This note presents an elementary numerical scheme for discretizing the Laplace-Beltrami operator on smooth manifolds, as a difference operator of arbitrary high order. The method is coordinate-free, and involves very little geometry.

We report on a theoretical study of photoluminescence (PL) of the resonant one-dimensional photonic crystals, or the near-Bragg quantum-well structures. The PL spectral intensity is found by including random sources in the equations for the exciton dielectric polarization and introducing the discrete Green’s function. The structures without and with the dielectric contrast are considered. The p...

Relatively little is known about the dynamic neural processes most critical to chronometric effects, and the ways in which task demands influence their various contributions to intelligence. The present study (n = 111) used the Hick paradigm to measure the effects of task complexity on decision time (DT) and event-related potentials (ERPs), and on their relations in turn with IQ. The surface La...

We derive explicit formulas for the eigenvalues and eigenvectors of the Discrete Laplacian on a rectangular grid for the standard finite difference and finite element methods in 1D, 2D, and 3D. Periodic, Dirichlet, Neumann, and mixed boundary conditions are all considered. We show how the higher dimensional operators can be written as sums of tensor products of one dimensional operators, and th...

The walk laplacian transform of genetic correlation matrices is described and the network display and interpretation of ensuing haplotype relations exemplified. Pharmacogenetic epistatic interactions can be decomposed using residual projections.

It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with some additional work, one can show finite time die-out for two types of systems of stochastic differential equations on the lattice Z d. For our first system, let 1/2 ≤ γ < 1, and consider non-negative solutions of du(t, x) = ∆u(t, x)dt + u γ (t, x)d...

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrödinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi [M1] and it involves a discrete Schrödinger operator H = −∆+q. The decay rates on the potential are less stringent than in [M1], since we require q ∈ l1,1. ...