arXiv preprints from January 1, 2026 through September 7, 2026 — 18:52:21 EST
For $π$ a finitely presented group, Hausmann and Weinberger defined $q(π) \in \mathbb Z$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $π$. This short note establishes that this minimum value in general differs depending on whether one minimizes over topological manifolds or only...
This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic...
Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-functions of certain CM Hecke eigenforms to Ramanujan's alternative bases...
In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(α+β)=\sum_{k=0}^n \left(\vphantom{\bigg|}\genfrac{}{}{0pt}{0}{n}{k}\right) p_k(α)p_{n-k}(β) $$ Some trivial representations of general...
We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Laurière and Tangpi, to reduce the convergence analysis to the study of...
In this paper we prove that, for a compact group $G$, a metrizable $G$-space is a $G$-ANR under the following asumptions: (1) if it dominates a $G$-ANR space through a fine $G$-homotopy equivalence; (2) if it is $G$-homotopy dense in a $G$-ANR; (3) if it contains a $G$-ANR as a $G$-homotopy dense subset; (4) if it is the inverse limit...
Denote by ${\mathcal P}_{g,d}^{r,k}$ the subset of the moduli space of curves of genus g consisting of those curves that have a linear series of degree d and dimension r for which the Petri map has kernel of dimension at least k. We show the existence of codimension k components of ${\mathcal P}_{g,d}^{r,k}$.
This paper improves the spectrally-filtered direct-insertion downscaling method for discrete-in-time data assimilation by introducing a relaxation parameter that overcomes a constraint on the observation frequency. Numerical simulations demonstrate that taking the relaxation parameter proportional to the time between observations...
Physics-informed neural networks based on automatic differentiation (AD-PINNs) and their finite-difference counterparts (FD-PINNs) are widely used for solving partial differential equations (PDEs), yet their analytical properties remain poorly understood. This work provides a unified mathematical foundation for both formulations....
Let $f(n)$ be the maximum sum of the sides of non-overlapping squares (or equilateral triangles) packed inside a unit square or (unit equilateral triangle). In this paper, we explore some properties of $f$ and examine how the square and triangle cases are similar. We prove that a conjecture of Erdős, which says that $f(k^2+1) = k$ for...
We introduce $\textsf{gradOL}$, the first gradient-based optimization framework for solving Chebyshev center problems, a fundamental challenge in optimal function learning and geometric optimization. $\textsf{gradOL}$ hinges on reformulating the semi-infinite problem as a finitary max-min optimization, making it amenable to...
Let $(M^m,g)$, $(N^n,h)$ be closed Riemannian manifolds, $m,n\geq 2$, with concave isoperimetric profiles and volumes $V_M$, $V_N$ respectively. We consider a one parameter family of product manifolds of the same volume, $(X,G_λ)=(M^m\times N^n,λ^{2n}g+ λ^{-2m}h)$, $λ>0$, and estimate a lower bound for their isoperimetric profile for...
This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+Δ^2 u-\int_τ^tg(t-s)Δ^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $τ=0$, was first considered by Cavalcanti (Discrete Contin. Dyn. Syst., 8(3), 675-695, 2002), where results of...
Matrix joint block-diagonalization (JBD) frequently arises from diverse applications such as independent component analysis, blind source separation, and common principal component analysis (CPCA), among others. Particularly, CPCA aims at joint diagonalization, i.e., each block size being $1$-by-$1$. This paper is concerned with {\em...
When the one-form is $Lip\left(γ-1\right) $ with $γ>p\geq 1$, we construct the integral of a branched $p$-rough path, which defines another branched $p$-rough path. We derive a quantitative bound for this integral and prove that it depends continuously on the driving branched rough path in rough path metric. Moreover, we prove that...
We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative structure of the scheme. The model can be trained both in a supervised setting with efficiently...
We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong...
In the 1960s, John Nash proposed a method to resolve singularities. Five decades of encouraging results could not prevent an unexpected ending: the method does not work in general. In this note (written in Spanish), we tell the story of the rise and fall of the Nash blowup.
We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a coarse ''subset'' of the Luna-Vust data of a spherical variety. Our smoothness criterion improves...
Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and $r\geq3$ are fixed. It is shown that for some positive constants $c$ and $d$, any integer $r\geq3$ and...
In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with Time-of-Use (TOU) energy tariffs and machine states, a variant of RCPSP for production scheduling, where energy price is part of the criteria and one highly energy-demanding machine can be in one of the following three states: proc, idle, or...
Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $σ$-algebra $\mfA\otimes\mfB$ and $\{(\mfA_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$. Then let $\mfA\dd\mfB$ be the $σ$-algebra generated $\mfA\otimes\mfB$ and by the family $\mcM:=\{E\subset{X\times{Y}}\colon...
We consider the notion of an exponential dichotomy with respect to a family of norms for an evolutionary family in a Banach space, and we characterize it by the admissibility of the pair $(L^p,L^q)$ for $p,q \in [1,\infty]$ with $p\ge q$. We then use this characterization to establish the robustness of an exponentially dichotomic...
Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of covers, the geometric and arithmetic setup is first reviewed, followed by the semi-modern...
Motivated by the desire of finding a geometric interpretation to the Yamabe equation on groups of Heisenberg type, we define a geometric structure on manifolds modelled locally on these groups, which we call contact structure of Heisenberg type. In the case of the Heisenberg group is equivalent to contact Riemannian manifolds. We...