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ΦI,−subscriptΦ𝐼\Phi_{I,-}roman_Φ start_POSTSUBSCRIPT italic_I , - end_POSTSUBSCRIPT is convex for each interval I𝐼Iitalic_I.

ΦΦ\Phiroman_Φ is slowly changing on intervals and ΨΨ\Psiroman_Ψ is uniformly doubling, and we do not need to

ΨΨ\Psiroman_Ψ is L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT dominating, uniformly doubling, and that

The family ΨΨ\Psiroman_Ψ is L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT dominating, uniformly doubling, and

ΨΨ\Psiroman_Ψ is L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT dominating and uniformly doubling.

B

(f∘g)⁢(xa,y)=f⁢(a,y)=1.𝑓𝑔subscript𝑥𝑎𝑦𝑓𝑎𝑦1(f\circ g)(x_{a},y)=f(a,y)=1.( italic_f ∘ italic_g ) ( italic_x start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_y ) = italic_f ( italic_a , italic_y ) = 1 .

The study of fuzzy set was initiated by Zadeh [1]delimited-[]1[1][ 1 ], since then a large number of mathematical structures like algebras, topological spaces, differential equations etc. has been fuzzified by many mathematicians. Rosenfield [2]delimited-[]2[2][ 2 ] introduced the concept of fuzzy subgroup with the assumption that subsets of the group are fuzzy. With this evolution of fuzzy group theory, various fuzzy counterparts of group theoretic concepts from classical group   theory were introduced by many authors. To list a few in this context, Bhattacharya and Mukharjee [3]delimited-[]3[3][ 3 ] introduced the concept of fuzzy cosets and fuzzy normal subgroups and proved fuzzy analogues of various group theoretic concepts. In [4]delimited-[]4[4][ 4 ], Mukharjee and Bhattacharya introduced notion of order of a fuzzy subgroup in a finite group, fuzzy abelian group and fuzzy solvable group. Based on fuzzy binary operations, Yuan and Lee [5]delimited-[]5[5][ 5 ], proposed a new kind of fuzzy group. Yao Bingxue [6]delimited-[]6[6][ 6 ] introduced the concept of fuzzy homomorphism. Motivated by this, we introduced the concept of fuzzy automorphism, fuzzy inner automorphism in groups, and fuzzy analogue of some standard results from classical group theory.

Associativity: Let f,g𝑓𝑔f,gitalic_f , italic_g and hℎhitalic_h are fuzzy automorphisms of a group G𝐺Gitalic_G. Then

Let fg1subscript𝑓subscript𝑔1f_{g_{1}}italic_f start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and fg2subscript𝑓subscript𝑔2f_{g_{2}}italic_f start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT be two fuzzy inner automorphisms of G𝐺Gitalic_G. Then

Let f𝑓fitalic_f and g𝑔gitalic_g be two fuzzy automorphisms of G𝐺Gitalic_G. Then for each y∈G𝑦𝐺y\in Gitalic_y ∈ italic_G and x1,x2∈Gsubscript𝑥1subscript𝑥2𝐺x_{1},x_{2}\in Gitalic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_G,

B

-2t}\binom{a-t}{i}\right)( FRACOP start_ARG italic_a end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a - 2 italic_t end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG italic_i end_ARG ) ) ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a - 2 italic_t end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_a - italic_t end_ARG start_ARG italic_i end_ARG ) ) used for (at)−1⁢∑i=0a−2⁢t(ti)⁢(a−ti)superscriptbinomial𝑎𝑡1superscriptsubscript𝑖0𝑎2𝑡binomial𝑡𝑖binomial𝑎𝑡𝑖\binom{a}{t}^{-1}\sum_{i=0}^{a-2t}\binom{t}{i}\binom{a-t}{i}( FRACOP start_ARG italic_a end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a - 2 italic_t end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG italic_i end_ARG ) ( FRACOP start_ARG italic_a - italic_t end_ARG start_ARG italic_i end_ARG ) is a fairly weak upper bound. So, our result may be able to be improved significantly with a better upper bound on (at)−1⁢∑i=0a−2⁢t(ti)⁢(a−ti)superscriptbinomial𝑎𝑡1superscriptsubscript𝑖0𝑎2𝑡binomial𝑡𝑖binomial𝑎𝑡𝑖\binom{a}{t}^{-1}\sum_{i=0}^{a-2t}\binom{t}{i}\binom{a-t}{i}( FRACOP start_ARG italic_a end_ARG start_ARG italic_t end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a - 2 italic_t end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_t end_ARG start_ARG italic_i end_ARG ) ( FRACOP start_ARG italic_a - italic_t end_ARG start_ARG italic_i end_ARG ). For a concrete application of Theorem 10, notice that when m=15𝑚15m=15italic_m = 15 and d=0.0959𝑑0.0959d=0.0959italic_d = 0.0959, the inequality in the hypothesis is satisfied. So, by Corollary 7 and Theorem 10, we have that 2.0959≤χD⁢P∗⁢(K2,15)≤2.6192.0959superscriptsubscript𝜒𝐷𝑃subscript𝐾2152.6192.0959\leq\chi_{{}_{DP}}^{*}(K_{2,15})\leq 2.6192.0959 ≤ italic_χ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D italic_P end_FLOATSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_K start_POSTSUBSCRIPT 2 , 15 end_POSTSUBSCRIPT ) ≤ 2.619.

In their 1979 paper Erdős et al. [6] asked: If G𝐺Gitalic_G is (a,b)𝑎𝑏(a,b)( italic_a , italic_b )-choosable and c,d∈ℕ𝑐𝑑ℕc,d\in\mathbb{N}italic_c , italic_d ∈ blackboard_N are such that c/d>a/b𝑐𝑑𝑎𝑏c/d>a/bitalic_c / italic_d > italic_a / italic_b, must G𝐺Gitalic_G be (c,d)𝑐𝑑(c,d)( italic_c , italic_d )-choosable? A negative answer to this question is given in [8]. Erdős et al. also asked: If G𝐺Gitalic_G is (a,b)𝑎𝑏(a,b)( italic_a , italic_b )-choosable, does it follow that G𝐺Gitalic_G is (a⁢t,b⁢t)𝑎𝑡𝑏𝑡(at,bt)( italic_a italic_t , italic_b italic_t )-choosable for each t∈ℕ𝑡ℕt\in\mathbb{N}italic_t ∈ blackboard_N? Tuza and Voigt [12] showed that the answer to this question is yes when a=2𝑎2a=2italic_a = 2 and b=1𝑏1b=1italic_b = 1. However, in general, a negative answer to this question was recently given in [4].It was shown that for each a≥4𝑎4a\geq 4italic_a ≥ 4, a graph that is (a,1)𝑎1(a,1)( italic_a , 1 )-choosable but not (2⁢a,2)2𝑎2(2a,2)( 2 italic_a , 2 )-choosable can be constructed. We briefly consider the fractional DP-coloring analogues of both of these questions below.

The answer to Question 11 is clearly yes when n=1𝑛1n=1italic_n = 1. If the answer is yes for some n≥2𝑛2n\geq 2italic_n ≥ 2, then Theorem 6 and Corollary 7 give the asymptotically tight result as the second partite set grows large. We could further ask whether, for some n≥2𝑛2n\geq 2italic_n ≥ 2, there is an m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N where we achieve χD⁢P∗⁢(Kn,m)=1/p∗subscriptsuperscript𝜒𝐷𝑃subscript𝐾𝑛𝑚1superscript𝑝\chi^{*}_{{}_{DP}}(K_{n,m})=1/p^{*}italic_χ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D italic_P end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT ) = 1 / italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT? If the answer to this follow-up question is yes it would give an example of a graph whose fractional DP-chromatic number is irrational, answering a question posed by Bernshteyn, Kostochka, and Zhu [3]. Question 11 could be extended to complete multipartite graphs where the sizes of all but one of the partite sets are held constant.

Acknowledgment. The authors would like to thank Doug West for helpful conversations regarding this paper.

In this paper all graphs are nonempty, finite, simple graphs unless otherwise noted. Generally speaking we follow West [14] for terminology and notation. The set of natural numbers is ℕ={1,2,3,…}ℕ123…\mathbb{N}=\{1,2,3,\ldots\}blackboard_N = { 1 , 2 , 3 , … }. Given a set A𝐴Aitalic_A, 𝒫⁢(A)𝒫𝐴\mathcal{P}(A)caligraphic_P ( italic_A ) is the power set of A𝐴Aitalic_A. For m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N, we write [m]delimited-[]𝑚[m][ italic_m ] for the set {1,2,…,m}12…𝑚\{1,2,\ldots,m\}{ 1 , 2 , … , italic_m }. If G𝐺Gitalic_G is a graph and S,U⊆V⁢(G)𝑆𝑈𝑉𝐺S,U\subseteq V(G)italic_S , italic_U ⊆ italic_V ( italic_G ), we use G⁢[S]𝐺delimited-[]𝑆G[S]italic_G [ italic_S ] for the subgraph of G𝐺Gitalic_G induced by S𝑆Sitalic_S, and we use EG⁢(S,U)subscript𝐸𝐺𝑆𝑈E_{G}(S,U)italic_E start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_S , italic_U ) for the subset of E⁢(G)𝐸𝐺E(G)italic_E ( italic_G ) with one endpoint in S𝑆Sitalic_S and the other endpoint in U𝑈Uitalic_U. For v∈V⁢(G)𝑣𝑉𝐺v\in V(G)italic_v ∈ italic_V ( italic_G ), we write dG⁢(v)subscript𝑑𝐺𝑣d_{G}(v)italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) for the degree of vertex v𝑣vitalic_v in the graph G𝐺Gitalic_G, and we write NG⁢(v)subscript𝑁𝐺𝑣N_{G}(v)italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) for the neighborhood of vertex v𝑣vitalic_v in the graph G𝐺Gitalic_G. Also, for S⊆V⁢(G)𝑆𝑉𝐺S\subseteq V(G)italic_S ⊆ italic_V ( italic_G ), we let NG⁢(S)=⋃v∈SNG⁢(v)subscript𝑁𝐺𝑆subscript𝑣𝑆subscript𝑁𝐺𝑣N_{G}(S)=\bigcup_{v\in S}N_{G}(v)italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_S ) = ⋃ start_POSTSUBSCRIPT italic_v ∈ italic_S end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ). A graph G𝐺Gitalic_G is d𝑑ditalic_d-degenerate if every subgraph of G𝐺Gitalic_G has a vertex of degree at most d𝑑ditalic_d. We use Kn,msubscript𝐾𝑛𝑚K_{n,m}italic_K start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT to denote the complete bipartite graphs with partite sets of size n𝑛nitalic_n and m𝑚mitalic_m. For a random variable X𝑋Xitalic_X, we use X∼B⁢(n,p)similar-to𝑋𝐵𝑛𝑝X\sim B(n,p)italic_X ∼ italic_B ( italic_n , italic_p ) to indicate that X𝑋Xitalic_X is binomially distributed with n𝑛nitalic_n trials each having probabilitiy of success p𝑝pitalic_p. For an event E𝐸Eitalic_E, we use E¯¯𝐸\overline{E}over¯ start_ARG italic_E end_ARG to denote the complement of E𝐸Eitalic_E.

C

However, we can also construct a derived formal moduli problem controling the deformation theory of a P∞subscript𝑃P_{\infty}italic_P start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebra A𝐴Aitalic_A directly in the ∞\infty∞-category P∞−A⁢l⁢gsubscript𝑃𝐴𝑙𝑔P_{\infty}-Algitalic_P start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT - italic_A italic_l italic_g of P∞subscript𝑃P_{\infty}italic_P start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebras (with quasi-isomorphisms as weak equivalences). This is not the same as deforming the morphism ψ𝜓\psiitalic_ψ (in a way precised below, the Maurer-Cartan elements are the same in both cases but the gauge equivalence relation differs).

To set up the appropriate framework for such a deformation theory, we introduce in Section 2 the notion of derived prestack group, which can be thought as a family of homotopy formal groups parametrized by a base space and apply this formalism to the deformation theory of algebras over properads. Briefly, one associates to A𝐴Aitalic_A its derived prestack group of homotopy automorphisms which is the ∞\infty∞-functor

As we explained, a derived prestack group G𝐺Gitalic_G gives rises to a family of derived formal groups parametrized by G⁢(𝕂)𝐺𝕂G(\mathbb{K})italic_G ( blackboard_K ).

In other words, a derived prestack group G𝐺Gitalic_G is a family of derived formal groups parametrized by G⁢(𝕂)𝐺𝕂G(\mathbb{K})italic_G ( blackboard_K ).

one associated to the derived prestack group of homotopy automorphisms of a P⁢o⁢i⁢sn𝑃𝑜𝑖subscript𝑠𝑛Pois_{n}italic_P italic_o italic_i italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-algebra:

A

2.2 Examples of 𝒪⁢(h2)𝒪superscriptℎ2\mathcal{O}(h^{2})caligraphic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) deviated surfaces where superconvergence fails

It has been, however, conjectured that the superconvergence of gradient recovery on general discretized surfaces, like ℳhsubscriptℳℎ\mathcal{M}_{h}caligraphic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT, may be proven if the vertices of ℳhsubscriptℳℎ\mathcal{M}_{h}caligraphic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT are in a 𝒪⁢(h2)𝒪superscriptℎ2\mathcal{O}(h^{2})caligraphic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) neighborhood of the corresponding vertices of ℳh∗subscriptsuperscriptℳℎ\mathcal{M}^{*}_{h}caligraphic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT. That is the following vertex-deviation condition

Especially, we show that barely with the condition (2.1), superconvergence rates of the recovered gradient may not be achieved as conjectured in [25].

We show in the next section that (3.2) provides the ingredient for proving the superconvergence of recovered gradient on deviated discretization of surfaces.

In this paper, we first construct some examples to show that there exist cases where the superconvergence of gradient recovery on surfaces is not guaranteed given barely the 𝒪⁢(h2)𝒪superscriptℎ2\mathcal{O}(h^{2})caligraphic_O ( italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) vertex condition. In particular, the examples show that data supercloseness does not guarantee superconvergence of the recovered gradient, in contrast to the exact nodal points case.

B

\int_{RB}\widehat{f_{h}}\beta\cdot\hat{n}\,f_{h}e^{H}\,d\sigma\,.divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT italic_R italic_B end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_β ⋅ over^ start_ARG italic_n end_ARG italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT italic_d italic_σ + ∫ start_POSTSUBSCRIPT italic_R italic_B end_POSTSUBSCRIPT over^ start_ARG italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT end_ARG italic_β ⋅ over^ start_ARG italic_n end_ARG italic_f start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT italic_d italic_σ .

We perform further analysis of these discrete entropy inequalities applied to the Boltzmann - Poisson system (1.1,1.5) for self consistent mean field charged transport, in the one space dimensional case, say x∈[0,L]𝑥0𝐿x\in[0,L]italic_x ∈ [ 0 , italic_L ], under the assumption for the electrostatic potential Φ⁢(x)Φ𝑥\Phi(x)roman_Φ ( italic_x ) solving the Poisson equation that it satisfies periodic boundary conditions as much as the probability density f⁢(x,p,t)𝑓𝑥𝑝𝑡f(x,p,t)italic_f ( italic_x , italic_p , italic_t ). These boundary conditions on the electrostatic potential can be viewed as having neutral charges in a neighborhood containing the endpoints {0;L}0𝐿\{0;L\}{ 0 ; italic_L } and zero potential bias, that is, the corresponding Poisson boundary value problem for the potential is

The calculation for internal edges and boundaries with periodic conditions both follow the same spirit, whereas the reflective boundary has to be treated separately as it will vanish by itself considering the inflow-outflow regions.

The numerical method we will study in this work in connection to our Boltzmann - Poisson system is the Discontinuous Galerkin (DG) Finite Element Method (FEM), to be explained in Section 3. It was proposed by Reed and Hill [7] for hyperbolic equations in the context of neutron transport. It is defined such that its numerics captures the mathematics of the hyperbolic transport by defining the so-called fluxes in such a way that the information propagates numerically in the same fashion as a hyperbolic equation propagates information analytically. In the particular context of electron collisional transport in semiconductors, the DG method has been used in works such as [4, 6, 5] after an evolution of the numerical methods used to solve it that transitioned from Upwind Finite Differences [1] to WENO schemes [2], [3] and finally to mainly two schools, one related to the aforementioned development on DG methods for Boltzmann - Poisson, and the other related to Spherical Harmonics Expansions,

Both internal edges and periodic boundaries are characterized by the fact that each edge has a pairing with another unique edge for which the conditions

B

Let X𝑋Xitalic_X be a symplectic 6-manifold with an integrally homologically trivial Lagrangian sphere S𝑆Sitalic_S. We fix a Weinstein neighborhood U𝑈Uitalic_U of S𝑆Sitalic_S such that there is a singular toric fibration on U𝑈Uitalic_U, as we described in Subsection 4.1. Topologically U𝑈Uitalic_U is isomorphic to S3×B3superscript𝑆3superscript𝐵3S^{3}\times B^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × italic_B start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT where B3superscript𝐵3B^{3}italic_B start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT is a 3-ball. We say U𝑈Uitalic_U is a round neighborhood if it is symplectomorphic to Dr⁢T∗⁢S3subscript𝐷𝑟superscript𝑇superscript𝑆3D_{r}T^{*}S^{3}italic_D start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT with respect to the round metric on the unit sphere S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. The preimages of four faces in the moment polytope are four 4-chains Ki=π−1⁢(Pi),i=1,2,3,4formulae-sequencesubscript𝐾𝑖superscript𝜋1subscript𝑃𝑖𝑖1234K_{i}=\pi^{-1}(P_{i}),i=1,2,3,4italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_i = 1 , 2 , 3 , 4. Each of them is homeomorphic to S3×[0,1]superscript𝑆301S^{3}\times[0,1]italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × [ 0 , 1 ] with two boundary components. Up to orientation ∂0(Ki)subscript0subscript𝐾𝑖\partial_{0}(K_{i})∂ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is the zero section S𝑆Sitalic_S and ∂1(Ki)subscript1subscript𝐾𝑖\partial_{1}(K_{i})∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is the generator of H3⁢(∂U;ℤ)subscript𝐻3𝑈ℤH_{3}(\partial U;\mathbb{Z})italic_H start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( ∂ italic_U ; blackboard_Z ). First we study some topological condition on S𝑆Sitalic_S to perform the conifold transition. Let V𝑉Vitalic_V be a small closed neighborhood containing X−U𝑋𝑈X-Uitalic_X - italic_U such that U∩V𝑈𝑉U\cap Vitalic_U ∩ italic_V is homeomorphic to S3×S2×[1−ϵ,1]superscript𝑆3superscript𝑆21italic-ϵ1S^{3}\times S^{2}\times[1-\epsilon,1]italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × [ 1 - italic_ϵ , 1 ].

Let X𝑋Xitalic_X be a closed symplectic 6-manifold with a Lagrangian 3-sphere S𝑆Sitalic_S. We say S𝑆Sitalic_S is integrally homologically trivial if the inclusion map i:H3⁢(S;ℤ)→H3⁢(X;ℤ):𝑖→subscript𝐻3𝑆ℤsubscript𝐻3𝑋ℤi:H_{3}(S;\mathbb{Z})\rightarrow H_{3}(X;\mathbb{Z})italic_i : italic_H start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_S ; blackboard_Z ) → italic_H start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_X ; blackboard_Z ) is a trivial map. For an oriented Lagrangian submanifold L𝐿Litalic_L in X𝑋Xitalic_X, we will study holomorphic disks with boundary on L𝐿Litalic_L and holomorphic cylinders with one boundary on L𝐿Litalic_L and the other on S𝑆Sitalic_S. The following condition is designed to make various moduli spaces behave nicely.

Let X𝑋Xitalic_X be a closed symplectic 6-manifold which contains an integrally homologically trivial Lagrangian 3-sphere S𝑆Sitalic_S. Consider the Lagrangian embedding

The Lagrangian sphere S𝑆Sitalic_S is homologically trivial in X𝑋Xitalic_X if and only if the inclusion

Let X𝑋Xitalic_X be a closed symplectic 6-manifold with an integrally homologically trivial Lagrangian 3-sphere S𝑆Sitalic_S, and let L𝐿Litalic_L be an oriented Lagrangian submanifold of X𝑋Xitalic_X satisfying Condition 1.1. For a 4-chain K𝐾Kitalic_K such that ∂K=S𝐾𝑆\partial K=S∂ italic_K = italic_S and K∩L=∅𝐾𝐿K\cap L=\emptysetitalic_K ∩ italic_L = ∅, let

C

L=0.italic_L = italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT over˙ start_ARG italic_x end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ⇒ caligraphic_H = italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT over˙ start_ARG italic_x end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - italic_L = 0 .

co-ordinate x0superscript𝑥0x^{0}italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT referred to as a cyclic co-ordinate,

\vec{p}\right|^{2}=\delta^{ij}p_{i}p_{j}( italic_A + | over→ start_ARG italic_p end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + ( italic_B + italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_D , where italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = | over→ start_ARG italic_x end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_δ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , | over→ start_ARG italic_p end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT

Upon parametrization wrt any one of the co-ordinates x0superscript𝑥0x^{0}italic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT,

co-ordinate x0=tsuperscript𝑥0𝑡x^{0}=titalic_x start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_t:

C

(defined for every ψ0∈L2⁢(Ω,ℂ4)subscript𝜓0superscript𝐿2Ωsuperscriptℂ4\psi_{0}\in L^{2}(\Omega,\mathbb{C}^{4})italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω , blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) with ‖ψ0‖=1normsubscript𝜓01\|\psi_{0}\|=1∥ italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ = 1) agrees with (4) and (6), based on (19), (20), and (21), whenever the expressions in (4) and (6) are well defined. E⁢(⋅)𝐸⋅E(\cdot)italic_E ( ⋅ ) has the property that for every ψ0∈L2⁢(Ω,ℂ4)subscript𝜓0superscript𝐿2Ωsuperscriptℂ4\psi_{0}\in L^{2}(\Omega,\mathbb{C}^{4})italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω , blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ), the restriction of the measure ⟨ψ0|E⁢(⋅)|ψ0⟩quantum-operator-productsubscript𝜓0𝐸⋅subscript𝜓0\langle\psi_{0}|E(\cdot)|\psi_{0}\rangle⟨ italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | italic_E ( ⋅ ) | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟩ to [0,∞)×∂Ω0Ω[0,\infty)\times\partial\Omega[ 0 , ∞ ) × ∂ roman_Ω is absolutely continuous relative to d⁢t⁢dd−1⁢𝐱𝑑𝑡superscript𝑑𝑑1𝐱dt\,d^{d-1}\boldsymbol{x}italic_d italic_t italic_d start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT bold_italic_x.

Our third theorem is an analog of Theorem 1 for the Dirac equation. In the version of the absorbing boundary rule for the Dirac equation on Ω⊂ℝ3Ωsuperscriptℝ3\Omega\subset\mathbb{R}^{3}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, described in [15], the Schrödinger equation (1) is replaced by the Dirac equation

For further discussion of the absorbing boundary rule, see [13, 14, 15]. For an overview of other proposals for the detection time distribution in quantum mechanics, see [8]. In Section 2, we describe our theorems. In Sections 3–7, we give the proofs.

