arXiv preprints from January 1, 2026 through September 8, 2026 — 13:06:38 EST
We study the complexity of the $KK$-equivalence relation on unital $C^*$-algebras, in the sense of descriptive set theory. We prove that $KK$-equivalence is analytic, which in turn shows that the set of separable $C^*$-algebras satisfying the UCT is analytic. This allows us to prove a game-theoretic refinement of the unital...
The point-thickness $θ'(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ of $G$ is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thickness of complete multipartite graphs. As a special case, we also obtain the point-thickness of complete graphs.
In this paper, we prove that if $\mathcal{A}$ is a unital separable $C^*$-algebra, $\mathcal{M}$ is a von Neumann algebra which has the Kirchberg's quotient weak expectation property (QWEP), and $φ:\, \mathcal{A}\rightarrow \mathcal{M}$ is a unital completely bounded representation, then there is an invertible operator $S\in...
An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the multidimensional torus is analyzed. The equations describe the dynamics of population species with repulsive or attractive interactions. The numerical scheme is based on a generalized Scharfetter-Gummel discretization of the nonlocal flux term. For...
This is the second of a series of three papers where we prove the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups and an Ichino--Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. The goal of this second paper is to compare the two relative trace formulae.
This paper studies a two-phase free boundary problem governed by the ElectroHydroDynamic equations, which describes a perfectly conducting, incompressible, irrotational fluid with gravity, surrounded by a dielectric gas. The interface separating fluid and gas is referred to as the free boundary. It is known that the free surface...
Studies of density matrices for random quantum states lead naturally to the fixed trace Laguerre ensemble in random matrix theory. Previous studies have uncovered explicit rational function formulas for moments of purity statistic (trace of the squared density matrix), and also a third order linear differential equation satisfied by...
In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures extend the twin group $T_n$ in close analogy to how the virtual singular braid monoid and virtual...
In this paper, we introduce a newly defined algebraic invariant for square matrices termed the \emph{Alternating Power Difference (APD)}. The APD is defined as the signed sum of the powers of diagonal sums along permutations of the symmetric group, distinguishing between even and odd permutations. It serves as a measure of the broken...
In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and...
The well-known Faber-Krahn theorem states that the ball has the lowest first Dirichlet eigenvalue among all domains of the same volume in $\mathbb{R}^n$. Leydold (Geom. Funct. Anal, 1997) gave the discrete version of Faber-Krahn inequality for regular trees with boundary. Bıyıko{ğ}lu and Leydold (J. Combin. Theory Ser. B, 2007)...
We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(ω,ω_1)$ and performs an Ord-length countable-support symmetric iteration. For fixed parameters $S:=A^ω$ and $T:=PowerSet(S)$ (as computed in...
Adaptive gradient methods, such as AdaGrad, have become fundamental tools in deep learning. Despite their widespread use, the asymptotic convergence of AdaGrad remains poorly understood in non-convex scenarios. In this work, we present the first rigorous asymptotic convergence analysis of AdaGrad-Norm for smooth non-convex...
Recently, Nath and Das investigated congruence properties for the second order mock theta function $B(q)$. In their paper, they asked for analytic proofs of three identities on the second order mock theta functions $A(q)$, $B(q)$ and $μ_2(q)$. In this paper, we settle Nath and Das' open problem by using the $(p, k)$-parametrization of...
Limit theorems of strong law of large numbers and central limit theorem types are obtained for the compositions of independent identically distributed random unitary channels.
We study nonexistence results and gradient estimates for solutions of \[ Δ_p v + a v^{q}=0 \] defined on complete Riemannian manifolds satisfying a \emph{$χ$-type Sobolev inequality}. We establish a Liouville theorem under the assumptions that the underlying manifold $(M,g)$ supports a \emph{$χ$-type Sobolev inequality} and that the...
The type of a complex projective plane curve has been recently introduced by T. Abe, P. Pokora and the first author. In the same paper they have studied the type two curves. In this paper we study plane curves of type three, with special attention to low degree curves and line arrangements.
We investigate the distribution of zeros of the independence polynomial ${\rm I}(G, x)$ for the family of Generalized Petersen graphs ${\rm GP}(n, k)$ in the complex plane. While the independence numbers and coefficients of these graphs have been studied, the global behavior of their roots remains largely unexplored. Using an exact...
We investigate static metrics on simple manifolds with compact boundary and establish an Obata-type rigidity theorem. We identify new sufficient geometric conditions under which the combined curvature map $g\mapsto (R_g, H_g)$ is a local surjection. Consequently, we demonstrate that in contrast to manifolds without boundary, where...
We investigate the analytic stability of wavelet frames in anisotropic Hardy spaces associated with expansive dilation matrices. The main result establishes a deterministic operator-norm lower bound on the reconstruction error of the mixed frame operator, uniformly across the Hardy range, whenever a real eigenvalue of the...
We demonstrate that the Cayley-Hamilton theorem is a derived consequence of a more fundamental dimensional constraint: the syzygy formed by the tensor product of two Levi-Civita symbols, which vanishes identically in m-dimensional space. By shifting perspective from the tensor A to the isotropic operators that induce A's invariants...
This article establishes sharp inverse and saturation statements for kernel-based approximation using finitely smooth Sobolev kernels on bounded Lipschitz regions. The analysis focuses on the superconvergence regime, for which direct statements have only recently been obtained. The resulting theory yields a one-to-one correspondence...
We study the filtering and smoothing problem for continuous-time linear Gaussian systems. While classical approaches such as the Kalman-Bucy filter and the Rauch-Tung-Striebel (RTS) smoother provide recursive formulas for the conditional mean and covariance, we present a pathwise perspective that characterizes the smoothing error...
This paper investigates a system of nonlinear reaction-diffusion equations modeling the industrial synthesis of ammonia. By applying Lie group analysis, we construct self-similar solutions and derive a reduced system of ordinary differential equations. Using comparison principles and barrier techniques, we establish sufficient...
Given a probability space $(S,Δ, \mathbb{P})$ and a separable metric space $(U,d)$, the $Ky~Fan$ metric $ρ(X,Y)$ on the space $\mathfrak{X}^0$ of equivalence classes of random variables (w.r.t. almost sure equality) formed from the set $\mathfrak{X}(U)$ of $U$-valued random variables is given by $ρ(X,Y)=\inf...