arXiv preprints from January 1, 2026 through July 20, 2026 — 23:33:22 EST
Existence, uniqueness and classification is established for plane waves supported by an irreversible reaction which is a smooth function of local reactant and product concentrations (or prey and predator populations). Rudimentary analytic techniques are used to guarantee a unique plane wave at every wavespeed $V>V_*$ above some...
The Ramsey number $R(s, t)$ is the smallest positive integer $n$ such that every graph on $n$ vertices contains either a clique of size $s$ or an independent set of size $t$. An $R(s,t)$-critical graph is a graph on $R(s,t)-1$ vertices that contains neither a clique of size $s$ nor an independent set of size $t$. It is known that...
Decentralized optimization has become a fundamental tool for large-scale learning systems; however, most existing methods rely on the classical Lipschitz smoothness assumption, which is often violated in problems with rapidly varying gradients. Motivated by this limitation, we study decentralized optimization under the generalized...
We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal left/right ideals which are not an ideal of $T$; in particular $T$ has at least $|R/M|+1$ distinct...
This paper investigates the Waring problem of harmonic polynomials. By characterizing the annihilating ideal of a homogeneous harmonic polynomial, i.e., a real binary form that is in the kernel of the Laplacian, we show that its Waring rank equals its degree. Moreover, we show that any linear form can appear in a minimal Waring...
This paper systematically investigates the optimal harvesting of a stochastic Lotka-Volterra competition model with periodic coefficients. Sufficient conditions for the extinction and persistence in the time average of each species are established. Using Khasminskii's stability theory with suitable Lyapunov functions, we establish...
Baseilhac-Benedetti, following ideas of Kashaev, introduced invariants of pseudo-Anosov homeomorphisms of punctured hyperbolic surfaces that depend on a complex root of unity of odd order. Around the same time, Bonahon-Liu introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. A little later, Dimofte...
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(α)\midα\in\mathbb{F}_{q}\}$ and denote the cardinality of this set as $N_{f}$. A much sharper bound for $N_{f}$ is established in this paper. In...
Geometric Quantization is a term used to describe a wide collection of techniques dating back to the 1960s in the work of Kirillov, Kostant, and Souriau, which take symplectic manifolds and produce complex vector spaces. The name comes from the natural interpretation of symplectic manifolds as the phase spaces of classical mechanical...
Stochastic dominance has been studied extensively, particularly in the finance and economics literature. In this paper, we obtain two results. First, necessary conditions for higher-order inverse stochastic dominance are developed. These conditions, which involve moment inequalities of the minimum order statistics, are analogous to...
This article aims to provide a comprehensive overview of sparse optimization, with a focus on both sparse signal recovery and sparse regularization techniques. We will begin by exploring the foundations of sparse optimization, delving into the mathematical tools and models that underpin sparse signal recovery and LASSO. We will then...
We study a family of fractional integral operators whose kernels satisfying an non-isotropic dilation have singularity on a coordinate subspace. A characterization is given for these operators bounded from the classical, atom decomposable $H^1$-Hardy space to $L^q$-spaces.
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a general theory for actions of the corresponding height-one group scheme G=SL_2[F]. Sending a point...
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work introduces persistent magnitude homology, an extension of magnitude homology that captures multi-scale...
We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial differential equations (PDEs) with discontinuous coefficients and non-smooth solutions. Unlike...
We establish a connection between Drinfeld modules and rank-metric codes, focusing on the case of semifield codes. Our method constructs rank-metric codes from linear subspaces of endomorphisms of a Drinfeld module acting on torsion submodules. We show that Sheekey's construction [She20] fits naturally into this framework, yielding a...
In this work, we study geodesic curvature of the boundary of a two dimensional Alexandrov space of curvature bounded below (CBB). We prove several comparison and globalization theorems for the geodesic curvature, generalizing the known results for curves in space of curvature bounded above (CBA) by Alexander and Bishop (Differ. Geom....
The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and Vinzant, and by Brändén and Huh. The weak form of Mason's conjecture was also generalized to a...
In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$, there exists a nonzero integer $k$ such that $k \cdot \mathbb{Z} \subseteq \{\, xy : (x,y) \in E -...
We establish noncommutative analogs of some well-known large deviation inequalities for noncommutative random variables. Firstly, for the noncommutative independent case, we characterize the uniformly exponential integrability of random variables in terms of large deviation inequalities. Secondly, for noncommutative martingale...
We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that is, an edge colouring with three colors) the answer is known to be negative. In the latter case, a...
We verify a conjecture proposed by X. Chen and Y. Shi, which arises from their study of the Green function on spheres in Euclidean space. More precisely, let $M\subset \mathbb{R}^3$ be a closed $C^{2}$ embedded surface and suppose that there exists a point $p\in M$ so that its Green function $G$ is of the form $G(p,q)=-\frac{1}{2π}...
We exhibit the symmetric monoidal $\infty$-category $\mathrm{GrCob}$ of graph cobordisms between spaces as a full $\infty$-subcategory of the $\infty$-category $\mathrm{Psh}(\mathrm{Gr})$ of presheaves over $\mathrm{Gr}$, where $\mathrm{Gr}$ is the symmetric monoidal $\infty$-category of graph cobordisms between finite sets. We also...
We carry out a semantic study of the constructive modal logic CK. We provide a categorical duality linking the algebraic and birelational semantics of the logic. We then use this to prove Sahlqvist style correspondence and completeness results, as well as a Goldblatt-Thomason style theorem on definability of classes of frames.
Freed (arXiv:hep-th/9712042) formulated special Kähler structures; in particular, the regular locus of the $\mathrm{SL}_2(\mathbb{C})$ Hitchin base $\mathcal{B}$ carries such a structure, while the associated metric $ω_{\mathrm{SK}}$ is singular along the discriminant locus $\mathcal{D}$. Baraglia-Huang (arXiv:1707.04975) computed its...