Related papers: Fourier restriction for the additive Brownian shee…
The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. If a set admits optimal $L^\infty$ bounds on its Fourier transform (that is, it is a Salem set), then it can…
A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the…
We obtain improved Fourier restriction estimate for the truncated cone using the method of polynomial partitioning in dimension $n\geq 3$, which in particular solves the cone restriction conjecture for $n=5$, and recovers the sharp range…
We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part we derive explicit solutions to two optimal stopping problems for the exponentially discounted…
This note records an asymptotic improvement on the known $L^p$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial partitioning method with recent geometric results regarding…
We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication…
Let $G$ be a graph and $h: E(G)\rightarrow [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\leq b$, a fractional $[a,b]$-factor of $G$ with the indicator function $h$ is a spanning subgraph with vertex set $V(G)$ and…
Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and…
We obtain an elementary invariance principle for multi-dimensional Brownian sheet where the underlying random fields are not necessarily independent or stationary. Possible applications include unit-root tests for spatial as well as panel…
We present a new formulation of the multipolar expansion of an exact boundary condition for the wave equation, which is truncated at the quadrupolar order. Using an auxiliary function, that is the solution of a wave equation on the sphere…
We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…
We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with…
It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods…
This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The…
We study the one-dimensional stationary solutions of an integro-differential equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising systems with Kac potentials, \cite{GiacominLebowitz}. We construct stationary solutions…