Showing 1–50 of 710 results for author: Zhao, Y

Supersaturation for Eventown via Generator Switching

Authors: Zicheng Han, Xiande Zhang, Yuhao Zhao

Abstract: An eventown family is a family of even-sized subsets of $[n]$ in which every two distinct members have an even-sized intersection. A classical theorem of Berlekamp and Graver shows that the maximum size of such a family is $2^{\lfloor n/2\rfloor}$. The supersaturation problem for eventown asks how many odd-intersection pairs must occur when this extremal bound is exceeded. For a family… ▽ More An eventown family is a family of even-sized subsets of $[n]$ in which every two distinct members have an even-sized intersection. A classical theorem of Berlekamp and Graver shows that the maximum size of such a family is $2^{\lfloor n/2\rfloor}$. The supersaturation problem for eventown asks how many odd-intersection pairs must occur when this extremal bound is exceeded. For a family $\mathcal F$ of even-sized subsets of $[n]$, let $e(\mathcal F)$ denote the number of unordered pairs whose intersection size is odd. O'Neill conjectured that if $|\mathcal F|=2^{\lfloor n/2\rfloor}+s$, then $e(\mathcal F)\ge s\,2^{\lfloor n/2\rfloor-1}$ for \[ 1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}. \] Previously, the conjecture was known for $s=1,2$, and, for $s\le 2^{\lfloor n/8\rfloor}/n$ with $n$ sufficiently large. We prove the conjectured bound for \[ 1\le s\le \frac{2^{\lfloor n/2\rfloor}}{26}, \] extending the known range to a fixed positive proportion of the extremal eventown size. The bound is sharp throughout this range. As further consequences, we derive a lower bound valid for arbitrary excess $s$, which improves the previously known estimate in an additional range. We also establish stability and removal results for families of extremal size satisfying $e(\mathcal F)<2^{\lfloor n/2\rfloor-1}$, showing that such a family is close to an extremal eventown family and can be made eventown by deleting a small number of its members. △ Less

Submitted 17 August, 2026; originally announced August 2026.

Comments: 12 pages

Derandomizing Karger's Contraction Algorithm for Matroids

Authors: Yu Cong, Chao Xu, Yajie Zhao

Abstract: Karger's randomized contraction algorithm finds a minimum-weight cocircuit of a matroid whenever the cogirth-density ratio is bounded. We prove that the same hypothesis yields a deterministic algorithm with the same exponent. If every contraction minor of rank at least $r_0$ of a matroid $M$ has cogirth-density ratio at most $c$, then a minimum-weight cocircuit of $M$ is computable deterministical… ▽ More Karger's randomized contraction algorithm finds a minimum-weight cocircuit of a matroid whenever the cogirth-density ratio is bounded. We prove that the same hypothesis yields a deterministic algorithm with the same exponent. If every contraction minor of rank at least $r_0$ of a matroid $M$ has cogirth-density ratio at most $c$, then a minimum-weight cocircuit of $M$ is computable deterministically in $m^{O(r_0)} n^{O(c)}$ time when the contraction minors of bounded rank have at most $m$ parallel classes, by an algorithm that knows neither $r_0$ nor $c$. As a consequence, we give a deterministic algorithm computing the cogirth of rank-$p$ perturbed graphic matroids in $2^{O(p^2)} n^{O(1)}$ time, fixed-parameter tractable in $p$, settling the cogirth side of a question of Geelen and Kapadia (2018). The extensions of the contraction method carry over deterministically: enumerating all near-minimum 1-cocycles, computing a minimum-weight $k$-cocycle, and computing the Pareto frontier under several positive criteria. △ Less

Submitted 17 August, 2026; originally announced August 2026.

A Curvature Gap for Minimal Submanifolds in Spheres

Authors: Fagui Li, Yuhang Zhao

Abstract: Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \] Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \] △ Less

Submitted 18 August, 2026; v1 submitted 17 August, 2026; originally announced August 2026.

Comments: 21 pages. Using AI to provide better inequality estimates and improve constants

MSC Class: 53C20; 53C24; 53C42

On the Number of Limit Cycles in Generalized Abel Equations with Coefficients Having the Chebyshev Property

Authors: Jianfeng Huang, Renhao Tian, Yulin Zhao

Abstract: This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the struct… ▽ More This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the structure of the family. Under some natural hypotheses and by means of first- and second-order analyses using Melnikov functions, we provide lower bounds for this maximum number. In contrast to previous work, no specific form for the coefficients is assumed. We then apply these estimates to Abel equations with trigonometric polynomial, polynomial, and hyperbolic coefficients. In the trigonometric polynomial case, we reestablish the results of Álvarez et al. (J. Math. Anal. Appl., 2008) and Huang et al. (SIAM J. Appl. Dyn. Syst., 2020), while in the polynomial case, we improve the classical lower bound given by Lins-Neto. △ Less

Submitted 16 August, 2026; originally announced August 2026.

Lascoux series, parking functions and noncrossing partitions

Authors: Alice L. L. Gao, Xin-Bei Liu, Arthur L. B. Yang, James J. Y. Zhao

Abstract: In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_σ(t)$ indexed by permutations $σ$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_σ(t)$ for the permutation $σ=[2,3,\ldots,n,1]$ or its inverse in terms of the des… ▽ More In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_σ(t)$ indexed by permutations $σ$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_σ(t)$ for the permutation $σ=[2,3,\ldots,n,1]$ or its inverse in terms of the descent statistic of parking functions of length $n-1$. Based on this progress on Lascoux's open problem, we find that the polynomial $\mathcal{E}_σ(t)$ for this special case coincides with the $h$-polynomial $h(Δ(\mathrm{NC}_W),t)$ of the order complex of the noncrossing partition lattice associated to the irreducible Coxeter group $W$ of type $A_{n-1}$. We are inspired by this coincidence to give an operator approach to $h(Δ(\mathrm{NC}_W),t)$ for any finite Coxeter group $W$. As an application, we completely solve an open problem on $h(Δ(\mathrm{NC}_W),t)$ which was proposed by Athanasiadis, Douvropoulos and Kalampogia-Evangelinou. For any $k$-divisible noncrossing partition poset $\mathrm{NC}^{(k)}_W$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(Δ(\mathrm{NC}^{(k)}_W),t)$. △ Less

Submitted 15 August, 2026; originally announced August 2026.

MSC Class: 05A20; 20F55; 26C10; 52B05

Same Resident Strains, Different Attractors: Opposite Local Growth Signs for a Rare Third Strain

Authors: Ruiwu Niu, Xincheng Shu, Ying Zhao, Jingyi Wang

Abstract: Most models assess whether a newly introduced pathogen strain can grow when rare by testing it against a steady endemic background. Resident strains, however, can have more than one long-term behavior under the same parameters. They may settle to a steady state or continue through recurring outbreaks, leaving different fractions of hosts with different infection histories. We study this possibilit… ▽ More Most models assess whether a newly introduced pathogen strain can grow when rare by testing it against a steady endemic background. Resident strains, however, can have more than one long-term behavior under the same parameters. They may settle to a steady state or continue through recurring outbreaks, leaving different fractions of hosts with different infection histories. We study this possibility in a stylized immune-history model. For two resident strains, we identify a parameter point where a stable endemic equilibrium and a stable outbreak cycle are both locally attracting. Numerical continuation maps a Hopf branch and a branch of folds of periodic orbits that bound a candidate bistability region, organized locally by a generalized Hopf (Bautin) point. We then embed the resident model exactly in a 20-state, three-strain system. When strain 3 infectiousness and total removal are common across four host-history classes, rare growth is determined by a weighted sum of the corresponding uninfected host fractions. The equilibrium and cycle yield different weighted sums, so one fixed strain 3 parameter set grows near one resident attractor but declines near the other. A second parameter set gives the reverse ordering, and both patterns persist over open regions of invader parameter space. Sixty full-model simulations with progressively smaller inocula reproduce these linear predictions across cycle phases and numerical solvers. Thus, knowing which resident strains are present may not suffice to predict initial invasion; the realized resident attractor can also matter. These results concern local growth from rarity, not eventual establishment, long-term coexistence, or post-invasion dynamics. △ Less

Submitted 14 August, 2026; originally announced August 2026.

