) −3(5) 3.99 –\nFRB 20190202B 2 0.31(1) 464.839(4) 66 −572.0(3) 571.3(2) 2(4) 20.42 –\nContinued on next pagePolarization properties of CHIME/FRB non-repeaters 11\nTable 1 – continued from previous page\nTNS Name ndown L/I DM obs,struct DM MW RM obs,FDF RM obs,QU RM MW PAχ2\nνfdepol\n(pc cm−3) (pc cm−3) (rad m−2) (rad m−2) (rad m−2)\nFRB 20190203A 16 0.64(2) 420.586(6) 44 −341.6(1) −337.8(1) −16(4) 2.39 –\nFRB 20190204B 64 0.85(5) 1464.842(6) 44 391.5(3) 391.7(3) −23(16) 2.67 –\nFRB 20190206A†64 0.57(2) 188.353(3) 45 −11.8(1) −0.30(6) 13(7) 3.05 –\nFRB 20190208C 4 0.73(3) 238.323(5) 44 −77.6(1) −80.7(1) −15(7) 0.88 –\nFRB 20190210B 4 0.81(1) 624.24(1) 76 −359.3(1) −359.42(9) 32(17) 25.44 –\nFRB 20190212B 4 0.73(2) 600.185(3) 43 175.8(2) −175.7(2) −10(3) 2.17 –\nFRB 20190213D 256 0.89(4) 1346.7(4) 47 −319.4(4) 318.3(5) −126(33) 3.33 –\nFRB 20190214C 128 0.84(3) 532.96(1) 42 1169.0(1) −1169.6(1) 8(5) 6.10 –\nFRB 20190217A 256 0.94(6) 798.14(4) 62 594.0(2) 595.1(3) 6(11) 2.83 0.8(1)\nFRB 20190224C 128 0.54(2) 497.12(2) 63 0.6(3) 0.8(3) 23(9) 6.42 –\nFRB 20190224D 2 0.81(2) 752.892(6) 44 −42.49(9) −41.86(9) −16(8) 24.68 1.04(8)\nFRB 20190226A 4 0.65(2) 601.546(7) 55 234.4(2) −234.1(1) 23(21) 2.48 0.96(7)\nFRB 20190303B 1 0.624(3) 193.429(5) 44 31.08(1) 31.074(3) −23(3) 3.55 –\nFRB 20190304A 8 0.57(2) 483.521(8) 44 −78.0(1) 78.0(1) −12(5) 1.83 –\nFRB 20190304B 64 1.03(5) 469.90(2) 41 39.5(2) −39.0(1) 5(1) 1.12 –\nFRB 20190320B 1 0.98(1) 489.501(8) 42 53.34(6) 53.36(6) 21(3) 1.91 0.99(3)\nFRB 20190320E 32 0.95(3) 299.09(2) 44 −75.1(1) 75.74(1) −17(7) 1.63 0.85(6)\nFRB 20190323B 1 0.911(5) 789.527(7) 43 229.46(2) 229.058(3) −15(8) 2.57 0.94(1)\nFRB 20190327A 8 0.48(1) 346.579(7) 44 11.2(1) 11.2(1) 61(19) 3.03 1.18(7)\nFRB 20190405B 64 0.46(3) 1113.72(7) 44 1.8(3) 2.0(3) −22(8) 5.90 –\nFRB 20190411C 1 0.804(4) 233.714(8) 43 −21.44(2) −20.42(2) −23(2) 149.03 –\nFRB 20190412A 16 0.74(2) 364.55(1) 43 −168.7(2) 167.58(7) −13(7) 10.82 1.0(1)\nFRB 20190417C 1 0.916(1) 320.266(4) 47 475.373(4) −475.4005(2) 5(11) 16.88 –\nFRB 20190419B∗∗8 0.23(1) 165.13(1) 44 4.1(3) −10.63(8) −31(10) – –\nFRB 20190423A 1 0.949(2) 242.600(8) 41 23.097(6) −22.933(7) 9(2) 8.37 –\nFRB 20190425A 1 0.949(3) 128.14(1) 44 57.278(4) 57.043(2) 49(15) 10.78 1.026(5)\nFRB 20190427A 32 0.61(3) 455.78(1) 72 −523.8(2) −524.1(3) 79(31) 1.61 1.0(1)\nFRB 20190430C‡16 1.04(2) 400.3(3) 45 −76.1(9) −70.4(5) 94(27) 1.17 –\nFRB 20190501B 16 0.88(2) 783.967(4) 43 −121.09(8) 121.35(9) 12(6) 3.80 –\nFRB 20190502B 256 0.50(3) 918.6(2) 42 126.7(7) 126.3(7) 50(8) 2.75 –\nFRB 20190502C 2 0.60(2) 396.878(9) 44 −35.5(2) 36.5(1) −13(5) 4.52 –\nFRB 20190519E 8 1.00(6) 693.622(7) 37 −17.0(3) −17.0(4) 20(5) 2.86 –\nFRB 20190519H 1 0.954(5) 1170.878(6) 45 −24.84(3) −25.03(3) −17(10) 3.32 –\nFRB 20190604G 16 0.33(1) 232.998(7) 47 364.4(3) 364.6(4) −8(4) 9.83 1.2(2)\nFRB 20190605C†1 0.319(7) 187.713(5) 43 −64.68(9) −0.10(4) −2(7) 4.28 –\nFRB 20190606B 128 0.86(4) 277.67(3) 44 16.5(1) 16.7(2) −28(10) 1.04 0.95(9)\nFRB 20190609A 8 0.86(3) 316.684(3) 45 42.4(1) −44.2(2) −16(11) 1.23 –\nFRB 20190609B 2 0.300(8) 292.174(7) 44 28.1(1) 29.7(1) −26(10) 4.18 –\nFRB 20190609C 2 1.04(6) 479.852(5) 66 −5.4(3) 5.1(3) −33(13) 4.14 –\nFRB 20190609D 64 0.91(5) 511.56(2) 51 −50.4(2) −48.6(2) 3(6) 4.36 –\nFRB 20190612A∗256 0.78(4) 433.14 43 −16.9(2) −18.7(5) −19(6) 1.09 –\nFRB 20190612B 1 0.92(2) 187.524(7) 43 2.05(6) −0.88(8) 5(6) 1.86 –\nFRB 20190613B 1 0.910(9) 285.088(5) 56 −22.18(3) −22.36(3) −11(18) 2.99 –\nFRB 20190614A 32 0.40(2) 1063.917(6) 44 −1.0(2) −1.7(2) −22(9) 1.37 –\nFRB 20190617A 1 0.869(2) 195.749(6) 44 −93.089(7) 94.068(1) −15(7) 48.31 –\nFRB 20190617B 32 0.45(1) 272.73(7) 58 1.7(2) 2.9(1) 45(14) 7.33 –\nContinued on next page12\nTable 1 – continued from previous page\nTNS Name ndown L/I DM obs,struct DM MW RM obs,FDF RM obs,QU RM MW PAχ2\nνfdepol\n(pc cm−3) (pc cm−3) (rad m−2) (rad m−2) (rad m−2)\nFRB 20190618A 1 0.550(6) 228.920(6) 44 −61.09(5) 62.958(1) −75(12) 2.66 –\nFRB 20190619B 64 0.48(3) 270.549(3) 44 −43.2(3) −43.1(1) −22(11) 1.33 –\nFRB 20190619C 8 0.97(3) 488.072(3) 47 −226.9(2) 226.9(3) −39(7) 0.97 1.0(1)\nFRB 20190621C 1 0.61(1) 570.342(7) 42 −0.18(6) −2.06(7) 1(5) 7.12 –\nFRB 20190621D 32 0.48(2) 647.32(4) 44 760.0(2) −759.2(2) −46(13) 3.71 –\nFRB 20190623A 16 0.86(4) 1082.16(1) 45 162.5(2) −161.54(7) 108(19) 1.41 –\nFRB 20190624B 1 0.802(1) 213.947(8) 45 −16.527(2) 4.00503(6) 9(14) 233.93 –\nFRB 20190627A 64 1.02(8) 404.3(1) 42 −48.3(6) 49.4(7) −10(6) 0.69 –\nFRB 20190627C 8 0.93(2) 968.50(1) 44 69.09(6) 68.47(6) 12(20) 1.40 1.09(4)\nFRB 20190628A 64 0.93(6) 745.790(8) 41 23.6(3) 23.7(3) 7(1) 1.23 0.9(1)\nFRB 20190628B 128 0.80(5) 407.99(2) 44 19.2(3) 19.7(3) −23(13) 3.91 –\nFRB 20190630B 64 0.98(1) 651.7(3) 46 6.65(4) 7.00(5) −205(102) 3.90 –\nFRB 20190630C 32 0.47(2) 1660.21(1) 45 641.0(4) 641.7(2) −8(8) 2.04 –\nFRB 20190630D 16 0.59(3) 323.540(3) 48 5.1(2) 4.8(2) 12(8) 12.94 –\nFRB 20190701A 16 0.92(5) 637.091(9) 44 −154.5(2) 154.3(2) 16(11) 2.28 1.1(1)\nFRB 20190701B 8 0.67(3) 749.093(8) 45 −534.3(2) −533.5(5) 4(10) 1.66 –\nFRB 20190701D 64 0.75(2) 933.32(3) 46 −138.67(8) 137.01(4) −20(10) 1.51 –\nUnpolarized FRBs\nFRB 20181219C 128 <0.21(3) 647.68(4) 43 – – −9(2) – –\nFRB 20181223C 32 <0.14(3) 112.45(1) 40 – – 7(6) – –\nFRB 20181229A 64 <0.11(3) 955.45(2) 44 – – 5(1) – –\nFRB 20181231A 256 <0.7(1) 1376.9(3) 44 – – −33(6) – –\nFRB 20181231C 128 <0.23(5) 556.03(2) 42 – – 11(5) – –\nFRB 20190103C 128 <0.14(2) 1349.3(1) 75 – – 33(22) – –\nFRB 20190115B 64 <0.10(4) 748.18(3) 45 – – −13(8) – –\nFRB 20190227A 8 <0.136(8) 394.031(8) 50 – – 24(11) – –\nFRB 20190320A 256 <0.16(4) 614.2(1) 49 – – −29(19) – –\nFRB 20190411B 256 <0.23(4) 1229.417(7) 40 – – 34(8) – –\nFRB 20190418A 64 <0.28(5) 184.473(3) 63 – – −40(24) – –\nFRB 20190423D 256 <0.20(4) 496(1) 45 – – −10(8) – –\nFRB 20190425B 4 <0.26(1) 1031.63(1) 44 – – −25(9) – –\nFRB 20190502A 16 <0.12(1) 625.74(1) 41 – – 4(2) – –\nFRB 20190518C 8 <0.14(2) 443.964(6) 45 – – 11(5) – –\nFRB 20190607B 128 <0.37(7) 289.331(2) 49 – – −1(12) – –\nFRB 20190608A 32 <0.21(4) 722.14(1) 43 – – −32(6) – –\nFRB 20190613A 128 <0.29(5) 714.71(3) 45 – – 49(14) – –\nFRB 20190614C 256 <0.26(7) 589.1(1) 45 – – −75(10) – –\nFRB 20190616A 8 <0.15(2) 212.511(5) 42 – – 6(2) – –\nFRB 20190617C 256 <0.6(7) 638.90(2) 47 – – 15(4) – –\nFRB 20190619A 32 <0.10(3) 899.82(1) 42 – – −12(3) – –\nFRB 20190621B 256 <0.22(7) 1061.14(2) 41 – – 16(4) – –\nFRB 20190622A 32 <0.28(6) 1122.807(9) 44 – – −20(9) – –\nFRB 20190623C 256 <0.25(8) 1049.94(1) 44 – – −17(8) – –\nFRB 20190624A 128 <0.23(4) 973.9(1) 42 – – −13(5) – –\nFRB 20190627D 256 <0.6(1) 2000.31(3) 45 – – −59(40) – –\nFRB 20190628C 256 <0.6(1) 1746.8(3) 46 – – 5(13) – –\nContinued on next pagePolarization properties of CHIME/FRB non-repeaters 13\nTable 1 – continued from previous page\nTNS Name ndown L/I DM obs,struct DM MW RM obs,FDF RM obs,QU RM MW PAχ2\nνfdepol\n(pc cm−3) (pc cm−3) (rad m−2) (rad m−2) (rad m−2)\nFRB 20190701C 64 <0.43(5) 973.79(1) 45 – – −21(10) – –\n∗DM was obtained by maximizing the S/N of the burst as a large fraction of the signal falls outside the time range of the\nsaved baseband data.\n∗∗Not enough points on the PA curve exceed the S /N(L)thresh = 5 requirement to derive a χ2\nνfit.\n†Corrected for instrumental polarization by re-fitting the RM obs,FDFto a secondary peak.\n‡A manual frequency mask was applied due to a few frequency channels containing severe radio frequency interference\nthat were not automatically flagged during raw data processing.\n3.2. Linear polarization\nWe plot a histogram of the L/Idistribution for the\n84 FRBs with well fit polarization properties (removing\nthe five FRBs that are corrected for instrumental polar-\nization) in the top left panel of Figure 4 in red and over-\nlay a Gaussian kernel density estimate (KDE). The L/I\ndistribution is skewed towards L/I= 1.0, with a mean\nL/Iof 0.764 (plotted as a dashed line in Figure 4) and\na median of 0 .805 (plotted as a dash-dotted line). For\nthe 29 FRBs with no significant polarized detection, we\nplace upper limits on their L/Igiven our 6 σdetection\nthreshold and plot them as a grey histogram. The mean\nand median L/I, after accounting for the unpolarized\nFRB upper limits, are L/I= 0.633 and L/I= 0.647,\nrespectively.\nThe normalized L/I distribution CDF is computed\nusing the Kaplan-Meier method and is presented in the\ntop right panel of Figure 4 as a red curve. This CDF in-\ncludes both the polarized FRBs (with a measured L/I)\nand the unpolarized FRBs (with a upper limit on L/I)\nand the 95% confidence interval (CI) is represented by\nthe red shaded region. For each repeating FRB in our\nsample, we take the median L/Iacross all their polar-\nized bursts and overlay a normalized CDF of the overall\nrepeating FRB L/Idistribution as a blue curve. The\nfull range of L/Ivalues for each repeater is depicted by\nthe blue shaded region. The mean and median of the\nrepeater L/Idistribution are 0 .681 and 0 .728, respec-\ntively (see Table 2). Note that we have not accounted\nfor individual bursts from repeating sources that are un-\npolarized, as they have not always been reported, and\nthe mean/median repeater L/Imay be slightly lower if\nwe were able to fully account for these bursts.\nWe apply the AD and KS tests to the polarized re-\npeating and non-repeating L/Idistributions to discern\nwhether they arise from the same underlying popula-\ntion. The summary statistics and associated p-values\nfrom these tests are listed in Table 3. We find no con-clusive evidence that the two polarized L/Idistributions\nare statistically different using the AD and KS tests.\nTaking into account the unpolarized non-repeaters, the\nPeto & Peto and Log-Rank tests return pPP= 0.962\nandpLR= 0.769, which suggests that there is no evi-\ndence for a dichotomy in repeater and non-repeater L/I\ndistributions.\n3.3. Rotation measure and dispersion measure\nIn the second and third row of the left columns of Fig-\nure 4, we plot the foreground subtracted |RMEG|and\nDM EGdistributions of our 89 non-repeating FRBs as a\nhistogram and overlay a Gaussian KDE. Note that we\nopt to analyze the |RMEG|here, as opposed to RM EG,\nbecause the sign of the RM is determined by the pro-\njected LoS direction of B∥, which we do not expect to\nhave any preferred orientation with the observer and,\ntherefore, does not provide any further insight regard-\ning the host environment for non-repeating FRBs. Sep-\narately, we provide a histogram of the RM EGdistribu-\ntion for our 89 non-repeating FRBs in Figure 5. As\nexpected, the RM EGdistribution is approximately sym-\nmetric about 0 rad m−2. Both distributions appear to\nbe approximately log-normally distributed. The mean,\nmedian, and 16th−84thpercentile range (PR) of both\nthe|RMEG|and DM EGdistributions are tabulated in\nTable 2.\nThe normalized CDFs of the non-repeater |RMEG|\nand DM EGdistributions are shown as red curves in the\nsecond and third row of the right panels of Figure 4. The\nbursts with no significant Ldetections are not plotted.\nAnalogous to Section 3.3, we take the median |RMEG|\nand DM EGacross all available bursts of each repeater as\ntheir representative value for the normalized CDFs (blue\ncurves). The mean and median values of |RMEG|and\nDM EGfor both repeaters and non-repeaters are tabu-\nlated in Table 2. The AD and KS test statistics and\np-values for repeating versus non-repeating |RMEG|and\nDM EGdistributions are listed in Table 3.\nFor|RMEG|, we find p-values of pAD= 0.172 and\npKS= 0.343 with the AD and KS tests, respectively.14\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n0.5\n 0.0 0.5\nTime [ms]400500600700800Frequency [MHz]FRB 20181209A\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n2\n 0 2\nTime [ms]400500600700800Frequency [MHz]FRB 20181213A\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]FRB 20181214C\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n1\n 0 1\nTime [ms]400500600700800Frequency [MHz]FRB 20181215B\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n1.0\n 0.5\n 0.0 0.5 1.0\nTime [ms]400500600700800Frequency [MHz]FRB 20181220A\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]FRB 20181221A\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]FRB 20181222E\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n10\n 5\n 0 5 10\nTime [ms]400500600700800Frequency [MHz]FRB 20181224E\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n1.0\n 0.5\n 0.0 0.5 1.0\nTime [ms]400500600700800Frequency [MHz]FRB 20181226D\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n2\n 1\n 0 1 2\nTime [ms]400500600700800Frequency [MHz]FRB 20181226E\n90\n45\n04590PA [deg]\n0.00.51.0Pol fracL/I\n|V|/IIntensityI\nL\nV\n10\n 5\n 0 5 10\nTime [ms]400500600700800Frequency [MHz]FRB 20181228B\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n2\n 1\n 0 1 2\nTime [ms]400500600700800Frequency [MHz]FRB 20181231B\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n5\n 0 5\nTime [ms]400500600700800Frequency [MHz]FRB 20190102A\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n2\n 0 2\nTime [ms]400500600700800Frequency [MHz]FRB 20190102B\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n1\n 0 1\nTime [ms]400500600700800Frequency [MHz]FRB 20190106B\n90\n45\n04590PA [deg]\n0.00.51.0Pol frac\nL/I\n|V|/IIntensityI\nL\nV\n1.0\n 0.5\n 0.0 0.5 1.0\nTime [ms]400500600700800Frequency [MHz]FRB 20190110A\nFigure 1. Total intensity waterfalls (de-dispersed to the associated DM obs,struct ) and temporal profiles of Stokes I(black line),\nL=p\nQ2+U2(red line), V(blue line), linear polarization fraction ( L/I; red circles), circular polarization fraction ( |V|/I; blue\ncircles), and linear polarization position angle (PA; black circles) for the first 16 FRBs from Table 1. All points plotted on the\nL/I,|V|/I, and PA profiles exceed a LS/N limit of 5 (see Appendix B for details). Some faint, low linear polarization FRBs,\ntherefore, may have very few (or no) points on their profiles, for example FRB 20181228B. Channels masked out due to radio\nfrequency interference are highlighted by red streaks on the left-hand side of the total intensity waterfall plots. As some of the\ndata have been downsampled by a factor of ndown >1, the resolution of the flagged radio frequency interference channels and\nStokes images do not always match exactly.Polarization properties of CHIME/FRB non-repeaters 15\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0102030405060Pol Intensity [S/N]250\n 200\n 150\n 100\n 50\n 00204060FRB 20181209A\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n05101520Pol Intensity [S/N]100\n 50\n 0 50 100 15001020FRB 20181213A\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0246810Pol Intensity [S/N]100\n 50\n 0 50 100 15005FRB 20181214C\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n020406080100Pol Intensity [S/N]100\n 50\n 0 50 100 150050FRB 20181215B\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0.02.55.07.510.012.515.0Pol Intensity [S/N]50\n 0 50 100 150 2000510FRB 20181220A\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0246810Pol Intensity [S/N]100\n 50\n 0 50 100 1500510FRB 20181221A\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n051015Pol Intensity [S/N]200\n 150\n 100\n 50\n 0 50051015FRB 20181222E\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0510152025Pol Intensity [S/N]100\n 50\n 0 50 100 15001020FRB 20181224E\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n010203040Pol Intensity [S/N]50\n 0 50 100 150 200020FRB 20181226D\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0.02.55.07.510.012.515.0Pol Intensity [S/N]150\n 100\n 50\n 0 50 1000510FRB 20181226E\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0.02.55.07.510.012.5Pol Intensity [S/N]150\n 100\n 50\n 0 50 1000510FRB 20181228B\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n010203040Pol Intensity [S/N]150\n 100\n 50\n 0 50 10002040FRB 20181231B\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0102030Pol Intensity [S/N]50 100 150 200 250 3000102030FRB 20190102A\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n051015Pol Intensity [S/N]150\n 100\n 50\n 0 50 100051015FRB 20190102B\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n0204060Pol Intensity [S/N]200\n 150\n 100\n 50\n 0 500204060FRB 20190106B\n2000\n 1500\n 1000\n 500\n 0 500 1000 1500 2000\n[radm2]\n01020304050Pol Intensity [S/N]150\n 100\n 50\n 0 50 10002040FRB 20190110A\nFigure 2. FDFs for the same sample of 16 FRBs presented in Figure 1 are shown as blue curves. The x-axis ( ϕ, in units of\nrad m−2) refers to the Faraday depth, which is an extension of RM obsin the scenario where we have multiple components of\nFaraday rotation in the same polarized signal. A zoomed in inset is presented centered at the position of the best fit RM obs,\nwhich is marked as a green dashed line. In the case where the instrumental polarization is the dominant feature in the FDF, a\ncomparable secondary peak is fit to derive the true RM obs(e.g., as seen in FRB 20181220A).\nThere is no statistical evidence for a dichotomy between\nthe two populations with respect to |RMEG|. In regards\nto the DM EGdistributions, we find pAD= 0.064 and\npKS= 0.290, which does not suggest a difference be-\ntween the two DM EGdistributions. These results, how-\never, are in contrast with CHIME/FRB Collaboration\net al. (2023a), who find a difference in the extragalac-\ntic DM of the repeating and non-repeating FRB pop-\nulations at the ∼99% level with the AD test and a∼96% level with the KS test. The cause for such a\ndisparity in DM EGresults between this work and those\nby CHIME/FRB Collaboration et al. (2023a) is two-\nfold. First, the memory buffer on CHIME/FRB base-\nband data is ∼20 seconds, which results in loss of fre-\nquency channels and hence sensitivity for FRBs with\nDM obs≳1000 pc cm−3, leading to a bias in sensitiv-\nity against high DM events in baseband data (reported\non here) that does not exist in intensity-only data (in16\n6\n 4\n 2\n 0 2 4 6\nTime [ms]400500600700800Frequency [MHz]Stokes I\n6\n 4\n 2\n 0 2 4 6\nTime [ms]400500600700800Frequency [MHz]Stokes Q\n6\n 4\n 2\n 0 2 4 6\nTime [ms]400500600700800Frequency [MHz]Stokes U\nFigure 3. Stokes I(left), Q(middle), and U(right) waterfalls for FRB 20190502A, which is unpolarized with an upper limit\nofL/I≤0.12. As in Figure 1, the red streaks to the left of each waterfall plot highlight frequency channels that are masked out\ndue to radio frequency interference. Here, the data have been downsampled by a factor of ndown = 16 and thus the resolution\nof the flagged radio frequency interference channels and Stokes images do not match exactly.\nwhich the dichotomy was seen). Secondly, the sensitiv-\nities of the AD and KS tests scale with the number of\ndata points in the input distributions. The intensity\ndata used by CHIME/FRB Collaboration et al. (2023a)\nis a substantially larger sample (305 non-repeaters and\n40 repeaters) than that used in this work. Both these\nbiases lead to repeaters and non-repeaters having more\nsimilar DM EGdistributions in this work than those seen\nby CHIME/FRB Collaboration et al. (2023a).\nFigure 6 shows DM EGplotted against |RMEG|for the\nrepeating and non-repeating FRBs. We apply Spear-\nman’s rank correlation coefficient to the log10(|RMEG|)\n– log10(DM EG) relation in three distinct sets of data: (i)\nonly the non-repeaters, (ii) only the repeaters, and (iii)\nthe combined repeater and non-repeater set. We find\nno statistically significant linear correlation in the re-\npeater only data, but in the non-repeater and combined\ndata sets we find marginal evidence for a positive mono-\ntonic correlations with pSR= 0.006 and pSR= 0.010,\nrespectively. The lack of correlation in the repeater only\nsample may be partially caused by the smaller sample\nsize when compared to the non-repeater data. The test\nresults are summarized in Table 4.\n3.4. LoS magnetic field lower limits\nFollowing the steps described in Section 2.6, we derive\nobserver frame lower limits on the mean magnetic field\nstrength of the host galaxy environment parallel to the\nLoS,|β|. Conducting the same analysis that we have ap-\nplied to L/I,|RMEG|, and DM EG, we plot the histogram\nand KDE of the |β|distribution in the bottom left panel\nof Figure 4 and the respective CDFs of the repeating and\nnon-repeating populations in the bottom right panel.\nSimilar to |RMEG|, the|β|distribution seems to be log-normally distributed, but with a more prominent tail\nextending towards lower values. We present the mean\nand median |β|for repeaters and non-repeaters in Table\n2. Further, we see marginal evidence for a dichotomy in\nthe|β|distributions with pAD= 0.017 and pKS= 0.042,\nrespectively, with repeating FRBs having higher |β|on\naverage. While both of these tests have relatively small\np-values, we caution against the overinterpretation of\nthese results for a few reasons which are discussed in\nSection 4.2.\nIn Section 3.3, we detailed why our baseband sam-\nple may be biased against high DM obsevents which,\nbased on the results by CHIME/FRB Collaboration\net al. (2023a), would preferentially be composed of non-\nrepeating events. If indeed an unbiased baseband sam-\nple resulted in non-repeaters having, on average, higher\nDM obsbut the same |RMobs|, then their |β|would be\nlower and the dichotomy in the |β|distributions would\nbe even more significant.\n3.5. FRB host rotation measure and LoS magnetic\nfield strength estimates\nWhile we do not have redshift information for indi-\nvidual FRBs, we attempt to apply a statistical correc-\ntion and derive |RMhost|and\f\f\nB∥,host\u000b\f\fdistributions\nfor our non-repeating sample. To do this, we draw\na DM hostdistribution matching the size of our polar-\nized non-repeating sample from a log-normal distribu-\ntion with a mean (1 .93/log10(e)) and standard devia-\ntion (0 .41/log10(e)) following Shin et al. (2023). Sub-\ntracting this DM hostfrom our DM EG, we obtain a\nDM IGM(z) distribution that is then used to compute\nazdistribution by assuming z∼DM IGM(z)/1000\n(Macquart et al. 2020). With a zdistribution inPolarization properties of CHIME/FRB non-repeaters 17\nFigure 4. (Left) Distribution of L/I,|RM EG|, DM EG, and|β|for the non-repeating FRBs with a significant linearly polarized\ndetection (89 in total but 84 for L/Iafter removing the instrumental polarization corrected FRBs). In the top row, we also\ninclude the L/Ilower limit distribution of the 29 unpolarized FRBs in grey. The dashed and dash-dotted black lines in each\npanel correspond to the respective mean and median values and a KDE of each distribution is overplotted. In the top left panel,\nthe solid and dotted lines represent the mean and median, respectively, when accounting for the 29 unpolarized non-repeating\nFRB L/I lower limits. (Right) The normalized L/I,|RM EG|, DM EG, and |β|CDFs for the repeating (solid blue line) and\nnon-repeating (solid red line) FRBs. For the non-repeating sources in the top right panel, we compute the CDF using the\nKaplan-Meier method and the 95% CI is shown as the shaded red region. For the repeating FRB sources in the top right panel,\nthe median value across all bursts is used and the full range of L/Iis encompassed in the shaded blue region.18\nFigure 5. The distribution of RM EGfor the 89 polarized\nnon-repeating FRBs in our sample. The mean and median\nvalues are presented as vertical dashed and dash-dotted lines,\nrespectively, and a Gaussian kernel density estimate is over-\nplotted as a red curve. The distribution appears to be sym-\nmetric about 0 rad m−2.\nTable 2. Mean, median, and the 16th−84thpercentile range\nof the L/I(only polarized FRBs), L/I(including unpolar-\nized upper limits), |RM EG|, DM EG, and|β|distributions for\nboth our non-repeating and repeating FRB samples.\nParameter Mean Median 16th−84thPR\nNon-repeating FRBs\nL/I 0.764 0 .805 [0 .545,0.967]\nL/I(with upper limits) 0.633 0.647 [ <0.234,0.949]\n|RM EG|(rad m−2) 135 .4 53 .8 [11 .2,241.9]\nDM EG(pc cm−3) 479 .1 389 .3 [191 .0,734.3]\n|β|(µG) 0 .386 0 .188 [0 .037,0.594]\nRepeating FRBs\nL/I 0.681 0 .728 [0 .240,0.939]\n|RM EG|(rad m−2) 541 .7 57 .2 [11 .2,592.8]\nDM EG(pc cm−3) 355 .8 301 .3 [135 .8,512.7]\n|β|(µG) 1 .539 0 .365 [0 .061,3.523]\nhand, we obtain |RMhost|=|RMEG|(1 + z)2and then\f\f\nB∥,host\u000b\f\f= 1.232|RMhost|/DM host. We repeat this\nprocess over 1 ,000 trials and derive the mean |RMhost|\nand\f\f\nB∥,host\u000b\f\fdistributions. We reinforce that we do\nnot apply this type of zcorrection to individual FRBs\nand only use it as a statistical correction to make con-\nclusions on the polarized, non-repeating population as a\nwhole.\nIn Figure 7, we plot the mean |RMhost|and\f\f\nB∥,host\u000b\f\f\ndistributions derived from this approach and contrast\nthem to the observer frame |RMEG|and|β|distribu-\ntions. As expected, both the |RMhost|and\f\f\nB∥,host\u000b\f\fTable 3. Summary of AD, KS, Peto & Peto, and Log-\nRank tests on whether the L/I,|RM EG|, DM EG, and|β|of\nnon-repeating and repeating FRBs are drawn from the same\nunderlying distribution. Recall that, without host galaxy\nredshifts, the values of |β|have an uncorrected dependence\non the distance of FRB sources. Both the test statistic and\nassociated significance level are reported for each test. The\nAD and KS tests are applied to only the 84 polarized FRBs\nin our sample while the Peto & Peto and Log-Rank tests\nadditionally factor in the L/I upper limits for the 29 un-\npolarized non-repeaters. The p-values that are <0.05 are\npresented in bold face.\nAD Test KS Test\nParameter SAD pAD SKS pKS\nL/I 0.179 ≥0.250 0.220 0.560\n|RM EG|(rad m−2) 0.687 0.172 0.263 0.343\nDM EG(pc cm−3) 1.71 0.064 0.277 0.290\n|β|(µG) 3.18 0.017 0.394 0.042\nPeto & Peto Log-Rank\nParameter pPP pLR\nL/I(with upper limits) 0.962 0.768\nFigure 6. The foreground subtracted |RM EG|plotted as\na function of the foreground subtracted DM, DM EGfor our\nsample of non-repeating FRBs (red circles) and repeating\nFRBs (blue squares). For the non-repeating FRBs, the mea-\nsurement uncertainties are plotted as black error bars and are\ntypically smaller than the markers. The shaded blue regions\nrepresent the intrinsic spread in the observed parameters over\ntime, which dominate the measurement uncertainties for re-\npeating sources.Polarization properties of CHIME/FRB non-repeaters 19\ndistributions are shifted to higher values compared\nto the observer frame |RMEG|and|β|distributions.\nThe mean and median of the |RMhost|distribution is\n302.7 rad m−2and 75 .9 rad m−2, respectively, with a\n16th−84thPR of [15 .0,494.3] rad m−2. The mean\nand median of the\f\f\nB∥,host\u000b\f\fdistribution is 4 .62µG\nand 0 .740µG, respectively, with a 16th−84thPR of\n[0.088,5.73]µG.\n3.6. PA variability\nIn Section 2.3, we describe a methodology to quan-\ntify the magnitude of PA variability for an FRB as a\nfunction of time using a reduced χ2\nνtest. We apply this\ntechnique to our sample of 89 non-repeating FRBs. A\nhistogram and KDE of the resulting reduced χ2\nνvalues\nare presented in Figure 8 and are included in Table 1.\nNote that one FRB in our sample (FRB 20190419B) did\nnot have a sufficient number of points on its PA curve\nto produce a reasonable χ2\nνfit due to a combination of\nlow S/N and low linear polarization. For this FRB, we\ndo not report a χ2\nνvalue and it is excluded from the cat-\negorization below. The mean and median χ2\nνare 9 .01\nand 2 .71, respectively, for the 88 non-repeating FRBs\nwith PA χ2\nνfits. We define a conservative PA variabil-\nity threshold of χ2\nν≥5, above which we consider the\nPA profile to be variable. We present four illustrative\nPA profiles in Figure 9 corresponding to four qualitative\narchetypes for PA behavior that we define below (in con-\nsonance with the classification system deployed by Sher-\nman et al. 2023a). In these archetypes, we also classify\nbursts based on whether they are single component or\nmulti-component (determined through visual inspection\nafter de-dispersing to the respective DM obs,struct).\n1.Single component, constant PA. This category en-\ncompasses all single-component FRBs in our non-\nrepeating sample that have χ2\nν<5 across their\nPA profiles. An example of this type of PA behav-\nior is presented in the first panel of Figure 9 (FRB\n20181226E), where the PA remains constant across\nthe∼1 ms burst duration. A single-component,\nconstant PA behavior is the most common out\nof the four qualitative archetypes we define, with\n50/88 FRBs falling under this categorization.\n2.Single component, variable PA. This second cat-\negory includes all single-component bursts with\nχ2\nν≥5. These FRBs display PA variations that\nare continuous as a function of time across a sin-\ngle burst. The second panel of Figure 9 shows one\nsuch example (FRB 20190425A) wherein the PA\nrises∼25 deg in 0 .35 ms. Most FRBs under this\numbrella are similar to FRB 20190425A in thatthey have modest PA variations and do not show,\nfor example, the large S-shaped swings typically\nseen in pulsar emission (Lorimer & Kramer 2012)\nand seen infrequently in FRBs (e.g., see Mckin-\nven et al. in preparation). This second category\nconstitutes only 9/88 FRBs in our sample.\n3.Multiple component, constant PA. Of the 29 multi-\ncomponent bursts, 19 display a constant PA, with\nχ2\nν<5, across their entire burst envelope. One\nexample of this behavior is shown in Figure 9; in\nthe third panel we see a constant PA across 3+\ncomponents spanning ∼1.7 ms (FRB 20190320B).\n4.Multiple components, variable PA. The final cate-\ngory describes all multi-component bursts whose\nPAs vary component-to-component, leading to\nχ2\nν≥5. This classification is distinct from the\nsingle component, variable PA category as the PA\nprofile, in this case, may sometimes remain con-\nstant within each component but the PA between\none or more components is variable and we do not\nsee a smooth, continuous change in the PA. For\nthe FRBs in this archetype it is possible that ei-\nther: (i) the PA variations are discontinuous be-\ntween components or (ii) the PA variation is con-\ntinuous but the emission bridging components is\ntoo faint to accurately measure PAs in that time\nrange. The fourth panel of Figure 9 shows the PA\nof FRB 20190224D which appears to have 3 −4\ncomponents, each with widths of 0 .05−0.25 ms.\nHere, the PA of the first component differs from\nthe rest by ∼45 deg, while the intra-component\nPA for each of them remains somewhat constant.\nIn total, 10/88 FRBs fall into under this classifi-\ncation.\nWe compare the L/I,|RMEG|, and |β|of the FRBs\nconstituting the four archetypes described above but\nfind no evidence for any differentiation between the sub-\npopulations. Further, we find no correlation or anti-\ncorrelation between the PA χ2\nνvalues and L/I and\n|RMEG|.\n3.6.1. Rotating vector model\nLike FRBs, pulsars display a rich phenomenology of\nbehavior in their PA evolution. However, the PA behav-\nior of pulsars is markedly different to that of FRBs; FRB\nPA curves tend to be flatter than those encountered in\ntime-averaged profiles of pulsars and excursions in the\nPA, when present, tend to be more erratic in FRBs. For\npulsars, a geometric model known as the rotating vector\nmodel (RVM; Radhakrishnan & Cooke 1969) is often in-\nvoked to explain the smooth swing in the PA over pulse20\nFigure 7. (Left) Mean |RM host|distribution (solid red line) of our non-repeating FRB sample after applying a redshift correction\nbased on the DM hostdistribution derived by Shin et al. (2023). The mean and median of the distribution are plotted as black\ndashed and dash-dotted lines, respectively. All 1000 individual trials, each with an independently drawn DM hostdistribution,\nare shown as faint red histograms in the background. The observer frame |RM EG|distribution with no redshift correction is\noverplotted (solid grey line) with its respective mean and median shown as grey dashed and dash-dotted lines, respectively.\n(Right) An analogous setup to the left panel but for comparing the rest frame\f\f\nB∥,host\u000b\f\fand observer frame lower limits, |β|.\nFigure 8. The distribution of χ2\nνvalues from comparing the\ntemporal PA variations of all 88 non-repeating FRBs against\na flat PA profile. The mean and median values are presented\nas vertical dashed and dash-dotted lines, respectively, and\na Gaussian kernel density estimate is overplotted as a red\ncurve. FRBs with multiple components are highlighted as a\nhatched histogram.\nphase as a projection effect of the neutron star’s rotating\ndipolar field. Over the years, polarimetric studies of pul-\nsars have demonstrated the RVM to be a powerful toolwith approximately ∼60% of the population displaying\nPA behavior that is well described by the RVM (e.g.,\nJohnston et al. 2023a). The RVM provides information\non important geometrical parameters, such as the incli-\nnation angle between the neutron star’s magnetic axis\nand rotation axis ( α) and the impact angle between the\nLoS and the magnetic axis ( γ; often referred to as β\nelsewhere).\nWhile growing evidence does exist for magnetospheric\norigins of some FRBs (e.g., Luo et al. 2020; Nimmo et al.\n2022b), the suitability of a geometric interpretation of\nthe RVM remains an open question. If the non-repeating\nFRB sample reported here does arise within the magne-\ntosphere of rotating neutrons stars then the substantial\ndifferences in PA behavior of FRBs and pulsars sug-\ngests that, at the very least, FRB emission occurs at\nvery different regions of the magnetosphere than that of\nthe pulsar sample. In an attempt to study this more\nsystematically, we fit the RVM to a subsample that dis-\nplays significant PA variations ( n= 19, i.e., the single\ncomponent, variable PA & multiple component variable\nPA archetypes from Section 3.6) and find best-fit re-\nduced χ2\nνvalues that are regularly between 2 −3. PA\ncurves and their associated best-fit RVMs are displayed\nin Figure 10 for a selection of this subsample with thePolarization properties of CHIME/FRB non-repeaters 21\nFigure 9. PA profiles of four FRBs in our sample (from top to bottom: FRB 20181226E, FRB 20190425A, FRB 20190320B, and\nFRB 20190224D) with distinct temporal behaviors exemplifying the archetypes presented in Section 3.6: (i) single component\nwith a constant PA, (ii) single component with a variable PA, (iii) multiple components with a constant PA, and (iv) multiple\ncomponents with a variable PA. Measurement uncertainties are presented as black lines but are, most of the time, smaller than\nthe marker size. The χ2\nνvalue from fitting the PA profile to a flat PA at 0 deg is listed at the top left of each panel. The text\nin the bottom left of each panel describes the classification of the PA variability into one of these four archetypes. The number\nof FRBs belonging to each archetype is presented in the top right corner as a fraction of the total polarized sample. Note that\nthere is one FRB (FRB 20190419B) for which we cannot derive a χ2\nνvalue as it does not have a sufficient number of data points\nin its PA profile that exceed the S /N(L)thresh = 5 S/N threshold, so it is not included in any of the four archetypes described\nhere.22\nlowest RVM χ2\nνvalues. Importantly, without informa-\ntion on the period of the source (in the rest frame of\nthe source), many geometric configurations can give rise\nto the observed PA behavior. This ambiguity is cap-\ntured by the four columns of Figure 10, which display\nthe best-fitting RVMs at four different assumed spin pe-\nriods corresponding to duty cycle trials of 5, 35, 65 &\n95%. While the fit quality obtained for this sample is\nnot inconsistent with that attained from pulsars that are\nconsidered well described by the RVM (e.g., Wang et al.\n2023), we caution against concluding this as proof of an\nRVM-like scenario operating in FRBs. Indeed, the RVM\ncan replicate a wide variety of PA behavior and is thus\nsusceptible to spurious “fits” that may potentially mis-\nlead the interpretation of the physical mechanism pro-\nducing the observed PA evolution. Good quality (but\nlikely spurious) RVM fits can be easily obtained in cases\nwhere the PA evolution is modest/quasi-linear and it is\nthus unsurprising that many of the FRBs in our sample\nwith the lowest χ2\nνare those exhibiting these properties.\nIn the absence of further information, namely the spin\nperiod of the source and detection of repetition, we are\nunable to make any strong claims of whether PA evo-\nlution of this sample can be described by an RVM-like\nscenario and which set of geometric α, γparameters are\npreferred. A continuous “S”-shaped PA swing, common\nto pulsars and recently observed in at least one nearby\nCHIME detected FRB (Mckinven et al., in prep.), has\nnot been observed in this sample. However, if we as-\nsume an RVM-like scenario for our sample, then the\noverwhelming preponderance of flat PA curves of our\nnon-repeating sample does imply a preference for ge-\nometries where αclusters near α= 0◦orα= 180◦.\nThis indicates a much higher degree of alignment/anti-\nalignment of the neutron star’s magnetic and rotation\naxis than what is commonly inferred for the pulsar pop-\nulation. This scenario remains quite speculative with\nthe data in hand, but is discussed further in Section 4.4.\n3.7. Depolarization\nEvidence for frequency-dependent depolarization has\nbeen seen in some prolific repeating FRBs and it has\nbeen speculated to arise from scattering in the magne-\ntoionic medium surrounding the FRB source (Feng et al.\n2022). In this Section, we investigate whether frequency-\ndependent depolarization is present in our non-repeating\nFRB sample to better inform whether or not this spec-\ntral depolarization is a ubiquitous feature across the en-\ntire FRB population.\n3.7.1. Depolarization within the CHIME/FRB band\nFirst, we attempt to constrain any changes in the lin-\near polarization fraction within the CHIME/FRB banditself, in the observer frame. For this purpose, we select\na subset of FRBs that emit over the full 400 MHz band-\nwidth of CHIME/FRB; in total this condition is met\nfor 23 FRBs. Note that this subset does not necessar-\nily cover all FRBs from the first CHIME/FRB catalog\n(CHIME/FRB Collaboration et al. 2021) with 400 MHz\nbandwidth for which baseband data exist. This is be-\ncause of the limited baseband buffer ( ∼20 seconds),\nwhich can lead to missing frequency channels for some\nheavily dispersed events. In addition, during the pre-\ncommissioning stage (up to 2018 August 27), the uncer-\ntainties in DM and FRB time of arrival were not cor-\nrectly accounted for in the baseband system, leading to\npartial or full loss of baseband data for some events.\nFor the subset of 23 FRBs with 400 MHz band-\nwidth in the baseband data, we derive a depolariza-\ntion ratio fdepol between the band-averaged linear po-\nlarization fraction at 400 −600 MHz, ( L/I)500, and at\n600−800 MHz, ( L/I)700,\nfdepol =(L/I)500\n(L/I)700. (21)\nNote that, while we refer to fdepol as a “depolarization\nratio”, it is possible to observe fdepol >1 (i.e., in that\ncase the polarization fraction increases towards lower\nfrequencies). Characterizing the depolarization in this\nmanner affords a model-independent determination of\nany decrease in L/Iacross the CHIME frequency range.\nIn Table 1, we present fdepol for all 23 of the FRBs\noccupying the full 400 −800 MHz band.\nThe top panel of Figure 11 shows ( L/I)500plotted\nas a function of ( L/I)700for these FRBs. In this\nspace, fdepol = 1 corresponds to ( L/I)500= (L/I)700\nand points along that line are not undergoing measur-\nable spectral depolarization across the CHIME/FRB\nband. FRBs below this line are depolarizing, while\nthose above it are experiencing increasing L/Iwith de-\ncreasing frequency. We also compare against the ex-\npected distribution of ( L/I)500and ( L/I)700if the ob-\nserved FRB population was undergoing RM scattering\ndue to multi-path propagation, following Equation 4,\nwith σRM∈[0.1,10] rad m−2and assuming an intrin-\nsicL/I= 1.0 at the time of emission. Here, it is im-\nportant to note that we are only sensitive to a limited\nrange of σRMvalues with our data. Setting a min-\nimum difference of |(L/I)700−(L/I)500| ≥ 0.05 and\nrequiring that at least ( L/I)700≥0.2, such that it is\npossible to detect a polarized signal, we would be able\nto detect 0 .5≲σRM≲5.0 rad m−2. In the bottom\npanel of Figure 11, we display the histogram of fdepol\nvalues. Most FRBs in this subset are consistent with\nfdepol = 1, meaning that their L/I is constant overPolarization properties of CHIME/FRB non-repeaters 23\n0 10\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=5%2=1.7\n=177.7o\n=27.1o\n50\n 0 50\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=35%2=1.4\n=0.2o\n=26.9o\n100\n 0 100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=65%2=1.1\n=0.2o\n=46.2o\n0 200\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=95%2=1.2\n=0.2o\n=55.4o\nFRB20190412A\n0 10\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=5%2=1.9\n=179.1o\n=40.8o\n50\n 0 50\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=35%2=1.8\n=179.8o\n=48.2o\n100\n 0 100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=65%2=2.9\n=0.2o\n=89.6o\n100\n 0100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=95%2=4.5\n=0.2o\n=89.8o\nFRB20190106B\n0 10\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=5%2=2.5\n=176.0o\n=9.2o\n50\n 0 50\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=35%2=2.4\n=0.2o\n=7.8o\n100\n 0 100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=65%2=2.4\n=0.2o\n=16.2o\n0 200\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=95%2=5.0\n=179.8o\n=18.9o\nFRB20190210B\n0 10\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=5%2=2.7\n=94.4o\n=0.0o\n50\n 0 50\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=35%2=2.6\n=0.7o\n=3.4o\n100\n 0 100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=65%2=2.5\n=0.5o\n=4.5o\n100\n 0100\nLongitude in Degrees90\n45\n04590PPA in degreesduty cycle=95%2=2.6\n=0.7o\n=7.0o\nFRB20190423A\nFigure 10. PA curves (black points) and associated best-fit RVM models (red) for a series of different assumed duty cycles\n(5,35,65,95%) for a sample of non-repeating FRBs (FRBs 20190412A, 20190106B, 20190210B, and 20190423A) that display\nsignificant PA variation. Quality of fits are indicated by reduced chi-squared values ( χ2\nν; top left corner) along with best-fit α, γ\nparameters. These four FRBs were chosen as they have the smallest RVM χ2\nνvalues from the analyzed sample. All four FRBs\nare associated with the multiple component, variable PA archetype.24\nthe CHIME/FRB band. Out of the 23 sources, 21\nhave less than a 20% change in their linear polariza-\ntion fraction between 500 and 700 MHz (i.e., they have\n0.8< fdepol <1.2). There are two notable outliers: (i)\nFRB 20190217A with fdepol = 0.8±0.1 and (ii) FRB\n20181214C with fdepol = 1.8±0.2.\nOne FRB, for which we only have data between\n400−650 MHz, that was labelled unpolarized (FRB\n20190227A) has unique polarimetric properties that may\nbe indicative of depolarization. The leftmost panel of\nFigure 12 shows the Stokes Idynamic spectra of FRB\n20190227A, which is composed of at least three compo-\nnents over ∼4 ms, though it is ambiguous whether the\nthird component is comprised of one or two subcompo-\nnents. Two of these components span ∼400−650 MHz,\nwhile the other component is only visible over ∼600−\n650 MHz in the baseband data. This narrowband com-\nponent displays some faint emission in Stokes Q(middle\npanel of Figure 12) and in Stokes U(rightmost panel of\nFigure 12), while the other components appear unpo-\nlarized. Considering only the narrowband component,\nwe find a polarized detection with L/I = 0.50±0.01\nand RM = 62 .9±0.9 rad m−2. Over the two broad-\nband, unpolarized components we place upper limits on\ntheir L/I of≤0.219 and ≤0.190, respectively. We\nnote that in the first unpolarized component, there is a\n∼2−3σpeak in the FDF at the same RM as the polar-\nized component. The lack of emission below ∼600 MHz\nmakes it difficult to constrain the scattering timescale\nof the polarized component but the burst widths of the\nfirst two components appear comparable at ≳600 MHz.\nLooking at the same FRB in the intensity data from\nthe first CHIME/FRB catalog (CHIME/FRB Collabo-\nration et al. 2021) shows that the burst emission extends\nup to ∼800 MHz, suggesting that the bursts may be\npart of a downward-drifting envelope that extends from\nhigher frequencies. However, without high time resolu-\ntion baseband data over the entire CHIME/FRB band,\nwe cannot determine the precise morphology of the sub-\nbursts at >650 MHz.\n3.7.2. Comparing with FRBs at different observing\nfrequencies\nHaving not found any strong depolarization within the\nCHIME/FRB band, we endeavor to expand our search\nto a broader frequency range by comparing our results\nto published FRB populations observed with other in-\nstruments. Recently, Sherman et al. (2023a) report\npolarization properties for 25 non-repeating FRBs, 20\nof which have a significant RM detection. Their ob-\nservations were conducted using the 110-antenna Deep\nSynoptic Array (DSA-110) between 1 .28 and 1 .53 GHz,\nwith a center observing frequency of ∼1.4 GHz. We\nFigure 11. (Top) The linear polarization fraction averaged\nover 400 −600 MHz plotted against the average fraction\nover 600 −800 MHz for 23 FRBs that emit over the full\n400 MHz CHIME/FRB band. The black dashed line indi-\ncated a depolarization ratio of 1 (i.e., no change in the linear\npolarization fraction over the observed band). Two outliers\nare highlighted: FRB 20181214C with fdepol = 1.8±0.2 in\nred and FRB 20190217A which has fdepol = 0.8±0.1 in\nblue. The grey dot-dashed line is the expected distribution\nof (L/I)500versus ( L/I)700due to RM scattering from multi-\npath propagation for a range of σRMbetween 0 .1−10 rad m−2\n(with σRM= 0.1 rad m−2being at the top of the curve and\nσRM= 10 rad m−2towards the origin). (Bottom) A his-\ntogram of the depolarization ratio fdepol with a dashed line\nindicating fdepol = 1.Polarization properties of CHIME/FRB non-repeaters 25\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]Stokes I\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]Stokes Q\n4\n 2\n 0 2 4\nTime [ms]400500600700800Frequency [MHz]Stokes U\nFigure 12. Stokes I(left), Q(middle), and U(right) waterfalls for FRB 20190227A which is composed of at least three\ncomponents. Dashed white lines are used to highlight the approximate start of each burst component. The red streaks to the\nleft of each waterfall plot highlight frequency channels that are masked out due to radio frequency interference. Note that the\ndata have been downsampled by a factor of ndown = 8 and thus the resolution of the flagged radio frequency interference channels\nand Stokes images do not match exactly. The first and third component are unpolarized with L/I≤0.219 and L/I≤0.190,\nrespectively, while the second component is polarized with L/I= 0.50±0.01 and has RM = 62 .9±0.9 rad m−2.\nnote that their set of FRB sources is completely dis-\ntinct from the sample presented in this work (i.e., there\nare no co-detected sources between the two sets of\ndata). In Figure 13, we compare the L/ICDFs for our\nCHIME/FRB non-repeating sample with that of Sher-\nman et al. (2023a) (both computed using the Kaplan-\nMeier method) and find that the L/Idistribution ob-\nserved with the DSA-110 is in close agreement with the\none seen in our data. Applying AD, KS, Peto & Peto\nand log-rank tests to the two distribution shows that\nthey likely arise from the same underlying distribution\n(pAD≥0.25,pKS= 0.83,pPP= 0.96 and pLR= 0.77).\nThe similarity between the two populations suggests\nthat there is no clear systematic depolarization between\n400 MHz and 1 .53 GHz. The consistent distribution of\nL/Iacross 400 MHz and 1 .53 GHz also suggests that\nL/Iis not very sensitive to redshift effects. However, it\nis possible that with larger samples, a dichotomy might\none day emerge.\n3.7.3. Beam Depolarization\nOne method by which FRBs may incur frequency-\ndependent depolarization is via multi-path propagation\nthrough inhomogeneous magneto-ionic media. For most\nof our sample, we are limited to one band-averaged\nL/Imeasurement across the emitting frequency range\nof each FRB. Therefore, the only way to uniformly\ncompute σRMfor our sample would be to impose that\nall FRBs are emitted with L/I = 1.0 and their mea-\nsured L/Iis solely the result of beam depolarization.\nThis process would provide poorly constrained upper\nlimits on σRMat best and is much better suited for\nFigure 13. L/I empirical CDFs, as computed using the\nKaplan-Meier method, of the non-repeating FRB population\nwith significant RM detections as observed at 400 −800 MHz\nusing CHIME/FRB in this work (red line; 84 polarized FRBs\nand 29 unpolarized FRBs) and at 1 .28−1.53 GHz with the\nDSA-110 (gold line; 20 polarized FRBs and 5 unpolarized\nFRBs; Sherman et al. 2023a). The 95% CIs in each respective\nCDF is plotted as a shaded red and golden region.\nFRBs that are observed over a large range of frequen-\ncies (e.g., Feng et al. 2022, who used observations of re-\npeating FRBs with observing frequencies ranging from\n∼0.1−5.0 GHz). If indeed FRBs are completely lin-\nearly polarized at the time they are emitted and the\nspread in the L/Idistribution is caused by beam depo-\nlarization, we should still observe a negative correlation\nbetween L/Iand|RMEG|. That is, FRBs whose emis-\nsion propagates through more dense and strongly mag-\nnetized environments should undergo a higher amount26\nof frequency-dependent depolarization from multi-path\npropagation.\nTo test this scenario, we plot L/I as a function of\n|RMEG|in Figure 14. Again, we apply Spearman’s rank\ncorrelation coefficient on the L/I− |RMEG|relation for\nonly the non-repeaters, only the repeaters, and the com-\nbined sample, respectively. We find no evidence for\na monotonic relationship between L/Iand|RMEG|in\nthe non-repeater only and combined data sets. We find\na marginally significant negative monotonic correlation\nbetween L/Iand|RMEG|in the repeater only data set\n(pSR= 0.002). Note, however, that the repeater data\nset is only comprised of 13 sources and therefore we do\nnot draw strong conclusions from this correlation. The\ncorresponding test statistic and p-values are reported in\nTable 4. For the non-repeating FRBs, this suggests that\nthere is not a strong relationship between the density\nand/or magnetic field strength of the FRB local envi-\nronment and the level of observed linear polarization.\nIn Figure 14, we are not able to account for the redshift\neffects on each FRB, which could smear out a potential\ncorrelation between L/Iand|RMEG|. However, based\non our previous results that show a consistent distribu-\ntion of L/Ibetween 400 MHz and 1 .53 GHz (e.g., see\nFigures 11 and 13), we do not expect L/Ito be very sen-\nsitive to redshift effects. Further, we plot the median\nand maximum shift in |RMEG|due to redshift correc-\ntions (based on our distribution of zfrom Section 3.5)\nin the lower left part of the plot. As a result, the loca-\ntion of most FRBs in this plot would not be drastically\naffected by a zcorrection and, thus, we argue that the\nL/I–|RMEG|correlation would not significantly change\neven if we were able to accurately derive |RMhost|for\neach FRB.\n3.8. Comparing with FRB burst rates\nIn this Section, we test whether the observed L/I,\n|RMEG|, or|β|values are correlated with the observed\nFRB burst rates (the bursts rates are obtained from\nCHIME/FRB Collaboration et al. 2023a, however, note\nthat not all FRBs in our sample have burst rate esti-\nmates available). We apply Spearman’s rank correlation\ncoefficient to each of the three pairs of correlations (burst\nrate– L/I, burst rate– |RMEG|, and burst rate– |β|), first\nfor only the non-repeaters, then for only the repeaters,\nand finally for the combined repeater plus non-repeater\ndata set. The Spearman’s rank correlation coefficients,\nSSR, and associated p-values, pSR, are reported in Table\n4. For the combined data set we furthermore calculate\nthe Kendall τcorrelation coefficient in the three tests\nwhere left-censored data are present. We find no evi-\ndence for a monotonic correlation between the observed\nFigure 14. Linear polarization fraction L/I for both the\nrepeating (blue) and non-repeating (red) sample of FRBs\nplotted as a function of their |RM EG|. Following the same\nsetup as Figure 6, the measurement uncertainties on the non-\nrepeating sample are denoted by black lines while the intrin-\nsic variation among repeater bursts, which are much larger\nthan the respective measurement uncertainties, are presented\nas blue shaded regions. Spearman’s rank correlation coeffi-\ncient suggests a marginal monotonic correlation for only the\nrepeating FRBs but no significant monotonic correlation for\nthe non-repeating FRBs or in the combined data set. Based\non Figures 11 and 13 we do not expect L/Ito be very sensi-\ntive to redshift effects. The median and maximum expected\nshift between |RM EG|and|RM host|due to redshift correc-\ntions (based on our distribution of zfrom Section 3.5) are\npresented in the lower left corner. Overall, a zcorrection\nwould not significantly change the L/I–|RM EG|correlation\nwe observe.\nburst rate and any of the three polarization properties\n(observed L/I,|RMEG|, and |β|) in the non-repeater\nonly data set, in the repeater only data set, or in the\ncombined data set. Again, as we are not able to correct\nfor redshift effects, the expected (1 + z) burst rate scal-\ning could smear out possible correlations. However, the\nmedian shift in burst rate is only a factor of ∼1.24. Fol-\nlowing the same argument as in Section 3.7.3, we do not\nbelieve a redshift correction would significantly change\nthe results of our correlation tests.\n4.DISCUSSION\n4.1. Typical polarization properties of FRBs\nOur sample of 128 non-repeating FRBs, 118 of which\nhave polarization information (89 polarized bursts andPolarization properties of CHIME/FRB non-repeaters 27\nTable 4. Summary of Spearman rank and Kendall τcorrela-\ntion coefficients and p-values testing whether a monotonic re-\nlationship exists between: log10(|RM EG|) and log10(DM EG),\n|RM EG|andL/I, burst rate and L/I, burst rate and |RM EG|,\nand burst rate and |β|for only the non-repeaters, only the\nrepeaters, and the combined repeater plus non-repeater data\nset. The Kendall τmethod is only used when data sets are\npartially left-censored, which is the case only for the last\nthree correlation tests. The p-values that are <0.05 are pre-\nsented in bold face.\nParameters SSR pSR\nNon-repeaters only\nlog10(|RM EG|) – log10(DM EG) 0 .287 0.006\n|RM EG|–L/I −0.052 0 .626\nBurst rate – L/I −0.086 0 .492\nBurst rate – |RM EG| − 0.214 0 .085\nBurst rate – |β| − 0.095 0 .447\nRepeaters only\nlog10(|RM EG|) – log10(DM EG) 0 .154 0 .616\n|RM EG|–L/I −0.770 0.002\nBurst rate – L/I 0.127 0 .709\nBurst rate – |RM EG| 0.218 0 .519\nBurst rate – |β| 0.127 0 .709\nNon-repeaters and repeaters\nlog10(|RM EG|) – log10(DM EG) 0 .252 0.010\n|RM EG|–L/I −0.189 0 .057\nBurst rate – L/I −0.157 0 .173\nBurst rate – |RM EG| − 0.077 0 .506\nBurst rate – |β| − 0.056 0 .630\nSKτ pKτ\nBurst rate – L/I −0.002 0 .973\nBurst rate – |RM EG| 0.052 0 .499\nBurst rate – |β| 0.080 0 .296\n29 unpolarized bursts), increases the total number of\nFRB sources with polarization properties by a factor of\n∼3. In the following subsections, we discuss the typical\npolarization properties of our sample and compare them\nto our understanding of FRB polarimetry prior to this\nwork (see Section 1 for a summary).\n4.1.1. Linear polarization\nIn the 400 −800 MHz band, we find that the mean and\nmedian observed levels of linear polarization are 63 .3%\nand 64 .7%, respectively, after accounting for unpolar-\nized upper limits (see Figure 4). The distribution of\nL/Iis skewed towards L/I= 1.0, with 14% of polar-\nized FRBs being consistent with 100% linear polariza-\ntion. There is, however, a significant spread in the L/I\ndistribution with 16% of FRBs being less than 50% lin-\nearly polarized. Furthermore, 29 FRBs did not reach the6σpolarization detection threshold and for these bursts\nwe place upper limits on L/I. Most of the unpolarized\nbursts have linear polarization upper limit constraints\nless than 30%, with the lowest constraints putting FRB\n20190115B and FRB 20190619A at less than 10% lin-\nearly polarized. This suggests either that there is a\nrange of possible linear polarization levels intrinsic to\nthe FRB emission mechanism or that many FRBs in\nour sample have been depolarized. We discuss the lat-\nter possibility further in Section 4.3, where we consider\nthe implications of the lack of evidence for depolariza-\ntion in our data..\n4.1.2. Magneto-ionic environment\nWe find that FRBs in our sample typically have a\nmoderate |RMEG|with a mean of 135 .4 rad m−2and\na median of 53 .8 rad m−2. Further, in Section 3.5,\nwe found that the distribution of |RMhost|is likely\nshifted a few tens of rad m−2higher based on ex-\npected redshift corrections. The majority of our sample\nhave|RMEG|in agreement with the median expected\n|RMEG|contribution from an FRB that is embedded\nrandomly within a MW like host galaxy (on the order\nof∼100−101rad m−2Hackstein et al. 2019), or one\noriginating near a star forming region in the host (on\nthe order of ∼101−102rad m−2Hackstein et al. 2019),\natz∼0.24 (the median expected redshift of our sam-\nple, as derived in Section 3.5). This result is in agree-\nment with the preponderance of FRB host galaxies be-\ning star forming, with the majority of these hosts trac-\ning the star forming main sequence for galaxies (Gordon\net al. 2023) and with Bhardwaj et al. (2023) who showed\nthat all hosts of localized nearby FRBs are spirals. The\npeak of the Galactic pulsar |RM|distribution is approx-\nimately ∼70 rad m−2, though the standard deviation is\nquite large as the distribution extends to ≳103rad m−2\n(based on the 1494 pulsars with RM measurements in\nthe Australia Telescope National Facility pulsar catalog\nV1.7.1 Manchester et al. 2005)6. The peak of both our\n|RMEG|and|RMhost|distributions are approximately\nconsistent with that of the Galactic pulsar distribution.\nWhile Galactic pulsar LoS do not probe the entire extent\nof the Galaxy, agreement between the pulsar and FRB\npopulations increases our confidence that FRB local en-\nvironments may be similar to those found in external,\nMW-like galaxies.\nThe highest |RMEG|in our sample is 1161 rad m−2for\nFRB 20190214C. It is the lone source with |RMEG| ≥\n103rad m−2, while only seven other sources exceed\n6The Galactic pulsar catalog is hosted online at\nhttps://www.atnf.csiro.au/research/pulsar/psrcat/28\n|RMEG|>500 rad m−2. We know that |RMEG| ≤\n|RMhost|and so, even without a redshift correction, this\nsubset of FRBs is consistent with the expectations for\na source embedded within a dense surrounding environ-\nment (e.g., a supernova remnant, which is expected to\nhave RM contributions up to ∼103rad m−2; Piro &\nGaensler 2018). It is possible that some other FRBs\nin our sample with lower |RMEG|also end up hav-\ning|RMhost|of this magnitude, but we cannot be cer-\ntain without their redshifts. Within our Galaxy, we\nhave seen some pulsars with similarly large |RM|s up\nto 103−104rad m−2originating near the dense Galac-\ntic center (e.g., pulsars PSRs J1746 −2849, J1746 −2850,\nJ1746−2856, and J1745 −2912; Abbate et al. 2023).\nTherefore, at least a few non-repeating FRBs originate\nin extremely dense and/or highly magnetized environ-\nments, though we cannot ascertain the specific proper-\nties of the environments from the |RMEG|measurements\nalone.\nOn the other hand, 14 of our 89 polarized FRBs have\n|RMEG|<10 rad m−2and, hence, are candidates for\nFRBs originating in clean magneto-ionic environments.\nOne of the reasons these are only candidates is that a\nredshift correction could place their |RMhost|values sub-\nstantially higher. There has been some evidence to sug-\ngest that the circumgalactic medium of galaxies could\ncontribute only a few rad m−2to the RM of background\nsources. In particular, Heesen et al. (2023) find an ex-\ncess RM of 3 .7 rad m−2for background polarized sources\nthat have an impact parameter of <100 kpc with nearby\ninclined galaxies. It is possible that FRBs inhabiting\nenvironments far away from the Galactic disk (e.g., the\ncircumgalactic medium and/or globular clusters) would\ntherefore have very small |RMEG|. However, even given\na redshift for the FRB host galaxy, our ability to confirm\nthe existence of an FRB in a clean magneto-ionic en-\nvironment (i.e., low density and/or weakly magnetized\nwith|RMEG| ∼0 rad m−2) is limited by the accuracy of\nthe current foreground MW RM map (Hutschenreuter\net al. 2022). This map has a resolution of 46 .9 arcmin2\nand relies on an RM catalog with a density of only\n∼1 source deg−2(primarily from the National Radio\nAstronomy Observatory Very Large Array Sky Survey\nTaylor et al. 2009). Even at high Galactic latitudes,\nthe MW RM contribution to extragalactic RMs varies\nby up to ∼10 rad m−2over degree angular scales (Op-\npermann et al. 2015; Hutschenreuter et al. 2022). To\naccurately compute the MW RM contribution towards\nFRBs, we therefore require a grid of RMs from back-\nground radio galaxies with a much higher sky density.\nThis will only become available for large portions of the\nsky with upcoming radio surveys such as the Polarisa-tion Sky Survey of the Universe’s Magnetism (POSSUM;\nGaensler et al. 2010). Otherwise, dedicated follow-up\nobservations around FRB sky positions are needed to\nconstruct sufficiently high density grids of RMs to de-\ntermine whether these FRBs originate in low density\nand/or weakly magnetized environments.\nWe find only a marginally significant pSRbetween\nlog10(|RMEG|) and log10(DM EG), and our results are in\nagreement with the same correlation seen by Mannings\net al. (2023) and Sherman et al. (2023b) using FRBs\nwith identified host galaxies. The marginal correlation\nmay suggest that while a fraction of DM EGoriginates\nfrom the same local environment as RM EG, a significant\namount of the DM EGis accumulated from other sources\n(e.g., the full extent of the host galaxy and the IGM).\nInterestingly, the correlation is stronger in non-repeaters\nand in the combined repeaters plus non-repeaters data\nsets than in only the repeaters. This may be due to\nthe smaller sample size of the repeating population,\nnamely that we do not have as many low |RMEG|or\nhigh DM EGrepeaters. Physically, this may imply that\nrepeating FRBs are embedded in environments that host\nstronger magnetic fields than non-repeaters and, there-\nfore, impart RMs that are disproportionately larger than\nthe DM contributed through that same medium. The\nmarginal dichotomy seen in |β|between repeaters and\nnon-repeaters lends some credence to this idea and is\ndiscussed further in Section 4.2.\nWe find typical |β|values of order ∼0.1−1µG\nwith values for some non-repeaters as low as ∼10−2−\n10−3µG. While it is possible for some FRBs to exist\nin clean environments (e.g., FRB 20200120E which is\nlocated in a globular cluster; Kirsten et al. 2022) or for\nsome FRBs to have magnetic fields that are oriented pre-\ndominantly in the plane of the sky rather than along the\nLoS, we do not expect typical magnetic field strengths of\n≲1µG for an FRB population that has, so far, been lo-\ncalized primarily to spiral galaxies (e.g., Bhardwaj et al.\n2023). For reference, Beck (2001) finds that the aver-\nage total magnetic field strength in the MW disk ranges\nfrom∼4µG (at 16 kpc from the Galactic center) to\n∼10µG (at 3 kpc from the Galactic center). Mean-\nwhile, upper limits for the LoS magnetic field strength\nof IGM filaments has been placed at ∼0.03µG (Ama-\nral et al. 2021). Therefore, many of the values of |β|\nderived in this work fall below the average total mag-\nnetic field strength in MW-like galaxies. However, there\nare still a handful of repeating and non-repeating FRBs\nwith|β|≳1µG that are roughly consistent with the\naverage total magnetic field strength in the outskirts of\nMW-like galaxies (assuming they have, on average, com-Polarization properties of CHIME/FRB non-repeaters 29\nparable magnetic field strengths in the plane of the sky\ndirection as well).\nApplying a statistical redshift correction to our polar-\nized non-repeating FRBs, we found a mean\f\f\nB∥,host\u000b\f\f\nof 4.62µG. As this is only the LoS component of the\nhost magnetic field strength, the total magnetic field\nstrength is likely larger, which makes this broadly con-\nsistent with the range of total magnetic field strengths in\nthe MW disk (Beck 2001). There is still a small fraction\nof the\f\f\nB∥,host\u000b\f\fdistribution in Figure 7 with values\n≲10−1µG. This low\f\f\nB∥,host\u000b\f\ftail can be explained\nby FRB LoS in which the magnetic field is largely in\nthe plane of the sky or a population of FRBs that orig-\ninate in environments that are much less magnetized\nthan those typically found in disks of MW-like galaxies.\n4.2. Comparing repeaters and non-repeaters\nThere has been evidence showing a dichotomy in\nthe observed properties of repeating and non-repeating\nFRBs, namely in regards to their burst duration and\nemitting bandwidth (Pleunis et al. 2021b; CHIME/FRB\nCollaboration et al. 2023a), and to a less significant ex-\ntent, in their DM EGdistributions (CHIME/FRB Col-\nlaboration et al. 2023a). This may be a selection effect,\nas it is easier to detect fainter repetitions from closer\n(i.e., on average lower DM) repeating FRB sources (see\nalso Gardenier et al. 2021; James 2023). On average, re-\npeating FRBs have larger burst widths, narrower emit-\nting bandwidths, and lower DM EGthan non-repeaters.\nIn this work, we compare 13 repeating and 89 non-\nrepeating polarized FRBs observed with CHIME/FRB\nto ascertain whether this observed dichotomy between\nthe two classes of FRBs extends to their polarization\nproperties.\nUsing the Peto & Peto and Log-Rank tests, we find no\ndifference in the L/ICDFs (accounting for the 29 L/I\nupper limits for non-repeaters) between non-repeating\nand repeating FRBs. However, note that we have not\nincorporated all repeating FRB bursts that are unpo-\nlarized into the L/ICDF for repeaters. Though we do\nnot suspect that this effect will dramatically change the\noutcome of our result as most repeaters in this sample\nconsist of only 1 −2 bursts with available baseband data.\nSo, we do not appear to be missing a large fraction of\nunpolarized bursts for most of these repeaters.\nNotably, the non-repeater distribution has a stronger\ntail towards lower |RMEG|, whereas only two repeat-\ning FRBs have |RMEG|<10 rad m−2. This may hint\ntowards the possibility that non-repeating FRBs more\nfrequently occupy clean environments than repeating\nFRBs, but proving that this is the case would require\na larger sample of repeating and non-repeating FRBsor a more accurate foreground MW RM map than is\ncurrently available. We observe no non-repeating FRBs\nwith extreme |RM|s as have been seen in some prolific\nrepeaters (FRB 20121102 and FRB 20190520B; Michilli\net al. 2018; Dai et al. 2022). In part, this may be due to\nintra-channel depolarization in CHIME/FRB (a signifi-\ncant change in the polarization angle within a single fre-\nquency channel), which causes a ∼50% depolarization\nat 600 MHz for |RM| ∼5000 rad m−2. FRBs with |RM|s\nexceeding this threshold should still be detectable using\nthe semi-coherent search developed by Mckinven et al.\n(2021). Specifically, Mckinven et al. (2021) use the semi-\ncoherent search to recover an RM of 1292 .6 rad m−2for\nFRB 20200917A, achieving a S/N boost of 13%, which\nis in agreement with the expected level of bandwidth\ndepolarization. However, the highest |RM|FRB de-\ntected by CHIME/FRB to date is FRB 20190417A with\n|RM|= 4429 .8 rad m−2(Mckinven et al. 2023b). The\nlack of very high |RM|detections for non-repeaters sug-\ngests that the population does not reside in the same\nextremely dense and highly magnetic environments as\nsome repeaters or that they undergo severe beam depo-\nlarization and become unpolarized in the CHIME/FRB\nband. We also cannot rule out that the lack of high\n|RM|detections with CHIME/FRB is exacerbated by\nthe instrumental bias from intra-channel depolarization.\nOverall, there is significant overlap between the |RMEG|\ndistributions of repeating and non-repeating FRBs, and\nwe see that both populations are broadly consistent with\nbeing drawn from the same underlying distribution.\nWe do see marginal evidence for a dichotomy in the\n|β|distributions of repeating and non-repeating FRBs\nat the 98 .3% (with the AD test) and 95 .8% (with the\nKS test) levels, with the non-repeaters having lower typ-\nical|β|than their repeating counterparts. This may hint\ntowards the idea that repeating FRBs, on average, orig-\ninate in more magnetized environments, which would\nbe consistent with the lack of very high |RMEG|non-\nrepeaters in our sample. However, we caution against\nthe over interpretation of this result for a few reasons.\nWe emphasize that |β|is only a lower limit on the\f\f\nB∥,host\u000b\f\fand a significant difference in the distribu-\ntion of |β|does not guarantee that such a divergence ex-\nists in the true\f\f\nB∥,host\u000b\f\f. The implicit assumption be-\nhind Equation 20 is that the RM and DM are imparted\nfrom the same medium. Devoid of any redshift infor-\nmation of the FRB host galaxies, our values of DM EG\nstill bear a substantial contribution from the IGM while\nRMEG, on average, traces the FRB host galaxy and\nlocal environment contributions. This means that |β|\nis sensitive to the distance to the FRB source, with\ncloser sources having a higher |β|. To better concep-30\ntualize this, consider two FRBs at redshifts za= 0.3\nandzb= 0.5, respectively, that are otherwise identical\n(assume for both FRBs that DM host= 100 pc cm−3\nand RM host= 100 rad m−2; implying that both have\f\f\nB∥,host\u000b\f\f∼1.232µG). Approximating the IGM con-\ntribution to the DM as DM IGM∼1000zpc cm−3\nbased on the observed DM- zrelation (Macquart et al.\n2020) gives DM IGM,a= 300 pc cm−3and DM IGM,b=\n500 pc cm−3. Scaling the observed DM hostand RM host\nwith redshift produces DM host(1 +za)−1= 77 pc cm−3,\nDM host(1 + zb)−1= 67 pc cm−3, RM host(1 + za)−2=\n59 rad m−2, and RM host(1+zb)−2= 44 rad m−2. There-\nfore, the |β|we would derive for these two FRBs would\nbe a factor of ∼2 different, solely due to the difference\nin their redshifts.\nWith this in mind, the apparent dichotomy in repeat-\ning and non-repeating FRB |β|should not be interpreted\nas definite evidence for two populations with distinct\nmagneto-ionic environments. Our results, instead, point\nto this scenario as being a possibility and the precise\nlocal magneto-ionic environments of repeating and non-\nrepeating FRBs warrant further exploration with a large\nsample of localized FRBs with associated host galaxies\nand redshifts, as will be available with the CHIME/FRB\nOutriggers in the near future (Mena-Parra et al. 2022).\nWe will further explore any potential dichotomies in\nFRB polarimetry using a larger sample of CHIME/FRB\nrepeaters in upcoming work by Ng et al. (in prepara-\ntion). This work will more than double the sample size\nof CHIME/FRB repeaters with polarization properties.\n4.3. Depolarization\n4.3.1. L/Ivariations across 400-800 MHz\nIf the large spread in L/Ivalues for the FRB pop-\nulation is indicative of widespread depolarization due\nto multi-path propagation, we would expect to see a\nsystematic difference in measured L/I distribution at\ndifferent observing frequencies. However, within the\nCHIME/FRB band, we see depolarization at the ∼20%\nlevel in only one source, FRB 20190217A. There also\nexists one outlier, FRB 20181214C, for which ( L/I)500\nis∼80% larger than ( L/I)700. This source, however, is\nvery faint and may suffer more from instrumental polar-\nization than many of the other FRBs. Similar analysis\non a much larger sample of FRBs is required to deter-\nmine whether FRB 20181214C is only an outlier or if\nthere truly exists a population of FRBs in which there\nis an increase in L/Iwith decreasing frequency. Aside\nfrom these two sources, all other FRBs from this subset\nhave 0 .8< fdepol <1.2. This indicates that most FRBs\nin the CHIME/FRB frequency band are not undergo-\ning a precipitous decrease in L/I, as might be expectedfrom Equation 4 or a power-law dependence for L/Iwith\nfrequency. Further, as shown in Figure 14, we find no\nevidence for a negative monotonic correlation between\n|RMEG|andL/Ifor the non-repeaters, as would be ex-\npected if the observed distribution of L/Iwere due to\ndepolarization from multi-path propagation.\n4.3.2. Comparing with an FRB population at 1.4 GHz\nExtending beyond the CHIME/FRB population,\nSherman et al. (2023a) recently presented a sample of\n25 FRBs with polarization properties observed in the\nrange 1 .28−1.53 GHz. Comparing our sample (which is\nat 400 −800 MHz) with that of Sherman et al. (2023a),\nwe find that the L/Idistributions are not statistically\ndifferent and the mean L/Iare in close agreement. Fur-\nthermore, the fraction of unpolarized events in both\nsamples are approximately in agreement; ∼25% of our\nsample does not have a significant Ldetection while\nthis number is ∼20% for the DSA-110 sample (al-\nthough they apply a higher significance threshold in\ntheir RM search, 9 σ, compared to the 6 σthreshold used\nin this work). This comparison, together with the pre-\ndominantly constant L/Iseen in the 23 FRBs within\nthe CHIME/FRB band, supports the conclusion that\nwidespread depolarization is not present in the observed\nFRB population. It is possible that most FRBs do not\nundergo a rapid decline in their L/Ibetween 1 .53 GHz\nand 400 MHz and instead this typically occurs at lower\nor higher frequencies. We should also remember that\nthe population observed in this work and that by Sher-\nman et al. (2023a) are two distinct sets of FRB sources\n(i.e., there are no co-detections of the same sources).\nTherefore, while a consistent mean L/Ibetween the two\npopulations points to a lack of depolarization on a pop-\nulation level, it is certainly possible that any individual\nsource could undergo frequency-dependent depolariza-\ntion between 400 MHz and 1 .53 GHz. As the DSA-110\nand CHIME/FRB sky coverage overlap significantly, a\nparticularly interesting avenue for future work would be\nto search for frequency-dependent depolarization specifi-\ncally in non-repeating FRBs co-detected by both instru-\nments.\nOn the other hand, Feng et al. (2022) show clear ex-\namples of individual repeating FRBs where depolariza-\ntion is evident over frequency ranges spanning a few\nhundred MHz to a few GHz. As derived in Section\n3.7.1, CHIME/FRB is only sensitive to a small range\nofσRMvalues within its frequency band. Hence, we re-\nquire large spectro-polarimetric FRB samples at higher\nobserving frequencies than 1 .53 GHz and at lower ob-\nserving frequencies than 400 MHz to better discern the\nprevalence of frequency-dependent depolarization acrossPolarization properties of CHIME/FRB non-repeaters 31\nthe entire FRB population. Further, most of the repeat-\ning FRBs for which Feng et al. (2022) fit a σRMvalue\nhave|RM|s≳500 rad m−2, values which are signifi-\ncantly higher than the typical |RMEG|that we find for\nnon-repeaters in our sample. It is possible that only\nFRBs embedded in very dense and/or strongly mag-\nnetized environments have a sufficiently large σRMto\nclearly show frequency-dependent depolarization at the\nobserving frequencies of CHIME/FRB and other FRB\nsurveys.\n4.3.3. A case study of FRB 20190227A\nWhile inspecting the unpolarized/depolarized FRBs\nwe came across FRB 20190227A, shown in Figure 12.\nThis FRBs is comprised of two broadband (400 −\n650 MHz) linearly unpolarized ( ≲20%) components and\none narrowband (600 −650 MHz; but note we have\nno baseband data >650 MHz) component that is lin-\nearly polarized at the ∼50% level with RM = 62 .9±\n0.9 rad m−2. These components are each separated by\n∼2 ms and, while the two broadband, unpolarized com-\nponents show scattering tails, the narrowband, polarized\ncomponent does not seem to extend to low enough fre-\nquencies to determine if it is scatter-broadened in the\nsame way.\nThere is a precedent for L/Ichanges from subburst to\nsubburst in the FRB population: for FRB 20181112A,\nwhich consists of a total of four components, there is\na visible decrease in L/Iof∼0.2 from the first com-\nponent to the third component (Cho et al. 2020). The\nlarge spread in the L/Idistribution of our non-repeating\nsample, in conjunction with the lack of population-wide\ndepolarization, suggests that FRBs are emitted at a\nrange of intrinsic L/Ivalues and that not all FRBs are\n100% linearly polarized. It is possible that the ≳30%\nchange in linear polarization between components in\nFRB 20190227A is explained by a change in the angle\nbetween our LoS and the center of an emitting beam.\nFor example, Desvignes et al. (2019) find that L/Iin-\ndeed decreases with increasing distance from the beam\ncenter for radio emission from pulsar PSR J1906+0746.\nAlternatively, we could attempt to explain the change\ninL/Ibetween components by a change in the circum-\nburst media through which the FRB propagates. If the\ndecrease in L/Iof≳0.3 between the unpolarized and\npolarized components is due to propagation effects, the\ncircumburst medium must have changed significantly on\na ∆t∼2 ms timescale. The length scale over which\nsuch a change could occur is c∆t≲600 km (ignoring\nany relativistic effects). The relative velocity of the\nemitting source to the observer, vsource , is likely ≪c\nso, assuming vsource = 500 km s−1(which is relativelyhigh for typical pulsar velocities; Verbunt et al. 2017),\nthe length scales for the evolving circumburst medium\nwould only be ∼1 km. While variations in scattering\ntimescales have been observed for other sources such as\nFRB 20190520B (over 2 .9 minutes; Ocker et al. 2023)\nand in the Crab pulsar (over 15 days; McKee et al. 2018),\nthese are both over timescales that are many orders of\nmagnitude longer than what would be required in the\ncase of FRB 20190227A. Predicated upon this and the\nextremely small length scales required in this scenario,\nwe believe it is unlikely that the decrease in L/Iseen\nbetween components of FRB 20190227A is caused by\npropagation effects in the circumburst medium. Instead,\nit is more likely that the subburst to subburst variation\ninL/Ioriginates from the FRB emission mechanism.\n4.4. Comparison to Galactic pulsars and magnetars\nSince at least some FRBs are likely produced by neu-\ntron stars (for arguments pointing towards magnetar ori-\ngins of FRBs, see Bochenek et al. 2020; CHIME/FRB\nCollaboration et al. 2020, 2022, ; Mckinven et al. in prepa-\nration), it is interesting to compare the polarized proper-\nties of FRBs with those of Galactic neutron stars, even\nthough the latter are less luminous in the radio band\nby many orders of magnitude. Whereas all FRBs are\ndetected directly, pulsars are often studied by means\nof their average or stacked pulse profiles over many ro-\ntational periods; the most direct comparison would be\nbetween FRBs and pulsar single pulses.\nRadio emission from canonical pulsars is understood\nto be produced by charged particles being accelerated\nnear the magnetic poles of the stars, typically leading\nto one observed pulse per rotational phase of the neu-\ntron star (or two, if the radio beams are broad and the\nbeam angle with respect to the LoS is favorable or if\nthe magnetic axis is offset towards 90 deg with respect\nto the rotational axis). The radio pulses are typically\nmoderately linearly polarized. About 60% of pulsar PA\ncurves are reproducible by a “rotating vector model,”\n(Johnston et al. 2023b) where, depending on impact pa-\nrameter and magnetic inclination angle, the observed\npulses exhibit a continuous swing in PA over the pulse\nbut can also show a more flat curve (Radhakrishnan &\nCooke 1969).\nIn Section 3.6.1, we fit the RVM to a non-repeating\nFRB subsample and found fits that generally match\nthe quality of that obtained from pulsars. Unlike\npulsars, the non-repeating FRB sample studied here\nhave unknown periods, thus making the RVM under-\nconstrained with equally plausible fits for a wide range\nof assumed periods and geometries (Mckinven et al. in\npreparation). The overwhelming preponderance of flat32\nPA curves of this sample is markedly different from\nthat of the pulsar population. If FRBs do originate\nwithin the dipolar magnetospheres of rotating neutron\nstars, the flatness of the PA curves indicates a prefer-\nence for RVM models for which the magnetic and ro-\ntation axes are preferentially aligned ( α= 0◦) or anti-\naligned ( α= 180◦) to a greater degree than that seen\nin the Galactic pulsar sample. Counter-intuitively, this\nclustering is seemingly at odds with a young progenitor\nscenario in which the rotation and magnetic axes would\nbe expected to be more randomly oriented (e.g., Gil &\nHan 1996), due to insufficient timescales for alignment\n(≳1 Myr) via electromagnetic torques (e.g., Young et al.\n2010), however, evidence for this alignment is lacking\n(Faucher-Gigu` ere & Kaspi 2006). Alternatively, the PA\ncurves displayed by FRBs may not be suitably inter-\npreted via the RVM, and alternative scenarios involving\nmore complex magnetic field configurations (e.g., mul-\ntipole models Qiu et al. 2023) and/or emission regions\nfar from the neutron star surface cannot be ruled out.\nIndeed, the recent observation of high luminosity radio\nbursts occurring at random rotational phases in Galac-\ntic FRB source SGR 1935+2154 (Zhu et al. 2023) is in-\nconsistent with the simple pulsar-like RVM framework\nexplored here.\nOne interesting avenue to pursue in future work would\nbe to classify the PA curves observed in the repeating\nFRB population and determine whether they also ex-\nhibit mostly constant PA profiles or if they have more\nvariable PA profiles.\n4.4.1. Galactic analogs for FRB polarization properties\nThe Crab pulsar is one of the brightest and most well-\nstudied pulsars, and demonstrates the variety of radio\nemission that oneneutron star can produce. Together\nwith other pulsars that have >100 kG magnetic fields\nnear the light cylinder, it does not adhere to the canoni-\ncal pulsar model and shows at least seven distinct radio\nemission components over one rotation that are likely\nproduced at different sites near the surface and in the\nouter magnetosphere (Eilek & Hankins 2016; Lyubarsky\n2019). While the Crab’s main pulse is only weakly po-\nlarized, individual nanoshots within the main pulse can\nbe strongly polarized with drastic polarization changes\nfrom nanoshot to nanoshot (Hankins et al. 2003). The\nhigh-frequency interpulse and two high-frequency com-\nponents, on the other hand, are 50%–100% linearly po-\nlarized (Moffett & Hankins 1999; Hankins et al. 2016).\nComponents do not show PA swings within a single com-\nponent, but the PA changes from component to compo-\nnent, similar to what is seen for FRBs with multiple\ncomponents and variable PAs (bottom panel of Fig. 9).Other young pulsars show similar PA jumps from com-\nponent to component (Weltevrede & Johnston 2008).\nRadio bursts from magnetically-powered magnetars\nare highly linearly polarized, with polarization fractions\n60%–100%, and display both constant and variable PA\nprofiles over the bursts (e.g., Kaspi & Beloborodov 2017;\nLower et al. 2021). These PAs are most similar to the\nsingle component constant and variable PA categories\nof FRBs in this work. Magnetar bursts arrive over a\nwide range of rotational phases, which is typically in-\nterpreted as the emission being produced higher in the\nneutron star’s magnetosphere, instead of in a narrow\nbeam near the magnetic poles.\nSensitive observations with the FAST telescope of the\nGalactic magnetar SGR 1935+2154 provide evidence for\ntwo emission modes existing at the same time in one ob-\nject (Zhu et al. 2023): low-luminosity pulsar-like pulses\nwere found to occur in a narrow ( ∼5%) region of the\nrotational phase, whereas FRB-like bursts that are eight\norders of magnitude brighter than the pulsar-like emis-\nsion, occur at random phases. The folded pulse profiles\nare 12%–65% linearly polarized and show no signs of PA\nswing across the integrated pulse.\nQualitatively, the variety in PA profiles and L/Ival-\nues observed for FRBs seems to be produced by SGR\n1935+2154 and the Crab pulsar, so it is not necessary\nto invoke novel populations of sources to explain dif-\nferent polarization properties seen in the FRB popula-\ntion, but it may be necessary to invoke different emission\nmechanisms. For a more quantitative result, Sherman\net al. (2023a) compared the polarised properties of vari-\nous classes of pulsars to FRBs and found that L/Ivalues\nof FRBs seem to match best the polarization properties\nof young pulsars, with characteristic ages <105years,\nbut are on average higher than that of the overall pop-\nulation of pulsars. So while FRBs are much more lu-\nminous than the radio emission from Galactic neutron\nstars, we find analogs to the polarization properties of\nthe FRB population among the most energetic neutron\nstars known in the Milky Way.\n4.5. Using FRB DMs and RMs to estimate redshifts\n4.5.1. Method and assumptions\nFRBs with available polarimetry may provide a\nunique opportunity to constrain redshifts for sources\nwithout a clear host galaxy association. In this Section,\nwe take 20 FRBs (15 non-repeaters and 5 repeaters)\nfrom the literature that have DMs, RMs, and confirmed\nhost galaxies with a measured redshift, z, and attempt\nto independently derive upper and lower limits on z\nusing only their respective DMs and RMs. The TNSPolarization properties of CHIME/FRB non-repeaters 33\nnames for all 20 FRBs are listed in Figure 15 and their\nDM, RM, and literature references are given in Table 5.\nWe start by deriving an upper limit on zusing the\nDM obsof each FRB. In deriving the upper limit, we want\nto be conservative and attribute as much of the FRB DM\nbudget as is reasonably possible to the DM IGM(z) com-\nponent. So, from the total observed DM of each FRB,\nwe subtract a conservative 0 .5 times the estimated MW\nDM contribution along their respective LoS, according\nto the YMW16 model. We assume that the remaining\nDM is all accumulated through the IGM (i.e., assuming\nDM halo= DM host= 0 pc cm−3). Using the Macquart\nrelation, we derive an upper limit for the FRB redshift\nas:\nzupper∼DM obs−0.5(DM disk)\n1000. (22)\nVariations of this technique have been broadly used to\ngetzupper limits for FRBs in the past. The novel\naspect that we propose is to then incorporate the FRB\nRMs to obtain a zlower limit.\nTo derive a lower limit, we need values for both DM obs\nand RM obsand impose some assumptions on the ob-\nserved parallel magnetic field strength of the host galaxy.\nIn this case, we want a lower limit and so we are more\ngenerous in attributing DM contributions to the MW.\nWe subtract 1 .5 times the estimated YMW16 MW DM\ncontribution along their respective LoS and also subtract\nDM halo= 30 pc cm−3. For the RM obs, we subtract the\nexpected MW RM using Hutschenreuter et al. (2022),\nas was done in Section 2.5 to get |RMEG|. We make\nan assumption on the minimum LoS averaged magnetic\nfield strength in the FRB host galaxy such that:\n\f\f\nB∥,host\u000b\f\f= 1.232|RMhost|\nDM host≥1µG. (23)\nWhile we choose a minimum of 1 µG in this work, this\nthreshold can be adjusted. Using Equation 23, we derive\nan upper limit on DM hostas:\nDM host≤1.232|RMEG|\n1µGpc cm−3. (24)\nSubtracting the DM hostupper limit from the remaining\nDM budget and assigning the remaining DM to the IGM\ncomponent, we use the Macquart relation to derive a\nlower limit on zas:\nzlower∼DM obs−1.5(DM disk)−DM halo−\u0010\n1.232|RMEG|\n1µG\u0011\n1000.\n(25)\n4.5.2. Results and interpretation of failure modes\nIn Figure 15, we compare the constraints from this\nmethod for 20 FRBs that have been associated with hostgalaxies for which a redshift is known. The main pur-\npose of this plot is to underscore the scenarios in which\nthe method of deriving zlower limits, zlower, does or\ndoes not work well. In almost all cases, the zupper\nlimit, zupper, falls above the measured zof the FRBs.\nFor the few cases where it does not, we suspect that\nthe LoS may be through a less dense region of the IGM\nsuch that the true DM IGM−zrelation is shallower than\nDM IGM∼1000×z.\nThe zlower provides a reasonable estimate (i.e., be-\ntween z= 0 and the measured zof the FRB host galaxy)\nfor 9/20 FRBs, as shown in the left part of the plot that\nis labelled “method works”.\nFor 4 /20 FRBs, we find that the zlower values over-\nestimate the measured z. All four of these FRB have\n|RM|<102rad m−2and, therefore, the DM hostre-\nquired to achieve a 1 µG strength LoS magnetic field is\nsmall and we could be overestimating the DM IGMcom-\nponent in these cases. Physically, this could be due to\nthe magnetic field being largely in the plane of the sky\nor encountering field reversals along the LoS that cause\nthe|RMhost|to be small compared to the DM hostac-\ncumulated while propagating through the host galaxy.\nAlternatively, these FRBs may have accumulated DM\nin excess of the mean DM IGM−zrelation by propagat-\ning through dense regions of the IGM or by intersecting\nstructures, such as galaxy halos or clusters. This would\nlead to the zlower values derived using Equation 25 over-\nestimating the true zof the FRB host galaxy. These\nFRBs are plotted in the middle section of Figure 15,\nlabelled “method overestimates z”.\nFor the last 7 /20 FRBs, the zlower values were found\nto be less than 0. A negative zis, of course, unphysi-\ncal in this context but we discuss the cause of this fail-\nure mode as it informs the conditions under which we\ncan apply the method outlined in the previous section.\nThese FRBs fall on the high end of the |RM|distribu-\ntion with six out of seven having |RM|>102rad m−2.\nThis group of FRBs also contains prolific repeaters that\nhave very large RM amplitudes (e.g., FRB 20121102A\nand FRB 20190520B). The cause of this failure mode is\nthat when the |RM|of a source is very high, we require a\nproportionally high DM hostto satisfy the assumed 1 µG\nLoS magnetic field strength. The required DM hostcan\nexceed the measured DM in some cases, leading to a neg-\native zlowerestimate. Based on this, we caution that this\ntechnique of deriving a zlower limit fails in cases where\nthere is a large |RM|contribution from the immediate\nenvironment around the FRB (e.g., a supernova rem-\nnant or wind nebula) as the LoS magnetic field strength\nin these regions likely far exceeds the 1 µG threshold.34\nThe FRBs in this category are on the right section of\nFigure 15, labelled “method fails”.\nOverall, we have shown that the combination of FRB\nDMs and RMs can be used to derive informative z\nupper limits and lower limits, assuming a minimum\nLoS magnetic field strength in the host galaxy, for\nsome FRBs where we have low to moderate |RM|(i.e.,\n∼101−103rad m−2). While there are clear failure\nmodes, we see that the method works for 45% of FRBs\nin our tests and could prove to be useful, especially on a\npopulation level, when there is no clear host galaxy as-\nsociation or if there is no photometric or spectroscopic\nredshift available for a host galaxy candidate. In the\nfuture, other source or burst properties, such as burst\nmorphology, could be investigated as an additional way\nof identifying FRBs for which this method is or is not\nwell suited.\n5.CONCLUSIONS\nWe have conducted polarimetric analyses for the 128\nnon-repeating fast radio bursts (FRBs) from the first\nbaseband catalog of the Canadian Hydrogen Intensity\nMapping Experiment FRB collaboration, more than\ndoubling the total number of FRB sources with mea-\nsured polarization properties to date. Of the 128 FRBs,\n89 have a significant ( >6σ) linearly polarized detec-\ntions and their polarized spectra are well-fit by the\nCHIME/FRB polarization pipeline. For these FRBs, we\nanalyze their observed time and frequency averaged lin-\near polarization fractions L/I, rotation measures (RMs),\ndispersion measures (DMs), polarization position angle\n(PA) profiles, and frequency-dependent depolarization\nbetween 400 −800 MHz, and further derive lower limits\non the magnetic field strength parallel to the line of sight\nin the FRB host environment, |β|. We apply a statistical\nredshift correction to derive a distribution of rest frame\nrotation measure amplitudes and average magnetic field\nstrengths parallel to the line of sight arising from the\nFRB host galaxies. Another 29 FRBs do not have sig-\nnificant RM detections and for these events we place\nupper limits on their linear polarization fractions based\non our 6 σdetection threshold. Our polarimetric sample\nis compared to published CHIME/FRB repeaters and\nFRB populations measured at other frequencies. The\nkey conclusions of this work are as follows:\n1. The median observed L/I of our non-repeating\nsample is 0 .647 and the distribution has a large\nspread, suggesting that the common assertion that\nFRBs are typically ∼100% linearly polarized\nis not always accurate. The median extragalac-\ntic component of the observed RM, |RMEG|, of\nour non-repeater sample is 53 .8 rad m−2. Ap-plying a statistical redshift correction, we get a\nmedian rest frame RM from the host galaxy of\n|RMhost|= 75.9 rad m−2. The observed |RMEG|\nand rest frame |RMhost|distributions are mostly\nconsistent with RM expectations from FRBs em-\nbedded in Milky Way-like galaxies or near star\nforming regions in their host galaxy. There are\n14 sources with |RMEG|<10 rad m−2, which are\ncandidates for FRBs originating from low density\nand/or weakly magnetized environments (or could\npossibly have high redshift host galaxies) but their\nverification requires a more accurate accounting of\nthe Milky Way RM contribution than is currently\navailable.\n2. We find no evidence for a dichotomy in the ob-\nserved L/I(including the unpolarized bursts) or\n|RMEG|distributions between repeaters and non-\nrepeaters. Note that, for the L/Istatistical tests,\nwe are not able to fully account for individual re-\npeater bursts that may have been unpolarized but\ndo not believe this significantly effects our results.\nWe do, however, see marginal evidence for non-\nrepeaters and repeaters having different |β|dis-\ntributions, suggesting that repeaters may origi-\nnate from more dense and/or more highly mag-\nnetized environments, but we also discuss biases\nthat could make this conclusion false.\n3. We define four archetypes for PA behavior based\non a reduced χ2\nνfit to a constant PA profile and the\nexistence of multiple components within the burst:\n(i) single component with constant PA (containing\n57% of the polarized non-repeating FRBs), (ii) sin-\ngle component with variable PA (10%), (iii) multi-\nple components with constant PA (22%), and (iv)\nmultiple components with variable PA (11%). Ra-\ndio bursts from magnetars, Crab pulses, and radio\nemission from some young pulsars provide useful\nGalactic analogs that have high typical L/Iand\ncan produce constant PAs. Specifically, some Crab\npulse components and young pulsars show simi-\nlar component-to-component PA jumps as seen for\nFRBs that fall under the “multiple components\nwith variable PA” archetype.\n4. In a subset of 23 bursts with 400 MHz emit-\nting bandwidths, 21 have a depolarization ra-\ntio between 700 MHz and 500 MHz of 0 .8<\nfdepol <1.2 and do not show any clear signs\nof frequency-dependent changes in L/I. Only\none FRB shows depolarization at the 20% level\nand one FRB shows a 80% increase in L/Iover\nthe CHIME/FRB band. Across all 89 polarizedPolarization properties of CHIME/FRB non-repeaters 35\nFigure 15. An experiment to test how well our method of deriving a zlower limit works on a sample of twenty FRBs that\nhave measured RM and are associated with host galaxies that have known z. The zof the twenty FRBs are plotted as stars\nand are colored by their respective measured log10(|RM obs|) (see Table 5 for details and references). The x-axis lists the names\nof each FRB. Independently derived redshift upper (blue bars) and lower (red bars) limits are overplotted for each FRB with\na shaded grey region encompassing the redshift range in between. In cases where the redshift limits are above or below the\nplotted range, triangular markers are used as indicators instead of bars. The plot is broken up into three sections by dashed\nblack lines: (i) “method works”, where the derived zlower limits are reasonable compared to the measured z; (ii) ‘method\noverestimates”, where the zlower limits higher than the independently measured z; and (iii) “method fails”, where the zlower\nlimits are negative.\nnon-repeating FRBs in our sample, we do not\nfind evidence for a negative monotonic correla-\ntion between |RMEG|andL/I, as might be ex-\npected from depolarization stemming from multi-\npath propagation. We also find similar L/Idis-\ntributions between non-repeating FRBs observed\nat 400 −800 MHz and those at 1 .4 GHz, further\nsupporting the lack of frequency-dependent depo-\nlarization seen in the CHIME/FRB band. In all,\nour findings suggest that there is no observable\nfrequency-dependent depolarization on the popu-\nlation level for non-repeating FRBs and the ob-\nserved scatter in their L/I distribution is likely\nintrinsic to the FRB emission mechanism.\n5. We derive a technique for obtaining lower limits\non FRB redshifts by using their observed DMs and\nRMs and making an assumption about the lower\nlimit of the host galaxy line of sight magnetic field\nstrength. Testing on a sample of FRBs with known\nredshifts, the technique appears to work best in\ncases of moderate |RMobs|values but has difficultywith FRBs that have either extremely low or high\n|RMobs|values. In cases where independent red-\nshift information is not available, this technique\ncan provide some loose constraints on the FRB\nhost redshift. However, it should be applied cau-\ntiously as peculiar lines of sight (e.g., extremely\nlow or high |RMobs|values, particularly under or\nover-dense regions of the intergalactic medium, in-\ntervening galaxies and/or clusters) are not always\nwell constrained.\nWhile our work expands the known FRB sample with\npolarized properties by a factor of ∼3, there are still\nmany promising avenues for future work to better under-\nstand FRB polarization. First, we require a higher res-\nolution Milky Way RM map to place tighter constraints\non the FRB host RM contribution. Doing so would en-\nable us to more carefully examine the subset of FRBs\nthat appear to be stemming from clean magneto-ionic\nenvironments that may be similar to, for example, FRB\n20200120E, which is located within a globular cluster\n(Kirsten et al. 2022). It would also be interesting to see36\nwhether the preponderance of of constant PA profiles\nobserved in non-repeating FRBs extends to the repeat-\ning FRB population or if repeaters exhibit more vari-\nable PA profiles. Another exciting prospect is studying\nthe spectro-polarimetry of potential co-detections be-\ntween the CHIME/FRB and the Deep Synoptic Array.\nDoing so would provide data spanning a range of fre-\nquencies between 400 MHz and 1 .53 GHz for individual\nnon-repeating FRBs and allow for much greater leverage\nin frequency when searching for spectral depolarization.\nAdditionally, the CHIME/FRB Outriggers (Mena-Parra\net al. 2022, ; Lanman et al. in preparation) are expected\nto provide host galaxy associations and redshift informa-\ntion for many FRBs in the near future, enabling precise\nstudies of magnetic fields in FRB host galaxies and the\nrelation between RM and redshift.\nACKNOWLEDGEMENTS\nWe acknowledge that CHIME is located on the\ntraditional, ancestral, and unceded territory of the\nSyilx/Okanagan people. We are grateful to the staff\nof the Dominion Radio Astrophysical Observatory,\nwhich is operated by the National Research Council\nof Canada. CHIME is funded by a grant from the\nCanada Foundation for Innovation (CFI) 2012 Leading\nEdge Fund (Project 31170) and by contributions from\nthe provinces of British Columbia, Qu´ ebec and Ontario.\nThe CHIME/FRB Project is funded by a grant from\nthe CFI 2015 Innovation Fund (Project 33213) and by\ncontributions from the provinces of British Columbia\nand Qu´ ebec, and by the Dunlap Institute for Astron-\nomy and Astrophysics at the University of Toronto. Ad-\nditional support was provided by the Canadian Insti-\ntute for Advanced Research (CIFAR), McGill Univer-\nsity and the Trottier Space Institute at McGill thanks\nto the Trottier Family Foundation, and the University\nof British Columbia. The baseband recording system\nfor CHIME/FRB is funded in part by a CFI John R.\nEvans Leaders Fund award to IHS. The University of\nToronto operates on the traditional land of the Huron-\nWendat, the Seneca, and most recently, the Mississaugasof the Credit River; we are grateful to have the oppor-\ntunity to work on this land. FRB research at UBC is\nsupported by an NSERC Discovery Grant and by CI-\nFAR. A.P. is funded by the NSERC Canada Gradu-\nate Scholarshops–Doctoral program. Z.P. was a Dun-\nlap Fellow and is supported by an NWO Veni fellowship\n(VI.Veni.222.295). B.M.G. acknowledges the support of\nthe Natural Sciences and Engineering Research Council\nof Canada (NSERC) through grant RGPIN-2022-03163,\nand of the Canada Research Chairs program. M.B. is\na McWilliams fellow and an International Astronomi-\ncal Union Gruber fellow. M.B. also receives support\nfrom the McWilliams seed grant. A.M.C. is funded by\nan NSERC Doctoral Postgraduate Scholarship. A.P.C.\nis a Vanier Canada Graduate Scholar. V.M.K. holds\nthe Lorne Trottier Chair in Astrophysics & Cosmology,\na Distinguished James McGill Professorship, and re-\nceives support from an NSERC Discovery grant (RGPIN\n228738-13), from an R. Howard Webster Foundation\nFellowship from CIFAR, and from the FRQNT CRAQ.\nC.L. is supported by NASA through the NASA Hubble\nFellowship grant HST-HF2-51536.001-A awarded by the\nSpace Telescope Science Institute, which is operated by\nthe Association of Universities for Research in Astron-\nomy, Inc., under NASA contract NAS5-26555. K.W.M.\nholds the Adam J. Burgasser Chair in Astrophysics\nand is supported by NSF grants (2008031, 2018490).\nK.N. is an MIT Kavli Fellow. A.B.P. is a Banting\nFellow, a McGill Space Institute (MSI) Fellow, and a\nFonds de Recherche du Quebec – Nature et Technolo-\ngies (FRQNT) postdoctoral fellow. K.S. is supported by\nthe NSF Graduate Research Fellowship Program.\nFacilities: CHIME\nSoftware: Astropy (Astropy Collaboration et al.\n2013, 2018, 2022), Matplotlib (Hunter 2007), NADA\n(Lopaka 2020), Nested Sampling (Skilling 2004), NumPy\n(Harris et al. 2020), PyGEDM (Price et al. 2021), RM-\nCLEAN (Heald et al. 2009), RM-synthesis (Brentjens &\nde Bruyn 2005), RM-tools (Purcell et al. 2020), SciPy\n(Virtanen et al. 2020), Seaborn (Waskom 2021).\nAPPENDIX\nA.POLARIZATION PROPERTIES OF FRBS FROM LITERATURE\nIn Table 5, we summarize the properties of the published polarized FRB sample in literature prior to this work. In\nFigure 16 and 17, we plot the log10(|RM|) and L/I, respectively, of each repeater from Table 5 as a violin plot to\nbetter visualize the spread in observed polarimetry of repeaters. The number of bursts included for each repeater is\nprovided as a second x-axis at the top of each plot.Polarization properties of CHIME/FRB non-repeaters 37\nFigure 16. Violin plot of the observed rotation measure magnitude for the 18 repeating FRBs currently published in the\nliterature (see Table 5). KDEs of the log10(|RM obs|) distribution for each repeater are shown in pink and truncated at the\nminimum and maximum observed values. All of the KDEs have been normalized such that the width of each violin plot marker\nis equal. A white dot highlights the median log10(|RM obs|) of each repeater with a grey box representing the interquartile range\nand vertical grey lines showing the full observed range of log10(|RM obs|). The number of polarized bursts for each repeater is\ngiven at the top of the plot.\nFigure 17. Analogous to Figure 16 for the observed L/Idistribution for the 18 repeating FRBs from Table 5.38\nTable 5 . Summary of published FRB polarization properties prior to this work. The FRB TNS name and whether is presented\nin column 1. Column 2 provides the observing frequency νobsin GHz and column 3 lists the total number of bursts with\npolarization properties for that source. Column 4 provides the measured DM range and column 5 provides the RM range\nfor each FRB source. Columns 6 and 7 show the time and frequency averaged linear polarization fraction, L/I, and circular\npolarization fraction, |V|/I, respectively. Note that in the case of repeaters, a range encompassing the 95% confidence interval\nfor the respective parameter is provided and these FRBs are highlighted in boldface. In some cases, a repeater may only have\na few bursts and the 95% confidence interval (computed assuming a Gaussian distribution) falls outside the range of the data;\nin these cases we instead report the minimum and maximum observed values for the given parameter and these are italicized.\nRecall that L/Imay be affected by the observing frequency, νobs, so, for some repeaters, a large range of L/Icould be due to\na wide span in νobs. We provide an expanded and machine readable version of this table that also includes all of the polarized\nnon-repeating FRBs listed in Table 1 online ( a link to the data release will be added upon acceptance ).\nFRB Name νobs Number of DM RM L/I |V|/I\n(GHz) bursts (pc cm−3) (rad m−2)\nFRB 20121102A 4.51,229 [560 .5,561.4] [74037 ,90998] 1 .0 ∼0.0\nFRB 20180301A 1.337 [516 .3,516.8] [536 .4,555.9] [0 .50,0.73] <0.10\nFRB 20180814A 0.641 189 .0 699 .8 0 .69 <0.20\nFRB 20180916B 0.45, 0.16, 0.87, 0.6855 [349 .0,349.2] [−112.9,−105.6] [0 .83,0.91] ∼0.0\nFRB 20181030A 0.643 103 .5 [36.4, 38.0] [0.60, 0.93] <0.20\nFRB 20181119A 0.643 [363.6, 364.3] [481.0, 1344] [0.47, 1.0] <0.20\nFRB 20190117A 0.642 [393.1, 395.8] [74.30, 76.31] [0.14, 0.20] <0.20\nFRB 20190208A 0.647 [579 .6,579.9] [ −14.56,24.66] [0 .81,0.94] <0.20\nFRB 20190213B 0.645 [301.4, 301.7] [-3.630, 1.060] [0.74, 1.0] <0.20\nFRB 20190222A 0.641 459 .6 571 .2 0 .25 <0.20\nFRB 20190303A 1.39, 0.6415 [221 .5,221.7] [−516.4,−388.3] [0 .48,0.73] <0.20\nFRB 20190417A 1.39, 0.646 1378 [4548 ,4729] [0 .40,0.80] <0.20\nFRB 20190520B 6.09, 2.410, 5.61116 [1201 ,1211] [ −6685,5830] [0 .33,0.51] [ ∼0.0,0.17]\nFRB 20190604A 0.641 552 .5 −17.80 0 .91 <0.20\nFRB 20190711A 1.3121 587 .87 10 .00 1 .0 ∼0.0\nFRB 20200120E 0.613, 1.4146 [87 .75,87.79] [ −44.76,−24.97] [0 .96,1.0] [ ∼0.0,0.11]\nFRB 20201124A 1.415, 2.416, 1.39,171148 [413 .1,413.2] [−594.5,−582.3] [0 .75,0.77] [0 .08,0.09]\nFRB 20220912A 0.618, 1.419, 1.3201087 [225 .1,235.1] [ −2.37,3.33] [0 .94,0.96] [0 .10,0.12]\nFRB 20110523A 0 .8211 623 .3 −186.1 0 .44 <0.30\nFRB 20150418A 1 .4221 776 .2 36 .00 0 .09 ∼0.0\nFRB 20150807A 1 .4231 266 .5 12 .00 0 .80 ∼0.0\nFRB 20150215A 1 .4241 1106 1 .600 0 .43 0 .03\nFRB 20160102A 1 .4251 2596 −220.6 0 .35 0 .06\nFRB 20171209A 1 .4261 1457 .4 121 .6 1 .0 0 .0\nFRB 20180311A 1 .4261 1570 .9 4 .8 0 .75 0 .11\nFRB 20180714A 1 .4261 1467 .92 −25.9 0 .91 0 .05\nFRB 20180924A 1 .3121 362 .2 22 .00 0 .90 0 .13\nFRB 20181112A 1 .3271 589 .3 10 .50 0 .92 <0.34a\nFRB 20190102C 1 .3121 364 .5 −105.0 0 .82 0 .05\nFRB 20190608B 1 .3121 340 .1 353 .0 0 .91 0 .09\nFRB 20190611B 1 .3121 332 .6 19 .00 0 .93 0 .15\nFRB 20191001A 0 .9281 507 .9 55 .5 0 .57 0 .05\nFRB 20191108A 1 .4291 588 .1 474 .0 0 .70 <0.10\nFRB 20201020A 1 .4301 398 .59 110 −b−b\nFRB 20220121B 1 .4311 313 .421c−4.60 0 .71 0 .51\nFRB 20220204A 1 .4311 612 .584c−11.0 0 .63 0 .07\nFRB 20220207C 1 .431,321 262 .3 162 .5 0 .75 0 .11\nContinued on next pagePolarization properties of CHIME/FRB non-repeaters 39\nTable 5 – continued from previous page\nFRB Name νobs Bursts DM RM L/I |V|/I\n(GHz) (pc cm−3) (rad m−2)\nFRB 20220208A 1 .4311 440 .73c−23.3 1 .0 0 .11\nFRB 20220307B 1 .431,321 499 .2 −947.2 0 .70 0 .10\nFRB 20220310F 1 .431,321 462 .2 11 .4 0 .61 0 .07\nFRB 20220319D 1 .431,321 111 .0 59 .9 0 .16 0 .03\nFRB 20220330D 1 .4311 467 .788c−122.2 0 .73 0 .03\nFRB 20220418A 1 .431,321 623 .5 6 .1 0 .64 0 .03\nFRB 20220424E 1 .4311 863 .932c129.0 0 .58 0 .25\nFRB 20220506D 1 .431,321 396 .9 −32.4 0 .95 0 .31\nFRB 20220509G 1 .431,321 269 .5 −109.0 0 .99 0 .05\nFRB 20220726A 1 .4311 686 .232c499.8 0 .94 0 .04\nFRB 20220825A 1 .431,321 651 .2 750 .2 0 .54 0 .07\nFRB 20220831A 1 .4311 1146 .14c772.1 0 .85 0 .02\nFRB 20220920A 1 .431,321 315 .0 −830.3 0 .89 0 .15\nFRB 20221012A 1 .431,321 442 .2 165 .7 0 .63 0 .01\nFRB 20221029A 1 .4311 1391 .75c−155.8 0 .86 0 .07\nFRB 20221101A 1 .4311 1475 .53c4670 0 .55 0 .11\nFRB 20221101B 1 .4311 491 .554c−32.2 0 .99 0 .19\n1Michilli et al. (2018);2Hilmarsson et al. (2021b);3Luo et al. (2020);4Mckinven et al. (2023b);5Chawla et al. (2020);\n6Pleunis et al. (2021a);7Sand et al. (2022);8Mckinven et al. (2023a);9Feng et al. (2022);10Dai et al. (2022);\n11Anna-Thomas et al. (2023);12Day et al. (2020);13Bhardwaj et al. (2021);14Nimmo et al. (2022b);\n15Hilmarsson et al. (2021a);16Kumar et al. (2022);17Xu et al. (2022);18Mckinven & CHIME/FRB Collaboration (2022);\n19Ravi et al. (2023);20Zhang et al. (2023);21Masui et al. (2015);22Keane et al. (2016);23Ravi et al. (2016);\n24Petroff et al. (2017);25Caleb et al. (2018);26Os lowski et al. (2019);27Cho et al. (2020);28Bhandari et al. (2020);\n29Connor et al. (2020);30Pastor-Marazuela et al. (2023);31Sherman et al. (2023a);32Sherman et al. (2023b);\na|V|/Iranges from ∼0.0 to 0 .34 across the burst;bPolarization calibration was corrupted by radio frequency interference,\ntherefore no L/IorV/Iare reported;cDM obtained from the TNS entry for the FRB. DM uncertainty is not reported.\nB.POLARIZATION POSITION ANGLE UNCERTAINTY ESTIMATES\nOur estimate of the PA uncertainty follows Vernstrom et al. (2019):\nσψ=1\n2Qobs,derotUobs,derot\nQ2\nobs,derot+U2\nobs,derot\"\u0012σQobs ,derot\nQobs,derot\u00132\u0012σUobs ,derot\nUobs,derot\u00132#1/2\n, (B1)\nwhere σQobs ,derotandσUobs ,derotare the measured rms noise in Qobs,derot andUobs,derot, respectively. We note that\nEquation B1 is an approximation for the true PA measurement uncertainties and becomes increasingly inaccurate\nfor lower S/N in L. Setting a minimum threshold on the S/N of L(S/N(L)thresh ) can help to ensure that we are\nonly presenting PA results for which we can correctly characterize the uncertainty. To determine a sensible value for\nS/N(L)thresh , we construct a simple simulation as follows.\nWe synthesize a set of polarized signals by taking 10000 draws of Stokes QsimandUsimfrom normal distributions\nwith means µQsimandµUsimand standard deviations σQsimandσQsimwith arbitrary units:\nQsim∼ N(µQsim, σQsim= 1), (B2)\nUsim∼ N(µUsim= 0.1, σQsim= 1). (B3)\nHere, we set σQsim=σQsim= 1, while progressively increasing µQsimin the range [0 .1,10] in intervals of 0 .1. This creates\na set of 100 synthetic polarized measurements (each with 10000 simulated points in Q−Uspace) in which most of the40\nFigure 18. The estimated PA uncertainty using Equation B1, σψ,est(dash-dotted line), compared to the uncertainty derived\nvia a brute force approach (solid line) as a function of the S/N in Lin our synthetic polarized measurements. The vertical\ndashed line indicates S /N(L)thresh = 5, which we apply to our FRB data.\nlinearly polarized signal arises from Stokes Qand the linearly polarized S/N grows incrementally with increasing µQsim.\nFor each synthetic polarized measurement, we compute a distribution of the PAs following Equation 14, assuming that\nour simulated QsimandUsimhave been de-rotated. Then we estimate the PA measurement uncertainty based on the\napproximation in Equation B1, σψ,est, and also via a brute force approach by taking the standard deviation of the\nresulting PA distribution, σψ,bf. Figure 18 shows σψ,est(dash-dotted line) and σψ,bf(solid line) as a function of S/N\ninL. We find that σψ,estandσψ,bfbegin to visibly diverge at a LS/N of ≲3, with a mean |σψ,est−σψ,bf| ∼0.1 deg\natLS/N of >3.\nTherefore, we set a conservative S /N(L)thresh = 5 in our analysis (depicted by the dashed line in Figure 18) and\nmask out any points in the PA, L,V,L/I, and|V|/Iprofiles that fall below this limit. This helps to ensure that the\nerror properties of the PAs that we analyze are well estimated, which is particularly important when considering PA\nvariations intrinsic to the source (e.g., in Section 2.3).\nREFERENCES\nAbbate, F., Noutsos, A., Desvignes, G., et al. 2023,\nMNRAS, 524, 2966, doi: 10.1093/mnras/stad2047\nAmaral, A. D., Vernstrom, T., & Gaensler, B. M. 2021,\nMNRAS, 503, 2913, doi: 10.1093/mnras/stab564\nAnderson, T., & Darling, D. 1954, Journal of the American\nStatistical Association, 49, 765\nAnna-Thomas, R., Connor, L., Dai, S., et al. 2023, Science,\n380, 599, doi: 10.1126/science.abo6526\nAstropy Collaboration, Robitaille, T. P., Tollerud, E. J.,\net al. 2013, A&A, 558, A33,\ndoi: 10.1051/0004-6361/201322068Astropy Collaboration, Price-Whelan, A. M., Sip˝ ocz, B. M.,\net al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f\nAstropy Collaboration, Price-Whelan, A. M., Lim, P. L.,\net al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74\nBeck, R. 2001, SSRv, 99, 243,\ndoi: 10.1023/A:1013805401252\nBeloborodov, A. M. 2020, ApJ, 896, 142,\ndoi: 10.3847/1538-4357/ab83eb\nBeniamini, P., Kumar, P., & Narayan, R. 2022, MNRAS,\n510, 4654, doi: 10.1093/mnras/stab3730Polarization properties of CHIME/FRB non-repeaters 41\nBethapudi, S., Spitler, L. G., Main, R. A., Li, D. Z., &\nWharton, R. S. 2023, MNRAS, 524, 3303,\ndoi: 10.1093/mnras/stad2009\nBhandari, S., Bannister, K. W., Lenc, E., et al. 2020, ApJL,\n901, L20, doi: 10.3847/2041-8213/abb462\nBhandari, S., Heintz, K. E., Aggarwal, K., et al. 2022, AJ,\n163, 69, doi: 10.3847/1538-3881/ac3aec\nBhardwaj, M., Gaensler, B. M., Kaspi, V. M., et al. 2021,\nApJL, 910, L18, doi: 10.3847/2041-8213/abeaa6\nBhardwaj, M., Michilli, D., Kirichenko, A. Y., et al. 2023,\narXiv e-prints, arXiv:2310.10018,\ndoi: 10.48550/arXiv.2310.10018\nBochenek, C. D., Ravi, V., Belov, K. V., et al. 2020,\nNature, 587, 59, doi: 10.1038/s41586-020-2872-x\nBrentjens, M. A., & de Bruyn, A. G. 2005, A&A, 441, 1217,\ndoi: 10.1051/0004-6361:20052990\nBurn, B. J. 1966, MNRAS, 133, 67,\ndoi: 10.1093/mnras/133.1.67\nCaleb, M., Keane, E. F., van Straten, W., et al. 2018,\nMNRAS, 478, 2046, doi: 10.1093/mnras/sty1137\nCarretti, E., Vacca, V., O’Sullivan, S. P., et al. 2022,\nMNRAS, 512, 945, doi: 10.1093/mnras/stac384\nChatterjee, S., Law, C. J., Wharton, R. S., et al. 2017,\nNature, 541, 58, doi: 10.1038/nature20797\nChawla, P., Andersen, B. C., Bhardwaj, M., et al. 2020,\nApJL, 896, L41, doi: 10.3847/2041-8213/ab96bf\nCHIME/FRB Collaboration, Andersen, B. C., Bandura,\nK. M., et al. 2020, Nature, 587, 54,\ndoi: 10.1038/s41586-020-2863-y\nCHIME/FRB Collaboration, Amiri, M., Andersen, B. C.,\net al. 2021, ApJS, 257, 59,\ndoi: 10.3847/1538-4365/ac33ab\nCHIME/FRB Collaboration, Andersen, B. C., Bandura, K.,\net al. 2023a, ApJ, 947, 83, doi: 10.3847/1538-4357/acc6c1\nCHIME/FRB Collaboration, Amiri, M., Andersen, B. C.,\net al. 2023b, arXiv e-prints, arXiv:2311.00111,\ndoi: 10.48550/arXiv.2311.00111\nCHIME/FRB Collaboration, Andersen, B. C., Bandura, K.,\nBhardwaj, M., et al. 2022, Nature, 607, 256,\ndoi: 10.1038/s41586-022-04841-8\nCho, H., Macquart, J.-P., Shannon, R. M., et al. 2020,\nApJL, 891, L38, doi: 10.3847/2041-8213/ab7824\nConnor, L., van Leeuwen, J., Oostrum, L. C., et al. 2020,\nMNRAS, 499, 4716, doi: 10.1093/mnras/staa3009\nCook, A. M., Bhardwaj, M., Gaensler, B. M., et al. 2023,\nApJ, 946, 58, doi: 10.3847/1538-4357/acbbd0\nDai, S., Feng, Y., Yang, Y. P., et al. 2022, arXiv e-prints,\narXiv:2203.08151. https://arxiv.org/abs/2203.08151\nDay, C. K., Deller, A. T., Shannon, R. M., et al. 2020,\nMNRAS, 497, 3335, doi: 10.1093/mnras/staa2138Desvignes, G., Kramer, M., Lee, K., et al. 2019, Science,\n365, 1013, doi: 10.1126/science.aav7272\nDolag, K., Gaensler, B. M., Beck, A. M., & Beck, M. C.\n2015, MNRAS, 451, 4277, doi: 10.1093/mnras/stv1190\nEilek, J. A., & Hankins, T. H. 2016, Journal of Plasma\nPhysics, 82, 635820302, doi: 10.1017/S002237781600043X\nEverett, J. E., & Weisberg, J. M. 2001, ApJ, 553, 341,\ndoi: 10.1086/320652\nFaucher-Gigu` ere, C.-A., & Kaspi, V. M. 2006, ApJ, 643,\n332, doi: 10.1086/501516\nFeigelson, E. D., & Babu, G. J. 2012, Modern Statistical\nMethods for Astronomy (Cambridge University Press),\ndoi: 10.48550/arXiv.1205.2064\nFeng, Y., Li, D., Yang, Y.-P., et al. 2022, Science, 375,\n1266, doi: 10.1126/science.abl7759\nGaensler, B. M., Landecker, T. L., Taylor, A. R., &\nPOSSUM Collaboration. 2010, in American Astronomical\nSociety Meeting Abstracts, Vol. 215, American\nAstronomical Society Meeting Abstracts #215, 470.13\nGajjar, V., Siemion, A. P. V., Price, D. C., et al. 2018,\nApJ, 863, 2, doi: 10.3847/1538-4357/aad005\nGardenier, D. W., Connor, L., van Leeuwen, J., Oostrum,\nL. C., & Petroff, E. 2021, A&A, 647, A30,\ndoi: 10.1051/0004-6361/202039626\nGardner, F. F., & Whiteoak, J. B. 1966, ARA&A, 4, 245,\ndoi: 10.1146/annurev.aa.04.090166.001333\nGil, J. A., & Han, J. L. 1996, ApJ, 458, 265,\ndoi: 10.1086/176809\nGordon, A. C., Fong, W.-f., Kilpatrick, C. D., et al. 2023,\nApJ, 954, 80, doi: 10.3847/1538-4357/ace5aa\nHackstein, S., Br¨ uggen, M., Vazza, F., Gaensler, B. M., &\nHeesen, V. 2019, MNRAS, 488, 4220,\ndoi: 10.1093/mnras/stz2033\nHankins, T. H., Eilek, J. A., & Jones, G. 2016, ApJ, 833,\n47, doi: 10.3847/1538-4357/833/1/47\nHankins, T. H., Kern, J. S., Weatherall, J. C., & Eilek,\nJ. A. 2003, Nature, 422, 141, doi: 10.1038/nature01477\nHarris, C. R., Millman, K. J., van der Walt, S. J., et al.\n2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2\nHeald, G., Braun, R., & Edmonds, R. 2009, A&A, 503, 409,\ndoi: 10.1051/0004-6361/200912240\nHeesen, V., O’Sullivan, S. P., Br¨ uggen, M., et al. 2023,\nA&A, 670, L23, doi: 10.1051/0004-6361/202346008\nHelsel, D. R. 2005, Nondetects and data analysis. Statistics\nfor censored environmental data. (Wiley-Interscience)\nHilmarsson, G. H., Spitler, L. G., Main, R. A., & Li, D. Z.\n2021a, MNRAS, 508, 5354, doi: 10.1093/mnras/stab2936\nHilmarsson, G. H., Michilli, D., Spitler, L. G., et al. 2021b,\nApJL, 908, L10, doi: 10.3847/2041-8213/abdec042\nHunter, J. D. 2007, Computing in Science & Engineering, 9,\n90, doi: 10.1109/MCSE.2007.55\nHutschenreuter, S., Anderson, C. S., Betti, S., et al. 2022,\nA&A, 657, A43, doi: 10.1051/0004-6361/202140486\nIbik, A. L., Drout, M. R., Gaensler, B. M., et al. 2023,\narXiv e-prints, arXiv:2304.02638,\ndoi: 10.48550/arXiv.2304.02638\nJames, C. W. 2023, PASA, 40, e057,\ndoi: 10.1017/pasa.2023.51\nJohnston, S., Kramer, M., Karastergiou, A., et al. 2023a,\nMNRAS, 520, 4801, doi: 10.1093/mnras/stac3636\n—. 2023b, MNRAS, 520, 4801,\ndoi: 10.1093/mnras/stac3636\nKaplan, E. L., & Meier, P. 1958, Journal of the American\nStatistical Association, 53, 457\nKaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55,\n261, doi: 10.1146/annurev-astro-081915-023329\nKeane, E. F., Johnston, S., Bhandari, S., et al. 2016,\nNature, 530, 453, doi: 10.1038/nature17140\nKendall, M. G. 1938, Biometrika, 30, 81\nKirsten, F., Marcote, B., Nimmo, K., et al. 2022, Nature,\n602, 585, doi: 10.1038/s41586-021-04354-w\nKolmogorov, A. 1956, Foundations of the theory of\nprobability (Chelsea Publishing Company)\nKumar, P., Lu, W., & Bhattacharya, M. 2017, MNRAS,\n468, 2726, doi: 10.1093/mnras/stx665\nKumar, P., Shannon, R. M., Lower, M. E., et al. 2022,\nMNRAS, 512, 3400, doi: 10.1093/mnras/stac683\nLam, M. T., Cordes, J. M., Chatterjee, S., et al. 2016, ApJ,\n821, 66, doi: 10.3847/0004-637X/821/1/66\nLaw, C. J., Sharma, K., Ravi, V., et al. 2023, arXiv e-prints,\narXiv:2307.03344, doi: 10.48550/arXiv.2307.03344\nLopaka, L. 2020, NADA: Nondetects and Data Analysis for\nEnvironmental Data.\nhttps://CRAN.R-project.org/package=NADA\nLorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic,\nD. J., & Crawford, F. 2007, Science, 318, 777,\ndoi: 10.1126/science.1147532\nLorimer, D. R., & Kramer, M. 2012, Handbook of Pulsar\nAstronomy (Cambridge University Press)\nLower, M. E., Johnston, S., Shannon, R. M., Bailes, M., &\nCamilo, F. 2021, MNRAS, 502, 127,\ndoi: 10.1093/mnras/staa3789\nLuo, R., Wang, B. J., Men, Y. P., et al. 2020, Nature, 586,\n693, doi: 10.1038/s41586-020-2827-2\nLyubarsky, Y. 2014, MNRAS, 442, L9,\ndoi: 10.1093/mnrasl/slu046\n—. 2019, MNRAS, 483, 1731, doi: 10.1093/mnras/sty3233\nLyutikov, M. 2021, ApJ, 922, 166,\ndoi: 10.3847/1538-4357/ac1b32Macquart, J. P., Prochaska, J. X., McQuinn, M., et al.\n2020, Nature, 581, 391, doi: 10.1038/s41586-020-2300-2\nManchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M.\n2005, AJ, 129, 1993, doi: 10.1086/428488\nMannings, A. G., Fong, W.-f., Simha, S., et al. 2021, ApJ,\n917, 75, doi: 10.3847/1538-4357/abff56\nMannings, A. G., Pakmor, R., Prochaska, J. X., et al. 2023,\nApJ, 954, 179, doi: 10.3847/1538-4357/ace7bb\nMantel, N. 1966, Cancer Chemotherapy Reports, 50, 163\nMarcote, B., Paragi, Z., Hessels, J. W. T., et al. 2017,\nApJL, 834, L8, doi: 10.3847/2041-8213/834/2/L8\nMasui, K., Lin, H.-H., Sievers, J., et al. 2015, Nature, 528,\n523, doi: 10.1038/nature15769\nMcKee, J. W., Lyne, A. G., Stappers, B. W., Bassa, C. G.,\n& Jordan, C. A. 2018, MNRAS, 479, 4216,\ndoi: 10.1093/mnras/sty1727\nMckinven, R., & CHIME/FRB Collaboration. 2022, The\nAstronomer’s Telegram, 15679, 1\nMckinven, R., Michilli, D., Masui, K., et al. 2021, ApJ, 920,\n138, doi: 10.3847/1538-4357/ac126a\nMckinven, R., Gaensler, B. M., Michilli, D., et al. 2023a,\nApJ, 950, 12, doi: 10.3847/1538-4357/acc65f\n—. 2023b, ApJ, 951, 82, doi: 10.3847/1538-4357/acd188\nMena-Parra, J., Leung, C., Cary, S., et al. 2022, AJ, 163,\n48, doi: 10.3847/1538-3881/ac397a\nMetzger, B. D., Margalit, B., & Sironi, L. 2019, MNRAS,\n485, 4091, doi: 10.1093/mnras/stz700\nMichilli, D., Seymour, A., Hessels, J. W. T., et al. 2018,\nNature, 553, 182, doi: 10.1038/nature25149\nMichilli, D., Masui, K. W., Mckinven, R., et al. 2021, ApJ,\n910, 147, doi: 10.3847/1538-4357/abe626\nMichilli, D., Bhardwaj, M., Brar, C., et al. 2023, ApJ, 950,\n134, doi: 10.3847/1538-4357/accf89\nMoffett, D. A., & Hankins, T. H. 1999, ApJ, 522, 1046,\ndoi: 10.1086/307654\nNg, C., Pandhi, A., Naidu, A., et al. 2020, MNRAS, 496,\n2836, doi: 10.1093/mnras/staa1658\nNimmo, K., Hewitt, D. M., Hessels, J. W. T., et al. 2022a,\nApJL, 927, L3, doi: 10.3847/2041-8213/ac540f\nNimmo, K., Hessels, J. W. T., Kirsten, F., et al. 2022b,\nNature Astronomy, 6, 393,\ndoi: 10.1038/s41550-021-01569-9\nNiu, C. H., Aggarwal, K., Li, D., et al. 2022, Nature, 606,\n873, doi: 10.1038/s41586-022-04755-5\nOcker, S. K., Cordes, J. M., Chatterjee, S., et al. 2023,\nMNRAS, 519, 821, doi: 10.1093/mnras/stac3547\nOppermann, N., Junklewitz, H., Greiner, M., et al. 2015,\nA&A, 575, A118, doi: 10.1051/0004-6361/201423995\nOs lowski, S., Shannon, R. M., Ravi, V., et al. 2019,\nMNRAS, 488, 868, doi: 10.1093/mnras/stz1751Polarization properties of CHIME/FRB non-repeaters 43\nParent, E., Chawla, P., Kaspi, V. M., et al. 2020, ApJ, 904,\n92, doi: 10.3847/1538-4357/abbdf6\nPastor-Marazuela, I., van Leeuwen, J., Bilous, A., et al.