数学

We present a fast and accurate potential theory-based method for the two-dimensional modified Helmholtz equation, treating the involved singular and nearly singular layer evaluations together with volume potentials within a single…

Let $X$ be a smooth irreducible projective variety of dimension $n\ge 3$ over an algebraically closed field of characteristic zero, polarized by a very ample line bundle $\OO_X(1)$. Let $\E$ be an Ulrich bundle on $X$. We prove that there…

Let \((a_n)_{n\ge1}\subset\mathbb{N}\) be a lacunary sequence, \(a_{n+1}\ge q a_n\) for \(q>1\). For \(x\in\mathbb{T}\), we study the maximal empty circular gap \(G_N(x)\) of the finite orbit \(\{a_1x,\ldots,a_Nx\}\). We prove that, for…

In this paper, we study regular second-order Sturm--Liouville difference equations using the discrete Pr\"ufer transformation. By representing solutions in amplitude and phase coordinates, we analyze an exact algebraic phase system that…

We consider the following derivative nonlinear Schr\"odinger equation with a single power-type perturbation \begin{equation*} i\partial_tu+\partial_x^2u+i|u|^2\partial_xu+b |u|^pu=0, \end{equation*} with $b\geq 0$ and $p\geq 4$. When $b=0$…

For a positive integer $h$, let $R_{A,h}(n)$ denote the number of ordered representations $n=s_1+\cdots+s_h$ with all $s_i\in A$. Let \[ B=\{0\}\cup\{m\ge 1:\text{ the base-4 expansion of }m\text{ begins with }1\text{ or }2\}. \] Shallit…

We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved…

This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the…

Marstrand's projection theorem states that the Hausdorff dimension of the orthogonal projection of a Borel set in the plane onto lines is constant almost surely. This property extends to other notions of dimension, such as box and packing…

Here, we show that if $m\ge 5$ is fixed and odd, then there are only finitely many Carmichael numbers of the form $2^np^m+1$ for positive integers $n$ and prime $p$.

In this paper, we develop a unified approach for various operators on Lie conformal algebras. Given a quasi-twilled Lie conformal algebra $(\Ep,\Vs,\Ws)$, we introduce two dual families of operators: \emph{right deformation maps}…

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different…

In this paper, we investigate the strong algebrability and $(\alpha,\beta)$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1< s< r< t\leq2$, we define $$H_s[0,1]=\{f\in…

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=h(u) \quad\mathrm{in}\ \mathbb{R}^{3},…

Let $B$ be a curve on an irrational ruled surface $X$. We prove that the logarithmic Kodaira dimension of $X-B$ equals the Iitaka dimension of $K_X+B$ and give a rough configuration of $B$ when the logarithmic Kodaira dimension of $X - B$…

This paper investigates the role of viscosity in the error upper bounds of a consistent splitting scheme for the Navier-Stokes equations proposed by Huang and Shen [5]. In their original analysis the viscosity is fixed to unity. By…

The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\sigma$ such that the spectral radius of the signed adjacency matrix $A_\sigma$ satisfies $\rho(A_\sigma)\le 2\sqrt{d-1}$. Bilu and Linial also proved the…

In this paper, we improve the arbitrary Banach space \(n \log n\) bound of Ivanisvili--Volberg \cite{IvanisviliVolberg2022} for the second order projection bound to the order \(\sqrt{n}\) bound. Moreover, we study the lower Riesz estimate…

We obtain an estimate of the rate of convergence on a set of full measure of partial sums of trigonometric Fourier series of functions from Lebesgue classes and construct a counterexample showing the order sharpness of this estimate. We…

This paper investigates the distribution and abundance of minimal measures (measures supported on minimal sets) in various dynamical systems, extending the well-known density results for general ergodic measures. We introduce the…