Tripartite Zarankiewicz numbers and norm graphs

Authors: Yantao Tang, Yi Zhao

Abstract: For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, Rónyai and Szabó, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{… ▽ More For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, Rónyai and Szabó, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{1-1/t}+o(1)\right)n^{2-1/t}. \] Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, \[ \operatorname{ex}(n,n,n,K_{s,t})\le \left(\frac{3}{2^{1/t}}(s-t+1)^{1/t}+o(1)\right)n^{2-1/t}. \] Together, these bounds recover the results for $t=2$, and give the new asymptotic formula \[ \operatorname{ex}(n,n,n,K_{3,3}) =\left(\frac{3}{\sqrt[3]{2}}+o(1)\right)n^{5/3}. \] Analogous results extend to $k$-partite graphs containing no $K_{s, t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically. △ Less

Submitted 11 August, 2026; originally announced August 2026.

Comments: 8 pages

MSC Class: 05C35 (Primary); 05C55; 05C25 (Secondary)

Bounds and constructions for constant/low-power error-correcting cooling codes

Authors: Shuangqing Liu, Tingting Tong, Menglong Zhang, Binwei Zhao, Yuhao Zhao

Abstract: The low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes, introduced in [IEEE Trans. Inf. Theory, 64 (2018), 3062--3085; 66 (2020), 4804--4818], respectively, are two coding schemes designed to simultaneously control the peak temperature and average power consumption of on-chip buses while providing error-correction capability for transmitted i… ▽ More The low-power error-correcting cooling (LPECC) codes and constant-power error-correcting cooling (CPECC) codes, introduced in [IEEE Trans. Inf. Theory, 64 (2018), 3062--3085; 66 (2020), 4804--4818], respectively, are two coding schemes designed to simultaneously control the peak temperature and average power consumption of on-chip buses while providing error-correction capability for transmitted information. This paper establishes new upper bounds for both $(n,1,w,w-2)$-CPECC codes and $(n,t,w,w-2)$-CPECC codes using graph-theoretic techniques, and constructs several new families of optimal CPECC codes using combinatorial configurations. Moreover, it completely resolves the conjecture concerning CPECC codes posed in [IEEE Trans. Inf. Theory, DOI: 10.1109/TIT.2026.3721101]. Finally, we derive a new upper bound for $(n,t,w,w-2)$-LPECC codes by probabilistic method, along with new optimal families, and establish the relationship between optimal $(n,t,w,w-2)$-LPECC codes and optimal $(n+1,t,w,w-2)$-CPECC codes. △ Less

Submitted 9 August, 2026; originally announced August 2026.

Bounds for moments of twisted quadratic characters of prime modulus

Authors: Peng Gao, Yuetong Zhao

Abstract: We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $χ_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the… ▽ More We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $χ_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the smoothed $m$-th moment for all integers $m\geq 4$. A matching lower bound for all even integers $m\geq 4$ shows that this bound is optimal. △ Less

Submitted 6 August, 2026; originally announced August 2026.

Comments: 23 pages

A superlogarithmic saving for Oddtown modulo composite numbers

Authors: Yuhao Zhao

Abstract: Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $ω(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell}(n)=n$. When $ω(\ell)\geq 2$, Bukh, Chao, and Zheng… ▽ More Let $f_{\ell}(n)$ be the largest size of a family $\mathcal{A}\subseteq2^{[n]}$ such that no member has size divisible by $\ell$, while the intersection of every two distinct members has size divisible by $\ell$, and let $ω(\ell)$ denote the number of distinct prime divisors of $\ell$. For any prime power $\ell$, the classical answer is $f_{\ell}(n)=n$. When $ω(\ell)\geq 2$, Bukh, Chao, and Zheng recently proved $ω(\ell)n-O_{\ell}\left(n^{\frac{ω(\ell)-2}{ω(\ell)-1}}(\log n)^{C_{\ell}}\right)\leq f_{\ell}(n)\leqω(\ell)n-2ω(\ell)\log n+11$ for some $C_{\ell}>0$. When $\ell$ has at least two distinct odd prime divisors, they further used Fourier analysis to improve the upper bound to $f_{\ell}(n)\leqω(\ell)n-(2ω(\ell)+\varepsilon_{\ell})\log n$ for some $\varepsilon_{\ell}>0$, provided that $n$ is sufficiently large in terms of $\ell$ . For every fixed $\ell$ with $ω(\ell)\geq2$, we prove \[ f_{\ell}(n)\leqω(\ell)n-Ω_{\ell}(\log n\log\log n) \] for large $n$. The upper bound relies on a submatrix lemma of Bhowmick, Dvir, and Lovett, which is based on the bounded-torsion polynomial Freiman--Ruzsa conjecture recently proved by Gowers, Green, Manners, and Tao. △ Less

Submitted 6 August, 2026; originally announced August 2026.

Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues

Authors: Jianquan Ge, Yuyang Zhao

Abstract: We construct two families of curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first, derived from the Einstein warped products, has two distinct Ricci eigenvalues and admits a local isometric immersion of minimum codimension two. The second, arising from the Riemannian Schwarzschild--Tangherlini manifold, has $k+1$ distinct Ricci eigenvalues and admits an isometric… ▽ More We construct two families of curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first, derived from the Einstein warped products, has two distinct Ricci eigenvalues and admits a local isometric immersion of minimum codimension two. The second, arising from the Riemannian Schwarzschild--Tangherlini manifold, has $k+1$ distinct Ricci eigenvalues and admits an isometric embedding of codimension $k+2$, which is the smallest within the adapted product class. △ Less

Submitted 4 August, 2026; originally announced August 2026.

MSC Class: 53C42; 53C40; 53C25; 53B25

Classification of Simple Cuspidal Modules over Nongraded Witt Lie Algebras

Authors: Genqiang Liu, Xiaoyao Zheng, Yufang Zhao

Abstract: For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. W… ▽ More For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}Δ)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(λ),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module. △ Less

Submitted 30 July, 2026; originally announced July 2026.