\n2023, A&A, 678, A149,\ndoi: 10.1051/0004-6361/202243339\nPeto, R., & Peto, J. 1972, Journal of the Royal Statistical\nSociety, 135, 185\nPetroff, E., Bailes, M., Barr, E. D., et al. 2015, MNRAS,\n447, 246, doi: 10.1093/mnras/stu2419\nPetroff, E., Burke-Spolaor, S., Keane, E. F., et al. 2017,\nMNRAS, 469, 4465, doi: 10.1093/mnras/stx1098\nPiro, A. L., & Gaensler, B. M. 2018, ApJ, 861, 150,\ndoi: 10.3847/1538-4357/aac9bc\nPleunis, Z., Michilli, D., Bassa, C. G., et al. 2021a, ApJL,\n911, L3, doi: 10.3847/2041-8213/abec72\nPleunis, Z., Good, D. C., Kaspi, V. M., et al. 2021b, ApJ,\n923, 1, doi: 10.3847/1538-4357/ac33ac\nPrice, D. C., Flynn, C., & Deller, A. 2021, PASA, 38, e038,\ndoi: 10.1017/pasa.2021.33\nPurcell, C. R., Van Eck, C. L., West, J., Sun, X. H., &\nGaensler, B. M. 2020, ASCL, 2005.003.\nhttp://ascl.net/2005.003\nQiu, J. L., Tong, H., & Wang, H. G. 2023, ApJ, 958, 78,\ndoi: 10.3847/1538-4357/ad003f\nR Core Team. 2020, R: A Language and Environment for\nStatistical Computing, R Foundation for Statistical\nComputing, Vienna, Austria.\nhttps://www.R-project.org/\nRadhakrishnan, V., & Cooke, D. J. 1969, Astrophys. Lett.,\n3, 225\nRavi, V., Shannon, R. M., Bailes, M., et al. 2016, Science,\n354, 1249, doi: 10.1126/science.aaf6807\nRavi, V., Catha, M., Chen, G., et al. 2023, ApJL, 949, L3,\ndoi: 10.3847/2041-8213/acc4b6\nSand, K. R., Faber, J. T., Gajjar, V., et al. 2022, ApJ, 932,\n98, doi: 10.3847/1538-4357/ac6cee\nSchnitzeler, D. H. F. M. 2010, MNRAS, 409, L99,\ndoi: 10.1111/j.1745-3933.2010.00957.x\nSeymour, A., Michilli, D., & Pleunis, Z. 2019, DM phase:\nAlgorithm for correcting dispersion of radio signals,\nAstrophysics Source Code Library, record ascl:1910.004.\nhttp://ascl.net/1910.004\nSherman, M. B., Connor, L., Ravi, V., et al. 2023a, arXiv\ne-prints, arXiv:2308.06813.\nhttps://arxiv.org/abs/2308.06813\n—. 2023b, ApJL, 957, L8, doi: 10.3847/2041-8213/ad0380\nShin, K., Masui, K. W., Bhardwaj, M., et al. 2023, ApJ,\n944, 105, doi: 10.3847/1538-4357/acaf06Skilling, J. 2004, in American Institute of Physics\nConference Series, Vol. 735, Bayesian Inference and\nMaximum Entropy Methods in Science and Engineering:\n24th International Workshop on Bayesian Inference and\nMaximum Entropy Methods in Science and Engineering,\ned. R. Fischer, R. Preuss, & U. V. Toussaint, 395–405,\ndoi: 10.1063/1.1835238\nSmirnov, N. 1948, The Annals of Mathematical Statistics,\n19, 279\nSobey, C., Bilous, A. V., Grießmeier, J. M., et al. 2019,\nMNRAS, 484, 3646, doi: 10.1093/mnras/stz214\nSpearman, C. 1904, The American Journal of Psychology,\n15, 201\nTaylor, A. R., Stil, J. M., & Sunstrum, C. 2009, ApJ, 702,\n1230, doi: 10.1088/0004-637X/702/2/1230\nTendulkar, S. P., Bassa, C. G., Cordes, J. M., et al. 2017,\nApJL, 834, L7, doi: 10.3847/2041-8213/834/2/L7\nVan Eck, C. L., Gaensler, B. M., Hutschenreuter, S., et al.\n2023, ApJS, 267, 28, doi: 10.3847/1538-4365/acda24\nVerbunt, F., Igoshev, A., & Cator, E. 2017, A&A, 608, A57,\ndoi: 10.1051/0004-6361/201731518\nVernstrom, T., Gaensler, B. M., Rudnick, L., & Andernach,\nH. 2019, ApJ, 878, 92, doi: 10.3847/1538-4357/ab1f83\nVirtanen, P., Gommers, R., Oliphant, T. E., et al. 2020,\nNature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2\nWang, P. F., Han, J. L., Xu, J., et al. 2023, Research in\nAstronomy and Astrophysics, 23, 104002,\ndoi: 10.1088/1674-4527/acea1f\nWaskom, M. L. 2021, Journal of Open Source Software, 6,\n3021, doi: 10.21105/joss.03021\nWeltevrede, P., & Johnston, S. 2008, MNRAS, 391, 1210,\ndoi: 10.1111/j.1365-2966.2008.13950.x\nXu, H., Niu, J. R., Chen, P., et al. 2022, Nature, 609, 685,\ndoi: 10.1038/s41586-022-05071-8\nYamasaki, S., & Totani, T. 2020, ApJ, 888, 105,\ndoi: 10.3847/1538-4357/ab58c4\nYang, Y.-P., & Zhang, B. 2018, ApJ, 868, 31,\ndoi: 10.3847/1538-4357/aae685\nYao, J. M., Manchester, R. N., & Wang, N. 2017, ApJ, 835,\n29, doi: 10.3847/1538-4357/835/1/29\nYoung, M. D. T., Chan, L. S., Burman, R. R., & Blair,\nD. G. 2010, MNRAS, 402, 1317,\ndoi: 10.1111/j.1365-2966.2009.15972.x\nZhang, Y.-K., Li, D., Zhang, B., et al. 2023, ApJ, 955, 142,\ndoi: 10.3847/1538-4357/aced0b\nZhu, W., Xu, H., Zhou, D., et al. 2023, Science Advances,\n9, eadf6198, doi: 10.1126/sciadv.adf6198" }, { "title": "2401.17426v1.Superiority_of_Multi_Head_Attention_in_In_Context_Linear_Regression.pdf", "content": "Superiority of Multi-Head Attention in In-Context Linear Regression\nYingqian Cui1Jie Ren1Pengfei He1Jiliang Tang1Yue Xing2\nAbstract\nWe present a theoretical analysis of the perfor-\nmance of transformer with softmax attention in\nin-context learning with linear regression tasks.\nWhile the existing literature predominantly fo-\ncuses on the convergence of transformers with\nsingle-/multi-head attention, our research centers\non comparing their performance. We conduct\nan exact theoretical analysis to demonstrate that\nmulti-head attention with a substantial embed-\nding dimension performs better than single-head\nattention. When the number of in-context exam-\nplesDincreases, the prediction loss using single-\n/multi-head attention is in O(1/D), and the one\nfor multi-head attention has a smaller multiplica-\ntive constant. In addition to the simplest data\ndistribution setting, we consider more scenarios,\ne.g., noisy labels, local examples, correlated fea-\ntures, and prior knowledge. We observe that,\nin general, multi-head attention is preferred over\nsingle-head attention. Our results verify the ef-\nfectiveness of the design of multi-head attention\nin the transformer architecture.\n1. Introduction\nIn-context learning (ICL) is a concept developed in natural\nlanguage processing (NLP). With the rise of transformer\narchitecture, NLP models become increasingly powerful\nand show their ability to learn new knowledge even without\ntuning the model parameters. Given prompts with several\nexamples, these models can generate improved responses,\nshowcasing their ability to adapt and ‘learn’ from the pro-\nvided context (Dong et al., 2022).\nThe mechanism of transformers has been widely studied in\nthe theoretical literature, with a main focus on linear atten-\ntion (Katharopoulos et al., 2020; Choromanski et al., 2020;\nSchlag et al., 2021; Liu et al., 2023; Ahn et al., 2023b),\n1Department of Computer Science and Engineering, Michi-\ngan State University, US.2Department of Statistics and Probabil-\nity, Michigan State Uniersity, US.. Correspondence to: Yue Xing\n.and emerging interest in the effectiveness and superiority\nof softmax attention function (Deng et al., 2023b;a; Trauger\nand Tewari, 2023; Hahn, 2020; Chiang and Cholak, 2022).\nIn recent literature, people have started to work on the theo-\nretical understanding of ICL, e.g., Zhang et al. (2023); Oy-\nmak et al. (2023); Li et al. (2023a); Huang et al. (2023);\nMahankali et al. (2023); Wu et al. (2023). Besides, V on Os-\nwald et al. (2023); Ahn et al. (2023a); Aky ¨urek et al.\n(2022); Zhang et al. (2023) explain how ICL learns gradi-\nent descent and linear regression models. Bai et al. (2023)\nstudies ICL in generalized linear models, ridge regression,\nand LASSO. Cheng et al. (2023) investigate the ability\nof transformers to conduct ICL on non-linear functions.\nBased on V on Oswald et al. (2023); Dai et al. (2023), ICL\ncan be connected with the gradient descent method.\nBesides, some other studies work on multi-head attention.\nFor example, Mahdavi et al. (2023) explored the memoriza-\ntion capacities of multi-head attention, and An et al. (2020)\nindicates a trade-off between the approximation accuracy\nand number of heads. Another work (Li et al., 2023b) stud-\nies the effectiveness of ReLU-activated transformers and\nshows the existence of multi-layer large transformers that\ncan conduct various regression tasks. In addition, the work\nof Deora et al. (2023) investigates the convergence and\ngeneralization performance of multi-head attention in clas-\nsification tasks.\nHowever, we notice that existing theoretical literature fo-\ncuses on either single-head or multi-head attention, and\nthere is limited theoretical understanding of their differ-\nence. This work bridges this gap by considering trans-\nformer with single/multi-head softmax attention to study\nits ICL performance in linear regression tasks. We provide\na clear comparison to quantify the superiority of multi-head\nattention over single-head attention. Different from Zhang\net al. (2023), we do not consider linear multi-attention be-\ncause linear-activated single-layer single-head attention is\nsufficient to learn linear regression tasks.\nOur contributions are summarized as follows:\nFirst, we study the transformer architecture and show the\neffectiveness of single-head attention with softmax activa-\ntion in ICL. We derive the exact prediction risk under the\nconsidered data generation model. (Section 4.2)\n1arXiv:2401.17426v1 [cs.LG] 30 Jan 2024Superiority of Multi-Head Attention in In-Context Linear Regression\nSecond, we show that multi-head attention is better than\nsingle-head attention by figuring out the exact prediction\nrisk of multi-head attention. With a high input embedding\ndimension, multi-head attention improves the flexibility of\nthe transformer and can obtain a better kernel for the linear\nregression task. (Section 4.3)\nFinally, we also investigate the scenarios where the training\ndata include prior knowledge, noisy responses, correlated\nfeatures, or local examples. While our analysis shows that\nin most scenarios, multi-head attention is preferred over\nsingle-head attention, we also reveal some interesting be-\nhaviors of ICL when the data consists of local examples\nor have prior knowledge. Specifically, we observe that (1)\nwhen there is a “strong” prior knowledge, predicting us-\ning this prior knowledge leads to good performance; (2)\nwhether local examples help or not depends on their dis-\ntance to the query. (Section 5)\nOur results provide a comprehensive understanding about\nthe impact of single-/multi-head attention on the perfor-\nmance of ICL. In addition, it also offers practical guid-\nance for selecting the efficient attention mechanism in real-\nworld applications. In particular, multi-head attention is\npreferred than single-head attention, and the total number\nof embedding dimensions should be much larger than the\nnumber of heads.\n2. Other Related Works\nIn addition to the aforementioned theoretical studies, we\nreview some empirical studies below:\nThe initial work utilized by Zhang et al. (2023) is done\nby Garg et al. (2022). They empirically show the effec-\ntiveness of the transformer in performing ICL, with perfor-\nmance matching the optimal least squares estimator. Fur-\nthermore, Aky ¨urek et al. (2022) demonstrate that the ICL\ndone by transformers implicitly applies standard learning\nalgorithms to conduct the in-context tasks.\nFollowing these works, Ahuja et al. (2023) extend the set-\nting of Garg et al. (2022) by considering a mixture of in-\ncontext tasks in the pre-training and demonstrating the abil-\nity of the transformer to resemble the effect of Bayesian\npredictor under the multi-task setting. Ravent ´os et al.\n(2023) empirically investigates how the diversity of the\ntasks in the pre-training dataset influences the performance\nof the transformer to do in-context tasks that are unseen in\nthe pre-training stage. Some other related works can also\nbe found in Fu et al. (2023); von Oswald et al. (2023); Shi\net al. (2022); Saparov and He (2022); Lu et al. (2021); Liu\net al. (2021); Work; Min et al. (2021a); Zhang et al. (2022);\nChen et al. (2022); Min et al. (2021b).3. Notations\nTo mathematically define ICL, instead of merely passing a\nquery xq∈Rd(or a test sample) to the transformer to make\na prediction, ICL passes a prompt, i.e., a few examples with\ntheir labels {(xi, yi)}i=1,...,D together with the query xq, to\nthe transformer. Using the prompt in the format of\nE=\u0012\nx1x2. . . x Dxq\ny1y2. . . y D0\u0013\n∈R(d+1)×(D+1),(1)\nthe transformer can learn from the examples to infer the\nprediction for xq. Following Zhang et al. (2023), we con-\nsider the following simplified neural network architecture\nf(E) =E+WoutH, (2)\nwhere Hdenotes the attention node and Woutrepresents\na fully-connected layer. Here, H=concat (H1, . . . , H h),\nwithhbeing the number of heads in the multi-head atten-\ntion. Each attention head Hjis given by\nHj=WV\njE·ϕ \n(WK\njE)⊤WQ\njE\nρj!\n(3)\nwhere ρjis a normalization factor, and the activation func-\ntionϕis the column-wise softmax function. For each j,\nWV\nj, WK\nj, WQ\nj∈Rm×d, and m=d/h. When h= 1,\nthe attention is single-head . When h >1, the structure is\ncalled multi-head attention.\nTo train the model, we fetch the last element of the last row\ninf(E)as the predicted value of yq(denote as byq), then\nminimize\nL(Ω) = E{xi},xq,θ(byq−yq)2, (4)\nwhere Ωis the set of parameters.\n4. Superiority of Multi-Head Attention\nIn this section, we introduce the assumptions, present\nthe optimal solution of single-head attention in ICL, and\ndemonstrate the superiority of multi-head attention.\n4.1. Assumptions\nBefore showing the main results, we first introduce the data\ngeneration model and configurations of the transformer:\nAssumption 4.1 (Data Generation Model) .In each\nprompt, the examples (xi, yi)and(xq, yq)are i.i.d. sam-\nples from the following noiseless regression model:\n• The “input” x∼N(0, Id).\n• The “output” y=θ⊤x.\n• The coefficients θare the same for the samples in\nthe same prompt and are different across different\nprompts. In addition, θ∼N(0, Id/d).\n2Superiority of Multi-Head Attention in In-Context Linear Regression\nAssumption 4.2 (Lazy Training) .We consider a lazy train-\ning scheme when deriving the optimal solution of the trans-\nformer. We first fix Wout,WVand optimize over the\nother parameters, and then figure out the best solution of\nWout, WV.\nAssumption 4.1 follows Zhang et al. (2023) on the data\ngeneration model. For simplicity, we use Gaussian distri-\nbution to avoid tedious discussions on potential heavy tail\nissues, and our proofs, in general, can be extended to other\ndata generation models.\nIn Assumption 4.2, we apply lazy training to the attention.\nAs mentioned by Huang et al. (2023), training all parame-\nters in a transformer is a non-convex problem. Assuming\nlazy training can simplify the analysis. However, it is im-\nportant to note that our conclusion, which states that the\noptimal solution of single-head attention has a worse ICL\nperformance than multi-head attention, is independent of\nthe lazy training assumption.\n4.2. Optimal Solution for Single-Head Attention\nIn this subsection, we figure out the optimal solution of\nsingle-head attention and summarize it in Theorem 4.1.\nTheorem 4.1 (Optimal Solution of Single-Head Attention) .\nUnder Assumption 4.1, 4.2, assume (1) there is infinite\ntraining prompts, (2) (WoutWV)d+1,:= (0, . . . , 0, v), and\n(3)(WK)⊤WQis in a format of\n(WK)⊤WQ=\u0014A0\nb0\u0015\n,\nthen when D→ ∞ , the loss value is\nL(A, b, v ) =1\ndtr((vA−Id)2) +v2∥b∥2E∥θ∥4+O(1/D),\nand the optimal solution satisfies that ∥vA−Id∥2\nF=\nO(1/D), and∥vb∥2=O(1/D). In addition, when tak-\ningA=Id/vandb= 0,\nL(Id/v,0, v) =v2\nD\u0012v2\nv2−2\u0013d\n2\n+1\nD\u0012v2\nv2−2\u0013d\n2+1\n+o(1/D).\n(5)\nDenoting the optimal solution as A∗,b∗, for any v2>2,\nL(A∗, b∗, v)−L(Id/v,0, v) =o(1/D).\nTheorem 4.1 shows the optimal solution of the single-head\nattention when fixing (WoutWV)d+1,:. To prove Theorem\n4.1, we use Taylor expansion to separate the denominator\nand numerator of the attention scores. Since there are in-\nfinitely many training samples, we directly calculate the ex-\npectation of the output. In addition, it is also observed that\nthe loss function is a quadratic function of Aandb. The\nformal proof can be found in Appendix A.1.In Theorem 4.1, we study the optimality of Aandbwhen\nkeeping vfixed. Generally, vaffects the prediction loss\nin two ways. First, as stated in Theorem 4.1, it is es-\nsential that v2>2. When taking v2≤2andA=\nId/v,exp(x⊤\nqAxq) = exp( ∥xq∥2/v2)≥exp(∥xq∥2/2),\nthusexp(x⊤\nqAxq)has no finite expectation, and the atten-\ntion score of (xq,0)towards itself becomes predominantly\nhigh. Second, when taking Taylor expansion on attention\nscores, we need the remainder terms to be negligible.\nRemark 4.1. In addition to the optimal solution in Theo-\nrem 4.1, since the prediction loss is approximately a convex\nfunction of (A, b), numerical methods such as gradient de-\nscent can successfully approximate the optimal solution.\nSimulation. While Theorem 4.1 presents the ICL perfor-\nmance of single-head attention given a fixed v, we also con-\nduct some simulation study to investigate the role of v. In\nthe simulation, we take different choices of (d, D, v )and\nset(A, b) = ( Id/v,0)to calculate the corresponding pre-\ndiction loss (MSE). We run 200k repetitions for each set-\nting to get an average and an error bar. The results are\nsummarized in Figure 1 and 2.\nFigure 1. ICL performance of single-head attention with (A, b) =\n(Id/v,0)andD= 1000 .\nFigure 2. ICL performance of single-head attention with (A, b) =\n(Id/v,0)andd= 5.\nThe figures show that the simulation of prediction loss\naligns well with theoretical values. Besides, there are two\nmain observations. First, with fixed (d, D), the MSE ex-\n3Superiority of Multi-Head Attention in In-Context Linear Regression\nhibits a U-shaped behavior as a function of v. In Figure 1,\nwhen dincreases, the optimal vincreases as well. Second,\nwhen fixing v, the MSE increases with d(Figure 1) and\ndecreases with D(Figure 2).\n4.3. Multi-Head Attention is Better\nWhile Section 4.2 shows the effectiveness of single-head\nattention, in this subsection, through deriving the exact per-\nformance, we show that multi-head attention is better than\nsingle-head attention.\nIn the implementation of Garg et al. (2022), a linear layer is\nused to transform Einto a space with a higher dimension\nbefore feeding the input into the transformer. In the last\nlayer of the transformer, another linear layer is added so\nthat the network outputs a single number. This increases\nthe flexibility of the transformer.\nWe denote the transformation matrix applied before the\ntransformer as Win∈Rp×(d+1)with p≥d+ 1. In\nsingle-head attention, introducing the linear layer does\nnot change the results. This is because the rank of\nW⊤\nin(WK)⊤WQWinis still d+ 1, meaning that the ad-\nditional layer does not enlarge the representational capac-\nity of single-head attention. In contrast, multi-head atten-\ntion benefits from the dimension increase provided by Win,\nwhich allows each head to learn more features and poten-\ntially improve predictions. To explain this, in single-head\nattention, there is only one attention score matrix, and all\nthe attention scores are non-negative. In contrast, we can\ncombine the attention scores from different heads in multi-\nhead attention so that some weights can negatively con-\ntribute to the final prediction. This flexibility is beneficial in\nlinear regression, as negative weights and positive weights\ntogether can provide a better fit for the data.\nWe consider a two-head attention in the following theorem\nto illustrate the superiority:\nTheorem 4.2 (Multi-head Attention is Better) .Consider a\ntwo-head attention with\n(WK\n1)⊤WQ\n1=\u0014A10\nb10\u0015\n,(WK\n2)⊤WQ\n2=\u0014A20\nb20\u0015\n.\nThe parameters WV\n1,WV\n2,WinandWoutsatisfy\nf(E)d+1,D+1=vmE d+1,:ϕ((WK\n1E)⊤WQ\n1E:,D+1)\n−vnE d+1,:ϕ((WK\n2E)⊤WQ\n2E:,D+1).\nThen the optimal solution satisfies that ∥vmA 1−\nvnA 2∥2\nF=O(1/D)and∥mb1−nb2∥2=O(1/D).\nConsidering a specific case when m= 2,n= 1,b1=\nb2= 0, and setting A1= (c/v)Idfor some 0< c < 1,\nwe find that A2= ((2 c−1)/v)Id. Consequently, for any\nv2>max{2c2,2(2c−1)2},\nL(A1, A2, b1, b2, v)=4v2\nD \u0012v2\nv2−2c2\u0013d\n2\n−\u0012v2\nv2−2c(2c−1)\u0013d\n2!\n+v2\nD\u0012v2\nv2−2(2c−1)2\u0013d\n2\n+(2c−1)2\nD\u0012v2\nv2−2(2c−1)2\u0013\u0012v2\nv2−2(2c−1)2\u0013d\n2\n−(8c−4)c\nD\u0012v2\nv2−2c(2c−1)\u0013\u0012v2\nv2−2c(2c−1)\u0013d\n2\n+4c2\nD\u0012v2\nv2−2c2\u0013\u0012v2\nv2−2c2\u0013d\n2\n+o(1/D).\nIn Theorem 4.2, the condition v2>max{2c2,2(2c−1)2}\nguarantees that E(x⊤\nqAxq)is finite. The proof of Theo-\nrem 4.2 is similar Theorem 4.1, and the main difficulty\nlies in the calculations regarding the cross terms of the two\nheads. Details of the proof can be found in Appendix A.2.\nIn addition to the formula in Theorem 4.2, the following\nproposition illustrates why the loss of multi-head attention\nis smaller than the optimal loss of single-head attention:\nProposition 4.1. Following the setting of Theorem 4.1 and\n4.2, multi-head attention can be reduced to single-head at-\ntention when taking c= 1, and c= 1 is not the optimal\nchoice for multi-head attention to achieve the minimal loss.\nThe proof of Proposition 4.1 and some simulations can the\nfound in Appendix A.3.\nSimulation. We also conduct some simulations to compare\nthe prediction loss of single- and multi-head attention. We\nuse the setting in Theorem 4.2, i.e., m= 2, n= 1 with\nA1= (c/v)IdandA2= (2c−1)Id/v.\nFrom Figure 3, we can see that the simulation result is close\nto the theoretical value for every choice of (c, v). In addi-\ntion, the MSE of multi-head attention is smaller than that\nof single-head attention.\nFigure 3. ICL performance of multi-head attention with (m, n ) =\n(2,1),(A1, A2, b1, b2) = (( c/v)Id,((2c−1)/v)Id,0,0), and\n(d, D ) = (5 ,1000) .\n4Superiority of Multi-Head Attention in In-Context Linear Regression\n5. Other Scenarios\nIn addition to the simplest scenario in Section 4.2 and Sec-\ntion 4.3, in this section, we relax the data generation model\nin Assumption 4.1 and discuss some other scenarios to\nunderstand the corresponding optimal solution for single-\nhead attention, and verify that multi-head attention again\ngives better ICL performance. In particular, we consider\nθwith a non-zero mean (prior knowledge, Section 5.1),\nnoisy response (Section 5.2), correlated features (Section\n5.3), and local examples xis given xq(Section 5.4).\n5.1. Prior Knowledge\nFrom the results in Section 4, the trained transformer only\nlearns to compare the similarity of different examples,\nrather than learning any particular knowledge from the\ndataset. In this subsection, we explore whether the trans-\nformer can learn prior knowledge from the training data\nwhere θis not fully random.\nAssumption 5.1. For each prompt, assume that θfollows\nθ0+N(0, σ2Id/d)for some ∥θ0∥= Θ(1) . The value of\nθ0is the same in all prompts.\nThe following theorem presents how the trained trans-\nformer learns θ0:\nTheorem 5.1. Denote (WoutWV)d+1,:= [u, v]for some\nvector uand value v. Assume there are infinite training\nprompts. Under Assumption 5.1, for single-head attention,\nwhen σ2= Θ(1) , the population loss is minimized when\n∥u∥2=O(1/D),∥b∥2=O(1/D), and∥vA−Id∥F=\nO(1/D). For the optimal solution (u∗, b∗, A∗)at a fixed v\nsuch that v2>2, the population loss is given by\nL(u∗, b∗, A∗, v)\n=E\u0012\nyq−(WoutWV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014\nxq\n0\u0015\u0013\u0013 2\n= (∥θ0∥2+σ2)Lnoprior (A∗, b∗, v) +o(1/D),\nwhere Lnoprior (A∗, b∗, v)denotes the optimal population\nloss in Theorem 4.1. When σ2=O(1/D), there exists\ninfinitely many choices of usuch that ∥u∥= Θ(1) and\nL(u, b, A, v ) = O(1/D). The specific conditions are in\nequation (10) in Appendix A.4. For multi-head attention,\nunder the same setting as Theorem 4.2, we denote the pop-\nulation loss in Theorem 4.2 as Lnoprior . Then, when taking\nu= 0,\nL(A1, A2, b1, b2, v)\n= (∥θ0∥2+σ2)Lnoprior (A1, A2, b1, b2, v) +o(1/D).\nThe proof of the theorem is done by computing the par-\ntial derivatives of the loss with respect to the parameters\nand identifying the points where the derivatives equal zero.More details are shown in Appendix A.4 together with\nsome simulation results.\nThere are several implications from Theorem 5.1. First,\nwhen the prior knowledge is weak, i.e., σ2= Θ(1) , the\nbest single-head attention does not learn θ0. Rather, it still\nmakes predictions by comparing the similarity between xq\nandxis. Second, when the prior knowledge is strong, i.e.,\nσ2=O(1/D), we can obtain good prediction performance\nwhen ulearns from θ0. Finally, multi-head attention can\nstill be better than single-head attention.\n5.2. Noisy Response\nWe consider linear regression tasks with noisy responses,\ni.e.,yi=x⊤\niθ+ϵiwithϵi∼N(0, σ2\nϵ). The following\ntheorem demonstrates the effect of the response noise.\nTheorem 5.2. Assume infinite training prompts and yi=\nx⊤\niθ+ϵiwithϵi∼N(0, σ2\nϵ). The optimal solution of\nsingle-head attention satisfies tr((Id−A/v)2) =O(1/D)\nand∥b∥2=O(1/D).\nWhen taking A=Id/v, where v2>2, and b= 0,\nL(Id/v,0, v)\n=σ2\nϵ+v2σ2\nϵ\nD\u0012v2\nv2−2\u0013d\n2\n+1\nDv4−v2\nv2−2\u0012v2\nv2−2\u0013d\n2\n+o(1/D).\nFor multi-head attention, taking the same parameter values\nas Theorem 4.2,\nL(A1, A2, b1, b2, v)\n=4v2(1 +σ2\nϵ)\nD \u0012v2\nv2−2c2\u0013d\n2\n−\u0012v2\nv2−2c(2c−1)\u0013d\n2!\n+v2(1 +σ2\nϵ)\nD\u0012v2\nv2−2(2c−1)2\u0013d\n2\n+σ2\nϵ\n+(2c−1)2\nD\u0012v2\nv2−2(2c−1)2\u0013\u0012v2\nv2−2(2c−1)2\u0013d\n2\n−(8c−4)c\nD\u0012v2\nv2−2c(2c−1)\u0013\u0012v2\nv2−2c(2c−1)\u0013d\n2\n+4c2\nD\u0012v2\nv2−2c2\u0013\u0012v2\nv2−2c2\u0013d\n2\n+o(1/D).\nThe proof of Theorem 5.2 is similar to that of Theorem 4.1\nand 4.2, which can be found in Appendix A.5. Theorem\n5.2 indicates that the existence of the noise ϵidoes not sig-\nnificantly change the optimal solution. For both single- and\nmulti-head attention, there are some additional terms in the\nprediction loss associated with σ2\nϵ.\nAnother difference from the noiseless case is the optimal v.\nSpecifically, with a larger σ2\nϵ, the optimal vshould ensure\n5Superiority of Multi-Head Attention in In-Context Linear Regression\nv2is smaller. To explain this, denoting wias the atten-\ntion score for each example i, andwqas the attention score\nfor itself, then the predicted value is byq=P\nivwi(x⊤\niθ+\nϵi) =P\nivwix⊤\niθ+P\nivwiϵi, and V ar(P\nivwiϵi) =\nv2σ2\nϵPw2\ni. Therefore, a smaller v2is required to achieve\na smaller variance of prediction.\nIn terms of the difference between single- and multi-head\nattention, from the theorem it is evident that multi-head at-\ntention is still superior to single-head attention.\n5.3. Correlated Features\nIn this subsection, we consider a scenario where xhas some\ncorrelated features, i.e. x∼N(0,Σ)for some general\nΣ∈Rd×d. The following theorem presents the ICL per-\nformance of the transformer in this situation.\nTheorem 5.3. Assume x∼N(0,Σ)and the read-in\nlayer is Win= Σ−1/2. For single-head attention, when\nE(x⊤\nqAxq)<∞, the optimal solution satisfies Eθ⊤(Id−\nvA)2θ=O(1/D)and∥b∥2E∥θ∥4=O(1/D)where\nθ∼N(0,Σ−1/2/d). For multi-head attention, the best\nICL performance is not worse than single-head attention.\nTo show Theorem 5.3, instead of directly deriving the loss\nstarting from correlated features, we show the equivalence\nof (1) the problem with correlated features and (2) the prob-\nlem with isotropic features and a new θdistribution. De-\ntailed discussions can be found in Appendix A.6.\nTheorem 5.3 implies some changes in the prediction loss\nwhen considering correlated features. In detail, following\nthe setting of Theorem 4.1, i.e., θ∼N(0, Id/d),Eθ⊤(Id−\nvA)2θ=tr((Id−vA)2). But in Theorem 5.3, the value\nofEθ⊤(Id−vA)2θdepends on the exact distribution of θ.\nHowever, similar to Theorem 4.1, we still have A=Id/v\nandb= 0close to the same optimal solution.\n5.4. Local Examples\nWhile ICL can learn from the examples chosen from the\nwhole population, we are also interested in its efficiency\nwhen the in-context samples are selected from the neigh-\nbors of xq.\nThe following two theorems indicate the prediction perfor-\nmance when the prompt is constructed with local examples.\nIn Theorem 5.4, we consider the scenario where xis are\nneighbors of xqin both training stage and inference stage.\nIn Theorem 5.5, we consider another scenario with distri-\nbution shift: xis are totally random in the training stage,\nand are neighbors of xqin the inference stage. We provide\nthe proof of the two theorems in Appendix 5.4 and 5.5\nTheorem 5.4. Assuming that for both training and test\nprompts, the in-context examples in the prompt are gen-\nerated from xi∼N(xq, σ2\nxId), and the response yi=x⊤\niθwithθ∼N(0, Id/d). Then when E(x⊤\nqAxq)<∞, the\noptimal solution of the single-head transformer satisfies\nv[σ2\nx(A+θb⊤) +Id]→Id,\nand the minimal population risk is\nL(A∗, b∗, v) =O(σ2\nx/D) +o(1/D).\nTheorem 5.4 indicates that the optimal solution for local\nexamples is different from the one when xis are fully ran-\ndom. We do not consider multi-head attention because: (1)\nifσ2\nx→0, the single-head attention is effective enough\nwith the overall prediction risk in o(1/D); (2) if σ2\nxis large\nenough, the signal xqis much smaller than the noise size\nσx, and the problem is similar to the scenario of Theorem\n4.1 and 4.2. Another observation is that, when taking dif-\nferent σ2\nxs in the training and inference stages, as long as\nσ2\nx=o(1)in the two stages, the ICL in the inference stage\ncan still achieve good performance.\nWhile the above result shows that a small distribution shift\ninσ2\nxdoes not hurt the ICL performance, the following the-\norem considers a large distribution shift:\nTheorem 5.5. Assume the training prompts are sampled\nin the same way as Theorem 4.1, i.e., xis are randomly\nselected from the whole population. Besides, in the infer-\nence stage, in each prompt, xq∼N(0, Id), and the other\nexamples xi∼N(xq, σ2\nxId)for some σ2\nx>0. Then for\nsingle-head attention, the prediction loss goes to zero only\nwhen σ2\nx+v−1 = 0 .\nWhile Theorem 5.4 demonstrates the benefit of local exam-\nples, Theorem 5.5 reveals that ICL may not be consistent\nwhen facing distribution shifts. From simulations in Sec-\ntion 6, the actual vobtained in training does not satisfy\nσ2\nx+v−1 = 0 . As a result, it is expected that ICL cannot\nperform well in such a scenario in general.\n6. Experiments\nWhile the simulations in previous sections directly calcu-\nlate the prediction loss of ICL given specific parameter\nweights, in this section, we conduct experiments starting\nfrom training the transformer. Due to the page limit, we\npostpone the experiments for noisy response and correlated\nfeatures to Appendix B.\n6.1. Experimental Settings\nWe modify the implementation of Garg et al. (2022) to con-\nduct the experiments. In particular, we change the input\nformat in Garg et al. (2022) and use the format Edefined\nin (1). In each training iteration, we generate a new batch of\n64 prompts to train the transformer. In terms of the loss to\nbe minimized during the training, we use the one defined\nin (4), i.e., we optimize the loss between yqandbyq. We\n6Superiority of Multi-Head Attention in In-Context Linear Regression\ntrain the transformers with 500k iterations and use Adam\noptimizer with 0.0001 learning rate.\nIn the inference stage, we randomly sample 1280 prompts\nto obtain the average and error bar of the loss. Instead of\nonly using xqto calculate the loss, for each in-context ex-\nample i∈[D], we also make the ICL prediction and calcu-\nlate the corresponding loss.\n6.2. Single-head vs Multi-head\nIn the experiment, we compare the performance of single-\nhead and multi-head attention. We set the input embedding\ndimension to p= 64 , the dimension of in-context examples\ntod= 5, and vary the number of heads for analysis. The\nresults are summarized in Figure 4.\nFigure 4 shows that single-head attention has a worse ICL\nperformance than multi-head attention. In addition, al-\nthough our theorems do not consider such a scenario, for\nmulti-head attention, when his too large so that p/h < d ,\nthe ICL performance can be affected. When taking h= 64 ,\nthe ICL performance gets worse.\nFigure 4. A comparison between single-head and multi-head with\nthe input embedding dimension p= 64 .\nIn addition to the ICL performance, we also conduct an-\nother experiment to examine (WK)⊤WQ. We remove\nthe read-in layer, train the transformer, and print out\n(WK)⊤WQ. We repeat the experiment 10 times to see\nthe value of (WK)⊤WQ. As in Theorem 4.1, for single-\nhead attention without the read-in layer, (WK)⊤WQis ex-\npected to be in the form of I/vwhen v2>2. In the 10\ntrials, 9 of them observe such a result, where 5 trials have\nv >0as in Figure 5 and 4 trials have v <0as in Figure 6.\nWe also visualize the attention score corresponding to these\ntwo cases in Figure 15 (See Appendix B). These results in-\ndicates that the theoretical global minimum is highly likely\nto be attained in the real practice of transformer training.\n6.3. Input Embedding Dimension\nAs mentioned in Section 4.3, increasing the input embed-\nding dimension pprovides the flexibility of multi-head at-\ntention to achieve better ICL performance. In this experi-\nFigure 5. An illustration of the matrix (WK)⊤WQfor the no\nread-in case. It is expected to be some kinds of αId. 4 of 10\ntrials are like this.\nFigure 6. An illustration of the matrix (WK)⊤WQfor the no\nread-in case. It is expected to be some kinds of αId. 5 of 10\ntrials are like this.\nment, we change pto examine the performance.\nIn Figure 7, we fix the dimension in each head ( p/h), and\nincrease h. We can observe that when the dimension is suf-\nficient, the increasing hleads to a smaller prediction loss.\nFigure 7. Results of fixing the dimension allocated in each head\n(p/h = 8), and increasing the number of heads.\nIn addition, we also run different p/h for different p.\nAs shown in Table 1, we can also see that for all p=\n64,128,256, the following setting gives good ICL perfor-\nmance: (1) p/h≥dand (2) his as large as possible.\n6.4. Prior Knowledge\nIn the experiment about prior knowledge, we study the\ninference-stage performance under different choices of θ.\n7Superiority of Multi-Head Attention in In-Context Linear Regression\nTable 1. Different choices of head.\np h p/h ICL p h p/h ICL\n61 6 0.41878\n641 64 0.18983\n2 3 0.29825 8 8 0.00769\n3 2 0.58036 16 4 0.01724\n6 1 0.56292 64 1 0.04899\n1281 128 0.16619\n2561 256 0.16141\n8 16 0.00577 8 32 0.00587\n16 8 0.00244 16 16 0.00144\n64 2 0.00611 64 4 0.00134\n128 1 0.01254 128 2 0.00159\n256 1 0.00549\nBefore training, we randomly generate a θ0∼N(0, Id).\nDuring the training, to generate one training prompt, we\ngenerate θ=θ0+N(0, σ2Id/d), and then generate the ex-\namples (xi, yi)based on θ. In the test stage, we generate\ndifferent prompts following different θ. The prediction re-\nsults can be found in Figure 8 for single-head attention and\nFigure 9 for multi-head attention with σ2= 1andα= 0.1.\nWe make the following observations. First, comparing Fig-\nure 8 with Figure 9, we note that multi-head attention gives\nbetter ICL performance than single-head attention. Second,\nas shown in Figure 8 and Figure 9, when θ∥θ0, a smaller\n∥θ∥implies better ICL performance. To explain this, since\nthe ICL loss is in O(1/D), a smaller ∥θ∥indicates less vari-\nation among the response of different examples; thus, the\nmultiplicative constant of the O(1/D)is smaller. Finally,\ncomparing η⊥θ0withη=θ0, although ∥θ∥forη⊥θ0\nis smaller, the ICL performance is worse. This observation\nimplies that the transformer learns the prior knowledge θ0.\nFigure 8. Head 1 prior knowledge. More results can be found\nfrom Figure 19 in Appendix B.3.\n6.5. Local Examples\nAs discussed in Theorem 5.4 and 5.5, when both the train-\ning and inference stage use local examples with the same\ndistribution (i.e., same σ2\nx), ICL leads to consistent predic-\ntions. When there is a large distribution shift, the prediction\nis not consistent.\nIn Table 2, we demonstrate the ICL performance in the in-\nference stage with local examples. As expected, the pre-\nFigure 9. Head 16 prior knowledge. More results can be found in\nAppendix B Figure 20.\ndiction is more accurate when the training and testing data\nhave the same distribution, with a diminishing σ2\nx. On the\nother hand, when training with fully random prompts (i.e.,\nnot local examples), the prediction is inconsistent.\nTable 2. ICL performance for local examples in the inference\nstage with/without distribution shift in the training data. “Fully\nrandom”: not local examples. More results can be found in Ap-\npendix 21.\nTraining TestingICL\n1 head 16 heads\nSame as testingσ2\nx= 120.01464 0.00285\nσ2\nx= 0.120.00049 0.00096\nσ2\nx= 0.0122.50e-05 9.79e-06\nFully randomσ2\nx= 120.29317 0.60400\nσ2\nx= 0.120.39023 1.23142\nσ2\nx= 0.0120.41253 1.12165\n7. Conclusion\nThis study explicitly calculates the ICL performance in lin-\near regression tasks to show that multi-head attention is\npreferred over single-head attention. In addition to the sim-\nplest case of noiseless regression, we extend the analysis to\nother scenarios. When the data contain prior knowledge,\na transformer that learns the prior knowledge can perform\nwell in ICL prediction. When the examples in the prompt\nare neighbors of xq, the ICL prediction can be very efficient\nif there is no distribution shift.\nThere are several future directions. First, our current study\nconsiders the case for large enough D. We may consider\nrelaxing this condition and studying the finite-example sce-\nnario. Second, although we consider different scenarios of\nthe data, we always consider linear regression tasks. We\nmay extend the analysis to other problems such as non-\nparametric models. Finally, we assume that the training\ndataset has almost infinite samples and directly study the\npopulation loss. We may extend it to a finite-prompt sce-\nnario and investigate the generalization performance.\n8Superiority of Multi-Head Attention in In-Context Linear Regression\n8. Impact Statements\nThis paper presents work whose goal is to advance the field\nof Machine Learning via deepening the theoretical under-\nstanding of existing neural network architectures. This pa-\nper does not introduce new methodology or new datasets.\nTherefore, there is no extra ethical impact or societal im-\nplication which is worth special emphasis here.\nReferences\nKwangjun Ahn, Xiang Cheng, Hadi Daneshmand, and Su-\nvrit Sra. Transformers learn to implement precondi-\ntioned gradient descent for in-context learning. arXiv\npreprint arXiv:2306.00297 , 2023a.\nKwangjun Ahn, Xiang Cheng, Minhak Song, Chulhee\nYun, Ali Jadbabaie, and Suvrit Sra. Linear attention is\n(maybe) all you need (to understand transformer opti-\nmization). arXiv preprint arXiv:2310.01082 , 2023b.\nKabir Ahuja, Madhur Panwar, and Navin Goyal. In-context\nlearning through the bayesian prism. arXiv preprint\narXiv:2306.04891 , 2023.\nEkin Aky ¨urek, Dale Schuurmans, Jacob Andreas, Tengyu\nMa, and Denny Zhou. What learning algorithm is in-\ncontext learning? investigations with linear models.\narXiv preprint arXiv:2211.15661 , 2022.\nBang An, Jie Lyu, Zhenyi Wang, Chunyuan Li, Chang-\nwei Hu, Fei Tan, Ruiyi Zhang, Yifan Hu, and\nChangyou Chen. Repulsive attention: Rethinking multi-\nhead attention as bayesian inference. arXiv preprint\narXiv:2009.09364 , 2020.\nYu Bai, Fan Chen, Huan Wang, Caiming Xiong, and Song\nMei. Transformers as statisticians: Provable in-context\nlearning with in-context algorithm selection. arXiv\npreprint arXiv:2306.04637 , 2023.\nMingda Chen, Jingfei Du, Ramakanth Pasunuru, Todor\nMihaylov, Srini Iyer, Veselin Stoyanov, and Zor-\nnitsa Kozareva. Improving in-context few-shot learn-\ning via self-supervised training. arXiv preprint\narXiv:2205.01703 , 2022.\nXiang Cheng, Yuxin Chen, and Suvrit Sra. Trans-\nformers implement functional gradient descent to learn\nnon-linear functions in context. arXiv preprint\narXiv:2312.06528 , 2023.\nDavid Chiang and Peter Cholak. Overcoming a the-\noretical limitation of self-attention. arXiv preprint\narXiv:2202.12172 , 2022.\nKrzysztof Choromanski, Valerii Likhosherstov, David Do-\nhan, Xingyou Song, Andreea Gane, Tamas Sarlos, PeterHawkins, Jared Davis, Afroz Mohiuddin, Lukasz Kaiser,\net al. Rethinking attention with performers. arXiv\npreprint arXiv:2009.14794 , 2020.\nDamai Dai, Yutao Sun, Li Dong, Yaru Hao, Shuming Ma,\nZhifang Sui, and Furu Wei. Why can gpt learn in-\ncontext? language models implicitly perform gradient\ndescent as meta-optimizers. In ICLR 2023 Workshop on\nMathematical and Empirical Understanding of Founda-\ntion Models , 2023.\nYichuan Deng, Zhihang Li, and Zhao Song. Attention\nscheme inspired softmax regression. arXiv preprint\narXiv:2304.10411 , 2023a.\nYichuan Deng, Zhao Song, and Tianyi Zhou. Superiority\nof softmax: Unveiling the performance edge over linear\nattention. arXiv preprint arXiv:2310.11685 , 2023b.\nPuneesh Deora, Rouzbeh Ghaderi, Hossein Taheri, and\nChristos Thrampoulidis. On the optimization and gen-\neralization of multi-head attention. arXiv preprint\narXiv:2310.12680 , 2023.\nQingxiu Dong, Lei Li, Damai Dai, Ce Zheng, Zhiyong\nWu, Baobao Chang, Xu Sun, Jingjing Xu, and Zhifang\nSui. A survey for in-context learning. arXiv preprint\narXiv:2301.00234 , 2022.\nDeqing Fu, Tian-Qi Chen, Robin Jia, and Vatsal Sharan.\nTransformers learn higher-order optimization methods\nfor in-context learning: A study with linear models.\narXiv preprint arXiv:2310.17086 , 2023.\nShivam Garg, Dimitris Tsipras, Percy S Liang, and Gregory\nValiant. What can transformers learn in-context? a case\nstudy of simple function classes. Advances in Neural In-\nformation Processing Systems , 35:30583–30598, 2022.\nMichael Hahn. Theoretical limitations of self-attention in\nneural sequence models. Transactions of the Association\nfor Computational Linguistics , 8:156–171, 2020.\nYu Huang, Yuan Cheng, and Yingbin Liang. In-\ncontext convergence of transformers. arXiv preprint\narXiv:2310.05249 , 2023.\nAngelos Katharopoulos, Apoorv Vyas, Nikolaos Pappas,\nand Franc ¸ois Fleuret. Transformers are rnns: Fast au-\ntoregressive transformers with linear attention. In Inter-\nnational conference on machine learning , pages 5156–\n5165. PMLR, 2020.\nHongkang Li, Meng Wang, Sijia Liu, and Pin-Yu Chen. A\ntheoretical understanding of shallow vision transform-\ners: Learning, generalization, and sample complexity.\narXiv preprint arXiv:2302.06015 , 2023a.\n9Superiority of Multi-Head Attention in In-Context Linear Regression\nYingcong Li, Muhammed Emrullah Ildiz, Dimitris Papail-\niopoulos, and Samet Oymak. Transformers as algo-\nrithms: Generalization and stability in in-context learn-\ning. 2023b.\nJiachang Liu, Dinghan Shen, Yizhe Zhang, Bill Dolan,\nLawrence Carin, and Weizhu Chen. What makes\ngood in-context examples for gpt- 3?arXiv preprint\narXiv:2101.06804 , 2021.\nLangming Liu, Liu Cai, Chi Zhang, Xiangyu Zhao, Jing-\ntong Gao, Wanyu Wang, Yifu Lv, Wenqi Fan, Yiqi Wang,\nMing He, et al. Linrec: Linear attention mechanism for\nlong-term sequential recommender systems. In Proceed-\nings of the 46th International ACM SIGIR Conference\non Research and Development in Information Retrieval ,\npages 289–299, 2023.\nYao Lu, Max Bartolo, Alastair Moore, Sebastian Riedel,\nand Pontus Stenetorp. Fantastically ordered prompts and\nwhere to find them: Overcoming few-shot prompt order\nsensitivity. arXiv preprint arXiv:2104.08786 , 2021.\nArvind Mahankali, Tatsunori B Hashimoto, and Tengyu\nMa. One step of gradient descent is provably the optimal\nin-context learner with one layer of linear self-attention.\narXiv preprint arXiv:2307.03576 , 2023.\nSadegh Mahdavi, Renjie Liao, and Christos Thram-\npoulidis. Memorization capacity of multi-head attention\nin transformers. arXiv preprint arXiv:2306.02010 , 2023.\nSewon Min, Mike Lewis, Hannaneh Hajishirzi, and Luke\nZettlemoyer. Noisy channel language model prompt-\ning for few-shot text classification. arXiv preprint\narXiv:2108.04106 , 2021a.\nSewon Min, Mike Lewis, Luke Zettlemoyer, and Hannaneh\nHajishirzi. Metaicl: Learning to learn in context. arXiv\npreprint arXiv:2110.15943 , 2021b.\nSamet Oymak, Ankit Singh Rawat, Mahdi Soltanolkotabi,\nand Christos Thrampoulidis. On the role of attention in\nprompt-tuning. arXiv preprint arXiv:2306.03435 , 2023.\nAllan Ravent ´os, Mansheej Paul, Feng Chen, and Surya\nGanguli. Pretraining task diversity and the emergence of\nnon-bayesian in-context learning for regression. arXiv\npreprint arXiv:2306.15063 , 2023.\nAbulhair Saparov and He He. Language models are greedy\nreasoners: A systematic formal analysis of chain-of-\nthought. arXiv preprint arXiv:2210.01240 , 2022.\nImanol Schlag, Kazuki Irie, and J ¨urgen Schmidhuber.\nLinear transformers are secretly fast weight program-\nmers. In International Conference on Machine Learning ,\npages 9355–9366. PMLR, 2021.Freda Shi, Mirac Suzgun, Markus Freitag, Xuezhi Wang,\nSuraj Srivats, Soroush V osoughi, Hyung Won Chung,\nYi Tay, Sebastian Ruder, Denny Zhou, et al. Lan-\nguage models are multilingual chain-of-thought reason-\ners.arXiv preprint arXiv:2210.03057 , 2022.\nJacob Trauger and Ambuj Tewari. Sequence length in-\ndependent norm-based generalization bounds for trans-\nformers. arXiv preprint arXiv:2310.13088 , 2023.\nJohannes V on Oswald, Eyvind Niklasson, Ettore Ran-\ndazzo, Jo ˜ao Sacramento, Alexander Mordvintsev, An-\ndrey Zhmoginov, and Max Vladymyrov. Transformers\nlearn in-context by gradient descent. In International\nConference on Machine Learning , pages 35151–35174.\nPMLR, 2023.\nJohannes von Oswald, Eyvind Niklasson, Maximilian\nSchlegel, Seijin Kobayashi, Nicolas Zucchet, Nino\nScherrer, Nolan Miller, Mark Sandler, Max Vladymy-\nrov, Razvan Pascanu, et al. Uncovering mesa-\noptimization algorithms in transformers. arXiv preprint\narXiv:2309.05858 , 2023.\nWhat Makes In-Context Learning Work. Rethinking the\nrole of demonstrations: What makes in-context learning\nwork?\nJingfeng Wu, Difan Zou, Zixiang Chen, Vladimir Braver-\nman, Quanquan Gu, and Peter L Bartlett. How many pre-\ntraining tasks are needed for in-context learning of linear\nregression? arXiv preprint arXiv:2310.08391 , 2023.\nRuiqi Zhang, Spencer Frei, and Peter L Bartlett. Trained\ntransformers learn linear models in-context. arXiv\npreprint arXiv:2306.09927 , 2023.\nYiming Zhang, Shi Feng, and Chenhao Tan. Active ex-\nample selection for in-context learning. arXiv preprint\narXiv:2211.04486 , 2022.\n10Superiority of Multi-Head Attention in In-Context Linear Regression\nA. Proofs\nA.1. Theorem 4.1\nProof of Theorem 4.1. When taking infinite many training samples (prompts), the loss function becomes\nE\u0012\nyq−(WoutWV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E\u0012\nyq−v\u0002y1, y2, . . . , y D,0\u0003\nϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)E{xi}i∈[D]\nyq−v\u0002y1, y2, . . . , y D,0\u0003\nϕ\n\nx⊤\n1Axq+y1b⊤xq\n. . .\nx⊤\nqAxq+ 0\n\n\n2\n=E(xq,θ)E{xi}i∈[D] \nyq−vPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!2\n=E(xq,θ)E{xi}i∈[D]\ny2\nq−2yq \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!\n| {z }\n=A1+ \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!2\n| {z }\n=A2\n.\nWhen D→ ∞ , we have\nE{xi,yi}i∈[D]A1\n=E{xi,yi}i∈[D](−2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)−DEexp(x1⊤Axq+y1b⊤xq) +DEexp(x1⊤Axq+y1b⊤xq)\n=E{xi,yi}i∈[D](−2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nexp(x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq)\n+E{xi,yi}i∈[D](2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\u0010X\nexp(x⊤\niAxq+yib⊤xq)−DEexp(x1⊤Axq+y1b⊤xq)\u0011\n−E{xi,yi}i∈[D](2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\u0010X\nexp(x⊤\niAxq+yib⊤xq)−DEexp(x1⊤Axq+y1b⊤xq)\u00112\n+o(1\nD)\n=A11+A12+A13+o(1\nD).\nWhen taking expectation w.r.t. {xi, yi}i∈[D], we have\nE{xi,yi}i∈[D]PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nexp(x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq)=E{x1,y1}DEθ⊤x1exp(x⊤\n1Axq+y1b⊤xq)\nexp(x⊤qAxq) +DEx1exp(x⊤\n1Axq+y1b⊤xq),\nwhere\nE{x1,y1}exp(x⊤\n1Axq+y1b⊤xq) = exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2),\nE{x1,y1}x1exp(x⊤Axq+y1b⊤xq) = E∂\n∂((A+θb⊤)xq)exp(x⊤\n1Axq+y1b⊤xq)\n= (A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2).\nTherefore, we have\nA11=E{x1,y1} \n−D(2vθ⊤xq)Eθ⊤x1exp(x⊤\n1Axq+y1b⊤xq)\nexp(x⊤qAxq) +DEx1exp(x⊤\n1Axq+y1b⊤xq)!\n11Superiority of Multi-Head Attention in In-Context Linear Regression\n=E{x1,y1} \n−D(2vθ⊤xq)Eθ⊤x1exp(x⊤\n1Axq+y1b⊤xq)\nDEx1exp(x⊤\n1Axq+y1b⊤xq)+D(2vθ⊤xq)Eθ⊤x1exp(x⊤\n1Axq+y1b⊤xq) exp( x⊤\nqAxq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))2!\n+E{x1,y1} \n−D(2vθ⊤xq)Eθ⊤x1exp(x⊤\n1Axq+y1b⊤xq) exp(2 x⊤\nqAxq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))3!\n| {z }\n=o(1/D)\n=−D(2vθ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)\n+D(2vθ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2) exp( x⊤\nqAxq)\nD2exp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq)+o(1\nD)\n=−(2vθ⊤xq)θ⊤(A+θb⊤)xq+(2vθ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+o(1\nD),\nA12=E{xi,yi}i∈[D](2vθ⊤xqPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq))(Pexp(x⊤\niAxq+yib⊤xq))\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n−E{xi,yi}i∈[D](2vθ⊤xqPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq))(DEexp(x1⊤Axq+y1b⊤xq))\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n=(2Dvθ⊤xq)Eθ⊤x1exp(2( x⊤\n1Axq+y1b⊤xq)))\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2+(2D(D−1)vθ⊤xqEx1,x2θ⊤x1exp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n−E{xi,yi}i∈[D](2vθ⊤xqPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq))(DEexp(x1⊤Axq+y1b⊤xq))\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n=A121+A122+A123,\nwhere\nA121 =4Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(2 x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(exp( x⊤qAxq) +Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2\n=4Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(2 x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2+o(1\nD)\n=4v\nD(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq) +o(1\nD),\nA122 =2(D(D−1)v(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n= 2 v(θ⊤xq)θ⊤(A+θb⊤)xq−4Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq) exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n−2Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2−2v(θ⊤xq)θ⊤(A+θb⊤)xqexp(2 x⊤\nqAxq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))2\n= 2 v(θ⊤xq)θ⊤(A+θb⊤)xq−4v(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq) exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2)\n(DEexp(x1⊤Axq+y1b⊤xq))2\n−(2Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(DEexp(x1⊤Axq+y1b⊤xq))2+o(1\nD)\n12Superiority of Multi-Head Attention in In-Context Linear Regression\n= 2 v(θ⊤xq)θ⊤(A+θb⊤)xq−4v(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\n(Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2))−2v\nD(θ⊤xq)θ⊤(A+θb⊤)xq+o(1\nD),\nA123 =−(2Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2))(DEexp(x1⊤Axq+y1b⊤xq))\nexp(x⊤qAxq) +Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2\n=−2D2v(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\nexp(x⊤qAxq) +Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2\n=−2v(θ⊤xq)θ⊤(A+θb⊤)xq+(4Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq) exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)/2)\n(exp( x⊤qAxq) +Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2+o(1\nD)\n=−2v(θ⊤xq)θ⊤(A+θb⊤)xq+(4Dv(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq) exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)/2)\n(Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)2+o(1\nD)\n=−2v(θ⊤xq)θ⊤(A+θb⊤)xq+4v(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+o(1\nD),\nand\nA13=−2D2v(θ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq−yib⊤xq)(Eexp(x1⊤Axq+y1b⊤xq))2\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n−2Dv(θ⊤xq)Ex1θ⊤x1exp(3 x1⊤Axq+ 3y1b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n−2D(D−1)(θ⊤xq)Ex1,x2θ⊤x1exp(x1⊤Axq+y1b⊤xq) exp(2 x2⊤Axq+ 2y2b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3)\n+4D2v(θ⊤xq)Ex1θ⊤x1exp(2 x1⊤Axq+ 2y1b⊤xq)Eexp(x1⊤Axq+y1b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n+4D2(D−1)v(θ⊤xq)Ex1,x2θ⊤x1exp(x1⊤Axq+y1b⊤xq) exp( x2⊤Axq+y2b⊤xq)Eexp(x1⊤Axq+y1b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n−4D(D−1)v(θ⊤xq)Ex1θ⊤x1exp(2 x1⊤Axq+ 2y1b⊤xq)Ex2exp(x2⊤Axq+y2b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n−2D(D−1)(D−2)v(θ⊤xq)Ex1θ⊤x1exp(x1⊤Axq+y1b⊤xq)Ex2exp(x2⊤Axq+y2b⊤xq)Ex3exp(x3⊤Axq+y3b⊤xq)\n(exp( x⊤qAxq) +DEexp(x1⊤Axq+y1b⊤xq))3\n=2v\nD(θ⊤xq)θ⊤(A+θb⊤)xq−2v\nD(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq).\nTo sum up, we have\nA1=A11+A121+A122+A123+A13\n=−(2vθ⊤xq)θ⊤(A+θb⊤)xq+(2vθ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)\n+2v\nD(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq).\nIn terms of the second-order term, since xis are independent of each other, we have\nE{xi}i∈[D]A2\n13Superiority of Multi-Head Attention in In-Context Linear Regression\n=E{xi}i∈[D] \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nDEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq)!2\n−2E{xi}i∈[D]\u0010\nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\u00112\n\u0000\nDEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq)\u00013\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u0011\n+3E{xi}i∈[D]\u0010\nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\u00112\n\u0000\nDEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq)\u00014\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u00112\n=Dv2Ex1θ⊤x1x⊤\n1θexp(2 x⊤\n1(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))2+D(D−1)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))2\n−E{xi}i∈[D]2D(D−1)v2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))3\n×\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u0011\n+E{xi}i∈[D]3D(D−1)v2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n×\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u00112\n+o(1\nD)\n=A21+A22+A23+A24.\nFor the terms A21toA24, we have\nA21=Dv2θ⊤(Id+ 4(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(2 x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2) + exp( x⊤qAxq))2\n=Dv2θ⊤(Id+ 4(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(2 x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n(Dexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2))2+o(1\nD)\n=v2\nDθ⊤(Id+ 4(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq) +o(1\nD),\nA22=D(D−1)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))2\n−2D(D−1)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq) exp( x⊤\nqAxq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))3+o(1\nD)\n=v2(1−1\nD)(θ⊤(A+θb⊤)xq)2−2v2(θ⊤(A+θb⊤)xq)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+o(1\nD),\nA23=4v2\nD(θ⊤(A+θb⊤)xq)2−8v2\nD(θ⊤(A+θb⊤)xq)2exp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq),\nand\nA24=3D3(D−1)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)(Ex1exp(x⊤\n1Axq+y1b⊤xq))2\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n+3D(D−1)(D−2)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)Ex3exp(2 x⊤\n3Axq+ 2y3b⊤xq)\n(DEx3exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n+12D(D−1)(D−2)v2Ex1,x2θ⊤x1x⊤\n2θexp(2 x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)(Ex1exp(x⊤\n1Axq+y1b⊤xq))\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n+3D(D−1)(D−2)(D−3)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n14Superiority of Multi-Head Attention in In-Context Linear Regression\n(Ex3,x4exp(x⊤\n3(A+θb⊤)xq+x⊤\n4(A+θb⊤)xq))\n−6D2(D−1)(D−2)v2Ex1,x2θ⊤x1x⊤\n2θexp(x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)(Ex1exp(x⊤\n1Axq+y1b⊤xq))2\n(DEx3exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n−12D2(D−1)v2Ex1,x2θ⊤x1x⊤\n2θexp(2 x⊤\n1(A+θb⊤)xq+x⊤\n2(A+θb⊤)xq)(Ex1exp(x⊤\n1Axq+y1b⊤xq))\n(DEx3exp(x⊤\n1Axq+y1b⊤xq) + exp( x⊤qAxq))4\n=−3v2\nD(θ⊤(A+θb⊤)xq)2+3v2\nD(θ⊤(A+θb⊤)xq)2exp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq) +o\u00121\nD\u0013\n.\nTo sum up,\nE{xi,yi}i∈[D]A2=v2\nDθ⊤(Id−(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2(θ⊤(A+θb⊤)xq)2−2v2(θ⊤(A+θb⊤)xq)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+o(1\nD).\nBased on the results of A1andA2, we have\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)(x⊤\nqθ)2+v2(θ⊤(A+θb⊤)xq)2−2v(x⊤\nqθ)(θ⊤(A+θb⊤)xq) +O\u00121\nD\u0013\n=E(xq,θ)\u0000\nθ⊤(v(A+θb⊤)−Id)xq\u00012+O\u00121\nD\u0013\n=E(xq,θ)h\u0000\nθ⊤(vA−Id)xq\u00012+v2(∥θ∥2b⊤xq)2+ 2vθ⊤(vA−Id)xqx⊤\nqb∥θ∥2i\n+O\u00121\nD\u0013\n=1\ndtr\u0000\n(vA−Id)2\u0001\n+v2∥b∥2E∥θ∥4+O\u00121\nD\u0013\n. (6)\nTherefore, to minimize the loss, the optimal Asatisfies tr\u0000\n(vA−Id)2\u0001\n=O(d/D), and∥b∥2=O(1/D).\nFurthermore, we have\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014\nxq\n0\u0015\u0013\u0013 2\n=E(xq,θ)\u0014\n(x⊤\nqθ)2+A1+A2\u0015\n=E(xq,θ)\u0014\n(x⊤\nqθ)2−(2vθ⊤xq)θ⊤(A+θb⊤)xq+2v\nD(θ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2\nDθ⊤(Id−(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq) +v2(θ⊤(A+θb⊤)xq)2\n+(2vθ⊤xq)θ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)−2v2(θ⊤(A+θb⊤)xq)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+o(1\nD)\u0015\n=1\ndtr\u0000\n(Av−Id)2\u0001\n+v2∥b∥2E∥θ∥4+Eθ\u0012v2\nDdet(Σ1)1\n2∥θ∥2+2v\nDdet(Σ1)1\n2θ⊤(A+θb⊤)Σ1θ+2v\nDdet(Σ2)1\n2θ⊤(A+θb⊤)Σ2θ\n−v2\nDdet(Σ1)1\n2θ⊤(A+θb⊤)Σ1(A+θb⊤)⊤θ−2v2\nDdet(Σ2)1\n2θ⊤(A+θb⊤)Σ2(A+θb⊤)⊤θ\u0013\n+o(1\nD), (7)\nwhere Σ1= (I−2(A+θb⊤)⊤(A+θb⊤))−1andΣ2= (I+ (A+θb⊤)⊤(A+θb⊤)−2A)−1.\n15Superiority of Multi-Head Attention in In-Context Linear Regression\nAssuming that A∗=Id\nv+ ∆ A,b∗= ∆ b, where ∆A=O(1√\nD)and∆b=O(1√\nD), we will have\nv2\nDdet(Σ1)1\n2∥θ∥2\f\f\nA=Id\nv+∆A,b=∆b\n=v2\nDdet\u0010\u0000\nI−2(Id/v+ ∆ A+θ∆⊤\nb)⊤(Id/v+ ∆ A+θ∆⊤\nb)\u0001−1\u00111\n2∥θ∥2\n=v2\nDdet \u0012\n(1−2\nv2)I−2\nv(∆A+θ∆⊤\nb)⊤−2\nv(∆A+θ∆⊤\nb)−2(∆A+θ∆⊤\nb)⊤(∆A+θ∆⊤\nb)\u0013−1!1\n2\n∥θ∥2\n=v2\nD(v2−2\nv2)d\n2det \u0012\n(1−2\nv2)I−2\nv(∆A+θ∆⊤\nb)⊤−2\nv(∆A+θ∆⊤\nb)−2(∆A+θ∆⊤\nb)⊤(∆A+θ∆⊤\nb)\u0013−1!\n∥θ∥2.\nTherefore,\nEθ\u0012v2\nDdet(Σ1)1\n2∥θ∥2\f\f\nA=Id\nv+∆A,b=∆b−v2\nDdet(Σ1)1\n2∥θ∥2\f\f\nA=Id\nv,b=0\u0013\n=Eθv2\nD(v2−2\nv2)d\n2\u0012\n−(v2\nv2−2)d+1tr(−2\nv(∆A+θ∆⊤\nb)⊤−2\nv(∆A+θ∆⊤\nb)−2(∆A+θ∆⊤\nb)⊤(∆A+θ∆⊤\nb))\u0013\n∥θ∥2\n=Eθv2\nD(v2\nv2−2)d\n2+1(4\nvtr(∆A) + 2∥∆A∥2\nF+ 2∥∆b∥2∥θ∥2)∥θ∥2=o(1\nD).\nFurthermore, we have:\nΣ2\f\f\nA=Id\nv+∆A,b=∆b=\u0000\nI+ (Id/v+ ∆ A+θ∆⊤\nb)⊤(Id/v+ ∆ A+θ∆⊤\nb)−2(Id/v+ ∆ A)\u0001−1\n=\u0012(v−1)2\nv2(I+v\n(v−1)2(∆A+θ∆⊤\nb) +v\n(v−1)2(∆A+θ∆⊤\nb)⊤−v2\n(v−1)22∆A+v2\n(v−1)2(∆A+θ∆⊤\nb)⊤(∆A+θ∆⊤\nb))\u0013−1\n=v2\n(v−1)2\u0012\nI−v\n(v−1)2(∆A+θ∆⊤\nb)−v\n(v−1)2(∆A+θ∆⊤\nb)⊤+v2\n(v−1)22∆A−v2\n(v−1)2(∆A+θ∆⊤\nb)⊤(∆A+θ∆⊤\nb))\u0013\n= Σ∗\n2.\nTherefore,\n−2v2\nD\u0010\ndet(Σ 2)1\n2θ⊤(A+θb⊤)Σ2(A+θb⊤)⊤θ\f\f\nA=Id\nv+∆A,b=∆b−det(Σ 2)1\n2θ⊤(A+θb⊤)Σ2(A+θb⊤)⊤θ\f\f\nA=Id\nv,b=0\u0011\n=−2v2\nD\u0012\ndet(Σ∗\n2)1\n2θ⊤(I/v+ ∆ A+θ∆⊤\nb)Σ∗\n2(I/v+ ∆ A+θ∆⊤\nb)⊤θ−det(Σ∗\n2)1\n21\n(v−1)2θ⊤θ\u0013\n−2v2\nD\u0012\ndet(Σ∗\n2)1\n21\n(v−1)2θ⊤θ−\u0014\ndet(v2\n(v−1)2I)1\n2\u00151\n(v−1)2θ⊤θ\u0013\n=o(1\nD).\nWe can obtain similar results for other terms. Therefore, we have L(A∗, b∗)−L(Id/v,0) = o(1\nD).\nWhen A=Id\nvandb= 0,\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)\u0014\n(1\nD(θ⊤xq)θ⊤xqexp(x⊤\nqxq/v2) +v2\nDθ⊤θexp(x⊤\nqxq/v2)\u0015\n=v2\nD(v2\nv2−2)d\n2+v2\nD(v2−2)(v2\nv2−2)d\n2+o(1\nD), (8)\nandvshould satisfies v2>2.\n16Superiority of Multi-Head Attention in In-Context Linear Regression\nA.2. Theorem 4.2\nProof of Theorem 4.2.\nE(yq−f(E)d+1,D+1)2\n=E\u0010\nyq−vmE d+1,:ϕ((WK\n1E)⊤WQ\n1E:,D+1) +vnE d+1,:ϕ((WK\n2E)⊤WQ\n2E:,D+1)\u00112\n=E\u0012\nyq−vm\u0002y1, y2, . . . , y D,0\u0003\nϕ\u0012\nE⊤(WK\n1)⊤WQ\n1\u0014xq\n0\u0015\u0013\n+vn\u0002y1, y2, . . . , y D,0\u0003\nϕ\u0012\nE⊤(WK\n2)⊤WQ\n2\u0014xq\n0\u0015\u0013\u0013 2\n=E \nyq−vmPD\ni=1θ⊤xiexp(x⊤\ni(A1+θb⊤\n1)xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)+vnPD\ni=1θ⊤xiexp(x⊤\ni(A2+θb⊤\n2)xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!2\n=E(xq,θ)E{xi}i∈[D]\ny2\nq+ \nvmPD\ni=1θ⊤xiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!2\n| {z }\nB1+ \nvnPD\ni=1θ⊤xiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!2\n| {z }\nB2\n\n+E(xq,θ)E{xi}i∈[D]\n−2yq \nvmPD\ni=1θ⊤xiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!\n| {z }\nB2+ 2yq \nvnPD\ni=1θ⊤xiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!\n| {z }\nB4\n.\n+E(xq,θ)E{xi}i∈[D]\n−2vmPD\ni=1θ⊤xiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)vnPD\ni=1θ⊤xiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)\n| {z }\nB5\n.\nSimilar to the E{xi}i∈[D]A2of A.1, we have\nE{xi}i∈[D]B1=v2m2\nDθ⊤(Id−(A1+θb⊤\n1)xqx⊤\nq(A1+θb⊤\n1)⊤)θexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)\n+v2m2(θ⊤(A1+θb⊤\n1)xq)2−2v2m2(θ⊤(A1+θb⊤\n1)xq)2exp (x⊤\nqA1xq)\nDexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)+o(1\nD),\nE{xi}i∈[D]B2=v2n2\nDθ⊤(Id−(A2+θb⊤\n2)xqx⊤\nq(A2+θb⊤\n2)⊤)θexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)\n+v2n2(θ⊤(A2+θb⊤\n2)xq)2−2v2n2(θ⊤(A2+θb⊤\n2)xq)2exp (x⊤\nqA2xq)\nDexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2)+o(1\nD),\nE{xi}i∈[D]B3=−(2mvθ⊤xq)θ⊤(A1+θb⊤\n1)xq+(2vmθ⊤xq)θ⊤(A1+θb⊤\n1)xqexp(x⊤\nqA1xq)\nDexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)\n+2mv\nD(θ⊤xq)θ⊤(A1+θb⊤\n1)xqexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq) +o(1\nD),\nE{xi}i∈[D]B4= +(2 nvθ⊤xq)θ⊤(A2+θb⊤\n2)xq−(2vnθ⊤xq)θ⊤(A2+θb⊤\n2)xqexp(x⊤\nqA2xq)\nDexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2)\n−2nv\nD(θ⊤xq)θ⊤(A2+θb+ 2⊤)xqexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq) +o(1\nD),\nand\nE{xi}i∈[D]B5\n17Superiority of Multi-Head Attention in In-Context Linear Regression\n=−2v2mnD (D−1)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))\n−2v2mnD (Ex1(θ⊤x1)2exp(x���\n1A1xq+y1b⊤\n1xq) exp( x⊤\n1A2xq+y1b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))\n+E{xi}i∈[D] \n2v2mnD (D−1)(θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))2!\n× DX\ni=1exp(x⊤\niA2xq+yib⊤\n2xq)−DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq)!\n+E{xi}i∈[D] \n2v2mnD (D−1)(θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))2(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))!\n× DX\ni=1exp(x⊤\niA1xq+yib⊤\n1xq)−DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq)!\n−E{xi}i∈[D]2v2mnD (D−1)(θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))2(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))2\n×\u0012DX\ni=1exp(x⊤\niA1xq+yib⊤\n1xq)−DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq)\u0013\n×\u0012DX\ni=1exp(x⊤\niA2xq+yib⊤\n2xq)−DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq)\u0013\n−E{xi}i∈[D]2v2mnD (D−1)(θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))3\n× DX\ni=1exp(x⊤\niA2xq+yib⊤\n2xq)−DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq)!2\n−E{xi}i∈[D]2v2mnD (D−1)(θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))3(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))\n× DX\ni=1exp(x⊤\niA1xq+yib⊤\n1xq)−DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq)!2\n+o(1\nD)\n=B51+B52+B53+B54+B55+B56+B57.\nFor the terms B51toB57, we have\nB51=−2v2mn(1−1\nD)θ⊤(A1+θb1)xqθ⊤(A2+θb2)xq+ 2v2mn1\nDexp(x⊤\nqA1xq)θ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\nexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)\n+2v2mn1\nDexp(x⊤\nqA2xq)θ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\nexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2),\nB52=−2v2mn\nDθ⊤(I+ (A1+θb⊤\n1+A2+θb⊤\n2)x⊤\nqxq(A1+θb⊤\n1+A2+θb⊤\n2)⊤)θ\nexp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2),\nB53=2v2mnD (D−1)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(2 x⊤\n2A2xq+ 2y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))2\n18Superiority of Multi-Head Attention in In-Context Linear Regression\n+2v2mnD (D−1)(Ex1θ⊤x1exp(x⊤\n1(A1+A2)xq+y1(b1+b2)⊤xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))2\n−4v2mnD (D−1)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq)Ex2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y2b⊤\n2xq) + exp( x⊤qA2xq))2\n=2v2mn\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\u0000\n2 exp( x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)−2\u0001\n+2v2mn\nDθ⊤(A1+θb1+A2+θb2)xqθ⊤(A2+θb2)xq\n×exp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2),\nB54=2v2mn\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\u0000\n2 exp( x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)−2\u0001\n+2v2mn\nDθ⊤(A1+θb1)xqθ⊤(A1+θb1+A2+θb2)xq\n×exp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2),\nB55=−2v2mnD3(D−1)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))(Ex2exp(x⊤\n2A2xq+y2b⊤\n2xq))2\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))3\n+4v2mnD2(D−1)(D−2)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))3\n×Ex2,x3exp(x⊤\n2A2xq+y2b⊤\n2xq+x⊤\n3A2xq+y3b⊤\n2xq)\n−2v2mnD (D−1)(D−2)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))3\nEx3exp(2 x⊤\n3A2xq+ 2y3b⊤\n2xq)\n−2v2mnD (D−1)(D−2)(D−3)(Ex1θ⊤x1exp(x⊤\n1A1xq+y1b⊤\n1xq)(Ex2θ⊤x2exp(x⊤\n2A2xq+y2b⊤\n2xq))\n(DEx1exp(x⊤\n1A1xq+y1b⊤\n1xq) + exp( x⊤qA1xq))(DEx2exp(x⊤\n2A2xq+y1b⊤\n2xq) + exp( x⊤qA2xq))3\nEx3exp(x⊤\n3A2xq+y3b⊤\n2xq+x⊤\n4A2xq+y4b⊤\n2xq)\n= 2 v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\n−2v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xqexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq),\nB56= 2 v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\n−2v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xqexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq),\nB57= 2 v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\n−2v2mn1\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xqexp\u0000\nx⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2\u0001\n.\nBased on the results of B1toB5, we have\nE(yq−f(E)d+1,D+1)2\n=E(xq,θ)E{xi}i∈[D](x⊤\nqθ)2+v2m2(θ⊤(A1+θb⊤\n1)xq)2+v2n2(θ⊤(A2+θb⊤\n2)xq)2−2vm(x⊤\nqθ)(θ⊤(A1+θb⊤\n1)xq)\n+2vn(x⊤\nqθ)(θ⊤(A2+θb⊤\n2)xq)−2v2mnθ⊤(A1+θb⊤\n1)xqθ⊤(A2+θb⊤\n2)xq+O(1\nD)\n19Superiority of Multi-Head Attention in In-Context Linear Regression\n= 1 +v2m2\ndtr(A1A⊤\n1) +v2m2∥b1∥2E∥θ∥4+v2n2\ndtr(A2A⊤\n2) +v2n2∥b2∥2E∥θ∥4−2vm\nd·tr(A1)\n+2vn\nd·tr(A2)−2v2mn\nd·tr(A1A⊤\n2)−2v2mnb⊤\n1b2E∥θ∥4+O(1\nD)\n=1\ndtr(vmA 1−vnA 2−I)2+v2(mb1−nb2)2E∥θ∥4+O(1\nD).\nTherefore, the optimal solutions satisfies ∥vmA 1−vnA 2∥2\nF=O(d\nD)and∥mb1−nb2∥2=O(1\nD).\nFurthermore, we have\nE(yq−f(E)d+1,D+1)2\n=E(xq,θ)\u0002\nx⊤\nqθ)2+B1+B2+B3+B4+B51+B52+B53+B54+B55+B56+B57\u0003\n=E(xq,θ)\"\n(x⊤\nqθ)2+v2m2\nDθ⊤(Id−(A1+θb⊤\n1)xqx⊤\nq(A1+θb⊤\n1)⊤)θexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)\n+v2n2\nDθ⊤(Id−(A2+θb⊤\n2)xqx⊤\nq(A2+θb⊤\n2)⊤)θexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)\n+v2m2(θ⊤(A1+θb⊤\n1)xq)2+v2n2(θ⊤(A2+θb⊤\n2)xq)2−(2mvθ⊤xq)θ⊤(A1+θb⊤\n1)xq+ (2nvθ⊤xq)θ⊤(A2+θb⊤\n2)xq\n−2v2m2(θ⊤(A1+θb⊤\n1)xq)2exp (x⊤\nqA1xq)\nDexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)−2v2n2(θ⊤(A2+θb⊤\n2)xq)2exp (x⊤\nqA2xq)\nDexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2)\n+(2vmθ⊤xq)θ⊤(A1+θb⊤\n1)xqexp(x⊤\nqA1xq)\nDexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)−(2vnθ⊤xq)θ⊤(A2+θb⊤\n2)xqexp(x⊤\nqA2xq)\nDexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2)\n+2mv\nD(θ⊤xq)θ⊤(A1+θb⊤\n1)xqexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)\n−2nv\nD(θ⊤xq)θ⊤(A2+θb⊤\n2)xqexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)−2v2mnθ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\n+2v2mn1\nDexp(x⊤\nqA1xq)θ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\nexp(x⊤q(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq/2)+ 2v2mn1\nDexp(x⊤\nqA2xq)θ⊤(A1+θb1)xqθ⊤(A2+θb2)xq\nexp(x⊤q(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq/2)\n−2v2mn\nDθ⊤θexp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2)\n−2v2mn\nDexp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2)\u0000\n(θ⊤(A1+θb1)⊤xqθ⊤(A2+θb2)⊤xq)\u0001\n+2v2mn\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xqexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)\n+2v2mn\nDθ⊤(A1+θb1)xqθ⊤(A2+θb2)xqexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)#\n+o(1\nD).\nWhen m, n, A 1, A2, b1, b2, vsatisfies vmA 1=vnA 2andmb1=mb2, we have\nE(yq−f(E)d+1,D+1)2\n=E(xq,θ)\u0014v2m2\nDexp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)∥θ∥2+v2n2\nDexp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)∥θ∥2\n−2v2mn\nDexp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2)∥θ∥2\n−2v2mn\nDexp(x⊤\nq(A1+θb⊤\n1)⊤(A2+θb⊤\n2)xq/2 +x⊤\nq(A2+θb⊤\n2)⊤(A1+θb⊤\n1)xq/2)\u0000\n(θ⊤(A1+θb1)⊤xqθ⊤(A2+θb2)⊤xq)\u0001\n+v2m2\nD(θ⊤(A1+θb⊤\n1)xq)2exp(x⊤\nq(A1+θb⊤\n1)⊤(A1+θb⊤\n1)xq)\n20Superiority of Multi-Head Attention in In-Context Linear Regression\n+v2n2\nD(θ⊤(A2+θb⊤\n2)xq)2exp(x⊤\nq(A2+θb⊤\n2)⊤(A2+θb⊤\n2)xq)\u0015\n+o(1\nD).\nTaking m= 2,n= 1,A1=c\nvI,A2=2c−1\nvIandb1=b2= 0,\nE(yq−f(E)d+1,D+1)2\n=4v2\nD\u0012\n(v2\nv2−2c2)d\n2−(v2\nv2−2c(2c−1))d\n2\u0013\n+v2\nD(v2\nv2−2(2c−1)2)d\n2\n+(2c−1)2\nD(v2\nv2−2(2c−1)2)(v2\nv2−2(2c−1)2)d\n2\n+4c2\nD(v2\nv2−2c2)(v2\nv2−2c2)d\n2−4(2c−1)c\nD(v2\nv2−2c(2c−1))(v2\nv2−2c(2c−1))d\n2+o(1\nD). (9)\nAssuming that v2>max{2c2,2(2c−1)2}, when 0< c < 1, we have\nE(yq−fmulti(E)d+1,D+1)2 O(1\nD), we have (cu+v)∥θ0∥2+cuv2σ2\n(c+2v)2=O(1\nD).\nE(yq−f(E)d+1,D+1)2=c2σ2\nd∥θ0∥2\u0000\n∥θ0∥2(c+ 2v)2+ 2v2σ2\u0001\nv2\u0010\n(c+v)2∥θ0∥4+σ2\nd∥θ0∥2(c+ 2v)2+v2σ42\nd+v2σ4(c+v)2\n(c+2v)2\u0011\n| {z }\n>0and>O(1\nD)+O(1\nD).\nTherefore, in order to minimize E(yq−f(E)d+1,D+1)2, it is required that cu=O(1√\nD). Then we have b=O(1√\nD).\n• When σ2≪O(1\nD): As long as A,b,uandvsatisfies\n2a1b+\u0000\nA⊤b1+ (vA−I)⊤b2\u0001\n= 0, (10)\nwe have E(yq−f(E)d+1,D+1)2=O(1\nD).\n24Superiority of Multi-Head Attention in In-Context Linear Regression\nFigure 11. Simultion: when A=I/v, the loss is minimized at c=0.\nWhen taking A=Id/v,b= 0andu= 0we have\nA11=o(1\nD),\nA12=−2E(θ⊤xqx⊤\nqθ) +2\nDEθ⊤xqx⊤\nqθexp(x⊤\nqxq/v)\nexp(x⊤qxq/2v2)+o(1\nD),\nA13+A14=2\nDEθ⊤xqx⊤\nqθexp(x⊤\nqxq/v2) +o(1\nD),\nA21=v2\nDEθ⊤θexp(x⊤\nqxq/v2) +Eθ⊤xqx⊤\nqθ−E2\nD\u0000\nθ⊤xqx⊤\nqθ\u0001\nexp (x⊤\nqxq/v)\nexp(x⊤qxq/2v2)−1\nDE\u0000\nθ⊤xqx⊤\nqθ\u0001\nexp(x⊤\nqxq/v2) +o(1\nD),\nA22=o(1\nD),\nA23=o(1\nD).\nAs a result,\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(θ⊤xqx⊤\nqθ) +A12+A13+A14+A21\n=1\nDEθ⊤xqx⊤\nqθexp(x⊤\nqxq/v2) +v2\nDEθ⊤θexp(x⊤\nqxq/v2) +o(1\nD).\n=v2(σ2+∥θ0∥2)\nD(v2\nv2−2)d\n2+v2(σ2+∥θ0∥2)\nD(v2−2)(v2\nv2−2)d\n2.\nFigure 12, 13 below demonstrate the theoretical values and the corresponding simulation results, which indicates that the\nsimulation of prediction loss aligns well with theoretical values.\nICL performance of multi-head attention\nE(yq−f(E)d+1,D+1)2\n=E\u0012\nyq−m\u0002\nu⊤x1+vy1, u⊤x2+vy2, . . . , u⊤xD+vyD, u⊤xq\u0003\nϕ\u0012\nE⊤(WK\n1)⊤WQ\n1\u0014xq\n0\u0015\u0013\n+n\u0002\nu⊤x1+vy1, u⊤x2+vy2, . . . , u⊤xD+vyD, u⊤xq\u0003\nϕ\u0012\nE⊤(WK\n2)⊤WQ\n2\u0014xq\n0\u0015\u0013\u0013 2\n25Superiority of Multi-Head Attention in In-Context Linear Regression\nFigure 12. ICL performance of single-head attention with prior knowledge, (A, b, u ) = (Id/v,0,0)andd= 5.\nFigure 13. ICL performance of single-head attention with prior knowledge, (A, b, u ) = (Id/v,0,0)andD= 1000 .\n=E\u0012\nyq−mPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A1+θb⊤\n1)xq) +u⊤xqexp(x⊤\nqA1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)\n+nPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A2+θb⊤\n2)xq) +u⊤xqexp(x⊤\nqA2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)\u00132\n=E\ny2\nq+ \nmPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A1+θb⊤\n1)xq) +u⊤xqexp(x⊤\nqA1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!2\n\n+E \nnPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A2+θb⊤\n2)xq) +u⊤xqexp(x⊤\nqA2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!2\n−E \n2yq \nmPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A1+θb⊤\n1)xq) +u⊤xqexp(x⊤\nqA1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!!\n+E \n2yq \nnPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A2+θb⊤\n2)xq) +u⊤xqexp(x⊤\nqA2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!!\n−E \nmPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A1+θb⊤\n1)xq) +u⊤xqexp(x⊤\nqA1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!\n×E \nnPD\ni=1(u⊤xi+vyi) exp( x⊤\ni(A2+θb⊤\n2)xq) +u⊤xqexp(x⊤\nqA2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!\n.\n26Superiority of Multi-Head Attention in In-Context Linear Regression\nWhen taking m= 2,n= 1,A1=c\nvI,A2=2c−1\nvIandu=b1=b2= 0, it becomes\nE(yq−f(E)d+1,D+1)2\n=σ2+∥θ0∥2+E\n \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!2\n+ \nvPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!2\n\n+E \n2yq \nvPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n−2yq \n2vPD\ni=1θ⊤xiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!!\n−E \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))2vPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n=σ2+∥θ0∥2+B1+B2+B3.\nThen we have\nB1=E\n \n2vPD\ni=1θ⊤xiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!2\n+ \nvPD\ni=1θ⊤xiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!2\n\n=4\nDv2∥θ∥2exp(c2∥xq∥2/v2) + 4c2(σ2+∥θ0∥2)−8c2(θ⊤xqx⊤\nqθ) exp ( c2∥xq∥2/v2)\nDexp(c2∥xq∥2/2v2)\n−4c\nD\u0000\nθ⊤xqx⊤\nqθ\u0001\nexp(c2∥xq∥2/v2) +1\nDv2∥θ∥2exp\u0000\n(2c−1)2∥xq∥2/v2\u0001\n+ (2c−1)2(σ2+∥θ0∥2)\n−2(2c−1)2(θ⊤xqx⊤\nqθ) exp (((2 c−1)2∥xq∥2/v2))\nDexp((2 c−1)2∥xq∥2/2v2)−2c−1\nD\u0000\nθ⊤xqx⊤\nqθ\u0001\nexp((2 c−1)2∥xq∥2/v2),\nB2=E \n2yq \nvPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n−2yq \n2vPD\ni=1θ⊤xiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!!\n= 2(2 c−1)θ⊤xqx⊤\nqθ−2(2c−1)θ⊤xqx⊤\nqθexp((2 c−1)∥xq∥2/v)\nDexp((2 c−1)2∥xq∥2/2v2)−2(2c−1)\nDθ⊤xqx⊤\nqθexp((2 c−1)2∥xq∥2/v2))\n−4cθ⊤xqx⊤\nqθ+4cθ⊤xqx⊤\nqθexp(c∥xq∥2/v)\nDexp(c2∥xq∥2/2v2)+4c\nDθ⊤xqx⊤\nqθexp(c2∥xq∥2/v2)),\nand\nB3=−4c(2c−1)(1−1\nD)θ⊤xqx⊤\nqθ+4\nDc(2c−1)θ⊤xqx⊤\nqθexp(c∥xq∥2/v)\nexp(c2∥xq∥2/2v2)+4\nDc(2c−1)θ⊤xqx⊤\nqθexp((2 c−1)∥xq∥2/v)\nexp((2 c−1)2∥xq∥2/2v2\n+4\nDc(2c−1)θ⊤xqx⊤\nqθ\u0000\n2 exp( c2∥xq∥2/v2)−2\u0001\n+4\nDc(2c−1)θ⊤xqx⊤\nqθ\u0000\n2 exp((2 c−1)2∥xq∥2/v2)−2\u0001\n−4v2\nDθ⊤θexp((2 c−1)c∥xq∥2/v2)−4\nDc(2c−1)(θ⊤xqx⊤\nqθ) exp((2 c−1)c∥xq∥2/v2)\n+4c2\nD(θ⊤xqx⊤\nqθ) exp( c2∥xq∥2/v2) +1\nD(2c−1)2(θ⊤xqx⊤\nqθ) exp((2 c−1)2∥xq∥2/v2),\nTo sum up, we have\nE(yq−f(E)d+1,D+1)2\n=4\nDv2∥θ∥2exp(c2∥xq∥2/v2) +1\nDv2∥θ∥2exp\u0000\n(2c−1)2∥xq∥2/v2\u0001\n+4\nDc2θ⊤xqx⊤\nqθexp(c2∥xq∥2/v2) +1\nD(2c−1)2θ⊤xqx⊤\nqθexp\u0000\n(2c−1)2∥xq∥2/v2\u0001\n27Superiority of Multi-Head Attention in In-Context Linear Regression\n+ 4 v2∥θ∥2exp((2 c−1)c∥xq∥2/v2)−4\nDc(2c−1)(θ⊤xqx⊤\nqθ) exp((2 c−1)c∥xq∥2/v2)\n=4v2(σ2+∥θ0∥2)\nD\u0012\n(v2\nv2−2c2)d\n2−(v2\nv2−2c(2c−1))d\n2\u0013\n+v2\nD(σ2+∥θ0∥2)(v2\nv2−2(2c−1)2)d\n2\n+(2c−1)2\nD(σ2+∥θ0∥2)(v2\nv2−2(2c−1)2)(v2\nv2−2(2c−1)2)d\n2\n+4c2\nD(σ2+∥θ0∥2)(v2\nv2−2c2)(v2\nv2−2c2)d\n2−4(2c−1)c\nD(σ2+∥θ0∥2)(v2\nv2−2c(2c−1))(v2\nv2−2c(2c−1))d\n2.\nFigure 14 below demonstrates the alignment between the theoretical values and the corresponding simulation results.\nFigure 14. ICL performance of multi-head attention with prior knowledge (A1, A2, b1, b2) = (( c/v)Id,((2c−1)/v)Id,0,0),(m, n ) =\n(2,1), and (d, D ) = (5 ,1000) .\nA.5. Noisy Response: Theorem 5.2\nProof of Theorem 5.2. The main logic of the proof is the same as Theorem 4.1 for single-head attention and Theorem 4.2\nfor multi-head attention.\nOptimal solution for single-head attention\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,yq)E{xi,yi}i∈[D]\nyq−v\u0002y1, y2, . . . , y D,0\u0003\nϕ\n\nx⊤\n1Axq+y1b⊤xq\n. . .\nx⊤\nqAxq+ 0\n\n\n2\n=E(xq,yq)E{xi,yi}i∈[D]\u0012\ny2\nq+ \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!2\n| {z }\n:=A1−2yq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!\n| {z }\n:=A2\u0013\n,\nwhere Ey2\nq= 1 + σ2\nϵ.\nWhen fixing xqandθ, the terms A1becomes\nE{xi,yi}i∈[D]A1\n=v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD2E2exp(x⊤\niAxq+yib⊤xq)\n−2v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD3E3exp(x⊤\niAxq+yib⊤xq)\"DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)#\n28Superiority of Multi-Head Attention in In-Context Linear Regression\n+3v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD4E4exp(x⊤\niAxq+yib⊤xq)\"DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)#2\n−2v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD3E3exp(x⊤\niAxq+yib⊤xq)exp(x⊤\nqAxq) +o\u00121\nD\u0013\n:=A11+A12+A13+A14+o\u00121\nD\u0013\n.\nTo figure out A11toA13, we know that\nEy2\niexp(x⊤\niAxq+yib⊤xq)2\n=E(x⊤\niθ+ϵi)2exp(2 x⊤\niAxq+ 2x⊤\niθb⊤xq+ 2ϵib⊤xq)\n=θ⊤\u0000\nI+ 4(Axq+θb⊤xq)(Axq+θb⊤xq)⊤\u0001\nθexp(2∥Axq+θb⊤xq∥2+ 2σ2\nϵ(b⊤xq)2)\n+8σϵ(θ⊤Axq)b⊤xqexp(2∥Axq+θb⊤xq∥2+ 2σ2\nϵ(b⊤xq)2)\n+σ2\nϵ(1 + 4 σ2\nϵ(b⊤xq)2) exp(2 ∥Axq+θb⊤xq∥2+ 2σ2\nϵ(b⊤xq)2),\nand\nEyiexp(x⊤\niAxq+yib⊤xq) = (θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq)) exp( ∥Axq+θb⊤xq∥2/2 +σ2\nϵ(b⊤xq)2/2).\nAs a result,\nA11=v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD2E2exp(x⊤\niAxq+yib⊤xq)\n=v2DEy2\niexp(x⊤\niAxq+yib⊤xq)2+D(D−1)E2yiexp(x⊤\niAxq+yib⊤xq)\nD2exp(∥Axq+θb⊤xq∥2+σ2ϵ(b⊤xq)2)\n=v21\nD\u0002\nθ⊤\u0000\nI+ 4(Axq+θb⊤xq)(Axq+θb⊤xq)⊤\u0001\nθ+ 8σϵ(θ⊤Axq)b⊤xq+σ2\nϵ(1 + 4 σ2\nϵ(b⊤xq)2)\u0003\n×exp(∥Axq+θb⊤xq∥2+σ2\nϵ(b⊤xq)2)\n+v2D−1\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2,\nA12\n=−2v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD3E3exp(x⊤\niAxq+yib⊤xq)\"DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)#\n=−2v2E2D(D−1)\nD3E3exp(x⊤\niAxq+yib⊤xq)\u0002\nEyiexp(x⊤\niAxq+yib⊤xq)Eyiexp(x⊤\niAxq+yib⊤xq)2\u0003\n+2v2E2D(D−1)\nD3E3exp(x⊤\niAxq+yib⊤xq)\u0002\nE2yiexp(x⊤\niAxq+yib⊤xq)Eexp(x⊤\niAxq+yib⊤xq))\u0003\n+o\u00121\nD\u0013\n=−4v2E2D(D−1)\nD3E3exp(x⊤\niAxq+yib⊤xq)(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2exp(5∥Axq+θb⊤xq∥2/2 + 5 σ2\nϵ(b⊤xq)2/2)\n+2v2E2D(D−1)\nD3E3exp(x⊤\niAxq+yib⊤xq)(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2exp(3∥Axq+θb⊤xq∥2/2 + 3 σ2\nϵ(b⊤xq)2/2)\n=−8v21\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2exp(∥Axq+θb⊤xq∥2+σ2\nϵ(b⊤xq)2)\n+4v21\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2+o\u00121\nD\u0013\n,\nand\nA13\n29Superiority of Multi-Head Attention in In-Context Linear Regression\n= 3 v2E{xi,yi}i∈[D](PD\ni=1yiexp(x⊤\niAxq+yib⊤xq))2\nD4E4exp(x⊤\niAxq+yib⊤xq)\"DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)#2\n= 3 v2E2yiexp(x⊤\niAxq+yib⊤xq)\nDE4exp(x⊤\niAxq+yib⊤xq)\u0002\nEexp(x⊤\niAxq+yib⊤xq)2−E2exp(x⊤\niAxq+yib⊤xq)\u0003\n+o\u00121\nD\u0013\n=3v2\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2\u0000\nexp(∥Axq+θb⊤xq∥2+σ2\nϵ(b⊤xq)2)−1\u0001\n+o\u00121\nD\u0013\n,\nwith\nA14=−2v21\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))2exp(x⊤\nqAxq− ∥Axq+θb⊤xq∥2/2−σ2\nϵ(b⊤xq)2/2).\nIn terms of A2, when fixing xqandθ, we have\nEyqE{xi,yi}i∈[D]A2\n=EyqE{xi,yi}i∈[D]2yq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!\n=E{xi,yi}i∈[D]2θ⊤xq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)\nDEexp(x⊤\niAxq+yib⊤xq)!\n−E{xi,yi}i∈[D]2θ⊤xq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)\nD2E2exp(x⊤\niAxq+yib⊤xq)! DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)!\n+E{xi,yi}i∈[D]2θ⊤xq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)\nD3E3exp(x⊤\niAxq+yib⊤xq)! DX\ni=1exp(x⊤\niAxq+yib⊤xq)−DEexp(x⊤\niAxq+yib⊤xq)!2\n−E{xi,yi}i∈[D]2θ⊤xq \nvPD\ni=1yiexp(x⊤\niAxq+yib⊤xq)\nD2E2exp(x⊤\niAxq+yib⊤xq)!\nexp(x⊤\nqAxq) +o\u00121\nD\u0013\n:=A21+A22+A23+A24.\nForA21toA24, we have\nA21= 2vθ⊤xq(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq)),\nA22=−2vθ⊤xq1\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))\u0000\n2 exp(∥Axq+θb⊤xq∥2+σ2\nϵ(b⊤xq)2)−1\u0001\n+o\u00121\nD\u0013\n,\nA23= 2vθ⊤xq1\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))\u0000\nexp(∥Axq+θb⊤xq∥2+σ2\nϵ(b⊤xq)2)−1\u0001\n+o\u00121\nD\u0013\n,\nA24=−2vθ⊤xq1\nD(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq)) exp( x⊤\nqAxq− ∥Axq+θb⊤xq∥2/2−σ2\nϵ(b⊤xq)2/2).\nInserting A11toA24intoA1andA2, we obtain\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E\u0000\nx⊤\nqθ−v(θ⊤Axq+∥θ∥2b⊤xq+σϵ(b⊤xq))\u00012+O\u00121\nD\u0013\n=1\ndtr\u0000\n(I−vA)2\u0001\n+E∥θ∥4∥b∥2+σ2\nϵ∥b∥2+O\u00121\nD\u0013\n. (11)\n30Superiority of Multi-Head Attention in In-Context Linear Regression\nAs a result, the optimal solution of Aandbsatisfies that tr\u0000\n(I−vA)2\u0001\n/d=O(1/D)and∥b∥2=O(1/D).\nIn addition, similar to Theorem 4.1, when taking A=Id/vandb= 0, we have\nA11=1\nDE\u0002\nv2∥θ∥2+ 4(x⊤\nqθ)2+v2σ2\nϵ\u0003\nexp(∥xq∥2/v2) +D−1\nD,\nA12=−8\nDE(x⊤\nqθ)2exp(∥xq∥2/v2) +4\nD,\nA13=−3\nD+3\nDE(x⊤\nqθ)2exp(∥xq∥2/v2),\nA14=−2\nDE(x⊤\nqθ)2exp(∥xq∥2/v− ∥xq∥2/v2/2),\nA21= 2 ,\nA22=−2\nDE(x⊤\nqθ)2(2 exp( ∥xq∥2/v2)−1),\nA23=2\nDE(x⊤\nqθ)2(exp(∥xq∥2/v2)−1),\nA24=−2\nDE(x⊤\nqθ)2exp(∥xq∥2/v− ∥xq∥2/v2/2).\nAs a result,\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n= 1 + σ2\nϵ+A11+A12+A13+A14−A21−A22−A23−A24\n=σ2\nϵ+v2(1 +σ2\nϵ)\nDEexp(∥xq∥2/v2) +1\nDE(x⊤\nqθ)2exp(∥xq∥2/v2) +o(1\nD).\nICL performance of multi-head attention\nE(yq−f(E)d+1,D+1)2\n=E\u0012\nyq−vm\u0002y1, y2, . . . , y D,0\u0003\nϕ\u0012\nE⊤(WK\n1)⊤WQ\n1\u0014xq\n0\u0015\u0013\n+vn\u0002y1, y2, . . . , y D,0\u0003\nϕ\u0012\nE⊤(WK\n2)⊤WQ\n2\u0014xq\n0\u0015\u0013\u0013 2\n=E \nyq−vmPD\ni=1yiexp(x⊤\ni(A1+θb⊤\n1)xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)+vnPD\ni=1yiexp(x⊤\ni(A2+θb⊤\n2)xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!2\n=E\ny2\nq+ \nvmPD\ni=1yiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!2\n−2yq \nvmPD\ni=1θ⊤xiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)!\n\n+E\n \nvnPD\ni=1yiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!2\n+ 2yq \nvnPD\ni=1yiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!\n\n−E \n2vnPD\ni=1yiexp(x⊤\niA1xq+yib⊤\n1xq)Pexp(x⊤\niA1xq+yib⊤\n1xq) + exp( x⊤qA1xq)vmPD\ni=1yiexp(x⊤\niA2xq+yib⊤\n2xq)Pexp(x⊤\niA2xq+yib⊤\n2xq) + exp( x⊤qA2xq)!\n.\nWhen taking m= 2,n= 1,A1=c\nvI,A2=2c−1\nvIandb1=b2= 0, it becomes\nE(yq−f(E)d+1,D+1)2\n= 1 + σ2\nϵ+E\n \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!2\n−2yq \n2vPD\ni=1θ⊤xiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!\n\n+E\n \nvPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!2\n+ 2yq \nvPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n\n31Superiority of Multi-Head Attention in In-Context Linear Regression\n−E \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))2vPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n:= 1 + σ2\nϵ+B1+B2+B3.\nSimilar to how we calculate A1andA2, forB1, the terms are similar. We follow the above proof and obtain\nA11=4\nDE\u0002\nv2∥θ∥2+ 4c2(x⊤\nqθ)2+v2σ2\nϵ\u0003\nexp(c2∥xq∥2/v2) +D−1\nD4c2,\nA12=−32c2\nDE(x⊤\nqθ)2exp(c2∥xq∥2/v2) +16c2\nD,\nA13=−12c2\nD+12c2\nDE(x⊤\nqθ)2exp(c2∥xq∥2/v2),\nA14=−8c2\nDE(x⊤\nqθ)2exp(c∥xq∥2/v−c2∥xq∥2/v2/2),\nA21= 4 c,\nA22=−4c\nDE(x⊤\nqθ)2(2 exp( c2∥xq∥2/v2)−1),\nA23=4c\nDE(x⊤\nqθ)2(exp( c2∥xq∥2/v2)−1),\nA24=−4c\nDE(x⊤\nqθ)2exp(c∥xq∥2/v−c2∥xq∥2/v2/2),\nthus\nB1=E\n \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!2\n−2yq \n2vPD\ni=1θ⊤xiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))!\n\n= 4 c2−4c+4v2(1 +σ2\nϵ)\nDEexp(c2∥xq∥2/v2)−4c2−4c\nDE(x⊤\nqθ)2exp(c2∥xq∥2/v2)\n−2(4c2−2c)\nDE(x⊤\nqθ)2exp(c∥xq∥2/v−c2∥xq∥2/v2/2).\nForB2, similarly, we obtain\nB2= (2 c−1)2+ 2(2 c−1) +v2(1 +σ2\nϵ)\nDEexp((2 c−1)2∥xq∥2/v2)\n−(2c−1)2+ 2(2 c−1)\nDE(x⊤\nqθ)2exp((2 c−1)2∥xq∥2/v2)\n−2((2c−1)2+ (2c−1))\nDE(x⊤\nqθ)2exp((2 c−1)∥xq∥2/v−(2c−1)(2c−1)2∥xq∥2/v2/2).\nIn terms of B3,\nB3\n=−E \n2vPD\ni=1yiexp(x⊤\nixq(c/v))Pexp(x⊤\nixq(c/v)) + exp( ∥xq∥2(c/v))2vPD\ni=1yiexp(x⊤\nixq(2c−1)/v)Pexp(x⊤\nixq(2c−1)/v) + exp( ∥xq∥2(2c−1)/v)!\n=−4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD2Eexp(x⊤\nixq(c/v))Eexp(x⊤\nixq(2c−1)/v)\n+4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD3E2exp(x⊤\nixq(c/v))Eexp(x⊤\nixq(2c−1)/v) DX\ni=1exp(x⊤\nixq(c/v))−DEexp(x⊤\nixq(c/v))!\n+4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD3Eexp(x⊤\nixq(c/v))E2exp(x⊤\nixq(2c−1)/v) DX\ni=1exp(x⊤\nixq(2c−1)/v)−DEexp(x⊤\nixq(2c−1)/v)!\n32Superiority of Multi-Head Attention in In-Context Linear Regression\n−4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD4E3exp(x⊤\nixq(c/v))Eexp(x⊤\nixq(2c−1)/v) DX\ni=1exp(x⊤\nixq(c/v))−DEexp(x⊤\nixq(c/v))!2\n−4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD4Eexp(x⊤\nixq(c/v))E3exp(x⊤\nixq(2c−1)/v)\n× DX\ni=1exp(x⊤\nixq(2c−1)/v)−DEexp(x⊤\nixq(2c−1)/v)!2\n−4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD4E2exp(x⊤\nixq(c/v))E2exp(x⊤\nixq(2c−1)/v)\n× DX\ni=1exp(x⊤\nixq(c/v))−DEexp(x⊤\nixq(c/v))! DX\ni=1exp(x⊤\nixq(2c−1)/v)−DEexp(x⊤\nixq(2c−1)/v)!\n+4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD3E2exp(x⊤\nixq(c/v))Eexp(x⊤\nixq(2c−1)/v)exp(∥xq∥2c/v)\n+4v2E\u0010PD\ni=1yiexp(x⊤\nixq(c/v))\u0011\u0010PD\ni=1yiexp(x⊤\nixq(2c−1)/v)\u0011\nD3Eexp(x⊤\nixq(c/v))E2exp(x⊤\nixq(2c−1)/v)exp(∥xq∥2(2c−1)/v) +o\u00121\nD\u0013\n:=E(B31+B32+B33+B34+B35+B36+B37+B38) +o\u00121\nD\u0013\n.\nForB31toB38, we have\nB31=−4c(2c−1)D−1\nD(x⊤\nqθ)2−4v21\nD\u0002\nθ⊤(Id+ (3c−1)2xqx⊤\nq/v2)θ+σ2\nϵ\u0003\nexp((3 c−1)2∥xq∥2/(2v2))\nexp(( c2+ (2c−1)2)∥xq∥2/(2v2))\n=−4c(2c−1)D−1\nD(x⊤\nqθ)2−4v21\nD\u0002\nθ⊤(Id+ (3c−1)2xqx⊤\nq/v2)θ+σ2\nϵ\u0003\nexp((2 c2−c)∥xq∥2/v2),\nB32= 41\nD(x⊤\nqθ)22c(2c−1) exp((4 c2+ (2c−1)2)∥xq∥2/(2v2))−c(2c−1) exp((2 c2+ (2c−1)2)∥xq∥2/(2v2))\nexp((2 c2+ (2c−1)2)∥xq∥2/(2v2))\n+41\nD(x⊤\nqθ)2c(3c−1) exp(( c2+ (3c−1)2)∥xq∥2/(2v2))−c(2c−1) exp(( c2+ 2(2 c−1)2)∥xq∥2/(2v2))\nexp((2 c2+ (2c−1)2)∥xq∥2/(2v2))+o\u00121\nD\u0013\n=4\nD(x⊤\nqθ)2[2c(2c−1)] exp( ∥xq∥2c2/v2)−8\nD(x⊤\nqθ)2[c(2c−1)]\n+4\nD(x⊤\nqθ)2[c(3c−1)] exp((2 c2−c)∥xq∥2/v2) +o\u00121\nD\u0013\n,\nB33= 41\nD(x⊤\nqθ)22c(2c−1) exp(( c2+ 4(2 c−1)2)∥xq∥2/(2v2))−c(2c−1) exp(( c2+ 2(2 c−1)2)∥xq∥2/(2v2))\nexp(( c2+ 2(2 c−1)2)∥xq∥2/(2v2))\n+41\nD(x⊤\nqθ)2(3c−1)(2c−1) exp(((3 c−1)2+ (2c−1)2)∥xq∥2/(2v2))−c(2c−1) exp(( c2+ 2(2 c−1)2)∥xq∥2/(2v2))\nexp(( c2+ 2(2 c−1)2)∥xq∥2/(2v2))\n+o\u00121\nD\u0013\n=4\nD(x⊤\nqθ)2[2c(2c−1)] exp( ∥xq∥2(2c−1)2/v2)−8v2\nD(x⊤\nqθ)2[c(2c−1)]\n+4\nD(x⊤\nqθ)2[(3c−1)(2c−1)] exp((2 c2−c)∥xq∥2/v2) +o\u00121\nD\u0013\n,\nB34=−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp(c2∥xq∥2/v2)−1\u0001\n+o\u00121\nD\u0013\n,\n33Superiority of Multi-Head Attention in In-Context Linear Regression\nB35=−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp((2 c−1)2∥xq∥2/v2)−1\u0001\n+o\u00121\nD\u0013\n,\nB36=−4c(2c−1)\nD(x⊤\nqθ)2\u0012exp((3 c−1)2∥xq∥2/(2v2))\nexp(c2∥xq∥2/(2v2)) exp((2 c−1)2∥xq∥2/(2v2))−1\u0013\n+o\u00121\nD\u0013\n=−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp((2 c2−c)∥xq∥2/v2)−1\u0001\n+o\u00121\nD\u0013\n,\nand\nB37=4\nDc(2c−1) exp( ∥xq∥2c/v− ∥xq∥2c2/(2v2)) +o\u00121\nD\u0013\n,\nB38=4\nDc(2c−1) exp( ∥xq∥2(2c−1)/v− ∥xq∥2(2c−1)2/(2v2)) +o\u00121\nD\u0013\n.\nPutting everything together, we have\nB3=B31+B32+B33+B34+B35+B36+B37+B38\n=−4c(2c−1)D−1\nD(x⊤\nqθ)2−4v21\nD\u0002\nθ⊤(Id+ (3c−1)2xqx⊤\nq/v2)θ+σ2\nϵ\u0003\nexp((2 c2−c)∥xq∥2/v2)\n+4\nD(x⊤\nqθ)2[2c(2c−1)] exp( ∥xq∥2c2/v2)−8\nD(x⊤\nqθ)2[c(2c−1)]\n+4\nD(x⊤\nqθ)2[c(3c−1)] exp((2 c2−c)∥xq∥2/v2)\n+4\nD(x⊤\nqθ)2[2c(2c−1)] exp( ∥xq∥2(2c−1)2/v2)−8\nD(x⊤\nqθ)2[c(2c−1)]\n+4\nD(x⊤\nqθ)2[(3c−1)(2c−1)] exp((2 c2−c)∥xq∥2/v2)\n−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp(c2∥xq∥2/v2)−1\u0001\n−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp((2 c−1)2∥xq∥2/v2)−1\u0001\n−4c(2c−1)\nD(x⊤\nqθ)2\u0000\nexp((2 c2−c)∥xq∥2/v2)−1\u0001\n+4\nDc(2c−1) exp( ∥xq∥2c/v− ∥xq∥2c2/(2v2))\n+4\nDc(2c−1) exp( ∥xq∥2(2c−1)/v− ∥xq∥2(2c−1)2/(2v2)) +o\u00121\nD\u0013\n=−(8c2−4c)(x⊤\nqθ)2−6\nD(x⊤\nqθ)2c(2c−1)−4v2\nD\u0002\n∥θ∥2+σ2\nϵ\u0003\nexp((2 c2−c)∥xq∥2/v2)\n+4\nD(x⊤\nqθ)2exp(∥xq∥2c2/v2) [2c(2c−1)−c(2c−1)]\n+4\nD(x⊤\nqθ)2exp(∥xq∥2(2c−1)2/v2) [2c(2c−1)−c(2c−1)]\n+4\nD(x⊤\nqθ)2exp(∥xq∥2(2c2−c)/v2) [−2c(2c−1)]\n+4\nDc(2c−1) exp( ∥xq∥2c/v− ∥xq∥2c2/(2v2))\n+4\nDc(2c−1) exp( ∥xq∥2(2c−1)/v− ∥xq∥2(2c−1)2/(2v2)) +o\u00121\nD\u0013\n=−4(2c2−c)(x⊤\nqθ)2−4v2\nD\u0002\n∥θ∥2+σ2\nϵ\u0003\nexp((2 c2−c)∥xq∥2/v2)\n34Superiority of Multi-Head Attention in In-Context Linear Regression\n+4\nD(x⊤\nqθ)2exp(∥xq∥2c2/v2) [c(2c−1)]\n+4\nD(x⊤\nqθ)2exp(∥xq∥2(2c−1)2/v2) [c(2c−1)]\n+4\nD(x⊤\nqθ)2exp(∥xq∥2(2c2−c)/v2) [−2c(2c−1)]\n+4\nDc(2c−1) exp( ∥xq∥2c/v− ∥xq∥2c2/(2v2))\n+4\nDc(2c−1) exp( ∥xq∥2(2c−1)/v− ∥xq∥2(2c−1)2/(2v2)) +o\u00121\nD\u0013\n.\nFinally,\nE(yq−f(E)d+1,D+1)2\n= 1 + σ2\nϵ+B1+B2+E(B31+B32+B33+B34+B35+B36+B37+B38) +o\u00121\nD\u0013\n= 1 + σ2\nϵ+ 4c2−4c+4v2(1 +σ2\nϵ)\nDEexp(c2∥xq∥2/v2)−4c2−4c\nDE(x⊤\nqθ)2exp(c2∥xq∥2/v2)\n−2(4c2−2c)\nDE(x⊤\nqθ)2exp(c∥xq∥2/v−c2∥xq∥2/v2/2)\n+(2c−1)2+ 2(2 c−1) +v2(1 +σ2\nϵ)\nDEexp((2 c−1)2∥xq∥2/v2)\n−(2c−1)2+ 2(2 c−1)\nDE(x⊤\nqθ)2exp((2 c−1)2∥xq∥2/v2)\n−2((2c−1)2+ (2c−1))\nDE(x⊤\nqθ)2exp((2 c−1)∥xq∥2/v−(2c−1)(2c−1)2∥xq∥2/v2/2)\n−E4(2c2−c)(x⊤\nqθ)2−E4v2\nD\u0002\n∥θ∥2+σ2\nϵ\u0003\nexp((2 c2−c)∥xq∥2/v2)\n+E4\nD(x⊤\nqθ)2exp(∥xq∥2c2/v2) [c(2c−1)]\n+E4\nD(x⊤\nqθ)2exp(∥xq∥2(2c−1)2/v2) [c(2c−1)]\n+E4\nD(x⊤\nqθ)2exp(∥xq∥2(2c2−c)/v2) [−2c(2c−1)]\n+E4\nDc(2c−1) exp( ∥xq∥2c/v− ∥xq∥2c2/(2v2))\n+E4\nDc(2c−1) exp( ∥xq∥2(2c−1)/v− ∥xq∥2(2c−1)2/(2v2)) +o\u00121\nD\u0013\n=σ2\nϵ+v2(1 +σ2\nϵ)\nDE\u0002\n4 exp( c2∥xq∥2/v2) + exp((2 c−1)2∥xq∥2/v2)−4 exp((2 c2−c)∥xq∥2/v2)\u0003\n+4c2\nD(x⊤\nqθ)2exp(∥xq∥2c2/v2) +(2c−1)2\nD(x⊤\nqθ)2exp(∥xq∥2(2c−1)2/v2)\n+4\nD(x⊤\nqθ)2exp(∥xq∥2(2c2−c)/v2) [−2c(2c−1)] + o\u00121\nD\u0013\n,\nthus\nE(yq−f(E)d+1,D+1)2\n=σ2\nϵ+4v2(1 +σ2\nϵ)\nD\u0012\n(v2\nv2−2c2)d\n2−(v2\nv2−2c(2c−1))d\n2\u0013\n+v2(1 +σ2\nϵ)\nD(v2\nv2−2(2c−1)2)d\n2\n+(2c−1)2\nD(v2\nv2−2(2c−1)2)(v2\nv2−2(2c−1)2)d\n2\n35Superiority of Multi-Head Attention in In-Context Linear Regression\n+4c2\nD(v2\nv2−2c2)(v2\nv2−2c2)d\n2−4(2c−1)c\nD(v2\nv2−2c(2c−1))(v2\nv2−2c(2c−1))d\n2+o(1\nD).\nA.6. Correlated Features: Theorem 5.3\nProof of Theorem 5.3. To figure out the optimal solution of single-head attention, we firstly transform the problem from\ncorrelated features to the problem with isotropic features with a new θdistribution. After transforming the problem, since\nTheorem 4.1 only utilize the distribution of θin its last derivation step, we can directly utilize the results in Theorem 4.1.\nTo transform correlated features, denote z∼N(0, Id)andx= Σ1/2z. Recall that the attention score is calculated as\nϕ\u0000\n(WKWinE)⊤(WQWinE)\u0001\n=ϕ\u0000\n(WKWinE)⊤(WQWinE)\u0001\nBased on Theorem 4.1, we have\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)(x⊤\nqθ)2+v2\nDθ⊤(Id−4(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2(1 +3\nD)(θ⊤(A+θb⊤)xq)2−2v2(θ⊤(A+θb⊤)xq)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)\n−3v2\nD(θ⊤(A+θb⊤)xq)2+3v2\nD(θ⊤(A+θb⊤)xq)2exp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n−2(x⊤\nqθ)\u0012\nvθ⊤(A+θb⊤)xq−vθ⊤(A+θb⊤)xqexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)+v\nDθ⊤(A+θb⊤)xq\n−2v\nDθ⊤(A+θb⊤)xqexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq−v\nD(θ⊤(A+θb⊤)xq)\n+v\nD(θ⊤(A+θb⊤)xq) exp( x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq))\u0013\n+o(1\nD),\nfrom which the optimal solution satisfies Eθ⊤(Id−vA)2θ=O(1/D)and∥b∥2E∥θ∥4=O(1/D)where θ∼\nN(0,Σ−1/2/d).\nFor multi-head attention, the same argument applies, and we can also transform the correlated features problem to isotropic\nfeatures with a new θdistribution. Further, due to the flexibility of multi-head attention, when each head is of full rank,\ni.e.,p/h > d , the performance of multi-head attention is not worse than single-head attention. There always exists some\nWoutsuch that the multi-head attention can be reduced to a single-head attention.\nA.7. Local Examples: Theorem 5.4 and 5.5\nA.7.1. T HEOREM 5.4\nProof of Theorem 5.4. The proof of Theorem 5.4 is almost the same as Theorem 4.1. The only difference is the change on\nthe distribution of the examples (xi, yi)s.\nWhen taking infinite many training prompts, the loss function becomes\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)E{xi}i∈[D]\ny2\nq−2yq \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!\n| {z }\n=A1+ \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!2\n| {z }\n=A2\n.\n36Superiority of Multi-Head Attention in In-Context Linear Regression\nForA1, we have\nE{xi,yi}i∈[D]A1\n=E{xi,yi}i∈[D](−2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nDEexp(x1⊤Axq+y1b⊤xq)+o(1\nD)\n+E{xi,yi}i∈[D](2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\n(DEexp(x1⊤Axq+y1b⊤xq))2\n×\u0010\nexp(x⊤\nqAxq) +X\nexp(x⊤\niAxq+yib⊤xq)−DEexp(x1⊤Axq+y1b⊤xq)\u0011\n−E{xi,yi}i∈[D](2vθ⊤xq)PD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nDEexp(x1⊤Axq+y1b⊤xq))3\u0010X\nexp(x⊤\niAxq+yib⊤xq)−DEexp(x1⊤Axq+y1b⊤xq)\u00112\n+o\u00121\nD\u0013\n=A11+A12+A13+o(1\nD).\nSince xi∼N(xq, σ2\nx), we have\nE{x1,y1}exp(x⊤\n1Axq+y1b⊤xq) = E{x1,y1}exp \u0012\nσxx1−xq\nσx\u0013⊤\n(A+θb⊤)xq!\n= exp\u00121\n2σ2\nxx⊤\nq(A+θb⊤)⊤(A+θb⊤)xq+x⊤\nq(A+θb⊤)xq\u0013\n,\nE{x1,y1}x1exp(x⊤Axq+y1b⊤xq) =\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\nexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq/2 +x⊤\nq(A+θb⊤)xq).\nTherefore,\nA11=E{x1,y1}\u0012\n−D(2vθ⊤xq)Eθ⊤x1exp(x⊤\n1Axq+y1b⊤xq)\nDEx1exp(x⊤\n1Axq+y1b⊤xq)\u0013\n=−(2vθ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n+o(1\nD).\nA12=E{xi,yi}i∈[D](2vθ⊤xqPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq))(exp( x⊤\nqAxq) +Pexp(x⊤\niAxq+yib⊤xq))\n(DEexp(x1⊤Axq+y1b⊤xq))2\n−E{xi,yi}i∈[D](2vθ⊤xqPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq))(DEexp(x1⊤Axq+y1b⊤xq))\n(DEexp(x1⊤Axq+y1b⊤xq))2\n=2v(θ⊤yq)\nDθ⊤\u0002\n2σ2\nx(A+θb⊤)xq+xq\u0003\nexp\u0000\nx⊤\nq(A+θb⊤)⊤(A+θb⊤)xq\u0001\n−2v\nD(θ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n+2v(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq).\nA13=2v\nD(θ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n−2v\nD(θ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\nexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq).\nTo sum up, we have\nA1=A11+A12+A13\n=−2(vθ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n+(2vθ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\nexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)\n37Superiority of Multi-Head Attention in In-Context Linear Regression\n+2v\nD(θ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq\u0003\nexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq).\nIn terms of A2, we have\nExix⊤\niexp(2 x⊤\ni(A+θb⊤)xq)\n=E\u0012xi−xq\nσx\u0013\u0012xi−xq\nσx\u0013⊤\nσ2\nxexp\u0012\n2σx(xi−xq)⊤(A+θb⊤)xq\nσx+ 2x⊤\nq(A+θb⊤)xq\u0013\n+xqx⊤\nqexp(σ2\nx\n2v2∥xq∥2+ 2∥xq∥2/v)\n=\u0000\nσ2\nx\u0000\nId+ 4σ2\nx(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤\u0001\n+xqx⊤\nq\u0001\nexp\u0012\n2σx(xi−xq)⊤(A+θb⊤)xq\nσx+ 2x⊤\nq(A+θb⊤)xq\u0013\n.\nsince xis are independent with each other, we have\nE{xi}i∈[D]A2\n=E{xi}i∈[D] \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\nDEx1exp(x⊤\n1Axq+y1b⊤xq)!2\n−2E{xi}i∈[D]\u0010\nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\u00112\n\u0000\nDEx1exp(x⊤\n1Axq+y1b⊤xq)\u00013\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u0011\n−2E{xi}i∈[D]\u0010\nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\u00112\n\u0000\nDEx1exp(x⊤\n1Axq+y1b⊤xq)\u00013exp(x⊤\nqAxq)\n+3E{xi}i∈[D]\u0010\nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)\u00112\n\u0000\nDEx1exp(x⊤\n1Axq+y1b⊤xq)\u00014\u0010X\nexp(x⊤\niAxq+yib⊤xq)−(DEx1exp(x⊤\n1Axq+y1b⊤xq))\u00112\n=v2DE(θ⊤xi)2exp(2 x⊤\niAxq+ 2yib⊤xq) +D(D−1)v2E2(θ⊤xi) exp( x⊤\niAxq+yib⊤xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))2\n−4D(D−1)v2E(θ⊤xi) exp( x⊤\niAxq+yib⊤xq)E(θ⊤xi) exp(2 x⊤\niAxq+ 2yib⊤xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))3\n+4D(D−1)v2E(θ⊤xi) exp( x⊤\niAxq+yib⊤xq)2Eexp(x⊤\niAxq+yib⊤xq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))3\n−D(D−1)v2E(θ⊤xi) exp( x⊤\niAxq+yib⊤xq)2exp(x⊤\nqAxq)\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))3\n+3D2(D−1)v2E2(θ⊤xi) exp( x⊤\niAxq+yib⊤xq)\u0002\nEexp(2 x⊤\niAxq+ 2yib⊤xq)−E2exp(x⊤\niAxq+yib⊤xq)\u0003\n(DEx1exp(x⊤\n1Axq+y1b⊤xq))4+o\u00121\nD\u0013\n=v2\nDθ⊤(σ2\nxId+ 4σ4\nx(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤+xqx⊤\nq)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2(θ⊤\u0002\nσ2\nx(A+��b⊤)xq+xq\u0003\n)2\n+v2\nDθ⊤(−4(2σ2\nx(A+θb⊤)xq+xq)(σ2\nx(A+θb⊤)xq+xq)⊤+ 3(σ2\nx(A+θb⊤)xq+xq)(σ2\nx(A+θb⊤)xq+xq)⊤)θ\n×exp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n−2v2(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq)+o(1\nD)\n=v2\nDθ⊤(σ2\nxId−σ4\nx(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n38Superiority of Multi-Head Attention in In-Context Linear Regression\n+v2(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2−2v2(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq)+o(1\nD)\nBased on the results of A1andA2, we have\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014\nxq\n0\u0015\u0013\u0013 2\n=E(xq,θ)\u0014\n(x⊤\nqθ)2+A1+A2\u0015\n=E(xq,θ)\u0014\n(x⊤\nqθ)2−2(vθ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n+(2vθ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\nexp(x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq/2)\n+2v\nD(θ⊤xq)θ⊤\u0002\nσ2\nx(A+θb⊤)xq\u0003\nexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2\nDθ⊤(σ2\nxId−σ4\nx(A+θb⊤)xqx⊤\nq(A+θb⊤)⊤)θexp(x⊤\nq(A+θb⊤)⊤(A+θb⊤)xq)\n+v2(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2−2v2(θ⊤\u0002\nσ2\nx(A+θb⊤)xq+xq\u0003\n)2exp (x⊤\nqAxq)\nDexp(x⊤q(A+θb⊤)⊤(A+θb⊤)xq)\u0015\n+o\u00121\nD\u0013\n.\nFrom the above formulation, one can see that the optimal solution satisfies\nv[σ2\nx(A+θb⊤) +Id]≈Id.\nTaking v= 1,A= 0d×d,b=0, we have\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014\nxq\n0\u0015\u0013\u0013 2\n=O\u0012σ2\nx\nD\u0013\n+o\u00121\nD\u0013\n.\nA.7.2. T HEOREM 5.5\nProof of Theorem 5.5. Recall that for single-head attention, we take A=Id/vandb= 0. Following the proof of Theorem\n4.1, the prediction risk becomes\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n=E(xq,θ)E{xi}i∈[D]\ny2\nq−2yq \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!\n+ \nvPD\ni=1θ⊤xiexp(x⊤\niAxq+yib⊤xq)Pexp(x⊤\niAxq+yib⊤xq) + exp( x⊤qAxq)!2\n\n=E(xq,θ)E{xi}i∈[D]\u0012\ny2\nq−2yq \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)Pexp(x⊤\nixq/v) + exp( x⊤qxq/v)!\n| {z }\n=A1+ \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)Pexp(x⊤\nixq/v) + exp( x⊤qxq/v)!2\n| {z }\n=A2\u0013\n.\nRecall that in the testing stage, xi∼N(xq, σ2\nx). In this case,\nEexp(x⊤\nixq/v) =Eexp\u0012(xi−xq)⊤xq\nσxvσx+∥xq∥2/v\u0013\n= exp\u0012σ2\nx\n2v2∥xq∥2+∥xq∥2/v\u0013\n,\nand\nExiexp(x⊤\nixq/v) =σ2\nx\nvxqexp\u0012σ2\nx\n2v2∥xq∥2+∥xq∥2/v\u0013\n+xqexp\u0012σ2\nx\n2v2∥xq∥2+∥xq∥2/v\u0013\n.\n39Superiority of Multi-Head Attention in In-Context Linear Regression\nConsequently, fixing xqandθ,\nEA1=−2yqE \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)Pexp(x⊤\nixq/v) + exp( x⊤qxq/v)!\n=−2yqE \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)\nEPexp(x⊤\nixq/v) + exp( x⊤qxq/v)!\n+2yqE \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)\n(EPexp(x⊤\nixq/v) + exp( x⊤qxq/v))2!\u0010X\nexp(x⊤\nixq/v)−EX\nexp(x⊤\nixq/v)\u0011\n−2yqE \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)\n(EPexp(x⊤\nixq/v) + exp( x⊤qxq/v))3!\u0010X\nexp(x⊤\nixq/v)−EX\nexp(x⊤\nixq/v)\u00112\n+o\u00121\nD\u0013\n:=A11+A12+A13,\nwhere\nA11=−2y2\nq\u0000\nσ2\nx+v\u0001Dexp(σ2\nx∥xq∥2/(2v2))\nDexp(σ2x∥xq∥2/(2v2)) + 1=−2y2\nq(σ2\nx+v) +O\u00121\nD\u0013\n,\nA12= 2y2\nq\u0012\n2σ2\nxexp\u00122σ2\nx\nv2∥xq∥2\u0013\n−σ2\nxexp\u0012σ2\nx\nv2∥xq∥2\u0013\u0013D\n(Dexp(σ2x∥xq∥2/(2v2)) + 1)2=O\u00121\nD\u0013\n,\nand\nA13=−2y2\nq(σ2\nx+v)Dexp(σ2\nx∥xq∥2/(2v2))\n(Dexp(σ2x∥xq∥2/(2v2)) + 1)3D\u0012\nexp\u00122σ2\nx\nv2∥xq∥2\u0013\n−exp\u0012σ2\nx\nv2∥xq∥2\u0013\u0013\n+o\u00121\nD\u0013\n=O\u00121\nD\u0013\n.\nForA2, when fixing xqandθ, we have\nA2=E \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)Pexp(x⊤\nixq/v) + exp( x⊤qxq/v)!2\n=E \nvPD\ni=1θ⊤xiexp(x⊤\nixq/v)\nEPexp(x⊤\nixq/v) + exp( x⊤qxq/v)!2\n+O\u00121\nD\u0013\n=D(D−1)\n(Dexp(σ2x∥xq∥2/(2v2)) + 1)2\u0002\n(σ2\nx+v)x⊤\nqθ\u00032exp\u0012σ2\nx\nv2∥xq∥2\u0013\n+O\u00121\nD\u0013\n= (σ2\nx+v)2y2\nq+ +O\u00121\nD\u0013\n.\nTo conclude, when fixing xqandθ, we obtain\nE\u0012\nyq−(WV\nd+1,:)⊤Eϕ\u0012\nE⊤(WK)⊤WQ\u0014xq\n0\u0015\u0013\u0013 2\n= (σ2\nx+v−1)2y2\nq+O\u00121\nD\u0013\n.\n40Superiority of Multi-Head Attention in In-Context Linear Regression\nB. Simulation and Experiment Details\nB.1. Visualization of Single-Head Attention Score\nBased on Theorem 4.1, the optimal Ais in the format of Id/v+o. As a result, there are two possible cases. (i) When\nv >0, the attention score of xqagainst itself is usually the largest one as x⊤\nqAxq=∥xq∥2/vis always positive. (ii) When\nv <0, the attention score of xqagainst itself is always small. Figure 15 shows these two cases correspondingly.\n(a)v > 0\n (b)v < 0\nFigure 15. Single-Head Attention Score for 10 tasks.\nB.2. Noisy Response and Correlated Features\nFor noisy response and correlated features, we conduct experiments to verify the effectiveness of multi-head attention. The\nresults for noisy label can be found in Figure 16. While the best prediction loss is away from zero, one can still see that\nwith sufficient input embedding dimension, multi-head attention improves the performance.\nFor correlated features, to generate Σ, we follow the procedure in Zhang et al. (2023) and take the diagonal elements\nfollowing exp(1) distribution. For the off diagonal elements, we take all of them as 0.1. From Figure 17 we can see that\nmulti-head attention with p/h > d is better than single-head attention.\n41Superiority of Multi-Head Attention in In-Context Linear Regression\n(a) No read-in layer\n (b)p= 6\n(c)p= 128\n (d)p= 256\nFigure 16. ICL performance with noisy responses.\nB.3. Other Figures\n42Superiority of Multi-Head Attention in In-Context Linear Regression\n(a) No read-in layer\n (b)p= 6\n(c)p= 128\n (d)p= 256\nFigure 17. ICL Performance with correlated features.\n43Superiority of Multi-Head Attention in In-Context Linear Regression\n(a)p= 6\n (b)p= 64\n(c)p= 128\n (d)p= 256\nFigure 18. Standard experiment.\n44Superiority of Multi-Head Attention in In-Context Linear Regression\n(a)σ= 0.01\n (b)σ= 0.05\n(c)σ= 0.1\n (d)σ= 0.5\nFigure 19. ICL performance given prior knowledge. Single-head attention. Different test method of prior knowlegde (random, parallel,\netc) in different subfigures. α= 0.1. Add description\n45Superiority of Multi-Head Attention in In-Context Linear Regression\n(a)σ= 0.01\n (b)σ= 0.05\n(c)σ= 0.1\n (d)σ= 0.5\nFigure 20. α= 0.1head=16 Different test method of prior knowlegde (random, parallel ..) in different subfigure, different sigma in\ndifferent line\n46Superiority of Multi-Head Attention in In-Context Linear Regression\n(a) Training and testing datasets follow the same distribu-\ntion. Single head.\n(b) Training and testing datasets follow the same distri-\nbution. 16 heads.\n(c) Training and testing datasets follow different distribu-\ntions. Single head.\n(d) Training and testing datasets follow different distribu-\ntions. 16 heads.\nFigure 21. ICL Performance with local examples.\n47" }, { "title": "2401.17437v1.Ultrafast_measurements_under_anisotropic_strain_reveal_near_equivalence_of_competing_charge_orders_in_TbTe__3_.pdf", "content": "Ultrafast measurements under anisotropic strain reveal near equivalence of competing\ncharge orders in TbTe 3\nSoyeun Kim,1, 2Gal Orenstein,1, 2Anisha G. Singh,2Ian R. Fisher,2David A. Reis,1, 2and Mariano Trigo1, 2\n1Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, United States\n2Stanford Institute for Materials and Energy Sciences,\nSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n(Dated: February 1, 2024)\nWe report ultrafast reflectivity measurements of the dynamics of the order parameter of the charge\ndensity wave (CDW) in TbTe 3under anisotropic strain. We observe an increase in the frequency\nof the amplitude mode with increasing tensile strain along the a-axis (which drives the lattice into\na > c , with aandcthe lattice constants), and similar behavior for tensile strain along c(c > a ).\nThis suggests that both strains stabilize the corresponding CDW order and further support the near\nequivalence of the CDW phases oriented in a- and c-axis, in spite of the orthorhombic space group.\nThe results were analyzed within the time-dependent Ginzburg-Landau framework, which agrees\nwell with the reflectivity dynamics. The strain dependence suggests that the orthogonal aandc\nCDWs are separately stabilized under the corresponding a > c andc > a strain, respectively. Our\nstudy presents an ultrafast approach to assess the stability of phases and order parameter dynamics\nin strained systems.\nA central problem in condensed matter physics is un-\nderstanding and controlling emergent phases in complex\nmaterials. Hydrostatic pressure and strain engineering,\nwhich can modify exchange couplings and inter-site hop-\nping energies, are particularly fruitful approaches for in-\nducing new phases [1–7]. Compared to hydrostatic pres-\nsure or epitaxial strain, uniaxial or highly anisotropic\nstrain can more effectively lower the lattice symmetry\nand induce new states with different broken symmetries.\nThis can have a qualitative impact on systems with com-\npeting phases, a prime example being nematic order in\nFeSe and its relation to superconductivity [8, 9].\nThe rare-earth tri-tellurides, RTe3(R= rare earth\nions), provide an attractive platform to realize uniaxial\nstrain-controlled phases and control of competing orders.\nThese materials exhibit two competing charge density\nwave (CDW) instabilities along two perpendicular crys-\ntallographic axes with almost identical lattice parameters\n(a= 4.3081(10) and c= 4.3136(10) for TbTe 3at room\ntemperature [10]). Chemical pressure [11, 12] as well as\nhydrostatic pressure [2, 3, 13] have been shown to tune\nthe CDW instability and even enhance superconductiv-\nity at high pressure [14], suggesting that the equilibrium\nphase can be tuned with strain. Since the CDWs are\nhosted in the near-square, quasi-two-dimensional Te lay-\ners [11], anisotropic strain in the a-cplane has the po-\ntential for tuning the near-degeneracy between the two\nperpendicular CDWs.\nIn this work, we present ultrafast reflectivity mea-\nsurements of the dynamics of the CDW order parame-\nter in TbTe 3under anisotropic strain at room temper-\nature. Unlike equilibrium measurements, ultrafast spec-\ntroscopy can provide crucial information on the single-\nparticle and collective modes of the CDW [15–17]. By\nvirtue of its nonequilibrium nature, pump-probe spec-\ntroscopy can unveil valuable information inaccessible byother means. For example, nonlinear dynamics of the\norder parameter under high excitation reveal the anhar-\nmonic shape of the potential energy [18–21], and provides\ndeeper insights into the broken symmetry phase.\nRecent experiments give support to the near equiv-\nalence of the two CDWs in RTe3. Ultrafast electron\ndiffraction (UED) studies have shown an emergent CDW\nwith wavevector in the adirection after photoexcitation\nquenches the equilibrium c-axis CDW [22]. Equilibrium\nx-ray diffraction measurements showed a similar rotation\nof the CDW wavevector by 90 degrees (from the c-axis to\nthea-axis) with tensile strain along the a-axis [23, 24].\nThese results suggest an intimate relationship between\nthea- and c-axes instabilities, a near equivalence between\nthese two directions, and an emergent tetragonal symme-\ntry proposed in ErTe 3[23].\nOur ultrafast reflectivity measurements on TbTe 3show\na stiffening of the CDW amplitude mode (AM) for a\nstrong enough tensile strain along the a-axis, which is\nknown to reorient the CDW wavevector by 90 degrees\nfrom the c- to the a-axis [23, 24]. This stiffening in-\ndicates that strain further stabilizes the order parameter\nof the rotated CDW for a > c . On the other hand, tensile\nstrain along the c-axis ( c > a ), also stiffens the original\nAM. These observations indicate a strong similarity be-\ntween the potential energy surface of the two CDWs and\nfurther supports the proposed near equivalence between\nthecandaorders.\nWe analyze the reflectivity dynamics within the frame-\nwork of the time-dependent Ginzburg-Landau model\n[16, 18, 19]. Our analysis of the coherent order param-\neter dynamics indicates that the system stiffens and de-\nvelops a deeper free energy minimum for either a/c < 1\nora/c > 1, which further stabilizes the candaorder, re-\nspectively. The near identical behavior for the two strain\nregimes is a signature of the near equivalence of the twoarXiv:2401.17437v1 [cond-mat.str-el] 30 Jan 20242\n0.00\n0.23\n0.37\n0.61\n0.72dL/Leff (%)(a)\nab\ncρ\nc > a a > c\nTbTe3platform\ntdelay\nprobe pumpdR(tdelay, dL/Leff)(b) (c)\nTeTb\nc\nbadhesive|Δ(t)|\nLeff+dL\naa\nc\n(d)\n0 2 4 6012dR (a.u.)\ntdelay (ps)\nFIG. 1. (a)The crystal structure of TbTe 3in the Cmcm space group, featuring nearly tetragonal Te bilayers that separate\nTbTe blocks. (b)A schematic representation of the charge density ( ρ) amplitude ( |∆|) in real space. The charge-density-wave\n(CDW) phase forms along the c-axis at room temperature, with a cross-section along the c-axis displayed at the top of the\nimage. (c)An illustration of the strain-dependent transient reflectivity setup. The sample is attached to the bowtie platform\nand experiences the anisotropic strain along the a-axis. Leffdenotes the effective distance, and dLthe applied displacement.\nTwo near-normal 800 nm pump and probe beams are incident onto the sample. (d)The transient time traces for various dL\nvalues, each displayed with an offset.\nCDWs.\nThe crystal structure of the bulk TbTe 3belongs to the\nCmcm space group (No. 63), and TbTe 3undergoes a\nCDW phase transition at TCDW = 336 K [11]. As in other\nseries of the RTe3family, the system is orthorhombic\nas depicted in Fig. 1(a). At room temperature, the c\nlattice constant is larger than aby 0.13 % and the CDW\nwavevector qCDW is oriented along the c-axis, as in Fig.\n1(b). The difference between aandcreduces at higher\ntemperatures and the CDW becomes unstable.\nThe small in-plane anisotropy is considered crucial for\ndetermining the direction of the CDW in RTe3. Recent\nx-ray diffraction studies on the anisotropic-strain depen-\ndence on ErTe 3and TbTe 3have demonstrated the CDW\ncan be reoriented by controlling the lattice constant ratio\na/c[23, 24]. Because applying strain does not remove the\nglide plane, the C4rotation symmetry remains absent,\nas in the pristine crystal. Nevertheless, x-ray diffraction\nmeasurements showed that the CDW wavevector rotates\nby 90◦with increasing a/c[23]. Resistivity measurements\nunder strain confirmed that the transition temperature\nincreases as a/cdeviates further from 1 for both a/c < 1\nanda/c > 1 [23, 24], suggesting that strain stabilizes the\nrespective coraCDW.\nTo manipulate the CDW state in TbTe 3, an anisotropic\nstrain field was applied along the a-axis using a CS-130\nRazorbill strain cell device [25]. As depicted in Fig. 1,\nthe bulk TbTe 3was attached to the neck of the bowtie-\nshaped platform, following the approach in the ErTe 3\nstudy [23]. The sample was cleaved to a thickness of 5-\n10µm and to expose a fresh surface and installed onto\nthe strain cell device for the reflectivity measurements.The effective length Leffof the bowtie neck where the\nstrain is active is 3.47 mm [23, 26], and the displacement\nchange dLin the strain cell device is obtained from the\ninternal capacitor equipped inside the cell [25]. It is note-\nworthy that for most values of dLwe used, the platform\napproaches or exceeds the plastic deformation limit of\nthe titanium bowtie. This indicate the actual strain ap-\nplied on the sample that affect the a/cmay not have a\nlinear relation with dL. Further details are provided in\nRef. [23] and in the Supplementary Material [27].\nThe transient reflectivity time traces were obtained\nfrom an optical pump-probe setup using a Coherent\nRegA laser with 800 nm wavelength at a 250 kHz repe-\ntition rate. The probe (pump) beam had a 1 /e2width\nof 40 (170) µm with a near-normal incident angle, which\npassed through a neutral density filter to change the flu-\nence. The polarization of probe and pump beams were\northogonal to each other. The photon energy 1.55 eV is\nmuch higher than the CDW optical gap of TbTe 3at room\ntemperature (220 meV) [12]. The pulse duration of the\npump and probe beams was around 70 fs at the sample\nposition, and the pump-probe delay was controlled with\na mechanical delay stage.\nFigure 1(d) presents the strain-dependent time traces\nof the transient reflectivity at room temperature. The\nscans were initiated from the maximum dL/L effand se-\nquentially lowered (bottom to top traces) to minimize\nthe extrinsic change in the reflectivity, such as crack for-\nmation.\nThe general features observed in the traces depicted\nin Fig. 1(d) align with findings from previous transient\nreflectivity studies on RTe3[15, 18, 19, 28]. The shape of3\n0.51.01.5\n4812Γ (THz)\n dL/Leff (%)η0Increasing a-axis tensile strain(b)\n(c)\n0.0 0.4 0.8\n0.0 0.4 0.8\n0 1 20123456dR (a.u.)\ntdelay (ps)0.72 %0.23\n0.37\n0.610.00 TDGL (a)\ndL/Leff = dL/Leff (%)\nFIG. 2. (a)Transient reflectivity traces under tensile strain\nalong the a-axis. Traces are offset vertically for clarity. The\nincident pump fluence was kept at 9 µJ/cm2for all traces.\nThe fit using the time-dependent Ginzburg-Landau (TDGL)\nmodel with β= 2.5 THz and Ω /2π= 2.2 THz are overlaid\nwith dashed lines. (b-c) Results of the fits (b) η0and (c) Γ.\nthe initial peak relates to the initial order parameter dy-\nnamics, and the rise time of the peak was interpreted\nas the time needed to fully suppress the CDW order\n[28]. Phenomenological Ginzburg-Landau models based\non quartic potentials successfully describe the trend in\ntransient reflectivity at different temperatures and over\na wide range of pump fluences, including at high fluence\nwhere nonlinearities in the lattice motion become impor-\ntant [18–20].\nWe now investigate how anisotropic strain affects the\norder parameter dynamics. When no strain is applied\n(top trace in Fig. 1(d); dL/L eff= 0 %), the time trace\nshows a smooth peak about 0.5 ps after the pump. A sec-\nond hump starts to appear at 1.2 ps and becomes sharper\nwith increasing dL/L eff(lower traces in Fig. 1(d)). Ad-\nditional oscillating features becomes discernible for time\nrange of about 1.5 to 6 ps, which are prominent at\ndL/L eff= 0.72% (see Fig. S3 in the Supplemental Mate-\nrials for details [27]).\nThe early dynamics of the reflectivity, i.e.,the initial\npeak and the second hump, are dominated by the CDW\namplitude mode [19] and dynamic critical slowing down\n[28]. After 1 .5 ps, the trace is dominated by coherent os-\ncillation of 1.65 THz [27] from an optical phonon coupled\nto the AM [15]. In our analysis below, we focus on the\nbehavior at t <1 ps, which reflect the order parameter\ndynamics in the dynamic critical slowing down regime.\nThe initial peak shows dramatic change with dL/L eff,\n0 1 2012345dR/Fpump (a.u.)\ntdelay (ps)\nIncreasing pump fluence TDGL (a)\n0 30 60\n0 30 60Fpump (μJ/cm2)0.51.01.5\n4812Γ (THz)η0(b)\n(c)\nFpump (μJ/cm2)9 μJ/cm2183570\nFpump =\nFIG. 3. (a)Fluence dependent time traces of transient re-\nflectivity under applied a-axis tensile strain ( dL/L eff= 0.72\n%). Each trace is normalized by fluence and offset vertically.\nThe fit using the TDGL with β= 2.5 THz and Ω /2π= 2.2\nTHz are overlaid with dashed lines. (b-c) Results of the fits\nfor (b) η0and (c) Γ.\nas shown in Fig. 2(a). With increasing tensile strain\nalong the a-axis (increasing dL/L eff), the peak position\nshifts to earlier in time, the peak intensity first increases\nand then decreases, and the peak sharpens. For traces\nwithout vertical offsets, see Fig. S3 in the Supplemental\nMaterials [27]. Importantly, the oscillations of the order\nparameter become more defined at higher dL/L eff(see\nsecond cycle at t= 1 ps in dL/L eff= 0.72 %). This\noccurs because of both a decrease in the damping and an\nincrease in the frequency of the AM at higher strain.\nThe rise time of the peak τis related to the period\nof the AM, which diverges (the AM frequency vanishes)\nat the critical point in equilibrium. Thus, the decrease\ninτand decrease in the AM damping (better-defined\nAM oscillations) with increasing dL/L effobserved here\nimplies that the order parameter becomes more robust\nwith increasing dL/L eff, and that the system is pushed\naway from the critical point by further stabilizing the\nordered phase. The initial peak dynamics showed a sim-\nilar CDW stabilizing trend when the tensile strain was\napplied along the c-axis (see Fig. S5 in Supplemental\nMaterials [27]).\nTo gain further insight into the strain-dependency in\nthe transient reflectivity, we analyze the traces using\nthe time-dependent Ginzburg-Landau (TDGL) model for\nsecond-order phase transitions, which has provided a\ngood phenomenological description of the CDW order pa-\nrameter dynamics in RTe3[18–20, 28, 29]. Following Ref.4\n[19], the model is constructed by assuming the real di-\nmensionless order parameter y, which is the amplitude of\nthe CDW distortion at qCDW. With normalizing factor\nx0, the effective potential can be represented with yas\nV(y) =ax2\n0\n4(2(η−1)y2+y4), (1)\nand the dynamics of yare governed by the equation of\nmotion\n1\na¨y+ [η(t)−1]y+y3+2Γ\na˙y= 0. (2)\nHere a= 95.54 THz2is related to the AM angular fre-\nquency Ω by a= Ω2/2, Γ is a phenomenological damp-\ning, and ηis a parameter that controls the stability of\nthe CDW [19]. In equilibrium, η=T/T CDW,i.e.,η= 0\nindicate the CDW ground state, and η > 1 represents\nthe free energy of the high-symmetry phase [18, 19]. To\nintroduce the transient change triggered by the optical\npump, we take η(t) =η0e−βtΘ(t), where Θ( t) is a step\nfunction and βis the decay rate of the electronic excita-\ntion [19, 27].\nWe now relate the order parameter y(t) to the reflec-\ntivity. The expansion of the reflectivity in terms of the\norder parameter is quadratic in y[1, 19] which by symme-\ntry must be insensitive to the sign of y. Thus we fit the\ntime traces using the function A(1−y2(t)),Abeing an\namplitude (See Fig. S1 in Supplemental Materials). We\nfixed Ω /2πto the AM frequency 2.2 THz in unstrained\nTbTe 3at low temperature [15]. To minimize the num-\nber of the parameters we fit only η0and Γ and kept the\nrelative zero time delay, A, and βfixed.\nFigure 2 presents the fit results of the TDGL model\nto reflectivity traces for the AM dynamics at t <2 ps.\nOverall, the fit curves in panel (a) capture the main fea-\ntures in each trace described earlier, including the trend\nwith increasing strain. The fitted η0values in (b) show a\ngradual decrease with increasing dL/L eff. The value of Γ\n(damping rate) also decreases with increasing strain, in-\ndicating that the order parameter is more harmonic (less\ndamped) and its dynamics last about 2.5 times longer in\nthe 0.72 % trace than for the 0 %. The results of the fits\nin Figs. 2(b,c) clearly show that the a-axis tensile strain\ndecreases η0and Γ. This signifies that the strain pushes\nthe potential energy into deeper double-well form (de-\ncreasing η) as shown schematically in Fig. 4(a), hence\nstabilizing the CDW, which for a > c is rotated with\nrespect to the unstrained CDW [23].\nAt higher pump fluence ( Fpump) an increase in ηcan\ndestabilize the CDW order [27–29]. In Fig. 3, we verified\nthe fluence dependence of the strain-stabilized CDW at\ndL/L eff= 0.72 %. The shape change in the initial peak\nwith increasing Fpump is analogous to the trend with de-\ncreasing dL/L effin Fig. 2. The gradual broadening (in-\ncreasing τ) and faster damping of the oscillation at high\nFpump again indicate that η0and Γ increase with Fpump.\na-axis tensile strain dL/Leffc > a a > c\na\nc\na > c\nPump fluence Fpump\n(b) (c)(a)\nV(y)0.0\n-0.5\n-1 0 1\nyFpump dL/Leff\nη = 0.00.51.01.5FIG. 4. (a)The Ginzburg-Landau potential V(y) shape for\nthe representative ηvalues. (b-c) Schematic diagrams of the\nCDW affected by increasing (b) the a-axis tensile strain and\n(c) pump fluence under strain. The arrows in (a)indicate the\nchange in the potential driven by dL/L efforFpump.\nThe values extracted from the TDGL fits are shown in\nFigs. 3(b,c) and are comparable with fluence dependence\nin prior measurements on SmTe 3[19].\nFigure 4 summarizes the behavior of the Ginzburg-\nLandau potential with the a-axis tensile strain and pump\nfluence, extracted from Figs. 2 and 3, respectively. As\nmentioned earlier, prior strain studies on ErTe 3and\nTbTe 3demonstrated that the CDW wavevector is ro-\ntated from the c- to the a-axis when a > c [23, 24]. As\nthe in-plane anisotropy grows, the reoriented CDW is\nfurther stabilized, making the potential well deeper and\nhardening the AM frequency, as evidenced by the ob-\nserved decrease in τand corresponding decrease in the\nfitted η0(Fig. 2 (b)) with increasing dL/L eff.\nThe fact that both CDWs can be modeled with sim-\nilar parameters within TDGL by using the same AM\nfrequency (Ω /2π= 2.2 THz) suggests that the original\n(unstrained) and reoriented CDW are virtually indistin-\nguishable and that they originate from the same free en-\nergy. This suggests they are manifestations of the same\nfree energy instability and further supports the interpre-\ntation that the original (with wavevector along c) and\nreoriented (with wavevector along a) CDW phases are\nnearly equivalent [23].\nTo conclude, we used ultrafast optical spectroscopy to\nprobe the dynamics of the order parameter in TbTe 3\nunder anisotropic strain. The response when a/c < 1\nanda/c > 1 is nearly identical and both CDWs can be\nmodeled with the same Ginzburg-Landau potential de-\nspite the intrinsically orthorhombic nature of the under-\nlying lattice, which further supports the near equivalence\nof the two CDWs. These results demonstrate a versa-5\ntile tool combining ultrafast spectroscopy with uniaxial\nstrain tuning device to probe the potential energy sur-\nfaces and low energy excitations of competing orders in\nquantum materials.\nS.K., V.K., A.G.S., I.R.F., D.A.R., and M.T. were\nsupported by the US Department of Energy, Office of\nScience, Office of Basic Energy Sciences through the Di-\nvision of Materials Sciences and Engineering under Con-\ntract No. DE-AC02-76SF00515. G.O. acknowledges sup-\nport from the Koret Foundation.\n[1] V. L. Ginzburg, Soviet Physics Uspekhi 5, 649–660\n(1963).\n[2] A. Sacchetti, E. Arcangeletti, A. Perucchi, L. Baldas-\nsarre, P. Postorino, S. Lupi, N. Ru, I. R. Fisher, and\nL. Degiorgi, Physical Review Letters 98, 026401 (2007).\n[3] D. A. Zocco, J. J. Hamlin, K. Grube, J.-H. Chu, H.-H.\nKuo, I. R. Fisher, and M. B. Maple, Physical Review B\n91, 205114 (2015).\n[4] D. G. Schlom, L.-Q. Chen, C.-B. Eom, K. M. Rabe, S. K.\nStreiffer, and J.-M. Triscone, Annual Review of Materials\nResearch 37, 589–626 (2007).\n[5] D. G. Schlom, L.-Q. Chen, C. J. Fennie, V. Gopalan,\nD. A. Muller, X. Pan, R. Ramesh, and R. Uecker, MRS\nBulletin 39, 118–130 (2014).