arXiv preprints from January 1, 2026 through September 9, 2026 — 22:35:12 EST
Building on the iHopf algebra realization of quasi-split universal iquantum groups developed in a prequel, we construct the dual canonical basis for a universal iquantum group of arbitrary finite type, which are further shown to be preserved by the ibraid group action; this recovers the results of Lu-Pan in ADE type obtained earlier...
We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator, and give...
Let $F$ be a non-Archimedean local field. For any irreducible smooth representation $π$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, we have an operation $D_{\mathfrak m}(π)$ to construct a simple quotient $τ$ of a Bernstein-Zelevinsky derivative of $π$. This article continues the previous one to study the following poset...
We investigate the strong-field limit of a charged particle in an electromagnetic field as a toy model for general covariant systems, establishing a novel connection between constrained Hamiltonian dynamics and noncommutative geometry. Starting from the action $S=\int dτ\, \dot{x}^i A_i(x)$, which represents the holonomy of the...
In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting...
We prove $\ell^p(\mathbb{Z}^n)-$estimates for long $r$-variational seminorm of two families of averages: discrete Birch-Magyar averages, for $r>max\{p,p'\}$ with $p>\frac{2c_{\mathfrak{R}}-2}{2c_{\mathfrak{R}}-3}$ and discrete Hardy-Littlewood type averages over certain algebraic varieties, for $r>max\{p,p'\}$ with $p>1$. Further, we...
We establish the global gradient bounds for weak solutions to the elliptic variational inequality with two-sided obstructions, associated with a $p(x)$-Laplacian type operator involving degenerate or singular matrix weights. Under the optimal regularity assumptions on the matrix-valued weight, suitable geometric flatness of the...
This paper establishes Lipschitz stability for the simultaneous recovery of a variable density coefficient and the initial displacement in a damped biharmonic wave equation. The data consist of the boundary Cauchy data for the Laplacian of the solution, \(Δu |_{\partial Ω}\) and \( \partial_{n}(Δu)|_{\partial Ω}.\) We first prove that...
In this paper, we provide some sufficient conditions for the compactness of weighted composition operators on Dirichlet space. Furthermore, we characterize the numerical range of certain classes of weighted composition operators on Dirichlet space and establish the criteria that guarantee the inclusion of zero within the numerical...
Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete state-dependent delays we construct a diffeomorphism on a neighbourhood of $X$ which takes $X$...
This paper establishes a rigorous mathematical framework for the Multi-Scale Negative Coupled System (MNCS), a dynamical model describing hierarchical state spaces with directed, sign-structured interactions. We address the stabilization of reaction-diffusion systems on bounded domains $Ω\subset \mathbb{R}^d$ ($d \le 3$) subject to...
In economics, construction of perfect models in a way that would be comparable to the standards customary in physical sciences is generally not feasible. In particular, the observed value for an economic equilibrium may deviate significantly from its model-based a priori expected value. Mathematically, the a posteriori observed...
Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that stochastically explore the search space and self-organize around global minima via a consensus-based...
We prove an inequality featuring three well-known functions from analysis, namely the cotangent, the Euler-Riemann zeta function, and the digamma function. Aside from a simple proof of our result, we give a conjectured strengthening. We offer various remarks about the origins of this problem.
For a graph $G$, the spectral radius $ρ(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $ρ(G)$ when $G$ contains a spanning complete bipartite graph. Using these lemmas and typical spectral method, we characterized the unique extremal graph with the maximum spectral radius among all...
A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex-deleted subgraphs, known as the deck of G. The Reconstruction Conjecture (RC) posits that every finite simple graph with at least three vertices is reconstructible. In this paper, we prove that the class of graphs with domination...
We study a new nonlinear system which contains a partial differential equation, a quasivariational inequality and a parabolic variational inequality in Banach spaces. We obtain the unique solvability of the coupled system under moderate conditions by using the Banach's fixed point theorem. We employ the main results to investigate a...
Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $φ$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $φ$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log...
We systematically investigate three different equivalence relations of connectedness: being connected by arcs, being connected by continua and being connected by chains of continua of decreasing diameter. The investigation is conducted from the point of view of Borel reductions, mainly on Polish spaces. All of the studied equivalence...
This work investigates the long-term distributional behavior of the reversible Selkov lattice systems defined on the set $\mathbb{Z}$ and driven by locally Lipschitz \emph{Lévy noises}, which possess two pairs of oppositely signed nonlinear terms and whose nonlinear couplings can grow polynomially with any order $p \geq 1$. Firstly,...
In this paper, we establish a Struwe type global compactness result for a class of nonlinear critical Hardy-Sobolev exponent problems driven by the fractional $p$-Laplace Hardy-Sobolev operator.
Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $δ_{k-1}(H)$ is the largest integer $t$ such that every $(k-1)$-subset of $V(H)$ is contained in at least $t$ edges of~$H$. The \emph{codegree Turán density} $γ(\mathcal{F})$ of a family $\mathcal{F}$ of $k$-graphs is the infimum of $γ> 0$ such that every...
In this paper we solve an open question formulated in the original paper of twisted skew group codes regarding when a twisted skew group code is checkable. Also, we prove that all ideals of dimension 3 over a twisted group algebra are abelian group codes, generalising another previous result over group algebras. Finally, we prove a...
We rigorously establish a formula for the correlation energy of a two-dimensional Fermi gas in the mean-field regime for potentials whose Fourier transform $\hat{V}$ satisfies $\hat{V}(\cdot) | \cdot | \in \ell^1$. Further, we establish the analogous upper bound for $\hat{V}(\cdot)^2 | \cdot |^{1 + \varepsilon} \in \ell^1$, which...
For a given variety of groups $\X$, we develop a systematic theory of $\CSX$-groups and $\XT$-groups, extending ideas proposed in \cite{Shah}. We analyze the interplay between these classes, describe their structural properties, and examine their connections with equational domains and residually $A$-free groups. Furthermore, we prove...