For an overview of the theory of boundary conditions for the Dirac equation, see [2]. For work on boundary conditions for the Dirac equation that lead to a self-adjoint Hamiltonian, see [5]. For a general characterization of the reflecting boundary conditions for the Dirac equation, as well as of the interior-boundary conditions, see [11].

where ψt:Ω→ℂ4:subscript𝜓𝑡→Ωsuperscriptℂ4\psi_{t}:\Omega\to\mathbb{C}^{4}italic_ψ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT : roman_Ω → blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT is spinor-valued and the potential V𝑉Vitalic_V may take values in the set Herm⁢(ℂ4)Hermsuperscriptℂ4\mathrm{Herm}(\mathbb{C}^{4})roman_Herm ( blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) of Hermitian (i.e., self-adjoint) complex 4×4444\times 44 × 4 matrices. The boundary condition (2) is replaced by the “semi-ideal absorbing boundary condition” [15] for the Dirac equation,

C

In other words, the source of non-injectiveness is the 0-skeleton of st⁢(τ,C)st𝜏𝐶\text{st}(\tau,C)st ( italic_τ , italic_C ), that is, the vertices. This lemma allows us to associate to any iterated stellar subvision of C𝐶Citalic_C a canonical cubical stellar subdivision. If, say, C𝐶Citalic_C is stellarly subdivided at a face τ𝜏\tauitalic_τ, then there is an associated cubical stellar subdivision of C𝐶Citalic_C carried out in τ𝜏\tauitalic_τ.

The confinement map is injective on st⁢(τ,C)>0stsuperscript𝜏𝐶absent0\text{st}(\tau,C)^{>0}st ( italic_τ , italic_C ) start_POSTSUPERSCRIPT > 0 end_POSTSUPERSCRIPT, that is, on the order filter and its preimage, the map is one to one as a map of posets.

Every cubical stellar subdivision of a PL cubical manifold X𝑋Xitalic_X can be achieved by cubical Pachner moves provided, assuming the link of the subdivided face is shellable.

Now if we consider a further stellar subdivision of the resulting complex C′superscript𝐶′C^{\prime}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, then the preimage of the subdivided face under the confinement map is a face (by Lemma 2.9) that is we can analogously perform a cubical stellar subvision on.

In other words, the source of non-injectiveness is the 0-skeleton of st⁢(τ,C)st𝜏𝐶\text{st}(\tau,C)st ( italic_τ , italic_C ), that is, the vertices. This lemma allows us to associate to any iterated stellar subvision of C𝐶Citalic_C a canonical cubical stellar subdivision. If, say, C𝐶Citalic_C is stellarly subdivided at a face τ𝜏\tauitalic_τ, then there is an associated cubical stellar subdivision of C𝐶Citalic_C carried out in τ𝜏\tauitalic_τ.

C

\xi)|^{2}d\xi,∥ italic_f ∥ start_POSTSUBSCRIPT over˙ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∫ | italic_ξ | start_POSTSUPERSCRIPT 2 italic_s start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | over^ start_ARG italic_f end_ARG ( italic_ξ ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_ξ ,

equation with log-supercritical defocusing nonlinearities of type |u|p⁢u⁢g⁢(|u|)superscript𝑢𝑝𝑢𝑔𝑢|u|^{p}ug(|u|)| italic_u | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_u italic_g ( | italic_u | ). The main result of [29] is that a robust global well-posedness and

theory of [28], replacing log-supercriticality by small power-type supercriticality for bounded sets of initial data, with the level of

It is well known (see, for instance, [8, 21] and [26]) that initial value problems of type (3) are equipped with a robust

technical subtleties, for instance in treating non-smooth nonlinearities, well understood in related energy-critical settings, see,

C

(since ⁢|F|=k⁢ (because ⁢F⁢ is a ⁢k⁢-element set))since 𝐹𝑘 (because 𝐹 is a 𝑘-element set)\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{since }\left|F\right|=k\text{

all }k\text{-element sets in }\mathcal{G}\right)⟺ ( italic_F ∈ caligraphic_G ) ( since caligraphic_G start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the set of all italic_k -element sets in caligraphic_G )

is an r𝑟ritalic_r-element subset of the n𝑛nitalic_n-element set E𝐸Eitalic_E. Thus, the set E𝐸Eitalic_E

(because }F\text{ is a }k\text{-element set)}\right)( since | italic_F | = italic_k (because italic_F is a italic_k -element set) )

(since ⁢|F|=k⁢ (because ⁢F⁢ is a ⁢k⁢-element set))since 𝐹𝑘 (because 𝐹 is a 𝑘-element set)\displaystyle\ \ \ \ \ \ \ \ \ \ \left(\text{since }\left|F\right|=k\text{

C

We examine the action of the transfer operator associated with a class of random dynamical systems. We establish appropriate anisotropic spaces by extending a spectral gap from a subshift of finite type to the transfer operator of the skew-product map. Following the general approach outlined in [19] and [20], we generalize the techniques and apply them to the system under study. In essence, we achieve a fixed point of the transfer operator of the skew-product within a specific anisotropic space by lifting to the RDS a known spectral gap from the base system.

We remark that, like in [19], we cannot appy the traditional Hennion, or Ionescu-Tulcea and Marinescu’s approach to our function spaces because there is no compact immersion of the strong space into the weak one. This comes from the fact that we are considering the same “dual of Lipschitz”distance in the contracting direction for both spaces.

This type of study is often carried out using the Ionescu-Tulcea and Marinescu Theorem by constructing a pair of suitable function spaces—a stronger space and an auxiliary weaker space—such that the action of the Perron-Frobenius operator on the stronger space exhibits a spectral gap (see [6], [35], [11], [17], and [40] for introductory texts). Commonly, these approaches achieve regularity through compact inclusion arguments. However, our approach differs; we obtain regularity by examining the behavior of the transfer operator in the neighbourhood of the constructed fixed point. Specifically, we achieve uniqueness and regularity as a result of convergence to the equilibrium and the presence of a spectral gap.

We examine the action of the transfer operator associated with a class of random dynamical systems. We establish appropriate anisotropic spaces by extending a spectral gap from a subshift of finite type to the transfer operator of the skew-product map. Following the general approach outlined in [19] and [20], we generalize the techniques and apply them to the system under study. In essence, we achieve a fixed point of the transfer operator of the skew-product within a specific anisotropic space by lifting to the RDS a known spectral gap from the base system.

Regarding relevant properties of invariant measures, assessing their degree of regularity is useful. This type of problem can be approached from different perspectives. For instance, if the invariant measure of the system is absolutely continuous with respect to a reference measure, its regularity can be evaluated based on the regularity of its Radon-Nikodym derivative (see [35]). On the other hand, in the absence of a reference measure, understanding the regularity of the disintegration with respect to a measurable partition can be valuable for various applications (see [19], [20], and [21]).

B

The discrete wave equation (2.5) trivially admits very fast simulations, a few examples are given in Fig. 5 and Fig. 6. Actually, constructing solutions to (2.5) is more memory-consuming than time-consuming if one wants to keep all binary digits so as not to lose control over cancellations, inherent to the wave equation in 2D. The pictures in Fig. 5 and, notably, in Fig. 6 are obtained by such exact simulations.

An equivalent characterization of space-like maximal surfaces in ℝ2,1superscriptℝ21\mathbb{R}^{2,1}blackboard_R start_POSTSUPERSCRIPT 2 , 1 end_POSTSUPERSCRIPT which we will be using is the harmonicity of coordinate functions under a conformal parametrization of the surface by a complex parameter; this follows from the Weierstrass–Enneper representation of maximal surfaces [18]. We stress that this conformal parameter is in general not given by the limit of the t-embedding. The relevance of maximal surfaces with respect to the initial probabilistic model is that finding such a conformal parametrization will yield the right conformal structure to describe the GFF fluctuations. Indeed, in the limit, the dimer coupling functions become holomorphic in this conformal variable.

It is shown in [7] that, if the graphs of the origami maps converge to a space-like maximal surface in ℝ2,1superscriptℝ21\mathbb{R}^{2,1}blackboard_R start_POSTSUPERSCRIPT 2 , 1 end_POSTSUPERSCRIPT as the mesh size tends to 00, and provided that the technical assumptions Exp-Fat and Lip (see [7, Section 1.2.3]) about the non-degeneracy of the t-embeddings and origami maps are satisfied, then the intrinsic conformal metric of the limiting surface provides the right parametrization of the liquid region, i.e. the parametrization in which the fluctuations are given by a flat Gaussian Free Field.

In recent developments [7, 8], the following idea arose: the conformal structure of dimer model fluctuations on abstract planar graphs can be understood via the so-called t-embeddings (which appeared under the name of Coulomb gauges in the preceding work [17]; see also [1]) and the associated origami maps; see Section 2 for definitions. The origami map of a t-embedding is a map from the plane to itself, so its graph is a two-dimensional piecewise linear surface in a four-dimensional space; following [7] we view the latter as the Minkowski space ℝ2,2superscriptℝ22\mathbb{R}^{2,2}blackboard_R start_POSTSUPERSCRIPT 2 , 2 end_POSTSUPERSCRIPT rather than as the Euclidean one. In the setup of this paper, the image of the origami map becomes one-dimensional in the small mesh size limit, hence the limiting surface lives in the three-dimensional Minkowski space ℝ2,1⊂ℝ2,2superscriptℝ21superscriptℝ22\mathbb{R}^{2,1}\subset\mathbb{R}^{2,2}blackboard_R start_POSTSUPERSCRIPT 2 , 1 end_POSTSUPERSCRIPT ⊂ blackboard_R start_POSTSUPERSCRIPT 2 , 2 end_POSTSUPERSCRIPT.

Despite some clear resemblance, we stress that the space-like maximal surface is not the same as the limiting height function, also known as the limit shape.

D

Thus encountering a free boundary in the nature is not unnatural. The problem in this paper is a large enough generalization to this physical phenomena which besides being a new addition to the literature can also serve as a note to bridge the problems in elliptic PDEs with algebraic topology.

The trend of applying the topics from algebraic topology in elliptic PDEs is not very old. We refer the readers to the works of [8, 11, 13, 23, 24] and the references therein. A rich literature survey has been done in the book due to Perera et al. [16] where the author has discussed problems of several variety involving the p𝑝pitalic_p-Laplace operators which could be studied using the Morse theory. The motivation for the current work has been drawn from the work due to Perera [18] who has considered a sublinear problem. We also refer the readers to the latest work due to Choudhuri-Repovš [9]. The treatment used to address the existence of at least one (or two) solution(s) to the approximating problem may be classical (section 3333, Theorems 3.3 and 3.5) but the result concerning the reguarity of the free boundary is very new and the question of existence of solution to the problem (1.1) has not been answered till now (section 4444, Lemma 4.1), to the best of my knowledge. A result due to Alt-Caffarelli [1] (section 4444, Lemma 4.2) were improved to the best possible extent to suit the purpose of the problem in this paper. Most importantly we have included a short subsection 2.12.12.12.1 on the fundamentals of algebraic topology for the benefit of the readers.

We begin by giving the relevant definitions and results besides defining the function space which will be used very frequently in the article.

We will be using the Palais-Smale condition which is a special type of compactness that can be referred to the Definition 5.5.15.5.15.5.15.5.1 in the book [19]. Following is the definition of Morse index which will be used in the subsequent sections.

Before we prove the existence of a solution to the problem (1.1), we sharpen a few tools which that be used in the proof. We observe that

B

In Section 5, we consider a generalisation of Ramanujan sums to Sp⁡(4,ℤ)Sp4ℤ\operatorname{Sp}(4,\mathbb{Z})roman_Sp ( 4 , blackboard_Z ), which appears in the Fourier coefficients of Eisenstein series. The construction of such exponential sums for the GL⁡(3)GL3\operatorname{GL}(3)roman_GL ( 3 ) case is due to Bump [5]. As in the GL⁡(3)GL3\operatorname{GL}(3)roman_GL ( 3 ) case, the degenerate sums reduce to classical Ramanujan sums, justifying the use of the term “generalisation”. The Dirichlet series associated to this Ramanujan sum is computed in Proposition 5.2.

This article is organised as follows, and we point out a number of auxiliary results of independent interest that are derived in the rest of the paper. In Section 2, we give for G=Sp⁡(4)𝐺Sp4G=\operatorname{Sp}(4)italic_G = roman_Sp ( 4 ) the explicit characterisations of the parabolic subgroups and other associated groups, which are used later on. In Section 3, we describe the coset representatives of (Pj∩Γ)\Γ\subscript𝑃𝑗ΓΓ(P_{j}\cap\Gamma)\backslash\Gamma( italic_P start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∩ roman_Γ ) \ roman_Γ, j∈{0,α,β}𝑗0𝛼𝛽j\in\left\{{0,\alpha,\beta}\right\}italic_j ∈ { 0 , italic_α , italic_β }, using Plücker coodinates as defined in [7]. We compute the explicit Bruhat decomposition in Section 3.2. Alternative expressions for Eisenstein series are given in Theorem 3.3.

To compute the Fourier coefficients E0,χ⁢(g,ν)subscript𝐸0𝜒𝑔𝜈E_{0,\chi}(g,\nu)italic_E start_POSTSUBSCRIPT 0 , italic_χ end_POSTSUBSCRIPT ( italic_g , italic_ν ), we break down the expression via Bruhat decomposition, and express them in terms of Whittaker functions. We have

In Section 6, we introduce the Whittaker functions on Sp⁡(4)Sp4\operatorname{Sp}(4)roman_Sp ( 4 ) in terms of Jacquet integrals [15]. We are then able to compute the Fourier coefficients of the minimal Eisenstein series in terms of these generalised Ramanujan sums and Whittaker functions, proving Theorem 1.1.

In Section 5, we consider a generalisation of Ramanujan sums to Sp⁡(4,ℤ)Sp4ℤ\operatorname{Sp}(4,\mathbb{Z})roman_Sp ( 4 , blackboard_Z ), which appears in the Fourier coefficients of Eisenstein series. The construction of such exponential sums for the GL⁡(3)GL3\operatorname{GL}(3)roman_GL ( 3 ) case is due to Bump [5]. As in the GL⁡(3)GL3\operatorname{GL}(3)roman_GL ( 3 ) case, the degenerate sums reduce to classical Ramanujan sums, justifying the use of the term “generalisation”. The Dirichlet series associated to this Ramanujan sum is computed in Proposition 5.2.

C

Vℝ⊗ℝℂsubscripttensor-productℝsubscript𝑉ℝℂV_{{\mathbb{R}}}\otimes_{{\mathbb{R}}}{\mathbb{C}}italic_V start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT ⊗ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT blackboard_C is the complexification of the vector space Vℝsubscript𝑉ℝV_{{\mathbb{R}}}italic_V start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT and hℎhitalic_h is the Coxeter number.

2.3. Graded ℂℂ{\mathbb{C}}blackboard_C-algebra structure on ℂ⁢[V]ℂdelimited-[]𝑉\mathbb{C}[V]blackboard_C [ italic_V ]

We give a graded ℂℂ{\mathbb{C}}blackboard_C-algebra structure on ℂ⁢[V]ℂdelimited-[]𝑉\mathbb{C}[V]blackboard_C [ italic_V ]

2. Graded ℂℂ{\mathbb{C}}blackboard_C-algebra structure on ℂ⁢[V]ℂdelimited-[]𝑉\mathbb{C}[V]blackboard_C [ italic_V ]

2.1. Graded ℂℂ{\mathbb{C}}blackboard_C-algebra structure on ℂ⁢[V]Gℂsuperscriptdelimited-[]𝑉𝐺\mathbb{C}[V]^{G}blackboard_C [ italic_V ] start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT

C

In view of the constraint (LABEL:C04-B), this theorem gives the almost sure convergence of uNsuperscript𝑢𝑁u^{N}italic_u start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT and μNsuperscript𝜇𝑁\mu^{N}italic_μ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT for any α<12𝛼12\alpha<\frac{1}{2}italic_α < divide start_ARG 1 end_ARG start_ARG 2 end_ARG by choosing p=2𝑝2p=2italic_p = 2 and β𝛽\betaitalic_β close to 1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG, provided the initial condition has enough regularity. On the other hand, the best possible rate of convergence one can get here is ξ=(18)−𝜉superscript18\xi=(\frac{1}{8})^{-}italic_ξ = ( divide start_ARG 1 end_ARG start_ARG 8 end_ARG ) start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT, by choosing p=2𝑝2p=2italic_p = 2, β=1−𝛽superscript1\beta=1^{-}italic_β = 1 start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT and α=(14)+𝛼superscript14\alpha=(\frac{1}{4})^{+}italic_α = ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT.

These inequalities are standard when 𝒟=ℝd𝒟superscriptℝ𝑑\mathcal{D}=\mathbb{R}^{d}caligraphic_D = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, see e.g. Eq. (2.2) in [25]. We prove them now for 𝒟=𝕋d𝒟superscript𝕋𝑑\mathcal{D}=\mathbb{T}^{d}caligraphic_D = blackboard_T start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT.

Similarly to [25, Eq. (2.3)], we obtain the following mild formulation for x∈ℝ𝑥ℝx\in\mathbb{R}italic_x ∈ blackboard_R:

Let us introduce a mollifier that will be used both to regularise the interaction kernel in the particle system and its empirical measure. Let V:ℝ→ℝ+:𝑉→ℝsubscriptℝV:\mathbb{R}\to\mathbb{R}_{+}italic_V : blackboard_R → blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT be a compactly supported, smooth probability density function. For α∈[0,1]𝛼01\alpha\in[0,1]italic_α ∈ [ 0 , 1 ] and any x∈ℝ𝑥ℝx\in\mathbb{R}italic_x ∈ blackboard_R, define

Similarly to [25, Eq. (2.3)], we obtain the following mild formulation for x∈𝕋d𝑥superscript𝕋𝑑x\in\mathbb{T}^{d}italic_x ∈ blackboard_T start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT:

B

(g−id.)(F)=(g−id.)(F~).(g-id.)(F)=(g-id.)(\widetilde{F}).( italic_g - italic_i italic_d . ) ( italic_F ) = ( italic_g - italic_i italic_d . ) ( over~ start_ARG italic_F end_ARG ) .

For V∈X~𝑉~𝑋V\in\widetilde{X}italic_V ∈ over~ start_ARG italic_X end_ARG, the followings are equivalent.

An element V∈X~𝑉~𝑋V\in\widetilde{X}italic_V ∈ over~ start_ARG italic_X end_ARG is called of “zero type” if

but for any V∈X~𝑉~𝑋V\in\widetilde{X}italic_V ∈ over~ start_ARG italic_X end_ARG, the subset f−1⁢(V)superscript𝑓1𝑉f^{-1}(V)italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_V )

I\}.over~ start_ARG italic_X end_ARG := { italic_V ⊂ over~ start_ARG italic_F end_ARG | over~ start_ARG italic_F end_ARG = italic_V ⊕ roman_rad italic_I } .

A

Thompson (2019) are very attractive, which also play important roles in the proofs of this paper. In the past proofs, the convexity and Lipschitz continuity of the Huber loss function were used, in other words, general properties of the loss function were used. In the proof of this paper, we use a specific property of the Huber loss function. (It changes the behavior at the threshold from a quadratic function to a linear function.) By such a careful analysis, we can give a sharper convergence rate than Nguyen and Tran (2012), even when the number of outliers is large.

First, we focus on the fact that the derivative of the loss function is zero at the minimizer. Next, we divide the derivative of the loss function into three parts via a specific property of the Huber function. These are shown in this subsection. Some properties related to the three terms are given in the subsequent sections. Combining the results, we can show the main theorem.

The robust and sparse estimation in linear regression has also been studied under the Huber’s contamination. Gao (2020) derived the minimax rate of the regression coefficient estimation, given by

First, the robust estimation of the mean and scatter matrix was examined under the Huber’s contamination.

Nguyen and Tran (2012) considered the case where the outputs were adversarially contaminated. The convergence rate is given by (1.3). It is slower than (1.5). This is because the adversarial contamination includes various types of contamination over the Huber’s contamination.

C

^{2}(D).⟨ italic_f , italic_g ⟩ start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) end_POSTSUBSCRIPT = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT italic_f ( italic_z ) over¯ start_ARG italic_g ( italic_z ) end_ARG italic_d italic_A ( italic_z ) , italic_f , italic_g ∈ italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) .

With this inner product A2⁢(D)superscript𝐴2𝐷A^{2}(D)italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) is the Hilbert space.

The Hilbert space A2⁢(D)superscript𝐴2𝐷A^{2}(D)italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) is separable, so it has a countable base {ϕj:j∈ℕ}conditional-setsubscriptitalic-ϕ𝑗𝑗ℕ\{\phi_{j}:j\in\mathbb{N}\}{ italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT : italic_j ∈ blackboard_N }. The Bergman kernel may be represented as

finite}\},italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) = { italic_f : italic_f is analytic mapping on italic_D and ∫ start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT | italic_f | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_A is finite } ,

be the Bergman space; d⁢A𝑑𝐴dAitalic_d italic_A is the area measure. In A2⁢(D)superscript𝐴2𝐷A^{2}(D)italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) one may introduce the inner product

A

Because H0⁢(X,ℚ¯p)≠0superscript𝐻0𝑋subscript¯ℚ𝑝0H^{0}(X,\overline{\mathbb{Q}}_{p})\neq 0italic_H start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X , over¯ start_ARG blackboard_Q end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ≠ 0, and ℚ¯psubscript¯ℚ𝑝\overline{\mathbb{Q}}_{p}over¯ start_ARG blackboard_Q end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT is a summand of R⁢ρ∗⁢(K′⊗ℒχ)𝑅subscript𝜌tensor-productsuperscript𝐾′subscriptℒ𝜒R\rho_{*}(K^{\prime}\otimes\mathcal{L}_{\chi})italic_R italic_ρ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊗ caligraphic_L start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT ), it follows that H0⁢(X,R⁢ρ∗⁢(K′⊗ℒχ))≠0superscript𝐻0𝑋𝑅subscript𝜌tensor-productsuperscript𝐾′subscriptℒ𝜒0H^{0}(X,R\rho_{*}(K^{\prime}\otimes\mathcal{L}_{\chi}))\neq 0italic_H start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X , italic_R italic_ρ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊗ caligraphic_L start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT ) ) ≠ 0.

We prove a group-theoretic relationship (Theorem 4.7) between the usual monodromy group of a typical A⁢[ℓn]𝐴delimited-[]superscriptℓ𝑛A[\ell^{n}]italic_A [ roman_ℓ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ]-eigenspace in the cohomology of a family of hypersurfaces and the Tannakian monodromy group of a typical member of the family of hypersurfaces. One can think of this as analogous to the relationship between the monodromy groups of the generic horizontal and vertical fibers of a family of varieties over (an open subset of) a product X1×X2subscript𝑋1subscript𝑋2X_{1}\times X_{2}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Using purely geometric arguments involving the results of [42] and [43], we show that the Tannakian monodromy group contains a classical group, and then using Theorem 4.7, we show that the usual monodromy group does as well.

of the normalizer of the classical group, such that any subgroup of the normalizer contains the classical group if and only if it has no invariants on any of these representations. Associated to each of these representations is a perverse sheaf on Aksubscript𝐴𝑘A_{k}italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. We prove Lemma 4.4 showing that the action of GalksubscriptGal𝑘\operatorname{Gal}_{k}roman_Gal start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT on the cohomology of a perverse sheaf, defined using a generic local system 𝖫𝖫\mathsf{L}sansserif_L, has invariants if and only if the perverse sheaf has a very special form. Using Lemma 4.1, we check that the relevant perverse sheaves do not have this special form unless the family of hypersurfaces over X𝑋Xitalic_X is constant, up to translation by a section of A𝐴Aitalic_A, completing the proof of Theorem 4.7.