\n[6] R. Xu, J. Huang, E. S. Barnard, S. S. Hong, P. Singh,\nE. K. Wong, T. Jansen, V. Harbola, J. Xiao, B. Y. Wang,\nS. Crossley, D. Lu, S. Liu, and H. Y. Hwang, Nature\nCommunications 11(2020).\n[7] C. Ji, B. Li, W. Liu, J. S. Smith, A. Majumdar, W. Luo,\nR. Ahuja, J. Shu, J. Wang, S. Sinogeikin, Y. Meng,\nV. B. Prakapenka, E. Greenberg, R. Xu, X. Huang,\nW. Yang, G. Shen, W. L. Mao, and H.-K. Mao, Nature\n573, 558–562 (2019).\n[8] M. Ghini, M. Bristow, J. C. A. Prentice, S. Sutherland,\nS. Sanna, A. A. Haghighirad, and A. I. Coldea, Physical\nReview B 103, 205139 (2021).\n[9] S. Hosoi, K. Matsuura, K. Ishida, H. Wang,\nY. Mizukami, T. Watashige, S. Kasahara, Y. Matsuda,\nand T. Shibauchi, Proceedings of the National Academy\nof Sciences 113, 8139–8143 (2016).\n[10] N. Ru, Charge density wave formation in rare-earth\ntritellurides , Ph.D. thesis, Stanford University (2008).\n[11] N. Ru, C. L. Condron, G. Y. Margulis, K. Y. Shin,\nJ. Laverock, S. B. Dugdale, M. F. Toney, and I. R. Fisher,\nPhysical Review B 77, 035114 (2008).\n[12] B. F. Hu, B. Cheng, R. H. Yuan, T. Dong, and N. L.\nWang, Physical Review B 90, 085105 (2014).\n[13] J. Kopaczek, H. Li, K. Yumigeta, R. Sailus, M. Y.\nSayyad, S. T. R. Moosavy, R. Kudrawiec, and S. Ton-\ngay, Journal of Materials Chemistry C 10, 11995–12000\n(2022).\n[14] J. J. Hamlin, D. A. Zocco, T. A. Sayles, M. B. Maple,\nJ. H. Chu, and I. R. Fisher, Physical Review Letters 102\n(2009).[15] R. V. Yusupov, T. Mertelj, J.-H. Chu, I. R. Fisher,\nand D. Mihailovic, Physical Review Letters 101, 246402\n(2008).\n[16] H. Schaefer, V. V. Kabanov, and J. Demsar, Physical\nReview B 89, 045106 (2014).\n[17] A. S. Disa, T. F. Nova, and A. Cavalleri, Nature Physics\n17, 1087 (2021).\n[18] R. Yusupov, T. Mertelj, V. V. Kabanov, S. Brazovskii,\nP. Kusar, J.-H. Chu, I. R. Fisher, and D. Mihailovic,\nNature Physics 6, 681 (2010).\n[19] M. Trigo, P. Giraldo-Gallo, M. E. Kozina, T. Henighan,\nM. P. Jiang, H. Liu, J. N. Clark, M. Chollet, J. M. Glow-\nnia, D. Zhu, T. Katayama, D. Leuenberger, P. S. Kirch-\nmann, I. R. Fisher, Z. X. Shen, and D. A. Reis, Physical\nReview B 99, 104111 (2019).\n[20] M. Trigo, P. Giraldo-Gallo, J. N. Clark, M. E. Koz-\nina, T. Henighan, M. P. Jiang, M. Chollet, I. R. Fisher,\nJ. M. Glownia, T. Katayama, P. S. Kirchmann, D. Leuen-\nberger, H. Liu, D. A. Reis, Z. X. Shen, and D. Zhu, Phys-\nical Review B 103, 054109 (2021).\n[21] G. Orenstein, R. A. Duncan, G. A. d. l. P. Munoz,\nY. Huang, V. Krapivin, Q. L. Nguyen, S. Teitelbaum,\nA. G. Singh, R. Mankowsky, H. Lemke, M. Sander,\nY. Deng, C. Arrell, I. R. Fisher, D. A. Reis, and M. Trigo,\narXiv , 2304.00168 (2023).\n[22] A. Kogar, A. Zong, P. E. Dolgirev, X. Shen, J. Straqua-\ndine, Y.-Q. Bie, X. Wang, T. Rohwer, I.-C. Tung,\nY. Yang, R. Li, J. Yang, S. Weathersby, S. Park, M. E.\nKozina, E. J. Sie, H. Wen, P. Jarillo-Herrero, I. R. Fisher,\nX. Wang, and N. Gedik, Nature Physics 16, 159–163\n(2019).\n[23] A. G. Singh, M. D. Bachmann, J. J. Sanchez, A. Pandey,\nA. Kapitulnik, J. W. Kim, P. J. Ryan, S. A. Kivelson,\nand I. R. Fisher, arXiv , 2306.14755 (2023).\n[24] A. Gallo-Frantz, A. Sinchenko, D. Ghoneim, L. Ortega,\nP. Godard, P.-O. Renault, P. Grigoriev, A. Hadj-Azzem,\nP. Monceau, D. Thiaudi` ere, E. Bellec, V. Jacques, and\nD. L. Bolloc’h, arXiv , 2306.15712 (2023).\n[25] Cryostrain cell. Data sheet: CS130, Razorbill Instru-\nments, 2023.\n[26] J. Park, J. M. Bartlett, H. M. L. Noad, A. L. Stern, M. E.\nBarber, M. K¨ onig, S. Hosoi, T. Shibauchi, A. P. Macken-\nzie, A. Steppke, and C. W. Hicks, Review of Scientific\nInstruments 91, 083902 (2020).\n[27] See Supplemental Material.\n[28] A. Zong, P. E. Dolgirev, A. Kogar, E. Erge¸ cen, M. B.\nYilmaz, Y.-Q. Bie, T. Rohwer, I.-C. Tung, J. Straqua-\ndine, X. Wang, Y. Yang, X. Shen, R. Li, J. Yang,\nS. Park, M. C. Hoffmann, B. K. Ofori-Okai, M. E. Koz-\nina, H. Wen, X. Wang, I. R. Fisher, P. Jarillo-Herrero,\nand N. Gedik, Physical Review Letters 123, 097601\n(2019).\n[29] J. Maklar, Y. W. Windsor, C. W. Nicholson, M. Pup-\npin, P. Walmsley, V. Esposito, M. Porer, J. Rittmann,\nD. Leuenberger, M. Kubli, M. Savoini, E. Abreu, S. L.\nJohnson, P. Beaud, G. Ingold, U. Staub, I. R. Fisher,\nR. Ernstorfer, M. Wolf, and L. Rettig, Nature Commu-\nnications 12, 2499 (2021)." }, { "title": "2401.17453v1.Flat__Uniformly_Expanding_Space_in_five_Exercises.pdf", "content": "arXiv:2401.17453v1 [physics.ed-ph] 30 Jan 2024Flat, Uniformly Expanding Space in five Exercises\nBernd A. Berg\nDepartment of Physics\nFlorida State University, Tallahassee, Florida 32306\nFebruary 1, 2024\nAbstract\nExercises with solutions are presented which should allow advanced u ndergraduate\nstudents to understand properties of a flat, uniformly expanding space. No knowledge of\ngeneral or special relativity is needed besides that the speed of ligh tcis a constant. The\nmaterial could be used stand alone or would fit well at the end of a tre atment of special\nrelativity,\nUsing a reasonable value for the Hubble constant H, trajectories of light are calculated\nwithin the model, followed by a calculation of the Hubble horizon, and th e synchronization\nof clocks is demonstrated.\nIntroduction\nWe consider flat space ( /vector x,t), where a common time is defined for all space points\n(synchronization of clocks can be achieved and is discussed in exerc ise 5). Transformations\nto other coordinate systems are not considered (the model is not invariant under general\ncoordinate transformations). Let us focus on two space points w hich are initially, i.e., at\ncommon time t= 0, located a distance x0>0 apart. The distance between them is assumed\nto increase according to\nx1(t) =x0exp(tH) (1)\ndue to the Hubble expansion of space. Here His the Hubble constant, which is a constant\nin space, and we take it also to be a constant in time. For sufficiently lar ge timetthe two\npoints separate faster than with the speed of light, which is often a n issue of confusion for\nstudents who just learned special relativity. Here the resolution is thatcdt, withcconstant\nis only one of two contributions to the infinitesimal motion of light.\nIt has to pointed out that our exercises aim at a qualitative underst anding, and we\ncannot expect quantitatively relevant estimates of physical obse rvables. Nevertheless we\nshall in numerical calculations use physical values for candH. Estimates for Hare in the\nrange [1] from 68 to 74 km/sperMpc. We use\nH= 71 km/s per Mpc . (2)\n1In the following we take the x-axis along the straight line which connects the two points.\nThe location of our first space point is by definition the zero of the ax is,x= 0 for all t. Our\nsecond space point lies at x1(t), given by Eq.(1), on the x-axis. At position x0=x1(0) we\nimagine a laser which emits at t= 0 an ultrashort photon ray towards the first space point\natx= 0, where a photon detector is located. There are no gravitationa l interactions as we\nare dealing with a model of pure space.\n2. Exercises and Solutions\nExercise 1: The Model.\nConsider the infinitesimal motion dxof an ultrashort photon ray that is at time t= 0\nemitted at space point two towards space point one. Find and verify the analytical solution\nfor the travel path x(t)\nSolution:\nFor the position of the photon ray we derive a differential equation f rom the infinitesimal\nmotion:\ndx=−cdt+x{exp[H(t+dt)]−exp[Ht]}=−cdt+xexp[Ht]{exp[Hdt]−1}\n=−cdt+xexp[Ht]Hdt⇒dx\ndt=−c+xHexp[tH], (3)\nwherecis the speed of light, and Hthe Hubble constant. The solution is (Mathematica)\nx(t) = exp/parenleftBig\netH/parenrightBig/parenleftbigg\nK1−c\nHEi/parenleftBig\n−etH/parenrightBig/parenrightbigg\n, (4)\nwhereK1is an integration constant, and Eiis the exponential integral\nEi(y) =/integraldisplayy\n−∞dt′et′\nt′=γ+ln(|y|)−∞/summationdisplay\nk=1yk\nkk!. (5)\nHereγ= limn→∞(−ln(n)+/summationtextn\nk=11/k) is the Euler-Mascheroni constant. In the following\nthe solution (4) is verified by differentiation. It is convenient to intro duce the variable\ny=etH>0 and to break Eq.(3) up into\ndx\ndt=dx\ndydy\ndt,wheredy\ndt=HetH=H yholds. (6)\nIt remains to calculate dx/dy. Forx(y) we get from Eq.(4)\nx(y) =ey/parenleftbigg\nK1−c\nHEi(−y)/parenrightbigg\n. (7)\nTheyderivative of the exponential integral Ei(−y) is\ndEi(−y)\ndy=−dEi(−y)\nd(−y)=−d\nd(−y)/integraldisplay−y\n−∞dt′et′\nt′=−e−y\n−y=e−y\ny. (8)\n2 0 5x109 1x1010 1.5x1010 2x1010 2.5x1010 3x1010\n 0 4x109 8x109 1.2x1010 1.6x1010t [y]\nx(t) [ly]x0 = 7.00x109 [ly]\nx0 = 8.00x109 [ly]\nx0 = 8,21x109 [ly]\nx0 = 8.22x109 [ly]\nx0 = 8.24x109 [ly]\nx0 = 8.30x109 [ly]\nFigure 1: Trajectories of light for H= 71 km/s per Mpc and various initial distances x0.\nSo we get\ndx\ndy=ey/parenleftbigg\nK1−c\nHEi(−y)/parenrightbigg\n−c\nHy. (9)\nMultiplying with dy/dt=Hyyields\ndx\ndydy\ndt=ey/parenleftbigg\nK1−c\nHEi(−y)/parenrightbigg\nHy−c . (10)\nInserting y=etHgives\ndx\ndt= exp/parenleftBig\netH/parenrightBig/parenleftbigg\nK1−c\nHEi(−etH)/parenrightbigg\nHexp(tH)−c=xHexp(tH)−c, (11)\nwhere Eq.(4) has been used backwards for the last equal sign.\nExercise 2: Trajectories of Light.\nUse numerical values for c, Handx0to obtain the x(t) trajectories of Fig.(1), where the\nunits are years [ y] and light years [ ly].\n3Solution:\nThe integrationconstant K1inEq.(4) isdetermined by theinitial conditionattime t= 0,\nx(0) =x0>0:\nx0=e1/parenleftbigg\nK1−c\nHEi(−1)/parenrightbigg\n⇒K1=x0\ne1+c\nHEi(−1), Ei(−1) =−0.219384... .(12)\nInserting K1, Eq.(4) becomes\nx(t) = exp/parenleftBig\netH/parenrightBig/parenleftbiggx0\ne1+c\nHEi(−1)−c\nHEi/parenleftBig\n−etH/parenrightBig/parenrightbigg\n. (13)\nTo plotx(t) we have to insert numbers for candH, We measure the time in units of years\n[y]. A light year [ ly] is defined as one year times the speed of light c. Therefore, the speed\nof light is one in units of years and light years,\nc= 1 [ly]/[y]. (14)\nFurthermore, we need to express Hof Eq.(2) in units of inverse years [1 /y]. One Mpc is\n3.262×106[ly]. Converting [ ly] to [km], the distance unit [ km] drops out of Eq.(2), and the\nvalue ofHbecomes\nH= 2.30222×10−18[1/s] = 7.2603×10−11[1/y], (15)\nwhere we have converted seconds [ s] to years [ y] in the last step. It is now straightforward\nto plotx(t) of Eq.(13) for various x0values, and so that the results agree with Fig.(1).\nExercise 3: Hubble Horizon.\nFind the largest value, xmax\n0, of the initial distance x0, so that the photon ray will reach\nits destination x(ta) = 0 for all initial values x0< xmax\n0. We call xmax\n0Hubble horizon and ta\narrival time. Plot the relationship between initial condition x0and arrival times ta.\nIn particular, consider ta= 13.7×109[y], i.e., the present estimate [2] of the age of\nthe universe, and calculate the corresponding x0. For these values indicate the vectors\n(0,ta)→(x0,ta) and (x0,ta)→(x0,0) in the figure. What is the distance between point\none and point two at time t=ta= 13.7×109[y]?\nSolution:\nIn Fig.(1) we see that trajectories with initial values x0≤8.21×109[ly] reach\ntheir destination x(ta) = 0 at some arrival time ta. To the contrary, for initial values\nx0≥8.22×109[ly] the curves turn around before reaching x= 0, and an arrival time ta\ndoes not exist. Hence, we have for the Hubble horizon 8 .21×109[ly]< xmax\n0<8.22×109[ly].\nTo get an accurate number for xmax\n0we sett=taandx(ta) = 0 in Eq.(13), and find the\nfollowing relation between taandx0:\nEi/parenleftBig\n−etaH/parenrightBig\n=Hx0\nce1+Ei(−1). (16)\n4 0 5x109 1x1010 1.5x1010 2x1010 2.5x1010 3x1010\n 0 2x109 4x109 6x109 8x109 1x1010ta [y]\nx0 [ly]x0max\nx0(ta)\nx0 = c ta \nFigure 2: Initial distance x0as function of the arrival time ta, wherexmax\n0is the Hubble\nhorizon. The arrows point from ta= 13.7×109[y] tox0= 7.499×109[ly].\nThe Hubble horizon is then obtained using the asymptotic behavior lim y→∞Ei(−y) = 0:\nxmax\n0=−ce1Ei(−1)\nH= 8.2138...×109[ly]. (17)\nThe relationship (16) between x0andtais plotted in Fig.(2). We see that for ta→ ∞the\nvaluexmax\n0of Eq.(17) is rapidly approached.\nThe value ta= 13.7×109ycorresponds to x0= 7.499×109[ly]. See the arrows inFig.(2).\nDuring the travel time the distance between the two space points in creases according to\nEq.(2):\nx1(ta) =x0exp(taH) = 20.473 [ly]. (18)\n5Exercise 4: Leading order small texpansion.\nTowards small tait is seen in Fig.(2) how the relation x0=ctbecomes a good\napproximation. Let us discuss this quantitatively: Expand x(t) of Eq.(13) to leading order in\nt. Transform the result into a first-order correction to the zero- order arrival time ta=x0/c,\nand compute the correction for x0= 1[ly].\nSolution:\nTo leading order in tEq.(13) reads\nx(t) = exp(1+ tH)/parenleftbiggx0\ne1+c\nHEi(−1)−c\nHEi(−1−tH)/parenrightbigg\n=e1(1+tH)/parenleftbiggx0\ne1+c\nHEi(−1)−c\nHEi(−1−tH)/parenrightbigg\n. (19)\nThe derivative of the exponential integral is given by Eq.(8). Definin gy=tHand using the\nresult of Eq.(8) we get\nEi(−1−y) =Ei(−1)−ydEi(−y)\nd(−y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=1=Ei(−1)+tHe−1. (20)\nTheEi(−1) terms cancel out in Eq.(19), and we arrive at\nx(t) =x0(1+tH)−ct (21)\nTransforming Eq.(21) into a correction to ta, we get\nta=x0\nc+x0H\nc2. (22)\nAssuming for x0the distance of 1[ ly], we obtain from the first term 1[ y] (remember c= 1\nin our units). Using Hfrom Eq.(15), we obtain from the second term the correction\n7.260−11[y] = 2.290−3[s].\nExercise 5: Synchronization of clocks.\nIn our model the time is globally defined. This requires a procedure to synchronize clocks\nat different space positions. The situation is as follows: An observer A emits at time zero\na photon ray in the direction of a second observer B. The photon ra y is reflected by B and\narrives at time ta>0 back at A. The entire information at hand is tafrom which A has to\ncalculate the time at which B reflected the signal. Show that this is pos sible.\nConclude this section with thenumerical example ta= 5×109y. This is a value forwhich\nthex0(ta) curve and the x0=ctaline of Fig.(2) have clearly separated. Draw a diagram of\nthe exchange of signals.\nSolution:\nWe denote by t1\nathe time the photon ray needs to travel from A to B, and by t2\nathe time\nit needs to travel back from B to A. So we have\n6 0 2x109 4x109 6x109 8x109\n 0 1x109 2x109 3x109AoBoAoBot [y]\nx [ly]\nFigure 3: Signal exchange A →B→A→B.\nta=t1\na+t2\na. (23)\nLet us start with the guess: tg1=t1\ng1+t2\ng1witht1\ng1=t2\ng1=ta/2. This is correct for H= 0 the\nsituation of special relativity. A would then tell B to set its clock to t1\na=ta/2 plus whatever\ntime has elapsed on B’s clock since receiving the signal (A and B are mac hines).\nHowever, for H >0 the guess is false because t1\ng1=ta/2 implies tg1> ta. Namely, assume\nthat the initial distance from A to B is x1\ng1, then the subsequent distance from B to A is\naccording to Eq.(1) x2\ng1=x1\ng1exp(t1\ng1H)> x1\ng1. As long as x2\ng1< xmax\n0there is a return signal\nthat needs a time t2\ng1> t1\ng1to travel from B to A. See Fig.(2). As t1\ng1=ta/2 it follows that\ntg1=t1\ng1+t2\ng1> ta.\nNow, with t1\ng1given we can calculate t2\ng1and, hence, tg1explicitly. As tg1(t1\ng1) is a\nmonotonically increasing function, we can compute t1\nawith the bisection method. The\nrelevant steps are outlined in the following.\nUsing Eq.(16), we find for the initial distance corresponding to the t ravel time t1\ng1:\nx0=x1\ng1=ce1\nH/bracketleftBig\nEi/parenleftBig\n−exp(t1\ng1H)/parenrightBig\n−Ei(−1)/bracketrightBig\n. (24)\n7For the return signal from B to A the initial distance is then x2\ng1=x1\ng1exp(t1\ng1H), which\nallows to calculate the return time t2\ng1using Eq.(16):\nexp(t2\ng1H) =−Ei−1/parenleftBiggHx2\ng1\nce1+Ei(−1)/parenrightBigg\n, (25)\nwhereEi−1is the inverse exponential integral.\nTogether with t1\ng1=ta/2 any value t1\ng2>0 withtg2=t1\ng2+t2\ng2< tawill give a starting\npointforthebisectionmethod. Asuitablechoiceis t1\ng2=t1\ng1−(tg1−ta), implying t1\ng2+t2\ng1=ta.\nAst1\ng1< t1\ng1impliest2\ng2< t2\ng1, we have tg2=t1\ng2+t2\ng2< ta, and the pair t1\ng2,t1\ng1provides staring\nvalues for the bisection method, continuing with t1\ng3= (t1\ng2+t1\ng1)/2, replacing in the next\nstep either t1\ng2(fortg3< ta) ort1\ng1(fortg3> ta) witht1\ng3, and so on. After completing its\ncalculation A sends the result to B, who then sets its clock according ly.\nForthenumericalexamplewehave t1\ng1=ta/2 = 2.5×109[y],andget tg1= 5.5661×109[y],\nt1\ng2=t1\ng1−(tg1−ta) = 1.93391×109[y]. Witht1\ng1andt1\ng2we have starting values for our\nbisection which proceeds with t1\ng3= (t1\ng1+t1\ng2)/2 = 2.21695×109[y]⇒tg3= 4.865869[y],\nthen replacing t1\ng2byt1\ng3because of tg3−ta<0, continuing with a new t1\ng3= (t1\ng1+t1\ng2)/2,\nand so on, till convergence is reached at ta=tg3= 2.271888×109[y]. A signals this result\nfort1\nato B, and B can set its clock accordingly. Figure (3) illustrates the ex change of signals.\nThere are deviations from straight lines which are not visible on the sc ale of this figure.\n3. Summary\nAsimplemodelofanexpandinguniverseisintroduced. Itallowsforav arietyofanalytical\ncalculation. They are intended for qualitative, pedagogical value an d not of quantitative\nrelevance. Results include:\nTrajectories of light. See Fig.(1) for examples.\nThe initial distance x0as function of the arrival time ta(or vice versa) and the Hubble\nhorizonxmax\n0are drawn in Fig.( 2).\nSynchronization of clocks to a common time is outlined in section 3, and a diagram of the\ncorresponding exchange of signals is given in Fig.(3).\nAcknowledgments: Ihave not seenthem theexercises presented hereintheliteratur e,\nbutcannot exclude thatsomeonehasthem(orvariationsofthem) worked outbefore. Please,\ncommunicate any pertinent information to my e-mail address: bber g@fsu.edu.\nReferences\n[1] E.g., NASA science https://science.nasa.gov/missions/hubble/ne w-hubble-constant-\nmeasurement-adds-to-mystery-of-universes-expansion-ra te/\n[2] E.g., NASA https://imagine.gsfc.nasa.gov/science/featured science/tenyear/age.html\n8" }, { "title": "2401.17507v1.On_skew_corner_free_sets.pdf", "content": "arXiv:2401.17507v1 [math.CO] 30 Jan 2024ON SKEW CORNER-FREE SETS\nCOSMIN POHOATA AND DMITRII ZAKHAROV\nAbstract. We construct skew corner-free sets in [ n]2of sizen5/4, thereby disproving a conjecture of\nKevin Pratt. We also show that any skew corner-free set in Fn\nq×Fn\nqmust have size at most q(2−c)n, for\nsome positive constant cwhich depends on q.\n1.Introduction\nMotivated by matrix multiplication algorithms, Pratt [8] a sked the following nice question: what is\nthe largest subset of [ n]2which does not contain ‘skew corners’, i.e. triples of point s of the form\n(1) ( x,y),(x,y+d),(x+d,y′)\nford/\\e}atio\\slash= 0 andx,y,y′arbitrary?\nWe call such sets skew corner-free sets . Puttingy=y′in the above, we get that such a set does\nnot contain regularcorners (x,y),(x+d,y),(x,y+d). Determining the maximum size of a subset\nof [n]2without regular corners is a well-studied problem in additi ve combinatorics. See for example\n[4] and the references therein for some background. Given th is connection to regular corner-free\nsets, Shkredov’s result from [9] immediately implies that a skew corner-free set S⊂[n]2satisfies\n|S|=O/parenleftbig\nn2/(loglogn)c/parenrightbig\n, for some absolute constant c>0. On the other hand, Petrov [6] constructed\na skew corner free set in [ n]2of size Ω(nlogn/√loglogn), and Pratt [8] conjectured that any such set\nin [n]2has sizeO(n1+ε) for anyε>0. Furthermore, Pratt showed that such a result would show th at\ncertain approaches to matrix multiplication cannot achiev e a running time of O(n2+ε). We provide a\nconstruction which disproves this prediction.\nTheorem 1.1. There exists a skew corner-free set S⊂[n]2of sizeΩ(n5/4).\nThe main idea is to take advantage of a well-known property of the (affine version) of the classical\nHermitian unital over F2\np2, object which was also used by Mattheus–Verstraete [5] in th e recent break-\nthrough lower bound construction for the Ramsey number R(4,t). We discuss the proof of Theorem\n1.1 in Section 2.\nWhile Theorem 1.1 shows skew corner-free sets in [ n]2may not always have size O(n1+ε) for any\nε >0, we believe that such sets should in the very least still hav e sizeO(n2−c) for some absolute\nconstantc>0. Establishing this fact already seems like an interesting problem, as there exist standard\nexamples of corner-free sets in [ n]2of sizen2−o(1). However, it does not seem that the usual Fourier\nanalytic methods can take advantage of the stronger conditi on of the set being skew corner free in\nany significant manner. In Section 3, we show a result in this s pirit for the finite field model of this\nproblem.\nTheorem 1.2. Letq/greaterorequalslant2be a prime and let S⊂Fn\nq×Fn\nqbe any set without triples of the form (1)\nwithx,y,y′∈Fn\nqandd∈Fn\nq\\0. Then,\n|S|/lessorequalslant3q(2−cq)n,\nDepartmentofMathematics, EmoryUniversity,Atlanta, GA. Email:cosmin.pohoata@emory.edu . Researchsupported\nby NSF Award DMS-2246659.\nDepartment of Mathematics, Massachusetts Institute of Tec hnology, Cambridge, MA. Email: zakhdm@mit.edu . Re-\nsearch supported by the Jane Street Graduate Fellowship.\n12 COSMIN POHOATA AND DMITRII ZAKHAROV\nwhere the exponent cqis a positive constant depending on qdefined as\nq1−cq= inf\n0 rc).(5)\nwhererrepresents the distance from the center of each\natom, and rcis the cutoff distance. βnandrnare the\nparameters, respectively. Radial functions with rc= 8.0\n˚A andβn= 1.0˚A−2andrn= (n−1)(rc−1.0)/11\nforn= 1,...,12 were used. We considered polyno-\nmial invariants up to third order characterizing neigh-\nboring atomic density based on spherical harmonics with\nthe maximum angular numbers of spherical harmonics\nl(2)\nmax=l(3)\nmax= 8. The polynomial models were then\nconstructed by the polynomial functions of the pair in-\nvariants and linear polynomial function of the polyno-\nmial invariants. We considered polynomial functions up\nto second order. The model coefficients were estimated\nfrom electronic total energies and forces by the linear\nridge regression method.\nD. First-principles calculation\nFor the first-principles calculations, we employed\nthe plane-wave basis projector augmented wave (PAW)\nmethod23within the framework of DFT as implemented\nin the VASP code.24–26The generalized gradient approx-\nimation (GGA) of Perdew, Burke, and Ernzerhof revised\nfor solids (PBEsol)27was used as the exchange corre-\nlation potential. To ensure high numerical accuracy in\ncomputing atomic forces, the projection operators were\napplied in reciprocal spaces and additional support grids\nwere employed for the evaluation of the augmentation\ncharges. Static dielectric constants and Born effective\ncharges were calculated with the conventional unit cells\nfrom density functional perturbation theory (DFPT) as\nimplemented in the VASP code.28,29\nA plane-wave energy cutoff of 520 eV was employed\nfor the supercell force calculations and 676 eV for the\nDFPT calculations. Reciprocal spaces of the zincblende-\nand rocksalt-type compounds were sampled by the half-\nshifted2×2×2meshesforthe2 ×2×2supercells, thehalf-\nshifted 1×1×1meshesforthe 4 ×4×4supercells, andthehalf-shifted8 ×8×8meshesfortheconventionalunitcells.\nReciprocal spaces of the wurtzite-type compounds were\nsampledbythe2 ×2×2meshesthatarehalf-shiftedalong\nthec∗axis for the 3 ×3×2 supercells, the 1 ×1×2 meshes\nthat are half-shifted along the c∗axis for the 5 ×5×3\nsupercells, andthe 12 ×12×8meshesthatarehalf-shifted\nalong the c∗axis for the unit cells.\nE. Automation of dataset preparation\nPerforming a large number of first-principles calcu-\nlations can be computationally intensive and may re-\nquire high-performance computing resources. This stage\nconsumes a significant amount of computational power\nthroughout the LTC calculation process. It is virtually\ninevitable that some of these calculations fail for vari-\nous reasons, such as reaching the maximum number of\nelectronic structure convergence iterations or encounter-\ning issues related to computer networks and hardware.\nAlthough the proportion of failed calculations was rela-\ntivelylow,wehavenotyetfullyautomatederrorrecovery\nfor all possible cases.\nWesystematicallyidentifiedcalculationfailuresandre-\nexecuted those calculations semi-manually with the as-\nsistance of the workflow system instead of attempting to\nfully automate all processes. After completing all the su-\npercell calculationsusing first-principlescalculations, the\ndataset for each compound required for the subsequent\nLTC calculation process was composed into a single com-\nputer file in a structured format.\nFor the systematic calculations of energies and forces\nin supercells using first-principles calculations, we uti-\nlizedthe AiiDAenvironment30–32in conjunctionwith the\nAiiDA-VASP plugin.33The advantage of using the work-\nflow automation system was not only the automation of\nsubmitting calculationjobs to high-performancecomput-\ners, but also the automated data storing of the calcula-\ntion results in a database systematically. The computed\ndata, stored within the AiiDA database, could be conve-\nniently accessed through the Python programming lan-\nguage. By writing a concise Python script, we were able\nto extract supercell energies, forces, and displacements\nfrom the AiiDA database on demand and convert this\ndata into the structured format required for immediate\nuse by the phono3py code.9,17\nF. Parameters for 103 binary compounds\n33 compounds for the wurtzite- and zincblende-type\nand 37 compounds for the rocksalt-type were used to\nevaluate the LCT calculation approach proposed in this\nstudy, and their chemical compositions are listed in Ta-\nblesIandII. Crystal structures of the wurtzite and\nzincblende types are similar, though their stacking or-\nders are different, much like the relationship between\nface-centered-cubicandhexagonal-close-packedstructure5\ntypes. Since it is of interest to explore their similarities\nand differences in calculations, as also studied in Ref. 9,\nthe compounds with the same chemical compositions for\nthe wurtzite and zincblende types were calculated. The\ntablesalsoprovideinformationon lattice parameters,the\nchoices of PAW datasets from the VASP package, and\nelectronic total energies of the elements that were sub-\ntracted from the total energies of the compounds used to\ntrain the polynomial MLPs.\nTABLE I. Lattice parameters, names of the VASP PAW-PBE\ndatasets, electronic total energies of the atoms used in thi s\nstudy for 33 wurtzite- and zincblende-type compounds. w-\na, w-c, and z-adenote the lattice parameters aandcof the\nwurtzite-type compounds, and aof the zincblende-type com-\npounds, respectively.\nw-aw-cz-aenergy (eV) energy (eV)\nAgI 4.56 7.45 6.44 (Ag pv)−0.233 (I) −0.182\nAlAs 4.00 6.58 5.67 (Al) −0.282 (As d)−0.989\nAlN 3.11 4.98 4.38 (Al) −0.282 (N) −1.905\nAlP 3.86 6.34 5.47 (Al) −0.282 (P) −1.140\nAlSb 4.35 7.16 6.17 (Al) −0.282 (Sb) −0.828\nBAs 3.35 5.55 4.77 (B) −0.359 (As d)−0.989\nBeO 2.70 4.38 3.80 (Be) −0.023 (O) −0.957\nBeS 3.41 5.63 4.84 (Be) −0.023 (S) −0.578\nBeSe 3.62 5.97 5.14 (Be) −0.023 (Se) −0.438\nBeTe 3.95 6.53 5.61 (Be) −0.023 (Te) −0.359\nBN 2.54 4.20 3.61 (B) −0.359 (N) −1.905\nBP 3.18 5.27 4.52 (B) −0.359 (P) −1.140\nCdS 4.13 6.72 5.84 (Cd) −0.021 (S) −0.578\nCdSe 4.31 7.03 6.09 (Cd) −0.021 (Se) −0.438\nCdTe 4.59 7.52 6.50 (Cd) −0.021 (Te) −0.359\nCuBr 3.92 6.48 5.56 (Cu pv)−0.274 (Br) −0.225\nCuCl 3.70 6.17 5.27 (Cu pv)−0.274 (Cl) −0.311\nCuH 2.81 4.44 3.93 (Cu pv)−0.274 (H) −0.946\nCuI 4.17 6.88 5.92 (Cu pv)−0.274 (I) −0.182\nGaAs 3.99 6.57 5.66 (Ga d)−0.286 (As d)−0.989\nGaN 3.18 5.18 4.50 (Ga d)−0.286 (N) −1.905\nGaP 3.83 6.31 5.44 (Ga d)−0.286 (P) −1.140\nGaSb 4.31 7.10 6.11 (Ga d)−0.286 (Sb) −0.828\nInAs 4.30 7.05 6.09 (In d)−0.264 (As d)−0.989\nInN 3.54 5.71 4.99 (In d)−0.264 (N) −1.905\nInP 4.15 6.81 5.88 (In d)−0.264 (P) −1.140\nInSb 4.60 7.56 6.52 (In d)−0.264 (Sb) −0.828\nMgTe 4.56 7.41 6.44 (Mg pv)−0.009 (Te) −0.359\nSiC 3.08 5.05 4.36 (Si) −0.522 (C) −1.340\nZnO 3.24 5.23 4.56 (Zn) −0.016 (O) −0.957\nZnS 3.79 6.21 5.36 (Zn) −0.016 (S) −0.578\nZnSe 3.98 6.54 5.64 (Zn) −0.016 (Se) −0.438\nZnTe 4.28 7.05 6.07 (Zn) −0.016 (Te) −0.359TABLE II. Lattice parameters a, names of the VASP PAW-\nPBE datasets, and electronic total energies of the atoms use d\nin this study for 37 rocksalt-type compounds.\na energy (eV) energy (eV)\nAgBr 5.67 (Ag pv)−0.233 (Br) −0.225\nAgCl 5.44 (Ag pv)−0.233 (Cl) −0.311\nBaO 5.53 (Ba sv)−0.035 (O) −0.957\nBaS 6.36 (Ba sv)−0.035 (S) −0.578\nBaSe 6.58 (Ba sv)−0.035 (Se) −0.438\nBaTe 6.97 (Ba sv)−0.035 (Te) −0.359\nCaO 4.77 (Ca pv)−0.010 (O) −0.957\nCaS 5.63 (Ca pv)−0.010 (S) −0.578\nCaSe 5.87 (Ca pv)−0.010 (Se) −0.438\nCaTe 6.30 (Ca pv)−0.010 (Te) −0.359\nCdO 4.71 (Cd) −0.021 (O) −0.957\nCsF 5.96 (Cs sv)−0.166 (F) −0.556\nKBr 6.59 (K pv)−0.182 (Br) −0.225\nKCl 6.29 (K pv)−0.182 (Cl) −0.311\nKF 5.37 (K pv)−0.182 (F) −0.556\nKH 5.63 (K pv)−0.182 (H) −0.946\nKI 7.05 (K pv)−0.182 (I) −0.182\nLiBr 5.41 (Li sv)−0.286 (Br) −0.225\nLiCl 5.06 (Li sv)−0.286 (Cl) −0.311\nLiF 4.00 (Li sv)−0.286 (F) −0.556\nLiH 3.97 (Li sv)−0.286 (H) −0.946\nLiI 5.90 (Li sv)−0.286 (I) −0.182\nMgO 4.22 (Mg pv)−0.009 (O) −0.957\nNaBr 5.93 (Na pv)−0.246 (Br) −0.225\nNaCl 5.60 (Na pv)−0.246 (Cl) −0.311\nNaF 4.63 (Na pv)−0.246 (F) −0.556\nNaH 4.79 (Na pv)−0.246 (H) −0.946\nNaI 6.41 (Na pv)−0.246 (I) −0.182\nPbS 5.90 (Pb d)−0.374 (S) −0.578\nPbSe 6.10 (Pb d)−0.374 (Se) −0.438\nPbTe 6.44 (Pb d)−0.374 (Te) −0.359\nRbBr 6.88 (Rb pv)−0.168 (Br) −0.225\nRbCl 6.58 (Rb pv)−0.168 (Cl) −0.311\nRbF 5.66 (Rb pv)−0.168 (F) −0.556\nRbH 5.95 (Rb pv)−0.168 (H) −0.946\nRbI 7.32 (Rb pv)−0.168 (I) −0.182\nSrO 5.13 (Sr sv)−0.032 (O) −0.957\nIV. RESULTS\nA. Choice of displacements and number of\nsupercells\nFor each compound, two distinct displacement-force\ndatasets that share the same supercell basis vectors were\nemployed to calculate LTCs. Energies and forces of the\nsupercells in the first dataset were computed using first-6\nprinciples calculations, while the polynomial MLPs were\nutilized for calculating forces in the second dataset. The\nfirstdatasetwasusedtotrainthepolynomialMLPs. The\nsecond dataset was employed to compute third-order su-\npercell force constants by fitting.\nTo investigate the performance of the polynomial\nMLPs in predicting LTC values, 100 supercells with ran-\ndom directional displacements were initially prepared as\nthe first dataset. Subsequently, the first 10, 20, 40, 60,\nand 80 supercells were selected from the list of100 super-\ncells as subsets. Using the displacement-force pairs and\nenergies of these supercells, the polynomial MLPs were\ntrained, and the last 20 supercells were reserved as test\ndata to optimize their ridge regularization parameters.\nFor the ease of use of the software package, we de-\ncided to employ a constant displacement distance, and\nto obtain reasonable LTC values, we chose a constant\ndisplacement distance of 0.03 ˚A. Interestingly, we found\nthat the polynomial MLPs performed well even with a\nrelatively large displacement distance, such as 0.1 ˚A. It\nis important to note that these factors are highly de-\npendent on the specific force calculators and calculation\nconfigurations used.\nWe utilized another displacement-force dataset that\nconsists of 400 supercells with random directional dis-\nplacements for the computation of third-order supercell\nforce constants. These supercell forces were calculated\nusing the trained polynomial MLPs, where the residual\nforces were subtracted. The root-mean-square errors of\nthe polynomialMLPs trainedon the 20supercells ranged\nfrom approximately5 .5×10−6to 1.4×10−3eV/˚A, which\nare expected to represent the same degree of numerical\nerrors in the displacement-force dataset.\nDue to the numerical smoothness of the polynomial\nMLPs for the force calculation with respect to positions\nofatomscomparedtothefirst-principlescalculationsem-\nployed in this study, we were able to choose a small\nconstant displacement distance of 0.001 ˚A. This bene-\nfits better convergence with smaller dataset when fitting\nthe supercell force constants by Eq. ( 3). For instance,\nin the case of a displacement distance of 0.03 ˚A, it was\nnecessary to employ 10000 supercells to achieve well con-\nverged LTC values for the 103 compounds. This suggests\nthat when high-order force constants are more relevant\nfor specific compounds, direct calculation of third-order\nsupercell force constants from the the displacement-force\ndataset through first-principles calculations may require\na large dataset to achieve convergence of LTC values.\nFor strongly anharmonic crystals, self-consistent\nphononmethods, whichwerenotconsideredinthisstudy,\nmay have to be employed to obtain physically more\nmeaningful results. In such cases, the use of the polyno-\nmial MLPs can also be beneficial to accelerate the LTC\ncalculations.B. Calculated LTCs\nIn Figs.2,3, and4, we present the calculated LTCs of\nthe 103 compounds at 300 K. We can see that datasets\nwith 20 supercells show good performance, at least for\nestimating LTC values roughly. In particular, the LTC\nvalues of most of the rocksalt-type compounds are well\nrepresented by these small datasets. The wurtzite- and\nzincblende-type compounds exhibit similar tendencies in\nLTCvalueswithrespecttodatasetsizesincethesecrystal\nstructures are similar. The datasets with 40 supercells\nyield LTC results that are roughly converged.\nLTC values at 300 K predicted by the conventional\napproach, which directly uses the displacement-force\ndataset obtained through first-principles calculations to\nfit third-order supercell force constants, are depicted\nby the horizontal dotted lines. The third-order super-\ncell force constants were computed by the linear regres-\nsion method as implemented in the symfc code21from\nthe first datasets with 100 supercells and 0.03 ˚A ran-\ndom directional displacements, which were those pre-\npared for training the polynomial MLPs, as explained in\nSec.IVA. In addition, LTC values with 400supercells for\nthe zincblende- and rocksalt-type compounds and those\nwith 400 and 2000 supercells for the wurtzite-type com-\npounds were also computed. These values are depicted\nas horizontal lines in Figs. 2,3, and4. For most of the\nzincblende- and rocksalt-type compounds, LTC values\nderived from datasets with 100 supercells are found to\nbe adequate when compared to those from 400 super-\ncells. However, for the wurtzite-type compounds, even\ndatasets with 400 supercells are insufficient.\nThe LTC values predicted for the wurtzite-type com-\npounds using polynomial MLPs tend to align with those\ncalculated directly from 2000 supercell datasets. This\nalignment emphasizes the utility and effectiveness of us-\ning polynomial MLPs in these cases.\nC. Comparison with conventional LTC calculation\nIn Fig. 5, the LTC values of the 103 compounds\ncalculated through the polynomial MLPs trained using\nthe 20 supercell datasets are compared with those cal-\nculated in the conventional approach using the same\nfinite-difference displacement-force datasets34as those\nemployed in Ref. 15. These datasets share the same unit\ncells and supercell sizes for each compound. The lat-\nter datasets for the wurtzite-, zincblende-, and rocksalt-\ntype compounds consist of 1254, 222, and 146 displace-\nments, respectively, with a displacement distance of 0.03\n˚A. These displacements were systematically introduced\nconsidering crystal symmetries35by using the phono3py\ncode.9,17In all these calculations, the same version of\nthe phono3py code9,17was utilized to calculate the LTCs\nfrom the respective supercell force constants. The results\ndemonstrate that the LTC values obtained through the\npolynomialMLPsconsistentlyagreewiththosepredicted7\nNumber of supercells Number of supercells Number of supercellsNumber of supercells Number of supercells Number of supercells020 40 60 80 1000123L TC (W/m-K) w-AgI\n020 40 60 80 100020 40 60 80 w-AlAs\n020 40 60 80 1000100200300w-AlN\n020 40 60 80 100020 40 60 80 w-AlP\n020 40 60 80 100020 40 60 w-AlSb\n020 40 60 80 100050010001500w-BAs\n020 40 60 80 1000100200300L TC (W/m-K) w-BeO\n020 40 60 80 100050 100150w-BeS\n020 40 60 80 1000100200300400w-BeSe\n020 40 60 80 1000100200w-BeT e\n020 40 60 80 10005001000w-BN \n020 40 60 80 1000200400w-BP \n020 40 60 80 1000510 15 20 L TC (W/m-K) w-CdS\n020 40 60 80 1000510 15 w-CdSe\n020 40 60 80 1000246w-CdT e\n020 40 60 80 10001234w-CuBr\n020 40 60 80 1000.00.51.01.52.0w-CuCl\n020 40 60 80 1000246w-CuH\n020 40 60 80 1000.02.55.07.510.0L TC (W/m-K) w-CuI\n020 40 60 80 100010 20 30 w-GaAs\n020 40 60 80 1000100200300w-GaN\n020 40 60 80 100050 100150w-GaP\n020 40 60 80 100010 20 30 w-GaSb\n020 40 60 80 100010 20 w-InAs\n020 40 60 80 100050 100L TC (W/m-K) w-InN\n020 40 60 80 100020 40 60 80 w-InP\n020 40 60 80 1000510 w-InSb\n020 40 60 80 1000.02.55.07.510.0w-MgT e\n020 40 60 80 1000200400w-SiC\n020 40 60 80 100020 40 60 w-ZnO\n020 40 60 80 100020 40 60 L TC (W/m-K) w-ZnS\n020 40 60 80 1000510 15 20 w-ZnSe\n020 40 60 80 100010 20 w-ZnT e\nFIG. 2. Filled circles show LTCs ( κ) of the 33 wurtzite-type compounds calculated at 300 K with r espect to the number of\nsupercells in the datasets used to train the polynomial MLPs . The LTC values are the averages of the diagonal elements, i. e.,\n(2κxx+κzz)/3. The horizontal dotted, dashed-dotted, and dashed lines d epict the LTC values calculated in the conventional\napproach from the datasets of 100, 400, and 2000 supercells w ithout using the polynomial MLPs, respectively.\nby the conventional approach.15D. Conclusion\nTo improve the efficiency of high-throughput LTC cal-\nculations, we developed methodologies and modular soft-\nware packages that utilize polynomial MLPs for com-\nputing LTC values based on first-principles calculation.8\nNumber of supercells Number of supercells Number of supercellsNumber of supercells Number of supercells Number of supercells020 40 60 80 10001234L TC (W/m-K) z-AgI\n020 40 60 80 100020 40 60 80 z-AlAs\n020 40 60 80 1000100200z-AlN\n020 40 60 80 100020 40 60 80 z-AlP\n020 40 60 80 100025 50 75 100z-AlSb\n020 40 60 80 100050010001500z-BAs\n020 40 60 80 1000100200300400L TC (W/m-K) z-BeO\n020 40 60 80 100050 100150200z-BeS\n020 40 60 80 1000200400z-BeSe\n020 40 60 80 1000100200300z-BeT e\n020 40 60 80 100050010001500z-BN \n020 40 60 80 1000200400z-BP \n020 40 60 80 100010 20 L TC (W/m-K) z-CdS\n020 40 60 80 1000510 15 z-CdSe\n020 40 60 80 1000.02.55.07.510.0z-CdT e\n020 40 60 80 1000.02.55.07.510.0z-CuBr\n020 40 60 80 100012z-CuCl\n020 40 60 80 100010 20 z-CuH\n020 40 60 80 1000.02.55.07.510.0L TC (W/m-K) z-CuI\n020 40 60 80 100010 20 30 40 z-GaAs\n020 40 60 80 1000100200300z-GaN\n020 40 60 80 100050 100150z-GaP\n020 40 60 80 100010 20 30 40 z-GaSb\n020 40 60 80 100010 20 30 z-InAs\n020 40 60 80 100025 50 75 100L TC (W/m-K) z-InN\n020 40 60 80 100025 50 75 100z-InP\n020 40 60 80 1000510 15 z-InSb\n020 40 60 80 1000510 z-MgT e\n020 40 60 80 1000200400z-SiC\n020 40 60 80 100020 40 60 z-ZnO\n020 40 60 80 100020 40 60 L TC (W/m-K) z-ZnS\n020 40 60 80 1000510 15 20 z-ZnSe\n020 40 60 80 100010 20 30 z-ZnT e\nFIG. 3. Filled circles show LTCs of the 33 zincblende-type co mpounds calculated at 300 K with respect to the number of\nsupercells in the datasets used to train the polynomial MLPs . The horizontal dotted and dashed-dotted lines depict the L TC\nvalues calculated in the conventional approach from the dat asets of 100 and 400 supercells without using the polynomial MLPs,\nrespectively.\nWe evaluated the feasibility of this computational ap-\nproach by calculating the LTCs of 103 compounds of\nwurtzite, zincblende, and rocksalt types. This approach\nwas benchmarked against our previously used conven-\ntional approach. We found that this approach signifi-\ncantlyreducescomputationaldemandswhilemaintaininga satisfactory accuracy level for LTC prediction. Apart\nfrom the initial stage of generating datasets using first-\nprinciples calculations, subsequent LTC calculation steps\nrequire minimal computational resources. This enables\nusers to calculate LTCs and various related physical val-\nues on standard computers, given access to high-quality9\nNumber of supercellsNumber of supercells Number of supercells Number of supercells Number of supercells Number of supercells020 40 60 80 1000.00.20.40.6L TC (W/m-K) r-AgBr\n020 40 60 80 1000.00.20.40.6r-AgCl\n020 40 60 80 1000123r-BaO\n020 40 60 80 10002468r-BaS\n020 40 60 80 1000510 r-BaSe\n020 40 60 80 1000510 r-BaT e\n020 40 60 80 100010 20 30 L TC (W/m-K) r-CaO\n020 40 60 80 100010 20 30 40 r-CaS\n020 40 60 80 1000510 15 20 r-CaSe\n020 40 60 80 1000510 r-CaT e\n020 40 60 80 1000.02.55.07.510.0r-CdO\n020 40 60 80 1000.00.51.01.52.0r-CsF\n020 40 60 80 1000123L TC (W/m-K) r-KBr\n020 40 60 80 10002468r-KCl \n020 40 60 80 10002468r-KF\n020 40 60 80 1000510 r-KH\n020 40 60 80 100012r-KI\n020 40 60 80 1000123r-LiBr\n020 40 60 80 1000246L TC (W/m-K) r-LiCl\n020 40 60 80 1000510 15 20 r-LiF\n020 40 60 80 100010 20 30 40 r-LiH\n020 40 60 80 100012r-LiI\n020 40 60 80 100020 40 60 r-MgO\n020 40 60 80 1000123r-NaBr\n020 40 60 80 10002468L TC (W/m-K) r-NaCl\n020 40 60 80 100010 20 r-NaF\n020 40 60 80 1000510 15 20 r-NaH\n020 40 60 80 100012r-NaI\n020 40 60 80 100012r-PbS\n020 40 60 80 100012r-PbSe\n020 40 60 80 1000123L TC (W/m-K) r-PbT e\n020 40 60 80 10001234r-RbBr\n020 40 60 80 1000123r-RbCl\n020 40 60 80 10001234r-RbF\n020 40 60 80 1000246r-RbH\n020 40 60 80 1000.00.51.01.52.0r-RbI\n020 40 60 80 1000510 L TC (W/m-K) r-SrO\nFIG. 4. Filled circles show LTCs of the 37 rocksalt-type comp ounds calculated at 300 K with respect to the number of\nsupercells in the datasets used to train the polynomial MLPs . The horizontal dotted and dashed-dotted lines depict the L TC\nvalues calculated in the conventional approach from the dat asets of 100 and 400 supercells without using the polynomial MLPs,\nrespectively.10\n 1 10 100 1000\n 1 10 100 1000AgBr AgCl BaO BaS BaSe BaTe CaO CaS \nCaSe \nCaTe \nCdO\nCsF KBr KCl KF KH \nKI LiBr LiCl LiF LiH \nLiI MgO \nNaBr NaCl NaF \nNaH \nNaI PbS PbSe PbTe RbBr\nRbClRbFRbH\nRbISrO \nAgI AlAs AlN \nAlP AlSb BAs \nBeO \nBeS BeSe \nBeTe BN \nBP \nCdS\nCdSe\nCdTe CuBr\nCuClCuH\nCuIGaAs GaN \nGaP \nGaSb \nInAsInN \nInP \nInSb\nMgTe SiC \nZnO ZnS \nZnSe ZnTe \nAgI AlAs AlN \nAlP \nAlSb BAs \nBeO \nBeS BeSe \nBeTe BN \nBP \nCdS\nCdSe\nCdTe\nCuBr\nCuClCuHCuIGaAs GaN \nGaP \nGaSb InAsInN \nInP \nInSb\nMgTe SiC \nZnO ZnS \nZnSe ZnTe \nLTC calculated by conventional approach (W/m∙s) LTC calculated through polynomial MLPs (W/m∙s) Wurtzite type \nZincblende type\nRocksalt type\nFIG. 5. Comparison between LTC values calculated\nthrough the polynomial MLPs and those by the conven-\ntional approach.15To train the polynomial MLPs for each\ncompound, we employed displacement-force pairs and ener-\ngies of 20 supercells obtained through first-principles cal cu-\nlations. For the latter LTCs, we used the finite-difference\ndisplacement-force datasets34from Ref. 15, comprising 1254,\n222, and 146 supercells for the wurtzite-, zincblende-, and\nrocksalt-type compounds, respectively, to fit third-order su-\npercell force constants.datasets. Our future plans include the computation and\ndistributionofsuchhigh-qualitydatasetsforawiderange\nof compounds.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumbers JP21K04632, JP22H01756, and JP19H05787.\n∗Author to whom any correspondence should be addressed. togo .atsushi@gmail.com.\n1R. E. Peierls, Ann. Phys. 395, 1055 (1929) .\n2R. E. Peierls, Quantum theory of solids (Oxford University\nPress, 2001).\n3J.M.Ziman, Electrons and phonons: the theory of transport phenomena in solids\n(Oxford University Press, 1960).\n4G. P. Srivastava, Physics of phonons (CRC Press, 1990).\n5T. Tadano, Y. Gohda, and S. Tsuneyuki,\nJ. Phys. Condens. Matter 26, 225402 (2014) .\n6J. Carrete, B. Vermeersch, A. Katre, A. van\nRoekeghem, T. Wang, G. K. Madsen, and N. Mingo,\nComput. Phys. Commun. 220, 351 (2017) .\n7A. Chernatynskiy and S. R. Phillpot,\nComput. Phys. Commun. 192, 196 (2015) .\n8O. Hellman, P. Steneteg, I. A. Abrikosov, and S. I. Simak,\nPhys. Rev. B 87, 104111 (2013) .\n9A. Togo, L. Chaput, and I. Tanaka, Phys. Rev. B 91,\n094306 (2015).\n10W. Li, J. Carrete, N. A. Katcho, and N. Mingo,\nComp. Phys. Commun. 185, 1747 (2014) .\n11F. Zhou, W. Nielson, Y. Xia, and V. Ozoli¸ nˇ s,\nPhys. Rev. Lett. 113, 185501 (2014) .\n12T. Tadano and S. Tsuneyuki,\nPhys. Rev. B 92, 054301 (2015) .13A. Seko, A. Togo, and I. Tanaka,\nPhys. Rev. B 99, 214108 (2019) .\n14A. Seko, J. Appl. Phys. 133, 011101 (2023) .\n15A. Togo, J. Phys. Soc. Jpn. 92, 012001 (2023) .\n16P. B. Allen and V. Perebeinos,\nPhys. Rev. B 98, 085427 (2018) .\n17A. Togo, L. Chaput, T. Tadano, and I. Tanaka,\nJ. Phys. Condens. Matter 35, 353001 (2023) .\n18R. M. Pick, M. H. Cohen, and R. M. Martin,\nPhys. Rev. B 1, 910 (1970) .\n19X. Gonze, J.-C. Charlier, D. C. Allan, and M. P. Teter,\nPhys. Rev. B 50, 13035 (1994).\n20X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997) .\n21A. Seko and A. Togo, “Symfc,”\nhttps://github.com/phonopy/symfc .\n22A. Togo and I. Tanaka, “ Spglib: a software library for\ncrystal symmetry search,” (2018), arXiv:1808.01590 .\n23P. E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n24G. Kresse, J. Non-Cryst. Solids 193, 222 (1995).\n25G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15\n(1996).\n26G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n27J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,\nG. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,\nPhys. Rev. Lett. 100, 136406 (2008).11\n28M. Gajdoˇ s, K. Hummer, G. Kresse, J. Furthm¨ uller, and\nF. Bechstedt, Phys. Rev. B 73, 045112 (2006) .\n29X. Wu, D. Vanderbilt, and D. R. Hamann,\nPhys. Rev. B 72, 035105 (2005) .\n30G. Pizzi, A. Cepellotti, R. Sabatini, N. Marzari, and\nB. Kozinsky, Comput. Mater. Sci. 111, 218 (2016).\n31S. P. Huber, S. Zoupanos, M. Uhrin, L. Talirz, L. Kahle,\nR. H¨ auselmann, D. Gresch, T. M¨ uller, A. V. Yakutovich,\nC. W. Andersen, F. F. Ramirez, C. S. Adorf, F. Gargiulo,\nS. Kumbhar, E. Passaro, C. Johnston, A. Merkys, A. Ce-pellotti, N. Mounet, N. Marzari, B. Kozinsky, and\nG. Pizzi, Sci. Data 7, 300 (2020) .\n32M. Uhrin, S. P. Huber, J. Yu, N. Marzari, and G. Pizzi,\nComputational Materials Science 187, 110086 (2021) .\n33https://github.com/aiida-vasp/aiida-vasp .\n34A. Togo, “Lattice thermal conductivity calculation\ndatasets for 103 binary compounds by finite displacement\nmethod,” https://doi.org/10.34968/nims.4370 .\n35L. Chaput, A. Togo, I. Tanaka, and G. Hug,\nPhys. Rev. B 84, 094302 (2011) ." }, { "title": "2401.17562v1.Phase_Transition_of_the_single_layer_vanadium_diselenide_on_Au_111__with_distinguished_electronic_structures.pdf", "content": "PhaseTransitionofthesingle-layervanadiumdiselenideonAu(111)withdistinguished\nelectronicstructures\nJinbangHu,XiansiWang,ChaoqinHuang,FeiSong,andJustinWWells\nJinbangHu\nDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,NO-7491\nTrondheim,Norway\nE-mail:jinbang.hu@ntnu.no\nXiansiWang\nHunanUniversity,Changsha,410082,China\nFeiSong,ChaoqinHuang\nShanghaiAdvancedResearchInstitute,ChineseAcademyofSciences,Shanghai201000,\nChina;\nJustinW.Wells\nDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,NO-7491\nTrondheim,Norway\nSemiconductorPhysics,DepartmentofPhysics,UniversityofOslo(UiO),NO-0371Oslo,\nNorway\nKeywords：phasetransition,2Dmaterial\nAbstract\nHerein,wereportthereversiblestructuretransitionofthesingle-layerVSe2grownonAu\n(111)withthealternatingthermalannealingandSereplenishment.Usingscanningtunnelling\nmicroscopy(STM)andangle-resolvedphotoemissionspectroscopy(ARPES),we\ndemonstratetheepitaxiallygrowthofhigh-qualityVSe2onAu(111)withtheoctahedral(1T)\nstructureandtheSe-vacancy-inducedstructuretransformationofVSe2fromthemetallic\nMorrie(1T)phasetothesemiconducting(2H)phasewithanincreasedin-planelattice\nconstant.WiththeconvincingagreementbetweentheexperimentalresultsandDFT\ncalculations,thenanostructurenearthegainboundaryinthedefectiveintermediatephaseis\nconfirmed,aswellasthereactionpathwaywithSegraduallydepletedatelevated\ntemperatures.ItisrevealedthattheSedensityoftheline-shapedefectsplaysanimportant\nroleintheformationofthe2Hdomainphase,duetotheincrementofthein-planelattice\nparameteraftertheSedesorption,andthebetterthermalstabilityof2Hphasecomparedtothe\n1Tphase.Importantly,thepropercontrolofthedensityofSecouldfeasiblymanipulatetheratiobetweenthe1Tand2Hphaseinthesteakdomain,whichisregardasagoodplatformfor\nthe2Dhomojunctionsinnanoelectronics.\n1.Introduction\nArtificialtwo-dimensional(2D)vanderWaalsstructures,showsgreatpromiseindesigning\nnovelelectronicandoptoelectronicdevices[1].Sofar,considerableeffortshavebeendevoted\nnotonlytoinvestigatingfundamentalpropertiesbutalsotoexploringtheengineering\nstrategiestomodifylocalphasetransitionsin2Dmaterials,withthegoalofrealizing\ncontrollabledesignofhomojunctions[2].Atomicallythin2Dmaterialssuchastransitionmetal\ndichalcogenides(TMDs)[3],graphene[4]andgrapheneanalogues[5]havealsobeencredited\nwithuniqueproperties,includingvalley-polarizedexciton[6],2Dferromagnetism[7],and\nroom-temperatureferroelectricity[8].Inthepost-Mooreera,avarietyofstrategiessuchas\nvapor-phasedeposition[9],laserirradiation[2a,10],chemicaldoping[11],anddoping\nengineering[12],havebeenadoptedforrealizingcontrollableconstructionof2D\nhomojunctions.\nApartfromtheheterojunctionsconstructedbytwodifferentmaterials,TMDswiththenatural\nmatchingofchemicalandelectronicstructuresindifferentpolymorphicphases(e.g.1T,2H)\nprovideaconvenientplatformforfabricating2Dhomojunctions,whichusuallyhavea\ncontinuousbandbendingandahigh-qualitycarriertransportandseparationattheinterface[2b].\nSpecifically,phase-engineeredmetallic1T-MoS2/semiconducting2H-MoS2[13]andmetallic\n1T′-MoTe2/semiconducting2H-MoTe2[2a]werereportedwithlow-resistancecontactsand\nimprovedcarriermobility.Hence,theactualdynamicalprocessofthetransformationbetween\n1Tand2Hphasesanditsboundarystructuresmustbecorroboratedinconstructing2D\nheterostructureshomojunctions.\nHere,wereportthesynthesisofepitaxialSLVSe2onAu(111)andprovidein-situ\nobservationsofthephasetransformationprocessfrompristinehomogeneous1Tphase\npassingthroughadefective2Hdomainphase,endinaVSestripephasedecoratedbySeline\nuponannealingatelevatedtemperature.Ourstudyappliesadiversesetofcharacterization\nMethods,combiningSTMwithARPEStoexhibitthedifferencebetween1Tand2Hphases.\nThroughouttheexpositionweusedensityfunctionaltheory(DFT)calculationstoaccountfor\ntheobservations.WediscussedthestructurereactionpathwaydeterminedbythelossofSe\ninducedin-planeincreasedlatticestructure,whichwillfacilitatefuturepreciselyartificial\ntwo-dimensional(2D)vanderWaalsstructuresasdesiredhomojunctions.\n2.Experimentdetailsandcalculationmethods\nAllsamplepreparationstepsandexperimentswereperformedunderultrahighvacuum\nconditionsbetterthan4×10−10mbar.TheAu(111)surfacewaspreparedbyrepeated\nsputteringandannealingcycles.Thequalityofthecleansurfacewasconfirmedbythe\nwell-knownAu-herringbonereconstructioninSTMobservationandtheintensityofthe\nsurfacestate,i.e.,theShockleystate(seeFigure.s1intheSupplementalMaterial).\nSubsequently,Se(purity99.9999%)andV(purity99.99%)wereco-depositedontotheclean\nsurfaceat300℃,yieldingapure2DMorriestructureataround1ML.Thentheprepared\nmonolayerVSe2wasfurtherannealingatelevatedtemperatureupto460℃,thedifferent\nstructureformationafterthesamplepreparationwasconfirmedbyLEEDandSTM\nobservation,andalsothecorrespondingbandstructuremeasurements.ThelabbasedARPESmeasurementswereperformedatRTusinganaberration-corrected,energy-filtered\nphotoemissionelectronmicroscope(EF-PEEM)(NanoESCAIII,ScientaOmicronGmbH)\nequippedwithafocusedheliumdischargelampprimarilygeneratingHeIphotonsathν=\n21.22eV.PassenergyEP=25eVanda0.5mmentranceslittotheenergyfilterwasused[14],\nyieldingnominalenergyandmomentumresolutionsofΔΕ=50meVandΔk=0.02Å−1.\nThedensityfunctionaltheory(DFT)calculationswereperformedusingtheViennaAb\ninitioSimulationPackage(VASP)[15].Theinteractionsbetweenthevalenceelectronsandion\ncoresweredescribedbytheprojectoraugmentedwavemethod[16].Theelectronexchangeand\ncorrelationenergywastreatedbythegeneralizedgradientapproximationwiththe\nPerdew-Burke-Ernzerhoffunctional[15,16b].Thekineticenergycutoffoftheplane-wavebasis\nwassetto500eVasdefault.A5×5×1supercellof1TphaseonaAu(111)substratewas\nconsideredtostudytheeffectofmoirésuperstructure.TheSTMsimulationsareperformed\nusingtheTersoff–Hamannapproach[17].Thebanddispersionforthefreestandingsinglelayer\n1T,2HandVSebucklingphasewerecalculatedwithoutspin-orbitalcoupling(SOC),\nrespectively.Allthestructureswereoptimizeduntiltheforcesontheatomswerelessthan20\nmeVÅ−1.Thevacuumlayeristhickerthan15Åappliedtoeliminatespuriousinteraction\nbetweentwoadjacentslabs.\n3.DiscussionFigure1.Evolutionofthecrystalmorphologyofsingle-layerVSe2onAu(111)atelevatedtemperature.Toprowshows(a)LEED\npattern(70eV),(b)STMobservationsandcorrespondingenergeticallyfavourableatomicmodel(c)ofepitaxiallygrownVSe2\nmonolayerat300C°,theSTMsimulationoverlaidonthetopsideofimage(c).Themiddlerowandthebottomrowshowfurther\nannealingto380C°(d-f)and460C°(g-h),respectively.Thecirclesin(b)showthepositionsofLEEDspotsforVSe2monlayerbefore\n(greencircle)andafterannealingto380C°(yellowcircle).Thecorrespondingunitcellsoftheselenideline-shapedreconstructionin\n(g-h)areindicatedbytherectangularanddashedparallelogram.Theblackdashedlinein(e,f,h,i)areusedtocomparetherelative\npositionoftopmostSe.Scanningparameters:(b)1V,1.0nA;(e)-1.2V,6.9nA;(h)-1.2V,8.4nA;\nFigure1depictsageneraloverviewofthreedifferentphasesobtainedinepitaxiallygrown\nVSe2monolayerat300℃,andsubsequentlyannealingto380℃and460℃asdemonstrated,\nrespectively,byLEEDpatternsandSTMimages.Initially,VSe2monolayerwasfabricatedby\nSeandVco-depositedontothecleansurfaceat300℃.FromtheLEEDpanelofFigure1a,a\nclearmoirépatternsurroundingthemainspotsoftheVSe2atomiclatticecoincidewiththe\nAu(1x1)diffractionspotindicatesapossiblecommensurationbetweena(6×6)\nreconstructionoftheAu(111)substratewiththeapparentmoirécorrugationwitha(5×5)\nsuperlatticeoftheas-grownVSe2,whichisinagreementwiththeSTMobservationshownin\nFigure1b.OnthebasisofthisSTMdata,theatomiclatticeconstantofthemoiré\nsuperstructurehasalatticeparameterof17.7Å,whiletheatomiclatticeparameteris3.45Å,astheunitcellmarkedbytheblackandgreenrhombusinFigure1b,respectively.Tofurther\ninvestigatetheatomicstructureofVSe2onAu(111),weemployedDFTcalculationsto\nconstructanoptimized5×5structuralmodelofthe1T-VSe2onAusubstrate,asdepictedin\nFigure1c.ThecorrespondingSTMsimulatedimageshowaperiodicmoiréstructure,in\nagreementwithSTMdata.Themodelof1T-VSe2appliedherebasedonourARPES\nmeasurement,whichwillbediscussedbelow.\nAnnealingthesamplesto380℃describedinmiddlerowofFigure1inducesthephase\ntransitionasevidencedfromtheLEEDpattern(Figure1d),inwhichshowsanadditional\nhexagonallattice(markedbyredcircle)withaslightlysmallerreciprocallatticeconstantthan\nthatofasgrown1Tphase(markedbygreencircle),andthecorrespondingmoiré\nspots(markedbyyellowarrow)wasmostlydisappeared.Meanwhile,theSTMimageof\nFigure1erevealsadifferentappearanceinthemoirépatternsandclearlyvisiblelinesofthe\ndefectdecoratedintheSLVSe2.Thedensityoftheline-shapeddefectsrelatestothe\nannealingtemperature,asdiscussedinRef[18]Weemphasizethatthisnewappearingmoiré\nstructure(markedbythewhiteribboninFigure1e)inthedefectivestructureofSLVSe2\noriginatesfromtheAu-herringbonereconstruction,whichcanbeclearlyclarifiedinFigure2a\nofref[18],sincethemoiréstructureindislocation-dominateddefectiveVSe2appearswithout\ncommensuratingrelationtotheline-shapeddefects.Wethendeducethenewappearingmoiré\nstructureiscausedbytheinteractionbetweenAusurfaceanddefectiveVSe2monolayer.\nMoreover,thetopmostSeatomseemstoberearrangedintwoadjacentsteak-shapeddomains,\nasindicatedbytheblackdashedlineinFigure1e.Thedashedlinegoesthroughthetopsite\nofSeinonesteak-shapeddomainwhilethroughthebridgesiteofSeatom.Weproposeda\npossiblestructuremodel(Figure1f)with1Tand2HphasecoexistinthedefectiveVSe2\nmonolayer,asthelineofSedefectmarkedbythegreyribbontoseparatetwodifferentphase,\nmuchlikewhathasbeenseenpreviouslyinSevacanciesrearrangementmediatethe\ntransformationfrom1Tdomainsto2Hdomains[9b,19].Thefurtherconfirmationofthe\nexistenceof2HphasewillbegiveninthediscussionofARPESdataandSTMobservationas\nbelow.\nThecompletedtransitiontoastripedphaseisseenafterannealingto460℃andisshownin\nbottomrowofFigure1.TheatomicallyresolvedSTMimage,showninFigure1h,revealsthe\nline-shapedrearrangementoftheSeatomwitharectangularunitcellfortwonearestbright\nlineofatoms,whileaparallelogramunitcellfortwobrightlinewiththeshiftofhalfaunit\ncellalongthedirectionofthelineofSeatoms.Theexperimentallyobservedshiftcanbe\nclearlyidentifiedbytheblackdashedguidelineinSTMimage1h.Correspondingly,The\nLEEDdataofstripedphaseinFigure1gexhibitsacomplicateddiffractionpatternwhichis\nmuchlikewhathasbeenobservedpreviouslyinsimilarlypreparedSLVS2onAu(111)[20].\nTakingintoaccounttheunitcellofthestructureidentifiedbySTM,thediffractionspots\n(markedbycolourfulcircles)inLEEDpatterncanbereproducedbythereciprocal\nrectangularunitcellsderivingfromthethreerotationaldomains.Moreover,theexpected\nLEEDspotsfromthereconstructionofSeatomsasbrightlineappearstobebroadeningalong\ntheshortsideoftherectangularunitcells,whichcanbeexplainedbytheparallelogramunit\ncellfortwobrightSelinewiththeshiftofhalfaunitcell,markedasdashedparallelogramin\nbothFigure1gand1h,respectively.BasedontheanalysisoftheLEEDdataandSTM\nobservation,anoptimizedstructuralmodelofVSewiththelineofSedecoratedonthetopas\nadefectivephaseofVSe2wasconstructed.Obviously,thetopmostSeatomintheline-shaped\narraycouldstackonthetopsiteorhcpsitewithregardtothebottomlayerofSeatom.TheparallelogramunitcellfortwobrightSelinewiththeshiftofhalfaunitcellrevealedinthe\nSTMobservationindicatesthecorrespondingtwoadjacentlineofSeatomhavethesame\nstackingstyle,whiletherectangularunitcellandtheratiobetweenthetwosideofrectangular\nindicatesoneoftheline-shapedSearraystacklike1TphaseandanotherlineofSeatomstack\nlike2Hphase.Hence,thetwodifferentstackstyleoftheSelinesrevealedherealsoindicates\nthepossibletransitionof1Tphaseto2Hphaseinlocalsteak-shapeddomainseparatedbythe\nlineoftheSedefects.Moreover,comparingtotheLEEDpatternoftheas-grownSLVSe2\nwithaperiodicmoiréstructure,theslightlyshrunkreciprocalrectangularunitcellindicates\nthatVSewithSelinestackedonthetophasanexpandedin-planelatticeconstantafteran\nobviousSedepletioncausedbyheatingtheVSe2layerinvacuum.\nFigure2.Electronicstructureofas-grownSLVSe2onAu(111)anduponheatingto380C°.Fermisurface(a)andbanddispersion(b)\nalongthehighsymmetrylinesoftheas-grownSLVSe2withthemoirésuperstructuremeasured(hν=21.2eV)atroomtemperature.\n(c)and(d)Sameas(a)and(b),butforthesteaked-VSe2(heatingto380C°)decoratedbylineofSedefect.Thedashedhexagonin(a)\nrepresentsthe2DBrillouinzone(BZ)withsymmetrypointlabelsoverlaid.Afewobservationsofparticularinterestareindicatedby\narrows.AnonmagneticcalculatedbandstructureoftheSL1T-VSe2without(e)andwith(f)Ausubstrateinducedmoiré\nsuperstructurebeingconsideredforthemodel.Thedifferentatomicorbitalcontributionsarecolorcoded.ARPESmeasurementshasbeenappliedtotracktheelectronicstructureoftheseveralphases\nobservedinSTMimageuponheating.Figures2aand2bshowthefermisurfaceandband\ndispersionalongthehighsymmetrylinesoftheas-grownSLVSe2,respectively.Thenew\nfeatures(pointedbyblackarrow)appeartobeperiodicwithrespecttotheunitcellofSL\nVSe2.Theoverallbanddispersionsobservedexperimentallyareentirelyconsistentwiththe\nreportedbandfeatureofSL1T-VSe2grownonaHOPGsubstrate[21]andthenonmagnetic\ncalculationoftheSL1T-VSe2,asshowninFigure2e.Basedontheorbital-projectedband\nstructuresoftheV3dandSe4porbitals,theFermisurfacewhichformsclosedflower-like\ncontourswasconfirmedtobedominatedbytheV3d-derivedbandcrossingFermilevel(Ef)as\nelectron-likeconductingband,andtheparabolicallydispersingbandmainlyfromSe4p\norbitalextendssteeplydownfromtheΓpointintothelowerlyingvalenceband.Wededuce\ntherelativelyfuzzyfeatureofthebanddispersionoriginatesfromtheperiodicmoiréstructure\nofSLVSe2,theDFTcalculationneglectingthespin-orbitcoupling(SOC)effectswas\nperformedusinga5x5supercellofSLVSe2onAusubstratewithaperiodicmoiréstructure,\nconcludedfromLEEDandSTMdata.Figure2fshowsthecalculatedbandstructurewiththe\ncontributionsfromV3dandSe4pstates.TheprojectedbandfeatureneartheEfshowsa\nsimilardispersionlikethesystemoffreestandingSLVSe2,butexhibitabroadeningcharacter,\nwhichindicatesthemoiréstructureinducesdistortionintheSLVSe2partlycontributestothe\nfuzzyfeature.\nUponheatingto380C°,Figures2cand2dshowthefermisurfaceandbanddispersionofthe\nsteakedphasedecoratedbylineofSedefect(showninFigure1e).Comparingwiththe\nARPESdatarecordedofas-grownhomogeneous1Tphase,thebandstructureoriginating\nfromtheSL1T-VSe2becomesweakerandseveralnewbandfeaturesappear,markedbythe\nblackarrowinFigure2d.TheweakbandfeatureneartheEfliketheas-grown1Tphase\nindicatestherearestilldomainof1TphaseremaininginthedefectiveSLVSe2network.\nHoweverthenewparabolicallybandappearswiththewholefeatureat2eVbelowtheEf,\nwhichisextremelydifferentfromtheparabolicallySe4p-derivedbandin1Tphase\ndispersingwiththebandtopneartheEf.Meanwhile,ourDFTcalculatedbandstructureof1T\nphasewithincreasedin-planelatticeconstantobservedindefectivesteakphaseindicatesthe\nenergypositionoftheparabolicallydispersingband(markedbyblackarrow)mainlyfromSe\n4porbitalseemstoberarelyinfluencedasacomparisonwiththebandstructureofthe\nhomogeneous1Tphase(Figures2).Basedonthenewparabolicallybandappearingdeeply\n2eVbelowtheEfwhiletheV3d-derivedbandofdefective1T-VSe2domainremainscrossing\nEfwithrarelyenergyshiftcausedbychargetransferfromtheAusubstrate,wethendeduce\nthisnewparabolicallybandfeatureoriginatesfromanewphaseuponannealing.Inaddition,\nthereisaparabolicallyelectronicpocketappearingaroundtheΓpointwiththebottomat-1.0\neV,whichisatypicallybandfeatureoftheShockleystatefromthecleanAu(111).The\nShockleystateobservedinARPESmeasurementindicatesthemoiréstructureof\ndislocation-dominateddefectiveVSe2recordedinSTMobservationoriginatesfromthe\ninteractionwiththetypicalherringbonereconstructionofAu(111)surface.Figure3.ElectronicstructureofdefectiveSLVSe2uponheatingto450C°(a-c)and460C°(d).Constant-energycontoursat-2.9eV\nwith2DBZofdefectiveSLVSe2showninblackdashedhexagon,(b)Constant-energycontoursat-1.45eVshowstheUmklapp\nscatteringprocessofAuspband.Theoriginal(scattering)Auspbandmarkedbythered(black)dashedhexagon.Thereciprocal\nlatticevectorsmarkedasredarrowindicateoneofthepossibleUmklappscatteringdirection.(c)bandstructurerecordedonthe\nhighsymmetrydirectionsmarkedbytheredlinein(a).(d)Sameas(c),butforstriped-VSe(heatingto460C°)decoratedbylineof\nSeonthetop.Thepositionofflatbandsin(c)and(d)areindicatedbyyellowdashedline.theV3dprojectedbandstructureoftheSL\n2H-VSe2(e)andSLVSe(f)withoutSOCbeingconsideredforthecalculation.\nTofurtherexploretheoriginofthenewbandfeatureinthesteakedVSe2,acomparativestudy\nbetweentheelectronicstructureofsteakedVSe2fromARPESmeasurementandthe\ncalculatedbandstructureoftheSL2H-VSe2canbeusedtoidentifythenewphaseformedin\nthedifferentdomainsseparatedbylineofSedefect.Figure3a-bshowstheconstant-energy\ncontoursat-2.9eVand-1.45eVbelowtheEf,thebandfeatureshowssixfoldsymmetry,which\nindicatesthedifferentdomainsinthesteakedVSe2hasasimilarhexagonallatticestructure.\nSpecificallyinFigure3b,acarefulanalysisofthehexagonshapedbandfeature,markedby\nthedashedline,revealstheUmklappscatteringprocessofAuspbandwiththereciprocal\nlatticevectors(redarrow)ofthesteakedphase.Basedononlyonetypeofreciprocallattice\nvectorwithsixfoldsymmetry,theproposed2DsurfaceBZofdefectiveSLVSe2was\noverlayedontheconstant-energycontoursat-2.9eVbyblackdashedhexagon(Figure3a).\nMoreover,comparingtothespbandfeaturefromAusubstrate(Figures1),wededucedthe\nlatticeparameterofthedomaininsteakedphaseishexagonalunitcellwithlatticevector3.65\nÅ,whichisinagreementwithanincreasedlatticeparameterobservedfromLEEDandSTMdata.However,afirst-principlescalculationsfromtheSL1T-VSe2withincreasedlattice\nvector3.65Å(Figures2c)hasconsistentlypredictedthebandfeaturesneartheEf,notthe\nnewparabolicallybandfeatureappearingdeeply2eVbelowEf.Henceapossiblecandidate,\n2H-VSe2,hasbeenconsidered,sinceconstant-energycontoursinFigure3aand3bconfirmed\nonlyonehexagonalunitcellinsteakedphase.Additionally,thesimilarphasetransitionhas\nbeenreportedinMoS2,MoSe2,MoTe2andPtSe2[2a,9b,19,22].Figure3cshowstheband\ndispersionalongthehighsymmetrydirections,markedbytheredlineinFigure3(a).The\nparabolicallybandswithtwobranchesalongtheΓ-MandΓ-Kdirection,labelledas'a'and\n'b',disperseseparatelyawayfromtheΓpoint.ThecalculatedbandsoftheSL2H-VSe2\nwithoutSOC,asdisplayedinFigure3e,exhibitaoverallgoodagreementwiththe\nexperimentalbands'a'and'b'inFigure3c.Aswediscussedabove,thetypicallyV3d-derived\nbandlosingpartiallyoftheirspectralweightremainscrossingEfconfirmstheexistenceof1T\nphaseinthesteakeddomain,andthenewclearbandfeaturedeeply2eVbelowEfcanbe\nexplainedbyourDFTpredictedbandstructureof2Hphasewithincreasedlatticeinsteadof\n1Tphase,wethendeducethephasetransitionfromasgrownhomogeneous1Tphaseto\ncoexistenceof1Tand2Hdomainsrandomlydistributinginthedefectivesteakedphase.Note\nthat,theclearV3dderivedbandfeatureof1TphaseneartheEfshowsrarelyenergyshift\nreferringtothecalculatedSL1Tphasewithoutsubstrate,whiletheV3dderivedbandof2H\nphaseislocatedataslightlyhigherbindingenergyof0.8eV.Thismayreflectaelectron\ndopingof2Hdomainsduetochargetransferfromthesubstrate,since1TphaseofVSe2has\nbeenreportedtobemetallicand2Hphaseshowslikeasemiconductor[23].\nAnnealinginUHVathighertemperatures460C°leadstoatransitionfromthesteakedphase\ntoastripedphaseinourSTMobservation.wespeculatethatitmightbeabucklingVSe\nnetworkdecoratedbylinesofSeatomwhichresultsfromlosingalmostalloftheSeatomat\nthetoplayerofVSe2athighertemperature.CorrespondinglyinFigure3d,theband\ndispersionalongthehighsymmetrydirectionshasbeenexperimentallyrecordedtostudy\nadditionalinformationoftheproposedVSenetwork.Obviously,twosimilarparabolicalband\ndispersetoamaximumenergyattheΓpointwithadirectgap0.45eV,whichisobviously\ndifferentfromthebanddispersionofthesteakedphaseinFigure3c.Anonmagnetic\ncalculationofthesimplifiedVSenetwork(Figure3f)withoutthedopingbylineofSeatoms\nrevealsabanddispersionquitesimilartoourARPESmeasurements,exceptthatthereone\nnon-dispersivebands(markedbyyellowdottedlines)at-2.6eVcoincidingwiththe\nmaximumofthelowerparabolicbandandanotheroneat-4.5eV.Sincethesenondispersive\nbandsremainingafteraphasetransitionfromfromthesteakedphasetoastripedphase,we\nassumetheseflatbandsmayoriginfromsurfaceresonantstatesthatmixwithbulkbands(or\nbulkbandsprojection)[24].Theoverallagreementbetweenexperimentalandcalculatedband\ndispersionsaswellasouranalysisofthemodelbasedonSTMobservationindicatesa\nbucklingVSenetworkformedwiththesignificantdepletionofalmostallSeattoplayerof\nVSe2uponheating.Comparingtotheinitialprepared1T-VSe2withthemoirésuperstructure,\nweexperimentallyfindanincreasingtendencyofthelatticeparameterduringthestructure\ntransformationinourARPESmeasurement-consistentwiththeobservationsfromLEEDand\nSTM.Notethat,theoverallbandstructureofVSestripedstructurerigidlyshiftinourARPES\nmeasurementasacomparisontothecalculationofSLVSewithouttheAusubstrateand\ndopingofSearraysconsidered,similartowhathasbeenobservedinhBNonIr(111)[25].Figure4.AtomicallyresolvedSTMoftwodifferentstripedpattern(a,d)uponheatingandSchematicillustrationofthestructural\ntransformationfrompure1TVSe2to1T/2HpatternedVSe(g-i).(a)STMobervationofthe1TsteakedphasedecoratedbythelineofSe\ndefect,(b)theproposedenergeticallyfavourableatomicmodelwithsimulatedSTMimageoverlaiedtothetoprightsideofthemodel.(c-d)\nsameas(a-b)butforthe2Hsteakedphase.TherelativeSeatompositionindifferentsteakdomainnearthesidesofline-shapeddefectis\nmarkedbyblackarrowin(a,b,d,e),respectively.FourshortlinenearthetopofSTMimage(a,d)usedtomarkthewidthoftheline-shaped\ndefect,correspondingtothegrayshadowin(b,e).(c)and(f)Enlargedimageoftheregionindicatedbytheredrectangleinimage(a)and(d).\nTheintensityprofilealongtheblackdashedlineisoverlaidontheimage.Scanningparameters:(a)-1.2V,6.9nA;(d)-1.2V,1.0nA.\nOnthebasisoftheaboveexperimentaldata,STMandARPESmeasurementsrevealthree\ndistinctcrystallinephases.Specifically,theas-grown1T-VSe2withperiodicmoiréstructure\nandthecompletetransitiontobucklingVSestripephasehasbeenclearlydiscussed.However,\ntheintermediatephase,dislocation-dominateddefectiveVSe2,hasbeenconfirmedtobea\ncoexistenceof1Tand2HdomainfromouranalysisofARPESresult.Gaininganunderstandingofthecoexistedsteakeddomainremainsunclearandisimportantforthe\nscenarioofartificialdesigningtwo-dimensional(2D)vanderWaalsstructuresbasedon\ntransformationmechanism.Here,atomicallyresolvedSTMimagerevealstwodifferent\nsteakedstructureinFigure4aand4d,togetherwiththeoptimizedconfigurationsinFig.4b\nand4e.Werefertothetwodifferentsteakedphaseas‘phaseIandII’showninFigure4aand\n4c,respectively.Atthefirstglimpse,thedomainofphaseIexhibitsbrightoneedge\nprotrusionsandfaintanotheredge,whichindicatesthestructurenearthetwosidesofthe\nsteakdomaininphaseIaredifferent.AndthephaseII,bycontrast,displaysbrighterinthe\nmiddlethanthetwosideoftheedges.Moreover,acarefulanalysistherelativeSeposition\nwithinthesteakdomain(indicatedbytheblackarrowinSTMimage),weexperimentallyfind\nthattheSeatomattheonesideofdefectivelineshowsahalfunitcellshiftalongthedirection\nofdefectivelineregardingtotheSeatomatanothersideinphaseI,whilenoobviousrelative\nshiftoftheSeatomatthetwosideofdefectivelineinphaseII.CombinedwiththelineofSe\ndefectinphaseIIdisplaymorenarrowthanphaseI,asindicatedbythefourshortlinenearthe\ntopofSTMimage,WededucetheadjacentsteakeddomainsinphaseIareseparatedbyone\nlineofSedefect,whiletheadjacentdomainsseparatedbytwoadjacentlineofSedefectin\nphaseII.\nInterestingly,comparingthetwoenlargedsteakeddomainphaseinFig.4(c)and(4f),the\nintensityprofileacrossthedomainintheupperlevelofthecorrespondingfigureshows\nobviousdifferencewherealternateSe-VpeaksareobservedonthedomainofphaseIand\nonlySepeaksareobservedonthephaseII,muchliketheasgrown1Tphase(Figure3s)and\nwhathasbeenseenpreviouslyinsimilarlypreparedSLMoS2,MoSe2andPtSe2[2a,9b,19,22].\nMoreover,theconsistenthexagonalandhoneycombfeaturesisalsoconfirmedinourSTM\nsimulationoffreestandingSL1Tand2Hphase,respectively(Figure3s).Combinedwithour\nARPESconfirmationofthecoexistenceof1Tand2Hphase,wededucedthesteakdomain\nwithlineofSedefectinphaseIis1Tphase,andphaseIIis2Hphase.Basedonour\ndiscussionoftherelativepositionofSeatomnearthelineshapeddefect,thecorresponding\nenergeticallyfavourableatomicmodelareproposedinFigure4band4e.Simulationofthe\nSTMimagebasedontheproposedmodel,showingoverallgoodagreementwiththeatomic\nresolutionSTMimage(Figure4aand4d),especiallyabambooshapedfeatureofthelineof\nSeatomsattheonesideofsteakdomaininphaseI(markedbytheredlineinFigure4a)\nreproducedinthesimulatedimage(toprightofFigure4b).Alikelyexplanationforthis\nabnormaldistortedfeatureattheboundaryofonesideofsteakdomainisthisarrayofSeatom\natatransitionstatebetween1Tphaseand2Hphase,duetoahigherelectroniccontribution\nfromVatomin2HphasedistributingbetweenthetwoadjacentSeatoms.\nAtomicmechanismofthephasetransitionfromthehomogeneous1Tphasetothestriped\nphaseofVSenetworkwiththelineofSedecoratedarepresentedinFigure4g-i.Specifically,\nthermalannealingofas-grown1TstructureVSe2filmatanelevatedtemperatureleadstothe\nlossoftop-layerSeatomstoformadefectivestructure,accompaniedbyanincreasedlattice\nconstant.AnincreasingdesorptionoftheSeatomsfromthetoplayerhasasignificantimpact\nontherearrangementoftheremainingatomsindifferentdomainphase.Thelocalsteak\ndomainseparatedbythelineofSedefectwouldkeepthe1TphasewhensmallamountSe\ndesorptionoccurs,sincethelatticeconstantrarelychanges.Furthermore,thearrangementof\nremainingSeatomstoformthedomainin2HphaseoccursafteranincreasingSedesorption,\nwhichindicatesthe2Hphaseismorestableinaslightlylargerlatticeunitcellthantheunit\ncellforhomogeneous1Tphase.Heatingto460℃leadstoacompletephasetransformation\nfromsteakphasetoaVSenetworkwithsmallamountofSedecoratedonthetop(Figure4i),\nandthisstructurecan,ofcourse,againbetransformedintopristinedefect-free1Tphaseby\naddingSeatomswhileannealingthesampleto300C°invacuum[18].Wethenconcludethat\nthedensityofthelineshapedSedefectsplaysanimportantroleintheformationof2Hdomainphase,duetothelatticeparameterofdefectivestructureincreasedafterSedesorption,\nandthethermalstabilityof2Hphaseisbetterthan1Tphasewithanincreasedlatticeunitcell.\nHence,propercontrolforthedensityofSeatomsbythermalannealingandreaddingSeatoms\ncouldadjusttheratiobetween1Tand2Hphaseinthesteakdomain.\n4.Conclusion\nInconclusion,wehavereportedphasetransitionfromtheinitialhomogeneous1Tphase\nfabricatedonAu(111)passingthroughadefective1T/2Hcoexistedintermediatespriorto\nformationofthefinalstripedVSenetwork.CharacterizationsbySTM,ARPES,andtheir\nagreementwithDFTcalculationselucidatedthedifferenceintheelectronicstructures\nbetween1Tphaseand2Hphase.Ourstudythedefectivestructureneartheboundarybetween\ntwoVSe2domainsrevealsthedynamicsofdefectmovementandrearrangementbehaviorof\ntheremainingSeatomatthetopsite.wefoundthatthedesorptionoftheSeatomleadstothe\ndefectivelatticeformedwithanincreasedlatticeparameters,whichismorestablefor2H\nphaseathightemperatures.Moreover,thereversiblephasetransitionpathwayindicatesthe\npropercontrolforthedensityofSeatomsbythermalannealingandreaddingSeatomscanbe\nregardasagoodmethodforartificialthe2DvanderWaalsstructuresashomojunctions.\nSupportingInformation\nSupportingInformationisavailablefromtheWileyOnlineLibraryorfromtheauthor.\nAcknowledgements\nThisworkwaspartlysupportedbytheResearchCouncilofNorway,ProjectsNo.324183,\nNo.315330,No.323766,andNo.262633.TheSTMmeasurementwasfinancially\nsupportedbytheNationalKeyResearchandDevelopmentProgramofChina\n(2021YFA1600800)andNationalNaturalScienceFoundationofChina(11874380,\n22002183).TheDFTcalculationsincollaborationwithX.S.Wangsupportedfromthe\nNationalNaturalScienceFoundationofChina(GrantsNo.11804045and12174093)andthe\nFundamentalResearchFundsfortheCentralUniversities.Reference\n[1]a)Q.H.Wang,K.Kalantar-Zadeh,A.Kis,J.N.Coleman,M.S.Strano,NatureNanotechnology\n2012,7,699;b)K.S.Novoselov,A.Mishchenko,A.Carvalho,A.H.CastroNeto,Science2016,353,\naac9439.\n[2]a)S.Cho,S.Kim,J.H.Kim,J.Zhao,J.Seok,D.H.Keum,J.Baik,D.-H.Choe,K.J.Chang,K.\nSuenaga,S.W.Kim,Y.H.Lee,H.Yang,Science2015,349,625;b)F.Wang,K.Pei,Y.Li,H.Li,T.Zhai,\nAdvancedMaterials2021,33,2005303.\n[3]S.Manzeli,D.Ovchinnikov,D.Pasquier,O.V.Yazyev,A.Kis,NatureReviewsMaterials2017,\n2,17033.\n[4]a)K.S.Novoselov,V.I.Fal′ko,L.Colombo,P.R.Gellert,M.G.Schwab,K.Kim,Nature2012,\n490,192;b)A.K.Geim,K.S.Novoselov,NatureMaterials2007,6,183.\n[5]F.Reis,G.Li,L.Dudy,M.Bauernfeind,S.Glass,W.Hanke,R.Thomale,J.Schäfer,R.Claessen,\nScience2017,357,287.\n[6]P.Rivera,K.L.Seyler,H.Yu,J.R.Schaibley,J.Yan,D.G.Mandrus,W.Yao,X.Xu,Science2016,\n351,688.\n[7]a)D.J.O’Hara,T.Zhu,A.H.Trout,A.S.Ahmed,Y.K.Luo,C.H.Lee,M.R.Brenner,S.Rajan,J.\nA.Gupta,D.W.McComb,R.K.Kawakami,NanoLetters2018,18,3125;b)Z.Zhang,J.Shang,C.Jiang,\nA.Rasmita,W.Gao,T.Yu,NanoLetters2019,19,3138.\n[8]a)S.Yuan,X.Luo,H.L.Chan,C.Xiao,Y.Dai,M.Xie,J.Hao,NatureCommunications2019,10,\n1775;b)K.Chang,J.Liu,H.Lin,N.Wang,K.Zhao,A.Zhang,F.Jin,Y.Zhong,X.Hu,W.Duan,Q.Zhang,\nL.Fu,Q.-K.Xue,X.Chen,S.-H.Ji,Science2016,353,274.\n[9]a)J.H.Sung,H.Heo,S.Si,Y.H.Kim,H.R.Noh,K.Song,J.Kim,C.-S.Lee,S.-Y.Seo,D.-H.Kim,\nH.K.Kim,H.W.Yeom,T.-H.Kim,S.-Y.Choi,J.S.Kim,M.-H.Jo,NatureNanotechnology2017,12,\n1064;b)X.Lin,J.C.Lu,Y.Shao,Y.Y.Zhang,X.Wu,J.B.Pan,L.Gao,S.Y.Zhu,K.Qian,Y.F.Zhang,D.L.\nBao,L.F.Li,Y.Q.Wang,Z.L.Liu,J.T.Sun,T.Lei,C.Liu,J.O.Wang,K.Ibrahim,D.N.Leonard,W.\nZhou,H.M.Guo,Y.L.Wang,S.X.Du,S.T.Pantelides,H.J.Gao,NatureMaterials2017,16,717.\n[10]Y.Yu,M.Ran,S.Zhou,R.Wang,F.Zhou,H.Li,L.Gan,M.Zhu,T.Zhai,AdvancedFunctional\nMaterials2019,29,1901012.\n[11]a)G.Eda,H.Yamaguchi,D.Voiry,T.Fujita,M.Chen,M.Chhowalla,NanoLett2011,11,5111;\nb)J.Wu,J.Peng,Y.Zhou,Y.Lin,X.Wen,J.Wu,Y.Zhao,Y.Guo,C.Wu,Y.Xie,JournaloftheAmerican\nChemicalSociety2019,141,592.\n[12]a)D.Xiang,T.Liu,J.Xu,J.Y.Tan,Z.Hu,B.Lei,Y.Zheng,J.Wu,A.H.C.Neto,L.Liu,W.Chen,\nNatureCommunications2018,9,2966;b)N.Liu,H.Tian,G.Schwartz,J.B.H.Tok,T.-L.Ren,Z.Bao,\nNanoLetters2014,14,3702.\n[13]R.Kappera,D.Voiry,S.E.Yalcin,B.Branch,G.Gupta,A.D.Mohite,M.Chhowalla,Nature\nMaterials2014,13,1128.\n[14]M.Escher,N.Weber,M.Merkel,B.Krömker,D.Funnemann,S.Schmidt,F.Reinert,F.\nForster,S.Hüfner,P.Bernhard,C.Ziethen,H.J.Elmers,G.Schönhense,JournalofElectron\nSpectroscopyandRelatedPhenomena2005,144-147,1179.\n[15]A.Görling,PhysicalReviewA1999,59,3359.\n[16]a)G.Kresse,D.Joubert,Physicalreviewb1999,59,1758;b)P.E.Blöchl,PhysicalReviewB\n1994,50,17953.\n[17]J.Tersoff,D.R.Hamann,PhysicalReviewB1985,31,805.\n[18]C.Huang,L.Xie,H.Zhang,H.Wang,J.Hu,Z.Liang,Z.Jiang,F.Song,Nanomaterials,\n10.3390/nano12152518\n[19]J.Lin,S.T.Pantelides,W.Zhou,ACSNano2015,9,5189.\n[20]F.Arnold,R.-M.Stan,S.K.Mahatha,H.E.Lund,D.Curcio,M.Dendzik,H.Bana,E.Travaglia,\nL.Bignardi,P.Lacovig,D.Lizzit,Z.Li,M.Bianchi,J.A.Miwa,M.Bremholm,S.Lizzit,P.Hofmann,C.E.\nSanders,2DMaterials2018,5,045009.\n[21]a)J.Feng,D.Biswas,A.Rajan,M.D.Watson,F.Mazzola,O.J.Clark,K.Underwood,I.\nMarković,M.McLaren,A.Hunter,D.M.Burn,L.B.Duffy,S.Barua,G.Balakrishnan,F.Bertran,P.LeFèvre,T.K.Kim,G.vanderLaan,T.Hesjedal,P.Wahl,P.D.C.King,NanoLetters2018,18,4493;b)Y.\nWang,J.Ren,J.Li,Y.Wang,H.Peng,P.Yu,W.Duan,S.Zhou,PhysicalReviewB2019,100,241404.\n[22]Y.-C.Lin,D.O.Dumcenco,Y.-S.Huang,K.Suenaga,NatureNanotechnology2014,9,391.\n[23]a)D.Li,X.Wang,C.-m.Kan,D.He,Z.Li,Q.Hao,H.Zhao,C.Wu,C.Jin,X.Cui,ACSApplied\nMaterials&Interfaces2020,12,25143;b)Y.Wang,Z.Sofer,J.Luxa,M.Pumera,AdvancedMaterials\nInterfaces2016,3,1600433;c)G.V.Pushkarev,V.G.Mazurenko,V.V.Mazurenko,D.W.Boukhvalov,\nPhysicalChemistryChemicalPhysics2019,21,22647.\n[24]A.Zangwill,PhysicsatSurfaces,CambridgeUniversityPress,Cambridge1988.\n[25]J.Cai,W.Jolie,C.C.Silva,M.Petrović,C.Schlueter,T.Michely,M.Kralj,T.-L.Lee,C.Busse,\nPhysicalReviewB2018,98,195443.Supportinginformation\nPhaseTransitionofthesingle-layervanadiumdiselenideonAu(111)withdistinguished\nelectronicstructures\nJinbangHu,XiansiWang,ChaoqinHuang,FeiSong,andJustinWWells\nJinbangHu\nDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,NO-7491\nTrondheim,Norway\nE-mail:jinbang.hu@ntnu.no\nXiansiWang\nHunanUniversity,Changsha,410082,China\nFeiSong,ChaoqinHuang\nShanghaiAdvancedResearchInstitute,ChineseAcademyofSciences,Shanghai201000,\nChina;\nJustinW.Wells\nDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,NO-7491\nTrondheim,Norway\nSemiconductorPhysics,DepartmentofPhysics,UniversityofOslo(UiO),NO-0371Oslo,\nNorwayFigure.S1(a)STMimageofcleanAu(111)surface.(b)ARPESdataofconstantenergycontours\nthroughoutthesurfaceBrillouinzoneat0.1eVbelowtheFermilevel.\nFigure.S2TheoreticallycalculatedbandstructureofmonolayerVSe2foranexpandedin-planelattice\nparameters(a)3.35Å,(b)3.55Å,(c)3.65Å,(d)3.75Å,therelativeenergypositionoftheparabolically\ndispersingbandaroundtheГpointismarkedbytheblackarrow.Figure.S3(a)Atomic-resolutionSTMimageofSLVSe2showingthehexagonallatticeofSeatomsin\nthetopmostsublayeroftheVSe2sandwich-typestructure.(b)Theintensityprofilealongtheblack\ndashedlinemarkedIn(a).(c)simulationoffreestanding1TVSe2and(d)2HVSe2.Scanning\nparameters:U=−1.2VandI=0.12nA." }, { "title": "2401.17568v1.Nonlinear_dispersion_relation_of_dust_acoustic_waves_using_the_Korteweg_de_Vries_model.pdf", "content": "Nonlinear dispersion relation of dust acoustic waves using the Korteweg-de\nVries model\nFarida Batool,1,a)Ajaz Mir,1, 2Sanat Tiwari,1,b)and Abhijit Sen2\n1)Indian Institute of Technology Jammu, Jammu, J&K, 181221, India\n2)Institute for Plasma Research, Gandhinagar, Gujarat, 382428, India\n(Dated: 1 February 2024)\nIn this brief communication, we present an exact analytic nonlinear dispersion relation (NLDR) for the dust\nacoustic waves using the Korteweg-de Vries (KdV) model. The NLDR agrees with the spectrum of spatio-\ntemporal evolution obtained from an exact solution as in Mir et al. [Phys. Plasmas 27, 113701 (2020)]. The\nNLDR also shows a reasonable match with the experimental data of Thompson et al. [Phys. Plasmas 4,\n2331 (1997)] in the long wavelength limit ( kλD≪1). We suggest that such nonlinear corrections should\nbe incorporated in the dispersion relation along with damping, streaming, and correlation effects in order to\nprovide a more realistic interpretation of experimental data.\nI. INTRODUCTION\nDust acoustic waves (DAWs) are analogs of ion\nacoustic waves in a dusty plasma that exist due to the\nbalance of charged dust inertia and plasma pressure1,2.\nThese waves can arise naturally3–7or can be excited by\nan external perturbation8. They are low-frequency waves\nin the range of a few tens of Hz with wavelengths of a\nfew tens of mm. Their characteristics slow timescales are\ndue to the heavy mass of the micron sized dust parti-\ncles. The experimentally observed space-time evolution\nof DAWs usually shows nonlinear features through the\nappearance of harmonics in the frequency and wave num-\nber domains9–11. The experimental data is frequently\ncompared with theoretically derived linear dispersion re-\nlations that customarily incorporate various linear contri-\nbutions arising from collisional/kinetic damping, particle\nstreaming and correlation effects. However since exper-\nimental data is based on measurements of finite ampli-\ntude waves its nature cannot be fully captured in a linear\nmodel based dispersion relation. In this paper we pro-\npose a nonlinear dispersion relation that is based on the\nKdV model and that can provide a better description of\nthe DAW data in the weakly nonlinear dynamical regime.\nDAWs show a sound wave nature in the long-\nwavelength limit4,5, a dispersive nature for short wave-\nlengths and a phase velocity reversal in the strongly cou-\npled regime12. As one of the fundamental modes of a\ndusty plasma, it has been an object of intense research\never since its theoretical prediction by Rao et al.1and\nits experimental identification thereafter by Barkan et\nal.13. Its linear properties were thoroughly explored in\nweak and strong coupling regimes using fluid models14,15,\nquasi-localized charge approximation16,17, molecular dy-\nnamics simulations17,18and in laboratory experiments13.\nA simple linear dispersion for DAW, ω/k≈λDωpd=\nCpd, was proposed by Rao et al.1in the long wavelength\na)Electronic mail: farida.batool@iitjammu.ac.in\nb)Electronic mail: sanat.tiwari@iitjammu.ac.inlimit kλD≪1. Here, Cpdis the dust sound speed\nin the medium, λDis the Debye length due to Boltz-\nmann ions and electrons, and ωpdis the characteristic\nfrequency of DAW. For experimental conditions effects\ndue to dust-neutral collisions, ion streaming, and other\ndamping mechanisms can be significant. Ruhunusiri et\nal.19have extensively explored the dispersion relation\nof DAW by retaining most of the above mentioned ef-\nfects except nonlinear corrections arising from the finite\nsize of the wave amplitude. In subsequent work, Goree\net al.20provided a kinetic dispersion relation for DAWs\nand compared it with data from the Plasma Kristall-4\n(PK-4) experiment on the International Space Station.\nInterestingly, we found that, this experimental data also\nshows features arising from non-linearity of the propa-\ngating wave. However, such nonlinear effects have not\nbeen incorporated so far in the context of a dispersion\nrelation.\nThe Korteweg-de Vries (KdV) equation successfully\nmodels low frequency weakly nonlinear wave propagation\nin a variety of media including dusty plasmas21,22. The\nconoidal waves and solitons are exact solutions of the\nKdV equation and have found applications in domains\nlike hydrodynamics23, oceanography24, plasmas2, nonlin-\near optics25and astrophysical systems26. Optical fiber\ncommunication is one such practical application where\nsignals in the form of soliton pulses are used to transmit\ninformation over long distances without suffering any dis-\ntortion or dissipation. While the KdV equation, derived\nusing a reductive perturbation technique, represents a re-\nduced dynamical form of the full fluid model, it retains\nthe essential features of nonlinearity and dispersion that\ninfluence the wave propagation. We have utilized the fact\nthat an exact solution is possible for the KdV equation\nto obtain an exact nonlinear dispersion relation (NLDR)\nfor DAW.\nWhile the linear dispersion relation (LDR) for KdV\nis trivial, the wave’s phase velocity and modes interac-\ntion drastically change in the nonlinear regime. A typical\nperturbation grows in amplitude and interacts nonlin-\nearly with others due to the spatiotemporal term of form\nn(x, t)∂n(x, t)/∂x ( a one-dimensional model). Due toarXiv:2401.17568v1 [physics.plasm-ph] 31 Jan 20242\nthe nonlinear interaction, there is a generation of har-\nmonics of fundamental modes in the medium that can be\nseen in the power spectrum of the exact analytic solu-\ntion and the numerical solution of the KdV equation27.\nIn this case, the functional form of the dispersion rela-\ntion appears to change from ω=f(k) toω=F(k, κ).\nHere, ω,k, and κare the frequency, wave number, and\nthe nonlinearity parameter associated with the cnoidal\nwave28. This paper has derived this functional form F\nfor the nonlinear dispersion relation.\nVarious generalizations of the KdV model exist in\nthe literature. These models are obtained by including\nthe medium’s different dissipative or dispersive mecha-\nnisms. As an example, the viscous damping leads to the\nKdV-Burgers equation29. Contrary to the KdV equation,\nthere is no known exact solution to these extended model\nforms of KdV. In all those cases, the numerical evolution\ndata reflects a change in the frequency due to the non-\nlinearity. Our results in this paper provide a benchmark\nby comparing the obtained NLDR with those obtained\nfrom exact analytic and numerical solutions for the KdV\nmodel. The obtained NLDR also facilitates the estima-\ntion of the phase velocity of a cnoidal or cnoidal-like non-\nlinear traveling solution.\nThe paper is organized as follows: The KdV equa-\ntion, its linear dispersion relation, analytic solution, and\nnumerical solution have been discussed in section II. The\nNLDR from the analytic and numerical spatiotemporal\nevolution of the KdV equation and the NLDR for the\nmodified KdV (mKdV) equation have been discussed in\nsection III. The theoretical NLDR from the KdV model\nhas been compared to the experimental observation of\nDAW in the long wavelength limit in section IV. The\nwork is summarized in section V with a mention of fu-\nture scope.\nII. KORTEWEG-DE VRIES MODEL\nThe KdV equation represents the weakly nonlinear\ndynamics of fluids in the absence of dissipating mecha-\nnisms and is written as1,2,30,31\n∂n(x, t)\n∂t+α n(x, t)∂n(x, t)\n∂x+β∂3n(x, t)\n∂x3= 0.(1)\nHere n(x, t) is the perturbed field of the fluid\nmedium and can be either density, velocity, potential\netc.αandβquantify nonlinearity and dispersion of the\nmedium, respectively. These parameters include all char-\nacteristic physical quantities of the medium, including\nthe equilibrium temperature, density, and pressure. The\ndensity, length, and time are normalized with the equi-\nlibrium dust density, nd0, Debye length, λDand dust\nplasma period, ω−1\npdrespectively. It should also be noted\nthat the perturbed density, n, can have values less than 1\nfor the physical scenario. The linear dispersion relation\nfor the KdV equation is\nωL=−βk3. (2)This dispersion relation is in a travelling frame moving\nin positive x-direction that was used to reduce the full\nset of fluid equations32into the Eq. (1). The frame ve-\nlocity is chosen as the sound speed in the medium. That\ntransformation shall be included for a complete disper-\nsion relation in a rest frame. We have provided the rest\nframe dispersion relation of the DAW as a case study in\nsection IV.\nA. Analytic solution of KdV\nThe balance between non-linearity and dispersion\ngives rise to the cnoidal wave solution of the KdV equa-\ntion and is given by23,28.\nn(x, t) =µcn2\"√µα\n2p\nβκ(κ+ 2)ξ(x, t);κ#\nξ(x, t) =\u0012\nx−κ+κ2−1\nκ(κ+ 2)αµt\u0013\n, (3)\nwhere cn and µare Jacobi elliptic function and amplitude\nof the cnoidal wave, respectively. κis the nonlinearity in-\ndex. While 0 < κ < 1 represents the cnoidal (solitary)\nwave train, the special case of κ= 1 gives the single\nsoliton solution. The solution in Eq. (3) can also be rep-\nresented in a travelling wave-like form as given by30,33\nn(x, t) =µcn2h\n2K(κ)\u0010x\nλ−ft\u0011\n;κi\n, (4)\nwhere K(κ) is the Jacobi complete elliptic integral of the\nfirst kind, λandfare the wave’s fundamental wavelength\nand frequency, respectively. Other secondary modes are\nthe harmonics of the fundamental mode.\nB. Numerical evolution of KdV\nWe have also developed a pseudo-spectral code34us-\ning FFTW3 library35to evolve the KdV equation numer-\nically. The code is validated against earlier established\nresults27,31. Here, we provide a pure sinusoidal per-\nturbation of the form n(x,0) = A0sin(k0x) to the KdV\nequation. It is then numerically evolved in both space\nand time. The left column of Figure 1(a, c, and e) shows\nthe spatial profiles of the wave at different times (t = 0,\n60, and 100 ω−1\npd). The right column of the same figure\n(b, d, and f) shows the respective power spectral density\n(PSD), which is obtained by taking the Fourier transform\nof the corresponding spatial wave profile. We can observe\nthat the initially provided sinusoidal wave distorts into\na nonlinear cnoidal waveform at later times. This wave\ndistortion is due to the nonlinear mixing of the modes\nand, hence, the generation of the harmonics, which we\ncan observe in the PSD. In order to get the temporal\nprofile of the wave, we collected the time series at a par-\nticular point in space, represented by subplot (g) and the3\ncorresponding Fourier transform by subplot (h). In the\nPSD, we observe the generation of harmonics besides the\nfundamental harmonic.\nAn analytic solution for the KdV equation is avail-\nable in Eq. (3). However, only numerical solutions can be\nobtained for other generalized KdV models, including ad-\nditional physical effects. Those numerical solutions can\nthen be Fourier transformed, and a nonlinear dispersion\nrelation can be obtained.\nFIG. 1. Spatio-temporal numerical evolution of sinusoidal\nperturbation in KdV equation (Eq. 1) with α= 2 and β= 1.\nThe initial condition is n(x,0) = A0sin(k0x), with A0= 0.8\nandk0= 7kmwhere km= 2π/Lxis the minimum wave vector\nassociated with a system of length Lx= 16π. All parameters\nare in normalized units.\nIII. NONLINEAR DISPERSION RELATION OF KDV\nA. Exact nonlinear dispersion relation\nComparing the solution of KdV Eq. (3) with Eq. (4),\nthe wavelength and frequency are given by28\nλ= 4K(κ)s\nβ(2κ+κ2)\nαµ(5)\nf=β\n4K(κ)\u0000\nκ2+κ−1\u0001\u0012αµ\nβ(2κ+κ2)\u00133/2\n.(6)\nThrough mathematical arrangements of Eq. (5) and\nEq. (6), the exact nonlinear dispersion relation for KdV\nequation can be written as\nω=4(K(κ))2\nπ2\u0000\nκ2+κ−1\u0001\nβk3. (7)\nThis relation converts to the linear dispersion relation\nas in Eq. (2) in the limit of κ→0. We have chosenparameters α= 2.0,β= 0.0667, κ= 0.98, and µ=\n0.25 for analytic solution of Eq. (7) without the loss of\ngenerality. However, choice of these parameter values has\nbeen mentioned explicitly as required.\nWe compare the linear and nonlinear dispersion re-\nlation to quantify the effect of nonlinearity. For this pur-\npose, we record the spatiotemporal evolution profile from\nthe Eq. (3). Fig. 2 includes all the evolution and disper-\nsion profiles. The subplot (A) shows the spatial evo-\nlution of the cnoidal wave at two different times 0 and\n1200 ω−1\npd. For a given time of 0 ω−1\npd, the PSD of the\nspatial profile gives the wavenumber associated with the\nwave and its higher harmonics due to the nonlinearity.\nThis is also true for any subsequent time in evolution in-\ncluding at time of 1200 ω−1\npd. Further, the subplot (C)\nshows the fundamental frequency and higher harmonics\nfor PSD of the time series of the cnoidal solution at a\ngiven location. Finally, the subplot (D) consolidates the\nPSD features of subplots (B, C) and compares the lin-\near and nonlinear dispersion relations using Eq. (2) and\nEq. (7), respectively. The NLDR passes through the fun-\ndamental frequency, wavenumber (first bright spot in the\nblack background of D) as obtained from Eq. (3). The\nextra-bright spots represent the harmonics, which repre-\nsent the signature of the nonlinearity. Besides this, we\ncan also observe that the NLDR shows an upward shift\nin the frequency from the LDR.\nB. Nonlinear dispersion relation from numerically evolved\nmodified KdV\nThe previous section delineates the importance of con-\nsidering the effect of nonlinearity while studying the\nDAW dispersion using the KdV equation and its ana-\nlytic solution. The effect of nonlinearity on dispersion\ncan be calculated numerically for either different variants\nof the KdV equation or using the full fluid model. These\nextended models may incorporate different physical pro-\ncesses such as supra-thermal electrons, relativistic effects,\nstrong coupling, visco-elastic effects, etc.36,37. These\nmodels may not have analytic solutions and hence require\nsolving them numerically. Here, we present one such\nwork by incorporating the dispersion relation obtained\nby numerically evolving the modified KdV equation. The\nequation we have chosen differs from the KdV equation\n(1) with the nonlinearity of the form n2∂n/∂x =n2∂xn.\nThe present dynamical model for dusty plasma can be\nobtained under the condition of dusty plasmas with vari-\nable temperatures36and due to the non-thermal ions37.\nFigure 3 (D) shows the NLDR of mKdV, which\nis obtained from the numerical spatiotemporal evolu-\ntion with initial perturbation of the from n(x,0) =\nA0cn2[2K(κ)x/λ;κ]. The parameters used for the evo-\nlution of the mKdV equation are provided in Table I.\nWe observe the fundamental frequency for the nonlinear\nwave to be 0 .016 compared to 0 .0045 for the linear wave4\nFIG. 2. The NLDR of KdV (Eq. (1)) obtained from analytical solution (Eq. (3)). (A) The spatial profiles of the analytical\nsolution of KdV at two different times, 0 and 1200. (B) and (C) depict PSDs of spatial and temporal profiles, respectively. (D)\nrepresents the comparison of LDR Eq. (2) (yellow circles), NLDR Eq. (7) (red pentagons), and NLDR obtained analytically\nfrom the spatio-temporal evolution of KdV (black background with few bright spots).\ncorresponding to wavenumber k. For reference and code\nvalidation purposes, we also evolved the KdV equation\nnumerically using the same initial condition, and their\nparameters are also mentioned in the same table. From\nFigure 3 (D), it is evident that there is a frequency in-\ncrease in comparison to the frequency obtained from the\nlinear dispersion relation. This frequency increase is sim-\nilar to what we obtained from the KdV model. This, in\nturn, supports our result that nonlinearity, or all forms\ndue to their physical origins, will show a frequency shift.\nThe magnitude of the frequency shift is governed by the\nnature of the nonlinearity of the medium for given phys-\nical conditions.\nTABLE I. Parameters for numerical simulations of the KdV\n(n∂xn) and the mKdV ( n2∂xn).\nModel αβ A0κ k km Lxf\nKdV 2 0.0670.250.986km2π/Lx16π0.020\nmKdV 11 0.0670.250.986km2π/Lx16π0.016IV. DISPERSION RELATION OF DUST ACOUSTIC\nWAVE\nAs a case study, we chose to discuss the nonlinear\ndispersion relation of DAW. Its linear dispersion relation\n(within cold dust approximation) is given by2,38\nω=kCpd\u0002\n1 + (kλD)2\u0003−1/2≈kCpd−1\n2(kλD)3ωpd.(8)\nYaroshenko et al.38provided a nonlinear analytic version\nof the DAW dispersion relation (for cold dust and no\ncharge fluctuations) as:\nω=kCpd(\n\u0002\n1 + (kλD)2\u0003−1/2+αir\neϕ\n2πTi)\nω≈kCpd(\u0014\n1−1\n2(kλD)2\u0015\n+αir\neϕ\n2πTi)\n(9)\nHere ϕis the wave amplitude, eis the electron charge,\nTiis the ion temperature, and αiis a parameter that\ndepends on the Jacobi complete elliptic integral of the\nfirstK(κ) and second E(κ) kinds. In comparison, the\nnonlinear dispersion relation we obtained from KdV can5\nFIG. 3. The NLDR of mKdV with nonlinearity of the form n2∂xn. (A) The spatial profiles of the numerical solution of\nmKdV at two different times, 0 and 2000 with initial perturbation n(x,0) = A0cn2[2K(κ)x/λ;κ]. (B) and (C) depict PSDs\nof spatial and temporal profiles, respectively. (D) represents the comparison of LDR of Eq. (2) (yellow circles) and NLDR\nobtained numerically from the spatio-temporal evolution of mKdV (black background with few bright spots).\nbe written in the stationary frame as:\nω=kCpd\u0014\n1 +4(K(κ))2\nπ2\u0000\nκ2+κ−1\u0001\nβ(kλD)2\u0015\n(10)\nThe NLDR of KdV (Eq. (10)) depends only on the Ja-\ncobi complete elliptic integral of first kind K(κ) and pa-\nrameter β. Parameters αandβcan be calculated from\nphysical plasma parameters as given in Liu et al.30.\nA. Comparison with experimental observation of DAW in\nlong wavelength limit kλD<<1\nWe compare the dispersion relation obtained from\nEq. (10) with the experimental result by Thompson\net al.8in long wavelength limit kλD<<1 and also with\nthe theory by Merlino et al.39, given below.\nk=ω(ω+q\n(ω2+ν2\nnd))1/2(11)\nifνnd/ω << 1, the above dispersion relation simplifies as\nω=kCpd(1−ν2\nnd/8k2C2\npd). (12)\nTo get the KdV dispersion (linear and nonlinear) from\nEq. (10), we calculate the value of βfrom experimentalparameters using relations derived by Tiwari et al.32.\nTo make a comparison with experimental data, we\nhave added an approximate dust-neutral drag νndas in\nEq. (12) to the Eq. (10) getting a form:\nω=kCpd\u0014\n1 +4(K(κ))2\nπ2\u0000\nκ2+κ−1\u0001\nβ(kλD)2−ν2\nnd/8k2C2\npd\u0015\n(13)\nUsing the plasma and damping parameters from the ref-\nerence experimental work, we calculate the NLDR for\nthree values of κ= 0,0.7, and 0 .9.\nThe comparison of experimental data with the pro-\nposed theory on NLDR is presented in Fig. 4. The sub-\nplot (a) shows the comparison of KdV dispersion with the\nexperiment (black hexagons) and with theory (red bold\nline, Eq. (11)). The KdV dispersion relation obtained\nfrom Eq. (10) is for three values of nonlinearity param-\neterκ(0, 0.7, and 0.9), which are represented by the\nblack dash-dotted, green dash-dash, and magenta dash-\ndotted lines, respectively. Please note that the disper-\nsion relation represented by Eq. (10) does not include\nthe dust-neutral collision term. To make it further real-\nistic to the reference experimental data, we include dust-\nneutral collision term from Eq. (12) to the Eq. (10). The\nNLDR then takes the form of Eq. (13). These modified6\nFIG. 4. Comparison of dispersion relation of DAW, KdV and effect of dust-neutral drag on KdV dispersion relation. (a)\nExperimental DAW (black hexagons)8, theoretical including dust-neutral drag (red bold)39, linearized KdV (black dash-dotted)\nκ= 0, NLDR KdV at κ= 0.7 (green dash-dash) and κ= 0.9 (magenta dash-dotted) (Eq. (10)). (b) Effect of drag ( νnd= 61 s−1)\non KdV dispersion relation corresponding to each case as in (a). Inset (L) shows the positive frequency shift as the nonlinearity\nincreased from κ= 0.7 to 0 .9. Inset (R) shows the magnified version of the NLDR of KdV at κ= 0.9 and the experimental\ndispersion data.\ndispersion relations are shown in subplot (b). Adding\nthe drag term, the nonlinear dispersion relation obtained\nfrom KdV matches well with the experiment.\nThe subplot Fig. 4(b) also has left and right in-\nsets. The left inset shows the frequency shift for a given\nwavenumber as the nonlinearity parameter κchanges\nfrom 0 .7 to 0 .9. The right inset magnifies the dispersion\nrelation, showing a reasonable match of the experimental\ndata with that of the theoretical model at κ= 0.9.\nV. SUMMARY\nIn this paper, we focus on the role of medium’s\nnonlinearity in estimating the dispersion relation of the\nwaves. We have demonstrated this using the KdV model.\nThrough our analysis, we could show a small positive fre-\nquency shift for a wave as it gets nonlinear. At higher\nwavenumbers, the shift is prominent.\nThe present work has the convenience of an exact\nanalytic solution for the KdV equation. However, numer-\nical solutions will be necessary for other systems where\nthe dynamics are governed by a modified KdV model or\nthrough the full set of fluid equations or lattice base dy-\nnamical models. In this present work, we have shown\na positive frequency shift for one such form of mKdVequation by evolving it numerically.\nFinally, connecting our work with dusty plasmas, we\nderived the nonlinear dispersion relation for DAWs using\nKdV model and then by transforming it back into the rest\nframe. We found results qualitatively similar to the dis-\npersion relation proposed by Yaroshenko et al.38. We also\nfound a good agreement between our theoretical NLDR\nand the experimental observations from Thompson et al.\n8in the long wavelength limit where the KdV-based dis-\npersion relation makes sense. However, we had to add the\neffect of drag as an artificial term to the NLDR obtained\nfrom analytic approaches. both with and without dust-\nneutral drag contribution and found a good agreement\nwith their experimental findings.\nACKNOWLEDGMENTS\nThis work was supported by the Indian Institute of\nTechnology Jammu Seed Grant No. SG0012. F. B.\nthanks the University Grants Commission (UGC) of In-\ndia for the PhD fellowship. A. S. thanks the Indian Na-\ntional Science Academy (INSA) for the INSA Honorary\nScientist position. We also thank R. Wani for the initial\ndiscussions and S. Sharma for providing experimental ref-\nerences on DAWs.7\nAPPENDIX\nFIG. 5. Spatio-temporal wavevector-frequency of cnoidal and\nsoliton of KdV Eq. (1). The time series of (a) cnoidal wave\nand (c) soliton are the exact solutions of the KdV equation.\nInsets show the frequency spectrum associated with their time\nseries. The dispersion relation for cnoidal wave and the soliton\nare shown in (b) and (d), respectively. The parameters are\nsame as for the Fig. 2.\nWe provide a comparative picture of the cnoidal and\nsoliton solutions in the time and frequency domains. It\nis common to see soliton (a single localized structure,\nFig. 5(c)) as a limit κ→1 of the generalized cnoidal wave\n(a wave-train of multiple localized structures, Fig. 5(a))\nwith 0 < κ < 1. Though it is less intuitive to see how\na discrete frequency spectrum (power lies at harmon-\nics, Fig. 5(b)) of a cnoidal wave approaches a contin-\nuous spectrum (power lies at all available frequencies,\nFig. 5(d)) for the soliton.\n1N. Rao, P. Shukla, and M. Yu, Planet. Space Sci. 38, 543 (1990).\n2P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma\nPhysic (Institute of Physics, Bristol, 2001).\n3J. Heinrich, S.-H. Kim, and R. L. Merlino, Phys. Rev. Lett. 103,\n115002 (2009).\n4R. L. Merlino, Phys. Plasmas 16, 124501 (2009).\n5T. M. Flanagan and J. Goree, Phys. Plasmas 17, 123702 (2010).\n6B. Tadsen, F. Greiner, S. Groth, and A. Piel, Phys. Plasmas 22,\n113701 (2015).\n7T. Deka, A. Boruah, S. K. Sharma, and H. Bailung, Phys. Plas-\nmas24, 093706 (2017).8C. Thompson, A. Barkan, N. D’Angelo, and R. L. Merlino, Phys.\nPlasmas 4, 2331 (1997).\n9S. Sharma, M. Sharma, G. V. Prakash, P. Srivastav, Y. Saxena,\nS. Bose, and S. Tiwari, AIP Advances , Accepted (2023).\n10P. Bandyopadhyay, G. Prasad, A. Sen, and P. K. Kaw, Phys.\nRev. Lett. 101, 065006 (2008).\n11L.-W. Teng, M.-C. Chang, Y.-P. Tseng, and L. I, Phys. Rev.\nLett. 103, 245005 (2009).\n12J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996).\n13A. Barkan, R. L. Merlino, and N. D’Angelo, Phys. Plasmas 2,\n3563 (1995).\n14P. K. Kaw and A. Sen, Physics of Plasmas 5, 3552 (1998).\n15A. Diaw and M. S. Murillo, Phys. Rev. E 92, 013107 (2015).\n16M. Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997).\n17Z. Donk´ o, G. J. Kalman, and P. Hartmann, J. Phys. Condens.\nMatter 20, 413101 (2008).\n18H. Ohta and S. Hamaguchi, Phys. Rev. Lett. 84, 6026 (2000).\n19W. D. Suranga Ruhunusiri and J. Goree, Phys. Plasmas 21,\n053702 (2014).\n20J. Goree, B. Liu, M. Y. Pustylnik, H. M. Thomas, V. E. Fortov,\nA. M. Lipaev, V. I. Molotkov, A. D. Usachev, O. F. Petrov,\nM. H. Thoma, E. Thomas, U. Konopka, and S. Prokopiev, Phys.\nPlasmas 27, 123701 (2020).\n21L. Debnath, Nonlinear Partial Differential Equations for Scien-\ntists and Engineers (Springer, New York, 2005).\n22A. Karczewska, P. Rozmej, and E. Infeld, Phys. Rev. E 90,\n012907 (2014).\n23P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cam-\nbridge University Press, Cambridge, 1989).\n24A. Costa, A. R. Osborne, D. T. Resio, S. Alessio, E. Chriv` ı,\nE. Saggese, K. Bellomo, and C. E. Long, Phys. Rev. Lett. 113,\n108501 (2014).\n25H. Leblond, Phys. Rev. A 78, 013807 (2008).\n26A. Coutant and S. Weinfurtner, Phys. Rev. D 97, 025005 (2018).\n27A. Mir, S. Tiwari, and A. Sen, Phys. Plasmas 29, 032303 (2022).\n28A. A. Mir, S. K. Tiwari, J. Goree, A. Sen, C. Crabtree, and\nG. Ganguli, Phys. Plasmas 27, 113701 (2020).\n29A. Mir, S. Tiwari, A. Sen, C. Crabtree, G. Ganguli, and J. Goree,\nPhys. Rev. E 107, 035202 (2023).\n30B. Liu, J. Goree, T. M. Flanagan, A. Sen, S. K. Tiwari, G. Gan-\nguli, and C. Crabtree, Phys. Plasmas 25, 113701 (2018).\n31A. Sen, S. Tiwari, S. Mishra, and P. Kaw, Adv. Space Res. 56,\n429 (2015).\n32S. K. Tiwari, A. Das, P. Kaw, and A. Sen, New J. Phys. 14,\n063008 (2012).\n33M. Dingemans, Water Wave Propagation Over Uneven Bottoms\n(World Scientific, Singapore, 1997).\n34J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, New\nYork, 2003).\n35M. Frigo and S. G. Johnson, Pro. IEEE 93, 216 (2005).\n36B. C. Kalita and S. Das, J. Plasma Phys. 83, 905830502 (2017).\n37M.-m. Lin and W.-s. Duan, Chaos, Solitons & Fractals 33, 1189\n(2007).\n38V. V. Yaroshenko, Phys. Rev. E 102, 023201 (2020).\n39R. L. Merlino, A. Barkan, C. Thompson, and N. D’Angelo, Phys.\nPlasmas 5, 1607 (1998)." }, { "title": "2401.17572v1.Null_geodesic_structure_for_the_Barriola_Vilenkin_spacetime_via__k__essence.pdf", "content": "arXiv:2401.17572v1 [gr-qc] 31 Jan 2024Null geodesic structure for the Barriola-Vilenkin spaceti me via k-essence\nBivash Majumder∗\nDepartment of Mathematics, Prabhat Kumar College\nContai, Purba Medinipur 721404, India\nSaibal Ray†\nCentre for Cosmology,\nAstrophysics and Space Science (CCASS), GLA University\nMathura 281406, Uttar Pradesh, India\nInstitute of Astronomy Space and Earth Science\nKolkata 700054, India\nGoutam Mannaa‡\nDepartment of Physics, Prabhat Kumar College\nContai, Purba Medinipur 721404, India\nInstitute of Astronomy Space and Earth Science\nKolkata 700054, India\naCorresponding author\nBased on the work of Chandrasekhar [ The Mathematical Theory of Black Holes, Oxford Univ.\nPress (1992) ], we investigate the null geodesic structure of the emergen t Barriola-Vilenkin spacetime\nin the context of k-essence theory. For k-essence, the emergent gravity metric is a one-to-one\ncorrespondence with the Barriola-Vilenkin (BV) metric con nected to the Schwarzschild background,\nwhere the global monopole charge is replaced by the dark ener gy density. This equivalence holds\nspecifically for a certain class of k-essence scalar fields that have been constructed by Gangopad hyay\nand Manna [Euro. Phys. Lett., 100, 49001, (2012)]. We have tr aced out different trajectories for\nnull geodesic in the presence of dark energy for the k-essence emergent Barriola-Vilenkin spacetime.\nIt is demonstrated that the outcomes deviate from the typica l Schwarzchild spacetime owing to the\nfundamental configuration with a constant dark energy densi ty.\nI. INTRODUCTION\nChandrasekhar [1] described the null geodesic struc-\nture of the Schwarzschild spacetime in his book on Black\nHoles. The author provides a detailed analysis of sev-\neral types of orbits, including radial geodesic orbits, crit-\nical orbits, orbits of the first kind, orbits of the second\nkind and orbits with imaginary eccentricities. The ac-\ncompanying pictures enhance the understanding of these\nconcepts.\nThe geodesic structure of the Schwarzschild spacetime\nhas also been addressed by Berti et al. [2]. In the present\ncontext, Majumder and colleagues [3] have provided a\ncomprehensive analysis of the temporal geodesic con-\nfiguration pertaining to the emergent Barriola-Vilenkin\nspacetime within the framework of k-essence. The ar-\nticle by Cruz et al. [4] gives an extensive review of the\nradial and non-radial trajectories associated with time-\nlike and null geodesics structures in the Schwarzschild\nanti-de Sitter spacetime.\nThe time-like and null geodesic structures with figures\nfor Bardeen spacetime have been shown in the ref. [5].\nThe solution of the Einstein equation governing the be-\n∗bivashmajumder@gmail.com\n†saibal.ray@gla.ac.in\n‡goutammanna.pkc@gmail.comhaviour of the fields outside the core of a monopole has\nbeen derived by Barriola and Vilenkin in 1989 [6]. The\nresulting black hole carries the global monopole charge\nwhere a global monopole falls under a Schwarzschild\nblack hole.\nThe following studies [7–13] offer an extensive inves-\ntigation of the k-essence theory, which proposes domi-\nnance of the kinetic energy over the potential energy. In\nthe work by the authors [14], it has been demonstrated\nthat the emergent gravity metric, denoted by ¯Gµν, ex-\nhibits conformal equivalence to the Barriola-Vilenkin\n(BV) metric [6] within the Schwarzschild background.\nThis equivalence is established for a specific configura-\ntion of the k-essence scalar field ( φ), which is based on\nthe Dirac-Born-Infeld (DBI) model [15–17]. Notably, in\nthis configuration, the global monopole charge replaced\nby the constant kinetic energy ( ˙φ2=K) of thek-essence\nscalar field. They have chosen a form of the Lagrangian\nasL=−V(φ)F(X) whereX=1\n2gµν∇µφ∇νφwith non-\ncanonical kinetic terms. The dynamical solutions of the\nk-essence equation of motion, which are not trivial, ex-\nhibit a spontaneous breaking of the Lorentz invariance\nand generate metric changes as a result of perturbations\nsurrounding these solutions. This behavior distinguishes\nk-essence from the relativistic field theories with canon-\nical kinetic components. The metric employed for these\nperturbations in the emergent or analogue curved space-\ntime differs from the conventional gravitational metric2\n[7–11].\nIn their recent publications, Manna et al. [18–27] have\nexploredarangeoftopics, includingthe thermodynamics\nof black holes, gravitational collapse, the relationship be-\ntweenk-essence and Vaidya geometry, features related to\nthe Raycahudhuri equation and modified theory of grav-\nity etc. These investigations have utilized the framework\nofk-essence geometry, whch has been shown to be con-\nsistent with the observational findings [28, 29].\nIt is important to remember that the conventional\n(canonical or standard) theories do not provide a com-\nprehensive understanding of every aspect of the physi-\ncal scenario. The provided information lacks clarity in\nexplaining the nature of dark matter, dark energy, the\nmechanisms behind the Big Bang, the disparity between\nmatter and antimatter, the cosmological constant prob-\nlem, the dimensions and configuration of the universe,\ncosmic inflation, the horizon problem, and other relevant\naspects within the realm of cosmos. The primary unre-\nsolved issue in the field of basic physics is the reconcilia-\ntion of gravity and quantum mechanics within a unified\ntheoretical framework. So, in this direction, there have\nbeen a lot of works initiated by several scientists, and\nstill work is to be done.\nIn this context, we shalldiscuss the significanceofnon-\nstandard theories such as the k-essence theory [30]. The\nk-essencemodelischaracterizedbythepresenceofanon-\ncanonical kinetic energy term in the Lagrangian, which\nhas the ability to produce cosmic acceleration at late\ntimes without relying on potential energy. An attrac-\ntor is defined as a point A that exerts an attractive force\non nearby points, causing them to gravitate towards it.\nThe categorization of attractor solutions for k-essence is\ndivided into two distinct categories, as shown in previous\nstudies [31–33]. The first one is trackersolution, in which\nk-essencemimicstheequationofstate(EOS)oftheback-\nground component in the Universe whereas in the second\nscenario, k-essence exhibits a preference towardsan EOS\nthat deviates from that of matter or radiation. As a\nresult of the attractor behavior, the k-essence model is\nalso insensitive to initial conditions. In contrast to the\nquintessence model, the k-essence field exclusively tracks\nthe radiation background, hence avoiding the need for\nfine-tuning that was present in the quintessence model.\nIn addition, the coincidence problem is resolved by the\npresence of an S-attractor that attracts shortly after the\nbeginning of the matter-dominated phase. However, the\nk-essence framework does not explain why vacuum en-\nergy is so small.\nNow we are going to explore the significance of the\nnon-canonical version of the Lagrangian within an alter-\nnative setting. The Lagrangian can be generally defined\nin either canonical or standard form as L=T−V, where\nTrepresents the kinetic energy and Vrepresents the po-\ntential energy of the system. However, as indicated by\nGoldstein and Rana [34, 35], the general form of the La-\ngrangian is non-canonical, whereas the canonical form is\nderived under specified conditions. The uniqueness ofthe functional form of Lis not guaranteed, as the Euler-\nLagrange equations of motion can be satisfied by many\nLagrangian choices [34, 35]. In addition, Raychaudhuri\n[36] highlights that when moving outside the realm of\nmechanics, the conventional notions of kinetic and po-\ntential energiesbecome inappropriate. Consequently, the\nequationL=T−Vloses its applicability. Since we al-\nready have the field equations and need to determine a\nLagrangian density to fix them, it may have started as a\nback calculation. Furthermore, it should be noted that\ntheclassicalnotionof L(=T−V)isnolongervalidwithin\nthe framework of special relativistic dynamics. Conse-\nquently, it may be asserted that the general form of the\nLagrangian is of a non-canonical nature [25].\nIn this work, our main motivation is thoroughly to\ninvestigate and trace out the null geodesic structures\nfor thek-essence emergent Barriola-Vilenkin (BV) type\nspacetime in the presence of dark energy based on the\nfollowing work [1], however not in the context of Jacobi\nmetric [37, 38].\nThe paper is organized as follows: In Section II, we\nhave briefly reviewed the k-essence geometry where the\nmetric¯Gµνcontains the dark energy field φ(i.e., the k-\nessence scalar field) which satisfies the emergent gravity\nequations of motion. In Section III, we have discussed\nabout the null geodesic structure for the k-essence emer-\ngent Barriola-Vilenkin (BV) type spacetime in the pres-\nence of dark energy and also traced out the trajectories\nby considering dark energy density in unit of critical den-\nsity [14, 18, 19] which is approximately 0 .7 [39–41]. The\nconclusion of our work is in the last Section IV.\nII. K-ESSENCE THEORY\nThe action ofthe k-essencegeometryis givenby [7–13]\nSk[φ,gµν] =/integraldisplay\nd4x√−gL(X,φ), (1)\nwhereX=1\n2gµν∇µφ∇νφand the energy-momentum\ntensor is\nTµν≡2√−gδSk\nδgµν=LX∇µφ∇νφ−gµνL,(2)\nwhereLX=dL\ndX,LXX=d2L\ndX2,Lφ=dL\ndφand∇µis the\ncovariant derivative defined with respect to the gravita-\ntional metric gµν.\nThe scalar field equation of motion (EOM) is [8, 9, 11]\n−1√−gδSk\nδφ=˜Gµν∇µ∇νφ+2XLXφ−Lφ= 0,(3)\nwith\n˜Gµν≡ LXgµν+LXX∇µφ∇νφ, (4)\nand 1+2XLXX\nLX>0. HereLX/negationslash= 0 forc2\nsto be positive\ndefinite.3\nUsing the conformal transformations Gµν≡cs\nL2x˜Gµν\nand¯Gµν≡cs\nLXGµν, withc2\ns(X,φ)≡(1+2XLXX\nLX)−1we\nhave [14, 18–22]\n¯Gµν=gµν−LXX\nLX+2XLXX∇µφ∇νφ. (5)\nIfLis not an explicit function of φthen the EOM (3)\nis reduces to\n−1√−gδSk\nδφ=¯Gµν∇µ∇νφ= 0. (6)\nIt should be noted that in the case of non-trivialspace-\ntime configurations of the field φ, the resulting metric\n¯Gµνdoes not generally exhibit conformal equivalence to\nthe metric gµν. The scalar field φexhibits distinct fea-\ntures compared to canonical scalar fields, and its local\ncausal structure differs from those described by the met-\nric tensorgµν.\nWe consider the DBI type Lagrangian as [14–19]\nL(X,φ) = 1−V(φ)√\n1−2X, (7)\nforV(φ) =V= constant and kinetic energy of φ >>\nV, i.e., (˙φ)2>> V. The presented Lagrangian form is\ncommonly seen in the context of k-essence fields, where\nthe dominance of kinetic energy over potential energy is\nprominent. Then c2\ns(X,φ) = 1−2X. For scalar fields\n∇µφ=∂µφ. Thus the effective metric (5) is\n¯Gµν=gµν−∂µφ∂νφ. (8)\nThe geodesic equation that corresponds to the k-\nessence theory, expressed in terms of the new Christof-\nfel connections denoted as ¯Γ, may be written as follows\n[14, 18, 19]:\nd2xα\ndλ2+¯Γα\nµνdxµ\ndλdxν\ndλ= 0, (9)\nwhereλis an affine parameter and\n¯Γα\nµν= Γα\nµν−1\n2(1−2X)/bracketleftBig\nδα\nµ∂νX+δα\nν∂µX/bracketrightBig\n.(10)\nIn k-essence geometry, it is worth noting that Ein-\nstein’s field equation can be expressed as\n¯Gµν=¯Rµν−1\n2¯Gµν¯R=κ¯Tµν, (11)\nwhereκ= 8πGis constant, ¯Rµνis Ricci tensor, ¯R(=\n¯Rµν¯Gµν) is the Ricci scalar and ¯Tµνis the energy-\nmomentum tensor of this geometry. The emergent\nenergy-momentum tensor ( ¯Tµν) can be determined to\nsolve the left-hand side of the emergent Einstein field\nequation (11).III. NULL GEODESICS FOR THE BARRIOLA\nVILENKIN TYPE EMERGENT SPACETIME\nThe authors [14] have demonstrated that the emergent\nmetric¯Gµν(8) precisely correlated with the Barriola-\nVilenkin (BV) metric, given a certain form of the k-\nessence scalar field. This correlation is obtained when\nthe standard gravitational metric gµνis assumed to be\nSchwarzschild. The global monopole charge has been\nsubstituted with the constant kinetic energy of the scalar\nfield.\nThek-essence emergent BV [14] type metric is\nds2= (1−2GM\nr−K)dt2−1\n(1−2GM\nr−K)dr2\n−r2dθ2−r2sin2θdΦ2\n= (β−2GM\nr)dt2−dr2\n(β−2GM\nr)−r2dθ2\n−r2sin2θdΦ2, (12)\nwhereKis the constant kinetic energy (i.e., the dark\nenergy density in unit of critical density [14, 18, 19]) of\nthek-essence scalar field and we define β= (1−K).\nThe range for Kis 0< K < 1. The expression for\nthek-essence scalar field is φ(r,t) =φ1(r) +φ2(t) =√\nK[r+ 2GM ln(r−2GM)] +√\nKt. The kinetic part\nof this field is ˙φ2≡˙φ2\n2=K. If we consider K= 0\nthen the k-essence theory is meaningless and if K= 1,\nthen (12) does not have a Newtonian limit [42]. Also\nif we consider K >1 then the signature of the above\nmetric(12)isill-definedandalsowehavethetotalenergy\ndensity (Ω matter+Ωradiation+Ωdark energy = 1) cannot\nexceed unity [14]. It is also worth mentioning that the k-\nessence emergent BV metric (12) can be found by solving\nthe emergent Einstein field equation (11).\nThe investigation of the geodesics equation in the\nemergent spacetime of the Barriola-Vilenkin type may\nbe accomplished by deriving it from the following La-\ngrangian (as discussed in [1–4]):\n2L= (β−2GM\nr)˙t2−1\n(β−2GM\nr)˙r2\n−r2˙θ2−(r2sin2θ)˙Φ2(13)\nwhere˙t=dt\ndτ,˙r=dr\ndτ,˙θ=dθ\ndτ,˙Φ =dΦ\ndτ,τis the\nproper time.\nTherefore, momenta are\npt= (β−2GM\nr)˙t, pr=1\n(β−2GM\nr)˙r,\npθ=r2˙θ, pΦ=r2sin2θ˙Φ (14)4\nand the Hamiltonian is\nH=pµ˙xµ−L\n=pt˙t−(pr˙r+pθ˙θ+pΦ˙Φ)−L\n=L. (15)\nFor spherically symmetric nature of the metric, here\nthe Lagrangian does not depend on tand Φ. So the\nequations of motions are ˙ pt= 0,˙pΦ= 0 which implies\nthat [3]\n/parenleftBig\nβ−2GM\nr/parenrightBig\n˙t= constant = E(say),(16)\nr2sin2θ˙Φ = constant , (17)\nd\ndτ/parenleftBig\nr2˙θ/parenrightBig\n= (r2sinθcosθ)˙Φ2. (18)\nIn [3], they also have considered that there is no con-\ntribution of potential energy since in the k-essence the-\nory, thecontributionofthe kineticenergypartdominates\nover the potential energy i.e., K.E. >> P.E. . Now to\nsimplify, we consider the motion in the equatorial plane\nθ=π\n2. Using the Eq. (17), we have\npΦ=r2˙Φ = constant = L(say), (19)\nwhereLis the angular momentum about an axis normal\nto the invariant plane. Finally using the Eqs. (16), (19)\nand (13) the Lagrangian becomes\n2L=E2\nβ−2M\nr−˙r2\nβ−2M\nr−L\nr2. (20)\nNow for time-like geodesic we consider 2 L= +1 and\nfor null geodesics 2 L= 0 . Here we only concentrate on\nthe null geodesics.\nA. Null Geodesic\nPuttingL= 0 in Eq. (20), we get\n/parenleftBigdr\ndτ/parenrightBig2\n+L2\nr2/parenleftBig\nβ−2M\nr/parenrightBig\n=E2, (21)\nwith\n/parenleftBig\nβ−2M\nr/parenrightBigdt\ndτ=EanddΦ\ndτ=L\nr2, (22)\nby using Eqs. (16) and (19). Now substituting r=1\nuin\nEq. (21), we obtain\n/parenleftBigdu\ndΦ/parenrightBig2\n= 2Mu3−βu2+1\nD2=f(u) (say),(23)\nandD=L\nE(say) which denotes the impact parameter.\nTo find different orbits, let us start with the equation\nf(u) = 2Mu3−u2+1\nD2= 0. (24)Let the roots of the above equation are u1,u2andu3\nthen we have\nu1+u2+u3=β\n2M, (25)\nu1u2+u2u3+u1u3= 0, (26)\nu1u2u3=−1\n2MD2. (27)\nNow using Descartes’ Rule of Sign, the Eq. (24) must\nhave one negative root, since M >0 andD2>0 and\nthe other two roots are real or complex. For simplicity\nwe assume that u1<0. In this situation one may note\nthat therearisethree differentcases, which areasfollows:\nCase-A: One root is negative and the others two roots\nare positive and equal, i.e., u2=u3. In this case we have\ntwo types of orbits: one is in the interval 0 M as\nK <1 and neglecting the solution u= 0.\nTherefore, we have\nu2=u3=1\n3¯M. (28)\nNowu2andu3are the roots of the Eq. (24), then we\nhave\nD=3√\n3¯M√β. (29)5\nHere we can see that the impact parameter Dfor null\ngeodesics in the presence of dark energy density is much\nhigher than the impact parameter for null geodesics in\nthe usual Schwarzschild background [1].\nNow substituting Eq. (28) in Eq. (25), we get\nu1=−1\n6¯M, u2=u3=1\n3¯MwhenD=3√\n3¯M√β(30)\nTherefore from Eq. (23)\ndu\ndΦ=−/radicalbigg\n2M(u+1\n6¯M)/parenleftBig\nu−1\n3¯M/parenrightBig\n.(31)\nHere we take the negative sign in the RHS, so that Φ\nmay increase. Now solving the above Eq. (31), we get\nu=−1\n6¯M+1\n2¯Mtanh2√β\n2(Φ−Φ0),(32)\nwhere Φ 0is a constant of integration. Again, Φ 0can be\nconsidered such a way that\ntanh2Φ0√β\n2=1\n3\nthen\nu= 0 when Φ = 0\nandu=1\n3¯Mwhen Φ→ ∞. (33)\nThus the orbits of first kind with the impact parame-\nterD=3√\n3√β¯Marrives from infinity and asymptotically\napproaches the circle r= 3¯Mas shown in Fig. 1.\nAgain, for the orbits of second kind, we substitute\nu=1\n3¯M1\n2¯Mtan2ξ\n2, (34)\nin Eq. (31), we get\ndξ\ndΦ=/radicalbig\nβsinξ\n2. (35)\nIn the above equation we consider the positive sign in\nthe RHS so that we may Φ increase. Now integrating Eq.\n(35) we get\nΦ = 2/radicalbig\nβlogtanξ\n4, (36)\nand therefore using the Eqs. (34) and (35)\nu=1\n3¯M+1\n¯M2eΦ√β\n/parenleftBig\neΦ√β−1/parenrightBig2, (37)\nwhich gives\nu→ ∞when Φ→0\nandu=1\n3¯Mwhen Φ→ ∞. (38)\nFIG. 1. The critical orbits of the first and second kind for\nthe impact parameter D=Dc=3√\n3¯M√βand for both cases\nM=9\n140andβ= 0.3.\nThereforestartingthe apheliondistance r= 3¯M, these\norbits of second kind plunges to singularity ( r= 0) as\nshown in Fig. 1 where M=9\n140andβ= 0.3.