arXiv preprints from January 1, 2026 through September 6, 2026 — 16:20:40 EST
Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $m \geq 1$. Let $η$ be a smooth $(0,1)$-tensor field on $M$. The divergence of $η$ is defined as $\text{div}_g(η):=g^{ij}(\nabla η)_{ij}$. Now let $Δ_g$ be a differential operator on $M$ that is given on functions by $Δ_g u = \text{div}_g \nabla u$. We will call $Δ_g$...
We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the eigenvalues of the adjacency matrix determine the graph isomorphism class among translation...
In this work, we exhibit several subclasses of weakly dicomplemented lattices (WDLs) based on their skeletons and dual skeletons. We investigate normal filters (resp. ideals) and show that the set of normal filters (resp. ideals) forms a complete lattice, which is not a sublattice of the lattice of all filters (ideals). The normal...
In this paper, we study contact structures supported by open book decompositions whose pages are four-punctured spheres. The paper is split into two parts. In the first part, we find infinitely many overtwisted, right-veering monodromies on the four-punctured sphere. This is done using the techniques developed by Ito-Kawamuro in the...
Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of certain type establishing the birationality, smoothness, and irreducibility of the corresponding...
We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and...
Standard fractional models on manifolds often conflate geometric anisotropy with medium heterogeneity. In this Letter, we overcome this rigidity by deriving the fundamental solution for a weighted space-time fractional ultrahyperbolic operator, denoted by $(-\Box_{φ,ω})^β$. Using a novel spectral approach based on the Weighted Fourier...
In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chvátal and Lovász showed every digraph contains a quasi-kernel. In 1976, P. L. Erdős and Székely conjectured that every sourceless digraph has a quasi-kernel of order at...
For a crystal group $Γ$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2(\mathbb{R}^n)$ is called $Γ$--shift invariant if, for every $f\in\mathcal{V}$, the shifts of $f$ by every element of $Γ$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $Γ$--shift invariant closed...
In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these...
We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends...
We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.
We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for...
We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship...
The problem of identifying the set of Dirichlet-to-Neumann (DtN) maps arising from conductivities on a smooth domain, among operators acting on functions on the boundary, is a challenging issue in the mathematical analysis of the Calderón inverse problem. This question is also relevant in specific applications since, as the inverse...
The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second...
This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a \textit{spherical \(t\)-design} if the average value of any polynomial of degree at most \(t\) over \(X_N\)...
In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in...
This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality...
The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results...
Let $(L, H)$ be closed subgroups of a locally compact group $G$. The pair $(L, H)$ is said to be proper if the action of $L$ on the homogeneous space $G/H$ is proper. We give a complete list of connected closed proper pairs in the affine transformation group of $\mathbb{R}^2$. This result extends Kobayashi's classification (1992) of...
We propose a novel method, Phase-IDENT, for identifying partial differential equations (PDEs) from noisy observations of dynamical systems that exhibit phase transitions. Such phenomena are prevalent in fluid dynamics and materials science, where they can be modeled mathematically as functions satisfying different PDEs within distinct...
Solutions to elliptic equations often exhibit higher regularity properties such as \emph{higher integrability}. That is, for instance, a solution $u$ to a system that a priori only satisfies $ u \in W^{1,r}$ is more regular and even in the Sobolev space $W^{1,s}$ for some $s>r$. Under additional constraints of the sign of specific...
For any modulus of continuity $ω$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^ω_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial...
We consider the dependence of non-zero Steklov eigenvalues on smooth perturbations of the domain boundary. We prove that these eigenvalues are generically simple under such boundary perturbations. This result complements our previous work on metric perturbations, thereby establishing generic simplicity Steklov eigenvalues under both...