In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensorsw...

where D = (−1, 1)× (−1, 1) , f(x, y) is a given Hölder continuous function in D, and φ(x, y) is an unknown function. The equation (1) has applications in the theory of aeroelasticity [1]. Note that the equation without logarithmic singularities was many times considered in different classes of functions. In the literature the solutions of the equation (1) in bounded domains [2, 5, 6, 9] as well...

We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator ∂t + ∆ 2 is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We ...

This research work is concerned with the eigenvalue problem for the integral operators which are obtained by linearization of a nonlocal evolution equation. The purpose of section II.A is to describe the nature of the problem and the objective of the project. The problem is related to the “stable solution” of the evolution equation which is the so-called “instanton” that describe the interface ...

The aim of this paper is to investigate the behavior as t ! 1 of solutions to the Cauchy problem u t ? u t ? u ? (b; ru) = r F (u); u(x; 0) = u 0 (x), where > 0 is a xed constant, t 0, x 2 IR n. First, we prove that if u is the solution to the linearized equation, i.e with rF(u) 0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long time behavior of ...

We propose a novel feature selection method based on quadratic mutual information which has its roots in Cauchy–Schwarz divergence and Renyi entropy. The method uses the direct estimation of quadratic mutual information from data samples using Gaussian kernel functions, and can detect second order non-linear relations. Its main advantages are: (i) unified analysis of discrete and continuous dat...

In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three–dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green’s representation formula, which are expressed in terms of ...

Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Lévy processes. These have (weak) Lévy-Itô decompositions and an associa...