Now that we have interpreted the existence of nontrivial monodromy invariants cohomologically, we can re-express the cohomology group in terms of shaves on A𝐴Aitalic_A, which will enable us to understand its dependence on χ𝜒\chiitalic_χ using the generic vanishing theorem. It follows from the Leray spectral sequence and the projection formula that

For the second step, the exceptional groups and spin groups are not too hard to eliminate, as they only occur for representations of very specific dimensions. The Tannakian dimension in our setting is the topological Euler characteristic of the hypersurface, which we have an explicit formula for. Comparing these, we can see in Lemma 3.12 that the problematic cases only occur for curves in an abelian surface, which are excluded by the assumption dim(A)≥3dimension𝐴3\dim(A)\geq 3roman_dim ( italic_A ) ≥ 3. The only remaining case, except for the good classical cases, is the case of a special linear group acting by a wedge power representation. For this representation, the Euler characteristic formula is not sufficient, but we are eventually able to rule this case out using a more sophisticated numerical invariant, the Hodge numbers. If the convolution monodromy group acts on the representation associated to H𝐻Hitalic_H by the k𝑘kitalic_k-th wedge power of an m𝑚mitalic_m-dimensional representation, we might expect that the Hodge structure on the cohomology of H𝐻Hitalic_H, or the cohomology of H𝐻Hitalic_H twisted by a rank one local system, is the k𝑘kitalic_k-th wedge power of an m𝑚mitalic_m-dimensional Hodge structure. This would place some restrictions on the Hodge numbers. We don’t prove this, but instead prove in Lemma 3.14 a p𝑝pitalic_p-adic Hodge-theoretic analogue, using the GalksubscriptGal𝑘\operatorname{Gal}_{k}roman_Gal start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-action discussed earlier. On the other hand, we can calculate the Hodge numbers of the cohomology of H𝐻Hitalic_H twisted by a rank one local system using the Hirzebruch-Riemann-Roch formula. Working this out gives a complicated set of combinatorial relations between the Hodge numbers of the original m𝑚mitalic_m-dimensional Hodge structure. By a lengthy combinatorial argument in Appendix B, we find all solutions of these relations, noting in particular that they occur only for abelian varieties of dimension less than four. This completes the proof.

C

It is obvious that every norm is also a quasinorm with the modulus of concavity equal to 1111 and that every quasinorm with the modulus of concavity equal to 1111 is also a norm.

It is a well-known fact that every norm defines a metrisable topology on X𝑋Xitalic_X and that it is continuous with respect to that topology. This is not true for quasinorms, but it can be remedied thanks to the Aoki–Rolewicz theorem which we list below. Further details can be found for example in [16] or in [2, Appendix H].

The direct consequence of this result is that every quasinorm defines a metrisable topology on X𝑋Xitalic_X and that the convergence in said topology is equivalent to the convergence with respect to the original quasinorm, in the sense that xn→x→subscript𝑥𝑛𝑥x_{n}\rightarrow xitalic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → italic_x in the induced topology if and only if limn→∞∥xn−x∥=0subscript→𝑛delimited-∥∥subscript𝑥𝑛𝑥0\lim_{n\rightarrow\infty}\lVert x_{n}-x\rVert=0roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_x ∥ = 0. Furthermore, we may naturally extend the concept of completeness to quasinormed spaces.

The proof is omitted since it is the same as in the classical case which can be found in [1, Chapter 1, Lemma 1.5].

Additionally, if ∥⋅∥Xsubscriptdelimited-∥∥⋅𝑋\lVert\cdot\rVert_{X}∥ ⋅ ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT is a Banach function norm then (2.1) holds with constant one. This is a classical result of Lorentz and Luxemburg, proof of which can be found for example in [1, Chapter 1, Theorem 2.7].

A

After proving a general dichotomy relative to a single splitting in §3.1, in §3.2 we focus on graphs that can be iteratively decomposed into pieces which are easily understood, such as chordal graphs. In the last section §3.3 we obtain an explicit formula for the rank of the Artin kernels and discuss some features of the BNS invariant of AΓsubscript𝐴ΓA_{\Gamma}italic_A start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT when ΓΓ\Gammaroman_Γ is a block graph.

Our results concern the ways in which such a splitting can induce a splitting of the Artin kernels associated to ΓΓ\Gammaroman_Γ, and are inspired by the work of Cashen and Levitt on free-by-cyclic groups (see [CL16]), in which analogous tame/wild dichotomies and rank formulas are obtained.

When ΓΓ\Gammaroman_Γ is a tree, the group AΓsubscript𝐴ΓA_{\Gamma}italic_A start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT is free-by-cyclic; indeed it follows from [D87] that the Bestvina-Brady group associated to a tree is a free group generated by the edges of ΓΓ\Gammaroman_Γ and the RAAG AΓsubscript𝐴ΓA_{\Gamma}italic_A start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT belongs to the class of groups studied by Cashen and Levitt in [CL16].

We would also like to warn the reader that while Artin kernels provide a natural generalization of the classical Bestvina-Brady groups, the name “generalized Bestvina-Brady groups” is nowadays mostly being used to refer to the groups introduced by Leary in [L18]; these groups are not subgroups of RAAGs in general.

Here we prove a general splitting result along a separating subgraph, which will later be used as a key tool for more specific applications. More precisely we propose an extension to Artin kernels of the main Proposition in [D87], which is about the classical Bestvina-Brady groups. This is inspired by an analogous dichotomy obtained by Cashen and Levitt in [CL16, Theorem 4.4] in the context of free-by-cyclic groups. It should be noted that Bestvina-Brady groups are free-by-cyclic precisely when the underlying graph is a tree. The main argument is based on the following orbit count lemma (which appears in [CL16, Lemma 4.5]), which we will apply to the action of Artin kernels on Bass-Serre trees associated to splittings of the underlying graph.

D

In this paper we study communication protocols that help two remote parties (Alice and Bob) to establish a common secret key, while the communication is done via a public channel. In most practical situations, this task is achieved with the Diffie–Hellman scheme, [8], or other computationally secure protocols based on the assumptions that the eavesdropper has only limited computational resources and that some specific problem (e.g., the computing of the discrete logarithm) is computationally hard. We, in contrast, address the information-theoretic version of this problem and assume no computational restriction on the power of the eavesdropper. In this setting, the problem of the common secret key agreement can be resolved only if Alice and Bob possess since the very beginning some correlated data. In this setting, the challenge is to extract from the correlated data available to Alice and Bob their mutual information and materialize it as a common secret key.

In some applications (see, e.g., [7, 14]) it is natural to assume that the eavesdropper is given a non-negligible a priori information about Alice’ and Bob’s inputs. For this setting we do not have a characterization of the optimal size of the key and the communication complexity of the protocol.

It is typical for modern theoretical cryptography that fundamental concepts and protocols (cryptographic primitives) can be studied separately from each other.

Besides the obvious theoretical interest, this setting is relevant for applications connected with quantum cryptography (see, e.g., [7, 14]) or biometrics (see the survey [15]).

Muchnik’s technique of conditional descriptions (see Proposition 2) and the lemma on the non-negativity of the triple mutual information for communication transcripts (see Lemma 7).

C

In this section we extend the family of binary sequences that we have presented in Section III to the k𝑘kitalic_k-symbols alphabet. We prove the following generalization of Theorem 2. Its proof is similar to proof of Theorem 2, see also [27, Theorem 3].

Let p>2𝑝2p>2italic_p > 2 be a prime number, d∈ℤ+𝑑superscriptℤd\in{\mathbb{Z}}^{+}italic_d ∈ blackboard_Z start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and p∤dnot-divides𝑝𝑑p\nmid ditalic_p ∤ italic_d. Let ΩdsubscriptΩ𝑑\Omega_{d}roman_Ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT be the set defined as

Let fβsubscript𝑓𝛽f_{\beta}italic_f start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT be an irreducible polynomial of degree d𝑑ditalic_d over the finite field 𝔽psubscript𝔽𝑝{\mathbb{F}}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT such that

Let d𝑑ditalic_d and p>2𝑝2p>2italic_p > 2 be distinct prime numbers and k𝑘kitalic_k be a positive integer such that

Let p>2𝑝2p>2italic_p > 2 be a prime number, d≥5𝑑5d\geq 5italic_d ≥ 5 and Ωp,dsubscriptΩ𝑝𝑑\Omega_{p,d}roman_Ω start_POSTSUBSCRIPT italic_p , italic_d end_POSTSUBSCRIPT be a set of irreducible polynomials over 𝔽psubscript𝔽𝑝{\mathbb{F}}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT of degree d𝑑ditalic_d defined as

C

Then 𝔤′superscript𝔤′\mathfrak{g}^{\prime}fraktur_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is an ideal of 𝔤𝔤\mathfrak{g}fraktur_g. Let G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be the analytic subgroup of

Then 𝔤′superscript𝔤′\mathfrak{g}^{\prime}fraktur_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is an ideal of 𝔤𝔤\mathfrak{g}fraktur_g. Let G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be the analytic subgroup of

G𝐺Gitalic_G associated with 𝔤′superscript𝔤′\mathfrak{g}^{\prime}fraktur_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, which is closed and has a compact center

and G𝐺Gitalic_G has compact center (i.e. 𝔷𝔭=0subscript𝔷𝔭0\mathfrak{z}_{\mathfrak{p}}=0fraktur_z start_POSTSUBSCRIPT fraktur_p end_POSTSUBSCRIPT = 0), so that X𝑋Xitalic_X is a

closed analytic subgroup of G𝐺Gitalic_G associated with 𝔤sssubscript𝔤ss\mathfrak{g}_{\mathrm{ss}}fraktur_g start_POSTSUBSCRIPT roman_ss end_POSTSUBSCRIPT

B

Here, |⋅||\cdot|| ⋅ | represents the cardinality of a set. The limit supremum for the counting dimension is taken over all possible cubes, with the side length ‖C‖norm𝐶||C||| | italic_C | | of the cubes going to infinity. Thus in considering the mass dimension, we always consider cubes with centers at the origin, while for the counting dimension, the limit supremum is taken over arbitrary cubes whose lengths are going to infinity(3)(3)(3)Even in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT we loosely refer to C𝐶Citalic_C as a cube.. Thus in particular, we always have D¯⁢(E)≤D⁢(E)¯𝐷𝐸𝐷𝐸\overline{D}(E)\leq D(E)over¯ start_ARG italic_D end_ARG ( italic_E ) ≤ italic_D ( italic_E ).

In a companion manuscript, we intend to study the Hausdorff dimensions of the exceptional sets of u𝑢uitalic_u-parameters that violate the inequality of Theorem 8, for specific 1111-separated sets E𝐸Eitalic_E. The proof of the Theorem 8 does not follow along the lines of the proof of the classical Marstrand slicing theorem, as Examples 1 and 2 in Section 3 demonstrate, and the exceptional sets also do not behave in classically expected ways.

Two prototypical examples of “fractal” sets in the integers are values of polynomials with integer coefficients at integer arguments, and restricted digit Cantor sets. The latter are integer Cantor sets with restrictions on the digits that are permissible to be used. These examples have been studied in detail in [Gla16],[Lima14], with the mass and counting dimensions as the natural dimensional parameters.

Recently there have been two proofs of this celebrated conjecture [Shm19, Wu19]. In the integer setting there are longstanding analogous transversality questions for the particular case of restricted digit Cantor sets. A discussion on this is found in the introduction in [Mor20]; there is a known conjecture [OEI20] about the set of integers A𝐴Aitalic_A that can be written in bases 2,3,4,523452,3,4,52 , 3 , 4 , 5 with digits only 0,1010,10 , 1 being {0,1,82000}0182000\{0,1,82000\}{ 0 , 1 , 82000 }. This is obviously a question about the intersections of multiple restricted-digit-Cantor-sets, which are sets respectively invariant under ×2,×3,×4,×5\times 2,\times 3,\times 4,\times 5× 2 , × 3 , × 4 , × 5 maps. Recently, Burrell and Yu [Yu19] showed that the above defined set A𝐴Aitalic_A is such that |A∩[0,N]|≤Cϵ⁢Nϵ𝐴0𝑁subscript𝐶italic-ϵsuperscript𝑁italic-ϵ|A\cap[0,N]|\leq C_{\epsilon}N^{\epsilon}| italic_A ∩ [ 0 , italic_N ] | ≤ italic_C start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT italic_N start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT for any positive ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, which shows that the intersection set has mass dimension 0. This is another illustration where the mass dimension is the natural dimensional quantifier. A well known conjecture of Erdős [Erd79] which states that except for finitely many natural numbers n𝑛nitalic_n, the integer 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT contains 1111 in its base 3333 expansion, can also be naturally formulated in terms of intersections of specific digit Cantor sets.

The formulation in the above corollary is how we immediately apply the theorem to specific subsets of ℕℕ\mathbb{N}blackboard_N such as polynomials or restricted-digit-Cantor-sets.

B

To express (𝒜K)subscript𝒜𝐾\Sha(\mathscr{A}_{K})( script_A start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) in terms of the group C⁢l⁢(Λ)𝐶𝑙ΛCl~{}(\Lambda)italic_C italic_l ( roman_Λ ),

represent the K-homology cycles. An elliptic operator can be twisted by a vector bundle, and the Fredholm index of the

Thus there are no Atiyah pairing in the case p=2𝑝2p=2italic_p = 2. The rest of the proof follows the Basis Theorem

Roughly speaking, the idea is this. To calculate the (𝒜K)subscript𝒜𝐾\Sha(\mathscr{A}_{K})( script_A start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ),

Roughly speaking, the cocycles in K𝐾Kitalic_K-theory are represented by the vector bundles. Atiyah proposed using elliptic operators to

D

{M})}\leqslant C\delta(\varepsilon)|\log(\delta(\varepsilon))|,∥ over~ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT - over~ start_ARG italic_T end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT start_OVERACCENT italic_ε → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_OVERACCENT start_ARG ⟶ end_ARG 0 and ∥ over~ start_ARG italic_T end_ARG start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT - over~ start_ARG italic_T end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ⩽ italic_C italic_δ ( italic_ε ) | roman_log ( italic_δ ( italic_ε ) ) | ,

In this section, we study the problem (2.5) for the case N=1𝑁1N=1italic_N = 1. In this scalar situation, we will have a family of Morse Smale problems with gap conditions in the eigenvalues. This property allows us to obtain a finite-dimensional invariant manifold and then the restriction of the flow in the attractors is given by ordinary differential equations in the Euclidian Space ℝMsuperscriptℝ𝑀\mathbb{R}^{M}blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT, where M=m𝑀𝑚M=mitalic_M = italic_m was defined in (3.1). Once this is done, we will use the results of the B to obtain a better rate of convergence than the previous section for the convergence of the attractors of (2.5).

Therefore, we have a Morse-Smale semigroup in ℝMsuperscriptℝ𝑀\mathbb{R}^{M}blackboard_R start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT and using techniques of shadowing see B, we are ready to prove the main result of this section.

In this section, we prove the main result of this work. We used all previous results to put the problem (1.1) in the conditions of Corollary A.3. Here, we consider the dimension N⩾2𝑁2N\geqslant 2italic_N ⩾ 2. The scalar case will be addressed in the next section.

In this section, we obtain the rate of convergence for families of linear and nonlinear semigroups. It is in this section that we see the arising of the exponent l𝑙litalic_l in the rate (1.7). The main difficulty is the presence of singularities in time when we use the variation of the constants formula due to the immersions between the fractional power spaces and the base space L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Our solution to this problem is to make an interpolation using appropriate exponents to improve the exponential decay of the linear semigroup.

B

[c⁢c′]⊗b⁢b′,tensor-productdelimited-[]𝑐superscript𝑐′𝑏superscript𝑏′\displaystyle[cc^{\prime}]\otimes bb^{\prime},[ italic_c italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ⊗ italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ,

Δ⁢([c]⊗b)=Δtensor-productdelimited-[]𝑐𝑏absent\displaystyle\Delta([c]\otimes b)=roman_Δ ( [ italic_c ] ⊗ italic_b ) =

▲⁢([c]⊗b)▲tensor-productdelimited-[]𝑐𝑏\displaystyle\blacktriangle([c]\otimes b)▲ ( [ italic_c ] ⊗ italic_b )

S⁢([c]⊗b)=𝑆tensor-productdelimited-[]𝑐𝑏absent\displaystyle S([c]\otimes b)=italic_S ( [ italic_c ] ⊗ italic_b ) =

▲⁢([c]⊗b)=▲tensor-productdelimited-[]𝑐𝑏absent\displaystyle\blacktriangle([c]\otimes b)=▲ ( [ italic_c ] ⊗ italic_b ) =

D

Let b=min⁡{|ϕi⁢(e)|:i≥1,e∈E⁢(Ti)}𝑏:subscriptitalic-ϕ𝑖𝑒formulae-sequence𝑖1𝑒𝐸subscript𝑇𝑖b=\min\{\lvert\phi_{i}(e)\rvert:i\geq 1,e\in E(T_{i})\}italic_b = roman_min { | italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_e ) | : italic_i ≥ 1 , italic_e ∈ italic_E ( italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) }.

Hence we may assume that h≥1ℎ1h\geq 1italic_h ≥ 1 and this lemma holds for all pairs lexicographically smaller (h,h−b)ℎℎ𝑏(h,h-b)( italic_h , italic_h - italic_b ).

We shall prove this theorem by induction on N+h−b−c𝑁ℎ𝑏𝑐N+h-b-citalic_N + italic_h - italic_b - italic_c.

By Lemma 6.3, to prove this lemma, we may restrict the problem to the components of Ti−Fisubscript𝑇𝑖subscript𝐹𝑖T_{i}-F_{i}italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.

We shall prove this lemma by induction on the lexicographic order of (h,h−b)ℎℎ𝑏(h,h-b)( italic_h , italic_h - italic_b ).

D

8.2.2. Estimate of 𝒮1,2subscript𝒮12\mathcal{S}_{1,2}caligraphic_S start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT

𝒮1,2subscript𝒮12\displaystyle\mathcal{S}_{1,2}caligraphic_S start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT

𝒮1,2subscript𝒮12\displaystyle\mathcal{S}_{1,2}caligraphic_S start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT

𝒮1,2subscript𝒮12\displaystyle\mathcal{S}_{1,2}caligraphic_S start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT

𝒮1,2subscript𝒮12\displaystyle\mathcal{S}_{1,2}caligraphic_S start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT

A

\mathbf{p}}-\mathbf{p}^{T}\mathbf{Q^{*}}\mathbf{p}\right)=\Theta(1)( bold_p start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_Qp roman_log divide start_ARG 1 end_ARG start_ARG bold_p start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_Qp end_ARG - bold_p start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT bold_p ) = roman_Θ ( 1 ).

In this section, we re-derive the well-known joint distributions of Laplace and KT probability assignments for completeness. Firstly we consider the Laplace probability assignment. Recalled that we defined

We first invoke a known property of stochastic block models (see, for example, [55, 56]). We include the proof here for completeness.

stochastic block models. Secondly, we present upper and lower bounds for the minimax redundancy of stochastic block models. Finally, we analyze the proposed compressor under the local weak convergence framework considered in [34, 42]. We show that the proposed compressor achieves the same performance guarantee (BC entropy) as the compressor in [34].

The rest of the paper is organized as follows. In Section I-A, we defined universality over a family of graph distributions, the stochastic block models and the minimax redundancy of a family of distributions. We present our main result in Section II, which is a polynomial-time graph compressor that is universal for a family containing most of the non-trivial stochastic block models. We describe the proposed graph compressor in Section II-A. In Section II-B, we state the main theorems of the paper—two theorems on

B

Dropping the last three non-negative terms in (Appendix A. Proof of the energy decay in SAV – Theorem 3.1), yields (3.25).

We adapt the analysis of time stepping in the thin-interface regime (0<ε≪1)0𝜀much-less-than1(0<\varepsilon\ll 1)( 0 < italic_ε ≪ 1 ) from [4]. We consider a general form of the Allen Cahn system

The ETD scheme is not competitive, requiring considerably more computational effort to achieve the same accuracy. A heuristic argument for this result, based upon scaling of trunction error in the thin-interface regime (ε≪1)\varepsilon\ll 1)italic_ε ≪ 1 ) is presented in Appendix B.

onset of defects, and comparison of time-stepping performance and computational efficiency. We summarize the performance in section 6. The appendixes provide proof of energy stability for the SAV scheme and a heuristic analysis of time-stepping for ETD and IMEX in the thin interface regime ε≪1much-less-than𝜀1\varepsilon\ll 1italic_ε ≪ 1.

We generalize this to the family of operators presented in (3.18), and find that the choice of β1+β2≈3subscript𝛽1subscript𝛽23\beta_{1}+\beta_{2}\approx 3italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≈ 3 provides optimal performance, with the choice β1=2subscript𝛽12\beta_{1}=2italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 and β2=1subscript𝛽21\beta_{2}=1italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 corresponding to the linearization about the spatially constant background state. These constant coefficient linear operators are trivially inverted in the spatially periodic setting considered herein. It certainly may not be the case that such a convenient and efficient linear-implicit operator is available in all systems. The ETD scheme does not seem to have competitive accuracy in the thin interface regime of the FCH system. The ETD formulation has been proven effective at handling linear stiffness. It places the higher-order differential operators into a semi-group where they are more stable to discretization error. However, as argued in Appendix B, the local truncation error in the ETD scheme seems to have poorer scaling with respect to interfacial thickness in the thin interface regime ε≪1much-less-than𝜀1\varepsilon\ll 1italic_ε ≪ 1 than IMEX type schemes. The large spatial gradients presented by the bilayer interfaces in the super-critical benchmark problem lead to an amplification of error in the ETD Runge-Kutta approximation.

A

4.2 Central Limit Theorems under Polynomially Increasing Nksubscript𝑁𝑘N_{k}italic_N start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.

In this part, we first establish the asymptotic normality of a time-varying linear recursion and provide the proof in Appendix C. This result will be applied in proving Theorems 5-7.

In the following lemma, we first give an upper bound on the expected mean-squared error of {xk}subscript𝑥𝑘\{x_{k}\}{ italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } for quadratic cost functions based on [49]. The proof can be found in Appendix A

In the following, we establish the central limit theorem of a linear recursion, for which the proof is provided in Appendix B. This result will be applied in the proof of Theorems 1-4.

Next, we establish an upper bound on the expected sub-optimality gap of the iterates generated by Algorithm 2. Its proof is similar to that in [3]; hence it is omitted here but included in the supplementary material for completeness. This is an important preliminary result to be used in the rate analysis of Algorithm 2.

