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The analysis of scattering from complex objects using surface integral equations is a challenging problem. Its resolution has wide ranging applications- from crack propagation to diagnostic medicine. The two ingredients of any integral…
Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time…
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit…
Despite the large amount of work done in quantum field theory in curved space-times, there are not great many results available for perturbative calculations of particle processes in these systems. Such processes are expected to be…
In this paper, we develop an analytical formulation for the Slepian spatial-spectral concentration problem on the sphere for a limited colatitude-longitude spatial region on the sphere, defined as the Cartesian product of a range of…
A new computational idea for continuum quantum field theories is outlined. This approach is based on the lattice source Galerkin methods developed by Garcia, Guralnik and Lawson. The method has many promising features including treating…
Parameter estimation in ordinary differential equations, although applied and refined in various fields of the quantitative sciences, is still confronted with a variety of difficulties. One major challenge is finding the global optimum of a…
We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings.…
We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. In contrast to previous studies, in which the domain occupied by the screen is assumed to be Lipschitz or smoother, we consider…
The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^\lambda v$ is reduced to…
We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of…
In this paper, we considered the problem of analytical continuation of the solution of the system equations of the moment theory of elasticity in spacious bounded domain from its values and values of its strains on part of the boundary of…
The Steklov problem on a compact Lipschitz domain is to find harmonic functions on the interior whose outward normal derivative on the boundary is some multiple (eigenvalue) of its trace on the boundary. These eigenvalues form the Steklov…
We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate…
We investigate quantum systems perturbed by noise in the form of repeated interactions between the system and the environment. As the number of interactions (aka time steps) tends to infinity, we show, following the works by Pellegrini,…
While new light sources allow for unprecedented resolution in experiments with X-rays, a theoretical understanding of the scattering cross-section is lacking. In the particular case of strongly correlated electron systems, numerical…
The goal of this work is to study the electromagnetic scattering problem of time-domain Maxwell's equations in an unbounded structure. An exact transparent boundary condition is developed to reformulate the scattering problem into an…
We explore the use of correlation with simple functions to get lower bounds for arithmetic quantities. In particular, we apply this idea to the power moments of the error term when counting visible lattice points in large spheres.
We apply Cauchy's interlacing theorem to derive some eigenvalue bounds to the chromatic number using the normalized Laplacian matrix, including a combinatorial characterization of when equality occurs. Further, we introduce some new…
It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation…