A Smoothing-Type Algorithm for Solving Linear Complementarity Problems with Strong Convergence Properties

In this paper, we construct an augmented system of the standard monotone linear complementarity problem (LCP), and establish the relations between the augmented system and the LCP. We present a smoothing-type algorithm for solving the augmented system. The algorithm is shown to be globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, if the LCP has a solution, then the algorithm either generates a maximal complementary solution of the LCP or detects correctly solvability of the LCP, and in the latter case, an existing smoothing-type algorithm can be directly applied to solve the LCP without any additional assumption and it generates a maximal complementary solution of the LCP; and that if the LCP is infeasible, then the algorithm detect correctly infeasibility of the LCP. To the best of our knowledge, such properties have not appeared in the existing literature for smoothing-type algorithms. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10571134), the Natural Science Foundation of Tianjin (Grant No. 07JCYBJC05200), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.     

George Boole and the origins of invariant theory

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.     

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.     

A VARIATIONAL EXPECTATION-MAXIMIZATION METHOD FOR THE INVERSE BLACK BODY RADIATION PROBLEM

The inverse black body radiation problem, which is to reconstruct the area temperature distribution from the measurement of power spectrum distribution, is a well-known ill-posed problem. In this paper, a variational expectation-maximization （EM） method is developed and its convergence is studied. Numerical experiments demonstrate that the variational EM method is more efficient and accurate than the traditional methods, including the Tikhonov regularization method, the Landweber method and the conjugate gradient method.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

A simple 1D model of inviscid fluid-solid interaction

We analyze a one-dimensional fluid-particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.     

Cn中有界域的Bergman核函数的零点问题

多复变数空间Cn中有界域的Bergman核函数的零点问题集中表现为陆启铿猜想.陆启铿猜想是波兰数学家M.Skwarczynski对陆启铿1966年的一篇文章中关于Bergman核函数的零点问题而命名的,至今已经40年了.该猜想已写入了多复变函数论的多本专著,引起很多数学家的兴趣而研究之,已经成为多复变函数论中的一个活跃的研究方向.本文简述了陆启铿猜想的最初含意,综述了迄今为止关于有界域的Bergman核函数有无零点的各种研究成果以及所用的思想和方法.特别对近来出现的陆启铿猜想的新研究领域进行了较详细的阐述并在最后提出了关于陆启铿猜想的6个Open Problems,希望国内的年轻数学家对陆启铿猜想感到兴趣而研究之.     

一类高维重特征方程Goursat的问题中的离散现象

§1 引言人们对具体实的重特征方程的了解还很不充分。,B,B首先举例指出重特征方程在解的光滑性问题中有所谓离散现象,即方程中的参数取一些离散的值时,解突然失去了光滑性。之后,Treves,F又给出一个例子,发现这种离散也出现在Cauchy问题解的唯一性问题中。王光寅等,陆柱家又进一步研究了较Treves,F.例子广泛的一类方程,指出Cauchy问题和Goursat问题的存在性中都有所谓离散现象。为了进一步揭示重     

Iterative proximal regularization of the modified Lagrangian functional for solving the quasi-variational Signorini inequality

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).     

广义Jacobi矩阵的逆特征问题

本文主要讨论广义Jacobi阵及多个特征对的广义Jacobi阵逆特征问题.通过相似变换将广义Jacobi阵变换为三对角对称矩阵,其特征不变、特征向量只作线性变换,再应用前人理论求得广义Jacobi阵元素ai,|bi|,|ci|有唯一解的充要条件及其具体表达式.