Two infinite families of counterexamples to the Stanley--Gasharov conjecture

Authors: David G. L. Wang, K. Zhang, T. Y. Zhao

Abstract: The Stanley--Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley--Gasharov conjecture, thereby answering a question of Matherne and… ▽ More The Stanley--Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley--Gasharov conjecture, thereby answering a question of Matherne and Morales. Every graph in the first family is a line graph, whereas no graph in the second family is a line graph. Prajapati further showed that the graph $G_2$, which has $12$ vertices and $21$ edges, is the smallest counterexample under the ordering that first compares the numbers of vertices and then the numbers of edges. We show that $G_2$ is also the smallest counterexample under the reverse ordering, which first compares the edge numbers and then the vertex numbers. Similarly, we exhibit a graph $Q$ with $13$ vertices and $27$ edges and show that $Q$ is the smallest counterexample that is not a line graph under each ordering. Our two infinite families are obtained from $G_2$ and $Q$, respectively, by adjoining a clique of order at least $4$ and connecting one of its vertices to a distinguished vertex of the original graph by a single edge. △ Less

Submitted 31 July, 2026; v1 submitted 29 July, 2026; originally announced July 2026.

Comments: 12 pages, 7 figures

MSC Class: Primary 05E05; Secondary 05C15; 05C76

Lower bounds on the strength of the determinant

Authors: Qiyuan Chen, Yuhao Zhao

Abstract: We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of… ▽ More We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section. △ Less

Submitted 23 July, 2026; originally announced July 2026.

Comments: 16 pages. Comments are welcome!

Shadowing property and transitivity of a set-valued map and its inverse limit

Authors: Yingcui Zhao, Lidong Wang

Abstract: We study the properties of shadowing, transitivity, weakly mixing, mixing, chain transitivity and chain mixing of a set-valued map and its generalized inverse limit. Concerning shadowing, we prove that for a surjective upper semi-continuous set-valued map $F$ on a compact metric space, $F$ has shadowing if and only if the shift map on the generalized inverse limit… ▽ More We study the properties of shadowing, transitivity, weakly mixing, mixing, chain transitivity and chain mixing of a set-valued map and its generalized inverse limit. Concerning shadowing, we prove that for a surjective upper semi-continuous set-valued map $F$ on a compact metric space, $F$ has shadowing if and only if the shift map on the generalized inverse limit $\underleftarrow{\lim}\,\underleftarrow{F}$ of its inverse set-valued map has shadowing; dually, $\underleftarrow{F}$ has shadowing if and only if the shift map on $\underleftarrow{\lim}F$ has shadowing. We further show that the shadowing of $F$ and that of $\underleftarrow{F}$ are always equivalent; consequently $F$, $\underleftarrow{F}$ and the shift maps on $\underleftarrow{\lim}\,\underleftarrow{F}$ and on $\underleftarrow{\lim}F$ all have shadowing simultaneously. In particular, $F$ has shadowing if and only if the shift map on its directly induced generalized inverse limit $\underleftarrow{\lim}F$ has shadowing. This strengthens a recent theorem established under continuity and openness assumptions. We show that if the shift map on the generalized inverse limit is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing), then the set-valued map is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing). For a set-valued map with shadowing, the properties of total transitivity, weak mixing, mixing, specification and chain mixing are mutually equivalent. △ Less

Submitted 19 July, 2026; originally announced July 2026.

Reconstruct the ambient noise source from the multi-frequency sparse correlation data

Authors: Hao Gu, Hongxia Guo, Xiang Xu, Yue Zhao

Abstract: In this paper, we develop a novel multi-frequency factorization method to reconstruct the spatial support of the ambient noise source. The proposed method only requires sparse correlation data and has low computational cost. Numerical experiments in two and three dimensions are presented to demonstrate the effectiveness of the proposed method. In this paper, we develop a novel multi-frequency factorization method to reconstruct the spatial support of the ambient noise source. The proposed method only requires sparse correlation data and has low computational cost. Numerical experiments in two and three dimensions are presented to demonstrate the effectiveness of the proposed method. △ Less

Submitted 15 July, 2026; originally announced July 2026.

Ensemble Controlled-Flow Filtering for Implicit Data Assimilation

Authors: Zhuoyuan Li, Yue Zhao, Ming Li

Abstract: Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is… ▽ More Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is defined as an energy tilt of the forecast distribution. We then propose the Ensemble Controlled-flow Filter (EnCF), which realizes this update through a stochastic controlled flow and learns the observation-dependent control by adjoint matching from terminal energy gradients. For simulator-defined observations, EnCF-LF learns a surrogate conditional energy from samples and applies the same controlled-flow solver. We prove ideal exactness, derive a one-step error decomposition, and establish non-accumulation of local errors under filter stability. Numerical results show that Kalman-type filters remain preferable for smooth additive-Gaussian observations, while the proposed methods are better suited to non-Gaussian, many-to-one, multimodal, and implicit observation models. △ Less

Submitted 14 July, 2026; originally announced July 2026.

Comments: 26 pages

MSC Class: 65C30; 65C05; 62M20; 93E11; 93E20

Optimal Weak-Type Estimates and Their Applications of Lifted Rough Maximal Operators

Authors: Dachun Yang, Wen Yuan, Yirui Zhao

Abstract: Let $n\in\mathbb N\cap[2,\infty)$ and $Ω\in L^1(\mathbb S^{n-1})$ with $Ω\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_θ^Ω\}_{θ\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent… ▽ More Let $n\in\mathbb N\cap[2,\infty)$ and $Ω\in L^1(\mathbb S^{n-1})$ with $Ω\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_θ^Ω\}_{θ\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{θ,λ\in(0,\infty)}λ^p \underset{{\mathcal M}^Ω_θ(f)(x,t) > λt^\fracγ{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{γ-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $γ\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $Ω\in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $γ\in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sjögren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_Ω$, arising from the method of rotation of Calderón and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals. △ Less

Submitted 9 July, 2026; originally announced July 2026.

MSC Class: Primary 42B25; Secondary 42B20; 42B30

The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

Authors: Fagui Li, Yuhang Zhao

Abstract: In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work\cite{Zhao2025}. If \(λ_ρ(Σ)\geqλ>0\) and \(S=|A|^2<1+λ\), then \(Σ\) is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincaré estimate giv… ▽ More In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work\cite{Zhao2025}. If \(λ_ρ(Σ)\geqλ>0\) and \(S=|A|^2<1+λ\), then \(Σ\) is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding--Xin and Brendle--Tsiamis weighted Poincaré estimate gives \(λ_ρ(Σ)\geq1/2\). Consequently, the pointwise upper pinching \(S<3/2\) forces \(Σ\) to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in \(\mathbb R^3\), we also obtain the endpoint case \(S\leq3/2\). These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding--Xin~\cite{DingXin2014}, Cheng--Wei~\cite{ChengWei2015}, and Lei--Xu--Xu~\cite{LeiXuXu2020}. △ Less

Submitted 3 August, 2026; v1 submitted 5 July, 2026; originally announced July 2026.

