Papers

We introduce a coordinate-wise version of the elephant random walk on the $k$-dimensional discrete hypercube $Q_k=\{-1,1\}^k$. At each global time step, one coordinate is selected uniformly at random and updated according to an elephant-type memory rule using only the past values of that coordinate. The resulting process is a nearest-neighbor walk with possible holding on the hypercube, but it is not Markovian on $Q_k$ because the transition probabilities depend on coordinate-wise empirical histories. We show that, when all memory parameters satisfy $p_i<1$, the coordinate-wise memory biases vanish almost surely. Consequently, the time-dependent transition kernels of the walk are asymptotically close in total variation to the memoryless coordinate-refresh kernel. Using this perturbation argument and the uniform ergodicity of the refresh chain, we prove a weak law of large numbers for bounded observables. We then study the Doob martingale associated with a bounded observable and prove a martingale central limit theorem and functional central limit theorem. The limiting variance is determined by the refresh kernel and the uniform measure on the hypercube, and is completely independent of the memory parameters $p_1,p_2,\dots,p_k.$

We establish the $C^1$-regularity of solutions to the obstacle problems associated with $p$-Laplacian type equations, where $1<p<\infty$. Specifically, we prove that the gradient of the solution is continuous under a Dini mean oscillation ($\mathsf{DMO}$) type condition on the data, which includes the coefficient matrix, the source term, and the obstacle function. This result relaxes the classical Dini continuity assumption on the data to a more general mean oscillation condition.

This paper investigates the robustness of quantum observability on compact hyperbolic surfaces under $L^2$ potential perturbations. Since $L^2$ regularity is strictly subcritical in dimension two, for every non-empty open set $\Omega\subset M$ and $T>0$, the solution of $(i\partial_t+\Delta-V)u=0$ satisfies the space-time observability estimate $$ \|u_0\|_{L^2(M)}^2 \leq C\int_0^T \|e^{-it(-\Delta+V)}u_0\|_{L^2(\Omega)}^2\,dt. $$ Our results provide a quantitative confirmation that the delocalization of high-energy quantum states on negatively curved manifolds is robust against $L^2$-class microscopic scattering. The proof combines the hyperbolic dynamics of the geodesic flow with semiclassical analysis. A key ingredient is an $L^4$ spectral cluster estimate with an arbitrarily small loss, obtained by exploiting Jacobi field analysis and Bourgain–Demeter $l^2$-decoupling. This estimate allows us to obtain refined spectral localized semiclassical Strichartz estimates adapted to rough potentials and to show that the potential's contribution vanishes in the propagation of semiclassical measures. The full-support property of invariant semiclassical measures on hyperbolic surfaces then yields the desired observability by contradiction. By the Hilbert Uniqueness Method, the corresponding internal controllability result follows.

We study a random compositional model for the growth of affine regions in deep piecewise-linear networks. The model is generated by i.i.d.\ perturbations of the symmetric height-one tent map, and the main observable is the number \(N_n\) of affine pieces after \(n\) layers. We prove the existence of a submultiplicative pressure for \(N_n\), yielding exponential upper bounds for both tails of \(n^{-1}\log N_n\). The same argument applies to abstract submultiplicative complexity observables and gives higher-dimensional extensions for convex-polytopal affine-cover counts and worst-line affine-piece counts. Since the true branch count has no matching supermultiplicative inequality, lower bounds require a separate certified construction. We introduce a finite-state defect process that records branches whose future splitting can be guaranteed, and use bridge words to obtain constructive upper-tail lower bounds. In a uniformly favorable small-noise regime, this process is governed by a companion matrix whose Perron root tends to \(2\), implying eventual exclusion of lower tails below \(\log 2-\xi\).

