arXiv preprints from January 1, 2026 through July 20, 2026 — 10:17:36 EST
We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is $Π^0_3$ and determine the exact complexities it can have. In...
In this paper, we extend the study of the quantum $k$-Bruhat order initiated in the work of Benedetti, Bergeron, Colmenarejo, Saliola, and Sottile concerning the quantum Murnaghan-Nakayama rule. Specifically, identifying maximal chains in intervals of the quantum $k$-Bruhat order with sequences of transpositions, we investigate a...
Let $H$ be a fixed graph whose edges are colored red and blue and let $β\in [0,1]$. Let $I(H, β)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the density of red edges in $G$ is $β$. This refines the problem of determining the semi-inducibility of $H$,...
Let p be any prime, and $p^(ν_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(ν_q(n!)) < p^(ν_p(n!))$ for all $n \ge n_0$ and all primes $q > p$. For twin primes $p$ and $q = p + 2$ it is proved that the minimal $n_0$ satisfying...
We study the stable translation length of homeomorphisms of a surface acting on the fine nonseparating curve graph and compare it to the stable translation lengths of its finite approximations - mapping classes relative to a finite invariant set - acting on the nonseparating curve graph. We prove that the stable translation length of...
We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial...
Given a projective variety $X$ and a very ample line bundle $\mathcal{L}$ on $X$, we classify for which $X$ and $\mathcal{L}$ the twisted syzygies and twisted dual syzygies bundles are Ulrich with respect to the polarizations $\mathcal{L}^a$. We obtain some partial results when considering an arbitrary polarization $H$.
A complex potential is a holomorphic function $Ω:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} = \overline{Ω'(z)}$. In this work, we generalize the concept of potential to the broader class of dynamical systems of...
We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{ő}s, and LeVeque \cite{DEL1963} and properties of the order of integers in the multiplicative groups. The...
We establish bounds on the mixing times of conjugacy-invariant random walks on finite nilpotent groups in terms of the mixing times of their projections onto the abelianization. This comparison framework shows that, in several natural cases of interest, the mixing behavior on a nilpotent group is governed by that of the projected walk...
Fusion rules of vertex operator algebra V_{L_2}^{S_4} are determined.
In this note, the author recalls the Calderon-Zygmund theory on the unit ball and derives the weak (1,1) boundedness of the projection for $\mathcal{H}$-harmonic Bergman space.
Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct projective $B$-modules $Q$ with $rank(Q)=p$, such that $x=[Q] -[B^p]\neq 0$ in $K_0(B)$ and the total...
In this paper, we present a Wachspress-based transfinite formulation on convex polygonal domains for exact enforcement of Dirichlet boundary conditions in physics-informed neural networks. This approach leverages prior advances in geometric design such as blending functions and transfinite interpolation over convex domains. For...
In this paper, we investigate the energy decay of the solution to a viscoelastic wave equation with variable exponents logarithmic nonlinearity and weak damping in a bounded domain. We establish an explicit general decay result under mild conditions on the relaxation function $g$. Furthermore, under the general assumption...
We show that a skew-gentle algebra is brick-finite if and only if it is representation-finite, generalizing Plamondon's original result for gentle algebras.
First-order fully implicit as well as implicit--explicit schemes for coupled elliptic-parabolic systems are discussed in [Ern and Meunier, ESAIM: M2AN, 2009] and [Altmann et al., Math.\ Comp., 2021], respectively. The extension of the analysis to higher-order (third-, fourth-, fifth-, and sixth-order) schemes is not straightforward...
How can we build surrogate solvers that train on small domains but scale to larger ones without intrusive access to PDE operators? Inspired by the Data-Driven Finite Element Method (DD-FEM) framework for modular data-driven solvers, we propose the Latent Space Element Method (LSEM), an element-based latent surrogate assembly approach...
This is the third and the last 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. In this paper, we compute the spectral expansions of these...
We prove an enumerative min-max theorem that relates the number of genus g minimal surfaces in 3-manifolds of positive Ricci curvature to topological properties of the set of embedded surfaces of genus $\leq g$, possibly with finitely many singularities. This completes a central component of our program of using topological methods to...
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.