\nAt this juncture let us discuss about the cone of avoid-\nancewhich at any point can be defined by the solution\nof Eq. (32) whose generators are null rays which passing\nthrough that point. We will establish in Case-C that the\nlight rays included in the cone are getting trapped after\ncrossing the horizon. Let Ψ be the half angle of the cone\nthen\ncotΨ =1\nrdr′\ndΦ, (39)\nwhere\ndr′=1/radicalBig\nβ−2M\nrdr, (40)\nwheredr′denotes an element of the proper length along\nthe generators of the cone.\nUsing Eqs. (39) and (40), we get\ndu\ndΦ=−u/radicalbig\nβcotΨ/radicalbig\n1−2¯Mu. (41)\nAgain substituting Eq. (41) in Eq. (31), we get\ncotΨ =/parenleftBig\nr\n3¯M−1/parenrightBig/radicalbigg/parenleftBig\nr\n6¯M+1/parenrightBig\n/radicalbigg/parenleftBig\nr\n2¯M−1/parenrightBig, (42)6\nwhich gives\nΨ∼3√\n3\nrwhenr→ ∞,\nΨ =π\n2whenr= 3¯M,\nand Ψ = 0 when r= 2¯M. (43)\nTherefore the cone of avoidance become narrower di-\nrected inward when r >3¯Mand the cone spread out\nfully atr= 3¯Mand directed outward when r <3¯M\nand again narrower when r→2¯M(see Fig. 2). This\nfigure is quite different in the presence of dark energy\ndensityK= 0.7 [28, 29, 39, 40] from usual Schwarzschild\nSpacetime [1] where the respective three case arise when\nr>3M,r= 3Mandr<3Msince¯M >M asβ= 0.3.\nFIG. 2. Cone of Avoidance at various distances\n2.Case-B : Orbits of the First and Second Kind\nHere we consider the second case where the one root\nis negative and the others two roots are positive and dis-\ntinct. Let us assume that u1<0, 00, (47)and foru1(P−6¯M)2, (50)\nSimplifying, we get\nP >3¯M. (51)\nSo, from the Eq. (49), we have\nD>3√\n3¯M√β=Dc(say). (52)\nTherefore, we can say that these kind of orbits are\nfound when the impact parameter is greater than3√\n3¯M√β.\nAgain, from Eqs. (24) and (44), we write\nf(u) = 2M/parenleftBig\nu−u1/parenrightBig/parenleftBig\nu−u2/parenrightBig/parenleftBig\nu−u3/parenrightBig\n.(53)\nSubstituting\nu=1\nP−Q−P+6¯M\n8¯MP(1+cosξ) (54)\nin Eq. (53) and using Eq. (23), we get\ndξ\ndΦ=/radicalbigg/parenleftBigQβ\nP/parenrightBig/radicalbigg\n1−k2sin2ξ\n2, (55)\nwherek2=Q−P−6¯M\n2Q.\nIn this above equation we consider the positive sign to\nkeep Φ increasing. Now integratingthe equation (55), we\nget\nΦ = 2/radicalBigg/parenleftBigP\nQβ/parenrightBig/bracketleftBig\nK(k)−F/parenleftBigξ\n2,k/parenrightBig/bracketrightBig\n, (56)\nwhereF/parenleftBig\nξ\n2,k/parenrightBig\nis the incomplete elliptic integral of the\nfirst kind and K(k) is the complete elliptic integral of the\nfirst kind. Therefore\nK(k) =/integraldisplayπ\n2\n0dz/radicalbig\n1−k2sin2z,\nF/parenleftBigξ\n2,k/parenrightBig\n=/integraldisplayξ\n2\n0dz/radicalbig\n1−k2sin2z,(57)\nwithξ\n2=z.7\nThus we have\nu=1\nPand Φ = 0 when ξ=π,\nu→0 and Φ →Φ∞whenξ=ξ∞,(58)\nwhere\nΦ∞= 2/radicalBigg/parenleftBigP\nQβ/parenrightBig/bracketleftBig\nK(k)−F/parenleftBigξ∞\n2,k/parenrightBig/bracketrightBig\n,\nand sin2ξ∞\n2=Q−P+2¯M\nQ−P+6¯M. (59)\nThus the range of ξisξ∞<ξ<π. So we can conclude\nthat starting from infinity (when Φ →Φ∞), the orbits of\nfirst kind asymptoticallyapproachesto r=P(when Φ =\n0) by spiralling around it. Now again in the presence of\ndark energy density K= 0.7, these ranges of Φ and rare\ndifferent since ¯M > M and similar in their orientation.\nIn Fig. 3, we have traced the orbits of first kind by\nconsidering P= 1.5,M=9\n140withβ= 0.3.\nTo obtain the orbits of the second kind, let us substi-\ntute\nu=1\nP+Q+P−6¯M\n4¯MPsec2χ\n2. (60)\nThen from the Eq. (23), we get\ndχ\ndΦ=/radicalbigg\nQβ\nP/bracketleftBig\n1−k2sin2χ\n2/bracketrightBig\n. (61)\nIn the above form we consider the positive sign in the\nRHS so that Φ may increase. Now integrating (61), one\nmay get\nΦ = 2/radicalBigg/parenleftBigP\nQβ/parenrightBig\nF/parenleftBigχ\n2,k/parenrightBig\n. (62)\nTherefore\nu=Q+P−2¯M\n4¯MPand Φ = 0 when χ= 0,\nu→ ∞and Φ = K(k) whenχ=π. (63)\nHere we have the range of χis 0< χ < φ . Thus\nstarting from the aphelion distance r=4¯MP\nQ+P−2¯Mwhen\nχ=π, the orbit of second kind plunges to the singularity\n(r= 0) see Fig. 3 where we consider M=9\n140andβ=\n0.3. Again the orientation of the orbits are similar with\ndifferent values in the presence of dark energy density\nK= 0.7 as we have seen in [1].\n3.Case-C : Orbits of with imaginary eccentricities\nand impact parameters less than 3√\n3¯M√β\nFinally, we are now discuss the nature of the orbits\nwith imaginary eccentricities ( ie) that is the two roots of\nFIG. 3. The orbits of the first and second kind for P= 1.5\nand for both cases M=9\n140andβ= 0.3.\nthe Eq. (24) are imaginary and the other one is negative.\nTo start with let us consider the roots of the Eq. (24) in\nthe form\nu1=1\n2¯M−2\nl, u2=1\nl(1+ie)and u 3=1\nl(1−ie).\n(64)\nNote that here we consider e >0, then from (23), we\nget\nf(u) = 2Mu3−βu2+2M/parenleftBige2−3\nl2+1\n2¯Ml/parenrightBig\nu\n−2M1+e2\nl2/parenleftBig1\n2¯M−2\nl/parenrightBig\n. (65)\nNow comparing this with the Eq. (23), we have\nl−¯M(3−e2) = 0, (66)\n1\n2MD2=/parenleftBig2\nl−1\n2¯M/parenrightBig1+e2\nl2.(67)\nTaking\nµ=M\nlβ(68)\nand then from Eq. (66), we get\ne2=3µ−1\nµ(69)\nand from Eq. (67)\nD2\n¯M2=1\nµβ(4µ−1)2. (70)\nSincee2>0, we have\nµ>1\n3(71)8\nand from the Eq. (70)\nD<3√\n3¯M√β. (72)\nThus to obtain these kind of orbits we must have the\nimpact parameter Dmust less than3√\n3¯M√βandµmust\ngreater than1\n3.\nNow if we consider the substitution\nu=1\nl/parenleftBig\n1+etanξ\n2/parenrightBig\n(73)\nin Eq. (23), then we have\n/parenleftBigdξ\ndΦ/parenrightBig2\n= 2β/bracketleftBig\n(6µ−1)cosξ+2µesinξ+(6µ−1)/bracketrightBig\n.\n(74)\nTo simplify things we substitute\nsin2ψ=1\n∆+6µ−1/bracketleftBig\n∆−2µesinξ−(6µ−1)cosξ/bracketrightBig\n(75)\nand then differentiating, we get\n/parenleftBigdψ\ndξ/parenrightBig2\n=∆\n2(∆+6µ−1)cos2ψ/bracketleftBig\n1−k2sin2ψ/bracketrightBig\n(76)\nwhere\nk2=∆+6µ−1\n2∆(77)\nand\n∆2= (6µ−1)2+4µ2e2. (78)\nCombining Eqs. (74) and (76)\n/parenleftBigdΦ\ndψ/parenrightBig2\n=1\n∆β(1−k2sin2ψ). (79)\nIntegrating (79), we get\nΦ =1√∆β/bracketleftBig\nK(k)−F/parenleftBig\nψ,k/parenrightBig/bracketrightBig\n. (80)\nTherefore:\nwhen ξ =π, u→ ∞andψ=−π\n2,π\n2\nand when ξ →ξ∞, u→0and ψ→sin−1∆+1\n∆+6µ−1\nwhere ξ ∞= 2tan−11\ne.\nThereforethese kind oforbits arearrivingfrom infinity\nwhenξ→ξ∞and plunging to singularity r= 0 when\nξ=π. In Figs. 4 and 5, we have traced the trajectories\nwithe= 0.0141i,l= 2.9998 ande= 0.01i,l= 5.9998\nrespectively with M= 0.3 andβ= 0.3.\nFIG. 4. The orbits of imaginary eccentricities with the impa ct\nparameter D < D c=3√\n3¯M√βfore= 0.0141i,l= 2.9998 and\nM= 0.3,β= 0.3\nFIG. 5. The orbits of imaginary eccentricities with the impa ct\nparameter D < D c=3√\n3¯M√βfore= 0.01i,l= 5.9998 and\nM= 0.3,β= 0.3\nB. Radial Geodesic\nFor radial geodesic the angular momentum is to be\nzero, therefore from the Eq. (21), we have\ndr\ndτ=±E (81)\nand from the Eq. (22)\n/parenleftBig\nβ−2M\nr/parenrightBigdt\ndτ=E\n⇒t=±r∗+constant ±, (82)9\nwhere\nr∗=1\nβ/bracketleftBig\nr+2¯Mlog/parenleftBigr\n2¯M−1/parenrightBig/bracketrightBig\n. (83)\nTherefore\nd\ndr∗=∆\nr2d\ndr, (84)\nwhere ∆ = βr2/parenleftBig\n1−2¯M\nr/parenrightBig\nis the horizon function.\nAgain, from Eq. (83)\nr→2¯M+0⇒r∗→ −∞ (85)\nand→ ∞ ⇒r∗→r\nβ, (86)\nand from Eq. (81),\nr=±Eτ+constant ±. (87)\nThisshowsthatthe radialgeodesictakesaninfinite co-\nordinatetime toarriveat the horizonforanobserverout-\nside the horizon even though the radial geodesic crosses\nthe horizon in its own proper time. This result is quite\nsimilar as in the Schwarzschild spacetime [1] since βis a\nconstant.\nIV. CONCLUSION\nAll conceivablepaths fornull geodesicsin the Barriola-\nVilenkin spacetime arising from k-essence have been\nsystematically determined. According to the scholarly\nwork authored by Chandrasekhar [1], the presence of\ndark energy density significantly alters the ranges seen\nin Schwarzschild spacetime, as described in the book.\nThe determination of critical orbits is possible by iden-\ntifying the condition under which the two roots of Eq.\n(24) are both positive and equal. In the case of critical\norbits of the first kind, it has been shown that when ini-\ntiated from an infinite distance, the orbits gradually ap-\nproach the value of r= 3¯Mby spiralling around it. On\nthe other hand, critical orbits of the second kind com-\nmence atr= 3¯Mand converge towards the singularity\nlocated at r= 0. In both cases the radius of the cir-\ncler= 3¯Mare much higher than the than radius thecircler= 3Mwhich can be seen in usual Schwarzschild\nspacetime. In our case the cone of avoidance, we found\nthat that the cone opens out fully when r= 3¯Mwhich\nis higher than usual Schwarzschild spacetime where it\nopens out fully when r= 3M.\nBy considering the two roots of the Eq. (24) are pos-\nitive and distinct, we have traced the orbits of the first\nand second kind where we found that the orbits of the\nfirst kind are arriving from infinity and asymptotically\napproaches to r=P(perihelion distance) and the orbits\nof the second kind are starting from r=4¯MP\nQ+P−2¯Mand\nplunge to the singularity. Here also we note that these\nkinds of orbits can be found only when the impact pa-\nrameter is greater than Dc=3√\n3¯M√β. It should be noted\nthat the value of Dcfor this case is much much greater\nthan theDcin the Schwarzschild Spacetime [1].\nWhen the impact parameter is less than Dc=3√\n3¯M√β,\ntheorbitsofimaginaryeccentricitiescanbetraced,where\narriving from infinity these kinds of the orbits are ap-\nproaching to singularity r= 0.\nForthe caseofradialgeodesics, we haveevaluated that\nthese kinds of geodesics take infinite co-ordinate time to\narrive at the horizon for an observer outside the horizon,\nalthough they crossthe horizonin their own proper time,\nwhich is similar to Schwarzschild spacetime.\nFinally, we conclude that although the orientation of\ntrajectories are quite similar for both the Schwarzschild\nspacetime and the k-essence emergent Barriola-Vilenkin\nspacetime but the ranges are much more different and\nthe value of the Dcis much higher due to the presence\nof dark energy density K.\nACKNOWLEDGMENTS\nG.M. acknowledges the DSTB, Government of West\nBengal, India for financial support through Grant Nos.\n856(Sanc.)/STBT-11012(26)/6/2021-ST SEC dated 3rd\nNovember 2023. S.R. is thankful to the Inter-University\nCentre for Astronomy and Astrophysics (IUCAA), Pune,\nIndia for providing Visiting Associateship under which a\npart of this work was carried out who also gratefully ac-\nknowledges the facilities under ICARD, Pune at CCASS,\nGLA University, Mathura.\n[1] S. Chandrasekhar, The Mathematical Theory of Black\nHoles, Chap. 3, Sec. 20, Ind. Edit. 2010, Oxford Univ.\nPress (1992).\n[2] E. Berti, A Black-Hole Primer: Particles, Waves,\nCritical Phenomena and Superradiant Instabilities ,\narXiv:1410.4481 (2014).\n[3] Majumder et al., Class. Quantum Grav. 37, (2020),\n115002.[4] N. Cruz et al., Class. Quantum Grav. 22, (2005),\n1167–1190.\n[5] S. Zhou et al., Int. Jour. Mod. Phys. D 21, (2012),\n1250077.\n[6] M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, (1989),\n341.\n[7] M. Visser et al., Gen. Rel. Grav. 34, (2002), 1719.\n[8] E. Babichev et al., Jour. H. E. Phys. 09, (2006), 061.\n[9] E. Babichev et al., Jour. H. E. Phys. 0802, (2008), 101.10\n[10] E. Babichev et al., WSPC-Proceedings, February 1\n(2008).\n[11] A. Vikman, K-essence: Cosmology, causality and\nEmergent Geometry , Dissertation an der Fakultat\nfur Physik,Arnold Sommerfeld Center for Theoretical\nPhysics, der Ludwig-Maximilians-Universitat Munchen,\nMunchen, den 29.08.2007 (2007).\n[12] R. J. Scherrer, Phys. Rev. Lett. 93, (2004), 011301.\n[13] L. P. Chimento, Phys. Rev. D 69, (2004), 123517.\n[14] D. Gangopadhyay and G. Manna, Euro. Phys. Lett. 100,\n(2012), 49001.\n[15] M. Born and L. Infeld, Proc. Roy. Soc. Lond A 144,\n(1934), 425.\n[16] W. Heisenberg, Zeit. Physik: A Hadrons and Nuclei 113,\n(1939), no.1-2.\n[17] P. A. M. Dirac, Proc. Roy. Soc. Lond. A 268, (1962), 57.\n[18] G. Manna and D. Gangopadhyay, Eur. Phys. J. C 74,\n(2014), 2811.\n[19] G. Manna and B. Majumder, Eur. Phys. J. C 79, (2019),\n553.\n[20] G. Manna et al., Eur. Phys. J. Plus 135, (2020), 107.\n[21] G. Manna et al., Phys. Rev. D 101, (2020), 124034.\n[22] G. Manna, Eur. Phys. J. C 80, (2020), 813.\n[23] S. Ray et al., Chinese Phys. C 46, (2022), 12, 125103.\n[24] G. Manna et al., Chinese Phys. C 47, (2023), 025101.\n[25] S. Das et al., Fortschr. Phys. 71, (2023), 2200193.\n[26] B. Majumder et al., Fortschr. Phys. 71, (2023), 2300133.\n[27] B. Majumder et al, Universe 9, (2023), 510.\n[28] P. A. R. Ade et al. (Planck Collaboration), Astro. & As-\ntrophys. 594, (2016), A14.[29] N. Aghanim et al. (Planck 2018 results. VI. Cosmologica l\nparameters, Planck Collaboration), Astro. & Astrophys.\n641, (2020), A6.\n[30] J. Yoo and Y. Watanabe, Int. Jour. Mod. Phys. D 21,\n(2012), 12, 1230002.\n[31] C. Armendariz-Picon et al., Phys. Rev. Lett. 85, (2000),\n4438.\n[32] C. Armendariz-Picon et al., Phys. Lett. B 458, (1999),\n2–3, 209-218.\n[33] J. U. Kang et al., Phys. Rev. D 76, (2007), 083511.\n[34] H. Goldstein, Classical Mechanics , 2nd Edition, Narosa\nPublishing House, India (2000).\n[35] N. C. Ranaand P. S. Joag, Classical Mechanics , McGraw\nHill Education (India) Private Limited, (1991).\n[36] A. K. Raychaudhuri, Classical Mechanics: A course of\nLectures, Oxford University Press (1983).\n[37] G. W. Gibbons, Class. Quantum Grav. 33, (2016),\n025004.\n[38] S. Chanda et al., Jour. Math. Phys. 58, (2017), 032503.\n[39] P. A. R. Ade et al. (Planck Collaboration), Astro. & As-\ntrophys. 571, (2014), A1.\n[40] P. A. R. Ade et al. (Planck Collaboration), Astro. & As-\ntrophys. 571, (2014), A16.\n[41] Y. Akrami et al. (Planck 2018 results. I. Overview and\nthecosmological legacy ofPlanck, PlanckCollaboration),\nAstro. & Astrophys. 641, (2020), A1.\n[42] B.F. Schutz, A First Course in General Relativity, Chap.\n8, sect. 4 , Cambridge University Press, Cambridge,\n(1985)." }, { "title": "2401.17676v1.Observer_based_Controller_Design_for_Oscillation_Damping_of_a_Novel_Suspended_Underactuated_Aerial_Platform.pdf", "content": "Observer-based Controller Design for Oscillation Damping of a Novel\nSuspended Underactuated Aerial Platform\nHemjyoti Das1, Minh Nhat Vu1,3, Tobias Egle1and Christian Ott1,2\nAbstract — In this work, we present a novel actuation strategy\nfor a suspended aerial platform. By utilizing an underactuation\napproach, we demonstrate the successful oscillation damping\nof the proposed platform, modeled as a spherical double\npendulum. A state estimator is designed in order to obtain the\ndeflection angles of the platform, which uses only onboard IMU\nmeasurements. The state estimator is an extended Kalman filter\n(EKF) with intermittent measurements obtained at different\nfrequencies. An optimal state feedback controller and a PD+\ncontroller are designed in order to dampen the oscillations of\nthe platform in the joint space and task space respectively. The\nproposed underactuated platform is found to be more energy-\nefficient than an omnidirectional platform and requires fewer\nactuators. The effectiveness of our proposed system is validated\nusing both simulations and experimental studies.\nI. I NTRODUCTION\nAerial robotic manipulation is a modern field of research\nthat involves manipulation performed using a flying base\n[1]. In recent years, this field of aerial robotic manipulation\nhas evolved significantly and has been utilized in numerous\napplications related to aerial inspection, construction, and\nload transportation [1]–[5]. Research on aerial manipulation\nis often centered around an unmanned aerial vehicle (UA V)\nintegrated with a robotic arm. In [6], [7], a fully actuated\nUA V is mounted with a 3 degrees of freedom (DoF) robotic\narm in order to perform collaborative human-handover tasks.\nAn impedance-based grasping task was demonstrated in [8]\nusing a UA V fitted with a 7-DoF Kuka robotic arm.\nThe manipulator that can be mounted on the aerial vehicle\nfor performing different tasks is often constrained by its\nweight. For instance, an ultra lightweight arm [9] of 3.2 kg\nwas developed and mounted on a medium-size multirotor\nplatform in order to perform manipulation tasks on power-\nlines. However, in order to mount a 15 kg manipulator arm,\na large scale helicopter system with a rotor-blade diameter\nof 3.7 meters was needed [10]. Such large dimensional\nrotors might be a concern for safety while operating in some\ncomplex environments. Moreover, the turbulence generated\nwhile operating in close proximity to the ground can be an\nadditional cause of concern for the safety of the aerial system\nand its surroundings. A lot of energy is often consumed to\ncompensate for gravity in aerial manipulator systems, which\n1Automation and Control Institute (ACIN), TU Wien, Gusshausstraße\n27-29, 1040, Vienna, Austria {hemjyoti.das, minh.vu,\ntobias.egle,christian.ott }@tuwien.ac.at\n2Institute of Robotics and Mechatronics, German Aerospace Center\n(DLR), Oberpfaffenhofen, Muenchener Strasse 20, 82234, Wessling, Ger-\nmany\n3Austrian Institute of Technology (AIT) GmbH, 1210, Vienna, Austria\nPropulsion UnitFlight Controller Unit Suspension Cables\nFig. 1: Proposed planar-thrust suspended aerial platform.\ncan limit the platform’s payload capacity and its total time\nof flight (ToF).\nIn order to mitigate the aforementioned issues, a sus-\npended aerial manipulator (SAM) was proposed at the DLR\n[11]. It consists of an omnidirectional multirotor system\nequipped with a 7-DoF Kuka robotic arm, which is sus-\npended from a carrier using a long cable. The gravity\ncompensation due to the suspension reduces the energy\nconsumption to a great extent and also ensures that the\nweight of the manipulator is not a limiting factor for its\ndesign. The SAM platform has been successfully utilized\nfor demonstrating a number of hierarchical task-priority\ncontrol applications [12]–[14]. In [15], a similar system was\nproposed which utilizes elastic suspension and can generate\na 6-DoF wrench. This system was further augmented with a\nnonlinear model predictive control (NMPC) and a computed\ntorque controller to assess its performance [16], [17].\nOne of the main challenges of such suspended aerial sys-\ntems is the pendulum motion caused due to the suspension.\nThe oscillation of the aerial base can arise due to its own\nmotion, or due to the motion of the manipulator. Other\nfactors such as wind gusts and other external disturbances\ncan also contribute to its oscillatory motion. The control of\noscillatory motion in free-flying multirotors with a cable-\nsuspended load has been studied previously in [18]–[21].\nHowever, the oscillatory motion in cable-suspended aerial\nplatforms requires different damping techniques and has\nbeen studied in [11], [22]. In [22], the double oscillations\nof the platform have been damped using optimal feedback\ncontrol. It utilizes a simplified model of a planar double\npendulum and considers the second oscillation angle to be\napproximately zero while framing the control law. Moreover,arXiv:2401.17676v1 [cs.RO] 31 Jan 2024the platform used in [11] and [14] have eight propulsion units\nto achieve omni-directionality which consumes additional\nenergy than an underactuated configuration. The damping\ncontrol of suspended platforms in [15]–[17] utilizes extero-\nceptive sensor feedback using motion-capture systems, which\nis unfavorable for outdoor environments.\nIn this paper, we propose two different actuation concepts:\na planar-thrust and a minimal-actuated configuration of a\nsuspended aerial platform, which is the main contribution\nof this work. These novel designs present a minimalist\napproach, as compared to some of the previous related\nworks [11], [14], [15]. By utilizing a minimalist approach,\nthe proposed system is found to be more energy-efficient\nand it requires less number of actuators, while successfully\ndamping the oscillations of the spherical double pendulum-\ntype system. A secondary contribution of this paper is the\ndesign of an extended Kalman filter (EKF) for the estimation\nof the oscillation angles of the suspended platform. The EKF\nrelies on intermittent observations from its onboard sensors\nthat are published at different frequencies. Finally, we discuss\nseveral control approaches for the proposed platforms. We\ndesign an optimal state feedback controller, which allows\ndamping of the oscillations of the platform. Additionally, a\nPD+ controller is also designed in order to control the task-\nspace dimensions. The proposed designs are evaluated using\nboth computer simulations and experimental tests.\nThe rest of the paper is organized as follows. Section II\npresents an overview of the suspended aerial system and\nexplains its dynamics. The observer and control law are\npresented in Section III and IV, respectively. The validation\nresults are presented in Section V. Finally, the conclusions\nand ideas for future work are given in Section VI.\nII. S YSTEM OVERVIEW AND MODELLING\nIn this section, we first introduce the system dynamics,\nfollowing which we review the omnidirectional suspended\naerial platform in section II-B. The novel underactuation\napproaches for the platform are then proposed in section II-C\nand II-D.\nA. System Dynamics\nWe model the suspended platform as a spherical double\npendulum as depicted in Fig. 2. The first spherical joint can\nbe decomposed into three rotational joints q1,q2andq3,\nwhich are the rotation of the first link with length L1about\nitsx,y, and zaxis, respectively. The second spherical joint\ncan be expressed as the rotations q4andq5about its xandy\naxis, respectively, with the length of the second link denoted\nbyL2. There is, however, no rotation about the zaxis for\nthe second joint due to mechanical constraints imposed due\nto the suspension, while assuming all the cables to behave\nas rigid links. The double pendulum system has two modes\nof oscillation, which are the low-frequency high-amplitude\nmode for the first spherical joint, whereas the second joint\nis dominated by high-frequency low-amplitude mode [11],\n[22]. The joint dynamics can be summarized as\nM(q)¨ q+C(q,˙ q)˙ q+g(q) =τ, (1)\n[q1, q2, q3]T\nL1\nL2z\nxy\n[q4, q5]T\nFig. 2: Suspended aerial platform represented as a spherical\ndouble pendulum.\nwhere q=\u0002q1q2q3q4q5\u0003Tis the configuration\nvector, M(q)is the inertia matrix and C(q,˙ q)˙ qis the\ncentrifugal/Coriolis term. g(q)is the gravity vector and τ\nis the torque applied at the joints. The torques exerted at\nthe joints τare related to the body wrenches uusing the\nJacobian J(q)asτ=J(q)Tu. Next, we transform these\nsystem dynamics to state-space form by defining the state\nvector as x=\u0002q ˙ q\u0003T. The nonlinear state-transition matrix\ncan be then expressed as\n˙ x=f(x,u) =\"˙ q\nM(q)−1\u0010\nJ(q)Tu−C(q,˙ q)˙ q−g(q)\u0011#\n.(2)\nNote that our work concerns the analysis of only the\nsuspended aerial base, and does not consider any mobile\nmanipulation system which will be studied in the future.\nB. Omnidirectional Platform\nIn an omnidirectional platform with unidirectional pro-\npellers [23], a 6 DoF wrench can be exerted in any direction\nwith only a positive thrust vector from the propulsion unit.\nThis wrench space includes the translational forces and the\nrotational moments about the center of the mass (COM) of\nthe base. In the following, the forces along the x,yandz\naxis are represented as Fx,FyandFzrespectively, while the\nrotational moments about the x,yandzaxis are represented\nasMx,MyandMzrespectively. The propulsion system used\nin this study is a combination of brushless DC motors and\nunidirectional propellers. The thrust vector generated by the\nmotor can be expressed as Fm. The wrench vector generated\nat the base, denoted as uo, can be related to the motor\nthrust using the allocation matrix Aoasuo=AoFm. The\nallocation matrix Aois obtained from the SAM platform\n[11] after scaling it down to the same size as our proposed\nplatforms. The SAM has eight motors installed at an angle\nthat is obtained by solving an optimization problem, which\nensures a balanced design with equal distribution of wrench\nbetween the thrusters. The omnidirectional platform is used\nas a baseline for comparison with our proposed designs.\nC. Planar-thrust Platform\nThe planar-thrust platform (Fig. 1) has six propulsion\nunits that can exert thrust only in the plane of its base.(a)M1\nM2\nM3M4\nM6\nM3 M5\nM1\nM4M2\n(b)\nFig. 3: Top-view of the proposed (a) planar-thrust platform\nwith six rotors and (b) minimal-actuated platform with four\nrotors. The arrows depict the direction of propulsion, whereas\nMiis used to denote the ithmotor.\nThese thrusts can generate a wrench space that comprises\nthe translational forces along its xandyaxis, and moment\nabout its zaxis. The wrench vector upcan be summarized\nas,\nup=\u0002FxFyMz\u0003T. (3)\nEven though this platform can exert only 3D wrench unlike\na 6D wrench possible with platforms in [11], [15], we still\nsuccessfully demonstrate the oscillation-damping of the base\nin section V. Moreover, besides requiring fewer actuators\nthan the omnidirectional design, we also show that such a\nplanar-thrust design will allow us to conserve its actuation\nenergy which serves as one of the main motivations for\nchoosing such a design. The total translational force that\ncan be exerted along the xandyaxis increases for such a\ndesign as compared to an omnidirectional design. This might\nbe beneficial for certain tasks that require the base to hold its\nposition at desired deflected positions, while simultaneously\ndamping its oscillations. The proposed planar-thrust design\nconsists of six unidirectional thrusters, with the installation\nangle of each motor being ±90 degrees. The direction of\neach thrust unit is highlighted by an arrow in Figure 3. The\nallocation-matrix Apfor this platform is given as\nAp=\n1 1 −0.5−0.5−0.5−0.5\n0 0 0 .86 0 .86−0.86−0.86\n0.4−0.4 0 .4−0.4−0.4 0 .4\n. (4)\nD. Minimial-Actuated Platform\nThe concept of minimal-actuated design (Fig. 3) is similar\nto the planar-thrust design, as both of them have the same\n3D wrench-space. However, the minimal-actuated platform\nconsists of only four rotors, which is the minimum number\nrequired for covering the entire 3D wrench-space. It is\nto be noted that the actuation units utilize unidirectional\npropellers1. The allocation matrix Amfor this platform is\ngiven as\nAm=\n0 1 0 −1\n−1 0 1 0\n−0.4 0.4−0.4 0.4\n. (5)\n1Using a bidirectional propeller will allow us to further reduce the number\nof actuation units to three in order to achieve a 3D wrench, which will be\nstudied in future.III. O BSERVER DESIGN\nIn this section, an extended Kalman filter (EKF) is de-\nsigned in order to estimate the joint angles qand velocities\n˙q. The EKF is of the continuous-discrete form [24] with\nintermittent measurements. The continuous-discrete version\nof the EKF is chosen because the measurements from the\nsensors are obtained at a discrete interval, while the process\nmodel in (2) is a continuous-time differential equation. The\nprediction step for the states xand its error covariance P\nare as follows,\n˙ x(t) =f(x(t),u(t)) (6a)\n˙P(t) =F(t)P(t) +P(t)F(t)T+Q(t), (6b)\nwhere F(t)is the Jacobian of f(t)with respect to x, while\nQ(t)is the process noise covariance. The measurement\nmodel used in the Kalman filter is given as,\nzk=h(xk) +vk, (7)\nwhere the subscript kdenotes the sampling instance, zk\nandh(xk)are the measurement vector and measurement\nmodel respectively and vkis the sensor noise covariance.\nThe measurements used in the EKF are the translational\nvelocity and the orientation of the suspended base, which\nare obtained directly from the flight-controller unit (FCU).\nThe selection of these measurement quantities ensures the\nobservability [25] of our system states. The optimal Kalman\ngain and the correction step for the state xand covariance\nPkare presented as follows,\nKk=Pk|k−1HT\nk\u0010\nHkPk|k−1HT\nk+Rk\u0011−1\n(8a)\nx(k|k) =x(k|k−1) +λiKk(zk−h(xk|k−1)) (8b)\nPk= (I−KkHk)P(k|k−1), (8c)\nwhere Rkis the covariance of the sensor noise vkandHkis\nthe derivative of the measurement model h(xk)with respect\nto the states. For a particular sampling instant, if the ith\nsensor publishes a new measurement then the variable λiis\none, and zero otherwise.\nIV. C ONTROLLER DESIGN\nA. Linear Quadratic Regulator (LQR)\nAn optimal state feedback control LQR [26] is designed\nwhich minimizes the following linear quadratic cost function,\nJ=Zt\n0\u0010\nx(t)TˆQx(t) +u(t)TˆRu(t)\u0011\ndt, (9)\nwhere ˆQis a positive semi-definite matrix which penalises\nthe state xwhereas the positive definite matrix ˆRpenalizes\nthe control input u. The state feedback control law can\nbe expressed as u=−kx, where the controller gain k\nis obtained by minimizing the cost function in (9). We\nchoose the gain kby linearizing the system at its origin.\nThe controllability matrix [25] is found to be full rank,\nfor both the omnidirectional and the proposed underactuated\nplatforms, which conveys that the chosen wrench vector uo\nandupare sufficient to control our system.B. PD+ Controller\nA PD+ control is designed in order to control the platform\nin the task space, which is described in this section.\n1) Omnidirectional Platform: We consider the task coor-\ndinates xoof the omnidirectional platform as\nxo=\u0002pxpyϕ θ ψ\u0003T, (10)\nwhere pxandpyare the inertial positions of the center of\nmass (COM) of the platform, while ϕ,θandψare its Euler\nangles. The task-coordinates are related to the joints using\nthe Jacobian Joas˙xo=Jo˙q. Next, transforming the system\ndynamics (1) to task space [27], we obtain\nΛ¨xo+µ˙xo+ρ=F, (11)\nwhere Λ,µ, andρare the task-space inertia, Coriolis matrix,\nand gravity respectively. The control input Fin task-space\nis then defined as2,\nF=Λ¨xd+µ˙xd+Kd(˙xd−˙xo) +Kp(xd−xo) +ρ,\n(12)\nwhere KpandKdare the respective coefficients correspond-\ning to the position and velocity error in task space. xd,˙ xdand\n¨ xddenote the desired task position, velocity and acceleration,\nrespectively. By choosing control law (12), we obtain a stable\nclosed-loop system dynamics as,\nΛ(¨xd−¨xo) + (µ+Kd) (˙xd−˙xo) +Kp(xd−xo) = 0 .\n(13)\nThe maximum thrust vector is denoted as Fmax, with the\nmaximum thrust of each unit being limited to 9 N. In order to\ngenerate only positive thrust from the motors, a least-squares\nproblem was solved to obtain the thrust Fmas,\nmin\nFm∥Fm∥2\n2\ns.t.JTAoFm=JT\noF,\n0≤Fm≤Fmax .(14)\n2) Planar-thrust and Minimial-actuated Plaforms: Due to\nthe reduced wrench-space, the task coordinates xuof the\nunderactuated platforms are chosen differently as follows,\nxu=\u0002\npxpyψ\u0003T. (15)\nA redundancy is involved with this selection of task space\nasxu∈R3whereas q∈R5. Therefore, we introduce the\nnullspace velocity vn=N˙qusing the nullspace operator N\n[28]. The task coordinates are related to the joints using the\ninertial Jacobian Juas˙xu=Ju˙q. The system dynamics is\nthen transformed into a decoupled task-space and nullspace\ncoordinates [29] as,\n\u0014\nΛx0\n0Λn\u0015\u0014¨xu\n˙vn\u0015\n+\u0014\nµxµxn\nµnxµn\u0015\u0014˙xu\nvn\u0015\n+\u0014\nρx\nρn\u0015\n=\u0014\nFx\nFn\u0015\n,\n(16)\nwhere the subscripts nandxrefer to quantities in the\nnullspace and taskspace, respectively. In order to obtain\n2If the task concerns damping the platform about the origin, then the\ncompensation of gravity is not needed as it has a stabilizing effect.stable closed-loop dynamics, the task-space controller Fxis\nchosen similar to (12) as2\nFx=Λe¨xd+µ˙xd+Kd(˙xd−˙xu) +Kp(xd−xu)\n+µxnvn+ρx.(17)\nThe task-space force Fxand the nullspace force Fnare\nrelated to the joint torque τas follows,\nτ=JT\nuFx+NTFn. (18)\nThe torque τis related to the body wrenches uasJTu.\nNext, we define the relation u=Bup, where upconsists\nof the selected wrench elements from the body-wrench u,\nwhich the underactuated platform can exert and Bis the\nunderactuation matrix. The wrench component upis applied\nby actuation units of the platform, which is obtained using\nthe following relation,\n\u0014\nup\nFn\u0015\n=h\nJTB−NTi−1\nJT\nuFx. (19)\nNotice that the chosen actuation architecture not only\ngenerates the task force but also an additional force Fnin\nthe nullspace dynamics.\nV. R ESULTS\nOur proposed novel underactuated platforms are evaluated\nusing both computer simulations and experimental studies,\nwhich will be discussed in this section.\nA. Numerical Simulations\nThe simulation model used in our studies describes the\nspherical double pendulum in Fig. 2. The initial joint-angles\nfor the simulation are chosen as q1= 0.15, q2= 0.2, q3=\n0.2,q4= 0 and q5= 0 radians. The initial joint velocities are\nall considered as zero. The length of each link of the double\npendulum is considered to be 0.75 m , in order to replicate\nthe actual experimental setup (Fig. 9). However, for outdoor\nexperiments, the length of the first link will be higher than\nthe second which will be studied in the future. The weight\nof the platform is found to be 4.06 kg , and its arm length is\n0.4 m. The principal inertia along the x,y, and zaxis are\nfound to be 0.0646, 0.0646, and 0.0682 kg m2, respectively.\nIn order to assume a similar comparison of the three\ndifferent platforms, the state-penalty matrix ˆQis chosen the\nsame for all platforms as\nˆQ= diag([200 ,200,20,0.01,0.01,50,50,1,0.01,0.01]) .\nTheˆRfor all the platforms is chosen as an identity\nmatrix. Moreover, the propulsion unit, mass, and inertia\nare considered to be the same for all three platforms. We\nobserve that all three platforms have successfully damped\nthe oscillations of the base, with a satisfactory settling\ntime (Fig. 4). However, for the second-spherical joint, the\nomnidirectional platform has a higher transient, as compared\nto the other platforms. The planar-thrust and the minimum-\nactuated platform demonstrate a similar behaviour for all five\njoints, with a high-frequency transient motion being observed\nfor the joints q4andq5, before it converges to equilibrium.Fig. 4: Joint-angles for the LQR-controlled system, using\nthe omnidirectional platform (denoted as OD), planar-thrust\nplatform (denoted as PT), and minimum-actuated platform\n(denoted as MA).\nFig. 5: Wrenches commanded by the LQR controller.\n(a)\n(b)\nFig. 6: Energy consumption for the (a) LQR-controlled\nsystem and (b) PD+ task-controlled system.\nThe wrenches commanded by the platform are shown\nin Fig. 5. We observe that the translational forces Fxand\nFyin the proposed planar-thrust and the minimum-actuated\nplatforms have both high-frequency as well as low-frequency\ncomponents, which is required to compensate for the two\nmodes of the spherical double pendulum. The wrench exerted\nby the omnidirectional platform is predominantly of lowerfrequency which is sufficient to damp the motion of the base.\nIn order to obtain an estimate of the energy consumption by\nthe platforms, we introduce the quantity ˆEas\nˆE=X\nNX\ni=1Fmi.pwmi\n∆t, (20)\nwhere Nis the number of actuators, Fmiis the thrust\ngenerated by ithmotor and pwmiis the PWM pulse issued\nto the motor i. The PWM pulse is used in computing\nˆEinstead of the motor speed, because they are directly\nproportional to each other, besides the absence of motor\nspeed sensing in the platform. ˆEfor the planar-thrust and\nthe minimal-actuated platform is found to be lower than the\nomnidirectional platform by 50.9 %and 52.8 %, respectively\n(Fig. 6).\nFig. 7: Task coordinates of the PD+ controlled system.\nFig. 8: Wrenches commanded by the PD+ controller.\nThe performance of the task-space controller and the\ncorresponding commanded wrenches are shown in Fig. 7 and\n8, respectively. In order to assume a similar comparison for\nthe different platforms, the same stiffness Kpis chosen for\nall of them. The Kpcorresponding to the states x,y, and\nψare chosen as 400, 400, and 100, respectively. The fully-\nactuated platform has two additional task spaces ϕandθ, and\ntheir corresponding Kpwere chosen as 100. The damping\ncoefficient Kdis chosen as twice the square root of Kp.\nThe performance for all the platforms is similar in termsFig. 9: Sequence of the experiments with the proposed planar-thrust platform being stabilized using the LQR controller.\nof the state’s evaluation. It is also observed that similar to\nthe previous case, the proposed platforms were more energy-\nefficient than the omnidirectional configuration (Fig. 6).\nB. Experimental Results\nFig. 10: Experimental results showing the joint-angles for\nthe LQR-controlled system using the planar-thrust platform.\nFig. 11: Experimental results for the planar-thrust platform\nwith the PD+ task-space controller.\nThe experiments are conducted using the planar-thrust\nplatform (Fig. 9). It comprises six brushless DC motors with\na rating of 380 KV and folding propellers with a diameter of\n15 inches. Such a configuration of the propulsion system is\nsufficient for our medium-size platform. The flight controller\nunit (FCU) is chosen as the Pixhawk Cube Orange which is\nlightweight and compact and includes three IMU sensors for\nadditional accuracy of our estimation algorithm. We utilize\nthe PX4 firmware [30] in order to interface the FCU, and\nour algorithm runs at a frequency of 500 Hz.\nThe experiments demonstrate the damping behaviour of\nthe proposed planar-thrust platform by stabilizing it from an\ninitial tilt angle (Fig. 9). For the LQR-controller, ˆRis chosen\nas identity matrix, while ˆQis chosen as\nˆQ= diag([100 ,100,0.1,0.01,0.01,1,1,0.1,0.0001,0.0001]) .A high-frequency oscillation for the joints q4andq5with\nan amplitude of around 0.01 radian is observed (Fig. 10).\nA similar phenomenon was also observed in the simulations\n(Fig. 4). This is possibly due to the low bandwidth of the\nmotor to counteract high-frequency oscillations. For the task-\nspace controlled system, Kpcorresponding to the states x,\ny, and ψare chosen as 25, 25, and 0.35 respectively, while\ntheir corresponding Kdare chosen as 10, 10 and 0.13. We\nobserve that the states are damped within 8 seconds (Fig. 11).\nHowever, there is a time delay in sensing the yaw angle ψ,\nwhich resulted in its slower convergence rate, as compared\nto the other states.\nVI. C ONCLUSION AND FUTURE WORK\nIn this paper, we proposed a novel planar-thrust and a\nminimum-actuated suspended aerial platform and compared\ntheir performance with an omnidirectional design. The pro-\nposed designs were found to be more energy-efficient and\nuse fewer actuators than the omnidirectional configuration,\nwhile successfully damping the spherical double pendulum-\ntype system. In order to validate and compare the platforms,\nwe designed optimal state-feedback and a PD+ controller\nfor damping the oscillations of the platform in the joint\nspace and task space, respectively. An intermittent extended\nKalman filter was designed using only its onboard IMU\nmeasurements, in order to provide state feedback to the\ncontrollers.\nIn the future, we plan to mount a robotic manipulator on\nthe platform base and perform aerial manipulation tasks with\nit. We would like to include additional sensors in order to ob-\ntain fast and drift-free reliable estimates of the yaw angle. It\nwould also be interesting to analyse the use of bi-directional\npropellers to further reduce the number of actuators for\nthe minimal configuration. We would also like to introduce\nlearning-based algorithms in order to improve the system\nmodeling while accounting for unknown disturbances, which\nwill further improve the tracking by the controller.\nREFERENCES\n[1] A. Ollero, M. Tognon, A. Suarez, D. Lee, and A. Franchi, “Past,\npresent, and future of aerial robotic manipulators,” IEEE Transactions\non Robotics , vol. 38, no. 1, pp. 626–645, 2021.\n[2] A. Ollero, G. Heredia, A. Franchi, G. Antonelli, K. Kondak, A. Sanfe-\nliu, A. Viguria, J. R. Martinez-de Dios, F. Pierri, J. Cort ´es,et al. , “The\naeroarms project: Aerial robots with advanced manipulation capabil-\nities for inspection and maintenance,” IEEE Robotics & Automation\nMagazine , vol. 25, no. 4, pp. 12–23, 2018.[3] M. Ryll, G. Muscio, F. Pierri, E. Cataldi, G. Antonelli, F. Caccavale,\nand A. Franchi, “6d physical interaction with a fully actuated aerial\nrobot,” in 2017 IEEE International Conference on Robotics and\nAutomation (ICRA) . IEEE, 2017, pp. 5190–5195.\n[4] A. Ollero and B. Siciliano, Aerial robotic manipulation . Springer,\n2019.\n[5] N. Staub, D. Bicego, Q. Sabl ´e, V . Arellano, S. Mishra, and A. Franchi,\n“Towards a flying assistant paradigm: the othex,” in 2018 IEEE\nInternational Conference on Robotics and Automation (ICRA) . IEEE,\n2018, pp. 6997–7002.\n[6] A. Afifi, G. Corsini, Q. Sable, Y . Aboudorra, D. Sidobre, and\nA. Franchi, “Physical human-aerial robot interaction and collabora-\ntion: Exploratory results and lessons learned,” in 2023 International\nConference on Unmanned Aircraft Systems (ICUAS) . IEEE, 2023,\npp. 956–962.\n[7] G. Corsini, M. Jacquet, H. Das, A. Afifi, D. Sidobre, and A. Franchi,\n“Nonlinear model predictive control for human-robot handover with\napplication to the aerial case,” in 2022 IEEE/RSJ International Con-\nference on Intelligent Robots and Systems (IROS) . IEEE, 2022, pp.\n7597–7604.\n[8] F. Huber, K. Kondak, K. Krieger, D. Sommer, M. Schwarzbach,\nM. Laiacker, I. Kossyk, S. Parusel, S. Haddadin, and A. Albu-\nSch¨affer, “First analysis and experiments in aerial manipulation using\nfully actuated redundant robot arm,” in 2013 IEEE/RSJ international\nconference on intelligent robots and systems . IEEE, 2013, pp. 3452–\n3457.\n[9] A. Suarez, S. R. Nekoo, and A. Ollero, “Ultra-lightweight anthro-\npomorphic dual-arm rolling robot for dexterous manipulation tasks\non linear infrastructures: A self-stabilizing system,” Mechatronics ,\nvol. 94, p. 103021, 2023.\n[10] K. Kondak, F. Huber, M. Schwarzbach, M. Laiacker, D. Sommer,\nM. Bejar, and A. Ollero, “Aerial manipulation robot composed of\nan autonomous helicopter and a 7 degrees of freedom industrial\nmanipulator,” in 2014 IEEE international conference on robotics and\nautomation (ICRA) . IEEE, 2014, pp. 2107–2112.\n[11] Y . S. Sarkisov, M. J. Kim, D. Bicego, D. Tsetserukou, C. Ott,\nA. Franchi, and K. Kondak, “Development of sam: cable-suspended\naerial manipulator,” in 2019 International Conference on Robotics and\nAutomation (ICRA) . IEEE, 2019, pp. 5323–5329.\n[12] A. Coelho, Y . S. Sarkisov, J. Lee, R. Balachandran, A. Franchi,\nK. Kondak, and C. Ott, “Hierarchical control of redundant aerial\nmanipulators with enhanced field of view,” in 2021 International\nConference on Unmanned Aircraft Systems (ICUAS) . IEEE, 2021,\npp. 994–1002.\n[13] C. Gabellieri, Y . S. Sarkisov, A. Coelho, L. Pallottino, K. Kondak,\nand M. J. Kim, “Compliance control of a cable-suspended aerial\nmanipulator using hierarchical control framework,” in 2020 IEEE/RSJ\nInternational Conference on Intelligent Robots and Systems (IROS) .\nIEEE, 2020, pp. 7196–7202.\n[14] Y . S. Sarkisov, A. Coelho, M. G. Santos, M. J. Kim, D. Tsetserukou,\nC. Ott, and K. Kondak, “Hierarchical whole-body control of the cable-\nsuspended aerial manipulator endowed with winch-based actuation,”\nin2023 IEEE International Conference on Robotics and Automation\n(ICRA) . IEEE, 2023, pp. 5366–5372.\n[15] A. Yi ˘git, G. Grappe, L. Cuvillon, S. Durand, and J. Gangloff, “Prelim-\ninary study of an aerial manipulator with elastic suspension,” in 2020\nIEEE International Conference on Robotics and Automation (ICRA) .\nIEEE, 2020, pp. 4287–4293.\n[16] A. Yi ˘git, M. A. Perozo, L. Cuvillon, S. Durand, and J. Gangloff,\n“Improving dynamics of an aerial manipulator with elastic suspension\nusing nonlinear model predictive control,” in 2021 IEEE International\nConference on Robotics and Automation (ICRA) . IEEE, 2021, pp.\n533–539.\n[17] A. Yi ˘git, M. A. Perozo, L. Cuvillon, S. Durand, and J. Gangloff,\n“Novel omnidirectional aerial manipulator with elastic suspension:\nDynamic control and experimental performance assessment,” IEEE\nRobotics and Automation Letters , vol. 6, no. 2, pp. 612–619, 2021.\n[18] K. Sreenath, T. Lee, and V . Kumar, “Geometric control and differential\nflatness of a quadrotor uav with a cable-suspended load,” in 52nd IEEE\nConference on Decision and Control . IEEE, 2013, pp. 2269–2274.\n[19] J. Zeng, P. Kotaru, and K. Sreenath, “Geometric control and differen-\ntial flatness of a quadrotor uav with load suspended from a pulley,” in\n2019 American Control Conference (ACC) . IEEE, 2019, pp. 2420–\n2427.[20] H. Das, “Dynamic inversion control of quadrotor with a suspended\nload,” IFAC-PapersOnLine , vol. 51, no. 1, pp. 172–177, 2018.\n[21] Z. Lv, Y . Wu, X.-M. Sun, and Q.-G. Wang, “Fixed-time control\nfor a quadrotor with a cable-suspended load,” IEEE Transactions on\nIntelligent Transportation Systems , vol. 23, no. 11, pp. 21 932–21 943,\n2022.\n[22] Y . S. Sarkisov, M. J. Kim, A. Coelho, D. Tsetserukou, C. Ott, and\nK. Kondak, “Optimal oscillation damping control of cable-suspended\naerial manipulator with a single imu sensor,” in 2020 IEEE Interna-\ntional Conference on Robotics and Automation (ICRA) . IEEE, 2020,\npp. 5349–5355.\n[23] M. Tognon and A. Franchi, “Omnidirectional aerial vehicles with\nunidirectional thrusters: Theory, optimal design, and control,” IEEE\nRobotics and Automation Letters , vol. 3, no. 3, pp. 2277–2282, 2018.\n[24] P. Frogerais, J.-J. Bellanger, and L. Senhadji, “Various ways to\ncompute the continuous-discrete extended kalman filter,” IEEE Trans-\nactions on Automatic Control , vol. 57, no. 4, pp. 1000–1004, 2011.\n[25] E. Kreindler and P. Sarachik, “On the concepts of controllability\nand observability of linear systems,” IEEE Transactions on Automatic\nControl , vol. 9, no. 2, pp. 129–136, 1964.\n[26] F. L. Lewis and V . L. Syrmos, “Optimal control,” 1995.\n[27] G. Garofalo, F. Beck, and C. Ott, “Task-space tracking control for\nunderactuated aerial manipulators,” in 2018 European Control Con-\nference (ECC) . IEEE, 2018, pp. 628–634.\n[28] J. Park, W. Chung, and Y . Youm, “On dynamical decoupling of\nkinematically redundant manipulators,” in Proceedings 1999 IEEE/RSJ\nInternational Conference on Intelligent Robots and Systems. Human\nand Environment Friendly Robots with High Intelligence and Emo-\ntional Quotients (Cat. No. 99CH36289) , vol. 3. IEEE, 1999, pp.\n1495–1500.\n[29] G. Garofalo, B. Henze, J. Englsberger, and C. Ott, “On the inertially\ndecoupled structure of the floating base robot dynamics,” IFAC-\nPapersOnLine , vol. 48, no. 1, pp. 322–327, 2015.\n[30] L. Meier, D. Honegger, and M. Pollefeys, “Px4: A node-based\nmultithreaded open source robotics framework for deeply embedded\nplatforms,” in 2015 IEEE international conference on robotics and\nautomation (ICRA) . IEEE, 2015, pp. 6235–6240." }, { "title": "2401.17689v2.Constraining_black_hole_parameters_with_the_precessing_jet_nozzle_of_M87_.pdf", "content": "arXiv:2401.17689v2 [gr-qc] 1 Feb 2024Constraining black hole parameters with the precessing jet nozzle of M87*\nShao-Wen Wei1,2∗, Yuan-Chuan Zou3†, Yu-Peng Zhang1,2‡, Yu-Xiao Liu1,2§\n1Lanzhou Center for Theoretical Physics, Key Laboratory of T heoretical Physics of Gansu Province,\nand Key Laboratory of Quantum Theory and Applications of MoE ,\nLanzhou University, Lanzhou, Gansu 730000, China,\n2Institute of Theoretical Physics &Research Center of Gravitation,\nLanzhou University, Lanzhou 730000, People’s Republic of C hina,\n3School of Physics, Huazhong University of Science and Techn ology, Wuhan, 430074, People’s Republic of China\nRecently, Cui et al. [Nature 621, 711 (2023)] reported that the jet nozzle of M87* exhibits a\nprecession with a period of approximately 11 years. This find ing strongly suggests that M87* is a\nspinning black hole with a tilted accretion disk. In this pap er, our aim is to utilize these observations\nto constrain the parameters of the black hole. Firstly, we in vestigate the properties of the tilted\ncircular orbits and the innermost stable circular orbits. T he corresponding angular momentum,\nenergy, and Carter constant for both prograde and retrograd e orbits are calculated. We find that,\ncompared to equatorial circular orbits, these quantities e xhibit significant differences for fixed tilt\nangles. Moreover, the Carter constant takes positive value sfor nonvanishingtilt angles. Notably, the\npresence of misalignment of the orbit angular momentum and b lack hole spin leads to a precession\neffect in these tilted circular orbits. We then make use of the se circular orbits to model the warp\nradius of the tilted accretion disk, which allows us to deter mine the corresponding precession period\nthrough the motion of massive particles. Further comparing with the observation of M87*, the\nrelationship between the black hole spin and the warp radius is given, through which if one of them\nis tested, the other one will be effectively determined. Addi tionally, our study establishes an upper\nbound on the warp radius of the accretion disk. These findings demonstrate that the precession of\nthe jet nozzle offers a promising approach for testing the phy sics of strong gravitational regions near\na supermassive black holes.\nPACS numbers: 04.70.Bw, 04.25.-g, 97.60.Lf\nI. INTRODUCTION\nAfewyearsago,theEventHorizonTelescope(EHT)Collaborationr eleasedthefirstdirectimageofthesupermassive\nblack hole located at the center of the M87 Galaxy [1]. This remarkab le achievement unveiled the distinctive ring\nstructure, which represents the characteristic features of th e black hole shadow, an extraordinary phenomenon arising\nfrom strong gravitationaleffects. Consequently, investigating t hese shadow patterns providesus with valuable insights\ninto the properties of supermassive black holes. By modeling M87* wit h the Kerr black hole, the EHT Collaboration\ndemonstrated the remarkable consistency between the observa tions of the black hole shadow and the predictions of\ngeneral relativity. Subsequently, EHT Collaboration also unveiled th e shadow of the supermassive black hole located\nat the center of our Milky Way [2]. These breakthrough observatio ns have sparked significant interest in the study of\nblack hole shadows, with the objective of constraining the paramet ers of these celestial objects [3]. However, due to\nthe precision required in capturing these images, this study remains a huge challenge.\nOn the other hand, the observation of M87* presents a distinctive opportunity to explore the relationship between\nthe relativistic jet and the black hole accretion disk. In a recent stu dy [4], the high-resolution imaging revealed a ring-\nlike structure with a diameter of approximately 8 .4 Schwarzschild radii at a wavelength of 3.5 mm. This remarkable\nphenomenon further signifies the connection between the jet and the accretion flow surrounding M87*.\n∗weishw@lzu.edu.cn, corresponding author\n†zouyc@hust.edu.cn\n‡zyp@lzu.edu.cn\n§liuyx@lzu.edu.cn2\nFIG. 1: Sketch figure of the tilted accretion disk (pink color ), jet axis (blue color), and black hole spin axis ( z-axis). Near the\nISCO, the disk is pulled back to the equatorial plane describ ed by the xyplane.\nVery recently, basing on radio observations spanning 22 years, Cu i et al. in Ref. [5] reported a remarkable finding\nthat there is a periodic variation in the position angle of the jet with a p eriod of approximately 11 years. This\nobservation can be attributed to the Lense-Thirring precession r esulting from the misalignment of the orbital angular\nmomentum and the black hole spin. Consequently, it strongly indicate s that M87* is a spinning black hole with a\ntilted accretion disk deviating from the equatorial plane. The half-o pening angle of the precession cone is estimated\nto be 1.25±0.18 degrees, while the angular velocity of precession is measured to b e 0.56±0.02 radians per year,\nleading to a precession period of 11 .24±0.47 years.\nThe study of tilted accretion disks has been extensively explored in p revious works [6, 7]. It is also related with\nquasi-periodic oscillations [8]. For clarity, we provide a schematic repr esentation of the tilted disk in Fig. 1. The\nsimulationfigurecanalsobefound inRef. [9]. Thefigureclearlyillustra testhepresenceofanangle ψjetthatquantifies\nthe misalignment between the orbital angular momentum and the blac k hole spin. At larger radial distances, the\naccretion disk, depicted in color, exhibits a tilt angle ψjet. As the radial distance decreases, the gravitational force\nof the black hole causes the tilt angle of the disk to decrease. Upon r eaching a characteristic radius known as the\nwarp radius, the disk is pulled back into the equatorial plane. Furthe rmore, as particles exceed the innermost stable\ncircular orbit (ISCO), they shall undergo a rapid fall into the black h ole, and disappear behind the event horizon.\nIn this paper, our purpose is to constrain the parameters of the b lack hole by utilizing the precession period, while\nmaking the following assumptions. Firstly, we assume that the motion of the disk particles at each radial distance\ncan be accurately described by the circular orbits with the same tilt a ngle. This assumption is well-justified as these\norbits tend to be circularized due to the friction within the disk. Seco ndly, we assume that the jet originates near\nthe warp radius and is oriented perpendicular to the accretion disk. It is important to note that the warp radius is\nlarger than the radius of the ISCO, and thus may not necessarily co incide with it. Lastly, as a third assumption, we\nconsider the precession axis provided in Ref. [5] as the axis of the bla ck hole spin.\nAt first, we study the tilted circular orbits and ISCOs through the m otion of a test massive particle in the Kerr\nblackhole background. The correspondingradius, angularmoment um, energy, andthe Carterconstant arecalculated.\nTheir behaviors with the tilt angle is also examined in detail. Then by makin g use of the equation of motion of the\nparticle, the angular velocity of the precession, as well as the perio d, for a circular orbit are obtained. Finally,\nmodeling the M87* with the Kerr black hole, we constrain the black hole parameter and the warp radius via the\nobserved period of the precession.\nThe present study is structured as follows. In Sec. II, we devote our study to the analysis of tilted circular orbits\nand ISCOs. Based on these results, Sec. III focuses on the calcu lation of precession for these orbits. In Section IV,\nwe employ the observed period of the precession to constrain the b lack hole spin and warp radius. Finally, our results\nare summarized and discussed in Section V.\nII. TILTED CIRCULAR ORBITS\nIn this section, we investigate the properties of the tilted circularo rbits for a massive particle within the background\nof a Kerr black hole, considering both cases with and without inclinatio n to the equatorial plane. The energy, angular\nmomentum, radius, and the corresponding Carter constant are c alculated.\nLet us start with the Kerr black hole. In the Boyer-Linquist coordin ates, the Kerr black hole can be described by3\nthe following line element\nds2=−∆\nρ2/parenleftbigg\ndt−asin2θdφ/parenrightbigg2\n+ρ2\n∆dr2+ρ2dθ2\n+sin2θ\nρ2/parenleftbigg\nadt−(r2+a2)dφ/parenrightbigg2\n, (1)\nwhere the metric functions are\nρ2=r2+a2cos2θ, (2)\n∆ =r2−2Mr+a2. (3)\nSolving ∆ = 0, it is easy to obtian the radii of the black hole horizons\nr±=M±/radicalbig\nM2−a2. (4)\nObviously, for a black hole with horizon, its spin must be a/M∈(0,1).\nUsing the Hamilton-Jacobi method, the geodesics of a massive part icle around the spinning Kerr black hole takes\nthe following forms\nρ2dr\ndτ=±/radicalbig\nR(r), (5)\nρ2dθ\ndτ=±/radicalbig\nΘ(θ), (6)\nρ2dφ\ndτ=−Pθ\nsin2θ+aPr\n∆, (7)\nρ2dt\ndτ=−aPθ+(r2+a2)Pr\n∆, (8)\nwhereτis the affine parameter along the geodesics, and\nPθ=aEsin2θ−l, (9)\nPr=E(r2+a2)−al, (10)\nR=P2\nr−∆(r2+Q+(aE−l)2), (11)\nΘ =Q+(aE−l)2−a2cos2θ−P2\nθ\nsin2θ. (12)\nThe symbols Eandlrepresent the energy and angular momentum per unit mass of the t est particle, respectively,\nand are associated with the Killing fields ∂tand∂φ. The Carter constant Q, which corresponds to the Killing-Yano\ntensor, is another conserved quantity along each geodesic.\nFor an orbit confined to the equatorial plane, we have θ=π\n2anddθ\ndτ= 0, resulting in a vanishing Carter constant\nQ= 0. Consequently, fortheorbitsdeviatingfromtheequatorialpla ne, theCarterconstantdoesnotvanish. However,\nit remains constant along the trajectory of a massive particle. Let us now focus on a geodesic in the θ-motion. By\nexamining Eqs. (6) and (12), it is apparent that the θ-motion of the massive particle exhibits symmetry about θ=π\n2.\nTherefore, for a bounded motion, the value of θmust be confined within the range [π\n2−ζ,π\n2+ζ], whereζ∈[0,π\n2]\nrepresents the maximum half-opening angle of the motion along the θdirection. For convenience, we refer to ζas the\ntilt angle. Considering the particle turns back at this point, we havedθ\ndτ= Θ = 0, which gives\nQ=l2tan2ζ+a2/parenleftbig\n1−E2/parenrightbig\nsin2ζ. (13)\nFor a geodesic with bounded θ-motion, the quantities landEremain constant. Additionally, if the maximum half-\nopening angle ζis given as a priority, the Carter constant will also be determined. Su bsequently, we can analyze the\nr-motion of the massive particle. In the following discussion, we specifi cally focus on circular orbits. On the other\nhand, according to equation (13), it can be observed that the Car ter constant vanishes for the equatorial motion with\nζ= 0, while it takes positive values for these orbits off the equatorial p lane with bounded energy E <1. Moreover,\nnegative values of the Carter constant indicate\nE >/radicalBigg\n1+l2\na2cos2ζ. (14)\nObviously, the energy E >1.4\n246810120246810\nr/Slash1Ml/Slash1M\n(a)246810120.900.951.00\nr/Slash1ME\n(b)2 4 6 8 1005101520\nr/Slash1MQ/Slash1M2\n(c)\n24681012/Minus10/Minus8/Minus6/Minus4/Minus20\nr/Slash1Ml/Slash1M\n(d)5 10 15 200.940.960.981.001.02\nr/Slash1ME\n(e)456789100510152025\nr/Slash1MQ/Slash1M2\n(f)\nFIG. 2: Angular momentum, energy, and Carter constant for th e circular orbits with a=0.8, and ζ=0,π\n4, andπ\n3. Upper row\nis for the prograde orbits and lower row is for the retrograde orbits. (a) l/M−r/M. (b)E−r/M. (c)Q/M2−r/M. (d)\nl/M−r/M. (e)E−r/M. (f)Q/M2−r/M.ζ=0,π\n4, andπ\n3from top to bottom for (a) and (e), and from bottom to top for\nother figures. Solid and dashed curves are for the stable and u nstabel circular orbits, respectively.\nA. Tilted circular orbits\nIt was recently demonstrated in Ref. [10] that tilted circular orbit s exist within Kerr black hole backgrounds, and\nthe precession of these orbits was obtained from the perspective of a local observer. However, the analysis of the\nenergy, angular momentum, and Carter constant associated with these orbits remains unexplored. In this section, we\naim to provide a comprehensive investigation of these quantities. Fu rthermore, we will examine the stability of these\ncircular orbits.\nFrom Eq. (5), the radial r-motion can be expressed as\n/parenleftbigg\nρ2dr\ndτ/parenrightbigg2\n+Veff= 0, (15)\nwhere the effective potential reads\nVeff=−R(r). (16)\nThe circular orbits require\nR=R′= 0, (17)\nwhere the prime symbol denotes the derivative with respect to r. From above equation, we can determine the energy\nand angular momentum corresponding to a circular orbit with a given r adiusr. The explicit form of these quantities\nis omitted for brevity. Additionally, the stability of the circular orbit is determined by the value of R′′. Specifically,\na negative value of R′′corresponds to a stable circular orbit, while a positive value indicates an unstable one.\nWe present the variations of the angular momentum, energy, and C arter constant as a function of the radius of\nthe circular orbits in Fig. 2. Notably, both the angular momentum and energy exhibit non-monotonic behaviors, as\nillustrated in Figs. 2(a), 2(b), 2(d), and 2(e). Taking the stability in to account, we represent stable circular orbits\nwith solid curves and unstable circular orbits with dashed curves. It is worth noting that these two types of orbits\nare connected by the ISCOs. The unstable circular orbits possess a minimum radius, leading the energy and angular\nmomentum to diverge towards positive infinity. In contrast, the st able circular orbits originate from the ISCO and\nextend towards radial infinity. It is worth noting that the energy o f the stable circular orbits is always bounded below5\n0Π\n8Π\n43Π\n8Π\n2/Minus4/Minus3/Minus2/Minus10123\nΖl/Slash1M\n(a)0Π\n8Π\n43Π\n8Π\n20.9540.9560.9580.960\nΖE\n(b)0Π\n8Π\n43Π\n8Π\n202468101214\nΖQ/Slash1M2\n(c)\nFIG. 3: Angular momentum, energy, and Carter constant as a fu nction of the tilt angle ζwitha/M= 0.8 andr/M= 10. (a)\nl/M−ζ. (b)E−ζ. (c)Q/M2−ζ. The red solid curves and blue dashed curves are for the progr ade and retrograde circular\norbits.\n1. Moreover, as the tilt angle ζincreases, the absolute value of the angular momentum ldecreases, and the energy\ndecreases for retrograde orbits but increases for prograde or bits. Notably, as illustrated in Figs. 2(c) and 2(f), the\nCarter constant Qalways vanishes for ζ= 0, while it takes positive values for non-zero ζ. In the case of unstable\ncircular orbits, Qdecreases as the radius increases, whereas it increases for stab le circular orbits. These findings\nprovide novel insights into the behavior of the Carter constant fo r the circular orbits.\nTo illustrate the behaviors of the angular momentum, energy, and C arter constant as a function of the tilt angle,\nwe present them in Fig. 3 for the specific case of a/M= 0.8 andr/M= 10, where only the stable circular orbits exist.\nFrom the figure, it is evident that the angular momentum l/Mdecreases with increasing tilt angle ζfor prograde\ncircular orbits, while it increases for retrograde circular orbits. Th e opposite trend is observed for the energy E,\nwhere it increases with ζfor prograde orbits and decreases for retrograde orbits. Nota bly, although the energy and\nangular momentum exhibit different values at ζ= 0, they tend to converge to the same values at ζ=π\n2. The results\ndepicted in Fig. 3(c) demonstrate that the Carter constant vanis hes for both prograde and retrograde circular orbits\nwhen the tilt angle ζis zero. Subsequently, Q/M2exhibits rapid increase for retrograde orbits. Notably, the Carte r\nconstant tends to converge to the same value as ζapproachesπ\n2. A notable characteristic of the retrograde orbits is\nthe non-monotonic behavior of the Carter constant Q/M2. Detailed calculations reveal that it decreases within the\nrangeζ∈(1.48,π\n2), reaching its maximum value of Q/M2= 14.30 atζ= 1.48. Similar behaviors can also be observed\nby varying the spin of the black hole.\nB. Tilted innermost stable circular orbits\nAs demonstrated earlier, the ISCO serves as a connection betwee n the stable and unstable circular orbits. In this\nsubsection, our attention will be directed towards these special o rbits characterized by a non-zero tilt angle ζ.\nFrom the effective potential, the ISCO can be determined by\nR=R′=R′′= 0. (18)\nEquivalently, by making use of the circular orbits, we can also solve on e of the following equation\n/parenleftbiggdl\ndr/parenrightbigg\na,ζ= 0,/parenleftbiggdE\ndr/parenrightbigg\na,ζ= 0,/parenleftbiggdQ\ndr/parenrightbigg\na,ζ= 0, (19)\nfor the ISCO.\nBy considering various values of the black hole spin, we present the r adius, angular momentum, energy, and Carter\nconstant in Fig. 4. Dashed and solid curves correspond to prograd e and retrograde orbits, respectively. It is evident\nthat the angular momentum, energy, and Carter constant exhibit similar behaviors to those observed in the stable\ncircular orbit depicted in Fig. 3. From the observationsin Fig. 4(a), it is evident that the radius of the progradeISCO\nis consistently smaller than 6 Mand increases as the tilt angle ζincreases. Conversely, for retrograde ISCOs with a\nnon-zero black hole spin, their radii decrease with the increasing of ζ. Notably, when ζreaches certain critical values,\nthe radius of the retrograde ISCO becomes smaller than 6 M, which corresponds to the scenario of a Schwarzschild\nblack hole. This behavior may be linked to the non-monotonic behavior of the Carter constant, as illustrated in Fig.\n4(d). Importantly, this characteristic of the radius differs signific antly from that of equatorial ISCOs.6\n0Π\n8Π\n43Π\n8Π\n2246810\nΖr/Slash1M\n(a)0Π\n8Π\n43Π\n8Π\n2/Minus4/Minus2024\nΖl/Slash1M\n(b)\n0Π\n8Π\n43Π\n8Π\n20.750.800.850.900.95\nΖE\n(c)0Π\n8Π\n43Π\n8Π\n2024681012\nΖQ/Slash1M2\n(d)\nFIG. 4: Characteristic quantities of the ISCO as a function o f the tilted angular ζ. (a)r/M−ζ. (b)l/M−ζ. (c)E−ζ. (d)\nQ/M2−ζ. Black hole spin a/M=-0.98, -0.8, -0.5, 0.5, 0.8, 0.98 for these thick curves fro m top to bottom in (a), (c), (d), and\nfrom bottom to top in (b). The solid thin curves are for the cas e witha/M= 0. Dashed and solid curves are for the retrograde\nand prograde ISCOs.\nFIG. 5: The trajectory of a massive particle moving along a ci rcular orbit with r/M= 10,a/M= 0.9, andζ=π\n4.\nIII. PRECESSION OF TILTED CIRCULAR ORBITS\nIn the case of equatorial circular orbits, the angular momentum is a ligned parallel to the black hole spin, resulting\nin the orbit being confined to the initial equatorial plane. However, w hen the circular orbits become tilted, an angle\nis formed between the directions of the orbital angular momentum a nd the black hole spin. This inclination gives\nrise to the precession, commonly referred to as Lense-Thirring pr ecession, causing the orbit plane to deviate from its\ninitial plane. In this section, our focus will be on investigating the pre cession of these circular orbits.\nToillustratethe precessionphenomenon, weconsideranexamplewit ha/M= 0.9andζ=π\n4. We plotthe trajectory\nof a massive particle orbiting the black hole at r/M= 10 in Fig. 5. The particle initially starts in the equatorial\nplane and gradually moves towards the north pole direction with a tilt a ngle. As it approaches θ=π\n2+ζ, the particle\nchanges its direction and crosses the equatorial plane until reach ingθ=π\n2−ζ. It is evident that the orbit is bounded\nbetweenθ=π\n2−ζandθ=π\n2+ζ. Notably, when the particle completes a loop along the θdirection, it does not\nreturn precisely to its initial location. There exists a difference along theφdirection, resulting in a precession of the7\n01002003004005006000Π2Π3Π4Π5Π6Π\nt/Slash1MΘ,Φ\n(a)01002003004005006000Π2Π3Π4Π5Π6Π\nt/Slash1MΘ,Φ\n(b)\nFIG. 6: Evolutions of the angular coordinates θ(bottom red curves) and φ(top blue curves) with r/M= 10 and a/M= 0.9.\n(a)ζ=π\n4. (b)ζ=4π\n9.\nθ1\nmax θ2\nmax θ3\nmax\nζ=π\n4t52.63957 263.35951 474.06870\nφ1.64887 8.25074 14.85215\nζ=4π\n9t50.73793 253.81609 456.87905\nφ1.65344 8.28467 14.91318\nTABLE I: Values of tandφwhen the particle crosses the maximal θfor these two tilted circular orbits described in Fig. 6.\norbital plane.\nTo provide a clear visualization, we present the θandφmotions in Fig. 6. In Fig. 6(a), we set the tilt angle\nasζ=π\n4. It is obvious that the θmotion is confined within finite values, while the φmotion increases boundlessly\nwith the coordinate time t. This behavior approximates a linear trend with slight deviations, indic ating that the\nparticle maintains a constant velocity in the φdirection. Furthermore, when considering a larger tilt angle, such a s\n4π\n9, the motion characteristics are displayed in Fig. 6(b). The θmotion remains bounded over time, but with notable\ndifference. It exhibits a step-like behavior, suggesting that the pa rticle experiences an abrupt change in its trajectory.\nThe slope of the curve represents the velocity in the φdirection, indicating that the particle attains higher velocities\nnear the north and south poles while having lower velocities near the e quatorial plane.\nThe precession angular velocity ωtof the circular orbit can be calculated with\nωt=∆φ−2π\nTθ, (20)\nwhereTθrepresents the period of the θmotion, while ∆ φquantifies the difference in angular φover one period of\ntheθmotion. It is important to note that the ωtdefined here is for the distant observer, as opposed to the local\nobserver defined in Ref. [10]. To extract the precession informatio n from the motions depicted in Fig. 6, we count the\ncoordinate time and the corresponding φwhen the particle crosses three successive maximum values of θ, as presented\nin Table I. Utilizing this data, we can calculate the precession angular v elocityωtfor each tilted circular orbit. For the\ntilted orbit with ζ=π\n4, we obtain ωt= 0.00151 and 0 .00151 for the two consecutive periodic motions in θ. Similarly,\nfor the orbit with ζ=4π\n9, we findωt= 0.00171 and 0 .00170. These results also indicate that the calculation error is\nwell controlled for our calculations.\nIV. CONSTRAINS OF M87*\nIn Ref. [5], Cui et al. reported a notable precession of the jet axis, with an observed period of approximately\n11 years. This observation provides an opportunity to constrain t he parameters of the black hole as expected. It is\nimportant to note that the sign of the precession angular velocity c annot be determined definitively. Therefore, in\nthis section, we consider both the prograde and retrograde case s to account for the possible scenarios.8\nFIG. 7: The period of the precession angular velocity Tfor the tilted ISCOs in the ζ-a/Mplane. Negative and positive a/M\nare for the retrograde and prograde cases, respectively.\nTo match the observable, we define the period of the precession fo r the tilted circular orbit as T=2π\nωt, or\nT=2π\nωtGM⊙\nc3/parenleftbiggM\nM⊙/parenrightbigg\n≈6.394×10−3×1\nωt/parenleftbiggM\nM⊙/parenrightbigg\n(year), (21)\nwhen the unit is restored.\nHere, we consider the location of the ISCO as the characteristic ra dius of the accretion disk. By varying the values\nof the black hole spin a/Mand the tilt angle, we determine the radius, energy, angular moment um, and the Carter\nconstant of the ISCO. Utilizing equations (5)-(8), we calculate the trajectory of the particle. Then, the precession\nangular velocity ωtand its corresponding period will be obtained.\nFollowing the above approach, we present the period of the preces sion angular velocity in Fig. 7 for the M87*\nblack hole with a mass of M= 6.5×109M⊙. Examining the ζ-a/Mplane, it becomes evident that the period\nremains predominantly below 5 years across most parameter region s. However, for retrograde ISCOs, their period\ncan extend to higher values, reaching up to 20 years when the black hole spin is extremely small. This behavior\nholds true regardless of the specific value of the tilt angle ζ, representing a universal result. In general, the precession\nangular velocity ωtexhibits a proportional relationship with the black hole spin, thereby explaining this result in a\nstraightforward manner.\nOn the other hand, the observations of the shadow and jet by the EHT suggest that M87* may possess a wide\naccretion disk, which indicates that it is inappropriate to describe th e disk by the ISCO. Fortunately, the study of\ntilted accretion disks reveals that the ISCO does not represent th e characteristic radius of the whole disk [6, 7]. As\nwe move to larger radii, the disk becomes increasingly tilted. However , as the radial distance decreases, the tilt angle\nof the accretion disk also reduces. Eventually, at the warp radius, the tilt angle vanishes. Moreover, as particles in\nthe disk approach the ISCO, they rapidly plunge into the black hole. D uring this process, it is nature to think that\nthe jet originates from the warp radius, and which is believed to lie with in the range of (6 M, 20M). Taking this\ninto consideration, our aim is to investigate the precession period at different radii of the circular orbits within the\naccretion disk.\nBased on the observations of M87*, the tilt angle is determined to be ζ= 1.25◦[5]. By varying the black hole\nspin and the radius of the circular orbits that characterize the acc retion disk, we calculate the precession period. The\nEHT observation reveals that the period corresponds to 11 .24±0.47 years. Consequently, in Fig. 8, we represent this\nobserved period in the a/M−r/Mplane. The solid curve corresponds to a period of T= 11.24 years. Additionally,\nthe blue and red dashed curves represent the upper and lower bou nds of the period, namely T= 10.77 and 11.71\nyears, respectively.\nUpon examining the figures, it becomes evident that, for each fixed period, the absolute value of the black hole\nspin increases with the warp radius. Conversely, for a fixed warp ra dius, the period decreases as the black hole spin\nincreases. Additionally, for a given black hole spin, the period of the p recession increases with the warp radius. Our\nprimary aim is to constrain the black hole spin via the observed preces sion period. However, the obtained results do\nnot appear to be optimistic. For instance, considering a warp radius ofr= 6M, the black hole spin a/Mcan be9\n6810121416/Minus1.0/Minus0.8/Minus0.6/Minus0.4/Minus0.20.0\nr/Slash1Ma/Slash1M\n(a)6 8 10 12 140.20.40.60.81.0\nr/Slash1Ma/Slash1M\n(b)\nFIG. 8: Constrains of the black hole spin and warp radius of th e accretion disk with the precessing period of the jet nozzle of\nM87*. The tilt angle ζ= 1.25◦. The solid curve corresponds the period T=11.24 years. The blue and red dashed curves are\nforT=10.77 and 11.71 years.\nreduced to 0.0604 and 0.0641 for the retrograde and prograde ca ses, respectively, which is quite close to the case of a\nnonspinning black hole. Nevertheless, there is an unexpected resu lt. We can establish an upper bound for the warp\nradius. In the case of prograde or retrograde tilted accretion dis ks, we can determine the maximum warp radius\nrmax= 14M,16M, (22)\nrespectively. As a result, by utilizing the observed period of the pre cession, we can determine the maximum warp\nradius for the tilted accretion disk. Additionally, if the observed war p radius falls within this range of a 2 Mdifference,\nit becomes possible to distinguish whether the accretiondisk is progr adeor retrograde. Consequently, this result holds\nsignificant value and warrants further testing through astronom ical observations.\nOn the other hand, we observe that the constraint imposed by the observation becomes tighter for smaller warp\nradii and lower black hole spins, while it becomes looser for highly spinnin g black holes and larger warp radii. Despite\nnot obtaining a stringent constraint on the black hole spin, our findin gs in Fig. 8 reveal the existence of a constrained\nregion, marked in green and yellow colors, within the parameter spac e. This strongly indicates that if the warp\nradius is determined through other astronomical observations, w e can subsequently determine the black hole spin.\nConversely, the warp radius can also be tested based on the black h ole spin. In order to provide convenience for future\napplications, we provide a high-precision fitting formula that relates the dimensionless black hole spin to the warp\nradius corresponding to a precession period of T= 11.24 years\na/M=−0.005742+0.001571(r/M)+0.000058(r/M)2−0.000341(r/M)3+0.000006(r/M)4,(23)\na/M= 0.450882−0.212997(r/M)+0.037536(r/M)2−0.002674(r/M)3+0.000091(r/M)4, (24)\nfor retrograde and prograde tilted accretion disks, respectively .\nV. DISCUSSIONS AND CONCLUSIONS\nIn this paper, we aimed to constrain the parameters of M87* by usin g the recent observation of the precessing jet\nnozzle [5]. In addition to the shadow, this observation presents ano ther promising approach for testing the properties\nof the supermassive black holes and investigating the physics within s trong gravitational regions.\nThe presence of a precessing jet axis in M87* suggests that its acc retion disk is not in the equatorial plane, but\ninstead exhibits a tilt angle of approximately 1.25◦[5]. Given that the accretion disk can be effectively described by\ncorresponding circular orbits, we initiated our investigation by exam ining the properties of the tilted circular orbits.\nThese orbits are confined within the θmotion. As the circular orbit represents a type of geodesic, the ra dius, angular\nmomentum, energy, and the Carter constant remain constant alo ng each tilted circular orbit. Utilizing the equations\nof motion for massive particles, we obtained the values for both the circular orbits and the ISCOs. Similar to the\nequatorial case, the circular orbits with small radii are found to be radially unstable, while those with larger radii are\nstable. However, a notable distinction is that the Carter constant no longer vanishes for these tilted circular orbits,\nbut instead maintains positive values. As expected, the presence o f a tilt angle results in the deviations of the angular\nmomentum, energy, and the Carter constant from those of equa torial circular orbits. Our findings reveal that as\nthe tilt angle increases, the angular momentum decreases while the e nergy increases for the prograde circular orbits.10\nConversely, this trend is reversed for the retrograde circular or bits. Moreover, for both prograde and retrograde\ncircular orbits, the Carter constant exhibits an increasing behavio r from a tilt angle of ζ= 0 and reaches a maximum\nnearζ=π\n2. Notably, a subtle feature is observed, wherein a slight decrease o ccurs in the vicinity of ζ=π\n2for the\nretrograde orbits.\nThe ISCO represents the last stable orbit of the massive particles b efore they rapidly fall towards the black hole.\nIt is widely accepted as the inner boundary of the accretion disk. In order to provide a more accurate description of\nthe disk, we also examined the behavior of the ISCOs when the tilt ang le is non-zero. Our results demonstrate that,\nfor each fixed black hole spin, the angular momentum, energy, and t he Carter constant exhibit similar trends as the\ntilt angle increases. However, the radius of the ISCO exhibits an intr iguing behavior. In the case of the equatorial\nISCO, the radius for either prograde or retrograde motion is alway s smaller or larger, respectively, than 6 M, which is\nthe value for a Schwarzschild black hole. However, with an increase in the tilt angle, the retrograde ISCO reveals a\nnotable feature: its radius becomes smaller than 6 M, indicating a distinct characteristic of tilted retrograde ISCOs.\nHaving obtained the characteristic quantities of the tilted circular o rbits, we proceeded to plot the trajectories for\nthe massive particles moving along these orbits. Due to the presenc e of the tilt angle, the orbital angular momentum\nand the black hole spin become misaligned. Consequently, the particle does not return to its initial location after\ncompleting one circular motion, but exhibits a discernible deviation. Fo r instance, following one complete circle in the\nθmotion, the particle’s φangle will experience a small shift. Exploiting this observation, we defi ned the precession\nangular velocity ωtas a measurement of the variation in the orbital plane. Consequent ly, the precession manifests\nas a periodic phenomenon with a period denoted as T, which precisely corresponds to the period observed in the\nprecessing jet nozzle [5].\nSubsequently, we assumed that the jet originates from the locatio n defined by the tilted ISCO and proceeded to\ncalculate the precession period for various black hole spins and tilt an gles. The results reveal that, in the majority of\nthe parameter space, the precession period is below five years. Ho wever, for extremely slowly spinning black holes, an\n11-year period can be reached. Consequently, based on this patt ern, if M87* possesses spin, it must be exceedingly\nsmall.\nOn the other hand, our study of the tilted accretion disk reveals th at the ISCO may be not the location where the\njet originates. Instead, at larger radii, the disk exhibits a significan t tilt, while with decreasing radius, the degree of\ntilt decreases. Remarkably, when the radius exceeds a certain thr eshold known as the warp radius, the gravitational\nforces pull the disk back towards the equatorial plane. Ultimately, at the equatorial ISCO, the accretion disk ceases\nto extend further. Consequently, it is natural to consider the wa rp radius as the origin of the jet, which is estimated\nto be within the range of 6 Mto 20M.\nUnder the assumption mentioned above, we revisited the precessio n period by setting ζ= 1.25◦. The numerical\nresults indicate that, with a warp radius at 6 Mand an expected period of T= 11.24 years, the possible black hole\nspin can be as low as 0.06. At first glance, it may seem challenging to est ablish a precise constraint on the black hole\nspin. However, this case might change for a large warp radius. We pr ovide an explicit relationship between the black\nhole spin and the warp radius, as illustrated in Fig. 8. Furthermore, w e derived the fitting formulas in (23) and (24).\nTherefore, if we can determine the warp radius through other obs ervations, we can precisely determine the black hole\nspin using such precession measurements. It is worth noting that t he reverse approach is also feasible.\nOn the other hand, while it may be challenging to constrain the black ho le spin precisely, it does impose an upper\nbound on the warp radius. In the case of prograde and retrograd e accretion disks, we find upper bounds of r/M= 14\nand 16, respectively. Notably, the difference between these boun ds can also serve as a potential distinguishing feature\nbetween the prograde and retrograde accretion disks.\nIn conclusion, our study provides valuable constraints on the para meters of M87* based on the recent observation\nof the precessing jet nozzle. This observation not only offers a pro mising insight into the characteristics of the\nsupermassive black hole and its surrounding accretion disk but also p resents an opportunity to test the physics such\nas the hidden dimensions [11] in strong gravitational regions near th e supermassive black holes.\nAcknowledgements\nWe warmly thank Prof. Weihua Lei for useful discussions. This work was supported by the National Natural\nScience Foundation of China (Grants No. 12075103, No. 11875151 , No.12105126, and No. 12247101) and the Major11\nScience and Technology Projects of Gansu Province.\n[1] K. Akiyama et al. [Event Horizon Telescope Collaboratio n],First M87 Event Horizon Telescope Results. I. The Shadow of\nthe Supermassive Black Hole , Astrophys. J. 875, L1 (2019), [arXiv:1906.11238 [astro-ph.GA]].\n[2] K. Akiyama et al. [Event Horizon Telescope], First Sagittarius A* Event Horizon Telescope Results. I. Th e Shadow of\nthe Supermassive Black Hole in the Center of the Milky Way , Astrophys. J. Lett. 930, L12 (2022), [arXiv:2311.08680\n[astro-ph.HE]].\n[3] P. Kocherlakota et al. [Event Horizon Telescope], Constraints on black-hole charges with the 2017 EHT observa tions of\nM87*, Phys. Rev. D 103, 104047 (2021), [arXiv:2311.08680 [astro-ph.HE]].\n[4] R.-S. Lu et al., A ring-like accretion structure in M87 connecting its black hole and jet , Nature 616, 686 (2023),\n[arXiv:2304.13252 [astro-ph.HE]].\n[5] Y. Cui et al, Precessing jet nozzle connecting to a spinning black hole in M87, Nature 621, 711 (2023), [arXiv:2310.09015\n[astro-ph.HE]].\n[6] J. A. Petterson, Twisted accretion disks. I. Derivation of the basic equatio ns, Astrophys. J. 214, 550 (1977).\n[7] E. C. Ostriker and J. J. Binney, Warped and tilted galactic discs , Mon. Not. Roy. Astron. Soc. 237, 785 (1989).\n[8] P. C. Fragile, G. J. Mathews, and J. R. Wilson, Bardeen-Petterson effect and quasi-periodic oscillations i nx-ray binaries ,\nAstrophys. J. 553, 955 (2001), [arXiv:astro-ph/0007478 [astro-ph]].\n[9] G. Lodato and D. Price On the diffusive propagation of warps in thin accretion discs , Mon. Not. R. Astron. Soc. 405, 1212\n(2010), [arXiv:1002.2973 [astro-ph.HE]].\n[10] A. M. A. Zahrani, Tilted Circular Orbits around a Kerr Black Hole , Phys. Rev. D 109, 024029 (2024), [arXiv:2312.12988\n[gr-qc]].\n[11] I. Banerjee, S. Chakraborty, and S. SenGupta, Silhouette of M87*: A new window to peek into the world of hidd en\ndimensions , Phys. Rev. D 101, 041301 (2020), [arXiv:1909.09385 [gr-qc]]." }, { "title": "2401.17720v1.Apéry_Acceleration_of_Continued_Fractions.pdf", "content": "arXiv:2401.17720v1 [math.NT] 31 Jan 2024Ap´ ery Acceleration of Continued Fractions\nHenri Cohen\nhenri.cohen2@free.fr\nFebruary 1, 2024\nAbstract\nWe explain in detail how to accelerate continued fractions ( for con-\nstants as well as for functions) using the method used by R. Ap ´ ery in his\nproof of the irrationality of ζ(3). We show in particular that this can be\napplied to a large number of continued fractions which can be found in\nthe literature, thus providing a large number of new continu ed fractions.\nAs examples, we give a new continued fraction for log(2) and f orζ(3), as\nwell as a simple proof of one due to Ramanujan.\n1 Introduction and Notation\nAfter R. Ap´ ery announced that he had proved the irrationality of ζ(3), the\nfirst explicit proof was found by D. Zagier using suitable telescoping s eries,\nin what was to become the celebrated Wilf–Zeilberger method (see [14 ]). It\nwas only a year later that Ap´ ery explained his method, see for insta nce [2]\nand [1], which was quite different. Although considerable effort was sp ent in\ntrying to generalize his method to prove the irrationality of other co nstants, the\nonly small successes were first, the discovery by G. Rhin and the au thor [5] of\na rapidly convergent continued fraction for ζ(4), unfortunately insufficient to\nprove its irrationality (which is of course well-known), and second, t he proof by\nA. Jeannin [8] of the irrationality of the sum of the inverses of the Fib onacci\nnumbers.\nNonetheless, an examination of the numerous continued fractions which can\nbe found in the literature, both for constants, but also for funct ions, it appears\nthat a large number can be accelerated using Ap´ ery’s technique, g iving usually\ncompletely new and interesting continued fractions, but without an y Diophan-\ntine consequences. A particularly simple example is as follows. The con tinued\nfraction\nlog(1+z) =z/(1+12z/(2−z+22z/(3−2z+32/(4−3z+42/(5−4z+···))))),\nvalid for instance for |z|<1, is trivially obtained from the Taylor expansion\nlog(1 +z) =z−z2/2 +z3/3− ···by Euler’s very classical transformation of\n1a series into a continued fraction, which converges at exactly the s ame speed\n(essentially in O(zn)). Applying Ap´ ery’s method, we obtain automatically the\nnew (but well-known) continued fraction\nlog(1+z) = 2z/(1(z+2)−12z2/(3(z+2)−22z2/(5(z+2)−32z2/(7(z+2)−···)))),\nwhich is just as simple but converges much faster, essentially in O((z2/(1 +√z+1)4)n) (for instance, for z= 1/2 this is approximately O(98−n) instead of\nO(2−n)).\nThe aim of this paper is to explain this method in great detail, together with\na large number of examples.\nTo abbreviate, we write simply “CF” instead of “continued fraction” . In\naddition, we use the following notation for a CF S, which may differ from\nnotation used in other papers in the literature:\nS=a(0)+b(0)/(a(1)+b(1)/(a(2)+b(2)/(a(3)+···))),\nand we denote as usual by p(n)/q(n) thenth convergent, so that p(0)/q(0) =\na(0)/1,p(1)/q(1) = (a(0)a(1) +b(0))/a(1), etc..., and for u=poru=qwe\nhaveu(n+1) =a(n+1)u(n)+b(n)u(n−1).\nWhena(n) andb(n) are polynomials A(n) andB(n) forn≥n0, we will\nwrite the continued fraction as\nS= ((a(0),a(1),...,a(n0−1),A(n)),(b(0),b(1),...,b(n0−1),B(n))).\nFor instance, the last continued fraction above will simply be written as\nlog(1+z) = ((0,(2n−1)(z+2)),(2z,−n2z2)).\nWe will consider exclusively CFs of that type.\n2 Bauer–Muir Acceleration\nContrary to general CFs, when a(n) andb(n) are polynomials for nlarge, it\nis quite simple to determine the speed of convergence, at least heur istically,\nsee [3] and [4]. When the convergence is fast, there is no need to impr ove the\nspeed of convergence. However, when the convergenceis slow, f or instance when\nS−p(n)/q(n) =O(n−α), it is useful if not necessaryto do so. The classicalidea,\ntheorized in particular by Bauer and Muir, is to use a so-called modification of\nthenthtailρ(n) of the CF, defined by ρ(n) =b(n)/(a(n+1)+b(n+1)/(a(n+\n2)+···)), so that\nS=a(0)+b(0)/(a(1)+b(1)/(a(2)+···+b(n−1)/(a(n)+ρ(n)))),and\nS=p(n)+ρ(n)p(n−1)\nq(n)+ρ(n)q(n−1).\n2Thus if we can find a function r(n) which approximates ρ(n), the quantity\n(p(n)+r(n)p(n−1))/(q(n)+r(n)q(n−1)) will be a better approximation to S\nthanp(n)/q(n).\nThis observation leads to the two fundamental formulas of Bauer– Muir ac-\nceleration. First, ρ(n) trivially satisfies the recursion ρ(n) =b(n)/(a(n+1)+\nρ(n+1)), in other words δ(n) :=ρ(n)(a(n+1)+ρ(n+1))−b(n) = 0. Thus,\nwe want a function r(n) such that\nd(n) :=r(n)(a(n+1)+r(n+1))−b(n)\nis as small as possible.\nSecond, itisimmediatetocreateanewCFwhoseconvergentsare p′(n)/q′(n)\nwithu′(n) =u(n)+r(n)u(n−1) foru=pandu=q: after a smallcomputation,\nwe find that such a CF is given by ( a′(n),b′(n)), with\na′(0) =a(0)+r(0), b′(0) =−d(0), a′(1) =a(1)+r(1),\na′(n) =a(n)+r(n)−r(n−2)d(n−1)/d(n−2) forn≥2, and\nb′(n) =b(n−1)d(n)/d(n−1) forn≥1.\nIt is also very easy to give a recipe for the choice of r(n), depending on the\ndifferent speeds of convergence of the CF. Let us give three exam ples.\nExample 1. Westartfromthestandardseries π/4 = 1−1/3+1/5−1/7+···,\nwhich is immediately transformed by Euler into the CF\nπ/4 = 1/(1+12/(2+32/(2+52/(2+72/(2+···))))),\nwhich using our notation is written π/4 = ((0,1,2),(1,(2n−1)2)). This CF of\ncourseconvergeslike the series, i.e., π/4−p(n)/q(n) =O(1/n). We now look for\nr(n) such that d(n) is small for nlarge, and we immediately find that there are\ntwopolynomialsolutions r(n) = 2n−3(givingd(n) =−4),andr(n) =−(2n−1)\n(givingd(n) = 0, which seems ideal!). Remembering that r(n) is supposed to\napproximate ρ(n) = (2n−1)2/(2+(2n+1)2/(2+···)), theonlyplausiblesolution\nisr(n) = 2n−3. We thus find the new CF π/4 = ((1,4,5,6),(−1,4,(2n−1)2)),\nwhich satisfies π/4−p(n)/q(n) =O(1/n3), so we indeed have accelerated our\nCF.\nIt is immediate to check that we can continue in this way, and obtain CF s\nwitha(n) = 4k+ 2,b(n) = (2n−1)2fornlarge, converging in O(1/n2k+1).\nThis will be essential in Ap´ ery’s method.\nExample 2. Here, we start from the standard series log(2) = −log(1−\n1/2) = (1/2)+(1/2)2/2+(1/2)3/3+···, which is immediately transformed by\nEuler into the CF log(2) = ((0 ,3n−1),(1,−2n2)), in other words\nlog(2) = 1/(2−2.12/(5−2.22/(8−2.32/(11−···)))).\nThis converges like the series: log(2) −p(n)/q(n) =O(1/(n2n)), so already\nrelatively fast. Choosing here r(n) = 1−ngives the new CF\nlog(2) = ((1 /2,3n),(1/2,−2n2)),\n3whichsatisfieslog(2) −p(n)/q(n) =O(1/n32n), soasmallpolynomialgain. This\ndoes not seem very interesting since the initial CF already converge s reasonably\nfast, but we will see that combined with Ap´ ery’s method leads to a se ries which\nconverges muchfaster, roughly in 1 /34n.\nExample 3. SetS=/integraltext∞\n0e−t/(1 +t)dt. We have the well-known CF\nS= 1/(2−12/(4−22/(6−32/(8−···)))), in other words S= ((2n),(1,−n2)),\nwith speed of convergence S−p(n)/q(n) =O(e−4√n). Usingr(n) =−n, we\nfind the new CF S= ((2n+1),(−1,−n(n+1))), in other words S= 1−1/(3−\n1.2/(5−2.3/(7−3.4/(9−4.5/(11−···))))). This has exactly the same speed\nof convergence, so no acceleration, but Bauer–Muir has given us a different and\ninteresting CF for the same quantity.\nImportant Remark. To keep things simple (and to be able to reasonably\niterate Bauer–Muir in Ap´ ery’s method), we must restrict to polyno mials or ra-\ntional functions with rational coefficients. It is easy to check that because of\nthis restriction, we can apply Bauer–Muir when the convergence is in O(1/nα)\n(sinceαwill always be rational), or in O(1/(Bnnα))whenBis a rational num-\nber, and some other types such as Example 3 above. In particular, it is not\napplicable if the convergence is in O(1/Bn) withBirrational.\n3 Ap´ ery’s method\nWe have seen above that for certain CFs it is possible to iterate the B auer–Muir\nprocess and still keep reasonably simple formulas. Ap´ ery’s method consists\nsimply in combining this with a diagonal process. More precisely, it proceeds as\nfollows. Let ( a(n),b(n)) be a CF with partial quotients ( p(n),q(n)). We define\nby abuse of notation a(n,0) =a(n),b(n,0) =b(n),p(n,0) =p(n), andq(n,0),\nand assume that at step lwe have a CF ( a(n,l),b(n,l)) with partial quotients\n(p(n,l),q(n,l)). We use Bauer–Muir with a sequence r(n,l) chosen such that\nd(n,l) =r(n,l)(r(n+1,l)+a(n+1,l))−b(n,l) is “small”, and we thus obtain a\nnew CF (a′(n,l),b′(n,l)) with partial quotients u′(n,l) =r(n,l)u(n,l)+u(n−\n1,l) foru=pandu=q. We do notsetu(n,l+ 1) =u′(n,l), but we shift n\nby 1, so we define u(n,l+1) =u′(n+1,l) =r(n+1,l)u(n+1,l)+u(n,l) for\nu=pandu=q, hencea(n,l+1) =a′(n+1,l) andb(n,l+1) =b′(n+1,l).\nNote that this shift by 1 is crucial. By induction this defines the 2-dime nsional\narrays (a(n,l),b(n,l),r(n,l),d(n,l),p(n,l),q(n,l)).\nSetting as usual u=poru=q, we summarize all the definitions and\nrecursions as follows, where to simplify notation, it is convenient to d efine\nR(n,l) =a(n+1,l)+r(n+1,l):\nu(n+1,l) =a(n+1,l)u(n,l)+b(n,l)u(n−1,l),\nu(n,l+1) =r(n+1,l)u(n+1,l)+u(n,l),\nR(n,l) =a(n+1,l)+r(n+1,l), d(n,l) =r(n,l)R(n,l)−b(n,l),\na(n,l+1) =R(n,l)−r(n−1,l)d(n,l)/d(n−1,l),\nb(n,l+1) =b(n,l)d(n+1,l)/d(n,l).\n4The game now consists in walkingin a regular or semi-regular manner in\nthe two-dimensional arrays u(n,l). The first idea is a diagonal walk, i.e., find\nthe CF corresponding to p(n,n)/q(n,n) or more generally p(n+m)/q(n+m)\nfor small fixed m. This is easily seen to involve annoying denominators, which\nin general give complicated formulas, but in the case where these de nominators\ncancel, these CFs are ideal and the most interesting. To avoid deno minators, we\nsimplyusea staircase walk,inotherwords p(n,n)/q(n,n),p(n+1,n)/q(n+1,n),\np(n+1,n+1)/q(n+1,n+1),etc... (or more generally with the first argument n\nandn+1replacedby n+morn+m+1), thusgivingCFswith( a(2n),b(2n))and\n(a(2n+1),b(2n+1)) given by different formulas, what I call period 2 CFs. In\naddition, by contraction it is immediate to check from the staircase CF whether\nthe diagonal one has simple coefficients. I call these CFs the Ap´ ery accelerates\nof the initial one, and are usually the ones which converge the faste st. The\nprecise formulas are as follows, where as usual udenotesporq:\nu(n+m+1,n+1) =R(n+1+m,n)u(n+m,n+1)\n−d(n+m+1,n)u(n+m,n),and\nu(n+m,n+1) =R(n+m,n)u(n+m,n)+b(n+m,n)u(n+m−1,n).\nFinally, we can also do a verticalwalk, i.e., find the CF corresponding to\np(0,n)/q(0,n) or more generally p(m,n)/q(m,n) for small fixed m. These will\nnot converge fast, but provide completely new CFs, and I call them the Ap´ ery\ndualsof the initial CF, since their corresponding 2-dimensional arrays will sim-\nply be the transpose of the initial ones. The precise formula is\nu(m,n+1) = (R(m+1,n−1)+r(m+1,n))u(m,n)−d(m+1,n−1)u(m,n−1).\nNote that in some cases the arrays are only defined for n≥lor 2n≥lfor\ninstance, so in that case the Ap´ ery dual is not defined, although t he correspond-\ning CF may converge, but usually not to (a M¨ obius transform of) th e initial\nlimitS.\nAn important remark must be made at this point. Since the initial term s\nof a CF can obey different formulas than the generic term, the form ulas that\nare given above for by Ap´ ery’s method are only valid for the generic terms, i.e.,\ngive (a(n),b(n)) only for n“large” (usually in fact n≥2). By the elementary\ntheory of CFs, this means that the limit of the Ap´ ery accelerate or of the Ap´ ery\ndual is of the form ( AS+B)/(CS+D), whereSis the initial value, and A,B,\nC, andDare small integers which can easily be found. One could avoid this by\nprecisely keeping track of each Bauer–Muir step, but this would be e xtremely\ncumbersome. We will see this at work in all the examples below.\n4 Basic Examples\n4.1log(2)andψ(z+1/2)−ψ(z)\nWe begin by the slowly convergent series log(2) = 1 −1/2+ 1/3−1/4+···,\nwhich by Euler is transformed into the CF log(2) = ((0 ,1),(1,n2)). Using\n5Bauer–Muir, we find the following 2-dimensional arrays:\n(a(n,l),b(n,l),r(n,l),d(n,l)) = (2l+1,n2,n−l−1,−(l+1)2).\nAfter correcting for the initial terms, we find the staircase Ap´ er y accelerate\nlog(2) = (( n),((1,1),(n2,(n+ 1)2))), using an evident notation for period 2\nCFs; luckily, the contraction of this is very simple and leads to the CF\nlog(2) = ((0 ,3(2n−1)),(2,−n2)) = 2/(3−12/(9−22/(15−32/(21−42/(27−···))))),\nwhich converges like (1 +√\n2)−4n, so quite fast. The vertical Ap´ ery dual is\n((0,1),(1,n2)), in other words the initial CF which is therefore self-dual.\nBut we can do more without any additional computation: in the above\nformulas,nis a formal variable, so if we simply change ninton+zforz∈C,\nthe same recursions are valid, so we can replace nbyn+zin the arrays. Now\n/summationdisplay\nn≥1(−1)n−1\nn+z=ψ(z/2+1)−ψ(z/2+1/2)\n2,\nwhereψ(z) is the logarithmic derivative of the gamma function, so the Eu-\nler transform of this gives the CF ( ψ(z/2 + 1)−ψ(z/2 + 1/2))/2 = ((0,z+\n1,1),(1,(n+z)2)), and using the formulas for Ap´ ery we deduce the period 2 CF\n(ψ(z/2+1)−ψ(z/2+1/2))/2 = ((0,2z+1,n+z),((1,1),((n+z)2,(n+1)2)))\nand after trivial modifications\nψ(z+1/2)−ψ(z) = ((0,4z−1,n+2z−1),((2,1),((n+2z−1)2,(n+1)2))).\nContraryto the case z= 0, the contractionofthis CF is not simple. However, in\nthe special case z= 3/4,b(2n) = ((2n/2)+1/2)2andb(2n+1)= (((2n+1)/2)+\n1/2)2are given by the same formula, so this simplifies to ψ(5/4)−ψ(3/4) =\n((0,2,n+ 1/2),(2,(n+ 1)2/4)), and since ψ(5/4)−ψ(3/4) = 4−π, after an\nevident simplification we obtain the following CF for π:\nπ= ((4,4,2n+1),(−4,(n+1)2)) = 4−4/(4+22/(5+32/(7+42/(9+52/(11+···))))),\nwhich converges in O((1+√\n2)−2n).\nIn addition, we find that for z/negationslash= 0 the initial CF is not self-dual, so after a\nsmall computation we find that the dual is still another CF:\nψ(z+1/2)−ψ(z) = ((0,4z−1),(2,n2)),\nwhich converges in O((−1)n/n4z−1), to be compared with our initial CF which\nconverged in O((−1)n/n).\nIn this basic example, we see how using Ap´ ery, a single series can lead to\nmany interesting CFs.\n64.2ζ(2) =π2/6andψ′(z)\nSince this is similar, we will be brief. Starting from ζ(2) =π2/6 = 1+1/22+\n1/32+···, by Euler we have the trivial CF ζ(2) = ((0,2n2−2n+1),(1,−n4)).\nUsing Bauer–Muir, we obtain the arrays\n(a(n,l),b(n,l)) = (2n2−2n+1+l2+l,−n4),and\n(r(n,l),d(n,l)) = (−n2+(l+1)n−(l+1)2/2,−(l+1)4/4)\nThe Ap´ ery accelerate can be nicely contracted, and leads to Ap´ e ry’s famous CF\nζ(2) = ((0,11n2−11n+3),(5,n4)),\nwhich converges in O((−1)n((1+√\n5)/2)−10n), and which can be used to prove\nthe (known) irrationalityof ζ(2). The Ap´ ery dual is the CF ((0 ,2n−1),(2,n4)),\nwhich corresponds to the alternating series ζ(2) = 2(1 −1/22+1/32−···).\nUsing the same recursions, one can also Ap´ ery accelerate the mor e general\nseries/summationtext\nn≥11/(n+z)2=ψ′(z+1), and obtain in this way interesting CFs for\nψ′(z).\nWe have seen that the alternating sum 1 −1/22+1/32−···is the dual of the\nsum with positive signs, so its Ap´ ery accelerate is the same. The cor responding\nfunction/summationtext\nn≥1(−1)n−1/(n+z)2gives interesting CFs for ψ′(z/2)/2−ψ′(z).\n4.3ζ(3)andψ′′(z)\nStarting from ζ(3) = 1+1 /23+1/33+···, by Euler we obtain the CF ζ(3) =\n((0,(2n−1)(n2−n+1)),(1,−n6)). Using Bauer–Muir, we find the arrays\n(a(n,l),b(n,l)) = (2n3−3n2+(4l2+4l+3)n−(2l2+2l+1),−n6),and\n(r(n,l),d(n,l)) = (−n3+2(l+1)n2−2(l+1)2n+(l+1)3,(l+1)6).\nThe Ap´ ery accelerate can be nicely contracted, and leads to Ap´ e ry’s famous CF\nζ(3) = ((0,(2n−1)(17n2−17n+5)),(6,−n6)),\nwhichconvergesin O((1+√\n2)−8n)andwhichheusedtogetherwithDiophantine\nresults to prove the irrationality of ζ(3). The initial CF is self-dual.\nThe correspondingfunction/summationtext\nn≥11/(n+z)3gives interesting CFs for ψ′′(z).\nIn particular, its Ap´ ery dual gives the CF\nζ(3,z+1) =−ψ′′(z+1)/2 = ((0,n3+(n−1)3+2z(z+1)(2n−1)),(1,−n6)),\nwith speed of convergence O(1/n4z+2), a formula due to Ramanujan. This is\npossibly one of the simplest proofs of this formula.\n74.4 An Example where Ap´ ery can be Iterated\nWe have mentioned that Bauer–Muir cannot reasonably be applied ite ratively\nwhentheconvergenceisin O(1/Bn)withBirrational. Inparticular,inthethree\nexamples above the Ap´ ery accelerates have B= (1+√\n2)4,B= ((1+√\n5)/2)10,\nandB= (1+√\n2)8, so there is no hope of applying Bauer–Muir and a fortiori\nAp´ ery once again. However, there are cases where it can be done , as in the\nfollowing example.\nWestartwiththefollowingCF:21/3= ((1/2,7n−5),(1,−4n(3n−2))),which\nconverges in 1 /((4/3)nn5/3), so already exponentially fast with B= 4/3. Using\nthe Ap´ ery formulas we obtain the period 2 CF ((4 n,4n+2),(−3n2+2n,−3n2−\n8n−5)), where we do not give the initial terms which have to be determine d at\nthe end (the complete CF is in fact (((1 /2,2),(4n,4n+2)),((1/2,−5),(−3n2+\n2n,−3n2−8n−5)))), but since we are not finished it is a waste of time to\ndo it here. This converges in 1 /(−3)n, which is already faster. To be able to\ncontinue, we need a period 1 CF, so we contract the above CF, and w e are in\nluck, the simplifications is still very simple: ((5(2 n−1)),(−(9n2−4))), which\nof course converges in O(1/(−3)2n) =O(1/9n) (the complete CF is in fact\n((1/2,3,5(2n−1)),(2,−(9n2−4)))).\nSince 9 is rational, we can try to use Ap´ ery once again, and indeed it w orks,\nand another miracle happens, the period 2 CF does not need to be co ntracted,\nit is already in fact a period 1 CF ((9(2 n−1)),(−(9n2−16))), which converges\ninO((−(1+√\n2)4)−n), and since B=−(1+√\n2)4is irrational, we cannot use\nAp´ ery anymore.\nTo finish and obtain the complete CF, we need to find the initial terms o f\nthe CF. To do this, we proceed as follows. We start from the (almost certainly\nwrong) period 1 CF obtained above, and compute its limit Lnumerically . Since\nthe convergenceis very fast, this is easy, and we find L=−7.2728···. We know\nthat 21/3= (AL+B)/(CL+D) for (hopefully small) integers A,B,C,D: to\nfind them, we use a linear dependence algorithm such as LLL on the nu mbers\n(1,L,21/3,21/3L), and we find immediately that 21/3= (−L−13)/(2L+ 10),\nat least numerically. Note that, even though this is a numerical henc e non-\nrigorous check, it would be easy but tedious to proveit. Now the CF begins as\n−9+16/(9+7/27+···), so we replace Lby−9+16/(9+x),wherexrepresents\nthetailoftheCFstartingat7 /(27+···), andwefind21/3= (x+13)/(2x+10)=\n1/2+4/(x+5). We thus create the CF ((1 /2,5,9(2n−1)),(4,−(9n2−16))),\nand we check that its limit is indeed 21/3, so\n21/3= ((1/2,5,9(2n−1)),(4,−(9n2−16))).\n5 Generalizations of Ap´ ery’s Method\n5.1 Case when r(n,l)is a rational function in l\nEven though we look by indeterminate coefficients or otherwise for r(n,l) as a\npolynomial in n, it may happen that as a function of lit is not a polynomial (as\n8in all the above examples), but a rational function of l. Consider for instance\nthe following (admittedly already rather complicated) example. Set\nG3=L(χ−3,2) =/summationdisplay\nn≥1(−3/n)\nn2= 1−1\n22+1\n42−1\n52+···\nIt is possible to prove that we have the following CF:\nG3= ((3/4,(2n−1)(9n2−9n+22)),(2/3,−9n4(9n2−1))),\nwhich converges in O(1/4n). Using Bauer–Muir, we find the arrays\n(a(n,l),b(n,l)) = ((2n−1)(9n2−9n+18(l+1)2+4),−9n4(9n2−1)),\nr(n,l) =−9n3+(18l+27)n2−(18l2+54l+40)n\n+(2/9)(3l+4)2(3l+5)2/(2l+3),and\nd(n,l) = (4/81)(3l+4)4(3l+5)4/(2l+3)2.\nWe see that r(n,l) (hence also d(n,l)) is a rational function in l. Note that the\nAp´ ery accelerate and Ap´ ery dual, while completely explicit, are rat her compli-\ncated, but the main purpose of this example was to show that r(n,l) can be a\nrational function in l.\n5.2 Need for a simplifier\nConsider the following CF for π, which converges as O(1/(4nn3/2)):\nπ= ((3,24,20n2+4n+1),(3,−8n(2n+1)3)).\nApplying Bauer–Muir, we find that the new a(n) has a denominator 2 n−1,\nwhich indicates that if we continue in order to find our 2-dimensional a rrays,\nthe degree of the denominator will increase, making the formulas un wieldy.\nHowever, recall that if ( a(n),b(n)) is a given CF, for any nonzero function t(n)\nwitht(0) = 1, the CF ( t(n)a(n),t(n)t(n+1)b(n)) will havethe same convergents\np(n)/q(n) (more precisely, both p(n) andq(n) will be multiplied by the same\nquantitytf(n) :=t!(n) =t(1)t(2)···t(n)). In our particular case, choosing\nt(n) = (2n−1)/(2n+1)not only gets rid ofthe denominator, does not introduce\nnew denominators, and also keeps constant the degrees of a(n) andb(n). Thus,\nwe generalize the Ap´ ery recursions given above as follows:\nu(n+1,l) =a(n+1,l)u(n,l)+b(n,l)u(n−1,l),\nu(n,l+1) =tf(n,l)(r(n+1,l)u(n+1,l)+u(n,l)),\nR(n,l) =a(n+1,l)+r(n+1,l), d(n,l) =r(n,l)R(n,l)−b(n,l),\ntf(n,l) =t(n,l)tf(n−1,l),\na(n,l+1) =t(n,l)(R(n,l)−r(n−1,l)d(n,l)/d(n−1,l)),\nb(n,l+1) =t(n,l)t(n+1,l)b(n,l)d(n+1,l)/d(n,l).\n9The formulas for the staircase walks are modified as follows:\nu(n+m+1,n+1) =t(n+1+m,n)R(n+1+m,n)u(n+m,n+1)\n−tf(n+m+1,n)d(n+m+1,n)u(n+m,n),and\nu(n+m,n+1) =tf(n+m,n)R(n+m,n)u(n+m,n)\n+tf(n+m,n)b(n+m,n)u(n+m−1,n),\nand the formula for the vertical walks is:\nu(m,n+1) = (t(m+1,n−1)R(m+1,n−1)+r(m+1,n))u(m,n)\n−t(m+1,n−1)d(m+1,n−1)u(m,n−1).\nComing back to our example of a CF for π, we find the arrays:\n(a(n,l),b(n,l)) = (20n2+(8l+4)n+1,−4(2n−l)(2n+1−l)(2n+1)2),\nr(n,l) =−4n2+(8l+4)n−(12l2+20l+9),\nd(n,l) =−144(2n−l+1)(l+1)3,and\n(t(n,l),tf(n,l)) = ((2n−l−1)/(2n−l+1),−(l−1)/(2n−l+1)).\nBecause of the denominator 2 n−l+1, the vertical walk gives a CF which is\nunrelated to the initial one (and in fact converges to a limit related to Catalan’s\nconstant). The staircase walks are OK since they us l=n+m, which is smaller\nthan 2nfornlarge, and give semi-complicated period 2 CFs for πconverging\nlike (−1)n/((1+√\n5)/2)5n.\n5.3 Need for a multiplier: the Case of ζ(4) =/summationtext\nn≥11/n4\nThe next variant that we will study is perhaps the most interesting ( of course\nin addition to the basic one which is already extremely useful), since it p rovides\nfor instance a nice CF for ζ(4) (found in 1980 by G. Rhin and the author using\nexactly this method), and a new CF for ζ(3).\nWe can try to proceed as for ζ(2) andζ(3). Applying Euler to the series, we\nobtain the trivial CF ζ(4) = ((0,2n4−4n3+6n2−4n+1),(1,−n8)). Applying\nBauer–Muir creates a denominator 28( n−1)2−9 ina(n) and and 28 n2−9 in\nb(n). We could try to use a simplifier, but the numerator of a(n) being a degree\n6 irreducible polynomial, this is hopeless.\nOur salvation comes from an a priori bad idea of using a “simplifier” (he re,\nthe opposite, a “complexifier”) on the initial CF, by multiplying the CF b y\n2n−1, in other words, using as initial series the more complicated\nζ(4) = ((0,(2n−1)(2n4−4n3+6n2−4n+1)),(1,−(2n−1)(2n+1)n8)).\nThe “miracle” is that when we now apply Bauer–Muir, the denominator is\nsimplyn−1, and one easily checks that the simplifier t(n) = (2n−2)/(2n)\n10works. And it is immediate to check that this continues, so that we ca n apply\nAp´ ery with simplifier to this more complicated CF. We find:\na(n,l) = (2n−1)/parenleftbigg\n2n4−4n3+6n2−4n+1+l(l+1)\n2(17n2−17n+5)/parenrightbigg\n,\nb(n,l) =−n6(4n2−l2)(4n2−(l+1)2)/4,\nr(n,l) = (2n+l)/parenleftbigg\n−n4+7\n2(l+1)n3−6(l+1)2n2+6(l+1)3n−3(l+1)4/parenrightbigg\n,\nd(n,l) =−9(l+1)8(4n2−l2),\nt(n,l) = (2n−l−2)/(2n−l),\ntf(n,l) =−l/(2n−l).\nOnce again since l <2nthe vertical walks are unrelated to ζ(4). On the\nother hand, the staircase walks are well-defined, and for m= 0 its contraction\nis reasonably simple and gives the CF\nζ(4) =π4/90 = ((0,3(2n−1)(3n2−3n+1)(15n2−15n+4)),(13,3n8(9n2−1))),\nwhich converges like O((2+√\n3)−6n), not sufficiently fast to have Diophantine\napplications, and as already mentioned, which was discovered by G. R hin and\nthe author in 1980, see [5].\nAsusual,wecanusethesamerecursionstofindCFsfor ψ′′′(z) = 6/summationtext\nn≥01/(n+\nz)4by replacing zbyn+z−1. This gives rather complicated formulas. The\nonly slightly interesting one is the following, whose limit I do not know, bu t\nwhich gives the asymptotic expansion ofψ′′′(z) asz→+∞:\nψ′′′(z)∼((0,(2n−1)(n4−2n3−2(z2−z−1)n2+(2z2−2z−1)n\n−(z−1)z(z2−z+1))),(4z−2,−n8(n2−(2z−1)2))).\nMore precisely, changing zinto 1/zand simplifying by using z4as simplifier,\nwe find a CF such that b(n) is divisible by z6forn≥1, so can be expanded in\na power series, and we find\nψ′′′(z) =2\nz3+3\nz4+2\nz5−1\nz7+4/3\nz9−3\nz11+10\nz13−691/15\nz15+···,\nwhich is indeed the asymptotic expansion of ψ′′′(z) for largez(note that the\ncoefficients of 1 /z2kvanish fork/negationslash= 2 as well as the telltale presence of 691, both\nsure signs of Bernoulli numbers).\n5.4 Need for a multiplier: a New CF for ζ(3)\nWe now give another example which leads to a new, but not especially int er-\nesting, CF for ζ(3). Here we start with the alternating series ζ(3) = (4/3)(1−\n1/23+ 1/33−1/43+···), which by Euler is equivalent to the CF ζ(3) =\n11((0,3n2−3n+1),(4/3,n6)). Once again, applying Bauer–Muir directly leads to\ncomplicated denominators. Exactly as for the case of ζ(4), we use a multiplier\nbefore applying Bauer–Muir.\nBefore giving the answer, I will explain how to find such factors witho ut\nguesswork. Let n+ube the unknown linear factor. Using it as a multiplier, we\nobtain the CF ((0 ,(n+u)(3n2−3n+ 1)),(u+1,(n+u)(n+u+1)n6)). We\nnow apply the Bauer–Muir formulas, keeping uas a formal variable. We find\nfor instance that a(n) =N(n,u)/D(n,u) for complicated polynomials Nand\nDin the variables nandu. Since we want this to simplify, we compute the\nresultant ofN(n,u) andD(n,u) with respect to the variable n. This gives a\npolynomial in ualone, and since we want uto be a rational number, we easily\nfind that the rational roots of this polynomial are u=−1/2 with multiplicity\n4, andu=−39/14 with multiplicity 6. Trying both, we find that u=−39/14\ngives a denominator of degree 2, probably difficult to simplify, while u=−1/2\ngives a denominator 2 n−3, which is promising.\nIndeed, using as multiplier 2 n−1 = 2(n−1/2), i.e., using the CF 3 ζ(3)/4 =\n((0,(2n−1)(3n2−3n+1)),(1,(2n−1)(2n+1)n6)), we find the following arrays:\n(a(n,l),b(n,l)) = ((2l+1)(2n−1)(3n2−3n+1),n6(4n2−(2l+1)2)),\nr(n,l) = (n−2(l+1))(2n+2l+1)(n2−2(l+1)n+4(l+1)2),\nd(n,l) =−64(l+1)6(4n2−(2l+1)2),and\n(t(n,l),tf(n,l)) = ((2n−(2l+3))/(2n−(2l+1)),−(2l+1)/(2n−(2l+1))).\nSince 2n−2l+1never vanishes, the Ap´ erydual is related to ζ(3), and indeed\nwe find that after simplification it gives the trivial CF coming by Euler fr om the\nseries with positive terms giving ζ(3), so totally uninteresting. On the contrary,\nthe Ap´ ery accelerate is a period 2 CF which can be reasonably simplifie d and\ngives the new CF:\nζ(3) = ((0,65n4−130n3+105n2−40n+6),(7,−4(16n2−1)n6)),\nwhich converges like O(1/64n), so quite fast, but not sufficiently fast to have\nany Diophantine application.\nNote that even though 64 is a rational number, applying Bauer–Muir to this\nCF leads to complicated denominators, so it is hopeless to try and use Ap´ ery\nonce again.\n5.5 Slowing Down Ap´ ery’s Method\nThemaingoalofAp´ ery’smethodistoaccelerateasmuchaspossible acontinued\nfraction. However, we have seen above that it is also very useful f or creating\nnew CFs. Thus, we can try to use sub-optimal Bauer–Muir iterations, together\nwith Ap´ ery’s diagonal or staircase walks, in order to find new CFs. W e give a\ndetailed example of how to proceed.\n12Consider the CF π2= ((0,2n2−2n+1),(6,−n4)), which is simply Euler’s\ntransformation of the series π2= 6/summationtext\nm≥11/m2. Using Ap´ ery’s method using\noptimal Bauer–Muirleads immediately to his famous CF for π2, as we haveseen\nin Section4.2. The first Bauer–Muiriterationisdone with r(n) =−n2+n−1/2,\nwhich is optimal since it leads to a constant d(n). Let us use instead r(n) =\n−n2+n+r0, for some unknown r0. To find a suitable r0, we would like the\nnexta(n) andb(n) to remain polynomials. Keeping r0as a formal variable, we\nthus compute d(n) =r(n)(a(n+1)+r(n+1))−b(n), thena′(n) =a(n+1)+\nr(n+1)−r(n−1)d(n)/d(n−1) andb′(n) =b(n)d(n+1)/d(n).\nBoth are rational functions in nandr0, and we want them to simplify, so we\nagaincomputethe resultant withrespectto nofthenumeratoranddenominator\nof both, thus obtaining two polynomials in r0, we compute their GCD, and\nfinally the roots, and we find that there are two possibilities r0=−1/2, which\nis the optimal one, and also r0= 0. We thus try r(n) =−n2+n, and we\nfindd(n) =n(which is of course not constant), and a′(n) = 2n2−n+ 1,\nb′(n) =−n3(n+ 1). We check that the corresponding CF has indeed been\naccelerated: it converges in O(1/n2) instead of O(1/n) (the optimal one with\nr0=−1/2 converges in O(1/n3)).\nOf course we now want to continue. Using the same method on this ne w\nCF, we now find that we have three possibilities for r0:r0=−2/3, 0, and\n2. The value r0=−2/3 corresponds to the optimal Bauer–Muir (i.e., with\nconstantd(n)), but the other two values are plausible, and there is no reason\nto choose one over the other. We decide (arbitrarily) to choose th e largestr0,\nwhich will probably give the slowest acceleration, and in this way we obt ain\nvery nice 2-dimensional arrays:\n(a(n,l),b(n,l)) = (2n2+(l−2)n+1,−n3(n+l)),\n(r(n,l),d(n,l)) = (−n2+n+l(l+1),(l+1)3(n+l)).\nThe vertical walk gives the initial CF, but the staircase walk can be nic ely\ncontracted into a diagonal walk, and after correcting for the initia l terms, we\nfind a new (but known) CF for π2:\nπ2= ((0,5n2−4n+1),(18,−2n3(2n−1))),\nwhich converges in O(1/(4nn3/2)).\nSinceB= 4 is rational, we can apply Ap´ ery’s method once again (slow or\nnot), but we do not find any new CF but “only” Ap´ ery’s CF.\nWe finish by giving another example of slow Ap´ ery, which leads to an ap -\nparently completely new CF for log(2). Applying Euler’s transformat ion to the\nseries log(2) = −log(1−1/2) = (1/2)+(1/2)2/2+(1/2)3/3+···gives the CF\nlog(2) = ((0 ,3n−1),(1,−2n2)), which of course converges at exactly the same\nspeed as the series, in O(1/(n2n)). We proceed exactly as before, in fact again\nwithr(n) =−n2+n+r0for an unknown r0, and we immediately find again\n13very nice 2-dimensional arrays:\n(a(n,l),b(n,l)) = (3n2−n+2l(2n−1),−2n3(n+2l+1)),\n(r(n,l),d(n,l)) = (−n2+n+2(l+1)(2l+1),4(n+2l+1)(n+2l+2)(l+1)2).\nThe vertical walk gives the interesting but known CF\nlog(2) = ((0 ,4n2−3n+1),(1/2,−2n3(2n+1)))\nwhich converges very slowly in O(1/n1/2). The “miracle” (which, as already\nmentioned, does not occur very often) is that the staircase walk c an again be\nnicely contracted into a diagonal walk, and after correcting for th e initial terms,\nwe find a new (and this time I believe completely new) CF for log(2):\nlog(2) = ((0 ,29n2−29n+8),(5,−6n2(9n2−1))),\nwhich converges in O(1/(27/2)n), which is rather fast.\nSince 27/2 is rational, we can try to apply Ap´ ery’s method once again (slow\nornot), buthereBauer–Muirintroducesdenominatorswhichcann otberemoved\nusing simplifiers.\nA similar procedure also gives the following new CF for log(2):\nlog(2) = ((0 ,59n2−59n+20),(5,−24n2(36n2−1))),\nwhich converges in O(1/(32/27)n).\nReferences\n[1] [Ape] R. Ap´ ery, Irrationalit´ e de ζ(2)etζ(3), Ast´ erisque 61(1979), 11–13.\n[2] [Bat-Oli] C. Batut and M. Olivier, Sur l’acc´ el´ eration de la convergence de\ncertaines fractions continues , S´ eminaireTh. des Nombres Bordeaux(1979–\n1980),9, 1–26.\n[3] [Bel-Coh] K. Belabas and H. Cohen, Numerical Algorithms for Number\nTheory using Pari/GP , AMS Math. Surveys and Monographs 254(2021).\n[4] [Coh] H. Cohen, Continued fractions of polynomial type: theory and ency-\nclopedic dictionary , 420p., in preparation.\n[5] [Coh-Rhi] H. Cohen and G. Rhin, Acc´ el´ eration de la convergence de\ncertaines r´ ecurrences lin´ eaires , S´ eminaire de Th. Nombres Bordeaux\n(1980/81), Expos´ e 16.\n[6] [Cuyt] A. Cuyt, V. Petersen, B. Verdonk, H. Waadeland, and W. Jones,\nHandbook of Continued Fractions for Special Functions , Springer Nether-\nlands (2008).\n14[7] [Jac] L. Jacobsen (ed.), Analytic theory of continued fractions III , Lecture\nNotes in Math. 1406, Springer (1989).\n[8] [Jea] R. Andr´ e-Jeannin, Irrationalit´ e de la somme des inverses de certaines\nsuites r´ ecurrentes , C. R. Acad. Sci. 308(1989), 539–541.\n[9] [Jon-Thr] W. Jones and W. Thron, Continued fractions. Analytic theory\nand applications , Enc. Math. and Appl., Addison–Wesley (1980).\n[10] [JTW] W. Jones, W. Thron, and H. Waadeland (eds), Analytic theory of\ncontinued fractions , Lecture Notes in Math. 932, Springer (1982).\n[11] [Kho] A. Khovanskii, The application of continued fractions , Noordhoff\n(1963).\n[12] [Khr] S. Khrushchev, Orthogonal polynomials and continued fractions,\nFrom Euler’s point of view , EncyclopediaMath.andAppl. 122, Cambridge\n(2008).\n[13] [Lor-Waa] L. Lorentzen and H. Waadeland, Continued fractions , Atlantis\nStudies in Math. 1(2008).\n[14] [PWZ] M. Petkovsek, H. Wilf, and D. Zeilberger, A=B, A. K. Peters\n(1996).\n[15] [Sti] T. Stieltjes, Recherches sur les fractions continues ,\nAnn. Fac. Sci. Toulouse 8(1894), J1–122; 9(1895), A1–47.\n[16] [Thr] W. Thron (ed.), Analytic theory of continued fractions II , Lecture\nNotes in Math. 1199, Springer (1986).\n[17] [Wal] H. Wall, Analytic theory of continued fractions , van Nostrand (1948).\n15" }, { "title": "2401.17756v1.Electromagnetic_form_factors_of_the_transition_from_the_Delta_to_the_nucleon.pdf", "content": "Electromagnetic form factors of the transition from the Delta to the nucleon\nMoh Moh Aung,1, 2Stefan Leupold,2Elisabetta Perotti,2, 3and Yupeng Yan1\n1School of Physics, Suranaree University of Technology, 111 University Avenue, Nakhon Ratchasima 30000, Thailand\n2Institutionen f¨ or fysik och astronomi, Uppsala universitet, Box 516, S-75120 Uppsala, Sweden\n3University of Colorado Boulder, Department of Electrical, Computer, and Energy Engineering, Boulder, USA\n(Dated: February 1, 2024)\nThe low-energy electromagnetic form factors of the ∆(1232)-to-nucleon transition are derived com-\nbining dispersion theory techniques and chiral perturbation theory. The form factors are expressed\nin terms of the well-understood pion vector form factor and pion-baryon scattering amplitudes.\nNucleon and Delta exchange terms and contact terms constitute the input for these pion-baryon\namplitudes. The framework is formulated for all form factors. When comparing to experimental\ndata in the spacelike region of e−N→e−∆ scattering, the focus lies on the numerically dominant\nmagnetic dipole transition form factor. Fitting two subtraction constants (one for the scattering\namplitude, one for the form factor) yields a very good description of this dominant form factor up to\nphoton virtualities of about 0.6 GeV. After determining the subtraction constants in the spacelike\nregion and at the photon point, respectively, predictions for the timelike region of Dalitz decays\n∆→N e+e−are presented.arXiv:2401.17756v1 [hep-ph] 31 Jan 20242\nI. INTRODUCTION AND SUMMARY\nOne of the interesting features of quantum field theory is the possibility that electrically neutral particles can\ninteract with photons. Even the Higgs boson as an “elementary” particle couples to two photons [1, 2]. In turn\nthis means that one can use electromagnetic processes to learn something about uncharged objects. Concerning\nthe neutron, its exploration went through several levels of understanding of its properties and structure, and this\nprocess has not come to an end yet. The neutron’s non-vanishing magnetic moment points to its substructure in\nterms of electrically charged objects, the quarks. Yet, the example of the two-photon decay of the Higgs boson tells\nthat charged constituents are not the only possibility to explain a non-trivial electromagnetic response of an object\ndescribed by quantum field theory. Also virtual many-body fluctuations can significantly contribute. Indeed the\nspatial charge distribution of the neutron [3] with a negatively charged surface suggests fluctuations into a negative\npion and a proton. Since the pion is so much lighter than other hadrons, a phenomenon caused by spontaneous\nchiral symmetry breaking, the large-distance electromagnetic response of the neutron is dominated by pion physics,\nirrespective of the detailed quark substructure. What really points to the composite nature of the neutron (and the\nproton) is the possibility to excite it to a higher-lying state. And also here electromagnetic processes come into play\nand pion physics is important. In this work we address the excitation of the neutron to the ∆0resonance. Of course,\nthis process is related to the excitation of the proton to ∆+via isospin symmetry, but we regard the electromagnetic\nexcitation of a neutral object as more intriguing. Therefore we will formulate many aspects of our framework in terms\nof neutrons and neutral Deltas. Yet, when comparing to data we utilize isospin symmetry and compare to the better\nmeasured reactions with protons and positive Delta states.\nElectromagnetic form factors (FFs) allow for a general characterization of composite objects, even when the intrinsic\nstructure of said objects is not yet fully understood. The FFs are functions of the momentum transfer squared q2,\nwhich means that they can be experimentally addressed in different kinematical regions, based on the choice of the\nreaction. Fig. 1 displays both the spacelike ( q2<0) and the timelike ( q2>0) regions. By scattering electrons on a\nbaryon, e−B1→e−B2, one has access to the spacelike region. The timelike region is explored via Dalitz decays into\na lighter baryon and an electron-positron pair, B1→B2e+e−, and via electron-positron annihilation into a baryon-\nantibaryon pair, e+e−→B1¯B2. The electromagnetic FFs are useful tools to investigate the intrinsic properties of\nhadrons. The interest in their determination is driven by the desire to better understand the properties of matter\n[4, 5].\nq2\n4m2e (m1−m2)2(m1+m2)20B1 B2\ne−e−B1 B2\ne−e+e−e+B1\n¯B2\nFIG. 1. Space- and timelike FFs can be accessed exploiting different reactions. Processes proceed from left to right. The\nmomentum of the virtual photon (wavy line) is denoted by q, the mass of electron e−and positron e+byme, the mass of\nbaryon Bibymi,i= 1,2.\nThis work focuses on the transition from the Delta to the nucleon and is the third in a series of similar studies.\nThe Uppsala (UU) group has previously considered other transition form factors (TFFs), namely the Σ0-Λ TFFs [6]\n(which are the only ones among the ground-state spin-1/2 baryons), and the Σ∗0-Λ TFFs (which involve a decuplet\nbaryon) [7]. These, as well as the ∆0-nTFFs, have in common that they are purely isovector FFs. However, the\nexperimental situation is rather different for these TFFs. The unstable nature of hyperons and therefore the major\ncomplication to obtain spacelike TFFs from fixed-target experiments1motivated our previous research program where\n1One would need a hyperon beam that scatters on the electron cloud of atoms in the target.3\nwe explored the possibility to learn about the spacelike region from dispersion theory and input from Dalitz decay\nmeasurements in the timelike region (see also [8]). In fact, the spacelike region is of primary interest from the point\nof view of hadron-structure studies. It is where the interpretation of the FFs as spatial distributions of e.g. electric\ncharge is possible [3, 9, 10].