Tensor equations involving both left and right tensor operators arise naturally in applications where multidimensional data are coupled through transformations acting from both sides. Motivated by the need to analyze such equations, this manuscript investigates a detailed solution analysis of $\mathcal{A}\ltimes \mathcal{X}\ltimes \mathcal{B} = \mathcal{C}$ via the semi-tensor product with the $t$-product. The manuscript provides a more general framework that includes, as a special case, the tensor equation $\mathcal{A}\ltimes\mathcal{X}=\mathcal{B}$ considered in \cite{J.FathitensorequationAX=BunderSTP}. Necessary and sufficient conditions are derived for the existence of vector and matrix valued solutions. Explicit compatibility conditions are established, and constructive Moore-Penrose-inverse-based algorithms are provided for the vector and matrix valued cases. The computational complexity and execution times are compared for Discrete Fourier Transform(DFT) and Fast Fourier Transform(FFT) based \(t\)-product implementations. The practical relevance of the proposed framework is further demonstrated through a color image deblurring application. Examples throughout illustrate the theoretical results.

We show that Mordell curves with arithmetic progressions in the $y$-coordinate of length $7$ have not been ruled out by previous work, and we give non-isomorphic families of Mordell curves with $y$-arithmetic progressions of length $6$. We also construct parametric families of elliptic curves of moderate rank, with subfamilies possessing rational points in arithmetic progression.

In this article, we investigate the cumulative residual extropy associated with linear consecutive k-out-of-n:G systems, which play an important role in reliability theory and engineering applications. We first derive explicit expressions for the proposed measure and examine the behavior of cumulative residual extropy under a variety of stochastic orderings. In addition, several bounds and meaningful results of the characterization are established. Moreover, we also introduced the dynamic version of the cumulative residual extropy and explored the relationship between the proposed dynamic version and the mean residual life function. Fromaninferentialperspective, wedevelopanonparametricestimationprocedure for the cumulative residual extropy and establish the corresponding consistency properties of the estimator. The finite-sample performance of the proposed estimator is further investigated through extensive Monte Carlo simulation studies under different parametric settings and validated through a real dataset.

We collect rank two difference-differential Lax pairs for classical Painlevé equations in the literature and put each in $2\times 2$ matrix form with the coefficient matrix of the spectral equation a degree two matrix polynomial. We describe and apply a general method to obtain integrals of motion in characteristic $p$ from these Lax pairs. For every relevant Painlevé equation, this leads to a countable list of integrals of motion, with one entry for each prime $p$.

In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training. This method generalizes the LSGD method for DeepONets. Since MIONet is the sum of the entrywise product of multiple branch networks and a trunk network, it can be viewed as a multilinear function with respect to the last layer parameters of each branch network. These sets of parameters can be optimized using the alternating least squares method, where we solve the LS system for a single branch network in turn. To handle the large-sized system matrix, we introduce Kronecker and Khatri-Rao products and tensor permutation matrices to factor the large matrix into small ones. Our method is compatible with a general type of $L^2$ loss with regularization terms for the last layer parameters of each branch, where linear operators can be applied to the MIONet output in each loss term.

In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers. Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons. Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.

Let $n$ be a positive integer. Given an $n$-rigid subcategory $\mathcal{M}$ of an algebraic triangulated category $\mathcal{T}$, we explicitly construct an extriangulated functor from the $(n+1)$-term subcategory of $\mathcal{T}$ generated by $\mathcal{M}$ to the full subcategory of $(n+1)$-term complexes in the bounded homotopy category $K^b(\mathcal{M})$, which restricts to the identity on $\mathcal{M}$. In the broader context of reduced $(n-1)$-Auslander extriangulated categories, we provide necessary and sufficient conditions for such a functor to be full, in which case it induces an equivalence of extriangulated categories modulo a certain ideal. Furthermore, we establish a mutation-compatible bijection between the silting subcategories of these categories. Finally, we apply these results to $n$-cluster tilting subcategories and $n$-cluster tilting objects in $(n+1)$-Calabi-Yau categories.

