arXiv preprints from January 1, 2026 through September 9, 2026 — 01:02:24 EST
The game of cops and robber has been studied for many years. Denoting $\mathsf{Forb}(4K_1)$ to be the family of all graphs that contain no induced subgraph isomorphic to $4K_1$ (e.g., with independence number less than $4$), we prove that for any $G\in\mathsf{Forb}(4K_1)$, we have $c(G)\leq 2$, where $c(\cdot)$ is the cop number. This...
We prove uniqueness of the maximal weak solutions to the supercooled Stefan problem in 1 dimension. This follows by showing that in 1 dimension, the optimal solution of the corresponding free target optimal transport problem given in \cite{GeneralDimensions}, is independent of the choice of the cost function. Moreover, we show that...
We introduce the mean Assouad dimension of a dynamical system, motivated by the Assouad dimension in fractal geometry. Using dimension interpolation, we further define the mean Assouad spectrum. This provides a new family of bi-Lipschitz invariants of dynamical systems. We study its basic properties and calculate it for several...
This paper explores the Fibonacci sequence and the Golden Ratio as organizing principles for visual composition and abstraction in painting. The author shows how recursive proportional systems, long associated with natural growth and aesthetic harmony, inform artistic structure and visual balance. The discussion traces Fibonacci-based...
In 2016, Dannan and Sitnik established the notable Damascus inequality, which features a symmetric structure under a multiplicative constraint. In this study, we consider the natural generalisation of this inequality by characterising all positive integers $m$ and $n$ such that the inequality...
Nica and Sprague [\textit{Am. Math. Mon., 2023}] derived a non-Archimedean version of the Gershgorin disk theorem. We derive a non-Archimedean version of the oval (of Cassini) theorem by Brauer [\textit{Duke Math. J., 1947}] which generalizes the Nica-Sprague disk theorem. We provide applications for bounding the zeros of polynomials...
This article is written in celebration of the 8th Kazakh-French Logical Colloquium. We expand on an unpublished research note of the second author. We record some results concerning local Keisler measures with respect to a formula which is stable in a model. We prove that in this context, every local Keisler measure on the associated...
It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite poset. We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of...
In this paper, we provide a sufficient condition on the non-existence of the common Kähler submanifolds between the complex Euclidean space and the ball bundles of some Hermitian vector bundles over Kähler manifolds. Then we get the non-existence theorems on several classes of ball bundles whose base spaces are Hermitian symmetric...
We establish characterization of $H^1$ Sobolev spaces by certain square functions, improving previous results.
This paper proposes a temporal two-grid compact difference (TTCD) scheme for solving the Benjamin-Bona-Mahony-Burgers (BBMB) equation with initial and periodic boundary conditions. The method consists of three main steps: first, solving a nonlinear system on a coarse time grid of size $τ_c$; then obtaining a coarse approximation on...
We introduce a one-parameter family of intermediate topological pressures for nonautonomous dynamical systems which interpolate between the Pesin-Pitskel topological pressure and the lower and upper capacity pressures. The construction is based on the Carathéodory-Pesin structure in which all admissible strings in a covering satisfy $...
In this paper we study the descriptive complexity of the topological orbit equvalence relation for some Borel classes of Cantor minimal systems. Specifically, we study the Borel class of all Cantor minimal systems with only finitely many ergodic measures, and show that the orbit equivalence for this class is Borel bireducible with the...
This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative viscosity solutions of the PDE $Δ_\infty^Nu=f(u)+g(u)|Du|^q$, where $0\le q\le 1$, and for a large class...
Chemical reaction networks are a widely accepted modeling framework for diverse science phenomena stemming from all disciplines of science, such as biochemistry, ecology, epidemiology, social and political science. In this paper we prove that every first order endotactic stochastic mass-action reaction system (SMART) is essential...
We define a grid graph $G$ as a Cartesian product of path-graphs $P_n$ or cycle-graphs $C_n$ as shown in Figure 1, and we ask, when can the edge set of a complete graph be expressed as a disjoint union of graphs isomorphic to $G$? That is, we are asking for which grid graphs a $G$-design exists, where a $G$-design is defined as a...
We present an NPT extension of Ewald summation with prolates (ESP), a spectrally accurate and scalable particle-mesh method for molecular dynamics simulations of periodic, charged systems. Building on the recently introduced ESP framework, this work focuses on rigorous and thermodynamically consistent pressure/stress evaluation in the...
Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently large $x$. This gives a solution with $γ= \frac{1}{36}$ to the Exercise 5.1 in Harman's...
In this work, we investigate the existence and orbital (in)stability of several branches of standing--wave solutions for the cubic nonlinear Schrödinger equation (NLS) posed on a looping--edge graph $\mathcal{G}$, consisting of a circle and a finite number $N$ of infinite half--lines attached to a common vertex. The model is endowed...
We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E, the resulting moduli space is isomorphic to some subscheme of the Quot-scheme parametrising...
We prove a sharp nonuniqueness result for the forced generalized SQG equation. First, this yields nonunique $\dot{H}^s$- energy solutions below the Miura-Ju class. In particular, this shows that the solutions constructed by Resnick and Marchand for the dissipative SQG equation are not necessarily unique. Second, this establishes...
Pisier's celebrated counterexample to Halmos's similarity-to-contractions problem was based on $2 \times 2$ upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz...
In this article, we study the fixed-point subgroups of the solvable Baumslag-Solitar groups $\BS(1,n)= \langle a, t \mid t a t^{-1} = a^{n} \rangle$, $n>1$ of automorphisms and endomorphisms. We also investigate the stabilizers of subgroups of $\BS(1,n)$, considered as subgroups of the group of automorphisms and submonoids of the...
In this paper we study parameter-dependent structured moment matrices with a canonical block form arising from weighted quadratic histopolation on simplicial meshes. For a strictly positive density on a simplex, we construct compatible face densities and an orthogonal decomposition of the quadratic polynomial space into face and...
In this paper, we introduce multiple zeta dagger values and special values of Carlitz multiple dagger polylogarithms, and study their properties. In particular, using these values, we construct a non-trivial involution on a certain quotient space of multiple zeta values in positive characteristic.