\nIn the present work we turn from hyperons to those low-lying baryons where the minimal quark content is provided\nsolely by the up and down quarks. These are the nucleon and the ∆(1232). Such studies will serve to scrutinize our\nmethods developed in the hyperon sector. The direct vector-isovector FFs of the nucleon have been addressed already\nin [11, 12]. Here we study the ∆-nucleon transitions. Of course, such studies are also interesting in their own right,\nirrespective of the exploration potential in the hyperon sector. Recall that the previous strategy has been to provide\npredictions for the low-energy spacelike TFFs of hyperons, for which there are no experimental data. This can be\nachieved provided that the TFFs are measured at least in the low-energy timelike region, and by means of theoretical\ntools like dispersion relations and chiral perturbation theory ( χPT). Thanks to dispersion relations, the TFFs can be\nanalytically continued into the experimentally not easily accessible region.\nRather opposite to the hyperon case, there are data for the ∆-nucleon TFFs in the spacelike region, see [5, 13–16]\nand further references therein. This fact makes us turn our strategy around compared to the hyperon case. Thus the\nscope of this paper is twofold. On the one hand, we want to make predictions for the ∆0-nTFFs (coinciding with\n∆+-pin the isospin limit) and in particular for Dalitz decays. On the other hand, we want to test the validity of our\nmethods against real data.\nOur framework [6, 7, 11, 12, 17] is based on dispersion theory, a model-independent approach justified by the\nfundamental properties of local relativistic quantum field theory: micro-causality, analyticity, unitarity and crossing\nsymmetry. Restricting ourselves to low energies allows for an effective-field-theory point of view. Only the low-\nlying degrees of freedom need to be considered explicitly, the short-distance physics is covered by contact terms\n(subtraction constants in the language of dispersion relations). The challenge is to include the longer-distance physics\nin a model-independent way. Effectively our framework includes the physics of pions, rho mesons, nucleons and Deltas.\nIn principle, χPT [18–24], the model-independent low-energy incarnation of QCD, includes nucleons and pions and\nallows for an extension to ∆ states. But it is difficult to include the rho meson in a model-independent way; see\nalso the corresponding discussions in [12, 25]. On the other hand, the rho meson is an elastic resonance of pion-pion\nscattering. The pion phase shifts are very well known [26, 27]. Dispersion theory allows for the systematic inclusion\nof phase shifts. In this way, the physics of the rho meson is covered in a model-independent fashion. Concerning\nelectromagnetic FFs at low energies, the virtual photon couples dominantly to the lightest degrees of freedom, the\npions. This is quantified by the pion vector form factor, which is also very well known [28]. The additional ingredient\nare pion-baryon scattering amplitudes. Since the physics of the rho meson is already covered by the dispersive part,\nthe pion-baryon scattering amplitudes need to cover the physics of the lightest baryon degrees of freedom. For the\nhyperon cases [6, 7], the UU group used χPT to obtain those hadronic amplitudes because there are no direct data\non pion-hyperon scattering. We follow here the same path.\nIn this work we use dispersive representations for the TFFs or linear combinations thereof. A comparison to previous\nworks is in order here. The reaction e−N→e−∆ is part of the physical reaction e−N→e−Nπ. Somewhat more\nformally this is e−→e−γ∗with subsequent γ∗N→∆→Nπ. Here the ∆ is typically extracted from a partial-wave\nanalysis since the Nπsystem does not only couple to the ∆ resonance. Dispersion theory for the Nπsystem has been\nused among other methods; for a review see [5]. In this way, TFFs have been extracted from data. But dispersion\ntheory has not been used for the TFFs themselves, i.e. it has not been used for the γ∗invariant mass instead of the\nNπinvariant mass. We fill this gap with our present exploratory study of the use of dispersion relations for the FFs\nof the Delta-to-nucleon transition.\nWe regard it also as important to stress what will notbe covered by our present work. We do not aim at a complete\ndescription of the (formal) reaction γ∗N→∆→Nπ. Instead we focus on the photon virtuality. This implies in\nparticular that we treat the ∆ as if it were an asymptotic state with the peak mass of the resonance. A fully dispersive\ntreatment of both virtualities of γ∗andNπis way beyond the scope of the present work. We note however that other\nworks (using χPT) deal with the ∆ as a pole in the complex plane [23, 24] and not as a stable state. Such a more\nrealistic treatment in dispersion theory instead of χPT would require a generalization of the optical theorem to the\ncomplex plane, which is also beyond the scope of the present work.\nIn addition, we note that one crucial input of our formalism are the scattering amplitudes ∆- πtoN-π. In line with\nour previous works [6, 7, 11, 12], we determine these amplitudes from χPT and unitarize them in the Muskhelishvili-\nOmn` es framework. This means that in the formal cross channel ∆- ¯Ntoπ-πthe pion rescattering is fully taken into\naccount by the pertinent pion phase shift, i.e. we have a framework for the formal reaction ∆ ¯N→ππ→ππthat\nrespects unitarity and analyticity. The only required input is the baryon dynamics. Following [6, 7, 11, 12], we cover\nthis aspect by χPT. This involves diagrams with an intermediate ∆ propagator corresponding to the formal scattering\nreaction ∆ π→∆→Nπ. Given that the ∆ is an elastic resonance in the N-πchannel, those ∆ exchange diagrams\nmight be seen as an approximation to the full rescattering into N-π, formally ∆ π→Nπ→Nπ. This means that4\nwe treat Nπintermediate states in the baryon-pion channels differently from ππintermediate states in the crossed\nchannel. Since the elastic nucleon-pion scattering amplitudes are very well known [29], there would be an alternative\nto our treatment of the scattering amplitudes in χPT. In principle, one could deal with such scattering amplitudes in\nthe very same way concerning direct and crossed channels. For instance, such a framework has been formulated for\nthe amplitudes η′-πtoη-πin [30]. Given that we treat the pion-pion rescattering dispersively, it would be appealing\nto treat also the pion-nucleon rescattering dispersively. On the other hand, we want to test the quality of our hyperon\ncalculations [6, 7] where no data exist for hyperon-pion scattering amplitudes. But it should be clear that our input,\nχPT (at next-to-leading order), is an approximation. Therefore one should not expect that we can obtain an accuracy\nat the percent level.\nIn view of all these possible future improvements — complete description of the reaction γ∗N→Nπand/or treating\nthe ∆ as a pole in the complex plane and/or treating not only pion-pion scattering but also pion-nucleon scattering\ndispersively — we regard the present work as a first exploratory study of our formalism. Consequently we will develop\nthe formalism for all TFFs but focus for the results on the dominant (magnetic) TFF. We will see that we obtain very\nencouraging results for the magnetic dipole TFF. With two subtraction constants — one for the TFF and one for the\nbaryon-meson scattering amplitude — fitted to the spacelike data, we can reproduce the magnetic TFF up to photon\nvirtualities of about 0.6 GeV in the spacelike region. The subtraction constants parametrize our ignorance of the\nshort-distance physics while the longer-range physics (pion rescattering in the 2-pion channel and nucleon and Delta\nexchange in the baryon-pion channels) is explicitly covered by our framework. The results are practically insensitive to\nthe phenomenological uncertainties of our other input parameters that are related to the longer-range physics and not\nfitted to the TFF data but obtained from other sources. The successful description of the spacelike data suggests that\nthe previous hyperon calculations have also a solid foundation and can be used to translate experimental information\nfrom the timelike to the spacelike region as proposed, for instance, in [6].\nAt low energies, the other two TFFs (electric and Coulomb quadrupole) are much smaller than the magnetic TFF\n(few-percent level) [5]. In view of our approximations and the exploratory nature of our present work, we do not\nexpect that we can reproduce these smaller TFFs very well. Still we will address the electric TFF in an extended\noutlook section and find that also these results are encouraging. Yet the main focus of the present work will lie on\nthe magnetic transition.\nThe rest of the paper is structured in the following way. In Section II, we introduce first TFFs that are free of\nkinematical constraints. We relate them to helicity amplitudes and to Jones-Scadron TFFs [31]. TFF ratios commonly\nused by the experimental groups are introduced for the spacelike region of ∆ production in electron-nucleon scattering.\nThe TFFs are also related to differential decay rates in the timelike region of Dalitz decays ∆ →N e+e−. In Section\nIII, we recall the main aspects of the dispersive framework that has been provided already in [7]. One interesting\naspect is the appearance of an anomalous cut. An appendix is reserved for the technical aspects. In practice, we use\nthe dispersion relations for Jones-Scadron TFFs, in particular for the dominant magnetic dipole TFF. In Section IV,\nwe specify the input from χPT at leading and next-to-leading order, used for the baryon-pion scattering amplitudes\nthat enter the dispersive framework. In Section V we discuss in detail the results for the magnetic TFF and related\nquantities. We provide predictions for Dalitz decay distributions. Section VI constitutes an extended outlook where\nwe explore whether our formalism is accurate enough to address one of the smaller Jones-Scadron TFFs, namely the\nelectric quadrupole TFF. Several appendices are added where we discuss technical aspects and details that would\ninterrupt too much the central line of reasoning in the main text.\nII. TRANSITION FORM FACTORS AND OBSERVABLES\nThe three ∆- nTFFs are defined in agreement with [7, 32, 33] as\n⟨0|jµ|∆¯n⟩=e¯vn(pn, λ) Γµν(p∆, pn)uν\n∆(p∆, σ) (1)\nwith\nΓµν(p∆, pn) :=−(γµqν− ̸q gµν)m∆γ5F1(q2)\n+ (pµ\n∆qν−p∆·q gµν)γ5F2(q2)\n+ (qµqν−q2gµν)γ5F3(q2) (2)\nandq:=p∆+pn. The neutral spin-3/2 Delta hyperon is denoted by ∆. Conventions for the spin-3/2 spinor\nuµcan be found in the Appendix B of Ref. [7]. The helicities of ∆ and ¯ nare denoted by σandλ, respectively.\nThe TFF definition of Eq. (1) reflects the fact that from a formal point of view we are interested in the process\n∆¯n→π+π−→γ∗. This is what enters our dispersive calculation. In practice, however, other kinematical regions5\nsuch as e+e−→γ∗→∆¯n,e−n→e−∆ and ∆ →nγ∗→n e+e−can be experimentally studied. In addition, the very\nsame TFFs enter e+e−→γ∗→∆+¯p,e−p→e−∆+and ∆+→pγ∗→p e+e−, respectively.\nFor the amplitude relevant for the Dalitz decay, ∆ →n γ∗, one finds\n⟨n|jµ|∆⟩=e¯un(pn, λ) Γµν(p∆,−pn)uν\n∆(p∆, σ). (3)\nThis leads to the very same expression as on the right-hand side of (2) but with q:=p∆−pn.\nFor later use we introduce the K¨ all´ en function\nλ(a, b, c ) :=a2+b2+c2−2(ab+bc+ac), (4)\nwhich appears frequently in the context of the kinematics of a two-body decay.\nWe construct linear combinations of F1,F2andF3, which correspond to TFFs with fixed helicity combinations.\nThese TFFs are denoted by Gm, with m=σ−λ= 0,±1, and are given by\nG−1(q2) := ( −mn(mn+m∆) +q2)F1(q2)\n+1\n2(m2\n∆−m2\nn+q2)F2(q2) +q2F3(q2)\nfor σ=−1\n2, λ= +1\n2, (5)\nG0(q2) := m2\n∆F1(q2) +m2\n∆F2(q2)\n+1\n2(m2\n∆−m2\nn+q2)F3(q2)\nfor σ= +1\n2, λ= +1\n2, (6)\nand\nG+1(q2) := m∆(mn+m∆)F1(q2)\n+1\n2(m2\n∆−m2\nn+q2)F2(q2) +q2F3(q2)\nfor σ= +3\n2, λ= +1\n2. (7)\nOn the one hand, a more straightforward study of the pion-loop contributions to the TFFs can be performed using\nthe fixed helicity combinations (5), (6) and (7). On the other hand, these helicity amplitudes are not independent\nquantities but satisfy kinematical constraints. At the production threshold q2= (mn+m∆)2, one has G+1=G−1=\n(m∆+mn)G0/m∆. At the decay threshold q2= (m∆−mn)2, one finds\nG+1+G−1= 2(m∆−mn)G0\nm∆. (8)\nSee also [33, 34] for further discussions. Those kinematical constraints can complicate the formulation of dispersion\nrelations [35, 36]. Nonetheless, we will follow the approach of [7] and formulate dispersion relations for helicity\namplitudes. In practice, the constraints at the production threshold are far away from the low-energy region we are\ninterested in. The constraint (8), however, is relevant. We will frequently come back to this issue.\nIn the literature one finds different conventions for the TFFs. In addition, there is a sign ambiguity related to the\nun-measurable overall phase of a quantum state |∆⟩. In [37], where the transition from nucleon to ∆ is considered,\nCarlson introduces TFFs (here labeled with “Ca”) which are related to our TFFs by2\nGCa\n−=ζQ−\n2mnG+1,\nGCa\n+=ζQ−\n2√\n3mnG−1,\nGCa\n0=ζQ Q −√\n6mnm∆G0 (9)\n2There is a mismatch between the conventions used in [37] and here. This is essentially based on the fact that we introduce our TFFs via\nthe coupling of a virtual timelike photon to a spin-3/2 baryon and a spin-1/2 antibaryon where the latter has helicity +1 /2. In [37] the\nTFFs are introduced via the coupling of a virtual spacelike photon to an incoming spin-1/2 baryon and an outgoing spin-3/2 baryon.\nThe former has helicity +1 /2. If one translates our case to the one in [37] our antibaryon turns to a baryon with helicity −1/2 and not\n+1/2. This sign change relates our TFF Gmto Carlson’s TFF GCa\n−mfor all m= 0,±1.6\nwith Q−:=p\nQ2+ (mn−m∆)2. Since one studies now reactions with q2<0, it is convenient to introduce Q2:=\n−q2>0 and Q:=p\nQ2. We will come back to the factor ζ=±1 below.\nIn Ref. [5] various conventions are related to each other, including the ones from [37]. With the help of (9) and [5]\nour TFFs can be easily related to any other TFF combinations and conventions. In particular, we provide here also\nthe relation to the Jones-Scadron TFFs [31] that are frequently used in the experimental analyses.\nIt turns out that it is convenient to introduce linear combinations of the TFFs Gm. This allows to single out a\nspecific combination that is much larger than the other two. This has been confirmed by experiment and is also in\nline with quark-model considerations and with QCD in the limit of a large number of quark colors [5, 38]. Thus it\nmakes sense to study first the dominant (and therefore best measured) structure with our dispersion relations. In a\nsecond step, one could look at the smaller quantities. We postpone a full discussion of these smaller quantities to\nfuture work, cf. also the corresponding discussion in Section I. But we provide here the dispersive formalism for all\nTFFs and explore also one of the smaller quantities to see what can be achieved with the present formalism.\nThe Jones-Scadron FFs are very well suited to separate the small quantities from the dominant contribution. In\nterms of our FFs, one finds\nG∗\nM=ζ√\n6mn\nmn+m∆(G−1−3G+1),\nG∗\nE=−ζ√\n6mn\nmn+m∆(G−1+G+1), (10)\nG∗\nC=−4ζ√\n6mn\nmn+m∆G0.\nAt low energies, the magnetic dipole TFF, G∗\nM, is much larger than the other two quantities, the electric and the\nCoulomb quadrupole TFF [5]. Thus the idea is to study first dispersion relations for G∗\nM(q2) (instead of G±1,\nseparately). After studying how well we can reproduce G∗\nM(−Q2) in the spacelike region, we will provide predictions\nfor the timelike region. Finally we will go through the same steps with G∗\nE(−Q2) but reserve the discussion of\nG∗\nC(−Q2) for future work.\nInterestingly, the kinematical constraint (8) relates the electric and Coulomb TFFs; a relation expressed by the\nSiegert theorem [34]:\nG∗\nC((m∆−mN)2) =2m∆\nm∆−mNG∗\nE((m∆−mN)2). (11)\nWe will come back to this relation in the context of dispersion relations at the end of Section III.\nWhen comparing to experimental results carried out in the spacelike region of electroproduction, there is a subtlety\nrelated to the fact that the TFFs of unstable resonances can be complex in the spacelike region [7]. On the other hand,\nthe experimental results constitute real-valued quantities [5, 13–16]. Thus we have to determine which quantities are\nactually meant by real-valued, published results denoted by G∗\nM,E,C . As we will show in Appendix A it is actually\nthe respective real part.\nTo adjust to the conventions used for the pion electroproduction we note that [5]\nReG∗\nM(−Q2) =8mnm∆\ne(mn+m∆)Q−r\n2πk∆Γ∆\n3ImM(3/2)\n1+ (12)\nwith Q−=p\nQ2+ (m∆−mN)2, the electric charge e=√\n4πα≈0.303, the ∆ decay width Γ ∆≈0.117 GeV, and the\nmomentum of the pion as produced in the ∆ rest frame, k∆=λ1/2(m2\n∆, m2\nn, m2\nπ)/(2m∆)≈0.229 GeV. The pertinent\npion electroproduction amplitude is denoted by M(3/2)\n1+.\nIt is common practice on the experimental side to study Im M(3/2)\n1+ and the multipole ratios REMandRSMinstead\nof the TFFs G∗\nM,E,C . Those multipole ratios are given by [5]\nREM=−ReG∗\nE\nReG∗\nM,\nRSM=−λ1/2(−Q2, m2\n∆, m2\nn)\n4m2\n∆ReG∗\nC\nReG∗\nM. (13)\nFinally we come back to the phase factor ζthat appears in (9) and (10). It has been introduced since the\nexperimental groups might use a different convention when extracting their helicity amplitudes or TFFs (see, e.g.,7\n[9]). In fact, when introducing a quantum field to represent the ∆ one has an undetermined quantum mechanical\nphase. This freedom might be used to make a choice for ζin (10) or(exclusive “or”) to make a choice for the sign of\nthe ∆- N-πcoupling constant hA; see Section IV below. In the following we will choose\nζ=−1. (14)\nThis means that we have to explore both sign options for the coupling constant hA. We will do that in a two-step\nprocedure. First, we will explore QCD for a large number of colors Ncto get a first indication of which sign of hA\nfits to the choice (14). Second, we will explore both options when comparing to electroproduction data. As we will\nsee, the sign choice suggested by large- NcQCD leads to a better description of data.\nIn the result sections we will also predict the radii that correspond to the TFFs. They are defined by [6]\n⟨r2⟩m:=6\nGm(0)dGm(q2)\ndq2\f\f\f\f\nq2=0. (15)\nIn principle, this slope of a form factor at the photon point can be measured in the spacelike and in the timelike\nregion. Concerning hyperons instead of Deltas and nucleons, it has been suggested in [6, 8] to measure the radius in\nthe timelike region using Dalitz decays. Valuable information about the structure of hyperons (TFFs in the spacelike\nregion) can be extracted in this way. Here we can go the opposite way and predict the radius and the Dalitz decay\ndistributions after exploring the spacelike region.\nThis brings us to the decay processes. The decay width of ∆ →nγis given by\nΓ∆→nγ=e2(m2\n∆−m2\nn)\n96πm3\n∆(m∆−mn)2\n×\u0000\n3|G+1(0)|2+|G−1(0)|2\u0001\n. (16)\nWe note in passing that the use of the constraint-free FFs, Fi, would generate interference terms. Therefore, helicity\namplitudes Gmare much more convenient here. The Jones-Scadron FFs share the same feature on account of the\nrelation\n3|G+1(q2)|2+|G−1(q2)|2=\n3\n2\u0012m∆+mn\nmn\u00132\u0000\n3|G∗\nE(q2)|2+|G∗\nM(q2)|2\u0001\n. (17)\nNext we provide the double differential decay rate for the Dalitz decay ∆ →n e+e−, first keeping the electron mass\nmeand then neglecting it:\ndΓ∆→n e+e−\ndq2d cos θ=\ne4\n(2π)396m3\n∆q2pzp\nq2\n2βe\u0000\n(m∆−mn)2−q2\u0001\n×\u0014\u0012\n1 + cos2θ+4m2\ne\nq2sin2θ\u0013\n×\u0000\n3|G+1(q2)|2+|G−1(q2)|2\u0001\n+ 4\u0012\nsin2θ+4m2\ne\nq2cos2θ\u0013q2\nm2\n∆|G0(q2)|2\u0015\n≈e4\n(2π)396m3\n∆q2pzp\nq2\n2βe\u0000\n(m∆−mn)2−q2\u0001\n×h\u0000\n1 + cos2θ\u0001\u0000\n3|G+1(q2)|2+|G−1(q2)|2\u0001\n+4q2\nm2\n∆sin2θ|G0(q2)|2i\n. (18)\nThe kinematical velocity factor associated to the electron is defined as\nβe:=s\n1−4m2e\nq2. (19)8\nIn Eq. (18) the angle θis taken between the electron and the neutron in the rest frame of the electron-positron pair.\nFor later use we also introduce a QED version of (18), which is supposed to describe the situation where the hyperon\nstructure is not resolved; see also the discussion in [33]. In practice we replace the TFF combinations by their q2= 0\nexpressions and make in this way also contact with the real photon case (16):\ndΓQED\n∆→n e+e−\ndq2d cos θ:=\ne4\n(2π)396m3\n∆q2pzp\nq2\n2βe\u0000\n(m∆−mn)2−q2\u0001\n×\u0012\n1 + cos2θ+4m2\ne\nq2sin2θ\u0013\n×\u0000\n3|G+1(0)|2+|G−1(0)|2\u0001\n. (20)\nPhenomenologically, it turns out that in the spacelike low-energy region, the TFFs of the electric and the Coulomb\nquadrupole are much smaller than the TFF of the magnetic dipole [5]. This property is also true for the timelike\nlow-energy region [38] (our results of Sections V and VI support this statement). If one neglects electric and Coulomb\nTFFs (and the electron mass), then (18) can be approximated by\ndΓ∆→n e+e−\ndq2d cos θ≈\ne4\n(2π)396m3\n∆q2pzp\nq2\n2βe\u0000\n(m∆−mn)2−q2\u0001\n×\u0000\n1 + cos2θ\u00013\n2\u0012m∆+mn\nmn\u00132\n|G∗\nM(q2)|2. (21)\nIn this approximation, the normalized angular distribution is trivial, i.e. independent of the (magnetic) TFF:\n1\nΓ∆→n e+e−dΓ∆→n e+e−\nd cos θ≈3\n8\u0000\n1 + cos2θ\u0001\n. (22)\nThus all relevant information is contained in the singly-differential dilepton-mass distribution dΓ ∆→n e+e−/dq2, which\nwe will predict in Subsection V B. The (trivial) angular distribution is in agreement with the findings of the HADES\ncollaboration [39].\nIII. DISPERSIVE MACHINERY\nFor completeness this section repeats the general framework presented in [6, 7] but with particular attention\ntowards the case studied here, the ∆- ntransition. In order to incorporate the pion rescattering effect, we use the\nOmn` es function,\nΩ(s) = exp\n\ns∞Z\n4m2πds′\nπδ(s′)\ns′(s′−s−iϵ)\n\n(23)\nwhere δdenotes the pion p-wave phase shift [26, 27].\nThe pion vector FF, FV\nπ, is taken from [11] (see also [12, 28, 40, 41]):\nFV\nπ(s) = (1 + αVs) Ω(s). (24)\nFor a value of αV= 0.12 GeV−2, equation (24) describes very well the pion vector FF data obtained from tau decays\n[42] for energies below 1 GeV [11].\nA. Dispersion relations\nAs motivated in [7], we expect that the three TFFs Gmintroduced in (5), (6), (7) and therefore also the Jones-\nScadron TFFs of (10) satisfy unsubtracted dispersion relations:\nGm(q2) =Zds\n2πidiscGm(s)\ns−q2(25)9\nform= 0,±1;M, E, C .\nThe low-energy behavior of the TFFs is determined by the lightest hadronic state that couples to the ∆¯ nsystem:\nthe two-pion state. Therefore, in complete analogy to [6], we can write:\nGm(q2) =1\n12π∞Z\n4m2πds\nπTm(s)p3\nc.m.(s)FV∗\nπ(s)\ns1/2(s−q2−iϵ)\n+Ganom\nm(q2) +. . . (26)\nwhere the ∆¯ nπ+π−scattering amplitudes are denoted by Tm. Here pc.m.denotes the modulus of the momenta of\nthe pions in the center-of-mass frame. The ellipsis stand for the infinitely many contributions coming from other\nintermediate states, such as four-pion [43], two-kaon or baryon-antibaryon states. None of these will be taken into\naccount in this work; they are regarded as negligible as long as the TFFs are studied at low energies. To enhance\nfurther the importance of the low-energy region in the dispersive integral, one can introduce additional subtractions.\nWe will come back to this aspect in Subsection III B. The “anomalous” contribution Ganom\nm will also be introduced\nlater.\nThe pion-baryon scattering amplitudes Tmare obtained in a two-step procedure. First we define the reduced\namplitudes:\nK±1(s) :=−3\n4πZ\n0dθsin2θ\n×M(s, θ,1/2±1,1/2)\n¯vn(−pz,1/2)γ5u1\n∆(pz,1/2±1)pc.m.,\nK0(s) :=−3\n2m2\n∆−m2\nn+s\n2sπZ\n0dθsinθcosθ\n×M(s, θ,1/2,1/2)\n¯vn(−pz,+1/2)γ5u3\n∆(pz,+1/2)pc.m.. (27)\nWe have introduced M(s, θ, σ, λ ) as the approximation to the Feynman amplitude for the reaction ∆ ¯ n→π+π−. In\npractice, M(s, θ, σ, λ ) does not include the rescattering effect of the pions, but this will be taken care of in a second\nstep. In addition, we want to distinguish conceptually between processes with left-hand cut structures and purely\npolynomial terms [44, 45]. Therefore, we will use the notation Kmto refer exclusively to the amplitudes that originate\nfrom the left-hand cut structures (and drop at high energies), while we will denote the polynomial terms by Pm. In\npractice we will determine Kmfrom the χPT tree-level expressions for nucleon and ∆ exchange [6, 7, 11, 12].\nPion rescattering is then taken into account by solving a Muskhelishvili-Omn` es equation [46, 47]. The result is\nTm(s) = Km(s) + Ω( s)Pm+Tanom\nm(s)\n+ Ω(s)sΛ2Z\n4m2πds′\nπKm(s′) sinδ(s′)\n|Ω(s′)|(s′−s−iϵ)s′. (28)\nNote that we have introduced a cutoff Λ. Since we have only the low-energy part under control, where the two-pion\nstate dominates, it is not reasonable to extend the integral into the uncontrolled high-energy region. In practice,\nthe two-pion state dominates the isovector channel up to about 1 GeV. Beyond this point, four-pion states might\nalso become important [28, 43]. To explore the uncertainties of our low-energy approximation we will vary the cutoff\nbetween 1 and 2 GeV.\nWe have used a once-subtracted dispersion relation in (28). For the polynomial Pmwe just take a constant (per\nchannel) that can be obtained from a fit to data. Note that in (28) this polynomial is multiplied by the Omn` es function\nΩ. The latter drops at high energies. Therefore the restriction of Pmto a constant has the additional feature that Tm\ndrops for high energies. Certainly a benefit for the integrand of the dispersion relation (26). Finally, Tanom(s) denotes\nan additional contribution to the amplitude, associated to the presence of an anomalous cut on the first Riemann\nsheet.\nIn Appendix B it is shown that the presence of an anomalous cut in the first Riemann sheet leads to a modification\nof the dispersion relations for both the amplitudes Tmand the TFFs Gm. These additional terms are denoted by\nTanom\nm andGanom\nm and are provided in Appendix B. As a consequence, the TFF integral in (26) becomes complex for10\nanyq2value. Without, it would be real below the two-pion threshold. Concretely, the anomalous pieces reflect the\nfact that the ∆ is unstable, hence the exchanged p-πpair can be on-shell and therefore contribute to the imaginary\npart. This made it necessary to specify in (12) and (13) that the respective real part (and not, e.g., the modulus)\nenters the equations. For the decay formulae (16) and (18), on the other hand, it is the respective modulus that\nappears, see also the corresponding discussion in Appendix A.\nSome comments about other inelasticities are in order. As motivated already in [7], we do not include the kaons as\nintermediate states. The two-kaon threshold starts at (2 mK)2≈1 GeV2, rather far away from the two-pion threshold\nwhich is located at (2 mπ)2≈0.08 GeV2. The latter is the most relevant from a dispersive point of view, since the\ninfluence of high-energy inelasticities is naturally suppressed for low values of q2. The branch point of the anomalous\ncut associated to the proton-pion-pion triangle lies in the vicinity, i.e. at s+≈(0.05−0.08i) GeV2. It is therefore\nalso important. Finally, note that there cannot be additional anomalous cuts since the reaction ∆¯ n→K¯Krequires\nthe exchange of hyperons, which are too heavy to satisfy the condition (B1).\nA second reason for not including the two-kaon state lies in the fact that the four-pion state seems to be more\nimportant than the two-kaon state [28, 43] (and both seem to be of minor importance at sufficiently low energies).\nThe two-kaon inelasticity (but not the four-pion states) has been included in a recent analysis of the TFFs of Σ0to\nΛ [48]. It is encouraging that it has been confirmed in [48] that the influence of the two-kaon states is minor.\nB. Subtracted dispersion relations\nA once-subtracted dispersion relation is used to enhance the importance of the low-energy region in the dispersive\nintegral:\nGm(q2) =Gm(0)\n+q2\n12πΛ2Z\n4m2πds\nπTm(s)p3\nc.m.(s)FV∗\nπ(s)\ns3/2(s−q2−iϵ)+Ganom\nm(q2) (29)\nform= 0,±1, M, E, C . In principle, the three subtraction constants Gm(0) are complex-valued, but at least the\nreal parts can be determined by experiment. This is a great advantage compared to the Σ∗-Λ case [7], where only\nunsubtracted dispersion relations could be used due to a lack of data. The subtraction constants encode the high-\nenergy contributions left out from our formalism; see also the corresponding discussions in [6, 7, 11, 12, 41]. The last,\n“anomalous” piece in (29) is given by\nGanom\nm(q2) =q2\n12π1Z\n0dxds′′(x)\ndx1\ns′′(x)−q2\n×fm(s′′(x))FV\nπ(s′′(x))\n−4 (−λ(s′′(x), m2\n∆, m2n))3/2. (30)\nMore technical details on this quantity are provided in Appendix B.\nFinally, we come back to the kinematical constraints (8), (11) that helicity amplitudes and Jones-Scadron TFFs\nmust satisfy. If we had a full coverage of all inelasticities (and not only two pions) in the dispersion relation (26) and if\nwe knew all scattering amplitudes up to very large energies, then the kinematical constraints would be automatically\nsatisfied. In practice, we have only control over the low-energy region with its dominance of the two-pion state.\nTherefore, the kinematical constraints will be violated to some extent. There are several ways how to deal with this\nissue. If one formulates dispersion relations for the constraint-free TFFs, F1,2,3, introduced in (2), then the helicity\namplitudes will be obtained from (5), (6), and (7). The kinematical constraints will then be automatically satisfied.\nThis is the path followed in [12, 17]. The disadvantage for the system studied here is the fact that it is then difficult\nto disentangle small and large TFFs. Technically it is also simpler to use helicity amplitudes for the hadronic input.\nA second option is the use of subtracted dispersion relations where the subtraction constants are subject to the\nkinematical constraints. Of course, one loses some freedom to parametrize the unknown high-energy physics.\nIn any case, the kinematical constraint (11) does not touch the dominant magnetic TFF but concerns solely the\nnumerically much smaller TFFs G∗\nEandG∗\nC. We leave it to future work to explore in detail which dispersion relations\nare best suited for these electric and Coulomb quadrupole TFFs.11\nIV. INPUT FROM CHIRAL PERTURBATION THEORY\nA. Effective Lagrangians\nThe leading-order (LO) chiral Lagrangian including the decuplet states is given by [5, 49–53]\nL(1)\nbaryon= tr\u0000¯B(i/D−m(8))B\u0001\n+¯Tµ\nabc(iγµναDα−γµνm(10)) (Tν)abc\n+D\n2tr(¯B γµγ5{uµ, B}) +F\n2tr(¯B γµγ5[uµ, B])\n+hA\n2√\n2\u0000\nϵade¯Tµ\nabc(uµ)b\ndBc\ne+ϵade¯Be\nc(uµ)d\nbTabc\nµ\u0001\n−HA\n4mRϵµναβ\u0000¯Tµ\nabc(DνTα)abd(uβ)c\nd\n+ (Dν¯Tα)abd(Tµ)abc(uβ)d\nc\u0001\n. (31)\nHere Bcontains the states of the baryon octet, Tcollects the baryon decuplet states and uµcontains the Goldstone\nboson fields. For further details we refer to [7, 52, 53].\nIn (31) m(8)(m(10)) denotes the mass of the baryon octet (decuplet) in the chiral limit. For the next-to-leading-\norder (NLO) calculation that we perform in the present work we use the physical masses [54] of all states. However,\nsince our accuracy is not high enough to resolve isospin breaking effects, we take one average mass for proton and\nneutron, e.g. mp≈mn≈mN= 0.939 GeV. We use for the ∆-resonance mass m∆= 1.232 GeV [54] and for the pion\nmass mπ= 0.13957 GeV.\nIn line with [7] we use Fπ= 92 .28 MeV and D= 0.80,F= 0.46, which implies for the pion-nucleon coupling\nconstant gA=F+D= 1.26. Sizes and signs of DandFhave been obtained from the weak semileptonic decays of\noctet baryons.\nFor the three-point coupling constants hAandHAwe use the two-flavor estimates for a large number of colors\n[55]:|hA| ≈3gA/√\n2≈2.67 and HA≈9gA/5≈2.27. This choice is slightly different from [7] where we used hAas\ndetermined from hyperon decays, cf. also the discussion in [52, 56]. In the present paper we explore solely the hadron\nsector of up and down quarks. Therefore two-flavor estimates are more reasonable. The size of hAfits rather well to\nits determination from the decay width of ∆ →Nπ[6, 7, 11, 52, 56].\nAs already pointed out, one has a free phase choice (sign choice) when introducing a decuplet field. One could use\nthis freedom to choose a positive or negative value for hA. Instead we have decided to use the freedom to adjust our\nTFFs to the experimental convention. This is the essence of the choice (14). Therefore we have to explore which sign\nour coupling constant hAshould have. As often in theoretical physics, the choices are: calculate the sign of hAor\nfit it to data. We will explore both options. As a calculational tool we have decided to use QCD for a large number\nof colors. This calculation is provided in Appendix C. When comparing to data in Section V we will see that the fit\nagrees with the sign prediction of large- NcQCD.\nWe take this as an encouragement to use even the numerical prediction for the other three-point coupling, HA.\nThere, the sign is not a convention. It is correlated to the sign of gA; see also the discussion in [57]. This is easy to\nsee when anticipating the calculations that we will present below. The reaction ∆ π→Nπcan proceed via nucleon\nexchange, where the amplitude is ∼hAgA, and also via ∆ exchange, where the amplitude is ∼HAhA. These two\namplitudes interfere. Thus the sign of HArelative to the sign of gAis not a matter of convention. On the other hand,\nthe sign of gAis determined from the “ vminus a” structure of the weak interaction (and a sign convention for the\nvector part).\nNow we turn to the Lagrangian of second order in the chiral counting. A complete and minimal NLO Lagrangian\nhas been presented in [52]. For our present purpose we need terms that lift the mass degeneracies that hold at LO\nand we need terms that provide interactions for ∆ π→nπ(or formally ∆¯ n→2π) with the two pions in a p-wave.\nThe relevant part of the NLO Lagrangian for the baryon octet sector reads [52, 58, 59]\nL(2)\n8=bχ,Dtr(¯B{χ+, B}) +bχ,Ftr(¯B[χ+, B]) (32)\nwith χ±=u†χu†±uχ†uandχ= 2B0(s+ip) obtained from the scalar source sand the pseudoscalar source p.\nThe low-energy constant B0is essentially the ratio of the light-quark condensate and the square of the pion-decay\nconstant; see, e.g. [19–22]. While at LO all baryon octet states are degenerate in mass, the NLO terms of (32) lift\nthis degeneracy and essentially move all masses to their respective physical values. Technically this is achieved if one\nreplaces the scalar source sby the quark mass matrix. Numerical results for the octet mass m(8)in (31) and the12\nsplitting parameters bχ,D/χ,F in (32) are given, for instance, in [60]. In practice we use the physical masses. Therefore\nwe do not specify these parameters here.\nThe relevant part of the NLO Lagrangian for the baryon decuplet sector reads [52]\nL(2)\n10=−dχ,(8)¯Tµ\nabc(χ+)c\ndγµν(Tν)abd. (33)\nIt provides a mass splitting for the decuplet baryons such that mΩ−mΞ∗=mΞ∗−mΣ∗=mΣ∗−m∆, in good\nagreement with phenomenology [54]. In the present work we only deal with the ∆ and, in practice, we use the\nphysical mass of the neutral ∆. In that way the physical thresholds are exactly reproduced.\nFor the formal reaction ∆0¯n→π+π−the relevant part of the NLO Lagrangian [52] is given by\nL(2)\n8−10→ −cF√\n3F2π¯nγµγ5∆0\nν\u0000\n∂µπ+∂νπ−−(µ↔ν)\u0001\n. (34)\nOne could fit cFto data when comparing calculation and experimental results for Re G∗\nMas given in (12). The\nlow-energy constant cFwill contribute to P∗\nM, see Appendix D. On the other hand, one can fit the constants Pmright\naway to data.\nTo judge the quality of our reproduction of data and of our predictions, it is important to understand how well\nour theory parameters are constrained and to check how much the observables are sensitive to parameter variations.\nTherefore, a brief summary of our theory parameters is in order. We use isospin averaged (but not three-flavor\naveraged) hadron masses throughout. The pion decay constant Fπand the axial charge of the nucleon gAare very\nwell known from weak decays. We will not explore parameter variations of masses or weak-decay constants.\nThe subtraction constants Gm(0) and Pmappearing in (29) and (28), respectively, will be fitted directly to spacelike\nTFF data. We will check, however, which value for G∗\nM(0) will be provided by an unsubtracted dispersion relation.\nThe remaining input parameters that are not fitted to TFF data are the ∆- N-πcoupling constant |hA|, the (p-wave)\n∆-∆- πcoupling constant HA, and the cutoff Λ that appears in the dispersive integrals (28) and (29). As already\nspelled out, we will vary Λ between 1 and 2 GeV. For the three-point coupling constants we will explore variations of\nabout 10% up and down, i.e.\n1≤Λ [GeV] ≤2,\n2.4≤ |hA| ≤2.9,\n2.0≤HA≤2.5. (35)\nB. Matrix elements\nThe ∆¯ nπ+π−tree-level amplitudes, i.e. χPT amplitudes up to (including) NLO, are calculated in a first step.\nThen, the reduced amplitudes are obtained applying the projection method presented in the appendix of Ref. [7]. The\ncalculations are performed using FeynCalc [61, 62]. Note that these amplitudes constitute the leading i.e. dominant\ncontribution; the notation NLO refers to the underlying chiral Lagrangian, whose tree-level contribution is equally\nimportant to that coming from the LO Lagrangian. Further comments on the power counting can be found in [7].\nThe Feynman matrix element for the reaction ∆0¯n→π+π−up to (including) NLO is given by\n−gAhA\n2√\n6F2π1\nu−m2p+iϵpµ\nπ−gµα¯vn/pπ+γ5(/p∆−/pπ−+mp)uα\n∆\n+hAHA\n3√\n6m∆F2πiϵλ\nναβpν\n∆pβ\nπ−pµ\nπ+¯vnSµλ(p∆−pπ−)uα\n∆\n−hAHA\n2√\n6m∆F2πiϵλ\nναβpν\n∆pβ\nπ+pµ\nπ−¯vnSµλ(p∆−pπ+)uα\n∆\n−cF√\n3F2π(pµ\nπ+pα\nπ−−pα\nπ+pµ\nπ−)gαβ¯vnγµγ5uβ\n∆. (36)\nHere Sµνdenotes the spin-3/2 propagator [63, 64]:\nSµν(p) :=−/p+m∆\np2−m2\n∆+iϵP3/2\nµν(p) +2\n3m2\n∆(/p+m∆)pµpν\np2\n−1\n3m∆pµpαγαν+γµαpαpν\np2. (37)13\nThe reduced amplitudes associated to the ∆±and proton exchange diagrams constitute the bare χPT input. They\nare given below in the form Km+Pm, where Pmare constant terms that are left out of the dispersive integrals. In\nline with [11] we have also found here that it is most reasonable to determine Pmby fits to data. But for completeness\nwe provide the χPT expressions for Pmin Appendix D.\nThe explicit expressions for the left-hand-cut contributions are\nK+1=−gAhA\n4√\n6F2π(C+1+D+1Roct.\ns)−5hAHA\n12√\n6F2π(E+1+F+1Rdec.\ns),\nK−1=−gAhA\n4√\n6F2π(C−1+D−1Roct.\ns)−5hAHA\n12���\n6F2π(E−1+F−1Rdec.\ns),\nK0=−gAhA\n4√\n6F2π(C0+D0Roct.\nd)−5hAHA\n12√\n6F2π(E0+F0Rdec.\nd) (38)\nwith\nRoct.\ns=−2Yp\nκ2 \n1− \n1−Y2\np\nκ2!\n|κ|\nYp\u0012\narctan\u0012|κ|\nYp\u0013\n+πΘ(sY−s)\u0013!\n,\nRoct.\nd=4\nκ2\u0012\n1−Yp\n|κ|\u0012\narctan\u0012|κ|\nYp\u0013\n+πΘ(sY−s)\u0013\u0013\n,\nRdec.\ns=−2Y∆\nκ2\u0012\n1−\u0012\n1−Y2\n∆\nκ2\u0013|κ|\nY∆arctan\u0012|κ|\nY∆\u0013\u0013\n,\nRdec.\nd=4\nκ2\u0012\n1−Y∆\n|κ|arctan\u0012|κ|\nY∆\u0013\u0013\n(39)\nand\nYp= 2m2\np−m2\n∆−m2\nn−2m2\nπ+s , (40)\nY∆=m2\n∆−m2\nn−2m2\nπ+s , (41)\nκ2=1\ns(s−4m2\nπ)λ(s, m2\n∆, m2\nn), (42)\nsY=m2\n∆+m2\nn+ 2m2\nπ−2m2\np. (43)\nFurther details are provided in Appendix B and in [7]. Note that κ2is negative in the range sdt< s < s st, i.e.\n|κ|=√\n−κ2. Only for negative κ2the expressions (39) are correct. For positive κ2one has log’s instead of arctan’s.\nIn practice, however, the integration boundaries of the dispersive integrals run from 4 m2\nπto Λ2= 4 GeV2which lies\nbelow the scattering threshold sst= (m∆+mn)2. It follows that one has to replace the arctan’s with the log’s only\nin the range 4 m2\nπ< s < s dt= (m∆−mn)2.\nThe coefficient functions in (38) are given by\nC+1=−2 (m∆−mn) (mn+mp)\ns−(m∆−mn)2, (44)\nC−1=−6 (m∆−mn) (mn+mp)\ns−(m∆−mn)2, (45)\nC0=(m∆+mn) (m∆+mp)\ns−3m∆(mn+mp)\ns−(m∆−mn)2, (46)14\nD+1= 3mp(mn+mp) +3 (m∆−mn) (mn+mp) (m2\nπ+m∆mn−m2\np)\ns−(m∆−mn)2, (47)\nD−1=3\nm∆(mn+mp) (m2\nπ−m2\n∆+m∆mp−m2\np)\n+9 (m∆−mn) (mn+mp) (m2\nπ+m∆mn−m2\np)\ns−(m∆−mn)2, (48)\nD0= 3mp(mn+mp)(m2\n∆−m∆mp−m2\nπ+m2\np)−9m∆(mn+mp)(m∆mn+m2\nπ−m2\np)2\ns−(m∆−mn)2\n+3(m∆+mn)(mp+mn)\ns\u0010\nm3\n∆mn−mp(m∆−mn)(m2\n∆+m2\nπ) + 2m2\n∆m2\nπ\n−m2\np\u0000\nm∆(m∆+mn) + 2m2\nπ\u0001\n+ 2m∆mnm2\nπ−m3\np(mn−m∆) +m4\nπ+m4\np\u0011\n, (49)\nE+1=(m∆−mn)\u0000\n(m∆+mn)2−m2\nπ\u0001\n3m∆(s−(m∆−mn)2), (50)\nE−1=(m∆−mn)\u0000\n(m∆+mn)2−m2\nπ\u0001\nm∆(s−(m∆−mn)2), (51)\nE0=−(m∆+mn)(2m2\n∆+ 2m∆mn−m2\nπ)\n6m∆s+(m∆+mn)2−m2\nπ\n2(s−(m∆−mn)2), (52)\nF+1=−3s\n2−m2\nπ(2m∆+ 3mn)\n2m∆+5(m∆+mn)2\n2\n+(m∆−mn)((m∆+mn)2−m2\nπ)(m2\n∆−m∆mn−m2\nπ)\n2m∆(s−(m∆−mn)2), (53)\nF−1=3s\n2+m2\nπ(m2\n∆+m∆mn−m2\nn) +m4\nπ\n2m2\n∆−5(m∆+mn)2\n2\n+3(m∆−mn)((m∆+mn)2−m2\nπ)(m2\n∆−m∆mn−m2\nπ)\n2m∆(s−(m∆−mn)2), (54)\nF0=3m2\n∆s\n2−m2\nπ(7m2\n∆−2m∆mn+ 2m2\nn) +m2\n∆(m∆+mn)2\n2+m4\nπ\n+4m2\n∆m2\nπ(m∆−2mn)(m∆+mn)2−m4\nπ(2m3\n∆+m2\n∆mn+m3\nn) +m6\nπ(m∆+mn)\n2m∆s\n+3((m∆+mn)2−m2\nπ)(m∆(mn−m∆) +m2\nπ)2\n2(s−(m∆−mn)2). (55)\nV. RESULTS\nA. Magnetic dipole transition form factor\nIn Fig. 2 we show the results for the quantity Im M(3/2)\n1+ using a subtracted or unsubtracted dispersion relation,\nrespectively, and varying the sign of hA. For all cases, P∗\nMis used as a fit parameter. We observe that the unsubtracted\ndispersion relation provides only a qualitative description of the data. Quantitatively reasonable results are obtained\nfor a subtracted dispersion relation. We also see that the sign choice hA>0 fits the data much better, in line with\nthe large- Ncconsiderations of Appendix C. In the following, we will use only hA>0. For the best fit, i.e. for the\nsubtracted dispersion relation using hA>0 we find P∗\nM= 406 .12 GeV−2. The overall size of this subtraction constant\nis completely natural. We can deduce from the formulae of Appendix D that the scale is set by 1 /F2\nπwhich is about\n102GeV−2.\nThe agreement between theory and experiment reaches up to aboutp\nQ2≈0.6 GeV. This is achieved by only two\nfit parameters, the subtraction constant P∗\nMfor the hadronic amplitude and the subtraction constant G∗\nM(0). Strictly15\nhA>0, unsubhA<0, subhA>0, sub\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7010203040\nQ2[GeV2]Im M1+(3/2)[10-3/mπ]\nFIG. 2. Im M(3/2)\n1+ as a function of Q2=−q2using an unsubtracted dispersion relation with hA>0 (blue, dashed line),\na subtracted dispersion relation with hA<0 (green, dotted), and a subtracted dispersion relation with hA>0 (red, dash-\ndotted). Data taken from BATES (blue) [13], MAMI (red) [14, 15], and CLAS (black) [16]. The values of the contact terms\nareP∗\nM≈367 GeV−2(blue), P∗\nM≈570 GeV−2(green), and P∗\nM≈406 GeV−2(red), respectively.\nspeaking, the latter is just fitted to the experimental value at the photon point [5, 54], not to the spacelike data of\nFig. 2. If we focus for a moment on the values around the photon point, we have to state that it is non-trivial to\nobtain a good reproduction of function value, slope and curvature(!) with only two fit parameters. Indeed, at low\nenergies, the whole shape of the curve suggested by the experimental points is reproduced very well.\nIn Fig. 3 we show the impact of parameter variations. In general, the results are very stable if one changes one of\nthe three-point coupling constants by 10%. It is very encouraging that our results are so robust. We do not study\nvariations of the cutoff Λ in the plots to keep the number of lines manageable. Below we will explore such variations\nwhen providing values for the various radii. For Figs. 2 and 3 we use Λ = 2 GeV. Quantitatively, we see that an\nincrease (decrease) in hAorHAleads to a decrease (increase) for Im M(3/2)\n1+ in this range of negative q2(i.e. positive\nQ2). This means that all three contributions (triangle diagrams with nucleon and with ∆ and the contact term P∗\nM)\nadd up constructively to the slope of G∗\nM(q2). We will see this more directly in Table I below.\nIn Fig. 4 we show real and imaginary part of G∗\nMfor space- and time-like q2. We have limited the latter region\nto the physical decay region where the Dalitz decay can take place. We stress that G∗\nM(q2) is notpurely real in the\nspacelike region q2<0. While the shape of the imaginary part is a prediction of our theory, one has to take the overall\nsize, i.e. the value at the photon point, with a grain of salt. A subtracted dispersion relation cannot predict Im G∗\nM(0),\nwhile the measurements of γN→∆ provide only the real part at the photon point. We use here the result from an\nunsubtracted dispersion relation. To judge the quality of this approximation, we provide also the value of Re G∗\nM(0)\nas obtained from the unsubtracted dispersion relation, but using the value P∗\nM= 406 .12 GeV−2coming from a fit of\nthe subtracted dispersion relation to the data; see Fig. 2. In this way we find G∗\nM(0) = 3 .3126 + 0 .0503i. The real\npart is only 10% away from the experimental value. This provides some support that the value for the imaginary part\nis reasonable. In particular, it suggests that the imaginary part is small (but not zero) in the spacelike region.\nIn the timelike region, Fig. 4 shows that the real part grows from about 3 at the photon point to about 4 at the\nend of the Dalitz decay region. This variation is comparable but somewhat larger than the result of the quark-model\ncalculation of [38]. But it also shows that one needs a resolution of 30% or better to distinguish in the Dalitz decay\nregion the true form factor from a constant.\nIn Table I, we display FF value (magnetic moment) and slope (radius) at the photon point and explore the impact of\nparameter variations. Note that since Re G∗\nM(0) is fitted to the data, it cannot change. The imaginary part, however,\nobtained from an unsubtracted dispersion relation, could change but it remains the same within the accuracy that\nwe display.\nMore interesting are the results for the radius. They are a pure prediction of our subtracted dispersion relations.\nFirst of all, we observe that the non-vanishing imaginary part has grown to a 10% effect, while it is less than 2% for16\nhA≈2.7,H A≈2.3hA≈2.9,H A≈2.3hA≈2.4,H A≈2.3hA≈2.7,H A≈2.5hA≈2.7,H A≈2.0\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7010203040\nQ2[GeV2]Im M1+(3/2)[10-3/mπ]\nFIG. 3. Same as Fig. 2 for a subtracted dispersion relation and hA>0 but varying the input parameters. From top to bottom:\nhAdecreased by 10%, central value for HA(green); HAdecreased by 10%, central value for hA(brown); central values for hA\nandHA(red, dash-dotted); HAincreased by 10%, central value for hA(orange); hAincreased by 10%, central value for HA\n(magenta).\nReGM*ImGM*\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.001234\nq2[GeV2]GM*\nFIG. 4. Real (red) and imaginary (black) part of the magnetic TFF in the space- and time-like region, −0.7 GeV2< q2<\n(m∆−mn)2. The curves correspond to the red curve in Fig. 2 obtained using hA>0 and the corresponding fit result\nP∗\nM= 406 .12 GeV−2with the cutoff at Λ = 2 GeV. The value Im G∗\nM(0) = 0 .0503 is obtained from an unsubtracted dispersion\nrelation.17\nG∗\nM(0) ⟨r2⟩∗\nM[GeV−2]\nΛ = 1 GeV 3.02 + 0 .05i19.03 + 2 .23i\nΛ = 2 GeV 3.02 + 0 .05i18.35 + 2 .19i\nhA≈2.93.02 + 0 .05i18.76 + 2 .43i\nhA≈2.43.02 + 0 .05i17.95 + 1 .94i\nHA≈2.53.02 + 0 .05i18.66 + 2 .19i\nHA≈2.03.02 + 0 .05i18.05 + 2 .19i\nTABLE I. Sensitivity of magnetic transition moment and magnetic radius with respect to parameter variations. We use\nΛ = 2 GeV and central values for hAandHAunless otherwise stated.\nthe magnetic moment. It remains to be seen if there is a way how to verify this effect experimentally. Presumably\none has to go away from the peak position of the Delta and carry out a full-fledged analysis of the variation in the\ninvariant mass of the pion-nucleon system that emerges from the produced Delta; see also the discussion in Appendix\nA. We have already stressed in the introductory section that this is far beyond the scope of the present paper. But\nthe non-trivial and quantitatively non-negligible imaginary part of the radius is certainly an intriguing result.\nThe sensitivity to parameter variations is mild, the largest impact is caused by a change in the cutoff. The size\nof the radius is completely reasonable. A size of 0 .85 fm would correspond to a squared radius of about 19 GeV−2.\nIncreasing one of the three-point coupling constants, i.e. increasing the importance of the nucleon or Delta exchange\ndiagram, leads to an increase of the radius. Thus all effects — nucleon exchange, Delta exchange (our proxy for\ncorrelated pion-nucleon pair exchange) and contact term (the short-distance physics) — add up constructively for the\nmagnetic radius. Of course, the radius can be addressed from the spacelike and the timelike side. For the latter we\nhave now full predictive power.\nB. Dalitz decay distribution\nGiven that the smaller form factors contribute only on the sub-percent level, we can predict the Dalitz decay\n∆→n e+e−based on the magnetic dipole TFF. The same idea has been utilized in [38] and has entered the\ncorresponding discussion in the experimental paper [39] by the HADES collaboration.\nIn Fig. 5 we plot both the single differential decay width dΓ /dq2for the Dalitz decay ∆ →ne+e−and the\ncorresponding QED case (20), in the kinematically allowed range 4 m2\ne< q2<(m∆−mn)2. The first is the angular\nintegral of (18) or its approximation (21), the second is a simplified version, blind to the q2-dependence of the TFF.\nThe discrepancy between these two curves is caused by the non-trivial internal structure of baryons and is therefore\nof great interest.\nWhat could already be anticipated from Fig. 4, given the limited variation in the timelike region, can now also\nbe seen fully quantitatively in Fig. 5. In order to resolve the difference between a non-trivial TFF and the pointlike\n(“QED”) case one needs a rather high resolution in the tail of the Dalitz distribution. At present, the experimental\naccuracy of the HADES experiment [39] is not high enough to resolve this difference. But our plot shows the accuracy\nthat is required. Variations of our parameters according to (35) lead to changes that lie within the line width of the\n“FFs” curve of Fig. 5.\nVI. EXTENDED OUTLOOK\nGiven the success of our model-independent framework based on dispersion theory and χPT concerning the nu-\nmerically dominant piece, the magnetic dipole TFF, it is interesting to check how well our formalism is doing for\nthe two smaller (quadrupole) TFFs. Yet we hesitate to provide a full-fledged analysis for the reasons spelled out\npreviously. In particular, our precision is limited by the following approximations: treating the Delta as if it was a\nstable asymptotic state; using the Delta exchange diagram as a proxy for pion-nucleon rescattering; using dispersion\nrelations for Jones-Scadron FFs instead of constraint-free FFs. Nonetheless, to satisfy our own curiosity we have\ncarried out an analysis of one of the smaller TFFs, namely the electric quadrupole. It shares with the magnetic dipole\nthe property that the subtraction constant G(0) can be determined at the photon point from the decay ∆ →Nγ.\nThe same is not true for the Coulomb quadrupole, which does not enter the decay formula (16). The reason is simply\nthe fact that a real photon cannot have vanishing helicity. Therefore G∗\nC∼G0cannot be populated for real photons.\nWe use the subtracted dispersion relation for G∗\nE(−Q2) and our previous result for G∗\nM(−Q2) to obtain REMfrom18\nFFs(q2)\nQED\n0.00 0.02 0.04 0.06 0.080123456\nq2[GeV2]dΓΔ→Νe-e+/dq2[10-5GeV-1]\nFIG. 5. Single-differential decay width for the ∆ →N e+e−Dalitz decay. The top curve, labeled “FFs ( q2)”, is the angular\nintegral of (18). The bottom curve is the QED analogue, given by (20).\n(13), using G∗\nE(0) and P∗\nEas fit parameters. The results are provided in Fig. 6. For the central values of hAandHA\nwe find P∗\nE≈10.24 GeV−2. We observe a fair agreement with the data up to about Q2≈0.5 GeV2, even though the\nquality is not as impressive as for the magnetic case displayed in Fig. 3. Variation of the input parameters is also\ndisplayed in Fig. 6 by the various lines. All lines lie close to each other, indicating again a very robust result. The\nnext thing to notice is the qualitative feature of a dip at around Q2≈0.12 GeV2. Interestingly, the data do not really\nshow this dip, but do not exclude it either. On the other hand, theoretical considerations indicate that this dip is\ncaused by pion-cloud effects [65, 66]. Of course, the pion-cloud physics is covered by our dispersive framework, which\nfeatures the fact that at low energies virtual photons couple dominantly to the lightest degrees of freedom, the pions\nas the Goldstone bosons of chiral symmetry breaking.\nIn Fig. 7 we provide the corresponding plot for G∗\nE(q2) itself, using central parameter values. A remarkable feature\nis a maximum of the real part, which lies very close to the photon point. As a consequence, close to the photon point\nthe shape of the ratio REMis dominated by the change of G∗\nM, leading to a drop of REMin Fig. 6. When moving\nfurther into the spacelike region, G∗\nE(q2) develops a slope, eventually a larger one than G∗\nM. This leads to a rise in\nthe ratio REM.\nAn extremum close to the photon point seems to be a generic feature of the smaller FFs of the Delta-to-nucleon\ntransition. In the advanced data parametrization of Ref. [67], taking into account all kinematical constraints, one sees\nthe occurrence of extrema in all helicity amplitudes. Only the particular combination of helicity amplitudes (10) that\nlead to the magnetic dipole TFF, Fig. 4, does not show this extremum.\nIt is also interesting to observe that our results for the real and imaginary part of G∗\nE(q2) have about the same size\nin the spacelike region. This is in sharp contrast to the magnetic case. The overall picture is that in the spacelike\nregion the real part of the magnetic dipole TFF is large, while all other quantities (the imaginary part of the magnetic\nTFF and both real and imaginary part of the electric TFF) are small in magnitude, but of comparable absolute size.\nThis has interesting consequences for the radius, provided in Table II. While for the magnetic case, Table I,\nthe respective real parts are always dominant relative to the imaginary parts, our formalism produces a very large\nimaginary part for the electric quadrupole radius. While parameter variations have quite some impact, the overall\nqualitative result is robust. At present, it is not clear if this is a physical effect or a deficiency of our approach. After\nall, we address here very small quantities, relative to the magnetic TFF. This requires high theoretical accuracy. If the\nlarge imaginary part is a true physical effect, it will be interesting to figure out how one can explore it experimentally.\nOverall, we regard our results for the magnetic TFF as solid predictions, but interpret our results for the much19\nhA≈2.7,H A≈2.3hA≈2.9,H A≈2.3hA≈2.4,H A≈2.3hA≈2.7,H A≈2.5hA≈2.7,H A≈2.0\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-12-10-8-6-4-20\nQ2[GeV2]REM[%]\nFIG. 6. The ratio REMas a function of Q2=−q2. The various lines display the impact of parameter variations. To the right\nfrom top to bottom: hAincreased by 10%, central value for HA(magenta); HAdecreased by 10%, central value for hA(brown);\ncentral values for hAandHA(red, dash-dotted); HAincreased by 10%, central value for hA(orange); hAdecreased by 10%,\ncentral value for HA(green). For the color code of the data see the figure caption of Fig. 2.\nReGE*\nImGE*\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0-0.06-0.04-0.020.000.020.040.060.08\nq2[GeV2]GE*\nFIG. 7. Same as Fig. 4 but for the electric TFF. The value of the contact term is P∗\nE= 10.24 GeV−2.\nsmaller electric TFF as a motivation for further studies and further improvements of the formalism.\nAcknowledgements: This work has been supported by the Swedish Research Council (Vetenskapsr˚ adet) (grant\nnumber 2019-04303) and by Thailand International Development Cooperation Agency (TICA), Thailand Science\nResearch and Innovation (TSRI), National Research Council of Thailand (NRCT) from the Royal Golden Jubilee\n(RGJ) Program, Center of Excellence in High Energy Physics and Astrophysics (CoE) at Suranaree University of20\nQuantity G∗\nE(0) ⟨r2⟩∗\nE[GeV−2]\nΛ = 1 GeV 0.07 + 0 .00i−8.01 + 28 .08i\nΛ = 2 GeV 0.07 + 0 .00i−8.69 + 27 .02i\nhA≈2.90.07 + 0 .00i−11.01 + 29 .75i\nhA≈2.40.07 + 0 .00i−6.38 + 24 .30i\nHA≈2.50.07 + 0 .00i−9.18 + 27 .03i\nHA≈2.00.07 + 0 .00i−8.21 + 27 .02i\nTABLE II. Same as Table I but for the electric quantities.\nTechnology, Swedish International Development Cooperation Agency (Sida) and International Science Programme\n(ISP) at Uppsala University under the Thai-Swedish Trilateral Development Cooperation Programme (contract no.\nPHD/0103/2560).\nAppendix A: Which quantities are really measured in the spacelike region?\nExperimentally the TFFs are extracted from the reaction e−N→e−∆→e−πNin the one-photon approximation,\ni.e. one studies formally the reaction γ∗N→∆→πN. Which events from all final-state πNpairs are used to\nreconstruct the ∆? Clearly, if one plots the number of events as a function of the invariant mass mπNof the final-\nstate πNpair, one observes a relatively broad bump with width Γ ∆that peaks at m∆. In principle, there are two\npossibilities how to continue:\na) One could take all events in the bump region, e.g. m∆−Γ∆/2< m πN< m ∆−Γ∆/2 (and subtract the background\nbelow the bump);3\nor b) one could take just the events from the bin at the peak position, i.e. mπN≈m∆.\nIt turns out that the second option is used by the experimentalists. This is spelled out explicitly, e.g., in [5] between\nequations (2.9) and (2.10).\nIn the following, we determine how the scattering amplitudes for γ∗N→∆→πNwith mπN≈m∆are related to\nour TFFs. In that way we are aiming at the justification of (12). Why is the real part used on the left-hand side?\nAnd not, e.g., the modulus? We stick to a one-loop calculation and treat vertices, in particular their spinor structure,\nvery schematically. The quantities F,TandPthat we introduce in the following are supposed to have no imaginary\nparts in the kinematical region that we study.\nThe “initial” γ∗Nsystem can couple to the ∆ directly or via a πNloop. We denote the direct coupling by Tand\nthe four-point γ∗N-πNstructure by F. The three-point coupling ∆- πNis denoted by P. Finally, the πNloop is\ngiven by\nL:=Zd4q\n(2π)41\nq2−m2π+iϵ1\n(p∆−q)2−m2\nN+iϵ. (A1)\nHere p∆is the ∆ momentum, i.e. the total momentum of the final πNpair. Using retarded propagators and/or Wick\nrotation, one can show\nIm(iL) =−pcm\n8πmπN(A2)\nwith the momentum of πandNin their center-of-mass frame,\npcm=1\n2mπNλ1/2(m2\nπN, m2\nπ, m2\nN). (A3)\nNote that at the peak position where mπN=m∆one has pcm=k∆with the latter being introduced after (12).\nA TFF, G, is obtained by\niG=iT+iFi2L iP (A4)\n3This is what is done in the timelike region of Dalitz decays [39].21\nwhich leads to\nG=T − F iLP. (A5)\nFor later use we determine\nReG=T − F Re(iL)P. (A6)\nThe Feynman matrix element Mfor the reaction γ∗N→∆→πNis obtained as\niM=iTiS iP+iFi2L iPiS iP. (A7)\nHere Sdenotes the full ∆ propagator\nS=1\np2\n∆−m2\n∆−Π≈1\np2\n∆−m2\n∆−iImΠ(A8)\nwith the self-energy Π [68] given by\n−iΠ =iPi2L iP=P2L . (A9)\nFor later use (we are aiming at a justification of (12)) we calculate\nImM=−T P ImS+F P2Im(iL S)\n=−T P ImS+F P2Re(iL) ImS\n+F P2Im(iL) ReS . (A10)\nUp to normalization issues, Mshould agree with M(3/2)\n1+, which appears in (12).\nThe width of the ∆ is practically given by its decay to πN. With the Breit-Wigner formula\nS≈1\np2\n∆−m2\n∆+im∆Γ∆(A11)\nthis implies\nm∆Γ∆=−ImΠ = −P2Im(iL). (A12)\nRecall that p∆denotes the total momentum of the final πNsystem; thus p2\n∆=m2\nπN. So far we have not specified\nmπN. But following the experimental procedure, the choice mπN=m∆leads to\nReS= 0, ImS=−1\nm∆Γ∆. (A13)\nFor this choice of the two-body mass, i.e. at the peak position of the ∆ bump, equation (A10) simplifies to\nImM=−T P ImS+F P2Re(iL) ImS\n= (−T+FRe(iL)P)PImS\n=−ReGPImS (A14)\nwhere we have used (A6) in the last step. We can invert this equation and use (A13), (A12), and (A2) to obtain\nReG=−ImM\nPImS=ImMm∆Γ∆\nP\n∼ImMm∆Γ∆s\n−Im(iL)\nm∆Γ∆∼ImMp\nk∆Γ∆. (A15)\nThis last version compares well with (12) concerning the appearance of the ∆ width and the phase-space factor ∼k∆.\nEquation (A15) specifies that it should be the real part of the TFF that appears on the left-hand side of (12). The\nprocedure to use the real parts is in line with [23].22\nAppendix B: Contributions from the anomalous cut\n1. General considerations about the analytic structure\nThe extra term Tanom(s) in (28) arises if the mass mexchof the exchanged state in a crossed channel is sufficiently\nlight to fulfill the anomalous threshold condition [69]:\nm2\nexch<1\n2\u0000\nm2\n∆+m2\nn−2m2\nπ\u0001\n. (B1)\nFor the formal reaction ∆0¯n→π+π−, one has to consider the exchange of the proton and ∆±resonances. The\ncondition (B1) does not hold for the ∆ exchange, but is satisfied for the proton exchange. In this case, the logarithm\nobtained from the partial-wave projection (27) has a cut in the complex splane that intersects with the unitarity cut.\nPart of this cut lies on the physical Riemann sheet. Dispersion relations have to be modified accordingly in order to\nproduce the correct results. In practice, the contour has to circumnavigate the new cut in addition to the familiar\ntwo-pion cut. As explored in [7], a dispersive representation that ignores such anomalous cut produces incomplete\nresults. To disentangle the cuts, one can define the cut of the logarithm such that it connects the branch point to the\nunitarity cut by a straight line. The additional contribution Tanom(s) takes care of the extra cut.\nTo be more concrete, we note that the p-wave partial-wave projection of type (27) for a t- oru-channel exchange\nprocess produces a term\nK(s) = g(s)−2f(s)\nY(s)κ2(s)\n+f(s)1\nκ3(s)logY(s) +κ(s)\nY(s)−κ(s)(B2)\nwith the functions Y,κandσdefined as\nY(s) :=s+ 2m2\nexch−m2\n1−m2\n2−2m2\nπ, (B3)\nκ(s) :=λ1/2(s, m2\n1, m2\n2)σ(s), (B4)\nand\nσ(s) :=r\n1−4m2π\ns. (B5)\nHere we choose m1=m∆,m2=mn. Specific formulae for the functions f(s),g(s) can be deduced from the matrix\nelements provided in Section IV. Obviously Eq. (B2) is ill-defined for Y(s) = 0. When the exchanged baryon is\nthe ∆ resonance, this point lies outside the integration path, bringing no additional complication to our formalism.\nHowever, in the case of proton exchange, this point is located on the unitarity cut, at sY= 0.675 GeV2. Therefore, the\nlogarithm needs an analytic continuation along the unitarity cut. For convenience one can consider different intervals\ndelimited by the following four points:\n•At the scattering threshold sst:= (m∆+mn)2we have κ= 0. Above this point, i.e. for sreal and larger than\nsst, there is the true scattering region. There, κis real and Yis positive and larger than κ. The logarithm in\n(B2) can be defined as the real-valued standard logarithm of positive numbers.\n•AtsY:=m2\n∆+m2\nn+ 2m2\nπ−2m2\npwe have Y= 0. For sreal and between sYandsstthe function κis purely\nimaginary and Yis still positive.\n•At the decay threshold4sdt:= (m∆−mn)2we have κ= 0. For sreal and between sdtandsYthe function κis\npurely imaginary and Yis negative.\n•Ats2π:= 4m2\nπwe have κ= 0. For sreal and between s2πandsdtthe function κis real and Yis negative.\n4Concerning the phrase “decay threshold” we note that for s <(m∆−mn)2the decay ∆ →n2πis possible.23\nWhen the exchanged baryon is the proton we have 0 < s2π< sdt< sY< sst. The function Kin (B2) that enters\nfinally (28) is then defined on the relevant part of the real axis by\nK(s) :=g(s)−2f(s)\nY(s)κ2(s)+f(s)\nκ3(s)logY(s) +κ(s)\nY(s)−κ(s)(B6)\nfors > s st, by\nK(s) :=g(s)−2f(s)\nY(s)κ2(s)+2f(s)\nκ2(s)|κ(s)|arctan|κ(s)|\nY(s)(B7)\nforsY< s < s st, by\nK(s) := g(s)−2f(s)\nY(s)κ2(s)\n+2f(s)\nκ2(s)|κ(s)|\u0012\narctan|κ(s)|\nY(s)+π\u0013\n(B8)\nforsdt< s < s Y, and by\nK(s) :=g(s)−2f(s)\nY(s)κ2(s)+f(s)\nκ3(s)\u0012\nlogY(s) +κ(s)\nY(s)−κ(s)+ 2iπ\u0013\n(B9)\nfors2π< s < s dt. Here the standard logarithm for positive real numbers is used and the arctan function with values\nbetween −π/2 and π/2.\nThe extra πin (B8) and 2 iπin (B9) ensure a smooth analytic continuation of K. However, such an extra piece\ncreates a singularity where κvanishes. This happens at the decay threshold sdt; see also the discussion in [70] and\nreferences therein. In principle, the singularity does not carry over to the FF and is of no physical significance. But\nit is numerically unpleasant to deal with explicitly. Therefore we have decided to modify the integration contour and\navoid the singularity altogether. This is better explained after finishing the discussion of the analytic structure of our\namplitudes. We will come back to the singularity in Subsection B 2.\nThe branch points of the logarithm in (B2) satisfy Y2(s) =κ2(s). They are located at\ns±=−1\n2m2\nexch+1\n2\u0000\nm2\n∆+m2\nn+ 2m2\nπ\u0001\n−m2\n∆m2\nn−m2\nπ(m2\n∆+m2\nn) +m4\nπ\n2m2\nexch\n∓λ1/2(m2\n∆, m2\nexch, m2\nπ)λ1/2(m2\nexch, m2\nn, m2\nπ)\n2m2\nexch. (B10)\nWe take s+as the solution that has a positive imaginary part for small values of m2\n∆. For the case m2\nexch=m2\np,\nthe trajectory of s+as a function of m2\n∆+iϵintersects with the unitarity cut where (B1) turns to an equality. In\nparticular, for the physical value of m2\n∆, the branch point s+is located in the lower half plane of the first Riemann\nsheet:\ns+=−1\n2m2\np+1\n2\u0000\nm2\n∆+m2\nn+ 2m2\nπ\u0001\n−m2\n∆m2\nn−m2\nπ(m2\n∆+m2\nn) +m4\nπ\n2m2p\n−iλ1/2(m2\n∆, m2\np, m2\nπ)\u0000\n−λ(m2\np, m2\nn, m2\nπ)\u00011/2\n2m2p(B11)\nwith positive square roots. This is the starting point for the definition of the anomalous contribution Tanomthat\nenters (28):\nTanom(s) = Ω( s)s1Z\n0dxds′(x)\ndx1\ns′(x)−s\n×2f(s′(x))\n(−λ(s′(x), m2\n∆, m2n))1/2κ2(s′(x))\n×t(s′(x))\nΩ(s′(x))s′(x)(B12)24\nwith the straight-line path\ns′(x) := (1 −x)s++x sc (B13)\nthat connects the branch point (B11) of the logarithm in (B2) to the point sc= 5m2\nπon the unitarity cut. We have\ndedicated Subsection B 2 below to motivate why we chose to connect the branch point s+to the point sc, instead of\nto the two-pion threshold point s2π, as originally done in [7]. But before turning to this issue, we specify the further\ningredients.\nAn analytic continuation of the scattering amplitude t(s) in the complex plane is needed for the anomalous part\nof Eq. (B12). We take from [71] the following expressions (extended to the complex plane). The approximation from\nχPT is given by\ntχPT(s)≈t2(s) +t4(s) (B14)\nand its unitarized version is\ntIAM(s) =t2\n2(s)\nt2(s)−t4(s)(B15)\nwith\nt2(s) =sσ2\n96πF2\n0, (B16)\nt4(s) =t2(s)\n48π2F2\n0\u0014\ns\u0012\n¯l+1\n3\u0013\n−15\n2m2\nπ−m4\nπ\n2s\u0010\n41−2Lσ\u0000\n73−25σ2\u0001\n+ 3L2\nσ\u0000\n5−32σ2+ 3σ4\u0001\u0011\u0015\n−ˆσ(s)t2\n2(s), (B17)\nLσ=1\nσ2\u00121\n2σlog1 +σ\n1−σ−1\u0013\n. (B18)\nThe function σ(s) is defined in (B5) and ˆ σ(s) by\nˆσ(z) :=r\n4m2π\nz−1. (B19)\nNote that the point schas been chosen close to the two-pion threshold such that the anomalous integral (B12) does\nnot run over the high-energy region where the representations (B14) and (B15) become unreliable.\nThe value for the pion decay constant in the chiral limit F0is taken from the ratio Fπ/F0= 1.064(7), where\nFπ= 92.28(9) MeV is the pion decay constant at the physical point. In the present work the low-energy constant\n¯lis adjusted such that the pion p-wave phase shifts from (B15) agrees with that from [27] at the point sc; see the\ndiscussion preceding (B31) below. In practice we use ¯l= 6.099. This compares well with previous choices. In the\noriginal paper [71], the low-energy constant ¯l= 5.73(8) has been adjusted such as to reproduce the position of the\npole of the ρ-meson resonance on the second Riemann sheet. In [7] ¯l= 6.47 had been used.\nFinally, we provide the anomalous piece of the TFFs:\nGanom\nm(q2) =1\n12π1Z\n0dxds′′(x)\ndx1\ns′′(x)−q2\n×fm(s′′(x))s′′(x)FV\nπ(s′′(x))\n−4 (−λ(s′′(x), m2\n∆, m2n))3/2, (B20)\nthis time with the straight-line path\ns′′(x) := (1 −x)s++x sY (B21)\nthat connects the branch point (B11) of the logarithm in (B2) to the point sYon the unitarity cut. The choice of\nthis integration path simplifies the calculations as explained in the next subsection.25\n2. Further modifications of the integration contour\nThe standard path of the branch cuts — from s+to the two-pion threshold s2π= 4m2\nπand then along the real\naxis to + ∞— involves an integrand that is singular at the decay threshold sdt= (m∆−mN)2. Even though this\nsingularity is integrable with the epsilon prescription m2\n∆→m2\n∆+iϵ[72], it is numerically easier to avoid this problem.\nLet the original integrals that we want to calculate be given by\nF(s) :=Z\nC+,2πdzJ1(z)\nz−s−iϵ′+∞Z\n4m2πds′J2(s′)\ns′−s−iϵ′(B22)\nwith the path C+,2πconnecting s+to the two-pion threshold. Consider a path along the closed triangle formed by\ns+, the two-pion threshold s2πand an arbitrary point scon the real axis above the decay threshold sdt. An integral\nover a function along this closed path vanishes, if this function is analytic inside of this triangle. This is the case for\nintegrands I(z) of the type\nI(z) =J1(z)\nz−s−iϵ′∼1\n[−λ(z, m2\n∆+iϵ, m2\nN)]3/21\nz−s−iϵ′. (B23)\nHere slies on the real axis and the square root function is defined with a cut along the negative real axis. With the\nϵprescription for the mass of the unstable ∆, the function −λ(z, m2\n∆+iϵ, m2\nN) adopts negative real values slightly\nabove the real axis (with real parts below sdtor above sst).\nThus instead of (B22) we can write\nF(s) =Z\nC+,cdzJ1(z)\nz−s−iϵ′+scZ\n4m2πds′J2(s′)−J1(s′)\ns′−s−iϵ′+∞Z\nscds′J2(s′)\ns′−s−iϵ′(B24)\nwith the path C+,cconnecting s+tosc. What we have used to obtain (B24) is\n0 =Z\nC+,2πdzJ1(z)\nz−s−iϵ′+scZ\n4m2πds′J1(s′)\ns′−s−iϵ′−Z\nC+,cdzJ1(z)\nz−s−iϵ′. (B25)\nThe difference J2(s′)−J1(s′) in (B24) involves just the standard logarithm/arctan without the extra term ∼2πi.\nThis difference diverges neither at the decay threshold sdtnor at the two-pion threshold s2π. To be slightly more\nspecific:\nJ2(s′)−J1(s′)∼log for s2π< s′< sdt,\nJ2(s′)−J1(s′)∼arctan for sdt< s′< sc,\nJ2(s′)∼arctan + π for sc< s′< sY,\nJ2(s′)∼arctan for sY< s′< sst,\nJ2(s′)∼log for sst< s′. (B26)\nHere sYdenotes the point where Y(s) vanishes. Obviously, the simplest choice would be sc=sY. Then we could use\nin (B24) the standard log or standard arctan along the whole real axis. But we will argue below that this is nota\ngood choice for sc.\nFor the calculation of (B24) the only numerically problematic point is at s=sc. Since this point is arbitrary, the\nresulting function F(s) must be smooth at this point. Schematically we can rewrite each of the integrals of (B24) into\nZ\ndzJ...(z)\nz−s−iϵ′=Z\ndzJ...(z)−J...(s)\nz−s−iϵ′+J...(s)Z\ndz1\nz−s−iϵ′. (B27)\nHere J...denotes J1,J2orJ1−J2. The first term on the right hand side of (B27) is smooth for any value of sif\nJ...(z)−J...(s)∼z−s for z→s . (B28)\nThe second term is proportional to J...(s) log( sc−s). Such a term diverges logarithmically for s→sc. If one takes\nall integrals of (B24) together, one obtains for the potentially divergent terms a sum proportional to\nJ1(s) log( sc−s) + (J2(s)−J1(s)) log( sc−s)−J2(s) log( sc−s) = 0 . (B29)26\nThus there is no numerical problem with (B24) if something like (B27) and (B29) is numerically implemented and if\n(B28) is satisfied.\nTo be more specific one needs in particular\nJ1(z)−J1(sc)∼z−sc for z→sc. (B30)\nHere zis a complex number on the line that connects s+with sc. All this resembles to some extent the discussion for\nthe two-pion threshold in [7]. We use in practice two versions for the two-pion scattering amplitude. One along the\nreal axis based on the measured phase shift, tps, and the other, tIAM, employed in the complex plane for the definition\nofJ1. Those two versions must agree at the connection s=scto make F(s) smooth, i.e. to ensure that (B30) is\nsatisfied. The latter is necessary, otherwise the cancellation (B29) does not happen. The crucial point is that J1(z) is\nobtained from tIAMwhile J1(sc) is obtained from tps. What needs to be done in practice is to readjust the low-energy\nconstant that appears in tIAMsuch that\nRet−1\nps(sc) = Re t−1\nIAM(sc) (B31)\nholds.\nAs spelled out in [7], we trust the complex-plane two-pion amplitude tIAMin the low-energy region. Thus scshould\nnot be chosen too large. More generally, the whole path C+,cshould lie in the low-energy region. In practice, the\nbranch point of the anomalous branch cut lies at s+≈(0.0458−0.0827i) GeV2. This is in size comparable to the\ntwo-pion threshold s2π≈0.0779 GeV2. On the other hand, the point where Y(s) vanishes lies at sY≈0.675 GeV2,\nwhich is not very small. Thus we should choose sclarger than the decay threshold sdt≈0.0858 GeV2, but much\nsmaller than sY. A convenient choice should be sc= 5m2\nπ≈0.0974 GeV2.\nAppendix C: Large- Ncrelations\nThis appendix has the purpose to provide an educated guess for the relative sign between the ∆-nucleon-pion\ncoupling constant hAand the TFFs provided by the experimental groups. This educated guess — based on calculations\nfor a large number of quark colors Nc— will be further supported by a direct comparison to the experimental results\nexploring both signs of hA. Still it is encouraging to have additional theoretical support for our final choice.\nWe extend the chiral Lagrangian such that it provides the interactions of baryons with Goldstone bosons and\nphotons up to (including) next-to-leading order [52]:\nD\n2tr(¯B γµγ5{uµ, B}) +F\n2tr(¯B γµγ5[uµ, B])\n+hA\n2√\n2\u0000\nϵade¯Be\nc(uµ)d\nbTabc\nµ+ϵade¯Tµ\nabc(uµ)b\ndBc\ne\u0001\n+bM,Dtr(¯B{fµν\n+, σµνB}) +bM,Ftr(¯B[fµν\n+, σµνB])\n+i cM\u0000\nϵade¯Be\ncγµγ5(fµν\n+)d\nbTabc\nν−ϵade(¯Tν)abcγµγ5(fµν\n+)b\ndBc\ne\u0001\n. (C1)\nIn the framework of [49, 73, 74] the baryons are treated as quasi non-relativistic fields. Four-component spinors Bare\nprojected on their two-component particle content Bvwhere vdenotes the baryon velocity. For simplicity we choose\nthe rest frame v= (1,0,0,0). Then the dominant components of Tµare the spatial components. They satisfy\nTk\nvσk= 0 (C2)\nwith the Pauli matrices σk. The dominant parts of the spinor matrices are given by\nγkγ5→σk,\nσij→ϵijkσk,\nγ0γ5≈0,\nσ0j≈0. (C3)\nEssentially one can treat the axial-vector field ukand the magnetic field ϵijkf+\nijon equal footing. Using (C3), the27\ninteraction terms (C1) can be approximated in the following way:\nD\n2tr(B†\nvσk{uk, Bv}) +F\n2tr(B†\nvσk[uk, Bv])\n+hA\n2√\n2\u0010\nϵade(B†\nv)e\nc(uk)d\nbTk,abc\nv +ϵadeTk†\nv,abc(uk)b\nd(Bv)c\ne\u0011\n+bM,Dtr(B†\nv{fij\n+, ϵijkσkBv}) +bM,Ftr(B†\nv[fij\n+, ϵijkσkBv])\n+i cM\u0010\nϵade(B†\nv)e\ncσj(fjk\n+)d\nbTk,abc\nv−ϵadeTk†\nv,abcσj(fjk\n+)b\nd(Bv)c\ne\u0011\n. (C4)\nReplacing ukbyϵijkf+\nij, one proceeds from the D(F) term to the bM,D(bM,F) term.\nThe relation between the hAterm and the cMterm is less obvious. Here the constraint (C2) comes into play. We\nfirst rewrite parts of the cMterms (flavor indices are suppressed; ordering matters for the spinor structure):\nσjfjk\n+Tk\nv=1\n2\u0000\nσjTk\nv−σkTj\nv\u0001\nfjk\n+=1\n2σlTm\nv\u0000\nδljδmk−δlkδmj\u0001\nfjk\n+\n=1\n2σlTm\nvϵlmnϵjknfjk\n+=−1\n4i[σm, σn]Tm\nvϵjknfjk\n+=−1\n4i σmσnTm\nvϵjknfjk\n+\n=−1\n4i{σm, σn}Tm\nvϵjknfjk\n+=−1\n4i2δnmTm\nvϵjknfjk\n+=−1\n2i Tm\nvϵjkmfjk\n+\n=−1\n2i Tk\nvϵijkfij\n+; (C5)\nTk†\nvσjfjk\n+=1\n2\u0000\nTk†\nvσj−Tj†\nvσk\u0001\nfjk\n+=1\n2Tm†\nvσl\u0000\nδljδmk−δlkδmj\u0001\nfjk\n+\n=1\n2Tm†\nvσlϵlmnϵjknfjk\n+=−1\n4i Tm†\nv[σm, σn]ϵjknfjk\n+=1\n4i Tm†\nvσnσmϵjknfjk\n+\n=1\n4i Tm†\nv{σn, σm}ϵjknfjk\n+=1\n4i Tm†\nv2δnmϵjknfjk\n+=1\n2i Tm†\nvϵjkmfjk\n+\n=1\n2i Tk†\nvϵijkfij\n+. (C6)\nThus we can rewrite (C4) into\nD\n2tr(B†\nvσk{uk, Bv}) +F\n2tr(B†\nvσk[uk, Bv])\n+bM,Dtr(B†\nvσk{ϵijkfij\n+, Bv}) +bM,Ftr(B†\nvσk[ϵijkfij\n+, Bv])\n+hA\n2√\n2\u0010\nϵade(B†\nv)e\ncTk,abc\nv (uk)d\nb+ϵadeTk†\nv,abc(uk)b\nd(Bv)c\ne\u0011\n+1\n2cM\u0010\nϵade(B†\nv)e\ncTk,abc\nvϵijk(fij\n+)d\nb+ϵadeTk†\nv,abcϵijk(fij\n+)b\nd(Bv)c\ne\u0011\n. (C7)\nTherefore the large- Ncformalism relates cM/√\n2 in the same way to bM,DandbM,Fas it relates hAtoDandF. In\nparticular, if conventions are chosen such that hAhas the same positive sign as DandF, then cMmust have the\nsame sign as bM,DandbM,F.\nAt next-to-leading order of baryon χPT, i.e. using (C1), the anomalous magnetic moments of proton and neutron\nare determined by bM,DandbM,F. The TFF F1is determined by cM.\nIt turns out that the respective signs of the magnetic moments of proton and neutron lead to positive values for\nbM,DandbM,F[60, 75]. Then the large- Ncconsiderations lead to a positive value for cM. In turn, this leads to a\npositive value for F1(0). Consequently, we obtain a negative value for G−1(0) from (5) and a positive value for G+1(0)\nfrom (7). Therefore the combination G−1(0)−3G+1(0) is negative. On the other hand, the usual convention for\nthe magnetic Jones-Scadron TFF is such that G∗\nM(0) is positive [5]. To obtain such a positive magnetic transition\nmoment requires an overall negative sign in (10) and therefore (14).28\nFinally, we note that these results are in complete agreement with the calculations of [7]. There is a relative minus\nsign in the flavor factors between the transitions Σ∗-Λ and ∆- Nas can be read off from table 3 of [52]. Thus the\npositive value of G−1(0) for the hyperon case in table I of [7] translates to a negative value for the ∆-nucleon case.\nThe negative value of G+1(0) for the hyperon case translates to a positive value for the ∆-nucleon case. All this agrees\nwith the analysis of the previous paragraph.\nTo summarize the large- Ncprediction: A positive value for hAleads to a negative value of ζin (14) and vice versa.\nA positive value for ζwould lead to a negative value for hA.\nIn practice, we demand (14). Then we expect to find a positive value of hA. But we will explore both options by a\ncomparison of our subtracted dispersion relation (29) to the experimental results for the q2dependence of G∗\nM(q2).\nAppendix D: Contact terms from chiral perturbation theory\nOur subtracted dispersion relations (29) and (28) contain undetermined subtraction constants. They parametrize\nour limited knowledge of the high-energy region. The same statement holds for the low-energy constants of effective\nfield theories. If we insist that all pion-baryon scattering amplitudes are obtained from χPT (plus pion rescattering)\nthen one can relate the polynomials Pmto contact terms emerging at various orders of the chiral power counting.\nOn the other hand, it has been found in [11] for the nucleon isovector FFs that the subtraction constants do not\nentirely agree with low-order estimates from χPT. We found the same for our case at hand. It is better to determine\nthe subtraction constants from data.\nFor completeness, however, we will present the results for the constants Pmthat are obtained from an NLO\ncalculation. The first step is to specify the three-point interactions. Modifications can be compensated by changes of\nthe contact terms (four-point interactions) [6, 11, 76].\nFor our purposes the interaction term proportional to HAeffectively reduces to\n+HA\n2mRFπϵµναβ¯Tµ\nabc∂ν(Tα)abd∂βΦc\nd (D1)\nwhere the resonance mass mRcorresponds in this case to m∆. Working with relativistic spin-3/2 Rarita-Schwinger\nfields is plagued by ambiguities how to deal with the spurious spin-1/2 components. In the present context the\ninteraction term ∼hAcauses not only the proper exchange of spin-3/2 resonances, but induces an additional contact\ninteraction. This unphysical contribution can be avoided by constructing interaction terms according to the Pascalutsa\ndescription [5, 55, 77, 78]. It boils down to the replacement\nTµ→ −1\nm∆ϵνµαβγ5γν∂αTβ. (D2)\nStrictly speaking this procedure induces an explicit flavor breaking, but such effects are anyway beyond leading order.\nTheHAterm of (D1) is already constructed such that only the spin-3/2 components contribute.\nWe will explore both the standard interaction term ∼hAfrom (31) and the corresponding one obtained by (D2).\nAs already discussed in [7], differences can be accounted for by contact interactions of the chiral Lagrangian at NLO\nand beyond.\nThe explicit expressions for the polynomial terms are\nP+1=−gAhA\n4√\n6F2π2−5hAHA\n12√\n6F2π5 (m∆+mn)\n6m∆,\nP−1=−gAhA\n4√\n6F2π2 (m∆−mn−mp)\nm∆\n−5hAHA\n12√\n6F2πs−2m2\nπ−(m∆+mn)(6m∆−mn)\n6m2\n∆\n≈ −gAhA\n4√\n6F2π2 (m∆−mn−mp)\nm∆\n−5hAHA\n12√\n6F2π−(m∆+mn)(6m∆−mn)\n6m2\n∆,\nP0=−gAhA\n4√\n6F2π−5hAHA\n12√\n6F2π3m∆−mn\n6m∆. (D3)29\nForP−1we dropped terms which are suppressed by two orders in the chiral counting.\nThe ∆¯ nπ+π−contact diagram produces the following polynomials:\nPNLO χPT\n+1 =−cFm∆+mn√\n3F2π,\nPNLO χPT\n0 =−cFm∆√\n3F2π,\nPNLO χPT\n−1 =cFmn(m∆+mn)−s√\n3F2πm∆\n≈cFmn(m∆+mn)√\n3F2πm∆. (D4)\nThe amplitudes (36) become slightly different when the Pascalutsa prescription is used: new contact terms pop up\nbut the pole terms and therefore (38) are not affected. In particular we have:\nPP\n+1=P+1\n−5hAHA\n36√\n6F2πm2\n∆((mn+m∆)(2m∆+ 3mn)−3s),\nPP\n0=P0+5hAHA\n36√\n6F2π,\nPP\n−1=P−1\n+5hAHA\n36√\n6F2πm2\n∆((mn+m∆)(3m∆+ 2mn)−2s).(D5)\nAs expected the proton exchange diagrams do not get any contribution since in that case the only ∆ baryon is the\nexternal one, which is on-shell.\n[1] G. Aad et al. (ATLAS), Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex].\n[2] S. Chatrchyan et al. (CMS), Phys. Lett. B 716, 30 (2012), arXiv:1207.7235 [hep-ex].\n[3] G. A. Miller, Phys. Rev. Lett. 99, 112001 (2007), arXiv:0705.2409 [nucl-th].\n[4] V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, Eur. Phys. J. A 51, 79 (2015), arXiv:1503.01452\n[nucl-ex].\n[5] V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys. Rept. 437, 125 (2007), arXiv:hep-ph/0609004 [hep-ph].\n[6] C. Granados, S. Leupold, and E. Perotti, Eur. Phys. J. A 53, 117 (2017), arXiv:1701.09130 [hep-ph].\n[7] O. Junker, S. Leupold, E. Perotti, and T. Vitos, Phys. Rev. C 101, 015206 (2020), arXiv:1910.07396 [hep-ph].\n[8] T. Husek and S. Leupold, Eur. Phys. J. C 80, 218 (2020), arXiv:1911.02571 [hep-ph].\n[9] L. Tiator, D. Drechsel, S. S. Kamalov, and M. Vanderhaeghen, Eur. Phys. J. ST 198, 141 (2011), arXiv:1109.6745 [nucl-th].\n[10] J. M. Alarc´ on, A. N. Hiller Blin, M. J. Vicente Vacas, and C. Weiss, Nucl. Phys. A 964, 18 (2017), arXiv:1703.04534\n[hep-ph].\n[11] S. Leupold, Eur. Phys. J. A 54, 1 (2018), arXiv:1707.09210 [hep-ph].\n[12] F. Alvarado, D. An, L. Alvarez-Ruso, and S. Leupold, Phys. Rev. D 108, 114021 (2023), arXiv:2310.07796 [hep-ph].\n[13] N. F. Sparveris et al. (OOPS), Phys. Rev. Lett. 94, 022003 (2005), arXiv:nucl-ex/0408003.\n[14] N. F. Sparveris et al. , Phys. Lett. B 651, 102 (2007), arXiv:nucl-ex/0611033.\n[15] S. Stave et al. , Eur. Phys. J. A 30, 471 (2006), arXiv:nucl-ex/0604013.\n[16] I. G. Aznauryan et al. (CLAS), Phys. Rev. C 80, 055203 (2009), arXiv:0909.2349 [nucl-ex].\n[17] D. An and S. Leupold, “Electromagnetic isovector form factors of the transition from the N∗(1520) to the nucleon,” (2024),\nin preparation.\n[18] S. Weinberg, Physica A 96, 327 (1979).\n[19] J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984).\n[20] J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985).\n[21] S. Scherer, Adv. Nucl. Phys. 27, 277 (2003), arXiv:hep-ph/0210398 [hep-ph].\n[22] S. Scherer and M. R. Schindler, Lect. Notes Phys. 830(2012), 10.1007/978-3-642-19254-8.\n[23] M. Hilt, T. Bauer, S. Scherer, and L. Tiator, Phys. Rev. C 97, 035205 (2018), arXiv:1712.08904 [nucl-th].\n[24] Y. ¨Unal, A. K¨ u¸ c¨ ukarslan, and S. Scherer, Phys. Rev. D 104, 094014 (2021), arXiv:2104.03125 [hep-ph].\n[25] C. Terschl¨ usen and S. Leupold, Phys. Rev. D 94, 014021 (2016), arXiv:1603.05524 [hep-ph].\n[26] G. Colangelo, J. Gasser, and H. Leutwyler, Nucl. Phys. B 603, 125 (2001), arXiv:hep-ph/0103088.30\n[27] R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira, and F. J. Yndurain, Phys. Rev. D 83, 074004 (2011),\narXiv:1102.2183 [hep-ph].\n[28] C. Hanhart, Phys. Lett. B 715, 170 (2012), arXiv:1203.6839 [hep-ph].\n[29] M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Phys. Rept. 625, 1 (2016), arXiv:1510.06039 [hep-ph].\n[30] T. Isken, B. Kubis, S. P. Schneider, and P. Stoffer, Eur. Phys. J. C 77, 489 (2017), arXiv:1705.04339 [hep-ph].\n[31] H. F. Jones and M. D. Scadron, Annals Phys. 81, 1 (1973).\n[32] J. G. K¨ orner and M. Kuroda, Phys. Rev. D 16, 2165 (1977).\n[33] N. Salone and S. Leupold, Eur. Phys. J. A 57, 183 (2021), arXiv:2104.05675 [hep-ph].\n[34] G. Ramalho, Phys. Rev. D 93, 113012 (2016), arXiv:1602.03832 [hep-ph].\n[35] W. A. Bardeen and W. K. Tung, Phys. Rev. 173, 1423 (1968), [Erratum: Phys. Rev. D 4, 3229 (1971)].\n[36] R. Tarrach, Nuovo Cim. A 28, 409 (1975).\n[37] C. E. Carlson, Phys. Rev. D 34, 2704 (1986).\n[38] G. Ramalho, M. T. Pena, J. Weil, H. van Hees, and U. Mosel, Phys. Rev. D 93, 033004 (2016), arXiv:1512.03764 [hep-ph].\n[39] J. Adamczewski-Musch et al. (HADES), Phys. Rev. C 95, 065205 (2017), arXiv:1703.07840 [nucl-ex].\n[40] C. Hanhart, A. Kup´ s´ c, U.-G. Meißner, F. Stollenwerk, and A. Wirzba, Eur. Phys. J. C 73, 2668 (2013), [Erratum: Eur.\nPhys. J. C 75, 242 (2015)], arXiv:1307.5654 [hep-ph].\n[41] M. Hoferichter, B. Kubis, J. Ruiz de Elvira, H.-W. Hammer, and U.-G. Meißner, Eur. Phys. J. A 52, 331 (2016),\narXiv:1609.06722 [hep-ph].\n[42] M. Fujikawa et al. (Belle), Phys. Rev. D 78, 072006 (2008), arXiv:0805.3773 [hep-ex].\n[43] F. Niecknig, B. Kubis, and S. P. Schneider, Eur. Phys. J. C 72, 2014 (2012), arXiv:1203.2501 [hep-ph].\n[44] R. Garcia-Martin and B. Moussallam, Eur. Phys. J. C 70, 155 (2010), arXiv:1006.5373 [hep-ph].\n[45] X.-W. Kang, B. Kubis, C. Hanhart, and U.-G. Meißner, Phys. Rev. D 89, 053015 (2014), arXiv:1312.1193 [hep-ph].\n[46] N. Muskhelishvili, Singular integral equations: Boundary problems of function theory and their application to mathematical\nphysics (Springer, 1958).\n[47] R. Omn` es, Nuovo Cim. 8, 316 (1958).\n[48] Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, Eur. Phys. J. A 59, 54 (2023), arXiv:2205.00850 [hep-ph].\n[49] E. E. Jenkins and A. V. Manohar, Phys. Lett. B 259, 353 (1991).\n[50] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A 700, 193 (2002), arXiv:nucl-th/0105042.\n[51] T. Ledwig, J. Martin Camalich, L. S. Geng, and M. J. Vicente Vacas, Phys. Rev. D 90, 054502 (2014), arXiv:1405.5456\n[hep-ph].\n[52] M. Holmberg and S. Leupold, Eur. Phys. J. A 54, 103 (2018), arXiv:1802.05168 [hep-ph].\n[53] C. J. G. Mommers and S. Leupold, Phys. Rev. D 106, 093001 (2022), arXiv:2208.11078 [hep-ph].\n[54] R. L. Workman et al. (Particle Data Group), PTEP 2022 , 083C01 (2022).\n[55] V. Pascalutsa and M. Vanderhaeghen, Phys. Lett. B 636, 31 (2006), arXiv:hep-ph/0511261.\n[56] M. Holmberg and S. Leupold, Phys. Rev. D 100, 114001 (2019), arXiv:1909.13562 [hep-ph].\n[57] M. Bertilsson and S. Leupold, (2023), arXiv:2311.07471 [hep-ph].\n[58] J. A. Oller, M. Verbeni, and J. Prades, JHEP 09, 079 (2006), arXiv:hep-ph/0608204 [hep-ph].\n[59] M. Frink and U.-G. Meißner, Eur. Phys. J. A 29, 255 (2006), arXiv:hep-ph/0609256.\n[60] B. Kubis and U.-G. Meißner, Eur. Phys. J. C 18, 747 (2001), arXiv:hep-ph/0010283.\n[61] Wolfram Research, Inc., Mathematica, Version 12.0 (Champaign, IL, 2019).\n[62] R. Mertig, M. B¨ ohm, and A. Denner, Computer Physics Communications 64, 345 (1991).\n[63] F. de Jong and R. Malfliet, Phys. Rev. C 46, 2567 (1992).\n[64] C. Hacker, N. Wies, J. Gegelia, and S. Scherer, Phys. Rev. C 72, 055203 (2005), arXiv:hep-ph/0505043.\n[65] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. D 76, 111501 (2007), arXiv:0711.0147 [hep-ph].\n[66] J. Friedrich and T. Walcher, Eur. Phys. J. A 17, 607 (2003), arXiv:hep-ph/0303054.\n[67] G. Eichmann and G. Ramalho, Phys. Rev. D 98, 093007 (2018), arXiv:1806.04579 [hep-ph].\n[68] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995).\n[69] R. Karplus, C. M. Sommerfield, and E. H. Wichmann, Phys. Rev. 111, 1187 (1958).\n[70] S. P. Schneider, B. Kubis, and F. Niecknig, Phys. Rev. D 86, 054013 (2012), arXiv:1206.3098 [hep-ph].\n[71] M. Dax, T. Isken, and B. Kubis, Eur. Phys. J. C 78, 859 (2018), arXiv:1808.08957 [hep-ph].\n[72] J. B. Bronzan and C. Kacser, Phys. Rev. 132, 2703 (1963).\n[73] R. F. Dashen and A. V. Manohar, Phys. Lett. B 315, 425 (1993), arXiv:hep-ph/9307241.\n[74] E. E. Jenkins, X.-d. Ji, and A. V. Manohar, Phys. Rev. Lett. 89, 242001 (2002), arXiv:hep-ph/0207092.\n[75] B. Kubis, Strong interactions and electromagnetism in low-energy hadron physics , Ph.D. thesis, Forschungszentrum J¨ ulich\n(2003).\n[76] V. Pascalutsa, Phys. Lett. B 503, 85 (2001), arXiv:hep-ph/0008026.\n[77] V. Pascalutsa and R. Timmermans, Phys. Rev. C 60, 042201 (1999), arXiv:nucl-th/9905065.\n[78] T. Ledwig, J. Martin-Camalich, V. Pascalutsa, and M. Vanderhaeghen, Phys. Rev. D 85, 034013 (2012), arXiv:1108.2523\n[hep-ph]." }, { "title": "2401.17758v2.Multimaterial_Inkjet_Printing_of_Mechanochromic_Materials.pdf", "content": "Multimaterial Inkjet Printing of Mechanochromic Materials\nMuriel Mauron1, Lucie Castens Vitanov1, C´ esar Michaud1, Rapha¨ el Wenger1,\nNicolas Muller1, Roseline Nussbaumer1, C´ eline Calvino2, Christoph Weder3,4,\nStephen Schrettl3,4,5, Gilbert Gugler1*, Derek Kiebala3,4,6*\n1iPrint Institute, HEIA-FR, HES-SO University of Applied Sciences and Arts Western\nSwitzerland, Fribourg, 1700, Switzerland.\n2Freiburg Center for Interactive Materials and Bioinspired Technologies, University of\nFreiburg, Freiburg, 79110, Germany.\n3Adolphe Merkle Institute, University of Fribourg, Fribourg, 1700, Switzerland.\n4National Competence Center in Research Bio-inspired Materials, University of Fribourg,\nFribourg, 1700, Switzerland.\n5TUM School of Life Sciences, Technical University of Munich, Freising, 85354, Germany.\n6*Department of Chemistry, Johannes Gutenberg University of Mainz, Mainz, 55128,\nGermany.\n*Corresponding author(s). E-mail(s): gilbert.gugler@hefr.ch; derek.kiebala@uni-mainz.de;\nAbstract\nInkjet printing technology achieves the precise deposition of liquid-phase materials viathe digitally\ncontrolled formation of picoliter-sized droplets. Beyond graphical printing, inkjet printing has been\nemployed for the deposition of separated drops on surfaces or the formation of continuous layers,\nwhich allows to construct materials gradients or periodic features that provide enhanced functionali-\nties. Here, we explore the use of multinozzle, drop-on-demand piezoelectric inkjet technology for the\nmanufacturing of mechanochromic materials, i.e.,materials that change their color or fluorescence in\nresponse to mechanical deformation. To accomplish this, suitable polyurethane polymers of differing\nhardness grades were tested with a range of organic solvents to formulate low-viscosity, inkjet-printable\nsolutions. Following their rheological characterization, two solutions comprised of “soft” and “hard”\npolyurethanes were selected for in-depth study. The solutions were imbibed with a mechanochromic\nadditive to yield fluorescent inks, which were either dropcast onto polymeric substrates or printed\nto form checkerboard patterns of alternating hardness using a lab-built, multimaterial inkjet plat-\nform. Fluorescence imaging and spectroscopy were used to identify different hardness grades in the\ndropcast and printed materials, as well as to monitor the responses of these gradient materials to\nmechanical deformation. The insights gained in this study are expected to facilitate the development\nof inkjet-printable, mechanochromic polymer materials for a wide range of applications.\nKeywords: inkjet printing, multimaterial jetting, mechanochromic materials, functional surfaces, hardness\ngradient\n1arXiv:2401.17758v2 [physics.app-ph] 28 Mar 20241 Introduction\nVoxelated matter, i.e., materials comprised of\nmodular, three-dimensional building blocks, has\nattracted growing interest due to the fact that\nsuch materials can be designed to exhibit tai-\nlored surface functionalities and gradient proper-\nties [1, 2]. Inkjet-based 3D printing is the method\nof choice for fabricating voxelated materials with\nhigh precision viathe digitally controlled forma-\ntion of picoliter-sized droplets, affording access to\na range of materials functionalities otherwise inac-\ncessible with conventional processing methods [1].\nFor example, it has been shown that cell growth\nand stem cell differentiation can be influenced by\nprinting gradients of biologically active materi-\nals [3, 4], and biomedical tomographic data sets\nhave been converted into topographical models\nviainkjet gradient printing [5]. Beyond biomedical\napplications, gradients of functional or mechan-\nical characteristics have been demonstrated in\n3D-printed metals [6], ceramics [7–9], as well as\npolymers [10]. This has been achieved by, for\nexample, reactive inkjet printing in which mul-\ntiple printheads eject pre-polymer reactants onto\nthe same substrate, which form the desired mate-\nrialin situ with well-resolved microscale features\n[11]. Moreover, advancements in printhead archi-\ntecture, optionally assisted by microfluidic mixers,\nhave enabled the printing of materials with gradi-\nent properties viaa technique known as multima-\nterial, multinozzle 3D printing, which has greatly\nexpanded the potential for generating new types\nof inkjet-printed materials [1, 2, 12].\nHere, we sought to harness the potential of\nmultimaterial printing to fabricate gradient prop-\nerty materials endowed with mechanochromic\nfunctionality, i.e.,the ability to change their color\nor fluorescence in response to mechanical defor-\nmation (Fig. 1a). Such mechanochromic materi-\nals are particularly useful for reporting stresses\nand damage in load-bearing materials viaan\neasily detectable and in some cases visually dis-\ncernible optical signal [13]. The fabrication of such\nmechanoresponsive polymer materials is made\npossible by the incorporation of a mechanophore,\nnamely a molecular entity that produces a defined\nresponse to external forces viathe breakage ofa labile bond [14]. Recently, some of us devel-\noped a macromolecular additive (tOPV) con-\nsisting of two excimer-forming oligo( p-phenylene-\nvinylene) (OPV) dyes connected viaa telechelic\npoly(ethylene- co-butylene) backbone ( Mn= 3,000\ng·mol·L−1) that is particularly convenient for con-\nferring mechanochromic behavior to a wide range\nof polymers [15]. Already when blended in very\nsmall quantities ( i.e., 0.2–1.0 wt%) into a host\npolymer, tOPV phase-separates from the matrix\nto form microscopic, fluorescent inclusions whose\nmechanical and optical properties are closely tied\nto that of the host matrix, while leaving the\nmechanical properties of the latter unaffected [16].\nThis blending process has been used to fabricate\nfilms comprised of polyurethane (PU), polyiso-\nprene (PI), poly(styrene- b-butadiene- b-styrene)\n(SBS) and poly( ϵ-caprolactone) (PCL) that dis-\nplay strain-dependent changes in fluorescence\ncolor [15, 16]. In fact, a change in emission\nwavelength of a tOPV-containing elastomeric\npolyurethane was spectroscopically discernible at\nstrains as low as 5% [17], which opens the possi-\nbility for sensing applications with unprecedented\nprecision. Taken together, the advantages offered\nby this mechanochromic additive make it an\nideal candidate for exploring the fabrication of\nmechanochromic materials viainkjet printing.\nThus, to pursue the inkjet printing of gradient-\nproperty, mechanochromic materials, we employed\na multimaterial printing system in which two\nprintheads are mounted in parallel and filled with\ninks containing polymers with two different hard-\nness grades. Droplets of both inks were printed in\nan alternating fashion on a substrate, enabling the\ncreation of a variable hardness, multilayer mate-\nrial (Fig. 1b-c). Imbibing the polymer inks with\nthe aforementioned mechanochromic additive was\npossible without the need for additional pre-\nprocessing or chemical modification, in contrast\nto polymer inks formulated with mechanochromic\nnanoparticles [18] or microgels [19]. This yielded\npatterned fluorescent films, which allowed us to\ninvestigate the mixing dynamics of the different\nmaterials at the voxel-voxel interface using fluores-\ncence imaging and spectroscopy. Finally, we show\nthat mechanical deformation of the substrate-\nsupported, gradient-hardness films changes their\nfluorescence in a quantifiable fashion, constitut-\ning an important step towards the realization of\ninkjet-printed, mechanochromic multimaterials.\n2Fig. 1 a) Operating principle of mechanochromic materials. When subjected to mechanical deformation, the color and/or\nfluorescence of the material changes from, for example, orange to green. b-c) Overview of how inkjet printing is used to\ndigitally control the composition of a continuous gradient between two materials. b) Top: Side view of a line printed with\ntwo printheads each jetting a different polymer-based ink. The ink deposited at each location is digitally controlled by\nselecting which printhead jets a drop as the substrate passes underneath them. Bottom left: Simplified top-view of a 2D\nsurface printed with two printheads to achieve a gradient. Bottom right: Industrial printheads typically feature hundreds\nto thousands of nozzles, allowing for much smoother gradients. c) Schematic depicting the deposition of multiple layers on\ntop of each other to create a three-dimensional gradient material.\n2 Materials and methods\n2.1 Ink formulation\nAs the basis for the mechanochromic poly-\nmer inks formulated in this study, commercially\navailable polyurethanes (PUs) of different hard-\nness, namely “soft” PU (PU-S, shore hardness\n35A) and “hard” PU (PU-H, shore hardness\n80A) were provided by BASF and Huntsmann,\nrespectively. 1,4-Dioxane solvent (also referred\nto as dioxane; 99%+ purity) was purchased\nfrom Acros Organics. The solvents 1-methyl-\n2-pyrrolidinone (NMP; 99%) and 2-butoxyethyl\nacetate (BCA; 99%) were purchased from\nSigma Aldrich. The telechelic oligo( p-phenylene-\nvinylene) (tOPV) mechanochromic additive used\nin the final ink formulations was prepared as pre-\nviously reported, omitting the final purificationstep [15]. Hence, a greater amount of tOPV was\nused for the ink formulations ( i.e.,5 wt% vs. poly-\nmer) in comparison to previous studies ( i.e.,0.2–1\nwt% vs. polymer) to ensure sufficient assembly of\ndifunctional tOPV in the printed samples.\nChemical compatibility tests were performed\nbetween the candidate solvents and the print-\nheads following the manufacturer’s instructions.\nTo carry out these tests, nine different parts of\nthe printhead were weighed, immersed individu-\nally in 30 mL of the solvent being tested in amber\nglass bottles, and placed in an oven at 60◦C for\nsix weeks. At two-week intervals, the parts were\ncleaned by immersion first in n-butyl lactate (99%,\nThermoScientific) and then in propan-2-ol (99%,\nThommen-Furler AG). After evaporation of the\ncleaning solvents, the parts were re-weighed and\nwere again immersed in the test solvent at 60◦C.\n3The data was sent back to the manufacturer, who\nanalyzed the changes in weight of each piece to\nassess compatibility between the solvent and the\nprinthead.\nThe rheological properties of the polymer solu-\ntions were analyzed with a Piezo Axial Vibrator\n(Tri-PAV) rheometer from TriJet at room temper-\nature from 100 Hz to 10 kHz to identify suitable\nsolvents and polymer concentrations for inkjet\nprinting [20]. The surface energy, contact angle\nand surface tension of the solutions were measured\nwith the OCA25 contact-angle and drop-shape\nmeasuring device from Dataphysics. The surface\nenergy of the substrate was calculated using the\ncontact angles of droplets ( V≈2µL) of water and\nethylene glycol, the Young Laplace equation, and\nthe Owens-Wendt-Rabel and Kaelble (OWRK)\nmodel. The static surface tension for droplets\n(V= 12–15 µL) of each ink was measured with\nthe same device using the pendant drop method.\n2.2 Inkjet printing\nTo enable simultaneous printing of multiple dif-\nferent PU inks, an inkjet printing platform was\nbuilt that incorporates two Seiko RC1536L print-\nheads, an ink distribution system, and a stage\nmovable along x-,y-, and z-axes, all of which\nis governed by a Beckhoff Programmable Logic\nController (PLC) (Fig. A1). The Seiko RC1536L\nprintheads, which features four rows of 384 nozzles\n(totaling 1536 available nozzles with a maximum\nresolution of 360 dpi), were selected for their gen-\nerally good chemical compatibility and suitability\nfor printing at relatively high viscosity ( i.e.,up to\n20 mPa ·s). The platform is capable of simultane-\nously printing materials in 2D with up to three\nprintheads, as well as 3D materials by sequen-\ntially printing multiple 2D layers as depicted in\nFig. 1c. The volume and speed of ink droplets\nexiting the printhead nozzles during the printing\nprocess were determined by acquiring images of\nthe droplets in flight using a camera (33GP031\nfrom The Imaging Source) with a 0.7–4.5X zoom\nlens (Model HY-180XA from Hayear), combined\nwith a strobing light source and a homemade\ntriggering system. A bespoke Matlab script was\nused for synchronizing the operations and extract-\ning key features from the images, such as the\ndrop speed, volume, and shape. The printheadsand their waveforms were controlled with driv-\ning electronics from Aewa and their dedicated\nAPRINT software. Selected inks containing the\nmechanochromic additive were deposited by drop-\ncasting or inkjet printing at room temperature on\na PU substrate fixed on a heating plate at 40◦C,\nfollowed by drying at 60◦C for 12 h. Round sub-\nstrates (55 mm diameter, 0.3 mm thickness) were\nproduced by Torson Injex by injection molding\nof Elastollan E565A12P (BASF), and rectangular\nsubstrates (38 ×20×60 mm) of the same material\nwere produced by injection molding at the iRap\nInstitute of the HEIA in Fribourg.\n2.3 Fluorescence spectroscopy and\nmicroscopy\nFluorescence changes in printed PU/tOPV films\nwere measured spectroscopically as a function of\nmechanical stress using an Ocean Optics USB\n4000 spectrometer connected to an Ocean Optics\nLS-450 LED light source with an excitation wave-\nlength of λex= 380 nm and an Ocean Optics\nQR230-7-XSR SMA 905 optical fiber. The extent\nof aggregation of the excimer-forming end groups\nof the tOPV macromolecule, which is influenced\nby processing conditions, changes in tempera-\nture, and the application of mechanical force,\nwere probed by measuring fluorescence spectra\nand determining the monomer-to-excimer emis-\nsion intensity ratio ( I510/I630). Mechanical defor-\nmation of tOPV-containing materials has been\nshown to reliably result in an increase in I510/I630,\nwhich occurs when the distance between adjacent\ntOPV emitters is increased [16, 17]. The value was\ncalculated from the solid-state fluorescence spec-\ntra by taking the intensity value at the maximum\nof the monomer emission peak ( λmax= 510 nm)\ndivided by the maximum of the excimer emission\npeak ( λmax= 630 nm). Samples were placed on\ntop of a piece of black paper, and the optical fiber\nwas oriented normal to the surface at a distance\nof 2 mm, resulting in a spot size of approximately\n800µm from which the diffuse reflectance was\nmeasured. Spectra were recorded before, after, or\nduring application of mechanical stress and fluo-\nrescence data were acquired using Stream Basic\nsoftware. Stress-strain data collected during these\nmeasurements was recorded by uniaxial deforma-\ntion of rectangular samples with dimensions of 30\n×5×1 mm (length ×width×thickness) at 100%\n4min−1using a Linkam TST350 microtenstile stage\nequipped with a 20 N load cell and controlled by\nthe accompanying Linksys32 software. A pre-load\nforce of 0.1–0.3 N was applied before initiating\ndeformation.\nConfocal microscopy images were acquired on\na Zeiss LSM 710 Meta confocal laser scanning\nmicroscope (CLSM) equipped with an Ar laser\n(max. power 25 mW) and a spectrometer allow-\ning acquisition of emitted light in two channels\nat nanometer precision within the visible range.\nAll data was recorded using a 63X/1.3NA oil\nlens at a lateral resolution of 132 nm. Channel 1\nrecorded emission between 461 and 525 nm ( i.e.,\nprimarily tOPV monomer emission), and chan-\nnel 2 recorded emission between 550 and 725 nm\n(i.e., primarily tOPV excimer emission). Both\nchannels used a laser excitation of 458 nm. For\na given region of interest (ROI) at the surface\nof the sample, the monomer-to-excimer ratio (for\nCLSM images denoted as IM/IE) was obtained\nby dividing the mean pixel intensity for the ROI\nin channel 1 by the mean intensity for the ROI\nin channel 2. While IM/IEvalues obtained by\nCLSM image analysis are not directly compara-\nble to the I510/I630values obtained by solid-state\nfluorescence spectroscopy, the trends in these val-\nues ( e.g., for PU of different hardness, or when\nsubjecting the films to mechanical force) mirror\neach other. CLSM images were acquired as 16-\nbit grayscale images, and false colors of green and\norange were applied to the monomer and excimer\nchannels, respectively, to match the visible flu-\norescence colors of the corresponding assembly\nstates of tOPV. Thus, combined-channel CLSM\nimages have a yellow-green appearance after false\ncoloring.\nPrinted PU/tOPV layers on Elastollan sub-\nstrates were imaged by cutting a 0.5 ×0.5 cm\nsquare piece and placing it on a glass slide. Two\ndrops of halogen-free, non-fluorescent Cargille\nImmersion Oil Type HF ( i.e., refractive index-\nmatching oil) were added to the sample, onto\nwhich a glass cover slip of approximately 0.17 mm\nthickness was placed. One drop of the same\nimmersion oil was then placed on the lens of the\nconfocal microscope, after which the glass slide\nand sample assembly was inverted such that the\ncover slip was facing downwards and placed on\nthe microscope stage. The lens was then raised so\nthat the oil came into contact with the glass coverslip, creating an air-free path for the excitation\nlaser of the laser confocal scanning microscope to\npass through the sample. Equibiaxial stretching\nwas carried out on samples using a custom-built\nstretching device that applied radial strain in all\ndirections (see Ref. [16] for a detailed description\nof the setup).\n3 Results and discussion\n3.1 Polymer solution testing\nAs a first step in the formulation of printable\nmechanochromic inks, solubility tests were car-\nried out with the PU polymers and the tOPV\nmechanochromic additive in a range of organic\nsolvents with varying hydrophobicity and boiling\npoints. Of the solvents tested, tOPV showed the\nbest solubility at 1 mg/mL in dioxane, tetrahydro-\nfuran (THF), N-methyl-2-pyrrolidinone (NMP),\nand mixtures of NMP and 2-butoxyethyl acetate\n(BCA) with at least 25 wt% of NMP. tOPV\nremained well-dissolved in these solutions at room\ntemperature over the course of 24 h as indicated by\nthe solutions’ green fluorescent color, thus serving\nas a good indicator that no tOPV would precipi-\ntate inside the printhead during the printing pro-\ncess. The other solvents tested that did not fully\ndissolve the PU (in the range of 15–20 mg/mL)\nor tOPV (at 1 mg/mL) include: n-hexane, xylene,\nisophorone, BCA (99%), methyl ethyl ketone,\nethyl acetate, methanol, ethanol, propanol, 1,5-\npentanediol, glycerol, ethylene glycol, propylene\nglycol, 2-methoxyethanol, and triethanolamine.\nAfter subjecting the different components of\nthe Seiko RC1536L printhead used in the lab-built\ninkjet printer to chemical compatibility tests (see\nSection 2.1 for details), the results showed diox-\nane and NMP/BCA 25/75 wt% to be the most\npromising solvent candidates (Table A1). Polymer\nsolutions were then made by dissolving pellets of\nPU-S and PU-H in dioxane and NMP/BCA 25/75\nwt% at 20 mg/mL, which was the concentration\nused for all further tests. To assess the printability\nof the different ink formulations, rheological mea-\nsurements were carried out on solutions of PU-S\nand PU-H without the tOPV additive, given that\nthe small amounts of tOPV to be added in the\nfinal inks would not affect the rheological prop-\nerties of the latter. The results showed that all\n5Fig. 2 Rheological measurements carried out on PU-S and\nPU-H at 20 mg/mL in both dioxane and NMP/BCA 25/75\nwt% solvents. Both (a) complex viscosity and (b) elasticity\n(G’/|G*|) were measured as a function of applied frequency\nat 25◦C.\nfour polymer solutions exhibit complex viscosi-\nties ranging from 3.5–8 mPa ·s between 20–3000\nHz applied frequency (Fig. 2a), well within the\nrange of 2–20 mPa ·s recommended for industrial\nprintheads [21, 22]. Moreover, the elasticity of\nthe solutions, defined as the ratio of the storage\nmodulus G’ to the absolute value of the com-\nplex modulus |G*|, was found to be sufficiently\nlow, i.e., near zero for the dioxane-based solu-\ntions and less than 0.1 for NMP/BCA 25/75 wt%\nup to 3 kHz (Fig. 2b), indicating that they could\nbe reliably jetted through the printhead nozzle.\nThus, dioxane was selected as the preferred solvent\nto formulate the mechanochromic inks based on\nthe lower elasticity of the dioxane-based PU solu-\ntions. Finally, printhead waveform optimizationwas performed by acquiring and analyzing real-\ntime images of polymer solution droplets ejected\nfrom the printhead nozzles (Fig. 3) and iterating\nover the waveform parameters until suitable drop\ncharacteristics were achieved. The waveforms of\nthe Seiko RC1536L printheads were initially opti-\nmized to achieve a minimum drop speed of 5 m/s\nand a drop size of 25 pL with the selected solu-\ntions. However, the quick drying of the solution\nin the nozzles prevented stable printing. Thus, a\nhigher waveform voltage was employed ( i.e., ca.\n28V instead of the 20 V initially used) to pro-\nduce faster-traveling droplets ( vdrop≈8 m/s)\nthat avoided precipitation of the polymer in the\nprinthead during printing.\nFig. 3 Real-time images of polymer solution droplets\nejected from seven adjacent nozzles in one row of the print-\nhead at (a) 20 V ( vdrop≈5 m/s) and at (b) 28 V ( vdrop≈\n8 m/s). Scale bars = 1 mm.\n3.2 Film-substrate interactions\nNext, we investigated the interaction of the diox-\nane polymer solutions with an Elastollan polymer\nfilm (thickness = 1 mm), which was to be used\nas the substrate for dropcasting and ultimately\nprinting the inks. To this end, we deposited a\n10µL drop of each solution onto the substrate\nand measured the surface tension of the solution\ndroplet, the solution-substrate contact angle, and\nthe surface energy of the substrate. The dioxane-\nbased polymer solutions were found to exhibit a\n6low static surface tension typical of solvent-based\ninks, namely 33.2 and 25.6 mN/m for PU-S and\nPU-H in dioxane, respectively, both of which fall\nbelow the 40 mN/m limit for inkjet drop ejec-\ntion [23]. The surface energy of the substrate was\nfound to be 33.9 mN/m, which exceeds the sur-\nface tension and indicates excellent wettability.\nFinally, the contact angles were determined to be\n21.8◦±0.9◦and 23.3◦±2◦for PU-S and PU-\nH in dioxane, respectively, further confirming that\nthese inks wet the substrate well.\n3.3 Mechanochromism of dropcast\nfilms\nTo endow these printable polymer solutions with\nmechanochromic properties, the tOPV additive\nwas dissolved in each solution at a concentration\nof 1 mg/mL to produce the final ink formulations\n(Ink A : PU-S [20 mg/mL] + tOPV [1 mg/mL] in\ndioxane, and Ink B : PU-H [20 mg/mL] + tOPV\n[1 mg/mL] in dioxane). The inks were dropcast\nonto Elastollan substrates and subsequently dried\nat 60◦C for 12 h in vacuo to yield thin films that\nexhibited a homogeneous yellow fluorescent color\nunder UV illumination, indicating proper assem-\nbly of the tOPV additive within the films [15, 17].\nThe two inks were dropcast next to each other\non the substrate in such a way that produced a\n“mixed region” or interface where the two inks\noverlapped (Fig. 4a).\nSolid-state fluorescence spectroscopy measure-\nments revealed that, in the as-prepared films,\nthe monomer-to-excimer ratio of “soft” ink A\n(I510/I630= 1.02) was lower than that of “hard”\nink B ( I510/I630= 1.18), and that of the mixed\nregion ( I510/I630= 1.13) was in between the\ntwo. Next, the substrate was subjected to uniax-\nial tensile deformation at a strain rate of 100%\nmin−1up to 250% strain (the resulting stress-\nstrain curve is shown in the inset in Fig. 4b),\nand the I510/I630for each region of the drop-\ncast film was monitored. Notably, the I510/I630\nfor all regions ( i.e., pure inks A and B, as well\nas the mixed region) increased sharply between\n0-50% strain, after which the I510/I630of each\nregion continued to increase in a linear fashion\nuntil the end of the test (Fig. 4b). This linear flu-\norescence response to uniaxial deformation closely\nmirrors that of previously reported PU/tOPV\nblend films that feature different slopes for low-\nFig. 4 a) Schematic depicting films made by dropcasting\ninks A and B onto the Elastollan substrate. Upper panel:\nRegions of pure ink A and ink B are indicated, as well as the\nmixed region or interface where the two inks overlap. Lower\npanel: Uniaxial stretching of the substrate caused the drop-\ncast films to deform along the stretching axis. b) Plot of the\nmonomer-to-excimer fluorescence ratio ( I510/I630) of the\npure and mixed ink regions as a function of applied strain.\nDashed lines serve as a guide to the eye. Inset: Stress-strain\ncurve of the Elastollan substrate measured during the test.\nvs. high-strain deformation [17], thus confirming\nthat force is transferred between the substrate and\ndropcast films. Notably, a greater initial increase\ninI510/I630, as well as a slightly steeper slope\nat high strains, is observed for the interfacial\nregion, suggesting that microscale domain bound-\naries between partially demixed PU-S and PU-H\nexert more local strain on the tOPV additive when\ndeformed. Taken together, these results confirm\nthat the PU/tOPV inks formulated herein can\nbe readily used to fabricate substrate-supported,\nmechanochromic films.\nAfter confirming the mechanochromism of\ndropcast films made from polymer inks A and B,\n7Fig. 5 a) Combined-channel 2D CLSM image of an interfacial region between inks A and B on the same film. b) IM/IE\nvalues determined as a function of x-position moving left to right along the dashed line in panel (a). c,d) Confocal microscopy\nimages of the mixing interface of inks A and B, which were dropcast onto an Elastollan substrate and subjected to equibiaxial\nstrains of (c) 30% and (d) 90%. Images were recorded in two channels, i.e.,the monomer (Ch 1: λem= 461–525 nm) and\nexcimer (Ch2: λem= 550–725 nm) emission channels and then merged to obtain the combined-channel images shown. At\neach strain, the monomer-to-excimer emission ratio ( IM/IE) was determined for the indicated regions of interest for inks\nA and B (labeled as A and B, respectively) by dividing the mean pixel intensity of channel 1 by that of channel 2. Scale\nbars = 20 µm.\nwe turned to confocal laser scanning microscopy\n(CLSM) to more closely study the morphology\nof the films, particularly at the mixing interface.\nTo this end, CLSM images were acquired. The\nmonomer and excimer emission of the dropcast\nfilms were recorded in two separate channels, thus\nallowing for the determination of the monomer-\nto-excimer emission ratio (denoted as IM/IEfor\nCLSM images) with micrometer-resolution for dif-\nferent areas of the film viaimage analysis (see\nSection 2.3 for details). Distinct areas comprised\nof inks A and B were readily identifiable by their\nfluorescence; namely, the area formed by “soft”\nink A consistently exhibited a lower IM/IEthan\nregions formed by “hard” ink B (Fig. 5). In\nsome regions, a quasi-linear decrease in IM/IE\nis observed across the mixing zone when moving\nfrom ink B (left side of the image in Fig. 5a) toink A (right side), indicative of a compositional\ngradient between the two inks (Fig. 5a,b). On the\nother hand, some areas of the mixing zone exhibit\na sharp transition between the two inks (Fig. 5c,d,\nFig. A2), likely stemming from limited miscibil-\nity of the two polymers and selective dewetting\nprocesses that occurred during drying. Moreover,\nlow-IM/IEfeatures were observed interspersed in\nregions of ink B and vice versa, pointing to a\ncomplex mixing behavior between the two inks\n(Fig. 5a,c,d).\nNext, the dropcast films were subjected to\nequibiaxial strain ( i.e., radial strain applied\nequally in all directions) using a custom-built\nstretching device (see Ref. [16] for details) and\nimaged viaCLSM. At each strain, parts of film\ncomprised of either ink A or B were identified,\nand the IM/IEwas determined for the indicated\n8regions by dividing the mean pixel intensity of the\nmonomer channel by that of the excimer channel.\nThe results show that, when increasing the equib-\niaxial strain from 30% to 90%, the IM/IEfor ink\nA increases from 0.98 to 1.09, and the IM/IEfor\nink B increases from 1.47 to 1.57 (Fig. 5c,d). These\nobservations are in agreement with the increase\ninI510/I630measured for both inks A and B via\nsolid-state fluorescence spectroscopy (Fig. 4b) and\nfurther corroborate that PU/tOPV films formed\nfrom the dioxane-based inks formulated herein\nexhibit mechanochromism.\n3.4 Inkjet printing trials\nFinally, we sought to employ our inkjet print-\ning setup to fabricate mechanochromic films with\ngradient mechanical properties. As a first test,\npolymer solutions A and B ( i.e., inks A and B\nwithout the tOPV additive) were printed in an\nalternating fashion onto a circular Elastollan sub-\nstrate to create a checkerboard pattern (Fig. 6a,b).\nMultiple passes were made by the printhead to\ndeposit a total of 20 layers, and a feature resolu-\ntion of 1 mm was obtained in the x,y-plane as seen\nby the naked eye. The same process was then car-\nried out with tOPV-containing inks A and B, and\nthe 20-layer printed pattern was imaged by CLSM\n(Fig. 6c). While the interface between the “soft”\nink A and “hard” ink B is clearly discernible, many\nsatellite droplets of the hard PU are visible at least\n0.5 mm into the soft PU region. This is largely a\nconsequence of the fact that, in order to avoid the\ninks drying in the printhead during deposition, a\nhigher waveform voltage was used to increase the\nspeed of the droplet ejection. Thus, the CLSM\nimages of the additive-containing, printed PU pat-\nterns reveal that the high voltage and speed of\nthe printing process diminish the accuracy and\ntherefore resolution of the printed features. More-\nover, many non-fluorescent areas are visible within\nthe hard PU region, indicating that this PU\nshows a greater tendency to dewet from the sub-\nstrate during printing and leave some portions\nof the substrate uncovered. To overcome these\nobstacles, further trials are needed to identify\nalternative polymer materials that are compati-\nble with inkjet-printable solvents and that, like\nfilms printed from ink B, do not show evidence\nof dewetting from the substrate when multiple-\npass printing is performed. Moreover, the useof a less volatile solvent could allow for slower\nprinting, which may avoid the generation of satel-\nlite droplets that breach the hard-soft polymer\ninterface.\nFig. 6 a) Image of a checkerboard pattern created by\nprinting polymer solutions of “soft” PU-S and “hard” PU-\nH (c(dioxane) = 20 mg/mL) in an alternating fashion on a\ncircular Elastollan substrate. Multiple passes were made by\nthe printhead to deposit a total of 20 layers. Scale bar = 10\nmm. b) Magnified view of a portion of the sample shown in\n(a). Scale bar = 2 mm. c) Combined-channel CLSM image\nof the intersection of two of the checkerboard squares com-\nprised of “soft” ink A (lower right region) and “hard” ink\nA (upper left region) that had been printed on the same\ntype of circular Elastollan substrate as shown in (a). Scale\nbar = 1 mm. Inset: Magnified CLSM image of the indi-\ncated interfacial region between the two inks (scale bar =\n50µm).\n4 Conclusion\nIn the present work, the printing of gradient-\nproperty, patterned polymer films with\n9mechanochromic sensing capability was investi-\ngated. In order to realize the fabrication of such\nfilms, a multimaterial inkjet printing platform\nwas developed that allows for the simultaneous\nprinting of two different polymer solutions into\nmultilayer voxels of alternating composition. After\nextensive testing was carried out on candidate\npolymers solutions to evaluate their printability,\n“hard” and “soft” PU solutions in dioxane were\nselected for their desirable rheological proper-\nties and compatibility with the printheads. A\nmechanochromic additive was mixed into the\nsolutions to obtain fluorescent polymer inks,\nwhich were either dropcast or printed in different\npatterns onto a stretchable polymer substrate.\nThe different inks were readily identifiable via\ntheir distinct fluorescence, and an in-depth evalu-\nation of the microscale features of the fluorescent\nfilms shed light on complex mixing behaviors at\nthe interface between the two inks. Importantly,\nwhen subjected to mechanical deformation, both\ndropcast and inkjet-printed, multimaterial films\nexhibited mechanochromic behavior that could be\ntracked individually for each voxel. The insights\ngained into the properties of inkjet-printable\npolymer inks and the microscale characterization\nof patterned polymers are expected to greatly\nfacilitate the development of tailor-made func-\ntional materials with complex property profiles\notherwise inaccessible by conventional processing\ntechniques.\nSupplementary information. Electronic\nSupplementary Information (ESI) available: Sup-\nplementary Figures A1-A2 and Supplementary\nTable A1.