Given any $\varepsilon > 0$, let $M(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$, with all entries independent and $\varepsilon$-balanced (lying in each residue class mod $p$ with probability at most $1-\varepsilon$). Wood proved that as $n \to \infty$ the distribution of $\mathrm{cok}(M(n))$ approaches Cohen and Lenstra's conjectured distribution of class groups. Given $\alpha,\beta >0$ such that $\alpha + \beta <1$, we prove that the distribution of $\mathrm{cok}(M(n))$ still approaches the Cohen–Lenstra distribution even if we weaken the hypothesis by allowing up to $\alpha n$ entries per column and up to $\beta n$ entries per row of $M(n)$ to not be $\varepsilon$-balanced. We also weaken the independence condition by allowing certain types of dependence between the entries of each column. In addition, we prove that, for any $\delta > 0$, the cokernels of random band matrices of width $\log(n)^{1+\delta}$ with $\varepsilon$-balanced entries in the band and arbitrary entries outside of it will also approach the Cohen–Lenstra distribution, which answers a question of Kang–Lee–Yu.

For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$

In this paper, we study the isoperimetric inequalities for the Robin eigenvalues of the weighted Laplacian with positive Robin parameter in the Euclidean space $\R^n$ and the hyperbolic space $\mathbb{H}^n$, respectively. More precisely, we prove that among all bounded Lipschitz domains with fixed weighted volume, the geodesic ball centered at the origin minimizes the first Robin eigenvalue of the weighted Laplacian, provided that the Robin parameter and the radial log-convex density satisfy suitable conditions. Furthermore, we show that the second Robin eigenvalue is bounded below by the first Robin eigenvalue of the geodesic ball centered at the origin with half the weighted volume. Our results extend classical Faber-Krahn inequalities to the setting of weighted spaces with log-convex densities. We also derive a lower bound for the second Robin eigenvalue in terms of the first eigenvalue of the centered ball with half the weighted volume.

Model Predictive Path Integral (MPPI) control is directly implementable on nonlinear systems because its online update requires only forward rollouts of the dynamics, not gradients, linearizations, or convex optimization. However, this algorithmic flexibility does not by itself provide a closed-loop stability certificate. This paper establishes such a certificate through a stability-inheritance argument. We assume that there exists a deterministic nonlinear MPC policy whose disturbance-free closed loop is certified by a Control Lyapunov Function terminal cost and a contraction metric, and we show that finite-sample MPPI inherits the nominal contraction when its sampling-based update approximates this reference policy with sufficient accuracy. The approximation error decomposes into a finite-temperature bias floor and a Monte Carlo term that vanishes at the inverse square-root rate in the sample count. Under an explicit small-gain condition, the resulting MPPI closed loop satisfies a finite-horizon, high-probability localized mean practical stability bound with residual floors due to MPPI approximation error, Gaussian process noise, and bad sampling events. The paper also gives an ISS-type restatement and a finite-horizon design procedure for choosing the localization set, temperature, and sample count.

Let $p$ be a prime number. We introduce a sparseness condition on the supports of $p$-adic Hahn series, and prove that this condition implies transcendence over $\breve{\mathbf Q}_p$, the completed maximal unramified extension of $\mathbf{Q}_p$. As an application, we prove the order-type conjecture of $\mathbf{Q}_p$-algebraic $p$-adic Hahn series with bounded support under the condition that the support has only finitely many accumulation points. All results in this paper have been fully formalized in the Lean theorem prover (v 4.31.0), building over Mathlib.

We study invariant measures for soliton systems described by Mealy automata. Motivated by recently introduced soliton models associated with 2-letter, 3-state Mealy automata, we formulate the time evolution induced by Mealy automata on bi-infinite configuration spaces. We provide sufficient conditions for the invariance of Bernoulli product measures and derive a criterion for the invariance of two-sided space-homogeneous Markov distributions. We then apply these general results to three soliton models, which can be interpreted as variants of the box-ball system (BBS). For two of these models, BBS-S(2) and BBS-V(2), we prove that Bernoulli product measures are invariant. For the remaining model, BBS-C(2), we establish a more general result: the invariance of two-sided space-homogeneous Markov distributions, which include Bernoulli product measures as a special case. Furthermore, for all three models, we compute the phase shift associated with the interaction of two solitons, as well as the velocity of an isolated soliton. Although the latter has already been studied previously, both quantities constitute fundamental characteristics for understanding the generalized hydrodynamics of these systems. These results provide a foundation for the study of invariant measures, generalized Gibbs ensembles, and generalized hydrodynamic behavior in Mealy-automaton soliton systems.

