Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind

In this paper, we suggest a convergence analysis for solving Fredholm integral equations of the first kind using Tikhonov regularization under supremum norm. We also provide an a priori parameter choice strategy for choosing the regularization parameter and obtain an error estimate.     

On the Geometrical Meaning of the Dynamic Equations in Analytical Mechanics

In this paper,the modern geometrical structure of analytical mechanics,the exteriordifferential forms and the geometrical meaning of dynamic equations are briefly discussed.     

对声波和弹性波传播模拟的Hamilton系统方法

建立了离散化网格上的准粒子体系，引入此体系的Hamilton系统描述，用来模拟声波和弹性波的传播.介绍了准粒子间相互作用的九点模型并给出互作用系数.证明了Hamilton系统方法 和声波、弹性波方程的关系，并给出两个方法中所使用物理量的关系.使用辛算法对给定的 介质模型进行数值模拟. 关键词： Hamilton系统方法 九点互作用模型 声波方程 弹性波方程 辛算法     

A stochastic model for risk management in global supply chain networks

With the increasing emphasis on supply chain vulnerabilities, effective mathematical tools for analyzing and understanding appropriate supply chain risk management are now attracting much attention. This paper presents a stochastic model of the multi-stage global supply chain network problem, incorporating a set of related risks, namely, supply, demand, exchange, and disruption. We provide a new solution methodology using the Moreau–Yosida regularization, and design an algorithm for treating the multi-stage global supply chain network problem with profit maximization and risk minimization objectives.     

On the regularization of an equation of the first kind with a multiple integration operator

The zero-order Tikhonov regularization method as applied to an equation of the first kind with a multiple differentiation operator is considered for the case when the solution belongs to a class from the domain of the adjoint operator. An estimate of the error of the approximate solution in the uniform metric is obtained, which is sharp with respect to the order, and the order is established. It is proved that the proposed method is optimal with respect to the order. Unimprovable estimates of the order of the modulus of continuity of the inverse operator are obtained.     

Approximation with polynomial kernels and SVM classifiers

This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L¹ space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels. Dedicated to Charlie Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 68T05, 62J02. Ding-Xuan Zhou: The first author is supported partially by the Research Grants Council of Hong Kong (Project No. CityU 103704).     

THE WEINSTEIN CONJECTURE ON C~1-SMOOTH HYPERSURFACE OF CONTACT TYPE

In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C1-smooth hypersurface of contact type.     

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier–Stokes equations with Q1/P0 element, by newly introducing a square‐type linear form into the variational divergence‐free constraint regularized with the global pressure jump (GPJ) method. The addition of the square‐type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non‐singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2‐D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and non‐leaky 3‐D driven cavity flows. Finally, the non‐leaky 3‐D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.