\nAcknowledgements. M.M., L.C.V., C.M.,\nR.W., R.N., N.M. and G.G. acknowledge financial\nsupport through funding from HES-SO University\nof Applied Sciences and Arts Western Switzerland,\nEngineering and Architecture, Grant SmartMat-\nJet 114624. S.S. and D.K. gratefully acknowledge\nfinancial support through the National Center of\nCompetence in Research (NCCR) Bio-inspired\nMaterials, a research instrument of the Swiss\nNational Science Foundation (SNF), and funding\nfrom the Adolphe Merkle Foundation.Statements and Declarations\nAuthor contributions. M.M., L.C.V., C.M.,\nand R.W. built the inkjet printing device; carried\nout the printhead compatibility testing, polymer\nsolution characterization, and substrate analysis;\noptimized the dropcasting protocol; and carried\nout the inkjet printing trials. D.K. carried out sol-\nubility testing and prepared the mechanochromic\ninks, measured the fluorescence of dropcast films\nduring tensile testing, and acquired and analyzed\nthe confocal microscopy images of printed films.\nC.C. synthesized the mechanochromic additive.\nS.S. and G.G. designed the original concept for\nthe study and provided guidance throughout the\nproject. D.K., M.M., R.N., R.W., and N.M. wrote\nthe manuscript, which was edited by C.C. All\nauthors have given approval to the final version of\nthe manuscript.\nData availability. The datasets generated and\nanalyzed during the current study are available\nfrom the corresponding author on reasonable\nrequest. The source data generated during this\nstudy will be deposited to the Zenodo repository\nafter acceptance of the article.\nCompeting interests. The authors declare no\ncompeting financial interests.\nReferences\n[1] Skylar-Scott, M.A., Mueller, J., Visser, C.W.,\nLewis, J.A.: Voxelated soft matter via mul-\ntimaterial multinozzle 3D printing. Nature\n575(7782), 330–335 (2019) https://doi.org/\n10.1038/s41586-019-1736-8\n[2] Doubrovski, E.L., Tsai, E.Y., Dikovsky,\nD., Geraedts, J.M.P., Herr, H., Oxman,\nN.: Voxel-based fabrication through mate-\nrial property mapping: A design method\nfor bitmap printing. Computer-Aided Design\n60, 3–13 (2015) https://doi.org/10.1016/j.\ncad.2014.05.010\n[3] Ilkhanizadeh, S., Teixeira, A., Hermanson,\nO.: Inkjet printing of macromolecules on\nhydrogels to steer neural stem cell differenti-\nation. Biomaterials 28(27), 3936–3943 (2007)\nhttps://doi.org/10.1016/j.biomaterials.2007.\n05.018\n10[4] Cai, K., Dong, H., Chen, C., Yang, L., Jandt,\nK.D., Deng, L.: Inkjet printing of laminin\ngradient to investigate endothelial cellular\nalignment. Colloids and Surfaces B: Biointer-\nfaces 72(2), 230–235 (2009) https://doi.org/\n10.1016/j.colsurfb.2009.04.008\n[5] Hosny, A., Keating, S.J., Dilley, J.D., Rip-\nley, B., Kelil, T., Pieper, S., Kolb, D., Bader,\nC., Pobloth, A.-M., Griffin, M., Nezafat,\nR., Duda, G., Chiocca, E.A., Stone, J.R.,\nMichaelson, J.S., Dean, M.N., Oxman, N.,\nWeaver, J.C.: From improved diagnostics to\npresurgical planning: high-resolution func-\ntionally graded multimaterial 3D printing\nof biomedical tomographic data sets. 3D\nPrinting and Additive Manufacturing 5(2),\n103–113 (2018) https://doi.org/10.1089/3dp.\n2017.0140\n[6] Godlinski, D., Morvan, S.: Steel parts with\ntailored material gradients by 3D-printing\nusing nano-particulate Ink. In: Materials Sci-\nence Forum vol. 492 - 493, pp. 679–684. Trans\nTech Publications Ltd., Stafa (2005). https:\n//doi.org/10.4028/0-87849-970-9.679\n[7] Mott, M., Evans, J.R.G.: Zirconia/alumina\nfunctionally graded material made by ceramic\nink jet printing. Materials Science and Engi-\nneering: A 271(1-2), 344–352 (1999) https:\n//doi.org/10.1016/S0921-5093(99)00266-X\n[8] Afzal, M.A.F., Kesarwani, P., Reddy, K.M.,\nKalmodia, S., Basu, B., Balani, K.: Function-\nally graded hydroxyapatite-alumina-zirconia\nbiocomposite: Synergy of toughness and bio-\ncompatibility. Materials Science and Engi-\nneering: C 32(5), 1164–1173 (2012) https:\n//doi.org/10.1016/j.msec.2012.03.003\n[9] Lee, J.-H., Kim, J.-H., Hwang, K.-T., Hwang,\nH.-J., Han, K.-S.: Digital inkjet printing in\nthree dimensions with multiple ceramic com-\npositions. Journal of the European Ceramic\nSociety 41(2), 1490–1497 (2021) https://doi.\norg/10.1016/j.jeurceramsoc.2020.09.044\n[10] Schaffner, M., Faber, J.A., Pianegonda, L.,\nR¨ uhs, P.A., Coulter, F., Studart, A.R.: 3D\nprinting of robotic soft actuators with pro-\ngrammable bioinspired architectures. NatureCommunications 9(1), 878 (2018) https://\ndoi.org/10.1038/s41467-018-03216-w\n[11] Schuster, F., Hirth, T., Weber, A.: Reac-\ntive inkjet printing of polyethylene glycol\nand isocyanate based inks to create porous\npolyurethane structures. Journal of Applied\nPolymer Science 136(3), 46977 (2019) https:\n//doi.org/10.1002/app.46977\n[12] Hardin, J.O., Ober, T.J., Valentine, A.D.,\nLewis, J.A.: Microfluidic printheads for\nmultimaterial 3D printing of viscoelastic\ninks. Advanced Materials 27(21), 3279–\n3284 (2015) https://doi.org/10.1002/adma.\n201500222\n[13] Traeger, H., Kiebala, D.J., Weder, C.,\nSchrettl, S.: From molecules to poly-\nmers—harnessing inter- and intramolecular\ninteractions to create mechanochromic mate-\nrials. Macromolecular Rapid Communica-\ntions 42(1), 2000573 (2021) https://doi.org/\n10.1002/marc.202000573\n[14] Deneke, N., Rencheck, M.L., Davis, C.S.: An\nengineer’s introduction to mechanophores.\nSoft Matter 16(27), 6230–6252 (2020) https:\n//doi.org/10.1039/D0SM00465K\n[15] Calvino, C., Sagara, Y., Buclin, V., Haehnel,\nA.P., Prado, A., Aeby, C., Simon, Y.C.,\nSchrettl, S., Weder, C.: Mechanorespon-\nsive, luminescent polymer blends based\non an excimer-forming telechelic macro-\nmolecule. Macromolecular Rapid Communi-\ncations 40(1), 1800705 (2019) https://doi.\norg/10.1002/marc.201800705\n[16] Kiebala, D.J., Style, R., Vanhecke, D.,\nCalvino, C., Weder, C., Schrettl, S.: Sub-\nmicrometer mechanochromic inclusions\nenable strain sensing in polymers. Advanced\nFunctional Materials 33(50), 2370297 (2023)\nhttps://doi.org/10.1002/adfm.202304938\n[17] Kiebala, D.J., Fan, Z., Calvino, C.,\nFehlmann, L., Schrettl, S., Weder,\nC.: Mechanoresponsive elastomers\nmade with excimer-forming telechelics.\nOrganic Materials 2(4), 313–322 (2020)\nhttps://doi.org/10.1055/s-0040-1721052\n11[18] Ogumi, K., Nagata, K., Takimoto, Y.,\nMishiba, K., Matsuo, Y.: Inkjet printing\nof mechanochromic fluorenylidene-acridane.\nScientific Reports 12(1), 16997 (2022) https:\n//doi.org/10.1038/s41598-022-21600-x\n[19] Xu, B., Wang, H., Luo, Z., Yang,\nJ., Wang, Z.: Multi-material 3D\nprinting of mechanochromic double\nnetwork hydrogels for on-demand pat-\nterning. ACS Applied Materials &\nInterfaces 15(8), 11122–11130 (2023)\nhttps://doi.org/10.1021/acsami.2c22564\n[20] Hussain, Z., Kiaee, Z., Nazarzadeh, M.,\nReichel, C., Tepner, S., Tuladhar, T., Jahn,\nM., Keding, R.: High-frequency rheologi-\ncal and piezo-voltage waveform characteriza-\ntion of inkjet-printed polymer-based dopant-\nsource inks. Micromachines 14(1), 80 (2022)\nhttps://doi.org/10.3390/mi14010080\n[21] Zapka, W. (ed.): Handbook of Industrial\nInkjet Printing: a Full System Approach, 1st\nedn. Wiley, ??? (2017). https://doi.org/10.\n1002/9783527687169\n[22] Tuladhar, T.: Measurement of complex rhe-\nology and jettability of inkjet inks. In: Zapka,\nW. (ed.) Inkjet Printing in Industry, 1st edn.,\npp. 655–693. Wiley, ??? (2022). https://doi.\norg/10.1002/9783527828074.ch28\n[23] He, B., Yang, S., Qin, Z., Wen, B., Zhang, C.:\nThe roles of wettability and surface tension\nin droplet formation during inkjet printing.\nScientific Reports 7(1), 11841 (2017) https:\n//doi.org/10.1038/s41598-017-12189-7\n12Appendix A\n13Table A1 Results of the compatibility tests carried out with different parts of the Seiko RC1536L printhead and the\nthree candidate solvents dioxane, NMP/BCA 25/75 wt%, and NMP/BCA 50/50 wt%. Changes in the mass of each\nprinthead component due to solvent uptake or degradation the components are reported as percentages (positive for\nincreased mass, negative for reduced mass) after immersing the components in the indicated solvent for 6 weeks at 60◦C.\nBold values fall outside of the manufacturer’s specification range and thus indicate insufficient compatibility.\nPart Material Mass increase (wt %) in:\nDioxane NMP/BCA (25/75) NMP/BCA (50/50)\n1 Plastic 1.1 0.2 0.7\n2 Metal 0.0 0.0 -0.1\n3 Metal 0.0 0.0 0.0\n4 Rubber 9.7 5.4 5.6\n5 Adhesive 2.9 10.2 41.7\n6 Adhesive 17.6 9.5 -100\n7 Adhesive 3.7 5.1 5.1\n8 Film -0.4 -0.7 -0.9\n9 Film 6.4 3.4 7.5\nFig. A1 Image of the inkjet printing platform employed in this study. The digitally controlled printing stage and ink\ndistribution system are visible on the left, and the dropwatching station is visible on the right.\n14Fig. A2 Combined-channel 3D CLSM image of an interfacial region between inks A and B, shown from three different\nviewing angles. These images revealed the dropcast film to be 9 µm thick.\n15" }, { "title": "2401.17764v1.The_GRAVITY_young_stellar_object_survey_XIII__Tracing_the_time_variable_asymmetric_disk_structure_in_the_inner_AU_of_the_Herbig_star_HD98922.pdf", "content": "Astronomy &Astrophysics manuscript no. aa46926-23_corr ©ESO 2024\nFebruary 1, 2024\nThe GRAVITY young stellar object survey\nXIII. Tracing the time-variable asymmetric disk structure in the inner AU of the\nHerbig star HD 98922\nGRA VITY Collaboration⋆: V . Ganci1,2, L. Labadie1, K. Perraut3, A. Wojtczak1, J. Kaufhold1, M. Benisty3, E.\nAlecian3, G. Bourdarot8, W. Brandner4, A. Caratti o Garatti4,5,6, C. Dougados3, R. Garcia Lopez4,5, J.\nSanchez-Bermudez4,9, A. Soulain3, A. Amorim12,14, J.-P. Berger3, P. Caselli7, Y . Clénet10, A. Drescher7, A. Eckart1,2,\nF. Eisenhauer7, M. Fabricius7, H. Feuchtgruber7, P. Garcia12,13, E. Gendron10, R. Genzel7, S. Gillessen7, S. Grant7, G.\nHeißel10, T. Henning4, M. Horrobin1, L. Jocou3, P. Kervella10, S. Lacour10, V . Lapeyrère10, J.-B. Le Bouquin3, P.\nLéna10, D. Lutz7, F. Mang7, N. Morujão12,13, T. Ott7, T. Paumard10, G. Perrin10, D. Ribeiro7, M. Sadun Bordoni7, S.\nScheithauer4, J. Shangguan7, T. Shimizu7, C. Straubmeier1, E. Sturm7, L. Tacconi7, E. van Dishoeck7,8, F. Vincent10,\nand J. Woillez11\n(Affiliations can be found after the references)\nReceived xx, xxxx; accepted xx, xxxx\nABSTRACT\nContext. Temporal variability in the photometric and spectroscopic properties of protoplanetary disks is common in young stellar objects. However,\nevidence pointing toward changes in their morphology over short timescales has only been found for a few sources, mainly due to a lack of high-\ncadence observations at high angular resolution. Understanding this type of variation could be important for our understanding of phenomena\nrelated to disk evolution.\nAims. We study the morphological variability of the innermost circumstellar environment of HD 98922, focusing on its dust and gas content.\nMethods. Multi-epoch observations of HD 98922 at milliarcsecond resolution with VLTI /GRA VITY in the K-band at low (R =20) and high\n(R=4000) spectral resolution are combined with VLTI /PIONIER archival data covering a total time span of 11 years. We interpret the interfero-\nmetric visibilities and spectral energy distribution with geometrical models and through radiative transfer techniques using the code MCMax. We\ninvestigated high-spectral-resolution quantities (visibilities and di fferential phases) to obtain information on the properties of the HI Brackett- γ\n(Brγ)-line-emitting region.\nResults. Comparing observations taken with similar (u,3)plane coverage, we find that the squared visibilities do not vary significantly, whereas we\nfind strong variability in the closure phases, suggesting temporal variations in the asymmetric brightness distribution associated to the disk. Our\nobservations are best fitted by a model of a crescent-like asymmetric dust feature located at ∼1 au and accounting for ∼70 % of the near-infrared\n(NIR) emission. The feature has an almost constant magnitude and orbits the central star with a possible sub-Keplerian period of ∼12 months,\nalthough a 9 month period is another, albeit less probable, solution. The radiative transfer models show that the emission originates from a small\namount of carbon-rich (25%) silicates, or quantum-heated particles located in a low-density region. Among di fferent possible scenarios, we favor\nhydrodynamical instabilities in the inner disk that can create a large vortex. The high spectral resolution di fferential phases in the Br γline show\nthat the hot-gas compact component is o ffset from the star and in some cases is located between the star and the crescent feature. The scale of\nthe emission does not favor magnetospheric accretion as a driving mechanism. The scenario of an asymmetric disk wind or a massive accreting\nsubstellar or planetary companion is discussed.\nConclusions. With this unique observational data set for HD 98922, we reveal morphological variability in the innermost 2 au of its disk region.\nThis property is possibly common to many other protoplanetary disks, but is not commonly observed due to a lack of high-cadence observation. It\nis therefore important to pursue this approach with other sources for which an extended dataset with PIONIER, GRA VITY , and possibly MATISSE\nis available.\nKey words. stars: pre-main sequence – protoplanetary disks – stars: variables: Herbig Ae /Be – stars: individual: HD 98922 – techniques: high\nangular resolution – techniques: interferometric\n1. Introduction\nIn the last decade, our knowledge of protoplanetary disks\naround young stars has grown considerably thanks to the\ndrastic improvement of observing facilities. It is nowadays\nestablished that such disks show di fferent substructures across\n⋆GRA VITY is developed in a collaboration by the Max Planck In-\nstitute for Extraterrestrial Physics, LESIA of the Paris Observatory, and\nIPAG of the Université Grenoble Alpes /CNRS, the Max Planck Institute\nfor Astronomy, the University of Cologne, the Centro de Astrofísica e\nGravitação, and the European Southern Observatory.the optical to submillimeter wavelength range, such as rings,\ngaps, spiral arms, vortices, warps, and shadows on scales of tens\nof au (Huang et al. (2018), Garufi et al. (2018) and references\ntherein). Thanks to long-baseline infrared interferometry, the\nmorphology of inner disks at scales of less than 1 au has been\nrevealed (e.g., Monnier et al. 2005; Eisner et al. 2014; Lazare ff\net al. 2017; GRA VITY Collaboration et al. 2019; Kluska et al.\n2020). Temporal photometric variability is a common property\nof YSOs (e.g., Kóspál et al. 2012; Rice et al. 2015; Wolk\net al. 2018; Guarcello et al. 2019; Robinson & Espaillat 2019)\nArticle number, page 1 of 45arXiv:2401.17764v1 [astro-ph.SR] 31 Jan 2024A&A proofs: manuscript no. aa46926-23_corr\nTable 1. HD 98922 stellar parameters\nParameter Unit Value Reference\nDistance pc 650 .9±8.8 1\nAge Myr [0 .2,0.7] 2\nM⋆ M⊙ [5.0,7.0] 2,3\nR⋆ R⊙ 11.45±0.36 3\nlog L⋆ L⊙ 3.16±0.02 3\nTeff K 10500±125 3\nlogg cm s−23.5±0.2 4\n[Fe/H] −0.5±0.2 4\n3sini km s−139.0±5.3 5\nProt d 4–8 5\nlog˙Macc M⊙yr−1[−7.0,−5.0] 6,3\nNotes. (1) Gaia Collaboration et al. (2021); (2) Garufi et al. (2022); (3)\nGuzmán-Díaz et al. (2021); (4) Caratti o Garatti et al. (2015); (5) Aarnio\net al. (2017); (6) Fairlamb et al. (2015).\nand may originate, for instance, from the presence of cool or\nhot spots on the stellar surface, variable accretion, or changes\nin the inner disk structure leading to partial occultation of\nthe central star and variable dimming of the system as in the\ncase of \"dippers\" or UX-Ori type objects. The question of\ntime-variable morphology of the inner disk has been tackled\nfor a handful of sources (e.g., Kluska et al. 2016; Kobus et al.\n2020; GRA VITY Collaboration et al. 2021b). The scarcity\nof such studies is mainly due to the fact that the long-period\nhigh-cadence observations of the same object required for such\nanalyses are seldom available. In this context, observations with\na large temporal baseline using the PIONIER (Le Bouquin et al.\n2011) and GRA VITY instruments (GRA VITY Collaboration\net al. 2017) at the VLTI provide a unique opportunity to probe\nthe origin of the variability in the brightness distribution of the\ninnermost regions of YSOs.\nIn the present paper, we study HD 98922, a B9Ve /A2III\nHerbig star (Hales et al. 2014; Caratti o Garatti et al. 2015)\ncharacterized by a spectral energy distribution (SED) with a\nhigh near-infrared (NIR) excess and a low far-infrared (FIR)\nexcess (Garufi et al. 2022). Classifications of the star and\nestimates of its parameters have shown discrepancies over the\nyears. Indeed, the distance estimates were drastically divergent\nbefore the Gaia era, with values going from ∼450 pc (Caratti\no Garatti et al. 2015) to ∼1150 pc (van Leeuwen 2007), which\naffected the derivation of parameters such as luminosity, mass,\nand age. For instance, Lee et al. (2016) suggested classification\nas a post-main sequence giant, but accurate Gaia parallax infor-\nmation supports the classification of HD 98922 as a Herbig Be\nstar (Vioque et al. 2018; Arun et al. 2019). Using VLT /SPHERE,\nGarufi et al. (2022) set a lower limit of at least 200 au in radius\nfor the physical extent of the dust disk. However, comparison\nwith an ALMA image suggests an even larger radius of ≲500 au\n(Garufi et al. 2022). At scales of smaller than 10 au, Menu\net al. (2015) constrained the N-band-emitting dust disk radius to\n∼7.2 au using VLTI /MIDI interferometric observations.\nFurthermore, the system shows a CO-rich circumstellar disk,\nwhich Hales et al. (2014) suggest is geometrically flat with\nan inner radius of ∼1 au, extending to ∼200 au. Other authors\ninstead suggested a system with a flared CO disk with an inner\nradius of∼5 au and a flattened dust disk (van der Plas et al.\n2015). This transitional disk scenario was suggested because of\nthe relatively strong polycyclic aromatic hydrocarbon (PAH)emission of HD 98922 and its similarities with two other\nobjects, HD 101412 and HD 95881, for which such a disk\nstructure was already proposed (Fedele et al. 2008; Verhoe ff\net al. 2010). At smaller spatial scales, a rotating gaseous disk\ninside the dust sublimation radius was proposed to explain the\n[OI] emission-line profiles (Acke et al. 2005), while a wind or\noutflow was suggested to explain the P Cygni profiles in the H α,\nSi II, and He I lines (Grady et al. 1996; Oudmaijer et al. 2011).\nFinally, the Br γemission line was found to arise from a compact\nregion of∼0.65 au in radius, possibly tracing magnetospheric\naccretion (Kraus et al. 2008) or a disk wind (Caratti o Garatti\net al. 2015).\nPrevious interferometric works (Lazare ffet al. 2017; GRA V-\nITY Collaboration et al. 2019; Kluska et al. 2020) revealed\nthe noncentrosymmetric brightness distribution of the inner\ndisk of HD 98922. Here, we exploit a VLTI /GRA VITY and\nVLTI /PIONIER multi-epoch data set to study the temporal\nmorphological variability of the innermost circumstellar envi-\nronment of this source, looking at the NIR continuum emission\nand the hot hydrogen gas. In Sect. 2, we present the observations\nand describe the interferometric data. In Sect. 3, we present\nthe methodology used to analyze the continuum data and our\nresults. In Sect. 4, we present the methodology used to analyze\nthe Brγ-emitting gas data and the ensuing results. In Sect. 5, we\npresent a new radiative transfer model for the source. In Sect. 6,\nwe discuss some possible interpretations of our results in detail,\nand finally, in Sect. 7, we summarize our main findings.\n2. Observations and data reduction\n2.1. The source\nHD 98922 is a∼6 M⊙pre-main sequence star of less than 1 Myr\nof age that shows a high accretion rate. We adopt the distance of\n650.9±8.8 pc from Gaia DR3 (Gaia Collaboration 2022), which\nwe assume throughout the paper. The most up-to-date stellar pa-\nrameters are listed in Table 1. HD 98922 is a group II source\n(Juhász et al. 2010) in the Meeus classification (Meeus et al.\n2001), which is typically associated with a flat disk morphol-\nogy. The K band emission was measured to be inside ∼1.5 au\n(GRA VITY Collaboration et al. 2019) and the H band emission\ninside∼1.2 au (Lazare ffet al. 2017).\n2.2. Observations\nHD 98922 was observed with VLTI /PIONIER (Le Bouquin et al.\n2011) using the four 1.8 m Auxiliary Telescopes (ATs) in 21 dif-\nferent epochs between 2011 and 2016. Data were obtained using\nsmall- to large-baseline configurations for di fferent epochs. The\ndata consist of low-spectral-resolution (R ≈40) interferometric\nobservables in the H band. The observations span a spatial fre-\nquency range between about 5 M λand 90 Mλwith a maximal\nangular resolution of λ/2B∼1.25 mas for the longest baseline\nof 138.7 m, which corresponds to 0.81 au at 650.9 pc. In total,\n45 files were acquired and four files were discarded due to bad\nweather conditions. The description of the data per epoch can be\nfound in Table A.1 along with the observation logs.\nHD 98922 was observed at 13 di fferent epochs between 2017\nand 2022 using GRA VITY , with the so-called astrometric, large-,\nmedium-, and small-baseline configurations. The data consist of\nhigh-spectral-resolution (R ≈4000) observables recorded by the\nscience channel (SC) detector across the K-band with individ-\nual integration times of 30 s, as well as low-spectral-resolution\nArticle number, page 2 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\n(R≈20) observables recorded using the fringe tracker (FT) de-\ntector. The spatial frequency ranges between about 5 M λand 65\nMλwith a maximal angular resolution of λ/2B∼1.72 mas for\nthe longest baseline of 129 m, which corresponds to 1.12 au at\nthe distance of HD 98922. Each observation block corresponds\nto 5 minutes of observing time on the object. In total, 97 files\nwere acquired. Detailed information on the GRA VITY observa-\ntions is given Table A.2. In summary, the PIONIER and GRA V-\nITY dataset spans an 11 year period from 2011 to 2022.\n2.3. Data reduction\nThe PIONIER data are archival reduced and calibrated data re-\ntrieved from the JMMC Optical interferometry DataBase1. The\nerrors bars were calculated using the reduction and calibration\npipeline. These range from 0.25◦to 10.1◦for the closure phases\nand from 0.001 to 0.18 for the squared visibilities, depending on\nthe atmospheric conditions.\nThe GRA VITY data were reduced and calibrated using the\nGRA VITY data reduction software (Lapeyrere et al. 2014). For\nthe low-resolution FT data, we discarded the first spectral chan-\nnel, which is typically a ffected by the metrology laser operating\nat 1.908µm. Following GRA VITY Collaboration et al. (2019),\nwe applied a floor value on the error bars of 2% for the squared\nvisibilities and 1◦on the closure phases as the error bars com-\nputed by the pipeline might be underestimated.\nFor the high-resolution science (SC) data, the pipeline produces\nsingle files containing the spectrum, the calibrated visibilities,\nand the calibrated di fferential phases. The observables for one\nepoch result from the averaging of N single files as listed in the\nfourth column of Table A.2. Each averaged file is wavelength-\ncalibrated using the position of the telluric absorption lines\nbracketing the Br γemission line and then corrected for the ra-\ndial velocity of the star and the Earth’s motion with respect to\nthe LSR. The wavelength calibration is discussed in more detail\nin Appendix C. The error bars are calculated by the reduction\npipeline for the single files and are propagated through the aver-\naging process. These range from 0.3% to 1% for the spectrum,\nfrom 0.001 to 0.01 for the squared visibilities, and from 0.5◦\nto 2◦for the di fferential phases, depending on the observation\nepoch.\n3. Results derived from the continuum\ninterferometric data\nThe reduced data are illustrated in Fig. 1 for a few selected\nepochs and are shown in full in Figs. A.1 and A.2. The plots\nshow the calibrated squared visibilities, the closure phases, and\nthe(u,3)coverage. PIONIER and GRA VITY spatially resolves\nthe H band and K band continuum emission of HD 98922 at all\nbaselines and epochs, with visibilities ranging between 0 and\n0.8. Clear closure-phase signals up to 20◦with PIONIER and\n40◦with GRA VITY are detected for all epochs. The data there-\nfore clearly suggest asymmetries in the brightness distribution\nof HD 98922 at spatial scales probed by the VLTI. Importantly,\nfor the configurations with a similar (u,3)coverage, we detect\nsignificant variations in the closure phases across the epochs. On\nthe contrary, we observe that the change in the corresponding\nvisibilities is not very strong, typically below V2∼0.05.\n1available at http://oidb.jmmc.fr/index.html3.1. Modeling methodology\nThe observational results show that, for the epochs using VLTI\nconfigurations with a similar (u,3)plane coverage, we see clear\nvariations in the interferometric quantities and in particular in\nthe closure phase signal. This indicates a noncentrosymmetric\nbrightness distribution that is temporally variable. We adopt a\nclassical approach in long-baseline interferometry based on the\nparametric fit of geometrical models to the squared visibilities\nand closure phase signals. As HD 98922 is known from previous\nworks to host a circumstellar disk (e.g., Kluska et al. (2020)), we\nfocus our methodology on the analysis of a disk-like parametric\nmodel.\n3.1.1. Choice of the model\nA geometrical disk model with an azimuthal modulation is one\npossible solution to account for the asymmetric brightness dis-\ntribution revealed by the nonzero closure phases (Lazare ffet al.\n2017); we adopt such a model here. We used chromatic geomet-\nric models that consist of a point-like central star, a scattered\nlight component (called halo), and a circumstellar environment\nto fit the continuum interferometric quantities. The star is as-\nsumed to be unresolved and the halo to be fully resolved with a\nvisibility of zero. The circumstellar emission is modeled through\nan azimuthally modulated wireframe (Lazare ffet al. 2017) with\na radial brightness distribution given by\nF(r)=1\n2πδ(r−ar)·\u0010\n1+mX\nj=1(cjcosjϕ+sjsinjϕ)\u0011\n, (1)\nwhereϕis the polar angle. The order of the azimuthal modula-\ntion is taken in our case to be m=1. The wireframe is convolved\nby an ellipsoid kernel that regulates the width of the ring-like\nemission, and whose visibility is given by Eq. 9 of Lazare ffet al.\n(2017). Hence, the model can describe from infinitesimally thin\nrings to very wide rings tending to ellipsoids. It is described by\nten parameters: the star flux contribution F s; the halo flux con-\ntribution F h; the spectral index of the circumstellar disk k c; the\nradius of the wireframe a rand of the Kernel a k; the inclination\ni; the position angle (PA); the weighted contribution of a Gaus-\nsian or Lorentzian distribution fLor; and the azimuthal modula-\ntion parameters c1ands1. More details on the relation between\nthe geometrical, physical, and fitted parameters is provided in\nSect. 3.6 of Lazare ffet al. (2017).\nThe total complex visibility of the system at spatial frequencies\n(u,3) and at wavelength λis therefore described by a linear com-\nbination of the three components as\nV(u,3,λ)=Fs(λ/λ 0)ks+Fc(λ/λ 0)kcVc(u,3)\n(Fs+Fh) (λ/λ 0)ks+Fc(λ/λ 0)kc, (2)\nwhere Vcis the complex visibility of the circumstellar en-\nvironment, and Fs,Fh, and Fcare the specific fractional\nflux contributions of the star, of the halo, and of the cir-\ncumstellar environment, respectively, at λ0=λK,0=2.15µm (the\nwavelength of the central spectral channel of the GRA VITY\nFT), or atλ0=λH,0=1.68µm (for PIONIER). The parameter\nks=dlogFλ,s/dlogλis the spectral index of the star assumed\nto be a black body at Teff=10500 K. This translates into a spec-\ntral index of ks=-3.645 atλK,0and -3.523 at λH,0. The parameter\nkc=dlogFλ,c/dlogλis the spectral index of the circumstellar\nenvironment.\nAt this point, we must reiterate an important convention of the\nArticle number, page 3 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. 1. HD 98922 PIONIER (first two rows) and GRA VITY (last two rows) data, squared visibilities (left panel), closure phases (central panel),\nand(u,3)plan coverage (right panel) for two epochs of the complete data set shown in Figs. A.1 and A.2. Colors refer to the di fferent spectral\nchannels.\nazimuthal modulation relevant for the correct interpretation of\nour results. The azimuthal modulation, which is parametrized\nwith the variables c1 and s1, has a PA of the peak emission given\nby the argument of the complex number c1+j.s1. The origin of\nthe angular position of the azimuthal modulation is the PA of the\ndisk (cf. Table 3) measured from north to east, with east to the\nleft. As an example, an azimuthal modulation described by c1 =1\nand s1 =1 in a disk with a PA =45◦will show a visual rendering\nwhere the azimuthal modulation peak emission appears at 90◦\ntowards east.\nThe model fitting consists in an initial minimization proce-\ndure with scipy.optimize.minimize using a sequential least\nsquares programming method to obtain an initial guess of thefree parameters, followed by a procedure based on a Markov\nchain Monte Carlo (MCMC; Foreman-Mackey et al. 2013) nu-\nmerical approach, which is robust against trapping in local min-\nima. We also report, when applicable, the uncertainty on the re-\nduced chi-square χ2\nrby computing the error on the mean of the\nstochastic variable Tidescribed in Eq. 8 in GRA VITY Collabo-\nration et al. (2021a).\n3.1.2. Global fit modeling\nAs the epochs are constituted by observations using di fferent ar-\nray configurations, di fferent epochs probe di fferent spatial fre-\nArticle number, page 4 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nTable 2. Results of the global fit for di fferent parameters tested for vari-\nability (Models #2 to #6). Model #1 corresponds to a nonvariable sys-\ntem. The uncertainty on the χ2\nris reported.\n# Model Var. par. Nfp χ2\nr(P)χ2\nr(G)\n1 SHRM1 – 10 17.2 ±0.8 21.0±0.9\n2 SHRM1 i 9+Ne 14.9±0.7 14.7±0.5\n3 SHRM1 F s 9+Ne 13.5±0.7 11.1±0.4\n4 SHRM1 PA 9 +Ne 12.4±0.6 12±0.5\n5 SHRM1 a r, ak 8+2Ne 12.0±0.6 7.7±0.3\n6 SHRM1 c 1, s1 8+2Ne 11.5±0.6 5.3±0.3\nNb of data points 3855 1803\nNotes . The model nomenclature (column 2) gives S =star, H =halo,\nRM1 =1storder azimuthal modulated ring; Nfp gives the number of\nfree parameters. Ne is the number of epochs, namely 20 for PIONIER\n(P) and 13 for GRA VITY (G)).\nquencies of the source. This may lead to the detection of spurious\nvariability in the model parameters, as shown in our test analysis\nin Appendix E. Therefore, instead of fitting each single epoch\nwith our ten-parameter model and searching for individual vari-\nability trends, we implement a so-called global fit of the data. In\nthis approach, we take advantage of the large temporal baseline\nof our data set to decipher which of the parameters tested could\nbe the dominant parameter causing the variability visible in the\ndata of Fig. 1. The global fit is obtained by forcing all the model\nparameters to be nonvariable over time, except for the parameter\n—or set of parameters— describing the feature to be tested for\nvariability.\n3.2. Temporal variability\nUsing the global fit approach on the continuum data, we explore\nwhich among the seven model parameters (star flux contribu-\ntion F s, ring inclination i, PA, characteristic size (a r,ak), and az-\nimuthal modulation (c 1,s1)) is most prone to cause the temporal\nvariability of the data visible in Fig. 1. The parameters describ-\ning the halo contribution F h, the spectral index k c, and the weight\nparameter fLor were not tested for temporal variability. We also\nreport for comparison the analysis result for a fully nonvariable\nsystem model. The results are shown in Table 2.\nFor the GRA VITY data set, we observe from the χ2\nranalysis\nthat the time-variable azimuthal modulation model #6 leads to\nthe smallest χ2\nrvalue. The nonvariable system (#1) gives the\npoorest fit. This is less marked for the PIONIER data set for\nwhich theχ2\nrvalues are larger. However, the trend in terms of\ndecreasingχ2\nrsuggests an analogous behavior for both the PI-\nONIER and GRA VITY data sets. This analysis indicates that,\namong the tested models, the one with a time-variable disk az-\nimuthal modulation (model #6) best describes our data, and that\nsuch asymmetry is probably the dominant e ffect in the variabil-\nity of the system, as opposed to other geometrical e ffects. The\nparameter values for the best-fit result of model #6 are shown in\nTable 3 and the corresponding marginal posterior distribution is\npresented in Appendices B.2 and B.3. The fit results for each\nindividual epoch of our PIONIER and GRA VITY data sets are\nshown in Appendix A.1.\n3.3. Disk azimuthal asymmetry\nOur best-fit model exhibits a crescent-like asymmetric feature\nin the disk resulting from the azimuthal modulation that varies\nin PA through the epochs and revolves around the central star.In Fig. 2, we present a subset of the continuum model images\ncorresponding to our fitted models and from which the geom-\netry, extent, and location of the asymmetry can be followed as\na function of time. The complete time sequence is presented in\nFigs. F.1 and F.2. The fit of the variable azimuthal modulation\nappears quite robust in Fig. B.4, with one single global mini-\nmum identified in the c1,s1diagrams. Figure 2 displays the az-\nimuthal uncertainties in the form a white-line cone as derived\nfrom the 3σerror bars on c1ands1. The azimuthal uncertainty\nis generally smaller (up to ∼10-20◦, depending on the configura-\ntion) in the K-band than in the H-band. In addition, the azimuthal\nposition is most poorly constrained with the small configuration\n(e.g., P2-P5-P6 with PIONIER) because the closure phase signal\nis marginal for the shortest baselines For epochs only separated\nby a few days at most and taken with the same array config-\nuration, the azimuthal locations of the asymmetric feature are\nconsistent in most cases within the error bars (e.g., P7-P8-P9-\nP10-P11 and P19-P20 with PIONIER, or G7-G8 with GRA V-\nITY). We also generally observe that the continuum emission\nappears azimuthally more compact in the H-band than in the K-\nband. The brightness contrast between the crescent-like feature\nand the corresponding centro-symmetric position in the disk is\nestimated from Figs. F.1 and F.2. The contrast is found to be ∼4\nin the K-band (ranging from ∼1.6 to 10 across the epochs, with\nσ∼2.8) and∼2.4 in the H-band (ranging from ∼1.3 to 3.3, with\nσ∼0.7). No specific trend is found in the temporal evolution of\nthe contrast. The dynamical properties of the inner disk feature\nrevealed by our observations are further discussed in Sect. 3.4.\n3.4. Orbital period of the crescent-like feature\nThe revealed azimuthal asymmetry in the inner disk of\nHD 98922 shows orbital motion around the central star. Assum-\ning this is the same asymmetric feature that is monitored with\nthe VLTI over the 11 year period in both the H and K bands, we\nattempt to investigate the orbital properties of the emission fea-\nture. We show in Fig. 3 the distribution of the time-variable PA\nof the emission feature. The small-configuration epochs P2, P5,\nP6, G6, and G10 are not taken into account as the small baselines\nprovide marginally weaker constraints on the azimuthal modu-\nlation because of the low level of closure phase signal detected.\nThe time origin corresponds to the first PIONIER observation of\nJune 2011. The gray line is a sine function corresponding to a\nuniform circular motion fitted to the data to derive a period es-\nTable 3. Nonvariable parameters of the azimuthal modulation global fit\ncontinuum model. F cis not a free parameter, but is obtained following\nFc=1-Fh-Fs. TheσMCMC error estimates derived through the MCMC\nfitting procedure are given by the 16thand 84thpercentiles of the sam-\nples in the MCMC marginalized distributions.\nPIONIER GRA VITY\nParameter Unit Value 3 σMCMC Value 3σMCMC\nFs % 23.09 0.39 22.69 0.42\nFh % 3.92 0.78 10.36 0.57\nFc % 72.99 1.17 66.95 0.99\nkc 3.47 0.18 -0.39 0.33\nar mas 0.93 0.06 2.02 0.06\nak mas 1.60 0.03 1.73 0.06\ni deg 0.67 2.01 34.15 1.08\nPA deg 106.9 1.80 122.9 1.50\nfLor 1.00 0.03 1.00 0.03\nχ2\nr 11.52 5.26\nArticle number, page 5 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. 2. Peak-normalized GRA VITY (top row) and PIONIER (bottom row) continuum model images. The dashed white lines represent the ±3σ\nuncertainty on the PA of the azimuthal modulation. The central object is not displayed but is marked with a star to enhance the circumstellar\nemission. North is up, east is to the left. See Appendix F for the full data set.\nFig. 3. Sine of the azimuthal modulation PAs as a function of time.\nThe markers represent the sine of the variable dusty feature PAs de-\npicted in Fig. 2, green for the PIONIER data and blue for the GRA V-\nITY one. The full gray line represents the best fit (12 .6±0.1 months)\nof the uniform circular motion PA expressed as a sine function. The\ndashed gray line represents the fit solution corresponding to a period of\n8.50±0.25 months.\ntimate. The best sine fit gives a period of 12.6 ±0.1 months. The\nχ2\nrvalue is large (111 ±51), which is due to the group of epochs\nat∼60 months that deviate from the best fit. If these epochs were\nfound to correspond to outliers and were removed, the fit would\ngive aχ2\nrvalue of 38±17, while the estimated period would re-\nmain the same (see Table 4). However, at this stage, there is no\nclear justification for the removal of these epochs.\nUsing Kepler’s law, we estimate the central mass of our target\nstar for a separation of the azimuthal asymmetry ranging from\n0.6 to 1.3 au based on the fitted ring annular radius a r, as well\nas for a range from 1.2 to 1.7 au based on the half-light ra-\ndius a, where a=(a2\nr+a2\nk)1/2following Lazare ffet al. (2017).From the derived orbital period of the azimuthal feature, we esti-\nmate a central mass of ∼1–3 M⊙depending on the separation re-\nported above. This is significantly lower than the literature value\nof∼6 M⊙, which was robustly established using UVES high-\nspectral-resolution observations (Caratti o Garatti et al. 2015).\nFor this higher mass, the estimated Keplerian orbit for a cir-\ncular motion would be 9.2 ±0.4 months, which is shorter than\nthat found with our measurement. It is noteworthy that our sine\nfit also shows a local minimum for a similar period of around\n∼8.5 months, although with a poorer χ2\nrvalue of∼350. We show\nthis sine fit for comparison as a dashed line in Fig. 3.\nSome caveats are worth mentioning here. The coverage of the\norbital motion is relatively sparse and a finer temporal sampling\nwould allow us to estimate the period with greater confidence.\nFurthermore, it should be noted that we have implemented here\nthe simplest case of a circular orbit, because our model #6 of\nTable 2 foresees a nonvariable ring radius ar. Accounting for\npossible eccentricity of the orbit of the crescent-like feature may\nimpact the derived orbital period, which is not explored in this\nwork. The orbital motion is further discussed in Sect. 6.1.3.\n3.5. Fractional flux and disk characteristic size\nIn H-band, we find fractional flux ratios F s, Fh, and F ccompara-\nble to those found by Lazare ffet al. (2017). In the K-band, we\nmeasure a stellar flux contribution of ∼23% (see Table 3). Dif-\nferent values for the K-band fractional stellar contribution based\non photometry and SED analysis are reported in the literature,\nranging from∼15 % (Kraus et al. 2008; Hales et al. 2014; GRA V-\nITY Collaboration et al. 2019) to ∼23 % (Caratti o Garatti et al.\n2015). However, when comparing these values, we must keep in\nArticle number, page 6 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nTable 4. Orbital period estimates of the dust azimuthal asymmetry\nPeriod χ2\nr M⋆ Method\n[months] [M ⊙]\n12.6±0.1 111±51 – (1)\n12.6±0.1 38±17 0.9±0.5 (2)\n12.6±0.1 38±17 3.0±0.8 (3)\nNotes: (1) period estimated from the fit of the time-variable PA of the\ncontinuum azimuthal feature; (2) same as (1) but with the group of\nepochs at 60 months removed. The mass of the central star is derived\nby assuming that the feature has a separation ranging from 0.6 to 1.3 au\nas given by the ring radius a r; (3) same as (1) but with the mass of the\ncentral star derived by assuming that the feature has a separation\nranging from 1.2 to 1.7 au as given by the half-light radius a.\nFig. 4. HD 98922 GRA VITY SC data for two di fferent epochs (one\nper column) with similar (u,3)plan coverage. From top to bot-\ntom: Wavelength-calibrated and continuum-normalized spectrum, total\nsquared visibilities (two panels), and total di fferential phases (two pan-\nels). Colors refer to the di fferent baselines.\nmind the di fferent distance estimates used in these previous stud-\nies. We also find a stronger halo contribution than GRA VITY\nCollaboration et al. (2019), which can be explained by the fact\nthat these latter authors constrained the circumstellar emission\nof HD 98922 in the K-band only with the astrometric configura-\ntion, whereas the short baselines of the small configuration are\nusually needed to better constrain the halo.\nThe radius of the ring disk emission in K-band is found to be\nar=2.02±0.1 mas (or 1.31±0.07 au), in good agreement with the\nestimates of Kraus et al. (2008) and Caratti o Garatti et al.\n(2015), of namely 2.2 mas and 1.6 mas, respectively. Our result is\nslightly larger than the estimate of a r=1.8 mas of GRA VITY Col-laboration et al. (2019), who only used two snapshots with the\nastrometric configuration. In the H-band, we find the circumstel-\nlar emission to be more compact than in K, with a characteristic\nradius of a r=0.93±0.1 mas (or 0.60±0.07 au), which is in line\nwith the estimate of 0.87 mas by Lazare ffet al. (2017). Kluska\net al. (2020) report a larger half-flux radius of 2.1 mas based on\nimage reconstruction, which evidences the impact of di fferent\nmodeling approaches. The K-to-H size ratio is further discussed\nin Sect. 6. The ratio a k/arbeing close to or larger than unity sug-\ngests a wide, smooth ring emission as opposed to a sharp edge.\nThe disk is known to have a low inclination, which is therefore\nmore di fficult to accurately constrain in the small angle range.\nThis applies to the PA as well (Fig. B.1).\n4. Results on the Br γ-line interferometric data\nThe GRA VITY line spectrum, the calibrated squared visibili-\nties, and the di fferential phases are shown in Figs. 4 and A.3\nfor 12 of the 13 epochs after zooming into the spectral re-\ngion between 2.164 and 2.168 µm. No high-spectral-resolution\ndata could be acquired on June 15, 2018 (epoch G3). From the\nhigh-spectral-resolution K band data, HD 98922 shows a slightly\nblueshifted (≈ − 23.5 km/s) single-peaked Br γemission line\nin all epochs. Considering the 3 Å spectral resolution, we can\nconsider the peak’s position of the continuum-normalized line\n(21659.7+0.8\n−1.3Å) to be constant through the epochs at our spec-\ntral resolution. The normalized peak flux varies between 1.17\nand 1.25 depending on the epoch, while the line width mea-\nsured at the peak’s 10% flux level ranges from 14 to 15 Å. The\ntotal squared visibilities in the Br γregion vary between ≈0\nand≈0.4 depending on the epoch and baseline configurations\n(medium or large), while they reach ≈0.7 for the small con-\nfiguration. Finally, the di fferential phase signals vary between\n−15◦and 25◦and have significantly di fferent shapes (flat, single-\npeaked, double-peaked, or S-shape) for the di fferent epochs and\nbaselines.\n4.1. Modeling methodology\nTo estimate the Br γgas region size, kinematics, and displace-\nment with respect to the continuum emission from the GRA V-\nITY SC visibilities and di fferential phases, we extrapolated\nthe pure-line contribution (marked with subscript L) from the\ninterferometric observables. Following Weigelt et al. (2011),\nthe pure-line interferometric quantities characterizing the gas-\nemitting region, the visibility VLand di fferential phase ϕL, are\nrelated as follows:\nF2\nLV2\nL=F2\ntotV2\ntot+F2\ncontV2\ncont−2FtotVtotFcontVcont·cosϕtot,(3)\nsinϕL=sinϕtotFtotVtot\nFLVL. (4)\nThe quantities reported in Eq. 3 and 4 refers to values inside\nthe Brγ-line spectral region. As the continuum quantities Fcont\nandVcontwithin the line are not directly measurable, they are\nestimated from the continuum near to the line region. However,\nhot Herbig stars exhibit a strong Br γphotospheric absorption\nfeature, which needs to be accounted for in order to retrieve the\ncorrect pure-line quantities. In the case where photospheric ab-\nsorption is present, knowledge from a model for the continuum\nArticle number, page 7 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. 5. Spatial and kinematic properties of the Br γ-line-emitting gas region overlaid to the continuum model for two epochs. The complete sequence\nis shown in Appendix G. The star is centered on the origin. The black cross and the white circle around it show the position of the continuum\nphotocenter and its uncertainty estimated from the continuum modeling. The red circle represents the extent of the gas-emitting region estimated\nat the peak of the line emission and centered at about ∼0 km s−1. The blue to red colored filled dots show the gas photocenter positions for the five\nspectral channels across the Br γline corresponding to velocities ranging from -100 km s−1to+100 km s−1, as color-coded in Fig. A.3.\nemission is required. One can show that\nVL=(VtotF′\nL/C)2+\u0010F′\ns\nF′\ncont(α−1)+V′\ncont\u00112\n\u0010\nF′\nL/C−α+β+γ\n1+β+γ\u00112\n−2 (VtotF′\nL/C)\u0010F′\ns\nF′\ncont(α−1)+V′\ncont\u0011\ncos(ϕtot)\n\u0010\nF′\nL/C−α+β+γ\n1+β+γ\u001121/2\n, (5)\nwhere the superscript (′) indicates that the quantities V′\ncontand\nF′\ncontare estimated outside the emission line spectral region, as\nopposed to Eq. 3. The line-to-continuum ratio F′\nL/Cis also nor-\nmalized to the nearby continuum value. The parameters βandγ\nare the disk-to-star β=Fc/Fsand halo-to-star γ=Fh/Fsflux ratios.\nThe parameter αdescribes the star photospheric absorption, with\n0<α⩽1, so that\nFL=Ftot−F′\ncont+Fs(1−α), (6)\nwhereαis equal to 1 when there is no absorption, and equal to\nzero when 100% of the star flux is absorbed (see Appendix C for\na description of the photospheric absorption model).\nSimilarly, Eq. 4 is written in terms of known quantities as\nsinϕL=sinϕtotVtot\nVL1\u0012\n1−α+β+γ\n1+β+γ1\nF′\nL/C\u0013. (7)\nFor a compact and marginally resolved gas emission component,\nwe derive the photocenter displacement along each baseline\nfrom the pure-line di fferential phases as in Lachaume (2003):\np=−ϕL\n2π·λ\nB, (8)\nwhere pis the projection on the baseline Bof the 2D photocen-\nter vector with its origin on the continuum photocenter of the\nsystem.4.2. Properties of the line-emitting region\nWe exploit the high-spectral resolution data of GRA VITY in the\nBrγ-line region to constrain the spatial scale and the kinematics\nof the hot-gas component. We used 10 out of 13 epochs for this\nanalysis: in addition to epoch G3, for which no high-spectral\nresolution data could be acquired, epochs G6 and G10 are not\nconsidered, as no di fferential phase signal was detected using\nthe small configuration.\nFrom the pure-line visibilities, we estimated the characteris-\ntic size of the gas-emitting region at the peak wavelength of\n2.1662µm using a simple Gaussian disk model, as well as a\nring model with 20% radial thickness for comparison. We ob-\ntain a Gaussian HWHM radius of 0.47 mas with a temporal\nstandard deviation of 0.04 mas. Similarly, we obtain a radius of\n0.50±0.05 mas for the thick ring. In both cases, the gas-emitting\ncomponent model is found to be consistent with a face-on orien-\ntation, as we find an inclination of 5◦±5◦. These values translate\ninto a physical radius of ∼0.3 au at a distance of 651 pc, in agree-\nment with the results of Caratti o Garatti et al. (2015).\nAs the estimation of the characteristic size of the hot-gas region\nhas been studied in previous works, we focus on the determina-\ntion of the location of the compact hot-gas component in relation\nto the star, and further explore whether there is a correlation be-\ntween the locations of the gas and the dust feature observed in\nthe continuum. This can be investigated through an analysis of\nthe interferometric di fferential phase, which provides precise in-\nformation about the spatial location of the photocenter of the gas\nemission component on angular scales that surpass the nominal\nresolution of the interferometer.\nThe detection of a di fferential phase signal as reported in\nFig. A.3 indicates that the photocenter positions of the contin-\nuum and gas emission components are not coincident. From the\ncontinuum-corrected (or pure-line) di fferential phases (Eq. 7),\nwe calculated the deprojected photocenter shifts of the hot-gas\nArticle number, page 8 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\ncomponent with respect to the continuum photocenter for few\nspectral channels around the 0 km /s velocity (Eq. 8) following\nthe formalism of Gravity Collaboration et al. (2023). The lo-\ncation of the continuum photocenter may di ffer from the posi-\ntion of the central star if the dust emission is noncentrosymmet-\nric. This is indeed the case for HD 98922, because our model-\ning of the continuum shows a strongly asymmetric time-variable\nbrightness distribution of the inner circumstellar disk. There-\nfore, the location of the continuum photocenter is the param-\neter that most a ffects the position of the Br γ-line photocenter\nwith respect to the central star. Figure G.1 shows the spatial lo-\ncation of the gas emission photocenter with respect to the contin-\nuum photocenter, and therefore with respect to the central star.\nFor most of the epochs, the bulk of the Br γemission appears\noffset —considering the retrieved characteristic size of the gas\nemission component— with respect to the stellar position by\nup to 0.5 mas. We also observe that the location of the compact\ngaseous component varies with time and that it is, on first order,\nconsistently found in an area between the central star and the\npeak of the dusty feature, following its orbital motion. Looking\ncloser at Fig. 5, we see that, for each epoch, the positions of the\nphotocenter across the Br γline are not spatially colocated but\nare distributed in a profile that qualitatively resembles a pattern\nof Keplerian motion.\n5. Physical properties of the disk\nIn an earlier work, Hales et al. (2014) proposed a model for the\ndisk of HD 98922 accounting for a distance of 507 pc and ex-\nploiting photometry data up to 160 µm. These authors proposed\na disk model with an inner radius located at 1.5 au. We revisit\nthis model in light of our new interferometric data.\n5.1. Disk structure\nWe used the radiative transfer code MCMax (Min et al. 2009)\nto constrain the disk density and temperature structure based on\narchival broadband photometry data and on the spatial structure\nevidenced by our interferometric measurements. We started from\na one-component model based on Hales et al. (2014). The inner\ndisk radius is 1.5 au, the outer radius is 320 au, the flaring in-\ndexγ=1, and the scale-height is 15 au at a radius of 100 au. We\nimplement a grain population based on DIANA standard dust\ngrains (Woitke et al. 2016) composed of 75% amorphous sili-\ncates (e.g., Mg 0.7Fe0.3SiO 3), 25% porosity, by volume, and ini-\ntially no amorphous carbon. The grain size aranges from 1 µm\nto 2200µm with a distribution of d n(a)∝a−3.5, as large grains\nappear to dominate in this source (Bouwman et al. 2001; van\nBoekel et al. 2003). The surface density profile is based on a\nmodified version from Li & Lunine (2003) with a power-law ex-\nponent of pΣ=−1.5, a gas-to-dust ratio of 100, and a dust disk\nmass of 2×10−5M⊙. The extinction is AV=0.5 mag (Guzmán-\nDíaz et al. 2021). The parameters of the central star are taken\nfrom Table 1.\nWith the revised stellar parameters, the stellar luminosity in-\ncreases by a factor ∼3 in comparison to Hales et al. (2014) and\nthe photospheric emission contribution results in overestimation\nof the K-band NIR flux in the SED by a factor ∼1.2. On the other\nhand, our GRA VITY and PIONIER observations show that dust\nis present at separations of less than 1.5 au. As the SED fitting\nprocess is a degenerated problem, we attempt to better constrain\nthe inner disk structure based on the consideration that a dust\ncomponent at ≲1 au, producing∼75%-80% NIR excess and re-\nsolved with the VLTI has to be accounted for. Simply moving theinner radius to∼0.6 au in the one-component model described\nabove is not possible because this leads to further overestima-\ntion of the NIR flux in the SED. We therefore revise this disk\nmodel and propose a two-component disk model in which the\ninner component has a lower surface density in comparison to\nthe outer component and an inner radius at 0.6 au.\nThe mass of the outer component is set to 2 ×10−5M⊙as in\nHales et al. (2014). This value can be compared to the rough\nmass estimate obtained from the archival ALMA photometry\npoint at 1.3 mm (ID: 2015.1.01600.S /PI Panic), which was not\nincluded in the work of these latter authors. We used the CARTA\nvisualization tool and estimate a flux density at 1.3 mm of\nFν=10.4 mJy for the unresolved HD 98922 source, with a typ-\nical conservative uncertainty of 10% . We compute the dust disk\nmass Mdfollowing Beckwith et al. (1990). Assuming optically\nthin emission, we use the relation\nFν≈κν2kTν2\nc2d2Md, (9)\nwhere Tis the dust mean temperature, κνthe dust absorption\ncoefficient at 1.3 mm, dthe distance, kthe Boltzmann constant,\nand cis the speed of light. A large uncertainty subsists on\nthe dust opacity value κνand on the mean temperature, and\ntherefore the mass value is only a crude approximation for\nfirst-order estimates. If we consider standard values of T=50 K,\nandκν(1.3 mm) =0.02 cm2/g (Beckwith et al. 1990), we obtain\na disk dust mass of Md∼0.013 M⊙. However, Woitke et al.\n(2016) underline the strong dependence of κνon the fraction of\namorphous carbon and grain size distribution, and report values\nas high asκν(1.3 mm)∼5-10 cm2/g. The latter value results in\na lower mass of Md∼5×10−5M⊙, which is comparable to that\nfound by Hales et al. (2014). Because of the uncertain estimate\nof the disk dust mass, we choose to adopt the value reported\nby the latter authors for the outer disk component. For the disk\ncomposition, we considered two approaches: in the first case, we\nincluded a 25% fraction of carbon grains (model CS) to quench\nthe strength of the silicate emission feature (van Boekel et al.\n2003); in the second case, we consider an alternative case of an\ninner component only composed of quantum-heated particles\n(model Q) strongly coupled to the gas (e.g., Kluska et al. 2018;\nGRA VITY Collaboration et al. 2021a).\nWe conducted a grid search for the inner disk dust mass M d,in\nand for the transition radius between the low- and high-surface-\ndensity disk R t(cf. Table 6) to match the SED as well as the NIR\nexcess (F c+Fh) estimated by interferometry (cf. Table 3). The\nother disk parameters were kept identical for the inner and outer\ncomponents. Table 5 shows our best result for the models CS\nand Q, respectively, for which our radiative transfer simulation\nproduces the SED displayed in the top left panel of Fig. 6. For\nbetter readability, only the case of model CS is shown because\nthe result for model Q is almost identical. Visually, these models\nprovide the best overlap between the model and the data, except\nin the FIR region where the photometry data are more severely\noverestimated by the model. The FIR emission traces the disk at\nlarge radii not immediately relevant in our study. For the model\nCS, a transition radius R tat∼5.5 au and an inner disk dust\nmass of∼6×10−9M⊙are derived from the radiative transfer\nmodeling (cf. Table 5). We note that our two best models CS\nand Q may still contain some level of degeneracy, because\nother parameters such as the scale height, the index of the\nsurface density power law, or the flaring index may influence\nthe NIR excess. However, we limited ourselves to the inner\nArticle number, page 9 of 45A&A proofs: manuscript no. aa46926-23_corr\nTable 5. HD 98922 RT models. The stellar parameters are from\nGuzmán-Díaz et al. (2021).\nStar\nParam. Unit Value Range\nT⋆ K 10500 Fixed\nR⋆ R⊙ 11.45 Fixed\nM⋆ M⊙ 7.0 Fixed\nd pc 650.9 Fixed\nAV mag 0.5 Fixed\nDisk Inner Component\nParam. Unit Model CS Model Q Range\nRin au 0.6 0.6 Fixed\nRt au 5.5 3.5 [2.5 - 7.5]\nMd,in M⊙ 6×10−92×10−12[10−13,−7]\nH100 au au 10.0 10.0 Fixed\nγ 1.0 1.0 Fixed\naminµm 1.0 0.006 Fixed\namaxµm 2200 0.006 Fixed\npΣ -1.5 -1.5 Fixed\nCarbon % 25 0 [0-35]\nDisk Outer Component\nParam. Unit Value Range\nRout au 320 Fixed\nMd,out M⊙ 2×10−5Fixed\nH100 au au 10.0 Fixed\nγ - 1.0 Fixed\namin µm 1.0 Fixed\namax µm 2200 Fixed\npΣ - -1.5 Fixed\nCarbon % 25 [0-35]\ndisk properties that can be directly constrained by the NIR flux\nratios and spatial confinement of the emission derived from our\ninterferometric measurements.\n5.2. Transition radius\nFor the adopted two-component disk model for HD 98922, the\nlow- to high-density transition radius R tis found to be located\nat∼5.5 au, with a lower-limit at 2.5 au below which the NIR\nexcess is systematically overestimated in the SED. Despite the\ndegeneracy between R tand M d,infor the determination of the\nNIR excess flux in the SED fit, our interferometric measure-\nments can set constrains on the inner radius, relative flux con-\ntribution, and compact spatial extent of the low-density compo-\nnent that dominates the NIR excess. Table 6 illustrates this point\nby comparing the relative flux contributions of the inner low-\nand outer high-density components inferred from radiative trans-\nTable 6. RT results for Model CS. The sign(∗)indicates that the relative\nflux contributions of the disk and halo components are added. LDC and\nHDC stand for low-density and high-density component, respectively.\nRt, M d,in 5.5 au, 6×10−9M⊙ 3.5 au, 3×10−9M⊙\nFlux SED Interf. SED Interf.\n(%) H K H K H K H K\nStar 30 16 22 23 28 14 22 23\nLDC 70 75 78(∗)77(∗)41 44 78(∗)77(∗)\nHDC 0 9 – – 31 42 – –fer to the estimates from the interferometric measurements, and\nthis for two values of R t, both giving a good fit of the SED. For\nsmaller values of R t, the NIR flux contribution of the outer high-\ndensity component increases, while that of the inner low-density\ncomponent decreases. Qualitatively, this would translate for our\ngeometrical models into wider rings, or into a higher contribu-\ntion of the halo component, which we only find to be less than\n10% in both bands. For instance, in the case R t=3.5 au, the in-\nnermost low-density component probed by GRA VITY and PIO-\nNIER contributes only up to ∼40%. However, it should be noted\nthat the proposed argument su ffers from some degree of uncer-\ntainty: as the modeling of our interferometric observables relies\non single-ring geometrical models rather than on radiative trans-\nfer images, the accuracy on the determination of R tremains lim-\nited. As the detailed interferometric modeling of radiative trans-\nfer images goes beyond the scope of this paper, we limit our-\nselves to propose that the structured disk of HD 98922 shows a\ndust density transition located no closer than ∼4-6 au to the cen-\ntral star. Estimating an upper value for R tis not feasible using\nonly the presented data because the NIR excess (set on the first\norder by R t, M d,inand M d,out) becomes a degenerated quantity\nfor excessively large values of R t.\n5.3. Surface density profile\nThe dust surface density profile we implement has a sharp\ndiscontinuity at R t, which does not correspond to a realistic\ncase, given that a continuous transition profile would be more\nphysically meaningful. Such sharp transitions are nonetheless\nproposed to di fferentiate between volatile-rich and volatile-free\nregions, as for instance in the case of the water snow line\n(Hayashi 1981; Lecar et al. 2006). More recent works have re-\nported structured inner disks with regions of di fferentiated sur-\nface density (Tatulli et al. 2011; Matter et al. 2016). We derive\na low- to high-density transition of two orders of magnitude at\nabout 5 au (Fig. 6), with a dust surface density Σdranging from\n∼2×10−4g cm−2to∼0.006 g cm−2in the low-density region. We\nemphasize that we cannot exclude that a gap is present in the disk\nof HD 98922 beyond ∼5 au, but mid-infrared (MIR) interfero-\nmetric data are necessary to explore this hypothesis. However,\nwe also note that HD 98922 is classified as a group II source in\nthe classification of Meeus et al. (2001) and based on the N-to-\nK size ratio (GRA VITY Collaboration et al. 2019), suggesting a\npreferentially flat, gap-free disk.\n5.4. Location and temperature of the dust\nThe dust emission in the H and K bands are expected to originate\nin a region close to the disk inner rim. In our case, the bulk\nof the emission is located at di fferent radii for the two bands.\nWe find the characteristic size of the K band is larger than that\nof the H band by a factor ∼1.4 to 2 when considering either\nthe ring annular radius a ror the half-light radius aparameters.\nThis trend is seen in previous works when comparing the results\nof Lazare ffet al. (2017) and GRA VITY Collaboration et al.\n(2019): out of 21 common sources, 17 show a larger K band by\na factor ranging from 1.1 to ∼3. At first glance, the di fference\nin resolution between GRA VITY (1.75 mas) and PIONIER\n(1.22 mas) by a factor 1.4 seems capable of explaining the fact\nthat PIONIER resolves a more compact emission, similarly to\nwhat observed for HD 163296 (GRA VITY Collaboration et al.\n2021b). A more physical, albeit simplistic, argument can be\nformulated based on the Wien’s temperature of the dust in a\nArticle number, page 10 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. 6. Radiative transfer modeling corresponding to Table 5. The top left panel shows the SED for Model CS, with the blue dashed line representing\nthe stellar black-body function and the red line showing the modeled total emission. The black bars represent the photometric data. The top right\npanel shows the derived dust surface density profile as a function of the distance from the star. The bottom left plot shows the dust density\nstructure, where the black dashed line represents the τ=1 surface at 2.2 µm, and the red lines represent, from left to right and for both components,\nthe density contours at 10−15, 10−16, 10−17, and 10−18g cm−3, respectively. The bottom right plot shows the dust temperature structure, where the\nblack lines represent, from left to right, the isothermal contours at 2300, 2000, 1700, 1500, and 1300 K, respectively.\ntemperature-gradient disk (i.e., T(r)=T0(r/r0)−q) emitting as\na blackbody. For a typical value of q=0.75 for a flat group II\ndisk, we derive a size ratio of rK/rH∼(1350/1750)1/0.75, or 1.41,\nin agreement with our findings.\nOne challenge posed by our results is the presence of H-\nband-emitting dust as close as ∼0.6 au to the star. In the\nscenario of a passively irradiated disk with an optically thin\ninner cavity (Muzerolle et al. 2004; Monnier et al. 2005), the\ndust temperature at this location, assuming black-body grain\nemitters (ϵ=1), is Tg∼2200 K, which is above the sublimation\ntemperature of standard silicates. This is also seen in Fig. 6\nfor the temperature structure of our carbon-rich disk model.\nThe problem of very hot dust grains inside the theoretical\nsublimation radius has been observed in other sources such\nas Z CMa, V1685 Cyg, MWC 297, and HD 190073 (Monnier\net al. 2005; Hone et al. 2017; Setterholm et al. 2018), invoking\nthe presence of refractory dust (e.g., corundum or iron), the\nsublimation temperature of which critically depends on the disk\ndensity (Kama et al. 2009). Alternatively, Benisty et al. (2010)\nproposed the possibility that a small fraction of highly refractory\ngraphite grains with high sublimation temperature ( Ts>2000 K)contribute to the NIR excess very close to HD 163296, and this\nmight apply to HD 98922 as well. Clearly, further radiative\ntransfer modeling is required to tackle this question by consider-\ning other values of aminandpΣ, as well as grain composition. We\nnote that adding amorphous carbon to our initial pure-silicate\nmodel was essential to decrease the dust temperature at 0.6 au\nfrom T>3000 K to T∼2300 K.\nAlternatively, for Herbig stars with masses similar to or larger\nthan that of HD 98922, the undersized inner disk is explained\nby the presence of optically thick gas in the inner cavity,\nwhich is found to be favored in systems with accretion rates of\n˙Macc≳10−8M⊙yr−1(Muzerolle et al. 2004), as for HD 98922.\nAnother possible explanation for the detection of excess\nemission very close to the star is the presence in the inner\nregion of a fraction of quantum heated particles (QHPs), which\ncan be stochastically heated by the strong UV radiation field\nfrom the central star, reaching temperatures higher than the\nequilibrium temperature and producing NIR continuum emis-\nsion. This scenario was proposed for HD 100453, HD 179218,\nand HD 141569 (Klarmann et al. 2017; Kluska et al. 2018;\nGRA VITY Collaboration et al. 2021a). Polycyclic aromatic\nhydrocarbons (PAHs), as an example of QHPs, are detected in\nArticle number, page 11 of 45A&A proofs: manuscript no. aa46926-23_corr\nHD 98922 (Geers et al. 2007; Acke et al. 2010). Interestingly,\nthe 6.2-to-11.3 µm feature ratio is estimated from Seok & Li\n(2017)2to be I6.2/I11.3∼3–4, with ratios larger than unity\npointing at predominantly unshielded ionized PAH species in\nthe disk inner regions that are directly exposed to the intense\nUV radiation field of the star (Maaskant et al. 2014). We tested\na radiative transfer model with a PAH-based inner component\n(Model Q, Table 5), which allows a good fit of the SED with a\nvery small mass of QHP grains (2 ×10−12M⊙) and a transition\nradius of 3.5 au. This value of R tis estimated as the balance\nbetween the increasing contribution of the outer high-density\ncomponent (HDC) to the NIR SED (for R t<3.5 au) and the\nincreasing contribution of the inner low-density component\n(LDC) to the SED at λ∼1.5µm (for R t>3.5 au and due to the\nblue spectral index of the QHP emission). However, in the\ncase of Model Q, the flux fractional contribution of the LDC\nis estimated to be only ∼5% in the K-band and 10% in the\nH-band.\nUltimately, a mixture of refractory grains and QHPs could\nexplain the presence of continuum emission as close as 0.6 au\nand detected in the H-band. The presence of predominantly\nionized PAH species close to the star is therefore not to be\ndiscarded and requires further work.\n6. Discussion\n6.1. The origin of the time-variable inner disk asymmetry\nOur analysis suggests a crescent-like asymmetric dust feature\nin the inner 1 au region of HD 98922 for which orbital motion\nis detected. Comparable azimuthal asymmetries are observed in\nother disks both at small and large scales: Varga et al. (2021)\nand GRA VITY Collaboration et al. (2021b) characterized a time-\nvariable crescent-like feature in the MIR and NIR in the in-\nnermost disk of HD 163296, whereas Ibrahim et al. (2023) di-\nrectly imaged the complex asymmetric and time-variable inner\nrim of HD 190073 at similar wavelengths. Using ALMA, Casas-\nsus et al. (2015) and van der Marel et al. (2013) reveal a strong\nazimuthal asymmetry at separations of >50 au in HD 142527\nand Oph IRS 48, respectively. Infrared and submillimeter obser-\nvations trace micron-sized and millimeter-sized dust grains, re-\nspectively, and vortices or e fficient dust traps are tentatively in-\nvoked for these sources.\nRegarding HD 98922, it is remarkable that Caratti o Garatti et al.\n(2015) also detect large-scale disk asymmetry at ∼40 au from\nthe central star. Imaged with SINFONI in the K-band, the emis-\nsion is associated with scattered light, and the authors exclude\nthe possibility that the arc-shaped structure is due to the pres-\nence of a possible stellar companion as close as ∼20 au to the\ncentral source and with a mass of ≥0.5 M⊙. Here, we explore a\nfew hypotheses as to the nature and dynamical evolution of the\nasymmetric structure in the innermost disk of HD 98922.\n6.1.1. Disk hydrodynamic instabilities\nInclination projection e ffects as in the case of FS CMa (Hofmann\net al. 2022; Kluska et al. 2020) are very unlikely to be able to ex-\nplain the azimuthal asymmetry in HD 98922. The strength of the\nclosure phase signal —despite the low disk inclination ( ≲30◦)—\nand the detected orbital motion suggest a physical e ffect in the\ndisk, possibly resulting from hydrodynamic instabilities.\nLocal perturbations in the inner disk may explain the variable\n2The ratio is uncertain due to incomplete wavelength coverageazimuthal asymmetry under the assumption of strong dust–gas\ncoupling. The Stokes parameter quantifies the degree of coupling\nand is given by Birnstiel et al. (2010):\nSt=ρs.a\nΣgπ\n2. (10)\nThe condition for dust–gas coupling in the Epstein regime,\nnamely St ≲1, is fulfilled for grain sizes of a≲2Σg/πρs. Consid-\nering a gas surface density at 1 au of 0.6 g cm−2(see Fig. 6) and\na bulk density of silicate-rich grains of ρs∼3 g.cm−3(Pollack\net al. 1994), we obtain a grain size of a≲1300µm, which sug-\ngests e fficient dust–gas coupling for most grain sizes assumed\nin our model CS (see Table 5). Under this hypothesis, a vortex\nresulting from disk instabilities could result in the observed\nover-density. The vortex might be formed through Rossby\nwave instability (RWI, Meheut et al. 2010), which could be\ninduced by the presence of a Jupiter-mass planetary companion.\nRWI-induced vortices occur preferentially in inviscid disks\n(typicallyα≲10−4) associated to a low-turbulence environment,\nwhich allows the vortex to survive for thousands of orbits. It is\ntherefore highly unlikely that we trace within the time baseline\nof our observations two unrelated over-densities that may have\ndeveloped in di fferent regions of the disk.\nRegarding the morphology of the crescent-like feature,\nboth Meheut et al. (2010) and Barge et al. (2017) show the\nformation of a vortex on one side of the disk, although the\nlatter authors explore this aspect in the outer disk, at 60 au. The\nlength /width aspect ratio of the simulated vortex in Meheut\net al. (2010) and Varga et al. (2021) appears to be large, with\nthe vortex covering at least one-third of the circumference.\nFrom our interferometric K-band observations, we model a\ncrescent-like structure with an azimuthal span comparable to\nthese authors (cf. Fig. 2). In Barge et al. (2017), the vortex\nappears azimuthally more compact. The latter authors include\ndust grains in their simulations and show that the contrast of the\nover-density is lowest for the smallest grains, with a value of\n∼10 for a size of 42 µm in the dust-density map. These latter\nauthors report a fainter contrast of ∼1.3 in the gas-density map\nfor which no dust is included. Similarly, these levels of contrast\ncompare well with the values derived from our geometrical\nmodeling (see Sect. 3.3).\nAn azimuthally asymmetric feature can also arise from\ndensity waves due to magnetorotational turbulence (Flock et al.\n2017). The observational signatures developed in this scenario\nalso include a vortex visible in the form of an arc that develops\nat the edge of the dead zone, with a brightness contrast of\ncomparable magnitude to that in the case of RWI.\nIt is noteworthy that, while the flux properties of the crescent-\nlike feature are derived from a pure geometrical modeling of our\ninterferometric data, the radiative transfer simulations predict\nan inner disk essentially optically thin at 2.2 µm due to the low\nsurface density. It would therefore be interesting to include a\nnonaxisymmetric model in future radiative transfer simulations\nin order to better constrain the influence of the grain properties\nand size on the azimuthal NIR contrast of the crescent-like\nfeature.\n6.1.2. Disk warp induced by a close companion\nAn alternative scenario that can explain our observations is the\npresence of a close companion inside the cavity of what then be-\ncomes a circumbinary (CB) disk. Ragusa et al. (2017) show from\nArticle number, page 12 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\n3D SPH gas and dust simulations that a crescent-like asymme-\ntry develops at the CB disk inner edge only (and not further out\nin the disk) due to the dynamics of the central pair. The con-\ntrast of the nonaxisymmetric feature depends on the mass ratio\nof the components q, with q⩾0.05 needed to produce a su ffi-\nciently contrasted over-density. This would translate in the case\nof HD 98922 into a companion mass of at least ∼0.3 M⊙at a sep-\naration of ≲1 mas.\nBaines et al. (2006) report a spectro-astrometric companion can-\ndidate with a separation of between 0.5′′and 3′′—that is, wider\nthan 325 au—, which is not reported by Garufi et al. (2022) from\nthe SPHERE total intensity images. Furthermore, the RUWE pa-\nrameter reported from Gaia measurements is 0.94, which means\nthat a single-star model provides, in principle, a good fit to\nthe astrometric observations. Finally, a comparison of the pho-\ntospheric lines obtained in 2005 and 2013 with the spectro-\ngraph /spectroplarimeter ESPaDOnS3between 0.3 and 1 µm does\nnot reveal any clear sign of variability in the profile or position\nof the line down to ∼10 km s−1(Alecian et al., private commu-\nnication). With the realistic hypothesis that the period of such\na companion does not coincide with the time lapse between the\ntwo ESPaDOnS epochs, we propose that no companion more\nmassive than 0.3 M ⊙at 0.1 au or 0.6 M ⊙at 0.5 au is seen in the\ninner cavity. Of course, the low inclination of the system and the\nlimited number of epochs mean that this preliminary conclusion\nrequires further strengthening.\nWhile the close binarity status of HD 98922 —in particular for\ncompanions lighter than 0.3 M ⊙— cannot be totally excluded in\nthe existing literature, we do not favor the idea that the presence\nof an inner stellar companion is inducing the crescent-like asym-\nmetry.\n6.1.3. Dynamics of the asymmetric feature\nThanks to our 11 year temporal baseline, we suggest that we\nare tracing the same azimuthal asymmetric feature possibly in-\nterpreted as a vortex in the inner disk at ∼1.0 au. One hypothe-\nsis resulting from our work is that the bright over-density could\nshow a sub-Keplerian motion with a period of ∼1 year. We em-\nphasize however that this hypothesis requires further assessment\nin the future, since our disk data modeling only accounts for cir-\ncular motion. Gas in protoplanetary disks is expected to orbit\nat sub-Keplerian rotation due to a radial pressure gradient with\na deviation from Keplerian rotation of η∝c2\ns/v2\nK. Even assum-\ning gas temperatures of a few thousand Kelvin at ∼1.0 au, the\ndeviation factor is η∼10−3, and would therefore not explain the\ndifference we find between Keplerian and sub-Keplerian motion.\nAlthough the scenario of a close binary inside the disk cavity is\nnot favored (Sect. 6.1.2), this is a case where more significant\nsub-Keplerian orbital velocity could arise. Ragusa et al. (2020)\nsuggest that disk eccentricity due to the presence of a companion\ncan result in the nesting of ellipses with a di fferent eccentricity\nleading to a slowly precessing over-density through which the\ndisk material passes. The caveat, in this case, is that the preces-\nsion timescale of the over-density is of the order of ∼100-1000\norbits, which should have resulted in a very steady feature con-\nsidering our temporal baseline.\nA more qualitative discussion concerns the structures formed\nin the disk by the presence of a planetary-mass companion. It\nis well known that the perturbing planet develops a pattern of\nwound spirals on either side of it (e.g., Kley 1999). The spiral\narms are corotating with the planet, or, in other words, they are\n3https: //www.cfht.hawaii.edu /Instruments /Spectroscopy /Espadons /stationary in the reference frame of the planet. If the crescent-\nlike feature in HD 98922 were associated with such a compan-\nion, this would lie at ∼1.8 au to match the orbital period of 12\nmonths, considering the literature value of 6 M ⊙for the mass of\nthe central star.\nWe underline that the question of the dynamics of asymmetric\nfeatures in the innermost disk regions will become an important\nquestion for future interferometric observations. It is remarkable\nto note that in their recent imaging campaign of HD 190073,\nIbrahim et al. (2023) present evidence that the detected sub-AU\nstructure rotates two times slower than Keplerian, pointing at dy-\nnamics e ffects from the outer disk.\n6.2. Kinematics of the hydrogen hot gas: Wind or\ncompanion?\nThe origin of the hydrogen Br γ-line in the system is debated:\nprevious works suggest various scenarios such as a stellar wind,\nan X-wind, or magnetospheric accretion (Kraus et al. 2008),\nwith the latter case being favored by these authors. On the\nother hand, strong variability over a timescale of a few days is\ninstead detected in the blueshifted absorption lobe of the Na I D\nand Balmer lines, which typically probes the inner dust-free\ncavity region (Aarnio et al. 2017). The blueshifted component\ncan be modeled with a disk wind extending from 0.17 au\nto 0.42-0.85 au and a wind mass-loss rate of 10−7M⊙yr−1.\nSimilarly, a disk-wind model extending from ∼0.1 au to∼1 au\nwith a wind mass-loss rate of 2 ×10−7M⊙along with an\nasymmetric continuum disk model can be successfully fitted to\nVLTI /AMBER Br γ-line interferometric data (Caratti o Garatti\net al. 2015), which is in agreement with our continuum results.\nAnother interpretation involves the presence of an accreting\nvery low-mass or planetary-mass companion. It is interesting\nto note that not only does the location of the compact Br γ-line\nemitting region change through the epochs, but so does the\nline luminosity. In the scenario of a young embedded planet,\nvariability of the line luminosity resulting from changes in the\naccretion rates can be seen during the orbital motion of the\nplanet (Szulágyi & Ercolano 2020).\nBy integrating the spectral line region through the observed\nwavelength-calibrated and continuum-normalized spectrum\n(Fig. A.3), we calculated the observed Br γ-line equivalent width\n(Wobs\nBrγ) for each epoch, from which the line equivalent width\ncorrected for photospheric absorption and veiling ( Wcorr\nBrγ) was\nderived (Table 7). Here, we discard the first two epochs G1\nand G2 where the line is strongly a ffected by the continuum\nnormalization and is therefore not highly reliable. We derive the\nflux density of the line FBrγin erg /sec/cm2by multiplying the\nspectral flux density of the nearby continuum,\nFcont,λ=F0,K10−0.4(mK−0.1AV), (11)\nwith the corrected equivalent width Wcorr\nBrγ. The K band zero-point\nflux is F0,K=4.28×10−11erg/s/cm2/Å (Rodrigo & Solano 2020).\nFinally, we compute the line luminosity as LBrγ=4πd2FBrγ.\nIn Table 7, we see that in the four-month period from March\n2019 (G4) to July 2019 (G8), the line luminosity LBrγis rel-\natively constant at ∼3.3×10−2L⊙, while after 6 months (G9) it\nincreases by∼10% and does not vary significantly for 11 months\n(G11), after which it increases again by ∼5% (G12) and again by\n∼4% in the following week (G13), for a total increase of ∼20%.\nWe note here that the values listed in Table 7 do not follow the\nknown linear relationship between equivalent width and peak\nArticle number, page 13 of 45A&A proofs: manuscript no. aa46926-23_corr\nTable 7. Properties of the Br γ-line emission. The relation between Wobs\nBrγ\nand Wcorr\nBrγfollows Eq. 2 of Grant et al. (2022).\nEpoch Wobs\nBrγWcorr\nBrγLBrγ\n[Å] [Å] [10−2L⊙]\nG4−1.69±0.06−2.90±0.06 3.34±0.11\nG5−1.56±0.15−2.77±0.15 3.19±0.19\nG6−1.63±0.13−2.84±0.13 3.27±0.17\nG7−1.59±0.24−2.80±0.24 3.23±0.29\nG8−1.59±0.07−2.80±0.07 3.23±0.12\nG9−1.93±0.11−3.14±0.11 3.62±0.16\nG10−1.96±0.15−3.17±0.15 3.65±0.20\nG11−1.97±0.14−3.18±0.14 3.67±0.19\nG12−2.14±0.10−3.35±0.10 3.86±0.16\nG13−2.32±0.10−3.53±0.10 4.07±0.16\nflux: W=A·Fpeak+B. This could be due to the fact that our\nWobs\nBrγvalues are measured by integrating the spectral line region\nusing the data themselves and not by integrating a Gaussian pro-\nfile fitted to the data points. Additionally, the relative errors on\nWandFpeakmay di ffer depending on the value of B. In any case,\nthe key insight captured by the data is that the variation in both\npeak intensity and in Wobs\nBrγis a positive one.\nUnder the hypothesis of an accreting planetary-mass compan-\nion, we use the empirical relation of Table 3 from Szulágyi &\nErcolano (2020) that relates the Br γ-line luminosity to the mass\nof the accreting planet Mp:\nlog LBrγ\nL⊙!\n=a·Mp+b, (12)\nwhere the coe fficients aandbdepend on the opacity model. Con-\nsidering the di fferent opacity cases, we derive a mass Mpranging\nbetween 10.3 MJand 12.6 MJ.\nA number of caveats must nevertheless be considered. First,\nthe value of Mpshould be treated as a loose estimate because\nthe simulation from Szulágyi & Ercolano (2020) accounts for a\nmodel with a central stellar mass of 1 M ⊙, a circumstellar envi-\nronment extending from 2 to 12.4 au, and a planet at 5.2 au. Fur-\nthermore, in this scenario, we make the implicit assumption that\nall the emission line flux is associated with the putative accreting\ncompanion, whereas a stellar component (e.g., in the form of a\nwind) may also contribute to the emission budget. In this sce-\nnario, the retrieved positions of the gas photocenter could result\nfrom the combined and weighted contributions from a stellar-\ndriven component and a companion-driven component. Based on\nthis argument, if it is a companion located further out at 1.8 au\nthat causes the azimuthal asymmetry (see Sect. 6.1.3), a suit-\nable weighting between the stellar-driven and companion-driven\nemission contributions could explain why the measured location\nof the combined photocenter is found between the star and the\ncrescent-like feature. It is not the goal of this study to explore\nthe case of a possible disk-embedded companion and its impact\non the size, location, and dynamics of the Br γ-line-emitting re-\ngion in detail; nonetheless, we speculate that a young accret-\ning planetary-mass companion is a possible scenario to consider,\nwith further modeling required.\n7. Summary\nWe present new multi-epoch VLTI /GRA VITY observations of\nHD 98922 which, coupled with VLTI /PIONIER archival data,\nform an 11 year observational-period data set for the system,accounting for a total of 33 di fferent epochs between 2011 and\n2022. This data set allows a unique interferometric study to test\nthe potential time variability of the innermost circumstellar mor-\nphology. We can summarize the main conclusions of our work\nas follows:\n–The system, which is spatially resolved by both instru-\nments at all epochs, shows temporal variability dominated by\nchanges in the asymmetric spatial distribution of brightness\nrather than in the flux ratio between the central star and its\ncircumstellar environment. This is supported by the signifi-\ncant variability in the nonzero closure phase signals for ob-\nservations obtained with comparable (u-3)coverage, whereas\nlittle time variability is observed in the continuum squared\nvisibilities.\n–Among the di fferent modeling scenarios that we tested, the\ndata are best explained by a crescent-like asymmetric dust\nfeature radially extending from ∼0.6 to∼2 au, which dom-\ninates the NIR excess. The revealed disk feature revolves\naround the central star with an estimated period of ∼1 year.\nThis could point to sub-Keplerian orbital motion, which is\nalso seen in the recently characterized system HD 190073;\nhowever, a shorter period cannot be excluded.\n–The innermost location of the warm emitting dust traced\nby the interferometric data ( ∼0.6 au) is coupled to revised\nradiative transfer models that suggest a radially structured\ndisk formed by an inner low-density component (optically\nthin at 2.2 µm) and an outer high-density component with\na transition between 3 to 5 au. The models favor either an\ninner region with a mass of 2 ×10−9M⊙made up of large,\ncarbon-enriched silicate grains, or the presence of very small\nstochastically heated particles. The high dust temperature re-\nquires the presence of some level of refractory dust.\n–The origin of the azimuthal asymmetry is preferentially con-\nnected to hydrodynamic e ffects, either induced by disk in-\nstabilities generating a vortex or by an undetected low-mass\n(substellar or planetary-mass) companion embedded in the\nlow-density region of the disk and launching a wound spi-\nral. An asymmetry resulting in disk fragmentation is unlikely\nconsidering the low surface density of ≲10 g cm−2, and we\ndo not find evidence for the presence of a close (0.1–0.5 au)\nstellar companion inside the inner dust rim that could drive\neccentricity in the central cavity resulting in a crescent-like\nstructure. However, we do believe that our data cannot com-\npletely rule out this hypothesis, in particular regarding a stel-\nlar companion less massive than ∼0.3 M⊙.\n–The high-resolution GRA VITY data show a single-peaked\nBrγemission line with a luminosity increasing by 20 % in 3\nyears. The location of the compact Br γ-line-emitting region\nis offset with respect to the central star and appears to be\nconsistently located inside the dust cavity and between the\ndusty feature and the star.\n–The interpretation of the interferometric observations of the\nhot, gaseous component in HD 98922 leaves some open\nquestions: while a stellocentric magnetospheric accretion\nscenario is not favored, a wider asymmetric disk wind is a\nqualitatively plausible scenario, though more advanced nu-\nmerical simulations would be needed to explain the time\nvariability in its spatial distribution. We also speculate that\na strongly accreting substellar or planetary companion with\na mass of larger than ∼10 M Jcould explain our measure-\nment of the Br γemission line. However, this would trigger\nfurther questions regarding its dynamics and connection to\nthe detected crescent-like structure in the inner disk.\nArticle number, page 14 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nAcknowledgements\nThe authors would like to thank Héloïse Meheut and François Mé-\nnard for the fruitful discussions on the inner disk of HD 98922. V .G.\nwas supported for this research through a stipend from the Interna-\ntional Max Planck Research School (IMPRS) for Astronomy and As-\ntrophysics at the Universities of Bonn and Cologne, and from the Bonn-\nCologne Graduate School of Physics and Astronomy (BCGS). A.C.G.\nhas been supported by PRIN-INAF MAIN-STREAM 2017 and PRIN-\nINAF 2019 (STRADE), P. This work is based on observations made\nwith ESO Telescopes at the La Silla Paranal Observatory under program\nIDs listed in Table A.1 and Table A.2. This work has made use of data\nfrom the European Space Agency (ESA) mission Gaia (https://www.\ncosmos.esa.int/gaia ), processed by the Gaia Data Processing and\nAnalysis Consortium (DPAC, https://www.cosmos.esa.int/web/\ngaia/dpac/consortium ). Funding for the DPAC has been provided\nby national institutions, in particular the institutions in the Gaia Mul-\ntilateral Agreement. We acknowledge the Gemini Observatory for the\nuse of the IR spectrum model of the atmospheric transmission above\nCerro Pachon. This research has made use of the model atmosphere\ngrid NeMo, provided by the Department of Astronomy of the Univer-\nsity of Vienna, Austria ( http://www.univie.ac.at/nemo/ ). NeMo\nwas funded by the Austrian Science Fonds. This research has made use\nof the Jean-Marie Mariotti Center Aspro . This research has made use\nof the Spanish Virtual Observatory ( https://svo.cab.inta-csic.\nes) project funded by MCIN /AEI/10.13039 /501100011033 /through\ngrant PID2020-112949GB-I00. This paper makes use of the following\nALMA data: ADS /JAO.ALMA#2015.1.01600.S. ALMA is a partner-\nship of ESO (representing its member states), NSF (USA) and NINS\n(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan),\nand KASI (Republic of Korea), in cooperation with the Republic of\nChile. The Joint ALMA Observatory is operated by ESO, AUI /NRAO\nand NAOJ. The authors thank the anonymous referee for assessing the\nquality of this work.\nReferences\nAarnio, A. N., Monnier, J. D., Harries, T. J., et al. 2017, ApJ, 848, 18\nAcke, B., Bouwman, J., Juhász, A., et al. 2010, ApJ, 718, 558\nAcke, B., van den Ancker, M. E., & Dullemond, C. P. 2005, A&A, 436, 209\nArun, R., Mathew, B., Manoj, P., et al. 2019, AJ, 157, 159\nBaines, D., Oudmaijer, R. D., Porter, J. M., & Pozzo, M. 2006, MNRAS, 367,\n737\nBarge, P., Ricci, L., Carilli, C. L., & Previn-Ratnasingam, R. 2017, A&A, 605,\nA122\nBeckwith, S. V . W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924\nBeichman, C. A., Neugebauer, G., Habing, H. J., Clegg, P. E., & Chester, T. J.\n1988, Infrared Astronomical Satellite (IRAS) Catalogs and Atlases.V olume\n1: Explanatory Supplement., V ol. 1\nBenisty, M., Natta, A., Isella, A., et al. 2010, A&A, 511, A74\nBirnstiel, T., Dullemond, C. P., & Brauer, F. 2010, A&A, 513, A79\nBouwman, J., Meeus, G., de Koter, A., et al. 2001, A&A, 375, 950\nCaratti o Garatti, A., Tambovtseva, L. V ., Garcia Lopez, R., et al. 2015, A&A,\n582, A44\nCasassus, S., Wright, C. M., Marino, S., et al. 2015, ApJ, 812, 126\nCutri, R. M. & et al. 2012, VizieR Online Data Catalog, II /311\nCutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data\nCatalog, II /246\nEisner, J. A., Hillenbrand, L. A., & Stone, J. M. 2014, MNRAS, 443, 1916\nEsa, . 1997, VizieR Online Data Catalog, I /239\nFairlamb, J. R., Oudmaijer, R. D., Mendigutía, I., Ilee, J. D., & van den Ancker,\nM. E. 2015, MNRAS, 453, 976\nFedele, D., van den Ancker, M. E., Acke, B., et al. 2008, A&A, 491, 809\nFlock, M., Fromang, S., Turner, N. J., & Benisty, M. 2017, ApJ, 835, 230\nForeman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125,\n306\nGaia Collaboration. 2020, VizieR Online Data Catalog, I /350\nGaia Collaboration. 2022, VizieR Online Data Catalog, I /355\nGaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1\nGaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1\nGarufi, A., Benisty, M., Pinilla, P., et al. 2018, A&A, 620, A94\nGarufi, A., Dominik, C., Ginski, C., et al. 2022, A&A, 658, A137\nGeers, V . C., van Dishoeck, E. F., Visser, R., et al. 2007, A&A, 476, 279\nGrady, C. A., Perez, M. R., Talavera, A., et al. 1996, A&AS, 120, 157Grant, S. L., Espaillat, C. C., Brittain, S., Scott-Joseph, C., & Calvet, N. 2022,\nApJ, 926, 229\nGRA VITY Collaboration, Abuter, R., Accardo, M., et al. 2017, A&A, 602, A94\nGRA VITY Collaboration, Ganci, V ., Labadie, L., et al. 2021a, A&A, 655, A112\nGRA VITY Collaboration, Perraut, K., Labadie, L., et al. 2019, A&A, 632, A53\nGRA VITY Collaboration, Sanchez-Bermudez, J., Caratti O Garatti, A., et al.\n2021b, A&A, 654, A97\nGravity Collaboration, Wojtczak, J. A., Labadie, L., et al. 2023, A&A, 669, A59\nGray, R. O. & Corbally, C. J. 1994, AJ, 107, 742\nGuarcello, M. G., Flaccomio, E., Micela, G., et al. 2019, A&A, 628, A74\nGuzmán-Díaz, J., Mendigutía, I., Montesinos, B., et al. 2021, A&A, 650, A182\nHales, A. S., De Gregorio-Monsalvo, I., Montesinos, B., et al. 2014, AJ, 148, 47\nHauck, B. & Mermilliod, M. 1998, A&AS, 129, 431\nHayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35\nHeiter, U., Kupka, F., van’t Veer-Menneret, C., et al. 2002, A&A, 392, 619\nHelou, G. & Walker, D. W. 1988, Infrared Astronomical Satellite (IRAS) Cata-\nlogs and Atlases: The Small Scale Structure Catalog, V ol. 7\nHofmann, K. H., Bensberg, A., Schertl, D., et al. 2022, A&A, 658, A81\nHøg, E., Fabricius, C., Makarov, V . V ., et al. 2000, A&A, 355, L27\nHone, E., Kraus, S., Kreplin, A., et al. 2017, A&A, 607, A17\nHuang, J., Andrews, S. M., Dullemond, C. P., et al. 2018, ApJ, 869, L42\nIbrahim, N., Monnier, J. D., Kraus, S., et al. 2023, arXiv e-prints,\narXiv:2302.06500\nIshihara, D., Onaka, T., Kataza, H., et al. 2010, A&A, 514, A1\nJuhász, A., Bouwman, J., Henning, T., et al. 2010, ApJ, 721, 431\nKama, M., Min, M., & Dominik, C. 2009, A&A, 506, 1199\nKlarmann, L., Benisty, M., Min, M., et al. 2017, A&A, 599, A80\nKley, W. 1999, MNRAS, 303, 696\nKluska, J., Benisty, M., Soulez, F., et al. 2016, A&A, 591, A82\nKluska, J., Berger, J. P., Malbet, F., et al. 2020, A&A, 636, A116\nKluska, J., Kraus, S., Davies, C. L., et al. 2018, ApJ, 855, 44\nKobus, J., Wolf, S., Ratzka, T., & Brunngräber, R. 2020, A&A, 642, A104\nKóspál, Á., Ábrahám, P., Acosta-Pulido, J. A., et al. 2012, ApJS, 201, 11\nKraus, S., Hofmann, K. H., Benisty, M., et al. 2008, A&A, 489, 1157\nLachaume, R. 2003, A&A, 400, 795\nLapeyrere, V ., Kervella, P., Lacour, S., et al. 2014, in Society of Photo-Optical\nInstrumentation Engineers (SPIE) Conference Series, V ol. 9146, Proc. SPIE,\n91462D\nLazare ff, B., Berger, J. P., Kluska, J., et al. 2017, A&A, 599, A85\nLe Bouquin, J.-B., Berger, J.-P., Lazare ff, B., et al. 2011, A&A, 535, A67\nLecar, M., Podolak, M., Sasselov, D., & Chiang, E. 2006, ApJ, 640, 1115\nLee, C.-D., Chen, W.-P., & Liu, S.-Y . 2016, A&A, 592, A130\nLi, A. & Lunine, J. I. 2003, ApJ, 594, 987\nLord, S. D. 1992, A new software tool for computing Earth’s atmospheric trans-\nmission of near- and far-infrared radiation, NASA Technical Memorandum\n103957\nMaaskant, K. M., Min, M., Waters, L. B. F. M., & Tielens, A. G. G. M. 2014,\nA&A, 563, A78\nMalfait, K., Waelkens, C., Waters, L. B. F. M., et al. 1998, A&A, 332, L25\nMatter, A., Labadie, L., Augereau, J. C., et al. 2016, A&A, 586, A11\nMeeus, G., Waters, L. B. F. M., Bouwman, J., et al. 2001, A&A, 365, 476\nMeheut, H., Casse, F., Varniere, P., & Tagger, M. 2010, A&A, 516, A31\nMenu, J., van Boekel, R., Henning, T., et al. 2015, A&A, 581, A107\nMin, M., Dullemond, C. P., Dominik, C., de Koter, A., & Hovenier, J. W. 2009,\nA&A, 497, 155\nMonnier, J. D., Millan-Gabet, R., Billmeier, R., et al. 2005, ApJ, 624, 832\nMuzerolle, J., D’Alessio, P., Calvet, N., & Hartmann, L. 2004, ApJ, 617, 406\nMyers, J. R., Sande, C. B., Miller, A. C., Warren, W. H., J., & Tracewell, D. A.\n2015, VizieR Online Data Catalog, V /145\nOudmaijer, R. D., van den Ancker, M. E., Baines, D., et al. 2011, Astronomische\nNachrichten, 332, 238\nPollack, J. B., Hollenbach, D., Beckwith, S., et al. 1994, ApJ, 421, 615\nRagusa, E., Alexander, R., Calcino, J., Hirsh, K., & Price, D. J. 2020, MNRAS,\n499, 3362\nRagusa, E., Dipierro, G., Lodato, G., Laibe, G., & Price, D. J. 2017, MNRAS,\n464, 1449\nRice, T. S., Reipurth, B., Wolk, S. J., Vaz, L. P., & Cross, N. J. G. 2015, AJ, 150,\n132\nRobinson, C. E. & Espaillat, C. C. 2019, ApJ, 874, 129\nRodrigo, C. & Solano, E. 2020, in XIV .0 Scientific Meeting (virtual) of the Span-\nish Astronomical Society, 182\nSeok, J. Y . & Li, A. 2017, ApJ, 835, 291\nSetterholm, B. R., Monnier, J. D., Davies, C. L., et al. 2018, ApJ, 869, 164\nSzulágyi, J. & Ercolano, B. 2020, ApJ, 902, 126\nTatulli, E., Benisty, M., Ménard, F., et al. 2011, A&A, 531, A1\nvan Boekel, R., Waters, L. B. F. M., Dominik, C., et al. 2003, A&A, 400, L21\nvan der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2013, Science, 340,\n1199\nvan der Plas, G., van den Ancker, M. E., Waters, L. B. F. M., & Dominik, C.\n2015, A&A, 574, A75\nArticle number, page 15 of 45A&A proofs: manuscript no. aa46926-23_corr\nvan Leeuwen, F. 2007, A&A, 474, 653\nVarga, J., Hogerheijde, M., van Boekel, R., et al. 2021, A&A, 647, A56\nVerhoe ff, A. P., Min, M., Acke, B., et al. 2010, A&A, 516, A48\nVioque, M., Oudmaijer, R. D., Baines, D., Mendigutía, I., & Pérez-Martínez, R.\n2018, A&A, 620, A128\nWeigelt, M., Baur, O., Reubelt, T., Sneeuw, N., & Roth, M. 2011, in ESA Special\nPublication, V ol. 696, 4th International GOCE User Workshop, 36\nWoitke, P., Min, M., Pinte, C., et al. 2016, A&A, 586, A103\nWolk, S. J., Günther, H. M., Poppenhaeger, K., et al. 2018, AJ, 155, 99\nYamamura, I., Makiuti, S., Ikeda, N., et al. 2010, VizieR Online Data Catalog,\nII/298\n1I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77,\n50937, Köln, Germany\ne-mail: ganci@ph1.uni-koeln.de\n2Max-Planck-Institute for Radio Astronomy, Auf dem Hügel 69,\n53121 Bonn, Germany\n3Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France\n4Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidel-\nberg, Germany\n5School of Physics, University College Dublin, Belfield, Dublin 4,\nIreland\n6INAF – Osservatorio Astronomico di Capodimonte, via Moiariello\n16, 80131 Napoli, Italy\n7Max Planck Institute for Extraterrestrial Physics, Giessenbach-\nstrasse, 85741 Garching bei München, Germany\n8Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Lei-\nden, The Netherlands\n9Instituto de Astronomía, Universidad Nacional Autónoma de Méx-\nico, Apdo. Postal 70264, Ciudad de México 04510, Mexico\n10LESIA, Observatoire de Paris, PSL Research University, CNRS,\nSorbonne Universités, UPMC Univ. Paris 06, Univ. Paris\nDiderot,Sorbonne Paris Cité, France\n11European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748,\nGarching bei München, Germany\n12CENTRA – Centro de Astrofísica e Gravitação, IST, Universidade\nde Lisboa, 1049-001 Lisboa, Portugal\n13Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto\nFrias, s /n, 4200-465 Porto, Portugal\n14Universidade de Lisboa – Faculdade de Ciências, Campo Grande,\n1749-016 Lisboa, Portugal\nArticle number, page 16 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nAppendix A: Logs and observation data\nArticle number, page 17 of 45A&A proofs: manuscript no. aa46926-23_corr\nTable A.1. Observation logs of the VLTI /PIONIER HD 98922 observations\nID Date UT Configuration N Calibrator Seeing [′′] Airmass τ0[ms] ID obs\nP1 03-06-2011 00:21 D0-G1-H0-I1 4 NDA 0.5-0.9 1.17-1.26 1.6-2.9 087.C-0458(C)\nP2 08-06-2011 23:28 A1-B2-C1-D0 2 NDA 0.8-1.3 1.15-1.16 2.0-2.7 087.C-0458(B)\nP3 25-03-2012 04:50 A1-G1-I1-K0 2 NDA 0.5-0.9 1.16-1.38 2.6-4.8 088.D-0828(B)\nP4 28-03-2012 05:22 A1-G1-I1-K0 2 NDA 1.4-1.6 1.22-1.28 1.3-1.4 088.D-0185(A)\nP5 20-12-2012 06:47 A1-B2-C1-D0 2 NDA 0.5-0.6 1.38-1.47 8.1-8.5 190.C-0963(C)\nP6 22-12-2012 09:13 A1-B2-C1-D0 1 NDA 0.8-0.9 1.15-1.16 8.1-9.5 190.C-0963(C)\nP7 26-01-2013 08:48 A1-G1-J3-K0 1 NDA 1.2-1.5 1.16-1.51 1.1-1.2 190.C-0963(A)\nP8 27-01-2013 07:11 A1-G1-J3-K0 2 NDA 0.9-1.0 1.14-1.15 2.7-2.8 190.C-0963(A)\nP9 28-01-2013 08:39 A1-G1-J3-K0 3 NDA 0.8-1.5 1.17-1.28 1.5-3.3 190.C-0963(A)\nP10 30-01-2013 07:25 A1-G1-J3-K0 1 NDA 1.6-1.3 1.30-1.80 1.5-1.6 190.C-0963(C)\nP11 31-01-2013 03:39 A1-G1-J3-K0 2 NDA 0.9-1.1 1.15-1.16 1.8-2.3 190.C-0963(A)\nP12 01-02-2013 04:35 A1-G1-J3-K0 3 NDA 0.8-1.1 1.18-1.34 2.0-3.4 190.C-0963(A)\nP13 17-02-2013 01:47 D0-G1-H0-I1 3 NDA 0.7-0.8 1.19-1.22 3.7-4.4 190.C-0963(B)\nP14 18-02-2013 03:27 D0-G1-H0-I1 4 NDA 0.5-0.9 1.14-1.52 3.2-5.4 190.C-0963(B)\nP15 19-02-2013 08:20 D0-G1-H0-I1 1 NDA 0.7-0.9 1.26-1.27 3.1-3.9 190.C-0963(B)\nP16 20-02-2013 02:36 D0-G1-H0-I1 2 NDA 0.7-1.0 1.41-1.49 3.8-5.2 190.C-0963(B)\nP17 23-06-2014 22:57 A1-G1-J3-K0 2 HD 98895 0.5-0.8 1.17-1.19 1.9-2.6 093.C-0559(D)\nP18 21-02-2016 06:08 D0-G2-J3-K0 1 HD 98895 1.4-1.9 1.13-1.14 0.9-1.4 096.C-0867(C)\nP19 01-03-2016 05:44 D0-G2-J3-K0 2 HD 98895 1.0-1.7 1.14-1.16 1.7-2.9 096.C-0867(D)\nP20 02-03-2016 06:18 D0-G2-J3-K0 2 HD 98895 0.8-1.0 1.16-1.20 1.9-2.3 096.C-0867(E)\nNotes. The date format is day-month-year. N denotes the number files that have been recorded on the target. NDA: no data available; These\narchival data were retrieved already calibrated from the JMMC Optical interferometry DataBase ( http://oidb.jmmc.fr/index.html ).\nTable A.2. Observation logs of the VLTI /GRA VITY HD 98922 observations\nID Date UT Configuration N Calibrator Seeing [′′] Airmass τ0[ms] ID obs\nG1 22-02-2017 07:13 A0-G1-J2-K0 [astro.] 14 HD 103125 0.7-1.7 1.18-1.70 NDA 098.C-0765(C)\nG2 19-03-2017 04:36 A0-G1-J2-K0 [astro.] 5 HD 100825 0.7-0.9 1.14-1.17 NDA 098.D-0488(A)\nG3 15-06-2018 00:09 D0-G2-J3-K0 [medium] 4 HD 100825 0.8-1.2 1.21-1.30 2.3-3.2 0101.C-0311(A)\nG4 19-03-2019 05:45 D0-G2-J3-K0 [medium] 6 HD 100825 0.5-0.6 1.20-1.27 3.5-5.9 0102.C-0408(D)\nG5 24-05-2019 01:45 A0-G1-J2-J3 [large] 6 HD 103125 0.6-0.9 1.22-1.29 1.8-3.2 0103.C-0347(C)\nG6 04-06-2019 23:21 A0-B2-C1-D0 [small] 7 HD 103125 0.8-1.1 1.14-1.16 2.8-4.1 0103.C-0347(B)\nG7 11-07-2019 23:22 D0-G2-J3-K0 [medium] 5 HD 103125 0.8-1.1 1.33-1.42 2.0-3.0 0103.C-0347(A)\nG8 13-07-2019 22:56 D0-G2-J3-K0 [medium] 7 HD 103125 0.3-0.5 1.29-1.41 1.3-1.4 0103.C-0347(A)\nG9 27-01-2020 05:56 A0-G2-J2-J3 [ ∗] 5 HD 103125 0.7-1.0 1.16-1.21 3.6-5.9 0104.C-0567(A)\nG10 04-02-2020 05:35 A0-B2-C1-D0 [small] 11 HD 103125 0.7-1.1 1.14-1.20 2.1-4.9 0104.C-0567(C)\nG11 23-12-2020 06:31 D0-G2-J3-K0 [medium] 13 HD 99311 0.7-1.9 1.20-1.47 2.1-4.6 106.212G.002\nG12 07-02-2022 07:43 D0-G2-J3-K0 [medium] 8 HD 103125 0.5-1.2 1.16-1.24 4.4-12.6 108.228Z.002\nG13 14-02-2022 06:11 A0-G1-J2-J3 [large] 6 HD 103125 NDA 1.14-1.15 NDA 108.228Z.001\nNotes. Same as A.1. The configuration names – small, medium, large, astrometric – are reported. The configuration A0-G2-J2-J3 [ ∗], not o ffered\non a regular basis, covers spatial frequencies comparable to the large configuration.\nTable A.3. Fit results per epoch for the model SHRM1 and combining the visibilities and the closure phases\nID P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14\nχ2\nr13.53 3.48 39.36 13.64 2.64 16.25 16.24 6.97 16.56 6.97 6.68 12.36 4.74 8.67\nID P15 P16 P17 P18 P19 P20 All epochs – – – – – – –\nχ2\nr 5.07 3.29 36.03 10.82 12.81 9.09 11.5 – – – – – – –\nID G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 All epochs\nχ2\nr11.41 5.0 9.96 2.53 24.29 2.89 1.19 1.04 5.54 2.42 2.79 1.14 2.36 5.30\nArticle number, page 18 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. A.1. HD 98922 PIONIER data, squared visibilities, closure phases, and (u,3)plan coverage for each epoch. Colors refer to the di fferent\nPIONIER spectral channels. Gray crosses represent the model described in Sect. 3.1 and Table 3. Gray circles in the bottom plots show the\nresiduals.\nArticle number, page 19 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. A.1. Continued.\nArticle number, page 20 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. A.1. Continued.\nArticle number, page 21 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. A.1. Continued.\nArticle number, page 22 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. A.2. HD 98922 GRA VITY FT data, squared visibilities, closure phases, and (u,3)plan coverage for each epoch. Colors refer to the di fferent\nGRA VITY spectral channels. Gray crosses represent the model described in Sect. 3.1, and Table 3. Gray circles in the bottom plots show the\nresiduals of the model fitting.\nArticle number, page 23 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. A.2. Continued.\nArticle number, page 24 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. A.2. Continued.\nArticle number, page 25 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. A.3. HD 98922 GRA VITY SC data for the di fferent epochs. For each epoch, top plots show the wavelength-calibrated and continuum-\nnormalized spectrum, left plots show the total squared visibilities, and right plots show the total di fferential phases. Circles represent the pure-line\nquantities. Colors refer to the di fferent baselines.\nArticle number, page 26 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. A.3. Continued.\nArticle number, page 27 of 45A&A proofs: manuscript no. aa46926-23_corr\nAppendix B: Global fit MCMC posterior distribution\nfunctions and azimuthal modulation parameters’\nχ2\nrmaps\nArticle number, page 28 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. B.1. Reduced chi-square χ2\nrcurves of each nontime-variable parameter from the azimuthal modulation global fit model. The first two rows,\nwhere the curves are blue, refer to the PIONIER data model, while the last two rows, where the curves are black, refer to the GRA VITY data\nmodel.\nArticle number, page 29 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. B.2. Global fit MCMC posterior distribution functions of the fitted parameters for the PIONIER data set. In the one-dimensional histograms,\nthe blue line identifies the median of the distribution, the dashed black lines identify the 16thand the 84thpercentiles.\nArticle number, page 30 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. B.3. Global fit MCMC posterior distribution functions of the fitted parameters for the GRA VITY data set. In the one-dimensional histograms,\nthe blue line identifies the median of the distribution, the dashed black lines identify the 16thand the 84thpercentiles.\nArticle number, page 31 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. B.4. Global fit azimuthal modulation parameters χ2\nrmaps. The red lines represent the 3 σerror bars of the parameters derived through the\nMCMC fitting procedure. Their intersections give the smallest χ2\nrfor a given epoch.\nArticle number, page 32 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nAppendix C: Spectrum wavelength calibration and\nstar photospheric absorption model\nThe continuum normalization of the GRA VITY science object\nspectrum of each epoch (average of all the observations blocks\nfiles that come from the four di fferent ATs for that epoch) was\ndone by fitting the slope of the raw spectrum and dividing\nthe latter by the resulting fit. The first step of the wavelength\ncalibration was done by comparing the observed wavelength\npositions of the telluric lines in the HD 98922 spectrum for\neach epoch with respect to the positions of the telluric lines\npresent in the IR spectrum of the atmospheric transmission\nabove Cerro Pachon. We used pre-computed ATRAN models\n(Lord 1992) available from the Gemini Observatory website\naccounting for 4 .3 mm water vapor column and an airmass of\n1.5. The atmospheric transmission spectrum was convolved by\na Gaussian with FHWM of 6 Å to have the same resolution as\nGRA VITY . The correction results in a blueshift between 0 and\n5 Å depending on the epoch. The HD 98922 spectrum was then\ncorrected for the star radial velocity ( −4.9 km/s) and its proper\nmotion with respect to the local standard of rest, with a value\nthat depends on the epoch of observations (between ≈−35 and\n≈2 km/s). The same corrections (telluric lines calibration, radial\nvelocity correction, and local standard of rest correction) were\nalso applied on the SC visibilities and di fferential phases for\neach epoch. The stellar atmospheric Br γabsorption was taken\ninto account through a model selected from the Vienna New\nModel Grid of Stellar Atmospheres (Heiter et al. 2002)4that best\nrepresents the star (see Table 1). The model accounts for a star\neffective temperature of 10500 K, a surface gravity logarithm of\n3.4, a metallicity of [Fe /H]=−0.5, and a microturbulence of 2.0\nkm/s, a common value for a Herbig star. Finally, we included in\nthe model the rotation broadening e ffect due to the rotation of\nthe star ( v sin i =39.0 km/s) and we convolved the final model\nby a Gaussian with FHWM of 6 Å to have the same resolution\nas GRA VITY using SPECTRUM (Gray & Corbally 1994).\nA systematic spectral shift of the line features was re-\ncently found in the GRA VITY SC data. The shift is seen when\ncomparing data from an individual telescope with respect to\nthe other telescopes, but also globally with regard to the known\nwavelength positions of the telluric lines in the K-band. It is\nthought that the shift is caused by the old grism of the science\nspectrometer which was upgraded in October 2019. GRA VITY\ndata taken after this month should therefore no longer be\naffected. To check if our results are a ffected by the shift, we\ncorrected the shift in the 19 March 2019 data following Gravity\nCollaboration et al. 2023, and compared the results with the\nones obtained without the correction. For a description of the\ncorrection we refer to Sect. 3.2 of Gravity Collaboration et al.\n2023. Figure C.1 shows the pure-line photocenter shift obtained\nwith the original data (top panel) and that obtained with the\ncorrected ones (bottom panel). We note a slight change in the\nshift when comparing photocenters of the same spectral channel.\nHowever, in this work we are not primarily interested in the\nrelative position of the photocenters for each spectral channel,\nbut rather in the overall location of the gas with respect to the\nstar and the dusty feature. This choice is also based on the fact\nthat our data were obtained with the ATs, meaning that we have\nonly 4 or 5 spectral channels across the line. Data obtained with\nthe UTs have instead around 12 spectral channels across the\n4available on the NeMo webpage (Ch. Stütz and E. Paunzen, http:\n//www.univie.ac.at/nemo/ )line, making these data more suitable for high-precision spectral\nanalysis. From this point of view, the di fferences between the\nresults obtained with the corrected and the uncorrected ATs\ndata is negligible. Therefore, we decided to not apply this\nhigh-precision spectral calibration to our data.\nFig. C.1. Difference between pure-line photocenters shift obtained with\ndata corrected with the GRA VITY data reduction software (Lapeyrere\net al. 2014, top panel), and with the Gravity Collaboration et al. (2023)\nhigh-precision spectral calibration (bottom panel). Same markers as in\nFig. 5.\nArticle number, page 33 of 45A&A proofs: manuscript no. aa46926-23_corr\nAppendix D: Photometric data\nArticle number, page 34 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nTable D.1. HD 98922 photometry data\nλ Fλ ∆Fλ Reference\n(µm) (erg cm−2s−1Å) (erg cm−2s−1Å)\n0.35 6.73×10−126.73×10−13Malfait et al. (1998)\n0.35 8.30×10−121.44×10−13Myers et al. (2015)\n0.42 1.27×10−116.79×10−14Esa (1997)\n0.50 8.05×10−122.35×10−14Gaia Collaboration et al. (2018)\n0.43 1.33×10−111.33×10−12Malfait et al. (1998)\n0.44 1.25×10−111.25×10−12Høg et al. (2000)\n0.53 7.82×10−127.42×10−14Esa (1997)\n0.55 7.17×10−123.30×10−13Malfait et al. (1998)\n0.55 7.39×10−127.39×10−13Høg et al. (2000)\n0.55 7.54×10−127.54×10−13Hauck & Mermilliod (1998)\n0.58 5.80×10−121.77×10−14Gaia Collaboration (2020)\n0.67 4.56×10−121.99×10−14Gaia Collaboration et al. (2018)\n0.76 3.04×10−121.55×10−14Gaia Collaboration (2020)\n0.77 3.01×10−121.51×10−14Gaia Collaboration et al. (2018)\n1.22 1.15×10−121.15×10−13Malfait et al. (1998)\n1.24 1.24×10−122.29×10−14Cutri et al. (2003)\n1.63 9.53×10−132.60×10−14Cutri et al. (2003)\n1.65 8.26×10−138.26×10−14Malfait et al. (1998)\n1.66 9.03×10−132.41×10−14Cutri et al. (2003)\n2.16 8.32×10−132.76×10−14Cutri et al. (2003)\n2.18 7.12×10−137.12×10−14Malfait et al. (1998)\n2.19 7.94×10−132.50×10−14Cutri et al. (2003)\n3.35 4.41×10−136.41×10−14Cutri & et al. (2012)\n3.55 4.99×10−134.99×10−14Malfait et al. (1998)\n4.60 4.35×10−135.67×10−15Cutri & et al. (2012)\n4.77 2.68×10−132.68×10−14Malfait et al. (1998)\n8.61 1.08×10−134.04×10−16Ishihara et al. (2010)\n12.0 6.34×10−146.34×10−15Beichman et al. (1988)\n18.4 1.91×10−148.86×10−17Ishihara et al. (2010)\n25.0 9.25×10−159.25×10−16Beichman et al. (1988)\n60.0 3.56×10−163.56×10−17Beichman et al. (1988)\n61.9 4.85×10−164.39×10−17Helou & Walker (1988)\n65.0 2.30×10−161.70×10−17Yamamura et al. (2010)\n70.0 2.19×10−161.10×10−17Hales et al. (2014)\n90.0 9.60×10−175.29×10−18Yamamura et al. (2010)\n140.0 1.73×10−177.98×10−18Yamamura et al. (2010)\n160.0 1.03×10−175.14×10−19Hales et al. (2014)\n1290.0 1.88×10−211.88×10−22ALMA Archive data\nArticle number, page 35 of 45A&A proofs: manuscript no. aa46926-23_corr\nAppendix E: Interferometric variability and UV\ncoverage\nAppendix E.1: Variability in the visibilities and closure phases\nOur data set on HD 98922 spans a 11 year time period for the\ncontinuum emission, which gives us a unique opportunity to\nmonitor intrinsic variability e ffects at a scale of 1au over an\nextended period of time. The time intervals between successive\nepochs range from 10 to 20 months for the PIONIER data\nset, and from 5 to 15 months for the GRA VITY one (see\nTable A.1 and Table A.2). Signatures of temporal variability in\ninterferometric data are ideally investigated with configurations\nhaving comparable (u,3)plane coverage. For our data set, this\nis illustrated in Fig. E.1, which provides a visual estimate of\ncomparable ( u,3) coverage planes; although a more accurate\nassessment requires the comparison of the full configurations in\nFig. A.1 and Fig. A.2.\nIn the top row of Fig. E.1, we report the length and PA of\nthe longest baseline for a given configuration for PIONIER\n(P) and GRA VITY (G). The nomenclature can be followed in\nTable A.1 and A.2. Following this approach, we note quali-\ntatively that a comparable ( u,3) plane coverage is found for\nthe following groups: (P1,15), (P2,6), (P3,4), (P7,8,9,10,17),\n(P11,12), (P13,14), and (P18,19,20). Depending on the group,\nthe di fference in PA of the longest baseline varies between ∼5◦\nand 20◦. For GRA VITY , the configurations with a similar ( u,3)\nplane coverage are (G3,4,7,8,12), (G6,10), and (G9,13). The\nconfigurations of the remaining epochs di ffer more strongly. The\ncentral and bottom rows of Fig. E.1 are constructed as follows:\nfirst, we visually identify for a group of similar configurations\nthe common spatial frequency for which a strong variation in\nthe closure phase (CP) signal is observed and report the CP\nvalue (Fig. E.1, central panels). For this same spatial frequency,\nwe report in the bottom panels the value of the squared visibility.\nFor those configurations with a similar (u,3)coverage, we\nare able to detect significant variations in the CP values across\nthe epochs. For instance, a clear di fference in the CP signal\nof∆ϕmax≈25◦is observed for G9 and G13, as well as in the\nmedium configuration epochs G3, G4, G7, G8, G12 where the\nCP signals vary between ϕmax≈ −7◦and≈3◦. Even though\nthe error bars are larger, the CP variability in the PIONIER\ndata set is also clearly observed. The epochs P1 and P15 show\na significant closure phase variation ( ∆ϕmax≈10◦), as does\nP17 with respect to P7, P8, P9, and P10 ( ∆ϕmax≈13◦). On the\ncontrary, when comparing observations taken with similar (u,3)\nplane coverage, we note that the variability of the corresponding\nsquared visibilities is typically smaller than V2∼0.05.\nArticle number, page 36 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. E.1. HD 98922 PIONIER and GRA VITY continuum data overview. Top row: Overview diagram summarizing Fig. A.1 and Fig. A.2 and\nshowing array configurations with comparable (u,3)coverage. Each line shows the spatial frequency and PA probed by the longest baseline of a\ngiven four-telescope configuration. Two epochs have the same color and line style if they show a similar (u,3)coverage. The gray lines refer to\nepochs for which the (u,3)coverage cannot be reasonably compared to any other. Numbers identify epochs following Table A.1 and A.2; Middle\nrow: Closure phase signal per epoch and instrument. The spatial frequency chosen is the one for which a strong variation in the CP signal between\nepochs with a comparable (u,3)plane coverage is observed. Bottom row: Squared visibilities signal for each epoch and instrument, for the same\nspatial frequency as in the middle row.\nArticle number, page 37 of 45A&A proofs: manuscript no. aa46926-23_corr\nAppendix E.2: Fit dependence on the (u,v) plane coverage\nIn the context of the study of spatial variability, we first veri-\nfied using synthetic data whether the fit of di fferent data sets re-\nlated to the same nonvariable system but obtained with di fferent\nbaseline configurations would lead to the same or to a di ffer-\nent solution. For this purpose, we generated with Aspro5three\nsynthetic data sets from an input image formed by a star and an\nazimuthally modulated ring observed with the GRA VITY small,\nmedium, and large configurations, respectively, on the 4 Febru-\nary 2020, 11 July 2019, and 14 February 2022. We then fitted\nthe di fferent synthetic data sets with the same parametric model\nand then compared the results obtained for each parameter.\nThe input model image, shown in Fig. E.2, corresponds to an az-\nimuthally modulated torus around a central star that contributes\n15% of the total flux of the system, and with no spatially resolved\nhalo contribution. The synthetic interferometric visibilities and\nclosure phases generated with Aspro are shown in Fig. E.3. The\nsame error bars as for the real data are adopted. The fit of the\nsynthetic data implements the nine freeparameters given in Ta-\nble E.1. The spectral indices of the star and the disk were not\nimplemented and were set to zero. Four cases are explored, cor-\nresponding to the individual small, medium, and large configu-\nrations plus a case where all the three configurations are com-\nbined.\nFrom the results in Table E.1, we find that the flux parameters\n(Fs, Fh, Fc) are the most a ffected by some spurious variabil-\nity due to the varying (u,3)coverage, with di fferences of up to\n∼30 %. In addition, the stellar contribution is slightly (large con-\nfiguration) to significantly (small configuration) overestimated\nwith respect to the input model. The geometrical parameters (a r,\nak) describing the characteristic size are found to be more in\nagreement (though not necessarily within the error bars), except\nfor the small configuration that only loosely constrains the so-\nlution. The inclination value is consistent within the error bars\nbetween the four cases, although modestly constrained by the\nsmall and medium configurations. Also, the results are found to\nbe generally consistent with a low-inclination disk. The same\nconclusion applies to the PA, with the small configuration ex-\npectedly failing to constrain this parameter. Finally, for the pa-\nrameters c1ands1describing the azimuthal modulation, the fit\nconverges towards very similar values for all four cases.\nSimply accounting for the error bars on the fitted parameters,\nthe modeling of the small-, medium-, and large-configuration\ndata sets results in a consistent value for the ring inclination,\nthe PA, c1,ands1, but clearly these parameters are more strin-\ngently constrained when going from the small to the large con-\nfiguration. In Fig. E.2, we compare the input model image to the\nparametric models resulting from the fit of the di fferent config-\nurations. Besides the di fferences found in the relative flux con-\ntributions (see Table E.1), we can visually observe that the size\nproperties are not in good agreement —in particular for the small\nconfiguration— when fitting the data of the di fferent configura-\ntions separately, although the general structure is retrieved. In\ncontrast, the azimuthal position of the disk asymmetry appears to\nbe well constrained. This simple test reminds us that the intrin-\nsic sparsity of infrared interferometric data implies that di ffer-\nent spatial frequencies are probed with di fferent configurations,\nwhich may result in a spurious variability of the fitted parameters\nthat may not reflect a physical temporal variability of the system.\nBased on this observation, we adopted here a di fferent strategy to\nmore robustly test the potential time-variability of the di fferent\nparameters describing our model: we fitted the full PIONIER (re-\n5Available at http://www.jmmc.fr/asproTable E.1. Results of the fit of the Aspro synthetic data for the small,\nmedium, and large configurations of GRA VITY . Uncertainties are re-\nported as 1 σerrors from the 16thand 84thpercentile of the MCMC\nmarginal distributions. The column Allrefers to the simultaneous fit of\nthe three data sets combined. F cis not a free parameter, but is obtained\naccording to Fc=1−Fh−Fs.\nAll Small Medium Large\nParameter Value Value Value Value\nFs[%] 22 .8+0.1\n−0.142.3+1.3\n−2.429.9+0.9\n−1.916.7+0.4\n−0.4\nFh[%] 0.3+0.1\n−0.10.9+0.2\n−0.30.3+0.7\n−0.233.6+3.7\n−4.2\nFc[%] 76 .9+0.2\n−0.256.8+1.5\n−2.769.8+1.6\n−2.149.7+4.1\n−4.6\nar[mas] 3 .46+0.01\n−0.014.52+0.20\n−0.153.76+0.30\n−0.293.52+0.06\n−0.06\nak[mas] 1 .75+0.01\n−0.010.25+0.40\n−0.081.05+0.23\n−0.191.59+0.07\n−0.07\ni[deg] 10 .7+1.0\n−1.111.8+10.8\n−8.18.5+5.2\n−4.115.1+3.2\n−4.4\nPA [deg] 0 .9+1.6\n−0.737.0+48.9\n−33.81.8+7.6\n−1.52.5+4.5\n−1.8\nfLor 0 .01+0.01\n−0.010.11+0.24\n−0.100.49+0.16\n−0.330.08+0.11\n−0.06\nc1 0.61+0.01\n−0.010.50+0.17\n−0.170.52+0.10\n−0.120.52+0.04\n−0.04\ns1 0.03+0.02\n−0.010.08+0.19\n−0.100.04+0.16\n−0.060.01+0.05\n−0.02\nχ2\nr 3.0±0.1 2.3±0.1 4.6±0.4 2.6±0.2\n 0 50 100 150 0 50 100 150Model\n 0 200 400 0 200 400All\n 0 200 400 0 200 400Large\n 0 200 400 0 200 400Medium\n 0 200 400 0 200 400Small\nFig. E.2. Aspro synthetic data test images for the small, medium, and\nlarge configurations of GRA VITY . The top-left model was used to gen-\nerate the synthetic visibilities and closure phases with Aspro . The re-\nsulting fitted models are shown for the respective configurations. The\nintensity scale is linear and ranges from zero to the peak pixel value\nof the asymmetry. The central star is not displayed and its position is\nmarked with the white cross. The images have a size of 20 ×20 mas. It is\nvisible from this figure and Table E.1 that the small configuration prop-\nerly constrains the absence of halo contribution in the system, but not\nthe disk morphology, as opposed to the large configuration.\nspectively GRA VITY) continuum data set by forcing all the free\nparameters to be constant across the di fferent epochs, except the\nparameter for which we wish to test the variability hypothesis.\nThroughout the paper we refer to this approach as the x global\nfit, where xis the selected time-variable parameter. This is moti-\nvated by the fact that a rich (u,3)coverage remains desirable to\nconstrain our model.\nArticle number, page 38 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. E.3. Aspro synthetic data (white markers), Vis2, closure phases, and (u,3)plane (left, center, and right panel, respectively) for the di fferent\nbaseline configurations. Blue markers represent the model described in Sect. E.2. Bottom panels show the residuals of the fitting process.\nArticle number, page 39 of 45A&A proofs: manuscript no. aa46926-23_corr\nArticle number, page 40 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nAppendix F: Visualisation of the continuum\ngeometrical models\nArticle number, page 41 of 45A&A proofs: manuscript no. aa46926-23_corr\nFig. F.1. Peak-normalized PIONIER continuum model images. The dashed white lines represent the ±3σMCMC uncertainties on the PA of the\nazimuthal modulation. A description of the model is given in Sect. 3. The central object is not displayed in order to enhance the circumstellar\nemission. The contrast reported in Sect. 3.3 is calculated from the peak-normalized brightness distribution map as the ratio between the brightest\npixel in the map and the corresponding centro-symmetric position in the disk. We report the standard deviation for the mean contrast value across\nthe di fferent PIONIER epochs.