Comments: 19 pages, any comments are welcome. We have revised the statement of the theorem and the corresponding proof

MSC Class: 53E10; 53C42; 58J50

Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities

Authors: Yuqi Zhao

Abstract: We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegativ… ▽ More We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegative constraint. For every admissible class $\mathcal{C}$, we prove two transfer principles. First, equality cases regularize optimally: if a non-matching graph $H$ is $\mathcal{C}$-Sidorenko, then every equality case $t(H,W)=p(W)^{e(H)}$ with $W\in\mathcal{C}$ is regular. Consequently, relative forcing is equivalent to relative regular-forcing for every non-matching $\mathcal{C}$-Sidorenko graph. Second, in the range $v(H)\le e(H)$, ordinary $\mathcal{C}$-Sidorenko is equivalent, as a universal property over $\mathcal{C}$, to the spectral inequality $t(H,W)\ge ρ(W)^{2e(H)-v(H)}p(W)^{v(H)-e(H)}$ for every non-zero $W\in\mathcal{C}$. The spectral transfer is obtained from a Perron-biased tensor regularization theorem detecting the Perron spectral radius on the exponential scale. We also prove quantitative near-equality variants and apply the framework to doubly nonnegative graphons and bounded doubly nonnegative kernels. This yields spectral equivalences for Sidorenko-good graphs in the range $v(F)\le e(F)$, and identifies Sidorenko-good forcing with regular-KNRS forcing for non-matching Sidorenko-good graphs. △ Less

Submitted 2 July, 2026; originally announced July 2026.

Flat minimal tori and Lu's second-gap conjecture

Authors: Fagui Li, Yuhang Zhao

Abstract: Lu conjectured that, for each dimension and codimension, there exists a positive gap above the first pinching value for the quantity $S+λ_2$ on closed minimal submanifolds of the unit sphere, where $S$ is the squared norm of the second fundamental form and $λ_2$ is the second eigenvalue of Lu's fundamental matrix. We disprove this second-gap conjecture for minimal surfaces in every codimension… ▽ More Lu conjectured that, for each dimension and codimension, there exists a positive gap above the first pinching value for the quantity $S+λ_2$ on closed minimal submanifolds of the unit sphere, where $S$ is the squared norm of the second fundamental form and $λ_2$ is the second eigenvalue of Lu's fundamental matrix. We disprove this second-gap conjecture for minimal surfaces in every codimension $q\ge3$. More precisely, in every odd codimension $q\ge3$ we construct linearly full closed embedded flat minimal tori with $S\equiv2$ for which the constant values of $S+λ_2$ are dense in $(2,3)$. Thus the first pinching value $2$ can be approached from above by closed embedded minimal surfaces, and no uniform second gap exists in codimension at least three. Together with the known positive results in codimensions one and two, our examples complete the codimension picture for Lu's second-gap problem for minimal surfaces: the conjecture holds precisely in codimensions $1$ and $2$, △ Less

Submitted 18 August, 2026; v1 submitted 29 June, 2026; originally announced June 2026.

Comments: 32 pages, any comments are welcome!

MSC Class: 53C42; 53C40; 53A10

$L^p$-form of the KNRS conjecture

Authors: Yuqi Zhao

Abstract: The Kohayakawa--Nagle--Rödl--Schacht conjecture predicts that locally dense graphs contain, asymptotically, at least as many homomorphic copies of any fixed graph as the random graph of the same edge density. We prove that every graph with at least one edge satisfies a natural $L^p$ relaxation of this conjecture in the graphon setting. More precisely, let $F$ be a graph with $m>0$ edges, and let… ▽ More The Kohayakawa--Nagle--Rödl--Schacht conjecture predicts that locally dense graphs contain, asymptotically, at least as many homomorphic copies of any fixed graph as the random graph of the same edge density. We prove that every graph with at least one edge satisfies a natural $L^p$ relaxation of this conjecture in the graphon setting. More precisely, let $F$ be a graph with $m>0$ edges, and let $n$ be the number of non-isolated vertices of $F$. If $$ p\ge \binom {n}{2}/m, $$ then for every $ρ$-locally dense graphon $W$, $$ t(F,W^{\circ p})\ge ρ^{pm}. $$ Equivalently, if $$ W_F(\mathbf x)=\prod_{ij\in E(F)}W(x_i,x_j), $$ then $$ \|W_F\|_{L^p}\ge ρ^{e(F)}. $$ The proof is based on a Hölder uniformization over vertex relabellings, in the spirit of Conlon--Lee. We also prove a more general comparison principle with edge-transitive KNRS supergraphs, yielding sharper exponents whenever $F$ embeds into an edge-transitive KNRS graph. Finally, positive-semidefinite methods give theta-subdivision results: Sidorenko-good graphs are closed under arbitrary uniform theta-subdivisions; the non-uniform theta theorem of Im--Li--Liu admits a Sidorenko-good lift, under the same divisibility assumptions, after removing the parity restriction; and uniform theta-subdivisions of KNRS graphs are regular-KNRS. △ Less

Submitted 29 June, 2026; originally announced June 2026.

Queues with Correlated Service Times -- the $M/M_D/c$ Model

Authors: Qihui Bu, Suman Thapa, Yiqiang Q. Zhao

Abstract: This paper studies multi-server queueing systems with correlated service times, modeled as the $M/M_D/c$ queue, which is a natural extension of the recent work by Thapa and Zhao \cite{Thapa-Zhao:2026}. In this model, arrivals follow a Poisson process, while service times across servers exhibit dependence captured by the Marshall--Olkin multivariate exponential distribution (MO-MVED). We first de… ▽ More This paper studies multi-server queueing systems with correlated service times, modeled as the $M/M_D/c$ queue, which is a natural extension of the recent work by Thapa and Zhao \cite{Thapa-Zhao:2026}. In this model, arrivals follow a Poisson process, while service times across servers exhibit dependence captured by the Marshall--Olkin multivariate exponential distribution (MO-MVED). We first develop a rigorous sample-path construction of the system and establish that the resulting queueing process is a continuous-time Markov chain. We then analyze the stationary behavior of the $M/M_D/c$ model. In the homogeneous case, we derive a complete solution via geometric tail structure and explicit boundary equations, recovering a tractable one-dimensional representation. In the heterogeneous case, we establish a general framework combining a geometric tail with a finite boundary system, and prove existence, uniqueness, and nonnegativity of the stationary distribution. The above results provide a unified analytic framework extending classical $M/M/c$ theory to correlated-service settings, and reveal how dependence among service times fundamentally affects system performance and structure. Beyond the $M/M_D/c$ model, We next study the interplay between Marshall--Olkin service dependence and queue-state Markovianity. On the one hand, Marshall--Olkin dependent service completions are shown to preserve Markovianity for a broad class of queueing systems. On the other hand, if a queueing process admits a Markovian state description without tracking service ages, residual service times, or service phases, then its service mechanism must satisfy a weak multivariate lack-of-memory property and consequently belongs to the Marshall--Olkin family. These results provide a probabilistic foundation for the use of Marshall--Olkin multivariate exponential service times in Markovian queueing models. △ Less

Submitted 23 June, 2026; originally announced June 2026.

Comments: 42 pages

MSC Class: 60K25; 60J27

Sidorenko Inequalities for Two-Sided Group Correlation Kernels

Authors: Yuqi Zhao

Abstract: Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups. Let $Γ$ be a finite group and let $f:Γ\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $Γ$ by… ▽ More Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups. Let $Γ$ be a finite group and let $f:Γ\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $Γ$ by $$\mathcal C_f(x,y)=|Γ|^{-1}\sum_{a_1,a_2\inΓ:\, xa_1=a_2y} f(a_1)f(a_2)=\mathbb{E}_{z\inΓ} f(x^{-1}z)f(zy^{-1}).$$ When $f=\mathbf{1}_A$, this is the normalized size of the intersection $xA\cap Ay$. We prove that, for every finite directed graph $F$, $$t(F,\mathcal C_f)\geq t(\overrightarrow{K_2},\mathcal C_f)^{e(F)}=(\mathbb{E}_{g\inΓ}f(g))^{2e(F)}.$$ Equivalently, if $W_f^\times(x,y)=f(xy)$ is the directed product Cayley kernel on $Γ$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^\times$. △ Less

Submitted 22 June, 2026; originally announced June 2026.

