arXiv preprints from January 1, 2026 through September 9, 2026 — 19:36:28 EST
This paper presents a comprehensive study of matrix Kloosterman sums, including their computational aspects, distributional behavior, and applications in cryptographic analysis. Building on the work of [Zelingher, 2023], we develop algorithms for evaluating these sums via Green's polynomials and establish a general framework for...
We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction...
We study optimal multiple weight assumptions in the weighted theory of multilinear Fourier multipliers and multilinear pseudo-differential operators. For multilinear Fourier multipliers, we revisit the weighted Hörmander-type theorem of Li and Sun, as a multilinear version of Kurtz and Wheeden, and show that their multiple weight...
We review Hilbert's classical analytical proof of the transcendence of the number $\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of $\mathrm{e}$ based on FPS. The first of them is a specialization of the 1990 proof by Beukers,...
RNA velocity is an important model that combines cellular spliced and unspliced RNA counts to infer dynamical properties of various regulatory functions. Despite its wide applicability and many variants used in practice, the model has not been adequately designed to directly account for both intracellular gene regulatory network...
We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,α)$, $F(p,q,s)$, and the non-derivative $M(p,q,s)$. For a harmonic $K$-quasiregular mapping $f=u+iv$, we first show that if the real part $u$ belongs to $Q_h(1,p,α)$ (with $α>-1$ and...
The double-exponential Sinc-collocation method is known as a super-accurate method for solving initial value problems of ordinary differential equations, for which the error decreases almost exponentially as a function of the number of sample points in the temporal direction, $N$. However, this method requires solving nonlinear...
How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$...
In this paper, we mainly study the Castelnuovo-Mumford regularity of the generalized binomial edge ideals of graphs. We show that this number can be any integer number from $2$ to $n-1$ where $n$ is the number of vertices in the underlying graph. We are able to show this, after giving some tight lower and upper bounds for the...
A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if $Ω\subset\mathbb{R}^{n+1}$ is a convex body such that for every $n$-dimensional subspace $M\subset\mathbb{R}^{n+1}$ the centroids of the sections $(x+M)\cap Ω$ are collinear, then $Ω$ is an ellipsoid. We study natural...
This paper studies phase and norm retrieval by subspaces. We first investigate norm retrieval by hyperplanes. We show that if $N$ hyperplanes $\{\varphi_i^\perp\}_{i=1}^N\subset \mathbb{R}^N$ allow norm retrieval and the vectors $\{\varphi_i\}_{i=1}^N$ are linearly independent, then these vectors must be an orthonormal basis for...
We discuss the low energy resolvent estimates for the Schrödinger operator with slowly decaying attractive potential. The main results are Rellich's theorem, the limiting absorption principle and Sommerfeld's uniqueness theorem. For the proofs we employ an elementary commutator method due to Ito--Skibsted, for which neither of...
In this paper, we study normalized solutions for the following critical Schrödinger-Bopp-Podolsky system: $$-Δu + q(x)φu = λu + |u|^{p-2}u + \bigl(I_α* |u|^{3+α}\bigr)|u|^{1+α}u,\quad \text{in } Ω_r,$$ $$-Δφ+ Δ^2φ= q(x)u^2, \ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \text{ in } Ω_r,$$ where $Ω_r \subset \mathbb R^3$ is a smooth...
We consider inverse eigenvalue problems for the perturbed Bessel operator in $L^{2}(0,1)$. (1) For the case where the angular-momentum quantum number $\ell\in\mathbb{N}\cup\{0\}$, we establish a uniqueness result for the inverse spectral problem by utilizing the closedness condition of a certain function system constructed based on...
It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. For example, each and every permutation of the $2\times2$ matrix coefficients $\{12,6,7,1\}$, generated by the Pythagorean triple $(5,12,13)$, yields a matrix with integer...
We present SaddleScape V1.0, a Python software package designed for the exploration and construction of solution landscapes in complex systems. The package implements the High-index Saddle Dynamics (HiSD) framework and its variants, including the Generalized HiSD for non-gradient systems and the Accelerated HiSD. SaddleScape V1.0...
Problem 20.21 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 20th Issue, 2022), contributed by M.~Conder and attributed to G.~Verret, asks whether there exists a finite group $G$ with two normal subgroups $K$ and $L$ of index $12$ such that $K \cong L$, but with non-isomorphic quotients $G/K \cong...
High-dimensional integration with respect to complex target measures remains a fundamental challenge in computational science. While Flow Matching (FM) offers a powerful paradigm for constructing continuous-time transport maps, its deployment in high-precision integration is severely limited by the discretization bias inherent to...
This paper investigates the wall structure of the space of stability conditions on Hirzebruch surfaces. Using the gluing construction of \cite{CP} and \cite{Uch} with respect to a fixed semiorthogonal decomposition, we focus on two main objectives: observing the intersection of the geometric chamber $U(S) \subset \mathrm{Stab}(Σ_e)$...
Given a compact cube complex $X$ that splits as a graph of virtually special cube complexes. Suppose that the fundamental groups of edge spaces are cyclonormal in the fundamental groups of adjacent vertex spaces. We show that $π_1X$ has finite stature with respect to vertex groups in the sense of Huang-Wise. In particular, when vertex...
We study the expectation-maximization (EM) algorithm for general latent-variable models under (i) distributional misspecification and (ii) nonidentifiability induced by a group action. We formulate EM on the quotient parameter space and measure error using an arbitrary integral probability metric (IPM). Our main results give (a) a...
The aim of this article is to use Banach lattice techniques to study coordinate systems in function spaces. We begin by proving that the greedy algorithm of a basis is order convergent if and only if a certain maximal inequality is satisfied. We then show that absolute frames need not admit a reconstruction algorithm with respect to...
Let $μ$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $μ$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for large classes of measures $μ$, the empirical distribution of zeros of the $k$-th derivative...
In 1987 Hiroshi Maehara conjectured that a graph can be represented by vectors considered adjacent when not orthogonal (a faithful orthogonal representation) in codimension the minimum degree of the graph. Without settling the conjecture, Làslò Lovàsz, Michael Saks, and Alexander Schrijver (LSS) showed that a codimension of vertex...
We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $Λ\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or...