A

Y^{\top}&0\end{pmatrix}.roman_Ξ ( italic_Y ) = ( start_ARG start_ROW start_CELL 0 end_CELL start_CELL italic_Y end_CELL end_ROW start_ROW start_CELL italic_Y start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW end_ARG ) .

where ∥⋅∥F\left\|\,\cdot\,\right\|_{\mathrm{F}}∥ ⋅ ∥ start_POSTSUBSCRIPT roman_F end_POSTSUBSCRIPT denotes the Frobenius norm, the first and second equalities follow from the continuity of f𝑓fitalic_f and eigenvalues λi⁢(⋅)subscript𝜆𝑖⋅\lambda_{i}(\cdot)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( ⋅ ), the third from Proposition 1, the fourth by the symmetry of ΣB\scaleobj⁢.8⁢12superscriptsubscriptΣ𝐵\scaleobj.812\Sigma_{B}^{{\scaleobj{.8}{\frac{1}{2}}}}roman_Σ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT .8 divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT and A^^𝐴\hat{A}over^ start_ARG italic_A end_ARG, and the fifth by the definition of K𝐾Kitalic_K.

There are also various generalizations of inequality (1) to the matrix setting, where the entries of vectors are replaced by the eigenvalues or singular values of matrices. One such example is the following result.

By the definition of singular values, we have that for any i=1,…,m𝑖1…𝑚i=1,\dots,mitalic_i = 1 , … , italic_m,

The following proposition shows that the singular values and singular vectors of Y𝑌Yitalic_Y are intimately related to the eigenvalues and eigenvectors of Ξ⁢(Y)Ξ𝑌\Xi(Y)roman_Ξ ( italic_Y ), respectively.

D

The “dual” of a segments-free ♄♄{\saturn}♄-Stokes graph gives an ideal triangulation of C𝐶Citalic_C with vertices in M𝑀Mitalic_M. In this situation, H1⁢(L,ℤ)subscript𝐻1𝐿ℤH_{1}(L,\mathbb{Z})italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_L , blackboard_Z ) is spanned by the following [BS15]:

The inverse image of the dotted line connecting v𝑣vitalic_v and w𝑤witalic_w along the projection πL:L→C:subscript𝜋𝐿→𝐿𝐶\pi_{L}\colon L\rightarrow Citalic_π start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT : italic_L → italic_C is a Voros edge cycle. We would like to compute the monodromy of the microlocalization of 𝒮ℳ♄subscriptsuperscript𝒮♄ℳ\mathcal{S}^{\saturn}_{\mathcal{M}}caligraphic_S start_POSTSUPERSCRIPT ♄ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT along the Voros edge cycle.

In an ♄♄{\saturn}♄-Stokes region of horizontal strip type, take a path connecting two turning points. This is lifted to a cycle in the spectral curve. We call this cycle, a Voros edge cycle.

Since γ𝛾\gammaitalic_γ is homotopy equivalent to the Voros edge cycle, we compute the monodoromy along γ𝛾\gammaitalic_γ.

For a cycle γ∈H1⁢(L,ℤ)𝛾subscript𝐻1𝐿ℤ\gamma\in H_{1}(L,\mathbb{Z})italic_γ ∈ italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_L , blackboard_Z ), the Voros symbol is defined by

B

T⁢dU⁢(η)=zβ⁢(z),𝑇subscript𝑑𝑈𝜂𝑧𝛽𝑧Td_{U}(\eta)=\frac{z}{\beta(z)},italic_T italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_η ) = divide start_ARG italic_z end_ARG start_ARG italic_β ( italic_z ) end_ARG ,

The Grothendieck’s version reduces to Hirzebruch’s one when Y𝑌Yitalic_Y is a point (see e.g. [20]). To describe its extension in the theory of complex cobordisms we recall that the characteristic Todd class T⁢dU⁢(ξ)𝑇subscript𝑑𝑈𝜉Td_{U}(\xi)italic_T italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_ξ ) of complex vector bundle over C⁢W𝐶𝑊CWitalic_C italic_W-complex X𝑋Xitalic_X with values in H∗⁢(X,ΩU⊗ℚ)superscript𝐻𝑋tensor-productsubscriptΩ𝑈ℚH^{*}(X,\Omega_{U}\otimes\mathbb{Q})italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X , roman_Ω start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ⊗ blackboard_Q ) is uniquely defined by the following properties (see [6]):

For any complex vector bundle ξ𝜉\xiitalic_ξ over C⁢W𝐶𝑊CWitalic_C italic_W-complex X𝑋Xitalic_X we have

The characteristic Todd class of complex vector bundle ξ𝜉\xiitalic_ξ over CW-complex X𝑋Xitalic_X is given by the formula

Let X𝑋Xitalic_X be a C⁢W𝐶𝑊CWitalic_C italic_W-complex and ξ→X→𝜉𝑋\xi\to Xitalic_ξ → italic_X be a complex vector bundle over X𝑋Xitalic_X.

B

χ𝜒\chiitalic_χ of N∩H𝑁𝐻N\cap Hitalic_N ∩ italic_H of degree one. In this section we will state and prove the main technical result allowing for this

From now on, let G𝐺Gitalic_G be a FAb compact p𝑝pitalic_p-adic analytic group and let N≤G𝑁𝐺N\leq Gitalic_N ≤ italic_G be a normal uniform subgroup. As in Section 5, let K≤G𝐾𝐺K\leq Gitalic_K ≤ italic_G be such that N≤K𝑁𝐾N\leq Kitalic_N ≤ italic_K and fix a pro-p𝑝pitalic_p Sylow subgroup Kpsubscript𝐾𝑝K_{p}italic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT of K𝐾Kitalic_K.

As in the previous section, let G𝐺Gitalic_G be a profinite group possessing a finite index normal pro-p𝑝pitalic_p subgroup N≤G𝑁𝐺N\leq Gitalic_N ≤ italic_G.

Let N𝑁Nitalic_N be a normal pro-p𝑝pitalic_p subgroup of a p𝑝pitalic_p-adic analytic group G𝐺Gitalic_G.

Let G𝐺Gitalic_G be a representation rigid profinite group, such that there exists a finite index normal pro-p𝑝pitalic_p subgroup N≤G𝑁𝐺N\leq Gitalic_N ≤ italic_G. For example, one can take G𝐺Gitalic_G to be FAb and compact p𝑝pitalic_p-adic analytic (see [9, Corollary 8.34]). For any K≤G𝐾𝐺K\leq Gitalic_K ≤ italic_G such that

B

We use the internal optimizer Adam in PyTorch with a learning rate l⁢r=0.01×0.5[j/1500]𝑙𝑟0.01superscript0.5delimited-[]𝑗1500lr=0.01\times 0.5^{[j/1500]}italic_l italic_r = 0.01 × 0.5 start_POSTSUPERSCRIPT [ italic_j / 1500 ] end_POSTSUPERSCRIPT, where j𝑗jitalic_j is the iterative times, and train the NN for 6000600060006000 epochs. After training, we achieve the train loss on 𝒟1trainsuperscriptsubscript𝒟1train\mathcal{D}_{1}^{\rm train}caligraphic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_train end_POSTSUPERSCRIPT being 1.3×10−31.3superscript1031.3\times 10^{-3}1.3 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and the test loss on 𝒟valsuperscript𝒟val\mathcal{D}^{\rm val}caligraphic_D start_POSTSUPERSCRIPT roman_val end_POSTSUPERSCRIPT being 1.6×10−31.6superscript1031.6\times 10^{-3}1.6 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT.

After the first round training, we find that the NN does not work well at some points, see e.g. Figure 2, by Monte Carlo test in Subsection 4.3.4. To improve the NN, we adaptively generate samples as in Subsection 4.3.3 with σ=0.1𝜎0.1\sigma=0.1italic_σ = 0.1, J1=5subscript𝐽15J_{1}=5italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 5 and L1=3subscript𝐿13L_{1}=3italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 3.

The training process takes approximately 5400540054005400 seconds on a ThinkPad T480s laptop without using GPU. Monte Carlo test as in Subsection 4.3.4 indicates that the trained NN works well.

Step 4. Model refinement. Based on the NN obtained in Step 2, we continue training the NN on the updated data set 𝒟2subscript𝒟2\mathcal{D}_{2}caligraphic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT as in Step 2. The training procedure stops once the NN satisfies a Monte Carlo test as in Subsection 4.3.4 below.

To obtain the error bounds as in Theorem 3.2, we use a Monte Carlo test. Specifically, we randomly select a certain number of initial points according to uniform distribution in ΩΩ\Omegaroman_Ω. Using the trained NN, pN⁢N⁢(θ,⋅)superscript𝑝𝑁𝑁𝜃⋅p^{NN}(\theta,\cdot)italic_p start_POSTSUPERSCRIPT italic_N italic_N end_POSTSUPERSCRIPT ( italic_θ , ⋅ ), we generate feedback control uN⁢Nsuperscript𝑢𝑁𝑁u^{NN}italic_u start_POSTSUPERSCRIPT italic_N italic_N end_POSTSUPERSCRIPT as (3.2), then check if the closed-loop trajectories is stabilized at these initial points by solving (3.4). If the trained NN works well, then the test error gives the error bounds in Theorem 3.2.

B

Suppose that Assumptions 1-3 hold. If ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0 is specified, CPCA ensures that every agent obtains an ϵitalic-ϵ\epsilonitalic_ϵ-optimal solution fe∗superscriptsubscript𝑓𝑒f_{e}^{*}italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT to problem (1), i.e., |fe∗−f∗|≤ϵsuperscriptsubscript𝑓𝑒superscript𝑓italic-ϵ\left|f_{e}^{*}-f^{*}\right|\leq\epsilon| italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_f start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | ≤ italic_ϵ.

We analyze the computational and communication complexities of the proposed algorithm. The following theorem establishes the orders of the numbers of zeroth-order queries (i.e., evaluations of values of local objective functions), inter-agent communication, and floating-point operations (i.e., flops) required for one agent. Suppose that one evaluation of the function value costs F0subscript𝐹0F_{0}italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT flops. In practice, this cost depends on the specific forms of objective functions[31].

The proof is provided in Appendix A-C. We summarize the costs of zeroth-order queries, inter-agent communication, and flops of the proposed algorithm in Table I. ∎

Table II compares the complexities of different algorithms. Though the algorithms in [21] and [24] handle multivariate problems, the comparison here is for the univariate case. In CPCA, the evaluations of function values are not required during iterations and are only performed when approximations are constructed. This design implies that the number of evaluations will not increase with the number of iterations. Hence, the cumulative costs of queries can be significantly reduced especially for large-scale networks. Due to the exchange of coefficient vectors, the number of elements communicated per iteration increases compared to existing algorithms. Nonetheless, the total communication cost (i.e. total communicated elements) can be acceptable given i) the decreased communication round and ii) the moderate degrees of approximations in practice, see Section V-A. Overall, CPCA is more suitable for problems where the number of rounds outweighs the number of elements in communication.

We characterize the complexities of zeroth-order queries, floating-point operations (i.e., flops111A flop is defined as one addition, subtraction, multiplication, or division of two floating-point numbers[31].), and inter-agent communication of the proposed algorithm. Thanks to its unique introduction of approximation and gradient-free iterations, CPCA is efficient in terms of communication rounds and queries, see the comparison in Section IV-B.

A

Suppose that p1,p2,..,pmp_{1},p_{2},..,p_{m}italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , . . , italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT are m𝑚mitalic_m integral polynomials and 0<δ<10𝛿10<\delta<10 < italic_δ < 1 any small real number. Then there exists a positive integer N⁢(m,δ)𝑁𝑚𝛿N(m,\delta)italic_N ( italic_m , italic_δ ) such that for all n≥N⁢(m,δ)𝑛𝑁𝑚𝛿n\geq N(m,\delta)italic_n ≥ italic_N ( italic_m , italic_δ ), and any set A⊂[n]𝐴delimited-[]𝑛A\subset[n]italic_A ⊂ [ italic_n ] of size at least δ⋅n⋅𝛿𝑛\delta\cdot nitalic_δ ⋅ italic_n, there exists a𝑎aitalic_a and 0<d≤n0𝑑𝑛0<d\leq n0 < italic_d ≤ italic_n such that the set {a}∪{a+pi⁢(d):1≤i≤m}𝑎conditional-set𝑎subscript𝑝𝑖𝑑1𝑖𝑚\{a\}\cup\{a+p_{i}(d):1\leq i\leq m\}{ italic_a } ∪ { italic_a + italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_d ) : 1 ≤ italic_i ≤ italic_m } is monochromatic.

Following Furstenberg, Katznelson’s original argument that showed that the multidimensional density Hales Jewett theorem follows from the density Hales Jewett theorem (Proposition 1.7 in [Pol12]), we show that the natural conjectured multidimensional analog of the polynomial density Hales Jewett theorem follows from the conjectured quadratic density Hales Jewett theorem.

Here we briefly outline how the conjectured Quadratic Density Hales Jewett theorem implies Sarkozy’s theorem[Sar78] which is a special case of the Polynomial Szemeredi theorem outlined above. By a similar argument to the one outlined below, one can also show that the Polynomial Density Hales Jewett theorem implies the Polynomial Szemeredi theorem.

There is a natural polynomial density Hales Jewett conjecture that generalizes Theorem 5, which we state below. It would be a generalization of both the Polynomial density Hales Jewett theorem as well as the Density Hales Jewett theorem. For k=2𝑘2k=2italic_k = 2, and only considering the quadratic term, we stated the corresponding version in 1.

This is the quadratic base case with k=2𝑘2k=2italic_k = 2 letters of the general Polynomial Density Hales Jewett conjecture stated later as 2. To state this conjecture formally, we first introduce some notation, which we essentially borrow from Walters [Wal00].

C

Let χ:ℝk→ℝ+:𝜒→superscriptℝ𝑘superscriptℝ\chi\colon\mathbb{R}^{k}\to\mathbb{R}^{+}italic_χ : blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT be a smooth cut-off function 0≤χ≤10𝜒10\leq\chi\leq 10 ≤ italic_χ ≤ 1, such that:

χT,R⁢(x):=χ⁢(T⁢(x)/R).assignsubscript𝜒𝑇𝑅𝑥𝜒𝑇𝑥𝑅\chi_{T,R}(x):=\chi(T(x)/R)\,.italic_χ start_POSTSUBSCRIPT italic_T , italic_R end_POSTSUBSCRIPT ( italic_x ) := italic_χ ( italic_T ( italic_x ) / italic_R ) .

and we shall often use the fact that |∇χ⁢(x)|2/χ⁢(x)≤2⁢sup‖D2⁢χ‖≤ρsuperscript∇𝜒𝑥2𝜒𝑥2supremumnormsuperscript𝐷2𝜒𝜌\lvert\nabla\chi(x)\rvert^{2}/\chi(x)\leq 2\sup\|D^{2}\chi\|\leq\rho| ∇ italic_χ ( italic_x ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_χ ( italic_x ) ≤ 2 roman_sup ∥ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_χ ∥ ≤ italic_ρ.

where G𝐺Gitalic_G is the positive, radial function G⁢(x′)=g⁢(|x′|)𝐺superscript𝑥′𝑔superscript𝑥′G(x^{\prime})=g(\lvert x^{\prime}\rvert)italic_G ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_g ( | italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ) defined by

χ⁢(x)𝜒𝑥\chi(x)italic_χ ( italic_x ) is a decreasing function of the radial variable r=|x|;𝑟𝑥r=\lvert x\rvert\,;italic_r = | italic_x | ;

D

The dimension of E⁢H¯b3⁢(𝒢Σ)superscriptsubscript¯𝐸𝐻𝑏3subscript𝒢Σ\overline{EH}_{b}^{3}(\mathcal{G}_{\Sigma})over¯ start_ARG italic_E italic_H end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( caligraphic_G start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT ) is uncountably infinite.

On the other hand, their result covers the case of surfaces with negative Euler characteristics. Therefore, in some sense, our results and theirs are complementary to each other in the case of 2-manifolds. Namely, we have the following theorem.

We also prove similar results for compact surfaces ΣΣ\Sigmaroman_Σ with non-negative Euler characteristic χ⁢(Σ)≥0𝜒Σ0\chi(\Sigma)\geq 0italic_χ ( roman_Σ ) ≥ 0. Let Bm⁢(Σ)subscript𝐵𝑚ΣB_{m}(\Sigma)italic_B start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( roman_Σ ) and Pm⁢(Σ)subscript𝑃𝑚ΣP_{m}(\Sigma)italic_P start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( roman_Σ ) denote the braid group and the pure braid group on a surface ΣΣ\Sigmaroman_Σ, respectively.

On the other hand, the bounded cohomology of non-positive curvature groups tends to be highly non-trivial. For example, the following theorem is known.

In this paper, we generalize Ishida’s result to higher-dimensional bounded cohomology for the case of three strands.

A

For d), let α∈C⁢Hi⁢(X)0𝛼𝐶superscript𝐻𝑖superscript𝑋0\alpha\in CH^{i}(X)^{0}italic_α ∈ italic_C italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_X ) start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and β∈C⁢Hd−i⁢(X)𝛽𝐶superscript𝐻𝑑𝑖𝑋\beta\in CH^{d-i}(X)italic_β ∈ italic_C italic_H start_POSTSUPERSCRIPT italic_d - italic_i end_POSTSUPERSCRIPT ( italic_X ). Choose an admissible π𝜋\piitalic_π. Writing [,][,][ , ] for the intersection product, we have [πη*⁢α,πη*⁢β]=0superscriptsubscript𝜋𝜂𝛼superscriptsubscript𝜋𝜂𝛽0[\pi_{\eta}^{*}\alpha,\pi_{\eta}^{*}\beta]=0[ italic_π start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_α , italic_π start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β ] = 0 by definition of C⁢Hi⁢(X1)0𝐶superscript𝐻𝑖superscriptsubscript𝑋10CH^{i}(X_{1})^{0}italic_C italic_H start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, hence [α,β]=0𝛼𝛽0[\alpha,\beta]=0[ italic_α , italic_β ] = 0.

To go further, we need to invert p𝑝pitalic_p in characteristic p𝑝pitalic_p; this is the object of the next subsections.

If we are in characteristic 00, the construction is finished since X𝑋Xitalic_X always admits a smooth model by resolution of singularities (Proposition 4.1). In characteristic p>0𝑝0p>0italic_p > 0, there turns out to be quite a bit of work to get a pairing in general after suitably inverting p𝑝pitalic_p, by using Gabber’s refinement of de Jong’s theorem: the general height pairing (4.1) is defined in Theorem 4.14; as said above, it makes sense in the category 𝐀𝐛⊗𝐙⁢[1/p]tensor-product𝐀𝐛𝐙delimited-[]1𝑝\operatorname{\mathbf{Ab}}\otimes\mathbf{Z}[1/p]bold_Ab ⊗ bold_Z [ 1 / italic_p ]. Functoriality and base change extend to this pairing (ibid.)

is recalled in §4.3. Here p𝑝pitalic_p is the exponential characteristic of k𝑘kitalic_k, so nothing is inverted in characteristic 00; the only reason to invert it in nonzero characteristic is a lack of resolution of singularities, see §4.1.

Suppose d=1𝑑1d=1italic_d = 1. In the pairing of Theorem 6.2 b), we have ⟨A⁢(k),A⁢(k)⟩⊆Pic0⁡(B)⁢{p}𝐴𝑘𝐴𝑘superscriptnormal-Pic0𝐵𝑝\langle A(k),A(k)\rangle\subseteq\operatorname{Pic}^{0}(B)\{p\}⟨ italic_A ( italic_k ) , italic_A ( italic_k ) ⟩ ⊆ roman_Pic start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_B ) { italic_p } in 𝐀𝐛⊗𝐐tensor-product𝐀𝐛𝐐\operatorname{\mathbf{Ab}}\otimes\mathbf{Q}bold_Ab ⊗ bold_Q, where p𝑝pitalic_p is the exponential characteristic of k𝑘kitalic_k.

A

We remark also that the expression of a statement as a first-order sentence in A⁢C⁢F0𝐴𝐶subscript𝐹0ACF_{0}italic_A italic_C italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a very important question. One of the main applications of this idea is Lefchetz’s Principle from algebraic geometry, since A⁢C⁢F0𝐴𝐶subscript𝐹0ACF_{0}italic_A italic_C italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a complete theory ([20]), in order to prove a statement for a variety over an algebraically closed field of zero characteristic, it suffices to show it for 𝗄=ℂ𝗄ℂ{\mathsf{k}}=\mathbb{C}sansserif_k = blackboard_C, where transcendental methods are appliable. The Gelfand-Kirillov Conjecture is obviously in the realm of noncommutative algebraic geometry, but as a illustration of our result, the validity of the Gelfand-Kirillov Conjecture for simple Lie algebras of type C or G, where it is still open, over any algebraically closed field of zero characteristic, is equivalent to its validity over ℂℂ\mathbb{C}blackboard_C, where we can use complex Lie group techniques and other analytic and geometric theories. We remark, however, that since of course not every geometric aspect can be expressed in first-order logic, there is a need to be careful (cf. [33])4 ††4 However, the general feeling that Lefchet’z holds for any reasonably simple statement in algebraic geometry has been formalized in the work [13].

J. S. was supported in part by FAPESP grant 2018/18146-5. The second author is grateful to M. Soares and S. Hoffmann Panucci.

The Conjecture was shown to be true for 𝔤⁢𝔩n𝔤subscript𝔩𝑛\mathfrak{gl}_{n}fraktur_g fraktur_l start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and 𝔰⁢𝔩n𝔰subscript𝔩𝑛\mathfrak{sl}_{n}fraktur_s fraktur_l start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, as well for nilpotent Lie algebras, in [16]. This later fact was generalized for solvable Lie algebras in [28], [23], [6]. Other cases considered in [30] [4], [31]; and certain modifications of it considered in [17] and [11]. The Conjecture was very influential in the development of Lie theory (cf. [12, Problem 3]) and, recently, a q𝑞qitalic_q-analogue of the Conjecture is studied in quantum group theory ([9, I.2.11, II.10.4]). It also became a paradigmatic example on the study of skew field of fractions of many Ore domains (cf. [15], [1], [2]). However, it was eventually shown that the Conjecture is false in general ([3]). Regarding simple Lie algebras, it was known to be true in the seminal work of Gelfand and Kirillov for type A𝐴Aitalic_A simple Lie algebras. This question was revisited by Premet in [32], where using reduction module prime techniques, he showed the Conjecture to be false for types B,D,E,F𝐵𝐷𝐸𝐹B,D,E,Fitalic_B , italic_D , italic_E , italic_F. About the types C,G𝐶𝐺C,Gitalic_C , italic_G, nothing is known at this moment.

In the 1966 the study of birational geometry of noncommutative objects began. In his adress at the 1966 ICM in Moscow, A. A. Kirillov proposed to classify, up to birational equivalence, the enveloping algebras U⁢(𝔤)𝑈𝔤U(\mathfrak{g})italic_U ( fraktur_g ) of finite dimensional algebraic Lie algebras 𝔤𝔤\mathfrak{g}fraktur_g when 𝗄𝗄\mathsf{k}sansserif_k is algebraically closed of zero characteristic. This means to find canonical division rings such that every skew field Frac⁡U⁢(𝔤)Frac𝑈𝔤\operatorname{Frac}\,U(\mathfrak{g})roman_Frac italic_U ( fraktur_g ) of the enveloping algebras, which are an Ore domain1 ††1 In general, it is not the case that a noncommutative domain can be embedded in a division ring, as shown by Malcev. Ore domains are an example of when a quotient division ring exists in a particularly nice form, cf. [25, Chapter 4]., is isomorphic to one of them.

