This article is devoted to study of the autoconvolution equations and generalized Mittag-Leffler functions. These types of equations are given in terms of the Laplace transform convolution of a function with itself. We state new classes of the autoconvolution equations of the first kind and show that the generalized Mittag-Leffler functions are solutions of these types of equations. In view of ...

Mesh smoothing is a method to remove noise or small scale features from large meshes, while still preserving the basic overall shape and important features of the original model. Most sophisticated smoothing methods rely on ideas from differential geometry. In this lecture we discuss some basic geometric concepts and introduce some simple smoothing algorithms; the paper presented at this lectur...

The well-studied methods of current source density analysis use the Laplacian transform to identify locations and relative magnitudes of current sources and sinks. The method is typically used in reduced one-dimensional form for electrophysiological measures, due to technical limitations. The present paper outlines a two-dimensional method in which simultaneous samples are recorded from multipl...

We introduce a curvature function for planar graphs to study the connection of curvature and geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite, locally similar to a tessellation and admits no cut locus. For negative curvature we prove empty interior of minimal bigons and explicit bounds for the growth of distance balls and Che...

in this paper we present a diferent proof of a well known asymptotic estimate for Laplace integrals. The novelty of our approach is that it emphasizes, and rigorously justifies, the appealing heuristic method of Laplace. As a bonus, we also obtain a simple and short proof of Watson’s Lemma. Let a be an element of the extended real number set [−∞,∞]. If lim x−→a f(x)/g(x) = 1 we write f(x) ∼ g(x...

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold H3/Γ are evaluated in the case in which the discrete group Γ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expan...

Alternative specifications of univariate asymmetric Laplace models are described and investigated. A more general mixture model is then introduced. Bivariate extensions of these models are discussed in some detail, with particular emphasis on associated parameter estimation strategies. Multivariate versions of the models are briefly introduced.

The full asymptotic expansion of the trace of the heat semi–group, tr(e−∆Ωt),where −∆Ω is the Dirichlet Laplace-Beltrami operator acting on L2(Ω) for geodesic spherical polygons Ω ⊂ S2, is derived in half–powers of t, and the coefficients determined explicitly. Let Ω be a non-empty open connected set in S = {x ∈ R : |x| = 1} with a piecewise geodesic boundary ∂Ω and interior angles γ1, · · · , ...

Applied researchers using kernel density estimation have worked with optimal bandwidth rules that invariably assumed the reference is Normal (optimal only if true underlying Normal). We offer four new rules-of-thumb based on other infinitely supported distributions: Logistic, Laplace, Student's t and Asymmetric Laplace. Additionally, we propose a psuedo rule-of-thumb (ROT) Gram-Charlier expansi...

It is well known that the Green function of the standard discrete Laplacian on l2(Zd), ∆stψ(n) = (2d) −1 ∑ |n−m|=1 ψ(m), exhibits a pathological behavior in dimension d ≥ 3. In particular, the estimate 〈δ0|(∆st − E − i0)δn〉 = O(|n|− d−1 2 ) fails for 0 < |E| < 1 − 2/d. This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested th...