arXiv preprints from January 1, 2026 through July 20, 2026 — 06:33:42 EST
This paper derives closed-form expressions for four-parameter Euler sums containing generalized harmonic numbers with rational arguments, expressed in terms of the polylogarithm and Lerch transcendent, and reducible (under admissible parameter choices) to Riemann zeta and Hurwitz zeta values. It further obtains closed forms for...
We present evidence for a conjectural relationship between absorption times for discrete Whittaker processes and maximal heights of non-intersecting Brownian bridges.
In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by...
In this paper, we give an explicit description about the second Hochschild cohomology groups of bipartite Brauer graph algebras with trivial grading. Based on this, we provide geometric interpretations of deformations associated to some standard cocycles in terms of the surface models of Brauer graph algebras.
This article focuses on smooth, projective, and geometrically integral varieties $X$ defined over a number field $k$ with torsion-free geometric Picard groups. We establish an isomorphism between the Brauer groups of $X$ and its symmetric products. As applications, we deduce the relationship between the Brauer--Manin obstruction to...
If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erdős-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $${\ell \choose 2}\leq \sum_{1\leq i<j\leq \ell}|F_i\cap F_j| $$ for any...
This paper studies the relationship between shadowing phenomena and Bohr chaos in dynamical systems. We provide sufficient conditions for Bohr chaos in terms of shadowing. By combining those conditions with the shadowing lemma, we obtain some results on Bohr chaos and hyperbolic sets. Our results also highlight some simple but...
These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy, in which analytic and geometric structures are introduced through explicit calculations and...
We consider stochastic particle dynamics on hypersurfaces represented in Monge gauge parametrization. Starting from the underlying Langevin system, we derive the surface Dean-Kawasaki (DK) equation and formulate it in the martingale sense. The resulting SPDE explicitly reflects the geometry of the hypersurface through the induced...
It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. We formulate a fixed-point-theoretic framework for the analysis of flat $\bar\partial$-connections that define holomorphic structures on line bundles over...
In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without...
We propose and analyze a second-order, dimension-split exponential time differencing Runge--Kutta scheme (ETD2RK-DS) for multidimensional reaction--diffusion equations in two and three spatial dimensions. Under mild assumptions on the nonlinear source term, we establish uniform stability bounds and prove second-order temporal...
In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of...
The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} μ(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+α}}\,d\y $$ with $1< α< 2$; here the function $μ(\x,\y)$ is $\Z^d$-periodic in the both variables,...
In 2023, Güntürk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^γ<+\infty$ for all $γ>0$. We extend their result to a finite...
Let $ρ_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $ρ_d(N)=\sum\limits_{k=1}^{d-1}π\left( \frac{N}{k} \right) +O_d(π(N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have...
We study discrete-time Markov Decision Processes (MDPs) on finite state-action spaces and analyze the stability of optimal policies and value functions in the long-run discounted risk-sensitive objective setting. Our analysis addresses robustness with respect to perturbations of the risk-aversion parameter and the discount factor, the...
We introduce and investigate single conflict coloring in the setting of r-uniform hypergraphs. We establish some basic properties of this hypergraph coloring model and study a probabilistic model of single conflict coloring where the conflicts for each edge are chosen randomly; in particular, we prove a sharp threshold-type result for...
Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case the computation can be modified in several aspects. We will discuss relations with previous...
We study a class of structured convex optimization problems, which have a two-block separable objective and nonlinear functional constraints as well as affine constraints that couple the two block variables. Such problems naturally arise from distributed resource allocation and constrained machine learning. To achieve high...
We study clinically motivated capacity-constrained treatment-adherence outreach through a belief-state restless multi-armed bandit model, in which each patient is a partially observed two-state Markov decision process and interventions induce reset-type belief dynamics. For the discounted criterion, partial conservation law...
This paper addresses the minimization of a finite sum of prox-convex functions under Lipschitz continuity of each component. We propose two variants of the splitting proximal point algorithms proposed in \cite{Bacak,Bertsekas}: one deterministic with a fixed update order, and one stochastic with random sampling, and we extend them...
Let $(Ω^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $Σ$, satisfying that ${\text{Ric}_Ω}\ge 0$ and $Σ$ is strictly convex, more precisely, its second fundamental form $h\ge cg_Σ$ for some positive constant $c$. Escobar {\cite{escobar1997geometry}} considered the first nonzero...
Let $A$ be a unital locally matrix algebra. Among the examples of such algebras are: (1) an infinite tensor product $\otimes M_{n_i}(\mathbb{F})$ of matrix algebras over a field $\mathbb{F}$, and (2) the Clifford algebra of a nondegenerate quadratic form on an infinite-dimensional vector space over an algebraically closed field of...
We discuss the design of ``wave packet systems'' that admit strong concentration properties in phase space. We make a connection between this problem and topics in signal processing related to the spectral behavior of spatial and frequency-limiting operators. The results have engineering applications in medical imaging, geophysics,...