Conjugacy Class Averages and Sidorenko's Conjecture

Authors: Yuqi Zhao

Abstract: Sidorenko's conjecture asserts that for every bipartite graph $H$ and every graph $G$, \[ t(H,G)\geq t(K_2,G)^{e(H)}. \] A result of Szegedy shows that, in order to prove the conjecture, it suffices to verify the corresponding inequality on a special family of highly symmetric bipartite Cayley type hosts arising from symmetric groups. Motivated by this reduction, we study Cayley type bipartite k… ▽ More Sidorenko's conjecture asserts that for every bipartite graph $H$ and every graph $G$, \[ t(H,G)\geq t(K_2,G)^{e(H)}. \] A result of Szegedy shows that, in order to prove the conjecture, it suffices to verify the corresponding inequality on a special family of highly symmetric bipartite Cayley type hosts arising from symmetric groups. Motivated by this reduction, we study Cayley type bipartite kernels associated with functions on finite groups and their conjugacy class averages. Our first result gives a reduction through conjugacy averaging: for a fixed bipartite graph $H$, if the $H$-density of each Cayley type host is at least the $H$-density of its conjugacy class average, then $H$ is strong Sidorenko, and hence Sidorenko. Our second result proves a Sidorenko-type inequality for 1-subdivision graphs on conjugacy-averaged Cayley kernels associated with arbitrary real-valued functions on finite groups. △ Less

Submitted 13 June, 2026; originally announced June 2026.

Gaussian-Weighted Curvature Gaps for Self-Shrinkers

Authors: Fagui Li, Yuhang Zhao

Abstract: In this paper, we prove lower bounds for the Gaussian-weighted \(L^2\)-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincaré inequalities arising from first-eigenvalue estimates of Ding--Xin and Brendle--Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap… ▽ More In this paper, we prove lower bounds for the Gaussian-weighted \(L^2\)-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincaré inequalities arising from first-eigenvalue estimates of Ding--Xin and Brendle--Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap theorem of Colding--Ilmanen--Minicozzi--White, yields a strict curvature gap for nonspherical closed shrinkers. In dimension two, we combine this estimate with the classification theorem and entropy gap theorem of Bernstein--Wang to obtain the corresponding gap statement for complete embedded self-shrinkers with polynomial volume growth. △ Less

Submitted 12 June, 2026; originally announced June 2026.

Comments: 11 pages, any comments are welcome!

MSC Class: 53E10; 53C42; 58J50

Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms

Authors: Jianquan Ge, Yuyang Zhao

Abstract: The classification of curvature homogeneous hypersurfaces in real space forms was established by Tsukada in 1988, with the remaining rank-two cases in $\mathbb{S}^4$ and $\mathbb{H}^4$ settled by Bryant-Florit-Ziller in 2025. It is obvious that curvature homogeneity implies constant Ricci eigenvalues. In this paper, we prove that for hypersurfaces in real space forms, the converse also holds: a co… ▽ More The classification of curvature homogeneous hypersurfaces in real space forms was established by Tsukada in 1988, with the remaining rank-two cases in $\mathbb{S}^4$ and $\mathbb{H}^4$ settled by Bryant-Florit-Ziller in 2025. It is obvious that curvature homogeneity implies constant Ricci eigenvalues. In this paper, we prove that for hypersurfaces in real space forms, the converse also holds: a connected hypersurface immersed in real space forms has constant Ricci eigenvalues if and only if it is curvature homogeneous. Hence, hypersurfaces with constant Ricci eigenvalues in real space forms are also classified, which, in particular, generalizes the classification of Einstein hypersurfaces obtained by Lawson for minimal hypersurfaces and by Ryan for general cases in 1969. Moreover, as a byproduct, curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues can not be isometrically immersed in any real space form of codimension one. Finally, we show that a hypersurface with constant Ricci eigenvalues is isoparametric if it is a complete hypersurface either in $\mathbb{S}^{n+1}$ or nonflat in $\mathbb{R}^{n+1}$ for $n \geq 3$; or if it is not of constant sectional curvature $-1$ in $\mathbb{H}^{n+1}$ for $n \geq 5$. △ Less

Submitted 11 June, 2026; originally announced June 2026.

Comments: 14 pages

A Unified Zeroth-Order Approach for Decentralized Minimax Optimization

Authors: Haoyuan Cai, Yike Zhao, Aleksandar Armacki, Jie Chen, Ali H. Sayed

Abstract: We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak--Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of o… ▽ More We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak--Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our \textbf{ZOMA} framework is a multi-level unification, along the following directions: (i) \emph{estimator} - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) \emph{bias correction} - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) \emph{acceleration} - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of \textbf{ZOMA} leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations. △ Less

Submitted 10 June, 2026; originally announced June 2026.

Stochastically evolving ellipsoids with symmetries

Authors: Elisha B. Abuya, Nihar Gargava, Yufei Zhao

Abstract: We prove that there is a universal constant $c > 0$ such that, along an infinite sequence of dimensions $N$, there are lattice sphere packings in $\mathbb{R}^N$ of density at least $c N^2 \log\log N \, 2^{-N}$, improving the previous best bound due to Klartag by a $\log\log N$ factor. The proof follows Klartag's stochastic ellipsoid evolution process, subject to the cyclotomic symmetries introduce… ▽ More We prove that there is a universal constant $c > 0$ such that, along an infinite sequence of dimensions $N$, there are lattice sphere packings in $\mathbb{R}^N$ of density at least $c N^2 \log\log N \, 2^{-N}$, improving the previous best bound due to Klartag by a $\log\log N$ factor. The proof follows Klartag's stochastic ellipsoid evolution process, subject to the cyclotomic symmetries introduced by Venkatesh. △ Less

Submitted 4 June, 2026; v1 submitted 3 June, 2026; originally announced June 2026.

Comments: Statement of AI use included. The first author is using a pseudonym. Chat log is in the ancillary files as a pdf

MSC Class: 11H06

MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers

Authors: Yanshun Zhao, Xiaoyu Peng, Jiamin Jiang, Congcong Zhu, Jingrun Chen

Abstract: Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and… ▽ More Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok. △ Less

Submitted 2 June, 2026; originally announced June 2026.

Comments: ICML2026

Exponential Time Differencing Schemes for a Phase-Field Model of Multicomponent Membranes

Authors: Wangbo Luo, Zhonghua Qiao, Yanxiang Zhao

Abstract: In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introd… ▽ More In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation. △ Less

Submitted 2 June, 2026; originally announced June 2026.

Comments: 25 pages, 6 figures

High-Velocity Inverse Scattering for Nonlinear Schrödinger Equations with Spatially Dependent Nonlinearities

Authors: Satoshi Masaki, Yaxin Zhao

Abstract: We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states gene… ▽ More We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions. △ Less

Submitted 12 June, 2026; v1 submitted 1 June, 2026; originally announced June 2026.