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. A biclique is a set of vertices inducing all possible pairs of opposite arcs. For a digraph $D$, define $\Delta(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We prove that, for every fixed integer $b\in\mathbb{N}$, every digraph $D$ with $\Delta(D) = \Delta$ being sufficiently large with respect to $b$ either contains a biclique whose size exceeds $\Delta-2b$ or has dichromatic number at most $\Delta-b$. This extends a classical result of Reed to the directed setting and supports a conjecture of the present authors. Furthermore, the theorem is tight, as for all integers $b$ and $\Delta\geq 3b$ there exists a digraph $D$ with $\Delta(D)= \Delta$, dichromatic number $\Delta-b+1$, and whose largest biclique has size $\Delta-2b+1$.

The paper aims to provide a categorification of the Brenti–Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545–556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of $(\mathbb{C}^r)^{\otimes n}$ and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.

We consider the recently constructed Segal–Sugawara vectors for the orthosymplectic Lie superalgebras. We calculate their images with respect to the Harish-Chandra homomorphism and extend this calculation to the associated Sugawara operators and Casimir elements. We also produce higher Gaudin Hamiltonians and elements of the quantum shift-of-argument subalgebras in the orthosymplectic enveloping algebra. In the Appendix, we review analogous results for the general linear Lie superalgebras.

Let \(G\) be a graph admitting a perfect matching. An edge is called a {\it \(k\)-matching edge} if it belongs to exactly \(k\) perfect matchings, and a {\it \(k^{+}\)-matching edge} if it belongs to at least \(k\) perfect matchings. Thus, an admissible edge is a \(1^{+}\)-matching edge, and a connected graph is matching covered if every edge is admissible. We call a connected graph {\it \(k\)-matching covered} if every edge is a \(k\)-matching edge; in particular, a \(2\)-matching covered graph is called matching double covered. Motivated by matching-covered graph theory and the Berge–Fulkerson conjecture (1970s), we introduce the class \(\mathfrak{B}\) of connected graphs in which every edge is either a \(1\)-matching edge or a \(2\)-matching edge, and no perfect matching contains edges of both types. In particular, every matching double covered graph belongs to \(\mathfrak{B}\). Using ear decompositions and tight-cut decompositions, we establish a complete structural characterization of graphs in \(\mathfrak{B}\). These characterizations reveal how restrictions on the number of perfect matchings containing each edge determine the global structure of the corresponding matching-covered graphs.

The projected linear system solver (PLSS), by incrementally appending columns to a random or deterministic sketching matrix, provides an attractive finite termination property for consistent linear systems. Nevertheless, a critical computational bottleneck of PLSS is accessing the whole coefficient matrix per iteration, making it prohibitive for extremely large-scale problems or applications with missing data. To alleviate this limitation, we propose a unified randomized PLSS (RPLSS) framework, built upon the tailored randomized row or column selection strategies that require only partial matrix information per iteration, for solving a general linear system, whether it is under- or overdetermined, and whether it is consistent or not. Within this framework, we develop a randomized Gaussian Kaczmarz method and its extended variant as row-action solvers, and randomized coordinate descent variants as column-action solvers. Theoretically, we prove that our methods inherit the finite termination property of PLSS, while achieving an exponential convergence rate, overcoming the sluggish convergence inherent in conventional randomized Kaczmarz and coordinate descent methods. Numerical experiments demonstrate the superiority of our method against state-of-the-art randomized methods, particularly in scenarios with large missing data.

