arXiv preprints from January 1, 2026 through September 5, 2026 — 00:31:52 EST
In this paper, we study the combinatorial object $\bar{b}^k_{r,s}(n)$ which counts the overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. We extend results of Chacon and Sellers for several families of $r,s$ and $k$. We also establish divisibility properties for...
We identify non-Gaussian corrections to the central limit theorem for the global and local laws, i.e. for Stieltjes transform of the empirical eigenvalue distribution of a large real symmetric or complex Hermitian Wigner matrix. We find that in the real case the rate of convergence in this CLT is substantially slower than in the...
We prove two anisotropic halfspace theorems for uniformly elliptic parametric integrands. In $\mathbb{R}^3$, every connected smooth properly embedded boundaryless anisotropic minimal surface for an even integrand is a plane if it lies in a halfspace. The proof replaces the catenoid in the Hoffman--Meeks argument by a strict exterior...
A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup.
We develop a geometric framework for conservative mechanical systems modeled on conic pseudo-Finslerian manifolds, extending classical Riemannian and semi-Riemannian mechanics to anisotropic geometries. In this setting, we define a Jacobi-type pseudo-Finslerian metric and demonstrate that motions of fixed energy correspond, up to...
Two permutations $σ,τ\in S_n$ are called disjoint if the composition $στ^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range...
The way numbers are represented strongly influences which arithmetic structures are easy to see. The \emph{prime clockwork} is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock...
Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every...
We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let $G_r$ denote the operator which inserts $r$ copies of a new largest letter. After the change of variables $δ=y-x$, $u=x/y$, and $E=u\partial_u$, we find that $$G_r=\frac{δ^r}{r!}E(E+1)\cdots(E+r-1).$$ Its generating series acts...
We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. Let $2<p<\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided \[ \sum_{j\in\mathbb Z} ...
The aim of this paper is to develop a framework for measure theory that avoids infinities and allows for the uniform treatment of positive and vector measures. Our approach is based on a modification of the notion of measure, which supplements the usual $σ$-additivity requirement with a suitable maximality condition. To each Hausdorff...
Let $G$ be a definable group in an NIP theory. We prove that every minimal proximal definable $G$-flow is strongly proximal. Consequently, the universal minimal proximal definable $G$-flow $Π^{\mathrm{def}}(G)$ coincides with the minimal strongly proximal definable $G$-flow $Π^{\mathrm{def}}_{\mathrm{s}}(G)$. Furthermore, for a...
We study a data-driven reflection control problem for a Brownian model with unknown drift and volatility. We first propose a learn-then-optimize (LTO) algorithm: it estimates the policy-relevant parameter during exploration, plugs the estimate into the optimality equation, and exploits the resulting policy---achieving an $O(\sqrt{T})$...
This paper is concerned with the precise asymptotics, as time goes to infinity, of a transport problem in a half plane coupled with a nonlinear nonlocal boundary condition. This system arises from a class of models for the spatial spread of epdemics, its space independent version being the classical Kermack-McKendrick model. Using...
We classify a class of abstract Bose field models in quantum field theory up to unitary equivalence. The class includes abstract free Bose field models, abstract van Hove--Miyatake models, and infrared-renormalized van Hove--Miyatake models. Moreover, as an application of our classification, we classify quadratic interaction models....
We study the asynchronous $\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} ...
We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+σ)|\,dσ\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ of Dirichlet series, \[ \left|a_1+\sum_{n=2}^\infty...
The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL...
We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric,...
We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well....
This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization, i.e., when the controller deliberately injects noise into the nominal control input. This appears counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive...
In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.
Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from...
We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and...
We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $Ω(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin, we prove an $Ω(n^{-1.6342})$ non-anytime lower bound and an $Ω(n^{-1.2408})$ anytime lower bound. These improve the recent...