狄氏型及其在数学物理中的应用

简介狄氏型理论。该理论已成为紧密联系解析位势论与马氏过程理论的强有力数学工具,并因此在数学与物理中有许多应用。     

An asymptotic derivation of weakly nonlinear ray theory

Using a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an errorO(ε²) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat’s theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.     

Beyond elementary catastrophe theory

We give a brief discussion of the relations between elementary catastrophe theory, general catastrophe theory, singularity theory, bifurcation theory, and topological dynamics. This is intended to clarify the status, and potential applicability, of “catastrophe theory,” a phrase used by different authors and at different times with different meanings. Catastrophe theory has often been criticized for (supposed) applicability only to gradient systems of differential equations; but properly speaking this criticism can apply only to the elementary version of the theory (where it is in any case wrong). Roughly speaking, elementary catastrophe theory deals with the singularities of real-valued functions, general catastrophe theory with singularities of flows. Between these lies singularity theory, which deals with vector-valued functions. All relate strongly to bifurcation theory and topological dynamics. The issue is more subtle than it appears to be, and we describe an example where elementary catastrophe theory has been used to solve a long-standing problem about nongradient flows: degenerate Hopf bifurcation.     

Riemann Zeros and Random Matrix Theory

In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.     

横观各向同性板的弹性理论和弹性改进理论及一种新的厚板理论

本文从横观各向同性体弹性力学位移形式的基本方程出发,考虑板面承受横向荷载,建立了横观各向同性板弯曲的弹性理论.并由此建立了一个在板的每边能满足三个边界条件的弹性改进理论和一种新的厚板理论.文中求得了周边简支多边形板的弹性改进理论解,数值结果与三维弹性理论精确解的结果非常接近.新的厚板理论和以往的中厚板理论的系统比较表明,我们提出的厚板理论最靠近弹性理论的结果.     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.     

环壳理论与直交异性板理论在计算三圆弧波纹膜片上的比较

有人用直交异性板的理论计算[1]波纹板(膜片)的弹性位移,所得结果曾与实验对照比较满意.但是并没有人认真分析波纹数目、形状对直交异性板的弹性位移和应力分布的数量上的影响,以致未能对用直交异性板理论计算波纹板的范围作明确的说明.以前只是较一般地说,直交异性板理论用于计算波纹数目较多的膜片弹性特性(弹性位移与外力的关系)较满意,计算应力误差较大.本文利用环壳理论[2][3]分析了对称和不对称三圆弧波纹膜片的位移与应力,并与直交异性理论所得结果比较,明确了直交异性板理论的应用范围.     

The complete finitely axiomatized theories of order are dense

We prove a conjecture of Lauchli and Leonard that every sentence of the theory of linear order which has a model, has a model with a finitely axiomatized theory.     

Reliability analysis in uncertain random system

Reliability analysis of a system based on probability theory has been widely studied and used. Nevertheless, it sometimes meets with one problem that the components of a system may have only few or even no samples, so that we cannot estimate their probability distributions via statistics. Then reliability analysis of a system based on uncertainty theory has been proposed. However, in a general system, some components of the system may have enough samples while some others may have no samples, so the reliability of the system cannot be analyzed simply based on probability theory or uncertainty theory. In order to deal with this type systems, this paper proposes a method of reliability analysis based on chance theory which is a generalization of both probability theory and uncertainty theory. In order to illustrate the method, some common systems are considered such as series system, parallel system, k-out-of-n system and bridge system.     

一种完备完全分配格上矩阵方程的求解方法

模糊理论已经在决策、模式识别、故障诊断、过程控制等许多方面得到了广泛应用，在这些应用问题求解中往往要遇到求解模糊关系方程。本文从完备完全分配格的角度给出了一种求解模糊关系方程极小解的完备解法，并用计算机实现了提出的算法，为实际问题求解计算机化打下了基础。     

Absolute Logarithmic Norm

In this paper we introduce and study a new concept of the absolute logarithmic norm, which has much in common with the classical definition of the logarithmic norm by S. M. Lozinskii. The the theory that we develop allows to obtain new facts from the Lyapunov stability theory for the systems of linear differential equations with constant coefficients. The presentation of the material relies heavily on the theory of off-diagonally nonnegative matrices arising from the Perron–Frobenius theory for nonnegative matrices.     

A minimalist two-level foundation for constructive mathematics

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin.One level is given by an intensional type theory, called Minimal type theory. This theory extends a previous version with collections.The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model.This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs” paradigm and acts as a programming language.     

Topological regluing of rational functions

Regluing is a surgery that helps to build topological models for rational functions. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston–Teichmüller theory. We will discuss a topological theory of regluing, and just trace a direction, in which a holomorphic theory can develop.     

Applying fuzzy weighted average approach to evaluate office layouts with Feng–Shui consideration

Office layout is an important issue, especially in China and the Asian countries, where the Feng–Shui theory frequently plays a vital role. Yet, in the literature, Feng–Shui theory has seldom been discussed. Another problem is the imprecise or vague satisfaction level of the linguistic expression used in this theory. In this article, the fuzzy set theory is applied to deal with this aspect of the problem. Using an improved and efficient fuzzy weighted average (EFWA) algorithm, which has been shown to be more advantageous than the existing FWA algorithms, an empirical study of an office-layout design problem with the consideration of Feng–Shui is presented to illustrate the EFWA approach. The results and the criteria developed, based on the interpretation of the Form school concept of the Feng–Shui are reported.     

完全二叉树的量词消去

量词消去法已经成为计算机科学和代数模型论中最有力的研究工具之一.本 文针对完全二叉树理论所独有的特性,给出了它的基本公式集,然后利用分布公式及 有限覆盖证明了完全二叉树的理论可以量词消去.     

光通信系统的信道容量

该文讨论了信息论中的一个重要概念信道容量的量子化描述，并证明了量子信道容量具有与经典通信理论中类似的性质．对传递数字特征态的二元光通信系统计算了信道容量．结果表明；为提高光通信系统的信息传输效率，应适当选择信号光以获得较大的信道容量．     

关于一些概率统计理论的一点认识

近来,林群从哲学的角度用公式相对真理/绝对真理=0.9阐述了微积分的内涵,并探究了计算数学、概率统计等学科领域乃至日常生活现象也同样蕴藏着该哲学公式.基于林群等在《概率论初步设想》中对大数定律和中心极限定理等概率统计学理论的哲学内涵分析,用一些数值算例展示概率统计学中这些常用理论所蕴含的这一哲学公式.     

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.     

不确定性信息处理的集对论思想与方法

集对论从主体与客体联系层次上较好地实现了不确定性信息的整体处理,集对论的同异反集合和联系数等概念弥补了模糊集合论的缺陷。本文从概念上探讨了集对论的基本原理和思想方法,并与模糊集合论做了对比分析,认为集对论体现了认识过程中的辩证思维,因而具有更强的方法论意义。     

代数表示论简介与综述

代数表示论是本世纪七十年代初兴起的代数学的一个新的分支。它的基本内容是研究一个Ａｒｔｉｎ代数上的模范畴。由于各国代数学家的共同努力，这一理论于最近二十年代有了异常迅猛的发展并逐步趋于完善。本文介绍了代数表示论的理论基础；几乎可裂序列；箭图和赋值箭图的表示；Ｃｏｘｅｔｅｒ函子；ＡＲ－箭图的覆盖及代数的Ｇａｌｏｉｓ覆盖。并简单介绍了在代数表示论中普遍应用的工具：Ｔｉｌｔ理论，以及著名的Ｄｐｏｚｇ定理证