Comments: 13 pages

MSC Class: Primary 35P25; Secondary 35Q55; 35R30; 35B40

Global well-posedness for 3D incompressible magneto-micropolar fluids without resistivity and spin viscosity in strip domains

Authors: Youyi Zhao

Abstract: The global existence of classical solutions to the 3D compressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain was recently established by Feng, Hong, and Zhu [Sci. China Math., 2024]. While Lin and Xiang [Sci. China Math., 2020] established global well-posedness for the 2D incompressible counterpart, the global well-posedness for the 3D incompressible… ▽ More The global existence of classical solutions to the 3D compressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain was recently established by Feng, Hong, and Zhu [Sci. China Math., 2024]. While Lin and Xiang [Sci. China Math., 2020] established global well-posedness for the 2D incompressible counterpart, the global well-posedness for the 3D incompressible case remains open. The analysis is rendered difficult by three major obstacles which are further compounded in the 3D case: the degeneracy induced by the lack of magnetic diffusion and spin viscosity; the coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure; and the interaction between the velocity, magnetic field, and pressure, where the pressure acts as a non-state variable. In this paper, by adapting the two-layer energy method of Guo and Tice [Arch. Ration. Mech. Anal., 2013] and the techniques employed in Feng et al., together with refined trace estimates, we overcome these difficulties and establish the global well-posedness of classical solutions to the 3D incompressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain. Moreover, we demonstrate the algebraic time-decay of solutions toward the equilibrium. △ Less

Submitted 20 May, 2026; originally announced May 2026.

Global well-posedness for 3D compressible and incompressible micropolar fluids without angular viscosity in strip domains

Authors: Youyi Zhao

Abstract: This paper investigates an initial-boundary value problem for three-dimensional (3D) micropolar fluids in a strip domain, including both the compressible and the (homogeneous and inhomogeneous) incompressible cases in the absence of angular viscosity. The analysis is rendered difficult by two major obstacles: the degeneracy induced by vanishing angular viscosity, and the strong coupling between mi… ▽ More This paper investigates an initial-boundary value problem for three-dimensional (3D) micropolar fluids in a strip domain, including both the compressible and the (homogeneous and inhomogeneous) incompressible cases in the absence of angular viscosity. The analysis is rendered difficult by two major obstacles: the degeneracy induced by vanishing angular viscosity, and the strong coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure. Moreover, the presence of physical boundaries in the strip domain further compounds these obstacles. While the global well-posedness of the 2D incompressible Cauchy problem has been established in the literature, no results are available for the 3D system and the initial-boundary value problem in both two and three dimensions, particularly in the compressible case. By exploiting the intrinsic structure of the system and establishing delicate energy estimates, we overcome these difficulties and prove the global well-posedness of strong solutions near equilibrium in a strip domain. △ Less

Submitted 24 July, 2026; v1 submitted 20 May, 2026; originally announced May 2026.

Relaxation via Separable Estimators: Arithmetic and Implementation

Authors: Yanlin Zha, Mario Eduardo Villanueva, Boris Houska, Benoît Chachuat

Abstract: This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in t… ▽ More This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation's tightness with its computational cost. △ Less

Submitted 11 May, 2026; originally announced May 2026.

MSC Class: 26B40; 41A30; 90C26; 90C59

Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs

Authors: Yuxuan Zhao, Yulong Lu

Abstract: We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a stron… ▽ More We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a strong solution in a Hölder space and using with a suitably constructed prior on the neural network weights, we prove that the posterior distribution concentrates around the exact solution at a near-minimax rate. Furthermore, the chosen prior is rate-adaptive: the posterior contracts at an (almost) optimal rate without prior knowledge of the smoothness level of the exact solution. Our results provide statistical guarantees for uncertainty quantification of PDEs via Bayesian PINNs. △ Less

Submitted 9 May, 2026; originally announced May 2026.

Higher-Order Flexible Configurations of Planar Parallel Manipulators Constructed by Averaging

Authors: Yudi Zhao, Georg Nawratil

Abstract: This paper investigates singular configurations of planar 3-RPR parallel manipulators, which result from applying the averaging technique to solution pairs of their direct kinematic problem. Without computing the zeros of the corresponding degree 6 polynomial we parametrize the input pairs and determine their relative orientation in a way that the flexion order of the averaged configurations incre… ▽ More This paper investigates singular configurations of planar 3-RPR parallel manipulators, which result from applying the averaging technique to solution pairs of their direct kinematic problem. Without computing the zeros of the corresponding degree 6 polynomial we parametrize the input pairs and determine their relative orientation in a way that the flexion order of the averaged configurations increases. Moreover, the obtained results are visualized for concrete examples. The presented methodology can also be used for studying the spherical and spatial analogues of planar 3-RPR parallel manipulators. △ Less

Submitted 4 May, 2026; originally announced May 2026.

Comments: This manuscript represents the full complementary version of the paper of the same title submitted to the International Conference on Geometry and Graphics 2026 (ICGG 2026)

MSC Class: 70B15

Finite-time blow-up in a class of chemotaxis systems with spatially heterogeneous diffusion sensitivity

Authors: Yashuang Zhao, Shijun Li, Shaopeng Xu

Abstract: \indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^β \bigtriangledown u)-\bigtriangledown\cdot(u^α \bigtriangledown v), 0=\bigtriangleup v-μ+u, \qquad μ:=\frac{1}{|Ω|}\int_Ωudx,\end{array}\right. \end{equation… ▽ More \indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^β \bigtriangledown u)-\bigtriangledown\cdot(u^α \bigtriangledown v), 0=\bigtriangleup v-μ+u, \qquad μ:=\frac{1}{|Ω|}\int_Ωudx,\end{array}\right. \end{equation} under homogeneous Neumann conditions in a ball $Ω=B_{R}(0)\subset \mathbb{R}^{n}$ with $α\ge 1$, $β>0$ and $n\ge 2$.\par \indent It is proved that any nonconstant nonnegative radial initial data $u_{0}\in C^θ(\overlineΩ)$, where $θ\in (0,1)$, there exists a radially symmetric classical solution of the system (0.1) in $(Ω\setminus \{ 0 \})\times (0,T)$ for some $T>0$; moreover, if the initial values $u_{0}\in C^{1+θ}(\overlineΩ)$ for some $θ\in (0,1)$ and satisfy a certain compatibility criterion and are radially decreasing, then this solution is bounded and unique in $(Ω\setminus \{ 0 \})\times (0,T^{*})$ with $T^{*}<T$.\par Finally, it is found that the initial mass corresponding to this parabolic-elliptic problem (0.1) is sufficiently concentrated to allow the solution to blow up in finite time. △ Less

Submitted 30 April, 2026; originally announced April 2026.

