SOME OF MY FAVORITE SOLVED AND UNSOLVED PROBLEMS IN GRAPH THEORY

Abstract

During my long life I published many papers with related titles. To keep this paper short I will not give proofs and will restrict myself to problems in graph theory, but I will try to give references and make these as complete as possible.

I will start with Turk type problems in extremal graph theory.     

线性回归诊断的若干问题

本文对于线性回归诊断提出了几种新的模型和方法。我们首次研究了方差加权和均值漂移的混合模型,得到了相应的诊断统计量。本文还引入了罚函数方法,并以此为工具,讨论了若干有偏估计的影响度量,最后,本文提出了基于重心的诊断统计量,对于识别异常点有较好的效果。     

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BYORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.     

Hopfield神经网络中的数学问题

Ｈｏｐｆｉｅｌｄ人工神经网络和多层网络模型中有许多基本的数学问题，其中，最重要的是存储容量的问题，即吸引子的个数，在以往的文章中，大多数都以概率模型为背景，本文从组合论的角度提供了几个问题，我们把Ｈｏｐｆｉｅｌｄ网络中的状态按连接矩阵所构成的变换分类，网络的容量就是类的个数，我们得出了一些基本结果，还有一些有意义的问题，如多层网络中的结果是否可交换等问题，还有待进一步研究。     

非齐次障碍问题的正则性结果

本文研究了非齐次椭圆方程的障碍问题,给出了二阶非齐次障碍问题解的定义,利用Poincar啨不等式,获得非齐次障碍问题的解及其导数的一些性质,填补了对非齐次障碍问题研究的空白.     

Banach空间中的一类非光滑齐次优化问题

本文主要研究了一类非光滑齐次优化问题(HOP).通过运用Clarke次微分的广义欧拉恒等式获得了使得(HOP)问题的最优解成为KKT点的充分条件并给出了(HOP)问题与(HOP)问题的KKT点及最优解之间的等价刻画.本文的结果是文[1]中已有结果的推广.文中还举例说明了结果的正确性.     

t 叉树和有序树的几个计数问题

一、引言树是计算机科学中广泛使用的一种数据结构。树的组合性质在程序设计和计算机算法复杂性分析中有重要的应用。文献[1—5]研究了 t 叉树和有序树的某些组合性质。在本文中,我们将着重考虑这两类树的计数问题。     

A NEW LOOK AT SOME CLASSIC PROBLEMS IN ORTHOGONAL POLYNOMIALS

<正> Until recently there were two kinds of orthogonality commonly accepted:that generatedby a real weight function and that generated by a complex weight function with integrationtaken along a path in the complex plane.The authors following R.D.Morton and A.M.Krall [10] research work are discussing the orthogonality from the point of view of distribu-tion theory.In doing so a third and new kind of orthogonality is pointed out.In some of the authors' forthcoming papers vrious kinds of orthogonality for polynomialsin several variables will be discussed [14]—[16],while the appropriate papers will deal withC.Brezinski's comprehensive very recent results [17] and with M.Picone's“total derivativeoperators”systems in the framework of D.Mangeron's polyvibrating or polywave equations,respectively.