The multi-mode approximation is presented to compute the interior wave function of Schrödinger equation. This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered. The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

In a recent paper, Pritchard, Krieble, and Duke (Psychophysiology, 33, 362–368, 1996) studied the validity of spatial embedding of electroencephalographic (EEG) data and rejected this method in favor of time-delay embedding. The present paper describes the nonlinear characterization of brain dynamics using either spatial or time-delay embedding. We discuss the arguments published in Pritchard et al. (1996) and demonstrate that the spatial embedding cannot be rejected on this basis. We also point out the limitations of both spatial and time-delay embeddings related to the spatial extension and the high-dimensional dynamics of brain activity.