Abstract— A high-accuracy economical iterative method is proposed for calculating the potential and strength of electric field in a three-dimensional inhomogeneous spatially periodic dielectric placed an initially uniform field. The idea underlying algorithm that represented as sum linear function correction, which can be expressed expansion eigenfunctions Laplace operator satisfy appropriate p...

It is shown how the recently developed Hadamard expansion procedure can be applied to the hyperasymptotic evaluation of Laplace-type integrals containing a large variable when the phase function has a cluster of close-lying saddle points. Themodification to this procedure that is requiredwhen the saddles in the cluster coalesce to form a single higher-order saddle is discussed.An example is als...

We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors of A − λI of order n − 1. By means of the Laplace Expansion Theorem, we show that if the geometric multiplicity of λ is k > 1, then the exterior product of k linearly eigenvectors corresponding to λ is un...

We present a solution to the inverse scattering problem for differential Laplace operators on metric noncompact graphs. We prove that for almost all boundary conditions (i) the scattering matrix uniquely determines the graph and its metric structure, (ii) the boundary conditions are determined uniquely up to trivial gauge transformations. The main ingredient of our approach is a combinatorial F...

In [1] and [2], Bernstein and Reznikov have introduced a new way of estimating the coefficients in the spectral expansion of φ2, where φ is a Maass cusp of norm 1 on a quotient Y = Γ\H of the Poincaré upper half-plane with finite volume. The question of obtaining the precise exponential decay of those coefficients had been posed by Selberg, and first solved by Good [5] (for holomorphic forms) a...

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues...

Approximate counting is an algorithm that provides a count of a huge number of objects within an error tolerance. The first detailed analysis of this algorithm was given by Flajolet. In this paper, we propose a new analysis via the Poisson-Laplace-Mellin approach, a method devised for analyzing shape parameters of digital search trees in a recent paper of Hwang et al. Our approach yields a diff...

This work is devoted to the nondestructive evaluation of materials using stimulated infrared thermography. The proposed approach provides simple analytical solutions to evaluate the lateral extent, and the thickness of a plane flaw in a three-dimensional (3D) heat transfer configuration. It is based on the application of a Laplace transform on the time variable t, then a double Fourier transfor...

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales r...

In [1], Bernyk et al. offer a power series and an integral representation for the density of S1, the maximum up to time 1, of a regular spectrally positive α-stable Lévy process. They also state the asymptotic behavior for large values of the density. A fact which was proved by Doney [4], by investigating the integral representation. In this note, we provide the asymptotic expansion of the dens...