By the above Theorem (the notation which we use here), X𝗄≠∅superscript𝑋𝗄X^{\mathsf{k}}\neq\varnothingitalic_X start_POSTSUPERSCRIPT sansserif_k end_POSTSUPERSCRIPT ≠ ∅. Hence X𝗄⁢(𝔸)≠∅superscript𝑋𝗄𝔸X^{\mathsf{k}}(\mathbb{A})\neq\varnothingitalic_X start_POSTSUPERSCRIPT sansserif_k end_POSTSUPERSCRIPT ( blackboard_A ) ≠ ∅, because X𝑋Xitalic_X is defined by polynomials with integer coefficients. The process of Theorem 2.7 is reversible, and relation (9) of it reffers to the ground field; hence, we can assume that elements in (†),(‡)†‡(\dagger),\,(\ddagger)( † ) , ( ‡ ) belongs to Frac⁡U⁢(𝔤𝔸,Σ)Frac𝑈subscript𝔤𝔸Σ\operatorname{Frac}\,U(\mathfrak{g}_{\mathbb{A},\Sigma})roman_Frac italic_U ( fraktur_g start_POSTSUBSCRIPT blackboard_A , roman_Σ end_POSTSUBSCRIPT ). Since the elements x1,…,xnsubscript𝑥1…subscript𝑥𝑛x_{1},\ldots,x_{n}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT also belong to 𝔤ℤ,Σsubscript𝔤ℤΣ\mathfrak{g}_{\mathbb{Z},\Sigma}fraktur_g start_POSTSUBSCRIPT blackboard_Z , roman_Σ end_POSTSUBSCRIPT and, by definition, are a basis of this vector space, then we can apply Proposition 2.6 to conclude that Gelfand-Kirillov Conjecture holds for U⁢(𝔤𝔸,Σ)𝑈subscript𝔤𝔸ΣU(\mathfrak{g}_{\mathbb{A},\Sigma})italic_U ( fraktur_g start_POSTSUBSCRIPT blackboard_A , roman_Σ end_POSTSUBSCRIPT ).

A

F⁢(L)=(π2−1)−L29⁢r2+o⁢(L2).𝐹𝐿𝜋21superscript𝐿29superscript𝑟2osuperscript𝐿2F(L)=\left(\frac{\pi}{2}-1\right)-\frac{L^{2}}{9r^{2}}+{\rm o}(L^{2}).italic_F ( italic_L ) = ( divide start_ARG italic_π end_ARG start_ARG 2 end_ARG - 1 ) - divide start_ARG italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 9 italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + roman_o ( italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .

Change variables and use the series expansions of tan\tanroman_tan and cot\cotroman_cot together with Proposition 3.1 to obtain the expansion

and use (4.6), (4.7), (4.2), and (4.9) to obtain the expansion up to quadratic terms in L𝐿Litalic_L of the Buffon probability for ℍksubscriptℍ𝑘\mathds{H}_{k}blackboard_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.

This section presents second order expansions in the parameter L𝐿Litalic_L for the probabilities given by (3.2) and (3.8). Zeroth-order terms give the probability in the planar case (which is independent of L𝐿Litalic_L), first-order terms vanish, and second-order terms depend on the Gaussian curvature of the space. The following subsections utilize the standard ‘little-o’ notation to simplify expressions, where o(L)o𝐿\operatorname*{o}(L)roman_o ( italic_L ) describes the behavior of a function as L𝐿Litalic_L tends to 00.

This expansion of F⁢(L)𝐹𝐿F(L)italic_F ( italic_L ) to quadratic order together with (4.1) gives the desired expansion for the Buffon probability.

D

The (boolean) ring ℛℛ\mathcal{R}caligraphic_R of definable sets contained in the subgroup ⟨X⟩delimited-⟨⟩𝑋\langle{X}\rangle⟨ italic_X ⟩ generated by X𝑋Xitalic_X, that is, the subcollection ℛℛ\mathcal{R}caligraphic_R is closed under finite unions and relative set-theoretic differences;

Note that the subgroup ⟨X⟩delimited-⟨⟩𝑋\langle{X}\rangle⟨ italic_X ⟩ generated by the subset X𝑋Xitalic_X need not be definable, but it is

The (boolean) ring ℛℛ\mathcal{R}caligraphic_R of definable sets contained in the subgroup ⟨X⟩delimited-⟨⟩𝑋\langle{X}\rangle⟨ italic_X ⟩ generated by X𝑋Xitalic_X, that is, the subcollection ℛℛ\mathcal{R}caligraphic_R is closed under finite unions and relative set-theoretic differences;

The subgroup ⟨X⟩delimited-⟨⟩𝑋\langle{X}\rangle⟨ italic_X ⟩ is definable if and only if μ⁢(⟨X⟩)𝜇delimited-⟨⟩𝑋\mu(\langle{X}\rangle)italic_μ ( ⟨ italic_X ⟩ )

Note that each coset of the subgroup ⟨X⟩M00subscriptsuperscriptdelimited-⟨⟩𝑋00𝑀\langle{X}\rangle^{00}_{M}⟨ italic_X ⟩ start_POSTSUPERSCRIPT 00 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT of Definition 2.5 is type-definable over M𝑀Mitalic_M and

A

We are particularly grateful towards Seidai Yasuda for giving us access to the newest version of his article with Go Yamashita.

The first named author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “K𝐾Kitalic_K-theory, algebraic cycles and motivic homotopy theory” when work on this paper was undertaken.

We thank Kenichi Bannai and Yoshinosuke Hirakawa for their helpful comments to the draft of this paper.

We would like to thank Kei Hagihara and Shuji Yamamoto for helpful discussions concerning extension groups of (φ,N)𝜑𝑁(\varphi,N)( italic_φ , italic_N )-modules.

We would like to thank Wiesława Nizioł for helpful discussions related to the subject of this paper.

D

S−1⁢(⋅)superscript𝑆1⋅S^{-1}(\ \cdot\ )italic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ⋅ ) is exact.

Assume A𝐴Aitalic_A is an algebra with a distinguished non-zero element a∈Z⁢(A)𝑎𝑍𝐴a\in Z(A)italic_a ∈ italic_Z ( italic_A ) and an

An algebra C𝐶Citalic_C together with a algebra monomorphism A→C→𝐴𝐶A\to Citalic_A → italic_C is called an algebra

an algebra with a fixed automorphism σ𝜎\sigmaitalic_σ. Thus we obtain a 𝕋𝕋\mathbb{T}blackboard_T-module algebra

h∈Hℎ𝐻h\in Hitalic_h ∈ italic_H. We also assume A𝐴Aitalic_A is an H𝐻Hitalic_H-module algebra, i.e. A𝐴Aitalic_A is an algebra and a left

B

2.5. Brownian motion in ℚpsubscriptℚ𝑝\mathds{Q}_{p}blackboard_Q start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT

Authors have traditionally referred to the type of processes that the present article studies as p𝑝pitalic_p-adic ultrametric diffusion processes [4, 9]. However, such processes may potentially be more general than the processes that the present work studies. Reference to the real time p𝑝pitalic_p-adic Brownian motion as a p𝑝pitalic_p-adic Brownian motion agrees with the general practice of referring to a real time analog of Brownian motion in a space other than ℝℝ\mathds{R}blackboard_R as a Brownian motion in the space, but it introduces some ambiguity with respect to the parameter space for those familiar with p𝑝pitalic_p-adic mathematical physics. In particular, Bikulov and Volovich introduced in their seminal work [8] a very different stochastic process, parameterized by a p𝑝pitalic_p-adic time variable, that they also referred to as p𝑝pitalic_p-adic Brownian motion. In some respects, such a process is the true analog of real Brownian motion. To clarify, the p𝑝pitalic_p-adic Brownian motion that the current paper studies is a stochastic process with a real time parameter and a p𝑝pitalic_p-adic state space.

The two principle ideas that motivate the study of p𝑝pitalic_p-adic mathematical physics are the idea that non-Archimedean physical models could describe the observed ultrametricity in certain complex systems and the idea that spacetime could have a non-Archimedean structure at extremely small distance and time scales. Ultrametric structures in spin glasses were already implicit in Parisi’s early works [19, 20]. Avetisov, Bikulov, and Kozyrev [3] and Parisi and Sourlas [21] independently proposed p𝑝pitalic_p-adic models to describe replica symmetry breaking. The understanding that there are fundamental limitations on physical measurements below the Planck scale motivated Volovich to introduce the Volovich hypothesis [27, Chapter 6], the idea that physical theories that deal with such ultramicroscopic measurements should involve non-Archimedean local fields [33]. The review of Dragovich, Khrennikov, Kozyrev, and Volovich [13] and the more recent work of Khrennikov, Kozyrev, and Zúñiga-Galindo [16] both give excellent accounts of the development of p𝑝pitalic_p-adic mathematical physics and list a variety of areas of potential application for the results of the current paper. A significant application of p𝑝pitalic_p-adic Brownian motion to physics that has experimental confirmation is the utilization by Avetisov, Bikulov, and Zubarev [5] and by Bikulov and Zubarev [10] of the Vladimirov equation in the description of protein molecule dynamics. The current work could also permit a study from a discrete perspective of the work of Khrennikov and Kochubei [15] on a p𝑝pitalic_p-adic analog of the porous medium equation, as well as its generalization by the previous two authors with Antoniouk [2]. It would also be of interest to extend the finite approximation of non-Archimedian quantum systems [6] to a discrete time setting.

See the books by Gouvêa [14] and by Ramakrishnan and Valenza [22] for further background on the p𝑝pitalic_p-adic numbers that this subsection summarizes. Refer to the book by Vladimirov, Volovich, and Zelenov [32] for both an introduction to p𝑝pitalic_p-adic mathematical physics and for many helpful examples of integration over ℚpsubscriptℚ𝑝\mathds{Q}_{p}blackboard_Q start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT.

For further background on parabolic equations over non-Archimedean fields and the theory of p𝑝pitalic_p-adic diffusion, refer to the books by Kochubei [18] and Zúñiga-Galindo [35]. The latter reference discusses diffusion processes of a more general type than those to be presently studied.

D

An example of a query that can be expressed in GQ, but not in Modeq can be found in [18] (Example 2).

We recall that, for a structure 𝔅𝔅\mathfrak{B}fraktur_B with a finite relational signature τ𝜏\tauitalic_τ, the constraint satisfaction problem for 𝔅𝔅\mathfrak{B}fraktur_B is the class of all finite τ𝜏\tauitalic_τ-structures that homomorphically map to 𝔅𝔅\mathfrak{B}fraktur_B.

Certain flag-and-check programs have been studied in the context of the complexity of constraint satisfaction under the name peek arc consistency, extending the famous (hyper-) arc consistency procedure [9]. In the following, τ𝜏\tauitalic_τ denotes a finite relational signature.

The following is based on an example from [22] of a structure such that the complement of the CSP of this structure cannot be solved by PAC, but can be solved by singleton arc consistency (SAC), and hence is in Datalog. We will show below that it is even in Modeq.

It is known that singleton linear arc consistency (SLAC) [36] captures the intersection of finite-domain CSPs with Datalog. It would be interesting to define an appropriate notion of singleton linear arc consistency for

B

We start with basic definitions and results in Section 2. We introduce the concepts of computability and the concepts of the zero-error capacity of noisy channels.

We will examine this in Section 3. We also compare the algorithmic relationship between Shannon’s characterization of the zero-error capacity of DMCs and Ahlswede’s characterization. We show that the zero-error capacity of noisy channels cannot be computed algorithmically.

Furthermore, we also investigate the relationship between the zero-error capacity of channels, the Shannon capacity of graphs, and Ahlswede’s

In summary, there are two approaches to estimate the zero-error capacity of noisy channels known today. Shannon couples the zero-error capacity of DMCs with the Shannon capacity of graphs.

Furthermore, we analyze the algorithmic relationship of Shannon’s characterization of the zero-error capacity of DMCs with Ahlswede’s characterization. We show that the zero-error capacity cannot be computed

B

\bm{\theta}\|_{\sharp}.roman_obj ( italic_B , bold_italic_θ ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - ⟨ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_B ⟩ - square-root start_ARG italic_n end_ARG italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ∥ italic_B ∥ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT + ∥ bold_italic_θ ∥ start_POSTSUBSCRIPT ♯ end_POSTSUBSCRIPT .

In the present paper, we also derive a sharper error bound than [65] about robust matrix compressed sensing and robust matrix completion under a weaker condition.

Therefore, in Section 4, we discuss the main theorem, adversarial matrix compressed sensing and adversarial matrix completion.

[65] dealt with robust matrix compressed sensing, robust matrix completion, trace regression with matrix decomposition.

[34] considered regression problems including matrix compressed sensing, lasso, matrix completion and reduced-rank regression with heavy-tailed covariates and random noises, proposing a new ‘shrinkage’ estimator.

C

When the bonding maps and index set are clear from context, the inverse system (Gα,pαα′)subscript𝐺𝛼superscriptsubscript𝑝𝛼superscript𝛼′(G_{\alpha},p_{\alpha}^{\alpha^{\prime}})( italic_G start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) may be denoted by (Gα)α∈𝒜subscriptsubscript𝐺𝛼𝛼𝒜(G_{\alpha})_{\alpha\in\mathcal{A}}( italic_G start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_α ∈ caligraphic_A end_POSTSUBSCRIPT or (Gα)αsubscriptsubscript𝐺𝛼𝛼(G_{\alpha})_{\alpha}( italic_G start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT or (Gα)subscript𝐺𝛼(G_{\alpha})( italic_G start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ).

An inverse system indexed by the natural numbers is an inverse sequence. An inverse system indexed by a singleton is a rudimentary system and is denoted by (G)𝐺(G)( italic_G ), where G𝐺Gitalic_G is its only term.

The inverse sequences in the second and third columns of (6.1) are stable since each is pro-isomorphic to a rudimentary inverse system.

The inverse limit of an inverse system of groups (Gα)subscript𝐺𝛼(G_{\alpha})( italic_G start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) is the group

Since the first row of (6.1) is an exact sequence of stable inverse systems, its inverse limit is again exact by Lemma 2.9.

A

While it is already clear that our approach is beneficial over Umeansubscript𝑈meanU_{\mathrm{mean}}italic_U start_POSTSUBSCRIPT roman_mean end_POSTSUBSCRIPT in terms of robust type I error control, we also claim that our test maintains a good power property. In particular we show that the asymptotic power of Tmeansubscript𝑇meanT_{\mathrm{mean}}italic_T start_POSTSUBSCRIPT roman_mean end_POSTSUBSCRIPT is only worse by a factor of 22\sqrt{2}square-root start_ARG 2 end_ARG than that of Umeansubscript𝑈meanU_{\mathrm{mean}}italic_U start_POSTSUBSCRIPT roman_mean end_POSTSUBSCRIPT under the setting where both test statistics converge to Gaussian. In a more general setting, Theorem 2.8 proves that our test is minimax rate optimal in terms of the L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT norm, which means that the power of the proposed test cannot be improved beyond a constant factor. Before we proceed, let us describe the following assumptions that facilitate our asymptotic power analysis.

We now study the problem of covariance testing under Gaussian assumptions as another application of our approach via sample-splitting and studentization, though we show how to relax the Gaussianity assumption later. Let X1,…,Xnsubscript𝑋1…subscript𝑋𝑛X_{1},\ldots,X_{n}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT be i.i.d. d𝑑ditalic_d-dimensional random vectors from a multivariate normal distribution N⁢(0,Σ)𝑁0ΣN(0,\Sigma)italic_N ( 0 , roman_Σ ). Given this Gaussian sample, we are concerned with testing whether

Gaussianity: X𝑋Xitalic_X has a multivariate Gaussian distribution N⁢(μ,Σ)𝑁𝜇ΣN(\mu,\Sigma)italic_N ( italic_μ , roman_Σ ).

\mu^{\top}\mu}{2\sqrt{\text{tr}(\Sigma^{2})}}\right)+o(1).blackboard_E start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT [ italic_ϕ start_POSTSUBSCRIPT roman_mean end_POSTSUBSCRIPT ] = roman_Φ ( italic_z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT + divide start_ARG italic_n italic_μ start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_μ end_ARG start_ARG 2 square-root start_ARG tr ( roman_Σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG end_ARG ) + italic_o ( 1 ) .

Local alternatives: μ⊤⁢μ=O⁢(d/n)superscript𝜇top𝜇𝑂𝑑𝑛\mu^{\top}\mu=O(\sqrt{d}/n)italic_μ start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT italic_μ = italic_O ( square-root start_ARG italic_d end_ARG / italic_n ).

B

}})}(\pi_{K_{I}})}.roman_E start_POSTSUPERSCRIPT italic_k , italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∞ , ⋂ start_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT roman_an , [ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] × [ italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] , italic_I , roman_Π start_POSTSUBSCRIPT roman_an , [ italic_t start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT , ∞ ] , italic_I , italic_A end_POSTSUBSCRIPT ( roman_Γ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT .

This implies that the cohomologies of the double complexes for this layer □msubscript□𝑚\square_{m}□ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT are actually preserved when we change the boxes. That is to say the totalization of this layer 𝐓𝐨𝐭𝐚𝐥⁢(□m)𝐓𝐨𝐭𝐚𝐥subscript□𝑚\mathbf{Total}(\square_{m})bold_Total ( □ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) will be quasi-isomorphic when we change boxes. Then we consider the double complex 𝐓𝐨𝐭𝐚𝐥⁢(□∙)𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}(\square_{\bullet})bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ), then a second stage of the comparison of the ∞\infty∞-pages of the spectral sequence will show eventually that the totalization 𝐓𝐨𝐭𝐚𝐥⁢(𝐓𝐨𝐭𝐚𝐥⁢(□∙))𝐓𝐨𝐭𝐚𝐥𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}\left(\mathbf{Total}(\square_{\bullet})\right)bold_Total ( bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) ) of 𝐓𝐨𝐭𝐚𝐥⁢(□∙)𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}(\square_{\bullet})bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) will have the isomorphic cohomology groups when we change boxes. Then for the remaining comparison we argue in the parallel way.

This implies that the cohomologies of the original double complexes are actually preserved when we change the boxes. Then for the remaining comparison we argue in the parallel way.

This implies that the cohomologies of the original double complexes are actually preserved when we change the boxes. Then for the remaining similar comparisons in this articles in the following discussion we argue in the parallel way.

This implies that the cohomologies of the double complexes for this layer □msubscript□𝑚\square_{m}□ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT are actually preserved when we change the boxes. That is to say the totalization of this layer 𝐓𝐨𝐭𝐚𝐥⁢(□m)𝐓𝐨𝐭𝐚𝐥subscript□𝑚\mathbf{Total}(\square_{m})bold_Total ( □ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) will be quasi-isomorphic when we change boxes. Then we consider the double complex 𝐓𝐨𝐭𝐚𝐥⁢(□∙)𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}(\square_{\bullet})bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ), then a second stage of the comparison of the ∞\infty∞-pages of the spectral sequence will show eventually that the totalization 𝐓𝐨𝐭𝐚𝐥⁢(𝐓𝐨𝐭𝐚𝐥⁢(□∙))𝐓𝐨𝐭𝐚𝐥𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}\left(\mathbf{Total}(\square_{\bullet})\right)bold_Total ( bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) ) of 𝐓𝐨𝐭𝐚𝐥⁢(□∙)𝐓𝐨𝐭𝐚𝐥subscript□∙\mathbf{Total}(\square_{\bullet})bold_Total ( □ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) will have the isomorphic cohomology groups when we change boxes. Then for the remaining comparison we argue in the parallel way.

B

For a graph ΓΓ\Gammaroman_Γ, the signless Laplacian Q⁢(Γ)=D⁢(Γ)+A⁢(Γ)𝑄Γ𝐷Γ𝐴ΓQ(\Gamma)=D(\Gamma)+A(\Gamma)italic_Q ( roman_Γ ) = italic_D ( roman_Γ ) + italic_A ( roman_Γ ) has received a growing amount of attention. When finding upper bounds for the Kirkohff Laplacian spectral radius of a graph, it turns out that signless Laplacian can be used since λn⁢(K⁢(Γ))≤λn⁢(Q⁢(Γ))subscript𝜆𝑛𝐾Γsubscript𝜆𝑛𝑄Γ\lambda_{n}(K(\Gamma))\leq\lambda_{n}(Q(\Gamma))italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K ( roman_Γ ) ) ≤ italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_Q ( roman_Γ ) ). This universal upper bound extends to more general settings of signed graphs [27] and 𝕋𝕋\mathbb{T}blackboard_T-gain graphs [20]. More recently, this has also been generalized to the setting of oriented hypergraphs [7]. This further generalizes to complex unit hypergraphs, where all of the above structures can be viewed as specializations of the first inequality in the following theorem.

Let G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be the underlying hypergraph of G𝐺Gitalic_G. Then,

Similarly, let 𝐲=(y1,…,yn)∈ℂn𝐲subscript𝑦1…subscript𝑦𝑛superscriptℂ𝑛\mathbf{y}=(y_{1},\ldots,y_{n})\in\mathbb{C}^{n}bold_y = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be a unit eigenvector of K⁢(G′)𝐾superscript𝐺′K(G^{\prime})italic_K ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) with corresponding eigenvalue λn⁢(K⁢(G′))subscript𝜆𝑛𝐾superscript𝐺′\lambda_{n}(K(G^{\prime}))italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ). Then

Let M∈ℂn×n𝑀superscriptℂ𝑛𝑛M\in\mathbb{C}^{n\times n}italic_M ∈ blackboard_C start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT be Hermitian. Then

The underlying hypergraph of G=(V,E,ℐ,ω)𝐺𝑉𝐸ℐ𝜔G=(V,E,\mathcal{I},\omega)italic_G = ( italic_V , italic_E , caligraphic_I , italic_ω ) is G′:=(V,E,ℐ,ω′)assignsuperscript𝐺′𝑉𝐸ℐsuperscript𝜔′G^{\prime}:=(V,E,\mathcal{I},\omega^{\prime})italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := ( italic_V , italic_E , caligraphic_I , italic_ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), where

A

\end{array}\right.{ start_ARRAY start_ROW start_CELL divide start_ARG ∂ italic_u end_ARG start_ARG ∂ italic_t end_ARG = divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_u , italic_x ∈ ( 0 , 2 italic_π ) , italic_t ≥ 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_u ( italic_x , 0 ) = italic_f ( italic_x ) end_CELL start_CELL end_CELL end_ROW end_ARRAY

For the non-periodic boundary conditions, as for the two-dimensional case, to approximate the values at the boundary points of the cube [0,π]×[0,π]×[0,π]0𝜋0𝜋0𝜋[0,\pi]\times[0,\pi]\times[0,\pi][ 0 , italic_π ] × [ 0 , italic_π ] × [ 0 , italic_π ], we perform an extrapolation to obtain the additional ghost points. The formulation for these extrapolations and the proofs are similar to the one- and two-dimensional cases.

The domain is divided into cells, I(j,k,l)subscript𝐼𝑗𝑘𝑙I_{(j,k,l)}italic_I start_POSTSUBSCRIPT ( italic_j , italic_k , italic_l ) end_POSTSUBSCRIPT, where

We assume the following mesh to cover the computational domain [0,2⁢π]02𝜋[0,2\pi][ 0 , 2 italic_π ], consisting of cells

The following operator includes the net contribution viewed as the difference between penalties from each side of the left border of the cell Ijsubscript𝐼𝑗I_{j}italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, the numerical flux applied to the same node at node xj−1/2subscript𝑥𝑗12x_{j-1/2}italic_x start_POSTSUBSCRIPT italic_j - 1 / 2 end_POSTSUBSCRIPT, and the contribution from the integration over both cells when we set v=u𝑣𝑢v=uitalic_v = italic_u.

