Here we develop a general purpose pre/post conditioner T, to solve an ill-posed system of linear equations, Ax = b. The conditioner T is obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to: Ty = x, where y is the vector of coefficients in the Trefftz expansion, and x is the boundary data at the discrete points on a unit circl...

Abstract The Laplace equation in three-dimensional Euclidean space is -separable bi-cyclide coordinates leading to harmonic functions expressed terms of Lamé–Wangerin called internal and external harmonics. An expansion for a fundamental solution Laplace’s products harmonics derived. In limiting cases this reduces known expansions bi-spherical prolate spheroidal coord...

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrödinger operators with a δ-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing C1-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are expl...

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in R, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Int...

The equations for EM dipole near and far radiation fields are formulated for the complex frequency domain with a Laplace Transform analysis for Hertzian dipoles. An s-domain pulsed current source function from ordinary circuit analysis is used in the expressions, and is augmented as needed to refine the pulse. This formulation allows a lucid pole-zero treatment of the field transfer function, y...

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

Abstract. We study a general transition operator, generated by a random walk on a graph X ; in particular we give necessary and sufficient condition on the matrix coefficient (1-step transition probablilities) to be a bounded operator from l(X) into itself. Moreover we characterize compact operators and we relate this property to the behaviour of the associated random walk. We give a necessary ...

It is well known that, under certain circumstances, discrete plane waves can propagate through lattices. Waves can also be generated by oscillating one point in the lattice: the corresponding solution of the governing partial difference equations is the discrete Green’s function, gmn. The far-field behaviour of gmn is obtained using three methods: textbook derivations are corrected and a formul...

The increasing use of learning videos in Higher Education (HE) have revolutionizing the traditional teaching environment. b-Mat@plicada is a b-Learning Mathematics course mainly composed educational that HE students Portuguese Institution can used for their study as complement face-to-face lectures. In previous research, an experiment was performed classroom context, where 49 watched video on M...