\nArticle number, page 42 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. F.2. Same as Fig. F.1 but for the GRA VITY data set.\nArticle number, page 43 of 45A&A proofs: manuscript no. aa46926-23_corr\nAppendix G: Pure-line photocenter displacements\nArticle number, page 44 of 45GRA VITY Collaboration: V . Ganci et al.: The GRA VITY young stellar object survey\nFig. G.1. GRA VITY continuum model images and Br γ-line-emitting gas region. The black filled dot shows the continuum (star +halo+disk)\nphotocenter. The white circle around that point gives the error on the continuum photocenter position estimated from the continuum modeling.\nThe red circle represents the gas region size estimated at the peak emission and is centered on the ∼0 km s−1point. The blue to red colored filled\ndots show the gas photocenters for di fferent spectral channels, as in Fig. A.3.\nArticle number, page 45 of 45" }, { "title": "2401.17792v1.Dimensional_Analysis_Theory_and_Molecular_Dynamics_Simulation_of_Polypropylene_Melt_Flow_during_Injection_Molding_Process.pdf", "content": "Dimensional Analysis Theory and Molecular Dynamics Simulation of Polypropylene\nMelt Flow during Injection Molding Process\nJinrong Zhang,1Dadong Yan,1,a)Li Peng,2and Xianbo Huang2\n1)Department of Physics, Beijing Normal University, Beijing 100875,\nChina\n2)National-certified Enterprise Technology Center, Kingfa\nScience and Technology Co., LTD., Guangzhou 510663,\nChina\nFlow marks are common surface defects that occur in injection-molded products.\nTheir formation may be related to the flow process of the melt in the mold.1–5Through\ndimensional analysis, we have discovered that the geometric shape of the flow field\nis controlled by specific dimensionless quantities. These quantities can be summa-\nrized as follows: geometric dimensionless quantities related to the shape of the mold,\nmaterial dimensionless quantities related to the melt and mold materials, and phys-\nical dimensionless quantities related to the flow. When the geometric shape of the\nmold changes proportionally, with the melt and mold material fixed, and the initial\ntemperature of the melt and mold fixed, the geometric shape of the flow field will be\nsolely controlled by the Weissenberg number Wi. IfWiis kept constant, changing\nthe injection speed, changing the relaxation time of the polypropylene melt, or scal-\ning the mold will result in similar geometric shapes of the flow field. If the size of the\nmold is not changed, the geometric shape of the flow field will be the same. Since\nthe dimensionless equation represents a similar system of all sizes, we verified the\nabove conclusion through molecular dynamics simulations at a smaller scale. After\nfurther improvement of the micro simulation system, there is a possibility of visualiz-\ning the formation process of flow marks. This would greatly aid in the advancement\nof theory and the elimination of flow marks in production and experiments. This\nwork also illustrates that the methodology of dimensional analysis plus molecular\ndynamics simulation may be applied to a wider range of other systems, scaling down\nlarge systems and thus significantly reducing their computational effort.\na)Electronic mail: yandd@bnu.edu.cn\n1arXiv:2401.17792v1 [cond-mat.soft] 31 Jan 2024I. INTRODUCTION\nPolypropylene (PP) mixtures made from melting and mixing ethylene propylene block\ncopolymer (EPBC), ethylene propylene rubber (EPR), and ethylene propylene rubber\n(EBR), or inorganic fillers such as talc powder, are widely used in various applications\ndue to their excellent mechanical properties.6–10Particularly in recent years, in the appli-\ncation of large structural materials like car bumpers, instrument panels, and door panels,\nthe trend of non-coating these molded products has increased, making it necessary for the\nmolded products to have a good surface appearance.2,11–19\nFlow marks are one of the representative surface appearance defects of injection-molded\nparts in various polymer materials, not limited to PP-type blends (including pure PP).1,2,14,15,20–32\nThey are surface defects characterized by wave streaks. The streaks are roughly perpendic-\nular to the direction of melt flow and form on the surface of the injection-molded device.20\nFlow marks can be classified as either same-phase type flow marks, where glossy and cloudy\nparts alternate and appear at the same position on the surface and inside of the molded part,\nor different-phase type flow marks, where the glossy and cloudy parts appear at different\nphases on the surface and inside. Flow marks are closely related to the unstable flow of the\nmelt.33–35\nGenerally, it has been said that flow marks do not appear easily in materials with large\ndie swells, but there are some materials in which flow marks do not appear easily even if\nthe die swell is small. For this reason, the elimination of flow marks in injection-molded\nproducts has conventionally been addressed by changing injection molding conditions such\nas injection temperature and speed, mold temperature, etc.13or molecular characteristics\nsuch as the molecular weight of the material and its distribution through repeated trial and\nerror. In recent years, in order to break away from the flow-mark elimination measures\nrelying on empirical methods, observations of resin flow using visualization molds have been\nactively conducted to elucidate the flow-mark generation mechanism.36,37These studies have\nshown that flow-mark generation is related to the unstable flow phenomenon at the material\nflow front during in-mold flow.12The flow mark generation is caused by the unstable flow of\nthe material in the mold, which is a phenomenon known as the ”flow front” phenomenon.\nHowever, no clear conclusion has yet been reached on the mechanism of flow mark generation\nand how it leads to flow mark formation.\n2In order to investigate the formation mechanism of flow marks, we want to visualize\nthe melt flow process that generates the formation of flow marks and at the same time\nobtain all the information about the flow field, the temperature field, and the polymer\nconformation and their evolution. The melt flow process is microscopic for experiments,\nmaking it difficult to observe without influencing the flow, and it is also difficult to obtain\nmicroscopic information about the evolution process. For mesoscopic simulations, such as\nfinite element analysis, information on polymer chain conformation and crystallization is\nlost. For molecular dynamics simulations, we can obtain this information, but the system is\ntoo macroscopic, resulting in an unmatched amount of calculations. Therefore, we would like\nto scale down the flow field equivalently (while ensuring it is large enough for the polymer\nchain size) and then visualize the flow process by means of molecular dynamics simulation\nand get all the information about the flow field, the temperature field, and the polymer\nconformation and their evolution.\nThrough dimensional analysis, we find the dimensionless quantities that control the geom-\netry of the flow field. These quantities can be categorized as follows: geometric dimensionless\nquantities related to the shape of the mold, material dimensionless quantities related to the\nmelt and mold materials, and physical dimensionless quantities related to the flow. When the\ngeometry of the mold changes proportionally, the melt and mold material are fixed, and the\ninitial temperature of the melt and mold is fixed, the geometric dimensionless quantities, the\nmaterial dimensionless quantities, and a part of the physical dimensionless quantities will re-\nmain constant, and the geometry of the flow field will be controlled only by the Weissenberg\nnumber Wi. IfWiis kept constant, changing the injection speed, changing the relaxation\ntime of the polypropylene melt, or scaling the mold will result in similar geometric shapes\nof the flow field. Theoretically following the above conclusions, we can scale the flow field in\nequal proportions, and then visualize the corresponding melt flow process, which generates\nflow marks, through molecular dynamics simulation. However, before doing so, we need to\nverify the correctness of the above conclusions. Since the dimensionless equations represent\nsimilar systems of all sizes, this paper verifies the above conclusions through a microscopic\nmolecular dynamics simulation with the same magnitude of dimensionless quantities as the\nexperimental injection molding process (the specific values are slightly different). The sim-\nulation results demonstrate that flow fields of the same Wisystems are similar, and the\nflow fields of different Wisystems are obviously different. The simulation results verify the\n3conclusion of dimensional analysis to some extent.\nII. THEORY AND METHODS\nA. Dimensional Analysis Theory\nSelect the characteristic quantity that has a significant impact on the system as the unit,\nv,ξ,τ, and the form before and after dimensionless transformation is as follows\nλ=f(v, ξ, τ, Z, w, L, R, kT, kT′, εLJ, εB, εA, εD, m, ξ′, t, N, n ) (1)\nλ\nvτ=f\u0012\n1,1,1,Z\nvτ,w\nvτ,L\nvτ,R\nvτ,kT\nv2ξτ,kT′\nv2ξτ,εLJ\nv2ξτ,εB\nv2ξτ,εA\nv2ξτ,εD\nv2ξτ,m\nξτ,ξ′\nξ,t\nτ, N, n\u0013\n(2)\nwhere the dependent variable λis the tiger stripe period or induced length (the dimensions\nare the same, the dimensional analysis is equivalent), vis the melt flow rate, ξis the friction\ncoefficient of the melt Kuhn unit, τis the polymer chain relaxation time, Zis the spacing\nbetween the upper and lower plates. wis the size of the injection port, Lis the total length\nof the mold, Ris the Kuhn unit size, kTis the Boltzmann constant and system temperature,\nkT′is the Boltzmann constant and mould temperature, εLJ,εB,εA,εDare the interactions\nbetween atoms, bond energy, bond angle energy, and dihedral angle energy, mis the element\nmass, ξ′is the friction coefficient of the melt Kuhn unit moving on the mold surface, tis the\nobservation time, Nis the number of chain elements, and nis the number of chains. Among\nthem, the number of independent variables needs to be equal to the degrees of freedom of\nthe system. So, we need to eliminate some independent variables that are not independent\n(t,N, and nhave been determined by the other variables)\nλ\nvτ=f\u0012kT\nv2ξτ,kT′\nv2ξτ,εLJ\nv2ξτ,εB\nv2ξτ,εA\nv2ξτ,εD\nv2ξτ,Z\nvτ,w\nvτ,L\nvτ,R\nvτ,m\nξτ,ξ′\nξ\u0013\n(3)\nAccording to the nature of dimensional analysis, since the function form is indefinite, we can\ncombine the multiplication or division of the dimensionless independent variables to separate\nthe pure geometric dimensionless quantities, the pure material dimensionless quantities and\nthe physical dimensionless quantities. Sort them and separate them with a semicolon\nλ\nvτ=f\u0012w\nZ,L\nZ,R\nZ;εLJ\nεB,εA\nεB,εD\nεB,ξ′\nξ;Wi,kT\nv2ξτ,εB\nv2ξτ,m\nξτ,T′\nT\u0013\n(4)\n4FIG. 1. Schematic diagram of the melt unit being sheared by the flow field.\nIf the dimensionless independent variable in the above equation is equal to that in the\nmacroscopic experiment, the scaling simulation can be carried out. Where\nWi=τ˙γ=2vτ\nZ(5)\nNext, invariants or small quantities in production will be eliminated. In the same experiment,\nthe mold geometry is the same, the related energy of melt polypropylene is the same, and\nthe friction coefficient of the melt and mold surface is the same. Therefore, after eliminating\nthe invariants, the equation becomes\nλ\nvτ=f\u0012\nWi,kT\nv2ξτ,εB\nv2ξτ,m\nξτ,T′\nT\u0013\n(6)\nIf we only consider the macroscopic flow field, then the above dimensional analysis is suffi-\ncient. However, if we consider the information of polymer chain orientation, then we need\nto consider the microscopic physical process.\nObserve the denominator of the second independent variable. It can be written as vξ×vτ.\nThe left side of the multiplication sign can be regarded as the magnitude of the friction of\na Kuhn unit subjected to melt shear, while the right side of the multiplication sign can be\nregarded as the distance traveled in relaxation time. Therefore, the denominator can be\nregarded as the work done by friction on the melt in relaxation time.\nSince the numerator of the fractional equation is the thermal kinetic energy of a Kuhn\nunit, the comparison of magnitudes should also correspond to the work done by friction\non a Kuhn unit. As shown in Fig. 1, the velocity of relative sliding between the units is\n5∆v=2R\nZv, and so the above equation should be transformed into\nkT\n∆v2ξτ=\u0012Z\nR\u00132\n×kT\nv2ξτ(7)\nDo the same forεLJ\nv2ξτ. Consider the entanglement effect38\nξchain=ξ×N3\nN2\ne(8)\nAfter correcting each dimensionless quantity, we observe the magnitude again. In the ex-\nperiments, the length of the polypropylene tangle is Ne= 32. The number of Kuhn units\nand the friction coefficient in the case of Ref.39areN= 1650 and ξ= 7×10−12Ns/m,\nrespectively. See Table I for the other parameters. Therefore,\nN2\ne\nN3×\u0012Z\n2R\u00132\n×kT\nv2ξτ≈322\n16503×\u001210−3\n2×10−9\u00132\n×10−23×5×102\n(10−1∼10−2)2×7×10−12×10−1≈101∼10−1\n(9)\nAccording to Ref.40,εLJ\nkT= 10−1\nN2\ne\nN3×\u0012Z\n2R\u00132\n×εLJ\nv2ξτ≈1∼10−2(10)\nKuhn unit mass m= 0.1878 kg /mol.39\nN2\ne\nN3×m\nξτ≈322\n16503×0.1878÷(6.02×1023)\n7×10−12×10−1≈10−19(11)\nMelt temperature T= 483 .15 K and mold temperature T= 323 .15 K.5\nT′\nT=323.15\n483.15≈0.7 (12)\nThe dimensionless number of mass is very small and can be eliminated. Using the Doi-\nEdwards crawling model,41,42we can obtain the relaxation time\nτ≈ξb2\nkT×N3\nNe(13)\nby replacing Eq. (9) with the relaxation time\nN2\ne\nN3×\u0012Z\n2R\u00132\n×kT\nv2ξτ=Ne×\u0012Z\n2vτ\u00132\n=Ne\nWi2(14)\nThe numerator and denominator have been determined by other quantities, and this inde-\npendent variable is equivalent to having been determined, and therefore can be eliminated.\n6Divide the independent variables, eliminate the two dimensionless numbers, and then mul-\ntiply both sides of Eq. (6) by Wi\nλ\nZ=f\u0012\nWi,εLJ\nkT,T′\nT\u0013\n(15)\nSince λrepresents an arbitrary length measure of the flow field, ifλ\nzremains constant\nfor different systems, then it indicates that the different systems are geometrically similar.\nTherefore, we can conclude that when the geometry of the mold changes in equal proportions,\nwhile the melt and mold material and the initial temperature of the melt and mold remain\nfixed, the geometry of the flow field will be solely controlled by the Weissenberg number Wi.\nIfWiis kept constant, changing the injection speed, changing the relaxation time of the\npolypropylene melt, or scaling the mold will result in similar geometric shapes of the flow\nfield. If the size of the mold is not changed, the geometric shape of the flow field will be the\nsame. Since the dimensionless equations represent similar systems of all sizes, we will then\nverify the above conclusions through molecular dynamics simulations at a smaller scale.\nB. Molecular Dynamics Simulation\nThe polymer consists of a bead-spring chain with a diameter of σand a degree of polymer-\nization of N (see Table I). The metal atoms are composed of beads with a diameter of σ, and\nthe interactions between all the beads in the system are modeled using the truncated-shifted\nLennard-Jones (LJ) potential\nULJ(r)=\n\n4εLJ\u0014\u0000σ\nr\u000112−\u0000σ\nr\u00016−\u0010\nσ\nrcut\u001112\n+\u0010\nσ\nrcut\u00116\u0015\n, r≤rcut\n0, r > r cut(16)\nThe cutoff distance for the bead-by-bead interaction rcut= 2.5σ. Referring to Ref. 5,\nwe take the Lennard-Jones interaction parameter for the intermelt interaction to be εLJ=\n0.1kBT, where kBis Boltzmann’s constant and Tis the absolute temperature. For metal-\npolymer interactions, if we want the flow field to be similar to the experimental one, then\nξ′\nξneeds to be equal to the experimental one. However, in order to verify the correctness\nof the dimension analysis, the exact interaction size is not needed, only the simulation and\nthe system that the dimensionless quantity is equal in magnitude, so in order to prevent\npenetration, the melt-metal interaction is taken as εLJ= 2.0kBT, which has been tested\n7ξ′\nξ≈20. In addition, to simplify the model we control that the metal atoms are stationary\nand there is no interaction between the metal atoms.\nBeads are bonded to form polymer chains, and all bonds in the system are represented\nby harmonic potentials:\nUharm =K(r−r0)2(17)\nwhere k= 300 kBT/σ2is the spring constant and r0= 1.2σis the bond length.\nFor simplicity, we first validate the conclusions of the dimensional analysis at the same\nmelt temperature as the mold temperature. Therefore, we simulate it in the NVT ensemble,\nwhere a constant temperature is maintained by coupling with a Langevin thermostat. In\nthis case, the motion equation of the ith bead is\nmd⃗ vi(t)\ndt=⃗Fi(t)−ξl⃗ vi(t) +⃗FR\ni(t) (18)\nWhere mrepresents the mass of the bead, and in our system, the mass of all beads is set to be\nuniform, ⃗ vi(t) denotes the bead velocity, and ⃗Fi(t) represents the net deterministic force act-\ning on the ith bead. The stochastic force ⃗FR\ni(t) has a zero mean value ⟨⃗FR\ni(t)⟩= 0, and has\naδ-functional correlations, ⟨⃗FR\ni(t)⃗FR\nj(t′)⟩= 6ξlkBTδijδ(t−t′). where τLJ=σ(m/ε LJ)1/2\nis the standard LJ time. The velocity-Verlet algorithm is utilized to numerically integrate\nthe motion equation with a time step ∆ t= 0.005τLJ. All molecular dynamics simulations\nare performed via the LAMMPS software package43. The unit friction coefficient ξis the\nsuperposition of the friction coefficient ξlin the Langevin thermostat and the friction coef-\nficient ξcof units colliding with each other. Therefore, by regulating the friction coefficient\nof the Langevin thermostat we can regulate the friction coefficient of the melt unit. We\nchose four different friction coefficients ξlis 40m/τ LJ, 26.7m/τ LJ, 17.8m/τ LJand 7 .9m/τ LJ,\ncorresponding to ξof 23 .2m/τ LJ, 16.7m/τ LJ, 11.6m/τ LJ, and 6 .2m/τ LJwhen the particle\nnumber density is 0.85. The value of ξlshrinks by 1 .5 times, 2 .25 times, and 5 .06 times,\nrespectively, and ξis approximately proportional to ξl.\nOur injection mold is shown in Fig. 2, where the polymer melt is injected through a\nthin tube on the left side, and a plate at the end is pushed downward at a constant speed\nso that the melt flows between the two metal plates. The green color represents metal\natoms, and the red and blue colors are the same polymer melt. The red and blue melt flow\nsuccessively from the thin tube into the two plates, so the dividing line between them can\nbe considered isochronous. Additionally, the atoms flowing through the center of the thin\n8FIG. 2. Structure and size of injection molded model in our simulation.\ntube are marked yellow, and a periodic curve is formed after flowing. The light green color\nat the bottom represents the front of flow during initial injection. The x-direction and z-\ndirection are fixed boundaries, and the y-direction is a periodic boundary. According to the\ncalculated quantities, we chose three system sizes (different systems satisfying geometrical\nsimilarity) with two-plate spacings Zof 40σ, 60σ, and 90 σ(the simulated system is a free-\nconnecting chain, and we would like to have one unit corresponding to one polypropylene\nKuhn unit, which has a size of 11 ˚A, so the two-plate spacing corresponds to the real system’s\napproximate size of 400 ˚A, 600 ˚A, and 900 ˚A). The maximum system geometric parameters\nare shown in Fig. 2, the length of the vertical pipe part Z′= 450 σ, the number of melt\nparticles is 1 ,541,788, and the number of metal particles is 58 ,444. The parameters of the\nother size systems are scaled proportionally.\nFor the simulated system, the relaxation time τof the polymer itself is not clearly defined.\nHowever, the system needs to satisfy the condition that each dimensionless independent vari-\nable is equal to that of the macroscopic experiment or is a very large or very small quantity,\nas in the macroscopic experiment. Hence, it is necessary to determine the relaxation time\nτof the polymer in advance. Using Eq. (13), we can calculate the relaxation time τfor\ndifferent polymerization degrees N. For systems where we need to control the relaxation\ntime, we can work backwards to calculate the degree of polymerization we need. To improve\nthe accuracy of selecting N, we also measure the relaxation times corresponding to different\n9Nby the unit mean square displacements versus time predicted by the crawling model. We\nuse the Ncomputed in the previous step as a baseline and increase or decrease Nuntil we\nfind the Nthat is closest to the target relaxation time. The relationship between the unit\nmean square displacement and time predicted by the crawling model is divided into four\nregions based on different power relationships, with respective exponents of 1/2, 1/4, 1/2,\nand 1. The turning point in the third and fourth regions is the crawling time τrep, which is\nthe relaxation time we need. We can find this turning point by fitting, and thus determine\nthe corresponding relaxation time. For the measurement of mean square displacements, we\nchoose a cube with a side length of 60 σas the simulation box, while the particle number\ndensity is 0 .85σ−3. For different degrees of polymerization, the particle number density is\nkept constant and ξlis set to 1 m/τ LJ, corresponding to ξbeing equal to 2 .1m/τ LJ, and all\nother parameter selections are the same as before. According to Ref.44, the central unit of\nthe chain was selected for measurement to improve accuracy.\nIn order to assess the degree of similarity or dissimilarity of the system, we quantified the\nsystem similarity S. As shown in Fig. 3, we divided the system into 240 chunks, of which\nthe number of chunks with yellow atoms is n(Fig. 3 is schematic, the actual division is\nmore dense). The coordinates of the zdirection of the center of mass of the yellow atoms\nin each block are calculated. The larger of the two compared systems is scaled down by a\nfactor of kso that the two systems are the same size (some adjustments are needed for the\ndifferent starting and ending positions of the yellow atoms in the x-direction of the different\nsystems). Then, calculate the root mean square error ∆ of the coordinates in the z-direction\nof the two systems, and we define the similarity S=1\n1+a∗∆. To keep the similarity within a\nsuitable interval, we choose a= 0.8.\n∆ =sPn\ni(z′\ni/k−zi)2\nn(19)\nIII. RESULTS AND DISCUSSION\nBefore verifying whether the flow field is controlled by Withrough simulation, we must\nfirst do some preparatory work. First, we consider that since there is a vertical pipe section,\nit is necessary to match not only the dimensionless quantity Wibetween the two plates\nbut also the dimensionless quantity kT/v2ξtinside the pipe (which can be obtained by a\n10FIG. 3. Demonstration of system division. Above is the system with Z= 60σ(a), below is the\nsystem with Z= 90σ(b). This is a schematic diagram, the actual division is more dense.\nsimilar dimensional analysis). Otherwise, for systems of different sizes, even if the invisible\nplate push velocity is scaled equally, the pipe outlet flow rate does not satisfy the scaling,\nwhich leads to the flow rate of the melt between the two plates not satisfying the scaling, as\nshown in Figs.4(a) and (b). In the macro system, because the size Z is large, so vandtare\nlarge, the above dimensionless quantity is very small and can be ignored, so the scaling of\nthe macro system should not consider this dimensionless quantity, but the volume reduced\nsimulation system needs to consider it. From Einstein’s formula D=kT/ξ :\nkT\nv2ξt=D\nv2t=Dt\nv2t2≈⟨[r(t)−r(0)]2⟩\n(vt)2(20)\nIt can be seen that this dimensionless quantity represents the ratio of the average distance\nmoved by the particle per unit time tto the distance pushed by the invisible plate. For\ndifferent systems, when this dimensionless quantity is equal, the average velocity of the\nparticle movement is proportional to the velocity pushed by the invisible plate, and the flow\nrate of the melt between the two plates will satisfy the scaling when the velocity pushed by\nthe invisible plate is scaled equivalently, as shown in Figs.6(a), (c) and (e).\nNext we consider the proportionality of the parameters between the different systems.\nSince we have controlled for equal proportional changes in mold geometry, constant melt\nand mold materials, and constant melt and mold temperatures, the dimensionless equations\n11FIG. 4. When we satisfy only the equality of Wi, but not the equality of kT/v2ξt, the velocity of\nthe plate push is scaled equivalently, but the flow rate at the pipe outlet is not scaled equivalently.\nThe left side shows the system with Z= 40σ, and the right side shows the system with Z= 60σ.\nare simplified to:\nλ\nZ=f\u0010vτ\nZ\u0011\n(21)\nOur goal is to prove that the flow field remains unchanged for any combination of values\nv,τ, and Z, as long as Wiremains constant. However, logically we only need to show\nthat this conclusion holds in two cases: 1. The shape of the flow field is constant when v\nandZare varied proportionally. 2. When vandτare varied inversely, the shape of the\nflow field remains constant. Because any combination of v,τ, and Zcan be obtained by\nproportionally changing vandZ, and inversely changing vandτ, as long as it satisfies the\nrequirement that Wiis invariant. Thus, if the conclusion holds in the above two cases, then\nit is proof that the conclusion holds in any case.\nSince we need to satisfy bothL\nZ,vτ\nZandkT\nv2ξtare unchanged, and L=vt. Recombining\nthese dimensionless numbers yieldst\nτ,vτ\nZandkT\nv2ξτare unchanged, while τ≈ξb2\nkT×N3\nNe, so for\nthe choice of the scaling relation of the parameters our only strategy is: 1. When vand\nZincrease proportionally, reduce ξby the square multiple of the increase in v(increasing\nTis equivalent to decreasing ξ), changing Nensures that τremains constant, and the\nprogram running time tremains constant. 2. When vandτare varied inversely, Zremains\nunchanged, and if vdecreases then ξincreases in the same proportion, τtherefore increases\nin the same proportion, and the program running time tincreases in the same proportion.\nTake the system we simulated as an example: The two-plate spacing Zfor the three\nsystems are 40 σ, 60σand 90 σ, respectively. When vandZincrease proportionally, Z\n12increases by 1.5, therefore vincreases by 1.5 and ξdecreases by 2.25. In order to ensure that\nτremains constant, we will increase the degree of polymerization, N. The exact value of\nthe degree of polymerization will be determined in the next steps. When vandτare varied\ninversely, vdecreases by a factor of 1.5, so ξincreases by a factor of 1.5, and τnaturally\nincreases by 1.5 without changing N.\nIn order to determine the size of Nwhen vandZare increased in equal proportion,\nwe first consider how much an increase in Nincreases the relaxation time by a factor of\n2.25 if the coefficient of friction remains constant, and then we choose this Nand reduce\nthe coefficient of friction by a factor of 2.25 to ensure that the relaxation time τremains\nunchanged in a close approximation. First, we can back-calculate using the Doi-Edwards\ncrawling model, see Eq. (13). For the smallest system Z= 40 σ, we choose N= 21 and\nN= 45, respectively, and calculate the degrees of polymerization to be N= 28 and N= 59\nforZ= 60 σ. For Z= 90 σ, the degree of polymerization is N= 37 and N= 77. By\nmeasuring the mean square displacement of the central unit of the polymer chain over\ntime44, we find that these degrees of polymerization correspond to relaxation times that\nincrease by a factor of less than 2.25. Therefore, we increase the degree of polymerization\nand measure the relaxation time until it increases by a factor of 2.25. As shown in Fig. 5,\nwe fit the mean-square displacement over time with four straight lines and divided four\nregions with different slopes based on the intersection of the lines. The slopes we obtained\nfor each region are slightly higher than those of the crawling model, similar to the results\nin the Ref.44. The turning points in the third and fourth regions are the relaxation times.\nThe relaxation time τ= 1740 τLJwhen the degree of polymerization N= 30. When the\nrelaxation time is expanded by a factor of 2 .25 to 3915 τLJ, the closest value of the relaxation\ntime is at N= 30. When it is further expanded by a factor of 2 .25 to 8809 τLJ, the closest\nvalue of the relaxation time is at N= 42. The relaxation time is τ= 8900 τLJwhen the\ndegree of polymerization is N= 45. When the relaxation time is expanded by a factor of\n2.25 to 20025 τLJ, the closest value of the relaxation time is at N= 63. When it is further\nexpanded by a factor of 2 .25 to 45056 τLJ, the closest value of the relaxation time is at\nN= 85. Ultimately, we determined that for Z= 60 σ, the degrees of polymerization are\nN= 42 and N= 63. For Z= 90σ, the degrees of polymerization are N= 59 and N= 85.\nAfter considering the entanglement effect and adding the dimensionless quantity in the\n13FIG. 5. The relationship between the mean square displacements of the center unit of the chain and\ntime, in logarithmic coordinates. The system temperature is 1 kBT, with the langevin thermostat\nfriction coefficient selection ξl= 1m/τ LJand measured ξ= 2.1m/τ LJ. The degree of polymerization\nfor each case is as follows: (a) N= 21, (b) N= 30, (c) N= 42, (d) N= 45, (e) N= 63, (f)\nN= 85. The straight lines represent the fit to the data for each region, and the key values show\ntheir slopes.\nvertical pipe, the complete dimensionless equation is:\nλ\nZ=f\u0012w\nZ,L\nZ,R\nZ;εA\nεB,εD\nεB,ξ′\nξ;Wi,εLJ\nkT,εB\nkT,kT\nv2ξτ,N2\nem\nN3ξτ\u0013\n(22)\nWith the above preparations, we are now able to determine the full parameters of the\nsystems. By choosing the parameters, we want to make each dimensionless independent\nvariable in Eq. (22) in the simulated system equal to that of the macroscopic experiment\nor be a very large or very small quantity as in the macroscopic experiment. The specific\nparameters are shown in Table I. In Table I, we compare the two simulated systems with\nZ= 40 σto the experiments, and the experimental parameters are taken from Ref.5,39.\nAccording to the above, the polymerization degree, friction coefficient, and relaxation time\nof the two simulation systems have been determined. To ensure that Wiis close to the\nexperimental value, we control the total number of steps for the invisible plate pushing from\nthe top of the pipe to the bottom of the pipe at a uniform speed to be 6,000,000 (the total\nnumber of steps changes proportionally when vandτvary inversely), and the average flow\nvelocity between the two plates is calculated to be v≈0.006. For the systems with N= 30\n14TABLE I. System parameters\nsystem N ξ τ v Wiw\nZL\nZR\nZεLJ\nkTεB\nkTξ′\nξkT\nv2ξtN2\nem\nN3ξτ\nexperiment 2000 10−120.1 0.01∼0.11∼200.540∼15010−60.1300 110−4\n∼10−610−19\nsimulation 1 21 23.2 19222 0.006 5.8 0.5 4.71\n400.130010 0.04 10−8\nsimulation 2 45 23.2 98324 0.006 29.5 0.5 4.71\n400.130010 0.04 10−10\nandN= 45, we have τ=23.2\n2.1×1740 = 19222 and τ=23.2\n2.1×8900 = 98324, respectively.\nFrom these values, we obtain Wi=2vτ\nZ= 5.8 and Wi=2vτ\nZ= 29.5. Here,w\nZrepresents the\nratio of the injection hole size (i.e., the size of the vertical tube) to the spacing between the\nplates. Horizontal injection molding hasw\nZ= 0.5 according to Ref.5, while vertical injection\nmolding in industrial production hasw\nZ= 4. Our model is based on vertical injection\nmolding. However, in order to enhance the periodic flow characteristics to facilitate the\ncomparison of different systems, we selecte the case ofw\nZ= 0.5 for simulation. ForL\nZ, we\nchose Las the length at which flow marks begin to appear in the experiment, and as the total\nlength of the injection molding in the simulation. It can be seen that the simulation requires\nan increase in the amount of melt in order to observe the flow corresponding to the flow\nmarks, which will be done in our future work. As forR\nZ, the experimental quantity is very\nsmall, but the simulation quantity is not sufficiently small, which will affect the similarity of\nthe system to some extent. Referring to Ref.40, we can obtainεLJ\nkTandεB\nkTsince the system\nconsists of freely connected chains, we do not consider the other interaction potentials. As\nforξ′\nξ, no experiments have been found to accurately measure this value. Therefore, we\nrefer to Ref.38. In ideal conditions, with a smooth solid surface and no chains attached to\nit, one expects that the friction kis comparable to what it is in a fluid of monomers. In\nthe simulation, this value is determined by the interaction between the melt and the metal.\nAs forkT\nv2ξtandN2\nem\nN3ξτ, they can be calculated using the aforementioned parameters.kT\nv2ξtis\na very small quantity for experiments and not sufficiently small for simulations, whileN2\nem\nN3ξτ\nis negligible in both experiments and simulations. Table II compares the parameters of the\ndifferent systems in the simulation for the two smallest simulated systems, the system after\nthe proportional change in vandZ, and the system after the inverse change in vandτ.\nThe units of each system are LJ units.\nBased on the above parameter selection, we carried out molecular dynamics simulations,\n15TABLE II. System parameters\nCommon parameters Wi≈6 Wi≈30\nZ v ξ l N N\n40 0.006 40 30 45\n60 0.009 17.8 42 63\n60 0.006 26.7 42 63\n90 0.0135 7.9 59 85\n90 0.006 17.8 59 85\nFIG. 6. The systems where vandZare proportional: (a)(c)(e) for the system Wi≈6, (b)(d)(f)\nfor the system Wi≈30. From top to bottom, ξlis 40m/τ LJfor (a)(b), 17 .8m/τ LJfor (c)(d), and\n7.9m/τ LJfor (e)(f), respectively.\nand the flow fields obtained are shown in Fig. 6 and Fig. 7. Qualitatively, we can observe\nthat the flow fields are similar for systems with the same Wiand significantly different for\nsystems with different Wi.\nThrough quantitative calculation of similarity, we obtained the results shown in Fig. 8.\nFrom Fig. 8, we can see that the system similarity ranges from high to low as follows:\nvandτchange inversely, vandZchange positively, and different Wi. When vandZ\nchange positively, the similarity is lower than when vandτchange inversely. There could\nbe several reasons for this: 1. The dimensionless quantity R/Z is an observable quantity\n16FIG. 7. The systems where vandτare inversely proportional: (a)(c)(e)(g) are the systems of\nWi≈6, (b)(d)(f)(h) are the systems of Wi≈30. From top to bottom, ξlis 17.8m/τ LJfor (a)(b),\n26.7m/τ LJfor (c)(d), 7 .9m/τ LJfor (e)(f), and 17 .8m/τ LJfor (g)(h), respectively.\n(1/40) in the simulation, but we do not change the particle size Rproportionally with Z\nbecause such a change would make the system equivalent in the simulation. Therefore, the\nchange in R/Z reduces the similarity of the system. Specifically, we can observe that the\nsimilarity is lower when the Zchanges from 40 σto 60 σ, but becomes higher when the Z\nchanges from 60 σto 90 σ. This is because, for a system with a larger Z, the value of R/Z\ntends to be smaller, resulting in a smaller effect on the system and thus making the systems\nmore similar. It is worth noting that this value is very small and can be ignored in macro\nsystems. 2. Due to ξchanges in different systems, it is necessary to keep τunchanged by\ncontrolling N. However, since Nis discrete and different values of Nhave a greate impact\non the relaxation time, it is impossible to accurately control the relaxation time, resulting in\nchanges in Wi. Conversely, when vandτvary inversely, since the plate thickness Zis the\nsame for the different systems, the dimensionless quantity R/Z, although it is an observable\nquantity, has a constant value. Therefore, it does not reduce the degree of similarity of the\nsystem, and the similarity does not increase due to the increase in size. Additionally, since\n17 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 50 60 70 80 90 100S\nZvt Wi=13\nvt Wi=30\nvz Wi=13vz Wi=30\ndifferent\ndifferent v=0.006FIG. 8. Similarity calculations for all the systems we compared. ”vt” corresponds to systems where\nvandτare inversely proportional. ”vz” corresponds to systems where vandZare proportional.\n”different” corresponds to systems where Wiare different.\nNdoes not change, Wiof different systems are closer to each other. In addition, when v\nandτare varied inversely, the fluctuations in the similarity of the different systems are due\nto the stochastic nature of the fluid motion.\nIV. SUMMARY AND REMARKS\nTo study the mechanism of flow marks in injection-molded products and eliminate them,\nwe would like to be able to visualize the flow process of the melt responsible for the formation\nof flow marks. However, this process is too macroscopic for molecular dynamics simulations\nand the computational effort cannot be matched. Therefore, we propose scaling down the\nflow field in equal proportions and then visualizing this process using molecular dynamics\nsimulations.\nUsing dimensional analysis, we find the dimensionless quantity that governs the geometry\nof this flow field. We conclude that when the mold geometry is proportionally varied, with\nfixed melt and mold materials and fixed initial temperatures of the melt and mold, the\nflow field geometry will be solely controlled by the Wiesenberg number Wi. Keeping Wi\nconstant, modifying the injection speed, adjusting the relaxation time of the polypropylene\nmelt, or scaling the mold will yield similar geometric shapes of the flow field.\nTheoretically, based on the above conclusions, we can scale the flow field proportionally\nand visualize the melt flow process that leads to the formation of flow marks using molec-\n18ular dynamics simulations. However, before doing so, we need to verify the correctness of\nthe above conclusions. We verified the above conclusions through microscopic molecular\ndynamics simulations with the same values of dimensionless quantities as the experimental\ninjection molding process (Some of these values differ from the experiment, such asw\nZ,L\nZ,ξ′\nξ,\netc. However, they are of the same order of magnitude as the experiment and thus have the\nsame dimensional analysis).\nBefore verifying whether the flow field is controlled by Withrough simulations, we made\nsome preparations. First, we consider that for the simulated system, we also need to match\nthe dimensionless quantity kT/v2ξtof the flow in the pipe so that the flow rate at the outlet\nof the pipeline can meet the scaling (for the experiments, this quantity is very small, and\nthe velocity driving the melt scales equally, the outlet flow rate automatically satisfies the\nscaling). Due to the existence of dimensionless numbers in the pipe, our only strategy for\nthe choice of scaling relationships for the simulation parameters is as follows: 1. When vand\nZincrease proportionally, reduce ξby the square multiple of the increase in v(increasing\nTis equivalent to decreasing ξ), changing Nensures that τremains constant, and the\nprogram running time tremains constant. 2. When vandτare varied inversely, Zremains\nunchanged, and if vdecreases, then ξincreases in the same proportion. Therefore, τincreases\nin the same proportion, and the program running time tincreases in the same proportion. To\ndetermine the value of Nwhen vandZincrease proportionally, the relationship between the\ndegree of polymerization and the relaxation time is obtained by measuring the mean-square\ndisplacement of the unit. We match the other adjustable parameters with the experiment\nto make each dimensionless quantity the same as the experimental value or be a very large\nor very small quantity as in the macroscopic experiment.\nThe simulation results show that the flow fields are similar for systems with the same Wi,\nand significantly different for systems with different Wi. Through quantitative calculation\nof similarity, we can see that the system similarity ranges from high to low as follows: v\nandτchange inversely, vandZchange positively, and different Wi. When vandZchange\npositively, the similarity is lower than when vandτchange inversely. This may be due to\nthe variation of the dimensionless quantity R/Z and the error caused by changing N. We\ncan see that the similarity is lower when Zchanges from 40 σto 60 σ, but becomes higher\nwhen Zchanges from 60 σto 90 σ. This shows that the effect of R/Z on the simulated\nsystem diminishes as this dimensionless number tends to be small. The results of the sim-\n19ulation validate the conclusion of the dimensional analysis to some extent. Furthermore,\nwe will improve the system to enhance its similarity, and select the same parameters as the\nexperimental injection molding process for molecular dynamics simulation to visualize the\nflow mark formation process. Since the formation of flow marks has an induction length,5\na longer simulation time will be required. After the system is completed, it will be possible\nto visualize the process of flow mark formation, which will help to advance the analytical\ntheory and eliminate the flow marks in production and experiments. This work also illus-\ntrates that the methodology of quantitative analysis plus simulation may be applied to a\nwider range of other systems, scaling down large systems and thus significantly reducing\ntheir computational effort.\nV. ACKNOWLEDGEMENTS\nFinancial support by the National Natural Science Foundation of China (grant nos.\nxxxxxx and xxxxxx) is gratefully acknowledged.\nVI. AUTHOR DECLARATIONS\nConflict of Interest\nThe authors have no conflicts of interest to disclose.\nVII. DATA AVAILABILITY\nThe data that support the findings of this study are available within the article.\nREFERENCES\n1K. Hirano, Y. Suetsugu, and T. Kanai, “Morphological analysis of the tiger stripe on\ninjection molding of polypropylene/ethylene-propylene rubber/talc blends dependent on\nbased polypropylene design,” J. Appl. Polym. Sci. 104, 192–199 (2007).\n2K. Hirano, S. Tamura, and T. Kanai, “Striped-pattern deterioration and morphological\nanalysis of injection moldings comprising polypropylene/ethylene α-olefin rubber blends.\ni. influence of ultraviolet irradiation,” J. Appl. Polym. Sci. 105, 2416–2426 (2007).\n203S. Maeda, “Tiger-striped flow mark of polypropylene alloys,” Nihon Reor. Gak. 49, 215–\n220 (2021).\n4S. Maeda, “Unusual tiger-striped flow mark pattern in injection moldings for polypropy-\nlene/talc compounds ii,” Nihon Reor. Gak. 49, 247–250 (2021).\n5S. Maeda, K. Fukunaga, and E. Kamei, “Flow mark in the injection molding of polypropy-\nlene/rubber/talc blends,” J. Society of Rheology Japan 35, 293 (2007).\n6J. Karger-Kocsis, “Polypropylene: Structure, blends and composites,” Polym. Polym.\nCompos. 2, 333–335 (1994).\n7J. Karger-Kocsis, Polypropylene: An AZ Reference , Vol. 2 (Springer Science & Business\nMedia, 2012).\n8T. Nomura, T. Nishio, H. Sato, and H. Sano, “Ipn-like structure generated by the control\nof micro crystal formation in the elastomer matrix,” Kobunshi Ronbunshu (Tokyo) 50,\n27–34 (1993).\n9M. Jin, K. Liu, H. Liu, Y. Zhang, H. Du, X. Li, and J. Zhang, “Effects of polyolefin\nelastomer and β-nucleating agent on morphological evolution of isotactic polypropylene\nunder an intensive shear rate,” Polym. Test. 39, 1–11 (2014).\n10Y. Shen, H. Tian, W. Pan, J. Feng, D. Wang, N. Ning, M. Tian, and L. Zhang, “Unex-\npected improvement of both mechanical strength and elasticity of epdm/pp thermoplastic\nvulcanizates by introducing β-nucleating agents,” Macromolecules 54, 2835–2843 (2021).\n11T. Nomura, T. Nishio, H. Sato, and H. Sano, “Structure of super olefine polymer,” Kobun-\nshi Ronbunshu (Tokyo) 50, 87–91 (1993).\n12H. Yokoi, “Visualization analysis of undesired phenomena on molded surface in injection\nmolding process,” J. Surface Finishing Society of Japan 51, 1182–1186 (2000).\n13N. Tanaka, Y. Fujioka, and H. Tanimoto, “Phenomenon and countermeasure for flow-mark\non injection molded bumper,” Seikei-Kakou 13, 549–553 (2001).\n14B. Patham, P. Papworth, K. Jayaraman, C. Shu, and M. D. Wolkowicz, “Flow marks in\ninjection molding of polypropylene and ethylene–propylene elastomer blends: Analysis of\nmorphology and rheology,” J. Appl. Polym. Sci. 96, 423–434 (2005).\n15K. Hirano, S. Tamura, Y. Obata, and T. Kanai, “Striped-pattern deterioration and mor-\nphological analysis of injection molding comprising polypropylene/ethylene– α-olefin rub-\nber blends. ii. influence of heating,” J. Appl. Polym. Sci. 108, 76–84 (2008).\n2116J. Li, J. Leng, Y. Jiang, and J. Zhang, “Experimental characterization of 3d printed pp/h-\nbn thermally conductive composites with highly oriented h-bn and the effects of filler size,”\nCompos. Part A: Appl. Sci. Manuf. 150, 106586 (2021).\n17P. Liu, H. Hao, H. Celio, J. Cui, M. Ren, Y. Wang, H. Dong, A. R. Chowdhury, T. Hutter,\nF. A. Perras, et al. , “Multifunctional separator allows stable cycling of potassium metal\nanodes and of potassium metal batteries,” Adv. Mater. 34, 2105855 (2022).\n18M. Liu, R. Hong, X. Gu, Q. Fu, and J. Zhang, “Remarkably improved impact fracture\ntoughness of isotactic polypropylene via combining the effects of shear layer-spherulites\nlayer alternated structure and thermal annealing,” Ind. Eng. Chem. Res. 58, 15069–15078\n(2019).\n19D. Zhang, L. Ding, T. Wu, F. Yang, F. Lan, Y. Cao, and M. Xiang, “Facile preparation\nof a lithium-ion battery separator with thermal shutdown function based on polypropy-\nlene/polyethylene microsphere composites,” Ind. Eng. Chem. Res. 60, 18530–18539 (2021).\n20M. Yoshii and H. Kuramoto, “Influence of mold temperature on transcription and signal\ndeterioration of injection-molded substrates for optical disks,” Kobunshi Ronbunshu 49,\n197–204 (1992).\n21M. Chang, “Surface defect formation in the injection molding of acrylonitrile-styrene-\nacrylate polymers,” Int. Polym. Process. 11, 76–81 (1996).\n22H. Hamada and H. Tsunasawa, “Correlation between flow mark and internal structure of\nthin pc/abs blend injection moldings,” J. Appl. Polym. Sci. 60, 353–362 (1996).\n23S. Hobbs, “The development of flow instabilities during the injection molding of multi-\ncomponent resins,” Polym. Eng. Sci. 36, 1489–1494 (1996).\n24M.-C. Heuzey, J. Dealy, D. Gao, and A. Garcia-Rejon, “The occurrence of flow marks\nduring injection molding of linear polyethylene,” Int. Polym. Process. 12, 403–411 (1997).\n25M. Lacrampe and J. Pabiot, “Defects in surface appearance of injection molded thermo-\nplastic parts–a review of some problems in surface gloss distribution,” J. Inject. Mold.\nTechnol. 4, 167 (2000).\n26C.-C. An and R.-H. Chen, “The experimental study on the defects occurrence of sl mold\nin injection molding,” J. Mater. Process. Technol. 201, 706–709 (2008).\n27L. Mathieu, L. Stockmann, J. Haudin, B. Monasse, M. Vincent, J.-M. Barthez, J.-Y.\nCharmeau, V. Durand, J.-P. Gazonnet, and D. Roux, “Flow marks in injection molding\nof pp: influence of processing conditions and formation in fountain flow,” Int. Polym.\n22Process. 16, 404–411 (2022).\n28S. Owada and H. Yokoi, “Visualization analysis of tiger-striped flow mark generation phe-\nnomena in injection molding,” in AIP Conference Proceedings , Vol. 1713 (AIP Publishing,\n2016).\n29R. Gahleitner, K. S. Niel, and S. Frank, “Optical measurement system for characterizing\nplastic surfaces,” in Image Processing: Machine Vision Applications , Vol. 6813 (SPIE,\n2008) pp. 244–251.\n30S. Kuroda, A. Mizutani, and H. Ito, “Effect of talc size on surface roughness and glossiness\nof polypropylene injection molding application to automotive plastics,” Polym. Eng. Sci.\n60, 132–139 (2020).\n31H.-T. Jeong and Y.-R. Kim, “Flow marks of polypropylene (pp) composites in the injection\nmolding,” J. Korean Appl. Sci. Technol. 32, 320–325 (2015).\n32G. Iannuzzi and M. Rigdahl, “Rheological properties of elastomer-modified polypropylene\nand their influence on the formation of flow marks,” J. Appl. Polym. Sci. 118, 604–610\n(2010).\n33J. Hu, Y. Zhang, W. Song, F. Zou, Z. Yan, Q. Fu, and J. Zhang, “Development of unstable\nflow and morphological analysis of flow marks in injection molding of polypropylene and\npolyolefin elastomer blend,” Ind. Eng. Chem. Res. 62, 13303–13313 (2023).\n34A. M. Grillet, A. C. Bogaerds, G. W. Peters, F. P. Baaijens, and M. Bulters, “Numerical\nanalysis of flow mark surface defects in injection molding flow,” J. Rheol. 46, 651–669\n(2002).\n35A. C. Bogaerds, M. A. Hulsen, G. W. Peters, and F. P. Baaijens, “Stability analysis of\ninjection molding flows,” J. Rheol. 48, 765–785 (2004).\n36Y. Murata, H. Yokoi, M. Nagaya, S. Nagami, and H. Watanabe, “Visual analysis of silver-\nstreak generation process by glass-inserted mold. glass insert kanagata ni yoru silver steak\nseisei katei no kaiseki,” Seisan Kenkyu (Monthly J. Inst. Ind. Sci., Univ. Tokyo);(Japan)\n45(1993).\n37H. Yokoi, N. Masuda, and H. Mitsuhata, “Visualization analysis of flow front behavior\nduring filling process of injection mold cavity by two-axis tracking system,” J. Mater.\nProcess. Technol. 130, 328–333 (2002).\n38F. Brochard and P. De Gennes, “Shear-dependent slippage at a polymer/solid interface,”\nLangmuir 8, 3033–3037 (1992).\n2339M. Zatloukal and J. Drabek, “Reduction of monomeric friction coefficient for linear iso-\ntactic polypropylene melts in very fast uniaxial extensional flow,” Physics of Fluids 33\n(2021).\n40X. Wei and T. Luo, “The effect of the block ratio on the thermal conductivity of amorphous\npolyethylene–polypropylene (pe–pp) diblock copolymers,” Phys. Chem. Chem. Phys. 20,\n20534–20539 (2018).\n41M. Doi, Introduction to polymer physics (Oxford university press, 1996).\n42M. Doi, S. F. Edwards, and S. F. Edwards, The theory of polymer dynamics , Vol. 73 (oxford\nuniversity press, 1988).\n43S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” J. Comput.\nPhys. 117, 1–19 (1995).\n44M. Schulz, R. Winkler, and P. Reineker, “Reptation of polymer chains: A combined monte\ncarlo and molecular-dynamics study,” Phys. Rev. B 48, 581 (1993).\n24" }, { "title": "2401.17824v1.A_Survey_of_Pre_trained_Language_Models_for_Processing_Scientific_Text.pdf", "content": "A Survey of Pre-trained Language Models for Processing Scientific Text\nXANH HO∗,National Institute of Informatics, Japan\nANH KHOA DUONG NGUYEN∗,Independent Researcher, Vietnam\nAN TUAN DAO∗, JUNFENG JIANG∗, and YUKI CHIDA∗,The University of Tokyo, Japan\nKAITO SUGIMOTO∗,NTT Communications Corporation, Japan\nHUY QUOC TO, University of Information Technology, Vietnam\nFLORIAN BOUDIN, JFLI, CNRS, National Institute of Informatics, Japan & LS2N, Université de Nantes, France\nAKIKO AIZAWA, National Institute of Informatics, Japan\nThe number of Language Models (LMs) dedicated to processing scientific text is on the rise. Keeping pace with the rapid growth\nof scientific LMs (SciLMs) has become a daunting task for researchers. To date, no comprehensive surveys on SciLMs have been\nundertaken, leaving this issue unaddressed. Given the constant stream of new SciLMs, appraising the state-of-the-art and how they\ncompare to each other remain largely unknown. This work fills that gap and provides a comprehensive review of SciLMs, including an\nextensive analysis of their effectiveness across different domains, tasks and datasets, and a discussion on the challenges that lie ahead.1\nCCS Concepts: •Computing methodologies →Natural language processing ;Natural language processing .\nAdditional Key Words and Phrases: Pre-trained language models, scientific text, comprehensive analysis, scientific language models\n(SciLMs)\nACM Reference Format:\nXanh Ho∗, Anh Khoa Duong Nguyen∗, An Tuan Dao∗, Junfeng Jiang∗, Yuki Chida∗, Kaito Sugimoto∗, Huy Quoc To, Florian Boudin,\nand Akiko Aizawa. 2024. A Survey of Pre-trained Language Models for Processing Scientific Text. 1, 1 (February 2024), 54 pages.\nhttps://doi.org/aaaaaaa.aaaaaaa\n1Resources are available at https://github.com/Alab-NII/Awesome-SciLM.\n∗The first six authors contributed equally to this research.\nAuthors’ addresses: Xanh Ho∗, National Institute of Informatics, Tokyo, Japan, xanh@nii.ac.jp; Anh Khoa Duong Nguyen∗, Independent Researcher,\nVietnam, dnanhkhoa@live.com; An Tuan Dao∗, dtan@nii.ac.jp; Junfeng Jiang∗, jiangjf@is.s.u-tokyo.ac.jp; Yuki Chida∗, chida@nii.ac.jp, The University\nof Tokyo, Tokyo, Japan; Kaito Sugimoto∗, NTT Communications Corporation, Tokyo, Japan, kaito.sugimoto@ntt.com; Huy Quoc To, University of\nInformation Technology, Ho Chi Minh, Vietnam, huytq@uit.edu.vn; Florian Boudin, JFLI, CNRS, National Institute of Informatics, Tokyo, Japan & and LS2N,\nUniversité de Nantes, Nantes, France, florian.boudin@univ-nantes.fr; Akiko Aizawa, National Institute of Informatics, Tokyo, Japan, aizawa@nii.ac.jp.\nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not\nmade or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components\nof this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to\nredistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.\n©2024 Association for Computing Machinery.\nManuscript submitted to ACM\nManuscript submitted to ACM 1arXiv:2401.17824v1 [cs.CL] 31 Jan 20242 Ho et al.\n1 INTRODUCTION\nThe introduction of Pre-trained Language Models (PLMs) [ 41,48,125,165,167,inter alia ] and Large Language Models\n(LLMs) [ 27,39,149,204,inter alia ] has had a profound impact on the landscape of NLP research [ 223], showcasing their\nremarkable effectiveness throughout a variety of NLP tasks [ 27,48]. This shift has prompted the development of LMs\nthat are capable of solving complex tasks, often involving language understanding, logical inference, or commonsense\nreasoning, in both general and specific domains. This is notably the case in the scientific domain, where many well-\nstudied tasks, such as Named Entity Recognition (NER), Relation Extraction (RE), Question-Answering (QA), document\nclassification, or summarization to name a few, have benefited from the utilization of LMs. One pivotal factor contributing\nto these successes is the abundant availability of scientific texts. For example, almost 2 million biomedical articles where\nadded to PubMed in 2022, contributing to a cumulative total of 36 million publications.2The ever increasing growth\nin the volume of scientific literature enables LMs to effectively learn and ingest scientific knowledge, fostering their\ncapability to excel in a wide array of tasks.\nHowever, despite the wealth of research on LMs for processing scientific texts (hereby reffered to as SciLMs), there is\na currently no comprehensive survey on this subject. Thus, a complete picture of the evolution of SciLMs over the past\nfew years is currently lacking, resulting in an unclear understanding of the actual state of progress in these models.\nThis paper aims to bridge this gap and offers the first comprehensive review of SciLMs. It provides the descriptions for\nover 110 models published in the last few years, conducts an extensive analysis of their effectiveness across different\ndomains, tasks and datasets, and initiates a discussion on the challenges that will likely shape future research.\nIn the remainder of this section, we first show the overall structure of our survey. We then outline the scope of\nour research, focusing on six aspects: time scope, target language models, target domains, target scientific text, target\nlanguages, and target modalities. We subsequently explain how we collected related papers. Following this, we clarify\nthe distinctions between our survey and existing surveys. Finally, we present an overview of the landscape of SciLMs\nover the past few years in the form of an evolutionary tree.\n1.1 Structure of the Paper\nFigure 1 shows the overall structure of our survey. We provide background information in Section 2, including details\nabout LM architectures and existing scientific tasks, as well as the distinctions between scientific text and text in other\ndomains. Next, in Section 3, we systematically review all existing PLMs and LLMs for processing scientific text from\n2019 to September 2023, analyzing their popularity based on three main aspects: domain, language, and size. After that,\nwe analyze the effectiveness of SciLMs by considering the performance changes over time across different tasks and\ndatasets in Section 4. We conclude by highlighting current challenges and open questions for future studies in Section 5.\n1.2 Survey Scope\nTime Scope. The BERT model was released in October 2018. Our focus is on exploring PLMs for processing scientific\ntext; therefore, we mainly consider papers released from 2019 to September 2023 .\nTarget LMs. Recently, most state-of-the-art (SOTA) approaches for NLP tasks are based on PLMs and LLMS. They\nhave made a significant impact on scientific research; for example, the release of GPT-4 [ 149] has changed the research\ndirections in NLP [ 122,223]. In the scope of this paper, we use the term ‘language models’ (LMs) to refer to both PLMs\nand LLM in general. We designate PLMs and LLMs proposed for processing scientific text as SciLMs. In specific cases\n2https://www.nlm.nih.gov/bsd/medline_pubmed_production_stats.html\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 3\nEffect of SciLMs ( Sec. 4 )\nChallenges\n(Sec. 5 )PLMs for Processing\nScientific TextPreliminary\nInformation ( Sec. 2 )LM Architectures\nScientific TextContent\nStyleScientific Tasks & Datasets\nSourceBasic Information\nBERT-based PerformanceExploring Task Performance\nExisting SciLMs\n(Sec. 3 )SciLMs & KBs\nMove Beyond Simple TasksMulti-modal SciLMsBuild Large SciLMsSciLMs\nAnalysisMulti-domainChemical\nOther Domains\nSizesLanguagesDomainsSciLMs for non-BiomedicalSciLMs for non-English\nReliable SciLMsEvaluation & ComparisonFoundation\nSciLMs\nEvaluation\nof SciLMsBiomedical\nFig. 1. Overall structure of our survey.\nwhere we aim to emphasize the effectiveness of LLMs, we may use two terms: LLMs and SciLLMs, instead of LMs\nand SciLMs. It is noted that we do not consider general LMs fine-tuning on scientific downstream tasks are SciLMs.\nFurthermore, we exclusively concentrate on neural network LMs due to their popularity and superior performance\ncompared to non-neural network ones on many tasks. For example, we do not consider n-gram [ 26], Conditional\nRandom Fields [104], and Hidden Markov Model [13] in our research.\nTarget Domains. The purpose of our survey is to explore LMs for processing scientific text; therefore, within the\nscope of this paper, we aim to cover as many scientific domains as possible. Specifically, the SciLMs in our survey\nspan various domains, including computer science (CS), biomedicine, chemistry, and mathematics. We remain open to\nextending the domains in our research as new SciLMs are proposed for additional fields.\nTarget Scientific Text. In this survey, we consider an LM to be a SciLM when its training data includes scientific\ntext. In specialized domains like chemistry, there are specific types of text that are used to represent and communicate\ninformation unique to that field such as the molecule structures in SMILE [ 229] or SEFIE [ 100]. The LMs that are trained\non these strings are also considered in this study.\nTarget Languages. Similar to ‘Target Domains’, we also aim to encompass as many available SciLMs in different\nlanguages as possible in order to conduct a comprehensive review of LMs for processing scientific text.\nTarget Modalities. In addition to the text in scientific papers, there are other types of information, such as images\nor tables. However, in the scope of this paper, we mainly focus on LMs for scientific text. For a more comprehensive\ndiscussion on multi-modal PLMs, we refer readers to an extensive survey available here [225, 253].\n1.3 Papers Collection\nDue to the rapid growth of the topic, after the first phase where we comprehensively obtain related papers (from 2019\nto February 2023). We also perform the second phase to update new SciLMs in our list.\nManuscript submitted to ACM4 Ho et al.\nPhase 1: From 2019 to February 2023. We use SciBERT [ 15] as a seed paper, then manually check cited papers of\nSciBERT. At the time of writing this paper,3SciBERT has 1,857 citations.4In addition, to ensure the coverage of our\nsurvey, we also use BERT as a seed paper, then manually check cited papers of BERT by using the function ‘Search\nwithin citing articles’ of Google Scholar with three keywords: ‘ scientific text ’, ‘scientific papers ’, and ‘ scientific articles ’.\nMoreover, when reading related papers and related survey papers, we also check mentioned papers in the paper, if we\nfind that we are missing any related papers, we add them to our study.\nPhase 2: From February 2023 to September 2023. In this phase, we only check the cited papers of SciBERT from\n2023 in Google Scholar. At the time of checking (September 13), SciBERT had 639 citations. We manually checked the\ntitles and/or abstracts of these cited papers to find the newly released SciLMs.\n1.4 Related Surveys\nHan et al . [66] , Kalyan et al . [91] , Wang et al . [220 ,223], and Zhao et al . [259] are the most similar papers to ours.\nSpecifically, both [ 66] and [ 223] delve into PLMs themselves, discussing related topics such as the history of PLMs and\nLM architectures. Meanwhile, both [ 91] and [ 220] focus on surveying PLMs in the biomedical domain. They summarize\nnumerous existing PLMs which we also cover in Section 3. However, our paper concentrates more on exploring PLMs\nacross all domains, not solely in the biomedical domain. Additionally, we include sections that compare scientific text\nwith text in other domains (Section 2.3) and analyse the effectiveness of SciLMs (Section 4). In contrast, Zhao et al . [259]\nsummarize newly released LLMs but do not emphasize scientific text as our paper does.\n1.5 Landscape of SciLMs\nFigure 2 presents a tree illustrating the landscape of SciLMs from 2019 to September 2023. We observe the following\npoints: (1) The tree is quite large and dense, indicating the existence of numerous proposed SciLMs during the period\nfrom 2019 to September 2023. Additionally, more and more models are proposed each year, indicating an increase in the\nnumber of models annually. (2) Most SciLMs are encoder-based models (91 models), and among these, BERT-based\nmodels are most commonly used. This suggests that research on SciLMs is still primarily focused on encoder-based\narchitectures and has not yet generalized to other architectures. (3) As depicted in the tree, many nodes are blue\n(biomedical domain nodes), indicating that the majority of SciLMs are proposed for the biomedical domain. For details\nabout all SciLMs in our study, as well as additional observations about existing SciLMs, we refer readers to Section 3.\n2 PRELIMINARY INFORMATION\nWe first briefly summarize existing LM architectures and other related information. We then introduce existing tasks in\nprocessing scientific text. Finally, we present the distinctions between scientific text and text in other domains.\n2.1 A Brief Summary of Existing LM Architectures\n2.1.1 The core backbone. Today, almost all LMs rely on the Transformer architecture [ 213]. Transformer is a type of\nneural network designed for sequence modeling. The core idea behind this architecture is to use self-attention mecha-\nnisms [ 36,119] without recurrent or convolutional networks. This allows the efficient computation of representations\nfor input and output sequences, regardless of their lengths.\n3February 2023\n4Due to the limitation of Google Scholar, our review is limited to the top 1,000 citations.\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 5\nS2ORC-SciBERTCharacterBERTChemBERTaOAG-BERTBioELECTRABio-cliMatBERTUTH-BERTMedRoBERTa.nlPubMedELECTRAClinical-LongformerBioLinkBERTViHealthBERTKM-BERTChemBERTa-2MaterialBERT(Yoshitake)ALIBERTGatortronCPT-BigBirdBioformerDrBERTTurkRadBERTCamemBERT-bioEriBERTa\nBio-ELECTRABioMegatronSapBERTKeBioLMEntityBERTMaterialsBERT(Shetty) ProtSTRAMM \nPathologyBERTbsc-bio-ehr-esScholarBERT\nLBERTCovidBERT\nClinicalBERT (Kexin)BlueBERTBEHRTEhrBERTBERT-MIMICNukeBERTMed-BERTouBioBERTMC-BERTexBERTSINA-BERTMathBERT(Peng)BatteryBERTRadBERTProcessBERTManuBERTPharmBERTBIOptimusKEBLM\nCySecBERT\nBioBERTUmlsBERTLightweight\nMatSciBERTSsciBERTSciEdBERT\nRoBERTaBio-LMNukeLMChemBERTSecureBERTVarMAE\nGPT-2SciGPT2MedGPTChemGPTBioGPTMed-PaLMK2\nT5SciFiveBioBARTCSL-T5ViPubmedT5BioMedLMBioReaderClinical-T5BioMedGPTClinicalGPTBioNART\nBioELMoG-BERTELECTRAMed\nFLAIRSMedBERTSciDEBERTa Clinical FlairDRAGONMOTORPattonGIT-Mol \nBioFLAIREncodersDecodersEnc.-Dec.OthersBERT-XMLAcademicRoBERTaBioberturkUCSF-BERT\nPubMedBERTSPECTERGreenBioBERTbert-for-radiology2019201820202021202220232024\nBioMedBERTCODERChestXRayBERT\nClinicalBERT (Emily)SciBERTClimateBERTGalactica\nClinical XLNetProteinBERTBioALBERTBRLTM\nBERTOpen-SourceClosed-SourceMathBERT(Shen)ClinicalTransformerClinicalT5\nFig. 2. Evolutionary tree of SciLMs. The nodes are color-coded based on their domains: blue for biomedical, pink for chemical,\nyellow for multi-domain, green for other domains, and gray for general domain models. The node is filled in white if the model\nis closed-source; otherwise, it is open-source. The English version of a model is used if it has multiple languages, and the most efficient\nvariant is used if a model has multiple variants. SciLMs that use continual pretraining are represented as children of the model whose\nweights they initialize. Only popular models are depicted as parent nodes in the tree for clarity. SciLMs trained from scratch are\nplaced as leaves in the rightmost branch.\nTransformer is comprised of two types of modules: the encoder and the decoder. Both modules are made up of\nstacked network layers, each consisting of a self-attention sub-layer and a feed-forward sub-layer. A notable feature of\nthe Transformer architecture lies in its multi-head attention mechanism, which enables the parallel computation of\ndifferent attention patterns.\n2.1.2 Types of architecture. Current LMs can be categorized into four distinct types: Encoder-Decoder style, Encoder-\nonly style, Decoder-only style, and other styles. The following subsections briefly introduce each type of LM.\nEncoder-Decoder style. LMs following the Encoder-Decoder style are based on the Transformer architecture and\ndiffer according to their pre-training objectives. For example, T5 [ 167] is trained on span corruption prediction task and\nBART [110] is trained on five tasks, which can be largely split into text infilling and sentence permutation.\nEncoder-only style. LMs of Encoder-only style are built upon the Transformer encoder and also differ according to\ntheir selected pre-training objectives. Masked Language Modeling (MLM), Next Sentence Prediction (NSP), Sentence\nOrder Prediction (SOP), and Replaced Token Prediction (RTP) are basic objectives. For instance, the origin of Encoder-\nonly LM is BERT [ 48] and it is trained by MLM and NSP. RoBERTa [ 125], which is intended to improve BERT, is trained\nonly MLM. ELECTRA [ 41] uses RTP with a smaller LM, generating token-replaced sentences and predicting the replaced\nManuscript submitted to ACM6 Ho et al.\ntoken. This type of LMs can utilize all of the information of the sentence by its nature (this feature is sometimes called\nbidirectional), and are often used in classification tasks.\nDecoder-only style. LMs of Decoder-only style are based on the Transformer decoder and its main pre-training\nobjective is Next Token Prediction (NTP). This kind of models are used for generative tasks such as QA, dialog generation,\nand so on.\nOther types of LMs. In this part, we present several LMs that appear in Tables 6, 14, or 15 but do not belong to the\nthree aforementioned types.\n(1)ELMo [ 159] is an LM that concatenates the representations of forward and backward LSTMs. ELMo introduces\nthe concept of contextualized word embeddings, which has later been standardized in Transformer-based LMs.\n(2)Flair [ 5] is a text embedding library supporting various types of LMs (including those that do not use the\nTransformer architecture).\n(3)Graph Neural Network (GNN) [ 65,233] is a type of neural networks for processing graph structures. GNNs are\noften combined with LMs to handle texts and knowledge graphs simultaneously.\n2.2 Existing Tasks and Datasets in Scientific Articles\nWe divide existing tasks in scientific articles into two sub-groups: scientific text mining andscientific text applica-\ntion. Figure 3 presents a summary of the tasks within each group in our classification.\n2.2.1 Scientific Text Mining. The purpose of this group is to mine existing scientific articles to extract knowledge, such\nas constructing a scientific dataset or a knowledge graph (KG) from unstructured text. We begin by discussing tasks\nrelated to KG construction, such as NER and RE. Following that, we present essential information on the list of existing\nscientific research datasets that we are aware of.\nKnowledge Graph Construction. There are many tasks related to KG construction. As listed in Figure 3, most\nof these tasks involve entities or relations within the KG, such as NER, entity linking, and RE. In Section 4.1, when\nwe analyse the effectiveness of SciLMs, we find that many tasks within this group are among the top 20 most popular\ntasks used for evaluation. Specifically, these tasks include NER, RE, PICO extraction, entity linking, disambiguation,\nText Mining Text Application\nKnowledge Graph Construction\nNamed entity recognition\nRelation extraction\nNamed entity disambiguation\nCoreference resolution\nEntity linking, ...\nScientific Dataset Construction\nAnalyzing the research data\nScience research corpus\nWorkflow scientific miningText Understanding\nScientific verification\nNatural language inference\nDocument analysis\nSemantic search\nCitation recommendation\nScientific reviewing, ...\nText Generation\nAutomatic related work generation\nAutomated evidence synthesis\nCitation text generation\nSummarization\nText Understanding & Generation\nQuestion answering\nFig. 3. Existing tasks in processing scientific text.\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 7\nand dependency parsing. We also observe that among the top 20 popular datasets used for evaluation, many belong to\nthe KG construction group. We provide important information related to these datasets in Table 1.\nTable 1. Details information of the popular datasets related to KG construction. (Sorted based on popularity in Section 4).\nRank Year Dataset Task Size Source Domain\n1 2014 NCBI-disease [51] NER 793 PubMed abstracts Biomedical\n2 2016 BC5CDR-disease [113] NER 1,500 PubMed abstracts Biomedical\n3 2004 JNLPBA [43] NER 2,404 MEDLINE abstracts Biomedical\n4 2016 ChemProt [101] RE 1,820 PubMed abstracts Biomedical\n5 2016 BC5CDR-chemical [113] NER 1,500 PubMed abstracts Biomedical\n7 2008 BC2GM [193] NER 20,000 MEDLINE sentences Biomedical\n8 2013 DDI [70] RE 1,025 MEDLINE abstracts Biomedical\n9 2011 i2b2 2010 [210] NER or RE 871 Patient reports Clinical\n11 2015 GAD [24] RE 5,330 PubMed abstracts Biomedical\n12 2015 BC4CHEMD [99] NER 10,000 PubMed abstracts Biomedical\n14 2013 Species-800 [151] NER 800 MEDLINE abstracts Biomedical\n15 2013 i2b2 2012 [196] NER 310 Clinical records Clinical\n17 2010 LINNAEUS [60] NER 153 PubMed articles Biomedical\n18 2018 EBM-NLP [147] PICO Extraction 5,000 MEDLINE abstracts Biomedical\nScientific Dataset Construction. With the rapid growth of the research community, numerous papers are published\nevery day. Therefore, collecting and processing existing research papers for downstream tasks or LMs plays a crucial\nrole when working with scientific text. Table 2 presents a list of scientific research datasets that we are aware of.\nTable 2. Existing scientific text datasets.\nYear Dataset Size Source Domain Available Information\n2009 AAN [164] 25K ACL Anthology Computational Linguistics Metadata, Citations\n2014 CiteSeer𝑋[30] + RefSeer [80] 1.0M CiteSeer𝑋+ DBLP Multi Metadata, Citations\n2018 CL Scholar [190] 40K ACL Anthology Computational Linguistics Metadata, Full-text\n2019 Bibliometric-Enhanced arXiv [176] 1M arXiv All domains in arXiv Metadata, Citations, Full-text\n2019 NLP4NLP [134] 65K 34 Conferences and Journals Speech and NLP Metadata, Citations\n2020 NLP Scholar [141] 45K ACL Anthology Computational Linguistics Metadata, Citations\n2020 S2ORC [127] 81.1M Semantic Scholar MultiMetadata, Full-text,\nCitations, Figures, Tables\n2022 D3 [217] 6.3M DBLP CS Metadata, Citations\n2022 NLP4NLP+5 [135] 90K 34 Conferences and Journals Speech and NLP Metadata, Citations\n2.2.2 Scientific Text Application. The purpose of this group is to focus on high-level tasks related to scientific under-\nstanding and scientific text generation. We further divide this group into three subcategories as follows.\nText Understanding. Scientific text understanding is often more challenging than general text understanding\nbecause it requires domain knowledge to comprehend specific terms. Currently, various tasks are used to test the\nunderstanding ability of models on scientific text, including scientific claim verification, natural language inference\n(NLI), document analysis, and paper evaluation. For more tasks, we refer readers to Figure 3.\nText Generation. This group emphasizes the ability to automatically generate scientific text. Tasks in this group\ninclude automatic related work generation, automated evidence synthesis, citation text generation, and summarization.\nText Understanding and Generation. We consider a QA task to belong to this group because it requires both\nunderstanding and generation abilities. From Section 4.1, we observe that only two QA datasets appear in the top 20\nManuscript submitted to ACM8 Ho et al.\npopular datasets used to evaluate SciLMs. However, these datasets do not appear as frequently when compared to\ndatasets from other tasks. Specifically, PubMedQA is used 10 times, BioASQ is used 7 times, while NCBI-disease is\nused 27 times. We argue that the QA task, which includes both testing understanding and generation, is an important\nevaluation criterion for future SciLMs. Therefore, we briefly summarize important information about existing QA\ndatasets in Table 13 (in Appendix A.1).\n2.3 Comparison between Scientific Text and Text in Other Domains\nScientific text has many special characteristics compared to texts in other domains. In this section, we will discuss it\nfrom three aspects: content, style, and source, to show the main differences between scientific text and text in other\ndomains and how these characteristics help SciLMs achieve superior performances in some scenarios. Note that we\nonly discuss texts, leaving other modalities like figures and tables aside, because it is out of the scope of this paper.\nTable 3 summarizes our comparison between scientific text and text in other domains.\nTable 3. Comparison between scientific text and text in other domains.\nAspects Scientific Text Text in Other Domains\nContent-Vocabulary : Contains domain specific terminologies -Vocabulary : Understandable by everyone\n-Knowledge : Advanced knowledge in the scientific domains -Knowledge : Common sense in the real world\n-Reasoning : Many statements require rigorous logical rea-\nsoning-Reasoning : Some statements contain shallow\nreasoning paths\n-Citation : Required in many cases -Citation : Voluntarily included\nStyle-Tone : Mainly for researchers. Formal, objective, faithful -Tone : Written for everyone. Informal, subjec-\ntive, sometimes emotional\n-Structure : Well-organized with rich structural information\n(e.g., title, abstract, sections, keywords, etc.)-Structure : Casual\n-Language : Long texts (e.g., books or articles) -Language : Short texts (e.g., reviews or tweets)\nSource-Amount : Growing rapidly, large-scale high-quality corpora\navailable. Not sufficient for training LLMs from scratch-Amount : Unlimited (Internet crawling)\n-Preprocessing : OCR or PDF parsing to extract texts from\npapers-Preprocessing : Careful filtering\n2.3.1 Content.\nVocabulary. According to the definition of scientific text [ 175], it is a type of written text that contains information\ndiscussing concepts, theories, or other series of topics that are based on scientific knowledge like medicine, biology, and\nchemistry. Therefore, compared to the text in other domains, scientific text usually has a larger vocabulary size including\nmany terminologies. For example, biomedical domain texts contain a considerable number of domain-specific proper\nnouns (e.g. BRCA1, c.248T>C) and terms (e.g. transcriptional, antimicrobial) [ 108], which are understood by most of the\nbiomedical researchers. Some of the vocabularies are new from before. Note that different from the new words from\nother domains like Social Network Sites (SNS), these new terminologies are usually followed by clear definitions and\ndetailed explanations described in some sections like ‘Introduction’ or can be found in their cited references. Therefore,\npre-training on self-contained scientific text is beneficial for new word discovery.\nBased on this characteristic, some LMs [ 15,111,127] that adopted pre-training from scratch can design tokenizers\nwith domain-specific vocabulary and learn better representations for terminologies. Some other works [ 132,199,242]\nchose to add new embeddings for extended vocabulary when performing continuous pre-training. Table 4 shows the\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 9\noverlapping rates between SciLMs and LMs in the general domain. We can see that the vocabulary overlapping rate\nbetween SciBERT and BERT is lower than that between SciBERT and PubMedBERT. This low Jaccard Index indicates\nthat many words or terminologies from scientific domains are not included in the vocabulary set used by LMs in the\ngeneral domain. Moreover, it also reflects that the distributions are different in scientific domains and the general\ndomain, which enhances the necessity of pre-training on scientific text.\nTable 4. Vocabulary Similarity Matrix (Jaccard Index).\nBERT SciBERT PubMedBERT\nBERT 1.00 0.28 0.25\nSciBERT 0.28 1.00 0.49\nPubMedBERT 0.25 0.49 1.00\nKnowledge. The main purpose of the scientific text is to share and report advanced research findings, theories,\nknowledge, and analysis to others who are specialized in related fields in a clear, understandable, and logical manner.\nResearchers usually catch up with the advanced development of science and obtain some remarkable knowledge for\nspecific tasks like drug discovery by reading published papers from journals and conferences. Therefore, to solve these\nprofessional and challenging tasks using LMs, we also need to pre-train them with scientific text that contains many\nadvanced technologies and new knowledge. For example, many domain-specific LMs pre-trained on scientific papers\n[4,64,69] can be applied to drug discovery and development, molecule synthesis, and materials discovery. When\nfine-tuning these LMs for downstream tasks in the same domain, it is easier to adapt and they usually perform better\nthan those LMs pre-trained only in the general domain.\nReasoning. Scientific text is usually complicated compared to general text. As we described in the previous section,\none of the core purposes of scientific papers is to propose something new to peers in their fields. Therefore, most of the\ncontents describe research findings with complicated reasoning processes. Also, some scientific papers contain complex\nformulas, which can be found in some mathematical and chemical papers. However, most of the current scientific\nLMs do not use such kind of information. Exploiting this information can yield better performances in understanding\ncomplex concepts and reasoning [200].\nCitation. In the scientific domain, supporting materials are required to be included in many cases. For example, in\na scientific paper, researchers are required to cite related literature to support their statements. Except for providing\nsupporting information, such citations also contain the relationship between the contents and the supporting materials.\nYasunaga et al . [244] utilized the information from the citations and introduced a novel training objective, document\nrelation prediction, to improve the language understanding ability of their model.\n2.3.2 Style.\nTone. The audiences of the scientific text are mainly researchers, scholars, and academics who are knowledgeable in\nthe specific field of research, whereas texts in other domains like news are available for everyone. Therefore, the prior\nconsideration in writing scientific texts is whether the texts are formal, objective, and faithful to convey information\nand support authors’ arguments with evidence. On the contrary, texts in other domains like blogs or novels may be\ninformal, subjective, and emotional that emphasize entertaining, persuading, or expressing authors’ personal opinions.\nTherefore, LMs pre-trained in scientific domains can learn to generate fluent and professional scientific texts for many\napplications including assisting academic writing [ 130]. Meanwhile, with the increasing push for open science and\nManuscript submitted to ACM10 Ho et al.\npublic knowledge, many researchers are also trying to promote their work to wider audiences now, including students,\njournalists, policymakers, and the general public. To achieve this goal, Scientific Text Simplification has become a\npromising research topic recently, which aims to simplify scientific texts for non-expert readers [54, 148].\nStructure. Scientific papers are well-organized with rich structural information, including title, abstract, content,\ntable, etc. For example, in the field of health sciences, Sollaci and Pereira [194] pointed out that a very standardized\nstructure is widely adopted in scientific writing, known as introduction, methods, results, and discussion (IMRAD).\nEspecially, some useful knowledge can be extracted from this structural information. For example, titles and abstracts\nusually provide rich semantics of the whole paper. Therefore, some work [ 140,185] extracted the abstract or title as\nthe summary of scientific papers for effective pre-training. Also, some work [ 200] adopted the keywords as a useful\nelement to filter undesired papers from pre-training corpora. It should be noted that texts from some other domains also\nhave structural information. For example, news articles are usually written in a certain way that readers are hooked by\nreading the first sentences, also known as lead sentences. Therefore, these sentences can be considered as a proxy for\nthe summary of a piece of news. However, these annotations may not always be reliable, and previous work on PLMs\nin the general domain usually overlooked this information.\nLanguage. Compared to text in other domains, scientific text is longer and usually contains multiple pages like books\nand novels. We select several popular pre-training corpora in the general domain and scientific domains and calculate\ntheir linguistic statistics. In the following analysis, we compute the statistics of all subsets of the Pile [ 59] containing 22\nsources from general and scientific domains. Since it is diverse and can be easily accessible, we believe it can serve as a\nrepresentative set of commonly used pre-training corpora. Note that different tokenizers can produce different statistics\nresults, like different numbers of tokens. Therefore, without loss of generality, we use the PunktSentenceTokenizer\nandNLTKWordTokenizer from the NLTK [ 18] library to segment documents into sentences based on punctuations and\nsegment sentences into words mainly based on space, respectively. The statistics can be found in Table 5.\nGenerally, we use crawled texts from the Internet to do pre-training [ 27,48,165]. We notice that sentences from the\nInternet usually have an average shorter length (25.94 words), which is not enough for pre-training an LM to solve\ntasks that require long-term consistency ability. Though sentences from GitHub and Ubuntu IRC have many words, we\nbelieve it is because there are many punctuations within a sentence, producing more ‘words’ than they actually have.\nBesides, we also calculate the depth of the syntactic tree and the readability of each sentence to show the complexities\nof scientific text and text in other domains. We adopt spacy [ 74] to compute the syntactic tree depth of sentences and\ncompute the Flesh-Kincaid Grade [ 97] to evaluate their readabilities. With these metrics, we see that the complexity of\nscientific text is similar to the texts from the Internet, but scientific text is much more complex than texts from other\ndomains like books, TV, news, email, etc, which makes them suitable for improving reasoning ability of SciLMs.\nFurthermore, scientific papers, especially peer-reviewed papers, are published after several manual discussions and\nproofreading, leading to extremely high quality compared to texts in other domains like SNS. Most of them merely\nhave grammar issues. Therefore, language models pre-trained with scientific text can also produce high-quality texts.\n2.3.3 Source.\nAmount. In the domain of scientific text, large-scale unsupervised corpora are freely available and the amount is\nstill growing rapidly. The PubMed Abstracts dataset and PMC Full-text articles that Lee et al . [108] used contain 4.5B\nand 13.5 tokens respectively, whereas the entire English Wikipedia contains only 2.5B tokens. Moreover, the PubMed\nsubset of a high-quality cleaned dataset, the Pile [ 59], contains 50B tokens, which is enough for training a medium-size\nLM (e.g., 2.7B) following the recommendation from Hoffmann et al. [71].\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 11\nTable 5. Statistics of some commonly used pre-training corpora.\nCorpus Domain #Word/Sentence Syntactic Tree Depth Readability #Word (B)General TextPile-CC\nWeb23.47 24.82 9.90 46.79\nOpenWebText2 24.27 25.10 9.85 23.73\nWikipedia (en) 24.87 29.69 13.05 10.23\nStackExchange 38.33 32.61 9.43 13.40\nAverage (micro) 25.94 26.53 10.16 Sum 94.15\nBooks3\nBook18.51 19.50 6.62 29.95\nBookCorpus2 14.48 16.98 5.44 3.89\nGutenberg (PG-19) 19.59 19.34 5.30 1.17\nAverage (micro) 18.10 19.21 6.44 Sum 35.01\nOpenSubtitles\nTV10.17 10.49 1.64 5.33\nYoutubeSubtitles 17.42 39.66 27.92 0.87\nAverage (micro) 11.19 14.58 5.33 Sum 6.20\nGithub Code 90.86 103.61 20.95 18.22\nUbuntu IRC Chat 40.83 29.98 9.47 1.15\nEuroParl Multilingual 26.78 22.67 11.79 1.87\nHackerNews News 21.24 25.24 9.89 1.19\nEnron Emails Email 21.71 22.52 6.66 0.25Scientific TextarXiv CS+Math+Physics 41.67 27.44 10.06 25.25\nPubMed Abstracts Biomed 24.16 25.43 14.02 6.59\nPubMed Central Biomed 33.42 28.76 11.81 34.47\nFreeLaw Law 20.15 19.43 6.31 13.99\nUSPTO Backgrounds Law 25.89 27.85 12.86 7.87\nDM Mathematics Math 19.22 12.66 1.84 6.55\nPhilPapers Philosophy 23.52 24.54 11.74 0.75\nNIH ExPorter Grant 27.97 29.93 16.12 0.72\nAverage (micro) Scientific 31.32 25.63 10.14 Sum 96.19\nBut it should also be aware that with the development of LLMs, the current amount for pretraining scientific LLMs\nfrom scratch may not be enough. Compared to texts in the general domain, with well-designed filtering strategies, texts\nfrom the Internet become a dominant resource for pre-training. For example, the filtered CommonCrawl that Brown\net al. [27] used contains 410B tokens, which is much more than the existing corpora in scientific domains. Furthermore,\nin some scientific fields, collecting enough high-quality data is still a challenge. For example, in nuclear science, Jain\net al. [81] only collected 7k internal reports. After preprocessing, they only obtained 8M words for language modeling\npre-training, which is limited. Therefore, how to explore more scientific texts for pre-training and how to pre-train an\nexcellent scientific foundation model with limited scientific texts is a promising direction in the near future. We leave a\ndetailed discussion of this challenge in Section 5.1.4.\nPreprocessing. Except for the biomedical domain, careful preprocessing is usually needed to obtain high-quality\nscientific texts. As for some old papers published many years ago, we need to perform OCR (Optical Character\nRecognition) or PDF parsing to extract texts. The most commonly used PDF parsing tool is Grobid [ 1], which was also\nused for preprocessing S2ORC dataset. Some other models (e.g., VILA [ 186]) and datasets (e.g., PubLayNet [ 260]) were\nalso proposed to support high-accuracy PDF parsing.\nManuscript submitted to ACM12 Ho et al.\n3 EXISTING LMS FOR PROCESSING SCIENTIFIC TEXT\nWe systematically organize and present 117 SciLMs in Tables 6, 14, and 15. Due to the space constraint, Tables 14\nand 15 are presented in Appendix B. We categorize surveyed SciLMs into four sections based on their pretraining\ncorpora: Biomedical, Chemical, Multi-domain, and Other Scientific Domains. Finally, we further analyze and discuss\nthe popularity of various aspects, such as data domain, language, and model size. In addition to exploring SciLMs, we\nalso provide an overview of LMs trained on non-scientific text during the same period. We summarize these models in\nAppendix B.1 to offer a more detailed understanding of the LM landscape.\n3.1 Biomedical Domain\nThis subsection provides a detailed summary of SciLMs specifically pretrained on biomedical corpora, ranging from\nwidely recognized sources such as PubMed, MIMIC-III, and ClinicalTrials, to COVID-19-related and manually constructed\ndatasets. Since the release of BERT [ 48], we have identified 85 existing SciLMs within the biomedical domain, showcasing\na diverse range of model architectures, pretraining objectives, and pretraining strategies. Over time, the architecture\nof these SciLMs has evolved from LSTM-based architectures to Transformer-based architectures such as BERT [ 48],\nALBERT [ 107], and RoBERTa [ 125]. Moreover, the size of these models has grown remarkably, starting from 12M to\nan impressive 540B parameters. Interestingly, models such as BERT and its variants with approximately 110M and\n340M parameters have become the preferred choices for biomedical research due to their cost-effectiveness and high\nperformance in downstream tasks [67].\n3.1.1 Bidirectional Language Modeling (Bi-LM). Bi-LM, a common pretraining objective before the rise of transformers,\ncombines forward and backward LMs to compute word probabilities based on previous and future words [220]. Since\n2019, a few studies have utilized LSTM-based architectures to pretrain LMs for the biomedical domain. For example,\nBioELMo [ 86] was pretrained from scratch using ELMo [ 159] architecture, while BioFLAIR [ 183] and Clinical Flair [ 173]\nwere pretrained using FLAIR [5] architecture with a continual pretraining strategy.\n3.1.2 Masked Language Modeling (MLM). MLM and NSP are two pretraining objectives introduced in the BERT\npaper [ 48]. MLM involves randomly masking tokens of input sequences and predicting the masked tokens with the\nmasked input. On the other hand, NSP aims to predict whether a given sentence is the next sentence.\nMLM-Based Models - Continual Pretraining. Since 2019, BERT architecture has become popular for PLMs in\nNLP. Several SciLMs have been developed using the BERT architecture, including BioBERT [ 108], BERT-MIMIC [ 189],\nClinical BERT (Emily) [ 6], ClinicalBERT (Kexin) [ 77], BlueBERT [ 158], BEHRT [ 114], EhrBERT [ 112], MC-BERT [ 255],\nexBERT [ 199], LBERT [ 227], CovidBERT [ 69], ChestXRayBERT [ 29], KM-BERT [ 96], and ClinicalTransformer [ 238].\nMC-BERT and KM-BERT were designed for Chinese and Korean languages, respectively. In addition to the original\nBERT, some studies have tried eliminating the NSP objective from the pretraining process. This modification is motivated\nby findings suggesting that the NSP objective could introduce unreliability and potentially hinder performance in\ndownstream tasks [ 88,107,125,240]. Notable studies in this context include UmlsBERT [ 137], SINA-BERT [ 198],\nPharmBERT [ 211], BIOptimus [ 156], CamemBERT-bio [ 203], and EntityBERT [ 118]. SINA-BERT was designed for\nPersian and uses Whole-Word Masking instead of Subword Masking. EntityBERT proposed a method to tag all entities\nin the input using XML tags, known as Entity-centric MLM [118].\nAttempts to utilize compact BERT-based models aim to address environmental concerns, enable real-time processing,\noffer lighter and faster alternatives, support edge computing, optimize parameter usage, overcome memory and speed\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 13\nTable 6. Existing LMs for scientific text. In the Training Objective column, MLM denotes Masked Language Modeling, NSP denotes\nNext Sentence Prediction, and SOP denotes Sentence Order Prediction. In the Type of Pre-training column, CPand FSdenote\nContinual Pretraining and From Scratch, respectively. In the Domain column, CSrepresents computer science, Biorepresents\nBiomedical domain, Chem represents Chemical domain, and Multi represents multiple domains. It is noted that the date information\nis chosen from the first date the paper appears on the Internet.\nNo. Date Model Architecture Training ObjectiveType of\nPre-trainingModel\nSizeDomain Pre-training Corpus\n1 2019/01 BioBERT [108] BERT MLM, NSP CP 110M Bio PubMed\n2 2019/02 BERT-MIMIC [189] BERT MLM, NSP CP 110M & 340M Bio MIMIC-III\n3 2019/03 SciBERT [15] BERT MLM, NSP CP & FS 110M Multi Semantic Scholar Corpus\n4 2019/04 BioELMo [86] ELMo Bi-LM FS 93.6M Bio PubMed\n5 2019/04 Clinical BERT (Emily) [6] BERT MLM, NSP CP 110M Bio MIMIC-III\n6 2019/04 ClinicalBERT (Kexin) [77] BERT MLM, NSP CP 110M Bio MIMIC-III\n7 2019/06 BlueBERT [158] BERT MLM, NSP CP 110M & 340M Bio PubMed + MIMIC-III\n8 2019/06 G-BERT [182]GNN\n+ BERTSelf-Prediction,\nDual-PredictionCP 3M Bio MIMIC-III\n9 2019/07 BEHRT [114] BERT MLM, NSP CP N/A Bio Clinical Practice Research Datalink\n10 2019/08 BioFLAIR [183] Flair Bi-LM CP N/A Bio PubMed\n11 2019/09 EhrBERT [112] BERT MLM, NSP CP 110M Bio Electronic Health Record Notes\n12 2019/11 S2ORC-SciBERT [127] BERT MLM, NSP FS 110M Multi S2ORC\n13 2019/12 Clinical XLNet [78] XLNetGeneralized Autoregressive\nPretrainingCP 110M Bio MIMIC-III\n14 2020/02 SciGPT2 [131] GPT2 LM CP 124M CS S2ORC\n15 2020/03 NukeBERT [81] BERT MLM, NSP CP 110M Chem NText\n16 2020/04 GreenBioBERT [163] BERTCBOW Word2Vec,\nWord Vector Space AlignmentCP 110M Bio PubMed + PMC\n17 2020/04 SPECTER [42] BERT Triple-Loss CP 110M Multi Semantic Scholar Corpus\n18 2020/05 BERT-XML [258] BERT MLM, NSP FS N/A Bio Electronic Health Record Notes\n19 2020/05 Bio-ELECTRA [150] ELECTRA Replaced Token Prediction FS 14M Bio PubMed + PMC\n20 2020/05 Med-BERT [168] BERTMLM,\nProlonged LOS PredictionFS 110M Bio Cerner Health Facts (Version 2017)\n21 2020/05 ouBioBERT [215] BERT MLM, NSP FS 110M Bio PubMed\n22 2020/07 PubMedBERT [61] BERTMLM, NSP,\nWhole-Word MaskingFS 110M Bio PubMed\n23 2020/08 MC-BERT [255] BERT MLM, NSP CP 110M & 340M BioChinese Biomedical Community QA\n+ Chinese Medical Encyclopedia\n+ Chinese Electric Medical Record\n24 2020/09 BioALBERT [145] ALBERT MLM, SOP CP 12M & 18M Bio PubMed + PMC\n25 2020/09 BRLTM [136] BERT MLM FS N/A Bio Private Electronic Health Record\n26 2020/10 BioMegatron [188] Megatron MLM, NSP CP & FS345M &\n800M & 1.2BBio PubMed + PMC\n27 2020/10 CharacterBERT [53] BERT + Character-CNN MLM, NSP FS 105M Bio MIMIC-III + PubMed\n28 2020/10 ChemBERTa [37] RoBERTa MLM FS 125M Chem SMILES from PubCHEM\n29 2020/10 ClinicalTransformer [238]BERT\nALBERT\nRoBERTa\nELECTRAMLM, NSP\nMLM, SOP\nMLM\nReplaced Token PredictionCP110M\n12M\n125M\n110MBio MIMIC-III\n30 2020/10 SapBERT [120] BERT Multi-Similarity Loss CP 110M Bio UMLS\n31 2020/10 UmlsBERT [137] BERT MLM CP 110M Bio MIMIC-III\n32 2020/11 bert-for-radiology [25] BERT MLM, NSP CP & FS 110M Bio Chest Radiograph Reports\n33 2020/11 Bio-LM [111] RoBERTa MLM FS 125M & 355M Bio PubMed + PMC + MIMIC-III\n34 2020/11 CODER [249]PubMedBERT\nmBERTContrastive Learning CP110M\n110MBio UMLS\n35 2020/11 exBERT [199] BERT MLM, NSP CP N/A Bio ClinicalKey + PMC\n36 2020/12 BioMedBERT [33] BERT MLM, NSP CP & FS 340M Bio BREATHE Dataset v1.0\n37 2020/12 LBERT [227] BERT MLM, NSP CP 110M Bio PubMed\n38 2021/03 OAG-BERT [124] BERT MLM FS 110M Multi AMiner + PubMed\n39 2021/04 CovidBERT [69] BERT MLM, NSP CP 110M Bio Covid-19 Related Corpora\n40 2021/04 ELECTRAMed [140] ELECTRA Replaced Token Prediction FS N/A Bio PubMed\n41 2021/04 KeBioLM [248] PubMedBERTMLM, Entity Detection,\nEntity LinkingCP 110M BioPubmed Docs\nfrom PubMedDS\nManuscript submitted to ACM14 Ho et al.\ncxonstraints, and enhance performance in NLP tasks. [ 107,178]. For instance, Lightweight Clinical Transformers [ 172]\nuses DistilBERT [ 178] architecture to distill knowledge during the pretraining phase, which reduces the size of the\nBERT model by 40% while retaining 97% of its language understanding capabilities. Similarly, BioALBERT [ 145]\nuses ALBERT [ 107] architecture, which is BERT-based but with much fewer parameters. BioALBERT has 12M-18M\nparameters, making it the smallest biomedical model in our survey. Additionally, It uses a self-supervised loss for SOP\nproposed in ALBERT, which helps maintain inter-sentence coherence.\nBERT-based alternatives, which are more advanced than BERT, have also been applied in the biomedical domain.\nSome works rely on Longformer [ 16] or BigBird [ 250] architectures for handling longer input. For instance, Clinical-\nLongformer [ 115], CPT-Longformer [ 52], and EriBERTa-Longformer [ 46] utilize the Longformer architecture, whereas\nClinical-BigBird [ 115] and CPT-BigBird [ 52] use the BigBird architecture. Additionally, RadBERT [ 237] uses RoBERTa\narchitecture [125], which extends BERT with changes to pretraining including dynamic masking and no NSP.\nSeveral specifically tailored architectures have been developed using BERT as the base model. DRAGON [ 243],\nG-BERT [ 182], KeBioLM [ 248], and CODER [ 249] focus on incorporating KGs through methods such as GNNs, KG\nLinking Tasks, or Contrastive Learning. Moreover, several LMs have been developed for specific use cases. These models\ninclude MOTOR [ 117] and RAMM [ 247], using additional Contrastive Learning and Image-Text Matching objectives for\npretraining multi-modal LMs. ViHealthBERT [ 139] incorporates Capitalized Prediction to improve NER for Vietnamese.\nGreenBioBERT [ 163] uses CBOW Word2Vec [ 138] and proposes Word Vector Space Alignment to expand wordpiece\nvectors of a general-domain PLM. SapBERT [ 120] presents Self-Alignment Pretraining to learn to self-align synonymous\nbiomedical entities. KEBLM [ 105] incorporates lightweight adapter modules to encode domain knowledge in different\nlocations of a backbone PLM. Finally, BioNART [ 11] proposes a non-autoregressive LM that enables fast text generation.\nMLM-Based Models - Pretraining From Scratch. Here, we also have several SciLMs that utilize the original\nBERT architecture. These include BRLTM [ 136], AliBERT [ 17], and Gatortron [ 239], where AliBERT is for French, while\nGatortron has a model size of 8.9B, which surpasses the average model size in this domain by more than 40 times. On\nthe other hand, various models without the NSP objective include BERT-XML [ 258], ouBioBERT [ 215], UTH-BERT [ 94],\nPathologyBERT [ 179], UCSF-BERT [ 197], PubMedBERT [ 61], and Bioformer [ 56], where UTH-BERT is for Japanese,\nand PubMedBERT uses Whole-Word Masking.\nSeveral other LMs were developed using variations of BERT, such as Bio-LM [ 111], MedRoBERTa.nl [ 214], bsc-bio-\nehr-es [ 32], DrBERT [ 103], EriBERTa [ 46], and Bio-cli [ 31]. These models were pretrained on biomedical data using the\nRoBERTa architecture. MedRoBERTa.nl, DrBERT, and Bio-cli were designed for Dutch, French, and Spanish, respectively.\nAdditionally, Bio-cli uses Whole-Word Masking instead of Subword Masking. SMedBERT [ 256] is an LM that was\npretrained on Chinese corpora. It incorporates deep structured semantics knowledge from neighboring structures\nof linked entities, which consists of entity types and relations. It utilizes objectives like Masked Neighbor Modeling,\nMasked Mention Modeling, MLM, and SOP.\nIn addition, there exist different custom architectures of BERT. For instance, CharacterBERT [ 53] eliminates the\nwordpiece [ 232] system and instead utilizes a CharacterCNN [ 257] module, similar to ELMo’s first layer representation,\nto represent any input token without splitting it into wordpieces. Med-BERT [168], on the other hand, incorporates a\ndomain-specific pretraining task to predict the prolonged length of stay in hospitals (Prolonged LOS). This task helps\nthe model learn more clinical and contextualized features for each input visit sequence and facilitates certain tasks.\nThe visit sequence refers to the order in which visits occur in a patient’s EHR data. Meanwhile, ProteinBERT [ 23]’s\npretraining combines language modeling with a novel task of Gene Ontology annotation prediction, enabling it to\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 15\ncapture a wide range of protein functions. Lastly, BioLinkBERT [ 244] takes advantage of document links to capture\nknowledge dependencies or connections across multiple documents.\nMLM-Based Models - Both Pretraining Strategies. We also consider SciLMs that have experimented with\nboth pretraining strategies. Some models, such as bert-for-radiology [ 25], BioMedBERT [ 33], Bioberturk [ 209], and\nTurkRadBERT [ 208], use the original BERT architecture - with Bioberturk and TurkRadBERT designed for the Turkish\nlanguage. BioMegatron [ 188], on the other hand, implements a simple and efficient intra-layer model parallel approach\nthat enables training transformer models with billions of parameters.\n3.1.3 Replaced Token Detection (RTP). Models such as BioELECTRA [ 92], Bio-ELECTRA [ 150], ELECTRAMed [ 140], and\nPubMedELECTRA [ 202] utilize ELECTRA [ 41]’s pretraining objectives to pretrain their models from scratch. ELECTRA\nis a model that replaces MLM with a more sample-efficient pretraining task called RTP. It trains two transformer models:\nthe generator and the discriminator. The generator replaces tokens in the sequence, and the discriminator tries to\nidentify the tokens replaced by the generator. BioELECTRA also found that the FS strategy performs better than the CP\nstrategy on most BLURB [61] and BLUE [158] benchmark tasks.\n3.1.4 Generation-Based Models. Several generation-based SciLMs have been developed based on auto-regressive\npretraining objectives. SciFive [ 162], ClinicalT5 [ 129], Clinical-T5 [ 109], BioReader [ 58], and ViPubmedT5 [ 160] are\nall based on T5 [ 167] architecture, which uses the pretraining objective of generating the given sequences in an auto-\nregressive way, taking the masked sequences as input. All models, except Clinical-T5, were pretrained based on an initial\nmodel weight rather than being pretrained from scratch. Also, Clinical-T5 has experimented with both pretraining\nstrategies. ClinicalGPT [ 222] is a BLOOM-based model [ 180] that employs rank-based training for reinforcement\nlearning with human feedback to improve performance further. In another work, BioReader uses a retrieval-enhanced\ntext-to-text LM for biomedical, which augments the input prompt by fetching and assembling relevant scientific literature\nchunks from a neural database with about 60 million tokens centered on PubMed [ 58]. ViPubmedT5, on the other\nhand, was pretrained on Vietnamese corpora. Moreover, BioBART [ 246] utilizes BART [ 110] architecture, a denoising\nautoencoder, to pretrain a biomedical text-to-text LM via continual pretraining. On the other hand, BiomedGPT [ 254],\nBioGPT [ 130], and BioMedLM [ 20] were pretrained from scratch using GPT [ 165] architecture, while MedGPT [ 98] uses\ncontinual pretraining. Finally, two biomedical PLMs that use an autoregressive Transformer are Clinical XLNet [ 78]\nand Med-PaLM [ 191]. Clinical XLNet was continuously pretrained using XLNet [ 240] architecture, which utilizes\nbidirectional contexts for masked word prediction. Unlike autoregressive models like GPT, XLNet considers all possible\npermutations of the input sequence, allowing it to capture dependencies between words in both directions and resulting\nin a better understanding of the context. Med-PaLM, on the other hand, is the largest LM specialized for the medical\ndomain, with 540 billion parameters. It is an instruction prompt-tuned version of Flan-PaLM [40].\n3.2 Chemical Domain\nWe now focus on SciLMs specialized in the chemical domain, summarizing those pretrained on corpora like PubChem,\nMaterial Science, NIMS Materials Database, ACS publications, or custom datasets. We also consider SciLMs that were\npretrained on domains relevant to Material Science, Nuclear, and Battery, as these domains are part of the chemical\ndomain [ 2,79,187,206]. Ultimately, we have gathered a total of 13 qualified SciLMs. Our survey shows a predominant\nreliance on the BERT architecture for PLMs in the chemical domain. Among the 13 surveyed models, 11 were based on\nBERT, featuring parameters ranging from 110M to 355M. This suggests limited diversity in model architecture choices\nfor LM pretraining in the chemical domain.\nManuscript submitted to ACM16 Ho et al.\nBERT-Based Models. NukeBERT [ 81] and ProcessBERT [ 93] were both pretrained from a BERT checkpoint. In\ncontrast, MaterialsBERT (Shetty) [ 187] was pretrained from a PubMedBERT [ 61] checkpoint with Whole-Word Masking\ninstead of Subword Masking, both of which aim to specialize in the chemical domain. Meanwhile, MaterialBERT\n(Yoshitake) [ 245] utilizes the BERT architecture to pretrain from scratch on chemical corpora. In addition, various\nstudies have been conducted using different variants of BERT, such as non-NSP BERT or RoBERTa. For example,\nMatSciBERT [ 64] and NukeLM [ 28] are non-NSP BERT models, while ChemBERT [ 62] and a variant of NukeLM use\nthe RoBERTa architecture to continually pretrain on domain-specific data. On the other hand, MatBERT [ 206] and\nChemBERTa [ 37] are models based on non-NSP BERT and RoBERTa, respectively, that use the pretraining from scratch\nstrategy. Additionally, ChemBERTa-2 [ 4], a variant of ChemBERTa, employs Multi-task Regression as an additional\nobjective. It is also worth mentioning that BatteryBERT [ 79], a non-NSP BERT model, utilizes both pretraining strategies.\nGeneration-Based Models. Several architectures have been developed for molecular modeling. One such model is\nChemGPT [ 57], a large chemical GPT-based model with over one billion parameters. It is used for generative molecular\nmodeling and was pretrained from scratch on datasets consisting of up to ten million unique molecules.\nMulti-Modal Models. GIT-Mol [ 121] is a multi-modal LLM that integrates the Graph, Image, and Text information.\nTo perform this task, Liu et al . [121] proposed a novel architecture called GIT-Former, which can map all modalities\ninto a unified latent space.\n3.3 Multi-domain\nThis subsection presents SciLMs pretrained on multi-domain corpora, incorporating text data from diverse domains.\nReferred to as ‘Multi-domain’ for simplicity, these models leverage abundant data, accommodate various domains, and\nallow further fine-tuning for specific domains. For instance, SciBERT [ 15] was pretrained on the full text of 1.14M\nbiomedical and CS papers from the Semantic Scholar corpus [ 127]. Since the introduction of BERT, we have identified\n11 multi-domain SciLMs. The majority use BERT or similar architectures, while two others adopt generation-based\narchitectures. Most models prefer a size of 110M parameters, except for Galactica [ 200], which ranges from 125M to\n120B, making it the largest in this domain.\nBERT-Based Models. Multiple works use the BERT architecture to create a general PLM for multiple domains. For\nexample, SciBERT [ 15] and S2ORC-SciBERT [ 127] pretrained their models from scratch with BERT’s recipe. Other works\nlike OAG-BERT [ 124] and ScholarBERT [ 72] eliminate the NSP pretraining objective due to its limited contribution\nto downstream task performance. AcademicRoBERTa [ 236] and VarMAE [ 75] employ the RoBERTa architecture.\nIt is worth mentioning that VarMAE uses the continual pretraining strategy, while OAG-BERT, ScholarBERT, and\nAcademicRoBERTa started from scratch. AcademicRoBERTa was built for the Japanese language. On the other hand,\nSciDEBERTa [ 82] further pretrained DeBERTa [ 68] with the science technology domain corpus. This transformer-based\narchitecture aims to improve BERT and RoBERTa models with two techniques: a disentangled attention mechanism\nand an enhanced mask decoder [68].\nSpecialized Architecture-Based Models. Some works customize pretraining objectives. For instance, SPECTER [ 42],\na new LM initialized with SciBERT [ 15], adds Triple-loss to learn high-quality document-level representations by\nincorporating citations. Another example is Patton [ 85], which employs continual pretraining and a GNN-nested\nTransformer architecture. Its architecture includes two pretraining objectives: Network-contextualized MLM and\nMasked Node Prediction, which enable the creation of an LLM to capture inter-document structure information.\nGeneration-Based Models. Several efforts have been made to pretrain multi-domain LLMs using generation-based\narchitectures. For instance, CSL-T5 [ 116] was pretrained from scratch using T5 for Chinese, while Galactica [ 200] is a\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 17\n120B Autoregressive LM designed to store, combine, and reason about scientific knowledge. The model was pretrained\nfrom scratch on a vast scientific corpus of papers, reference materials, knowledge bases, and other sources.\n3.4 Other Scientific Domains\nIn addition to the well-explored domains mentioned earlier, our survey provides a comprehensive overview of SciLMs\npretrained in less commonly explored domains such as Climate, Computer Science, Cybersecurity, Geoscience, Manufac-\nturing, Math, Protein, Science Education, and Social Science. Across these domains, we observed a prevailing preference\nfor a model size of 110M, with the exception of one LLM containing 7B parameters.\nBERT-Based Models. There are different models based on the BERT architecture, some of which use the original\nBERT while others use advanced variants like non-NSP BERT and RoBERTa. For example, CySecBERT [ 14], SsciB-\nERT [ 185], ManuBERT [ 102], and SciEdBERT [ 126] are continually pretrained SciLMs designed for Cybersecurity,\nSocial Science, Manufacturing, and Science Education, respectively. While CySecBERT and SsciBERT use the original\nBERT architecture, ManuBERT and SciEdBERT use BERT without the NSP objective. Additionally, ClimateBert [ 228],\nSecureBERT [ 3], and MathBERT (Shen) [ 184] take advantage of the RoBERTa architecture to further pretrain on Climate,\nCybersecurity, and Math corpora, respectively. Note that ClimateBert is a distilled version of the RoBERTa-base model\nthat follows the same training procedure as DistilBERT.\nSpecialized Architecture-Based Models. There exist various BERT-based models specifically designed for specific\nuse cases. MathBERT (Peng) [ 157] is an example of a continual pretrained BERT-based model that utilizes MLM, Context\nCorrespondence Prediction, and Masked Substructure Prediction to learn representations and capture semantic-level\nstructural information of mathematical formulas. On the other hand, ProtST [ 234] is a framework built upon the BERT\narchitecture for multi-modality learning of protein sequences and biomedical texts. It includes tasks like Masked Protein\nModeling, Contrastive Learning, and Multi-modal Masked Prediction to incorporate different granularities of protein\nproperty information into a protein LM. ProtST was continuously pretrained on Protein corpora.\nGeneration-Based Models. There are several GPT-based models available. SciGPT2 [ 131] is a model specifically\ndesigned for the CS domain, and it was created through a process of continued pretraining of GPT2 [ 166] model.\nAnother example is K2 [ 47], which is an LLaMA model [ 204] comprising 7B parameters, and it was further pre-trained\non a text corpus specific to Geoscience.\n3.5 Summary and Discussion\nThis subsection discusses the popularity of LLMs specifically designed for processing scientific text. Our analysis takes\ninto account various factors such as domain, language, and model size. We present our findings using statistical data\nobtained from our survey on SciLMs, which are presented in Table 8, Table 7, and Figure 4.\n3.5.1 Domain-wise Distribution of SciLMs. Table 8 provides an overview of SciLM distribution across different domains.\nThe Biomedical domain has the highest number of existing models, with 85 in total. This dominance is due to the vast\namount of scientific literature available in the biomedical field, with PubMed being a prime example, accounting for\n17.5% of The Pile dataset [ 59]. The availability of vast and high-quality data within a specific field, such as the biomedical\ndomain, has made it easier to develop domain-specific LLMs that perform well on downstream domain-specific NLP\ntasks [ 177,220]. Consequently, researchers have been motivated to create pretrained LLMs for the biomedical domain,\nleading to the growth of SciLMs in this field. The Chemical domain has 13 existing models, making it the second-highest\nnumber of models among all other domains. Interestingly, there is a significant overlap between the Chemical and\nManuscript submitted to ACM18 Ho et al.\nTable 7. Distribution of SciLMs across different\nlanguages.\nLanguage # Model Names\nDutch 1 MedRoBERTa.nl\nKorean 1 KM-BERT\nPersian 1 SINA-BERT\nTurkish 2Bioberturk,\nTurkRadBERT\nVietnamese 2ViHealthBERT,\nViPubmedT5\nFrench 3AliBERT, DrBERT,\nCamemBERT-bio\nJapanese 3ouBioBERT,\nUTH-BERT,\nAcademicRoBERTa\nSpanish 3Bio-cli,\nEriBERTa,\nbsc-bio-ehr-es\nChinese 4MC-BERT, CSL-T5,\nSMedBERT,\nClinicalGPT\nEnglish 97The remaining models\nin Tables 6, 14 and 15Table 8. Distribution of SciLMs across different domains.\nDomain #\nBiomedical 85\nChemical 13\nMulti-domain 11\nCybersecurity and Math 2 per domain\nManufaturing, Computer Science,\nClimate, Protein, Social Science,\nGeoscience, and Science Education1 per domain\n3M\n12M\n43M\n110M\n340M\n770M\n2.7B\n8.9B\n30B\n120B\n540B\nModel Size (log scale)15102030405060Occurrences\nFig. 4. Distribution of model sizes.\nBiomedical domains, as seen in various Biomedical or Chemical datasets such as BC5CDR, JNLPBA, BC4CHEMD, and\nothers. This overlap presents an opportunity to leverage the vast amount of data available in the Biomedical domain to\nfacilitate the development of more effective LMs for chemistry-related tasks [ 143]. Besides, there are numerous potential\napplications for LM development in the Chemical domain, such as autonomous chemical research, drug discovery,\nmaterials design, and exploration of chemical space [ 19,22,142,192]. These emphasize the importance of language\nprocessing in chemistry-related research. There are also 11 multi-domain models that aim to cater to a broader range of\nscientific domains. Moreover, pretraining LLMs with mixtures of domains can enhance their ability to generalize to\ndifferent tasks and datasets [ 10,200,226]. By absorbing a wide range of knowledge, these models can gain a better\nunderstanding of multiple topics, resulting in better performance and greater versatility for diverse downstream tasks.\nOther domains like Cybersecurity, Math, Climate, CS, Geoscience, Manufacturing, Protein, Science Education, and\nSocial Science each have one or two models, indicating potential areas for future research and development.\n3.5.2 Language-wise Distribution of SciLMs. Table 7 shows SciLM prevalence across languages. English dominates\nwith 97 models, underscoring its role as the primary language for scientific communication. Other languages, such\nas Chinese, Spanish, Japanese, and French, also have a considerable presence, with multiple models developed for\nscientific text processing. However, Dutch, Korean, Persian, Turkish, and Vietnamese have fewer dedicated models,\nindicating that the need for scientific text processing in these languages has only recently attracted attention from the\ncommunity. This diversification in language usage underlines the global nature of scientific research and underscores\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 19\nTable 9. Examples of grouping task names.\nOriginal Name Grouped Name Original Name Grouped Name\nInformation Retrieval\nRetrievalDialogue\nDialogue Medical Question Retrieval Clinical Dialogue\nMathematical Information Retrieval Medical Conversation\nText Generation\nGenerationSentiment AnalysisSentiment\nAnalysisFormula Headline Generation Medical Sentiment Analysis\nKeyword Generation Sentiment Labeling\nQuestion AnsweringQuestion\nAnsweringDocument Multi-label Classification\nClassificationVisual Question Answering Text Classification\nMedical Visual Question Answering Discipline Classification\nthe importance of multilingual LMs to cater to researchers from diverse linguistic backgrounds. It is worth noting that\nscientific texts can come in various forms, such as medical records, theses, articles, speeches, textbooks, and books,\nwhich are often written in specialized technical language or non-English language for their intended audience.\n3.5.3 Distribution of Model Sizes. The distribution of model sizes is shown in Figure 4. Models within the size range of\n100M to less than 400M are the most preferred, with a total of 103 models. Among these, models with sizes of 110M and\n125M account for 58 and 11 models, respectively. This popularity can be attributed to the balance between efficiency and\neffectiveness. Many of these models are based on established architectures like BERT-Base and BERT-Large, allowing\nresearchers to leverage prior work while balancing computational cost and performance. There are 13 models with\nsizes less than 100M, which are likely favored for their cost-effectiveness and suitability for on-device applications.\nResearchers usually opt for these models when computational resources are limited, focusing on tasks where lighter\nmodels suffice. There are 6 relatively large models with sizes from 700M to less than 1B. These models offer enhanced\ncapabilities in handling complex scientific language nuances; however, their size presents challenges regarding training\ncost and computational requirements. Recent developments have seen a surge in attempts to build large-scale LMs\nwith sizes ranging from 1B to 540B, represented by 11 models. These models represent the cutting edge of language\nprocessing techniques, but they also pose significant challenges, including the need for vast corpora, extensive training\ntime, and substantial computational resources. While these models offer remarkable performance, their feasibility and\npracticality for widespread adoption in the scientific community remain topics of active research and debate.\n4 EFFECTIVENESS OF LMS FOR PROCESSING SCIENTIFIC TEXT\nIn this section, we first present the basic information related to tasks and datasets used by SciLMs, highlighting the\ntop 20 popular tasks and datasets (Section 4.1). We then explore the performance changes over time for five tasks:\nNER, Classification, RE, QA, and NLI (Section 4.2). Additionally, we discuss the detailed information about models that\noutperform previous ones or achieve the SOTA, such as the number of tasks and datasets they used. Finally, we zoom in\na bit closer to the performance when the architecture of the model is fixed (Section 4.3).\n4.1 Basic Information\nDue to the disparities in writing styles and terminologies among scientific papers, many tasks and datasets are labelled\nwith different names, such as ‘relation extraction’ and ‘relation classification’, or the EU-ADR dataset [ 212] can be\nwritten as ‘EU-ADR’ (in Lee et al . [108] ) or ‘EUADR’ (in Naseem et al . [145] ). To ensure consistency in our analysis, we\nManuscript submitted to ACM20 Ho et al.\nnormalize the names of both tasks and datasets. If the task names differ but the dataset names are similar, we carefully\nreview the task names to determine whether they should be grouped or kept separate. After this tedious manual process,\nwe found out that many different task names still look similar. Therefore, we rely on heuristics to group task names.\nSpecifically, we rely on cue words such as ‘classification’ to categorize name tasks into our list of predefined task names.\nFor instance, if the task name contains ‘classification’ (e.g., ‘sequence classification’), we categorize it as a classification\ntask. Table 9 presents examples of groups of task names produced by our method. It is noted that this method cannot\ngroup all different tasks perfectly, for example, the task ‘Relationship explanation task’ (in Luu et al . [131] ) is a type of\ngeneration task but the task name does not include the word ‘generation’. For simplicity, we keep it with its original\ntask name. All the details of task and dataset names of all SciLMs are presented in Table 16 in Appendix C.1.\nTable 10. Top-20 popular tasks and the number of SciLMs\nevaluated for each respective task.\nTask Name #\n1 Named Entity Recognition 58\n2 Classification 47\n3 Relation Extraction 33\n4 Question Answering 31\n5 Natural Language Inference 20\n6 Sentence Similarity 8\n7 Summarization 8\n8 PICO Extraction 7\n9 Retrieval 6\n10 Generation 6\n11 Sentiment Analysis 4\n12 Regression 4\n13 Recommendation 3\n14 Entity Linking 3\n15 Disambiguation 3\n16 Intrinsic Evaluation 3\n17 Dialogue 3\n18 Dependency Parsing 2\n19 Disease Prediction 2\n20 Citation Prediction 2Table 11. Top-20 popular datasets, along with the number of SciLMs evaluated\nfor each respective dataset and information on the task names.\nDataset Name # Task Names\n1 NCBI-disease 27 NER (23) or EN (1) or EL (3)\n2 BC5CDR-disease 21 NER (19) or EL (2)\n3 JNLPBA 21 NER\n4 ChemProt 21 RE\n5 BC5CDR-chemical 19 NER (18) or EL (1)\n6 MedNLI 18 NLI\n7 BC2GM 16 NER\n8 DDI 15 RE\n9 i2b2 2010 14 NER (9) or RE (5)\n10 HOC 13 Document Multi-label Classification\n11 GAD 12 RE\n12 BC4CHEMD 10 NER\n13 PubMedQA 10 QA\n14 Species-800 8 NER\n15 i2b2 2012 8 NER\n16 BC5CDR 8 NER\n17 LINNAEUS 7 NER\n18 EBM-NLP 7 PICO Extraction\n19 BIOSSES 7 Sentence Similarity\n20 BioASQ 7 QA\nIn summary, after grouping, there are 79 tasks and 337 datasets used to evaluate 117 SciLMs in our survey. It is worth\nnoting that these 79 tasks could be further grouped if we carefully examine the details. However, we find that this\nprocess is not necessary, so we choose to skip it. Tables 10 and 11 present the top-20 most popular tasks and datasets,\nand the number of SciLMs evaluated for each respective task and dataset. We observe that NER, Classification, RE, QA,\nand NLI emerge as the top five most popular tasks. For specifics, the top five datasets for the NER task are NCBI-disease,\nBC5CDR-disease, JNLPBA, BC5CDR-chemical, and BC2GM. Regarding the RE task, ChemProt and DDI stand out as the\ntop two datasets. MedNLI claims the top spot for the NLI task, while HOC leads as the most popular dataset for the\nDocument Multi-label Classification task, which is grouped under the classification task. For the QA task, PubMedQA\nand BioASQ are recognized as the two most popular datasets, although fewer SciLMs have been evaluated on these\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 21\ncompared to other datasets. In the next subsection, we delve deeper into exploring the task performance of the SciLMs\non these popular tasks and datasets.