数学
We establish three local-to-global propagation results for Dirichlet--to--Neumann maps. First, in a general geometric setting, we show that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component…
We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in…
Circular operators have been studied extensively since the work of R. Gellar, who conjectured that every circular operator on a complex separable Hilbert space is strongly circular. In this short note, we show that circularity and strong…
This article concerns Wiener amalgam spaces % , recalls their basic properties and provides some hints about their usefulness in various branches of Harmonic Analysis. Despite the fact that the underlying construction principles % of Wiener…
For an integer $q\ge 2$ and a graph $F$ satisfying $q\mid e(F)$, the zero-sum Ramsey number $R(F,\mathbb Z_q)$ is the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \mathbb Z_q$ contains a copy of $F$ whose edge-label…
We describe an elementary sequential realization of the Banach Gelfand triple (S0(R^d), L2(R^d), S0'(R^d)). Here S0(R^d) is a Segal algebra of test functions, L2(R^d) is the usual Hilbert space, and S0'(R^d) is its dual space of mild…
In this review, we offer a comprehensive survey of emerging techniques in gradient-based optimization, with a particular emphasis on the interplay between ordinary differential equation (ODE) perspectives and their extensions into discrete…
We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the…
Given a graph $G$, the domination number $\gamma(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $\rho(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The…
We propose second-order-in-time parametric finite element methods for surface diffusion of closed curves in two dimensions and closed surfaces in three dimensions. The construction is based on exact geometric variation identities along a…
This study focuses on a mathematical model of Alzheimer's disease involving $\beta$-amyloid, cellular prion protein and their complex. The global asymptotic stability of the model indicates that the complex continues to induce neuronal…
A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$ \left\| \sum_{k=1}^m A_k\right\| \le \sqrt{2} \left\| \sum_{k=1}^m |A_k|_{\qsym}\right\| $$ for every unitarily invariant…
Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong S\'{a}rk\"{o}zy-type result: if $A\subseteq [N]$ has no nonzero…
The Moore graph of degree $57$, if one exists, is the remaining open case of the Hoffman-Singleton classification in diameter two. Although its existence remains open, substantial restrictions on the automorphism group of such a graph are…
We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means.…
We study the two-dimensional incompressible viscous non-resistive magnetohydrodynamics in the periodic strip $\mathbb T\times\mathbb R$, subject to a smooth sheared background magnetic field $(\xi(x_2),0)^{\top}$, where $\xi(x_2)$ is…
We introduce an operator-theoretic framework for long-range operators over general dynamical systems with analytic hopping and small potential. By establishing a partially hyperbolic splitting on the fibered solution bundle, we define the…
We present an a priori error analysis of consistent-loss PINNs for stationary convection-diffusion equations on curved level-set domains. The standard mean-squared interior loss fails in the convection-dominated regime: the solution…
In this paper, based on the concept of weighted distribution, we introduce a kind of new approximations for sums of lognormal random variables, such that they are both comonotonic and moment matching. Numerical results show that the…
Gromov's simplicial volume is a fundamental invariant measuring the topological complexity of a manifold. A conjecture of Gromov predicts that every closed manifold admitting a metric of positive scalar curvature has vanishing simplicial…