We study the nonlinear stability of an equilibrium with a background magnetic field for the three-dimensional compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$, in the strongly anisotropic regime where the velocity is dissipated only in the horizontal directions and the magnetic field is diffused in a single direction. We prove that for initial data sufficiently close to the equilibrium in a Sobolev space, the system admits a unique global-in-time solution that remains close to the equilibrium and enjoys quantitative dissipation estimates. The proof overcomes the severe lack of dissipation through two mechanisms: the background magnetic field is shown to generate enhanced dissipation for the magnetic field and the density, while a nonlinear cancellation mechanism is devised to resolve the loss of vertical derivatives caused by the compressible coupling.

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

We solve a classical problem posed by F. Klein and studied by A. Hurwitz concerning the construction of linear ordinary differential equations associated with modular transformations of fixed degree. For every odd integer $N\ge 3$ (respectively, even integer $N\ge 4$), we construct a canonical invariant model of the modular curve $X(N)=\mathbb{H}/\Gamma(N)$ (respectively, $X_H(N)=\mathbb{H}/H(N)$ where $H(N)=\Gamma(N)\cap\Gamma_0^0(2N)$), together with a linear ordinary differential equation with rational coefficients whose Schwarz map parametrizes this model and whose projective monodromy group is the finite quotient $PSL_2(\mathbb{Z})/\tilde{\Gamma}(N)$ (respectively, $PSL_2(\mathbb{Z})/\tilde{H}(N)$). The construction is expressed in terms of invariant projective geometry and Picard-Vessiot theory and yields equations that are canonical up to projective equivalence. In this framework, Hurwitz's classical equation for degree $7$ appears as a special case of a general mechanism. The results place Klein's question within the modern theory of algebraic linear ordinary differential equations and provide a uniform geometric realization of modular transformation groups as projective differential Galois groups. As an application, we construct an explicit example of a linear ordinary differential equation associated with $X(9)$.

Let $G$ be a locally compact group and let $\widetilde{G}$ be a central extension that splits over a maximal compact subgroup $K$ of $G$. We derive an explicit cocycle that lifts the natural action of $G$ on the homogeneous space $G/K$ to an action of $\widetilde{G}$. As an application, for a non-Archimedean local field $F$, we construct a connected locally finite tree on which the metaplectic covers of $\operatorname{GL}_2(F)$ act by automorphisms, providing a geometric analog of the Bruhat–Tits tree of $\operatorname{GL}_2(F)$. Furthermore, under suitable transitivity assumptions, we prove that $(\widetilde{G},\widetilde{K})$ is a Gelfand pair. Finally, we describe the associated parabolic and contraction subgroups with respect to $\widetilde{G}$ from the perspective of the geometry of the constructed tree.

In this paper, we establish the power convexity of solutions to the complex Monge-Ampère equation $\det(u_{i\overline{j}})=1$ in a convex set in the complex 2 dimensional Euclidean space. Even the convexity result is only for 2 dimension, our approach is applicable to all dimensions.

The Stochastic-Oracle Turing Machine (SOTM) framework models AI-augmented computation as the interaction of a probabilistic Turing machine with an oracle whose responses are drawn from context-dependent distributions. This paper studies what an SOTM can achieve under two oracle-response schemes: in a cached-response oracle, each distinct query receives one response that is reused on later calls to the same query, while in a fresh-response oracle, each call returns an independent response. In both schemes, the SOTM first computes from its input and internal random source to generate its first query, then proceeds adaptively, computing from its query-response transcript (the record of queries issued and responses received) to generate each subsequent query or produce a final output. Cached responses impose two transcript-based ceilings on achievable performance: a correct-identification ceiling governed by the total variation distance between the transcript distributions induced by the hidden states of the oracle, and an output quality ceiling equal to the expected score of the best output the SOTM can compute from the transcript. Fresh responses can raise these ceilings by allowing repeated calls to accumulate independent evidence toward correct or high-quality outputs. In the binary single-informative-query case, the error probability decreases exponentially in the number of calls to the same query at the Chernoff rate. For output quality, query-count bounds characterize threshold stopping when the score function is incorporated as part of the SOTM, and majority-based amplification bounds characterize the binary candidate-output model when it is not. Together, the results identify how response reuse, transcript information, and access to the score function determine what an SOTM can compute and at what token cost.

We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.