C

Table 4: The sample standard deviation of the MESES for each nonparametric estimator fitted to the simulated continuous regression function data.

Table 5: The mean of the MESEs for each nonparametric estimator fitted to the simulated continuous data with a jump in the regression function.

Table 3: The mean of the MESEs for each nonparametric estimator fitted to the simulated continuous regression function data.

Table 1: The mean of the MESEs for each nonparametric estimator fitted to the simulated piecewise constant regression function data.

Table 6: The sample standard deviation of the MESES for each nonparametric estimator fitted to the simulated continuous data with a jump in the regression function.

A

As a result, a tremendous amount of work has gone into the development of effective methods for this problem;

on nonhomogeneous random matrices (Seginer, 2000; Latała, 2005; Schuett and Riemer, 2013; Bandeira and Van Handel, 2016; Van Handel, 2017; Latała et al., 2018).

This requires showing a generalized Marčenko-Pastur law under relaxed independence conditions, related to, but different from those of Hui and Pan (2010); Wei et al. (2016); Bryson et al. (2021),

(Buja and Eyuboglu, 1992; Bai et al., 2018; Hu et al., 2020; Fan et al., 2020; Ke et al., 2021; Landa et al., 2022).

(Kapetanios, 2004, 2010; Kritchman and Nadler, 2009; Onatski, 2010; Lam and Yao, 2012; Ahn and Horenstein, 2013; Passemier and Yao, 2014; Li et al., 2017).

D

\mathbf{K}=\mathbf{H}(\mathbf{R}).bold_xx start_POSTSUPERSCRIPT ⊺ end_POSTSUPERSCRIPT ( bold_KR ) start_POSTSUPERSCRIPT ⊺ end_POSTSUPERSCRIPT bold_QK = bold_H ( bold_R ) .

𝐇𝐇\mathbf{H}bold_H as the matrix acting on 𝐕𝐞𝐜⁡(𝐑)𝐕𝐞𝐜𝐑\operatorname{\mathbf{Vec}}(\mathbf{R})bold_Vec ( bold_R ) we have

In order to represent 𝐇𝐇\mathbf{H}bold_H as a matrix, we apply the operator 𝐕𝐞𝐜𝐕𝐞𝐜\operatorname{\mathbf{Vec}}bold_Vec to both sides of the above equation to get

We provide bounds on the eigenvalues (from below and above) of 𝐇𝐇\mathbf{H}bold_H which are necessary

where 𝐕𝐞𝐜𝐕𝐞𝐜\operatorname{\mathbf{Vec}}bold_Vec takes a matrix and outputs its columns stacked. Considering contributions from each 𝐱(n)superscript𝐱𝑛\mathbf{x}^{(n)}bold_x start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT,

B

Similarly, if M𝑀Mitalic_M is 2222-dimensional we can replace M𝑀Mitalic_M by T2×Msuperscript𝑇2𝑀T^{2}\times Mitalic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_M.

Put n~:=dim(N)assign~𝑛dimension𝑁\tilde{n}:=\dim(N)over~ start_ARG italic_n end_ARG := roman_dim ( italic_N ).

Taking traces, we conclude Hg⁢(λ)=λ−1⁢dimM−1dimW+dimN−1⁢Hgb>0subscript𝐻𝑔𝜆superscript𝜆1dimension𝑀1dimension𝑊dimension𝑁1subscript𝐻subscript𝑔𝑏0H_{g(\lambda)}=\lambda^{-1}\frac{\dim M-1}{\dim W+\dim N-1}H_{g_{b}}>0italic_H start_POSTSUBSCRIPT italic_g ( italic_λ ) end_POSTSUBSCRIPT = italic_λ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT divide start_ARG roman_dim italic_M - 1 end_ARG start_ARG roman_dim italic_W + roman_dim italic_N - 1 end_ARG italic_H start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT > 0 for all λ>0𝜆0\lambda>0italic_λ > 0.

Pick a Riemannian metric gWsubscript𝑔𝑊g_{W}italic_g start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT on W𝑊Witalic_W with IIgW=C⋅(gW)0subscriptIIsubscript𝑔𝑊⋅𝐶subscriptsubscript𝑔𝑊0\mathrm{II}_{g_{W}}=C\cdot(g_{W})_{0}roman_II start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_C ⋅ ( italic_g start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT for some constant C>dimF+dimW−1dimW−1⋅h0𝐶⋅dimension𝐹dimension𝑊1dimension𝑊1subscriptℎ0C>\frac{\dim F+\dim W-1}{\dim W-1}\cdot h_{0}italic_C > divide start_ARG roman_dim italic_F + roman_dim italic_W - 1 end_ARG start_ARG roman_dim italic_W - 1 end_ARG ⋅ italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT.

Thus there is no loss of generality in assuming that n=dim(M)≥4𝑛dimension𝑀4n=\dim(M)\geq 4italic_n = roman_dim ( italic_M ) ≥ 4.

D

There exists a finite abelian group scheme G𝐺Gitalic_G over k𝑘kitalic_k and a ramified G𝐺Gitalic_G-cover X→S→𝑋𝑆X\to Sitalic_X → italic_S with birational building datum (𝒜,ℒ)𝒜ℒ(\mathcal{A},\mathcal{L})( caligraphic_A , caligraphic_L );

-1.0pt{\emph{iv}}}}_{\mathscr{X}_{\textsf{ét}}}caligraphic_E : italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT → start_OPFUNCTION script_D iv end_OPFUNCTION start_POSTSUBSCRIPT script_X start_POSTSUBSCRIPT ét end_POSTSUBSCRIPT end_POSTSUBSCRIPT be the universal Deligne–Faltings object. Any vector bundle on 𝒢s¯subscript𝒢¯𝑠\mathcal{G}_{\bar{s}}caligraphic_G start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT is a direct sum of ℰλ|𝒢s¯evaluated-atsubscriptℰ𝜆subscript𝒢¯𝑠\mathcal{E}_{\lambda}|_{\mathcal{G}_{\bar{s}}}caligraphic_E start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT | start_POSTSUBSCRIPT caligraphic_G start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT for λ∈𝒜s¯𝜆subscript𝒜¯𝑠\lambda\in\mathcal{A}_{\bar{s}}italic_λ ∈ caligraphic_A start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT. By Proposition [BB17, Proposition 3.18], (1) holds if and only if every vector bundle on 𝒢s¯≅B⁢𝒜s¯subscript𝒢¯𝑠𝐵subscript𝒜¯𝑠\mathcal{G}_{\bar{s}}\cong B\mathcal{A}_{\bar{s}}caligraphic_G start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT ≅ italic_B caligraphic_A start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT is a quotient of a subbundle of ℱ|𝒢s¯evaluated-atℱsubscript𝒢¯𝑠\mathcal{F}|_{\mathcal{G}_{\bar{s}}}caligraphic_F | start_POSTSUBSCRIPT caligraphic_G start_POSTSUBSCRIPT over¯ start_ARG italic_s end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT,

For every geometric point s¯¯𝑠\bar{s}over¯ start_ARG italic_s end_ARG in the branch locus, we have that

For every geometric point s¯¯𝑠\bar{s}over¯ start_ARG italic_s end_ARG in the branch locus, we have that

Let 𝒫→𝒬→𝒫𝒬\mathcal{P}\to\mathcal{Q}caligraphic_P → caligraphic_Q be a homomorphism of fine sheaves of monoids. The stack S𝒫,𝒬,ℒsubscript𝑆𝒫𝒬ℒS_{\mathcal{P},\mathcal{Q},\mathcal{L}}italic_S start_POSTSUBSCRIPT caligraphic_P , caligraphic_Q , caligraphic_L end_POSTSUBSCRIPT is flat over S𝑆Sitalic_S if for every geometric point x¯∈S¯𝑥𝑆\bar{x}\in Sover¯ start_ARG italic_x end_ARG ∈ italic_S, the morphism 𝒫x¯a→𝒬x¯a→superscriptsubscript𝒫¯𝑥𝑎superscriptsubscript𝒬¯𝑥𝑎\mathcal{P}_{\bar{x}}^{a}\to\mathcal{Q}_{\bar{x}}^{a}caligraphic_P start_POSTSUBSCRIPT over¯ start_ARG italic_x end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT → caligraphic_Q start_POSTSUBSCRIPT over¯ start_ARG italic_x end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT is integral.

B

Θ⁢Λ:𝖡⟶𝖡:ΘΛ⟶𝖡𝖡\Theta\Lambda\colon\mathsf{B}\longrightarrow\mathsf{B}roman_Θ roman_Λ : sansserif_B ⟶ sansserif_B is the identity functor).

Part (c): the full subcategory Λ⁢(𝖡)⊂𝖠Λ𝖡𝖠\Lambda(\mathsf{B})\subset\mathsf{A}roman_Λ ( sansserif_B ) ⊂ sansserif_A is closed

Then the essential image Λ⁢(𝖡)⊂𝖠Λ𝖡𝖠\Lambda(\mathsf{B})\subset\mathsf{A}roman_Λ ( sansserif_B ) ⊂ sansserif_A of the functor

right adjoint Λ:𝖡⟶𝖠:Λ⟶𝖡𝖠\Lambda\colon\mathsf{B}\longrightarrow\mathsf{A}roman_Λ : sansserif_B ⟶ sansserif_A (so the essential image

Λ⁢(𝖡)⊂𝖠Λ𝖡𝖠\Lambda(\mathsf{B})\subset\mathsf{A}roman_Λ ( sansserif_B ) ⊂ sansserif_A of the functor ΛΛ\Lambdaroman_Λ is a reflective

B

\mathcal{H}}^{\prime}))_{H}.( caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ) ∈ italic_π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( fraktur_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( caligraphic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT . Let T⁢(t)𝑇𝑡T(t)italic_T ( italic_t ) and S⁢(t)𝑆𝑡S(t)italic_S ( italic_t ) be the associated unitary operators of ℒ1⁢(t)subscriptℒ1𝑡{\mathcal{L}}_{1}(t)caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) and ℒ2⁢(t)subscriptℒ2𝑡{\mathcal{L}}_{2}(t)caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) respectively. Then,

Mashγ⁢(ℒ1⁢(t),ℒ2⁢(t))=Mash−γ⁢(ℒ2⁢(t),ℒ1⁢(t)).subscriptsuperscriptMas𝛾ℎsubscriptℒ1𝑡subscriptℒ2𝑡subscriptsuperscriptMas𝛾ℎsubscriptℒ2𝑡subscriptℒ1𝑡{\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))={%

=Mash⁢((ℒ1⁢(t),ℒ2⁢(t))+Mash⁢(ℒ1′⁢(t),ℒ2′⁢(t))).absentsubscriptMasℎsubscriptℒ1𝑡subscriptℒ2𝑡subscriptMasℎsubscriptsuperscriptℒ′1𝑡subscriptsuperscriptℒ′2𝑡\displaystyle={\mathrm{Mas}}_{h}(({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))+{%

Mash⁢(ℒ1⁢(t),ℒ2⁢(t))=wh⁢(T∗⁢(t)⁢S⁢(t))=w⁢(a⁢T∗⁢(t)⁢S⁢(t))subscriptMasℎsubscriptℒ1𝑡subscriptℒ2𝑡subscript𝑤ℎsuperscript𝑇𝑡𝑆𝑡𝑤𝑎superscript𝑇𝑡𝑆𝑡{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))=w_{h}(T^{*}(t)S(%

Mash⁢(ℒ1⁢(t),ℒ2⁢(t))=Mash⁢(ℒ1′⁢(t),ℒ2′⁢(t)).subscriptMasℎsubscriptℒ1𝑡subscriptℒ2𝑡subscriptMasℎsubscriptsuperscriptℒ′1𝑡subscriptsuperscriptℒ′2𝑡{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))={\mathrm{Mas}}_{%

C

Using the Jacobson-Morozov Theorem 3.3 and Mal’cev Theorem [CM, Theorem 3.4.12], it follows that nilpotent elements are real in simple Lie algebra over ℂℂ\mathbb{C}blackboard_C.

Here we will construct an element g∈Sp⁢(n,ℂ)𝑔Sp𝑛ℂg\in{\rm Sp}(n,\mathbb{C})italic_g ∈ roman_Sp ( italic_n , blackboard_C ) so that g⁢X=−X⁢g𝑔𝑋𝑋𝑔gX=-Xgitalic_g italic_X = - italic_X italic_g for nilpotent X∈𝔰⁢𝔭⁢(n,ℂ)𝑋𝔰𝔭𝑛ℂX\in\mathfrak{s}\mathfrak{p}(n,\mathbb{C})italic_X ∈ fraktur_s fraktur_p ( italic_n , blackboard_C ).

We will provide an elementary proof of the fact that every unipotent element in a complex semi-simple Lie group G𝐺Gitalic_G is real without using Jacobson-Morozov Theorem, see Lemma 4.2.

Moreover, we will construct an explicit element in G𝐺Gitalic_G corresponding to the nilpotent element X∈𝔤𝑋𝔤X\in\mathfrak{g}italic_X ∈ fraktur_g which conjugates X𝑋Xitalic_X and −X𝑋-X- italic_X.

We will provide an elementary proof without using those results and also construct an explicit conjugating element for the nilpotent element. The following basic lemma is useful for this.

D

\prime}(D)}{2}\right)\frac{1}{d_{WR}(D)}∑ start_POSTSUBSCRIPT piece-adjacent junction discs italic_D end_POSTSUBSCRIPT ( 1 - divide start_ARG italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_D ) end_ARG start_ARG 2 end_ARG ) divide start_ARG 1 end_ARG start_ARG italic_d start_POSTSUBSCRIPT italic_W italic_R end_POSTSUBSCRIPT ( italic_D ) end_ARG

between $P$ and some junction disc $D$ along WR-arc}}1≤ - divide start_ARG 1 end_ARG start_ARG 4 end_ARG ∑ start_POSTSUBSCRIPT pieces italic_P end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT incidences between italic_P and some junction disc italic_D along WR-arc end_POSTSUBSCRIPT 1

∑pieces P∑incidences between P and some junction disc D along WR-arcQ⁢(D)subscriptpieces Psubscriptincidences between P and some junction disc D along WR-arc𝑄𝐷\displaystyle\sum_{\text{pieces $P$}}\sum_{\text{incidences between $P$ and %

∑incidences between D and WR-arc-sides1subscriptincidences between D and WR-arc-sides1\displaystyle\sum_{\text{incidences between $D$ and WR-arc-sides}}1∑ start_POSTSUBSCRIPT incidences between italic_D and WR-arc-sides end_POSTSUBSCRIPT 1

some junction disc $D$ along WR-arc}}Q(D)∑ start_POSTSUBSCRIPT pieces italic_P end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT incidences between italic_P and some junction disc italic_D along WR-arc end_POSTSUBSCRIPT italic_Q ( italic_D )

C

χ⁢(Iω⁢(Y^y|Σ))=−y.𝜒superscript𝐼𝜔conditionalsubscript^𝑌𝑦Σ𝑦\chi(I^{\omega}(\widehat{Y}_{y}|\Sigma))=-y.italic_χ ( italic_I start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( over^ start_ARG italic_Y end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | roman_Σ ) ) = - italic_y .

For a surgery slope y1/x1subscript𝑦1subscript𝑥1y_{1}/x_{1}italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with y1≥0subscript𝑦10y_{1}\geq 0italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ 0, suppose its continued fraction is

If y>1𝑦1y>1italic_y > 1, we have y>z≥1𝑦𝑧1y>z\geq 1italic_y > italic_z ≥ 1. If z=1𝑧1z=1italic_z = 1, then we have an exact triangle from Axiom (A2)

Finally, when y>z>1𝑦𝑧1y>z>1italic_y > italic_z > 1, suppose the continued fraction of −y/z𝑦𝑧-y/z- italic_y / italic_z is

y=y′+y′′⁢and⁢z=z′+z′′.𝑦superscript𝑦′superscript𝑦′′and𝑧superscript𝑧′superscript𝑧′′y=y^{\prime}+y^{\prime\prime}~{}{\rm and~{}}z=z^{\prime}+z^{\prime\prime}.italic_y = italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_y start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT roman_and italic_z = italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_z start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT .

C

Generalizing Property (2)(a) of Theorem 11 in [BW11, §3.4] to the case of general n𝑛nitalic_n is accomplished by using the quantum left and right matrices 𝐋ωsuperscript𝐋𝜔\mathbf{L}^{\omega}bold_L start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT and 𝐑ωsuperscript𝐑𝜔\mathbf{R}^{\omega}bold_R start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT, with coefficients in the Fock–Goncharov quantum torus 𝒯nω⁢(𝔗)superscriptsubscript𝒯𝑛𝜔𝔗\mathscr{T}_{n}^{\omega}(\mathfrak{T})script_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( fraktur_T ) for a triangle 𝔗𝔗\mathfrak{T}fraktur_T in the ideal triangulation λ𝜆\lambdaitalic_λ, appearing earlier in Theorem 16.

Similarly, to an extended right-moving arc γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG, as in Figure 13, we associate a quantum right matrix 𝐑ωsuperscript𝐑𝜔\mathbf{R}^{\omega}bold_R start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT in Mn⁢(𝒯nω⁢(𝔗))subscriptM𝑛superscriptsubscript𝒯𝑛𝜔𝔗\mathrm{M}_{n}(\mathscr{T}_{n}^{\omega}(\mathfrak{T}))roman_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( script_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( fraktur_T ) ) by the formula

Let 𝔗𝔗\mathfrak{T}fraktur_T be a triangle. An extended left-moving arc γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG is similar to a left-moving arc, from §2.6, except that it extends all the way to the two distinct edges of the triangle 𝔗𝔗\mathfrak{T}fraktur_T; see Figure 13. We think of an extended left-moving arc γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG as the concatenation of ‘half’ of an edge-crossing arc γ¯11/2subscriptsuperscript¯𝛾121\overline{\gamma}^{1/2}_{1}over¯ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT together with a left-moving arc γ¯2subscript¯𝛾2\overline{\gamma}_{2}over¯ start_ARG italic_γ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT together with another half of an edge-crossing arc γ¯31/2subscriptsuperscript¯𝛾123\overline{\gamma}^{1/2}_{3}over¯ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, as indicated in Figure 13; compare Remark 11. We refer to these halves of edge-crossing arcs as half-edge-crossing arcs. Similarly, we define extended right-moving arcs γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG.

To an extended left-moving arc γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG, as in Figure 13, we associate a quantum left matrix 𝐋ωsuperscript𝐋𝜔\mathbf{L}^{\omega}bold_L start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT in Mn⁢(𝒯nω⁢(𝔗))subscriptM𝑛superscriptsubscript𝒯𝑛𝜔𝔗\mathrm{M}_{n}(\mathscr{T}_{n}^{\omega}(\mathfrak{T}))roman_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( script_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT ( fraktur_T ) ) by the formula

Consider a single extended left-moving or right-moving arc crossing the triangle between two distinct boundary edges, such as those shown in Figure 13; see §3.3.2.

D

The ℤαsubscriptℤ𝛼\mathbb{Z}_{\alpha}blackboard_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT-action on M𝑀Mitalic_M is free away from finite points and {x1,…,xn}⊂X:=M/ℤαsubscript𝑥1…subscript𝑥𝑛𝑋assign𝑀subscriptℤ𝛼\{x_{1},\dots,x_{n}\}\subset X:=M/\mathbb{Z}_{\alpha}{ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } ⊂ italic_X := italic_M / blackboard_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT are singularities such that for each i𝑖iitalic_i a neighbourhood of xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in X𝑋Xitalic_X is c⁢L⁢(ai,bi)𝑐𝐿subscript𝑎𝑖subscript𝑏𝑖cL(a_{i},b_{i})italic_c italic_L ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), a cone over a Lens space where aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT divides α𝛼\alphaitalic_α and bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is coprime to aisubscript𝑎𝑖a_{i}italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.

In this section we sketch the construction of bubble tree compactification for 4-manifolds. Details can be found in [8].

In Section 3, we introduce the equivariant Taubes gluing construction by following the methods in Section 7.2 of [12].

We have all the data we need to do the Taubes’ gluing construction. For λisubscript𝜆𝑖\lambda_{i}italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT small enough, define the pre-glued approximated ASD connection

It is obvious that the Taubes’ gluing construction can be applied to elements in GLT⁢(X)subscriptGL𝑇𝑋\textbf{GL}_{T}(X)GL start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_X ) with zero gluing parameters.

B

Our results show that in general, there cannot be a simple recursive closed form formula for the channel reliability function in a very precise interval.

Holger Boche thanks Martin Bossert for discussions and questions on the theory of the channel reliability function and on questions about the trustworthiness of numerical simulations on digital computers of the channel reliability function. Holger Boche also thanks Vince Poor and Martin Bossert for discussions at ISIT 2019 in Paris. These discussions initiated the research work whose results are presented in this paper. The authors acknowledge the financial support by the Federal Ministry of Education and Research

task to determine the behavior of the channel reliability function on the entire interval (0,C)⊂ℝ0𝐶ℝ(0,C)\subset{\mathbb{R}}( 0 , italic_C ) ⊂ blackboard_R; a common approach is to consider sequences of upper and lower bounds on E⁢(R)𝐸𝑅E(R)italic_E ( italic_R ). The first results in this direction were published in [42].

In this section we give the basic definitions and results from computability theory that we need for this work.

Some results from this paper were presented at the IEEE International Symposium on Information Theory in Espoo, as noted in [13].

D

\Bigr{)}=\xi_{i}(s^{*}(\tilde{\theta})).roman_Ξ start_POSTSUBSCRIPT italic_P , italic_θ end_POSTSUBSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_P , italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( divide start_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG start_ARG italic_i ! end_ARG ) ) = italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( over~ start_ARG italic_θ end_ARG ) ) .

In particular, stable pushforward operations of degree k𝑘kitalic_k for principal G𝐺Gitalic_G-bundles with F𝐹Fitalic_F-orientation correspond bijectively to elements Ξclak={Ξclam|m+k}m⩾0superscriptsubscriptΞcla𝑘subscriptsuperscriptsubscriptΞclaconditional𝑚𝑚𝑘𝑚0\Xi_{\mathrm{cla}}^{k}=\bigl{\{}\Xi_{\mathrm{cla}}^{m|m+k}\bigr{\}}_{m%

natural in P𝑃Pitalic_P and θ.𝜃\theta.italic_θ . In general, these are hard to classify. This simplifies once we introduce stable pushforward operations whose maps are defined for all m𝑚mitalic_m with fixed degree k=n−m𝑘𝑛𝑚k=n-mitalic_k = italic_n - italic_m and which satisfy the analogue of (1.5).

In particular, even-degree stable pushforward operations are determined by their values on the trivial bundle with arbitrary orientations θ.𝜃\theta.italic_θ .

This proves (3.10) for P𝑃Pitalic_P trivial; the general case follows by naturality, using the isomorphism (π,γP,s):P→B×B⁢U⁢(1).:𝜋subscript𝛾𝑃𝑠→𝑃𝐵𝐵U1(\pi,\gamma_{P,s})\colon P\to B\times{B\mathrm{U}(1)}.( italic_π , italic_γ start_POSTSUBSCRIPT italic_P , italic_s end_POSTSUBSCRIPT ) : italic_P → italic_B × italic_B roman_U ( 1 ) . The last claim in (a) is obvious, as Ξcla2⁢esubscriptsuperscriptΞ2𝑒cla\Xi^{2e}_{\mathrm{cla}}roman_Ξ start_POSTSUPERSCRIPT 2 italic_e end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_cla end_POSTSUBSCRIPT is given by its coefficients ξisubscript𝜉𝑖\xi_{i}italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT which are determined entirely by their values on arbitrary K-theory classes, which are equivalently orientations on the trivial bundle.

