Symmetric Eighth Algebraic Order Methods with Minimal Phase-Lag for the Numerical Solution of the Schrödinger Equation

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.     

On Solvability of a Boundary Value Problem for Singular Integral Equations

This paper deals with the solvability of boundary value problems for singular integral equations of the form (i)-(ii).By an algebraic method we reduce the problem (i)-(ii) to a system of linear algebraic equations which gives all solutions in a closed form.AMS Subject Classification: 47G05, 45GO5, 45E05     

Existence of solutions to an anisotropic phase‐field model

We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability and instability of relativistic electrons in classical electromagnetic fields

The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields, is already unstable when a, the fine structure constant, is too large, it is noteworthy that the combination of the two is still stableprovided the projection onto the positive energy states of the Dirac operator, whichdefines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here. This paper is dedicated to Bernard Jancovici on the occasion of his 65th birthday.     

多元Kantorovich算子L_p范数下的渐近展开

本文给出了高维单纯形上Kantorovich算子在Lp（1＜P＜∞）范数下的高阶渐近展开公式．     

Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires

We consider general Schrödinger operators on domains of Riemannian manifolds with possibly exponential volume growth. We prove sharp large time Gaussian upper bounds. These bounds are then used to prove new Lp-Lp estimates for the corresponding semigroups. Applications to semi-linear parabolic equations are given.     

Parametric Method for Global Optimization

This paper considers constrained and unconstrained parametric global optimization problems in a real Hilbert space. We assume that the gradient of the cost functional is Lipschitz continuous but not smooth. A suitable choice of parameters implies the linear or superlinear (supergeometric) convergence of the iterative method. From the numerical experiments, we conclude that our algorithm is faster than other existing algorithms for continuous but nonsmooth problems, when applied to unconstrained global optimization problems. However, because we solve 2n + 1 subproblems for a large number n of independent variables, our algorithm is somewhat slower than other algorithms, when applied to constrained global optimization.This work was partially supported by the NATO Outreach Fellowship - Mathematics 219.33.We thank Professor Hans D. Mittelmann, Arizona State University, for cooperation and support.     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

Classical Mechanics Without Determinism

Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrödinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical interpretation is sufficient to predict all measurable results of classical mechanics. In the classical case, the wave function that satisfies a linear equation is positive, which is the main source of the fundamental difference between classical and quantum mechanics.     

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf)=conv(suppf) holds for all compactly supported smooth functions f so that conv(suppf) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.