\n4.2 Exploring Task Performance\nIn this section, we first present charts for the five most popular tasks to visualize how SciLM performance changes over\ntime. We then analyze the list of SciLMs that outperform previous models or achieve SOTA results.\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-scoreBioALBERTAverage\nBest\nFig. 5. Average performance changes in the NER task. These scores are the average from the five NER datasets: NCBI-disease,\nBC5CDR-disease, JNLPBA, BC5CDR-chemical, and BC2GM.\n4.2.1 Performance Changes Over Time.\nNER Task. As shown in Tables 10 and 11, NER is the most popular task used to evaluate SciLMs. There are eleven\npopular NER datasets among the top-20 datasets used by SciLMs to evaluate their performance, and we use five of\nthem for drawing charts. From the Table 16 (in Appendix C.1), we obtain a list of SciLMs that were evaluated on\nthese five NER datasets. Subsequently, we assess the performance of the following SciLMs: BioBERT, GreenBioBERT,\nPubMedBERT, BioALBERT, Bio-LM, BioMedBERT, KeBioLM, SciFive, BioELECTRA, PubMedELECTRA, BioLinkBERT,\nBioReader, Bioformer, and BIOptimus. For clarity, we display the average performance (F1-score) changes of all five\ndatasets in Figure 5. Performance changes for each of the five datasets are detailed in Figure 13 (in Appendix C.2).\nWe note a consistent pattern in the average performance changes across the five datasets. In September 2020,\nBioALBERT achieved the highest score with an average F1-score of 94.7.5To date, the performance of BioALBERT\non these NER datasets remains unmatched by any of the proposed models. Our observations raise two main research\nquestions related to the NER task and dataset. The first question is: Does it imply that the NER task is already solved\nby current SciLMs? We believe that the answer is no. Based on the detailed results in Figure 13 (in Appendix C.2), we\nobserve that the F1-score of BioALBERT on the JNLPBA dataset is only 84.0. This suggests that the high scores on\nother datasets may be due to overfitting between the training data of BioALBERT and these NER datasets. Additionally,\nit would be interesting to evaluate the models further, such as assessing their performance on adversarial sets. The\nsecond question is: Why are newly proposed models unable to surpass the performance of BioALBERT? We believe there\nare several reasons for this, and here we discuss two that we consider are most important: (1) The architecture of\nlater SciLMs differs; they may experiment with alternative architectures, such as using T5 for SciFive, ELECTRA for\nBioELECTRA and PubMedELECTRA. (2) The research focus varies; subsequent studies may explore additional methods\n5We searched for papers discussing the highest scores achieved by BioALBERT, but we couldn’t find any. Additionally, BioALBERT has released its model\nweights on the GitHub repository. Therefore, we defer the in-depth analysis of fair comparisons and reliable results for BioALBERT to future studies.\nManuscript submitted to ACM22 Ho et al.\nfor solving the task rather than solely aiming for the highest performance. For instance, Yasunaga et al . [244] proposed\nnew types of LMs by incorporating link information between documents into the training dataset and loss. Additionally,\nPavlova and Makhlouf [156] introduces a SciLM by pre-training with a curriculum learning schedule.\n2019/06 2020/05 2020/09 2020/11 2021/05 2022/03 2022/12\nDate707580859095100HOC - F1-scoreBioALBERT BioLinkBERTHOC\nBest\n2020/07 2021/06 2021/12 2022/03 2022/09 2022/12\nDate707580859095100HOC - Micro F1-scoreBioELECTRABioLinkBERT BioGPTHOC\nBest\nFig. 6. Performance changes in the HOC dataset. Left: measure by F1-score ; Right: measure by Micro F1-score .\nClassification Task. As shown in Table 10, the classification task is the second most popular task. However, there\nare many different types of classification tasks, such as citation intent classification (e.g., ACL-ARC [ 90]) or formula topic\nclassification (e.g., TopicMath-100K [ 157]). This explains why only one classification dataset, namely HOC, appears in\nthe top 20 most popular datasets used to evaluate SciLMs. HOC denotes ‘Hallmarks of Cancer’; the HOC dataset consists\nof 1,499 cancer-related PubMed abstracts that have been annotated by experts. It includes 10 classes, each corresponding\nto one of the hallmarks of cancer. This is a multi-label classification task, and we note that the F1-score and micro\nF1-score are commonly used for comparison. We observe that only BioLinkBERT obtains both scores. Therefore, we\ncreate two charts for two lists of SciLMs. The performance changes of the HOC dataset are presented in Figure 6. On\nthe left side, they are evaluated using F1-score, with the list of SciLMs as follows: BlueBERT, ouBioBERT, BioALBERT,\nBio-LM, SciFive, BioLinkBERT, and BioReader. On the right side, they are evaluated using micro F1-score, with the\nlist of SciLMs as follows: PubMedBERT, BioELECTRA, PubMedELECTRA, BioLinkBERT, BioGPT, and clinicalT5. We\nbelieve that it is hard to draw any reliable conclusion when some models show scores on one metric, and others show\nscores on another type of metric.\n2019/06 2020/05 2020/09 2020/10 2020/11 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02\nDate707580859095100ChemProt - F1; DDI - Micro F1ouBioBERTBioALBERTSciFive\nouBioBERTBioALBERTBioReader BioformerChemProt\nBest\nDDI\nBest\nFig. 7. Performance changes in the RE task. ChemProt is evaluated with F1-score, while DDI uses Micro F1-score.\nRE Task. The third popular task is RE. Three popular RE datasets (ChemProt, DDI, and GAD) appear in the top\n20 datasets. However, after merging the list of models evaluated on these three datasets, the number is quite small,\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 23\nwith only 9 remaining models. Therefore, we only draw a chart for the first two datasets, ChemProt and DDI, with\nthe following list of SciLMs: BlueBERT, ouBioBERT, BioALBERT, CharacterBERT, Bio-LM, KeBioLM, ELECTRAMed,\nSciFive, BioELECTRA, PubMedELECTRA, BioLinkBERT, BioReader, and Bioformer.6Figure 7 presents the performance\nchanges of these two datasets. Similar to the NER task, BioALBERT achieved SOTA results in September 2020. However,\nfor the RE task, there are proposed models that have surpassed the score of BioALBERT. Specifically, on the ChemProt\ndataset, SciFive achieved SOTA in May 2021 and still holds the SOTA title. In the case of the DDI dataset, BioReader\nachieved SOTA in December 2022, and Bioformer achieved SOTA in February 2023.\n2020/07 2021/06 2021/12 2022/03 2022/09 2022/10 2022/11 2022/12 2023/07\nDate60657075808590PubMedQA - AccuracyBioELECTRAPubMedELECTRABioLinkBERTBioGPTMed-PaLMPubMedQA\nBest\nFig. 8. Performance changes in PubMedQA. The range is from 60 to 90 , which differs from other tasks due to space constraints.\nQA Task. There are only two QA datasets (PubMedQA and BioASQ) in the top 20 popular datasets. However,\nthese two datasets are not as common as others, such as MedNLI or HOC. Few SciLMs are evaluated on the BioASQ\ndataset; therefore, we have decided to only draw a chart for the PubMedQA dataset. Figure 8 presents the performance\nchanges in PubMedQA. We observe that the performance of SciLMs on PubMedQA shows a gradual improvement over\ntime. BioELECTRA surpasses the score of PubMedBERT (55.8) and achieves better performance (64.0). Subsequently,\nPubMedELECTRA surpasses the score of BioELECTRA, demonstrating even better performance (67.6). By utilizing\ncitation link information in training the SciLM, BioLinkBERT outperforms all previous models and obtains the new best\nscore (72.2) in March 2022. However, most of these models lack the ability to generate. By using the new generation\nmodel, GPT, Luo et al . [130] proposed BioGPT and achieved a new SOTA result in the PubMedQA dataset, reaching\n78.2. Afterward, several proposed SciLMs were introduced, but their scores are still lower than the score of BioGPT.\nRecently, Singhal et al . [191] introduced Med-PaLM by performing instruction prompt tuning on the Flan-PaLM model.\nHowever, the improvement here is smaller than the improvement from BioLinkBERT to BioGPT.\nNLI Task. The last popular task is NLI. MedNLI was the sole dataset for the NLI task focused on processing scientific\ntext in English until recently when Jullien et al . [89] introduced the NLI4CT dataset. Additionally, there are other NLI\ndatasets for different languages, such as ViMedNLI [ 160] for Vietnamese. MedNLI has been evaluated by many models\nfrom April 2019 until the present. Figure 9 presents the performance changes in the MedNLI dataset. As shown in the\nfigure, in April 2019, Clinical BERT (Emily) achieved the SOTA score of 82.7 on the MedNLI dataset. Subsequently, in\nJune 2019, BlueBERT surpassed the score of Clinical BERT (Emily) and achieved a new SOTA of 84.0. CharacterBERT\nsurpassed the scores of BlueBERT and achieved SOTA in November 2020 (86.1). One month later, Bio-LM established a\nnew SOTA with a score of 88.5. Recently, GatorTron surpassed Bio-LM, achieving a new SOTA with a score of 90.2.\n6LBERT is excluded due to scores being on a different scale from other SciLMs. PubMedBERT is omitted as its F1 score for ChemProt is unavailable.\nManuscript submitted to ACM24 Ho et al.\n2019/04 2019/06 2020/05 2020/09 2020/10 2020/11 2021/05 2021/06 2022/01 2022/10 2022/12 2023/02 2023/05 2023/07\nDate707580859095100MedNLI - AccuracyClinical BERT (E)BlueBERTCharacterBERTBio-LMGatorTronMedNLI\nBest\nFig. 9. Performance changes in the MedNLI dataset.\n1 2 3 4 5 6 7 8\nNumber of T asks01020304050Frequency49 SciLMs\nAll SciLMs\n1 2 3 4 5 6 7 8 9 10 12 14\nNumber of Datasets0369121518Frequency49 SciLMs\nAll SciLMs\nFig. 10. Histogram of number of tasks and datasets of all SciLMs and 49 SciLMs that outperform previous models.\n4.2.2 Number of Models Outperform Previous Models or Achieve SOTA Results. It is difficult and time-consuming to\nprecisely obtain the number of SOTA SciLMs on all different datasets separately. In each proposed SciLM paper, the\nauthors often mention whether their model achieves SOTA results or outperforms previous models. Therefore, we\nutilize this information for analysis. If the SciLM outperforms previous models or achieves SOTA results, we add the\nhighlight word (All) at the end of the column ‘Datasets’ for each model in Table 16 in Appendix C.1. In summary,\nthere are 49 SciLMs that outperform previous models or achieve SOTA results in the list of 117 SciLMs. However, this\nnumber alone does not provide a comprehensive understanding of the effectiveness of SciLMs. In some cases, SciLMs\nonly evaluate on one dataset or one task, making it unfair for comparison. To effectively represent this number, we\nconducted a statistical analysis to count how many datasets and tasks these 49 SciLMs used in their evaluations. Figure\n10 displays histograms for the number of tasks (Left) and datasets (Right) among these 49 SciLMs (green columns). As\ndepicted in the figure, many SciLMs conducted their evaluation on only one task (26 out of 49 SciLMs) or on only a\nfew datasets (8 out of 49 on one dataset, 14 out of 49 on two datasets). These numbers raise two main issues: (1) the\ngeneralization ability of proposed models remain unclear and (2) comparing the performance of proposed models may\nnot be meaningful. For the first issue, if the model is only evaluated on one task, it indicates that the abilities of the\nmodel on other tasks have not been fully evaluated yet. Regarding the second issue, if the comparison is performed\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 25\non two or three datasets, and it’s quite domain-specific, then even if the model achieves SOTA scores or outperforms\nprevious models, it’s challenging to draw any reliable conclusions in this case.\nMotivated by the above observations and to get a better understanding of the gravity of the identified issues, we have\nalso created histograms for both the number of tasks and datasets in the complete list of 117 SciLMs. These histograms\nare the pink columns in Figure 10. As depicted in the figure, many SciLMs evaluated their models on a limited number\nof tasks and datasets. This raises concerns about the reliability of the evaluation conducted on these SciLMs, which is\ndiscussed in Section 5.2.1.\n4.3 Variations in BERT-based Models Performance Across Tasks\nIn this section, we extensively explore variations in performance over time with fixed model architectures.\n2019/01 2019/03 2019/11 2020/04 2020/07 2020/12 2022/03 2022/05 2023/02 2023/07\nDate707580859095F1-score\nS2ORC-SciBERTPubMedBERTBioLinkBERT\nJNLPBA\nBest\nNCBI-disease\nBest\n2019/01 2019/06 2020/04 2020/05 2020/07 2020/12 2022/03 2023/02 2023/07\nDate707580859095F1-scoreBioLinkBERT\nBC5CDR-disease\nBest\nBC5CDR-chemical\nBest\nFig. 11. Details of performance changes for four NER datasets, JNLPBA, NCBI-disease, BC5CDR-disease, and BC5CDR-chemical\nwhen we fix the architecture information of the models.\nNER Task. From the list of SciLMs in Section 4.2.1, we select the subset of models utilizing the BERT architecture.\nWe find that only seven of these models evaluate their performance on BC2GM. Therefore, in this part, we only draw\ncharts for four NER datasets: JNLPBA, NCBI-disease, BC5CDR-disease, and BC5CDR-chemical. The list of SciLMs for the\nJNLPBA and NCBI-disease datasets is: BioBERT, SciBERT, S2ORC-SciBERT, GreenBioBERT, PubMedBERT, BioMedBERT,\nBioLinkBERT, ScholarBERT, Bioformer, and BIOptimus. The list of SciLMs for the BC5CDR-disease and BC5CDR-\nchemical datasets is: BioBERT, BlueBERT, GreenBioBERT, ouBioBERT, PubMedBERT, BioMedBERT, BioLinkBERT,\nBioformer, and BIOptimus. Figure 11 details performance changes for JNLPBA, NCBI-disease, BC5CDR-disease, and\nBC5CDR-chemical NER datasets with fixed model architectures. For the NCBI-disease and BC5CDR-disease datasets\n(orange lines), we observe that no proposed models have surpassed the performance of BioBERT, introduced in January\n2019. For the BC5CDR-chemical dataset, BioLinkBERT, proposed in March 2022, can improve the performance of\nBioBERT, but the improvement is small. For the JNLPBA dataset, we observe a slow but steady progress in performance\nchanges over time. Specifically, S2ORC-SciBERT, proposed in November 2019, slightly improves the performance of\nBioBERT (77.5 to 77.7). Following that, PubMedBERT, introduced in July 2020, outperforms S2ORC-SciBERT by a large\nmargin, increasing from 77.7 to 79.1. In March 2022, BioLinkBERT outperforms all previous BERT-based models and\nobtains the best score until now with an 80.1 F1-score when compared with BERT-based models only.\nClassification Task. We observe that there are only three BERT-based models (BlueBERT, ouBioBERT, and Bi-\noLinkBERT) using F1 score, while there are two BERT-based models (PubMedBERT and BioLinkBERT) using micro F1\nscore for evaluation. Therefore, we have decided not to draw charts for the HOC dataset. In terms of model performance,\nthere is an improvement from BlueBERT (87.3 F1-score) to BioLinkBERT (88.1 F1-score).\nManuscript submitted to ACM26 Ho et al.\nQA Task. Similar to the classification task, there are also only three BERT-based models (PubMedBERT, BioLinkBERT,\nand KEBLM) for the PubMedQA dataset. Therefore, we do not draw charts for it. We observe an improvement from\nPubMedBERT (55.8% accuracy) to BioLinkBERT (72.2% accuracy), but KEBLM shows a performance drop with only\n68.0% accuracy. The main reason may be that KEBLM focuses on proposing models that can incorporate information\nfrom multiple types of knowledge, instead of relying solely on unstructured text.\n2019/06 2020/05 2020/10 2022/03 2023/02\nDate707580859095F1-score\nouBioBERTBioLinkBERTouBioBERTCharacterBERT BioLinkBERTBioformerChemProt\nBest\nDDI\nBest\n2019/04 2019/06 2020/05 2020/10 2022/10 2022/12 2023/07\nDate707580859095MedNLI - AccuracyBlueBERTCharacterBERTUCSF-BERTGatorTronMedNLI\nBest\nFig. 12. Details of performance changes for the RE datasets (Left) and the MedNLI dataset (Right) of BERT-based SciLMs.\nRE Task. From the list of SciLMs in Section 4.2.1, we retain only the models that utilize the BERT architecture. This\nyields five SciLMs: BlueBERT, ouBioBERT, CharacterBERT, BioLinkBERT, and Bioformer. Figure 12 (Left) illustrates the\nperformance changes for the DDI and ChemProt datasets. We observe a gradual performance improvement over the past\nfour years in the RE task. Specifically, for the DDI dataset, BioBERT first outperforms BlueBERT, and then CharacterBERT\nsurpasses BioBERT. After that, BioLinkBERT and Bioformer also demonstrate improvements over the previous SciLMs.\nChemProt shows a quite similar pattern to the DDI dataset. However, the performance of CharacterBERT and ouBioBERT\nis similar (75.5 F1-score), and Bioformer does not outperform BioLinkBERT on the ChemProt dataset.\nNLI Task. Similar to previous tasks, we only retain the models that utilize the BERT architecture. This results in\neight SciLMs: Clinical BERT (Emily), BlueBERT, ouBioBERT, UmlsBERT, CharacterBERT, UCSF-BERT, GatorTron, and\nKEBLM. Figure 12 (Right) illustrates the performance changes for the MedNLI dataset. We observe a clear performance\nimprovement in the MedNLI dataset from April 2019 to December 2022. BlueBERT surpasses Clinical BERT (Emily),\nfollowed by CharacterBERT improving over all previous SciLMs. Subsequently, UCSF-BERT and GatorTron also\ndemonstrate improvement over all previous SciLMs. Currently, GatorTron stands as the best BERT-architecture model\nfor the MedNLI dataset.\n5 CURRENT CHALLENGES AND FUTURE DIRECTIONS\n5.1 Foundation SciLMs\n5.1.1 SciLMs for non-English Language. Study on multilingual and monolingual models for non-English languages has\nreceived significant attention in recent couples of years. Such LMs attempt to address the limitations in the solving of\nNLP tasks for non-English languages [ 106]. The development of monolingual PLMs for other languages also witnessed a\nmassive increase. This evolution improves the performance of PLMs in various downstream tasks, such as classification,\nsummarization, and machine-reading comprehension in languages other than English. As a result, many benchmarking\ndatasets and evaluations are performed in low-resource languages and have benefited the NLP research community.\nWith respect to scientific text, most documents are written in English; therefore, SciLMs are initially designed to\nhandle only English text. Few attempts have been made to investigate the performance of multilingual SciLMs in other\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 27\nlanguages. Table 7 summarizes the current SciLMs for different languages. We find that nine languages other than\nEnglish have pre-trained SciLMs. Chinese is the language spoken most among them; therefore, it receives a lot of\nattention from researchers. French, Japanese, and Spanish are also popular languages for which SciLMs have been\nevaluated. The lack of studies in other languages is probably due to the lack of large-scale scientific datasets. One possible\nexplanation is that most research articles and scientific reports are written in English, making it a time-consuming task\nto collect and create datasets for non-English languages. The development of machine translation models is rapidly\nadvancing [ 224] and can be integrated into future SciLMs. However, as far as we know, models for low-resource\nlanguages are not able to capture scientific phrases and academic writing styles; hence, it is essential to conduct more\nresearch on multilingual or non-English SciLMs.\n5.1.2 SciLMs for non-Biomedical Domain. The landscape of SciLMs extends beyond the biomedical domain to encompass\nvarious scientific disciplines. In the chemical domain, there are 13 specialized SciLMs, primarily based on the BERT\narchitecture. The survey expands to less commonly explored domains such as Climate, CS, Cybersecurity, Geoscience,\nManufacturing, Math, Protein, Science Education, and Social Science. In the multi-domain category, models like SciBERT,\nS2ORC-SciBERT, OAG-BERT, ScholarBERT, AcademicRoBERTa, and VarMAE are designed to handle diverse domains.\nDespite progress, there is a significant gap in SciLM representation across scientific domains. While the biomedical\ndomain has 85 identified models, other domains often have only one or two dedicated models, leading to concerns\nabout limited generalizability ,neglect of domain-specific nuances , and impediments to domain-specific applications . The\ndominance of biomedical SciLMs raises questions about their generalizability across diverse scientific disciplines,\npotentially lacking contextual understanding for accurate representation in other fields.\nTo address these challenges, strategies include encouraging domain-specific research collaboration, open access\nto specialized datasets, incorporating transfer learning techniques, and establishing shared evaluation benchmarks.\nCollaboration between NLP researchers and domain experts can foster the development of SciLMs tailored to specific\nscientific domains. Open access to specialized datasets and leveraging transfer learning techniques allow adaptation to\nspecific domains, even with limited data. Shared benchmarks incentivize researchers, encouraging contributions to\nSciLM development across various domains and advancing research in scientific disciplines.\n5.1.3 Integrating Knowledge into SciLMs. The exploration of integrating external knowledge, specifically Knowledge\nBases (KBs), into SciLMs fills a crucial gap in the existing literature [ 153,221]. KBs play a pivotal role in enhancing\nLMs’ capabilities within the scientific domain, providing structured information retrieval, domain-specific precision,\ncontextual enrichment, informed reasoning, and task performance improvement. Tailored to scientific disciplines, KBs\noffer comprehensive knowledge coverage, enriching the context for LMs.\nTable 12. SciLMs with Knowledge Integrating. The No. column referred to Tables 6, 14, and 15.\nNo. Model Domain Arch. Base-model KBs Pretraining Task KBs Used\n30 SapBERT [120] Bio En PubMedBERT Synonyms Clustering UMLS\n31 UmlsBERT [137] Bio En ClinicalBERT (Emily) CUI Words Connecting UMLS\n34 CODER [249] Bio En PubMedBERT Contrastive Learning UMLS\n41 KeBioLM [248] Bio En PubMedBERT KG Embeddings (TransE [21]) UMLS\n45 ProteinBERT [23] Bio En ProteinBERT Gene Ontology Prediction UniRef90\n84 DRAGON [243] Bio Others BioLinkBERT-Large MLM, KG Link Prediction UMLS\nTable 12 summarizes key models, such as UmlsBERT, ProteinBERT, DRAGON, KeBioLM, CODER, and SapBERT,\neach tailored to specific domains and pretraining tasks. The integration process involves methodologies categorized\nManuscript submitted to ACM28 Ho et al.\ninto key approaches, such as integrating knowledge into the training objective and integrating knowledge into LM\ninputs. UmlsBERT, KeBioLM, CODER, and DRAGON exemplify the former, embedding knowledge directly into the\nlearning process during pretraining. ProteinBERT, on the other hand, aligns more closely with the latter, incorporating\nexternal knowledge, such as Gene Ontology annotations, into the LM inputs to enhance context and semantics.\nChallenges in knowledge integration include knowledge noise ,domain mismatch ,interpretability , and coverage issue .\nKnowledge noise refers to challenges stemming from irrelevant or noisy information in KBs, encompassing outdated or\nincorrect data, ambiguous terms, and irrelevant concepts, which can significantly impact the precision of SciLMs, posing\nspecific challenges in scientific domains where accuracy and precision are critical. Domain mismatch addresses disparities\nbetween KB language and scientific text nuances, requiring navigation for effective integration. Interpretability concerns\nmaintaining transparency in decision-making post-integration, crucial for validating reliability. The coverage issue stems\nfrom the limited size of KBs, necessitating strategies to handle gaps in knowledge for accurate predictions. Successfully\novercoming these challenges is pivotal for enhancing SciLM efficacy in processing scientific text, allowing for reliable\nand precise outcomes.\n5.1.4 Build Large SciLMs. As shown in Figure 4, the majority of existing SciLMs have less than 1B parameters (i.e. BERT-\nlevel). One reason is that BERT-based SciLMs perform relatively well in various downstream tasks with limited budgets.\nAnother reason is that building larger SciLMs requires much more computation resources and data. Galactica [ 200] is\nthe first attempt to scale SciLM up to 100B+ parameters. However, training such a large model requires a significantly\nlarger amount of data and computation resources compared to BERT-like models. SciBERT was pre-trained with a\nsingle TPU v3 with 8 cores (similar to 2 A100 GPUs), whereas Taylor et al . [200] used 128 A100 GPUs to pre-train their\nGalactica-120B. Therefore, an effective solution for pre-training SciLMs is a big challenge these days, especially for\nresearchers who work in a university.\nSince the existence of high-quality open-sourced LLMs [ 204,205], some researchers pay much attention on continual\npre-training with these LLMs with extra scientific texts, relaxing the need for collecting large-scale scientific corpora\nand reducing the need for tons of computation resources as we train SciLMs from scratch. Therefore, effective continual\npre-training is a promising direction for building large SciLMs. Meditron [ 35] included a small proportion of the original\npre-training corpus used by Llama-2 [ 205] to avoid forgetting when continual pre-training on data in the medical\ndomain. Some attempts have been made to guide the continual pretraining in the general domain [ 63], however, practical\nmethods designed specifically for the scientific domain are still rare.\nAs found by Beltagy et al . [15] , training SciLMs from scratch can benefit from designing domain-specific tokenizers\nfor scientific domains, achieving better performances. Compared to other languages, there have been already a lot of\nEnglish scientific texts for pre-training SciLMs. However, if we want to train a large SciLM (i.e. with more than 100B\nparameters) from scratch, the scale of existing data may not be enough. Detailedly, SciBERT used only 3.17B tokens\nduring pre-training [ 15], whereas Galactica consumed 450B tokens in total by repeating 106B tokens for approximately\nfour epochs, which are 140 times more than those for training BERT-like models. According to the Chinchilla Scaling\nLaws [ 71], LLMs with 63B parameters require 1.4T tokens for pre-training, and language models pre-trained with\ndeduplicated texts perform better generalization ability, whereas the scientific texts in The Pile [ 59] contain much less\ntokens (96B tokens) than the recommended amount. Therefore, how to collect enough scientific texts for pre-training\nlarge SciLMs still remains a challenge these days.\n5.1.5 Multi-modal SciLMs. LMs capable of handling both language and non-language information such as image, audio,\nand video, have received considerable attention in recent years [ 128,225,235]. Notably, vision-and-language models\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 29\npre-trained on vast amounts of language and image data have achieved significant success in various tasks, such as image\ncaptioning [ 195] and image generation from text instructions [ 252]. Lately, with the successful deployment of LLMs\nlike GPT-4 [ 149], many studies are emphasizing on training adapters that can transform non-language information to\nbe treated in the same embedding space as language [ 44,262]. Such architectures are expected to handle non-language\ndata while retaining the extensive problem-solving capabilities of LLMs.\nIn the scientific domain, the advent of multi-modal models is also gaining momentum. Multi-modal SciLMs can be\nbuilt by doing additional training on mono-modal or multi-modal PLMs on general domains, and thus take advantage\nof the robust performance of general-domain models. However, there are challenges yet to be overcome. For instance,\nin scientific domains, there is less data available compared to the general domain [ 117,247], which makes it difficult to\nsufficiently train or fine-tune the multi-modal SciLMs. In addition, in scientific domains, models handling more than\ntwo modalities are anticipated. Typically, scientific papers include many different types of information, such as tables,\nequations, figures, and codes. Therefore, their multi-modal integration into SciLMs should be considered as a crucial\nstep forward. Also, biomedical SciLMs that incorporate a wide spectrum of data, such as CT, MRI, and ultrasound, are\ndesirable [117]. However, research dealing with more than three modalities is relatively sparse.\nAddressing these challenges requires strategies to increase the amount of available training data, including PDFs and\nLaTeX files. It should be encouraged to explore data augmentation techniques and learning methods that integrate\nexternal knowledge. In addition, the recent upsurge in LLMs signifies the need to develop multi-modal SciLMs based on\npublicly available LLMs (e.g., Llama 2 [ 205]) in scientific domains as well. Building such models will bring us closer to\nfully realizing the immense potential of multi-modal models in academic research.\n5.2 Evaluating the Effectiveness, Efficiency, and Trustworthiness of SciLMs\n5.2.1 Issues with Evaluation and Comparison. As discussed in Section 4.2.2, many SciLMs evaluate their models on a\nlimited number of tasks and datasets. This raises issues related to both evaluation and comparison. First, concerning\nevaluation, when proposed SciLMs assess their models on only one or a few tasks, it implies that their model is not\ncomprehensively tested, and its performance may only be effective for some tasks while performing poorly on others.\nThis can be explained by the fact that most existing SciLMs are based on an encoder-based architecture, such as BERT\n[48], rather than being text-to-text models like T5 [ 167]. Therefore, their models are not easily adaptable for evaluation\nacross various NLP tasks. With regard to the second issue of comparison, if we only compare different SciLMs on one\nor a few datasets, it becomes challenging to draw any reliable conclusions.\nOne promising direction to enhance the evaluation and comparison of different models is to create a benchmark\ncomprising various tasks and datasets. In the general domain, GLUE [ 219] and SuperGLUE [ 218] were introduced as\nstandardized benchmarks for comparison. Motivated by this, in the biomedical domain, Peng et al . [158] and Gu et al .\n[61] introduced the BLUE and the BLURB benchmarks, respectively. The BLUE benchmark comprises five different\ntasks with ten datasets across these tasks, while the BLURB benchmark encompasses six different tasks with thirteen\ndatasets across these tasks. However, we observe that currently, there are only a few SciLMs in our list of 117 SciLMs\nthat conduct evaluations on these benchmarks (for BLUE, the models are: BlueBERT, ouBioBERT, BioALBERT, and\nBioELECTRA; for BLURB, the models are: PubMedBERT, BioELECTRA, PubMedELECTRA, and BioLinkBERT). Perhaps\nbecause many datasets in these benchmarks already yield high scores, researchers may be less motivated to evaluate on\nthem. However, we believe that creating benchmarks with diverse tasks and datasets is a promising direction for future\nresearch to enable fair and reliable comparisons. We encourage future researchers to develop benchmarks with more\ntasks and datasets, even across different domains.\nManuscript submitted to ACM30 Ho et al.\n5.2.2 Move Beyond Simple Tasks. As shown in Tables 10 and 11, most existing SciLMs focus their evaluation on simple\ntasks in NLP, such as NER and RE. In the top-20 popular datasets used to evaluate SciLMs, there are eleven NER datasets,\nbut there is only one NLI dataset and two QA datasets. As we know, NER and RE are basic tasks in NLP, while NLI and\nQA emphasize language understanding, testing the models’ broader comprehension skills. However, datasets for these\ntwo tasks are not commonly used.\nWith the current issues, we suggest that future work on SciLMs should shift their focus to the evaluation of more\ncomplex understanding tasks in NLP, such as NLI and QA. To accomplish this, the first step is to create additional\ndatasets specifically designed for the NLI and QA tasks, serving as benchmarks for meaningful comparisons among\nSciLMs. Learning from general domain, there are some directions that we can go. For example, for the QA task, we can\npropose datasets for testing different skills, such as reasoning over multiple documents [ 230,241] and conversational\nQA [ 38,169]. In the case of the NLI task, there has been only one dataset dedicated to English scientific text over the\npast few years: the MedNLI dataset [ 174]. Recently Jullien et al . [89] introduce an NLI4CT dataset for clinical trial\nreports. One notable feature of the dataset is that it is the first dataset with interpretation for the NLI task in processing\nscientific text. However, the size of the dataset is quite small, with only 2,400 instances. We believe that introducing\nmore NLI datasets with larger sizes and containing adversarial samples would be helpful for evaluating SciLMs.\n5.2.3 Reliable SciLMs. In addition to the issues regarding evaluation and comparison, we also need to pay much\nattention to three other aspects —robustness, generalization, and explanation— to obtain more reliable SciLMs.\nIn terms of robustness, we observe that not many SciLMs undergo evaluations on various types of adversarial\ntests. As seen in general domain tasks, many models exhibit strong performance on original datasets but experience a\nsignificant drop in performance on adversarial versions of those datasets [ 83,84,170]. Therefore, to ensure robustness,\nit is essential to test SciLMs on adversarial tests during model development.\nFor the second aspect, generalization, we also observe a similar situation to that of robustness, where not many\nproposed SciLMs consider testing the generalization of their models. It is noted that there are multiple ways to define\nthe generalization ability of models. In this research, we simplify the definition by only considering the ability to\ngeneralize from one dataset to another dataset within the same task, whether in the same domain or a different domain.\nAnother concern regarding generalization ability is the lack of available datasets for testing models across tasks. For\nexample, in the NLI task, only one dataset, MedNLI, is available (fortunately, recently Jullien et al . [89] introduce an\nNLI4CT dataset). As a result, researchers cannot test the generalization ability of their models–for instance, training\non one dataset and conducting evaluations on another dataset. We suggest that future studies focus on evaluating\nmodels across multiple datasets with different distributions in the training set. By doing so, we can obtain a clearer\nunderstanding of the generalization ability of the models.\nFor the third aspect, we also observe a lack of research on SciLMs that emphasizes the aspect of explanation. In the\nNLP domain, explanations are considered as the reasons for “why [input] is assigned [label],” and they are crucial for\nensuring the reliability of the models [ 231]. However, this point is not well-discussed and emphasized in the scientific\ndomain. For example, to the best of our knowledge, there are currently only four datasets in the scientific domain\nthat include explanation information—PubMedQA [87] (long answer can be considered as explanation), SciFact [216],\nQASPER [ 45], and NLI4CT [ 89]. We believe that, to enhance the explanation ability of SciLMs, more datasets dedicated\nto explanation should be proposed for use in evaluating and analyzing the models’ capabilities.\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 31\n6 CONCLUSION\nWe surveyed existing LMs for processing scientific text. Specifically, we reviewed 117 SciLMs across various scientific\ndomains, languages and architectures. We presented an extensive analysis of these SciLMs, highlighting the general bias\nof previous work towards the biomedical domain and BERT-based encoders, and the uprising of non-English SciLMs. We\nalso provided new insights on the current SOTA performance of SciLMs across commonly used tasks and datasets, and\nits evolution through the recent years. Finally, we discussed challenges and showed the two potential directions. The\nfirst direction is about establishing foundational SciLMs, for which we provided practical recommendations to improve\ntheir performance, such as integrating knowledge bases, developing larger models, or leveraging the multi-modal\ncontent present in scientific papers. Additionally, we proposed actionable recommendations on the development of\nSciLMs for low-resource languages, extending beyond the English language and biomedical domains. The second\ndirection is about improving the evaluation of SciLMs, where we proposed to incorporate a broader range of tasks,\ndatasets, and more challenging evaluation criteria. We also underscored the urgent need to establish standardized\nbenchmarks across various domains and tasks, ensuring a fair and reliable comparison between proposed SciLMs.\nREFERENCES\n[1] 2008–2023. GROBID. https://github.com/kermitt2/grobid.\n[2]Anurag Acharya, Sai Munikoti, Aaron Hellinger, Sara Smith, Sridevi Wagle, and Sameera Horawalavithana. 2023. NuclearQA: A Human-Made\nBenchmark for Language Models for the Nuclear Domain. arXiv:2310.10920\n[3]Ehsan Aghaei, Xi Niu, Waseem G. Shadid, and Ehab Al-Shaer. 2022. SecureBERT: A Domain-Specific Language Model for Cybersecurity. In\nSecureComm . 39–56.\n[4]Walid Ahmad, Elana Simon, Seyone Chithrananda, Gabriel Grand, and Bharath Ramsundar. 2022. ChemBERTa-2: Towards Chemical Foundation\nModels. arXiv:2209.01712\n[5]Alan Akbik, Tanja Bergmann, Duncan Blythe, Kashif Rasul, Stefan Schweter, and Roland Vollgraf. 2019. FLAIR: An Easy-to-Use Framework for\nState-of-the-Art NLP. In NAACL (Demonstrations) . 54–59.\n[6]Emily Alsentzer, John Murphy, William Boag, Wei-Hung Weng, Di Jindi, Tristan Naumann, and Matthew McDermott. 2019. Publicly Available\nClinical BERT Embeddings. In Clinical Natural Language Processing Workshop . 72–78.\n[7]Sequoia R Andrade and Hannah S Walsh. 2023. SafeAeroBERT: Towards a Safety-Informed Aerospace-Specific Language Model. In AIAA AVIATION\n2023 Forum . 3437.\n[8] Dogu Araci. 2019. FinBERT: Financial Sentiment Analysis with Pre-trained Language Models. arXiv:1908.10063\n[9]Mario Aragon, Adrian Pastor Lopez Monroy, Luis Gonzalez, David E. Losada, and Manuel Montes. 2023. DisorBERT: A Double Domain Adaptation\nModel for Detecting Signs of Mental Disorders in Social Media. In ACL. 15305–15318.\n[10] Kristjan Arumae, Qing Sun, and Parminder Bhatia. 2020. An Empirical Investigation Towards Efficient Multi-Domain Language Model Pre-training.\nInEMNLP . 4854–4864.\n[11] Masaki Asada and Makoto Miwa. 2023. BioNART: A Biomedical Non-AutoRegressive Transformer for Natural Language Generation. In BioNLP .\n[12] Marco Basaldella, Fangyu Liu, Ehsan Shareghi, and Nigel Collier. 2020. COMETA: A Corpus for Medical Entity Linking in the Social Media. In\nEMNLP . 3122–3137.\n[13] Leonard E Baum and Ted Petrie. 1966. Statistical inference for probabilistic functions of finite state Markov chains. The annals of mathematical\nstatistics (1966), 1554–1563.\n[14] Markus Bayer, Philipp Kuehn, Ramin Shanehsaz, and Christian Reuter. 2022. CySecBERT: A Domain-Adapted Language Model for the Cybersecurity\nDomain. arXiv:2212.02974\n[15] Iz Beltagy, Kyle Lo, and Arman Cohan. 2019. SciBERT: A Pretrained Language Model for Scientific Text. In EMNLP-IJCNLP . 3615–3620.\n[16] Iz Beltagy, Matthew E. Peters, and Arman Cohan. 2020. Longformer: The Long-Document Transformer. arXiv:2004.05150\n[17] Aman Berhe, Guillaume Draznieks, Vincent Martenot, Valentin Masdeu, Lucas Davy, and Jean-Daniel Zucker. 2023. AliBERT: A Pre-trained\nLanguage Model for French Biomedical Text. In BioNLP . 223–236.\n[18] Steven Bird, Ewan Klein, and Edward Loper. 2009. Natural language processing with Python: analyzing text with the natural language toolkit . O’Reilly\nMedia, Inc.\n[19] Daniil A Boiko, Robert MacKnight, Ben Kline, and Gabe Gomes. 2023. Autonomous chemical research with large language models. Nature (2023).\n[20] Elliot Bolton, David Hall, Michihiro Yasunaga, Tony Lee, Chris Manning, and Percy Liang. 2023. BioMedLM. https://crfm.stanford.edu/2022/12/15/\nbiomedlm.html.\nManuscript submitted to ACM32 Ho et al.\n[21] Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. 2013. Translating embeddings for modeling\nmulti-relational data. NIPS (2013).\n[22] Andres M Bran, Sam Cox, Oliver Schilter, Carlo Baldassari, Andrew D White, and Philippe Schwaller. 2023. ChemCrow: Augmenting large-language\nmodels with chemistry tools. arXiv:2304.05376\n[23] Nadav Brandes, Dan Ofer, Yam Peleg, Nadav Rappoport, and Michal Linial. 2022. ProteinBERT: a universal deep-learning model of protein sequence\nand function. Bioinformatics (2022), 2102–2110.\n[24] Àlex Bravo, Janet Piñero, Núria Queralt-Rosinach, Michael Rautschka, and Laura I Furlong. 2015. Extraction of relations between genes and\ndiseases from text and large-scale data analysis: Implications for translational research. BMC Bioinformatics (2015).\n[25] Keno K Bressem, Lisa C Adams, Robert A Gaudin, Daniel Troltzsch, Bernd Hamm, Marcus R Makowski, et al .2020. Highly accurate classification of\nchest radiographic reports using a deep learning natural language model pre-trained on 3.8 million text reports. Bioinformatics (2020), 5255–5261.\n[26] Peter F. Brown, Vincent J. Della Pietra, Peter V. deSouza, Jenifer C. Lai, and Robert L. Mercer. 1992. Class-Based n-gram Models of Natural Language.\nComputational Linguistics (1992), 467–480.\n[27] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry,\nAmanda Askell, et al. 2020. Language Models are Few-Shot Learners. In NeurIPS .\n[28] Lee Burke, Karl Pazdernik, Daniel Fortin, Benjamin Wilson, Rustam Goychayev, and John Mattingly. 2021. NukeLM: Pre-Trained and Fine-Tuned\nLanguage Models for the Nuclear and Energy Domains. arXiv:2105.12192\n[29] Xiaoyan Cai, Sen Liu, Junwei Han, Libin Yang, Zhenguo Liu, and Tianming Liu. 2023. ChestXRayBERT: A Pretrained Language Model for Chest\nRadiology Report Summarization. IEEE Transactions on Multimedia (2023), 845–855.\n[30] Cornelia Caragea, Jian Wu, Alina Ciobanu, Kyle Williams, Juan Fernández-Ramírez, Hung-Hsuan Chen, Zhaohui Wu, and Lee Giles. 2014. CiteSeerx:\nA Scholarly Big Dataset. In Advances in Information Retrieval . 311–322.\n[31] Casimiro Pio Carrino, Jordi Armengol-Estapé, Asier Gutiérrez-Fandiño, Joan Llop-Palao, Marc Pàmies, et al .2021. Biomedical and Clinical Language\nModels for Spanish: On the Benefits of Domain-Specific Pretraining in a Mid-Resource Scenario. arXiv:2109.03570\n[32] Casimiro Pio Carrino, Joan Llop, Marc Pàmies, Asier Gutiérrez-Fandiño, Jordi Armengol-Estapé, Joaquín Silveira-Ocampo, Alfonso Valencia, Aitor\nGonzalez-Agirre, and Marta Villegas. 2022. Pretrained Biomedical Language Models for Clinical NLP in Spanish. In BioNLP . 193–199.\n[33] Souradip Chakraborty, Ekaba Bisong, Shweta Bhatt, Thomas Wagner, Riley Elliott, and Francesco Mosconi. 2020. BioMedBERT: A Pre-trained\nBiomedical Language Model for QA and IR. In COLING . 669–679.\n[34] Chetan Chandra, Xiao Jing, Mayank V Bendarkar, Kshitij Sawant, Lidya Elias, Michelle Kirby, and Dimitri N Mavris. 2023. Aviation-BERT: A\nPreliminary Aviation-Specific Natural Language Model. In AIAA AVIATION 2023 Forum . 3436.\n[35] Zeming Chen, Alejandro Hernández Cano, Angelika Romanou, Antoine Bonnet, Kyle Matoba, Francesco Salvi, Matteo Pagliardini, Simin Fan,\nAndreas Köpf, Amirkeivan Mohtashami, et al .2023. MEDITRON-70B: Scaling Medical Pretraining for Large Language Models. arXiv:2311.16079\n[36] Jianpeng Cheng, Li Dong, and Mirella Lapata. 2016. Long Short-Term Memory-Networks for Machine Reading. In EMNLP . 551–561.\n[37] Seyone Chithrananda, Gabriel Grand, and Bharath Ramsundar. 2020. ChemBERTa: Large-Scale Self-Supervised Pretraining for Molecular Property\nPrediction. arXiv:2010.09885\n[38] Eunsol Choi, He He, Mohit Iyyer, Mark Yatskar, Wen-tau Yih, Yejin Choi, Percy Liang, and Luke Zettlemoyer. 2018. QuAC: Question Answering in\nContext. In EMNLP . 2174–2184.\n[39] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles\nSutton, Sebastian Gehrmann, et al. 2023. PaLM: Scaling Language Modeling with Pathways. J. Mach. Learn. Res. (2023), 240:1–240:113.\n[40] Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma,\net al. 2022. Scaling Instruction-Finetuned Language Models. arXiv:2210.11416\n[41] Kevin Clark, Minh-Thang Luong, Quoc V. Le, and Christopher D. Manning. 2020. ELECTRA: Pre-training Text Encoders as Discriminators Rather\nThan Generators. In ICLR .\n[42] Arman Cohan, Sergey Feldman, Iz Beltagy, Doug Downey, and Daniel Weld. 2020. SPECTER: Document-level Representation Learning using\nCitation-informed Transformers. In ACL. 2270–2282.\n[43] Nigel Collier, Tomoko Ohta, Yoshimasa Tsuruoka, Yuka Tateisi, and Jin-Dong Kim. 2004. Introduction to the Bio-entity Recognition Task at\nJNLPBA. In NLPBA/BioNLP . 73–78.\n[44] Wenliang Dai, Junnan Li, Dongxu Li, Anthony Meng Huat Tiong, Junqi Zhao, Weisheng Wang, Boyang Li, Pascale Fung, and Steven Hoi. 2023.\nInstructBLIP: Towards General-purpose Vision-Language Models with Instruction Tuning. arXiv:2305.06500\n[45] Pradeep Dasigi, Kyle Lo, Iz Beltagy, Arman Cohan, Noah A. Smith, and Matt Gardner. 2021. A Dataset of Information-Seeking Questions and\nAnswers Anchored in Research Papers. In NAACL . 4599–4610.\n[46] Iker de la Iglesia, Aitziber Atutxa, Koldo Gojenola, and Ander Barrena. 2023. EriBERTa: A Bilingual Pre-Trained Language Model for Clinical\nNatural Language Processing. arXiv:2306.07373\n[47] Cheng Deng, Tianhang Zhang, Zhongmou He, Yi Xu, Qiyuan Chen, Yuanyuan Shi, Luoyi Fu, Weinan Zhang, Xinbing Wang, Chenghu Zhou,\nZhouhan Lin, and Junxian He. 2023. K2: A Foundation Language Model for Geoscience Knowledge Understanding and Utilization. arXiv:2306.05064\n[48] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of Deep Bidirectional Transformers for Language\nUnderstanding. In NAACL . 4171–4186.\n[49] Bhuwan Dhingra, Danish Danish, and Dheeraj Rajagopal. 2018. Simple and Effective Semi-Supervised Question Answering. In NAACL . 582–587.\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 33\n[50] Amber Dood, Blair Winograd, Solaire Finkenstaedt-Quinn, Anne Gere, and Ginger Shultz. 2022. PeerBERT: Automated Characterization of Peer\nReview Comments across Courses. In LAK22 . 492–499.\n[51] Rezarta Islamaj Doğan, Robert Leaman, and Zhiyong Lu. 2014. NCBI disease corpus: A resource for disease name recognition and concept\nnormalization. J. of Biomedical Informatics (2014), 1–10.\n[52] Louisa Ebert, Sebastian Huettemann, and Roland M. Mueller. 2023. Hey Article, What Are You About? Question Answering for Information\nSystems Articles through Transformer Models for Long Sequences. In HICSS .\n[53] Hicham El Boukkouri, Olivier Ferret, Thomas Lavergne, Hiroshi Noji, Pierre Zweigenbaum, and Jun’ichi Tsujii. 2020. CharacterBERT: Reconciling\nELMo and BERT for Word-Level Open-Vocabulary Representations From Characters. In COLING . 6903–6915.\n[54] Björn Engelmann, Fabian Haak, Christin Katharina Kreutz, Narjes Nikzad-Khasmakhi, and Philipp Schaer. 2023. Text Simplification of Scientific\nTexts for Non-Expert Readers. In SimpleText@CLEF-2023 . 2987–2998.\n[55] Jessica López Espejel, Mahaman Sanoussi Yahaya Alassan, Walid Dahhane, and El Hassane Ettifouri. 2023. JaCoText: A Pretrained Model for Java\nCode-Text Generation. arXiv:2303.12869\n[56] Li Fang, Qingyu Chen, Chih-Hsuan Wei, Zhiyong Lu, and Kai Wang. 2023. Bioformer: an efficient transformer language model for biomedical text\nmining. arXiv:2302.01588\n[57] Nathan C. Frey, Ryan Soklaski, Simon Axelrod, Siddharth Samsi, Rafael Gómez-Bombarelli, Connor W. Coley, and Vijay Gadepally. 2023. Neural\nscaling of deep chemical models. Nat. Mac. Intell. (2023), 1297–1305.\n[58] Giacomo Frisoni, Miki Mizutani, Gianluca Moro, and Lorenzo Valgimigli. 2022. BioReader: a Retrieval-Enhanced Text-to-Text Transformer for\nBiomedical Literature. In EMNLP . 5770–5793.\n[59] Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima,\nShawn Presser, and Connor Leahy. 2020. The Pile: An 800GB Dataset of Diverse Text for Language Modeling. arXiv:2101.00027\n[60] Martin Gerner, G. Nenadic, and Casey M. Bergman. 2010. LINNAEUS: A species name identification system for biomedical literature. BMC\nBioinformatics (2010), 85 – 85.\n[61] Yu Gu, Robert Tinn, Hao Cheng, Michael Lucas, Naoto Usuyama, Xiaodong Liu, Tristan Naumann, Jianfeng Gao, and Hoifung Poon. 2021.\nDomain-Specific Language Model Pretraining for Biomedical Natural Language Processing. ACM Trans. Comput. Healthcare (2021).\n[62] Jiang Guo, A. Santiago Ibanez-Lopez, Hanyu Gao, Victor Quach, Connor W. Coley, Klavs F. Jensen, and Regina Barzilay. 2022. Automated Chemical\nReaction Extraction from Scientific Literature. J. of Chemical Information and Modeling (2022), 2035–2045.\n[63] Kshitij Gupta, Benjamin Thérien, Adam Ibrahim, Mats Leon Richter, Quentin Gregory Anthony, Eugene Belilovsky, Irina Rish, and Timothée\nLesort. 2023. Continual Pre-Training of Large Language Models: How to re-warm your model?. In ES-FoMo@ICML2023 .\n[64] Tanishq Gupta, Mohd Zaki, NM Anoop Krishnan, and Mausam. 2022. MatSciBERT: A materials domain language model for text mining and\ninformation extraction. npj Computational Materials (2022), 102.\n[65] William L. Hamilton, Rex Ying, and Jure Leskovec. 2017. Representation Learning on Graphs: Methods and Applications. IEEE Data Eng. Bull.\n[66] Xu Han, Zhengyan Zhang, Ning Ding, Yuxian Gu, Xiao Liu, Yuqi Huo, Jiezhong Qiu, Yuan Yao, Ao Zhang, Liang Zhang, et al .2021. Pre-trained\nmodels: Past, present and future. AI Open (2021), 225–250.\n[67] Yaru Hao, Li Dong, Furu Wei, and Ke Xu. 2019. Visualizing and Understanding the Effectiveness of BERT. In EMNLP-IJCNLP . 4143–4152.\n[68] Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. 2021. Deberta: decoding-Enhanced Bert with Disentangled Attention. In ICLR .\n[69] Shashank Hebbar and Ying Xie. 2021. CovidBERT-Biomedical Relation Extraction for Covid-19. FLAIRS (2021).\n[70] María Herrero-Zazo, Isabel Segura-Bedmar, Paloma Martínez, and Thierry Declerck. 2013. The DDI corpus: An annotated corpus with pharmaco-\nlogical substances and drug–drug interactions. J. of Biomedical Informatics (2013), 914–920.\n[71] Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks,\nJohannes Welbl, Aidan Clark, et al. 2022. Training Compute-Optimal Large Language Models. arXiv:2203.15556\n[72] Zhi Hong, Aswathy Ajith, James Pauloski, Eamon Duede, Kyle Chard, and Ian Foster. 2023. The Diminishing Returns of Masked Language Models\nto Science. In Findings of ACL . 1270–1283.\n[73] Dong Hongyuan, Che Wanxiang, He Xiaoyu, Zheng Guidong, and Wen Junjie. 2023. FinBART: A Pre-trained Seq2seq Language Model for Chinese\nFinancial Tasks. In Chinese National Conference on Computational Linguistics . 906–917.\n[74] Matthew Honnibal, Ines Montani, Sofie Van Landeghem, and Adriane Boyd. 2020. spaCy: Industrial-strength NLP in Python. (2020).\n[75] Dou Hu, Xiaolong Hou, Xiyang Du, Mengyuan Zhou, Lianxin Jiang, Yang Mo, and Xiaofeng Shi. 2022. VarMAE: Pre-training of Variational Masked\nAutoencoder for Domain-adaptive Language Understanding. In Findings of EMNLP . 6276–6286.\n[76] Yibo Hu, MohammadSaleh Hosseini, Erick Skorupa Parolin, Javier Osorio, Latifur Khan, Patrick Brandt, and Vito D’Orazio. 2022. ConfliBERT: A\nPre-trained Language Model for Political Conflict and Violence. In NAACL . 5469–5482.\n[77] Kexin Huang, Jaan Altosaar, and Rajesh Ranganath. 2020. ClinicalBERT: Modeling Clinical Notes and Predicting Hospital Readmission.\narXiv:1904.05342\n[78] Kexin Huang, Abhishek Singh, Sitong Chen, Edward Moseley, Chih-Ying Deng, Naomi George, and Charolotta Lindvall. 2020. Clinical XLNet:\nModeling Sequential Clinical Notes and Predicting Prolonged Mechanical Ventilation. In Clinical Natural Language Processing Workshop . 94–100.\n[79] Shu Huang and Jacqueline M. Cole. 2022. BatteryBERT: A Pretrained Language Model for Battery Database Enhancement. J. of Chemical Information\nand Modeling (2022), 6365–6377.\nManuscript submitted to ACM34 Ho et al.\n[80] Wenyi Huang, Zhaohui Wu, Chen Liang, Prasenjit Mitra, and C. Lee Giles. 2015. A Neural Probabilistic Model for Context Based Citation\nRecommendation. In AAAI .\n[81] Ayush Jain, Dr. N. M. Meenachi, and Dr. B. Venkatraman. 2020. NukeBERT: A Pre-trained language model for Low Resource Nuclear Domain.\narXiv:2003.13821\n[82] Yuna Jeong and Eunhui Kim. 2022. SciDeBERTa: Learning DeBERTa for Science and Technology Documents and Fine-tuning Information Extraction\nTasks. IEEE Access (2022).\n[83] Robin Jia and Percy Liang. 2017. Adversarial Examples for Evaluating Reading Comprehension Systems. In EMNLP . 2021–2031.\n[84] Yichen Jiang and Mohit Bansal. 2019. Avoiding Reasoning Shortcuts: Adversarial Evaluation, Training, and Model Development for Multi-Hop QA.\nInACL. 2726–2736.\n[85] Bowen Jin, Wentao Zhang, Yu Zhang, Yu Meng, Xinyang Zhang, Qi Zhu, and Jiawei Han. 2023. Patton: Language Model Pretraining on Text-Rich\nNetworks. In ACL. 7005–7020.\n[86] Qiao Jin, Bhuwan Dhingra, William Cohen, and Xinghua Lu. 2019. Probing Biomedical Embeddings from Language Models. In RepEval . 82–89.\n[87] Qiao Jin, Bhuwan Dhingra, Zhengping Liu, William Cohen, and Xinghua Lu. 2019. PubMedQA: A Dataset for Biomedical Research Question\nAnswering. In EMNLP-IJCNLP . 2567–2577.\n[88] Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S Weld, Luke Zettlemoyer, and Omer Levy. 2020. SpanBERT: Improving Pre-training by Representing\nand Predicting Spans. TACL (2020), 64–77.\n[89] Mael Jullien, Marco Valentino, Hannah Frost, Paul O’Regan, Dónal Landers, and Andre Freitas. 2023. NLI4CT: Multi-Evidence Natural Language\nInference for Clinical Trial Reports. In EMNLP . 16745–16764.\n[90] David Jurgens, Srijan Kumar, Raine Hoover, Dan McFarland, and Dan Jurafsky. 2018. Measuring the Evolution of a Scientific Field through Citation\nFrames. TACL (2018), 391–406.\n[91] Katikapalli Subramanyam Kalyan, Ajit Rajasekharan, and Sivanesan Sangeetha. 2022. AMMU: A Survey of Transformer-Based Biomedical\nPretrained Language Models. J. of Biomedical Informatics (2022).\n[92] Kamal raj Kanakarajan, Bhuvana Kundumani, and Malaikannan Sankarasubbu. 2021. BioELECTRA:Pretrained Biomedical text Encoder using\nDiscriminators. In BioNLP . 143–154.\n[93] Shota Kato, Kazuki Kanegami, and Manabu Kano. 2022. ProcessBERT: A Pre-trained Language Model for Judging Equivalence of Variable Definitions\nin Process Models*. IFAC-PapersOnLine (2022), 957–962.\n[94] Yoshimasa Kawazoe, Daisaku Shibata, Emiko Shinohara, Eiji Aramaki, and Kazuhiko Ohe. 2021. A clinical specific BERT developed using a huge\nJapanese clinical text corpus. PLOS ONE (2021).\n[95] Yash Khare, Viraj Bagal, Minesh Mathew, Adithi Devi, U Deva Priyakumar, and CV Jawahar. 2021. Mmbert: Multimodal bert pretraining for\nimproved medical vqa. In ISBI. 1033–1036.\n[96] Yoojoong Kim, Jong-Ho Kim, Jeong Moon Lee, Moon Joung Jang, Yun Jin Yum, Seongtae Kim, Unsub Shin, Young-Min Kim, Hyung Joon Joo, and\nSanghoun Song. 2022. A pre-trained BERT for Korean medical natural language processing. Sci. Rep. (2022), 13847.\n[97] J Peter Kincaid, Robert P Fishburne Jr, Richard L Rogers, and Brad S Chissom. 1975. Derivation of new readability formulas (automated readability\nindex, fog count and flesch reading ease formula) for navy enlisted personnel. (1975).\n[98] Zeljko Kraljevic, Anthony Shek, Daniel Bean, Rebecca Bendayan, James Teo, and Richard Dobson. 2021. MedGPT: Medical Concept Prediction\nfrom Clinical Narratives. arXiv:2107.03134\n[99] Martin Krallinger, Florian Leitner, Obdulia Rabal, Miguel Vazquez, Julen Oyarzabal, and Alfonso Valencia. 2015. CHEMDNER: The drugs and\nchemical names extraction challenge. J. Cheminformatics (2015), S1.\n[100] Mario Krenn, Florian Häse, AkshatKumar Nigam, Pascal Friederich, and Alán Aspuru-Guzik. 2020. Self-referencing embedded strings (SELFIES): A\n100% robust molecular string representation. Mach. Learn. Sci. Technol. (2020), 45024.\n[101] Jens Kringelum, Sonny Kim Kjaerulff, Søren Brunak, Ole Lund, Tudor I. Oprea, and Olivier Taboureau. 2016. ChemProt-3.0: a global chemical\nbiology diseases mapping. Database (2016).\n[102] Aman Kumar, Binil Starly, Collin Lynch, and Collin Lynch. 2023. ManuBERT: A Pretrained Manufacturing Science Language Representation\nModel. SSRN (2023).\n[103] Yanis Labrak, Adrien Bazoge, Richard Dufour, Mickael Rouvier, Emmanuel Morin, Béatrice Daille, and Pierre-Antoine Gourraud. 2023. DrBERT: A\nRobust Pre-trained Model in French for Biomedical and Clinical domains. In ACL. 16207–16221.\n[104] John D. Lafferty, Andrew McCallum, and Fernando C. N. Pereira. 2001. Conditional Random Fields: Probabilistic Models for Segmenting and\nLabeling Sequence Data. In ICML . 282–289.\n[105] Tuan Manh Lai, ChengXiang Zhai, and Heng Ji. 2023. KEBLM: Knowledge-Enhanced Biomedical Language Models. J. of Biomedical Informatics .\n[106] Viet Lai, Nghia Ngo, Amir Pouran Ben Veyseh, Hieu Man, Franck Dernoncourt, Trung Bui, and Thien Nguyen. 2023. ChatGPT Beyond English:\nTowards a Comprehensive Evaluation of Large Language Models in Multilingual Learning. In Findings of EMNLP . 13171–13189.\n[107] Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2020. ALBERT: A Lite BERT for Self-supervised\nLearning of Language Representations. arXiv:1909.11942\n[108] Jinhyuk Lee, Wonjin Yoon, Sungdong Kim, Donghyeon Kim, Sunkyu Kim, Chan Ho So, and Jaewoo Kang. 2020. BioBERT: a pre-trained biomedical\nlanguage representation model for biomedical text mining. Bioinformatics (2020), 1234–1240.\n[109] Eric Lehman and Alistair Johnson. 2023. Clinical-T5: Large Language Models Built Using MIMIC Clinical Text. PhysioNet (2023).\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 35\n[110] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. 2020.\nBART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension. In ACL. 7871–7880.\n[111] Patrick Lewis, Myle Ott, Jingfei Du, and Veselin Stoyanov. 2020. Pretrained Language Models for Biomedical and Clinical Tasks: Understanding\nand Extending the State-of-the-Art. In Clinical Natural Language Processing Workshop . 146–157.\n[112] Fei Li, Yonghao Jin, Weisong Liu, Bhanu Pratap Singh Rawat, Pengshan Cai, and Hong Yu. 2019. Fine-Tuning Bidirectional Encoder Representations\nFrom Transformers (BERT)–Based Models on Large-Scale Electronic Health Record Notes: An Empirical Study. JMIR Med Inform (2019).\n[113] Jiao Li, Yueping Sun, Robin J. Johnson, Daniela Sciaky, Chih-Hsuan Wei, Robert Leaman, Allan Peter Davis, Carolyn J. Mattingly, Thomas C.\nWiegers, and Zhiyong Lu. 2016. BioCreative V CDR task corpus: a resource for chemical disease relation extraction. Database (2016).\n[114] Y. Li, S. Rao, J. Solares, A. Hassaine, R. Ramakrishnan, D. Canoy, Y. Zhu, K. Rahimi, and G. Salimi-Khorshidi. 2020. BEHRT: Transformer for\nElectronic Health Records. Sci Rep (2020).\n[115] Yikuan Li, Ramsey M. Wehbe, Faraz S. Ahmad, Hanyin Wang, and Yuan Luo. 2022. Clinical-Longformer and Clinical-BigBird: Transformers for\nlong clinical sequences. arXiv:2201.11838\n[116] Yudong Li, Yuqing Zhang, Zhe Zhao, Linlin Shen, Weijie Liu, Weiquan Mao, and Hui Zhang. 2022. CSL: A Large-scale Chinese Scientific Literature\nDataset. In COLING . 3917–3923.\n[117] Bingqian Lin, Zicong Chen, Mingjie Li, Haokun Lin, Hang Xu, Yi Zhu, Jianzhuang Liu, Wenjia Cai, Lei Yang, Shen Zhao, et al .2023. Towards\nMedical Artificial General Intelligence via Knowledge-Enhanced Multimodal Pretraining. arXiv:2304.14204\n[118] Chen Lin, Timothy Miller, Dmitriy Dligach, Steven Bethard, and Guergana Savova. 2021. EntityBERT: Entity-centric Masking Strategy for Model\nPretraining for the Clinical Domain. In BioNLP . 191–201.\n[119] Zhouhan Lin, Minwei Feng, Cícero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen Zhou, and Yoshua Bengio. 2017. A Structured Self-Attentive\nSentence Embedding. In ICLR .\n[120] Fangyu Liu, Ehsan Shareghi, Zaiqiao Meng, Marco Basaldella, and Nigel Collier. 2021. Self-Alignment Pretraining for Biomedical Entity Represen-\ntations. In NAACL . 4228–4238.\n[121] Pengfei Liu, Yiming Ren, and Zhixiang Ren. 2023. GIT-Mol: A Multi-modal Large Language Model for Molecular Science with Graph, Image, and\nText. arXiv:2308.06911\n[122] Pengfei Liu, Weizhe Yuan, Jinlan Fu, Zhengbao Jiang, Hiroaki Hayashi, and Graham Neubig. 2023. Pre-Train, Prompt, and Predict: A Systematic\nSurvey of Prompting Methods in Natural Language Processing. ACM Comput. Surv. (2023).\n[123] Shixuan Liu, Chen Peng, Chao Wang, Xiangyan Chen, and Shuangyong Song. 2023. icsBERTs: Optimizing Pre-trained Language Models in\nIntelligent Customer Service. In INNS DLIA@IJCNN 2023 . 127–136.\n[124] Xiao Liu, Da Yin, Jingnan Zheng, Xingjian Zhang, Peng Zhang, Hongxia Yang, Yuxiao Dong, and Jie Tang. 2022. OAG-BERT: Towards a Unified\nBackbone Language Model for Academic Knowledge Services. In KDD . 3418–3428.\n[125] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.\nRoBERTa: A Robustly Optimized BERT Pretraining Approach. arXiv:1907.11692\n[126] Zhengliang Liu, Xinyu He, Lei Liu, Tianming Liu, and Xiaoming Zhai. 2023. Context Matters: A Strategy to Pre-train Language Model for Science\nEducation. In AIED . 666–674.\n[127] Kyle Lo, Lucy Lu Wang, Mark Neumann, Rodney Kinney, and Daniel Weld. 2020. S2ORC: The Semantic Scholar Open Research Corpus. In ACL.\n[128] Siqu Long, Feiqi Cao, Soyeon Caren Han, and Haiqin Yang. 2022. Vision-and-Language Pretrained Models: A Survey. In IJCAI . 5530–5537.\n[129] Qiuhao Lu, Dejing Dou, and Thien Nguyen. 2022. ClinicalT5: A Generative Language Model for Clinical Text. In Findings of EMNLP . 5436–5443.\n[130] Renqian Luo, Liai Sun, Yingce Xia, Tao Qin, Sheng Zhang, Hoifung Poon, and Tie-Yan Liu. 2022. BioGPT: generative pre-trained transformer for\nbiomedical text generation and mining. Briefings in Bioinformatics (2022).\n[131] Kelvin Luu, Xinyi Wu, Rik Koncel-Kedziorski, Kyle Lo, Isabel Cachola, and Noah A. Smith. 2021. Explaining Relationships Between Scientific\nDocuments. In ACL-IJCNLP . 2130–2144.\n[132] Ruotian Ma, Yiding Tan, Xin Zhou, Xuanting Chen, Di Liang, Sirui Wang, Wei Wu, and Tao Gui. 2022. Searching for Optimal Subword Tokenization\nin Cross-domain NER. In IJCAI . 4289–4295.\n[133] Enrique Manjavacas Arevalo and Lauren Fonteyn. 2021. MacBERTh: Development and Evaluation of a Historically Pre-trained Language Model\nfor English (1450-1950). In Workshop on Natural Language Processing for Digital Humanities . 23–36.\n[134] Joseph Mariani, Gil Francopoulo, and Patrick Paroubek. 2019. The NLP4NLP Corpus (I): 50 Years of Publication, Collaboration and Citation in\nSpeech and Language Processing. Frontiers in Research Metrics and Analytics (2019).\n[135] Joseph Mariani, Gil Francopoulo, Patrick Paroubek, and Frederic Vernier. 2022. NLP4NLP+5: The Deep (R)evolution in Speech and Language\nProcessing. Frontiers in Research Metrics and Analytics (2022).\n[136] Yiwen Meng, William Speier, Michael K. Ong, and Corey W. Arnold. 2021. Bidirectional Representation Learning From Transformers Using\nMultimodal Electronic Health Record Data to Predict Depression. IEEE J. of Biomedical and Health Informatics (2021), 3121–3129.\n[137] George Michalopoulos, Yuanxin Wang, Hussam Kaka, Helen Chen, and Alexander Wong. 2021. UmlsBERT: Clinical Domain Knowledge\nAugmentation of Contextual Embeddings Using the Unified Medical Language System Metathesaurus. In NAACL . 1744–1753.\n[138] Tomás Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. 2013. Efficient Estimation of Word Representations in Vector Space. In ICLR .\n[139] Nguyen Minh, Vu Hoang Tran, Vu Hoang, Huy Duc Ta, Trung Huu Bui, and Steven Quoc Hung Truong. 2022. ViHealthBERT: Pre-trained Language\nModels for Vietnamese in Health Text Mining. In LREC . 328–337.\nManuscript submitted to ACM36 Ho et al.\n[140] Giacomo Miolo, Giulio Mantoan, and Carlotta Orsenigo. 2021. ELECTRAMed: a new pre-trained language representation model for biomedical\nNLP. arXiv:2104.09585\n[141] Saif M. Mohammad. 2020. NLP Scholar: A Dataset for Examining the State of NLP Research. In LREC . 868–877.\n[142] Michael Moret, Irene Pachon Angona, Leandro Cotos, Shen Yan, Kenneth Atz, Cyrill Brunner, Martin Baumgartner, Francesca Grisoni, and Gisbert\nSchneider. 2023. Leveraging molecular structure and bioactivity with chemical language models for de novo drug design. Nature Communications .\n[143] Tsendsuren Munkhdalai, Meijing Li, Khuyagbaatar Batsuren, Hyeon Ah Park, Nak Hyeon Choi, and Keun Ho Ryu. 2015. Incorporating domain\nknowledge in chemical and biomedical named entity recognition with word representations. J. of cheminformatics (2015), 1–8.\n[144] Martin Müller, Marcel Salathé, and Per E. Kummervold. 2023. COVID-Twitter-BERT: A natural language processing model to analyse COVID-19\ncontent on Twitter. Frontiers in Artificial Intelligence (2023).\n[145] Usman Naseem, Adam Dunn, Matloob Khushi, and Jinman Kim. 2022. Benchmarking for biomedical natural language processing tasks with a\ndomain specific ALBERT. BMC Bioinformatics (2022).\n[146] Joel Niklaus and Daniele Giofre. 2023. Can we Pretrain a SotA Legal Language Model on a Budget From Scratch?. In SustaiNLP . 158–182.\n[147] Benjamin Nye, Junyi Jessy Li, Roma Patel, Yinfei Yang, Iain Marshall, Ani Nenkova, and Byron Wallace. 2018. A Corpus with Multi-Level\nAnnotations of Patients, Interventions and Outcomes to Support Language Processing for Medical Literature. In ACL. 197–207.\n[148] Brian Ondov, Kush Attal, and Dina Demner-Fushman. 2022. A survey of automated methods for biomedical text simplification. JAMIA (2022).\n[149] OpenAI. 2023. GPT-4 Technical Report. arXiv:2303.08774\n[150] Ibrahim Burak Ozyurt. 2020. On the effectiveness of small, discriminatively pre-trained language representation models for biomedical text mining.\nInWorkshop on Scholarly Document Processing . 104–112.\n[151] Evangelos Pafilis, Sune Frankild, Lucia Fanini, Sarah Faulwetter, Christina Pavloudi, Katerina Vasileiadou, C. Arvanitidis, and Lars Jensen. 2013.\nThe SPECIES and ORGANISMS resources for fast and accurate identification of taxonomic names in text. PloS one (2013).\n[152] Anusri Pampari, Preethi Raghavan, Jennifer Liang, and Jian Peng. 2018. emrQA: A Large Corpus for Question Answering on Electronic Medical\nRecords. In EMNLP . 2357–2368.\n[153] Shirui Pan, Linhao Luo, Yufei Wang, Chen Chen, Jiapu Wang, and Xindong Wu. 2024. Unifying large language models and knowledge graphs: A\nroadmap. IEEE Transactions on Knowledge and Data Engineering (2024).\n[154] Dimitris Pappas, Ion Androutsopoulos, and Haris Papageorgiou. 2018. BioRead: A New Dataset for Biomedical Reading Comprehension. In LREC .\n[155] Dimitris Pappas, Petros Stavropoulos, Ion Androutsopoulos, and Ryan McDonald. 2020. BioMRC: A Dataset for Biomedical Machine Reading\nComprehension. In BioNLP . 140–149.\n[156] Vera Pavlova and Mohammed Makhlouf. 2023. BIOptimus: Pre-training an Optimal Biomedical Language Model with Curriculum Learning for\nNamed Entity Recognition. In BioNLP . 337–349.\n[157] Shuai Peng, Ke Yuan, Liangcai Gao, and Zhi Tang. 2021. MathBERT: A Pre-Trained Model for Mathematical Formula Understanding. arXiv:2105.00377\n[158] Yifan Peng, Shankai Yan, and Zhiyong Lu. 2019. Transfer Learning in Biomedical Natural Language Processing: An Evaluation of BERT and ELMo\non Ten Benchmarking Datasets. In BioNLP . 58–65.\n[159] Matthew E. Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. 2018. Deep Contextualized\nWord Representations. In NAACL . 2227–2237.\n[160] Long Phan, Tai Dang, Hieu Tran, Trieu H. Trinh, Vy Phan, Lam D. Chau, and Minh-Thang Luong. 2023. Enriching Biomedical Knowledge for\nLow-resource Language Through Large-scale Translation. In EACL . 3131–3142.\n[161] Long Phan, Hieu Tran, Daniel Le, Hieu Nguyen, James Annibal, Alec Peltekian, and Yanfang Ye. 2021. CoTexT: Multi-task Learning with Code-Text\nTransformer. In NLP4Prog . 40–47.\n[162] Long N. Phan, James T. Anibal, Hieu Tran, Shaurya Chanana, Erol Bahadroglu, Alec Peltekian, and Grégoire Altan-Bonnet. 2021. SciFive: a\ntext-to-text transformer model for biomedical literature. arXiv:2106.03598\n[163] Nina Poerner, Ulli Waltinger, and Hinrich Schütze. 2020. Inexpensive Domain Adaptation of Pretrained Language Models: Case Studies on\nBiomedical NER and Covid-19 QA. In Findings of EMNLP . 1482–1490.\n[164] Dragomir R. Radev, Pradeep Muthukrishnan, and Vahed Qazvinian. 2009. The ACL Anthology Network Corpus. In NLPIR4DL . 54–61.\n[165] Alec Radford and Karthik Narasimhan. 2018. Improving Language Understanding by Generative Pre-Training.\n[166] Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019. Language Models are Unsupervised Multitask Learners.\n[167] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020. Exploring\nthe Limits of Transfer Learning with a Unified Text-to-Text Transformer. JMLR (2020), 1–67.\n[168] Laila Rasmy, Yang Xiang, Ziqian Xie, Cui Tao, and Degui Zhi. 2020. Med-BERT: pretrained contextualized embeddings on large-scale structured\nelectronic health records for disease prediction. NPJ Digital Medicine (2020).\n[169] Siva Reddy, Danqi Chen, and Christopher D. Manning. 2019. CoQA: A Conversational Question Answering Challenge. TACL (2019), 249–266.\n[170] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. 2018. Semantically Equivalent Adversarial Rules for Debugging NLP models. In ACL.\n[171] Rafael BM Rodrigues, Pedro IM Privatto, Gustavo José de Sousa, Rafael P Murari, Luis CS Afonso, João P Papa, et al .2022. PetroBERT: A Domain\nAdaptation Language Model for Oil and Gas Applications in Portuguese. In PROPOR . 101–109.\n[172] Omid Rohanian, Mohammadmahdi Nouriborji, Hannah Jauncey, Samaneh Kouchaki, ISARIC Clinical Characterisation Group, Lei Clifton, Laura\nMerson, and David A. Clifton. 2023. Lightweight Transformers for Clinical Natural Language Processing. arXiv:2302.04725\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 37\n[173] Matías Rojas, Jocelyn Dunstan, and Fabián Villena. 2022. Clinical Flair: A Pre-Trained Language Model for Spanish Clinical Natural Language\nProcessing. In Clinical Natural Language Processing Workshop . 87–92.\n[174] Alexey Romanov and Chaitanya Shivade. 2018. Lessons from Natural Language Inference in the Clinical Domain. In EMNLP . 1586–1596.\n[175] David R Russell and Viviana Cortes. 2012. Academic and scientific texts: the same or different communities? In University writing: Selves and texts\nin academic societies . 1–17.\n[176] Tarek Saier and Michael Färber. 2019. Bibliometric-Enhanced arXiv: A Data Set for Paper-Based and Citation-Based Tasks. In BIR@ECIR .\n[177] Chris Sanchez and Zheyuan Zhang. 2022. The Effects of In-domain Corpus Size on pre-training BERT. arXiv:2212.07914\n[178] Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. 2019. DistilBERT, a distilled version of BERT: smaller, faster, cheaper and\nlighter. In EMC2@NeurIPS 2019 .\n[179] Thiago Santos, Amara Tariq, Susmita Das, Kavyasree Vayalpati, Geoffrey H. Smith, Hari Trivedi, and Imon Banerjee. 2022. PathologyBERT -\nPre-trained Vs. A New Transformer Language Model for Pathology Domain. In AMIA .\n[180] Teven Le Scao, Angela Fan, Christopher Akiki, Ellie Pavlick, Suzana Ilić, Daniel Hesslow, Roman Castagné, Alexandra Sasha Luccioni, François\nYvon, et al. 2022. Bloom: A 176b-parameter open-access multilingual language model. arXiv preprint arXiv:2211.05100 (2022).\n[181] Stefan Schweter, Luisa März, Katharina Schmid, and Erion Çano. 2022. hmBERT: Historical Multilingual Language Models for Named Entity\nRecognition. In CLEF . 1109–1129.\n[182] Junyuan Shang, Tengfei Ma, Cao Xiao, and Jimeng Sun. 2019. Pre-training of Graph Augmented Transformers for Medication Recommendation. In\nIJCAI . 5953–5959.\n[183] Shreyas Sharma and Ron Daniel Jr au2. 2019. BioFLAIR: Pretrained Pooled Contextualized Embeddings for Biomedical Sequence Labeling Tasks.\narXiv:1908.05760\n[184] Jia Tracy Shen, Michiharu Yamashita, Ethan Prihar, Neil Heffernan, Xintao Wu, Ben Graff, and Dongwon Lee. 2023. MathBERT: A Pre-trained\nLanguage Model for General NLP Tasks in Mathematics Education. arXiv:2106.07340\n[185] Si Shen, Jiangfeng Liu, Litao Lin, Ying Huang, Lin Zhang, Chang Liu, Yutong Feng, and Dongbo Wang. 2022. SsciBERT: A Pre-Trained Language\nModel for Social Science Texts. Scientometrics (2022), 1241–1263.\n[186] Zejiang Shen, Kyle Lo, Lucy Lu Wang, Bailey Kuehl, Daniel S. Weld, and Doug Downey. 2022. VILA: Improving Structured Content Extraction\nfrom Scientific PDFs Using Visual Layout Groups. TACL (2022), 376–392.\n[187] Pranav Shetty, Arunkumar Chitteth Rajan, Chris Kuenneth, Sonakshi Gupta, Lakshmi Prerana Panchumarti, et al .2023. A general-purpose material\nproperty data extraction pipeline from large polymer corpora using natural language processing. npj Computational Materials (2023).\n[188] Hoo-Chang Shin, Yang Zhang, Evelina Bakhturina, Raul Puri, Mostofa Patwary, Mohammad Shoeybi, and Raghav Mani. 2020. BioMegatron: Larger\nBiomedical Domain Language Model. In EMNLP . 4700–4706.\n[189] Yuqi Si, Jingqi Wang, Hua Xu, and Kirk Roberts. 2019. Enhancing clinical concept extraction with contextual embeddings. JAMIA (2019), 1297–1304.\n[190] Mayank Singh, Pradeep Dogga, Sohan Patro, Dhiraj Barnwal, Ritam Dutt, Rajarshi Haldar, Pawan Goyal, and Animesh Mukherjee. 2018. CL\nScholar: The ACL Anthology Knowledge Graph Miner. In NAACL . 16–20.\n[191] Karan Singhal, Shekoofeh Azizi, Tao Tu, S Sara Mahdavi, Jason Wei, Hyung Won Chung, Nathan Scales, Ajay Tanwani, Heather Cole-Lewis,\nStephen Pfohl, et al. 2023. Large language models encode clinical knowledge. Nature (2023), 1–9.\n[192] Michael A Skinnider, R Greg Stacey, David S Wishart, and Leonard J Foster. 2021. Chemical language models enable navigation in sparsely\npopulated chemical space. Nature Machine Intelligence (2021), 759–770.\n[193] Larry Smith, Lorraine K Tanabe, Rie Johnson nee Ando, Cheng-Ju Kuo, I-Fang Chung, Chun-Nan Hsu, Yu-Shi Lin, Roman Klinger, Christoph M\nFriedrich, Kuzman Ganchev, et al. 2008. Overview of BioCreative II gene mention recognition. Genome Biology (2008).\n[194] Luciana B Sollaci and Mauricio G Pereira. 2004. The introduction, methods, results, and discussion (IMRAD) structure: a fifty-year survey. J. of the\nmedical library association (2004), 364.\n[195] Matteo Stefanini, Marcella Cornia, Lorenzo Baraldi, Silvia Cascianelli, Giuseppe Fiameni, and Rita Cucchiara. 2023. From Show to Tell: A Survey\non Deep Learning-Based Image Captioning. IEEE Trans. Pattern Anal. Mach. Intell. (2023), 539–559.\n[196] Weiyi Sun, Anna Rumshisky, and Özlem Uzuner. 2013. Evaluating temporal relations in clinical text: 2012 i2b2 Challenge. JAMIA (2013), 806–813.\n[197] Madhumita Sushil, Dana Ludwig, Atul J. Butte, and Vivek A. Rudrapatna. 2022. Developing a general-purpose clinical language inference model\nfrom a large corpus of clinical notes. arXiv:2210.06566\n[198] Nasrin Taghizadeh, Ehsan Doostmohammadi, Elham Seifossadat, Hamid R. Rabiee, and Maedeh S. Tahaei. 2021. SINA-BERT: A pre-trained\nLanguage Model for Analysis of Medical Texts in Persian. arXiv:2104.07613\n[199] Wen Tai, H. T. Kung, Xin Dong, Marcus Comiter, and Chang-Fu Kuo. 2020. exBERT: Extending Pre-trained Models with Domain-specific Vocabulary\nUnder Constrained Training Resources. In Findings of EMNLP . 1433–1439.\n[200] Ross Taylor, Marcin Kardas, Guillem Cucurull, Thomas Scialom, Anthony Hartshorn, Elvis Saravia, Andrew Poulton, Viktor Kerkez, and Robert\nStojnic. 2022. Galactica: A Large Language Model for Science. arXiv:2211.09085\n[201] Huishuang Tian, Kexin Yang, Dayiheng Liu, and Jiancheng Lv. 2021. AnchiBERT: A Pre-Trained Model for Ancient Chinese Language Understanding\nand Generation. In IJCNN . 1–8.\n[202] Robert Tinn, Hao Cheng, Yu Gu, Naoto Usuyama, Xiaodong Liu, Tristan Naumann, Jianfeng Gao, and Hoifung Poon. 2023. Fine-tuning large\nneural language models for biomedical natural language processing. Patterns (2023), 100729.\nManuscript submitted to ACM38 Ho et al.\n[203] Rian Touchent, Laurent Romary, and Eric de la Clergerie. 2023. CamemBERT-bio: a Tasty French Language Model Better for your Health.\narXiv:2306.15550\n[204] Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric\nHambro, Faisal Azhar, et al. 2023. LLaMA: Open and Efficient Foundation Language Models. arXiv:2302.13971\n[205] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava,\nShruti Bhosale, et al. 2023. Llama 2: Open Foundation and Fine-Tuned Chat Models. arXiv:2307.09288\n[206] Amalie Trewartha, Nicholas Walker, Haoyan Huo, Sanghoon Lee, Kevin Cruse, John Dagdelen, et al .2022. Quantifying the advantage of\ndomain-specific pre-training on named entity recognition tasks in materials science. Patterns (2022), 100488.\n[207] George Tsatsaronis, Georgios Balikas, Prodromos Malakasiotis, Ioannis Partalas, Matthias Zschunke, Michael R Alvers, Dirk Weissenborn, et al .\n2015. An overview of the BIOASQ large-scale biomedical semantic indexing and question answering competition. BMC Bioinformatics (2015), 138.\n[208] Hazal Türkmen, Oguz Dikenelli, Cenk Eraslan, Mehmet Calli, and Suha Ozbek. 2023. Harnessing the Power of BERT in the Turkish Clinical\nDomain: Pretraining Approaches for Limited Data Scenarios. In Clinical Natural Language Processing Workshop . 161–170.\n[209] Hazal Turkmen, Oguz Dikenelli, Cenk Eraslan, and Mehmet Cem Callı. 2022. Bioberturk: Exploring Turkish Biomedical Language Model\nDevelopment Strategies in Low Resource Setting.\n[210] Özlem Uzuner, Brett R South, Shuying Shen, and Scott L DuVall. 2011. 2010 i2b2/VA challenge on concepts, assertions, and relations in clinical text.\nJAMIA (2011), 552–556.\n[211] Taha ValizadehAslani, Yiwen Shi, Ping Ren, Jing Wang, Yi Zhang, Meng Hu, Liang Zhao, and Hualou Liang. 2023. PharmBERT: a domain-specific\nBERT model for drug labels. Briefings in Bioinformatics (2023).\n[212] Erik M. van Mulligen, Annie Fourrier-Reglat, David Gurwitz, Mariam Molokhia, Ainhoa Nieto, Gianluca Trifiro, Jan A. Kors, and Laura I. Furlong.\n2012. The EU-ADR corpus: Annotated drugs, diseases, targets, and their relationships. J. of Biomedical Informatics (2012), 879–884.\n[213] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is\nall you need. NIPS (2017).\n[214] Stella Verkijk and Piek Vossen. 2021. MedRoBERTa.nl: A Language Model for Dutch Electronic Health Records. CLIN Journal (2021), 141–159.\n[215] Shoya Wada, Toshihiro Takeda, Shiro Manabe, Shozo Konishi, Jun Kamohara, and Yasushi Matsumura. 2021. Pre-training technique to localize\nmedical BERT and enhance biomedical BERT. arXiv:2005.07202\n[216] David Wadden, Shanchuan Lin, Kyle Lo, Lucy Lu Wang, Madeleine van Zuylen, Arman Cohan, and Hannaneh Hajishirzi. 2020. Fact or Fiction:\nVerifying Scientific Claims. In EMNLP . 7534–7550.\n[217] Jan Philip Wahle, Terry Ruas, Saif Mohammad, and Bela Gipp. 2022. D3: A Massive Dataset of Scholarly Metadata for Analyzing the State of\nComputer Science Research. In LREC . 2642–2651.\n[218] Alex Wang, Yada Pruksachatkun, Nikita Nangia, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel Bowman. 2019. SuperGLUE:\nA Stickier Benchmark for General-Purpose Language Understanding Systems. In NeurIPS .\n[219] Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. 2019. GLUE: A Multi-Task Benchmark and Analysis\nPlatform for Natural Language Understanding. In ICLR .\n[220] Benyou Wang, Qianqian Xie, Jiahuan Pei, Zhihong Chen, Prayag Tiwari, Zhao Li, and Jie Fu. 2024. Pre-trained Language Models in Biomedical\nDomain: A Systematic Survey. ACM Comput. Surv. (2024), 55:1–55:52.\n[221] Chenguang Wang, Xiao Liu, and Dawn Song. 2020. Language Models are Open Knowledge Graphs. arXiv:2010.11967\n[222] Guangyu Wang, Guoxing Yang, Zongxin Du, Longjun Fan, and Xiaohu Li. 2023. ClinicalGPT: Large Language Models Finetuned with Diverse\nMedical Data and Comprehensive Evaluation. arXiv:2306.09968\n[223] Haifeng Wang, Jiwei Li, Hua Wu, Eduard Hovy, and Yu Sun. 2022. Pre-Trained Language Models and Their Applications. Engineering (2022).\n[224] Haifeng Wang, Hua Wu, Zhongjun He, Liang Huang, and Kenneth Ward Church. 2022. Progress in Machine Translation. Engineering (2022).\n[225] Xiao Wang, Guangyao Chen, Guangwu Qian, Pengcheng Gao, Xiao-Yong Wei, Yaowei Wang, Yonghong Tian, and Wen Gao. 2023. Large-scale\nMulti-modal Pre-trained Models: A Comprehensive Survey. Mach. Intell. Res. (2023), 447–482.\n[226] Zige Wang, Wanjun Zhong, Yufei Wang, Qi Zhu, Fei Mi, Baojun Wang, Lifeng Shang, Xin Jiang, and Qun Liu. 2023. Data Management For Large\nLanguage Models: A Survey. arXiv:2312.01700\n[227] Neha Warikoo, Yung-Chun Chang, and Wen-Lian Hsu. 2020. LBERT: Lexically-aware Transformers based Bidirectional Encoder Representation\nmodel for learning Universal Bio-Entity Relations. Bioinformatics (2020).\n[228] Nicolas Webersinke, Mathias Kraus, Julia Anna Bingler, and Markus Leippold. 2022. ClimateBert: A Pretrained Language Model for Climate-Related\nText. arXiv:2110.12010\n[229] David Weininger. 1988. SMILES, a chemical language and information system. 1. Introduction to methodology and encoding rules. J. of Chemical\nInformation and Computer Sciences (1988), 31–36.\n[230] Johannes Welbl, Pontus Stenetorp, and Sebastian Riedel. 2018. Constructing Datasets for Multi-hop Reading Comprehension Across Documents.\nTACL (2018), 287–302.\n[231] Sarah Wiegreffe and Ana Marasovic. 2021. Teach Me to Explain: A Review of Datasets for Explainable Natural Language Processing. In NeurIPS .\n[232] Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus\nMacherey, et al .2016. Google’s Neural Machine Translation System: Bridging the Gap between Human and Machine Translation. arXiv:1609.08144\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 39\n[233] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. 2021. A Comprehensive Survey on Graph Neural\nNetworks. IEEE Trans. Neural Networks Learn. Syst. (2021), 4–24.\n[234] Minghao Xu, Xinyu Yuan, Santiago Miret, and Jian Tang. 2023. ProtST: Multi-Modality Learning of Protein Sequences and Biomedical Texts. In\nICML . 38749–38767.\n[235] Peng Xu, Xiatian Zhu, and David A. Clifton. 2023. Multimodal Learning With Transformers: A Survey. IEEE Trans. Pattern Anal. Mach. Intell.\n[236] Hiroki Yamauchi, Tomoyuki Kajiwara, Marie Katsurai, Ikki Ohmukai, and Takashi Ninomiya. 2022. A Japanese Masked Language Model for\nAcademic Domain. In Workshop on Scholarly Document Processing . 152–157.\n[237] An Yan, Julian McAuley, Xing Lu, Jiang Du, Eric Y Chang, Amilcare Gentili, and Chun-Nan Hsu. 2022. RadBERT: Adapting transformer-based\nlanguage models to radiology. Radiology: Artificial Intelligence (2022).\n[238] Xi Yang, Jiang Bian, William R Hogan, and Yonghui Wu. 2020. Clinical concept extraction using transformers. JAMIA (2020), 1935–1942.\n[239] Xi Yang, Aokun Chen, Nima PourNejatian, Hoo Chang Shin, Kaleb E Smith, Christopher Parisien, Colin Compas, Cheryl Martin, Anthony B Costa,\nMona G Flores, et al. 2022. A large language model for electronic health records. NPJ Digit. Med. (2022), 194.\n[240] Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G. Carbonell, Ruslan Salakhutdinov, and Quoc V. Le. 2019. XLNet: Generalized Autoregressive\nPretraining for Language Understanding. In NeurIPS . 5754–5764.\n[241] Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William Cohen, Ruslan Salakhutdinov, and Christopher D. Manning. 2018. HotpotQA: A\nDataset for Diverse, Explainable Multi-hop Question Answering. In EMNLP . 2369–2380.\n[242] Yunzhi Yao, Shaohan Huang, Wenhui Wang, Li Dong, and Furu Wei. 2021. Adapt-and-Distill: Developing Small, Fast and Effective Pretrained\nLanguage Models for Domains. In Findings of ACL-IJCNLP . 460–470.\n[243] Michihiro Yasunaga, Antoine Bosselut, Hongyu Ren, Xikun Zhang, Christopher D Manning, Percy Liang, and Jure Leskovec. 2022. Deep Bidirectional\nLanguage-Knowledge Graph Pretraining. In NeurIPS .\n[244] Michihiro Yasunaga, Jure Leskovec, and Percy Liang. 2022. LinkBERT: Pretraining Language Models with Document Links. In ACL. 8003–8016.\n[245] Michiko Yoshitake, Fumitaka Sato, Hiroyuki Kawano, and Hiroshi Teraoka. 2022. MaterialBERT for natural language processing of materials\nscience texts. Science and Technology of Advanced Materials: Methods (2022), 372–380.\n[246] Hongyi Yuan, Zheng Yuan, Ruyi Gan, Jiaxing Zhang, Yutao Xie, and Sheng Yu. 2022. BioBART: Pretraining and Evaluation of A Biomedical\nGenerative Language Model. In BioNLP . 97–109.\n[247] Zheng Yuan, Qiao Jin, Chuanqi Tan, Zhengyun Zhao, Hongyi Yuan, Fei Huang, and Songfang Huang. 2023. RAMM: Retrieval-augmented Biomedical\nVisual Question Answering with Multi-modal Pre-training. In ACM MM . 547–556.\n[248] Zheng Yuan, Yijia Liu, Chuanqi Tan, Songfang Huang, and Fei Huang. 2021. Improving Biomedical Pretrained Language Models with Knowledge.\nInBioNLP . 180–190.\n[249] Zheng Yuan, Zhengyun Zhao, Haixia Sun, Jiao Li, Fei Wang, and Sheng Yu. 2022. CODER: Knowledge-infused cross-lingual medical term embedding\nfor term normalization. J. Biomed. Informatics (2022), 103983.\n[250] Manzil Zaheer, Guru Guruganesh, Kumar Avinava Dubey, Joshua Ainslie, Chris Alberti, Santiago Ontanon, Philip Pham, Anirudh Ravula, Qifan\nWang, Li Yang, et al. 2020. Big bird: Transformers for longer sequences. NeurIPS (2020), 17283–17297.\n[251] Wei Zeng, Xiaozhe Ren, Teng Su, Hui Wang, Yi Liao, Zhiwei Wang, Xin Jiang, ZhenZhang Yang, Kaisheng Wang, Xiaoda Zhang, et al .2021.\nPanGu- 𝛼: Large-scale Autoregressive Pretrained Chinese Language Models with Auto-parallel Computation. arXiv:2104.12369\n[252] Chenshuang Zhang, Chaoning Zhang, Mengchun Zhang, and In So Kweon. 2023. Text-to-image Diffusion Models in Generative AI: A Survey.\narXiv:2303.07909\n[253] Jingyi Zhang, Jiaxing Huang, Sheng Jin, and Shijian Lu. 2023. Vision-Language Models for Vision Tasks: A Survey. arXiv:2304.00685\n[254] Kai Zhang, Jun Yu, Zhiling Yan, Yixin Liu, Eashan Adhikarla, Sunyang Fu, Xun Chen, Chen Chen, Yuyin Zhou, Xiang Li, et al .2023. BiomedGPT: A\nUnified and Generalist Biomedical Generative Pre-trained Transformer for Vision, Language, and Multimodal Tasks. arXiv:2305.17100\n[255] Ningyu Zhang, Qianghuai Jia, Kangping Yin, Liang Dong, Feng Gao, and Nengwei Hua. 2020. Conceptualized Representation Learning for Chinese\nBiomedical Text Mining. arXiv:2008.10813\n[256] Taolin Zhang, Zerui Cai, Chengyu Wang, Minghui Qiu, Bite Yang, and Xiaofeng He. 2021. SMedBERT: A Knowledge-Enhanced Pre-trained\nLanguage Model with Structured Semantics for Medical Text Mining. In ACL-IJCNLP . 5882–5893.\n[257] Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015. Character-level convolutional networks for text classification. NIPS (2015).\n[258] Zachariah Zhang, Jingshu Liu, and Narges Razavian. 2020. BERT-XML: Large Scale Automated ICD Coding Using BERT Pretraining. In Clinical\nNatural Language Processing Workshop . 24–34.\n[259] Wayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, et al .\n2023. A Survey of Large Language Models. arXiv:2303.18223\n[260] Xu Zhong, Jianbin Tang, and Antonio Jimeno Yepes. 2019. PubLayNet: Largest Dataset Ever for Document Layout Analysis. In ICDAR . 1015–1022.\n[261] Sicheng Zhou, Nan Wang, Liwei Wang, Hongfang Liu, and Rui Zhang. 2022. CancerBERT: a cancer domain-specific language model for extracting\nbreast cancer phenotypes from electronic health records. JAMIA (2022), 1208–1216.\n[262] Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mohamed Elhoseiny. 2023. MiniGPT-4: Enhancing Vision-Language Understanding with\nAdvanced Large Language Models. arXiv:2304.10592\n[263] Hongyin Zhu, Hao Peng, Zhiheng Lyu, Lei Hou, Juanzi Li, and Jinghui Xiao. 2023. Pre-training language model incorporating domain-specific\nheterogeneous knowledge into a unified representation. Expert Systems with Applications (2023), 119369.\nManuscript submitted to ACM40 Ho et al.\nTable 13. Existing question answering datasets.\nYear Dataset Answer Style Size Domain Question Source Level\n2015 BioASQ [207] Yes/No, Extractive 3,743 Biomedical Expert Full paper\n2018 Biomed-Cloze [49] Cloze-style 1M Biomedical Automated Passage\n2018 emrQA [152] Extractive 400K Clinical documents Automated Clinical note\n2018 BioRead [154] Multiple-choice 16.4M Biomedical Automated 900 tokens\n2018 MedHop [230] Multiple-choice 2,508 Molecular biology Automated Abstract\n2019 PubmedQA [87] Yes/no/maybe 273.5K Biomedical Expert + Automated Abstract\n2020 BioMRC [155] Multiple-choice 821K Biomedical Automated Abstract\n2021 QASPER [45]Extractive, Abstractive,\nYes/No, Unanswerable5,049 NLP papers NLP practitioner Full paper\nA PRELIMINARY INFORMATION\nA.1 Existing Tasks and Datasets in Scientific Articles\nWe present important information of existing QA datasets in Table 13.\nB EXISTING LMS FOR PROCESSING SCIENTIFIC TEXT\nB.1 Non-Scientific Domains\nWhile our primary focus remains on SciLMs tailored for scientific text, it is essential to briefly summarize LMs trained\non non-scientific text during the same period. By providing a concise overview of these models, we hope to expand our\nknowledge of the overall LM landscape.\nAccording to our survey, we have found 21 LMs that fall under this category. Most of these models were built upon\nthe BERT or similar architecture. Moreover, we observed that models with a size of 110-125M are typically preferred.\nBERT-Based Models. There are 9 non-SciLMs based on the BERT architecture, which have been trained with\n110-125M parameters for various domains, including General, Bio, Financial, Aviation, Historical, and Political. Among\nthem, CT-BERT [ 144], FinBERT [ 8], CancerBERT [ 261], Bioreddit-BERT [ 12], Aviation-BERT [ 34], and SafeAeroBERT [ 7]\nwere continually pretrained on previous work’s checkpoints. In contrast, MacBERTh [ 133] and HmBERT [ 181] were\npretrained from scratch using their own corpora. ConfliBERT [76] was pretrained using both strategies.\nNon-NSP BERT-based models such as PetroBERT [ 171] and AnchiBERT [ 201] were continually pretrained using\nmodel weights from previous work. They were built for the petroleum domain in the Portuguese language and the\nAncient Chinese domain, respectively. PeerBERT [ 50] is a RoBERTa-based model pretrained from scratch on Peer\nReview Comments.\nSpecialized Architecture-Based Models. Several models combine MLM with additional objectives to solve specific\ntasks. For instance, MMBERT [ 95] is a multi-modal BERT model that utilizes a CNN to capture image and text features\nfor medical domain visual QA tasks. It was pretrained from scratch on a corpus of images and text. DisorBERT [ 9],\non the other hand, explores lexical knowledge to continually pretrain the BERT model to pay more attention to\nwords related to mental disorders. icsBERTs [ 123] adapts Multi-Task Learning and Sentence Structural Objective to\nimprove the performance of existing PLMs in the Chinese business domain. Moreover, TravelBERT [ 263] is a continued\npretraining LM aiming to learn entity and topic knowledge by incorporating triple classification objective and title\nmatching objective, respectively. The model was developed for the Chinese language in the tourism domain. Finally,\nBudgetLongformer [146] uses LongFormer to train from scratch on legal data.\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 41\nTable 14. A continuation of the table from Table 6. In the Training Objective column, MLM denotes Masked Language Modeling,\nNSP denotes Next Sentence Prediction, and SOP denotes Sentence Order Prediction. In the Type of Pre-training column, CPandFS\ndenote Continual Pretraining and From Scratch, respectively. In the Domain column, CSrepresents computer science, Biorepresents\nBiomedical domain, Chem represents Chemical domain, and Multi represents multiple domains. It is noted that the date information\nis chosen from the first date the paper appears on the Internet.\nNo. Date Model Architecture Training ObjectiveType of\nPre-trainingModel\nSizeDomain Pre-training Corpus\n42 2021/04 SINA-BERT [198] BERT MLM CP 110M BioManually Collected\n(2.8M Documents)\n43 2021/05 MathBERT (Peng) [157] BERTMLM, Masked Substructure Prediction,\nContext Correspondence PredictionCP 110M MathFormula-Context Pairs\nExtracted from arXiv\n44 2021/05 NukeLM [28]BERT\nRoBERTa-Base\nRoBERTa-LargeMLM CP110M\n125M\n355MChemU. S. Department of\nEnergy Office Scientific and\nTechnical Information Database\n45 2021/05 ProteinBERT [23] BERT Corrupted Token, Annotation Prediction FS 16M Bio UniProtKB/UniRef90 + GO\n46 2021/05 SciFive [162] T5 Span Corruption Prediction CP220M\n& 770MBio PubMed + PMC\n47 2021/06 BioELECTRA [92] ELECTRA Replaced Token Prediction CP & FS 110M Bio PubMed + PMC\n48 2021/06 ChemBERT [62] RoBERTa MLM CP 110M Chem ACS Publications\n49 2021/06 EntityBERT [118] BERT Entity-Centric MLM CP 110M Bio MIMIC\n50 2021/06 MathBERT (Shen) [184] RoBERTa MLM CP 110M MathMathematics Curricula\n+ Mathematics Textbooks\n+ Mathematics Course Syllabi\n+ Mathematics Paper Abstracts\n51 2021/07 MedGPT [98]GPT2 + GLU\n+ RotaryEmbedLM CP N/A Bio Electronic Health Record Notes\n52 2021/08 SMedBERT [256] SMedBERTMasked Neighbor Modeling, SOP,\nMasked Mention Modeling, MLMFS N/A Bio DXY BBS (Bulletin Board System)\n53 2021/09 Bio-cli [31] RoBERTaMLM, Subword Masking\nor Whole Word MaskingFS 125M Bio Clinical Corpus + Biomedical Corpus\n54 2021/09 MatSciBERT [64] BERT MLM CP 110M Chem Material Science Corpus\n55 2021/10 ClimateBert [228] DistilROBERTA MLM CP 66M ClimateNews Articles + Research Abstracts\n+ Corporate Climate Reports\n56 2021/10 MatBERT [206] BERT MLM FS 110M ChemPeer-Reviewed Materials\nScience Journal Articles\n57 2021/11 UTH-BERT [94] BERT MLM, NSP FS 110M BioClinical Texts from The\nUniversity of Tokyo Hospital\n58 2021/12 ChestXRayBERT [29] BERT MLM, NSP CP 110M Bio 20617 Scientific Papers in PMC\n59 2021/12 MedRoBERTa.nl [214] RoBERTa MLM FS 123M Bio Dutch Hospital Notes, from EHRs\n60 2021/12 PubMedELECTRA [202] ELECTRA Replaced Token Prediction FS110M\n& 335MBio PubMed\n61 2022/01 Clinical-Longformer [115]BigBird\nLongformerMLM CP166M\n149MBio MIMIC-III\n62 2022/03 BioLinkBERT [244] BERTMLM,\nDocument Relation PredictionFS110M\n& 340MBio PubMed + PMC\n63 2022/04 BioBART [246] BARTText Infilling,\nSentence PermutationCP140M\n& 400MBio PubMed\n64 2022/04 SecureBERT [3] RoBERTa MLM CP 125M CybersecurityManually Collected\n(98,411 Documents)\n65 2022/05 BatteryBERT [79] BERT MLM CP & FS 110M ChemBattery-Related Scientific Papers\nfrom RSC, Elsevier, Springer\n66 2022/05 bsc-bio-ehr-es [32] RoBERTa MLM FS 125M BioElectronic Health Record Corpus\n+ Biomedical Corpus\n67 2022/05 ChemGPT [57] GPT Autoregressive LM FS 1B Chem SMILES from PubCHEM\n68 2022/05 PathologyBERT [179] BERT MLM, NSP FS 110M BioManually Labeled Pathology Reports\n+ Unlabeled Pathology Reports from EUH\n69 2022/05 ScholarBERT [72] BERT MLM FS 770M Multi Public Resource Dataset\n70 2022/06 RadBERT [237] RoBERTa MLM CP 110M Bio Radiology Reports\n71 2022/06 SciDeBERTa [82] DeBERTa MLM CP N/A Multi S2ORC\n72 2022/06 SsciBERT [185] BERT MLM, NSP CP 110M Social Science Collected from Web of Science\n73 2022/06 ViHealthBERT [139] PhoBERTMLM, NSP,\nCapitalized PredictionCP 110M BioOSCAR Dataset\n+ Their Text Mining Corpus\n74 2022/07 Clinical Flair [173] Flair Character-Level Bi-LM CP N/A Bio Free-Text Diagnostic Suspicions\n75 2022/08 KM-BERT [96] BERT MLM, NSP CP 99M BioMedical Textbooks\n+ Health Information News\n+ Medical Research Articles\n76 2022/08 MaterialsBERT (Shetty) [187] PubMedBERT MLM, NSP, Whole-Word Masking CP 110M Chem Polymer Scholar\n77 2022/08 ProcessBERT [93] BERT MLM, NSP CP 110M Chem ChemECorpus\nManuscript submitted to ACM42 Ho et al.\nTable 15. A continuation of the table from Table 6 and Table 14.\nNo. Date Model Architecture Training ObjectiveType of\nPre-trainingModel\nSizeDomain Pre-training Corpus\n78 2022/09 BioGPT [130] GPT Autogressive LM FS347M\n& 1.5BBio PubMed\n79 2022/09 ChemBERTa-2 [4] RoBERTa MLM, Multi-Task Regression FS 125M Chem SMILES from PubCHEM\n80 2022/09 CSL-T5 [116] T5Fill-in-the-blank-style\nDenoising ObjectiveFS 220M Multi Chinese Academic Journals\n81 2022/09 MaterialBERT (Yoshitake) [245] BERT MLM, NSP FS 110M Chem Scientific Articles from NIMS\n82 2022/10 AcademicRoBERTa [236] RoBERTa MLM FS 125M Multi CiNii Articles\n83 2022/10 Bioberturk [209] BERT MLM, NSP CP & FS N/A BioTurkish Medical Articles\n+ Turkish Radiology Thesis\n84 2022/10 DRAGON [243] GreaseLM MLM, KG Link Prediction CP 360M Bio PubMed + UMLS KG\n85 2022/10 UCSF-BERT [197] BERT MLM, NSP FS 135M Bio Clinical Notes of UCSF Health\n86 2022/10 ViPubmedT5 [160] T5 Spans-masking Learning CP 220M Bio ViPubmed\n87 2022/11 Galactica [200] GPT Autogressive LM FS125M\n& 1.3B & 6.7B\n& 30B & 120BMultiPapers, Reference Material, Knowledge Bases,\nCommon Crawl, Code, Datasets\n88 2022/11 VarMAE [75] RoBERTa MLM CP 110M MultiSemantic Scholar Corpus\n+ Finance-Related Online Platforms\n89 2022/12 AliBERT [17] BERT MLM FS 110M BioDrug Database 550MB + Cochrane 27MB +\nArticles 4300MB + Thesis 300MB + RCP 2200MB\n90 2022/12 BioMedLM [20] GPT2 Autogressive LM FS 2.7B Bio PubMed + PMC\n91 2022/12 BioReader [58] T5 + RETRO MLM CP 229.5M Bio PubMed\n92 2022/12 ClinicalT5 [129] T5 Span-mask Denoising Objective CP220M\n& 770MBio MIMIC-III\n93 2022/12 CySecBERT [14] BERT MLM, NSP CP 110M CybersecurityBlog Data + arXiv Data + Twitter Data\n+ National Vulnerability Database\n94 2022/12 GatorTron [239] BERT MLM FS 8.9B BioDe-identified Clinical Notes\n+ PubMed + Wikipedia\n95 2022/12 Med-PaLM [191] Flan-PaLM Instruction Prompt Tuning CP 540B Bio N/A\n96 2023/01 Clinical-T5 [109] T5Fill-in-the-blank-style\nDenoising ObjectiveCP & FS220M\n& 770MBio MIMIC-III + MIMIC-IV\n97 2023/01 CPT-BigBird [52]BigBird\nLongformerMLM CP166M\n149MBio Senior Scholars’ Basket of Journals\n98 2023/01 ProtST [234] BERTMasked Protein Modeling\n+ Contrastive Learning\n+ Multi-modal Masked PredictionCP & FS N/A Protein ProtDescribe\n99 2023/01 SciEdBERT [126] BERT MLM CP 110M Science EducationThe Academic Journal Dataset\n+ Students’ Responses\n100 2023/02 Bioformer [56] BERT MLM, NSP FS 43M Bio PubMed + PMC\n101 2023/02Lightweight Clinical\nTransformers [172]DistilBERTMLM,\nKnowledge Distillation ObjectiveCP15M & 18M\n& 25M & 65MClinical MIMIC-III\n102 2023/03 ManuBERT [102] BERT MLM CP110M\n& 126MManufaturingManufacturing Process Journals\n+ Six Commonly Used Textbooks\n103 2023/03 RAMM [247] PubmedBERTMLM, Image-Text Matching,\nImage-Text Contrastive LearningCP N/A Bio ROCO, MIMIC-CXR, PMCPM\n104 2023/04 DrBERT [103] RoBERTa MLM FS 110M BioOnline Sources\n+ Private Hospital Stays Reports\n105 2023/04 MOTOR [117] BLIPMLM, Image-Text Matching,\nImage-Text Contrastive LearningCP N/A Bio MIMIC-CXR\n106 2023/05 BiomedGPT [254] BARTText Infilling,\nSentence PermutationFS33M\n& 93M\n& 182MBioCheXpert + PathVQA + Retinal Fundus\n+ International Skin Imaging Collaboration\n+ Peir Gross + CytoImageNet + MIMIC-III\n+ DeepLesion + NCBI BioNLP Corpus\n+ MedICaT + PubMed + SLAKE + IU X-ray\n107 2023/05 Patton [85]GNN-nested\nTransformerNetwork-contextualized MLM,\nMasked Node PredictionCP N/A MultiAcademic Networks from Microsoft Academic Graph,\nE-commerce Networks from Amazon\n108 2023/05 TurkRadBERT [208] BERT MLM, NSP CP & FS 110M BioGeneral Turkish Corpus + Head CT Reports\n+ Turkish Biomedical Corpus\n+ Turkish Electronic Radiology Theses\n109 2023/06 CamemBERT-bio [203] CamemBERT Whole Word Masking CP 111M Bio Biomed-fr: ISTEX, CLEAR, and E3C\n110 2023/06 ClinicalGPT [222] BLOOMSupervised Fine Tuning\nRank-based TrainingInstruction Tuning CP Bio Supervised Task Datasets\n111 2023/06 EriBERTa [46]RoBERTa\nLongformerMLM CP & FS125M\n149MBioMIMIC-III, EMEA, ClinicalTrials, PubMed,\nSNOMED-CT, SPACCC, UFAL, Wikipedia Med,\nPrivate Clinical Documents, Medical Crawler\n112 2023/06 K2 [47] LLaMA Cosine Loss CP 7B GeoscienceGeoscience-Related Wikipedia Pages,\nOpen-access Geoscience Papers,\nGeoscience Paper’s Abstracts\n113 2023/06 PharmBERT [211] BERT MLM CP 110M Bio DailyMed\n114 2023/07 BioNART [11] BERT Connectionist Temporal Classification CP 110M Bio PubMed/MEDLINE\n115 2023/07 BIOptimus [156] BERT MLM CP 110M Bio PubMed\n116 2023/07 KEBLM [105] BERTMLM, Ranking Objective,\nContrastive LearningCP N/A Bio UMLS, PubChem, MSI\n117 2023/08 GIT-Mol [121] BLIP2Xmodal-Text Matching,\nXmodal-Text Contrastive LearningN/A 700M Chem PubChem\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 43\nGeneration-Based Models. Several autoregressive Transformer architectures have also been employed. For instance,\nFinBART [ 73] is a BART-based model pretrained from scratch on its Chinese-language financial data. Similarly,\nJaCoText [ 55] was pretrained from CoTexT [ 161] checkpoints, basically T5-Base and T5-Large for the Code domain.\nAfter the release of GPT-3 [ 27], there has been a rise in the development of large-scale LMs. For example, PANGU- 𝛼\n[251] is a large-scale autoregressive pretrained Chinese LM built for the General domain with increasing magnitude of\nparameter sizes, namely 2.6B, 13B, and 200B. The model introduces a query layer on top of the stacked Transformer\nlayers to explicitly induce the expected output, which is the prediction of the next token in the pretraining stage of the\nautoregressive model.\nC EFFECTIVENESS OF LMS FOR PROCESSING SCIENTIFIC TEXT\nC.1 Details about Tasks and Datasets of SciLMs\nTable 16. Tasks and Datasets are used in existing LMs for scientific text. The underlying datasets are those in which the proposed\nSciLMs outperform previous models or achieve SOTA results.\nNo. Model Tasks Datasets\n1 BioBERTNamed Entity RecognitionNCBI-disease ,BC5CDR-disease ,BC5CDR-chemical ,BC4CHEMD ,\nBC2GM , JNLPBA, LINNAEUS, i2b2 2010, Species-800\nRelation Extraction ChemProt , GAD, EU-ADR\nQuestion Answering BioASQ 5b-factoid , BioASQ 6b-factoid , BioASQ 4b-factoid\n2 BERT-MIMIC Named Entity Recognition i2b2 2010 ,i2b2 2012 ,SemEval 2014 Task 7 ,SemEval 2015 Task 14\n(All)\n3 SciBERTNamed Entity Recognition BC5CDR , SciERC , JNLPBA, NCBI-disease\nPICO Extraction EBM-NLP\nText Classification ACL-ARC , SciCite , Paper Field\nRelation Extraction ChemProt , SciERC\nDependency Parsing GENIA-LAS, GENIA-UAS\n4 BioELMoNamed Entity Recognition BC2GM (Probing task), CoNLL 2003\nNatural Language Inference MedNLI (Probing task), SNLI\n5 Clinical BERT (E)Named Entity Recognition i2b2 2006, i2b2 2010, i2b2 2012, i2b2 2014\nNatural Language Inference MedNLI\n6 ClinicalBERT (K) Readmission Prediction 30-day hospital readmission prediction\n7 BlueBERTNamed Entity Recognition BC5CDR-disease , BC5CDR-chemical , ShARe/CLEF\nRelation Extraction DDI , ChemProt , i2b2 2010\nSentence Similarity BIOSSES , MedSTS\nDocument Multi-label Classification HOC\nNatural Language Inference MedNLI (All)\n8 G-BERT Medication Recommendation Task Medication Recommendation Task\n9 BEHRT Disease Prediction Next Visit , Next 6M , Next 12M (All)\nManuscript submitted to ACM44 Ho et al.\n10 BioFLAIR Named Entity Recognition BC5CDR ,Species-800 , NCBI-disease, LINNAEUS, JNLPBA, BC5CDR-\ndisease\n11 EhrBERT Entity Normalization MADE 1.0 , NCBI-disease , CDR (All)\n12 S2ORC-SciBERTNamed Entity Recognition BC5CDR, SciERC, JNLPBA, NCBI-disease\nPICO Extraction EBM-NLP\nText Classification ACL-ARC, SciCite, Paper Field\nRelation Extraction ChemProt, SciERC\nDependency Parsing GENIA-LAS, GENIA-UAS\n13 Clinical XLNet Prolonged Mechanical Ventilation\nPredictionPMV , Mortality (All)\n14 SciGPT2 Relationship Explanation Task Relationship Explanation Task\n15 NukeBERT Question Answering NQUAD (All)\n16 GreenBioBERT Named Entity RecognitionBC5CDR-chemical ,BC4CHEMD ,Species-800 , BC5CDR-disease,\nNCBI-disease, BC2GM, JNLPBA, LINNAEUS\n17 SPECTERText Classification MeSH , MAG\nCitation Prediction Co-Cite , Cite\nUser Activity Prediction Co-Views , Co-Reads\nPaper Recommendation Task Paper Recommendation Task\n18 BERT-XML ICD Classification ICD Classification All\n19 Bio-ELECTRANamed Entity Recognition BC2GM, BC4CHEMD, NCBI-disease, LINNAEUS\nRelation Extraction GAD, ChemProt\nQuestion Answering BioASQ 8b-yes/no , BioASQ 5b based , BioASQ 8b-factoid\n20 Med-BERT Disease PredictionHeart failure in diabetes patients (DHF-Cerner), Pancreatic cancer\n(PaCa-Cerner, PaCa-Truven) (All)\n21 ouBioBERTNamed Entity Recognition BC5CDR-disease , BC5CDR-chemical, ShARe/CLEF\nRelation Extraction DDI , ChemProt , i2b2 2010\nSentence Similarity BIOSSES , MedSTS\nDocument Multi-label Classification HOC\nNatural Language Inference MedNLI\n22 PubMedBERTNamed Entity Recognition BC5CDR-disease ,BC5CDR-chemical ,JNLPBA ,BC2GM , NCBI-\ndisease\nRelation Extraction ChemProt , DDI , GAD\nPICO Extraction EBM PICO\nSentence Similarity BIOSSES\nDocument Multi-label Classification HOC\nQuestion Answering BioASQ , PubMedQA\n23 MC-BERTNamed Entity Recognition cEHRNER , cMedQANER\nQuestion Answering cMedQNLI , cMeQA\nText Classification cMedTC\nIntent Classification cMedIC\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 45\nParaphrase Identification cMedQQ\nInformation Retrieval cMedIR (All)\n24 BioALBERTNamed Entity RecognitionNCBI-disease ,BC5CDR ,BC4CHEMD ,BC2GM ,JNLPBA ,LNNAEUS ,\nSpecies-800 , Share/Clefe\nRelation Extraction ChemProt , DDI , i2b2 , GAD, Euadr\nSentence Similarity BIOSSES , MedSTS\nDocument Multi-label Classification HOC\nQuestion Answering BioASQ\nNatural Language Inference MedNLI\n25 BRLTM Depression Prediction Depression Prediction (All)\n26 BioMegatronNamed Entity Recognition BC5CDR-chemical , BC5CDR-disease , NCBI-disease\nRelation Extraction ChemProt\nQuestion Answering BioASQ 7b-factoid (All)\n27 CharacterBERTNamed Entity Recognition i2b2 2010\nRelation Extraction DDI , ChemProt\nSentence Similarity ClinicalSTS\nNatural Language Inference MedNLI\n28 ChemBERTa MoleculeNet Task BBBP, ClinTox, HIV, Tox21\n29 ClinicalTransformer Named Entity Recognition i2b2 2010 , i2b2 2012 , n2c2 2018 (All)\n30 SapBERT Entity LinkingNCBI-disease ,BC5CDR-disease ,BC5CDR-chemical ,MedMentions ,\nAskAPatient , COMETA (All)\n31 UmlsBERTNamed Entity Recognition i2b2 2010 , i2b2 2012\nNatural Language Inference MedNLI\nDe-Identification i2b2 2006 , i2b2 2014\n32 bert-for-radiology Text Classification Binary Classifier ,Classification of CT Reports , Multi-label Classifi-\ncation\n33 Bio-LMNamed Entity RecognitionBC5CDR-chemical ,BC5CDR-disease ,JNLPBA ,NCBI-disease ,\nBC4CHEMD ,BC2GM ,LINNAEUS ,Species-800 ,i2b2-2010 ,\ni2b2-2012\nRelation Extraction ChemProt , DDI , i2b2 2010 , GAD, EU-ADR\nDe-Identification i2b2 2014\nDocument Multi-label Classification HOC\nNatural Language Inference MedNLI\n34 CODERTerm Normalization Cadec , PsyTar , MANTRA\nMedical Embeddings MCSM\nRelation Extraction DDBRC (CODER-Eng version )\n35 exBERTNamed Entity Recognition MTL-Bioinformatics-2016\nRelation Extraction MTL-Bioinformatics-2016 (All)\n36 BioMedBERTNamed Entity Recognition NCBI-disease, BC5CDR-chemical, BC5CDR-disease, BC4CHEMD,\nBC2GM, JNLPBA\nManuscript submitted to ACM46 Ho et al.\nRelation Extraction GAD, EU-ADR\nQuestion Answering BioASQ 5b-factoid ,BioASQ 6b-factoid ,BioASQ 7b-factoid , BioASQ\n4b-factoid\n37 LBERT Relation ExtractionHPDR50 , IEPA , EU-ADR , MedLine ,\nBioNLP Shared Task 2011 corpus , ChemProt, AIMed, BioInfer,\nGAD, LLL\n38 OAG-BERTAuthor Name Disambiguation whoiswho-v1\nScientific Literature Retrieval OAG-QA\nPaper Recommendation SciDocs (Recommendation task)\nUser Activity Prediction Co-Views , Co-Reads\nEntity Graph Completion CS Dataset\nFields-of-study Tagging MAG\nVenue Prediction arXiv\nAffiliation Prediction arXiv (All)\n39 CovidBERTRelation Extraction BioCreative V\nText Classification DisGeNet Database (All)\n40 ELECTRAMedNamed Entity Recognition BC5CDR , NCBI-disease, JNLPBA\nRelation Extraction ChemProt, DDI\nQuestion Answering BioASQ 7b-factoid\n41 KeBioLMNamed Entity Recognition NCBI-disease ,BC5CDR-chemical ,BC2GM ,JNLPBA , BC5CDR-\ndisease\nRelation Extraction GAD , ChemProt , DDI\n42 SINA-BERTFill-in-the-Blank 10k-sentences\nText Classification 4k-samples\nMedical Sentiment Analysis 5k-comments\nMedical Question Retrieval 200k-pairs (All)\n43 MathBERT (P)Mathematical Information Retrieval NTCIR-12 MathIR\nFormula Topic Classification TopicMath-100K\nFormula Headline Generation EXEQ-300K (All)\n44 NukeLMOSTI Multi-class Prediction OSTI subject categories\nOSTI Binary Prediction OSTI custom (All)\n45 ProteinBERTProtein Structure Secondary Structure, Disorder, Remote Homology, Fold Classes\nPost-translational Modifications Neuropeptide Cleavage, Major PTMs, Signal Peptide\nBiophysical Properties Stability , Fluorescence\n46 SciFiveNamed Entity RecognitionJNLPBA ,Species-800 ,BC5CDR-disease , BC5CDR-chemical, NCBI-\ndisease, BC4CHEMD, BC2GM\nRelation Extraction ChemProt , DDI\nNatural Language Inference MedNLI\nDocument Multi-label Classification HOC\nQuestion Answering BioASQ 4b-factoid , BioASQ 5b-factoid , BioASQ 6b-factoid\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 47\n47 BioELECTRANamed Entity Recognition BC5CDR-chemical ,BC5CDR-disease ,JNLPBA ,NCBI-disease ,\nBC2GM , ShARe/CLEFE\nRelation Extraction ChemProt , DDI , GAD , i2b2-2010\nPICO Extraction EBM-NLP\nSentence Similarity BIOSSES , ClinicalSTS\nDocument Multi-label Classification HOC\nQuestion Answering PubMedQA , BioASQ\nNatural Language Inference MedNLI (All)\n48 ChemBERTReaction Role Labeling Reaction Role Labeling\nProduct Extraction Product Extraction (All)\n49 EntityBERTTemporal Relation Extraction THYME+\nDocument Time Relation Classifica-\ntionTHYME+\nNegation Detection i2b2, MiPACQ, Seed, Strat\nQuestion Answering PubMedQA\n50 MathBERT (S)Knowledge Component Prediction ASSISTments\nAuto-grading ASSISTments\nKnowledge Tracing Correctness Pre-\ndictionASSISTments (All)\n51 MedGPT Next Disorder Prediction MIMIC-III , KTH (All)\n52 SMedBERTNamed Entity Recognition cMedQANER , DXY-NER\nRelation Extraction DXY-RE , CHIP-RE\nQuestion Matching cMedQQ\nIntrinsic Evaluation D1 , D2, D3\nQuestion Answering cMedQA , WebMedQA\nNatural Language Inference cMedQNLI (All)\n53 Bio-cli Named Entity Recognition PharmaCoNER , CANTEMIST, ICTUSnet\n54 MatSciBERTNamed Entity Recognition SOFC , SOFC-Slot , Matscholar NER\nText Classification Glass vs. Non-glass Dataset\nRelation Extraction Materials Synthesis Procedures Dataset (All)\n55 ClimateBertText Classification Climate-related Classification\nSentiment Analysis Climate-related Paragraphs (All)\n56 MatBERT Named Entity RecognitionSolid-state Materials Abstracts , Inorganic Doping Abstracts ,\nGold Nanoparticle Synthesis (All)\n57 UTH-BERT MedWeb Task (8 labels) MedWeb (All)\n58 ChestXRayBERT Summarization OPEN-I , MIMIC-CXR (All)\n59 MedRoBERTa.nlMedical Odd-one-out Similarity Task Medical odd-one-out Similarity Task\nICF Classification Task ICF Classification Task\nNamed Entity Recognition (Dutch) CoNLL 2002\nManuscript submitted to ACM48 Ho et al.\n60 PubMedELECTRANamed Entity Recognition BC5CDR-chemical, JNLPBA, NCBI-disease, BC5CDR-disease,\nBC2GM\nRelation Extraction GAD , ChemProt, DDI\nPICO Extraction EBM-NLP\nSentence Similarity BIOSSES\nDocument Multi-label Classification HOC\nQuestion Answering PubMedQA, BioASQ\n61 Clinical-LongformerNamed Entity Recognition i2b2 2006 , i2b2 2010 , i2b2 2012 , i2b2 2014\nText Classification MIMIC-AKI , OpenI , OHSUMed\nQuestion Answering emrQA\nNatural Language Inference MedNLI\n62 BioLinkBERTNamed Entity Recognition BC5CDR-chemical ,BC5CDR-disease ,JNLPBA ,NCBI-disease ,\nBC2GM\nRelation Extraction GAD , ChemProt , DDI\nPICO Extraction EBM-NLP\nSentence Similarity BIOSSES\nDocument Multi-label Classification HOC\nQuestion Answering PubMedQA , BioASQ (All)\n63 BioBARTNamed Entity Recognition ShARe13, ShARe14, CADEC, GENIA\nEntity Linking BC5CDR , AskAPatient , COMETA , MedMentions, NCBI-disease\nSummarization MeQSum ,MEDIQA-ANS , iCliniq, HealthCareMagic, MEDIQA-QS,\nMEDIQA-MAS\nDialogue CovidDialog\n64 SecureBERTNamed Entity Recognition MalwareTextDB\nSentiment Analysis Rotten Tomatoes\n65 BatteryBERTQuestion Answering Retional Data Extraction, Device Component Classification\nText Classification Battery Abstract Classification\n66 bsc-bio-ehr-es Named Entity Recognition PharmaCoNER , CANTEMIST , ICTUSnet (All)\n67 ChemGPT N/A N/A\n68 PathologyBERT Cancer Severity Classification Breast Cancer Diagnose Severity Classification (All)\n69 ScholarBERTNamed Entity RecognitionScienceExam , SciERC, NCBI-disease, ChemDNER, Matscholar NER,\nBC5CDR, JNLPBA, Coleridge\nRelation Extraction SciERC , ChemProt, Paper Field\n70 RadBERTText Classification Abnormal Sentence Classification\nReport Coding AAA , BI-RADS , Lung-RADS , Abnormal , Alert\nSummarization Summarization (All)\n71 SciDeBERTaNamed Entity Recognition SciERC , GENIA\nRelation Extraction SciERC\nCo-reference Resolution GENIA , SciERC\n72 SsciBERTNamed Entity Recognition Scientometrics\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 49\nIdentification Results of Abstract\nStructuresIdentification Results of Abstract Structures\nDiscipline Classification JCR Social Science (All)\n73 ViHealthBERTNamed Entity Recognition COVID-19 , ViMQ\nAcronym Disambiguation Task acrDrAid\nSummarization FAQ Summarization (All)\n74 Clinical Flair Named Entity Recognition CANTEMIST , Clinical Trials , NUBes , PharmaCoNER\n75 KM-BERTNamed Entity Recognition Korean NER\nSemantic Textual Similarity Korean MedSTS (All)\n76 MaterialsBERT (S) Named Entity Recognition A polymer dataset annotated by the authors (All)\n77 ProcessBERT Text Classification Phrases manually annotated dataset (All)\n78 BioGPTGeneration Self-created Text Generation Dataset\nRelation Extraction BC5CDR , KD-DTI , DDI\nDocument Multi-label Classification HOC\nQuestion Answering PubMedQA\n79 ChemBERTa-2Classification BBBP , SR-p53 , BACE, ClinTox\nRegression Clearance , Delaney , Lipo , BACE\n80 CSL-T5Summarization Predict Title from Abstract (Created by authors)\nGeneration Chinese Keyword Generation Task (Created by authors)\nText Classification CTG , DCP (Created by authors) (All)\n81 MaterialBERT (Y)Visualization of Materials Visualization of Materials\nText Classification CoLA\n82 AcademicRoBERTa Text Classification Author Identification , Category Classification (All)\n83 Bioberturk Radiology Report classification Impressions , Findings (All)\n84 DRAGON Question Answering MedQA , PubMedQA , BioASQ (All)\n85 UCSF-BERTNamed Entity Recognition i2b2 2010 , i2b2 2012\nRelation Extraction i2b2 2010\nNatural Language Inference MedNLI\nICD9-top50 Coding ICD9-top50 Coding\nTherapeutic Class Prediction Therapeutic Class Prediction\n86 ViPubmedT5Natural Language Inference ViMedNLI\nAcronym Disambiguation Task acrDrAid\nSummarization FAQ Summarization\n87 GalacticaGeneral Capabilities 57 task selection from BIG-bench\nDownstream Scientific NLP MMLU benchmark , other popular scientific QA benchmarks\nCitation Prediction Citation Accuracy (PWC Citations, Extended Citations, Contextual\nCitations)\nChemical Understanding IUPAC Name Prediction , MoleculeNet\nToxicity and Bias Bias and Stereotypes (CrowS-Pairs, StereoSet, Toxicity), TruthfulQA\nManuscript submitted to ACM50 Ho et al.\nKnowledge ProbesLaTeX Equations , Domain Probes ( AminoProbe , BioLAMA,\nChemical Reactions ,Galaxy Clusters ,Mineral Groups ), Reasoning\n(MMLU Mathematics , MATH)\nBiological UnderstandingSequence Validation Perplexity , Protein Keyword Prediction ,\nProtein Function Description\n88 VarMAENamed Entity Recognition JNLPBA , IEE\nText Classification ACL-ARC , SciCite , OIR , MTC\nPICO Extraction EBM-NLP\nText Matching PSM (All)\n89 AliBERT Named Entity Recognition BioNER , QUAERO (All)\n90 BioMedLMText Generation MeQSum\nQuestion Answering MedQA , PubMedQA, and BioASQ\n91 BioReaderNamed Entity RecognitionBC4CHEMD ,Species-800 , JNLPBA, NCBI-disease, BC5CDR-disease,\nBC5CDR-chemical, BC2GM\nRelation Extraction ChemProt, DDI\nDocument Multi-label Classification HOC\nQuestion Answering BioASQ 4b-factoid, BioASQ 5b-factoid, BioASQ 6b-factoid, MedQA-\nUSMLE\nNatural Language Inference MedNLI\n92 ClinicalT5Named Entity Recognition NCBI-disease , BC5CDR-disease\nIntrinsic Evaluation UMNSRS-Sim, UMNSRS-Rel\nDocument Multi-label Classification HOC\nNatural Language Inference MedNLI\n93 CySecBERTNamed Entity Recognition NVD\nWord Similarity Task Word Similarity\nClustering Task Log4J dataset\nClassification CySecAlert , MS Exchange\nCombined Task SuperGLUE\n94 GatorTronNamed Entity Recognition i2b2 2010 , i2b2 2012 , n2c2 2018\nRelation Extraction n2c2 2018\nSemantic Textual Similarity n2c2 2019\nNatural Language Inference MedNLI\nMedical Question Answering emrQA (All)\n95 Med-PaLM Question AnsweringMedQA-USMLE ,MedMCQA ,PubMedQA , MMLU, LiveQA, Medica-\ntionQA, HealthSearchQA\n96 Clinical-T5 N/A N/A\n97 CPT-BigBird Question Answering SQuAD, IS-QA\n98 ProtSTProtein Localization Prediction Bin , Sub\nFitness Landscape Prediction 𝛽-lac, AAV , Thermo , Flu, Sta\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 51\nProtein Function Annotation EC , GO-BP , GO-MF , GO-CC (All)\n99 SciEdBERTAutomatic Scoring of Students’ Writ-\nten Responses4T, 7T(All)\n100 BioformerNamed Entity RecognitionBC5CDR-disease, NCBI-disease, BC5CDR-chemical, BC4CHEMD,\nBC2GM, JNLPBA, LINNAEUS, Species-800\nRelation Extraction ChemProt, DDI, EU-ADR, GAD\nDocument Multi-label Classification HOC, BioCreative-LitCovid\n101Lightweight\nClinical\nTransformersNamed Entity Recognition i2b2 2010 , i2b2 2012, i2b2 2014\nRelation Extraction i2b2 2010\nNatural Language Inference MedNLI\nSequence Classification Isaric Clinical Notes (ICN)\n102 ManuBERT Named Entity Recognition FabNER\n103 RAMM Visual Question Answering Med-VQA 2019 , VQA-Med 2021 , VQARAD , SLAKE (All)\n104 DrBERTNamed Entity RecognitionMUSCA-DET , QUAERO-EMEA , QUAERO-MEDLINE ,\nAcute heart failure (aHF) , Medical Report\nText Classification aHF , MUSCA-DET , Technical Specialties Sorting\nPOS Tagging ESSAIS , CAS\nQuestion Answering FrenchMedMCQA\n105 MOTORMedical Report Generation IU-Xray\nImage Report Retrieval MIMIC-CXR\nDiagnosis Classification MIMIC-CXR , ChestX-ray14\nVisual Question Answering SLAKE , VQARAD\n106 BiomedGPTImage Classification MedMNIST v2\nNatural Language Inference MedNLI\nSummarization MeQSum, iCliniq, HealthCareMagic\nImage Captioning Peir Gross , IU X-ray, ROCO\nVisual Question Answering SLAKE , PathVQA , VQARAD\n107 PattonClassification PATTON ( Mathematics, Clothes, Sports ), SciPATTON\n(Geology, Economics )\nRetrieval PATTON ( Clothes, Sports ), SciPATTON\n(Mathematics, Geology, Economics )\nReranking PATTON ( Clothes, Sports ), SciPATTON\n(Mathematics, Geology, Economics )\nLink Prediction SciPATTON (Mathematics, Geology, Economics )\n108 TurkRadBERT Text Classification Turkish head CT reports\n109 CamemBERT-bio Named Entity Recognition CAS1 , CAS2 , QUAERO-EMEA , QUAERO-MEDLINE, E3C\n110 ClinicalGPTMedical Conversation MedDialog\nDiagnosis MD-EHR\nMedical Question Answering cMedQA2 , MEDQA-MCMLE\nManuscript submitted to ACM52 Ho et al.\n111 EriBERTaNamed Entity RecognitionBC5CDR-disease, BC5CDR-chemical, JNLPBA, NCBI-disease,\nBC4CHEM, DIANN, Treatment\nNamed Entity Recognition (Spanish) PharmaCoNER , Cantemist-NER\n112 K2 Question Answering NPEE , APTest\n113 PharmBERTDrug LabelingDetection of adverse drug reaction ,Extraction of drug–drug interaction ,\nADME classification\nSentiment Labeling SST-2\n114 BioNARTSummarization iCliniq, HealthCareMagic\nClinical Dialogue CovidDialog\n115 BIOptimus Named Entity Recognition BC5-chemical , NCBI-disease , BC2GM , BC5-disease, JNLPBA\n116 KEBLMEntity Linking COMETA , NCBI-disease , BC5CDR-chemical, BC5CDR-disease\nNatural Language Inference MedNLI\nQuestion Answering PubMedQA\n117 GIT-MolCaptioning Molecule Captioning , Molecule Image Captioning\nMolecule Generation Molecule Generation\nMolecular Property Prediction ToxCast , Sider , ClinTox , Bace , BBBP , Tox21\nC.2 Exploring Task Performance Details\nManuscript submitted to ACMA Survey of Pre-trained Language Models for Processing Scientific Text 53\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-scoreBioALBERTNCBI-d\nBest\n(a) NCBI-disease\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-scoreBioALBERT BC5CDR-d\nBest (b) BC5CDR-disease\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-score\nPubMedBERTBioALBERTJNLPBA\nBest\n(c) JNLPBA\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-scoreBioALBERT BC5CDR-c\nBest (d) BC5CDR-chemical\n2019/01 2020/04 2020/07 2020/09 2020/11 2020/12 2021/04 2021/05 2021/06 2021/12 2022/03 2022/12 2023/02 2023/07\nDate707580859095100F1-scoreBioALBERTBC2GM\nBest\n(e) BC2GM\nFig. 13. Details of performance changes for five NER datasets: (a) NCBI-disease, (b) BC5CDR-disease, (c) JNLPBA, (d) BC5CDR-\nchemical, and (e) BC2GM.\nD LIST OF ABBREVIATIONS USED IN THIS PAPER\nManuscript submitted to ACM54 Ho et al.\nTable 17. List of abbreviations used in this paper.\nAbbreviation Details Defined in Section\nPLMs Pre-trained Language Models 1. Introduction\nLLMs Large Language Models 1. Introduction\nNER Named Entity Recognition 1. Introduction\nRE Relation Extraction 1. Introduction\nQA Question-Answering 1. Introduction\nSOTA state-of-the-art 1. Introduction\nSciLMs LMs for processing scientific texts 1. Introduction\nMLM Masked Language Modeling 2.1. Existing LM Architectures\nNSP Next Sentence Prediction 2.1. Existing LM Architectures\nSOP Sentence Order Prediction 2.1. Existing LM Architectures\nRTP Replaced Token Prediction 2.1. Existing LM Architectures\nNTP Next Token Prediction 2.1. Existing LM Architectures\nGNN Graph Neural Network 2.1. Existing LM Architectures\nKG Knowledge Graph 2.2. Existing Tasks\nNLI Natural Language Inference 2.2. Existing Tasks\nSNS Social Network Sites 2.3. Scientific Text\nBi-LM Bidirectional Language Modeling 3. Existing SciLMs\nKBs Knowledge Bases 5.1. Foundation SciLMs\nManuscript submitted to ACM" }] \ No newline at end of file