C

Loosely speaking, almost any dynamical system whose nonlinearity is exclusively due to discrete logics can be written in PWA form [24].

We have demonstrated the applicability of the proposed framework to control systems: many optimal control problems are interpretable as SPPs in GCS and, in our tests, the proposed formulation outperforms state-of-the-art techniques for their solution.

Among the many applications of PWA systems, we have automotive [6], power electronics [20], and robotics [33].

(c) Solution of our convex relaxation. The relaxation gap is 20%percent2020\%20 %, and the MICP is solved in 7.17.17.17.1s.

The growth of the space dimension to n=20𝑛20n=20italic_n = 20 increases the size of our programs, and also loosens the relaxation.

B

πi∈𝖦𝖯n⁢(F)subscript𝜋𝑖subscript𝖦𝖯𝑛𝐹\pi_{i}\in\mathsf{GP}_{n}(F)italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ sansserif_GP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_F ) (resp. with πisubscript𝜋𝑖\pi_{i}italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT or −πi∈𝖯n⁢Fsubscript𝜋𝑖subscript𝖯𝑛𝐹-\pi_{i}\in\mathsf{P}_{n}F- italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ sansserif_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_F).

From the Witt equivalence ⟨⟨x⁢y⟩⟩=⟨⟨x⟩⟩+x⁢⟨⟨y⟩⟩delimited-⟨⟩delimited-⟨⟩𝑥𝑦delimited-⟨⟩delimited-⟨⟩𝑥𝑥delimited-⟨⟩delimited-⟨⟩𝑦\langle\!\langle xy\rangle\!\rangle=\langle\!\langle x\rangle\!\rangle+x%

Let F𝐹Fitalic_F be a field of finite level s⁢(F)=2s𝑠𝐹superscript2𝑠s(F)=2^{s}italic_s ( italic_F ) = 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT and let π𝜋\piitalic_π be the m𝑚mitalic_m-fold Pfister form ⟨⟨−1,…,−1⟩⟩delimited-⟨⟩1…1\langle\!\langle-1,\ldots,-1\rangle\!\rangle⟨ ⟨ - 1 , … , - 1 ⟩ ⟩.

By convention, we denote by ⟨1⟩delimited-⟨⟩1\langle 1\rangle⟨ 1 ⟩ the unique 00-fold Pfister form and

The nonreal case of (c) follows since in this case, the m𝑚mitalic_m-fold Pfister form ⟨⟨−1,…,−1⟩⟩delimited-⟨⟩1…1\langle\!\langle-1,\ldots,-1\rangle\!\rangle⟨ ⟨ - 1 , … , - 1 ⟩ ⟩ is clearly universal.

C

We consider a single series f𝑓fitalic_f for the purpose of the introduction. For a general statement see Theorem 5.15.

Note that by the reconstruction lemma [63, Thm. 5.7.1], Ben-Zvi [30, Thm. 4.4.1] and Kac [58, Thm. 4.5] these formulae are sufficient for determining all fields.

Taking g𝑔gitalic_g to be trivial and f𝑓fitalic_f to be the Segre class, we relate Theorem 1 to [76, Thm. 3, Thm. 6, Thm. 14]. A further analogous statement in K𝐾Kitalic_K-theory is stated in 5.15 leading to a natural question.

Let X𝑋Xitalic_X now be a Calabi–Yau fourfold. The following theorem is the topological version of [41, Thm. 1.1], [52, Thm. 5.19]§§§Joyce’s work is currently unfinished so we expect that the label of the theorem in the final version will change. extending it also to pairs.

The next proposition follows the arguments in the proof of Gross [41, Thm. 4.7, Thm. 4.15] and the remark below it.

B

The point φ⁢(x)𝜑𝑥\varphi(x)italic_φ ( italic_x ) is contained in every irreducible component of 𝔖ssubscript𝔖𝑠\mathfrak{S}_{s}fraktur_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT and 𝔊ssubscript𝔊𝑠\mathfrak{G}_{s}fraktur_G start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT.

Let 𝔘′superscript𝔘′\mathfrak{U}^{\prime}fraktur_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be an open neighborhood of x𝑥xitalic_x such that there exists an étale morphism

that 𝔘𝔘\mathfrak{U}fraktur_U is an open affine subset of 𝔛𝔛\mathfrak{X}fraktur_X such that there exists an étale morphism

Let x𝑥xitalic_x be the generic point of 𝒱𝒱\mathcal{V}caligraphic_V and 𝔘𝔘\mathfrak{U}fraktur_U be an open neighborhood of x𝑥xitalic_x in 𝔛𝔛\mathfrak{X}fraktur_X such that

Let 𝔘𝔘\mathfrak{U}fraktur_U resp. 𝔘′superscript𝔘′\mathfrak{U}^{\prime}fraktur_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be an open neighborhood of x𝑥xitalic_x such that

A

Take q=q⁢(D,r,Q,ϵ22)𝑞𝑞𝐷𝑟𝑄superscriptitalic-ϵ22q=q(D,r,Q,\frac{\epsilon^{2}}{2})italic_q = italic_q ( italic_D , italic_r , italic_Q , divide start_ARG italic_ϵ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ) as in Theorem 4.4. Now apply the measure increment argument (Lemma 3.3) to pass to a Tksuperscript𝑇𝑘T^{k}italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT-ergodic component ν𝜈\nuitalic_ν of μ𝜇\muitalic_μ such that B𝐵Bitalic_B is (q,δ)𝑞𝛿(q,\delta)( italic_q , italic_δ )-equidistributed where δ=112⁢ϵ4𝛿112superscriptitalic-ϵ4\delta=\frac{1}{12}\epsilon^{4}italic_δ = divide start_ARG 1 end_ARG start_ARG 12 end_ARG italic_ϵ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and k≤k0⁢(q,ϵ)≤qlog⁡(ϵ−1)/log⁡(1+δ)𝑘subscript𝑘0𝑞italic-ϵsuperscript𝑞superscriptitalic-ϵ11𝛿k\leq k_{0}(q,\epsilon)\leq q^{\log(\epsilon^{-1})/\log(1+\delta)}italic_k ≤ italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_q , italic_ϵ ) ≤ italic_q start_POSTSUPERSCRIPT roman_log ( italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) / roman_log ( 1 + italic_δ ) end_POSTSUPERSCRIPT. So we may apply Theorem 4.4 to the Tk:ℤr↷(X,ν):superscript𝑇𝑘↷superscriptℤ𝑟𝑋𝜈T^{k}:\mathbb{Z}^{r}\curvearrowright(X,\nu)italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT : blackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ↷ ( italic_X , italic_ν ) and get the desired conclusion (note that ν⁢(B′)>0𝜈superscript𝐵′0\nu(B^{\prime})>0italic_ν ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) > 0 implies μ⁢(B′)>0𝜇superscript𝐵′0\mu(B^{\prime})>0italic_μ ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) > 0). ∎

We now show how to prove Theorem B on uniform recurrence of unipotent actions from Theorem C on uniform polynomial recurrence. This will amount to constructing appropriate polynomials from sufficiently nice sets of unipotent elements as given in the hypothesis of Theorem B.

We now demonstrate how to deduce Theorem A from Theorem B by showing how the hypothesis of Theorem B is satisfied by the companion matrices in 𝔰⁢𝔩d⁢(ℤ)𝔰subscript𝔩𝑑ℤ\mathfrak{sl}_{d}(\mathbb{Z})fraktur_s fraktur_l start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( blackboard_Z ). This technique will be easily generalized to SO⁡(F)SO𝐹\operatorname{SO}(F)roman_SO ( italic_F ) for the quadratic form F⁢(x,y,z)=x2−y2−z2𝐹𝑥𝑦𝑧superscript𝑥2superscript𝑦2superscript𝑧2F(x,y,z)=x^{2}-y^{2}-z^{2}italic_F ( italic_x , italic_y , italic_z ) = italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Moreover, if we were allowed to change k𝑘kitalic_k (and even insist the bound k≤ϵ−1+1𝑘superscriptitalic-ϵ11k\leq\epsilon^{-1}+1italic_k ≤ italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT + 1) for different polynomials P𝑃Pitalic_P then the result trivially follows from the Poincaré Recurrence theorem. We cannot afford these relaxations in the proof of Theorem B from Theorem C given in Section 5.

We now prove the Polynomial Bogolyubov Theorem (Theorem 1.4). By use of Furstenberg’s correspondence principle [7] it is enough to show the following result.

A

We would like to express our gratitude to the anonymous referee who help us to correct, improve and clarify our manuscript.

The number sdepth⁡(M)sdepth𝑀\operatorname{sdepth}(M)roman_sdepth ( italic_M ) is called the Stanley depth of M𝑀Mitalic_M. Herzog Vlădoiu and Zheng [8] proved

In Section 3 of our paper we study the Hilbert depth of S/I𝑆𝐼S/Iitalic_S / italic_I, where I𝐼Iitalic_I is a squarefree monomial ideal with

The second author was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI,

The Stanley conjecture was disproved by Duval et. al [6], in the case M=I/J𝑀𝐼𝐽M=I/Jitalic_M = italic_I / italic_J, where (0)≠J⊂I⊂S0𝐽𝐼𝑆(0)\neq J\subset I\subset S( 0 ) ≠ italic_J ⊂ italic_I ⊂ italic_S

C

Let K𝐾Kitalic_K be a complete discrete valuation field of equal characteristic p>0𝑝0p>0italic_p > 0. Assume that the residue field k𝑘kitalic_k of K𝐾Kitalic_K is perfect. Let L/K𝐿𝐾L/Kitalic_L / italic_K be a finite Galois extension of K𝐾Kitalic_K. Then we can define a filtration of the Galois group, called ramification groups, as in [7], IV, §3.

Our results are related to a paper [2], in which Abrashkin calculated the upper ramification groups of a profinite extension Kp/Ksubscript𝐾𝑝𝐾K_{p}/Kitalic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_K over some local field K𝐾Kitalic_K of equal characteristic p𝑝pitalic_p. The extension Kp/Ksubscript𝐾𝑝𝐾K_{p}/Kitalic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_K considered in his paper is the composition of all finite Galois p𝑝pitalic_p-extension over K𝐾Kitalic_K of nilpotency class less than p𝑝pitalic_p. In [2], the upper ramification groups were expressed in terms of generators of the Galois group of Kp/Ksubscript𝐾𝑝𝐾K_{p}/Kitalic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_K. Thus we can compute the ramification groups of some subextension L/K𝐿𝐾L/Kitalic_L / italic_K of Kp/Ksubscript𝐾𝑝𝐾K_{p}/Kitalic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_K using the results of [2], if we know which generators generate the subgroup Gal⁡(Kp/L)Galsubscript𝐾𝑝𝐿\operatorname{Gal}(K_{p}/L)roman_Gal ( italic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_L ) of Gal(Kp/K\operatorname{Gal}(K_{p}/Kroman_Gal ( italic_K start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / italic_K).

In this paper, by contrast, we express the ramification jumps of the extension Mn/Ksubscript𝑀𝑛𝐾M_{n}/Kitalic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT / italic_K defined in Definition 1.8 using the coefficients of a defining equation of Mn/Ksubscript𝑀𝑛𝐾M_{n}/Kitalic_M start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT / italic_K. We evaluate the values of the ramification jumps by using some elementary equalities and inequalities, such as the ones proven in Lemma 1.10 and Lemma 1.11.

On the other hand, when the extension is no longer abelian, the ramification jumps are not necessarily integer. Furthermore, to the best of the author’s knowledge, the jumps have not been calculated explicitly in most cases. In this paper, we present the values of the upper ramification jumps for some of the finite non-abelian extensions, expecting that our results provide a foothold for obtaining more general conclusions in the future.

This filtration is a convenient tool for studying the wild ramification. If Gal⁡(L/K)Gal𝐿𝐾\operatorname{Gal}(L/K)roman_Gal ( italic_L / italic_K ) is abelian, the Hasse-Arf theorem ensures that the upper ramification jumps are integer. Moreover, we can find the concrete values of these jumps using results by Brylinski [4].

D

Then, the optimal solution 𝛃^^𝛃\hat{\bm{\beta}}over^ start_ARG bold_italic_β end_ARG satisfies ‖𝛃^−𝛃∗‖2<r1subscriptnorm^𝛃superscript𝛃2subscript𝑟1\|\hat{\bm{\beta}}-\bm{\beta}^{*}\|_{2}<r_{1}∥ over^ start_ARG bold_italic_β end_ARG - bold_italic_β start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

We show that (4.2)-(4.6) in Proposition 4.1 are satisfied with probability at least 1−5⁢δ15𝛿1-5\delta1 - 5 italic_δ under Assumption 3.1. Then, we see that Theorem 3.1 holds.

Propositions 4.2, 4.3, 4.4 and 4.5 insist that the conditions in the main proposition are satisfied with high probability.

As we refer in the Introduction, there are many algorithms to estimate mean with existence on outliers and many of them have the properties such that (2.1). However, the algorithm in [6] derived dimension free bound at the first time with polynomial time conplexity and our slightly modified algorithm inherit the properties. We note that, the algorithm in [6] have another fascinating points such that high breakdown point and so on.

for 𝐮𝐮\mathbf{u}bold_u such that ‖𝐮‖∞≤csubscriptnorm𝐮𝑐\|\mathbf{u}\|_{\infty}\leq c∥ bold_u ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≤ italic_c for sum numerical constant c𝑐citalic_c. In the proof of our main proposition (Proposition 4.1), the evaluation of (2.1) appears.

B

This intuition depends on one’s basic intuition about the nature of set existence, which seems to come in three relevant varieties, which I will call the iterative conception, the lasso, and the Platonist.777I do not claim that my analysis is relevant to some very different conceptions: positive and related theories such as those of Malitz and Barwise & Moss; ill-founded set theories without a universal set, such as Forti & Honsell’s and Aczel’s; and the type theory of Whitehead & Russell’s Principia Mathematica. For each of these conceptions, a supposed intuition of the existence of the Russell Set does not withstand scrutiny: the supposed Comprehension intuition does not cohere with the conception. A fourth category, which I discuss first, is restrictive repair attempts without a conspicuous intuition behind them, for instance as in Frege’s hurried addition (below) to his Grundgesetze in reaction to Russell’s letter, or Quine’s New Foundations [Quine 1937].

To Frege’s credit, his reaction to Russell’s letter describing his paradox was acceptance rather than denial; he rushed into print, however, a hurried restriction (without a philosophical justification) to his Axiom V (which in modern terminology is a higher-order generalization of the unrestricted Axiom of Comprehension) [Frege 1893 & 1903] “Afterword.” This attempt is now generally called Frege’s Way Out.

In 1937, Quine published his New Foundations set theory ([Quine 1937]), which has a syntactically-restricted Axiom of Comprehension, also without a philosophical justification.

The Russell Paradox: The Unrestricted Axiom Schema of Comprehension leads immediately to a contradiction.

Before presenting those, for historical and motivational context, I note the Unrestricted Axiom Schema of Comprehension, which I do not assert, and which leads immediately to the Russell Paradox ([Frege 1893 & 1903], “Afterword”). This Schema is the simplification and restriction to first-order set theory of Frege’s more general Axiom V ([Frege 1893 & 1903], §20 & §47). Conventional Zermelo-Fraenkel set theory’s Axiom Schema of Separation restricts the set whose existence is posited by the axiom to being a subset of an existing set; Quine’s New Foundations [Quine 1937] has a Comprehension Schema with a syntactic restriction on the formula ϕitalic-ϕ\phiitalic_ϕ.

A

σ1⁢(0)=limx→0+a1⁢(c)+s⁢(0)⁢a2⁢(c),c=(x,s⁢(0)⁢x).formulae-sequencesubscript𝜎10subscript→𝑥superscript0subscript𝑎1𝑐𝑠0subscript𝑎2𝑐𝑐𝑥𝑠0𝑥\sigma_{1}(0)=\lim_{x\to 0^{+}}a_{1}(c)+s(0)a_{2}(c),\quad c=(x,s(0)x).italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) = roman_lim start_POSTSUBSCRIPT italic_x → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_c ) + italic_s ( 0 ) italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_c ) , italic_c = ( italic_x , italic_s ( 0 ) italic_x ) .

In view of Steps 1 and 2, this shows that σ1⁢(0)subscript𝜎10\sigma_{1}(0)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) can be

value σ1⁢(0)subscript𝜎10\sigma_{1}(0)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) at this point.

5.1 Recovering σ1⁢(0)subscript𝜎10\sigma_{1}(0)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 )

σ1⁢(0)subscript𝜎10\sigma_{1}(0)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ). We can then apply Proposition

A

\operatorname{O}(U).∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_T ≤ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_T + italic_U end_CELL end_ROW start_ROW start_CELL italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ( italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ) = divide start_ARG italic_U end_ARG start_ARG 2 italic_π end_ARG roman_log roman_log divide start_ARG italic_T end_ARG start_ARG 2 italic_π end_ARG + roman_O ( italic_U ) and ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_T ≤ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_T + italic_U end_CELL end_ROW start_ROW start_CELL italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_O ( italic_U ) .

In this article, we obtain the explicit constant in the error terms of Levinson and Montgomery’s results  [14, Theorem 5, Corollary of Theorem 4] as well as Ki and Lee’s results  [12, Theorem 3 and Theorem 2].

In 1989, Conrey [5] proved that more than two-fifths of the zeros of the Riemann zeta function are on the critical line. To achieve this he showed the following result.

In [14], Levinson and Montgomery have used the well-known Littlewood lemma on the function zk⁢(s):=(−1)k⁢2s⁢(log⁡2)−k⁢ζ(k)⁢(s)assignsubscript𝑧𝑘𝑠superscript1𝑘superscript2𝑠superscriptlog2𝑘superscript𝜁𝑘𝑠z_{k}(s):=(-1)^{k}2^{s}(\operatorname{log}{2})^{-k}\zeta^{(k)}(s)italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_s ) := ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( roman_log 2 ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT italic_ζ start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ( italic_s ) and concluded their results. Ki and Lee [12] applied the Littlewood lemma on a function which is a linear combination of derivatives of ζ𝜁\zetaitalic_ζ and by using an upper bound estimate of a fractional moment of logarithmic derivatives of ζ𝜁\zetaitalic_ζ, proved their unconditional result. In our case, we consider the function zk⁢(s)⁢Φ⁢(s)subscript𝑧𝑘𝑠Φ𝑠z_{k}(s)\Phi(s)italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_s ) roman_Φ ( italic_s ) in the place of zk⁢(s)subscript𝑧𝑘𝑠z_{k}(s)italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_s ), where Φ⁢(s)Φ𝑠\Phi(s)roman_Φ ( italic_s ) is a mollifier and apply Littlewood lemma, this refines the explicit constant of the error terms. The main technique of our method is the asymptotic estimate of the mean square of the product of finite order derivatives of the Riemann zeta function with mollifier within short intervals. This mean square result obtain here in connection with the technique developed in our previous paper [9], which motivated form the work of Selberg [21] and Levinson [13].

Recently, Ki and Lee [12] extended the above unconditional result (2). Precisely, they showed that for

D

)}(\omega)\xi^{0}_{l}(\omega)italic_ξ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ω ) = italic_ξ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_ω ) end_POSTSUBSCRIPT = ∑ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l = 1 end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_l ) end_POSTSUBSCRIPT ( italic_ω ) italic_ξ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ( italic_ω ) for any k∈ℕ𝑘ℕk\in\mathbb{N}italic_k ∈ blackboard_N and any ω∈Ω𝜔Ω\omega\in\Omegaitalic_ω ∈ roman_Ω (as usual, (nk=l)subscript𝑛𝑘𝑙(n_{k}=l)( italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_l ) denotes the set {ω∈Ω:nk⁢(ω)=l}conditional-set𝜔Ωsubscript𝑛𝑘𝜔𝑙\{\omega\in\Omega:n_{k}(\omega)=l\}{ italic_ω ∈ roman_Ω : italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_ω ) = italic_l } for any k𝑘kitalic_k and l𝑙litalic_l in ℕℕ\mathbb{N}blackboard_N). If we use ξisubscript𝜉𝑖\xi_{i}italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for the equivalence class of ξi0subscriptsuperscript𝜉0𝑖\xi^{0}_{i}italic_ξ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for each i∈ℕ𝑖ℕi\in\mathbb{N}italic_i ∈ blackboard_N (similarly, ξnksubscript𝜉subscript𝑛𝑘\xi_{n_{k}}italic_ξ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT for the equivalence class of ξnk0subscriptsuperscript𝜉0subscript𝑛𝑘\xi^{0}_{n_{k}}italic_ξ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT for each k∈ℕ𝑘ℕk\in\mathbb{N}italic_k ∈ blackboard_N), then ξnk=∑l=1∞I~(nk=l)⁢ξlsubscript𝜉subscript𝑛𝑘subscriptsuperscript𝑙1subscript~𝐼subscript𝑛𝑘𝑙subscript𝜉𝑙\xi_{n_{k}}=\sum^{\infty}_{l=1}\tilde{I}_{(n_{k}=l)}\xi_{l}italic_ξ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l = 1 end_POSTSUBSCRIPT over~ start_ARG italic_I end_ARG start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_l ) end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT for each k∈ℕ𝑘ℕk\in\mathbb{N}italic_k ∈ blackboard_N. Here, recall that a set G𝐺Gitalic_G of an R⁢N𝑅𝑁RNitalic_R italic_N module (E,∥⋅∥)(E,\|\cdot\|)( italic_E , ∥ ⋅ ∥ ) with base (Ω,ℱ,P)Ωℱ𝑃(\Omega,\mathcal{F},P)( roman_Ω , caligraphic_F , italic_P ) is said to be a.s. bounded (or, bounded with respect to the L0superscript𝐿0L^{0}italic_L start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT–norm ∥⋅∥\|\cdot\|∥ ⋅ ∥) if there exists ξ∈L+0⁢(ℱ)𝜉subscriptsuperscript𝐿0ℱ\xi\in L^{0}_{+}(\mathcal{F})italic_ξ ∈ italic_L start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( caligraphic_F ) such that ‖x‖≤ξnorm𝑥𝜉\|x\|\leq\xi∥ italic_x ∥ ≤ italic_ξ for any x𝑥xitalic_x in G𝐺Gitalic_G (namely, ⋁{∥x∥:x∈G}∈L+0(ℱ)\bigvee\{\|x\|:x\in G\}\in L^{0}_{+}(\mathcal{F})⋁ { ∥ italic_x ∥ : italic_x ∈ italic_G } ∈ italic_L start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( caligraphic_F )). Therefore, the randomized Bolzano–Weierstrass theorem can be restated as: any a.s. bounded nonempty subset of L0⁢(ℱ,ℝd)superscript𝐿0ℱsuperscriptℝ𝑑L^{0}(\mathcal{F},\mathbb{R}^{d})italic_L start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( caligraphic_F , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) is random relatively sequentially compact in the sense of Definition 1.10, below.

