arXiv preprints from January 1, 2026 through July 21, 2026 — 20:52:57 EST
We discuss the initial value problem for the Einstein equations in Hitchin's generalised geometry for the case of closed divergence (which correspond to the equations of motion in the bosonic part of the NS-NS sector in type II ten-dimensional supergravity) and establish the existence of a maximal globally hyperbolic development...
Greedy Sampling Methods (GSMs) are widely used to construct approximate solutions of Configuration Optimization Problems (COPs), where a loss functional is minimized over finite configurations of points in a compact domain. While effective in practice, deterministic convergence analyses of greedy-type algorithms are often restrictive...
Let $ (T(t))_{t\geq0} $ be a positive $C_{0}$-semigroup with generator $A$ on a C$^*$-algebra or on the predual of a W$^*$-algebra. Then the growth bound $ω_{0}$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the spectral bound of $A$, is a spectral value.
In this paper, we employ contour integration and residue calculus to derive explicit parity formulas for (cyclotomic) multiple zeta values (MZVs). A key innovation lies in applying double shuffle regularization to the contour integrals, which leads to two distinct regularized parity formulas-one via shuffle and one via stuffle...
This paper investigates the finite-time blow-up phenomena to a quasilinear two-species chemotaxis system with two chemicals \begin{align}\tag{$\star$} \begin{cases} u_t = \nabla \cdot \left(D_1(u) \nabla u\right) - \nabla \cdot \left(u \nabla v\right), & x \in Ω, \ t > 0, 0 = Δv - μ_2 + w, \quad μ_2=\fint_Ωw, & x \in Ω, \ t > 0,...
We establish a comparison principle for entire solutions of the Allen--Cahn equation whose nodal sets, possibly singular, are asymptotic to a regular minimizing hypercone. We show that inclusion of the positive phases enforces a global ordering of the solutions. As a consequence, the positive phase uniquely determines the solution,...
The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis number at most 2, and more generally, graphs embedded on a fixed surface of bounded genus are...
In this article, we study the value distribution theory for the first derivative of the logarithmic derivative of Dirichlet $L$-functions, generalizing certain results of Ihara, Matsumoto et al. related to ``$M$-functions'' for $σ= \operatorname{Re}(s) > 1$. We then discuss the main obstruction toward generalization to higher derivatives.
We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of...
For a compact and connected Lie group $G$, we present an explicit construction of an $\mathbb{S}^1$-gerbe over the differentiable stack $[G/G]$ in the framework of $\mathbb{S}^1$-central extensions of Lie groupoids. This gives a complete proof of the construction outlined earlier by Behrend--Xu--Zhang, together with an explicit proof...
We consider random integer partitions $λ$ that follow the Poissonized Plancherel measure of parameter $t^2$. Using Riemann$-$Hilbert techniques, we establish the asymptotics of the multiplicative averages $$Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\mathrm{e}^{η(λ_i-i+\frac{1}{2}-s)}\right)^{-1} \right] $$ for fixed $η>0$ in...
We prove that the Kadison-Kastler and Christensen distances are stable under the Banach space injective tensor product (resp., the Banach space projective tensor product) of a Banach space with any unital commutative $C^*$-algebra (resp., of a $C^*$-algebra with any unital $C^*$-algebra). Apart from these stability results, we make...
The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript, we study the limit of their Minkowski gauges under two natural asymptotic regimes: \\...
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results is usually not guaranteed. We propose a new method for the rigorous integration of parabolic...
Node coloring is the task of assigning colors to the nodes of a graph such that no two adjacent nodes have the same color, while using as few colors as possible. It is the most widely studied instance of graph coloring and of central importance in graph theory; major results include the Four Color Theorem and work on the...
Previous work of the author and N. Reshetikhin defines an invariant $\operatorname{Z}_{N}^ψ(K, ρ, μ)$ of a knot $K$, a representation $ρ: π_{1}(S^{3} \setminus K) \to \operatorname{SL}_2(\mathbb{C})$, and a logarithm $μ$ of a meridian eigenvalue of $ρ$. It can be interpreted as a geometric twist of the Kasahev invariant or as a...
We consider selected aspects of (non-Archimedean) quantum mathematics and non-Archimedean (quantum) computation.
We show that the classification up to homeomorphism of closed topological nonorientable 4-manifolds with fundamental group of order 2 due to Hambleton-Kreck-Teichner can be used to classify a large set of such 4-manifolds with cyclic fundamental group of order 2p for every odd $p > 1$. This is done through a simple cut-and-paste...
We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the potentials, under the assumption that the transport map solving the unregularised problem is...
We introduce and analyze a **memory-weighted velocity operator** \(\mathscr{V}_{α,β}\) as a mathematical framework for describing rates of change in systems with time-varying, power-law memory. The operator employs two independent continuous exponents \(α(t)\) and \(β(t)\) that separately weight past state increments and elapsed time...
Let $k$ be a natural number with $k\ge 2$, and let $\varepsilon>0$. We consider the number $V_k^*(P)$ of integral solutions of the system of simultaneous Diophantine equations \[ x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k), \] with $1\le x_i,y_i\le P$ $(1\le i\le k+1)$. Writing $L_k^*(P)$ for...
Inspired by Bruggesser-Mani's line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order. This gives a new proof of Björner's...
In our previous work, we introduced the concept of a \emph{spectral pair} for a half-line Schrödinger operator with a \emph{complex} bounded potential $q$, serving as a substitute for the spectral measure in a non-self-adjoint setting. In this paper, we study the case of $q \in L^1(\mathbb{R}_+)$. We derive explicit formulas for the...
We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schrödinger equations of the form $i\partial_t u + Δu + |x|^{-b}|u|^2 u = 0$ in the range $b\in(0,\tfrac12)$. By modifying the...
In a ball $Ω\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = Δv - uv, \end{array} \right. \] is considered under no-flux boundary conditions for $u$, and for prescribed constant positive boundary...