The category of Whittaker modules over the Cartan Type Lie algebra $\bar{S}_2$

Authors: Xiaoyao Zheng, Yufang Zhao, Genqiang Liu

Abstract: The Lie algebra $\bar{S}_2$ of polynomial vector fields on $\mathbb{C}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker $\bar{S}_2$-modules that are locally finite over $\text{span}\{\frac{\partial}{\partial t_1}, \frac{\partial}{\partial t_2}\}$. We first show that each block $Ω^{\widetilde{S}_2}_{\mathbf{a}}$ of the category of… ▽ More The Lie algebra $\bar{S}_2$ of polynomial vector fields on $\mathbb{C}^2$ with constant divergence is an important Cartan type Lie algebra. In this paper, we study Whittaker $\bar{S}_2$-modules that are locally finite over $\text{span}\{\frac{\partial}{\partial t_1}, \frac{\partial}{\partial t_2}\}$. We first show that each block $Ω^{\widetilde{S}_2}_{\mathbf{a}}$ of the category of $(A_2, \bar{S}_2)$-Whittaker modules with finite-dimensional Whittaker vector spaces is equivalent to the category of finite-dimensional modules over the parabolic subalgebra $\bar{S}_2^{\geq 0}$. Then we classify all simple Whittaker $\bar{S}_2$-modules in every block $Ω^{\bar{S}_2}_{\mathbf{a}}$ . Finally, we establish an equivalence between $Ω^{\bar{S}_2}_{\mathbf{1}}$ and the category $H_{\mathbf{1}}$-fmod of finite-dimensional modules over an associative algebra $H_{\mathbf{1}}$, whose generators are also determined. △ Less

Submitted 17 June, 2026; v1 submitted 27 April, 2026; originally announced April 2026.

PRP, HS and LS Conjugate Gradient Methods for Interval-Valued Multiobjective Optimization Problems

Authors: Tapas Mondal, Debdulal Ghosh, Zai-Yun Peng, Yong Zhao

Abstract: In this article, we develop an efficient algorithm based on three special variants of the nonlinear conjugate gradient method, namely, the Polak--Ribiere--Polyak, Hestenes--Stiefel, and Liu--Story schemes for computing Pareto critical points in unconstrained interval-valued multiobjective optimization problems. The proposed algorithm incorporates a Wolfe line search strategy to determine a suitabl… ▽ More In this article, we develop an efficient algorithm based on three special variants of the nonlinear conjugate gradient method, namely, the Polak--Ribiere--Polyak, Hestenes--Stiefel, and Liu--Story schemes for computing Pareto critical points in unconstrained interval-valued multiobjective optimization problems. The proposed algorithm incorporates a Wolfe line search strategy to determine a suitable range of step size that satisfies the standard Wolfe conditions. For each of the proposed variants of the nonlinear conjugate gradient method, we establish rigorous global convergence results under appropriate assumptions. To demonstrate the effectiveness of the proposed methods, we conduct numerical experiments on a set of benchmark test problems and present a comprehensive performance profile analysis. △ Less

Submitted 27 April, 2026; originally announced April 2026.

Comments: 26 pages, 3 figures, and 2 tables

Formulae for the Drazin inverse of Modified Tensors via the Einstein Product

Authors: Yue Zhao, Daochang Zhang, Dijana Mosic

Abstract: This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literat… ▽ More This paper establishes exact expressions for the Drazin inverse of the modified tensor $\mathcal A-\mathcal C*_N\mathcal D^D*_N\mathcal B$ via the Einstein product, formulated using the Drazin inverse of $\mathcal A$ and the generalized Schur complement $\mathcal D-\mathcal B*_N\mathcal A^{D}*_N\mathcal C$, providing a comprehensive generalization and unification of existing results in the literature for the case when the tensors are of order two. Furthermore, the findings reduce to the classical Sherman-Morrison-Woodbury formula in the special case of second-order tensors. Finally, we give an example to illustrate our new explicit expression. △ Less

Submitted 23 April, 2026; originally announced April 2026.

Towards Fully Parameter-Free Stochastic Optimization: Grid Search with Self-Bounding Analysis

Authors: Yuheng Zhao, Yu-Hu Yan, Amit Attia, Tomer Koren, Lijun Zhang, Peng Zhao

Abstract: Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such method… ▽ More Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such methods as ``partially parameter-free''. In this work, we target achieving ``fully parameter-free'' methods, i.e., the algorithmic inputs do not need to satisfy any unverifiable condition related to the true problem parameters. We propose a powerful and general grid search framework, named \textsc{Grasp}, with a novel self-bounding analysis technique that effectively determines the search ranges of parameters, in contrast to previous work. Our method demonstrates generality in: (i) the non-convex case, where we propose a fully parameter-free method that achieves near-optimal convergence rate, up to logarithmic factors; (ii) the convex case, where our parameter-free methods are competitive with strong performance in terms of acceleration and universality. Finally, we contribute a sharper guarantee for the model ensemble, a final step of the grid search framework, under interpolated variance characterization. △ Less

Submitted 18 April, 2026; originally announced April 2026.

Well-posedness of the compressible boundary layer equations with data in the Gevrey class

Authors: Ya-Guang Wang, Yi-Lei Zhao

Abstract: This paper is devoted to the study of the compressible boundary layer equations in the Gevrey-2 solution space. Compared to the classical Prandtl equation, the additional complexity arises from the strong interaction between viscous layer and thermal layer. By introducing new auxiliary functions and observing the cancellation mechanism to overcome the loss of derivatives, we show the local existen… ▽ More This paper is devoted to the study of the compressible boundary layer equations in the Gevrey-2 solution space. Compared to the classical Prandtl equation, the additional complexity arises from the strong interaction between viscous layer and thermal layer. By introducing new auxiliary functions and observing the cancellation mechanism to overcome the loss of derivatives, we show the local existence and uniqueness of the solution in the Gevrey-2 space in the tangential variable and Sobolev regularity in the normal variable by using a direct energy method. △ Less

Submitted 17 April, 2026; originally announced April 2026.

MSC Class: 35Q35; 35M13; 35B65; 76N20

GTH Algorithm, Censored Markov Chains, and $RG$-Factorization in Block-Form

Authors: Qihui Bu, Yiqiang Q. Zhao

Abstract: In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance quantities in both theory and applications. This algorithm later became the well-known GTH algorithm (or the state-reduction method) in the literature, becoming one of t… ▽ More In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance quantities in both theory and applications. This algorithm later became the well-known GTH algorithm (or the state-reduction method) in the literature, becoming one of the standard algorithms in applied probability. Later, this algorithm was extended to deal with the stationary distributions of block-structured Markov chains with repeating rows. In this paper, we focus on the block-form GTH algorithm and organize it into two parts. In the first part, we connect the block-form GTH algorithm to censored Markov chains and the block-form $RG$-factorization. We show that the forward block-elimination and back block-form substitution of the block-form GTH algorithm are equivalent to solving a system formulated using the $RG$-factorization in two steps. We also show that this connection remains valid when the block-form GTH algorithm is extended to infinite-state Markov chains. It is well known that censoring an infinite-state Markov chain to a finite state space yields a stationary distribution that provides a best approximation to the stationary distribution of the original infinite-state Markov chain. In the second part, we first derive an explicit expression for the censored Markov chain from the infinite state space to a finite space for Markov chains of $M/G/1$ type. Based on this expression, we propose a renormalized approximated censored transition matrix (RA-CM). The resulting stationary distribution is shown to be asymptotically optimal in terms of approximation error. We compare the approximation error of the RA-CM with the error arising from the last-block-column augmentation. △ Less

Submitted 15 April, 2026; originally announced April 2026.