According to the randomized Bolzano–Weierstrass theorem [45], G𝐺Gitalic_G in Corollary 1.14 is random sequentially compact, while an a.s. sequentially continuous mapping is obviously random sequentially continuous. So Corollary 1.14 is also a special case of Theorem 1.12.

Just as pointed out in Remark 2.2, Theorem 2.3 remains valid for a d𝑑ditalic_d–σ𝜎\sigmaitalic_σ–stable random metric space. When (Ω,ℱ,P)Ωℱ𝑃(\Omega,\mathcal{F},P)( roman_Ω , caligraphic_F , italic_P ) is trivial Theorem 2.3 reduce to the classical Hausdorff theorem. It is quite obvious that the notion of random total boundedness is an outgrowth of the locally L0superscript𝐿0L^{0}italic_L start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT–convex topology, whereas the notion of random sequential compactness is an outgrowth of the (ε,λ)𝜀𝜆(\varepsilon,\lambda)( italic_ε , italic_λ )–topology. Such results as Theorem 2.3 are most useful in random functional analysis since it establishes a connection between some basic results derived from the two kinds of topologies, and in particular, they will play a crucial role in constructing the Schauder projection and in establishing Schauder fixed point theorem in R⁢N𝑅𝑁RNitalic_R italic_N modules by simultaneously considering the two kinds of topologies, see the proofs of Lemma 3.8, Lemma 3.9 and Theorem 1.12 for details.

In fact, with the randomized Bolzano–Weierstrass theorem stated above it is not difficult for one to present the notion of a random sequentially compact set in an R⁢N𝑅𝑁RNitalic_R italic_N module. For example, in [30] Guo presented such a notion and conjectured the following Theorem 1.12, which is the best possible result we can prove regarding the theorem in this paper.

To outline the key second step in the proof of Theorem 1.12, first recall the idea of the proof of the classical Schauder fixed point theorem. The classical Hausdorff theorem states that a nonempty subset in a normed space (or more generally, a metric space) is sequentially compact iff it is both totally bounded and complete. It is the total boundedness of a compact convex set that allows Schauder to define the Schauder projection on a finite ε𝜀\varepsilonitalic_ε–net of the convex set and further define the Schauder approximation for a compact mapping so that the classical Brouwer fixed point theorem can be employed to prove the classical Schauder fixed point theorem, see [24, p.117] for details. Since a random sequentially compact set in an R⁢N𝑅𝑁RNitalic_R italic_N module is, generally, not compact, the classical notion of a total boundedness is impossible to fit together with the notion of random sequential compactness in order for us to establish a random version of the Hausdorff theorem. For this, we introduce the notion of random total boundedness in Section 2, and further prove that a σ𝜎\sigmaitalic_σ–stable set is random sequentially compact iff it is both random totally bounded and 𝒯ε,λsubscript𝒯𝜀𝜆\mathcal{T}_{\varepsilon,\lambda}caligraphic_T start_POSTSUBSCRIPT italic_ε , italic_λ end_POSTSUBSCRIPT–complete (see Theorem 2.3 of this paper). Based on the random Hausdorff theorem, we can first decompose the mapping T𝑇Titalic_T in Theorem 1.12 into a sequence {Tn,n∈ℕ}subscript𝑇𝑛𝑛ℕ\{T_{n},n\in\mathbb{N}\}{ italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_n ∈ blackboard_N } of local mappings according to some countable partition {An,n∈ℕ}subscript𝐴𝑛𝑛ℕ\{A_{n},n\in\mathbb{N}\}{ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_n ∈ blackboard_N } of ΩΩ\Omegaroman_Ω to ℱℱ\mathcal{F}caligraphic_F, then construct a random Schauder approximation for each Tnsubscript𝑇𝑛T_{n}italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT in a subtle manner, namely, in a manner rather different from the classical Schauder approximation. See the construction before Lemma 3.8 of this paper so that we can obtain an approximate fixed point xnsubscript𝑥𝑛x_{n}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT for each Tnsubscript𝑇𝑛T_{n}italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, at the same time ∑nI~An⁢xnsubscript𝑛subscript~𝐼subscript𝐴𝑛subscript𝑥𝑛\sum_{n}\tilde{I}_{A_{n}}x_{n}∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT over~ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is exactly an approximate fixed point of T𝑇Titalic_T, and eventually achieve the proof of Theorem 1.12.

C

If S∈[q]𝑆delimited-[]𝑞S\in[q]italic_S ∈ [ italic_q ] then Ψ(S;n)↓=Φn(n)\Psi(S;n)\downarrow=\Phi_{n}(n)roman_Ψ ( italic_S ; italic_n ) ↓ = roman_Φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_n ).

^{-n}}\big{)}∀ italic_n ( italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is prefix-free and italic_μ ( italic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ≤ 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT )

Case 2: ∃n⁢f⁢(n)↑↑𝑛𝑓𝑛absent\exists n\ f(n)\uparrow∃ italic_n italic_f ( italic_n ) ↑. By compactness,

Φn(n)↓⇒f(n)≠Φn(n)\Phi_{n}(n)\downarrow\ \Rightarrow\ f(n)\neq\Phi_{n}(n)roman_Φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_n ) ↓ ⇒ italic_f ( italic_n ) ≠ roman_Φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_n ),

If f𝑓fitalic_f is total then ∃n⁢f⁢(n)=Φn⁢(n)↓𝑛𝑓𝑛subscriptΦ𝑛𝑛↓absent\exists n\ f(n)=\Phi_{n}(n)\downarrow∃ italic_n italic_f ( italic_n ) = roman_Φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_n ) ↓.

B

Since |ci−cj|≥9⁢ηsubscript𝑐𝑖subscript𝑐𝑗9𝜂|c_{i}-c_{j}|\geq 9\eta| italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ≥ 9 italic_η for every i≠j𝑖𝑗i\neq jitalic_i ≠ italic_j, this yields the desired result, up to a redefinition of ξ𝜉\xiitalic_ξ.

To conclude, we give a counterpart to Theorem 1.1 in the case when the two balls are sufficiently far apart.

The main motivation for introducing the continuous version of SON clustering is that it allows us to uncover the asymptotic behavior of the discrete problem in (1.1) when the number of datapoints N𝑁Nitalic_N becomes very large. In particular, we will study the “stochastic ball model,” which has become a common testbed in the analysis of clustering algorithms, see for instance Nellore and Ward (2015); Awasthi et al. (2015); Iguchi et al. (2017); Li et al. (2020); De Rosa and Khajavirad (2022). That is, we suppose that we are given a large number of points sampled independently at random, each being distributed according to the uniform measure on the union of two disjoint balls of unit radius, and ask whether SON clustering allows us to identify the presence of the two balls. Surprisingly, we find that if d≥2𝑑2d\geq 2italic_d ≥ 2 and the balls are too close to each other, then the algorithm will typically fail to do so.

Our experiments were performed using the algorithm of Jiang and Vavasis (Preprint, 2020). This algorithm provides a certificate that the ouput clustering is correct. When λ𝜆\lambdaitalic_λ is very close to a value at which the number of clusters changes, limitations on computer time and numerical accuracy make it difficult to perform the calculations to sufficient accuracy to obtain the certificate. In particular, for situations such as that described by Theorem 1.1, the SON clustering algorithm becomes numerically very challenging to resolve for λ𝜆\lambdaitalic_λ close to λcsubscript𝜆𝑐\lambda_{c}italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, while the clustering structures that are produced for other values of λ𝜆\lambdaitalic_λ are not the expected partition into two parts. This further clarifies how the SON algorithm fails to resolve this clustering problem successfully in practice.

Shortly after we posted the first version of this article, Nguyen and Mamitsuka (Preprint, 2021) derived several results on the properties of the optimal clusters. Our framework allows us to recover one of their main results in the measure-valued setting. The following proposition, which is analogous to Theorem 3 of Nguyen and Mamitsuka (Preprint, 2021), states that each cluster is contained in a ball centered at the centroid of the cluster and of radius λ𝜆\lambdaitalic_λ times the total mass of the cluster; and that the centroids of the different clusters are sufficiently far apart from one another that these balls do not intersect.

A

A crucial case in the proof of the latter, when M𝑀Mitalic_M has a spanning-clique restriction, is treated in Section 8.

In this section, we place structure on k𝑘kitalic_k-element extensions of a clique with k≥3𝑘3k\geq 3italic_k ≥ 3, in three cases.

Prior to that, Sections 5, 6, and 7 consider 1111-, 2222-, and k𝑘kitalic_k-element extensions of a clique.

In this section, we show that, up to isomorphism, there are only two rank-r𝑟ritalic_r non-trivial single-element extensions of a clique that have no minor isomorphic to U2,5subscript𝑈25U_{2,5}italic_U start_POSTSUBSCRIPT 2 , 5 end_POSTSUBSCRIPT, F7subscript𝐹7F_{7}italic_F start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT, or R9subscript𝑅9R_{9}italic_R start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT.

We now show that, up to isomorphism, there are only two rank-r𝑟ritalic_r non-trivial single-element extensions of a clique that have no minor isomorphic to U2,5subscript𝑈25U_{2,5}italic_U start_POSTSUBSCRIPT 2 , 5 end_POSTSUBSCRIPT, F7subscript𝐹7F_{7}italic_F start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT, or R9subscript𝑅9R_{9}italic_R start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT.

B

In the usual case when ℙn⁢ℙn⁢𝐤⋆=Ω⁢(1)subscriptℙ𝑛subscriptℙ𝑛subscript𝐤⋆Ω1\mathbb{P}_{n}\mathbb{P}_{n}{\mathbf{k}_{\star}}=\Omega(1)blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT bold_k start_POSTSUBSCRIPT ⋆ end_POSTSUBSCRIPT = roman_Ω ( 1 ), this error bound does not decay to 00 with n𝑛nitalic_n.

Paige et al. [66] analyze the impact of approximating a kernel in super-sampling with a reservoir but do not analyze the quality of the constructed MMD coreset.

For the conditionally positive definite energy distance kernel, Mak and Joseph [56] establish that an optimal coreset of size n12superscript𝑛12n^{\frac{1}{2}}italic_n start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT has o⁢(n−14)𝑜superscript𝑛14o(n^{-\frac{1}{4}})italic_o ( italic_n start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG end_POSTSUPERSCRIPT ) MMD but do not provide a construction; in addition, Mak and Joseph [56] propose two support points convex-concave procedures for constructing MMD coresets but do not establish their optimality and do not analyze their quality.

In each of the twelve experiments, KT significantly improves both the rate of decay and the order of magnitude of mean MMD, in line with the guarantees of Thm. 1.

Our first main result, proved in App. E, bounds the MMD of a kernel thinning coreset in terms of the input 2 and kernel 6 radii, the combined radii

A

}_{1}^{*})_{\theta}( caligraphic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , caligraphic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = ( caligraphic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT.

Using Lemma 6.10, we can prove the embeddings in (6.1) for the sequence structures associated with the γ𝛾\gammaitalic_γ-interpolation method introduced in Example 3.3iv. Example 6.9 follows by combining the following proposition with Theorem 6.1 and Proposition 3.25.

We now turn to reiteration for the complex interpolation method. In contrast to the previous subsection, in which we only recovered the already known reiteration theorem for the real interpolation method, we will obtain various new reiteration results for the complex interpolation method. In particular, we will deduce the following example from Theorem 6.1:

Using Proposition 3.14 we immediately obtain duality results for the methods introduced in Example 3.3. In the following example we will use some Banach space geometry.

the following proposition, in which we prove the embeddings in (6.1) for the sequence structures associated with the complex interpolation method (see Example 3.3ii).

C

Its motivation is to be used as a substitute in applications where a polyhedral terminal set is intractable, such as in the case of large systems or in many robust MPC applications.

The contribution of the article, when compared with [19] and [22], is in how the ADMM algorithm is posed to deal with the terminal ellipsoidal constraint (instead of the terminal equality constraints used in [19] and [22]).

laxMPC. The sparse ADMM-based solver presented in [19] for formulation [19, Eq. (9)], which is exactly the same as (1) but without a terminal constraint. This solver also uses the procedure from [19] to deal with the equality and box constraints. We included it to show the effect that the addition of the terminal quadratic constraint has on the performance of the solver.

The proposed ADMM-based solver deals with the ellipsoidal constraint by modifying the ADMM equality constraints so that the projection step related to the ellipsoidal constraint results in an optimization problem with a simple explicit solution.

The main contribution is to show how a simple linear transformation of a particular part of the equality constraints of the ADMM optimization problem leads to an explicit solution of the projection step related to the ellipsoidal constraints.

C

On the other hand, both the MM-FEM and MM-SUPG perform well with N=32768𝑁32768N=32768italic_N = 32768 elements in the mesh. This is supported by the H1superscript𝐻1H^{1}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-seminorm errors in Table 1 and cap view in Figure 5.

Even with smaller Δ⁢tΔ𝑡\Delta troman_Δ italic_t as 5×10−35superscript1035\times 10^{-3}5 × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT, we see in Figure 2 that MM-FEM still has layers at the outflow, where MM-SUPG performs much better.

where numerical values are recorded in Tables 3 and 4. Again, both MM-FEM and MM-SUPG perform much better than FM-FEM and FM-SUPG with larger N𝑁Nitalic_N.

The L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-norm errors in Table 2 shows that MM-SUPG performs a bit better than MM-FEM for this example.

On the other hand, both the MM-FEM and MM-SUPG perform well with N=32768𝑁32768N=32768italic_N = 32768 elements in the mesh. This is supported by the H1superscript𝐻1H^{1}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-seminorm errors in Table 1 and cap view in Figure 5.

C

The ν𝜈\nuitalic_ν-random walk on G𝐺Gitalic_G is defined in the following way. First let (Xi)i=1∞superscriptsubscriptsubscript𝑋𝑖𝑖1(X_{i})_{i=1}^{\infty}( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT be the i.i.d. process with distribution ν𝜈\nuitalic_ν. We set Zi=X1⋅…⋅Xisubscript𝑍𝑖⋅subscript𝑋1…subscript𝑋𝑖Z_{i}=X_{1}\cdot\ldots\cdot X_{i}italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ … ⋅ italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Process (Zi)subscript𝑍𝑖(Z_{i})( italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is called the right ν𝜈\nuitalic_ν-random walk on G𝐺Gitalic_G. Similarly, we can define the left random walk by setting Zi′=Xi⋅…⋅X1superscriptsubscript𝑍𝑖′⋅subscript𝑋𝑖…subscript𝑋1Z_{i}^{\prime}=X_{i}\cdot\ldots\cdot X_{1}italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ … ⋅ italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

Now we will show that the tail boundary for the ν𝜈\nuitalic_ν-random walk is nontrivial. For this we will construct a tail-measurable function τ𝜏\tauitalic_τ and show that its image is not essentially a one-point set. We will also demonstrate that the natural G𝐺Gitalic_G-action on the boundary of the ν𝜈\nuitalic_ν-random walk on G𝐺Gitalic_G is essentially free. Moreover, we will show that function τ𝜏\tauitalic_τ actually defines the boundary (its image could be identified with the factor-space of the Poisson-Furstenberg boundary).

Let G𝐺Gitalic_G be a group. Let ν𝜈\nuitalic_ν be a non-degenerate probability measure on G𝐺Gitalic_G, we will denote ∂(G,ν)𝐺𝜈\partial(G,\nu)∂ ( italic_G , italic_ν ) the Poisson-Furstenberg(= tail in this case) boundary of the ν𝜈\nuitalic_ν-random walk.

By default, random walk will mean right random walk. We will restrict ourselves to non-degenerate measures on groups.

It is trivial to see that instead of left random walks, we may consider right random walks with opposite measures.

C

In particular, assuming Theorem C is false, Theorem D produces a (𝒵⁢A)𝒵𝐴(\mathcal{Z}A)( caligraphic_Z italic_A )-invariant Borel probability measure with non-zero Lyapunov exponents. By Proposition 2.1 this measure is G𝐺Gitalic_G-invariant, contradicting Lemma 2.2.

To motivate the outline in the next subsection, we recall the proof of [BFH-SLnZ, Theorem B]—the analogue of theorem C

We end the introduction by reducing theorem B to the first technical result of the paper, theorem C below. The proof of theorem B from theorem C follows the same outline as in [BFH].

We now outline the proof of theorem D and highlight the new tools developed in this paper when compared with the proof of [BFH-SLnZ, Theorem B]. For lattices of higher ℚℚ\mathbb{Q}blackboard_Q-rank, the proof of theorem D has the same main lines as in [BFH-SLnZ] though with considerable additional technical difficulty, particularly in the case where the ℚℚ\mathbb{Q}blackboard_Q-rank is 2222. In the case of ℚℚ\mathbb{Q}blackboard_Q-rank-1111, there are genuinely new complications that require ideas not in any way present in [BFH-SLnZ].

by C1+βsuperscript𝐶1𝛽C^{1+\beta}italic_C start_POSTSUPERSCRIPT 1 + italic_β end_POSTSUPERSCRIPT diffeomorphism, the proof of [BFH, Theorem 2.4] can be adapted to establish an analogue of theorem 1.8 to obtain a continuous invariant Riemannian metric. For actions by C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT diffeomorphisms, an analogue of theorem 1.8 is obtained in [BrownDamjanovicZhang, Proposition 5].

A

Hence, δta⁢[μ]⁢(nB⁢a,nG⁢a)subscriptsuperscript𝛿𝑎𝑡delimited-[]𝜇superscript𝑛B𝑎superscript𝑛G𝑎\delta^{a}_{t}[{\mu}]({{n}^{\texttt{B}a},{n}^{\texttt{G}a}})italic_δ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT [ italic_μ ] ( italic_n start_POSTSUPERSCRIPT B italic_a end_POSTSUPERSCRIPT , italic_n start_POSTSUPERSCRIPT G italic_a end_POSTSUPERSCRIPT ) quantify the value of exploration.

The characterization of the optimality gap presented in this paper, while focused on a specific problem, is a step toward better understanding decomposition methods. While it does not fully explain why DeCo works so well, it is a step into that direction as it provides a visual interpretation of the gap between the upper bound provided by the relaxed solution and the performance of DeCo.

The offline phase of the DeCo algorithm is the minimization of the upper bound V0r⁢[μ]subscriptsuperscript𝑉𝑟0delimited-[]𝜇V^{r}_{0}[\mu]italic_V start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ italic_μ ] with respect to μ𝜇\muitalic_μ (see (11m)),

\mathbb{N}}\times{\mathbb{N}}italic_n = ( italic_n start_POSTSUPERSCRIPT B italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_n start_POSTSUPERSCRIPT G italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT ∈ ∏ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT blackboard_N × blackboard_N at time t𝑡titalic_t, the DeCo algorithm selects an arm that maximizes the

It is insightful to observe that the selected arm during the DeCo online phase (see Equation (11n)) is the one that maximizes

D

{4}).italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - 2 square-root start_ARG italic_m italic_τ end_ARG | italic_ξ | + italic_O ( | italic_ξ | start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) and italic_λ start_POSTSUBSCRIPT roman_I end_POSTSUBSCRIPT - square-root start_ARG 1 - italic_m italic_τ end_ARG | italic_ξ | = italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | italic_ξ | start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_O ( | italic_ξ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) .

The solution ψ^normal-^𝜓\widehat{\psi}over^ start_ARG italic_ψ end_ARG to the Cauchy problem (3.4) satisfies the following pointwise estimates in the Fourier space:

Benefit from the determined initial condition (3.3), the solution to the Cauchy problem (1.1) is uniquely given by

Therefore, benefit from the conjugate roots, the unique solution to the Cauchy problem (3.4) is given by

The solution ψ^normal-^𝜓\widehat{\psi}over^ start_ARG italic_ψ end_ARG to the Cauchy problem (3.4) satisfies the following pointwise estimate in the Fourier space:

C

We note that Jn+1subscript𝐽𝑛1J_{n+1}italic_J start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT is a strong deformation retract of Jnsubscript𝐽𝑛J_{n}italic_J start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.

Δ⁢(↑⁡A)Δ↑𝐴\Delta(\operatorname{\uparrow}A)roman_Δ ( ↑ italic_A ) in Δ⁢(𝒳)Δ𝒳\Delta({\mathcal{X}})roman_Δ ( caligraphic_X ), f−1⁢(↑⁡A)superscript𝑓1↑𝐴f^{-1}(\operatorname{\uparrow}A)italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ↑ italic_A )

The set Δ⁢(↑⁡A)Δ↑𝐴\Delta(\operatorname{\uparrow}A)roman_Δ ( ↑ italic_A ) is a compact polyhedral subset of

When A𝐴Aitalic_A is a compact polyhedral subset of X1subscript𝑋1X_{1}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, the inclusion map Δ⁢(↑⁡A)↪f−1⁢(↑⁡A)↪Δ↑𝐴superscript𝑓1↑𝐴\Delta(\operatorname{\uparrow}A)\hookrightarrow f^{-1}(\operatorname{\uparrow}A)roman_Δ ( ↑ italic_A ) ↪ italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ↑ italic_A ) is a homotopy equivalence.

Δ⁢(↑⁡A)Δ↑𝐴\Delta(\operatorname{\uparrow}A)roman_Δ ( ↑ italic_A ) is clearly decomposable as the union of three compact

B

H⁢(4)⊊Nsp⁢(4)𝐻4subscript𝑁sp4H(4)\subsetneq N_{\operatorname{sp}}(4)italic_H ( 4 ) ⊊ italic_N start_POSTSUBSCRIPT roman_sp end_POSTSUBSCRIPT ( 4 )

\qquad E^{\prime}\colon\href https://www.lmfdb.org/EllipticCurve/Q/1728/n/4italic_E : italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - 1 and italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 2

^{3}\!\cdot 179^{3}\!\cdot 409^{3}/79^{16}- 2 start_POSTSUPERSCRIPT 21 end_POSTSUPERSCRIPT ⋅ 3 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 5 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 7 ⋅ 13 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 23 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 41 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 179 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 409 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / 79 start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT

2573/28superscript2573superscript28257^{3}/2^{8}257 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / 2 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT

{3}\!\cdot 283^{3}\!\cdot 929/(5^{13}\!\cdot 61^{13})2 start_POSTSUPERSCRIPT 18 end_POSTSUPERSCRIPT ⋅ 3 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 13 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ⋅ 127 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 139 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 157 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 283 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ⋅ 929 / ( 5 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT ⋅ 61 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT )

C

When comparable, that is for the case of equi-oriented quivers, an interval basis is equivalent to the canonical form.

Indeed, the interval decomposition implies that ℳℳ\mathcal{M}caligraphic_M decomposes into a direct sum of interval modules of the form

Decomposing a persistence module via an interval basis consists of retrieving, given a persistence module ℳℳ\mathcal{M}caligraphic_M, an interval basis v1,…,vNsubscript𝑣1…subscript𝑣𝑁{v_{1},\dots,v_{N}}italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT, where N𝑁Nitalic_N equals the number of interval modules in the interval decomposition of Definition 1.

On the contrary, in our parallel decomposition algorithm in Section 4, we take advantage of the independence properties under the action of x𝑥xitalic_x of the interval basis in order to propose a parallel approach with output-dependent complexity.

Specifically, an interval basis encodes data of a canonical form in a compressed way. Indeed, in an interval basis, we represent a single generator per interval belonging to the interval decomposition and the structure maps are the original ones, implicitly encoded by the action of x𝑥xitalic_x.

D