Comments: dedicated to Dr. Winfried Grassmann

MSC Class: 60-08; 60J22

Ordinary Least Squares is a Special Case of Transformer

Authors: Xiaojun Tan, Yuchen Zhao

Abstract: The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter better describes Transformer's basic nature: Ordinary Least Squares (OLS) is a special case of the single-layer Linear Transformer. Using the spectral decomposition… ▽ More The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter better describes Transformer's basic nature: Ordinary Least Squares (OLS) is a special case of the single-layer Linear Transformer. Using the spectral decomposition of the empirical covariance matrix, we construct a specific parameter setting where the attention mechanism's forward pass becomes mathematically equivalent to the OLS closed-form projection. This means attention can solve the problem in one forward pass, not by iterating. Building upon this prototypical case, we further uncover a decoupled slow and fast memory mechanism within Transformers. Finally, the evolution from our established linear prototype to standard Transformers is discussed. This progression facilitates the transition of the Hopfield energy function from linear to exponential memory capacity, thereby establishing a clear continuity between modern deep architectures and classical statistical inference. △ Less

Submitted 15 April, 2026; originally announced April 2026.

Stable maps, multiplicities, and compactified Jacobians

Authors: Yifan Zhao

Abstract: Let $C$ be a complex projective integral curve with planar singularities. In this note, we study numerical relations among its versal deformation space, moduli space of stable maps, and compactified Jacobian. In particular, we correct a statement by Fantechi--Göttsche--van Straten on the multiplicity of the $δ$-constant stratum of the versal deformation space at $[C]$. We also give a necessary and… ▽ More Let $C$ be a complex projective integral curve with planar singularities. In this note, we study numerical relations among its versal deformation space, moduli space of stable maps, and compactified Jacobian. In particular, we correct a statement by Fantechi--Göttsche--van Straten on the multiplicity of the $δ$-constant stratum of the versal deformation space at $[C]$. We also give a necessary and sufficient condition for the original claim to hold. △ Less

Submitted 7 April, 2026; originally announced April 2026.

Comments: 10 pages

Reinforcement Learning for Speculative Trading under Exploratory Framework

Authors: Yun Zhao, Alex S. L. Tse, Harry Zheng

Abstract: We study a speculative trading problem within the exploratory reinforcement learning (RL) framework of Wang et al. [2020]. The problem is formulated as a sequential optimal stopping problem over entry and exit times under general utility function and price process. We first consider a relaxed version of the problem in which the stopping times are modeled by the jump times of Cox processes driven b… ▽ More We study a speculative trading problem within the exploratory reinforcement learning (RL) framework of Wang et al. [2020]. The problem is formulated as a sequential optimal stopping problem over entry and exit times under general utility function and price process. We first consider a relaxed version of the problem in which the stopping times are modeled by the jump times of Cox processes driven by bounded, non-randomized intensity controls. Under the exploratory formulation, the agent's randomized control is characterized via the probability measure over the jump intensities, and their objective function is regularized by Shannon's differential entropy. This yields a system of the exploratory HJB equations and Gibbs distributions in closed-form as the optimal policy. Error estimates and convergence of the RL objective to the value function of the original problem are established. Finally, an RL algorithm is designed, and its implementation is showcased in a pairs-trading application. △ Less

Submitted 2 April, 2026; originally announced April 2026.

Comments: 37 pages, 14 figures

MSC Class: 93E20; 60G40; 91G80; 60J60; 91G60; 68T07; 93E35

From molecular dynamics to kinetic models: data-driven generalized collision operators in 1D3V plasmas

Authors: Yue Zhao, Guosheng Fu, Huan Lei

Abstract: We present a data-driven approach for constructing generalized collisional kinetic models for inhomogeneous plasmas in one-dimensional physical space and three-dimensional velocity space (1D-3V). The collision operator is directly learned from micro-scale molecular dynamics (MD) and accurately accounts for the unresolved particle interactions over a broad range of plasma conditions. Unlike the sta… ▽ More We present a data-driven approach for constructing generalized collisional kinetic models for inhomogeneous plasmas in one-dimensional physical space and three-dimensional velocity space (1D-3V). The collision operator is directly learned from micro-scale molecular dynamics (MD) and accurately accounts for the unresolved particle interactions over a broad range of plasma conditions. Unlike the standard Landau operator, the present operator takes an anisotropic, non-stationary form that captures the heterogeneous collisional energy transfer arising from the many-body interactions, which is crucial for plasma kinetics beyond the weakly coupled regime. Efficient numerical evaluation is achieved through a low-rank tensor representation with $O(N \log N)$ computational complexity. The constructed kinetic equation strictly preserves conservation laws and physical constraints and therefore, enables us to develop an explicit second-order, energy-conserving scheme that ensures fully discrete conservation of mass and total energy. Numerical results demonstrate that the present model accurately predicts both transport coefficients and several 1D-3V kinetic processes compared with MD simulations across a broad range of densities and temperatures in spatially inhomogeneous settings. This work provides a systematic pathway for bridging micro-scale MD and inhomogeneous plasma kinetic descriptions where empirical models show limitation. △ Less

Submitted 29 March, 2026; originally announced March 2026.

SDP Feasibility Problems and sos Representation Ranks for OT-FKM Type Isoparametric Polynomials

Authors: Jianquan Ge, Kai Jia, Yuyang Zhao

Abstract: Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underl… ▽ More Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underlying Clifford system, and in the sos cases we obtain quantitative rank bounds for sos representations, with rigidity when m >= 3. △ Less

Submitted 17 June, 2026; v1 submitted 22 March, 2026; originally announced March 2026.

Comments: 31 pages; removed the auxiliary computer-algebra appendix and replaced the verification of the (6,8) case by a self-contained blockwise matrix computation; results unchanged

MSC Class: 53C40; 14P99; 90C22; 15A63

The Erdős-Ginzburg-Ziv theorem constant of finite groups

Authors: Yang Zhao, Guoqing Wang

Abstract: Let $G$ be a multiplicatively written finite group of order $n$. The Erdős-Ginzburg-Ziv Theorem constant of the group $G$, denoted $\mathsf E(G)$, is defined as the smallest positive integer $\ell$ with the following property: for any given sequence $(g_1,\ldots,g_{\ell})$ over $G$, there exist $n$ distinct integers $i_1,\ldots,i_n\in \{1,\ldots,\ell\}$ such that the product of… ▽ More Let $G$ be a multiplicatively written finite group of order $n$. The Erdős-Ginzburg-Ziv Theorem constant of the group $G$, denoted $\mathsf E(G)$, is defined as the smallest positive integer $\ell$ with the following property: for any given sequence $(g_1,\ldots,g_{\ell})$ over $G$, there exist $n$ distinct integers $i_1,\ldots,i_n\in \{1,\ldots,\ell\}$ such that the product of $g_{i_1},\ldots,g_{i_n}$, in some order, is the identity element of $G$. The Erdős-Ginzburg-Ziv Theorem constant originates from the celebrated additive theorem proved by Erdős, Ginzburg and Ziv in 1961, which amounts to proving $\mathsf E(G)\leq 2|G|-1$ holds in case that $G$ is abelian. It is also well-known that $\mathsf E(G)=2|G|-1$ holds for all finite cyclic groups. In 2010, Gao and Li [J. Pure Appl. Algebra] conjectured that $\mathsf E(G)\leq \frac{3|G|}{2}$ for every finite non-cyclic group $G$. In this paper, we confirm the conjecture for all non-cyclic groups $G$ whose order is not divisible by four, and characterize the groups achieving the equality $\mathsf E(G)=\frac{3|G|}{2}$ as those with a cyclic subgroup of index two. △ Less

Submitted 22 March, 2026; originally announced March 2026.

Comments: 13 pages